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\section{Introduction}
Irrotational dust spacetimes have been widely studied, in
particular as models for the late universe, and as arenas for
the evolution of density perturbations and gravity wave
perturbations. In linearised theory, i.e. where the irrotational
dust spacetime is close to a Friedmann--Robertson--Walker dust
spacetime, gravity wave perturbations are usually
characterised by
transverse traceless tensor modes.
In terms of the covariant and gauge--invariant
perturbation formalism initiated by Hawking \cite{h}
and developed by Ellis and Bruni \cite{eb},
these perturbations are described by the
electric and magnetic Weyl tensors, given respectively by
\begin{equation}
E_{ab}=C_{acbd}u^c u^d\,,\quad H_{ab}={\textstyle{1\over2}}\eta_{acde}u^e
C^{cd}{}{}_{bf}u^f
\label{eh}
\end{equation}
where $C_{abcd}$ is the Weyl tensor, $\eta_{abcd}$ is the
spacetime permutation tensor, and $u^a$ is the dust four--velocity.
In the so--called `silent universe' case
$H_{ab}=0$, no information is exchanged between neighbouring
particles, also in the exact nonlinear case. Gravity wave
perturbations require nonzero $H_{ab}$, which is
divergence--free in the linearised case \cite{led}, \cite{he},
\cite{b}.
A crucial question for the analysis of gravity waves
interacting with matter is whether
the properties of the linearised perturbations are
in line with those of the exact nonlinear theory.
Lesame et al. \cite{led} used the covariant formalism
and then specialised to a shear tetrad, in order to
study this question. They concluded that in the nonlinear case,
the only solutions with $\mbox{div}\,H=0$ are those with $H_{ab}=0$
--- thus indicating a linearisation instability, with potentially
serious implications for standard analyses of gravity waves, as
pointed out in \cite{m}, \cite{ma}.
It is shown here that the argument of \cite{led}
does not in fact prove that
$\mbox{div}\,H=0$ implies
$H_{ab}=0$.
The error in \cite{led} is traced to an incorrect sign
in the Weyl tensor decomposition (see below).\footnote{The authors of
\cite{led} are in agreement about the error and its implication
(private communication).}
The same covariant formalism is used here, but with modifications
that
lead to simplification and greater clarity. This improved
covariant formalism renders the equations more transparent, and
together with the new identities derived via the formalism,
it facilitates a fully covariant analysis,
not requiring
lengthy tetrad calculations such as those
used in \cite{led}.
The improved formalism
is presented in Section II, and the identities that are crucial for
covariant analysis are given in the appendix.
In Section III, a covariant derivation is given to show
that {\em in the generic case of irrotational dust
spacetimes, the constraint equations are
preserved under evolution.}
A by--product of the
argument is the identification of the
error in \cite{led}.
In a companion paper \cite{mel},
we use the covariant formalism of Section III
to show that when $\mbox{div}\,H=0$,
no further conditions are generated. In particular, $H_{ab}$ {\em is
not forced to vanish, and there is not a linearisation instability.}
A specific example is presented in
Section IV, where
it is shown that Bianchi type V spacetimes
include cases in which
$\mbox{div}\,H=0$ but $H_{ab}\neq0$.
\section{The covariant formalism for propagation and
constraint equations}
The notation and conventions are based on
those of \cite{led}, \cite{e1};
in particular $8\pi G=1=c$, round brackets enclosing indices
denote symmetrisation and square brackets denote
anti--symmetrisation. Curvature tensor conventions are given in
the appendix.
Considerable simplification and streamlining results from
the following definitions: the projected permutation tensor
(compare \cite{e3}, \cite{mes}),
\begin{equation}
\varepsilon_{abc}=\eta_{abcd}u^d
\label{d1}
\end{equation}
the projected, symmetric and trace--free part of a tensor,
\begin{equation}
S_{<ab>}=h_a{}^c h_b{}^d S_{(cd)}-
{\textstyle{1\over3}}S_{cd}h^{cd} h_{ab}
\label{d2}
\end{equation}
where $h_{ab}=g_{ab}+u_au_b$ is the spatial projector
and $g_{ab}$ is the metric,
the projected spatial covariant derivative (compare \cite{e2},
\cite{eb}, \cite{mes}),
\begin{equation}
\mbox{D}_a S^{c\cdots d}{}{}{}{}_{e\cdots f}=h_a{}^b h^c{}_p \cdots
h^d{}_q h_e{}^r \cdots h_f{}^s \nabla_b
S^{p\cdots q}{}{}{}_{r\cdots s}
\label{d3}
\end{equation}
and the covariant spatial curl of a tensor,
\begin{equation}
\mbox{curl}\, S_{ab}=\varepsilon_{cd(a}\mbox{D}^c S_{b)}{}^d
\label{d4}
\end{equation}
Note that
$$
S_{ab}=S_{(ab)}\quad\Rightarrow\quad\mbox{curl}\, S_{ab}=\mbox{curl}\, S_{<ab>}
$$
since $\mbox{curl}\,(fh_{ab})=0$ for any $f$.
The covariant spatial divergence of $S_{ab}$ is
$$(\mbox{div}\,S)_a=\mbox{D}^b S_{ab}$$
The covariant spatial curl of a vector is
$$
\mbox{curl}\, S_a=\varepsilon_{abc}\mbox{D}^bS^c
$$
Covariant analysis of propagation and constraint equations
involves frequent use of a number of algebraic and differential
identities governing the above quantities. In particular, one
requires commutation rules for spatial and time derivatives.
The necessary identities are collected for convenience in the
appendix, which includes a simplification of
known results and a number of new results.
The Einstein, Ricci and Bianchi equations may be covariantly split
into propagation and constraint equations \cite{e1}.
The propagation equations given in
\cite{led} for irrotational dust are simplified by the present
notation, and become
\begin{eqnarray}
\dot{\rho}+\Theta\rho &=& 0
\label{p1}\\
\dot{\Theta}+{\textstyle{1\over3}}\Theta^2 &=& -{\textstyle{1\over2}}\rho
-\sigma_{ab}\sigma^{ab}
\label{p2}\\
\dot{\sigma}_{ab}+{\textstyle{2\over3}}\Theta\sigma_{ab}+\sigma_{c<a}
\sigma_{b>}{}^c &=& -E_{ab}
\label{p3}\\
\dot{E}_{ab}+\Theta E_{ab}-3\sigma_{c<a}E_{b>}{}^c &=&
\mbox{curl}\, H_{ab}-{\textstyle{1\over2}}\rho\sigma_{ab}
\label{p4}\\
\dot{H}_{ab}+\Theta H_{ab}-3\sigma_{c<a}H_{b>}{}^c &=& -\mbox{curl}\, E_{ab}
\label{p5}
\end{eqnarray}
while the constraint equations become
\begin{eqnarray}
\mbox{D}^b\sigma_{ab} &=& {\textstyle{2\over3}}\mbox{D}_a \Theta
\label{c1}\\
\mbox{curl}\, \sigma_{ab}&=& H_{ab}
\label{c2}\\
\mbox{D}^b E_{ab} &=& {\textstyle{1\over3}}\mbox{D}_a \rho +
\varepsilon_{abc}\sigma^b{}_d H^{cd}
\label{c3}\\
\mbox{D}^b H_{ab} &=& -\varepsilon_{abc}\sigma^b{}_d E^{cd}
\label{c4}
\end{eqnarray}
A dot denotes a covariant derivative along $u^a$, $\rho$ is the
dust energy density, $\Theta$ its rate of
expansion, and $\sigma_{ab}$ its shear. Equations (\ref{p4}),
(\ref{p5}), (\ref{c3}) and (\ref{c4}) display the analogy with
Maxwell's theory. The FRW case is covariantly characterised by
$$
\mbox{D}_a\rho=0=\mbox{D}_a\Theta\,,\quad\sigma_{ab}=E_{ab}=H_{ab}=0
$$
and in the linearised case of an almost FRW spacetime, these gradients
and tensors are first order of smallness.
The dynamical fields in these equations are the scalars $\rho$ and
$\Theta$, and the
tensors $\sigma_{ab}$,
$E_{ab}$ and $H_{ab}$, which all satisfy $S_{ab}=S_{<ab>}$. The
metric $h_{ab}$ of the spatial
surfaces orthogonal to $u^a$ is implicitly
also involved in the equations as a dynamical field. Its propagation
equation is simply the identity $\dot{h}_{ab}=0$,
and its constraint equation is the identity $\mbox{D}_a h_{bc}=0$ --
see (\ref{a4}). The Gauss--Codacci equations for the Ricci curvature
of the spatial surfaces \cite{e1}
\begin{eqnarray}
R^*_{ab}-{\textstyle{1\over3}}R^*h_{ab} &=& -\dot{\sigma}_{ab}-\Theta
\sigma_{ab} \nonumber\\
R^* &=&-{\textstyle{2\over3}}\Theta^2+\sigma_{ab}\sigma^{ab}+2\rho \label{r1}
\end{eqnarray}
have not been included, since the curvature is algebraically
determined by the other fields,
as follows from (\ref{p3}):
\begin{equation}
R^*_{ab}=E_{ab}-{\textstyle{1\over3}}\Theta\sigma_{ab}+\sigma_{ca}
\sigma_b{}^c+{\textstyle{2\over3}}\left(\rho-{\textstyle{1\over3}}\Theta^2\right)
h_{ab}
\label{r2}\end{equation}
The contracted Bianchi identities for the 3--surfaces \cite{e1}
$$
\mbox{D}^b R^*_{ab}={\textstyle{1\over2}}\mbox{D}_a R^*
$$
reduce to the Bianchi constraint (\ref{c3}) on using (\ref{c1}),
(\ref{c2}) and the identity (\ref{a13}) in (\ref{r1}) and
(\ref{r2}). Consequently, these identities do not impose any new
constraints.
By the constraint (\ref{c2}), one can in principle eliminate $H_{ab}$.
However, this leads to second--order derivatives in the propagation
equations (\ref{p4}) and (\ref{p5}). It seems preferable to maintain
$H_{ab}$ as a basic field.
One interesting use of (\ref{c2}) is in
decoupling the shear from the Weyl tensor.
Taking the time derivative of
the shear propagation equation (\ref{p3}), using the propagation
equation (\ref{p4}) and the constraint (\ref{c2}), together with
the identity (\ref{a16}), one gets
\begin{eqnarray}
&&-\mbox{D}^2\sigma_{ab}+\ddot{\sigma}_{ab}+{\textstyle{5\over3}}\Theta
\dot{\sigma}_{ab}-{\textstyle{1\over3}}\dot{\Theta}\sigma_{ab}+
{\textstyle{3\over2}}\mbox{D}_{<a}\mbox{D}^c\sigma_{b>c} \nonumber\\
&&{}=4\Theta\sigma_{c<a}\sigma_{b>}{}^c+6\sigma^{cd}\sigma_{c<a}
\sigma_{b>d}-2\sigma^{de}\sigma_{de}h_{c<a}\sigma_{b>}{}^c+
4\sigma_{c<a}\dot{\sigma}_{b>}{}^c
\label{s}\end{eqnarray}
where $\mbox{D}^2=\mbox{D}^a \mbox{D}_a$ is the covariant Laplacian.
This is {\em the exact nonlinear generalisation of the linearised
wave equation for shear perturbations} derived in \cite{he}.
In the linearised
case, the right hand side of (\ref{s}) vanishes, leading to a
wave equation governing the propagation of shear perturbations in
an almost FRW dust spacetime:
$$
-\mbox{D}^2\sigma_{ab}+\ddot{\sigma}_{ab}+{\textstyle{5\over3}}\Theta
\dot{\sigma}_{ab}-{\textstyle{1\over3}}\dot{\Theta}\sigma_{ab}+
{\textstyle{3\over2}}\mbox{D}_{<a}\mbox{D}^c\sigma_{b>c} \approx 0
$$
As suggested by comparison of (\ref{c2}) and (\ref{c4}), and
confirmed by the identity (\ref{a14}), div~curl is {\em not} zero,
unlike its Euclidean vector counterpart. Indeed, the divergence of
(\ref{c2}) reproduces (\ref{c4}), on using the (vector) curl
of (\ref{c1}) and
the identities
(\ref{a2}), (\ref{a8}) and (\ref{a14}):
\begin{equation}
\mbox{div (\ref{c2}) and curl (\ref{c1})}\quad\rightarrow\quad
\mbox{(\ref{c4})}
\label{i1}\end{equation}
Further
differential relations amongst the propagation and constraint
equations are
\begin{eqnarray}
\mbox{curl (\ref{p3}) and (\ref{c1}) and (\ref{c2}) and
(\ref{c2})$^{\displaystyle{\cdot}}$}\quad
& \rightarrow & \quad\mbox{(\ref{p5})} \label{i2}\\
\mbox{grad (\ref{p2}) and div (\ref{p3}) and (\ref{c1}) and
(\ref{c1})$^{\displaystyle{\cdot}}$ and
(\ref{c2})}\quad & \rightarrow & \quad \mbox{(\ref{c3})} \label{i3}
\end{eqnarray}
where the identities (\ref{a7}), (\ref{a11.}), (\ref{a13}),
(\ref{a13.}) and (\ref{a15}) have been used.
Consistency
conditions may arise
to preserve the constraint equations under
propagation along $u^a$ \cite{led}, \cite{he}.
In the general
case, i.e. without imposing any assumptions about
$H_{ab}$ or other quantities, the constraints are
preserved under evolution.
This is shown in the next section, and forms the
basis for analysing special cases, such as
$\mbox{div}\,H=0$.
\section{Evolving the constraints: general case}
Denote the constraint equations (\ref{c1}) --- (\ref{c4}) by
${\cal C}^A=0$, where
$$
{\cal C}^A=\left(\mbox{D}^b\sigma_{ab}-{\textstyle{2\over3}}\mbox{D}_a\Theta\,,\,
\mbox{curl}\,\sigma_{ab}-H_{ab}\,,\,\cdots\right)
$$
and $A={\bf 1},\cdots, {\bf 4}$.
The evolution of ${\cal C}^A$ along $u^a$ leads to a
system of equations $\dot{{\cal C}}^A={\cal F}^A
({\cal C}^B)$, where ${\cal F}^A$ do not contain
time derivatives, since these are eliminated via the propagation
equations and suitable identities. Explicitly, one obtains after
lengthy calculations the following:
\begin{eqnarray}
\dot{{\cal C}}^{\bf 1}{}_a&=&-\Theta{\cal C}^{\bf 1}{}_a+2\varepsilon_a{}^{bc}
\sigma_b{}^d{\cal C}^{\bf 2}{}_{cd}-{\cal C}^{\bf 3}{}_a
\label{pc1}\\
\dot{{\cal C}}^{\bf 2}{}_{ab}&=&-\Theta{\cal C}^{\bf 2}{}_{ab}
-\varepsilon^{cd}{}{}_{(a}\sigma_{b)c}{\cal C}^{\bf 1}{}_d
\label{pc2}\\
\dot{{\cal C}}^{\bf 3}{}_a&=&-{\textstyle{4\over3}}\Theta{\cal C}^{\bf 3}{}_a
+{\textstyle{1\over2}}\sigma_a{}^b{\cal C}^{\bf 3}{}_b-{\textstyle{1\over2}}\rho
{\cal C}^{\bf 1}{}_a \nonumber\\
&&{}+{\textstyle{3\over2}}E_a{}^b{\cal C}^{\bf 1}{}_b
-\varepsilon_a{}^{bc}E_b{}^d{\cal C}^{\bf 2}
{}_{cd}+{\textstyle{1\over2}}\mbox{curl}\,{\cal C}^{\bf 4}{}_a
\label{pc3}\\
\dot{{\cal C}}^{\bf 4}{}_a&=&-{\textstyle{4\over3}}\Theta{\cal C}^{\bf 4}{}_a
+{\textstyle{1\over2}}\sigma_a{}^b{\cal C}^{\bf 4}{}_b
\nonumber\\
&&{}+{\textstyle{3\over2}}H_a{}^b{\cal C}^{\bf 1}{}_b
-\varepsilon_a{}^{bc}H_b{}^d{\cal C}^{\bf 2}
{}_{cd}-{\textstyle{1\over2}}\mbox{curl}\,{\cal C}^{\bf 3}{}_a
\label{pc4}
\end{eqnarray}
For completeness, the following list of equations used in the
derivation is given:\\
Equation
(\ref{pc1}) requires (\ref{a7}), (\ref{a11.}), (\ref{p2}), (\ref{p3}),
(\ref{c1}), (\ref{c2}), (\ref{c3}), (\ref{a13}) -- where (\ref{a13})
is needed to eliminate the following term from the right hand side
of (\ref{pc1}):
\begin{eqnarray*}
&&\varepsilon_{abc}\sigma^b{}_d\,\mbox{curl}\,\sigma^{cd}
-\sigma^{bc}\mbox{D}_a \sigma_{bc}\\
&&{}+\sigma^{bc}
\mbox{D}_c \sigma_{ab}+{\textstyle{1\over2}}\sigma_{ac}\mbox{D}_b\sigma^{bc} \equiv0
\end{eqnarray*}
Equation
(\ref{pc2}) requires (\ref{a15}), (\ref{p3}), (\ref{p5}), (\ref{c1}),
(\ref{c2}), (\ref{a3.}) -- where (\ref{a3.}) is needed to eliminate
the following term from the right hand side of (\ref{pc2}):
$$
\varepsilon_{cd(a}\left\{\mbox{D}^c\left[\sigma_{b)}{}^e\sigma^d{}_e\right]+
\mbox{D}^e\left[\sigma_{b)}{}^d\sigma^c{}_e\right]\right\}\equiv0
$$
Equation
(\ref{pc3}) requires (\ref{a11.}), (\ref{p1}), (\ref{p4}), (\ref{p5}),
(\ref{a14}), (\ref{a3}), (\ref{c1}), (\ref{c3}), (\ref{c4}),
(\ref{a13}) -- where (\ref{a13}) is needed to eliminate the
following term from the right hand side of (\ref{pc3}):
\begin{eqnarray*}
&& {\textstyle{1\over2}}\sigma_{ab}\mbox{D}_c E^{bc}
+\varepsilon_{abc}E^b{}_d\, \mbox{curl}\,\sigma^{cd}\\
& &{}+\varepsilon_{abc}\sigma^b{}_d
\,\mbox{curl}\, E^{cd}
+{\textstyle{1\over2}}E_{ab}\mbox{D}_c\sigma^{bc}+E^{bc}\mbox{D}_b\sigma_{ac}\\
& &{}+\sigma^{bc}\mbox{D}_b E_{ac}-
\mbox{D}_a\left(\sigma^{bc}E_{bc}\right)\equiv 0
\end{eqnarray*}
Equation
(\ref{pc4}) requires (\ref{a11.}), (\ref{p3}), (\ref{p4}), (\ref{p5}),
(\ref{a14}), (\ref{a13}), (\ref{c1}), (\ref{c2}), (\ref{c3}),
(\ref{c4}).
In \cite{led}, a sign error in the Weyl tensor decomposition
(\ref{a5}) led to spurious consistency conditions arising from
the evolution of (\ref{c1}), (\ref{c2}). The evolution
of the Bianchi constraints (\ref{c3}), (\ref{c4})
was not considered in \cite{led}.
Now suppose that the constraints
are satisfied on an initial spatial surface $\{t=t_0\}$, i.e.
\begin{equation}
{\cal C}^A\Big|_{t_0}=0
\label{i}\end{equation}
where
$t$ is proper time along the dust worldlines. Then by
(\ref{pc1}) -- (\ref{pc4}), it follows that the
constraints are satisfied for all time, since ${\cal C}^A=0$ is
a solution for the given initial data. Since the system is linear,
this solution is unique.
This establishes that the constraint equations are preserved under
evolution. However, it does not prove existence of solutions to
the constraints in the generic case
--- only that if solutions exist, then they evolve
consistently. The question of existence is currently under
investigation. One would like to show explicitly how a metric
is constructed from given initial data in the covariant formalism.
This involves in particular considering whether the
constraints generate new constraints, i.e. whether they are
integrable as they stand, or whether there are implicit
integrability conditions. The relation (\ref{i1}) is part of the
answer to this question, in that it shows how, within any
$\{t=\mbox{ const}\}$ surface, the constraint ${\cal C}^{\bf 4}$
is satisfied if ${\cal C}^{\bf 1}$ and ${\cal C}^{\bf 2}$ are
satisfied. Specifically, (\ref{i1}) shows that
\begin{equation}
{\cal C}^{\bf 4}{}_a={\textstyle{1\over2}}\mbox{curl}\,{\cal C}^{\bf 1}
{}_a-\mbox{D}^b{\cal C}^{\bf 2}{}_{ab}
\label{i4}\end{equation}
Hence, if one takes ${\cal C}^{\bf 1}$ as determining
$\mbox{grad}\,\Theta$,
${\cal C}^{\bf 2}$ as defining $H$ and ${\cal C}^{\bf 3}$ as
determining $\mbox{grad}\,\rho$, the constraint equations are
consistent with each other because ${\cal C}^{\bf 4}$ then follows.
Thus if there exists a solution to the constraints on
$\{t=t_0\}$, then it is consistent and it evolves consistently.
In the next section, Bianchi type V spacetimes are shown to provide
a concrete example of existence and consistency in the case
$$
\mbox{div}\,E\neq 0\neq\mbox{curl}\, E\,,\quad\mbox{div}\,H=0\neq\mbox{curl}\, H\,,\quad
\mbox{grad}\,\rho=0=\mbox{grad}\,\Theta
$$
\section{Spacetimes with $\mbox{div}\,H=0\neq H$}
Suppose now that the magnetic Weyl tensor is divergence--free, a
necessary condition for gravity waves:
\begin{equation}
\mbox{div}\,H=0\quad\Leftrightarrow\quad [\sigma,E]=0
\label{dh}\end{equation}
where $[S,V]$ is the index--free notation for the covariant commutator
of tensors [see (\ref{a2})], and the equivalence follows from
the constraint (\ref{c4}).
Using the covariant
formalism of Section III, it can be shown \cite{mel} that (\ref{dh})
is preserved under evolution without generating further conditions.
In particular, (\ref{dh}) does not force $H_{ab}=0$ -- as shown by
the following explicit example.
First note that by (\ref{r2}) and (\ref{dh}):
$$
R^*_{ab}={\textstyle{1\over3}}R^*h_{ab}\quad\Rightarrow\quad
[\sigma,R^*]=0\quad\Rightarrow\quad\mbox{div}\,H=0
$$
i.e., {\em irrotational dust spacetimes
have $\mbox{div}\,H=0$ if $R^*_{ab}$ is isotropic.}
Now the example arises from the class of irrotational spatially
homogeneous spacetimes,
comprehensively analysed and classified by Ellis and MacCallum
\cite{em}.
According to Theorem 7.1 of \cite{em}, the only non--FRW
spatially homogeneous spacetimes
with $R^*_{ab}$ isotropic are Bianchi type I and
(non--axisymmetric) Bianchi type V. The former have $H_{ab}=0$.
For the latter, using
the shear eigenframe $\{{\bf e}_a\}$ of \cite{em}
\begin{equation}
\sigma_{ab} = \sigma_{22}\,\mbox{diag}(0,0,1,-1) \label{b0}
\end{equation}
Using (\ref{r1}) and (\ref{r2}) with (\ref{b0}), one
obtains
\begin{eqnarray}
E_{ab} &=& {\textstyle{1\over3}}\Theta\sigma_{ab}-\sigma_{c<a}
\sigma_{b>}{}^c \nonumber\\
&=&{\textstyle{1\over3}}
\sigma_{22}\,\mbox{diag}\left(0,2\sigma_{22},\Theta-\sigma_{22},
-\Theta-\sigma_{22}\right) \label{b0'}
\end{eqnarray}
in agreement with \cite{em}.\footnote{Note that
$E_{ab}$ in \cite{em} is the negative of $E_{ab}$ defined
in (\ref{eh}).}
The tetrad forms of div and curl
for type V are (compare \cite{vu}):
\begin{eqnarray}
\mbox{D}^b S_{ab}&=&\partial_b S_a{}^b-
3a^b S_{ab} \label{b2}\\
\mbox{curl}\, S_{ab} &=& \varepsilon_{cd(a}\partial^c
S_{b)}{}^d+\varepsilon_{cd(a}S_{b)}{}^c a^d \label{b3}
\end{eqnarray}
where $S_{ab}=S_{<ab>}$, $a_b=a\delta_b{}^1$
($a$ is the type V Lie algebra parameter) and
$\partial_a f$ is the directional derivative of $f$
along ${\bf e}_a$. Using (\ref{b3}) and (\ref{c2}):
\begin{eqnarray}
H_{ab} &=& \mbox{curl}\,\sigma_{ab}\nonumber\\
&=&-2a\sigma_{22}\delta_{(a}{}^2\delta_{b)}{}^3
\label{b1}\end{eqnarray}
Hence:\\ {\em Irrotational Bianchi V dust spacetimes in general
satisfy} $\mbox{div}\,H=0\neq H$.
Using (\ref{b0})---(\ref{b1}), one obtains
\begin{eqnarray}
\mbox{D}^bH_{ab}&=&0 \label{v1}\\
\mbox{curl}\, H_{ab}&=& -a^2\sigma_{ab} \label{v2}\\
\mbox{curl}\,\c H_{ab}&=& -a^2H_{ab} \label{v3}\\
\mbox{D}^bE_{ab} &=& -\sigma_{bc}\sigma^{bc}a_a \label{v4}\\
\mbox{curl}\, E_{ab} &=&{\textstyle{1\over3}}\Theta H_{ab} \label{v5}
\end{eqnarray}
Although (\ref{v1}) is a necessary condition for gravity waves,
it is not sufficient, and (\ref{b0'}) and (\ref{b1}) show that
$E_{ab}$ and $H_{ab}$ decay with the shear, so that
the type V solutions cannot be interpreted as gravity waves.
Nevertheless, these solutions do establish the existence of
spacetimes with $\mbox{div}\,H=0\neq H$.
This supplements the known result that the only spatially homogeneous
irrotational dust spacetimes with $H_{ab}=0$ are FRW, Bianchi types
I and VI$_{-1}$ $(n^a{}_a=0)$, and Kantowski--Sachs \cite{bmp}.
When $H_{ab}=0$, (\ref{b0}) and (\ref{b1}) show that $\sigma_{ab}=0$,
in which case the type V solution reduces to FRW.\\
A final remark concerns the special case $H_{ab}=0$, i.e. the
silent universes. The considerations of this paper show that the
consistency analysis of silent universes undertaken in \cite{lde}
needs to be re--examined. This is a further topic currently under
investigation. It seems likely that the silent universes, in the
full nonlinear theory, are {\em not} in general consistent.
\acknowledgements
Thanks to the referee for very helpful comments, and
to George Ellis, William Lesame and Henk van Elst
for very useful discussions.
This research was supported by grants from Portsmouth, Natal and
Cape Town Universities. Natal University, and especially Sunil
Maharaj, provided warm hospitality while part of this research
was done.
|
proofpile-arXiv_065-600
|
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
The Kadomtsev-Petviashvili (KP) equation and the whole KP
hierarchy of equations are significant parts of the theory
of integrable equations. They arise in various fields of physics
from hydrodynamics to string theory. They are also the tools to solve
several problems in mathematics from the differential geometry
of surfaces to an algebraic geometry.
The KP hierarchy has been described and studied within the framework of
different approaches.The Sato approach \cite{Sato}
(see also \cite{Jimbo}-\cite{Segal}) and the
$\bar{\partial}$-dressing method \cite{dbar1}-\cite{dbar4} are, perhaps,
the most beautiful and powerful among them. The infinite-dimensional
Grassmannian, pseudo-differential operators, Hirota bilinear
identity and $\tau$-function are the basic ingredients
of the Sato approach which is in essence algebraic.
In contrast, the $\bar{\partial}$-method, based on the nonlocal
$\bar{\partial}$-problem for the wave function, mostly uses the analytic properties
of the wave function. These two approaches look like completely different.
On the other hand, each of them has its own advantages.
So one could expect that their marriage could be rather profitable.
A bridge between the Sato approach and the $\bar{\partial}$-dressing method has been
established by the observation that the Hirota bilinear identity can be
derived from the $\bar{\partial}$-dressing method \cite{dbar4}, \cite{Carroll}.
Elements of the
approach which combines the characteristic features of both methods, namely,
the Hirota bilinear identity from the Sato approach and the analytic
properties of solutions from the $\bar{\partial}$-dressing method have been considered
in \cite{DZM}, \cite{NLS}.
This paper is devoted to the analytic-bilinear approach to integrable
hierarchies. It is based on the generalized Hirota bilinear identity for
the wave function with simple analytic properties (Cauchy-Baker-Akhiezer
(CBA) function). This approach allows us to derive generalized hierarchies
of integrable equations in terms of the CBA function
$\psi(\lambda,\mu,{\bf x})$ in a compact, finite form. Such a generalized
equations contain integrable equations in their usual form, their
modified partners and corresponding linear problems.
The generalized Hirota bilinear identity provides various functional
equations for the CBA function. The resolution of some of them leads to the
introduction of the $\tau$-function, while others give rise to the addition
theorems for the $\tau$-function. The properties of the $\tau$-function,
such as the associated closed 1-form and its global definition are also
arise in a simple manner
within the approach under consideration.
Generic discrete transformations of the CBA function and of the $\tau$-
function are presented in the determinant form. These transformations are
the generalized form of the usual iterated Darboux transformations.
The generalized KP hierarchy is presented also in the `moving'
frame depending on the parameter. This generalized KP hierarchy
written in the different `moving' frames contains the Darboux system
of equations.
In addition to the usual infinite-dimensional symmetries the generalized KP
hierarchy possess the symmetries given by the Combescure transformations.
The invariants of these symmetry transformations have the compact forms in
terms of the CBA function. The generalized KP hierarchy written in terms of
these invariants coincide with the usual KP hierarchy , mKP hierarchy and
SM-KP hierarchy.
The present paper is devoted to the one-component KP hierarchy.
The authors plan to consider multi-component KP hierarchy and
the 2-dimensional Toda lattice in subsequent papers.
The paper is organized as follows. Generalized Hirota identity
is introduced in section 2. The generalized KP hierarchy is derived in
section 3. Combescure symmetry transformations are discussed
in section 4. Generalized KP
hierarchy in the `moving' frame is considered in section 5. The $\tau$-function is introduced in section 6. The addition formulae for the $\tau $-function are also obtained here. Transformations of the CBA function and
the $\tau$-function given by the determinant formulae and the Darboux
transformations are discussed in section 7. Closed one-form variational
formulae for
the $\tau $-function are presented in section 8.
\section{Generalized Hirota identity}
The famous Hirota bilinear identity provides us condensed
and compact form of the integrable hierarchies.
Here we will derive the generalized Hirota bilinear identity
in frame of the $\bar{\partial}$-dressing method.
The $\bar{\partial}$-dressing method (see \cite{dbar1} - \cite{dbar4})
is based on the nonlocal $\bar{\partial}$-problem
of the form
\begin{eqnarray}
\bar{\partial}_{\lambda}(\chi({\bf x} ,\lambda)-\eta({\bf x},\lambda))=
\int\!\!\!\int_{\bf C}\: d\mu\wedge d\bar{\mu}\chi(\mu)g^{-1}(\mu)
R(\mu,\lambda)g(\lambda))
\label{dpr} \\
(\chi({\bf x} ,\lambda)-\eta({\bf x},\lambda))_
{|\lambda|\rightarrow\infty}\rightarrow 0.\nonumber
\end{eqnarray}
where $\lambda \in {\bf C},$ $ \bar{\partial}_{\lambda}={\partial /
\partial \bar{\lambda}}$,
$\eta({\bf x},\lambda)$ is a rational function
of $\lambda$ (normalization). In general case
the function $\chi(\lambda)$
and the kernel $R(\lambda,\mu)$ are matrix-valued
functions.
The dependence of the solution
$\chi(\lambda)$ of the problem (\ref{dpr}) on the dynamical
variables
is hidden in the function $g(\lambda)$.
Here we will consider only the case of continuous variables,
for which
$g_i=\exp(K_i x_i)$,
where $K_i(\lambda)$ are, in general, matrix meromorphic
functions.
It is assumed that the problem (\ref{dpr}) is uniquely solvable.
The $\bar{\partial}$-dressing method allows us to construct and solve
wide classes of nonlinear PDEs which correspond to the different
choice of the functions $K_i(\lambda)$ .
Here we will assume
that the kernel $R(\lambda,\mu)$
is equal to zero in some open subset $G$ of
the complex plane
with respect to $\lambda$ and to $\mu$.
This subset should typically include all zeroes
and poles of the considered class of functions
$g(\lambda)$ and a neighborhood of infinity.
In this case the solution of the problem (1)
normalized by $\eta$
is the function \[\chi(\lambda)=
\eta({\bf x},\lambda)+\varphi({\bf x},\lambda),\]
where $\eta(\lambda)$ is a rational function
of $\lambda$ (normalization), all poles of $\eta(\lambda)$
belong to $G$,
$\varphi(\lambda)$
decreases as $\lambda \rightarrow \infty$ and is
{\em analytic} in $G$.
The special class of solutions of the $\bar{\partial}$ problem
(\ref{dpr}) normalized by $(\lambda-\mu)^{-1}$
($\eta(\lambda)=(\lambda-\mu)^{-1},\quad \mu\in G$)
is of particular importance for the whole $\bar{\partial}$-dressing method
\cite{dbar2}.
Let us consider the $\bar{\partial}$-problem (\ref{dpr}) for such solutions
and corresponding dual problem for the dual function
$\chi^{\ast}(\lambda,\mu)$:
\begin{eqnarray}
\bar{\partial}_{\lambda}\chi(\lambda,\mu)
=2\pi i \delta(\lambda-\mu)+
\int\!\!\!\int_{\bf C} d\nu\wedge d\bar{\nu}\chi(\nu,\mu)g_1(\nu)
R(\nu,\lambda)g_1(\nu)^{-1},\nonumber\\
\bar{\partial}_{\lambda}\chi^{\ast}(\lambda,\mu)
=-2\pi i \delta(\lambda-\mu)-
\int\!\!\!\int_{\bf C} d\nu\wedge d\bar{\nu}g_2(\nu) R(\lambda,\nu)
g_2(\nu)^{-1}\chi^{\ast}(\nu,\mu).
\end{eqnarray}
where $g_1=g(\lambda,{\bf x})$, $g_2=g(\lambda,{\bf x'})$
After simple calculations, one gets
\begin{eqnarray}
\int\!\!\!\int_{G}d\nu\wedge d\bar{\nu}
{\partial\over\partial\bar{\nu}}\left(
\chi(\nu,\lambda;g_1)g_1(\nu)g_2(\nu)^{-1}
\chi^{\ast}(\nu,\mu;g_2)\right)=\nonumber\\
\int_{\partial G} \chi(\nu,\lambda;g_1)g_1(\nu)g_2(\nu)^{-1}
\chi^{\ast}(\nu,\mu;g_2)d\nu=0\,.
\label{HIROTA0}
\end{eqnarray}
In the
particular case $g_1=g_2$ from
(\ref{HIROTA0}) it follows
that in $\bar{G}$ the function $\chi(\lambda,\mu)$
is equal to $-\chi^{\ast}(\mu,\lambda)$
(see also \cite{ManZen}).
So finally we have
\begin{equation}
\int_{\partial G} \chi(\nu,\mu;g_1)g_1(\nu)g_2^{-1}(\nu)
\chi(\lambda,\nu;g_2)d\nu=0\,.
\label{HIROTA}
\end{equation}
Here the function $\chi(\lambda,\mu)$
possesses the following analytical properties:
$$
{\bar{\partial}}_{\lambda}\chi(\lambda,\mu)=
2\pi {\rm i}\delta(\lambda-\mu),
\quad
-{\bar{\partial}}_{\mu}
\chi(\lambda,\mu)=2\pi {\rm i}\delta(\lambda-\mu),
$$
where $\delta(\lambda-\mu)$ is a $\delta$-function,
or, in other words,
$\chi\rightarrow (\lambda-\mu)^{-1}$
as $\lambda
\rightarrow\mu$ and
$\chi(\lambda,\mu)$ is analytic function of
both variables $\lambda,\mu$ for $\lambda\neq\mu$.
In the particular case $\lambda=\mu=0$ the relation (\ref{HIROTA})
is nothing but the usual Hirota identity for the KP wave function
$\chi(\nu,0,{\bf x})$ and the dual KP wave function
$\chi^{\ast}(\nu,0,{\bf x'})$ (see e.g. \cite{Sato}-\cite{Segal}).
Just in this case
(for $\lambda=\mu=\infty$ ) the Hirota bilinear identity has been derived
from the $\bar{\partial}$-problem in \cite{Carroll}.
The identity (\ref{HIROTA}) represents itself the generalization of
the Hirota identity which is bilocal with respect to the dynamical
variables ${\bf x}$ and the spectral variables $(\lambda,\mu)$.
It can be considered as the point of departure without any reference
to the $\bar{\partial}$-dressing method. Namely, the generalized Hirota
bilinear relation (\ref{HIROTA}) is the starting point of the
analytic-bilinear approach which we will consider in this paper.
The double bilocality (with respect to ${\bf x}$ and
$(\lambda,\mu)$ provides us an additional freedom which will allow
us to represent integrable hierarchies in a unified and condensed form.
Introducing the function $\psi(\lambda,\mu,{\bf x})$
via
\begin{equation}
\psi(\lambda,\mu;g)=g^{-1}(\mu)\chi(\lambda,\mu;g)g(\lambda)\,,
\label{substitution}
\end{equation}
one gets another form of the generalized Hirota equation
\begin{equation}
\int_{\partial G} \psi(\nu,\mu;g_1)
\psi(\lambda,\nu;g_2)d\nu=0\,.
\label{HIROTA1}
\end{equation}
Note that in the framework of algebro-geometric technique
the function $\psi(\lambda,\mu)$ corresponds to the
Cauchy-Baker-Akhiezer kernel on the Riemann surface
(see \cite{Orlov}). We will refer to the function
$\psi(\lambda,\mu)$ as the Cauchy-Baker-Akhiezer (CBA)
function.
We will assume in what follows that we are able to find
solutions for the relation (\ref{HIROTA}) somehow.
In particular, it can be done by the
$\bar{\partial}$-dressing method.
The general setting of the problem
of solving (\ref{HIROTA}) requires some modification of
Segal-Wilson Grassmannian approach \cite{Segal}.
Let us consider two
linear spaces $W(g)$ and $\widetilde W(g)$
defined by the function $\chi(\lambda,\mu)$
(satisfying (\ref{HIROTA}))
via equations connected with equation (\ref{HIROTA})
\begin{eqnarray}
\int_{\partial G} f(\nu;g)\chi(\lambda,\nu;g)
d\nu=0,
\label{W}\\
\int_{\partial G}\chi(\nu,\mu;g)h(\nu;g)
d\nu=0 ,
\label{W'}
\end{eqnarray}
here
$f(\lambda)\in W$, $h(\lambda)\in \widetilde W$; $f(\lambda)$,
$h(\lambda)$ are defined in $\bar G$.
It follows from the definition of
linear spaces $W,\;\widetilde W$ that
\begin{eqnarray}
f(\lambda)&=&2\pi {\rm i}\int\!\!\!\int_G
\eta(\nu)\chi(\lambda,\nu)d\nu\wedge d\bar{\nu},\quad
\eta(\nu)=\left({\partial\over
\partial\bar\nu}
f(\nu)\right),\nonumber\\
h(\mu)&=-&2\pi {\rm i}\int\!\!\!
\int_G \chi(\nu,\mu)\widetilde\eta(\nu) d\nu\wedge d\bar{\nu},\quad
\widetilde\eta(\nu)=\left({\partial\over \partial\bar\nu}
h(\nu)\right).
\label{basis}
\end{eqnarray}
These formulae in some sense provide an expansion of the
functions $f\,,h$ in terms of the basic function $\chi(\lambda,\mu)$.
The formulae (\ref{basis}) readily imply that linear spaces
$W,\; \widetilde W$ are transversal to the space of holomorphic functions
in $G$ (transversality property).
{}From the other point of view, these formulae define a map of
the space of functions (distributions)
on $\bar G$ $\eta,\;\widetilde\eta$
to the spaces
$W$, $\widetilde W$. We will call
$\eta$ ($\widetilde\eta$) a {\em normalization} of the
corresponding function belonging to $W$ ($\widetilde W$).
The dynamics of linear spaces $W,\;\widetilde W$ looks very
simple
\begin{equation}
W(g)=W_0g^{-1};\quad \widetilde W(g)=g\widetilde W_0,
\label{dynamics}
\end{equation}
here $W_0=W(g=1)$, $\widetilde W_0=\widetilde W(g=1)$
(the formulae (\ref{dynamics})
follow from identity (\ref{HIROTA}) and the formulae (\ref{basis})).
To introduce a dependence on several variables (may be of different
type), one should consider a product of corresponding functions $g(\lambda)$
(all of them commute).
The formula (\ref{HIROTA}) is a basic tool for our construction.
Analytic properties of the CBA kernel accompanied with the
different choices of the functions $g_1$ and $g_2$ will provide
us various compact and useful relations.
\section{Generalized KP hierarchy}
In the present paper we will consider the scalar KP hierarchy.
It is generated by the generalized Hirota formula (\ref{HIROTA})
where $G$ is a unit disk
and
\begin{equation}
g({\bf x},\lambda)
=\exp\left(\sum_{i=1}^{\infty}x_i\lambda^{-i}\right)\,.
\label{gKP}
\end{equation}
Let us consider the formula (\ref{HIROTA}) with
\begin{eqnarray}
g_1 g_2^{-1}=
g({\bf x},\nu)g^{-1}({\bf x'},\nu)
=\exp\left(\sum_{i=1}^{\infty}(x_i-x_i')\nu^{-i}\right)=
{\nu-a\over\nu-b}\,.
\label{gKP1}
\end{eqnarray}
where $a$ and $b$ are arbitrary complex parameters,
$a\,,b\in G$. Since $\log(1-\epsilon)
=\sum_{i=1}^{\infty}\epsilon^i/i$, one has
$$
x'_i-x_i={1\over i}a^i-{1\over i}b^i\,.
$$
Substituting the expression (\ref{gKP1}) into
(\ref{HIROTA}), one gets
\begin{eqnarray}
&&\left({\mu-a\over\mu-b}\right)
\chi(\lambda,\mu,{\bf x}+[a])-
\left({\lambda-a\over\lambda-b}\right)
\chi(\lambda,\mu,{\bf x}+[b])+\nonumber\\
\nonumber\\
&&(b-a) \chi(\lambda,b,{\bf x}+[a])\chi(b,\mu,{\bf x}+[b])=0,
\quad \lambda\neq\mu
\label{KPbasic0}
\end{eqnarray}
where ${\bf x}+[a]=x_i+[a]_i\,,0 \leq i<\infty\,,[a]_i
={1\over i} a^i$.
Equation (\ref{KPbasic0}) is the simplest functional equation
for $\chi(\lambda,\mu,{\bf x})$ which follows from
the generalized Hirota equation (\ref{HIROTA}).
Residues of the l.h.s. of (\ref{KPbasic0}) at the
poles $\mu=b$ and $\lambda=b$ vanish identically.
Evaluating the l.h.s. of (\ref{KPbasic0}) at
$\mu=a$ and $\lambda=a$, one gets the equations
\begin{eqnarray}
\left({\lambda-a\over\lambda-b}\right)
\chi(\lambda,a,{\bf x}+[b])&=&
(b-a) \chi(\lambda,b,{\bf x}+[a])\chi(b,a,{\bf x}+[b])\,,
\label{KPbasic01}\\
\left({\mu-a\over\mu-b}\right)
\chi(a,\mu,{\bf x}+[a])&=&
(a-b)\chi(a,b,{\bf x}+[a])\chi(b,\mu,{\bf x}+[b])\,,
\label{KPbasic02}\\
a\neq b\,.&&\nonumber
\end{eqnarray}
Equations (\ref{KPbasic01}) and (\ref{KPbasic02})
imply that
\begin{eqnarray}
&&(\lambda-\mu)\chi(\lambda,\mu,{\bf x})=\nonumber\\
\nonumber\\
&&{(\lambda-a)\chi(\lambda,a,{\bf x}-[a]+[\mu])\over
(\mu-a)\chi(\mu,a,{\bf x}-[a]+[\mu])}=
{(a-\mu)\chi(a,\mu,{\bf x}+[a]-[\lambda])\over
(a-\lambda)\chi(a,\lambda,{\bf x}+[a]-[\lambda])}\,.
\label{KPbasic03}
\end{eqnarray}
Since $(\mu-a)\chi(\mu,a)\rightarrow 1$ as $\mu-a\rightarrow 0$,
one gets from (\ref{KPbasic03})
\begin{equation}
(\lambda-\mu)^2\chi(\lambda,\mu,{\bf x}+[\lambda])
\chi(\mu,\lambda,{\bf x}+[\mu])=-1\,.
\label{KPbasic04}
\end{equation}
We will solve the equations
(\ref{KPbasic01}) -(\ref{KPbasic04}) in the
next section. Now, let us consider the particular form of equation
(\ref{KPbasic0}) for $b=0$. In terms of the CBA function
it reads
\begin{equation}
\psi(\lambda,\mu,{\bf x}+[a])-\psi(\lambda,\mu,{\bf x})=
a \psi(\lambda,0,{\bf x}+[a])\psi(0,\mu,{\bf x});\quad
x'_i-x_i={1\over i} a^i\,.
\label{KPbasic}
\end{equation}
This equation is a condensed
finite form of the whole KP-mKP hierarchy.
Indeed, the expansion of this relation over $a$ generates
the KP-mKP hierarchies (and dual hierarchies)
and linear problems for them.
To demonstrate this, let us take the first three equations given by the
expansion of (\ref{KPbasic}) over~$ a$
\begin{eqnarray}
a:&\;\;&\psi(\lambda,\mu,{\bf x})_x=
\psi(\lambda,0,{\bf x})\psi(0,\mu,{\bf x})\,,
\label{KPbasic1}\\
a^2:&\;\;&\psi(\lambda,\mu,{\bf x})_y=
\psi(\lambda,0,{\bf x})_x\psi(0,\mu,{\bf x})-
\psi(\lambda,0,{\bf x})\psi(0,\mu,{\bf x})_x\,,
\label{KPbasic2}\\
a^3:&\;\;&\psi(\lambda,\mu,{\bf x})_t=
{1\over 4}\psi(\lambda,\mu,{\bf x})_{xxx} -
{3\over 4}\psi(\lambda,0,{\bf x})_x\psi(0,\mu,{\bf x})_x+\nonumber\\
&&{3\over 4}\left ( \psi(\lambda,0,{\bf x})_y\psi(0,\mu,{\bf x})-
\psi(\lambda,0,{\bf x})\psi(0,\mu,{\bf x})_y\right )
\label{KPbasic3} \\
&&x=x_1;\quad y=x_2;\quad t=x_3\,.\nonumber
\end{eqnarray}
In the order $ a^2$ equation (\ref{KPbasic})
gives rise equivalently to the equations
\begin{eqnarray}
\psi(\lambda,\mu,{\bf x})_y
-\psi(\lambda,\mu,{\bf x})_{xx}&=&-
2\psi(\lambda,0,{\bf x})\psi(0,\mu,{\bf x})_x\,,
\label{KPbasic2+}\\
\psi(\lambda,\mu,{\bf x})_y
+\psi(\lambda,\mu,{\bf x})_{xx}&=&
2\psi(\lambda,0,{\bf x})_x\psi(0,\mu,{\bf x})\,,
\label{KPbasic2-}
\end{eqnarray}
Evaluating the first equation at $\mu=0$, the second
at $\lambda=0$ one gets
\begin{eqnarray}
f({\bf x})_y-f({\bf x})_{xx}&=&
u({\bf x})f({\bf x}),\label{LKP}\\
\widetilde f({\bf x})_y+\widetilde f({\bf x})_{xx}&=&-
u({\bf x})\widetilde f({\bf x})
\label{LKPd}
\end{eqnarray}
where $u({\bf x})=-2\psi(0,0)_x$ and
$$f=\int\psi(\lambda,0)
\rho(\lambda)d\lambda\,,$$
$$\widetilde f=\int\widetilde\rho(\mu)\psi(0,\mu)
d\mu\,;$$
$\rho(\lambda)$ and $\widetilde\rho(\mu)$ are some
arbitrary functions.
In a similar manner, one obtains from (\ref{KPbasic1})-
(\ref{KPbasic3}) the equations
\begin{eqnarray}
f_t-f_{xxx}={3\over 2}u f_x+
{3\over 4}(u_x+\partial_x^{-1}u_y)f\,,
\label{AKP}\\
\widetilde f_t-\widetilde f_{xxx}={3\over 2}u \widetilde f_x+
{3\over 4}(u_x-\partial_x^{-1}u_y)\widetilde f\,.
\label{AKPd}
\end{eqnarray}
Both the linear system (\ref{LKP}), (\ref{AKP}) for the
wave function $f$ and the linear system (\ref{LKPd}), (\ref{AKPd})
for the
wave function $\widetilde f$ give rise to the same
KP equation
\begin{equation}
u_t={1\over 4}u_{xxx}+{3\over2}uu_x+{3\over4}\partial_x^{-1}u_{yy}\,.
\end{equation}
To derive linear problems for the mKP and dual mKP
equations, we integrate equations
(\ref{KPbasic1}), (\ref{KPbasic2+}), (\ref{KPbasic2-})
and (\ref{KPbasic3})
with the two arbitrary functions $\rho(\lambda)$, $\widetilde\rho(\mu)$
\begin{eqnarray}
\Phi({\bf x})_x&=&
f({\bf x})\widetilde f ({\bf x})\,,\\
\Phi({\bf x})_y-\Phi({\bf x})_{xx}&=&-
2f({\bf x})\widetilde f({\bf x})_x\,,\\
\Phi({\bf x})_y+\Phi({\bf x})_{xx}&=&
2f({\bf x})_x\widetilde f({\bf x})\,,\\
\Phi({\bf x})_t-\Phi({\bf x})_{xxx} &=&-
{3\over2}f({\bf x})_x\widetilde f({\bf x})_x-
{3\over4}(f({\bf x})\widetilde f({\bf x})_y-
f({\bf x})_y\widetilde f({\bf x}))
\end{eqnarray}
where
$$\Phi=\int\!\!\!\int
\widetilde\rho(\mu)\psi(\lambda,\mu)
\rho(\lambda)d\lambda\,d\mu\,.$$
Using the first equation to exclude $f$ from the second
(and $\widetilde f$ from the third), we obtain
\begin{eqnarray}
\Phi_y-\Phi_{xx}&=&
v({\bf x})\Phi_x\,,
\label{mKPlinear+}\\
\Phi_y+\Phi_{xx}&=&
-\widetilde v({\bf x})\Phi_x
\label{mKPlinear-}
\end{eqnarray}
where $v=-2{\widetilde f({\bf x})_x\over\widetilde f({\bf x})}$,
$\widetilde v=2{f({\bf x})_x\over f({\bf x})}$.
Similarly, one gets from (\ref{KPbasic3})
\begin{eqnarray}
\Phi_t-\Phi_{xxx}&=&
{3\over2}v({\bf x})\Phi_{xx}+
{3\over4}(v_x+v^2+\partial_x^{-1}v_y)\Phi_x\,,
\label{mKPA+}\\
\Phi_t-\Phi_{xxx}&=&
{3\over2}\widetilde v({\bf x})\Phi_{xx}+
{3\over4}(\widetilde v_x+v^2-\partial_x^{-1}\widetilde v_y)\Phi_x\,.
\label{mKPA-}
\end{eqnarray}
The system (\ref{mKPlinear+}), (\ref{mKPA+}) gives rise
to the mKP equation
\begin{equation}
v_t=v_{xxx}+{3\over4} v^2v_x +3v_x\partial_x^{-1}v_y+
3\partial_x^{-1}v_{yy}\,,
\label{mKP}
\end{equation}
while the system
(\ref{mKPlinear-}), (\ref{mKPA-}) leads to the dual mKP equation,
which is obtained from the (\ref{mKP}) by the substitution
$v\rightarrow\widetilde v$, $t\rightarrow -t$, $y\rightarrow -y$,
$x\rightarrow -x$.
So the function $\Phi$ is simultaneously a wave function
for the mKP and dual mKP linear problems with different potentials,
defined by the dual KP (KP) wave functions.
Using the equation (\ref{KPbasic3}) and relations (\ref{mKPlinear+})
and (\ref{mKPlinear-}), one also obtains an equation for
the function $\Phi$
\begin{eqnarray}
\Phi_t-{1\over4}\Phi_{xxx}-{3\over8}
{\Phi_y^2-\Phi_{xx}^2\over\Phi_x}+
{3\over 4}\Phi_x W_y&=&0 ,\quad W_x={\Phi_y\over\Phi_x}\,.
\label{singman}
\end{eqnarray}
This equation first arose in Painleve analysis of the KP
equation as a singularity manifold equation \cite{Weiss}.
It is tedious but absolutely straightforward check that
the expansion of (\ref{KPbasic}) in higher orders of $a$
generates\\
{\bf 1}) the whole hierarchy of KP singularity manifold equations
for $\psi(\lambda,\mu)$ (or $\Phi({\bf x})$\\
{\bf 2}) the hierarchy of linear problems for the mKP and dual
mKP equations, where $\psi(\lambda,\mu)$ (or $\Phi({\bf x})$
is the common wave function and
$v=-2(\log\psi(0,\lambda,{\bf x}))_x$,
$\widetilde v=-2(\log\psi(\lambda,0,{\bf x}))_x$
are the potentials\\
{\bf 3}) mKp hierarchy for $v$ and dual mKP hierarchy for
$\widetilde v$\\
{\bf 4}) the hierarchies of KP linear problems for $\psi(\lambda,0,{\bf x})$
and dual KP linear problems for $\widetilde\psi(\lambda,0,{\bf x})$\\
and, finally\\
{\bf 5}) the KP hierarchy of equations for
$u=-2\psi(0,0)_x$.
Note also one interesting consequence of the formula
(\ref{KPbasic04})
\begin{equation}
\chi(0,\lambda,{\bf x})=-{1\over\lambda^2}
\chi^{-1}(\lambda,0,{\bf x}+[\lambda])\,.
\end{equation}
\section{KP hierarchy in the `moving frame'. Darboux equations as the
horizontal
subhierarchy}
Now let us consider the expansion of the l.h.s. of (\ref{KPbasic0})
over $\epsilon=a-b$, where $\epsilon\rightarrow 0$. In the first
order in $\epsilon$ one gets
\begin{eqnarray}
\Delta_1(b)\psi(\lambda,\mu,{\bf x})=\psi(b,\mu,{\bf x})
\psi(\lambda,b,{\bf x})
\end{eqnarray}
where $$\Delta_1(b)=\sum_{n=1}^{\infty}b^{n-1}
{\partial\over\partial x_n}.$$ In the higher orders in
$\epsilon$ one obtains the hierarchy of equations of the form
(\ref{KPbasic1})-(\ref{KPbasic3}) and their higher analogues
with the substitution $\psi(\lambda,0,{\bf x})\rightarrow
\psi(\lambda,b,{\bf x})$, $\psi(0,\mu,{\bf x})\rightarrow
\psi(b,\mu,{\bf x})$ and $${\partial\over\partial x_i}
\rightarrow \Delta_i(b)=\sum_{n=i}^{\infty}
{n!\over n(n-i)!i!}
b^{n-i}{\partial\over\partial x_i}.$$
Such a substitution is in fact nothing
but the change of dynamical variables (or the coordinates on
the group of functions $g$). Indeed, it is not difficult to show
that $ \Delta_i(b)={\partial\over\partial x_i(b)}$,
where the dynamical variables $x_i(b)$ are defined by the relation
\begin{eqnarray}
\sum_{i=1}^{\infty}{x_i(b)\over (\lambda-b)^i}=
\sum_{i=1}^{\infty}{x_i\over (\lambda)^i}\,.
\end{eqnarray}
It is clear that
\begin{eqnarray}
\left [{\partial\over\partial x_i(\lambda')},
{\partial\over\partial x_i(\lambda)}\right]=
0\,.
\end{eqnarray}
Note one interesting property of the derivatives
${\partial\over\partial x_i(b)}$, namely
\begin{eqnarray}
\left [{\partial\over\partial \lambda},
{\partial\over\partial x_i(\lambda)}\right]=
(i+1){\partial\over\partial x_{i+1}(\lambda)}\,.
\end{eqnarray}
So the operator ${\partial\over\partial \lambda}$
is a `mastersymmetry' for all vector fields
${\partial\over\partial x_i(\lambda)}$.
The expansion of equation (\ref{KPbasic0})
up to the third order in $\epsilon $ gives
the equations
\begin{equation}
\frac \partial {\partial x_1(b)}\psi (\lambda ,\mu ,x(b))=\psi (b,\mu
,x(b))\psi (\lambda ,b,(x(b)),
\label{m1}
\end{equation}
\begin{equation}
\frac \partial {\partial x_2(b)}\psi (\lambda ,\mu ,x(b))=\frac \partial
{\partial x_1(b)}\psi (\lambda ,b)\cdot \psi (b,\mu )-\psi (\lambda ,b)\cdot
\frac \partial {\partial x_1(b)}\psi (b,\mu ),
\label{m2}
\end{equation}
\begin{eqnarray}
\frac \partial {\partial x_3(b)}\psi (\lambda ,\mu ,x(b))=\frac 14\frac{%
\partial ^3}{\partial x_1(b)^3}\psi (\lambda ,\mu )-\frac 34\frac \partial
{\partial x_1(b)}\psi (\lambda ,b)\cdot \frac \partial {\partial x_1(b)}\psi
(b,\mu )+
\nonumber\\
+\frac 34\left( \frac \partial {\partial x_2(b)}\psi (\lambda ,b)\cdot \psi
(b,\mu )-\psi (\lambda ,b)\cdot \frac \partial {\partial x_2(b)}\psi (b,\mu
)\right) .
\label{m3}
\end{eqnarray}
The analogues of equations (\ref{KPbasic2+}),
(\ref{KPbasic2-}) have the form
\begin{equation}
\frac \partial
{\partial x_2(b)}\psi (\lambda ,\mu ,x(b))-\frac{\partial ^2}{%
\partial x_1(b)^2}\psi (\lambda ,\mu )+2\psi (\lambda ,b)\frac \partial
{\partial x_1(b)}\psi (b,\mu )=0,
\label{m2+}
\end{equation}
\begin{equation}
\frac \partial {\partial x_2(b)}
\psi (\lambda ,\mu ,x(b))+\frac{\partial ^2}{%
\partial x_1(b)^2}\psi (\lambda ,\mu )-2\frac \partial {\partial x_1(b)}\psi
(\lambda ,b)\cdot \psi (b,\mu )=0.
\label{m2-}
\end{equation}
Equations (\ref{m1})-(\ref{m3}) and
higher equations again give rise to the generalized KP
hierarchy but now in coordinates $x_i(b),i=1,2,3...$. For such KP hierarchy
written in the `moving'
frame the parameter b is an arbitrary one, but fixed.
Let us consider now equations of the type (\ref{m1})
written for several values of
$b$. We denote $x_1(b_\alpha )=\xi _\alpha ,\alpha =1,2,...,n.$ Equations
(\ref{m1})
taken for $b=b_\alpha ,\lambda =b_\beta ,\mu =b_\gamma (\alpha \neq \beta
\neq \gamma ),$ look like
\begin{equation}
\frac \partial {\partial \xi _\alpha }\psi _{\beta \gamma }=\psi _{\beta
\alpha }\psi _{\alpha \gamma }\,,\quad\alpha \neq \beta \neq \gamma
\label{m5}
\end{equation}
where $\psi _{\alpha \beta }=\psi (b_\alpha ,b_{\beta ,}x).$ The system
(\ref{m5})
is just well-known system of $N^2-N$ resonantly interacting waves.
Integrating equations (\ref{m1})
over $\mu $ with the function $\rho (\mu )$ and
evaluating the result at $b=b_\alpha ,\gamma =b_\beta $, one gets
\begin{equation}
\frac{\partial f_\beta }{\partial \xi _\alpha }=\psi _{\beta \alpha
}f_\alpha\,,\quad(\alpha \neq \beta )
\label{m6}
\end{equation}
where $f_\beta =\int d\mu \psi (b_{\beta ,}\mu )\rho (\mu )$. Analogously
one gets
\begin{equation}
\frac{\partial f_\beta ^{*}}{\partial \xi _\alpha }=\psi _{\alpha \beta
}f_\alpha ^{*}\,,\quad(\alpha \neq \beta )
\label{m7}
\end{equation}
where $f_\beta ^{*}=\int d\lambda \psi (\lambda ,b_\beta )\rho ^{*}(\lambda
) $ and $\rho ^{*}(\lambda )$ is an arbitrary function. The systems
(\ref{m6})
and (\ref{m7})
are the linear problem and dual linear problem for equations (\ref{m5}),
respectively.
Expressing $\psi _{\alpha \beta }$ via
$f_\alpha $ and $f_\alpha ^{*}$, one
gets from (\ref{m6})
and (\ref{m7}) (using (\ref{m5}))
the same system for $f_\alpha $ and $f_\alpha ^{*}$
\begin{equation}
\frac{\partial ^2H_\alpha }{\partial \xi _\beta \partial \xi _\gamma }=\frac
1{H_\beta }\frac{\partial H_\beta }{\partial \xi _\gamma }\frac{\partial
H_\alpha }{\partial \xi _\beta }+\frac 1{H_\gamma }
\frac{\partial H_\gamma }{%
\partial \xi _\beta }\frac{\partial H_\alpha }{\partial \xi _\gamma }%
\,,\quad(\alpha \neq \beta \neq \gamma \neq \alpha ).
\label{Darboux}
\end{equation}
The system (\ref{Darboux})
is the Darboux system which was introduced for the first time
in the differential geometry of surfaces [14] and then was rediscovered in
the matrix form within the $\partial -$dressing method in the paper [5].
Note that the Darboux equations in the variables of the type $x_1(b_\alpha )$
have appeared also in the paper
\cite{Nijhoff} within completely different approach.
One can treat the Darboux equations (\ref{Darboux})
with different n as the horizontal
subhierarchy of the whole generalized KP hierarchy. Note that equations
(\ref{m1})-(\ref{m3})
and their higher analogues give rise to the higher resonantly interacting
waves equations.
\section{Combescure symmetry transformations for the
generalized KP hierarchy}
Let us consider now the symmetries of the equations
derived above.
All the higher equations of the hierarchy are, as usual,
the symmetries of each member of the hierarchy. Here we will
discuss another type of symmetries.
Since $\rho(\lambda)$ and $\widetilde\rho(\mu)$ are arbitrary functions,
equation (\ref{singman}) and the hierarchy (\ref{KPbasic})
possess the symmetry transformation
$$\Phi(\rho(\lambda),\widetilde\rho(\mu))\rightarrow \Phi'=
\Phi(\rho'(\lambda),\widetilde\rho'(\mu))\,.$$
This transformation is, in fact, the transformation which
changes the normalization of the wave functions. The fact
that such transformations are connected with the so-called
Combescure transformations,known for a long time in differential
geometry, was pointed out in \cite{DZM}.
The Combescure transformation was introduced last century
within the study of the transformation properties of
surfaces (see e.g. \cite{Darboux}, \cite{GEOMA}). It is a transformation
of surface such that the tangent vector at a given point of the surface
remains parallel. The Combescure transformation is essentially
different from the well-known B\"acklund and Darboux transformations.
The Combescure transformation plays an important role in the theory of the
systems of hydrodynamical type \cite{Tsarev}. It is also of great interest
for the theory of (2+1)-dimensional integrable systems \cite{Kon2}.
Combescure symmetry transformations
are essential part of the analytic-bilinear approach.
The Combescure transformation can be
characterized in terms of the
corresponding invariants. The simplest of
these invariants for the mKP equation is just the potential of
the KP equation L-operator expressed through the
wave function
\begin{eqnarray}
u&=&{f({\bf x})_y-f({\bf x})_{xx} \over f({\bf x})}\,,
\label{invar1}\\
u&=&{\widetilde f({\bf x})_y-\widetilde f({\bf x})_{xx}
\over \widetilde f({\bf x})}\,,
\label{invar2}
\end{eqnarray}
or, in terms of the solution for the mKP (dual mKP) equation
\begin{eqnarray}
v'_y+v'_{xx}-{1\over 2}((v')^2)_x&=&v_y+v_{xx}-{1\over 2}(v^2)_x\,,
\label{invar01}\\
\widetilde v'_y-\widetilde v'_{xx}-{1\over 2}((\widetilde v')^2)_x&=&
\widetilde v_y-\widetilde v_{xx}-{1\over 2}(\widetilde v^2)_x\,.
\label{invar02}
\end{eqnarray}
The solutions of the mKP equations are transformed only by
a subgroup of the Combescure symmetry group corresponding
to the change of the weight function $\widetilde\rho(\mu)$
(left subgroup)
and they are invariant under the action of the subgroup
corresponding to $\rho(\lambda)$ (vice versa for the dual
mKP).
All the hierarchy of the Combescure transformation
invariants is given by the expansion over $\epsilon$
near the point ${\bf x}$ of the
relation (\ref{KPbasic}) rewritten in the form
\begin{eqnarray}
{\partial \over \partial \epsilon}
\left ({\widetilde f({\bf x}')-\widetilde f({\bf x})\over
\epsilon \widetilde f({\bf x})}\right ) &=&
-{1\over2}{\partial \over \partial \epsilon}
\partial^{-1}_{x'} u({\bf x}'),\quad
x'_i-x_i={1\over i}\epsilon^i;\\
{\partial \over \partial \epsilon}
\left ({f({\bf x})-f({\bf x'})\over
\epsilon f({\bf x})}\right ) &=&
{1\over2}{\partial \over \partial \epsilon}
\partial^{-1}_{x'} u({\bf x}')\,,\quad
x'_i-x_i=-{1\over i}\epsilon^i.
\end{eqnarray}
The expansion of the left part of these relations gives the
Combescure transformation invariants in terms of the wave functions
$\widetilde f$, $f$. To express them in terms of mKP equation (dual mKP
equation) solution, one should use the formulae
\begin{eqnarray}
v&=&-2{{\widetilde f}_x\over \widetilde f}\,,
\quad \widetilde f=\exp(-{1\over2}\partial_x^{-1}v);\\
\widetilde v&=&2{{f}_x\over f},
\quad f=\exp({1\over2}\partial_x^{-1}\widetilde v)\,.
\end{eqnarray}
It is also possible to consider special Combescure transformations
keeping invariant the KP equation (dual KP equation) wave functions
(i.e. solutions for the dual mKP (mKP) equations). The first invariants
of this type are
\begin{eqnarray}
{\Phi'_x({\bf x}) \over \widetilde f'({\bf x})}
&=&{\Phi_x({\bf x}) \over \widetilde f({\bf x})}\,,\\
{\Phi'_x({\bf x}) \over f'({\bf x})}
&=&{\Phi_x({\bf x}) \over f({\bf x})}\,.
\end{eqnarray}
All the hierarchy of the invariants of this type is generated
by the expansion of the left part of the following relations over
$\epsilon$
\begin{eqnarray}
\left ({\Phi({\bf x}')-\Phi({\bf x})\over
\widetilde f({\bf x})}\right ) &=&
\epsilon f({\bf x}')\,,\quad
x'_i-x_i={1\over i}\epsilon^i;\\
\left ({\Phi({\bf x})-\Phi({\bf x'})\over
f({\bf x})}\right ) &=&
\epsilon \widetilde f({\bf x}')\,,\quad
x'_i-x_i=-{1\over i}\epsilon^i\,.
\end{eqnarray}
Now let us consider the equation (\ref{singman}) and all the
hierarchy given by the relation (\ref{KPbasic}).
This equation admits the Combescure group of symmetry transformations
$\Phi(\rho(\lambda),\widetilde\rho(\mu))\rightarrow \Phi'=
\Phi(\rho'(\lambda),\widetilde\rho'(\mu))$
consisting of two subgroups (right and left Combescure
transformations). These subgroups have the following invariants
\begin{equation}
v={\Phi_y-\Phi_{xx}\over\Phi_x}
\label{Comb+}
\end{equation}
and
\begin{equation}
\widetilde v ={\Phi_y+\Phi_{xx}\over\Phi_x}\,.
\label{Comb-}
\end{equation}
{}From (\ref{mKPlinear+}), (\ref{mKPlinear-}) it follows that they
just obey the mKP and dual mKP equation respectively.
The invariant for the full Combescure transformation can be
obtained by the substitution of the expression for $v$ via $\Phi$
(\ref{Comb+}) to the formula (\ref{invar01}). It reads
\begin{equation}
u=\partial_x^{-1}\left({\Phi_y\over\Phi_x}\right)_y-
{\Phi_{xxx}\over\Phi_x} + {\Phi_{xx}^2-\Phi_y^2\over 2 \Phi_x^2}\,.
\label{invarfull}
\end{equation}
{}From (\ref{LKP}), (\ref{LKPd}), (\ref{invar1}), (\ref{invar2}),
(\ref{invar01}), (\ref{invar02}) it follows that $u$ solves
the KP equation.
So there is an interesting connection between equation (\ref{singman}),
mKP-dual mKP equations and KP equation. Equation (\ref{singman})
is the unifying equation. It possesses a Combescure symmetry
transformations group. After the factorization of equation
(\ref{singman}) with respect to one of the subgroups (right or left),
one gets the mKP or dual mKP equation in terms of the invariants
for the subgroup (\ref{Comb+}), (\ref{Comb-}).
The factorization of equation (\ref{singman})
with respect to the full Combescure transformations group
gives rise to the KP equation in terms of the invariant
of group (\ref{invarfull}).
In other words, the invariant of equation (\ref{singman})
under the full Combescure group is described by the KP equation,
while the invariants under the action of its right and left subgroups
are described by the mKP or dual mKP equations.
\section{$\tau$-function and addition formulae}
Now we will analyze the functional equations
(\ref{KPbasic0})-(\ref{KPbasic04}). Equation
(\ref{KPbasic03}), evaluated at $\mu=0$
for some $a=a_0$ gives
\begin{eqnarray}
\lambda\chi(\lambda,0,{\bf x})=
{(\lambda-a)\chi(\lambda,a_0,{\bf x}-[a_0])\over
(-a)\chi(\mu,a,{\bf x}-[a_0])}=
{Z(\lambda\,,{\bf x})\over Z(0\,,{\bf x})}
\label{KPbasic003}
\end{eqnarray}
where we denote $Z(\lambda\,,{\bf x})=
(\lambda-a_0)\chi(\lambda,a_0,{\bf x}-[a_0])$.
Substituting the expression (\ref{KPbasic003}) into equation
(\ref{KPbasic03}), we get
\begin{eqnarray}
(\lambda-\mu)\chi(\lambda,\mu,{\bf x})=
{Z(\lambda\,,{\bf x}+[\mu])\over Z(\mu\,,{\bf x}+[\mu])}\,.
\label{KPbasic004}
\end{eqnarray}
It is easy to check that in virtue of (\ref{KPbasic004})
equation (\ref{KPbasic01}) is satisfied identically,
while equation (\ref{KPbasic02}) takes the form
\begin{equation}
R(a,\lambda)R(\lambda,b)R(b,a)=
R(a,b)R(b,\lambda)R(\lambda,a)
\label{tri}
\end{equation}
where $R(a,b,{\bf x})=Z(a,x+[a]+[b])$.
In terms of $R(a,b)$ we have
\begin{eqnarray}
(\lambda-\mu)\chi(\lambda,\mu,{\bf x})=
{R(\lambda,\mu,{\bf x}-[\lambda])\over R(\mu,\mu,{\bf x}-[\mu])}\,.
\label{KPbasic005}
\end{eqnarray}
Thus the problem of resolving equations (\ref{KPbasic01}),
(\ref{KPbasic02}) is reduced to the single functional
equation (\ref{tri}), which is of the form of the triangle
(Yang-Baxter) equation, well-known in the quantum theory of
solvable models (see e.g. \cite{triangle}).
{}From the definition of $R(a,b,{\bf x})$ it follows that it has
a certain special structure. Indeed, since $Z(a,{\bf x})=
R(a,b,{\bf x}-[a]-[b])$, one has $R(a,b,{\bf x}-[a]-[b])=R(a,0,x-[a])$.
Consequently $R(a,b,{\bf x})=R(a,0,x+[b])$.
So we should solve the triangle equation (\ref{tri})
within the class of $R$ of the form
$R(a,b,{\bf x})=\Xi_a({\bf x}+[b])$, where $\Xi_a({\bf x})$
is some function.
Taking the logarithm of both parts of (\ref{tri}),
one gets
\begin{equation}
\Theta(a,\lambda)+\Theta(\lambda,b)+\Theta(b,a)=
\Theta(a,b)+\Theta(b,\lambda)+\Theta(\lambda,a)
\label{trilog}
\end{equation}
where $\Theta=\log R$.
Representing $\Theta$ as $\Theta(a,b,{\bf x})=\Theta_+(a,b,{\bf x})+
\Theta_-(a,b,{\bf x})$, where $\Theta_+$ and $\Theta_-$ are
respectively symmetric and antisymmetric parts of $\Theta$,
one easily concludes that $\Theta_+$ solves (\ref{trilog})
identically while $\Theta_-$ satisfies the equation
\begin{equation}
\Theta_-(a,\lambda)+\Theta_-(\lambda,b)+\Theta_-(b,a)=0\,.
\label{trilog1}
\end{equation}
Taking equation (\ref{trilog1}) at $b=0$, one gets
\begin{equation}
\Theta_-(a,\lambda)=\Theta_-(0,\lambda)-\Theta_-(0,a)\,.
\label{trilog2}
\end{equation}
Then since $\Theta(a,b,{\bf x})$ (as $R(a,b,{\bf x})$)
has the form $\Theta(a,b,{\bf x})=Z_a({\bf x}+[b])$,
where $Z_a$ are some functions, it follows from
(\ref{trilog2})
that
$$
\Theta_-(a,\lambda)=Z_{0-}({\bf x}+[\lambda])-
Z_{0-}({\bf x}+[a])\,.
$$
Then for the symmetric part of $\Theta_+$ one has
$Z_{a+}({\bf x}+[b])=Z_{b+}({\bf x}+[a])$.
Taking $b=0$, one concludes that
$Z_{a+}({\bf x})=Z_{0+}({\bf x}+[a])$
So
$\Theta_+(a,b)=Z_{0+}({\bf x}+[a]+[b])$
Thus general
solution of(\ref{trilog}) has the form
$$
\Theta(a,b,{\bf x})= Z_{0+}({\bf x}+[a]+[b])+
Z_{0-}({\bf x}+[b])-Z_{0-}({\bf x}+[a])\,.
$$
Consequently, the general solution of (\ref{tri}) for our class
of $R(a,b,{\bf x})$ reads
\begin{equation}
R(a,b,{\bf x})=R_s({\bf x}+[a]+[b])
{\tau({\bf x}+[b])\over\tau({\bf x}+[a])}
\label{trilog3}
\end{equation}
where $R_s$ and $\tau$ are arbitrary functions.
Substituting now the expression (\ref{trilog3})
into the expression (\ref{KPbasic005}), we get
\begin{equation}
\chi(\lambda,\mu,{\bf x})={1\over(\lambda-\mu)}
{\tau({\bf x}-[\lambda]+[\mu])
\over \tau({\bf x})}
\label{tauform}
\end{equation}
This formula coincides with the formula introduced
in the paper \cite{NLS} in a completely different context.
Note that in our approach the function $\tau({\bf x})$ is still
an arbitrary function giving a general solution of
the functional equations (\ref{KPbasic01}), (\ref{KPbasic02})
through the formula (\ref{tauform}).
Now we will use the general equation (\ref{KPbasic0}).
Substituting (\ref{tauform}) into
(\ref{KPbasic0}), one gets
\begin{eqnarray}
&&(a-\mu)(\lambda-b)\tau({\bf x}+[a]+[\mu])
\tau({\bf x}+[\lambda]+[b])+\nonumber\\
&&(\lambda-a)(b-\mu)\tau({\bf x}+[\lambda]+[a])
\tau({\bf x}+[b]+[\mu])+\nonumber\\
&&(b-a)(\lambda-\mu)\tau({\bf x}+[b]+[a])
\tau({\bf x}+[\lambda]+[\mu])
=0\,.
\label{taubasic}
\end{eqnarray}
It is nothing but the simplest addition formula for the
$\tau$-function derived in \cite{Sato},
which is closely connected
with the Fay's trisecant formula \cite{Fay}.
Generalized Hirota identity gives rise also to other
addition formulae from \cite{Sato}.
Indeed, let us choose in (\ref{HIROTA})
\begin{equation}
g({\bf x})g^{-1}({\bf x'})=\prod_{\alpha=1}^n
{\nu-a_{\alpha}\over \nu-b_{\alpha}}
\end{equation}
where $n$ is an arbitrary integer and
${\bf x'}-{\bf x}=\sum_{\alpha=1}^n [a_{\alpha}]-[b_{\alpha}]$.
In this case equation (\ref{HIROTA}) gives
\begin{eqnarray}
\prod_{\alpha=1}^n{\mu-a_{\alpha}\over\mu-b_{\alpha}}
\chi\left(\lambda,\mu,{\bf x}+\sum_{\alpha=1}^n[a_{\alpha}]\right)-
\prod_{\alpha=1}^n {\lambda-a_{\alpha}\over\lambda-b_{\alpha}}
\chi\left(\lambda,\mu,{\bf x}+\sum_{\alpha=1}^n[b_{\alpha}]\right)+&&\nonumber\\
\sum_{\alpha=1}^n
(b_{\alpha}-a_{\alpha})\prod_{\gamma\,,\gamma\neq\alpha}
{b_{\alpha}-a_{\gamma}\over b_{\alpha}-b_{\gamma}}
\chi\left(\lambda,b_{\alpha},{\bf x}+\sum_{\delta=1}^n[a_{\delta}]\right)
\chi\left(b_{\alpha},\mu,{\bf x}+\sum_{\delta=1}^n[b_{\delta}]\right)=0\,.&&
\label{KPbasic00}
\end{eqnarray}
Substituting the expression (\ref{tauform}) into (\ref{KPbasic00})
and shifting ${\bf x}\rightarrow {\bf x}+[\lambda]$, one gets
\begin{eqnarray}
&&
\prod_{\alpha=1}^n{(\mu-a_{\alpha})\over(\mu-b_{\alpha})(\mu-\lambda)}
\tau\left({\bf x}+[\mu]+\sum_{\gamma=1}^n[a_{\gamma}]\right)
\tau\left({\bf x}+[\lambda]+\sum_{\gamma=1}^n[b_{\gamma}]\right)-\nonumber\\
&&
\prod_{\alpha=1}^n
{(\lambda-a_{\alpha})\over(\lambda-b_{\alpha})(\lambda-\mu)}
\tau\left({\bf x}+[\mu]+\sum_{\gamma=1}^n[b_{\gamma}]\right)
\tau\left({\bf x}+[\lambda]+\sum_{\gamma=1}^n[a_{\gamma}]\right)
+\nonumber\\
&&
\sum_{\alpha=1}^n
{(b_{\alpha}-a_{\alpha})\over(b_{\alpha}-\mu)(\lambda-b_{\alpha})}
\prod_{\gamma\,,\gamma\neq\alpha}
{b_{\alpha}-a_{\gamma}\over b_{\alpha}-b_{\gamma}}
\times\nonumber\\&&
\tau\left({\bf x}+[b_{\alpha}]+\sum_{\beta=1}^n[a_{\beta}]\right)
\tau\left({\bf x}+[\lambda]+[\mu]+
\sum_{\beta\,,\beta\neq\alpha}[b_{\beta}]\right)
=0\,,
\label{taubasic00}
\end{eqnarray}
It is not difficult to check (after some renotations) that equation
(\ref{taubasic00}) coincide with the Pl\"ucker's relations
for universal Grassmannian manifold (see Theorem 3 of \cite{Sato}).
That means, according to the Theorems 1-3 of the
paper \cite{Sato} that $\tau$ is the $\tau$ function of the
KP hierarchy. Note that the formulae (\ref{tauform}),
(\ref{taubasic00}) provide solutions simultaneously for the KP,
mKP (dual mKP) and SM-KP hierarchies.
\section{Determinant formulae for transformations.}
The analytic-bilinear approach allows to represent in a compact from not
only the integrable hierarchies but also rather general transformations
acting in the space of solutions.
We will consider here the transformations which are equivalent to the action
of an arbitrary meromorphic function $g(\lambda )$ on the CBA function $\chi
(\lambda ,\mu ,x)$. Let $g(\lambda )$ be a meromorphic function in $G$ which
has simple poles at the points $a_i\;(i=1,2,...,n)$ and simple zeros at the
points $b_i\;(i=1,2,...,n)$, i.e. $g(\lambda )=\prod_{i=1}^n\frac{\lambda -a_i}{
\lambda -b_i}$. To construct the transformed CBA function $\chi ^{\dagger
}(\lambda ,\mu )$ it sufficient to find a solution of the equation
\[
\int_{\partial G}\chi (\nu ,\mu )g(\nu )\chi ^{\dagger }(\lambda ,\nu )d\nu
=0
\]
where $\chi ^{\dagger }$ has the same normalization $((\lambda -\mu )^{-1})$
as $\chi $ .
The simplest way to find $\chi ^{\dagger }$ consists in the use the
following consequence of equation (\ref{HIROTA})
\[
\int_{\partial G}\int_{\partial G}d\nu\, d\rho \chi (\nu ,\mu )g(\nu )\chi
^{\dagger }(\rho ,\nu ,g)g^{-1}(\rho )\chi (\lambda ,\rho )=0.
\]
Using this formula, one finds
\begin{equation}
\chi ^{\dagger }(\lambda ,\mu )=g^{-1}(\lambda )g(\mu )\frac{\det \Delta
_{n+1}}{\det \Delta _n}
\label{determinant}
\end{equation}
where
\[
\Delta _{m\,,\alpha \beta }
=\chi (a_\alpha ,b_\beta ),\quad\alpha ,\beta =1,2,...,m
\]
and $a_{n+1}=\lambda$, $b_{n+1}=\mu $.
The formula (\ref{determinant})
defines the generic discrete transformation of $\chi $. In
terms of the $\tau -$function this transformation has the form
\[
\frac{\tau ^{\dagger }(x-[\lambda ]+[\mu ])}{\tau ^{\dagger }(x)}=(\lambda
-\mu )g^{-1}(\lambda )g(\mu )\frac{\det F_{n+1}}{\tau (x)\det F_n}
\]
where
\[
F_{n\,,\alpha \beta }
=\frac{\tau (x-[a_\alpha ]+[b_\beta ])}{a_\alpha -b_\beta }
\]
In the simplest case $n=1$ we have $(b_1=b\,,a_1=a)$
\begin{equation}
\chi ^{\dagger }(\lambda ,\mu )=\frac{(\lambda -b)(\mu -a)}
{(\lambda -a)(\mu
-b)}\frac{
\left|
\begin{array}{cc}
\chi (a,b) & \chi (a,\mu ) \\
\chi (\lambda ,b) & \chi (\lambda ,\mu )
\end{array}
\right|
}{\chi (a,b)}
\label{determinant1}
\end{equation}
and
\[
\frac{\tau ^{\dagger }(x-[\lambda ]+[\mu ])}{\tau ^{\dagger }(x)}=\frac{%
(\lambda -\mu )(a-b)(\lambda -b)(\mu -a)}{(\lambda -a)(\mu -b)}\frac{
\left|
\begin{array}{cc}
\frac{\tau (x-[a]+[b])}{a-b} & \frac{\tau (x-[a]+[\mu ])}{a-\mu } \\
\frac{\tau (x-[\lambda ]+[b])}{\lambda -b}
& \frac{\tau (x-[\lambda ]+[\mu ])%
}{\lambda -\mu }
\end{array}
\right|
}{\tau (x)\tau (x-[a]+[b])}
\]
where $|A|=\det A$.
One can represent the transformations
(\ref{determinant}) also in terms of the function $
\psi (\lambda ,\mu )$ . In that form
the determinant formulae (\ref{determinant}) taken at $%
\lambda =0$ or $\mu =0$ are very similar to the determinant formulae for the
iterated Darboux transformations
(see e.g. \cite{Matveev}). The formulae
(\ref{determinant}) give us the
Darboux transformations for all subhierarchies (KP, mKP, dual mKP, SM-KP
hierarchies) of the generalized KP hierarchy.
The determinant formulae (\ref{determinant})
provide us also the multilinear relations for
for the $\tau$-function. Since $\tau^{\dagger}({\bf x})=\tau({\bf x}+
\sum_{i=1}^{n}(-[a_i]+[b_i]))$, the formula (\ref{determinant}) is the
$n$-linear relation for the $\tau$-function. It is easy to check that
in the simplest case $n=1$ this formula gives the simplest
addition formula (\ref{taubasic}). At $n=2$ the formula
looks like
\begin{eqnarray}
&&\tau({\bf x})\tau({\bf x}-[\lambda]+[\mu]+[b_1]-[a_1]+[b_2]-[a_2])
\times
\left|
\begin{array}{cc}
{\tau({\bf x}-[a_1]+[b_1])\over a_1-b_1}&
{\tau({\bf x}-[a_1]+[b_2])\over a_1-b_2}\\
\\
{\tau({\bf x}-[a_2]+[b_1])\over a_2-b_1}&
{\tau({\bf x}-[a_2]+[b_2])\over a_2-b_2}
\end{array}
\right|=\nonumber\\
&&{(\lambda-\mu)(\mu-a_1)(\mu-a_2)(\lambda-b_1)(\lambda-b_2)\over
(\mu-b_1)(\mu-b_2)(\lambda-a_1)(\lambda-a_2)}\times
\nonumber\\
\nonumber\\
&&\tau({\bf x}+[b_1]-[a_1]+[b_2]-[a_2])\times
\left|
\begin{array}{ccc}
{\tau({\bf x}-[\lambda]+[\mu])\over \lambda-\mu}&
{\tau({\bf x}-[\lambda]+[b_1])\over \lambda-b_1}&
{\tau({\bf x}-[\lambda]+[b_2])\over \lambda-b_2}\\ \\
{\tau({\bf x}-[a_1]+[\mu])\over a_1-\mu}&
{\tau({\bf x}-[a_1]+[b_1])\over a_1-b_1}&
{\tau({\bf x}-[a_1]+[b_2])\over a_1-b_2}\\ \\
{\tau({\bf x}-[a_2]+[\mu])\over a_2-\mu}&
{\tau({\bf x}-[a_2]+[b_1])\over a_2-b_1}&
{\tau({\bf x}-[a_2]+[b_2])\over a_2-b_2}
\end{array}
\right|
\label{3det}
\end{eqnarray}
It is cumbersome but straightforward check that the relation
(\ref{3det}) is satisfied due to the higher addition formulae
(\ref{taubasic00}) with $n=2$.
\section{Global expression for the $\tau$-function and the closed one-form}
In this section we will investigate in what sense the formula
(\ref{tauform}) defines the $\tau$-function. Using this formula,
it is possible to prove that the $\tau$-function is defined
through the CBA function in terms of the closed 1-form, which
can be written both for differentials of the
variables ${\bf x}$ and
for variations of the function $g$ ($g$ is defined on the
unit circle).
Moreover, it is possible
to show that the formula (\ref{tauform}) and the definition
in terms of the closed 1-form are equivalent.
For calculation in terms of variations, it is more
convenient to use the formula (\ref{tauform})
in the form
\begin{equation}
\chi(\lambda,\mu)={\tau\left(g(\nu)\times
\left({\nu-\lambda\over\nu-\mu}\right)\right)
\over \tau(g(\nu))(\lambda-\mu)}
\label{tauform1}
\end{equation}
(see \cite{NLS}).
Differentiating (\ref{tauform1}) with respect to $\lambda$,
one obtains
\begin{equation}
-{1\over\tau(g)}\oint {\delta \tau\left(g(\nu)\times
\left({\nu-\lambda\over\nu-\mu}\right)\right)
\over\delta \log g}{1\over\nu-\lambda}d\nu
={\partial\over \partial \lambda}(\lambda-\mu)
\chi(\lambda,\mu;g)\,,
\end{equation}
or for $\lambda=\mu$
\begin{equation}
-{1\over\tau(g)}\oint {\delta \tau(g(\nu))
\over\delta \log g}{1\over\nu-\lambda}d\nu
=
\chi_r(\lambda,\lambda;g)
\end{equation}
where $\chi_r(\lambda,\mu;g)=\chi(\lambda,\mu;g)-
(\lambda-\mu)^{-1}$.
This equation can be rewritten in the form
\begin{equation}
-{1\over\tau(g)}\oint {\delta \tau(g(\nu))
\over\delta \log g}{1\over\nu-\lambda}d\nu
=\oint
\chi_r(\nu,\nu;g){1\over\nu-\lambda}d\nu\,.
\label{formvar1}
\end{equation}
Relation (\ref{formvar1}) implies that the functionals
$-{1\over\tau(g)}{\delta \tau(g(\nu))\over\delta \log g}
$ and
$\chi_r(\nu,\nu;g)$
are identical for the class of functions analytic
outside the unit circle and decreasing at infinity.
Thus for ${\delta g\over g}$ belonging to this class
one has
\begin{equation}
-\delta \log \tau(g(\nu))=
\oint
\chi_r(\nu,\nu;g){\delta g(\nu)\over g(\nu)} d\nu\,.
\label{formvar}
\end{equation}
This expression defines a variational 1-form defining the
$\tau$-function. It is easy to prove using the identity (\ref{HIROTA})
that this form is
{\em closed} .
Indeed, according to (\ref{HIROTA})
\begin{eqnarray}
\delta \chi(\lambda,\mu;g)=
\oint \chi(\nu,\mu;g){\delta g(\nu)\over g(\nu)}
\chi(\lambda,\nu;g)d\nu\,,\nonumber\\
\delta \chi_r(\lambda,\lambda;g)=
\oint\chi_r(\nu,\lambda;g){\delta g(\nu)\over g(\nu)}
\chi_r(\lambda,\nu;g)d\nu\,.
\label{variation}
\end{eqnarray}
So the variation
of the (\ref{formvar}) gives
\begin{eqnarray}
-\delta^2 \log \tau(g)=
\oint\!\!\!\oint
\chi_r(\nu,\lambda;g)\chi_r(\lambda,\nu;g){\delta' g(\nu)\over g(\nu)}
{\delta g(\lambda)\over g(\lambda)} d\nu\,d\lambda\,.
\end{eqnarray}
The symmetry of the kernel of second variation with respect to
$\lambda$, $\nu$ implies that the form (\ref{formvar})
is closed.
So the formula (\ref{formvar}) gives the definition of the
$\tau$-function in terms of the closed 1-form. For the standard KP
coordinates
$$
{\delta g(\lambda)\over g(\lambda)}=
\sum_{n=1}^{\infty} {dx_n \over \lambda^n}\,.
$$
This formula allows us to obtain a closed 1-form in terms
of $dx_n$
\begin{equation}
-\delta \log \tau(g(\nu))=
\sum_{n=0}^{\infty}
\left.{\partial^n\over\partial\nu^n}
\chi_r(\nu,\nu;g)\right|_{\nu=0}dx_n,.
\label{formvar4}
\end{equation}
For $x=x_1$ this formula immediately gives the standard formula
$$
{\partial^2\over\partial x^2 }\log\tau={1\over 2}u
$$
where $u$ is a solution for the KP equation.
In fact it is possible to prove that the function $\tau$ defined
as the solution of the relation (\ref{formvar})
satisfies the global formula (\ref{tauform}).
To do this, we will show using the formula (\ref{formvar})
that the derivatives of
difference of logarithms
of the l.h.s. and the r.h.s.
of the expression (\ref{tauform1}) with respect to
$\lambda\,,\mu\,,\bar{\lambda}\,,\bar{\mu}$ and the
variation with respect to $g$ are equal to zero
for arbitrary $\lambda\,,\mu\,,g$.
That means
that l.h.s. and r.h.s. of (\ref{tauform}) could differ only
by the factor, and the normalization of the function $\chi$
implies that this factor is equal to 1.
First we will calculate the derivative with respect to $\lambda$
\begin{eqnarray}
{\partial\over \partial \lambda}\left(\log
\chi(\lambda,\mu)(\lambda-\mu)-\log\tau\left(g(\nu)\times
\left({\nu-\lambda\over\nu-\mu}\right)\right)
+\log\tau(g(\nu))\right)=\nonumber\\
{\partial\over \partial \lambda}\log
\chi(\lambda,\mu)(\lambda-\mu)+
\oint
\chi_r\left(\nu,\nu;g(\nu'))\times
\left({\nu'-\lambda\over\nu'-\mu}\right)\right)
{1\over \lambda-\nu} d\nu=\nonumber\\
{\partial\over \partial \lambda}\log
\chi(\lambda,\mu)(\lambda-\mu)+
\chi_r\left(\lambda,\lambda;g(\nu)\times
\left({\nu-\lambda\over\nu-\mu}\right)\right)
\label{derivative}
\end{eqnarray}
The second term can be found in terms of $\chi(\lambda,\mu;g)$
using the determinant formula (\ref{determinant1}).
The formula (\ref{determinant1}) gives
\begin{eqnarray}
\chi(\lambda,\mu;g\times {\nu-\lambda\over\nu-\mu'})
={(\lambda-\mu')(\mu-\lambda)\over\mu-\mu'}
{\det\left(\begin{array}{cc}
\chi(\lambda,\mu;g)&\chi(\lambda,\mu';g)\\
{\partial\over \partial \lambda}\chi(\lambda,\mu;g)
&{\partial\over \partial \lambda}\chi(\lambda,\mu';g)
\end{array}\right)\over\chi(b,a;g)}\,.
\end{eqnarray}
Taking the regular part of this formula at $\mu=\lambda$, one
immediately obtains that the expression (\ref{derivative})
is equal to zero.
The case with the derivative over $\mu$ is analogous;
derivatives over $\bar{\lambda}$ and $\bar{\mu}$ immediately
give zero.
So now we proceed to the calculation of variation
\begin{eqnarray}
&&\delta\left(\log
\chi(\lambda,\mu)(\lambda-\mu)-\log\tau\left(g(\nu)\times
\left({\nu-\lambda\over\nu-\mu}\right)\right)
+\log\tau(g(\nu))\right)=\nonumber\\
&&{1\over\chi(\lambda,\mu)}
\oint \chi(\nu,\mu;g){\delta g(\nu)\over g(\nu)}
\chi(\lambda,\nu;g)d\nu-
\oint
\chi_r(\nu,\nu;g(\nu)\times{\lambda-\nu\over \mu-\nu})
{\delta g(\nu)\over g(\nu)} d\nu+\nonumber\\
&&\oint
\chi(\nu,\nu;g){\delta g(\nu)\over g(\nu)} d\nu
\end{eqnarray}
(we have used (\ref{variation}) and (\ref{formvar})).
Using the determinant formula (\ref{determinant})
to transform the second term, one concludes
that the variation is equal to zero.
Below we will give a brief sketch of derivation of
1-form in terms of Baker-Akhiezer function.
Differentiation of (\ref{tauform}) (with shifted arguments)
with respect to $\lambda$ and $\mu$ gives
\begin{eqnarray}
&&
\left({\partial\over\partial \lambda}+
{\partial\over\partial x_1(\lambda)}\right)
\log[(\lambda-\mu)\chi(\lambda,\mu,{\bf x})]=
-{\partial\over\partial x_1(\lambda)}\log \tau({\bf x})
\label{form1}\,,\\
&&
\left({\partial\over\partial \mu}-
{\partial\over\partial x_1(\mu)}\right)
\log[(\lambda-\mu)\chi(\lambda,\mu,{\bf x})]=
{\partial\over\partial x_1(\mu)}\log \tau({\bf x})
\label{form2}
\end{eqnarray}
where ${\partial\over\partial x_1(\lambda)}=
\sum_{n=1}^{\infty}\lambda^{n-1}{\partial\over\partial x_n}$.
Equation (\ref{form1}) implies that
$$
{\partial\over\partial x_n}\log \tau({\bf x})=\left .
-{1\over(n-1)!}{\partial^{n-1}\over\partial \lambda^{n-1}}
\left(\left({\partial\over\partial \lambda}+
{\partial\over\partial x_1(\lambda)}\right)
\log[(\lambda-\mu)\chi(\lambda,\mu,{\bf x})]\right)\right|_{\lambda=0}\,.
$$
Hence, we have the closed 1-form $\omega$
$$
\omega({\bf x}\,,{\bf dx})=
$$
\begin{eqnarray}
\left .
-\sum_{n=1}^{\infty}
{1\over(n-1)!}{\partial^{n-1}\over\partial \lambda^{n-1}}
\left(\left({\partial\over\partial \lambda}+
{\partial\over\partial x_1(\lambda)}\right)
\log[(\lambda-\mu)\chi(\lambda,\mu,{\bf x})]\right)\right|_{\lambda=0}
dx_n &&
\label{form}\,.
\end{eqnarray}
and $\omega=d\log \tau$. Similar expression for $\omega$
can be obtained using (\ref{form2}). At $\mu=0$ the
formula (\ref{form}) is equivalent to the formula found
in \cite{Date}.
\subsection*{Acknowledgments}
The first author (LB) is grateful to the
Dipartimento di Fisica dell'Universit\`a
and Sezione INFN, Lecce, for hospitality and support;
(LB) also acknowledges partial support from the
Russian Foundation for Basic Research under grant
No 96-01-00841.
|
proofpile-arXiv_065-601
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\subsection{Susceptibility of unconstrained and constrained electron
systems}
We now present the basic formulas defining the magnetic
susceptibility and then compare the unconstrained magnetic response with
the susceptibility obtained by confining the electron gas to a finite
region to illustrate the subject of our studies in this paper.
Let us consider a noninteracting electron gas confined in a volume
(area in two dimensions) $A$ at temperature $T$ under a magnetic
field $H$. The magnetic moment of the system in statistical equilibrium
is given by the thermodynamic relation
\begin{equation}
{\cal M} = - \left(\frac{\partial \Omega}{\partial H}
\right)_{T,\mu}
\label{eq:magngc}
\end{equation}
where $\Omega(T,\mu,H)$ is the thermodynamic potential, and $\mu$ the
chemical potential of the electron gas. The
differential magnetic susceptibility is defined by
\begin{equation}
\chi^{\rm \scriptscriptstyle GC} = \frac{1}{A} \
\left(\frac{\partial {\cal M}}{\partial H} \right)_{T,\mu} =
- \frac{1}{A} \left(\frac{\partial^{2}\Omega}{\partial H^{2}}
\right)_{T,\mu} \ .
\label{eq:susgc}
\end{equation}
The notation with the superscript GC is used in order to emphasize the
fact that we are working in the grand canonical ensemble. The choice of the
ensemble in the macroscopic limit of $N$ and $A$ $\rightarrow \infty$ is a
matter of
convenience. As it is well known by now \cite{BM,Imry,ensemble}
the equivalence
between the ensembles may break down in the
mesoscopic regime that interests us,
and this point will be thoroughly discussed in the remaining of the paper.
However, for the purpose of this didactical introduction
we will work in the
grand canonical ensemble studying the magnetic response of
electron systems with fixed chemical potentials.
The calculation advantages of the GC ensemble arise from the
simple form of the thermodynamic potential
\begin{equation}
\Omega(T,\mu,H) = - \frac{1}{\beta} \int \ {\rm d} E \ d(E) \
\ln{(1+\exp{[\beta(\mu\!-\!E)]})} \ ,
\label{eq:therpot}
\end{equation}
in terms of the single--particle density of states
\begin{equation}
d(E) = {\sf g_s} \sum_{\lambda} \delta (E-E_{\lambda}) \ .
\label{DOS}
\end{equation}
The factor ${\sf g_s} \! = \! 2$ takes into account spin degeneracy,
$E_{\lambda}$ are the eigenenergies of the system.
The magnetic susceptibility is directly extracted from the knowledge of the
density of states. The case of a free electron gas is particularly simple
since the electron eigenstates are Landau states with energies
\begin{equation}
E_{k} = \hbar w \ (k+1/2) \hspace{2cm} k=0,1,2,\ldots
\label{LL}
\end{equation}
and degeneracies ${\sf g_s} \Phi/\Phi_0$. The cyclotron frequency is $w=eH/mc$,
$\Phi = H A$ is the flux through an area $A$, and $\Phi_0 = hc/e$ is the
elemental flux quantum.
Throughout this work we will neglect the Zeeman splitting term due
to the electron spin. It can however be incorporated easily when
spin-orbit coupling is negligible \cite{Harsh}.
Landau's derivation of the magnetic susceptibility of a free
electron system arising from
the quantization condition (\ref{LL}) can be found for the
three--dimensional case in standard textbooks \cite{LanLip,Peierls}.
The two--dimensional case \cite{Shoe,vanLthesis} follows upon
the same lines.
In the following we present a sketch of the latter which will be
useful towards a semiclassical understanding of the problem.
($H$ is now the component of the field perpendicular to the plane
of the electrons.)
By the use of the Poisson summation formula the density of states
related to the quantization condition (\ref{LL}) can be written as
\begin{equation}
d(E) = {\sf g_s} \frac{mA}{2 \pi \hbar^{2}}
+ {\sf g_s} \frac{mA}{\pi \hbar^{2}}
\sum_{n=1}^{\infty} (-1)^{n}
\cos{\left(\frac{2\pi n E}{\hbar w}\right)} \ .
\label{DOSLL}
\end{equation}
This decomposition is usually interpreted as coming from the Weyl term
(given by the volume of the energy manifold in phase space) and the
contribution of cyclotron orbits (second term, strongly energy dependent).
We stress though that in the bottom of the spectra, from which the
Landau diamagnetic component originates, this distinction is
essentially meaningless.
In the case of a degenerate electron gas
with a weak field such that $\hbar w \ll k_{B}T \ll \mu$
the energy integral (\ref {eq:therpot}) is easily performed resulting in
\begin{equation}
\Omega(\mu) \simeq \bar \Omega(\mu) =
- {\sf g_s} \frac{mA}{2 \pi \hbar^{2}} \ \frac{\mu^{2}}{2} +
{\sf g_s} \frac{e^2}{24 \pi mc^2} \ \frac{AH^2}{2} \ ,
\label{therpotwf}
\end{equation}
where $\bar \Omega$ is the smooth part (in energy) of the thermodynamic
potential. (Note that the second term of Eq.~(\ref{therpotwf})
comes nevertheless from the integral of the rapidly oscillating term
of the density of states.)
Thus, we obtain the two-dimensional diamagnetic Landau susceptibility
\begin{equation}
-\chi_{\scriptscriptstyle L} = - \frac{{\sf g_s} e^2}{24 \pi mc^2} \ .
\label{susLand}
\end{equation}
For high magnetic fields, $k_{B}T \ll \hbar w$, the energy integrals are
slightly more complicated than before since the rapidly oscillating
component of $\Omega$ is not negligible any longer. This latter
can be computed (see Appendix \ref{app:convolution} for
the treatment of similar cases) as
\begin{equation} \label{therpotdHvA}
{\Omega^{\rm osc}} = {\sf g_s} \left(\frac{m A}{\pi \hbar^2}\right)
\sum_{n=1}^{\infty} (-1)^{n}
\left(\frac{\hbar w}{2 \pi n}\right)^{2}
\cos{\left(\frac{2\pi n \mu}{\hbar w}\right)}
R_T(n) \ ,
\end{equation}
where $R_T(n)$ is a temperature dependent damping factor
\begin{equation}
R_T(n) = \frac{2 \pi^2 n k_{B}T/\hbar w}
{\sinh{(2 \pi^2 n k_{B}T/\hbar w)}} \ .
\label{R:dHvA}
\end{equation}
With $\Omega = \bar \Omega + {\Omega^{\rm osc}}$, we have the Landau and de Haas--van
Alphen contributions to the magnetic susceptibility
\begin{equation}
\frac{\chi^{\rm \scriptscriptstyle GC}}{\chi_{\scriptscriptstyle L}} =- 1 - 24 \left(\frac{\mu}{\hbar w}\right)^{2}
\sum_{n=1}^{\infty} (-1)^{n}
\cos{\left(\frac{2\pi n \mu}{\hbar w}\right)}
R_T(n) \ .
\label{intro:susdHvA}
\end{equation}
The second term exhibits the characteristic oscillations
with period $1/H$ and is exponentially damped with temperature (and the
summation index $n$).\footnote{For high fields we cannot in
principle separate
the orbital and spin effects. The de Haas--van Alphen
oscillations are given
only by the orbital component, that is the only one that interests
us for our model of spinless electrons.}
While going from the bulk two-dimensional case (macroscopic regime) to the
constrained case (ballistic mesoscopic) two important changes
take place: i) the confining energy appears as a relevant scale and
Eq.~(\ref {LL}) no longer provides the quantization condition; ii) since we
are not in the thermodynamic limit of $N$ and $A \rightarrow \infty$, the
constraint of a constant number of electrons in [isolated] microstructures
is no longer equivalent to having a fixed chemical potential.
These two effects will be thoroughly discussed in the paper.
For didactical purposes we restrict
ourselves in this introductory section to only the changes
(i) due to the confinement, and we anticipate some of the results
that will be later discussed in detail.
We imagine a mesoscopic square of size $a$ connected to an electron
reservoir with chemical potential $\mu$. Direct numerical diagonalization
in the presence of a magnetic field (Fig.~\ref{f1}a) allows us to
obtain $\chi^{\rm \scriptscriptstyle GC}$ (solid line in
Fig.~\ref{f1}b). In the high field region ($2r_c\!<\!a$, \ we note
$r_c= v_{\scriptscriptstyle F} / \omega$ the cyclotron radius)
the characteristic de Haas~-~van Alphen oscillations are obtained,
although not with the amplitude expected from calculations in the bulk
(Eq.~(\ref {intro:susdHvA})).
For lower fields the discrepancy between our numerical results and the bulk
Landau diamagnetism is quite striking. Thus, confining deeply alters the
orbital response of an electron gas. Without entering into details at this
point we remark the fact that the whole curve is quite well reproduced
by a finite-temperature semiclassical theory (dashed line) that
takes into account
only one type of trajectory (see insets) in each of the three regimes:
a) the interference-like regime, dominated by the shortest trajectories
with the largest enclosed area for squares at zero magnetic field;
b) the transition regime dominated by the bending of bouncing--ball
trajectories between parallel
sides of the square;
c) the de Haas~-~van Alphen regime dominated by
cyclotron orbits. It is remarkable how an exceedingly
complicated spectrum as that of Fig.~\ref{f1}a can be understood within
such a simple semiclassical picture once finite temperature acts as a
filter selecting only few types of trajectories.
\subsection {Overview of this work}
The purpose of this paper is to provide an [essentially comprehensive]
theory of the orbital magnetic properties of non-interacting spinless
electrons in the mesoscopic ballistic regime. We restrict ourselves
to the clean limit, where the different behavior of the magnetic
response arises as a geometrical effect (shape of the microstructure).
We will make extensive use of semiclassical techniques since they appear
to be perfectly suited for these problems.
For the smooth components (such as in Eq.~(\ref{therpotwf})) we will
use the general techniques developed by Wigner to obtain higher $\hbar$
corrections to the Weyl term which are field dependent.
For the oscillating components, we will rely on the so called
semiclassical trace formulas, which provide simple and intuitive
expressions for the density of states as a sum over Fourier-like
components associated to closed classical orbits.
In this respect it will be seen that the nature of the classical
dynamics, i.e.\ integrable versus chaotic (and more precisely
existence versus absence of continuous families of periodic orbits),
plays a major role. Although we will present a complete
formalism for both cases, our main emphasis, and in particular all
the examples treated explicitly, will concern integrable
geometries.
The reason for this choice is twofold. First, as we will make clear
in the sequel, one expects a much larger magnetic response
for integrable systems than for chaotic ones, yielding a
more striking effect easier to observe.
The second point is that, contrarily
to what might seem natural a priori, integrable geometries
present a few conceptual difficulties in their treatment which are not
present for chaotic systems. Indeed integrable systems lack
of structural stability, which means that under any small
perturbation (such as the one provided by the presence of a magnetic
field) they generically do not remain integrable.
Chaotic systems on the contrary remain chaotic
under a small perturbation. Therefore, as shown in Ref.\cite{aga94},
the Gutzwiller trace formula \cite{gutz_book,gut71},
valid for chaotic
systems, can be used at finite fields without further complications.
For integrable geometries however, the Berry-Tabor \cite{ber76,ber77}
or Balian-Bloch \cite{bal69} trace formulae valid for integrable systems
usually do not apply in the presence of a perturbing magnetic field.
It will therefore be necessary, following Ozorio de Almeida
\cite{ozor86,ozor:book}, to consider the more complicated case of nearly
integrable systems, which we will do in detail here.
To perform this program, the present work is organized as follows.
In the next section we present the thermodynamic formalism
appropriate for working in the canonical and grand canonical ensembles,
stressing its semiclassical interpretation and incorporating the changes
due to the constancy of the number of electrons in the experimentally
relevant microstructures.
In Sec.~\ref{sec:LanGen} we consider the smooth magnetic response
and show that the Landau diamagnetism is present
in any confined geometry at arbitrary fields.
In Sec.~\ref{sec:integrable} we address the
magnetic response (susceptibility and persistent currents) in the
simplest possible geometries: circles and rings billiards that are
integrable with and without magnetic field.
In Sec.~\ref{sec:square} we present the calculation of the
magnetic susceptibility for the experimentally relevant case of the square
billiard \cite{levy93} whose integrability at zero field is broken by the
effect of an
applied magnetic field.
An initial study along these lines was presented in Refs.~\cite{URJ95}
and \cite{JRU94} and independently proposed by von
Oppen \cite{vO95}. This geometry and the corresponding experiment have
also been analyzed from a completely different point of view by
Gefen, Braun and Montambaux \cite{gef94} stressing the importance of the
residual disorder (see also Ref.~\cite{altgef95}).
We consider in Sec.~\ref{sec:general} the generic magnetic response
of both integrable and chaotic geometries, stressing the similarities and
differences in their behavior and calculating the line-shape of the
average magnetization in generic chaotic systems.
In Sec.~\ref{sec:highB} we demonstrate how the semiclassical formalism
we have developed applies not only to the weak--field limit,
but also to higher field and in particular to the high field
regime of the de Haas~-~van Alphen oscillations. We treat
explicitly the example of the square geometry, including an
intermediate field regime dominated by bouncing--ball orbits as depicted
in Fig.~\ref{f1}.
We discuss our conclusions and their experimental relevance
in Sec.~\ref{sec:concl}. The modifications of our results due to the
effect of a weak disordered potential are discussed in a separate
publication \cite{rod2000}.
To keep the focus on the physical concepts developed in the text,
a few technical derivations have been relegated to some
appendices. Appendix~\ref{app:convolution} presents
the generic case of the convolution of a
rapidly oscillating function with the derivative of the Fermi
function.
Appendix~\ref{app:wigner} gives the calculation of the first
field-dependent term of the heat Kernel in an $\hbar$ expansion.
In Appendix~\ref{app:ring_g} we compute the action integrals
associated with the dynamics of circular and ring billiards
needed to define the energy
manifold in action space. Appendix~\ref{app:D_M} presents the calculation
of the prefactor of the Green function for an integrable system,
while in
Appendix~\ref{app:highB} we show how to compute the semiclassical Green
function at a focal point, and apply the obtained result to the
particular case of cyclotron motion.
\newpage
\section {Thermodynamic formalism}
\label{sec:TherFor}
One main subject of the present work is the introduction of semiclassical
concepts into the thermodynamics of mesoscopic systems.
In this section we provide the basic formalism allowing one to obtain
the thermodynamic properties (grand potential, free energy)
from the quasiclassically calculated single--particle density of states
and hence the susceptibility. We begin with general
definitions and relations between grand canonical and canonical
quantities.
For a system of electrons in a volume (area in two dimensions)
$A$ connected to a reservoir of particles with chemical potential
$\mu$ (grand canonical ensemble) the magnetic susceptibility
is obtained, as given by Eq.~(\ref{eq:susgc}), as
\[
\chi^{\rm \scriptscriptstyle GC} = - \frac{1}{A}
\left(\frac{\partial^{2}\Omega}{\partial H^{2}} \right)_{T,\mu} \ .
\]
$\Omega(T,\mu,H)$ is the thermodynamic potential, which can be
expressed for non--interacting electrons in terms of the single--particle
density of states through Eq.~(\ref{eq:therpot}).
For actual microstructures, the number $\bf N$ of particles
inside the device might be large but is fixed in contrast to the
chemical potential $\mu$.
As discussed in the introduction, it will be necessary in some cases,
namely when considering the average susceptibility of an ensemble of
microstructures, to take explicitly into account the conservation
of $\bf N$, and to work within the canonical ensemble.
For such systems with a fixed number $\bf N$ of particles, the relevant
thermodynamic function is not the grand potential $\Omega$, but its
Legendre transform, the free energy\footnote{In standard
thermodynamics, Eq.~(\ref{eq:free}) just represents the definition of the
grand potential. It should be borne in mind however that from a
statistical physics point of view this is not an exact relation,
but the result of a stationary--phase evaluation of the average over
the occupation number, valid only when $k_BT$ is larger than the
typical level spacing. Therefore, we are entitled
to use this relation in the mesoscopic regime that interests us, but
not in the microscopic regime, where features on the scale of a
mean spacing become relevant.}
\begin{equation} \label{eq:free}
F(T,H,{\bf N}) = \mu {\bf N} + \Omega(T,H,\mu) \ .
\end{equation}
In particular, the magnetic susceptibility of a system of $\bf N$ electrons
is
\begin{equation} \label{eq:sus}
\chi = - \frac{1}{A}
\left(\frac{\partial^{2}F}{\partial H^{2}} \right)_{T,{\bf N}}
\ .
\end{equation}
Except for the calculation of the Landau contribution performed in
the following section all the computations of the magnetic response of the
microstructures to be considered
will involve two clearly separated parts. In the first one
the (oscillating part of the) density of states will be
calculated semiclassically. Depending on the underlying
classical dynamics (integrable versus chaotic, with or without breaking
of the invariant tori, with or without focal points, etc.), the
results as well as their derivation will vary noticeably.
In the second stage the integrals over energy yielding the desired
thermodynamic properties have to be performed in a leading order in $\hbar$
approximation. To avoid tedious repetitions,
we shall consider here in some detail this part of the calculation
of the thermodynamic properties, and refer without many additional
comments to the results obtained in this section whenever needed.
We begin with the grand canonical quantities which exhibit
the simplest expressions in terms of the density of states.
In a second subsection we shall consider the canonical ensemble following
closely the approaches presented in Refs.~\cite{ensemble}.
\subsection{Grand canonical properties}
We begin with the standard definition, Eq.~(\ref{DOS}) of the density
of states
\[ d(E) = {\sf g_s} \sum_{\lambda} \delta(E-E_{\lambda}) \ , \]
(${\sf g_s} = 2$ is the spin degeneracy, $E_{\lambda}$ the eigenenergies)
and its successive energy integrals. They are the energy staircase
\begin{equation}
n(E) = \int_0^{E} {\rm d} E' \ d(E') \ ,
\label{eq:roughn}
\end{equation}
and the grand potential at zero temperature
\begin{equation}
\omega (E) = - \int_0^E {\rm d} E' \ n(E') \ .
\label{eq:rougho}
\end{equation}
These are purely quantum mechanical quantities, depending only on the
eigenstates $E_\lambda$ of the system.
At finite temperature the corresponding
quantities are obtained by convolution with the derivative
$f'(E-\mu)$ of the Fermi distribution function
\begin{equation} \label{eq:fermi}
f(E-\mu) = \frac{1}{(1 + \exp [\beta (E-\mu)])} \ .
\end{equation}
We then have
\begin{mathletters}
\label{allsmooths}
\begin{eqnarray}
D (\mu) & = & - \int_0^{\infty} {\rm d} E \ d(E) \ f'(E\!-\!\mu) \ ,
\label{smootha} \\
N (\mu) & = & - \int_0^{\infty} {\rm d} E \ n(E) \ f'(E\!-\!\mu) \ ,
\label{smoothb} \\
\Omega (\mu) & = &
- \int_0^{\infty} {\rm d} E \ \omega(E) \ f'(E\!-\!\mu)
\ .
\label{smoothc}
\end{eqnarray}
\end{mathletters}
Integration by parts leads to the standard definition
(\ref{eq:therpot}) of the grand potential and the mean number of
particles in the GCE with a chemical potential $\mu$, {\em i.e.}
\begin{equation}
N(\mu) = \int_0^{\infty} {\rm d} E \ d(E) \ f(E\!-\!\mu) \ .
\label{eq:mGCE}
\end{equation}
That means that the thermodynamic properties (\ref{smoothb})--(\ref{smoothc})
are obtained by performing the energy integrations
(\ref{eq:roughn})--(\ref{eq:rougho}) with the Fermi function as a weighting factor.
In the following the separation of the above quantum mechanical and
thermodynamic expressions into smooth (noted with a ``$\; \bar{~~} \;$'')
and oscillating (noted with the superscript ``$\; ^{\rm osc} \;$'') parts
is going to play a major role. It has its origin in the well-known decomposition
of the density of states as
\begin{equation} \label{eq:dosc1}
d(E) = \bar d(E) + {d^{\rm osc}}(E) \; .
\end{equation}
This decomposition has a rigorous meaning only in the semiclassical
($E \rightarrow
\infty$) regime for which the scales of variation of $\bar d$ and ${d^{\rm osc}}$
decouple. To leading order in $\hbar$, the mean component $\bar d(E)$
is the Weyl term reflecting the volume of accessible classical
phase space at energy $E$ (zero-length trajectories), while ${d^{\rm osc}}(E)$ is
given as a sum over periodic
orbits (Gutzwiller and Berry-Tabor trace formulas) \cite{gutz_book}.
Generically, it will be expressed as a sum
\begin{equation} \label{eq:dosc:gen}
{d^{\rm osc}}(E) = \sum_t d_t(E) \qquad ; \qquad
d_t(E) = A_t(E) \, \sin \left(S_t(E)/\hbar + \nu_t \right) \; .
\end{equation}
running over periodic orbits labeled by $t$
where $S_t$ is the action integral along the orbit $t$, $A_t(E)$
is a slowly
varying prefactor and $\nu_t$ a constant phase.\footnote{When considering
systems whose integrability is broken by a perturbing magnetic field,
we shall stress the necessity to consider families of recurrent --~but not
periodic~-- orbits of the perturbed system. This will, however, not
affect the discussion which follows.}
Using the expression~(\ref{eq:dosc:gen}) for ${d^{\rm osc}}$ in
Eqs.~(\ref{eq:roughn}) and (\ref{eq:rougho}), ${n^{\rm osc}}$ and ${\omega^{\rm osc}}$ are
obtained to
leading order in $\hbar$ as
\begin{equation} \label{eq:rough_osc:def}
{n^{\rm osc}}(E) = \int^E {\rm d} E' \ {d^{\rm osc}}(E') \qquad ; \qquad
{\omega^{\rm osc}}(E) = - \int^E {\rm d} E' \ {n^{\rm osc}}(E') \ .
\end{equation}
The lower bounds are not specified because the constants of
integration are determined by the constraint that ${n^{\rm osc}}$ and ${\omega^{\rm osc}}$
must have zero
mean values. (It should be borne in mind that semiclassical
expressions like
(\ref{eq:dosc1}), and those that will follow, are not applicable at the
bottom of the spectrum.)
In a leading $\hbar$ calculation the integration over energy in
Eq.~(\ref{eq:rough_osc:def}) has to be applied only to
the rapidly oscillating part of each periodic orbit contribution $d_t$.
Noting moreover that if $S_t(E)$ is the action along a periodic orbit, then
$\tau_t(E) \equiv dS_t / dE$ is the period of the orbit, one has in a
leading $\hbar$ approximation
\begin{equation} \label{eq:primitive}
\int^E A_t(E') \, \sin \left( S_t(E')/\hbar +\nu_t \right) dE'
= \frac{-\hbar}{\tau_t(E)} \,
A_t(E) \, \cos \left( S_t(E)/\hbar +\nu_t \right)
\end{equation}
as can be checked by differentiating both sides
of Eq.~(\ref{eq:primitive}).
In order to emphasis that the integration over energy merely yields
a multiplication by
$(- \hbar/\tau)$, we use the notation $(i_\otimes \cdot d_t)$ to assign
the contribution $d_t$ of a periodic orbit after shift of the phase by
$\pi/2$, {\em i.e.}
$(i_\otimes \cdot [B \, \sin(S/\hbar)]) = B \, \cos (S/\hbar)$. We get
\begin{eqnarray}
{n^{\rm osc}}(E) = \sum_t n_t(E) \qquad & ; & \qquad
n_t(E) = \frac{- \hbar}{\tau_t(E)} \, (i_\otimes \cdot d_t(E)) \; ,
\label{eq:rough_oscn} \\
{\omega^{\rm osc}}(E) = \sum_t \omega_t(E) \qquad & ; & \qquad
\omega_t(E) = \left( \frac{\hbar}{\tau_t(E)} \right)^2 d_t(E) \; .
\label{eq:rough_osco}
\end{eqnarray}
The thermodynamic functions ${D^{\rm osc}}(\mu)$, ${N^{\rm osc}}(\mu)$
and ${\Omega^{\rm osc}}(\mu)$ are then obtained
by application of Eqs.~(\ref{allsmooths}) in which the full
functions are replaced by their oscillating component.
The resulting integrals involve the convolution of functions
(${d^{\rm osc}}(E)$, ${n^{\rm osc}}(E)$ or ${\omega^{\rm osc}}(E)$) oscillating [locally
around $\mu$] with a frequency
$\tau(\mu) / (2\pi\hbar)$, with the derivative of the Fermi factor
$f'(E-\mu)$ being smooth on the scale of $\beta^{-1} = k_{\rm B} T$.
One can therefore already anticipate that this convolution yields an
exponential damping of the periodic orbit contribution
whenever $\tau(\mu) \gg \hbar \beta$.
As shown in appendix~\ref{app:convolution}
the temperature smoothing gives rise to
an additional factor for each periodic orbit contribution,
\begin{equation} \label{sec2:RT}
R_T(\tau_t) = \frac{\tau_t/\tau_c}{\sinh (\tau_t/\tau_c)}
\qquad ; \qquad
\tau_c = \frac{\hbar\beta}{\pi} \; ,
\end{equation}
in a leading $\hbar$ and $\beta^{-1}$ approximation
(without any assumption concerning the order the limits are taken).
In this way we obtain relations between the following useful
thermodynamic functions and the semiclassical density of states:
\begin{mathletters} \label{eq:smooth_osc:all}
\begin{eqnarray}
{D^{\rm osc}}(\mu) = \sum_t D_t(\mu) \qquad & ; & \qquad
D_t(\mu) = R_T(\tau_t) \, d_t(\mu) \qquad ,
\label{eq:smooth_oscd} \\
{N^{\rm osc}}(\mu) = \sum_t N_t(\mu) \qquad & ; & \qquad
N_t(\mu) = R_T(\tau_t) \, \left( \frac{-\hbar}{\tau_t} \right)
(i_\otimes \cdot d_t(\mu)) \; , \label{eq:smooth_oscn} \\
{\Omega^{\rm osc}}(\mu) = \sum_t \Omega_t(\mu) \qquad & ; & \qquad
\Omega_t(\mu) = R_T(\tau_t) \,
\left( \frac{\hbar}{\tau_t} \right)^2 d_t(\mu) \; .
\label{eq:smooth_osco}
\end{eqnarray}
\end{mathletters}
At very low temperature,
$R_T \simeq 1 - [(\tau_t \pi) /( \hbar \beta)]^2 / 6$ which,
for billiard-like systems where $\tau_t = L_t/v_F$
(with $L_t$ being the length of
the orbit and $v_{\scriptscriptstyle F}$ the Fermi velocity),
simply gives the standard Sommerfeld--expansion $R_T \simeq 1
- [(L_t \pi) /( \hbar \beta v_{\scriptscriptstyle F})]^2 / 6$.
For long trajectories or high temperature
it yields an exponential suppression and therefore the only
trajectories contributing significantly to the thermodynamic
functions are those with $\tau_t \leq \tau_c$.
Thus, temperature smoothing has a noticeable effect on the oscillating
quantities since it effectively suppresses the higher harmonics,
which are associated with long classical orbits
in a semiclassical treatment. On the contrary, for a degenerate electron gas
($\beta \mu \gg 1$), finite temperature has no effect on the mean
quantities. Temperature is then the tuning parameter for passing from
$d(E)$ at $T=0$ to
$\bar D(E) = \bar d(E)$ at large temperatures (by the progressive
reduction of ${d^{\rm osc}}$). Similar considerations hold for the energy
staircase and the grand potential.
The oscillatory part of the semiclassical
susceptibility in the grand--canonical ensemble is finally obtained
from Eq.~(\ref{eq:susgc}) by replacing $\Omega$ by ${\Omega^{\rm osc}}$.
\subsection{Canonical ensemble}
\label{sec:Canonical}
Let us now consider the
susceptibility in the canonical ensemble, appropriate for systems with
a fixed number of particles. We follow Imry's derivation
for persistent currents in ensembles of disordered rings \cite{Imry}.
The only important difference
is that we will take averages over the size and the Fermi energy of
ballistic structures instead of averages over impurity realizations.
We will stress the semiclassical interpretation that will be at the
heart of our work, and highlight some of its subtleties.
As mentioned in the introduction the definition Eq.~(\ref{eq:sus})
of the susceptibility $\chi$ is equivalent to $\chi^{\rm \scriptscriptstyle GC}$ up to $1/{\bf N}$
({\em i.e.}~$\hbar$ corrections). Therefore,
in the macroscopic limit of ${\bf N} \rightarrow \infty$ the choice of the
ensemble in which the calculations are done is unimportant. On the
other hand,
in the mesoscopic regime of small structures (with large but finite
$\bf N$) we have to consider such corrections if we want to take advantage
of the computational simplicity of the Grand Canonical Ensemble (GCE). The
difference between the two definitions is particularly important when
the GCE
result is zero as it is the case for the
ensemble average of $\chi^{\rm \scriptscriptstyle GC}$. The evaluation of the corrective terms can be
obtained from the relationship Eq.~(\ref{eq:free}) between the thermodynamic
functions $F({\bf N})$ and $\Omega(\mu)$%
\footnote{In the following we will only write the $\bf N$ dependence of
$F$ and the $\mu$ dependence of $\Omega$, assuming always the $T$ and
$H$ dependence of both functions.} and the relation $N(\mu) = {\bf N}$.
In the case of finite systems the previous implicit
relation is difficult to invert. However, when $\bf N$ is large we can use
the decomposition of $N(\mu)$ in a smooth part $\bar N(\mu)$
and a small component ${N^{\rm osc}}(\mu)$ that fluctuates around the secular
part, and we can perturbatively treat the previous implicit relation.
The contribution of a given orbit to ${d^{\rm osc}}$ is always of lower order
in $\hbar$ than $\bar d$ as can be checked for the various examples
we are going to consider and by inspection of semiclassical trace formulae.
However, since there are infinitely many of such contributions,
we obtain ${d^{\rm osc}}$ and $\bar d$ to be of the same order when adding them up.
(This must be the case since the quantum mechanical $d(E)$ is a sum
of $\delta$ peaks.) Thus, we cannot use ${d^{\rm osc}} / \bar d$ as a small
expansion parameter.
On the other hand, finite temperature provides an exponential cutoff
in the length of the trajectories contributing to ${D^{\rm osc}}$
so that only a finite number of them must be taken into account.
Therefore, ${D^{\rm osc}}$ is of lower order in $\hbar$ than $\bar D$, and
in the semiclassical regime it is possible to expand
the free energy $F$ with respect to the small parameter ${D^{\rm osc}} / \bar D$.
The use of a temperature smoothed density of states
Eq.~(\ref{smootha}) closely follows the
Balian and Bloch approach \cite{bal69}, where, due to the
exponential proliferation of orbits and the impossibility of
exchanging the infinite time and semiclassical limits, the semiclassical
techniques based on trace formulae are considered meaningful only when applied to
smoothed quantities.
The decomposition of $D(E)$ is depicted in Fig.~\ref{f2}, where
we have taken $\bar D (\simeq \bar d)$ to be energy independent,
corresponding to the two-dimensional (potential free) case.
For a perturbative treatment of the mentioned implicit relation
we define a mean chemical potential $\bar \mu$ by the condition of
accommodating $\bf N$ electrons to the mean number of states
\begin{equation} \label{eq:mub_def}
{\bf N} = N(\mu) = \bar N (\bar \mu) \ .
\end{equation}
Expanding this relation to first order in ${D^{\rm osc}}/\bar D$, and employing that
${\rm d} N/{\rm d} \mu = D$, one has
\begin{equation} \label{deltamu}
\Delta \mu \equiv \mu - \bar \mu
\simeq - \ \frac{1}{\bar D (\bar \mu)} \ {N^{\rm osc}} (\bar \mu) \; .
\end{equation}
\noindent The physical
interpretation of $\Delta \mu$ is very clear from Fig.~\ref{f2}: The
shaded area represents the number of electrons in the system and it is
equal to the product $\bar{D} \times \bar{\mu}$.
Expanding the relationship (\ref{eq:free}) to second order
in $\Delta \mu$,
\begin{equation}
F({\bf N}) \simeq (\bar \mu + \Delta \mu) {\bf N}
+ \Omega (\bar \mu)
- N (\bar \mu) \Delta \mu
- D (\bar \mu) \ \frac{\Delta \mu^2}{2} \ ,
\label{eq:secor}
\end{equation}
using the decomposition of $\Omega(\bar \mu)$ and $N(\bar \mu)$
into mean and oscillating parts and
eliminating $\Delta \mu$ (Eq.~(\ref{deltamu})) in the second
order term,
one obtains the expansion of
the free energy to second order in ${D^{\rm osc}} / \bar D$ \cite{Imry,ensemble}
\begin{equation} \label{eq:fd}
F({\bf N}) \simeq F^{0} + \Delta F^{(1)} +\Delta F^{(2)} \ ,
\end{equation}
with
\begin{mathletters}
\label{allDF}
\begin{eqnarray}
F^{0} & = & \bar \mu {\bf N} + \bar \Omega(\bar \mu) \; ,
\label{DF0} \\
\Delta F^{(1)} & = & {\Omega^{\rm osc}} (\bar \mu) \; ,
\label{DF1} \\
\displaystyle
\Delta F^{(2)} & = & \frac{1}{2 \bar D (\bar \mu)} \
\left( {N^{\rm osc}} (\bar \mu) \right)^{2} \ .
\label{DF2}
\end{eqnarray}
\end{mathletters}
Then $\Delta F^{(1)}$ and $\Delta F^{(2)}$ can be expressed in terms
of the oscillating part of the density of states by means of
Eqs.~(\ref{eq:smooth_oscn}) and (\ref{eq:smooth_osco}).
The first two terms
$F^{0} + \Delta F^{(1)}$ yield the magnetic response
calculated in the GCE with an effective
chemical potential $\bar \mu$. The first ``canonical correction''
$\Delta F^{(2)}$ has a grand canonical form since it is expressed in
terms of a temperature smoothed integral of the density of states
(Eq.~(\ref{eq:mGCE})) for a fixed chemical potential $\bar \mu$.
It is convenient to use the expansion (\ref{eq:fd}) in the calculation of
the magnetic susceptibility of a system with a fixed number of particles
because the leading $\hbar$ contribution to $\bar N (\bar \mu)$ has
no magnetic field dependence, independent of the precise system under consideration.
Therefore, {\em at this level of approximation}, keeping $\bf N$ constant
in Eq.~(\ref{eq:sus}) when taking the derivative with respect to the
magnetic field
amounts to keep $\bar \mu$ constant. Since
$F^{(0)}$ is field independent in a leading order semiclassical expansion
the weak-field susceptibility of a given mesoscopic sample will be dominated
by $\Delta F^{(1)}$. However, when
considering ensembles of mesoscopic devices, with slightly different sizes or
electron fillings, $\Delta F^{(1)}$ (and its associated
contribution to the susceptibility) averages to zero due to its oscillatory
behavior independently of the order in $\hbar$ up to which it is
calculated.\footnote{In the following, we
shall always calculate $\Delta F^{(1)}$ in a leading order $\hbar$
approximation. Higher order corrections to $\Delta F^{(1)}$ may
be of the same order as $\Delta F^{(2)}$ but will average to zero
under ensemble averaging.} Then we must consider the next order
term $\Delta F^{(2)}$.
As mentioned above, we will essentially work in the semiclassical
regime (to leading order in $\hbar$) where
$F^{0}$ is field independent. However, in the following section we
will examine the next
order $\hbar$ correction to $\bar \Omega (\bar \mu)$ (and to $F^0$),
demonstrating that its field dependence gives rise to the standard
Landau diamagnetism, independent of any confinement.
\newpage
\section{Landau susceptibility}
\label{sec:LanGen}
In the previous section we showed that the various quantum mechanical
({\em i.e.} $d(E)$, $n(E)$, $\omega(E)$) and thermodynamic ({\em i.e.}
$D(\mu)$, $N(\mu)$, $\Omega(\mu)$) properties of a mesoscopic system can
be decomposed into smooth and fluctuating parts. In the semiclassical limit,
where the Fermi wavelength is much smaller than the system size, each of
these quantities allows an asymptotic expansion in powers of $\hbar$. For
most of the purposes it is sufficient to consider only leading order terms
while higher order corrections must only be added if the former vanish for some reason.
This is the case for the smooth part ${\bar \Omega}(\mu)$
of the grand potential, which is the dominant term at any temperature,
but is magnetic field independent to leading order in $\hbar$. The
present section will be the only part of our work where higher $\hbar$
corrections are considered. We will show that they give rise to the
standard Landau susceptibility.
Our derivation relies neither, on the quantum side, on the existence of
Landau levels, nor, on the classical side, on
boundary trajectories or the presence of circular
cyclotronic orbits fitting into the confinement potential.
This shows that the Landau
susceptibility is a property of mesoscopic devices as well as
infinite systems, being the dominant contribution at sufficiently high
temperature\footnote{Analog results have been independently obtained
by S.D.~Prado {\em et al.} \cite{Prado}. The Wigner distribution function
was previously used by R.\ Kubo \cite{Kubo64} in the study of
Landau diamagnetism.}.
We consider a $d$-dimensional ($d=2,3$) system of electrons governed
by the quantum Hamiltonian
\begin{equation} \label{eq:quantH}
{\hat {\cal H}} = \frac{1}{2m} \ \left(\hat {\bf p} -
\frac{e}{c} {\bf A}(\hat {\bf q})\right)^2 \
+ \ V(\hat {\bf q}) \; ,
\end{equation}
where ${\bf A}$ is the vector potential generating the magnetic field $H$
and $V({\bf q})$ is the potential which confines the electrons
in some region of the space. This region can a priori have any dimension, and
it can be smaller that the cyclotron radius. We will only assume in the
following that $V({\bf q})$ is {\em smooth} on the scale of a Fermi
wavelength, so that semiclassical asymptotic results can be used.
In billiards the effect of {\em hard} boundaries on the susceptibility
is negligible compared to the Landau bulk term \cite{rob86,antoine}, and
therefore the results obtained below apply there, too.
There exist general techniques to compute the semiclassical expansion of
the mean part of the density of states (or of its integrated versions
Eqs.~(\ref{eq:roughn}), (\ref{eq:rougho})) up to arbitrary
order in $\hbar$. The most complete approach, which allows one
to take into account the effect of sharp boundaries, can be found in
the work of Seeley \cite{seeley}. However, assuming the smoothness of
$V({\bf q})$ allows us to follow the standard approach introduced by Wigner
in 1932 \cite{wig32} which is based on the notion of the Wigner
transform of an operator.
As a starting point we consider the Laplace transform of the level
density (or heat Kernel),
\begin{equation} \label{partition}
Z(\lambda ) = \int_0^\infty {\rm d} E \ e^{-\lambda E} \ d(E)
\ = \ {\sf g_s} \ {\rm Tr} (e^{-\lambda \hat{{\cal H}}} ) \; ,
\end{equation}
where ${\sf g_s}=2$ takes into account the spin degeneracy.
In appendix~\ref{app:wigner} we apply after a brief description the
technique to calculate the first two terms
of the expansion of $Z(\lambda)$ with respect to $\lambda$. They
yield under the inverse transformation the first two terms
of the expansion of $d(E)$ in powers of $\hbar$. The
oscillating part ${d^{\rm osc}}(E)$ of $d(E)$ is not included in this procedure
since it is associated with exponentially
small terms in $Z(\lambda)$, that is, $Z(\lambda) \simeq \bar Z(\lambda)$
for $\lambda \simeq 0$. This well known property can be easily seen from
the integral treated in appendix~\ref{app:convolution} by identifying
$\beta$ with $\lambda$ and using the exponential form of the distribution
function in the classical limit of high temperatures ($\beta \mu \ll 1)$.
Noting ${\cal H}({\bf q},{\bf p})$ the classical Hamiltonian corresponding to
Eq.~(\ref{eq:quantH}), the leading order [Weyl] contribution to $Z(\lambda)$
is given by Eq.~(\ref{ZW}),
\begin{equation} \label{ZWeyl}
Z_{\rm W}(\lambda)
= \frac{{\sf g_s}}{(2\pi \hbar )^d} \ \int {\rm d} {\bf q} {\rm d} {\bf p} \,
\exp \left( -\lambda {\cal H}({\bf q},{\bf p})
\right) \; ,
\end{equation}
and the inverse Laplace transform yields the familiar result
\begin{equation} \label{dWeyl}
d_{\rm W}(E) =
\bar d_{\rm W}(E) = \frac{{\sf g_s}}{(2\pi \hbar )^d} \ \int {\rm d} {\bf q} {\rm d} {\bf p} \,
\delta \left( E - {\cal H}({\bf q},{\bf p}) \right) \; .
\end{equation}
In the above integrals, the substitution
\begin{equation} \label{change}
{\bf p} \rightarrow {\bf p}' = {\bf p} - \frac{e}{c} {\bf A}
\end{equation}
eliminates any field dependence. Therefore
\begin{equation} \label{eq:defwweyl}
\omega_{\rm W}(E) \ = \
\bar \omega_{\rm W}(E) \ = \ - \int_0^E {\rm d} E' \int_0^{E'}
{\rm d} E'' \
d_{\rm W}(E'') \; ,
\end{equation}
as well as the leading term
$\bar \Omega_{\rm W}(\mu) $ of the grand potential (obtained
in the high temperature limit of Eq.~(\ref{smoothc})),
are field independent. This is the reason for the absence of orbital
magnetism in classical mechanics.
To observe a field dependence, one must consider the first
correcting term of $Z(\lambda)$ which, as shown in
appendix~\ref{app:wigner} (Eq.~(\ref{Z1})), is given by
\begin{equation} \label{Z_1}
Z_1(\lambda,H) = - \lambda^2 \ \frac{\mu_B^2 H^2}{6} \ Z_{\rm W}
+ Z_1^0 \; .
\end{equation}
Here, $\mu_B = (e\hbar) / (2mc)$ is the Bohr magneton, and $Z_1^0 =
Z_1({\scriptstyle H\!=\!0})$ is a field independent term that we will drop
from now on since it does not contribute to the susceptibility.
The integrated functions $n(E)$ and $\omega(E)$ can be obtained from
their Laplace
transforms
\begin{equation} \label{eq:wmu}
n(\lambda) = \frac{Z(\lambda)}{\lambda} \; , \qquad
w(\lambda) = - \frac{Z(\lambda)}{\lambda^2} \; .
\end{equation}
Then the first correction to the zero-temperature grand potential is
\begin{equation}
\omega_1(E) \ = \
\bar \omega_1(E) \ = \frac{\mu_B^2 H^2}{6} \ \bar d_{\rm W}(E) \; .
\end{equation}
After convolution with the derivative of the Fermi
function (Eq.~(\ref{smoothc})) we obtain the first corrective term of
the grand potential
\begin{equation}
\Omega_1 (\mu) =
\bar \Omega_1 (\mu) = \frac{\mu_B^2 H^2}{6} \ \bar D_{\rm W}(\mu)
\; .
\end{equation}
In the grand canonical ensemble, the above
equation readily gives the leading contribution to the susceptibility
\begin{equation} \label{eq:landau}
\bar \chi^{\rm \scriptscriptstyle GC} = -\frac{\mu_B^2}{3 A} \
\bar D_{\rm W} \; ,
\end{equation}
coming from the mean part of the grand potential. In Eq.~(\ref{eq:landau})
$A$ is the confining volume (area for $d=2$) of the electrons.
Noting that $\bar D_{\rm W} = {\rm d} \bar N_{\rm W} / {\rm d} \mu$,
one recognizes the familiar result of Landau \cite{Land}.
For systems without potential (bulk, or billiard systems), it gives
in the degenerate case $(\beta \mu \gg 1)$ in two, respectively,
three dimensions
\begin{equation} \label{eq:Lan23}
\bar \chi^{\rm \scriptscriptstyle GC}_{2d} = - \frac{{\sf g_s} e^2}{24 \pi m c^2} \ , \qquad
\bar \chi^{\rm \scriptscriptstyle GC}_{3d} = - \frac{{\sf g_s} e^2 k_{\scriptscriptstyle F}}{24 \pi^2 m c^2} \; .
\end{equation}
In the non--degenerate limit the susceptibility is
\begin{equation} \label{eq:Lannondeg}
\bar \chi^{\rm \scriptscriptstyle GC} = -\frac{\mu_B^2}{3 A} \ \frac{{\bf N}}{k_B T} \; .
\end{equation}
The temperature independence in the degenerate regime and the power--law decay
in the non-degenerate limit cause the dominance of the Landau contribution
at high temperatures since, as mentioned in the previous section (and
demonstrated in appendix~\ref{app:convolution}), the
contributions from $\Delta F^{(1)}$ and $\Delta F^{(2)}$ (Eqs.~(\ref{DF1}) and
(\ref{DF2})) are exponentially damped by temperature.
The Landau diamagnetism is
usually derived for free
electrons or for a quadratic confining potential \cite{LanLip,Peierls}.
We have provided here its
generalization to any confining potential
(including systems smaller than the cyclotron radius).
For a system with fixed number $\bf N$ of electrons, defining a
Weyl chemical
potential $\mu_{\rm W}$ by
\begin{equation}
{\bf N} = \bar N_{\rm W} (\mu_{\rm W})
\end{equation}
and following the same procedure as in Sec.~\ref{sec:Canonical}
one can write
\begin{equation}
F^{(0)}({\bf N}) \simeq F_{\rm W} + \bar \Omega_1(\mu_{\rm W}) \; ,
\end{equation}
where both $\mu_{\rm W}$ and
\begin{equation}
F_{\rm W} = \mu_{\rm W} {\bf N} +
\bar \Omega_{\rm W}(\mu_{\rm W})
\end{equation}
are field independent. Therefore, the smooth part of the free energy
gives the same contribution than Eq.~(\ref{eq:landau}): We
recover the Landau diamagnetic response in the canonical ensemble, too.
At the end of this section we would like to comment on the
case of free electrons in two dimensions. Since
Eq.~(\ref{DOSLL}) represents an exact formula for the density of states,
$\bar d(E) = ({\sf g_s} mA)/(2 \pi \hbar^{2})$ can be
interpreted as the exact mean density of states, and
${d^{\rm osc}}(E) = ({\sf g_s} mA)/(\pi \hbar^{2}) \sum_{n=1}^{\infty} (-1)^{n}
\cos{\left((2\pi n E)/(\hbar w)\right)}$ as the exact oscillating part.
However, $\omega(E)$ being obtained by integrating $d(E)$ twice, has a
mean value which, in addition to $ -\bar d \, E^2/2$, contains
the term $(\mu_B^2 H^2/6) \bar d$ yielding the Landau susceptibility.
In the usual derivation, this term comes from the integration of
${d^{\rm osc}}(E)$, more precisely from the boundary contribution at $E=0$
(i.e., from levels too close to the ground state in order
to properly separate the mean value from oscillating parts).
One should be aware that $\bar \omega(E)$ cannot
be defined by Eq.~(\ref{eq:defwweyl}) as
soon as non leading terms are considered. For this reason
some care was required for the definitions of the
last section (see the discussion around
Eqs.~(\ref{eq:rough_osc:def})-(\ref{eq:smooth_osc:all})).
\newpage
\section{Systems integrable at arbitrary fields}
\label{sec:integrable}
In the remainder of this work we will provide semiclassical approximations
for the corrective free-energy terms $\Delta F^{(1)}$ and $\Delta F^{(2)}$
(see Eq.~(\ref{eq:fd})) and their associated magnetic responses
for systems that react differently under the influence of an applied
field. We will be mainly working in the
weak-field regime (except in section~\ref{sec:highB}),
where the magnetic field
acts as a perturbation almost without altering the classical dynamics.
In this regime the nature of the zero-field dynamics ({\em i.e.\ }
integrable vs.\ chaotic, or more precisely, the organization of periodic
orbits in phase space) becomes the dominant factor determining the
behavior and magnitude of the magnetic susceptibility.
For systems which are integrable at zero field
the generic situation is that the magnetic field breaks the integrability
(as any perturbation will do).
It is necessary in that case to develop semiclassical
methods allowing to deal with nearly, but not exactly, integrable systems.
This question will be addressed in sections~\ref{sec:square} and
\ref{sec:general}.
There exist however ``non generic'' systems where the classical dynamics
remains integrable in the presence of the magnetic field. Due to their
rotational symmetry, circles
and rings (which are the geometries used in many experiments) fall
into this
category. In these cases (and similarly for the Bohm-Aharonov flux
\cite{RivO}) the Berry-Tabor semiclassical trace formula
\cite{ber76,ber77} provides the appropriate path to calculate semiclassically the
oscillating part of the density of states ${d^{\rm osc}}$, including its field
dependence. Thus, $\Delta F^{(1)}$ and
$\Delta F^{(2)}$, and their respective contributions to the susceptibility,
can be deduced.
This is the program we perform in this section, treating
specifically the example of circular and ring billiards.
The magnetic susceptibility of the circular billiard can be calculated from
its exact quantum mechanical solution in terms of Bessel functions
\cite{Dingle52,Bog,von_thesis}. The magnetic response of long cylinders
\cite{Kulik,RiGe} and
narrow rings \cite{RiGe} (the two nontrivial generalizations of
one-dimensional rings) can be calculated by neglecting the curvature of
the circle and solving the Schr\"{o}dinger equation for a rectangle with periodic
boundary conditions. Our semiclassical derivation provides an intuitive
and unifying approach to the magnetic response of circular billiards
and rings of any thickness (for individual systems as well as ensembles)
and establishes the range of validity of
previous studies. Moreover,
we present it for completeness since it provides a pedagogical introduction
to the more complicated (``generic") cases of the following sections.
\subsection{Oscillating density of states for weak field}
\label{sec:GenInt}
By definition, a classical Hamiltonian ${\cal H}({\bf p},{\bf q})$ is integrable
if there exist as many constants of motion in involution as degrees of
freedom. For bounded systems, this implies (see e.g.
\cite{arnold:book}) that all trajectories are trapped
on torus-like manifolds (invariant tori), each of which
can be labeled by the action integrals
\begin{equation} \label{eq:action}
I_i = \frac{1}{2\pi} \oint_{{\cal C}_i} {\bf p} \, {\rm d} {\bf q}
\qquad (i=1,2) \; ,
\end{equation}
taken along two independent paths ${\cal C}_1$ and ${\cal C}_2$ on
the torus. (We are dealing with two degrees of freedom.)
It is moreover possible to perform a canonical transformation from the
original $({\bf p},{\bf q})$ variables to the action-angle variables $({\bf I}, \phi)$
where ${\bf I} = (I_1,I_2)$ and $\phi = (\varphi_1, \varphi_2)$ with
$\varphi_1,\varphi_2$ in $[0,2\pi]$. Because both, $I_1$ and $I_2$, are
constants of motion, the Hamiltonian ${\cal H}(I_1,I_2)$ expressed in
action-angle variables depends only on the actions.
For a given torus we note $\nu_i = \partial {\cal H} / \partial I_i$
$(i=1,2)$ the angular frequencies, and $\alpha \equiv \nu_1/\nu_2$
the rotation number.
A torus is said to be ``resonant'' when its rotation number
is rational ($\alpha = u_1/u_2$ where $u_1$ and $u_2$ are coprime
integers). In that case all the orbits on the torus are periodic, and
the torus itself constitutes a one-parameter family of periodic orbits,
each member of the family having the same period and action.
The families of periodic orbits can be labeled by the two
integers $(M_1,M_2) = (r u_1, r u_2)$ where $(u_1,u_2)$ specifies the
primitive orbits and $r$ is the number of repetitions.
$M_i$ ($i=1,2$) is thus the winding number of $\varphi_i$
before the
orbits close themselves. The pair ${\bf M} = (M_1, M_2)$ has been coined
the ``topology'' of the orbits by Berry and Tabor.
For two-dimensional systems, the Berry-Tabor formula can be cast in
the form \cite{ber76,ber77}
\begin{equation} \label{BT}
{d^{\rm osc}}(E) = \sum_{{\bf M} \neq (0,0),\epsilon} d_{{\bf M},\epsilon}(E) \ ,
\end{equation}
with
\begin{equation} \label{BTT}
d_{{\bf M},\epsilon}(E) =
\frac{{\sf g_s} \ \tau_{{\bf M}}}{\pi \hbar^{3/2} M_2^{3/2}
\left| g_E^{''}(I_{1}^{{\bf M}})
\right|^{1/2}}
\ \cos{\left(\frac{S_{{\bf M},\epsilon}}{\hbar} - \hat\eta_{{\bf M}} \
\frac{\pi}{2} +
\gamma \ \frac{\pi}{4} \right)} \ .
\end{equation}
The sum in Eq.\ (\ref{BT}) runs over all families of closed orbits at energy $E$,
labeled by their topology ${\bf M}$ (in the first quadrant, that is $M_1$
and $M_2$ are positive integers), and, except for self-retracing orbits,
by an additional index $\epsilon$ specifying tori related to each other
through time reversal symmetry and therefore having the same topology.
${\sf g_s}$ represents the spin degeneracy factor, while
$S_{{\bf M},\epsilon}$ and $\tau_{{\bf M}}$ are, respectively,
the action integral and the period along the periodic trajectories
of the family ${\bf M}$. $\hat\eta_{{\bf M}}$ is
the Maslov index which counts the number of caustics
of the invariant torus encountered by the trajectories.
For billiard systems with Dirichlet boundary conditions, we will
also take into account in $\hat\eta_{{\bf M}}$ the phase $\pi$
acquired at each bounce of the trajectory on the hard walls
(and still refer to $\hat\eta_{{\bf M}}$ as the Maslov index, although
slightly improperly).
The energy surface $E$ in action space whose implicit form is
${\cal H}(I_1,I_2) = E$, is explicitly defined by the function
$I_2 = g_E(I_1)$.
We note ${\bf I}^{\bf M} = (I^{\bf M}_1,I^{\bf M}_2)$ the action variables of the torus
where the periodic orbits of topology ${\bf M}$ live. They are determined
by the resonant-torus condition
\begin{equation} \label{eq:resonant}
\alpha = - \left. \frac{d g_E(I_1)}{d I_1}
\right|_{I_1=I^{\bf M}_1} \ =
\ \frac{M_1}{M_2} \; ,
\end{equation}
where the first equality arises from the differentiation of
${\cal H}(I_1,g_E(I_1)) = E$ with respect to $I_1$.
Finally, the last contribution to the phase is given by
$\gamma = {\rm sgn}(g''_E(I^{\bf M}_1))$.
The [first] derivation of the Berry-Tabor trace formula \cite{ber76}
follows very
similar lines as the treatment of the density of states performed in the
introduction for the macroscopic Landau susceptibility. The EBK (Einstein,
Brillouin, Keller) quantization condition is used instead of the exact
form (\ref{LL}) of the Landau levels, followed by the
application of the Poisson summation rule. While in the latter case
this procedure leads to the
exact sum of Eq.~(\ref{DOSLL}), the Berry-Tabor formula is obtained
(similar to the treatment of de Haas~--~van Alphen oscillations
for a non-spherical Fermi surface) after
a stationary-phase approximation valid in the semiclassical limit where
$S \gg \hbar$ (with a stationary-phase condition according to
Eq.~(\ref{eq:resonant})).
Given a two-dimensional electron system whose classical Hamiltonian
\begin{equation} \label{eq:classH}
{\cal H}({\bf p},{\bf q}) = \frac{1}{2m} \
\left({\bf p} - \frac{e}{c} {\bf A}( {\bf q}) \right)^2
+ V( {\bf q})
\end{equation}
remains integrable for finite values of the transverse field
$H {\hat z} = \nabla \times {\bf A}$, the magnetic response can be obtained,
in principle, from the calculation of the various quantities involved
in the Berry-Tabor formula at finite fields.
However, for weak fields, one can use the fact that the
field dependence of each contribution $d_{\bf M}$ to the oscillating
part of the density of states is essentially due to the
modification of the classical action, since this latter is multiplied
by the large factor $1/\hbar$, while the field dependence of the periods
and the curvatures of the energy manifold can be neglected.
Therefore, in this regime we will use for $\tau_{\bf M}$ and $g_E$ the
values $\tau^0_{\bf M}$ and $g^0_E$ at zero field
and consider the first order correction $\delta S$ to the unperturbed
action $S^0_{\bf M}$. A general result in classical mechanics
\cite{ozor:book,boh95} states that the change ({\em at constant energy})
in the action integral along a closed orbit under the
effect of a parameter $\lambda$ of the Hamiltonian is given by
\begin{equation} \label{theorem}
\left(\frac{\partial S}{\partial \lambda} \right)_E =
- \oint {\rm d} t \ \frac{\partial {\cal H}}{\partial \lambda} \ ,
\end{equation}
where the integral is taken along the {\em unperturbed} trajectory.
Therefore, if the Hamiltonian has the form of Eq.~(\ref{eq:classH}),
classical perturbation theory yields for small magnetic fields $H$,
\begin{equation} \label{dS}
\delta S = \frac{e}{c} \ H {\cal A}_{\epsilon} \; ,
\end{equation}
where ${\cal A}_{\epsilon}$ is the
directed area enclosed by the unperturbed orbit.
This expansion is valid for magnetic fields
low enough, or energies high enough, such that the cyclotron radius of the
electrons is much larger than the typical size of the structure
($r_c=mcv/eH \gg a$, which is, e.g., the case for electrons at the Fermi
energy in the
experiments of Refs. \cite{levy93,BenMailly}). In this case we neglect the
change in the classical dynamics and consider the effect of the applied
field only through the change of the action integral.
For a generic integrable system there is no reason, a priori, that
all the orbits of a given family ${\bf M}$ should enclose the same area.
However, as pointed out above, a characteristic feature of integrable
systems is that the action is a constant for all the periodic orbits
of a given resonant torus. Therefore, the fact that a system remains
integrable under the effect of a constant magnetic field implies
(because of Eq.~(\ref{dS})) that all the orbits of a
family enclose the same absolute area ${\cal A}_{{\bf M},\epsilon}$.
Moreover, since the system is time-reversal invariant
at zero field, each closed orbit
(${\bf M},\epsilon$) enclosing an area ${\cal A}_{{\bf M},\epsilon}$ is associated
with a time-reversed partner having exactly the same
characteristics except for an opposite enclosed area (if the orbit
is its own time reversal, ${\cal A}_{\bf M} = 0$).
Grouping time-reversal trajectories in Eq.~(\ref{BT}) at $H\!=\!0$ we
have
\begin{equation} \label{eq:dmstrhe0}
d^0_{{\bf M}}(E) = \left\{ \begin{array}{ll} \displaystyle
d^0_{{\bf M},\epsilon}(E) & \qquad \qquad \mbox{for self-retracing orbits}\\
\displaystyle \sum_{\epsilon=\pm 1} d^0_{{\bf M},\epsilon}(E) = 2 \ d^0_{{\bf M},\epsilon}(E)
& \qquad \qquad \mbox{for non self-retracing orbits}\\
\end{array} \right.\; .
\end{equation}
For weak fields the contribution of self-retracing orbits is unaltered and therefore
they do not contribute to the magnetic response. For the non
self-retracing ones we have
\begin{equation} \label{eq:lowBd}
d_{\bf M}(E,H) = \sum_{\epsilon=\pm 1} d_{{\bf M},\epsilon}(E,H) = d^0_{{\bf M}}(E) \
\cos{\left(\frac{eH}{\hbar c} {\cal A}_{{\bf M}} \right)}
\; \; , \; \; {\cal A}_{\bf M} = |{\cal A}_{{\bf M},\epsilon}| \; .
\end{equation}
\noindent This is the basic relation to be used in the examples that follow.
\subsection{Circular billiards}
\label{sec:circle}
We now apply the preceding considerations to a
two-dimensional gas of electrons
moving in a circular billiard of radius $a$ (where the potential $V({\bf q}) $
is zero in the region $|{\bf q}| < a$ and infinite outside it). Thus we deal
with vanishing wavefunctions at the boundary (Dirichlet
boundary condition).
In billiards without magnetic field the magnitude
$p$ of the momentum is conserved,
and it is convenient to introduce the wave number,
\begin{equation} \label{eq:k}
k = \frac{p}{\hbar} = \frac{\sqrt{2mE}}{\hbar}
\end{equation}
\noindent since at $H\!=\!0$ the time-of-flight and the action-integral
of a given trajectory
can be simply expressed in terms of its length $L$ as
\begin{equation} \label{eq:SandT}
\tau^0 = \frac{m}{p} \ L \; , \qquad
\frac{S^0}{\hbar} = k L \; .
\end{equation}
Following Keller and Rubinow \cite{keller}, we calculate the action
integrals ${\bf I}=(I_1,I_2)$ by using the independent paths ${\cal C}_1$
and ${\cal C}_2$ displayed
in Fig.~\ref{fig:circle}(a). The function $g_E$ is given by
(see \cite{keller} and
Appendix \ref{app:ring_g})
\begin{equation} \label{circle:gE}
g_E(I_1) = \frac{1}{\pi} \left\{ \left[(pa)^2-I_1^2\right]^{1/2}
- \ I_1 \ \arccos\left(\frac{I_1}{pa}\right) \right\} \ ,
\end{equation}
where $I_1$ is interpreted as the angular momentum and bounded by
$0 \leq I_1 < pa$.
The periodic orbits of the circular billiard are labeled by
the topology ${\bf M} = (M_1,M_2)$,
where $M_1$ is the number of turns around the circle until
coming to the initial point after $M_2$ bounces. (Obviously
$M_2 \ge 2 M_1$.) Elementary geometry yields for the length of
the topology-${\bf M}$ trajectories
\begin{equation} \label{lengthM}
L_{{\bf M}}=2M_2a \ \sin{\delta} \ ,
\end{equation}
\noindent where $\delta = \pi M_1/M_2$.
The resonant-torus condition,
Eq.~(\ref{eq:resonant}), allows us to obtain
${\bf I}^{\bf M}$ as
\begin{mathletters} \label{allIMs}
\begin{equation} \label{IMa}
I_1^{\bf M} = p a \cos{\delta} \; ,
\end{equation}
\begin{equation} \label{IMb}
I_2^{\bf M} = \frac{p a}{\pi} \left\{\sin{\delta}
\ - \ \delta \
\cos{\delta} \right\} \; .
\end{equation}
\end{mathletters}
The Maslov index of the topology-${\bf M}$ trajectories is
$\hat\eta_{\bf M} = 3 M_2$
($M_2$ bounces, each of them giving a dephasing of $\pi$, and $M_2$
encounters with the caustic per period).
We therefore have all the ingredients necessary to calculate the oscillating
part of the density of states at
zero field: For the non self-retracing trajectories we obtain
\begin{equation} \label{dgwgzf}
d^0_{{\bf M}}(E) = \sqrt{\frac{2}{\pi}} \
\frac{{\sf g_s} m L_{ {\bf M}}^{3/2}}{\hbar^2}
\ \frac{1}{k^{1/2}M_{2}^{2}} \ \cos{\left(k
L_{{\bf M}}\ + \frac{\pi}{4} - \frac{3\pi}{2} M_{2} \right)} \; .
\end{equation}
The contribution of a self-retracing orbit is just one half of the contribution
(\ref{dgwgzf}). Its field dependent counterpart is obtained from
Eq.~(\ref{eq:lowBd}) with the area enclosed by the periodic orbits given by
\begin{equation} \label{areaMg}
{\cal A}_{{\bf M}} = \frac{M_2 a^2}{2} \sin{2\delta} \ .
\end{equation}
The bouncing-ball trajectories $M_2= 2 M_1$ (with zero angular momentum)
are self-retracing and have no
enclosed area; thus they do not contribute to the low field susceptibility.
Using Eqs.~(\ref{DF1}) and (\ref{eq:smooth_osco}), and noting
$k_{\scriptscriptstyle F} = k(\bar \mu) = (2/a) \ ({\bar N}({\bar \mu})/{\sf g_s})^{1/2}$ the
Fermi wave vector, we obtain the contribution to the magnetic
susceptibility associated with $\Delta F^{(1)}$:
\begin{eqnarray}
\frac{\chi^{(1)}}{\chi_L} & = &
\frac{48}{\sqrt{2 \pi}} \ (k_{\scriptscriptstyle F} a)^{3/2} \; \times
\label{chicir1} \\
& \times &
\sum_{M_1,M_2 > 2M_1} \frac{({\cal A}_{{\bf M}}/a^2)^2}{(L_{{\bf M}}/a)^{1/2}} \
\frac{1}{M_2^2}
\cos{\left(k_{\scriptscriptstyle F} L_{{\bf M}}\ + \frac{\pi}{4} - \frac{3\pi}{2}M_2\right)}
\cos{\left(\frac{eH}{\hbar c} {\cal A}_{{\bf M}} \right)}
\ R_T(L_{{\bf M}}) \ . \nonumber
\end{eqnarray}
Since we are working with billiards, the temperature factor $R_T$ is
given in terms of the trajectory length $L_{\bf M}$ by Eq.~(\ref{R_factor2})
and the characteristic cut-off length
$L_c = \hbar v_{\scriptscriptstyle F} \beta/\pi $. For
$M_2 \gg M_1$ we have $L_{\bf M} \simeq 2 \pi M_1 a$ and
${\cal A}_{\bf M} \simeq \pi M_1 a^2$, independent of $M_2$.
Performing the summation over the index $M_2$ (for fixed value of
$M_1$) by taking the length and area
dependent terms outside the sum we are left with a rapidly convergent
series (whose general term is $(-1)^{M_2}/M_2^2$). We can therefore
truncate the series after the first few terms. In Fig.~\ref{fig:chi_circle}
the sum (\ref{chicir1}) is evaluated numerically at zero field
(solid line) for a cut-off length $L_c = 6 a$ which selects only
the first ($M_1=1$) harmonic,
and the beating between the first few periodic orbits is obtained
as a function of wave-vector $k_{\scriptscriptstyle F}$. With only the first two primitive orbits
($M_2=3$ and $4$, dashed line) we give a good account of $\chi^{(1)}$ for
most of the $k$-interval. Taking the first four primitive orbits
suffices to reproduce the
whole sum. The short period in $k_{\scriptscriptstyle F}$ corresponds
approximately to the circle perimeter $L=2\pi a$.
Going to lower temperatures gives an overall increase of
the susceptibility but does not modify the structure of the
first harmonic contribution
since the length of the whispering-gallery trajectories is bounded
by $L$. However, for larger values of $L_c$ higher harmonics, namely
up to $M_1$ of the order of $L_c/2\pi a$, will be observed.
The predominance of
the first few trajectories also appears in the beating as a function
of magnetic field (not shown) that results from the evaluation of
(\ref{chicir1}) at finite fields.
>From Fig.~\ref{fig:chi_circle} we see that the susceptibility of a circular
billiard oscillates as a function of the number of electrons (or $k_{\scriptscriptstyle F}$)
taking paramagnetic and diamagnetic values. Its overall magnitude is
much larger than the two-dimensional Landau susceptibility and grows as
$(k_{\scriptscriptstyle F} a)^{3/2}$. We will later show (Sec.~\ref{sec:general})
that this finite-size increase with respect
to the bulk value is distinctive of systems that are integrable at
zero field. In order to characterize the typical value of the magnetic
susceptibility we define
\begin{equation}
\label{chicir1t}
\chi^{({\rm t})} \ = \ \left[ \ \overline{(\chi^{(1)})^2} \ \right]^{1/2}
\end{equation}
\noindent where, as in section \ref{sec:TherFor},
the average is over a $k_{\scriptscriptstyle F} a$ interval
classically negligible ($\Delta(k_{\scriptscriptstyle F} a) \ll k_{\scriptscriptstyle F} a$)
but quantum mechanically large ($\Delta(k_{\scriptscriptstyle F} a) \gg 2\pi$), so that
off-diagonal terms $\cos(k_{\scriptscriptstyle F} L_{\bf M}) \cos(k_{\scriptscriptstyle F} L_{{\bf M}'})$
with ${\bf M} \neq {\bf M}'$ vanish under averaging. A remark is in order
here because at fixed $M_1$, $L_{\bf M}$ goes to $2\pi M_1 a$ as
$M_2$ goes to $\infty$, and $(L_{(M_1,M_2)} - L_{(M_1,M'_2)})$
can be made arbitrarily small by increasing $M_2$ and $M'_2$.
Therefore, for any interval of $k_{\scriptscriptstyle F} a$ over which the
average is taken, some non-diagonal terms should remain
unaffected. Nevertheless, because of the rapid decay of the
contribution with $M_2$, these non-diagonal terms can be neglected in
practice for the experimentally relevant temperatures.
The typical zero-field susceptibility
of the circular billiard is then given by
\begin{equation}
\label{chicir1td}
\frac{\chi^{({\rm t})}(H\!=\!0)}{\chi_L} \
\simeq \ \frac{48}{\sqrt{2 \pi}} \ (k_{\scriptscriptstyle F} a)^{3/2} \
\left[ \frac{1}{2} \sum_{M_1,M_2>2M_1}
\frac{({\cal A}_{{\bf M}}/a^2)^4}{L_{{\bf M}}/a} \ \frac{R^{2}_T(L_{{\bf M}})}{M_2^4}
\right]^{1/2} \ .
\end{equation}
Numerical evaluation of the first harmonic ($M_1\!=\!1$) from (\ref{chicir1t}) on the
$k_{\scriptscriptstyle F} a$ interval of Fig.~\ref{fig:chi_circle} with $L_c = 6a$ gives
$2.20 (k_{\scriptscriptstyle F} a)^{3/2} \chi_L$ (dotted horizontal line),
while Eq.~(\ref{chicir1td}) restricted to $M_2\!\le\!6$
yields $2.16 (k_{\scriptscriptstyle F} a)^{3/2} \chi_L$
illustrating the smallness of the off-diagonal and large-$M_2$ terms.
For an ensemble made of circular billiards with a dispersion in size
or in the number of electrons such that
$\Delta (k_{\scriptscriptstyle F} a) > 2 \pi $, the term $\chi^{(1)}$ yields a vanishing
contribution to the average susceptibility. In such a case it is necessary
to go to the next-order free-energy term $\Delta F^{(2)}$, whose
associated contribution $\chi^{(2)}$ yields the average susceptibility
by means of Eqs.~(\ref{eq:sus}) and (\ref{DF2}). For the same
reason as above one can show that only diagonal
terms of $({N^{\rm osc}})^2$ survive the $k_{\scriptscriptstyle F} a$ average, in spite of the degeneracy
of the length of the closed orbits as $M_2$ goes to $\infty$. One therefore has
\begin{equation} \label{chicir}
\frac{\overline{\chi}}{\chi_L} =
\frac{48}{\pi} \ k_{\scriptscriptstyle F} a \ \sum_{M_1,M_2 > 2M_1}
\frac{(A_{\bf M}/a^2)^2 (L_{\bf M}/a) }{M_2^4} \
\cos{\left(\frac{2eH}{\hbar c} {\cal A}_{{\bf M}} \right)}
\ R_T^{2}(L_{\bf M}) \ .
\end{equation}
Again, the terms generally decay rapidly with $M_2$ (as $1/M_2^4$), and
for a cutoff length $L_c$ selecting only the terms with $M_1=1$
the total amplitude at zero field ($5.2 k_{\scriptscriptstyle F} a$) can be obtained
from the first few lowest terms. The low field
susceptibility of an ensemble of circular billiards is paramagnetic
and increases linearly with $k_{\scriptscriptstyle F} a$. As for the $\chi^{(1)}$
contribution, we will show in the sequel that this behavior does
not necessitate the integrability at finite fields, but rests only
upon the integrability at zero field.
Up to now there have not been measurements of the magnetic response
of electrons in circular billiards (individual or ensembles). Our
typical (Eq.~(\ref{chicir1td})) or average (Eq.~(\ref{chicir}))
susceptibilities exhibit a large enhancement with respect to the bulk
values (by powers of $k_{\scriptscriptstyle F} a$). Thus it should
be possible to detect experimentally these finite-size effects.
\subsection {Rings}
The magnetic response of small rings can be calculated along the same
lines as
in the case of the circles. The ring geometry deserves special interest since
it is the preferred configuration for persistent current
measurements. In a
ring geometry at $H\!=\!0$ we have two types of periodic orbits:
those which do not touch the inner disk (type-I),
and those which do hit it (type-II).
(See Fig.~\ref{fig:circle} of Appendix \ref{app:ring_g}; we note
by $a$ and $b$ respectively the outer and inner radius of the ring.)
The function
$g_{E}(I_1)$ has two branches corresponding to the interval to which the
angular momentum $I_1$ belongs. For $pb<I_1<pa$, (type-I trajectories)
$g_{E}$ has the same form (\ref{circle:gE}) as for the circle, while for
$0 \leq I_1 <pb$, (type-II trajectories) we show in
Appendix \ref{app:ring_g} that
\begin{equation} \label{ring:gE}
g_E(I_1) = \frac{1}{\pi} \left\{ \left[(pa)^2-I_1^2\right]^{1/2} -
\left[(pb)^2-I_1^2\right]^{1/2} \ - \
I_1 \left[\arccos\left(\frac{I_1}{pa}\right) -
\arccos\left(\frac{I_1}{pb}\right)\right] \right\} \ .
\end{equation}
The type-I trajectories are labeled in the same way as for the circle
by the topology ${\bf M}=(M_1,M_2)$ representing the number of turns $M_1$
around the inner circle until returning to the initial point after $M_2$
bounces on the outer circle. We therefore obtain the
resonant-tori condition
Eqs.~(\ref{allIMs}) and the same contribution (\ref{dgwgzf}) to the
oscillating part of the density of states as in the case of the circle.
The only difference is that in the Berry-Tabor trace formula
(Eq.~(\ref{BT})) the sum corresponding to type-I trajectories is now
restricted to $M_2 \geq \hat M_2(M_1) = {\rm Int}[M_1 \pi/\arccos{r}]$.
We note by
${\rm Int}$ the integer-part function and $r=b/a$.
We stress the fact that the minimum value of $M_2$ is itself a
function of $M_1$. The previous restriction
can also be expressed as $\cos{\delta} > r$, with
$\delta = \pi M_1/M_2$.
Type-II trajectories can be labeled by the
topology ${\bf M}=(M_1,M_2)$, where $M_1$ is the number of turns
around the inner circle in coming to the initial point after $M_2$
bounces on the {\em outer} circle. We have the same restriction
$M_2 \geq \hat M_2(M_1) $ as for type-I trajectories, and we can use
$\hat\eta_{{\bf M}}=0$
since there are $2M_2$ bounces with the hard walls and no encounters
with the caustic. From (\ref{ring:gE}) we obtain the resonant-torus
condition
\begin{mathletters} \label{allIMRs}
\begin{equation} \label{IMRa}
I_1^{\bf M} = p b \
\frac{\sin{\delta}}{\sqrt{1+r^2-2r\cos{\delta}}} \; ,
\end{equation}
\begin{equation} \label{IMRb}
I_2^{\bf M} = \frac{p a}{\pi} \left\{
\sqrt{1+r^2-2r\cos{\delta}} \ - \
\frac{r \delta \sin{\delta}}{\sqrt{1+r^2-2r\cos{\delta}}}
\right\} \; .
\end{equation}
\end{mathletters}
The $H\!=\!0$ contribution to the oscillating part of the density of states
from non self-retracing type-II trajectories with topology ${\bf M}$ is given by
\begin{equation}
\tilde{d}^0_{{\bf M}}(E) = 4 \sqrt{\frac{2}{\pi}} \
\frac{{\sf g_s} a^2 m}{\hbar^2 } \
\frac{\left[(1-r\cos{\delta})(r\cos{\delta}-r^2) \right]^{1/2}}
{\left( k \tilde L_{\bf M} \right)^{1/2}}
\ \sin{\left(k \tilde{L}_{{\bf M}}\ + \frac{\pi}{4} \right)} \ ,
\label{dgwgzft1}
\end{equation}
while its length is
\begin{equation} \label{lenghtMt1}
\tilde{L}_{{\bf M}}=2M_2a \ \sqrt{1+r^2-2r\cos{\delta}} \ .
\end{equation}
The small field dependence follows from Eq.~(\ref{eq:lowBd}) using
the enclosed area
\begin{equation} \label{areaMt1}
\tilde A_{\bf M} =
M_2 ab \ \sin{\delta} \ .
\end{equation}
In the case of annular geometries it is customary to characterize the magnetic
moment ${\cal M}$ of the ring by the persistent current
\begin{equation} \label{eq:percu}
I = \frac{c}{A} \ {\cal M} =
- c \left(\frac{\partial F}{\partial \Phi} \right)_{T,N} \ .
\end{equation}
In order to pass from the applied magnetic field $H$ to the
flux $\Phi$ we use the area $A$ of the outer circle ($\Phi=A H$, \
$A=\pi a^2$) as defining area. (For thin rings, all periodic orbits with
the same repetition number $M_1$ enclose approximately the same
flux $M_1 \Phi$.) Applying Eqs.~(\ref{eq:smooth_osco})--(\ref{allDF}),
and calling $I_0=e v_{\scriptscriptstyle F}/2\pi a$ the typical current of one-dimensional
electrons at the Fermi energy, the persistent current of a ring billiard
can be expressed as the sum of two contributions corresponding to both
types of trajectories:
\begin{equation} \label{I1}
\frac{I^{(1)}}{I_0} =
{\sf g_s} \ (k_{\scriptscriptstyle F} a)^{1/2} \sum_{M_1,M_2 \geq \hat M_2}
\left\{ {\cal I}^{(1)}_{{\bf M},I} \ \sin{\left(\frac{eH}{\hbar c} {\cal A}_{{\bf M}} \right)}
\ R_T(L_{{\bf M}}) \ +
{\cal I}^{(1)}_{{\bf M},II} \ \sin{\left(\frac{eH}{\hbar c} \tilde {\cal A}_{\bf M} \right)}
R_T(\tilde{L}_{{\bf M}}) \right\} \ ,
\end{equation}
\begin{mathletters} \label{allI1Rs}
\begin{equation} \label{I1Ra}
{\cal I}^{(1)}_{{\bf M},I} =
2\sqrt{\frac{2}{\pi}} \ \frac{1}{M_2^2}
\ \frac{({\cal A}_{\bf M}/a^2)}{(L_{{\bf M}}/a)^{1/2}}
\ \cos{\left(k_{\scriptscriptstyle F} L_{\bf M} + \frac{\pi}{4} - \frac{3\pi}{2} M_2\right)}
\ ,
\end{equation}
\begin{equation} \label{I1Rb}
{\cal I}^{(1)}_{{\bf M},II} = 8 \sqrt{\frac{2}{\pi}}
\frac{(\tilde {\cal A}_{\bf M}/a^2)}{(\tilde{L}_{{\bf M}}/a)^{5/2}}
\left[(1-r\cos{\delta})(r\cos{\delta}-r^2) \right]^{1/2}
\sin{\left(k \tilde{L}_{{\bf M}}\ + \frac{\pi}{4} \right)}
\; .
\end{equation}
\end{mathletters}
In Fig.~\ref{fig:chi_ring} we present the first harmonic
$I_{1}^{(1)}$ of the persistent current for a thin ring and a cut-off
length $L_c=6a$ (solid line). (I.e., we are considering the winding number $M_1=1$.)
The contribution of type-I trajectories (dashed line) is similar as in
the case of the
circle: a rapidly convergent sum showing as a function of $k_{\scriptscriptstyle F}$ the
beating between the first two trajectories ($\hat M_2$ and $\hat M_2+1$). On the
other hand, Eq.~(\ref{I1Rb}) shows that the trajectories with low
values of $M_2$ (i.e.~$M_2 \sim \hat M_2$) contributing
to ${\cal I}^{(1)}_{{\bf M},II}$ have negligible weight due to the small
stability prefactor caused by the defocusing effect exerted by the
inner disk ($\cos {\delta} \simeq r$). The sum is dominated by
trajectories with $M_2 > \hat M_2$ and therefore we loose the
previous beating structure in the total $I_1^{(1)}$.
The short period in $k_{\scriptscriptstyle F}$ still corresponds to the circle perimeter $L$.
As in the previous subsection, we characterize
the typical value of the magnetic response by averaging
$(I^{(1)})^2$ over a $k_{\scriptscriptstyle F} a$-interval containing many oscillations,
but yet negligible on the classical scale.
\begin{equation} \label{i1t}
I^{({\rm t})} \ = \ \left[ \ \overline{\left( I^{(1)} \right)^2}
\ \right]^{1/2} \ .
\end{equation}
In the same way as for the circular billiard, one can in practice consider
that, despite the degeneracy in the length of type-I trajectories for
large $M_2$, only diagonal terms (in both index ${\bf M}$ and
trajectory-type) survive the averaging for large enough
$\Delta(k_{\scriptscriptstyle F} a)$. Therefore
\begin{eqnarray} \label{I1td}
\frac{I^{({\rm t})}}{I_0} & \simeq & {\sf g_s} (k_{\scriptscriptstyle F} a)^{1/2}
\sum_{M_1,M_2 \geq \hat M_2}
\left[ \left({\cal I}^{({\rm t})}_{{\bf M},I} \right)^2
\sin^2{\left(\frac{eH}{\hbar c} {\cal A}_{{\bf M}} \right)} R_T^2(L_{{\bf M}}) \right. \nonumber + \\
& + & \left. \left( {\cal I}^{({\rm t})}_{{\bf M},II} \right)^2
\sin^2{\left(\frac{eH}{\hbar c} \tilde {\cal A}_{\bf M} \right)} R_T^2(\tilde{L}_{{\bf M}})
\right]^{1/2} \ ,
\end{eqnarray}
\noindent where $({\cal I}^{({\rm t})}_{{\bf M},I})^2$ and
$({\cal I}^{({\rm t})}_{{\bf M},II})^2$ are obtained
from Eqs.~(\ref{allI1Rs}) simply by replacing the average of
$\cos^2(k_{\scriptscriptstyle F} L_{\bf M} + \pi/4 -3M_2\pi/2)$ and
$\sin^2(k_{\scriptscriptstyle F} \tilde L_{\bf M} + \pi/4)$ by $1/2$.
In Fig.~\ref{fig:chi_ty} we present the typical persistent current and its
two contributions for various ratios $r=b/a$ and cut-off lengths $L_c$ for
the first harmonic ($M_1\!=\!1$).
The contribution ${\cal I}^{({\rm t})}_{{\bf M},I}$ of type-I trajectories dominates
for small $r$ (where the inner circle is not important and we
recover the magnetic response of the circular billiard) while type-II
trajectories take over for narrow rings. The crossover $r$ depends on
temperature through $L_c$ due to the different dependence of the
trajectory length on ${\bf M}$ (Eqs.~(\ref{lengthM}) and (\ref{lenghtMt1}))
for both types of trajectories.
As in the case of $\chi^{(1)}$ for the circular billiard, $I^{(1)}$
gives a vanishing contribution to
the persistent current of an ensemble of rings with different sizes or
electron fillings as soon as the dispersion in $k_{\scriptscriptstyle F} a$ is of the order
of $2 \pi$. We therefore need to go to the term $\Delta F^{(2)}$ in
the free-energy expansion, which is obtained (see Eq.~(\ref{DF2})) from
\begin{equation} \label{noscring}
{N^{\rm osc}} (\bar \mu) = \sum_{M_1,M_2 \geq \hat M_2}
\left\{ N_{{\bf M},I}(\bar \mu)+N_{{\bf M},II}(\bar \mu) \right\} \ ,
\end{equation}
where $N_{{\bf M},I}(\bar \mu)$ and $N_{{\bf M},II}(\bar \mu)$ are given in terms
of the respective contributions to the field dependent density of
states through Eq.~(\ref{eq:smooth_oscn}).
For an ensemble with a large dispersion of
sizes only diagonal terms survive the average and we have (with
$\bar D = {\sf g_s} m A (1-r^2) / (2\pi\hbar^2)$)
\begin{equation} \label{i2av}
\frac{\overline{I^{(2)}}}{I_0} =
{\sf g_s} \ \sum_{M_1,M_2 \geq \hat M_2}
\left\{ \overline{{\cal I}^{(2)}_{{\bf M},I}} \
\sin{\left(\frac{2 eH}{\hbar c} {\cal A}_{{\bf M}} \right)}
\ R_T^2(L_{{\bf M}}) +
\overline{{\cal I}^{(2)}_{{\bf M},II}}
\ \sin{\left(\frac{2eH}{\hbar c} \tilde {\cal A}_{\bf M} \right)}
\ R_T^2(\tilde{L}_{{\bf M}}) \right\} \ ,
\end{equation}
\begin{mathletters} \label{allI2Rs}
\begin{equation} \label{I2Ra}
\overline{{\cal I}^{(2)}_{{\bf M},I}} =
\frac{2}{\pi} \ \frac{1}{M_2^4}
\ \left(\frac{L_{{\bf M}}}{a}\right) \
\left(\frac{{\cal A}_{\bf M}}{a^2}\right)\ \frac{1}{1-r^2} \
\ ,
\end{equation}
\begin{equation} \label{I2Rb}
\overline{{\cal I}^{(2)}_{{\bf M},II}} = \frac{32}{\pi} \
\frac{\left( \tilde {\cal A}_{\bf M}/a^2 \right)}{\left( \tilde
L_{\bf M} / a \right)^3}
\ \frac{(1-r \cos{\delta})(r\cos{\delta}-r^2)}{1-r^2}
\; .
\end{equation}
\end{mathletters}
The $k_{\scriptscriptstyle F}$ dependence of the average persistent current is linear (through $I_0$),
similarly to the case of the average susceptibility of an ensemble of circular
billiards.
\subsubsection*{Thin rings}
In the case of thin rings ($a \simeq b$, \ $r \simeq 1$)
further approximations can be performed on Eqs.~(\ref{allI1Rs})
and (\ref{allI2Rs}) using $(1-r)$ as
a small parameter, giving more compact and meaningful expressions
for the typical and average persistent currents. Since in addition
this is the configuration used in the experiment of
Ref.~\cite{BenMailly}, we shall consider more closely
this limiting case. First, we note that $\hat \delta = \arccos{r}
\simeq \sqrt{2(1-r)} \ll 1$. Thus
\begin{equation} \label{mhat}
\hat M_2 = {\rm Int}\left[\frac{\pi M_1}{\hat \delta}\right]
\simeq \frac{\pi}{\sqrt{2}} \ \frac{M_1}{\sqrt{1-r}}
\gg M_1 \; ,
\end{equation}
and for $M_2 \ge \hat M_2$, the area and length of contributing orbits
can be approximated by
\begin{equation} \label{ALTRs}
{\cal A}_{\bf M} \simeq \tilde {\cal A}_{\bf M} \simeq M_1 A = M_1 \pi a^2
\qquad ; \qquad
L_{\bf M} \simeq M_1L = M_1 2 \pi a \; .
\end{equation}
For the length of type-II trajectories we have
$\tilde{L}_{{\bf M}} \simeq M_1 L$ for $M_2 \simeq \hat M_2$, and
${\tilde L}_{\bf M} \simeq 2 M_2(a-b)$ when $ M_2 \gg \hat M_2$.
All trajectories with winding number $M_1$ enclose approximately the same
flux $M_1 \Phi$, and the field dependent terms in Eq.~(\ref{I1})
may be replaced by $\sin{(2 \pi M_1 \Phi/\Phi_0)}$.
There is therefore no difference between the case that we study (where
a uniform magnetic field $H$ is applied) and the ideal case of a flux line
$\Phi$ through the inner circle of the ring.
The length dependent factors $R_T^2$ can also be taken outside the sum
over $M_2$ since the main contribution of type-II trajectories comes from
$M_2 \simeq \pi M_1/[5^{1/6}(1-r)^{2/3}]$.
Even if these $M_2$'s are much larger than $\hat M_2$, their associated
${\tilde L}_{\bf M}$ are still
of the order of $M_1 L$ to leading order in $1-r$.
Turning now to the typical and ensemble average currents,
it should be stressed that for narrow rings it is necessary
to go to fairly large energies before an average on a scale
being quantum mechanically large but classically small is possible.
Indeed, one has for both types of trajectories
$k_{\scriptscriptstyle F}(L_{\hat M_2+1}-L_{\hat M_2}) \simeq k_{\scriptscriptstyle F}(\tilde L_{\hat M_2+1} -
\tilde L_{\hat M_2}) \simeq (4\sqrt{2}/3) \pi {\cal N} \sqrt{1-r}$,
where ${\cal N}=k_{\scriptscriptstyle F} (a-b)/\pi$ is the number of transverse occupied
channels. Therefore, ${\cal N}$ should be much larger than
$(1-r)^{-1/2}$ if one wants to assume $\Delta (k_{\scriptscriptstyle F} a)$ sufficiently
large to average out all non-diagonal terms without
violating the condition $\Delta(k_{\scriptscriptstyle F} a) \ll k_{\scriptscriptstyle F} a$.
Supposing the previous condition is met, and
introducing the typical amplitudes ${\cal J}^{({\rm t})}_{M_1,I}$ and
${\cal J}^{({\rm t})}_{M_1,II}$ of each harmonic, we write
\begin{equation}
\frac{I^{({\rm t})}}{I_0} =
{\sf g_s} \ \left[ \sum_{M_1} \left\{
\left( {\cal J}^{({\rm t})}_{M_1,I} \right)^2
+ \left( {\cal J}^{({\rm t})}_{M_1,II} \right)^2 \right\}
\sin^2\left(2\pi M_1\frac{\Phi}{\Phi_0} \right)
R^2_T(M_1 L) \right]^{1/2} \ ,
\end{equation}
\begin{mathletters} \label{allJts}
\begin{equation} \label{Jta}
\left({\cal J}^{({\rm t})}_{M_1,I}\right)^2 =
k_{\scriptscriptstyle F} a \ \sum_{M_2 \geq \hat M_2} \left({\cal I}^{({\rm t})}_{{\bf M},I}\right)^2
= 2 k_{\scriptscriptstyle F} a M_1 \left[ \sum_{M_2 \geq \hat M_2}
\frac{1}{M_2^4} \right] \; ,
\end{equation}
\begin{equation}
\label{Jtb}
\left({\cal J}^{({\rm t})}_{M_1,II}\right)^2 =
k_{\scriptscriptstyle F} a \ \sum_{M_2 \geq \hat M_2} \left({\cal I}^{({\rm t})}_{{\bf M},II}\right)^2 =
2 \pi \ k_{\scriptscriptstyle F} a M_1^2 \left[\sum_{M_2 \geq \hat M_2}
\frac{(1-r)^2 - \delta^4/4}
{M_2^5((1-r)^2+\delta^2)^{5/2}} \right] \ .
\end{equation}
\end{mathletters}
Since $\hat M_2 \gg 1$ we can convert the previous sums into integrals and obtain
\begin{mathletters} \label{allJtlos}
\begin{equation} \label{Jtloa}
\left({\cal J}^{({\rm t})}_{M_1,I}\right)^2 \simeq \frac{4 \sqrt{2}}{3 (\pi M_1)^2} \
{\cal N} \ (1-r)^{1/2} \ .
\end{equation}
\begin{equation}
\label{Jtlob}
\left({\cal J}^{({\rm t})}_{M_1,II}\right)^2 \simeq \frac{4}{3 (\pi M_1)^2}
{\cal N} \ \left(1 - \sqrt{2} (1-r)^{1/2} \right) \ .
\end{equation}
\end{mathletters}
In leading order in $1-r$ the persistent current is dominated by type-II trajectories
(independent of the temperature) and given by
\begin{equation} \label{I1TRn5}
\frac{I^{({\rm t})}}{I_0} = \frac{2}{\pi\sqrt{3}} \ {\sf g_s} \
\sqrt{\cal N} \left[\sum_{M_1} \frac{1}{M_1^2}
\ \sin^2{\left(2 \pi M_1\frac{\Phi}{\Phi_0}\right)} \ R_T^2(M_1 L)
\right]^{1/2} \ ,
\end{equation}
consistent with the result of Ref.~\cite{RiGe}. For the next order term
the contribution from type-I trajectories is cancelled by that of type-II
resulting in the relatively flat character of the curves for $I^{({\rm t})}$
in Fig.~\ref{fig:chi_ty}.
For the current of an ensemble of thin rings, the calculations are similar
to those of Eqs.~(\ref{allJtlos}), and in leading order in
$1-r$ we obtain:
\begin{equation}
\frac{\overline{I^{(2)}}}{I_0} =
{\sf g_s} \ \sum_{M_1} \left\{
\overline{{\cal J}^{({2})}_{M_1,I}}
+ \overline{{\cal J}^{({2})}_{M_1,II}} \right\}
\sin\left(4\pi M_1\frac{\Phi}{\Phi_0} \right)
R^2_T(M_1 L) \ ,
\end{equation}
\begin{mathletters} \label{allI3Rs}
\begin{equation} \label{I3Ra}
\overline{{\cal J}^{({2})}_{M_1,I}} =
\sum_{M_2 \geq \hat M_2} \overline{{\cal I}^{({2})}_{{\bf M},I}} =
\frac{4\sqrt{2}}{3\pi^2}
\ \sqrt{1-r} \frac{1}{M_1}
\ ,
\end{equation}
\begin{equation} \label{I3Rb}
\overline{{\cal J}^{({2})}_{M_1,II}} =
\sum_{M_2 \geq \hat M_2} \overline{{\cal I}^{({2})}_{{\bf M},II}} =
\frac{2}{\pi^2} \
\left( 1 - \frac{2\sqrt{2}}{3} \sqrt{1-r} \right)
\frac{1}{M_1}
\; .
\end{equation}
\end{mathletters}
\noindent Type-II trajectories once again dominate the average magnetic
response of thin rings and the amplitude for the first harmonic is
$\overline{I_{1}^{(2)}}/I_0 \simeq (2{\sf g_s}/\pi^2) \sin{(4 \pi \Phi/\Phi_0)}
R_T^2(L)$,
independently of the number of transverse channels ${\cal N}$. The average
persistent current shows the halfing of the flux period with respect to
$I^{(1)}$
characteristic for ensemble results (as found in the disordered case and
consistently with the results for averages in the following sections).
\subsubsection*{Comparison with Experiment}
Persistent currents have been measured by Mailly, Chapelier and Benoit
\cite{BenMailly} in a thin
semiconductor ring (with effective outer and inner radii $a=1.43 \mu m$ and
$b=1.27 \mu m$) in the ballistic and phase-coherent regime ($l=11 \mu m$
and
$L_{\Phi}=25 \mu m$). The Fermi velocity is $v_{\scriptscriptstyle F}=2.6\times 10^7 cm/s$ and
therefore the number of occupied channels is ${\cal N} \simeq 4$.
The quoted temperature of $T=15 mK$ makes the temperature factor irrelevant
for the first harmonic ($L_c \simeq 30a$, $R_T(L) \simeq 1$).
The magnetic response exhibits an
$hc/e$ flux periodicity and changes from diamagnetic to paramagnetic
by changing the microscopic configuration, consistently with
Eqs.~(\ref{I1})-(\ref{allI1Rs}). Unfortunately, the sensitivity is not high
enough in order to test the signal averaging with these microscopic changes.
The typical persistent current was found to be $4 nA$, while Eq.~(\ref{I1TRn5})
and Ref.~\cite{RiGe} would yield $7 nA$. The difference between the theoretical
and measured values is not significant given the experimental uncertainties as
discussed in Refs.~\cite{BenMailly} and \cite{von_thesis}.
Moreover, as we stressed above,
a very large $k_{\scriptscriptstyle F} a$ interval is needed for the average of $(I^{(1)})^2$
in order to recover $I^{({\rm t})}$; otherwise we expect large
statistical fluctuations. As in the case of the susceptibility of squares
that we analyze in the next section, residual disorder (reducing the magnetic
response without altering the physical picture) and interactions may be
necessary in order to attempt a detailed comparison with the experiment.
Clearly, new experiments on individual rings of
various thickness and on ensembles of ballistic rings
would be helpful in order to test the ideas of the present section.
\newpage
\section {Simple Regular Geometries: the Square}
\label{sec:square}
The circular and annular billiards studied in section~\ref{sec:integrable}
have the remarkable property that, due to their rotational symmetry,
they remain integrable under the application of a magnetic field.
However, for a generic integrable system (a {\em regular} geometry)
any perturbation breaks the integrability of the dynamics. Moreover,
the periodic orbits which are playing the central role in the semiclassical
trace formulas are most strongly affected by the perturbation. Indeed,
the Poincar\'e-Birkhoff theorem \cite{arnold:book} states that as soon as the
magnetic field is turned on, all resonant tori (i.e.~all
families of periodic orbits) are instantaneously broken, leaving only two
isolated periodic orbits (one stable and one unstable). It is therefore no longer
possible to use the Berry-Tabor semiclassical trace formula to
calculate the oscillating part of the density of states for finite field
since it is based on a sum over resonant tori, which do not exist any further.
One has therefore to devise a semiclassical technique allowing to
calculate ${d^{\rm osc}}(E)$ for nearly, but not completely, integrable systems.
To achieve this, it is necessary to go back to the basic equations from
which the standard semiclassical trace formulae (Gutzwiller \cite{gut71},
Balian-Bloch \cite{bal69}, Berry-Tabor \cite{ber77}) are derived.
The density of states $d(E)$, Eq.~(\ref{DOS}), is related to the trace of the
energy dependent Green function $G({\bf q},{\bf q'};E)$ by
\begin{equation} \label{eq:traceG}
d(E) = - \frac{{\sf g_s}}{\pi} \ {\rm Im} \ {\cal G}(E) \; ,
\qquad \qquad
{\cal G}(E) = \int {\rm d} {\bf q} \,
G ({\bf q},{\bf q};E) \; ,
\end{equation}
where again the factor ${\sf g_s}=2$ comes from the spin degeneracy.
$G({\bf q},{\bf q'};E)$ has a singularity (logarithmic in two dimensions)
when ${\bf r} \rightarrow {\bf r}'$ which just gives the smooth [Weyl] part $\bar d(E)$
of the density of states in a leading order semiclassical expansion. However,
in order to consider only the oscillating part of $d(E)$
one can use the semiclassical approximation of the Green function
\cite{gutz_book}
\begin{equation} \label{eq:green}
G_E^{\text sc}({\bf q},{\bf q'};E) =
\frac{1}{i \hbar} \frac{1}{\sqrt{ 2 i \pi \hbar}}
\sum_t D_t \exp{\left[\frac{i}{\hbar} S_t - i
\eta_t\frac{\pi}{2}\right]}
\end{equation}
where the sum runs over all classical trajectories $t$ joining
${\bf q}$ and ${\bf q'}$ {\em at energy $E$}.
$S_t$ is the action along the trajectory $t$,
$D_t$ a determinant involving second derivatives of the action
(the general expression of which is given in appendix~\ref{app:D_M})
and $\eta_t$ is the Maslov index of the trajectory, i.e. the number of
focal points encountered when traveling from ${\bf q}$ to ${\bf q}'$.
As in section~\ref{sec:integrable}, we shall also take into account
in $\eta_t$ the phase $\pi$ acquired at
each reflection at the wall of a billiard with Dirichlet boundary conditions.
By taking the trace (\ref{eq:traceG}) the sum in Eq.~(\ref{eq:green})
becomes a sum over all orbits closed in configuration (i.e.~${\bf q}$) space,
to which we will refer in the following as {\em recurrent} orbits.
The standard route to obtain ${d^{\rm osc}}$ is to
evaluate this integral by stationary-phase approximation.
This selects the trajectories which are not only closed in configuration
space $({\bf r}'\!=\!{\bf r})$, but also closed in phase space
($\bf p'\! =\! \bf p$), i.e. {\em periodic} orbits.
When these latter are [well] isolated the Gutzwiller Trace Formula \cite{gut71}
is obtained. For integrable systems, all recurrent orbits are in fact periodic
since the action variables are constants of motion.
Periodic orbits appear in continuous
families associated with resonant tori. All orbits of a family have the same
action and period, and one can calculate the density of states
using the Berry-Tabor Formula as described in
the previous section. For systems such as the square billiard, the
physical effect which generates the susceptibility comes along with
the breaking of the
rational tori, so that just ignoring this, i.e.~using the Berry-Tabor Formula,
is certainly inadequate. On the other hand, for $H \rightarrow 0$ the
remaining orbits are not sufficiently well isolated to apply the Gutzwiller
Trace Formula. Therefore, as stated before, we need a uniform treatment of the perturbing
field, in which not only the orbits being closed in phase space are taken
in account, but also the orbits closed in configuration space which can be
traced back to a periodic orbit when $H \rightarrow 0$.
In this section we show how this can be performed in the particular case of
a square billiard. Because of the simplicity of its geometry,
the integrals involved in the trace Eq.~(\ref{eq:traceG})
can be performed exactly for weak magnetic fields.
Moreover, the square geometry deserves special interest since it was the first
microstructure experimentally realized to measure the magnetic response
in the ballistic regime.
We present here a semiclassical approach addressing the
physical explanation of the experimental findings of Ref.~\cite{levy93},
which have pointed the way
for the ongoing research. In order to obtain semiclassical expressions
for the susceptibility of individual and ensembles of squares
we will proceed as outlined in section~\ref{sec:TherFor}:
We will calculate the density
of states and use the decomposition of the susceptibility according to
Eq.~(\ref{eq:fd}) into contributions corresponding to $\Delta F^{(1)}$
and $\Delta F^{(2)}$. In section~\ref{sec:general} we present the
theory for a generic integrable system perturbed by a magnetic field,
generalizing the results of this section.
\subsection{Oscillating density of states for weak field}
\label{seq:square1}
To start with, we consider a square billiard (of side $a$) in
the absence of a field. Each family of periodic orbits can be labeled by
the topology ${\bf M} = (M_x, M_y)$ where $M_x$ and $M_y$ are the
number of bounces occurring on the bottom and left side
of the billiard (see Fig.~\ref{fig:fam11}).
The length of the periodic orbits for all members of a family is
\begin{equation} \label{square:length}
L_{\bf M} = 2 a \sqrt{M_x^2 + M_y^2} \ .
\end{equation}
\noindent The unperturbed action along
the trajectory is, as for any billiard system,
$S^0_{\bf M}/\hbar = k L_{\bf M} $ where $k$ is the wavenumber.
The Maslov indices are $\eta_{\bf M} = 4 (M_x + M_y)$,
and we will omit them from now on since they only yield a dephasing of
a multiple of $2\pi$. Finally the unperturbed determinant reduces to
\begin{equation} \label{eq:square:DM}
D_{\bf M} = \frac{m}{\sqrt{\hbar k L_{\bf M}}} \; .
\end{equation}
One way to obtain this result is to use the method of images
(see Fig.~\ref{fig:image})
and express the exact Green function $G({\bf q},{\bf q}';E)$ in terms of the
free Green function $G^0({\bf q},{\bf q}';E)$ as \cite{bal69,gut71}
\begin{equation} \label{image}
G({\bf q},{\bf q}';E) = G^0({\bf q},{\bf q}';E)
+ \sum_{{\bf q}_i} \epsilon_i G^0({\bf q}_i,{\bf q}') \ ,
\end{equation}
where the ${\bf q}_i$ represent all the mirror images of ${\bf q}$ by any combination
of symmetry across a side of the square, and $\epsilon_i = +1$ or $-1$
depending on whether one needs an even or odd number of symmetries to map
${\bf q}$ on ${\bf q}_i$. $G^0({\bf q},{\bf q}';E)$ gives the above mentioned
logarithmic singularity of $G$ when ${\bf q}' \rightarrow {\bf q}$, but the
long range asymptotic behavior of the two-dimensional free Green function
\begin{equation}
G^0({\bf q}_i,{\bf q}') \simeq
\frac{1}{i \hbar} \frac{m}{\sqrt{ 2 i \pi \hbar}}
\frac{ \exp(ik|{\bf q}'-{\bf q}_i|)}{ \sqrt{\hbar k |{\bf q}'-{\bf q}_i|}}
\end{equation}
can be used for all other terms (images).
For sufficiently weak magnetic fields, one may follow the same
approach as in the previous section, keeping in Eq.~(\ref{eq:green}) the
zero'th order approximation for the prefactor $D_{\bf M}$,
and using the first-order correction $\delta S$ to the action which, as
expressed by Eq.~(\ref{dS}), is proportional to the area enclosed
by the unperturbed trajectory. Here however, as is the generic case (and
contrary to circular or annular geometries) the area enclosed by an orbit
varies within a family.
Let us consider the contribution to the density of states
of the family of recurrent trajectories which for $H\! \rightarrow \!0$ tends
to the family of shortest periodic orbits with non-zero enclosed area,
that plays a crucial role in determining the magnetic response,
as already recognized in Ref.~\cite{levy93}.
For $H\!=\!0$, this family consists in the set of orbits which, say, start
with an angle of 45 degrees with respect to the boundary on
the bottom side of the billiard at a distance $x_0$ ($0 \leq x_0 \leq a$)
from its left corner, bounce once on each side of the square before returning
to their initial position (family ${\bf M} = (1,1)$, see Fig.~\ref{fig:fam11}(a)).
It is convenient to use as configuration space coordinates $x_0$ which labels the
trajectory, the distance $s$ along the trajectory, and the index
$\epsilon = \pm 1$ which specifies the direction in which the trajectory is
traversed. In this way, each point ${\bf q}$ is counted four times
corresponding to the four sheets of the invariant torus.
The enclosed area ${\cal A}_{\epsilon}(x_0,s)$ obviously does not depend on $s$ and is
given by
\begin{equation} \label{area}
{\cal A}_{\epsilon}(x_0) = \epsilon \ 2 \ x_0 \ (a-x_0) \; .
\end{equation}
Periodic orbits are those paths for which the action is extremal
(${\bf \nabla} S = {\bf p'} - {\bf p} = 0$). Therefore Eqs.~(\ref{dS})
and (\ref{area}) illustrate the contents of the Poincar\'e-Birkhoff
theorem, that for any non-zero field only the two trajectories
corresponding to $x_0 = a/2$ remain periodic (one stable, one unstable
according to the two possible directions of traversal).
The contribution of the family (1,1) to ${d^{\rm osc}}(E)$ is $d_{11}(E) = - ({\sf g_s}/\pi)
\ {\rm Im} \ {\cal G}_{11}(E)$. Inserting Eqs.~(\ref{area}) and
(\ref{dS}) into the integral of Eq.~(\ref{eq:traceG}) we have
\begin{equation} \label{dg11}
{\cal G}_{11}(H) =
\frac{1}{i \hbar} \frac{1}{\sqrt{ 2 i \pi \hbar}}
\int_0^{L_{11}} d s \left(\frac{ d y}{d s} \right)
\int_0^{a} d x_0 \sum_{\epsilon=\pm 1} D_{11}
\exp{\left[i k L_{11} + i \frac{2e\epsilon}{\hbar c}
H x_0(a-x_0) \right]} \; .
\end{equation}
The contribution to the density of states of the family (1,1) factorizes
into an unperturbed (Berry-Tabor-like) term and a field dependent
factor
\begin{equation} \label{dosc11}
d_{11}(E,H) = d^0_{11}(E) \ {\cal C}(H)
\end{equation}
with
\begin{equation} \label{dosc11nf}
d^0_{11} \equiv d_{11}(H\!=\!0) =
\frac{4 {\sf g_s}}{\pi} \ \frac{ m a^2}{\hbar^2 (2\pi k L_{11})^{1/2}}
\ \sin{\left(kL_{11}\!+\!\frac{\pi}{4}\right)} \ ,
\end{equation}
and
\begin{equation}
\label{Csimple}
{\cal C}(H) = \frac{1}{a}
\int_0^{a} {\rm d} x_0 \cos \left( \frac{2e}{\hbar c} H x_0 (a-x_0)
\right)
=
\frac{1}{\sqrt{2 \varphi}}
\left[ \cos(\pi \varphi) {\text C}(\sqrt{\pi \varphi}) +
\sin(\pi \varphi) {\text S}(\sqrt{\pi \varphi}) \right]
\; .
\end{equation}
$\text C$ and $\text S$ respectively denote the cosine and sine
Fresnel integrals \cite{Gradshtein}, and
\begin{equation} \label{eq:varphi}
\varphi = \frac{Ha^2}{\Phi_0}
\end{equation}
is the total flux through the square measured in units of the flux quantum
($\Phi_0 = hc/e$).
For the circular and annular geometries, the field dependence
of the density of states, and therefore the susceptibility, was related to
the dephasing between time reversal families of orbits. Here,
Eq.~(\ref{Csimple}) expresses that the dependence of ${d^{\rm osc}}$ on the
field is also
determined by the field induced decoherence of different orbits
{\em within} a given family.
As soon as $\varphi$ reaches a value close to one, the Fresnel
integrals can be replaced by their asymptotic value $1/2$,
which amounts to evaluate ${\cal C}(\varphi)$ by stationary phase, i.e.
\begin{equation} \label{eq:statphas}
{\cal C}^{\rm S} (\varphi) = \frac{\cos(\pi \varphi \! - \! \pi/4)}{
\sqrt{4\varphi}} \; .
\end{equation}
This means that for $\varphi>1$ the dominant contribution
to ${\cal C}(\varphi)$ comes from the neighborhood
of the two surviving periodic orbits ($x_0 = a/2, \epsilon = \pm 1$),
and the oscillations of ${\cal C}(\varphi)$ are related to the successive
dephasing and rephasing of these orbits. In fact, one would have obtained just
$d_{11}^{\rm S} = d_{11}^0 {\cal C}^{\rm S} (\varphi)$ by evaluating the
contribution to the density of states of the two surviving periodic orbits
using the Gutzwiller trace formula
with a first-order classical perturbative evaluation of
the actions and stability matrices.
${\cal C}^{\rm S} (\varphi)$ however diverges when $H \! \rightarrow \! 0$,
while the full expression Eq.~(\ref{Csimple}) simply gives ${\cal C}(0)=1$.
To compute the contribution $d_{\bf M}$ of longer trajectories, it
is worthwhile to write $(M_x, M_y)$ as $(r u_x, r u_y)$, where $u_x$ and $u_y$
are coprime integers labeling the primitive orbits and $r$ is the number
of repetitions. As illustrated in Fig.~\ref{fig:fam11}(b), for any orbit
of the family the square can be decomposed into $u_x \times u_y$ cells, such
that the algebraic area enclosed by the trajectory inside two adjacent
cells exactly compensate. Therefore, keeping $x_0$ as a label of the
orbit (with $x_0 \in [0, a/u_x]$ to avoid double counting),
the total area enclosed by the trajectory $(ru_x,ru_y)$ is
\begin{equation} \label{eq:Anxny}
{\cal A}_{\bf M} = \left\{ \begin{array}{cl} \displaystyle
0 & \qquad \qquad \mbox{$u_x$ or $u_y$ even}\\
\displaystyle r \frac{{\cal A}_{\epsilon}(u_x x_0)}{u_x u_y}
& \qquad \qquad \mbox{$u_x$ and $u_y$ odd}\\
\end{array} \right. \; ,
\end{equation}
where ${\cal A}_{\epsilon}(x_0)$ is given by Eq.~(\ref{area}). From the above
equation, and proceeding in the same way as for the orbit $(1,1)$
Eq.~(\ref{dosc11}) can be generalized to
\begin{equation} \label{square:GM}
d_{\bf M}(E,H) =
\left\{ \begin{array}{ll} \displaystyle
d^0_{\bf M}(E) \ & \qquad \qquad \mbox{$u_x$ or $u_y$ even}\\
\displaystyle
d^0_{\bf M}(E) \ {\cal C}\left(\frac{r\varphi}{u_xu_y}\right)
& \qquad \qquad \mbox{$u_x$ and $u_y$ odd}\\
\end{array} \right. \; ,
\end{equation}
where ${\cal C}(\varphi)$ is given by Eq.~(\ref{Csimple}) and
$d^0_{\bf M} \equiv d_{\bf M}(H\!=\!0) $ is the zero-field contribution of the
family ${\bf M}$
\begin{equation} \label{square:G0M}
d^0_{\bf M} =
\frac{4 {\sf g_s}}{\pi} \ \frac{ m a^2}{\hbar^2 (2\pi k L_{\bf M})^{1/2}}
\ \sin{\left(kL_{\bf M}\!+\!\frac{\pi}{4}\right)} \ .
\end{equation}
\subsection{The susceptibility: individual samples vs. ensemble averages}
\label{sec:chisemicl}
For clarity of the presentation we will calculate in a first stage the
susceptibility contribution of the family (1,1) of the shortest flux enclosing
orbits only. This corresponds to the temperature regime of the experiment
Ref.~\cite{levy93} where the characteristic
length $L_c$ given by Eq.~(\ref{R_factor2}) is of the order of
$L_{11}$, the length
of the shortest orbits, and contributions of all longer orbits are eliminated
due to temperature damping. In the next subsection we will state the results
valid at arbitrary temperature by taking into account the contribution of
longer orbits.
>From the expressions (\ref{dosc11}) and (\ref{dosc11nf}) of the
contributions of the family $(1,1)$ to ${d^{\rm osc}}(E,H)$ one obtains
the corresponding contribution to $\Delta F^{(1)}$
(Eqs.~(\ref{eq:smooth_osco}) and (\ref{DF1})) as
\begin{equation}
\label{square:F11}
\Delta F^{(1)}_{11}(H) = \frac{{\sf g_s}\hbar^2}{m}
\left( \frac{2^3 a}{\pi^3 L^5_{11}} \right)^{1/2} (k_{\scriptscriptstyle F} a)^{3/2} \
\sin{\left(k_{\scriptscriptstyle F} L_{11}\!+\!\frac{\pi}{4}\right)} \
{\cal C}(H) R_T(L_{11}) \; .
\end{equation}
$R_T(L_{11})$ is the temperature dependent
reduction factor Eq.~(\ref{R_factor2}), valid for billiard systems.
The field-dependent factor ${\cal C}(\varphi)$ is given by
Eq.~(\ref{Csimple}).
Taking the derivatives with respect to the
magnetic field, we have [for $L_c \simeq L_{11}$]
\begin{equation} \label{square:chi1}
\frac{\chi^{(1)}}{\chi_{\scriptscriptstyle L}} = -\frac{3}{(\sqrt{2}\pi)^{5/2}} \
(k_{\scriptscriptstyle F} a)^{3/2} \ \sin{\left(k_{\scriptscriptstyle F} L_{11}+
\frac{\pi}{4}\right)}\frac{{\rm d}^2 \ {\cal C}}{{\rm d} \varphi^2}
\ R_T(L_{11}) \ .
\end{equation}
The susceptibility of a given square oscillates as a function
of the Fermi energy and can be paramagnetic or
diamagnetic (see Fig.~\ref{fig:chi1}(a)).
Since we are considering only one kind of trajectory the typical
susceptibility $\chi^{({\rm t})}$ (with the definition (\ref{chicir1t}))
is simply proportional to the prefactor of $\chi^{(1)}$. Therefore, it
is of the order of $(k_F a)^{3/2}$, which is much larger than
the Landau susceptibility $\chi_{\scriptscriptstyle L}$.
As shown in Fig.~\ref{fig:chi1}b (solid line)
$\chi^{(1)}$ exhibits also (by means of
$\partial^2 {\cal C}/\partial \varphi^2$) oscillations as a function of the
flux at a given number of electrons in the square.
The divergent susceptibility
obtained from ${\cal C}^{\rm S}$ (dashed line) provides a good
description of $\chi^{(1)}$ for $\varphi \stackrel{>}{\sim} 1$.
For a measurement made on an ensemble of squares of different sizes $a$,
$\chi^{(1)}$ vanishes under averaging if the dispersion of
$k_{\scriptscriptstyle F} L_{11}$
across the ensemble is larger than $2\pi$. In that case
the average susceptibility is given by the contribution to $\Delta F^{(2)}$
arising from the $(1,1)$ family (Eq.~(\ref{DF2})).
Proceeding in a similar way as for the first--order
term, the contribution of the family $(1,1)$ to the integrated density
${N^{\rm osc}}$ is given by Eq.~(\ref{eq:smooth_oscn}) as
\begin{equation} \label{square:N11}
N_{11}(\bar \mu,H) = - {\sf g_s}
\left( \frac{2^3 a^3}{\pi^3 L^3_{11}} \right)^{1/2}
(k_{\scriptscriptstyle F} a)^{1/2} \
\cos{\left(k_{\scriptscriptstyle F} L_{11}\!+\!\frac{\pi}{4}\right)} \
{\cal C}(H) R_T(L_{11}) \; .
\end{equation}
To calculate $\chi^{(2)}$ we have to consider
$\Delta F^{(2)} = ({N^{\rm osc}})^2/2\bar{D}$ (with
$\bar D = ({\sf g_s} m a^2)/(2\pi\hbar^2)$), and in particular the term
\begin{equation}
\frac{(N_{11}(\bar \mu,H))^2}{2\bar D} =
\frac{{\sf g_s}\hbar^2}{(\sqrt{2})^3 \pi^2 m a^2} \ k_{\scriptscriptstyle F} a
\ \cos^{2}{\left(k_{\scriptscriptstyle F} L_{11}+\frac{\pi}{4}\right)}
\ {\cal C}^2 (\varphi) \ R^2_T(L_{11}) \ .
\end{equation}
This contribution is of lower order in $k_{\scriptscriptstyle F} a$ than that of
$\Delta F_{11}^{(1)}$,
but its sign does not change as a function of the phase $k_{\scriptscriptstyle F} L_{11}$.
Therefore the squared cosine survives the ensemble average\footnote{
Beside the orbits (1,1) the orbits (1,0) and (0,1) which are even shorter
contribute to $\Delta F^{(1)}$ in the limit $L_c \sim L_{11}$. Since
they do not enclose any flux the second derivative of $\Delta F_{10}^{(1)}$
with respect to $H$, i.e. $\chi_{10}^{(1)}$ can be neglected for small
fields. However, they enter into $\chi^{(2)}$ by means of the cross
products
$(N_{10}+N_{01})N_{11}$ in $({N^{\rm osc}})^2$. Nevertheless, they play no role for
the averaged $\overline{\chi^{(2)}}$ because $N_{10}$ and $N_{11}$
do not oscillate with the same frequency and therefore their product
averages out.}
and we obtain, performing the derivatives with respect to $\varphi$
(still in the regime $L_c \simeq L_{11}$),
\begin{equation} \label{square:chi}
\frac{\overline{\chi^{(2)}}}{\chi_{\scriptscriptstyle L}} =
- \frac{3}{(\sqrt{2}\pi)^3} \ k_{\scriptscriptstyle F} a
\ \frac{{\rm d}^2 {\cal C}^2}{{\rm d} \varphi^2} \ R_T^2(L_{11})
\; .\end{equation}
The total averaged susceptibility is therefore
\[ \overline{\chi} = - \chi_{\scriptscriptstyle L}
+ \overline{\chi^{(2)}} \; ,\]
since, as seen in section~\ref{sec:LanGen}, one has also to include
the diamagnetic (bulk) ``Landau term'' $-\chi_{\scriptscriptstyle L}$
arising
from $\hbar$ corrections to $F^0$. In the regime $L \simeq L_c$ we are
considering here, $\chi_{\scriptscriptstyle L}$ is negligible
with respect to $\overline{\chi^{(2)}}$ as $\hbar \rightarrow 0$,
and one can use $\overline{\chi} \simeq \overline{\chi^{(2)}}$.
Note however that when $L_c \ll L$, Eqs.~(\ref{square:chi1}) and
(\ref{square:chi}) remain valid but $\chi^{(1)}$ as well
as $\chi^{(2)}$ is exponentially suppressed. In this ``trivial'' regime
$\chi$ (and thus $\overline{\chi}$) reduces to the Landau susceptibility,
and becomes independent of the underlying classical dynamics. The linear
dependence of the average susceptibility on $k_{\scriptscriptstyle F}$ is shown in Fig.~\ref{fig:chi2}a.
Since ${\cal C}$ has its absolute maximum at $\varphi\!=\!0$, the average
zero-field susceptibility is paramagnetic and attains a maximum value
of \cite{URJ95,vO95}
\begin{equation} \label{square:chizf}
\overline{\chi^{(2)}}({H\!=\!0}) =
\frac{4\sqrt{2}}{5\pi} \ k_{\scriptscriptstyle F} a \ \chi_{\scriptscriptstyle L} \ R^2_T(L_{11}) \; .
\end{equation}
For small fields the average susceptibility (thin solid line,
Fig.~\ref{fig:chi2}b)
has an overall decay as $1/\varphi$ and oscillates in sign on the scale
of one flux quantum through the sample. As in the disordered case \cite{BM}
the period of the field oscillations of the average is half of that of the
individual systems (see Fig.~\ref{fig:chi1}(b)).
In our case the difference can be traced to the
$ {\cal C}^2$ dependence that appears in Eq.~(\ref{square:chi}) in contrast to
the simple ${\cal C}$ dependence of Eq.~(\ref{square:chi1}).
For an ensemble with a wide distribution of lengths
(as in Ref.~\cite{levy93}) an average $\langle \cdots \rangle$
on a classical scale (i.e.~$\Delta a /a \not \ll 1$) rather
than on a quantum scale ($\Delta (k_{\scriptscriptstyle F} a) \simeq 2\pi$) needs
to be performed, and the dependence of ${\cal C}$
on $a$ (through $\varphi$) has to be considered. Since the scale
of variation of ${\cal C}$ with $a$ is much slower than that of
$\sin^2{(k_{\scriptscriptstyle F}L_{11})}$
we can effectively separate the two averages and obtain the total
mean by averaging the local mean:
\begin{equation} \label{clasav}
\langle \chi \rangle =
\int d a \ \overline{\chi} \ P(a) \; ,
\end{equation}
\noindent where the quantum average $\overline{\chi}$ is given by Eq.~(\ref{square:chi}) and
$P(a)$ is the probability distribution of sizes $a$. Taking for $P(a)$ a Gaussian
distribution with a $30\%$ dispersion we obtain the thick solid line of Fig.~\ref{fig:chi2}b.
The low-field oscillations with respect to
$\varphi$ are suppressed under the second average, while the zero-field
behavior remains unchanged.
The expected value for the susceptibility measured in an ensemble of $n$ squares is
$n\langle \chi \rangle \propto n k_{\scriptscriptstyle F} a$, with a {\em large} statistical dispersion of
$\sqrt{n} \chi^{\rm (t)} \propto \sqrt{n} (k_{\scriptscriptstyle F} a)^{3/2}$. However, for experiments
like the one of Ref.~\cite{levy93} where $n \simeq 10^5 \gg k_{\scriptscriptstyle F} a \simeq 10^2$, it
is not possible to obtain a diamagnetic response by a statistical fluctuation.
\subsection{Contribution of longer orbits}
\label{sec:longorbit}
In the zero temperature limit\footnote{It should be kept in mind however
that the expansion in Eq.~(\ref{eq:fd}) is a priori not valid when
$T\rightarrow 0$.} or more generally if one is interested in results
valid at any temperature, it is
necessary to take also into account the contribution of longer trajectories.
This can be done following exactly the same lines as for the contribution
of the family (1,1). From Eqs.~(\ref{square:GM}) and (\ref{square:G0M})
one obtains the contribution of the family
${\bf M} = (M_x,M_y) = (r u_x, r u_y)$, (where $ u_x$ and $u_y$ are coprime)
to $\Delta F^{(1)}$
\begin{equation}
\label{square:FM}
\Delta F^{(1)}_{{\bf M}}(H) = \frac{{\sf g_s}\hbar^2}{m}
\left( \frac{2^3 a}{\pi^3 L^5_{{\bf M}}} \right)^{1/2} (k_{\scriptscriptstyle F} a)^{3/2} \
\sin{\left(k_{\scriptscriptstyle F} L_{{\bf M}}\!+\!\frac{\pi}{4}\right)} \
{\cal C}_{\bf M}(\varphi) R_T(L_{{\bf M}}) \; ,
\end{equation}
where
\begin{equation} \label{square:CbM}
{\cal C}_{\bf M}(\varphi) =
\left\{ \begin{array}{ll} \displaystyle
1 \ & \qquad \qquad \mbox{$u_x$ or $u_y$ even}\\
\displaystyle
{\cal C}\left(\frac{r\varphi}{u_xu_y}\right)
& \qquad \qquad \mbox{$u_x$ and $u_y$ odd}\\
\end{array} \right. \; .
\end{equation}
$L_{\bf M}$ and the function ${\cal C} (\varphi)$ are given respectively by
Eqs.~(\ref{square:length}) and (\ref{Csimple}). In order to get
$\chi^{(1)}$ we have to take the second derivative of ${\cal C}_{\bf M}$
with respect to the magnetic field. This yields zero if either $u_x$
or $u_y$ is even and a factor $r^2/(u_x u_y)^2$, if both are odd.
We therefore obtain
\begin{equation} \label{square:chi1_tot}
\frac{\chi^{(1)}}{\chi_{\scriptscriptstyle L}} = -\frac{3}{\pi^{5/2}} \ (k_{\scriptscriptstyle F} a)^{3/2}
\sum_r \! \! \sum_{\begin{array}{c} \scriptstyle u_x, \; u_y \\
\scriptstyle {\rm odd} \end{array}}
\frac{1}{r^{1/2}(u_x^2+u_y^2)^{5/4} (u_x u_y)^2}
\sin{\left(k_{\scriptscriptstyle F} L_{{\bf M}}+\frac{\pi}{4}\right)}
\ {\cal C}'' \! \left(\frac{r \varphi}{u_x u_y} \right)
\ R_T(L_{{\bf M}}) \ ,
\end{equation}
valid at any temperature.
The low temperature result for $\chi^{(2)}$ follows in essentially the same
way, but taking the average is made rather intricate in the case of a square
(as compared for instance to a rectangle) because of the degeneracies in
the lengths of the particular orbits of this system.
Indeed, there are infinitely many integers
which can be decomposed in at least two different ways into sums of two squares.
For instance, $ 11^2 + 7^2 = 13^2 + 1^2 = 170$. As a consequence,
$L_{11,7} = L_{13,1}$, and
$ \overline{ N_{11,7} N_{13,1} } \neq 0$. An explicit formula
for $\overline{\chi^{(2)}}$ therefore
requires to handle correctly all the non--diagonal terms containing orbits of degenerated
lengths which do not average to zero.
This leads to a number theoretical problem (i.e.\ characterizing all numbers
which decomposition as the sum of two squares is not unique), with which we
do not deal and which moreover will be seen
to be of no practical relevance. Therefore, instead of considering
a square, we will give the expression for $\overline{\chi^{(2)}}$ for
a rectangle of area $a^2$ and of horizontal and vertical lengths
$a\cdot e$ and $a\cdot e^{-1}$. In that case,
all the formulae given in section~\ref{seq:square1} remain valid.
As the only difference one has now
\[ L_{\bf M} = 2 a \sqrt{(M_x/e)^2 + (M_y e)^2} \]
instead of Eq.~(\ref{square:length}), which does not give rise to length degeneracies
if, as we will suppose, $e^4$ is irrational.
Noting that the prefactor of
$ N_{\bf M}^2$ depends as $L_{\bf M}^{-3}$ on the length of the orbit
(instead of $L_{\bf M}^{-5/2}$ for $\Delta F^{(1)}_{\bf M}$),
one obtains for the canonical correction to the susceptibility
\begin{equation} \label{square:chi_tot}
\frac{\overline{\chi^{(2)}}}{\chi_{\scriptscriptstyle L}} =
- \frac{3}{\pi^3} \ k_{\scriptscriptstyle F} a
\sum_r \! \! \sum_{\begin{array}{c} \scriptstyle u_x, u_y \\
\scriptstyle {\rm odd} \end{array}}
\frac{1}{r\left((u_x/e)^2+(u_y e)^2\right)^{3/2} (u_x u_y)^2}
\ \ ({\cal C}^2)'' \! \left(\frac{r\varphi}{u_x u_y}\right)
\ R_T^2(L_{{\bf M}}) \; .
\end{equation}
The equations (\ref{square:chi1_tot}) and (\ref{square:chi_tot}) show that
even at zero temperature the strong
flux cancelation typical for the square (or rectangular) geometry
generates a very small prefactor
$1 / (r^{1/2} (u_x^2+u_y^2)^{5/4} (u_x u_y)^2)$ for $\chi^{(1)}$ (square
geometry) and $ 1 / (r \left((u_x/e)^2+(u_y e)^2\right)^{3/2} (u_x u_y)^2 )$
for $\chi^{(2)}$ (rectangular geometry). For the second shortest contributing
primitive orbit, ${\bf M} = (1,3)$, this yields for instance for
$\chi^{(1)}$ a damping of $1/(9 \times 10^{5/4}) \simeq 0.0062$.
For $\overline{\chi^{(2)}}$ the multiplicative factor is even smaller.
In practice only the repetitions $(r,r)$ of the family (1,1)
will contribute significantly to the susceptibility, and
one can use Eqs.~(\ref{square:chi1_tot}) and (\ref{square:chi_tot})
keeping only the term $u_x = u_y = 1$ of the second summation.
As a consequence, all the complications due to the
degeneracies in the length of the orbits for the square are
of no practical importance (Eq.~(\ref{square:chi_tot}) restricted
to $u_x = u_y = 1$ can be used for the square with $e=1$),
showing why their detailed treatment was not necessary.
As illustrated in Fig.~\ref{fig:repetition} for $\overline{\chi^{(2)}}$,
the repetitions of the orbit (1,1) are yielding a diverging susceptibility
at zero field when the temperature goes to zero, but barely affect the
result even as $T \rightarrow 0$ for finite $H$, where the contributions of the
repetitions do no longer add coherently.
\subsection{Numerical calculations}
\label{sec:numeric}
As a check of our semiclassical results we calculated quantum mechanically
the orbital susceptibility of spinless particles in
a square potential well $[-a/2,a/2]$ in an homogeneous magnetic
field. Within the symmetric gauge ${\bf A} = H (-y/2, x/2, 0)$ the
corresponding
Hamiltonian in scaled units $\tilde x=x/a$ and
$\tilde E=(m a^2/\hbar^2) E$ reads
\begin{equation}
\tilde {\cal H} = -\frac{1}{2}
\left(\frac{\partial^2}{\partial \tilde x^2}
+ \frac{\partial^2}{\partial \tilde y^2}\right)
- i\pi \, \varphi
\left(\tilde y \frac{\partial}{\partial \tilde x}
- \tilde x \frac{\partial}{\partial \tilde y}\right)
+ \frac{\pi^2}{2} \varphi^2 ({\tilde x}^2 + {\tilde y}^2) \ ,
\label{eq:hamilton}
\end{equation}
with the normalized flux $\varphi$ defined as in Eq.~(\ref{eq:varphi}).
Taking into account the invariance of the Hamiltonian (\ref{eq:hamilton})
with respect to rotations by $\pi, \pi/2$ we use linear
combinations of plane-waves which
are eigenfunctions of the parity operators
${\bf P}_\pi$, ${\bf P}_{\pi/2}$, respectively. Omitting the tilde in
order to simplify the notation, they read
\begin{eqnarray}
& & \sqrt{2} [S_n(x) C_m(y) \pm i C_m(x) S_n(y)]
\hspace{5mm} ; \hspace{5mm} (P_\pi = -1) \; , \\
& & \begin{array}{l}
\sqrt{2} [C_n(x) C_m(y) \pm C_m(x) C_n(y)] \\
\sqrt{2}i\, [S_n(x) S_m(y) \pm S_m(x) S_n(y) ]
\end{array}
\hspace{5mm} ; \hspace{5mm} (P_\pi = +1)
\end{eqnarray}
with $S_n(u) = \sin(n \pi u )$, $n$ even, and $C_m(u) = \cos(m \pi u )$,
$m$ odd, obeying Dirichlet boundary conditions. In this representation
the resulting matrix equation
is real symmetric and decomposes into four blocks representing the
different symmetry classes. By diagonalization we calculated the first
3000 eigenenergies taking into account up to 2500 basis functions
for each symmetry class. A typical energy level diagram
of the symmetry class $(P_\pi,P_{\pi/2})=(1,1)$ as a function of
the magnetic field is shown in Fig.~\ref{f1}(a). In between the two
separable limiting cases $\varphi = 0$ and $\varphi \longrightarrow \infty$
the spectrum exhibits a complex structure typical for a non--integrable
system which classical dynamics is at least partly chaotic.
We calculate numerically the {\em grand-canonical} susceptibility
(Eq.~(\ref{eq:susgc}), Fig.~\ref{f1}b) from
\begin{equation}
\chi^{\rm \scriptscriptstyle GC}(\mu) = -\frac{{\sf g_s}}{a^2}\frac{\partial^2}{\partial H^2}\,
\sum_{i=1}^\infty \, \frac{\epsilon_i }{1+\exp[\beta(\epsilon_i-\mu)]}
\end{equation}
where ${\sf g_s}$ accounts for the spin degeneracy and $\epsilon_i$ denotes the
single particle energies.
However, in order to address the experiment of Ref.~\cite{levy93} and
to compare with the semiclassical approach of the preceeding subsection
we have to work in the canonical ensemble.
At $T=0$ the free energy $F$ reduces to the total energy and the
canonical susceptibility (Eq.~(\ref{eq:sus})) is
given as the sum
\begin{equation}
\chi(\!T=\!0) = -\frac{{\sf g_s}}{a^2} \sum_{i=1}^N \,
\frac{\partial^2\, \epsilon_i }{\partial H^2}
\end{equation}
over the curvatures of the $N$ single particle energies
$\epsilon_i$. The susceptibility is therefore dominated by large
paramagnetic singularities each time the highest occupied state undergoes a
level crossing with a state of a different symmetry class or a narrow avoided
crossing with a state of the same symmetry.
This makes $T=0$ susceptibility spectra of quasi--integrable billiards (with
nearly exact level crossings) or systems with spectra composed of energy levels
from different symmetry classes (as it is the case for the square)
looking much more erratic than those of chaotic systems with stronger level
repulsion \cite{NakTho88}.
The $T=0$ peaks are compensated
once the next higher state at a (quasi) crossing is considered, and therefore
disappear at finite temperature when the occupation of nearly degenerated
states becomes almost the same. Thus finite temperature regularizes the
singular behavior of $\chi$ at $T=0$ and of course describes
the physical situation. We obtain the canonical susceptibility at finite $T$
from
\begin{equation}
\chi = \frac{{\sf g_s}}{a^2 \beta}\,\frac{\partial^2}{\partial H^2} \ln Z_N(\beta) \ .
\label{chiqm}
\end{equation}
The canonical partition function $Z_N(\beta)$ is given by
\begin{equation}
Z_N(\beta) = \sum_{\{\alpha\}} \; \exp[-\beta E_\alpha(N)]
\label{Zcanonical}
\end{equation}
with
\begin{equation}
E_\alpha(N) = \sum_{i=1}^\infty \, \epsilon_i \; n_i^\alpha \hspace{5mm} ,
\hspace{5mm} N = \sum_{i=1}^\infty \; n_i^\alpha \; .
\label{Ealpha}
\end{equation}
The $n_i^\alpha \in \{0,1\}$ describe the occupation
of the single particle energy-levels. A direct numerical computation of
the canonical partition function becomes extremely
time consuming at finite temperature. We approximate the sum in
Eq.~(\ref{Zcanonical}) which runs over all (infinitely many)
occupation distributions $\{\alpha\}$ for $N$
electrons by a finite sum $Z_N(M;\beta)$ over all possibilities to distribute
$N$ particles over the first $M$ levels with $M \ge N$ sufficiently large.
Following Brack et al.~\cite{bra91} we calculate $Z_N(M;\beta)$ recursively
using
\begin{equation}
Z_N(M;\beta) = Z_N(M-1;\beta) +
Z_{N-1}(M-1;\beta) \exp(-\beta \epsilon_M)
\label{recrelation}
\end{equation}
with initial conditions
\begin{equation}
Z_0(M;\beta) \equiv 1 \hspace{3mm} , \hspace{3mm} Z_N(N-1;\beta)
\equiv 0
\end{equation}
and increase $M$ until convergence of $Z_N(M,\beta)$, i.e.
the difference between $Z_N(M;\beta)$ and $Z_N(M-1;\beta)$ is negligible.
This recursive algorithm reduces the number of algebraic operations to
calculate $Z_N$ drastically and is fast and accurate even if $k_{\scriptscriptstyle B} T$ is of
the order of 10 or 20 times the mean level spacing, i.e., in a regime
where a direct calculation of $Z_N$ is not feasible.
\subsection{Comparison between numerical and semiclassical results}
\label{sec:num}
Our numerical results for the susceptibility of individual and ensembles
of squares are displayed as the dashed lines in Figs.~\ref{fig:chi1} and
\ref{fig:chi2} and are in excellent agreement with the semiclassical
predictions of Sec.~\ref{sec:chisemicl}. Fig.~\ref{fig:chi1}a shows
the numerical result for the canonical susceptibility and the
semiclassical leading order contribution $\chi^{(1)}_{11}$ at zero field
as a function of $k_F a$ ($\sqrt{4\pi N/{\sf g_s}}$ in terms of the number of
electrons). The temperature $k_{\scriptscriptstyle B} T$ is equal to five times the mean level
spacing $\Delta$ of the single particle spectrum. The quantum result oscillates
with a period $\pi/\sqrt{2}$ as semiclassically expected (Eq.~(\ref{square:chi1}))
indicating the dominant effect of the fundamental orbits of length $L_{11} =
2\sqrt{2}a$. The semiclassical amplitudes (solid line)
are slightly smaller than the numerics because only the shortest orbits
are included.
Fig.~\ref{fig:chi1}b shows the flux dependence of $\chi$ for a fixed number of electrons
$N \approx 1100 {\sf g_s}$.
The semiclassical prediction (Eq.~(\ref{square:chi1}), solid curve)
is again in considerable agreement with the quantum result while the
analytical result (Eq.~(\ref{eq:statphas}), dashed line)
from stationary phase integration yields an (unphysical)
divergence for $\varphi \rightarrow 0$ as discussed in Sec.~\ref{sec:chisemicl}.
For the numerical calculations we can perform the ensemble average directly and
we obtain the averages on the quantum scale (thin dashed line, Fig.~\ref{fig:chi2}b)
or classical scale (thick dashed line) by taking a Gaussian distribution
of sizes with respectively a small or large $\Delta a/a$ dispersion.
Fig.~\ref{fig:chi2}(a) depicts the $k_F a$ dependence of
$\overline{ \chi }$ assuming a Gaussian distribution of lengths $a$ with a
standard deviation $\Delta a/a \approx 0.1$ for each of the
three temperatures $k_{\scriptscriptstyle B} T/\Delta = 2,3,5$. The dashed curves are the ensemble
averages of the quantum mechanically calculated {\em entire} canonical
susceptibility $\overline{ \chi}$. The dotted lines are the
{\em exact} (numerical) results for the averaged term $\overline{
\chi^{(2)}_{\rm qm}} = (\overline{ {N^{\rm osc}}_{\rm qm})^2}/2\Delta $.
They are nearly indistinguishable (on the scale of the figure) from
the {\em semiclassical} approximation of Eq.~(\ref{square:chi}) (solid line).
Although a small flux $\varphi \approx 0.15$
has been chosen (here the contribution from the next longer orbits
$(2,2)$ nearly vanishes) the precision of the semiclassical approximation
based on the fundamental orbits (1,1) is striking. The difference between
the results for $\overline{ \chi }$ and
$\overline{ \chi^{(2)} }$ gives an estimate
for the precision of the thermodynamic expansion Eq.~(\ref{eq:fd}).
The convoluted semiclassical result has been shifted
additionally by $-\chi_{\scriptscriptstyle L}$ to account for the diamagnetic Landau
contribution and is again in close
agreement with the numerical result of the averaged susceptibility
$\bar{ \chi }$.
\subsection {Comparison with the experiment}
\label{sec:experiment}
In a recent experiment, L\'evy {\em et al.} \cite{levy93} measured
the magnetic response of an {\em ensemble} of $10^5$ microscopic
billiards of square geometry lithographically defined on a high mobility
GaAs heterojunction. The size of the squares is on average $a=4.5 \mu m$,
but has a large variation (estimated between 10 and 30\%) between the center
and the border of the array. Each square can be considered as phase-coherent
and ballistic since the phase-coherence length and elastic mean free path are
estimated, respectively, to be between 15 and 40 $\mu m$ and between
5 and 10 $\mu m$.
Therefore, it is worthwhile to compare the observed magnetic response
with the prediction of our clean model of non-interacting electrons,
to see whether this simple picture contains the main physical input
to understand the experimental observations, although one should control in addition
that the residual impurities do not alter fundamentally the magnetic response
of the system. This is the subject of a forthcoming article \cite{rod2000}. Ongoing
calculations including (weak) disorder indeed indicate that the underlying
physical picture remains correct.
At a qualitative level, a large paramagnetic peak at zero field
has been observed in Ref.~\cite{levy93}, two orders of magnitude larger
than the Landau susceptibility, decreasing on a scale of approximately
one flux quantum through each square. Since there is a large dispersion
of sizes we do not observe the field oscillations of the quantum average
(\ref{square:chi}), but the comparison has to be established with the
classical average results Eq.~(\ref{clasav}). The corresponding
results from our semiclassical calculations
(Eq.~(\ref{square:chi},\ref{clasav}))
and the full quantum calculations are shown in Fig.~\ref{fig:chi2}b) as
the thick full, respectively dashed, lines (denoted by $\langle \chi\rangle$ in the
figure). The offset in the semiclassical curve with respect to the
quantum mechanical curve is due to the Landau
susceptibility $\chi_{\scriptscriptstyle L}$ and additional effects from bouncing--ball orbits
(see section \ref{sec:highB} A) not included in the semiclassical trace.
Our theoretical results for the flux dependence of the average $\langle \chi \rangle$
with respect to a wide distribution in the size of the squares
agree on the whole with the experiment. However, the
diamagnetic response for $\langle \chi\rangle$
that we obtain for $\varphi \approx 0.5$ is not
observed experimentally, indicating that there may be a more important
size-dispersion than estimated. As will be discussed in more detail in
section~\ref{sec:general}, a very large distribution of lengths
enhances the effect of the breaking of time reversal invariance due
to the magnetic field, yielding a vanishing average response at
{\em finite field} and a paramagnetic susceptibility at {\em zero
field} decaying on a field scale $\Phi_0$
by the dephasing of the contribution of time reversal symmetric orbits
to the density of states.
More quantitatively, the experiment of Ref.~\cite{levy93}
yielded a paramagnetic susceptibility at $H\!=\!0$ with a
value of approximately 100 (with an uncertainty of a factor of 4)
in units of $\chi_{\scriptscriptstyle L}$. The two electron densities
considered in the experiment are $10^{11}$ and
$3 \! \times \! 10^{11} {\rm cm}^{-2}$ corresponding
to approximately $10^4$ occupied levels per square. Therefore our
semiclassical approximation is well justified. For a temperature of $40mK$
the factor $4\sqrt{2}/(5\pi) k_{\scriptscriptstyle F} a R_T^2(L_{11})$ from
Eq.~(\ref{square:chizf}) gives zero field susceptibility values of 60
and 170, respectively, in reasonable agreement with the measurements.
In order to attempt a more
detailed comparison with the measurements we need to incorporate
the suppression of the clean susceptibility by the residual
disorder, which depends on the strength and correlation length
of the impurity potential \cite{rod2000}.
The field scale for the decrease of
$\langle \chi(\varphi) \rangle$ is of the order of one flux quantum
through each square, in agreement with our theoretical findings.
The temperature dependence experimentally observed seems however less
drastic than the theoretical prediction.
\newpage
\section{Generic integrable and chaotic systems}
\label{sec:general}
In sections~\ref{sec:integrable} and \ref{sec:square} we have studied
in detail specific geometries of conceptual as well as experimental relevance.
In particular, we have demonstrated the degree
of accuracy of our semiclassical approach by a careful comparison with
exact quantum results. The aim of the present section is to take
a broader point of view and to give more general
semiclassical implications concerning the magnetic properties of ballistic
quantum dots. We shall first consider the weak--field behavior of
generic integrable systems, generalizing the results of the previous
section. We focus on weak fields because only this regime
is affected by the integrability of the dynamics at zero
field. The case of systems which remain integrable at
arbitrary field strength was discussed in section~\ref{sec:integrable}.
In the second stage we shall turn to chaotic systems
(at weak as well as finite fields) and finally finish the section by
discussing the similarity and differences of the magnetic response
for the various cases of classical stability.
\subsection{Generic integrable systems}
\label{sec:gen_int}
We consider the generic magnetic response
of two--dimensional integrable systems perturbed by a weak magnetic field
breaking the integrability.
The Eqs.~(\ref{eq:smooth_osc:all}) and (\ref{allDF}), which relate
the thermodynamic functions $\Delta F^{(1)}$ and $\Delta F^{(2)}$
to the oscillating part ${d^{\rm osc}}(E)$ of the density of states, are general
relations which apply in particular here. The main difficulty
is therefore to obtain semiclassical uniform approximations
for ${d^{\rm osc}}(E)$ interpolating between
the zero field regime, for which the Berry-Tabor Formula \cite{ber76,ber77}
(suitable for integrable systems) applies, and higher fields
(still classically perturbative however),
for which the periodic orbits which have survived under the perturbation
are sufficiently well isolated in order to use the Gutzwiller trace formula
\cite{gut71}.
This problem of computing for a generic system the oscillating part of the
density of states in the nearly but not exactly integrable regime
has been addressed by Ozorio de Almeida \cite{ozor86,ozor:book}.
We are going to
follow this approach for the case of a perturbation by a magnetic field.
However, for the sake of completeness and in order to define their regime
of validity, we will give a brief derivation of the basic results needed.
This is the subject of section~\ref{sec:6A1}.
In section~\ref{sec:6A2} we then deduce the grand-canonical
and canonical contributions to the susceptibility.
\subsubsection{Perturbation theory for magnetic fields}
\label{sec:6A1}
Let $\hat{\cal H} (\hat {\bf p},\hat{\bf q})$ be a quantum Hamiltonian which
classical analogue can be expressed as
\begin{equation} \label{perturbed_H}
{\cal H}({\bf p},{\bf q}) =
{\cal H}^0\left({\bf p} \! - \! \frac{e}{c}{\bf A},{\bf q}\right) \; .
\end{equation}
${\cal H}^0({\bf p},{\bf q})$ is the Hamiltonian describing the motion in
the absence of a magnetic field and ${\bf A}$ is the vector potential
generating
a uniform magnetic field $H$. ${\cal H}^0$ is supposed to be integrable
which permits to define action-angle coordinates $({\bf I},{\bf \varphi})$,
$\varphi_1, \varphi_2 \in [0,2\pi]$ such that at zero field
the Hamiltonian ${\cal H}^0(I_1,I_2)$ depends only on the actions.
To compute ${d^{\rm osc}}(E)$ we start from the same basic equations as for the
square geometry. In the weak--field regime which we are considering, the only
recurrent trajectories of the sum Eq.~(\ref{eq:green}) which
contribute noticeably to the trace Eq.~(\ref{eq:traceG}) are those which
merge into periodic orbits of the unperturbed Hamiltonian as
$H \rightarrow 0$.
Considering only these contributions, which we can label by the topology
${\bf M}$ of the unperturbed periodic orbits, and dropping the Weyl
part of the trace ${\cal G}(E)$ of the Green function we can write
\begin{equation} \label{contribution}
{\cal G}(E) \simeq \sum_{{\bf \scriptscriptstyle M}} {\cal G}_{{\bf \scriptscriptstyle M}} \; ,
\qquad
{\cal G}_{{\bf \scriptscriptstyle M}}(E) = \frac{1}{i \hbar} \frac{1}{\sqrt{ 2 i \pi \hbar}}
\int {\rm d} q_1 {\rm d} q_2 \, D_{{\bf \scriptscriptstyle M}}
\exp{\left[\frac{i}{\hbar} S_{{\bf \scriptscriptstyle M}} -
i\eta_{{\bf \scriptscriptstyle M}}\frac{\pi}{2} \right]} \ .
\end{equation}
Let us now focus on the contribution ${\cal G}_{{\bf \scriptscriptstyle M}}$ of the family of
closed orbits $\bf M$. For sufficiently low fields we will employ (as
in sections~\ref{sec:integrable} and \ref{sec:square})
that the change in the semiclassical
Green function by changing $H$ is essentially given by the
modification of the phase, $S_M/\hbar$ being large in the semiclassical
limit. The variation in the determinant $D_{\bf \scriptscriptstyle M}$ can usually be neglected.
Therefore, in the evaluation of the integral in Eq.~(\ref{contribution})
one should keep the (unperturbed) zero'th order approximation for $D_{\bf \scriptscriptstyle M}$
and evaluate the action up to the first order correction.
For the action this yields
\begin{equation}
S_{\bf \scriptscriptstyle M}({\bf q},{\bf q}) = S^0_{\bf \scriptscriptstyle M} + \delta S_{\bf \scriptscriptstyle M}({\bf q},{\bf q})
\end{equation}
with
\begin{equation} \label{S0}
S^0_{{\bf \scriptscriptstyle M}} = \oint_{\rm orbit} {\bf p} \cdot d {\bf q}
= \oint_{\rm orbit} {\bf I} \cdot d {\bf \varphi}
= 2\pi I_{{\bf \scriptscriptstyle M}} \cdot {\bf M} \; ,
\end{equation}
noting $\bf I_{\bf \scriptscriptstyle M}$ the action coordinates of the periodic
orbit family ${\bf M}$ at $H=0$.
The contribution $\delta S_{\bf \scriptscriptstyle M}$ is expressed in terms of the
area enclosed by the
{\em unperturbed} orbit by means of Eq.~(\ref{dS}).
$S^0_{\bf \scriptscriptstyle M}$ is constant for all members of the family, but
$\delta S$ generically depends on the trajectory on which
the point ${\bf q}$ lies. However,
the area enclosed by the orbit and thus $\delta S_{\bf \scriptscriptstyle M}$ does not change when
varying $\bf q$ along the orbits. It is therefore convenient to use a
coordinate system such that one coordinate is constant along
the unperturbed trajectory.
Writing ${\bf M} = (r u_1,r u_2)$ where $u_1$ and $u_2$ are
coprime integers, this is provided explicitly by the standard canonical
transformation $({\bf I},{\bf \varphi}) \rightarrow ({\bf J},{\bf \theta})$
generated by $F_2 ({\bf J},{\bf \varphi}) =
(u_2 \varphi_1 -u_1 \varphi_2)J_1 + \varphi_2 J_2$~:
\begin{equation} \label{canonical_transform}
\begin{array}{ll}
\theta_1 = u_2 \varphi_1 - u_1 \varphi_2 \
\ \ & J_1 = I_1 / u_2 \\
\theta_2 = \varphi_2 & J_2 = I_2 + (u_1/u_2) I_1
\end{array} \; ,
\end{equation}
for which $\theta_1$ is constant along a trajectory {\em on the torus
$\bf I_{{\bf \scriptscriptstyle M}}$}. Then $\theta_1$ specifies
the trajectory and $\theta_2$ the position on the trajectory.
For a square geometry, $\theta_1$ and $\theta_2$ are up to a dilatation,
respectively, the variables $x_0$ and $s$ introduced in
section~\ref{sec:square}.
$\theta_2$ should be taken in the range $[0,2\pi u_2]$ (rather
than $[0,2\pi]$) to ensure that the transformation
Eq.~(\ref{canonical_transform}) constitutes a one to one correspondence.
After substituting ${\bf q}$ by ${\bf \theta}$ in the
integral of Eq.~(\ref{contribution}), $\delta S$ depends only
on $\theta_1$, but no longer on $\theta_2$. One can moreover
show (see appendix~\ref{app:D_M})
the following relation for the zero field approximation of the
determinant $D_{\bf \scriptscriptstyle M}$:
\begin{equation} \label{eq:DM_relation}
D_{\bf \scriptscriptstyle M} \cdot
\left| \left( \frac{\partial {\bf q}}{\partial \theta} \right) \right|
= \frac{1}{\dot{\theta}_2}
\frac{1}{ \left| 2 \pi r u_2^3 g_E^{''} \right|^{1/2}} \; ,
\end{equation}
where $I_2 = g_E(I_1)$ is the function introduced in
section~\ref{sec:integrable} to describe the energy surface $E$. From
Eq.~(\ref{contribution}) and (\ref{eq:DM_relation}) one gets
\begin{equation} \label{BT_green}
{\cal G}_{{\bf \scriptscriptstyle M}}(E) = \frac{1}{i \hbar} \frac{1}{\sqrt{ 2 i \pi \hbar}}
\frac{1}{\left| 2\pi r u_2^3 g_E^{''} \right|^{1/2}}
\exp{\left[\frac{i}{\hbar} S^0_{{\bf \scriptscriptstyle M}} -
i \eta_{{\bf \scriptscriptstyle M}}\frac{\pi}{2} \right]}
\int_0^{2\pi u_2} \frac{{\rm d} \theta_2 }{\dot{\theta}_2}
\int_0^{2\pi} {\rm d} \theta_1
\exp{\left[\frac{i}{\hbar} \delta S (\theta_1) \right]} \; .
\end{equation}
The integral over $\theta_2$ is the period $\tau_{\bf \scriptscriptstyle M} / r$ of the primitive
periodic orbit. In the absence of a field the integral over $\theta_1$
is simply $2\pi$ which gives
\begin{equation} \label{BT_GM}
{\cal G}^0_{\bf \scriptscriptstyle M}(E) = -
\frac{i \tau_{\bf \scriptscriptstyle M}}{\hbar^{3/2} M_2^{3/2}
\left| g_E^{''} \right|^{1/2}}
\exp i \left[ \frac{S^0_{\bf \scriptscriptstyle M}}{\hbar}
- \eta_{\bf \scriptscriptstyle M} \frac{\pi}{2} - \frac{\pi}{4} \right] \; .
\end{equation}
$d^0_{\bf \scriptscriptstyle M}(E)$, the zero field contribution of the orbits of
topology ${\bf M}$ to the oscillating part of the density of states,
is obtained from Eq.~(\ref{BT_GM}) as
$d^0_{\bf \scriptscriptstyle M}(E) = -({\sf g_s}/\pi) {\rm Im} {\cal G}^0_{\bf \scriptscriptstyle M}(E)$.
Therefore, except for the evaluation of the Maslov indices that we have
disregarded here, one recovers in this way for the integrable limit
the Berry-Tabor formula Eq.~(\ref{BTT}) of a two-dimensional
system (as we have used in section~\ref{sec:integrable}).
Inspection of Eq.~(\ref{BT_green}) for weak magnetic fields shows that,
upon perturbation, ${\cal G}_{\bf \scriptscriptstyle M}$ is just given by the product of
the unperturbed result ${\cal G}^0_{\bf \scriptscriptstyle M}$ and a factor
\begin{equation} \label{reduction1}
\tilde {\cal C}_{\bf \scriptscriptstyle M}(H) = \frac{1}{2\pi} \int_0^{2\pi} {\rm d} \theta_1 \,
\exp \left[ 2 i \pi \frac{H {\cal A}_{\bf \scriptscriptstyle M}(\theta_1) }{ \Phi_0} \right] \;.
\end{equation}
This accounts for the small dephasing between different
closed (in configuration space) orbits of topology $\bf M$
due to the fact that the
resonant torus on which they are living is slightly broken by the
perturbation. (An orbit of topology $\bf M$ closed in configuration
space is then generally not periodic, i.e.\ closed in phase space.)
Supposing the unperturbed motion to be time reversal invariant,
it can be seen
moreover that only the real part of $\tilde {\cal C}_{\bf \scriptscriptstyle M}(H)$ has
to be considered:
The function ${\cal A}_{\bf \scriptscriptstyle M} (\theta_1)$ is defined for the unperturbed
system. Therefore, the time reversed of a trajectory labeled by $\theta_1$
is a periodic orbit of the unperturbed system which encloses an area
$-{\cal A}_{\bf \scriptscriptstyle M}(\theta_1)$. Its contribution cancels the imaginary part of
$\exp [ 2 i \pi H {\cal A}_{\bf \scriptscriptstyle M}(\theta_1) / \Phi_0 ]$, and one can use
\begin{equation} \label{reduction2}
{\cal C}_{\bf \scriptscriptstyle M}(H) = \frac{1}{2\pi} \int_0^{2\pi} {\rm d} \theta_1 \,
\cos \left[ 2 \pi \frac{H {\cal A}_{\bf \scriptscriptstyle M}(\theta_1) }{ \Phi_0} \right] \;
\end{equation}
instead of $\tilde {\cal C}_{\bf \scriptscriptstyle M}(H)$. Since ${\cal C}_{\bf \scriptscriptstyle M}(H)$ is real, one obtains
from Eq.~(\ref{eq:traceG})
\begin{equation} \label{uniform_d}
{d^{\rm osc}}(E) = \sum_{{\bf \scriptscriptstyle M} \neq 0} {\cal C}_{\bf \scriptscriptstyle M}(H) d^0_{\bf \scriptscriptstyle M}(E) \; ,
\end{equation}
where $d^0_{\bf \scriptscriptstyle M}(E)$ is the zero--field contribution given by the
Berry-Tabor expression of Eq.~(\ref{BTT}). At zero field we obviously
have ${\cal C}_{\bf \scriptscriptstyle M}(0) = 1$. At sufficiently large field, the integral
(\ref{reduction1}) can be evaluated using stationary phase
approximation.\footnote{To be precise the ratio $HA/\Phi_0$ rather
than the field must be large. Formally, one has to
consider not the $H\rightarrow \infty$ limit, which is incompatible
with the classical perturbation scheme, but an
$\hbar \mbox{ ({\em i.e.}~$\Phi_0$) } \rightarrow 0$
limiting process, which
does not change the classical mechanics. In practice this means that the
fluxes considered are large on a quantum scale, but still small on the
classical scale. This is achieved at high enough energies.}
${\cal C}_{\bf \scriptscriptstyle M}$ can be expressed as a sum over all extrema of
${\cal A}_{\bf \scriptscriptstyle M}(\theta_1)$
({\em i.e.}~of $\delta S$). These are all the periodic orbits which
survive under the perturbation. It can be seen \cite{gri95} that,
in this approximation, Eq.~(\ref{uniform_d}) yields exactly the
Gutzwiller trace formula for which the actions, periods
and stabilities of the periodic orbits are evaluated using classical
perturbation theory. Eq.~(\ref{uniform_d}) thus provides
an interpolation between the Berry-Tabor and Gutzwiller formulae.
The functions ${\cal A}_{\bf \scriptscriptstyle M} (\theta_1)$, and therefore ${\cal C}_{\bf \scriptscriptstyle M} (H)$, are system
and trajectory dependent. One can, however, gain some further understanding of
the perturbative regime by following again Ozorio de Almeida and
writing ${\cal A}_{\bf \scriptscriptstyle M} (\theta_1)$ in term of its Fourier series
\begin{equation} \label{fourier_s}
{\cal A}_{\bf \scriptscriptstyle M} = \sum_{n=0}^\infty {\cal A}^{(n)}_{\bf \scriptscriptstyle M}
\sin(n \theta_1 - \gamma^{(n)}) \ .
\end{equation}
If ${\cal A}_{\bf \scriptscriptstyle M}$ is a smooth function of $\theta_1$, the coefficients
${\cal A}_{\bf \scriptscriptstyle M}^{(n)}$ are usually rapidly decaying functions of $n$.
For systems where one can neglect all harmonics higher than the first
one, the integral Eq.~(\ref{reduction2}) can be performed, and
it is possible to distinguish two types
of functions ${\cal C}_{\bf \scriptscriptstyle M} (H)$, depending on the symmetry properties of the
unperturbed family of orbits under time reversal.
Indeed, one may encounter
two different situations depending on whether the torus ${\bf I_{\bf \scriptscriptstyle M}}$ is
time reversal invariant (e.g.~square geometry) or
has a distinct partner ${\bf I_{\bf \scriptscriptstyle M}}^*$
in phase space which is its counterpart under time reversal (e.g.~circular
geometry). In the former case, the origin of the angles $\theta_1$ can
be chosen such that ${\cal A}_{\bf \scriptscriptstyle M} (\theta_1)$ is an antisymmetric
function, while
in the latter case it can be in principle any real function of $\theta_1$.%
\footnote{Note in the former case $\tilde {\cal C}_{\bf \scriptscriptstyle M} = {\cal C}_{\bf \scriptscriptstyle M}$, while in the
latter $\tilde {\cal C}_{\bf \scriptscriptstyle M} \neq {\cal C}_{\bf \scriptscriptstyle M}$ but ${\cal G}_{\bf \scriptscriptstyle M} + {\cal G}_{{\bf \scriptscriptstyle M}^*}
= {\cal G}^0_{\bf \scriptscriptstyle M} {\cal C}_{\bf \scriptscriptstyle M}(H) + {\cal G}^0_{{\bf \scriptscriptstyle M}^*} {\cal C}_{{\bf \scriptscriptstyle M}^*}(H)$.}
If $I_{\bf \scriptscriptstyle M}$ is time reversal invariant,
${\cal A}_{\bf \scriptscriptstyle M} (-\theta_1) = - {\cal A}_{\bf \scriptscriptstyle M} (\theta_1)$ implies that
${\cal A}_{\bf \scriptscriptstyle M}^{(0)} = 0$ (as well as all the phases $\gamma^{(n)}$).
In this case
\begin{equation} \label{square_like}
{\cal C}(H) \simeq J_0(2\pi H {\cal A}_{\bf \scriptscriptstyle M}^{(1)}/\Phi_0) \; .
\end{equation}
It is interesting to compare the approximation of ${\cal C}(H)$ given by the
above Bessel function with the exact integral Eq.~(\ref{Csimple}) obtained
in section~\ref{sec:square} for the shortest family (${\bf M} = (1,1)$)
of the square geometry.
Noting that $\theta_1 = \epsilon \pi x_0 / a$ and using Eq.~(\ref{area}),
the Fourier coefficients ${\cal A}_{11}^{(n)}$ of ${\cal A}_{11} (\theta_1)$
are given by
\begin{equation}
{\cal A}_{11}^{(n)} = \left\{ \begin{array}{l l}
\displaystyle \frac{16}{(n\pi)^3} a^2 &
\qquad \mbox{$n$ odd} \; ,\\
0 & \qquad \mbox{$n$ even} \; .
\end{array} \right.
\end{equation}
Keeping only the first harmonic of ${\cal A}_{11} (\theta_1)$ amounts
to approximate
the function ${\cal C}(\varphi)$ of Eq.~(\ref{Csimple}) by
$J_0 (32 \varphi / \pi^2)$ which, as seen in
Fig.~\ref{FresnelvsBessel}, is an excellent approximation.
If the torus $\bf I_{\bf \scriptscriptstyle M}$ is not its own time reversal,
${\cal A}_{\bf \scriptscriptstyle M} (\theta_1)$ is not constrained to be an antisymmetric
function, and in particular ${\cal A}_{\bf \scriptscriptstyle M}^{(0)}$ is usually non zero.
Neglecting, as above,
all harmonics of ${\cal A}_{\bf \scriptscriptstyle M} (\theta_1)$
except the first gives
\begin{equation} \label{circle_like}
{\cal C}_{\bf \scriptscriptstyle M}(H) =
\cos \left( 2\pi \frac{H {\cal A}_{\bf \scriptscriptstyle M}^{(0)}}{\Phi_0} \right)
J_0 \left( 2\pi \frac{H {\cal A}_{\bf \scriptscriptstyle M}^{(1)}}{\Phi_0} \right) \; .
\end{equation}
If moreover ${\cal A}_{\bf \scriptscriptstyle M}^{(1)} \ll {\cal A}_{\bf \scriptscriptstyle M}^{(0)}$, then the field
oscillation frequency is
essentially given by the mean area ${\cal A}_{\bf \scriptscriptstyle M}^{(0)}$ enclosed by the orbits
of the family while the overall decrease is determined by the
first harmonic coefficient ${\cal A}_{\bf \scriptscriptstyle M}^{(1)}$. The circular billiard
can be regarded as a particular case where ${\cal A}_{\bf \scriptscriptstyle M}^{(0)}$ is
non zero while ${\cal A}_{\bf \scriptscriptstyle M}^{(1)}$ as well as all other coefficients vanish.
\subsubsection{Magnetic susceptibility for a generic integrable system}
\label{sec:6A2}
>From the expression (\ref{uniform_d}) of the oscillating part of the
density of states the contributions $\chi^{(1)}$ and $\chi^{(2)}$ to the
susceptibility are obtained by the application of
Eqs.~(\ref{eq:smooth_osc:all})
and (\ref{allDF}), which express $\Delta F^{(1)}$ and $\Delta F^{(2)}$
in terms of ${d^{\rm osc}}(E,H)$. Taking twice the field derivative
according to Eq.~(\ref{eq:sus}) and introducing the dimensionless quantities
\[ {\cal C}''_{\bf \scriptscriptstyle M}(H) \equiv \left( \frac{\Phi_0}{2\pi A} \right)^2
\frac{d^2 {\cal C}_{\bf \scriptscriptstyle M}}{d H^2}
\qquad ; \qquad
({\cal C}^2)''_{\bf \scriptscriptstyle M}(H) \equiv \left( \frac{\Phi_0}{2\pi A} \right)^2
\frac{d^2 {\cal C}^2_{\bf \scriptscriptstyle M}}{d H^2} \; , \]
($A$ is the total area of the system) one obtains for the grand
canonical contribution to the susceptibility
\begin{equation} \label{gen:chi1}
\frac{\chi^{(1)}}{\chi_{\scriptscriptstyle L}} =- 24 \pi m A
\sum_{{\bf M}} \frac{R_T(\tau_{\bf \scriptscriptstyle M}) }{\tau_{\bf \scriptscriptstyle M}^2}
\, \frac{d^0_{\bf \scriptscriptstyle M}(\mu)}{{\sf g_s}}\, {\cal C}''_{\bf \scriptscriptstyle M}(H) \; .
\end{equation}
If one assumes moreover that there are no degeneracies in the length
of the orbits, one has for the averaged canonical correction
\begin{eqnarray} \label{eq:chi2_int}
\frac{\overline{ \chi^{(2)} }}{\chi_{\scriptscriptstyle L}} & = &
- 24 \pi^2 \hbar^2
\, \sum_{{\bf M}} \frac{R^2_T(\tau_{\bf \scriptscriptstyle M}) }{\tau_{\bf \scriptscriptstyle M}^2}
\, \frac{\overline{ (d^0_{\bf \scriptscriptstyle M}(E))^2 }}{{\sf g_s}^2}
\, ({\cal C}^2)''_{\bf \scriptscriptstyle M}(H) \label{gen:chi2} \\
& = & - \frac{12}{\hbar} \sum_{\bf M}
\frac{R^2_T(\tau_{\bf \scriptscriptstyle M}) }{M^3_2 |g''_\mu({\bf I}_{\bf M})|}
\, ({\cal C}^2)''_{\bf \scriptscriptstyle M}(H)
\nonumber \; .
\end{eqnarray}
The field--dependent component of $\overline{\chi^{(2)} }$ for weak fields
is given by
\[
({\cal C}^2)''_{\bf \scriptscriptstyle M}(H\!=\!0) =
- \frac{1}{2\pi A^2} \int_0^{2\pi} d\theta_1 \, A_{\bf \scriptscriptstyle M}^2(\theta_1) \; ,
\]
which is always negative. Therefore, for an ensemble of integrable
structures the magnetic response is always paramagnetic at zero
field. We shall come back to this point in the last part of this
section.
\subsection{Generic chaotic systems}
\label{sec:gen_chaos}
Let us now consider generic chaotic systems, more generally, systems where
all the periodic orbits are sufficiently
isolated that the trace of the semiclassical Green function
Eq.~(\ref{eq:traceG}) can be evaluated within stationary phase
approximation. In this case the Gutzwiller trace formula
provides the appropriate path to calculate the oscillating part
of the density of states (with or without magnetic field).
The Gutzwiller trace formula expresses the oscillating part of
the density of states as a sum over all [here isolated] periodic
orbits $t$ as \cite{gutz_book}
\begin{equation} \label{gutz}
{d^{\rm osc}}(E,H) = \sum_t d_t \qquad ; \qquad
d_t(E,H) = \frac{1}{\pi \hbar}
\frac{\tau_t}{r_t |{\rm det}(M_t-I)|^{1/2}}
\cos (\frac{S_t}{\hbar} - \sigma_t \frac{\pi}{2}) \; .
\end{equation}
$S_t$ is the action along the orbit $t$, $\tau_t$ the period
of the orbit, $M_t$ the stability matrix, $\sigma_t$
its Maslov index, and $r_t$ the number of repetitions of the
full trajectory along the primitive orbit.
All these classical quantities generally depend on
energy and magnetic field. If, as considered above for the integrable
case, one is interested in the magnetic response to weak field,
one can express $d_t(E,H)$ in terms of the characteristics of
the orbits at zero field by taking into
account the field dependence only in the actions. Proceeding in
exactly the same way as in section~\ref{sec:GenInt}, i.e.~grouping
together the contributions of time--reverse symmetrical orbits,
one obtains the same relation as Eq.~(\ref{eq:lowBd}) \cite{aga94,Prado}:
\begin{equation} \label{chaotic_d}
d_t(E,H) = d^0_{t}
\cos \left[ 2 \pi \frac{H {\cal A}^0_{t}}{ \Phi_0} \right] \; .
\end{equation}
$d^0_{t}$ is the zero--field contribution of the orbit, obtained
from Eq.~(\ref{gutz}) at $H=0$, and ${\cal A}^0_{t}$ is the enclosed area
of the {\em unperturbed} orbit. In the case of a generic integrable
system, the zero field regime played a peculiar role: except for
the circular
and annular geometries which remain integrable at all fields, a generic
integrable system looses its integrability under the effect of a perturbing
magneti field. For chaotic geometries on the contrary, the zero field
behavior is not substantially different from that at finite fields
(as far as
the stability of the dynamics is concerned). Since we are discussing the
general semiclassical formalism of chaotic systems without referring
to specific examples we do not need to restrict ourselves to weak
fields. Within this generic framework the chaotic geometries have
the same conceptual simplicity as
the systems which remain integrable at arbitrary field studied in
section~\ref{sec:integrable}. Namely Eq.~(\ref{gutz}) applies
independently of the field, and for derivatives with respect to the
field one can use
\begin{equation} \label{eq:dSdH}
\frac{\partial S_t(H)}{\partial H} = \frac{e}{c}
{\cal A}_t(H) \ ,
\end{equation}
where ${\cal A}_t(H)$ is the area enclosed by the trajectory $t$ at the
considered field. Therefore the computation of the contribution
$\chi^{(1)}$ and $\chi^{(2)}$ to the susceptibility follows essentially along
the same lines as described in
section~\ref{sec:integrable}: $\Delta F^{(1)}$ and $\Delta F^{(2)}$
are given by Eqs.~(\ref{eq:smooth_osc:all}) and
(\ref{allDF}), and to leading order in $\hbar$ the derivatives with
respect to the field should be applied only to the rapidly
varying term. As a consequence, taking twice the derivative
of the contribution of the orbit $t$ to $\Delta F^{(1)}$ merely amounts to
a multiplication by a factor $(e{\cal A}_t)^2/(c\hbar)^2$, yielding
\begin{equation} \label{eq:chi1_c}
\frac{\chi^{(1)}}{\chi_{\scriptscriptstyle L}} = 24\pi m A \sum_t
\frac{R_T(\tau_t)}{\tau_t^2} \left( \frac{{\cal A}_t}{A} \right)^2
\frac{d_t(\mu)}{{\sf g_s}} \; ,
\end{equation}
where $d_t$ is given by Eq.~(\ref{gutz}). Note that
Eq.~(\ref{eq:chi1_c}) applies also to systems which remain
integrable at all fields provided the Berry-Tabor formula
Eq.~(\ref{BTT}) is used instead of the Gutzwiller one. For chaotic as well
as for integrable systems, $\chi^{(1)}$ can be paramagnetic or diamagnetic
with equal probability. The response of an ensemble of structures is given
by $\Delta F^{(2)}$, which can be calculated as a double
sum over all pairs of orbits
\begin{eqnarray}
\label{eq:chichi}
\frac{\chi^{(2)}}{\chi_{\scriptscriptstyle L}}
& = & 24 \sum_{tt'} \frac{R_T(\tau_t) R_T(\tau_{t'})}
{r_tr'_t|{\rm det}(M_t-I)\,{\rm det}(M_{t'}-I)|^{1/2}}
\left[ \left( \frac{{\cal A}_t - {\cal A}_{t'}}{A} \right)^2
\cos\left(\frac{S_t -S_{t'}}{\hbar} -
(\sigma_t-\sigma_{t'})\frac{\pi}{2} \right)
- \right. \label{eq:chi2_c} \nonumber \\
& & - \qquad \qquad \left.
\left( \frac{{\cal A}_t + {\cal A}_{t'}}{A} \right)^2
\cos\left(\frac{S_t + S_{t'}}{\hbar} -
(\sigma_t+\sigma_{t'})\frac{\pi}{2} \right) \right]
\; .
\end{eqnarray}
Here some remarks are in order. Due to the exponential proliferation
of closed orbits in chaotic systems off--diagonal terms should be
considered at low temperatures since near--degeneracies in the actions
of long orbits may appear, so that their contributions do not average out.
However, at sufficiently high temperatures where only short
periodic orbits are relevant, off--diagonal terms (of orbits not related
by time reversal symmetry) are eliminated
upon averaging. At finite field where time--reversal symmetry is broken
(more precisely, when no anti-unitary symmetry is preserved) only the
terms with $t'=t$ survive the averaging process, and
(at the order of $\hbar$ considered)
$\overline{ \chi^{(2)} }$ vanishes since then
${\cal A}_t = {\cal A}_{t'}$. The origin of the weak--field response
for an ensemble is a consequence of time--reversal symmetry since
non--diagonal terms involving an orbit
and its time reversal have an action sufficiently close to
survive the average process but an area of opposite sign.
Indeed, assuming (in the weak--field regime) an ensemble average such
that only diagonal and time reversal related terms are not affected,
Eq.~(\ref{eq:chichi}) reduces to
\begin{equation}
\label{eq:chichi_zero}
\frac{\overline{\chi^{(2)}}_D}{\chi_{\scriptscriptstyle L}}
= 24 \sum_{t} \frac{R^2_T(\tau^0_t)}{r_t^2|{\rm det}(M^0_t-I)|}
\left( \frac{2{\cal A}^0_t}{A} \right)^2
\cos \left(\frac{4\pi A^0_t H}{\Phi_0} \right)
\; . \end{equation}
At zero field the cosine of the surviving terms in
Eq.~(\ref{eq:chichi_zero}) is one and their prefactors positive.
This merely reflects that the dephasing of time reversal
orbits due to the perturbing magnetic field necessarily induces
on average a decrease of the amplitude of ${N^{\rm osc}}$, and therefore
by means of Eq.~(\ref{DF2}) a {\em paramagnetic} susceptibility.
For extremely large distributions in systems size, such as those
discussed in section~\ref{sec:experiment}, even the oscillating patterns
of Eq.~(\ref{eq:chichi_zero}) due to the subsequent rephasing
and dephasing of the time reversal orbits contributions vanish upon
smoothing.
In this case, only the paramagnetic response related to the original
dephasing is observed, and the average susceptibility reaches zero
as soon as $4\pi A^0_t H / \Phi_0$ is of the order of $2\pi$
for all trajectories.
\subsubsection*{Magnetization line-shape for chaotic systems}
The expressions we have obtained up to now in
this subsection do not require the system to be actually chaotic,
but only that periodic orbits are isolated. They should therefore
be valid also for the contribution of isolated
orbits in mixed systems, where the phase space contains both
regular and chaotic regions. This includes for instance
the contributions of elliptic, i.e.\ stable orbits,
provided they are not close to any bifurcation and the surrounding
island of stability is large enough.
For geometries being actually chaotic it is however possible
to proceed further and to derive a general expression for
the line-shape of the field dependent susceptibility,
if the temperature is low enough.
For temperatures such that the cutoff time $\tau_c$ of
the damping factor $R_T(\tau_t)$ is of the order of
the period of the fundamental periodic orbits, the average susceptibility
will be dominated by the shortest orbits, whose characteristics
are largely system dependent. However, for higher $\tau_c$ a large
number of trajectories will contribute to $\overline{ \chi^{(2)} }_D$,
and a statistical treatment of the sum on the r.h.s.\ of
Eq.~(\ref{eq:chichi_zero}) is possible, yielding an {\em universal}
line-shape for the average susceptibility.
For sake of clarity, we discuss
here only the case of billiard like structures, but the following
developments can be generalized in a straightforward way to any
kind of potentials.
Two basic ingredients are required here in addition to
Eq.~(\ref{eq:chichi_zero}) to obtain the magnetization peak
line-shape. The first
one is the semiclassical sum rule derived by Hannay and
Ozorio de Almeida \cite{han84}, which states that in sums like
Eq.~(\ref{eq:chichi_zero}) the two effects of an exponential decrease
in the prefactors on the one hand and the exponential
proliferation of orbits on the other hand cancel each other
yielding
\begin{equation} \label{eq:sumrule}
\overline{ \sum_t \frac{\delta(\tau_t - \tau)}
{|{\rm det}(M_t-I)|} } = \frac{1}{\tau} \; .
\end{equation}
(Note, that in the above sum the contributions of orbits with
number of repetitions $r_t > 1$ are neglected.)
To be valid, this equation requires that the periodic
orbits are uniformly distributed in phase space which will
only be achieved for sufficiently large $\tau$. For
billiards the periods are given, up to a
multiplication by the Fermi speed, by the length of the
orbits and the periods $\tau$ in Eq.~(\ref{eq:sumrule}) can be replaced
by the lengths $L$. We call $L_1^*$ the characteristic length for which
periodic orbits can be taken as uniformly distributed in phase
space. Typically, $L_1^*$ is not much larger than the shortest
period of the system.
The second ingredient is the distribution of area enclosed
by the trajectories. For chaotic systems, this distribution
has a generic form \cite{Chaost,dor91}.
Namely the probability $P_N(\Theta)$ for a trajectory to
enclose an algebraic area $\Theta$ after $N$ bounces on the
boundaries of the billiard is given by
\begin{equation} \label{eq:Adist1}
P_N(\Theta) = \frac{1}{\sqrt{2\pi N \sigma_N}}
\exp\left(-\frac{\Theta^2}{2 N
\sigma_N}\right) \; .
\end{equation}
This result actually follows from a general argument
\cite{dor91}
which in our case can be stated as
follows: With a proper choice of the origin, the area swept by the
ray vector for a given bounce
is characterized by a distribution, with zero mean value
and a width $\sigma_N$ which define the parameter
of the distribution Eq.~(\ref{eq:Adist1}). For a strongly
chaotic system, successive bounces can be taken as independent
events, which by means of the central limit theorem yield the distribution
Eq.~(\ref{eq:Adist1}). Denoting $\bar{ L }$ the average
distance between two successive reflections and
$\sigma_L=\sigma_N/\bar{ L }$, this is equivalent to
\begin{equation} \label{eq:Adist2}
P_L(\Theta) = \frac{1}{\sqrt{2\pi L \sigma_L}}
\exp\left(-\frac{\Theta^2}{2 L \sigma_L}\right) \; .
\end{equation}
Now $P_L(\Theta)$ is the distribution of enclosed areas
for trajectories of length $L$, and the above equation is valid
for $L$ larger than a characteristic value $L^*_2$, which again
is of the order of the shortest closed orbit's length.
For temperature sufficiently low so that $L_c > L^*_1,L^*_2$,
Eqs~(\ref{eq:sumrule}) and (\ref{eq:Adist2}) can be used to
replace the sum over periodic orbits Eq.~(\ref{eq:chichi_zero})
by the integral
\begin{equation}
\frac{ \overline{ \chi^{(2)} }_D }{\chi_{\scriptscriptstyle L}} =
24 \int_0^\infty \frac{dL}{L} \int_{-\infty}^{+\infty} d\Theta
P_L(\Theta) \,
R^2_T(L) \left( \frac{4\Theta^2}{A^2} \right) \!
\cos \left( \frac{4\pi \Theta H}{\Phi_0} \right) \; .
\end{equation}
Performing the Gaussian integral over $\Theta$, and introducing the
dimensionless factor $\xi = 2\pi H \sqrt{\sigma_L L_c}/\Phi_0$, one
obtains the average susceptibility as
\begin{equation} \label{eq:lineshape}
\frac{ \overline{ \chi^{(2)} }_D }{\chi_{\scriptscriptstyle L}} =
96 \left(\frac{\sigma_L L_c}{A^2}\right) \,
{\sf F}(\xi) \,
\end{equation}
where the function ${\sf F}(\xi)$ is defined as
\begin{equation} \label{eq:Fzeta}
{\sf F}(\xi) = \int_0^\infty \left(\frac{x}{\sinh{x}} \right)^2
(1 - 4 \xi^2 x^2) \exp(-2 \xi^2 x) \, dx\;\; ; \hspace{1cm} x = L/L_c \; .
\end{equation}
The quadrature cannot be performed analytically (in a closed expression) for
arbitrary $\xi$\footnote{Using for $R_T(L)$ the asymptotic expression
$R_T(L) = 2 (L/L_c) \exp(-L/L_c)$, valid for $L>L_c=\hbar \beta
v_F/\pi$, yields ${\sf F}(\xi) = (1-5\xi^2)/(1+\xi^2)^4$, but the
contribution of the range $L \leq L_c$ is of the same order.},
but it can easily be calculated numerically. As seen in
Fig.~\ref{fig:Fzeta}, ${\sf F}(\xi)$ has a maximum at $\xi=0$ with a
half-width $\Delta \xi \simeq 0.252$. Expansion of ${\sf F}(\xi)$ for small
$\xi$ yields ${\sf F}(\xi) \approx \pi^2/6 -( 3 \zeta(3) + 2 \pi^4/15) \xi^2$
(where $\zeta(x)$ is the Zeta--function).
Denoting $\Lambda = \sigma_L L_c / A^2$,
the susceptibility at zero field is thus given by
\begin{equation} \label{eq:hight}
\frac{ \overline{ \chi^{(2)} }_D }{\chi_{\scriptscriptstyle L}} (H\!=\!0) =
16 \pi^2 \Lambda \; ,
\end{equation}
and the value half-width $\Delta \Phi$ by
\begin{equation} \label{eq:width}
\frac{\Delta\Phi}{\Phi_0} = \frac{\Delta\xi}{2\pi} \Lambda^{-1/2}
\; .
\end{equation}
The experimental observation of Eq.~(\ref{eq:lineshape}) would
be a very stringent confirmation for the applicability of the whole
semiclassical picture developed here. However, two remarks are in order:
(i)
it is experimentally usually rather difficult to make a clear cut
distinction between the function ${\sf F}(\xi)$ we obtained and,
say, a Lorentzian shape. Therefore,
the temperature dependence (through $L_c$) of both the height
and, more surprisingly, the
width of the magnetization peak should be observable rather than
the precise functional form of Eq.~(\ref{eq:lineshape}).
The physical picture underlying these
results is that at a given temperature, the cutoff length $L_c$
determines the length of the orbits providing the main contribution
to the susceptibility. The smaller the temperature, the larger
$L_c$ and the longer the contributing orbits. The typical areas
enclosed by these orbits thus increase, making them more sensitive to
the magnetic field and yielding a larger susceptibility at zero
field and a smaller width since time reversal invariance is more
rapidly destroyed. The precise temperature dependence of the height
and the width (and their relationship, which might be useful when
$\sigma_L$ is unknown) is given by Eqs.~(\ref{eq:hight}) and
(\ref{eq:width}).
(ii)
It should be
borne in mind that Eq.~(\ref{eq:lineshape}) gives only the contribution
of the diagonal part of $\overline{ \chi^{(2)} }$, but does not
take into account the contribution of pairs of orbits which are not
related by time reversal symmetry. Moreover, the statistical approach
used implies that fairly long orbits are contributing to the susceptibility,
which because of the exponential proliferation of such orbits should
yield an increasing number of quasidegeneracies in their length.
Therefore, to smooth out these non-diagonal term, one should a priori
require that the smoothing is taken on a very large range of $(k_{\scriptscriptstyle F} a)$.
In practice however, and as will be discussed in more detail in
\cite{rod2000}, the smooth disorder characteristic of the GaAs/AlGaAs
heterostructures for which this kind of experiments are
done will actually be responsible for the cancelation of the
non-diagonal terms {\em without affecting} (for small enough disorder)
{\em the contribution we have calculated}.%
\footnote{ Without entering into
any details, the reason for this is the following. For smooth
disorder, one should distinguish between an ``elastic mean free path''
$l$, and a transport mean free path $l_T$ which is much larger than
$l$. For small disorder, $l_T$ can be assumed infinite,
but long orbits will usually be longer than $l$.
As a consequence, the action of each orbit is going to acquire a random
phase from sample to sample, which is decorrelated for different orbits,
but is the same for time reversal symmetric orbits. Thus the diagonal
contribution we have calculated will not be affected, but non-diagonal
terms will be strongly suppressed.}
The effects of non-diagonal terms should therefore be noticeably
less important in actual systems that it might appear in a clean
model.
\subsection{Integrable vs. chaotic geometries}
\label{sec:gen_comp}
The magnetic responses of chaotic and integrable systems have
similarities and differences with respect to their treatment as well as to the resulting
susceptibility. The most remarkable similarity is the paramagnetic character of the
average susceptibility, while the magnitude of this response greatly differs for
both types of geometries. Concerning their treatment the differences
arise form the lack of structural stability of integrable systems
under a perturbing magnetic field. Indeed, for non generic integrable
systems
such as the ring or circular billiards which remain integrable at
all fields, the structure of the obtained equations are, except
for the use of the Berry-Tabor trace formula instead of the Gutzwiller
trace formula, the same as those for the chaotic systems.
For generic integrable systems however, the breaking of invariant
tori requires a more careful treatment yielding slightly less
transparent, though essentially similar expressions.
\subsubsection{Paramagnetic character of the average susceptibility}
Because of this formal similarity, the qualitative behavior of the
magnetic response is also quite the same for generic chaotic and integrable
systems. The susceptibility of a single structure can be paramagnetic or
diamagnetic and changes sign with a periodicity in $k_{\scriptscriptstyle F} a$ of the
order of $2\pi$. On the other hand, the average
susceptibility for an ensemble of microstructure is, as expressed by
Eqs.~(\ref{eq:chi2_int}) and (\ref{eq:chichi_zero}),
paramagnetic at zero field independent of the kind of dynamics
considered. Indeed Eq.~(\ref{DF2}) states that
$\overline{\Delta F^{(2)}}$ is, up to a multiplicative factor,
the variance of the [temperature
smoothed] number of states for a given chemical potential $\mu$.
In integrable and chaotic systems
the basic mechanism involved is that the magnetic field reduces
the degree of symmetry of the system, which as a general result
lowers this variance. Therefore the $\overline{\Delta F^{(2)}}$
necessarily decreases when the magnetic field is applied and the average
susceptibility is paramagnetic at zero field.
There are some differences worth being considered. First, for
chaotic systems the only symmetry existing at zero field
is the time reversal invariance, while for integrable
systems the breaking of time reversal invariance {\em and}
the breaking of invariant tori together reduces
the amplitude of ${N^{\rm osc}}(E)$. For chaotic systems
{\em the paramagnetic character of the ensemble susceptibility arises
as naturally as the negative sign of the magnetoresistance in
coherent microstructures}. The situation is similar to a random matrix point
of view, where the ensembles modeling the fluctuations of time reversal
invariant systems are known to be less rigid (in the sense that
the fluctuation of the number of states in any given stretch of energy
is larger) compared to the case where time reversal invariance is broken.
The transition from one symmetry class to the other can be understood by the
introduction of generalized ensembles whose validity can be justified
semiclassically \cite{boh95}. It is however important to recognize that
even for the chaotic case we do not have the standard GOE-GUE
transition \cite{RevBoh} since (\ref{DF2}) involves the integration over
a large energy interval. We are therefore not in the universal,
but in the ``saturation" regime where $({N^{\rm osc}}(E))^2$ is given by the
shortest periodic orbits.
Secondly, for chaotic systems and for temperatures sufficiently
low that a large number of orbits contribute to the susceptibility,
it is possible --- similar as in the weak localization effect in
electric transport \cite{Chaost} --- to derive a universal shape
of the magnetization peak. This is
not possible for integrable systems, which do not
naturally lend themselves to a statistical treatment.
\subsubsection{Typical magnitude of the magnetic susceptibility}
Even if there are some analogies between the magnetic response of chaotic
and integrable systems (especially when the latter remain integrable at
finite fields), the {\em magnitude} of the susceptibility exhibits
significant differences.
The contribution of an orbit to the Gutzwiller formula
for two--dimensional systems is half an order in $\hbar$ smaller than a
term in the Berry-Tabor formula for the integrable case.
More generally, in the case of $f$ degrees of freedom,
the $\hbar$ dependence of the Berry--Tabor formula is
$\hbar^{-(1+f)/2}$ being the same as in the semiclassical Green function.
The Gutzwiller formula is obtained by performing the trace integral of the
Green function by stationary phase in $f-1$ directions,
each of which yielding a factor $\hbar^{1/2}$. This results in an entire
$\hbar^{-1}$ behavior independent of $f$ for a chaotic system.
Important consequences therefore arise for the case of two--dimensional
billiards of typical size $a$ at temperatures such
that only the first few shortest orbits are significantly
contributing to the
free energy, and gives rise to a different
parametrical $k_{\scriptscriptstyle F} a$ characteristic of integrable and chaotic systems.
The $k_{\scriptscriptstyle F} a$ behavior of the density of states and susceptibility
for individual systems as well as ensemble averages is displayed in
Table \ref{tab:I}. While the magnetic response of chaotic systems
results from {\em isolated} periodic orbits, it is the existence of
{\em families} of flux enclosing orbits in quasi-- or partly integrable
systems which is reflected in a parametrically different dependence
of their magnetization and susceptibility on $k_{\scriptscriptstyle F} a$ (or $\sqrt{N}$ in terms
of the number of electrons). The difference is especially drastic for
ensemble averages where we expect a $k_{\scriptscriptstyle F} a$ independent
response $\bar{ \chi}$
for a chaotic system while the averaged susceptibility for integrable
systems, e.g. the ensemble of square potential wells in the experiment
discussed in section \ref{sec:square}, increases linearly in $k_{\scriptscriptstyle F} a$. Under
the conditions of that measurement \cite{levy93} the enhancement
should be of
the order of 100 compared to an ensemble of chaotic quantum dots.
We therefore suggested \cite{URJ95} to use the different parametrical
behavior of the magnetic response as a tool in order to unambiguously
distinguish (experimentally) chaotic and integrable dynamics in
quantum dots.
We stress that this criterion is not based on the long time behavior of the
chaotic dynamics but on
short time properties, namely the existence of families of orbits
contributing in phase to the trace of the Green function of
integrable systems.
\newpage
\section{Non--perturbative fields:\ bouncing--ball-- and
de Haas--van Alphen--oscillations}
\label{sec:highB}
Up to now we have essentially focused on mesoscopic effects in the weak
magnetic field regime where the classical cyclotron radius
$r_c$ is large compared to the typical size $a$ of the
system, i.e.
\begin{equation}
\frac{r_c}{a} = \frac{c \hbar\, k}{e \, H \, a} \gg 1 \; .
\label{classcond}
\end{equation}
Then, electron trajectories can be considered as straight lines
between bounces and the
dominant effect of the magnetic field enters as a semiclassical phase
in terms of the enclosed flux. Nevertheless,
as shown in Fig.~\ref{f1} in the introduction (for the case of a
square) the low--field oscillations of $\chi$ are accurately described
by {\em classical} perturbation theory in terms of the family (11)
of unperturbed orbits (left inset in Fig.~\ref{f1}(b)). They persist
up to field strengths $\varphi \approx 10$ which is by orders of magnitude larger
than the typical flux scale which describes the breakdown
of first order {\em quantum} perturbation theory, i.e., magnetic fluxes
where the first avoided level crossings appear. Due to condition
(\ref{classcond}) the relevant classical ``small'' parameter is $H/k_{\scriptscriptstyle F}$.
The semiclassical ``weak--field'' regime increases with
increasing Fermi energy.
In this section we will go beyond this (classically) perturbative
regime and discuss microstructures under larger fields, where the magnetic
response reflects the interplay between the scale of the confining
energy and the scale of the magnetic field energy $\hbar\omega_c$ on the
quantum level. Classically, non--perturbative fields affect
the motion not only through a change of the actions (by means of the
enclosed flux), but additionally due to the bending of the trajectories.
A priori, the semiclassical approach we used for weak magnetic fields applies
also to this case without any difference: Oscillating components
of the single--particle density of states can be related to periodic
(or nearly periodic) orbits by taking the trace of the semiclassical Green
function. The magnetic response is then obtained from integration over the energy
and taking the derivatives with respect to the magnetic field.
These operations correspond to the multiplication
by the inverse of the period of the orbit, by the damping factor $R_T$
and by the area enclosed by the orbit. Three field regimes
(weak ($a \ll r_c$), intermediate ($a \simeq r_c$), and high
($a \geq 2 r_c$) fields) can be clearly distinguished as is illustrated in
Fig.~\ref{f1}(b) for the square geometry. The distinction of the
three regimes appears not because they deserve a fundamentally different
semiclassical treatment, but simply because of some salient features
of the classical dynamics associated to each of these regimes.
In the high--field regime, most of the orbits simply follow
a cyclotron motion. In that case, the system behaves essentially as an
infinite system, and one recovers the well known de Haas-van Alphen oscillations
for $\chi^{(1)}$. We shall moreover see below that within our semiclassical
approach, the destruction of some of the cyclotronic orbits due to reflections at
the boundaries can be taken into account, allowing to handle
correctly the crossover regime where $a \geq 2 r_c$ but $r_c$ is not yet
negligible with respect to $a$.
While the high field ($a \gg r_c$) classical dynamics is generally
(quasi) integrable the dynamics in the intermediate field regime is always
mixed (in the sense that chaotic and regular motion coexists in phase space)
except for particular cases of systems with rotational symmetry
which remain integrable independent of the magnetic field.
In contrast to that, systems in the small field regime can exhibit
any degree of chaoticity {\em in the zero field limit}.
Indeed, there is a large variety of geometries for which the
motion of the electrons in the absence of a magnetic field is either
integrable, or completely chaotic. Therefore, increasing the field starting
from an integrable (respectively chaotic) configuration at $H=0$, the
intermediate field regime will be characterized by an increase
(respectively
a decrease) of the degree of chaos of the classical dynamics, which
will noticeably affect the magnetic response of the system. However,
if the zero--field configuration already shows a mixed dynamics (which is
generically the case), the only noticeable difference
between the weak and intermediate field regime will consist in the
complete lost of time reversal symmetry and naturally its consequences
on $\overline{ \chi^{(2)} }$ as discussed in section~\ref{sec:general}.
In addition, for some particular geometries, namely those for which the
boundary contains some pieces of parallel straight lines, the intermediate
field susceptibility will be characterized by the dominating influence
of {\em bouncing--ball orbits}, periodic electron motion due to reflection
between opposite boundaries. Fig.~\ref{f1}(b)
depicts a whole scan of the magnetic susceptibility of
a square from zero flux up to flux $\varphi = 55$ $(3r_c \approx a$).
We can see there, and we will discuss in detail below, that there are
--- besides the small--field oscillations due to orbits (11)
--- two well separated regimes
of susceptibility oscillations: The intermediate field regime
($2r_c > a$) reflects quantized {\em bouncing--ball periodic orbits}
(second inset) and the oscillations in the strong
field regime ($2r_c < a$) which, as mentioned above,
are related to {\em cyclotron orbits} (right inset).
Although the results to be reported are of quite general nature
we will discuss them quantitatively for the case of
square microstructures.
We study individual squares and perform our analysis within the grand
canonical formalism.
\subsection{Intermediate fields: Bouncing-ball magnetism}
\label{sec:interfield}
The full line in Fig.~\ref{fig:chibb}(a) shows the quantum mechanically
calculated (see section~\ref{sec:square}.D)
grand canonical susceptibility for small and
intermediate fluxes at a Fermi energy corresponding to $\sim$2100 enclosed electrons
in a square at a temperature such that $k_BT/\Delta = 8$.
The semiclassical result $\chi^{(1)}_{(11)}$
from the family (11) (Eq.~(\ref{square:chi1})) shown as the dashed--dotted line
(with negative offset) in Fig.~\ref{fig:chibb}(a) exhibits
the onset of deviations from the quantum result
with respect to phase and amplitude starting at $\varphi \approx 8$
($r_c \approx 2a$)
indicating the breakdown of the family (11) of straight line
orbits. With increasing flux we enter into a regime where the
non--integrability of the system manifests itself in
a complex structured energy level diagram (see Fig.~\ref{f1}(a)) on the
quantum level and in a mixed classical phase space \cite{rob86b} of
co--existing regular and chaotic motion. However, besides the variety
of isolated stable and unstable periodic orbits there remains a family
of orbits with specular reflections only on opposite sides of the square.
We will denote these periodic orbits shown in Fig.~\ref{fig:bbschema} which are
known as ``bouncing--ball'' orbits in billiards without magnetic
field by $(M_x,0)$ and $(0,M_y)$ according to the labeling introduced
in section~\ref{sec:square}.A. ($M_x$ and $M_y$ are the number of bounces at the
bottom and left side of the square.) These orbits form families
which can be parameterized, e.g., for the case $(M_x,0)$ in terms of the
point of reflection $x_0$ at the bottom of the square. We thus expect
--- as in the case of the families $(M_x,M_y)$ in section~\ref{sec:square} ---
in the semiclassical limit a parametrical dependence on $k_{\scriptscriptstyle F} a$
of the related susceptibilities which should strongly dominate the contributions
of the co--existing isolated periodic orbits.
We present our semiclassical calculation of the susceptibility contribution
related to bouncing--ball orbits for the primitive periodic
orbits, i.e., $(M_x,0) = (1,0)$ and generalize our results at the end
to the case of arbitrary repetitions. We proceed as in section~\ref{sec:square}
for the derivation of $\chi^{(1)}_{11}$. However, while those
calculations were performed in the limit of a small magnetic field (assuming
$H$--independent classical amplitudes and shapes of the trajectories (11))
we now have to consider explicitly the field dependence of the
classical motion. The contribution to the diagonal part of the Green
function of a recurring path starting at a point
${\bf q}$ on a bouncing--ball orbit reads
\begin{equation}
{\cal G}_{10} ({\bf q, q'=q};E,H) =
\frac{1}{i\hbar \sqrt{2\pi i\hbar}} \, D_{10} \,
\exp\left[i\left(\frac{S_{10}}{\hbar} -
\eta_{10}\frac{\pi}{2}\right) \right]
\ . \label{eq:Gbb}
\end{equation}
Simple geometry yields for its length, enclosed area, and action
\begin{equation}
L_{10}(H) = \frac{2a \zeta}{\sin\zeta} \quad; \qquad
A_{10}(H) = -(2\zeta-\sin 2\zeta) \, r_c^2 \quad; \qquad
\frac{S_{10}}{\hbar}=
k \left(L_{10} + \frac{A_{10}(H)}{r_c(H)}\right) \; ;
\label{eq:LAS10}
\end{equation}
where $\zeta$, the angle between the tangent to a bouncing--ball trajectory
at the point of reflection and the normal to the side, is given by (see
Fig.~\ref{fig:bbschema})
\begin{equation}
\sin\zeta = \frac{a}{2 r_c} \; .
\label{eq:beta}
\end{equation}
The Maslov index $\eta_{10}$ is 4 and will be therefore omitted from now on.
As in section~\ref{sec:square}, we will use as configuration space coordinates
the couple ${\bf q}=(x_0,s)$, where $x_0$ labels the abscissa of the last
intersection of the trajectory with the lower side of the square (see Fig.\
\ref{fig:bbschema}) and
$s$ is the distance along the trajectory. This choice has the advantage that
$D_{10} (x_0,s)$ is constant, and therefore taking the trace of the Green
function merely amounts to a multiplication
by the size of the integration domain. As discussed in more detail
in Appendix~\ref{app:D_M}, the semiclassical amplitude $D_{10}$ is
given by \cite{gutz_book}
\begin{equation}
D_{10}({\bf q, q'=q}) = \frac{1}{|\dot{s}|} \, \left|
\frac{\partial x_0'}{\partial p_{x_0}}
\right|^{-\frac{1}{2}}_{{x_0}' = x_0} \; ,
\label{eq:D10:def}
\end{equation}
where $ (x_0,p_{x_0}) \rightarrow (x_0', p'_{x_0} )$ is the Poincar\'e map
between two successive reflections on the lower side of the billiard.
Noting $u_{x_0} = (p_{x_0} - e A_x/c)/(\hbar k)$ ($u_{x_0}$ is the
projection of the unit vector parallel to the initial velocity
on the $x$ axis) one obtains from simple geometrical considerations
\begin{eqnarray}
p_{x_0}' & = & p_{x_0} \nonumber \\
x_0' & = & x_0 + 2 r_c
\left( \sqrt{1 - ( u_{x_0} \! - \! {a}/{r_c} )^2} -
\sqrt{1 - (u_{x_0})^2} \right) \; .
\label{eq:bb:map}
\end{eqnarray}
For the periodic orbits, $x_0'= x_0$ implies that $u_{x_0} = a/2 r_c =
\sin \zeta$, and therefore
\begin{equation}
D_{10}({\bf q, q'=q})
= \frac{1}{|\dot{s}|} \, \sqrt{\frac{\hbar k \cos\zeta}{2 a}}
\label{eq:D10}
\end{equation}
which reduces to Eq.~(\ref{eq:square:DM}) in the limit $H=0$ ($\zeta=0$).
For the contribution of the whole family (1,0) we must perform the trace
integral Eq.~(\ref{eq:traceG}).
The integral over $s$ gives as usual a multiplication by the period
\[ \tau_{10} = \frac{L_{10}}{\hbar k/m} \]
of the orbit. Moreover, since neither the actions $S_{10}$, nor the
amplitude $D_{10}$ depend on $x_0$, the $x_0$-component of the trace
integral simply yields a length factor
\begin{equation}
l(H) = a\left(1 - \tan\frac{\zeta}{2} \right)
\label{eq:l}
\end{equation}
(see Fig.~\ref{fig:bbschema}) which describes the magnetic field dependent
effective range for the lower reflection points of bouncing--ball
trajectories (1,0).
$l(H)$ vanishes for magnetic fields corresponding to $2 r_c = a$.
We therefore obtain for the
bouncing--ball contribution $d_{10}=-({\sf g_s} / \pi) $Im${\cal G}_{10}$
to the density of states
\begin{equation}
d_{10}(E,H) = - \frac{2{\sf g_s} }{(2\pi \hbar)^{3/2}}
l(H) L_{10} D_{10}
\sin\left(\frac{S_{10}}{\hbar}+ \frac{\pi}{4}\right) \; .
\label{eq:d10}
\end{equation}
In order to compute the contribution $\chi^{(1)}_{10}$ to the
(grand canonical) susceptibility we first have to calculate
$\Delta F_{10}^{(1)}$ by performing the energy integral
Eq.~(\ref{eq:smooth_osco}), and then
to take twice the derivative with respect to the magnetic field. In a leading $\hbar$
calculation, integrals and derivative should again be applied only on
the rapidly oscillating part of $d_{10}$. Noting moreover that
Eq.~(\ref{dS}) is not restricted to perturbation around $H=0$,
i.e.~that at any field
\[ \frac{\partial S_{10}}{\partial H} = \frac{e}{c} A_{10} \; , \]
we therefore obtain in the same way as we did for Eq.~(\ref{eq:chi1_c})
\begin{equation} \label{eq:chi110}
\chi^{(1)}_{10} = \frac{1}{a^2}
\left( \frac{e A_{10}}{c \tau_{10}} \right)^2
d_{10}(\mu,H) R_T (L_{10}) \; .
\label{eq:Chi10}
\end{equation}
Inserting the expressions Eqs.~(\ref{eq:LAS10}), (\ref{eq:l})
and (\ref{eq:D10}) into Eqs.~(\ref{eq:d10}) and (\ref{eq:Chi10}),
we finally have $\chi^{(1)}_{10}$ explicitly in terms of $\zeta$ as
\begin{eqnarray}
\frac{\chi^{(1)}_{10}}{\chi_L} & = &
\frac{3 }{8 \pi^{1/2}} (k_{\scriptscriptstyle F} a)^\frac{3}{2} \,
\frac{\sqrt{\cos \zeta} (\sin \zeta+\cos \zeta -1)}{\zeta}
\frac{ (2 \zeta - \sin (2 \zeta) )^2}{\sin^4\zeta} \times
\label{eq:chibb} \\
& & \qquad \qquad \qquad \times
\sin\left(\frac{S_{10}}{\hbar}+ \frac{\pi}{4}\right)
\, R_T (L_{10}) \; . \nonumber
\end{eqnarray}
The entire bouncing--ball susceptibility $(\chi^{(1)}_{10}+\chi^{(1)}_{01})/
\chi_L = 2 \chi^{(1)}_{10}/\chi_L$ according to Eq.~(\ref{eq:chibb})
is shown in Fig.~\ref{fig:chibb}(a) as the dashed line. At fluxes up to
$\varphi \approx 15$ it just explains the low frequency shift in the
oscillations of the quantum result indicating that the overall small field
susceptibility is well approximated by $\chi_{11} + \chi_{10}+ \chi_{01}$.
For fluxes between $\varphi \approx 15$ ($r_c=1.2 a$) up to $\varphi\approx 37$
(the limit where $r_c = a/2$, i.e., the last bouncing--ball orbits vanish)
the magnetic response is entirely governed by bouncing--ball periodic motion
and the agreement between the semiclassical prediction and the
full quantum result is excellent.
The flux dependence of the actions $S_{10}$ (see Eq.~(\ref{eq:LAS10}))
is rather complicated. However, an expansion for
$a/r_c = 2\pi \varphi/(k_{\scriptscriptstyle F} a) \ll 1$ yields a quadratic dependence
on $\varphi$
\begin{equation}
\frac{S_{10}}{\hbar} \simeq 2\, k_{\scriptscriptstyle F} \, a \left[1 -
\frac{1}{24} \left(\frac{2\pi\varphi}{k_{\scriptscriptstyle F} a} \right)^2 \right] \; .
\label{eq:S10exp}
\end{equation}
The susceptibility from Eq.~(\ref{eq:chibb}) with $S_{10}$ according to
Eq.~(\ref{eq:S10exp}) is shown as dotted curve in Fig.~\ref{fig:chibb}(a).
It agrees well at moderate fields and runs out of phase at a flux
corresponding to $a/r_c > 1$. While the period of the $\chi_{11}$ small field
oscillations is nearly constant with respect to $\varphi$ we find a
quadratic $\varphi$ characteristic for the oscillations in the
intermediate regime which turns
into a $1/\varphi$ behavior in the strong field regime (see next subsection).
To show that the agreement between the semiclassical (dashed) curve and the
quantum result is not an artefact of the particular number of electrons
chosen, Fig.~\ref{fig:chibb}(b) depicts semiclassical and quantum
bouncing--ball oscillations for $k_B T/\Delta=7$ and
at a different Fermi energy corresponding to $\sim$1400 electrons.
With decreasing Fermi energy the upper limit $r_c=a/2$ (or
$k_{\scriptscriptstyle F} a/(2\pi\varphi)=1/2$)
of the bouncing--ball oscillations is shifted towards smaller fluxes
($\varphi \approx 30$ in Fig.~\ref{fig:chibb}(b)) and the
number of oscillations shrinks. The oscillations for $\varphi > 30$
belong already to the strong field regime discussed in the next subsection.
Up to know we discussed the magnetic response of the family of primitive
orbits (1,0) and (0,1) which completely describes the intermediate
field regime at rather high temperatures corresponding to a
temperature cutoff length in the order of the system size.
At low temperatures we have to include contributions
from higher repetitions $(r,0)$, $(0,r)$ along bouncing--ball paths.
$L_{r0}$ and $A_{r0}$ have a linear $r$-dependence, and from
the Poincar\'e map Eq.~(\ref{eq:bb:map}), one obtains that
$D_{r0}=r^{-1/2} D_{10}$. Therefore
\begin{eqnarray}
\frac{\chi^{(1)}}{\chi_L} & = &
\frac{1}{\chi_L} \, \sum_{r=1}^\infty \,
(\chi^{(1)}_{r0} +\chi^{(1)}_{0r}) \nonumber \\
& = &
\frac{3 }{4 \pi^{1/2}} (k_{\scriptscriptstyle F} a)^\frac{3}{2} \,
\frac{\sqrt{\cos \zeta} (\sin \zeta+\cos \zeta -1)}{\zeta}
\frac{ (2 \zeta - \sin (2 \zeta) )^2}{\sin^4 \zeta} \times
\label{eq:chibbr} \\
& & \qquad \qquad \times
\sum_{r=1}^\infty \, r^{-{1}/{2}} \,
\sin\left(r\, \frac{S_{10}}{\hbar}+ \frac{\pi}{4}\right) \,
R_T (r\, L_{10})\; . \nonumber
\end{eqnarray}
Fig.~\ref{fig:chibb}(c) shows the susceptibility at the same Fermi energy as in
Fig.~\ref{fig:chibb}(b) but at a significantly lower temperature
$k_B T/\Delta = 2$.
The bouncing--ball peaks are much higher and new peaks related to long
periodic orbits differing from the bouncing--ball ones appear.
However, the bouncing--ball peak heights and even their shape
(which is no longer sinusoidal and symmetrical with respect
to $\chi = 0$) is
well reproduced by the analytical sum Eq.~(\ref{eq:chibbr}) showing the
correct temperature characteristic of the semiclassical theory.
The $k_{\scriptscriptstyle F} a$ behavior of the bouncing--ball susceptibility at a fixed flux
is not as simple as in the case of the weak--field oscillations (where
$\chi^{(1)}_{11} \sim (k_{\scriptscriptstyle F} a)^{3/2}$) since the angle $\zeta$ occurring
in the
prefactor in Eq.~(\ref{eq:chibb}) depends on $k_{\scriptscriptstyle F} a$ and the action is
non--linear in $k_{\scriptscriptstyle F} a$. Nevertheless, the overall oscillatory behavior
is similar as for example in Fig.~\ref{fig:chi1}(a).
However, at a given non--zero magnetic field the
classically relevant parameter Eq.~(\ref{classcond}) changes with energy.
Therefore, by increasing the Fermi energy beginning at the ground state
one generally passes from the strong field regime (at small energies
or high field strengths, see next section) to
the bouncing--ball regime and will finally reach the regime of oscillations
related
to the family (11). A unique behavior of periodic orbit oscillations
is only expected by changing magnetic field and Fermi energy
simultaneously in order to keep the classical parameter
Eq.~(\ref{classcond}) which determines the classical phase
space of the microstructure constant. Such a technique is
known as {\em scaled energy spectroscopy} in the context of atomic
spectra \cite{ERWS88}.
Bouncing--ball oscillations are expected to exist in general in
microstructures with parts of their opposite boundaries
being parallel and in spherical symmetrical microstructures as the disk
discussed in section \ref{sec:integrable}. (In the latter case the oscillations
should be even stronger than in the square since the effective length $l(H)$
(Eq.~(\ref{eq:l})) is not reduced with increasing magnetic field.)
An investigation of rectangular billiards for instance shows a splitting
of the frequencies of oscillations related to orbits $(M_x,0)$ and $(0,M_y)$
due to the different lengths of the orbits in $x$ and $y$ direction.
\subsection{Strong field regime}
At large magnetic field strengths or small energy the spectrum of a square
potential well exhibits the Landau fan corresponding to bulk--like
Landau states being almost unaffected by the system boundaries, while
surface affected states fill the gaps between the Landau levels and
condensate successively into the Landau channels with increasing
magnetic field (see, e.g., Fig.~\ref{f1}(a)). This spectral characteristic
corresponds to susceptibility oscillations which emerge with increasing
amplitude for fluxes corresponding to $r_c < a/2$, for instance for
$\varphi > 40$ in Fig.~\ref{f1}(b). They are shown in more detail in
Fig.~\ref{fig:dhva} where the full line depicts the numerical quantum result.
These susceptibility oscillations
exhibit the same period $\sim 1/H$ as de Haas--van Alphen bulk
oscillations but differ in amplitude, because here the cyclotron radius is not
negligible compared to the system size.
For the bulk or in the extreme high field regime $r_c \ll a$, where
quantum mechanically the influence of the boundaries of
the microstructure on the position of the quantum levels can
be neglected, an expression for the susceptibility
is most easily obtained by Poisson summation of the quantum density of states
as was briefly sketched in the introduction following standard textbooks
\cite{LanLip}. One then obtains the bulk magnetism as given
by Eq.~(\ref{intro:susdHvA}). It may be interesting to note however
that a semiclassical interpretation of this equation follows naturally
from an analysis similar to the one we followed throughout this paper.
In this case only one type of primitive periodic orbits exists,
the cyclotron orbits with length, enclosed area, and action given by
\begin{equation}
L_0(H) = 2 \pi r_c \quad; \qquad
A_0(H) = -\pi r_c^2 \quad; \qquad
\frac{S_0}{\hbar}= k L_0 + \frac{e}{c\hbar} H A_0 = k \pi r_c \; .
\label{eq:LASdHvA}
\end{equation}
Moreover, the trajectory passes through a focal point after each half
traversal along the cyclotron orbit. Therefore, using $\eta_n = 2n$ for
the Maslov indices and omitting the Weyl part of $G$, one obtains from
Eq.~(\ref{eq:green}) a semiclassical expression for the diagonal part
of the Green function
\begin{equation} \label{sec7:G_dHvA}
G({\bf r} , {\bf r}' \! = \! {\bf r}) =
\frac{1}{i\hbar \sqrt{2\pi i\hbar}} \sum_n (-1)^n D_n \,
\exp(i n \pi k r_c ) \; ,
\end{equation}
in which the main structure of Eq.~(\ref{intro:susdHvA}) is already apparent.
A direct evaluation of the amplitude $D_n$ in configuration space is
however complicated here by the fact that all trajectories
starting at some point ${\bf r}$ refocus precisely at ${\bf r}$
(focal point). Therefore, an expression like Eq.~(\ref{eq:D10:def})
for $D_n$ is divergent and cannot be used.
A method to overcome this problem by working with a Green function
$\tilde{G} (x,y;p_x',y')$ in momentum representation
for the $x'$ direction instead of $G(x,y;x',y')$ is described in
appendix~\ref{app:highB}. It yields (see Eq.~(\ref{app:G_dHvA}))
\begin{equation}
\frac{D_n}{i\hbar \sqrt{2\pi i\hbar}} =
\frac{m}{i \hbar^2} \; .
\end{equation}
Inserting this expression in Eq.~(\ref{sec7:G_dHvA}) we obtain
the oscillating part of density of states
\begin{equation}
{d^{\rm osc}}(E;H) = \sum_n d_n(E,H) = \frac{{\sf g_s} A m}{\pi \hbar^2}
\sum_n (-1)^n \cos(n \pi k r_c ) \; ,
\end{equation}
from which the de Haas--van Alphen susceptibility Eq.~(\ref{intro:susdHvA})
is obtained by using
\begin{equation}
\chi^{(1)} = \frac{1}{A}
\left( \frac{e A_0}{c \tau_0} \right)^2
\sum_n d_n(\mu,H) R_T (n L_0) \; .
\end{equation}
(with $\tau_0 = L_0/ v_{\scriptscriptstyle F}$) which applies for the
same reasons as Eq.~(\ref{eq:chi110}).
For an infinite system, this direct semiclassical approach to
the susceptibility therefore yields the same result as
the Poisson summation.
For billiard systems, it allows moreover to take correctly into
account the fact that the trajectories too close to the boundary
do not follow a cyclotron motion. Indeed, as seen in
appendix~\ref{app:highB}, the contribution of cyclotron orbits to
the susceptibility Eq.~(\ref{intro:susdHvA}) has to be modified when
$r_c$ is not negligible compared to $a$ by the introduction of a
multiplicative factor $s(H)$. It accounts for the effect that the family
of periodic cyclotron orbits (not affected by the boundaries)
which can be parameterized by the positions of the orbit centers is
diminished with decreasing field since the minimal distance between orbit
center and boundary must be at least $r_c$.
One therefore obtains for a billiard like quantum dot
\begin{equation}
\frac{\chi^{GC}_{cyc}}{\chi_L} = - 6 s(H) \,
(k_{\scriptscriptstyle F} r_c)^2
\sum_{n=1}^\infty \, (-1)^n \, R_T(2\pi n r_c) \,
\cos \left(n \pi k_{\scriptscriptstyle F} r_c \right) \; ,
\label{sec7:susdHvA}
\end{equation}
where $s(H)$ is given by Eq.~(\ref{app:s(H)}). In the case of the square
we find for the area reduction factor
\begin{equation} \label{eq:s(H)}
s(H) = \left(1-2\frac{r_c}{a}\right)^2 \,
\Theta\left(1-2\frac{r_c}{a}\right) \; ,
\end{equation}
$\Theta$ being the Heavyside step function. The last cyclotron orbit
disappears at a field where $r_c = a/2$, i.e. $s(\varphi)=0$ which happens near
$\varphi \approx 38$ in Fig.~\ref{fig:dhva}. There the dashed line
showing the semiclassical expression (\ref{sec7:susdHvA}) is in good
agreement with our numerical results and reproduces the decrease
in the amplitudes of the de Haas--van--Alphen oscillations when
approaching $\varphi(r_c=a/2)$ from the strong field limit. This
behavior is specific for quantum dots and does not occur in the
two--dimensional bulk. Corresponding bulk de Haas--van Alphen
oscillations under the same conditions as for the curves in
Fig.~\ref{fig:dhva} have (nearly constant) amplitudes in the order
of $\chi/\chi_L \approx 3000$.
The semiclassical curve which only reflects
the contribution from unperturbed cyclotron orbits agrees with the
numerical curve (representing the complete system)
even in spectral regions which show a complex variety
of levels between the Landau manifolds (see Fig.~\ref{f1}). Due to temperature
cutoff and since angular momentum is not
conserved in the square the corresponding edge or whispering gallery orbits
are mostly chaotic and do not show up in the magnetic response.
The strong de Haas--van Alphen like oscillations manifest the dominant
influence of
the family of cyclotron orbits. In related work on the
magnetization of a (angular momentum conserving) circular disk in the
quantum Hall effect regime Sivan and Imry \cite{SivanImry} observed
additional high frequency oscillations related to
whispering gallery orbits superimposed on the de Haas--van Alphen oscillations.
\newpage
\section{Conclusion}
\label{sec:concl}
In this work we have studied orbital magnetism and persistent currents
of small mesoscopic samples in the ballistic regime.
Within a model of non-interacting electrons we have provided
a comprehensive semiclassical description of these phenomena
based on the semiclassical trace formalism initiated by Gutzwiller,
Balian, and Bloch. We have moreover
treated in detail a few examples of experimental
relevance such as the square, circle and ring geometries.
The global picture that emerges from our study can be summarized
as follows. The magnetic
response is obtained from the variation of the thermodynamic
potential (or the free energy) under an applied magnetic field
and therefore, in a non-interacting model, from the knowledge of the
single-particle density of states.
The semiclassical formalism naturally leads to a separate
treatment of the smooth (in energy) component of the density of
states (or its integrated versions) and of its rapidly
oscillating part. The former is related to the local properties
of the energy manifold, while the latter is associated with the dynamical
properties of the system, more precisely to its periodic
(or nearly periodic) orbits. For the smooth component
we have shown that, despite the leading (Weyl) term
in an $\hbar$ expansion is independent of the field,
higher order terms can be computed and give rise to the
standard Landau diamagnetism for any confined electron system
at arbitrary magnetic fields. In the high temperature regime,
where the rapidly oscillating component of the density of states
is suppressed by the rounding of the Fermi surface,
the magnetic response reduces to the Landau diamagnetism.
On the other hand, for the temperatures of experimental relevance the
contribution coming from the oscillating part of the density
of states is much larger than the Landau term and dominates
the magnetic response. Similarly to the case of diffusive
systems, the susceptibility of a ballistic sample in contact with
a particle reservoir with chemical potential $\mu$ can be paramagnetic or
diamagnetic (depending on $\mu$) with equal probability. The fact
that the samples are isolated (with respect to electron transfer)
forces us to work in the canonical ensemble. Because of the
breaking of time reversal invariance occurring when the field is
turned on, this results, for essentially the same reason as in
the diffusive regime, in a small paramagnetic asymmetry for the
probability distribution of the susceptibility of a given sample.
For generic integrable systems, this effect is reinforced by the
breaking of invariant tori, which acts concurrently with the lost
of time reversal invariance.
The asymmetry disappears for a flux $\Delta \Phi$ inside the system
which is of the order of the quantum flux $\Phi_0$ at a
temperature selecting only the first few shortest orbits contributions,
but may be smaller for lower temperature.
Measuring the magnetic response of an ensemble of
structures with a large dispersion in the size or the number of
electrons
magnifies this asymmetry and yields a total response [per
structure] which is paramagnetic and much smaller than the
typical susceptibility for a flux smaller than $\Delta \Phi$, and zero
for larger flux. For ensembles with only microscopic differences
between the individual structures (i.e.~$\Delta(k_{\scriptscriptstyle F} a)
\geq 2 \pi$, but still $\Delta a/a \ll 1$ and $\Delta {\bf N}/{\bf
N} \ll 1$) further oscillating patterns in the average susceptibility
should be observed for larger fields.
Since the oscillating part of the density of states is semiclassically
related to the classical periodic orbits, the nature of the classical
dynamics quite naturally plays a major role in the determination
of the amplitude of the magnetic response. Indeed, for a system
in which continuous families of periodic orbits are present, these
orbits contribute in phase to the density of states, yielding
much larger fluctuations of the density of states than for systems
possessing only isolated orbits, and therefore much larger magnetic
response. Families of periodic orbits are characteristic for
integrable systems, while for chaotic systems the periodic orbits are usually
isolated. This different behavior can therefore be referred to
as the hallmark for the distinction between integrable and chaotic
systems. It should be borne in mind however that this difference
is due to short-time properties, namely the existence or absence of
families of orbits, rather than to long-time properties such as
exponential divergence of orbits. In this respect, some atypical
chaotic systems, such as the Sinai billiard for instance, may show
a magnetic response typical for an integrable system because of the
existence of marginally stable families of orbits.
The importance of classical mechanics can be illustrated in the
[experimentally
relevant] case of two-dimensional billiard-like quantum dots in the
weak-field regime. If the system is chaotic, more precisely
if the periodic trajectories are isolated, the typical susceptibility
scales as $(k_{\scriptscriptstyle F} a) \chi_{\scriptscriptstyle L}$, where $k_{\scriptscriptstyle F}$ is the Fermi wave number and
$a$ the typical size of the dot. By comparison, the typical
susceptibility of an integrable
system scales with $(k_{\scriptscriptstyle F} a )^{3/2} \chi_{\scriptscriptstyle L}$. This characteristic behavior of
integrable systems is found in the generic case (like the square) where
the magnetic field breaks the integrability as well as in the non-generic
case (like the disk) where the system remains integrable at finite
fields. The difference due to the nature of the classical mechanics is
even stronger for measurements on ensembles of structures since
one obtains a $(k_{\scriptscriptstyle F} a) \chi_{\scriptscriptstyle L}$ dependence for integrable systems and
no dependence on $(k_{\scriptscriptstyle F} a)$ for the chaotic ones.
The same parametric dependences are
obtained for the persistent currents in integrable and chaotic
multiply-connected geometries.
Therefore, the nature
of the dynamics yields an order-of-magnitude difference in the
magnetic response of integrable and chaotic systems, which should be
easy to observe experimentally (especially for ensemble measurements).
Finally, for systems with mixed dynamics, for which the phase-space
is characterized by the coexistence of regular and chaotic
motion, the magnetic response should be dominated by the nearly integrable
regions of phase-space. This gives rise to a $(k_{\scriptscriptstyle F} a )^{3/2} \chi_{\scriptscriptstyle L}$
dependence for the typical susceptibility as long as some families
of periodic orbits remain sufficiently unperturbed. The precise calculation
of the prefactor may however present some complications that we have not
considered here (the general semiclassical treatment of
mixed systems remains an open problem) and should
depend on the fraction of phase-space being integrable.
The semiclassical approach we are using not only allows a global
understanding of the magnetic response of ballistic devices,
but also provides precise predictions when specific systems
are considered.
The detailed comparison between exact quantum calculations
and semiclassical results for the square geometry
demonstrates indeed that the semiclassical predictions are
extremely accurate. This has been shown in section~\ref{sec:square}
for weak fields, such that the trajectories are essentially unaffected
by the magnetic field, and also in section~\ref{sec:highB}
for fields large enough to yield a cyclotron radius of the
order of the typical size of the structure (where the bending of the
classical trajectories has to be taken into account). For intermediate
fields we have identified a new regime where the magnetic susceptibility
is dominated by bouncing-ball trajectories that alternate between
opposite sides of the structure (enclosing flux due to their bending).
For high fields the electrons move on cyclotron orbits and we have
recovered the de Haas~-~van Alphen oscillations (with finite-size
corrections that we calculated semiclassically).
In order to understand the success of the semiclassical approach, it
should be kept in mind
that the lack of translational invariance characteristic for
the ballistic regime, where the shape of the device
plays an important role, complicates the application of other
approximation schemes as, e.g., diagrammatic expansions. Therefore,
except for very specific cases where exact quantum calculations
are possible, and unless one is satisfied by direct numerical
calculations,
some semiclassical ideas have to be implemented to deal with such
problems. Moreover, from a more practical point of view, the
semiclassical trace
formalism we have used appears perfectly adapted to deal
with thermodynamic quantities such as the Grand
Potential $\Omega(\mu)$ or its first and second derivatives
$N(\mu)$ and $D(\mu)$. Indeed, the beauty of this approach is
that the oscillating part of the density of states
is directly expressed in terms of Fourier-like components, each of which
is associated with a periodic (or nearly periodic) orbit. The
thermodynamic properties are obtained from their purely quantal
(or zero temperature) analogues $\omega$, $n$ and $d$ by temperature
smoothing, which merely amounts to multiply each oscillating
component by a temperature-dependent damping factor. For all fields
(high, intermediate, or weak), this factor depends
only on the ratio of the {\em period} $\tau$ of the
corresponding orbit and the temperature-dependent
cutoff time $\tau_c = \beta \hbar / \pi$
and suppresses exponentially the contribution of orbits with period
longer than $\tau_c$.
As a consequence, not only the effect of temperature is taken into
account in an intuitive transparent way, but in addition only the shortest
periodic orbits have to be considered in the semiclassical
expansion. All the problems concerning the convergence of trace formulae and
the validity of semiclassical propagation of the wave function
for very long times are of no importance here. One therefore avoids
most of the problems which plague the field of quantum chaos
when semiclassical trace formulae are used to resolve the spectrum
on a mean-spacing scale. Mesoscopic physics is usually
concerned with the properties of the spectrum on an energy-scale large
compared to the mean-spacing. In the spirit of
the work of Balian and Bloch \cite{bal69}, this is the situation
for which the semiclassical trace formalism is especially appropriate.
Having stressed the success of the semiclassical approach in dealing
with our model of non-interacting electrons evolving in a clean
medium, it is worthwhile to consider in more detail
how the above picture should be modified when going closer to the real
world, and incorporating the effects of residual disorder,
electron-electron or electron-phonon interactions.
As stressed in the introduction, the first of these points is
relatively harmless because of finite temperature smoothing.
The restriction to short periodic orbits actually justifies an approach to
the ballistic regime using a model for clean systems since long diffusive
trajectories do not contribute to the finite-temperature susceptibility.
Indeed, careful numerical and semiclassical studies of the effect of small
residual disorder \cite{rod2000} show that, except for a possible
reduction of the magnetic response, the above description of the orbital
magnetism of ballistic systems remains essentially unaltered. In
particular, the mechanism proposed by Gefen et al.\ \cite{gef94}
is not borne out by the numerical simulations at the temperatures of
experimental relevance.
For smooth disorder, such as presumably prevails in the
systems of Refs.~\cite{levy93} and \cite{BenMailly},
the magnetic response is decreased by the
dephasing of nearby trajectories in a way that depends on its strength
and the ratio between the correlation length and the size of the
structure \cite{rod2000}, but diffusive trajectories can be seen to
be absolutely irrelevant if the elastic mean free path is larger than
the size of the structure. The precise knowledge of this reduction is
however needed in order to make a decisive comparison with the experimental
results of Ref.~\cite{levy93}.
At the low temperatures of the experiments the inelastic mean free path
of the electrons is much larger than the system size since electron-phonon
interactions are suppressed. On the other hand,
the effect of electron-electron
interactions on the magnetic response is a much more controversial
point. In particular, it has been invoked to be the necessary mechanism
to obtain the measured values \cite{inter} for the problem of persistent
currents in disorder metals. In a first approximation to the
experimental conditions that we investigated in this work we would
infer that
electron-electron interactions are not crucial since the screening
length is much smaller than the size of the samples and since the
2-d renormalization of the effective mass at these electron densities
is only about 11\% \cite{JDS}. Clearly the two previous criteria
will not be satisfied in smaller structures, and the possibility
that electron-electron interactions express themselves through
a mechanism for which these estimates are not relevant remains
open even in the experimental realizations we consider.
Contrarily to the effect of disorder, which can be implemented
within a semiclassical framework without essential difficulties,
a semiclassical treatment of the electron-electron interaction
still remains an open problem. However, the genuine effects
that we have found within our semiclassical approach for the
clean model of non--interacting electrons should prevail in more
sophisticated theories. We think that the rich variety of
possible experimental configurations for ballistic devices
(the shape and the size can nowadays be chosen at will)
provides an ideal testing ground for these more complete approaches.
We hope that the work presented here will stimulate
experimental and theoretical activity addressing the magnetic
response of ballistic microstructures.
\section*{Acknowledgments}
We acknowledge helpful discussions with H.~Baranger, O.~Bohigas, Y.~Gefen,
M.~Gutzwiller, L.~L\'evy, F.~von~Oppen, N.~Pavloff, B.~Shapiro, and
H.~Weidenm\"{u}ller. We are particularly indebted to H.~Baranger for
continuous support and a careful reading of the manuscript, and to
O.~Bohigas for forcing us not to stop until getting to the
bones of the problem. We thank B.~Mehlig for communicating us Ref. \cite{Kubo64}.
KR and RAJ acknowledge support from the
``Coop\'eration CNRS/DFG" (EB/EUR-94/41).
KR thanks the A. von Humboldt foundation for financial support.
The Division de Physique Th\'eorique is ``Unit\'e de recherche des
Universit\'es Paris~11 et Paris~6 associ\'ee au C.N.R.S.''.
\newpage
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proofpile-arXiv_065-602
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\section{Introduction}
\setcounter{equation}{0}
$q$-deformation of oscillator algebra
was first introduced by Biedenharn (B) \cite{bied}
and Macfarlane (M) \cite{macf} in the context of
oscillator realization of quantum algebra $su_q(2)$ \cite{skly}.
They obtained their $q$-deformation of $su(2)$ algebra
by using the double oscillators realization of Jordan-Schwinger
type. Later, Kulish and Damaskinsky \cite{Kulda}
used a single oscillator realization of $su(1,1)$ to
obtain the $q$-deformed algebra
for a special value of Casimir constant equal to $-{3 \over 16}$.
Another oscillator realization of $su(1,1)$ and its
$q$-deformation was obtained for
Calogero-type oscillator \cite{cho}.
This oscillator
does not satisfy the Heisenberg algebra but a modified one
due to exchange operator, {\it i.e.\/}, Dunkle operator.
$q$-deformation of this oscillator was also achieved
and gave a different commutation relation from those of
B and M.
In this paper, we consider the $q$-deformation of
a single oscillator representation
of $su(1,1)$ and $su(2)$ algebras and their coherent states
in terms of Holstein-Primakoff (HP) \cite{hols}
and Dyson (D) \cite{dyso} realizations which
appear in many applications from spin density wave in
condensed matter physics \cite{kitt} to nuclear physics \cite{rowe}.
In spite of the existing literature \cite{bied1}
on the $q$-deformation and $q$-coherent states \cite{gong} of
HP and D realizations of those algebras,
we find that a comprehensive study on
the relations among the various realizations is still lacking,
especially the relations among the measures of the $q$-coherent
state in the resolution of unity. We are going to resolve the issues
by studying other types of $q$-deformations suited to HP and D
realizations, starting with the "symmetric" $q$-deformation of the
Lie algebras developed by Curtright and Zachos \cite{curt}.
We find that this procedure is instructive since we can obtain the
B, M, $q$-deformed anyonic
oscillators, and their $q$-coherent states
naturally. Also, $q$-deformation of Fock-Bargmann (FB) type
can be realized in $q$-derivatives.
This paper is organized as follows.
In section II, we obtain $su_q(1,1)$ in HP and D
realizations in terms of the B, M,
and anyonic type oscillators.
We construct the $q$-coherent states
and compare the measure in each case.
We also present the FB representation in $q$-derivative.
In Section III, the same analysis is performed for $su_q(2)$ case.
Section IV contains the conclusion and discussion.
Before going into details, we briefly explain our notation.
In the oscillator representation of the Heisenberg-Weyl algebra,
\begin{equation}
[a_-, a_+] =1\,,
\label{heisenberg}
\end{equation}
we introduce a number eigenstate $|n>$
of number operator $N=a_+ a_-$.
We require
$|0>$ be annihilated by $a_-$,
$a_- |0> = 0$.
Explicitly, the creation and annihilation operators
act on the ket,
\begin{equation}
a_-|n> = n |n-1>\,
\quad
a_+ |n> = |n+1>\,.
\label{anrelation}
\end{equation}
It should be noted that the normalization of the ket
is not fixed yet, which will result
from Hermitian property of the generators of
$su_q(1,1)$ and $su_q(2)$. We use
FB holomorphic representation of the
oscillator algebra, in which
$ < \xi | n> = \xi^n$ and
\begin{equation}
{d \over d \xi} < \xi| n> = < \xi | a_- | n> \,
\quad
\xi < \xi| n> = < \xi | a_+ | n> \, .
\end{equation}
\def\arabic{section}.\arabic{equation}{\arabic{section}.\arabic{equation}}
\section{$su_q(1,1)$ and coherent state}
\setcounter{equation}{0}
$su(1,1)$ satisfies the algebra,
\begin{equation}
[ K_0, K_{\pm}]=\pm K_{\pm}, \quad
[ K_+, K_-]=-2K_0,
\label{su11algebr}
\end{equation}
and Casimir invariant is expressed as
$ C=K_0(K_0-1)-K_+K_-\,$.
To have a connection with the oscillator algebra,
we require the number eigenstate $|n>$ be
an eigenstate of $K_0$,
\begin{equation}
K_0 |n> = (k_0 + n) |n>\,.
\end{equation}
Here, $k_0$ is assumed to be
a positive integer or half odd integer
and $|0>$ is also annihilated by $K_-$.
This representation gives the Casimir number
$k_0(k_0 -1)$.
For definiteness, we will assume that
the $su(1,1)$ algebra act on the ket $|n>$ as
\begin{equation}
K_- |n> = n | n-1>\,\quad
K_+ |n> = (n + 2 k_0) |n + 1>\,.
\label{k-ladder}
\end{equation}
This convention is consistent with
the holomorphic first-order differential operator
representation for $su(1,1)$ given in \cite{pere}
\begin{equation}
\hat K_+ (\xi) = \xi^2 {d \over d \xi} + 2 k_0 \xi\,,\quad
\hat K_-(\xi) = {d \over d \xi} \,,\quad
\hat K_0 (\xi) = \xi {d \over d \xi} + k_0 \,.
\label{su11holo}
\end{equation}
Since $
\hat K_i(\xi) < \xi| n> = < \xi | K_i | n>
$, we can check the relation in Eq.~(\ref{k-ladder}) holds.
$q$-deformed algebra $su_q(1,1)$ is given as \cite{skly}
\begin{equation}
[Q_0, Q_{\pm}]=\pm Q_{\pm}\,,\quad
[ Q_+, Q_-]=-[2Q_0]_{q}\,,
\label{su11algebra}
\end{equation}
where the $q$-deformation is defined as
\begin{equation}
[x]_q\equiv \frac{q^x-q^{-x}}{q-q^{-1}}\,.
\end{equation}
$q$-deformed Casimir invariant is given by
$C_q = [Q_0]_q[Q_0 - 1]_q - Q_+ Q_-$.
One can obtain the explicit form of the $q$-deformed
generators following \cite{curt},
\begin{equation}
Q_0=K_0\,,\quad
Q_-=K_-f(K_0)\,,\quad
Q_+=f(K_0)K_+\,.
\label{su11qrealization}
\end{equation}
Noting a useful identity
$ -[2K_0]_{q}=g(K_0)-g(K_0+1)$
where
$g(K_0)=[K_0-k_0]_{q}[K_0+k_0-1]_{q}\,$
and
$g(K_0=k_0)=0\,$,
we may identify $g(K_0)$ as $f(K_0)^2 (K_0(K_0-1)-C)$,
\begin{equation}
f(K_0) = \sqrt{[K-k_0]_q [K_0 + k_0 -1]_q
\over (K-k_0) (K + k_0 -1)}
= \sqrt{[N]_q [N + 2k_0 -1]_q
\over N (N + 2k_0 -1)}\,.
\label{fk0}
\end{equation}
This realization is not the unique choice.
In the following, we give some other examples
which will result in the $q$-deformed oscillator algebra
or the $q$-deformed FB representation.
In addition, independent of the realization
we are requiring the conjugate
relations,
\begin{equation}
Q_-^{\dagger} = Q_+\,,\quad
Q_0^{\dagger} = Q_0\,.
\label{conjugate}
\end{equation}
This will determine the norm of each state
for the given realization and the resolution of
unity for coherent state.
\subsection{B oscillator realization.}
(1) D type.
Let us consider the realization,
\begin{equation}
Q_0=K_0 = N+ k_0\,,\quad
Q_-=K_- \sqrt{[N]_q \over N} \,,\quad
Q_+=\sqrt{[N]_q \over N}
{[N + 2k_0 -1]_q \over (N+ 2k_0 -1)} K_+\,.
\label{su11-ID}
\end{equation}
This is obtained if we re-scale
$Q_+$ and $Q_-$ in Eq.~(\ref{su11qrealization}).
This ladder operators act on the ket as
\begin{equation}
Q_- |n>_D = \sqrt{n [n]_q} |n-1>_D \,,\qquad
Q_+ |n>_D = \sqrt{ [n+1]_q \over n+1} \, [n+ 2k_0]_q\, |n+1>_D\,.
\end{equation}
Eq.~(\ref{su11-ID}) becomes
an oscillator realization if we interpret it as
\begin{equation}
Q_0 = N+ k_0\,,\quad
Q_- = (a_q)_- \,,\quad
Q_+ = [N + 2k_0-1 ]_q (a_q)_+ \,,
\end{equation}
and identify $(a_q)_\pm$ as
\begin{equation}
(a_q)_- = a_- \sqrt{[N]_q \over N}\,,\quad
(a_q)_+ = \sqrt{[N]_q \over N}\,\, a_+ \,.
\label{a-Bied}
\end{equation}
$(a_q)_\pm$ satisfy the $q$-deformed
oscillator algebra of B type,
\begin{equation}
(a_q)_- (a_q)_+ - q (a_q)_+ (a_q)_- = q^{-N}\,.
\label{Biedenharn}
\end{equation}
$q$-deformed coherent state is defined by
( {\it \`a la} Perelomov' coherent
state \cite{pere})
\begin{equation}
|z>_D = e_q^{\bar z Q_+} |0>_D
=\sum_{n=0}^\infty \bar z^n
\sqrt{1 \over [n]_q! n!}
{[n+2k_0-1]_q! \over [2k_0 -1]_q! } \,\, |n>_D\,,
\label{su11qcoherent}
\end{equation}
where we use the $q$-deformed exponential function.
The subscript D stands for D type.
(Note that we do not add a normalization constant
in this definition since this will introduce $z$ in addition to
$\bar z$).
Conjugate relation, Eq.~(\ref{conjugate}) gives the normalization
of the number eigenstate,
\begin{equation}
{}_D\!\!<n|n>_D = {n! [2k_0 -1]_q! \over [n+2k_0 -1]_q!}\,.
\label{normalization}
\end{equation}
This normalization provides us the resolution of unity as
\begin{equation}
I = \sum_{n=0}^{\infty}
{[n + 2k_0 -1]_q! \over n! [ 2k_0 -1]_q!}
|n>_D\,{}_D\!\!<n|
= \int d^2_q z \, G(z)\, |z>_D\,{}_D\!\!<z|\,,
\end{equation}
and the measure $G(z)$ is given as
\begin{equation}
G(z)=\left\{ \begin{array}{ll}
{[2k_0 -1]_q \over \pi} (1 - |z|^2)_q^{2k_0 -2}&
\text{for } 2k_0= \text{integer} >1 \\
{|z|^2 \over \pi} & \text{for } k_0= 1
\end{array}\right.
\label{su11qmeasure}
\end{equation}
where the $q$-deformed function is defined as
$(1-x)^n _q = \sum_{m=0}^n {[n]_q! \over [m]_q! [n-m]_q!} (-x)^m$.
(For $k_0= {1 \over 2}$, see below Eq. (\ref{aq-measure})).
Here, the two dimensional integration is defined as
\begin{equation}
d^2_q z \equiv {1 \over 2} d\theta\,\, d_q |z|^2 \,.
\label{integ}
\end{equation}
The angular integration is an
ordinary integration, $0 \le \theta \le 2\pi$.
The radial part is a $q$-integration, which is the inverse
operation of $q$-derivative defined as
\begin{equation}
{d \over d_q z} f(z) = {f(qz) - f(q^{-1}z) \over z(q- q^{-1})}\,.
\label{qderivative}
\end{equation}
One can check that $I$ commutes with the $su_q(1,1)$ generators.
We note that in this Hilbert space,
$(a_q)_+$ is not an adjoint of $(a_q)_-$.
To have the conjugation property
between $(a_q)_-$ and $(a_q)_+$
as well as between $Q_-$ and $Q_+$,
we may resort to HP realization.
Therefore, we need to compare the quantities in different
realization.
It turns out to be useful
to express the quantities in
unit normalized eigenstate basis $|n)$ instead of
$|n>$.
For later comparison, we give the explicit expression;
\begin{equation}
Q_- |n)_D = \sqrt{[n]_q [n+2 k_0 -1]_q} |n-1)_D\,,\quad
Q_+ |n)_D = \sqrt{[n+1]_q [n+2 k_0 ]_q} |n+1)_D\,.
\label{Qrelation}
\end{equation}
And the coherent state in Eq.~(\ref{su11qcoherent}) becomes
\begin{equation}
|z>_D = e_q^{\bar z Q_+} |0>_D
=\sum_{n=0}^\infty \bar z^n
\sqrt{ [n+2k_0-1]_q! \over [n]_q! [2k_0 -1]_q! } \,\, |n)_D\,.
\label{su11qcoherent-0}
\end{equation}
(2) HP type.
Let us consider the realization in terms of $q$-deformed oscillator
in Eq.~(\ref{a-Bied}) to have the HP realization,
\begin{eqnarray}
&&Q_0= N+ k_0\,,
\nonumber \\
&& Q_-=K_- \sqrt{[N]_q [N+2k_0-1]_q\over N }
= (a_q)_- \sqrt{[N +2k_0 -1]_q} \,,
\nonumber \\
&&Q_+=\sqrt{[N]_q [N + 2k_0 -1]_q \over N}
{1 \over (N+ 2k_0 -1)} K_+
= \sqrt{[N + 2k_0 -1]_q} (a_q)_+\,.
\label{su11-IHP}
\end{eqnarray}
The ladder operators act on the ket as
\begin{eqnarray}
Q_- |n>_H &=&
\sqrt{ n [n]_q [n+2k_0 -1]_q } |n-1>_H \nonumber\\
Q_+ |n>_H &=& \sqrt{ [n+1]_q [n+2k_0]_q \over n+1}
\, |n+1>_H.
\label{QHP-relation}
\end{eqnarray}
The subscript $H$ stands for HP.
Conjugate relation between $Q_\pm$'s requires
the normalization in this
Hilbert space,\\
${}_H\!\!<n|n>_H = n!$,
which is different from the previous one,
Eq.~(\ref{normalization}).
This is not surprising since
two Hilbert spaces are different.
If we use the unit normalized ket $|n)_H$,
then the relation given in Eq.~(\ref{QHP-relation}) becomes
exactly the same form given in Eq.~(\ref{Qrelation})
with subscript D replaced by subscript H, and the $q$-deformed
coherent state corresponding
to Eq.~(\ref{su11qcoherent}) is given as
\begin{equation}
|z>_H = e_q^{\bar z Q_+} |0>_H
=\sum_{n=0}^\infty \bar z^n
\sqrt{[n + 2k_0 -1]_q! \over [n]_q! [2k_0 -1]_q!}
\,\, |n)_H
\end{equation}
in terms of the normalized ket $|n)_H$.
This is again exactly the same form given
in Eq.~(\ref{su11qcoherent-0}).
Therefore, the resolution of unity for the coherent state
is expressed in terms of the same measure $G(z)$
in Eq.~(\ref{su11qmeasure}),
even though two realizations look so different at first sight.
Note that in this HP realization,
the conjugate relation
between $a_+^\dagger = a_-$ is satisfied automatically,
since
\begin{equation}
(a_q)_- |n)_H = \sqrt{[n]_q}\,|n-1)_H\,\quad
(a_q)_+ |n)_H = \sqrt{[n+1]_q} |n+1)_H \,.
\label{aq-relation}
\end{equation}
Therefore, one can equally use a new coherent state,
$q$-deformed version of Glauber coherent state \cite{klau},
\begin{equation}
|z]_H = e^{\bar z a_+}_q |0>_H\,.
\label{qGlaubercoherent}
\end{equation}
Explicit form of this coherent state is given as
$ |z]_H = \sum_{n=0}^{\infty} {\bar z^n \over [n]_q!} |n)_H\,$.
The resolution of unity is given as
\begin{equation}
I = \sum_{n=0} ^\infty |n)_H\,{}_H\!(n|
= \int d^2_q z\, g(z) | z]_H\, {}_H\![ z|\,
\end{equation}
where $g(z)$ is given by
\begin{equation}
g(z) = {1 \over \pi} e_q^{- |z|^2}\,,
\label{aq-measure}
\end{equation}
and the domain is over an infinite plane.
Unlike the D case,
this holds for any value of $k_0$.
Therefore,
for $k_0={1 \over 2}$,
one can use the $q$-analogue of Glauber
coherent state.
For other value of $k_0$,
one can use the Bargmann measure defined in
Eq.~(\ref{aq-measure}) or
Liouville measure in
Eq.~(\ref{su11qmeasure})
depending on the definition of coherent state.
\subsection{M oscillator realization.}
(1) D type.
We consider a different realization from the B
type oscillator realization,
\begin{eqnarray}
&&Q_0=K_0= N + k_0 \,,
\nonumber\\
&&Q_-=K_- \sqrt{[N]_q \over N}q^{N-2 \over 2}= (b_q)_-\,,
\nonumber\\
&&Q_+=q^{-{N -1 \over 2}} \sqrt{[N]_q \over N}
{[N + 2k_0 -1]_q \over (N+ 2k_0 -1)} K_+
=q^{-(N-1)} [N + 2k_0 -1]_q (b_q)_+ \,,
\label{su11-IID}
\end{eqnarray}
with new oscillator given as
\begin{equation}
(b_q)_- = (a_q)_- q^{N-1 \over 2}
= a_- \sqrt{\{N\}_q \over N}\,,\quad
(b_q)_+ = q^{N-1 \over 2}(a_q)_+
= \sqrt{\{N\}_q \over N} a_+ \,,
\label{aq-Mac}
\end{equation}
where we introduce a new definition of $q$-number,
\begin{equation}
\{x\}_q = {q^{2x} - 1 \over q^2 -1} = [x]_q \,\, q^{x-1}\,.
\end{equation}
This oscillator realization gives the $q$-deformed
oscillator algebra of M type,
\begin{equation}
(b_q)_- (b_q)_+ - q^2 (b_q)_+ (b_q)_- = 1\,.
\label{Macfarlane}
\end{equation}
In terms of this realization, the ladder operators act
on the ket as
\begin{equation}
Q_- |n> = \sqrt{n \{n\}_q} |n-1> \,,\qquad
Q_+ |n> = q^{-n }\sqrt{ \{n+1\}_q \over n+1} \,
[n+ 2k_0]_q\, |n+1>\,.
\end{equation}
In the following, for notational simplicity,
we will delete the
subscript on the ket which distinguishes the Hilbert space,
since there is no possibility of confusion.
The conjugate relation between $Q_\pm$'s
gives the normalization,
\begin{equation}
<n|n> = { n! [ 2k_0 -1]_q! \over [n + 2k_0 -1]_q!}
q^{n(n-1) \over 2} \,.
\end{equation}
In terms of unit normalized ket $|n)$,
we have the canonical operator relations
for $Q_\pm$ as in Eq.~(\ref{Qrelation}).
In addition, $q$-deformed coherent state
for $su_q(1,1)$ has the same form as in Eq.~(\ref{su11qcoherent-0})
and therefore, the measure $G(z)$
in Eq.(\ref{su11qmeasure})
is used for the resolution of unity
for the coherent state.
(2) HP type.
Another HP type realization is given as
\begin{eqnarray}
&&Q_0= N+ k_0\,,
\nonumber \\
&&Q_- = (b_q)_- \sqrt{q^{-(N-1)} [N +2k_0 -1]_q} \,,
\nonumber \\
&&Q_+=\sqrt{q^{-(N-1)} [N + 2k_0 -1]_q} (b_q)_+\,,
\label{su11-IIHP}
\end{eqnarray}
with $b_q$'s defined in Eq.~(\ref{aq-Mac}).
However, these generators coincide with the one given in
HP type of the B oscillator realization,
Eq.~(\ref{su11-IHP}).
That is, the Hilbert space is exactly same for both cases
as far as $su_q(1,1)$ is concerned.
Therefore, the $su_q(1,1)$ $q$-deformed coherent state
in terms of the normalized ket $|n)$
is exactly the same form given in Eq.~(\ref{su11qcoherent-0})
and the same measure $G(z)$
in Eq.~(\ref{su11qmeasure})
is used for the resolution of unity.
On the other hand, from the oscillator point of view,
one can define a new coherent state since
the conjugate relation
between $(b_q)_+^\dagger = (b_q)_-$ is satisfied automatically;
\begin{equation}
(b_q)_- |n) = \sqrt{\{n\}_q}\,|n-1)\,\quad
(b_q)_+ |n) = \sqrt{\{n+1\}_q} |n+1) \,.
\label{dagger}
\end{equation}
Let us define
another version of $q$-deformed Glauber coherent state as
\begin{equation}
|z\} = E_q^{\bar z (b_q)_+}|0>
= \sum_{n=0}^{\infty} {\bar z^n \over \{n\}_q!} |n)\,,
\label{newqcoherent}
\end{equation}
where $q$-deformed exponential function $E_q^x$ differs
from $e_q^x$ in that $q$-number $[n]_q$ is replaced by
$\{n\}_q$,
\begin{equation}
E_q^x = \sum_{n=0}^\infty {x^n \over \{n\}_q!}
\end{equation}
The resolution of unity is given as
\begin{equation}
I
= \sum_{n=0} ^\infty |n)\,(n|
= \int d^2_q z\, h(z) | z\}\, \{ z|\,.
\end{equation}
where $h(z)$ is given by
\begin{equation}
h(z) = {1 \over \pi} E_q^{- |z|^2}\,,
\label{bq-measure}
\end{equation}
and the domain is over an infinite plane.
\subsection{$q$-anyonic oscillator realization}
Let us consider the HP type realization again.
As seen in the previous section, one can have the
B oscillator Eq.~(\ref{su11-IHP})
or M oscillator Eq.~(\ref{su11-IIHP})
from the same $q$-deformed form of the $Q_i$'s.
We give another useful form of oscillator realization,
$q$-anyonic oscillator.
Since D and HP type realizations are
now trivially connected, we present only HP
realization which maintains the conjugate condition for
oscillator algebra also.
Let us put $Q_i$'s as
\begin{eqnarray}
&&Q_0= N+ k_0\,,
\nonumber \\
&& Q_-=K_- \sqrt{[N]_q [N+2k_0-1]_q\over N }
= (A_q)_- \sqrt{ (A_q)_+ (A_q)_+ + 2[k_0 - {1 \over 2}]_q} \,,
\nonumber \\
&&Q_+=\sqrt{[N]_q [N + 2k_0 -1]_q \over N}
{1 \over (N+ 2k_0 -1)} K_+
= \sqrt{ (A_q)_+ (A_q)_+ + 2[k_0 - {1 \over 2}]_q}\,\, (A_q)_+ \,,
\end{eqnarray}
Then we have the $q$-deformed oscillator as
\begin{equation}
(A_q)_-= a_- \,\, \sqrt{[N+ k_0 - {1\over 2} ]_q -
[k_0 - {1 \over 2}]_q
\over N}\,,
\quad
(A_q)_+= \sqrt{[N+ k_0 - {1\over 2} ]_q - [k_0 - {1 \over 2}]_q
\over N}\,\, a_+\,.
\end{equation}
Its commutation relation looks complicated,
\begin{equation}
((A_q)_- (A_q)_+ + [k_0 - {1 \over 2}]_q)
-q ((A_q)_+ (A_q)_- + [k_0 - {1 \over 2}]_q)
= q^{-(N + k_0 + { 1\over 2})}\,.
\end{equation}
However, the meaning of this commutation relation becomes clear if
we rewrite the relation in M's form,
\begin{equation}
(B_q)_- (B_q)_+ - q^2 (B_q)_+ (B_q)_- =1 \,,
\label{paraMac}
\end{equation}
by identifying
\begin{eqnarray}
(B_q)_- (B_q)_+ = q^{N + k_0 - {1 \over 2}}[N +k_0 + {1 \over 2}]_q
= q^{N + k_0 - {1 \over 2}}
\,\,((A_q)_- (A_q)_+ + [k_0 - {1 \over 2}]_q)\,,
\nonumber\\
(B_q)_+ (B_q)_- = q^{N + k_0 - {3 \over 2}}[N+ k_0 - {1 \over 2}]_q
= q^{N + k_0 - {3 \over 2}}
\,\,((A_q)_+ (A_q)_- + [k_0 - {1 \over 2}]_q)\,.
\label{Bquad}
\end{eqnarray}
In this realization,
the vacuum $|0>$ is not annihilated by $(B_q)_-$
unless $k_0 = {1 \over2}$, since
\begin{equation}
(B_q)_+ (B_q)_- |0> = \{k_0 - {1 \over 2}\}_q |0>\,.
\end{equation}
This feature reflects the fact that this realization corresponds
to the $q$-deformed non-trivial one dimensional analogue of
anyon which appears in two dimensional oscillator representation
with $k_0$ being related with statistical
parameter in anyon physics \cite{chorim}.
The conjugate relation between $(B_q)_-$ and $(B_q)_+$
can be seen formally at the operator level in Eq.~(\ref{Bquad})
since the conjugate relation between $(A_q)_-$ and $(A_q)_+$ does
hold. However, the fact that $(B_q)_-$ does not annihilate the
vacuum $|0>$ implies that one cannot define a proper Hilbert space.
Therefore, the measure of the coherent state
of the Glauber type for the $B_q$ oscillator system cannot be defined.
On the other hand, the measure of the $q$-deformed
coherent state of Perelomov type is given in Eq.~(\ref{su11qmeasure}).
One may also define the coherent state of the
Glauber type in terms of
the $A_q$ oscillator, whose explicit form of the measure
turns out to be very complicated and will not be reproduced here.
As far as $B_q$ oscillator is concerned, we may
construct a Hilbert space from a new vacuum which is
annihilated by $(B_q)_-$. Then, since the commutation
relation Eq.~(\ref{paraMac}) is the same form as
in Eq.~(\ref{Macfarlane}),
the generators act on the
new Hilbert space as in Eq.~(\ref{dagger}).
In this case, one can contruct the $q$-deformed Glauber type
coherent state and the measure is given in Eq.~(\ref{bq-measure}).
However, this representation has nothing to do with the
anyonic representation mentioned above.
We comment in passing that there is another well-known
one dimensional oscillator representation for anyon type;
Calogero oscillator system, which turns out to be
the realization \cite{cho,mac} of parabose system \cite{gree}.
Its $q$-deformed realization does not
satisfy the commutation relation of
M type Eq.~(\ref{paraMac}).
The explicit measure for the $q$-deformed
coherent state of the Glauber type in this case is
already known \cite{cho}.
\subsection{FB realization with symmetric $q$-derivative}
Let us consider a realization,
\begin{equation}
Q_0=K_0 = N+ k_0\,,\quad
Q_-=K_- {[N]_q \over N} \,,\quad
Q_+= {[N + 2k_0 -1]_q \over (N+ 2k_0 -1)} K_+\,.
\label{su11-IIIB}
\end{equation}
These generators act on the ket as
\begin{equation}
Q_- |n> = [n]_q |n-1> \,,\qquad
Q_+ |n> = [n+ 2k_0]_q\, |n+1>\,.
\end{equation}
Conjugate relation between $Q_\pm$'s
requires the normalization
of the number eigenstate,
\begin{equation}
<n|n> = {[n]_q! [2k_0 -1]_q! \over [n+2k_0 -1]_q!}.
\end{equation}
In terms of the unit normalized ket $|n)$,
we reproduce the same form of
$su_q(1,1)$ coherent state and
resolution of unity as seen in the previous subsections, A and B.
What makes this realization different from the previous ones
is that it gives a natural $q$-deformation of the FB representation
of $su(1,1)$. By using $<\xi|n>= \xi^n$, we have
\begin{equation}
\hat Q_+ (\xi) = \xi [\xi {d \over d \xi} + 2 k_0 ]_q \,,\quad
\hat Q_-(\xi) = {d \over d_q \xi} \,,\quad
\hat Q_0 (\xi) = \xi {d \over d \xi} + k_0 \,.
\end{equation}
The $q$-derivative in $\hat Q_-$ is
defined in Eq. (\ref{qderivative}).
This implies that the oscillator realization is given as
\begin{equation}
(a_q)_-(\xi) = {d \over d_q \xi} \,,\quad
(a_q)_+(\xi) = \xi \,,
\label{axi}
\end{equation}
which satisfies the $q$-deformed oscillator algebra
of B type, Eq.~(\ref{Biedenharn}).
\subsection{FB realization with a-symmetric $q$-derivative}
We may consider a little modified version of Eq.~(\ref{su11-IIIB}),
\begin{equation}
Q_0=K_0 = N+ k_0\,,\quad
Q_-=K_- {[N]_q \over N}q^{N-1} \,,\quad
Q_+= q^{-(N-1)}{[N + 2k_0 -1]_q \over (N+ 2k_0 -1)} K_+\,.
\label{su11-IIIFB}
\end{equation}
Then the generators act on the ket as
\begin{equation}
Q_- |n> = \{n\}_q |n-1> \,,\qquad
Q_+ |n> = q^{-n} [n+ 2k_0]_q\, |n+1>\,.
\end{equation}
Conjugate relation between $Q_\pm$'s
requires the normalization
of the ket as,
\begin{equation}
<n|n> = q^{n(n+2k_0-2)}
{\{n\}_q! \{2k_0 -1\}_q! \over \{n+2k_0 -1\}_q!}
\end{equation}
One can easily check that in this Hilbert space,
the same form of $su_q(1,1)$ coherent state and
resolution of unity are reproduced
as in the previous sections
if we use the unit normalized ket $|n)$.
We have a similar
$q$-deformation of the FB representation
of $su(1,1)$ as in the previous section,
\begin{equation}
\hat Q_0 (\xi) = \xi {d \over d \xi} + k_0 \,,\quad
\hat Q_-(\xi) = {D \over D_q \xi} \,,\quad
\hat Q_+ (\xi) = \xi q^{-(2 \xi {d \over d \xi} + 2k_0 -1)}
\{\xi {d \over d \xi} + 2 k_0 \}_q \,.
\end{equation}
The derivative in $\hat Q_-$ is replaced by a new $q$-derivative
which is given by
\begin{equation}
{D \over D_q z} f(z) = {f(q^2z) - f(z) \over z(q^2- 1)}\,,
\end{equation}
This implies that the oscillator realization is given by
\begin{equation}
(b_q)_-(\xi) = {D \over D_q \xi} \,,\quad
(b_q)_+(\xi) = \xi \,,
\end{equation}
which satisfies the $q$-deformed oscillator algebra
of M type, Eq.~(\ref{Macfarlane}).
\def\arabic{section}.\arabic{equation}{\arabic{section}.\arabic{equation}}
\section{$su_q(2)$ and coherent state}
\setcounter{equation}{0}
$su_q(2)$ and its coherent state can be studied
in close analogy with
the previous section and therefore,
we will describe briefly about B oscillator
realization only.
$su(2)$ satisfies the algebra,
\begin{equation}
[ K_3, K_{\pm}]=\pm K_{\pm}, \quad
[ K_+, K_-]=2K_3,
\label{su2algebra}
\end{equation}
and Casimir operator is given as
$C=K_3(K_3+1)+K_-K_+.$ In addition,
\begin{equation}
K_- |n> = n |n-1>\,, \quad
K_+ |n> = (J-n) |n+1>\,.
\end{equation}
The Hilbert space is finite dimensional with dimension
$J +1 $ where
\begin{equation}
K_-\vert n=0>=0\,,\quad
K_+ \vert n=J> = 0\,,
\end{equation}
where $J$ is an integer.
Since $|n>$ is an eigenstate of $K_3$,
\begin{equation}
K_3 |n>= (n - {J \over 2})|n>\,,
\end{equation}
we have the Casimir constant,
$C= {J \over 2}({J \over 2} + 1)$.
$q$-deformed $su(2)$ algebra is given as \cite{skly}
\begin{equation}
[ Q_3, Q_{\pm}]=\pm Q_{\pm}\,,\quad
[Q_+, Q_-]=[2 Q_3]_{q^2}\,.
\end{equation}
We require conjugate relation $Q_-^{\dagger} = Q_+$,
$Q_3^{\dagger} = Q_3$
independent of the realization.
Repeating the same procedure as $su(1,1)$ case, we find
\begin{equation}
Q_3= K_3\,,\quad
Q_-= K_- F(K_3)\,,\quad
Q_+ = F(K_3) K_+\,.
\label{su2qcom}
\end{equation}
where
\begin{equation}
F(K_3)=
\sqrt{[{J \over 2} + K_3]_q [{J \over 2} +1 - K_3]_q
\over ({J \over 2} + K_3)({J \over 2} +1 - K_3)}
= \sqrt{[N]_q [J +1 - N]_q \over N(J +1 - N)}\,.
\end{equation}
\subsection{D type of B oscillator representation.}
\begin{equation}
Q_3= K_3 = {J \over 2} - N\,,\quad
Q_-= K_- \sqrt{[N]_q \over N}\,,\quad
Q_+ = \sqrt{[N]_q \over N} {[J+1-N]_q \over (J+1-N)} K_+\,.
\label{su2-D}
\end{equation}
These generators act on the ket as
\begin{equation}
Q_- |n> = \sqrt{n [n]_q} |n-1> \,,\qquad
Q_+ |n> = \sqrt{ [n+1]_q \over n+1} \, [J-n]_q\, |n+1>\,.
\end{equation}
We get an oscillator realization if
\begin{equation}
Q_0 = {J \over 2} - N\,,\quad
Q_- = (a_q)_- \,,\quad
Q_+ = [J - N+1 ]_q (a_q)_+ \,.
\end{equation}
$(a_q)_\pm$ is as defined in Eq.~(\ref{a-Bied}).
Introducing unit normalized ket,
$ |n) = \sqrt{J! \over n1 (J-n)!}\,|n>\,$,
we have the canonical operator relations of $su_q(2)$.
\begin{equation}
Q_- |n) = \sqrt{[n]_q [J+1-n ]_q} |n-1)\,,\quad
Q_+ |n) = \sqrt{[n+1]_q [J-n ]_q} |n+1)\,.
\label{su2Qrelation}
\end{equation}
$q$-deformed coherent state is given as
\begin{equation}
|z> = e_q^{\bar z Q_-} |J>
=\sum_{n=0}^\infty \bar z^n
\sqrt{[J]_q! \over [n]_q! [J-n]_q!}\,|n)\,.
\label{su2qcoherent}
\end{equation}
Resolution of unity is expressed as
\begin{equation}
I = \sum_{n=0}^{J}
|n)\,(n|
= \int d^2_q z \, H(z)\, |z>\,<z|\,,
\end{equation}
and the measure is given as
\begin{equation}
H(z)= {[J+1]_q \over \pi} {1 \over (1 + |z|^2)_q^{2+J}}\,.
\label{su2qmeasure}
\end{equation}
One can check that $I$ commutes with the $su_q(2)$ generators.
\subsection{HP
type of B oscillator representation.}
\begin{eqnarray}
&&Q_3= K_3 = {J \over 2} - N\,,
\nonumber \\
&&Q_-= K_- \sqrt{[N]_q [J+1-N]_q \over N}
= (a_q)_-\sqrt{[J+1-N]_q}\,,
\nonumber \\
&&Q_+ = \sqrt{[N]_q [J+1-N]_q \over N}
{1 \over (J+1-N)} K_+
= \sqrt{[J+1-N]_q} (a_q)_+ \,.
\label{su2-HP}
\end{eqnarray}
$(a_q)_\pm$ is defined in Eq.~(\ref{a-Bied}).
The ladder operators act on the ket as
\begin{equation}
Q_- |n> = \sqrt{n [n]_q [J +1 -n]_q} |n-1> \,,\qquad
Q_+ |n> = \sqrt{ [n+1]_q [J-n]_q \over n+1} \, |n+1>\,.
\end{equation}
Using the normalized ket,
$|n) = \sqrt{1\over n! }\,|n>\,$,
we have the canonical
ladder operator realization as in Eq.~(\ref{su2Qrelation})
and $q$-deformed coherent state is given as
\begin{equation}
|z> = e_q^{\bar z Q_-} |J>
=\sum_{n=0}^\infty \bar z^n
\sqrt{[J]_q! \over [n]_q! [J-n]_q!}\,|n)\,.
\end{equation}
Therefore, the measure $H(z)$
given in Eq.(\ref{su2qmeasure})
is used for the resolution of unity.
Because of the conjugate relation
between $(a_q)_+$ and $(a_q)_-$,
we can equally consider the $q$-coherent state
of finite Glauber coherent state.
However, the Hilbert space is finite dimensional, so
one has to modify the definition of the coherent
state from the $su(1,1)$ case,
Eq.~(\ref{newqcoherent});
\begin{equation}
|z>> = e^{\bar z (a_q)_-}_q |J)
= \sum_{n=0}^{\infty} \bar z^n
\sqrt{[J]_q! \over [n]_q! [J-n]_q!}\,\, |n)\,.
\end{equation}
It is interesting to note that the
oscillator coherent state
reproduces the same form of $su_q(2)$ coherent state
given in Eq.~(\ref{su2qcoherent}).
This is because the Hilbert space is finite dimensional
in contrast with $su_q(1,1)$ case.
\def\arabic{section}.\arabic{equation}{\arabic{section}.\arabic{equation}}
\section{conclusion}
\setcounter{equation}{0}
We have presented and compared various
type of oscillator algebra realizations of
$su_q(1,1)$ and $su_q(2)$ algebras, and their coherent states.
For $su_q(1,1)$, if we impose the conjugate condition
for the generators, the Perelomov $q$-coherent states has
a common measure in the resolution of unity independently
of the explicit forms of realization. Another type of
$q$-coherent state, the Glauber type is considered in the HP
realization since $a_-$ and $a_+$ are automatically conjugate
to each other. The explicit measure for this type of $q$-coherent
state depends on the oscillator realization such as
B or M type; the Liouville type measure
defined in Eq.~(\ref{su11qmeasure}) or
the Bargmann type in Eq.~(\ref{aq-measure}),
or the other Bargmann type in Eq.~(\ref{bq-measure}).
In addition, it is shown that
the explicit forms of the generators of $su_q(1,1)$ can be
modified such that $q$-anyonic oscillator and
various definition of $q$-derivatives can be
accommodated in the realizations.
$su_q(2)$ shares much of the same results with $su_q(1,1)$.
However, in HP realization, the finite Glauber $q$-coherent
state does not have the Bargmann measure, but has the
Liouville measure. The difference comes from the finiteness
of the dimension of the Hilbert space. Therefore, the measure
of coherent state in $su_q(2)$ is distinguishable from that
of the oscillator coherent state on a plane.
We also presented two different types of FB realization which
provide two different definitions of $q$-derivative and
$q$-integration such that we can describe their $q$-deformed
oscillators algebra in a natural and simple fashion.
We conclude with a couple of remarks.
The D representations can be extended
to the $SU(N)$ case \cite{oh955}.
The HP version in the
$SU(N)$ case can also be constructed \cite{rand}.
In addition, its $q$-deformation was considered in \cite{sunfu}.
It would be interesting to go through the same analysis in this
higher case, especially in connection with FB realization.
$q$-deformed FB representation will be useful for evaluating
the $q$-deformed version of the path integral \cite{baul}.
In our approach, $q$-deformation of FB representation
is understood in terms of the oscillator representation
and the role of the $q$-derivatives are illustrated.
However, $q$-integration is performed essentially
for one dimensional direction, radial part.
Angular part is treated as an ordinary integration
Eq. (\ref{integ}). So our
resolution of unity cannot be used directly in evaluating
the $q$-deformed version of path integral at this stage.
To overcome the shortcomings, one has to fully develop $q$-deformed
higher dimensional integral in terms of non-commuting numbers.
We expect that this direction of research should
accommodate $q$-calculus on plane and sphere \cite{wess}.
\acknowledgments
We like to thank Professors
J. Wess, J. Klauder and
V. Manko and Dr. K. H. Cho for useful conversations.
This work is supported by the KOSEF
through the CTP at SNU and the project number
(94-1400-04-01-3, 96-0702-04-01-3),
and by the Ministry of Education through the
RIBS (BSRI/96-1419,96-2434).
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proofpile-arXiv_065-603
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
Study of the Coulomb blockade and charging effects in the transport
properties of semiconductor systems is peculiarly suitable to
investigation through self-consistent electronic structure techniques.
While the orthodox theory \cite{Lik}, in parameterizing the energy of
the system in terms of capacitances, is strongly applicable to metal
systems, the much larger ratio of Fermi wavelength to system size,
$\lambda_F / L$, in mesoscopic semiconductor devices, requires
investigation of the interplay of quantum mechanics and charging.
In the first step beyond the orthodox theory, the ``constant
interaction'' model of the Coulomb blockade supplemented the
capacitance parameters, which were retained to characterize the gross
electrostatic contributions to the energy, with non-interacting quantum
levels of the dots and leads of the mesoscopic device \cite{Ruskies,Been}.
This theory was successful in explaining some of the fundamental features,
specifically the periodicity, of Coulomb oscillations in the conductance
of a source-dot-drain-gate system with varying gate voltage. Other
effects, however, such as variations in oscillation amplitudes, were not
explained.
In this paper we employ density functional (DF) theory to
compute the self-consistently changing
effective single particle levels of a lateral $GaAs-AlGaAs$
quantum dot, as a function of gate voltages,
temperature $T$, and dot electron number $N$ \cite{RComm}.
We also compute the total system free energy from the results of
the self-consistent calculation. We are then able to calculate
the device conductance in the linear bias regime
without any adjustable parameters. Here we consider
only weak ($\stackrel{\sim}{<} 0.1 \; T$) magnetic fields
in order to study the effects of breaking time-reversal
symmetry. We will present results for
the edge state regime in a subsequent publication
\cite{lp2}.
We include donor layer disorder in the calculation and present
results for the statistics of level spacings and partial
level widths due to tunneling to the leads. Recently we have employed
Monte-Carlo variable range hopping simulations to consider
the effect of Coulomb regulated ordering
of ions in the donor layer on the mode characteristics of split-gate
quantum {\it wires} \cite{BR2}. The results of those simulations
are here applied to quantum dot electronic structure.
A major innovation in this calculation is our method for
determining the two dimensional electron gas (2DEG) charge
density.
At each iteration of the self-consistent calculation, at
each point in the $x-y$ plane we determine the subbands $\epsilon_n (x,y)$ and
wave functions $\xi^{xy}_n (z)$ in the $z$ (growth) direction.
The full three dimensional density is then
determined by a solution of the multi-component 2D Schr\"{o}dinger
equation and/or 2D Thomas-Fermi approximation.
Among the many approximation in the calculation are the following.
We use the local density approximation (LDA) for exchange-correlation (XC),
specifically the parameterized form of Stern and Das Sarma \cite{SternDas}.
While the LDA is difficult to justify in small
($N \sim 50-100$) quantum dots it is empirically known to give
good results in atomic and molecular systems where the density
is also changing appreciably on the scale of the Fermi wavelength
\cite{Slater}.
In reducing the 3D Schr\"{o}dinger equation to a multi-component 2D
equation we cutoff
the expansion in subbands, often taking only the
lowest subband into account.
We also cutoff the wavefunctions by placing another artificial $AlGaAs$
interface at a certain depth (typically $200 \; \stackrel{0}{A}$) away from
the first interface, thereby ensuring the existence of subbands at all points
in the $x-y$ plane. Generally the subband
energy of this bare square well is much smaller than the triangular
binding to the interface in all but those regions which are very nearly
depleted.
The dot electron states in the zero magnetic field regime
are simply treated as spin degenerate. For $B \ne 0$ an
unrenormalized Land\'{e} g-factor of $-0.44$ is used.
We employ the effective mass
approximation uncritically and ignore the effective mass difference
between $GaAs$ and $AlGaAs$ ($m^* = 0.067 \; m_0$). Similarly we take
the background dielectric constant to be that of pure $GaAs$ ($\kappa =
12.5$) thereby ignoring image effects (in the $AlGaAs$).
We ignore interface grading and
treat the interface as a sharp potential step. These effects have been
treated in other calculations of self-consistent electronic structure
for $GaAs-AlGaAs$ devices \cite{SternDas} and have generally been found
to be small.
We mostly employ effective atomic units wherein
$1 \; Ry^* = m^* e^4/2 \hbar^2 \kappa^2 \approx 5.8 \; meV$ and
$1 \; a_B^* = \hbar^2 \kappa/m^* e^2 \approx 100 \; \stackrel{0}{A}$.
The structure of the paper is as follows.
In section II we first discuss the calculation of the electronic structure,
focusing on those features which are new to our method. Further subsections
then consider the treatment of discrete
ion charge and disorder, calculation of the total
dot free energy from the self-consistent electronic structure results,
calculation of the source-dot-drain conductance in the linear
regime and calculation of the dot capacitance matrix.
Section III provides new results which are further subdivided
into basic electrostatic properties, properties of the
effective single electron spectra, statistics of level
spacings and widths and conductance in the Coulomb oscillation
regime. Section IV summarizes the principal conclusions
which we derive from the calculations.
\section{Calculations}
\subsection{Quantum dot self-consistent electronic structure}
We consider a
lateral quantum dot patterned on a 2DEG heterojunction via metallic surface
gates (Fig. \ref{fig1}). At a semiclassical level, other gate
geometries, such as a simple point contact or a multiple dot system, can be
treated with the same method \cite{BR2,G-res}. However, a
full 3D solution of Schr\"{o}dinger's equation, even employing our subband
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig1.eps}
\vspace*{3mm}
\caption{Schematic of device used in calculation. The
$z$-subband structure throughout the plane are
calculated at each iteration of the self-consistency loop. Most
results presented with gate variation assume that both
the upper and lower pins of the relevant gate are simultaneously
varied. \label{fig1} }
\end{minipage}
\end{figure}
expansion procedure for the $z$ direction,
is only tractable in the current method when a region with a small
number of electrons ($N \le 100$) is quantum mechanically isolated, such as
in a quantum dot.
\subsubsection{Poisson equation and Newton's method}
In principal, a self-consistent solution is obtained by iterating the
solution of Poisson's equation and {\it some} method for calculating the
charge density (see following sections II.A.2 and II.A.3). In practise, we follow
Kumar {\it et al} \cite{Kumar}
and use an ${\cal N}$-dimensional Newton's method for finding the
zeroes of the functional
$\vec{F}(\vec{\phi}) \equiv {\bf \Delta} \cdot \vec{\phi} + \vec{\rho}(\vec{\phi})
+ \vec{q}$; where the potential, $\phi_i$, and density,
$\rho_i$, on the ${\cal N}$ discrete lattice
sites (${\cal N} \sim 100,000$) are written as vectors, $\vec{\phi}$
and $\vec{\rho}$.
The vector $\vec{q}$ represents the inhomogeneous contribution from
any Dirichlet boundary conditions, ${\bf \Delta}$ is the Laplacian
(note that here it is a matrix, not a differential operator), modified
for boundary conditions. Innovations for treating the Jacobian
$\partial \rho_i / \partial \phi_j$
beyond 3D Thomas-Fermi,
and for rapidly evaluating the mixing parameter~$t$ (see Ref. \cite{Kumar})
are discussed below.
The Poisson grid spans a rectangular solid and hence the boundary conditions
on six surfaces must be supplied. Wide regions of the source and drain
must be included in order to apply Neumann boundary conditions
on these ($x = $ constant) interfaces, so a non-uniform mesh is essential.
It is also possible to apply Dirichlet boundary conditions on these interfaces
using the ungated wafer (one dimensional) potential profile calculated
off-line \cite{AFS}.
In this case, failure to include sufficiently wide lead regions shows
up as induced charge on these surfaces (non-vanishing electric field).
To keep the total induced charge on all surfaces below $0.5$ electron,
lead regions of $\sim 5 \; \mu m$
are necessary, assuming a surface gate to 2DEG
distance (i.e. $AlGaAs$ thickness) of $1000 \stackrel{0}{A}$. In other words
we need an aspect ratio of $50:1$.
We note that we ignore background compensation and merely
assume that the Fermi level is pinned at some fixed depth
(``$z_{\infty}$'' $\sim 2.5 \; \mu m$)
into the $GaAs$ at the donor level.
The donor energy for $GaAs$ is taken as $1 \; Ry^*$ below
the conduction band. In the source and drain regions,
the potential of the 2DEG Fermi surface is fixed by the desired
(input) lead voltage.
We apply Neumann boundary conditions at the $y = $ constant surfaces.
The $z=0$ surface of
the device has Dirichlet conditions on the gated regions (voltage equal to the
relevant desired gate voltage) and Neumann conditions,
$\partial \phi / \partial n = 0$,
elsewhere.
This is equivalent to the ``frozen surface'' approximation
of \cite{JHD2}, further assuming a high dielectric constant for
the semiconductor relative to air. Further discussion of this
semiconductor-air boundary condition can be found in Ref. \cite{JHD2}.
\subsubsection{Charge density, quasi-2D treatment}
The charge density {\it within} the Poisson grid (i.e. not surface
gate charge) includes the 2DEG electrons and the ions in the
donor layer.
The treatment of discreteness, order and disorder in the donor ionic
charge $\vec{\rho}_{ion}$ has been discussed in Ref. \cite{BR2}
in regards to quantum wire electronic structure.
Some further relevant remarks are made below in section II.B.
As noted above, we take advantage of the quasi-2D nature of the
electrons at the $GaAs-AlGaAs$ interface to simplify the calculation for
their contribution to
the total charge. Given $\vec{\phi}$, we begin by solving Schr\"{o}dinger's
Eq. in the $z$-direction {\it at every point} in the $x-y$ plane,
\begin{equation}
[-\frac{\partial^2}{\partial z^2} + V_B (z) + e \phi(x,y,z)] \xi^{xy}_n (z)
= \epsilon_n (x,y) \xi^{xy}_n (z) \label{eq:eqz}
\end{equation}
where $V_B(z)$ is the potential due to the conduction band offset between
$GaAs$ and $Al_x Ga_{1-x} As$. We generally employ fast Fourier
transform with $16$ or $32$ subbands.
In order that there be a discrete spectrum at each point in the
$x-y$ plane, it is convenient to take $V_B(z)$ as a {\it square well} potential
(Fig. \ref{fig1}). That is, we effectively cutoff the wave
function with a second barrier,
typically $200 \stackrel{0}{A}$ from the primary interface. In undepleted
regions the potential is still basically triangular and only the tail of
the wave function is affected. However, near the border between depleted and
undepleted regions the artificial second barrier will introduce some error
into the electron density. This is because
as a depletion region is approached, the binding {\it electric
field} at the 2DEG interface (slope of the triangular potential)
reduces, in addition to the interface potential itself rising.
Consequently, all subbands become degenerate and {\it near the edge
electrons are three dimensional} \cite{McEuenrecent}. We have checked that this
departure from interface confinement, and in general in-plane
gradients of $\xi^{x,y}_n (z)$
contribute negligibly to quantum dot level energies.
However, theoretical descriptions of 2DEG edges commonly assume perfect
confinement of electrons in a plane.
In particular the description of edge excitations in the quantum Hall
effect regime in terms of a chiral Luttinger liquid
\cite{Wen} may be complicated in real samples by the emergence
of this vanishing energy scale and collective modes related to it.
Assuming only a single $z$-subband now and dropping the index $n$,
we determine the
charge distribution in the $x-y$ plane from the effective potential
$\epsilon (x,y)$,
employing a 2D Thomas-Fermi approximation for the charge in the leads and
solving a 2D Schr\"{o}dinger equation in the dot. In order that the dot
states
be well defined, the QPC saddle points must be classically inaccessible.
(If this
is not the case it is still possible to use a Thomas-Fermi approximation
throughout the
plane for the charge density \cite{BR2,G-res}).
In the dot, the density is determined from the eigenstates by
filling states according to a Fermi distribution
either to a prescribed ``quasi-Fermi energy'' of the dot, or to a fixed number
of electrons. It has been pointed out that a Fermi distribution for the level
occupancies in the dot is an inaccurate approximation to the correct grand
canonical ensemble distribution \cite{Been}. Nonetheless, for small dots
($N \stackrel{<}{\sim} 15$)
Jovanovic {\it et al.} \cite{Jovanovic} have shown that, regarding the
filling factor, the discrepancy
between a Fermi function evaluation and that of the full grand canonical
ensemble
is $\sim 5\%$ at half filling and significantly smaller away from
the Fermi surface. As $N$ increases the discrepancy should become smaller.
\subsubsection{Solution of Schr\"{o}dinger's equation in the dot}
To solve the effective 2D Schr\"{o}dinger's equation in the dot,
\begin{equation}
(-{\bf \nabla}^2 + \epsilon({\bf x}) ) f ({\bf x}) = E f ({\bf x}) \label{eq:eqE}
\end{equation}
we set the 2D potentials throughout the {\it leads} to their values at the
saddle points,
thereby ensuring that
the wave functions decay uniformly into the leads. Thus the energy of the higher
lying states will be shifted upward slightly.
In seeking a basis in which to expand the solution of Eq. \ref{eq:eqE} we
must consider the approximate shape of the potential. The quantum dots which we
model here
are lithographically approximately square in shape. However
the potential at the 2DEG
level and also the effective 2-D potential $\epsilon(r,\theta)$, (now in
polar coordinates) are to lowest
order azimuthally symmetric. The {\it radial} dependence of the potential is
weakly parabolic across the center.
Near the perimeter higher order terms become important
(cf. figure \ref{fig3}b and Eq. \ref{eq:phi}).
As the choice of a good basis is not completely clear, we have tried two
different sets of functions: Bessel functions and the so-called Darwin-Fock (DF)
states \cite{Darwin}.
The details of the solution for the eigenfunctions and eigenvalues differ
significantly whether we use the Bessel functions or the DF states.
The Bessel function case is largely numerical whereas the DF functions together
with polynomial fitting of the azimuthally symmetric part of
the radial potential allow a considerable amount
of the work to be done analytically. Further, neither of the two bases comes
particularly close to fitting the somewhat eccentric shape of the actual dot
potential. It is therefore gratifying that comparing the eigenvalues determined
from the two bases when reasonable cutoffs are used, we find for up to the
$50^{th}$ eigenenergy agreement to three significant
figures, or to within roughly $5 \; micro \; eV$.
\subsubsection{Summary and efficiency}
To summarize the calculation, we begin by choosing the device dimensions
such as the gate pattern, the ionized donor charge density and its
location relative to the 2DEG, the aluminum concentration for the height
of the barrier, and the thickness of the $AlGaAs$ layer. We construct
non-uniform grids in $x$, $y$ and $z$ that best fit the
device within a total of about $10^5$ points.
Gate voltages, temperature, source-drain voltages, and
either the electron number $N$ or the quasi-Fermi energy of the dot
are inputs. The iteration scheme begins
with a guess of $\vec{\phi}^{(0)}$. The
1-D Schr\"{o}dinger equation is solved at each point in the $x-y$ plane and
an effective 2-D potential $\epsilon(x,y)$ for one or
at most two subbands is thereby determined.
Taking $|\xi^{xy}_n (z)|^2$ for the $z$-dependence of the charge density,
we compute the 2D dependence in the leads using
a 2D Thomas-Fermi approximation and in the dot by solving Schr\"{o}dinger's equation
and filling the computed states according to a Fermi distribution. We
compute $\vec{F} (\vec{\phi}^{(0)})$,
which is a measure of how far we are from self-consistency, and solve for
$\delta \vec{\phi}$, the potential increment, using a mixing parameter $t$.
This gives the next estimate for the potential $\vec{\phi}^{(1)}$. The
procedure is iterated and convergence is
gauged by the norm of $F$.
In practise there are many tricks which one uses to hasten (or
even obtain !) convergence. First, we use a scheme
developed by Bank and Rose \cite{Bank,Kumar} to search for an
optimal mixing parameter $t$.
Repeated calculation of Schr\"{o}dinger's equation, which is very
costly, is in principle required in the search for $t$.
Far from convergence the Thomas-Fermi approximation can be used
in the dot as well as the leads. Nearer to convergence we find that
diagonalizing $t \; \delta \vec{\phi}$ in a basis of
about ten states near the Fermi surface, treating the charge in the other
filled states as inert, is highly efficient.
Periodically the full solution of Schr\"{o}dinger's equation
is employed to update the wave functions.
The wave function information is also used to make a better estimate
of $\partial \rho_i / \partial \phi_i$. The 3D Thomas-Fermi method
for estimating this quantity does not account for the fact that
the change in density at a given grid point will be most strongly
influenced by the changes in the occupancies of the partially
filled states at the Fermi surface. Thus use of these wave functions
greatly improves the speed of the calculation.
\subsection{Disorder}
Evidence of Coulombic {\it ordering}
of the donor charge in a modulation doping layer adjacent to a 2DEG
has recently accumulated \cite{Buks}.
When the fraction ${\cal F}$ of ionized donors among all donors is less
than unity, redistribution of the ionized sites through hopping can lead
to ordering of the donor layer charge \cite{Efros,BR2}.
In this paper we consider the effects of donor charge distribution on
the statistical properties of quantum dot level spectra, in particular the
unfolded level spacings, and on the connection coefficients to the
leads $\Gamma_p$ of the individual states (see below). These dot
properties are calculated with ensembles of donor charge which
range from completely random (identical to ${\cal F}=1$, no ion
re-ordering possible) to highly ordered (${\cal F} \sim 1/10$).
For a discussion of the glass-like properties of the donor layer
and the Monte-Carlo variable range hopping calculation which
is used to generate ordered ion ensembles, see Refs. \cite{BR2} and
\cite{ISQM2}.
Note that hopping is assumed to take place at temperatures ($\sim 160 \, K$)
much higher than the sub-liquid Helium temperatures at which the dot electronic
structure is calculated. Thus the ionic charge distributions
generated in the Monte-Carlo calculation are, for
the purposes of the 2DEG electronic structure calculation,
considered fixed space charges which are specifically
not treated as being in thermal equilibrium
with the 2DEG.
The region where the donor charge can be taken as discrete is limited by
grid spacing and hence computation time.
In the wide lead regions and wide region lateral to the
dot the donor charge
is always treated as ``jellium.'' Also,
to serve as a baseline, we calculate the
dot structure with jellium across the dot region as well.
We introduce the term ``quiet dot''
to denote this case.
\subsection{Free energy}
To calculate the total interacting free energy we
begin from the
semi-classical expression
\begin{eqnarray}
F(&\{n_p\},Q_i,V_i) = \sum_p n_p \varepsilon_p^0 +
\frac{1}{2} \sum_i^M Q_i V_i \nonumber \\
& - \sum_{i \ne dot} \int dt \; V_i (t) I_i (t) \label{eq:cl}
\end{eqnarray}
where $n_p$ are the occupancies of non-interacting dot energy
levels $\varepsilon_p^0$;
$Q_i$ and $V_i$ are the charges and voltages of the $M$ distinct
``elements'' into which we divide the system: dot, leads and gates.
$I_i$ are the currents supplied by power supplies to the elements.
The {\it self-consistent} energy levels for the electrons
in the dot are $\varepsilon_p = < \psi_p \mid - \nabla^2 + V_B (z) +e \phi ({\bf r})
\mid \psi_p >$. A sum over these levels double counts
the electron-electron interaction. Thus, for the terms in Eq. \ref{eq:cl}
relating to the dot, we make the replacement:
\begin{eqnarray}
& \sum_p n_p \varepsilon_p^0 + \frac{1}{2} Q_{dot} V_{dot} \rightarrow
\sum_p n_p \varepsilon_p \nonumber \\
& - \frac{1}{2} \int d{\bf r} \rho_{dot}({\bf r})
\phi ({\bf r}) + \frac{1}{2} \int d{\bf r} \rho_{ion}({\bf r}) \phi ({\bf r})
\end{eqnarray}
where $\rho_{dot}({\bf r})$ refers only to the charge in the dot
states and $\rho_{ion}({\bf r})$ refers to all the charge in the donor layer.
We have demonstrated \cite{BR1,MCO} that
previous investigations \cite{Been,vanH} had failed to correctly include the
work from the power supplies, particularly to the source and drain leads,
in the energy balance for tunneling between leads and dot in the
Coulomb blockade regime. Here, we assume a low impedance environment
which allows us to make the replacement:
\begin{equation}
\frac{1}{2} \sum_{i \ne dot} Q_i V_i -
\sum_{i \ne dot} \int dt \; V_i (t) I_i (t) \rightarrow
- \frac{1}{2} \sum_{i \ne dot} Q_i V_i.
\end{equation}
The charges on the gates are determined from the gradient of the potential
at the various surface regions, the voltages being given.
Including only the classical electrostatic energy of the leads,
the total free energy is \cite{RComm}:
\begin{eqnarray}
& F(\{n_p\},N,V_i) = \sum_{p} n_p
\varepsilon_{p} - \frac{1}{2} \int d{\bf r}
\rho_{dot}({\bf r}) \phi ({\bf r}) \nonumber \\
& + \frac{1}{2} \int d{\bf r} \rho_{ion}({\bf r}) \phi ({\bf r})
- \frac{1}{2} \sum_{i \; \epsilon \; leads} \; \int d{\bf r}
\rho_i ({\bf r}) \phi ({\bf r}) \nonumber \\
& - \frac{1}{2} \sum_{i \; \epsilon \; gates} Q_i V_i \label{eq:free}
\end{eqnarray}
where the energy levels, density, potential and induced charges are
implicitly functions of $N$ and the
applied gate voltages $V_i$. Note that the occupation number dependence of
these terms is ignored. In the $T=0$ limit the electrons occupy the
lowest $N$ states of the dot, and the free energy is denoted
$F_0 (N,V_i)$.
\subsection{Conductance}
The master
equation formula for the linear source-drain conductance though the
dot, derived by several authors \cite{Been,Ruskies,Meir}
for the case of a fixed dot spectrum, is modified to the self-consistently
determined free energy case as follows \cite{RComm}:
\begin{eqnarray}
& G(V_g) = \displaystyle{\frac{e^2}{k_B T} \sum_{\{ n_{i} \} }}
P_{eq}( \{ n_{i} \} ) \sum_{p} \delta_{n_{p},0}
\displaystyle{\frac{\Gamma_p^s \Gamma_p^d}{\Gamma_p^s +
\Gamma_p^d}} \nonumber \\
& \times f(F(\{n_i+p\},N+1,V_g)
- F(\{n_i\},N,V_g) - \mu ) \label{eq:cond}
\end{eqnarray}
where the first sum is over dot level occupation configurations and the second
is over dot levels. The equilibrium probability distribution
$P_{eq} ( \{ n_i \} )$ is
given by the Gibbs distribution,
\begin{equation}
P_{eq} ( \{ n_i \} ) = \frac{1}{Z} exp[- \beta (F(\{n_i\},N,V_g) - \mu)]
\end{equation}
and the partition function is
\begin{equation}
Z \equiv \sum_{\{ n_{i} \} } exp[- \beta (F( \{ n_i \} ,N,V_g) - \mu)]
\end{equation}
note that the sum on occupation configurations, $\{ n_{i} \}$, includes
implicitly a sum on $N$.
In Eq. \ref{eq:cond} $f$ is the Fermi function, $\mu$ is the
electrochemical potential of the source and drain and $\Gamma_p^{s(d)}$
are the elastic couplings of level $p$ to source (drain).
The notation $\{ n_i + p \}$ denotes
the set of occupancies $\{ n_i \}$ with the $p^{th}$ level, previously
empty by assumption, filled. In Eq. \ref{eq:cond} it is assumed that
only a single gate voltage, $V_g$
(the ``plunger gate'', cf. Fig. \ref{fig1}), is varied.
\subsection{Tunneling coefficients}
The elastic couplings in Eq. \ref{eq:cond} are calculated from the
self-consistent wave functions \cite{Bardeen}:
\begin{equation}
\hbar \Gamma_{np} = 4 \kappa^2 W_n^2 (a,b) \; \left| \int dy \;
f_p (x_b,y) \chi^*_n (x_b,y) \right|^2 \label{eq:tun}
\end{equation}
where $f_p (x_b,y)$ is the two dimensional part of the $p^{th}$ wave function
evaluated at the midpoint of the barrier, $x_b$, and $\chi^*_n (x_b,y)$
is the $n^{th}$ channel wavefunction decaying into the barrier from the
leads, $W_n(a,b)$ is the barrier penetration factor between
the classical turning point in the lead and the point $x_b$,
for channel $n$ computed in the WKB approximation, and $\kappa$ is the
wave vector at the matching point. Though the channels are 1D we use
the two dimensional density of states characteristic of the wide
2DEG region \cite{Matveev}.
\subsection{Capacitance}
Quantum dot system electrostatic energies are commonly estimated on the
basis of a capacitance model \cite{various}. When the self-consistent level energies
and potential are known the total free energy can be computed without
reference to capacitances. However, the widespread use of this model
and the ease with which capacitances can be calculated from our
self-consistent results (see below) encourages
a discussion.
For a collection of $N$
metal elements with charges $Q_i$ and voltages $V_j$
the capacitance matrix, defined by \cite{BR1,meandYasu}
$Q_i = \sum_{j=1}^{N} C_{ij} V_j$, can be written in
terms of the Green's function $G_D ({\bf x,x^{\prime}})$
for Laplace's equation
satisfying Dirichlet boundary conditions
on the element surfaces:
\begin{equation}
C_{ij} = \frac{1}{4 \pi^2} \int d \Omega_i \int d \Omega_j
\hat{n}_j \cdot \vec{\nabla}_x (\hat{n}_i \cdot \vec{\nabla}_{x^\prime}
G_D ({\bf x,x^{\prime}}))
\end{equation}
where the integrals are over element surfaces with $\hat{n}_j$
the outward directed normal.
In a system with an element of size $L$ not much greater than the screening
length $\lambda_s$, the voltage of the component, and hence the
capacitance, is not well defined \cite{meandYasu,Buttikercap}.
In this case, as discussed in reference \cite{meandYasu},
the capacitance can no longer be written in terms of the solution
of Poisson's equation alone, but must take account of the
full self-consistent determination of the $i^{th}$
charge distribution $\rho_i({\bf x})$
from the $j^{th}$ potential $\phi_j({\bf x})$ $\forall i,j$.
In general the capacitance will
then become a kernel in an integral relation. A relationship
of this kind has recently been derived in terms of the
Linhard screening function by B\"{u}ttiker \cite{Buttikercap}.
To compute the dot self-capacitance from the calculated self-consistent
electronic structure we have three separate procedures.
In all three cases we
vary the Fermi energy of the dot by some small amount to change
the net charge in the dot. This requires that the QPCs be closed.
For the first method
the total charge variation of the dot is divided by the
change in the electrostatic potential minimum of
the dot. This is taken as the dot self-capacitance $C_{dd}$.
A second procedure for the dot self-capacitance is to
divide the change in the dot charge simply
by the fixed, imposed change of the Fermi energy. This result is
denoted $C_{dd}^\prime$. Since the change in the potential
minimum of the dot is not always equal to the change of
the Fermi energy these results are not identical.
Finally, we can fit the computed free energy $F(N,V_g)$ to a
parabola in $N$ at each $V_g$. If the quadratic term is
$\alpha N^2$ then the final form for the self-capacitance is
$C_{dd}^{\prime \prime} = 1/(2 \alpha)$ (primes are {\it not}
derivatives here). This form, which
also serves as
a consistency check on our functional for the energy,
is generally quite close to the first form and
we present no results for it.
For the capacitances between dot and gates or leads, the extra
dot charge (produced by increasing the Fermi energy in the dot)
is screened in the gates and the leads so that the net
charge inside the system (including that on the gated boundaries)
remains zero. The fraction of the charge screened in a
particular element gives that element's capacitance
to the dot as a fraction of $C_{dd}$.
\section{Results}
We consider only a small subspace of the huge available
parameter space. For the results presented here we have fixed
the nominal 2DEG density to $1.4 \times 10^{11} \; cm^{-2}$ and
the aluminum concentration of the barrier to $0.3$. The lithographic
gate pattern is shown in figure \ref{fig1}, as is the growth profile
(including our artificial second barrier). Some results are
presented with a variation of the total thickness $t$ of the
AlGaAs (Fig. \ref{fig1}).
To interpret the results we note the following considerations.
Hohenberg-Kohn-Sham theory provides only that the ground
state energy of an interacting electron system can be
written as a functional of the density \cite{HKS1,HKS2}.
The single particle eigenvalues $\varepsilon_p$ have, strictly
speaking, no
physical meaning. However, as pointed out by Slater
\cite{Slater}, the usefulness of DF theory depends to
some extent on being able to interpret the energies
and wave functions as some kind of single particle
spectrum. In the Coulomb blockade regime it is particularly
important to be clear what that interpretation, and its
limitations, are.
A distinction is commonly made between the addition
spectrum and the excitation spectrum for quantum dots
\cite{McEuen,Ashoori}. Differences between our effective
single particle eigenvalues represent an approximation
to the excitation spectrum. As a specific
example, in the
absence of depolarization and excitonic effects
the first single
particle excitation from the $N$-electron ground state
with gate voltages $V_i$
is $\varepsilon_{N+1}(N,V_i)-\varepsilon_{N}(N,V_i)$.
The addition spectrum, on the other hand,
depends on the energy difference between the
ground states of the dot {\it interacting
with its environment} at two different $N$.
Thus, in our formalism, the addition spectrum
is given by differences in $F(\{n_p\},N,V_i)$ at
neighboring $N$, possibly further modulated by excitations,
i.e. differences
in the occupation numbers $\{ n_p \}$.
In contrast to experiment, the electronic structure can
be determined for arbitrary $N$ and $V_i$ (so long as
the dot is closed). This includes both non-integer
$N$ as well as values which are far from equilibrium
(differing chemical potential) with the leads. The
``resonance curve'' \cite{RComm} is given by the $N$
which minimizes $F_0 (N,V_g)$ at each $V_g$ (gates other than
the plunger gate are assumed fixed). This occurs when
the chemical potential of the
dot equals those of the leads (which are taken as equal
to one another and represent the energy zero)
and gives the most probable electron number. Results
presented below as a function of varying gate voltage,
particularly the spectra in Figs. \ref{fig10} and \ref{fig14},
are assumed to be along the resonance curve.
\subsection{Electrostatics}
Figure \ref{fig3}a shows an example of a
potential profile along with a corresponding
density plot for a quiet dot containing $62$ electrons. The basic
potential/density configuration, as well as the
capacitances are highly robust. These data are computed completely in the
2D Thomas-Fermi approximation, single $z$-subband,
at $T = 0.1 \; K$. Solution of Schr\"{o}dinger's
equation or variation of $T$ result in only subtle changes. The depletion region
spreading is roughly $100 \; nm$. Figure \ref{fig3}b shows a set of potential
and density profiles along the y-direction (transverse to the current direction)
in steps of $3.3 \; a_B^*$ in $x$, from the QPC saddle point to the dot center.
Note that the density at the dot center is only about $65 \%$ of the ungated 2DEG
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig3ab.eps}
\vspace*{3mm}
\epsfxsize=8cm
\epsfbox{LPfig3bb.eps}
\vspace*{3mm}
\caption{(a) Contour plot for density and potential, quiet dot, TF.
Isolines in
potential spaced at $\sim 0.1 \; Ry^*$ up to $0.5 \; Ry^*$
above Fermi level, after which much more widely. Density isoline
spacing $\sim 0.01 \; a_B^{* \, -2}$, maximum density
$\sim 0.1 \; a_B^{* \, -2}$. Ripples near QPCs are finite grid size
effect; plotted $x-y$ mesh shows every other grid line.
(b) Transverse (y-direction) half-profiles of density and potential
corresponding to (a), taken at $3.3 \; a_B^*$ intervals
from dot center. Uppermost potential trace, entirely above Fermi surface,
is in QPC ($x \approx 54 \; a_B^*$ in Fig. 2a) where density is zero.
Density is scaled to nominal
2DEG value $0.14 \; a_B^{* \, -2} \approx 1.4 \times 10^{11} \; cm^{-2}$.
\label{fig3} }
\end{minipage}
\end{figure}
density. Correspondingly the potential at the center is above the floor of the
ungated 2DEG ($\sim -0.9 \; Ry^*$).
We discuss a simple model for the potential shape of a
circular quantum dot below (Sec. III.B.1). Here we note only
that the radial potential can be regarded as parabolic to lowest
order with quartic and higher order corrections whose influence
increases near the perimeter. In Thomas-Fermi studies on larger
dots \cite{MCO,G-res} with a comparable aspect ratio we find that
the potential and density achieve only $90 \; \%$ of their ungated 2DEG value
nearly $200 \; nm$ from the gate. Regarding classical billiard calculations for
gated structures therefore \cite{chaos1,Been2,Bird,Ferry} even in the
absence of impurities it is difficult to see how the ``classical''
Hamiltonian at the 2DEG level can
be even approximately integrable unless the lithographic
gate pattern is azimuthally symmetric \cite{square}.
The importance of the remote ionized impurity distribution is demonstrated in
figure \ref{fig4} which shows a quantum dot with randomly placed ionized
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig4.eps}
\vspace*{3mm}
\caption{Contour plots of dot showing ion placement for disordered case
(left) and ordered (${\cal F}=1/5$)
ion distribution, TF. Isolines at $0.08 \; Ry^*$ up to
Fermi surface, wider thereafter. Gate voltages and locations identical
in the two cases. Note particularly position of right QPC determined
by ions in disordered case. \label{fig4} }
\end{minipage}
\end{figure}
donors on
the left and with ions which have been allowed to reach quasi-equilibrium via
variable range hopping, on the right. In both cases the total ion number in
the area of the dot is fixed. The example shown here for
the ordered case assumes, in the variable range hopping calculation,
one ion for every five donors (${\cal F}=1/5$). As in Ref. \cite{BR2}
we have, for simplicity, ignored the negative $U$ model for the donor
impurities (DX centers), which is still controversial
\cite{Buks,Mooney,Yamaguchi}.
If the negative $U$ model, at some barrier aluminum concentration,
is correct, the most ordered ion distributions will occur for ${\cal F}=1/2$,
as opposed to the neutral DX picture employed here, where ordering increases
monotonically as ${\cal F}$ decreases \cite{Heiblum_private}.
For these assumptions figure \ref{fig5} indicates that ionic ordering
substantially reduces the potential fluctuations relative to the
completely disordered case, even for relatively large ${\cal F}$.
Here, using ensembles
of dots with varying ${\cal F}$ we compare the effective 2D potential
with a quiet dot (jellium donor layer)
at the same gate voltages and same dot electron
number. The distribution of the potential deviation is computed as:
\begin{equation}
P(\Delta V) = \frac{1}{SN^2} \sum_s \sum_{i,j} \delta(\Delta V -
[V_{\cal F} (x_i,y_j) - V_{qd} (x_i,y_j)])
\end{equation}
where $s$ labels samples (different ion distributions),
typically up to $S=10$, $N$ is the total number of $x$ or $y$ grid points
in the dot ($\sim 50$), and ``qd'' stands for quiet dot.
The distributions for all ${\cal F}$ are asymmetric (Fig. \ref{fig5}).
Although the means are indistinguishably close to zero, the probability for
large potential hills resulting from disorder is greater than for deep
depressions. Also, the distributions for points above the Fermi surface
(dashed lines)
are broader by an order of magnitude (in standard deviation) than below, due to
screening. Finally, saturation as
${\cal F} \rightarrow 0$ (inset Fig. \ref{fig5}) shows that even if the
ions are arranged in a Wigner
crystal (the limiting case at ${\cal F} = 0$), potential
fluctuations would be expected in comparison with ionic jellium.
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig5.eps}
\vspace*{3mm}
\caption{Histograms of deviation of effective 2D potential from
quiet dot values at the same $x,y$ point and same gate voltages,
for several ion to donor ratios ${\cal F}$, TF. Solid lines are
statistics for points below Fermi surface, dashed lines, showing
substantially more variation, above. ${\cal F}=1$ is completely
random (disordered) case. Distributions uniformly asymmetric,
positive potential deviations from quiet dot case being more
likely, but means are very close to zero. Inset shows standard
deviation of histograms versus ${\cal F}$, triangle below,
squares above Fermi level. \label{fig5} }
\end{minipage}
\end{figure}
The success of the capacitance model in describing experimental results
of charging phenomena in mesoscopic systems has been remarkable \cite{various}.
For our calculations as well, even the simplest
formulations for the capacitance tend to produce smoothly varying results
when gate voltages or dot charge are varied. Figure \ref{fig6} shows the trend
of the dot self-capacitances with $V_g$. Also shown are the equilibrium
dot electron number $N$ and the minimum of the dot potential $V_{min}$ as
functions of $V_g$. Note here that $V_{min}$ is the minimum of the 3D
electrostatic potential rather than the effective 2D potential which is
presented elsewhere (such as in Figs. \ref{fig3} and \ref{fig4}).
That $C_{dd}$ generally decreases as the dot becomes
smaller is not surprising and has been discussed elsewhere \cite{ep2ds10}.
All three forms of $C_{dd}$ are roughly in agreement giving a
value $\sim 2 \; fF$ (the capacitance as calculated from the
free energy is not shown). The fluctuations result from variations
in the quantized level energies as the dot size and shape are changed by
$V_g$. Note that {\it numerical} error is indiscernible on the
scale of the figure. The pronounced collapse of $C_{dd}^{\prime}$
near $V_g = -1.15 \; V$, which is expanded in the
upper panel, shows the presence of a region where the change
of $N$ with $E_F$ is greatly suppressed. Since the change of $V_{min}$
with $E_F$ is similarly suppressed there is no corresponding
anomaly in $C_{dd}$. Interestingly, the capacitance computed from the free energy
also reveals no deep anomaly.
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig6.eps}
\vspace*{3mm}
\caption{Dot self-capacitances, equilibrium electron number and
potential minimum as a function of plunger gate voltage (lower).
{\it Numerical} uncertainty is indiscernible, so variations
of $C_{dd}$ are real and related to spectrum. $C_{dd}^{\prime}$
calculated using $\Delta E_F$ rather than $\Delta V_{min}$, so
strong anomaly near $-1.15 \; V$ due to rigidity of $N$.
Upper panel: expanded view of capacitances near anomaly;
cf. spectrum, Fig. 9. \label{fig6} }
\end{minipage}
\end{figure}
The anomaly at $V_g = -1.15 \; V$ and also the fluctuation in
the electrostatic properties near $-1.1 \; V$ are related to a
shell structure in the spectrum which we discuss below.
A frequently encountered model for the classical charge
distribution in a quantum dot is the circular conducting disk
with a parabolic confining potential \cite{Shikin,Chklovskii}.
It can be shown (solving, for example, Poisson's equation
in oblate spheroidal coordinates)
that for such a model the 2D charge distribution
in the dot goes as
\begin{equation}
n(r) = n(0)(1-r^2/R^2)^{1/2} \label{eq:circ}
\end{equation}
where $R$ is the dot radius and $n(0)=3N/2 \pi R^2$ is the
density at the dot center.
The ``external'' confining potential is assumed to
go as $V(r)=V_0 + kr^2/2$ and $R$ is related to $N$
through
\begin{equation}
R = \frac{3 \pi}{4} \frac{e^2}{\kappa k} N
\end{equation}
where $\kappa$ is the dielectric constant \cite{Shikin}.
To justify this model, the authors of Ref. \cite{Shikin}
claim that the calculations of Kumar {\it et al.} \cite{Kumar}
show that ``the confinement...has a nearly parabolic
form for the external confining potential ({\it sic}).''
This is incorrect. What Kumar {\it et al.}'s calculations shows
is that (for $N \stackrel{<}{\sim} 12$) the {\it self-consistent} potential,
which includes the potential from the electrons themselves,
is approximately cutoff parabolic. The {\it external} confining
potential, as it is used in Ref. \cite{Shikin},
would be that produced by the donor layer charge and the
charge on the surface gates only. We introduce a simple model
(see III.B.1 below) wherein this confining potential charge is replaced
by a circular disk of positive charge whose
density is fixed by the doping density and whose radius is determined by
the number of electrons {\it in the dot}. The gates can be
thought of as merely cancelling the donor charge outside that
radius. The essential point, then, is this: adding
electrons to the dot decreases the (negative) charge on the gates and
therefore increases the radius. One can make the assumption, as in Ref.
\cite{Shikin}, that the external potential is parabolic, but it is a
mistake to treat that parabolicity, $k$, as independent of $N$.
This is illustrated in
figure \ref{fig7} where we have plotted contours for the {\it change} in
the 2D density, as $E_F$ is incrementally increased,
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig7.eps}
\vspace*{3mm}
\caption{Grey scale of density change as Fermi energy in dot is raised
relative to leads, Thomas-Fermi (TF). Total change in $N$ about $1.4$
electrons. Screening charge, white region, in leads is positive.
White curve gives profile along line bisecting dot, scaled
to average change of $N$ per unit area. Right panel
shows model of Ref. $~^{44}$ where confining potential
has fixed parabolicity. Note that this model drastically
underestimates degree to which charge is added
to perimeter. \label{fig7} }
\end{minipage}
\end{figure}
as determined self-consistently (Thomas-Fermi everywhere, left panel)
and as determined from Eq. \ref{eq:circ}. The white curves display
the density change profiles across the central axis of the dot.
The total change in $N$ is the same in both cases, but
clearly the model of Eq. \ref{eq:circ} underestimates
the degree to which new charge is added mostly to the
perimeter.
Recently the question of charging energy renormalization via
tunneling as the conductance $G_0$ through a QPC
approaches unity has received much attention \cite{Matveev2,Halperin,Kane}.
In a recent experiment employing two dots in series a
splitting of the Coulomb oscillation peaks has been observed
as the central QPC (between the two dots) is lowered
\cite{Westervelt}. Perturbation theory for small $G_0$
and a model which treats the decaying channel between
the dots as a Luttinger liquid for $G_0 \rightarrow 1 \, (e^2/h)$
lead to expressions for the peak splitting which is
linear in $G_0$ in the former case and goes
as $(1-G_0)ln(1-G_0)$ in the latter case.
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig8.eps}
\vspace*{3mm}
\caption{Variation of dot capacitances with QPC voltage (gates
$1$ and $4$ in figure 1). Solid
lines for $V_{L(R)}$ are effective 2D potential for
left (right) saddle point (right hand scale).
$C_{\Sigma}(A)$ and $C_{\Sigma}(B)$
are dot self-capacitances (cf. Fig. 5) computed using $\Delta V_{min}$
and $\Delta E_F$ respectively. ``Source'' is (arbitrarily)
outside {\it left}
saddle point. Note that $V_L$ goes practically to zero
but the dot capacitance to the source only marginally
increases relative to dot to drain capacitance. Capacitances for QPC
and plunger are for a single finger only in each case.
Anomaly related to dot reconstruction also visible here as
QPC voltage is changed. \label{fig8} }
\end{minipage}
\end{figure}
A crucial assumption of the model, however, is that the
``bare'' capacitance, specifically that between the
dots $C_{d1-d2}$, remains approximately independent
of the height of the QPC, even when an open channel
connects the two dots. Thus the mechanism of the peak
splitting is assumed to be qualitatively different
from a model which predicts peak splitting entirely
on an electrostatic basis when the inter-dot capacitance
increases greatly \cite{Ruzin}. The independence
of $C_{d1-d2}$ from the QPC potential is plausible insofar
as most electrons, even when a channel is open, are
below the QPC saddle points and hence localized on either
one dot or the other. Further, if the screening length
is short and if the channel itself does not accommodate
a significant fraction of the electrons, there is
little ambiguity in retaining $C_{d1-d2}$ to
describe the gross electrostatic interaction of the dots,
even when the dots are {\it connected} at the Fermi level.
In figure \ref{fig8} we present evidence for this theory by
showing the capacitance between a dot and the {\it leads}
as the QPC voltage is reduced. In the figure $V_{L(R)}$
is the effective 2D potential of the left (right)
saddle point as the left QPC gate voltages $V_{QPC}$ only are varied. The
dot is nearly open when the QPC voltages (both pins on the left) reach
$\sim -1.34 \; V$. The results here use the full quantum
mechanical solution (without the LDA exchange-correlation energy),
however the electrons in the lead continue to be treated
with a 2D TF approximation. The dot ``reconstruction''
seen in figure~\ref{fig5} is visible
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig9.eps}
\vspace*{3mm}
\caption{$AlGaAs$ thickness dependence of capacitances (lower).
Self-capacitance decreases as gates get closer to 2DEG. Upper panel
shows that, for smaller $t$ the potential confinement is
steeper and charge more compact, hence smaller $C_{dd}$.
$t_1=5.25, \, t_2=7.5, \, t_3=9,$ and $t_4=12 \, a_B^*$.
Relative capacitance from dot to gates and leads fairly
insensitive to $t$. \label{fig9} }
\end{minipage}
\end{figure}
here also around $V_{QPC}=-1.365 \; V$.
Note that the right saddle point is sympathetically affected when
we change this left QPC. While the effect is faint, $\sim 5 \%$ of the
change of the left saddle,
the sensitivity of tunneling to saddle point voltage (see also below)
has resulted in this kind of cross-talk being problematical
for experimentalists. The figure also shows that the
capacitance between the dot and one lead exceeds that to
a (single) QPC gate or even to a plunger gate. However the
most important result of the figure is to show that
the dot to lead capacitance is largely insensitive to
QPC voltage. When the left QPC is as closed as
the right ($V_{QPC} \sim -1.375 \; V$) the
capacitances to the source and drain are equal.
But even near the open condition the capacitance to
the left lead (arbitrarily the ``source'') only
exceeds that to the drain (which is still closed)
minutely. Therefore the assumptions of a ``bare'' capacitance
which remains constant even as contact is made with a lead
(or, in the experiment, another dot) seems to be very well
founded.
As noted above, the interaction between a gate and the
2DEG depends upon the distance of the gates from the 2DEG,
i.e., the $AlGaAs$ thickness $t$. In figure \ref{fig9} we show that,
as we decrease $t$, simultaneously changing the
gate voltages such that $N$ and the saddle point potentials
remain constant, the total dot
capacitance also decreases, but the distribution
of the dot capacitance between leads, gates and (not shown)
back gate change only moderately. That gates closer to the 2DEG
plane should produce dots of lower capacitance is made
clear in the upper panel of the figure, which shows the
potential and density profile (using TF) near a depletion
region at the side of the dot
at varying $t$ and constant gate voltage. For smaller $t$
the depletion region is widened but the density
achieves its ungated 2DEG value (here $0.14 \; a_B^{* \, -2}$)
more quickly; a potential closer to hard walled is realized.
In the presence of stronger confinement the capacitance
decreases
and the charging energy increases.
The profile of the tunnel barriers and the barrier penetration
factors are also dependent on $t$. However we postpone
a discussion of this until the section on tunneling
coefficients.
\subsection{Spectrum}
The bulk electrostatic properties of a dot are, to first
approximation, independent of whether a Thomas-Fermi
approximation is used or Schr\"{o}dinger's equation
is solved. A notable exception to this is the
fluctuation in the capacitances. Figure \ref{fig10} shows
the plunger gate voltage dependence of the energy
levels. The Fermi level of the dot is kept constant
and equal to that of the leads (it is the energy zero).
Hence as the gate voltage
increases (becomes less negative) $N$ increases.
Since the QPCs lie along the $x$-axis, the dot is never
fully symmetric with respect to interchange of $x$ and
$y$, however the most symmetric configuration occurs for
$V_g \sim -1.16 \; V$, towards the right side of the plot.
The levels clearly group into quasi-shells with gaps between.
The number of states per shell follows the degeneracy of a 2D parabolic
potential, i.e. 1,2,3,4,... degenerate levels per shell
(ignoring spin). There is a pronounced
tendency for the levels to cluster at the Fermi surface,
here given by $E=0$, which we discuss below.
\subsubsection{Shell structure}
Shell structure in atoms arises from the approximate
constancy of individual electron
angular momenta, and degeneracy with respect to $z$-projection.
Since in two dimensions the angular momentum $m$ is fixed in
the $z$ (transverse) direction, the isotropy of space is broken
and the only remaining manifest degeneracy, and this only for
azimuthally symmetric dots, is with respect to $\pm z$.
A two dimensional parabolic potential, in the absence of
magnetic field, possesses an accidental degeneracy for
which a shell structure is recovered.
We have shown above that modelling a quantum dot as a
classical, conducting layer in an {\it external} parabolic potential
$kr^2/2$, where $k$ is independent of the number of electrons
in the dot, ignores the image charge in the surface gates
forming the dot and therefore fails to properly describe
the evolving charge distribution as electrons are added to the dot.
A more realistic model, which {\it explains} the approximate parabolicity
of the {\it self-consistent} potential, and hence the
apparent shell structure, is illustrated in figure \ref{fig11}.
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig10.eps}
\vspace*{3mm}
\caption{Electronic spectrum showing level grouping into
shells for quiet dot (Hartree), quiet dot with LDA
exchange-correlation, disordered sample ${\cal F}=1$ and
ordered sample ${\cal F}=1/5$. Range of gate voltage in latter
three is from $V_g = -1.142$ to $-1.17 \; V$. \label{fig10} }
\end{minipage}
\end{figure}
The basic electrostatic structure of a quantum dot,
in the simplest approximation, can be represented by two circular disks,
of radius $R$ and
homogeneous charge density $\sigma_0$, separated by a distance $a$.
The positive charge outside $R$ is assumed to be cancelled by the
surface gates. This approximation will be best for
surface gates very close to the donor layer (i.e. small $t$).
Larger $AlGaAs$ thicknesses will require a non-abrupt termination
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{DCD11.eps}
\vspace*{3mm}
\caption{Schematic for simple two charge disk model of quantum dot.
Positive charge outside radius $R$ taken to be uniformly
cancelled by gates, electric charge in 2DEG mirrors positive
charge. Resultant radial potential in 2DEG plane, Eq. 15,
dominated by parabolic term inside $R$. \label{fig11} }
\end{minipage}
\end{figure}
of the positive charge. In either case, the electronic charge is assumed
in the classical limit to screen the background charge as nearly
as possible. This is similar to the postulate in which wide parabolic
quantum wells are expected to produce approximately homogeneous layers
of electronic charge \cite{parabola}.
A simple calculation for the radial potential (for $a<R$)
in the electron layer ($z=0$)
gives, for the first few terms:
\begin{eqnarray}
& \phi(r)= \frac{2 Ne}{\kappa R} [\sqrt{1-a/R} - 1 +
\frac{3}{8} \frac{a^2}{R^2} \frac{r^2}{R^2} \nonumber \\
& - \frac{15}{32} \frac{a^4}{R^4} \frac{r^2}{R^2} +
\frac{45}{128} \frac{a^2}{R^2} \frac{r^4}{R^4} + \cdots ] \label{eq:phi}
\end{eqnarray}
where $Ne = \pi R^2 \sigma_0$ and $\kappa$ is the background dielectric
constant. While the coefficient of the quartic term is
comparable to that of the parabolic term, the dependences are scaled
by the dot radius $R$. Hence, the accidental degeneracy of
the parabolic potential is broken only by coupling via the
quartic term near the dot perimeter. This picture clearly
agrees with the full self-consistent results wherein the
parabolic degeneracy is observed for low lying states and
a spreading of the previously degenerate states occurs nearer
to the Fermi surface.
Comparison (not shown) of the potential computed from Eq. \ref{eq:phi} and
the radial potential profile (lowest curve, Fig. \ref{fig3}b) from the
full self-consistent structure, shows
good agreement for overall shape. However the former
is about $25 \%$ smaller (same $N$) indicating that the sharp cutoff
of the positive charge is, for these parameters, too extreme.
However Eq. \ref{eq:phi}
improves for larger $N$ and/or smaller $t$.
The wavefunction moduli squared associated with the Fig. \ref{fig10}
quiet dot levels for $V_g \sim -1.16 \; V$, $N \approx 54$ are shown
schematically for levels $1$ through $10$ in figure \ref{fig12}, and
for levels $11$ through $35$ in figure \ref{fig13}.
The lowest level in a shell is, for the higher shells,
typically the most circularly symmetric. When the last
member of a shell depopulates with $V_g$ the inner
shells expand outward, as can be seen near $V_g = -1.15 \; V$
(Fig. \ref{fig10})
where level $p=29$ depopulates. Since to begin filling a new shell
requires the inward compression of the other shells
and hence more energy, the
capacitance decreases in a step when a shell is depopulated.
The shell structure should have two distinct
signatures in the standard (electrostatic) Coulomb
oscillation experiment \cite{various}.
First, since the self-capacitance drops appreciably
(figure \ref{fig6}) when the last member of a shell depopulates,
here $N$ goes from $57$ to $56$, a concomitant discrete rise
in the activation energy in the minimum between Coulomb
oscillations can be predicted.
Second, envelope modulation of peak heights \cite{RComm}
occurs when excited dot states are thermally accessible
as channels for transport, as opposed to the $T=0$ case where
the only channel is through the first open state
above the Fermi surface (i.e. the $N+1^{st}$ state).
When $N$ is in the middle of a shell of closely spaced, spin
degenerate levels, the entropy of the dot, $k_B ln \Omega $,
where $\Omega$ is the number of states accessible to the
dot, is sharply peaked. For example, for six electrons
occupying six spin degenerate levels (i.e. twelve altogether)
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig12.eps}
\vspace*{3mm}
\caption{Schematic showing the first ten levels of quiet dot.
Shell structure consistent with $n+m=$ constant, where
$n$ and $m$ are nodes in $x$ and $y$. Lower energy states
show rectangular symmetry. \label{fig12}}
\vspace*{6mm}
\epsfxsize=8cm
\epsfbox{wvQDs.eps}
\vspace*{3mm}
\caption{Levels $11$ through $35$ (each spin degenerate) of
quiet dot, Hartree. Circular symmetry increases with energy. States
elongated in $x$ (horizontal) most connected to leads. \label{fig13}}
\end{minipage}
\end{figure}
all within $k_B T$ of the Fermi surface, the number of channels
available for transport is $924$. For eleven electrons in the
shell, however, the number of channels reduces to $12$.
Consequently, minima and peaks of envelope modulation
(see also figure \ref{fig22} below)
of CB oscillations which are frequently observed are clear
evidence of level bunching, if not an organized shell structure.
Recently experimental evidence has accumulated for the
existence of a shell structure as observed by
inelastic light scattering \cite{Lockwood} and via Coulomb oscillation
peak positions in transport through
extremely small ($N \sim 0-30$) vertical
quantum dots \cite{Tarucha}. Interestingly, a {\it classical}
treatment, via Monte-Carlo molecular dynamics simulation \cite{Peeters}
also predicts a shell structure. Here,
the effect of the neutralizing positive background
are assumed to produce a parabolic confining potential.
A similar assumption is made in Ref. \cite{Akera} which
analyzes a vertical structure similar to that of Ref. \cite{Tarucha}.
We believe that continued advances in fabrication will result
in further emphasis on such invariant, as opposed to merely statistical,
properties of dot spectra.
As noted above, there is a strong tendency for levels at the
Fermi surface to ``lock.'' Such an effect has been described by
Sun {\it et al.} \cite{Sun} in the case of subband levels for
parallel quantum wires.
In dots, the effect
can be viewed as electrostatic pressure on the
individual wavefunctions thereby shifting
level energies in such a way as to produce level
{\it occupancies} which minimize the
total energy. Insofar as a given set of level occupancies
is electrostatically most favorable, level locking
is a temperature dependent effect which increases as
$T$ is lowered. This self-consistent modification of
the level energies can also be viewed as an excitonic
correction to excitation energies.
The difference between the cases of a quantum dot and that
of parallel wires is one of localized versus extended systems.
It is well known that, unlike Hartree-Fock theory, wherein
self-interaction is completely cancelled since the direct and
exchange terms have the same kernel $1/|{\bf r} - {\bf r^{\prime}}|$,
in Hartree theory and even density functional theory in
the LDA, uncorrected self-interaction remains \cite{Perdew}.
While it is reasonable to expect that excited states will have their energies
corrected downward by the remnants of an excitonic effect, we expect
that LDA and especially Hartree calculations will generally overestimate this
tendency to the extent that corrections for self-interaction are
not complete.
The panel labelled ``xc'' in figure \ref{fig10} illustrates the preceding point.
In contrast to the large panel (on the left) these results have
had the XC potential in LDA included. The differences
between Hartree and LDA are generally subtle, but here the clustering
of the levels at the Fermi surface is clearly mitigated by the
inclusion of XC. The approximate parabolic degeneracy is
evidently not broken by LDA, however, and the shell structure
remains intact.
Similarly for xc, the capacitances
also show anomalies near the same gate
voltages, where shells depopulate, as in figure \ref{fig5},
which is pure Hartree.
The two remaining panels in figure \ref{fig10} illustrate the effects of
disorder and ordering in the donor layer (XC not included).
As with the ``xc'' panel, $V_g$ is varied between $-1.142$ and $-1.17 \; V$.
The ``disorder'' panel represents a single fixed distribution of
ions placed at random in the donor layer as discussed above.
Similarly, the ``order'' panel represents a single ordered distribution
generated from a random distribution via the Monte-Carlo simulation
\cite{BR2,ISQM2}; here ${\cal F} = 1/5$ (cf. two panels of Fig. \ref{fig4}).
The shell structure, which is completely destroyed for fully
random donor placement (see also Fig. \ref{fig15}), is almost perfectly
recovered in the
ordered case. In both cases the energies are uniformly shifted upwards
relative to the quiet dot by virtue of the discreteness of donor
charge (cf. also discussion of Fig. \ref{fig5} above). Closer examination
of the disordered spectrum shows considerably more level repulsion
than the other cases.
The application of a small magnetic field, roughly a single
flux quantum through the dot, has a dramatic impact on both
the spectrum, figure \ref{fig14}, and the wave functions, figure \ref{fig15},
top. The magnetic field dependence of the levels (not shown) up to $0.1 \; T$
exhibits shell splitting according to azimuthal quantum number as
\begin{figure}
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig14b.eps}
\vspace*{3mm}
\caption{$V_g$ dependence at fixed $B$ ($0.05 \; T$)
of level energies, quiet dot. Multiple
re-constructions seen as levels depopulate.
Homogeneous level spacing related
to uniformity of Coulomb oscillation peak heights in a magnetic
field. \label{fig14} }
\end{minipage}
\end{figure}
well as level anti-crossing. By $0.05 \; T$ level spacing (Fig. \ref{fig14})
is substantially more uniform than $B=0$,
Fig. \ref{fig10}. Furthermore, while
the $B=0$ quiet dot displays reconstruction due to the depopulation of
shells at $V_g \approx -1.15$ and $-1.1 \; V$,
the $B=0.05 \; T$ results show a similar pattern, a step in
the levels, repeated
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{wv2.eps}
\vspace*{3mm}
\caption{Levels $31$ through $35$ for (from bottom) quiet dot with LDA
for XC, Hartree for disordered dot, Hartree for ordered dot
${\cal F}=1/5$ and $B=0.05 \; T$. XC changes ordering
of some levels, but has very little influence on states.
Ordered case recovers much of quiet dot symmetry. Small $B$
changes states altogether. \label{fig15}}
\end{minipage}
\end{figure}
many times in the same gate voltage range. The physical meaning of this
is clear. The magnetic field principally serves to remove the
azimuthal dependence of the mod squared of the wave functions
(Fig. \ref{fig15}). In a magnetic field, the
states at the Fermi surface also tend to be at the dot perimeter.
Depopulation of an electron in a magnetic field, like depopulation of
the last member of a shell for $B=0$, therefore
removes charge from the perimeter of the dot and a self-consistent
expansion of the remaining states outward occurs.
\subsection{Statistical properties}
\subsubsection{Level spacings}
The statistical spectral properties of quantum systems
whose classical Hamiltonian is chaotic are believed to
obey the predictions of random matrix theory (RMT) \cite{Andreev}.
Arguments for this conjecture however invariably treat
the Hamiltonian as a large finite matrix with averaging
taken only near the band center. Additionally, an
often un-clearly stated assumption is that the system in question can
be treated {\it semi-classically}, that is, in some sense the action is
large on the scale of Planck's constant and the wavelength {\it of all
relevant states} is short on the scale of the system size.
Clearly, for small quantum dots these assumptions are violated.
RMT predictions apply to level spacings $S$
and to transition amplitudes (for the ``exterior problem,''
level widths $\Gamma$) \cite{Brody}. RMT is also
applied to scattering matrices in investigations of
transport properties of
quantum wires \cite{Slevin}.
Ergodicity for chaotic systems is the claim that variation of
some external parameter $X$ will sweep the Hamiltonian rapidly through
its entire Hilbert space, whereupon energy averaging
and ensemble (i.e. $X$) averaging produce identical statistics.
In our study $X$ is either the set of gate voltages, the
magnetic field or the impurity configuration
and we consider the statistics of the lowest
lying $45$ levels (spin is ignored here). Care must also be taken
in removing the secular variations of the spacings or widths
with energy, the so-called unfolding.
According to RMT level repulsion leads to statistics of level
spacings which are given by the ``Rayleigh distribution:''
\begin{equation}
P(S)=\frac{\pi S}{2D} exp(-\pi S^2/4 D^2) \label{eq:stat}
\end{equation}
where $D$ is the mean local spacing \cite{Brody,Wigner}.
Figure \ref{fig16} shows the calculated histogram for the level spacings
for the quiet dot as well as for disordered, ordered and
ordered with $B=0.05 \; T$ cases. Statistics are generated from
(symmetrical) plunger gate variation,
in steps of $0.001 \; V$, over a range of $0.1 \; V$, employing the spacings
between the lowest $45$ levels; thus about $4500$ data points.
Deviation from the Rayleigh distribution is evident. An important
feature of our dot is symmetry under inversion through both
axes bisecting
the dot. It is well known that groups of states which
are\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig16.eps}
\vspace*{3mm}
\caption{Histograms of level spacings, normalized to local level spacing.
Dark curve represents Rayleigh distribution. Black bars (main panel)
include all states, white bars only for states that are completely
even under $x$ or $y$ inversion. Insets: disordered panel
recapitulates Rayleigh distribution, both ordered and $B \ne 0$
marginally but significantly different. \label{fig16} }
\end{minipage}
\end{figure}
un-coupled will, when plotted together, show a Poisson distribution for
the spacings rather than the level repulsion of Eq. \ref{eq:stat}.
Thus we have also plotted (white bars) the statistics for
those states which are totally even in
parity. While the probability of degeneracy decreases,
a $\chi^2$ test shows that the distribution remains
substantially removed from the Rayleigh form.
In contrast to this, the disordered case shows remarkable
agreement with the RMT prediction. As with the spectrum
in figure \ref{fig10} we use a single ion distribution. However we
also find (not shown) that fixing the gate voltage and varying the random
ion distributions results in nearly the same statistics.
When the ions are allowed to order the level statistics again
deviate from the RMT model. This is somewhat surprising
since Fig. \ref{fig5} shows that, even for ${\cal F}= 1/5$, the standard
deviation
of the effective 2D potential below the Fermi surface from the
quiet dot case, $\sim 0.05 \; Ry^*$, is still substantially greater than
the mean level spacing $\sim 0.02 \; Ry^*$. We have recently
shown that, as ${\cal F}$ goes from unity to zero, a continuous transition
from the level repulsion of Eq. \ref{eq:stat} to a Poisson
distribution of level spacings results \cite{NanoMes}.
Finally, the application of
a magnetic field strong enough to break time-reversal symmetry
clearly reduces the incidence of very small spacings, but the
distribution is still significantly different from RMT.
\subsubsection{Level widths}
In Eq. \ref{eq:tun} we defined $W_n(a,b)$ as the barrier penetration factor
from the classically accessible region of the lead to the matching point in
the barrier, for the $n^{th}$ channel. The
penetration factor {\it completely} through the barrier, $P_n \equiv W_n(a,c)$
where $c$ is the classical turning point on the dot side of the barrier,
is plotted as a function of QPC voltage in figure \ref{fig17}.
$P_n$ is simply the WKB penetration for a given channel with
a given self-consistent barrier profile, and
can be computed at any energy. Here we have computed it
at energies coincident with the dot levels. Therefore the dashes
recapitulate the level structure, spaced now not in energy but in
``bare'' partial width. The {\it actual} width of a level
depends upon the wave
function for that state (cf. Eq. \ref{eq:tun}).
For energies above the barrier $ln(P)=0$.
The solid lines represent $P$ {\it at the Fermi surface} computed for
three different $AlGaAs$ thicknesses $t$ (as in figure \ref{fig9}) and
for both $n=1$ and $n=2$ (the dashes are computed for $t=12 \; a_B^*$).
The QPC voltage is given relative to
values at which $P$ for $n=1$ is the same for all three $t$
(hence the top three solid lines converge at $\Delta V_{QPC} = 0$).
Quite surprisingly $t$ has very little influence on the trend of
$P$ with QPC voltage. Note that the ratio of
barrier penetration between the second and first channels $P_2/P_1$
decreases substantially with increasing $t$ since the saddle
profile becomes wider for more distant gates. Even for $t=7.5 \; a_B^*$
however, penetration via the second channel is about a factor
of five smaller than via $n=1$.
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig17.eps}
\vspace*{3mm}
\caption{Barrier penetration factors from classical turning point in lead
to turning point in dot at same energy, as a function
of QPC voltage offset. $P$ evaluated at energies of
states in quiet dot for $AlGaAs$ thickness $t=12 \; a_B^*$.
Solid lines indicate barrier penetration
at Fermi level. Upper three lines for first channel, $t = 7.5,9.0,12.0 \ a_B^*$
respectively. Lower three lines for second channel, same $t$.
$\Delta V_{QPC}$ zero set such that first channel conducts
equally at the Fermi surface for all $t$. \label{fig17}}
\end{minipage}
\end{figure}
Figure \ref{fig18} shows the partial width for tunneling via $n=1$ through the
barrier, now using the full Eq. \ref{eq:tun}, for the quiet dot.
The barriers here are fairly wide.
While this strikingly coherent structure is quickly destroyed
by discretely localized donors even when donor ordering is allowed,
the pattern is nonetheless highly informative. The principal division
between upper and lower states is based on parity. States which
are odd with respect to the axis bisecting the QPC should in fact
have identically zero partial width (that they don't is evidence of
numerical error, mostly imperfect convergence).\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig18.eps}
\vspace*{3mm}
\caption{Partial widths (through first channel) for tunneling to the leads,
quiet dot. Numbers indicate ordinate of wave functions,
Figs. 11 and 12.
Weakly connected states zero by parity (non-zero only
through numerical error). \label{fig18}}
\end{minipage}
\end{figure}
Note that {\it this}
division
is largely preserved for discrete but ordered ions.
The widest states (largest $\Gamma$) are
labelled with their level index for comparison with their
wave functions in Figs. \ref{fig13} and \ref{fig14}. Comparison shows they represent the states
which are aligned along the direction of current flow. Thus in
each shell there are likely to be a spread of tunneling coefficients,
that is, two members of the same shell will not have the same $\Gamma$.
Statistics of the level partial widths are shown in figure \ref{fig19},
here normalized
to their local mean values. While the statistics for the quiet dot
are in substantial disagreement with RMT it is clear that discreteness
of the ion charge, even ordered, largely restores ergodicity.
The RMT prediction, the ``Porter-Thomas'' (PT) distribution, is also plotted.
For non-zero $B$, panels (b) and (c), the predicted distribution
is $\chi_2^2$ rather than PT. Even the completely disordered case (e)
retains a fraction of vanishing partial width states. Since in our
case the zero width states result from residual reflection symmetry,
it would be interesting to compare the data from references
\cite{Chang} and \cite{Marcus}, which employ nominally symmetric
and non-symmetric dots respectively, to see if the incidence of zero
width states shows a statistically significant difference.
One further statistical feature which we calculate is the
autocorrelation function of the level widths as an external parameter
$X$ is varied:
\begin{equation}
C(\Delta X) = \\
\frac{\sum_{i,j} \delta \Gamma_i(X_j)
\delta \Gamma_i(X_j + \Delta X)}
{\sqrt{\sum_{i,j} \delta \Gamma_i(X_j)^2}
\sqrt{\sum_{i,j} \delta \Gamma_i(X_j + \Delta X)^2}} \label{eq:auto}
\end{equation}
where $\delta \Gamma_i(X) \equiv \Gamma_i(X)-\bar{\Gamma}_i(X)$, and
where
$\bar{\Gamma}(X)$ is again the {\it local} average, over levels at fixed
$X$, of the level widths. Note that the sum on $i$ is over levels and the sum on
$j$ is over starting values of $X$.
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig19.eps}
\vspace*{3mm}
\caption{Statistics of unfolded partial level widths, first channel only,
(a) quiet dot showing large weight near zero due to parity, (b) and (c)
have $B=0.05 \; T$, quiet dot and disordered, respectively. Remnant
of peak at small coupling remains. Dark line represents $\chi_2^2$
distribution predicted by RMT. (d) and (e) are ordered and disordered
with $B=0$. Ordered case differs significantly from Porter-Thomas
distribution plotted in black here. \label{fig19}}
\end{minipage}
\end{figure}
In figure \ref{fig20} we show the autocorrelation function for varying magnetic
field (cf. Ref. \cite{Marcus}, figure \ref{fig4}). The sample is ordered,
${\cal F}=1/5$.
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig20.eps}
\vspace*{3mm}
\caption{Autocorrelation function for level partial widths;
ordered, ${\cal F}=1/5$, averaged over $B$ starting point and
all $45$ levels. Range of $B$ is only $0-0.1 \; T$, so statistics
are weaker to the right. Pronounced anti-correlation near
$0.03 \; T$ in contradiction with RMT. \label{fig20}}
\end{minipage}
\end{figure}
Our range of $B$ only encompasses
$[0,0.1] \; T$ in steps of $0.005 \; T$, so we have here averaged over
all levels (i.e. $i=1-45$). The crucial feature, which has been
noted in Refs. \cite{Marcus} and, for conductance correlation
in open dots in \cite{Bird2}, is that the correlation function becomes
negative, in contradiction with a recent prediction based on
RMT \cite{Alhassid}.
Indeed, as noted by Bird {\it et al.} \cite{Bird2},
an oscillatory structure seems to emerge in the data.
Comparison with calculation here is hampered since the statistics
are less good as $B$ increases.
Nonetheless, the RMT prediction is clearly erroneous.
We speculate that the basis of the discrepancy is in the
assumption \cite{Alhassid} that $C(\Delta X)=C(-\Delta X)$.
Given this assumption \cite{Ferry2} the correlation
becomes positive definite. Physically this means that,
regardless of whether $B$ is positive or negative,
the self-correlation of a level width will be independent
of whether $\Delta B$ is positive or negative. This implies
that the level widths should be independent of the
absolute value of $B$, or any even powers of $B$, at
least to lowest order in $\Delta B/B$.
For real quantum dot systems this assumption is inapplicable.
Similar behaviour is observed with $X$ taken as the (plunger) gate voltage,
for which we have considerably more calculated results, Fig. \ref{fig21}.
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig21.eps}
\vspace*{3mm}
\caption{Autocorrelation function with $V_g$, averaged over groups
of $15$ levels (upper panel). Number indicates center of (contiguous)
range of averaged values. Dashed line is average of all states.
Lower panel is grey scale for autocorrelation of individual
levels averaged only over $V_g$ starting point. Black is $1.0$
and white is $-1.0$. Data suggests that behaviour of
autocorrelation is sensitive to {\it which} levels are averaged.
\label{fig21}}
\end{minipage}
\end{figure}
The upper panel is the analogue of Fig. \ref{fig20}, only we have broken the
average on levels into separate groups of fifteen levels centered on
the level listed on the figure (e.g., the ``$28$'' denotes a sum
in equ. \ref{eq:auto} of $i=21,35$). the lower panel shows the autocorrelation
as a grey scale for the individual levels (averaging performed
only over starting $V_g$). The very low lying levels, up to $\sim 10$, remain
self-correlated across the entire range of gate voltage. This simply
indicates that the correlation field is level dependent. However,
rather than becoming uniformly grey in a Lorentzian fashion, as
predicted by RMT \cite{Alhassid}, individual
levels tend to be strongly correlated or anti-correlated with their
original values, and the disappearance of correlation only occurs
as an average over levels.
Again we expect that the explanation for this behaviour lies in the
shell structure. Coulomb interaction prevents states which are nearby in
energy from having common spatial distributions.
Thus in a given range of energy, when one state is strongly connected
to the leads, other states are less likely to be. Further, the ordering
of states appears to survive at least a small amount of disorder in
the ion configuration.
\subsection{Conductance}
The final topic we consider here is the Coulomb oscillation conductance of
the dot. We will here focus on the temperature dependence \cite{RComm},
although statistical properties related to ion ordering are also
interesting.
We have shown in Ref. \cite{RComm} that
detailed temperature dependence of
Coulomb oscillation amplitudes can be employed as a form
of quantum dot spectroscopy.
Roughly, in the low $T$ limit the peak heights give the
individual level connection coefficients and, as temperature is
raised activated conductance {\it at the peaks} depends on the
nearest level spacings at the Fermi surface. In this regard
we have explained envelope modulation of peak heights, which
had previously not been understood, as clear
evidence of thermal activation involving tunneling through
excited states of the dot \cite{RComm}.
Figure \ref{fig22}a
shows the conductance as a function of plunger gate voltage
for the ordered dot at $T=250 \; mK$.
Note that the magnitude of the conductance is small because the
coupling coefficients
are evaluated with relatively wide barriers for numerical reasons.
Over this range the dot $N$
depopulates from $62$ (far left) to $39$. The level spacings and tunneling
coefficients are all changing with $V_g$. At low temperature a given peak
height is determined mostly by the coupling to the first empty dot
level ($\Gamma_{N+1}$) and by the spacings between the $N^{th}$ level
and the nearest other level (above or below). The relative importance of
the $\Gamma$'s and the level spacings can obviously vary.
In this example, Figs. \ref{fig22}a and \ref{fig22}b suggest that
peak heights correlate more strongly with the level spacings.
The double envelope coincides with the Fermi level passing through
two shells. In general, the DOS fluctuations embodied in the shell
structure and the
observation (above) that within a shell a spreading of the $\Gamma$'s
(with a most strongly coupled level) results from Coulomb interaction
provide the two fundamental bases of envelope modulation.
\begin{figure}[hbt]
\setlength{\parindent}{0.0in}
\begin{minipage}{\linewidth}
\epsfxsize=8cm
\epsfbox{LPfig22ab.eps}
\vspace*{3mm}
\epsfxsize=8cm
\epsfbox{LPfig22b.eps}
\vspace*{3mm}
\caption{(a) Conductance versus $V_g$ for ordered dot, $T= 0.25 \; K$.
(b) Fermi surface level spacing and
tunneling coefficient at resonance. Conductance in (a) correlates
somewhat more strongly with smaller level spacing than with larger
$\Gamma$. \label{fig22}}
\end{minipage}
\end{figure}
Finally, we typically find
that, when peak heights are plotted as a function of
temperature (not shown) some peaks retain
activated conductance down to $T=10 \; mK$.
Since the dot which we are modelling is small on the scale
of currently fabricated structures, this study suggests that
claims to have reached the regime where all Coulomb oscillations
represent tunneling through a single dot level are questionable.
\section{Conclusions}
We have presented extensive data from calculations on the electronic
structure of lateral $GaAs-AlGaAs$ quantum dots, with electron
number in the range of $N=50-100$. Among the principal conclusions which we
reach are the following.
The electrostatic profile of the dot is determined by metal gates
at fixed voltage rather than a fixed space charge. As a consequence of
this the model of the dot as a conducting disk with fixed,
``external,'' parabolic confinement is incorrect.
Charge added to the dot resides much more at the dot
perimeter than this model predicts.
The assumption of complete disorder in the donor layer is probably
overly pessimistic. In such a case the 2DEG
electrostatic profile is completely dominated by the ions
and it is difficult to see how workable structures could be
fabricated at all. The presence of even a small degree of
ordering in the donor layer, which can be experimentally
modified by a back gate, dramatically reduces potential
fluctuations at the 2DEG level.
Dot energy levels show a shell structure
which is robust to ordered donor layer ions, though for complete
disorder it appears to break up. The shell
structure is responsible for variations in the capacitance with gate
voltage as well as envelope modulation of Coulomb oscillation peaks.
The claims that Coulomb oscillation data through
currently fabricated lateral quantum dots shows unambiguous
transport through single levels are questionable, though
some oscillations will saturate at a higher temperature than others.
The capacitance between the dot and a lead increases only
very slightly as the QPC barrier is reduced. Thus the electrostatic
energy between dot and leads is dominated by charge below the Fermi
surface and splitting of oscillation peaks through double dot
structures \cite{Westervelt} is undoubtedly a result of tunneling.
Finally, chaos is well known to be mitigated in quantum
systems where barrier penetration is non-negligible \cite{Smilansky}.
Insofar as non-inegrability of the underlying classical
Hamiltonian is being used as the justification for an
assumption of ergodicity \cite{Jalabert} in quantum dots, our
results suggest that further success in comparison with real
(i.e. experimental) systems will occur only when account is
taken in, for example, the level velocity \cite{Alhassid,Simons},
of the correlating influences of quantum mechanics.
\acknowledgements
I wish to express my thanks for benefit I have gained in conversations
with many colleagues. These include but are not limited to:
Arvind Kumar, S. Das Sarma, Frank Stern, J. P. Bird, Crispin Barnes,
Yasuhiro Tokura, B. I. Halperin, Catherine Crouch, R. M. Westervelt,
Holger F. Hofmann, Y. Aoyagi,
K. K. Likharev, C. Marcus and D. K. Ferry. I am also grateful for support
from T. Sugano, Y. Horikoshi, and S. Tarucha.
Computational support from the Fujitsu VPP500 Supercomputer
and the Riken Computer Center is also
gratefully acknowledged.
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proofpile-arXiv_065-604
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\section{Introduction and results}
The interstellar medium (ISM) is a gas essentially formed by atomic (HI)
and molecular ($H_2$) hydrogen, distributed in cold ($T \sim 5-50 K$)
clouds, in a very inhomogeneous and fragmented structure.
These clouds are confined in the galactic plane
and in particular along the spiral arms. They are distributed in
a hierarchy of structures, of observed masses from
$1\; M_{\odot}$ to $10^6 M_{\odot}$. The morphology and
kinematics of these structures are traced by radio astronomical
observations of the HI hyperfine line at the wavelength of 21cm, and of
the rotational lines of the CO molecule (the fundamental line being
at 2.6mm in wavelength), and many other less abundant molecules.
Structures have been measured directly in emission from
0.01pc to 100pc, and there is some evidence in VLBI (very long based
interferometry) HI absorption of structures as low as $10^{-4}\; pc = 20$ AU
(3 $10^{14}\; cm$). The mean density of structures is roughly inversely
proportional to their sizes, and vary
between $10$ and $10^{5} \; atoms/cm^3$ (significantly above the
mean density of the ISM which is about
$0.1 \; atoms/cm^3$ or $1.6 \; 10^{-25}\; g/cm^3$ ).
Observations of the ISM revealed remarkable relations between the mass,
the radius and velocity dispersion of the various regions, as first
noticed by Larson \cite{larson}, and since then confirmed by many other
independent observations (see for example ref.\cite{obser}).
From a compilation of well established samples of data for many different
types of molecular clouds of maximum linear dimension (size) $ R $,
mass fluctuation $ \Delta M$ and internal velocity dispersion $
\Delta v$ in each region:
\begin{equation}\label{vobser}
\Delta M (R) \sim R^{d_H} \quad , \quad \Delta v
\sim R^q \; ,
\end{equation}
over a large range of cloud sizes, with $ 10^{-4}\; - \; 10^{-2}
\; pc \; \leq R \leq 100\; pc, \;$
\begin{equation}\label{expos}
1.4 \leq d_H \leq 2 , \; 0.3 \leq q \leq
0.6 \; .
\end{equation}
These {\bf scaling} relations indicate a hierarchical structure for the
molecular clouds which is independent of the scale over the above
cited range; above $100$ pc in size, corresponding to giant molecular clouds,
larger structures will be destroyed by galactic shear.
These relations appear to be {\bf universal}, the exponents
$d_H , \; q$ are almost constant over all scales of the Galaxy, and
whatever be
the observed molecule or element. These properties of interstellar cold
gas are supported first at all from observations (and for many different
tracers of cloud structures: dark globules using $^{13}$CO, since the
more abundant isotopic species $^{12}$CO is highly optically thick,
dark cloud cores using $HCN$ or $CS$ as density tracers,
giant molecular clouds using $^{12}$CO, HI to trace more diffuse gas,
and even cold dust emission in the far-infrared).
Nearby molecular clouds are observed to be fragmented and
self-similar in projection over a range of scales and densities of
at least $10^4$, and perhaps up to $10^6$.
The physical origin as well as the interpretation of the scaling relations
(\ref{vobser}) are not theoretically understood. The theoretical
derivation of these
relations has been the subject of many proposals and controversial
discussions. It is not our aim here to account for all the proposed models
of the ISM and we refer the reader to refs.\cite{obser} for a review.
The physics of the ISM is complex, especially when we consider the violent
perturbations brought by star formation. Energy is then poured into
the ISM either mechanically through supernovae explosions, stellar winds,
bipolar gas flows, etc.. or radiatively through star light, heating or
ionising the medium, directly or through heated dust. Relative velocities
between the various fragments of the ISM exceed their internal thermal
speeds, shock fronts develop and are highly dissipative; radiative cooling
is very efficient, so that globally the ISM might be considered
isothermal on large-scales.
Whatever the diversity of the processes, the universality of the
scaling relations suggests a common mechanism underlying the physics.
We propose that self-gravity is the main force at the origin of the
structures, that can be perturbed locally by heating sources.
Observations are compatible with virialised structures at all scales.
Moreover, it has been suggested that the molecular clouds ensemble is
in isothermal equilibrium with the cosmic background radiation at $T \sim 3 K$
in the outer parts of galaxies, devoid of any star and heating
sources \cite{pcm}. This colder isothermal medium might represent the ideal
frame to understand the role of self-gravity in shaping the hierarchical
structures. Our aim is to show that the scaling laws obtained are then
quite stable to perturbations.
Till now, no theoretical derivation of the scaling laws
eq.(\ref{vobser}) has been
provided in which the values of the exponents are {\bf obtained}
from the theory
(and not just taken from outside or as a starting input or hypothesis).
The aim of these authors is to develop a theory of the cold ISM. A first
step in this goal is to provide a theoretical derivation of the scaling
laws eq.(\ref{vobser}), in which the values of the exponents $d_H , \; q$ are
{\bf obtained} from the theory.
For this purpose, we will implement for the ISM the powerful tool of field
theory and the Wilson's approach to critical phenomena \cite{kgw}.
We consider a gas of non-relativistic atoms interacting with each other
through Newtonian gravity and which are in thermal
equilibrium at temperature $ T $.
We work in the grand canonical ensemble, allowing for a variable
number of particles $N$.
Then, we show that this system
is exactly equivalent to a field theory of a single scalar field
$\phi({\vec x})$ with
exponential interaction. We express the grand canonical partition function
$ {\cal Z} $ as
\begin{equation}\label{zetafi}
{\cal Z} = \int\int\; {\cal D}\phi\; e^{-S[\phi(.)]} \; ,
\end{equation}
where
\begin{eqnarray}\label{SmuyT}
S[\phi(.)] & \equiv & {1\over{T_{eff}}}\;
\int d^3x \left[ \frac12(\nabla\phi)^2 \; - \mu^2 \;
e^{\phi({\vec x})}\right] \; , \cr \cr
T_{eff} &=& 4\pi \; {{G\; m^2}\over {T}} \quad , \quad
\mu^2 = \sqrt{2\over {\pi}}\; z\; G \, m^{7/2} \, \sqrt{T} \; ,
\end{eqnarray}
$ m $ stands for the mass of the atoms and $ z $ for the fugacity.
We show that in the $\phi$-field language, the particle density
expresses as
\begin{equation}\label{denfi}
<\rho({\vec r})> = -{1 \over {T_{eff}}}\;<\nabla^2 \phi({\vec r})>=
{{\mu^2}\over{T_{eff}}} \; <e^{\phi({\vec r})}> \; .
\end{equation}
where $ <\ldots > $ means functional average over $ \phi(.) $
with statistical weight $ e^{-S[\phi(.)]} $. Density correlators are
written as
\begin{eqnarray}\label{correI}
C({\vec r_1},{\vec r_2})&\equiv&
<\rho({\vec r_1})\rho({\vec r_2}) > -<\rho({\vec r_1})><\rho({\vec r_2}) >
\cr \cr
&=& {{\mu^4}\over{T_{eff}}^2} \; \left[
<e^{\phi({\vec r_1})} \; e^{\phi({\vec r_2})}> -
<e^{\phi({\vec r_1})}> \; <e^{\phi({\vec r_2})}> \right]\; .
\end{eqnarray}
The $\phi$-field defined by eqs.(\ref{zetafi})-(\ref{SmuyT}) has remarkable
properties under scale transformations
$$
{\vec x} \to {\vec x}_{\lambda} \equiv \lambda{\vec x} \; ,
$$
where $\lambda$ is an arbitrary real number. For any solution $
\phi({\vec x}) $ of the stationary point equations,
\begin{equation}\label{eqMovI}
\nabla^2\phi({\vec x}) + \mu^2 \; e^{\phi({\vec x})} = 0 \; ,
\end{equation}
there is a family of dilated solutions of the same equation (\ref{eqMovI}),
given by
$$
\phi_{\lambda}({\vec x}) \equiv \phi(\lambda{\vec x}) +\log\lambda^2
\; .
$$
In addition, $ S[\phi_{\lambda}(.)] = \lambda^{2-D} \; S[\phi(.)] $.
\bigskip
We study the field theory (\ref{zetafi})-(\ref{SmuyT}) both
perturbatively and non-perturbatively.
The computation of the thermal fluctuations through the evaluation of
the functional integral eq.(\ref{zetafi}) is quite non-trivial. We
use the scaling property as a guiding principle. In order to built a
perturbation theory in the dimensionless coupling $ g \equiv
\sqrt{\mu \, T_{eff}} $ we look for stationary points of
eq.(\ref{SmuyT}). We compute the density correlator eq.(\ref{correI}) to
leading order in $ g $. For large distances it behaves as
\begin{equation}\label{coRa}
C({\vec r_1},{\vec r_2}) \buildrel{ | {\vec r_1} - {\vec r_2}|\to
\infty}\over = {{ \mu^4 }\over {32\, \pi^2 \;
| {\vec r_1} - {\vec r_2}|^2}} + O\left( \; | {\vec
r_1} - {\vec r_2}|^{-3}\right)\; .
\end{equation}
We analyze further this theory with the renormalization group
approach. Such non-perturbative approach is the more powerful
framework to
derive scaling behaviours in field theory \cite{kgw,dg,nn}.
We show that the mass contained in a region of volume $ V =R^3 $ scales as
$$
<M(R)> = m \; \int^R
<e^{\phi({\vec x})}> \; d^3x \simeq m \, V \, a +m \, {K
\over{1-\alpha}}\; R^{ \frac1{\nu}} + \ldots\; ,
$$
and the mass fluctuation, $ (\Delta M(R))^2 = <M^2>-<M>^2 $,
scales as
$$
\Delta M(R) \sim R^{d_H}\; .
$$
Here $ \nu $ is the correlation length critical exponent for the
$\phi$-theory (\ref{zetafi}) and $ a $ and $ K $ are constants. Moreover,
\begin{equation}\label{SdensiI}
<\rho({\vec r})> = m a \; + m \, {K
\over{4\pi\nu(1-\alpha)}}\;r^{ \frac1{\nu}-3} \quad {\rm for}\; r
\; {\rm of ~ order}\;\sim R \; .
\end{equation}
The scaling exponent $ \nu $ can be identified with the inverse
Haussdorf (fractal) dimension $d_H$ of the system
$$
d_H = \frac1{\nu} \; .
$$
In this way, $ \Delta M\sim R^{d_H} $ according to the usual
definition of fractal dimensions \cite{sta}.
From the renormalization group analysis,
the density-density correlators (\ref{correI}) result to be,
\begin{equation}\label{corI}
C({\vec r_1},{\vec r_2})\sim |{\vec r_1} -{\vec r_2}|^{\frac2{\nu} -6} \; .
\end{equation}
Computing the average gravitational potential energy and using the
virial theorem yields for the velocity dispersion,
$$
\Delta v \sim R^{\frac12(\frac1{\nu} -1)} \; .
$$
This gives a new scaling relation between the exponents $ d_H $ and $ q $
$$
q =\frac12\left(\frac1{\nu} -1\right) =\frac12(d_H -1) \; .
$$
The perturbative calculation (\ref{coRa}) yields the mean field value
for $ \nu $ \cite{ll}. That is,
\begin{equation}\label{meanF}
\nu= \frac12 \quad , \quad d_H = 2 \quad {\rm and } \quad q = \frac12 \; .
\end{equation}
We find scaling behaviour in the $\phi$-theory for a {\bf continuum set} of
values of $\mu^2$ and $ T_{eff} $.
The renormalization group transformation amounts to replace
the parameters $ \mu^2 $ and $ T_{eff} $
in $ \beta\, H $ and $ S[\phi(.)] $ by the effective ones at the scale
$ L $ in question.
The renormalization group approach applied to a
{\bf single} component scalar field in three space dimensions
indicates that the long distance critical behaviour is governed by the
(non-perturbative) Ising fixed point \cite{kgw,dg,nn}.
Very probably, there are no further fixed points \cite{grexa}.
The scaling exponents associated to the Ising fixed point are
\begin{equation}\label{Isint}
\nu = 0.631... \quad , \quad d_H = 1.585... \quad {\rm and} \quad
q = 0.293...\; \; .
\end{equation}
Both the mean field (\ref{meanF}) and the Ising (\ref{Isint})
numerical values are compatible
with the present observational values (\ref{vobser}) - (\ref{expos}).
\bigskip
The theory presented here also predicts a power-law behaviour for
the two-points ISM density correlation function (see eq.(\ref{corI}),
$ 2 d_H - 6 = - 2.830\ldots$, for the Ising fixed point
and $ 2 d_H - 6 = - 2 $ for the mean field exponents),
that should be compared with observations. Previous attempts to
derive correlation functions from observations were not entirely conclusive,
because of lack of dynamical range \cite{klein}, but much more extended maps of
the ISM could be available soon to test our theory. In addition, we predict
an independent exponent for the gravitational
potential correlations ($ \sim r^{-1-\eta} $, where
$ \eta_{Ising}=0.037\ldots $ and $ \eta_{mean ~ field} = 0 $
\cite{dg}), which could be checked through
gravitational lenses observations in front of quasars.
\bigskip
The mass parameter $\mu $ [see eq.(\ref{SmuyT})] in the $\phi$-theory
turns to coincide at the tree level with the inverse of the Jeans length
$$
\mu = \sqrt{12 \over {\pi}}\; { 1 \over {d_J}} \; .
$$
We find that in the scaling domain the Jeans distance $ d_J $ grows
as $ <d_J> \sim R $. This shows that the Jeans distance {\bf scales}
with the {\bf size} of the system and therefore the instability is
present for all sizes $ R $. Had $ d_J $ being of order larger than
$ R $, the Jeans instability would be absent.
\bigskip
The gravitational gas in thermal equilibrium explains quantitatively
the observed scaling laws in the ISM. This fact does not exclude
turbulent phenomena in the ISM.
Fluid flows (including turbulent regimes) are probably relevant
in the dynamics (time dependent processes) of the ISM. As usual in critical
phenomena \cite{kgw,dg}, the equilibrium scaling laws can be understood
for the ISM without dwelling with the dynamics.
A further step in the study of the ISM will be to include the
dynamical (time dependent) description within the field theory
approach presented in this paper.
\bigskip
If the ISM is considered as a flow, the Reynolds number $Re_{ISM}$ on
scales $L \sim 100$pc has a very high value of the order of $10^6$.
This led to the suggestion that the ISM (and the universe in general)
could be {\bf modelled} as a turbulent flow \cite{weisz}.
(Larson \cite{larson} first observed that the
exponent in the power-law relation for the velocity dispersion is not greatly
different from the Kolmogorov value $1/3$ for subsonic turbulence).
It must be noticed that the turbulence hypothesis for the ISM is based on
the comparison of the ISM with the results known for incompressible
flows. However, the physical conditions in the ISM are
very different from those of incompressible flows in the laboratory.
(And the
study of ISM turbulence needs more complete and enlarged investigation
than those performed until now based in the concepts of flow turbulence
in the laboratory).
Besides the facts that the ISM exhibits large density fluctuations on all
scales, and the observed fluctuations are highly supersonic, (thus the
ISM can not viewed as an `incompressible' and `subsonic' flow),
and besides other differences, an essential feature to point out is that
the long-range self-gravity interaction present in the ISM is completely
absent in the studies of flow turbulence.
In any case, in a satisfactory theory of the ISM,
it should be possible to extract the behaviours of
the ISM (be turbulent or whatever) from the theory
as a result, instead to be introduced as a starting input or hypothesis.
This paper is organized as follows. In section II we develop the field
theory approach to the gravitational gas. A short distance cutoff is
naturally present here and prevents zero distance gravitational
collapse singularities (which would be unphysical in the present
case). Here, the cutoff theory is physically meaningful. The
gravitational gas is also treated in a $D$-dimensional space.
In section III we study the scaling behaviour and thermal fluctuations
both in perturbation theory and non-perturbatively (renormalization
group approach). $g^2 \equiv \mu\, T_{eff} $ acts as the dimensionless
coupling constant for the non-linear fluctuations of the field $\phi$.
We show that these fluctuations are massless and that the theory
scales (behaves critically) for a continuous range of values $ \mu^2
\; T_{eff} $. Thus, changing $ \mu^2 $ and $ T_{eff} $ keeps the
theory at {\bf criticality}. The renormalization group analysis made
in section III confirm such results. We also treat (sect. III.E) the
two dimensional case making contact with random surfaces and their
fractal dimensions.
Discussion and remarks are presented in section IV. External gravity forces to
the gas like stars are shown {\bf not} to affect the scaling behaviour
of the gas. That is, the scaling exponents $ q , \; d_H $ are solely
governed by fixed points and hence, they are stable under gravitational
perturbations.
In addition, we generalize the $\phi$-theory to a gas formed by
several types of atoms with different masses and fugacities. Again, the
scaling exponents are shown to be identical to the gravitational gas
formed of identical atoms.
The differences between the critical
behaviour of the gravitational gas and those in spin models (and other
statistical models in the same universality class) are also pointed
out in sec. IV.
\section{Field theory approach to the gravitational gas}
Let us consider a gas of non-relativistic atoms with mass $m$ interacting
only through Newtonian gravity and which are in thermal
equilibrium at temperature $ T \equiv \beta^{-1} $.
We shall work in the grand canonical ensemble, allowing for a variable
number of particles $N$.
The grand partition function of the system can be written as
\begin{equation}\label{gfp}
{\cal Z} = \sum_{N=0}^{\infty}\; {{z^N}\over{N!}}\; \int\ldots \int
\prod_{l=1}^N\;{{d^3p_l\, d^3q_l}\over{(2\pi)^3}}\; e^{- \beta H_N}
\end{equation}
where
\begin{equation}\label{hami3}
H_N = \sum_{l=1}^N\;{{p_l^2}\over{2m}} - G \, m^2 \sum_{1\leq l < j\leq N}
{1 \over { |{\vec q}_l - {\vec q}_j|}}
\end{equation}
$G$ is Newton's constant and $z$ is the fugacity.
The integrals over the momenta $p_l, \; (1 \leq l \leq N) $
can be performed explicitly in eq.(\ref{gfp})
using
$$
\int\;{{d^3p}\over{(2\pi)^3}}\; e^{- {{\beta p^2}\over{2m}}} =
\left({m \over{2\pi \beta}}\right)^{3/2}
$$
We thus find,
\begin{equation}\label{gfp2}
\displaystyle{
{\cal Z} = \sum_{N=0}^{\infty}\; {1 \over{N!}}\;
\left [ z\left({m \over{2\pi \beta}}\right)^{3/2}\right]^N
\; \int\ldots \int
\prod_{l=1}^N d^3q_l\;\; e^{ \beta G \, m^2 \sum_{1\leq l < j\leq N}
{1 \over { |{\vec q}_l - {\vec q}_j|}} }}
\end{equation}
We proceed now to recast this many-body problem into a field theoretical
form \cite{origen,stra,sam,kh}.
Let us define the density
\begin{equation}\label{defro}
\rho({\vec r})= \sum_{j=1}^N\; \delta({\vec r}- {\vec q}_j)\; ,
\end{equation}
such that, we can rewrite the potential energy in eq.(\ref{gfp2}) as
\begin{equation}\label{PotE}
\frac12 \, \beta G \, m^2 \sum_{1\leq l \neq j\leq N}
{1 \over { |{\vec q}_l - {\vec q}_j|}} = \frac12\, \beta \, G \, m^2
\int_{ | {\vec x} - {\vec y}|> a}\;
{{d^3x\, d^3y}\over { | {\vec x} - {\vec y}|}}\; \rho({\vec x})
\rho({\vec y}) \; .
\end{equation}
The cutoff $ a $ in the r.h.s. is introduced in order to avoid
self-interacting divergent terms. However, such divergent terms would
contribute to ${\cal Z}$ by
an infinite multiplicative factor that can be factored out.
By using
$$
\nabla^2 { 1 \over { | {\vec x} - {\vec y}|}}= -4\pi \; \delta( {\vec
x} - {\vec y}) \; ,
$$
and partial integration we can now represent the exponent of the
potential energy eq.(\ref{PotE}) as a functional integral\cite{stra}
\begin{equation}\label{reprf}
e^{ \frac12\, \beta G \, m^2
\int \;
{{d^3x\, d^3y}\over { | {\vec x} - {\vec y}|}}\; \rho({\vec x})
\rho({\vec y})} = \int\int\; {\cal D}\xi \; e^{ -\frac12\int d^3x \; (\nabla
\xi)^2 \; + \; 2 m \sqrt{\pi G\beta}\; \int d^3x \; \xi({\vec x})\;
\rho({\vec x}) }
\end{equation}
Inserting this expression into eq.(\ref{gfp2}) and using
eq.(\ref{defro}) yields
\begin{eqnarray}\label{gfp3}
{\cal Z} &=& \sum_{N=0}^{\infty}\; {1 \over{N!}}\;
\left [ z\left({m \over{2\pi \beta}}\right)^{3/2}\right]^N\;
\int\int\; {\cal D}\xi \; e^{ -\frac12\int d^3x \; (\nabla \xi)^2}
\; \int\ldots \int
\prod_{l=1}^N d^3q_l\; \; e^{ 2 m \sqrt{\pi G\beta}\; \sum_{l=1}^N
\xi({\vec q}_l)} \cr \cr
&=& \int\int\; {\cal D}\xi \; e^{ -\frac12\int d^3x \;(\nabla \xi)^2}\;
\sum_{N=0}^{\infty}\; {1 \over{N!}}\;
\left [ z\left({m \over{2\pi \beta}}\right)^{3/2}\right]^N\;
\left[ \int d^3q \; e^{ 2 m \sqrt{\pi G\beta}\;\xi({\vec q})}
\right]^N \cr \cr
&=& \int\int\; {\cal D}\xi \; e^{ -\int d^3x \left[ \frac12(\nabla \xi)^2\;
- z \left({m \over{2\pi \beta}}\right)^{3/2}\; e^{ 2 m \sqrt{\pi
G\beta}\;\xi({\vec x})}\right]} \; \quad .
\end{eqnarray}
It is convenient to introduce the dimensionless field
\begin{equation}
\phi({\vec x}) \equiv 2 m \sqrt{\pi G\beta}\;\xi({\vec x}) \; .
\end{equation}
Then,
\begin{equation}\label{zfi}
{\cal Z} = \int\int\; {\cal D}\phi\; e^{ -{1\over{T_{eff}}}\;
\int d^3x \left[ \frac12(\nabla\phi)^2 \; - \mu^2 \; e^{\phi({\vec
x})}\right]}\; ,
\end{equation}
where
\begin{equation}\label{muyT}
\mu^2 = \sqrt{2\over {\pi}}\; z\; G \, m^{7/2} \, \sqrt{T}
\quad , \quad T_{eff} = 4\pi \; {{G\; m^2}\over {T}} \; .
\end{equation}
The partition function for the gas of particles in gravitational
interaction has been transformed into the partition function for a single
scalar field $\phi({\vec x})$ with {\bf local} action
\begin{equation}\label{acci}
S[\phi(.)] \equiv {1\over{T_{eff}}}\;
\int d^3x \left[ \frac12(\nabla\phi)^2 \; - \mu^2 \; e^{\phi({\vec
x})}\right] \; .
\end{equation}
The $\phi$ field exhibits an exponential self-interaction $ - \mu^2
\; e^{\phi({\vec x})} $.
Notice that the effective
temperature $ T_{eff} $ for the $\phi$-field partition function
turns out to be {\bf inversely}
proportional to $ T $ whereas the characteristic length $\mu^{-1}$ behaves
as $ \sim T ^{-1/4}$. This is a duality-type mapping between the two models.
It must be noticed that the term $ - \mu^2 \; e^{\phi({\vec x})} $
makes the $\phi$-field energy density unbounded from
below. Actually, the initial Hamiltonian (\ref{gfp}) is also unbounded from
below. This unboundness physically originates in the attractive
character of the gravitational force. Including a short-distance
cutoff [see sec. 2A, below] eliminates the zero distance singularity
and hence the possibility of zero-distance
collapse which is unphysical in the present context.
We therefore expect meaningful physical results in the
cutoff theory. Moreover, assuming zero boundary conditions for
$\phi({\vec r})$ at $ r \to \infty $ shows that the derivatives of
$\phi$ must also be large if $ e^\phi$ is large. Hence, the term $
\frac12(\nabla\phi)^2 $ may stabilize the energy.
The action (\ref{acci}) defines a non-renormalizable field
theory for any number of dimensions $ D > 2 $ [see
eq.(\ref{zfiD})]. This is a further reason to keep the short-distance
cutoff non-zero.
\bigskip
Let us compute now the statistical average value of the density
$\rho({\vec r})$ which in the grand canonical ensemble is given by
\begin{equation}
<\rho({\vec r})> = {\cal Z}^{-1}\; \sum_{N=0}^{\infty}\; {1 \over{N!}}\;
\left [ z\left({m \over{2\pi \beta}}\right)^{3/2}\right]^N
\; \int\ldots \int
\prod_{l=1}^N d^3q_l\; \; \rho({\vec r}) \;
e^{ \frac12\, \beta G \, m^2 \sum_{1\leq l \neq j\leq N}
{1 \over { |{\vec q}_l - {\vec q}_j|}} }\; .
\end{equation}
As usual in the functional integral calculations,
it is convenient to introduce sources in the partition function (\ref{zfi})
in order to compute average values of fields
\begin{equation}\label{zfiJ}
{\cal Z}[J(.)] \equiv \int\int\; {\cal D}\phi\; e^{ -{1\over{T_{eff}}}\;
\int d^3x \left[ \frac12(\nabla
\phi)^2 \; - \mu^2 \; e^{\phi({\vec x})}\; \right]
+\int d^3x \;J({\vec x})\; \phi({\vec x}) \; }\; .
\end{equation}
The average value of $ \phi({\vec r}) $ then writes as
\begin{equation}
< \phi({\vec r})> = {{\delta \log{\cal Z} }\over{\delta J({\vec r})}}\; .
\end{equation}
In order to compute $<\rho({\vec r})>$ it is useful to introduce
\begin{equation}
{\cal V}[J(.)] \equiv \frac12 \,\beta G \, m^2
\int_{ | {\vec x} - {\vec y}|> a }\;
{{d^3x\, d^3y}\over { | {\vec x} - {\vec y}|}}\;
\left[ \rho({\vec x})+ \;J({\vec x})\;\right]
\left[\rho({\vec y})+ \;J({\vec y})\;\right]\; .
\end{equation}
Then, we have
$$
\rho({\vec r}) \; e^{{\cal V}[0]} = -{1 \over{T_{eff}}} \; \nabla^2_{\vec
r} \left({{\delta}\over{\delta J({\vec r})}} e^{{\cal V}[J(.)]}
\right)|_{J=0}\; .
$$
By following the same steps as in eqs.(\ref{reprf})-(\ref{gfp3}), we find
\begin{eqnarray}
<\rho({\vec r})> &=& -{1 \over{T_{eff}}} \; \nabla^2_{\vec
r} \left({{\delta}\over{\delta J({\vec r})}}
\sum_{N=0}^{\infty}\; {1 \over{N!}}\;
\left [ z\left({m \over{2\pi \beta}}\right)^{3/2}\right]^N
\; \; {\cal Z}[0]^{-1} \right.\cr \cr
\int\int \; {\cal D}\xi & & \left. e^{ -\int d^3x\left[\frac12 \;
(\nabla \xi)^2
- 2 m \sqrt{\pi G\beta}\;\xi({\vec x})\; J({\vec x})\right]}\;
\; \int\ldots \int
\prod_{l=1}^N d^3q_l\; \; e^{ 2 m \sqrt{\pi G\beta}\; \sum_{l=1}^N
\xi({\vec q}_l)}\right)|_{J=0} \cr\cr
&=& -{1 \over{T_{eff}}} \; \nabla^2_{\vec
r} \left({{\delta}\over{\delta J({\vec r})}}\;\log {\cal Z}[J(.)]\right)|_{J=0}
\quad .
\end{eqnarray}
Performing the derivatives in the last formula yields
\begin{equation}
<\rho({\vec r})> = - {1 \over {T_{eff}}}\; \int\int\; {\cal D}\phi\; \;
\nabla^2 \phi({\vec r})\;
e^{-{1\over{T_{eff}}}\; \int d^3x \left[ \frac12(\nabla
\phi)^2 \; - \mu^2 \; e^{\phi({\vec x})}\;\right]}\; {\cal Z}[0]^{-1}\; .
\end{equation}
One can analogously prove that $ \rho({\vec r}) $ inserted in any
correlator becomes $ -{1 \over {T_{eff}}}\; \nabla^2 \phi({\vec r}) $
in the $\phi$-field language. Therefore, we can express the particle density
operator as
\begin{equation}\label{rouno}
\rho({\vec r}) = -{1 \over {T_{eff}}}\; \nabla^2 \phi({\vec r}) \; .
\end{equation}
Let us now derive the field theoretical equations of motion. Since the
functional integral of a total functional derivative identically
vanishes, we can write
$$
\int\int\; {\cal D}\phi\; \;\left[ - {{\delta S
}\over{\delta \phi({\vec r})}} + J({\vec r}) \right] e^{-S[\phi(.)] +
\int d^3x \;J({\vec x})\; \phi({\vec x}) \; } = 0
$$
We get from eq.(\ref{acci})
$$
{{\delta S}\over{\delta \phi({\vec r})}} = - {1\over{T_{eff}}}\;
\left[ \nabla^2\phi({\vec r}) \; + \mu^2 \; e^{\phi({\vec
r})}\right]\; .
$$
Thus, setting $ J({\vec r}) \equiv 0 $,
\begin{equation}\label{ecmov}
< \nabla^2\phi({\vec r}) > + \; \mu^2 \; <e^{\phi({\vec r})}> = 0
\end{equation}
Now, combining eqs.(\ref{rouno}) and (\ref{ecmov}) yields
\begin{equation}\label{densi}
<\rho({\vec r})>={{\mu^2}\over{T_{eff}}} \; <e^{\phi({\vec r})}> \; .
\end{equation}
\bigskip
By using eq.(\ref{rouno}), the gravitational potential at the point $ \vec r $
$$
U( \vec r ) = -G m \int {{d^3x} \over { | {\vec x} - {\vec r}|}}\;
\rho({\vec x}) \; ,
$$
can be expressed as
\begin{equation}\label{Ufi}
U( \vec r ) = - {T \over m}\; \phi( \vec r ) \; .
\end{equation}
We can analogously express the correlation functions as
\begin{eqnarray}\label{corre}
C({\vec r_1},{\vec r_2})&\equiv&
<\rho({\vec r_1})\rho({\vec r_2}) > -<\rho({\vec r_1})><\rho({\vec r_2}) >
\cr \cr
&=& \left({1 \over{T_{eff}}} \right)^2\; \nabla^2_{\vec r_1}\;
\nabla^2_{\vec r_2} \;
\left({{\delta}\over{\delta J({\vec r_1})}}\;{{\delta}\over{\delta
J({\vec r_2})}}\; \log{\cal Z}[J(.)]\right)|_{J=0} \; .
\end{eqnarray}
This can be also written as
\begin{equation}\label{corr2}
C({\vec r_1},{\vec r_2}) = {{\mu^4}\over{T_{eff}}^2} \; \left[
<e^{\phi({\vec r_1})} \; e^{\phi({\vec r_2})}> -
<e^{\phi({\vec r_1})}> \; <e^{\phi({\vec r_2})}> \right]\; .
\end{equation}
\subsection{Short distances cutoff}
A simple short distance regularization of the Newtonian force for the
two-body potential is
$$
v_a({\vec r}) = -{{G m^2} \over r}\; [ 1 - \theta(a-r) ] \; ,
$$
$ \theta(x)$ being the step function. The cutoff $ a $ can be chosen of
the order of atomic distances but its actual value is unessential.
The $N$-particle regularized Hamiltonian takes then the form
\begin{equation}\label{hamiR}
H_N = \sum_{l=1}^N\;{{p_l^2}\over{2m}} + \frac12\,
\sum_{1\leq l, j\leq N} \; v_a({\vec q}_l - {\vec q}_j) \; .
\end{equation}
Notice that now we can include in the sum terms with $l = j $
since $ v_a(0) = 0 $.
The steps from eq.(\ref{hami3}) to eq.(\ref{zfi}) can be just
repeated by using now the regularized $v_a({\vec r})$. Notice that we
must use now the inverse operator of $ v_a({\vec r}) $ instead of that
of $ 1/r , \; \left[ -\frac1{4\pi}\nabla^2 \right] $ , previously used.
We now find,
\begin{equation}\label{zfiR}
{\cal Z}_a = \int\int\; {\cal D}\phi\; e^{ -{1\over{T_{eff}}}\;
\int d^3x \left[ \frac12\phi K_a \phi \; - \mu^2 \; e^{\phi({\vec
x})}\right]}\; ,
\end{equation}
i. e.
\begin{equation}\label{Sregu}
S_a[\phi(.)] = {1\over{T_{eff}}}\;
\int d^3x \left[ \frac12\phi K_a \phi \; - \mu^2 \; e^{\phi({\vec
x})}\right] \; ,
\end{equation}
where $ K_a $ is the inverse operator of $ v_a $,
\begin{eqnarray}
K_a \phi ({\vec r}) &=& \int K_a ({\vec r}-{\vec r}\, ') \;
\phi ({\vec r}\,')\; d^3r' \cr \cr
\int K_a({\vec r}-{\vec r}\,'')\; &{1 \over {4\pi}}&\;
{ {1 - \theta(a- |{\vec r}\,''-{\vec r}\,'|
) }\over {|{\vec r}\,''-{\vec r}\,'|}}\; d^3r'' =
\delta ({\vec r}-{\vec r}\,')\nonumber
\end{eqnarray}
$ K_a ({\vec r})$ admits the Fourier representation,
$$
K_a ({\vec r}) = V.P.\int {{d^3p}\over {(2\pi)^3}}\; {{p^2\; e^{i {\vec
p}.{\vec r}}}\over {\cos pa}}\; .
$$
Actually, $ K_a ({\vec r}) = 0 $ for $r \neq 0$. $ K_a ({\vec r})$ has
the following asymptotic expansion in powers of the cutoff $ a^2 $
\begin{equation}\label{deska}
K_a ({\vec r}) = -\nabla^2 \delta ({\vec r}) + {{a^2}\over 2} \;
\nabla^4 \delta ({\vec r}) + O(a^4) \; ,
\end{equation}
and then
\begin{equation}\label{Sar}
S_a[\phi(.)] = S[\phi(.)]
+ {{a^2}\over 2} \; \int d^3x\; (\nabla^2\phi)^2 + O(a^4) \; .
\end{equation}
As we see, the high orders in $ a^2 $ are irrelevant operators
which do not affect the scaling behaviour, as is well known from
renormalization group arguments. For $a \to 0$, the action (\ref{acci}) is
recovered.
\subsection{D-dimensional generalization}
This approach generalizes immediately to $D$-dimensional space where
the Hamiltonian (\ref{hami3}) takes then the form
\begin{equation}\label{hamiD}
H_N = \sum_{l=1}^N\;{{p_l^2}\over{2m}} - G \, m^2 \sum_{1\leq l < j\leq N}
{1 \over { |{\vec q}_l - {\vec q}_j|^{D-2}}},\quad {\rm for}\; D \neq 2
\end{equation}
and
\begin{equation}\label{hami2}
H_N = \sum_{l=1}^N\;{{p_l^2}\over{2m}} - G \, m^2 \sum_{1\leq l < j\leq N}
\log{1 \over { |{\vec q}_l - {\vec q}_j|}}, \quad {\rm at}\; D= 2\; .
\end{equation}
The steps from eq.(\ref{gfp}) to (\ref{zfi}) can be trivially
generalized with the help of the relation
\begin{equation}\label{Dgreen}
\nabla^2 { 1 \over { | {\vec x} - {\vec y}|^{D-2}}}= -C_D \; \delta( {\vec
x} - {\vec y}) \;
\end{equation}
in $D$-dimensions and
$$
\nabla^2 \log{ 1 \over { | {\vec x} - {\vec y}|}}= -C_2 \; \delta( {\vec
x} - {\vec y}) \;
$$
at $ D= 2$.
Here,
\begin{equation}
C_D \equiv (D-2)\, {{2 \pi^{D/2}}\over {\Gamma(\frac{D}{2})}}\; \;
{\rm for}~~D\neq 2 \quad {\rm and}~~ C_2 \equiv 2\pi\; .
\end{equation}
We finally obtain as a generalization of eq.(\ref{zfi}),
\begin{equation}\label{zfiD}
{\cal Z} = \int\int\; {\cal D}\phi\; e^{ -{1\over{T_{eff}}}\;
\int d^Dx \left[ \frac12(\nabla\phi)^2 \; - \mu^2 \; e^{\phi({\vec
x})}\right]}\; ,
\end{equation}
where
\begin{equation}\label{paramD}
\mu^2 = {{C_D} \over {(2\pi)^{D/2}}}\;
z\; G \, m^{2+D/2} \, T^{D/2-1}
\quad , \quad T_{eff} = C_D \; {{G\; m^2}\over {T}} \; .
\end{equation}
We have then transformed
the partition function for the $D$-dimensional
gas of particles in gravitational interaction into the partition
function for a
scalar field $\phi$ with exponential interaction.
The effective
temperature $ T_{eff} $ for the $\phi$-field partition function is
{\bf inversely}
proportional to $ T $ for {\bf any} space dimension. The characteristic length
$\mu^{-1}$ behaves as $ \sim T^{-(D-2)/4} $.
\section{Scaling behaviour}
We derive here the scaling behaviour of the $\phi$ field following the
general renormalization group
arguments in the theory of critical phenomena \cite{kgw,dg}
\subsection{Classical Scale Invariance}
Let us investigate how the action (\ref{acci}) transforms under scale
transformations
\begin{equation}\label{trafoS}
{\vec x} \to {\vec x}_{\lambda} \equiv \lambda{\vec x} \; ,
\end{equation}
where $\lambda$ is an arbitrary real number.
In $D$-dimensions the action takes the form
\begin{equation}\label{acciD}
S[\phi(.)] \equiv {1\over{T_{eff}}}\;
\int d^D x \left[ \frac12(\nabla\phi)^2 \; - \mu^2 \; e^{\phi({\vec
x})}\right] \; .
\end{equation}
We define the scale transformed field $\phi_{\lambda}({\vec x})$ as follows
\begin{equation}\label{filam}
\phi_{\lambda}({\vec x}) \equiv \phi(\lambda{\vec x}) +\log\lambda^2
\; .
\end{equation}
Hence,
$$
(\nabla\phi_{\lambda}({\vec x}))^2 = \lambda^2 \; (\nabla_{x_{\lambda}}
\phi({\vec x}_{\lambda}))^2
\quad , \quad e^{\phi_{\lambda}({\vec x})}= \lambda^2 \;
e^{\phi({\vec x}_{\lambda})}
$$
We find upon changing the integration variable in eq.(\ref{acciD})
from $ {\vec x} $ to $ {\vec x}_{\lambda} $
\begin{equation}\label{covdil}
S[\phi_{\lambda}(.)] = \lambda^{2-D} \; S[\phi(.)]
\end{equation}
We thus see that the action (\ref{acciD}) {\bf scales} under dilatations
in spite of the
fact that it contains the dimensionful parameter $ \mu^2 $.
This remarkable scaling property is of course a consequence of the
scale behaviour of the gravitational interaction (\ref{hamiD}).
In particular, in $ D = 2 $
the action (\ref{acciD}) is scale invariant. In such
special case, it is moreover conformal invariant.
\bigskip
The (Noether) current associated to the scale transformations
(\ref{trafoS}) is
\begin{equation}
J_i({\vec x}) = x_j\; T_{ i j} ({\vec x}) + 2 \;\nabla_i\phi({\vec x})\; ,
\end{equation}
where $ T_{ij} ({\vec x}) $ is the stress tensor
$$
T_{ i j} ({\vec x}) = \nabla_i\phi({\vec x}) \; \nabla_j\phi({\vec x})
- \delta_{ij} \; L
$$
and $L \equiv \frac12(\nabla\phi)^2 \; - \mu^2 \; e^{\phi({\vec x})}
$ stands for the action density. That is,
$$
J_i({\vec x}) = ({\vec x}. \nabla\phi + 2)\; \nabla_i\phi({\vec x})-
x_i \; \left[ \frac12(\nabla\phi)^2 \; - \mu^2 \; e^{\phi({\vec x})}\right]
$$
By using the classical equation of motion (\ref{eqMov}), we then find
$$
\nabla_i J_i({\vec x}) = (2 - D) L \; .
$$
This non-zero divergence is due to the variation of the action under
dilatations [eq. (\ref{covdil})].
\bigskip
If $\phi({\vec x})$ is a stationary point of the action (\ref{acciD}):
\begin{equation}\label{eqMov}
\nabla^2\phi({\vec x}) + \mu^2 \; e^{\phi({\vec x})} = 0 \; ,
\end{equation}
then $ \phi_{\lambda}({\vec x}) $ [defined by eq.(\ref{filam})] is
also a stationary point:
$$
\nabla^2\phi_{\lambda}({\vec x}) + \mu^2 \; e^{\phi_{\lambda}({\vec
x})} = 0 \; .
$$
A rotationally invariant stationary point is given by
\begin{equation}\label{fic}
\phi^c(r) = \log{{2(D-2)}\over { \mu^2 r^2}} \; .
\end{equation}
This singular solution is {\bf invariant} under the scale
transformations (\ref{filam}). That is
$$
\phi^c_{\lambda}(r) =\phi^c(r) \; .
$$
Eq.(\ref{fic}) is dilatation and rotation invariant.
It provides the {\bf most symmetric} stationary point of
the action. Notice that there are no constant stationary solutions
besides the singular solution $ \phi_0 = -\infty $.
The introduction of the short distance cutoff $a$, eq.(\ref{hamiR}),
spoils the scale behaviour (\ref{covdil}). For the cutoff theory
from eqs.(\ref{Sregu}) and (\ref{trafoS})-(\ref{filam}),
we have instead
$$
S_a[\phi_{\lambda}(.)] = \lambda^{2-D} \; S_{\lambda a}[\phi(.)] \; .
$$
For $ r \sim a $, eq.(\ref{fic}) does not hold anymore for the
spherically symmetric solution $\phi^c(r)$. For small $ r $ and $ a $,
using eqs.(\ref{Sregu}-\ref{Sar}) we have
\begin{equation}\label{fica}
\phi^c(r) \buildrel{r\to 0}\over = -{{ \mu^2 r^2}\over {2 D}}
+ O(r^2, r^2 a^2)\; .
\end{equation}
That is, $\phi^c(r)$ is regular at $r = 0$ in the presence of the
cutoff $a$.
\subsection{Thermal Fluctuations}
In this section we compute the partition function eqs.(\ref{zfi}) and
(\ref{zfiJ}) by saddle point methods.
Eq.(\ref{eqMov}) admits only one constant stationary solution
\begin{equation}\label{fiS}
\phi_0 = -\infty \; .
\end{equation}
In order to make such solution finite we now introduce a
regularization term $ \; \epsilon \, \mu^2 \, \phi({\vec x}) $ with $
\epsilon << 1 $ in the action $ S $ [eq.(\ref{acci})]. This
corresponds to an action density
\begin{equation}\label{densac}
L = \frac12(\nabla\phi)^2 \; + \; u(\phi)
\end{equation}
where
$$
u(\phi) = - \mu^2 \; e^{\phi({\vec x})} + \epsilon \; \mu^2 \;
\phi({\vec x}) \; .
$$
This extra term can be obtained by adding a small constant term $
-\epsilon \; \mu^2/T_{eff} $ to $\rho({\vec x})$ in eqs.(\ref{defro}) -
(\ref{reprf}). This is a simple way to make $ \phi_0 $ finite.
We get in this way a constant stationary point at $ \phi_0 =
\log\epsilon $ where $ u'(\phi_0) = 0 $. However, scale invariance is
broken since $ u''(\phi_0) = - \epsilon \; \mu^2 \neq 0 $. We can add
a second regularization term to $ \; \frac12 \, \delta \, \mu^2 \;
\phi({\vec x})^2 \; $ to $ L $, (with $ \delta << 1 $) in order to enforce $
u''(\phi_0) = 0 $. This quadratic term amounts to a long-range
shielding of the gravitational force.
We finally set:
$$
u(\phi) = - \mu^2 \left[ e^{\phi({\vec x})} - \epsilon \;
\phi({\vec x}) - \frac12 \; \delta \; \phi({\vec x})^2 \right]
\; ,
$$
where the two regularization parameters $ \epsilon$ and $ \delta $ are
related by
$$
\epsilon( \delta ) = \delta [1 - \log \delta] \; ,
$$
and the stationary point has the value
$$
\phi_0 =\log\delta \; .
$$
Expanding around $ \phi_0 $
$$
\phi({\vec x}) = \phi_0 + g \; \chi({\vec x})
$$
where $ g \equiv \sqrt{\mu^{D-2} \, T_{eff}} $ and
$ \chi({\vec x}) $ is the fluctuation field, yields
\begin{equation}\label{fiinfi}
\frac{1}{g^2}\; L = \frac12 \; (\nabla\chi)^2 \;
- {{\mu^2 \delta}\over {g^2}} \left[ e^{g \chi} -1 - g \; \chi -
\frac12 \; g^2 \; \chi^2 \right]
\end{equation}
We see
perturbatively in $g$ that $ \chi({\vec x}) $ is a {\bf massless} field.
\bigskip
Concerning the boundary conditions, we must consider the system inside
a large sphere of radius $R \; ( 10^{-4}\; - \; 10^{-2}
\; pc \; \leq R \leq 100\; pc )$. That is, all
integrals are computed over such large sphere.
\bigskip
Using eq.(\ref{rouno}) the particle density takes now the form
$$
\rho({\vec r}) = -{1 \over {T_{eff}}}\; \nabla^2 \phi({\vec r}) =
-{g \over {T_{eff}}}\; \nabla^2 \chi({\vec r}) = {{\mu^2
\delta}\over { T_{eff} }} \left[ e^{g \chi({\vec r})} -1 - g
\chi({\vec r}) \right] \; .
$$
It is convenient to renormalize the particle density
by its stationary value $ \delta = e^{\phi_0} $,
\begin{equation}\label{renro}
\rho({\vec r})_{ren} \equiv \frac{1}{\delta} \; \rho({\vec r}) =
{{\mu^D}\over {g^2 }} \left[ e^{g \chi({\vec r})} -1 - g \chi({\vec
r}) \right] \; .
\end{equation}
We see that in the $ \delta \to 0 $ limit the interaction in
eq.(\ref{fiinfi}) vanishes. No infrared divergences appear in the
Feynman graphs calculations, since we work on a
very large but finite volume of size $ R $. Hence, in the $\delta \to
0 $ limit, the whole perturbation series around $ \phi_0 $ reduces to the
zeroth order term.
The constant saddle point $ \phi_0 $ fails to catch up the whole field
theory content. In fact, more information arises perturbing around the
stationary point $ \phi^c(r) $ given by eq.(\ref{fic}) \cite{fut}.
Using eqs.(\ref{corr2}), (\ref{Dgreen}), (\ref{fiinfi}) and
(\ref{renro}) we obtain for the density correlator in the $ \delta
\to 0 $ limit,
$$
C({\vec r_1},{\vec r_2}) = {{\mu^{2 D}}\over {g^4}}\;
\left\{ \exp\!\left[{{g^2}\over { C_D\; \,
\left( \mu \, | {\vec r_1} - {\vec r_2}|\right)^{D-2}}} \right] -1
- {{g^2}\over { C_D \; \left( \mu \, | {\vec r_1} - {\vec
r_2}|\right)^{D-2}}} \right\} \; .
$$
For large distances, we find
\begin{equation}\label{corrasi}
C({\vec r_1},{\vec r_2}) \buildrel{ | {\vec r_1} - {\vec r_2}|\to
\infty}\over = {{ \mu^4 }\over {2\, C_D^2 \;
| {\vec r_1} - {\vec r_2}|^{2(D-2)}}} + O\left( \; | {\vec
r_1} - {\vec r_2}|^{-3(D-2)}\right)\; .
\end{equation}
That is, the $\phi$-field theory {\bf scales}. Namely, the
theory behaves critically for a {\bf continuum set} of values of $\mu$ and
$ T_{eff} $.
Notice that the
density correlator $C({\vec r_1},{\vec r_2})$ behaves for large
distances as the correlator of $ \chi({\vec r})^2 $. This stems from
the fact that $ \chi({\vec r})^2 $ is the most relevant operator in
the series expansion of the density (\ref{renro})
\begin{equation}\label{denser}
\rho({\vec r})_{ren} = \frac12 \; \mu^D \; \chi({\vec r})^2 + O(\chi^3)\; .
\end{equation}
As remarked above, the constant stationary point $ \phi_0 = \log\delta \to
-\infty $ only produces the zeroth order of perturbation theory.
More information arises perturbing around the stationary point $ \phi^c(r) $
given by eq.(\ref{fic}) \cite{fut}.
\subsection{Renormalization Group Finite Size Scaling Analysis}
As is well known \cite{kgw,dg,nn}, physical quantities for {\bf
infinite} volume systems diverge at the critical point as $ \Lambda $ to a
negative power. $ \Lambda $ measures the distance to the critical
point. (In condensed matter and spin systems, $ \Lambda $ is
proportional to the temperature minus the critical temperature \cite{dg,nn}).
One has for the correlation length $ \xi $,
$$
\xi( \Lambda ) \sim \Lambda^{-\nu} \; ,
$$
and for the specific heat (per unit volume) $ {\cal C} $,
\begin{equation}\label{calor}
{\cal C} \sim \Lambda^{-\alpha} \; .
\end{equation}
Correlation functions scale at criticality. For example, the
scalar field $\phi$ (which in spin systems describes the magnetization)
scales as,
$$
<\phi({\vec r})\phi(0)> \sim r^{-1-\eta} \; .
$$
The critical exponents $\nu, \;\alpha $ and $ \eta $ are pure numbers
that depend only on the universality class \cite{kgw,dg,nn}.
For a {\bf finite} volume system, all physical quantities are {\bf
finite} at the critical point. Indeed, for a system whose size $ R $
is large, the physical magnitudes
take large values at the critical point. Thus, for large $ R $, one can
use the infinite volume theory to treat finite size systems at
criticality. In particular, the correlation length provides the
relevant physical length $ \xi \sim R $. This implies that
\begin{equation}\label{fss}
\Lambda \sim R^{-1/\nu} \; .
\end{equation}
We can apply these concepts to the $\phi$-theory
since, as we have seen in the previous section, it
exhibits scaling in a finite volume $\sim R^3 $.
Namely, the two points correlation function exhibits a power-like
behaviour in perturbation theory as shown by eq.(\ref{corrasi}). This happens
for a {\bf continuum set} of values of
$T_{eff}$ and $\mu^2$. Therefore, changing $\mu^2/T_{eff}$ keeps the
theory in the scaling region.
At the point $ \mu^2/T_{eff} = 0 $, the partition function $ {\cal Z}
$ is singular. From eq.(\ref{muyT}), we shall thus identify
\begin{equation}\label{zcritico}
\Lambda \equiv {{\mu^2}\over{T_{eff}}} = z\,
\left({{mT}\over{2\pi}}\right)^{3/2} \; .
\end{equation}
Notice that the critical point $ \Lambda = 0 $, corresponds to zero
fugacity.
Thus, the partition function in the scaling regime can be written as
\begin{equation}\label{Zsca1}
{\cal Z}(\Lambda) =
\int\int\; {\cal D}\phi\; e^{ -S^* + \Lambda
\int d^Dx \; e^{\phi({\vec x})}\;}\; ,
\end{equation}
where $S^*$ stands for the action (\ref{acci}) at the critical point
$\Lambda = 0 $.
We define the renormalized mass density as
\begin{equation}\label {dfensi}
m\, \rho({\vec x})_{ren} \equiv m\, \, e^{\phi({\vec x})}
\end{equation}
and we identify it with the energy density in the renormalization
group. [Also called the `thermal perturbation operator'].
This identification follows from the fact that they are the most
relevant positive definite operators. Moreover, such identification is
supported by the perturbative result (\ref{denser}).
In the scaling regime we have \cite{dg} for the logarithm of the
partition function
\begin{equation}\label{Zsca2}
{1 \over V} \; \log{\cal Z}(\Lambda) = {K \over{(2-\alpha)(1-\alpha)}}\;
\Lambda^{2-\alpha} + F(\Lambda) \; ,
\end{equation}
where $ F(\Lambda) $ is an analytic
function of $ \Lambda $ around the origin
$$
F(\Lambda) = F_0 + a \; \Lambda + \frac12 \, b \; \Lambda^2 + \ldots \; .
$$
$ V = R^D $ stands for the volume and $ F_0, \; K, \; a $ and $ b $
are constants.
Calculating the logarithmic derivative of ${\cal Z}(\Lambda)$ with
respect to $ \Lambda $ from eqs.(\ref{Zsca1}) and from (\ref{Zsca2})
and equating the results yields
\begin{equation}\label{masaR}
{1 \over V} \;{{\partial}\over{\partial\Lambda}}\log{\cal Z}(\Lambda)=
a + {K \over{1-\alpha}}\,
\Lambda^{1-\alpha}
+ \ldots = {1 \over V} \int d^Dx \; <e^{\phi({\vec x})}>\; .
\end{equation}
where we used the scaling
relation $ \alpha = 2 - \nu D $ \cite{dg,nn}.
We can apply here finite size scaling arguments and
replace $\Lambda$ by $\sim R^{-\frac{1}{\nu}}$ [eq.(\ref{fss})],
$$
{{\partial}\over{\partial\Lambda}}\log{\cal Z}(\Lambda)= V \, a +
{K \over{1-\alpha}}\, R^{1/\nu} + \ldots\; .
$$
Recalling eq.(\ref{dfensi}), we can express the mass contained in a region
of size $ R $ as
\begin{equation}\label{defM}
M(R) = m \int^R e^{\phi({\vec x})} \; d^Dx \; .
\end{equation}
Using eq.(\ref{masaR}) we find
$$
<M(R)> = m \, V \, a +m \, {K \over{1-\alpha}}\; R^{ \frac1{\nu}} +
\ldots\; .
$$
and
\begin{equation}\label{Sdensi}
<\rho({\vec r})> = m a \; +m \, {K
\over{\nu(1-\alpha)\Omega_D}}\;r^{ \frac1{\nu}-D} \quad {\rm for}\; r
\; {\rm of ~ order}\;\sim R .
\end{equation}
where $ \Omega_D $ is the surface of the unit sphere in $D$-dimensions.
The energy density correlation function is known in general in the
scaling region (see refs.\cite{dg} -\cite{nn}).
We can therefore write for the density-density correlators
(\ref{corre}) in $ D $ space dimensions
\begin{equation}\label{corrG}
C({\vec r_1},{\vec r_2})\sim |{\vec r_1} -{\vec r_2}|^{\frac2{\nu} -2D} \; .
\end{equation}
where both $ {\vec r_1} $ and $ {\vec r_2} $ are inside the
finite volume $ \sim R^D $.
The perturbative calculation (\ref{corrasi}) matches
with this result for $ \nu = \frac12 $. That is, the mean field
value for the exponent $ \nu $.
Let us now compute the second derivative of $ \log{\cal Z}(\Lambda) $
with respect to $\Lambda$ in two ways. We find from eq.(\ref{Zsca2})
$$
{{\partial^2}\over{\partial\Lambda^2}}\log{\cal Z}(\Lambda)= V\left[
\Lambda^{-\alpha} \, K + b + \ldots \right] \; .
$$
We get from eq.(\ref{Zsca1}),
\begin{equation}\label{flucM}
{{\partial^2}\over{\partial\Lambda^2}}\log{\cal Z}(\Lambda)=
\int d^Dx\; d^Dy\; C({\vec x},{\vec y}) \sim R^D \int^R
{{ d^3x}\over{x^{2D - 2d_H}}} \sim \Lambda^{-2}\sim R^D \;
\Lambda^{-\alpha}
\end{equation}
where we used eq.(\ref{fss}), eq.(\ref{corrG}) and the scaling
relation $ \alpha = 2 - \nu D $ \cite{dg,nn}. We
conclude that the scaling
behaviours, eq.(\ref{Zsca2}) for the partition function,
eq.(\ref{calor}) for the specific heat and eq.(\ref{corrG}) for the
two points correlator are consistent.
In addition, eqs.(\ref{defM}) and (\ref{flucM})
yield for the mass fluctuations squared
$$
(\Delta M(R))^2 \equiv \; <M^2> -<M>^2 \; \sim
\int d^Dx\; d^Dy\; C({\vec x},{\vec y}) \sim R^{2d_H}\; .
$$
Hence,
\begin{equation}\label{Msca}
\Delta M(R) \sim R^{d_H}\; .
\end{equation}
\bigskip
The scaling exponent $\nu$ can be identified with the inverse
Haussdorf (fractal) dimension $d_H$ of the system
$$
d_H = \frac1{\nu} \; .
$$
In this way, $ \Delta M \sim R^{d_H} $ according to the usual
definition of fractal dimensions \cite{sta}.
\medskip
Using eq.(\ref{corrG}) we can compute the average potential energy
in three space dimensions as
$$
< {\cal V}> = \frac12 \,\beta \, G \, m^2
\int_{ | {\vec x} - {\vec y}|> a }^R \;
{{d^3x\, d^3y}\over { | {\vec x} - {\vec y}|}}\; C({\vec x},{\vec y})
\sim R^{\frac2{\nu} -1} \; .
$$
From here and eq.(\ref{Msca}) we get as virial estimate for the atoms
kinetic energy
$$
<v^2> = {{< {\cal V}>}\over {< \Delta M(R)>}} \sim R^{\frac1{\nu} -1} \; .
$$
This corresponds to a velocity dispersion
\begin{equation}\label{Vsca}
\Delta v \sim R^{\frac12(\frac1{\nu} -1)} \; .
\end{equation}
That is, we predict [see eq.(\ref{vobser})] a new scaling relation
$$
q =\frac12\left(\frac1{\nu} -1\right) =\frac12(d_H -1) \; .
$$
\bigskip
The calculation of the critical amplitudes [that is, the coefficients in
front of the powers of $ R $ in eqs.(\ref{corrG}), (\ref{Msca}) and
(\ref{Vsca})] is beyond the scope of the present paper \cite{fut}.
\subsection{Values of the scaling exponents and the fractal dimensions}
The scaling exponents $ \nu , \; \alpha $ considered in sec IIIC can
be computed through the renormalization group approach. The case of a
{\bf single} component scalar field has been extensively studied
in the literature \cite{dg,nn,grexa}. Very probably, there is an
unique, infrared stable fixed point in three space dimensions: the
Ising model fixed point. Such non-perturbative fixed point is
reached in the long scale regime independently of the initial shape of
the interaction $ u(\phi) $ [eq.(\ref{densac})] \cite{grexa}.
The numerical values of the scaling exponents associated to the
Ising model fixed point are
\begin{equation}\label{Ising}
\nu = 0.631\ldots \quad , \quad d_H = 1.585\ldots \quad , \quad \eta =
0.037\ldots \quad {\rm and} \quad \alpha = 0.107\ldots \; \; .
\end{equation}
\bigskip
In the $\phi$ field model there are two dimensionful parameters: $\mu$
and $T_{eff}$. The dimensionless combination
\begin{equation}\label{defg}
g^2 = \mu \, T_{eff} = (8 \pi)^{3/4}\; \sqrt{z} \; \; {{G^{3/2}\;
m^{15/4}}\over T^{3/4}}
\end{equation}
acts as the coupling constant for the non-linear fluctuations of the
field $\phi$.
Let us consider a gas formed by neutral hydrogen at thermal equilibrium with
the cosmic microwave background. We set $ T = 2.73\, K $ and estimate
the fugacity $ z $ using the ideal gas value
$$
z = \left( {{2\pi}\over {m T } }\right)^{3/2}\; \rho \; .
$$
Here we use $ \rho = \delta_0 $ atoms cm$^{-3}$ for the ISM density and
$ \delta_0 \simeq 10^{10} $. Eq. eqs.(\ref{muyT}) yields
\begin{equation}\label{valN}
{1 \over {\mu }} = 2.7 \; {1 \over { \sqrt{\delta_0}}}\; {\rm AU} \sim
30 \; {\rm AU} \quad {\rm and} \quad
g^2 = \mu \, T_{eff} = 4.9 \; 10^{-58} \; \sqrt{\delta_0} \sim 5 \,10^{-53}
\; .
\end{equation}
This extremely low value for $g^2 $
suggests that the perturbative calculation [sec. IIIB] may apply here
yielding the mean field values for the exponents, i. e.
\begin{equation}\label{campM}
\nu = 1/2 \quad , \quad d_H = 2 \quad , \quad \eta =0 \quad {\rm
and } \quad \alpha = 0 \; .
\end{equation}
That is, the effective coupling constant grows with
the scale according to the renormalization group flow (towards the
Ising fixed point). Now, if the extremely low value of the initial
coupling eq.(\ref{valN}) applies, the perturbative result (mean field)
will hold for many scales (the effective $ g $ grows roughly as the
length).
$ \mu^{-1} $ indicates the order of the smallest distance where the
scaling regime applies. A safe
lower bound supported by observations is around $20$ AU $\sim 3.\,
10^{14}$ cm , in agreement with our estimate.
\bigskip
Our theoretical predictions for $ \Delta M(R) $ and $ \Delta v $
[eqs.(\ref{Msca}) and (\ref{Vsca})] both for
the Ising eq.(\ref{Ising}) and for the mean field values
eq.(\ref{campM}), are in agreement with the astronomical
observations [eq.(\ref{vobser})]. The present observational bounds on
the data are larger than the difference between the mean field and
Ising values of the exponents $ d_H $ and $ q $.
Further theoretical work in the $\phi$-theory will determine whether
the scaling behaviour is given by the mean field or by the Ising fixed
point \cite{fut}.
\subsection{The two dimensional gas and random surfaces fractal dimensions}
In the two dimensional case ($D=2$) the partition function
(\ref{zfiD}) describes the Liouville model that arises in string
theory\cite{poly} and in the theory of random surfaces
(also called two-dimensional quantum gravity).
For strings in $c$-dimensional Euclidean space the partition function
takes the form\cite{poly}
\begin{equation}\label{zfiL}
{\cal Z}_c = \int\int\; {\cal D}\phi\; e^{ -{{26-c}\over{24\pi}}\;
\int d^2x \left[ \frac12(\nabla\phi)^2 \; + \mu^2 \; e^{\phi({\vec
x})}\right]}\; .
\end{equation}
This coincides with eq.(\ref{zfiD}) at $D=2$ provided we flip the sign
of $ \mu^2 $ and identify the parameters (\ref{paramD}) as follows,
\begin{equation}
T = G m^2\; {{26-c}\over 12}\quad , \quad \mu^2 = z G m^3 \; .
\end{equation}
Ref.\cite{amb} states that $ d_H = 4 $ for $ c \leq 1 $, $ d_H = 3 $ for
$ c = 2 $ and $d_H = 2$ for
$ c \geq 4 $. In our context this means
$$
d_H =2 \;\; {\rm for}~~ T \leq \frac{25}{12} \; G m^2 \quad , \quad
d_H = 3 \;\; {\rm for}~~ T = 2\, G m^2 \quad
{\rm and} \quad
d_H =4 \;\; {\rm for}~~ T \geq \frac{11}6 \; G m^2 \; .
$$
For $ c \to \infty , \; g^2 \to 0 $ and we can use the perturbative
result (\ref{corrasi}) yielding $ \nu = \frac12 , \; d_H = 2 $
in agreement with the above discussion for $ c \geq 4 $.
\subsection{Stationary points and the Jeans length}
The stationary points of the $\phi$-field partition function (\ref{zfi})
are given by the non-linear partial differential equation
$$
\nabla^2\phi = -\mu^2\, e^{\phi({\vec x})} \; .
$$
In terms of the gravitational potential $U({\vec x})$ [see eq. (\ref{Ufi})],
this takes the form
\begin{equation}\label{equih}
\nabla^2U({\vec r}) = 4 \pi G \, z \, m
\left({{mT}\over{2\pi}}\right)^{3/2} \, e^{ - \frac{m}{T}\,U({\vec r})} \; .
\end{equation}
This corresponds to the Poisson equation for a thermal matter distribution
fulfilling an ideal gas in hydrostatic equilibrium,
as can be seen as follows \cite{sas}.
The hydrostatic equilibrium condition
$$
\nabla P({\vec r}) = - m \, \rho({\vec r}) \; \nabla U({\vec r})\; ,
$$
where $ P({\vec r}) $ stands for the pressure, combined with the
equation of state for the ideal gas
$$
P = T \rho \; ,
$$
yields for the particle density
$$
\rho({\vec r}) = \rho_0 \; e^{ - \frac{m}{T}\,U({\vec r})} \; ,
$$
where $ \rho_0 $ is a constant. Inserting this relation into the
Poisson equation
$$
\nabla^2U({\vec r}) = 4 \pi G\, m \, \rho({\vec r})
$$
yields eq.(\ref{equih}) with
\begin{equation} \label{RO0}
\rho_0 = z \,\left({{mT}\over{2\pi}}
\right)^{3/2} \; .
\end{equation}
For large $ r $, eq.(\ref{equih}) gives a density decaying as $
r^{-2} $ ,
\begin{equation}
\rho({\vec r}) \buildrel{r\to \infty}\over = {T\over{2\pi G m}}\,
\frac1{r^2} \,\left[ 1 + O\left(\frac1{\sqrt{r}} \right) \right]
\quad , \quad U({\vec r}) \buildrel{r\to \infty}\over =
\frac{T}{m}\;\log\left[{{2\pi G \rho_0}\over T}\; r^2\right] +
O\left(\frac1{\sqrt{r}} \right) \; .
\end{equation}
Notice that this density, which describes a single stationary solution,
decays for large $r$ {\bf faster} than the density (\ref{Sdensi}) governed by
thermal fluctuations.
\bigskip
Spherically symmetric solutions of eq.(\ref{equih}) has been studied
in detail \cite{chandra}.
The small fluctuations around such isothermal spherical solutions
as well as the stability problem were studied in \cite{kh}.
\bigskip
The Jeans distance is in this context,
\begin{equation}\label{distaJ}
d_J \equiv \sqrt{ 3 T \over m}\; {1 \over{\sqrt{G\, m \, \rho_0}}} =
{{ \sqrt{ 3}\; (2\pi)^{3/4}}\over{ \sqrt{z\, G}\; m^{7/4}\; T^{1/4}}}
\; .
\end{equation}
This distance precisely coincides with $ \mu^{-1} $ [see eq.(\ref{muyT})] up to
an inessential numerical coefficient ($\sqrt{12/\pi}$). Hence, $
\mu $, the only dimensionful parameter in the $\phi$-theory can
be interpreted as the inverse of the Jeans distance.
We want to notice that in the critical regime, $ d_J $ grows as
\begin{equation}\label{escaA}
d_J \sim R^{d_H/2} \; ,
\end{equation}
since $ \rho_0 = \Lambda \sim R^{-d_H} $ vanishes as
can be seen from eqs.(\ref{fss}), (\ref{zcritico}) and
(\ref{distaJ}). In this tree level estimate we should use for consistency
the mean field value $ d_H = 2 $, which yields $ d_J \sim R$.
This shows that the Jeans distance is of the order of the {\bf size} of the
system. The Jeans distance {\bf scales} and the instability is
therefore present for all sizes $ R $.
Had $ d_J $ being of order larger than $ R $, the Jeans instability
would be absent.
The fact that the Jeans instability is present {\bf precisely}
at $ d_J \sim R $ is probably essential to the scaling regime and to
the self-similar (fractal) structure of the gravitational gas.
The dimensionless coupling constant $ g^2 $ can be written from
eqs.(\ref{zcritico}) and (\ref{defg}) as
$$
g^2 = \left( 2m \sqrt{{\pi \, G}\over T}\right)^3 \sqrt{\Lambda}\; .
$$
Hence, the tree level coupling scales as
$$
g^2 \sim R^{-1} \; .
$$
Direct perturbative calculations explicitly exhibit such scaling behaviour
\cite{fut}.
We can express $ g^2 $ in terms of $ d_J $ and $ \rho_0 $ as follows,
$$
g^2 = {{(12 \pi)^{3/2}}\over { \rho_0 \; d_J^3}} = {{\pi^2
\mu^3}\over { \rho_0}}
\; .
$$
This shows that $ g^2 $ is, at the tree level, the inverse of the
number of particles inside a Jeans volume.
Eq.(\ref{escaA}) applies to the tree level Jeans length or tree level
$ \mu^{-1} $. We can furthermore estimate the Jeans length using the
renormalization group behaviour of the physical quantities derived in
sec. III.C. Setting,
$$
<d_J> = {{<\Delta v>}\over { \sqrt{G\, m\, <\Delta \rho>}}} \; ,
$$
we find from eqs.(\ref{Sdensi}) and (\ref{Vsca}),
$$
<d_J> \sim R \; .
$$
Namely, we find again that the Jeans length grows as the size $ R $.
\section{Discussion}
In previous sections we ignored gravitational forces external to the
gas like stars etc. Adding a fixed external mass density $
\rho_{ext}({\vec r}) $ amounts to introduce an external source
$$
J({\vec r}) = - T_{eff}\; \rho_{ext}({\vec r})\; ,
$$
in eq.(\ref{zfiJ}). Such term will obviously affect correlation
functions, the mass density, etc. except when we look at the scaling
behaviour which is governed by the critical point.
That is, the values we find for the scaling exponents $ d_H $ and $ q
$ are {\bf stable} under external perturbations.
\bigskip
We considered all atoms with the same mass in the gravitational gas.
It is easy to generalize the transformation into the $\phi$-field
presented in section II for a mixture of several kinds of atoms. Let
us consider $ n $ species of atoms with
masses $ m_a, \; 1 \leq a \leq n $. Repeating the steps from
eq.(\ref{gfp}) to (\ref{acci})
yields again a field theory with a single scalar field but the
action now takes the form
\begin{equation}\label{gasM}
S[\phi(.)] \equiv {1\over{T_{eff}}}\;
\int d^3x \left[ \frac12(\nabla\phi)^2 \; - \sum_{a=1}^n \; \mu_a^2 \;
e^{{{m_a}\over m}\, \phi({\vec x})}\right] \; ,
\end{equation}
where
$$
\mu_a^2 = \sqrt{2\over {\pi}}\; z_a \; G \, m_a^{3/2} \, m^2 \, \sqrt{T}
\; ,
$$
and $ m $ is just a reference mass.
Correlation functions, mass densities and other observables will
obviously depend on the number of species, their masses and fugacities
but it is easy to see that the fixed points and scaling exponents are
exactly the {\bf same} as for the $\phi$-field theory (\ref{zfi})-(\ref{muyT}).
\bigskip
We want to notice that there is an important difference between the
behaviour of the gravitational gas and the spin models (and all other
statistical models in the same universality class). For the
gravitational gas we find scaling behaviour for a {\bf full range} of
temperatures and couplings. For spin models scaling only appears
at the critical value of the temperature. At the critical temperature
the correlation length $ \xi $ is infinite and the theory is
massless.
For temperatures near the critical one, i. e. in the critical
domain, $ \xi $ is finite (although very
large compared with the lattice spacing) and the correlation functions
decrease as $ \sim e^{ - r/\xi} $ for large distances $ r $.
Fluctuations of the relevant operators support perturbations which can
be interpreted as massive excitations. Such
(massive) behaviour does not appear for the gravitational gas. The ISM
correlators scale exhibiting power-law behaviour. This feature is
connected with the scale invariant character of the Newtonian force
and its infinite range.
\bigskip
The hypothesis of strict thermal equilibrium does not apply to the ISM as
a whole where temperatures range from $ 5 $ to $ 50 $ K and even $ 1000 $ K.
However, since the scaling behaviour is independent of the temperature,
it applies to {\bf each} region of the ISM in thermal equilibrium.
Therefore, our theory applies provided thermal equilibrium holds
in regions or clouds.
We have developed here the theory of a gravitationally interacting
ensemble of bodies at a fixed temperature. In a real situation like the ISM,
gravitational perturbations from external masses,
as well as other perturbations are present.
We have shown that the scaling solution is stable
with respect to the gravitational perturbations. It is well known that
solutions based on a fixed point are generally quite robust.
Our theory especially applies to the interstellar medium far from
star forming regions, which can be locally far from thermal equilibrium,
and where ionised gas at 10$^4$K together with coronal gas at 10$^6$K
can coexist with the cold interstellar medium. In the outer parts of
galaxies, devoid of star formation, the ideal isothermal conditions
are met \cite{pcm}. Inside the Galaxy, large regions satisfy also
the near isothermal criterium, and these are precisely the regions
where scaling
laws are the best verified. Globally over the Galaxy, the fraction
of the gas in the hot ionised phase represents a negligible mass,
a few percents, although occupying a significant volume. Hence, this
hot ionised gas is a perturbation which may not change the fixed point
behaviour of the thermal self-gravitating gas.
\bigskip
In ref.\cite{pm} a connection between a gravitational gas of galaxies
in an expanding universe and the
Ising model is conjectured. However, the unproven identification made
in ref.\cite{pm} of the mass density contrast with the Ising spin leads to
scaling exponents different from ours.
\acknowledgements
H J de V and N S thank D. Boyanovsky and M. D'Attanasio for discussions.
|
proofpile-arXiv_065-605
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\section{INTRODUCTION}
\noindent
The evolution and the final fate of stellar structures is mainly
governed by the amount of original mass. According to such a
plain evidence, the large amount of present theoretical interpretations
of the evolutionary status of stars and stellar systems is providing
tight constrains on the mass of the investigated objects. An independent
estimation of stellar masses would be of course of paramount interest
since it would represent a {\em prima facie} evidence for the physical
reliability of the adopted evolutionary scenario. As it is
well known, radial pulsating
structures offer such an opportunity for the simple reason that the
pulsations are mainly governed by gravity. On this simple basis,
the periods should depend, as they actually do,
on pulsator masses and radii (i.e. on the stellar parameters M, L and Te).
Jorgensen \& Petersen (1967) originally suggested that the occurrence
of double-mode pulsators could give the unique opportunity to provide
a straightforward evaluation of pulsator masses taking into account only
the ratio between the fundamental ($P_0$~) and the first overtone
periods ($P_1$~).
\noindent
According to this scenario, Petersen (1973)
introduced the diagram $P_1 / P_0$~ vs. $P_0$~ (hereinafter referred to as PD)
as a suitable tool for estimating the actual mass value of double-mode
pulsators. The application of this procedure to
RR Lyrae stars dates back to Cox, King \& Hodson (1980).
On the basis of the PD they found a mass value of about $M/M_{\odot}$=0.65 for
the only double-mode RR Lyrae (RRd) known at that time (AQ Leonis,
Jerzykiewicz \& Wenzel 1977). Since then the discovery of new RRd
variables in several Galactic globular clusters, in the field
(Clement et al. 1986, hereinafter referred to as C86)
and in dwarf spheroidal galaxies (Nemec 1985a, Kaluzny et al. 1995) has
brought on an interesting discussion for constraining their
pulsational and evolutionary characteristics. Cox, Hodson
\& Clancy (1983, hereinafter referred to as CHC)
investigated the PD for RRd pulsators in the metal poor Oosterhoff II
cluster M15 and derived a pulsational mass of the order of
M/${M_{\odot}}$ =0.65. The same authors suggested a mass of the order
of $M/M_{\odot}$= 0.55 for the two RRd pulsators belonging to M3, the prototype
of intermediate metallicity Oosterhoff type I (Oo I) clusters.
These results were subsequently confirmed by Nemec (1985b) and by
C86 who found similar mass values for RRd variables in the Oo
I cluster IC4499.
\noindent
However, for the quoted HB pulsators current evolutionary theories foresee
larger masses, namely $M/M_{\odot}$$\simeq$0.8 and $\simeq$0.65 for
Oo II and Oo I clusters respectively (see Bono et al. 1996).
Such a disturbing discrepancy between the masses of RRd variables determined
from pulsational and from evolutionary theories was settled as soon as
Cox (1991) found that pulsational models incorporating new and updated
opacity evaluations were able to reconcile pulsational
and evolutionary predictions. The settling of this long-standing
discrepancy was thus regarded as an evidence
for the reliability of the new opacity tables.
\noindent
In this paper we
present a new investigation of the Petersen approach which
discloses some unexpected results and sheds new light on the matter.
We show that opacity, as originally suggested by Cox (1991,1995), is the
key physical ingredient which produces the disagreement between
pulsational and evolutionary masses. However, we also find that this
discrepancy, even by using old opacities, can be consistently removed
either by adopting a much finer spatial resolution in linear computations
or by relying on detailed nonlinear models. Nevertheless, an exhaustive
solution to the problem of RR Lyrae masses can only be achieved if both
new opacities and full amplitude, detailed, nonlinear, nonlocal and
time-dependent convective models are taken into account.
\section{MASSES AND LUMINOSITIES OF RRd VARIABLES}
\noindent
During the last few years we have carried out an extensive survey of
limiting amplitude, nonlinear models of RR Lyrae variables (Bono
\& Stellingwerf 1994, hereinafter referred to as BS). The main purpose
of this project is to examine
the dependence of modal stability and pulsation behavior on astrophysical
parameters (for complete details see Bono et al. 1996). As a by-product of
this investigation we revisited the problem of
pulsator masses by investigating the dependence of the Petersen
diagram on the various assumptions governing theoretical calculations.
\noindent
The sequences of static envelope models were analyzed in the linear
nonadiabatic approximation (Castor 1971) and each model was required to
cover the outer 90\% of the stellar photospheric radii. The outer boundary
condition was typically fixed at an optical depth of the order of 0.001.
The linear models were constructed by neglecting convection and by adopting
the analytical approximation of old Los Alamos "King" opacity tables
provided by Stellingwerf (1975a,b). On the basis of these assumptions
a typical {\em coarse} model is characterized by 100-150 zones and few
percents of the total stellar mass. Complete details of the mass ratio
between consecutive zones and the method adopted for constraining the
hydrogen ionization region are given in Stellingwerf (1975a) and BS.
\noindent
As a starting point, Fig. 1 shows the theoretical PD obtained for
selected values of stellar masses and luminosities. The models plotted
in this figure present a stable {\em linear} limit cycle in the first
two modes. To understand the meaning of theoretical data displayed in
this figure, we recall
that linear models provide evaluations of periods independently of the
actual limit cycle stability of a given mode. As a consequence, we have
to bear in mind that a rather large
amount of data in similar figures should be regarded as unphysical,
since they supply the ratio $P_1 / P_0$~ even where either the fundamental or
the first overtone modes present an unstable {\em nonlinear} limit cycle.
\noindent
The period ratios of M15 RRd variables plotted in Fig. 1 were evaluated
taking into account different estimates (Nemec 1985b; Kovacs, Shlosman
\& Buchler 1986; Clement \& Walker 1990; Purdue et al. 1995).
The error bar plotted in the lower right corner is referred to these
measurements.
The comparison between theoretical models and observational data,
shown in the above figure, clearly supports previous results given in
the literature under similar theoretical assumptions and discloses
the occurrence of the "mass discrepancy problem". At the same time,
Fig. 1 shows
that at a given fundamental period the period ratio $P_1 / P_0$~ appears largely
independent of the assumed luminosity level.
\noindent
In order to investigate the dependence of linear periods on the spatial
resolution previously adopted, a new set of linear {\em detailed} models
have been
computed by adopting the prescriptions suggested by BS. The number of
zones for these new sequences of models is increased by roughly a factor
of two with respect to the {\em coarse} ones and ranges from 200 to 300.
Fig. 2 shows the results of these new computations, disclosing that
the "mass discrepancy problem" appears affected also by the method adopted
to discretize the physical structure of the static envelope model.
As a matter of fact, we find that periods provided by linear, nonadiabatic,
radiative models constructed with a finer spatial resolution partially
remove the degeneracy of the luminosity levels. Moreover, as a most
relevant point, these calculations now suggest that the mass value of
Oo II RRd variables should be of the order of $M/M_{\odot}$= 0.8, whereas
Oo I RRd variables should increase to about $M/M_{\odot}$= 0.70, in much
better agreement with evolutionary prescriptions (see Bono et al. 1996).
\noindent
However, BS have already shown that linear
periods are only a first, though good approximation of the
pulsational periods obtained from a more appropriate
nonlinear treatment of the pulsation. Thus the problem arises if
linear predictions about RRd masses are preserved in the nonlinear
approach. To properly address this fundamental theoretical question,
Fig. 3 displays the results of several sequences of nonlinear, nonlocal
and time-dependent convective models constructed by assuming the same
equation of state and the same opacities adopted in the linear regime.
According to the negligible influence of
spatial resolution on nonlinear limiting amplitude characteristics and
modal stability (BS and references therein) in order to speed up the
calculations required by the nonlinear approach only {\em coarse} static
envelope models were taken into account.
\noindent
The dynamical behavior of the envelope models was examined for the first
two modes and the static structures were forced out of equilibrium by
perturbing the linear radial eigenfunctions with a constant velocity
amplitude of 20 km$s^{-1}$. The method adopted for initiating nonlinear
models unavoidably introduces a spurious component of both periodic
and nonperiodic fluctuations which are superimposed to the pure radial
motions. As a consequence, before the dynamical behavior approaches
the limit cycle stability it is necessary to carry out extensive
calculations. The fundamental and first overtone sequences have been
integrated in time for at least 2,000 periods. The models located close
to the fundamental blue edge and to the first overtone red edge were
followed for a longer time interval (2,000-6,000 periods) since in these
regions of the instability strip before the dynamical motions approach
their asymptotic behavior a switch-over to a different mode could
take place even after several thousand periods.
The integration is generally stopped as soon as the nonlinear work
term is vanishing and the pulsational amplitudes present a periodic
similarity of the order of $10^{-(4 \div 5)}$.
\noindent
Therefore it turns out that the decrease of theoretical points plotted in
Fig. 3 is tightly connected with the morphology of the "OR region", since
were taken into account only envelope models which present stable nonlinear
limit cycles both in the fundamental and in the first overtone modes.
Moreover, data in Fig. 3 reveal that the nonlinear PD differs intrinsically
from the canonical linear PD. In fact in this new context the
spurious theoretical points connected with models which present a
unique stable limit cycle (fundamental or first overtone)
have obviously disappeared. A direct interesting consequence of this
new theoretical scenario is that the comparison between nonlinear
periods and observational data can now give useful information on both
stellar masses and luminosities of the pulsators. The reader interested
to a thorough analysis concerning
the evaluation of these parameters on the basis of RRd variables belonging
to both Oo I and Oo II clusters is also referred to Cox (1995) and
Walker (1995). In the evaluation of masses we eventually find that
nonlinear results do not fully support linear indications.
In fact, on the basis of nonlinear periods we obtain a stellar mass
of $M/M_{\odot}$$\simeq$0.7 for Oo II cluster pulsators, whereas for
the RRd variables in IC4499 we estimate a mass of the order of
$M/M_{\odot}$=0.60.
As a consequence, the agreement found by relying on linear detailed models
has to be regarded as an artifact of the computational procedure.
Moreover, for M15 and M68 pulsators we find a luminosity
around \lsun$\simeq$1.8, which appears somewhat larger than the
currently accepted evolutionary predictions.
\noindent
Bearing in mind the present scenario, we now take into account the
effects of the new opacities provided by Rogers \& Iglesias (1992)
for temperatures higher than $10^4$ $^oK$ and by Alexander \& Ferguson
(1994) for lower temperatures. The reader interested in the method
adopted for handling the new opacity tables is referred to Bono, Incerpi
\& Marconi (1996).
For the sake of conciseness,
we briefly quote the mass evaluations obtained from linear computations:
$M/M_{\odot}$= 0.72, 0.60 (coarse models) and $M/M_{\odot}$= 0.78, 0.65 (detailed models)
for pulsators in Oo II and Oo I clusters respectively.
Fig. 4 shows nonlinear periods based on updated radiative opacities.
The comparison with observational data is now pointing out a promising
theoretical scenario since it predicts a stellar mass slightly greater
than $M/M_{\odot}$= 0.8 for RRd pulsators in Oo II clusters and a mass value
around $M/M_{\odot}$= 0.65 for RRd variables in IC4499.
Both results are now in excellent agreement, within the error bar, with
canonical evolutionary predictions.
Data plotted in Fig. 4 also suggest a luminosity level of the order
of \lsun $\simeq$1.7 for Oo II RRd variables, whereas the corresponding
luminosity level for RRd variables in IC4499 falls between the computed
luminosity levels at \lsun =1.61 and 1.72. The overall good agreement
with evolutionary predictions, presented in Bono et al. (1996), shows
that thanks to the updated physical input both pulsational and evolutionary
theories converge to form a homogeneous scenario concerning
the long debated question of RR Lyrae luminosity in globular clusters.
\noindent
Finally, it is worth noting that the two RRd variables in M3 appear
slightly more massive and more luminous than RRd variables in IC4499.
According to current metallicity estimates for these clusters
($[Fe/H]_{M3}$=-1.7, $[Fe/H]_{IC4499}$=-1.5), even this finding appears
again in satisfactory agreement with the evolutionary prescriptions.
\section{CONCLUSIONS}
\noindent
In this paper we have revisited the approach based on the PD
for determining the masses of RRd variables. It is shown that the
pulsator masses evaluated through the comparison between periods obtained
in a linear, nonadiabatic, radiative regime and observational data might
be affected by substantial systematic errors. On the other hand, the periods
provided by the surveys of nonlinear, nonlocal and time-dependent convective
models point out that even though the discrepancy between linear and
nonlinear periods has often been considered negligible, it plays a key role
for properly defining the location of double-mode pulsators inside the
Petersen diagram ($P_1 / P_0$~ vs. $P_0$~).
\noindent
As a most relevant point,
we found that a nonlinear Petersen diagram constructed taking
simultaneously into account both nonlinear models and new
radiative opacities provides valuable constraints not only
on the stellar masses but also on the luminosities of RRd
variables. The pulsational masses and luminosities of
double-mode pulsators obtained in this new theoretical framework confirm
the results recently provided by Cox (1995).
The comparison with observational data of RRd variables in both Oo I and
Oo II galactic globular clusters discloses a satisfactory agreement
with current evolutionary and pulsational predictions. At the same time,
this agreement supplies a new piece of evidence against the suggested
anomaly of HB star luminosities.
Further applications of this new approach for constraining the physical
parameters of RRd variables belonging to the Galactic field, the central
region of LMC and to dwarf spheroidal galaxies are under way.
\noindent
It is a pleasure to thank A. Cox as referee for several valuable comments
and for the pertinence of his suggestions on the original version of this
paper. This work was partially supported by MURST, CNR-GNA and ASI.
\pagebreak
|
proofpile-arXiv_065-606
|
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\section{Introduction}
In the first part of this paper we shall investigate a
special case of relative continuity of symplectically adjoint maps
of a symplectic space. By this, we mean the following.
Suppose that $(S,\sigma)$ is a symplectic space, i.e.\ $S$
is a real-linear vector space with an anti-symmetric,
non-degenerate bilinear form $\sigma$ (the symplectic form).
A pair $V,W$ of linear maps of $S$ will be called
{\it symplectically adjoint} if
$\sigma(V\phi,\psi) = \sigma(\phi,W\psi)$ for all $\phi,\psi \in S$.
Let $\mu$ and $\mu'$ be two
scalar products on $S$ and assume that,
for each pair $V,W$ of symplectically adjoint
linear maps of $(S,\sigma)$, the boundedness
of both $V$ and $W$ with respect to $\mu$
implies their boundedness with respect to $\mu'$.
Such a situation we refer to as {\it relative $\mu - \mu'$
continuity of symplectically adjoint maps} (of $(S,\sigma)$).
A particular example of symplectically adjoint maps is
provided by the pair $T,T^{-1}$ whenever $T$ is a symplectomorphism
of $(S,\sigma)$. (Recall that a symplectomorphism of $(S,\sigma)$
is a bijective linear map $T : S \to S$ which preserves the
symplectic form, $\sigma(T\phi,T\psi) = \sigma(\phi,\psi)$ for
all $\phi,\psi \in S$.)
In the more specialized case to be considered in the present work,
which will soon be indicated to be relevant in applications,
we show that a certain distinguished relation between a
scalar product $\mu$ on $S$
and a second one, $\mu'$,
is sufficient for the relative $\mu - \mu'$
continuity of symplectically adjoint maps.
(We give further details in Chapter 2,
and in the next paragraph.)
The result will be applied in Chapter 3 to answer
a couple of open questions
concerning the algebraic structure of the quantum theory of the free
scalar field in arbitrary globally hyperbolic spacetimes:
the local definiteness, local primarity and Haag-duality
in representations of the local observable algebras
induced by quasifree Hadamard states,
as well as the determination of the type of the local
von Neumann algebras in such representations.
Technically, what needs to be proved in our approach to this problem
is the continuity of the temporal evolution of the Cauchy-data of
solutions of the scalar Klein-Gordon equation
\begin{equation}
(\nabla^a \nabla_a + r)\varphi = 0
\end{equation}
in a globally hyperbolic spacetime with respect to a certain
topology on the Cauchy-data space.
(Here, $\nabla$ is the covariant derivative of the metric $g$
on the spacetime, and $r$ an arbitrary realvalued,
smooth function.)
The Cauchy-data space is a symplectic space on which the said
temporal evolution is realized by symplectomorphisms. It
turns out that the classical ``energy-norm'' of solutions
of (1.1), which is given by a scalar
product $\mu_0$ on the Cauchy-data space, and the
topology relevant for the required continuity statement
(the ``Hadamard one-particle space norm''), induced by a
scalar product $\mu_1$ on the Cauchy-data space, are precisely
in the relation for which our result on relative $\mu_0 - \mu_1$
continuity of symplectically adjoint maps applies. Since the continuity
of the Cauchy-data evolution in the classical energy norm,
i.e.\ $\mu_0$, is well-known, the desired continuity in the
$\mu_1$-topology follows.
The argument just described may be viewed as the prime example of
application of the relative continuity result. In fact,
the relation between $\mu_0$ and $\mu_1$ is abstracted from
the relation between the classical energy-norm and the
one-particle space norms arising from ``frequency-splitting'' procedures
in the canonical quantization of (linear) fields.
This relation has been made precise
in a recent paper by Chmielowski [11]. It provides the
starting point for our investigation in Chapter 2, where
we shall see that one can associate
with a dominating scalar product $\mu \equiv \mu_0$ on
$S$ in a canonical way a positive, symmetric operator
$|R_{\mu}|$ on the $\mu$-completion of $S$, and a family of scalar
products $\mu_s$, $s > 0$, on $S$, defined as $\mu$ with
$|R_{\mu}|^s$ as an operator kernel. Using abstract
interpolation, it will be shown that
then relative $\mu_0 - \mu_s$ continuity of symplectically adjoint maps
holds for all $0 \leq s \leq 2$. The relative
$\mu_0 - \mu_1$ continuity arises as
a special case.
In fact, it turns out that the indicated interpolation
argument may even be extended to an apparently more general
situation from which the relative $\mu_0 - \mu_s$ continuity
of symplectically adjoint maps derives as a corollary, see
Theorem 2.2.
Chapter 3 will be concerned with the application of the result
of Thm.\ 2.2 as indicated above. In the preparatory Section
3.1, some notions of general relativity will be summarized, along
with the introduction of some notation. Section 3.2 contains a brief
synopsis of the notions of local definiteness, local primarity and
Haag-duality in the the context of quantum field theory in curved
spacetime. In Section 3.3 we present the $C^*$-algebraic quantization of the
KG-field obeying (1.1) on a globally hyperbolic spacetime, following [16].
Quasifree Hadamard states will be described in Section 3.4 according
to the definition given in [45]. In the same section we briefly summarize
some properties of Hadamard two-point functions, and derive, in
Proposition 3.5, the result concerning the continuity of the
Cauchy-data evolution maps in the topology of the Hadamard two-point
functions which was mentioned above. It will be seen in the last Section
3.5 that this leads, in combination with results obtained earlier
[64,65,66], to Theorem 3.6 establishing detailed properties of the algebraic
structure of the local von Neumann observable algebras in representations
induced by quasifree Hadamard states of the Klein-Gordon field over
an arbitrary globally hyperbolic spacetime.
\section{Relative Continuity of Symplectically Adjoint Maps}
\setcounter{equation}{0}
Let $(S,\sigma)$ be a symplectic space. A (real-linear) scalar
product $\mu$ on $S$ is said to {\it dominate} $\sigma$
if the estimate
\begin{equation}
|\sigma(\phi,\psi)|^2 \leq 4 \cdot \mu(\phi,\phi)\,\mu(\psi,\psi)\,,
\quad \phi,\psi \in S\,,
\end{equation}
holds; the set of all scalar products on $S$ which dominate $\sigma$
will be denoted by ${\sf q}(S,\sigma)$.
Given $\mu \in {\sf q}(S,\sigma)$, we write $H_{\mu} \equiv
\overline{S}^{\mu}$ for the completion of $S$ with respect to the
topology induced by $\mu$, and denote by $\sigma_{\mu}$ the
$\mu$-continuous extension, guaranteed to uniquely exist by (2.1),
of $\sigma$ to $H_{\mu}$. The estimate (2.1) then extends to
$\sigma_{\mu}$ and all $\phi,\psi \in H_{\mu}$. This entails
that there is a uniquely determined, $\mu$-bounded linear
operator $R_{\mu} : H_{\mu} \to H_{\mu}$ with the property
\begin{equation}
\sigma_{\mu}(x,y) = 2\,\mu(x,R_{\mu}y)\,, \quad x,y \in H_{\mu}\,.
\end{equation}
The antisymmetry of $\sigma_{\mu}$ entails for the
$\mu$-adjoint $R_{\mu}^*$ of $R_{\mu}$
\begin{equation}
R_{\mu}^* = - R_{\mu}\,,
\end{equation}
and by (2.1) one finds that the operator norm of $R_{\mu}$
is bounded by 1, $||\,R_{\mu}\,|| \leq 1$.
The operator $R_{\mu}$ will be called the {\it polarizator} of $\mu$.
In passing, two things should be noticed here:
\\[6pt]
(1) $R_{\mu}|S$ is injective since $\sigma$ is a non-degenerate
bilinear form on $S$, but $R_{\mu}$ need not be injective on
on all of $H_{\mu}$, as $\sigma_{\mu}$ may be degenerate.
\\[6pt]
(2) In general, it is not the case that $R_{\mu}(S) \subset S$.
\\[6pt]
Further properties of $R_{\mu}$ will be explored below.
Let us first focus on two significant subsets of ${\sf q}(S,\sigma)$ which
are intrinsically characterized by properties of the
corresponding $\sigma_{\mu}$ or, equivalently, the $R_{\mu}$.
The first is ${\sf pr}(S,\sigma)$, called the set of {\it primary}
scalar products on $(S,\sigma)$, where $\mu \in {\sf q}(S,\sigma)$ is
in ${\sf pr}(S,\sigma)$ if $\sigma_{\mu}$ is a symplectic form
(i.e.\ non-degenerate) on $H_{\mu}$. In view of (2.2) and
(2.3), one can see that this is equivalent to either
(and hence, both) of the following conditions:
\begin{itemize}
\item[(i)] \quad $R_{\mu}$ is injective,
\item[(ii)] \quad $R_{\mu}(H_{\mu})$ is dense in $H_{\mu}$.
\end{itemize}
The second important subset of ${\sf q}(S,\sigma)$ is denoted by
${\sf pu}(S,\sigma)$ and defined as consisting of those $\mu \in {\sf q}(S,\sigma)$
which satisfy the {\it saturation property}
\begin{equation}
\mu(\phi,\phi) = \sup_{\psi \in S\backslash \{0\} } \,
\frac{|\sigma(\phi,\psi)|^2}{4 \mu(\psi,\psi) } \,,\ \ \ \psi \in S \,.
\end{equation}
The set ${\sf pu}(S,\sigma)$ will be called the set of {\it pure} scalar
products on $(S,\sigma)$. It is straightforward to check that
$\mu \in {\sf pu}(S,\sigma)$ if and only if $R_{\mu}$ is a unitary
anti-involution, or complex structure, i.e.\
$R_{\mu}^{-1} = R_{\mu}^*$, $R_{\mu}^2 = - 1$. Hence
${\sf pu}(S,\sigma) \subset {\sf pr}(S,\sigma)$.
\\[10pt]
Our terminology reflects well-known relations between properties of
quasifree states on the (CCR-) Weyl-algebra of a symplectic space
$(S,\sigma)$ and properties of $\sigma$-dominating scalar products
on $S$, which we shall briefly recapitulate. We refer to
[1,3,5,45,49] and also references quoted therein for proofs
and further discussion of the following statements.
Given a symplectic space $(S,\sigma)$, one can associate with it
uniquely (up to $C^*$-algebraic equivalence) a $C^*$-algebra
${\cal A}[S,\sigma]$, which is generated by a family of unitary elements
$W(\phi)$, $\phi \in S$, satisfying the canonical commutation
relations (CCR) in exponentiated form,
\begin{equation}
W(\phi)W(\psi) = {\rm e}^{-i\sigma(\phi,\psi)/2}W(\phi + \psi)\,,
\quad \phi,\psi \in S\,.
\end{equation}
The algebra ${\cal A}[S,\sigma]$ is called the {\it Weyl-algebra}, or
{\it CCR-algebra}, of $(S,\sigma)$. It is not difficult to see that
if $\mu \in {\sf q}(S,\sigma)$, then one can define a state (i.e., a positive,
normalized linear functional) $\omega_{\mu}$ on ${\cal A}[S,\sigma]$ by setting
\begin{equation}
\omega_{\mu} (W(\phi)) : = {\rm e}^{- \mu(\phi,\phi)/2}\,, \quad \phi \in S\,.
\end{equation}
Any state on the Weyl-algebra ${\cal A}[S,\sigma]$ which can be realized in this way
is called a {\it quasifree state}. Conversely, given any quasifree
state $\omega_{\mu}$ on ${\cal A}[S,\sigma]$, one can recover its $\mu \in {\sf q}(S,\sigma)$ as
\begin{equation}
\mu(\phi,\psi) = 2 {\sf Re}\left. \frac{\partial}{\partial t}
\frac{\partial}{\partial \tau} \right|_{t = \tau = 0}
\omega_{\mu} (W(t\phi)W(\tau \psi))\,, \quad \phi,\psi \in S\,.
\end{equation}
So there is a one-to-one correspondence between quasifree states
on ${\cal A}[S,\sigma]$ and dominating scalar products on $(S,\sigma)$.
\\[10pt]
Let us now recall the subsequent terminology. To a state $\omega$
on a $C^*$-algebra $\cal B$ there corresponds (uniquely up to
unitary equivalence) a triple $({\cal H}_{\omega},\pi_{\omega},\Omega_{\omega})$,
called the GNS-representation of $\omega$ (see e.g.\ [5]), characterized by
the following properties: ${\cal H}_{\omega}$ is a complex Hilbertspace,
$\pi_{\omega}$ is a representation of $\cal B$ by bounded linear operators
on ${\cal H}_{\omega}$ with cyclic vector $\Omega_{\omega}$, and
$\omega(B) = \langle \Omega_{\omega},\pi_{\omega}(B)\Omega_{\omega}
\rangle $ for all $B \in \cal B$.
Hence one is led to associate with $\omega$ and
$\cal B$ naturally
the $\omega$-induced von Neumann algebra $\pi_{\omega}({\cal B})^-$,
where the bar means taking the closure with respect to the weak
operator topology in the set of bounded linear operators on ${\cal H}_{\omega}$.
One refers to $\omega$ (resp., $\pi_{\omega}$) as {\it primary}
if $\pi_{\omega}({\cal B})^- \cap \pi_{\omega}({\cal B})' = {\bf C} \cdot 1$
(so the center of $\pi_{\omega}({\cal B})^-$ is trivial), where the
prime denotes taking the commutant, and as {\it pure} if
$\pi_{\omega}({\cal B})' = {\bf C}\cdot 1$ (i.e.\ $\pi_{\omega}$ is
irreducible --- this is equivalent to the statement that $\omega$
is not a (non-trivial) convex sum of different states).
In the case where $\omega_{\mu}$ is a quasifree state on a Weyl-algebra
${\cal A}[S,\sigma]$, it is known that (cf.\ [1,49])
\begin{itemize}
\item[(I)] $\omega_{\mu}$ is primary if and only if $\mu \in {\sf pr}(S,\sigma)$,
\item[(II)] $\omega_{\mu}$ is pure if and only if $\mu \in {\sf pu}(S,\sigma)$.
\end{itemize}
${}$\\
We return to the investigation of the properties of the polarizator
$R_{\mu}$ for a dominating scalar product $\mu$ on a symplectic space
$(S,\sigma)$. It possesses a polar decomposition
\begin{equation}
R_{\mu} = U_{\mu} |R_{\mu}|
\end{equation}
on the Hilbertspace $(H_{\mu},\mu)$, where $U_{\mu}$ is an isometry
and $|R_{\mu}|$ is symmetric and has non-negative spectrum. Since
$R_{\mu}^* = - R_{\mu}$, $R_{\mu}$ is normal and thus
$|R_{\mu}|$ and $U_{\mu}$ commute. Moreover, one has
$|R_{\mu}| U_{\mu}^* = - U_{\mu} |R_{\mu}|$, and hence $|R_{\mu}|$ and $U_{\mu}^*$ commute
as well. One readily observes that $(U_{\mu}^* + U_{\mu})|R_{\mu}| = 0$.
The commutativity can by the spectral calculus be generalized to the
statement that, whenever $f$ is a real-valued, continuous function
on the real line, then
\begin{equation}
[f(|R_{\mu}|),U_{\mu}] = 0 = [f(|R_{\mu}|),U_{\mu}^*] \,,
\end{equation}
where the brackets denote the commutator.
In a recent work [11], Chmielowski noticed that if one defines
for $\mu \in {\sf q}(S,\sigma)$ the bilinear form
\begin{equation}
\tilde{\mu}(\phi,\psi) := \mu (\phi,|R_{\mu}| \psi)\,, \quad \phi,\psi \in S,
\end{equation}
then it holds that $\tilde{\mu} \in {\sf pu}(S,\sigma)$. The proof of this is straightforward.
That $\tilde{\mu}$ dominates $\sigma$ will be seen in Proposition 2.1 below.
To check the saturation property (2.4) for $\tilde{\mu}$, it suffices to
observe that for given $\phi \in H_{\mu}$, the inequality in the
following chain of expressions:
\begin{eqnarray*}
\frac{1}{4} | \sigma_{\mu}(\phi,\psi) |^2 & = & |\mu(\phi,U_{\mu} |R_{\mu}| \psi)|^2
\ = \ |\mu(\phi,-U_{\mu}^*|R_{\mu}|\psi) |^2 \\
& = & |\mu(|R_{\mu}|^{1/2}U_{\mu}\phi,|R_{\mu}|^{1/2}\psi)|^2
\\
& \leq & \mu(|R_{\mu}|^{1/2}U_{\mu}\phi,|R_{\mu}|^{1/2}U_{\mu}\phi) \cdot \mu(|R_{\mu}|^{1/2}\psi,
|R_{\mu}|^{1/2}\psi) \nonumber
\end{eqnarray*}
is saturated and becomes an equality upon choosing $\psi \in H_{\mu}$
so that $|R_{\mu}|^{1/2}\psi$ is parallel to $|R_{\mu}|^{1/2}U_{\mu} \phi$.
Therefore one obtains for all $\phi \in S$
\begin{eqnarray*}
\sup_{\psi \in S\backslash\{0\}}\, \frac{|\sigma(\phi,\psi)|^2}
{4 \mu(\psi,|R_{\mu}| \psi) } & = & \mu(|R_{\mu}|^{1/2}U_{\mu} \phi,|R_{\mu}|^{1/2}U_{\mu} \phi)
\\
& = & \mu(U_{\mu}|R_{\mu}|^{1/2}\phi,U_{\mu} |R_{\mu}|^{1/2} \phi) \\
& = & \tilde{\mu}(\phi,\phi)\,,
\end{eqnarray*}
which is the required saturation property.
Following Chmielowski, the scalar product $\tilde{\mu}$ on $S$ associated with
$\mu \in {\sf q}(S,\sigma)$ will be called the {\it purification} of $\mu$.
It appears natural to associate with $\mu \in {\sf q}(S,\sigma)$ the family $\mu_s$,
$s > 0$, of symmetric bilinear forms on $S$ given by
\begin{equation}
\mu_s(\phi,\psi) := \mu(\phi,|R_{\mu}|^s \psi)\,, \quad \phi,\psi \in S\,.
\end{equation}
We will use the convention that $\mu_0 = \mu$.
Observe that $\tilde{\mu} = \mu_1$. The subsequent proposition ensues.
\begin{Proposition}
${}$\\[6pt]
(a) $\mu_s$ is a scalar product on $S$ for each $s \geq 0$. \\[6pt]
(b) $\mu_s$ dominates $\sigma$ for $0 \leq s \leq 1$. \\[6pt]
(c) Suppose that there is some $s \in (0,1)$ such that $\mu_s \in {\sf pu}(S,\sigma)$.
Then $\mu_r = \mu_1$ for all $r > 0$. If it is in addition assumed
that $\mu \in {\sf pr}(S,\sigma)$, then it follows that $\mu_r = \mu_1$ for all
$r \geq 0$, i.e.\ in particular $\mu = \tilde{\mu}$. \\[6pt]
(d) If $\mu_s \in {\sf q}(S,\sigma)$ for some $s > 1$, then $\mu_r = \mu_1$ for
all $r > 0$. Assuming additionally $\mu \in {\sf pr}(S,\sigma)$, one obtains
$\mu_r = \mu_1$ for all $r \geq 0$, entailing $\mu = \tilde{\mu}$.\\[6pt]
(e) The purifications of the $\mu_s$, $0 < s < 1$, are equal
to $\tilde{\mu}$: We have $\widetilde{\mu_s} = \tilde{\mu} = \mu_1$ for all
$0 < s < 1$.
\end{Proposition}
{\it Proof.} (a) According to (b), $\mu_s$ dominates $\sigma$ for
each $0 \leq s \leq 1$, thus it is a scalar product whenever $s$ is
in that range. However, it is known that
$\mu(\phi,|R_{\mu}|^s \phi) \geq \mu(\phi,|R_{\mu}| \phi)^s$ for all vectors
$\phi \in H_{\mu}$ of unit length ($\mu(\phi,\phi) = 1$) and
$1 \leq s < \infty$, cf.\ [60 (p.\ 20)]. This shows that
$\mu_s(\phi,\phi) \neq 0$ for all nonzero $\phi$ in $S$, $s \geq 0$.
\\[6pt]
(b) For $s$ in the indicated range there holds the following estimate:
\begin{eqnarray*}
\frac{1}{4} |\sigma(\phi,\psi)|^2 & = & |\mu(\phi,U_{\mu}|R_{\mu}| \psi)|^2
\ = \ |\mu(\phi,-U_{\mu}^*|R_{\mu}| \psi )|^2 \\
& = & | \mu(|R_{\mu}|^{s/2}U_{\mu} \phi, |R_{\mu}|^{1 - s/2} \psi) |^2 \\
& \leq & \mu(U_{\mu} |R_{\mu}|^{s/2}\phi,U_{\mu}|R_{\mu}|^{s/2} \phi)
\cdot \mu(|R_{\mu}|^{s/2}\psi,|R_{\mu}|^{2(1-s)}|R_{\mu}|^{s/2} \psi) \\
& \leq & \mu_s(\phi,\phi)\cdot \mu_s(\psi,\psi)\,, \quad \phi,\psi \in S\,.
\end{eqnarray*}
Here, we have used that $|R_{\mu}|^{2(1-s)} \leq 1$.
\\[6pt]
(c) If $(\phi_n)$ is a $\mu$-Cauchy-sequence in $H_{\mu}$, then it is,
by continuity of $|R_{\mu}|^{s/2}$, also a $\mu_s$-Cauchy-sequence in
$H_s$, the $\mu_s$-completion of $S$. Via this identification, we obtain
an embedding $j : H_{\mu} \to H_s$. Notice that $j(\psi) = \psi$
for all $\psi \in S$, so $j$ has dense range; however, one has
\begin{equation}
\mu_s(j(\phi),j(\psi)) = \mu(\phi,|R_{\mu}|^s \psi)
\end{equation}
for all $\phi,\psi \in H_{\mu}$. Therefore $j$ need not be injective.
Now let $R_s$ be the polarizator of $\mu_s$. Then we have
\begin{eqnarray*}
2\mu_s(j(\phi),R_s j(\psi))\ = \ \sigma_{\mu}(\phi,\psi) & = &
2 \mu(\phi,R_{\mu}\psi) \\
& = & 2 \mu(\phi,|R_{\mu}|^sU_{\mu}|R_{\mu}|^{1-s}\psi) \\
& = & 2 \mu_s(j(\phi),j(U_{\mu}|R_{\mu}|^{1-s})\psi)
\,,\quad \phi,\psi \in H_{\mu}\,.
\end{eqnarray*}
This yields
\begin{equation}
R_s {\mbox{\footnotesize $\circ$}} j = j {\mbox{\footnotesize $\circ$}} U_{\mu}|R_{\mu}|^{1-s}
\end{equation}
on $H_{\mu}$. Since by assumption $\mu_s$ is pure, we have
$R_s^2 = -1$ on $H_s$, and thus
$$ j = - R_s j U_{\mu}|R_{\mu}|^{1-s} = - j(U_{\mu}|R_{\mu}|^{1-s})^2 \,.$$
By (2.12) we may conclude
$$ |R_{\mu}|^{2s} = - U_{\mu} |R_{\mu}| U_{\mu} |R_{\mu}| = U_{\mu}^*U_{\mu}|R_{\mu}|^2 = |R_{\mu}|^2 \,, $$
which entails $|R_{\mu}|^s = |R_{\mu}|$. Since $|R_{\mu}| \leq 1$, we see that for
$s \leq r \leq 1$ we have
$$ |R_{\mu}| = |R_{\mu}|^s \geq |R_{\mu}|^r \geq |R_{\mu}| \,,$$
hence $|R_{\mu}|^r = |R_{\mu}|$ for $s \geq r \geq 1$. Whence $|R_{\mu}|^r = |R_{\mu}|$
for all $r > 0$. This proves the first part of the statement.
For the second part we observe that $\mu \in {\sf pr}(S,\sigma)$ implies that
$|R_{\mu}|$, and hence also $|R_{\mu}|^s$ for $0 < s < 1$, is injective. Then the
equation $|R_{\mu}|^s = |R_{\mu}|$ implies that $|R_{\mu}|^s(|R_{\mu}|^{1-s} - 1) = 0$,
and by the injectivity of $|R_{\mu}|^s$ we may conclude $|R_{\mu}|^{1-s} =1$.
Since $s$ was assumed to be strictly less than 1, it follows that
$|R_{\mu}|^r = 1$ for all $r \geq 0$; in particular, $|R_{\mu}| =1$.
\\[6pt]
(d) Assume that $\mu_s$ dominates $\sigma$ for some $s > 1$, i.e.\ it
holds that
$$ 4|\mu(\phi,U_{\mu}|R_{\mu}|\psi)|^2 = |\sigma_{\mu}(\phi,\psi)|^2
\leq 4\cdot \mu(\phi,|R_{\mu}|^s\phi)\cdot \mu(\psi,|R_{\mu}|^s\psi)\,, \quad \phi,\psi
\in H_{\mu}\,, $$
which implies, choosing $\phi = U_{\mu} \psi$, the estimate
$$ \mu(\psi,|R_{\mu}| \psi) \leq \mu(\psi,|R_{\mu}|^s \psi) \,,
\quad \psi \in H_{\mu}\,,$$
i.e.\ $|R_{\mu}| \leq |R_{\mu}|^s$. On the other hand, $|R_{\mu}| \geq |R_{\mu}|^r \geq |R_{\mu}|^s$
holds for all $1 \leq r \leq s$ since $|R_{\mu}| \leq 1$. This implies
$|R_{\mu}|^r = |R_{\mu}|$ for all $r > 0$. For the second part of the statement one
uses the same argument as given in (c). \\[6pt]
(e) In view of (2.13) it holds that
\begin{eqnarray*}
|R_s|^2j & = & - R_s^2 j\ =\ - R_s j U_{\mu} |R_{\mu}|^{1-s} \\
& = & - j U_{\mu} |R_{\mu}|^{1-s}U_{\mu}|R_{\mu}|^{1-s}\ =\ - j U_{\mu}^2 (|R_{\mu}|^{1-s})^2 \,.
\end{eqnarray*}
Iterating this one has for all $n \in {\bf N}$
$$ |R_s|^{2n} j = (-1)^n j U_{\mu}^{2n}(|R_{\mu}|^{1-s})^{2n}\,. $$
Inserting this into relation (2.12) yields for all $n \in {\bf N}$
\begin{eqnarray}
\mu_s(j(\phi),|R_s|^{2n}j(\psi)) & = & \mu(\phi,
|R_{\mu}|^s (-1)^n U_{\mu}^{2n}(|R_{\mu}|^{1-s})^{2n}\psi) \\
& = & \mu(\phi,|R_{\mu}|^s(|R_{\mu}|^{1-s})^{2n}\psi)\,,\quad \phi,\psi \in H_{\mu}\,.
\nonumber
\end{eqnarray}
For the last equality we used that $U_{\mu}$ commutes with $|R_s|^s$
and $U_{\mu}^2|R_{\mu}| = - |R_{\mu}|$. Now let $(P_n)$ be a sequence of polynomials
on the intervall $[0,1]$ converging uniformly to the square root
function on $[0,1]$. From (2.14) we infer that
$$ \mu_s(j(\phi),P_n(|R_s|^2)j(\psi)) = \mu(\phi,|R_{\mu}|^s P_n((|R_{\mu}|^{1-s})^2)
\psi)\,, \quad \phi, \psi \in H_{\mu} $$
for all $n \in {\bf N}$, which in the limit $n \to \infty$ gives
$$ \mu_s(j(\phi),|R_s|j(\psi)) = \mu(\phi,|R_{\mu}|\psi)\,, \quad
\phi,\psi \in H_{\mu}\,, $$
as desired. $\Box$
\\[10pt]
Proposition 2.1 underlines the special role of
$\tilde{\mu} = \mu_1$. Clearly, one has $\tilde{\mu} = \mu$ iff $\mu \in {\sf pu}(S,\sigma)$.
Chmielowski has proved another interesting connection between
$\mu$ and $\tilde{\mu}$ which we briefly mention here. Suppose that
$\{T_t\}$ is a one-parametric group of symplectomorphisms of
$(S,\sigma)$, and let $\{\alpha_t\}$ be the automorphism group
on ${\cal A}[S,\sigma]$ induced by it via $\alpha_t(W(\phi)) = W(T_t\phi)$,
$\phi \in S,\ t \in {\bf R}$. An $\{\alpha_t\}$-invariant quasifree
state $\omega_{\mu}$ on ${\cal A}[S,\sigma]$ is called {\it regular} if the unitary
group which implements $\{\alpha_t\}$ in the GNS-representation
$({\cal H}_{\mu},\pi_{\mu},\Omega_{\mu})$ of $\omega_{\mu}$ is strongly
continuous and leaves no non-zero vector in the one-particle space
of ${\cal H}_{\mu}$ invariant. Here, the one-particle space is spanned
by all vectors of the form $\left. \frac{d}{dt} \right|_{t = 0}
\pi_{\mu}(W(t\phi))\Omega_{\mu}$, $\phi \in S$.
It is proved in [11] that, if $\omega_{\mu}$ is a regular quasifree
KMS-state for $\{\alpha_t\}$, then $\omega_{\tilde{\mu}}$ is the unique
regular quasifree groundstate for $\{\alpha_t\}$. As explained in
[11], the passage from $\mu$ to $\tilde{\mu}$ can be seen as a rigorous
form of ``frequency-splitting'' methods employed in the canonical
quantization of classical fields for which $\mu$ is induced
by the classical energy norm. We shall come back to this in the
concrete example of the Klein-Gordon field in Sec.\ 3.4.
It should be noted that the purification map $\tilde{\cdot} :
{\sf q}(S,\sigma) \to {\sf pu}(S,\sigma)$, $\mu \mapsto \tilde{\mu}$, assigns to a quasifree state
$\omega_{\mu}$ on ${\cal A}[S,\sigma]$ the pure quasifree state $\omega_{\tilde{\mu}}$
which is again a state on ${\cal A}[S,\sigma]$. This is different from the
well-known procedure of assigning to a state $\omega$ on a
$C^*$-algebra ${\cal A}$, whose GNS representation is primary,
a pure state $\omega_0$ on ${\cal A}^{\circ} \otimes
{\cal A}$.
(${\cal A}^{\circ}$ denotes the opposite algebra of ${\cal A}$,
cf.\ [75].) That procedure was introduced by Woronowicz and is an
abstract version of similar constructions for quasifree states on
CCR- or CAR-algebras [45,54,75]. Whether the purification map
$\omega_{\mu} \mapsto \omega_{\tilde{\mu}}$ can be generalized from quasifree states
on CCR-algebras to a procedure of assigning to (a suitable class of)
states on a generic $C^*$-algebra pure states on that same algebra,
is in principle an interesting question, which however we shall not
investigate here.
\begin{Theorem} ${}$\\[6pt]
(a) Let $H$ be a (real or complex) Hilbertspace with
scalar product $\mu(\,.\,,\,.\,)$, $R$ a (not necessarily bounded) normal
operator in $H$, and $V,W$ two $\mu$-bounded linear operators on $H$
which are $R$-adjoint, i.e.\ they satisfy
\begin{equation}
W{\rm dom}(R) \subset {\rm dom}(R) \quad {\it and} \quad V^*R = R W \quad
{\rm on \ \ dom}(R) \,.
\end{equation}
Denote by $\mu_s$ the Hermitean form on ${\rm dom}(|R|^{s/2})$ given by
$$ \mu_s(x,y) := \mu(|R|^{s/2}x,|R|^{s/2} y)\,, \quad
x,y \in {\rm dom}(|R|^{s/2}),\ 0 \leq s \leq 2\,.$$ We write
$||\,.\,||_0 := ||\,.\,||_{\mu} := \mu(\,.\,,\,.\,)^{1/2}$ and
$||\,.\,||_s := \mu_s(\,.\,,\,.\,)^{1/2}$ for the corresponding semi-norms.
Then it holds for all $0 \leq s \leq 2$ that
$$ V{\rm dom}(|R|^{s/2}) \subset {\rm dom}(|R|^{s/2}) \quad
{\it and} \quad W{\rm dom}(|R|^{s/2}) \subset {\rm dom}(|R|^{s/2}) \,, $$
and $V$ and $W$ are $\mu_s$-bounded for $0 \leq s \leq 2$.
More precisely, the estimates
\begin{equation}
||\,Vx\,||_0 \leq v\,||\,x\,||_0 \quad {\rm and} \quad
||\,Wx\,||_0 \leq w\,||\,x\,||_0\,, \quad x \in H\,,
\end{equation}
with suitable constants $v,w > 0$, imply that
\begin{equation}
||\,Vx\,||_s \leq w^{s/2}v^{1 -s/2}\,||\,x\,||_s \quad {\rm and} \quad
||\,Wx\,||_s \leq v^{s/2}w^{1-s/2}\,||\,x\,||_s \,,
\end{equation}
for all
$ x \in {\rm dom}(|R|^{s/2})$ and
$0 \leq s \leq 2$.
\\[6pt]
(b)\ \ \ (Corollary of (a))\ \ \ \
Let $(S,\sigma)$ be a symplectic space, $\mu \in {\sf q}(S,\sigma)$ a dominating
scalar product on $(S,\sigma)$, and $\mu_s$, $0 < s \leq 2$, the
scalar products on $S$ defined in (2.11). Then we have relative
$\mu-\mu_s$ continuity of each pair
$V,W$ of symplectically adjoint linear maps of $(S,\sigma)$
for all $0 < s \leq 2$. More precisely, for each pair $V,W$ of
symplectically adjoint linear maps of $(S,\sigma)$, the estimates
(2.16) for all $x \in S$ imply (2.17) for all $x \in S$.
\end{Theorem}
{\it Remark.} (i) In view of the fact that the operator $R$
of part (a) of the Theorem may be unbounded, part (b) can be
extended to situations where it is not assumed that the scalar
product $\mu$ on $S$ dominates the symplectic form $\sigma$.
\\[6pt]
(ii) When it is additionally assumed that $V = T$ and $W = T^{-1}$
with symplectomorphisms $T$ of $(S,\sigma)$, we refer in that
case to the situation of relative continuity of the pairs
$V,W$ as relative continuity of symplectomorphisms.
In Example 2.3 after the proof of Thm.\ 2.2 we show that
relative $\tilde{\mu} - \mu$ continuity of symplectomorphisms fails in general.
Also, it is not the case that relative $\mu - \mu'$ continuity
of symplectomorphisms holds if $\mu'$ is an arbitrary element
in ${\sf pu}(S,\sigma)$ which is dominated by $\mu$ ($||\,\phi\,||_{\mu'} \leq
{\rm const.}||\,\phi\,||_{\mu}$, $\phi \in S$), see Example 2.4
below. This shows that the special relation between $\mu$ and $\tilde{\mu}$
(resp., $\mu$ and the $\mu_s$) expressed in (2.11,2.15) is important
for the derivation of the Theorem.
\\[10pt]
{\it Proof of Theorem 2.2.} (a)\ \ \ In a first step, let it be
supposed that $R$ is bounded.
From the assumed relation (2.15)
and its adjoint relation $R^*V = W^* R^*$ we obtain, for $\epsilon' > 0$
arbitrarily chosen,
\begin{eqnarray*}
V^* (|R|^2 + \epsilon' 1) V & = & V^*RR^* V + \epsilon' V^*V \ =
\ RWW^*R^* + \epsilon' V^*V \\
& \leq & w^2 |R|^2 + \epsilon' v^21 \ \leq \ w^2(|R|^2 + \epsilon 1)
\end{eqnarray*}
with $\epsilon := \epsilon'v^2/w^2$.
This entails for the operator norms
$$ ||\,(|R|^2 + \epsilon' 1 )^{1/2}V \,||
\ \leq\ w\,||\,(|R|^2 + \epsilon 1)^{1/2}\,|| \,, $$
and since $(|R|^2 + \epsilon 1)^{1/2}$ has a bounded inverse,
$$ ||\,(|R|^2 + \epsilon' 1)^{1/2} V
(|R|^2 + \epsilon 1 )^{-1/2} \,||\ \leq\ w\,. $$
On the other hand, one clearly has
$$ ||\,(|R|^2 + \epsilon' 1)^0 V (|R|^2 + \epsilon 1)^0\,||
\ =\ ||\,V\,||\ \leq\ v\,. $$
Now these estimates are preserved if $R$ and $V$
are replaced by their complexified versions on the complexified
Hilbertspace $H \oplus iH = {\bf C} \otimes H$.
Thus, identifying if necessary
$R$ and $V$ with their complexifications, a standard interpolation
argument (see Appendix A) can be applied to yield
$$ ||\,(|R|^2 + \epsilon' 1)^{\alpha} V
(|R|^2 + \epsilon 1)^{-\alpha} \,||\ \leq\ w^{2\alpha}
v^{1 - 2\alpha} $$
for all $0 \leq \alpha \leq 1/2$. Notice that this inequality
holds uniformly in $\epsilon' > 0$. Therefore we may conclude that
$$ ||\,|R|^{2\alpha}Vx \,||_0\ \leq\ w^{2\alpha}v^{1 - 2\alpha}
\,||\,|R|^{2\alpha}x\,||_0
\,, \quad x \in H\,,\ 0 \leq \alpha \leq 1/2\,,$$
which is the required estimate for $V$. The analogous bound for
$W$ is obtained through replacing $V$ by $W$ in the
given arguments.
Now we have to extend the argument to the case that $R$ is unbounded.
Without restriction of generality we may assume that the Hilbertspace
$H$ is complex, otherwise we complexify it and with it all the
operators $R$,$V$,$W$, as above, thereby preserving their assumed properties.
Then let $E$ be the spectral measure of $R$, and denote by
$R_r$ the operator $E(B_r)RE(B_r)$ where $B_r := \{z \in {\bf C}:
|z| \leq r\}$, $r > 0$. Similarly define $V_r$ and $W_r$. From the
assumptions it is seen that $V_r^*R_r = R_rW_r$ holds for all
$r >0$. Applying the reasoning of the first step we arrive, for each
$0 \leq s \leq 2$, at the bounds
$$ ||\,V_r x\,||_s \leq w^{s/2}v^{1-s/2}\,||\,x\,||_s \quad {\rm and}
\quad ||\,W_r x\,||_s \leq v^{s/2}w^{1-s/2}\,||\,x\,||_s \,,$$
which hold uniformly in $r >0$ for all $x \in {\rm dom}(|R|^{s/2})$.
From this, the statement of the Proposition follows.\\[6pt]
(b) This is just an application of (a), identifying $H_{\mu}$ with $H$,
$R_{\mu}$ with $R$ and $V,W$ with their bounded extensions to $H_{\mu}$.
$\Box$
\\[10pt]
{\bf Example 2.3} We exhibit a symplectic space $(S,\sigma)$
with $\mu \in {\sf pr}(S,\sigma)$ and a symplectomorphism $T$ of $(S,\sigma)$
where $T$ and $T^{-1}$ are continuous with respect to $\tilde{\mu}$,
but not with respect to $\mu$. \\[6pt]
Let $S := {\cal S}({\bf R},{\bf C})$, the Schwartz space of rapidly decreasing
testfunctions on ${\bf R}$, viewed as real-linear space.
By $\langle \phi,\psi \rangle := \int \overline{\phi} \psi \,dx$
we denote the standard $L^2$ scalar product. As a symplectic form on
$S$ we choose
$$ \sigma(\phi,\psi) := 2 {\sf Im}\langle \phi,\psi \rangle\,, \quad
\phi,\psi \in S\,. $$
Now define on $S$ the strictly positive, essentially selfadjoint operator
$A\phi := -\frac{d^2}{dx^2}\phi + \phi$,
$\phi \in S$, in $L^2({\bf R})$. Its closure
will again be denoted by $A$; it is bounded below by $1$.
A real-linear scalar product $\mu$ will be defined on $S$ by
$$ \mu(\phi,\psi) := {\sf Re}\langle A\phi,\psi \rangle\,, \quad \phi,\psi \in
S. $$
Since $A$ has lower bound $1$, clearly $\mu$ dominates $\sigma$, and
one easily obtains $R_{\mu} = - i A^{-1}$, $|R_{\mu}| = A^{-1}$.
Hence $\mu \in {\sf pr}(S,\sigma)$ and
$$ \tilde{\mu}(\phi,\psi) = {\sf Re}\langle \phi,\psi \rangle\,,
\quad \phi,\psi \in S\,.$$
Now consider the operator
$$ T : S \to S\,, \quad \ \ (T\phi)(x) := {\rm e}^{-ix^2}\phi(x)\,,
\ \ \ x \in {\bf R},\ \phi \in S\,. $$
Obviously $T$ leaves the $L^2$ scalar product invariant, and hence also
$\sigma$ and $\tilde{\mu}$. The inverse of $T$ is just $(T^{-1}\phi)(x) =
{\rm e}^{i x^2}\phi(x)$, which of course leaves $\sigma$ and $\tilde{\mu}$
invariant as well. However, $T$ is not continuous with respect to $\mu$.
To see this, let $\phi \in S$ be some non-vanishing smooth function
with compact support, and define
$$ \phi_n(x) := \phi(x -n)\,, \quad x \in {\bf R}, \ n \in {\bf N}\,. $$
Then $\mu(\phi_n,\phi_n) = {\rm const.} > 0$ for all $n \in {\bf N}$.
We will show that $\mu(T\phi_n,T\phi_n)$ diverges for $n \to \infty$.
We have
\begin{eqnarray}
\mu(T\phi_n,T\phi_n) & = & \langle A T\phi_n,T\phi_n \rangle
\geq \int \overline{(T\phi_n)'}(T\phi_n)' \, dx \\
& \geq & \int (4x^2|\phi_n(x)|^2 + |\phi_n'(x)|^2)\, dx
- \int 4 |x \phi_n'(x)\phi_n(x)|\,dx \,,\nonumber
\end{eqnarray}
where the primes indicate derivatives and we have used that
$$ |(T\phi_n)'(x)|^2 = 4x^2|\phi_n(x)|^2 + |\phi_n'(x)|^2
+ 4\cdot {\sf Im}(ix \overline{\phi_n}(x)\phi_n'
(x))\,. $$
Using
a substitution of variables, one can see that in the last term
of (2.18) the positive integral grows like $n^2$ for large $n$, thus
dominating eventually the negative integral which grows only like $n$.
So $\mu(T\phi_n,T\phi_n) \to \infty$ for $n \to \infty$, showing that
$T$ is not $\mu$-bounded.
\\[10pt]
{\bf Example 2.4} We give an example of a symplectic space
$(S,\sigma)$, a $\mu \in {\sf pr}(S,\sigma)$ and a $\mu' \in {\sf pu}(S,\sigma)$, where
$\mu$ dominates $\mu'$ and where there is a symplectomorphism $T$
of $(S,\sigma)$ which together with its inverse is $\mu$-bounded,
but not $\mu'$-bounded.\\[6pt]
We take $(S,\sigma)$ as in the previous example and write for each
$\phi \in S$, $\phi_0 := {\sf Re}\phi$ and $\phi_1 := {\sf Im}\phi$.
The real scalar product $\mu$ will be defined by
$$ \mu(\phi,\psi) := \langle\phi_0,A\psi_0\rangle + \langle \phi_1,
\psi_1 \rangle \,, \quad \phi,\psi \in S\,, $$
where the operator $A$ is the same as in the example before. Since its
lower bound is $1$, $\mu$ dominates $\sigma$, and it is not difficult
to see that $\mu$ is even primary. The real-linear scalar product
$\mu'$ will be taken to be
$$ \mu'(\phi,\psi) = {\sf Re}\langle \phi,\psi \rangle\,, \quad
\phi,\psi \in S\,.$$
We know from the example above that $\mu' \in {\sf pu}(S,\sigma)$. Also, it is
clear that $\mu'$ is dominated by $\mu$. Now consider the
real-linear map $T: S \to S$ given by
$$ T(\phi_0 + i\phi_1) := A^{-1/2} \phi_1 - i A^{1/2}\phi_0\,, \quad
\phi \in S\,.$$
One checks easily that this map is bijective with $T^{-1} = - T$,
and that $T$ preserves the symplectic form $\sigma$. Also,
$||\,.\,||_{\mu}$ is preserved by $T$ since
$$ \mu(T\phi,T\phi) = \langle \phi_1,\phi_1 \rangle +
\langle A^{1/2}\phi_0,A^{1/2}\phi_0 \rangle = \mu(\phi,\phi)\,,
\quad \phi \in S\,.$$
On the other hand, we have for each $\phi \in S$
$$ \mu'(T\phi,T\phi) = \langle \phi_1,A\phi_1 \rangle + \langle \phi_0,
A^{-1}\phi_0 \rangle \,, $$
and this expression is not bounded by a ($\phi$-independent) constant times
$\mu'(\phi,\phi)$, since $A$ is unbounded with respect to the
$L^2$-norm.
\newpage
\section{The Algebraic Structure of Hadamard Vacuum Representations}
\setcounter{equation}{0}
${}$
\\[20pt]
{\bf 3.1 Summary of Notions from Spacetime-Geometry}
\\[16pt]
We recall that a spacetime manifold consists of a pair
$(M,g)$, where $M$ is a smooth, paracompact, four-dimensional
manifold without boundaries, and $g$ is a Lorentzian metric for $M$
with signature $(+ - -\, - )$. (Cf.\ [33,52,70], see these references
also for further discussion of the notions to follow.)
It will be assumed that $(M,g)$ is time-orientable, and moreover,
globally hyperbolic. The latter means that $(M,g)$ possesses
Cauchy-surfaces, where by a Cauchy-surface
we always mean a {\it smooth}, spacelike
hypersurface which is intersected exactly once by each inextendable
causal curve in $M$. It can be shown [15,28] that this is
equivalent to the statement that $M$ can be smoothly foliated in
Cauchy-surfaces. Here, a foliation of $M$ in Cauchy-surfaces is
a diffeomorphism $F: {\bf R} \times \Sigma \to M$, where $\Sigma$ is a
smooth 3-manifold so that $F(\{t\} \times \Sigma)$ is, for each
$t \in {\bf R}$, a Cauchy-surface, and the curves
$t \mapsto F(t,q)$
are timelike for all $q \in\Sigma$.
(One can even show that, if global hyperbolicity had been
defined by requiring only the existence of a non necessarily
smooth or spacelike Cauchy-surface (i.e.\ a topological
hypersurface which is intersected exactly once by each
inextendable causal curve), then it is still true that a globally
hyperbolic spacetime can be smoothly foliated in Cauchy-surfaces,
see [15,28].)
We shall also be interested in
ultrastatic globally hyperbolic spacetimes.
A globally hyperbolic spacetime
is said to be {\it ultrastatic} if a foliation
$F : {\bf R} \times \Sigma \to M$ in Cauchy-surfaces can be found so that
$F_*g$ has the form $dt^2 \oplus (- \gamma)$ with a complete
($t$-independent) Riemannian metric $\gamma$ on $\Sigma$.
This particular foliation will then be called a {\it natural foliation}
of the ultrastatic spacetime. (An ultrastatic spacetime may posses
more than one natural foliation, think e.g.\ of Minkowski-spacetime.)
The notation for the causal sets and domains of dependence
will be recalled: Given a spacetime $(M,g)$ and
${\cal O} \subset M$, the set $J^{\pm}({\cal O})$ (causal future/past of
${\cal O}$) consists of all points $p \in M$ which can be reached by
future/past directed causal curves emanating from ${\cal O}$. The set
$D^{\pm}({\cal O})$ (future/past domain of dependence of ${\cal O}$) is defined
as consisting of all $p \in J^{\pm}({\cal O})$ such that every
past/future inextendible causal curve starting at $p$ intersects ${\cal O}$.
One writes $J({\cal O}) := J^+({\cal O}) \cup J^-({\cal O})$ and $D({\cal O}) :=
D^+({\cal O}) \cup D^-({\cal O})$. They are called the {\it causal set}, and
the {\it domain of dependence}, respectively, of ${\cal O}$.
For ${\cal O} \subset M$, we denote by ${\cal O}^{\perp} := {\rm int}(M
\backslash J({\cal O}))$ the {\it causal complement} of ${\cal O}$,
i.e.\ the largest {\it open}
set of points which cannot be connected to ${\cal O}$ by any causal curve.
A set of the form ${\cal O}_G := {\rm int}\,D(G)$, where $G$ is a subset
of some Cauchy-surface $\Sigma$ in $(M,g)$, will be referred to
as the {\it diamond based on} $G$; we shall also say that
$G$ is the {\it base} of ${\cal O}_G$. We note that if ${\cal O}_G$ is a
diamond, then ${\cal O}_G^{\perp}$ is again a diamond, based on
$\Sigma \backslash \overline{G}$.
A diamond will be called
{\it regular} if $G$ is an open, relatively compact subset of
$\Sigma$ and if the boundary $\partial G$ of $G$ is contained in
the union of finitely many smooth, two-dimensional submanifolds
of $\Sigma$.
Following [45], we say that an open neighbourhood $N$ of a
Cauchy-surface $\Sigma$ in $(M,g)$ is a {\it causal normal neighbourhood}
of $\Sigma$ if (1) $\Sigma$ is a Cauchy-surface for $N$, and
(2) for each pair of points $p,q \in N$ with $p \in J^+(q)$, there
is a convex normal neighbourhood ${\cal O} \subset M$ such that
$J^-(p) \cap J^+(q) \subset {\cal O}$. Lemma 2.2 of [45] asserts the
existence of causal normal neighbourhoods for any Cauchy-surface
$\Sigma$.
\\[20pt]
{\bf 3.2 Some Structural Aspects of
Quantum Field Theory in Curved Spacetime}
\\[16pt]
In the present subsection, we shall address some of the problems
one faces in the formulation of quantum field theory in curved
spacetime, and explain the notions of local definiteness, local
primarity, and Haag-duality. In doing so, we follow our presentation
in [67] quite closely. Standard general references related to the
subsequent discussion are [26,31,45,71].
Quantum field theory in curved spacetime (QFT in CST, for short)
means that one considers quantum field theory means that one considers
quantum fields propagating in a (classical) curved background spacetime
manifold $(M,g)$. In general, such a spacetime need not possess
any symmetries, and so one cannot tie the notion of ``particles''
or ``vacuum'' to spacetime symmetries, as one does in quantum field
theory in Minkowski spacetime. Therefore, the problem of
how to characterize the physical states arises. For the discussion
of this problem, the setting of algebraic quantum field theory is
particularly well suited. Let us thus summarize some of the relevant
concepts of algebraic QFT in CST. Let a spacetime manifold
$(M,g)$ be given. The observables of a quantum system (e.g.\ a quantum
field) situated in $(M,g)$ then have the basic structure of a map
${\cal O} \to {\cal A(O)}$, which assigns to each open, relatively compact
subset ${\cal O}$ of $M$ a $C^*$-algebra ${\cal A(O)}$,\footnote{
Throughout the paper, $C^*$-algebras are assumed to be unital, i.e.\
to possess a unit element, denoted by ${ 1}$. It is further
assumed that the unit element is the same for all the ${\cal A(O)}$.}
with the properties:\footnote{where $[{\cal A}({\cal O}_1),{\cal A}({\cal O}_2)]
= \{A_1A_2 - A_2A_1 : A_j \in {\cal A}({\cal O}_j),\ j =1,2 \}$.}
\begin{equation}
{\it Isotony:}\quad \quad {\cal O}_1 \subset {\cal O}_2
\Rightarrow {\cal A}({\cal O}_1) \subset {\cal A}({\cal O}_2)
\end{equation}
\begin{equation}
{\it Locality:} \quad \quad {\cal O}_1 \subset {\cal O}_2^{\perp}
\Rightarrow [{\cal A}({\cal O}_1),{\cal A}({\cal O}_2)] = \{0 \} \,.
\end{equation}
A map ${\cal O} \to {\cal A(O)}$ having these properties is called a {\it net
of local observable algebras} over $(M,g)$. We recall that the
conditions of locality and isotony are motivated by the idea
that each ${\cal A(O)}$ is the $C^*$-algebra formed by the observables
which can be measured within the spacetime region ${\cal O}$ on the
system. We refer to [31] and references given there for further
discussion.
The collection of all open, relatively compact subsets of
$M$ is directed with respect to set-inclusion, and so we can, in view
of (3.1), form the smallest $C^*$-algebra ${\cal A} := \overline{
\bigcup_{{\cal O}}{\cal A(O)}}^{||\,.\,||}$ which contains all local algebras
${\cal A(O)}$.
For the description of a system we need not only observables but also
states. The set ${\cal A}^{*+}_1$ of all positive, normalized linear
functionals on ${\cal A}$ is mathematically referred to as the set
of {\it states} on ${\cal A}$, but not all elements of ${\cal A}^{*+}_1$
represent physically realizable states of the system. Therefore, given
a local net of observable algebras ${\cal O} \to {\cal A(O)}$ for a physical
system over $(M,g)$, one must specify the set of physically relevant states
${\cal S}$, which is a suitable subset of ${\cal A}^{*+}_1$.
We have already mentioned in Chapter 2 that every state $\omega \in
{\cal A}^{*+}_1$ determines canonically its GNS representation
$({\cal H}_{\omega},\pi_{\omega},\Omega_{\omega})$ and thereby induces
a net of von Neumann algebras (operator algebras on ${\cal H}_{\omega}$)
$$ {\cal O} \to {\cal R}_{\omega}({\cal O}) := \pi_{\omega}({\cal O})^- \,. $$
Some of the mathematical properties of the GNS representations, and of
the induced nets of von Neumann algebras, of states $\omega$ on
${\cal A}$ can naturally be interpreted physically. Thus one obtains
constraints on the states $\omega$ which are to be viewed as
physical states. Following this line of thought, Haag, Narnhofer
and Stein [32] formulated what they called the ``principle of local
definiteness'', consisting of the following three conditions
to be obeyed by any collection ${\cal S}$ of physical states.
\\[10pt]
{\bf Local Definiteness:} ${}\ \ \bigcap_{{\cal O} \owns p}
{\cal R}_{\omega}({\cal O}) = {\bf C} \cdot { 1}$ for all $\omega \in {\cal S}$
and all $p \in M$.
\\[6pt]
{\bf Local Primarity:} \ \ For each $\omega \in {\cal S}$, ${\cal R}_{\omega}
({\cal O})$ is a factor.
\\[6pt]
{\bf Local Quasiequivalence:} For each pair $\omega_1,\omega_2 \in {\cal S}$
and each relatively compact, open ${\cal O} \subset M$, the representations
$\pi_{\omega_1} | {\cal A(O)}$ and $\pi_{\omega_2} | {\cal A(O)}$ of ${\cal A(O)}$ are
quasiequivalent.
\\[10pt]
{\it Remarks.} (i) We recall (cf.\ the first Remark in Section 2) that
${\cal R}_{\omega}({\cal O})$ is a factor if ${\cal R}_{\omega}({\cal O}) \cap {\cal R}_{\omega}
({\cal O})' = {\bf C} \cdot { 1}$ where the prime means taking the
commutant. We have not stated in the formulation of local primarity
for which regions ${\cal O}$ the algebra ${\cal R}_{\omega}({\cal O})$ is required to be
a factor. The regions ${\cal O}$ should be taken from a class of subsets of
$M$ which forms a base for the topology.
\\[6pt]
(ii) Quasiequivalence of representations means unitary equivalence up to
multiplicity. Another characterization of quasiequivalence is to say
that the folia of the representations coincide, where the
{\it folium} of
a representation $\pi$ is defined as the set of all $\omega \in
{\cal A}^{*+}_1$ which can be represented as $\omega(A) = tr(\rho\,\pi(A))$
with a density matrix $\rho$ on the representation Hilbertspace of $\pi$.
\\[6pt]
(iii) Local definiteness and quasiequivalence together express
that physical states have finite (spatio-temporal) energy-density
with respect to each other, and local primarity and quasiequivalence
rule out local macroscopic observables and local superselection
rules. We refer to [31] for further discussion and background
material.
A further, important property which one expects to be satisfied
for physical states $\omega \in {\cal S}$ whose GNS representations are
irreducible \footnote{It is easy to see that,
in the presence of local primarity, Haag-duality will be
violated if $\pi_{\omega}$ is not irreducible.} is
\\[10pt]
{\bf Haag-Duality:} \ \ ${\cal R}_{\omega}({\cal O}^{\perp})' = {\cal R}_{\omega}({\cal O})$, \\
which should hold for the causally complete regions ${\cal O}$, i.e.\ those
satisfying $({\cal O}^{\perp})^{\perp} = {\cal O}$, where ${\cal R}_{\omega}({\cal O}^{\perp})$
is defined as the von Neumann algebra generated by all the ${\cal R}_{\omega}
({\cal O}_1)$ so that $\overline{{\cal O}_1} \subset {\cal O}^{\perp}$.
\\[10pt]
We comment that Haag-duality means that the von Neumann algebra
${\cal R}_{\omega}({\cal O})$ of local observables is maximal in the sense
that no further observables can be added without violating the
condition of locality. It is worth mentioning here that the condition
of Haag-duality plays an important role in the theory of superselection
sectors in algebraic quantum field theory in Minkowski spacetime
[31,59]. For local nets of observables generated by Wightman fields
on Minkowski spacetime it follows from the results of Bisognano and
Wichmann [4] that a weaker condition of ``wedge-duality'' is
always fulfilled, which allows one to pass to a new, potentially
larger local net (the ``dual net'') which satisfies Haag-duality.
In quantum field theory in Minkowski-spacetime where one is given
a vacuum state $\omega_0$, one can define the set of physical states
${\cal S}$ simply as the set of all states on ${\cal A}$ which are locally
quasiequivalent (i.e., the GNS representations of the states are
locally quasiequivalent to the vacuum-representation) to $\omega_0$.
It is obvious that local quasiequivalence then holds for ${\cal S}$.
Also, local definiteness holds in this case, as was proved by Wightman
[72]. If Haag-duality holds in the vacuum representation (which,
as indicated above, can be assumed to hold quite generally), then it
does not follow automatically that all pure states locally quasiequivalent
to $\omega_0$ will also have GNS representations fulfilling Haag-duality;
however, it follows once some regularity conditions are satisfied
which have been checked in certain quantum field models [19,61].
So far there seems to be no general physically motivated criterion
enforcing local primarity of a quantum field theory in algebraic
formulation in Minkowski spacetime. But it is known that many quantum
field theoretical models satisfy local primarity.
For QFT in CST we do in general not know what a vacuum state is and so
${\cal S}$ cannot be defined in the same way as just described. Yet in some
cases (for some quantum field models) there may be a set
${\cal S}_0 \subset {\cal A}^{*+}_1$ of distinguished states, and if this class
of states satisfies the four conditions listed above, then the set
${\cal S}$, defined as consisting of all states $\omega_1 \in {\cal A}^{*+}_1$
which are locally quasiequivalent to any (and hence all) $\omega
\in {\cal S}_0$, is a good candidate for the set of physical states.
For the free scalar Klein-Gordon field (KG-field) on
a globally hyperbolic spacetime, the following classes of
states have been suggested as distinguished, physically
reasonable states \footnote{The following list is not meant
to be complete, it comprises some prominent families of states
of the KG-field over a generic class of spacetimes for which
mathematically sound results are known. Likewise, the indicated
references are by no means exhaustive.}
\begin{itemize}
\item[(1)] (quasifree) states fulfilling local stability
[3,22,31,32]
\item[(2)] (quasifree) states fulfilling the wave front set (or microlocal)
spectrum condition [6,47,55]
\item[(3)] quasifree Hadamard states [12,68,45]
\item[(4)] adiabatic vacua [38,48,53]
\end{itemize}
The list is ordered in such a way that the less restrictive
condition preceeds the stronger one. There are a couple of
comments to be made here. First of all, the specifications
(3) and (4) make use of the information that one deals with
the KG-field (or at any rate, a free field obeying a linear
equation of motion of hyperbolic character), while the
conditions (1) and (2) do not require such input and are
applicable to general -- possibly interacting -- quantum
fields over curved spacetimes.
(It should however be mentioned that only for the KG-field (2)
is known to be stronger than (1). The relation between (1) and (2)
for more general theories is not settled.)
The conditions imposed on the classes of
states (1), (2) and (3) are related in that they are ultralocal remnants
of the spectrum condition requiring a certain regularity of the
short distance behaviour of the respective states which can be
formulated in generic spacetimes. The class of states (4) is
more special and can only be defined for the KG-field (or other
linear fields) propagating in Robertson-Walker-type spacetimes.
Here a distinguished choice of a time-variable can be made,
and the restriction imposed on adiabatic vacua is a regularity
condition on their spectral behaviour with respect to that
special choice of time. (A somewhat stronger formulation of
local stability has been proposed in [34].)
It has been found by Radzikowski [55] that for quasifree states of the
KG-field over generic globally hyperbolic spacetimes
the classes (2) and (3) coincide. The microlocal spectrum condition
is further refined and applied in [6,47]. Recently it was
proved by Junker [38] that adiabatic vacua of the KG-field
in Robertson-Walker spacetimes fulfill the microlocal spectrum
condition and thus are, in fact, quasifree Hadamard states.
The notion of the microlocal spectrum condition and the just mentioned
results related to it draw on pseudodifferential operator
techniques, particularly the notion of the wave front set, see [20,36,37].
Quasifree Hadamard states of the KG-field (see definition
in Sec.\ 3.4 below)
have been investigated for quite some time. One of the early
studies of these states is [12]. The importance of these
states, especially in the context of the semiclassical
Einstein equation, is stressed in [68]. Other significant
references include [24,25] and, in particular, [45] where,
apparently for the first time, a satisfactory definition of
the notion of a globally Hadamard state is given, cf.\
Section 3.4 for more details. In [66] it is proved that the
class of quasifree Hadamard states of the KG-field fulfills
local quasiequivalence in generic globally hyperbolic spacetimes
and local definiteness, local primarity and Haag-duality
for the case of ultrastatic globally hyperbolic spacetimes.
As was outlined in the beginning, the purpose of the present
chapter is to obtain these latter results also for arbitrary
globally hyperbolic spacetimes which are not necessarily
ultrastatic. It turns out that some of our previous results
can be sharpened, e.g.\ the local quasiequivalence specializes
in most cases to local unitary equivalence, cf.\ Thm.\ 3.6.
For a couple of other
results about the algebraic structure of the KG-field as well as other
fields over curved spacetimes we refer to
[2,6,15,16,17,40,41,46,63,64,65,66,74].
\\[24pt]
{\bf 3.3 The Klein-Gordon Field}
\\[18pt]
In the present section we summarize the quantization of the
classical KG-field over a globally hyperbolic spacetime in the
$C^*$-algebraic formalism. This follows in major parts the
the work of Dimock [16], cf.\ also references given there.
Let $(M,g)$ be a globally hyperbolic spacetime. The KG-equation
with potential term $r$ is
\begin{equation}
(\nabla^a \nabla_a + r) \varphi = 0
\end{equation}
where $\nabla$ is the Levi-Civita derivative of the metric $g$,
the potential function
$r \in C^{\infty}(M,{\bf R})$ is arbitrary but fixed, and the
sought for solutions $\varphi$ are smooth and real-valued.
Making use of the fact that $(M,g)$ is globally hyperbolic
and drawing on earlier results by Leray, it is shown in
[16] that there are two uniquely determined, continuous
\footnote{ With respect to the usual locally convex topologies
on $C_0^{\infty}(M,{\bf R})$ and $C^{\infty}(M,{\bf R})$, cf.\
[13].}
linear maps $E^{\pm}: C_0^{\infty}(M,{\bf R}) \to C^{\infty}(M,{\bf R})$
with the properties
$$ (\nabla^a \nabla_a + r)E^{\pm}f = f = E^{\pm}(\nabla^a
\nabla_a + r)f\,,\quad f \in C_0^{\infty}(M,{\bf R})\,, $$
and
$$ {\rm supp}(E^{\pm}f) \subset J^{\pm}({\rm supp}(f))\,,\quad
f \in C_0^{\infty}(M,{\bf R})\,. $$
The maps $E^{\pm}$ are called the advanced(+)/retarded(--)
fundamental solutions of the KG-equation with potential term
$r$ in $(M,g)$, and their difference $E := E^+ - E^-$ is referred
to as the {\it propagator} of the KG-equation.
One can moreover show that the Cauchy-problem for the KG-equation
is well-posed. That is to say, if $\Sigma$ is any Cauchy-surface
in $(M,g)$, and
$u_0 \oplus u_1 \in C_0^{\infty}(M,{\bf R}) \oplus
C_0^{\infty}(M,{\bf R})$ is any pair of Cauchy-data on $\Sigma$,
then there exists precisely one smooth solution $\varphi$
of the KG-equation (3.3) having the property that
\begin{equation}
P_{\Sigma}(\varphi) := \varphi | \Sigma \oplus
n^a \nabla_a \varphi|\Sigma = u_0 \oplus u_1\,.
\end{equation}
The vectorfield $n^a$ in (3.4) is the future-pointing
unit normalfield of $\Sigma$.
Furthermore, one has ``finite propagation speed'', i.e.\
when the supports of $u_0$ and $u_1$ are contained in a subset
$G$ of $\Sigma$, then ${\rm supp}(\varphi) \subset J(G)$. Notice that
compactness of $G$ implies that $J(G) \cap \Sigma'$ is compact
for any Cauchy-surface $\Sigma'$.
The well-posedness of the
Cauchy-problem is a consequence of the classical energy-estimate
for solutions of second order hyperbolic partial differential
equations, cf.\ e.g.\ [33]. To formulate it, we introduce further
notation. Let $\Sigma$ be a Cauchy-surface for $(M,g)$, and
$\gamma_{\Sigma}$ the Riemannian metric, induced by the ambient
Lorentzian metric, on $\Sigma$. Then denote the Laplacian
operator on $C_0^{\infty}(\Sigma,{\bf R})$ corresponding to
$\gamma_{\Sigma}$ by $\Delta_{\gamma_{\Sigma}}$, and define the
{\it classical energy scalar product} on
$C_0^{\infty}(\Sigma,{\bf R}) \oplus C_0^{\infty}(\Sigma,{\bf R})$ by
\begin{equation}
\mu_{\Sigma}^E(u_0 \oplus u_1,
v_0 \oplus v_1 ) := \int_{\Sigma} (u_0 (- \Delta_{\gamma_{\Sigma}} + 1)v_0
+ u_1 v_1) \, d\eta_{\Sigma} \,,
\end{equation}
where $d\eta_{\Sigma}$ is the metric-induced volume measure on
$\Sigma$. As a special case of the energy estimate
presented in [33] one then obtains
\begin{Lemma}
(Classical energy estimate for the KG-field.)
Let $\Sigma_1$ and $\Sigma_2$ be a pair of Cauchy-surfaces
in $(M,g)$ and $G$ a compact subset of $\Sigma_1$. Then there
are two positive constants $c_1$ and $c_2$ so that there
holds the estimate
\begin{equation}
c_1\,\mu^E_{\Sigma_1}(P_{\Sigma_1}(\varphi),P_{\Sigma_1}
(\varphi)) \leq \mu^E_{\Sigma_2}(P_{\Sigma_2}(\varphi),
P_{\Sigma_2}(\varphi)) \leq
c_2 \, \mu^E_{\Sigma_1}(P_{\Sigma_1}(\varphi),P_{\Sigma_1}
(\varphi))
\end{equation}
for all solutions $\varphi$ of the KG-equation (3.3)
which have the property that the supports of the Cauchy-data
$P_{\Sigma_1}(\varphi)$ are contained in $G$.
\footnote{ {\rm The formulation given here is to some extend
more general than the one appearing in [33] where it is
assumed that $\Sigma_1$ and $\Sigma_2$ are members of a foliation.
However, the more general formulation can be reduced to that case.}}
\end{Lemma}
We shall now indicate that the space of smooth solutions of the
KG-equation (3.3) has the structure of a symplectic space,
locally as well as globally, which comes in several equivalent
versions. To be more specific, observe first that the
Cauchy-data space
$$ {\cal D}_{\Sigma} := C_0^{\infty}(\Sigma,{\bf R}) \oplus C_0^{\infty}(\Sigma,{\bf R}) $$
of an arbitrary given Cauchy-surface $\Sigma$ in $(M,g)$
carries a symplectic form
$$ \delta_{\Sigma}(u_0 \oplus u_1, v_0 \oplus v_1)
:= \int_{\Sigma}(u_0v_1 - v_0u_1)\,d\eta_{\Sigma}\,.$$
It will also be observed that this symplectic form
is dominated by the classical energy scalar product
$\mu^E_{\Sigma}$.
Another symplectic space is $S$, the set
of all real-valued $C^{\infty}$-solutions $\varphi$ of the
KG-equation (3.3) with the property that, given any Cauchy-surface
$\Sigma$ in $(M,g)$, their Cauchy-data $P_{\Sigma}(\varphi)$
have compact support on $\Sigma$. The symplectic form on
$S$ is given by
$$ \sigma(\varphi,\psi) := \int_{\Sigma}(\varphi n^a \nabla_a\psi
-\psi n^a \nabla_a \varphi)\,d\eta_{\Sigma} $$
which is independent of the choice of the Cauchy-surface $\Sigma$
on the right hand side over which the integral is formed;
$n^a$ is again the future-pointing unit normalfield of $\Sigma$.
One clearly finds that for each Cauchy-surface $\Sigma$ the
map $P_{\Sigma} : S \to {\cal D}_{\Sigma}$ establishes a symplectomorphism
between the symplectic spaces $(S,\sigma)$ and $({\cal D}_{\Sigma},\delta_{\Sigma})$.
A third symplectic space equivalent to the previous ones is obtained
as the quotient $K := C_0^{\infty}(M,{\bf R}) /{\rm ker}(E)$ with symplectic form
$$ \kappa([f],[h]) := \int_M f(Eh)\,d\eta \,, \quad
f,h \in C_0^{\infty}(M,{\bf R})\,, $$
where $[\,.\,]$ is the quotient map $C_0^{\infty}(M,{\bf R}) \to K$ and
$d\eta$ is the metric-induced volume measure on $M$.
Then define for any open subset ${\cal O} \subset M$ with compact
closure the set $K({\cal O}) := [C_0^{\infty}({\cal O},{\bf R})]$.
One can see that the space $K$ has naturally the structure of
an isotonous, local net ${\cal O} \to K({\cal O})$ of subspaces, where
locality means that the symplectic form $\kappa([f],[h])$
vanishes for $[f] \in K({\cal O})$ and $[h] \in K({\cal O}_1)$
whenever ${\cal O}_1 \subset {\cal O}^{\perp}$.
Dimock has proved in [16 (Lemma A.3)] that moreover there holds
\begin{equation}
K({\cal O}_G) \subset K(N)
\end{equation}
for all open neighbourhoods $N$ (in $M$) of $G$, whenever
${\cal O}_G$ is a diamond. Using this, one obtains that the map
$(K,\kappa) \to (S,\sigma)$ given by $[f] \mapsto Ef$ is
surjective, and by Lemma A.1 in [16], it is even a
symplectomorphism. Clearly,
$(K({\cal O}_G),\kappa|K({\cal O}_G))$ is a symplectic subspace
of $(K,\kappa)$ for each diamond ${\cal O}_G$
in $(M,g)$. For any such diamond one
then obtains, upon viewing it
(or its connected components separately), equipped with the appropriate
restriction of the spacetime metric $g$, as a globally
hyperbolic spacetime in its own right, local versions
of the just introduced symplectic spaces and the symplectomorphisms
between them. More precisely, if we denote by $S({\cal O}_G)$
the set of all smooth solutions of the KG-equation (3.3)
with the property that their Cauchy-data on $\Sigma$ are
compactly supported in $G$, then the map
$P_{\Sigma}$ restricts to a symplectomorphism $(S({\cal O}_G),
\sigma|S({\cal O}_G)) \to ({\cal D}_{G},\delta_{G})$,
$\varphi \mapsto P_{\Sigma}(\varphi)$. Likewise, the
symplectomorphism $[f] \mapsto Ef$ restricts to a symplectomorphism
$(K({\cal O}_G),\kappa|K({\cal O}_G)) \to
(S({\cal O}_G),\sigma|S({\cal O}_G))$.
To the symplectic space $(K,\kappa)$ we can now associate its
Weyl-algebra ${\cal A}[K,\kappa]$, cf.\ Chapter 2. Using the
afforementioned local net-structure of the symplectic space
$(K,\kappa)$, one arrives at the following result.
\begin{Proposition}
{\rm [16]}. Let $(M,g)$ be a globally hyperbolic
spacetime, and $(K,\kappa)$ the symplectic space, constructed
as above, for the KG-eqn.\ with smooth potential term $r$ on $(M,g)$.
Its Weyl-algebra ${\cal A}[K,\kappa]$ will be called the {\em Weyl-algebra
of the KG-field with potential term $r$ over} $(M,g)$. Define
for each open, relatively compact ${\cal O} \subset M$, the set ${\cal A}({\cal O})$
as the $C^*$-subalgebra of ${\cal A}[K,\kappa]$ generated by all the
Weyl-operators $W([f])$, $[f] \in K({\cal O})$. Then
${\cal O} \to {\cal A}({\cal O})$ is a net of $C^*$-algebras fulfilling
isotony (3.1) and locality (3.2), and moreover {\em primitive
causality}, i.e.\
\begin{equation}
{\cal A}({\cal O}_G) \subset {\cal A}(N)
\end{equation}
for all neighbourhoods $N$ (in $M$) of $G$, whenever ${\cal O}_G$ is
a (relatively compact) diamond.
\end{Proposition}
It is worth recalling (cf.\ [5]) that the Weyl-algebras
corresponding to symplectically equivalent spaces are
canonically isomorphic in the following way: Let
$W(x)$, $x \in K$ denote the Weyl-generators of ${\cal A}[K,\kappa]$
and $W_S(\varphi)$, $\varphi \in S$, the Weyl-generators
of ${\cal A}[S,\sigma]$. Furthermore, let $T$ be a symplectomorphism
between $(K,\kappa)$ and $(S,\sigma)$.
Then there is a uniquely determined
$C^*$-algebraic isomorphism $\alpha_T : {\cal A}[K,\kappa] \to
{\cal A}[S,\sigma]$ given by $\alpha_T(W(x)) = W_S(Tx)$, $x \in K$.
This shows that if we had associated e.g.\ with $(S,\sigma)$
the Weyl-algebra ${\cal A}[S,\sigma]$ as the algebra of quantum observables
of the KG-field over $(M,g)$, we would have obtained an
equivalent net of observable algebras
(connected to the previous one by a net isomorphism,
see [3,16]), rendering the same
physical information.
\\[24pt]
{\bf 3.4 Hadamard States}
\\[18pt]
We have indicated above that quasifree Hadamard states
are distinguished by their short-distance behaviour which
allows the definition of expectation values of energy-momentum
observables with reasonable properties [26,68,69,71]. If
$\omega_{\mu}$ is a quasifree state on the Weyl-algebra
${\cal A}[K,\kappa]$, then we call
$$ \lambda(x,y) := \mu(x,y) + \frac{i}{2}\kappa(x,y)\,, \quad
x,y \in K\,, $$
its {\it two-point function} and
$$ \Lambda(f,h) := \lambda([f],[h])\,, \quad f,h \in C_0^{\infty}(M,{\bf R})\,, $$
its {\it spatio-temporal} two-point function.
In Chapter 2 we have seen that a quasifree state is entirely
determined through specifying $\mu \in {\sf q}(K,\kappa)$, which
is equivalent to the specification of the two-point function
$\lambda$. Sometimes the notation $\lambda_{\omega}$
or $\lambda_{\mu}$ will be used to indicate the quasifree
state $\omega$ or the dominating scalar product $\mu$ which is
determined by $\lambda$.
For a quasifree Hadamard state, the spatio-temporal two-point
function is of a special form, called Hadamard form. The
definition of Hadamard form which we give here follows that due to
Kay and Wald [45]. Let $N$ is a causal normal neighbourhood
of a Cauchy-surface $\Sigma$ in $(M,g)$. Then a smooth function
$\chi : N \times N \to [0,1]$ is called {\it $N$-regularizing}
if it has the following property: There is an open
neighbourhood, $\Omega_*$, in $N \times N$ of the set of
pairs of causally related points in $N$ such that
$\overline{\Omega_*}$ is contained in a set $\Omega$ to
be described presently, and $\chi \equiv 1$ on $\Omega_*$
while $\chi \equiv 0$ outside of $\overline{\Omega}$. Here,
$\Omega$ is an open neighbourhood in $M \times M$ of the
set of those $(p,q) \in M \times M$ which are causally related
and have the property that (1) $J^+(p) \cap J^-(q)$ and
$J^+(q) \cap J^-(p)$ are contained within a convex normal
neighbourhood, and (2) $s(p,q)$, the square of the geodesic
distance between $p$ and $q$, is a well-defined, smooth
function on $\Omega$. (One observes that there are always
sets $\Omega$ of this type which contain a neighbourhood of the
diagonal in $M \times M$, and that an $N$-regularizing function
depends on the choice of the pair of sets $\Omega_*,\Omega$
with the stated properties.) It is not difficult to check that
$N$-regularizing functions always exist for any causal normal
neighbourhood; a proof of that is e.g.\ given in [55].
Then denote by $U$ the square root of the VanVleck-Morette determinant,
and by $v_m$, $m \in {\bf N}_0$ the sequence determined by the
Hadamard recursion relations for the KG-equation (3.3),
see [23,27] and also [30] for their definition.
They are all smooth functions on $\Omega$.\footnote{For any
choice of $\Omega$ with the properties just described.}
Now set for $n \in {\bf N}$,
$$ V^{(n)}(p,q) := \sum_{m = 0}^n v_m(p,q)(s(p,q))^m \,,
\quad (p,q) \in \Omega\,, $$
and, given a smooth time-function $T: M \to {\bf R}$ increasing
towards the future, define for all $\epsilon > 0$ and
$(p,q) \in \Omega$,
$$ Q_T(p,q;\epsilon) := s(p,q) - 2 i\epsilon (T(p) - T(q)) -
\epsilon^2 \,,$$
and
$$G^{T,n}_{\epsilon}(p,q) := \frac{1}{4\pi^2}\left(
\frac{U(p,q)}{Q_T(p,q;\epsilon)} + V^{(n)}(p,q)ln(Q_T(p,q;
\epsilon)) \right) \,, $$
where $ln$ is the principal branch of the logarithm.
With this notation, one can give the
\begin{Definition}{\rm [45]}. A ${\bf C}$-valued
bilinear form $\Lambda$ on $C_0^{\infty}(M,{\bf R})$ is called an {\em Hadamard form}
if, for a suitable choice of a causal normal neighbourhood
$N$ of some Cauchy-surface $\Sigma$, and for suitable choices of an
$N$-regularizing function $\chi$ and a future-increasing
time-function $T$ on $M$, there exists a sequence
$H^{(n)} \in C^n(N \times N)$, so that
\begin{equation}
\Lambda(f,h) = \lim_{\epsilon \to 0+}
\int_{M\times M} \Lambda^{T,n}_{\epsilon}(p,q)f(p)h(q)\,
d\eta(p) \,d\eta(q)
\end{equation}
for all $f,h \in C_0^{\infty}(N,{\bf R})$, where
\footnote{ The set $\Omega$ on which the functions forming
$G^{T,n}_{\epsilon}$ are defined and smooth is here to
coincide with the $\Omega$ with respect to which $\chi$ is
defined.}
\begin{equation}
\Lambda^{T,n}_{\epsilon}(p,q) := \chi(p,q)G^{T,n}_{\epsilon}(p,q)
+ H^{(n)}(p,q)\,,
\end{equation}
and if, moreover, $\Lambda$ is a global bi-parametrix of
the KG-equation (3.3), i.e.\ it satisfies
$$ \Lambda((\nabla^a\nabla_a + r)f,h) = B_1(f,h)\quad {\it and}
\quad \Lambda(f,(\nabla^a\nabla_a + r)h) = B_2(f,h) $$
for all $f,h \in C_0^{\infty}(M)$, where $B_1$ and $B_2$ are
given by smooth integral kernels on $M \times M$.\footnote{
We point out that statement (b) of Prop.\ 3.4 is wrong if
the assumption that $\Lambda$ is a global bi-parametrix is not made.
In this respect, Def.\ C.1 of [66] is imprecisely formulated as
the said assumption is not stated. There, like in several other
references, it has been implicitely assumed that $\Lambda$ is a
two point function and thus a bi-solution of (3.3), i.e. a
bi-parametrix with $B_1 = B_2 \equiv 0$.}
\end{Definition}
Based on results of [24,25], it is shown in [45] that this is
a reasonable definition. The findings of these works will be
collected in the following
\begin{Proposition} ${}$\\[6pt]
(a) If $\Lambda$ is of Hadamard form on a causal normal neighbourhood
$ N$ of a Cauchy-surface $\Sigma$ for some choice of a time-function
$T$ and some $N$-regularizing function $\chi$ (i.e.\ that
(3.9),(3.10) hold with suitable $H^{(n)} \in C^n(N \times
N)$), then so it is for any other time-function $T'$ and
$N$-regularizing $\chi'$. (This means that these changes
can be compensated by choosing another sequence $H'^{(n)} \in
C^n( N \times N)$.)
\\[6pt]
(b) (Causal Propagation Property of the Hadamard Form)\\
If $\Lambda$ is of Hadamard form on a causal normal neighbourhood
$ N$ of some Cauchy-surface $\Sigma$, then it is of Hadamard form
in any causal normal neighbourhood $ N'$ of any other Cauchy-surface
$\Sigma'$.
\\[6pt]
(c) Any $\Lambda$ of Hadamard form is a regular kernel distribution
on $C_0^{\infty}(M \times M)$.
\\[6pt]
(d) There exist pure, quasifree Hadamard states (these will be
referred to as {\em Hadamard vacua}) on the Weyl-algebra ${\cal A}[K,\kappa]$ of the
KG-field in any globally hyperbolic spacetime. The family of quasifree
Hadamard states on ${\cal A}[K,\kappa]$ spans an infinite-dimensional subspace
of the continuous dual space of ${\cal A}[K,\kappa]$.
\\[6pt]
(e) The dominating scalar products $\mu$ on $K$ arising from quasifree
Hadamard states $\omega_{\mu}$ induce locally the same topology,
i.e.\ if $\mu$ and $\mu'$ are arbitrary such scalar products and
${\cal O} \subset M$ is open and relatively compact, then there are two
positive constants $a,a'$ such that
$$ a\, \mu([f],[f]) \leq \mu'([f],[f]) \leq a'\,\mu([f],[f])\,,
\quad [f] \in K({\cal O})\,.$$
\end{Proposition}
{\it Remark.} Observe that this definition of Hadamard form rules out
the occurence of spacelike singularities, meaning that the Hadamard form
$\Lambda$ is, when tested on functions $f,h$ in (3.9) whose
supports are acausally separated, given by a $C^{\infty}$-kernel.
For that reason, the definition of Hadamard form as stated
above is also called {\it global} Hadamard form (cf.\ [45]).
A weaker definition of Hadamard form would be to prescribe (3.9),(3.10)
only for sets $N$ which, e.g., are members of an open covering of $M$
by convex normal neighbourhoods, and thereby to require the Hadamard form
locally. In the case that $\Lambda$ is the spatio-temporal two-point
function of a state on ${\cal A}[K,\kappa]$ and thus dominates the symplectic
form $\kappa$ ($|\kappa([f],[h])|^2 \leq 4\,\Lambda(f,f)\Lambda(h,h)$),
it was recently proved by Radzikowski that if $\Lambda$ is locally
of Hadamard form, then it is already globally of Hadamard form [56].
However, if $\Lambda$ doesn't dominate $\kappa$, this need not hold
[29,51,56]. Radzikowski's proof makes use of a characterization
of Hadamard forms in terms of their wave front sets which was mentioned
above. A definition of Hadamard form which is less technical in appearence
has recently been given in [44].
We should add that the usual Minkowski-vacuum of the free scalar
field with constant, non-negative potential
term is, of course, an Hadamard vacuum.
This holds, more generally, also for ultrastatic spacetimes, see below.
\\[10pt]
{\it Notes on the proof of Proposition 3.4.}
The property (a) is proved in [45]. The argument for (b) is
essentially contained in [25] and in the generality stated here it is
completed in [45]. An alternative proof using the
``propagation of singularities theorem'' for hyperbolic differential
equations is presented in [55].
Also property (c) is proved in [45 (Appendix B)] (cf.\
[66 (Prop.\ C.2)]). The existence of Hadamard vacua (d)
is proved in [24] (cf.\ also [45]); the stated Corollary has
been observed in [66] (and, in slightly different formulation,
already in [24]). Statement (e) has been shown to hold in
[66 (Prop.\ 3.8)].
\\[10pt]
In order to prepare the formulation of the next result, in which we will
apply our result of Chapter 2, we need to collect some more notation.
Suppose that we are given a quasifree state $\omega_{\mu}$ on
the Weyl-algebra ${\cal A}[K,\kappa]$ of the KG-field over some
globally hyperbolic spacetime $(M,g)$, and that $\Sigma$ is a
Cauchy-surface in that spacetime. Then we denote by $\mu_{\Sigma}$
the dominating scalar product on $({\cal D}_{\Sigma},\delta_{\Sigma})$
which is, using the symplectomorphism between $(K,\kappa)$ and
$({\cal D}_{\Sigma},\delta_{\Sigma})$, induced by the dominating scalar
product $\mu$ on $(K,\kappa)$, i.e.\
\begin{equation}
\mu_{\Sigma}(P_{\Sigma}Ef,P_{\Sigma}Eh) = \mu([f],[h])\,, \quad
[f],[h] \in K\,.
\end{equation}
Conversely, to any $\mu_{\Sigma} \in {\sf q}({\cal D}_{\Sigma},\delta_{\Sigma})$
there corresponds via (3.11) a $\mu \in {\sf q}(K,\kappa)$.
Next, consider a complete Riemannian manifold $(\Sigma,\gamma)$, with
corresponding Laplacian $\Delta_{\gamma}$, and as before, consider the
operator $ -\Delta_{\gamma} +1$ on $C_0^{\infty}(\Sigma,{\bf R})$.
Owing to the completeness
of $(\Sigma,\gamma)$ this operator is,
together with all its powers, essentially selfadjoint in
$L^2_{{\bf R}}(\Sigma,d\eta_{\gamma})$ [10],
and we denote its selfadjoint extension
by $A_{\gamma}$. Then one can introduce the
{Sobolev scalar products} of $m$-th order,
$$ \langle u,v \rangle_{\gamma,m} := \langle u, A_{\gamma}^m v \rangle\,,
\quad u,v \in C_0^{\infty}(\Sigma,{\bf R}),\ m \in {\bf R}\,, $$
where on the right hand side is the scalar product of $L^2_{{\bf R}}(\Sigma,
d\eta_{\gamma})$. The completion of $C_0^{\infty}(\Sigma,{\bf R})$
in the topology of $\langle\,.\,,\,.\,\rangle_{\gamma,m}$
will be denoted by $H_m(\Sigma,\gamma)$.
It turns out that the topology of $H_m(\Sigma,\gamma)$ is locally
independent of the complete Riemannian metric $\gamma$, and that
composition with diffeomorphisms and multiplication with smooth,
compactly supported functions are continuous operations on these
Sobolev spaces. (See Appendix B for precise formulations of these
statements.) Therefore, whenever $G \subset \Sigma$ is open and
relatively compact, the topology which $\langle \,.\,,\,.\, \rangle_{m,\gamma}$
induces on $C_0^{\infty}(G,{\bf R})$ is independent of the particular
complete Riemannian metric $\gamma$, and we shall refer to the
topology which is thus locally induced on $C_0^{\infty}(\Sigma,{\bf R})$
simply as the (local) {\it $H_m$-topology.}
Let us now suppose that we have an ultrastatic spacetime $(\tilde{M},\tilde{\gamma})$,
given in a natural foliation as $({\bf R} \times \tilde{\Sigma},dt^2 \otimes (-\gamma))$
where $(\tilde{\Sigma},\gamma)$ is a complete Riemannian manifold. We shall
identify $\tilde{\Sigma}$ and $\{0\} \times \tilde{\Sigma}$. Consider again
$A_{\gamma}$ = selfadjoint extension of $- \Delta_{\gamma} + 1$ on
$C_0^{\infty}(\tilde{\Sigma},{\bf R})$ in $L^2_{{\bf R}}(\tilde{\Sigma},d\eta_{\gamma})$ with
$\Delta_{\gamma}$ = Laplacian of $(\tilde{\Sigma},\gamma)$, and the scalar product
$\mu^{\circ}_{\tilde{\Sigma}}$ on ${\cal D}_{\tilde{\Sigma}}$ given by
\begin{eqnarray}
\mu^{\circ}_{\tilde{\Sigma}}(u_0 \oplus u_1,v_0 \oplus v_1) & := & \frac{1}{2} \left (
\langle u_0,A_{\gamma}^{1/2}v_0 \rangle
+ \langle u_1,A_{\gamma}^{-1/2}v_1 \rangle \right) \\
& = & \frac{1}{2} \left( \langle u_0,v_0 \rangle_{\gamma,1/2}
+ \langle u_1,v_1 \rangle_{\gamma,-1/2} \right) \nonumber
\end{eqnarray}
for all $u_0 \oplus u_1,v_0 \oplus v_1 \in {\cal D}_{\tilde{\Sigma}}$. It is now straightforward to
check that $\mu^{\circ}_{\tilde{\Sigma}} \in {\sf pu}({\cal D}_{\tilde{\Sigma}},\delta_{\tilde{\Sigma}})$,
in fact, $\mu^{\circ}_{\tilde{\Sigma}}$ is the purification of the classical energy
scalar product $\mu^E_{\tilde{\Sigma}}$ defined in eqn.\ (3.5). (We refer to
[11] for discussion, and also the treatment of more general situations
along similar lines.) What is furthermore central for the derivation of
the next result is that $\mu^{\circ}_{\tilde{\Sigma}}$ corresponds (via (3.11)) to
an Hadamard vacuum $\omega^{\circ}$ on the Weyl-algebra
of the KG-field with potential term $r \equiv 1$ over the ultrastatic
spacetime $({\bf R} \times \tilde{\Sigma},dt^2 \oplus (-\gamma))$. This has been
proved in [24]. The state $\omega^{\circ}$ is called the
{\it ultrastatic vacuum} for the said KG-field over
$({\bf R} \times \tilde{\Sigma} ,dt^2 \oplus (-\gamma))$; it is the unique pure,
quasifree ground state on the corresponding Weyl-algebra for the
time-translations $(t,q) \mapsto (t + t',q)$ on that ultrastatic
spacetime with respect to the chosen natural foliation (cf.\ [40,42]).
\\[6pt]
{\it Remark.} The passage from $\mu^E_{\tilde{\Sigma}}$ to $\mu^{\circ}_{\tilde{\Sigma}}$,
where $\mu^{\circ}_{\tilde{\Sigma}}$ is the purification of the classical
energy scalar product, may be viewed as a refined form of
``frequency-splitting'' procedures (or Hamiltonian diagonalization),
in order to obtain pure dominating scalar products and hence, pure states
of the KG-field in curved spacetimes, see [11]. However, in the case
that $\tilde{\Sigma}$ is not a Cauchy-surface lying in the natural foliation of
an ultrastatic spacetime, but an arbitrary Cauchy-surface in an
arbitrary globally hyperbolic spacetime, the $\mu^{\circ}_{\tilde{\Sigma}}$
may fail to correspond to a quasifree Hadamard state --- even though,
as the following Proposition demonstrates, $\mu^{\circ}_{\tilde{\Sigma}}$ gives
locally on the Cauchy-data space ${\cal D}_{\tilde{\Sigma}}$ the same topology
as the dominating scalar products induced on it by any quasifree
Hadamard state. More seriously, $\mu^{\circ}_{\tilde{\Sigma}}$ may even correspond to
a state which is no longer locally quasiequivalent to any
quasifree Hadamard state. For an explicit example demonstrating
this in a closed Robertson-Walker universe, and for additional
discussion, we refer to Sec.\ 3.6 in [38].
\\[6pt]
We shall say that a map $T : {\cal D}_{\Sigma} \to {\cal D}_{\Sigma'}$,
with $\Sigma,\Sigma'$ Cauchy-surfaces, is {\it locally continuous} if,
for any open, locally compact $G \subset \Sigma$, the restriction of
$T$ to $C_0^{\infty}(G,{\bf R}) \oplus C_0^{\infty}(G,{\bf R})$ is continuous
(with respect to the topologies under consideration).
\begin{Proposition}
Let $\omega_{\mu}$ be a quasifree Hadamard state on the Weyl-algebra
${\cal A}[K,\kappa]$ of the KG-field with smooth potential term $r$ over the
globally hyperbolic spacetime $(M,g)$, and $\Sigma,\Sigma'$ two
Cauchy-surfaces in $(M,g)$.
Then the Cauchy-data evolution map
\begin{equation}
T_{\Sigma',\Sigma} : = P_{\Sigma'} {\mbox{\footnotesize $\circ$}} P_{\Sigma}^{-1} :
{\cal D}_{\Sigma} \to {\cal D}_{\Sigma'}
\end{equation}
is locally continuous in the $H_{\tau} \oplus H_{\tau -1}$-topology,
$0 \leq \tau \leq 1$, on the Cauchy-data spaces, and the topology
induced by $\mu_{\Sigma}$ on ${\cal D}_{\Sigma}$ coincides locally
(i.e.\ on each $C_0^{\infty}(G,{\bf R}) \oplus C_0^{\infty}(G,{\bf R})$
for $G \subset \Sigma$ open and relatively compact) with the
$H_{1/2} \oplus H_{-1/2}$-topology.
\end{Proposition}
{\it Remarks.} (i) Observe that the continuity statement is
reasonably formulated since, as a consequence of the support
properties of solutions of the KG-equation with Cauchy-data
of compact support (``finite propagation speed'') it holds that
for each open, relatively compact $G \subset \Sigma$ there is
an open, relatively compact $G' \subset \Sigma'$ with
$T_{\Sigma',\Sigma}(C_0^{\infty}(G,{\bf R}) \oplus C_0^{\infty}(G,{\bf R}))
\subset C_0^{\infty}(G',{\bf R}) \oplus C_0^{\infty}(G',{\bf R})$.
\\[6pt]
(ii) For $\tau =1$, the continuity statement is just the
classical energy estimate.
It should be mentioned here that the claimed continuity can
also be obtained by other methods. For instance, Moreno [50]
proves, under more restrictive assumptions on $\Sigma$ and $\Sigma'$
(among which is their compactness), the continuity of $T_{\Sigma',\Sigma}$
in the topology of $H_{\tau} \oplus H_{\tau -1}$ for all $\tau \in {\bf R}$,
by employing an abstract energy estimate for first order hyperbolic equations
(under suitable circumstances, the KG-equation can be brought into this form).
We feel, however, that our method, using the results of Chapter 2, is
physically more appealing and emphasizes much better the ``invariant''
structures involved, quite in keeping with the general approach to quantum
field theory.
\\[10pt]
{\it Proof of Proposition 3.5.} We note that there is a diffeomorphism
$\Psi : \Sigma \to \Sigma'$. To see this, observe that we may pick
a foliation $F : {\bf R} \times \tilde{\Sigma} \to M$ of $M$ in Cauchy-surfaces. Then
for each $q \in \tilde{\Sigma}$, the curves $t \mapsto F(t,q)$ are inextendible,
timelike curves in $(M,g)$. Each such curve intersects $\Sigma$ exactly
once, at the parameter value $t = \tau(q)$. Hence $\Sigma$ is the set
$\{F(\tau(q),q) : q \in \tilde{\Sigma}\}$. As $F$ is a diffeomorphism and
$\tau: \tilde{\Sigma} \to {\bf R}$ must be $C^{\infty}$ since, by assumption,
$\Sigma$ is a smooth hypersurface in $M$, one can see that $\Sigma$
and $\tilde{\Sigma}$ are diffeomorphic. The same argument shows that
$\Sigma'$ and $\tilde{\Sigma}$ and therefore, $\Sigma$ and $\Sigma'$, are
diffeomorphic.
Now let us first assume that the $g$-induced Riemannian metrics
$\gamma_{\Sigma}$ and $\gamma_{\Sigma'}$ on $\Sigma$, resp.\
$\Sigma'$, are complete. Let $d\eta$ and $d\eta'$ be the induced
volume measures on $\Sigma$ and $\Sigma'$, respectively. The $\Psi$-transformed
measure of $d\eta$ on $\Sigma'$, $\Psi^*d\eta$, is given through
\begin{equation}
\int_{\Sigma} (u {\mbox{\footnotesize $\circ$}} \Psi) \,d\eta = \int_{\Sigma'} u\,(\Psi^*d\eta)\,,
\quad u \in C_0^{\infty}(\Sigma')\,.
\end{equation}
Then the Radon-Nikodym derivative $(\rho(q))^2 :=(\Psi^*d\eta/d\eta')(q)$,
$q \in \Sigma'$, is a smooth, strictly positive function on $\Sigma'$,
and it is now easy to check that the linear map
$$ \vartheta : ({\cal D}_{\Sigma},\delta_{\Sigma})
\to ({\cal D}_{\Sigma'},\delta_{\Sigma'})\,, \quad
u_0 \oplus u_1 \mapsto \rho \cdot (u_0 {\mbox{\footnotesize $\circ$}} \Psi^{-1}) \oplus \rho \cdot
(u_1 {\mbox{\footnotesize $\circ$}} \Psi^{-1}) \,, $$
is a symplectomorphism. Moreover, by the result given in Appendix
B, $\vartheta$ and its inverse are locally continuous maps in the
$H_s \oplus H_t$-topologies on both Cauchy-data spaces, for all
$s,t \in {\bf R}$.
By the energy estimate, $T_{\Sigma',\Sigma}$ is locally continuous
with respect to the $H_1 \oplus H_0$-topology on the Cauchy-data
spaces, and the same holds for the inverse $(T_{\Sigma',\Sigma})^{-1}
= T_{\Sigma,\Sigma'}$. Hence, the map
$\Theta := \vartheta^{-1} {\mbox{\footnotesize $\circ$}} T_{\Sigma',\Sigma}$ is a symplectomorphism
of $({\cal D}_{\Sigma},\delta_{\Sigma})$, and $\Theta$ together with its inverse
is locally continuous in the $H_1 \oplus H_0$-topology on ${\cal D}_{\Sigma}$.
Here we made use of Remark (i) above. Now pick two sets $G$ and $G'$
as in Remark (i), then there is some open, relatively compact neighbourhood
$\tilde{G}$ of $\Psi^{-1}(G') \cup G$ in $\Sigma$. We can choose a smooth,
real-valued function $\chi$ compactly supported on $\Sigma$ with $\chi \equiv
1$ on $\tilde{G}$. It is then straightforward to check that the maps
$\chi {\mbox{\footnotesize $\circ$}} \Theta {\mbox{\footnotesize $\circ$}} \chi$ and $\chi {\mbox{\footnotesize $\circ$}} \Theta^{-1} {\mbox{\footnotesize $\circ$}} \chi$
($\chi$ to be interpreted as multiplication with $\chi$) is a pair
of symplectically adjoint maps on $({\cal D}_{\Sigma},\delta_{\Sigma})$ which are bounded
with respect to the $H_1 \oplus H_0$-topology, i.e.\ with respect to the
norm of $\mu_{\Sigma}^E$. At this point we use Theorem 2.2(b) and consequently
$\chi {\mbox{\footnotesize $\circ$}} \Theta {\mbox{\footnotesize $\circ$}} \chi$ and $\chi {\mbox{\footnotesize $\circ$}} \Theta^{-1}{\mbox{\footnotesize $\circ$}} \chi$ are
continuous with respect to the norms of the $(\mu^E_{\Sigma})_s$,
$0 \leq s \leq 2$. Inspection shows that
$$ (\mu^E_{\Sigma})_s (u_0 \oplus u_1,v_0 \oplus v_1) =
\frac{1}{2} \left( \langle u_0,A_{\gamma_{\Sigma}}^{1-s/2}v_0 \rangle
+ \langle u_1,A_{\gamma_{\Sigma}}^{-s/2}v_1 \rangle \right)
$$
for $0 \leq s \leq 2$. From this it is now easy to see that
$\Theta$ restricted to $C_0^{\infty}(G,{\bf R}) \oplus C_0^{\infty}(G,{\bf R})$
is continuous in the topology of $H_{\tau} \oplus H_{\tau -1}$,
$0 \leq \tau \leq 1$, since
$\chi {\mbox{\footnotesize $\circ$}} \Theta {\mbox{\footnotesize $\circ$}} \chi(u_0 \oplus u_1) = \Theta(u_0 \oplus u_1)$ for all
$u_0 \oplus u_1 \in C_0^{\infty}(G,{\bf R}) \oplus C_0^{\infty}(G,{\bf R})$
by the choice of $\chi$. Using that $\Theta = \vartheta^{-1}{\mbox{\footnotesize $\circ$}} T_{
\Sigma',\Sigma}$ and that $\vartheta$ is locally continuous with respect to
all the $H_s \oplus H_t$-topologies, $s,t \in {\bf R}$, on the Cauchy-data
spaces, we deduce that that $T_{\Sigma',\Sigma}$ is locally continuous
in the $H_{\tau} \oplus H_{\tau -1}$-topology, $0 \leq \tau \leq 1$,
as claimed.
If the $g$-induced Riemannian metrics $\gamma_{\Sigma}$, $\gamma_{\Sigma'}$
are not complete, one can make them into complete ones
$\hat{\gamma}_{\Sigma} := f \cdot \gamma_{\Sigma}$, $\hat{\gamma}_{\Sigma'}
:= h \cdot \gamma_{\Sigma'}$ by multiplying them with suitable smooth,
strictly positive functions $f$ on $\Sigma$ and $h$ on $\Sigma'$ [14].
Let $d\hat{\eta}$ and $d\hat{\eta}'$ be the volume measures corresponding
to the new metrics. Consider then the density functions
$(\phi_1)^2 := (d\eta/d\hat{\eta})$,
$(\phi_2)^2 := (d\hat{\eta}'/d\eta')$,
which are $C^{\infty}$ and strictly positive, and define
$({\cal D}_{\Sigma},\hat{\delta}_{\Sigma})$, $({\cal D}_{\Sigma'},\hat{\delta}_{\Sigma'})$
and $\hat{\vartheta}$ like their unhatted counterparts but with
$d\hat{\eta}$ and $d\hat{\eta}'$ in place of $d\eta$ and $d\eta'$.
Likewise define $\hat{\mu}^E_{\Sigma}$ with respect to $\hat{\gamma}_{\Sigma}$.
Then $\hat{T}_{\Sigma',\Sigma} := \phi_2 {\mbox{\footnotesize $\circ$}} T_{\Sigma',\Sigma}
{\mbox{\footnotesize $\circ$}} \phi_1$ (understanding that $\phi_1,\phi_2$ act as
multiplication operators) and its inverse are symplectomorphisms
between $({\cal D}_{\Sigma},\hat{\delta}_{\Sigma})$ and
$({\cal D}_{\Sigma'},\hat{\delta}_{\Sigma'})$ which are locally
continuous in the $H_1 \oplus H_0$-topology. Now we can apply the
argument above showing that $\hat{\Theta} = \hat{\vartheta}^{-1} {\mbox{\footnotesize $\circ$}}
\hat{T}_{\Sigma',\Sigma}$ and, hence, $\hat{T}_{\Sigma',\Sigma}$ is
locally continuous in the $H_{\tau} \oplus H_{\tau -1}$-topology for
$0 \leq \tau \leq 1$. The same follows then for
$T_{\Sigma',\Sigma} = \phi_2^{-1} {\mbox{\footnotesize $\circ$}} \hat{T}_{\Sigma',\Sigma}
{\mbox{\footnotesize $\circ$}} \phi_1^{-1}$.
For the proof of the second part of the statement, we note first
that in [24] it is shown that there exists another globally
hyperbolic spacetime $(\hat{M},\hat{g})$ of the form
$\hat{M} = {\bf R} \times \Sigma$ with the following properties:
\\[6pt]
(1) $\Sigma_0 : = \{0\} \times \Sigma$ is a Cauchy-surface in $(\hat{M},
\hat{g})$, and a causal normal neighbourhood $N$ of $\Sigma$ in $M$
coincides with a causal normal neigbourhood $\hat{N}$ of
$\Sigma_{0}$ in $\hat{M}$, in such a way that $\Sigma = \Sigma_0$
and $g = \hat{g}$ on $N$.
\\[6pt]
(2) For some $t_0 < 0$, the $(-\infty,t_0) \times \Sigma$-part of
$\hat{M}$ lies properly to the past of $\hat{N}$, and on that part,
$\hat{g}$ takes the form $dt^2 \oplus (- \gamma)$ where
$\gamma$ is a complete Riemannian metric on $\Sigma$.
\\[6pt]
This means that $(\hat{M},\hat{g})$ is a globally hyperbolic
spacetime which equals $(M,g)$ on a causal normal neighbourhood of $\Sigma$
and becomes ultrastatic to the past of it.
Then consider the Weyl-algebra ${\cal A}[\hat{K},\hat{\kappa}]$ of the
KG-field with potential term $\hat{r}$ over $(\hat{M},\hat{g})$, where
$\hat{r} \in C_0^{\infty}(\hat{M},{\bf R})$ agrees with $r$ on the
neighbourhood $\hat{N} = N$ and is identically equal to $1$ on the
$(-\infty,t_0) \times \Sigma$-part of $\hat{M}$. Now observe that
the propagators $E$ and $\hat{E}$ of the respective KG-equations
on $(M,g)$ and $(\hat{M},\hat{g})$ coincide when restricted to
$C_0^{\infty}(N,{\bf R})$. Therefore one obtains an identification map
$$ [f] = f + {\rm ker}(E) \mapsto [f]\,\hat{{}} = f + {\rm ker}(\hat{E}) \,,
\quad f \in C_0^{\infty}(N,{\bf R}) \,,$$
between $K(N)$ and $\hat{K}(\hat{N})$ which preserves the
symplectic forms $\kappa$ and $\hat{\kappa}$. Without danger we may
write this identification as an equality,
$K(N) = \hat{K}(\hat{N})$.
This identification map between $(K(N),\kappa|K(N))$
and $(\hat{K}(\hat{N}),\hat{\kappa}|\hat{K}(\hat{N}))$ lifts to
a $C^*$-algebraic isomorphism between the corresponding
Weyl-algebras
\begin{eqnarray}
{\cal A}[K(N),\kappa|K(N)]& =& {\cal A}[\hat{K}(\hat{N}),\hat{\kappa}|
\hat{K}(\hat{N})]\,, \nonumber \\
W([f])& =& \hat{W}([f]\,\hat{{}}\,)\,,\ \ \
f \in C_0^{\infty}(N,{\bf R})\,.
\end{eqnarray}
Here we followed our just indicated convention to abbreviate
this identification as an equality. Now we have
$D(N) = M$ in $(M,g)$ and $D(\hat{N}) = \hat{M}$ in
$(\hat{M},\hat{g})$, implying that $K(N) = K$ and
$\hat{K}(\hat{N}) = \hat{K}$. Hence ${\cal A}[K(N),\kappa|K(N)] =
{\cal A}[K,\kappa]$ and the same for the ``hatted'' objects.
Thus (3.15) gives rise to an identification between
${\cal A}[K,\kappa]$ and ${\cal A}[\hat{K},\hat{\kappa}]$, and so the
quasifree Hadamard state $\omega_{\mu}$ induces a quasifree
state $\omega_{\hat{\mu}}$ on ${\cal A}[\hat{K},\hat{\kappa}]$
with
\begin{equation}
\hat{\mu}([f]\,\hat{{}},[h]\,\hat{{}}\,) = \mu([f],[h])\,, \quad
f,h \in C_0^{\infty}(N,{\bf R}) \,.
\end{equation}
This state is also an Hadamard state since we have
\begin{eqnarray*}
\Lambda(f,h)& =& \mu([f],[h]) + \frac{i}{2}\kappa([f],[h]) \\
& = & \hat{\mu}([f]\,\hat{{}}\,,[h]\,\hat{{}}\,) + \frac{i}{2}
\hat{\kappa}([f]\,\hat{{}}\,,[h]\,\hat{{}}\,)\,, \quad f,h \in
C_0^{\infty}(N,{\bf R})\,,
\end{eqnarray*}
and $\Lambda$ is, by assumption, of Hadamard form.
However, due to the causal propagation property of the
Hadamard form this means that $\hat{\mu}$ is the dominating
scalar product on $(\hat{K},\hat{\kappa})$ of a quasifree
Hadamard state on ${\cal A}[\hat{K},\hat{\kappa}]$.
Now choose some $t < t_0$, and let $\Sigma_t = \{t\} \times
\Sigma$ be the Cauchy-surface in the ultrastatic part
of $(\hat{M},\hat{g})$ corresponding to this value of the
time-parameter of the natural foliation. As remarked above,
the scalar product
\begin{equation}
\mu^{\circ}_{\Sigma_t}(u_0 \oplus u_1,v_0 \oplus v_1)
= \frac{1}{2}\left( \langle u_0,v_0 \rangle_{\gamma,1/2}
+ \langle u_1,v_1 \rangle_{\gamma,-1/2} \right)\,,
\quad u_0 \oplus u_1,v_0 \oplus v_1 \in {\cal D}_{\Sigma_t} \,,
\end{equation}
is the dominating scalar product on
$({\cal D}_{\Sigma_t},\delta_{\Sigma_t})$ corresponding to the
ultrastatic vacuum state $\omega^{\circ}$ over the
ultrastatic part of $(\hat{M},\hat{g})$, which is an Hadamard
vacuum. Since the dominating scalar products of all
quasifree Hadamard states yield locally the same topology
(Prop.\ 3.4(e)), it follows that the dominating scalar product
$\hat{\mu}_{\Sigma_t}$ on $({\cal D}_{\Sigma_{t}},\delta_{\Sigma_t})$,
which is induced (cf.\ (3.11)) by the the dominating scalar product
of $\hat{\mu}$ of the quasifree Hadamard state $\omega_{\hat{\mu}}$,
endows ${\cal D}_{\Sigma_t}$ locally with the same topology
as does $\mu^{\circ}_{\Sigma_t}$. As can be read off from (3.17),
this is the local $H_{1/2} \oplus H_{-1/2}$-topology.
To complete the argument, we note that (cf.\ (3.11,3.13))
$$ \hat{\mu}_{\Sigma_0}(u_0 \oplus u_1,v_0 \oplus v_1) =
\hat{\mu}_{\Sigma_t}(T_{\Sigma_t,\Sigma_0}(u_0 \oplus u_1),T_{\Sigma_t,\Sigma_0}
(v_0 \oplus v_1))\,, \quad u_0 \oplus u_1,v_0 \oplus v_1 \in {\cal D}_{\Sigma_0}\,.$$
But since $\hat{\mu}_{\Sigma_t}$ induces locally the
$H_{1/2} \oplus H_{-1/2}$-topology and since the symplectomorphism
$T_{\Sigma_t,\Sigma_0}$ as well as its inverse are locally continuous
on the Cauchy-data spaces in the $H_{1/2}\oplus H_{-1/2}$-topology,
the last equality entails that $\hat{\mu}_{\Sigma_0}$ induces the
local $H_{1/2} \oplus H_{-1/2}$-topology on ${\cal D}_{\Sigma_0}$.
In view of (3.16), the Proposition is now proved. $\Box$
\\[24pt]
{\bf 3.5 Local Definiteness, Local Primarity,
Haag-Duality, etc.}
\\[18pt]
In this section we prove Theorem 3.6 below on the algebraic
structure of the GNS-representations associated with quasifree
Hadamard states on the CCR-algebra of the KG-field on an
arbitrary globally hyperbolic spacetime $(M,g)$. The
results appearing therein extend our previous work [64,65,66].
Let $(M,g)$ be a globally hyperbolic spacetime.
We recall that a subset ${\cal O}$ of $M$
is called a { regular diamond} if it is of the form
${\cal O} = {\cal O}_G = {\rm int}\,D(G)$ where
$G$ is an open, relatively compact subset of some Cauchy-surface
$\Sigma$ in $(M,g)$ having the property that the boundary
$\partial G$ of $G$ is contained in the
union of finitely many smooth, closed, two-dimensional submanifolds
of $\Sigma$. We also recall the notation ${\cal R}_{\omega}({\cal O})
= \pi_{\omega}({\cal A}({\cal O}))^-$ for the local von Neumann algebras in
the GNS-representation of a state $\omega$. The $C^*$-algebraic
net of observable algebras ${\cal O} \to {\cal A}({\cal O})$
will be understood as being that associated
with the KG-field in Prop.\ 3.2.
\begin{Theorem}
Let $(M,g)$ be a globally hyperbolic spacetime and
${\cal A}[K,\kappa]$ the Weyl-algebra of the KG-field with smooth, real-valued
potential function $r$ over $(M,g)$. Suppose that
$\omega$ and $\omega_1$ are two quasifree Hadamard states on
${\cal A}[K,\kappa]$. Then the following statements hold.
\\[6pt]
(a) The GNS-Hilbertspace ${\cal H}_{\omega}$
of $\omega$ is infinite dimensional and separable.
\\[6pt]
(b) The restrictions of the GNS-representations $\pi_{\omega}|{\cal A(O)}$
and $\pi_{\omega_1}|{\cal A(O)}$ of any open, relatively compact
${\cal O} \subset M$ are quasiequivalent. They are even unitarily
equivalent when ${\cal O}^{\perp}$ is non-void.
\\[6pt]
(c) For each $p \in M$ we have local definiteness,
$$ \bigcap_{{\cal O} \owns p} {\cal R}_{\omega}({\cal O}) = {\bf C} \cdot 1\, . $$
More generally, whenever $C \subset M$ is the subset of a compact
set which is contained in the union of finitely many smooth, closed,
two-dimensional submanifolds of an arbitrary Cauchy-surface
$\Sigma$ in $M$,
then
\begin{equation}
\bigcap_{{\cal O} \supset C} {\cal R}_{\omega}({\cal O}) = {\bf C} \cdot 1\,.
\end{equation}
\\[6pt]
(d) Let ${\cal O}$ and ${\cal O}_1$ be two relatively compact diamonds, based
on Cauchy-surfaces $\Sigma$ and $\Sigma_1$, respectively, such
that $\overline{{\cal O}} \subset {\cal O}_1$. Then the split-property
holds for the pair ${\cal R}_{\omega}({\cal O})$ and
${\cal R}_{\omega}({\cal O}_1)$, i.e.\ there exists a type ${\rm I}_{\infty}$
factor $\cal N$ such that one has the inclusion
$$ {\cal R}_{\omega}({\cal O}) \subset {\cal N} \subset {\cal R}_{\omega}
({\cal O}_1) \,. $$
\\[6pt]
(e) Inner and outer regularity
\begin{equation}
{\cal R}_{\omega}({\cal O}) = \left( \bigcup_{\overline{{\cal O}_I} \subset {\cal O}}
{\cal R}_{\omega}({\cal O}_I) \right) '' =
\bigcap_{{\cal O}_1 \supset \overline{{\cal O}}} {\cal R}_{\omega}({\cal O}_1)
\end{equation}
holds for all regular diamonds ${\cal O}$.
\\[6pt]
(f) If $\omega$ is pure (an Hadamard vacuum), then we have Haag-Duality
$$ {\cal R}_{\omega}({\cal O})' = {\cal R}_{\omega}({\cal O}^{\perp}) $$
for all regular diamonds ${\cal O}$. (By the same arguments as in {\rm
[65 (Prop.\ 6)]}, Haag-Duality extends to all pure (but not necessarily
quasifree or Hadamard) states $\omega$ which are locally normal
(hence, by (d), locally quasiequivalent) to any Hadamard vacuum.)
\\[6pt]
(g) Local primarity holds for all regular diamonds, that is, for
each regular diamond ${\cal O}$, ${\cal R}_{\omega}({\cal O})$ is a factor.
Moreover, ${\cal R}_{\omega}({\cal O})$ is isomorphic to the unique
hyperfinite type ${\rm III}_1$ factor if ${\cal O}^{\perp}$
is non-void. In this case, ${\cal R}_{\omega}({\cal O}^{\perp})$ is
also hyperfinite and of type ${\rm III}_1$, and if $\omega$ is
pure, ${\cal R}_{\omega}({\cal O}^{\perp})$ is again a factor.
Otherwise, if ${\cal O}^{\perp} = \emptyset$, then
${\cal R}_{\omega}({\cal O})$ is a type ${\rm I}_{\infty}$ factor.
\end{Theorem}
{\it Proof.} The key point in the proof is that, by results which
for the cases relevant here are to large extend due to Araki [1],
the above statement can be equivalently translated into statements
about the structure of the one-particle space, i.e.\ essentially the
symplectic space $(K,\kappa)$ equipped with the scalar product
$\lambda_{\omega}$. We shall use, however, the formalism of [40,45].
Following that, given a symplectic space $(K,\kappa)$ and
$\mu \in {\sf q}(K,\kappa)$ one calls a real linear map
${\bf k}: K \to H$ a {\it one-particle Hilbertspace structure} for
$\mu$ if (1) $H$ is a complex Hilbertspace, (2) the complex linear
span of ${\bf k}(K)$ is dense in $H$ and (3)
$$ \langle {\bf k}(x),{\bf k}(y) \rangle = \lambda_{\mu}(x,y)
= \mu(x,y) + \frac{i}{2}\kappa(x,y) $$
for all $x,y \in K$. It can then be shown (cf.\ [45 (Appendix A)])
that the GNS-representation of the quasifree state $\omega_{\mu}$
on ${\cal A}[K,\kappa]$ may be realized in the following way:
${\cal H}_{\omega_{\mu}} = F_s(H)$, the Bosonic Fock-space over the one-particle
space $H$, $\Omega_{\omega_{\mu}}$ = the Fock-vacuum, and
$$ \pi_{\omega_{\mu}}(W(x)) = {\rm e}^{i(a({\bf k}(x)) +
a^+({\bf k}(x)))^-}\,, \quad
x \in K\, ,$$
where $a(\,.\,)$ and $a^+(\,.\,)$ are the Bosonic annihilation and
creation operators, respectively.
Now it is useful to define the symplectic complement
$F^{\tt v} := \{\chi \in H : {\sf Im}\,\langle \chi,\phi \rangle = 0
\ \ \forall \phi \in F \}$ for $F \subset H$, since it is known
that
\begin{itemize}
\item[(i)] ${\cal R}_{\omega_{\mu}}({\cal O})$ is a factor \ \ \ iff\ \ \
$ {\bf k}(K({\cal O}))^- \cap {\bf k}(K({\cal O}))^{\tt v} = \{0\}$,
\item[(ii)] ${\cal R}_{\omega_{\mu}}({\cal O})' = {\cal R}_{\omega_{\mu}}
({\cal O}^{\perp})$\ \ \ iff\ \ \
${\bf k}(K({\cal O}))^{\tt v} = {\bf k}(K({\cal O}^{\perp}))^-$,
\item[(iii)] $\bigcap_{{\cal O} \supset C} {\cal R}_{\omega_{\mu}}({\cal O})
= {\bf C} \cdot 1$ \ \ \ iff\ \ \
$\bigcap_{{\cal O} \supset C}{\bf k}(K({\cal O}))^- = \{0\}\,,$
\end{itemize}
cf.\ [1,21,35,49,58].
After these preparations we can commence with the proof of the various
statements of our Theorem.
\\[6pt]
(a) Let ${\bf k}: K \to H$ be the one-particle Hilbertspace structure
of $\omega$. The local one-particle spaces ${\bf k}(K({\cal O}_G))^-$ of
regular diamonds ${\cal O}_G$ based on $G \subset \Sigma$ are topologically
isomorphic to the completions of $C_0^{\infty}(G,{\bf R}) \oplus
C_0^{\infty}(G,{\bf R})$ in the $H_{1/2} \oplus H_{-1/2}$-topology and
these are separable. Hence ${\bf k}(K)^-$, which is generated by a
countable set ${\bf k}(K({\cal O}_{G_n}))$, for $G_n$ a sequence of
locally compact subsets of $\Sigma$ eventually exhausting $\Sigma$,
is also separable. The same holds then for the one-particle
Hilbertspace $H$ in which the complex span of ${\bf k}(K)$ is
dense, and thus separability is implied for ${\cal H}_{\omega} = F_s(H)$.
The infinite-dimensionality is clear.
\\[6pt]
(b) The local quasiequivalence has been proved in [66] and we refer to
that reference for further details. We just indicate that the
proof makes use of the fact that the difference $\Lambda - \Lambda_1$
of the spatio-temporal two-point functions of any pair of
quasifree Hadamard states is on each causal normal neighbourhood
of any Cauchy-surface given by a smooth integral kernel ---
as can be directly read off from the Hadamard form --- and this turns
out to be sufficient for local quasiequivalence. The statement
about the unitary equivalence can be inferred from (g) below,
since it is known that every $*$-preserving isomorphism between
von Neumann algebras of type III acting on separable Hilbertspaces
is given by the adjoint action of a unitary operator which maps
the Hilbertspaces onto each other. See e.g.\ Thm.\ 7.2.9 and
Prop.\ 9.1.6 in [39].
\\[6pt]
(c) Here one uses that there exist Hadamard vacua, i.e.\ pure
quasifree Hadamard states $\omega_{\mu}$. Since by Prop.\ 3.4
the topology of $\mu_{\Sigma}$ in ${\cal D}_{\Sigma}$ is locally that of
$H_{1/2} \oplus H_{-1/2}$, one can show as in [66 (Chp.\ 4 and
Appendix)] that under the stated hypotheses about $C$ it holds
that $\bigcap_{{\cal O} \supset C} {\bf k}(K({\cal O}))^- = \{0\}$ for the
one-particle Hilbertspace structures of Hadamard vacua. From
the local equivalence of the topologies induced by the dominating
scalar products of all quasifree Hadamard states (Prop.\ 3.4(e)),
this extends to the one-particle structures of all quasifree
Hadamard states. By (iii), this yields the statement (c).
\\[6pt]
(d) This is proved in [65] under the additional assumption that
the potential term $r$ is a positive constant. (The result was
formulated in [65] under the hypothesis that $\Sigma = \Sigma_1$,
but it is clear that the present statement without this hypothesis
is an immediate generalization.) To obtain the general case
one needs in the spacetime deformation argument of [65]
the modification that the potential term $\hat{r}$ of the KG-field
on the new spacetime $(\hat{M},\hat{g})$ is equal to a positive
constant on its ultrastatic part while being equal to $r$ in a
neighbourhood of $\Sigma$. We have used that procedure already in
the proof of Prop.\ 3.5, see also the proof of (f) below where
precisely the said modification will be carried out in more detail.
\\[6pt]
(e) Inner regularity follows simply from the definition of the
${\cal A}({\cal O})$; one deduces that for each $A \in {\cal A}({\cal O})$ and each
$\epsilon > 0$ there exists some $\overline{{\cal O}_I} \subset {\cal O}$
and $A_{\epsilon} \in {\cal A}({\cal O}_I)$ so that
$||\,A - A_{\epsilon}\,|| < \epsilon$. It is easy to see that
inner regularity is a consequence of this property.
So we focus now on the outer regularity.
Let ${\cal O} = {\cal O}_G$ be based on the subset $G$ of the Cauchy-surface
$\Sigma$. Consider the symplectic space $({\cal D}_{\Sigma},\delta_{\Sigma})$
and the dominating scalar product $\mu_{\Sigma}$ induced by $\mu
\in {\sf q}({\cal D}_{\Sigma},\delta_{\Sigma})$, where $\omega_{\mu} = \omega$;
the corresponding one-particle Hilbertspace structure we denote by
${\bf k}_{\Sigma}: {\cal D}_{\Sigma} \to H_{\Sigma}$. Then we denote by
${\cal W}({\bf k}_{\Sigma}({\cal D}_G))$ the von Neumann algebra in
$B(F_s(H_{\Sigma}))$ generated by the unitary groups of the
operators $(a({\bf k}_{\Sigma}(u_0 \oplus u_1)) + a^+({\bf k}_{\Sigma}(u_0 \oplus u_1)))^-$
where $u_0 \oplus u_1$ ranges over ${\cal D}_G := C_0^{\infty}(G,{\bf R}) \oplus
C_0^{\infty}(G,{\bf R})$. So ${\cal W}({\bf k}_{\Sigma}({\cal D}_G)) =
{\cal R}_{\omega}({\cal O}_G)$. It holds
generally that $\bigcap_{G_1 \supset \overline{G}} {\cal W}({\bf k}_{\Sigma}
({\cal D}_{G_1})) = {\cal W}(\bigcap_{G_1 \supset \overline{G}}
{\bf k}_{\Sigma}({\cal D}_{G_1})^-)$ [1], hence, to establish outer
regularity, we must show that
\begin{equation}
\bigcap_{G_1 \supset \overline{G}} {\bf k}_{\Sigma}({\cal D}_{G_1})^-
= {\bf k}_{\Sigma}({\cal D}_G)^-\,.
\end{equation}
In [65] we have proved that the ultrastatic vacuum $\omega^{\circ}$
of the KG-field with potential term $\equiv 1$ over the ultrastatic
spacetime $(M^{\circ},g^{\circ}) = ({\bf R} \times \Sigma,dt^2 \oplus
(-\gamma))$ (where $\gamma$ is any complete Riemannian metric on
$\Sigma$) satisfies Haag-duality. That means, we have
\begin{equation}
{\cal R}^{\circ}_{\omega^{\circ}}({\cal O}_{\circ})' =
{\cal R}^{\circ}_{\omega^{\circ}}({\cal O}_{\circ}^{\perp})
\end{equation}
for any regular diamond ${\cal O}_{\circ}$ in $(M^{\circ},g^{\circ})$
which is based on any of the Cauchy-surfaces $\{t\}\times \Sigma$ in
the natural foliation, and we have put a ``$\circ$'' on the local
von Neumann algebras to indicate that they refer to a KG-field
over $(M^{\circ},g^{\circ})$. But since we have inner regularity
for ${\cal R}^{\circ}_{\omega^{\circ}}({\cal O}_{\circ}^{\perp})$ ---
by the very definition --- the outer regularity of ${\cal R}^{\circ}
_{\omega^{\circ}}({\cal O}_{\circ})$ follows from the Haag-duality (3.21).
Translated into conditions on the one-particle Hilbertspace
structure ${\bf k}^{\circ}_{\Sigma} : {\cal D}_{\Sigma} \to H^{\circ}_{\Sigma}$
of $\omega^{\circ}$, this means that the equality
\begin{equation}
\bigcap_{G_1 \supset \overline{G}} {\bf k}^{\circ}_{\Sigma}
({\cal D}_{G_1})^- = {\bf k}^{\circ}_{\Sigma}({\cal D}_G)^-
\end{equation}
holds. Now we know from Prop.\ 3.5 that $\mu_{\Sigma}$ induces
locally the $H_{1/2} \oplus H_{-1/2}$-topology on ${\cal D}_{\Sigma}$. However,
this coincides with the topology locally induced by $\mu^{\circ}_{\Sigma}$
on ${\cal D}_{\Sigma}$ (cf.\ (3.11)) --- even though $\mu^{\circ}_{\Sigma}$ may,
in general, not be viewed as corresponding to an Hadamard vacuum
of the KG-field over $(M,g)$. Thus the required relation (3.20)
is implied by (3.22).
\\[6pt]
(f) In view of outer regularity it is enough to show that, given
any ${\cal O}_1 \supset \overline{{\cal O}}$, it holds that
\begin{equation}
{\cal R}_{\omega}({\cal O}^{\perp})' \subset {\cal R}_{\omega}({\cal O}_1)\,.
\end{equation}
The demonstration of this property relies on a spacetime deformation
argument similar to that used in the proof of Prop.\ 3.5. Let
$G$ be the base of ${\cal O}$ on the Cauchy-surface $\Sigma$ in $(M,g)$.
Then, given any other open, relatively compact subset $G_1$ of
$\Sigma$ with $\overline{G} \subset G_1$, we have shown in
[65] that there exists an ultrastatic spacetime $(\hat{M},\hat{g})$
with the properties (1) and (2) in the proof of Prop.\ 3.5, and with
the additional property that there is some $t < t_0$ such that
$$ \left( {\rm int}\,\hat{J}(G) \cap \Sigma_t \right )^- \subset
{\rm int}\, \hat{D}(G_1) \cap \Sigma_t\,.$$
Here, $\Sigma_t = \{t\} \times \Sigma$ are the Cauchy-surfaces in the
natural foliation of the ultrastatic part of $(\hat{M},\hat{g})$.
The hats indicate that the causal set and the domain of dependence
are to be taken in $(\hat{M},\hat{g})$. This implies that we can find
some regular diamond ${\cal O}^t := {\rm int}\hat{D}(S^t)$ in
$(\hat{M},\hat{g})$ based on a subset $S^t$ of $\Sigma_t$ which
satisfies
\begin{equation}
\left( {\rm int}\, \hat{J}(G) \cap \Sigma_t \right)^-
\subset S^t \subset
{\rm int}\,\hat{D}(G_1) \cap \Sigma_t \,.
\end{equation}
Setting $\hat{{\cal O}} := {\rm int}\, \hat{D}(G)$ and
$\hat{{\cal O}}_1 := {\rm int}\,\hat{D}(G_1)$, one derives from (3.24)
the relations
\begin{equation}
\hat{{\cal O}} \subset {\cal O}^t \subset \hat{{\cal O}}_1
\,.
\end{equation}
These are equivalent to
\begin{equation}
\hat{{\cal O}}_1^{\perp} \subset ({\cal O}^t)^{\perp} \subset \hat{{\cal O}}^{\perp}
\end{equation}
where $\perp$ is the causal complementation in $(\hat{M},\hat{g})$.
Now as in the proof of Prop.\ 3.5, the given Hadamard vacuum $\omega$
on the Weyl-algebra ${\cal A}[K,\kappa]$ of the KG-field over $(M,g)$
induces an Hadamard vacuum $\hat{\omega}$ on the Weyl-algebra
${\cal A}[\hat{K},\hat{\kappa}]$ of the KG-field over $(\hat{M},\hat{g})$
whose potential term $\hat{r}$ is $1$ on the ultrastatic
part of $(\hat{M},\hat{g})$. Then by Prop.\ 6 in [65] we have
Haag-duality
\begin{equation}
\hat{\cal R}_{\hat{\omega}}(\hat{{\cal O}_t}^{\perp}) ' =
\hat{\cal R}_{\hat{\omega}}(\hat{{\cal O}_t})
\end{equation}
for all regular diamonds $\hat{{\cal O}_t}$ with base on
$\Sigma_t$; we have put hats on the von Neumann algebras
to indicate that they refer to ${\cal A}[\hat{K},\hat{\kappa}]$.
(This was proved in [65] assuming that $(\hat{M},\hat{g})$
is globally ultrastatic. However, with the same argument, based on
primitive causality, as we use it next to pass from (3.28) to
(3.30), one can easily establish that (3.27) holds if only
$\Sigma_t$ is, as here, a member in the natural foliation of the
ultrastatic part of $(\hat{M},\hat{g})$.)
Since ${\cal O}^t$ is a regular diamond based on $\Sigma_t$, we obtain
$$\hat{\cal R}_{\hat{\omega}}(({\cal O}^t)^{\perp})' =
\hat{\cal R}_{\hat{\omega}}({\cal O}^t) $$
and thus, in view of (3.25) and (3.26),
\begin{equation}
\hat{\cal R}_{\hat{\omega}}(\hat{{\cal O}}^{\perp})'
\subset \hat{\cal R}_{\hat{\omega}}(({\cal O}^t)^{\perp})'
= \hat{\cal R}_{\hat{\omega}}({\cal O}^t) \subset
\hat{\cal R}_{\hat{\omega}}(\hat{{\cal O}}_1)\,.
\end{equation}
Now recall (see proof of Prop.\ 3.5) that $(\hat{M},\hat{g})$
coincides with $(M,g)$ on a causal normal neighbourhood $N$ of
$\Sigma$. Primitive causality (Prop.\ 3.2) then entails
\begin{equation}
\hat{\cal R}_{\hat{\omega}}(\hat{{\cal O}}^{\perp} \cap N)'
\subset \hat{\cal R}_{\hat{\omega}}(\hat{{\cal O}}_1 \cap N) \,.
\end{equation}
On the other hand, $\hat{{\cal O}}^{\perp} = {\rm int} \hat{D}(\Sigma
\backslash G)$ and $\hat{{\cal O}}_1$ are diamonds in $(\hat{M},\hat{g})$
based on $\Sigma$. Since $(M,g)$ and $(\hat{M},\hat{g})$
coincide on the causal normal neighbourhood $N$ of $\Sigma$,
one obtains that ${\rm int}\,D(\tilde{G}) \cap N =
{\rm int}\, \hat{D}(\tilde{G}) \cap N$ for all $\tilde{G} \in
\Sigma$. Hence, with ${\cal O} = {\rm int}\,D(G)$,
${\cal O}_1 = {\rm int}\, D(G_1)$ (in $(M,g)$), we have that (3.23) entails
$$ {\cal R}_{\omega}({\cal O}^{\perp} \cap N)' \subset
{\cal R}_{\omega}({\cal O}_1 \cap N)
$$
(cf.\ the proof of Prop.\ 3.5) where the causal complement $\perp$
is now taken in $(M,g)$. Using primitive causality once more,
we deduce that
\begin{equation}
{\cal R}_{\omega}({\cal O}^{\perp})' \subset {\cal R}_{\omega}({\cal O}_1)\,.
\end{equation}
The open, relatively compact subset $G_1$ of $\Sigma$ was
arbitrary up to the constraint $\overline{G} \subset G_1$.
Therefore, we arrive at the conclusion that the required inclusion
(3.23) holds of all ${\cal O}_1 \supset \overline{{\cal O}}$.
\\[6pt]
(g) Let $\Sigma$ be the Cauchy-surface on which ${\cal O}$ is based.
For the local primarity one uses, as in (c), the existence of
Hadamard vacua $\omega_{\mu}$ and the fact (Prop.\ 3.5) that
$\mu_{\Sigma}$ induces locally the $H_{1/2} \oplus H_{-1/2}$-topology;
then one may use the arguments of [66 (Chp.\ 4 and Appendix)]
to show that due to the regularity of the boundary
$\partial G$ of the base $G$ of ${\cal O}$
there holds
$$ {\bf k}(K({\cal O}))^- \cap {\bf k}(K({\cal O}))^{\tt v} = \{ 0 \}$$
for the one-particle Hilbertspace structures of Hadamard vacua.
As in the proof of (c), this can be carried over to the
one-particle structures of all quasifree Hadamard states since
they induce locally on the one-particle spaces the same topology,
see [66 (Chp.\ 4)]. We note that for Hadamard vacua the local
primarity can also be established using (3.18) together with Haag-duality
and primitive causality purely at the algebraic level, without
having to appeal to the one-particle structures.
The type ${\rm III}_1$-property of ${\cal R}_{\omega}({\cal O})$ is then
derived using Thm.\ 16.2.18 in [3] (see also [73]).
We note that for some points $p$ in the boundary $\partial G$ of $G$, ${\cal O}$
admits domains which are what is in Sect.\ 16.2.4 of [3] called
``$\beta_p$-causal sets'', as a consequence of the regularity of
$\partial G$ and the assumption ${\cal O}^{\perp} \neq \emptyset$.
We further note that it is straightforward to prove that
the quasifree Hadamard states of the KG-field over $(M,g)$
possess at each point in $M$ scaling limits (in the sense of
Sect.\ 16.2.4 in [3], see also [22,32]) which are equal to the
theory of the massless KG-field in Minkowski-spacetime. Together
with (a) and (c) of the present Theorem this shows that the
the assumptions of Thm.\ 16.2.18 in [3] are fulfilled, and the
${\cal R}_{\omega}({\cal O})$ are type ${\rm III}_1$-factors for all
regular diamonds ${\cal O}$ with ${\cal O}^{\perp} \neq \emptyset$.
The hyperfiniteness follows from the split-property (d) and the
regularity (e), cf.\ Prop.\ 17.2.1 in [3]. The same arguments
may be applied to ${\cal R}_{\omega}({\cal O}^{\perp})$, yielding
its type ${\rm III}_1$-property (meaning that in its central
decomposition only type ${\rm III}_1$-factors occur) and
hyperfiniteness. If $\omega$ is an Hadamard vacuum, then
${\cal R}_{\omega}({\cal O}^{\perp}) = {\cal R}_{\omega}({\cal O})'$ is
a factor unitarily equivalent to ${\cal R}_{\omega}({\cal O})$.
For the last statement note that ${\cal O}^{\perp} = \emptyset$ implies
that the spacetime has a compact Cauchy-surface on which ${\cal O}$
is based. In this case ${\cal R}_{\omega}({\cal O}) =
\pi_{\omega}({\cal A}[K,\kappa])''$ (use the regularity of $\partial G$,
and (c), (e) and primitive causality). But since $\omega$ is
quasiequivalent to any Hadamard vacuum by the relative compactness of
${\cal O}$, ${\cal R}_{\omega}({\cal O}) = \pi_{\omega}({\cal A}[K,\kappa])''$
is a type ${\rm I}_{\infty}$-factor. $\Box$
\\[10pt]
We end this section and therefore, this work, with a few concluding
remarks.
First we note that the split-property signifies a strong notion of
statistical independence. It can be deduced from constraints on the
phase-space behaviour (``nuclearity'') of the considered quantum
field theory. We refer to [9,31] for further information and
also to [62] for a review, as a discussion of these issues lies
beyond the scope of of this article. The same applies to a discussion
of the property of the local von Neumann algebras ${\cal R}_{\omega}({\cal O})$
to be hyperfinite and of type ${\rm III}_1$. We only mention that
for quantum field theories on Minkowski spacetime it can be established
under very general (model-independent)
conditions that the local (von Neumann) observable
algebras are hyperfinite and of type ${\rm III}_1$, and refer the reader to
[7] and references cited therein. However, the property of the
local von Neumann algebras to be of type ${\rm III}_1$, together with
the separability of the GNS-Hilbertspace ${\cal H}_{\omega}$, has an
important consequence which we would like to point out (we have
used it implicitly already in the proof of Thm.\ 3.6(b)):
${\cal H}_{\omega}$ contains a dense subset ${\sf ts}({\cal H}_{\omega})$ of vectors
which are cyclic and separating for all ${\cal R}_{\omega}({\cal O})$
whenever ${\cal O}$ is a diamond with ${\cal O}^{\perp} \neq \emptyset$. But
so far it has only been established in special cases that $\Omega_{\omega}
\in {\sf ts}({\cal H}_{\omega})$, see [64]. At any rate, when
$\Omega \in {\sf ts}({\cal H}_{\omega})$ one may consider for a pair of
regular diamonds ${\cal O}_1,{\cal O}_2$ with $\overline{{\cal O}_1} \subset {\cal O}_2$
and ${\cal O}_2^{\perp}$ nonvoid the modular operator $\Delta_2$
of ${\cal R}_{\omega}({\cal O}_2)$,$\Omega$ (cf.\ [39]). The split property
and the factoriality of ${\cal R}_{\omega}({\cal O}_1)$ and ${\cal R}_{\omega}
({\cal O}_2)$ imply the that the map
\begin{equation}
\Xi_{1,2} : A \mapsto \Delta^{1/4}_2 A \Omega\,, \quad A \in
{\cal R}_{\omega}({\cal O}_1)\,,
\end{equation}
is compact [8]. As explained in [8],
``modular compactness'' or ``modular nuclearity'' may be viewed
as suitable generalizations of ``energy compactness'' or
``energy nuclearity'' to curved spacetimes as notions to measure
the phase-space behaviour of a quantum field theory
(see also [65]). Thus an interesting
question would be if the maps (3.31) are even nuclear.
Summarizing it can be said that Thm.\ 3.6 shows that the nets of von
Neumann observable algebras of the KG-field over a globally hyperbolic
spacetime in the representations of quasifree Hadamard states have
all the properties one would expect for physically reasonable
representations. This supports the point of view that quasifree
Hadamard states appear to be a good choice for physical states
of the KG-field over a globally hyperbolic spacetime. Similar results
are expected to hold also for other linear fields.
Finally, the reader will have noticed that we have been considering
exclusively the quantum theory of a KG-field on a {\it globally hyperbolic}
spacetime. For recent developments concerning quantum fields in the
background of non-globally hyperbolic spacetimes, we
refer to [44] and references cited there.
\\[24pt]
{\bf Acknowledgements.} I would like to thank D.\ Buchholz for
valueable comments on a very early draft of Chapter 2. Moreover,
I would like to thank C.\ D'Antoni, R.\ Longo, J.\ Roberts
and L.\ Zsido for
their hospitiality, and their interest in quantum field theory
in curved spacetimes. I also appreciated conversations with R.\ Conti,
D.\ Guido and L.\ Tuset on various parts of the material of the
present work.
\\[28pt]
\noindent
{\Large {\bf Appendix}}
\\[24pt]
{\bf Appendix A}
\\[18pt]
For the sake of completeness, we include here the interpolation argument
in the form we use it in the proof of Theorem 2.2 and in Appendix B
below. It is a standard argument based on Hadamard's three-line-theorem,
cf.\ Chapter IX in [57].
\\[10pt]
{\bf Lemma A.1}
{\it
Let ${\cal F},{\cal H}$ be complex Hilbertspaces, $X$ and $Y$ two non-negative,
injective, selfadjoint operators in ${\cal F}$ and ${\cal H}$, respectively,
and $Q$ a bounded linear operator ${\cal H} \to {\cal F}$
such that $Q{\rm Ran}(Y) \subset {\rm dom}(X)$.
Suppose that the operator $XQY$ admits a
bounded extension $T :{\cal H} \to {\cal F}$. Then for all $0 \leq \tau \leq 1$,
it holds that $Q{\rm Ran}(Y^{\tau}) \subset {\rm dom}(X^{\tau})$, and
the operators $X^{\tau}QY^{\tau}$ are bounded by $||\,T\,||^{\tau}
||\,Q\,||^{1 - \tau}$. }
\\[10pt]
{\it Proof.} The operators $\ln(X)$ and $\ln(Y)$ are (densely defined)
selfadjoint operators. Let the vectors $x$ and $y$ belong to the
spectral subspaces of $\ln(X)$ and $\ln(Y)$, respectively, corresponding
to an arbitrary finite intervall. Then the functions
${\bf C} \owns z \mapsto {\rm e}^{z\ln(X)}x$ and
${\bf C} \owns z \mapsto {\rm e}^{z\ln(Y)}y$ are holomorphic. Moreover,
${\rm e}^{\tau \ln(X)}x = X^{\tau}x$ and ${\rm e}^{\tau \ln(Y)}y =
Y^{\tau}y$ for all real $\tau$. Consider the function
$$ F(z) := \langle {\rm e}^{\overline{z}\ln(X)}x,Q{\rm e}^{z\ln(Y)}y
\rangle_{{\cal F}} \,.$$
It is easy to see that this function is holomorphic on ${\bf C}$, and
also that the function is uniformly bounded for $z$ in the
strip $\{z : 0 \leq {\sf Re}\,z \leq 1 \}$.
For $z = 1 + it$, $t \in {\bf R}$, one has
$$ |F(z)| = |\langle {\rm e}^{-it\ln(X)}x,XQY{\rm e}^{it\ln(Y)}y \rangle_{{\cal F}} |
\leq ||\,T\,||\,||\,x\,||_{{\cal F}}||\,y\,||_{{\cal H}} \,,$$
and for $z = it$, $t \in {\bf R}$,
$$ |F(z)| = |\langle {\rm e}^{-it\ln(X)}x,Q{\rm e}^{it\ln(Y)}y \rangle_{{\cal F}} |
\leq ||\,Q\,||\,||\,x\,||_{{\cal F}}||\,y\,||_{{\cal H}} \,.$$
By Hadamard's three-line-theorem, it follows that for all $z = \tau + it$
in the said strip there holds the bound
$$ |F(\tau + it)| \leq ||\,T\,||^{\tau}||\,Q\,||^{1 - \tau}||\,x\,||_{{\cal F}}
||\,y\,||_{{\cal H}}\,.$$
As $x$ and $y$ were arbitrary members of the finite spectral intervall
subspaces, the last estimate extends to all $x$ and $y$ lying in cores
for the operators
$X^{\tau}$ and $Y^{\tau}$, from which the
the claimed statement follows. $\Box$
\\[24pt]
{\bf Appendix B}
\\[18pt]
For the convenience of the reader we collect here two well-known
results about Sobolev norms on manifolds which are used in
the proof of Proposition 3.5. The notation is as follows.
$\Sigma$ and $\Sigma'$ will denote smooth, finite dimensional manifolds
(connected, paracompact, Hausdorff); $\gamma$ and $\gamma'$ are
complete Riemannian metrics on $\Sigma$ and $\Sigma'$, respectively.
Their induced volume measures are denoted by $d\eta$ and $d\eta'$.
We abbreviate by $A_{\gamma}$ the selfadjoint extension
in $L^2(\Sigma,d\eta)$
of the operator $-\Delta_{\gamma} +1$ on $C_0^{\infty}(\Sigma)$,
where $\Delta_{\gamma}$ is the Laplace-Beltrami operator on $(\Sigma,\gamma)$;
note that [10] contains a proof that $(-\Delta_{\gamma} + 1)^k$
is essentially selfadjoint on $C_0^{\infty}(\Sigma)$ for all $k \in {\bf N}$.
$A'$ will be defined similarly with respect to the corresponding
objects of $(\Sigma',\gamma')$. As in the main text, the
$m$-th Sobolev scalar product is $\langle u,v \rangle_{\gamma,m}
= \langle u,A_{\gamma}^{m}v \rangle$ for $u,v \in C_0^{\infty}(\Sigma)$ and
$m \in {\bf R}$, where $\langle\,.\,,\,.\, \rangle$ is the scalar product
of $L^2(\Sigma,d\eta)$. Anagolously we define $\langle\,.\,,\,.\,\rangle_{
\gamma',m}$. For the corresponding norms we write $||\,.\,||_{\gamma,m}$,
resp., $||\,.\,||_{\gamma',m}$.
\\[10pt]
{\bf Lemma B.1}
{\it (a) Let $\chi \in C_0^{\infty}(\Sigma)$. Then there is for each
$m \in {\bf R}$ a constant $c_m$ so that
$$ ||\,\chi u \,||_{\gamma,m} \leq c_m ||\, u \,||_{\gamma,m}\,,
\quad u \in C_0^{\infty}(\Sigma) \,.$$
\\[6pt]
(b) Let $\phi \in C^{\infty}(\Sigma)$ be strictly positive and
$G \subset \Sigma$ open and relatively compact. Then there are
for each $m \in {\bf R}$ two positive constants $\beta_1,\beta_2$ so that
$$ \beta_1||\,\phi u\,||_{\gamma,m} \leq ||\,u\,||_{\gamma,m}
\leq \beta_2||\,\phi u\,||_{\gamma,m}\,, \quad u \in C_0^{\infty}(G)\,.$$ }
{\it Proof.} (a) We may suppose that $\chi$ is real-valued (otherwise
we treat real and imaginary parts separately). A tedious but straightforward
calculation shows that the claimed estimate is fulfilled for all $m =2k$,
$k \in {\bf N}_0$. Hence $A^k \chi A^{-k}$ extends to a bounded operator
on $L^2(\Sigma,d\eta)$, and the same is true of the adjoint
$A^{-k}\chi A^k$.
Thus by the interpolation argument, cf.\ Lemma A.1,
$A^{\tau k} \chi A^{-\tau k}$ is bounded for all $-1 \leq \tau \leq 1$.
This yields the stated estimate.
\\[6pt]
(b) This is a simple corollary of (a). For the first estimate, note
that we may replace $\phi$ by a smooth function with compact support.
Then note that the second estimate is equivalent to
$||\,\phi^{-1}v\,||_{\gamma,m} \leq \beta_2||\,v\,||_{\gamma,m}$,
$v \in C_0^{\infty}(G)$, and again we use that instead of $\phi^{-1}$
we may take a smooth function of compact support. $\Box$
\\[10pt]
{\bf Lemma B.2}
{\it
Let $(\Sigma,\gamma)$ and $(\Sigma',\gamma')$ be two complete
Riemannian manifolds, $N$ and $N'$ two open subsets of $\Sigma$ and
$\Sigma'$, respectively, and $\Psi : N \to N'$ a diffeomorphism.
Given $m \in {\bf R}$ and some open, relatively compact subset $G$ of
$\Sigma$ with $\overline{G} \subset N$, there are two positive
constants $b_1,b_2$ such that
$$ b_1||\,u\,||_{\gamma,m} \leq ||\,\Psi^*u\,||_{\gamma',m}
\leq b_2||\,u\,||_{\gamma,m} \,, \quad u \in C_0^{\infty}(G)\,,$$
where $\Psi^*u := u {\mbox{\footnotesize $\circ$}} \Psi^{-1}$. }
\\[10pt]
{\it Proof.} Again it is elementary to check that such a result
is true for $m = 2k$ with $k \in {\bf N}_0$. One infers that, choosing
$\chi \in C_0^{\infty}(N)$ with $\chi|G \equiv 1$ and setting
$\chi' := \Psi^*\chi$, there is for each $k \in {\bf N}_0$ a
positive constant $b$ fulfilling
$$ ||\,A^k\chi\Psi_*\chi'v\,||_{\gamma,0} \leq
b\,||\,(A')^kv\,||_{\gamma',0}\,, \quad v \in C_0^{\infty}(\Sigma')\,;$$
here $\Psi_*v := v {\mbox{\footnotesize $\circ$}} \Psi$. Therefore,
$$ A^k{\mbox{\footnotesize $\circ$}} \chi{\mbox{\footnotesize $\circ$}} \Psi_*{\mbox{\footnotesize $\circ$}} \chi'{\mbox{\footnotesize $\circ$}} (A')^{-k} $$
extends to a bounded operator $L^2(\Sigma',d\eta') \to L^2(\Sigma,d\eta)$
for each $k \in {\bf N}_0$. Interchanging the roles of $A$ and $A'$, one
obtains that also
$$ (A')^k{\mbox{\footnotesize $\circ$}} \chi'{\mbox{\footnotesize $\circ$}} \Psi^*{\mbox{\footnotesize $\circ$}} \chi{\mbox{\footnotesize $\circ$}} A^{-k} $$
extends, for each $k \in {\bf N}_0$, to a bounded operator
$L^2(\Sigma,d\eta) \to L^2(\Sigma',d\eta')$. The boundedness transfers
to the adjoints of these two operators. Observe then that for
$(\Psi_*)^{\dagger}$, the adjoint of $\Psi_*$, we have
$(\Psi_*)^{\dagger} = \rho^2{\mbox{\footnotesize $\circ$}} (\Psi^*)$ on $C_0^{\infty}(N)$, and
similarly, for the adjoint $(\Psi^*)^{\dagger}$ of $\Psi^*$ we have
$(\Psi^*)^{\dagger} = \Psi_* {\mbox{\footnotesize $\circ$}} \rho^{-2}$ on $C_0^{\infty}(N')$,
where $\rho^2 = \Psi^*d\eta/d\eta'$ is a smooth density function
on $N'$, cf.\ eqn.\ (3.14).
It can now easily be worked out that the interpolation argument
of Lemma A.1 yields again the claimed result.
\begin{flushright}
$\Box$
\end{flushright}
{\small
|
proofpile-arXiv_065-607
|
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
}
|
\section{INTRODUCTION}
\label{section1}
Since the discovery of high-temperature superconductivity in copper-oxide
based materials, experimental studies have revealed a lot of evidence that
interaction between electrons and lattice vibrations
plays an important role in these compounds.\cite{Lattice} The changes
in position and width of the phonon peaks
below superconducting transition temperature measured by Raman
spectroscopy\cite{Thomsen} and by neutron scattering techniques\cite{Mook},
the existence of the isotope effect which varies
as a function of both doping and rare earth ion substitution\cite{Franck},
clearly demonstrate
that coupling between charge carriers and phonon modes in high-$T_c$ superconductors
is not negligible.
Nevertheless, theoretical studies of the effects of electron-phonon
interaction on the dynamics of the charge carriers in electron-correlated
systems are
far from complete. The main problem for proper treatment of
electron-phonon interaction in high-$T_c$ superconductors
is that charge carrier motion in an antiferromagnetic background (or spin-fluctuating
background at higher doping concentration)
is strongly affected by electron-magnon interactions itself.
\cite{Auerbach,Schmitt-Rink:1988,Kane,Martinez:1991,Igarashi,Liu:1992}
Including the electron-phonon interaction deals with quasiparticles renormalized
by interaction with magnons.\cite{Ramsak:1992}
As a consequence the renormalized bandwidth becomes comparable
to phonon frequencies and the `classical' Migdal-Eliashberg approach
to the electron-phonon
problem\cite{Migdal:1958}
in metallic systems seems beyond the region of application.
In the present paper, we add a Fr\"{o}hlich term to the $t-J$ model
Hamiltonian to study
the quasiparticle properties in the presence of hole-phonon
interaction.
This is particularly interesting, since it is widely believed that
the $t-J$ model incorporates essential features of
electron systems with strong local repulsion between electrons,
characteristic of high T$_{c}$ copper oxides.\cite{Anderson}
We examine the properties of this model
at low doping concentrations (near half-filling)
where analytical and numerical results for the $t-J$ model are
well-known.
The optical phonon mode in the
copper-oxide plane is assumed to have a substantial influence
on the dynamics of
doped holes in cuprate compounds.
In the past, quasiparticle properties
in the presence of electron-phonon interaction
were studied by Engelsberg and Schrieffer \cite{Engelsberg:1963}
for a weakly correlated model of conventional metal.
The authors
examined the spectral density function $A(\vec{k},\omega)$ of an
electron
for phonon spectra of the Einstein and Debye forms.
They found that spectral density function
exhibits several branches of
excitations rather than a single branch of a dressed electron.
Our calculations of the quasiparticle energy dispersion
show the same
for the present model of a strongly correlated
system with hole-phonon interaction
\cite{Kyung:19961}.
Recently the mass renormalization
due to a coupling to optical phonons was studied for strongly
correlated electrons by Ram\u{s}ak {\em et al.}
\cite{Ramsak:1992}.
The authors found that the phonon-induced mass
renormalization of a single hole that propagates in the $t-J$ model on
the
scale $2J$ is much larger than that in the corresponding uncorrelated model
for $J \agt t$.
The mass enhancement, increasing with $J/t$ ratio, is due to the slow
motion of a spin
polaron, which makes hole-phonon interaction more effective.
The hole excursions
from the center of the spin polaron are restricted to a few lattice
sites and the bandwidth is on scale $t^{2}/J$, as long as $J/t$ is large.
On the other hand, when $J\ll t$
the confining
antiferromagnetic potential becomes weak and a hole performs large radius
incoherent excursions on scale $t$. Therefore, the mass enhancement
induced by the hole-phonon
interaction becomes weaker and the coherent part of the spectral weight
tends to zero.
In our paper, we study
various quasiparticle properties as well as
the mass renormalization due to hole-phonon
interaction in the limit of a finite but small concentration of doped holes.
The experimentally
measured optical phonon frequencies \cite{Falck:1993}
are close to 50 $meV$ and typical values of $J$ are
of the order of 100 $meV$.
We assume the optical phonon frequency, $\Omega$,
to be $\Omega=0.5J$.
The $t-J$ model Hamiltonian appended with Fr\"ohlich term
will be denoted as the $t-J-g$ model.
Our paper is organized as follows.
In Sec.~\ref{section2}, the quasiparticle residue and the mass
renormalization constant are calculated for $t-J-g$ model.
The incoherent part of the spectrum and Luttinger's theorem are studied in
Sec.~\ref{section3}.
Vertex corrections due to the hole-phonon interaction are analyzed
in Sec.~\ref{section4}.
Section \ref{section5} presents the optical conductivity in the
$t-J-g$ model.
A summary is given in Sec.~VI.
\section{FORMULATION AND QUASI-PARTICLE SPECTRAL WEIGHT}
\label{section2}
To analyze the interaction between optical phonons
and charge carriers in the copper oxide planes, we consider a two-dimensional
$t-J$ model Hamiltonian appended by a Fr\"{o}hlich term \cite{Mahan:1986}
with the dispersionless (optical)
phonon mode $\Omega$
and the coupling constant $g$. Using linear spin-wave approximation through
Holstein-Primakoff transformation\cite{Primakoff} and spinless-fermion/
Schwinger-boson representation\cite{Auerbach} of the electron operators
on a site $i$ with spin $\sigma$
$c_{i\sigma}$: $c_{i\downarrow}^{\dagger} \rightarrow h_i S_i^-$ and
$c_{i\uparrow} \rightarrow h_i^{\dagger}$ the Hamiltonian can be written:
\begin{eqnarray}
H_{tJg} & = & -t\sum_{\langle i,j\rangle}\left[h_{i}h^{+}_{j}S^{-}_{j}+
S^{-}_{i}h_{i}h^{+}_{j}+h.c.\right]
+J\sum_{\langle i,j\rangle}\rho^{e}_{i}\vec{S}_{i}\cdot
\vec{S}_{j}\rho^{e}_{j}
\nonumber \\
& - & \mu\sum_{i}\rho^{h}_{i}+
g\sum_{i}h^{+}_{i}h_{i}(b_{i} + b^{+}_{i})+
\Omega\sum_{i}b^{+}_{i}b_{i} \; ,
\label{eq:226}
\end{eqnarray}
where $h_i$ is a hole creation operator on site $i$ and $\vec{S}_{i}$
is the on-site electron spin operator.
($S^{-}_{i}$ turns down the spin of the electron on site $i$.)
The summation is over the nearest neighbor
sites $\langle i,j\rangle$ of
the square
lattice, $b^{+}_{i}$ is an optical phonon creation operator at site $i$.
$\rho^{h}_{i}\equiv h_{i}^{+}h_{i}$ is the local density
operator of the spinless
hole, while $\rho^{e}_{i}=1-\rho^{h}_{i}$ is the local
density operator of electron
under the single occupancy constraint.
Further, $\mu$ is
the chemical
potential related to the concentration of the doped holes:
$\langle\rho^{h}_{i}\rangle = x$ (at half-filling $x=0$).
Since $t>J$ would correspond to a situation in cuprates,
the usual perturbation
approximation does not work in the strongly coupled $t-J$ model.
However, it has been for some time
known that the self-consistent noncrossing approximation for
the hole self-energy gives a fairly accurate result because of the
small vertex corrections. Namely, it was explained in Ref.~\cite{Sush},
that the leading correction to the hole-spin wave vertex
vanishes as a consequence of
the electron (hole) spin conservation in the $t-J$ model.
Mathematically, this fact
is reflected in the special symmetry of the hole-spin wave interaction
vertices
$M_{\vec{k},\vec{q}}$ and $N_{\vec{k},\vec{q}}$, which leads to the identical
vanishing of non-maximally crossing hole-spin wave vertex corrections
\cite{Liu:1992}.
However, it is uncertain whether or not the vertex corrections due to
the hole-phonon interaction are negligible in a strongly correlated
electron system. Thus, in the first place we assume that they are so, and
later in Sec.~\ref{section4} this assumption will be
supported by calculating the
lowest order vertex corrections.
To study the influence of
hole-phonon coupling on the quasiparticle properties we treat the
Hamiltonian within the noncrossing approximation
for spin wave and phonon interactions
\cite{Ramsak:1992}.
Fig.~\ref{fig110} shows
four possible noncrossing diagrams contributing to the self-energy:
spin wave emitting and absorbing diagrams, and
phonon emitting and absorbing ones.
The thick solid line is
the dressed hole Green's function, while
the unperturbed hole propagator is denoted by the thin solid line.
The dashed and wavy lines
stand for the
spin wave and phonon propagators, respectively.
Here $M$, $N$ are the hole-spin wave interaction vertices, while $g$ is
the hole-phonon vertex.
After summing over the intermediate frequency,
the hole self-energy at zero temperature becomes
\begin{eqnarray}
\Sigma^{R}(\vec{k},\omega) = \sum_{\vec{q}} \{ M_{\vec{k},
\vec{q}}^{2}\int_{0}^{\infty}dy\frac{A(\vec{k}-\vec{q},y)}
{\omega-y-E_{\vec{q}}+i\delta}
+ N_{\vec{k},\vec{q}}^{2}\int_{-\infty}^{0}dy\frac{A(
\vec{k}-\vec{q},y)}{\omega-y+E_{\vec{q}}+i\delta}
\nonumber \\
+\ \ g^{2}\int_{0}^{\infty}dy\frac{A(\vec{k}-\vec{q},y)}{
\omega-y-\Omega+i\delta}
+ g^{2}\int_{-\infty}^{0}dy\frac{A(\vec{k}-\vec{q},y)}{
\omega-y+\Omega+i\delta} \} \; , \label{eq:236}
\end{eqnarray}
where the superscript $R$ indicates retarded functions which are
analytic in the upper
half-plane of $\omega$, $E_{\vec{q}}$ is the spin wave energy, and
$A(\vec{k},\omega)$ is the
spectral density function of the interacting hole propagator given as
\begin{equation}
A(\vec{k},\omega) \; = \; -\frac{1}{\pi}ImG^{R}(\vec{k},\omega) \; .
\label{eq:234}
\end{equation}
The hole Green's function is then obtained from the self-energy
\begin{equation}
G^{R}(\vec{k},\omega) \; = \; \frac{1}{\omega-\Sigma^{R}(\vec{k},
\omega)+\mu+i\delta} \; .
\label{eq:235}
\end{equation}
The self-consistent integral equation, Eq.~\ref{eq:236}, is solved
numerically (See Ref.~\cite{Kyung:19961} for
detail).
Table~\ref{table2} shows the quasiparticle residues at three different
$\vec{k}$ points and the energy shift due to both the hole-spin wave and
hole-phonon interactions.
$a_{(\pi/2,\pi/2)}^{calc}$ in the fifth column is the quasiparticle residue
at the point $(\pi/2,\pi/2)$ calculated
by perturbation theory.
The numerically computed mass renormalization factor due to
the hole-phonon interaction is $\lambda_{num}$.
As expected, the quasiparticle residue decreases as the coupling constant
increases. At the $(0,0)$ point the hole loses its quasiparticle property more
rapidly, as
the vanishingly small spectral weight indicates. This is because
the state at the $(0,0)$ point is located at a high hole energy region
and hence it
is more vulnerable to additional decay from the hole-phonon
interaction.
The energy shift
due to the hole-phonon interaction is generally small ($-0.975$ for
$g=2.0J$), compared with that
from the pure hole-spin wave interaction $(-5.378)$.
This suggests that a perturbative
treatment for the hole-phonon interaction is possible.
According to Eq.~\ref{eq:236}, the hole self-energy is
composed of two terms, one from the hole-spin wave interaction and
the other from the hole-phonon interaction. Thus the self-energy can be written
as $\Sigma(\vec{k},\omega)=\Sigma_{m}(\vec{k},\omega)
+\Sigma_{g}(\vec{k},\omega)$, where the former comes from the
hole-spin wave interaction and the latter from the hole-phonon interaction.
If we define $X$ and $Y$ as follows
\begin{eqnarray*}
X & = & 1 - \frac{\partial}{\partial \omega}
Re\Sigma_{m}(\vec{k},\omega){\mid}_{g=0} \; ,
\\
Y & = & - \frac{\partial}{\partial \omega}
Re\left(\Sigma_{g}(\vec{k},\omega)+
\Sigma_{m}(\vec{k},\omega)-
\Sigma_{m}(\vec{k},\omega){\mid}_{g=0}\right) \; ,
\end{eqnarray*}
the quasiparticle residue can be approximated by
\begin{eqnarray}
a_{\vec{k}} & = & \frac{1}{X+Y}
= \frac{1}{X}[1-(\frac{Y}{X})+(\frac{Y}{X})^{2}+\cdots]
\nonumber \\
& \approx & \frac{1}{X}[1-(\frac{Y}{X})+(\frac{Y}{X})^{2}] \; .
\label{eq:522}
\end{eqnarray}
The $X$, $Y$ are determined by
\begin{eqnarray*}
X & = & \frac{1}{a_{\vec{k}}(g=0)} \; ,
\\
Y & = & \frac{1}{a_{\vec{k}}} - \frac{1}{a_{\vec{k}}(g=0)} \; .
\end{eqnarray*}
Substituting $X$ and $Y$ into Eq.~\ref{eq:522} leads to the
$a_{(\pi/2,\pi/2)}^{calc}$ in the fifth column of Table~\ref{table2}.
For coupling constants less than $1.5J$, agreement with the numerical
results is quite satisfactory. However, the perturbative
treatment based on
the two term expansion breaks down for $g>1.5J$, since $(Y/X)$
increases significantly for the strong hole-phonon interaction.
In fact, $Y$ is the approximate hole-phonon mass renormalization constant
$\lambda$ evaluated numerically on the basis of the $t-J$ model Hamiltonian.
This is listed as $\lambda_{num}$ in the sixth column.
The mass renormalization constant for a small coupling constant
can also be computed from perturbation theory.
We used a similar method employed by
Ram\u{s}ak {\em et al.}
\cite{Ramsak:1992}.
Since the hole-phonon coupling constant is small compared with the
hole-spin wave interaction strength, the effective mass of the hole can
be computed using the lowest-order perturbation correction to the
hole self-energy,
\begin{equation}
\omega_{\vec{k}}(g)-\omega_{\vec{k}}=\frac{1}{(2\pi)^{2}}\int
d^{2}q\frac{a_{\vec{k}-\vec{q}}g^{2}}
{\omega_{\vec{k}}-\omega_{\vec{k}-\vec{q}}
-\Omega} \; ,
\label{eq:523}
\end{equation}
where $\omega_{\vec{k}}(g)$ and $\omega_{\vec{k}}$ are the quasiparticle
dispersion functions in the presence of the hole-phonon interaction and
in its absence, respectively. Approximately, $\omega_{\vec{k}}$
is given by
\[ \omega_{\vec{k}} = \frac{(\vec{k}-(\pm\pi/2,\pm\pi/2))^{2}}
{2m_{eff}} \; , \]
where $m_{eff}$ is the averaged effective mass of the hole near the bottom of
the energy dispersion function, which is taken as $3.36/t$ according to
Martinez {\em et al.} \cite{Martinez:1991}. From the second derivative
around the points
$\vec{k}=(\pm\pi/2,\pm\pi/2)$, we arrive at
\begin{eqnarray}
\frac{1}{m_{eff}(g)}-\frac{1}{m_{eff}} & \approx &
-\frac{16\bar{a}g^{2}m_{eff}}{\pi}\int_{0}^{\pi}dq
\frac{q^{3}}{(q^{2}+2m_{eff}\Omega)^{3}}
\nonumber \\
& = & -\frac{1}{m_{eff}}
\frac{2\bar{a}g^{2}m_{eff}}{\pi\Omega}
[4\int_{0}^{y_{m}}dy
\frac{y^{3}}{(y^{2}+1)^{3}}] \ ,
\label{eq:524}
\end{eqnarray}
where $y_{m}$ is given by $\pi/\sqrt{2m_{eff}\Omega}$ and
$\bar{a}$ is the averaged quasiparticle residue near the point
$(\pi/2,\pi/2)$.
Hence,
\begin{eqnarray}
\lambda_{eff} & = & \frac{m_{eff}(g)}{m_{eff}} - 1
\approx \frac{1}{1-1.5\bar{a}g^{2}m_{eff}/(\pi\Omega)} - 1
\nonumber \\
& \approx & 1.5\bar{a}g^{2}m_{eff}/(\pi\Omega) \; ,
\label{eq:5241}
\end{eqnarray}
where a small $\lambda_{eff}$ is assumed.
For $g/J = 0.5$ and 1.0, the substitution of $\bar{a} \approx 0.34$
gives 0.127 and 0.822, respectively.
The former value is favorably compared with 0.102, the numerically
calculated mass renormalization constant for $g/J = 0.5$.
Clearly $g/J = 1.0$ is too strong to apply perturbation theory,
because of the too large effective mass, $m_{eff}=3.36/t$.
Besides which, the large value of
$\sqrt{2m_{eff}\Omega}$, namely, 1.16 makes the calculation
very sensitive to the
upper bound, $y_{m}$, for the integration, leading to an additional difficulty.
A straightforward calculation of $\lambda_{eff}$ for noninteracting
electrons yields 0.016, 0.068, 0.167 and 0.342 for $g/J=0.5$,
1.0, 1.5 and 2.0, respectively. In this case, the above mentioned
difficulties do not occur because of the small effective mass,
$m_{eff}=1/2t$.
The calculation shows that the mass renormalization factor is
more enhanced
for the strongly correlated electrons
than the factor for the noninteracting
electron system. This enhancement factor which is 6.4 for $g/J$ = 0.5
is substantially large, compared
with 3.5 which
Ram\u{s}ak {\em et al.} \cite{Ramsak:1992} reported.
The discrepancy between these two values originates from the
use of different definitions for the mass enhancement parameter.
The authors of this paper obtained the enhancement factor
by explicitly computing the change in the curvature of the quasiparticle
dispersion
along the line $(\pi/2,\pi/2)-(0,0)$, namely, $\lambda_{\parallel}=
m_{\parallel}(g)/m_{\parallel}-1$, while the effective
mass in the present calculation,
$\lambda_{eff}=\sqrt{
m_{\parallel}(g)m_{\perp}(g)/m_{\parallel}m_{\perp}}-1$,
is averaged out
around the point $(\pi/2,\pi/2)$.
This indicates even larger mass enhancement along the line
$(0,\pi)-(0,0)$, which is consistent with the observation
that the hole moves slowest along this direction.
\section{INCOHERENT SPECTRUM AND LUTTINGER'S
THEOREM}
\label{section3}
Because of the additional scattering channel, we expect that
there are more spectra associated with the optical phonon excitations
above the quasiparticle pole in the spectral density function.
Fig.~\ref{fig111} shows the spectral density function for four different
hole-phonon coupling constants $g$.
As $g$ increases, the spin wave peaks become more suppressed, whereas the
phonon peaks become stronger.
Since the coupling strength for the hole-spin wave interaction is much
larger than that for the hole-phonon interaction, the weak phonon
features appear on top of the spin wave peaks as a satellite structure.
For $g \geq 1.5J$, the phonon induced peaks
are even sharper and larger in height than the
spin wave peaks. Especially the appearance of the multiple
phonon peaks below the chemical potential for $g=2.0J$, compared
with the spectral density function for $g=0$, is clearly noticeable.
The spikes due to finite size effects are gradually
reduced, since those artificial peaks get smeared out due to the
additional decay induced by the hole-phonon interaction.
Fig.~\ref{fig112} presents the hole density of states for various
coupling constants.
The general behavior for the hole density of states is similar to that for the
spectral density function, since in most of the Brillouin zone the
spectral density function is very similar to that at the point ($\pi/2,\pi/2$).
As the hole-phonon coupling constant increases, the position of the
strongest phonon peak right above the Fermi energy ($\omega = 0$)
is shifted upward, although this is not well pronounced due to finite
size effects.
This is understood based on a (lattice) polaronic formation.
For the strong hole-phonon coupling constant, the
hole is surrounded by an increasing number of phonons.
Hence, approximately 2 phonons are involved in the formation of the polaron
for $g=2.0J$, since the position of the peak is close to $1.0J$ and
the optical phonon frequency is $0.5J$.
The figure for $g=0$ shows, however, 2.5 spin waves with energy $2J$
(i.e. from the top of the spin wave band where the density of spin wave
states is sharply peaked) participate in a magnetic polaron.
The incoherent spectrum below the Fermi energy in the density of states
is crucial in satisfying Luttinger's theorem
\cite{Kyung:19961}, as far as the quasiparticle (coherent) contribution to the
density of the occupied states (at $T=0$) is substantially suppressed due to
the strong hole-spin wave (and phonon) coupling.
The momentum distribution function is shown in Fig.~\ref{fig113}.
We chose a much larger cluster ($240 \times 240$) for $n(\vec k)$ calculation,
than the original cluster ($24 \times 24$) used for
the self-consistent calculation
of the self-energy in the $\vec{k}$ points along
the line $(\pi/2,\pi/2)-(0,\pi)$ or
$S-Y$. This provides detailed information about the
behavior of $n(\vec{k})$ in the
vicinity of the Fermi surface.
The four very elongated ellipses in the inset denote the Fermi surface.
The distribution function shows a sharp drop
{\em at the same $\vec{k}$ point} for all the coupling constants we have
studied, as seen in the figure.
We also compared the doping concentration $x$ computed from
the spectral density function, with
the ratio of the number of $\vec{k}$ states inside the Fermi surface
to the number in the entire (antiferromagnetic) Brillouin zone.
The former yields $x=0.030$, 0.031, 0.032, 0.030, 0.030 for
$g=2.0J$, $1.5J$, $1.0J$, $0.5J$, $0$, respectively, while
the latter shows 0.032. They agree with each other within less than
$4 \%$ on the average.
These two features numerically verify Luttinger's theorem
in the $t-J-g$ model for
a small doping concentration.
As the coupling constant increases, the distribution function inside the
Fermi surface decreases gradually from 0.36 to 0.23.
This is due to the reduced quasiparticle residue for the strong
hole-phonon interaction, as seen in Table~\ref{table2}.
At the same time, however,
some density of the occupied states also appears outside the Fermi surface.
This is because in the $t-J-g$
model the spectral density function $A(\vec{k},\omega)$ at the $\vec{k}$
points
outside the Fermi surface possesses
a strong incoherent tail below the chemical potential.
\section{VERTEX CORRECTIONS}
\label{section4}
Since quasiparticles move coherently on a reduced energy scale $2J$,
the Fermi energy is quite small for a small doping case, i.e.\,
almost on the order of the phonon frequency or even less, which signals a
possible breakdown of the standard strong (phonon) coupling
theory. In the present section, the Migdal-type vertex corrections
are studied in the $t-J-g$ model. Below
$k$ is defined as $(\vec{k},ik_{n})$ where $k_{n}$ is a
Matsubara frequency.
Hence a summation over $k$ means the summation over both
momenta $\vec{k}$ and Matsubara frequencies $k_{n}$.
The lowest order vertex corrections to the hole-phonon interaction
in Fig.~\ref{fig114} can be written as $g\Gamma(k,k+q)$, where
\begin{equation}
\Gamma(k,k+q) = -\frac{1}{\beta}\sum_{k'}G(k')G(k'+q)B(k-k') \; .
\label{eq:541}
\end{equation}
$B(k-k')$ is the Green's function of the optical phonon.
Using the spectral representation for the hole Green's function
and converting the summation over Matsubara frequencies
into a contour integration
leads to
\begin{eqnarray}
\Gamma(k,k+q) & = & g^{2}\sum_{\vec{k}'}\int\int d\omega d\omega'
A(\vec{k}',\omega)A(\vec{k}'+\vec{q},\omega')
\frac{1}{\omega-\omega'} \hspace{3.0cm}
\nonumber \\
& \times & \{ \frac{1}{-ik_{n}+\omega+\Omega}
[F(\omega)-N(\Omega)-1]
- \frac{1}{-ik_{n}+\omega-\Omega}
[F(\omega)+N(\Omega)] \hspace{1.0cm}
\nonumber \\
& - & \frac{1}{-ik_{n}+\omega'+\Omega}
[F(\omega')-N(\Omega)-1]
+ \frac{1}{-ik_{n}+\omega'-\Omega}
[F(\omega')+N(\Omega)] \} \; ,
\label{eq:543}
\end{eqnarray}
where $F(\omega)$ and $N(\omega)$ are the Fermi and Bose distribution functions
respectively, and the $iq_{n} \rightarrow 0$ limit is taken first for
numerical simplicity. Since the numerical computation for
$q \ne 0$ shows a similar result to $q=0$ case, we restrict
ourselves to the latter case.
By taking the $T=0$ and $\vec{q} \rightarrow 0$ limits and
the analytic continuation $ik_{n} \rightarrow k_{0}+i\delta$ as well as
by noting that
$\Theta(x)=1-\Theta(-x)$ and that (an integral is in the principal value
sense)
\[ \int d\omega'\frac{A(\vec{k},\omega')}{\omega
-\omega'} = ReG(\vec{k},\omega) \; , \]
we arrive at
\begin{eqnarray}
\Gamma(k_{0}) & = & 2g^{2}\sum_{\vec{k}}\int d\omega
A(\vec{k},\omega)ReG(\vec{k},\omega)
\nonumber \\
& \times & \{ \frac{\Theta(\omega)}
{-k_{0}+\omega+\Omega-i\delta}
+ \frac{\Theta(-\omega)}
{-k_{0}+\omega-\Omega-i\delta} \}
\nonumber \\
& = & 2g^{2}\sum_{\vec{k}}\int d\omega
A(\vec{k},\omega)ReG(\vec{k},\omega)
\nonumber \\
& \times & \frac{1}
{-k_{0}+\omega+\Omega sign(\omega)-i\delta} \ .
\label{eq:545}
\end{eqnarray}
Therefore, the real and imaginary parts of the lowest order
vertex correction are found as
\begin{eqnarray}
Re\Gamma(k_{0}) & = & 2g^{2}\sum_{\vec{k}}\int_{-\infty}^{\infty} d\omega
A(\vec{k},\omega)ReG(\vec{k},\omega)
\nonumber \\
& \times & \frac{1}
{-k_{0}+\omega+\Omega sign(\omega) } \ ,
\label{eq:546}
\end{eqnarray}
and
\begin{eqnarray}
Im\Gamma(k_{0}) = \left\{ \begin{array}{lll}
2\pi g^{2}\sum_{\vec{k}}
A(\vec{k},k_{0}+\Omega)ReG(\vec{k},k_{0}+\Omega)
& \mbox{if $k_{0} < -\Omega$}
\nonumber \\
0
& \mbox{if $-\Omega < k_{0} < \Omega$}
\nonumber \\
2\pi g^{2}\sum_{\vec{k}}
A(\vec{k},k_{0}-\Omega)ReG(\vec{k},k_{0}-\Omega)
& \mbox{if $k_{0} > \Omega$} \; .
\end{array}
\right.
\label{eq:547}
\end{eqnarray}
Fig.~\ref{fig115} and Fig.~\ref{fig116} show the real and
imaginary parts of the lowest order vertex correction for several
hole-phonon coupling constants, respectively.
As the coupling strength
increases, the real and imaginary parts of the vertex correction grow.
In spite of the expectation
that the Migdal approximation may break down due to the small hole band
width $2J$ determined from the $t-J$ model \cite{Ramsak:1992},
the vertex correction is
much smaller than unity for up to $g \sim 2J$.
For a noninteracting electron system, first order vertex correction
has been known to be of the order of $\omega_{D}/E_{F}$
\cite{Migdal:1958}, where $\omega_{D}$ is the Debye frequency and
$E_{F}$ is the Fermi energy.
But, the adiabatic argument valid for a weakly interacting system
breaks down for
strongly correlated electrons, since the Fermi velocity is comparable
to or even less than the phonon phase velocity in a considerable part of
the Brillouin zone.
According to Eq.~\ref{eq:546}, first order vertex correction is
roughly proportional to the square of the quasiparticle residue.
This indicates a possibility that the significant
renormalization $(0.2-0.3)$ of the quasiparticle residue for a strongly
correlated electron system makes the vertex correction much reduced,
thereby accounting for the small vertex correction from the hole-phonon
interaction in the $t-J-g$ model.
Hence, the present calculation numerically corroborates
using the noncrossing approximation for the self-energy both in the
hole-magnon and hole-phonon interactions.
\section{OPTICAL CONDUCTIVITY IN THE $\lowercase{t} - J -
\lowercase{g}$ MODEL}
\label{section5}
The current operator after Bogoliubov transformations of the spin
wave operators becomes \cite{Kyung:19961}
\begin{equation}
J_{x}(\vec{q}) = \frac{2et}{N}\sum_{\vec{k},\vec{p}}
h_{\vec{k}}h^{+}_{\vec{p}} [
C_{\vec{k},\vec{p}}\alpha^{+}_{\vec{k}-\vec{p}-\vec{q}}
+ D_{\vec{k},\vec{p}}\alpha_{-\vec{k}+\vec{p}+\vec{q}} ] \; ,
\label{eq:616}
\end{equation}
where the bare current vertices $C_{\vec{k},\vec{p}},D_{\vec{k},\vec{p}}$
are defined as
\begin{eqnarray}
C_{\vec{k},\vec{p}} & = & u_{\vec{k}-\vec{p}}\sin p_{x}
+v_{\vec{k}-\vec{p}}\sin k_{x} \; ,
\nonumber \\
D_{\vec{k},\vec{p}} & = & v_{\vec{k}-\vec{p}}\sin p_{x}
+u_{\vec{k}-\vec{p}}\sin k_{x} \; .
\label{eq:617}
\end{eqnarray}
Due to the special nature of the interaction vertices, it was
established that the lowest order contribution to the optical
conductivity dominates at $\omega \neq 0$ \cite{Kyung:19961}.
Hence, we consider only the lowest order
diagrams in the present study.
The lowest order contribution comes from two diagrams in
Fig.~\ref{fig117} owing to the structure of the current
operator Eq.~\ref{eq:616}.
In the combined limits of zero temperature $(T \rightarrow 0)$
and long wavelength
electromagnetic radiation $(\vec{q} \rightarrow 0)$,
the optical conductivity becomes
\begin{eqnarray}
\sigma_{1}(q_{0}) = \frac{\pi}{q_{0}}(\frac{2t}{N})^{2}
\int_{0}^{q_{0}} d\omega
\sum_{\vec{k}}A(\vec{k},\omega-q_{0})
\nonumber \\
\times \sum_{\vec{p}}
C_{\vec{k},\vec{p}}^{2}
A(\vec{p},\omega-E_{\vec{k}-\vec{p}}) \; .
\label{eq:628}
\end{eqnarray}
Fig.~\ref{fig118} presents the optical
conductivity for the $t-J-g$ model, as the hole-phonon coupling
constant varies.
As expected from the corresponding spectral density function, the
contribution to the conductivity from the multi-magnon excitations
(``string structure'') decreases,
while strong
absorption appears right above the $2J$ peak. This new peak
in the absorption comes from
the hole-phonon interaction, as can be seen in the corresponding spectral
density function. Generally, the peak at
$2J$ peak and higher energy
incoherent spin wave peaks
are suppressed and broadened in the presence of the
strong hole-phonon interaction. This may be associated with the
growth of {\em featureless} spectral weight at the mid-infrared region, when
the CuO$_{2}$ plane is doped with charge carriers \cite{Uchida}.
\section{CONCLUSION}
\label{section6}
The influence of the hole-phonon interaction on various physical
quantities was studied within the noncrossing
approximation for the spin wave and optical phonon interactions on the
same footing.
As the hole-phonon coupling
constant $g$ increases,
the quasiparticle residue is further reduced
and spin wave peaks in the spectral density function and optical conductivity
are more suppressed.
Phonon peaks in the spectral density function $A(\vec k,\omega)$, instead,
grow more pronounced around the quasiparticle pole at a
low energy on the scale of $\Omega$ .
A sharp drop in the hole momentum distribution function is found for
all the hole-phonon coupling constants we have studied.
The invariance of the volume enclosed by the Fermi surface for all
the chosen hole-phonon coupling constants
numerically verifies Luttinger's theorem for doped holes in
the $t-J-g$ model.
Our numerical estimate of the lowest order vertex corrections to the
hole-phonon coupling vertex $g$ due to the
hole-phonon interaction, gives relatively small values $\ll 1$.
This means that electron-phonon vertex
corrections are not
important and Migdal's approximation can be used
in calculating the hole self-energy. The smallness of the effective
hole bandwidth of order $2J$, is compensated by suppressed quasiparticle
residues.
Due to the presence of additional phonon induced absorption,
$2J$ and higher hole-multi-spin wave peaks in optical conductivity
are suppressed and broadened.
\acknowledgments
This work has been supported by the Center for
Superconductivity Research
of the University of Maryland at College Park and by NASA Grant
NAG3-1395. The authors are grateful to P. B. Allen and V. J. Emery for
discussions.
S. I. Mukhin is grateful to colleagues at the Department of Physics
of the University
of Maryland for their warm hospitality during his stay at College Park.
|
proofpile-arXiv_065-608
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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|
\section{Introduction}
Evolution of the physical conditions of the universe, galaxies and stars
can be described in terms of the increase or decrease of hundreds of
elemental abundances of atomic nuclides. They originate from the primordial
nucleosynthesis about fifteen billion years ago and the subsequent
production/destruction cycle in stars and ejection into the intergalactic
space. It is therefore inevitable and even fundamental to study the nuclear
processes in several astrophysical sites for a deep understanding of the
evolution of the universe.
Cosmologically, the primordial nucleosynthesis provides a unique method
to determine the average universal mass--density parameter $\Omega_B$.
Although the {\em homogeneous big--bang model\/} for primordial nucleosynthesis
predicts $\Omega_B\,h_{50}^{2} \sim 0.04$,
X--ray observations of dense clusters have indicated that $\Omega_B$
could be as large as $\leq 0.15$.
Recent MACHO detections also suggest that there exist more baryons
in our Galaxy than ever expected. There is clearly a serious potential
conflict between these observations and the theoretical prediction
in the homogeneous big--bang model. The situation is even crucial
if high deuterium abundances, which were detected in Lyman--$\alpha$
absorption systems along the line of sight to high red--shifted quasars,
are presumed to be primordial.
On the other hand, an {\em inhomogeneous big--bang model\/}, which allows inhomogeneous
baryon density distribution, can predict $\Omega_B\,h_{50}^{2} \sim 0.1 - 0.2$.
Among the possible observable signatures of baryon
inhomogeneous cosmologies are the high abundances of heavier elements
than lithium such as beryllium and boron\,\cite{kajino90}.
In an environment of baryon inhomogeneous distribution, neutrons can easily
diffuse out of the fluctuations to form high density proton--rich
and low density neutron--rich regions, where a lot of proton/neutron--rich
radioactive isotopes can help produce the intermediate--to--heavy mass elements.
Another astrophysical site where the neutron--rich isotopes may play
a significant role in nucleosynthesis is the $\alpha$--process occurring
in supernovae.
The nucleosynthesis in the high--entropy bubble is thought to proceed as
follows. Due to the high temperature, the previously produced nuclei up
to iron will be destroyed again by photodisintegration. At temperatures
of about 10$^{10}$\,K the nuclei would be dismantled into their
constituents, protons and neutrons. At slightly lower temperatures one is
still left with $\alpha$--particles. During the subsequent cooling of
the plasma the nucleons will recombine again, first to $\alpha$--particles,
then to heavier nuclei.
Depending on the exact temperatures, densities
and the neutron excess, quite different abundance distributions can be
produced in this {\em $\alpha$--rich freeze--out\/} (sometimes also called
{\em $\alpha$--process\/}). Temperature and density are
dropping quickly in the adiabatically expanding high--entropy bubble.
This will hinder the recombination of $\alpha$--particles into heavy nuclei,
leading in some scenarios to a high neutron density for an r--process,
at the end of the $\alpha$--process after freeze--out of charged particle reactions.
These astrophysical motivations have led us to critically study the
the role of radiative neutron capture reactions by neutron--rich Li--
and Be--isotopes theoretically in explosive nucleosynthesis.
Since it is the focus in recent years to study the the nuclear reactions
dynamics by the use of radioactive nuclear beams, our theoretical studies
are also being tested experimentally.
\section{Calculation of Radiative--Capture Cross Sections}
Nuclear burning in explosive astrophysical environments produces
unstable nuclei which can again be targets for subsequent reactions. In
addition, it involves a
very large number of stable nuclei which are not yet fully explored
by experiments. Thus, it is necessary to be able to predict reaction
cross sections and thermonuclear rates with the aid of theoretical models.
In astrophysically relevant nuclear reactions two important reaction
mechanisms take place. These two mechanisms are compound--nucleus reactions
(CN) and direct reactions (DI).
The reaction mechanism and therefore
also the reaction model depends on the number of levels in the CN.
If one is considering only a few CN resonances the R--matrix theory
is appropriate.
In the case of a single resonance the R--matrix theory reduces to the
simple phenomenological Breit--Wigner
formula. If the level density of the CN is so high that there are
many overlapping resonances,
the CN mechanism will dominate and the statistical
HF--model can be applied. Finally, if there are no CN resonances
in a certain energy interval the DI mechanism dominates and
one can use DI models, like Direct Capture (DC).
In the case of a single isolated resonance the resonant part
of the cross section
is given by the well--known Breit--Wigner
formula\,\cite{Bre36,Bla62}:
\begin{equation}
\label{BW}
\sigma_{\rm r}(E) =
\frac{\pi \hbar^2}{2 \mu E}
\frac{\left(2J+1\right)}{\left(2j_{\rm p}+1\right)\left(2j_{\rm t}+1\right)}
\frac{\Gamma_{\rm in} \Gamma_{\rm out}}
{\left(E_{\rm r} - E\right)^2 + \frac{\Gamma_{\rm tot}^2}{4}} \quad ,
\end{equation}
where $J$ is the angular momentum quantum number and
$E_{\rm r}$ the resonance energy.
The partial widths of the entrance and exit channels
are $\Gamma_{\rm in}$ and $\Gamma_{\rm out}$, respectively.
The total width $\Gamma_{\rm tot}$
is the sum over the partial widths of all channels.
One important aspect is that
the particle width $\Gamma_{\rm p}$ can be related to
spectroscopic factors $S$ and the single--particle width $\Gamma_{\rm s.p.}$
by\,\cite{wie82,her95}
\begin{equation}
\label{SF}
\Gamma_{\rm p} = C^2 S \Gamma_{\rm s.p.} \quad,
\end{equation}
where $C$ is the isospin Clebsch--Gordan coefficient.
The single--particle width $\Gamma_{\rm s.p.}$ can be calculated
from the scattering phase shifts of a scattering potential with
the potential depth determined by matching the resonance energy.
The nonresonant part of the cross section can be obtained
using the DC model\,\cite{kim87,obe91,moh93}:
\begin{equation}
\label{NR}
\sigma^{\rm nr} = \sum_{c} \: C^{2} S_c\sigma^{\rm DC}_c \quad .
\end{equation}
The sum extends over all bound states
in the final nuclei. The DC cross sections $\sigma^{\rm DC}_c$ are
essentially
determined by the overlap of the scattering wave function
in the entrance channel, the bound--state wave function
in the exit channel and the multipole transition--operator.
The total cross section can be calculated by summing
over the resonant (Eq.~\ref{BW}) and nonresonant parts
(Eq.~\ref{NR}) of the cross section (if the
widths of the resonances are broad, also an
interference term has to be added).
For both parts the spectroscopic factors have to be known. They can be obtained
from other reactions, e.g., the spectroscopic factors
necessary for calculating
A(n,$\gamma$)B can be extracted
from the reaction A(d,p)B. The $\gamma$--widths can be
extracted from reduced electromagnetic transition
strengths.
For unstable nuclei
where only limited or even no
experimental information is available, the
spectroscopic factors and electromagnetic transition strengths
can also be extracted from nuclear structure models like the shell model (SM).
The most important ingredients in the potential models are the wave functions
for the scattering and bound states in the entrance and exit channels.
This is the case for the DC cross sections $\sigma^{\rm DC}_c$ in
Eq.~\ref{NR} as well as for the calculation of the single--particle width
$\Gamma_i$ in Eq.~\ref{SF}.
For the calculation of these wave functions we use real folding potentials
which are given by\,\cite{obe91,kob84}
\begin{equation}
\label{FO}
V(R) =
\lambda\,V_{\rm F}(R)
=
\lambda\,\int\int \rho_a({\bf r}_1)\rho_A({\bf r}_2)\,
v_{\rm eff}\,(E,\rho_a,\rho_A,s)\,{\rm d}{\bf r}_1{\rm d}{\bf r}_2 \quad ,
\end{equation}
with $\lambda$ being a potential strength parameter close
to unity, and $s = |{\bf R} + {\bf r}_2 - {\bf r}_1|$,
where $R$ is the separation of the centers of mass of the
projectile and the target nucleus.
The density can been derived from measured
charge distributions\,\cite{vri87} or from nuclear structure models (e.g.,
Hartree--Fock calculations) and the effective nucleon--nucleon
interaction $v_{\rm eff}$
has been taken in the DDM3Y parametrization\,\cite{kob84}.
The imaginary part of the potential
is very small because of the small flux into other reaction channels
and can be neglected in most cases involving neutron capture
by neutron--rich target nuclei.
\section{Reaction Rates for Li-- and Be--Isotopes}
The parameters for the resonant and nonresonant contributions to the
reaction rates are listed in Tables \ref{res} and \ref{nres},
respectively. In the tables we give experimental values if
available. Otherwise the excitation energies, spectroscopic
factors, neutron-- and $\gamma$--widths were calculated with the
shell model. We used the code OXBASH\,\cite{bro84} for the calculations.
For normal parity states we employed the interaction
(8--16)POT of Cohen and Kurath\,\cite{coh65}.
For nonnormal parity states we used the WBN interaction of
Warburton and Brown\,\cite{war92}.
With Eq.~\ref{BW} the resonant reaction rate can be derived as
\begin{eqnarray}
\label{resrate}
N_{\rm A}\left\langle\sigma v\right\rangle_{\rm r} & = & 1.54
\times 10^5 \mu^{-3/2} T_9^{-3/2}\\\nonumber
&& \sum_i {\omega \gamma_i \exp(-11.605 E_{\rm r} / T_9)\,{\rm cm}^3
\,{\rm mole}^{-1}\,{\rm s}^{-1}} \quad ,
\end{eqnarray}
where $T_9$ is the temperature in $10^9$K, $E_r$ the resonance energy
in the c.m.~system (in MeV), and the resonance strength $\omega \gamma$ (in eV) is
given by
\begin{equation}
\omega \gamma = \frac{2J+1}{(2j_{\rm p}+1)(2j{\rm _t}+1)} \frac{\Gamma_{\rm in}
\Gamma_{\rm out}}{\Gamma_{\rm tot}} \quad .
\end{equation}
The partial widths
of the entrance and exit channel, $\Gamma_{\rm in}$ and $\Gamma_{\rm out}$,
are in the case of (n,$\gamma$)--reactions the neutron-- and
$\gamma$--widths. Since the neutron width is usually much larger than
the $\gamma$--width, the total width $\Gamma_{\rm tot}$ is practically
identical with the neutron--width.
In Table \ref{res} we list the excitation energies, resonance energies,
neutron-- and $\gamma$--widths and the resonance strengths of the
resonances.
The nonresonant capture cross section is parametrized as
\begin{equation}
\sigma_{\rm nr} (E)=
A / \sqrt{E} + B \sqrt{E} -C E^D\: ,
\end{equation}
with $[A]={\rm \mu b\, MeV^{1/2}}$,
$[B]={\rm \mu b \, MeV^{-1/2}}$, and $[C]={\rm \mu b\, MeV^{-D}}$.
The parameters $A, B, C$ and $D$ are listed in Table \ref{nres}.
Using this equation, we obtain for the reaction rate
\begin{eqnarray}
\label{nresrate}
N_{\rm A}\left\langle\sigma v\right\rangle_{\rm nr} & = & \bigg(
836.565 A\mu^{-1/2}+108.130 B \mu^{-1/2} T_9 \nonumber \\
& & -277.097 C\mu^{-1/2}
\frac{\Gamma(2+D)}{\left.11.605^D\right.}
T_9^{D+1/2}\bigg)\,{\rm cm^3\, s^{-1}\,mole^{-1}}\; ,
\end{eqnarray}
where $\mu$ ist the reduced mass in units of the atomic mass unit
and $\Gamma(z)$
is the Euler gamma function.
The total reaction rate is given as the sum of the resonant (Eq.~\ref{resrate}
and nonresonant (Eq.~\ref{nresrate}) part.
\begin{table}
\caption[RES]{\label{res} Resonance parameters}
\begin{tabular}{|lrrrrrr|}
\hline
Reaction &
\multicolumn{1}{c}{$E_{\rm x}$} &
\multicolumn{1}{c}{$E_{\rm n}$} &
\multicolumn{1}{c}{$J^{\pi}$} &
\multicolumn{1}{c}{$\Gamma_{\rm n}$} &
\multicolumn{1}{c}{$\Gamma_{\gamma}$} &
\multicolumn{1}{c|}{$\omega\gamma$} \\
& \multicolumn{1}{c}{(MeV)} &
\multicolumn{1}{c}{(MeV)} &&
\multicolumn{1}{c}{(eV)} &
\multicolumn{1}{c}{(eV)} &
\multicolumn{1}{c|}{(eV)} \\
\hline
$^7$Li(n,$\gamma$)$^8$Li & 2.26 & 0.227 & $3^+$ & $3.1 \times 10^4$ &
$0.07$ & $0.061$ \\
$^8$Li(n,$\gamma$)$^9$Li & 4.31 & 0.247 & $5/2^-$ & $1 \times 10^5$ &
0.11 & $0.066$ \\
$^9$Be(n,$\gamma$)$^{10}$Be & 7.371 & 0.559 & $3^-$ & $1.57 \times 10^4$ &
0.661 & 0.578 \\
& 7.542 & 0.73 & $2^+$ & $6.3 \times 10^3$ &
0.814 & 0.509 \\
\hline
\end{tabular}
\end{table}
\begin{table}[htb]
\caption[di cross section]{\label{nres}Parametrization of the nonresonant cross
section (see text).}
\begin{center}
\begin{tabular}{|llrrrrr|}\hline
& \multicolumn{1}{c}{$A$}& \multicolumn{1}{c}{$B$} &
\multicolumn{1}{c}{$C$} & \multicolumn{1}{c}{$D$}
& \multicolumn{2}{c|}{$\sigma_{\rm nr}({\rm\mu b})\:\:{\rm at}\:\:{\rm 30\, keV}$}
\\ \hline
& & & & & \multicolumn{1}{c}{This}
& \multicolumn{1}{c|}{Rauscher}
\\
& & & & &
\multicolumn{1}{c}{work} &
\multicolumn{1}{c|}{{\em et al.}\cite{rau94}} \\
\hline
${\rm ^{7}Li(n,\gamma)^{8}Li}$ & $6.755^{\rm a}$
& \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---}
& \multicolumn{1}{c}{---} & \multicolumn{1}{c}{$39.000$}
& \multicolumn{1}{c|}{---} \\
${\rm ^{8}Li(n,\gamma)^{9}Li}$ & $2.909$ & \multicolumn{1}{c}{---}
& \multicolumn{1}{c}{---} &\multicolumn{1}{c}{---}
& \multicolumn{1}{c}{$16.795$} & $30.392$
\\
${\rm ^{9}Be(n,\gamma)^{10}Be}$ & $1.147^{\rm a}$ & $11.000$
& $6.815$ & $0.962$
& $8.294$ & $6.622$ \\
${\rm ^{10}Be(n,\gamma)^{11}Be}$ & $0.132$ & $24.000$ & $15.725$ & $0.914$
& $4.281$ & $3.943$ \\
${\rm ^{11}Be(n,\gamma)^{12}Be}$ & \multicolumn{1}{c}{---}
& $7.000$ & $4.851$ & $0.887$
& $0.996$ & $2.373$ \\ \hline
\end{tabular}
\end{center}
\small
$^{\rm a}$extracted from experimental thermal cross section\,\cite{sea92}
\end{table}
\subsection{$^7$Li(n,$\gamma$)$^8$Li}
The cross section of the reaction $^7$Li(n,$\gamma$)$^8$Li is
well known (see, e.g.,\,\cite{bla96}). The cross section is
dominated by s--wave capture to the $^8$Li ground state and
a resonance at 227\,keV neutron energy. Using the spectroscopic
factors of Cohen and Kurath\,\cite{coh67} yields a thermal
cross section of $8.2 \times 10^{-2}$\,b, which is a factor
1.8 higher than the experimental value of $4.54 \times 10^{-2}$\,b.
The shell model calculation is purely p--shell and does not
include excitations to other oscillator shells. Therefore the
spectroscopic amplitude of 0.977 for a p$_{3/2}$--transition to
the ground state of $^8$Li might be too high.
For the resonance, however, we find excellent agreement between
calculation and experiment. The calculated width --- using the
folding potential and spectroscopic amplitudes from
Cohen and Kurath\,\cite{coh67}
--- is 28.9\,keV, almost identical to the known value of $31 \pm 7$\,keV.
\subsection{$^8$Li(n,$\gamma$)$^9$Li}
The resonance at 247 keV is a $5/2^-$ state\,\cite{van84}.
With a total width of 100\,keV the resonance strength
is determined by the $\gamma$--width which was previously
estimated with 0.56\,eV\,\cite{mal88}. A shell model calculation
yielded a width $\Gamma_{\gamma} = 0.11$\,eV. Therefore the
resonance strength is a factor 5 smaller than previously
assumed\,\cite{rau94}.
The calculated thermal cross section, resulting from s--wave
capture to the ground state and first excited state in $^9$Li,
is 1.94 $\times 10^{-2}$\,b and is smaller than the value
of $3.51 \times 10^{-2}$\,b given by Rauscher {\em et al.}\,\cite{rau94}.
\subsection{$^9$Be(n,$\gamma$)$^{10}$Be}
Like in the reaction $^7$Li(n,$\gamma$)$^8$Li the spectroscopic
factors of Cohen and Kurath\,\cite{coh67} are a little too high.
The thermal cross section is dominated by the transition to the
$^{10}$Be ground state with a theoretical spectroscopic factor
of 2.36. With this value the calculated thermal cross section
is $1.06 \times 10^{-2}$\,b, compared to experimental cross section
of $7.6 \times 10^{-3}$\,b. With the spectroscopic factor given by
Mughabghab\,\cite{mug85} of 1.45 the calculated cross section
would be close to the experimental value. For high temperatures the
p--wave capture to excited states has to be taken into account.
Two resonances are known at 559\,keV and 730\,keV. The total widths
are known experimentally. We have calculated the $\gamma$--widths
which were only estimated previously. Both resonance strengths are larger
than the previous estimates, for the 559\,keV resonance the
enhancement is one order of magnitude.
With the higher resonance strengths and the p--wave contribution the
reaction rate is clearly higher compared to Ref.~\cite{rau94}.
\subsection{$^{10}$Be(n,$\gamma$)$^{11}$Be}
Cross section and reaction rate of this reaction were recently
determined experimentally with the help of the inverse
Coulomb dissociation\,\cite{men96}. They supported their experimental
values by a direct capture calculation. In order to reproduce the
experimental data they enhanced the spectroscopic factors to the
$^{11}$Be ground state by 20\%. Our calculation confirms the results.
Using the spectroscopic factors from the (d,p)--reaction\,\cite{ajz90}
the calculated cross section is a little smaller than the experimental.
The results are grossly different from the rate given in Ref.~\cite{rau94}.
\subsection{$^{11}$Be(n,$\gamma$)$^{12}$Be}
There is no resonant contribution to the reaction rate. The transition
is a p--wave capture from the $1/2^+$ ground state of $^{11}$Be
to the ground state of $^{12}$Be and the $0^+$ state at 2.7\,MeV
excitation energy, while the transition to the $2^+$ state at 2.1\,MeV
is negligible.
\section{Discussion}
The new reaction rates could change the reaction flow in the inhomogeneous big
bang nucleosynthesis. The smaller rate for $^8$Li(n,$\gamma$)$^9$Li could mean
that the main reaction flow will proceed through the reaction $^8$Li($\alpha$,n)$^{11}$B.
The higher rate for $^9$Be(n,$\gamma$)$^{10}$Be might give more importance to this
reaction. Detailed network calculations with the new rates are planned for the
near future.
\section*{Acknowledgments}
We thank the Fonds zur F\"orderung der
wissenschaftlichen Forschung in \"Osterr\-reich (project S7307--AST)
and the \"Ostereichische Nationalbank (project 5054)
for their support.
\section*{References}
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proofpile-arXiv_065-609
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
Recently, there has been a revival of interest in the area of exactly
solvable models in one and higher dimensions. A celebrated example of a
solvable many-body system is the well-known Calogero-Sutherland model
(CSM) in one dimension \cite{C69,S71,S72}. The model has found wide
application in areas as diverse as quantum chaos and Fractional
Statistics. The particles in the CSM are confined in a one-body oscillator
potential or on the rim of a circle, and interact with each other through
a two-body potential which varies as the inverse-square of the distance
between particles. The CSM and its variants in one dimension, like the
Haldane-Shastry model for spin chains \cite{H88}, have provided the
paradigms to analyze more complicated interacting systems. A
characteristic feature of the CSM is the structure of the highly
correlated wave function. The correlations are built into the exact wave
function through a Jastrow factor $(x_i -x_j)^{\lambda}
|x_i-x_j|^{\alpha}$ for any pair of particles denoted by $i,j$. The
exponents on the correlator are related to the strength of the
inverse-square interaction. Notice that this factor is antisymmetric
(symmetric) in particle labels for $\lambda=1(0)$ and vanishes as the two
particles approach each other. A generalization of this in two dimensions
is to be found in Laughlin's trial wave function \cite{L83} where the
correlations are built in through the factor $(z_i - z_j)$, where $z_i$
are the particle coordinates in complex notation. The corresponding
Hamiltonian for which the Laughlin wave function is an exact ground state
has not been analyzed to the same degree of detail as the CSM. It is known
that it is the ground state for a Hamiltonian describing spin polarized
electrons in the lowest Landau level with a short-range repulsive
interaction \cite{TK85}. It is also known that such correlations are
present in the exact ground state of a spin Hamiltonian\cite{L89} in
two-dimensions. The anyon Hamiltonian\cite{LM79} in two-space dimensions
is another example where the Jastrow correlation appears\cite{WU84}. While
the two-anyon problem is exactly solvable, the many-anyon problem is not.
For a system of anyons confined in an oscillator potential many exact
solutions and their properties are known but, unlike the CSM in
one-dimension, the analytical solution of the full many-body problem is
not tractable\cite{exact}. It is therefore of great interest to find
models analogous to the CSM in higher dimensions.
In a recent paper\cite{MBS96}, three of us proposed a model in two-space
dimensions with nontrivial two- and three-body interactions which could
be solved exactly for the ground states and some excited states. It
betrayed some similarity to both CSM in one-dimension and the anyonic
model in two-dimensions through the spectrum. The model was devised by noting
that in two dimensions there exists another form of the pair
correlator with which a Jastrow-type many-body wave function may be
constructed, namely
\beq
X_{ij} ~=~ x_i y_j ~-~ x_j y_i ~.
\label{xij}
\eeq
The correlation is by definition antisymmetric and goes to zero as two
particles approach each other. In addition, it introduces zeros in the
wave function whenever the relative angle between the two particles goes
to zero or $\pi$. The difference with the Jastrow-Laughlin form is also
significant; $X_{ij}$ in (\ref{xij}) is a pseudo-scalar. Unlike the Laughlin
type of correlation, it does not impart any angular momentum to the two
dimensional wave function . One important drawback of this correlation is
that it is not translationally invariant unless the radial degrees of
freedom is frozen. The model Hamiltonian has solutions which have this
correlation built in. Intuitively the correlation can be understood easily
by imagining objects with associated "arrows". The arrows cannot be
oriented either parallel or anti-parallel to each other.
The model has some interesting features and it would be of great
interest to find physical systems which incorporate these features.
In this paper we elaborate on our earlier results \cite{MBS96} and
present several
new results. In Sect. II, we discuss the many-body Hamiltonian and display
some of the exact solutions and their structure. The similarities between
the spectrum of these exact solutions and the spectrum of CSM is quite
remarkable. Further, when projected on to a circle the model reduces to a
variant of the trigonometric Sutherland model. In this limit
the model also has translational invariance. In Sect. III, we discuss
the two-body problem in detail and show that the solutions of the two-body
problem are described by the Heun equation. In particular, the spectrum
becomes very simple for large values of the interaction strength. The singular
interaction discussed in this paper requires a careful treatment
in the region near $X_{ij} =0$; this is discussed in the Appendix \cite{S58}.
Sect. IV contains a discussion and summary.
\section{Many-body Hamiltonian and some exact solutions}
For the sake of completeness we recall first the Hamiltonian and some
of its properties proposed earlier\cite{MBS96}. We also clarify
some points which were not made explicitly clear in the earlier
paper.
The $N$-particle Hamiltonian which we consider is given by
\beq
H~=~ -{\hbar^2\over {2m}} ~\sum_{i=1}^{N} {\vec \nabla}^2_i + {{m\omega^2}
\over 2} ~\sum_{i=1}^{N} {\vec r}_i^2 + \frac{\hbar^2}{2m} g_1 ~
\sum_{\stackrel{i,j}{ (i\ne j)}}^{N} \frac{{\vec r}_j^2}{X_{ij}^2}
+ \frac{\hbar^2}{2m} g_2 ~\sum_{\stackrel{i,j,k}{ (i\ne j\ne k)}}^{N}
\frac{{\vec r}_j \cdot {\vec r}_k}{X_{ij}X_{ik}} ~,
\eeq
where $X_{ij}$ is given by (\ref{xij}); $g_1$ and $g_2$ are
dimensionless coupling strengths of the two- and three-body interactions
respectively. While $g_1$ and $g_2$ can be independent of each
other in general, for the type of solutions involving the correlator in
(\ref{xij}) they are not. We will specify their relationship shortly. The
particles are confined in a one-body oscillator
confinement potential. The Hamiltonian is rotationally
invariant and manifestly symmetric in all particle indices. As in the
CSM, we may scale
away the mass $m$ and oscillator frequency $\omega$ by scaling all distances
$\vec r_i \rightarrow \sqrt{m\omega/\hbar}~ \vec r_i$, and measuring the
energy in units of $\hbar \omega$. This is done by setting
$\hbar =m=\omega =1$. In these units the Hamiltonian is given by
\beq
H~=~ -{1\over 2} ~\sum_{i=1}^{N} {\vec \nabla}^2_i +{1\over 2} ~\sum_{i=1}^{N}
{\vec r}_i^2 + \frac{g_1}{2} ~\sum_{\stackrel{i,j}{(i\ne j)}}^N
\frac{{\vec r}_j^2}{X_{ij}^2} + \frac{g_2}{2} ~\sum_{\stackrel{i,j,k}{(i\ne
j\ne k)}}^N \frac{{\vec r}_j \cdot {\vec r}_k}{X_{ij}X_{ik}} ~.
\label{ham2}
\eeq
Note that the total angular momentum operator
\beq L ~=~ \sum_i ~(x_i p_{y_i} - y_i p_{x_i})
\label{angmom}
\eeq
commutes with the Hamiltonian since it is rotationally invariant, and may
therefore be used to label the states. The Hamiltonian is invariant under
parity $x\rightarrow -x$ and $y\rightarrow y$. In addition, for any $i$,
the Hamiltonian is invariant under the transformation $\vec r_i
\rightarrow -\vec r_i$ and $\vec r_k \rightarrow \vec r_k$ for all $k \neq i$.
This $D_{2N}$ invariance is special to this system, and we are not aware of
any other interacting
many-body Hamiltonian which has this symmetry. The consequences of this will
be discussed explicitly in the two-body problem where this is related
to the supersymmetric properties of the system.
We will consider both bosonic and fermionic systems governed by the Hamiltonian
(\ref{ham2}), i.e., wave functions which are totally symmetric and
antisymmetric respectively. It will turn out that certain calculations (for
example, in the two-body problem) simplify if we do not impose any symmetry to
begin with.
\subsection{The exact bosonic ground state}
We first obtain the exact bosonic ground state of this Hamiltonian. As an
ansatz for the ground state wave function, consider a solution of the form
\beq
\Psi_0 (x_i,y_i)~=~ \prod_{i<j}^N ~|X_{ij}|^{g}~
\exp ~(-{1\over 2} \sum_{i=1}^N {\vec r}_i^{~2}) ~.
\label{ansatz}
\eeq
Clearly $\Psi_0$ correctly incorporates the
behavior of the wave function in the asymptotic region $|\vec r_i|
\rightarrow \infty$, and $\Psi_0$ is regular for $g \ge 0$. In general
we insist that our solutions have this asymptotic form; the conditions under
which this is valid will be specified later. The eigenvalue equation now
takes the form
\beq
H \Psi_0 ~=~[\frac{1}{2} (g_1-g(g-1)) \sum_{\stackrel{i,j}{(i\ne j)}}^{N}
\frac{{\vec r}_j^2}{X_{ij}^2} + \frac{1}{2} (g_2-g^2) \sum_{\stackrel{i,j,k}
{(i\ne j\ne k)}}^{N} \frac{{\vec r}_j \cdot {\vec r}_k}{X_{ij}X_{ik}} +
gN(N-1) +N] \Psi_0 ~.
\eeq
Therefore $\Psi_0$ is the exact many-body ground state for an arbitrary
number of particles of the Hamiltonian if
\beq
g_1 ~=~ g(g-1) \quad {\rm and} \quad g_2 ~=~ g^2 ~.
\label{g12}
\eeq
Since $g \ge 0$, we have $g_1 \ge -1/4$ and $g_2$ is positive
definite. Note that the range of $g_1$ is identical to the one obtained in
the CSM. The ground state energy is now given by
\beq
E_0 ~=~ N ~+~ gN(N-1) ~.
\label{egs}
\eeq
Note that this has exactly the form of the ground state energy of the CSM.
Since $g$ determines both $g_1$ and $g_2$ uniquely, we will regard $g$ as
the fundamental parameter of the Hamiltonian which determines the strength of
the interaction. In other words, we demand that the ground state should be
given by (\ref{ansatz}), and we {\it define} the Hamiltonian to ensure that.
It turns out that such a definition requires some special care in the
vicinity of $X_{ij} =0$ (called the ultraviolet region below). The Appendix
will discuss this for the case of two particles. Note that in the two-particle
case, the Hamiltonian only contains the parameter $g_1 = g(g-1)$ and not $g_2$.
As a result, for every value of $g_1$ in the range $-1/4 <g_1 < 0$,
the bosonic ground state energy as given by $E_0 = 2 +2g$ has two possible
values; these two possibilities correspond to different potentials in the
ultraviolet region. This is somewhat unusual but it is not uncommon for
singular potentials. The same thing also happens in the CSM even for the
$N$-body problem; see for example Ref. \cite{MS94}. We discuss this issue
in detail in the Appendix where we show that the ultraviolet
regularization is determined by the parameter $g$ rather than by $g_1$.
We emphasize that our objective here is not to find the general solutions
for arbitrary $g_1$ and $g_2$, but to find a Hamiltonian whose solutions
have the novel correlation in Eq. (\ref{xij}) built in. In general, if
$g_1$ and $g_2$ are independent, the Hamiltonian will have a ground state
different from the one given above. Our procedure is
therefore similar to the many-anyon problem where also there are two- and
three-body interactions, but the strengths are related to a single
parameter. With the form of $g_1$ and $g_2$ given in (\ref{g12}),
the solution found above is indeed the lowest energy state.
A neat way of proving that we have indeed obtained the ground state
can be given using the method of operators \cite{P92}. To this end, define
the operators
\bea
Q_{x_i} ~&=&~ p_{x_i} ~-~ ix_i ~+~ i~g ~\sum_{j(j\ne i)}
\frac{y_j}{X_{ij}} ~, \nonumber \\
Q_{y_i} ~&=&~ p_{y_i} ~-~ iy_i ~-~ i~g ~\sum_{j(j\ne i)}
\frac{x_j}{X_{ij}} ~, \nonumber \\
Q_{x_i}^{\dag} ~&=&~ p_{x_i} ~+~ ix_i ~-~ i~g ~\sum_{j(j\ne i)}
\frac{y_j}{X_{ij}} ~, \nonumber \\
Q_{y_i}^{\dag} ~&=&~ p_{y_i} ~+~ iy_i ~+~ i~g ~\sum_{j(j\ne i)}
\frac{x_j}{X_{ij}} ~.
\eea
It is easy to see that the $Q$'s annihilate the ground state in Eq.
(\ref{ansatz}), $$Q_{x_i}\Psi_0 = 0 \quad {\rm and} \quad Q_{y_i}\Psi_0 =0.$$
The Hamiltonian can now be recast in terms of these operators as
\beq
\frac{1}{2} ~\sum_i ~\left[ Q_{x_i}^{\dag} Q_{x_i} + Q_{y_i}^{\dag} Q_{y_i}
\right] ~=~ H ~-~ E_0 ~,
\eeq
where $E_0$ is given by Eq. (\ref{egs}). Clearly the operator on the left hand
side is positive definite and annihilates the ground state wave function given
by Eq. (\ref{ansatz}). Therefore $E_0$ must be the minimum energy that an
eigenstate can have.
As we remarked earlier, the ground state of the Hamiltonian is bosonic.
The ground state of the Hamiltonian for a fermionic system is not easy to
determine analytically (for $g > 0$). The problem here is analogous to a
similar problem in the many-anyon Hamiltonian \cite{KM91,anyon}. In Sect. III,
we will determine the fermionic ground state energy for two particles both
numerically and to first order in $g$ using perturbation theory near $g=0$
and show that it has quite unusual behavior.
\subsection{Spectrum of excited states}
While we have not been able to find the complete excited state spectrum of
the model, the eigenvalue equation for a general excited state may be
obtained as follows. From the asymptotic properties of the solutions of
the Hamiltonian in Eq. (\ref{ham2}), it is clear that $\Psi$ has the
general structure
\beq
\Psi (x_i,y_i) ~=~ \Psi_0 (x_i,y_i) ~\Phi(x_i,y_i) ~,
\eeq
where $\Psi_0$ is the ground state wave function. Obviously if $\Phi$ is
a constant we recover the ground state. In general $\Phi$ satisfies the
eigenvalue equation
\beq
[~-{1\over 2} ~\sum_{i=1}^{N}{\vec \nabla}_i^2 ~+~\sum_{i=1}^{N} {\vec r}_i
\cdot {\vec \nabla}_i ~+~ g ~\sum_{\stackrel{i,j}{(i\ne j)}} \frac{1}{X_{ij}}
(x_j \frac{\partial}{\partial y_i}-y_j \frac{\partial}{\partial x_i})~] ~\Phi ~
=~ (E - E_0) ~\Phi ~.
\label{hamphi}
\eeq
It is interesting to note that while $g_1$ is zero both at $g=0$ and $1$, the
term containing $g$ in the above expression is zero only when $g=0$. This
is so because of the boundary condition that the wave
functions must vanish as $|X_{ij}|^g$ for nonzero $g$.
We first discuss the exact solutions of the above differential equation.
This is easily done by defining the complex coordinates
\beq
z ~=~ x+iy \quad {\rm and} \quad z^* ~=~ x-iy ~,
\eeq
and their partial derivatives
\bea
\partial ~=~ \partial/\partial z ~=~ \frac{1}{2} ~(\partial /\partial x -i
\partial/\partial y)~, \nonumber \\
\partial^* ~=~ \partial/\partial z^* ~=~ \frac{1}{2} ~(\partial /\partial x +i
\partial/\partial y) ~.
\eea
In these coordinates, the differential equation for $\Phi$ reduces to
${\tilde H} \Phi = (E-E_0) \Phi$, where
\beq
{\tilde H} ~=~ -2\sum_i \partial_i\partial_i^* ~+~ \sum_i (z_i\partial_i
+z_i^*\partial_i^*) ~+~ 2g\sum_{\stackrel{i,j}{(i\ne
j)}}\frac{z_j\partial_i-z_j^*\partial_i^*}{z_i z_j^*-z_j z_i^*} ~.
\label{hcomplex}
\eeq
In addition, $\Phi$ is an eigenstate of the total angular momentum operator,
$L\Phi = l\Phi$. We can now classify some exact solution according to their
angular momentum.
\noindent
(a) $l=0$ solutions: Define an auxiliary parameter
$$ t ~=~ \sum_i z_i z_i^* ~,$$
and let $\Phi = \Phi(t)$. This has zero total angular momentum.
The differential equation for $\Phi$ reduces to
\beq
t\frac{d^2 \Phi}{dt^2} +(b-t)\frac{d\Phi}{dt} -a\Phi ~=~ 0 ~,
\eeq
where $b=E_0$ and $a=(E_0-E)/2$; $E_0$ is the energy of the ground state. The
allowed solutions are the regular confluent hypergeometric functions \cite{LL}
\beq
\Phi(t) ~=~ M(a,b,t) ~.
\eeq
Normalizability imposes the restriction $ a= -n_r $, where $n_r$ is
a positive integer; then $\Phi(t)$ is a polynomial of degree $n_r$ (the
subscript 'r' denotes radial excitations as discussed later). The
corresponding eigenvalues are
\beq
E~=~ E_0 + 2n_r ~.
\eeq
This class of solutions was discussed before in \cite{MBS96}.
\noindent
(b) $l > 0$ solutions: Let
$$ t_z ~=~ \sum_i z_i^2 ~,$$
and let $\Phi = \Phi(t_z)$.
The total angular momentum is not zero. All the mixed
derivative terms in Eq. (\ref{hcomplex}) drop out, and we get the
differential equation
\beq
2t_z\frac{d\Phi}{dt_z} ~=~ (E-E_0)\Phi.
\eeq
This is the well known Euler equation whose solutions are just monomials
in $t_z$. The solution is given by
\beq
\Phi(t_z) ~=~ t^m_z ~,
\eeq
and the total angular momentum is $l=2m$. The eigenvalues are
\beq
E ~=~ E_0 +2m ~=~ E_0 +l ~.
\eeq
\noindent
(c) $l < 0$ exact solutions: Let
$$ t_{z^*}=\sum_i (z_i^*)^2 ~,$$
and let $\Phi = \Phi(t_{z^*})$. Once again the
differential equation for $\Phi$ reduces to
\beq
2t_{z^*}\frac{d\Phi}{dt_{z^*}} ~=~ (E-E_0)\Phi ~.
\eeq
This is similar to the previous case. The solution is given by
\beq
\Phi(t_{z^*}) ~=~ t_{z^*}^m ~,
\eeq
and the total angular momentum is $l=-2m$. The eigenvalues are
\beq
E ~=~ E_0 +2m ~=~ E_0 -l ~.
\eeq
\noindent
(d) Tower of excited states: One can now combine solutions of a given $l$ in
cases (b) or (c) with the solutions in (a), and get a new class of excited
states. Let us define
\beq
\Phi (z_i,z_i^*) ~=~ \Phi_1(t) \Phi_2(t_z) ~,
\eeq
where $\Phi_1$ is the solution with $l=0$, $\Phi_2$ is the solution
with $l> 0$, and $t$ and $t_z$ have been defined before. The
differential equation for $\Phi$ is again a confluent hypergeometric
equation given by
\beq
t\frac{d^2 \Phi}{dt^2} +(b-t)\frac{d\Phi}{dt} -a\Phi ~=~ 0 ~,
\eeq
where $b=E_0+2m$ and $a=(E_0+2m-E)/2$. The energy eigenvalues are then
given by
\beq
E ~=~ E_0 + 2n_r + 2m ~=~ E_0 +2n_r +l ~.
\eeq
One may repeat the procedure to obtain exact solutions for a tower of
excited states with $l<0$ solutions. As we shall see below, the
existence of the tower is a general result applicable to all excited
states of which the exact solutions shown above form a subset. We notice
that these solutions bear a remarkable resemblance to the many-anyon
system where a similar structure exists for the known class of
exact solutions \cite{exact}.
\noindent
(e) A general class of excited states: One can combine the solutions of all the
three classes (a), (b) and (c) to obtain an even more general class of
solutions. Consider the polynomial
\beq
P(n_1,n_2,n_3) ~=~ t^{n_1}~ t_z^{n_2}~ t_{z^*}^{n_3} ~,
\eeq
where the $n_i$ are non-negative integers. Using the form in (\ref{hcomplex}),
one can show that
\bea
{\tilde H} P(n_1,n_2,n_3) ~=~ &2& (n_1 + n_2 + n_3) P(n_1,n_2,n_3) ~-~ 8 n_2
n_3 P(n_1+1,n_2-1,n_3-1) \nonumber \\
&-&~ 2 n_1 [n_1 + 2 n_2 + 2 n_3 + g N (N-1)] P(n_1-1,n_2,n_3) ~.
\eea
Using this one can show that there is an exact polynomial solution, whose
highest degree term is $P(n_1,n_2,n_3)$. The energy of this solution is
\beq
E ~=~ E_0 ~+~ 2 (n_1 + n_2 + n_3) ~,
\eeq
and the angular momentum is $l = 2 (n_2 - n_3)$.
While there may be more exact solutions, we do not know of a simple way of
solving for them. We can however glean some general features as follows.
The coordinates $(x_i , y_i)$ can be separated into one `radial' coordinate
$t = \sum_i {\vec r}_i^2$ as above and $2N-1$ `angular' coordinates
collectively
denoted by $\Omega_i$(say). Then, the Eq. (\ref{hamphi}) can be expressed as
\beq
t {{\partial^2 \Phi} \over {\partial t^2}} ~+~ (E_0 - t) {{\partial \Phi}
\over {\partial t}} ~-~ \frac{1}{t} ~{\cal L} ~\Phi ~+~ \frac{1}{2} (E-E_0)~
\Phi ~=~ 0 ~,
\eeq
where ${\cal L} = {\cal D}_2 + g {\cal D}_1$, and ${\cal D}_n$ is an
$n^{th}$-order differential operator which only acts on functions of the angles
$\Omega_i$. In particular, ${\cal D}_2$ is the Laplacian on a sphere of
dimension $2N-1$. Next we note that the $\Phi$ can be factorized in the
form
\beq
\Phi (x_i , y_i) ~=~ R(t) ~Y(\Omega_i) ~,
\eeq
where $Y$, generalized spherical harmonic defined on the $2N-1$
dimensional sphere $S^{2N-1}$,
satisfies the eigenvalue equation ${\cal L} Y = \lambda Y$. (This is the hard
part of the spectral problem, to find the eigenvalues $\lambda$). We now define
\beq
\mu ~=~ {\sqrt {(E_0 -1 )^2 + 4 g \lambda}} ~-~ (E_0 -1) ~.
\eeq
Further if we write $R(t) = t^{\mu /2} {\tilde R} (t)$, then $\tilde R$
satisfies a confluent hypergeometric equation
\beq
t \frac{d^2 {\tilde R}}{dt^2} ~+~ (b -t) \frac{d{\tilde R}}{dt} ~-~ a
{\tilde R} ~=~ 0 ~.
\eeq
where $b= E_0 + \mu$ and $a=(E_0+ \mu -E)/2$. The admissible solutions are
the regular confluent hypergeometric functions, ${\tilde R} (t)=M(a,b,t)$.
Normalizability imposes the restriction $a=-n_r$, where $n_r$ is a
positive integer. Then ${\tilde R} (t)$ is a polynomial of degree $n_r$, and it
has $n_r$ nodes. The energy of this state is given by $E = E_0 + \mu + 2n_r$.
We see that for a given value of $\mu$, there is an infinite tower of energy
eigenvalues separated by a spacing of $2$. As remarked earlier, this is
reminiscent of what happens
in the case of anyons. The tower structure and the angular momentum
are useful in organizing a numerical or analytical study of the energy
spectrum. Since the radial quantum number $n_r$ and the angular momentum $l$
are integers, they cannot change as the parameter $g$ is varied continuously.
\subsection{Relation to Sutherland model}
It may be of interest to note that the model reduces to a variant of the
Sutherland model\cite{S71} in one dimension. In this limit, therefore the
model is exactly solvable. Restricting the particles to
move along the perimeter of a unit circle in the Hamiltonian (\ref{ham2})
without the confinement potential, we get
\beq
H~=~ -{1\over 2} \sum_{i=1}^{N}{\partial^2\over \partial \theta_i^2} +
\frac{g_1}{2} \sum_{\stackrel{i,j}{ (i\ne j)}}^{N} \frac{1}{\sin^2(\theta_i
-\theta_j)} + \frac{g_2}{2} \sum_{\stackrel{i,j,k}{ (i\ne
j\ne k})}^{N} [1+\cot(\theta_i-\theta_j)\cot(\theta_i -\theta_k)] ~,
\label{ham5}
\eeq
since $X_{ij}=-\sin (\theta_i -\theta_j)$ now. Using the identity
\beq
\sum_{\stackrel{i,j,k}{ (i\ne j\ne k)}}^{N}
\cot(\theta_i-\theta_j)\cot(\theta_i-\theta_k) ~=~ -~ \frac{N(N-1)(N-2)}{3} ~,
\eeq
we immediately recover an analog of the trigonometric Sutherland model, but
shifted by the constant $g_2~ N(N-1)(N-2)/3$. Note, however, that the
potential in (\ref{ham5}) depends on the function $\sin(\theta_i
-\theta_j)$, rather than the chord-length which is proportional to
$\sin[(\theta_i -\theta_j)/2]$ . Interestingly the wave function has
twice the periodicity of the Sutherland model solutions- the wave
function vanishes whenever the particles are at diametrically opposite
points on a circle or at the same point.
\section{The two-body problem: Complete solution}
While we have not been able to solve the many-body problem completely, the
two-body problem in our model is exactly solvable. We demonstrate this by
going
over to the hyperspherical formalism first proposed in two dimensions by
Kilpatrick and Larsen \cite{KL87}(see also \cite{KM91}). We
discuss some of the properties of
the two-body spectrum. We also explicitly show that the two-body problem is
integrable. It is important to note that the two-particle interaction is
sufficiently singular that a careful treatment is required in order
to define the problem completely consistently; this is described in the
Appendix.
The two-body Hamiltonian is given by
\beq
H ~=~ -\frac{1}{2}[{\vec \nabla}_1^2 +{\vec \nabla}_2^2] ~+~ \frac{1}{2}
[{\vec r}_1^2+{\vec r}_2^2] ~ +~ \frac{g_1}{2}
\frac{{\vec r}_1^2+{\vec r}_2^2}{X^2} ~,
\eeq
where $X =x_1 y_2 - x_2 y_1$. The two-body
problem is best solved in the hyperspherical coordinate system which allows
a parameterization of the coordinates ${\vec r}_1, {\vec r}_2$ in terms of three
angles and one length, $(R,\theta,\phi,\psi)$ as follows:
\bea
x_1 + i y_1 ~&=&~ R~ (\cos \theta ~\cos \phi -
i\sin \theta ~\sin \phi) ~\exp (i\psi) ~, \nonumber \\
x_2 + i y_2 ~&=&~ R ~(\cos \theta ~\sin(\phi) +
i\sin \theta ~\cos \phi) ~\exp (i\psi) ~.
\eea
We may regard $(R,\theta,\phi)$ as the body-fixed coordinates which are
transformed to the space-fixed system by an overall rotation of $\psi$.
For a fixed $R$, these coordinates define a sphere in four-dimensions within
the following intervals:
\bea
-\pi/4 \le & \theta & \le \pi/4 ~, \nonumber \\
-\pi/2 \le & \phi & \le \pi/2 ~, \nonumber \\
-\pi~~ \le & \psi & \le \pi ~.
\eea
Exchange of two particles is achieved by
\bea
\theta & & \rightarrow -~ \theta ~, \nonumber \\
{\rm and} \quad \phi & & \rightarrow \pi/2 - \phi ~, \quad \psi \rightarrow
\psi \quad {\rm if} \quad \phi > 0 ~, \nonumber \\
{\rm and} \quad \phi & & \rightarrow - \pi/2 - \phi ~, \quad \psi \rightarrow
\pi + \psi \quad {\rm if} \quad \phi < 0 ~.
\label{exchange}
\eea
With this choice of coordinates, the radial coordinate becomes
\beq
R^2 ~=~ r_1^2+r_2^2
\eeq
which is the radius of the sphere in four dimensions. Also,
\beq
X ~=~ x_1 y_2 -x_2 y_1 = R^2 \sin (2\theta) /2 ~.
\eeq
Notice that $X$ depends only on $R$ and $\theta$. Therefore
the two-body interaction in the Hamiltonian is independent of
the angles $\phi$ and $\psi$. The integrals of motion of the system may be
constructed in terms of these new coordinates. The angular momentum
operator is given by
\beq
L ~=~ \sum_i (x_i p_{y_i} - y_i p_{x_i}) ~=~ -i\frac{\partial}{\partial \psi}
\eeq
which commutes with the Hamiltonian. There
exists another constant of motion given by
\beq
Q ~=~ i ~[~ x_2 \frac{\partial}{\partial x_1} + y_2 \frac{\partial}{\partial
y_1} - x_1 \frac{\partial}{\partial x_2} - y_1 \frac{\partial}{\partial
y_2} ~] ~=~ -i\frac{\partial}{\partial \phi} ~.
\eeq
Since Q is antisymmetric, acting on a symmetric state produces an
antisymmetric state and vice versa. We therefore refer to this as a
supersymmetry operator (SUSY). The operator $Q$ is similar to the SUSY
operator discovered in the many-anyon problem by Sen \cite{susy}.
Note that
the differential operator for both angular momentum and the SUSY
operators has a very simple form in the hyperspherical coordinates. The
states can therefore be labeled by the quantum numbers associated with
these two operators which we denote by $l$ and $q$ respectively. With
SUSY, the two-body problem is integrable. (The four constants of motion
are the Hamiltonian $H$, the angular part of $H$, $L$ and $Q$). Note that we
have $QX=0$ which makes calculations simple. It is easy to check that
the bosonic ground state of the Hamiltonian has the quantum numbers $l$ and
$q$ of the angular momentum and SUSY operators equal to zero.
We would like to emphasize that the eigenstates of the SUSY operator $Q$ are
neither symmetric nor antisymmetric, unless the eigenvalue $q=0$. After
finding a simultaneous eigenstate of $H$, $L$ and $Q$, we can separate it into
symmetric (bosonic) and antisymmetric (fermionic) parts. These parts
are individually eigenstates of $Q^2$ but not of $Q$. Specifically, we have
$Q \Psi_B = q \Psi_F$ and $Q \Psi_F = q \Psi_B$, where $B$ and $F$ denote
bosonic and fermionic states respectively. Then $\Psi_B \pm \Psi_F$ are
eigenstates of $Q$, while $\Psi_B$ and $\Psi_F$ are eigenstates of $Q^2$.
The two-body Hamiltonian in terms of the hyperspherical coordinates is
given by
\beq
H ~=~ -\frac{1}{2} ~[~ \frac{\partial^2}{\partial R^2} +\frac{3}{R}
\frac{\partial}{\partial R} -\frac{\Lambda^2}{R^2} - R^2 ~] ~+~ g_1
\frac{2}{R^2 \sin^2(2\theta)} ~,
\label{twoham}
\eeq
where the operator $\Lambda^2$ is the Laplacian on the sphere $S^3$ and is
given by
\beq
- \Lambda^2 ~=~ \frac{\partial^2}{\partial \theta^2}
- \frac{2\sin(2\theta)}{\cos(2\theta)}\frac{\partial}{\partial \theta}
+ \frac{1}{\cos^2(2\theta)} ~\Bigl[ ~\frac{\partial^2}{\partial \phi^2}
+ 2\sin(2\theta) \frac{\partial^2}{\partial \phi\partial
\psi} + \frac{\partial^2}{\partial \psi^2} ~\Bigr] ~.
\eeq
The interaction in the Hamiltonian is independent of the angles $\phi,
\psi$ and depends only on $R,\theta$. The operators $L$ and $Q$
commute with the Hamiltonian since they commute with the noninteracting ($g=0$)
Hamiltonian. We thus label the states with the eigenvalues of these
operators for all $g_1$. Each of these states is four-fold degenerate:
Under parity, $L \rightarrow -L$ and $ Q \rightarrow Q$ and the
Hamiltonian is invariant under parity. Therefore the
states labeled by quantum numbers $(l,q)$ have the same energy as $(-l,q)$.
The Hamiltonian is also invariant under the transformation $\vec
r_1 \rightarrow -\vec r_1$ and $\vec r_2 \rightarrow \vec r_2$. This is a
discrete symmetry peculiar to this system.
Under this transformation $L \rightarrow L$ and $ Q \rightarrow -Q$.
Therefore the states labeled by quantum numbers $(l,q)$ have the same
energy as $(l,-q)$. Combining the two we get the four-fold degeneracy of
the states. Later we will find that the states with $(l,q)$ have the same
energy as $(q,l)$ since interchanging $q$ and $l$ leaves the differential
equation invariant; therefore the energy of these two states must be the same.
We thus have an eight-fold degeneracy for the levels for which $|q|$ and $|l|$
are nonzero and different from each other. Note that this degeneracy is a
subset of the degeneracy of the noninteracting system. If $|l|=|q|$ is
nonzero, we have a four-fold degeneracy. Finally, there is a four-fold
degeneracy between the states $(\pm l,0)$ and $(0, \pm l)$ if $l \ne 0$.
\subsection{Solutions of the eigenvalue equation}
We are now interested in solving the eigenvalue equation given by
$ H\Psi = E\Psi$. Following the remarks made in the previous subsection,
we may in general write
\beq
\Psi ~=~ F(R) ~\Phi(\theta, \phi, \psi) ~.
\eeq
The eigenvalue equation separates into angular and radial
equations. The angular equation is given by
\beq
(~ \Lambda^2 + \frac{4g_1}{\sin^2(2\theta)} ~) ~\Phi ~=~ \beta (\beta + 2) ~
\Phi ~,
\label{angle}
\eeq
where $\beta \ge -1$, and the radial equation is given by
\beq
\frac{d^2F}{dR^2} +\frac{3}{R} \frac{dF}{dR} +(~ 2E - R^2 - \frac{\beta
(\beta +2)}{R^2} ~) ~F ~=~0 ~.
\eeq
The radial equation can be easily solved using the methods outlined in
the last section of \cite{LL}. The solution is given by
\beq
F(R) ~=~ R^{\beta} M(a,b,R^2) \exp{(-R^2/2)} ~,
\eeq
where $b=\beta+2 $ and $a=(\beta+2-E)/2$ and $M(a,b,R^2)$ is the confluent
hypergeometric function. Demanding that $a = -n_r$ where $n_r$ is an integer,
the energy is given by
\beq
E ~=~ 2n_r + \beta +2 ~.
\label{eigenval}
\eeq
Note that $\beta$ is still unknown and has to be obtained by solving the
angular equation. Nevertheless the tower structure of the eigenvalues
built on radial excitation of the ground states is obvious from the above.
The angular equation (\ref{angle}) may be solved with the ansatz
\beq
\Phi (\theta ,\phi ,\psi) ~=~ P(x) ~\exp (iq\phi) ~\exp (il\psi) ~,
\eeq
where $x =\sin(2\theta)$ and $l,q$ are the state labels in terms of the
integer valued eigenvalues of the angular momentum and SUSY operators. The
angular equation then reduces to a differential equation in a
single variable $x$ for the function $P(x)$:
\beq
(1-x^2) \frac{d^2 P}{dx^2}
-2x\frac{dP}{dx}-\frac{1}{4(1-x^2)} ~[~ q^2+2x q l+l^2 ~]P
-\frac{g_1}{x^2}P+\frac{\beta (\beta +2)}{4}P ~=~ 0 ~.
\label{px}
\eeq
Note that the equation has four regular singularities at $x=0, 1,
-1,\infty$ (the singularity at $\infty$ does not play any role since $x$
is bounded). Therefore the solution is of the form \beq
P(x) ~=~ |x|^a (1-x)^b(1+x)^c \Theta^{a,b,c}(x) ~.
\eeq
One can now fix $a,b,c$ to cancel the singularities. We find that
\beq
b ~=~ \frac{|l+q|}{4} \quad {\rm and} \quad c ~=~ \frac{|l-q|}{4} ~.
\eeq
Since $l$ and $q$ are integer valued, the values of $b$ and $c$ are
restricted. The other exponent $a$ is given by
\beq
a(a-1)~=~g_1 ~=~ g(g-1) \quad {\rm with} \quad a ~\ge ~0 ~,
\label{a2g}
\eeq
where we have already defined $g_1$ through Eq. (\ref{g12}) in terms of $g$.
Note that we have used the symbol $a$ instead of $g$. As shown in the
Appendix, we have to take $a=g$ if $g \ge 1/2$. But if $g < 1/2$, we have to
generally consider a linear superposition of solutions with $a$ equal to $g$
and $1-g$ (more on this later).
We finally arrive at the required differential equation from which the
eigenvalues are determined,
\bea
(1-x^2)\frac{d^2\Theta}{d x^2} &+& 2[a/x -(b-c) -(a+b+c+1)x] ~\frac{d
\Theta}{d x} \nonumber \\
&+& ~[~ \frac{(\beta +1)^2}{4} -(a+b+c+1/2)^2 +2a(c-b)/x ~] ~\Theta ~=~0 ~.
\label{heun}
\eea
For $g=0$, the solutions are simply Jacobi polynomials and the full solution
for the angular part is given in terms of the spherical harmonics on a
four-dimensional sphere. In general, this differential equation is known as
the Heun equation whose solutions $\Theta^{a,b,c}(x)$ are characterized by the
so-called $P$-symbols \cite{bateman}. The Heun equation is exactly solvable if
either $l$ or $q$ vanishes, i.e., if $b=c$ as discussed in the next
subsection. The equation is also exactly solvable at an infinite number of
isolated points in the space of parameters $(a,b,c)$. These are isolated
points because if we vary $a$ slightly away from any one of them, the
equation
is not exactly solvable. Note that $b,c$ take discrete values and cannot be
varied continuously.
\subsection{Polynomial solutions}
Let us first consider a class of solutions which are polynomials in $x$. We
may then write
\beq
\Theta(x) ~=~ \sum_{k=0}^p ~C_k x^k ~,
\eeq
where we may define $C_0 = 1$.
Substituting this in the differential equation for $\Theta$, we
see that the $C_k's$ satisfy a three-term recursion relation given by
\bea
(k+2)(k+1+2a) ~C_{k+2} ~&-&~ 2(b-c)(k+1+a) ~C_{k+1} \nonumber \\
&+& ~[~ \frac{(\beta +1)^2}{4} -(a+b+c+k+1/2)^2 ~] ~C_k ~=~ 0 ~,
\label{recursion}
\eea
which is in general difficult to solve.
However there are two special cases when polynomial solutions are possible.
(i) For $b=c$, this reduces to a
two-term recursion relation which can be easily solved to obtain all the
energy levels. This is an example of a Conditionally Exactly
Solvable (CES) problem \cite{D93} in which the full spectrum is exactly
solvable for some special condition (like $b=c$ here). (ii) The other case is
when the coefficient of $C_k$ is zero with $k=p$ (where $p \ge 1$), i.e.
\beq
E ~=~ 2n_r+2a+2b+2c+2p+2 ~,
\label{E}
\eeq
in which case one has a polynomial solution of degree p. This is an example
of a Quasi-Exactly Solvable (QES) problem when only a finite number of states
are exactly solvable for some given values of the parameters. As far as we
are aware, this is the first example where both CES and QES solutions exist in
the same problem. We now discuss both types of solutions in detail.
\noindent
(i) CES-type Solutions: To see the solutions explicitly, define $$y ~=~
x^2 ~.$$ In terms of the variable $y$ the differential equation is written as,
\beq
y(1-y)\frac{d^2\Theta}{d y^2} + 2 ~[~ a' - (b'+ c'
+1)y ~]\frac{d \Theta}{d y} - b'c'\Theta ~=~ 0 ~,
\eeq
where
\bea
a' & ~=~ & a+1/2 ~, \nonumber \\
b' & ~=~ &\frac{1}{2}[a+2b+1+\frac{\beta}{2}] ~, \nonumber \\
c' & ~=~ &\frac{1}{2}[a+2b-\frac{\beta}{2}] ~.
\eea
This is now a hypergeometric equation whose solutions are given by
$F(b',c',a'; y)$. Note that we need to concentrate only on the solutions for
$0\le y \le 1$. The hypergeometric series terminates
whenever $b'$ or $c'$ is a negative integer or zero. In our case $b'$ is
always positive hence the solutions are given when $c' =-m$, where $m$ is
an integer. Therefore $\beta =~ 2m+2a+4b$. The energy eigenvalues are obtained
by substituting this in Eq. (\ref{eigenval}),
\beq
E ~=~ 2n_r+2a+4b+2m+2 ~.
\eeq
Since $b=c$, we must have either $l=0$ or $q=0$. The energy varies linearly
with $a$ as in the case of the exact solutions of the many-body problem. The
expression for $a$ in terms of $g$ will be clarified in the next subsection.
It might be tempting to conclude that all polynomial solutions vary linearly
with $g$. In fact that is not so as can be seen from the following examples.
\noindent
(ii) QES-type Solutions: We can have polynomial solutions of degree $p$
(where $p \ge 1$) if $b$, $c$
and $g$ satisfy some specific relations. Let us consider the case
\beq
\Theta(x) ~=~ 1+\delta x ~.
\eeq
This is a solution if
\beq
a ~=~ \frac{b+c}{(b-c)^2-1} -1
\label{constraint1}
\eeq
and $\delta = b-c$. We have implicitly assumed that $b$ is not equal to
$c$ since otherwise this solution is trivial and of CES type. (Once again,
$a$ can be equal to either $g$ or
$1-g$ as discussed in the previous subsection). Then $\beta = 2a+2b+2c+2$,
and the full spectrum is given by
\beq
E ~=~ 2n_r+2a+2b+2c+4 ~.
\eeq
We should point out that a solution of this kind is only possible for fairly
large values of $l$ and $q$; the minimum values needed are $|l|=|q|=3$, in
which case $a=1/5$.
Similarly, there is a polynomial solution of the form
\beq
\Theta (x) = 1 + \delta x + \epsilon x^2 ~,
\eeq
if
\bea
a ~&=&~ 2 \alpha - \frac{3}{2} + {\sqrt {4 \alpha^2 - \alpha + \frac{1}{4}}} ~,
\nonumber \\
{\rm where} \quad \alpha ~&=&~ \frac{b+c}{(b-c)^2 -4} ~.
\label{constraint2}
\eea
The spectrum is given by
\beq
E ~=~ 2n_r+2a+2b+2c+6 ~.
\eeq
These expressions for the energies are nonlinear in $a$ because of the
constraints (\ref{constraint1}) or (\ref{constraint2}). We should however
caution that these are isolated solutions since $b$ and $c$ can only take
discrete values; hence the above solutions do not vary smoothly with $a$.
In general, solutions similar to the above may be constructed for every degree
$p$ of the polynomial; the corresponding energies are given by Eq. (\ref{E})
where $a$ is given by a function of $b$ and $c$ which can be derived by
solving $p+1$ recursion relations obtained from Eqs. (\ref{recursion}) by
setting $k=-1,0,1,...,p-1$.
\subsection{Numerical analysis}
We now consider the numerical solution of some low-lying states of the
two-body problem since the polynomial solutions described in the previous
subsection do not exhaust the full spectrum.
The noninteracting limit of the system is $g=0$ where we
have the solutions corresponding to a four-dimensional oscillator. These are
simply the spherical harmonics on a four-sphere $Y_{k,l,q}$, where
$k=0,1,2,\cdots$ and $|l|,|q| \leq k$ label the states. When the
degeneracy of these states is taken into account, all the states in the
noninteracting limit $g=0$ are completely specified. We now demand that the
wave functions and energy levels should vary continuously with the parameter
$g$ which is the interaction strength. We also require that the wave functions
should not diverge at any value of $x$ in the interval $[-1,1]$.
For given values of $b \ne c$, we can numerically find the energy levels
in two different ways. We can diagonalize the differential operator in
(\ref{px}) in the basis of the noninteracting ($g=0$) states, or we can solve
the differential equation (\ref{px}) or (\ref{heun}) directly for each
state. We have used both methods and will present the results below.
In order to proceed further, it is necessary
to clarify the dependence of $a$ on $g$ in Eq. (\ref{a2g}).
By the arguments given in the Appendix, $a=g$ if $g \ge 1/2$. If $g < 1/2$,
$a$ can be equal to either $g$ or $1-g$; for any given state, we choose the
value which is continuously connected to the noninteracting solution at $g=0$.
Namely, any solution of the above kind must, at $g=0$, go either as $1$ or
as $x$ near $x=0$; we then choose $a=g$ or $a=1-g$ in the two cases
respectively for $0 < g < 1/2$. In particular, an exact solution which,
near $x=0$, goes as $1$ at $g=0$ will go as $|x|^g$ for all $g > 0$. An exact
solution which goes as $x$ at $g=0$ will go as $|x|^{1-g}$ upto $g=1/2$ and
then as $|x|^g$ for $g > 1/2$. As a result solutions of this form are
discontinuous at $g=1/2$. In general, when solving numerically for the
non-exact solutions, we have to allow a superposition of both $|x|^g$ and
$|x|^{1-g}$. For $g \ge 1/2$, however, $a$ must be equal to $g$.
When solving the differential equations (\ref{px}) or
(\ref{heun}), we have to consider the two
regions separately. According to the rules discussed in the Appendix, for
$0 \le g < 1/2$, we take the function $P(x)$ to go as
\beq
P(x) ~=~ |x|^g ~+~ d ~{\rm sgn} (x) ~|x|^{1-g} ~,
\label{pxg}
\eeq
near $x=0$, and we vary both the coefficient $d$ as well as $\beta$ in Eq.
(\ref{px}) till we find that the solution of (\ref{px}) does not diverge
at $x = \pm 1$. Both $d$ and $\beta$ depend on $g$; it is possible to
determine the limiting values $d(0)$ and $d(1/2)$ as follows. At $g=0$,
suppose that the Jacobi polynomial is normalized so that $P=1$ and $P^{\prime}
=C_J$ at $x=0$. On the other hand, from Eq. (\ref{recursion}), we see that
$C_1 = b-c$ for any nonzero $g$, no matter how small. By taking the limit
$g \rightarrow 0$ in Eq. (\ref{pxg}), we therefore find that $d(0)=C_J -b+c$.
At the other end, $d(1/2)=\pm 1$ because, as we will show next, $P(x)$ must
vanish for either $x \ge 0$ or for $x \le 0$ for all $g \ge 1/2$ (except for
the exact polynomial solutions discussed previously).
For $g \ge 1/2$, we must take the solution of the differential equation
to go either as (a) $P(x) = 0$ for $x \le 0$ and $\sim x^g$
for $x$ small and positive, or as (b) $P(x) = 0$ for $x \ge 0$ and $\sim
(-x)^g$ for $x$ small and negative. In either case, we vary the energy till we
find that the solution of (\ref{heun}) (with $a=g$) does not diverge at
$x=1$ and $-1$ for (a) and (b) respectively. It is interesting to note that for
each such solution, $P(x)$ vanishes identically in one of the half-intervals
$[-1,0]$ or $[0,1]$. Note also that if $\Theta^{a,b,c} (x)$ is a solution
which is only nonzero for $x \ge 0$, then $\Theta^{a,c,b} (x)= \Theta^{a,b,c}
(-x)$ will be a solution which is only nonzero for $x \le 0$; further, the two
solutions will have the same energy.
In contrast to the above method for solving the Heun equation (\ref{heun})
directly, the numerical diagonalization procedure for finding the eigenvalues
involves solving the eigenvalue equation (\ref{angle}). The basis for
diagonalization is provided by the eigenstates of the Laplacian
$\Lambda^2$ on
$S^3$, namely the spherical harmonics on a four-sphere $Y_{k,l,q}$. For
non-zero interaction strengths, the singular interaction is handled by
multiplying the non-interacting eigenstates by $|x|^g$. The resulting basis is
nonorthogonal, and the diagonalization procedure is fairly straightforward
though cumbersome. We have truncated the basis such that the highest
energy state in the basis is $100\hbar\omega$. The results for the low
lying states in the spectrum obtained through both
methods are displayed in Fig. 1 (for $0\leq g \leq 1$)and Fig. 2 (for $g\geq
1$). For $g < 1/2$, it is more convenient to use the diagonalization procedure
since the direct solution of Eq. (\ref{px}) requires one to numerically fix
two separate parameters $d$ and $\beta$. On the other hand, it is easier to
solve the Heun equation (\ref{heun}) for $g \ge 1/2$ since one has to fix only
one parameter $\beta$. In general we have used both methods to arrive at the
spectrum of low-lying states. For small values of g ($<1$), however, the
solutions
of the differential equation produce eigenvalues which are somewhat smaller
than the ones obtained by the diagonalization procedure. For large values of
$g$, however, there is no perceptible difference between the results from the
two methods.
Fig. 1 shows the energies for some values of $l$ and $q$. Each level is
labeled by $(l,q;D)$, where $D$ is the degeneracy of the level away from
the noninteracting limit; the degeneracy is computed by counting the allowed
values of $\pm l$ and $\pm q$ using the parity and supersymmetry
transformations for a given level. A subscript 'r' on the label $(l,q,D)$
denotes the radial excitation which is simply inferred from the existence of
the towers. The bosonic ground state has
the predicted behavior for all $g$; it is linear with a slope $2$ as a
function of $g$. The corresponding wave function goes as $|x|^g$ as $x
\rightarrow 0$. In contrast, the level $(0,0;1)$ starting at $E=4$ has
an entirely different behavior . It is exactly
solvable for all $g$. According to the previous discussion, $dE/dg =-2$ for
$g<1/2$ (with the wave function going as $|x|^{1-g}$ for small $x$) and
$dE/dg =2 $ for $g>1/2$ (with the wave function going as $|x|^g$); thus
$dE/dg$ is discontinuous at $g=1/2$.
At $g=0$, the slopes $dE/dg$ for all the levels can be calculated using
first-order perturbation theory as shown in the next subsection. For {\it
large} values $g$, we find that all the energies converge to $2g$ plus even
integers as shown in Fig. 2. This amazing behavior can be understood using
the WKB method as shown in subsection E.
\subsection{Perturbation theory around $g=0$}
It is interesting to use perturbation theory to calculate the changes in
energy from $g=0$, and to compare the results with the numerical analysis. We
will only describe first order perturbation theory here, and the example we
will consider is the fermionic ground state which is doubly degenerate for
$N=2$.
In general, naive perturbation theory fails at $g=0$ because most $g=0$
eigenstates do not vanish as $X_{ij} \rightarrow 0$; hence the expectation
value of $1/X_{ij}^2$ diverges. This problem can be tackled by using a special
kind of perturbation theory first devised for anyons \cite{pert}. We will
first describe the idea for $N$ particles and then specialize to $N=2$.
Instead of solving the equation
$H \Psi = E \Psi$, where H is given in Eq. (\ref{ham2}), we perform a
similarity transformation to ${\tilde H} = X_N^{-g} H X_N^g$ and ${\tilde \Psi}
= X_N^{-g} \Psi$, where $$X_N \equiv \prod_{i<j} |X_{ij} |^g ~.$$ We then
find that ${\tilde H} = H_0 + {\tilde V}$, where
$H_0$ is the noninteracting Hamiltonian (with $g_1=g_2=0$), and
\beq
{\tilde V} ~=~ g \sum_{i \ne j} ~\frac{1}{X_{ij}} ~(x_j
\frac{\partial}{\partial y_i} - y_j \frac{\partial}{\partial x_i}) ~.
\eeq
The first order changes in energy may now be obtained by calculating the
expectation values (or matrix elements, in the case of degenerate states) of
$\tilde V$ in the zeroth order (noninteracting) eigenstates. These expectation
values can be shown to be convergent for all states. Note that $\tilde V$ only
contains two-body terms. Although $\tilde V$ is not hermitian, it is guaranteed
that its expectation values are real because the original problem has a
hermitian Hamiltonian $H$.
For $N=2$, we find that
\beq
{\tilde V} ~=~ -2g ~(~ \frac{1}{R} \frac{\partial}{\partial R} + \frac{\cot 2
\theta}{R^2} \frac {\partial}{\partial \theta} ~)
\label{pertv}
\eeq
in hyperspherical coordinates.
Note that $\tilde V$ commutes with both $L$ and $Q$, so that we only have to
consider its matrix elements within a particular block labeled by the
eigenvalues $l$ and $q$. Let us now use (\ref{pertv}) to compute the first
order change in the states which have $E=3$ at $g=0$. There are four such
states, with $l=\pm 1$ and $q= \pm 1$ (labeled $(1,1;4)$ in Fig. 1); two of
these states are actually the
ground states of the two-fermion system. Due to parity and SUSY, these four
states remain degenerate for all $g$. Hence it is sufficient to calculate the
first order change in the state with, say, $(l,q)=(1,1)$. Since this state is
unique at $g=0$, we only need to do non-degenerate perturbation theory with
${\tilde V}$. The normalized wave function for this state is
\beq
\Psi ~=~ \frac{1}{\pi {\sqrt 2}} ~(\cos \theta - \sin \theta) ~\exp [i(\phi +
\psi)] ~R \exp [-R^2 /2] ~.
\eeq
We now obtain the expectation value
\beq
\int_0^{\infty} R^3 ~dR ~\int_{-\pi/4}^{\pi/4} \cos (2\theta) ~d\theta ~
\int_{-\pi/2}^{\pi/2} d\phi ~\int_{-\pi}^{\pi} d\psi ~\Psi^* ~{\tilde V}
\Psi ~=~ g ~.
\eeq
We can see from Fig. 1 that this gives the correct first order expression
for the energy $E=3+g$ near $g=0$ for the states labeled $(1,1;4)$; their
first radial excitations $(1,1;4)_r$ therefore have $E=5+g$.
We can similarly calculate the first order expressions for the energies
near $g=0$ for
all the other levels shown in Fig. 1. We find that $E=2+2g$ for the lower state
labeled $(0,0;1)$ (i.e. the bosonic ground state); $E=4+2g$ for its
radial excitation $(0,0;1)_r$, and the states $(2,0;4)$; $E=4-2g$ for the
upper state $(0,0;1)$; $E=4$ for the states $(2,2;4)$; $E=5-g/2$ for the
states $(1,1;4)$ and $(3,3;4)$; and $E=5+3g/2$ for the states $(3,1;8)$.
\subsection{Large-$g$ perturbation theory}
We can study the solutions of Eq. (\ref{heun}) for large values of $g$ by using
an expansion in $1/g$. For any value of $b$ and $c$, we will
only study the lowest energy $E$, and we will calculate the leading order
terms in $E$ and the wave function $\Theta (x)$. We first note that the terms
of order $g^2$ in (\ref{heun}) can be satisfied only if $E=2g +O(1)$. Next, we
assume that $E$ and $\Theta$ have WKB expansions \cite{wkb} of the form
\bea
E ~&=&~ 2g ~+~ 2b ~+~ 2c ~+~ 2 ~+~ f_0 ~+~ \frac{f_1}{g} ~+~ O(1/g^2) ~,
\nonumber \\
\Theta ~&=&~ \exp [ ~w_0 (x) ~+~ \frac{w_1(x)}{g} ~] ~.
\eea
The boundary condition $\Theta =1$ implies that $w_0 (0) = w_1 (0) =0$.
To order $g$, Eq. (\ref{heun}) gives the first order differential equation
\beq
(1 - x^2) ~\frac{d w_0}{dx} ~=~ b-c-\frac{f_0}{2} x ~.
\eeq
We now look at solutions which are
nonzero only for $x \ge 0$. We demand that $\Theta$
should neither diverge nor vanish (since the lowest energy solution should
be node less) anywhere in the range $0 \le x \le 1$. Hence the functions $w_0$
and $w_1$ should not diverge to $\infty$ or $-\infty$ in that range. This
fixes $f_0 =2(b-c)$, so that
\bea
E ~&=&~ 2g ~+~ 4b ~+~ 2 ~=~ 2g ~+~ |l+q| ~+~ 2 ~, \nonumber \\
{\rm and} \quad \Theta ~&=&~ (1 ~+~ x)^{b-c} ~.
\label{wkb1}
\eea
Similarly, there are solutions which are nonzero only for $x \le 0$. These have
\bea
E ~&=&~ 2g ~+~ 4c ~+~ 2 ~=~ 2g ~+~ |l-q| ~+~ 2 ~, \nonumber \\
{\rm and} \quad \Theta ~&=&~ (1 ~-~ x)^{c-b} ~.
\label{wkb2}
\eea
We now go to order $1$ in Eq. (\ref{heun}). For solutions which are nonzero
only for $x \ge 0$, we find that
\bea
E ~&=&~ 2g ~+~ 4b ~+~ 2 ~+~ \frac{c^2 - b^2}{g} ~=~ 2g ~+~ |l+q| ~+~ 2 ~-~
\frac{lq}{4g} ~, \nonumber \\
{\rm and} \quad \Theta ~&=&~ (1 ~+~ x)^{b-c} ~\exp [~ \frac{(b-c)(b+c+2)}{2g} ~
( \ln (1+x) ~-~ \frac{x}{1+x} ) ~] ~.
\label{wkb3}
\eea
We can similarly find solutions which are nonzero only for $x \le 0$, by
changing $x \rightarrow -x$ and interchanging $b \leftrightarrow c$, i.e.
$l+q \rightarrow l-q$ and $lq \rightarrow -lq$, in Eq. (\ref{wkb3}).
We see from Fig. 2 that these formulae correctly describe the leading
behavior of $E$. In fact, the large-$g$ behavior is already visible in Fig.
1, for some states, as we approach $g=1$. The various levels shown in that
figure have the
following WKB energies; $E=2g+2$ for both the $(0,0;1)$ states (one of these
is the bosonic ground state and the other is the fermionic ground state for
$g>1/2$ as discussed later); $E=2g+2+1/4g$ for the states $(1,1;4)$;
$E=2g+2+1/g$ for the states
$(2,2;4)$; $E=2g+2+9/4g$ for the states $(3,3;4)$; $E=2g+4-1/4g$ for the
states $(1,1;4)$; $E=2g+4$ for the radial excitation $(0,0;1)_r$ and the
states $(2,0;4)$; $E=2g+4+1/4g$ for the radial excitations $(1,1;4)_r$; and $E=
2g+4+3/4g$ for the states $(3,1;8)$. We have also checked that the leading
order wave functions in Eqs. (\ref{wkb3}) agree remarkably well with the
correct wave functions $\Theta (x)$ obtained by solving the Heun equation
(\ref{heun}) even if $g$ is not very large.
It is easy to see from Eqs. (\ref{wkb1}-\ref{wkb2}) that for large $g$, the
ground state and also the excited states become infinitely degenerate. This is
so because one can choose the quantum numbers $l$ and $q$ in infinitely many
ways such that the energies are the same as $g \rightarrow \infty$. Further,
the spacings now become twice the spacing at $g=0$ since $l$ and $q$ have the
same parity mod $2$.
The large-$g$ behavior therefore displays a remarkable similarity to the
problem of a particle in an uniform magnetic field where the Landau level
spacing is twice the cyclotron frequency, and each level is infinitely
degenerate.
\subsection{Fermionic Ground State Energy}
The fermionic ground state energy has a very unusual behavior as can be seen
from Fig. 1. For $0 < g < 0.367$, the ground state energy
monotonically and nonlinearly increases from $3$ to $3.266$ along
the curve labeled $(1,1;4)$. Beyond this point, for $0.367 < g <0.5$,
the ground state energy monotonically and linearly decreases from $3.266$ to
$3$ along the upper curve $(0,0;1)$ satisfying $E=4-2g$. For $g \ge 1/2$, the
fermionic and bosonic ground state energies are identical and are given by the
curve $(0,0;1)$ which satisfies $E_0 = 2 + 2g$, i.e., both the ground states
monotonically increase with $g$. Thus the fermionic
ground state consists of three pieces as a function of $g$, while the bosonic
ground state is given by the single line $E_0 = 2 + 2g$ for all $g \ge 0$.
For two particles, one can understand why the fermionic and bosonic ground
state energies are identical for $g > 1/2$ as follows. In this range of $g$,
the ultraviolet potential near $x=0$ is infinite and it prevents tunneling
between the regions $x>0$ and $x<0$ (see the Appendix). For two identical
particles, an exchange necessarily takes us from a region with $x>0$ to
a region with $x<0$ according to (\ref{exchange}). If tunneling between the
two regions is forbidden, it becomes impossible to compare the phases of the
wave function of a given configuration of the two particles and the wave
function of the exchanged configuration. Thus it is impossible to distinguish
bosons from fermions if $g>1/2$, and their energy levels must be identical.
It is possible that the same argument will go through for more than two
particles; however we need to understand the ultraviolet regularization of the
three-body interactions properly in order to prove that rigorously. If the
argument holds, then we would have the interesting result that the
$N$-fermion ground state energy is also given by (\ref{egs}) for $g>1/2$,
while it may show one or more level crossings for $g<1/2$.
\section{Discussion and Summary}
To summarize, we have studied a two-dimensional Hamiltonian whose eigenstates
have a novel two-particle correlation. We have shown the existence of several
classes of exact solutions in the many-body problem. We have analyzed the
two-particle problem in detail and shown that it is completely solvable by
reducing it to an ordinary differential equation in one variable which can be
solved exactly for a subset of states and numerically otherwise. The
two-body problem is integrable since there are four
constants of motion in involution. We have also discussed perturbation theory
for both small and large coupling strengths. In the strong interaction limit,
the system simplifies and bears a remarkable resemblance to the Landau level
structure.
We have also clarified in the Appendix the ultraviolet prescription
which is required to make sense of an inverse-square (singular) potential
especially at small coupling strengths. In
particular, we emphasize that it is in general not sufficient to specify that
the wave functions are regular and square integrable to obtain an energy
spectrum uniquely when dealing with singular interactions. In some domains of
the coupling strength, we also need to specify the ultraviolet regularization
to make complete sense of the results. We do this by demanding that as the
parameter $g \rightarrow 0$, the energy levels should smoothly approach the
known noninteracting levels. We believe
that this discussion is quite general and may have a wider applicability
to Hamiltonians with singular interactions.
Interesting problems for the future would be to extend this analysis to more
than two particles, and to find an application of our model to some physical
system. Recently, we have come to know that our model has been generalized to
three (and higher) dimensions with novel three-body
(and many-body) correlations \cite{G96}.
Three of us (RKB, JL and MVNM) would like to acknowledge financial support
from NSERC (Canada). RKB would like to acknowledge the hospitality at the
Institute of Physics, Bhubaneswar where part of this work was done.
MVNM thanks the Department of Physics and Astronomy, McMaster University
for hospitality.
\vskip 1 true cm
\centerline{\bf Appendix}
We begin directly from Eq. (\ref{px}). Given the real number $g \ge 0$
satisfying
$g_1 = g(g-1)$, $P(x)$ could go, as $|x| \rightarrow 0$, as either $|x|^g$
or $|x|^{1-g}$ or even as a general superposition of the two powers.
We therefore need to define the problem more carefully in order to pick out
a desired solution \cite{S58}.
As mentioned above in the text, we demand the following. Firstly, the limit
$g=0$ should give all the noninteracting two-particle solutions, both bosonic
and fermionic. Secondly, all the wave functions and energies $E$ should be
{\it continuous} functions of $g$, but the
first derivative of $E$ need not be continuous (indeed $dE/dg$ is not always
continuous at $g=1/2$ as we saw earlier). Finally, for $g > 1$, the wave
function should go as $|x|^g$, and not as $|x|^{1-g}$ which diverges at $x=0$.
>From these three requirements, it is clear that for $g \ge 1/2$, the wave
functions must go purely as $|x|^g$, whereas for $g < 1/2$, the wave function
could go either as $|x|^g$ or $|x|^{1-g}$ or a superposition of the two.
We will now show that we can satisfy the above requirements if we redefine the
problem with a different potential
in an {\it ultraviolet} region $|x| < x_o$. We take the potential to be
\bea
V(x) ~&=&~ \frac{g(g-1)}{x^2} \quad {\rm for} \quad |x| > x_o ~,
\nonumber \\
&=&~ \frac{u^2}{x_o^2} \quad {\rm for} \quad |x| < x_o ~, \nonumber \\
{\rm where} \quad u ~\tanh u ~&=&~ g ~\quad {\rm if} \quad 0 \le
g < 1/2~, \nonumber \\
{\rm and} \quad u ~&=&~ \infty ~\quad {\rm if} \quad g \ge 1/2 ~.
\label{vx}
\eea
Eventually, of course, we have to take the limit $x_o \rightarrow 0$ to recover
our original problem. Note that the potential in the ultraviolet region is
not symmetric under $g \rightarrow 1-g$ for $g \le 1$. Hence the energy
spectrum does not have this symmetry.
To see why Eqs. (\ref{vx}) work, we note that the wave function, for $|x|$
slightly greater than $x_o$ (where $x_o$ is much smaller than any physical
length scales like the width of the harmonic oscillator potential), is
generally given by
\bea
P(x) ~&=&~ x^g ~+~ d_+ ~x^{1-g} \quad {\rm if} \quad x > x_o ~, \nonumber \\
{\rm and} \quad P(x) ~&=&~ (-x)^g ~+~ d_- ~(-x)^{1-g} \quad {\rm if} \quad
x < - x_o ~.
\eea
(For the exceptional case $g=1/2$, we have to replace $|x|^g$ and $|x|^{1-g}$
by $|x|^{1/2}$ and $|x|^{1/2} \ln |x|$ respectively).
Now consider the first case in (\ref{vx}), i.e., $0 \le g < 1/2$. Since the
energy $E$ is much less than the potential in the inside region $|x| < x_o$
(this is necessarily true for any finite value of $E$ as $x_o \rightarrow 0$),
the wave function in that region is given by
\beq
P(x) ~\simeq ~ \cosh ~[~ (~{u \over x_o} + O(x_o) ~)~ x ~+~ \delta ~]~,
\eeq
where $\delta$ can be a complex number, and the term of $O(x_o)$ arises from
the energy $E$ which is much less than $(u /x_o)^2$. We now match the
wave function and its first derivative, or, more simply, the ratio $P^{\prime}
(x)/P(x)$ at $x= x_o \pm \epsilon$ and at $x = -x_o \pm \epsilon$, where
$\epsilon$ is an infinitesimal number. We then find three possibilities.
\noindent
(i) The wave function may be even about $x=0$. Then $\delta =0$, and $d_+ =
d_-$ must vanish as $x_o^{1+2g}$ as $x_o \rightarrow 0$. (The behavior of
$d_{\pm}$ can be deduced by equating the terms of $O(x_o^{-1})$ and $O(x_o)$
in $P^{\prime} /P$ at $x=x_o \pm$). In the limit $x_o \rightarrow 0$, therefore,
the wave function goes purely as $|x|^g$.
\noindent
(ii) The wave function may be odd about $x=0$. Then $\delta = i
\pi /2$, and $d_+ = d_-$ must
diverge as $x_o^{2g-1}$ as $x_o \rightarrow 0$. The wave function is
proportional to ${\rm sgn} (x) ~|x|^{1-g}$ in that limit.
\noindent
(iii) In the general asymmetric case, we find that we must have $\delta$ of
order $x_o^{1-2g}$, and $d_+ = - d_- = d$ of O(1). (This is found by equating
terms of $O(x_o^{-1})$ and $O(x_o^{-2g})$ in $P^{\prime} /P$ at $x=\pm x_o$).
The wave function is therefore a superposition of the form
\beq
P(x) ~=~ |x|^g ~+~ d ~ {\rm sgn} (x) ~|x|^{1-g} ~.
\eeq
The cases (i) and (ii) arise if either $l$ or $q$ is zero in Eq. (\ref{px}),
since the equation is invariant under $x \rightarrow -x$ in that case. This
is precisely when $b=c$ and the equation is exactly solvable. We thus see
that the even solutions go as $|x|^g$, while the odd solutions go as
$|x|^{1-g}$. If neither $l$ nor $q$ is zero, i.e. $b \ne c$, we have case
(iii) where a superposition of the two powers are required.
The second case in (\ref{vx}), i.e. $g \ge 1/2$, is relatively simpler to
analyze since the wave function must be zero in the inside region $|x| \le
x_o$. On imposing this condition on the wave function in the outside region, we
see that both $d_+$ and $d_-$ must vanish as $x_o \rightarrow 0$. Hence the
wave function will go purely as $|x|^g$ in that limit. However, since there
is no tunneling possible through the infinite barrier separating $x > x_o$
from $x < - x_o$, we will generally have wave functions which are nonzero
only for $x > x_o$ or only for $x < -x_o$. This is indeed true as we saw
earlier for the solutions of the Heun equation for $g > 1/2$.
We would like to emphasize that the relation $u \tanh u = g$ in
Eq. (\ref{vx}) is absolutely essential in order to have the possibility of
$P(x) \sim |x|^g$ for $g < 1/2$. If $u$ were to take any other value,
we would find that $P(x)$ necessarily goes as $|x|^{1-g}$ in the limit $x_o
\rightarrow 0$. A similar fine tuning of $u$ is necessary in the CSM for $g <
1/2$. Incidentally, the strongly repulsive potential in the
ultraviolet region explains the peculiar result that the bosonic ground state
energy increases monotonically with $g$ even though the potential away from
the ultraviolet region becomes more and more attractive as $g$ goes from $0$
to $1/2$. One can show from Eq. (\ref{vx}) that the integrated potential
$\int_{-1}^11 dx V(x)$ is actually {\it positive} and large if $x_o$ is
small, and it increases as $g$ varies from $0$ to $1/2$.
Several comments are in order at this stage.
\noindent
(i) A similar fine tuning is also required in the CSM if $g <1/2$ is to be
allowed as has been done by several people \cite{MS94}.
Historically, both Calogero \cite{C69} and Sutherland \cite{S71} restricted
themselves to $g>1/2$. In a sense they could do that since the free fermion
limit corresponds to $g = 1$; thereby they avoided the problem with $g<1/2$.
However, one cannot reach the free bosonic limit smoothly in that case.
Both these authors believed $g<1/2$ to be unphysical because they chose a
particular regularization. What we have argued here
is that one can choose an alternative regularization (called the resonance
condition by Sutherland) which allows one to go continuously all the way upto
$g=0$ and hence reach the free bosonic limit continuously.
We have not seen this being clearly stated in
the CSM literature before, although Scarf \cite{S58} discusses this issue in
a different problem containing the inverse-square potential.
It is worth noting that CSM has only two-body interactions. Henca the entire
discussion here is also valid in the many-body case in CSM.
This is in
contrast to our problem where, for $N>2$, one also has to analyze the
ultraviolet regularization of the three-body interactions.
\noindent
(ii) One important consequence of our regularization is that many of the
states have discontinuities in $dE/dg$ at $g=1/2$; the fermionic ground state
also has a level crossing at $g=0.367$. Further, for each value
of $g_1 <0$, there are two possible ground states since the ultraviolet
regularization depends on $g$ and not on $g_1$.
\noindent
(iii) Put differently, our Hamiltonian $H$ has several self-adjoint extensions
(SAE) for each value of $g_1$. What we have done is to choose a particular
SAE for $g_1>0$ and
two different SAE for $g_1<0$. As a result, we have found that for every
value of $g_1$ in the range $-1/4<g_1<0$, there are two possible ground state
energies since, as seen above, the SAE depends on $g$ rather than on $g_1$.
Actually, there is an even more general SAE possible for any value of
$g_1$ where another real parameter (besides $g$) has to be introduced; however
we shall not discuss that here.
|
proofpile-arXiv_065-610
|
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\section{INTRODUCTION}
\subsection{Statement of result} In this note we prove the
following:
\begin{th}\label{main}
Let $\Gamma\subset PSL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})$ be a congruence subgroup, and $X_\Gamma$ the
corresponding modular curve. Let $D_\Gamma = [PSL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}):\Gamma]$ and let
$d_\bfc(X_\Gamma)$ be the $\bfc$-gonality of
$X_\Gamma$. Then $${7\over 800} D_\Gamma \leq
d_\bfc(X_\Gamma).$$
For $\Gamma = \Gamma_0(N)$ we have that $d_\bfc(X_{\Gamma_0(N)})$ is bounded
below by
${7\over {800}}\cdot N$.
Similarly, we
obtain a quadratic lower bound in $N$ for $d_\bfc(X_{\Gamma_1(N)})$.
\end{th}
\subsection{Remarks}
The proof, which was included in the author's thesis \cite{thesis}, follows
closely a suggestion of N. Elkies. In the exposition here
many details were added to the argument in \cite{thesis}.
We utilize the work \cite{liyau} of P. Li
and S. T. Yau
on conformal volumes, as well as the known bound on the leading nontrivial
eigenvalue of the non-euclidean Laplacian $\lambda_1\geq {{21}\over {100}}$
\cite{lrs}. If Selberg's eigenvalue
conjecture is true, the constant $7/800$ above may be replaced by $1/96$.
Since, by the Gauss - Bonnet formula, the genus $g(X_\Gamma)$ is bounded by
$D_\Gamma/12+1$ (indeed the difference is
$o(D_\Gamma)$), we may rewrite the inequality above in the
slightly weaker form $${{21}\over {200} } (g(X_\Gamma)-1) \leq
d_\bfc(X_\Gamma).$$
For an analogous result about Shimura curves, see theorem \ref{shimura} below.
It should be noted (as was pointed out by P. Sarnak) that the gonality
has an {\em upper} bound of the same
type. For the $\bfc$-gonality, by Brill-Noether theory \cite{kl} we
have $d_\bfc(X_\Gamma) \leq 1+\left[{{g+1}\over 2}\right]$. If, instead,
one is interested in the gonality over the field of definition of $X_\Ga$, one
can use the canonical linear series to obtain the upper bound $2g-2$ if $g>1$,
and in the few cases where $g=1$ one can use the morphism to $X(1)$ and get the
upper bound
$D_\Gamma$.
\subsection{Acknowledgements} As mentioned above, I am indebted to Noam Elkies
for the main idea. The question was first brought to my attention in a letter
by S. Kamienny. The result first appeared in my thesis under the supervision of
Prof. J. Harris. Thanks are due to David Rohrlich and Glenn Stevens who set me
straight on some details, and to Peter Sarnak for helpful suggestions.
\section{Setup and proof}
\subsection{Gonality}\label{gonal} Let $C$ be a smooth, projective, absolutely
irreducible
algebraic curve over a field $K$. Define the
$K$-{\bf gonality} $d_K(C)$ of $C$ to be the minimum degree of a finite
$K$-morphism $f:C\rightarrow} \newcommand{\dar}{\downarrow \bfp^1_K$. Clearly if $K\subset L$ then
$d_K(C)\geq d_L(C\times_KL)$, and equality must hold whenever $K$ is
algebraically closed.
\subsection{Congruence subgroups and modular curves}
By a {\bf congruence subgroup} $\Gamma\subset PSL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})$ we mean that for
some $N$, $\Gamma$
contains the principal congruence subgroup $\Ga(N)$ of $2\times 2$ integer
matrices congruent to the identity modulo $N$.
Since $PSL_2({\Bbb{R}}} \newcommand{\bfh}{{\Bbb{H}})$ acts on $\bfh= \{z = x+iy|y>0\}$ via fractional linear
transformations, we may let $Y_\Gamma = \Gamma\setminus \bfh$. It is well known that $Y_\Gamma$
may
be compactified by adding finitely many points, called {\bf cusps}, to obtain a
compact Riemann surface $X_\Ga$, which we call the {\bf modular curve}
corresponding to $\Ga$.
\subsection{The Poincar\'e metric}
The upper half plane $\bfh$ carries the Poincar\'e metric $ds^2 = {{dx^2 +
dy^2}\over {y^2}}$, which is $PSL_2({\Bbb{R}}} \newcommand{\bfh}{{\Bbb{H}})$ - invariant. The
corresponding area form is given by ${{dx \, dy} \over {y^2}}$.
Away from a finite set $T$
consisting the cusps and possibly some elliptic fixed points, the metric
descends to a Riemannian metric on $X_\Ga {\,\,^{_\setminus}\,\,} T$, of finite
area.
We denote the area measure by $d\mu$.
We will accordingly call a quadratic differential $ds^2$ a {\bf singular
metric} if it is a Riemannian metric away from finitely many points, and has
finite area. Thus the Poincar\'e metric gives rise to a singular metric on
$X_\Ga$.
\subsection{The Laplacian} It is natural to consider the Hilbert space
$L_2(\Gamma\setminus \bfh)= L_2(X_\Gamma)$, where the $L_2$ pairing is taken with respect to the
Poincar\'e metric. The Laplace-Beltrami operator associated with the metric
$$ \Delta = -y^2({{\partial^2}\over {\partial x^2}} +{{\partial^2}\over
{\partial y^2}})$$ gives rise to a self adjoint unbounded operator on $
L_2(X_\Gamma)$, which is in fact positive semidefinite.
The kernel of $\Delta$ consists of the constant functions. In contrast with the
case of a genuine Riemannian metric on a compact manifold, the spectrum of
$\Delta$ is not discrete (see e.g. \cite{hejhal}, VI\S 9, VII\S 2, VIII\S 5).
The continuous spectrum is $\{\lambda\geq 1/4\} \subset {\Bbb{R}}} \newcommand{\bfh}{{\Bbb{H}}$, and is fully
accounted for by an integral formula involving Eisenstein series $E(z,s)$ for
$Re(s) = 1/2$. The discrete part of the spectrum is given by $\lambda_0=0$
corresponding to the constants, and $0< \lambda_1 < \lambda_2...$ corresponding
to the so called {\bf cuspidal} eigenvectors.
\subsection{Selberg's conjecture} The question, what is $\lambda_1$ turns out
to
be a fundamental one. Selberg \cite{selberg} has shown that $\lambda_1\geq
3/16$ and conjectured that $\lambda_1\geq 1/4$. Recently, Luo, Rudnick and
Sarnak \cite{lrs} showed that $\lambda_1\geq 0.21$ (note that $3/16 < 0.21 <
1/4$).
Since the continuous spectrum is known to be $\lambda\geq 1/4$, denote by
$\lambda_1' = \min(\lambda_1, 1/4)$. The value of $\lambda_1'$ has the
following characterization:
Let $g$ be a nonzero continuous, piecewise differentiable function on $X_\Ga$
such that $\nabla g$ is square integrable with respect to $\mu$, and
$\int_{X_\Ga} g d\mu=0$. Then (identifying $X_\Ga$ with $\Gamma\setminus \bfh$) we have
$$\int_{\Gamma\setminus \bfh} \left(\,\left({{\partial g}\over {\partial x}}\right)^2
+\left({{\partial
g}\over {\partial y}}\right)^2 \,\right) dx \, dy \geq \lambda_1' \int_{\Gamma\setminus \bfh}
g^2
{{dx \, dy} \over {y^2}}.$$
This is, in fact, the way Selberg originally stated his result.
\subsection{Conformal area}
Let $C$ be a compact Riemann surface. Following
\cite{liyau}, we define the {\bf conformal area}, or the first conformal
volume $A_c(C)$ to be the infimum of $\int_C f^* d\mu_0$, where
$f:C\rightarrow} \newcommand{\dar}{\downarrow\bfp^1_\bfc$ runs over all nonconstant conformal mappings, and
where $d\mu_0$ is the $SO_3$-invariant area element on the Riemann sphere.
Using the conformal property of homotheties in $\bfp^1$, Li and Yau show
easily that $$A_c(C) \leq 4\pi\cdot d_\bfc(C).$$
On the other hand, given a Riemannian metric on $C$, let $A(C)$ be the area of
$C$. Using an elegant fixed point argument, Li and Yau obtain (\cite{liyau},
Theorem 1) $$\lambda_1 A(C) \leq 2A_c(C).$$
Their proof works word for word in the case of our singular metric on
$X_\Ga$, once we replace $\lambda_1$ by $\lambda_1'$. All that is
needed is, first, the characterization of $\lambda_1'$ discussed above, and
second, the
fact that differentiable functions on $X_\Gamma$ have a square-integrable
gradient. The latter follows since $\int_{X_\Gamma}|\nabla g|^2d\mu$ is
invariant under conformal change of the metric, therefore it may be calculated
using a regular metric, and thus is finite.
\subsection{Conclusion of the proof} Since the Poincar\'e metric on $X_\Ga$ is
pulled back from $X_{PSL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})}=X(1)$,
we have $A(X_\Ga)=D_\Gamma \cdot A(X(1)) = D_\Gamma \cdot\pi/3$.
Combine this with the inequalities of Li and Yau, and obtain the first part of
the theorem. Now note that $[PSL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}):\Gamma_0(N)]$ is at least $N$, and
similarly $[PSL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}):\Gamma_1(N)]$ is quadratic in $N$ (between $6(N/\pi)^2$
and $N^2$), and obtain the second part.
\qed
\subsection{An analogous result for Shimura curves} As was pointed out by
P. Sarnak, we have the following:
\begin{th}\label{shimura}
Let $D$ be an indefinite quaternion algebra over ${\Bbb{Q}}} \newcommand{\bff}{{\Bbb{F}}$, and
let $G$ be the group of units of norm 1 in some order of $D$. Let
$\Gamma\subset G$ be a subgroup of
finite index, and let $X_\Gamma=\Gamma \setminus \bfh$ be the corresponding
Shimura curve. Then $${{21}\over {200} } (g(X_\Gamma)-1) \leq
d_\bfc(X_\Gamma).$$
\end{th}
{\bf Proof.} Since $X_\Gamma$ is compact, every automorphic form $g$ appearing
in $L^2(X_\Gamma)$ is cuspidal. It follows from the Jacquet - Langlands
correspondence (see \cite{gelbart}, Theorem 10.1 and Remark 10.4) that unless
$g$ is the constant function, there exists a cuspidal automorphic form for some
congruence subgroup in $SL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})$ which has the same eigenvalue with respect
to the non-euclidean Laplacian. Therefore
$\lambda_1 \geq 0.21$ holds for $X_\Gamma$.
The results of Li and Yau give $\lambda_1 A(X_\Gamma) \leq 8\pi\cdot
d_\bfc(X_\Gamma)$, and the Gauss - Bonnet formula gives $4\pi(g(X_\Gamma))-1)
\leq A(X_\Gamma)$ (the difference coming from elliptic fixed points). Combining
the three inequalities we obtain the result. \qed
The author was informed that the results of \cite{lrs} were generalized by
Rudnick and Sarnak to
cuspidal automorphic forms on $GL_2$ over an arbitrary number field
$F$. Therefore Theorem \ref{shimura} holds for $D$ a quaternion algebra over a
totally real
field, which is indefinite at exactly one infinite place.
\section{Applications and remarks}
\subsection{${\Bbb{Q}}} \newcommand{\bff}{{\Bbb{F}}$-gonality and rational torsion on elliptic curves}
Let $C$ be a curve as in \ref{gonal}. Recall \cite{ah} that a point
$P\in C$ is called {\bf a point of degree $d$} if $[K(P):K]=d$. Suppose $C$ has
infinitely many points of degree $d$. By taking Galois orbits on the $d$-th
symmetric power of $C$ we have that ${\operatorname{Sym}}^d(C)(K)$ is infinite. Let
$W_d(C)\subset Pic^d(C)$ be the image of ${\operatorname{Sym}}^d(C)(K)$ by the Abel-Jacobi
map. In \cite{ah}
it was noted that in this situation either $d_K(C) \leq d$, or $W_d(C)(K)$
is infinite.
Now assume $K$ is a number field. By a celebrated theorem of Faltings
\cite{fal}, if $W_d(C)(K)$ is infinite then $W_d(C)\subset Pic^d(C)$ contains
a positive dimensional translate
of an abelian variety, and the simple lemma 1 of \cite{ah}
implies that $d_K(C)\leq 2d$ (\cite{thesis}, theorem 9). The latter
conclusion was also obtained by G. Frey in \cite{frey}.
We now restrict attention to the case where $K={\Bbb{Q}}} \newcommand{\bff}{{\Bbb{F}}$ and $C = X_0(N)$.
In
\cite{thesis}, Theorem 12, as well as in \cite{frey}, it was noted that a lower
bound on the ${\Bbb{Q}}} \newcommand{\bff}{{\Bbb{F}}$-gonality, such as given by theorem \ref{main}, implies
that there exists a constant $m(d)$ (in fact, $m=230d$ will do),
such that if $N> m(d)$ then $X_0(N)$ (and thus also $X_1(N)$) has finitely many
points of degree $d$. In section 1 of \cite{km}, Kamienny and Mazur showd that
this
reduces the uniform boundedness conjecture on torsion points on elliptic curves
to bounding rational torsion of prime degree. The conjecture was finally proved
by L. Merel in \cite{merel}.
It should be remarked that, since for this application one only needs a lower
bound on the ${\Bbb{Q}}} \newcommand{\bff}{{\Bbb{F}}$-gonality of $X_0(N)$, one can use other methods, such as
Ogg's method \cite{ogg}. This is indeed the method used by Frey in \cite{frey},
although the bound obtained is not linear.
For points of low degree, one can use the main results of \cite{ah} with Ogg's
method to slightly improve the bound on $N$ (see \cite{hs} and \cite{thesis},
2.5).
For another arithmetic application of the lower bound on teh $\bfc$-gonality,
regarding pairs of elliptic curves with with isomorphic mod $N$
representations, see Frey \cite{frey2}.
\subsection{Torsion points: the function field case}
Recently, there has been renewed interest in the question of $\bfc$-gonality of
modular curves. In their paper \cite{ns}, K. V. Nguyen and M.-H. Saito used
algebraic techniques to give a lower bound on
the gonality. Although their bound is a bit weaker than ours, their methods are
of interest on their own right: they combine Ogg's method with a Castelnuovo
type bound. They pointed out that given any such bound, one obtains a
function field analogue of the strong uniform boundedness theorem about
torsion on elliptic curves, namely: given a non-isotrivial elliptic curve over
the function field of a complex curve $B$, the size of the torsion
subgroup is bounded solely in terms of the gonality of $B$. This result is
strikingly analogous to a recent result of P. Pacelli (\cite{p}, Theorem 1.3):
assuming Lang's conjecture on rational curves on varieties of general type,
the number of non-constant points on a curve $C$ of genus $>1$
over the function field of $B$ is bounded solely in terms of the genus of $C$
and the gonality of $B$.
|
proofpile-arXiv_065-611
|
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
}
|
\section{Introduction}
Systems producing absorption in the spectra of distant quasars offer
an excellent probe of the early Universe. At high redshifts, they
easily outnumber other observed tracers of cosmic structure, including
both normal and active galaxies. Mounting evidence that
the high column density absorbers are young galaxies links relatively
pristine baryonic matter to highly evolved objects at the present day.
The amount of atomic hydrogen in damped Ly$\alpha$\ (DLA) absorbers
at $z \sim 3$ is comparable to the mass in stars at the current epoch (Wolfe
1988), and two DLA\ systems are known to have radial extents $\mathrel{\copy\simgreatbox}
10 h^{-1}$ kpc (Briggs et al.\ 1989; Wolfe et al.\ 1993). Photometry of
damped absorbers supports the view that they are high-redshift
galaxies (Djorgovski et al.\ 1996; Fontana et al.\ 1996).
At somewhat lower column densities and redshifts,
deep imaging and spectroscopy indicate that Lyman
limit systems are associated with lines of sight passing near bright
galaxies (Yanny 1990; Lanzetta \& Bowen 1990; Bergeron \& Boiss\'e
1991; Steidel, Dickinson, \& Persson 1994) or galaxy clusters
(Lanzetta, Webb, \& Barcons 1996).
The interpretation of quasar absorption systems has undergone
something of a revolution during the past two years, with the
recognition that they may be gas aggregating into nonlinear structures
in hierarchical models like those invoked to account for the observed galaxy
distribution (e.g., Cen {et al.\/} 1994; Petitjean, Mucket, \&
Kates 1995; Zhang, Anninos, \& Norman 1995; Hernquist et al.\
1996; Miralda-Escud\'e et al.\ 1996). In particular, Katz et al.\
(1996; hereafter KWHM) used
simulations that evolve baryons and dark matter in the presence of a
background radiation field to show that high column density absorbers
arise naturally in a cold dark matter (CDM) universe from radiatively
cooled gas in galaxy-sized halos, supporting the notion that damped
Ly$\alpha$\ systems are a byproduct of galaxy formation. Together with the
results of Hernquist et al.\ (1996) for the Ly$\alpha$\ forest, the column
density distribution predicted by KWHM matches existing data
reasonably well, but it falls below the observations by factors $\approx
2$ and $\approx 8$ for DLA\ and Lyman limit absorbers, respectively.
This discrepancy can be attributed at least
partly to resolution effects in the simulations. Owing to
computational expense, the KWHM simulation could not resolve halos
with circular velocities below $v_c \approx 100$ km s$^{-1}$. However,
higher resolution simulations of localized regions by Quinn, Katz, \&
Efstathiou (1996; hereafter QKE)
indicate that halos down to $v_c \approx 35$
km s$^{-1}$\ can host damped absorbers, so clearly the number of high
column density systems found by KWHM is artificially depressed
by the finite resolution of their simulation.
In this paper,
we overcome this numerical limitation using a two-step correction
procedure. First, we employ the Press \& Schechter (1974) algorithm
to correct the KWHM data by extending the halo mass function to
values lower than could be resolved by their simulation. Then, we
account for absorption by gas in these halos from a relation between
the absorption cross section for a halo and its circular velocity.
This relation is established by fitting both the KWHM data and
high-resolution simulations that use the QKE initial conditions
and the KWHM background radiation field.
These additional simulations examine localized
regions around low mass objects with sufficient resolution to resolve
halos down to $v_c \approx 35$ km s$^{-1}$.
Heating by the UV background prevents the collapse and cooling of gas
in smaller halos (QKE; Thoul \& Weinberg 1996).
The high-resolution volumes are small and
were chosen in a non-random way, so they cannot be used directly to
infer the number of DLA\ and Lyman limit systems. By convolving the
absorption area vs.\ circular velocity relation with the halo mass function
given by the Press-Schechter method, we can predict the absorption at
any mass scale, effectively extending the dynamic range of the
simulations down to the lowest mass objects that produce high
column density absorption.
We also present another calculation, similar
to that in KWHM but including star formation, to quantify the effects
of gas depletion on high column density absorption.
\section{Simulations and Methods}
\label{secSimulation}
Our primary simulation, the same as that used by KWHM, follows the
evolution of a periodic cube whose edges measure 22.22 Mpc in comoving
units. This region was drawn randomly from a CDM universe with
$\Omega=1$, $h \equiv H_0/100$ km s$^{-1}$\ Mpc$^{-1}=0.5$, baryon
density $\Omega_b=0.05$, and power spectrum normalization
$\sigma_8=0.7$. A uniform background radiation field was imposed to
mimic the expected ultraviolet (UV) output of quasars, with a spectrum
of the form $J(\nu) = J_0(\nu_0/\nu) F(z)$, where $\nu_0$ is the
Lyman-limit frequency, $J_0=10^{-22}$ erg s$^{-1}$ cm$^{-2}$ sr$^{-1}$
Hz$^{-1}$, and $F(z)=0$ if $z>6$, $F(z)=4/(1+z)$ if $3 \le z \le 6$,
and $F(z)=1$ if $2<z<3$. The simulations employ $64^3$ gas and $64^3$
dark-matter particles, with a gravitational softening length of 20
comoving kpc (13 comoving kpc equivalent Plummer softening). The
particle mass is $1.45 \times 10^8 M_\odot$ and $2.8 \times 10^9
M_\odot$ for gas and dark matter, respectively. Detailed descriptions
of the simulation code and the radiation physics can be found in
Hernquist \& Katz (1989) and Katz, Weinberg, \& Hernquist (1996;
hereafter KWH). The low column density absorption in this simulation
is discussed by Hernquist et al.\ (1996), and the galaxy population is
discussed by Weinberg, Hernquist, \& Katz (1996).
We also employ two simulations that have the same initial conditions,
cosmological parameters, and numerical parameters as QKE but the UV
background spectrum given above. These comprise smaller, 10 Mpc
periodic volumes (with $\Omega=1$, $h=0.5$, $\Omega_b=0.05$ as
before), which are evolved using a hierarchical grid of particles in
the initial conditions. The central region forms a collapsed object
that is represented using a large number of low mass particles, while
regions further away are modeled using a small number of more massive
particles. A simulation of the same volume as QKE would require
$256^3$ particles of each species to match the resolution of the
central region throughout; the nesting technique allows us to achieve
high-resolution locally while preserving the cosmological context of
the calculation.
QKE find that a photoionizing background suppresses the collapse
and cooling of gas in halos with circular velocities
$v_c \mathrel{\rlap{\lower 3pt\hbox{$\mathchar"218$} 35$ km s$^{-1}$. Thoul \& Weinberg (1996) find a similar
cutoff in much higher resolution, spherically symmetric calculations.
Hence, it should be possible to estimate the amount of gas capable of
producing DLA\ and Lyman limit absorption by accounting for
halos down to this cutoff in $v_c$.
Both QKE and Thoul \& Weinberg (1996) find that photoionization
has little effect on the amount of gas that cools in halos
with $v_c \mathrel{\copy\simgreatbox} 60$ km s$^{-1}$, consistent with the results of
Navarro \& Steinmetz (1996) and Weinberg et al.\ (1996).
The current generation of hydrodynamic simulations lacks
the dynamic range necessary to represent halos over the entire range
$35 < v_c \mathrel{\rlap{\lower 3pt\hbox{$\mathchar"218$} 300$ km s$^{-1}$. To overcome this limitation, we use
the approximation developed by Press \& Schechter (1974), who give the
following analytic estimate for the number density of halos of mass
$M$ at redshift $z$:
\begin{equation}
N(M,z) dM = \sqrt{2\over \pi} {\rho_0\over M}
{\delta_c\over \sigma_0} \left({\gamma R_f\over R_*}\right)^2
\exp{\left({-\delta_c^2\over 2\sigma_0^2}\right)} dM ,
\label{PSnumber}
\end{equation}
where $\rho_0$ is the mean comoving density, $R_f$ is the Gaussian
filter radius corresponding to mass
$M= (2\pi)^{3/2} \rho_0 R_f^3$, and $\delta_c$ is
the critical linear density contrast that corresponds to
gravitational collapse. The
parameters $\sigma_0$, $\gamma$ and $R_*$ are related to moments of
the power spectrum (Bardeen {et al.\/} 1986). Equation (\ref{PSnumber}) can
be integrated from $M$ to infinity to yield the number density
of objects above a given mass.
In what follows, for comparison with our simulations, we use the CDM
transfer function given by Bardeen {et al.\/} (1986).
To determine the number of DLA\ and Lyman limit systems per
unit redshift, we first fix the parameters in the Press-Schechter
algorithm so that it reproduces the mass function of our 22.22 Mpc
simulations. Then, we use the 22.22 Mpc and 10 Mpc
simulations together to fit a relation between the circular velocity
of a halo and its cross section for producing DLA\
or Lyman limit absorption.
To identify halos in the simulations, we apply a friends-of-friends
algorithm with a linking length equal to the mean interparticle
separation on an isodensity contour of an isothermal sphere with an
overdensity $177$, $b= (177 n/3)^{-1/3}$ where $n$ is the particle number
density.
We also apply the algorithm of Stadel {et al.\/} (1996;
see also KWH and
http://www-hpcc.astro.washington.edu/tools/DENMAX)
to the cold gas particles
in the same volume to locate regions of collapsed gas capable of
producing Lyman limit and damped Ly$\alpha$\ absorption. A region of gas is
considered a potential absorber only if it contains at least four
gas particles that are mutually bound, have a smoothed overdensity
$\rho_g/\bar\rho_g > 170$, and a temperature $T < 30,000$ K.
All of the gas concentrations found by this method
are associated with a friends-of-friends halo,
even at $z=4$. We match each absorber with its
parent halo and discard halos that contain no absorbers.
For each of the halos that contains a cold gas concentration, we
determine the radius of the sphere centered on the most tightly bound
particle within which the average density is equal to 177 times the
mean background density. We characterize halo masses and circular
velocities by their values at this radius. This method of quantifying
the properties of halos in the simulations corresponds to that
envisioned in the Press-Schechter approximation, which is based on the
spherical collapse model. We find that the mass distribution of halos
in the simulations is best fit using the Press-Schechter form with a
Gaussian filter and $\delta_c = 1.69$. Many workers have instead used
a top-hat filter, with $M_f=(4 \pi/3) \rho_0 R_f^3$ ({\it cf.\/} Ma 1996; Ma
\& Bertschinger 1994; Mo \& Miralda-Escud\'e 1994; Mo {et al.\/} 1996), or a
Gaussian filter with a modified relation between filter radius and
associated mass, $M_f=6 \pi^2 \rho_0 R_f^3$ (Lacey \& Cole 1994), with
similar values for $\delta_c$. However, these studies used the halo
masses as returned by the friends-of-friends algorithm itself, and if
we do this we also find that top-hat or modified Gaussian filters
provide good fits to the mass function for $\delta_c \approx 1.7$.
The combination $\delta_c=1.69$, Gaussian filter, and
$M_f=(2\pi)^{3/2} \rho_0 R_f^3$ is appropriate for our definition of
halo masses within overdensity 177 spheres. Including or excluding
the ``absorberless'' halos in our mass function does not change the
results above $v_c=100$ km s$^{-1}$\ because all halos above this
circular velocity contain at least one absorber.
We calculate HI column densities for the halos by encompassing each
halo with a sphere which is centered on the most tightly bound
gas particle and is of a sufficient size to contain all gas particles
which may contribute to absorption within the halo. We
project the gas
distribution within this sphere onto a uniform grid of cell size 5.43
comoving kpc, equal to the highest resolution achieved anywhere in the
22.22 Mpc simulation.
Using the method of KWHM, we calculate an initial HI column
density for each gridpoint assuming that the gas is optically thin,
then apply a self-shielding correction to yield a true HI column
density (see KWHM for details). For each halo we compute the
projected area over which it produces damped absorption, with $N_{\rm HI} >
10^{20.3} \;\cdunits$, and Lyman limit absorption, with $N_{\rm HI} >
10^{17.2}\;\cdunits$. For simplicity, we project all halos from a
single direction, though we obtain a similar fit of absorption area to
circular velocity if we project randomly in the $x$, $y$, and $z$
directions or average the projections in $x$, $y$, and $z$.
\begin{figure}
\vglue-0.65in
\plottwo{f1a.eps}{f1b.eps} \\
\vglue-0.2in
\plottwo{f1c.eps}{f1d.eps} \\
\vglue-0.2in
\plottwo{f1e.eps}{f1f.eps}
\vglue-0.26in
\caption{Comoving absorbing area in kpc$^2$ vs. circular velocity
$v_c$ in km s$^{-1}$\ for halos in the 22.22 Mpc simulation
(skeletal points) and the 10 Mpc simulations (open circles).
Left hand panels show the area for DLA absorption,
$N_{\rm HI} \geq 10^{20.3}\;\cdunits$, and right hand panels
for Lyman limit absorption, $N_{\rm HI} \geq 10^{17.2}\;\cdunits.$
The number of vertices in the skeletal points corresponds
to the number of gas concentrations in the halo. The solid line shows the
fitted smooth relation of equation~(\ref{avc}), with
parameter values listed in Table 1.}
\label{figVAplot}
\end{figure}
Figure~\ref{figVAplot} shows the cross section for damped
absorption (left hand panels) and
Lyman limit absorption (right hand panels)
as a function of circular velocity for each of
our halos, at redshifts 2, 3, and 4.
The open circles at low $v_c$ represent
halos from the 10 Mpc, high-resolution runs. Other
points refer to the 22.22 Mpc simulation, and the number of vertices
in each symbol indicates the number of absorbers (i.e., distinct regions
of cold, collapsed gas) within each halo. For these halos
there are two competing effects that determine the trend between
absorption cross section and circular velocity.
Higher mass halos have deeper potential
wells, so concentrations of cold gas contract further, explaining the
downward trend in cross section with circular velocity exhibited by
points with a fixed number of vertices. However,
more massive halos tend to harbor more than one
concentration of gas, increasing their absorption cross section.
The overall trend in Figure 1 is that
halos of higher circular velocities on average have larger absorption
cross sections.
The solid lines in Figure~\ref{figVAplot} show a smooth function
$\alpha_z(v_c)$ fitted to the relation between absorption area
and circular velocity. We will need this function for our
Press-Schechter correction procedure below. As a functional form
we adopt a linear relation between ${\rm log}\,\alpha$ and
${\rm log}\,v_c$ with a damping factor $1-\exp(-(v_c-35)/12)$,
which reflects the suppression of gas cooling in low $v_c$ halos.
We bin the data points in intervals of 0.15 in ${\rm log}\,v_c$,
compute the mean and standard deviation of ${\rm log}\,\alpha$
in each bin, and determine the parameters of the smooth
relation by $\chi^2$ minimization. Fitting binned data rather
than individual halos gives more appropriate weight to the relatively
rare, high $v_c$ halos. Table 1 lists the fitted values
of $A$ and $B$ for the functional relation
\begin{equation}
{\rm log}\,\alpha = (A\,{\rm log}\,v_c + B)(1-\exp(-(v_c-35)/12)),
\label{avc}
\end{equation}
with $\alpha$ in comoving kpc$^2$, $v_c$ in km s$^{-1}$, and
base-10 logarithms.
We determine values separately for DLA\ and Lyman limit
absorption and for each redshift. Figure~\ref{figVAplot}
shows that there is substantial scatter about this
mean relation, and our adopted functional form is rather arbitrary,
but we will see shortly that this characterization of the
$\alpha_z(v_c)$ relation suffices for our purposes.
\begin{table}
\begin{tabular}{lllll}
\tableline\tableline
\multicolumn{1}{c}{$z$} & \multicolumn{1}{c}{$A_{\rm DLA}$} &
\multicolumn{1}{c}{$B_{\rm DLA}$}& \multicolumn{1}{c}{$B_{\rm LL}$} &
\multicolumn{1}{c}{$B_{\rm LL}$} \\ \tableline
2.0& 2.32& -1.87 & 2.70 & -2.13 \\
3.0& 2.94& -3.03 & 3.21 & -2.96 \\
4.0& 2.84& -2.63 & 3.02 & -2.28 \\ \tableline\tableline
\end{tabular}
\caption{Fitted parameter values for $\alpha_z(v_c)$, with
the functional form in equation~(\ref{avc}).}
\label{tabalpha}
\end{table}
The observable quantity that we would like to test the CDM model
against is $n(z)$, the number of DLA\ or Lyman limit absorbers
per unit redshift interval along a random line of sight.
We can estimate this from the projected HI map of the 22.22 Mpc
simulation as in KWHM, by dividing the fractional area that has
projected column density above the DLA\ or Lyman limit threshold
by the depth of the box in redshift. However, because the
simulation does not resolve gas cooling in halos with
$v_c \mathrel{\copy\simlessbox} 100$ km s$^{-1}$, this procedure really yields
estimates of $n(z,100\;\vunits)$, where $n(z,v_c)$
denotes the number of absorbers per unit redshift produced
by halos with circular velocity greater than $v_c$.
Since halos with $35\;\vunits < v_c < 100\;\vunits$ can
harbor DLA\ and Lyman limit absorbers,
$n(z,100\;\vunits)$ is only a lower limit to the observable
quantity $n(z)$.
We have now assembled the tools to fix this problem, for the
Press-Schechter formula~(\ref{PSnumber}) tells us the number
density of halos as a function of circular velocity and the
relation $\alpha_z(v_c)$ tells us how much absorption these
halos produce. Equation~(\ref{PSnumber}) is given in terms
of the mass $M$; since we define the halo mass within a sphere
of overdensity 177, the corresponding circular velocity is
\begin{equation}
v_c = (GM/R_{177})^{1/2} =
\left[GM^{2/3} \left({4\pi \over 3} 177 \rho_c\right)^{1/3}\right]^{1/2} =
117~ \left({M \over 10^{11} M_\odot}\right)^{1/3}
\left({1+z \over 4}\right)^{1/2} \; \vunits.
\label{vcM}
\end{equation}
Thus,
\begin{equation}
n(z,v_c)= {dr \over dz} \int_M^{\infty} N(M',z)
\alpha_z(v_c) dM',
\label{nofzM}
\end{equation}
where $N(M',z)$ is taken from equation~(\ref{PSnumber}), and
equation~(\ref{vcM}) is used to convert between $v_c$ and $M$
as necessary. Multiplying the comoving number density of halos by
the comoving absorption area yields a number of absorbers per
comoving distance, and multiplying by $dr/dz$, the derivative of
comoving distance with respect to redshift, converts to a number
per unit redshift.
Figure~\ref{figNZplot} shows $n(z,v_c)$ computed from
equation~(\ref{nofzM}) using our fitted relations $\alpha_z(v_c)$.
Starting from high $v_c$, the abundance first rises steeply with
decreasing $v_c$ because of the increasing number of halos,
but it flattens at low $v_c$ because of the suppression of gas
cooling in small halos. Points with error bars show $n(z,v_c)$
obtained directly from the halos in the 22.22 Mpc simulation.
The curves fit these points quite well --- they are, of course,
constructed to do so, but the agreement shows that our full
procedure, including the details of the Press-Schechter calibration
and fitting for $\alpha_z(v_c)$, is able to reproduce the original
numerical results in the regime where halos are resolved.
We can therefore be fairly confident in using this method to
extrapolate to $n(z,0) = n(z)$, the incidence of high column
density absorption produced by gas in all halos, thus
incorporating the new information provided by the high-resolution simulations.
These values of $n(z)$, the $y$-intercepts of the curves in
the panels of Figure~\ref{figNZplot}, are the principal numerical
results of this paper. We will compare them to observations in
the next section.
Table 2 lists the values of $n(z)$ determined by this procedure
at $z=2$, 3, and 4. It also lists the correction factors that
must be applied to the quantities $n(z,100\;\vunits)$ obtainable
by the KWHM procedure in order to get the total abundance
$n(z)=n(z,0)$. In all cases, roughly half of the absorption
occurs in halos with $v_c > 100\;\vunits$ and half in the
more common but smaller halos with lower circular velocities.
\begin{figure}
\vglue-0.65in
\plottwo{f2a.eps}{f2b.eps} \\
\vglue-0.2in
\plottwo{f2c.eps}{f2d.eps} \\
\vglue-0.2in
\plottwo{f2e.eps}{f2f.eps}
\vglue-0.23in
\caption{Incidence of DLA (left) and Lyman limit (right)
absorption at $z=2,$ 3, and 4. Curves show $n(z,v_c)$,
the number of absorbers per unit redshift arising in halos
with circular velocity greater than $v_c$, computed from
equation~(\ref{nofzM}). The $y$-intercepts show the
incidence of absorption produced by all halos. Points with $N^{1/2}$ error
bars show numerical results from the 22.22 Mpc simulation.}
\label{figNZplot}
\end{figure}
\section{Comparison to Observations}
\label{secResults}
\begin{table}
\begin{tabular}{lllcllclllcll} \tableline\tableline
\multicolumn{6}{c}{Damped Ly$\alpha$\ } && \multicolumn{6}{c}{Lyman Limit}
\\ \cline{1-6} \cline{8-13}
\multicolumn{3}{c}{Calculated}&&\multicolumn{2}{c}{Observed}&&
\multicolumn{3}{c}{Calculated}&&\multicolumn{2}{c}{Observed}\\
z&\multicolumn{1}{c}{$n(z)$}&\multicolumn{1}{c}{$F_C$}&&
\multicolumn{1}{c}{$z$}&\multicolumn{1}{c}{$n(z)$} &&
z&\multicolumn{1}{c}{$n(z)$}&\multicolumn{1}{c}{$F_C$}&&
\multicolumn{1}{c}{$z$}&\multicolumn{1}{c}{$n(z)$}
\\ \cline{1-3}\cline{5-6}\cline{8-10}\cline{12-13}
2& 0.17857 & 2.05&& $1.75\pm 0.25$& $0.14\pm 0.073$ &&
2& 0.59586 & 1.74&& $0.90\pm 0.5$& $0.65\pm 0.25$ \\
3& 0.17411 & 1.91&& $2.5\pm 0.5$& $0.18\pm 0.039$ &&
3& 0.72439 & 1.81&& $2.95\pm 0.6$& $2.08\pm 0.35$ \\
& & && $3.25\pm 0.25$& $0.21\pm 0.10$ &&
& & && & \\
4& 0.19422 & 2.54&& $4.1\pm 0.6$& $0.47\pm 0.17$ &&
4& 1.00660 & 2.31&& $4.15\pm 0.6$& $3.45\pm 0.95$ \\ \tableline\tableline
\end{tabular}
\caption{The incidence $n(z)$ of DLA\ and Lyman limit absorption for
the $\Omega=1$ CDM model, computed by our calibrated
Press-Schechter procedure. Observational values are taken from
Storrie-Lombardi {et al.\/} (1996) for DLA\ absorption and from
Storrie-Lombardi {et al.\/} (1994) for Lyman limit absorption.
Also listed is $F_C$, the correction factor by which the KWHM results
for $n(z,100\;\vunits)$ must be multiplied to obtain the
absorption $n(z)$ produced by all halos.}
\label{tabResults}
\end{table}
\begin{figure}
\epsfxsize=6.5truein
\centerline{\epsfbox[18 144 590 718]{f3.eps}}
\caption{
\label{figObsComp}
Incidence of DLA\ and Lyman limit absorption as a function of
redshift. Triangles and squares show the resolution-corrected
theoretical predictions for DLA\ and Lyman limit absorption,
respectively. The upper error crosses represent the Lyman limit
data of Storrie-Lombardi {et al.\/} (1994), with $1\sigma$ and $2\sigma$
abundance errors shown. The smooth curve shows their fitted power law.
The lower set of error crosses and solid curve represent the DLA\
data of Storrie-Lombardi {et al.\/} (1996), with $1\sigma$ and $2\sigma$
errors. The dotted error crosses and curve show the data, $1\sigma$
errors, and fit from Wolfe {et al.\/} (1995).}
\end{figure}
Figure~\ref{figObsComp} compares our derived values of $n(z)$ to
observational estimates of the incidence of damped Ly$\alpha$
absorption, taken from Storrie-Lombardi {et al.\/} (1996) and
Wolfe {et al.\/} (1995), and Lyman limit absorption, taken from
Storrie-Lombardi {et al.\/} (1994).
The theoretical predictions and observed values are listed in Table 2.
The resolution correction increases the predicted $n(z)$ values
relative to those of KWHM by about a factor of two, leading to
quite good agreement with the observed abundance of DLA\ absorbers
at $z=2$ and 3. At $z=4$ the predicted abundance is $1.6\sigma$
below the Storrie-Lombardi {et al.\/} (1996) data. Since there are
systematic as well as statistical uncertainties in this observational
estimate --- in particular, it includes candidate DLA\ systems
that have not yet been confirmed by Echelle spectroscopy ---
we regard this level of agreement as acceptable.
The situation for Lyman limit absorption is quite different.
Here the theoretical predictions fall systematically below the
observed abundances, by about a factor of three.
The correction for unresolved halos reduces the discrepancy found
by KWHM, but it does not remove it.
The deficit of Lyman limit systems could reflect a failing of
the CDM model considered here, or it could indicate the presence
in the real universe of an additional population of Lyman limit
absorbers that are not resolved by our simulations.
We discuss this issue further in \S~\ref{secSummary}
\section{Effects of Star Formation}
\label{secStars}
The simulations examined in the previous section do not allow
conversion of gas into stars, and one might worry that depletion
of the atomic gas supply by star formation would substantially
reduce the predicted abundance of DLA\ absorbers.
We investigate this issue by analyzing a simulation identical to the
KWHM run considered above except that it incorporates star formation.
The algorithm, a modified form of that introduced by Katz (1992),
is described in detail by KWH; we summarize it here.
A gas particle becomes ``eligible'' to form stars if (a)
the local hydrogen density exceeds 0.1 cm$^{-3}$ (similar to that of
neutral hydrogen clouds in the interstellar medium), (b)
the local overdensity exceeds the virial overdensity,
and (c) the particle resides in a converging flow that is Jeans-unstable.
Star formation takes place
gradually, with a star formation rate that depends on an
assumed efficiency for conversion of gas into stars and on the local
collapse timescale (the maximum of the local dynamical timescale and
the local cooling timescale).
We set the efficiency parameter defined by KWH to $c_*=0.1$,
though the tests in KWH show that results are insensitive to
an order-of-magnitude change in $c_*$.
Until the gas mass of such a particle
falls below 5\% of its original mass, it is categorized as a
``star-gas'' particle. Thereafter, it is treated as a collisionless
star particle. This gradual transition overcomes computational
difficulties associated with alternative implementations of
star formation, such as
the artificial reduction in resolution caused by rapid removal of
collisionless gas particles from converging flows, or the
spawning of a large number of extra particles that slow the
computations and consume memory.
When stars form, we add supernova feedback energy to the
surrounding gas in the form of heat, assuming that
each supernova yields $10^{51}$ ergs and that all stars greater than
$8M_\odot$ go supernova.
We add this energy gradually, with an exponential decay time of
2 $\times 10^7$ years, the approximate lifetime of an $8M_\odot$ star.
Thermal energy deposited in the dense, rapidly cooling gas is quickly radiated
away, so although feedback has some effect in our simulation, the
impact is usually not dramatic.
\begin{figure}
\epsfysize=5.0truein
\centerline{\epsfbox{f4.eps}}
\caption{
\label{starfig}
The column density distribution $f(N_{\rm HI})$ --- the number of absorbers
per unit redshift per linear interval of $N_{\rm HI}$ --- for simulations
with and without star formation.
Histograms show the simulation results at $z=2$ (solid), $z=3$ (dotted),
and $z=4$ (dashed). Heavier lines represent the simulation
without star formation and lighter lines the simulation with
star formation.
}
\end{figure}
Figure~\ref{starfig} shows column density distributions
for the simulations with and without star formation at $z = 2$, 3, and 4;
$f(N_{\rm HI})$ is the number of absorbers per unit redshift per linear
interval of column density. Star formation alters $f(N_{\rm HI})$ only at
column densities greater than $10^{22}$ cm$^{-2}$, higher than any observed
column density.
Star formation does affect the amount of cold, collapsed gas, however.
The simulation without star formation
yields an $\Omega$ in cold, collapsed gas, i.e.\ gas with
$\rho/\bar\rho > 1000$ and $T<30,000$K, of (6.5, 3.6, 1.7)$\times 10^{-3}$
at $z = (2, 3, 4)$. In the simulation with star formation,
the $\Omega$ in cold, collapsed gas is (3.4, 2.3, 1.2)$\times 10^{-3}$
at $z = (2, 3, 4)$, while the $\Omega$ in stars is
(3.1, 1.2, 0.4)$\times 10^{-3}$,
making a total $\Omega$ in collapsed galactic
baryons of (6.5, 3.5, 1.6)$\times 10^{-3}$, just slightly below the
simulation without star formation. Hence, star formation simply
converts very high column density gas into stars while
affecting little else.
It does not significantly alter the predicted values of $n(z)$ given
previously because absorbers with $N_{\rm HI} \geq 10^{22}\;\cdunits$
are a small fraction of all DLA\ absorbers.
All of the distributions in Figure~\ref{starfig} show a clear
flattening in the column density range
$10^{18.5}\;\cdunits \leq N_{\rm HI} \leq 10^{20.5}\;\cdunits$.
This flattening reflects the onset of self-shielding.
A small range of total hydrogen column densities maps into a wider
range of neutral hydrogen column densities because the neutral
fraction rises rapidly with column density as self-shielding
becomes important. While the optical depth to Lyman limit photons
is one at $N_{\rm HI} = 10^{17.2}\;\cdunits$, self-shielding does not
become strong until significantly higher column densities because
higher frequency photons have a lower ionization cross section and
can still penetrate the cloud.
\section{Summary}
\label{secSummary}
The finite resolution of numerical simulations affects their predictions
for the abundance $n(z)$ of DLA\ and Lyman limit absorption systems.
It is not currently feasible to simulate a volume large enough to
contain a representative population of high circular velocity halos
while maintaining enough resolution to accurately model the smallest
halos ($v_c \approx 35\;\vunits$) that can harbor such systems.
We have therefore devised a method that integrates results from high-
and low-resolution simulations to obtain accurate predictions for $n(z)$.
We use the simulations to determine the relation between absorption
cross section and halo circular velocity over the full range
of relevant circular velocities, then combine this relation with
the Press-Schechter formula for halo abundance --- itself calibrated
against the simulated halo population --- to compute $n(z)$ via
equation~(\ref{nofzM}).
As a method to correct for finite resolution, this technique should
be quite reliable, and it can be applied to other cosmological models
once the appropriate simulations are available for calibrating
$\alpha_z(v_c)$. In the absence of these simulations, one can
make the plausible but uncertain assumption that the relation between
absorbing area and halo circular velocity is similar from one
model to another, then combine $\alpha_z(v_c)$ from this study
with the Press-Schechter halo abundance for other models to predict $n(z)$.
We apply this approach to a number of popular cosmological scenarios
in a separate paper (Gardner {et al.\/} 1996).
While it is less secure than the resolution-corrected numerical
approach of this paper, it is an improvement over existing
semi-analytic calculations of DLA\ abundances
(e.g., Mo \& Miralda-Escud\'e 1994; Kauffmann \& Charlot 1994;
Ma \& Bertschinger 1994; Klypin {et al.\/} 1995),
which usually assume
that {\it all} gas within the halo virial radius cools and becomes neutral,
and which either assume a form and scale for the collapsed gas
distribution or compare to observations only through the
atomic gas density parameter $\Omega_g$, which is sensitive mainly
to the very highest column density systems.
Our resolution correction increases the incidence of
DLA\ and Lyman limit absorption in the CDM model by about a factor
of two, relative to the results of KWHM. This increase brings the
predicted abundance of DLA\ absorbers into quite good agreement
with observations at $z=2$ and 3,
indicating that the high redshift galaxies that form in the CDM
model can account naturally for the observed damped Ly$\alpha$\ absorption.
At $z=4$ the predicted $n(z)$ is $1.6\sigma$ (a factor 2.4)
below a recent observational estimate. However, many of
the systems that contribute to this data point have not yet been confirmed
by high-resolution spectroscopy, so the estimate may decrease with
future observations.
The underprediction of Lyman limit absorption in the simulations is
more dramatic, nearly a factor of three at $z=2$, 3, and 4.
This discrepancy could represent a
failure of the CDM model with our adopted parameters
($\Omega=1$, $h=0.5$, $\Omega_b=0.05$, $\sigma_8=0.7$),
though most of the popular alternatives to standard CDM have
less small scale power and therefore fare at least as badly in this regard.
An alternative possibility is that most Lyman limit
absorption occurs in structures far below the resolution scale of
even our high-resolution, individual object simulations.
For example, Mo \& Miralda-Escud\'e (1996) propose that most Lyman limit
systems are low mass ($\sim 10^5 M_\odot$) clouds formed by thermal
instabilities in galactic halo gas.
We could also be underestimating Lyman limit absorption if some of it
arises in partially collapsed structures --- sheets or filaments ---
that are not accounted for by the Press-Schechter halo formula.
While the KWHM simulation includes such structures, it may underestimate
their numbers in regions of low background density, where its spatial
resolution is degraded, and the QKE simulations select high density
regions from the outset. High resolution
simulations focused on underdense regions could investigate this
possibility. At lower redshifts Lyman limit absorption is
always associated with normal galaxies (Steidel {et al.\/} 1994; Lanzetta {et al.\/} 1996),
but this is not necessarily the case at high redshifts.
In addition to resolution-corrected estimates of $n(z)$, our
results provide some insights into the physical nature of DLA\ absorbers.
As shown in Figure~\ref{figNZplot}, roughly half of the absorbers
reside in halos with circular velocities greater than $100\;\vunits$
and half in halos with $35\;\vunits \leq v_c \leq 100\; \vunits$.
High resolution spectroscopy of metal-line absorption in damped
systems (e.g., Wolfe {et al.\/} 1994) may be able to test this prediction
over the next few years, and future simulations can provide predictions
for other cosmological models. We find that halos with
$v_c \geq 150 \;\vunits$ frequently host more than one gas concentration
(Figure~\ref{figVAplot}), so imaging observations might often
reveal multiple objects close to the line of sight.
At $z\geq 2$, star formation and feedback --- at least as implemented
in our simulations --- have virtually no effect on the predicted
numbers of Lyman limit and DLA\ absorbers. Roughly half of the
cold, collapsed gas is converted to stars by $z=2$, but this
affects the absorption statistics only at $N_{\rm HI} \geq 10^{22} \;\cdunits$.
Depletion of the gas supply by star formation may account for the
absence of observed systems with column densities in this range,
though the number expected in existing surveys would be small in any case.
At lower redshifts, the effects of gas depletion may extend to lower
column densities. For $\Omega=1$ and $h=0.5$,
there are just over a billion years between $z=4$ and $z=2$,
but there are over two billion years between $z=2$ and $z=1$
and over eight billion years from $z=1$ to the present.
Assuming a roughly constant star formation rate in disk galaxies,
most of the depletion of DLA\ gas would occur at low redshifts.
Ongoing searches for DLA\ absorbers are improving the observational
constraints on their abundance at high redshift, and
follow-up spectroscopic studies of their metal-line absorption
and imaging studies of associated Ly$\alpha$\ and continuum emission
are beginning to yield important insights into their physical
properties. Multi-color searches for ``Lyman-break'' galaxies are
beginning to reveal the population of ``normal'' high redshift galaxies,
which are the likely sources of most DLA\ absorption.
In the hierarchical clustering framework, the abundance,
properties, and clustering of these objects depend on the
amount of power in the primordial fluctuation spectrum on galactic
mass scales, which in turn depends on the nature of dark matter,
on the mechanism that produces the fluctuations, and on
cosmological parameters such as $\Omega$, $h$, and $\Omega_b$.
The initial fluctuations on galactic scales are difficult to
constrain with local observations because much larger structures
(e.g., galaxy clusters) have since collapsed. The comparison
between rapidly improving high redshift data and numerical
simulations like those used here opens a new window for testing
cosmological models, and we expect that it will take us much further
towards understanding the origin of quasar absorbers, high
redshift galaxies, and the galaxies that we observe today.
\acknowledgments
This work was supported in part by the San Diego, Pittsburgh, and Illinois
supercomputer centers, the Alfred P. Sloan Foundation, NASA Theory
Grants NAGW-2422, NAGW-2523, NAG5-2882, and NAG5-3111, NASA HPCC/ESS Grant
NAG5-2213, NASA grant NAG5-1618, and the NSF under Grant ASC 93-18185
and the Presidential Faculty Fellows Program.
|
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{
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
}
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\subsection*{Introduction}
In this talk I will discuss some results of a next-to-leading order
event generator for hadronic three jet production. I will begin
by briefly outlining the procedure for performing next-to-leading
order jet calculations. I will then present a status report on the
progress of this calculation.
\subsection*{Next-to-Leading Order Jet Calculations}
The calculation of three jet production at next-to-leading order
combines two-to-three parton scattering to one loop with Born level
two-to-four parton scattering. Both of these contributions are
singular. Only the sum of the two contributions is finite and
physically meaningful. The one loop two-to-three parton amplitudes
contain infrared singularities arising from the presence of nearly
on-shell massless partons in the loops. The Born
level two-to-four parton amplitudes are also infrared singular,
diverging when one of the partons is very soft or when two partons are
highly collinear.
The origin of the singularities concerns parton resolvability. If a
parton becomes very soft, or if two partons are highly collinear, it
becomes impossible to resolve all final state partons from one
another. The four parton final
state looks instead like a three parton final state. (In fact, of
course, individual partons can never be resolved, and are only
observed as jets of hadrons.) By imposing a resolvabilty criterion
one can define an infrared region of phase space. By using asymptotic
approximations of soft or collinear matrix elements\cite{BGb} within
that region, one can integrate out the unresolved parton and obtain
effective three body matrix elements with poles that exactly cancel
those of the one loop matrix elements.
This is known as the phase space slicing method.\cite{GG,GGK} The
infrared region is defined by the
arbitrary resolution parameter $s_{min}$. If the invariant mass
$s_{ij}$ of two partons labelled $i$ and $j$ is larger than $s_{min}$,
the partons are said to be resolved from one another (although they
may yet be clustered into the same jet), otherwise partons $i$ and $j$
are said to be unresolved from one another.
If there is only one pair of unresolved partons $i$ and $j$, then
those partons are said to be collinear. If there
is some parton $i$ which in unresolvable from two or more partons
$j,k,\dots$, then parton $i$ is said to be soft.
Using phase space slicing, the singularities are removed from the
two-to-four parton scattering process and added to the one-loop
two-to-three process, cancelling the singularities.
However, since the boundary of the infrared region was defined by the
arbitrary parameter $s_{min}$, the slicing procedure induces
logarithmic $s_{min}$ dependence in both sub-processes. The
cancellation of the $s_{min}$ dependence in the sum of the two
processes provides an important cross check on the calculation.
The resolution parameter $s_{min}$ is completely arbitrary and is
independent of the jet clustering algorithm. This
allows us to use a variety of jet algorithms and
facilitates comparison with experiment. In principle, $s_{min}$ can
take any value, but in practice must lie within a finite range. If
$s_{min}$ is too large, it forces partons to be clustered that the
jet algorithm would otherwise leave unclustered. If $s_{min}$ is too
small, the logarithms of $s_{min}$ become large. The magnitude of the
cancellation between the two sub-processes grows, requiring increased
computer time to obtain the cancellation to a given statistical
accuracy.
\subsection*{Progress Report}
At this time, we have developed a working event generator for pure
gluon scattering. We thus combine the one-loop virtual cross section
for $gg \rightarrow ggg$ scattering\cite{BDKa} with the Born level
cross section for $gg \rightarrow gggg$. This development is a
significant step towards completing an event generator for the full
three jet cross section at next to leading order, since all essential
components such as phase space generators, jet clustering algorithms,
phase space slicing, etc., must be working properly. In principle,
the completion of the project simply involves adding the remaining matrix
elements to the existing program structure.
In Figure~\ref{fig:smin}, I show the $s_{min}$ dependence of the total
cross section and of the two component subprocesses. Clearly, the
calculation is well behaved for any value of $s_{min}$ below
approximately $30$ GeV${}^2$. Above that value, the resolution
parameter interferes with the jet algorithm and forces excessive
parton clustering. As expected, good statistical accuracy is more
difficult to obtain at small values of $s_{min}$.
\begin{figure}[t]
\epsfxsize=\hsize\epsfbox{sigvsminC.ps}
\epsfxsize=\hsize\epsfbox{sigvsminD.ps}
\caption{a) $s_{min}$ dependence of the total cross section
(center), $gg\rightarrow gggg$ sub-process (top) and $gg\rightarrow
ggg$ sub-process (bottom). b) Expanded view of the $s_{min}$
dependence of the total cross section.}
\label{fig:smin}
\end{figure}
\section*{Acknowledgments} Fermilab is operated by Universities
Research Association, Inc., under contract DE-AC02-76CH03000 with the
U.S. Department of Energy.
\section*{References}
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{
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
In this talk we will describe how to calculate the Wilson loop $W(\Gamma)$
determining the spin dependent, velocity dependent heavy quark potential $V_{q
\bar q}$ using the assumption of electric-magnetic duality; namely, that the
long distance physics of Yang Mills theory depending upon strongly coupled
gauge potentials $A_\mu$ is the same as the long distance physics of a dual
theory describing the interactions of weakly coupled dual potentials $C_\mu$
and monopole fields $B_i$. To calculate $V_{q \bar q}$ at long distances we
replace $W(\Gamma)$ by $W_{\mbox{\scriptsize eff}}(\Gamma)$ a functional integral over the
variables of the dual theory \cite{calc}. Because the long distance fluctuations of the
dual variables are small we can use a semi-classical expansion to evaluate
$W_{\mbox{\scriptsize eff}}$. The classical approximation gives the dual superconductor picture
of confinement \cite{mand} and the semi-classical corrections lead to an effective string
theory \cite{string}. We first review electric-magnetic duality in electrodynamics.
\section {Electric-Magnetic Duality in Electrodynamics}
Consider a pair of particles with charges $e(-e)$ moving along trajectories
$\vec z_1(t) (\vec z_2 (t))$
in a relativistic medium having
dielectric constant $\epsilon$. The trajectories $\vec z_1(t)(\vec z_2(t))$ define world lines $\Gamma_1(\Gamma_2)$ running from
$t_i$ to $t_f(t_f$ to $t_i)$. The world lines $\Gamma_1(\Gamma_2)$,
along with two straight lines at fixed time connecting $\vec y_1$ to
$\vec y_2$ and $\vec x_1$ to $\vec x_2$, then make up a
closed contour $\Gamma$ (See Fig.1). The
current density $j^\mu(x)$ then has
the form
\begin{equation}
j^\mu (x) = e \oint_\Gamma dz^\mu \delta (x - z).
\label{eq:2.1}
\end{equation}
\begin{figure}
\leavevmode
\centering
\psfig{file=fig.eps,height=2.2in}
\caption{The loop $\Gamma$.}
\end{figure}
In the usual $A_\mu$ (electric) description this system is described by a
Lagrangian
\begin{equation}
{\cal L}_A(j) = - {\epsilon\over 4} (\partial_\alpha A_\beta - \partial_\beta
A_\alpha)^2 - j^\alpha A_\alpha.
\label{eq:2.2}
\end{equation}
\noindent
Then
\begin{equation}
\int dx {\cal L}_A (j) = - \int dx {\epsilon (\partial_\mu A_\nu -
\partial_\nu A_\mu)^2\over 4} - e \oint_\Gamma dz^\mu A_\mu (z).
\label{eq:2.3}
\end{equation}
\noindent
The functional integral defining $W(\Gamma)$ in electrodynamics is
\begin{equation}
W(\Gamma)={\int {\cal D} A_\mu e^{i\int dx [{\cal L}_A (j) + {\cal
L}_{GF}]}\over \int {\cal D} A_\mu e^{i\int dx [{\cal L}_A (j = 0) + {\cal
L}_{GF}]}},
\label{eq:2.4}
\end{equation}
\noindent
where ${\cal L}_{GF}$ is a gauge fixing term.
The spin independent electron positron potential $V_{e^+ e^-}
(\vec R, {\dot{\vec z}}_1, {\dot{\vec z}}_2 )$
is obtained from the expansion of $i$ log
$W(\Gamma)$ to second order in the velocities ${\dot{\vec z}}_1$ and
${\dot{\vec z}}_2$ by the equation:
\begin{equation}
i \log W(\Gamma) = \int_{t_{i}}^{t_{f}} dt
V_{e^+ e^-} (\vec R, {\dot{\vec z}}_1, {\dot{\vec z}}_2 ) \,,
\label{eq:2.5}
\end{equation}
\noindent
where $\vec R = \vec z_1(t) - \vec z_2 (t)$. To higher order in the
velocities, $i$ log $W(\Gamma)$ cannot be written in the above form and the
concept of a potential is not defined because of the occurence of radiation.
Eq.(\ref{eq:2.5}) does not include contributions of closed loops of
electron positron pairs to $V_{e^+e^-}$.
The integral (\ref{eq:2.4}) is gaussian and has the value
\begin{equation}
W(\Gamma) = e^{-{ie^{2}\over 2}\int_\Gamma dx^\mu \int_{\Gamma}
dx^{\prime\nu}
{D_{\mu\nu}(x - x')\over \epsilon}},
\label{eq:2.6}
\end{equation}
\noindent
where $D_{\mu\nu}$ is the free photon propagator.
Letting $\epsilon = 1$ and expanding $i \log W(\Gamma)$ to second order in the
velocities gives: \cite{darwin}
\begin{equation}
V_{e^+e^-} = - {e^2 \over 4\pi R} + {1\over 2} {e^2 \over 4\pi R} \left[{\dot{\vec z}}_1
\cdot {\dot{\vec z}}_2 + {({\dot{\vec z}}_1 \cdot \vec R ) ( {\dot{\vec z}}_2
\cdot \vec R ) \over R^2 }\right] \equiv V_{D} .
\label{eq:2.7}
\end{equation}
\noindent
Furthermore the spin dependent electron positron potential
$V_{e^+e^-}^{\mbox{\scriptsize spin}}$ is determined by the expectation value $\langle \langle
F_{\mu \nu} \rangle \rangle_{Maxwell}$ of the electromagnetic field in the
presence of the external current (\ref{eq:2.1}).
In the dual description first we write the inhomogeneous Maxwell equations in
the form:
\begin{equation}
-\partial^\beta {\epsilon_{\alpha\beta\sigma\lambda} G^{\sigma \lambda}\over
2}
= j_\alpha ,
\label{eq:2.8}
\end{equation}
\noindent
where $G_{\mu\nu}$ is the dual field tensor composed of the electric
displacement vector $\vec D$ and the magnetic field vector $\vec H$:
\begin{equation}
G_{0k} \equiv H_k ,\qquad G_{\ell m} \equiv \epsilon_{\ell mn} D^n.
\label{eq:2.9}
\end{equation}
Next attach a line $L$ of polarization charge between the electron positron
pair. As the charges move the line $L$ sweeps out a surface $y^{\alpha}
\left(\sigma, \tau \right)$ bounded by $\Gamma$ (the Dirac
sheet) and generates
the Dirac polarization tensor $G_{\mu \nu}^S \left( x \right)$: \cite{dirac}
\begin{equation}
G_{\mu\nu}^S (x) = - e \epsilon_{\mu\nu \alpha\beta} \int d\sigma \int d \tau
{\partial y^\alpha\over\partial\sigma} {\partial y^\beta\over\partial\tau}
\delta (x - y(\sigma,\tau)).
\label{eq:2.10}
\end{equation}
\noindent
The current density (\ref{eq:2.1}) can then be written in the form: \cite{dirac}
\begin{equation}
- \partial^\beta {\epsilon_{\alpha\beta\sigma\lambda}
G^{S\sigma\lambda}(x)
\over 2}=j_\alpha (x) ,
\label{eq:2.11}
\end{equation}
\noindent
and the solution of the inhomogeneous Maxwell equations (\ref{eq:2.8}) is
\begin{equation}
G_{\mu\nu} = \partial_\mu C_\nu - \partial_\nu C_\mu + G_{\mu\nu}^S,
\label{eq:2.12}
\end{equation}
\noindent
which defines the magnetic variables (the dual potentials $C_\mu$).
The homogeneous Maxwell equations for $\vec E$ and $\vec B$, written in
the form
\begin{equation}
\partial^\alpha (\mu G_{\alpha\beta}) = 0,
\label{eq:2.13}
\end{equation}
\noindent
where $\mu = {1\over\epsilon}$ is the magnetic susceptibility, become
dynamical equations for the dual potentials, and can be obtained
by varying $C_\mu$ in the Lagrangian
\begin{equation}
{\cal L}_C (G_{\mu\nu}^S) = - {1\over 4} \mu G_{\mu\nu} G^{\mu\nu} \,,
\label{eq:2.14}
\end{equation}
\noindent
where $G_{\mu\nu}$ is given by (\ref{eq:2.12}). This Lagrangian
provides the
dual (magnetic) description of the Maxwell theory (\ref{eq:2.2}).
In the dual description the Wilson loop $W(\Gamma)$
is given by
\begin{equation}
W (\Gamma) = {\int {\cal D}C_\mu e^{i\int dx [{\cal L}_C (G_{\mu\nu}^S) +
{\cal L}_{GF}]}\over \int {\cal D} C_\mu e^{i\int dx [{\cal L}_C
(G_{\mu\nu}^S= 0) + {\cal L}_{GF}]}}.
\label{eq:2.15}
\end{equation}
The functional integral (\ref{eq:2.15}) is also Gaussian and has the value (\ref{eq:2.6}) with
$1 \over \epsilon$ replaced by $\mu$. We then have two equivalent
descriptions at all distances of the electromagnetic interaction of two
charged particles.
Note from (\ref{eq:2.2}) and (\ref{eq:2.14}) that the equations
\begin{equation}
\epsilon= {1 \over g_{el}^2}, \hspace{.25in}
\mu= {1 \over g_{mag}^2}
\label{eq:2.16}
\end{equation}
\noindent
define electric and magnetic coupling constants. If the wave number dependent
dielectric constant $\epsilon \rightarrow 0$ at long distances, then $g_{el}
\rightarrow \infty$ and the Maxwell potentials $A_\mu$ are strongly coupled.
By contrast, $g_{mag} \rightarrow 0$, and the dual potentials are weakly
coupled at large distances.
\section{The Heavy Quark Potential in QCD}
The heavy quark potential $V_{q \bar q}$ is determined by the Wilson loop $W
(\Gamma)$ of Yang Mills theory:
\begin{equation}
W (\Gamma) = {\int {\cal D} Ae^{iS_{YM}(A)} tr P\exp (-ie \oint_\Gamma dx^\mu
A_\mu (x))\over \int {\cal D} Ae^{iS_{YM} (A)}}.
\label{eq:3.1}
\end{equation}
\noindent
(See Fig.1)
As usual $A_\mu (x) = {1\over 2}
\lambda_a A_\mu^a (x)$, $tr$ means the trace over color indices, $P$
prescribes the ordering of the color matrices according to the direction fixed
on the loop and $S_{YM}(A)$ is the Yang--Mills action including a gauge fixing
term. We have denoted the Yang--Mills coupling constant by e, i.e.,
\begin{equation}
\alpha_s = {e^2\over 4\pi}.
\label{eq:3.2}
\end{equation}
The spin independent part $V ( \vec R, {\dot{\vec z}}_1, {\dot{\vec z}}_2)$
of $V_{q \bar q}$ is obtained from (\ref{eq:3.1}) by the QCD analogue of
(\ref{eq:2.5}):
\begin{equation}
i \log W (\Gamma) = \int_{t_i}^{t_f} dt V (\vec R, {\dot{\vec z}}_1, {\dot{\vec z}}_2 ).
\label{eq:3.3}
\end{equation}
The spin dependent heavy quark potential $V^{\mbox{\scriptsize spin}}$ is a sum of terms
depending upon quark spin matrices, masses, and momenta: \cite{calc}
\begin{equation}
V^{\mbox{\scriptsize spin}}= V_{LS}^{MAG} + V_{Thomas} + V_{Darwin} + V_{SS},
\label{eq:3.4}
\end{equation}
\noindent
where the notation indicates the physical significance of the individual
terms (MAG denotes magnetic). Each term in (\ref{eq:3.4}) can be obtained from a
corresponding term in $V_{e^+e^-}^{\mbox{\scriptsize spin}}$ by making the replacement
\begin{equation}
\langle \langle F_{\mu \nu} \left(z_1 \right) \rangle \rangle_{Maxwell} \longrightarrow \langle \langle F_{\mu \nu} \left(z_1 \right) \rangle \rangle_{YM} ,
\label{eq:3.5}
\end{equation}
\noindent
where
\begin{equation}
\langle \langle F_{\mu \nu} \left(z_1 \right) \rangle \rangle_{YM} \equiv {\int {\cal D} Ae^{iS_{YM}(A)} tr P \exp [-ie \oint_\Gamma dx^\mu
A_\mu (x)] F_{\mu \nu} \left(z_1 \right) \over \int {\cal D} Ae^{iS_{YM}(A)} tr P \exp [-ie \oint_
\Gamma dx^{\mu} A_\mu (x)]} ,
\label{eq:3.6}
\end{equation}
\noindent
and
\begin{equation}
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - ie [A_\mu, A_\nu],
\label{eq:3.7}
\end{equation}
\noindent
i.e. $\langle\!\langle F_{\mu\nu}(x)\rangle\!\rangle_{YM}$ is the expectation value of
the Yang--Mills field tensor in the presence of a quark and anti--quark moving
along classical trajectories $\vec z_1 (t)$ and $\vec z_2 (t)$ respectively.
The calculation of the heavy quark potential is then reduced to the evaluation
of functional integrals of Yang Mills theory. Because of the strong coupling
at long distances all field configurations can give important contributions to
(\ref{eq:3.1}) and (\ref{eq:3.6}) for large loops $\Gamma$ and there is no simple description
in terms of Yang Mills potentials.
\section{The Dual Description of Long Distance Yang-Mills Theory}
The dual theory described here is a concrete realization of the Mandelstam \linebreak 't Hooft \cite{mand} dual
superconductor picture of confinement. A dual Meissner effect prevents the
electric color flux from spreading out as the distance $R$ between the quark
anti-quark pair increases. As a result a linear potential develops which
confines the quarks in hadrons. Such a dual picture is suggested
by the solution of a truncated set of Dyson
equations of Yang Mills theory \cite{nph81} which gives an effective dielectric constant
$\epsilon(q)\rightarrow q^2/M^2$ as $q^2 \rightarrow 0$ ($M$ is an
undetermined mass scale). As a consequence $\mu = {1 \over \epsilon}
\rightarrow {M^2 \over q^2}$ as $q^2 \rightarrow 0$ so that the dual gluon
becomes massive as is characteristic of dual superconductivity. However, such
a truncation cannot be justified in the strongly coupled domain and duality in
Yang Mills theory remains an hypothesis.
On the other hand, there has been a recent revival of interest in
electric-magnetic duality due to the work of Seiberg and Witten \cite{npb94} on
supersymmetric $N=2$ Yang Mills theory and Seiberg \cite{npb95} on $N=1$ supersymmetric
QCD. The long distance physics of these models, which are asymptotically
free, is described by weakly coupled dual gauge theories. These examples of
non-Abelian gauge theories for which duality can can be inferred provide new
motivation for the duality hypothesis for Yang Mills theory.
The dual theory is
described by an effective Lagrangian density ${\cal L}_{\mbox{\scriptsize eff}}$ in which the fundamental
variables are an octet of dual potentials ${\bf C_{\mu}}$ coupled minimally to
three octets of scalar Higgs fields ${\bf B}_i$ carrying magnetic color
charge \cite{calc,physrev68}. (The gauge coupling constant of the dual theory $g = {2 \pi \over e}$). The Higgs potential
has a minimum at non-zero values ${\bf B}_{0i}$ which have the color
structure
\begin{equation}
{\bf B}_{01} = B_0 \lambda_7 ,
\quad {\bf B}_{02} = B_0 (-\lambda_5),\quad {\bf
B}_{03} = B_0 \lambda_2.
\label{eq:4.1}
\end{equation}
\noindent
The three matrices $\lambda_7, - \lambda_5$ and $\lambda_2$ transform as a
$j=1$ irreducible representation of an $SU(2)$ subgroup of $SU(3)$ and as
there is no $SU(3)$ transformation which leaves all three ${\bf B}_{0i}$
invariant the dual $SU(3)$ gauge symmetry is completely broken and the eight
Goldstone bosons become the longitudinal components of the now massive
$\bf{C}_\mu$.
The basic manifestation of the dual superconducting properties of ${\cal L}_{\mbox{\scriptsize eff}}$ is
that it generates classical equations of motion having solutions \cite{physrev90} carrying
a unit of $Z_3$ flux confined in a narrow tube along the $z$ axis
(corresponding to having quark sources at $z = \pm \infty$). (These solutions
are dual to Abrikosov-Nielsen-Olesen magnetic vortex solutions \cite{abrikosov} in a
superconductor). Before writing ${\cal L}_{\mbox{\scriptsize eff}}$ we briefly describe these classical
solutions. The monopole fields
${\bf B}_i$ have the form : \cite{calc}
\begin{eqnarray}
{\bf B}_1 & = & B_1(x) \lambda_7 + \bar B_1(x)(-\lambda_6)\,,
\nonumber \\
{\bf B}_2 & = & B_2(x)(-\lambda_5) + \bar B_2(x) \lambda_4 \,,
\label{eq:4.2}
\\
{\bf B}_3 & = & B_3(x)\lambda_2 + \bar B_3(x)(-\lambda_1) \,.
\nonumber
\end{eqnarray}
\noindent
We denote
\begin{equation}
\phi_i(x) = B_i(x) - i \bar B_i(x) \,,
\label{eq:4.3}
\end{equation}
\noindent
and look for solutions where the dual potential is proportional to the
hypercharge matrix $Y = {\lambda_8 \over \sqrt{3}}$,
\begin{equation}
{\bf C}_\mu = C_\mu Y ,
\label{eq:4.4}
\end{equation}
\noindent
and where
\begin{equation}
\phi_1(x) = \phi_2(x) \equiv \phi(x), \hspace{.25in}
\phi_3(x) = B_3(x).
\label{eq:4.5}
\end{equation}
\noindent
At large distances from the center of the flux tube in cylindrical
coordinates $\rho,\theta,z$ the boundary conditions are:
\begin{equation}
\vec C \rightarrow - {\hat e_\theta \over g \rho},\quad ~\phi \rightarrow
B_0 e^{i\theta},\quad B_3 \rightarrow B_0, ~\quad \mbox{as} \ ~ \rho \rightarrow \infty .
\label{eq:4.6}
\end{equation}
The non-vanishing of $B_0$ produces a color monopole current confining the
electric color flux. The line integral of the dual potential around a large
loop surrounding the $z$ axis measures this flux, and the
boundary condition (\ref{eq:4.6}) for $\vec{\bf C}$ gives
\begin{equation}
e^{- ig \oint_{loop} \vec {\bf C} \cdot d \vec \ell} = e^{2\pi i Y} =
e^{2\pi\left({i\over 3}\right)},
\label{eq:4.7}
\end{equation}
\noindent
which manifests the unit of $Z_3$ flux in the tube.
The energy per unit
length in this flux tube gives the string
tension $\sigma$: \cite{physrev90}
\begin{equation}
\sigma \sim 24B_0^2 .
\label{eq:4.8}
\end{equation}
The field $\phi (\vec x)$ vanishes at the center of the flux tube. By contrast
$B_3(\vec{x})$ does not couple to quarks and remains close to its vacuum value
for all $\vec x$. For simplicity in the rest of this talk we set
$B_3(x) = B_0$, in which case ${\cal L}_{\mbox{\scriptsize eff}}$ reduces to the Abelian Higgs model.
To couple ${\bf C}_\mu$ to a $q \bar q$ pair separated by a finite distance we
represent quark sources by a Dirac string tensor ${\bf G}_{\mu \nu}^S$. We
choose the dual potential to have the same color structure (\ref{eq:4.4}) as the
flux tube solution. Then ${\bf G}_{\mu \nu}^S$ must
also be proportional to the hypercharge matrix,
\begin{equation}
{\bf G}_{\mu \nu}^S = YG_{\mu \nu}^S ,
\label{eq:4.9}
\end{equation}
\noindent
where $G_{\mu \nu}^S$ is given by (\ref{eq:2.10}), so that one unit of $Z_3$ flux
flows along the Dirac string connecting the quark and anti-quark. With the
ansaetze (\ref{eq:4.9}) and (\ref{eq:4.2})- (\ref{eq:4.5}) along with the simplification
$B_3(x)=B_0$ , the Lagrangian density ${\cal L}_{\mbox{\scriptsize eff}}\left( G_{\mu \nu}^S \right)$ coupling
dual potentials to classical quark sources moving along trajectories $\vec
z_1(t)$ and $\vec z_2(t)$ assumes the form:
\begin{equation}
{\cal L}_{\mbox{\scriptsize eff}} ( G_{\mu \nu}^S ) = - {4 \over 3} {\left( G_{\mu \nu}G^{\mu \nu} \right) \over 4} + { 8 | (\partial_\mu - igC_\mu) \phi |^2 \over 2} - {100 \over 3} \lambda \left( |\phi |^2 - B_0^2 \right)^2 ,
\label{eq:4.10}
\end{equation}
\noindent
where
\begin{equation}
G_{\mu \nu} = \partial_\mu C_\nu - \partial_\nu C_\mu + G_{\mu \nu}^S \,,
\label{eq:4.11}
\end{equation}
\noindent
and
\begin{equation}
g= {2 \pi \over e}.
\label{eq:4.12}
\end{equation}
The first term in ${\cal L}_{\mbox{\scriptsize eff}}$ is the coupling of dual potentials to
quarks, the second is the coupling of the dual potentials to monopole fields
$\phi$, while the third term is the quartic self coupling of the monopole
fields. The numerical factors in (\ref{eq:4.10}) arise from inserting the color
structures (\ref{eq:4.2})- (\ref{eq:4.5}) in the original non-Abelian form of ${\cal L}_{\mbox{\scriptsize eff}}$.
By a suitable redefinition of $\phi$ and $\lambda$ the last two terms can be
written in the standard form of the Abelian Higgs model, while the color
factor $4 \over 3$ in the first term is a consequence of (\ref{eq:4.4}) and (\ref{eq:4.9}),
which combined with the boundary condition (\ref{eq:4.6}) provides the unit of $Z_3$
flux.
We find from (\ref{eq:4.10}) the following values of the dual gluon mass $M$ and the
monopole mass $M_\phi$ :
\begin{equation}
M^2 = 6g^2B_0^2 \hspace{.15in}, \hspace{.25in} M_\phi^2 = {100 \lambda \over 3} B_0^2 .
\label{eq:4.13}
\end{equation}
\noindent
The quantity $g^2/\lambda$ plays the role of a Landau-Ginzburg parameter. Its
value can be estimated by relating the difference between the energy density
at a large distance from the flux tube and the energy density at its center to
the gluon condensate.\cite{physrev90} This procedure gives $g^2/\lambda \simeq 5$. There
remain two free parameters in ${\cal L}_{\mbox{\scriptsize eff}}$, which we take to be
$\alpha_s = { e^2 \over 4 \pi} = {\pi \over g^2 }$ and the string tension
$\sigma$.
We denote by $W_{\mbox{\scriptsize eff}}(\Gamma)$ the Wilson loop of the dual theory,
i.e.,
\begin{equation}
W_{\mbox{\scriptsize eff}} (\Gamma) =
{
\int {\cal D} C_\mu {\cal D} \phi
e ^ {i \int dx [ {\cal L}_{\mbox{\scriptsize eff}} (G_{\mu\nu}^S) + {\cal L}_{GF} ] }
\over
\int {\cal D} C_\mu {\cal D} \phi
e ^ {i \int dx [ {\cal L}_{\mbox{\scriptsize eff}} (G_{\mu\nu}^S=0) + {\cal L}_{GF} ] }
}.
\label{eq:4.14}
\end{equation}
\noindent
\noindent
The functional integral $W_{\mbox{\scriptsize eff}}(\Gamma)$ determines in the effective dual
theory the same physical quantity as $W(\Gamma)$ in Yang-Mills theory, namely
the action for a quark anti-quark pair moving along classical trajectories.
The coupling of dual potentials to Dirac strings in ${\cal L}_{\mbox{\scriptsize eff}} \left(G_{\mu \nu}^S \right)$
plays the role in eq.(\ref{eq:4.14}) for
$W_{\mbox{\scriptsize eff}}(\Gamma)$ of the Wilson loop $Pe^{-ie \oint_\Gamma dx^\mu
A_\mu(x)}$ in eq.(\ref{eq:3.1}) for $W(\Gamma)$.
The assumption that the dual theory describes the long distance $q \bar q$
interaction in Yang-Mills theory then takes the form:
\begin{equation}
W(\Gamma) = W_{\mbox{\scriptsize eff}}(\Gamma), ~{\rm for ~large ~loops ~\Gamma}.
\label{eq:4.15}
\end{equation}
\noindent
Large loops mean that the size $R$ of the loop is large compared to the
inverse of $M$ and $M_\phi$. Since the dual theory is weakly
coupled at large distances we can evaluate $W_{\mbox{\scriptsize eff}}(\Gamma)$ via a
semi-classical expansion to which the classical configuration of dual
potentials and monopoles gives the leading contribution. Furthermore using (\ref{eq:4.15}), we can relate the
expectation value (\ref{eq:3.6}) of the Yang Mills Field tensor at the position of a quark to the corresponding
expectation value of the dual field tensor in the effective theory: \cite{calc}
\begin{equation}
\langle \langle F_{\mu\nu}(z_1)\rangle\rangle_{YM} ={4 \over 3} \langle
\langle \hat G_{\mu \nu}(z_1) \rangle\rangle_{\mbox{\scriptsize eff}} ,
\label{eq:4.16}
\end{equation}
\noindent
where
\begin{equation}
\hat G_{\mu\nu}(x) \equiv {1 \over 2} \epsilon_{\mu \nu \lambda \sigma}
G^{\lambda \sigma}(x),
\label{eq:4.17}
\end{equation}
\noindent
and
\begin{equation}
\langle\!\langle G^{\mu \nu}(z_1)\rangle\!\rangle_{\mbox{\scriptsize eff}} \equiv {\int {\cal
D}
C_\mu {\cal D} \phi e^{i\int dx
({\cal L}_{\mbox{\scriptsize eff}} (G_{\mu\nu}^S) + {\cal L}_{GF})} G^{\mu \nu} (z_1) \over
\int
{\cal D} C_\mu {\cal D} \phi e^{i\int dx
({\cal L}_{\mbox{\scriptsize eff}}
(G_{\mu\nu}^S) + {\cal L}_{GF})}}.
\label{eq:4.18}
\end{equation}
To obtain the spin independent heavy quark potential $V({\vec R},{\dot{\vec z}}_1, {\dot{\vec z}}_2
)$ in the dual theory we replace $W(\Gamma)$ by $W_{\mbox{\scriptsize eff}}(\Gamma)$ in eq.(\ref{eq:3.3}). This
expresses the spin independent heavy quark potential in terms of the zero order and quadratic
terms in the expansion of $i \log W_{\mbox{\scriptsize eff}}(\Gamma)$ for small velocities ${\dot{\vec z}}_1$ and ${\dot{\vec z}}_2$.
The corresponding spin dependent potential in the dual theory is obtained by
making the replacement
\begin{equation}
\langle \langle F_{\mu \nu} (z_1) \rangle \rangle_{YM} \longrightarrow \hspace{.15in}
{4 \over 3} \langle \langle \hat
G_{\mu \nu} (z_1) \rangle \rangle_{\mbox{\scriptsize eff}} ,
\label{eq:4.19}
\end{equation}
\noindent
in the expressions in eq.(\ref{eq:3.4}) for $V^{\mbox{\scriptsize spin}}$.
\section{The Classical Approximation to the Dual Theory}
In the classical approximation all quantities are replaced by their classical
values
\begin{equation}
\langle\langle G_{\mu \nu}(x) \rangle\rangle_{\mbox{\scriptsize eff}} = G_{\mu \nu} (x),\hspace{.25in} i \log W_{\mbox{\scriptsize eff}} = - \int dx {\cal L}_{\mbox{\scriptsize eff}}(G_{\mu \nu}^S ),
\label{eq:5.1}
\end{equation}
\noindent
where $G_{\mu \nu}$ and ${\cal L}_{\mbox{\scriptsize eff}}\left(G_{\mu \nu}^S \right)$ are
evaluated at the solution of the classical equations of motion:
\begin{equation}
\partial^\alpha \left(\partial_\alpha C_\beta - \partial_\beta C_\alpha
\right) = - \partial^\alpha G_{\alpha \beta}^S + j_\beta ^{MON},
\label{eq:5.2}
\end{equation}
\begin{equation}
\left( \partial_\mu - igC_\mu \right)^2 \phi = - {200 \lambda \over 3} \phi
\left(|\phi |^2 - B_0^2 \right) ,
\label{eq:5.3}
\end{equation}
\noindent
where the monopole current $j_{\mu}^{MON}$ is
\begin{equation}
j_{\mu}^{MON} = - 3ig[\phi^* \left(\partial_\mu -
igC_\mu \right) \phi - \phi \left(\partial_\mu + igC_\mu \right) \phi^*].
\label{eq:5.4}
\end{equation}
\noindent
The boundary conditions on $\phi$ are:
\begin{equation}
\phi(x) \rightarrow 0, \quad \mbox{as} \ x \rightarrow y\left(\sigma , \tau \right); \qquad
\phi(x) \rightarrow B_0 \,, \quad \mbox{as} \ x \rightarrow \infty.
\label{eq:5.5}
\end{equation}
\noindent
The vanishing of $\phi(x)$ on the Dirac sheet $y^{\mu}(\sigma,\tau)$
produces a flux tube with energy concentrated in the
neighborhood of the string connecting the quark anti-quark pair. Using the minimum energy solution corresponding to a straight
line string , we evaluate $i
\log W_{\mbox{\scriptsize eff}}$ to second order in the velocities ${\dot{\vec z}}_1,$ and
${\dot{\vec z}}_2$ and obtain
the spin independent heavy quark
potential.
At large separations $V (\vec R, {\dot{\vec z}}_1, {\dot{\vec z}}_2 )$ is linear
in $R$ since the monopole current screens the color field of the quarks
so that a color electric Abrikosov-Nielsen-Olesen vortex forms between
the moving $q \bar q$ pair. For the case of circular motion,
$({\dot{\vec z}}_i \cdot \vec R = 0$,
$ {\dot{\vec z}}_2 = - {\dot{\vec z}}_1 )$, we find:
\begin{equation}
V \rightarrow \sigma R
\left[ 1 - A { (\dot{\vec z_1} \times \vec R)^2 \over R^2} \right] \,,
\quad \mbox{as} \ R \rightarrow \infty \,,
\label{eq:5.6}
\end{equation}
\noindent
where
\begin{equation}
A \simeq .21 \sigma .
\label{eq:5.7}
\end{equation}
\noindent
The constant $A$ determines the long distance moment of inertia $I(R)$ of the
rotating flux tube:
\begin{equation}
\lim_{R\rightarrow\infty} I(R) = {1\over 2} (AR)R^2.
\label{eq:5.8}
\end{equation}
At small separations the color field generated by the quarks expels the
monopole condensate from the region between them and as
$R \rightarrow 0$, $V$ approaches the one gluon
exchange result, ${4 \over 3} V_D$. See eq.(\ref{eq:2.7}).
As the simplest application of this potential, we add relativistic kinetic
energy terms to obtain a classical Lagrangian, and calculate classically the energy and angular momentum of $q \bar q$ circular orbits, which are those which have the largest angular momentum $J$ for a given energy. We find \cite{95proc} a Regge trajectory $J$ as a function of $E^2$ which for large $E^2$ becomes linear with slope $\alpha^\prime = J/E^2 = 1/8\sigma \left(1-A/\sigma \right)$. Then (\ref{eq:5.7}) gives $\alpha^\prime \approx 1/6.3 \sigma$, which is close to the string model relation $\alpha^\prime = {1 \over 2 \pi \sigma}$. This comparison shows how at the classical level a string model emerges when the velocity dependence of the $q \bar q$ potential is included.
To calculate the spin dependent heavy quark potential we use (\ref{eq:4.19}) and (\ref{eq:5.1}) to evaluate $V^{\mbox{\scriptsize spin}}$
(\ref{eq:3.4}) in the classical approximation to the dual theory.
The resulting expressions are given in reference 1. Here we discuss
only the result for the spin-spin interaction $V_{SS}^{\mbox{\scriptsize spin}}$ between the color magnetic moments of the quark anti-quark pair.
This magnetic dipole interaction is determined by the gradient of the Greens function $G \left( \vec x, \vec x^\prime \right)$ describing the interaction
of monopoles:
\begin{equation}
V_{SS}^{\mbox{\scriptsize spin}} = {4\over 3} {e^2\over m_1 m_2}
\Bigg\{ (\vec S_1 \cdot \vec S_2)
\delta (\vec z_1 - \vec z_2) - {(\vec S_1 \cdot \vec\nabla)
(\vec S_2 \cdot \vec \nabla^\prime)} G (\vec x, \vec x^\prime)
\Bigg|_{\vec x= \vec z_1, \vec x^\prime= \vec z_2 }\Bigg\}.
\label{eq:5.9}
\end{equation}
$G$ satisfies the following equation obtained from eq.(\ref{eq:5.2}) for $C^0$ :
\begin{equation}
[-\bigtriangledown^2 + 6g^2 \phi^2(\vec x)]G \left(\vec x, \vec x^\prime \right) = \delta \left( \vec{x} - \vec{x}^\prime \right)\,,
\label{eq:5.10}
\end{equation}
\noindent
where $\phi (\vec x)$ is the static monopole field. ($\phi(x)$ is real so that the monopole charge density $j^0(x) = 6g^2
\phi^2(x)C^0$.) Since $\phi (\vec x)$ approaches its vacuum value $B_0$ as $\vec x
\rightarrow \infty$, $G$ vanishes exponentially at large distances:
\begin{equation}
G(\vec x, \vec x')
\mathrel {\mathop {\longrightarrow}_{\vec x \to \infty}}
{e^{-M |\vec x - \vec x'|}\over 4\pi |\vec x - \vec x'|}\,.
\label{eq:5.11}
\end{equation}
\noindent
See eq.(\ref{eq:4.13}).
The dual Higgs mechanism then produces the long distance
Yukawa potential (\ref{eq:5.11}) between
monopoles along with the linear potential (\ref{eq:5.6}) between quarks. The resulting
suppression of the color magnetic interaction between quarks is an unambiguous
prediction of electric-magnetic duality.
\section{Fluctuations of the Flux Tube and Effective String Theory}
To evaluate the contributions to $W_{\mbox{\scriptsize eff}}$ arising
from fluctuations of the shape and length of the flux tube we must
integrate over field configurations generated by all strings connecting
the $q \bar q$ pair. This amounts to doing a functional integral over
all polarization tensors $G_{\mu \nu}^S(x)$. Similar integrals have
recently been carried out by Akhmedov et al. \cite{string} in the case
$\lambda \rightarrow \infty$. By changing from field variables to
string variables ,the functional integral over $G_{\mu \nu}^S(x)$ is
replaced by a functional integral over corresponding world sheets
$y^\mu \left(\sigma,\tau \right)$, multipled by an appropriate Jacobian
and there results \cite{string} an effective string theory free from
the conformal anomaly \cite{plb81} Such techniques if extended to
finite $\lambda $ could be applied to $W_{\mbox{\scriptsize eff}}$ to
obtain a corresponding effective string theory . The leading long
distance contribution to the static potential due to fluctuations of
the string which is independent of the details of the string theory
would then have the universal value $ - {\pi \over 12R} $.
\cite{npb81}
\section{Conclusion}
We have obtained an expression for the heavy quark potential $V_{q\bar
q}$ in terms of an effective Wilson loop $W_{\mbox{\scriptsize eff}}
(\Gamma)$ determined by the dynamics of a dual theory which is weakly
coupled at long distances. The classical approximation gives the
leading long distance contribution to $W_{\mbox{\scriptsize
eff}}(\Gamma)$ and yields a velocity dependent spin dependent heavy
quark potential which for large $R$ becomes linear in $R$ and which for
small $R$ approaches lowest order perturbative QCD. The dual theory
cannot describe QCD at shorter distances, where radiative corrections
giving rise to asymptotic freedom become important. At such distances
the dual potentials are strongly coupled and the dual description is no
longer appropriate.
As a final remark we note that the dual theory is an $SU(3)$ gauge
theory, like the original Yang-Mills gauge theory. However, the
coupling to quarks selected out only Abelian configurations of the dual
potential. Therefore, our results for the $q\bar q$ interaction do not
depend upon the details of the dual gauge group and should be regarded
more as consequences of the general dual superconductor picture rather
than of our particular realization of it.
\section{Acknowledgments}
I would like to thank N. Brambilla and G. M. Prosperi for the opportunity to
attend this conference and for their important contributions
to the work presented here.
\section{References}
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\section{Introduction}
The strong CP problem is one of the most intriguing issues of modern particle
physics. The additional term in the QCD Lagrangian
\begin{equation}
{\cal L}= \theta\fr{g^2_3}{16\pi^2} G^a_{\mu\nu}\mbox{$\tilde{G}$}^a_{\mu\nu}
\end{equation}
violates P and CP symmetries \cite{theta}. In the electroweak theory, the
diagonalization of the quark mass matrices $M_u$ and $M_d$ involves chiral
rotations and brings the additional contribution to the theta term:
\begin{equation}
\bar{\theta}=\theta+arg(det M_uM_d)
\end{equation}
The current experimental limits on the electric dipole moment (EDM) of the
neutron put a severe constraint on the $\bar{\theta}$ parameter. The chiral
algebra calculation of the neutron EDM induced by the theta term \cite{CDVW}
gives the following prediction:
\begin{equation}
d_n\simeq 3.6\times10^{-16}\bar{\theta}\,e\cdot cm.
\end{equation}
Together with the current neutron EDM constraints it implies the limit
$\bar{\theta}<10^{-10}$. Bearing in mind other alternative ways to calculate EDM
and the big diversity of the results (See, for example the review \cite{Chang})
we shall assume here the following milder limit for $\bar{\theta}$:
\begin{equation}
\bar{\theta}<10^{-9}.
\label{eq:limit}
\end{equation}
The extreme smallness of $\bar{\theta}$ could be explained theoretically in
different manners. The most popular solution for strong CP problem is to allow
the dynamical relaxation of $\bar{\theta}$ through the axion mechanism
\cite{PQ}. Since no axion, visible or invisible, is found so far, one has to
consider other alternative ways to obtain naturally small $\bar{\theta}$
\cite{LR,NB}.
\section{Radiative corrections to $\bar{\theta}$ }
In recent works Kuchimanchi \cite{K} and Mohapatra and Rasin \cite{MR1},
\cite{MR2} proposed a solution for the strong CP-problem in the framework of the
supersymmetric models conserving parity. The theta parameter in the Lagrangian
is simply set to zero above some scale $M_{W_R}$ where parity and CP are the
exact symmetries of the theory. After spontaneous symmetry breaking, at the
scale where $W_R$ becomes massive, the $\bar{\theta}$ parameter picks up no
contribution from $arg(det M_uM_d)$. This is because the minimum of the
superpotential corresponds to the real vacuum expectations values of scalar
fields which leads to hermitean mass matrices \cite{K,MR1,MR2}. It does not
mean, however, that the strong CP problem is solved; the theta term can be
generated through radiative corrections if there is a CP-violating source in the
theory.
It is clear that to ensure these radiative corrections $\bar{\theta}_{rad}$ to
satisfy the limit (\ref{eq:limit}) and thus to solve the CP-problem completely,
one has to eliminate all extra sources of CP-violation beyond the
Kobayashi-Maskawa (KM) phase. The latter provides a {\em minimal} content of
CP-violation. If the contribution to $\bar{\theta}$ from KM phase happens to be
large, this means that one cannot obtain the viable solution to the strong CP
problem without fine tuning. This question was studied in the framework of pure
SM \cite{EG,Kh}, where radiative corrections to $\bar{\theta}$ arise first in
the order $\mbox{$\alpha$}_sG_F^2m_c^2m_s^2$ times the CP-odd KM invariant \cite{Kh}, and in
the MSSM with the KM mechanism of CP-violation \cite{DGH} where the result also
is found to be much smaller than $10^{-9}$.
Here we address the same question to the generic left-right supersymmetric model
and calculate the theta term unduced by the radiative corrections through the KM
type of CP-violation. The simple estimate of the upper limit for $\bar{\theta}$,
$\bar{\theta}<\fr{\mbox{$\alpha$}_s}{64\pi^3}
\mbox{Im}(V^*_{td}V_{tb}V^*_{cb}V_{cd})\times \,\log(M_{W_R}/M_{W_L})\sim 10^{-8}$,
presented in the work \cite{MR2} is not satisfactory because it can predict the
electric dipole moment of the neutron one order of magnitude above the present
experimental limit. This estimate does not take into account the dependence of
the quark masses which should be associated with the KM-type of CP-violation.
As usual, the potentially large CP-violating effects emerge through the one loop
induced by quark-squark-gluino interaction. Following the works \cite{K,MR1,MR2}
we take all new CP-violating phases specific for supersymmetric models to be
equal to zero as the result of the parity conservation at $\Lambda_{GUT}$ scale:
\begin{equation}
A=A^*;\; B=B^*;\; m_{\lambda_i}=m_{\lambda_i}^*;\; \mu_{ij}= \mu_{ij}^*.
\end{equation}
The left-right symmetry imposed on the interaction of the quarks with Higgs
bidoublets $\Phi_1$ and $\Phi_2$ requires the hermiticity of the Yukawa
matrices:
\begin{eqnarray}
{\cal L}_Y=Y_1\bar{Q}_L \Phi_1 Q_R + Y_2 \bar{Q}_L \Phi_2
Q_R\,+\,H.c.\nonumber\\
Y_1=Y_1^\dagger;\;\;Y_2=Y_2^\dagger
\end{eqnarray}
We do not specify here the particular content of the Higgs sector giving masses
to $M_{W_R}$ in order to obtain the maximum of generality. The reality of the
vacuum expecation values (VEV's) for Higgs bidoublets $\Phi_1$ and $\Phi_2$,
\begin{equation}
\langle\Phi_1\rangle =
\left(\begin{array}{cc}\kappa_1&0\\0&\kappa'_1\end{array}\right);\;
\langle\Phi_2\rangle =
\left(\begin{array}{cc}\kappa'_2&0\\0&\kappa_2\end{array}\right),
\end{equation}
corresponds to the minimum of the superpotential \cite{K,MR1,MR2}. It ensures
the hermiticity of the mass matrices $M_u$ and $M_d$ and provides the same KM
matrices for left- and right-handed charged currents. To get the simplest
relations between mass matrices and Yukawa couplings and to avoid the problems
with flavour changing neutral currents, we assume for the moment that
$\kappa_1'=\kappa_2'=0$. Then $M_u$ and $M_d$ read as follows:
\begin{equation}
M_u=\kappa_1Y_1\equiv \kappa_u\lambda_u;\;\;M_d=
\kappa_2Y_2^\dagger\equiv\kappa_d\lambda_d,
\end{equation}
where $\kappa_u$ and $\kappa_d$, $\lambda_u$ and $\lambda_d$ are introduced from
matter of convenience. As in the MSSM there is one additional free parameter,
$\tan\beta=\kappa_u/\kappa_d$.
Let us now turn to the squark mass sector. The mass matrix for the down type
squarks has the following general form:
\begin{equation}
(\tilde{D}_L^\dagger\;
\tilde{D}_R^\dagger)
\left(
\begin{array}{cc}
m_L^2+c_u \lambda_u^2+ c_d \lambda_d^2&{\cal A}_d\\
{\cal A}_d^\dagger &m_R^2+c_u' \lambda_u^2+ c_d' \lambda_d^2 ,
\end{array}
\right)
\left(\begin{array}{c}
\tilde{D}_L\\
\tilde{D}_R
\end{array}\right),
\label{eq:mass}
\end{equation}
where ${\cal A}_d=(A-\mu\tan\beta)(M_d+a_d\lambda_d^2M_d+
a_u\lambda_u^2M_d+a'_uM_d\lambda_u^2)$.\newline
The coefficients $c_u,\, c_u',\,c_d,\, c_d',\, a_d,\, a_u,\,a'_u$ appear either
at the tree level or in the one-loop renormalization from $\Lambda_{GUT}$. The
obvious requirement of the L-R symmetry is:
\begin{equation}
m_L=m_R,\; c_d= c_d',\; c_u= c_u'\; a_u= a_u'.
\label{eq:LR}
\end{equation}
As a result the mass matrix (\ref{eq:mass}) differs from that of the MSSM where
$c_u'=0$ and $a_u'=0$. The values of all these coefficients depend on many
additional parameters and we simply assume here the following estimate: $
c_u\sim c_u'\sim m_{susy}^2(16\pi^2)^{-1}\ln(\Lambda_{GUT}^2/M_{W_R}^2)\sim
{\cal O}(m_{susy}^2)$.
Let us now estimate the CP-violating mass term for quarks induced by the
squark-gluino loop. The characteristic loop momenta are of order $m_{susy}$ and
as a first approximation we can expand the propagators of squarks in series of
the fermion U-quark Yukawa couplings. This expansion has the following simple
form:
\begin{equation}
(A-\mu \tan\beta)\sum_{n,m}\fr{c_u^nc_u'^m\left(V^
\dagger \lambda_u^{2n}
(VM_dV^\dagger+a_u\lambda_u^2VM_dV^\dagger
+a'_uVM_dV^\dagger \lambda_u^2)
\lambda_u^{2m}V\right)_{ii}}{(p^2-m_L^2)^{n+1}(p^2-m_R^2)^{m+1}},
\label{eq:nm}
\end{equation}
where $V$ is the usual KM matrix and the subscript $ii$ denotes the projection
on the initial flavour $i$. We have droped also all $c_d$-proportional terms as
they are further suppressed by the D-quark Yukawa couplings. It is clear that if
the conditions of the left-right symmetry (\ref{eq:LR}) are held, the expression
(\ref{eq:nm}) is purely CP-conserving. In other words, in the mass eigenstate
basis, the mixing matrices in the quark-squark-gluino couplings are identical
for left- and right-handed particles and the CP-violating phase drops out at the
one-loop level. However the further running of the mass parameters from the
scale of parity violation down to the electroweak scale necessarily implies the
departure from the exact relations (\ref{eq:LR}). As a result of that, the
CP-violation can be developed, and the lowest-order term where it arises is
$\lambda_t^4\lambda_c^2$. The explicit extraction of the CP-violating part from
Eq. (\ref{eq:nm}) for the external $d$-flavour leads to the following
expression:
\begin{eqnarray}
(A-\mu
\tan\beta)\mbox{Im}(V^*_{td}V_{tb}V^*_{cb}V_{cd})\lambda_c^2\lambda_t^4(m_b-m_s)
\times\nonumber\\ \left[\fr{2a_uc_u(c_u-c_u')+2c_u^2(a_u-a_u')}{(p^2-m^2)^4}+
\fr{2a_uc_u^2(m_R^2-m^2_L)+c_u^2(c_u-c_u')}
{(p^2-m^2)^5} +\fr{c_u^3(m_R^2-m^2_L)}
{(p^2-m^2)^6}\right]
\end{eqnarray}
The differences between the coefficients $c_u$ and $c_u'$, $m_L$ and $m_R$
cannot be calculated without the knowledge of all masses below $M_{W_R}$. For
our purposes, however, it is sufficient to use the reliable estimate for mass
difference $m_L^2-m_R^2\sim
m_{susy}^2 6g_2^2(16\pi^2)^{-1}\ln(M_{W_R}^2/M_{W_L}^2)$ and similar relations
for other coefficients. Combining together all these factors and performing the
trivial integration, we arrive to the following form of the CP-violating quark
masses:
\begin{eqnarray}
{\cal L}_5 \sim\mbox{Im}(V^*_{td}V_{tb}V^*_{cb}V_{cd})\fr{\mbox{$\alpha$}_s}{4\pi}
\fr{3\mbox{$\alpha$}_w}{2\pi}\mbox{ln}\fr{M^2_{W_R}}{M^2_{W_L}}\lambda_c^2
\lambda_t^4 \fr{m_{\tilde{G}}(A-\mu\tan\beta)}{m_{susy}^2}
F(m_{\tilde{G}^2}/m^2_{susy})\times\nonumber\\
\left[(m_b-m_s)\bar{d}i\gamma_5d +(m_d-m_b)\bar{s}i\gamma_5s +
(m_s-m_d)\bar{b}i\gamma_5b \right]
\label{eq:g5}
\end{eqnarray}
The exact form of the function $F$ is not important to us and we can take it
$F\sim {\cal O}(1)$. All three CP-odd masses are suppressed by the square of the
charm quark Yukawa coupling as it should be. To sufficient accuracy we can take
also $\lambda_t^\simeq 1$ because no $\lambda_t$-expansion can be made.
The analogous calculation of the radiatively induced CP-violating gluino mass
term yields the following result:
\begin{equation}
m_{\tilde{G}}-m_{\tilde{G}}*\sim
i\mbox{Im}(V^*_{td}V_{tb}V^*_{cb}V_{cd})\fr{\mbox{$\alpha$}_s}{4\pi}
\fr{3\mbox{$\alpha$}_w}{2\pi}\mbox{ln}\fr{M^2_{W_R}}{M^2_{W_L}}
\fr{(A-\mu\tan\beta)m_b^2\lambda_c^2\lambda_s}{m_{susy}^2}
\end{equation}
Due to the additional suppressions by the D-quark masses, this imaginary gluino
mass gives just a negligible contribution to the theta-term. The main
contribution to $\bar{\theta}$ comes from $\bar{d}i\mbox{$\gamma_{5}$} d$-operator:
\begin{equation}
\bar{\theta}\sim \mbox{Im}(V^*_{td}V_{tb}V^*_{cb}V_{cd})\fr{\mbox{$\alpha$}_s}{4\pi}
\fr{3\mbox{$\alpha$}_w}{2\pi}\mbox{ln}\fr{M^2_{W_R}}{M^2_{W_L}}
\fr{m_{\tilde{G}}(A-\mu\tan\beta)}{m_{susy}^2}
\lambda_c^2\fr{m_b}{m_d}
\label{eq:anal}
\end{equation}
The interesting feature of this formula is a sort of "chiral" enhancement
$m_b/m_d$ which is natural in the framework of the left-right model and simply
impossible in MSSM where the chirality flip is always proportional to the mass
of the external fermion. (This formula is valid only for the situation when the
coefficient in front of $\bar{d}i\gamma_{5}d$ is much smaller than $m_d$).
Substituting the numbers into (\ref{eq:anal}), we get the following estimate for
the theta term developed in the generic left-right supersymmetric model with the
KM-source of CP-violation:
\begin{equation}
|\theta|\sim 10^{-9}\left\{\begin{array}{c}
\tan\beta\; \;\;\;\mbox{for}\;
\tan\beta\gg1\\
{\cal O}(1)\;\;\;\; \mbox{for}\;
\tan\beta\sim 1\\
\tan^{-2}\beta\; \;\;\;\mbox{for}
\;\tan\beta\ll 1
\end{array}\right.
\label{eq:answ}
\end{equation}
When obtaining (\ref{eq:answ}) out of Eq. (\ref{eq:anal}), we took
$\mbox{Im}(V^*_{td}V_{tb}V^*_{cb}V_{cd})\simeq 2.5\cdot 10^{-5}$; $m_{\tilde{G}}\sim
|A|\sim |\mu| \sim m_{susy}$ and $\mbox{ln}(M^2_{W_R}/M^2_{W_L})\simeq 7$.
\section{Discussion}
The common wisdom that the KM mechanism always gives the negligibly small
contribution to the CP-violating flavour-conserving observables apparently is
not true in the case of the left-right supersymmetric model. We have shown that
the radiative corrections to the $\bar{\theta}$-parameter in the generic
left-right supersymmetric model are large, just about the edge of the current
experimental constraint. The only contribution to theta term comes from the
radiative corrections to the $d$-quark mass. The main difference of our answer
(\ref{eq:answ}) in comparison with the simple estimate quoted in \cite{MR2} is
in the additional multiplier $\lambda_c^2m_b/m_d$ which is of the order
$5\cdot10^{-2}$ for $\kappa_1\sim \kappa_2$ and $ \kappa_1'= \kappa_2'=0$. In
this domain of the parameter space, the radiatively induced $\bar{\theta}$ is
hard but not impossible to reconcile with the current experimental limit. One
way for that would be to make the ratio
$m_{\tilde{G}}(A-\mu\tan\beta)/m_{susy}^2$ reasonably small, of order $10^{-1}$.
It turns out that the value of $\bar{\theta}$ is very sensitive to the
relations between different VEV's of the model. Thus, the Eq. (\ref{eq:answ})
suggests that both small and large $\kappa_1/\kappa_2$ are almost excluded. In
the more general formulation of the model $\kappa_1',$ $ \kappa_2'$ also differ
from zero. In that case we observe another contribution to $\bar{\theta}$ which
is suppressed only by the first power of the charm quark Yukawa coupling. This
contribution comes from the cubic term $a_u(\lambda_u^2M_d+M_d\lambda_u^2)\simeq
a_u(\kappa'_1/\kappa^3)M_uM_dM_u$ in the mixing of the left- and right-handed
squarks. The overall factor $\lambda_c^2$ in the estimate (\ref{eq:anal}) is
then substituted for $\lambda_c\kappa'_1/\kappa_1$. To keep this contribution in
agreement with the experimental limit, one has to assume that
$\kappa'_1/\kappa_1<10^{-2}$. This constraint is held even in the limit of very
large $m_{susy}$ and $M_{W_{R}}$ where many other phenomenological constraints
(such as the flavour changing neutral currents) are trivially satisfied. In the
limit $M_{W_{R}}\longrightarrow\infty$ the squark mass matrix keeps the
nonvanishing remnants of the left-right symmetry resembling the case of the
supersymmetric $SO(10)$ models \cite{SO10} where the radiative corrections to
$\theta$ are also known to be large (the last Ref. in \cite{SO10}).
If the strong CP-problem is cured by the axion, CP-violating mass term
(\ref{eq:g5}) has no effect on the physical observables. The EDM of the neutron,
in this case, originates from operators of dimension bigger than 4, such as EDM
of quarks, color EDM of quarks, etc. We give a crude estimate for the EDM of the
neutron using the size of coefficient in front of $\bar{d}\gamma_5d$ in
(\ref{eq:g5}) multiplied by $e/m_{susy}^2$ which is of the order $10^{-29}e\cdot
cm$ for $m_{susy}$ taken close to electroweak scale.
I would like to thank C. Burgess, G. Couture, M. Frank, C. Hamzaoui, H. K\"onig
and A. Zhitnitsky for many helpful discussions. This work is supported by NATO
Science Fellowship, N.S.E.R.C., grant \# 189 630 and Russian Foundation for
Basic Research, grant \# 95-02-04436-a.
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\section{Introduction}}
\newcommand{\section{Conclusions}}{\section{Conclusions}}
\newcommand{\section*{Acknowledgments}}{\section*{Acknowledgments}}
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\renewcommand{\thepage}{S.~De Leo and K.~Abdel-Khalek ~-~ pag.~\arabic{page}}
\newcommand{\mbox{\boldmath $\cal C$}}{\mbox{\boldmath $\cal C$}}
\newcommand{\mbox{\boldmath $\cal H$}}{\mbox{\boldmath $\cal H$}}
\newcommand{\mbox{\boldmath $\cal O$}}{\mbox{\boldmath $\cal O$}}
\newcommand{\mbox{\boldmath $\cal R$}}{\mbox{\boldmath $\cal R$}}
\newcommand{GL(8, \rea )}{GL(8, \mbox{\boldmath $\cal R$} )}
\newcommand{GL(4, \co )}{GL(4, \mbox{\boldmath $\cal C$} )}
\title{OCTONIONIC DIRAC EQUATION}
\author{Stefano De Leo$^{(a,b)}$ and
Khaled Abdel-Khalek$^{(a)}$}
\address{$^{(a)}$~Dipartimento di Fisica - Universit\`a di Lecce\\
$^{(b)}$~Istituto Nazionale di Fisica Nucleare, Sezione di Lecce\\
- Lecce, 73100, Italy -}
\date{Revised Version, July 1996}
\draft
\begin{document}
\maketitle
\begin{abstract}
In order to obtain a consistent formulation of octonionic quantum mechanics
(OQM), we introduce left-right barred operators. Such operators enable us to
find the translation rules between octonionic numbers and $8\times 8$ real
matrices (a translation is also given for $4\times 4$ complex matrices).
We develop an octonionic relativistic free wave
equation, linear in the derivatives. Even if the wave functions are only
one-component we show that four independent solutions, corresponding to
those of the Dirac equation, exist.
\end{abstract}
\renewcommand{\thefootnote}{\sharp\arabic{footnote}}
\section{Introduction}
From the sixties onwards, there has been renewed and intense interest in
the use of octonions in physics~\cite{gur1}. The octonionic algebra has been
in fact linked with a number of interesting subjects: structure
of interactions~\cite{pais}, $SU(3)$ color symmetry and quark
confinement~\cite{gur2,mor}, standard model gauge group~\cite{dix},
exceptional GUT groups~\cite{gur3}, Dirac-Clifford algebra~\cite{edm},
nonassociative Yang-Mills theories~\cite{jos1,jos2}, space-time symmetries in
ten dimensions~\cite{dav}, supersymmetry and supergravity
theories~\cite{sup1,sup2}. Moreover, the recent successful application of
quaternionic numbers in quantum mechanics~\cite{adl,adl1,qua1,qua2,qua3},
in particular in formulating a quaternionic Dirac
equation~\cite{dir1,dir2,dir3,dir4}, suggests
going one step further and using octonions as underlying numerical field.
In this work, we overcome the problems due to the nonassociativity of the
octonionic algebra by introducing left-right barred operators
(which will be sometimes called barred octonions). Such operators
complete the mathematical material introduced in the recent papers
of Joshi {\it et al.}~\cite{jos1,jos2}. Then, we
investigate their relations to $GL(8, \rea )$ and $GL(4, \co )$. Establishing this
relation we find
interesting translation rules, which gives us the opportunity to formulate
a consistent OQM.
The philosophy behind the translation can be concisely expressed by the
following sentence: ``There exists at least one version of octonionic
quantum mechanics where the standard quantum mechanics is reproduced''.
The use of a complex scalar product (complex geometry)~\cite{hor}
will be the main tool to obtain OQM.
We wish to stress that translation rules don't imply that our octonionic
quantum world (with complex geometry) is equivalent to the standard quantum
world. When translation
fails the two worlds are not equivalent. An interesting case
can be supersymmetry~\cite{rk}.
Similar translation rules, between quaternionic quantum mechanics (QQM)
with complex geometry and standard quantum mechanics, have been recently
found~\cite{qua2}. As an application, such rules can be exploited in
reformulating in a natural way the electroweak sector of the standard
model~\cite{qua3}.
In section II, we discuss octonionic algebra and introduce barred
operators. Then, in Section III, we investigate
the relation between barred octonions and $8\times 8$ real matrices.
In this section, we also give the translation rules between octonionic barred
operators and $GL(4, \co )$, which will be very useful in formulating our OQM
(full details of the mathematical material appear elsewhere~\cite{jmp}).
In section IV, we explicitly develop an
octonionic Dirac equation and suggest possible difference between complex
and octonionic quantum theories. In the final section we draw our conclusions.
\section{Octonionic barred operators}
We can characterize the algebras $\mbox{\boldmath $\cal R$}$, $\mbox{\boldmath $\cal C$}$, $\mbox{\boldmath $\cal H$}$ and
$\mbox{\boldmath $\cal O$}$ by the concept of {\tt division algebra} (in which one has no nonzero
divisors of zero). Octonions, which locate a nonassociative division algebra,
can be represented by seven imaginary units $(e_1 , \ldots ,e_7 )$
and $e_0\equiv 1$:
\begin{equation}
{\cal O} =r_{0}+\sum_{m=1}^{7} r_{m}e_{m}
\quad\quad (~r_{0,...,7}~~ \mbox{reals}~) \quad .
\end{equation}
These seven imaginary units, $e_{m}$, obey the noncommutative
and nonassociative algebra
\begin{equation}
e_{m}e_{n}=-\delta_{mn}+ \epsilon_{mnp}e_{p} \quad\quad
(~\mbox{{\footnotesize $m, \; n, \; p =1,..., 7$}}~)
\quad ,
\end{equation}
with $\epsilon_{mnp}$ totally antisymmetric and equal to unity for the seven
combinations
$123, \; 145, \; 176, \; 246, \; 257, \; 347 \; \mbox{and} \; 365$.
The norm, $N({\cal O})$, for the octonions is defined by
\begin{equation}
N({\cal O})=({\cal O}^{\dag}{\cal O})^{\frac{1}{2}}=
({\cal O}{\cal O}^{\dag})^{\frac{1}{2}}=
(r_{0}^{2}+ ... + r_{7}^{2})^{\frac{1}{2}} \quad ,
\end{equation}
with the octonionic conjugate $o^{\dag}$ given by
\begin{equation}
{\cal O}^{\dag}=r_{0}-\sum_{m=1}^{7} r_{m}e_{m} \quad .
\end{equation}
The inverse is then
\begin{equation}
{\cal O}^{-1}={\cal O}^{\dag}/N({\cal O}) \quad\quad (~{\cal O}\neq 0~) \quad .
\end{equation}
We can define an {\tt associator} (analogous to the usual algebraic
commutator) as follows
\bel{ass}
\{x, \; y , \; z\}\equiv (xy)z-x(yz) \quad ,
\end{equation}
where, in each term on the right-hand, we must, first of all, perform
the multiplication in brackets.
Note that for real, complex and quaternionic numbers the associator is
trivially null. For octonionic imaginary units we have
\bel{eqass}
\{e_{m}, \; e_{n}, \; e_{p} \}\equiv(e_{m}e_{n})e_{p}-e_{m}(e_{n}e_{p})=
2 \epsilon_{mnps} e_{s} \quad ,
\end{equation}
with $\epsilon_{mnps}$ totally antisymmetric and equal to unity for the seven
combinations
\[
1247, \; 1265, \; 2345, \; 2376, \; 3146, \; 3157 \; \mbox{and} \; 4567 \quad .
\]
Working with octonionic numbers the associator~(\ref{ass}) is in general
non-vanishing, however, the ``alternative condition'' is fulfilled
\bel{rul}
\{ x, \; y, \; z\}+\{ z, \; y, \; x\}=0 \quad .
\end{equation}
In 1989, writing a quaternionic Dirac equation~\cite{dir2}, Rotelli introduced
a {\tt barred} momentum operator
\begin{equation}
-\bfm{\partial}\mid i \quad\quad [~(-\bfm{\partial}\mid i)\psi\equiv -\bfm{\partial}\psi i~] \quad .
\end{equation}
In a recent paper~\cite{qua2}, based upon the Rotelli operators,
{\tt partially barred quaternions}
\begin{equation}
q+p\mid i \quad\quad [~q, \; p \in \mbox{\boldmath $\cal H$}~] \quad ,
\end{equation}
have been used to formulate a quaternionic quantum mechanics.
A complete generalization for quaternionic numbers is represented by
the following barred operators
\begin{equation}
q_{1} + q_{2}\mid i + q_{3}\mid j + q_{4}\mid k \quad\quad
[~q_{1,...,4} \in \mbox{\boldmath $\cal H$}~] \quad ,
\end{equation}
which we call {\tt fully barred quaternions}, or simply barred
quaternions. They, with their 16 linearly independent elements, form
a basis of $GL(4, \mbox{\boldmath $\cal R$} )$ and are successfully used
to reformulate Lorentz space-time transformations~\cite{rel} and write down a
one-component Dirac equation~\cite{dir4}.
Thus, it seems to us natural to investigate the existence of
{\tt barred octonions}
\begin{equation}
{\cal O}_{0}+ \sum_{m=1}^{7} {\cal O}_{m}\mid e_{m} \quad\quad
[~ {\cal O}_{0,...,7} ~~ \mbox{octonions}~] \quad .
\end{equation}
Nevertheless, we must observe that an octonionic {\tt barred} operator,
\bfm{a\mid b}, which acts on octonionic wave functions, $\psi$,
\[ [~a\mid b~]~\psi \equiv a\psi b \quad , \]
is not a well defined object. For $a\neq b$ the triple product $a\psi b$
could be either $(a\psi)b$ or $a(\psi b)$. So, in order to avoid the
ambiguity due to the nonassociativity of
the octonionic numbers, we need to define left/right-barred
operators. We will indicate {\tt left-barred} operators by
\bfm{a~)~b}, with $a$ and $b$ which represent octonionic numbers. They
act on octonionic functions $\psi$ as follows
\begin{mathletters}
\begin{equation}
[~a~)~b~]~\psi = (a\psi)b \quad .
\end{equation}
In similar way we can introduce {\tt right-barred} operators, defined by
\bfm{a~(~b} ,
\begin{equation}
[~a~(~b~]~\psi = a(\psi b) \quad .
\end{equation}
\end{mathletters}
Obviously, there are barred-operators in which the nonassociativity is not
of relevance, like
\[ 1~)~a = 1~(~a \equiv 1\mid a \quad . \]
Furthermore, from eq.~(\ref{rul}), we have
\[ \{ x, \; y, \; x\}=0 \quad ,\]
so
\[ a~)~a = a~(~a \equiv a\mid a \quad .\]
Besides, it is possible to prove, by eq.~(\ref{rul}),
that each right-barred operator can be
expressed by a suitable combination of left-barred operators. For further
details, the reader can consult the mathematical paper~\cite{jmp}. So we can
represent the most general octonionic operator by only 64 left-barred
objects
\bel{go}
{\cal O}_{0}+\sum_{m=1}^{7} {\cal O}_{m}~)~e_{m} \quad\quad
[~{\cal O}_{0, ...,7}~~ \mbox{octonions}~] \quad .
\end{equation}
This suggests a correspondence between our barred
octonions and
$GL(8, \mbox{\boldmath $\cal R$})$ (a complete discussion about the
above-mentioned relationship is given in the following section).
\section{Translation Rules}
In order to explain the idea of translation, let us look
explicitly at the action of the operators $1\mid e_1$ and $e_2$, on a generic
octonionic function $\varphi$
\begin{equation}
\varphi = \varphi_0
+ e_1 \varphi_1 + e_2 \varphi_2 + e_3 \varphi_3
+ e_4 \varphi_4 + e_5 \varphi_5 + e_6 \varphi_6 + e_7 \varphi_7
\quad [~\varphi_{0,\dots ,7} \in \mbox{\boldmath $\cal R$}~] \quad .
\end{equation}
We have
\begin{mathletters}
\beal{opa}
[~1\mid e_{1}~]~\varphi ~ \equiv ~\varphi e_1 & ~=~ &
e_1 \varphi_0 - \varphi_1 - e_3 \varphi_2 + e_2 \varphi_3
- e_5 \varphi_4 + e_4 \varphi_5 + e_7 \varphi_6 - e_6 \varphi_7 \quad , \\
e_{2}\varphi & ~=~ & e_2 \varphi_0 - e_3 \varphi_1 - \varphi_2 + e_1 \varphi_3
+ e_6 \varphi_4 + e_7 \varphi_5 - e_4 \varphi_6 - e_5 \varphi_7
\quad .
\end{eqnarray}
\end{mathletters}
If we represent our octonionic function $\varphi$ by the following real column
vector
\begin{equation}
\varphi ~ \leftrightarrow ~ \left( \begin{array}{c}
\varphi_0\\
\varphi_1\\
\varphi_2\\
\varphi_3\\
\varphi_4\\
\varphi_5\\
\varphi_6\\
\varphi_7
\end{array}
\right) \quad ,
\end{equation}
we can rewrite the eqs.~(\ref{opa}-b) in matrix form,
\begin{mathletters}
\begin{eqnarray}
\left(
\begin{array}{cccccccc}
0 & $-1$ & 0 & 0 & 0 & 0 & 0 &0\\
1 & 0 & 0 & 0 & 0 & 0 & 0 &0\\
0 & 0 & 0 & 1 & 0 & 0 & 0 &0\\
0 & 0 & $-1$& 0 & 0 & 0 & 0 &0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 &0\\
0 & 0 & 0 & 0 &$ -1$& 0 & 0 &0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 &$-1$\\
0 & 0 & 0 & 0 & 0 & 0 & 1 &0
\end{array}
\right)
\left( \begin{array}{c}
\varphi_0\\
\varphi_1\\
\varphi_2\\
\varphi_3\\
\varphi_4\\
\varphi_5\\
\varphi_6\\
\varphi_7
\end{array}
\right) & = &
\left( \begin{array}{c}
$-$\varphi_1\\
\varphi_0\\
\varphi_3\\
$-$\varphi_2\\
\varphi_5\\
$-$\varphi_4\\
$-$\varphi_7\\
\varphi_6
\end{array} \right) \quad , \\
\left(
\begin{array}{cccccccc}
0 & 0 &$-1$ & 0 & 0 & 0 & 0 &0\\
0 & 0 & 0 & 1 & 0 & 0 & 0 &0\\
1 & 0 & 0 & 0 & 0 & 0 & 0 &0\\
0 & $-1$& 0 & 0 & 0 & 0 & 0 &0\\
0 & 0 & 0 & 0 & 0 & 0 & $-1$&0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 &$-1$\\
0 & 0 & 0 & 0 & 1 & 0 & 0 &0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 &0
\end{array}
\right)
\left( \begin{array}{c}
\varphi_0\\
\varphi_1\\
\varphi_2\\
\varphi_3\\
\varphi_4\\
\varphi_5\\
\varphi_6\\
\varphi_7
\end{array}
\right) & = &
\left( \begin{array}{c}
$-$\varphi_2\\
\varphi_3\\
\varphi_0\\
$-$\varphi_1\\
$-$\varphi_6 \\
$-$\varphi_7\\
\varphi_4\\
\varphi_5
\end{array} \right) \quad .
\end{eqnarray}
\end{mathletters}
In this way we can immediately obtain a real matrix representation for the
octonionic barred operators $1\mid e_{1}$ and $e_{2}$. Following this
procedure we can construct the complete set of translation rules~\cite{jmp}.
Let us now discuss of the relation between octonions and complex matrices.
Complex groups play a critical role in physics. No one can deny the importance
of $U(1, \mbox{\boldmath $\cal C$})$ or $SU(2, \mbox{\boldmath $\cal C$})$. In relativistic
quantum mechanics, $GL(4, \co )$ is essential in writing the
Dirac equation. Having $GL(8, \rea )$, we should be able
to extract its subgroup $GL(4, \co )$. So, we can
translate the famous Dirac-gamma matrices and
write down a new octonionic Dirac equation.
If we analyse the action of left-barred operators on our octonionic wave
functions
\begin{equation}
\psi = \psi_{1} + e_{2} \psi_{2} + e_{4} \psi_{3} + e_{6} \psi_{4} \quad\quad
[~\psi_{1, ..., 4} \in \bfm{\cal C}(1, \; e_{1})~] \quad ,
\end{equation}
we find, for example,
\begin{eqnarray*}
& e_{2}\psi & ~=~ -\psi_{2} + e_{2} \psi_{1} -
e_{4} \psi_{4}^{*} + e_{6} \psi_{3}^{*} \quad ,\\
~[~e_{3}~)~e_{1}~]~\psi ~\equiv ~ & (e_{3}\psi) e_{1} & ~=~
\psi_{2} + e_{2} \psi_{1} + e_{4} \psi_{4}^{*} - e_{6} \psi_{3}^{*} \quad .
\end{eqnarray*}
Obviously, the previous operators $e_2$ or $e_3~)~e_1$ cannot be
represented by matrices, nevertheless we note that their combined action
gives us
\[
e_{2}\psi + (e_{3}\psi)e_{1} = 2 e_{2}\psi_{1} \quad ,
\]
and it allows us to represent the octonionic barred operator
\begin{mathletters}
\begin{equation}
e_{2} \; + \; e_{3}~)~e_{1} \quad ,
\end{equation}
by the $4\times 4$ complex matrix
\begin{equation}
\bamq{0}{0}{0}{0} 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array} \right) \quad .
\end{equation}
\end{mathletters}
Following this procedure we can represent a generic $4\times 4$ complex
matrix by octonionic barred operators.
In Appendix B
we give the full basis of $GL(4, \co )$ in terms of octonionic
left-barred operators. It is clear that, only, particular
combinations of left-barred operators is allowed to
reproduce the associative matrix algebra. In order to make
our discussion smooth, we refer the interested reader to the
mathematical paper~\cite{jmp}.
We can quickly relate
$1\mid e_1$ with the complex matrix $i\openone_{4 \times 4}$ which will be
relevant to an {\tt appropriate} definition for the octonionic momentum
operator~\cite{oqm}. The operator $1\mid e_{1}$ (represented by the matrix
$i \openone_{4\times 4}$) commutes with all operators which can be translated
by $4\times 4$ complex matrices. This is not generally true
for a generic octonionic operator. For example, we can show that the
operator $1\mid e_{1}$ doesn't commute with $e_{2}$, explicitly
\begin{mathletters}
\begin{eqnarray}
e_2 ~ \{ ~[~1\mid e_1 ~] ~\psi ~\} &~\equiv e_{2}(\psi e_{1}) & ~=~ -e_1 \psi_2 - e_3 \psi_1 -
e_5 \psi_4^* - e_7 \psi_3^* \quad ,\\
~[~1\mid e_1 ~] ~ \{ e_2 ~\psi ~\} & ~\equiv (e_{2} \psi) e_{1} & ~=~ -e_1 \psi_2 - e_3 \psi_1 +
e_5 \psi_4^* + e_7 \psi_3^*
\quad .
\end{eqnarray}
\end{mathletters}
The interpretation is simple: $e_{2}$ cannot be represented by a $4\times 4$
complex matrix.
We conclude this section by showing explicitly an octonionic representation
for the Dirac gamma-matrices~\cite{itz}:\\
\pagebreak
\begin{center}
{\tt Dirac representation,}
\begin{mathletters}
\beal{odgm1}
\gamma^{0} & = & \frac{1}{3} -\frac{2}{3} \sum_{m=1}^{3} e_{m}\mid e_{m} +
\frac{1}{3} \sum_{n=4}^{7} e_{n} \mid e_{n} \quad ,\\
\gamma^{1} & = & -\frac{2}{3} e_{6} -\frac{1}{3}\mid e_{6} + e_{5}~)~e_{3} -
e_{3}~)~e_{5} - \frac{1}{3} \sum_{p, \; s =1}^{7} \epsilon_{ps6} e_{p}~)~e_{s}
\quad ,\\
\gamma^{2} & = & -\frac{2}{3} e_{7} -\frac{1}{3}\mid e_{7} + e_{3}~)~e_{4} -
e_{4}~)~e_{3} - \frac{1}{3} \sum_{p, \; s =1}^{7} \epsilon_{ps7} e_{p}~)~e_{s}
\quad ,\\
\gamma^{3} & = & -\frac{2}{3} e_{4} -\frac{1}{3}\mid e_{4} + e_{7}~)~e_{3} -
e_{3}~)~e_{7} - \frac{1}{3} \sum_{p, \; s =1}^{7} \epsilon_{ps4} e_{p}~)~e_{s}
\quad ;
\end{eqnarray}
\end{mathletters}
\end{center}
\section{Octonionic Dirac Equation}
In the previous section we have given the gamma-matrices in three different
octonionic representations. Obviously, we can investigate the
possibility of having a more simpler representation for our octonionic
$\gamma^{\mu}$-matrices, without translation.
Why not
\[ e_{1} \; , \quad e_{2} \; , \quad e_{3} \quad \mbox{and} \quad e_{4}\mid e_{4} ~~~~\]
or
\[ e_{1} \; , \quad e_{2} \; , \quad e_{3} \quad \mbox{and} \quad e_{4}~)~e_{1} \quad ?\]
Apparently, they represent suitable choices. Nevertheless, the octonionic
world is full of hidden traps and so we must proceed with prudence.
Let us start from the standard Dirac equation
\begin{equation}
\gamma^\nu p_{\nu} \psi=m\psi \quad ,
\end{equation}
(we discuss the momentum operator in the paper of ref.~\cite{oqm},
here, $p_{\nu}$ represents the ``real'' eigenvalue of the
momentum operator) and apply
$\gamma^{\mu} p_{\mu}$ to our equation
\begin{equation}
\gamma^{\mu} p_{\mu}(\gamma^{\nu} p_{\nu} \psi)=m \gamma^{\mu} p_{\mu} \psi \quad .
\end{equation}
The previous equation can be concisely rewritten as
\begin{equation}
p^{\mu} p_{\nu} \gamma^{\mu} (\gamma^{\nu} \psi)=m^{2} \psi \quad .
\end{equation}
Requiring that each component of $\psi$ satisfy the standard Klein-Gordon
equation we find the Dirac condition, which becomes in the octonionic world
\bel{odc}
\gamma^{\mu}(\gamma^{\nu}\psi)+\gamma^{\nu}(\gamma^{\mu}\psi)=2g^{\mu \nu} \psi \quad ,
\end{equation}
(where the parenthesis are relevant because of the octonions nonassociative
nature). Using octonionic numbers and no barred operators we can obtain,
from~(\ref{odc}), the standard Dirac condition
\bel{sdc}
\{ \gamma^{\mu}, \; \gamma^{\nu} \} = 2 g^{\mu \nu} \quad .
\end{equation}
In fact, recalling the associator property
[which follows from eq.~(\ref{eqass})]
\[ \{a, \; b, \; \psi \} = - \{b, \; a, \; \psi \} \quad\quad
[~a, \; b \quad \mbox{octonionic numbers}~] \quad , \]
we quickly find the following correspondence relation
\[ (ab+ba)\psi=a(b\psi)+b(a\psi) \quad . \]
We have no problem to write down three suitable gamma-matrices which
satisfy the Dirac condition~(\ref{sdc}),
\begin{equation}
(\gamma^{1}, \; \gamma^{2}, \; \gamma^{3}) \equiv (e_{1}, \; e_{2}, \; e_{3}) \quad ,
\end{equation}
but, barred operators like
\[ e_{4}\mid e_{4} \quad \mbox{or} \quad e_{4}~)~e_{1} \]
cannot represent the matrix $\gamma^{0}$. After straightforward algebraic
manipulations, one can prove that
the barred operator, $e_{4}\mid e_{4}$, doesn't anticommute
with $e_{1}$,
\begin{eqnarray}
e_{1}(e_{4}\psi e_{4})+e_{4}(e_{1}\psi)e_{4} & = &
-2 (e_{3}\psi_2 + e_7 \psi_4 ) \neq 0 \quad\quad
[~\psi=\psi_1 + e_2 \psi_2 +e_4 \psi_3 + e_6 \psi_4 ~] \quad ,
\end{eqnarray}
whereas $e_{4}~)~e_{1}$ anticommutes with $e_1$
\begin{mathletters}
\begin{eqnarray}
e_{1}[(e_{4}\psi) e_{1}]+[e_{4}(e_{1}\psi)]e_{1} & = & 0 \quad ,
\end{eqnarray}
but we know that $\gamma_0^2=1$, whereas
\begin{eqnarray}
\{ e_{4}[(e_{4}\psi) e_{1} ] \} e_1 & = & \psi_1 -e_2 \psi_2 +e_4 \psi_3
-e_6 \psi_4 \neq \psi \quad .
\end{eqnarray}
\end{mathletters}
Thus, we must be satisfied with the octonionic representations given in the
previous section.
We recall that the appropriate momentum operator in OQM with complex
geometry~\cite{oqm} is
\[
{\cal P}^{\mu} \equiv \partial^{\mu} \mid e_{1} \quad .
\]
Thus, the octonionic Dirac equation, in covariant form, is given by
\bel{ode}
\gamma^{\mu}(\partial_{\mu}\psi e_{1})=m\psi \quad ,
\end{equation}
where $\gamma^{\mu}$ are represented by octonionic barred
operators~(\ref{odgm1}-d). We can now proceed in the standard manner.
Plane wave solutions exist [${\bf p}~(\equiv -\bfm{\partial} \mid e_{1}$)
commutes with a generic octonionic Hamiltonian] and are of the form
\begin{equation}
\psi({\bf x}, \; t) = [~u_{1}({\bf p})+e_{2}u_{2}({\bf p})+
e_{4}u_{3}({\bf p})+e_{6}u_{4}({\bf p})~]~e^{-pxe_{1}} \quad\quad
[~u_{1, ... , 4} \in \bfm{\cal C}(1, \; e_{1})~] \quad .
\end{equation}
Let's start with
\[{\bf p} \equiv (0, \; 0, \; p_{z}) \quad , \]
from~(\ref{ode}), we have
\bel{ode2}
E(\gamma^{0} \psi) - p_{z}(\gamma^{3} \psi) =m\psi \quad .
\end{equation}
Using the explicit form of the octonionic operators $\gamma^{0, \; 3}$ and
extracting their action (see appendix A) we find
\bel{appc}
E(u_{1}+e_{2}u_{2}-e_{4}u_{3}-e_{6}u_{4})
-p_{z}(u_{3}-e_{2}u_{4}-e_{4}u_{1}+e_{6}u_{2})
=m(u_{1}+e_{2}u_{2}+e_{4}u_{3}+e_{6}u_{4})
\end{equation}
From~(\ref{appc}), we derive four complex equations:
\begin{eqnarray*}
(E-m)u_{1} & = & +p_{z}u_{3} \quad ,\\
(E-m)u_{2} & = & -p_{z}u_{4} \quad ,\\
(E+m)u_{3} & = & +p_{z}u_{1} \quad ,\\
(E+m)u_{4} & = & -p_{z}u_{2} \quad .
\end{eqnarray*}
After simple algebraic manipulations we find the following octonionic
Dirac solutions:
\begin{eqnarray*}
E=+\vert E \vert & ~~~~~~~~u^{(1)}=N \left( 1 + e_{4}
\frac{p_{z}}{\vert E \vert +m} \right) \quad ,
\quad u^{(2)}= N \left( e_{2}-e_{6} \frac{p_{z}}{\vert E \vert +m} \right)
=u^{(1)}e_{2} \quad ; \\
E=-\vert E \vert & ~~~~~~~~u^{(3)}=N \left(\frac{p_{z}}{\vert E \vert +m}
- e_{4} \right) \quad ,
\quad u^{(4)}=N \left(e_{2}\frac{p_{z}}{\vert E \vert +m} +e_{6} \right)
=u^{(3)}e_{2} \quad ,
\end{eqnarray*}
with $N$ real normalization constant. Setting the norm to
$2\vert E \vert$, we find
\[ N=(\vert E \vert + m)^{\frac{1}{2}} \quad . \]
We now observe (as for the quaternionic Dirac equation) a difference with
respect to the standard Dirac equation.
Working in our representation~(\ref{odgm1}-d) and introducing the
octonionic spinor
\[ \bar{u}\equiv(\gamma_{0} u)^{+}= u_{1}^{*}-e_{2}u_{2}+
e_{4}u_{3}+e_{6}u_{4} \quad\quad [~u= u_{1}+e_{2}u_{2}+
e_{4}u_{3}+e_{6}u_{4}~] \quad ,\]
we have
\begin{equation}
\bar{u}^{(1)}u^{(1)}=u^{(1)}\bar{u}^{(1)}=
\bar{u}^{(2)}u^{(2)}=u^{(2)}\bar{u}^{(2)}=2(m+e_{4}p_{z}) \quad .
\end{equation}
Thus we find
\begin{mathletters}
\bel{os}
u^{(1)}\bar{u}^{(1)}+u^{(2)}\bar{u}^{(2)}=4(m+e_{4}p_{z}) \quad ,
\end{equation}
instead of the expected relation
\bel{cs}
u^{(1)}\bar{u}^{(1)}+u^{(2)}\bar{u}^{(2)}=\gamma^{0} E - \gamma^{3} p_{z} + m \quad .
\end{equation}
\end{mathletters}
Furthermore, the previous difference is compensated if we compare the
complex projection of~(\ref{os}) with the trace of~(\ref{cs})
\begin{equation}
[~(u^{(1)}\bar{u}^{(1)}+u^{(2)}\bar{u}^{(2)})^{OQM}~]_{c}~\equiv~
Tr~[~(u^{(1)}\bar{u}^{(1)}+u^{(2)}\bar{u}^{(2)})^{CQM}~]~=~4m \quad ,
\end{equation}
which suggest to redefine the trace as ``complex'' trace.
We know that spinor relations like~(\ref{os}-b) are relevant in
perturbation calculus, so the previous results suggest to analyze quantum
electrodynamics in order to investigate possible differences between
complex and octonionic quantum field. This could represent the aim of
a future work.
\section{Conclusions}
In the physical literature, we find a method to partially overcome the issues
relating to the octonions nonassociativity. Some people introduces a ``new''
imaginary units ``~$i=\sqrt{-1}$~'' which commutes with all others
octonionic imaginary units, $e_{m}$. The new field is often called
{\tt complexified octonionic field}. Different papers have been written in
such a formalism: Quark Structure and Octonions~\cite{gur2},
Octonions, Quark and QCD~\cite{mor}, Dirac-Clifford algebra~\cite{edm},
Octonions and Isospin~\cite{pen}, and so on.
In literature we also find a Dirac equation formulation by {\tt complexified}
octonions with an embarrassing doubling of solutions: {\sl ``... the wave
functions $\tilde{\psi}$ is not a column matrix, but must be taken as an
octonion. $\tilde{\psi}$ therefore consists of eight wave functions, rather
than the four wave functions of the Dirac equation''}~\cite{pen}.
In this paper we have presented an alternative
way to look at the octonionic world.
No new imaginary unit is necessary to formulate in a consistent way an
octonionic quantum mechanics.
Nevertheless complexified ring division algebras have been used in
interesting works of Morita~\cite{mor2} to formulate the whole standard
model.
Having a nonassociative algebra needs special care. In this work, we
introduced a ``trick'' which allowed us to manipulate octonions without useless
efforts. We summarize the more important results found in previous sections:
\begin{center}
{\tt P - Physical Contents :}
\end{center}
{\tt P1} - We emphasize that a characteristic of our formalism is the
{\em absolute need of a complex scalar product} (in QQM the use of a
complex geometry is not obligatory and thus a question of choice).
Using a complex geometry we
overcame the hermiticity problem and gave the appropriate and unique
definition of momentum operator;
{\tt P2} - A positive feature of this octonionic version of quantum
mechanics, is the appearance of all four standard Dirac free-particle
solutions notwithstanding the one-component structure of the wave functions.
We have the following situation for the division algebras:
\begin{center}
\begin{tabular}{lcccl}
{\sf field} :~~~ & ~~complex,~~ & ~~quaternions,~~ & ~~octonions,~~& \\
{\sf Dirac Equation} :~~~ & $4\times 4$, & $2\times 2$, & $1\times 1$ &
~~~{\footnotesize ( matrix dimension ) ;}
\end{tabular}
\end{center}
{\tt P3} - Many physical result can be reobtained by translation, so we
have one version of octonionic quantum mechanics where the standard
quantum mechanics could be reproduced. This represents for the authors a first
fundamental step towards an octonionic world. We remark that
our translation will not be possible in
all situations, so it is only partial, consistent with the fact that the
octonionic version could provide additional physical predictions.
\begin{center}
{\tt I - Further Investigations :}
\end{center}
We list some open questions for future investigations,
whose study lead to further insights.
{\tt I4} - The reproduction in octonionic calculations of the standard QED
results will be a nontrivial objective, due to the explicit differences in
certain spinorial identities (see section IV). We are going to study
this problem in a forthcoming paper;
{\tt I5} - A very attractive point is to try to treat the strong field
by octonions, and then to formulate in a suitable manner a standard
model, based on our octonionic dynamical Dirac equation.
We conclude emphasizing that the core of our paper is surely represented by
absolute need of adopting a complex geometry within a quantum octonionic
world.
\section*{Appendix A\\
$\gamma^{0, \; 3}$-action on octonionic spinors}
In the following tables, we explicitly show the action on the octonionic
spinor
\[
u=u_{1}+e_{2}u_{2}+e_{4}u_{3}+e_{6}u_{4} \quad\quad [~u_{1,...,4} \in
\bfm{\cal C}(1, \; e_{1})~] \quad , \]
of the barred operators which appear in $\gamma^{0}$ and $\gamma^{3}$. Using such
tables, after straightforwards algebraic manipulations we find
\begin{eqnarray*}
\gamma^{0}u & ~=~ & u_{1}+e_{2}u_{2}-e_{4}u_{3}-e_{6}u_{4} \quad ,\\
\gamma^{3}u & ~=~ & u_{3}-e_{2}u_{4}-e_{4}u_{1}+e_{6}u_{2} \quad .
\end{eqnarray*}
\vs{.5cm}
\begin{center}
\begin{tabular}{l|rrrr}
& & & & \\
$\gamma^{0}$-action~~~ &
$~~~~~~~u_{1}$ & $~~~~~~~e_{2}u_{2}$ & $~~~~~~~e_{4}u_{3}$ &
$~~~~~~~e_{6}u_{4}$\\
& & & & \\
\hline \hline
& & & & \\
$e_{1}\mid e_{1}$ &
$-u_{1}$ & $e_{2}u_{2}$ & $e_{4}u_{3}$ & $e_{6}u_{4}$\\
$e_{2}\mid e_{2}$ &
$-u_{1}^{*}$ & $-e_{2}u_{2}^{*}$ & $e_{4}u_{3}$ & $e_{6}u_{4}$\\
$e_{3}\mid e_{3}$ &
$-u_{1}^{*}$ & $e_{2}u_{2}^{*}$ & $e_{4}u_{3}$ & $e_{6}u_{4}$\\
$e_{4}\mid e_{4}$ &
$-u_{1}^{*}$ & $e_{2}u_{2}^{*}$ & $-e_{4}u_{3}^{*}$ & $e_{6}u_{4}$\\
$e_{5}\mid e_{5}$ &
$-u_{1}^{*}$ & $e_{2}u_{2}$ & $e_{4}u_{3}^{*}$ & $e_{6}u_{4}$\\
$e_{6}\mid e_{6}$ &
$-u_{1}^{*}$ & $e_{2}u_{2}$ & $e_{4}u_{3}$ & $-e_{6}u_{4}^{*}$\\
$e_{7}\mid e_{7}$ &
$-u_{1}^{*}$ & $e_{2}u_{2}$ & $e_{4}u_{3}$ & $e_{6}u_{4}^{*}$
\end{tabular}
\end{center}
\vs{.5cm}
\begin{center}
\begin{tabular}{l|rrrr}
& & & & \\
$\gamma^{3}$-action~~~ &
$~~~~~~~u_{1}$ & $~~~~~~~e_{2}u_{2}$ & $~~~~~~~e_{4}u_{3}$ &
$~~~~~~~e_{6}u_{4}$\\
& & & & \\
\hline \hline
& & & & \\
$e_{4}$ &
$e_{4}u_{1}$ & $-e_{6}u_{2}^{*}$ & $-u_{3}$ & $e_{2}u_{4}$\\
$1\mid e_{4}$ &
$e_{4}u_{1}^{*}$ & $e_{6}u_{2}^{*}$ & $-u_{3}^{*}$ & $-e_{2}u_{4}^{*}$\\
$e_{7}~)~e_{3}$ &
$e_{4}u_{1}^{*}$ & $e_{6}u_{2}$ &
$u_{3}$ & $-e_{2}u_{4}^{*}$\\
$e_{3}~)~e_{7}$ &
$-e_{4}u_{1}^{*}$ & $-e_{6}u_{2}^{*}$ &
$-u_{3}$ & $e_{2}u_{4}$\\
$e_{6}~)~e_{2}$ &
$e_{4}u_{1}^{*}$ & $-e_{6}u_{2}$ &
$u_{3}$ & $-e_{2}u_{4}^{*}$\\
$e_{2}~)~e_{6}$ &
$-e_{4}u_{1}^{*}$ & $-e_{6}u_{2}^{*}$ &
$-u_{3}$ & $-e_{2}u_{4}$\\
$e_{5}~)~e_{1}$ &
$e_{4}u_{1}$ & $e_{6}u_{2}^{*}$ & $u_{3}$ &
$-e_{2}u_{4}^{*}$\\
$e_{1}~)~e_{5}$ &
$-e_{4}u_{1}^{*}$ & $-e_{6}u_{2}^{*}$ & $-u_{3}^{*}$ &
$e_{2}u_{4}^{*}$
\end{tabular}
\end{center}
\pagebreak
\section*{Appendix B}
In the following charts we establish the connection between $4\times 4$
complex matrices and octonionic left/right-barred operators. We indicate with
${\cal R}_{mn}$ (${\cal C}_{mn}$) the $4\times 4$ real (complex) matrices
with 1 ($i$) in $mn$-element and zeros elsewhere.\\
\begin{center}
{\tt $4 \times 4$ complex matrices and left-barred operators:}
\end{center}
\begin{eqnarray*}
{\cal R}_{11} & ~\leftrightarrow~ & \frac{1}{2}~[~1-e_{1}\mid e_{1}~] \\
{\cal R}_{12} & ~\leftrightarrow~ & \frac{1}{6}~[~2 e_{1}~)~e_{3} + e_{3}~)~e_{1} -
2 \mid e_{2} - e_{2} + e_{4}~)~e_{6} - e_{6}~)~e_{4} +
e_{5}~)~e_{7} - e_{7}~)~e_{5} ~] \\
{\cal R}_{13} & ~\leftrightarrow~ & \frac{1}{6}~[~2 e_{1}~)~e_{5} + e_{5}~)~e_{1} -
2 \mid e_{4} - e_{4} + e_{6}~)~e_{2} - e_{2}~)~e_{6} +
e_{7}~)~e_{3} - e_{3}~)~e_{7} ~] \\
{\cal R}_{14} & ~\leftrightarrow~ & \frac{1}{6}~[~2 e_{1}~)~e_{7} + e_{7}~)~e_{1} -
2 \mid e_{6} - e_{6} + e_{2}~)~e_{4} - e_{4}~)~e_{2} +
e_{5}~)~e_{3} - e_{3}~)~e_{5} ~] \\
{\cal R}_{21} & ~\leftrightarrow~ & \frac{1}{2}~[~e_{2} + e_{3}~)~e_{1} ~] \\
{\cal R}_{22} & ~\leftrightarrow~ & \frac{1}{6}~[~1+e_{1}\mid e_{1}+e_{4}\mid e_{4}+
e_{5}\mid e_{5}+e_{6}\mid e_{6}+e_{7}\mid e_{7}~] -
\frac{1}{3}~[~e_{2}\mid e_{2}+e_{3}\mid e_{3}~]\\
{\cal R}_{23} & ~\leftrightarrow~ & \frac{1}{2}~[~-e_{2}~)~e_{4} - e_{3}~)~e_{5} ~] \\
{\cal R}_{24} & ~\leftrightarrow~ & \frac{1}{2}~[~e_{3}~)~e_{7} - e_{2}~)~e_{6} ~] \\
{\cal R}_{31} & ~\leftrightarrow~ & \frac{1}{2}~[~e_{4} + e_{5}~)~e_{1} ~] \\
{\cal R}_{32} & ~\leftrightarrow~ & \frac{1}{2}~[~-e_{5}~)~e_{3} - e_{4}~)~e_{2} ~] \\
{\cal R}_{33} & ~\leftrightarrow~ & \frac{1}{6}~[~1+e_{1}\mid e_{1}+e_{2}\mid e_{2}+
e_{3}\mid e_{3}+e_{6}\mid e_{6}+e_{7}\mid e_{7}~] -
\frac{1}{3}~[~e_{4}\mid e_{4}+e_{5}\mid e_{5}~]\\
{\cal R}_{34} & ~\leftrightarrow~ & \frac{1}{2}~[~e_{5}~)~e_{7} - e_{4}~)~e_{6} ~] \\
{\cal R}_{41} & ~\leftrightarrow~ & \frac{1}{2}~[~e_{6} - e_{7}~)~e_{1} ~] \\
{\cal R}_{42} & ~\leftrightarrow~ & \frac{1}{2}~[~e_{7}~)~e_{3} - e_{6}~)~e_{2} ~] \\
{\cal R}_{43} & ~\leftrightarrow~ & \frac{1}{2}~[~e_{7}~)~e_{5} - e_{6}~)~e_{4} ~] \\
{\cal R}_{44} & ~\leftrightarrow~ & \frac{1}{6}~[~1+e_{1}\mid e_{1}+e_{2}\mid e_{2}+
e_{3}\mid e_{3}+e_{4}\mid e_{4}+e_{5}\mid e_{5}~] -
\frac{1}{3}~[~e_{6}\mid e_{6}+e_{7}\mid e_{7}~]\\
{\cal C}_{11} & ~\leftrightarrow~ & \frac{1}{2}~[~1\mid e_{1}+e_{1}~] \\
{\cal C}_{12} & ~\leftrightarrow~ & \frac{1}{6}~[~-2 e_{1}~)~e_{2} - e_{3} - 2 \mid e_{3}
- e_{2}~)~e_{1} + e_{4}~)~e_{7} + e_{6}~)~e_{5} - e_{5}~)~e_{6} -
e_{7}~)~e_{4} ~] \\
{\cal C}_{13} & ~\leftrightarrow~ & \frac{1}{6}~[~-2 e_{1}~)~e_{4} - e_{5} - 2 \mid e_{5}
- e_{4}~)~e_{1} - e_{6}~)~e_{3} - e_{2}~)~e_{7} + e_{7}~)~e_{2}
+ e_{3}~)~e_{6} ~] \\
{\cal C}_{14} & ~\leftrightarrow~ & \frac{1}{6}~[~-2 e_{1}~)~e_{6} + e_{7} + 2 \mid e_{7}
- e_{6}~)~e_{1} - e_{2}~)~e_{5} +
e_{4}~)~e_{3} + e_{5}~)~e_{2} - e_{3}~)~e_{4} ~] \\
{\cal C}_{21} & ~\leftrightarrow~ & \frac{1}{2}~[~-e_{3} + e_{2}~)~e_{1} ~] \\
{\cal C}_{22} & ~\leftrightarrow ~ & \frac{1}{6}~[~1\mid e_{1} - e_{1} + e_{4}~)~e_{5}
-e_{5}~)~e_{4}-e_{6}~)~ e_{7}+e_{7}~)~ e_{6}~] -
\frac{1}{3}~[~e_{2}~)~ e_{3}-e_{3}~)~e_{2}~]\\
{\cal C}_{23} & ~\leftrightarrow~ & \frac{1}{2}~[~ - e_{2}~)~e_{5} + e_{3}~)~e_{4} ~] \\
{\cal C}_{24} & ~\leftrightarrow~ & \frac{1}{2}~[~ e_{3}~)~e_{6} + e_{2}~)~e_{7}~] \\
{\cal C}_{31} & ~\leftrightarrow~ & \frac{1}{2}~[~- e_{5} + e_{4}~)~e_{1} ~] \\
{\cal C}_{32} & ~\leftrightarrow~ & \frac{1}{2}~[~e_{5}~)~e_{2} - e_{4}~)~e_{3} ~] \\
{\cal C}_{33} & ~\leftrightarrow~ & \frac{1}{6}~[~1 \mid e_{1} - e_{1} +e_{2}~)~ e_{3}
-e_{3}~)~ e_{2}-e_{6}~)~ e_{7}+e_{7}~)~e_{6}~] -
\frac{1}{3}~[~e_{4}~)~ e_{5}-e_{5}~)~e_{4}~]\\
{\cal C}_{34} & ~\leftrightarrow~ & \frac{1}{2}~[~e_{5}~)~e_{6} + e_{4}~)~e_{7} ~] \\
{\cal C}_{41} & ~\leftrightarrow~ & \frac{1}{2}~[~ e_{7} + e_{6}~)~e_{1} ~] \\
{\cal C}_{42} & ~\leftrightarrow~ & \frac{1}{2}~[~- e_{7}~)~e_{2} - e_{6}~)~e_{3} ~] \\
{\cal C}_{43} & ~\leftrightarrow~ & \frac{1}{2}~[~- e_{7}~)~e_{4} -e_{6}~)~e_{5} ~] \\
{\cal C}_{44} & ~\leftrightarrow~ & \frac{1}{6}~[~1\mid e_{1} - e_{1} +e_{2}~)~e_{3}
-e_{3}~)~ e_{2}+e_{4}~)~ e_{5}-e_{5}~)~ e_{4}~] -
\frac{1}{3}~[~e_{7}~)~ e_{6}-e_{6}~)~ e_{7}~]
\end{eqnarray*}
\\
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\bibitem{oqm}
S.~De Leo and K.~Abdel-Khalek, {\em Octonionic Quantum Mechanics and
Complex Geometry}, Prog.~Theor.~Phys. (submitted).
\bibitem{itz}
The eqs.~(\ref{odgm1}-d)
represent the octonionic
counterpart of the complex matrices given on pag.~49 of the book:\\
C.~Itzykson and J.~B.~Zuber, {\it Quantum Field Theory}
(McGraw-Hill, New York, 1985).
\bibitem{pen}
R.~Penney, \nxd{B3}{95}{71}.
\bibitem{mor2}
K.~Morita, \pxxa{67}{1860}{81}; \xxx{68}{2159}{82}; \xxx{70}{1648}{83};
\xxx{72}{1056}{84}; \xxx{73}{999}{84}; \xxx{75}{220}{85}; \xxx{90}{219}{93}.
\end{references}
\end{document} %
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proofpile-arXiv_065-616
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{
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
The field of CMB anisotropies has become one of the main
testing grounds for the theories of structure formation and early universe.
Since the first detection by COBE satellite \cite{smoot}
there have been several new detections on smaller
angular scales (see \cite{reviews} for a recent review).
There is hope that future
experiments such as MAP \cite{map} and COBRAS/SAMBA \cite{cobra}
will accurately measure the anisotropies over the whole sky
with a fraction of a degree angular resolution, which will help to
determine several cosmological parameters with an unprecedented accuracy
\cite{jungman}.
Not all of the cosmological
parameters can be accurately determined by the CMB temperature
measurements. On large angular scales
cosmic variance (finite number of multipole moments on the sky)
limits our ability to extract useful
information from the observational data. If a certain parameter
only shows its signature on large angular scales then the
accuracy with which it can be determined is limited. For example,
contribution from primordial gravity waves, if present, will only
be important on large angular scales. Because both scalar and
tensor modes contribute to the temperature anisotropy one cannot
accurately separate them
if only a small number of independent realizations
(multipoles) contain a significant contribution from tensor modes.
Similarly,
reionization tends to uniformly suppress the temperature
anisotropies for all
but the lowest multipole moments and is thus almost degenerate with
the amplitude
\cite{jungman,bond94}.
It is clear from previous discussion that additional information
will be needed to constrain some of the cosmological
parameters. While the epoch of reionization could in principle be
determined through the high redshift observations, primordial gravity
waves can only be detected at present from CMB observations.
It has
been long recognized that there is additional information present
in the CMB data in the form of linear polarization
\cite{bond87,crittenden93,prev,coul,critt,zh95}.
Polarization
could be particularly useful for constraining the epoch and
degree of reionization because the amplitude is significantly increased
and has a characteristic signature \cite{zal}.
Recently it was also shown that density perturbations (scalar modes)
do not contribute to polarization for a
certain combination of Stokes
parameters, in contrast with the primordial gravity waves
\cite{uros,letter,kks}, which can therefore in principle be detected even
for very small amplitudes.
Polarization information which will potentially become
available
with the next generation of experiments will thus provide
significant additional information that will help to
constrain the underlying cosmological model.
Previous work on polarization has been
restricted to the small scale limit
(e.g. \cite{crittenden93,prev,coul,uros,kosowsky96,polnarev}).
The correlation
functions and corresponding power spectra
were calculated for the Stokes $Q$ and $U$
parameters, which are defined with respect to a fixed coordinate
system in the sky. While such a coordinate system is well defined over a
small patch in the sky, it becomes ambiguous once
the whole sky is considered
because one cannot define a rotationally invariant orthogonal basis on a
sphere. Note that this is not problematic if one is only considering
cross-correlation function
between polarization and temperature \cite{critt,coul},
where one can fix $Q$ or $U$ at a
given point and
average over temperature, which is rotationally invariant. However, if
one wants to analyze the auto-correlation function of polarization or
perform directly the power spectrum analysis on the data
(which, as argued in \cite{uros}, is more efficient in terms of
extracting the signal from the data)
then a more general analysis of polarization is required.
A related problem is the calculation of rotationally invariant power spectrum.
Although it is relatively simple to
calculate $Q$ and $U$ in the coordinate system where the wavevector
describing the perturbation is aligned with the $z$ axis, superposition
of the different modes becomes complicated because $Q$ and $U$ have to be
rotated to a common frame before the superposition can be done.
Only in the small scale limit can this rotation be simply expressed
\cite{uros}, so
that the
power spectra can be calculated.
However, as argued above, this is not
the regime where polarization can make most significant impact
in breaking the
parameter degeneracies caused by cosmic variance. A more general
method that would allow to analyze polarization over the whole sky
has been lacking so-far.
In this paper we present a complete all-sky analysis of polarization
and its corresponding power spectra. In section \S 2 we
expand polarization in the sky in
spin-weighted harmonics \cite{goldberg67,np}, which form a complete
and orthonormal system of tensor functions on the sphere.
Recently, an alternative expansion in tensor harmonics
has been presented \cite{kks}. Our approach differs both in the
way we expand polarization on a sphere and in the way we solve
for the theoretical power spectra.
We use the line of sight integral solution
of the photon Boltzmann equation \cite{sz}
to obtain the correct expressions for the polarization-polarization and
temperature-polarization power spectra both for scalar (\S 3) and
tensor (\S 4) modes.
In contrast with previous work the expressions presented
here are valid for any angular scale and in \S 5 we
show how they reduce to the corresponding small scale expressions.
In section \S 6 we discuss how to generate and analyze all-sky
maps of polarization and what is the accuracy with which one can
reconstruct the various power spectra when cosmic variance
and noise are included. This is followed by discussion and
conclusions in \S 7.
For completeness we review in Appendix the basic properties of spin-weighted
functions. All the calculations in this paper are restricted to a flat
geometry.
\section{Stokes parameters and spin-s spherical harmonics}
The CMB radiation field is characterized by a $2\times 2$ intensity
tensor $I_{ij}$. The Stokes parameters $Q$ and $U$ are defined as
$Q=(I_{11}-I_{22})/4$ and $U=I_{12}/2$, while the temperature
anisotropy is
given by $T=(I_{11}+I_{22})/4$. In principle the fourth
Stokes parameter $V$ that describes circular polarization would also
be needed, but in cosmology it can be ignored because it cannot
be generated through Thomson scattering.
While the temperature is invariant
under a right handed rotation in the plane perpendicular to direction
$\hat{\bi{n}}$,
$Q$ and $U$ transform under rotation by an angle $\psi$ as
\begin{eqnarray}
Q^{\prime}&=&Q\cos 2\psi + U\sin 2\psi \nonumber \\
U^{\prime}&=&-Q\sin 2\psi + U\cos 2\psi
\label{QUtrans}
\end{eqnarray}
where ${\bf \hat{e_1}}^{\prime}=\cos \psi{\bf \hat{e_1}}+\sin\psi{\bf
\hat{e_2}}$
and ${\hat{\bf e_2}}^{\prime}=-\sin \psi{\bf \hat{e_1}}+\cos\psi{\bf
\hat{e_2}}$.
This means we can construct two quantities from the Stokes $Q$
and $U$ parameters that have
a definite value of spin
(see Appendix for a review of spin-weighted functions and their properties),
\begin{equation}
(Q\pm iU)'(\hat{\bi{n}})=e^{\mp 2i\psi}(Q\pm iU)(\hat{\bi{n}}).
\end{equation}
We may therefore expand each of the quantities
in the appropriate
spin-weighted basis
\begin{eqnarray}
T(\hat{\bi{n}})&=&\sum_{lm} a_{T,lm} Y_{lm}(\hat{\bi{n}}) \nonumber \\
(Q+iU)(\hat{\bi{n}})&=&\sum_{lm}
a_{2,lm}\;_2Y_{lm}(\hat{\bi{n}}) \nonumber \\
(Q-iU)(\hat{\bi{n}})&=&\sum_{lm}
a_{-2,lm}\;_{-2}Y_{lm}(\hat{\bi{n}}).
\label{Pexpansion}
\end{eqnarray}
$Q$ and $U$ are defined at a given direction $\bi{n}$
with respect to the spherical coordinate system $(\hat{{\bf e}}_\theta,
\hat{{\bf e}}_\phi)$.
Using the first equation in (\ref{propYs}) one can show that
the expansion coefficients for the polarization variables
satisfy $a_{-2,lm}^*=a_{2,l-m}$. For temperature the relation is
$a_{T,lm}^*=a_{T,l-m}$.
The main difficulty when computing the power spectrum of
polarization in the
past originated in the fact that the Stokes parameters
are not invariant under
rotations in the plane perpendicular to $\hat{\bi{n}}$. While
$Q$ and $U$ are easily calculated in a
coordinate system where the wavevector $\bi k$ is
parallel to $\hat{\bi{z}}$,
the superposition
of the different modes is complicated by the behaviour of $Q$ and $U$
under rotations (equation \ref{QUtrans}). For each wavevector
$\bi k$ and direction on the
sky $\hat{\bi{n}}$ one has to rotate the $Q$ and $U$ parameters from the
$\bi{k}$ and $\hat{\bi{n}}$
dependent basis into a fixed basis on the sky. Only in the
small scale limit is this process well defined, which is why this
approximation has always been assumed in previous work
\cite{crittenden93,prev,coul,uros,kosowsky96}.
However, one can use the spin raising and lowering operators
$\;\raise1.0pt\hbox{$'$}\hskip-6pt\partial\;$ and $\baredth$ defined in Appendix
to obtain spin zero quantities. These
have the advantage of being {\it rotationally invariant}
like the temperature and no ambiguities connected with the
rotation of coordinate system arise. Acting twice with
$\;\raise1.0pt\hbox{$'$}\hskip-6pt\partial\;$, $\baredth$ on $Q\pm iU$ in equation (\ref{Pexpansion}) leads to
\begin{eqnarray}
\baredth^2(Q+iU)(\hat{\bi{n}})&=&
\sum_{lm} \left[{(l+2)! \over (l-2)!}\right]^{1/2}
a_{2,lm}Y_{lm}(\hat{\bi{n}})
\nonumber \\
\;\raise1.0pt\hbox{$'$}\hskip-6pt\partial\;^2(Q-iU)(\hat{\bi{n}})&=&\sum_{lm} \left[{(l+2)! \over (l-2)!}\right]^{1/2}
a_{-2,lm}Y_{lm}(\hat{\bi{n}}).
\end{eqnarray}
The expressions for the expansion coefficients are
\begin{eqnarray}
a_{T,lm}&=&\int d\Omega\; Y_{lm}^{*}(\hat{\bi{n}}) T(\hat{\bi{n}})
\nonumber \\
a_{2,lm}&=&\int d\Omega \;_2Y_{lm}^{*}(\hat{\bi{n}}) (Q+iU)(\hat{\bi{n}})
\nonumber \\
&=&\left[{(l+2)! \over (l-2)!}\right]^{-1/2}
\int d\Omega\; Y_{lm}^{*}(\hat{\bi{n}})
\baredth^2 (Q+iU)(\hat{\bi{n}})
\nonumber \\
a_{-2,lm}&=&\int d\Omega \;_{-2}Y_{lm}^{*}(\hat{\bi{n}}) (Q-iU)(\hat{\bi{n}})
\nonumber \\
&=&\left[{(l+2)! \over (l-2)!}\right]^{-1/2}
\int d\Omega\; Y_{lm}^{*}(\hat{\bi{n}})\;\raise1.0pt\hbox{$'$}\hskip-6pt\partial\;^2 (Q-iU)(\hat{\bi{n}}).
\label{alm}
\end{eqnarray}
Instead of $a_{2,lm}$, $a_{-2,lm}$ it is convenient to introduce their
linear combinations
\cite{np}
\begin{eqnarray}
a_{E,lm}=-(a_{2,lm}+a_{-2,lm})/2 \nonumber \\
a_{B,lm}=i(a_{2,lm}-a_{-2,lm})/2.
\label{aeb}
\end{eqnarray}
These two combinations
behave differently under parity transformation:
while $E$ remains unchanged $B$ changes the sign \cite{np}, in
analogy
with electric and magnetic fields.
The sign convention in equation (\ref{aeb}) makes
these expressions consistent with those defined
previously in the small scale limit \cite{uros}.
To characterize the statistics of the CMB perturbations
only four power spectra are needed,
those for $T$, $E$, $B$ and the cross correlation between $T$ and $E$.
The cross correlation between $B$ and $E$ or $B$ and
$T$ vanishes because
$B$ has the opposite parity of $T$ and $E$.
We will
show this explicitly for scalar and tensor modes in the following
sections.
The power spectra are defined as the rotationally invariant quantities
\begin{eqnarray}
C_{Tl}&=&{1\over 2l+1}\sum_m \langle a_{T,lm}^{*} a_{T,lm}\rangle
\nonumber \\
C_{El}&=&{1\over 2l+1}\sum_m \langle a_{E,lm}^{*} a_{E,lm}\rangle
\nonumber \\
C_{Bl}&=&{1\over 2l+1}\sum_m \langle a_{B,lm}^{*} a_{B,lm}\rangle
\nonumber \\
C_{Cl}&=&{1\over 2l+1}\sum_m \langle a_{T,lm}^{*}a_{E,lm}\rangle
\label{Cls}
\end{eqnarray}
in terms of which,
\begin{eqnarray}
\langle a_{T,l^\prime m^\prime}^{*} a_{T,lm}\rangle&=&
C_{Tl} \delta_{l^\prime l} \delta_{m^\prime m} \nonumber \\
\langle a_{E,l^\prime m^\prime}^{*} a_{E,lm}\rangle&=&
C_{El} \delta_{l^\prime l} \delta_{m^\prime m} \nonumber \\
\langle a_{B,l^\prime m^\prime}^{*} a_{B,lm}\rangle&=&
C_{Bl} \delta_{l^\prime l} \delta_{m^\prime m} \nonumber \\
\langle a_{T,l^\prime m^\prime}^{*} a_{E,lm}\rangle&=&
C_{Cl} \delta_{l^\prime l} \delta_{m^\prime m} \nonumber \\
\langle a_{B,l^\prime m^\prime}^{*} a_{E,lm}\rangle&=&
\langle a_{B,l^\prime m^\prime}^{*} a_{T,lm}\rangle=
0.
\label{stat}
\end{eqnarray}
For real space calculations it is useful to introduce
two scalar quantities $\tilde{E}(\hat{\bi{n}})$ and $\tilde{B}(\hat{\bi{n}})$
defined as
\begin{eqnarray}
\tilde{E}(\hat{{\bi n}})&\equiv&
-{1\over 2}\left[\baredth^2(Q+iU)+\;\raise1.0pt\hbox{$'$}\hskip-6pt\partial\;^2(Q-iU)\right]
\nonumber \\
&=&\sum_{lm}\left[{(l+2)! \over (l-2)!}\right]^{1/2}
a_{E,lm}Y_{lm}(\hat{{\bi n}}) \nonumber \\
\tilde{B}(\hat{\bi n})&\equiv&{i\over 2}
\left[\baredth^2(Q+iU)-\;\raise1.0pt\hbox{$'$}\hskip-6pt\partial\;^2(Q-iU)\right]
\nonumber \\
&=&\sum_{lm}\left[{(l+2)! \over (l-2)!}\right]^{1/2}
a_{B,lm}Y_{lm}(\hat{\bi n})
\label{EBexpansions}
\end{eqnarray}
These variables have the advantage of being rotationally invariant
and easy to calculate in real space. These are not rotationally invariant
versions of
$Q$ and $U$, because $\;\raise1.0pt\hbox{$'$}\hskip-6pt\partial\;^2$ and $\baredth^2$ are differential
operators and are more closely related to the rotationally invariant
Laplacian of $Q$ and $U$.
In $l$ space the two are simply related as
\begin{equation}
a_{(\tilde{E},\tilde{B}),lm}=\left[{(l+2)! \over (l-2)!}\right]^{1/2}
a_{(E,B),lm}.
\label{eblm}
\end{equation}
\section{Power Spectrum of Scalar Modes}
The usual starting point for solving the radiation transfer
is the Boltzmann equation. We will expand
the perturbations in Fourier modes characterized by wavevector $\bi{k}$.
For a given Fourier mode we can work in the
coordinate system where
$\bi{k} \parallel \hat{\bi{z}}$ and
$(\hat{{\bf e}}_1,\hat{{\bf e}}_2)=(\hat{{\bf e}}_\theta,
\hat{{\bf e}}_\phi)$.
For each plane wave the scattering can be described as the transport through
a plane
parallel medium \cite{chandra,kaiser}.
Because of azimuthal symmetry only $Q$
Stokes parameter is generated in this frame and its amplitude
only depends on the
angle between the photon direction and wavevector,
$\mu=\hat{\bi{n}}\cdot\hat{\bi{k}}$.
The Stokes parameters for this mode are
$Q=\Delta_P^{(S)}(\tau,k,\mu)$ and $U=0$,
where the superscript $S$ denotes
scalar modes,
while the temperature anisotropy is denoted with
$\Delta_T^{(S)}(\tau,k,\mu)$. The Boltzmann equation
can be written in the synchronous gauge as \cite{bond87,mabert}
\begin{eqnarray}
\dot\Delta_T^{(S)} +ik\mu \Delta_T^{(S)}
&=&-{1\over 6}\dot h-{1\over 6}(\dot h+6\dot\eta)
P_2(\mu) +\dot\kappa\left[-\Delta_T^{(S)} +
\Delta_{T0}^{(S)} +i\mu v_b +{1\over 2}P_2(\mu)\Pi
\right] \nonumber \\
\dot\Delta_P^{(S)} +ik\mu \Delta_P^{(S)} &=& \dot\kappa \left[
-\Delta_P^{(S)} +
{1\over 2} [1-P_2(\mu)] \Pi\right] \nonumber \\
\Pi&=&\Delta_{T2}^{(S)}
+\Delta_{P2}^{(S)}+
\Delta_{P0}^{(S)}.
\label{Boltzmann}
\end{eqnarray}
Here the derivatives are taken with respect to the conformal time $\tau$.
The differential optical depth for Thomson scattering is denoted as
$\dot{\kappa}=an_ex_e\sigma_T$, where $a(\tau)$
is the expansion factor normalized
to unity today, $n_e$ is the electron density, $x_e$ is the ionization
fraction and $\sigma_T$ is the Thomson cross section. The total optical
depth at time $\tau$ is obtained by integrating $\dot{\kappa}$,
$\kappa(\tau)=\int_\tau^{\tau_0}\dot{\kappa}(\tau) d\tau$.
The sources in these equations involve
the multipole moments of temperature and polarization, which
are defined as $ \Delta(k,\mu)=\sum_l(2l+1)(-i)^{l}\Delta_l(k)P_l(\mu)$,
where $P_l(\mu)$ is the Legendre polynomial of order $l$.
Temperature anisotropies have additional sources
in metric perturbations $h$ and $\eta$
and in baryon velocity term $v_b$.
To obtain the complete solution we need to evolve the anisotropies
until the present epoch and integrate over all
the Fourier modes,
\begin{eqnarray}
T^{(S)}(\hat{\bi{n}})&=&\int d^3 \bi{k} \xi(\bi{k})\Delta_T^{(S)}(\tau=\tau_0,k,\mu) \nonumber \\
(Q^{(S)}+iU^{(S)})(\hat{\bi{n}})
&=&\int d^3 \bi{k} \xi(\bi{k})e^{-2i\phi_{k,n}}\Delta_P^{(S)}
(\tau=\tau_0,k,\mu) \nonumber \\
(Q^{(S)}-iU^{(S)})(\hat{\bi{n}})
&=&\int d^3 \bi{k} \xi(\bi{k})e^{2i\phi_{k,n}}\Delta_P^{(S)}
(\tau=\tau_0,k,\mu),
\end{eqnarray}
where
$\phi_{k,n}$ is the angle needed to rotate
the $\bi{k}$ and $\hat{\bi{n}}$ dependent basis to a
fixed frame in the sky. This rotation
was a source of complications in previous
attempts to characterize the CMB polarization. We will avoid it in what
follows by working with the rotationally invariant quantities.
We introduced $\xi(\bi{k})$, which
is a random variable used to characterize the initial
amplitude of the mode. It has the following statistical property
\begin{equation}
\langle \xi^{*}(\bi{k_1})\xi(\bi{k_2})
\rangle=
P_\phi(k)\delta(\bi{k_1}- \bi{k_2}),
\end{equation}
where $P_\phi(k)$ is the initial power spectrum.
To obtain the power spectrum we
integrate the Boltzmann equation (\ref{Boltzmann})
along the line of sight \cite{sz}
\begin{eqnarray}
\Delta_T^{(S)}(\tau_0,k,\mu) &=&
\int_0^{\tau_0} d\tau e^{ix \mu} S_T^{(S)}(k,\tau) \nonumber \\
\Delta_P^{(S)}(\tau_0,k,\mu) &=& {3 \over 4}(1-\mu^2)\int_0^{\tau_0} d\tau
e^{ix \mu}g(\tau)\Pi(k,\tau) \nonumber \\
S_T^{(S)}(k,\tau)&=&g\left(\Delta_{T,0}+2 \dot{\alpha}
+{\dot{v_b} \over k}+{\Pi \over 4 }
+{3\ddot{\Pi}\over 4k^2 }\right)\nonumber \\
&+& e^{-\kappa}(\dot{\eta}+\ddot{\alpha})
+\dot{g}\left(\alpha+{v_b \over k}+{3\dot{\Pi}\over 4k^2 }\right)
+{3 \ddot{g}\Pi \over
4k^2} \nonumber \\
\Pi&=&\Delta_{T2}^{(S)}
+\Delta_{P2}^{(S)}+
\Delta_{P0}^{(S)},
\label{integsolsc}
\end{eqnarray}
where $x=k (\tau_0 - \tau)$
and $\alpha=(\dot h + 6 \dot \eta)/2k^2$.
We have introduced the visibility function $g(\tau)=\dot{\kappa}
{\rm exp}(-\kappa)$. Its peak
defines the epoch of recombination, which gives the
dominant contribution to the CMB anisotropies.
Because in the $\bi{k}\parallel \hat{\bi{z}}$ coordinate
frame $U=0$ and $Q$ is only
a function of $\mu$ it follows from equation
(\ref{operators1}) that
$\baredth^2(Q+iU)=\;\raise1.0pt\hbox{$'$}\hskip-6pt\partial\;^2(Q-iU)$, so that ${}_2a_{lm}={}_{-2}a_{lm}$.
Scalar modes thus contribute only to the $E$ combination and
$B$ vanishes identically.
Acting with the spin raising operator
twice
on the integral solution for $\Delta_P^{(S)}$ (equation
\ref{integsolsc}) leads to the following
expressions for the scalar polarization $\tilde{E}$
\begin{eqnarray}
\Delta_{\tilde{E}}^{(S)}(\tau_0,k,\mu)&=&-{3 \over 4}
\int_0^{\tau_0} d\tau g(\tau)\Pi(\tau,k)
\; \partial^2_{\mu} \left[(1-\mu^2)^2 e^{ix\mu}
\right] \nonumber \\
&=&{3 \over 4}\int_0^{\tau_0} d\tau g(\tau)\Pi(\tau,k)\; (1+\partial_x^2)^2
\left(x^2 e^{ix\mu} \right).
\label{tilEs}
\end{eqnarray}
The power spectra defined in equation
(\ref{Cls}) are rotationally invariant quantities
so they can be calculated in the frame
where $\bi{k} \parallel \hat{\bi{z}}$
for each Fourier mode and then integrated over all the modes,
as different modes are
statistically independent. The present day
amplitude for each mode depends both
on its evolution and on its
initial amplitude.
For temperature anisotropy $T$ it is given by \cite{sz}
\begin{eqnarray}
C_{Tl}^{(S)}&=&
{1 \over 2l+1} \int d^3\bi{k} P_\phi(k)
\sum_m \left|\int d\Omega Y^*_{lm}(\hat{\bi{n}})
\int_0^{\tau_0} d\tau
S^{(S)}_T(k,\tau)
\; e^{ix\mu}\right|^2
\nonumber \\
&=&(4\pi)^2\int k^2dkP_\phi(k)\left[
\int_0^{\tau_0} d\tau
S^{(S)}_T(k,\tau)j_l(x) \right]^2
\end{eqnarray}
where $j_l(x)$ is the spherical Bessel function of order $l$ and we
used that in the $\bi{k} \parallel \hat{\bi{z}}$ frame
$\int d\Omega Y^*_{lm}(\hat{\bi{n}})\; e^{ix\mu}=
\sqrt{4\pi(2l+1)}i^l j_l(x) \delta_{m0}$.
For the spectrum of $E$ polarization the calculation is
similar. Equation (\ref{tilEs}) is used to compute
the power spectrum of $\tilde{E}$ which combined with
equation (\ref{eblm}) gives
\begin{eqnarray}
C_{El}^{(S)}&=&
{1 \over 2l+1} {(l-2)! \over (l+2)!}
\int d^3\bi{k} P_\phi(k)\sum_m \left|{3 \over 4}\int_0^{\tau_0} d\Omega Y^*_{lm}(\hat{\bi{n}})
\int_0^{\tau_0} d\tau
g(\tau)\Pi(k,\tau)
\; ([1+\partial_x^2]^2 (x^2e^{ix\mu})\right|^2
\nonumber \\
&=&
(4\pi)^2{(l-2)! \over (l+2)!}
\int k^2dk P_\phi(k)\left(
{3 \over 4}\int_0^{\tau_0} d\tau
g(\tau)\Pi(\tau,k)
\; ([1+\partial_x^2]^2 [x^2j_l(x)]\right)^2
\nonumber \\
&=&(4\pi)^2{(l+2)! \over (l-2)!}\int k^2dkP_\phi(k)\left[
{3 \over 4}\int_0^{\tau_0} d\tau
g(\tau)\Pi(\tau,k){j_l(x) \over x^2}\right]^2.
\end{eqnarray}
To obtain
the last expression we used the differential equation satisfied
by the spherical Bessel functions, $j_l^{\prime \prime}+2j_l^{\prime}/x+
[1-l(l+1)/x^2]j_l=0$.
If we introduce
\begin{eqnarray}
\Delta^{(S)}_{Tl}(k)&=&\int_0^{\tau_0} d\tau
S^{(S)}_{T}(k,\tau) j_l(x) \nonumber \\
\Delta^{(S)}_{El}(k)&=&\sqrt{(l+2)! \over (l-2)!}\int_0^{\tau_0} d\tau
S^{(S)}_{E}(k,\tau) j_l(x) \nonumber \\
S^{(S)}_E(k\tau)&=&
{3g(\tau)\Pi(\tau,k) \over 4 x^2},
\label{es}
\end{eqnarray}
then the power spectra for $T$ and $E$ and their cross-correlation
are simply given by
\begin{eqnarray}
C_{T,El}^{(S)}&=&
(4\pi)^2\int k^2dkP_\phi(k)\Big[\Delta^{(S)}_{T,El}(k)\Big]^2
\nonumber \\
C_{Cl}^{(S)}&=&
(4\pi)^2\int k^2dkP_\phi(k)\Delta^{(S)}_{Tl}(k)
\Delta^{(S)}_{El}(k).
\label{esc}
\end{eqnarray}
Equations (\ref{es}) and (\ref{esc}) are the main results of this section.
\section{Power spectrum of tensor modes}
The method of analysis used in previous section for scalar polarization
can be used for tensor modes as well.
The situation is somewhat more complicated here because
for each Fourier mode
gravity waves have two independent polarizations usually
denoted with $+$ and $\times$. For our purposes it is convenient to
rotate this combination and work with the following two linear
combinations,
\begin{eqnarray}
\xi^1 &=&(\xi^+ - i \xi^\times)/ \sqrt{2} \nonumber \\
\xi^2 &=&(\xi^+ + i \xi^\times)/ \sqrt{2}
\end{eqnarray}
where $\xi$'s are independent random variables
used to characterize the statistics of the gravity
waves. These variables have the following statistical
properties
\begin{equation}
\langle \xi^{1*}(\bi{k_1})\xi^1(\bi{k_2})
\rangle=\langle \xi^{2*}(\bi{k_1})\xi^2(\bi{k_2})
\rangle=
{P_h(k)\over 2}\delta(\bi{k_1}- \bi{k_2}),
\; \langle \xi^{1*}(\bi{k_1})\xi^2(\bi{k_2})
\rangle=0
\label{statxi}
\end{equation}
where $P_h(k)$ is the primordial power spectrum of
the gravity waves.
In the coordinate
frame where $\hat{\bi{k}} \parallel \hat{\bi{z}}$ and
$({\bf e}_1,{\bf e}_2)=({\bf e}_\theta,{\bf e}_\phi)$
tensor perturbations can be decomposed as
\cite{kosowsky96,polnarev},
\begin{eqnarray}
\Delta_T^{(T)}(\tau,\hat{\bi{n}},\bi{k}) &=& \left[(1-\mu^2)e^{2i\phi}\xi^1(\bi{k})
+ (1-\mu^2)e^{-2i\phi}\xi^2(\bi{k})\right]
\tilde{\Delta}_T^{(T)}(\tau,\mu,k) \nonumber \\
(\Delta_Q^{(T)}+i\Delta_U^{(T)})
(\tau,\hat{\bi{n}},\bi{k}) &=& \left[(1-\mu)^2 e^{2i\phi}\xi^1(\bi{k})
+ (1+\mu)^2e^{-2i\phi}\xi^2(\bi{k})\right]
\tilde{\Delta}_P^{(T)}(\tau,\mu,k)\nonumber \\
(\Delta_Q^{(T)}-i\Delta_U^{(T)})(\tau,\hat{\bi{n}},\bi{k})
&=& \left[(1+\mu)^2 e^{2i\phi}\xi^1(\bi{k})
+ (1-\mu)^2e^{-2i\phi}\xi^2(\bi{k})\right]
\tilde{\Delta}_P^{(T)}(\tau,\mu,k),
\label{deconten}
\end{eqnarray}
where
$\tilde{\Delta}_T^{(T)}$
and $\tilde{\Delta}_P^{(T)}$ are the variables introduced by
Polnarev to describe the temperature and polarization perturbations
generated by gravity waves.
They satisfy the
following Boltzmann equation \cite{crittenden93,polnarev}
\begin{eqnarray}
&\dot{\tilde{\Delta}}_T^{(T)}& +ik\mu \tilde{\Delta}_T^{(T)}
=-\dot h
-\dot\kappa[\tilde{\Delta}_T^{(T)}-\Psi
] \nonumber \\
&\dot{\tilde{\Delta}}_P^{(T)}&
+ik\mu \tilde{\Delta}_P^{(T)} = -\dot\kappa [\tilde{\Delta}_P^{(T)} +
\Psi ] \nonumber \\
&\Psi & \equiv \Biggl\lbrack
{1\over10}\tilde{\Delta}_{T0}^{(T)}
+{1\over 7}
\tilde {\Delta}_{T2}^{(T)}+ {3\over70}
\tilde{\Delta}_{T4}^{(T)}
-{3\over 5}\tilde{\Delta}_{P0}^{(T)}
+{6\over 7}\tilde{\Delta}_{P2}^{(T)}
-{3\over 70}
\tilde{\Delta}_{P4}^{(T)} \Biggr\rbrack.
\label{BoltzmannT}
\end{eqnarray}
Just like in the scalar case these equations can be integrated along the
line of sight to give
\begin{eqnarray}
\Delta_T^{(T)}(\tau_0,\hat{\bi n},{\bi k}) &=&
\left[(1-\mu^2)e^{2i\phi}\xi^1({\bi k})
+ (1-\mu^2)e^{-2i\phi}\xi^2({\bi k})\right]
\int_0^{\tau_0} d\tau e^{ix \mu} S_T^{(T)}(k,\tau) \nonumber \\
(\Delta_Q^{(T)}+i\Delta_U^{(T)})
(\tau_0,\hat{\bi n},{\bi k}) &=& \left[(1-\mu)^2 e^{2i\phi}\xi^1({\bi k})
+ (1+\mu)^2e^{-2i\phi}\xi^2({\bi k})\right]
\int_0^{\tau_0} d\tau
e^{ix \mu} S_P^{(T)}(k,\tau) \nonumber \\
(\Delta_Q^{(T)}-i\Delta_U^{(T)})
(\tau_0,\hat{\bi n},{\bi k}) &=& \left[(1+\mu)^2 e^{2i\phi}\xi^1({\bi k})
+ (1-\mu)^2e^{-2i\phi}\xi^2({\bi k})\right]
\int_0^{\tau_0} d\tau
e^{ix \mu} S_P^{(T)}(k,\tau)
\label{integsolten}
\end{eqnarray}
where
\begin{eqnarray}
S_T^{(T)}(k,\tau) &=& -\dot he^{-\kappa}+g\Psi \nonumber \\
S_P^{(T)}(k,\tau) &=& -g\Psi .
\label{sourten}
\end{eqnarray}
Acting twice with the spin raising and lowering operators on the
terms with $\xi^1$ gives
\begin{eqnarray}
\baredth^2(\Delta_Q^{(T)}+i\Delta_Q^{(T)})(\tau_0,\hat{\bi{n}},\bi{k})&=&
\xi^1(\bi{k})e^{2i\phi}\int_0^{\tau_0} d\tau
S_P^{(T)}(k,\tau)\left(-\partial \mu + {2 \over 1-\mu^2}\right)^2 \left[
(1-\mu^2) (1-\mu)^2 e^{ix\mu}\right] \nonumber \\
&=&\xi^1(\bi{k})e^{2i\phi}\int_0^{\tau_0} d\tau
S_P^{(T)}(k,\tau)[-{\hat{\cal E}}(x)-i{\hat{\cal B}}(x)]\left[
(1-\mu^2) e^{ix\mu}\right] \nonumber \\
\;\raise1.0pt\hbox{$'$}\hskip-6pt\partial\;^2(\Delta_Q^{(T)}-i\Delta_Q^{(T)})(\tau_0,\hat{\bi{n}},\bi{k})&=&
\xi^1(\bi{k})e^{2i\phi}\int_0^{\tau_0} d\tau
S_P^{(T)}(k,\tau)\left(-\partial \mu - {2 \over 1-\mu^2}\right)^2 \left[
(1-\mu^2) (1+\mu)^2 e^{ix\mu}\right] \nonumber \\
&=&\xi^1e^{2i\phi}(\bi{k})\int_0^{\tau_0} d\tau
S_P^{(T)}(k,\tau)[-{\hat{\cal E}}(x)+i{\hat{\cal B}}(x)]\left[
(1-\mu^2) e^{ix\mu}\right] \nonumber \\
\end{eqnarray}
where we introduced operators
${\hat{\cal E}}(x)=-12+x^2[1-\partial_x^2]-8x\partial_x $ and
${\hat{\cal B}}(x)=8x+2x^2\partial_x$. Expressions for
the terms proportional to $\xi^2$ can be obtained analogously.
For tensor modes all three quantities $\Delta_T^{(T)}$,
$\Delta_{\tilde{E}}^{(T)}$ and
$\Delta_{\tilde{B}}^{(T)}$ are non-vanishing and given by
\begin{eqnarray}
\Delta_T^{(T)}
(\tau_0,\hat{\bi{n}},\bi{k})&=&\Big[(1-\mu^2)e^{2i\phi}\xi^1(\bi{k})+
(1-\mu^2)e^{-2i\phi}\xi^2(\bi{k})\Big]
\int_0^{\tau_0} d\tau S_T^{(T)}(\tau,k)\; e^{ix\mu}
\nonumber \\
\Delta_{\tilde{E}}^{(T)}
(\tau_0,\hat{\bi{n}},\bi{k})&=&\Big[(1-\mu^2)e^{2i\phi}\xi^1(\bi{k})+
(1-\mu^2)e^{-2i\phi}\xi^2(\bi{k})\Big]{\hat{\cal E}}(x)
\int_0^{\tau_0} d\tau S_P^{(T)}(\tau,k)\; e^{ix\mu}
\nonumber \\
\Delta_{\tilde{B}}^{(T)}
(\tau_0,\hat{\bi{n}},\bi{k})&=&\Big[(1-\mu^2)e^{2i\phi}\xi^1(\bi{k})-
(1-\mu^2)e^{-2i\phi}\xi^2(\bi{k})\Big]{\hat{\cal B}}(x)
\int_0^{\tau_0} d\tau S_P^{(T)}(\tau,k)\; e^{ix\mu}.
\label{tebT}
\end{eqnarray}
From these expressions and equations (\ref{aeb}), (\ref{statxi})
one can explicitly show that
$B$ does not cross correlate with either $T$ or $E$.
The temperature power spectrum can be obtained easily in this
formulation,
\begin{eqnarray}
C_{Tl}^{(T)}&=&
{4\pi \over 2l+1} \int k^2dk P_h(k)\sum_m \left|\int d\Omega
Y^*_{lm}(\hat{\bi{n}})
\int_0^{\tau_0} d\tau
S_T^{(T)}(k,\tau)
\; (1-\mu^2) e^{2i\phi}e^{ix\mu}\right|^2
\nonumber \\
&=& 4\pi^2{(l-2)!\over (l+2)!}\int k^2dk P_h(k)\left|
\int_0^{\tau_0} d\tau S_T^{(T)}(k,\tau)
\int_{-1}^1 d\mu P^2_l(\mu)
\; (1-\mu^2) e^{ix\mu}\right|^2
\nonumber \\
&=& 4\pi^2{(l-2)!\over (l+2)!}\int k^2dk P_h(k)\left|
\int_0^{\tau_0} d\tau S_T^{(T)}(k,\tau)
\int_{-1}^1 d\mu {d^2 \over d\mu^2} P_l(\mu)
\; (1-\mu^2)^2 e^{ix\mu}\right|^2
\nonumber \\
&=& 4\pi^2{(l-2)!\over (l+2)!}\int k^2dk P_h(k)\left|
\int_0^{\tau_0} d\tau S_T^{(T)}(k,\tau)
\int_{-1}^1 d\mu {d^2 \over d\mu^2} P_l(\mu)
\; (1+\partial_x^2)^2 e^{ix\mu}\right|^2
\nonumber \\
&=& 4\pi^2{(l-2)!\over (l+2)!}\int k^2dk P_h(k)\left|
\int_0^{\tau_0} d\tau S_T^{(T)}(k,\tau)
\int_{-1}^1 d\mu P_l(\mu)
\; (1+\partial_x^2)^2 (x^2 e^{ix\mu})\right|^2
\nonumber \\
&=&(4\pi)^2{(l+2)!\over (l-2)!}\int k^2 dk P_h(k)\left|
\int_0^{\tau_0} d\tau
S_T^{(T)}(k,\tau) {j_l(x)\over x^2}\right|^2,
\end{eqnarray}
where we used
$Y_{lm}=[(2l+1)(l-m)!/(4\pi)(l+m)!]^{1/2}P_l^{m}(\mu)
e^{im\phi}$ and $P_l^m(\mu)=(-1)^m
(1-\mu^2)^{m/2}{d^m \over d\mu^m}P_l(\mu)$.
Note that the calculation involved in the last step is the
same as for the scalar polarization. The final expression agrees with
the expression given in \cite{sz}, which was obtained using
the radial decomposition of the tensor eigenfunctions \cite{abbott}.
Although the final result is not new,
the simplicity of the derivation presented here demonstrates the
utility of this approach and will in fact be used to derive
tensor polarization power spectra.
The expressions for the $E$ and $B$ power spectra are now easy to
derive by noting that the angular dependence
for $\Delta_{\tilde{E}}^{(T)}$ and
$\Delta_{\tilde{B}}^{(T)}$ in (\ref{tebT}) are equal
to those for $\Delta_T^{(T)}$.
The expressions only differ in
the $\hat{\cal E}$ and $\hat{\cal B}$ operators
that can be applied after the
angular integrals are done.
This way we obtain using equation (\ref{eblm})
\begin{eqnarray}
C_{El}^{(T)}&=&
(4\pi)^2\int k^2dk P_h(k)\left|
\int_0^{\tau_0} d\tau
S_P^{(T)}(k,\tau) {\hat{\cal E}}(x){j_l(x)\over x^2}\right|^2 \nonumber \\
&=&
(4\pi)^2\int k^2dkP_h(k)\left(
\int_0^{\tau_0} d\tau
S_P^{(T)}(k,\tau)\Big[-j_l(x)+j_l''(x)+{2j_l(x) \over x^2}
+{4j_l'(x) \over x}\Big]\right)^2 \nonumber \\
C_{Bl}^{(T)}&=&
(4\pi)^2\int k^2dk P_h(k)\left|
\int_0^{\tau_0} d\tau
S_P^{(T)}(k,\tau) {\hat{\cal B}}(x){j_l(x)\over x^2}\right|^2 \nonumber \\
\nonumber \\
&=&
(4\pi)^2\int k^2dkP_h(k)\left(
\int_0^{\tau_0} d\tau
S_P^{(T)}(k,\tau)\Big[2j_l'(x)
+{4j_l \over x}\Big]\right)^2
\end{eqnarray}
For computational purposes it is convenient to
further simplify these expressions by integrating by parts the
derivatives $j_l'(x)$ and $j_l''(x)$.
This finally leads to
\begin{eqnarray}
\Delta_{Tl}^{(T)}&=&\sqrt{(l+2)! \over (l-2)!}\int_0^{\tau_0} d\tau S_T^{(T)}(k,\tau){j_l(x) \over x^2} \nonumber \\
\Delta_{E,Bl}^{(T)}&=&\int_0^{\tau_0} d\tau S_{E,B}^{(T)}(k,\tau)j_l(x) \nonumber \\
S_E^{(T)}(k,\tau)&=&g\left(\Psi-{\ddot{\Psi}\over k^2}+{2\Psi \over x^2}
-{\dot{\Psi}\over kx}\right)-\dot{g}\left({2\dot{\Psi}\over k^2}+
{4 \Psi \over kx}\right)-2\ddot{g}{\Psi \over k^2}
\nonumber \\
S_B^{(T)}(k,\tau)&=&g\left({4\Psi \over x}+{2\dot{\Psi}\over k}\right)+
2\dot{g} {\Psi \over k}.
\label{et}
\end{eqnarray}
The power spectra are given by
\begin{eqnarray}
C_{Xl}^{(T)}&=&
(4\pi)^2\int k^2dkP_h(k)\Big[\Delta^{(T)}_{Xl}(k)\Big]^2
\nonumber \\
C_{Cl}^{(T)}&=&
(4\pi)^2\int k^2dkP_h(k)\Delta^{(T)}_{Tl}(k)
\Delta^{(T)}_{El}(k),
\label{est}
\end{eqnarray}
where $X$ stands for $T$, $E$ or $B$.
Equations (\ref{et}) and (\ref{est}) are the main results of this section.
\section{Small scale limit}
In this section we derive the expressions for polarization
in the small scale limit. The purpose of this section is to make
a connection with previous work on this subject
\cite{crittenden93,prev,uros,kosowsky96} and to provide an estimate
on the validity of the small scale approximation.
In the small scale limit one considers only directions in the sky
$\hat{\bi{n}}$ which are close to
$\hat{\bi{z}} $, in which case instead of spherical decomposition
one may use a plane wave expansion.
For temperature anisotropies
we replace
\begin{equation}
\sum_{lm}a_{T,lm}Y_{lm}(\hat{\bi{n}}) \longrightarrow
\int d^2\bi{l} T(\bi{l})e^{i\bi{l}\cdot\bi{\theta}},
\end{equation}
so that
\begin{equation}
T(\hat{\bi{n}})=(2\pi)^{-2}\int d^2 \bi{l}\;\;
T(\bi{l})e^{i\bi{l} \cdot \bi{\theta}}.
\end{equation}
To expand $s=\pm 2$ weighted
functions we use
\begin{eqnarray}
_2Y_{lm}=
\left[{(l-2)!\over (l+2)!}\right]^{1\over 2}\;\raise1.0pt\hbox{$'$}\hskip-6pt\partial\;^2 Y_{lm}
&\longrightarrow&(2\pi)^{-2}{1\over l^2}\;\raise1.0pt\hbox{$'$}\hskip-6pt\partial\;^2 e^{i\bi{l} \cdot \bi{\theta}}
\nonumber \\
_{-2}Y_{lm}=
\left[{(l-2)!\over (l+s)!}\right]^{1\over 2}(
\baredth^{2} Y_{lm}
&\longrightarrow&(2\pi)^{-2}{1\over l^2}\baredth^2
e^{i\bi{l} \cdot \bi{\theta}},
\end{eqnarray}
which leads to the following expression
\begin{eqnarray}
(Q+iU)(\hat{\bi{n}})&=&-(2\pi)^2\int d^2 \bi{l}\;\;
[E(\bi{l})+iB(\bi{l})] {1\over l^2}\;\raise1.0pt\hbox{$'$}\hskip-6pt\partial\;^2
e^{i\bi{l} \cdot \bi{\theta}} \nonumber \\
(Q-iU)(\hat{\bi{n}})&=&-(2\pi)^2\int d^2 \bi{l}\;\;
[E(\bi{l})-iB(\bi{l})]
{1\over l^2}\baredth^2
e^{i\bi{l} \cdot \bi{\theta}}.
\label{SSL1}
\end{eqnarray}
From equation (\ref{edth}) we obtain in the small scale limit
\begin{eqnarray}
{1\over l^2}\;\raise1.0pt\hbox{$'$}\hskip-6pt\partial\;^2
e^{i\bi{l} \cdot \bi{\theta}}&=& - e^{-2i(\phi-\phi_{l})}
e^{i\bi{l} \cdot \bi{\theta}} \nonumber \\
{1\over l^2}\baredth^2
e^{i\bi{l} \cdot \bi{\theta}}&=& - e^{2i(\phi-\phi_{l})}
e^{i\bi{l} \cdot \bi{\theta}} \nonumber \\
\label{SSL2}
\end{eqnarray}
where $(l_x+il_y)=le^{i\phi_{l}}$.
The above expression was derived in the spherical basis where
$\hat{{\bi e}}_1=\hat{{\bi e}}_{\theta}$ and $\hat{{\bi e}}_2
=\hat{{\bi e}}_{\phi}$,
but in the small scale limit one can define a fixed basis in the sky
perpendicular to $\hat{\bi{z}}$,
$\hat{{\bi e}}_1'=\hat{{\bi e}}_{x}$ and $\hat{{\bi e}}_2'
=\hat{{\bi e}}_{y}$.
The Stokes parameters in the two coordinate
systems are related by
\begin{eqnarray}
(Q+iU)'&=&e^{-2i\phi}(Q+iU)\nonumber \\
(Q-iU)'&=&e^{2i\phi}(Q-iU).
\label{SSL3}
\end{eqnarray}
Combining equations (\ref{SSL1}-\ref{SSL3}) we find
\begin{eqnarray}
Q'(\bi{\theta})&=&(2\pi)^{-2}\int d^2 \bi{l}\;\;
[E(\bi{l}) \cos(2\phi_{l})
-B(\bi{l}) \sin(2\phi_{l})]
e^{i\bi{l} \cdot \bi{\theta}}
\nonumber \\
U'(\bi{\theta})&=&(2\pi)^{-2}\int d^2 \bi{l}\;\;
[E(\bi{l}) \sin(2\phi_{l})
+B(\bi{l}) \cos(2\phi_{l})]
e^{i\bi{l} \cdot \bi{\theta}}.
\label{QUreal}
\end{eqnarray}
These relations agree with those given in \cite{uros}, which were
derived in the small scale approximation. As already shown there,
power spectra and correlation functions for $Q$ and $U$ used in
previous work on this subject \cite{crittenden93,prev,kosowsky96}
can be simply derived from these
expressions. Of course,
for scalar modes $B^{(S)}(\bi{l})=0$, while for the tensor
modes both $E^{(T)}(\bi{l})$ and $B^{(T)}(\bi{l})$ combinations contribute.
The expressions for $Q$ and $U$ (equation \ref{QUreal})
are easier to compute
in the small scale limit
than the general expressions
presented in this paper
(equation \ref{Pexpansion}), because
Fourier analysis allows one to use Fast Fourier Transform
techniques. In addition, the characteristic signature of scalar polarization
is simple to understand in this limit and can in principle be directly
observed with the interferometer measurements \cite{uros}.
On the other hand, the exact power spectra derived in this
paper (equations \ref{es},
\ref{esc} and \ref{et}, \ref{est}) are as
simple or even simpler to compute with the integral approach
than their small scale analogs.
Note that this need not be the case if one uses the standard approach
where Boltzmann equation is first expanded in a hierarchical system of
coupled differential equations \cite{bond87}.
In Fig. \ref{fig1} we compare the exact power spectrum (solid lines)
with the one derived in the small scale approximation (dashed lines),
both for scalar $E$ (a) and tensor $E$ (b) and $B$ (c) combinations.
The two models are standard CDM with and without reionization. The latter
boosts the amplitude of polarization on large scales.
The integral solution for scalar polarization in the small scale approximation
was given in \cite{sz} and is actually more complicated that the
exact expression presented in this paper.
In the reionized case the small scale approximation
agrees well with exact calculation even at very large scales, while in
the standard recombination scenario
there are significant differences for $l<30$.
Even though the relative error is large in this case,
the overall amplitude
on these scales is probably too small to be observed.
For tensors the small scale approximation
results in equation (\ref{et}) without the terms that contain $x^{-1}$ or
$x^{-2}$. Because $j_l(x) \sim 0$ for $x<l$
these terms are suppressed by $l^{-1}$ and
$l^{-2}$, respectively, and are negligible compared to other terms
for large $l$.
The small scale approximation agrees well with the exact
calculation for $B$ combination (Fig. \ref{fig1}c), specially for
the no-reionization model. For the $E$ combination the agreement is
worse and there are notable discrepancies between the two even at
$l \sim 100$. We conclude that
although the small scale expressions for the power spectrum
can provide a good
approximation in certain models, there
is no reason to use these instead of the exact expressions.
The exact
integral solution for the power spectrum requires no additional
computational expense
compared to the small scale approximation and it should be used whenever
accurate theoretical predictions are required.
\section{Analysis of all-sky maps}
In this section we discuss issues related to simulating and
analyzing all-sky polarization and temperature
maps. This should be specially useful
for future satellite missions \cite{map,cobra}, which will measure
temperature anisotropies and polarization over
the whole sky
with a high angular resolution.
Such an all-sky analysis will be of particular importance
if reionization and tensor fluctuations
are important, in which case polarization will give
useful information on large angular scales, where Fourier analysis
(i.e. division of
the sky into locally flat patches) is not possible. In addition, it is
important
to know how to simulate an all-sky map which preserves proper correlations
between neighboring patches of the sky and with which small scale
analysis can be tested for possible biases.
To make an all-sky map we need to generate the multipole moments
$a_{T,lm}$, $a_{E,lm}$ and $a_{B,lm}$.
This can be done by a generalization of the method given
in \cite{uros}. For each $l$ one diagonalizes
the correlation matrix $M_{11}=C_{Tl}$,
$M_{22}=C_{El}$, $M_{12}=M_{21}=C_{Cl}$ and generates
from a normalized gaussian distribution two pairs of
random numbers (for real and imaginary components of $a_{l\pm m}$).
Each pair is multiplied with the square root of
eigenvalues of $M$ and rotated back to the original frame.
This gives a realization of $a_{T,l\pm m}$ and $a_{E,l\pm m}$ with correct
cross-correlation properties. For $a_{B,l\pm m}$ the procedure is simpler,
because it does not cross-correlate with either $T$ or $E$, so a pair
of gaussian random variables is multiplied with $C_{Bl}^{1/2}$ to
make a realization of $a_{B,l\pm m}$. Of course, for scalars $a_{B,lm}=0$.
Once $a_{E,lm}$ and $a_{B,lm}$ are generated we can form their linear
combinations $a_{2,lm}$ and $a_{-2,lm}$, which are equal in the scalar case.
Finally, to make a map of $Q(\hat{\bi n})$ and $U(\hat{\bi n})$ in the
sky we perform the sum in equation (\ref{Pexpansion}), using the explicit
form of
spin-weighted harmonics ${}_sY_{lm}(\hat{\bi n})$
(equation \ref{expl}).
To reconstruct the polarization
power spectrum from a map of $Q(\hat{\bi n})$ and
$U(\hat{\bi n})$ one first combines them in $Q+iU$ and $Q-iU$ to obtain
spin $\pm 2$ quantities. Performing the integral
over ${}_{\pm 2}Y_{lm}$ (equation \ref{alm})
projects out ${}_{\pm 2}a_{lm}$, from which $a_{E,lm}$ and $a_{B,lm}$
can be obtained.
Once we have the multipole moments we can
construct various power spectrum estimators and analyze their
variances. In the case of full sky coverage one may generalize the
approach in \cite{knox95} to estimate the variance in the power spectrum
estimator in the presence of noise. We will assume that we are given a
map of temperature
and polarization with $N_{pix}$ pixels and
that the noise is uncorrelated from pixel to pixel
and also between $T$, $Q$ and $U$.
The rms noise in the temperature is
$\sigma_T$ and that in $Q$ and $U$ is $\sigma_P$. If temperature and
polarization are obtained from the same experiment by adding and
subtracting the intensities between two orthogonal polarizations then
the rms noise in temperature and polarization
are related by $\sigma_T^2=\sigma_P^2/2$
\cite{uros}.
Under these conditions and using the orthogonality
of the $\;_sY_{lm}$ we obtain the statistical property of noise,
\begin{eqnarray}
\langle (a_{T,lm}^{{\rm noise}})^{*}a^{{\rm noise}}_{T,l^{\prime}m^{\prime}}\rangle
&=& {4\pi \sigma_T^2 \over N_{pix}}
\delta_{l l^{\prime}} \delta_{m m^{\prime}}
\nonumber \\
\langle (a^{{\rm noise}}_{2,lm})^{*}a^{{\rm noise}}_{2,l^{\prime}m^{\prime}}\rangle
&=& {8\pi \sigma_P^2 \over N_{pix}}
\delta_{l l^{\prime}} \delta_{m m^{\prime}}
\nonumber \\
\langle (a^{{\rm noise}}_{-2,lm})^{*}a^{{\rm noise}}_{-2,l^{\prime}m^{\prime}}\rangle
&=& {8\pi \sigma_P^2 \over N_{pix}}
\delta_{l l^{\prime}} \delta_{m m^{\prime}}
\nonumber \\
\langle (a^{{\rm noise}}_{-2,lm})^{*}a^{{\rm noise}}_{2,l^{\prime}m^{\prime}}\rangle
&=& 0,
\end{eqnarray}
where by assumption there are no correlations between the noise in
temperature and polarization.
With these and equations
(\ref{aeb},\ref{stat}) we find
\begin{eqnarray}
\langle a_{T,lm}^{*}a_{T,l^{\prime}m^{\prime}}\rangle
&=& (C_{Tl} e^{-l^2 \sigma_b^2} + w_T^{-1})
\delta_{l l^{\prime}} \delta_{m m^{\prime}}
\nonumber \\
\langle a_{E,lm}^{*}a_{E,l^{\prime}m^{\prime}}\rangle
&=& (C_{El} e^{-l^2 \sigma_b^2} + w_P^{-1})
\delta_{l l^{\prime}} \delta_{m m^{\prime}}
\nonumber \\
\langle a_{B,lm}^{*}a_{B,l^{\prime}m^{\prime}}\rangle
&=& (C_{Bl} e^{-l^2 \sigma_b^2} +w_P^{-1})
\delta_{l l^{\prime}} \delta_{m m^{\prime}}
\nonumber \\
\langle a_{E,lm}^{*}a_{T,l^{\prime}m^{\prime}}\rangle
&=& C_{Cl} e^{-l^2 \sigma_b^2}
\delta_{l l^{\prime}} \delta_{m m^{\prime}}
\nonumber \\
\langle a_{B,l^\prime m^\prime}^{*} a_{E,lm}\rangle&=&
\langle a_{B,l^\prime m^\prime}^{*} a_{T,lm}\rangle=
0.
\label{almvar}
\end{eqnarray}
For simplicity we characterized the beam smearing by
$e^{l^2 \sigma_b /2}$ where $\sigma_b$ is the gaussian size of the beam
and we defined $w_{T,P}^{-1}=4\pi\sigma_{T,P}^2/N$ \cite{uros,knox95}.
The estimator for the temperature power spectrum is \cite{knox95},
\begin{eqnarray}
\hat{C}_{Tl}&=&\left[\sum_m{ |a_{T,lm}|^2 \over 2l+1} - w^{-1}_T
\right]e^{l^2\sigma_b^2}
\end{eqnarray}
Similarly for polarization and cross correlation the optimal
estimators are given by \cite{uros}
\begin{eqnarray}
\hat{C}_{El}&=&\left[\sum_m{ |a_{E,lm}|^2 \over 2l+1} - w^{-1}_P
\right]e^{l^2\sigma_b^2}\nonumber \\
\hat{C}_{Bl}&=&\left[\sum_m{ |a_{B,lm}|^2 \over 2l+1}- w^{-1}_P
\right]e^{l^2\sigma_b^2}\nonumber \\
\hat{C}_{Cl}&=&\left[\sum_m{ (a_{E,lm}^{*}a_{T,lm}+a_{E,lm}a_{T,lm}^{*})
\over 2(2l+1)}\right]e^{l^2 \sigma_b^2}.
\end{eqnarray}
The covariance matrix between the different estimators,
${\rm Cov }(\hat{X}\hat{X}^{\prime})=\langle (\hat X - \langle \hat X \rangle)
(\hat X^{\prime} - \langle \hat X^{\prime} \rangle)\rangle$
is easily calculated using equation (\ref{almvar}).
The diagonal terms are given by
\begin{eqnarray}
{\rm Cov }(\hat{C}_{Tl}^2)&=&{2\over 2l+1}(\hat{C}_{Tl}+
w_T^{-1}e^{l^2 \sigma_b^2})^2
\nonumber \\
{\rm Cov }(\hat{C}_{El}^2)&=&{2\over 2l+1}(\hat{C}_{El}+
w_P^{-1}e^{l^2 \sigma_b^2})^2
\nonumber \\
{\rm Cov }(\hat{C}_{Bl}^2)&=&{2\over 2l+1}(\hat{C}_{Bl}+
w_P^{-1}e^{l^2 \sigma_b^2})^2
\nonumber \\
{\rm Cov }(\hat{C}_{Cl}^2)&=&{1\over 2l+1}\left[\hat{C}_{Cl}^2+
(\hat{C}_{Tl}+w_T^{-1}e^{l^2 \sigma_b^2})
(\hat{C}_{El}+w_P^{-1}e^{l^2 \sigma_b^2})\right].
\end{eqnarray}
The non-zero off diagonal terms are
\begin{eqnarray}
{\rm Cov }(\hat{C}_{Tl}\hat{C}_{El})&=&{2\over 2l+1}\hat{C}_{Cl}^2
\nonumber \\
{\rm Cov }(\hat{C}_{Tl}\hat{C}_{Cl})&=&{2\over 2l+1}\hat{C}_{Cl}
(\hat{C}_{Tl}+w_T^{-1}e^{l^2 \sigma_b^2})
\nonumber \\
{\rm Cov }(\hat{C}_{El}\hat{C}_{Cl})&=&{2\over 2l+1}\hat{C}_{Cl}
(\hat{C}_{El}+w_P^{-1}e^{l^2 \sigma_b^2}).
\end{eqnarray}
These expressions agree in the small scale limit with those given in
\cite{uros}. Note that the theoretical analysis is more
complicated if all four power spectrum estimators are used to deduce
the underlying cosmological model. For example, to test the sensitivity of
the spectrum on the underlying parameter one uses the Fisher information
matrix approach \cite{jungman}. If only temperature information is
given then for each $l$ a derivative of the temperature
spectrum with respect to the parameter under investigation is computed
and this information is then summed over all $l$ weighted
by ${\rm Cov }^{-1}(\hat{C}_{Tl}^2)$.
In the more general case discussed here instead of a single derivative
we have a vector of four derivatives and the
weighting is given by the inverse of the covariance matrix,
\begin{equation}
\alpha_{ij}=\sum_l \sum_{X,Y}{\partial \hat{C}_{Xl} \over \partial s_i}
{\rm Cov}^{-1}(\hat{C}_{Xl}\hat{C}_{Yl}){\partial \hat{C}_{Yl} \over \partial s_j},
\end{equation}
where $\alpha_{ij}$ is the Fisher information or curvature
matrix, ${\rm Cov}^{-1}$ is the inverse of the covariance matrix,
$s_i$ are the cosmological parameters one would like to
estimate and $X,Y$ stands for $T,E,B,C$. For each $l$ one has to
invert the covariance matrix and sum over $X$ and $Y$,
which makes the numerical evaluation of this expression somewhat
more involved.
\section{Conclusions}
In this paper
we developed the formalism for an all-sky analysis of polarization
using the theory of spin-weighted functions.
We show that one can define rotationally invariant
electric and magnetic-type parity fields $E$ and $B$
from the usual $Q$ and $U$ Stokes parameters.
A complete statistical
characterization of CMB anisotropies
requires four correlation functions,
the auto-correlations of $T$, $E$ and $B$ and the cross-correlation
between $E$ and $T$. The pseudo-scalar nature of $B$
makes its cross-correlation with $T$ and $E$ vanish.
For scalar modes $B$ field vanishes.
Intuitive understanding of these results
can be obtained by considering polarization created by each plane
wave given by direction $\bi{k}$. Photon propagation
can be described by scattering through a plane-parallel medium.
The cross-section only depends on the angle between photon
direction $\bi{\hat{n}}$ and $\bi{k}$, so for a local coordinate system
oriented in this direction only $Q$ Stokes parameter will be
generated, while $U$ will vanish by symmetry arguments \cite{chandra}.
In the real universe one has to consider a superposition of plane waves
so this property does not hold in real space. However, by performing
the analog of a plane wave expansion on the sphere this property becomes
valid again and leads to the vanishing of $B$ in the scalar case.
For tensor perturbations this is not true even in this
$\bi{k}$ dependent frame, because each plane
wave consists of two different independent ``polarization'' states, which
depend not only on the direction of plane wave, but also
on the azimuthal angle perpendicular to $\bi{k}$. The symmetry
above is thus explicitly broken. Both $Q$ and $U$ are generated
in this frame and, equivalently, both $E$ and $B$ are generated in general.
Combining the formalism of spin-weighted functions and
the line of sight solution of the Boltzmann equation
we obtained the exact expressions for the power spectra both
for scalar and tensor modes. We present their numerical evaluations
for a representative set of models. A numerical implementation of the
solution is publicly available and can be obtained from the
authors \cite{cmbfast}.
We also compared the exact solutions to their analogs
in the small scale approximation obtained previously. While the latter
are accurate for all but the largest angular scales, the simple form
of the exact solution suggests that the small scale approximation
should be replaced with the exact solution for all calculations.
If both scalars and tensors are contributing to a particular
combination then the power spectrum for that combination is obtained
by adding the individual contributions. Cross-correlation terms
between different types of perturbations vanish after the integration
over azimuthal
angle $\phi$ both for the temperature and for the $E$ and $B$ polarization,
as can be seen from equations (\ref{tilEs}) and (\ref{tebT}).
This result holds even for the defect models, where the same
source generates scalar, vector and tensor perturbations.
In summary,
future CMB satellite missions will produce all-sky maps of polarization
and these
maps will have to be analyzed using techniques similar to the one
presented in this paper. Polarization measurements have the sensitivity
to certain cosmological parameters which is not achievable from the
temperature
measurements alone. This sensitivity is particularly important on
large angular scales, where previously used approximations break down
and have to be replaced with the exact expressions for the polarization
power spectra presented in this paper.
\acknowledgments
We would like to thank D. Spergel for helpful discussions.
U.S. acknowledges
useful discussions with M. Kamionkowski, A.
Kosowsky and A. Stebbins.
|
proofpile-arXiv_065-617
|
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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|
\section{Introduction}
\label{intro}
In two previous papers, we (Wu et al.\ 1993, 1996, hereafter WCFHS93, WCHFLS96)
described HST FOS observations of the UV spectrum of the
SM star (Schweizer \& Middleditch 1980) which lies
behind and close to the projected center of the remnant of SN1006.
In the present paper we offer a theoretical interpretation of
the broad silicon and iron UV absorption features observed with HST.
These features are almost certainly caused by supernova ejecta in SN1006,
as originally proposed by Wu et al.\ (1983),
who first observed the features with IUE.
Detailed theoretical analysis of the \ion{Fe}{2} features observed
with IUE has been presented previously by
Hamilton \& Fesen (1988, hereafter HF88).
The main purpose of that paper was to try to explain the apparent
conflict between the low
$\approx 0.015 \,{\rmn M}_{\sun}$
mass of \ion{Fe}{2} inferred from the IUE observations of SN1006
(Fesen et al.\ 1988)
with the expected presence of several tenths of a solar mass of iron
(H\"{o}flich \& Khokhlov 1996)
in this suspected Type~Ia remnant
(Minkowski 1966; Schaefer 1996).
HF88 demonstrated that ambient UV starlight and UV and x-ray emission
from reverse-shocked ejecta could photoionize unshocked iron
mainly to \ion{Fe}{3}, \ion{Fe}{4}, and \ion{Fe}{5},
resolving the conflict.
Recently
Blair, Long \& Raymond (1996) used the Hopkins Ultraviolet Telescope (HUT)
to measure \ion{Fe}{3} 1123\,\AA\ absorption in the spectrum of the SM star.
They found \ion{Fe}{3}/\ion{Fe}{2} $= 1.1 \pm 0.9$,
which neither confirms, nor excludes,
the ratio \ion{Fe}{3}/\ion{Fe}{2} $= 2.6$ predicted by HF88.
The HST spectra, particularly the silicon features,
prove to be a rich source of information
beyond the reach of IUE's capabilities.
In the first half of this paper,
Section~\ref{silicon},
we analyze the Si absorption features.
We find (\S\ref{red}) that
the profile of the redshifted \ion{Si}{2} 1260\,\AA\ feature,
with its sharp red edge and Gaussian blue edge,
appears to be attributable to the presence
of both unshocked and shocked silicon.
We then develop a chain of inferences,
first about the reverse shock (\S\ref{jump}) and collisionless heating
(\S\ref{collisionless}),
then about the column density and mass (\S\S\ref{mass}, \ref{Simass}),
purity (\S\ref{purity}), and ionization state (\S\ref{SiIII+IV})
of the silicon.
We argue (\S\ref{blue}) that the ambient interstellar density on the far of
SN1006 is anomalously low compared to density around the rest of the remnant,
which explains the high velocity of the redshifted Si,
the absence of corresponding blueshifted Si (\S\ref{blueion}),
and some other observational puzzles.
In the second half of the paper,
Section~\ref{iron},
we discuss the broad \ion{Fe}{2} absorption features.
WCFHS93 reported blueshifted \ion{Fe}{2} absorption up to $\sim - 8000 \,{\rmn km}\,{\rmn s}^{-1}$.
Finding the presence of such high velocity blueshifted absorption
difficult to understand in the light of other observational evidence,
we detail a reanalysis of the \ion{Fe}{2} features in \S\ref{reanalysis}.
We conclude (\S\ref{fe2sec}) that the evidence for high velocity blueshifted
absorption is not compelling,
and we propose (\S\ref{nearside}) that the sharp blue edge on the
\ion{Fe}{2} profiles at $- 4200 \,{\rmn km}\,{\rmn s}^{-1}$ represents the position of the
reverse shock on the near side.
In the remainder of the Section,
\S\S\ref{shockedFe}-\ref{where},
we address the issue of the ionization state and mass of the iron.
We attempt in this paper to construct a consistent theoretical picture,
but there remain some discrepancies,
and we highlight these in Section~\ref{worries}.
Section~\ref{summary} summarizes the conclusions.
\section{Silicon}
\label{silicon}
There are three possibilities for the origin of the broad
Si absorption features in the spectrum of the SM star,
if it is accepted that these features arise from the remnant of SN1006.
The first possibility is that the absorption arises from cool, dense,
fast moving knots of ejecta, as suggested by Fesen et al.\ (1988)
and further discussed by Fesen \& Hamilton (1988).
However, there has been no significant change in the features over 12 years,
from the original detection of the features with IUE in 1982 (Wu et al.\ 1983),
through their re-observation with IUE in 1986 and 1988 (Fesen \& Hamilton 1988),
up to the FOS observation with HST in 1994 (WCHFLS96).
Furthermore,
the relative strengths of the redshifted
\ion{Si}{2} 1260\,\AA, 1304\,\AA, and 1527\,\AA\ features
are approximately proportional to their oscillator strengths
times wavelengths, indicating that the lines are not saturated.
This constancy in time and lack of saturation
argues against the absorption features being caused by small, dense knots.
We do not consider this hypothesis further in this paper.
A second possibility is that the Si absorption is from shocked ejecta
in which the collisional ionization timescale is so long that
the observed low ionization species
\ion{Si}{2}, \ion{Si}{3}, and \ion{Si}{4} can survive.
At first sight,
the high $\sim 5000\,{\rmn km}\,{\rmn s}^{-1}$ velocity of the observed absorption
makes this possibility seem unlikely,
because shocked ejecta should be moving no faster than the velocity
of gas behind the interstellar shock,
which in the NW sector of the remnant can be inferred
from the $2310\,{\rmn km}\,{\rmn s}^{-1}$ FWHM of the Balmer broad line emission
to be
$1800$-$2400\,{\rmn km}\,{\rmn s}^{-1}$
(this is $3/4$ of the shock velocity),
depending on the extent to which electron-ion equilibration takes place
(Kirshner, Winkler \& Chevalier 1987;
Long, Blair \& van den Bergh 1988;
Smith et al.\ 1991;
Raymond, Blair \& Long 1995).
However, below we will conclude that it is likely that
much, in fact most, of the Si absorption is from shocked ejecta,
and that the ISM surrounding SN1006 may be quite inhomogeneous.
A third possibility is that the Si absorption arises from
unshocked supernova ejecta freely expanding in SN1006,
which is consistent with the high velocity of the absorption.
The low ionization state of the Si,
predominantly \ion{Si}{2}, with some \ion{Si}{3} and \ion{Si}{4},
is at least qualitatively consistent with the expectations of models
in which the unshocked ejecta are photoionized by ambient UV starlight
and by UV and x-ray emission from shocked ejecta
(HF88).
Neutral Si is neither observed, e.g.\ at \ion{Si}{1} 1845\,\AA,
nor expected, since it should be quickly ($\sim 20 \, {\rm yr}$)
photoionized by ambient UV starlight.
Recombination is negligible at the low densities here.
At the outset therefore, this possibility seems most likely,
and we pursue the idea further in the next subsection.
Fesen et al.\ (1988)
pointed out that the redshifted \ion{Si}{2} 1260\,\AA\ feature
(at $\sim 1280$\,\AA) in the IUE data
appeared to be somewhat too strong compared to the weaker redshifted
\ion{Si}{2} 1527\,\AA\ feature.
The discrepancy appears to be confirmed by the HST observations (WCHFLS96).
Fesen et al.\ proposed that some of the \ion{Si}{2} 1260\,\AA\ feature
may come from
\ion{S}{2} 1260, 1254, 1251\,\AA\
redshifted by $\approx 800 \,{\rmn km}\,{\rmn s}^{-1}$ relative to the \ion{Si}{2},
a possibility also addressed by WCHFLS96.
In the present paper we regard the possibility of any significant contribution
from \ion{S}{2} as unlikely,
notwithstanding the discrepancy between the \ion{Si}{2} profiles.
The oscillator strengths of the
\ion{S}{2} 1260, 1254, 1251\,\AA\
lines are
$f = 0.01624$, 0.01088, 0.005453,
whose combined strength is only 1/30 of the oscillator strength
$f = 1.007$ of the \ion{Si}{2} 1260\,\AA\ line
(Morton 1991).
In models of Type~Ia supernovae, such as SN1006 is believed to be,
silicon and sulfur occur typically in the same region of space,
with a relative abundance of $\mbox{Si} : \mbox{S} \approx 2 : 1$
(e.g.\ Nomoto, Thielemann \& Yokoi 1984).
Thus \ion{S}{2} might be expected to contribute only $\sim 1/60$ of the
optical depth of \ion{Si}{2} in the 1260\,\AA\ feature,
assuming a similar ionization state of Si and S.
In the remainder of this paper we ignore any possible contribution
of \ion{S}{2} to the \ion{Si}{2} 1260\,\AA\ feature.
In this we follow Wu et al.'s (1983) original identification of
the 1280\,\AA\ absorption feature as redshifted Si.
\subsection{The redshifted \protect\ion{Si}{2} 1260\,\AA\ feature}
\label{red}
\begin{figure}[tb]
\epsfbox[170 262 415 525]{si1260.ps}
\caption[1]{
HST G130H spectrum relative to the stellar continuum around
the redshifted \protect\ion{Si}{2} 1260.4221\,\AA\ feature,
showing the best fit Gaussian profile of shocked \protect\ion{Si}{2},
and the residual unshocked \protect\ion{Si}{2}.
Upper axis shows velocity in the rest frame of the \protect\ion{Si}{2} line.
Measured parameters of the feature are given in Table~\protect\ref{redtab}.
As elsewhere in this paper,
we assume a stellar continuum which is linear in $\log F$-$\log \lambda$,
and fit the continuum to ostensibly uncontaminated regions around the line.
The uncertainties in the parameters given in Table~\protect\ref{redtab}
include uncertainty from placement of the continuum.
The adopted stellar continuum is
$\log F =
\log ( 4.1 \times 10^{-14} \,{\rmn erg}\,{\rmn s}^{-1}\,{\rmn cm}^{-2}\,{\rm \AA}^{-1} )
- 2.3 \log (\lambda/1260\,{\rm \AA})$.
\label{si1260}
}
\end{figure}
\begin{deluxetable}{lr}
\tablewidth{0pt}
\tablecaption{Parameters measured from redshifted
\protect\ion{Si}{2} 1260\,\AA\ feature
\label{redtab}}
\tablehead{\colhead{Parameter} & \colhead{Value}}
\startdata
Expansion velocity into reverse shock & $7070 \pm 50 \,{\rmn km}\,{\rmn s}^{-1}$ \nl
Mean velocity of shocked \protect\ion{Si}{2} & $5050 \pm 60 \,{\rmn km}\,{\rmn s}^{-1}$ \nl
Dispersion of shocked \protect\ion{Si}{2} & $1240 \pm 40 \,{\rmn km}\,{\rmn s}^{-1}$ \nl
Reverse shock velocity & $2860 \pm 100 \,{\rmn km}\,{\rmn s}^{-1}$ \nl
Lower edge of unshocked \protect\ion{Si}{2} & $5600 \pm 100 \,{\rmn km}\,{\rmn s}^{-1}$ \nl
Preshock density of \protect\ion{Si}{2} &
$5.4 \pm 0.7 \times 10^{-5} \,{\rmn cm}^{-3} $ \nl
Column density of shocked \protect\ion{Si}{2} &
$9.0 \pm 0.3 \times 10^{14} \,{\rmn cm}^{-2}$ \nl
Column density of unshocked \protect\ion{Si}{2} &
$1.5 \pm 0.2 \times 10^{14} \,{\rmn cm}^{-2}$ \nl
Column density of all \protect\ion{Si}{2} &
$10.5 \pm 0.1 \times 10^{14} \,{\rmn cm}^{-2}$ \nl
Mass of shocked \protect\ion{Si}{2}
& $0.127 \pm 0.006 \,{\rmn M}_{\sun}$ \nl
Mass of unshocked \protect\ion{Si}{2} & $0.017 \pm 0.002 \,{\rmn M}_{\sun}$ \nl
\enddata
\tablecomments{
Masses assume spherical symmetry.
For simplicity,
velocities and masses have not been adjusted for the small offset
of the SM star from the projected center of the remnant.
}
\end{deluxetable}
Given that SN1006 shows a well-developed interstellar blast wave
in both radio
(Reynolds \& Gilmore 1986, 1993)
and x-rays
(Koyama et al.\ 1995;
Willingale et al.\ 1996),
it is inevitable that a reverse shock must be propagating into
any unshocked ejecta.
If the observed UV Si absorption is from unshocked ejecta,
then there should be a sharp cutoff in the line profile
at the expansion velocity of the reverse shock,
because the shock should `instantaneously' decelerate the ejecta to
lower bulk velocity.
In fact the \ion{Si}{2} 1260\,\AA\ feature does show a steep red edge
at $7070 \pm 50 \,{\rmn km}\,{\rmn s}^{-1}$,
albeit with a possible tail to higher velocities.
Tentatively,
we take the presence of the steep red edge as evidence
that at least some of the \ion{Si}{2} is unshocked.
Once shocked, how long will \ion{Si}{2} last before being collisionally
ionized to higher levels?
The ionization timescale of \ion{Si}{2} entering the reverse shock
can be inferred from the optical depth of the
\ion{Si}{2} 1260\,\AA\ absorption just inside its steep red edge
at $7070\,{\rmn km}\,{\rmn s}^{-1}$,
which putatively represents the position of the reverse shock.
In freely expanding ejecta where radius equals velocity times age,
$r = v t$,
the column density per unit velocity $\dd N/\dd v$ of any species
is equal to the density $n = \dd N/\dd r$ times age $t$.
For freely expanding ejecta,
the optical depth $\tau$
in a line of wavelength $\lambda$ and oscillator strength $f$
is then proportional to $n t$:
\begin{equation}
\label{tau}
\tau
= {\pi e^2 \over m_e c} f \lambda {\dd N \over \dd v}
= {\pi e^2 \over m_e c} f \lambda n t
\ .
\end{equation}
The optical depth of the \ion{Si}{2} 1260.4221\,\AA\ line
($f = 1.007$, Morton 1991)
just inside its steep red edge is $\tau \approx 1$.
This implies, from equation (\ref{tau}),
a preshock \ion{Si}{2} density times age of
$n_\SiII^{\rm presh} t = 3.0 \times 10^6 {\rmn cm}^{-3} {\rmn s}$.
The postshock \ion{Si}{2} density times age would then be 4 times higher
for a strong shock,
$n_\SiII t = 1.2 \times 10^7 {\rmn cm}^{-3} {\rmn s}$.
At a collisional ionization rate for \ion{Si}{2} of
$\langle \sigma v \rangle_\SiII = 6 \times 10^{-8} {\rmn cm}^3 {\rmn s}^{-1}$
(Lennon et al.\ 1988;
see subsection \ref{purity} below for further discussion of this rate),
the ratio of the \ion{Si}{2} ionization timescale
$t_\SiII \equiv ( n_e \langle \sigma v \rangle_\SiII )^{-1}$
to the age of the remnant, $t$, is
(the following estimate is revised below, equation [\ref{tion'}])
\begin{equation}
\label{tion}
{t_\SiII \over t}
= {1 \over n_e t \langle \sigma v \rangle_\SiII}
= {n_\SiII \over n_e} {1 \over n_\SiII t \langle \sigma v \rangle_\SiII}
= 1.4 {n_\SiII \over n_e}
\ .
\end{equation}
Since the ratio $n_e / n_\SiII$ of electron to \ion{Si}{2} density
in the postshock gas should be greater than but of order unity,
this estimate (\ref{tion})
indicates that the collisional timescale of \ion{Si}{2} is of
the order of the age of the remnant.
It follows that shocked \ion{Si}{2} is likely also to contribute to the
observed \ion{Si}{2} absorption.
Shocked \ion{Si}{2} will be decelerated by the reverse shock to lower
velocities than the freely expanding unshocked ejecta.
Shocked \ion{Si}{2} should have a broad thermal profile,
unlike the unshocked \ion{Si}{2}.
Examining the redshifted \ion{Si}{2} 1260\,\AA\ profile,
we see that the blue edge extends down to about $+ 2500\,{\rmn km}\,{\rmn s}^{-1}$,
with a shape which looks Gaussian.
Fitting the blue edge to a Gaussian,
we find a best fit to a Gaussian centered at $5050\,{\rmn km}\,{\rmn s}^{-1}$,
with a dispersion (standard deviation) of $\sigma = 1240\,{\rmn km}\,{\rmn s}^{-1}$.
This fit is shown in Figure~\ref{si1260}.
Having started from the point of view that the \ion{Si}{2} was likely
to be unshocked, we were surprised to see that,
according to the fit, it is shocked \ion{Si}{2} which causes most of the
absorption,
although an appreciable quantity of unshocked Si is also present,
at velocities extending upwards from $5600\,{\rmn km}\,{\rmn s}^{-1}$.
The slight tail of \ion{Si}{2} absorption to high velocities
$> 7070\,{\rmn km}\,{\rmn s}^{-1}$ is naturally produced by the tail of the Gaussian profile
of the shocked \ion{Si}{2}.
The estimate (\ref{tion}) of the collisional ionization timescale of
\ion{Si}{2} presumed that all the \ion{Si}{2} 1260\,\AA\ absorption
was from unshocked \ion{Si}{2},
whereas the picture now is that only some of the absorption
is from unshocked \ion{Si}{2}.
According to the fit in Figure~\ref{si1260},
the optical depth of unshocked \ion{Si}{2} at the reverse shock front
is $\tau = 0.56 \pm 0.07$,
a little over half that adopted in estimate (\ref{tion}),
so a revised estimate of the ionization timescale of \ion{Si}{2} is
not quite double that of the original estimate (\ref{tion}):
\begin{equation}
\label{tion'}
{t_\SiII \over t} =
2.5 {n_\SiII \over n_e}
\ .
\end{equation}
Evidently the conclusion remains that the ionization timescale of \ion{Si}{2}
is comparable to the age of the remnant.
\subsection{Shock jump conditions}
\label{jump}
The fitted profile of the \ion{Si}{2} 1260\,\AA\ feature in Figure~\ref{si1260}
includes both unshocked and shocked components.
The consistency of the fitted parameters can be checked against
the jump conditions for a strong shock.
The shock jump conditions predict that
the three-dimensional velocity dispersion $3^{1/2} \sigma$ of the ions
should be related to
the deceleration $\Delta v$ of the shocked gas
by energy conservation
\begin{equation}
\label{Dv}
3^{1/2} \sigma
=
\Delta v
\end{equation}
provided that all the shock energy goes into ions.
The observed dispersion is
\begin{equation}
\label{sigmaobs}
3^{1/2} \sigma = 3^{1/2} \times ( 1240 \pm 40 \,{\rmn km}\,{\rmn s}^{-1} )
= 2140 \pm 70 \,{\rmn km}\,{\rmn s}^{-1}
\end{equation}
while the observed deceleration is
\begin{eqnarray}
\label{Dvobs}
\Delta v &=& ( 7070 \pm 50 \,{\rmn km}\,{\rmn s}^{-1} ) - ( 5050 \pm 60 \,{\rmn km}\,{\rmn s}^{-1} )
\nonumber \\
&=& 2020 \pm 80 \,{\rmn km}\,{\rmn s}^{-1}
\ .
\end{eqnarray}
These agree remarkably well,
encouraging us to believe that this interpretation is along the right lines.
The reverse shock velocity, $v_s$, corresponding to the observed dispersion is
\begin{equation}
\label{vs}
v_s = (16/3)^{1/2} \sigma = 2860 \pm 100 \,{\rmn km}\,{\rmn s}^{-1}
\ .
\end{equation}
We prefer to infer the shock velocity from the observed dispersion rather than
from the observed deceleration $\Delta v$,
since the latter may underestimate the true deceleration, if, as is likely,
the shocked Si is moving on average slightly faster than the immediate
postshock gas (see below).
The predicted equality (\ref{Dv}) between the deceleration and ion dispersion
holds provided
that all the shock energy goes into ions,
and that the bulk velocity and dispersion of the ions in the shocked gas
are equal to their postshock values.
Since the shocked \ion{Si}{2} can last for a time comparable to the age
of the remnant (equation [\ref{tion'}]),
it cannot be assumed automatically that the bulk velocity and dispersion of the
observed \ion{Si}{2} ions are necessarily equal to those immediately
behind the reverse shock front.
We discuss first the issue of the bulk velocity,
then the dispersion,
and finally the question of collisionless heating
in subsection \ref{collisionless}.
Consider first the bulk velocity of the shocked ions.
In realistic hydrodynamic models,
the velocity of shocked gas increases outward from the reverse shock.
Indeed, the fact that
the observed dispersion $3^{1/2} \sigma$
is larger than
the observed deceleration $\Delta v$
by $120 \pm 130 \,{\rmn km}\,{\rmn s}^{-1}$
(the uncertainty here takes into account the correlation between
the uncertainties in $\sigma$ and $\Delta v$)
is consistent with the notion that the shocked \ion{Si}{2} is moving on average
$120 \pm 130 \,{\rmn km}\,{\rmn s}^{-1}$ faster than the immediate postshock gas.
This modest velocity is consistent with expectations from
one-dimensional hydrodynamic simulations appropriate to SN1006
(see HF88, Fig.~2),
according to which
shocked ejecta remain relatively close to the reverse shock front.
However,
the deceleration of ejecta is generally Rayleigh-Taylor unstable
(e.g.\ Chevalier, Blondin \& Emmering 1992),
which instabilities could have caused shocked \ion{Si}{2} to appear
at velocities many hundred ${\rmn km}\,{\rmn s}^{-1}$ faster than the immediate postshock gas.
Since the observations do not show this,
it suggests, though by no means proves,
either that Rayleigh-Taylor instabilities are not very important,
or perhaps that the line of sight through SN1006 to the SM star
happens to lie between Rayleigh-Taylor plumes.
What about the ion dispersion?
If there were a range of ion dispersions in the shocked gas,
then the line profile would tend to be more peaked and have broader wings
than a simple Gaussian.
The observed profile of the blue edge of the redshifted \ion{Si}{2} 1260\,\AA\
feature is consistent with a Gaussian, which suggests, again weakly,
that the ion dispersion does not vary by a large factor
over the bulk of the shocked \ion{Si}{2}.
While the observations agree well with the simplest possible interpretation,
it is certainly possible to arrange situations
in which a combination of Rayleigh-Taylor instabilities,
spatially varying ion dispersion,
and collisionless heating conspire to produce fortuitous agreement
of the observations with the jump condition (\ref{Dv}).
\subsection{Collisionless heating}
\label{collisionless}
The shock jump condition (\ref{Dv}) is valid provided that all the shock
energy goes into the ions,
rather than into electrons, magnetic fields, or relativistic particles.
The timescale for equilibration by Coulomb collisions between
electrons and ions or between ions and ions
is much longer than the age of SN1006.
However,
collisionless processes in the shock may also transfer energy,
and the extent to which these may accomplish equilibration remains uncertain
(see Laming et al.\ 1996 for a recent review).
The prevailing weight of evidence favors little if any collisionless heating
of electrons in the fast shocks in SN1006.
Raymond et al.\ (1995)
find similar velocity widths in emission lines of
\ion{H}{1} (Ly~$\beta$), \ion{He}{2}, \ion{C}{4}, \ion{N}{5}, and \ion{O}{6}
observed by HUT from the interstellar shock along the NW sector of SN1006.
They conclude that there has been little equilibration between ions,
though this does not rule out substantial electron heating.
From the same data,
Laming et al.\ (1996)
argue that the ratio of \ion{C}{4} (which is excited mainly by protons)
to \ion{He}{2} (which is excited mainly by electrons)
is again consistent with little or no electron-ion equilibration.
Koyama et al.\ (1995) present spectral and imaging evidence from ASCA
that the high energy component of the x-ray spectrum of SN1006
is nonthermal synchrotron radiation,
obviating the need for collisionless electron heating
(see also Reynolds 1996).
The results reported here, equations (\ref{sigmaobs}) and (\ref{Dvobs}),
tend to support the conclusion
that virtually all the shock energy is deposited into the ions.
If some of the shock energy were chaneled into electrons,
magnetic fields, or relativistic particles,
then the ion dispersion would be lower than predicted by
the observed deceleration,
whereas the opposite is observed --- the ion dispersion is slightly higher.
\subsection{Column density and mass of \protect\ion{Si}{2}}
\label{mass}
The numbers given here are summarized in Table~\ref{redtab}.
Quoted uncertainties in column densities and masses here and
throughout this paper ignore uncertainties in oscillator strengths.
The uncertainties in the oscillator strengths of \ion{Si}{2} 1260\,\AA\
is 17\%, and of \ion{Si}{3} 1206\,\AA\
and \ion{Si}{4} 1394, 1403\,\AA\ are 10\% (Morton 1991).
The column density of shocked and unshocked \ion{Si}{2}
follows from integrating over the line profiles shown in
Figure~\ref{si1260}.
The column density $N_\SiII^{\rm sh}$ of shocked \ion{Si}{2} is
\begin{equation}
\label{NSiIIshk}
N_\SiII^{\rm sh}
= {m_e c \over \pi e^2 f \lambda} (2\pi)^{1/2} \sigma \tau_0
= 9.0 \pm 0.3 \times 10^{14} {\rmn cm}^{-2}
\end{equation}
where $\tau_0 = 0.98 \pm 0.02$ is the optical depth at line center of the
fitted Gaussian profile of the shocked \ion{Si}{2},
and the factor $(2\pi)^{1/2} \sigma$ comes from integrating over
the Gaussian profile.
The profile of unshocked \ion{Si}{2} is the residual after
the shocked \ion{Si}{2} is subtracted from the total,
and we measure its column density $N_\SiII^{\rm unsh}$ by integrating
the unshocked profile over the velocity range $5600$-$7200 \,{\rmn km}\,{\rmn s}^{-1}$:
\begin{equation}
\label{NSiIIunshk}
N_\SiII^{\rm unsh}
= {m_e c \over \pi e^2 f \lambda} \!\int\! \tau^{\rm unsh} \dd v
= 1.5 \pm 0.2 \times 10^{14} {\rmn cm}^{-2}
\end{equation}
where the uncertainty is largely from uncertainty in the fit
to the shocked \ion{Si}{2}.
Integrating the full \ion{Si}{2} profile over $1000$-$8200 \,{\rmn km}\,{\rmn s}^{-1}$
yields the total column density of \ion{Si}{2}
\begin{equation}
\label{NSiIItot}
N_\SiII
= 10.5 \pm 0.1 \times 10^{14} {\rmn cm}^{-2}
\end{equation}
where the uncertainty is from photon counts,
and is smaller than the uncertainties in either of the shocked or
unshocked \ion{Si}{2} column densities individually
(because there is some uncertainty in allocating the total column density
between the shocked and unshocked components).
The corresponding masses of unshocked and shocked \ion{Si}{2}
can also be inferred,
if it is assumed that the Si was originally ejected spherically symmetrically.
In subsection \ref{blue}
we will argue that the absence of blueshifted \ion{Si}{2} absorption can be
explained if the reverse shock has passed entirely through the Si on the
near side of SN1006, and Si has been collisionally ionized to high ion stages.
Thus the absence of blueshifted \ion{Si}{2} need not conflict with
spherical symmetry of Si ejected in the supernova explosion.
If the shocked ejecta are taken to lie in a thin shell at a
free expansion radius of $v = 7200 \,{\rmn km}\,{\rmn s}^{-1}$
(slightly outside the position of reverse shock ---
cf.\ the argument in paragraph 3 of subsection \ref{jump}),
then the mass of shocked \ion{Si}{2} is
\begin{equation}
\label{MSiIIshk}
M_\SiII^{\rm sh}
= 4\pi m_\SiII (v t)^2 N_\SiII^{\rm sh}
= 0.127 \pm 0.006 \,{\rmn M}_{\sun}
\end{equation}
where the uncertainty includes only the uncertainty in the column density
of shocked \ion{Si}{2}.
This value should perhaps be modified to
$M_\SiII^{\rm sh} = 0.13 \pm 0.01 \,{\rmn M}_{\sun}$
to allow for uncertainty in the radial position of the shocked \ion{Si}{2}.
The mass of unshocked \ion{Si}{2} is an integral over the unshocked
line profile
\begin{equation}
M_\SiII^{\rm unsh}
= 4\pi m_\SiII t^2 \!\int\! v^2 \dd N_\SiII^{\rm unsh}
= 0.017 \pm 0.002 \,{\rmn M}_{\sun}
\end{equation}
where the uncertainty is largely from uncertainty in the fit
to the shocked \ion{Si}{2}.
\subsection{Purity of shocked Si}
\label{purity}
We show below
that the observed column density of \ion{Si}{2}
is close to the column density which would be predicted under the simple
assumption of steady state collisional ionization downstream of the shock
(note that recombination is negligible).
Now the assumption of steady state ionization is surely false,
since the timescale to ionize \ion{Si}{2} is comparable to the age of
the remnant, equation (\ref{tion'}).
Below we will argue that the effect of non-steady state is generally such
as to reduce the column density below the steady state value.
The constraint that the observed column density should be less than
or comparable to the steady state value then leads to an upper limit on the
electron to ion ratio, equation (\ref{nela}).
The fact that this upper limit is not much greater than unity
leads to the interesting conclusion that the bulk of the observed
shocked \ion{Si}{2} must be fairly pure,
since admixtures of other elements would increase the electron to ion ratio
above the limit.
If the postshock number density of \ion{Si}{2} ions is $n_\SiII$
(which is 4 times the preshock number density),
then at shock velocity $v_s$
the number of \ion{Si}{2} ions entering the reverse shock per unit area
and time is
$n_\SiII v_s/4$.
The \ion{Si}{2} ions are collisionally ionized by electrons in the shocked gas
at a rate $n_e \langle \sigma v \rangle_\SiII$ ionizations per unit time.
For $v_s = 2860 \,{\rmn km}\,{\rmn s}^{-1}$ and
$\langle \sigma v \rangle_\SiII = 6.1 \times 10^{-8} {\rmn cm}^3 {\rmn s}^{-1}$,
it follows that
the column density $N_\SiII^{\rm steady}$ of shocked \ion{Si}{2}
in steady state should be
\begin{equation}
\label{NSiIIsteady}
N_\SiII^{\rm steady} =
{n_\SiII v_s \over 4 n_e \langle \sigma v \rangle_\SiII}
= 1.2 \times 10^{15} {\rmn cm}^{-2} {n_\SiII \over n_e}
\ .
\end{equation}
If, as will be argued below,
the actual (non-steady state) column density (\ref{NSiIIshk})
of shocked \ion{Si}{2}
is less than or comparable to the steady state value (\ref{NSiIIsteady}),
then the mean ratio of electron to \ion{Si}{2} density in the shocked gas
satisfies
\begin{equation}
\label{nela}
{n_e \over n_\SiII} \la 1.3
\ .
\end{equation}
The ratio must of course also satisfy $n_e / n_\SiII \ge 1$,
since each \ion{Si}{2} ion itself donates one electron.
Such a low value (\ref{nela}) of the electron to \ion{Si}{2} ratio in the
shocked \ion{Si}{2} would indicate that the bulk of the shocked \ion{Si}{2}
must be of a rather high degree of purity,
since the presence of significant quantities of other elements or
higher ionization states of Si would increase the number of electrons
per \ion{Si}{2} ion above the limit (\ref{nela}).
Indeed, even if the \ion{Si}{2} entering the shock were initially pure,
ionization to \ion{Si}{3} and higher states would release
additional electrons.
In steady state, the mean electron to \ion{Si}{2} ratio experienced
by \ion{Si}{2} during its ionization from an initially pure \ion{Si}{2}
state is $n_e / n_\SiII = 1.5$,
already slightly larger than the limit (\ref{nela}).
The limit (\ref{nela}) on the electron to \ion{Si}{2} ratio
is so low as to make it difficult to include even modest quantities
of other elements with the silicon.
This may be a problem.
While Si is generally the most abundant element
in Si-rich material produced by explosive nucleosynthesis,
other elements, notably sulfur, usually accompany the silicon.
For example, the deflagrated white dwarf model W7 of
Nomoto et al.\ (1984)
contains $0.16 \,{\rmn M}_{\sun}$ of Si
mostly in a layer which is about 60\% Si, 30\% S by mass.
At this elemental abundance,
and assuming similar ionization states for all elements,
the expected mean electron to \ion{Si}{2} ratio in steady state would be
$n_e / n_\SiII = 1.5 / 0.6 = 2.5$,
almost twice the limit given by equation (\ref{nela}).
Because of this potential difficulty, we discuss carefully below
how robust is the constraint (\ref{nela}).
First we address the accuracy of the predicted value (\ref{NSiIIsteady})
of the steady state column density,
and then we discuss the non-steady state case.
The electron to \ion{Si}{2} ratio could be increased
if the predicted steady state column density (\ref{NSiIIsteady}) were increased,
either by increasing the shock velocity $v_s$,
or by reducing the collisional ionization rate
$\langle \sigma v \rangle_\SiII$.
We consider the former first, then the latter.
The shock velocity $v_s = 2860 \,{\rmn km}\,{\rmn s}^{-1}$ is inferred from the observed
ion dispersion in the \ion{Si}{2} 1260\,\AA\ line, equation (\ref{vs}).
This should give a fair estimate of the mean shock velocity of the
observed shocked \ion{Si}{2},
except for a small correction from the fact that
the ion dispersion (temperature) in the past would have been higher because
of adiabatic expansion of the remnant.
Moffet, Goss \& Reynolds (1993)
find that the global radius $R$ of the radio remnant is currently increasing
with time according to $R \propto t^{0.48 \pm 0.13}$.
If the ambient ISM is assumed to be uniform,
this indicates that the pressure in the remnant is varying with time as
$P \propto ( R / t )^2 \propto t^{-1.04 \pm 0.26}$,
hence that the temperature is varying in Lagrangian gas elements as
$T \propto P^{2/5} \propto t^{-0.42 \pm 0.10}$,
hence that the ion dispersion is varying as
$\sigma \propto T^{1/2} \propto t^{-0.21 \pm 0.05}$,
a rather weak function of time.
If one supposes that the observed \ion{Si}{2} was shocked
on average when SN1006 was say half its current age,
then the dispersion, hence the reverse shock velocity,
could have been $\sim 20\%$ higher than at present.
The steady state column density (\ref{NSiIIsteady}) would then be
$\sim 20\%$ higher,
and the constraint on the electron to \ion{Si}{2} ratio (\ref{nela})
would be relaxed slightly to
$n_e / n_\SiII \la 1.5$.
The collisional ionization rate
$\langle \sigma v \rangle_\SiII = 6.1 \times 10^{-8} {\rmn cm}^3 {\rmn s}^{-1}$
used in equation (\ref{NSiIIsteady}) comes from integrating the
cross sections of Lennon et al.\ (1988)
over a Maxwellian distribution of electrons at a temperature of 83\,eV.
The quoted error in the cross-sections is 60\%,
the large uncertainty arising from the fact
that the cross-sections for \ion{Si}{2} are derived from
isoelectronic scaling rather than from real data.
Reducing the ionization rate by 0.2 dex
would relax the constraint (\ref{nela}) to
$n_e / n_\SiII \la 2.0$.
The temperature of 83\,eV used in the collisional ionization rate above
is the temperature reached by electrons as a result
of Coulomb collisions with \ion{Si}{2} ions
over the collisional ionization timescale of \ion{Si}{2}.
The assumption here that there is no collisionless heating of electrons
in the shock is in accordance with the arguments given
in subsection \ref{collisionless}.
Actually,
the ionization rate of \ion{Si}{2} by electron impact,
as derived from the cross-sections of Lennon et al.,
has a broad maximum at an electron temperature of $\sim 200$\,eV,
and varies over only $5$-$7 \times 10^{-8} {\rmn cm}^3 {\rmn s}^{-1}$
for electron temperatures 40-1000\,eV.
Thus uncertainty in the electron temperature does not lead to much
uncertainty in the collisional ionization rate,
unless there is substantial collisionless heating
of electrons to temperatures much higher than 1\,keV.
We have argued against significant collisionless electron heating
in subsection~\ref{collisionless}.
However, if
collisionless electron heating to temperatures greater than 1\,keV did occur,
it would imply both a higher shock velocity,
since the observed ion dispersion would underestimate the shock energy,
and a lower collisional ionization rate,
both of which act to increase the steady state column density
(\ref{NSiIIsteady}).
Thus collisionless electron heating, if it occurs,
would allow a larger electron to \ion{Si}{2} ratio than
given by equation (\ref{nela}).
We now turn to the argument that in a non-steady state situation, as here,
the column density of shocked \ion{Si}{2} is likely to be less
than the steady state column density (\ref{NSiIIsteady}),
which leads to the constraint (\ref{nela}) on the
mean electron to \ion{Si}{2} ratio in the shocked \ion{Si}{2}.
In the first place,
simply truncating the Si at some point downstream of the shock
will give a lower column density than the steady state value.
Secondly,
geometric effects tend to reduce the column density below the steady
state value.
That is,
the column density of \ion{Si}{2} is diluted by the squared ratio
$(r_s/r)^2$ of the original radius $r_s$ of the gas at the time it was
shocked to the present radius $r$ ($> r_s$) of this shocked gas.
Thirdly,
if the density profile of shocked Si at the present time increases outwards,
then the faster ionization of the denser, earlier shocked, gas
reduces its column density per interval of ionization time,
and the net column density is again lower than steady state
(a more rigorous demonstration of this is given in the Appendix).
Conversely,
if the density profile of shocked Si decreases outwards,
then the net column density can be higher than steady state,
but only if the flow is allowed to continue for
sufficiently longer than a collisional ionization time.
However, according to equation (\ref{tion'})
the ionization timescale at the present \ion{Si}{2} density
is comparable to the age of the remnant,
and the ionization timescale would be longer at lower density,
so there is not much room for increasing the column density this way either.
So is there any way that the actual column density of \ion{Si}{2}
could be higher than the steady state value?
Clearly yes, given sufficient freedom with the density profile of shocked Si.
For example,
one possibility is that there is a `hump' in the shocked
\ion{Si}{2} density profile, such that the density increases outward of the
present position of the reverse shock front, but then declines at larger radii.
Some tuning is required to ensure that
the density on both near and far sides of the hump
is high enough to produce significant column density,
but not so high as to ionize Si above \ion{Si}{2}.
The higher the column density, the more fine-tuning is required.
We thus conclude that while some violation of the limit
(\ref{nela}) on the mean electron to \ion{Si}{2} ratio in the shocked
\ion{Si}{2} is possible,
greater violations, exceeding say a factor of two, are less likely.
It then follows that the bulk of the shocked \ion{Si}{2}
is likely to be of a fairly high degree of purity.
In particular,
there is unlikely to be much iron mixed in with the shocked silicon,
a conclusion which is consistent with the absence of \ion{Fe}{2}
absorption with the same profile as the shocked \ion{Si}{2},
as discussed in subsection~\ref{shockedFe}.
To avoid misunderstanding,
this statement refers only to the shocked \ion{Si}{2}:
iron could be mixed with the unshocked Si,
and indeed the absorption profile of \ion{Fe}{2},
Figure~\ref{rho} below,
does suggest that there is some Fe mixed with unshocked Si.
\subsection{\protect\ion{Si}{3} and \protect\ion{Si}{4} line profiles}
\label{SiIII+IV}
Given that shocked \ion{Si}{2} apparently persists
for a time comparable to its ionization time,
it is difficult to avoid producing an appreciable quantity of
\ion{Si}{3} and \ion{Si}{4} as the result of collisional ionization of
\ion{Si}{2} in the shocked ejecta.
We thus conclude that it is likely that most of the observed
\ion{Si}{3} and \ion{Si}{4}
absorption arises from shocked ejecta.
This is consistent with the observed line profiles,
as will now be discussed.
\begin{figure}[tb]
\epsfbox[170 262 415 525]{si3.ps}
\caption[1]{
HST G130H spectrum (solid line) relative to the adopted stellar continuum
around the redshifted \ion{Si}{3} 1206.500\,\AA\
($f = 1.669$) feature.
Upper axis shows velocity in the rest frame of the \ion{Si}{3} line.
The position and width of the fitted Gaussian profile of \ion{Si}{3}
(dotted line)
have been constrained to be the same as that of the
\ion{Si}{2} 1260\,\AA\ feature.
Also shown is the residual (dashed line) after subtraction both of
the \ion{Si}{3} fitted Gaussian,
and of the contribution of redshifted
\ion{Si}{2} 1193, 1190\,\AA\ (dotted line)
assumed to have the same profile as the \ion{Si}{2} 1260\,\AA\ feature.
The residual shows,
besides Ly\,$\alpha$ emission and absorption,
a weak indication of absorption by unshocked \ion{Si}{3}
at velocities $5500$-$7000 \,{\rmn km}\,{\rmn s}^{-1}$.
The adopted stellar continuum is the same as that
for the \ion{Si}{2} 1260\,\AA\ feature in Figure~\ref{si1260}.
\label{si3}
}
\end{figure}
\begin{figure}[tb]
\epsfbox[170 262 415 525]{si4.ps}
\caption[1]{
HST G130H spectrum relative to the adopted stellar continuum around the
redshifted \ion{Si}{4} 1393.755, 1402.770\,\AA\ ($f = 0.5140$, 0.2553) feature.
Upper axis shows velocity in the rest frame
of the weighted mean wavelength of the feature.
Dotted lines show the best fit to a Gaussian pair with line center and
dispersion constrained to be that of \ion{Si}{2} 1260\,\AA\ for each
component of the doublet,
and with optical depths fixed equal to the ratio of oscillator strengths
times wavelengths of the doublet.
The best fit dispersion of the \ion{Si}{4} is $1700 \pm 100 \,{\rmn km}\,{\rmn s}^{-1}$,
which is $4.5 \sigma$ larger than the $1240 \,{\rmn km}\,{\rmn s}^{-1}$ dispersion
of the fit shown here.
There is no evidence of unshocked \ion{Si}{4}.
The adopted stellar continuum is
$\log F =
\log ( 2.9 \times 10^{-14} \,{\rmn erg}\,{\rmn s}^{-1}\,{\rmn cm}^{-2}\,{\rm \AA}^{-1} )
- 1.4 \log (\lambda/1397\,{\rm \AA})$.
\label{si4}
}
\end{figure}
Figures~\ref{si3} and \ref{si4} show fits to
the redshifted
\ion{Si}{3} 1206\,\AA\
and \ion{Si}{4} 1394, 1403\,\AA\ features
using as templates the shocked and unshocked profiles of
\ion{Si}{2} 1260\,\AA\ shown in Figure~\ref{si1260}.
Table~\ref{sitab} gives
the fitted column densities of shocked and unshocked
\ion{Si}{3} and \ion{Si}{4},
expressed relative to the best fit column density of shocked and unshocked
\ion{Si}{2} given in Table~\ref{redtab}.
The \ion{Si}{3} 1206\,\AA\ profile appears to be mostly shocked.
There is some indication of unshocked \ion{Si}{3}
over $5500$-$7000 \,{\rmn km}\,{\rmn s}^{-1}$ at the $2 \sigma$ level,
Table~\ref{sitab},
as suggested by the residual profile after subtraction of shocked \ion{Si}{3}
plotted in Figure~\ref{si3}.
The dispersion of the fitted Gaussian profile of the \ion{Si}{3},
if allowed to be a free parameter,
is $1290 \pm 60 \,{\rmn km}\,{\rmn s}^{-1}$ if unshocked \ion{Si}{3} is excluded,
or $1210 \pm 60 \,{\rmn km}\,{\rmn s}^{-1}$ if unshocked \ion{Si}{3} is admitted,
which are in good agreement with the $1240 \pm 40 \,{\rmn km}\,{\rmn s}^{-1}$ dispersion
of the \ion{Si}{2}.
The profile of the
\ion{Si}{4} 1394, 1403\,\AA\ feature, Figure~\ref{si4},
is consistent with containing no unshocked \ion{Si}{4}.
If the line center and width of the Gaussian pair fitted to
the \ion{Si}{4} 1394, 1403\,\AA\ doublet are allowed to be free,
then the center remains close to $5050 \,{\rmn km}\,{\rmn s}^{-1}$,
the same as for \ion{Si}{2} 1260\,\AA,
but the best fit dispersion of the \ion{Si}{4} is
$1700 \pm 100 \,{\rmn km}\,{\rmn s}^{-1}$,
which is $4.5 \sigma$ higher than the $1240 \,{\rmn km}\,{\rmn s}^{-1}$ dispersion of
\ion{Si}{2} 1260\,\AA.
The broader dispersion suggests that the \ion{Si}{4}
may be in slightly lower density gas than the \ion{Si}{2},
since the pressure is presumably the same for both.
It is not clear however that the observed difference in velocity width
between \ion{Si}{4} and \ion{Si}{2} is real.
One problem is that the continuum around \ion{Si}{4} appears less
well defined than for the \ion{Si}{2} and \ion{Si}{3} features.
As elsewhere in this paper,
we assume a stellar continuum which is linear in $\log F$-$\log \lambda$,
but in fact there is a hint of curvature, a large scale depression in the
continuum around \ion{Si}{4} 1394, 1403\,\AA\ (see WCHFLS96, Figure~1).
If we have systematically misjudged the continuum, then it is possible that we
have underestimated the uncertainties in the parameters of \ion{Si}{4} given
Table~\ref{sitab}, perhaps by as much as a factor of 2.
The Gaussian profile shown in Figure~\ref{si4} is constrained to
have the same center and $1240 \,{\rmn km}\,{\rmn s}^{-1}$ dispersion as the
\ion{Si}{2} 1260\,\AA\ feature.
Visually, at least, the fit appears satisfactory.
\begin{deluxetable}{lccc}
\tablewidth{0pt}
\tablecaption{Column densities of shocked and unshocked \protect\ion{Si}{3}
and \protect\ion{Si}{4}, relative to best fit column densities of
\protect\ion{Si}{2}
\label{sitab}}
\tablehead{& \colhead{\protect\ion{Si}{2}} & \colhead{\protect\ion{Si}{3}} &
\colhead{\protect\ion{Si}{4}}}
\startdata
Shocked & $1 \pm 0.03$ & $0.43 \pm 0.02$ & $0.41 \pm 0.02$ \nl
Unshocked & $1 \pm 0.13$ & $0.065 \pm 0.035$ & $0.02 \pm 0.07$ \nl
\enddata
\tablecomments{
Absolute column densities of \protect\ion{Si}{2} are given in
Table~\protect\ref{redtab}.
Column densities of \protect\ion{Si}{4} relative to \protect\ion{Si}{2}
are for fits in which the dispersion of \protect\ion{Si}{4} is constrained to
be that of \protect\ion{Si}{2} 1260\,\AA, namely $1240 \,{\rmn km}\,{\rmn s}^{-1}$.
The column densities of \protect\ion{Si}{4} become
$0.46 \pm 0.02$ (shocked) and $-0.06 \pm 0.06$ (unshocked)
if instead the dispersion of the \protect\ion{Si}{4} is taken to have the best
fit value $1700 \pm 100 \,{\rmn km}\,{\rmn s}^{-1}$.
}
\end{deluxetable}
\subsection{Total Si mass}
\label{Simass}
In subsection~\ref{purity} we showed that the observed column density
of shocked \ion{Si}{2} is close to the theoretical steady state value.
Is the same also true for \ion{Si}{3} and \ion{Si}{4}?
The answer is no.
In steady state,
the predicted column densities of shocked ionic species are inversely
proportional to their respective collisional ionization rates
$\langle \sigma v \rangle$,
modified by an appropriate electron to ion ratio
(cf.\ equation [\ref{NSiIIsteady}]).
Adopting rates
$6.1 \times 10^{-8} {\rmn cm}^3 {\rmn s}^{-1}$,
$2.2 \times 10^{-8} {\rmn cm}^3 {\rmn s}^{-1}$,
$1.1 \times 10^{-8} {\rmn cm}^3 {\rmn s}^{-1}$
for \ion{Si}{2}, \ion{Si}{3}, \ion{Si}{4} respectively
(Lennon et al.\ 1988),
and assuming a nominal $i$ electrons per ion for Si$^{+i}$,
yields relative column densities in steady state
\begin{equation}
\label{NSisteady}
N_\SiII : N_\SiIII : N_\SiIV
= 1 : 1.4 : 1.8
\ .
\end{equation}
By comparison the observed shocked column densities are
$N_\SiII : N_\SiIII : N_\SiIV = 1 : 0.43 : 0.41$,
according to Table~\ref{sitab}.
Evidently
the observed abundances of shocked \ion{Si}{3} and \ion{Si}{4}
relative to \ion{Si}{2} are several times
less than predicted in steady state.
As discussed in subsection~\ref{purity},
there are several ways to reduce the column density below the steady state
value, of which the most obvious is to truncate the column density,
as is strongly suggested by the fact that the ionization timescales
of \ion{Si}{3} and \ion{Si}{4} are becoming long compared to the age
of the remnant.
In fact these ionization timescales are in precisely the same ratio
(cf.\ equation [\ref{tion}])
as the steady state column densities (\ref{NSisteady})
\begin{equation}
\label{tionSi}
t_\SiII : t_\SiIII : t_\SiIV
= 1 : 1.4 : 1.8
\end{equation}
and it has already been seen that the ionization timescale $t_\SiII$
of \ion{Si}{2} is comparable to the age of the remnant, equation (\ref{tion'}).
If it is true that it is the long ionization times which cause
the column densities of \ion{Si}{3} and \ion{Si}{4}
to be lower than steady state,
then this suggests that there may be little Si in higher ionization states
in the shocked gas on the far side of SN1006.
Thus it appears plausible that we are observing in UV absorption
essentially all the Si there is on the far side of SN1006
along the line of sight to the background SM star.
To avoid confusion, it should be understood that this statement refers
specifically to Si on the far side along this particular
line of sight.
Higher ionization states of Si are indicated by the observed
Si x-ray line emission
(Koyama et al.\ 1995),
which could arise from denser shocked gas in other parts of the remnant.
The mass of Si can be inferred from the observed column densities of
\ion{Si}{2}, \ion{Si}{3}, and \ion{Si}{4},
if it is assumed that silicon was ejected spherically symmetrically
by the supernova explosion.
We will argue in subsection~\ref{blue}
that the absence of blueshifted absorbing Si is not inconsistent
with spherical symmetry.
Taking the shocked and unshocked masses of \ion{Si}{2} from Table~\ref{redtab},
and the ratios of
\ion{Si}{2}, \ion{Si}{3}, and \ion{Si}{4}
from Table~\ref{sitab},
yields a total inferred Si mass of
\begin{eqnarray}
M_{\rmn Si}
&=& 0.127 ( 1 + 0.43 + 0.41 ) {\rmn M}_{\sun} + 0.017 ( 1 + 0.065 ) {\rmn M}_{\sun}
\nonumber \\
&=& 0.25 \pm 0.01 \,{\rmn M}_{\sun}
\ .
\label{MSi}
\end{eqnarray}
This is comparable to the $0.16\,{\rmn M}_{\sun}$ of Si in model W7 of
Nomoto et al.\ (1984).
It is also consistent with the $0.20\,{\rmn M}_{\sun}$ of Si inferred from
the strength of the Si\,K line observed with ASCA
(Koyama et al.\ 1995).
Koyama et al.\ do not quote a mass,
but they do state that the measured surface brightness of the Si\,K line
is 5 times higher than that in the model\footnote{
Hamilton et al.\ took the strength of the Si\,K line
to be (the upper limit to) that measured from
the Einstein Solid State Spectrometer
(Becker et al.\ 1980),
so there is a discrepancy between the ASCA and SSS data.
However, Hamilton et al.\ also noted in their Figure~6 a marked discrepancy
between the SSS and HEAO-1 data of Galas, Venkatesan \& Garmire (1982),
so it is reasonable to suspect an error in the normalization of the SSS data.
}
of Hamilton, Sarazin \& Szymkowiak (1986; see also Hamilton et al.\ 1985),
which had $0.04\,{\rmn M}_{\sun}$ of Si.
\subsection{No blueshifted Si --- evidence for an inhomogeneous ISM?}
\label{blue}
There is no evidence for blueshifted Si absorption in the UV spectrum.
At best, there is a possible hint of a broad shallow depression
around $\sim - 5000 \,{\rmn km}\,{\rmn s}^{-1}$, from 1365\,\AA\ to 1390\,\AA,
on the blue side of the
\ion{Si}{4} 1394, 1403\,\AA\ line
(see WCHFLS96, Figure~1).
The possibility that some high velocity blueshifted \ion{Si}{2} 1260\,\AA\
is hidden in the red wing of \ion{Si}{3} 1206\,\AA\ is excluded by the
absence of corresponding blueshifted \ion{Si}{2} 1527\,\AA.
There are two possible reasons for the asymmetry in the observed Si absorption.
One is that there was an intrinsic asymmetry in the supernova explosion.
According to Garcia-Senz \& Woosley (1995),
the nuclear runaway that culminates in the explosion of a nearly
Chandrasekhar mass white dwarf begins as stable convective carbon burning,
and ignition is likely to occur off-center at one or more points
over a volume of the order of a pressure scale height.
The subsequent propagation of the convectively driven burning front is
Rayleigh-Taylor unstable
(Livne 1993;
Khokhlov 1995;
Niemeyer \& Hillebrandt 1995),
although Arnett \& Livne (1994) find that in delayed detonation models
the second, detonation, phase of the explosion tends to restore
spherical symmetry.
Thus asymmetry in the explosion is possible,
perhaps even likely, as an explanation of the asymmetry in the Si absorption,
especially since the absorption samples only a narrow line of sight
through SN1006 to the background SM star.
However, we do not pursue this possibility further,
in part because in abandoning spherical symmetry we lose any
predictive power,
and in part because there is another explanation which
is more attractive because it resolves some other observational problems.
The other possible cause of the asymmetry in the Si absorption
is that the ISM around SN1006 is inhomogeneous,
with the ISM on the far side of SN1006 having a significantly lower density
than the near side.
According to this hypothesis,
the low density on the far side is such that
the reverse shock on the far side has reached inward only to
a free expansion radius of $7070 \,{\rmn km}\,{\rmn s}^{-1}$,
whereas the higher density on the near side is such that
the reverse shock on the near side has passed entirely through the Si layer,
and Si has been collisionally ionized to stages higher than \ion{Si}{4},
making it unobservable in UV absorption.
A serious objection to the reverse shock on the near side being
farther in than the reverse shock on the far side
is the observation by WCFHS93
of blueshifted \ion{Fe}{2}
absorption certainly to velocities $-7000 \,{\rmn km}\,{\rmn s}^{-1}$,
perhaps to velocities $-9000 \,{\rmn km}\,{\rmn s}^{-1}$.
We will review the observational evidence for such high velocity
blueshifted \ion{Fe}{2} in Section \ref{iron} below,
where we will argue that the evidence is not compelling.
An inhomogeneous ISM around SN1006 is indicated by other observations.
Wu et al.\ (1983) and subsequent authors have remarked on the difficulty of
fitting the observed high velocity ($\sim 7000 \,{\rmn km}\,{\rmn s}^{-1}$) Si
within the confines of the observed interstellar blast wave,
if spherical symmetry is assumed.
At a distance of $1.8 \pm 0.3 \,{\rmn kpc}$ (Laming et al.\ 1996),
the remnant's observed $15'$ radius
(Reynolds \& Gilmore 1986, 1993; see also Willingale et al.\ 1996)
at 980 years old\footnote{
Note there is a 23 year light travel time across one radius of the remnant,
so really we are seeing the far side at an age 23 years younger,
and the near side 23 years older, than the mean age.
}
corresponds to a free expansion radius
of $7700 \pm 1300 \,{\rmn km}\,{\rmn s}^{-1}$.
The difficulty is resolved if
the remnant of SN1006 bulges out on the far side,
because of the lower density there.
A second piece of evidence suggesting inhomogeneity
is the high $5050 \,{\rmn km}\,{\rmn s}^{-1}$ velocity of the shocked ejecta
behind the reverse shock on the far side of SN1006
inferred from the present observations,
compared with the $1800$-$2400 \,{\rmn km}\,{\rmn s}^{-1}$
(the lower velocity corresponds to no collisionless electron heating,
which is the preferred case)
velocity of shocked gas behind the interstellar shock
inferred from H\,$\alpha$ and UV emission line widths along the
NW sector of the remnant
(Kirshner et al.\ 1987;
Long et al.\ 1988;
Smith et al.\ 1991;
Raymond et al.\ 1995).
These two velocities, $5050 \,{\rmn km}\,{\rmn s}^{-1}$ versus $1800$-$2400 \,{\rmn km}\,{\rmn s}^{-1}$,
appear incompatible,
especially as the velocity of shocked gas is expected in realistic
hydrodynamic models to increase outwards from the reverse shock
to the interstellar shock
(see for example HF88, Fig.~2).
The incompatibility is resolved if the ISM around SN1006 is inhomogeneous,
with the density on the far side of SN1006 being substantially lower
than the density at the NW edge.
A final piece of evidence supporting inhomogeneity
comes from the observation of Si\,K line emission in x-rays
(Koyama et al.\ 1995).
This emission is likely to be from ejecta, since the inferred abundance
of Si is stated to be an order of magnitude higher than that of O, Ne, or Fe.
To ionize Si to high ionization states in the age of SN1006,
and to produce Si\,K line emission at the observed luminosity,
requires densities
$n_{\rmn Si} \ga 10^{-2} \,{\rmn cm}^{-3}$
(i.e.\ $n_e \ga 10^{-1} \,{\rmn cm}^{-3}$ since the Si is highly ionized),
substantially higher than the postshock density of
$n_\SiII = 2 \times 10^{-4} \,{\rmn cm}^{-3}$
deduced here from the redshifted \ion{Si}{2} 1260\,\AA\ absorption profile.
It is difficult to see how the required high Si density could be achieved
if the reverse shock everywhere around the remnant is at a radius as
high as $7070 \,{\rmn km}\,{\rmn s}^{-1}$,
whereas higher densities would occur naturally if
the reverse shock around most of the remnant were farther in,
since then ejecta would have been shocked at earlier times when
its density ($\propto t^{-3}$ in free expansion) was higher.
All these arguments point to the notion that the ISM density on the far
side of SN1006 is anomalously low compared to the density around the
rest of the remnant.
\subsection{Ionization of Si on the near side}
\label{blueion}
In Section~\ref{iron}, we will argue that the \ion{Fe}{2}
absorption profiles suggest that the reverse shock on the near side
of SN1006 may be at a free expansion radius of $4200 \,{\rmn km}\,{\rmn s}^{-1}$
(as with the Hubble expansion of the Universe, it is often convenient
to think in a comoving frame expanding freely with the ejecta,
so that a free expansion velocity $v = r/t$ can be thought of as a radius).
Here we estimate whether Si could have ionized beyond \ion{Si}{4},
as required by the absence of observed blueshifted Si absorption,
if the reverse shock on the near side is indeed at $4200 \,{\rmn km}\,{\rmn s}^{-1}$.
In our first estimate,
we find that blueshifted \ion{Si}{4} {\em should}\/ be observable,
with a column density $\sim 40\%$ that of the observed redshifted \ion{Si}{4}
absorption.
However, the column density is somewhat sensitive to the assumptions made,
and it is not hard to bring the column density down below observable
levels.
The line profile of any blueshifted absorbing Si would have
a width comparable to that of the observed broad redshifted Si absorption,
but the centroid would be shifted to lower velocities,
to $\sim -2000 \,{\rmn km}\,{\rmn s}^{-1}$,
if it assumed that the reverse shock velocity on the near side
is comparable to that, $v_s = 2860 \,{\rmn km}\,{\rmn s}^{-1}$, observed on the far side.
To avoid being observed at say $3 \sigma$,
the column density of blueshifted \ion{Si}{4} should be less than
$0.6 \times 10^{14} \,{\rmn cm}^{-2}$.
To estimate the ionization of Si,
it is necessary to adopt a hydrodynamic model.
Now one interesting aspect of hydrodynamical models of deflagrated white dwarfs
expanding into a uniform ambient medium
is that the reverse shock velocity $v_s$ remains almost constant in time
(this conclusion is based on hydrodynamic simulations carried out by HF88).
In model W7 of Nomoto et al.\ (1984),
the reverse shock velocity varies (non-monotonically)
between $3300 \,{\rmn km}\,{\rmn s}^{-1}$ and $5200 \,{\rmn km}\,{\rmn s}^{-1}$
as it propagates inward from a free expansion radius of $13000 \,{\rmn km}\,{\rmn s}^{-1}$
to $700 \,{\rmn km}\,{\rmn s}^{-1}$,
after which the shock velocity accelerates.
Similarly in model CDTG7 (Woosley 1987, private communication),
which is similar to model CDTG5 of Woosley \& Weaver (1987),
the reverse shock velocity varies between $3200 \,{\rmn km}\,{\rmn s}^{-1}$ and $4100 \,{\rmn km}\,{\rmn s}^{-1}$
as it propagates inward from a free expansion radius of $10000 \,{\rmn km}\,{\rmn s}^{-1}$
to $1500 \,{\rmn km}\,{\rmn s}^{-1}$.
These numbers do not depend on the density of the ambient medium,
although they do depend on the ambient density being uniform.
If the reverse shock velocity $v_s$ remains constant in time,
then the radius $r$ of the reverse shock evolves with time $t$
according to $\dd r/\dd t = r/t - v_s$,
from which it follows that the free expansion radius $r/t$
of the reverse shock varies with time as
\begin{equation}
\label{rt}
{r \over t} = v_s \ln left ( {t_\ast \over t} right )
\end{equation}
where $t_\ast$ is the age at which the reverse shock eventually hits
the center of the remnant.
The assumption that the reverse shock $v_s$ is constant in time
may not be correct for SN1006,
but it provides a convenient simplification to estimate the ionization of Si
on the near side.
Let us first estimate the ionization state of Si which was originally
at a free expansion radius of $7070 \,{\rmn km}\,{\rmn s}^{-1}$,
the current location of the reverse shock on the far side.
If the reverse shock on the near side is currently at a free expansion radius of
$r/t = 4200 \,{\rmn km}\,{\rmn s}^{-1}$,
and if it has moved at a constant $v_s = 2860 \,{\rmn km}\,{\rmn s}^{-1}$,
the measured value on the far side
(recall that the reverse shock velocity is independent of the ambient density,
for uniform ambient density),
then it would have passed through a free expansion radius of $7070 \,{\rmn km}\,{\rmn s}^{-1}$
when the age $t_s$ of SN1006 was
$t_s / t = \exp [ ( 4200 \,{\rmn km}\,{\rmn s}^{-1} - 7070 \,{\rmn km}\,{\rmn s}^{-1} ) / 2860 \,{\rmn km}\,{\rmn s}^{-1} ] = 0.37$
times its present age $t$,
according to equation (\ref{rt}).
The postshock density of Si ions at that time would have been
$(t / t_s)^3 = 20$ times higher than the presently observed postshock
density of $n_\SiII = 2.2 \times 10^{-4} \,{\rmn cm}^{-3}$.
The ion density in the parcel of gas shocked at that time has been decreasing
because of adiabatic expansion.
The rate of decrease of density by adiabatic expansion can be inferred from the
observed global rate of expansion of the remnant
(Moffet, Goss \& Reynolds 1993)
\begin{equation}
R \propto t^\alpha
\ ,\ \ \
\alpha = {0.48 \pm 0.13}
\ ,
\end{equation}
which for an assumed uniform ambient density would imply that the pressure is
decreasing as
$P \propto ( R / t )^2 \propto t^{2 \alpha - 2}$,
hence that the density in Lagrangian gas elements is varying as
$n \propto P^{3/5} \propto t^{6(\alpha - 1)/5}$.
The current ionization time
$\tau \equiv \int_{t_s}^t n \,\dd t$
of the parcel of gas
originally at free expansion radius $7070 \,{\rmn km}\,{\rmn s}^{-1}$
which was shocked at an age $t_s / t = 0.37$
is then
\begin{eqnarray}
\tau
&=& {n_\SiII t \over (6 \alpha - 1)/5} \left( {t \over t_s} \right)^2
\left[ \left( {t \over t_s} \right)^{(6 \alpha - 1)/5} - 1 \right]
\nonumber \\
&=& 6.0 \times 10^7 \,{\rmn cm}^{-3}\,{\rmn s}
\label{tauf}
\end{eqnarray}
where $n_\SiII t = 6.6 \times 10^6 \,{\rmn cm}^{-3}\,{\rmn s}$ is the present
postshock density of \ion{Si}{2} ions times age
at the radius $7070 \,{\rmn km}\,{\rmn s}^{-1}$.
If the Si is assumed unmixed with other elements and initially singly ionized,
then the ionization time (\ref{tauf}) yields ion fractions
5\% \ion{Si}{3}, 34\% \ion{Si}{4}, and the remaining 61\% in higher stages.
Since this ionization state is close to (just past) the peak
in the \ion{Si}{4} fraction, it should be a good approximation
to estimate the column density of \ion{Si}{4} by expanding locally about
the conditions at $7070 \,{\rmn km}\,{\rmn s}^{-1}$.
The expected column density is
the steady state column density,
multiplied by a geometric factor,
and further multiplied by a `density profile' factor
$[1 + \gamma \tau / (n_s t_s)]^{-1}$,
as shown in the Appendix,
equation (\ref{dNdtau}).
The steady column density is
$N_\SiIV^{\rm steady}$
= $n_\SiIV v_s / ( 4 n_e \langle \sigma v \rangle_\SiIV )$
= $22 \times 10^{14} \,{\rmn cm}^{-2}$
assuming a nominal $n_e/n_\SiIV = 3$.
The geometric factor is
$(7070 t_s / 4200 t)^2 = 0.38$,
which is the squared ratio of the radius of the gas at the time $t_s$
it was shocked to its radius at the present time $t$.
For the density profile factor,
equation (\ref{tauf}) gives
$\tau/(n_s t_s) = [(t/t_s)^{(6\alpha-1)/5} - 1]/[(6\alpha-1)/5] = 1.22$,
while the logarithmic slope $\gamma = - \partial \ln n / \partial \ln t_s |_t$
of the shocked density profile, equation (\ref{gamma}), is
$\gamma$ =
$3 + v_s \partial \ln n^{\rm unsh} / \partial (r/t) + 6(\alpha - 1)/5$
= $3.7$,
the 3 coming from free expansion,
the $v_s \partial \ln n^{\rm unsh} / \partial (r/t) = 1.37$
from the observed unshocked \ion{Si}{2} density profile
(see eq.~[\ref{nSiquad}]) at $7070 \,{\rmn km}\,{\rmn s}^{-1}$
along with equation (\ref{rt}),
and the $6(\alpha - 1)/5 = -0.62$ from the reduction in density caused
by adiabatic expansion.
The resulting density profile factor
is $[1 + \gamma \tau / (n_s t_s)]^{-1} = 0.18$.
The expected column density of blueshifted \ion{Si}{4} is then
$N_\SiIV = 1.5 \times 10^{14} \,{\rmn cm}^{-2}$,
which is 40\% of the observed column density of redshifted \ion{Si}{4},
and 2.5 times the minimum ($3 \sigma$) observable column density.
Thus under a reasonable set of simplifying assumptions,
there should be an observable column density of blueshifted \ion{Si}{4},
contrary to observation.
However,
the expected column density is sensitive to the assumed parameters.
For example, reducing the shock velocity on the near side by 20\%
to $2300 \,{\rmn km}\,{\rmn s}^{-1}$ reduces the column density by a factor 2.5 to
the observable limit
$N_\SiIV = 0.6 \times 10^{14} \,{\rmn cm}^{-2}$.
Whether the shock velocity on the near side is less or more than that
on the far side depends on the unshocked density profile of ejecta
(generally, a shorter exponential scale length of unshocked density with
velocity yields lower shock velocities).
The expected column density is also sensitive to the shocked density profile,
as might be guessed from the fact that the density profile factor of $0.18$
estimated above differs substantially from unity.
Alternatively,
the column density of \ion{Si}{4} could be reduced below the
observable limit by mixing the Si with a comparable mass of other elements,
such as iron, since this would increase the number of electrons
per silicon ion, causing more rapid ionization.
The possibility that there is iron mixed with Si at velocities
$\la 7070 \,{\rmn km}\,{\rmn s}^{-1}$ gains some support from the observed density
profile of \ion{Fe}{2}, shown in Figure~\ref{rho} below.
Note this does not conflict with the argument in subsection~\ref{purity}
that most of the shocked Si
(which was originally at higher free expansion velocities)
is probably fairly pure,
with little admixture of other elements such as iron.
\section{Iron}
\label{iron}
We have argued above that an attractive explanation for
the presence of redshifted Si absorption and absence of blueshifted absorption
is that the ISM on the near side of SN1006 is much denser than that on the
far side,
so that a reverse shock has already passed all the way through the Si layer
on the near side, ionizing it to high ionization stages,
whereas the reverse shock is still moving through the Si layer on the far side.
This picture appears to conflict with our previously reported result
(WCFHS93),
according to which blueshifted \ion{Fe}{2} is present to velocities
$\sim - 8000 \,{\rmn km}\,{\rmn s}^{-1}$.
In this Section we reexamine the \ion{Fe}{2} absorption lines to see
how robust is this result.
In the average broad \ion{Fe}{2} profile shown in Figure~3 of WCFHS93,
redshifted \ion{Fe}{2} seems to extend up to about $7000 \,{\rmn km}\,{\rmn s}^{-1}$,
but not much farther.
This is consistent with the argument of the present paper,
which is that the reverse shock on the far side of SN1006 lies at
$7070 \,{\rmn km}\,{\rmn s}^{-1}$.
The problem lies on the blueshifted side of the \ion{Fe}{2} profile,
which appears to extend clearly to $- 7000 \,{\rmn km}\,{\rmn s}^{-1}$,
possibly to $- 9000 \,{\rmn km}\,{\rmn s}^{-1}$.
Figure~2 of WCFHS93 shows separately the two broad
\ion{Fe}{2} 2383, 2344, 2374\,\AA\ and
\ion{Fe}{2} 2600, 2587\,\AA\ features.
The \ion{Fe}{2} 2600, 2587\,\AA\ feature
appears to have a fairly abrupt blue edge,
although there is perhaps a tail to higher velocities
depending on where the continuum is placed.
The blue edge is at a velocity of $-4200 \,{\rmn km}\,{\rmn s}^{-1}$
with respect to the stronger 2600\,\AA\ component of the doublet,
and the same edge appears at this velocity in the average \ion{Fe}{2}
profile shown in WCFHS93's Figure~3.
We will argue that this edge plausibly represents the position of
the reverse shock on the near side of SN1006.
In contrast to \ion{Fe}{2} 2600\,\AA,
the deconvolved profile of the \ion{Fe}{2} 2383\,\AA\ feature,
plotted in the bottom curve of WCFHS93's Figure~2,
shows blueshifted absorption clearly extending to
$\la 7000 \,{\rmn km}\,{\rmn s}^{-1}$.
We note that the second strongest component of the triplet, 2344\,\AA,
with $1/3$ the oscillator strength of the principal 2383\,\AA\ component,
lies at $- 4900 \,{\rmn km}\,{\rmn s}^{-1}$ blueward of the principal line,
and uncertainty involved in removing the secondary component in the
deconvolution procedure could tend to obscure any sharp blue edge at
$- 4200 \,{\rmn km}\,{\rmn s}^{-1}$ on the principal component.
\subsection{\protect\ion{Fe}{2} analysis}
\label{reanalysis}
\begin{figure*}[tb]
\begin{minipage}{175mm}
\epsfbox[25 256 542 513]{fluxfe.ps}
\caption[1]{
G190H and G270H spectra,
with the calibrated spectra at bottom,
and the dereddened spectra at top.
Also shown are
the continua linear in $\log F$-$\log \lambda$ (solid lines)
adopted in the present paper
for each of the \ion{Fe}{2} 2600\,\AA\ and \ion{Fe}{2} 2383\,\AA\ features,
and a continuum quadratic in $F$-$\lambda$ (dashed line) similar
(but not identical, because of the slightly different reddening)
to that adopted by WCFHS93.
The linear continua are
$\log F =
\log ( 1.89 \times 10^{-14} \,{\rmn erg}\,{\rmn s}^{-1}\,{\rmn cm}^{-2}\,{\rm \AA}^{-1} )
- 2.7 \log (\lambda/2383\,{\rm \AA})$
for the \ion{Fe}{2} 2383\,\AA\ feature, and
$\log F =
\log ( 1.38 \times 10^{-14} \,{\rmn erg}\,{\rmn s}^{-1}\,{\rmn cm}^{-2}\,{\rm \AA}^{-1} )
- 3.65 \log (\lambda/2600\,{\rm \AA})$
for the \ion{Fe}{2} 2600\,\AA\ feature.
\label{fluxfe}
}
\end{minipage}
\end{figure*}
In this subsection we present details of a reanalysis of the
\ion{Fe}{2} absorption lines in the HST G190H and G270H spectra
originally analyzed by WCFHS93.
The observations are described by WCFHS93,
and here we describe the differences between
the analysis here and that of WCFHS93.
In carrying out the reanalysis,
we paid particular attention to procedures
which might affect the blue wing of the \ion{Fe}{2} 2383\,\AA\ feature.
The G190H and G270H spectra overlap over the wavelength range
2222-2330\,\AA,
and we merged the two spectra in this region using inverse variance weighting,
whereas WCFHS93 chose to abut the spectra at 2277\,\AA.
In merging the spectra we interpolated the G270H data to the same bin size
as the G190 spectrum, which has higher resolution
(2\,\AA\ versus 2.8\,\AA),
and higher signal to noise ratio than the G270H spectrum
in the overlap region.
According to the FOS handbook,
there is contamination at the level of a few percent in the G190H spectrum
above 2300\,\AA\ from second order,
but, given the absence of strong features over 1150-1165\,\AA,
we accept this contamination in the interests of obtaining higher signal
to noise ratio.
The merged spectrum is noticeably less choppy than the G270H spectrum alone
in the overlap region.
The 2200\,\AA\ extinction bump is close to the blue wing of
the \ion{Fe}{2} 2383\,\AA\ feature,
so we re-examined the reddening correction.
In practice, the changes made here
had little effect on the profile of the \ion{Fe}{2} 2383\,\AA\ feature.
We dereddened the G190H and G270H spectra using the extinction curve of
Cardelli, Clayton \& Mathis (1989),
adopting $E_{B-V} = 0.119$, which is the best fitting value
determined by Blair et al.\ (1996) from HUT data,
and $R \equiv A_V / E_{B-V} = 3.0$.
The value of $E_{B-V}$ is slightly higher than the value
$E_{B-V} = 0.1 \pm 0.02$ measured by WCFHS93
using the extinction curve of Savage \& Mathis (1979).
WCFHS93 comment that their dereddening leaves a bump from
1900\,\AA\ to 2100\,\AA\ and a shallow trough from 2100 to 2300\,\AA.
We find the same difficulty here:
the slightly higher value of $E_{B-V}$ adopted here does slightly better
at removing the 2200\,\AA\ depression,
but at the expense of producing a bigger bump at 2000\,\AA.
The choice of $R$ makes little difference,
but lower values help to reduce the bump marginally.
The value $R = 3.0$ adopted here is slightly below the
`standard' value $R = 3.1$.
WCFHS93 fitted the continuum flux to a quadratic function of wavelength.
The simple quadratic form does impressively well in fitting the
dereddened spectrum over the full range 1600\,\AA\ to 3300\,\AA\
(cf.\ Figure~\ref{fluxfe} and WCFHS93 Figure~2).
However, the quadratic form does not fit perfectly,
and there remains a residual discrepancy which is not well fitted
by a low order polynomial,
and which may possibly result from imperfect modeling of the extinction,
especially around the 2200\,\AA\ bump.
The imperfection mainly affects the \ion{Fe}{2} 2383\,\AA\ feature:
the quadratic continuum appears too steep compared to the `true' continuum
around this feature.
Here we resolve the difficulty by the expedient of fitting
the dereddened continua around each of the two broad \ion{Fe}{2} features
separately, to two different linear functions in $\log F$-$\log \lambda$.
The adopted continua are shown in Figure~\ref{fluxfe}.
An important difference between the present analysis and that of
WCFHS93 is in the treatment of narrow lines.
WCFHS93's procedure was to identify all lines with an equivalent width,
defined relative to a local continuum,
greater than 3 times the expected error.
WCFHS93 then subtracted the best fitting Gaussian for each such line,
treating the position, width, and strength of each line as free parameters.
Here we adopt a different policy,
requiring that the positions, widths, and strengths of narrow
lines conform to prior expectation.
That is,
for each identified narrow line we subtract a Gaussian profile in which
the wavelength is set equal to the expected wavelength
(modulo an overall $+ 36 \,{\rmn km}\,{\rmn s}^{-1}$ shift for all lines),
the width is set equal to the instrumental resolution
(2.8\,\AA\ FWHM for G270H),
and the strengths are required to be mutually consistent
with other narrow lines of the same ion.
The relevant narrow lines are those in and near the
broad \ion{Fe}{2} features.
In the \ion{Fe}{2} 2383, 2344, 2374\,\AA\ feature,
besides the narrow (presumed interstellar) components of the \ion{Fe}{2}
lines themselves,
we identify the narrow line at 2298\,\AA\
as stellar \ion{C}{3} 2297.578\,\AA\
(Bruhweiler, Kondo \& McCluskey 1981).
WCFHS93 subtracted an unidentified narrow line at 2316\,\AA,
but the G190H spectrum does not confirm the reality of this line
in the G270H spectrum,
and here we leave it unsubtracted.
In the \ion{Fe}{2} 2600, 2587\,\AA\ feature,
besides the narrow \ion{Fe}{2} lines themselves,
we identify narrow lines of
\ion{Mn}{2} 2577, 2594, 2606\,\AA,
as did WCFHS93.
The mean velocity shift of the three most prominent narrow \ion{Fe}{2} lines,
those at 2383\,\AA, 2600\,\AA, and 2587\,\AA,
is $+ 36 \pm 24 \,{\rmn km}\,{\rmn s}^{-1}$, and we adopt this velocity shift for all the narrow
\ion{Fe}{2} and \ion{Mn}{2} lines.
We allow the stellar \ion{C}{3} 2298\,\AA\ line its own
best fit velocity shift of $+ 19 \,{\rmn km}\,{\rmn s}^{-1}$,
since there is no reason to assume that the stellar and interstellar
velocities coincide exactly.
The observed equivalent widths
of the \ion{Mn}{2} lines are approximately
proportional to their oscillator strengths times wavelengths,
which suggests the lines are unsaturated,
so we fix the ratios of the fitted \ion{Mn}{2} lines at
their unsaturated values.
Of the five narrow \ion{Fe}{2} lines,
the two principal lines
\ion{Fe}{2} 2383\,\AA\ and \ion{Fe}{2} 2600\,\AA,
and also the line with the fourth largest oscillator strength,
\ion{Fe}{2} 2587\,\AA,
have approximately equal observed equivalent widths (in velocity units)
relative to a local continuum,
$W_{2383}$, $W_{2600}$, $W_{2587}$ =
$85 \pm 6 \,{\rmn km}\,{\rmn s}^{-1}$, $64 \pm 6 \,{\rmn km}\,{\rmn s}^{-1}$, $62 \pm 6 \,{\rmn km}\,{\rmn s}^{-1}$
respectively
(at fixed centroid and dispersion),
which suggests the lines are saturated.
The fifth and weakest line, \ion{Fe}{2} 2374\,\AA,
has an observed equivalent width about half that of the strong lines,
which is consistent with the weak line being marginally optically thin
and the strong lines again being saturated.
For these four lines we allow the strength of the fitted line to take its
best fit value,
since they are mutually consistent within the uncertainties.
The line with the third largest oscillator strength,
\ion{Fe}{2} 2344\,\AA,
appears anomalous, since the observed line has an equivalent width
less than $1/4$ that of the strong lines,
or $1/2$ that of the intrinsically weaker \ion{Fe}{2} 2374\,\AA\ line.
In the fit,
we force the equivalent width of the anomalous \ion{Fe}{2} 2344\,\AA\
narrow line to be the saturated value measured from the
\ion{Fe}{2} 2600\,\AA\ and \ion{Fe}{2} 2587\,\AA\ lines,
multiplied by $0.8$ to allow for a $2 \sigma$ uncertainty in this value.
The fit gives the impression that the anomalous \ion{Fe}{2} 2344\,\AA\ line
is oversubtracted,
but the effect is to bring the profile of the deconvolved broad
\ion{Fe}{2} 2383\,\AA\ line into closer agreement with
that of \ion{Fe}{2} 2600\,\AA.
\begin{figure*}[tb]
\begin{minipage}{175mm}
\epsfbox[52 252 540 533]{fe2.ps}
\caption[1]{
G270H spectrum showing at left the
\ion{Fe}{2} 2382.765, 2344.214, 2374.4612\,\AA\
($f = 0.3006$, 0.1097, 0.02818;
we ignore an unobserved fourth component \ion{Fe}{2} 2367.5905\,\AA\
with $f = 1.6 \times 10^{-4}$) feature,
and at right the
\ion{Fe}{2} 2600.1729, 2586.6500\,\AA\
($f = 0.2239$, 0.06457)
feature
(Morton 1991).
Below 2330\,\AA, the spectrum is an inverse-variance-weighted merger
of G190H and G270H spectra.
The lower curve shows the dereddened spectrum,
scaled to a continuum,
with narrow interstellar \ion{Fe}{2},
stellar \ion{C}{3} 2297.578\,\AA,
and interstellar \ion{Mn}{2} 2576.877, 2594.499, 2606.462\,\AA\
($f = 0.3508$, 0.2710, 0.1927)
lines subtracted
as indicated by dashed lines.
The upper curve (offset by $\log 1.15$)
shows the deconvolved spectra, after removal of
the weaker components of the broad \ion{Fe}{2} lines.
The velocity scales shown on the upper axis
are with respect to the rest frames of the principal component of each
of the features, the
\ion{Fe}{2} 2383\,\AA\ line on the left,
and the
\ion{Fe}{2} 2600\,\AA\ line on the right.
The adopted continua are as shown in Figure~\protect\ref{fluxfe}.
\label{fe2}
}
\end{minipage}
\end{figure*}
We deconvolved the broad
\ion{Fe}{2} 2383, 2344, 2374\,\AA\ and \ion{Fe}{2} 2600, 2587\,\AA\
features by subtracting the contributions from the weaker components,
using the following analytic procedure.
In a two component line,
the observed optical depth $\tau ( v )$
at velocity $v$ with respect to the principal component
is a sum of the line profile $\phi ( v )$ of the principal component
and the line profile $\epsilon \phi ( v + \Delta v )$ of the secondary
component, where
$\Delta v$ is the velocity shift of the secondary relative to the principal
component,
and $\epsilon = f_2 \lambda_2 / ( f_1 \lambda_1 ) < 1$
is the ratio of oscillator strengths times wavelengths:
\begin{equation}
\label{tauv}
\tau ( v ) = \phi ( v ) + \epsilon \phi ( v + \Delta v )
\ .
\end{equation}
Equation (\ref{tauv}) inverts to
\begin{equation}
\label{phi}
\phi ( v ) = \tau ( v ) - \epsilon \tau ( v + \Delta v )
+ \epsilon^2 \tau ( v + 2 \Delta v )
+ \cdots
\end{equation}
which can conveniently be solved iteratively by
\begin{eqnarray}
\label{phin}
\phi_1 ( v ) &=& \tau ( v ) - \epsilon \tau ( v + \Delta v )
\ , \nonumber \\
\phi_{n+1} ( v )
&=& \phi_n ( v ) + \epsilon^{2^n} \phi_n ( v + 2^n \Delta v )
\ .
\end{eqnarray}
The iterative procedure converges rapidly to the solution,
$\phi_n \rightarrow \phi$, as $n$ increases;
we stop at $\phi_3$.
To avoid irrelevant parts of the spectrum outside the line profile
from propagating through the solution,
we set the optical depth to zero, $\tau ( v ) = 0$ in equation (\ref{phin}),
at velocities $v > 10000\,{\rmn km}\,{\rmn s}^{-1}$.
The above procedure works for a two component line such as
\ion{Fe}{2} 2600, 2587\,\AA,
and a slightly more complicated generalization works for a three component
line such as
\ion{Fe}{2} 2383, 2344, 2374\,\AA.
\subsection{\protect\ion{Fe}{2} line profiles}
\label{fe2sec}
Figure~\ref{fe2} shows the results of our reanalysis.
The upper curves in the Figure show the deconvolved
\ion{Fe}{2} 2383\,\AA\ and \ion{Fe}{2} 2600\,\AA\ line profiles,
and these deconvolved profiles agree well with each other.
The deconvolved \ion{Fe}{2} 2600\,\AA\ profile here also agrees well with
that of WCFHS93.
However, the revised \ion{Fe}{2} 2383\,\AA\ profile
no longer shows compelling evidence
for high velocity blueshifted absorption beyond $- 4500 \,{\rmn km}\,{\rmn s}^{-1}$,
although the presence of some absorption is not excluded.
What causes the difference between the \ion{Fe}{2} 2383\,\AA\ line profile
shown in Figure~\ref{fe2} versus that of WCFHS93?
One factor is that we adopt different continua,
as illustrated in Figure~\ref{fluxfe}.
WCFHS93's single quadratic fit to the continuum over the entire spectrum
is certainly more elegant than the two separate linear fits
to each of the two broad \ion{Fe}{2} features which we adopt here.
The advantage of the fit here is that it removes the apparent tilt in the
\ion{Fe}{2} 2383\,\AA\ line profile left by the quadratic fit,
evident in Figures~2 and 3 of WCFHS93.
However, the major difference between the two analyses is
the subtraction here of the narrow \ion{Fe}{2} 2344\,\AA\ interstellar line
with a strength $0.8$ times the saturated line strength observed
in the \ion{Fe}{2} 2600\,\AA\ and 2587\,\AA\ narrow lines.
By comparison, WCFHS93 subtracted the narrow \ion{Fe}{2} 2344\,\AA\ line
using the observed strength of the line, which is anomalously weak compared
to the other four narrow \ion{Fe}{2} lines.
\begin{figure}[tb]
\epsfbox[162 270 410 498]{rho.ps}
\caption[1]{
Inferred density profile of ejecta in SN1006.
The \ion{Fe}{2} profile is from the mean of the two deconvolved
broad \ion{Fe}{2} absorption features shown in Figure~\protect\ref{fe2}.
The dotted line at $+5600$-$7070 \,{\rmn km}\,{\rmn s}^{-1}$ is the profile of unshocked
\ion{Si}{2} from the redshifted \ion{Si}{2} 1260\,\AA\ absorption feature
in Figure~\protect\ref{si1260}.
The dashed line above $7070 \,{\rmn km}\,{\rmn s}^{-1}$ is a plausible extrapolation
of the total Si density before it was shocked:
it is a quadratic function of velocity, equation (\protect\ref{nSiquad}),
which approximately reproduces the observed profile of unshocked \ion{Si}{2},
and which contains the observed total Si mass of $0.25 \,{\rmn M}_{\sun}$
(assuming spherical symmetry),
equation~(\protect\ref{MSi}).
\label{rho}
}
\end{figure}
\begin{deluxetable}{lr}
\tablewidth{0pt}
\tablecaption{Parameters measured from \protect\ion{Fe}{2} profile
\label{fe2tab}}
\tablehead{\colhead{Parameter} & \colhead{Value}}
\startdata
Position of reverse shock on near side & $-4200 \pm 100 \,{\rmn km}\,{\rmn s}^{-1}$ \nl
Column density of \protect\ion{Fe}{2} &
$10.8 \pm 0.9 \times 10^{14} \,{\rmn cm}^{-2}$ \nl
Mass of \protect\ion{Fe}{2} up to $7070\,{\rmn km}\,{\rmn s}^{-1}$ & $0.029 \pm 0.004 \,{\rmn M}_{\sun}$ \nl
\enddata
\tablecomments{
\protect\ion{Fe}{2} mass is from red side of profile,
and assumes spherical symmetry.
}
\end{deluxetable}
Figure~\ref{rho} shows the \ion{Fe}{2} density inferred
from the mean of the two deconvolved \ion{Fe}{2} features.
The \ion{Fe}{2} column density inferred from the mean profile,
integrated from $- 4500 \,{\rmn km}\,{\rmn s}^{-1}$ to $+ 7100 \,{\rmn km}\,{\rmn s}^{-1}$,
is
\begin{equation}
\label{NFeII}
N_\FeII = 10.8 \pm 0.9 \times 10^{14} \,{\rmn cm}^{-2}
\ .
\end{equation}
Most of the uncertainty,
based here on the scatter between the two deconvolved profiles,
comes from the blue side of the profile:
the column density on the blue side
from $- 4500 \,{\rmn km}\,{\rmn s}^{-1}$ to $0 \,{\rmn km}\,{\rmn s}^{-1}$ is
$N_\FeII = 5.2 \pm 0.8 \times 10^{14} \,{\rmn cm}^{-2}$,
while the column density on the red side
from $0 \,{\rmn km}\,{\rmn s}^{-1}$ to $+ 7100 \,{\rmn km}\,{\rmn s}^{-1}$ is
$N_\FeII = 5.6 \pm 0.2 \times 10^{14} \,{\rmn cm}^{-2}$.
In estimating the mass of \ion{Fe}{2}
from the density profile in Figure~\ref{rho},
we take into account the small correction which results from the fact that
the SM star is offset by $2'.45 \pm 0'.25$ southward from the projected center
of the $15'$ radius remnant
(Schweizer \& Middleditch 1980).
The offset corresponds to a free expansion velocity of
$v_\perp = 1300 \pm 250 \,{\rmn km}\,{\rmn s}^{-1}$
at the $1.8 \pm 0.3 \,{\rmn kpc}$ distance of the remnant.
If spherical symmetry is assumed,
then the mass is an integral over the density $\rho(v)$
at line-of-sight velocity $v$:
\begin{equation}
\label{M}
M =
M ( <\! v_\perp )
+
t^3 \! \int_{0}^{v_{\max}} \! \rho(v)
(v^2 + v_\perp^2)^{1/2} \, v \dd v
\end{equation}
where $M ( <\! v_\perp )$ is the mass inside the free-expansion velocity
$v_\perp$.
At a constant central density of
$\rho_\FeII = 0.005 \times 10^{-24} \,{\rmn gm}\,{\rmn cm}^{-3}$,
the mass inside $v_\perp = 1300 \,{\rmn km}\,{\rmn s}^{-1}$ would be
$M_\FeII(<\! v_\perp) = 0.0007 \,{\rmn M}_{\sun}$,
and the actual \ion{Fe}{2} mass is probably slightly higher,
given that the density is increasing mildly inward.
The masses given below, equation (\ref{MFeII}), include a fixed
$M_\FeII(<\! v_\perp) = 0.001 \,{\rmn M}_{\sun}$.
The factor $(v^2 + v_\perp^2)^{1/2}$ rather than $v$ in equation (\ref{M})
increases the \ion{Fe}{2} masses by a further $0.002 \,{\rmn M}_{\sun}$,
so the masses quoted in equation (\ref{MFeII}) are altogether $0.003 \,{\rmn M}_{\sun}$
larger than they would be if no adjustment for the offset of the SM star
were applied.
The total mass of \ion{Fe}{2} inferred from the cleaner, red side
of the profile, assuming spherical symmetry, is then
\begin{equation}
\label{MFeII}
M_\FeII = \left\{
\begin{array}{cl}
0.0156 \pm 0.0009 \,{\rmn M}_{\sun} & ( v \leq 4200 \,{\rmn km}\,{\rmn s}^{-1} ) \\
0.0195 \pm 0.0013 \,{\rmn M}_{\sun} & ( v \leq 5000 \,{\rmn km}\,{\rmn s}^{-1} ) \\
0.029 \pm 0.004 \,{\rmn M}_{\sun} & ( v \leq 7070 \,{\rmn km}\,{\rmn s}^{-1} ) \ .
\end{array}
\right.
\end{equation}
The uncertainties here are based on the scatter between the two
deconvolved \ion{Fe}{2} profiles,
and do not included systematic uncertainties arising from placement
of the continuum,
which mostly affects the outer, high velocity parts of the profile.
WCFHS93 obtained
$M_\FeII = 0.014 \,{\rmn M}_{\sun}$ from the red side of the mean \ion{Fe}{2} profile,
which is lower than the
$M_\FeII = 0.029 \,{\rmn M}_{\sun}$ obtained here mainly because
of the different placement of the continuum
(see Figure~\ref{fluxfe}),
and to a small degree because of
the adjustment applied here for the offset of the SM star.
Figure~\ref{rho} also shows for comparison
the density of unshocked Si.
Below $7070 \,{\rmn km}\,{\rmn s}^{-1}$,
the Si density profile is just the unshocked profile inferred from
the \ion{Si}{2} 1260\,\AA\ absorption, Figure~\ref{si1260}.
Above $7070 \,{\rmn km}\,{\rmn s}^{-1}$,
the Si density is a plausible extrapolation which is consistent
with observational constraints:
it is a quadratic function of the free expansion velocity $v$
\begin{eqnarray}
n_{\rmn Si} &=&
0.00413 \times 10^{-24} \,{\rmn gm}\,{\rmn cm}^{-3}\,
\nonumber \\
& & \times \
{(v - 5600 \,{\rmn km}\,{\rmn s}^{-1}) ( 12000 \,{\rmn km}\,{\rmn s}^{-1} - v ) \over (3200 \,{\rmn km}\,{\rmn s}^{-1})^2}
\label{nSiquad}
\end{eqnarray}
which approximately reproduces the observed profile of unshocked \ion{Si}{2},
and which contains, on the assumption of spherical symmetry,
a total Si mass of $0.25 \,{\rmn M}_{\sun}$,
in accordance with equation~(\ref{MSi}).
\subsection{Reverse shock on the near side}
\label{nearside}
If the reanalysis of the \ion{Fe}{2} lines here is accepted,
then it is natural to interpret the sharp blue edge on the
broad \ion{Fe}{2} lines at $-4200 \,{\rmn km}\,{\rmn s}^{-1}$
as representing the free expansion radius of the reverse shock on the
near side of SN1006.
This identification is not as convincing as the identification
of the sharp red edge on the \ion{Si}{2} 1260\,\AA\ feature
as representing the radius of the reverse shock on the far side
at $7070 \,{\rmn km}\,{\rmn s}^{-1}$.
\begin{figure}[tb]
\epsfbox[190 312 440 479]{pic.ps}
\caption[1]{
Schematic diagram, approximately to scale,
of the structure of the remnant of SN1006 inferred in this paper.
The remnant on the far side bulges out because of the low interstellar
density there.
Shaded regions represent silicon ejecta, both shocked and unshocked.
Iron, both shocked and unshocked, lies inside the silicon.
The background SM star is offset slightly from the projected
center of the remnant.
\label{pic}
}
\end{figure}
Figure~\ref{pic} illustrates schematically
the inferred structure of the remnant of SN1006.
The picture is intended to be approximately to scale,
and in it the diameter of SN1006 along the line of sight
is roughly 20\% larger than the diameter transverse to the line of sight.
By comparison, the diameter of the observed radio and x-ray remnant
varies by 10\%,
from a minimum of $30'$ to a maximum of $33'$
(Reynolds \& Gilmore 1986, 1993)
or $34'$ (Willingale et al.\ 1996).
As already discussed in subsections~\ref{blue} and \ref{blueion},
if the position of the reverse shock on the near side at $4200 \,{\rmn km}\,{\rmn s}^{-1}$
is typical of the rest of the remnant,
while the $7070 \,{\rmn km}\,{\rmn s}^{-1}$ position of the reverse shock on the far side
is anomalously high because the interstellar density on the far side is low,
then it would resolve several observational puzzles.
In summary, these observational puzzles are:
(1) how to fit gas expanding at $\sim 7000 \,{\rmn km}\,{\rmn s}^{-1}$ within the confines
of the interstellar shock
(answer: the remnant bulges out on the far side because of the low density);
(2) how the $5050 \,{\rmn km}\,{\rmn s}^{-1}$ velocity of shocked Si on the far side could
be so much higher than the $1800 \,{\rmn km}\,{\rmn s}^{-1}$ velocity
(assuming no collisionless electron heating)
of gas behind the interstellar shock along the NW filament
(answer: velocities on the far side are anomalously high because
the interstellar density there is anomalously low);
(3) how to achieve Si densities $n_{\rmn Si} \ga 10^{-2} \,{\rmn cm}^{-2}$
necessary to produce the observed Si x-ray emission,
compared to the postshock Si density of $2.2 \times 10^{-4} \,{\rmn cm}^{-2}$
measured from the \ion{Si}{2} absorption on the far side
(answer: gas shocked at earlier times is denser because density
decreases as $\rho \propto t^{-3}$ in free expansion);
and (4) why there is strong redshifted Si absorption but no blueshifted
absorption (answer: Si on the near side has been shocked and collisionally
ionized above \ion{Si}{4}).
As regards the second of these problems,
if the reverse shock on the near side is indeed at $4200 \,{\rmn km}\,{\rmn s}^{-1}$,
then the velocity of reverse-shocked gas on the near side
would be of order $2000 \,{\rmn km}\,{\rmn s}^{-1}$,
much more in keeping with the $1800 \,{\rmn km}\,{\rmn s}^{-1}$ velocity
of shocked gas in the NW filament.
\subsection{Contribution of shocked Fe to absorption}
\label{shockedFe}
In subsections~\ref{red} and \ref{SiIII+IV}
we concluded that most of the observed absorption by Si ions
is from shocked Si.
Does shocked Fe also contribute to the observed broad \ion{Fe}{2}
absorption profiles?
The answer, on both observational and theoretical grounds,
is probably not much.
On the red side of the \ion{Fe}{2} profile,
shocked \ion{Fe}{2} would have a Gaussian line profile
centered at $5050 \,{\rmn km}\,{\rmn s}^{-1}$, the same as observed for shocked Si.
No absorption with this profile is observed,
Figure~\ref{fe2} or \ref{rho}.
Since the collisional ionization rates of \ion{Fe}{2} and \ion{Si}{2}
are similar (Lennon et al.\ 1988),
the absence of such \ion{Fe}{2} absorption
implies that there cannot be much iron mixed in with the shocked Si.
This is consistent with the argument in subsection~\ref{purity},
that the bulk of the shocked \ion{Si}{2} must be fairly pure,
unmixed with other elements.
On the other hand the observed \ion{Fe}{2} profile, Figure~\ref{rho},
does suggest the presence of some \ion{Fe}{2} mixed with unshocked Si,
at velocities $\la 7070 \,{\rmn km}\,{\rmn s}^{-1}$.
The picture then is that there is Fe mixed with Si at lower velocities,
but not at higher velocities.
This is consistent with SN\,Ia models, such as W7 of Nomoto et al.\ (1984),
in which the transition from the inner Fe-rich layer to the Si-rich layer
is gradual rather than abrupt.
On the blue side of the \ion{Fe}{2} profile,
if the reverse shock is at $-4200 \,{\rmn km}\,{\rmn s}^{-1}$,
then shocked \ion{Fe}{2} should have a Gaussian profile centered at
$\sim - 2000 \,{\rmn km}\,{\rmn s}^{-1}$,
with a width comparable to that of the broad redshifted Si features.
While some such absorption may be present,
the similarity between the blueshifted and redshifted sides of the
\ion{Fe}{2} absorption suggests that the contribution
to the blue side from shocked \ion{Fe}{2} is not large.
This is consistent with expectation from
the density profile of unshocked \ion{Fe}{2} on the red side.
The mass of \ion{Fe}{2} at velocities $4200$-$7070 \,{\rmn km}\,{\rmn s}^{-1}$
inferred from the red side of the profile
on the assumption of spherical symmetry
is $0.013 \,{\rmn M}_{\sun}$.
If this mass of \ion{Fe}{2} is supposed shocked on the blue side
and placed at the reverse shock radius of $- 4200 \,{\rmn km}\,{\rmn s}^{-1}$,
the resulting Fe column density is
$1.3 \times 10^{14} \,{\rmn cm}^{-2}$,
which translates into a peak Fe density of
$0.0013 \times 10^{-24} \,{\rmn gm}\,{\rmn cm}^{-3}$
at velocity $-2200 \,{\rmn km}\,{\rmn s}^{-1}$,
for an assumed dispersion of $1240 \,{\rmn km}\,{\rmn s}^{-1}$
like that of the redshifted Si features.
This density of shocked Fe is low enough that it makes only
a minor contribution to the \ion{Fe}{2} profile in Figure~\ref{rho}.
In practice, collisional ionization of shocked \ion{Fe}{2}
reduces its contribution further.
For pure iron with an initial ionization state
of, say, 50\% \ion{Fe}{2}, 50\% \ion{Fe}{3}
(see subsection~\ref{photoion}),
we find that the column density of shocked \ion{Fe}{2} is reduced by
a factor 0.6 to
$0.8 \times 10^{14} \,{\rmn cm}^{-2}$,
which translates into a peak \ion{Fe}{2} density of
$0.0008 \times 10^{-24} \,{\rmn gm}\,{\rmn cm}^{-3}$
at velocity $-2200 \,{\rmn km}\,{\rmn s}^{-1}$.
The shocked column density would be even lower if the initial ionization
state is higher, or if there are other elements mixed in with the Fe,
since a higher electron to \ion{Fe}{2} ratio would
make collisional ionization faster.
If the initial ionization state of the iron is as high as proposed by HF88,
then the shocked \ion{Fe}{2} column density of Fe could be as low as
$0.1 \times 10^{14} \,{\rmn cm}^{-2}$,
for a peak density of
$0.0001 \times 10^{-24} \,{\rmn gm}\,{\rmn cm}^{-3}$
at velocity $-2200 \,{\rmn km}\,{\rmn s}^{-1}$
in Figure~\ref{rho}.
\subsection{Photoionization}
\label{photoion}
The mass of \ion{Fe}{2} inferred here from the \ion{Fe}{2} profile is,
according to equation~(\ref{MFeII}),
$0.0195 \pm 0.0013 \,{\rmn M}_{\sun}$ up to $5000 \,{\rmn km}\,{\rmn s}^{-1}$,
and $0.029 \pm 0.004 \,{\rmn M}_{\sun}$ up to $7070 \,{\rmn km}\,{\rmn s}^{-1}$.
Historical and circumstantial evidence suggests that SN1006 was
a Type~Ia
(Minkowski 1966; Schaefer 1996).
Exploded white dwarf models of SN\,Ia
predict that several tenths of a solar mass of iron ejecta should be present,
as required to explain SN\,Ia light curves
(H\"{o}flich \& Khokhlov 1996).
Thus,
as emphasised by Fesen et al. (1988) and by HF88,
the observed mass of \ion{Fe}{2} in SN1006 is only a fraction
($\la 1/10$)
of the expected total Fe mass.
Hamilton \& Sarazin (1984)
pointed out that unshocked SN ejecta will be subject to photoionization
by UV starlight and by UV and x-ray emission from the reverse shock.
Recombination is negligible at the low densities here.
HF88 presented detailed calculations of the time-dependent
photoionization of unshocked ejecta in SN1006,
using deflagrated white dwarf models W7 of Nomoto et al.\ (1984),
and CDTG7 of Woosley (1987, private communication),
evolved by hydrodynamic simulation into a uniform interstellar medium.
HF88 found that in these models most of the unshocked Fe was photoionized
to \ion{Fe}{3}, \ion{Fe}{4}, and \ion{Fe}{5}.
While model W7 produced considerably more \ion{Fe}{2} than observed in SN1006,
model CDTG7, which is less centrally concentrated than W7,
produced an \ion{Fe}{2} profile in excellent agreement with the IUE
\ion{Fe}{2} 2600\,\AA\ feature.
HF88 concluded that several tenths of a solar mass of unshocked Fe
could be present at velocities $\la 5000 \,{\rmn km}\,{\rmn s}^{-1}$ in SN1006,
as predicted by Type~Ia supernova models.
However,
the low ionization state of unshocked Si
inferred from the present HST observations
does not support the high ionization state of Fe advocated by HF88.
According to the ion fractions given in Table~\ref{sitab},
unshocked Si is $92 \pm 7\%$ \ion{Si}{2}.
By comparison,
HF88 argued that unshocked Fe is only $\sim 10\%$ \ion{Fe}{2}.
We now show that these ionization states of \ion{Si}{2} and \ion{Fe}{2}
are not mutually consistent.
The ionizations of unshocked \ion{Si}{2} and \ion{Fe}{2} are related
according to their relative photoionization cross-sections,
and by the energy distribution of the photoionizing radiation.
Neutral Si and Fe can be neglected here,
since they have ionization potentials below the Lyman limit,
and are quickly ionized by UV starlight
(\ion{Si}{1} in $\sim 20 \,{\rmn yr}$, \ion{Fe}{1} in $\sim 100 \,{\rmn yr}$
if the starlight is comparable to that in the solar neighborhood),
once the ejecta start to become optically thin to photoionizing radiation,
at $\sim 100$-$200 \,{\rmn yr}$.
\begin{figure}[tb]
\epsfbox[173 272 423 498]{anu.ps}
\caption[1]{
Photoionization cross-sections of \ion{Si}{2} and \ion{Fe}{2}
(Reilman \& Manson 1979, adapted to include autoionizing photoionization
as described by HF88).
The ionization state of unshocked ejecta depends on photoionization,
and the plotted photoionization cross-sections are important in relating
the ionization state of unshocked iron,
characterized by the ratio \ion{Fe}{2}/Fe,
to the ionization state of unshocked silicon,
characterized by the ratio \ion{Si}{2}/Si.
\label{anu}
}
\end{figure}
Photoionization cross-sections of \ion{Si}{2} and \ion{Fe}{2},
taken from Reilman \& Manson (1979) and adapted to include
autoionizing photoionization as described by HF88,
are shown in Figure~\ref{anu}.
The Figure shows that
the photoionization cross-section of \ion{Fe}{2}
is about an order of magnitude larger than that of \ion{Si}{2}
from the ionization potential up to the L-shell (autoionizing) photoionization
threshold of \ion{Si}{2} at 100\,eV,
above which the photoionization cross-sections are about equal,
until the L-shell threshold of \ion{Fe}{2} at 700\,eV.
According to HF88 (Table~6, together with Table~2 of Hamilton \& Sarazin 1984),
much of the photoionizing emission from the reverse shock is in the UV
below 100\,eV.
However, there is also some soft x-ray emission above 100\,eV,
which is important for \ion{Si}{2} because
its L-shell photoionization cross-section is larger than
that of the valence shell.
Averaging over the photoionizing photons tabulated by HF88,
we find that the effective photoionization cross-section of \ion{Fe}{2}
is about 5 times that of \ion{Si}{2},
which is true whether the source of the emission in the reverse shock
is oxygen, silicon, or iron.
If the effective photoionization cross-section of \ion{Fe}{2}
is 5 times that of \ion{Si}{2},
then an unshocked \ion{Si}{2} fraction of $0.92 \pm 0.07$
would predict an unshocked \ion{Fe}{2} fraction of
$( 0.92 \pm 0.07 )^5 = 0.66^{+ 0.29}_{- 0.22}$,
considerably larger than the desired unshocked \ion{Fe}{2} fraction
of $0.1$.
The $3 \sigma$ lower limit on the \ion{Si}{2} fraction is $0.71$,
which would predict an unshocked \ion{Fe}{2} fraction of
$( 0.71 )^5 = 0.18$,
closer to but still higher than desired.
A higher ionization state of Fe compared to Si might be achieved if
the photoionizing emission could be concentrated entirely in the UV
below 100\,eV,
since then the effective photoionization cross-section of \ion{Fe}{2}
would be about 10 times that of \ion{Si}{2},
as illustrated in Figure~\ref{anu}.
This could occur if the photoionizing emission were mainly from He.
In this case,
the predicted unshocked \ion{Fe}{2} fraction would be
$( 0.92 \pm 0.07 )^{10} = 0.43^{+ 0.47}_{- 0.27}$,
with a $3 \sigma$ lower limit of
$( 0.71 )^{10} = 0.03$,
which is satisfactory.
To achieve this relatively high level of Fe ionization
requires that there be little soft x-ray emission from heavy elements.
It is not clear whether this is possible,
given the observed x-ray emission from oxygen and silicon
(Koyama et al.\ 1995).
Thus the low ionization state of unshocked Si inferred
in this paper is difficult to reconcile with the expected presence of
several tenths of a solar mass of Fe at velocities $\la 5000 \,{\rmn km}\,{\rmn s}^{-1}$.
Is it possible that the unshocked Si is substantially more ionized
than we have inferred?
From Tables~\ref{redtab} and \ref{sitab},
the total column densities of Si ions, shocked and unshocked, are in the ratio
\begin{equation}
\label{NSitot}
N_\SiII : N_\SiIII : N_\SiIV =
1 : 0.39 : 0.36
\ .
\end{equation}
If the ionization state of the unshocked Si were this high,
then a high ionization state of Fe would be allowed,
and the problem would be resolved.
In fact, in photoionization trials similar to those described by HF88,
we find that the unshocked Si fractions predicted by the CDTG7 model
are close to the ratio (\ref{NSitot}).
However,
in subsection~\ref{SiIII+IV}
we argued both theoretically and from the observed line profiles
that most of the observed \ion{Si}{3} and \ion{Si}{4} absorption
is from shocked Si.
Might this be wrong,
and could in fact much or most of the absorption be from unshocked Si?
And is then our interpretation of the \ion{Si}{2} 1260\,\AA\ profile
as arising mostly from shocked \ion{Si}{2} also faulty?
If so, then much of the tapestry of reasoning in this paper begins to
unravel.
For example,
we must regard as merely coincidental the agreement,
found in subsection~\ref{jump},
of the measured parameters of the \ion{Si}{2} 1260\,\AA\ profile
with the energy shock jump condition, equations~(\ref{Dv})-(\ref{Dvobs}).
We must also conclude that the observed asymmetry between the red and
blueshifted Si absorption arises from asymmetry in the initial explosion,
not (or not all) from asymmetry in the ambient ISM.
For if the blue edge of the redshifted
\ion{Si}{2} and \ion{Si}{4} features
(the blue edge of redshifted \ion{Si}{3} is obscured by Ly\,$\alpha$)
arises from unshocked Si extending down to velocities $+ 2500 \,{\rmn km}\,{\rmn s}^{-1}$
(see Figs.~\ref{si1260} and \ref{si4}),
then there should be, on the assumption of spherical symmetry,
corresponding blueshifted Si absorption outward of $- 2500 \,{\rmn km}\,{\rmn s}^{-1}$,
which is not seen.
\subsection{Where's the iron?}
\label{where}
H\"{o}flich \& Khokhlov's (1996) Table~1 presents a survey of 37 SN\,Ia models,
encompassing all currently discussed explosion scenarios.
In their models, the ejected mass of $^{56}{\rmn Ni}$,
which decays radioactively to iron,
ranges from $0.10 \,{\rmn M}_{\sun}$ to $1.07 \,{\rmn M}_{\sun}$.
Models yielding `acceptable' fits to the sample of 26 SN\,Ia considered by
H\"{o}flich \& Khokhlov
have ejected $^{56}{\rmn Ni}$ masses of
$0.49$-$0.83 \,{\rmn M}_{\sun}$ for normal SN\,Ia,
and $0.10$-$0.18 \,{\rmn M}_{\sun}$ for subluminous SN\,Ia.
In the subluminous models,
a comparable amount of Fe is ejected along with the $^{56}{\rmn Ni}$
(H\"{o}flich, Khokhlov \& Wheeler 1995),
so the total iron ejected in these cases is $\approx 0.2$-$0.3 \,{\rmn M}_{\sun}$.
In the previous subsection,
we argued that the low ionization state of unshocked Si
inferred from the present observations suggests
that the ionization state of unshocked Fe is also likely to be low.
Specifically, if the unshocked \ion{Si}{2} fraction is
$\mbox{\ion{Si}{2}/Si} = 0.92 \pm 0.07$,
from Table~\ref{redtab},
then the predicted unshocked \ion{Fe}{2} fraction is
$\mbox{\ion{Fe}{2}/Fe} = 0.66^{+ 0.29}_{- 0.22}$.
Correcting the \ion{Fe}{2} mass of
$M_\FeII = 0.029 \pm 0.004 \,{\rmn M}_{\sun}$ up to $7070 \,{\rmn km}\,{\rmn s}^{-1}$,
equation~(\ref{MFeII}),
for the ionization state of the Fe
yields a total inferred Fe mass of
$M_{\rmn Fe} = 0.044^{+ 0.022}_{- 0.013} \,{\rmn M}_{\sun}$
up to $7070 \,{\rmn km}\,{\rmn s}^{-1}$,
with a $3 \sigma$ upper limit of $M_{\rmn Fe} < 0.16 \,{\rmn M}_{\sun}$.
These Fe masses are lower than predicted by either normal or
subluminous models of SN\,Ia.
A low ionization state of Fe is supported by the HUT observations of
Blair et al.\ (1996),
who looked for \ion{Fe}{3} 1123\,\AA\ absorption in the background SM star.
If there is a large mass of Fe in SN1006,
significantly larger than the observed \ion{Fe}{2} mass,
then certainly \ion{Fe}{3} should be more abundant than \ion{Fe}{2},
and from detailed models HF88 predicted
$\mbox{\ion{Fe}{3}/\ion{Fe}{2}} = 2.6$.
Blair et al.'s best fit is
$\mbox{\ion{Fe}{3}/\ion{Fe}{2}} = 1.1 \pm 0.9$,
and their $3 \sigma$ upper limit is
$\mbox{\ion{Fe}{3}/\ion{Fe}{2}} < 3.8$.
This result does not support, though it does not yet definitely exclude,
HF88's prediction.
Neither of the above two observational evidences favoring a low ionization
state of Fe, hence a low mass of Fe in SN1006, is yet definitive.
To settle the issue will require re-observation of the \ion{Fe}{3} 1123\,\AA\
line at a higher signal to noise ratio.
The Far Ultraviolet Space Explorer (FUSE) should accomplish this.
\section{Worries}
\label{worries}
In this paper we have attempted to present a consistent
theoretical interpretation of the broad Si and Fe absorption features
in SN1006.
While the overall picture appears to fit together nicely,
the pieces of the jigsaw do not fit perfectly everywhere.
In this Section we highlight the ill fits.
What causes the discrepancy between the profiles of the redshifted
\ion{Si}{2} 1260\,\AA\ and \ion{Si}{2} 1527\,\AA\ features?
This discrepancy was originally pointed out and discussed for the IUE data
by Fesen et al.\ (1988),
and the discrepancy remains in the HST data (WCHFLS96).
The discrepancy is especially worrying for the present paper
because the excess in the \ion{Si}{2} 1260\,\AA\ profile
compared to \ion{Si}{2} 1527\,\AA\ (see WCHFLS96, Figure~2)
looks a lot like what we
have interpreted here as the unshocked component of the
\ion{Si}{2} 1260\,\AA\ absorption (Figure~\ref{si1260}).
We have argued that the redshifted
\ion{Si}{2}, \ion{Si}{3}, and \ion{Si}{4}
absorption is caused mostly by shocked Si,
yet it is not clear that the observed relative column densities
are naturally reproduced in collisionally ionized gas.
Specifically, one might expect relatively more \ion{Si}{3},
or relatively less \ion{Si}{2} or \ion{Si}{4} in the shocked Si.
On the other hand the observed relative column densities of Si
are naturally reproduced in unshocked, photoionized gas.
Is our interpretation at fault?
The best fit dispersion of the redshifted \ion{Si}{4} feature,
Figure~\ref{si4},
is $1700 \pm 100 \,{\rmn km}\,{\rmn s}^{-1}$, which is $4.5 \sigma$ larger than that
of the \ion{Si}{2} 1260\,\AA\ feature, $1240 \pm 40 \,{\rmn km}\,{\rmn s}^{-1}$.
What causes this discrepancy?
Does the density of shocked Si vary a little or a lot?
In subsection~\ref{red} we argued that the Gaussian profile of the
shocked Si suggests little temperature variation,
hence little density variation.
On the other hand the unshocked Si density profile below $7070 \,{\rmn km}\,{\rmn s}^{-1}$
(Fig.~\ref{si1260})
suggests a density profile increasing steeply outwards,
and there are other clues hinting at the same thing:
the large column density of shocked \ion{Si}{2}, subsection~\ref{purity},
and the need for a high density to ionize Si on the near side above
\ion{Si}{4}, subsection~\ref{blueion}.
Is there an inconsistency here?
In subsection~\ref{purity} we argued that the high observed column density
of shocked \ion{Si}{2} indicates a low mean electron to \ion{Si}{2} ratio,
$n_e / n_\SiII \la 1.3$, equation (\ref{nela}).
Higher ratios would cause more rapid collisional ionization of \ion{Si}{2},
reducing the column density below what is observed.
This limit on the electron to \ion{Si}{2} ratio is
satisfactory as it stands
(the limit is somewhat soft, and at least it is not less than 1),
but it is uncomfortably low,
making it difficult to admit even modest quantities of other elements,
such as sulfur, which might be expected to be mixed with the silicon.
Is there a problem here,
and if so have we perhaps overestimated the contribution of shocked
compared to unshocked Si in the \ion{Si}{2} absorption?
In subsection~\ref{blueion}
we estimated that if the reverse shock on the near side of SN1006
is at $- 4200 \,{\rmn km}\,{\rmn s}^{-1}$,
then under a `simplest' set of assumptions
there should be an observable column density of blueshifted \ion{Si}{4},
contrary to observation.
However, we also showed that the predicted column density is sensitive
to the assumptions, and that it is not difficult to bring the column
density of \ion{Si}{4} below observable levels.
Is this explanation adequate,
or does the absence of blueshifted Si absorption
hint at asymmetry in the supernova explosion?
Is the sharp blue edge on the \ion{Fe}{2} features at $- 4200 \,{\rmn km}\,{\rmn s}^{-1}$ real?
Further observations of the \ion{Fe}{2} features at higher resolution
would be helpful in deciding this issue.
Notwithstanding our reanalysis of the \ion{Fe}{2} features,
there remains some suggestion of high velocity blueshifted absorption
outside $- 4200 \,{\rmn km}\,{\rmn s}^{-1}$, perhaps to $- 7000 \,{\rmn km}\,{\rmn s}^{-1}$ or even farther,
Figures~\ref{fe2} and \ref{rho}.
Is this absorption real?
If so,
then the arguments of subsection~\ref{blue} fail,
and the absence of blueshifted Si absorption must be attributed
to intrinsic asymmetry in the initial supernova explosion.
Finally, there is the problem discussed in subsection~\ref{where}:
where is the iron?
\section{Summary}
\label{summary}
We have presented a consistent
interpretation of the broad Si and Fe absorption features
observed in SN1006 against the background SM star
(Schweizer \& Middleditch 1980).
We have argued that the strong redshifted \ion{Si}{2} 1260\,\AA\ absorption
feature arises from both unshocked and shocked Si,
with the sharp red edge of the feature at $7070 \,{\rmn km}\,{\rmn s}^{-1}$ representing
the free expansion radius of the reverse shock on the far side of SN1006,
and the Gaussian blue edge signifying shocked Si
(Fig.~\ref{si1260}).
Fitting to the \ion{Si}{2} 1260\,\AA\ line profile yields three velocities,
the position of the reverse shock,
and the velocity and dispersion of the shocked gas,
permitting a test of the energy jump condition for a strong shock.
The measured velocities satisfy the condition remarkably well,
equations~(\ref{Dv})-(\ref{Dvobs}).
The \ion{Si}{2} 1260\,\AA\ line thus provides direct evidence
for the existence of a strong shock under highly collisionless conditions.
The energy jump condition is satisfied provided that virtually all the shock
energy goes into ions.
This evidence suggests little or no collisionless heating of electrons
in the shock,
in agreement with recent evidence from UV line widths and strengths
(Raymond et al.\ 1995; Laming et al.\ 1996).
The observed column density of shocked \ion{Si}{2} is close to the column
density expected for steady state collisional ionization behind a shock,
provided that the electron to \ion{Si}{2} ion ratio is low.
From the low electron to \ion{Si}{2} ratio, we have argued
that the shocked Si is probably of a fairly high degree of purity,
with little admixture of other elements.
More directly,
the absence of \ion{Fe}{2} absorption with the same line profile as
the shocked Si indicates that there is little Fe mixed with the shocked Si.
On the other hand,
there is some indication of absorption by \ion{Fe}{2}
at the velocity 5600-$7070 \,{\rmn km}\,{\rmn s}^{-1}$ of the unshocked \ion{Si}{2},
which suggests that some Fe is mixed with Si in the lower velocity region
of the Si layer.
We have proposed that the ambient interstellar density on the far side
of SN1006 is anomalously low compared to the density around the rest of
the remnant,
so that the remnant bulges out on the far side (Fig.~\ref{pic}).
This would explain several observational puzzles.
Firstly,
it would explain the absence of blueshifted Si absorption
matching the observed redshifted Si absorption.
If the interstellar density on the near side is substantially larger than
on the far side,
then the reverse shock on the near side would be further in,
so that all the Si on the near side could have been shocked and collisionally
ionized above \ion{Si}{4}, making it unobservable in absorption.
Secondly,
if the velocity on the far side is anomalously high because of the low
interstellar density there,
it would resolve the problem noted by Wu et al.\ (1983) and
subsequent authors of how to fit gas expanding at $\sim 7000 \,{\rmn km}\,{\rmn s}^{-1}$
within the confines of the interstellar shock.
Thirdly,
a low density on the far side
would explain how the $5050 \,{\rmn km}\,{\rmn s}^{-1}$ velocity of shocked Si there
could be so much higher than the $1800 \,{\rmn km}\,{\rmn s}^{-1}$ velocity
(assuming no collisionless electron heating)
of gas behind the interstellar shock along the NW filament
(Smith et al.\ 1991;
Raymond et al.\ 1995).
Finally,
the density of Si on the far side inferred from the Si absorption profiles
is one or two orders of magnitude too low to yield Si x-ray emission
at the observed level
(Koyama et al.\ 1995).
Again, an anomalously low density on the far side is indicated.
The notion that the reverse shock on the near side
has moved inward much farther, to lower velocities, than on the far side
conflicts with our earlier conclusion (WCFHS93)
that there is blueshifted \ion{Fe}{2} absorption to velocities
$\sim - 8000 \,{\rmn km}\,{\rmn s}^{-1}$.
Reanalyzing the \ion{Fe}{2} data,
we find that the evidence for such high velocity blueshifted \ion{Fe}{2}
absorption is not compelling.
In the WCFHS93 analysis,
the main evidence for high velocity blueshifted \ion{Fe}{2} comes from the
\ion{Fe}{2} 2383, 2344, 2374\,\AA\ feature.
However, the 2344\,\AA\ component, which lies at $- 4900 \,{\rmn km}\,{\rmn s}^{-1}$
relative to the principal 2383\,\AA\ component,
confuses interpretation of the blue wing of the feature.
The \ion{Fe}{2} 2600, 2587\,\AA\ feature is cleaner,
and it shows a sharp blue edge at $- 4200 \,{\rmn km}\,{\rmn s}^{-1}$,
which we interpret as representing the free expansion radius of the
reverse shock on the near side of SN1006.
In our reanalysis of the \ion{Fe}{2} features,
we adopt a rigorous approach to the subtraction of narrow
interstellar and stellar lines,
requiring that lines subtracted have the correct positions and dispersions,
and have mutually consistent strengths.
In particular,
we subtract the narrow \ion{Fe}{2} 2344\,\AA\ line with a strength
consistent with the other narrow \ion{Fe}{2} lines,
which strength is substantially greater than the apparent strength.
Subtraction of this line introduces a sharp blue edge on the
deconvolved \ion{Fe}{2} 2383, 2344, 2374\,\AA\ feature
at the same place, $\approx - 4200 \,{\rmn km}\,{\rmn s}^{-1}$,
as the \ion{Fe}{2} 2600, 2587\,\AA\ feature.
The resulting deconvolved \ion{Fe}{2} profiles
(Fig.~\ref{fe2})
are in good agreement with each other.
The mass and velocity distribution of Si and Fe inferred in this paper
provides useful information for modeling the remnant of SN1006
(see Fig.~\ref{rho}).
Freely expanding unshocked Si on the far side extends from a low velocity of
$5600 \pm 100 \,{\rmn km}\,{\rmn s}^{-1}$ up to the position of the reverse shock at $7070 \,{\rmn km}\,{\rmn s}^{-1}$.
Above this velocity the Si is shocked,
and information about its detailed velocity distribution
before being shocked is lost.
The total mass of Si, both unshocked and shocked,
inferred from the \ion{Si}{2}, \ion{Si}{3}, and \ion{Si}{4} lines
is $M_{\rmn Si} = 0.25 \pm 0.01 \,{\rmn M}_{\sun}$,
on the assumption of spherical symmetry.
We have argued that the observed broad \ion{Fe}{2} absorption arises
almost entirely from unshocked freely expanding Fe.
The mass of \ion{Fe}{2} inferred from the cleaner, red side of the
mean \ion{Fe}{2} profile is
$M_\FeII = 0.0195 \pm 0.0013 \,{\rmn M}_{\sun}$ up to $5000 \,{\rmn km}\,{\rmn s}^{-1}$,
and $M_\FeII = 0.029 \pm 0.004 \,{\rmn M}_{\sun}$ up to $7070 \,{\rmn km}\,{\rmn s}^{-1}$,
again on the assumption of spherical symmetry.
These masses include a small positive adjustment ($0.003 \,{\rmn M}_{\sun}$)
resulting from
the offset of the SM star from the projected center of the remnant.
Our analysis of the Si lines indicates a low ionization state for
the unshocked silicon, with \ion{Si}{2}/Si = $0.92 \pm 0.07$.
Such a low state would imply a correspondingly low ionization state of
unshocked iron,
with \ion{Fe}{2}/Fe = $0.66^{+ 0.29}_{- 0.22}$.
If this is correct,
then the total mass of Fe up to $7070 \,{\rmn km}\,{\rmn s}^{-1}$ is
$M_{\rmn Fe} = 0.044^{+ 0.022}_{- 0.013} \,{\rmn M}_{\sun}$
with a $3 \sigma$ upper limit of $M_{\rmn Fe} < 0.16 \,{\rmn M}_{\sun}$.
The absence of \ion{Fe}{2} absorption with a profile like that of
the shocked \ion{Si}{2} suggests that there is not much more Fe
at higher velocities.
Such a low mass of Fe conflicts with the expectation that there should
be several tenths of a solar mass of Fe in this suspected Type~Ia remnant.
A low ionization state of Fe and a correspondingly low Fe mass
is consistent with the low
\ion{Fe}{3}/\ion{Fe}{2} $= 1.1 \pm 0.9$
ratio measured by Blair et al.\ (1996) from HUT observations of the
\ion{Fe}{3} 1123\,\AA\ line in the spectrum of the SM star.
However,
neither the present observations nor the HUT data are yet conclusive.
Re-observation of the \ion{Fe}{3} 1123\,\AA\ line at higher signal to noise
ratio with FUSE will be important in determining the ionization state of
unshocked Fe in SN1006,
and in resolving the question, Where's the iron?
\acknowledgements
We would like to thank Bill Blair for helpful correspondence on the HUT data,
and Graham Parker and Mike Shull for advice
on respectively stellar and interstellar lines.
Support for this work was provided by NASA through grant number
GO-3621
from the Space Telescope Science Institute,
which is operated by AURA, Inc., under NASA contract NAS 5-26555.
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\section{Introduction}
The main purpose of this work is to apply the induced geometry approach to N=1
supergravity to the construction of N=1 SYM theory on a curved superspace.
This approach was
introduced in paper \cite{Gstr} . We refer
the reader to that paper for all details about the geometric constructions we
use in the present one. However we give all necessary definitions and the paper
can be read independently. The main benefit of induced geometry approach is
that it
does not require any additional constraints like those one has to impose
on a curvature and torsion tensors in the standard formulation of supergravity
(see \cite{BW} for a good exposition). The induced SCR-structure on a superspace
incorporates all the necessary constraints and seems to be a more natural geometric
construction then the Lorentz connections of the conventional approach.
The induced geometry approach to N=1 supergravity was further developed in papers
\cite{gN1}, \cite{gN1 (2)}, \cite{auxdim}. In paper \cite{Rosly letter} the application to
the construction of SYM theory over curved superspace was proposed. In the present
work we develop another point of view on this problem.
The paper is organized as follows. In section 2 we give some auxiliary results about
the integration over integral surfaces defined by $(0,2)$-dimensional distribution.
The derivation of these results is postponed until appendix A.
Then we give the description of a curved superspace and CR-structure on it within
the framework of induced geometry approach. After that we explain how the results
about the integration over integral surfaces can be applied to the construction of a
chiral projecting operator.
In section 3 we introduce some more geometric notions and define the gauge fields
as sections of CR-bundles over the superspace. Then we formulate the main result
and explain all the ingredients. The derivation of the Lagrangian is given in
appendix B. We finish this section by considering the proper reality
conditions imposed on the fields used in the construction. As a result of these restrictions
the set of fields reduces to the standard one. This procedure is similar to the one
introduced in \cite{Wessart}.
In section 4 we compare our constructions to the conventional ones.
\section{Integration over integral surfaces determined by
(0,2)-dimensional distributions }
Let $\cal M$ be a real $(m,n)$-dimensional supermanifold.
Consider a pair of odd vector fields
$E_{\alpha}$ (where $\alpha=1,2$) which are
closed with respect to the anticommutator, i.e.
\begin{equation} \label{anticom}
\{E_{\alpha},E_{\beta}\}=c_{\alpha\beta}^{\enspace \enspace \gamma}E_{\gamma}
\end {equation}
where $c_{\alpha\beta}^{\enspace \enspace \gamma}$ are some odd functions on
$\cal M$. By a super version of the Frobenius theorem,
condition (\ref{anticom}) means that the vector fields $E_{\alpha}$
determine an integrable distribution, i.e. for every point in
$\cal M$ there exists a $(0,2)$-dimensional surface $\Sigma$
going through it such that its tangent plane at each point coincides
with the one spanned by $E_{\alpha}$
(equivalently our distribution defines a foliation which has the
surfaces $\Sigma$ as leaves).
We are interested in the functions on $\cal M$ which are stable under
the action of the vector fields $E_{\alpha}$, i.e.
the functions $\Phi$ such that $E_{\alpha}\Phi=0$. Here and below we use
the same symbol for vector fields and for the first order differential
operators corresponding to them.
On every surface $\Sigma$ we have a natural volume element
defined by the requirement that the value of the volume form taken on the
fields $E_{1}$ and $E_{2}$ is equal to one.
Given an arbitrary function $\Phi$ one can get a function with
the property we want by integrating $\Phi$ over the leaves $\Sigma$.
We will denote this operation by $\Box_{E} \Phi$ when applied to $\Phi$.
The explicit formula for $\Box_{E}$
in terms of given
$E_{\alpha}$ and functions $c_{\alpha\beta}^{\enspace \enspace \gamma}$ reads
as
\begin{eqnarray} \label{mainformula}
\Box_{E}\Phi&=&
E^{\alpha}E_{\alpha}\Phi+c^{\alpha \enspace \sigma}_{\enspace
\sigma}E_{\alpha}\Phi + \nonumber\\
& & +\Phi\left( \frac{2}{3}E^{\alpha}
c_{\alpha\sigma}^{\enspace \enspace \sigma}+
\frac{1}{3}c^{\alpha \enspace \sigma}_{\enspace \sigma}
c_{\alpha \beta}^{\enspace \enspace \beta} +
\frac{1}{6}c_{\alpha \beta}^{\enspace \enspace \sigma}
c_{\sigma}^{\enspace \alpha \beta} +
\frac{1}{12}c_{\alpha\beta\sigma}c^{\alpha\beta\sigma} \right)
\end{eqnarray}
Here we are raising indices by means of the spinor metric tensor
$\epsilon^{\alpha\beta}$ ($\epsilon^{\alpha\beta}=
-\epsilon^{\beta\alpha}, \epsilon^{12}=1$) and lowering by means of
the inverse matrix $\epsilon_{\alpha\beta}$.
We give a proof of this formula in appendix A.
It is worth noting that the expression in parentheses in
formula (\ref{mainformula}) is just the volume of the leaf evaluated with
respect to the volume element specified above.
In order to get a function which is invariant with respect to a given
integrable distribution it is not necessary to use the natural volume element
related to the chosen pair of vector fields when integrating.
One can perform the integration using any volume element as well.
But since it differs from the natural volume element by multiplication by
some function, all possible freedom is reflected by the following formula
for a generic operation $\Box$:
\begin{equation} \label{freedom}
\Box\Phi = \Box_{\rho}\Phi\equiv \Box_{E}(\rho\Phi)
\end{equation}
where $\rho$ is some function. In short, this freedom is the same as modifying
the initial function by multiplying it by some fixed function.
Now let us consider the case of a complex supermanifold $\cal M$ with a pair of
complex vector fields $E_{\alpha}$ on it
satisfying the integrability condition (\ref{anticom}). In this case we have to
modify the consideration above slightly.
Formula (\ref{mainformula}) again defines an operation yielding a
$E_{\alpha}$-invariant function if one assumes that $E_{\alpha}$ is a
holomorphic vector field and $\Phi$ is a holomorphic function.
As before by the Frobenius theorem we have an integral complex
surface $\Sigma_{\mbox{\rm C}}$ going through every point in $\cal M$.
Our vector fields define a natural holomorphic volume element on
$\Sigma_{\mbox{\rm C}}$.
For a generic real submanifold
$\Sigma_{\mbox{\rm R}}$, a real basis in a tangent space to
$\Sigma_{\mbox{\rm R}}$ can be considered as a
complex basis in a tangent space to $\Sigma_{\mbox{\rm C}}$.
This means that the holomorphic volume form determines a nondegenerate volume
element in a tangent space to $\Sigma_{\mbox{\rm R}}$.
Given such a generic submanifold
$\Sigma_{\mbox{\rm R}}\subset \Sigma_{\mbox{\rm C}}$, one can perform an
integration of a holomorphic function over it.
One can show that when we have a purely odd supermanifold $\Sigma_{\mbox{\rm C}}$,
the result of integration does not depend on the particular choice
of $\Sigma_{\mbox{\rm R}}$. Therefore we see that in the case of a complex
manifold, the operation $\Box_{E_{\alpha}}$
(formula (\ref{mainformula})) can be also interpreted in terms of integration
over the leaves.
We are interested in applications of formula (\ref{mainformula}) in the
framework of the induced geometry approach to the description of curved
superspace. In this approach a curved superspace is described as a generic real
$(4,4)$-dimensional surface $\Omega$ embedded into ${\rm C}^{4|2}$
(see \cite{Gstr} for details). The complex structure on ${\rm C}^{4|2}$
induces a CR-structure on $\Omega$. This means that a complex plane is singled
out in every tangent space to $\Omega$.
Namely if $T_{z}(\Omega)$ is the real $(4,4)$-dimensional tangent space at a
point $z$ and $J$ denotes the linear operator given by the multiplication
by $i$, then $T_{z}(\Omega)\cap JT_{z}(\Omega)$ is the maximal complex subspace
contained in $T_{z}(\Omega)$. It is not difficult to figure out that this
complex subspace is of dimension $(0,2)$. One can choose vector fields
\begin{equation} \label{vfields}
E_{\alpha}, {\bar E}_{\dot \alpha}, E_{c}\
(\alpha,\dot \alpha=1,2;c=1,\cdots,4)
\end{equation}
tangent to $\Omega$ such that the fields $E_{\alpha}$ form a (complex)
basis of the complex subspace at each point,
the fields ${\bar E}_{\dot \alpha}$ define a basis in the complex conjugate
plane and the fields $E_{c}$ complete $E_{\alpha}, {\bar E}_{\dot \alpha}$
to a (real) basis of the whole tangent space.
The (anti)commutator of two vector fields tangent to $\Omega$ is also a vector
field tangent to $\Omega$. Thus we have
\begin{equation} \label{C}
[E_{A},E_{B}\}=c_{AB}^{\enspace \enspace D}E_{D}
\end{equation}
where $c_{AB}^{\enspace \enspace D}$ are some functions and the indices take on
the values of the indices $\alpha, \dot \alpha, c$. The fact that our
CR-structure defined on $\Omega$ is induced by
a $GL(4,2|{\rm C})$-structure in the ambient space implies
\begin{equation} \label{intcr}
\{E_{\alpha},E_{\beta}\}=c_{\alpha \beta}^{\enspace \enspace \gamma}E_{\gamma}
\end{equation}
and the corresponding complex conjugate equations.
This means that we are dealing with an integrable CR-structure on $\Omega$.
We call a function $\Phi$ defined on $\Omega$ chiral if
$\bar E_{\dot \alpha}\Phi=0$. A function $\Phi^{+}$ is called antichiral if
$E_{\alpha}\Phi^{+}=0$. Note that the restriction to $\Omega$ of any holomorphic
function defined in some neighborhood of $\Omega$ in ${\rm C}^{4|2}$
is a chiral function (the converse is also true in some sense, see \cite{Gstr}).
Formula (\ref{intcr}) looks exactly like formula (\ref{anticom}). The only
difference is that in the case at hand the complex fields $E_{\alpha}$ are
defined on a real manifold $\Omega$. Formula (\ref{mainformula}) can be used
to construct an (anti)chiral function from an arbitrary given one.
Moreover one can give an interpretation of formula (\ref{mainformula}) similar
to those we gave for the purely real and complex cases using the
complexification of $\Omega$.
\section{Formulation of N=1 SUYM in curved superspace
in terms of induced geometry and CR-bundles}
To construct the Lagrangian of N=1 super-Yang-Mills theory in the induced
geometry approach, first we need to say more about induced CR-structures and
introduce some useful geometric notions.
Note that the basis (\ref{vfields}) of tangent vectors defining a CR-structure
on $\Omega$ is fixed up to linear transformations of the form
\begin{eqnarray}
E'_{a}=g_{a}^{b}E_{b}+g_{a}^{\beta}E_{\beta}+\bar g_{a}^{\dot \beta}
\bar E_{\dot \beta} \nonumber \\
E'_{\alpha}=g_{\alpha}^{\beta}E_{\beta}, \, \,
\bar E'_{\dot \alpha}=\bar g_{\dot \alpha}^{\dot \beta}\bar E_{\dot \beta}
\label{scrtr} \\
\end{eqnarray}
where $(g_{a}^{b})$ is a real matrix and $(\bar g_{\dot \alpha}^{\dot \beta})$
is the complex conjugate matrix of $(g_{\alpha}^{\beta}E_{\beta})$.
If $\rm C^{4|2}$ is equipped with a volume element one can choose
$E_{a}, E_{\alpha}$ to be a unimodular complex basis in the tangent space
to $\rm C^{4|2}$. This allows one to restrict the transformations (\ref{scrtr})
by the requirement
\begin{equation} \label{scrdet}
\det(g_{a}^{b})=\det(g_{\alpha}^{\beta})
\end{equation}
In this case we say that there is an induced SCR-structure on $\Omega$.
>From now on we will assume that the
basis (\ref{vfields}) defines a SCR-structure.
For the functions $c_{AB}^{\enspace \enspace D}$ defined by formula (\ref{C})
in the case of induced SCR-structure we have the following identities
\begin{equation} \label{scrid}
c_{\alpha \beta}^{\enspace \enspace \dot \gamma}=c_{\alpha \beta}^{\enspace \enspace d}=0,\enspace
c_{\alpha \dot \beta}^{\enspace \enspace \dot \beta}=c_{\alpha d}^{\enspace \enspace d}
\end{equation}
and the corresponding complex conjugate ones.
We define the Levi matrix of the surface $\Omega$
by the expression
$$
\Gamma^{a}_{b}=
i\bar \sigma^{\alpha \dot \beta}_{b}c_{\alpha \beta}^{a}
$$
where $\bar \sigma_{b}$ are the Pauli matrices for $b=1,2,3$ and the identity
matrix for $b=0$.
The matrix $\Gamma^{a}_{b}$ coincides with the matrix of the Levi form defined
in the standard way (see \cite{Gstr}).
To construct a Yang-Mills theory on $\Omega \subset {\rm C}^{4|2}$,
we start with two complex vector bundles $\cal F$ and ${\cal F}^{+}$
with structure group $G$.
Denote the Lie algebra corresponding to $G$ by $\cal G$.
Trivializing our bundles we can represent their sections $\Phi$ and $\Phi^{+}$
locally as vector functions, called fields.
They describe matter and charge conjugated matter respectively. Gauge
transformations correspond to the change of trivialization. They have the form
\begin{equation} \label{gaugefi}
\Phi'=e^{i\Lambda}\Phi \qquad (\Phi^{+})'=e^{-i\bar \Lambda}\Phi^{+}
\end{equation}
where $\Lambda$ and $\bar \Lambda$ are some functions
(sections of corresponding homomorphism
bundles) with values in the representation of the Lie algebra $\cal G$
corresponding to the field $\Phi$.
We want to stress the fact that for now we consider $\cal F$ and ${\cal F}^{+}$
separately, not requiring them to be complex conjugate bundles (the functions
$\Lambda$, $\bar \Lambda$ in (\ref{gaugefi}) are also independent).
By SUYM fields we understand two pairs of semiconnections
$$
\nabla_{\alpha}\Phi^{+} \equiv (E_{\alpha} + {\cal A}_{\alpha})\Phi^{+} \qquad
\bar \nabla_{\dot \alpha}\Phi \equiv (\bar E_{\dot \alpha} +
{\cal A}_{\dot \alpha})\Phi
$$
restricted by the conditions
\begin{equation} \label{zerocurv}
\{\nabla_{\alpha},\nabla_{\beta}\}=
c_{\alpha \beta}^{\enspace \enspace \gamma}\nabla_{\gamma}\qquad
\{\bar \nabla_{\dot \alpha},\bar \nabla_{\dot \beta}\}=
c_{\dot \alpha \dot \beta}^{\enspace \enspace \dot
\gamma}\bar \nabla_{\dot \gamma}
\end{equation}
These conditions mean that the corresponding semiconnections have vanishing
curvature. It can be shown (see for example \cite{auxdim}) that
semiconnections $\bar \nabla_{\dot \alpha}$ satisfying (\ref{zerocurv})
determine a CR-bundle structure on ${\cal F}^{+}$, i.e. ${\cal F}^{+}$
can be pasted together from trivial bundles by chiral gluing functions.
One can define the chiral sections as those annihilated by
$\bar \nabla_{\dot \alpha}$. Then condition (\ref{zerocurv}) guarantees that
there are sufficiently many of them. Analogously, given
$\nabla_{\alpha}$ satisfying (\ref{zerocurv}), one obtains a
$\overline{\rm CR}$-bundle. By $\overline{\rm CR}$-bundle we mean a bundle whose
gluing functions are antichiral. Moreover, one can take this property as a
definition of CR and $\overline{\rm CR}$-bundles (see \cite{auxdim} for
details). Thus the basic geometrical objects we start with are the surface
$\Omega \subset {\rm C}^{4|2}$, the CR-bundle $\cal F^{+}$ and the
$\overline{\rm CR}$-bundle $\cal F$, both defined over $\Omega$.
The solutions to the zero curvature equations (\ref{zerocurv}) can be written
locally as
\begin{equation} \label{UU}
{\cal A}_{\alpha}=e^{U}E_{\alpha}e^{-U} \qquad {\cal A}_{\dot \alpha}=
e^{-\tilde U}\bar E_{\dot \alpha}e^{\tilde U}
\end{equation}
where $U$ and $\tilde U$ are some $\cal G$-valued fields.
Note that the fields $e^{-U}$ and $e^{\tilde U}$ are determined by (\ref{UU})
only up to the left multiplication by arbitrary antichiral and chiral fields
respectively.
If we want to write a gauge invariant Lagrange function
for chiral fields we will immediately encounter the difficulty in writing the
kinetic term, which for the case of free chiral fields
is simply $\Phi^{+}\Phi$. This difficulty is due to the fact that in the case
at hand these fields are sections of different bundles. Therefore we are
forced to identify CR and $\overline{\rm CR}$ bundles choosing a section of
the bundle ${\cal F}^{+}\otimes{\cal F}^{*}$.
Here ${\cal F}^{*}$ is the dual to the bundle ${\cal F}$. This section we denote by
$e^{V}$. Under the gauge transformations the field $e^{V}$ transforms in the
following way
\begin{equation} \label{gaugeV}
e^{V'}=e^{-i\Lambda^{t}}e^{V}e^{i\bar \Lambda}
\end{equation}
Now we can take the gauge invariant combination $\Phi^{+}e^{V}\Phi$ as a kinetic
density term.
Our next goal is to describe a gauge invariant theory in terms of the fields
$\Phi_{i}, \Phi_{i}^{+}, e^{-U}, e^{\tilde U}, e^{V}$ defined on our curved
superspace $\Omega$.
This will be done in a manifestly SCR-covariant way, i.e. independently of the
choice of basis vector fields (\ref{vfields}) up to local SCR-transformations
(\ref{scrtr}). Instead of the customary Lorentz connections in our construction
of the Lagrangian, we use only objects defined by the internal geometry of the
superspace $\Omega$, namely the Levi matrix $\Gamma$ and the functions
$c_{AB}^{\enspace \enspace D}$.
We postpone the details of this construction until appendix B.
Now we want to formulate the main result. The Lagrangian has the following form
\begin{eqnarray}
{\cal S}&=&
\int dV\left[\frac{1}{k}(\det\Gamma)^{-1}(\Box_{\bar {\cal D}}e^{G}E^{\alpha}
e^{-G}) (\Box_{\bar {\cal D}}e^{G}E_{\alpha}e^{-G})\right] + \nonumber \\
&&+ \int dV\left[\bar \Box_{\bar E}\left| \frac{1}{4}\det\Gamma
\right|^{-\frac{1}{3}}\Phi^{+}e^{V}\Phi\right] + \nonumber \\
&& + \int dV\left[a_{i}\Phi_{i}+\frac{1}{2}m_{ij}\Phi_{i}\Phi_{j}+
\frac{1}{3}g_{ijk}\Phi_{i}\Phi_{j}\Phi_{k} \right] + h.c. \label{scrlag}
\end{eqnarray}
Here as in flat space the N=1 SUYM Lagrangian contains a Lagrangian of gauge
fields, a kinetic term of chiral fields and a term describing
the interaction between chiral fields.
In (\ref{scrlag}) we are using the following notations: $k$ is a coupling
constant, $e^{G}=e^{\tilde U}e^{-V}e^{-U^{t}}$, $a_{i}, m_{ij}, g_{ijk}$ are
coupling constants which must be chosen in a way that ensures the gauge
invariance of matter-matter interaction, $\Box_{\bar E}$ is a chiral projector
whose general form was described in section 2, and $\Box_{\bar {\cal D}}$ is a
"covariant" chiral projector which is constructed as $\Box_{\bar E}$
with derivatives $\bar E_{\alpha}$ replaced by "covariant" derivatives
$\bar {\cal D}_{\dot \alpha}$. More precisely, $\bar {\cal D}_{\dot \alpha}$
acts on an arbitrary tensor $V$ carrying the spinor index $\alpha$ in the
following way
$$
{\bar {\cal D}_{\dot \alpha}}V_{\beta} = \bar E_{\dot \alpha}V_{\beta} -
{\check c}_{\dot \alpha, \dot \sigma \beta}^{\enspace \enspace \enspace
\, \dot \sigma \sigma} V_{\sigma}
$$
where ${\check c}_{\dot \alpha, \dot \sigma \beta}^{\enspace \enspace
\enspace \, \dot \sigma \sigma}$
are some functions which can be expressed in terms of ${c_{AB}}^{D}$
(see formulae (\ref{crel}), (\ref{twobas}), (\ref{a}) in appendix B).
Note that the integrands in (\ref{scrlag}) are chiral functions (for the
first term this fact follows from its construction, which is explained in
details in appendix B).
The integration in (\ref{scrlag}) should be understood as a chiral integration.
If a chiral function can be extended to a holomorphic function in some domain in
$\rm C^{4|2}$ where it can be integrated with the holomorphic volume element
over a real (4,2)-submanifold contained in $\Omega$.
We have constructed a Lagrangian depending on the field $V$,
the specific combination of the fields
$e^{U}, e^{\tilde U}, e^{V}$ which we denoted by $e^G$ , and matter fields
$\Phi_{i}, \Phi_{i}^{+}$. All these fields are complex. In order to perform
a functional integration over the fields $U, \tilde U, V, \Phi_{i}$ one has to
restrict these fields to a real surface in functional space. Once this surface
is chosen this will restrict our large gauge group to a smaller one.
The real surface we choose is given by the equations
\begin{equation} \label{realcond}
\bar \Phi_{i}=\Phi_{i}^{+} , \, V=\bar V^{t} , \, -\bar U=\tilde U
\end{equation}
where the upper bar denotes complex conjugation.
The gauge transformations preserving these reality conditions are of the form
(\ref{gaugefi}), (\ref{gaugeV}) where $\Lambda$ and $\bar \Lambda$ are complex
conjugates of each other and $T=\bar S^{+}\equiv S $. Still we have a rather
large gauge group. Let us do a partial gauge fixing by requiring that $e^{U}=1$.
By reality conditions (\ref{realcond}) this implies $e^{-\tilde U}=1$ and
therefore $e^{-G}=e^{V}$, which means that our Lagrangian (\ref{scrlag})
contains only the field $V$ in this partial gauge fixing. The remaining gauge
group contains the transformations with $i\Lambda=S^{+}, \, -i\bar \Lambda=S$,
i.e. the antichiral transformations of the fields $\Phi_{i}$
and the corresponding complex conjugate chiral transformations of the fields
$\Phi_{i}^{+}$.
\section{Comparison with the conventional approach}
In this section we want to compare our constructions with the conventional
(Wess-Zumino) approach to supergravity
and super Yang-Mills theory in curved superspace which is presented in \cite{BW}
in great detail.
We start with a comparison of formula (\ref{mainformula}) with the standard
formula for (anti)chiral projection operators (see \cite{BW}, chapter 19).
But first let us recall briefly the main constituents of the conventional
approach.
In this approach we have a connection defined on a tangent bundle over
$(4,4)$-dimensional real superspace with the Lorentz group as structure group.
Another dynamical variable in this approach is a vielbein $E^{M}_{A}$ which
defines the covariant derivatives ${\cal D}_{M}$ on the tangent bundle and also
identifies the Lorentz bundle with the tangent one allowing to transform
world indices into Lorentz indices and vice-versa. Here we are working
with Lorentz indices. Under a certain set of constraints
(see \cite{BW}) the vector fields
$E_{\alpha}=E_{\alpha}^{M}\partial_{M}$ are closed with respect to the
anticommutator. Moreover the functions
$c_{\alpha\beta}^{\enspace \enspace \gamma}$ defining the anticommutation
relations satisfy the following condition
\begin{equation} \label{connect}
c_{\alpha\beta}^{\enspace \enspace \gamma}=
\omega_{\alpha\beta}^{\enspace \enspace \gamma}+
\omega_{\beta\alpha}^{\enspace \enspace \gamma}
\end{equation}
where $\omega_{\alpha\beta}^{\enspace \enspace \gamma}$ are the connection
coefficients for the covariant differentiation of spinor fields, having only
Lorentz indices.
The antichiral projector acts on arbitrary function $\Phi$ as follows
\begin{equation} \label{8R}
({\cal D}^{\alpha}{\cal D}_{\alpha} - 8R^{+})\Phi
\end{equation}
and gives an antichiral function as a result. Here
$R^{+}=\frac{1}{24}R_{\alpha\beta}^{\enspace \enspace \alpha \beta}$
denotes the invariant obtained from the curvature tensor
$$
R_{\alpha\beta\delta}^{\enspace \enspace \enspace \gamma}=
E_{\alpha}\omega_{\beta\delta}^{\enspace\enspace\gamma}
+E_{\beta}\omega_{\alpha\delta}^{\enspace\enspace\gamma}+
\omega_{\alpha\sigma}^{\enspace\enspace\gamma}
\omega_{\beta\delta}^{\enspace\enspace\sigma}+
\omega_{\beta\sigma}^{\enspace\enspace\gamma}
\omega_{\alpha\delta}^{\enspace\enspace\sigma}-
c_{\alpha\beta}^{\enspace\enspace\sigma}
\omega_{\sigma\delta}^{\enspace\enspace\gamma}
$$
After the contraction of indices this gives the following expression for
$-8R^{+}$
\begin{equation} \label{R(omega)}
-8R^{+}=-(E_{\alpha}\omega_{\beta}^{\enspace\alpha\beta}+
E_{\beta}\omega_{\alpha}^{\enspace\alpha\beta}+
\omega_{\beta\sigma}^{\enspace\enspace\beta}
\omega_{\alpha}^{\enspace\alpha\sigma})+
\omega^{\delta\alpha\beta}\omega_{\alpha\beta\delta}
\end{equation}
The induced geometry approach to supergravity has been shown to be equivalent
to the Wess-Zumino one (\cite{Gstr}). Thus one can use formula
(\ref{mainformula}) to obtain (\ref{8R}).
We identify the pair of vector fields $E_{\alpha}$ appearing in the Wess-Zumino
approach with those in definition of the $\overline{\rm CR}$-structure induced on
$\Omega$ (i.e. complex conjugate to the corresponding CR-structure).
The formulae (\ref{8R}) and
(\ref{mainformula}) must be equivalent at least up to the freedom described by
formula (\ref{freedom}). Indeed as one can easily check, substituting relation
(\ref{connect}) in (\ref{mainformula}), we will get exactly formula (\ref{8R})
with the term $-8R^{+}$ expressed as in (\ref{R(omega)}).
If one starts with the induced geometry approach then in order to get the Lorentz
gauge group one has to require the following gauge condition
$$
{c_{\alpha, \dot \beta}}^{a}=2i\sigma_{\alpha \dot \beta}^{a}
$$
where $\sigma_{\alpha \dot \beta}^{a} $ are the Pauli matrices.
This condition fixes the SCR-basis
(\ref{vfields}) up to transformations of the form (\ref{scrtr}) where
$\det(g_{a}^{b})=\det(g_{\alpha}^{\beta})=
\det(\bar g_{\dot \alpha}^{\dot \beta})=1$, i.e. up to the Lorentz
transformations. In this gauge $4(\det\Gamma)^{-1}=1$, which as it is shown in
appendix B (see formulae (\ref{important}) and (\ref{conjimp})) implies
$$
{{\check c}_{\gamma, \sigma \dot \sigma}}^{\enspace \enspace \enspace \,
\sigma \dot \sigma} =
{{\check c}_{\dot \gamma, \dot \sigma \sigma}}^{\enspace \enspace \enspace
\, \dot \sigma \sigma} = 0
$$
Moreover, the quantities
$-{\check c}_{\dot \alpha, \dot \sigma \beta}^{\enspace
\enspace \enspace \, \dot \sigma \sigma},
-{\check c}_{ \alpha, \sigma \dot \beta}^{\enspace \enspace \enspace \,
\sigma \dot \sigma} $
transform now as coefficients of the Lorentz connection (as can be seen from
(\ref{trcheckc}) for Lorentz transformations). Thus it seems reasonable to
identify $-{\check c}_{\dot \alpha, \dot \sigma \beta}^{\enspace
\enspace \enspace \, \dot \sigma \sigma}$ and $
-{\check c}_{ \alpha, \sigma \dot \beta}^{\enspace \enspace \enspace \,
\sigma \dot \sigma}$
with the connection coefficients ${\omega_{\dot \alpha}}^{\beta}_{\sigma}$ and
${\omega_{\alpha}}_{\dot \beta}^{\dot \sigma}$ respectively, from the
conventional approach. The last assumption implies
$$
\Box_{\bar {\cal D}}V_{\alpha} =
(\bar {\cal D}_{\dot \gamma}\bar {\cal D}^{\dot \gamma} -
8R)V_{\alpha}
$$
and the corresponding complex conjugate identity. This shows that in the gauge
specified above, the Yang-Mills Lagrangian term
from (\ref{scrlag}) reduces to the standard one, which is written in terms of
field strengths $\bar W_{\dot \alpha}$.
Indeed, the identities $-{\check c}_{\dot \alpha, \dot
\sigma \beta}^{\enspace \enspace \enspace \, \dot \sigma \sigma}
= {\omega_{\dot \alpha}}^{\beta}_{\sigma},
-{\check c}_{ \alpha, \sigma \dot \beta}^{\enspace \enspace
\enspace \, \sigma \dot \sigma} =
{\omega_{\alpha}}_{\dot \beta}^{\dot \sigma}$
are true. One can derive them from the standard set of torsion constraints
\begin{eqnarray}
&&T_{\underline{\alpha} \underline{\beta} }^{\underline{\gamma}} = 0, \,
T_{\alpha \beta}^{a}=T_{\dot \alpha \dot \beta}^{a} = 0
\nonumber \\
&& T_{\alpha \dot \beta}^{a}=2i\sigma_{\alpha \dot \beta}^{a}, \,
T_{\underline{\alpha} b}^{a}=0 , \,
T_{ab}^{c}=0
\end{eqnarray}
where $\underline{\alpha}$ denotes either $\alpha$ or $\dot \alpha$.
Finally, the volume element $dV$ we used in (\ref{scrlag}) is nothing but the
chiral volume element of the conventional approach, usually
denoted as ${\cal E}d^{2}\Theta$. This completes the derivation of the
conventional picture from the induced geometry one.
|
proofpile-arXiv_065-619
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
}
|
\section{Introduction}\label{intro}
The first detection of hydrogen molecules in space came from a
distinctive pattern of absorption features that appeared in a low
resolution uv spectrum of $\xi$~Per recorded by a spectrometer on a
sounding rocket \markcite{1007} (Carruthers 1970). Starting with that
pioneering discovery, the Lyman and Werner bands of H$_2$ in the spectra
of early-type stars have led us down a trail of new discoveries about
this most abundant molecule in space. Progressively more refined
observations by the {\it Copernicus} satellite have given us a
fundamental understanding on this molecule's abundances in various
diffuse cloud environments \markcite{1002, 1176, 1141} (Spitzer et al.
1973; York 1976; Savage et al. 1977), how rapidly it is created and
destroyed in space \markcite{1276} (Jura 1974), and the amount of
rotational excitation that is found in different circumstances
\markcite{1212, 1015, 1772} (Spitzer \& Cochran 1973; Spitzer, Cochran,
\& Hirshfeld 1974; Morton \& Dinerstein 1976). The observed populations
in excited rotational levels have in turn led to theoretical
interpretations about how this excitation is influenced by such
conditions as the local gas density, temperature and the flux of uv
pumping radiation from nearby stars \markcite{1762, 1925, 1277} (Spitzer
\& Zweibel 1974; Jura 1975a, b). Many of the highlights of these
investigations have been reviewed by Spitzer \& Jenkins \markcite{1326}
(1975) and Shull \& Beckwith \markcite{2406} (1982). The Lyman and
Werner bands of H$_2$ can even be used to learn more about the
properties of very distant gas systems whose absorption lines appear in
quasar spectra \markcite{1078, 287} (Foltz, Chaffee, \& Black 1988;
Songaila \& Cowie 1995), although the frequency of finding these H$_2$
features is generally quite low \markcite{2297} (Levshakov et al. 1992).
In addition to the general conclusions just mentioned, there were some
intriguing details that came from the observations of uv absorption
lines. The early surveys by the {\it Copernicus} satellite indicated
that toward a number of stars the H$_2$ features became broader as the
rotational quantum number $J$ increased \markcite{1212, 1015} (Spitzer
\& Cochran 1973; Spitzer, Cochran, \& Hirshfeld 1974). An initial
suggestion by Spitzer \& Cochran \markcite{1212} (1973) was that the
extra broadening of the higher $J$ levels could arise from new molecules
that had a large kinetic energy that was liberated as they formed and
left the grain surfaces. However, a more detailed investigation by
Spitzer \& Morton \markcite{1213} (1976) showed that, as a rule, the
cases that exhibited the line broadening with increasing $J$ were
actually composed of two components that had different rotational
excitations and a velocity separation that was marginally resolved by
the instrument. In general, they found that the component with a more
negative radial velocity was relatively inconspicuous at low $J$, but
due to its higher rotation temperature it became more important at
higher $J$ and made the composite profile look broader.
By interpreting the rotational populations from the standpoint of
theories on collisional and uv pumping, Spitzer \& Morton
\markcite{1213} (1976) found a consistent pattern where the components
with the most negative velocity in each case had extraordinarily large
local densities and exposure to unusually high uv pumping fluxes. They
proposed that these components arose from thin, dense sheets of
H$_2$-bearing material in the cold, compressed regions that followed
shock fronts coming toward us. These fronts supposedly came from either
the supersonic expansions of the stars' H~II regions or perhaps from the
blast waves caused by supernova explosions in the stellar associations.
Now, some twenty years after the original investigations with the {\it
Copernicus} satellite, we have an opportunity to study once again the
behavior of the H$_2$ profiles, but this time with a wavelength
resolution that can cleanly separate the components. We report here the
results of an investigation of H$_2$ toward $\zeta$~Ori~A, one of the
stars studied earlier that showed the intriguing behavior with the H$_2$
components discussed above. Once again, the concept of the H$_2$
residing in the dense gas behind a shock comes out as a central theme in
the interpretation, but our description of the configuration given in
\S\ref{shock} is very different from that offered by Spitzer \& Morton
\markcite{1213} (1976).
\section{Observations}\label{obs}
The Lyman and Werner band absorptions of H$_2$ in the spectrum of
$\zeta$~Ori~A were observed with the Interstellar Medium Absorption
Profile Spectrograph (IMAPS). IMAPS is an objective-grating echelle
spectrograph that was developed in the 1980's as a sounding rocket
instrument \markcite{1390, 1367} (Jenkins et al. 1988, 1989) and was
recently reconfigured to fly in orbit. It can record the spectrum of a
star over the wavelength region 950$-$1150\AA\ at a resolving power of
about 200,000.\footnote{The observations reported here had a resolution
that fell short of this figure, for reasons that are given in
\S\ref{wl_scale}.} This instrument flew on the ORFEUS-SPAS carrier
launched on 12 September 1993 by the Space Shuttle flight STS-51.
Jenkins, et al. \markcite{340} (1996) have presented a detailed
description of IMAPS, how it performed during this mission and how the
data were reduced. Their article is especially useful for pointing out
special problems with the data that were mostly overcome in the
reduction. It also shows an image of a portion of the echelle spectrum
of $\zeta$~Ori~A.
The total exposure time on $\zeta$~Ori was 2412~s, divided among 63
frames, each of which covered \onequarter\ of the echelle's free
spectral range. Spectra were extracted using an optimal extraction
routine described by Jenkins, et al. \markcite{340} (1996), and
different measurements of the intensity at any given wavelength were
combined with weights proportional to their respective inverse squares
of the errors. Samples of some very restricted parts of the final
spectrum are shown in Fig.~\ref{two_spec}, where lines from $J$ = 0, 1,
3 and 5 may be seen.
\placefigure{two_spec}
\begin{figure}
\plotone{h2zori_fig1.ps}
\caption{Sample wavelength intervals covering four H$_2$
lines in the spectrum of $\zeta$~Ori recorded by IMAPS. The intensity
scale is in photons per detector pixel. Except for the transition out
of $J=5$, two prominent velocity components are seen in each H$_2$ line,
one at a heliocentric radial velocity of about $-$1~km~s$^{-1}$ (labeled
with marks that identify the transitions) and another at
+25~km~s$^{-1}$. Detector pixels are oversampled by a factor of two in
this figure. Dashed lines show the amplitude of ten times the expected
standard deviation of the points resulting from photon counting
statistics.\label{two_spec}}
\end{figure}
\section{Data Reduction}\label{reduction}
\subsection{Wavelength Scale and Resolution}\label{wl_scale}
Since IMAPS is an objective-grating instrument, there is no way that we
can use an internal line emission light source to provide a calibration
of the wavelength scale. However, as explained in Jenkins, et al.
\markcite{340} (1996), we have an accurate knowledge of how the apparent
detector coordinates map into real geometrical coordinates on the image
plane, and we also know the focal length of the cross-disperser grating
and the angles of incidence and diffraction for the echelle grating.
The only unknown parameter that we must measure is a zero offset that is
driven by the pointing of IMAPS relative to the target. We determined
this offset by measuring the positions of telluric absorption features
of O~I in excited fine-structure levels. These features are rarely seen
in the interstellar medium, but there is enough atmospheric oxygen above
the orbital altitude of 295~km to produce the absorption features in all
of our spectra.
To obtain a wavelength scale that would give heliocentric
velocities\footnote{To obtain the LSR velocity in the direction of
$\zeta$~Ori~A, one should subtract 17.5~km~s$^{-1}$ from the
heliocentric velocity. Differential galactic rotation at an assumed
distance of 450~pc to $\zeta$~Ori~A should cause undisturbed gases in
the general vicinity of the star to move at 4.5~km~s$^{-1}$ with respect
to the LSR if the galaxy has a constant rotation velocity of
220~km~s$^{-1}$ and $R_0=8.5$ kpc \protect\markcite{1681} (Gunn, Knapp,
\& Tremaine 1979). Thus, any feature appearing at a heliocentric
velocity of 22.0~km~s$^{-1}$ should be approximately in the rest frame
of gaseous material in the vicinity of our target.} for all of our
lines, we adjusted the zero offset so that the telluric features
appeared at +27.0~km~s$^{-1}$, a value that was appropriate for the
viewing direction and time of our observations. The general accuracy of
our wavelength scale is indicated by the fact that oxygen lines in 4
different multiplets all gave velocities within 0.5~km~s$^{-1}$ of the
average. Also, for H$_2$ lines out of a given $J$ level that had
roughly comparable transition strengths, the dispersion of measured
velocities was about 0.5~km~s$^{-1}$. The measured position of the
strongest component (for all $J$ levels) of 24.5~km~s$^{-1}$ compares
favorably with the heliocentric velocity of a strong, but complex
absorption feature of Na~I centered on 24~km~s$^{-1}$ \markcite{262}
(Welty, Hobbs, \& Kulkarni 1994).
The excited O~I lines can also be used to give an indication of the
wavelength resolution of our observations. We measured equivalent
widths of 10.5 and 7.25m\AA\ for the O~I$^*$ and O~I$^{**}$ lines at
1040.9 and 1041.7\AA, respectively. For the applicable densities and
temperatures of the Earth's upper atmosphere, the occupation of the
singly excited level (O~I$^*$) should be 3 times that of the doubly
excited level (O~I$^{**}$), i.e., their relative numbers are governed by
just their statistical weights $g$. Making use of this fact allows us
to apply a standard curve of growth analysis to derive
log~N(O~I$^{**}$)~=~14.19 and $b=0.99~{\rm km~s}^{-1}$ (equivalent to a
doppler broadening for $T=950$K).\footnote{These results agree very well
with predictions of the MSIS-86 model of the Earth's thermosphere
\protect\markcite{3267} (Hedin 1987) for the column density and
temperature along a sight line above our orbital altitude and at a
moderate zenith angle (40\arcdeg).} The observed profiles have widths
that correspond to $b=3.0~{\rm km~s}^{-1}$, which leads to the
conclusion that the instrumental spread function is equivalent to a
profile with
\begin{equation}\label{binst}
b_{\rm inst}=\sqrt{3.0^2-1.25^2}=2.7~{\rm km~s}^{-1}
\end{equation}
(The representative $b$ for the excited O~I lines has been elevated to
1.25~km~s$^{-1}$ to account for the small broadening caused by
saturation).
The wavelength resolving power that we obtained is lower than what is
achievable in principle with IMAPS and the pointing stability of the
spacecraft. We attribute the degradation to small motions of the
echelle grating during the exposures, caused by a sticky bearing that
relieved mechanical stresses at random times. The magnitude and
character of this effect is discussed in detail by Jenkins, et al.
\markcite{340} (1996).
\subsection{Absorption Line Measurements}\label{line_meas}
We used the MSLAP analysis program\footnote{MSLAP is a third-generation
program developed for NASA. MSLAP is copyrighted by Charles L. Joseph
and Edward B. Jenkins.} to define the continuum level $I_0$ and
re-express the intensities $I(v)$ in terms of the apparent absorption
optical depths $\tau_a$ as a function of radial velocity,
\begin{equation}\label{tau_a}
\tau_a(v) = \ln \Bigl( {I_0\over I(v)}\Bigr)~.
\end{equation}
For the ideal case where the instrument can resolve the finest details
in velocity, $\tau_a(v)$ usually gives an accurate depiction of a
differential column density per unit velocity through the relation
\begin{equation}\label{N_a}
N_a(v) = 3.768\times 10^{14}{\tau_a(v)\over f\lambda}{\rm cm}^{-2}({\rm
km~s}^{-1})^{-1}~,
\end{equation}
where $f$ is the transition's $f$-value and $\lambda$ is expressed in
\AA. However, if there are saturated, fine-scale details that are not
resolved, the true optical depths $\tau(v)$ averaged over velocity will
be underestimated, and one will miscalculate the true column density
$N(v)$. One can ascertain that this is happening if the application of
Eq.~\ref{N_a} for weaker lines indicates the presence of more material
than from the strong ones \markcite{110,3184} (Savage \& Sembach 1991;
Jenkins 1996). As will be evident in \S\ref{unres_sat}, this appears to
happen for the strongest features of H$_2$ in the $J$ = 0, 1 and 2
levels of rotational excitation.
For $J$ levels 0 through 3, we were able to draw together the results
for many different absorption lines, each going to different rotational
and vibrational levels in the upper electronic states,
2p$\sigma\,B\,^1\Sigma_u^+$ and 2p$\pi\,C\,^1\Pi_u$. In so doing, it
was important to keep track of the errors in the measured $I(v)$ and
combine redundant information at each velocity in a manner that lowered
the error in the final result. To achieve this goal, we evaluated for
every individual velocity point the $\chi^2$ from a summation over the
separate transitions,
\begin{equation}\label{chi_sq}
\chi^2(\tau_a)=\sum \biggl( {\exp(-\tau_a) - I/I_0\over
\epsilon(I/I_0)}\biggr)^2~.
\end{equation}
The expected errors in intensity $\epsilon(I/I_0)$ represented a
combination of several sources of error: (1) the noise in the individual
measurements of $I$, (2) an error in the placement of the continuum
$I_0$, and (3) an error in the adopted value of zero spectral intensity
(which is a finite value of real intensity extracted from the echelle
order). The errors in $I$ (item 1) were measured from the dispersion of
residual intensities on either side of the adopted continuum at points
well removed in velocity from the absorption feature. This error
generally becomes larger at progressively shorter wavelengths, because
the sensitivity of IMAPS decreases. (Variations of sensitivity also
result from being away from the center of the echelle blaze function.)
In every case, the noise errors were assumed to be the same magnitude at
low $I/I_0$ at the centers of lines because statistical fluctuations in
the background illumination are important. (Generally, the background
was about as large as $I_0$, so the noise amplitude would decrease only
by a factor of $\sqrt{2}$.) In a number of cases, the computed S/N was
higher than 50 (see Tables~\ref{j0table}$-$\ref{j3table}). Because
there might be some residual systematic errors that we have not
recognized, we felt that it was unwarranted to assume that these cases
had the full reliability as indicated by the calculation of S/N, when
compared with other measurements at lower S/N. To account for this, we
uniformly adopted an estimate for the relative noise level consistent
with the value
\begin{equation}\label{noise}
{\rm adopted~S/N} = 1/\sqrt{({\rm computed~S/N})^{-2} + 50^{-2}}.
\end{equation}
The error in $I_0$ (item 2 in the above paragraph) represents the
uncertainty of the continuum level that arises from a pure vertical
translation that would be permitted by the noise in the many intensity
measurements that define $I_0$. It does {\it not} include errors in the
adopted curvature of the continuum [see a discussion of this issue in
the appendix of Sembach \& Savage \markcite{181} (1992)]. For most
cases, the curvature was almost nonexistent. The error in the adopted
background level (item 3) was judged from the dispersion of residual
intensities of saturated atomic lines elsewhere in the spectrum. At
every velocity point, the worst combinations of the systematic errors
(i.e., both the adopted continuum and background levels are
simultaneously too high or, alternatively, too low) were combined in
quadrature with the random intensity errors (item 1), as modified in
Eq.~\ref{noise}, to arrive at the net $\epsilon(I/I_0)$.
Tables~\ref{j0table}$-$\ref{j3table} show the transitions for the four
lowest rotational levels of H$_2$ covered in our spectrum of
$\zeta$~Ori. Laboratory wavelengths are taken from the calculated
values of Abgrall, et al. \markcite{280} (1993a) for the Lyman band
system and Abgrall, et al. \markcite{281} (1993b) for the Werner bands.
Transition $f$-values are from Abgrall \& Roueff \markcite{2066} (1989).
The listed values of S/N are those computed as described above, but
without the modification from Eq.~\ref{noise}.
All of the lines for $J$ = 4 were too weak to measure. Only one line
from $J$ = 5 was strong enough to be useful (the Werner 0$-$0\,Q(5) line
at 1017.831\AA\ with $\log (f\lambda)$ = 1.39), although weaker lines
showed very noisy profiles that were consistent with this line.
Many lines (or certain portions thereof) were unsuitable for
measurement. These lines and the reasons for their rejection are
discussed in the endnotes of the tables. Table~\ref{j3table} omits some
lines that are far too weak to consider.
\placetable{j0table}
\placetable{j1table}
\placetable{j2table}
\placetable{j3table}
\begin{deluxetable}{
r
r
r
r
}
\small
\tablewidth{400pt}
\tablecaption{Lines from J=0\label{j0table}}
\tablehead{
\colhead{Ident.\tablenotemark{a}} &
\colhead{$\lambda$ (\AA)} &
\colhead{Log ($f\lambda$)} &
\colhead{S/N} }
\startdata
0$-$0 R(0)\tablenotemark{b}&1108.127&0.275&50\nl
1$-$0 R(0)\phm{\/}&1092.195&0.802&77\nl
2$-$0 R(0)\phm{\/}&1077.140&1.111&46\nl
3$-$0 R(0)\phm{\/}&1062.882&1.282&45\nl
4$-$0 R(0)\phm{\/}&1049.367&1.383&30\nl
5$-$0 R(0)\phm{\/}&1036.545&1.447&39\nl
6$-$0 R(0)\phm{\/}&1024.372&1.473&36\nl
7$-$0 R(0)\phm{\/}&1012.810&1.483&81\nl
8$-$0 R(0)\phm{\/}&1001.821&1.432&32\nl
9$-$0 R(0)\tablenotemark{c}&991.376&1.411&\nodata\nl
10$-$0 R(0)\tablenotemark{d}&981.437&1.314&23\nl
11$-$0 R(0)\tablenotemark{e}&971.985&1.289&\nodata\nl
12$-$0 R(0)\tablenotemark{d}&962.977&1.098&14\nl
13$-$0 R(0)\tablenotemark{d}&954.412&1.126&20\nl
W 0$-$0 R(0)\tablenotemark{f}&1008.552&1.647&31\nl
W 1$-$0 R(0)\tablenotemark{g}&985.631&1.833&\nodata\nl
W 2$-$0 R(0)\tablenotemark{h}&964.981&1.823&\nodata\nl
\enddata
\tablenotetext{a}{All transitions are in the
2p$\sigma\,B\,^1\Sigma_u^+\leftarrow {\rm X}\,^1\Sigma_g^+$ Lyman band
system, unless preceded with a ``W'' which refers to the
2p$\pi\,C\,^1\Pi_u\leftarrow {\rm X}\,^1\Sigma_g^+$ Werner bands.}
\tablenotetext{b}{Not used in the composite profile, because components
1 and 2 were too weak compared with the noise. For component 3, this
was the weakest line and had the least susceptibility to errors from
saturated substructures. This line was used to define the preferred
value for $N_{\rm total}$ with Method~A (see \S\protect\ref{method_A}).}
\tablenotetext{c}{Not considered because this line had interference from
the W~1$-$0~P(3) line.}
\tablenotetext{d}{Not included in the composite profile because the S/N
was significantly inferior to those of other lines of comparable log
($f\lambda$).}
\tablenotetext{e}{Stellar flux severely attenuated by the Ly-$\gamma$
feature.}
\tablenotetext{f}{Not included; there is serious interference from the
W~0$-$0~R(1) line.}
\tablenotetext{g}{Not included; there is serious interference from the
W~1$-$0~R(1) line.}
\tablenotetext{h}{Not included; there is serious interference from the
W~2$-$0~R(1) line.}
\end{deluxetable}
\begin{deluxetable}{
r
r
r
r
}
\small
\tablewidth{400pt}
\tablecaption{Lines from J=1\label{j1table}}
\tablehead{
\colhead{Ident.\tablenotemark{a}} &
\colhead{$\lambda$ (\AA)} &
\colhead{Log ($f\lambda$)} &
\colhead{S/N} }
\startdata
0$-$0 P(1)\tablenotemark{b}&1110.062&$-$0.191&46\nl
1$-$0 P(1)\tablenotemark{c}&1094.052&0.340&40\nl
2$-$0 P(1)\phm{\/}&1078.927&0.624&33\nl
3$-$0 P(1)\phm{\/}&1064.606&0.805&48\nl
4$-$0 P(1)\phm{\/}&1051.033&0.902&48\nl
5$-$0 P(1)\phm{\/}&1038.157&0.956&78\nl
6$-$0 P(1)\tablenotemark{d}&1025.934&0.970&\nodata\nl
7$-$0 P(1)\phm{\/}&1014.325&0.960&62\nl
8$-$0 P(1)\phm{\/}&1003.294&0.931&19\nl
9$-$0 P(1)\tablenotemark{e}&992.808&0.883&14\nl
10$-$0 P(1)\tablenotemark{e}&982.834&0.825&14\nl
11$-$0 P(1)\tablenotemark{e}&973.344&0.759&3\nl
12$-$0 P(1)\tablenotemark{e}&964.310&0.683&12\nl
13$-$0 P(1)\tablenotemark{e}&955.707&0.604&9\nl
W 0$-$0 Q(1)\phm{\/}&1009.770&1.384&36\nl
W 1$-$0 Q(1)\tablenotemark{f}&986.796&1.551&8\nl
W 2$-$0 Q(1)\tablenotemark{f}&966.093&1.529&10\nl
0$-$0 R(1)\tablenotemark{g}&1108.632&0.086&39\nl
1$-$0 R(1)\tablenotemark{h}&1092.732&0.618&69\nl
2$-$0 R(1)\phm{\/}&1077.700&0.919&55\nl
3$-$0 R(1)\phm{\/}&1063.460&1.106&59\nl
4$-$0 R(1)\phm{\/}&1049.960&1.225&64\nl
5$-$0 R(1)\phm{\/}&1037.149&1.271&56\nl
6$-$0 R(1)\tablenotemark{i}&1024.986&1.312&11\nl
7$-$0 R(1)\phm{\/}&1013.434&1.307&48\nl
8$-$0 R(1)\phm{\/}&1002.449&1.256&16\nl
9$-$0 R(1)\tablenotemark{j}&992.013&1.252&19\nl
10$-$0 R(1)\tablenotemark{e}&982.072&1.138&14\nl
11$-$0 R(1)\tablenotemark{e}&972.631&1.134&5\nl
12$-$0 R(1)\tablenotemark{e}&963.606&0.829&12\nl
13$-$0 R(1)\tablenotemark{e}&955.064&0.971&9\tablebreak
W 0$-$0 R(1)\tablenotemark{k}&1008.498&1.326&\nodata\nl
W 1$-$0 R(1)\tablenotemark{l}&985.642&1.512&\nodata\nl
W 2$-$0 R(1)\tablenotemark{m}&965.061&1.529&\nodata\nl
\enddata
\tablenotetext{a}{All transitions are in the
2p$\sigma\,B\,^1\Sigma_u^+\leftarrow {\rm X}\,^1\Sigma_g^+$ Lyman band
system, unless preceded with a ``W'' which refers to the
2p$\pi\,C\,^1\Pi_u\leftarrow {\rm X}\,^1\Sigma_g^+$ Werner bands.}
\tablenotetext{b}{For component 3, this was the weakest line and had the
least susceptibility to errors from saturated substructures. This line
was used to define the preferred value for $N_{\rm total}$. Component 1
of the 0$-$0~R(2) is near this feature, but it is not close and strong
enough to compromise the measurement of $N_{\rm total}$ with Method~A
(\S\protect\ref{method_A}). We did not use the line in the composite
profile however.}
\tablenotetext{c}{Not used in the composite profile because of
interference from the 1$-$0~R(2) line. This interference did not
compromise our use of the line for obtaining a measurement of
Component~3 using Method~B (\S\protect\ref{method_B}).}
\tablenotetext{d}{Stellar flux severely attenuated by the Ly-$\beta$
feature.}
\tablenotetext{e}{Not included in the composite profile because the S/N
was significantly inferior to those of other lines of comparable log
($f\lambda$).}
\tablenotetext{f}{S/N too low to use this line, even though its $\log
(f\lambda)$ is large.}
\tablenotetext{g}{Components 1 and 2 too weak to measure, hence not
included in composite profile. For Component~3, this line was used in
Method~B (\S\protect\ref{method_B}).}
\tablenotetext{h}{Possible interference from 1092.620 and 1092.990\AA\
lines of S~I, hence not included in composite profile.}
\tablenotetext{i}{On a wing of the stellar Ly-$\beta$, hence the S/N is
low. Line not used in the composite profile.}
\tablenotetext{j}{Not included in the composite profile because the
error array shows erratic behavior.}
\tablenotetext{k}{This line has interference from the W~0$-$0~R(0) and
8$-$0~P(3) lines. It was not used.}
\tablenotetext{l}{This line has interference from the W~1$-$0~R(0) line.
It was not used.}
\tablenotetext{m}{This line has interference from the W~2$-$0~R(0) line.
It was not used.}
\end{deluxetable}
\begin{deluxetable}{
r
r
r
r
}
\small
\tablewidth{400pt}
\tablecaption{Lines from J=2\label{j2table}}
\tablehead{
\colhead{Ident.\tablenotemark{a}} &
\colhead{$\lambda$ (\AA)} &
\colhead{Log ($f\lambda$)} &
\colhead{S/N} }
\startdata
0$-$0 P(2)\tablenotemark{b}&1112.495&$-$0.109&39\nl
1$-$0 P(2)\tablenotemark{c}&1096.438&0.420&51\nl
2$-$0 P(2)\phm{\/}&1081.266&0.706&55\nl
3$-$0 P(2)\tablenotemark{d}&1066.900&0.879&65\nl
4$-$0 P(2)\phm{\/}&1053.284&0.982&35\nl
5$-$0 P(2)\phm{\/}&1040.366&1.017&38\nl
6$-$0 P(2)\tablenotemark{e}&1028.104&1.053&13\nl
7$-$0 P(2)\tablenotemark{f}&1016.458&1.007&34\nl
8$-$0 P(2)\phm{\/}&1005.390&0.998&29\nl
9$-$0 P(2)\tablenotemark{g}&944.871&0.937&18\nl
10$-$0 P(2)\tablenotemark{h}&984.862&0.907&5\nl
11$-$0 P(2)\tablenotemark{h}&975.344&0.809&7\nl
12$-$0 P(2)\tablenotemark{h}&966.273&0.798&13\nl
13$-$0 P(2)\tablenotemark{h}&957.650&0.662&12\nl
W 0$-$0 P(2)\tablenotemark{i}&1012.169&0.746&23\nl
W 1$-$0 P(2)\tablenotemark{h}&989.086&0.904&8\nl
W 2$-$0 P(2)\tablenotemark{h}&968.292&0.843&14\nl
W 0$-$0 Q(2)\phm{\/}&1010.938&1.385&29\nl
W 1$-$0 Q(2)\tablenotemark{j}&987.972&1.551&7\nl
W 2$-$0 Q(2)\tablenotemark{j}&967.279&1.530&11\nl
0$-$0 R(2)\tablenotemark{k}&1110.119&0.018&45\nl
1$-$0 R(2)\tablenotemark{l}&1094.243&0.558&56\nl
2$-$0 R(2)\tablenotemark{c}&1079.226&0.866&35\nl
3$-$0 R(2)\phm{\/}&1064.994&1.069&53\nl
4$-$0 R(2)\phm{\/}&1051.498&1.168&76\nl
5$-$0 R(2)\phm{\/}&1038.689&1.221&72\nl
6$-$0 R(2)\tablenotemark{m}&1026.526&1.267&\nodata\nl
7$-$0 R(2)\phm{\/}&1014.974&1.285&52\nl
8$-$0 R(2)\phm{\/}&1003.982&1.232&40\nl
9$-$0 R(2)\phm{\/}&993.547&1.228&20\nl
10$-$0 R(2)\phm{\/}&983.589&1.072&18\nl
11$-$0 R(2)\tablenotemark{h}&974.156&1.103&4\nl
12$-$0 R(2)\tablenotemark{n}&965.044&0.161&\nodata\nl
13$-$0 R(2)\tablenotemark{h}&956.577&0.940&10\nl
W 0$-$0 R(2)\phm{\/}&1009.024&1.208&32\nl
W 1$-$0 R(2)\tablenotemark{h}&986.241&1.409&3\tablebreak
W 2$-$0 R(2)\phm{\/}&965.791&1.490&16\nl
\enddata
\tablenotetext{a}{All transitions are in the
2p$\sigma\,B\,^1\Sigma_u^+\leftarrow {\rm X}\,^1\Sigma_g^+$ Lyman band
system, unless preceded with a ``W'' which refers to the
2p$\pi\,C\,^1\Pi_u\leftarrow {\rm X}\,^1\Sigma_g^+$ Werner bands.}
\tablenotetext{b}{Component 1 of this line is too weak to see above the
noise, and Component 3 has interference from Component 1 of the
0$-$0~R(3) line. Hence this transition is not useful.}
\tablenotetext{c}{In constructing the composite profile, we used only
the velocity interval covering Component 3 because Components 1 and 2
are completely buried in the noise.}
\tablenotetext{d}{Component 1 feature seems to be absent for some reason
that is not understood. Perhaps an unidentified feature on the edge of
this component makes it unrecognizable.}
\tablenotetext{e}{Stellar flux severely attenuated by the Ly-$\beta$
feature. This line was not used because the S/N was too low.}
\tablenotetext{f}{Component 1 was badly corrupted by an unidentified
line. Only the region around Component~3 was used.}
\tablenotetext{g}{The nearby W~1$-$0~Q(5) line makes the continuum
uncertain. Thus, we did not use the 9$-$0 P(2) line.}
\tablenotetext{h}{Not included in the composite profile because the S/N
was significantly inferior to those of other lines of comparable log
($f\lambda$).}
\tablenotetext{i}{This line was not used because it might be corrupted
by the presence of the 1012.502\AA\ line of S~III at $-$80~km~s$^{-1}$.}
\tablenotetext{j}{S/N too low to use this line, even though its $\log
(f\lambda)$ is large.}
\tablenotetext{k}{We used only the portion covered by Component 3, since
Component 1 of this line has serious interference from Component 3 of
the 0$-$0~P(1) line.}
\tablenotetext{l}{We used only the portion covered by Component 3, since
the continuum just to the left of Component 1 is compromised by the
presence of Component 3 of the 1$-$0~P(1) line.}
\tablenotetext{m}{Stellar flux severely attenuated by the Ly-$\beta$
feature. This line was not used.}
\tablenotetext{n}{There is interference from the N~I line at 965.041\AA.
Hence this line was not used.}
\end{deluxetable}
\begin{deluxetable}{
r
r
r
r
}
\small
\tablewidth{400pt}
\tablecaption{Lines from J=3\label{j3table}}
\tablehead{
\colhead{Ident.\tablenotemark{a}} &
\colhead{$\lambda$ (\AA)} &
\colhead{Log ($f\lambda$)} &
\colhead{S/N} }
\startdata
0$-$0 P(3)\tablenotemark{b}&1115.895&$-$0.083&45\nl
1$-$0 P(3)\tablenotemark{b}&1099.787&0.439&31\nl
2$-$0 P(3)\tablenotemark{c}&1084.561&0.734&\nodata\nl
3$-$0 P(3)\phm{\/}&1070.141&0.910&26\nl
4$-$0 P(3)\phm{\/}&1056.472&1.006&56\nl
5$-$0 P(3)\phm{\/}&1043.502&1.060&48\nl
6$-$0 P(3)\phm{\/}&1031.192&1.055&41\nl
7$-$0 P(3)\phm{\/}&1019.500&1.050&57\nl
8$-$0 P(3)\tablenotemark{d}&1008.383&1.004&\nodata\nl
9$-$0 P(3)\tablenotemark{e}&997.824&0.944&18\nl
10$-$0 P(3)\tablenotemark{e}&987.768&0.944&10\nl
11$-$0 P(3)\tablenotemark{e}&978.217&0.817&20\nl
12$-$0 P(3)\tablenotemark{e,f}&969.089&0.895&10\nl
13$-$0 P(3)\tablenotemark{e}&960.449&0.673&12\nl
W 0$-$0 P(3)\phm{\/}&1014.504&0.920&54\nl
W 1$-$0 P(3)\tablenotemark{g}&991.378&1.075&\nodata\nl
W 2$-$0 P(3)\tablenotemark{e}&970.560&0.974&10\nl
W 0$-$0 Q(3)\phm{\/}&1012.680&1.386&31\nl
W 1$-$0 Q(3)\tablenotemark{h}&989.728&1.564&\nodata\nl
W 2$-$0 Q(3)\tablenotemark{i,j}&969.047&1.530&8\nl
0$-$0 R(3)\tablenotemark{k}&1112.582&$-$0.024&\nodata\nl
1$-$0 R(3)\tablenotemark{l}&1096.725&0.531&\nodata\nl
2$-$0 R(3)\phm{\/}&1081.712&0.840&47\nl
3$-$0 R(3)\phm{\/}&1067.479&1.028&42\nl
4$-$0 R(3)\phm{\/}&1053.976&1.137&37\nl
5$-$0 R(3)\phm{\/}&1041.157&1.222&49\nl
6$-$0 R(3)\phm{\/}&1028.985&1.243&24\nl
7$-$0 R(3)\phm{\/}&1017.422&1.263&35\nl
8$-$0 R(3)\phm{\/}&1006.411&1.207&18\nl
9$-$0 R(3)\phm{\/}&995.970&1.229&33\nl
10$-$0 R(3)\tablenotemark{e}&985.962&0.908&4\nl
11$-$0 R(3)\tablenotemark{m}&976.551&1.104&\nodata\nl
12$-$0 R(3)\tablenotemark{e}&967.673&1.347&10\nl
13$-$0 R(3)\tablenotemark{e}&958.945&0.931&10\nl
W 0$-$0 R(3)\tablenotemark{n}&1010.129&1.151&40\nl
W 1$-$0 R(3)\tablenotemark{i}&987.445&1.409&6\tablebreak
W 2$-$0 R(3)\tablenotemark{e}&966.780&0.883&16\nl
\enddata
\tablenotetext{a}{All transitions are in the
2p$\sigma\,B\,^1\Sigma_u^+\leftarrow {\rm X}\,^1\Sigma_g^+$ Lyman band
system, unless preceded with a ``W'' which refers to the
2p$\pi\,C\,^1\Pi_u\leftarrow {\rm X}\,^1\Sigma_g^+$ Werner bands.}
\tablenotetext{b}{This line is too weak to show up above the noise. It
was not used in constructing the composite profile.}
\tablenotetext{c}{This line could not be used because it has serious
interference from the 1084.562 and 1084.580\AA\ lines from an excited
fine-structure level of N~II.}
\tablenotetext{d}{Not used since this line has interference from the
W~0$-$0\,R(1) line.}
\tablenotetext{e}{Not included in the composite profile because the S/N
was significantly inferior to those of other lines of comparable log
($f\lambda$).}
\tablenotetext{f}{Not used since this line has interference from the
W~2$-$0\,Q(3) line.}
\tablenotetext{g}{Not used since this line has interference from the
9$-$0\,R(0) line.}
\tablenotetext{h}{Line is submerged in a deep stellar line of N~III at
989.8\AA. Thus, it could not be used.}
\tablenotetext{i}{S/N too low to use this line, even though its log
($f\lambda$) is large.}
\tablenotetext{j}{Not used since this line has interference from the
12$-$0\,P(3) line.}
\tablenotetext{k}{This line could not be used because it has
interference from the 0$-$0\,P(2) line.}
\tablenotetext{l}{This line could not be used because it has
interference from the 1096.877\AA\ line of Fe~II.}
\tablenotetext{m}{The left-hand side of Component~1 has interference
from Component~3 of the line of O~I at 976.448\AA. This line could not
be used even for Component~3 because the continuum level was uncertain.}
\tablenotetext{n}{Line inadvertently omitted. The omission was
discovered long after the combined analysis had been completed.}
\end{deluxetable}
\clearpage
\section{Results}\label{results}
Figs.~\ref{j0fig}$-$\ref{j5fig} show gray-scale representations of
$\chi^2-\chi_{\rm min}^2$ as a function of $\log N_a(v)$ and the
heliocentric radial velocity $v$. The minimum value $\chi_{\rm min}^2$
was determined at each velocity, and our representation that shows how
rapidly $\chi^2$ increases on either side of the most probable $\log
N_a(v)$ (i.e., the value where $\chi_{\rm min}^2$ is achieved) is a
valid measure of the relative confidence of the result \markcite{1666}
(Lampton, Margon, \& Bowyer 1976). Since we are measuring a single
parameter, the $\chi^2$ distribution function with 1 degree of freedom
is appropriate, and thus, for example, 95\% of the time we expect the
true intensity to fall within a band where $\chi^2-\chi_{\rm min}^2 <
3.8$, i.e., the ``$\pm 2\sigma$'' zone. To improve on the range of the
display without sacrificing detail for low values of $\chi^2-\chi_{\rm
min}^2$, the actual darknesses in the figures and their matching
calibration squares on the right are scaled to the quantity
$\log(1+\chi^2-\chi_{\rm min}^2)$. Measurements at velocities separated
by more than a single detector pixel (equivalent to 1.25 km~s$^{-1}$)
should be statistically independent.\footnote{This statement is not
strictly true, since single photoevents that fall near the border of two
pixels will contribute a signal to each one. However, the width of one
pixel is a reasonable gauge for distance between nearly independent
measurements if one wants to judge the significance of the $\chi^2$'s.}
This separation is less than the wavelength resolving power however.
Thus, reasonable assumptions about the required continuity of the
profiles for adjacent velocities can, in principle, restrict the range
of allowable departures from the minimum $\chi^2$ even further than the
formal confidence limits.
\placefigure{j0fig}
\placefigure{j1fig}
\placefigure{j2fig}
\placefigure{j3fig}
\placefigure{j5fig}
\begin{figure}
\plotone{h2zori_fig2.ps}
\caption{A composite of 8 absorption profiles from H$_2$ in
the $J$ = 0 rotational level. Transitions listed in
Table~\protect\ref{j0table} were used, except where noted. Shades of
gray, as indicated by the boxes on the right, map out the changes in
$\chi^2-\chi_{\rm min}^2$ as a function of $\log N_a(v)$ for each value
of $v$. For reasons discussed in \S\protect\ref{unres_sat} the strong
peak on the right-hand side probably understates the true amount of
H$_2$ that is really present.\label{j0fig}}
\end{figure}
\begin{figure}
\plotone{h2zori_fig3.ps}
\caption{Same as for Fig.~\protect\ref{j0fig}, except that
the applicable transitions, listed in Table~\protect\ref{j1table}, are
from the $J$ = 1 level. Thirteen transitions were combined to make this
figure. As with Fig.~\protect\ref{j0fig}, the rightmost, strong peak
probably under-represents the true amount of H$_2$.\label{j1fig}}
\end{figure}
\begin{figure}
\plotone{h2zori_fig4.ps}
\caption{Same as for Fig.~\protect\ref{j0fig}, except that
the applicable transitions, listed in Table~\protect\ref{j2table}, are
from the $J$ = 2 level. A total of 19 transitions were used to
construct this figure, but because of interference problems only 14 of
them covered Components 1 and 2.\label{j2fig}}
\end{figure}
\begin{figure}
\plotone{h2zori_fig5.ps}
\caption{Same as for Fig.~\protect\ref{j0fig}, except that
the 15 applicable transitions, listed in Table~\protect\ref{j3table},
are from the $J$ = 3 level. Unlike the cases for $J$ = 0, 1 or 2, the
right-hand peak does not show any disparities in the height from one
transition to another, indicating that the representation is probably
correct.\label{j3fig}}
\end{figure}
\begin{figure}
\plotone{h2zori_fig6.ps}
\caption{Same as for Fig.~\protect\ref{j0fig}, except that
only one transition, the Werner 0$-$0~Q(5) line, was used.
\label{j5fig}}
\end{figure}
The profiles that appear in Figs.~\ref{j0fig}$-$\ref{j5fig} indicate
that there are two prominent peaks in H$_2$ absorption, with the
left-hand one holding molecules with a higher rotational temperature
than the one on the right. This effect, one that creates dramatic
differences in the relative sizes of the two peaks with changing $J$,
was noted earlier by Spitzer, et al. \markcite{1015} (1974). There is
also some H$_2$ that spans the velocities between these two peaks. For
the purposes of making some general statements about the H$_2$, we
identify the material that falls in the ranges $-$15 to +5, +5 to +15,
and +15 to +35 km~s$^{-1}$ as Components 1, 2 and 3, respectively.
While some residual absorption seems to appear outside the ranges of the
3 components, we are not sure of its reality. Some transitions seemed
to show convincing extra absorption at these large velocities, while
others did not.
Component~1 shows a clear broadening as the profiles progress from $J=0$
to 5. Precise determinations of this effect and the accompanying
uncertainties in measurement will be presented in \S\ref{prof_changes}.
The widths of the profiles for Component~3 also seem to increase with
$J$, but the effect is not as dramatic as that shown for Component~1.
We are reluctant to present any formal analysis of the broadening for
Component~3 because we believe the $N_a(v)$ profile shapes misrepresent
the true distributions of molecules with velocity for $J=0$, 1 and 2,
for reasons given in \S\ref{unres_sat}. As a rough indication of the
trend, we state here only that the {\it apparent} profile widths are
4.5, 5.8, 5.8 and $7.7~{\rm km~s}^{-1}$ (FWHM) for $J=0$, 1, 2 and 3,
respectively. These results for this component only partly agree with
the finding by Spitzer, et al. \markcite{1015} (1974) that the velocity
width of molecules in the $J=1$ state is higher than those in both $J=0$
or $J=2$. The latter conclusions were based on differences in the $b$
parameters of the curves of growth for the lines.
Table~\ref{comp_summary} lists our values for the column densities $\int
N_a(v)\,dv$, obtained for profiles that follow the valley of minimum
$\chi^2$. Exceptions to this way of measuring $N({\rm H}_2)$ are
discussed in \S\ref{unres_sat} below. We also list in the table the
results that were obtained by Spitzer, et al. \markcite{1015} (1974)
and Spitzer \& Morton \markcite{1213} (1976). With only two significant
exceptions, our results seem to be in satisfactory agreement with these
previous determinations. One of the discrepancies is the difference
between our determination $\log N(J=5)=13.70$ for Component~1, compared
with the value of 13.32 found by Spitzer, et al. \markcite{1015} (1974).
We note that latter was based on lines that had special problems: either
the lines had discrepant velocities or the components could not be
resolved. The second discrepancy is between our value of $\log
N(J=0)=15.09$ (Method~A discussed in \S\ref{method_A}) or 14.79
(Method~B given in \S\ref{method_B}) for Component~3 and the value 15.77
found by Spitzer \& Morton \markcite{1213} (1976) from an observation of
just the Lyman 4$-$0\,R(0) line. However, this line is very badly
saturated (the central optical depth must be about 12 with {\it our}
value of $\log N$ and $b=2~{\rm km~s}^{-1}$), and thus it is not
suitable, by itself, for measuring a column density.
\placetable{comp_summary}
\begin{deluxetable}{
c
c
c
c
}
\small
\tablewidth{0pt}
\tablecaption{Log Column Densities\tablenotemark{a}~~and Rotational
Temperatures\label{comp_summary}}
\tablehead{
\colhead{} &
\colhead{Component 1} &
\colhead{Component 2\tablenotemark{b}} &
\colhead{Component 3} \\
\colhead{$J$} &
\colhead{($-15 < v < +5$ km~s$^{-1}$)} &
\colhead{($+5 < v < +15$ km~s$^{-1}$)} &
\colhead{($+15 < v < +35$ km~s$^{-1}$)} }
\startdata
0&13.53 (13.46B\tablenotemark{c}, 13.48\tablenotemark{d}~)&12.86
(13.23\tablenotemark{d}~)&15.09\tablenotemark{e}, 14.79\tablenotemark{f}
(15.21A\tablenotemark{c}, 15.77\tablenotemark{d}~)\nl
1&13.96 (14.15B\tablenotemark{c}, 14.20\tablenotemark{d}~)&13.44
(13.85\tablenotemark{d}~)&15.72\tablenotemark{e}, 15.69\tablenotemark{f}
(15.43B\tablenotemark{c}~)\nl
2&13.64 (13.64B\tablenotemark{c}, 13.68\tablenotemark{d}~)&13.27
(13.36\tablenotemark{d}~)&14.78\tablenotemark{e}, 14.66\tablenotemark{f}
(14.74B\tablenotemark{c}, 14.87\tablenotemark{d}~)\nl
3&13.99 (14.05A\tablenotemark{c}, 14.08\tablenotemark{d}~)&13.55
(13.69\tablenotemark{d}~)&14.19 (14.14A\tablenotemark{c},
14.34\tablenotemark{d}~)\nl
4&\nodata (13.22A\tablenotemark{c}, 13.11\tablenotemark{d}~)&\nodata
($-\infty$\tablenotemark{d}~)&\nodata (12.95A\tablenotemark{c},
12.85\tablenotemark{d}~)\nl
5&13.70 (13.32A\tablenotemark{c},
13.45\tablenotemark{d}~)&13.13\tablenotemark{g}
(12.48\tablenotemark{d}~)&13.21
($<$12.79\tablenotemark{c},$-\infty$\tablenotemark{d}~)\nl
Total&14.52&14.01&15.86\tablenotemark{e}, 15.79\tablenotemark{f}\nl
Rot. Temp.\tablenotemark{h}&950K&960K&320K\tablenotemark{e},
340K\tablenotemark{f}\nl
\enddata
\tablenotetext{a}{Numbers in parentheses are from earlier {\it
Copernicus} results reported by Spitzer, Cochran \& Hirshfeld
\markcite{1015} (1974) and Spitzer \& Morton \markcite{1213} (1976) for
comparison with our results (and to fill in for $J$ = 4).}
\tablenotetext{b}{Not really a distinct component, but rather material
that seems to bridge the gap between Components 1 and 3.}
\tablenotetext{c}{From Spitzer, Cochran \& Hirshfeld \markcite{1015}
(1974), with errors A = 0.04$-$0.09 and B = 0.10$-$0.19.}
\tablenotetext{d}{From Spitzer \& Morton \markcite{1213} (1976).}
\tablenotetext{e}{Derived from Method A discussed in
\S\protect\ref{method_A}.}
\tablenotetext{f}{Derived from Method B discussed in
\S\protect\ref{method_B}.}
\tablenotetext{g}{Not a distinct component (see
Fig.~\protect\ref{j5fig}). The number given is a formal integration
over the specified velocity range and represents the right-hand wing of
the very broad component centered near the velocity of Component~1.}
\tablenotetext{h}{From the reciprocal of the slope of the best fit to
$\ln [N(J)/g(J)]$ {\it vs.\/} $E_J$, excluding $J=4$.}
\end{deluxetable}
\subsection{Unresolved Saturated Substructures in Component
3}\label{unres_sat}
For the right-hand peaks (Component 3) in $J$ = 0, 1 and 2, the weakest
transitions show more H$_2$ than indicated in Figs.~\ref{j0fig} to
\ref{j2fig}, which are based on generally much stronger transitions.
This behavior reveals the presence of very narrow substructures in
Component 3 that are saturated and not resolved by the instrument.
Jenkins \markcite{3184} (1996) has shown how one may take any pair of
lines (of different strength) that show a discrepancy in their values of
$N_a(v)$, as evaluated from Eqs.~\ref{tau_a} and \ref{N_a}, and evaluate
a correction to $\tau_a(v)$ of the weaker line that compensates for the
under-representation of the smoothed real optical depths $\tau(v)$. In
effect, this correction is a method of extrapolating the two distorted
$N_a(v)$'s to a profile that one would expect to see if the line's
transition strength was so low that the unresolved structures had their
maximum (unsmoothed) $\tau (v)\ll 1$.
Unfortunately, we found that for each of the three lowest $J$ levels,
different pairs of lines yielded inconsistent results. In each case, an
application of the analysis of the first and second weakest lines gave
column densities considerably larger than the same procedure applied to
the second and third weakest lines. We list below a number of
conjectures about the possible cause(s) for this effect:
\begin{enumerate}
\item The functional forms of the distributions of subcomponent
amplitudes and velocity widths are so bizarre, and other conditions are
exactly right, that the assumptions behind the workings of the
correction procedure are not valid. As outlined by Jenkins
\markcite{1355,3184} (1986, 1996), these distributions would need to be
very badly behaved.
\item We have underestimated the magnitudes of the errors in the
determinations of scattered light in the spectrum, which then reflect on
the true levels of the zero-intensity baselines and, consequently, the
values of $\tau_a(v)$ near maximum absorption.
\item The transition $f$-values that we have adopted are wrong. The
sense of the error would be such that the weakest lines are actually
somewhat stronger than assumed, relative to the $f$-values of the next
two stronger lines. Another alternative is that the second and third
strongest lines are much closer together in their $f$-values than those
that were adopted.
\end{enumerate}
While we can not rigorously rule out possibilities (1) and (2) above, we
feel that they are unlikely to apply. Regarding possibility (3), the
$f$-values are the product of theoretical calculations, and to our
knowledge only some of the stronger transitions have been verified
experimentally \markcite{3186} (Liu et al. 1995). It is interesting to
see if there is any observational evidence outside of the results
reported here that might back up the notion that alternative (3) is the
correct explanation.
We are aware of two potentially useful examples where the weakest
members of the Lyman series have been seen in the spectra of
astronomical sources. One is in a survey of many stars by Spitzer,
Cochran \& Hirshfeld \markcite{1015} (1974),\footnote{There are many
papers that report observations of H$_2$ made by the {\it Copernicus}
satellite. Oddly enough, the paper by Spitzer et al. \markcite{1015}
(1974) is the only one that includes measurements of the weakest lines.}
and another is an array of H$_2$ absorption features identified by
Levshakov \& Varshalovich \markcite{3318} (1985) and Foltz, et al.
\markcite{1078} (1988) at $z$ = 2.811 in the spectrum of the quasar
PKS~0528$-$250. The quasar absorption lines have subsequently been
observed at much higher resolution by Songaila \& Cowie \markcite{287}
(1995) using the Keck Telescope.
In the survey of Spitzer et al. \markcite{1015} (1974), the only target
that showed lines from $J$=0 that were not on or very close to the flat
portion of the curve of growth (or had an uncertain measurement of the
Lyman 0$-$0\,R(0) line) was 30 CMa. The 10m\AA\ equivalent width
measured for this line is above a downward extrapolation of the the
trend from the stronger lines. If the line's value of $\log f\lambda$
were raised by 0.28 in relation with the others, the measured line
strength would fall on their adopted curve of growth. Unfortunately, we
can not apply the same test for the Lyman 0$-$0\,P(1) or
0$-$0\,R(2)lines, the two weakest lines that we could use here for the
next higher $J$ levels, because these lines were not observed by Spitzer
et al.
The H$_2$ lines that appear in the spectrum PKS~0528$-$250 are created
by a heavy-element gas system that is moving at only 2000 to 3000
km~s$^{-1}$ with respect to the quasar (and hence one that is not very
far away from the quasar). The overall widths of the H$_2$ lines of
about 20~km~s$^{-1}$ were resolved in the R = 36,000 spectrum of
Songaila \& Cowie \markcite{287} (1995), but the shallow Lyman 0$-$0,
1$-$0 and 2$-$0\,R(0) features showed a strengthening that was far less
than the changes in their relative $f$-values. Songaila \& Cowie
interpreted this behavior as the result of saturation in the lines if
they consisted of a clump of 5 unresolved, very narrow features, each
with $b$ = 1.5 km~s$^{-1}$, distributed over the observed velocity
extent of the absorption. One might question how plausible it is to
find gas clouds with such a small velocity dispersion that could cover a
significant fraction the large physical dimension of the
continuum-emitting region of the quasar. As an alternative, we might
accept the notion that the lines do not contain unresolved saturated
components, but instead, that the real change in the $f$-values is less
than assumed.
Finally, we turn to our own observations. In our recording of the Lyman
0$-$0\,R(0) line in our spectrum of $\zeta$~Ori, the amplitude of the
$\tau_a(v)$ profile of Component~1 (about $4\sigma$ above the noise), in
relation to that of Component~3, is not much different than what may be
seen in the next stronger line, 1$-$0\,R(0). If significant distortion
caused by unresolved, saturated substructures were occurring for
Component 3 in the latter, the size difference for the two components
would be diminished, contrary to what we see in the data. If one were
to say that the difference in $\log (f\lambda)$ for the two lines were
smaller by 0.4, we would obtain $N_a(v)$'s that were consistent with
each other.
We regard the evidence cited above as suggestive, but certainly not
conclusive, evidence that our problems with the disparity of answers for
$N_a(v)$ might be caused by incorrect relative $f$-values. Even if this
conjecture is correct, we still do not know whether the stronger or
weaker $f$-values need to be revised. In view these uncertainties, we
chose to derive $N(v)$ for Component~3 by two different methods,
Method~A and Method~B, outlined in the following two subsections. Total
column densities $\int N(v)\,dv$ derived each way are listed in
Table~\ref{comp_summary}.
\subsubsection{Method A}\label{method_A}
Method~A invokes the working assumption that the adopted $f$-value for
the weakest line is about right, and that there is a problem with the
somewhat stronger lines. If this is correct, then our only recourse is
to derive $N_a(v)$ from this one line through the use of
Eqs.~\ref{tau_a} and \ref{N_a} and assume that the correction for
unresolved saturated substructures is small. For $J=0$, 1 and 2, we
used the Lyman 0$-$0\,R(0), 0$-$0\,P(1) and 0$-$0\,R(2) lines,
respectively. (The weakest line for $J=2$, 0$-$0\,P(2) could not be
used; see note $b$ of Table~\ref{j2table}.)
\subsubsection{Method B}\label{method_B}
Here we assume that the published $f$-value for the weakest line is too
small, but that the values for the next two stronger lines are correct.
We then derive corrections for $\tau_a(v)$ for the weaker line using the
method of Jenkins \markcite{3184} (1996). While the errors in this
extrapolation method can be large, especially after one considers the
effects of the systematic deviations discussed earlier [items (2) and
(3) covered in \S\ref{line_meas}], under the present circumstances they
are probably not much worse than the arbitrariness in the choice of
whether Method~B is any better than Method~A or some other way to derive
$N_a(v)$. Lyman band line pairs used for this method were 1$-$0\,R(0)
and 2$-$0\,R(0) for $J=0$, 0$-$0\,R(1) and 1$-$0\,P(1) for $J=1$, and
1$-$0\,P(2) and 1$-$0\,R(2) for $J=2$.
\subsection{Profile Changes with $J$ for
Component~1}\label{prof_changes}
Figures \ref{j0fig} to \ref{j5fig} show very clearly that the profiles
for Component~1 have widths that progressively increase as the
rotational quantum numbers go from $J=0$ to $J=5$.
Figure~\ref{j0-j5fig} shows a consolidation of the results from
Figs.~\ref{j0fig} to \ref{j5fig}: the valleys of $\chi^2-\chi^2_{\rm
min}$ are depicted as lines [now in a linear representation for
$N_a({\rm H}_2)$], and the profiles are stacked vertically to make
comparisons for different $J$ in Component~1 more clear. In addition to
showing the changes in profile widths, this figure also shows that there
is a small ($\sim 1~{\rm km~s}^{-1}$) shift toward negative velocities
with increasing $J$ up to $J=3$, followed by a more substantial shift
for $J=5$.
\placefigure{j0-j5fig}
\begin{figure}
\plotone{h2zori_fig7.ps}
\caption{Plots of $\log N_a(v)$ versus $v$ scaled such that
the heights of the peaks for Component~1 are nearly identical. To
facilitate comparisons of widths and velocity centers across different
$J$ levels, the two vertical, dotted lines mark the half amplitude
points of the $J=0$ profile.\label{j0-j5fig}}
\end{figure}
A simple, approximate way to express numerically the information shown
in Fig.~\ref{j0-j5fig} is to assume that most of the H$_2$ at each $J$
level has a one-dimensional distribution of velocity that is a Gaussian
function characterized by a peak value for $N(v)$, $N_{\rm max}$, a
central velocity, $v_0$, and a dispersion parameter, $b$. We can then
ascertain what combinations of these 3 parameters give an acceptable fit
to the data as defined, for example, by values $\chi^2-\chi^2_{\rm min}
< 7.8$ that lead to a 95\% confidence limit. We carried out this study
with $\chi^2$'s, of the type displayed in Figs.~\ref{j0fig} to
\ref{j5fig}, summed over velocity points spaced 1.6~km~s$^{-1}$ apart to
assure statistical independence. Table~\ref{chi2} summarizes the
results of that investigation. The quantities $v_{\rm min}$ and $v_{\rm
max}$ are the velocity limits over which the fits were evaluated. The
error bounds are defined only by the $\chi^2$ limits and do not include
systematic errors, such as those that arise from errors in $f$-values or
our overall adopted zero-point reference for radial velocities. For
given $J$ levels, there are small differences between the preferred
$\log (N_{\rm max}\sqrt{\pi}b)$ and the log column densities given in
Table~\ref{comp_summary} caused by real departures from the Gaussian
approximations ($J=5$ shows the largest deviation, 0.08 dex, as one
would expect from the asymmetrical appearance shown in
Fig.~\ref{j0-j5fig}).
\placetable{chi2}
\begin{deluxetable}{
c
c
c
c
c
c
}
\tablewidth{0pt}
\tablecaption{Gaussian Fits to Component~1\label{chi2}}
\tablehead{
\colhead{} &
\colhead{$J=0$} &
\colhead{$J=1$} &
\colhead{$J=2$} &
\colhead{$J=3$} &
\colhead{$J=5$}
}
\startdata
$v_{\rm max}$ (km~s$^{-1}$)&+4&+4&+5&+5&+7\nl
$v_{\rm min}$&$-$5&$-$6&$-$8&$-$8&$-$14\nl
\tablevspace{15pt}
Largest $\log N_{\rm max}~[{\rm cm}^{-2}({\rm
km~s}^{-1})^{-1}]$&12.82&13.12&12.68&13.00&12.66\nl
Most probable $\log N_{\rm max}$&12.77&13.09&12.62&12.96&12.56\nl
Smallest $\log N_{\rm max}$&12.72&13.06&12.58&12.94&12.46\nl
\tablevspace{15pt}
Largest $v_0$
(km~s$^{-1}$)\tablenotemark{a}&$-$0.3&$-$0.9&$-$1.0&$-$1.0&$-$1.0\nl
Most probable $v_0$&$-$0.5&$-$1.0&$-$1.5&$-$1.3&$-$2.9\nl
Smallest $v_0$&$-$0.7&$-$1.2&$-$2.0&$-$1.6&$-$4.4\nl
\tablevspace{15pt}
Largest $b$ (km~s$^{-1}$)\tablenotemark{b}&3.2&4.2&7.0&6.8&14\nl
Most probable $b$&2.9&3.9&6.0&6.5&9.4\nl
Smallest $b$&2.6&3.8&5.2&6.0&7.2\nl
\enddata
\tablenotetext{a}{Heliocentric radal velocity of the profile's center.}
\tablenotetext{b}{Includes instrumental broadening and registration
errors (see \S\protect\ref{prof_changes}). Hence, the real $b$ should
equal about $\sqrt{b_{\rm obs}^2-(2.8~{\rm km~s}^{-1})^2}$.}
\end{deluxetable}
To determine the real widths of the profiles, one must subtract in
quadrature two sources of broadening in the observations. First, there
is the instrumental broadening of each line in the spectrum that we
recorded, as discussed in \S\ref{wl_scale}. Adding to this effect are
the small errors in registration of the lines, as they are combined to
create the $\chi^2-\chi^2_{\rm min}$ plots (Figs.~\ref{j0fig} to
\ref{j5fig}). From the apparent dispersion of line centers at a given
$J$, we estimate the rms registration error to be 0.5~km~s$^{-1}$. We
estimate that the effective $b$ parameter for these two effects combined
should be about 2.8~km~s$^{-1}$, and thus the formula given in note $a$
of Table~\ref{chi2} should be applied to obtain a best estimate for the
true $b$ of each H$_2$ profile (the results for the lowest $J$ levels
will not be very accurate, since $b_{\rm obs}$ is only slightly greater
than 2.8~km~s$^{-1}$).
The results shown in Fig.~\ref{j0-j5fig} and Table~\ref{chi2} show two
distinct trends of the profiles with increasing $J$. First, the most
probable values for the widths $b$ increase in a steady progression from
$J=0$ to $J=5$. Second, the most probable central velocities $v_0$
become steadily more negative with increasing $J$, except for an
apparent reversal between $J=2$ and $J=3$ that is much smaller than our
errors. It is hard to imagine that systematic errors in the
observations could result in these trends. The absorption lines for
different $J$ levels appear in random locations in the spectral image
formats, so any changes in the spectral resolution or distortions in our
wavelength scale should affect all $J$ levels almost equally.
\section{Discussion}\label{discussion}
\subsection{Preliminary Remarks}\label{prelim}
The information given in Table~\ref{comp_summary} shows that the 3
molecular hydrogen velocity components toward $\zeta$~Ori~A have
populations in different $J$ levels that, to a reasonable approximation,
conform to a single rotational excitation temperature in each case.
This behavior seems to reflect what has been observed elsewhere in the
diffuse interstellar medium. For instance, in their survey of 28 lines
of sight, Spitzer, et al. \markcite{1015} (1974) found that for
components that had $N(J=0)\lesssim 10^{15}{\rm cm}^{-2}$, a single
excitation temperature gave a satisfactory fit to all of the observable
$J$ levels. By contrast, one generally finds for much higher column
densities that there is bifurcation to two temperatures, depending on
the $J$ levels [see, e.g., Fig.~2 of Spitzer \& Cochran \markcite{1212}
(1973)]. This is a consequence of the local density being high enough
to insure that collisions dominate over radiative processes at low to
intermediate $J$ and thus couple the level populations to the local
kinetic temperature, whereas for higher $J$ the optical pumping can take
over and yield a somewhat higher temperature. For cases where the total
column densities are exceptionally low [$N({\rm H}_2)\approx 10^{13}{\rm
cm}^{-2}$ for such stars as $\zeta$~Pup, $\gamma^2$~Vel and $\tau$~Sco],
the rotation temperatures can be as high as about 1000K. This behavior
is consistent with what we found for our Components~1 and 2. Our
Component~3 has a somewhat lower excitation temperature, but one that is
in accord with other lines of sight that have $N({\rm H}_2)\approx
10^{15}{\rm cm}^{-2}$ in the sample of Spitzer, et al. \markcite{1015}
(1974).
It is when we go beyond the information conveyed by just the column
densities and study changes in the profiles for different $J$ that we
uncover some unusual behavior. Here, we focus on Component~1, where the
widths and velocity centroids show clear, progressive changes with
rotational excitation. While Component~3 also shows some broadening
with increasing $J$, the magnitude of the effect is less, and it is
harder to quantify because there are probably unresolved, saturated
structures that distort the $N_a(v)$ profiles. The changes in
broadening with $J$ are inconsistent with a simple picture that, for the
most diffuse clouds, the excitation of molecular hydrogen is caused by
optical pumping out of primarily the $J=0$ and 1 levels by uv starlight
photons in an optically thin medium.
We might momentarily consider an explanation where the strength of the
optical pumping could change with velocity, by virtue of some shielding
in the cores of some of the strongest pumping lines. However, in the
simplest case we can envision, one where the light from $\zeta$~Ori
dominates in the pumping, the shielding is not strong enough to make
this effect work. For example, in Component~1 we found $\log N({\rm
H}_2)=13.53$ (Table~\ref{comp_summary}) and a largest possible {\it real
value\/}\footnote{See note $b$ of Table~\protect\ref{chi2}} of
$b=1.55~{\rm km~s}^{-1}$ for molecules in the $J=0$ level. We would
need to have a pumping line from $J=0$ with a characteristic strength
$\log f\lambda=3.0$ to create an absorption profile $1-I(v)/I_0$ that is
saturated enough to have it appear, after a convolution with our
instrumental profile, as broad as the observed $N_a(v)$ for molecules in
the $J=2$ state.\footnote{This simple proof is a conservative one, since
it neglects other processes that tend to make the $J=2$ profile as
narrow as that for $J=0$, such as pumping from many other, much weaker
lines or the coupling of molecules in the $J=2$ state with the kinetic
motions of the gas through elastic collisions.} In reality, the
strongest lines out of $J=0$ have $\log f\lambda$ only slightly greater
than 1.8 (see Table~\ref{j0table}). Likewise, the width of the $N_a(v)$
profile for $J=3$ can only be matched with a pumping line out of $J=1$
with $\log f\lambda=2.0$, again a value that is much higher than any of
the actual lines out of this level (see Table~\ref{j1table}). Thus, if
we are to hold on to the notion that line shielding could be an
important mechanism, we must abandon the idea that $\zeta$~Ori is the
source of pumping photons.
We could, of course, adopt a more imaginative approach and propose that
light from another star is responsible for the pumping. Then, we could
envision that a significant concentration of H$_2$ just off our line of
sight could be shielding (at selective velocities) the radiation for the
molecules that we can observe. While this could conceivably explain why
the profiles for $J>1$ look different from those of $J=0$ or 1, it does
not address the problem that the profile for $J=1$ disagrees with that
of $J=0$. (The coupling of these two levels by optical pumping is very
weak.) As indicated by the numbers in Table~\ref{chi2}, both the
velocity widths and their centroids for these lowest two levels differ
by more than the measurement errors.
Another means for achieving a significant amount of rotational
excitation is heating due to the passage of a shock --- one that is slow
enough not to destroy the H$_2$ \markcite{2812} (Aannestad \& Field
1973). Superficially, we might have imagined that Component~1 is a
shocked portion of the gas that was originally in Component~3, but that
is now moving more toward us, relatively speaking. However this picture
is in conflict with the change in velocity centroids with $J$, for the
gas would be expected to speed up as it cools in the postshock zone
where radiative cooling occurs. Our observations indicate that the
cooler (rear) part of this zone that should emphasize the lower $J$
levels is actually traveling more slowly.
From the above argument on the velocity shift, it is clear that if we
are to invoke a shock as the explanation for the profile changes, we
must consider one that is headed in a direction away from us. If this
is so, we run into the problem that we are unable to see any H$_2$ ahead
of this shock, i.e, at velocities more negative than Component~1. Thus,
instead of creating a picture where existing molecules are accelerated
and heated by a shock, we must turn to the idea that perhaps the
molecules are formed for the first time in the dense, compressed
postshock zone, out of what was originally atomic gas undergoing cooling
and recombination. In this case, one would look for a shock velocity
that is relatively large, so that the compression is sufficient to raise
the density to a level where molecules can be formed at a fast rate.
\subsection{Evidence of a Shock that could be Forming
H$_2$}\label{shock}
\subsubsection{Preshock Gas}\label{preshock_gas}
There is some independent evidence from atomic absorption lines that we
could be viewing a bow shock created by the obstruction of a flow of
high velocity gas coming toward us, perhaps a stellar wind or a
wind-driven shell \markcite{1696} (Weaver et al. 1977). A reasonable
candidate for this obstruction is a cloud that is responsible for the
low-ionization atomic features that can be seen near $v=0~{\rm
km~s}^{-1}$.
In the IMAPS spectrum of $\zeta$~Ori~A, there are some strong
transitions of C~II (1036.337\AA) and N~II (1083.990\AA) that show
absorption peaks at $-$94~km~s$^{-1}$, plus a smaller amount of material
at slightly lower velocities \markcite{2956} (Jenkins 1995). Features
from doubly ionized species are also present at about the same velocity,
i.e., C~III, N~III, Si~III, S~III \markcite{1151} (Cowie, Songaila, \&
York 1979) and Al~III (medium resolution GHRS spectrum in the HST
archive\footnote{Exposure identification: Z165040DM.}). Absorption by
strong transitions of O~I and N~I are not seen at $-$94~km~s$^{-1}$
however. The moderately high state of ionization of this rapidly moving
gas, a condition similar to that found for high velocity gas in front of
23~Ori by Trapero et al. \markcite{356} (1996), may result from either
photoionization by uv radiation from the Orion stars or collisional
ionization at a temperature somewhat greater than $10^4$K.
Figure~\ref{CIV} shows spectra that we recovered from the HST
archive\footnote{Again, a medium resolution GHRS spectrum: Exposure
identification: Z1650307T.} in the vicinity of the C~IV doublet (1548.2,
1550.8\AA). We determined an upper limit $\log N({\rm C~IV})<12.2$ at
$v\approx -90~{\rm km~s}^{-1}$. When this result is compared with the
determination $\log N({\rm C~II})=13.84$ \markcite{1151} (Cowie,
Songaila, \& York 1979) or 13.82 (IMAPS spectrum), we find that
$T<20,000$K if we use the collisional ionization curves of Benjamin \&
Shapiro \markcite{327} (1996) for a gas that is cooling isobarically.
(A similar argument arises from an upper limit for N~V/N~II, but the
resulting constraint on the temperature is weaker.) There is
considerably more Si~III than Si~II in the high velocity gas
\markcite{1151} (Cowie, Songaila, \& York 1979), but this is may be due
to photoionization. Thus, to derive a lower limit for the temperature
of the gas, we must use a typical equilibrium temperature for an H~II
region, somewhere in the range $8,000 < T < 12,000$K \markcite{1855}
(Osterbrock 1989).
\placefigure{CIV}
\begin{figure}
\plotone{h2zori_fig8.ps}
\caption{A medium resolution (R = 20,000) recording of the
C~IV doublet in a spectrum of $\zeta$~Ori~A taken with the G160M grating
of GHRS on the Hubble Space Telescope. A correction of +0.06\AA\ has
been added to the wavelength scale of calibrated file for exposure
Z1650307T to reflect the offset in a 1554.9285\AA\ calibration line that
may be seen in exposure Z1650306T.\label{CIV}}
\end{figure}
\subsubsection{Immediate Postshock Gas}\label{immediate_PS}
Figure~\ref{CIV} shows that there is a broad absorption from C~IV
centered at a velocity of about $-36~{\rm km~s}^{-1}$, in addition to a
narrower peak at about +20~km~s$^{-1}$. The equivalent widths of 50 and
22~m\AA\ for the broad, negative velocity components for the transitions
at 1548.2 and 1550.8\AA, respectively, indicate that $\log N({\rm
C~IV})=13.1$, a value that is in conflict with the upper limit $\log
N({\rm C~IV})=11.9$ obtained by Cowie, et al. \markcite{1151} (1979).
Absorptions by Si~III (1206.5\AA) and Al~III (1854.7\AA) are also
evident at $-20$ and $-15~{\rm km~s}^{-1}$,
respectively.\footnote{Archive exposure identifications: Z165040CM and
Z165040DM.} We propose that these high ionization components arise from
collisionally ionized gas behind the shock front. (Ultraviolet
radiation from the shock front also helps to increase the ionization of
the downstream gas.) The width of the C~IV feature shown in
Fig.~\ref{CIV} reflects the effects of thermal doppler broadening,
instrumental smearing, and the change in velocity as the gas cools to
the lowest temperature that holds any appreciable C~IV.
\subsubsection{Properties of the Shock}\label{shock_properties}
We return to our conjecture that the preshock gas flow is being
intercepted by an obstacle at $v\approx 0~{\rm km~s}^{-1}$, and thus the
front itself is at this velocity. (While this assumption is not backed
up by independent evidence, it is nevertheless a basic premise behind
our relating the atomic absorption line data to our interpretation in
\S\ref{warm_formation} and \S\ref{cool_formation} of how the H$_2$ in
Component~1 is formed in a region where there is a large compression and
a temperature that is considerably lower than that of the immediate
postshock gas.) The fact that the C~IV feature does not appear at
\onequarter\ times that of the high velocity (preshock) C~II and N~II
features indicates that the compression ratio is less than the value 4.0
for strong shocks with an adiabatic index $\gamma=5/3$. This is
probably a consequence of either the ordinary or Alfv\'en Mach numbers
(or both) not being very high. For example, if the preshock magnetic
field and density were 5$\mu$G and $n_0=0.1~{\rm cm}^{-3}$, the Alfv\'en
speed would be 29~km~s$^{-1}$. For $T=20,000$K, the ordinary sound
speed would be 21~km~s$^{-1}$, and under these conditions the
compression ratio would be only 2.67 [cf. Eq.~2.19 of Draine \& McKee
\markcite{187} (1993)] if the magnetic field lines are perpendicular to
the shock normal. This value is close to the ratio of velocities of the
preshock and postshock components, $(-94~{\rm km~s}^{-1})/(-36~{\rm
km~s}^{-1})=2.6$. The immediate postshock temperature would be about
$2.3\times 10^5$K.
Our simple picture of a shock that is moderated by a transverse magnetic
field adequately explains the velocity difference between the two atomic
components, but it fails when we try to fit the kinematics of the much
cooler gas where we find H$_2$. If we follow the material in the
postshock flow to the point that radiative cooling has lowered the
temperature to that of the preshock gas or below, we expect to have a
final compression ratio equal to 3.7, i.e., the number that we would
expect for an ``isothermal shock'' [cf. Eq.~2.27 of Draine \& McKee
\markcite{187} (1993)]. This limited amount of compression would mean
that the cool, H$_2$-bearing gas would appear at a velocity of
$(-94~{\rm km~s}^{-1})/3.7$ = $-25~{\rm km~s}^{-1}$, a value that is
clearly inconsistent with what we observe.
A resolution of the inconsistency between the kinematics noted above and
the theoretical picture of a shock dominated by magnetic pressure could
be obtained if, instead of having the initial magnetic field lines
perpendicular to the shock normal, the field orientation is nearly
parallel to the direction of the flow. (Intuitively, this arrangement
seems more plausible, since the field lines are likely to be dragged
along by the gas.) The picture than can then evolve to the more complex
situation where there is a ``switch-on'' shock, giving an initial
moderate compression and a sudden deflection of the velocity flow and
direction of the field lines. As described by Spitzer \markcite{2832,
33} (1990a, b), this phase may then be followed by a downstream
``switch-off'' shock that redirects the flow and field lines to be
perpendicular to the front and allows further compression of the gas up
to values equal to the square of the shock's ordinary Mach number, i.e.,
the compression produced by a strong shock without a magnetic field.
In order to obtain a solution for a switch-on shock, one must satisfy
the constraint that the Alfv\'en speed must be greater than slightly
more than half of the shock speed [cf. Eq. 2.21 of Draine \& McKee
\markcite{187} (1993)]. Thus, we must at least double the Alfv\'en
speed of the previous example by either raising the preshock magnetic
field, lowering the density, or both. If this speed equalled $58~{\rm
km~s}^{-1}$, the compression ratio in the switch-on region should be
$(94/58)^2=2.6$, i.e., the square of the Alfv\'en Mach number, a value
that is again very close to our observed ratio of gas velocities on
either side of the front. [There is a complication in deriving a
compression ratio from an observation taken at some arbitrary viewing
direction through a switch-on shock. Behind the front, the gas acquires
a velocity vector component that is parallel to the front. For an
inclined line of sight, this component can either add to or subtract
from the projection of the component perpendicular to the front, which
is the quantity that must be compared to the (again projected) preshock
velocity vector when one wants to obtain a compression ratio. However
in our situation it seems reasonable to suppose that a wind from
$\zeta$~Ori is the ultimate source of high velocity gas, and this in
turn implies that the shock front is likely to be nearly perpendicular
to the line of sight.] While this picture is still rather speculative,
we will adopt the view that, through the mechanism of the switch-off
shock, the magnetic fields do not play a significant role in limiting
the amount of compression at the low temperatures where H$_2$ could
form.
One additional piece of information is a limit on the preshock density
$n_0$. Cowie, et al. \markcite{1151} (1979) obtained an upper limit for
the electron density $n_{\rm e}<0.3~{\rm cm}^{-3}$ from the lack of a
detectable absorption feature from C~II in an excited fine-structure
level (assuming $T=10^4$K). Since there is virtually no absorption seen
for lines of N~I or O~I at the high velocities in front of the shock, we
can be confident that the hydrogen is almost fully ionized and thus the
limit for $n_{\rm e}$ applies to the total density. For the purposes of
argument in the discussions that follow, we shall adopt a value
$n_0=0.1~{\rm cm}^{-3}$, as we have done earlier.
\subsection{Formation of H$_2$ in a Warm Zone}\label{warm_formation}
\subsubsection{Reactions and their Rate Constants}\label{reactions}
In the light of evidence from the atomic lines that a standing shock may
be present, we move on to explore in a semiquantitative way the
prospects that H$_2$ forming behind this front could explain our
observations. For several reasons, we expect that an initial zone where
$T \gtrsim 10^4$K will produce no appreciable H$_2$. At these
temperatures the gas is mostly ionized, and for $T > 18,000$K collisions
with electrons will dissociate H$_2$ very rapidly (Draine \& Bertoldi,
in preparation). Furthermore, the column density of material at
$T>6500$K is not large because the cooling rate is high. As soon as the
gas has reached 6500K, there is a significant, abrupt reduction in the
cooling rate while there is still some heating of the gas by ionizing
radiation produced by the much hotter, upstream material. These effects
create a plateau in the general decrease of temperature with postshock
distance [see Fig.~3 of Shull \& McKee \markcite{1800} (1979)].
The 6500K plateau, extending over a length of approximately $2\times
10^{16}n_0^{-1}{\rm cm}$, seems to be a favorable location for
synthesizing the initial contribution of H$_2$ that we could be viewing
in the upper $J$ levels. Its velocity with respect to much cooler gas
should be about $(-36~{\rm km~s}^{-1})\times (6500{\rm K})/(2.3\times
10^5{\rm K})=-1.0~{\rm km~s}^{-1}$ if the conditions are approximately
isobaric. This velocity difference is consistent, to within the
observational errors, with the shift between the peaks at $J=5$ and
$J=0$, with the latter emphasizing molecules in the material that has
cooled much further and come nearly to a halt. Considering that the
fractional ionization over the temperature range $6500 > T > 2000$K is
$0.5\gtrsim n_{\rm e}/n_{\rm H}\gtrsim 0.03$ \markcite{1800} (Shull \&
McKee 1979), we anticipate that potentially important sources of H$_2$
arise from either the formation of a negative hydrogen ion,
\begin{eqnarray}\label{c8}
{\rm H} + e&\rightarrow &{\rm H}^- + h\nu\nonumber\\
C_{\ref{c8}}&=&1.0\times 10^{-15}T_3\exp(-T_3/7)~{\rm cm}^3{\rm s}^{-1}
\end{eqnarray}
($T_3$ is the gas's temperature in units of $10^3$K) followed by the
associative detachment,
\begin{eqnarray}\label{c9}
{\rm H}^- + {\rm H}&\rightarrow &{\rm H}_2 + e\nonumber\\
C_{\ref{c9}}&=&1.3\times 10^{-9}{\rm cm}^3{\rm s}^{-1}
\end{eqnarray}
or the production of H$_2^+$ by radiative association,
\begin{eqnarray}\label{c2}
{\rm H} + {\rm H}^+&\rightarrow &{\rm H}_2^+ + h\nu\nonumber\\
C_{\ref{c2}}&=&4.1\times 10^{-17}{\rm cm}^3{\rm s}^{-1}
\end{eqnarray}
followed by its reaction with neutral atoms,
\begin{eqnarray}\label{c3}
{\rm H}_2^+ + {\rm H}&\rightarrow &{\rm H}_2 + {\rm H}^+\nonumber\\
C_{\ref{c3}}&=&1.0\times 10^{-10}{\rm cm}^3{\rm s}^{-1}
\end{eqnarray}
\markcite{2611, 3272} (Black 1978; Black, Porter, \& Dalgarno 1981).
The rate constants for the above reactions (plus the destruction
reactions \ref{c10} and \ref{c5} below) are the same as those adopted by
Culhane \& McCray \markcite{3157} (1995) in their study of H$_2$
production in a supernova envelope. Later, as the gas becomes cooler,
denser and mostly neutral, we expect that the formation of H$_2$ on the
surfaces of dust grains,
\begin{eqnarray}\label{cg}
2{\rm H} + {\rm grain}&\rightarrow & {\rm H}_2 + {\rm grain}\nonumber\\
\lbrack {\rm applicable~to}~n({\rm H})^2\rbrack
~~C_{\ref{cg}}&=&{10^{-16}T_3^{0.5}\over
1+1.3T_3^{0.5}+2T_3+8T_3^2}~{\rm cm}^3{\rm s}^{-1}
\end{eqnarray}
should start to become more important \markcite{3190} (Hollenbach \&
McKee 1979). We will address this possibility in
\S\ref{cool_formation}.
In order to evaluate the effectiveness of reactions \ref{c9} and
\ref{c3} in producing H$_2$ in the warm gas, we must consider the most
important destruction processes that counteract the production of the
feedstocks H$^-$ (reaction~\ref{c8}) and H$_2^+$ (reaction~\ref{c2}).
Radiative dissociation of H$^-$ by uv starlight photons (i.e., the
reverse of reaction~\ref{c8}),
\begin{eqnarray}\label{c8-1}
{\rm H}^- + h\nu&\rightarrow & {\rm H} + e\nonumber\\
\beta_{\ref{c8-1}}&=& 1.9\times 10^{-7}{\rm s}^{-1}
\end{eqnarray}
is generally the most important mechanism for limiting the eventual
production of H$_2$ in partially ionized regions of the interstellar
medium. The value for $\beta_{\ref{c8-1}}$ is adopted from an estimate
for this rate of destruction in our part of the Galaxy by Fitzpatrick \&
Spitzer \markcite{2701} (1994). Less important ways of destroying H$^-$
include recombination with protons,
\begin{eqnarray}\label{c10}
{\rm H}^- + {\rm H}^+&\rightarrow &2{\rm H}\nonumber\\
C_{\ref{c10}}&=&7\times 10^{-8}T_3^{-0.4}{\rm cm}^3{\rm s}^{-1}
\end{eqnarray}
and, of course, the production of H$_2$ (reaction~\ref{c9}). H$_2^+$ is
destroyed by the reaction with electrons,
\begin{eqnarray}\label{c5}
{\rm H}_2^+ + e&\rightarrow &2{\rm H}\nonumber\\
C_{\ref{c5}}&=& 1.4\times 10^{-7}T_3^{-0.4}{\rm cm}^3{\rm s}^{-1}
\end{eqnarray}
and the creation of H$_2$ in reaction~\ref{c3}. We can safely disregard
the interaction of H$_2^+$ with H$_2$,
\begin{eqnarray}\label{c4}
{\rm H}_2^+ + {\rm H}_2&\rightarrow &{\rm H}_3^+ + {\rm H}\nonumber\\
C_{\ref{c4}}&=& 2.1\times 10^{-9}{\rm cm}^3{\rm s}^{-1}
\end{eqnarray}
because $C_{\ref{c4}}n({\rm H}_2)\ll C_{\ref{c5}}n(e)+C_{\ref{c3}}n(H)$.
Finally, our end product H$_2$ is destroyed by photodissociation,
\begin{eqnarray}\label{c17}
{\rm H}_2 + h\nu&\rightarrow & 2{\rm H}\nonumber\\
({\rm optically~thin})~~\beta_{\ref{c17}}&=&3.4\times 10^{-11}{\rm
s}^{-1}~.
\end{eqnarray}
Our adopted general value for $\beta_{\ref{c17}}$ makes use of Jura's
\markcite{1276} (1974) calculation of $\beta_{\ref{c17}}=5.4\times
10^{-11}{\rm s}^{-1}$ for a flux of $4\pi J_\lambda=2.4\times
10^{-6}{\rm erg~cm}^{-2}{\rm s}^{-1}{\rm \AA}^{-1}$ at 1000\AA, but
rescaled to a local flux of $1.5\times 10^{-6}{\rm erg~cm}^{-2}{\rm
s}^{-1}{\rm \AA}^{-1}$ calculated by Mezger, et al. \markcite{3273}
(1982). A large fraction of this background may come from sources that
are behind or within cloud complexes containing H$_2$. If this is true,
the stellar radiation in the cores of the most important Werner and
Lyman lines is converted to radiation at other wavelengths via
fluorescence \markcite{2042} (Black \& van Dishoeck 1987), leading to a
lower value for $\beta_{\ref{c17}}$. The reduction in
$\beta_{\ref{c17}}$ caused by self shielding of material within
Component~1 is small: for $\log N({\rm H}_2)=14.52$ and $b=3~{\rm
km~s}^{-1}$ it is only 32\% \markcite{349} (Draine \& Bertoldi 1996).
We will discuss in \S\ref{reconciliation} how much the photodissociation
of H$_2$ could be increased by the gas's proximity to the hot, bright
stars in the Orion association.
\subsubsection{Expected Amount of H$_2$}\label{expected_h2}
We now investigate whether or not it is plausible that the above
reactions can produce the approximate order of magnitude of H$_2$ that
we observe in the higher $J$ levels of Component~1. For the condition
that the preshock density $n_0 = 0.1~{\rm cm}^{-1}$ (\S\ref{shock}), we
expect that the time scale for perceptible changes in temperature and
ionization when $6500 > T > 2000$K is about $4\times 10^{11}{\rm s}$, a
value that is much greater than the equilibration time scales
$\beta_{\ref{c17}}^{-1}$ for the production of H$_2$,
$[\beta_{\ref{c8-1}} + C_{\ref{c10}}n({\rm H}^+) + C_{\ref{c9}}n({\rm
H})]^{-1}$ for H$^-$, or $[C_{\ref{c5}}n(e) + C_{\ref{c3}}n({\rm
H})]^{-1}$ for H$_2^+$. Thus, for a total density $n_{\rm H} \equiv
n({\rm H}^+) + n({\rm H})$ and a fractional ionization $f=n({\rm
H}^+)/n_{\rm H}$ the density of H$_2$ at any particular location is
given by a straightforward equilibrium equation
\begin{mathletters}
\begin{equation}\label{h2_equilibrium}
n({\rm H}_2)=f(1-f)^2n_{\rm H}^2[F({\rm H}^-)+F({\rm H}_2^+)+F({\rm
grain})]/\beta_{\ref{c17}}
\end{equation}
with
\begin{equation}\label{FH-}
F({\rm H}^-)={C_{\ref{c9}}C_{\ref{c8}}\over \beta_{\ref{c8-1}}/n_{\rm H}
+ C_{\ref{c10}}f + C_{\ref{c9}}(1-f)}~,
\end{equation}
\begin{equation}\label{FH2+}
F({\rm H}_2^+)={C_{\ref{c3}}C_{\ref{c2}}\over C_{\ref{c5}}f +
C_{\ref{c3}}(1-f)}~,
\end{equation}
and
\begin{equation}\label{Fgrain}
F({\rm grain})=C_{\ref{cg}}/f
\end{equation}
\end{mathletters}
In order to make an initial estimate for the amount of H$_2$ that could
arise from the warm, partly ionized gas, we must evaluate the integral
of the right-hand side of Eq.~\ref{h2_equilibrium} through the relevant
part of the cooling, postshock flow. The structure of this region is
dependent on several parameters that are poorly known and whose effects
will be discussed in \S\ref{reconciliation}. As a starting point,
however, we can define a template for the behavior of $f$, $n_{\rm H}$
and $T$ with distance by adopting the information displayed by Shull \&
McKee \markcite{1800} (1979) for a 100~${\rm km~s}^{-1}$ shock with
$n_0=10~{\rm cm}^{-3}$ and solar abundances for the heavy elements
(their Model E displayed in Fig.~3). To convert to our assumed
$n_0=0.1~{\rm cm}^{-3}$, we scale their densities $n({\rm H})$ and
$n({\rm H}^+)$ down by a factor of 100 and the distance scale up by the
same factor.
Over all temperatures, we discover that $F({\rm H}_2^+)$ is at least 100
times smaller than $F({\rm H}^-)$, and hence this term is not
significant for our result. $F({\rm grain})$ is negligible compared to
$F({\rm H}^-)$ at high temperatures, but its importance increases as the
temperatures decrease: the two terms equal each other at $T=2000$K, and
$F({\rm grain})=3.7F({\rm H}^-)$ at 1000K. Within the $F({\rm H}^-)$
term, the terms for photodestruction and recombination with H$^+$ in the
denominator are about equal at $T=6500$K, but the photodestruction
becomes much more important at lower temperatures.
The integral of the predicted $n({\rm H}_2)$ (Eq.~\ref{h2_equilibrium})
over a path that extends down to $T=2000$K equals $3.6\times 10^{13}{\rm
cm}^{-2}$. This value is substantially lower than the amount of H$_2$
that we observed in the higher $J$ levels in Component~1
[$\sum_{J=2}^5N({\rm H}_2)=2\times 10^{14}{\rm cm}^{-2}$; see
Table~\ref{comp_summary}].
\subsubsection{Ways to Reconcile the Expected and Observed
H$_2$}\label{reconciliation}
There are several effects that can cause significant deviations from the
simple prediction for $N({\rm H}_2)$ given above. First, if we accept
the principle that the origin of the preshock flow at $-94~{\rm
km~s}^{-1}$ is from either a stellar wind produced by $\zeta$~Ori (plus
perhaps other stars in the association) or some explosive event in
Orion, we must then acknowledge that the H$_2$ production zone is
probably not very distant from this group of stars that produce a very
strong uv flux. As a consequence, we must anticipate that
$\beta_{\ref{c17}}$ could be increased far above that for the general
interstellar medium given in Eq.~\ref{c17}. Eq.~\ref{h2_equilibrium}
shows that this will give a reduction in the expected yield of H$_2$ in
direct proportion to this increase. (For a given enhancement of
$\beta_{\ref{c17}}$, we expect that the increase in $\beta_{\ref{c8-1}}$
will be very much less because the cross section for this process is
primarily in the visible part of the spectrum where the contrast above
the general background is relatively small.) Working in the opposite
direction, however, is the fact that the stars' Lyman limit fluxes will
supplement the ionizing radiation produced by the hot part of the shock
front, thus providing heating and photoionization rates above those
given in the model. The resulting higher level of $f$ and the increase
of the length of the warm gas zone will result in an increase in the
expected $N({\rm H}_2)$.
To see how important these effects might be, we can make some crude
estimates for the relevant increases in the uv fluxes. In the vicinity
of 1000\AA\, i.e., the spectral region containing the most important
transitions that ultimately result in photodissociation of H$_2$, the
fluxes from $\epsilon$ and $\sigma$~Ori at the Earth are $5.6\times
10^{-8}$ and $1.8\times 10^{-8}{\rm erg~cm}^{-2}{\rm s}^{-1}{\rm
\AA}^{-1}$, respectively \markcite{1043} (Holberg et al. 1982). We can
assume that other very luminous stars that might make important
contributions, such as $\delta$, $\zeta$, $\kappa$ and $\iota$~Ori, have
uv fluxes consistent with that of $\epsilon$~Ori after a scaling
according to the differences in visual magnitudes. The probable
distance of the H$_2$ from the stars is probably somewhere in the range
60 to 140 pc, as indicated by various measures of the transverse
dimensions of shell-like structures seen around the Orion association
\markcite{2674} (Goudis 1982) (and assuming that the Orion association
is at a distance of 450 pc from us). If we compare the far-uv
extinction differences for $\delta$ and $\epsilon$~Ori reported by
Jenkins, Savage \& Spitzer \markcite{1063} (1986) to these stars' color
excesses E(B$-$V) = 0.075, we infer from the uv extinction formulae of
Cardelli, Clayton \& Mathis \markcite{3280} (1989) that $R_{\rm V}$=4.6
and, again using their formulae, that $A_{\rm 1000\AA}=0.96~{\rm Mag.}$
In the absence of such extinction, these plus the other stars should
produce a net flux $F_{\rm 1000\AA}=1.0\times 10^{-5}r_{100}^{-2}~{\rm
erg~cm}^{-2}{\rm s}^{-1}{\rm \AA}^{-1}$, where $r_{100}$ is the distance
away from the stars divided by 100 pc. With $r_{100}=1$,
$\beta_{\ref{c17}}$ is enhanced over the value in Eq.~\ref{c17} by a
factor of 7.
Stars in the Orion association produce about $3.8\times 10^{49}$ Lyman
limit photons ${\rm s}^{-1}$, and only a small fraction of this flux is
consumed by the ionization of hydrogen in the immediate vicinity of the
stars \markcite{3279} (Reynolds \& Ogden 1979). From this estimate, one
may conclude that the ionizing flux of $\sim 10^6{\rm
photons~cm}^{-2}{\rm s}^{-1}$ radiated by the immediate postshock gas
\markcite{1800} (Shull \& McKee 1979) could be enhanced by a factor
approaching $30r_{100}^{-2}$, thus increasing the thickness of the
region over which there is a significant degree of ionization and
heating.
Another parameter that can influence the length of the zone where
reactions \ref{c9} and \ref{c3} are important is the relative abundances
of heavy elements. Here, the cooling is almost entirely from the
radiation of energy by forbidden, semi-forbidden and fine-structure
lines from metals -- see Fig.~2 of Shull \& McKee \markcite{1800}
(1979). If these elements are depleted below the solar abundance ratio
because of grain formation, the length of the warm H$_2$ production zone
must increase \markcite{3319} (Shull \& Draine 1987). It is unlikely
that the grains will been completely destroyed as they passed through a
$90~{\rm km~s}^{-1}$ shock \markcite{2783} (Jones et al. 1994).
Finally, it is important to realize that the outcome for $N({\rm H}_2)$
should scale roughly in proportion to $n_0^2$. The reason for this is
that over most of the path, we found that $\beta_{\ref{c8}}/n_{\rm H}$
was the most important term within denominator of the dominant
production factor $F({\rm H}^-)$. This in turn makes $n({\rm H}_2)$
scale in proportion to $n_{\rm H}^3$ almost everywhere (note that $\int
n_{\rm H}dl$ does not vary with $n_0$).
\subsubsection{Independent Information from an Observation of
Si~II$^*$}\label{siII}
It is important to look for other absorption line data that can help to
narrow the uncertainties in the key parameters discussed above. One
such indicator is the column density of ionized silicon in an excited
fine-structure level of its ground electronic state (denoted as
Si~II$^*$). This excited level is populated by collisions with
electrons, and the balance of this excitation with the level's radiative
decay (and collisional de-excitations) results in a fractional abundance
\begin{equation}\label{siII*ratio}
\log\left( {N({\rm Si~II}^*)\over N({\rm Si~II})}\right) = \log n(e) -
0.5\log T_3 - 2.54
\end{equation}
\markcite{2389} (Keenan et al. 1985). In conditions where the hydrogen
is only partially ionized, we expect that $n({\rm Si}^{++})/n{(\rm
Si}^+)$ will be much less than $n({\rm H}^+)/n({\rm H})$ because ionized
Si has a larger recombination coefficient \markcite{111} (Aldrovandi \&
P\'equignot 1973) and a smaller photoionization cross section
\markcite{1874} (Reilman \& Manson 1979) (its ionization potential of
16.34~eV is also greater than that of hydrogen). Thus, for situations
where $f$ is not very near 1.0, it is reasonably safe to assume that
virtually all of the Si is singly ionized. If, for the moment, we also
assume that the Si to H abundance ratio is equal to the solar value, we
expect that
\begin{equation}\label{siII*abund}
n({\rm Si~II}^*)=10^{-7}fn_{\rm H}^2T_3^{-1/2}~{\rm cm}^{-3}~.
\end{equation}
As we did for H$_2$, we can integrate the expression for $n({\rm
Si~II}^*)$ through the modeled cooling zone to find an expectation for
the column density $N({\rm Si~II}^*)=5.8\times 10^{11}{\rm cm}^{-2}$.
A very weak absorption feature caused by Si~II$^*$ at approximately the
same velocity as our Component~1 can be seen in a medium resolution HST
spectrum\footnote{Archive exposure identification Z165030GT} of
$\zeta$~Ori~A that covers the very strong transition at 1264.730\AA.
Our measurement of this line's equivalent width was $13.4\pm 3$m\AA,
leading to $N({\rm Si~II}^*)=1.0\times 10^{12}{\rm cm}^{-2}$, a
result\footnote{From the equivalent width of 3.4m\AA\ (no error stated)
for the 1194.49\AA\ line of Si~II$^*$ reported by Drake \& Pottasch
\markcite{1803} (1977), one obtains a somewhat lower value, $\log N({\rm
Si~II}^*)=11.6$} that is almost twice the prediction stated above.
From our result for Si~II$^*$, we conclude that the combined effect of
the stars' ionizing flux and a possible increase in $n_0$ over our
assumed value of $0.1~{\rm cm}^{-3}$ could raise $\int n(e)n_{\rm H}dl$
by not much more than a factor of two. However, we have no sensitivity
to the possibility that metals are depleted since the decrease in the
abundance of Si would be approximately compensated by the increase in
the characteristic length for the zone to cool (assuming the primary
coolants and Si are depleted by about the same amount). Thus, it is
still possible that the our calculation based on a model with solar
abundances will result in an inappropriate (i.e., too low) value for the
expected $N({\rm H}_2)$.
\subsubsection{Coupling of the Rotational Temperature to
Collisions}\label{coupling}
One remaining task is to establish that the conditions in the
H$_2$-formation zone are such that collisional excitation of the higher
$J$ levels can overcome the tendency for the molecules to move to other
states through either radiative decay or the absorption of uv photons.
Tawara et al. \markcite{3266} (1990) summarize the collision cross
sections as a function of energy for excitations $J=0\rightarrow 2$ and
$J=1\rightarrow 3$ by electrons. We calculate that these cross sections
should give a rate constant of about $1\times 10^{-10}T_3^{3/2}{\rm
cm}^3{\rm s}^{-1}$ for $T_3\gtrsim 2$. Thus, for $n_e\gtrsim 1~{\rm
cm}^{-3}$ and $T_3\approx 5$ the collisions can dominate over radiative
transition rates of about $3\times 10^{-10}~{\rm s}^{-1}$ for $J=2$ and
3. To collisionally populate $J=5$ which can decay at a rate of
$1\times 10^{-8}~{\rm s}^{-1}$ to $J=3$, we would need to have
$n_e\gtrsim 10~{\rm cm}^{-3}$ just to match the radiative rate, assuming
that the collisional rate constant is not significantly lower than what
we calculated for $J=2$ and 3.
\subsection{Further Formation of H$_2$ in a Cool
Zone}\label{cool_formation}
Additional formation of H$_2$ molecules probably takes place in gas that
has cooled well below 2000K and is nearly fully recombined. We are
unable to distinguish between this gas and the material that was
originally present as an obstruction to the high velocity flow to create
the bow shock. To obtain an approximate measure of the total amount of
cool, mostly neutral gas, we determined $\int N_a(v)dv$ over the
velocity interval $-10 < v < +5~{\rm km~s}^{-1}$ for the N~I line at
1134.165\AA\ which does not appear to be very strongly saturated. The
relative ionization of nitrogen should be close to that of hydrogen
\markcite{1927} (Butler \& Dalgarno 1979), and this element is not
strongly depleted in the interstellar medium \markcite{14} (Hibbert,
Dufton, \& Keenan 1985). Our conclusion that $\log N({\rm N~I})=14.78$
leads to an inferred value for $\log N({\rm H})$ equal to 18.73.
According to a model\footnote{This model is not exactly applicable to
our situation, since it has a compression ratio of 4 instead of our
value of 2.6} for a 90~km~s$^{-1}$ shock of Shull \& McKee
\markcite{1800} (1979), $\log N({\rm H})=18.40$ is the amount of
material that accumulates by the time $T$ reaches 1000K. Hence, from
our measure of the total N(H) (but indeed an approximate one), we
estimate that the amount of gas at $T<1000$K is about comparable to that
at the higher temperatures.
An insight on the conditions in the cool, neutral zone is provided by
the populations of excited fine-structure levels of C~I. Jenkins \&
Shaya \markcite{1034} (1979) found that $\log p/k=4.1$ in the part of
our Component~1 that carries most of the neutral carbon atoms. If we
take as a representative temperature $T=300$K, the local density should
be $n_{\rm H}\approx 40~{\rm cm}^{-3}$ and $C_{\ref{cg}}=1.8\times
10^{-17}{\rm cm}^3{\rm s}^{-1}$. With the general interstellar value
for $\beta_{\ref{c17}}$, we expect an equilibrium concentration $n({\rm
H}_2)/n_{\rm H}=2\times 10^{-5}$. When we multiply this number by our
estimate $N({\rm H})=3\times 10^{18}{\rm cm}^{-2}$, we find that we
should expect to observe $\log N({\rm H}_2)=13.8$, a value that, after
considering the crudeness of our calculations, is acceptably close to
our actual measurements of H$_2$ in $J=0$ and 1, the levels that arise
primarily from the coolest gas. If $\beta_{\ref{c17}}$ is significantly
enhanced by radiation from the Orion stars (\S\ref{reconciliation}), we
would then have difficulty explaining the observations.
\section{Summary}\label{summary}
We have observed over 50 absorption features in the Lyman and Werner
bands of H$_2$ in the uv spectrum of $\zeta$~Ori~A. An important aspect
of our spectrum is that it had sufficient resolution to detect in one of
the velocity components (our Component~1) some important changes in the
one-dimensional velocity distributions of the molecules with changing
rotational excitation $J$. The main focus of our investigation has been
to find an explanation for this result, since it is a departure from the
usual expectation that the rotational excitation comes from uv pumping,
an effect that would make the profiles look identical. A smaller amount
of broadening for higher $J$ is also seen for Component~3, a component
that has much more H$_2$ than Component~1.
In Component~1, we have found that as $J$ increases from 0 to 5 there is
a steady increase in the width of the velocity profile, combined with a
small drift of the profile's center toward more negative velocities. We
have shown that the pumping lines are not strong enough to make
differential shielding in the line cores a satisfactory explanation for
the apparent broadening of the $J$ levels that are populated by such
pumping. While one might resort to an explanation that unseen,
additional H$_2$ could be shielding light from a uv source (or sources)
other than $\zeta$~Ori, we feel that this interpretation is implausible,
and, moreover, it does not adequately explain the differences that we
see between the profiles of $J=0$ and 1.
One could always argue that the absorption that we identify as
Component~1 is really a chance superposition of two, physically
unrelated regions that have different rotation temperatures and central
velocities.\footnote{There is a good way to illustrate how this creates
the effect that we see in Component~1. Imagine that we recorded the
H$_2$ lines at a resolution that was so low that Complexes 1 and 3 were
not quite resolved from each other. We would then see features that got
broader with increasing $J$, and their centers would shift toward the
left. This is qualitatively exactly the same effect that we see on a
much smaller velocity scale within Component~1.} While not impossible,
this interpretation is unattractive. It requires a nearly exact
coincidence of the two regions' velocity centers to make up a component
that stands out from the rest of the H$_2$ absorption and, at the same
time, shows smoothly changing properties with $J$ at our velocity
resolution. We feel that the most acceptable interpretation is the
existence of a coherent region of gas that, for some particular reason
that has a rational explanation, has systematic changes in the
properties of the material within it that could produce the effects that
we observe. One phenomenon that fits this picture is the organized
change in temperature and velocity for gas that is cooling and
recombining in the flow behind a shock front. The excess width of the
higher $J$ lines could arise from both higher kinetic temperatures and
some velocity shear caused by the steady compression of the gas as it
cools.
Our concept of a shock is supported by evidence from atomic absorption
lines in the spectrum of $\zeta$~Ori~A. We see features that we can
identify with both the preshock medium and the immediate postshock gas
that is very hot. If this interpretation is correct (and not a
misguided attempt to assign a significant relationship between atomic
components with different levels of ionization at very different
velocities), we can use the atomic features to learn much more about the
shock's general properties.
We start with the expectation that the coolest molecular material, that
which shows up in the lowest $J$ levels at $v=-1~{\rm km~s}^{-1}$, is in
a region containing gas that is very strongly compressed and thus nearly
at rest with respect to the shock front. The atomic features of C~II
and N~II that we identify with the preshock flow appear at a velocity of
$-94~{\rm km~s}^{-1}$ with respect to the cool H$_2$. Hence this is the
value that we normally associate with the ``shock velocity.'' This
preshock gas also shows up in the lines of C~III, N~III, Si~III, S~III
and Al~III. An upper limit to its temperature $T<20,000$K results from
the apparent lack of C~IV that would arise from collisional ionization
at slightly higher temperatures. The temperature could be as low as
typical H~II region temperatures ($8,000<T<12,000$K) if uv photons are
the main source of ionization.
Absorption features from more highly ionized gas at around $-36~{\rm
km~s}^{-1}$ that show up in the C~IV doublet indicate that the initial
compression factor is only 2.6, a value that is significantly lower than
the usual 4.0 expected for a shock with a high Mach number. Reasonable
numbers for the preshock density, temperature and magnetic field
strength ($0.1~{\rm cm}^{-3}$, 20,000K and $10\mu$G) can explain this
lower compression factor and establish a switch-on shock. However,
except for a limit $n_0<0.3~{\rm cm}^{-3}$ that comes from the lack of
C~II$^*$ absorption, we have no independent information that can
distinguish between these somewhat arbitrary assignments and other,
equally acceptable combinations.
To overcome the problem that there seems to be a low initial compression
of the gas but eventually the densities are allowed to increase to the
point that H$_2$ can form, we invoke the concept of an oblique magnetic
shock, where the theoretical models outline the existence of two
discontinuities, a ``switch on'' front and a ``switch off'' front.
However, we do not attempt to explore the validity of this picture in
any detail.
Neglecting complications that are introduced by the oblique shock
picture, we expect that as the gas flow cools to temperatures
significantly below the immediate postshock temperature, it decelerates
and begins to show ions that have ionization potentials below that of
C~IV, such as Si~III (at $-20~{\rm km~s}^{-1}$) and Al~III ($-15~{\rm
km~s}^{-1}$). At temperatures somewhat below $10^4$K, the gas should be
still partially ionized and at a density $n_{\rm H}>60n_0$. These
conditions favor the production of H$_2$ via the formation of H$^-$ and
its subsequent reaction with H to produce the molecule plus an electron,
rather than the usual formation on grains that dominates in cool clouds.
Our observation of the Si~II$^*$ absorption feature at a velocity near
$0~{\rm km~s}^{-1}$ indicates that it is unlikely that $n_0\ll 0.1~{\rm
cm}^{-3}$, and thus the density in the molecule forming region is high
enough to insure that the photodetachment of H$^-$ does not deplete this
feed material to the point that the expected abundance of H$_2$ is well
below the amount that we observe.
It is possible that the uv flux from the Orion stars could enhance the
H$_2$ photodissociation rate $\beta_{\ref{c17}}$ to a level that is far
above that which applies to the average level in our part of the Galaxy.
If this is true, then we must make a downward revision to our prediction
that $\log N({\rm H}_2)=13.56$. At the same time, however, additional
ionizing photons from the stars could lengthen the warm H$_2$ production
zone, and this effect may gain back a large amount of the lost H$_2$. A
reduction in the metal abundance in the gas may also lengthen the zone,
giving a further increase in the expected H$_2$.
As the gas compresses further and becomes almost fully neutral, the
H$^-$ production must yield to grain surface reactions as the most
important source of molecules. Using information from the C~I
fine-structure excitation, we can infer that the density in the cool gas
is sufficient to give $n({\rm H}_2)/n_{\rm H}$ equal to about half of
what we observed, if we assume that most of the H$_2$ in the $J=0$ and 1
states comes from the cool region.
On the basis of a diverse collection of evidence and some rough
quantitative calculations, we have synthesized a general description of
the cooling gas behind the shock and have shown that H$_2$ production
within it could plausibly explain the unusual behavior in the profiles
that we observed. Obviously, if one had the benefit of detailed shock
models that incorporated the relevant magnetohydrodynamic, atomic and
molecular physics, it would be possible to substantiate this picture (or
perhaps uncover some inconsistencies?) and narrow the uncertainties in
various key parameters. Also, more detailed models should allow one to
address certain questions that are more difficult to answer, such as
whether or not more complex chemical reactions play an important role in
modifying the production of H$_2$; we have identified only a few good
prospects. For instance, is there enough Ly-$\alpha$ radiation produced
in the front (or in the H~II region ahead of it) to make the formation
by excited atom radiative association (i.e., H($n=1$) + H($n=2$)
$\rightarrow$ H$_2$ + $h\nu$) an important additional production route
\markcite{1923} (Latter \& Black 1991)? On the observational side, we
expect to see very soon a vast improvement in the amount and quality of
data on atomic absorption lines toward $\zeta$~Ori~A. Very recently,
the GHRS echelle spectrograph on HST obtained observations of various
atomic lines at extraordinarily good resolution and S/N.
\acknowledgments
Support for flying IMAPS on the ORFEUS-SPAS-I mission and the research
reported here came from NASA Grant NAG5-616 to Princeton University.
The ORFEUS-SPAS project was a joint undertaking of the US and German
space agencies, NASA and DARA. The successful execution of our
observations was the product of efforts over many years by engineering
teams at Princeton University Observatory, Ball Aerospace Systems Group
(the industrial subcontractor for the IMAPS instrument) and Daimler-Benz
Aerospace (the German firm that built the ASTRO-SPAS spacecraft and
conducted mission operations). Most of the development of the data
reduction software was done by EBJ shortly after the mission, while he
was supported by a research award for senior U.S. scientists from the
Alexander von Humboldt Foundation and was a guest at the Institut f\"ur
Astronomie und Astrophysik in T\"ubingen. We are grateful to B.~T.
Draine for valuable advice about the different alternatives for
interstellar shocks. B.~T. Draine, L. Spitzer, and J.~H. Black supplied
useful comments on an early draft of this paper. Some of the
conclusions about atomic absorption features are based on observations
made with the NASA/ESA Hubble Space Telescope, obtained from the data
archive at the Space Telescope Science Institute. STScI is operated by
AURA under NASA contract NAS 5-26555.
\newpage
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\section{Introduction}
At the heart of many phenomena in condensed matter physics is the interplay
between the charge and spin degrees of freedom of interacting electrons. The
impact of the magnetic ordering and fluctuations on the charge correlations or
the effect of the phase separation on the spin correlations, for example, are
important issues in the study of strongly correlated electron systems. One of
the simplest scenarios in which these questions can be formulated transparently
and investigated systematically comprises two successive approximations of the
Hubbard model with very strong on-site repulsion. They are known under the
names $t$-$J$ and $t$-$J_z$ models.\cite{ZR88
Here we consider a one-dimensional (1D)
lattice.\cite{HM91,AW91,PS91,OLSA91,KY90,BBO91,THM95,YO91} In both models
the assumption is that the Hubbard on-site repulsion is so strong that double
occupancy of electrons on any site of the lattice may as well be prohibited
completely. This constraint is formally incorporated into the two models by
dressing the fermion operators of the standard hopping term with projection
operators:
\begin{equation}\label{I.1}
H_t = -t\sum_{\sigma =\uparrow ,\downarrow }\sum_{l} \left\{
\tilde c_{l,\sigma }^{\dagger }\tilde c_{l+1,\sigma }\ +
\tilde c_{l+1,\sigma }^{\dagger }\tilde c_{l,\sigma }\right\}
\end{equation}
with $\tilde c_{l,\sigma } =c_{l,\sigma }(1-n_{l,-\sigma })$, $n_l=n_{l,\uparrow
}+n_{l,\downarrow }$, $n_{l,\sigma }=c_{l,\sigma }^{\dagger }c_{l,\sigma }$.
In the $t$-$J$ model the Hubbard interaction is further taken into account by an
isotropic antiferromagnetic exchange coupling between electrons on
nearest-neighbor sites:
\begin{equation}\label{I.2}
H_{t\text{-}J} = H_t +
J\sum_{l}\left\{{\bf S}_l\cdot {\bf S}_{l+1}-\frac 14 n_ln_{l+1}\right\}
\end{equation}
with $S_l^z=\frac 12(n_{l,\uparrow}-n_{l,\downarrow })$, $S_l^{+}=
\tilde c_{l,\uparrow }^{\dagger }\tilde c_{l,\downarrow }$, and $S_l^{-}=
\tilde c_{l,\downarrow}^{\dagger }\tilde c_{l,\uparrow }$.
In the $t$-$J_z$ model the isotropic exchange interaction is replaced by an
Ising coupling:
\begin{equation}\label{I.3}
H_{t\text{-}J_z}=H_t +
J_z \sum_{l} \left\{ S_l^z S_{l+1}^z - \frac 14 n_l n_{l+1} \right\}.
\end{equation}
The absence of spin-flip terms in $H_{t\text{-}J_z}$ introduces additional
invariants (not present in $H_{t\text{-}J}$) for the spin configurations of
eigenstates and thus alters the relationship between charge and spin
correlations considerably. All results presented here will be for
one-quarter-filled bands ($N_e=N/2$ electrons on a lattice of $N$ sites).
For weak exchange coupling, both models have a Luttinger liquid ground
state. For stronger coupling, electron-hole phase separation sets in. Phase
separation is primarily a transition of the charge degrees of freedom. Here it
is driven by an interaction of the spin degrees of freedom, and it is
accompanied by a magnetic transition. The degree of spin ordering in the
phase-separated state depends on the presence ($t\text{-}J$) or absence
$(t\text{-}J_z)$ of spin-flip terms in the interaction.
Detailed information on the charge and spin fluctuations in $H_{t\text{-}J}$ and
$H_{t\text{-}J_z}$ is contained in the dynamic charge structure factor
$S_{nn}(q,\omega)$ and in the dynamic spin structure factor $S_{zz}(q,\omega)$,
i.e. in the quantity
\begin{equation}\label{I.4}
S_{AA}(q,\omega )\equiv\int\limits_{-\infty}^{+\infty }dt
e^{i\omega t}\langle A_q(t)A_{-q}\rangle,
\end{equation}
where $A_q$ stands for the fluctuation operators
\begin{equation}\label{I.5}
n_q = N^{-\frac{1}{2}} \sum_l e^{-iql}n_l,\quad
S_q^z = N^{-\frac{1}{2}} \sum_l e^{-iql}S_l^z.
\end{equation}
The degree of spin and charge ordering in the ground state is also reflected in
the equal-time charge correlation function $\langle n_ln_{l+m}\rangle$ and spin
correlation function $\langle S_l^zS_{l+m}^z \rangle$ and in their Fourier
transforms, the structure factors $S_{nn}(q)\equiv\langle n_q n_{-q} \rangle$
and $S_{zz}(q)\equiv \langle S_q^z S_{-q}^z\rangle$.
In the following we investigate the $T=0$ charge and spin fluctuations of the
two models $H_{t\text{-}J}$ and $H_{t\text{-}J_z}$ in three different regimes
with the calculational tools adapted to the situation: the limit of zero
exchange coupling (Sec. \ref{II}), the Luttinger liquid state
(Sec. \ref{Sec:LLS}), and the phase-separated state (Sec. \ref{Sec:PS}).
\section{Free lattice fermions}\label{II}
\subsection{Charge correlations and dynamics}
The tight-binding Hamiltonian (\ref{I.1}) has a highly spin-degenerate ground
state. The charge correlations are independent of the spin configurations and,
therefore, equivalent to those of a system of spinless lattice fermions,
\begin{equation}\label{II.1}
H_t^\prime = -t \sum_l \left\{
c_l^\dagger c_{l+1} + c_{l+1}^\dagger c_l
\right\}.
\end{equation}
This Hamiltonian has been well studied in the context of the 1D $s=1/2$ $XX$
model,
\begin{equation}\label{II.2}
H_{XX} = -J_\perp \sum_l \left\{ S_l^x S_{l+1}^x + S_l^y S_{l+1}^y \right\},
\end{equation}
which, for $J_\perp = 2t$, becomes (\ref{II.1}) via Jordan-Wigner
transformation.\cite{LSM61,K62} The equal-time charge correlation function of
$H_t$ (or $H_t^\prime$) exhibits power-law decay,
\begin{equation}\label{II.3}
\langle n_ln_{l+m}\rangle-\langle n_l\rangle\langle n_{l+m}\rangle =
{{\cos (\pi m)-1} \over {2\pi ^2 m^2}},
\end{equation}
and the charge structure factor has the form
\begin{equation}\label{II.4}
S_{nn}(q) - {N \over 4}\delta_{q,0} = {{|q|} \over {2\pi}}.
\end{equation}
The dynamic charge structure factor, which is equivalent to the $zz$ dynamic
spin structure factor of (\ref{II.2}) reads (for
$N\rightarrow\infty$):\cite{N67
\begin{eqnarray}\label{II.5}
S_{nn}(q,\omega )
&=&
\pi^2\delta(q)\delta(\omega)
\nonumber\\ &+&
\frac{2\Theta\biglb(\omega -2t\sin q\bigrb)
\Theta\biglb(4t\sin(q/2)-\omega\bigrb)}
{\sqrt{16t^2\sin ^2(q/2)-\omega^2}}.
\end{eqnarray}
\subsection{Spin correlations}
The charge-spin decoupling as is manifest in the product nature of the
ground-state wave functions of $H_{t\text{-}J_z}$ at $J_z/t=0^+$ and
$H_{t\text{-}J}$ at $J/t=0^+$ was shown to lead to a factorization in the spin
correlation function.\cite{PS91,PS90,OS90} We can write
\begin{equation}\label{II.6}
\langle S_l^z S_{l+m}^z\rangle = \sum_{j=2}^{m+1} C(j-1) P(m,j),
\end{equation}
where $C(m)\equiv\langle S_l^zS_{l+m}^z\rangle_{LS}$ is the correlation function
in the ground state of a system of $N_e$ localized spins with antiferromagnetic
Heisenberg $(t$-$J)$ or Ising ($t$-$J_z$) coupling, and
\[
P(m,j) \equiv \langle n_l n_{l+m}\delta_{j,N_m}\rangle, \quad
N_m \equiv \sum_{i=l}^{l+m} n_i
\]
is the probability of finding $j$ electrons on sites $l, l+1,\ldots,l+m$ with no
holes at the end points of the interval. This expression can be brought into the
form
\begin{eqnarray}\label{II.8}
\langle S_l^z S_{l+m}^z\rangle = &&{-1 \over 4N_e} \sum_{k\neq 0}
{S(k) \over \sin^2(k/2)}
\nonumber\\ && ~~
\times[D_m(k)-2D_{m-1}(k)+D_{m-2}(k)],
\end{eqnarray}
\begin{eqnarray}
S(k) &=& \sum_{j=1}^{N_e}e^{ikj}C(j), \label{II.9}
\\
D_m(k) &=& \left\langle \exp\left(-ik\sum_{l=0}^m n_l\right)\right\rangle,
\nonumber
\end{eqnarray}
where $S(k)$ for $k=(2\pi/N_e)n$, $n=0,\ldots,N_e-1$ is the static structure
factor for the localized spins, and the $D_m(k)$ are many-fermion expectation
values, which are expressible as determinants of dimension $m+1$:\cite{PS91}
\[
D_m(k)=\left|\delta_{ij}-{(1+e^{-ik}) \over 2N_e}
{\sin[\pi(i-j)/2] \over \sin[\pi(i-j)/2N_e]}\right|_{i,j=0,\ldots,m}.
\]
In $H_{t\text{-}J_z}$ we have $C(m)\!=\!{1\over 4}(-1)^m$, i.e.
$S(k)\!=\!(N_e/4)\delta_{k,\pi}$, reflecting the (invariant) alternating up-down
sequence of successive electron spins. Expression (\ref{II.8}) can
then be evaluated in closed form:
\begin{mathletters}\label{II.12}
\begin{equation}
\langle S_{l}^z S_{l + 2n }^z \rangle = \frac{(-1)^n}{2\pi^2}
\prod_{i = 1}^{n-1} P_i^2,
\end{equation}
\begin{equation}\label{statzzodd}
\langle S_{l}^z S_{l+2n+1}^z \rangle =
-\frac{1}{2}\left(
\langle S_{l}^z S_{l+2n}^z \rangle +
\langle S_{l}^z S_{l+2n+2}^z \rangle \right)
\end{equation}
\end{mathletters}
with
\[
P_i = \frac{2}{\pi}\prod_{j=1}^i
\left(1 - \frac{1}{4j^2} \right)^{-1}.
\]
The leading terms of the long-distance asymptotic expansion are\cite{note6}
\begin{eqnarray}\label{II.15}
\langle S_{l}^zS_{l+m}^z\rangle_{t\text{-}J_z}
&&
\stackrel{m\to\infty}{\longrightarrow}
\frac{A^2}{4\sqrt{2}}\frac{1}{\sqrt{|m|}}
\nonumber\\ && \hspace*{-6mm}
\times \left[
\left(1-\frac{1}{8}\frac{1}{m^2}\right)\cos \frac{m\pi}{2}
-\frac{1}{2m} \sin \frac{m\pi}{2}
\right]
\end{eqnarray}
with $A = 2^{1/12}\exp[3\zeta^\prime(-1)] = 0.64500\ldots$. The structure of
$D_m(\pi)$ is very similar to that of the $xx$ spin correlation function of
$H_{XX}$.\cite{LSM61,M68,MPS83} Its leading asymptotic term has the form
$\langle S_l^x S_{l+m}^x\rangle_{XX} \sim (A^2/2\sqrt{2})m^{-1/2}$.
In $H_{t\text{-}J}$ the spin-flip terms weaken the spin correlations at
$J/t=0^+$. The function $S(k)$ in (\ref{II.8}) is determined via (\ref{II.9})
by the spin correlation function of the 1D $s=1/2$ Heisenberg antiferromagnet
($XXX$ model). Its leading asymptotic term reads\cite{SFS89} $C(m) \sim
\Gamma(-1)^mm^{-1}(\ln m)^{1/2}$ with amplitude $\Gamma\simeq 0.125(15)$ as
estimated from finite-chain data.\cite{note9} The leading asymptotic term of the
$t$-$J$ spin correlation function inferred from (\ref{II.8}) has the
form\cite{PS90}
\begin{equation}\label{II.16}
\langle S_l^z S_{l+m}^z\rangle_{t\text{-}J}
\sim \Gamma A^2\sqrt{2}{\cos(\pi m/2) \frac{(\ln m)^{1/2}}{m^{3/2}}}.
\end{equation}
The $t$-$J$ and $t$-$J_z$ spin structure factors $S_{zz}(q)$ inferred from the
results presented here will be presented and discussed in Sec. III.E.
For an intuitive understanding of the $q=\pi$ charge density wave in the ground
state at $J_z/t=0^{+}$ and $J/t=0^{+}$, we note that the hopping term opposes
electron clustering. In the absence of the exchange term, which favors
clustering of electrons with opposite spin, the hopping effectively causes an
electron repulsion. This is reflected in the power-law decay (\ref{II.3}) of
the charge correlation function, specifically in the term which oscillates with a
period equal to twice the lattice constant ($q=4k_F=\pi$). In this state, an
electron is more likely to have a hole next to it than another electron.
How does this affect the spin correlations? Recall that the ground state of
$H_{t\text{-}J_z}$ at $J_z/t=0^{+}$ is characterized by an (invariant)
alternating spin sequence. In a perfect electron cluster this sequence would
amount to saturated N\'eel ordering ($q=\pi $), but here it is destroyed by a
distribution of holes. Spin long-range order exists only in a topological
sense. However, some amount of actual spin ordering survives by virtue of the
effective electron repulsion in the form of the algebraically decaying term
(\ref{II.15}) in the spin correlation function with a wavelength equal to four
times the lattice constant ($q=2k_F=\pi/2$).
A similar argument obtains for the $t$-$J$ model. Since its ground state at
$J/t=0^{+}$ contains all spin sequences with $S_T^z=0$, not just the alternating
ones, the resulting $q=\pi /2$ oscillations (\ref{II.16}) in the spin
correlation function decay more rapidly than in the $t$-$J_z$ case.\cite{note1}
\subsection{Spin dynamics}
Expression (\ref{II.6}) cannot be generalized straightforwardly for the
calculation of {\em dynamic} spin correlations, the principal reason being that
the number of electrons between any two lattice sites is not invariant under
time evolution. However, in the $t$-$J_z$ case we can determine the function
$\langle S_l^z(t)S_{l+m}^z\rangle$ on a slight detour. We use open boundary
conditions and write
\begin{equation}\label{II.17}
S_{l}^z = - {1 \over 2}\sigma_L \prod_{i=1}^l (-1)^{n_i} n_l, \nonumber
\end{equation}
where $\sigma_L=\pm 1$ denotes the spin direction of the leftmost particle in
the chain, which {\em is} an invariant under time evolution. The time-dependent
two-spin correlation function of the open-ended $t$-$J_z$ chain is then related
to the following many-fermion correlation function:
\begin{eqnarray*}
\langle S_l^z &(t)& S_{l+m}^z \rangle =
{1\over 4}\left\langle n_l(t)\prod_{i=1}^l (-1)^{n_i(t)}\right.
\left.\prod_{j=1}^{l+m} (-1)^{n_j} n_{l+m}\right\rangle
\nonumber \\ &=&
\left\langle c_l^{\dag}(t) c_l(t) A_1(t) B_1(t)\right.\!
A_2(t) B_2(t) \cdots A_l(t) B_l(t)
\nonumber \\ && ~~
\times A_1 B_1 A_2 B_2 \cdots A_{l+m}\!
\left.B_{l+m}c_{l+m}^{\dag}c_{l+m} \right\rangle
\end{eqnarray*}
with $ A_l\equiv c_l^{\dag} +c_l,\; B_l\equiv c_l^{\dag} -c_l$. In order to
extract the bulk behavior of $\langle S_l^z(t)S_{l+m}^z \rangle$ from this
expression, we must choose both sites $l$ and $l+m$ sufficiently far from the
boundaries.
The numerical evaluation of this function via Pfaffians
shows\cite{SVM92,SNM95,note5} that the leading long-time asymptotic term
describes uniform power-law decay, $\langle S_l^z(t) S_{l+2n}^z\rangle \sim
t^{-1/2}$, for even distances and (more rapid) oscillatory power law decay, $
\langle S_l^z(t) S_{l+2n+1}^z\rangle \sim e^{-2it}t^{-\alpha},\; \alpha\gtrsim
1$, for odd distances. Moreover, we have found compelling numerical evidence
that the relation (\ref{statzzodd}) can be generalized to time-dependent
correlation functions in the bulk limit $l\to\infty$.
From our data in conjunction with the long-distance asymptotic result
(\ref{II.15}) we predict that the leading term for large distances and long
times has the form\cite{note:t-energy}
\begin{equation}\label{tasymptjz}
\langle S_l^z(t)S_{l+m}^z\rangle_{t\text{-}J_z}
\sim {1 \over 4}{A^2/\sqrt{2} \over (m^2-4t^2)^{1/4}}
\cos\frac{\pi m}{2}.
\end{equation}
The corresponding asymptotic result in the $XX$ model is well
established:\cite{MPS83,VT78}
\begin{equation}\label{II.19}
\langle S_l^x(t)S_{l+m}^x\rangle_{XX} \sim {1 \over 4}{A^2\sqrt{2} \over
(m^2-J_\perp^2t^2)^{1/4}}. \nonumber
\end{equation}
The asymptotic behavior (\ref{tasymptjz}) of the dynamic spin correlation
function implies that the dynamic spin structure factor has a divergent
infrared singularity at $q=\pi/2$: $S_{zz}(\pi/2,\omega)_{t\text{-}J_z} \sim
\omega^{-1/2}$. Further evidence for this singularity and for a corresponding
singularity in $S_{zz}(q,\omega)_{t\text{-}J}$ will be presented in
Sec.~III.F.
\section{Luttinger liquid state}\label{Sec:LLS}
Turning on the exchange interaction in $H_{t\text{-}J}$ and $H_{t\text{-}J_z}$,
which is attractive for electrons with unlike spins and zero otherwise, alters
the charge and spin correlations in the ground state gradually over the range of
stability of the Luttinger liquid state.
In the $t$-$J_z$ model, where successive electrons on the lattice have opposite
spins, the exchange coupling counteracts the effectively repulsive force of the
hopping term and thus gradually weakens the enhanced $q=\pi$ charge and
$q=\pi/2$ spin correlations. We shall see that the repulsive and attractive
forces reach a perfect balance at $J_z/t=4^-$. Here the distribution of
electrons (or holes) is completely random. All charge pair correlations vanish
identically and all spin pair correlations too, except those between
nearest-neighbor sites. This state marks the boundary of the Luttinger liquid
phase. At $J_z/t>4$ the attractive nature of the resulting force between
electrons produces new but different charge and spin correlations in the form of
charge long-range order at $q=0^+$ (phase separation) and spin long-range order
at $q=\pi $ (antiferromagnetism).
In the $t$-$J$ model the disordering and reordering tendencies are similar, but
the exchange interaction with spin-flip processes included is no longer
uniformly attractive. At no point in parameter space do the attractive and
repulsive forces cancel each other and produce a random distribution of
electrons. A sort of balance between these forces exists at $J/t=2$, which is
reflected in the observation\cite{YO91} that the ground-state is particularly
well represented by a Gutzwiller wave function at this coupling strength. Charge
and spin correlations exhibit power-law decay at the endpoint, $J/t\simeq 3.2$,
of the Luttinger liquid phase. Here the attractive forces start to prevail on
account of sufficiently strong antiferromagnetic short-range correlations and
lead to phase separation, but the spin correlations continue to decay to zero
asymptotically at large distances.
One characteristic signature of a Luttinger liquid is the occurrence of infrared
singularities with interaction-dependent exponents in dynamic structure factors.
In the following we present direct evidence for interaction-dependent infrared
singularities in the dynamic charge and spin structure factors of
$H_{t\text{-}J_z}$ and $H_{t\text{-}J}$. We employ the recursion
method\cite{rm} in combination with techniques of continued-fraction analysis
recently developed in the context of magnetic
insulators.\cite{VM94,VZSM94,VZMS95,FKMW96}
The recursion algorithm in the present context is based on an orthogonal
expansion of the wave function $|\Psi_q^A(t)\rangle \equiv A_q(-t)|\phi\rangle$
with $A_q$ as defined in (\ref{I.5}). It produces (after some intermediate
steps) a sequence of continued-fraction coefficients
$\Delta^A_1(q),\Delta^A_2(q),\ldots$ for the relaxation function,
\[
c_0^{AA}(q,z) = \frac{1}{\displaystyle z + \frac{\Delta^{A}_1(q)}
{\displaystyle
z + \frac{\Delta^{A}_2(q)}{\displaystyle z + \ldots } } }\;, \nonumber
\]
which is the Laplace transform of the symmetrized correlation function
$\Re\langle A_q(t)A_{-q}\rangle/\langle A_qA_{-q}\rangle$. The $T=0$ dynamic
structure factor (\ref{I.4}) is then obtained via
\[
S_{AA}(q,\omega) = 4\langle A_qA_{-q}\rangle\Theta(\omega)\lim
\limits_{\varepsilon \rightarrow 0}
\Re [c_{0}^{AA} (q, \varepsilon - i\omega)] \;. \nonumber
\]
For some aspects of this study, we benefit from the close relationship of the
two itinerant electron models $H_{t\text{-}J_z}$ and $H_{t\text{-}J}$ with the
1D $s=1/2$ $XXZ$ model,
\[
H_{XXZ}=H_{XX}-J_\parallel\sum_l S_l^zS_{l+1}^z, \nonumber
\]
a model for localized electron spins. The equivalence of $H_{t\text{-}J_z}$ and
$H_{XXZ}$ for $J_\parallel=J_z/2$ and $J_\perp=2t$ was pointed out and used
before.\cite{BBO91,PS91} Depending on the boundary conditions, it can be
formulated as a homomorphism between eigenstates belonging to specific invariant
subspaces of the two models. The mapping assigns to any up spin and down spin
in $H_{XXZ}$ an electron and a hole, respectively, in $H_{t\text{-}J_z}$. The
spin sequence of the electrons in the subspace of interest here is fixed, namely
alternatingly up and down. The importance of this mapping derives from the fact
that the ground state properties of $H_{XXZ}$ have been analyzed in great
detail.\cite{YY66,DG66,LP75}
The $T=0$ dynamic charge structure factor $S_{nn}(q,\omega )$ of
$H_{t\text{-}J_z}$ is thus equivalent to the $T=0$ dynamic spin structure factor
$S_{zz}(q,\omega)$ of $H_{XXZ}$ throughout the Luttinger liquid phase, and we
shall take advantage of the results from previous studies of $XXZ$ spin
dynamics.\cite{SSG82,BM82} The spin dynamics of $H_{t\text{-}J_z}$ is not
related to any known dynamical properties of $H_{XXZ}$.
\subsection{Charge structure factor}
Certain dominant features of the dynamic charge structure factor
$S_{nn}(q,\omega)$ are related to known properties of the static charge
structure factor. Figure \ref{F1} displays finite-$N$ data of $S_{nn}(q)$ for
various coupling strengths in the Luttinger liquid phase of (a)
$H_{t\text{-}J_z}$ and (b) $H_{t\text{-}J}$.
The alignment of the data points on a sloped straight line in the free-electron
limit represents the exact result (\ref{II.4}), which is common to both models.
The persistent linear behavior at small $q$ for nonzero coupling reflects an
asymptotic term of the form $\sim A_0m^{-2}$ in the charge correlation function
$\langle n_ln_{l+m}\rangle$, while the progressive weakening of the cusp
singularity at $q=\pi$ reflects an asymptotic term of the form $\sim A_1\cos(\pi
m)/m^{\eta_\rho}$ with a coupling-dependent charge correlation exponent
$\eta_\rho$. For $H_{t\text{-}J_z}$ this exponent is exactly
known:\cite{LP75}
\begin{equation}\label{III.4}
\eta _\rho = 2/[1-(2/\pi)\arcsin (J_z/4t)]\;.
\end{equation}
No exact result exists for the $t$-$J$ case, but the prediction is that the
charge correlation exponent varies over the same range of values,\cite{OLSA91}
i.e. between $\eta_\rho=2$ at $J/t=0$ and $\eta_\rho=\infty$ at $J/t\simeq 3.2$.
For $J/t \gtrsim 1$, the data in Fig.~\ref{F1}(b) indicate the presence of a
third cusp singularity in $S_{nn}(q)$, namely at $q=\pi/2$, which reflects the
third asymptotic term, $\sim A_2\cos(\pi m/2)/m^{1+\eta_\rho/4}$, predicted for
the $t$-$J$ charge correlations.\cite{note3} No corresponding singularity is
indicated in the data of Fig.~\ref{F1}(a), nor is any corresponding asymptotic
term predicted in the $XXZ$ spin correlations.
At the endpoint of the Luttinger liquid phase ($J_z/t=4$), the $t$-$J_z$
ground-state wave function has the form
\begin{eqnarray}\label{III.5}
|\phi_0\rangle
&&
=\sum_{1\leq l_1<l_2<\ldots<l_{N/2}\leq N}
{N\choose N/2}^{-1/2}|l_1,\ldots ,l_{N/2}\rangle
\nonumber\\ &&~~~~~~~~~~~~~~
\times \frac 1{\sqrt{2}}\left\{
|\uparrow\downarrow\uparrow\ldots\rangle -
|\downarrow\uparrow\downarrow\ldots\rangle\right\}\;,
\end{eqnarray}
where $|l_1,\ldots ,l_{N/2}\rangle$ specifies the variable charge positions.
The electrons are distributed completely at random on the lattice, while the
sequence of spin orientations is frozen in a perfect up-down pattern. This state
is non-degenerate for finite $N$, and its energy per site is $N$-independent:
$E_0/N\!=\!-t$. For $N\!\rightarrow\!\infty$, the $t\text{-}J_z$ charge
correlations disappear completely, $\langle n_l n_{l+m}\rangle-\langle
n_l\rangle \langle n_{l+m}\rangle=\delta_{m,0}/4$ as is indicated by the
finite-$N$ data for $J_z/t=4$ in Fig.~\ref{F1}(a): $S_{nn}(q) -
(N/4)\delta_{q,0}=[N/4(N-1)](1-\delta_{q,0})$. The $t$-$J$ charge correlations,
by contrast, seem to persist at $J/t\simeq 3.2$.
\subsection{Charge dynamics (weak-coupling regime)}
Expression (\ref{II.5}) for the $T=0$ dynamic charge structure factor
$S_{nn}(q,\omega)$ of $H_t$ is modified differently under the influence of a
$J_z$-type or a $J$-type exchange interaction. Within the Luttinger liquid
phase we distinguish two regimes for the charge dynamics: a {\it weak-coupling}
regime and a {\it strong-coupling} regime as identified in the context of the
recursion method.\cite{VZMS95}
In the framework of weak-coupling approaches, the dynamically dominant
excitation spectrum of $S_{nn}(q,\omega)$ is confined to a continuum as in
(\ref{II.5}) but with modified boundaries and a rearranged spectral-weight
distribution. Moreover, a discrete branch of excitations appears outside the
continuum. A weak-coupling continued-fraction (WCCF) analysis for
$S_{nn}(\pi,\omega)$ of $H_{t\text{-}J}$ and, in disguise, also of
$H_{t\text{-}J_z}$, namely in the form of $S_{zz}(\pi,\omega)$ for $H_{XXZ}$ was
reported in Ref. \onlinecite{VZMS95}. Without repeating any part of that
analysis we recall here those results which are important in the present
context.
The renormalized bandwidth $\omega_0$ of the dynamic charge structure factor
$S_{nn}(\pi,\omega)$ versus the coupling constant as obtained from a WCCF
analysis is shown in the main plot of Fig.~\ref{F2} for both the $t$-$J_z$ model
($\Box$) and the $t$-$J$ model ($\circ$). In the $XXZ$ context, $\omega_0$ is
the bandwidth of the 2-spinon continuum, which is exactly known.\cite{DG66}
Translated into $t$-$J_z$ terms, the expression reads
\begin{equation}\label{III.8}
\omega_0/2t=(\pi/\mu)\sin\mu, \quad \cos\mu=-J_z/4t
\end{equation}
and is represented by the solid line. Comparison with our data confirms the
reliability of the WCCF analysis.
Our bandwidth data for the $t$-$J$ model can be compared with numerical results
of Ogata {\it et al.}\cite{OLSA91} for the charge velocity $v_c$ as derived from the
numerical analysis of finite chains. The underlying assumption is that the
relation $\omega_0=2v_c$, which is exact in $H_{t\text{-}J_z}$, also holds for
the $H_{t\text{-}J}$ model. The $t\text{-}J$ charge-velocity results of
Ref. \onlinecite{OLSA91} over the entire range of the Luttinger liquid phase are
shown as full circles connected by a dashed line in the inset. The solid line
represents the exact $t\text{-}J_z$ charge velocity $v_c=\omega_0/2$ with
$\omega_0$ from (\ref{III.8}).
The dashed line in the main plot is the $t$-$J$ bandwidth prediction inferred
from the data of Ref. \onlinecite{OLSA91}. It is in near perfect agreement with
the WCCF data ($\circ$). The open squares in the inset show the WCCF data over a
wider range of coupling strengths. The renormalized bandwidth $\omega_0$ will
shrink to zero at the endpoint of the Luttinger liquid phase, and the spectral
weight will gradually be transferred from the shrinking continuum to states of a
different nature at higher energies.
\subsection{Infrared exponent}
In the Luttinger liquid phase, the dynamic charge structure factor has
an infrared singularity with an exponent related to the charge correlation
exponent:
\begin{equation}\label{III.9}
S_{nn}(\pi,\omega)\sim \omega^{\beta_\rho},\;\;
\beta_\rho = \eta_\rho - 2\;.
\end{equation}
The WCCF analysis yields specific predictions for $\beta_\rho$ in both models.
Our results plotted versus coupling constant are shown in the inset to
Fig.~\ref{F3} for $H_{t\text{-}J_z}$ ($\Box$) and $H_{t\text{-}J}$
($\circ$). The solid line represents the exact $t$-$J_z$ result inferred from
(\ref{III.4}).
We observe that the WCCF prediction for the infrared exponent ($\Box$) rises
somewhat more slowly from zero with increasing coupling than the exact
result. The solid line in the main plot depicts the inverse square of the exact
$t$-$J_z$ correlation exponent (\ref{III.4}) over the entire range of the
Luttinger liquid phase. The open squares represent the WCCF data for
$2+\beta_\rho=\eta_\rho$ extended to stronger coupling. For $H_{t\text{-}J}$
the correlation exponent is not exactly known. The solid circles interpolated
by the dashed line represent the prediction for $\eta_\rho $ of Ogata et
al.\cite{OLSA91} based on a finite-size analysis. The dashed line in the inset
is inferred from the same data. It agrees reasonably well with the WCCF data for
$\beta_\rho$ ($\circ$).
The solid and long-dashed curves in the main plot suggest the intriguing
possibility that the exponents $\eta_\rho$ of the two models have the same
dependence on the scaled coupling constants $J_z/J_z^{(c)}$ with $J_z^{(c)}=4t$
and $J/J^{(c)}$ with $J^{(c)}\simeq 3.2t$. The short-dashed line represents the
exact $t$-$J_z$ result (\ref{III.4}) thus transcribed for $H_{t\text{-}J}$. Its
deviation from the data of Ogata {\it et al.} are very small throughout the
Luttinger liquid phase.
In Ref. \onlinecite{VZMS95} we carried out a WCCF reconstruction of the function
$S_{nn}(\pi,\omega)$ for the $t$-$J$ model and the $t$-$J_z$ model (alias $XXZ$
model).\cite{note2} The observed spectral-weight distributions of both models
consisted of a gapless continuum with a cusp-like infrared singularity
($\beta_\rho >0$), a shrinking bandwidth $(\omega_0/2t<2)$, and a lone discrete
state outside the continuum near its upper boundary.
\subsection{Charge dynamics (strong-coupling regime)}
What happens to the dynamic charge structure factor $S_{nn}(q,\omega)$ as the
exchange interaction is increased beyond the weak-coupling regime of the
Luttinger liquid phase? For the $t$-$J_z$ case the answer can be inferred from
known results for the spin dynamics of $H_{XXZ}$.\cite{SSG82,BM82} The continuum
of charge excitations with sine-like boundaries
\[
\epsilon_L(q)={\pi t\sin\mu \over \mu}|\sin q|, \quad
\epsilon_U(q)=2\epsilon_L(q/2),
\]
continues to shrink to lower and lower energies, and discrete branches of
excitations
\[
\epsilon_n(q)={2\pi t\sin\mu \over \mu\sin y_n}\sin{q \over 2}
\sqrt{\sin^2{q \over 2}+\sin^2y_n\cos^2{q \over 2}}
\]
with $y_n=(\pi n/2\mu)(\pi-\mu)$ emerge successively at $\mu=\pi/(1+1/n)$ from
the upper continuum boundary.\cite{JKM73,BM82} All these excitations carry some
spectral weight, at least for finite $N$, but most of the spectral weight in
$S_{nn}(q,\omega)$ is transferred from the shrinking continuum to the top
branch, the one already present in the WCCF reconstruction.\cite{VZMS95}
At the endpoint of the Luttinger liquid phase $J_z/t=4$, the continuum states
have been replaced by a series of branches $\epsilon_n(q)=(2t/n)(1-\cos q)$,
$n=1,2,\ldots$, all the spectral weight is carried by the top branch $(n=1)$,
and the dynamic charge structure factor reduces to the single-mode form
\[
S_{nn}(q,\omega)=\pi^2\delta(q)\delta(\omega)
+{\pi \over 2}\delta\left(\omega-J_z\sin^2{q \over 2}\right).
\]
In the framework of the recursion method applied to the exact finite-size ground
state (\ref{III.5}), this simple result follows from a spontaneously terminating
continued fraction with coefficients $\Delta_1(q)=J_z^2\sin^4(q/2),
\Delta_2(q)=0$.
The dynamically relevant charge excitation spectrum of $H_{t\text{-}J}$, which
has an even more complex structure, will be presented in a separate study. In
this case, exact results exist only at one point $(J/t=2)$ in the strong
coupling regime.\cite{BBO91}
\subsection{Spin structure factor}
The long-distance asymptotic behavior of the $t$-$J$ spin correlation function
in the Luttinger liquid phase was predicted to be governed by two leading
power-law terms of the form\cite{HM91,AW91,PS91,OLSA91}
\begin{equation}\label{III.13}
\langle S_l^zS_{l+m}^z\rangle_{t\text{-}J}\sim
B_1\frac 1{m^2} + B_2\frac{\cos(\pi m/2)}{m^{\eta_\rho/4+1}}\;,
\end{equation}
where $\eta_\rho$ is the charge correlation exponent discussed previously. The
open circles in Fig.~\ref{F4}(a) depict the spin structure factor
$S_{zz}(q)_{t\text{-}J}$ for $J/t=0^+$ of a system with $N=56$ sites as inferred
via numerical Fourier transform from the results for the spin correlation
function presented in Sec. II. The two asymptotic terms of (\ref{III.13}) are
reflected, respectively, in the linear behavior at small $q$ and in the pointed
maximum at $q=\pi/2$. The latter turns into a square-root cusp as
$N\rightarrow\infty$. The extrapolated maximum is
$S_{zz}(\pi/2)_{t\text{-}J}=0.28(1)$ (indicated by a $+$ symbol). The
extrapolated slope at $q=0$ is $S_{zz}(q)_{t\text{-}J}/q=0.0847(20)$. The
observed smooth minimum at $q=\pi$ suggests that $S_{zz}(q)_{t\text{-}J}$,
unlike $S_{nn}(q)_{t\text{-}J}$, has no singularity there. The extrapolated
value is $S_{zz}(\pi)_{t\text{-}J}=0.127019(2)$.
The predictions of (\ref{III.13}) that the linear behavior in
$S_{zz}(q)_{t\text{-}J}$ at small $q$ persists throughout the Luttinger liquid
phase and that the cusp singularity at $q=\pi/2$ weakens with increasing $J/t$
and disappears at the onset of phase separation are consistent with our result
for $J/t=3.2$, plotted in Fig.~\ref{F4}(b). The open circles suggest a smooth
curve which rises linearly from zero at $q=0$. The smooth extremum at $q=\pi$
has turned from a minimum at $J/t=0^+$ into a maximum at $J/t=3.2$.
The solid line in Fig.~\ref{F4}(a) represents $S_{zz}(q)_{t\text{-}J_z}$ for the
free-fermion case $J_z/t=0^+$ as obtained from Fourier transforming
(\ref{II.12}). It differs from the corresponding $t$-$J$ result ($\circ$) mainly
in three aspects: (i) the rise from zero at small $q$ is quadratic instead of
linear, reflecting non-singular behavior at $q=0$, i.e. the absence of a
non-oscillatory power-law asymptotic term in $\langle
S_l^zS_{l+m}^z\rangle_{t\text{-}J_z}$; (ii) the singularity at $q=\pi/2$ is
divergent: $\sim |q-\pi/2|^{-1/2}$; (iii) the smooth local minimum at $q=\pi$
has a slightly higher value, $S_{zz}(\pi)_{t\text{-}J_z}\simeq 0.129$.
Over the range of the Luttinger liquid phase, the asymptotic term in $\langle
S_l^zS_{l+m}^z\rangle_{t\text{-}J_z}$ which governs the singularity in
$S_{zz}(q)_{t\text{-}J_z}$ at $q=\pi/2$ is of the form $\sim~B_2\cos(\pi
m/2)/m^{\eta_\rho/4}$. As in the $t$-$J$ case, the singularity weakens
gradually and then disappears at the transition point, $J_z/t=4$. The
finite-$N$ result of $S_{zz}(q)_{t\text{-}J_z}$ at $J_z/t=4$, ($\bullet$) in
Fig.~\ref{F4}(b), indeed suggests a curve with no
singularities. This is confirmed by the exact result,
\begin{equation}\label{III.14}
S_{zz}(q)_{t\text{-}J_z}={1 \over 8}(1-\cos q)\;,
\end{equation}
inferred from the exact ground-state wave function (\ref{III.5}) for
$N\rightarrow\infty$. It reflects a spin correlation function which vanishes for
all distances beyond nearest neighbors.
\subsection{Spin dynamics}
Under mild assumptions, which have been tested for $H_{t\text{-}J_z}$ at
$J_z/t=0^{+}$, the following properties of the dynamic spin structure factors
$S_{zz}(q,\omega)$ of $H_{t\text{-}J}$ or $H_{t\text{-}J_z}$ can be inferred
from the singularity structure of $S_{zz}(q)$: (i) The excitation spectrum in
$S_{zz}(q,\omega)$ is gapless at $q=\pi/2$. (ii) The spectral-weight
distribution at the critical wave number $q=\pi/2$ has a singularity of the
form:
\[
S_{zz}\left({\pi \over 2},\omega\right)_{t\text{-}J_z}\sim
\omega^{{\eta_\rho\over 4}-2}\;,\;\;
S_{zz}\left({\pi\over 2},\omega\right)_{t\text{-}J}
\sim \omega^{{\eta_\rho\over 4}-1}\;.
\]
In the weak-coupling limit $(\eta_\rho =2)$, this yields $\sim \omega^{-3/2}$
for $H_{t\text{-}J_z}$ and $\sim \omega^{-1/2}$ for $H_{t\text{-}J}$. In both
cases, the infrared exponent increases with increasing coupling. A landmark
change in $S_{zz}(\pi,\omega)$ occurs at the point where the infrared exponent
switches sign (from negative to positive). In the $t$-$J_z$ case this happens
for $\eta_\rho =8$ and in the $t$-$J$ case for $\eta_\rho =4$. According to the
data displayed in Fig.~\ref{F3}, this corresponds to the coupling strengths
$J_z/t=3.6955\ldots$ and $J/t\simeq 2.3$, respectively.
The dynamic spin structure factor $S_{zz}(q,\omega)_{t\text{-}J_z}$ as obtained
via the recursion method combined with a strong-coupling continued-fraction
(SCCF) analysis\cite{VM94,VZSM94} is plotted in Fig.~\ref{F5} as a continuous
function of $\omega$ and a discrete function of $q=2\pi m/N$, $m=0,\ldots ,N/2$
with $N=12$ for coupling strengths $J_z/t=0^{+},2,3,4$. This function has a
non-generic $(q \leftrightarrow \pi-q)$ symmetry, which obtains for the
dynamically relevant excitation spectrum and for the line shapes, but not for
the integrated intensity. In the weak-coupling limit, $J_z/t=0^{+}$, the
spectral weight in $S_{zz}(q,\omega)$ is dominated by fairly well defined
excitations at all wave numbers. The dynamically relevant dispersion is $|\cos
q|$-like.
With $J_z/t$ increasing toward the endpoint of the Luttinger liquid phase, the
following changes can be observed in $S_{zz}(q,\omega )$: The peaks at $q\neq
\pi /2$ gradually grow in width and move toward lower frequencies. The $|\cos
q|$-like dispersion of the peak positions stays largely intact, but the
amplitude shrinks steadily. The central peak at the critical wave number
$q=\pi/2$ starts out with large intensity and slowly weakens with increasing
coupling. Between $J_z/t=3$ and $J_z/t=4$, it turns rather quickly into a broad
peak, signaling the expected change in sign of the infrared exponent.
The dynamically relevant dispersion of the dominant spin fluctuations as
determined by the peak positions in our SCCF data for $S_{zz}(q,\omega)$ is
shown in Fig.~\ref{F6} for several values of $J_z/t.$ The linear initial rise
from zero at $q=\pi/2$ is typical of a Luttinger liquid. The amplitude of the
$|\cos q|$-like dispersion decreases with increasing $J_z/t$ and approaches zero
at the transition to phase separation. At the same time, the line shapes of
$S_{zz}(q,\omega)_{t\text{-}J_z}$ tend to broaden considerably. These trends
are not shared with the $t$-$J$ spin excitations as we shall see.
The SCCF analysis indicates that the Luttinger liquid phase of the $t$-$J$ model
can be divided into two regimes with distinct spin dynamical properties. For
coupling strengths $0<J/t\lesssim 1$, the function
$S_{zz}(q,\omega)_{t\text{-}J}$, which is plotted in Fig.~\ref{F8}, exhibits
some similarities with the corresponding $t$-$J_z$ results. The main
commonality is a well-defined spin mode at not too small wave numbers with a
$|\cos q|$-like dispersion. This dispersion is displayed in the main plot of
Fig.~\ref{F9} for different $J/t$-values within this first regime of the
Luttinger liquid phase.
However, even in the common features, the differences cannot be overlooked: (i)
The $(q\leftrightarrow \pi-q)$ symmetry in the line shapes of
$S_{zz}(q,\omega)_{t\text{-}J_z}$ is absent in $S_{zz}(q,\omega)_{t\text{-}J}$.
(ii) The amplitude of the $|\cos q|$-like dispersion grows with increasing
$J/t$, contrary to the trend observed in Fig.~\ref{F6} for the corresponding
$t$-$J_z$ spin dispersion. (iii) The gradual upward shift of the peak position
in $S_{zz}(\pi,\omega )_{t\text{-}J}$ is accompanied by a significant increase
in line width (see inset to Fig.~\ref{F10}). Over the range $0\leq J/t\lesssim
1.25$, the trend of the $q=\pi$ spin mode is opposite to what one expects under
the influence of an antiferromagnetic exchange interaction of increasing
strength. (iv) The intensity of the central peak in
$S_{zz}(\pi/2,\omega)_{t\text{-}J}$ is considerably weaker than in in
$S_{zz}(\pi/2,\omega)_{t\text{-}J_z}$. The peak turns shallow and disappears
quickly with increasing coupling (see Fig.~\ref{F10}, main plot). This
observation is in accord with the proposed dependences of the infrared exponents
on the coupling constants. (v) The linear dispersion of the dynamically
relevant spin excitations have markedly different slopes above and below the
critical wave number $q=\pi/2$ (Fig.~\ref{F9}, main plot). At long wavelengths
the spectral weight in $S_{zz}(q,\omega)_{t\text{-}J}$ is concentrated at much
lower frequencies than in $S_{zz}(q,\omega)_{t\text{-}J_z}$.\cite{note8}
As the coupling strength increases past the value $J/t\simeq 0.75$, the spin
modes which dominate $S_{zz}(q,\omega)_{t\text{-}J}$ in the first regime of the
Luttinger liquid phase broaden rapidly and lose their distinctiveness. There is
a crossover region between the first and second regime, which roughly comprises
the coupling range $1\lesssim J/t\lesssim 2$. Over that range, the spin dynamic
structure factor tends to be governed by complicated structures with rapidly
moving peaks.
At the end of the crossover region, a new type of spin mode with an entirely
different kind of dispersion has gained prominence in
$S_{zz}(q,\omega)_{t\text{-}J}$, and it stays dominant throughout the remainder
of the Luttinger liquid phase. This is illustrated in Fig.~\ref{F11} for three
$J/t$-values in the second regime of the Luttinger liquid phase. The dispersion
of these new spin modes gradually evolves with increasing coupling strength as
shown in the inset to Fig.~\ref{F9}. Note that the frequency has been rescaled
by $J$ both here and in Fig.~\ref{F11}. At $J/t\lesssim 2.0$ the dispersion has
a smooth maximum at $q=\pi $ and seems to approach zero linearly as
$q\rightarrow 0$. As $J/t$ increases toward the transition point, the peak
positions in $S_{zz}(q,\omega)_{t\text{-}J}$ gradually shift to lower values of
$\omega/J$, most rapidly at $q$ near $\pi$.
\section{Phase separation}\label{Sec:PS}
The transition from the Luttinger liquid phase to a phase-separated state in
$H_{t\text{-}J_z}$ takes place at $J_z/t=4$. The equivalent $XXZ$ model
undergoes a discontinuous transition to a state with ferromagnetic long-range
order at the corresponding parameter value ($J_{\parallel}/J_{\perp}=1$). The
ground state at the transition is non-critical and degenerate even for finite
$N$. The $XXZ$ order parameter, $\overline{M}=N^{-1}\sum_lS_l^z,$ commutes with
$H_{XXZ}$.
Notwithstanding the exact mapping, the transition of $H_{t\text{-}J_z}$ at
$J_z/t=4$ is of a different kind. Only one of the $N+1$ vectors which make up
the degenerate $XXZ$ ground state at $J_{\parallel}/J_{\perp }=1$ is contained
in the invariant subspace that also includes the $t$-$J_z$ ground state. The
other vectors correspond to $t$-$J_z$ states with different numbers $N_e$ of
electrons. The $t$-$J_z$ ground state at $J_z/t=4$ for fixed $N_e=N/2$ is
non-degenerate and represented by the wave function $|\phi_0\rangle$ as given in
(\ref{III.5}).
The fully phase-separated state as represented by the wave function
\begin{eqnarray}\label{IV.6}
|\phi _1\rangle \equiv
&&
\frac 1{\sqrt{2N}}\sum_{l_1=1}^N|l_1,l_1+1,\ldots,l_1+N/2-1\rangle
\nonumber\\ && ~~~~~~~~
\times\left\{ |\uparrow \downarrow \uparrow \ldots \rangle
\pm|\downarrow \uparrow \downarrow \ldots \rangle \right\}
\end{eqnarray}
has an energy expectation value at $J_z/t=4,\langle E_1\rangle=-t(N-2)$, which
exceeds the finite-$N$ ground-state energy, $E_0=-tN$, pertaining to
$|\phi_0\rangle$. However, by comparing the $J_z$-dependence of the energy
expectation values (per site) of the two wave functions $|\phi_0\rangle$ and
$|\phi_1\rangle$,
\begin{eqnarray*}
\tilde{e}_0\equiv \frac{1}{N}\langle\phi_0|H_{t\text{-}J_z}|\phi_0\rangle
&=&
-t -\frac{1}{2}\!\left(\frac{J_z}{4}-t\right)\!\left(1-\frac{1}{N-1}\right),
\\
\tilde{e}_1\equiv \frac{1}{N}\langle\phi_1|H_{t\text{-}J_z}|\phi_1\rangle
&=&
-\frac{J_z}{4}\left(1-\frac{2}{N}\right),
\end{eqnarray*}
in the vicinity of the transition, $J_z/t=4(1+\epsilon)$, we obtain
\[
\tilde{e}_0-\tilde{e}_1
\stackrel{N\to\infty}{\longrightarrow}\frac{\epsilon}{2t},
\]
which implies that the two levels cross at $J_z/t=4$ in the infinite system.
The transition to phase separation in $H_{t\text{-}J_z}$ is characterized by
the charge and spin order parameters,
\[
Q_\rho = \frac{1}{N} \sum_{l=1}^Ne^{i2\pi l/N}n_l\;, \quad
Q_\sigma = \frac{1}{N} \sum_{l=1}^Ne^{i\pi l }S_l^z\;.
\]
Neither operator commutes with $H_{t\text{-}J_z}$. The phase-separated state of
$H_{t\text{-}J_z}$ is characterized, for $N\rightarrow \infty $, by a broken
translational symmetry, $\langle Q_\rho \rangle \neq 0$, and a broken spin-flip
symmetry, $\langle Q_\sigma \rangle \neq 0$.
In the $t$-$J$ model, the transition to the phase-separated state, which takes
place at $J/t\simeq 3.2,$ produces charge long-range order, $\langle Q_\rho
\rangle \neq 0$, but is not accompanied by the onset of spin long-range order,
$\langle Q_\sigma \rangle =0$. The similarities in the charge correlations and
the differences in the spin correlations of the two models are evident in the
finite-size static charge and spin structure factors.
\subsection{Charge structure factor}
The vanishing charge correlations in the finite-size $t$-$J_z$ ground state at
the onset of phase separation ($J_z/t=4)$ is reflected in the flat charge
structure factor $S_{nn}(q)$ as shown in Fig.~11(a). The corresponding $t$-$J$
result for $J/t\simeq 3.2$ as shown in Fig.~11(b) indicates that correlated
charge fluctuations do exist at the transition.
With the exchange coupling increasing beyond the transition point, the charge
structure factors of the two models become more and more alike and reflect the
characteristic signature of phase separation. Phase separation is associated
with an enhancement of $S_{nn}(q)$ in the long-wavelength limit. Because of
charge conservation, this enhancement is manifest, in a finite system, not at
$q=0$ but at $q=2\pi/N$. It is conspicuously present in the data for couplings
$J_z/t=4.5$ and $J/t=3.5$, not far beyond the transition point.
The charge correlation function for the fully phase separated state, as
represented by the wave function (\ref{IV.6}), is a triangular
function,\cite{note4} $\langle n_ln_{l+m}\rangle =1/2-|m|/N,\; |m|\leq N/2$.
This translates into a charge structure factor of the form
\begin{equation}\label{IV.8}
S_{nn}(q)={\frac N4}\delta _{q,0}+\frac{{1+\cos (Nq/2)}}{{N(1-\cos q)}}
(1-\delta _{q,0}),
\end{equation}
as shown (for $N=12$) by the full diamonds in Fig.~\ref{F12}. This function
vanishes for all wave numbers $q=2\pi l/N$ with even $l$ and increases
monotonically with decreasing odd $l$. The data in Fig.~\ref{F12} suggest that
the phase separation is nearly complete before the exchange coupling has reached
twice the value at the transition. In the $t\text{-}J_z$ case, we already know
that complete phase separation is established (for $N\to\infty$) right at the
transition.
\subsection{Spin structure factor}
The extremely short-ranged spin correlations in the $t\text{-}J_z$ ground state
(\ref{III.5}) for $N\rightarrow \infty $ are reflected by the static spin
structure factor (\ref{III.14}). For finite $N$ the spin correlations at
distances $|n|\geq 2$ do not vanish identically. An exponential decay is
observed instead with a correlation length that disappears as $N\rightarrow
\infty $. Hence the difference between (\ref{III.14}) and the finite-$N$ data
depicted in Fig.~12(a) ($\bullet$). The $t$-$J$ spin structure factor near the
transition $(J/t\simeq 3.2)$ has a similar $q$-dependence except at small $q$,
where it tends to zero linearly instead of quadratically.
Whereas the charge structure factors of the two models become more and more
alike as the exchange coupling increases in the phase-separated state
(Fig.~\ref{F12}), divergent trends are observed in the respective spin structure
factors, on account of the fact that the $t$-$J_z$ model supports spin
long-range order, and the $t$-$J$ model does not.
The fully phase-separated state of the $t$-$J_z$ model is at the same time fully
N\'{e}el ordered. The spin correlation function in the state (\ref{IV.6}) reads
$\langle S_l^zS_{l+m}^z\rangle=\frac{1}{4}(-1)^m(1/2-|m|/N),\: |m|\leq N/2$
and the corresponding spin structure factor has the form
\begin{equation}\label{IV.11}
S_{zz}(q)= \frac{N}{16}\delta_{q,\pi}+
\frac{1-\cos [N(\pi-q)/2]}{4N[1-\cos (\pi -q)]}(1-\delta_{q,\pi}).
\end{equation}
The function (\ref{IV.11}) vanishes (for even $N/2$) at all wave numbers $q=2\pi
l/N$ with even $l$, just as (\ref{IV.8}) did. The exception is the wave number
$q=\pi $, where $S_{zz}(q)$ assumes its largest value.
The $t$-$J$ spin structure factor evolves quite differently in the presence of
increasing phase separation as is illustrated in Fig.~12(b). The electron
clustering produces in this case the Heisenberg antiferromagnet, whose ground
state is known to stay critical with respect to spin fluctuations. The spin
structure factor of that model is known to be a monotonically increasing
function of $q$, which grows linearly from zero at small $q$ and (for
$N\rightarrow \infty $) diverges logarithmically at $q=\pi$.\cite{SFS89}
\subsection{Spin dynamics ($t$-$J$ model)}
The charge long-range order in the phase-separated state freezes out the charge
fluctuations in both models, and the accompanying spin long-range order in the
$t$-$J_z$ model freezes out the spin fluctuations too. What remains strong are
the spin fluctuations in the $t$-$J$ model.
At the transition to phase separation ($J/t\simeq 3.2$), the $q=\pi $ spin mode
in $S_{zz}(q,\omega )_{t\text{-}J}$ does not go soft. However, the gradual
electron clustering tendency in conjunction with the continued strengthening of
the antiferromagnetic exchange interaction brings about a softening in frequency
and an enhancement in intensity of the order-parameter fluctuations associated
with N\'{e}el order. Both effects can be observed in the reconstructed dynamic
spin structure factors at $J/t=3.25,4.0,5.0$ as shown in Figs.~11(c), 13(a), and
13(b).
A close-up view of the gradual transformation of the $q=\pi $ mode is shown in
Fig.~14(a). For sufficiently strong exchange coupling, the function $S_{zz}(\pi
,\omega )_{t\text{-}J}$ will be characterized by a strong i.e. nonintegrable
infrared divergence, $\sim \sqrt{-\ln\omega}/\omega$,\cite{BCK96} which
characterizes the order-parameter fluctuations of the 1D $s=1/2$ $XXX$
antiferromagnet.
Figure 14(b) shows the gradual change in line shape and shift in peak position
of the function $S_{zz}(\pi /2,\omega )_{t\text{-}J}$ in the phase-separated
state. The peak, which starts out relatively broad at the transition, shrinks in
width, loses somewhat in intensity, and moves to a higher frequency. For
$J/t\gtrsim 5.0$ it settles at $\omega /J\simeq \pi /2$ in agreement with the
lower boundary, $\omega_L(q)=(\pi J/2)|\sin q|$, at $q=\pi/2$ of the 2-spinon
continuum. The width has shrunk to a value consistent with the width of the
2-spinon continuum at that wave number.
In the inset to Fig.~14 we show the evolution of the dynamically relevant
dispersion for $S_{zz}(q,\omega )_{t\text{-}J}$ in the phase-separated state, as
determined by the peak positions of our data obtained via SCCF
reconstruction. The dashed line represents the exact lower threshold of the
2-spinon continuum. The shift of the peak positions in our data is directed
toward that asymptotic position at all wave numbers for sufficiently large
$J/t$.
\acknowledgments
This work was supported by the U.\ S. National Science Foundation, Grant
DMR-93-12252, and the Max-Kade Foundation. Computations were carried out on
supercomputers at the National Center for Supercomputing Applications,
University of Illinois at Urbana-Champaign.
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\section{Introduction}\label{Introduction}
For over a 100 years various approaches have been used to study the
foundations of statistical mechanics. That this subject continues to
be of interest \cite{Lebowitz} indicates that no completely
satisfactory description exists; this is true, both, for classical and
quantum statistical mechanics. The question of how macroscopic
irreversibility arises from reversible microscopic dynamics continues
to be of a topic of discussion and is reviewed in
Ref.~\cite{Lebowitz}.
For quantum statistical mechanics the goal to show that a system can
be described by a microcanonical density matrix, which for some given
energy $E$ is
\begin{equation}\label{densitymatrix}
\mbox{\boldmath $\rho$}_{i,j}(E)=\frac{\delta_{i,j}}{N(E)}\, ,
\end{equation}
where $N(E)$ is the number of degenerate eigenstates with energy
$E$. It would be desirable to obtain the completely mixed state
represented by the density matrix of eq.~(\ref{densitymatrix}) as a
time development of a pure initial state; this is of course impossible
under unitary time evolution and one has to resort to various
averaging assumptions. In the most straight forward ensemble
``derivation'' of this result one essentially assumes the answer by
postulating random phases and equal {\it \`{a} priori} probabilities
for the eigenstates of some Hamiltonian. More recently developments
in the theory of chaotic systems have lead to a different ways of
obtaining quantum statistical mechanics \cite{chaos}. An old approach
to (\ref{densitymatrix}) is through the Pauli master equation
\cite{Cohen},
\begin{equation}\label{Pauli}
\frac{d{\cal P}_i(t)}{dt}=\sum_n\left[t_{i\leftarrow n}{\cal P}_n(t)
-t_{n\leftarrow i}{\cal P}_i(t)\right]\, ,
\end{equation}
where ${\cal P}_i(t)$ is the probability at time $t$ of the system
being in the state $i$ and $t_{i\leftarrow n}$ is the rate per unit
time to go from state $n$ to state $i$. In order to obtain
eq.~(\ref{Pauli}) certain assumptions have to hold\cite{Cohen}:
\begin{itemize}
\item[(i)] A repeated randomness assumptions insuring that the off
diagonal elements of $\mbox{\boldmath $\rho$}_{i,j}$ vanish. This is
an assumption on the states that are accessible as the system
evolves.
\item[(ii)] The interaction potential that causes the mixing of states
is assumed to be weak as to permit the use of first order perturbation
theory.
\item[(iii)] There is an inherent clash in that the ${\cal P}_i$'s
refer to discrete states and yet one has to use continuum
normalization in order to obtain a Dirac delta function in the
energies.
\end{itemize}
Conditions (i) and (ii) can be weakened for certain classes of
interactions \cite{Fujita} (also see L. Van Hove's in
Ref. \cite{Cohen}).
In this presentation we shall consider a nonisolated, finite but
large, system that is allowed to interact with its environment; by
this we mean that from time to time the system is exposed to a low
energy, low intensity packet of some quanta. These quanta are
spatially unconstrained (continuum normalized) and after a certain
characteristic interaction time $\tau$ the system settles into a new
state or density matrix. After intervals larger than this
characteristic time, this process is repeated. The above qualitative
terms, low intensity, low energy and time interval, will be made
precise and summarized in Sec. \ref{conclusion}. The quanta in
these pulses will be referred to as ``photons''. We shall show that
with one assumption on the interaction of these packets with the
system, {\em but independent of any assumptions on the accessibility
of the system itself}, microcanonical equilibrium,
eq.~(\ref{densitymatrix}), will be reached. Contrary to the approach
to equilibrium for isolated systems, in this case equilibrium is
attained by ``entangling'' the system with photons and then looking at
expectation values of operators that are diagonal in the photon
variables. Even though we may start out with a pure system-photon
state, the system itself will be in a mixed state after the
interactions have ceased. No recourse is made to perturbation theory;
the interactions are treated exactly.
The system, its interactions and time evolution are presented in
Sec. \ref{interactions} while the time evolution of density
matrices is discussed in Sec. \ref{t-evol}; an assumption on the
randomness of phases of certain amplitudes is presented and discussed
in this section. How the system approaches equilibrium is shown in
\ref{Equilibrium}; although we do not get a master equation the
approach to equilibrium is Markovian \cite{Cohen}, governed by a
Chapman-Kolmogorov \cite{Cohen} equation. The relations among various
energy and time scales that the system and perturbing quanta have to
satisfy are given in Sec. \ref{conclusion} where the origin of time
irreversibility and the absence of recurrence is discussed.
\section{Interactions and time evolution of states}\label{interactions}
The energy levels $E_{\alpha}$ of the system are highly degenerate
with states $|\alpha, i\rangle$ .The energies of the photons in the
packets are taken to be too small to cause transitions between states
of different energy; these interactions can, however, cause
transitions between the degenerate levels within a given $E_{\alpha}$;
for convenience we take the energy $E_{\alpha}=0$ and drop the quantum
number $\alpha$ in the description of the states. As usual the total
Hamiltonian will be split into two parts, $H=H_0+H_1$; eigenstes of
$H_0$ that are of interest are $|i,{\mbox{\boldmath $k$}}\rangle$ with
\begin{equation}\label{freeH}
H_0|i,{\mbox{\boldmath $k$}}\rangle =
\omega(k)|i,{\mbox{\boldmath
$k$}}\rangle\, ;
\end{equation}
the specific dispersion relation for $\omega(k)$ is unimportant. $H_1$
induces transitions between $|i,{\mbox{\boldmath $k$}}\rangle
\leftrightarrow |j,\mbox{\boldmath $k$}'\rangle$. Whether this
Hamiltonian is time reversal invariant or not is immaterial to
subsequent developments.
Let $|i,{\mbox{\boldmath $k$}}\rangle_H$ be that eigenstate of the
total Hamiltonian that approaches $|i,{\mbox{\boldmath $k$}}\rangle$
for large negative times; we use the notation $|\ \rangle_H$ for an
eigenstate of the total Hamiltonian, whereas $|\ \rangle$, without the
subscript $H$, for an eigenstate of $H_0$. This state satisfies the
Lippmann-Schwinger equation \cite{GW}
\begin{equation}\label{L-S} |i,{\mbox{\boldmath
$k$}}\rangle_H=|i,{\mbox{\boldmath $k$}}\rangle
+\frac{1}{\omega(k)-H_0+i\epsilon}H_1|i,{\mbox{\boldmath
$k$}}\rangle_H\, .
\end{equation} The overlap with an eigenstate of the free
Hamiltonian is
\begin{equation}\label{overlap}
\langle j,\mbox{\boldmath $p$}|i,\mbox{\boldmath $k$}\rangle_H =
\delta_{i,j}\delta({\mbox{\boldmath $k$}}-{\mbox{\boldmath $p$}})+
\frac{1}{\omega(k)-\omega(p)+i\epsilon}
{\cal T}_{j,i}(\mbox{\boldmath $p$},\mbox{\boldmath $k$})\, ,
\end{equation} where
\begin{equation}\label{defT} {\cal T}_{j,i}(\mbox{\boldmath
$p$},\mbox{\boldmath
$k$})=
\langle j,\mbox{\boldmath $p$}|H_1|i,\mbox{\boldmath $k$}\rangle_H
\end{equation} is the scattering amplitude for the transition
$|i,{\mbox{\boldmath $k$}}\rangle\to |j,{\mbox{\boldmath
$p$}}\rangle$and satisfies the off shell unitarity relations
\begin{eqnarray}\label{unitarity1}
{\cal T}_{j,i}(\mbox{\boldmath
$p$},\mbox{\boldmath $k$})-
{\cal T}^*_{i,j}(\mbox{\boldmath $k$},\mbox{\boldmath $p$})&=&
\int d\mbox{\boldmath$q$}
\left[\frac{1}{\omega(q)-\omega(p)+i\epsilon}-
\frac{1}{\omega(q)-\omega(k)-i\epsilon}\right]
{\cal T}_{j,n}(\mbox{\boldmath $p$},\mbox{\boldmath $q$})
{\cal T}^*_{i,n}(\mbox{\boldmath $k$},\mbox{\boldmath $q$})\, ,
\nonumber\\
&=&\int d\mbox{\boldmath$q$}
\left[\frac{1}{\omega(q)-\omega(p)+i\epsilon}-
\frac{1}{\omega(q)-\omega(k)-i\epsilon}\right]
{\cal T}_{n,i}(\mbox{\boldmath $q$},\mbox{\boldmath $k$})
{\cal T}^*_{n,j}(\mbox{\boldmath $q$},\mbox{\boldmath $p$})\, .
\end{eqnarray}
We ant to describe the time evolution of a state that at $t=0$
started out
as $|i,\mbox{\boldmath
$k$}\rangle$; more specifically we want the overlap at time $t$ with
the state $|j,\mbox{\boldmath $p$}\rangle$.
\begin{equation}\label{timeevol1}
\langle j,\mbox{\boldmath $p$}|e^{-iHt}|i,{\mbox{\boldmath
$k$}}\rangle = [\delta_{i,j}\delta(\mbox{\boldmath $p$}
-\mbox{\boldmath $k$})+
i{\cal R}_{j,i}(\mbox{\boldmath $p$},\mbox{\boldmath $k$};t)]
e^{-i\omega(p)t}\, ;
\end{equation} in the above
\begin{eqnarray}\label{timeevol2} i{\cal R}_{j,i}(\mbox{\boldmath
$p$},\mbox{\boldmath $k$};t)&=&
\frac{1}{\omega(p)-\omega(k)-i\epsilon}\left[
-{\cal T}_{j,i}(\mbox{\boldmath $p$},\mbox{\boldmath $k$})
e^{i\left[\omega(p)-\omega(k)\right]t}+
{\cal T}^*_{i,j}(\mbox{\boldmath $k$},\mbox{\boldmath $p$})
\right]\nonumber\\
&&+\int d\mbox{\boldmath$q$}e^{i\left[\omega(p)-\omega(q)\right]t}
\frac{{\cal T}_{j,n}(\mbox{\boldmath $p$},\mbox{\boldmath $q$})}
{\omega(q)-\omega(p)+i\epsilon}\,
\frac{{\cal T}^*_{i,n}(\mbox{\boldmath $k$},\mbox{\boldmath $q$})}
{\omega(q)-\omega(k)-i\epsilon}\, .
\end{eqnarray} For the definition of ${\cal R}$ in eq.~(\ref{timeevol1})
we have
pulled out the factor $\exp\left[-i\omega(p)\right]$ for later
convenience.
Eq.~(\ref{unitarity1}) or (\ref{timeevol1}) implies unitarity relations for
${\cal R}$
\begin{eqnarray}\label{unitarity2} -i\left[{\cal R}_{j,i}
(\mbox{\boldmath $k$}',\mbox{\boldmath $k$};t)
-{\cal R}^*_{i,j}(\mbox{\boldmath $k$},\mbox{\boldmath $k$}';t)
\right ]&=&
\sum_n\int d\mbox{\boldmath$q$}
{\cal R}_{j,n}(\mbox{\boldmath $k$}',\mbox{\boldmath $q$};t)
{\cal R}^*_{i,n}(\mbox{\boldmath $k$},\mbox{\boldmath $q$};t)\, ,
\nonumber\\ &=&\sum_n\int d\mbox{\boldmath$q$}
{\cal R}_{n,i}(\mbox{\boldmath $q$},\mbox{\boldmath $k$};t)
{\cal R}^*_{n,j}(\mbox{\boldmath $q$},\mbox{\boldmath $k$}';t)\, .
\end{eqnarray}
\section{Time evolution of the density matrix}\label{t-evol}
We shall be interested in operators ${\cal O}$ be that have
off diagonal matrix elements between the different $|i\rangle$'s and
are diagonal in the photon subspace. For any state
\begin{equation}
|S\rangle=\sum_i\int d\mbox{\boldmath
$q$}\phi_i(\mbox{\boldmath
$q$}) |i,\mbox{\boldmath $q$}\rangle\, ,
\end{equation}
the expectation value of ${\cal O}$ is
\begin{equation}\label{entangle1}
\langle S|{\cal O}|S\rangle=
\sum_{i,j}\langle j|{\cal O}|i\rangle
\mbox{\boldmath $\rho$}_{i,j}\, ,
\end{equation}
with a density matrix
\begin{equation}\label{entangledensmatrix}
\mbox{\boldmath $\rho$}_{i,j}=\int d\mbox{\boldmath $q$}
\phi_i(\mbox{\boldmath $q$})\phi_j^*(\mbox{\boldmath $q$})\, .
\end{equation}
\subsection{Evolution of \mbox{\boldmath $\rho$} entangled with a wave
packet of photons}
Suppose that at time $t=0$ our system, described by a density
matrix $\mbox{\boldmath $\rho$}_{j,i}(0)$, is exposed to a photon state.
The total density matrix, for the system plus photons is
\begin{equation}\label{t=0densmatrix}
\mbox{\boldmath $\rho$}_T=\sum_{i,j} \int d\mbox{\boldmath $k$}
\, d\mbox{\boldmath $k$}' \psi(\mbox{\boldmath $k$})
\psi^*(\mbox{\boldmath $k$}') |j,\mbox{\boldmath $k$} \rangle
\mbox{\boldmath $\rho$}_{j,i}(0)
\langle i,\mbox{\boldmath $k$}'|\, ,
\end{equation}
where $\psi(k)$ describes the photon wave packet scattered of the mixed
state at $t=0$; $\int d\mbox{\boldmath $k$} |\psi(\mbox{\boldmath
$k$})|^2=1$. Using eq.~(\ref{timeevol1}) and eq.~(\ref{timeevol2}) the
density matrix at time $t$, after summing over the photon states, is
\begin{eqnarray}\label{t=tdensmatrix}
\mbox{\boldmath $\rho$}_{j,i}(t)=\mbox{\boldmath $\rho$}_{j,i}(0) &+&\int
d\mbox{\boldmath $k$}\, d\mbox{\boldmath $k$}'
\psi(\mbox{\boldmath $k$})\psi^*(\mbox{\boldmath $k$}')
\Big [i\sum_n{\cal R}_{j,n}(\mbox{\boldmath
$k$}',\mbox{\boldmath $k$};t)
\mbox{\boldmath $\rho$}_{n,i}(0)\nonumber\\&-& i\sum_m{\cal
R}^*_{i,m}(\mbox{\boldmath $k$},\mbox{\boldmath $k$}';t)
\mbox{\boldmath $\rho$}_{jm}(0)+
\int d\mbox{\boldmath $p$}\sum_{n,m} {\cal R}_{j,n}(\mbox{\boldmath
$p$},\mbox{\boldmath $k$};t) {\cal R}^*_{i,m}(\mbox{\boldmath
$p$},\mbox{\boldmath $k$}';t)
\mbox{\boldmath $\rho$}_{n,m}(0)
\Big ] .\nonumber\\
\end{eqnarray}
In subsequent discussions we shall need some randomness condition on the
phases of the ${\cal R}$'s. It is unlikely that such a condition could
be valid for all times; we shall try for ones that may hold at large
times. For short duration pulses we expect that after some
characteristic collision time $\tau$, as for example the inverse of
the width of a resonance in resonance dominated scattering
\cite{t-3/2}, the system will settle down and
\begin{equation}\label{asympt1}
\lim_{t\to\infty}\int d\mbox{\boldmath $k$}\psi(\mbox{\boldmath $k$})
{\cal R}_{j,i}(\mbox{\boldmath $p$},\mbox{\boldmath $k$};t)=
{\cal R}_{j,i}(\mbox{\boldmath $p$})\, ;
\end{equation}
${\cal R}_{j,i}(\mbox{\boldmath $p$})$ depends implicitly on the wave
function
$\psi$ and, using the definition in eq.~(\ref{timeevol2}) we find
\begin{equation}\label{asymptot1'} {\cal R}_{j,i}(\mbox{\boldmath $p$})=
\int d\mbox{\boldmath $k$}\psi(\mbox{\boldmath $k$})
\frac{-i}{\omega(p)-\omega(k)-i\epsilon}{\cal T}^*_{i,j}(
\mbox{\boldmath $k$},\mbox{\boldmath $p$})\, .
\end{equation}
For further developments, this explicit form will not be needed. We
also define
\begin{equation}\label{asymptot2}
{\cal R}_{j,i}=\int d\mbox{\boldmath
$k$}\psi^*(\mbox{\boldmath $k$}') {\cal R}_{j,i}(\mbox{\boldmath
$k$}')\, .
\end{equation}
These, in turn, satisfy the unitarity relations
\begin{equation}\label{unitarity3}
-i\left({\cal R}_{j,i}-{\cal
R}^*_{i,j}\right)=
\sum_n\int d\mbox{\boldmath $p$}{\cal R}_{n,i}(\mbox{\boldmath $p$})
{\cal R}^*_{n,j}(\mbox{\boldmath $p$})\, .
\end{equation}
In particular, we find
\begin{equation}\label{positivity}
\mbox{\rm Im}\, {\cal R}_{j,j}>0\, .
\end{equation}
The evolution of the density matrix, eq.~(\ref{t=tdensmatrix}), may
be expressed as
\begin{equation}\label{t=tdensmatrix'}
\mbox{\boldmath $\rho$}_{j,i}(t)=\mbox{\boldmath $\rho$}_{j,i}(0)+
i\sum_n\left[{\cal R}_{j,n}\mbox{\boldmath $\rho$}_{n,i}(0)-
{\cal R}^*_{i,n}\mbox{\boldmath $\rho$}_{j,n}(0)\right] +\sum_{m,n}\int
d\mbox{\boldmath $p$}{\cal R}_{j,n}(\mbox{\boldmath $p$})
{\cal R}^*_{i,m}(\mbox{\boldmath $p$})
\mbox{\boldmath $\rho$}_{n,m}(0)\, .
\end{equation}
\subsection{Randomness Assumption}\label{Randomness Assumptions}
In order to proceed further we must impose a crucial condition on the
${\cal R}$'s:
\begin{equation}\label{assumption}
\int d\mbox{\boldmath $p$} {\cal
R}_{j,n}(\mbox{\boldmath $p$}){\cal R}^*_{i,m}(\mbox{\boldmath $p$})
=0\ \ \mbox{for}\ i\ne j\ \ \mbox{and}\ m\ne n.
\end{equation}
This results from the assumption that, as we integrate over
\mbox{\boldmath $p$}, the phases of ${\cal R}_{j,n}(\mbox{\boldmath
$p$})$ fluctuate rapidly. {\em It should be emphasized that this is an
assumptions on the dynamics of the system and not on states or density
matrices of the system at any particular time.}
The above and unitarity relation, eq.~(\ref{unitarity3}), lead to
\begin{equation}\label{randomconseq1}
\left({\cal R}_{j,i}-{\cal R}^*_{i,j}\right)=0\,\ \ \mbox{\rm for}\ \
i\ne j\, ;
\end{equation}
which together with eq.~(\ref{positivity}) implies
\begin{equation}\label{Rdecomp}
{\cal R}={\cal R}_H+i{\cal D}
\end{equation}
with ${\cal R}_H$ Hermitian and ${\cal D}$ a diagonal matrix with
positive elements. As a matter of fact, there always exists a basis
of the states $|i\rangle$ where such a decomposition of ${\cal R}$
holds. Any matrix can be written as a sum of a Hermitian and an
anti-Hermitian one and we go to the basis where the anti-Hermitian part
is diagonal. Although eq.~(\ref{assumption}) implies
eq.~(\ref{Rdecomp}), the inverse is not true and eq.~(\ref{assumption})
remains an assumption. We work in the basis where this
decomposition holds. The time evolution of the density matrix becomes
\begin{equation}\label{timeevol3}
\mbox{\boldmath $\rho$}_{j,i}(t)=\mbox{\boldmath $\rho$}_{j,i}(0)+
i\left[{\cal R}_H,\mbox{\boldmath $\rho$}(0)\right]_{j,i}-
\left[{\cal D},\mbox{\boldmath $\rho$}(0)\right]_{j,i}
+\delta_{i,j}\sum_n\int
d\mbox{\boldmath $p$}|{\cal R}_{i,n}(\mbox{\boldmath $p$})|^2
\mbox{\boldmath $\rho$}_{n,n}(0)\, ;
\end{equation}
$[A,B]$ is the commutator of the matrices $A$ and $B$.
\section{Approach to Equilibrium}\label{Equilibrium}
At this point we have to require the ${\cal R}_{j,i}$'s to be small;
how this is achieved will be made precise in Sec.
\ref{conclusion}. We solve eq.~(\ref{timeevol3}) to first order in the
${\cal R}$'s by first rewriting it as
\begin{equation}\label{timeevol4}
\mbox{\boldmath {$\tilde\rho$}}_{j,i}(t)=\mbox{\boldmath $\rho$}_{j,i}(0)
-\left[{\cal D},\mbox{\boldmath $\rho$}(0)\right]_{j,i}
+\delta_{i,j}\sum_n\int
d\mbox{\boldmath $p$}|{\cal R}_{i,n}(\mbox{\boldmath $p$})|^2
\mbox{\boldmath $\rho$}_{n,n}(0)\, ,
\end{equation}
with
\begin{equation}
\mbox{\boldmath {$\tilde\rho$}}_{j,i}(t)=
\sum_{n,m}\left[1-i{\cal R}\right]_{j,n}\mbox{\boldmath $\rho$}_{n,m}(t)
\left[1+i{\cal R}\right]_{m,i}\, .
\end{equation}
To the order we are working $1-i{\cal R}$ is a unitary matrix and
$\mbox{\boldmath
{$\tilde\rho$}}_{j,i}(t)$ is the density matrix in a basis rotated
from the one we started out at $t=0$.
Let us first look at eq.~(\ref{timeevol4}) for $i\ne j$.
\begin{equation}
\mbox{\boldmath {$\tilde\rho$}}_{j,i}(t)=\left[\mbox{\bf $1$}
-\left({\cal D}_{i,i}+{\cal D}_{j,j}\right)\right]
\mbox{\boldmath $\rho$}_{j,i}(0)\, .
\end{equation}
The off diagonal elements of the unitarily rotated density matrix at
time $t$ are smaller than the corresponding matrix element at $t=0$;
\begin{equation}\label{t=ta}
|\mbox{\boldmath {$\tilde\rho$}}_{j,i}(t)|\le
|\mbox{\boldmath $\rho$}_{j,i}(0)|\, .
\end{equation}
For the case $i=j$ we have
\begin{equation}\label{t=tb}
\mbox{\boldmath {$\tilde\rho$}}_{i,i}(t)=
\mbox{\boldmath $\rho$}_{i,i}(0)+i\left({\cal R}_{i,i}-
{\cal R}^*_{i,i}\right)\mbox{\boldmath $\rho$}_{i,i}(0)+\sum_n\int
d\mbox{\boldmath $p$}|{\cal R}_{i,n}(\mbox{\boldmath $p$})|^2
\mbox{\boldmath $\rho$}_{n,n}(0)\, .
\end{equation}
The coefficient of $\mbox{\boldmath $\rho$}_{n,n}(0)$ in the last term of
this equation may be identified with the transition probability for
$|n\rangle$ to evolve into $|i\rangle$,
\begin{equation}\label{transprob1}
{\cal W}_{i\leftarrow n}=\int d\mbox{\boldmath $p$}
|{\cal R}_{i,n}(\mbox{\boldmath $p$})|^2\, .
\end{equation}
Using the unitarity relation, eq.~(\ref{unitarity3}), we find
\begin{equation}
i({\cal R}_{i,i}-{\cal R}^*_{i,i})=-\sum_n{\cal W}_{n\leftarrow i}
\end{equation}
and the evolution equation for this case becomes
\begin{equation}\label{CK}
\mbox{\boldmath {$\tilde\rho$}}_{i,i}(t)=
\mbox{\boldmath $\rho$}_{i,i}(0)-\sum_n W_{n\leftarrow i}
\mbox{\boldmath $\rho$}_{i,i}(0)
+\sum_n W_{i\leftarrow n}\mbox{\boldmath $\rho$}_{i,i}(0)\, .
\end{equation}
Even if we include the unitary rotations this equation is of the
Chapman-Kolmogorov type \cite{Cohen} and for the ${\cal
W}_{i\leftarrow n}$'s not too large, will directly yield
microcanonical equilibrium. This follows from the observation that
the matrix
\begin{equation}
M_{i,j}=-\delta_{i,j}\sum_n {\cal W}_{n\leftarrow i}
+{\cal W}_{i,j}
\end{equation}
has one eigenvalue equal to zero and all others are negative which in
turn is obtained by showing the function $H(\tau)=-\sum_i{\cal P}_i(\tau)
\ln{\cal P}_i(\tau)$ satisfies a Boltzmann H-theorem, $dH/d\tau\ge 0$
with the probabilities being functions of the auxiliary variable
$\tau$ and (c.f. eq~(\ref{Pauli}))
\begin{equation}
\frac{d{\cal P}_i}{d\tau}=\sum_j M_{i,j}{\cal P}_j\, .
\end{equation}
Let $v^{(0)}_i\sim (1,1,\cdots,1 )$ be the eigenvector of $M_{i,j}$
with eigenvalue zero and $v^{(\alpha)}_i$ be all the others.
\begin{equation}
\mbox{\boldmath $\rho$}_{i,i}=c_0v^{(0)}_i
+\sum_{\alpha}c_{\alpha}v^{(\alpha)}_i\, .
\end{equation}
At the end of the interval $t$ $c_0$ hasn't changed and the other
$c_{\alpha}$'s have all decreased in magnitude. We find that repeated
pulses will drive off diagonal elements to zero and the diagonal ones
to the same constant or the final density matrix will be as in
eq.~(\ref{densitymatrix}).
\section{Concluding Remarks}\label{conclusion}
We have to consider three characteristic energies: (i)
$\Delta_\alpha$, the difference in the energy levels $E_\alpha$ of
the system, (ii) $\delta_\omega$, the energy spread of the impinging
wave packets, and (iii) $\gamma$, the inverse of the interaction time
$\tau$. These quantities have to satisfy
\begin{equation}\label{characenergies}
\Delta_\alpha\gg\delta_\omega\gg\gamma\, .
\end{equation}
The first of these inequalities insures that there will be no
transitions between the different energy levels of the system; for
such transition the energy denominator in eq.~(\ref{overlap})
would have a typical magnitude of $\Delta_\alpha$ as opposed to
$\delta_\omega$ for intra level transitions. Due to the second
inequality the ``arrival period'' of the pulse is much shorter than
the interaction time and for times greater than $\tau$ the system and
the photons propagate separately. In addition the repetition time of
these pulses, $T$, must satisfy
\begin{equation}\label{charactimes}
T\gg\frac{1}{\gamma}\, ;
\end{equation}
this guarantees that as a new packet interacts with the system it is
unencumbered by photons from the previous pulses; for example $T$
must be large enough to allow any possible system-photon resonances to
decay.
Till now, all this has been worked out for the system interacting
with a {\em one} photon packet; the random impulses from the outside
are likely to contain many photons and we extend our results to
$N$ photons where the states are $\int\prod_{n=1}^N d\mbox{\bf k}_n\,
\psi_n(\mbox{\bf k}_n)|i;\mbox{\bf k}_1,\mbox{\bf k}_2\cdots
\mbox{\bf k}_N\rangle$. In the case the packets $\psi_n({\bf k})$
are different for different $n$'s this expression should be
appropriately symmetrized; doing this explicitly would just lead
to unnecessary notational complications. The scattering amplitude
is not necessarily a sum of two body amplitudes and
the rest of the development is as the previous one. We identify a
weak, or low intensity pulse as one with few photons and an intense
one with a large number. We require that $N$ be sufficiently small
that the ${\cal R}_{j,i}$'s are small compared to one; $N$ should
not be too small as the time to reach equilibrium would become very
large.
With these conditions and the assumption about interactions made in
eq.~(\ref{assumption}) satisfied, repeats interactions of the system
with external quanta will drive it to equilibrium.
Time irreversibility is built in right at the start in the choice of
the sign of the $i\epsilon$ term in eq.~(\ref{L-S}) The
continuous energy levels of the photons preclude any recurrences; had
the photons also been quantized in a finite volume the limit
considered in eq.~(\ref{asympt1}) would not have existed.
\nobreak
|
proofpile-arXiv_065-623
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{
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
}
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\section{Introduction}
Since our spacetime may ultimately prove to be discrete, field theories on
discrete spacetimes have developed an interest in and of themselves \cite
{DiscreteST}. In particular, field theories on two-dimensional discrete
spacetimes can be interpreted as string theories with a discrete string
worldsheet and a continuous target space. Exact solutions for such theories
can often be obtained even after quantization (see, for example, \cite
{DiscreteStrings}). Yet another approach to discretizing spacetime and
constructing a theory of gravity based on stochastic properties of random
lattices is advocated by R. Sorkin \cite{Sorkin}. Investigation of field
theories on discrete spacetime might therefore contribute to our
understanding of discrete quantum picture of the Universe.
We shall focus on the connection between the discrete-spacetime and the
continuous-spacetime versions of the same field theory. This work was
motivated by the (somewhat ``mysterious'') fact noted in Ref.\ \cite{Alex}
that a particular discrete version of the two-dimensional wave equation
gives {\em exact} solutions of the underlying continuous equation. In other
words, the exact solution of the free scalar field theory coincides, on the
lattice points, with the solution of the lattice version of the same theory.
The main intent of this paper is to understand the origin of the exact
equivalence of continuous field theories and their discrete counterparts, as
well as to find more general classes of (classical) field theories that are
exactly solved by properly chosen discretizations. The origin of the exact
solution property of discretized equations turns out to be the discrete
conformal symmetry of the models \cite{Henkel}. The exact solution property
of the massless bosonic model has a direct application in numerical
simulations of strings (which was the original context of Ref.\ \cite{Alex}%
). A quantized version of the discrete conformal symmetry may prove useful
in string theory.
The paper is organized as follows. We first present the exact solution
property of the discretized two-dimensional wave equation. Then, in Sec. \ref
{SecDCI}, an analysis of the discrete equations leads us to the concept of
discrete conformal invariance (DCI), and we give general conditions for a
given discrete field equation to possess DCI. In Sec. \ref{SecGen} we
present a general description of discrete models with DCI, and show that
they all yield exact solutions for the corresponding (first and/or
second-order) differential equations; these solutions are always
algebraically factorized. Further examples of equations with DCI based on
Lie groups and semi-groups are presented in Sec. \ref{Examples}. Conclusions
follow in Sec. \ref{SecConcl}.
\section{Exact solution property of the wave equation}
The two-dimensional wave equation came out of numerical modeling of cosmic
strings \cite{Alex}. The evolution equations for the target space variables $%
X^\mu \left( \tau ,\sigma \right) $ are
\begin{equation}
X_{\tau \tau }^\mu -X_{\sigma \sigma }^\mu =0, \label{WaveEqu}
\end{equation}
with gauge conditions
\begin{equation}
\left( X_\tau ^\mu \pm X_\sigma ^\mu \right) ^2=0. \label{WEGaugeFix}
\end{equation}
Here, the subscripts $\tau $ and $\sigma $ denote differentiation with
respect to world-sheet coordinates.
The string world-sheet can be discretized to a rectangular lattice with
coordinates $\left( \tau ,\sigma \right) $ and steps $\Delta \tau ,$ $\Delta
\sigma $. A common second-order discretization of (\ref{WaveEqu}) would be
\begin{equation}
\frac{X^\mu \left( \tau +\Delta \tau ,\sigma \right) -2X^\mu \left( \tau
,\sigma \right) +X^\mu \left( \tau -\Delta \tau ,\sigma \right) }{\Delta
\tau ^2}-\frac{X^\mu \left( \tau ,\sigma +\Delta \sigma \right) -2X^\mu
\left( \tau ,\sigma \right) +X^\mu \left( \tau ,\sigma -\Delta \sigma
\right) }{\Delta \sigma ^2}=0. \label{CommonDiscr}
\end{equation}
However, a special choice of discretization steps, $\Delta \tau =\Delta
\sigma \equiv \Delta $, was adopted in \cite{Alex}, and it was noted that
the resulting discrete equations are not only simpler,
\begin{equation}
X^\mu \left( \tau +\Delta ,\sigma \right) +X^\mu \left( \tau -\Delta ,\sigma
\right) -X^\mu \left( \tau ,\sigma +\Delta \right) -X^\mu \left( \tau
,\sigma -\Delta \right) =0, \label{DWaveEqu}
\end{equation}
\begin{equation}
\left[ X^\mu \left( \tau +\Delta ,\sigma \right) -X^\mu \left( \tau ,\sigma
\pm \Delta \right) \right] ^2=0, \label{DWEGaugeFix}
\end{equation}
but in addition exactly solve the continuous equation (\ref{WaveEqu}).
Namely, one can easily check that the general solution of (\ref{WaveEqu}),
\begin{equation}
X^\mu \left( \tau ,\sigma \right) =a_{+}^\mu \left( \tau -\sigma \right)
+a_{-}^\mu \left( \tau +\sigma \right) , \label{WESol}
\end{equation}
(where $a_{+}^\mu $ and $a_{-}^\mu $ are vector functions of one argument,
which are determined from the appropriate boundary conditions) exactly
satisfies the discretized equation (\ref{DWaveEqu}). The gauge conditions (%
\ref{DWEGaugeFix}) are then reduced to constraints on $a_{\pm }^\mu $,
\begin{equation}
\left[ a_{\pm }^\mu \left( x+\Delta \right) -a_{\pm }^\mu \left( x-\Delta
\right) \right] ^2=0. \label{DWESolConstr}
\end{equation}
To try to understand why this happens, let us examine the discretization (%
\ref{DWaveEqu}) in more detail. Expanding (\ref{DWaveEqu}) in powers of the
discretization step $\Delta $, we notice that the discrete equation (\ref
{DWaveEqu}), on solutions of $\Box X=0$, is satisfied to all orders in $%
\Delta $:
\begin{eqnarray}
X\left( \tau +\Delta ,\sigma \right) +X\left( \tau -\Delta ,\sigma \right)
-X\left( \tau ,\sigma +\Delta \right) -X\left( \tau ,\sigma -\Delta \right)
&=& \nonumber \\
\left( \Box X\right) \Delta ^2+\sum_{n=2}^\infty \left( \frac{\partial ^{2n}X%
}{\partial \tau ^{2n}}-\frac{\partial ^{2n}X}{\partial \sigma ^{2n}}\right)
\frac{2\Delta ^{2n}}{\left( 2n\right) !} &=&0, \label{DWESeries}
\end{eqnarray}
since
\[
\frac{\partial ^{2n}X}{\partial \tau ^{2n}}-\frac{\partial ^{2n}X}{\partial
\sigma ^{2n}}=\Box ^nX=0.
\]
Approximation to all orders means exact solution in the following sense. One
can show that
\begin{equation}
X^\mu \left( \tau ,\sigma \right) =a_{+}^\mu \left( \tau -\sigma \right)
+a_{-}^\mu \left( \tau +\sigma \right) , \label{DWESol}
\end{equation}
where $a_{+}^\mu $ and $a_{-}^\mu $ are now arbitrary vector functions of
one discrete argument, is in fact the general solution of (\ref{DWaveEqu}).
Given a boundary value problem for the continuous equation (\ref{WaveEqu}),
we can restrict the boundary conditions to the discrete boundary points and
find the discrete functions $a_{\pm }^\mu $. Then, from the form of the
solution (\ref{DWESol}) it is clear that $X^\mu \left( \tau ,\sigma \right) $
will coincide, on lattice points, with the solution of the original boundary
value problem. This is what we call the exact solution property of the
discretization (\ref{DWaveEqu}).
Like in the continuous case, the lines $\tau \pm \sigma =const$ are
characteristics of Eq.\ (\ref{DWaveEqu}). It is easily checked that the
gauge conditions (\ref{WEGaugeFix}) are compatible with (\ref{DWaveEqu}), in
the sense that if the discretized gauge conditions (\ref{DWEGaugeFix}) hold
at some point $\left( \tau ,\sigma \right) $, then the evolution equation (%
\ref{DWaveEqu}) guarantees that they hold everywhere along the
characteristic that intersects the point $\left( \tau ,\sigma \right) $.
However, the exact solutions of the equations (\ref{WaveEqu})$-$(\ref
{WEGaugeFix}) do not necessarily satisfy (\ref{DWEGaugeFix}) for any
discretization step $\Delta $, because the continuous constraint (\ref
{WEGaugeFix}) imposed on the derivative of $a_{\pm }^\mu $ is non-linear and
does not integrate to a discrete constraint (\ref{DWESolConstr}). In this
sense, the discrete constraints (\ref{DWEGaugeFix}) provide only an
approximate solution of (\ref{WEGaugeFix}).
We have shown that a particular discretization (\ref{DWaveEqu}) of the wave
equation exactly solves\ the corresponding continuum equation, regardless of
the magnitude of the lattice step $\Delta $. The remaining part of this work
is intended to explain and explore this unexpected phenomenon. The
``mysterious'' cancellation of all higher-order terms in (\ref{DWESeries})
is due to conformal invariance, as will be explored in the next two sections.
\section{Discrete conformal invariance of the wave equation}
\label{SecDCI}
Since the exact solution property is independent of the discretization step $%
\Delta $, we are motivated to explore some kind of scale invariance in the
model. We first notice that Eq.\ (\ref{DWaveEqu}) does not mix the values of
the field on ``even'' and ``odd'' sublattices, each sublattice consisting of
points with $\tau +\sigma $ even or odd. Therefore we shall further consider
only one of these two sublattices, which is itself a square lattice in the
lightcone coordinates $\xi _{\pm }\equiv \tau \pm \sigma $. In Fig.\ 1 we
drew four adjacent squares, each square corresponding to a group of four
adjacent points on the lightcone lattice with the field values $X^\mu $
related by (\ref{DWaveEqu}), for instance
\[
X^\mu \left( a^{\prime }\right) +X^\mu \left( b^{\prime }\right) -X^\mu
\left( c^{\prime }\right) -X^\mu \left( c\right) =0.
\]
By adding together the corresponding equations of motion for all four
squares in Fig.\ 1, it is straightforward to show that the same equation
actually holds for points $a$, $b$, $c$, and $d$ that lie at the vertices of
a larger $2\times 2$ square, i.e.
\begin{equation}
X^\mu \left( a\right) +X^\mu \left( b\right) -X^\mu \left( c\right) -X^\mu
\left( d\right) =0.
\end{equation}
In a similar fashion we find that the discrete equation (\ref{DWaveEqu})
remains valid if we replace the discretization step $\Delta $ by its
multiple $n\Delta $, which means removing all but $n$-th points from the
lattice, and effectively stretching the lattice scale by a factor of $n$. We
could call this fact {\em scale invariance} of the discrete equations. In
fact, an even more general kind of invariance holds: namely, Eq.\ (\ref
{DWaveEqu}) retains its form when applied to the four vertices of any
rectangle on the lightcone lattice (i.e. with sides parallel to the null
directions; such a rectangle would have its sides parallel to the $\xi _{\pm
}$ axes). One can prove this by noticing that, by adding Eq.\ (\ref{DWaveEqu}%
) applied to points $a-e-a^{\prime }-c^{\prime }$ and $a^{\prime }-c^{\prime
}-b^{\prime }-c$, one obtains
\[
X^\mu \left( a\right) +X^\mu \left( b^{\prime }\right) -X^\mu \left(
e\right) -X^\mu \left( c\right) =0,
\]
which is the same equation applied to points $a-b^{\prime }-e^{\prime }-c$
which lie at vertices of a rectangle rather than a square. The same
procedure shows that Eq.\ (\ref{DWaveEqu}) applies to points $f-a^{\prime
}-b-c$ as well. After deriving (\ref{DWaveEqu}) for the ``elementary'' $%
2\times 1$ rectangles, it is clear that, for any given rectangle, one only
needs to add together the relation (\ref{DWaveEqu}) for all squares inside
it to obtain the same relation for the vertices of the rectangle.
The fact that Eq.\ (\ref{DWaveEqu}) applies to vertices of any lattice
rectangle means that if we remove all points lying on any number of lines $%
\tau +\sigma =const$ and $\tau -\sigma =const$ from the lattice, the
discrete equations will still apply to the remaining points. In lightcone
coordinates, such a transformation amounts to replacing the (discrete)
coordinates $\xi _{\pm }$ by functions of themselves:
\begin{equation}
\tilde \xi ^{+}=f^{+}\left( \xi ^{+}\right) ,\quad \tilde \xi
^{-}=f^{-}\left( \xi ^{-}\right) . \label{DCTrans}
\end{equation}
Here, the functions $f^{\pm }\left( \xi ^{\pm }\right) $ are arbitrary
monotonically increasing discrete-valued functions of one discrete argument.
The monotonicity of these functions is necessary to preserve causal
relations on the spacetime.
We immediately notice a similarity between (\ref{DCTrans}) and conformal
transformations in a two-dimensional pseudo-Euclidean space. As is well
known, the wave equation (\ref{WaveEqu}) allows arbitrary conformal
transformations of the world-sheet coordinates,
\begin{equation}
\tilde \xi ^{+}=f^{+}\left( \xi ^{+}\right) ,\quad \tilde \xi
^{-}=f^{-}\left( \xi ^{-}\right) . \label{twodCoordChange}
\end{equation}
This is the most general form of the coordinate transformation that
preserves the pseudo-Euclidean metric
\begin{equation}
ds^2=d\tau ^2-d\sigma ^2=d\xi _{+}d\xi _{-}
\end{equation}
up to a conformal factor. Transformations (\ref{DCTrans}) are obviously the
discrete analog of conformal transformations (\ref{twodCoordChange}) on the
lightcone lattice.
We have, therefore, found that the discrete wave equation (\ref{DWaveEqu})
is invariant under the discrete conformal transformations (\ref{DCTrans}).
We shall refer to this as the {\em discrete conformal invariance} (DCI) of
the discrete wave equation. The natural question is then whether there exist
other equations with the property of DCI, and whether such equations also
deliver exact solutions of their continuous limits. In the remaining
sections, we will answer this question in the positive, establishing a
relation between the exact solution property and DCI.
Note that the constraint equations (\ref{DWEGaugeFix}) do not preserve their
form under discrete conformal transformations. For instance, if the
constraints hold for some step size $\Delta $,
\[
\left[ X^\mu \left( \tau +\Delta ,\sigma \right) -X^\mu \left( \tau ,\sigma
+\Delta \right) \right] ^2=0,\quad \left[ X^\mu \left( \tau ,\sigma +\Delta
\right) -X^\mu \left( \tau -\Delta ,\sigma +2\Delta \right) \right] ^2=0,
\]
it does not follow in general that they also hold for step size $2\Delta $:
\[
\left[ X^\mu \left( \tau +\Delta ,\sigma \right) -X^\mu \left( \tau -\Delta
,\sigma +2\Delta \right) \right] ^2\neq 0.
\]
Accordingly, as we have seen in the previous Section, the discrete
constraint equations do not exactly solve their continuous counterparts.
\section{A general form of a DCI field theory}
\label{SecGen}To try to generalize Eq.\ (\ref{DWaveEqu}), we write the
generic discrete evolution equation on a square lightcone lattice as
\begin{equation}
X\left( \tau +\Delta ,\sigma \right) =F\left[ X\left( \tau ,\sigma +\Delta
\right) ,X\left( \tau ,\sigma -\Delta \right) ,X\left( \tau -\Delta ,\sigma
\right) \right] , \label{DEvol}
\end{equation}
where $F$ is an unknown function. As we have found in the previous Section,
the property of DCI will be satisfied if (\ref{DEvol}) is invariant under
the ``elementary'' conformal transformations, that is, if the relation (\ref
{DEvol}) is valid when applied to the vertices of all $2\times 1$ lattice
rectangles. This requirement can be written as two functional conditions on $%
F$:
\begin{mathletters}
\label{DCIConds}
\begin{eqnarray}
F\left( a,F\left( b,c,d\right) ,b\right) &=&F\left( a,c,d\right) , \\
F\left( F\left( a,c,d\right) ,b,c\right) &=&F\left( a,b,d\right) .
\end{eqnarray}
Here, $a$, $b$, $c$, and $d$ are arbitrary field values which may be scalar
or vector (or even belong to a non-linear manifold of a Lie group, as in our
examples below), and $F$ is a similarly valued function. Before we try to
solve these conditions for $F$, we would like to show that for any function $%
F$ satisfying (\ref{DCIConds}), the discrete evolution equation (\ref{DEvol}%
) exactly solves its continuous limit.
The continuous limit of (\ref{DEvol}) is obtained by expanding it in powers
of $\Delta $, for instance
\end{mathletters}
\begin{equation}
X\left( \tau ,\sigma +\Delta \right) =X_0+X_\sigma \Delta +\frac{X_{\sigma
\sigma }}2\Delta ^2+O\left( \Delta ^3\right) ,
\end{equation}
and with the assumption that $F\left( X_0,X_0,X_0\right) =X_0$, which is
natural if we suppose that a constant function $X=X_0$ must be a solution of
(\ref{DEvol}), we obtain the following equations corresponding to first and
second powers of $\Delta $:
\begin{equation}
X_\tau =F_1X_\sigma -F_2X_\sigma -F_3X_\tau , \label{ContEqu1}
\end{equation}
\begin{equation}
X_{\tau \tau }=\left( F_1+F_2\right) X_{\sigma \sigma }+F_3X_{\tau \tau
}+\left( F_{11}+F_{22}-2F_{12}\right) X_\sigma X_\sigma +\left(
F_{23}-F_{31}\right) \left( X_\sigma X_\tau +X_\tau X_\sigma \right)
+F_{33}X_\tau X_\tau . \label{ContEqu2}
\end{equation}
Here, $F_i$ and $F_{ij}$ are derivatives of $F$ with respect to its three
numbered arguments (which are in general vector arguments, but we suppressed
the indices in the above equations). One can easily verify that for $F\left(
a,b,c\right) =a+b-c$, these equations give the usual wave equation (\ref
{WaveEqu}).
The derivatives $F_i$ and $F_{ii}$ are constrained by Eqs.\ (\ref{DCIConds}%
). For example, the first derivatives satisfy
\begin{equation}
F_1=F_1F_1,\quad F_2=F_2F_2,\quad F_3=-F_1F_2, \label{FirstDerivs}
\end{equation}
where $F_i$ are understood as linear operators on the tangent target space.
It follows that $F_i$ are projection operators (i.e. operators $P$ that obey
$P^2=P$) with eigenvalues $0$ and $1$ only.
Now we shall show that the equations (\ref{ContEqu1})$-$(\ref{ContEqu2}) are
exactly solved by (\ref{DEvol}) if the DCI\ conditions (\ref{DCIConds})
hold. Denote the exact solution of the continuous equations (\ref{ContEqu1})$%
-$(\ref{ContEqu2}) by $X_e\left( \tau ,\sigma \right) $, and the solution of
the discrete equation (\ref{DEvol}) by $X_d\left( \tau ,\sigma \right) $.
Here, $\tau $ and $\sigma $ are discrete lattice coordinates, and we assume
that $X_e=X_d$ on the lines $\tau \pm \sigma =0$ (Fig.\ 2). Since Eqs.\ (\ref
{ContEqu1})$-$(\ref{ContEqu2}) were obtained from (\ref{DEvol}) by expansion
in $\Delta $ up to third-order terms, $X_e$ satisfies the discrete equation (%
\ref{DEvol}) up to $O\left( \Delta ^3\right) $, i.e.
\begin{equation}
\left. X_d-X_e\right| _{\tau =2\Delta ,\sigma =0}=C\left( \Delta \right)
\Delta ^3,\quad C\left( \Delta \right) =C_0+O\left( \Delta \right) ,
\label{CubeError}
\end{equation}
in the limit of small $\Delta $. Now we shall scale up the lattice by an
arbitrarily chosen factor of $n$. As we have seen in the previous Section,
from DCI it follows that Eq.\ (\ref{DEvol}) applies also to the vertices of
an $n\times n$ square on the lightcone lattice, and this is a consequence of
using Eq.\ (\ref{DEvol}) $n^2$ times, once for each ``elementary'' square.
At each elementary square, we can replace the field values $X_d$ by $X_e$
and introduce an error of $C\left( \Delta \right) \Delta ^3$. An error of $%
\delta X$ in $X$ entails an error of not more than $3\delta X$ in $F\left(
X,X,X\right) $, because the first derivatives of $F$ are operators with
eigenvalues of $0$ and $1$ only. Therefore, replacing $X_d$ by $X_e$ in the $%
n\times n$ square entails an error in $X\left( \tau =2n,\sigma =0\right) $
of not more than $3n^2C\left( \Delta \right) \Delta ^3$:
\begin{equation}
\left. X_d-X_e\right| _{\tau =2n\Delta ,\sigma =0}\leq 3n^2C\left( \Delta
\right) \Delta ^3.
\end{equation}
However, we can also apply (\ref{CubeError}) directly to the $n\times n$
square to obtain
\begin{equation}
\left. X_d-X_e\right| _{\tau =2n\Delta ,\sigma =0}=C\left( n\Delta \right)
n^3\Delta ^3.
\end{equation}
It means that
\[
C\left( n\Delta \right) \leq \frac 3nC\left( \Delta \right) ,
\]
which forces $C\left( \Delta \right) =0$ since $C\left( \Delta \right) $ is
a polynomial in $\Delta $. Therefore, the approximation (\ref{CubeError}) is
in fact exact.
This argument shows that it was in fact only necessary to expand (\ref{DEvol}%
) up to second order in $\Delta $, and all higher-order terms will lead to
differential equations which are consequences of (\ref{ContEqu1})$-$(\ref
{ContEqu2}), just as we have seen in the case of the wave equation. It also
shows that the differential equations corresponding to given DCI field
equations (\ref{DEvol}) are always second-order or lower.
Using the evolution equation (\ref{DEvol}), one can write the exact solution
of the continuous equations for boundary conditions on the lightcone $\tau
\pm \sigma =0$. To find $X\left( \tau ,\sigma \right) $ at a point $\left(
\tau ,\sigma \right) $ within the future lightcone of the origin, we
construct a lattice that has $\left( \tau ,\sigma \right) $ as one of its
points, and then apply Eq.\ (\ref{DEvol}) to the rectangle $\left( \tau
,\sigma \right) -\left( \frac{\tau +\sigma }2,\frac{\tau +\sigma }2\right)
-\left( \frac{\tau -\sigma }2,-\frac{\tau -\sigma }2\right) -\left(
0,0\right) $ and explicitly write $X\left( \tau ,\sigma \right) $ through
the boundary values:
\begin{equation}
X\left( \tau ,\sigma \right) =F\left( X\left( \frac{\tau +\sigma }2,\frac{%
\tau +\sigma }2\right) ,X\left( \left( \frac{\tau -\sigma }2,-\frac{\tau
-\sigma }2\right) \right) ,X\left( 0,0\right) \right) . \label{ExactSol}
\end{equation}
Now we try to describe a general class of functions $F$ satisfying (\ref
{DCIConds}). First, we find by combining Eqs.\ (\ref{DCIConds}) that
\begin{equation}
F\left( a,F\left( d,b,c\right) ,c\right) =F\left( F\left( a,d,c\right)
,b,c\right) .
\end{equation}
If we denote
\begin{equation}
a*_cb\equiv F\left( a,b,c\right) , \label{Multi}
\end{equation}
the above will look like an associative law for a binary operation $*_c$:
\begin{equation}
a*_c\left( d*_cb\right) =\left( a*_cd\right) *_cb.
\end{equation}
The condition $F\left( a,a,a\right) =a$ looks like a unity law
\begin{equation}
a*_aa=a,
\end{equation}
although it doesn't follow from (\ref{DCIConds}) that $a*_ab=b$ for all $b$.
However, if we suppose that (\ref{Multi}) is actually an operation of group
multiplication, with the usual unity law
\begin{equation}
a*_ab=b\text{ for all }b,
\end{equation}
and the inverse operation $b^{-1}$, then it is shown in the Appendix that
there exists a reparametrization $r$ of the field values such that the
operation $*_c$ is written as
\begin{equation}
r\left( a*_cb\right) =r\left( a\right) *r\left( c^{-1}\right) *r\left(
b\right) , \label{GenM}
\end{equation}
where by $*$ we denote the group multiplication in the group of field
values. Note that although Eq.\ (\ref{GenM}) is not symmetric with respect
to interchange of $a$ and $b$, such interchange is equivalent to a
reparametrization $r\left( a\right) =a^{-1}$ (where $a^{-1}$ is the group
inverse of $a$).
We arrive at a picture of a field equation of the type (\ref{DEvol}),
derived from a group multiplication in an arbitrary Lie group. We shall
assume that any needed reparametrization is already effected, and that the
evolution is directly given by (\ref{DEvol}) with
\begin{equation}
F\left( a,b,c\right) =a*c^{-1}*b. \label{GenF}
\end{equation}
For example, if we consider a vector space as an Abelian group with addition
of vectors as the operation $*$, we again obtain the formula
\begin{equation}
F\left( a,b,c\right) =a-c+b
\end{equation}
for the discrete wave equation.
In view of Eq.\ (\ref{GenF}), the exact solution (\ref{ExactSol}) has an
algebraically factorized form, as a product of functions of the lightcone
coordinates:
\begin{equation}
X\left( \tau ,\sigma \right) =a\left( \tau +\sigma \right) *b\left( \tau
-\sigma \right) \label{ESolGroup}
\end{equation}
(the constant $c^{-1}$ is absorbed by either of the functions). This is a
characteristic feature of the DCI equations we are concerned with.
The continuous limit of the discrete equations (\ref{DEvol})$-$(\ref{GenF})\
can also be described in terms of an arbitrary Lie group $G$ as a target
space; it is a Wess-Zumino-Witten (WZW)-type model \cite{WZW}. If $X$ is a
field with values in a (matrix) group $G$ defined on a two-dimensional
manifold $M$, the action functional of the WZW model is defined by
\begin{equation}
L=\frac 1{2\lambda }\int_MTr\left( X^{-1}\partial _aX\cdot X^{-1}\partial
^aX\right) d^2M+\frac k{24\pi }\int_B\epsilon ^{abc}Tr\left( X^{-1}\partial
_aX\cdot X^{-1}\partial _bX\cdot X^{-1}\partial _cX\right) d^3B,
\end{equation}
where $B$ is an auxiliary $3$-dimensional manifold whose boundary is $M$,
and $Tr$ is the matrix trace operation. For a specific choice $k=\pm \frac 4%
\lambda $, the equations of motion in the lightcone coordinates become
\begin{equation}
\partial _{\mp }\left( X^{-1}\partial _{\pm }X\right) =0,
\end{equation}
with the general solution
\begin{equation}
X\left( \xi _{+},\xi _{-}\right) =X_{+}\left( \xi _{+}\right) \cdot
X_{-}\left( \xi _{-}\right) \text{ or }X_{-}\left( \xi _{-}\right) \cdot
X_{+}\left( \xi _{+}\right) , \label{WZWSol}
\end{equation}
with $X_{\pm }$ being arbitrary $G$-valued functions. It is immediately seen
that the solution (\ref{WZWSol}) is the same as (\ref{ESolGroup}), if we
(naturally) choose $*$ to be the group multiplication in $G$.
\section{Examples of field theories with DCI}
\label{Examples}In this Section we present some examples to illustrate the
constructions of Sec. \ref{SecGen} and to give a physical interpretation of
the field theories arising from them.
\subsection{Interacting scalar fields}
As was noted in the previous section, we can obtain the discrete wave
equation by using the general formula (\ref{GenF}) on a vector space
considered as an additive group of vectors. Since all commutative Lie groups
are locally isomorphic to a vector space, we should take a non-commutative
group to find a less trivial example. The simplest non-commutative Lie group
has two parameters and can be realized by matrices of the form
\begin{equation}
\left\{ \alpha ,a\right\} \equiv \left(
\begin{array}{cc}
e^\alpha & a \\
0 & 1
\end{array}
\right) . \label{MatrixTwod}
\end{equation}
The composition law of the group is
\begin{equation}
\left(
\begin{array}{cc}
e^\alpha & a \\
0 & 1
\end{array}
\right) \left(
\begin{array}{cc}
e^\beta & b \\
0 & 1
\end{array}
\right) =\left(
\begin{array}{cc}
e^{\alpha +\beta } & a+e^\alpha b \\
0 & 1
\end{array}
\right) ,
\end{equation}
or, written more compactly,
\begin{equation}
\left\{ \alpha ,a\right\} \left\{ \beta ,b\right\} =\left\{ \alpha +\beta
,a+e^\alpha b\right\} . \label{twodMC}
\end{equation}
The inverse element is given by $\left\{ \alpha ,a\right\} ^{-1}=\left\{
-\alpha ,-e^{-\alpha }a\right\} $.
A calculation shows that the continuous limit of the discrete evolution
equation (\ref{GenF}) with the composition law (\ref{twodMC}) is (in
lightcone coordinates $\xi _{\pm }$)
\begin{mathletters}
\label{twodME}
\begin{eqnarray}
\frac{\partial ^2\alpha }{\partial \xi _{+}\partial \xi _{-}} &=&0,
\label{twodMEalpha} \\
\frac{\partial ^2a}{\partial \xi _{+}\partial \xi _{-}} &=&\frac{\partial
\alpha }{\partial \xi _{+}}\frac{\partial a}{\partial \xi _{-}}.
\end{eqnarray}
Note that the non-commutativity of the group leads to asymmetry with respect
to the spatial reflection (i.e. interchange of $\xi _{+}$ and $\xi _{-}$),
but at the same time spatial reflection is equivalent to reparametrization $%
\alpha \rightarrow -\alpha ,$ $a\rightarrow -a\exp \left( -\alpha \right) $
which corresponds to taking the inverse matrix to (\ref{MatrixTwod}).
The exact solution of (\ref{twodME}) is obtained from the general formula (%
\ref{ExactSol}):
\end{mathletters}
\begin{eqnarray}
\alpha \left( \tau ,\sigma \right) &=&\alpha _{+}\left( \tau +\sigma \right)
+\alpha _{-}\left( \tau -\sigma \right) , \\
a\left( \tau ,\sigma \right) &=&a_{+}\left( \tau +\sigma \right) +e^{\alpha
_{+}\left( \tau +\sigma \right) }a_{-}\left( \tau -\sigma \right) ,
\end{eqnarray}
with arbitrary functions $\alpha _{\pm }$ and $a_{\pm }$.
The equations (\ref{twodME}) were obtained from a two-dimensional group and
describe a pair of coupled scalar fields. More generally, one may start with
an arbitrary $n$-dimensional non-commutative group $G$ and construct the
corresponding discrete and continuous equations describing $n$ coupled
scalar fields. The group structure will then be reflected in the coupling of
the fields: for example, if the group $G$ has a commutative subgroup $H$,
then the corresponding parameters will satisfy a free wave equation. This
can be easily seen from the example above: the elements of the form $\left\{
\alpha ,a=0\right\} $ form a commutative subgroup, and the corresponding
equation (\ref{twodMEalpha}) for $\alpha $ is a wave equation.
\subsection{Fermionic fields: the discrete Dirac equation}
So far, we have been dealing with scalar fields. Now we shall consider the
possibility of DCI equations describing fermions. The 2-dimensional massless
Dirac equation for the two-component fermionic field $\psi \left( \xi
^{+},\xi ^{-}\right) $ is
\begin{equation}
\gamma ^a\partial _a\psi =0, \label{twodDirac}
\end{equation}
where the corresponding Dirac matrices satisfy the usual relations of a
Clifford algebra $\left\{ \gamma ^a,\gamma ^b\right\} =2g^{ab}$ and can be
chosen as
\begin{equation}
\gamma ^{-}=\left(
\begin{array}{cc}
0 & 0 \\
1 & 0
\end{array}
\right) ,\quad \gamma ^{+}=\left(
\begin{array}{cc}
0 & 1 \\
0 & 0
\end{array}
\right) . \label{twodDiracGammas}
\end{equation}
Here, the metric $g^{ab}$ in the lightcone coordinates $\xi ^a$ is $%
g^{+-}=g^{-+}=1$, $g^{++}=g^{--}=0$.
With the choice (\ref{twodDiracGammas}), Eq.\ (\ref{twodDirac}) becomes
\begin{equation}
\partial _{+}\psi _1=0,\quad \partial _{-}\psi _2=0, \label{twodDiracLC}
\end{equation}
where $\psi _{1,2}$ are left- and right-moving components of the field $\psi
\left( \xi ^{+},\xi ^{-}\right) $.
The corresponding lattice equations are
\begin{mathletters}
\label{twodDiracD}
\begin{eqnarray}
\psi _1\left( \tau +\Delta ,\sigma \right) -\psi _1\left( \tau ,\sigma
+\Delta \right) &=&0, \\
\psi _2\left( \tau +\Delta ,\sigma \right) -\psi _2\left( \tau ,\sigma
-\Delta \right) &=&0.
\end{eqnarray}
Their solution is
\end{mathletters}
\begin{equation}
\psi _1=f_1\left( \tau +\sigma \right) ,\quad \psi _2=f_2\left( \tau -\sigma
\right) ,
\end{equation}
which is also an exact solution of the continuous equations (\ref
{twodDiracLC}). (Here, $f_{1,2}$ are functions of discrete argument
determined by boundary conditions.)
Since we again discovered a case of an exact solution, we naturally try to
write the lattice equations (\ref{twodDiracD}) in the form (\ref{DEvol})$-$(%
\ref{GenF}). This is possible if we define the multiplication operation $*$
by
\begin{equation}
\left(
\begin{array}{c}
\psi _1 \\
\psi _2
\end{array}
\right) *\left(
\begin{array}{c}
\phi _1 \\
\phi _2
\end{array}
\right) =\left(
\begin{array}{c}
\phi _1 \\
\psi _2
\end{array}
\right) .
\end{equation}
(Note how the left- and right-handed components of the field propagate to
the left and to the right of the multiplication sign.) Such a multiplication
operation is associative, but does not allow a unity element and is not
invertible, which would make it impossible to write $a*c^{-1}*b$ as in (\ref
{GenF}). However, this operation has the property that $a*x*b=a*b$
regardless of the value of $x$, and therefore we can disregard $c^{-1}$ in (%
\ref{GenF}). A matrix representation of this multiplication operation can be
defined by
\begin{equation}
\left(
\begin{array}{c}
\psi _1 \\
\psi _2
\end{array}
\right) \equiv \left(
\begin{array}{cccc}
1 & \psi _1 & & \\
0 & 0 & & \\
& & 1 & 0 \\
& & \psi _2 & 0
\end{array}
\right) . \label{MatrixSemigroup}
\end{equation}
Since the derivations of Sec. \ref{SecGen} only use the associativity of the
multiplication operation $*$, all our considerations apply also to cases
where this operation does not have an inverse, such as in the case of {\em %
semigroups} \cite{Algebra}. We shall be interested in a specific example of
the semigroup structure represented in (\ref{MatrixSemigroup}), where $\psi
_1$ and $\psi _2$ can be, in general, multi-component fields; we shall refer
to such a structure as a ``fermionic semigroup''.
\subsection{Coupled bosons and fermions}
Heuristically, a Lie group generates bosons and a fermionic semigroup
generates fermions in DCI field theories. The direct product of a group and
a semigroup would result in a theory describing uncoupled bosons and
fermions. An example of a model containing coupled bosons and fermions can
be obtained from a semigroup built as a {\em semi-direct product} of a group
and a fermionic semigroup. A semi-direct (or ``twisted'') product of two
semi-groups $S$ and $G$ can be defined if $S$ {\em acts} on $G$, i.e. if for
each $s\in S$ there is a map $s:G\rightarrow G$ such that
\begin{equation}
s\left( g_1*g_2\right) =s\left( g_1\right) *s\left( g_2\right)
\label{MapHom}
\end{equation}
and
\begin{equation}
s_1\left( s_2\left( g\right) \right) =\left( s_1*s_2\right) \left( g\right) .
\label{HomMap}
\end{equation}
(Here, the multiplication denotes the respective semi-group operation in $S$
or $G$, where appropriate.) The semi-direct product of $S$ and $G$ is the
set of pairs $\left\{ s,g\right\} $ with the multiplication defined by
\begin{equation}
\left\{ s_1,g_1\right\} *\left\{ s_2,g_2\right\} \equiv \left\{
s_1*s_2,g_1*s_1\left( g_2\right) \right\} . \label{SemiMult}
\end{equation}
One can easily check that this operation is associative. Of course, a group
is also a semigroup, and the semi-direct product construction can be applied
to two groups or to a group and a semigroup as well.
The existence of an associative binary operation on the target space is
really all we need to build a DCI field theory. We can obtain a generic
theory of this kind containing both bosons and fermions by taking the
fermionic semigroup $S$ and some Lie group $G$ on which $S$ acts. Such pairs
$\left( S,G\right) $ can be constructed for arbitrary dimensions of $S$ and $%
G$. We shall, to illustrate this construction, couple the two previous
examples and arrive to a model with two interacting bosons and three
fermions.
To do this, we take $S$ to be the simplest fermionic semigroup of (\ref
{MatrixSemigroup}). As $G$ we choose a group like one represented by (\ref
{MatrixTwod}), but with two parameters $a_{1,2}$:
\begin{equation}
\left\{ \alpha ,a_1,a_2\right\} \equiv \left(
\begin{array}{ccc}
e^\alpha & a_1 & a_2 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}
\right) .
\end{equation}
To couple bosons with fermions, we need to introduce some action of $S$ on $%
G $. For example, we can multiply the column of parameters $a_{1,2}$ of the
group $G$ by the right-moving part of the matrix $s$,
\begin{equation}
\left(
\begin{array}{cc}
1 & 0 \\
\psi _2 & 0
\end{array}
\right) \left(
\begin{array}{c}
a_1 \\
a_2
\end{array}
\right) =\left(
\begin{array}{c}
a_1 \\
\psi _2a_1
\end{array}
\right) ,
\end{equation}
which defines the action of $\left\{ \psi _1,\psi _2\right\} $ on $\left\{
\alpha ,a_1,a_2\right\} $ as
\begin{equation}
\left(
\begin{array}{c}
\psi _1 \\
\psi _2
\end{array}
\right) \left( \left\{ \alpha ,a_1,a_2\right\} \right) \equiv \left\{ \alpha
,a_1,\psi _2\alpha _1\right\} .
\end{equation}
One can check that the conditions (\ref{MapHom})$-$(\ref{HomMap}) hold for
such an action. The multiplication law of the resulting five-parametric
semigroup is
\begin{equation}
\left\{ \psi _1,\psi _2;\alpha ,a_1,a_2\right\} *\left\{ \phi _1,\phi
_2;\beta ,b_1,b_2\right\} =\left\{ \phi _1,\psi _2;\alpha +\beta
,a_1+e^\alpha b_1,a_2+e^\alpha \psi _2b_1\right\}
\end{equation}
with a matrix realization
\begin{equation}
\left\{ \psi _1,\psi _2;\alpha ,a_1,a_2\right\} \equiv \left(
\begin{array}{cccccc}
e^\alpha & & & & & \\
& e^\alpha & 0 & a_1 & & \\
& e^\alpha \psi _2 & 0 & a_2 & & \\
& 0 & 0 & 1 & & \\
& & & & 1 & \psi _1 \\
& & & & 0 & 0
\end{array}
\right) .
\end{equation}
The multiplication in this semigroup does not allow an inverse operation.
Nevertheless, the problem with $c^{-1}$ in the general formula (\ref{GenF})
is circumvented because we have chosen the twisting of the semigroup and the
group in such a way as to make the product $a*c*b$ independent of the $\psi
_{1,2}$ components of $c$.
The continuous limit equations in this model are
\begin{eqnarray*}
\partial _{+-}\alpha &=&0, \\
\partial _{+}\psi _1 &=&0,\quad \partial _{-}\psi _2=0, \\
\partial _{+-}a_1 &=&\partial _{+}a_1\partial _{-}\alpha , \\
\partial _{+}a_2 &=&\psi _2\partial _{+}a_1.
\end{eqnarray*}
As can be seen from the above, $\psi _{1,2}$ and $a_2$ are fermions, while $%
\alpha $ and $a_1$ are bosons, $a_1$ is coupled to $\alpha $ and $a_2$ is
coupled to $\psi _2$ and $a_1$. (Here, we formally refer to the fields with
first-order equations of motion as ``fermionic''.) Similar models can be
constructed for a larger number of coupled bosonic and fermionic fields.
Note that the field $a_2$ which was a boson in the $\{\alpha ,a_1,a_2\}$
model, became a coupled fermion after we added the fermionic sector.
\section{Conclusions}
\label{SecConcl}
We have explored the phenomenon of the exact solution of continuous
equations by their discretizations. We formulated the property of discrete
conformal invariance (DCI), and showed that any system of lattice equations
possessing DCI delivers exact solutions of its continuous limit. In this
sense, the conformal invariance is the cause of the exact solution property;
the continuous limit equations must also be conformally invariant (although
not all conformally invariant equations are exactly solved by any
discretizations). We found a class of lattice equations, based on Lie group
target space, with the property of DCI; their continuous limit corresponds
to a theory of WZW bosons. We also found a more general class of theories
based on semigroups describing scalar fields and fermions, which can be in
general nonlinearly coupled to each other.
In all these models, solutions to boundary value problems can be written
explicitly (see Eq.\ (\ref{ExactSol})) using the multiplication law of the
group or semigroup at hand. Expressed in this fashion through the boundary
conditions on a lightcone, the solutions are always algebraically factorized.
In case of the wave equation (\ref{WaveEqu}) with gauge constraints (\ref
{WEGaugeFix}), it was found that the discretized constraint equations (\ref
{DWEGaugeFix}) are also exactly solved by certain solutions of the
discretized wave equation, however they are not conformally invariant and,
correspondingly, the exact solutions of the equations (\ref{WaveEqu})$-$(\ref
{WEGaugeFix}) do not necessarily satisfy (\ref{DWEGaugeFix}) for any
discretization step $\Delta $. The ``compatibility'' of the discretized
equations (\ref{DWaveEqu}) and (\ref{DWEGaugeFix}) is perhaps due to the
simple algebraic form of the solutions (\ref{DWESol}).
The author is grateful to Alex Vilenkin for suggesting the problem and for
comments on the manuscript, and to Itzhak Bars, Arvind Borde, Oleg Gleizer,
Leonid Positselsky, and Washington Taylor for helpful and inspiring
discussions.
\section*{Appendix}
Here we show that if the binary operation $a*_cb$ on target space $V$ is not
only associative but is actually a group multiplication in some group $G$,
then the target space can be reparametrized so that the operation $*_c$
becomes, in terms of the group multiplication $*$,
\begin{equation}
a*_cb=a*c^{-1}*b.
\end{equation}
By assumption, for each $c\in V$ there is a one-to-one map $g_c:V\rightarrow
G$ from the target space to the group $G$ such that
\begin{equation}
g_c\left( a*_cb\right) =g_c\left( a\right) *g_c\left( b\right) .
\end{equation}
The first condition of (\ref{DCIConds}) can be written as
\[
a*_b\left( b*_dc\right) =a*_dc,
\]
or, if we take $g_b$ of both parts,
\[
g_b\left( a\right) *g_b\left( b*_dc\right) =g_bg_d^{-1}\left( g_d\left(
a\right) *g_d\left( c\right) \right) .
\]
If we now denote $g_d\left( a\right) \equiv x$ and $g_d\left( c\right)
\equiv y$, where $x$ and $y$ are elements of $G$, this relation becomes
\begin{equation}
g_bg_d^{-1}\left( x\right) *g_bg_d^{-1}\left( g_d\left( b\right) *y\right)
=g_bg_d^{-1}\left( x*y\right) . \label{rel2}
\end{equation}
This shows that $g_bg_d^{-1}$ is almost a group homomorphism, and we can
make it one if we define
\[
h\left( b,d\right) \left( x\right) \equiv g_bg_d^{-1}\left( g_d\left(
b\right) *x\right) ,
\]
where $h\left( b,d\right) $ is a map $G\rightarrow G$. After this (\ref{rel2}%
) becomes
\[
h\left( b,d\right) \left( x\right) *h\left( b,d\right) \left( y\right)
=h\left( b,d\right) \left( x*y\right)
\]
for all $x,y\in G$. Now, obviously $h\left( a,a\right) $ is the identity
map, and $h\left( a,b\right) h\left( b,c\right) =h\left( a,c\right) $ for
all $a,b,c\in V$. This means that $h\left( a,b\right) $ can be expressed as
\[
h\left( a,b\right) =\lambda \left( a\right) \left[ \lambda \left( b\right)
\right] ^{-1},
\]
where $\lambda \left( a\right) $ is an appropriately chosen homomorphism of $%
G$, and $\left[ \lambda \left( b\right) \right] ^{-1}$ is the inverse of the
homomorphism $\lambda \left( b\right) $. For example, we could choose an
arbitrary element $a_0\in V$ and define a map $\lambda :V\rightarrow $Hom$G$
by
\[
\lambda \left( a\right) \equiv h\left( a,a_0\right) .
\]
Now, if we modify the function $g_b\left( a\right) $ by a $\lambda $
transformation:
\[
\tilde g_b\left( a\right) \equiv \left[ \lambda \left( b\right) \right]
^{-1}g_b\left( a\right) ,
\]
then we obtain
\[
\tilde g_c\left( b\right) *\tilde g_b\left( a\right) =\tilde g_c\left(
a\right) ,
\]
which similarly means that $\tilde g_b\left( a\right) $ is of the form
\[
\tilde g_b\left( a\right) =r\left( b\right) *\left[ r\left( a\right) \right]
^{-1},
\]
where $r:V\rightarrow G$ is an appropriately chosen 1-to-1 map, and $\left[
r\left( a\right) \right] ^{-1}$ is the group inverse of $r\left( a\right) $.
Finally, we can put the pieces together and find that
\begin{equation}
F\left( a,b,c\right) =g_c^{-1}\left( g_c\left( a\right) *g_c\left( b\right)
\right) =r^{-1}\left( r\left( a\right) *\left[ r\left( c\right) \right]
^{-1}*r\left( b\right) \right) ,
\end{equation}
which is the desired result.
|
proofpile-arXiv_065-624
|
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
}
|
\section{Introduction}
In the present paper we give a Lie algebraic and differential
geometry derivation of a wide class of multidimensional nonlinear
systems. The systems under consideration are generated by
the zero curvature condition for a connection on a trivial principal
fiber bundle $M \times G \to M$, constrained by the relevant grading
condition. Here $M$ is either the real manifold ${\Bbb R}^{2d}$, or
the complex manifold ${\Bbb C}^d$, $G$ is a complex Lie group,
whose Lie algebra ${\frak g}$ is endowed with a ${\Bbb
Z}$--gradation. We call the arising systems of partial differential
equations the multidimensional Toda type systems. From the physical
point of view, they describe Toda type fields coupled to matter
fields, all of them living on $2d$--dimensional space. Analogously to the
two dimensional situation, with an appropriate In\"on\"u--Wigner
contraction procedure, one can exclude for our systems the back
reaction of the matter fields on the Toda fields.
For the two dimensional case and the finite dimensional Lie algebra
${\frak g}$, connections taking values in the local part of ${\frak
g}$ lead to abelian and nonabelian conformal Toda systems and their
affine deformations for the affine ${\frak g}$, see \cite{LSa92} and
references therein, and also \cite{RSa94,RSa96} for differential and
algebraic geometry background of such systems. For the connection
with values in higher grading subspaces of ${\frak g}$ one deals with
systems discussed in \cite{GSa95,FGGS95}.
In higher dimensions our systems, under some additional
specialisations, contain as particular cases the Cecotti--Vafa type
equations \cite{CVa91}, see also \cite{Dub93}; and those of
Gervais--Matsuo \cite{GMa93} which represent some reduction of a
generalised WZNW model. Note that some of the arising systems are
related to classical problems of differential geometry, coinciding
with the well known completely integrable Bourlet type equations
\cite{Dar10,Bia24,Ami81} and those sometimes called multidimensional
generalisation of the sine--Gordon and wave equations, see, for
example, \cite{Ami81,TTe80,Sav86,ABT86}.
In the paper by the integrability of a system of partial differential
equations we mean the existence of a constructive procedure to obtain
its general solution. Following the lines of
\cite{LSa92,RSa94,GSa95,RSa96}, we formulate the integration scheme for the
multidimensional Toda type systems. In accordance with this scheme,
the multidimensional Toda type and matter type fields are
reconstructed from some mappings which we call integration data. In
the case when $M$ is ${\Bbb C}^d$, the integration data are divided
into holomorphic and antiholomorphic ones; when $M$ is ${\Bbb
R}^{2d}$ they depend on one or another half of the independent
variables. Moreover, in a multidimensional case the integration data
are submitted to the relevant integrability conditions which are
absent in the two dimensional situation. These conditions split into
two systems of multidimensional nonlinear equations for integration
data. If the integrability conditions are integrable systems, then
the corresponding multidimensional Toda type system is also
integrable. We show that in this case any solution of our systems
can be obtained using the proposed integration scheme. It is also
investigated when different sets of integration data give the same
solution.
Note that the results obtained in the present paper can be extended in a
natural way to the case of supergroups.
\section{Derivation of equations}\label{de}
In this section we give a derivation of some class of
multidimensional nonlinear equations. Our strategy here is a direct
generalisation of the method which was used to obtain the Toda type
equations in two dimensional case \cite{LSa92,RSa94,GSa95,RSa96}. It
consists of the following main steps. We consider a general flat
connection on a trivial principal fiber bundle and suppose that the
corresponding Lie algebra is endowed with a ${\Bbb Z}$--gradation.
Then we impose on the connection some grading conditions and prove
that an appropriate gauge transformation allows to bring it to the
form parametrised by a set of Toda type and matter type fields. The
zero curvature condition for such a connection is equivalent to a set
of equations for the fields, which are called the multidimensional
Toda type equations. In principle, the form of the equations in
question can be postulated. However, the derivation given below
suggests also the method of solving these equations, which is
explicitly formulated and discussed in section \ref{cgs}.
\subsection{Flat connections and gauge transformations}
Let $M$ be the manifold ${\Bbb R}^{2d}$ or the manifold ${\Bbb C}^d$.
Denote by $z^{-i}$, $z^{+i}$, $i = 1, \ldots, d$, the standard
coordinates on $M$. In the case when $M$ is ${\Bbb C}^d$ we suppose
that $z^{+i} = \mbar{z^{-i}}$. Let $G$ be a complex connected matrix
Lie group. The generalisation of the construction given below to the
case of a general finite dimensional Lie group is straightforward,
see in this connection \cite{RSa94,RSa96} where such a generalisation was
done for the case of two dimensional space $M$. The general
discussion given below can be also well applied to infinite
dimensional Lie groups. Consider the trivial principal fiber
$G$--bundle $M \times G \to M$. Denote by ${\frak g}$ the Lie algebra
of $G$. It is well known that there is a bijective correspondence
between connection forms on $M \times G \to G$ and ${\frak
g}$--valued 1--forms on $M$. Having in mind this correspondence, we
call a ${\frak g}$--valued 1--form on $M$ a connection form, or
simply a connection. The curvature 2--form of a connection $\omega$
is determined by the 2--form $\Omega$ on $M$, related to $\omega$ by
the formula
\[
\Omega = d\omega + \omega \wedge \omega,
\]
and the connection $\omega$ is flat if and only if
\begin{equation}
d\omega + \omega \wedge \omega = 0. \label{16}
\end{equation}
Relation (\ref{16}) is called the {\it zero curvature condition}.
Let $\varphi$ be a mapping from $M$ to $G$. The connection
$\omega$ of the form
\[
\omega = \varphi^{-1} d \varphi
\]
satisfies the zero curvature condition. In this case one says that
the connection $\omega$ is generated by the mapping $\varphi$. Since
the manifold $M$ is simply connected, any flat connection is
generated by some mapping $\varphi: M \to G$.
The gauge transformations of a connection in the case under
consideration are described by smooth mappings from $M$ to $G$. Here
for any mapping $\psi: M \to G$, the gauge transformed connection
$\omega^\psi$ is given by
\begin{equation}
\omega^\psi = \psi^{-1} \omega \psi + \psi^{-1} d \psi. \label{17}
\end{equation}
Clearly, the zero curvature condition is invariant with respect to
the gauge transformations. In other words, if a connection satisfies
this condition, then the gauge transformed connection also satisfies
this condition. Actually, if a flat connection $\omega$ is generated
by a mapping $\varphi$ then the gauge transformed connection
$\omega^\psi$ is generated by the mapping $\varphi \psi$. It is
convenient to call the gauge transformations defined by (\ref{17}),
{\it $G$--gauge transformations}.
In what follows we deal with a general connection $\omega$ satisfying
the zero curvature condition. Write for $\omega$ the representation
\[
\omega = \sum_{i=1}^d (\omega_{-i} dz^{-i} + \omega_{+i} dz^{+i}),
\]
where $\omega_{\pm i}$ are some mappings from $M$ to ${\frak g}$,
called the components of $\omega$. In terms of $\omega_{\pm i}$ the
zero curvature condition takes the form
\begin{eqnarray}
&\partial_{-i} \omega_{-j} - \partial_{-j} \omega_{-i} +
[\omega_{-i},
\omega_{-j}] = 0,& \label{18} \\
&\partial_{+i} \omega_{+j} - \partial_{+j} \omega_{+i} +
[\omega_{+i},
\omega_{+j}] = 0,& \label{19} \\
&\partial_{-i} \omega_{+j} - \partial_{+j} \omega_{-i} +
[\omega_{-i},
\omega_{+j}] = 0.& \label{20}
\end{eqnarray}
Here and in what follows we use the notation
\[
\partial_{-i} = \partial/\partial z^{-i}, \qquad \partial_{+i} =
\partial/\partial z^{+i}.
\]
Choosing a basis in ${\frak g}$ and treating the components of the
expansion of $\omega_{\pm i}$ over this basis as fields, we can
consider the zero curvature condition as a nonlinear system of
partial differential equations for the fields. Since any flat
connection can be gauge transformed to zero, system
(\ref{18})--(\ref{20}) is, in a sense, trivial. From the other hand,
we
can obtain from (\ref{18})--(\ref{20}) nontrivial integrable systems
by imposing some gauge noninvariant constraints on the connection
$\omega$. Consider one of the methods to impose the constraints in
question, which is, in fact, a direct generalisation of the
group--algebraic approach \cite{LSa92,RSa94,GSa95,RSa96} which was used
successfully in two dimensional case ($d=1$).
\subsection{${\Bbb Z}$--gradations and modified Gauss decomposition}
Suppose that the Lie algebra ${\frak g}$ is a ${\Bbb Z}$--graded Lie
algebra. This means that ${\frak g}$ is represented as the direct sum
\begin{equation}
{\frak g} = \bigoplus_{m \in {\Bbb Z}} {\frak g}_m, \label{1}
\end{equation}
where the subspaces ${\frak g}_m$ satisfy the condition
\[
[{\frak g}_m, {\frak g}_n] \subset {\frak g}_{m+n}
\]
for all $m, n \in {\Bbb Z}$. It is clear that the subspaces ${\frak
g}_0$ and
\[
\widetilde {\frak n}_- = \bigoplus_{m < 0} {\frak g}_m,
\qquad \widetilde {\frak n}_+ = \bigoplus_{m > 0} {\frak g}_m
\]
are subalgebras of ${\frak g}$. Denoting the subalgebra
${\frak g}_0$ by $\widetilde {\frak h}$, we write the generalised
triangle decomposition for ${\frak g}$,
\[
{\frak g} = \widetilde {\frak n}_- \oplus \widetilde {\frak h} \oplus
\widetilde {\frak n}_+.
\]
Here and in what follows we use tildes to have the notations
different from ones usually used for the case of the canonical
gradation of a complex semisimple Lie algebra. Note, that this
gradation is closely related to the so called principal
three--dimensional subalgebra of the Lie algebra under consideration
\cite{Bou75,RSa96}.
Denote by $\widetilde H$ and by $\widetilde N_\pm$ the connected Lie
subgroups corresponding to the subalgebras $\widetilde {\frak h}$ and
$\widetilde {\frak n}_\pm$. Suppose that $\widetilde H$ and
$\widetilde N_\pm$ are closed subgroups of $G$ and, moreover,
\begin{eqnarray}
&\widetilde H \cap \widetilde N_\pm = \{e\}, \qquad \widetilde N_-
\cap \widetilde N_+ = \{e\},& \label{67} \\
&\widetilde N_- \cap \widetilde H \widetilde N_+ = \{e\}, \qquad
\widetilde N_- \widetilde H \cap \widetilde N_+ = \{e\}.& \label{68}
\end{eqnarray}
where $e$ is the unit element of $G$. This is true, in particular,
for the reductive Lie groups, see, for example, \cite{Hum75}. The
set $\widetilde N_- \widetilde H \widetilde N_+$ is an open subset of
$G$. Suppose that
\[
G = \mbar{\widetilde N_- \widetilde H \widetilde N_+}.
\]
This is again true, in particular, for the reductive Lie groups.
Thus,
for an element $a$ belonging to the dense subset of $G$, one has the
following, convenient for our aims, decomposition:
\begin{equation}
a = n_- h n_+^{-1}, \label{69}
\end{equation}
where $n_\pm \in \widetilde N_\pm$ and $h \in \widetilde H$.
Decomposition (\ref{69}) is called the {\it Gauss decomposition}. Due
to
(\ref{67}) and (\ref{68}), this decomposition is unique. Actually,
(\ref{69}) is one of the possible forms of the Gauss decomposition.
Taking the elements belonging to the subgroups $\widetilde N_\pm$ and
$\widetilde H$ in different orders we get different types of the
Gauss
decompositions valid in the corresponding dense subsets of $G$. In
particular, below, besides of decomposition (\ref{69}), we will
often use the Gauss decompositions of the forms
\begin{equation}
a = m_- n_+ h_+, \qquad a = m_+ n_- h_-, \label{70}
\end{equation}
where $m_\pm \in \widetilde N_\pm$, $n_\pm \in \widetilde N_\pm$ and
$h_\pm \in \widetilde H$. The main disadvantage of any form of the
Gauss decomposition is that not any element of $G$ possesses such a
decomposition. To overcome this difficulty, let us consider so called
modified Gauss decompositions. They are based on the following almost
trivial remark. If an element $a \in G$ does not admit the Gauss
decomposition of some form, then, subjecting $a$ to some left shift
in $G$, we can easily get an element admitting that decomposition.
So, in particular, we can say that any element of $G$ can be
represented in forms (\ref{70}) where $m_\pm \in a_\pm \widetilde
N_\pm$ for some elements $a_\pm \in G$, $n_\pm \in \widetilde N_\pm$
and $h_\pm \in \widetilde H$. If the elements $a_\pm$ are fixed, then
decompositions (\ref{70}) are unique. We call the Gauss
decompositions obtained in such a way, the {\it modified Gauss
decompositions} \cite{RSa94,RSa96}.
Let $\varphi: M \to G$ be an arbitrary mapping and $p$ be an
arbitrary point of $M$. Suppose that $a_\pm$ are such elements of $G$
that the element $\varphi(p)$ admits the modified Gauss
decompositions (\ref{70}). It can be easily shown that for any point
$p'$ belonging to some neighborhood of $p$, the element $\varphi(p')$
admits the modified Gauss decompositions (\ref{70}) for the same
choice of the elements $a_\pm$ \cite{RSa94,RSa96}. In other words, any
mapping $\varphi: M \to G$ has the following local decompositions
\begin{equation}
\varphi = \mu_+ \nu_- \eta_-, \qquad \varphi = \mu_- \nu_+ \eta_+,
\label{2}
\end{equation}
where the mappings $\mu_\pm$ take values in $a_\pm \widetilde N_\pm$
for some elements $a_\pm \in G$, the mappings $\nu_\pm$ take values
in $\widetilde N_\pm$, and the mappings $\eta_\pm$ take values in
$\widetilde H$. It is also clear that the mappings $\mu_+^{-1}
\partial_{\pm i} \mu_+$ take values in $\widetilde {\frak n}_+$,
while the mappings $\mu_-^{-1} \partial_{\pm i} \mu_-$ take values in
$\widetilde {\frak n}_-$.
\subsection{Grading conditions}
The first condition we impose on the connection $\omega$ is that the
components $\omega_{-i}$ take values in $\widetilde {\frak n}_-
\oplus
\widetilde {\frak h}$, and the components $\omega_{+i}$ take values
in
$\widetilde {\frak h} \oplus \widetilde {\frak n}_+$. We call this
condition
the {\it general grading condition}.
Let a mapping $\varphi: M \to G$ generates the connection $\omega$;
in other words, $\omega = \varphi^{-1} d \varphi$. Using
respectively the first and the second equalities from (\ref{2}),
we can write the following representations for the connection
components $\omega_{-i}$ and $\omega_{+i}$:
\begin{eqnarray}
&&\omega_{-i} = \eta^{-1}_- \nu^{-1}_- (\mu^{-1}_+ \partial_{-i}
\mu_+)
\nu_- \eta_- + \eta^{-1}_- (\nu^{-1}_- \partial_{-i} \nu_-) \eta_- +
\eta^{-1}_- \partial_{-i} \eta_-, \label{6} \\
&&\omega_{+i} = \eta^{-1}_+ \nu^{-1}_+ (\mu^{-1}_- \partial_{+i}
\mu_-)
\nu_+ \eta_+ + \eta^{-1}_+ (\nu^{-1}_+ \partial_{+i} \nu_+) \eta_+ +
\eta^{-1}_+ \partial_{+i} \eta_+. \label{7}
\end{eqnarray}
{}From these relations it follows that the connection $\omega$
satisfies the general grading condition if and only if
\begin{equation}
\partial_{\pm i} \mu_\mp = 0. \label{8}
\end{equation}
When $M = {\Bbb R}^{2d}$ these equalities mean that $\mu_-$ depends
only on coordinates $z^{-i}$, and $\mu_+$ depends only on coordinates
$z^{+i}$. When $M = {\Bbb C}^d$ they mean that $\mu_-$ is a
holomorphic mapping, and $\mu_+$ is an antiholomorphic one. For a
discussion of the differential geometry meaning of the general
grading condition, which is here actually the same as for two
dimensional case, we refer the reader to \cite{RSa94,RSa96}.
Perform now a further specification of the grading condition. Define
the subspaces $\widetilde {\frak m}_{\pm i}$ of $\widetilde {\frak
n}_\pm$ by
\[
\widetilde {\frak m}_{-i} = \bigoplus_{-l_{-i} \le m \le -1} {\frak
g}_m, \qquad \widetilde {\frak m}_{+i} = \bigoplus_{1 \le m \le
l_{+i}} {\frak g}_m,
\]
where $l_{\pm i}$ are some positive integers. Let us require that
the connection components $\omega_{-i}$ take values in the subspace
$\widetilde {\frak m}_{-i} \oplus \widetilde {\frak h}$, and the
components $\omega_{+i}$ take values in $\widetilde {\frak h} \oplus
\widetilde {\frak m}_{+i}$. We call such a requirement the {\it
specified grading condition}. Using the modified Gauss decompositions
(\ref{2}), one gets
\begin{eqnarray}
&&\omega_{-i} = \eta^{-1}_+ \nu^{-1}_+ (\mu^{-1}_- \partial_{-i}
\mu_-)
\nu_+ \eta_+ + \eta^{-1}_+ (\nu^{-1}_+ \partial_{-i} \nu_+) \eta_+ +
\eta^{-1}_+ \partial_{-i} \eta_+, \label{3} \\
&&\omega_{+i} = \eta^{-1}_- \nu^{-1}_- (\mu^{-1}_+ \partial_{+i}
\mu_+)
\nu_- \eta_- + \eta^{-1}_- (\nu^{-1}_- \partial_{+i} \nu_-) \eta_- +
\eta^{-1}_- \partial_{+i} \eta_-. \label{4}
\end{eqnarray}
Here the second equality from (\ref{2}) was used for $\omega_{-i}$
and the first one for $\omega_{+i}$. From relations (\ref{3}) and
(\ref{4}) we conclude that the connection $\omega$ satisfies the
specified grading condition if and only if the mappings $\mu_-^{-1}
\partial_{-i} \mu_-$ take values in $\widetilde {\frak m}_{-i}$, and
the mappings $\mu_+^{-1} \partial_{+i} \mu_+$ take values in
$\widetilde {\frak m}_{+i}$.
It is clear that the general grading condition and the specified
grading condition are not invariant under the action of an arbitrary
$G$--gauge transformation, but they are invariant under the action of
gauge transformations (\ref{17}) with the mapping $\psi$ taking
values in the subgroup $\widetilde H$. In other words, the system
arising from the zero curvature condition for the connection
satisfying the specified grading condition still possesses some gauge
symmetry. Below we call a gauge transformation (\ref{17}) with the
mapping $\psi$ taking values in $\widetilde H$ an {\it $\widetilde
H$--gauge transformations}. Let us impose now one more restriction on
the connection and use the $\widetilde H$--gauge symmetry to bring it
to the form generating equations free of the $\widetilde H$--gauge
invariance.
\subsection{Final form of connection}
Taking into account the specified grading condition, we write the
following representation for the components of the connection
$\omega$:
\[
\omega_{-i} = \sum_{m = 0}^{-l_{-i}} \omega_{-i, m}, \qquad
\omega_{+i} = \sum_{m = 0}^{l_{+i}} \omega_{+i, m},
\]
where the mappings $\omega_{\pm i, m}$ take values in ${\frak g}_{\pm
m}$. There is a similar decomposition for the mappings $\mu_\pm^{-1}
\partial_{\pm i} \mu_\pm$:
\[
\mu^{-1}_- \partial_{-i} \mu_- = \sum_{m = -1}^{-l_{-i}} \lambda_{-i,
m}, \qquad \mu^{-1}_+ \partial_{+i} \mu_+ = \sum_{m = 1}^{l_{+i}}
\lambda_{+i, m}.
\]
{}From (\ref{3}) and (\ref{4}) it follows that
\begin{equation}
\omega_{\pm i, \pm l_{\pm i}} = \eta_\mp^{-1} \lambda_{\pm i, \pm
l_{\pm i}}
\eta_\mp. \label{5}
\end{equation}
The last restriction we impose on the connection $\omega$ is
formulated as follows. Let $c_{\pm i}$ be some fixed elements of the
subspaces ${\frak g}_{\pm l_{\pm i}}$ satisfying the relations
\begin{equation}
[c_{-i}, c_{-j}] = 0, \qquad [c_{+i}, c_{+j}] = 0. \label{39}
\end{equation}
Require that the mappings $\omega_{\pm i, \pm l_{\pm i}}$ have the
form
\begin{equation}
\omega_{\pm i, \pm l_{\pm i}} = \eta_\mp^{-1} \gamma_\pm c_{\pm i}
\gamma_\pm^{-1} \eta_\mp \label{21}
\end{equation}
for some mappings $\gamma_\pm: M \to \widetilde H$. A connection
which
satisfies the grading condition and relation (\ref{21}) is called an
{\it admissible connection}. Similarly, a mapping from $M$ to $G$
generating an admissible connection is called {\it an admissible
mapping}.
Taking into account (\ref{5}), we conclude that
\begin{equation}
\lambda_{\pm i, \pm l_{\pm i}} = \gamma_\pm c_{\pm i}
\gamma^{-1}_\pm.
\label{11}
\end{equation}
Denote by $\widetilde H_-$ and $\widetilde H_+$ the isotropy
subgroups of the sets formed by the elements $c_{-i}$ and $c_{+i}$,
respectively. It is clear that the mappings $\gamma_\pm$ are defined
up to multiplication from the right side by mappings taking values in
$\widetilde H_\pm$. In any case, at least locally, we can choose the
mappings $\gamma_\pm$ in such a way that
\begin{equation}
\partial_{\mp i} \gamma_\pm = 0. \label{38}
\end{equation}
In what follows we use such a choice for the mappings $\gamma_\pm$.
Let us show now that there exists a local $\widetilde H$--gauge
transformation that brings an admissible connection to the connection
$\omega$ with the components of the form
\begin{eqnarray}
&\omega_{-i} = \gamma^{-1} \partial_{-i} \gamma + \sum_{m =
-1}^{-l_{-i} + 1} \upsilon_{-i,m} + c_{-i},& \label{14} \\
&\omega_{+i} = \gamma^{-1} \left(\sum_{m = 1}^{l_{+i} -1}
\upsilon_{+i, m} +
c_{+i} \right) \gamma,& \label{15}
\end{eqnarray}
where $\gamma$ is some mapping from $M$ to $\widetilde H$, and
$\upsilon_{\pm i, m}$ are mappings taking values in ${\frak g}_{\pm
m}$.
To prove the above statement, note first that taking into account
(\ref{8}), we get from (\ref{6}) and (\ref{7}) the following
relations
\begin{eqnarray}
&&\omega_{-i} = \eta^{-1}_- (\nu^{-1}_- \partial_{-i} \nu_-) \eta_-
+
\eta^{-1}_- \partial_{-i} \eta_-, \label{9} \\
&&\omega_{+i} = \eta^{-1}_+ (\nu^{-1}_+ \partial_{+i} \nu_+) \eta_+ +
\eta^{-1}_+ \partial_{+i} \eta_+. \label{10}
\end{eqnarray}
Comparing (\ref{9}) and (\ref{3}), we come to the relation
\[
\nu_-^{-1} \partial_{-i} \nu_- = \left[ \eta \nu_+^{-1} (\mu_-^{-1}
\partial_{-i} \mu_-) \nu_+ \eta^{-1} \right]_{\widetilde {\frak
n}_-},
\]
where
\begin{equation}
\eta = \eta_- \eta_+^{-1}. \label{36}
\end{equation}
Hence, the mappings $\nu_-^{-1} \partial_{-i} \nu_-$ take values in
subspaces ${\frak m}_{- i}$ and we can represent them in the form
\[
\nu_-^{-1} \partial_{-i} \nu_- = \eta \gamma_- \left( \sum_{m =
-1}^{-l_{-i}} \upsilon_{-i,m} \right) \gamma_-^{-1} \eta^{-1},
\]
with the mappings $\upsilon_{-i, m}$ taking values in ${\frak g}_{-
m}$. Substituting this representation into (\ref{9}), we obtain
\[
\omega_{-i} = \eta_+^{-1} \gamma_- \left( \sum_{m = -1}^{-l_{-i}}
\upsilon_{-i, m} \right) \gamma_-^{-1} \eta_+ + \eta_-^{-1}
\partial_{-i} \eta_-.
\]
{}From (\ref{5}) and (\ref{11}) it follows that $\upsilon_{-i, -l_{-i}}
= c_{-i}$. Therefore,
\begin{equation}
\omega_{-i} = \eta_+^{-1} \gamma_- \left(c_{-i} + \sum_{m =
-1}^{-l_{-i}
+ 1} \upsilon_{-i, m} \right) \gamma_-^{-1} \eta_+ + \eta_-^{-1}
\partial_{-i} \eta_-. \label{12}
\end{equation}
Similarly, using (\ref{10}) and (\ref{4}), we conclude that
\[
\nu_+^{-1} \partial_{+i} \nu_+ = \left[ \eta^{-1} \nu_-^{-1}
(\mu_+^{-1}
\partial_{+i} \mu_+) \nu_- \eta \right]_{\widetilde {\frak n}_+}.
\]
Therefore we can write for $\nu_+^{-1} \partial_{+i} \nu_+$ the
representation
\[
\nu_+^{-1} \partial_{+i} \nu_+ = \eta^{-1} \gamma_+ \left( \sum_{m =
1}^{l_{+i}} \upsilon_{+i,m} \right) \gamma_+^{-1} \eta,
\]
where the mappings $\upsilon_{+i, m}$ take values in ${\frak g}_m$.
Taking into account (\ref{10}), we get
\[
\omega_{+i} = \eta_-^{-1} \gamma_+ \left( \sum_{m = 1}^{l_{+i}}
\upsilon_{+i, m} \right) \gamma_+^{-1} \eta_- + \eta_+^{-1}
\partial_{+i} \eta_+.
\]
Using again (\ref{5}) and (\ref{11}), we obtain $\upsilon_{+i,
l_{+i}} = c_{+i}$. Therefore, the following relation is valid:
\begin{equation}
\omega_{+i} = \eta_-^{-1} \gamma_+ \left(\sum_{m = 1}^{l_{+i} -1}
\upsilon_{+i, m} + c_{+i} \right) \gamma_+^{-1} \eta_- +
\eta_+^{-1} \partial_{+i} \eta_+. \label{13}
\end{equation}
Taking into account (\ref{12}) and (\ref{13}) and performing the
gauge transformation defined by the mapping $\eta_+^{-1} \gamma_-$,
we
arrive at the connection with the components of the form given by
(\ref{14}) and (\ref{15}) with
\begin{equation}
\gamma = \gamma_+^{-1} \eta \gamma_-. \label{40}
\end{equation}
Note that the connection with components (\ref{14}), (\ref{15}) is
generated by the mapping
\begin{equation}
\varphi = \mu_+ \nu_- \eta \gamma_- = \mu_- \nu_+ \gamma_-.
\label{95}
\end{equation}
\subsection{Multidimensional Toda type equations}
The equations for the mappings $\gamma$ and $\upsilon_{\pm i,m}$,
which result from the zero curvature condition (\ref{18})--(\ref{20})
with the connection components of form (\ref{14}), (\ref{15}), will
be
called {\it multidimensional Toda type equations}, or {\it
multidimensional Toda type systems}. It is natural to call the
functions parametrising the mappings $\gamma$ and $\upsilon_{\pm
i,m}$, {\it Toda type} and {\it matter type fields}, respectively.
The multidimensional Toda type equations are invariant with respect
to the remarkable symmetry transformations
\begin{equation}
\gamma' = \xi_+^{-1} \gamma \xi_-, \qquad \upsilon'_{\pm i} =
\xi_\pm^{-1} \upsilon_{\pm i} \xi_\pm, \label{91}
\end{equation}
where $\xi_\pm$ are arbitrary mappings taking values in the isotropy
subgroups $\widetilde H_\pm$ of the sets formed by the elements
$c_{-i}$ and $c_{+i}$, and satisfying the relations
\begin{equation}
\partial_\mp \xi_\pm = 0. \label{92}
\end{equation}
Indeed, it can be easily verified that the connection components of
form (\ref{14}), (\ref{15}) constructed with the mappings $\gamma$,
$\upsilon_{\pm i}$ and $\gamma'$, $\upsilon'_{\pm i}$ are connected
by the $\widetilde H$--gauge transformation generated by the mapping
$\xi_-$. Therefore, if the mappings $\gamma$, $\upsilon_{\pm i}$
satisfy the multidimensional Toda type equations, then the mappings
$\gamma'$, $\upsilon'_{\pm i}$ given by (\ref{91}) satisfy the same
equations. Note that, because the mappings $\xi_\pm$ are subjected to
(\ref{92}), transformations (\ref{91}) are not {\it gauge} symmetry
transformations of the multidimensional Toda type equations.
Let us make one more useful remark. Let $h_\pm$ be some fixed
elements of $\widetilde H$, and mappings $\gamma$, $\upsilon_{\pm i}$
satisfy the multidimensional Toda type equations generated by the
connection with the components of form (\ref{14}), (\ref{15}). It is
not difficult to get convinced that the mappings
\[
\gamma' = h_+^{-1} \gamma h_-, \qquad \upsilon'_{\pm i} = h_\pm^{-1}
\upsilon_{\pm i} h_\pm
\]
satisfy the multidimensional Toda type equations where instead of
$c_{\pm i}$ one uses the elements
\[
c'_{\pm i} = h_\pm^{-1} c_{\pm i} h_\pm.
\]
In such a sense, the multidimensional Toda type equations determined
by the elements $c_{\pm i}$ and $c'_{\pm i}$ which are connected by
the above relation, are equivalent.
Let us write the general form of the multidimensional Toda type
equations for $l_{-i} = l_{+i} = 1$ and $l_{-i} = l_{+i} = 2$. The
cases with other choices of $l_{\pm i}$ can be treated similarly.
Consider first the case $l_{-i} = l_{+i} = 1$. Here the connection
components have the form
\[
\omega_{-i} = \gamma^{-1} \partial_{-i} \gamma + c_{-i}, \qquad
\omega_{+i} = \gamma^{-1} c_{+i} \gamma.
\]
Equations (\ref{18}) are equivalent here to the following ones:
\begin{eqnarray}
&{}[c_{-i}, \gamma^{-1} \partial_{-j} \gamma] + [\gamma^{-1}
\partial_{-i} \gamma, c_{-j}] = 0,& \label{25} \\
&\partial_{-i} (\gamma^{-1} \partial_{-j} \gamma) - \partial_{-j}
(\gamma^{-1} \partial_{-i} \gamma) + [\gamma^{-1} \partial_{-i}
\gamma, \gamma^{-1} \partial_{-j} \gamma] = 0.& \label{26}
\end{eqnarray}
Equations (\ref{26}) are satisfied by any mapping $\gamma$;
equations (\ref{25}) can be identically rewritten as
\begin{equation}
\partial_{-i} (\gamma c_{-j} \gamma^{-1}) = \partial_{-j} (\gamma
c_{-i} \gamma^{-1}). \label{22}
\end{equation}
Analogously, equations (\ref{19}) read
\begin{equation}
\partial_{+i} (\gamma^{-1} c_{+j} \gamma) = \partial_{+j}
(\gamma^{-1}
c_{+i} \gamma). \label{23}
\end{equation}
Finally, we easily get convinced that equations (\ref{20}) can be
written as
\begin{equation}
\partial_{+j} (\gamma^{-1} \partial_{-i} \gamma) = [c_{-i},
\gamma^{-1}
c_{+j} \gamma]. \label{24}
\end{equation}
Thus, the zero curvature condition in the case under consideration is
equivalent to equations (\ref{22})--(\ref{24}). In the two
dimensional
case equations (\ref{22}) and (\ref{23}) are absent, and equations
(\ref{24}) take the form
\[
\partial_+ (\gamma^{-1} \partial_- \gamma) = [c_-, \gamma^{-1}
c_+ \gamma].
\]
If the Lie group $G$ is semisimple, then using the canonical
gradation of the corresponding Lie algebra ${\frak g}$, we get the
well known abelian Toda equations; noncanonical gradations lead to
various nonabelian Toda systems.
Proceed now to the case $l_{-i} = l_{+i} = 2$. Here the connection
components are
\[
\omega_{-i} = \gamma^{-1} \partial_{-i} \gamma + \upsilon_{-i} +
c_{-i}, \qquad \omega_{+i} = \gamma^{-1} (\upsilon_{+i} + c_{+i})
\gamma,
\]
where we have denoted $\upsilon_{\pm i, \pm 1}$ simply by
$\upsilon_{\pm i}$. Equations (\ref{18}) take the form
\begin{eqnarray}
&[c_{-i}, \upsilon_{-j}] = [c_{-j}, \upsilon_{-i}],& \label{27} \\
&\partial_{-i}(\gamma c_{-j} \gamma^{-1}) - \partial_{-j} (\gamma
c_{-i} \gamma^{-1}) = [\gamma \upsilon_{-j} \gamma^{-1}, \gamma
\upsilon_{-i} \gamma^{-1}],& \label{28} \\
&\partial_{-i} (\gamma \upsilon_{-j} \gamma^{-1}) = \partial_{-j}
(\gamma \upsilon_{-i} \gamma^{-1}).& \label{29}
\end{eqnarray}
The similar system of equations follows from (\ref{19}),
\begin{eqnarray}
&[c_{+i}, \upsilon_{+j}] = [c_{+j}, \upsilon_{+i}],& \label{30} \\
&\partial_{+i}(\gamma^{-1} c_{+j} \gamma) - \partial_{+j}
(\gamma^{-1} c_{+i} \gamma) = [\gamma^{-1} \upsilon_{+j} \gamma,
\gamma^{-1} \upsilon_{+i} \gamma],& \label{31} \\
&\partial_{+i} (\gamma^{-1} \upsilon_{+j} \gamma) = \partial_{+j}
(\gamma^{-1} \upsilon_{+i} \gamma).& \label{32}
\end{eqnarray}
After some calculations we get from (\ref{20}) the equations
\begin{eqnarray}
&\partial_{-i} \upsilon_{+j} = [c_{+j}, \gamma \upsilon_{-i}
\gamma^{-1}],& \label{33} \\
&\partial_{+j} \upsilon_{-i} = [c_{-i}, \gamma^{-1} \upsilon_{+j}
\gamma],& \label{34} \\
&\partial_{+j} (\gamma^{-1} \partial_{-i} \gamma) = [c_{-i},
\gamma^{-1}
c_{+j} \gamma] + [\upsilon_{-i}, \gamma^{-1} \upsilon_{+j} \gamma].&
\label{35}
\end{eqnarray}
Thus, in the case $l_{-i} = l_{+i} = 2$ the zero curvature condition
is equivalent to the system of equations (\ref{27})--(\ref{35}). In
the two dimensional case we come to the equations
\begin{eqnarray*}
&\partial_- \upsilon_+ = [c_+, \gamma \upsilon_-
\gamma^{-1}], \qquad \partial_+ \upsilon_- = [c_-, \gamma^{-1}
\upsilon_+ \gamma],& \\
&\partial_+ (\gamma^{-1} \partial_- \gamma) = [c_-, \gamma^{-1}
c_+ \gamma] + [\upsilon_-, \gamma^{-1} \upsilon_+ \gamma],&
\end{eqnarray*}
which represent the simplest case of higher grading Toda systems
\cite{GSa95}.
\section{Construction of general solution}\label{cgs}
{}From the consideration presented above it follows that any admissible
mapping generates local solutions of the corresponding
multidimensional Toda type equations. Thus, if we were to be able to
construct admissible mappings, we could construct solutions of the
multidimensional Toda type equations. It is worth to note here that
the solutions in questions are determined by the mappings $\mu_\pm$,
$\nu_\pm$ entering Gauss decompositions (\ref{2}), and from the
mapping $\eta$ which is defined via
(\ref{36}) by the mappings $\eta_\pm$ entering the same
decomposition. So, the problem is to find the mappings $\mu_\pm$,
$\nu_\pm$ and $\eta$ arising from admissible mappings by means of Gauss
decompositions (\ref{2}) and relation (\ref{36}). It appears that
this problem has a remarkably simple solution.
Recall that a mapping $\varphi: M \to G$ is admissible if and
only if the mappings $\mu_\pm$ entering Gauss decompositions
(\ref{2}) satisfy conditions (\ref{8}), and the mappings
$\mu_\pm^{-1}
\partial_{\pm i} \mu_\pm$ have the form
\begin{eqnarray}
&\mu^{-1}_- \partial_{-i} \mu_- = \gamma_- c_{-i} \gamma_-^{-1} +
\sum_{m = -1}^{-l_{-i} + 1 } \lambda_{-i, m},& \label{41} \\
&\mu^{-1}_+ \partial_{+i} \mu_+ = \sum_{m = 1}^{l_{+i} - 1}
\lambda_{+i, m} + \gamma_+ c_{+i} \gamma_+^{-1}.& \label{42}
\end{eqnarray}
Here $\gamma_\pm$ are some mappings taking values in $\widetilde H$
and satisfying conditions (\ref{38}); the mappings $\lambda_{\pm i,
m}$ take values in ${\frak g}_{\pm m}$; and $c_{\pm i}$ are the fixed
elements of the subspaces ${\frak g}_{\pm l_\pm}$, which satisfy
relations (\ref{39}).
{}From the other hand, the mappings $\mu_\pm$ uniquely determine the
mappings $\nu_\pm$ and $\eta$. Indeed, from (\ref{2}) one gets
\begin{equation}
\mu_+^{-1} \mu_- = \nu_- \eta \nu_+^{-1}. \label{37}
\end{equation}
Relation (\ref{37}) can be considered as the Gauss decomposition of
the mapping $\mu_+^{-1} \mu_-$ induced by the Gauss decomposition
(\ref{69}). Hence, the mappings $\mu_\pm$ uniquely determine the
mappings $\nu_\pm$ and $\eta$.
Taking all these remarks into account we propose the following
procedure for obtaining solutions to the multidimensional Toda type
equations.
\subsection{Integration scheme}\label{is}
Let $\gamma_\pm$ be some mappings taking values in $\widetilde H$,
and $\lambda_{\pm i, m}$ be some mappings taking values in ${\frak
g}_{\pm m}$. Here it is supposed that
\begin{equation}
\partial_{\mp i} \gamma_\pm = 0, \qquad \partial_{\mp i} \lambda_{\pm
j, m} = 0. \label{56}
\end{equation}
Consider (\ref{41}) and (\ref{42}) as a system of partial
differential equations for the mappings $\mu_\pm$ and try to solve
it. Since we are going to use the mappings $\mu_\pm$ for
construction of admissible mappings, we have to deal only with
solutions of equations (\ref{41}) and (\ref{42}) which satisfy
relations (\ref{8}). The latter are equivalent to the following ones:
\begin{eqnarray}
&\mu_-^{-1} \partial_{+ i} \mu_- = 0,& \label{93} \\
&\mu_+^{-1} \partial_{- i} \mu_+ = 0.& \label{94}
\end{eqnarray}
So, we have to solve the system consisting of
equations (\ref{41}), (\ref{42}) and (\ref{93}), (\ref{94}).
Certainly, it is possible to solve this system if and only if the
corresponding integrability conditions are satisfied. The right hand sides of
equations (\ref{41}), (\ref{93}) and (\ref{42}), (\ref{94}) can be
interpreted as components of flat connections on the trivial
principal fiber bundle $M \times G \to M$. Therefore, the
integrability conditions of equations (\ref{41}), (\ref{93}) and
(\ref{42}), (\ref{94}) look as the zero curvature condition for these
connections. In particular, for the case $l_{-i} = l_{+i} = 2$ the
integrability conditions are
\begin{eqnarray*}
&\partial_{\pm i} \lambda_{\pm j} = \partial_{\pm j} \lambda_{\pm
i},& \\
&\partial_{\pm i}(\gamma_\pm c_{\pm j} \gamma_\pm^{-1}) -
\partial_{\pm
j}(\gamma_\pm c_{\pm i} \gamma_\pm^{-1}) = [\lambda_{\pm j},
\lambda_{\pm
i}],& \\
&[\lambda_{\pm i}, \gamma_\pm c_{\pm j} \gamma_\pm ^{-1}] =
[\lambda_{\pm j}, \gamma_\pm c_{\pm i} \gamma_\pm ^{-1}],&
\end{eqnarray*}
where we have denoted $\lambda_{\pm i, 1}$ simply by $\lambda_{\pm
i}$.
In general, the integrability conditions can be considered as two
systems of partial nonlinear differential equations for the mappings
$\gamma_-$, $\lambda_{-i, m}$ and $\gamma_+$, $\lambda_{+i, m}$,
respectively. The multidimensional Toda type equations are integrable
if and only if these systems are integrable. In any case, if we
succeed to find a solution of the integrability conditions, we can
construct the corresponding solution of the multidimensional Toda
type equations. A set of mappings $\gamma_\pm$ and $\lambda_{\pm i,
m}$ satisfying (\ref{56}) and the corresponding integrability
conditions will be called {\it integration data}. It is clear that
for any set of integration data the solution of equations (\ref{41}),
(\ref{93}) and (\ref{42}), (\ref{94}) is fixed by the initial
conditions which are constant elements of the group $G$. More
precisely, let $p$ be some fixed point of $M$ and $a_\pm$ be some
fixed elements of $G$. Then there exists a unique solution of
equations (\ref{41}), (\ref{93}) and (\ref{42}), (\ref{94})
satisfying the conditions
\begin{equation}
\mu_\pm (p) = a_\pm. \label{72}
\end{equation}
It is not difficult to show that the mappings $\mu_\pm$ satisfying
the equations under consideration and initial conditions (\ref{72})
take values in $a_\pm \widetilde N_\pm$. Note that in the two
dimensional case the integrability conditions become trivial.
The next natural step is to use Gauss decomposition (\ref{37}) to
obtain the mappings $\nu_\pm$ and $\eta$. In general, solving
equations (\ref{41}), (\ref{93}) and (\ref{42}), (\ref{94}), we get
the mappings $\mu_\pm$, for which the mapping $\mu_+^{-1} \mu_-$ may
have not the Gauss decomposition of form (\ref{37}) at some points of
$M$. In such a case one comes to solutions of the
multidimensional Toda type equations with some irregularities.
Having found the mappings $\mu_\pm$ and $\eta$, one uses (\ref{40})
and the relations
\begin{eqnarray}
&\sum_{m = -1}^{-l_{-i}} \upsilon_{-i,m} = \gamma_-^{-1} \eta^{-1}
(\nu_-^{-1} \partial_{-i} \nu_-) \eta \gamma_-,& \label{52} \\
&\sum_{m = 1}^{l_{+i}} \upsilon_{+i,m} = \gamma_+^{-1} \eta
(\nu_+^{-1} \partial_{+i} \nu_+) \eta^{-1} \gamma_+& \label{53}
\end{eqnarray}
to construct the mappings $\gamma$ and $\upsilon_{\pm i, m}$. Show
that these mappings satisfy the multidimensional Toda type equations.
To this end consider the mapping
\[
\varphi = \mu_+ \nu_- \eta \gamma_- = \mu_- \nu_+ \gamma_-,
\]
whose form is actually suggested by (\ref{95}). The mapping $\varphi$
is admissible. Moreover, using formulas of section \ref{de}, it is
not difficult to demonstrate that it generates the connection with
components of form (\ref{14}) and (\ref{15}), where the mappings
$\gamma$ and $\upsilon_{\pm i, m}$ are defined by the above
construction. Since this connection is certainly flat, the mappings
$\gamma$ and $\upsilon_{\pm i, m}$ satisfy the multidimensional Toda
type equations.
\subsection{Generality of solution}\label{gs}
Prove now that any solution of the multidimensional Toda type
equations can be obtained by the integration scheme described above.
Let $\gamma: M \to \widetilde H$ and $\upsilon_{\pm i, m}: M \to
{\frak g}_{\pm m}$ be arbitrary mappings satisfying the
multidimensional Toda type equations. We have to show that there
exists a set of integration data leading, by the above integration
scheme, to the mappings $\gamma$ and $\upsilon_{\pm i, m}$.
Using $\gamma$ and $\upsilon_{\pm i, m}$, construct the
connection with the components given by (\ref{14}) and (\ref{15}).
Since this connection is flat and admissible, there exists an
admissible mapping $\varphi: M \to G$ which generates it. Write for
$\varphi$ local Gauss decompositions (\ref{2}). The mappings
$\mu_\pm$ entering these decompositions satisfy relations (\ref{8}).
Since the mapping $\varphi$ is admissible, we have expansions
(\ref{41}), (\ref{42}). It is convenient to write them in
the form
\begin{eqnarray}
&\mu^{-1}_- \partial_{-i} \mu_- = \gamma'_- c_{-i} \gamma_-^{\prime
-1} +
\sum_{m = -1}^{-l_{-i} + 1 } \lambda_{-i, m},& \label{54} \\
&\mu^{-1}_+ \partial_{+i} \mu_+ = \sum_{m = 1}^{l_{+i} - 1}
\lambda_{+i, m} + \gamma'_+ c_{+i} \gamma_+^{\prime -1},& \label{55}
\end{eqnarray}
where we use primes because, in general, the mappings $\gamma'_\pm$
are not yet the mappings leading to the considered solution of the
multidimensional Toda type equations. Choose the mappings
$\gamma'_\pm$ in such a way that
\[
\partial_{\mp i} \gamma'_\pm = 0.
\]
Formulas (\ref{12}) and (\ref{13}) take in our case the
form
\begin{eqnarray}
&\omega_{-i} = \eta_+^{-1} \gamma'_- \left(c_{-i} + \sum_{m =
-1}^{-l_{-i}
+ 1} \upsilon'_{-i, m} \right) \gamma_-^{\prime -1} \eta_+ +
\eta_-^{-1} \partial_{-i} \eta_-,& \label{43} \\
&\omega_{+i} = \eta_-^{-1} \gamma'_+ \left(\sum_{m = 1}^{l_{+i} -1}
\upsilon'_{+i, m} + c_{+i} \right) \gamma_+^{\prime -1} \eta_- +
\eta_+^{-1} \partial_{+i} \eta_+,& \label{44}
\end{eqnarray}
where the mappings $\upsilon'_{\pm i, m}$ are defined by the
relations
\begin{eqnarray}
&\sum_{m = -1}^{-l_{-i}} \upsilon'_{-i,m} = \gamma_-^{\prime -1}
\eta^{-1} (\nu_-^{-1} \partial_{-i} \nu_-) \eta \gamma'_-,&
\label{50} \\
&\sum_{m = 1}^{l_{+i}} \upsilon'_{+i,m} = \gamma_+^{\prime -1} \eta
(\nu_+^{-1} \partial_{+i} \nu_+) \eta^{-1} \gamma'_+.& \label{51}
\end{eqnarray}
{}From (\ref{44}) and (\ref{15}) it follows that the mapping $\eta_+$
satisfies the relation
\[
\partial_{+ i} \eta_+ = 0.
\]
Therefore, for the mapping
\[
\xi_- = \gamma_-^{\prime -1} \eta_+
\]
one has
\[
\partial_{+ i} \xi_- = 0.
\]
Comparing (\ref{43}) and (\ref{14}) one sees that the mapping $\xi_-$
takes values in $\widetilde H_-$. Relation (\ref{95}) suggests to define
\begin{equation}
\gamma_- = \eta_+, \label{45}
\end{equation}
thereof
\[
\gamma_- = \gamma'_- \xi_-.
\]
Further, from (\ref{43}) and (\ref{14}) we conclude that
\[
\partial_{-i} (\eta_- \gamma^{-1}) = 0,
\]
and, hence, for the mapping
\[
\xi_+ = \gamma_+^{\prime -1} \eta_- \gamma^{-1}
\]
one has
\[
\partial_{- i} \xi_+ = 0.
\]
Comparing (\ref{44}) and (\ref{15}), we see that the mapping $\xi_+$
takes values in $\widetilde H_+$. Denoting
\begin{equation}
\gamma_+ = \eta_- \gamma^{-1}, \label{49}
\end{equation}
we get
\[
\gamma_+ = \gamma'_+ \xi_+.
\]
Show now that the mappings $\gamma_\pm$ we have just defined, and the
mappings $\lambda_{\pm i, m}$ determined by relations (\ref{54}),
(\ref{55}), are the sought for mappings leading to the considered
solution of the multidimensional Toda type equations.
Indeed, since the mappings $\xi_\pm$ take values in $\widetilde
H_\pm$, one gets from (\ref{54}) and (\ref{55}) that the mappings
$\mu_\pm$ can be considered as solutions of equations (\ref{41}) and
(\ref{42}). Further, the mappings $\nu_\pm$ and $\eta = \eta_-
\eta_+^{-1}$ can be treated as the mappings obtained from the Gauss
decomposition (\ref{37}). Relations (\ref{45}) and (\ref{49}) imply
that the mapping $\gamma$ is given by (\ref{40}). Now, from
(\ref{43}), (\ref{44}) and (\ref{14}), (\ref{15}) it follows that
\[
\upsilon_{\pm i, m} = \xi_\pm^{-1} \upsilon'_{\pm i, m} \xi_\pm.
\]
Taking into account (\ref{50}) and (\ref{51}), we finally see that
the mappings $\upsilon_{\pm i, m}$ satisfy relations (\ref{52}) and
(\ref{53}). Thus, any solution of the multidimensional Toda type
equations can be locally obtained by the above integration scheme.
\subsection{Dependence of solution on integration data}
It appears that different sets of integration data can give the same
solution of the multidimensional Toda type equations. Consider this
problem in detail. Let $\gamma_\pm$, $\lambda_{\pm i, m}$ and
$\gamma'_\pm$, $\lambda'_{\pm i, m}$ be two sets of mappings
satisfying the integrability conditions of the equations determining
the corresponding mappings $\mu_\pm$ and $\mu'_\pm$. Suppose that
the solutions $\gamma$, $\upsilon_{\pm i, m}$ and $\gamma'$,
$\upsilon'_{\pm i, m}$ obtained by the above procedure coincide. In
this case the corresponding connections $\omega$ and $\omega'$ also
coincide. As it follows from the discussion given in section
\ref{is}, these connections are generated by the mappings $\varphi$
and $\varphi'$ defined as
\begin{equation}
\varphi = \mu_- \nu_+ \gamma_- = \mu_+ \nu_- \eta \gamma_-, \qquad
\varphi' = \mu'_- \nu'_+ \gamma'_- = \mu'_+ \nu'_- \eta' \gamma'_-.
\label{59}
\end{equation}
Since the connections $\omega$ and $\omega'$ coincide, we have
\[
\varphi' = a \varphi
\]
for some element $a \in G$. Hence, from (\ref{59}) it follows that
\[
\mu'_- \nu'_+ \gamma'_- = a \mu_- \nu_+ \gamma_-.
\]
This equality can be rewritten as
\begin{equation}
\mu'_- = a \mu_- \chi_+ \psi_+, \label{57}
\end{equation}
where the mappings $\chi_+$ and $\psi_+$ are defined by
\[
\chi_+ = \nu_+ \gamma_- \gamma_-^{\prime -1} \nu_+^{\prime -1}
\gamma'_- \gamma_-^{-1}, \qquad \psi_+ = \gamma_- \gamma_-^{\prime
-1}.
\]
Note that the mapping $\chi_+$ takes values in $\widetilde N_+$ and
the mapping $\psi_+$ takes values in $\widetilde H$. Moreover, one
has
\begin{equation}
\partial_{+i} \chi_+ = 0, \qquad \partial_{+i} \psi_+ = 0. \label{61}
\end{equation}
Similarly, from the equality
\[
\mu'_+ \nu'_- \eta' \gamma'_- = a \mu_+ \nu_- \eta \gamma_-
\]
we get the relation
\begin{equation}
\mu'_+ = a \mu_+ \chi_- \psi_-, \label{58}
\end{equation}
with the mappings $\chi_-$ and $\psi_-$ given by
\[
\chi_- = \nu_- \eta \gamma_- \gamma_-^{\prime -1} \eta^{\prime -1}
\nu_-^{\prime -1} \eta' \gamma'_- \gamma_-^{-1} \eta^{-1}, \qquad
\psi_- = \eta \gamma_- \gamma_-^{\prime -1} \eta^{\prime -1}.
\]
Here the mapping $\chi_-$ take values in $\widetilde N_-$, the
mapping $\psi_-$ take values in $\widetilde H$, and one has
\begin{equation}
\partial_{-i} \chi_- = 0, \qquad \partial_{-i} \psi_- = 0.
\label{62}
\end{equation}
Now using the Gauss decompositions
\[
\mu_+^{-1} \mu_- = \nu_- \eta \nu_+^{-1}, \qquad
\mu_+^{\prime -1} \mu'_- = \nu'_- \eta' \nu_+^{\prime -1}
\]
and relations (\ref{57}), (\ref{58}), one comes to the equalities
\begin{equation}
\eta' = \psi_-^{-1} \eta \psi_+, \qquad \nu'_\pm = \psi_\pm^{-1}
\chi_\pm^{-1} \nu_\pm \psi_\pm. \label{63}
\end{equation}
Further, from the definition of the mapping $\psi_+$ one gets
\begin{equation}
\gamma'_- = \psi_+^{-1} \gamma_-. \label{65}
\end{equation}
Since $\gamma' = \gamma$, we can write
\[
\gamma_+^{\prime -1} \eta' \gamma'_- = \gamma_+^{-1} \eta \gamma_-,
\]
therefore,
\begin{equation}
\gamma_+' = \psi_-^{-1} \gamma_+. \label{66}
\end{equation}
Equalities (\ref{57}) and (\ref{58}) give the relation
\begin{eqnarray}
\lefteqn{\mu_\pm^{\prime -1} \partial_{\pm i} \mu'_\pm} \nonumber \\
&=& \psi_\mp^{-1} \chi_\mp^{-1} (\mu_\pm^{-1} \partial_{\pm i}
\mu_\pm) \chi_\mp \psi_\mp + \psi_\mp^{-1} (\chi_\mp^{-1}
\partial_{\pm i} \chi_\mp) \psi_\mp + \psi_\mp^{-1} \partial_{\pm
i} \psi_\mp, \hspace{3.em} \label{60}
\end{eqnarray}
which implies
\begin{equation}
\sum_{m = \pm 1}^{\pm l_\pm \mp 1} \lambda'_{\pm i, m} =
\left[ \psi_\mp^{-1} \chi_\mp^{-1} \left( \sum_{m = \pm 1}^{\pm l_\pm
\mp
1} \lambda_{\pm i, m} \right) \chi_\mp \psi_\mp \right]_{\widetilde
{\frak n}_\pm}. \label{71}
\end{equation}
Let again $\gamma_\pm$, $\lambda_{\pm i, m}$ and $\gamma'_\pm$,
$\lambda'_{\pm i, m}$ be two sets of mappings satisfying the
integrability conditions of the equations determining the
corresponding mappings $\mu_\pm$ and $\mu'_\pm$. Denote by $\gamma$,
$\upsilon_{\pm i, m}$ and by $\gamma'$, $\upsilon'_{\pm i, m}$ the
corresponding solutions of the multidimensional Toda type equations.
Suppose that the mappings $\mu_\pm$ and $\mu'_\pm$ are connected by
relations (\ref{57}) and (\ref{58}) where the mappings $\chi_\pm$
take values in $\widetilde N_\pm$ and the mappings $\psi_\pm$ take
values in $\widetilde H$. It is not difficult to get convinced that
the mappings $\chi_\pm$ and $\psi_\pm$ satisfy relations (\ref{61})
and (\ref{62}). It is also clear that in the case under
consideration relations (\ref{63}) and (\ref{60}) are valid. From
(\ref{60}) it follows that
\[
\gamma'_\pm c_{\pm i} \gamma_\pm^{\prime -1} = \psi_\mp^{-1}
\gamma_\pm c_{\pm i} \gamma_\pm^{-1} \psi_\mp.
\]
Therefore, one has
\begin{equation}
\gamma'_\pm = \psi_\mp^{-1} \gamma_\pm \xi_\pm, \label{64}
\end{equation}
where the mappings $\xi_\pm$ take values in $\widetilde H_\pm$.
Taking into account (\ref{63}), we get
\[
\gamma' = \xi_+^{-1} \gamma \xi_-.
\]
Using now (\ref{52}), (\ref{53}) and the similar relations for the
mappings $\upsilon'_{\pm i, m}$, we come to the relations
\[
\upsilon'_{\pm i, m} = \xi_\pm^{-1} \upsilon_{\pm i, m} \xi_\pm.
\]
If instead of (\ref{64}) one has (\ref{65}) and (\ref{66}), then
$\gamma' = \gamma$ and $\upsilon'_{\pm i, m} = \upsilon_{\pm i, m}$.
Thus, the sets $\gamma_\pm$, $\lambda_{\pm i, m}$ and $\gamma'_\pm$,
$\lambda'_{\pm i, m}$ give the same solution of the multidimensional
Toda type equations if and only if the corresponding mappings
$\mu_\pm$ and $\mu'_\pm$ are connected by relations (\ref{57}),
(\ref{58}) and equalities (\ref{65}), (\ref{66}) are valid.
Let now $\gamma_\pm$ and $\lambda_{\pm i, m}$ be a set of integration
data, and $\mu_\pm$ be the solution of equations (\ref{41}),
(\ref{93}) and (\ref{42}), (\ref{94}) specified by initial conditions
(\ref{72}). Suppose that the mappings $\mu_\pm$ admit the Gauss
decompositions
\begin{equation}
\mu_\pm = \mu'_\pm \nu'_\mp \eta'_\mp. \label{73}
\end{equation}
where the mappings $\mu'_\pm$ take values in $a'_\pm \widetilde
N_\pm$, the mappings $\nu'_\pm$ take values in $\widetilde N_\pm$ and
the mappings $\eta'_\pm$ take values in $\widetilde H$.
Note that if $a_\pm \widetilde N_\pm = a'_\pm \widetilde N_\pm$, then
$\mu'_\pm = \mu_\pm$. Equalities (\ref{73}) imply that
the mappings $\mu_\pm$ and $\mu'_\pm$ are connected by relations
(\ref{57}) and (\ref{58}) with $a = e$ and
\[
\chi_\pm = \eta_\pm^{\prime -1} \nu_\pm^{\prime -1} \eta'_\pm, \qquad
\psi_\pm = \eta_\pm^{\prime -1}.
\]
{}From (\ref{60}) it follows that the mappings $\gamma'_\pm$ and
$\lambda'_{\pm i, m}$ given by (\ref{65}), (\ref{66}) and (\ref{71})
generate the mappings $\mu'_\pm$ as a solution of equations
(\ref{41}), (\ref{42}). It is clear that in the case under
consideration the solutions of the multidimensional Toda type
equations, obtained using the mappings $\gamma_\pm$, $\lambda_{\pm i,
m}$ and $\gamma'_\pm$, $\lambda'_{\pm i, m}$, coincide. Certainly, we
must use here the appropriate initial conditions for the mappings
$\mu_\pm$ and $\mu'_\pm$. Thus, we see that the solution
of the
multidimensional Toda equation, which is determined by the mappings
$\gamma_\pm$, $\lambda_{\pm i, m}$ and by the corresponding mappings
$\mu_\pm$ taking values in $a_\pm \widetilde N_\pm$, can be also
obtained starting from some mappings $\gamma'_\pm$, $\lambda'_{\pm i,
m}$ and the corresponding mappings $\mu'_{\pm}$ taking values in
$a'_\pm \widetilde N_\pm$. The above construction fails when the
mappings $\mu_\pm$ do not admit Gauss decomposition (\ref{73}).
Roughly speaking, almost all solutions of the multidimensional Toda
type equations can be obtained by the method described in the present
section if we will use only the mappings $\mu_\pm$ taking values in
the
sets $a_\pm \widetilde N_\pm$ for some fixed elements $a_\pm \in G$.
In particular, we can consider only the mappings $\mu_\pm$ taking
values in $\widetilde N_\pm$.
Summarising our consideration, describe once more the procedure for
obtaining the general solution to the multidimensional Toda type
equations. We start with the mappings $\gamma_\pm$ and $\lambda_{\pm
i, m}$ which satisfy (\ref{56}) and the integrability conditions of
equations (\ref{41}), (\ref{93}) and (\ref{42}), (\ref{94}).
Integrating these equations, we get the mappings $\mu_\pm$. Further,
Gauss decomposition (\ref{37}) gives the mappings $\eta$ and
$\nu_\pm$. Finally, using (\ref{40}), (\ref{52}) and (\ref{53}), we
obtain the mappings $\gamma$ and $\upsilon_{\pm i, m}$ which satisfy
the multidimensional Toda type equations. Any solution can be
obtained by using this procedure. Two sets of mappings $\gamma_\pm$,
$\lambda_{\pm i, m}$ and $\gamma'_\pm$, $\lambda'_{\pm i, m}$ give
the same solution if and only if the corresponding mappings $\mu_\pm$
and $\mu'_\pm$ are connected by relations (\ref{57}), (\ref{58}) and
equalities (\ref{65}), (\ref{66}) are valid. Almost all solutions of
the multidimensional Toda type equations can be obtained using the
mappings $\mu_\pm$ taking values in the subgroups $\widetilde N_\pm$.
\subsection{Automorphisms and reduction}\label{ar}
Let $\Sigma$ be an automorphism of the Lie group $G$, and $\sigma$ be
the corresponding automorphism of the Lie algebra ${\frak g}$.
Suppose that
\begin{equation}
\sigma({\frak g}_m) = {\frak g}_m. \label{115}
\end{equation}
In this case
\begin{equation}
\Sigma(\widetilde H) = \widetilde H, \qquad \Sigma(\widetilde N_\pm) =
\widetilde N_\pm. \label{111}
\end{equation}
Suppose additionally that
\begin{equation}
\sigma(c_{\pm i}) = c_{\pm i}. \label{116}
\end{equation}
It is easy to show now that if mappings $\gamma$ and $\upsilon_{\pm
i, m}$ satisfy the multidimensional Toda type equations, then the
mappings $\Sigma \circ \gamma$ and $\sigma \circ
\upsilon_{\pm i, m}$ satisfy the same equations. In such a situation
we can consider the subset of the solutions satisfying the conditions
\begin{equation}
\Sigma \circ \gamma = \gamma, \qquad \sigma \circ \upsilon_{\pm i,
m} = \upsilon_{\pm i, m}. \label{109}
\end{equation}
It is customary to call the transition to some subset of the
solutions of a given system of equations a reduction of the system.
Below we discuss a method to obtain solutions of
the multidimensional Toda type system satisfying relations (\ref{109}).
Introduce first some notations and give a few definitions.
Denote by $\widehat G$ the subgroup of $G$ formed by the elements
invariant with respect to the automorphism $\Sigma$. In other words,
\[
\widehat G = \{a \in G \mid \Sigma(a) = a\}.
\]
The subgroup $\widehat G$ is a closed subgroup of $G$. Therefore,
$\widehat G$ is a Lie subgroup of $G$. It is clear that the
subalgebra $\widehat {\frak g}$ of the Lie algebra ${\frak g}$,
defined by
\[
\widehat {\frak g} = \{x \in {\frak g} \mid \sigma(x) = x\},
\]
is the Lie algebra of $\widehat G$. The Lie algebra $\widehat{\frak
g}$ is a ${\Bbb Z}$--graded subalgebra of ${\frak g}$:
\[
\widehat {\frak g} = \bigoplus_{m \in {\Bbb Z}} \widehat{\frak g}_m,
\]
where
\[
\widehat {\frak g}_m = \{x \in {\frak g}_m \mid \sigma(x) = x\}.
\]
Define now the following Lie subgroups of $\widehat G$,
\[
\widehat{\widetilde H} = \{a \in \widetilde H \mid \Sigma(a) = a\},
\qquad \widehat{\widetilde N}_\pm = \{a \in \widetilde N_\pm \mid
\Sigma(a) = a\}.
\]
Using the definitions given above, we can reformulate conditions
(\ref{109}) by saying that the mapping $\gamma$ takes value in
$\widehat{\widetilde H}$, and the mappings $\upsilon_{\pm i, m}$ take
values in $\widehat {\frak g}_m$.
Let $a$ be an arbitrary element of $\widehat G$. Consider $a$ as an
element of $G$ and suppose that it has the Gauss decomposition
(\ref{69}). Then from the equality $\Sigma(a) = a$, we get the relation
\[
\Sigma(n_-) \Sigma(h) \Sigma(n_+^{-1}) = n_- h n_+^{-1}.
\]
Taking into account (\ref{111}) and the uniqueness of the Gauss
decomposition (\ref{69}), we conclude that
\[
\Sigma(h) = h, \qquad \Sigma(n_\pm) = n_\pm.
\]
Thus, the elements of some dense subset of $\widehat G$
possess the Gauss decomposition (\ref{69}) with $h \in
\widehat{\widetilde H}$, $n_\pm \in \widehat{\widetilde N}_\pm$,
and this decomposition is unique. Similarly, one can get convinced
that any element of $\widehat G$ has the modified Gauss
decompositions (\ref{70}) with $m_\pm \in a_\pm \widehat{\widetilde
N}_\pm$ for some elements $a_\pm \in \widehat G$, $n_\pm \in
\widehat{\widetilde N}_\pm$ and $h_\pm \in \widehat{\widetilde H}$.
To obtain solutions of the multidimensional Toda type equations
satisfying (\ref{109}), we start with the mappings $\gamma_\pm$ and
$\lambda_{\pm i, m}$ which satisfy the corresponding integrability
conditions and the relations similar to
(\ref{109}):
\begin{equation}
\Sigma \circ \gamma_\pm = \gamma_\pm, \qquad \sigma \circ \lambda_{\pm
i, m} = \lambda_{\pm i, m}. \label{112}
\end{equation}
In this case, for any solution of equations (\ref{41}), (\ref{93}) and
(\ref{42}), (\ref{94}) one has
\[
\sigma \circ (\mu_\pm^{-1} \partial_{\pm i} \mu_\pm) = \mu_\pm^{-1}
\partial_{\pm i} \mu_\pm, \qquad
\sigma \circ (\mu_\pm^{-1} \partial_{\mp i} \mu_\pm) = \mu_\pm^{-1}
\partial_{\mp i} \mu_\pm.
\]
{}From these relations it follows that
\begin{equation}
\Sigma \circ \mu_\pm = b_\pm \mu_\pm, \label{110}
\end{equation}
where $b_\pm$ are some elements of $G$. Recall that a solution of
equations (\ref{41}), (\ref{93}) and (\ref{42}), (\ref{94}) is
uniquely specified by conditions (\ref{72}). If the elements $a_\pm$
entering these conditions belong to the group $\widehat G$, then
instead of (\ref{110}), we get for the corresponding mappings
$\mu_\pm$ the relations
\[
\Sigma \circ \mu_\pm = \mu_\pm.
\]
For such mappings $\mu_\pm$ the Gauss decomposition (\ref{37}) gives
the mappings $\eta$ and $\nu_\pm$ which satisfy the equalities
\[
\Sigma \circ \eta = \eta, \qquad \Sigma \circ \nu_\pm = \nu_\pm.
\]
It is not difficult to get convinced that the corresponding solution
of the multidimensional Toda type equations satisfies (\ref{109}).
Show now that any solution of the multidimensional Toda type
equations satisfying (\ref{109}) can be obtained in such a way.
Let mappings $\gamma$ and $\upsilon_{\pm i, m}$ satisfy the
multidimensional Toda type equations and equalities (\ref{109}) are
valid. In this case, for the flat connection $\omega$ with the components
defined by (\ref{14}) and (\ref{15}), one has
\[
\sigma \circ \omega = \omega.
\]
Therefore, a mapping $\varphi: M \to G$ generating the connection
$\omega$ satisfies, in general, the relation
\[
\Sigma \circ \varphi = b \varphi,
\]
where $b$ is some element of $G$. However, if for some point $p \in
M$, one has $\varphi(p) \in \widehat G$, then we have the relation
\begin{equation}
\Sigma \circ \varphi = \varphi. \label{113}
\end{equation}
Since the mapping $\varphi$ is defined up to the multiplication from the
left hand side by an arbitrary element of $G$, it is clear that we
can always choose this mapping in such a way that it satisfies
(\ref{113}). Take such a mapping $\varphi$ and construct for it the
local Gauss decompositions (\ref{2}) where the mappings $\mu_\pm$
take values in the sets $a_\pm \widehat{\widetilde N}_\pm$ for some
$a_\pm \in \widehat G$, the mappings $\nu_\pm$ take values in
$\widehat{\widetilde N}_\pm$, and the mappings $\eta_\pm$ take values
in $\widehat{\widetilde H}$. In particular, one has
\begin{equation}
\Sigma \circ \mu_\pm = \mu_\pm. \label{114}
\end{equation}
As it follows from the consideration performed in section \ref{gs},
the mappings $\mu_\pm$ can be treated as solutions of equations
(\ref{41}), (\ref{93}) and (\ref{42}), (\ref{94}) for some mappings
$\lambda_{\pm i, m}$ and the mappings $\gamma_\pm$ given by
(\ref{45}), (\ref{49}). Clearly, in this case
\[
\Sigma \circ \gamma_\pm = \gamma_\pm,
\]
and from (\ref{114}) it follows that
\[
\Sigma \circ \lambda_{\pm i, m} = \lambda_{\pm i, m}.
\]
Moreover, the mappings $\gamma_\pm$ and $\lambda_{\pm i, m}$ are
integration data leading to the considered solution of the
multidimensional Toda type equations. Thus, if we start with
mappings $\gamma_\pm$ and $\lambda_{\pm i, m}$ which satisfy the
integrability conditions and relations (\ref{112}), use the
mappings $\mu_\pm$ specified by conditions (\ref{72}) with $a_\pm \in
\widehat G$, we get a solution satisfying (\ref{109}), and any such a
solution can be obtained in this way.
Let now $\Sigma$ be an antiautomorphism of $G$, and $\sigma$ be the
corresponding antiautomorphism of ${\frak g}$. In this case we again
suppose the validity of the relations $\sigma({\frak g}_m) = {\frak
g}_m$ which imply that $\Sigma(\widetilde H) = \widetilde H$ and
$\Sigma(\widetilde N_\pm) = \widetilde N_\pm$. However, instead of
(\ref{116}), we suppose that
\[
\sigma(c_{\pm i}) = - c_{\pm i}.
\]
One can easily get convinced that if the mappings $\gamma$ and
$\upsilon_{\pm i, m}$ satisfy the multidimensional Toda type
equations, then the mappings $(\Sigma \circ \gamma)^{-1}$ and
$-\sigma \circ \upsilon_{\pm i, m}$ also satisfy these equations.
Therefore, it is natural to consider the reduction to the mappings
satisfying the conditions
\[
\Sigma \circ \gamma = \gamma^{-1}, \qquad \sigma \circ \upsilon_{\pm
i, m} = - \upsilon_{\pm i, m}.
\]
The subgroup $\widehat G$ is defined now as
\begin{equation}
\widehat G = \{a \in G \mid \Sigma(a) = a^{-1} \}. \label{134}
\end{equation}
To get the general solution of the reduced system, we should start
with the integration data $\gamma_\pm$ and $\lambda_{\pm i, m}$
which satisfy the relations
\[
\Sigma \circ \gamma_\pm = \gamma_\pm^{-1}, \qquad \sigma \circ
\lambda_{\pm i, m} = - \lambda_{\pm i, m},
\]
and use the mappings $\mu_\pm$ specified by conditions (\ref{72})
with $a_\pm$ belonging to the subgroup $\widehat G$ defined by (\ref{134}).
One can also consider reductions based on antiholomorphic
automorphisms of $G$ and on the corresponding antilinear
automorphisms of ${\frak g}$. In this way it is possible to introduce
the notion of `real' solutions to multidimensional Toda type system.
We refer the reader to the discussion of this problem given in
\cite{RSa94,RSa96} for the two dimensional case. The generalisation to the
multidimensional case is straightforward.
\section{Examples}
\subsection{Generalised WZNW equations}
The simplest example of the multidimensional Toda type equations is
the so called generalised Wess--Zumino--Novikov--Witten (WZNW)
equations \cite{GMa93}. Let $G$ be an arbitrary complex connected
matrix Lie group. Consider the Lie algebra ${\frak g}$ of $G$ as a
${\Bbb Z}$--graded Lie algebra ${\frak g} = {\frak g}_{-1} \oplus
{\frak g}_0 \oplus {\frak g}_{+1}$, where ${\frak g}_0 = {\frak g}$
and ${\frak g}_{\pm 1} = \{0\}$. In this case the subgroup
$\widetilde H$ coincides with the whole Lie group $G$, and the
subgroups $\widetilde N_\pm$ are trivial. So, the mapping $\gamma$
parametrising the connection components of form (\ref{14}),
(\ref{15}), takes values in $G$. The only possible choice for the
elements $c_{\pm i}$ is $c_{\pm i} = 0$, and equations
(\ref{22})--(\ref{24}) take the form
\[
\partial_{+j} (\gamma^{-1} \partial_{-i} \gamma) = 0,
\]
which can be also rewritten as
\[
\partial_{-i}(\partial_{+j} \gamma \gamma^{-1}) = 0.
\]
These are the equations which are called in \cite{GMa93} the {\it
generalised WZNW equations}. They are, in a sense, trivial and can
be easily solved. However, in a direct analogy with two dimensional
case, see, for example, \cite{FORTW92}, it is possible to consider
the multidimensional Toda type equations as reductions of the
generalised WZNW equations.
Let us show how our general integration scheme works in this simplest
case. We start with the mappings $\gamma_\pm$ which take values in
$\widetilde H = G$ and
satisfy the
relations
\[
\partial_{\mp i} \gamma_\pm = 0.
\]
For the mappings $\mu_\pm$ we easily find
\[
\mu_\pm = a_\pm,
\]
where $a_\pm$ are some arbitrary elements of $G$. The Gauss decomposition
(\ref{37}) gives $\eta = a_+^{-1} a_-$, and for the general solution
of the generalised WZNW equations we have
\[
\gamma = \gamma_+^{-1} a_+^{-1} a_- \gamma_-.
\]
It is clear that the freedom to choose different elements $a_\pm$ is
redundant, and one can put $a_\pm = e$, which gives the usual
expression for the general solution
\[
\gamma = \gamma_+^{-1} \gamma_-.
\]
\subsection{Example based on Lie group ${\rm GL}(m, {\Bbb C})$}
Recall that the Lie group ${\rm GL}(m, {\Bbb C})$ consists of all
nondegenerate $m \by m$ complex matrices. This group is reductive.
We identify the Lie algebra of ${\rm GL}(m, {\Bbb C})$ with the Lie algebra
${\frak gl}(m, {\Bbb C})$.
Introduce the following ${\Bbb Z}$--gradation of ${\frak gl}(m, \Bbb C)$.
Let $n$ and $k$ be some positive integers such that $m = n + k$.
Consider a general element $x$ of ${\frak gl}(m, {\Bbb C})$ as a $2
\by 2$ block matrix
\[
x = \left( \begin{array}{cc}
A & B \\
C & D
\end{array} \right),
\]
where $A$ is an $n \by n$ matrix, $B$ is an $n \by k$ matrix,
$C$ is a $k \by n$ matrix, and $D$ is a $k \by k$ matrix.
Define the subspace ${\frak g}_0$ as the subspace of ${\frak gl}(m,
{\Bbb C})$, consisting of all block diagonal matrices, the subspaces
${\frak g}_{-1}$ and ${\frak g}_{+1}$ as the subspaces formed by all
strictly lower and upper triangular block matrices, respectively.
Consider the multidimensional Toda type equations
(\ref{22})--(\ref{24}) which correspond to the choice $l_{-i} =
l_{+i} = 1$. In our case the general form of the elements $c_{\pm i}$ is
\[
c_{-i} = \left(\begin{array}{cc}
0 & 0 \\
C_{-i} & 0
\end{array} \right), \qquad
c_{+i} = \left(\begin{array}{cc}
0 & C_{+i} \\
0 & 0
\end{array} \right),
\]
where $C_{-i}$ are $k \by n$ matrices, and $C_{+i}$ are $n \by k$
matrices. Since ${\frak g}_{\pm 2} = \{0\}$, then conditions
(\ref{39}) are satisfied. The subgroup $\widetilde H$ is isomorphic
to the group ${\rm GL}(n, {\Bbb C}) \times {\rm GL}(k, {\Bbb C})$, and
the mapping $\gamma$ has the block diagonal form
\[
\gamma = \left( \begin{array}{cc}
\beta_1 & 0 \\
0 & \beta_2
\end{array} \right),
\]
where the mappings $\beta_1$ and $\beta_2$ take values in ${\rm
GL}(n, {\Bbb C})$ and ${\rm GL}(k, {\Bbb C})$, respectively.
It is not difficult to show that
\[
\gamma c_{-i} \gamma^{-1} = \left( \begin{array}{cc}
0 & 0 \\
\beta_2 C_{-i} \beta_1^{-1} & 0
\end{array} \right);
\]
hence, equations (\ref{22}) take the following form:
\begin{equation}
\partial_{-i} (\beta_2 C_{-j} \beta_1^{-1}) = \partial_{-j} (\beta_2
C_{-i} \beta_1^{-1}). \label{96}
\end{equation}
Similarly, using the relation
\[
\gamma^{-1} c_{+i} \gamma = \left( \begin{array}{cc}
0 & \beta_1^{-1} C_{+i} \beta_2 \\
0 & 0
\end{array} \right),
\]
we represent equations (\ref{23}) as
\begin{equation}
\partial_{+i} (\beta_1^{-1} C_{+j} \beta_2 ) =
\partial_{+j} (\beta_1^{-1} C_{+i} \beta_2). \label{97}
\end{equation}
Finally, equations (\ref{24}) take the form
\begin{eqnarray}
&\partial_{+j} (\beta_1^{-1} \partial_{-i} \beta_1) = - \beta_1^{-1}
C_{+j} \beta_2 C_{-i},& \label{98} \\
&\partial_{+j} (\beta_2^{-1} \partial_{-i} \beta_2) = C_{-i}
\beta_1^{-1} C_{+j} \beta_2. \label{99}
\end{eqnarray}
In accordance with our integration scheme, to construct the general
solution for equations (\ref{96})--(\ref{99}) we should start with
the mappings $\gamma_{\pm}$ which take values in $\widetilde H$ and
satisfy (\ref{56}). Write for these mappings the block matrix
representation
\[
\gamma_\pm = \left( \begin{array}{cc}
\beta_{\pm 1} & 0 \\
0 & \beta_{\pm 2}
\end{array} \right).
\]
Recall that almost all solutions of the multidimensional Toda type
equations can be obtained using the mappings $\mu_\pm$ taking values
in the subgroups $\widetilde N_\pm$. Therefore, we choose these
mappings in the form
\[
\mu_- = \left( \begin{array}{cc}
I_n & 0 \\
\mu_{-21} & I_k
\end{array} \right), \qquad
\mu_+ = \left( \begin{array}{cc}
I_n & \mu_{+12} \\
0 & I_k
\end{array} \right),
\]
where $\mu_{-21}$ and $\mu_{+12}$ take values in the spaces of $k \by
n$ and $n \by k$ matrices, respectively. Equations (\ref{41}),
(\ref{93}) and (\ref{42}), (\ref{94}) are reduced now to the equations
\begin{eqnarray}
&\partial_{-i} \mu_{-21} = \beta_{-2} C_{-i} \beta_{-1}^{-1}, \qquad
\partial_{+i} \mu_{-21} = 0,& \label{102} \\
&\partial_{+i} \mu_{+12} = \beta_{+1} C_{+i} \beta_{+2}^{-1}, \qquad
\partial_{-i} \mu_{+12} = 0.& \label{103}
\end{eqnarray}
The corresponding integrability conditions are
\begin{eqnarray}
&\partial_{-i} (\beta_{-2} C_{-j} \beta_{-1}^{-1}) = \partial_{-j}
(\beta_{-2} C_{-i} \beta_{-1}^{-1}),& \label{100} \\
&\partial_{+i} (\beta_{+1} C_{+j} \beta_{+2}^{-1}) = \partial_{+j}
(\beta_{+1} C_{+i} \beta_{+2}^{-1}).& \label{101}
\end{eqnarray}
Here we will not study the problem of solving the integrability
conditions for a general choice of $n$, $k$ and $C_{\pm i}$. In the
end of this section we discuss a case when it is quite easy to find
explicitly all the mappings $\gamma_\pm$ satisfying the integrability
conditions, while now we will continue the consideration of the
integration procedure for the general case.
Suppose that the mappings $\gamma_\pm$ satisfy the integrability
conditions and we have found the corresponding mappings $\mu_\pm$.
Determine from the Gauss decomposition (\ref{37}) the mappings
$\nu_\pm$ and $\eta$. Actually, in the case under consideration we
need only the mapping $\eta$. Using for the mappings $\nu_-$,
$\nu_+$ and $\eta$ the following representations
\[
\nu_- = \left( \begin{array}{cc}
I_n & 0 \\
\nu_{-21} & I_k
\end{array} \right), \qquad
\nu_+ = \left( \begin{array}{cc}
I_n & \nu_{+12} \\
0 & I_k
\end{array} \right), \qquad
\eta = \left( \begin{array}{cc}
\eta_{11} & 0 \\
0 & \eta_{22}
\end{array} \right),
\]
we find that
\begin{eqnarray*}
&\nu_{-21} = \mu_{-21} (I_n - \mu_{+12} \mu_{-21})^{-1}, \qquad
\nu_{+12} = (I_n - \mu_{+12} \mu_{-21})^{-1} \mu_{+12}, \\
&\eta_{11} = I_n - \mu_{+12} \mu_{-21}, \qquad
\eta_{22} = I_k + \mu_{-21} (I_n - \mu_{+12} \mu_{-21})^{-1}
\mu_{+12}.
\end{eqnarray*}
It is worth to note here that the mapping $\mu_+^{-1} \mu_-$ has the
Gauss decomposition (\ref{37}) only at those points $p$ of $M$, for
which
\begin{equation}
\det \left( I_n - \mu_{+12}(p) \mu_{-21}(p) \right) \ne 0. \label{104}
\end{equation}
Now, using relation (\ref{40}), we get for the general solution of
system (\ref{96})--(\ref{99}) the following expression:
\begin{eqnarray*}
&\beta_1 = \beta_{+1}^{-1} (I_n - \mu_{+12} \mu_{-21}) \beta_{-1},& \\
&\beta_2 = \beta_{+2}^{-1} (I_k + \mu_{-21} (I_n - \mu_{+12}
\mu_{-21})^{-1} \mu_{+12}) \beta_{-2}.&
\end{eqnarray*}
Consider now the case when $n = m-1$. In this case $\beta_1$ takes
values in ${\rm GL}(n, {\Bbb C})$, $\beta_2$ is a complex function,
$C_{-i}$ and $C_{+i}$ are $1 \by n$ and $n \by 1$ matrices,
respectively. Suppose that the dimension of the manifold $M$ is equal
to $2n$ and define $C_{\pm i}$ by
\[
(C_{\pm i})_r = \delta_{ir}.
\]
System (\ref{96})--(\ref{99}) takes now the form
\begin{eqnarray}
&\partial_{-i} (\beta_2 (\beta_1^{-1})_{jr}) = \partial_{-j} (\beta_2
(\beta_1^{-1})_{ir}),& \label{105} \\
&\partial_{+i}((\beta_1^{-1})_{rj} \beta_2) =
\partial_{+j}((\beta_1^{-1})_{ri} \beta_2),& \label{106} \\
&\partial_{+j}(\beta_1^{-1} \partial_{-i} \beta_1)_{rs} = -
(\beta_1^{-1})_{rj} \beta_2 \delta_{is},& \label{107} \\
&\partial_{+j}(\beta_2^{-1}\partial_{-i} \beta_2) =
(\beta_1^{-1})_{ij} \beta_2,& \label{108}
\end{eqnarray}
and the integrability conditions (\ref{100}), (\ref{101}) can be
rewritten as
\begin{eqnarray*}
&\partial_{-i}(\beta_{-2}(\beta_{-1}^{-1})_{jr}) =
\partial_{-j}(\beta_{-2}(\beta_{-1}^{-1})_{ir}),& \\
&\partial_{+i}((\beta_{+1})_{rj} \beta_{+2}^{-1}) =
\partial_{+j}((\beta_{+1})_{ri} \beta_{+2}^{-1}).&
\end{eqnarray*}
The general solution for these integrability conditions is
\begin{eqnarray*}
&(\beta_{-1}^{-1})_{ir} = U_- \partial_{-i} V_{-r}, \qquad
\beta_{-2}^{-1} = U_-,& \\
&(\beta_{+1})_{ri} = U_+ \partial_{+i} V_{+r}, \qquad \beta_{+2} =
U_+.
\end{eqnarray*}
Here $U_\pm$ and $V_{\pm r}$ are arbitrary functions satisfying
the conditions
\[
\partial_{\mp} U_\pm = 0, \qquad \partial_\mp V_{\pm r} = 0.
\]
Moreover, for any point $p$ of $M$ one should have
\[
U_\pm(p) \ne 0, \qquad \det (\partial_{\pm i} V_{\pm r}(p)) \ne 0.
\]
The general solution of equations (\ref{102}), (\ref{103}) is
\[
\mu_{-21} = V_-, \qquad \mu_{+12} = V_+,
\]
where $V_-$ is the $1 \by n$ matrix valued function constructed with
the functions $V_{-r}$, and $V_+$ is the $n \by 1$ matrix valued
function constructed with the functions $V_{+r}$. Thus, we have
\[
\eta_{11} = I_n - V_+ V_-.
\]
In the case under consideration, condition (\ref{104}) which
guarantees the existence of the Gauss decomposition (\ref{37}), is
equivalent to
\[
1 - V_-(p) V_+(p) \ne 0.
\]
When this condition is satisfied, one has
\[
(I_n - \mu_{+12} \mu_{-21})^{-1} = (I_n - V_+ V_-)^{-1} = I_n +
\frac{1}{1 - V_- V_+} V_+ V_-,
\]
and, therefore,
\[
\eta_{22} = \frac{1}{1 - V_- V_+}.
\]
Taking the above remarks into account, we come to the following
expressions for the general solution of system (\ref{105})--(\ref{108}):
\begin{eqnarray*}
&(\beta_1^{-1})_{ij} = - U_+ U_- (1 - V_- V_+) \partial_{-i}
\partial_{+j} \ln (1 - V_- V_+),& \\
&\beta_2^{-1} = U_+ U_- (1 - V_- V_+).&
\end{eqnarray*}
\subsection{Cecotti--Vafa type equations}
In this example we discuss the multidimensional Toda system
associated with the loop group ${\cal L}({\rm GL}(m, {\Bbb C}))$
which is an infinite dimensional Lie group defined as the group of
smooth mappings from the circle $S^1$ to the Lie group ${\rm GL}(m,
{\Bbb C})$. We think of the circle as consisting of complex numbers
$\zeta$ of modulus one. The Lie algebra of ${\cal L}({\rm GL}(m,
{\Bbb C}))$ is the Lie algebra ${\cal L}({\frak gl}(m, {\Bbb C}))$
consisting of smooth mappings from $S^1$ to the Lie algebra ${\frak
gl}(m, {\Bbb C})$.
In the previous section we considered some class of ${\Bbb
Z}$--gradations of the Lie algebra ${\frak gl}(m, {\Bbb C})$ based on
the representation of $m$ as the sum of two positive integers $n$ and
$k$. Any such a gradation can be extended to a ${\Bbb
Z}$--gradation of the loop algebra ${\cal L}({\rm GL}(m, {\Bbb C})$.
Here we restrict ourselves to the case $m = 2n$.
In this case the element
\[
q = \left(\begin{array}{cc}
I_n & 0 \\
0 & -I_n
\end{array} \right)
\]
of ${\frak gl}(2n, {\Bbb C})$ is the grading operator of the ${\Bbb
Z}$--gradation under consideration. This means that an element $x$ of
${\frak gl}(2n, {\Bbb C})$ belongs to the subspace ${\frak g}_k$ if
and only if $[q, x] = k x$. Using the operator $q$, we introduce the
following ${\Bbb Z}$--gradation of ${\cal L}({\frak gl}(2n, {\Bbb
C}))$. The subspace ${\frak g}_k$ of ${\cal L}({\frak gl}(2n, {\Bbb
C}))$ is defined as the subspace formed by the elements $x(\zeta)$ of
${\cal L}({\frak gl}(2n, {\Bbb C}))$ satisfying the relation
\[
[q, x(\zeta)] + 2 \zeta \frac{dx(\zeta)}{d\zeta} = k x(\zeta).
\]
In particular, the subspaces ${\frak g}_0$, ${\frak g}_{-1}$ and
${\frak g}_{+1}$ of ${\cal L}({\frak gl}(2n, {\Bbb C}))$
consist respectively of the elements
\[
x(\zeta) = \left(\begin{array}{cc}
A & 0 \\
0 & D
\end{array} \right), \qquad
x(\zeta) = \left(\begin{array}{cc}
0 & \zeta^{-1} B \\
C & 0
\end{array} \right), \qquad
x(\zeta) = \left(\begin{array}{cc}
0 & B \\
\zeta C & 0
\end{array} \right),
\]
where $A$, $B$, $C$ and $D$ are arbitrary $n \by n$ matrices which do not
depend on $\zeta$.
Consider the multidimensional Toda type equations
(\ref{22})--(\ref{24}) which correspond to the choice $l_{-i} =
l_{+i} = 1$. In this case the general form of the elements $c_{\pm
i}$ is
\[
c_{-i} = \left(\begin{array}{cc}
0 & \zeta^{-1} B_{-i} \\
C_{-i} & 0
\end{array} \right), \qquad
c_{+i} = \left(\begin{array}{cc}
0 & C_{+i} \\
\zeta B_{+i} & 0
\end{array} \right).
\]
To satisfy conditions (\ref{39}) we should have
\[
B_{\pm i} C_{\pm j} - B_{\pm j} C_{\pm i} = 0, \quad C_{\pm i} B_{\pm
j} - C_{\pm j} B_{\pm i} = 0.
\]
The subgroup $\widetilde H$ is isomorphic to the group ${\rm GL}(n,
{\Bbb C}) \times {\rm GL}(n, {\Bbb C})$, and the mapping $\gamma$ has
the block diagonal form
\[
\gamma = \left( \begin{array}{cc}
\beta_1 & 0 \\
0 & \beta_2
\end{array} \right),
\]
where $\beta_1$ and $\beta_2$ take values in ${\rm GL}(n, {\Bbb C})$.
Hence, one obtains
\[
\gamma c_{-i} \gamma^{-1} = \left( \begin{array}{cc}
0 & \zeta^{-1} \beta_1 B_{-i} \beta_2^{-1} \\
\beta_2 C_{-j} \beta_1^{-1} & 0
\end{array} \right),
\]
and comes to following explicit expressions for equations (\ref{22}):
\begin{eqnarray}
&\partial_{-i} (\beta_1 B_{-j} \beta_2^{-1}) =
\partial_{-j} (\beta_1 B_{-i} \beta_2^{-1}),& \label{74}\\
&\partial_{-i} (\beta_2 C_{-j} \beta_1^{-1}) =
\partial_{-j} (\beta_2 C_{-i} \beta_1^{-1}).&\label{75}
\end{eqnarray}
Similarly, using the relation
\[
\gamma^{-1} c_{+i} \gamma = \left( \begin{array}{cc}
0 & \beta_1^{-1} C_{+i} \beta_2 \\
\zeta \beta_2^{-1} B_{+i} \beta_1 & 0
\end{array} \right),
\]
we can represent equations (\ref{23}) as
\begin{eqnarray}
&\partial_{+i} (\beta_1^{-1} C_{+j} \beta_2 ) =
\partial_{+j} (\beta_1^{-1} C_{+i} \beta_2),& \label{76}\\
&\partial_{+i} (\beta_2^{-1} B_{+j} \beta_1) =
\partial_{+j} (\beta_2^{-1} B_{+i} \beta_1).&\label{77}
\end{eqnarray}
Finally, equations (\ref{24}) take the form
\begin{eqnarray}
&\partial_{+j} (\beta_1^{-1} \partial_{-i} \beta_1) =
B_{-i} \beta_2^{-1} B_{+j} \beta_1 - \beta_1^{-1} C_{+j} \beta_2
C_{-i},& \label{78}\\
&\partial_{+j} (\beta_2^{-1} \partial_{-i} \beta_2) =
C_{-i} \beta_1^{-1} C_{+j} \beta_2 - \beta_2^{-1} B_{+j} \beta_1
B_{-i}.&\label{79}
\end{eqnarray}
System (\ref{74})--(\ref{79}) admits two interesting reductions,
which can be defined with the help of the general scheme described in
section \ref{ar}. Represent an arbitrary element $a(\zeta)$ of
${\cal L}({\rm GL}(2n, {\Bbb C}))$ in the block form,
\[
a(\zeta) = \left( \begin{array}{cc}
A(\zeta) & B(\zeta) \\
C(\zeta) & D(\zeta)
\end{array} \right),
\]
and define an automorphism $\Sigma$ of ${\cal L}({\rm GL}(2n, {\Bbb
C}))$ by
\[
\Sigma(a(\zeta)) = \left( \begin{array}{cc}
D(\zeta) & \zeta^{-1} C(\zeta) \\
\zeta B(\zeta) & A(\zeta)
\end{array} \right).
\]
It is clear that the corresponding automorphism $\sigma$ of ${\cal
L}({\frak gl}(2n, {\Bbb C}))$ is defined by the relation of the same
form. In the case under consideration relation (\ref{115}) is valid.
Suppose that $B_{\pm i} = C_{\pm i}$, then relation (\ref{116}) is
also valid. Therefore, we can consider the reduction of system
(\ref{74})--(\ref{79}) to the case when the mapping $\gamma$
satisfies the equality $\Sigma \circ \gamma = \gamma$ which can be
written as $\beta_1 = \beta_2$. The reduced system looks as
\begin{eqnarray}
&\partial_{-i}(\beta C_{-j} \beta^{-1}) = \partial_{-j}(\beta C_{-i}
\beta^{-1}),& \label{117} \\
&\partial_{+i}(\beta^{-1} C_{+j} \beta) = \partial_{+j}(\beta^{-1}
C_{+i} \beta),& \label{118} \\
&\partial_{+j}(\beta^{-1} \partial_{-i} \beta) = [C_{-i}, \beta^{-1}
C_{+j} \beta],& \label{119}
\end{eqnarray}
where we have denoted $\beta = \beta_1 = \beta_2$.
The next reduction is connected with an antiautomorphism $\Sigma$ of
the group ${\cal L}({\rm GL}(2n, {\Bbb C}))$ given by
\[
\Sigma(a(\zeta)) = \left( \begin{array}{cc}
A(\zeta)^t & -\zeta^{-1} C(\zeta)^t \\
-\zeta B(\zeta)^t & D(\zeta)^t
\end{array} \right).
\]
The corresponding antiautomorphism of ${\cal L}({\frak gl}(2n, {\Bbb
C}))$ is defined by the same formula. It is evident that
$\sigma({\frak g}_k) = {\frak g}_k$. Suppose that $B_{\pm i} =
C^t_{\pm i}$, then $\sigma(c_{\pm i}) = - c_{\pm i}$, and one can
consider the reduction of system (\ref{74})--(\ref{79}) to the case
when the mapping $\gamma$ satisfies the equality $\Sigma \circ \gamma
= \gamma^{-1}$ which is equivalent to the equalities $\beta_1^t =
\beta_1^{-1}$, $\beta_2^t = \beta_2^{-1}$. The reduced system of
equations can be written as
\begin{eqnarray}
&\partial_{-i}(\beta_2 C_{-j} \beta_1^t) = \partial_{-j} (\beta_2
C_{-i} \beta_1^t),& \label{123} \\
&\partial_{+i}(\beta_1^t C_{+j} \beta_2) = \partial_{+j} (\beta_1^t
C_{+i} \beta_2),& \label{124} \\
&\partial_{+j}(\beta_1^t \partial_{-i} \beta_1) = C_{-i}^t \beta_2^t
C_{+j}^t \beta_1 - \beta_1^t C_{+j} \beta_2 C_{-i},& \label{125} \\
&\partial_{+j}(\beta_2^t \partial_{-i} \beta_2) = C_{-i} \beta_1^t
C_{+j} \beta_2 - \beta_2^t C_{+j}^t \beta_1 C_{-i}^t.& \label{126}
\end{eqnarray}
If simultaneously $B_{\pm i} = C_{\pm i}$ and $B_{\pm i} = C_{\pm
i}^t$, one can perform both reductions. Here the reduced system has
form (\ref{117})--(\ref{119}) where the mapping $\beta$ take values
in the complex orthogonal group ${\rm O}(n, {\Bbb C})$. These are
exactly the equations considered by S. Cecotti and C. Vafa \cite{CVa91}.
As it was shown by B. A. Dubrovin \cite{Dub93} for $C_{-i} = C_{+i} =
C_i$ with
\[
(C_i)_{jk} = \delta_{ij} \delta_{jk},
\]
the Cecotti--Vafa equations are connected with some well known
equations in differential geometry. Actually, in \cite{Dub93} the
case $M = {\Bbb C}^n$ was considered and an additional restriction
$\beta^\dagger = \beta$ was imposed. Here equation
(\ref{118}) can be obtained from equation (\ref{117}) by hermitian
conjugation, and the system under consideration consists of equations
(\ref{117}) and (\ref{119}) only. Rewrite equation (\ref{117}) in the
form
\[
[\beta^{-1} \partial_{-i} \beta, C_j] = [\beta^{-1} \partial_{-j}
\beta, C_i].
\]
{}From this equation it follows that for some matrix valued mapping $b =
(b_{ij})$, such that $b_{ij} = b_{ji}$, the relation
\begin{equation}
\beta^{-1} \partial_{-i} \beta = [C_i, b] \label{120}
\end{equation}
is valid. In fact, the right hand side of relation (\ref{120}) does
not contain the diagonal matrix elements of $b$, while the other
matrix elements of $B$ are uniquely determined by the left hand side of
(\ref{120}). Furthermore, relation (\ref{120}) implies that the
mapping $b$ satisfies the equation
\begin{equation}
\partial_{-i} [C_j, b] - \partial_{-j} [C_i, b] + [[C_i, b], [C_j,
b]] = 0. \label{121}
\end{equation}
{}From the other hand, if some mapping $b$ satisfies equation
(\ref{121}), then there is a mapping $\beta$ connected with $b$ by
relation (\ref{120}), and such a mapping $\beta$ satisfies equation
(\ref{117}). Therefore, system (\ref{117}), (\ref{119}) is equivalent
to the system which consist of equations (\ref{120}), (\ref{121})
and the equation
\begin{equation}
\partial_{+j} [C_i, b] = [C_i, \beta^{-1} C_j \beta] \label{122}
\end{equation}
which follows from (\ref{120}) and (\ref{119}). Using the concrete
form of the matrices $C_i$, one can write the system (\ref{120}),
(\ref{121}) and (\ref{122}) as
\begin{eqnarray}
&\partial_{-k} b_{ji} = b_{jk} b_{ki}, \qquad \mbox{$i$, $j$, $k$
distinct};& \label{127} \\
&\sum_{k=1}^n \partial_{-k} b_{ij} = 0; \qquad i \neq j;& \label{128}
\\
&\partial_{-i} \beta_{jk} = b_{ik} \beta_{ji}, \qquad i \neq j;&
\label{129} \\
&\sum_{k=1}^n \partial_{-k} \beta_{ij} = 0;& \label{130} \\
&\partial_{+k} b_{ij} = \beta_{ki} \beta_{kj}, \qquad i \neq j.&
\label{131}
\end{eqnarray}
Equations (\ref{127}), (\ref{128}) have the form of equations which
provide vanishing of the curvature of the diagonal metric with
symmetric rotation coefficients $b_{ij}$ \cite{Dar10,Bia24}. Recall
that such a metric is called a Egoroff metric. Note that the
transition from system (\ref{117}), (\ref{119}) to system
(\ref{127})--(\ref{131}) is not very useful for obtaining solutions
of (\ref{117}), (\ref{119}). A more constructive way here is to use
the integration scheme described in section \ref{cgs}. Let us
discuss the corresponding procedure for a more general system
(\ref{123})--(\ref{126}) with $C_{-i} = C_{+i} = C_i$.
The integrations data for system (\ref{123})--(\ref{126}) consist of
the mappings $\gamma_\pm$ having the following block diagonal form
\[
\gamma_\pm = \left( \begin{array}{cc}
\beta_{\pm 1} & 0 \\
0 & \beta_{\pm 2}
\end{array} \right).
\]
As it follows from the discussion given in section \ref{ar}, the
mappings $\beta_{\pm 1}$ and $\beta_{\pm 2}$ must satisfy the conditions
\[
\beta_{\pm 1}^t = \beta_{\pm 1}^{-1}, \qquad \beta_{\pm 2}^t =
\beta_{\pm 2}^{-1}.
\]
The corresponding integrability conditions have the form
\begin{equation}
\partial_{\pm i}(\beta_{\pm 2} C_{j} \beta_{\pm 1}^t) =
\partial_{\pm j}(\beta_{\pm 2} C_{i} \beta_{\pm 1}^t). \label{133}
\end{equation}
Rewriting these conditions as
\[
\beta_{\pm 2}^t \partial_{\pm i} \beta_{\pm 2} C_j - C_j
\beta_{\pm 1}^t \partial_{\pm i} \beta_{\pm 1} = \beta_{\pm 2}^t
\partial_{\pm j} \beta_{\pm 2} C_i - C_i \beta_{\pm 1}^t
\partial_{\pm j} \beta_{\pm 1},
\]
we can get convinced that for some matrix valued mappings $b_\pm$ one
has
\begin{equation}
\beta_{\pm 1}^t \partial_{\pm i} \beta_{\pm 1 } = C_i b_\pm
- b_\pm^t C_i, \qquad \beta_{\pm 2}^t \partial_{\pm i}
\beta_{\pm 2 } = C_i b_\pm^t - b_\pm C_i. \label{132}
\end{equation}
{}From these relations it follows that the mappings $b_\pm$ satisfy the
equations
\begin{eqnarray}
&\partial_{\pm i} (b_\pm)_{ji} + \partial_{\pm j} (b_\pm)_{ij} +
\sum_{k \neq i,j} (b_\pm)_{ik} (b_\pm)_{jk} = 0, \quad i \neq j;&
\label{83} \\
&\partial_{\pm k} (b_\pm)_{ji} = (b_\pm)_{jk} (b_\pm)_{ki}, \qquad
\mbox{$i$, $j$, $k$ distinct};& \label{84} \\
&\partial_{\pm i} (b_\pm)_{ij} + \partial_{\pm j} (b_\pm)_{ji} +
\sum_{k \neq i,j} (b_\pm)_{ki} (b_\pm)_{kj} = 0, \qquad i \neq j.&
\label{85}
\end{eqnarray}
Conversely, if we have some mappings $b_\pm$ which satisfy equations
(\ref{83})--(\ref{85}), then there exist mappings $\beta_{\pm 1}$ and
$\beta_{\pm 2}$ connected with $b_\pm$ by (\ref{132}) and satisfying
the integrability conditions (\ref{133}).
System (\ref{83})--(\ref{85}) represents a limiting case of the
completely integrable Bourlet equations \cite{Dar10,Bia24} arising after
an appropriate In\"on\"u--Wigner contraction of the corresponding Lie
algebra \cite{Sav86}. Sometimes this system is called the
multidimensional generalised wave equations, while equation
(\ref{119}) is called the generalised sine--Gordon equation
\cite{Ami81,TTe80,ABT86}.
\section{Outlook}
Due to the algebraic and geometrical clearness of the equations
discussed in the paper, we are firmly convinced that, in time, they
will be quite relevant for a number of concrete applications in
classical and quantum field theories, statistical mechanics and
condensed matter physics. In support of this opinion we would like to
remind a remarkable role of some special classes of the equations
under consideration here.
Namely, in the framework of the standard abelian and nonabelian,
conformal and affine Toda fields coupled to matter fields, some
interesting physical phenomena which possibly can be described on the
base of corresponding equations, are mentioned in
\cite{GSa95,FGGS95}. In particular, from the point of view of
nonperturbative aspects of quantum field theories, they might be very
useful for understanding the quantum theory of solitons, some
confinement mechanisms for the quantum chromodynamics,
electron--phonon systems, etc. Furthermore, the Cecotti--Vafa
equations \cite{CVa91} of topological--antitopological fusion, which,
as partial differential equations, are, in general, multidimensional
ones, describe ground state metric of two dimensional $N=2$
supersymmetric quantum field theory. As it was shown in \cite{CVa93},
they are closely related to those for the correlators of the massive
two dimensional Ising model, see, for example, \cite{KIB93} and
references therein. This link is clarified in \cite{CVa93}, in
particular, in terms of the isomonodromic deformation in spirit of
the holonomic field theory developed by the Japanese school, see, for
example, \cite{SMJ80}.
The authors are very grateful to J.--L.~Gervais and B. A. Dubrovin
for the interesting discussions. Research supported in part by the
Russian Foundation for Basic Research under grant no. 95--01--00125a.
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{LIGHT QUARK MASSES}
\label{s_mq}
The masses of light quarks $m_u$, $m_d$, and $m_s$ are three of the
least well known parameters of the Standard Model. These quark
masses have to be inferred from the masses of low lying hadrons.
\hbox{$ \chi $PT}\ relates the masses of pseudoscalar mesons to $m_u,\ m_d$, and
$m_s$, however, the presence of the unknown scale $\mu$ in ${\cal
L}_{\hbox{$ \chi $PT}}$ implies that only ratios of quark masses can be determined.
For example $2m_s/(m_u + m_d) \equiv m_s/\mbar= 25$ at lowest order,
and $31$ at next order \cite{gasserPR,Donoghue}. Latest estimates using QCD
sum rules give $m_u + m_d = 12(1) \MeV$ \cite{BPR95}. However, as
discussed in \cite{rSUMRULE96BGM}, a reanalysis of sum rules shows
that far more experimental information on the hadronic spectral
function is needed before sum rules can give reliable estimates.
Thus, lattice QCD is currently the most promising approach.
To extract $a$, $\mbar$, $m_s$ we fit the global data as
\begin{eqnarray}
M_{PS} &=& B_{PS} (m_1 + m_2)/2 \nonumber \\
M_{V} &=& A_{V} + B_{V} (m_1 + m_2)/2 \ ,
\label{e:chiralfits}
\end{eqnarray}
for each value of the lattice parameters, $\beta$, $n_f$, fermion
action. From $B_{PS}, A_V, B_V$ we determine the three desired
quantities; the lattice scale $a$ using $M_\rho$, $\mbar$ using
$M_\pi^2 / M_\rho^2$, and $m_s$ in three different ways using $M_K, \
M_{K^*}, \ M_\phi$. Throughout the analysis we assume that $\phi$ is a
pure $s \bar s$ state. Note that using
Eq.~\ref{e:chiralfits} means that we can predict
only one independent quark mass from the pseudoscalar data, which we
choose to be $\mbar$. The reason for this truncation is that in most
cases the data for $M_\pi$ and $M_\rho$ exist at only $2-4$ values of
``light'' quark masses in the range $0.3m_s - 2m_s$. In this
restricted range of quark masses the existing data do not show any
significant deviation from linearity. One thus has to use $M_V$ in
order to extract $m_s$. Details of our analysis
and of the global data used are given in \cite{rMq96LANL}.
For Wilson fermions the lattice quark mass, defined at scale $q^*$, is
taken to be $m_L(q^*) a = ({1 / 2\kappa} - {1/ 2 \kappa_c} )$. For
staggered fermions $m_L(q^*) = m_0$, the input mass. The \MSbar\ mass
at scale $\mu$ is $\mmsbar(\mu) = Z_m(\mu a) m_L(a)$, where $Z_m$ is
the mass renormalization constant relating the lattice and the
continuum regularization schemes at scale $\mu$, and $\lambda =
g^2/16\pi^2$. In calculating $Z_m$, $a\ la$ Lepage-Mackenzie, we use
$\alphamsbar$ for the lattice coupling, use ``horizontal'' matching,
$i.e.$ $\mu = q^*= 1/a$, and do tadpole subtraction. We find that the
results are insensitive to the choice of $q^*$ in the range $0.86/a -
\pi/a$ and to whether or not tadpole subtraction is done. Once
$\mmsbar(\mu)$\ has been calculated, its value at any other scale $Q$
is given by the two loop running. We quote all results at $Q=2 \GeV$.
We extrapolate the lattice masses to $a=0$ using the lowest order
corrections (Wilson are $O(a)$ and Staggered are $O(a^2)$). In the quenched
fits we omit points at the stronger couplings ($a > 0.5 \GeV^{-1}$)
because we use only the leading correction in the extrapolation to
$a=0$, and because the perturbative matching becomes less reliable as
$\beta$ is decreased. The bottom line is that we find that the leading
corrections give a good fit to the data, and in the $a =0$ limit the
two different fermion formulations give consistent results. Our final
results are summarized in Table \ref{t_m}.
\begin{table}
\caption{Summary of results in $\MeV$ in $\MSbar$ scheme at $\mu=2\ \GeV$.
The label $W(0)$ stands for Wilson with $n_f=0$. An additional
uncertainty of $\sim 10\%$ due to the uncertainty in the lattice scale
$a$ is suppressed.\looseness=-1}
\input {t_m}
\label{t_m}
\end{table}
The global data for $\mbar$ and the extrapolations to $a=0$ for Wilson
are shown in Fig.~\ref{f_mbar}. Using the average of quenched estimates
given in Table~\ref{t_m} we get
\begin{equation}
\mbar(\MSbar,2 \GeV) = 3.2(4)(3) \MeV \quad {\rm (quenched)} .
\end{equation}
where the first error estimate is the larger of the two extrapolation
errors, and the second is that due to the uncertainty in the scale $a$.
\begin{figure}[t]
\hbox{\hskip15bp\epsfxsize=0.9\hsize \epsfbox {f_mbar.ps}}
\figcaption{$\mbar(\MSbar,2 \GeV)$ extracted using $M_\pi$ data with the scale
set by $M_\rho$.}
\label{f_mbar}
\end{figure}
The pattern of $O(a)$ corrections in the unquenched data ($n_f =2$) is
not clear and we only consider data for $\beta \ge 5.4$. The strongest
statement we can make is qualitative; at any given value of the
lattice spacing, the $n_f=2$ data lies below the quenched result.
Taking the existing data at face value, we find that the average of
the Wilson and staggered values are the same for the choices $\beta
\ge 5.4$, $\beta \ge 5.5$, or $\beta \ge 5.6$. We therefore take this
average
\begin{equation}
\mbar(2 \GeV) \approx 2.7 \MeV \qquad (n_f=2 {\rm\ flavors}) \ ,
\end{equation}
as the current estimate. To obtain a value in the physical case
of $n_f=3$, the best we can do is to assume a behavior linear in $n_f$.
In which case extrapolating the $n_f=0$ and $2$ data gives
\begin{equation}
\mbar(2 \GeV) \approx 2.5 \MeV \qquad (n_f=3 {\rm\ flavors}) .
\end{equation}
We stress that this extrapolation in $n_f$ is extremely preliminary.
We determine $m_s$ using the three different mass-ratios,
$M_K^2/M_\pi^2$, \ $M_{K^*} / M_\rho $, and $M_{\phi} / M_\rho$.
Using a linear fit to the pseudo-scalar data constrains ${m_s(M_K) =
25 \mbar }$. Using the vector mesons $M_K^*$ and $ M_\phi$ gives
independent estimates. The quenched data and extrapolation to $a=0$
of $m_s(M_\phi)$ are shown in Fig.~\ref{f_msphiQ}. The average of
Wilson and staggered values are
$m_s(M_\phi) = 96(10) \MeV $ and $m_s(M_{K^*}) = 82(20) \MeV$
where the errors are taken to be the larger of Wilson/staggered data.
From these we get our final estimate
\begin{equation}
m_s = 90(15) \MeV \qquad ({\rm quenched}) \ .
\end{equation}
The $n_f=2$ data shows a pattern similar to that for $\mbar$.
Therefore, we again take the average of values quoted in Table~\ref{t_m}
to get
\begin{equation}
m_s = 70(11) \MeV \qquad (n_f=2) \ .
\end{equation}
The error estimate reflects the spread in the data.
\begin{figure}[t]
\hbox{\hskip15bp\epsfxsize=0.9\hsize \epsfbox {f_msphiQ.ps}}
\figcaption{Comparison of $m_s(\MSbar,2 \GeV)$ extracted using $M_\phi$
for the quenched Wilson and staggered theories.}
\label{f_msphiQ}
\end{figure}
Qualitatively, the data show three consistent patterns. First, agreement
between Wilson and Staggered values. Second, for a
given value of $a$ the $n_f=2$ results are smaller than those in the
quenched approximation. Lastly, the ratio $\mbar / m_s(M_\phi)$ is in
good agreement with the next-to-leading-order predictions of chiral
perturbation theory for both the $n_f=0$ and $2$ estimates.
It is obvious that more lattice data are needed to resolve the
behavior of the unquenched results. However, the surprise of this
analysis is that both the quenched and $n_f=2$ values are small and
lie at the very bottom of the range predicted by phenomenological
analyses \cite{gasserPR}.
\section{CP VIOLATION and $\epsilon'/\epsilon$}
\label{s_CP}
A detailed analysis of 4-fermion matrix elements with quenched Wilson
fermions at $\beta=6.0$ is presented in \cite{rBK96LANL}. The
methodology, based on the expansion of the matrix elements in powers of
the quark mass and momentum, is discussed in \cite{rBK95LAT}. Our
estimates in the NDR scheme at $\mu=2\ \GeV$ are
\begin{eqnarray}
B_K &=& 0.68(4) \ , \nonumber \\
B_D &=& 0.78(1) \ , \nonumber \\
B_7^{3/2} &=& 0.58(2) \ , \nonumber \\
B_8^{3/2} &=& 0.81(3) \ .
\end{eqnarray}
The errors quoted are statistical. The major remaining sources of
errors in these estimates are lattice discretization and quenching.
To exhibit the dependence of the Standard Model (SM) prediction
of $\epsilon'/\epsilon$ on the light quark masses and the $B$ parameters
we write
\begin{equation}
\epsilon'/\epsilon = A \bigg(c_0 + c_6 B_6^{1/2} M_r + c_8 B_8^{3/2} M_r \bigg) \ ,
\end{equation}
where $M_r = (158\MeV/(m_s + m_d))^2$. For reasonable choices of SM
parameters Buras \etal\ estimate
$A = 1.3\times 10^{-4}$, $c_0 = - 1.3$, $c_6 = 7.9$, $c_8 = - 4.3$
\cite{rCP96Buras}. Thus, to a good approximation
$\epsilon'/\epsilon \propto M_r$; and increases as $B_8^{3/2}$
decreases. As a result, our estimates of $\mbar, m_s, B_8^{3/2}$
increase $\epsilon'/\epsilon $ by roughly a factor of three compared
to previous analysis, $i.e.$ from $3.6\times10^{-4}$ to $\sim
10.4\times10^{-4}$. This revised estimate lies in between the Fermilab
E731 ($7.4(5.9)\times10^{-4}$) and CERN NA31 ($23(7)\times10^{-4}$)
measurements and provides a scenario in which direct CP violation can
be explained within the Standard Model.
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\section*{Summary}
A striking similarity between the rise with energy ($\sqrt{s}$) of the
charged particle multiplicity in \ee and
the rise of \f at HERA is observed.
To the best of our knowledge, this similarity has not been noted before.
For $Q^2 \geq 5$ GeV$^2$ and $10^{-4}<x<0.05$,
the phenomenologically successful
MLLA expression for the average multiplicity in \ee
collisions, with the transformation $ s \rightarrow 1/x$, and adding
a QCD inspired
\q dependence, describes the HERA data on \f at small $x$ very well.
The result suggests that both deep inelastic small-$x$ scattering
and \ee annihilation can be adequately described
by angular ordered QCD radiation in an essentially free phase space.
\section*{Acknowledgement}
We thank V. Khoze, L. Lipatov and W. Ochs for stimulating discussions.
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\section{Introduction}
The mechanisms, by which proteins adopt well-defined three
dimensional topological structures, have been extensively investigated
theoretically \cite{Bryn95,Dill,Hinds94,Karp92,Thirum94,Wol}
as well as experimentally \cite{Bald,Rad}.
The major intellectual impetus for these
studies originate in the so called Levinthal paradox \cite{Levin}.
Since the number of conformations of even relatively short proteins is
astronomically large Levinthal suggested that it would be impossible for
proteins to reach the native conformation by a random search through all
the available conformation space. In the last few years several groups
\cite{Bryn95,Dill,Hinds94,Karp92,Thirum94} have provided,
largely complimentary, possible theoretical resolutions to the Levinthal
paradox. The unifying theme that emerges from these studies, all of which
are based on certain minimal model representations of proteins
\cite{Amara,Bryn89,Chan93,Chan94,Cov,Hinds92,Honey90,Gar88,Leop,Sali94a,Sali94b,Shakh89,Shakh91,Skol90,Skol91,Zwan95,Zwan92}
is that due to
certain intrinsic preference for native structures proteins efficiently
explore the underlying rough energy landscape. Explicit computations of
the energy landscape for certain lattice models \cite{Cam95}
reveal that foldable sequences (those that reach the native
conformations on a relatively fast time scales, which for real proteins is
typically of the order of a second) have relatively small free energy
barriers. These and related studies \cite{Chan93,Leop,Chan95}
have emphasized the
importance of the connectivity
between various low energy states in determining the kinetic foldability
of proteins. Thus, it appears that in order to fully elucidate the folding
kinetics of proteins it is necessary to understand not only the low energy
spectrum, but also how the various states are connected.
It is of interest to wonder if natural proteins have been designed so that
the requirements of kinetic foldability and stability have been
simultaneously satisfied and if so how are they encoded in the primary
sequence. If this is the case, then it follows that because protein
folding is a self-assembly process the kinetic foldability of proteins
should be described in terms of the properties of the sequence itself.
This argument would indicate that certain intrinsic thermodynamic
characteristics associated with the sequence may well determine the
overall kinetic accessibility of the native conformation. The minimal
models are particularly suitable for addressing this question. For these
models the folding kinetic rates for every sequence can be precisely
calculated for small enough values of \(N\) - the number of beads in the
model polypeptide chain. In addition the energy spectrum for small values
of \(N\) can be explicitly enumerated for lattice models and for moderate
sized proteins it can be computed by simulation methods. Thus, these
models afford a systematic investigation of the factors governing folding
rates.
It is perhaps useful right at the outset to say a few words about the
lattice representation of polypeptide chains. The energies employed in the
minimal models (or in other knowledge based schemes) should be thought of
as estimates of potentials of mean force after other irrelevant degrees of
freedom are integrated out. In principle, this leaves one with an
effective potential surface involving only the protein coordinates. In the
minimal models one further coarse grains this force field by eliminating
all coordinates except, perhaps, those associated with centers of the
residues. The lattice models further confine these centers to the
vertices of a chosen lattice. These arguments show that we can at best
expect only qualitative themes to emerge from these studies.
Nevertheless, these simulations together with other theoretical ideas have
provided testable predictions for the kinetics of refolding of proteins.
By using the random energy model (REM), originally introduced as the
simplest mean field spin glass model \cite{Der}, as a caricature of
proteins it has been proposed that the dual requirement (kinetic
accessibility of the native conformation as well as the stability) can be
satisfied if the ratio of the folding transition temperature \(T_{f}\) to
an equilibrium glass transition temperature \(T_{g,eq}\), at which the
entropy vanishes in REM, is maximized \cite{Bryn95,Gold92}. (It has been
noted that in order to use this
criterion in lattice models \(T_{g,eq}\) has to be replaced by a kinetic
glass transition temperature \(T_{g,kin}\) \cite{Socci94}.
It would be desirable to clarify the relationship between \(T_{g,eq}\)
and \(T_{g,kin}\)).
Based partially on lattice simulations of proteins a plausible
relationship between folding rates and the ratio of \(T_{f}\) to the
collapse transition temperature \(T_{\theta}\) was conjectured a couple of
years ago. In particular, Camacho and Thirumalai have argued that the fast
folding sequences have small values of \(\sigma =
(T_{\theta}-T_{f})/T_{\theta}\) \cite{Cam93}. The
theoretical reason for such an expectation has been given recently
\cite{Thirum95}. The advantage of the criterion, based on the
smallness of \(\sigma \) to classify fast and slow folding sequences, is
that both \(T_{\theta}\) and \(T_{f}\) are readily calculable from
equilibrium properties. More importantly, one can deduce \(T_{\theta}\)
and \(T_{f}\) directly from experiments.
More recently, Sali {\em et al.} \cite{Sali94a,Sali94b} have forcefully
asserted that for the class of minimal models of the sort described here
the \underline{necessary and sufficient} condition for folding (within a
preset time scale in Monte Carlo simulations) is that the native
conformation be separated from the first excited state by a large gap
(which is presumably measured in the units of \(k_{B}T\) with \(k_{B}\)
being the Boltzmann constant and \(T\) the temperature). However, their
studies are incomplete and rest on untested assumptions. They restricted
their conformation space to only a search among all compact structures of
a 27-bead heteropolymer. More importantly, they did not provide for their
model the dependence of the folding times as a function of the gap to
establish the kinetic foldability of any sequence. Thus there is no direct
evidence of the dependence of the folding times for various sequences and
the gap \(\Delta = E_{1}-E_{0}\) - which in the original study has been
stated as a mathematical theorem. Notice that this gap has been defined as
the difference between the two lowest energy levels assuming that both
these correspond to compact conformations. It is, in fact, relatively
straightforward to provide counter examples to this criterion \cite{Cam95}
casting serious doubts on the general validity of the
strict relationship between gap and folding times. Moreover, it is
extremely difficult, if not impossible, to obtain the value of the gap
for proteins
either experimentally or theoretically. Thus, the practical utility of
this criterion for models other than lattice systems is limited at best.
The major purpose of this study is to critically examine the
various properties of sequences (all of which have a unique ground state)
that determine the kinetic accessibility and stability of lattice
representations of proteins. This is done by calculating the folding rates
and thermodynamic properties using \(N=15\) for a number of sequences. The
qualitative lesson from this case is verified by studying a smaller number
of sequences for the much studied case of \(N=27\). The rest of the paper
is organized as follows: In Sec. (II) the complete details of the model as
well as simulation methods are discussed. The results of this study for a
variety of cases are given in Sec. (III). The paper is concluded in Sec.
(IV) with a discussion.
\section{Description of the Model and Simulation Techniques}
\subsection{Model}
We model proteins as chains of \(N\) successively connected beads
(residues) located at the sites of an infinite cubic lattice (Figs.
(1,2)). To satisfy
self-avoiding condition we impose the restriction that each lattice site
can be occupied only once (or remain free). The length of the bond between
two residues is fixed and is equal to the lattice spacing \(a=1\). Any
conformation (structure, in lattice terms), which the protein can adopt,
is described by \({\bf r}_{i}\) (\(i=1,..,N\)) vectors with discrete
coordinates \(x_{i}, y_{i}, z_{i} = 0,1,2...\), which are the positions of
residues on the lattice. We assume that the only interactions,
contributing to the total energy of a protein structure, are those
that arise due to the interactions between
residues that are far apart along a chain. We further assume that the
interactions are short ranged and can be represented by topological
contacts between residues. A topological contact is formed when two
nonbonded beads \(i\) and \(j\) (\(\mid i-j \mid \geq 3\)) are
nearest-neighbors on the cubic lattice, i.e., \(\mid{\bf r}_{i} - {\bf
r}_{j}\mid=a\). Thus, the total energy of a protein \(E\) is given by the
sum of the energies assigned to topological contacts found in a structure,
i.e
\begin{equation}
E = \sum_{i<j+3} B_{ij} \delta(r_{ij} - a),
\end{equation}
where \(B_{ij}\) is the interaction energy
between \(i\)th and \(j\)th residues, which form a topological contact,
\(r_{ij} = \mid{\bf r}_{i} - {\bf r}_{j}\mid\) is the distance between
them, and \(\delta(0) = 1\) and 0 otherwise. In order to take into
account the heterogeneity of interactions found in real proteins
we assume that the interaction
energies \(B_{ij}\) have a Gaussian distribution of the form
\cite{Sali94a,Sali94b,Shakh89,Shakh91}
\begin{equation}
P(B_{ij}) = \frac{1}{(2\pi B^{2})^{1/2}}\exp\biggl(-
\frac{(B_{ij}-B_{0})^{2}}{2B^{2}}\biggr),
\end{equation}
where \(B_{0}\) is the mean value and \(B\) is the standard deviation.
This model that has been extensively investigated theoretically
\cite{Sali94a,Sali94b,Shakh91,Shakh90}.
It should be stressed that the models studied here and similar lattice
models are at best caricatures of real proteins \cite{White}. The only
objective of
these studies should be to obtain qualitative behavior which hopefully
shed light on the experiments. This, of course, requires extrapolating
from these model systems to the behavior expected in proteins in terms of
experimentally variable parameters. A tentative proposal for achieving
this has recently been given \cite{Thirum95}.
\subsection{Choice of \(B_{0}\)}
Since the actual energy scales are not known, we set \(B\) in Eq. (2)
equal to \(1\) and thus all energies are expressed in the units of
\(B\). In contrast, we will demonstrate that the precise value of
\(B_{0}\) (or more precisely the ratio \(B_{0}/B\)) plays a crucial role.
Negative values of \(B_{0}\) favor
random collapse of the chain as the temperature is lowered. In addition,
the mean value \(B_{0}\) controls the nature of conformations that
constitute the low energy part of the spectrum. The extensive full
enumeration study of the conformational space for different sequences of
various lengths \(N\) indicates that as the mean value \(B_{0}\) decreases
structures with maximum number of topological contacts (compact
structures, CS) start to dominate among conformations with minimum
energies \cite{Sali94b,Klim}. Furthermore, a relationship
can be
obtained between the value of \(B_{0}\) and the ratio of hydrophilic and
hydrophobic residues in a sequence.
This would be relatively straightforward for random site models of
the sort introduced recently \cite{Gar94}.
For the random bond case,
that is the subject
of this and numerous previous studies, the computation of the fraction of
hydrophobic residues is somewhat ambiguous. Nevertheless, the following
procedure can be used to calculate their fraction.
Since it is
natural to identify negative interaction energies as those between
hydrophobic residues and positive ones to be between hydrophilic residues,
one can specify boundary energies \(B_{H}\) and \(B_{P}\)
(\(B_{H}=-B_{P}\)) in such a way that the energies \(B_{ij}\) below \(B_{H}\)
corresponds to hydrophobic interactions and the energies above \(B_{P}\)
pertain to hydrophilic interactions. The energies \(B_{ij}\), lying
between these boundaries, are associated with mixed interactions. If the
number of hydrophobic and hydrophilic residues in a sequences is
\(N_{H}\) and \(N_{P}\), respectively, (\(N_{H}+N_{P}=N\)), the fraction
of hydrophobic energies \(\lambda _{H}\) among \(B_{ij}\) is roughly
\((N_{H}/N)^{2}\).
This fraction can be also obtained by integrating the distribution (2) from
infinity to the energy \(B_{H}\). The relationship between
\(\lambda _{H}\) and \(B_{0}\) may be obtained as
\begin{equation}
\lambda _{H} \simeq (N_{H}/N)^{2} = \int _{-\infty}^{B_{H}}
P(B_{ij})dB_{ij}.
\end{equation}
The precise value of \(B_{H}\)
(and \(B_{P}\)) can easily be determined if one considers the case with
\(B_{0}=0\), for which \(N_{H}=N_{P}\).
Using Eqs. (2) and (3) we find that \(B_{H}=-0.675\) (for this
particular value of \(B_{0}\) one quarter of all energies \(B_{ij}\) are
below the boundary energy \(B_{H}\) and one quarter - above \(B_{P}\)).
It is known that in natural proteins hydrophobic residues make up
approximately 54 percent of all residues in a sequence \cite{Miller}. For the
distribution in Eq. (2) this implies that the mean value \(B_{0}\) should
be approximately \(-0.1\). Most of our simulations have been performed
with this value of \(B_{0}\).
We have also performed a study of the sequences with \(B_{0}=-2.0\)
that in the language of sequence composition means that hydrophobic
residues constitute about 94 percent of all residues. The motivation for
choosing this value of \(B_{0}\) is the following. For \(B_{0}=-2.0\) the
low energy spectrum becomes more sparse \cite{Sali94b}, because as
mentioned above the
main contribution comes from CS, whose total number is considerably less
than the number of conformations with any other number of topological
contacts \(c\). Specifically, for
\(N=15\) the number of conformations having \(c=11\) is 3,848, \(c=10\) -
17,040, \(c=9\) - 97,216, \(c=8\) - 313,868 etc. Studying the folding
rates of the sequences having different \(B_{0}\) enables us to assess the
role of the available conformation space and the connectivity between
various states in determining the kinetics of the folding process. The
choice of \(B_{0}=-2.0\) also allows us to compare directly our results to
previous studies found in the literature \cite{Sali94a,Sali94b}.
\subsection{Choice of \(N\)}
In order to fully characterize the folding scenarios it is
necessary to understand the kinetics of approach to a native conformation
in these models
as a function of \(N\) and temperature. It has been shown recently that
the folding of real proteins depends critically on \(N\), the
characteristic temperatures of the polypeptide chain (\(T_{\theta},
T_{f}\), and perhaps the kinetic glass transition temperature \(T_{g}\)),
viscosity, surface tension \(\gamma\) etc. \cite{Thirum95}. Thus in order to
make the results
of the minimal models relevant to proteins it is imperative to vary \(N\)
in the simulations.
Although one would like to understand the kinetic behavior of
foldable heteropolymers for sufficiently large \(N\) this is currently
computationally difficult. In the present study we have chosen \(N=15\)
and \(N=27\). We chose \(N=15\) because for this value one can perform
detailed kinetic study by including {\em all conformations} (compact and
noncompact). A detailed study for three dimensional (3D) lattice models
comparable to that undertaken for two dimensional (2D) systems has never
been done \cite{Dill,Chan93,Chan94,Binder}. With this value of \(N\)
one of the
limitations of the study of Sali {\em et al.}, who restricted themselves
to compact structures only, can be overcome. From any theoretical
perspective the qualitative difference in results between \(N=15\) and
\(N=27\) should be insignificant. This is certainly the experience in
simulations of polymeric systems \cite{Binder}. Thus, we
expect that the
qualitative aspects of the kinetics of folding should be quite similar for
\(N=15\) and \(27\). This is, in fact, the case.
One might naively think that for
\(N=15\) the total number of conformations is not enough for folding
times to exceed the Levinthal time, which is roughly the number of
conformations of the polypeptide chain.
The basis for this argument is that
the conformation space of \(N=15\) is considerably less than for \(N=27\).
The total number of conformations
of the chain of \(N\) residues \(C_{N}\) or equivalently the number of all
possible self-avoiding walks of \(N-1\) steps on a cubic lattice is
\cite{Chan90}
\begin{equation}
C_{N} \approx a(N-1)^{\gamma-1} Z_{\text{eff}}^{N-1}
\end{equation}
where \(Z_{\text{eff}}=4.684\), the universal exponent
\(\gamma\approx1.16\), and \(a=o(1)\). For \(N=15\) and 27 \(C_{N}\) is
approximately \(7.77\times10^{8}\) and \(4.6\times10^{17}\), respectively.
Full enumeration (FE) of SAW which are {\em unrelated} by symmetry for
\(N=15\)
gives \(C_{15}^{FE}=93,250,730\), which differs approximately from the
number of {\em all} SAW by a factor of 48. Thus, \(C_{15}\) obtained
from Eq. (4) and
\(C_{15}^{FE}\) are consistent. The chain of 15 residues can form 3,848
CS, which belong to 3x3x2 or 4x2x2 tetragonals, whereas 27-mer chain
adopts 103,346 CS with 28 topological contacts
\cite{Shakh90,Chan90}; all
these CS are confined to 3x3x3 cube.
From the above enumerations of the conformations it might be tempting to
speculate that
virtually all sequences for \(N=15\) with a unique ground state should
fold on the Levinthal time scale of \(9.3 \times 10^{7}\) Monte Carlo
steps (MCS). However, we find that for some of the thirty two sequences
examined the maximum folding time can be larger than \(10^{9}\) MCS
depending upon several characteristics (see below). Thus, even if the
chain samples one conformation per MCS the bottlenecks in the energy
surface can prevent the chain from reaching the native conformation. This
implies that the number of conformations \underline{alone} cannot
determine folding times \cite{Chan95}. In fact, in the case of
the models of disulfide bonded
proteins it has been explicitly demonstrated that a significant reduction
of available conformation space does not guarantee a decrease in folding
times \cite{Cam95}. Thus, kinetic
foldability is determined by several factors and
hence explicit studies of \(N=15\), where full enumeration of all
conformations is possible, should help us gain insights into the folding
of small proteins. A comparison of the results for small values of \(N\)
is also useful in assessing finite size effects.
The arguments given above together with explicit computations
given here reject claims \cite{KarpSali} that the
study of short chains (\(N < 27\) in
three or two dimensions) are not of significance in illustrating the
qualitative behavior of protein folding kinetics. Numerous studies have
shown that it is not merely the size of the conformation space, but the
connectivity between conformations, i.e. the nature of the underlying
energy landscape that allows one to distinguish between foldable and
non-foldable sequences \cite{Chan95}. The obvious advantage of
\(N=15\) is
that systematic thermodynamic and kinetic studies (already performed for 2D
chains of \(N\) up to 30) can be undertaken for 3D chains.
\subsection{Correlation Functions}
For probing the thermodynamics and kinetics of protein folding we use
the overlap function (considered here as an order parameter), which is
defined as \cite{Cam93}
\begin{equation}
\chi = 1 - \frac{1}{N^{2}-3N+2} \sum_{i\neq
j,j\pm 1} \delta(r_{ij} - r_{ij}^{N}),
\end{equation}
where \(r_{ij}^{N}\) refers to the coordinates of the native state. This
function
measures structural similarity between the native state and the state of
interest: the smaller the value of \(\chi\) becomes the larger a given
structure resembles the native one. Additional structural and
kinetic information can be obtained using
the function \(Q\), which counts the relative
number of native-like topological contacts in a structure
\cite{Bryn89,Sali94a,Sali94b,Skol90}
\begin{equation}
Q = \frac{c_{n}}{c_{n}^{tot}},
\end{equation}
where \(c_{n}\) is the number of native-like contacts in a given structure
and \(c_{n}^{tot}\) is the total number of contacts found in the native
structure.
We have calculated the relevant thermodynamic properties such as the
total energy \(<E>\), the specific heat \(C_{v}\) with
\begin{equation}
C_{v}=\frac{<E^{2}>-<E>^{2}}{T^{2}},
\end{equation}
the function \(<Q>\), and the Boltzmann probability of being in the native
state
\begin{equation}
P(E_{0})=\frac{\exp(-\beta E_{0})}{Z},
\end{equation}
where \(Z=\sum_{i} \exp(-\beta E_{i})\) and \(\beta=1/(k_{B}T)\) (\(k_{B}\)
is set to 1 in our simulations). The brackets \(< ... >\) indicate the
thermodynamic averages. In
addition, the overlap function and the fluctuations in \(<\chi>\), namely
\begin{equation}
\Delta \chi = <\chi ^{2}> - <\chi >^{2}
\end{equation}
were also calculated. The thermodynamic characteristics of the system can be
exactly calculated for each sequence by exhaustively enumerating the
various symmetry unrelated conformations for small enough values of
\(N\). In particular, we calculated these quantities exactly for
\(N=15\). For \(N=27\) we used slow cooling Monte Carlo method to calculate
the appropriate quantities of interest. The annealing simulation
procedure is discussed in Appendix A.
The parameter that distinguishes fast folding and slow
folding sequences appears to be \(\sigma =
(T_{\theta}-T_{f})/T_{\theta}\), where \(T_{\theta}\) is the collapse
transition temperature and \(T_{f}\) is the folding transition
temperature \cite{Cam93,Thirum95}.
It is known that even in these finite
sized
systems \(T_{\theta}\) can be estimated using the peak in the temperature
dependence of \(C_{v}\) (cf. Eq. (7)) \cite{Skol91,Cam93,Honey92}.
We have shown in previous
studies involving both lattice and off-lattice models that the temperature
dependence of the fluctuations in the overlap function, which serves as an
order parameter, can be used to calculate \(T_{f}\)
\cite{Cam93,Guo}. In particular,
\(T_{f}\) corresponds to the peak in the function \(\Delta \chi \).
For most sequences \(T_{\theta}\) and \(T_{f}\) are sufficiently
well separated that an unambiguous determination is possible by a
straightforward computation of the temperature dependence of \(C_{v}\) and
\(\Delta \chi \). However, we have generated five sequences (out of 32
for \(N=15, B_{0}=-0.1\)), for which \(C_{v}\) or \(\Delta \chi \)
appear not to have well-defined single maximum due to specific arrangement
of the energy states. For example, sequence 32 shows two maxima in the
dependence \(C_{v}(T)\) at \(T_{1}=0.28\) and \(T_{2}=0.73\) of different
amplitudes \(C_{v}(T_{1})=8.25\) and \(C_{v}(T_{2})=13.22\), respectively.
In this case, we defined \(T_{\theta}\) as a weighted average over the
temperatures \(T_{1}\) and \(T_{2}\)
\begin{equation}
T_{\theta}=\frac{C_{v}(T_{1})T_{1}+
C_{v}(T_{2})T_{2}}{C_{v}(T_{1})+C_{v}(T_{2})}.
\end{equation}
In other instances (e.g., for sequence 10), we found that although the
dependence \(C_{v}(T)\) has a single maximum at \(T_{max}\), it also has
the interval \((T',T'')\) not including \(T_{max}\), wherein the
derivative \(\frac{dC_{v}}{dT}\) again approaches almost zero value that
gives essentially unsymmetric form to the peak of specific heat. We have
also applied Eq. (10) for calculating \(T_{\theta}\) for such sequences by
setting \(T_{1}=T_{max}\) and \(T_{2}\neq T_{max}\) corresponds to the
temperature, at which \(|\frac{dC_{v}}{dT}|\) has the smallest value
within \((T',T'')\).
There are other ways of calculating \(T_{\theta}\) and \(T_{f}\). For
example, \(T_{\theta}\) could be directly inferred from the
temperature dependence of the radius of gyration of the polypeptide
chain. It has been shown in our earlier work on off-lattice models
\cite{Honey92}
that the resulting values of \(T_{\theta}\) coincide with those
obtained from the peak in the specific heat. The folding transition
temperature is often associated with the midpoint of the temperature
dependence of the probability of being in the native
conformation. This estimate of \(T_{f}\) is in good agreement with the
peak position of the temperature dependence of \(\Delta \chi \). In
general, different order parameters can be used to calculate
\(T_{f}\). The resulting values are fairly consistent with each
other.
\subsection{Sequence Design}
To create a database of different sequences for \(N=15\) we
generated 60 random sequences, using the mean value \(B_{0}=-0.1\) and 9
random sequences, using \(B_{0}=-2.0\). For \(N=27\) we generated 15
random sequences with \(B_{0}=-0.1\) and 2 with \(B_{0}=-2.0\). Note that
the computational procedures for 15-mer and 27-mer sequences are similar,
except that thermodynamic quantities for \(N=27\) are calculated from slow
cooling Monte Carlo simulations. By enumerating all possible
conformations for \(N=15\) we determined the energy spectrum for each
sequence. The program for enumerating all the structures a protein can
adopt on a cubic lattice is based on the Martin algorithm
\cite{Martin}
that is supplemented by the procedure that rejects all structures related
by symmetry. This algorithm allows us to determine the lowest (native)
energy state \(E_{0}\), its degeneracy \(g\), the coordinates of
corresponding structure(s), and the number of topological contacts \(c\)
for each sequence. The energy levels for 10 sequences are summarized in
Fig. (3) for \(N=15, B_{0}=-0.1\) and in Fig. (4) for \(N=27, B_{0}=-0.1\).
The spectra for \(N=15\) were obtained by enumerating all possible
conformations of the chain and arranging them in increasing order of
energy. For \(N=27\), on the other hand, the spectra for the various
sequences are obtained by slow cooling Monte Carlo method, the details of
which are reported in Appendix B. The results in Fig. (4) for \(N=27\)
are instructive. For each sequence we show two columns. The left column
gives the spectrum calculated by numerical method, whereas the right
column is the spectrum that would be obtained if only the compact
structures were retained. A comparison of the two columns for various
sequences clearly reveals that for a majority of sequences the low lying
energy levels are, in fact, noncompact. Thus, from this figure we would
conclude that these noncompact structures would make significant
contributions to various thermodynamic properties. This figure also shows
that for the sequences whose native conformation is compact,
the energy gap \(\Delta _{CS}\) calculated using compact structures
spectrum alone exceeds the true gap. This appears to be a general result
and can be
understood by noting that the lowest energy excitations for such
sequences are created by
flipping surface bonds. The resulting structure would be noncompact
and its energy would be lower or equal to that of other compact
structures.
Thus, it should in general be true that when the native conformation is
compact, \(\Delta _{CS} \geq \Delta\). Since there are large scale
motions that are expected to be involved in protein folding the
physically relevant energy scale should be the stability gap
\cite{Honey90,GuoHoney}. There is no straightforward relationship between the
stability gap and \(\Delta _{CS}\) or \(\Delta \).
We
rejected all sequences with \underline{nonunique} ground state from further
analysis.
In order to determine folding times for a range of
\(\sigma \) \((=(T_{\theta}-T_{f})/T_{\theta})\) values sequences with
varying
spectral characteristics are required. It is known that a generic randomly
generated sequence (even with unique native state) does not fold
rapidly \cite{Bryn95,Dill,Karp92}. Thus, to expand the
database of the sequences we used a
technique proposed by Shakhnovich and Gutin
\cite{Shakh93,Shakh94} to
create a set of
optimized sequences. We should stress that in our study this was used as
merely a technical device to generate sequences that span a rather wide
range of \(\sigma \). Here we will briefly present the idea of this
scheme \cite{Shakh93,Shakh94}.
One selects an arbitrary ('target') structure corresponding to a given
initial random sequence. Then standard Monte Carlo algorithm is applied
in the sequence space. The first step of this scheme can be described in
the following way. Two energies \(B_{ij}\) and \(B_{kl}\) of the initial
('old') sequence picked at random are interchanged, so that the
topological contact between residues \(k\),\(l\) has the interaction
energy \(B_{kl}'=B_{ij}\) and the contact between residues \(i\),\(j\) -
the interaction energy \(B_{ij}'=B_{kl}\). Thus, a new probe sequence,
which differs from the initial one by the energies \(B_{ij}'\) and
\(B_{kl}'\), is produced and is subject to Metropolis criterion. To do
this, the energy of the new sequence \(E_{new}\) fitted to the target
structure is calculated and compared with \(E_{old}\) of the initial
sequence. If the new sequence provides lower energy at the target
structure, it is unconditionally accepted and \(B_{ij}=B_{ij}'\),
\(B_{kl}=B_{kl}'\). If \(E_{old}<E_{new}\), the new sequence is accepted
with the Boltzmann probability \(P=\exp(-(E_{new}-E_{old})/T)\). If the
new sequence is rejected, the initial sequence is restored. This
permutation procedure, which does not alter the composition of the
sequences, is repeated \(n\) times. The control parameter is temperature,
and if it is sufficiently low, a series of sequences are quickly generated
after relatively small number of MCS (\(10^{4}\)), whose energies
when fitted to the target structure are remarkably low. By employing the
full enumeration procedure (or Monte Carlo simulations for \(N=27\)) one can
verify
that these energies are actually the lowest ones in the spectrum. In this
manner, a series of new 'optimized' sequences are created. An application
of optimization scheme allowed us to investigate essentially wider range
of values of the parameter \(\sigma\) than can be done by analyzing
sequences produced at random.
In all, for \(N=15\) we chose 32 sequences with \(B_{0}=-0.1\) and
9 sequences with \(B_{0}=-2.0\) for detailed study. Among these with
\(B_{0}=-0.1\), 15 sequences were random and 17 - optimized. The native
structures of these sequences were CS (e.g., structure (a) in Fig. (1)) as
well non-CS (e.g., structure (b)) and included
from \(8\) to \(11\) topological contacts. For \(B_{0}=-2.0\) all
sequences were optimized and as suspected the native conformations
were
all CS \cite{Sali94b}. For \(N=27\) we analyzed 15
optimized sequences
with \(B_{0}=-0.1\) and 2 with \(B_{0}=-2.0\). As for \(N=15\) their
native
conformations were CS as well as non-CS with 21-28
topological contacts (Fig. (2)).
We believe that this choice of sequences is sufficient and we do not
expect qualitatively different behavior for this model, if a larger
database is selected.
Since our objective is to compare the rates of folding for
different sequences it is desirable to subject them to identical folding
conditions. The equilibrium value of \(<\chi >\) measures the extent to
which the conformation at a given temperature is exactly equivalent to a
microscopic conformation, namely the native state. At sufficiently low
temperature \(<\chi >\) would approach zero, but the folding time may be
far too long. We chose to run our Monte Carlo simulations at a sequence
dependent simulation temperature \(T_{s}\) that
is subject to two conditions:
(a) \(T_{s}\) \underline{be less than} \(T_{f}\) for a specified sequence
so that the native conformation has the highest occupation probability; (b)
the value of \(<\chi(T=T_{s}) >\) be a constant for all sequences, i.e.
\begin{equation}
<\chi(T=T_{s}) >=\alpha .
\end{equation}
In our simulations we choose \(\alpha =0.21\) and this value was low
enough so that \(T_{s}/T_{f} < 1 \) for all the sequences examined. This
general procedure for selecting the simulation temperatures has been
previously used in
the literature \cite{Sali94b,Cam95}.
For \(N=15\) \(T_{s}\) is precisely determined using Eq. (11)
because \(<\chi(T=T_{s}) >\) can be calculated exactly using the full
enumeration procedure. The simulation temperatures for \(N=27\) are
calculated using the protocol described in Appendix A.
\subsection{Monte Carlo Simulations and Interpretation of Folding Kinetics}
In the present work we used standard Monte Carlo (MC) algorithm
for studying folding of different sequences to their native states. The
local simulation dynamics includes the following moves (Fig. (5)): (i)
corner moves, which flip the position of \(j\)th residue across the
diagonal of the square formed by bonds \((j-1,j)\) and \((j,j+1)\); (ii)
crankshaft rotations, which involve changing the positions of two
successively connected beads \(j+1\) and \(j+2\) (positions of \(j\) and
\(j+1\) beads, which are nearest neighbors on a lattice, remain
unchanged); (iii) rotations of end beads, in which the end bead (\(1\) or
\(N\)) moves to any of 5 adjacent sites to the beads \(2\) or \(N-1\) (the
sites previously occupied by the beads \(1\) or \(N\) are not considered
as possible new sites). This particular set of moves has already been
applied in Monte Carlo simulations \cite{Sali94b,Socci94}; for
a discussion of the dependence of the kinetic results on the move set
used in the Monte Carlo simulations see Ref. 13. In
order to ensure maximum efficiency of
exploring conformation space it is reasonable to assign different
probabilities for the moves (i), (iii) and the move (ii). After probing
several values we found that the probability \(p=0.2\) for the moves (i)
and (iii) (involving single residue) provides the best efficiency of
searching a native state of a test sequence by Monte Carlo algorithm.
Qualitatively, it is clear that very small probability of
moves (ii) depletes the ability of a chain to sample different
conformations, while excessively high probability of their occurrence may
deteriorate the ability to accept moves that would lead to acquisition of
the last few native contacts when the chain is near the native
conformation.
For clarity of presentation let us now describe one step of the
Monte Carlo algorithm. In the beginning, the type of move (single bead
move ((i) or (iii)) or crankshaft rotation (ii)) is selected at random
taking into account the probability \(p\) introduced above. After this a
bead in the chain is chosen at random and the possibility of performing
the selected move depending on the local configuration of the chain is
established as follows. If there is a chain turn at the bead
\(j\) selected for
move (i) or in the case of move (ii) the beads \(j,j+3\) are lattice
nearest neighbors, a move is accomplished; otherwise, one must return to
selection of move type. Then a self-avoidance criterion is applied: if the
move results in double occupancy of lattice sites, it is rejected and a
new selection of move type has to be made. If self-avoidance criterion is
satisfied, the new conformation is adopted and Metropolis criterion is
used, i.e., the energies of old and new conformations, \(E_{old}\),
\(E_{new}\), are compared. If new conformation has lower energy, the move
is accepted and a Monte Carlo step is completed. If \(E_{new}\) is higher
than \(E_{old}\), then the Boltzmann probability
\(P=\exp(-(E_{new}-E_{old})/T)\) is calculated and compared with a random
number \(\xi \) (\( 0 < \xi < 1\)). If \(\xi \) is smaller than \(P\), the
move is accepted and
MC step is completed. Otherwise, the move is rejected and the previous MC
step is counted as a new one. The fraction of accepted steps on average
constitutes 5-15 percents for the entire trajectory. In principle, using
ergodic measures this percentage can be adjusted {\em a priori} to
maximize sampling rate \cite{Mount}.
The initial conformations of all trajectories correspond to random
extended coil ('infinite temperature' conformation). After a sudden
temperature quench the chain dynamics was monitored for approximately
\(10^{5}\) to
\(5\) x \(10^{7}\) MC steps (MCS), depending on a folding kinetics of a
given sequence. In order to obtain the kinetics of folding for a
particular sequence the dynamics was averaged over \(M\) independent
initial conditions. For example, \(<\chi (t)>\) is calculated as
\begin{equation}
<\chi (t)> = \frac{1}{M} \sum_{i=1}^{M} \chi _{i}(t) ,
\end{equation}
where \(\chi _{i}(t)\) is the value of \(\chi \) for the \(i^{\text{th}}\)
trajectory at time \(t\). Another important probe of folding kinetics is
the fraction of trajectories \(P_{u}(t)\) which have not yet reached
the native conformation at time \(t\)
\begin{equation}
P_{u}(t)=1 - \int_{0}^{t} P_{fp}(s)ds,
\end{equation}
where \(P_{fp}(s)\) is the probability of the first passage to the native
structure at time \(s\) defined as
\begin{equation}
P_{fp}(s) = \frac{1}{M} \sum_{i=1}^{M} \delta (s - \tau _{1i})
\end{equation}
In Eq. (14) \(\tau _{1i}\) denotes the first passage time for the
\(i^{\text{th}}\) trajectory. Similarly, other quantities were
calculated. Typically, the number of trajectories \(M\) used in the
averaging varied from 100 to 800. We find that for smaller values one
cannot get reliable results at all. If \(M\) as small as \(10\) is used
\cite{Sali94b},
one can obtain qualitatively incorrect results. The precise choice of
\(M\) (which is sequence dependent) was determined by the condition that
the resulting kinetic and thermodynamic properties should not change
significantly with subsequent increase in \(M\). The tolerance used in
determining \(M\) was that the various quantities of interest converge to
within 5 percent.
\subsection{Computation of Folding Rates}
The most important goal of this paper is to obtain folding times
for sequences at various temperatures. These times (or
equivalently the folding rates) were calculated by analyzing the time
dependent behavior of the dynamic quantities. It is clear that \(<\chi
(t)>\) provides the most microscopic description of the kinetics of
approach to the native state. The folding times reported here were
obtained
by a quantitative analysis of \(<\chi (t)>\). For all sequences we find
that after a transition time \(<\chi (t)>\) can be fitted as a sum of
exponentials, i.e.,
\begin{equation}
<\chi (t)> = a_{1} \exp(-\frac{t}{\tau_{TINC}})+a_{2}
\exp(-\frac{t}{\tau_{f}})
\end{equation}
In most cases, biexponential fit like in Eq. (15) gave the best
approximation to the computed kinetic curves. However, for some sequences
it was found that a single or three exponential fit were more suitable.
The interpretation of the amplitudes \(a_{1}, a_{2} \) and the time
constants \(\tau_{TINC}, \tau_{f} \) are discussed in Sec. (III.A.3) . It
must be
noted that the kinetic curves for very slow folding sequences (those with
large values of \(\sigma \)) do not reach the required equilibrium values.
However, even in these instances (6 out of 32 for \(N=15, B_{0}=-0.1\))
our simulations were long enough to observe the transition to the native
conformation so that an accurate estimate of the folding time can be made.
It should be noted that the trends in folding times remain unchanged if
the mean first passage time is substituted for \(\tau _{f}\) (or \(\tau
_{TINC}\)) in Eq. (15).
\subsection{Monitoring Intermediates in Folding Process}
Since the underlying energy landscape in proteins is thought to be
rugged \cite{Bryn95,Dill,Thirum94} it is likely that there
are low energy basins of attraction, in which the protein can get trapped
in for arbitrary long times. Explicit construction of such landscape in
lattice models, albeit
in two dimensions, has revealed the presence of such states as important
kinetic intermediates in certain folding pathways \cite{Cam95,Cam93}. In our
simulations we
have used the following strategy to describe the nature of intermediates
in the folding of the various sequences. We divided each trajectory into
two parts: the first part starts at the beginning of the trajectory
(\(t=0\)) and ends when the native structure is formed for the first time,
i.e. when the first passage time \(\tau_{1i}\) for the \(i^{\text{th}}\)
trajectory is reached. We labeled the trajectory for \(0 \leq t \leq
\tau_{1i}\) as the relaxation part.
The remaining portion of the
trajectory for \(\tau_{1i} \leq t \leq t_{max}\) is referred to as the
fluctuation part, where \(t_{max}\) is the maximum time for which the
computations are done for a given sequence.
Using the trajectories corresponding to the relaxation regime we
calculated for each sequence the probability of occurrence of the low
energy states \(E_{k}\), i.e.
\begin{equation}
P_{r}(E_{k})=\frac{1}{M}\sum_{i=1}^{M}\frac{1}{\tau_{1i}}
\int_{t=0}^{\tau_{1i}} \delta(E_{i}(t)-E_{k})dt,
\end{equation}
where \(E_{i}(t)\) is the energy at the \(i^{\text{th}}\) trajectory at
the time \(t\).
The state with the energy \(E_{k}\), which has the largest value of
\(P_{r}\), is defined to be a kinetic intermediate for a given sequence.
We also calculated the probability that \underline{this state} occurs in the
fluctuation parts of the trajectories.
The fluctuation probability that the kinetic intermediates with the
energy \(E_{k}\) (for conformations other than the native one there
could be more than one structure with the same
energy) are visited after the transition to the native conformation is
defined as
\begin{equation}
P_{fl}(E_{k})=\frac{1}{M}\sum_{i=1}^{M}\frac{1}{t_{max}-\tau_{1i}}
\int_{t=\tau_{1i}}^{t_{max}} \delta(E_{i}(t)-E_{k})dt,
\end{equation}
where \(t_{max}\) is the maximum time of simulation.
This quantity was calculated to monitor if the chain, after reaching the
native conformation,
makes a transition to the same intermediate which was visited with
overwhelming probability on the way to the native conformation.
\section{Results}
Since this section describes in complete detail the results for a
variety of cases making it quite lengthy we provide a brief summary of its
contents. The general
methodology described in the previous section has been used to study in
extreme detail the kinetics of folding for \(N=15\). For this value of
\(N\) we have considered two values of \(B_{0}\) which sets the overall
strength of the hydrophobic interactions. We have chosen \(B_{0}=-0.1\)
and \(B_{0}=-2.0\). The former choice is a bit more realistic, while the
latter was chosen to contrast the role of CS versus non-CS in determining
the thermodynamics and kinetics of folding. As emphasized before the value
of \(N=15\) is about the largest value of \(N\) for which exact
enumeration studies are possible in three dimensions and for which the
kinetics can be precisely determined in terms of all allowed conformations
being explored. The results for \(N=15\) and for \(B_{0}=-0.1\) and
\(B_{0}=-2.0\) are presented in Sec. (III.A). In Sec. (III.B) we present the
results for \(N=27\), for which we have also chosen \(B_{0}=-0.1\). The
thermodynamic properties with \(B_{0}=-2.0\) are discussed as well. A
comparison of \(N=15\) and \(N=27\) shows very similar qualitative
behavior.
\subsection{\(N = 15\): \(B_{0}=-0.1\) and \(B_{0}=-2.0\)}
\subsubsection{Thermodynamic Characteristics}
The two relevant temperatures \(T_{\theta}\) and \(T_{f}\) for each
sequence are computed from the temperature dependence of \(C_{v}\) and
\(\Delta \chi \), respectively.
In addition we have computed \(<Q>\) and \(<\chi >\) as a function of
temperature. The midpoints in the graphs of these quantities can
sometimes be used to obtain an estimate of \(T_{f}\). In general,
\(T_{f}\) obtained from the peak of \( \Delta \chi \) is smaller than
that determined from the midpoint of \(<\chi >\) or \(<Q>\).
The plots of \(<\chi >\), \( \Delta \chi \) ,\(<Q>\), and \(C_{v}\) as a
function of temperature are displayed in Fig. (6) for the sequence labeled
14. From the graphs of \(C_{v}\) and \( \Delta \chi \)
the collapse transition temperature \(T_{\theta}\) and the folding
transition temperature \(T_{f}\) are found to be 0.65 and 0.45
respectively. The simulation temperature, \(T_{s}\), is calculated using
Eq. (11) and in this case it turns out to be 0.38. In Fig. (7a) we present
the dependence of \(T_{s}\) on the crucial parameter \( \sigma =(T_\theta
-T_f)/T_\theta \). In addition we also display the correlation between
\(T_{s}\) and the energy gap \(\Delta \) (Fig. (7b)). From these figures it
appears that \(T_{s}\) correlates well with \(\sigma \).
Thus as far as the simulation
temperature is concerned it appears that \(T_{s}\) is decreasing function
of \(\sigma \). This implies that if \(\sigma \) is small then the
simulation temperature can be made higher and fast folding can therefore be
expected. This is further quantified in Sec. (III.A.4). We should emphasize
that this correlation is only statistical in the sense that large
(small) values of \(\sigma\) yield small (large) values of \(T_{s}\).
However given two values of \(\sigma\) that are closely spaced it is not
possible to predict the precise values of \(T_{s}\).
Since the dimensionless parameter \(\sigma \) serves to distinguish between
foldable sequences and those that do not reach their native conformation
on a reasonable time scale it is interesting to see if \(\sigma \) can
correlate with the spectrum of the underlying energy function. It has been
argued that the only important parameter that is both necessary and
sufficient to account for foldability is the gap
\(\Delta \) \cite{Sali94a,Sali94b,KarpSali}.
The plot of \(\sigma \) as a
function of \(\Delta \) is shown
in Fig. (8a). In the lower panel (Fig. (8b)) we plot \(\sigma \) as a
function of the dimensionless parameter \(\Delta /T_{s}\). This figure
shows very clearly the lack of correlation between \(\sigma \) and
\(\Delta \). Thus, it is seen that no clear correspondence exists even in
these models between \(\sigma \) and \(\Delta \). This, of course, is not
surprising because \(T_{\theta}\) is determined by the entire energy
spectrum - most notably the higher energy non-compact structures. The
results in Fig. (8a) show that large values of \(\sigma \) appear to
correspond well with small values of \(\Delta \). On the other hand small
values of \(\sigma \) simultaneously corresponds to both small as well as
large values of \(\Delta \).
\subsubsection{Contribution to Thermodynamic Properties from Non-compact
Structures}
The number of non-compact structures even for small values of \(N\) (such
as 15 and 27) far exceeds that of compact structures.
It has already been mentioned that full enumeration of all self-avoiding
structures in three dimensions becomes increasingly difficult for \(N >
15\). Thus in the literature it has been explicitly assumed that, in
general, the native conformation in this model is maximally compact
and that the thermodynamic properties can be determined
using the spectrum of compact structures alone \cite{Sali94a,Sali94b,Karp95}.
It
has also been argued that the only relevant aspect of the energy spectrum
that
determines both kinetics and thermodynamics of folding in these models is
the gap defined as
\begin{equation}
\Delta = E_{1} - E_{0},
\end{equation}
where \(E_{0}\) and \(E_{1}\) are the lowest energy and the
energy of the first
excited state in the spectrum. Since we have
enumerated all possible conformations for \(N=15\) this can be explicitly
checked by comparing various thermodynamic quantities computed exactly
with those obtained by
including only the contribution from compact structures. This has been done
for several
sequences and the typical results for two sequences are shown in Figs. (9)
and (10). In Fig. (9) we present the results for \(<\chi >\) and
\(\Delta \chi \) for \(B_{0}=-0.1\). The relatively small but
realistic value of
\(B_{0}\) makes the comparison between these quantities calculated using
CS alone and the values calculated using full enumeration least
favorable. From Fig. (9a) we find that \(T_{s}\) found from full
enumeration is roughly one half that obtained using compact structures
enumeration (CSE) alone. In fact,
\(T_{s}^{CSE}\) \underline{exceeds} the collapse transition temperature
\(T_{\theta}\)
which implies that if simulations are performed at this temperature the
native conformation will not be stable at all. Fig. (9b) shows that
\(T_{f}\) (the folding transition temperature) is once again one half
that of \(T_{f}^{CSE}\), indicating the importance of non-compact
structures in this case.
In Fig. (10) we show the behavior of \(<\chi > (T)\) and
\(\Delta \chi (T) \) for \(B_{0}=-2.0\) for another sequence. In this case
the agreement between the exact results and that calculated using CS
alone is significantly better. The difference between the two is roughly
on the order of ten percent. These calculations clearly show that
even in these models the importance of non-compact
structures is dependent upon the value of \(B_{0}\). Only for large values
of \(\mid B_{0} \mid \) the low energy spectrum is dominated by CS alone.
\subsubsection{Folding Kinetics: Kinetics of Approach to the Native
Conformation}
We begin with a discussion of the approach to the native state starting
from an ensemble of disordered conformations. The kinetics of reaching the
native state was monitored by studying the time dependence of \(<\chi
(t)>\) averaged over several initial conditions. For all the sequences
that we have examined we find that \(<\chi (t)>\) can be fit by a sum of
exponentials after a transient time. In our previous studies using
off-lattice models we had shown
that in general for a foldable sequence a fraction of initial population
of molecules reaches the native state directly without encountering any
discernible intermediates \cite{Guo,ThirumGuo}. This is the case in these
models as well, thus
further supporting the conclusions of our earlier work which was based on
Langevin
simulations of off-lattice models. Since the data base analyzed here is
more extensive, covering a wide span of \(\sigma \), we can further classify
the meaning of the various exponential terms that arise in the time
dependence of the overlap parameter. In order to classify the various
sequences in terms of the rapidity of folding to the native conformation
we have used the parameter \(\sigma \) as a discrimination factor. \\
{\bf (a) Fast folding sequences (\(\sigma \lesssim 0.1\)):}\\
For these sequences the structural overlap function \(<\chi (t) >\) for
\(t\) greater than a transient time is adequately fit by a single
exponential, i.e.,
\begin{equation}
<\chi (t) > \simeq a_{f} \exp(-t/t_{TINC}).
\end{equation}
In these cases the folding appears to be a two state all-or-none process
and \(\tau _{f} \approx \tau _{TINC}\), where \(\tau _{TINC}\) is the
time scale of topology inducing nucleation collapse (TINC).
The folding and the collapse is
almost synchronous. It has been shown in other studies that the
folding for these sequences proceeds by a TINC mechanism
\cite{Guo,ThirumGuo,Abk1,Abk2,Otzen}. In these studies the TINC
mechanism was established by studying the microscopic dynamics of the
trajectories. We found that once a critical number of contacts is
formed (corresponding to a nucleus) the native conformation is reached
rapidly.
This fast folding is clearly observed when \(\sigma\) is less that 0.1. \\
{\bf (b) Moderate folding sequences (\(0.1 \lesssim \sigma \lesssim
0.6\)):}\\
These are sequences when a single
exponential fit cannot describe time course of the overlap function
\(<\chi (t) >\). We find that after an initial time, \(<\chi (t) >\) is
well fit by two exponentials. The interpretation of the fast and slow
processes have been given elsewhere in our studies of continuum models
using Langevin simulations \cite{Guo,ThirumGuo}.
The range of \(\sigma \) that
characterizes
moderate folding is \(0.1 \lesssim \sigma \lesssim 0.6\). The onset of the
intermediate values of
\(\sigma\) is easy to obtain. This can be inferred by the smallest value
of \(\sigma\) for which biexponential fit of \(<\chi (t) >\) is
required to describe the approach to the native conformation. \\
{\bf (c) Slow folding sequences (\(\sigma \gtrsim 0.6\)):}\\
These are sequences with \( \sigma \gtrsim 0.6 \). When
\(\sigma \) gets close to unity, these sequences do not fold on
any reasonable simulation time. If \(\sigma \) is greater than almost 0.6
we once again find that multiexponential fit to \(<\chi (t) >\) is needed.
The boundary between moderate and slow folding sequences is rather
arbitrary. In both these cases we find that the various stages of folding
can be described by a three stage multipathway mechanism (TSMM):
The initial stage is characterized by random collapse, the second stage
corresponds to the kinetic ordering regime in which the search among
compact structures leads to native-like intermediates
\cite{Cam93,Guo}. The final stage
corresponds to activated transition from one of the native-like structures
to the native state, which is the rate
determining step
for folding. Thus, the transition states occur close to
the native conformation as was shown some time ago in off-lattice
simulations \cite{Honey90,Honey92}. Our earlier lattice and
off-lattice studies describe in detail the evidence for the
TSMM. Analysis of the trajectories probing the approach to the native
state (measured by \(\chi (t)\)) for the sequences studied exhibits
similar behavior.
\subsubsection{Dependence of Folding Times on \(\sigma \)}
In an earlier two dimensional lattice simulations we have suggested that
the sequences that fold fast appear to have small values of \(\sigma =
(T_{\theta } - T_{f})/T_{\theta }\) \cite{Cam93}. The reason for expecting
the relationship between \(\sigma \) and the folding time \(\tau_{f}\) has
been given recently \cite{Thirum95}. Physically if
\(\sigma \) is small then
\(T_{\theta } \approx T_{f}\) and all possible transient structures that
are explored by the chain on its way towards the native state are of
relatively high free energy. Consequently, all the structures that are
likely to act as \underline{traps or intermediates are effectively
destabilized} and thus folding to the native structure is rapid. For
large \(\sigma \) \(T_{f}\) and \(T_{\theta }\) are well separated
and hence the chain searches many compact globular states in a rough
energy landscape that leads to slow folding. When \(\sigma \) is small
the folding process and the collapse is synchronous and this leads to
fast folding.
Similar conclusions have been reached for 2D square lattice proteins
\cite{Cam93,Betan}.
In order
to test the possible relationship between the folding time and \(\sigma
\) we have calculated \(\tau_{f}\) for the
database of sequences generated by the method described in Sec. (II). The
results of this simulations are plotted in Fig. (11) for \(N=15\) and
\(B_{0}=-0.1\) (solid circles), \(B_{0}=-2.0\) (open circles). The figure
shows that the folding time \(\tau_{f}\) correlates statistically
extremely well with the intrinsically thermodynamic parameter \(\sigma \).
The sequences span a range of \(\sigma \) and consequently meaningful
conclusions can be made. In fact, sequences with small values of \(\sigma
\) fold extremely rapidly (fast folding sequences): for \(\sigma \)
smaller than 0.1, \(\tau_{f}\) hardly exceeds \(10^{4}\) MCS. Most
sequences (i.e., having \(\sigma \) between 0.15 and 0.6) have
intermediate folding rates extending from roughly \(10^{5}\) to \(10^{7}\)
MCS (moderate folding sequences). The sequences with the largest values of
\(\sigma \) (greater than 0.6) are slow folding sequences, whose typical
folding times are above \(10^{7}\) MCS. Thus, in the range of \(\sigma \)
examined the folding rate changes by about four to five orders of
magnitude. It is important to point out that all fast folding sequences
are optimized, whereas moderate folding sequences also include random
ones. Slow folding sequences are exclusively random. This distinction
based on folding times and its dependence on \(\sigma \) was used in
classification of the sequences in the discussion in Sec. (III.A.3).
\subsubsection{Relationship between \(\tau_{f}\) and \(\Delta\)}
Sali {\em et al.} have recently asserted (without providing explicit
calculation of folding times) that sequences that fold rapidly and whose
native conformation is also stable are characterized by large
gap \cite{Sali94a,Sali94b}.
The gap in their model is defined using Eq. (18). In
order to check
this claim we plot \(\tau_{f}\) as a function of \(\Delta \) in Fig.
(12) for \(N=15\). The corresponding plot for \(N = 27\)
is shown in the next section. Fig. (12) shows that this parameter
is of little relevance when used to classify folding rates of
different sequences. It appears that the sequences with large gaps
\(\Delta \) usually fold rather rapidly (about \(10^{4}-10^{5}\) MCS).
However, sequences having a small energy gap \(\Delta \) can have either very
small folding times (below \(10^{4}\) MCS) or do not reach native state
even after \(10^{8}-10^{9}\) MCS. For example, from Fig. (12) it is clear
that if \(\tau_{f}\) is fixed at
\(10^{5}\) MCS then we can, in principle, generate a large number of
sequences with very small values of \(\Delta \) to very large
\(\Delta \) all with roughly the same \(\tau_{f}\). Thus,
\(\Delta \) alone cannot be used to discriminate between fast and
slow folding sequences. The existence of large values of \(\Delta/k_{B}T\)
being a criterion for stability follows from Boltzmann's law with proteins
being no exception.
\subsubsection{Kinetic Events in the Folding Process}
We have systematically investigated the microscopic processes that are
involved in the folding of several sequences. This has been done by using
the methodology for monitoring the intermediates described in Eqs. (16) and
(17). We discuss the results of this study for the various sequences using
the classification in terms of the parameter \(\sigma \). \\
{\bf (a) Fast folders (\(\sigma \lesssim 0.1\)):} In this case there are no
well defined
intermediates in the sense that the chain gets trapped in a conformation
that is distinct from the native state for any length of time. For these
fast folding sequences we find that the native conformation is reached by
essentially a TINC mechanism,
i.e. once a certain
number of critical native contacts is established then the native state
is reached rapidly \cite{Guo,ThirumGuo,Abk1}.
By using a combination of
Eqs. (16) and (17) we find that
for fast folders the chain frequently visits the nearest low lying energy
conformations even after reaching the native structure. For these
sequences these low lying states are almost native-like (have about 90
percent of native contacts). In terms of the underlying energy landscape
it is clear that they belong to the same basin of attraction as the
native conformation. \\
{\bf (b) Moderate and slow folders (\(\sigma \gtrsim 0.1\)):} These
sequences appear to have well defined intermediates and
their significant role makes folding in this range of \(\sigma \) quite
distinct from the fast folders. In Fig. (13) we plot \(P_{r}\)
and \(P_{fl}\) (see Eqs. (16,17)) for a variety of sequences. The sequences
are arranged in
order of increasing folding time. Recall \(P_{r}\) corresponds to the
average probability that the
intermediate with the energy \(E_{k}\) has the highest probability of
occurrence before the
native conformation is reached for the first time and \(P_{fl}\) is the
average probability that this state is revisited after the native state is
reached. This graph shows several striking features: (i) For moderate
folders there is a finite probability of the chain revisiting the same
intermediate that it sampled in the approach to the native conformation.
This seldom happens in the sequences that fold slowly. It is
clear that slow folding sequences have well defined intermediates which
are not visited after the chain reaches the native conformation. These
results suggest that the rate determining step in slow folding sequences
is the transition from one of these intermediates to the native state.
This involves overcoming a substantial free energy barrier. The
existence of this barrier also prevents frequent excursions from the
native state. (ii) It is of interest to probe the nature of intermediates
that are encountered in the folding process. In Figs. (14a) and (14b) we
show,
respectively, the fraction of native contacts in the most populated
intermediates and the corresponding overlap \(\chi _{k}\) with
native conformation for all sequences. For both fast and moderate folders
it is clear that
the states that are sampled have great structural similarity to the native
conformation. In fact, in this case these conformations have roughly 80
percent of the native contacts. However, slow folding sequences have only
about 50 percent of native contacts in the intermediate structures. It
also turns out that in the case of moderate folders the most populated
intermediates prior to formation of native conformation is most often the
first excited state whereas in slow folding sequences it is the higher
excited states that have the largest probability of occurring. (iii) The
rate of formation of the intermediates can be ascertained by examining
Fig. (14c), in which the ratio of the mean time to reach the intermediate
\(\tau _{k}\) to the folding time \(\tau _{fp}\) is plotted.
Intermediates for fast and moderate folding sequences
usually occur at later stages of folding than for slow folding sequences.
In fact, for the latter cases the ration \(\tau _{k}/\tau _{fp}\) can be
as low as 0.1. This implies that these relatively stable misfolded
structures are formed
relatively early in the folding process and these off-pathway processes
therefore slow down the folding considerably. It also follows that the
rate determining step for slow folders occurs late in the folding process
implying that the transition states are closer to the native structure
\cite{Honey92}.
\subsection{\(N = 27\): \(B_{0}=-0.1\) and \(B_{0}=-2.0\)}
For \(N = 27\) we have generated 15 sequences with \(B_{0}=-0.1\)
using the optimized design procedure described in Sec. (II.E). In this case
all of the sequences have been optimized so that \(\sigma \) is in the
range of \(\sigma \lesssim 0.12\). We have studied the thermodynamic
properties of few sequences with \(B_{0}=-2.0\). Since we have already
established that non-compact structures make significant contribution to
both the thermodynamic and kinetic properties for \(N=15\) it is
necessary to include them in studying the case of \(N=27\) as well. The
number of non-compact structures for \(N=27\) is of the order of
\(10^{18}\) and their exact enumeration is impossible. We have,
therefore, used slow cooling Monte Carlo method (see Appendix A) to
calculate the thermodynamic properties in this case. The calculation
of the kinetic processes
have been done as before for \(N=15\). We discuss there results
below. Since the qualitative behavior remains the same we provide a less
detailed account for this case.
\subsubsection{Thermodynamic Characteristics}
As before we have determined \(T_{\theta }\) and \(T_{f}\) from computing
the temperature dependence of \(C_{v}\) and \(\Delta \chi\). The
simulation temperature was found by requiring that the overlap function
\(<\chi (T_{s})> = 0.21\). These temperatures were calculated using
Monte Carlo simulations. It is interesting to compare the results for the
overlap function obtained from MC simulations with that calculated using
CSE only (Figs. (15,16)). The results for the temperature
dependence of \(\Delta \chi\) for one sequence with \(B_{0}=-0.1\) is
given in Fig. (15b) and in Fig. (16b) \(\Delta \chi\) as a function of
\(T\) is plotted for another sequence with \(B_{0}=-2.0\). The dotted
line in these figures are the results obtained using the contribution of
compact structures only. Both these sequences have large values of the
energy gaps \(\Delta \). These figures show dramatically that the
neglect of noncompact
structures leads to serious errors in the determination of \(\Delta
\chi\). Similar discrepancies are found for other thermodynamic
quantities as well. The errors in the estimate of \(T_{f}\) in these two
sequences are about a factor of 2 - 3. In fact, in both cases the
estimate of \(T_{f}^{CSE}\) obtained using compact structures alone
\underline{exceeds}
the collapse transition temperature \(T_{\theta }\). The neglect of
noncompact structures, even for \(B_{0}=-2.0\), results in more serious
errors than for \(N=15\). In any event for both \(N=15\) and \(N=27\)
restriction to compact structures alone can lead to incorrect results for
thermodynamic properties.
It is interesting that the discrepancy between the simulation
temperature \(T_{s}^{MC}\) and \(T_{s}^{CSE}\) is even more dramatic for
\(N=27\) at the value of \(B_{0}=-0.1\) (Fig. 15a). In particular, for the
sequence 61 for which \(\Delta \chi (T)\) is displayed in Fig. (15b)
\(T_{s}^{CSE}=3.19\), which is even larger than \(T_{\theta
}^{MC}=1.22\), exceeds \(T_{s}^{MC}=1.17\) by almost a factor of three.
As expected for \(B_{0}=-2.0\) the discrepancy is somewhat smaller but is
still very significant (see Fig. (16a)). In this case \(T_{s}^{CSE}=3.60\)
is almost twice as large as \(T_{s}^{MC}=2.06\). The value of \(T_{\theta }\)
for this sequence is \(T_{\theta }^{MC} = 2.14\). This implies
that, if the simulation temperatures \(T_{s}^{CSE}\) are used, the only
conformations that are thermodynamically relevant are the random
coil ones. These observations clearly demonstrate that the use of only
compact structures for calculating thermodynamic quantities is in general
totally flawed and would lead to incorrect evaluation of the folding
rates for both \(B_{0}=-0.1\) and \(B_{0}=-2.0\) for any \(N\). It is
worth noting that a similar conclusion has been reached
for a two letter code model with \(N = 27\) \cite{Socci95}.
In Fig. (17) we collect the results for the ratio of the simulation
temperature calculated using compact structures \(T_{s}^{CSE}\) to that
computed using all available conformations \(T_{s}\). For the case
of \( N = 27\) we have already emphasized that the simulation temperature
(denoted as \(T_{s}^{MC}\)) is calculated using Monte Carlo simulations
the details of which are examined in Appendix A. In Fig. (17a) we show the
ratio \(T_{s}^{CSE}/T_{s}^{FE}\) for \(N=15, B_{0}=-0.1\). It is clear
that except for one sequence this ratio is greater than unity and is
large as 2.5. The results \(T_{s}^{CSE}/T_{s}^{MC}\) for \( N = 27,
B_{0}=-0.1\) are displayed in Fig. (17b). Here the effects are even more
dramatic. In all cases this ratio exceed 2.0 implying that noncompact
structures make significant contributions in determining thermodynamic
properties.
A further consequence of using only compact structures to determine
\(T_{s}\) (as has been done elsewhere \cite{Sali94a,Sali94b})
is that at these
high temperatures the native conformation has no stability. It is because
of the very large values of \(T_{s}^{CSE}\), Sali {\em et al.} find that in
many cases their native conformations are not well populated. In fact, in
most cases the probability of being in the native conformation is less
than 0.1.
\subsubsection{Folding Kinetics: \(N=27\) and \(B_{0}=-0.1\)}
Since the time scales for folding for \(N=27\) are quite long we have
restricted ourselves to determining the folding rates for optimized
sequences only. Thus, we have examined 15 sequences with characteristic
temperatures \(T_{\theta }\) and \(T_{f}\) that provide the values of
\(\sigma \) being less than about 0.12. These sequences, according to the
classification derived at detailed study of \(N=15\), would all be the
fast folders. Thus, we expect that most of these sequences would
reach
the native conformation extremely rapidly.
In these instances folding would appear to be a two state
all-or-none process and the time dependence of \(<\chi (t)>\) should be
exponential. All these expectations are borne out. In Fig. (18a) we show
\(<\chi (t)>\) for one of the sequences from fifteen examined. It is
obvious that \(<\chi (t)>\) is well fit by a single exponential process.
For some sequences we do find that \(<\chi (t)>\) can be better fitted by
a sum of two exponentials. Thus, for \(N=27\) even for these small values
of \(\sigma \) we find that these sequences should be classified as
moderate folders (Fig. (18b)). This is not surprising because as \(N\)
increases the
probability of forming misfolded structure also increases. The boundaries
differentiating the fast and moderate folders depend on the sequence
length. For large values of \(N\) the range of \(\sigma \) over which the
sequences behave as fast folders decreases. Consequently, the partition
factor \(\Phi (T)\), which is the fraction of initial population of
molecules that reaches the native conformation {\em via }
TINC mechanism, decreases.
\subsubsection{Dependence of Folding Times on \(\sigma \),
\(\Delta \), and \(\Delta _{CS}\) }
The dependence of the \(\tau_{f}\) on \(\sigma \) is shown in Fig. (19).
Even though we have examined only a small range of \(\sigma \) the general
trend that \(\tau_{f}\) is well correlated with \(\sigma \) is clear.
Considering that one has statistical errors in determining both \(\sigma
\) and \(\tau_{f}\) the observed correlation between these quantities is,
in fact, remarkable. In this limited range of \(\sigma \) the folding time
changes by nearly a factor of 300 indicating that small changes in
\(\sigma \) (which is an intrinsic property of the sequence) can lead to
rather large changes in \(\tau_{f}\). The behavior of \(\tau_{f}\) on
\(\Delta \) is shown in Fig. (20). The trend one notices is the same as in
the case of \(N=15\). In fact, the lack of any correlation between
\(\tau_{f}\) and \(\Delta \) is even more apparent here. As in the case
for \(N=15\) it is possible to generate sequences with arbitrary values of
the gap that would all have roughly the same folding time. Notice that
although \(\sigma \) covers only a small range the gap extends over a
much wider interval. This also implicitly indicates no dependence of
\(\sigma \) on \(\Delta \).
If there would be any plausible relation between \(\tau _{f}\) and
\(\Delta \) it is clear that \(\Delta \) has to be expressed in terms of
a suitable dimensionless parameter. Unfortunately the proponents of the
energy gap idea \cite{Sali94a,Sali94b}
as the discriminator of folding sequences have used
varying definitions of \(\Delta \) in different papers without providing
the precise way this is to be made dimensionless. The energy parameter
that would make \(\Delta \) dimensionless cannot be \(B\), the standard
deviation in the distribution of contact energies which merely sets the
energy scale,
because this would mean this criterion would apply only to this model.
The gap \(\Delta \) in Figs. (12) and (20) is
measured in
units of \(B\). A very natural way to make \(\Delta \) dimensionless is
to divide it by \(k_{B}T\) which is in this case \(k_{B}T_{s}\). In
Fig. (21) we have presented the folding times \(\tau _{f}\) as a function
of \(\Delta /T_{s}\). The upper panel is for \(N = 15\), while the lower
one is for \(N = 27\). These figures demonstrate even more
dramatically the irrelevance of \(\Delta /T_{s}\) as a parameter in
determining folding times. In fact, this figure is almost like a scatter
plot. Thus, it is clear that energy gap alone (measured in any suitable
units)
does not determine folding times in these models.
From this it follows that the classification of sequences into fast and
slow folders cannot be done using the value of \(\Delta \) (measured in
any reasonable units) alone.
In the literature it has been forcefully asserted that foldability of
sequences in this class of models is determined by \(\Delta _{CS}\),
the energy gap for the ensemble of compact structures
\cite{Sali94a,Sali94b}. Plotting \(\tau
_{f}\) as a function of \(\Delta _{CS}\) for our model appears to be
somewhat ambiguous because about half of the sequences have
non-compact native conformations. In Fig. (22) we plot \(\tau _{f}\)
as a function of \(\Delta _{CS}\). The upper panel is for \(N=15\)
and the lower panel is for \(N=27\). It is clear from this figure
that there is no useful correlation between \(\tau _{f}\) and \(\Delta
_{CS}\). It appears that one can generate sequences with a range of
\(\Delta _{CS}\) all of which have roughly the same folding times.
\subsection{Kinetic Accessibility and Stability of Native Conformation}
It is well known that many natural proteins reach their native conformation
quite rapidly without forming any detectable intermediates. However,
proteins are only marginally
stable in the sense the equilibration constant \(K\) for the reaction
\begin{equation}
U \rightleftharpoons F
\end{equation}
is only between \(10^{4} - 10^{7}\). In Eq. (20) {\em U} refers to the
denaturated
unfolded conformations, and {\em F} is the folded native state. Thus,
\(\Delta G = G_{F} - G_{U} = -k_{B}T ln K\) is in the range of
\(-12k_{B}T\) to \(-18k_{B}T\). While this is not as large as one observes
in typical chemical reactions involving cleavage of bonds it is still
sufficiently large so that the
native conformation is overwhelmingly populated relative to the ensemble
of \underline{unfolded} states. The Helmholtz free energy of
stabilization of the
folded state with respect to the ensemble of denaturated conformations
can be written as
\begin{equation}
\beta \Delta F \simeq -\Delta E + S_{U}
\end{equation}
where \(\Delta E\) is the stabilization energy, \(S_{U}\) is the entropy
of the ensemble of the unfolded structures. We have assumed that the
conformational entropy associated with the native conformation is
negligible. If the
ensemble of denaturated structures corresponds to self-avoiding random walks
\(S_{U}\)
for lattice models can be estimated using Eq. (4). For cubic lattice
\(Z_{eff}=4.684\), \(\gamma = 1.16\), and thus \(\beta \Delta F \simeq
-\Delta E +42\) for \(N=27\). If we insist that \(\beta \Delta F \approx
10\) this would imply that \(\Delta E (T) \approx -52 k_{B}T\). The same
calculations would show that \(\Delta E (T) \approx -20 k_{B}T\) if the
ensemble of structures in the denaturated states are essentially compact
structures. This estimate is a bit more realistic. These estimates show that
the native conformation is
overwhelmingly populated under appropriate conditions relative to the
compact structures. More recent
experimental studies involving denaturant induced unfolding of few
proteins have been used to probe the "spectrum" of low free energy
conformations \cite{Bai}. These tools,
while being relatively primitive, suggest that typically the equilibrium
intermediates are also about \(6-8 k_{B}T\) higher in free energy than
the native conformation. Thus, even in these cases, under native
conditions, the native state is overwhelmingly (\(\geq 0.9\)) occupied.
Most of the simulations we have discussed so far have been done at
temperatures below \(T_{f}\) but high enough that the \(<\chi (T_{s})>\)
is as large as possible. This, of course, has been done for computational
reasons and the constant value of \(<\chi (T_{s})>\) has been chosen so
that the properties of different sequences can be compared on equal
footing. However, at these simulation temperatures \(T_{s}\) the
probability of occupation of the native conformation \(P_{nat}(T_{s})\)
varies between \(0.2\) and \(0.6\), which is significantly smaller than
what is observed in real proteins. Notice that the calculations of
Sali {\em et al.} \cite{Sali94a,Sali94b} have been done at such elevated
temperatures that
the probability of occupation of native conformation for two hundred
sequences examined is usually between
\(0.01 - 0.05\) and none exceeds \(0.4\). Thus, these authors,
although have stated the criterion for simultaneously satisfying kinetic
accessibility and stability (this follows from Boltzmann`s law), did not
provide \underline{any} computational or theoretical evidence that
\underline{any} of
their sequences obeyed the stated criterion at any temperature.
In light of the above arguments it is necessary to use a different
criterion for the choice of \(T_{s}\) which would ensure stability of the
native proteins. Accordingly, we have performed simulations for a few
sequences with \(N=15\) and \(B_{0}=-0.1\) at the temperatures which are
determined using the following equation
\begin{equation}
\eta (T_{s \eta}) = 1 - P_{nat}(T_{s \eta}) = c
\end{equation}
where \(P_{nat}(T_{s \eta}) \) is the probability of the chain being in
the native conformation at \(T=T_{s \eta}\). The constant \(c\) was
chosen to be equal to \(0.1\), which implies that probability of the
chain being in the native conformation is \(0.9\). In order to present
the contrast between the kinetic behavior of the sequences at
temperatures chosen using Eq. (22) \(T_{s \eta}\) we chose three sequences
one from fast folders (\(\sigma \lesssim 0.1\)), one from moderate folders
(\(0.1 \lesssim \sigma \lesssim 0.6\)), and one from slow folders (\(\sigma
\gtrsim 0.6\)). The
ratios of the simulation temperatures for these sequences \(T_{s \eta}\)
determined using Eq. (22) to \(T_{s}\) obtained using the overlap criterion
are approximately \(0.5\), \(0.6\), and \(0.6\) for fast, moderate, and
slow folders, respectively. It appears that stability of the
native conformation (occupancy of this state \(\geq 0.8\)) for both
\underline{fast}
and \underline{moderate} folders can be achieved, if \underline{the
simulation
temperature \(T_{s}\) is taken to be one half of the folding temperature
\(T_{f}\)}.
The quantity \(\eta (T)\) should only be the function of \(T/T_{s \eta}\)
if the gap between the native conformation and the first excited state
is large. More precisely, we expect this to be valid for all temperatures
such that \(k_{B}T \ll \Delta\), where \(\Delta\) is the gap, separating
the energy of the native conformation and the first excited state. To see
this \(\eta (T)\) can be written as \begin{equation}
\eta (T) \simeq \frac{\exp(-\frac{\Delta}{k_{B}T} )}{1 +
\exp(-\frac{\Delta} {k_{B}T})}
\end{equation}
for \(\Delta/k_{B}T \gtrsim 1\). The temperature \(T_{s \eta}\) is
determined from Eq. (22) and thus \(\eta (T)\) becomes
\begin{equation}
\eta (T) = f(T/T_{s \eta}) \simeq \frac{y^{1/\tau}}{1+y^{1/\tau}},
\end{equation}
where \(y=c/(1-c)\) and \(\tau = T/T_{s \eta }\). This is confirmed in Fig.
(23), where plots of
\(\eta \) as a function of \(T/T_{s \eta}\) are shown for three sequences.
Two of them follow the behavior in Eq. (24) for \(T/T_{s \eta} < 1.5\),
whereas the sequence with small gap (\(\Delta/k_{B}T_{s \eta} < 1\)) does not.
The behavior of \(<\chi (t)>\) and the fraction of unfolded molecules
\(P_{u}(t)\) as a
function of time for two of the sequences is shown in Figs. (24) and
(25).
The temperature for the third sequence was so low for Eq. (22) to be
satisfied that the folding time for this sequence was estimated to be in
excess of \(10^{10}\) MCS. By studying these graphs we draw the following
conclusions: (i) The overlap function \(\chi (t)\) at the low temperature
(Fig. (24a))
is clearly biexponential, whereas at \(T=T_{s}\) (see Eq. (11)) the folding
process was an all-or-none process. This is because the very small
barrier (\(\delta E^{\ddagger} \simeq k_{B}T_{s}\)) becomes discernible at
\(T
\simeq T_{s \eta}\). This fact alone would yield to the prediction that
\(\tau _{f}\) at \(T_{s \eta}\) should be about a factor of \(10\)
larger. This is consistent with simulation results. So the emergence of
the second component in \(P_{u}(t)\) (Fig. (24b)) is due to the activated
transition
from the first excited state to the native conformation. The behavior of
\(\chi (t)\) for the moderate folding sequence is qualitatively similar to
that at \(T=T_{s}\) except that the time constants are larger because
\(T_{s \eta} \approx \frac{1}{2}T_{s}\) (Fig. (25)). (ii) We find that the
ratio of
the folding times for the three sequences at \(T=T_{s \eta}\) is roughly
the same as found at \(T=T_{s}\). This suggest that although the overall
folding times have increased considerably we do not expect to see
qualitative differences in the fundamental conclusion regarding the
statistical correlation between \(\tau _{f}\) and \(\sigma \). (iii)
Examination of the fraction of unfolded molecules \(P_{u}(t)\) shows that
the biexponential behavior is consistent with the kinetic
partitioning mechanism \cite{Guo}.
A fraction of the molecules, \(\Phi (T)\), reaches the native
state very rapidly without forming any intermediates {\em via} TINC
mechanism, while the remainder follows a more complex kinetic
mechanism. The partition factor \(\Phi (T)\) is nearly unity for fast
folding sequences at \(T=T_{f}\) leading to a two state behavior, whereas
at low temperatures \(\Phi (T) < 1\). For fast folding sequences shown in
Fig. (24b) \(\Phi (T=T_{s \eta})\) is approximately \(0.4\). In the case of
moderate folders \(\Phi (T)\) is always less than one for all \(T <
T_{f}\). This is affirmed in Fig. (25b), where we find that \(\Phi (T=T_{s
\eta})\) is approximately \(0.19\) for the sequence with \(\sigma =
0.19\).
\section{Conclusions and Discussion}
In recent years there have been numerous studies of lattice models of
proteins, in which the protein is modeled as a self-avoiding walk on a
three dimensional cubic lattice. A variety of interactions between the
beads on this lattice has been studied. In this work we have carried out
a systematic investigation of the kinetics and thermodynamics of a
heteropolymer chain confined to a cubic lattice under a variety of
conditions. We have, as majority of the studies have in the past, used a
random bond model to specify the interaction between the beads.
Specifically the interactions between the beads are chosen from a Gaussian
distribution of energies with a non-zero value of the mean \(B_{0}\). In
order to obtain a coherent picture of folding in this highly simplified
representation of proteins we have studied the kinetic and thermodynamic
behavior for two values of \(N\) (the number of beads in the chain),
namely \(N = 15\) and \(N = 27\). For \(N = 15\) the thermodynamic
characteristics can be exactly calculated because all possible
conformations in this case can be exhaustively enumerated, whereas for \(N
= 27\) all the quantities of interest have to be obtained by Monte Carlo
simulations. In addition the mean hydrophobic interaction \(B_{0}\) has
also been varied in this work. The kinetics of folding for a number of
sequences have been obtained by Monte Carlo simulations. The work reported
here allows us to assess the various factors that govern folding in this
model. In addition the variation in parameters can be used to address some
of the proposals made earlier in the literature.
The exact calculations for \(N = 15\) of the thermodynamic quantities by
enumerating all the conformations clearly demonstrate the importance of
non-compact structures in determining accurately the characteristic
properties of the protein chain. For \(B_{0} = -0.1\), which is a
realistic value for proteins (see Sec. (II.B)), the values of the
temperatures
\(T_{\theta}\) (the collapse transition temperature) and \(T_{f}\) (the
folding transition temperature) are much higher, if only the compact
structure are included. This, in turn, makes the simulation temperature
\(T_{s}\) (see Eq. (11)) also quite high. In fact for most sequences
\(T_{s}\) determined using only the compact conformations exceeds
\(T_{\theta}\). Although the discrepancies between \(T_{s}\) determined
using CS and all the conformations for \(B_{0} = -2.0\) (a rather
unrealistic value for proteins) is lesser the resulting kinetics is
significantly affected. However, for \(N = 27\) the differences between
\(T_{s}\) found from CSE and that determined from Monte Carlo simulations
are very
significant even for \(B_{0} = -2.0\). The two values in some instances
differ by a factor of two. Thus if the simulation temperature is chosen
using only compact structures that is much more
convenient, then one obtains a value for \(T_{s}\) that often exceeds the
collapse transition temperature. This fact alone explains why the
probability of occupancy of the
native state was very small (in many cases less than \(0.1\)) in the
previously reported studies in the literature \cite{Sali94a,Sali94b}. It has
been pointed out by Chan \cite{Chan95} that the high temperatures used by
Sali {\em et al.} \cite{Sali94a,Sali94b} results in the
equilibrium population of the native conformation for 'folding sequences'
of only \(0.01 - 0.05\) with none exceeding \(0.4\). This was not
appreciated by Sali {\em et al.} because they used a very
crude criterion for folding. In particular a sequence was designated as a
folding sequence if at the temperature of simulation the native
conformation was reached once (a first passage time) at least four times
among ten independent trajectories of maximum duration of \(50\times
10^{6}\) MCS \cite{Sali94b}. Our studies
indicate that ten trajectories is absolutely
inadequate for obtaining even qualitatively reliable estimate of
\underline{any} property of interest.
We had shown earlier in lattice and off-lattice studies
\cite{Cam95,Cam93,Guo} that the two temperatures that are intrinsic
to a given sequence are \(T_{\theta}\) and \(T_{f}\). This general
thermodynamic behavior for protein-like heteropolymers has been confirmed
recently \cite{Socci95}.
In addition the
temperature of simulation (or experiment) should be below \(T_{f}\) so
that the folded state is the most stable. Theoretical arguments
suggest \cite{Thirum95} that the parameter \(\sigma = (T_{\theta} -
T_{f})/T_{\theta}\) is
a useful indicator of kinetic foldability in the models of the sort
considered here, and perhaps in real proteins as well. By examining the
kinetics of approach to the native conformation we have found that roughly
(this holds good for \(N = 15\)) the kinetic foldability of sequences can
be divided into three classes depending on the value of \(\sigma \). For
\(N = 15\) we
find that fast folders have \(\sigma \) less than about \(0.1\), moderate
folders have \(\sigma \) in the range of approximately \(0.1\) and
\(0.6\), while \(\sigma \) values greater than about \(0.6\) correspond to
slow folders. It should be emphasized that the ranges for these three
classes of sequences depend on the length of the chain: longer chains
have smaller range of \(\sigma \) for fast folding. Fast folding sequences
can kinetically access their ground
state at relatively higher temperature compared to slow folding sequences.
The kinetics of approach to the native state differs significantly between
fast folding sequences and those that are moderate folding sequences. In
the former case the native conformation is reached via a TINC
mechanism and there are no detectable intermediates. In
this case the folding appears as an all-or-none process. The
kinetics is essentially exponential. On the other hand the moderate
folding sequences reach the native conformation (predominantly) via a
three stage multipathway mechanism as was reported in our several earlier
studies.
The time dependence of the approach to the native conformation is
very revealing. For moderate folding sequences the simulation temperature
is lower than for sequences with small \(\sigma \). Thus if approach to
the native conformation with the same value of the overlap function is
examined we find that the moderate folding sequences follow the kinetic
partitioning mechanism (KPM) \cite{Guo,ThirumGuo}. This implies that a
fraction of the initial
population of molecules reaches the native state via the TINC
mechanism while the remaining one follows the three stage multipathway
process. The partition factor, \(\Phi (T)\), that determines the fraction
that follows the fast process is sequence and temperature dependent. Thus
even fast folding sequences, which would have \(\Phi (T)\) close to unity at
higher temperature, would have fractional values less than unity
at lower temperature.
This is clearly seen in Fig. (24b). This work suggests that KPM should be a
generic feature of foldable proteins. The partition factor can be altered
by mutations, temperature, and by changing other external factors such as
pH.
We have explicitly calculated the folding times for a number of sequences
for the various parameters (different \(N\) and \(B_{0}\)) values. This is
of great interest to examine whether there is any intrinsic property of
the sequences that can be used to predict if a particular sequence folds
rapidly or not. It has been argued based on the random energy model for
proteins that folding sequences should have as large a value of
\(T_{f}/T_{g}\) as possible, where \(T_{g}\) is an equilibrium glass
transition temperature \cite{Bryn95,Gold92}. Kinetic studies of lattice
models of proteins
suggest that this criterion may be satisfied for foldable sequences
\underline{provided}
\(T_{g}\) is substituted by a kinetic glass transition temperature which
is defined as the temperature at which folding time scale exceeds a
certain arbitrary value \cite{Sali94b}. Scaling
arguments have been used to suggest that
there should exist a correlation between folding times and \(\sigma \)
\cite{Cam93,Thirum95}.
More recently it has been emphatically stated in the form of a theorem
(without the benefit of explicit computations) that the necessary and
sufficient condition for rapid folding in the models studied here is that
there should be a large gap (presumably measured in units of \(k_{B}T\))
between the lowest energy levels. The calculations of folding times
reported here clearly show that there is no useful correlation between the
gaps and folding times for any parameter values that we have investigated.
In fact for a specified folding time one can engineer sequences with both
large and small values of the gap. Thus the precise value of the gap alone
cannot discriminate between folding sequences. On the other hand there is
a good correlation between the folding times for sequences and \(\sigma
\). It is clear that sequences with small values of \(\sigma \) have short
folding times, while those with larger values have higher folding times.
It should be emphasized here that the criterion based on \(\sigma \)
should only be used to predict trends in the folding times.
In this sense this criterion
should be used in a qualitative manner. The major advantage of showing the
correlation between \(\sigma \) and the folding times is that \(\sigma \)
can be experimentally determined. The folding transition temperature is
nominally associated with the midpoint of the denaturation curve while
\(T_{\theta}\) is the temperature at which the protein resembles a random
coil.
In the usual discussion of protein folding only the issue of kinetic
foldability of sequences are raised. Since natural proteins are
relatively stable (this does not imply that the protein does not undergo
fluctuations in the native conformation) with respect to both the
equilibrium intermediates and the ensemble of unfolded conformations it
is imperative to devise the criterion for simultaneously satisfying kinetic
accessibility of the native conformation and the associated stability.
This paper for the first time has provided an answer to this issue. It is
clear from our study that fast folders have small values of \(\sigma \),
and consequently designing proteins \cite{Cam93,Fer} using this criterion
assures kinetic
accessibility of the native conformation at relatively high temperature.
For example, if \(T_{\theta }\) is taken to be about \(60^{\circ }C\) then
the fast folders would reach the native conformation rapidly even at
temperatures as \(55^{\circ}C\) (\(\sigma \approx 0.1\)). However, at
these temperatures the native conformation may not be very stable, i.e.
the probability of being in the native conformation may be less than 0.5.
On the other hand, if these fast folders are maintained at \(T \approx
(0.5 - 0.6) T_{f} \approx (30-35)^{\circ} C \) for \(\sigma
\approx 0.1\) and \(T_{\theta } \approx 60^{\circ}C\) then both kinetic
accessibility as well as stability can be simultaneously satisfied.
Our simulations also suggest that the dual criterion can also be
satisfied by using moderate folders. In these cases we find that the
native state is reached relatively rapidly at low enough temperatures
(\(T \approx (25 - 30)^{\circ} C\) assuming \(T_{\theta} \approx
60^{\circ}C\)) at which the excursions to other conformations are not
very likely. In the extreme case of very slow folders we find (see Fig.
(26)) that the average fluctuation probability of leaving the native
conformation after initially reaching it is small, i.e. the
stability condition is easily satisfied. In these cases however the
kinetic accessibility is, in general, not satisfied. From these
observations it follows that in order to satisfy the dual criterion it is
desirable to engineer fast folders (small values of \(\sigma \)) and
perform folding at temperatures around \((0.5-0.6)T_{f}\).
Alternatively, one can use moderate folders at temperatures around
\(0.8T_{f}\). Since moderate folders are more easily generated (by a
random process) it is tempting to suggest that many natural proteins
specifically large single domain proteins may be moderate folders.
Finally, we address the applicability of the results obtained here to real
proteins. Since many features that are known to be important in real
proteins (such as side chains, the possibility of forming secondary
structures etc.) are not contained in these models the direct applicability
to real proteins is not clear. Nevertheless simulations based on other more
realistic minimal models together with theoretical arguments can be used to
suggest that the scenarios that have emerged from this and other related
studies should be observed experimentally. In particular it appears that the
kinetic partitioning mechanism which for most generic sequences is a
convolution of the topology inducing nucleation collapse mechanism
and the three
stage multipathway kinetics should be a very general feature of protein
folding in vitro. The theoretical ideas developed based on these minimal
models also suggest that the partition factor \(\Phi (T)\) is a property of
the intrinsic
sequence as well as external factors such as temperature, pH etc. In fact
recent experiments on chymotrypsin inhibitor 2 (CI2) suggest that this
KPM has indeed been observed
\cite{Otzen}. In this
particular case Otzen {\em et al.} have observed that CI2 reaches the
native
state immediately following collapse. This, in the picture suggested here
and elsewhere \cite{Guo},
would imply that in the case of CI2 under the
conditions of
their experiment (\(T = 25^{\circ}C\) and \(\text{pH} = 6.2\)) the native
conformation is
accessed via a TINC mechanism. Moreover, the
partition factor
\(\Phi (T = 25^{\circ}C)\) is close to unity making CI2 under these
conditions a
fast folder. If we assume that \(T=25^{\circ}C \approx T_{f}/2\) then it
follows that for CI2 the value of \(\sigma \) is roughly 0.15. On the other
hand these authors have also noted that for barnase the rate limiting step
comes closer to the native state involving the rearrangement of the
hydrophobic core. They suggest in the form of a figure (see Fig. (3) of
Ref. 52) that barnase follows a three stage kinetics
with the rate determining
step being the final stage. In terms of the physical picture suggested here
this can be interpreted to mean that for barnase \(\Phi (T)\) is small making
it either a moderate folder or even a slow folder. If this were the case our
theoretical picture would suggest \(\sigma \) for barnase is bigger which
is a
consequence of the fact that it is a larger protein. These observations are
consistent with the experimental conclusions of Otzen {\em et al.}
which are
perhaps the first experiments that seem to provide some confirmation of the
theoretical ideas that have emerged from the minimal model studies. It is
clear that a more detailed comparison of the entire kinetics for various
proteins under differing experimental conditions is required to fully
validate the general conclusions based on the kinetic partitioning mechanism
together with the classification of sequences based on the values of the
parameter \(\sigma \).
\acknowledgments
This work was supported by a grant from the Airforce Office of Scientific
Research (through grant number F496209410106) and the National Science
Foundation. One of us (DT) is grateful to
P.G. Wolynes and J.N. Onuchic for a number of interesting discussions.
|
proofpile-arXiv_065-628
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\section{Introduction}
\setcounter{equation}0
Early interest in lower-dimensional black hole physics \cite{early}
has grown into a rich and fruitful field of research. The main motivation
for this is that the salient problems of quantum black holes, such as loss
of information and the endpoint of quantum evaporation, can be more
easily understood in some simple low-dimensional models
than directly in four dimensions \cite{1}. Several interesting
2D candidates have been explored to this end
which share many common features with their four-dimensional
cousins \cite{2}. This is intriguing since the
one-loop quantum effective action in two dimensions is exactly
known, in the form of the Polyakov-Liouville term, giving rise to
the hope that the semiclassical treatment
of quantum black holes in two dimensions can be done explicitly
(see reviews \cite{1}).
The black hole in three-dimensional gravity discovered by
Ba$\tilde{n}$ados,
Teitelboim and Zanelli (BTZ) \cite{3} has features that are
even more realistic than its two-dimensional counterparts.
It is similar to the Kerr black hole, being
characterized by mass $M$ and angular momentum $J$ and having an event
horizon and
(for $J\neq 0$) an inner horizon \cite{4,rev1,rev2,rev3}.
This solution naturally appears
as the final stage of collapsing matter \cite{5}.
In contrast to the Kerr solution it is asymptotically Anti-de Sitter
rather than asymptotically flat. Geometrically, the BTZ
black hole is obtained from 3D Anti-de Sitter (AdS$_3$)
spacetime by performing some identifications. Although
quantum field theory on curved three-dimensional manifolds
is not as well understood as in two dimensions, the large symmetry
of the BTZ geometry and its relation to AdS$_3$ allow one to
obtain some precise results when field is quantized
on this background. The Green's function
and quantum stress tensor for the conformally coupled scalar field
and the resultant back reaction were
calculated in \cite{6,7,8}.
The possibility that black hole entropy might have
a statistical explanation remains an intriguing issue, and there has
been much recent activity towards its resolution via a variety
of approaches (for a review see \cite{9}). One such proposal is
that the Bekenstein-Hawking entropy
is completely generated by quantum fields propagating
in the black hole background. Originally it was belived that
UV-divergent quantum corrections associated with such fields
to the Bekenstein-Hawking expression play a fundamental role in the statistical
interpretation of the entropy. However, it was subsequently
realized that these divergent corrections can be associated with
those that arise from the standard UV-renormalization
of the gravitational couplings in the effective action \cite{10}, \cite{J},
\cite{SS}, \cite{FS}, \cite{SS2}, \cite{MS}.
The idea of complete generation of the
entropy by quantum matter in the spirit of induced gravity \cite{J}
encountered the problem of an appropriate statistical treatment
of the entropy of non-minimally-coupled matter \cite{Kab}, \cite{FFZ}
(see, however, another realization of this idea within the superstring
paradigm \cite{D}). At the same time, it was argued in number of
papers \cite{SS1,FWS,F1,Zas} that UV-finite quantum corrections to the
Bekenstein-Hawking
entropy might be even more important than the UV-infinite ones. They
could provide essential modifications of the thermodynamics of a hole
at late stages of the evaporation when quantum effects come to play.
Relatively little work has been done concerning
the quantum aspects of the entropy for the BTZ black hole.
Carlip \cite{Carlip} has shown that the
appropriate quantization of 3D gravity represented in the Chern-Simons
form yields a set of boundary states at the horizon. These can be counted
using methods of Wess-Zumino-Witten theory. Remarkably,
the logarithm of their number gives the classical Bekenstein-Hawking
formula. This is the unique case of a statistical explanation of
black hole entropy. Unfortunately, it is essentially based on features
peculiar to three-dimensional gravity and its extension to four dimensions
is not straightforward.
An investigation of the thermodynamics of quantum scalar fields on
the BTZ background \cite{Ichinose} concluded
that the divergent terms in
the entropy are not always due to the existence of the outer horizon
(i.e. the leading term in the quantum entropy is not proportional to the
area of the
outer horizon) and depend upon the regularization method. This
conclusion seems to be in disagreement with the
expectations based on the study of the problem in two and
four dimensions.
In this paper we systematically calculate
the heat kernel, effective action and quantum entropy of scalar matter
for the BTZ black hole. The relevant operator is
$(\Box+\xi /l^2)$, where
$\xi$ is an arbitrary constant
and $1/l^2$ is the cosmological constant appearing in the BTZ solution.
Since we are interested in the
thermodynamic aspects we consider the Euclidean BTZ
geometry with a conical singularity at the horizon as the background.
In the process of getting the heat kernel and effective action on this
singular geometry we proceed in steps, first calculating quantities
explicitly for AdS$_3$, then
the regular BTZ instanton and finally the conical BTZ
instanton. The entropy is calculated by differentiating the effective
action with respect to the angular deficit at the horizon.
It contains both UV-divergent and UV-finite terms. The analysis of the
divergences shows that they are
explicitly renormalized by renormalization of Newton's constant in
accordance with general arguments \cite{FS}.
We find the structure of the UV-finite terms in the entropy to be
particularly interesting. These terms, negligible
for large outer horizon area $A_+$, behave logarithmically at small
$A_+$. Hence they should become important at late stages
of black hole evaporation.
The paper is organized as follows. In Section 2 we briefly review
the Euclidean BTZ geometry, omitting details that
appear in earlier work \cite{3,4,rev1,rev2,rev3}. We discuss
in section 3 various forms of the metric for 3D Anti-de Sitter space
giving expressions for the geodesic distance
on AdS$_3$ that are relevant for our purposes.
We solve explicitly the heat kernel equation and find the Green's function
on AdS$_3$ as a function of the geodesic distance.
In Section 4 we calculate explicitly the trace of
heat kernel and the effective action on the regular and singular
Euclidean BTZ instantons. The quantum entropy is the subject of Section 5
and in Section 6 we provide some concluding remarks.
\bigskip
\section{Sketch of BTZ black hole geometry}
\setcounter{equation}0
We start with the black hole metric written in a form
that makes it similar to the four-dimensional Kerr metric.
Since we are interested in its thermodynamic behaviour,
we write the metric in the Euclidean form:
\begin{equation}
ds^2=f(r)d\tau^2+f^{-1}(r) dr^2+r^2(d\phi+N(r)d\tau)^2~~,
\label{1}
\end{equation}
where the metric function $f(r)$ reads
\begin{equation}
f(r)={r^2\over l^2}-{j^2\over r^2}-m={(r^2-r_+^2)(r^2+|r_-|^2)\over l^2 r^2}
\label{2}
\end{equation}
and we use the notation
\begin{equation}
r^2_+={ml^2\over 2}(1+\sqrt{1+({2j\over m l})^2})~,
~~|r_-|^2={ml^2\over 2}(\sqrt{1+({2j\over m l})^2}-1)~~
\label{3}
\end{equation}
where we note the useful identity
$$
r_+|r_-|=jl~~
$$
for future reference.
The function $N(r)$ in (\ref{1}) is
\begin{equation}
N(r)=-{j\over r^2}~~.
\label{5}
\end{equation}
In order to transform the metric (\ref{1}) to Lorentzian singnature we
need to make the transformation:
$\tau\rightarrow \imath ~t$, $j\rightarrow -\imath ~j$.
Then we have that
\begin{eqnarray}
&&r_+\rightarrow r^L_+=\sqrt{ml^2\over 2}~\left(1+\sqrt{1-({2j\over m l})^2}~\right)^{1/2}~~, \nonumber \\
&&|r_-|\rightarrow \imath ~r_-^L=\sqrt{ml^2\over 2}~\left(1-\sqrt{1-({2j\over m l})^2}~\right)^{1/2}~~,
\label{?}
\end{eqnarray}
where $r^L_+$ and $r^L_-$ are the values in the Lorentzian space-time. These
are the respective radii of the outer and inner horizons of the
Lorentzian black hole in $(2+1)$ dimensions. Therefore
we must always apply the transformation (\ref{?}) after carrying out
all calculations in the Euclidean geometry in order to
obtain the result for the Lorentzian black hole.
The Lorentzian version of the metric (\ref{1}) describes a black hole with mass $m$ and angular
momentum $J=2j$ \cite{3}, \cite{4}.
Introducing
\begin{equation}
\beta_H\equiv {2\over f'(r_+)}={r_+ l^2 \over r^2_++|r_-|^2}
\label{6}
\end{equation}
we find that in the $(\tau , r)$ sector of the metric (\ref{1}) there is
no conical singularity at the horizon if the Euclidean time $\tau$
is periodic with period $2\pi\beta_H$.
The quantity $T_H=(2\pi\beta_H)^{-1}$ is the
Hawking temperature of the hole.
The horizon $\Sigma$ is a one-dimensional space with metric
$
ds^2_{\Sigma}=l^2d\psi^2~~,
$
where $\psi={r_+\over l} \phi-{|r_-|\over l^2}\tau$ is a natural coordinate on the horizon.
Looking at the metric (\ref{1}) one can conclude that there is no
constraint on the periodicity of the ``angle'' variable $\phi$ (or $\psi$).
This is in contrast to the four-dimensional black hole, for which
the angle $\phi$ in the spherical line
element $(d\theta^2+\sin^2\theta d\phi^2)$ varies between the limits
$0\leq \phi\leq 2\pi$ in order to avoid the appearance of the
conical singularities at the poles of the sphere.
However, following tradition we will assume that the metric
(\ref{1}) is periodic in $\phi$, with limits
$0\leq \phi\leq 2\pi$. This means that $\Sigma$ is a circle with
length (``area'') $A_+=2\pi r_+$.
There are a number of other useful forms for the metric
(\ref{1}). It is very important for our considerations that (\ref{1})
is obtained from the metric of three-dimensional Anti-de Sitter
space by making certain identifications along the trajectories of
its Killing vectors. In order to find the appropriate
metric for the 3D Anti-de Sitter space (denoted below by
$H_3$)
we consider a four-dimensional flat space with
metric
\begin{equation}
ds^2=dX^2_1-dT^2_1+dX^2_2+dT^2_2~~.
\label{7}
\end{equation}
AdS$_3$ ($H_3$) is defined as a subspace defined by the equation
\begin{equation}
X^2_1-T^2_1+X^2_2+T^2_2=-l^2~~.
\label{8}
\end{equation}
Introducing the coordinates $(\psi , \theta , \chi )$
\begin{eqnarray}
&&X_1={l\over \cos \chi}\sinh \psi~,~~ T_1={l\over \cos \chi}\cosh \psi \nonumber \\
&&X_2=l~ \tan \chi \cos \theta ~,~~T_2=l~ \tan \chi \sin \theta
\label{9}
\end{eqnarray}
the metric on $H_3$ reads
\begin{equation}
ds^2_{H_3}={l^2\over \cos^2 \chi }(d\psi^2+d\chi^2+\sin^2 \chi d\theta^2 )~~.
\label{10}
\end{equation}
It is easy to see that under the
coordinate transformation
\begin{eqnarray}
&&\psi={r_+\over l} \phi-{|r_-|\over l^2}\tau~, ~~\theta={r_+\over l} \tau
+{|r_-|\over l^2}\phi \nonumber \\
&&\cos \chi =({r^2_++|r_-|^2 \over r^2+|r_-|^2})^{1/2}~~
\label{11}
\end{eqnarray}
the metric (\ref{1}) coincides with (\ref{10}).
In the next section we will derive a few other forms of the metric
on $H_3$ which are useful in the context of calculation of the heat
kernel and Green's function on $H_3$.
The BTZ black hole ($B_3$) described by the metric (\ref{1})
is obtained from AdS$_3$
with metric (\ref{10}) by making the
following identifications:
\noindent
$i).$ $(\psi , \theta , \chi ) \rightarrow (\psi , \theta+2\pi , \chi )$.
This means that $(\phi , \tau , r) \rightarrow (\phi+\Phi , \tau+T^{-1}_H ,
r)$, where $\Phi=T^{-1}_Hj r^{-2}_+$.
\noindent
$ii).$ $(\psi , \theta , \chi ) \rightarrow (\psi+2\pi {r_+\over l} , \theta+2\pi
{|r_-|\over l} , \chi )$, which is the analog
of $(\phi , \tau , r) \rightarrow (\phi+2\pi , \tau , r)$.
The coordinate $\chi$ is the analog of the radial coordinate $r$.
It has the range $0\leq \chi\leq {\pi\over 2}$.
The point $\chi=0$ is the horizon ($r=r_+$) while $\chi={\pi\over 2}$
lies at infinity. Geometrically, $i)$ means that there is no conical
singularity at the horizon, which is easily seen from (\ref{10}).
A section of BTZ black hole at fixed $\chi$ is illustrated in Fig.1
for the non-rotating ($|r_-|=0$) and rotating cases.
The opposite sides of the quadrangle in Fig.1 are identified.
Therefore, the whole section looks like a torus. In the rotating
case the torus is deformed with deformation parameter $\gamma$, where
$\tan \gamma={r_+\over |r_-|}$.
The whole space $B_3$ is a region between two semispheres with
$R=\exp( \psi )$ being radius, $\chi$ playing the role of azimuthal angle
and $\theta$ being the orbital angle. The boundaries of the region are
identified according to $ii)$.
\bigskip
\section{3D Anti-de Sitter space: geometry, heat kernel and Green's function}
\setcounter{equation}0
{\bf 3.1 Metric on $H_3$}
3D Anti-de Sitter space ($H_3$) is defined as a 3-dimensional subspace of
the flat four-dimensional space-time with metric
\begin{equation}
ds^2=dX_1^2-dT^2_1+dX^2_2+dT^2_2
\label{2.1}
\end{equation}
satisfying the constraint
\begin{equation}
X_1^2-T^2_1+X^2_2+T^2_2=-l^2~~.
\label{2.2}
\end{equation}
We are interested in AdS$_3$, which has Euclidean signature. This is
easily done by appropriately choosing the signature in (\ref{2.1}),
(\ref{2.2}). The induced metric has a number of different
representations depending on the choice
of the coordinates on AdS$_3$. Below we consider two such choices.
{\bf A.} Resolve equation (\ref{2.2}) as follows:
\begin{eqnarray}
&&X_1=l \cosh \rho \sinh \psi~,~~T _1=l \cosh \rho \cosh \psi \nonumber \\
&&X_2=l \sinh \rho \cos \theta~,~~T _1=l \sinh \rho \sin \theta~~.
\label{2.3}
\end{eqnarray}
The variables $(\rho , \psi , \theta )$ can be considered as coordinates
on $H_3$. They are closely related to the system $(\chi , \psi , \theta )$
via the transformation $\cos \chi=\cosh^{-1}\rho$.
Note that the section of $H_3$ corresponding to a fixed $\rho$ is a
two-dimensional torus. The induced metric then
takes the following form:
\begin{equation}
ds^2_{H_3}=l^2 \left( d\rho^2+\cosh^2\rho d\psi^2+\sinh^2\rho d\theta^2 \right)
~~.
\label{2.4}
\end{equation}
The BTZ black hole metric is then obtained from (\ref{2.4}) by making the
identifications
$\theta\rightarrow \theta +2\pi$ and $\psi\rightarrow \psi+2\pi{r_+\over l}$,
$\theta\rightarrow\theta+2\pi{|r_-|\over l}$.
{\bf B.} Another way to resolve the constraint (\ref{2.2}) is
by employing the transformation
\begin{eqnarray}
&&X_1=l\sinh (\sigma /l) \cos \lambda ~,~~T_1=l \cosh (\sigma /l) \nonumber \\
&&X_2=l\sinh (\sigma /l) \sin\lambda\sin\phi~,~~T_2=l \sinh (\sigma /l) \sin\lambda\cos\phi~~.
\label{2.5}
\end{eqnarray}
The section $\sigma=const$ of $H_3$ is a two-dimensional sphere. The
induced metric in the coordinates $(\sigma , \lambda , \phi )$
takes the form
\begin{equation}
ds^2_{H^3}= d\sigma^2+l^2\sinh^2(\sigma /l) (d\lambda^2+\sin^2\lambda d\phi^2)
\label{2.6}
\end{equation}
from which one can easily see that $H_3$ is a
hyperbolic version of the metric on the 3-sphere
\begin{equation}
ds^2_{S^3}= d\sigma^2+l^2\sin^2(\sigma /l) (d\lambda^2+\sin^2\lambda d\phi^2)~~,
\label{2.7}
\end{equation}
allowing us to making use of our experience with the 3-sphere
in understanding the geometry of $H_3$.
\bigskip
{\bf 3.2 Geodesic distance on $H_3$ }
An important fact equally applicable both to $S_3$ and $H_3$ is the
following. Consider two different points on $S_3$ ($H_3$).
Then we can choose the coordinate system
$(\sigma , \lambda , \phi )$ such that one of the points lies at the origin
($\sigma=0$) and the other point lies on the radius
$(\sigma , \lambda=0 , \phi )$.
This radial trajectory joining the two points is a geodesic.
Moreover, the geodesic distance between these two points coincides with
$\sigma$. More generally, for the
metric (\ref{2.6}), (\ref{2.7}) the geodesic distance between two
points with equal values of $\lambda$ and $\phi$
($\lambda =\lambda '~,~~\phi=\phi '$)
is given by $|\sigma-\sigma '|=\Delta\sigma$.
In order to find the geodesic distance in the coordinate system
$(\rho , \psi , \theta)$ (\ref{2.3}) consider the following trick.
The two points $M$ and $M'$ in the
embedding four-dimensional space determine the vectors
${\bf a}$ and ${\bf a'}$ starting from the origin:
\begin{eqnarray}
&&{\bf a}=l \cosh \rho \sinh \psi ~{\bf x_1}+l\cosh\rho \cosh \psi ~{\bf t_1}
+l\sinh\rho \cos\theta ~{\bf x_2} +l\sinh\rho \sin \theta ~ {\bf t_2}
\nonumber \\
&&{\bf a'}=l \cosh \rho ' \sinh \psi ' ~{\bf x_1}+l\cosh\rho ' \cosh \psi ' ~{\bf t_1}
+l\sinh\rho ' \cos\theta ' ~ {\bf x_2} +l\sinh\rho ' \sin \theta ' ~ {\bf t_2}~~,
\nonumber \\
&&
\label{2.8}
\end{eqnarray}
where $({\bf t_1}, {\bf x_1}, {\bf t_2}, {\bf x_2 })$ is an
orthonormal basis of vectors in
the space (\ref{2.1}):
\begin{equation}
-({\bf t_1 } , {\bf t_1 })=({\bf x_1} , {\bf x_1 })
=({\bf t_2 } , {\bf t_2 })=
({\bf x_2 } , {\bf x_2 })=1~~.
\label{2.9}
\end{equation}
For the scalar product of $\bf a$ and $\bf a '$ we have
\begin{equation}
({\bf a }, {\bf a'})=l^2 \left(-\cosh^2
\rho \cosh \Delta \psi+\sinh^2\rho \cos\Delta \theta \right) ~~,
\label{2.10}
\end{equation}
where $\Delta\psi=\psi-\psi '~,~~\Delta\theta=\theta-\theta '$
and for simplicity we assumed that $\rho=\rho '$.
The scalar product $({\bf a }, {\bf a'})$ is invariant quantity not dependent on a
concrete choice of coordinates. Therefore, we can calculate it
in the coordinate system ($\sigma , \lambda , \phi$).
In this system we have
\begin{eqnarray}
&&{\bf a}=l\cosh (\sigma /l) ~{\bf t_1'}+l\cosh (\sigma /l) ~{\bf x_1'}\nonumber \\
&&{\bf a'}=l\cosh (\sigma '/l) '~{\bf t_1'}+
l\cosh (\sigma ' /l) ~{\bf x_1'}~~.
\label{2.11}
\end{eqnarray}
The new basis $({\bf t_1'}, {\bf x_1'}, {\bf t_2'}, {\bf x_2' })$
is obtained from the old basis $({\bf t_1}, {\bf x_1}, {\bf t_2}, {\bf x_2 })$
by some orhogonal rotation. Therefore, it satisfies the same
identities (\ref{2.9}). In new basis we have
\begin{equation}
({\bf a }, {\bf a'})=-l^2\cosh {\Delta \sigma \over l}~~.
\label{2.12}
\end{equation}
As we explained above $\Delta\sigma$ is the geodesic distance between $M$
and $M'$. Equating (\ref{2.10}) and (\ref{2.12}) we finally obtain
the expression for the geodesic distance in
terms of the coordinates $(\rho , \psi , \theta )$:
\begin{equation}
\cosh {\Delta \sigma \over l}=\cosh^2 \rho \cosh \Delta \psi-\sinh^2\rho \cos\Delta
\theta
\label{2.13}
\end{equation}
or alternatively, after some short manipulations
\begin{equation}
\sinh^2{\Delta\sigma \over 2l}=\cosh^2\rho\sinh^2{\Delta\psi\over 2}+\sinh^2\rho \sin^2{\Delta\theta \over 2}~~.
\label{2.14}
\end{equation}
For small $\rho <<1$ and $\Delta \psi <<1$ from (\ref{2.14}) we get
\begin{equation}
\Delta\sigma ^2=l^2 \left( \Delta \psi^2+4\rho^2\sin^2{\Delta \theta \over 2}
\right)
\label{2.15}
\end{equation}
what coincides with the result for 3D flat space in cylindrical coordinates.
Note, that $\Delta \sigma$ in (\ref{2.13}), (\ref{2.14}) is the intrinsic
geodesic distance on $H_3$. It is worth comparing with the
chordal four-dimensional distance $\Sigma$ between the points $M$ and $M'$
measured in the imbedding 4D space. In the coordinate system
(\ref{2.3}) we obtain
\begin{eqnarray}
&&\Sigma^2\equiv \sum (X-X')^2=l^2 ( \cosh^2\rho (\sinh \psi -\sinh \psi ')^2
-\cosh^2\rho (\cosh \psi-\cosh \psi ')^2\nonumber \\
&&+\sinh^2\rho (\cos\theta-\cos\theta ')^2
+\sinh^2\rho (\sin \theta -\sin \theta ')^2 )~~.
\label{2.16}
\end{eqnarray}
After simplification we obtain
\begin{equation}
\Sigma^2=4l^2\sinh^2{\Delta \sigma \over 2l}~~.
\label{2.17}
\end{equation}
Consider now the point $M''$ which is antipodal to the point $M'$. It is
obtained from $M'$ by antipodal transformation $X'\rightarrow -X'$
(in the coordinates $(\chi , \theta , \psi )$ the antipode has
coordinates $(\pi-\chi , \theta , \psi )$).
The point $M''$ lies in the lower ``semisphere'' of the space $H_3$.
For some applications we will need the chordal distance
$\hat{\Sigma}$ bewteen points $M$ and $M''$:
$\hat{\Sigma}^2=\sum (X+X')^2$, where we find
\begin{equation}
\hat{\Sigma}^2=-4l^2\cosh^2{\Delta\sigma \over 2l}~~.
\label{2.18}
\end{equation}
Here $\Delta\sigma$ is the geodesic distance between $M$ and $M'$.
\bigskip
{\bf 3.3 Heat kernel and Green's function}
Consider on $H_3$ the heat kernel equation
\begin{eqnarray}
&&(\partial_s-\Box-\xi /l^2) K(x,x',s)=0 \nonumber \\
&&K(x,x',s=0)=\delta (x,x')~~,
\label{2.19}
\end{eqnarray}
where $s$ is a proper time variable.
The operator $(\Box +\xi /l^2)$ on $H_3$ or $B_3$ can be equivalently
represented in the form of non-minimal coupling $(\Box-{\xi\over 6}R)$.
For $\xi={3\over 4}$ this operator would be conformal invariant.
This equivalence, however, is no longer valid for the space $B_3^\alpha$
which has a conical singularity. This is because the scalar curvature on
a conical space has a $\delta$-function-like contribution
due to a singularity that is additional to the regular value of the
curvature.
The $\delta$-function in the operator $(\Box -{\xi \over 6}R)$
has been shown \cite{SS2} to non-trivially modify the regular heat kernel.
In order to avoid the problem of dealing with this perculiarity
we will not make use of this form of the operator and will treat the
term $\xi /l^2$ as just a constant that is unrelated to
the curvature of space-time.
The function
$K(x,x',s)$ satisfying (\ref{2.19}) can be found as some function of the
geodesic distance $\sigma$ between the points $x$ and $x'$. The simplest
way to do this is to use the coordinate system
$(\sigma , \lambda ,\phi )$ with the metric (\ref{2.6}) when
both points lie on the radius:
$\lambda=\lambda ', \phi=\phi '$. Then the Laplace
operator $\Box=\nabla^{\mu} \nabla_{\mu}$ has only the ``radial'' part:
\begin{equation}
\Box={1\over l^2\sinh^2 {\sigma\over l}} \partial_{\sigma}\sinh^2 ({\sigma\over l})
\partial_{\sigma}= {1\over l^2\sinh {\sigma\over l}} \partial^2_{\sigma}\sinh
({\sigma\over l}) -l^{-2}~~.
\label{2.20}
\end{equation}
Equation (\ref{2.19}) is then easily solved and the solution takes the
form
\begin{equation}
K_{H_3}(\sigma , s)={1\over (4\pi s)^{3/2}}{\sigma /l \over \sinh (\sigma / l)}
e^{-{\sigma^2\over 4s}-\mu{s\over l^2}}~~,
\label{2.21}
\end{equation}
where $\mu=1-\xi$.
In the conformal case we have $\xi=3/4$
and $\mu=1/4$. The heat kernel (\ref{2.21}) was first found by
Dowker and Critchley \cite{Dow-Crit} for $S_3$
(for which $\sinh (\sigma /l)$ is replaced by $\sin (\sigma /l)$)
and then was extended to the hyperbolic space $H_3$ by Camporesi
\cite{Camporesi}.
Knowing the heat kernel function $K(\sigma , s)$ we can find the
Green's function
$G(x, x')$ as follows
$$
G(x,x')=\int_{0}^{+\infty}ds~ K(x,x',s)~~.
$$
Applying this to the heat kernel (\ref{2.21}) and using the integral
\begin{equation}
\int_0^{\infty}{ds\over s^{3/2}}e^{-bs-{a^2\over 4s}}={2\sqrt{\pi}\over a}
\left(\cosh (
\sqrt{b}a)-\sinh ( \sqrt{b}a)\right)
\label{*}
\end{equation}
the Green's function on $H_3$ reads
\begin{equation}
G_{H_3}(x,x')={1\over 4\pi}{1\over l \sinh ({\sigma \over l})}\left(\cosh (\sqrt{\mu}{\sigma
\over l})-\sinh (\sqrt{\mu}{\sigma
\over l}) \right)~~,
\label{2.22}
\end{equation}
where $\sigma$ is the instrinsic geodesic distance on $H_3$ between $x$ and
$x'$. It is important to observe that the function $G_{H_3}(x,x')$ vanishes
when $\cosh (\sqrt{\mu}{\sigma
\over l})=\sinh (\sqrt{\mu}{\sigma
\over l})$. This happens when $\sigma (x, x')=\infty$, i.e. one of the
points lies on the equator $(\chi={\pi \over 2})$.
This fact is important in view of the arguments of \cite{IS} that the
correct quantization on a non-globally hyperbolic space, like AdS$_3$,
requires the
fixing of some boundary condition for a quantum field at infinity.
The Green's function (\ref{2.22}) constructed by means of the heat
kernel (\ref{2.21}) automatically satisfies the Dirichlet boundary
condition and thus provides for us the correct quantization on $H_3$.
To our knowledge, the form (\ref{2.22})
of the Green's function on $H_3$ is not known in the current literature.
A special case occurs when $\xi=3/4$ and $\mu=1/4$, for which
the operator
$(\Box+\xi /l^2 )\equiv (\Box-{1\over 8}R)$ is conformally
invariant. In this case we get
\begin{equation}
G_{H_3}={1\over 4\pi} \left( {1\over 2l\sinh {\sigma \over 2l}}-
{1\over 2l\cosh {\sigma \over 2l}}\right)
\label{2.23}
\end{equation}
for the Green's function.
Using (\ref{2.16}) and (\ref{2.17}) we observe that (\ref{2.23}) has a
nice form in terms of the chordal distance in the imbedding space:
\begin{equation}
G_{H_3}(x,x')={1\over 4\pi} \left({1\over |\Sigma|}-{1\over |\hat{\Sigma}|}
\right)={1\over 4\pi} \left({1\over |X-X'|}-{1\over |X+X'|}
\right)~~.
\label{2.24}
\end{equation}
The Green's function for the conformal case in the form (\ref{2.24})
was reported by Steif \cite{6}.
\bigskip
\section{Heat kernel on the Euclidean BTZ instanton}
\setcounter{equation}0
\medskip
{\bf 4.1 Regular BTZ instanton}
As was explained in Section 2 the regular Euclidean
BTZ instanton ($B_3$) may be obtained from $H_3$
by a combination of identifications which in the coordinates
$(\rho , \theta , \psi )$ are
$i).~~\theta\rightarrow \theta+2\pi
$
$
ii). ~~\theta\rightarrow \theta+2\pi {|r_-|\over l}~,~~\psi\rightarrow\psi+
2\pi{r_+\over l}
$
Therefore, the heat kernel $K_{B_3}$
on the BTZ instanton $B_3$ is constructed via the heat kernel $K_{H_3}$ on $H_3$ as infinite sum over images
\begin{equation}
K_{B_3}(x,x',s)=\sum_{n=-\infty}^{+\infty} K_{H_3}(\rho ,~ \rho ',~
\psi-\psi '+2\pi{r_+\over l}n,~\theta-\theta '+2\pi{|r_-|\over l}n)~~.
\label{3.1}
\end{equation}
Using the path integral representation of heat kernel we would say that
the $n=0$ term in (\ref{3.1}) is due to the direct way of connecting
points $x$ and $x'$ in the path integral. On the other hand,
the $n\neq 0$ terms are due to uncontractible winding paths that
go $n$ times around the circle. Note that $K_{H_3}$ automatically has
the periodicity given in $i)$.
Therefore the sum over images in (\ref{3.1}) provides us with the
periodicity $ii)$.
Assuming that $\rho=\rho '$ it can be represented in the form
\begin{eqnarray}
&&K_{B_3}=\sum_{n=-\infty}^{\infty}K_{H_3}(\sigma_n,s)~,\nonumber \\
&&\cosh {\sigma_n\over l}=\cosh^2\rho\cosh\Delta\psi_n-\sinh^2\rho
\cos \Delta \theta_n~, \nonumber \\
&&\Delta\psi_n=\psi-\psi '+2\pi{r_+\over l}n~,~~\Delta\theta_n=\theta-\theta '+2\pi{|r_-|\over l}n~~,
\label{3.2}
\end{eqnarray}
where $K_{H_3}(\sigma , s)$ takes the form (\ref{2.20}).
For the further applications consider the integral
\begin{equation}
Tr_{w}K_{B_3}\equiv \int_{B_3} K_{B_3}(\rho=\rho ',
\psi=\psi ', \theta=\theta '+w) ~d\mu_x~~,
\label{3.3}
\end{equation}
where $d\mu_x=l^3\cosh\rho\sinh\rho d\rho
\theta d \psi$ is the measure on $B_3$.
Note that volume of $B_3$
$$V_{B_3}=\int_{B_3}d\mu_x=l^3\int_0^{2\pi}d\theta
\int_0^{2\pi r_+\over l}d\psi\int_0^{+\infty}\cosh\rho\sinh\rho d \rho
$$
is infinite and so does not depend on $|r_-|$.
This is just a simple consequence of the geometrical
fact that the two quadrangles in Fig.1 have the same area.
The integration in (\ref{3.3}) can be easily performed if
for a fixed $n$ we change the variable
$\rho\rightarrow \bar{\sigma}_n=\sigma_n/l$ (see Eqs.(\ref{3.2}),
(\ref{2.13})) with the corresponding change of integration measure
$$
\cosh \rho \sinh \rho d\rho={1\over 2}{\sinh\bar{\sigma}_n d\bar{\sigma}_n
\over (\cosh\Delta\psi_n-\cos\Delta\theta_n )}=
{1\over 4}{\sinh\bar{\sigma}_n d\bar{\sigma}_n
\over (\sinh^2{\Delta\psi_n\over 2}+\sin^2{\Delta\theta_n \over 2})}~~.
$$
Then after integration Eq.(\ref{3.3}) reads
\begin{eqnarray}
&&Tr_w K_{B_3}= V_w~{e^{-\mu\bar{s}}\over (4\pi \bar{s})^{3/2}}
+(2\pi)~({2\pi r_+\over l})~{e^{-\mu\bar{s}}\over (4\pi \bar{s})^{3/2}}~
\bar{s}~\sum_{n=1}^{\infty}~{e^{-{\Delta\psi^2_n\over 4\bar{s}}}\over (\sinh^2{\Delta\psi_n\over 2}+\sin^2{\Delta\theta_n \over 2})}~~,\nonumber \\
&&V_w=
\left\{
\begin{array}{ll}
{V_{B_3}\over l^3} & {\rm if\ } w=0 ~~, \\
(2\pi)({2\pi r_+\over l}) {1\over \sin^2{w\over 2}} {\bar{s}\over 2} & {\rm if\ } w\neq 0 ~~,
\end{array}
\right.
\label{3.4}
\end{eqnarray}
where we defined $\bar{s}=s/l^2~,~~\Delta\psi_n={2\pi r_+\over l}n~,~~
\Delta\theta_n=w+{2\pi|r_-|\over l}n$.
The knowledge of the heat kernel allows us to calculate the effective action on $B_3$:
\begin{eqnarray}
&& W_{eff}[B_3]=-{1\over 2}\int_{\epsilon^2}^{\infty}{ds\over s} Tr_{w=0}K_{B_3}
\nonumber \\
&&=W_{div}[B_3]-\sum_{n=1}^{\infty}{1\over 4n}~{e^{-\sqrt{\mu}\bar{A}_+n}
\over (\sinh^2{\bar{A}_+n\over 2}+\sin^2{|\bar{A}_-|n\over 2})}~~,
\label{3.5}
\end{eqnarray}
where $\bar{A}_+=A_+/l$ and $|\bar{A}_-|=|A_-|/l$ and the divergent part
of the action takes the form
\begin{eqnarray}
&&W_{div}[B_3]=-{1\over 2}{1\over (4\pi)^{3/2}}V_{B_3}\int_{\epsilon^2}^{\infty}
{ds\over s^{5/2}}e^{-\mu s}\nonumber \\
&&=-{1\over (4\pi)^{3/2}}V_{B_3}~ ({1\over 3\epsilon^3}-{
\mu^2\over \epsilon}+{2\over 3}\mu^{3/2}\sqrt{\pi}+O(\epsilon ))~~,
\label{3.5'}
\end{eqnarray}
where we used (\ref{*}) to carry out the integration over $s$
in (\ref{3.5}).
Remarkably, the expression (\ref{3.5}) is invariant under transformation:
$|\bar{A}_-|\rightarrow |\bar{A}_-|+2\pi k$. As discussed in \cite{rev2}
this is a consequence of the invariance of $B_3$ under large
diffeomorphisms corresponding to Dehn twists: the identifications $i)$
and $ii)$ determining the geometry of $B_3$ are
unchanged if we replace
$r_+\rightarrow r_+~,~~|r_-|\rightarrow |r_-|+kl$ for any integer $k$.
This invariance appears only for the Euclidean black hole and disappears
when we make the Lorentzian continuation (see discussion below).
The first quantum correction to the action due to quantization of the
three-dimensional gravity
itself was discussed in \cite{rev2}. In this case the correction was
shown to be determined by only
quantity $2\pi (\sinh^2 {\bar{A}_+\over 2}+\sin^2{|\bar{A}_|\over 2})$
related with holonomies of the BTZ instanton.
\bigskip
{\bf 4.2 BTZ instanton with Conical Singularity}
The conical BTZ instanton $(B_3^{\alpha})$ is obtained
from $H_3$ by the replacing the identification $i)$ as follows:
$i').~\theta\rightarrow \theta +2\pi\alpha
$
\noindent
and not changing the identification $ii)$. For $\alpha \neq 1$ the space
$B_3^\alpha$ has a conical singularity at the horizon ($\rho=0$).
The heat kernel on $B^{\alpha}_3$ is constructed via the heat kernel
on the regular instanton $B_3$ by means of the Sommerfeld formula \cite{Som},
\cite{Dow}:
\begin{equation}
K_{B_3^{\alpha}}(x,x',s)=K_{B_3}(x,x',s)+{1\over 4\pi\alpha}
\int_{\Gamma}\cot {w\over 2\alpha}~~K_{B_3}(\theta-\theta '+w,s)~~dw~~,
\label{3.6}
\end{equation}
where $K_{B_3}$ is the heat kernel (\ref{3.1}). The contour $\Gamma$ in
(\ref{3.6}) consists of two vertical
lines, going from $(-\pi+\imath \infty )$ to $(-\pi-\imath \infty )$
and from $(\pi-\imath \infty )$ to $(\pi+\imath \infty )$ and
intersecting the real axis between the poles of the $\cot {w\over 2\alpha}$:
$-2\pi\alpha,~0$ and $0,~+2\pi\alpha$ respectively.
For $\alpha=1$ the integrand in (\ref{3.6}) is
a $2\pi$-periodic function and the contributions from these two vertical
lines (at a fixed distance $2\pi$ along the real axis)
cancel each other.
Applying (\ref{3.6}) to the heat kernel (\ref{3.4}) on $B_3$ we get
\begin{eqnarray}
&&TrK_{B^{\alpha}_3}=TrK_{B_3}+(2\pi\alpha)~({2\pi r_+\over l})~{e^{-\mu\bar{s}}\over (4\pi \bar{s})^{3/2}}~{\bar{s}\over 2}~[~{\imath\over 4\pi\alpha}
\int_{\Gamma}{\cot {w\over 2\alpha}~dw\over \sin^2{w\over 2}}\nonumber \\
&&+
\sum_{n=1}^{\infty}e^{-{\Delta\psi^2_n\over 4\bar{s}}}{\imath\over 4\pi\alpha}
\int_{\Gamma}{\cot {w\over 2\alpha}~dw\over \sinh^2{\Delta\psi_n\over 2}+\sin^2
({w\over 2}+{\pi |r_-|\over l}n)}~]~~.
\label{3.7}
\end{eqnarray}
for the trace of the heat kernel on $B_3^\alpha$.
Note, that the first term comes from the $n=0$
term (the direct paths) in the sum (\ref{3.1}),
(\ref{3.2})
while the other one corresponds to $n\neq 0$ (winding paths).
Only the $n=0$ term leads
to appearance of UV divergences (if $s\rightarrow 0$). The
term due to winding paths ($n\neq 0$)
is regular in the limit $s\rightarrow 0$ due
to the factor $e^{-{\Delta\psi^2_n \over 4\bar{s}}}$.
To analyze (\ref{3.7}) we shall consider the rotating and non-rotating
cases separately.
\noindent
{\bf Non-rotating black hole ($J=0,~|r_-|=0$)}
For this case the contour integrals in (\ref{3.7})
are calculated as follows (see (\ref{A1}), (\ref{A2}))
\begin{equation}
{\imath\over 4\pi\alpha}
\int_{\Gamma}{\cot {w\over 2\alpha}~dw\over \sin^2{w\over 2}}={1\over 3}({1
\over \alpha^2}-1)\equiv 2c_2(\alpha )~~,
\label{3.8}
\end{equation}
\begin{equation}
{\imath\over 4\pi\alpha}
\int_{\Gamma}{\cot {w\over 2\alpha}~dw\over \sinh^2{\Delta\psi_n\over 2}+
\sin^2{w\over 2}}={1\over \sinh^2{\Delta\psi_n\over 2}}\left( {1\over \alpha}
{\tanh {\Delta\psi_n\over 2}\over \tanh {\Delta\psi_n\over 2\alpha}}-1
\right)
\label{3.9}
\end{equation}
Therefore, taking into account that $Tr K_{B_3}$ is given by (\ref{3.4}) multiplied
by $\alpha$ we get for the trace (\ref{3.7}):
\begin{eqnarray}
&&Tr K_{B_3^{\alpha}}=\left( {V_{B_3^{\alpha}}\over l^3}+{A_+\over l}(2\pi\alpha )
c_2(\alpha )~\bar{s}~ \right){e^{-\mu\bar{s}}\over (4\pi \bar{s})^{3/2}}
\nonumber \\
&&+2\pi ~ {e^{-\mu\bar{s}}\over (4\pi \bar{s})^{3/2}}~{A_+\over l}~\bar{s}~
\sum_{n=1}^{\infty}~{\tanh {\Delta\psi_n\over 2}\over \tanh {\Delta\psi_n\over 2\alpha}}
~~{e^{-{\Delta\psi^2_n\over 4\bar{s}}}\over \sinh^2{\Delta\psi_n\over 2}}~~,
\label{3.10}
\end{eqnarray}
where $\Delta\psi_n={A_+\over l}n~,~~A_+=2\pi r_+$.
\noindent
{\bf Rotating black hole ($J\neq 0,~|r_-|\neq 0$)}
When rotation is present we have for the contour integral in (\ref{3.7})
(see (\ref{A5})):
\begin{eqnarray}
&&{\imath\over 4\pi\alpha}
\int_{\Gamma}~{\cot {w\over 2\alpha}~dw\over \sinh^2{\Delta\psi_n\over 2}+
\sin^2({w\over 2}+{\gamma_n\over 2})} \nonumber \\
&&={1\over \alpha}~{\sinh {\Delta\psi_n\over \alpha}\over \sinh\Delta\psi_n}~
{1\over (\sinh^2{\Delta\psi_n\over \alpha}+\sin^2{ [ \gamma_n ] \over 2\alpha})}-
{1\over (\sinh^2{\Delta\psi_n}+\sin^2{\gamma_n\over 2})}
\label{3.12}
\end{eqnarray}
where $[\gamma ]=\gamma-\pi k,~|[\gamma ]|<\pi$.
Then we obtain for the heat kernel on $B^{\alpha}_3$:
\begin{eqnarray}
&&Tr K_{B_3^{\alpha}}=\left( {V_{B_3^{\alpha}}\over l^3}+{A_+\over l}(2\pi\alpha )
c_2(\alpha )~\bar{s}~ \right){e^{-\mu\bar{s}}\over (4\pi \bar{s})^{3/2}}
\nonumber \\
&&+2\pi {e^{-\mu\bar{s}}\over (4\pi \bar{s})^{3/2}}{A_+\over l}~\bar{s}~
\sum_{n=1}^{\infty}{\sinh {\Delta\psi_n\over \alpha}\over \sinh {\Delta\psi_n}}
~~{e^{-{\Delta\psi^2_n\over 4\bar{s}}}\over (\sinh^2{\Delta\psi_n\over 2\alpha}+\sin^2{[\gamma_n]\over 2\alpha})}~~,
\label{3.13}
\end{eqnarray}
where $\gamma_n=|A_-|n/l$ and $\Delta \psi_n=A_+n/l$. Remarkably,
(\ref{3.13}) has the periodicity
$\gamma_n\rightarrow \gamma_n+2\pi\alpha$ or equivalently $|A_-|n/l
\rightarrow |A_-|n/l +2\pi\alpha$.
As discussed in Section 2, any result obtained for the Euclidean
black hole must be analytically continued to Lorentzian values of the
parameters by means of (\ref{?}).
For the non-rotating black hole this is rather straightforward.
It simply means that the area $A_+$ of the Euclidean horizon becomes
the area of the horizon in the Lorentzian space-time.
For a rotating black hole the procedure is more subtle. From (\ref{?})
we must also transform $|A_-|$ which after
analytic continuation becomes imaginary ($|A_-|\rightarrow \imath A_-$),
where $A_-$ is area of the lower horizon of the Lorentzian black hole.
Doing this continuation in the left hand side of the contour integral
(\ref{3.12}) we find that the right hand side becomes
\begin{equation}
\sin^2( {\imath\gamma_n\over 2})=-\sinh^2{\gamma_n\over 2}~,~~
\sin^2({[\imath\gamma_n]\over 2\alpha})=-\sinh^2{\gamma_n\over 2\alpha}~~,
\label{??}
\end{equation}
where $\gamma_n=A_- n/l$. Below we are assuming this kind of substitution
when we are applying our formulas to the Lorentzian black hole.
We see that after the continuation we lose periodicity with respect
to $\gamma_n$.
It should be noted that there is only
a small group of conical spaces for which the
heat kernel is known explicitly \cite{con}. (The small $s$ expansion
for the heat kernel on conical spaces has been more widely studied, and
a rather general result that the coefficients of this expansion contain
terms (additional to the standard ones) due to the
conical singularity only and are defined on the singular subspace $\Sigma$
has recently been obtained \cite{DF}, \cite{Dowker}.)
However, no black hole geometry among these special cases were known.
In (\ref{3.13}) we have an exact result for a rather
non-trivial example of a black hole with rotation,
providing us with an exciting possibility to
learn something new about black holes. We consider some of these
issues in the context of black hole thermodynamics in the next section.
\bigskip
\noindent
{\bf Small $s$ Expansion of the Heat Kernel}
As we can see from Eqs.(\ref{3.10}), (\ref{3.13}) the trace of the heat
kernel on the conical space $B^{\alpha}_3$ both for the rotating and non-rotating cases has the form
\begin{equation}
Tr K_{B_3^{\alpha}}=\left( {V_{B_3^{\alpha}}\over l^3}+{A_+\over l}(2\pi\alpha )
c_2(\alpha )~\bar{s}~ \right){e^{-\mu\bar{s}}\over (4\pi \bar{s})^{3/2}}~ +~ES~~,
\label{a}
\end{equation}
where $ES$ stands for exponentially small terms which
behave as $e^{-{1\over s}}$ in the limit $s\rightarrow 0$.
So, for small $s$ we get the asymptotic formula
\begin{equation}
Tr K_{B_3^{\alpha}}={1\over (4\pi s)^{3/2}} \left(V_{B_3^{\alpha}}+
(-{1\over l^2}V_{B_3^{\alpha}}+{\xi \over l^2}V_{B_3^{\alpha}}+
A_+~(2\pi\alpha )
c_2(\alpha ))~s~ +O(s^2)~\right)
\label{b}
\end{equation}
where $\mu=1-\xi $.
The asymptotic behavior of the heat kernel on various manifolds
is well known and the asymptotic expressions
are derived in terms of geometrical invariants of the manifold.
For the operator $(\Box +X)$, where $X$ is some scalar function,
on a $d$-dimensional manifold $M^\alpha$ with conical singularity
whose angular deficit is $\delta=2\pi (1-\alpha )$
at the surface $\Sigma$,
the corresponding expression reads
\begin{equation}
Tr K_{M^{\alpha}}={1\over (4\pi s)^{d/2}} (a_0+a_1~s+O(s^2))~~,
\label{c}
\end{equation}
where
\begin{equation}
a_0=\int_{M^{\alpha}}1~~,~~a_1=\int_{M^{\alpha}}({1\over 6}R+X)~+~(2\pi\alpha )c_2(\alpha )\int_{\Sigma} ~1~~.
\label{d}
\end{equation}
The volume part of the coefficients is standard \cite{BD} while the surface
part in $a_1$ is due to the conical singularity according to \cite{DF}.
One can see that (\ref{b}) exactly reproduces (\ref{c})-(\ref{d})
for operator $(\Box +\xi /l^2 )$ since for the case under consideration
we have
$R=-6/l^2$. Note, that in (\ref{a}), (\ref{b}) we do not obtain the usual term
$\int_{\partial M}k$ due to extrinsic curvature $k$ of boundary $\partial M$.
This term does not appear in our case since we calculate the heat kernel
for spaces with boundary lying at infinity where the boundary term
is divergent. But, it would certainly appear if we deal with a boundary staying
at a finite distance. Also, in the expressions (\ref{a}), (\ref{b})
we do not observe a contribution due to extrinsic curvature of the horizon
surface. According to arguments by Dowker \cite{Dowker} such a contribution
to the heat kernel occurs for generic conical space.
However, in the case
under consideration the extrinsic curvature of the horizon precisely vanishes.
We observed \cite{MS} the similar phenomenon for charged Kerr black hole
in four dimensions.
\bigskip
\noindent
{\bf Effective action and renormalization}
For the effective action we immedately obtain that
\begin{eqnarray}
&&W_{eff}[B^{\alpha}_3]=-{1\over 2}\int_{\epsilon^2}^{\infty} {ds\over s} Tr K_{B^3_{\alpha}}
\nonumber \\
&&=W_{div}[B^{\alpha}_3]- \sum_{n=1}^{\infty}{1\over 4n}~~{\sinh ({\bar{A}_+
\over \alpha}n)\over \sinh (\bar{A}_+n) }~~ {e^{-\sqrt{\mu}\bar{A}_+ n}\over
(\sinh^2{\bar{A}_+ n\over 2\alpha}+\sin^2{[|\bar{A}_-|n]\over 2\alpha})}~~,
\label{3.14}
\end{eqnarray}
where the divergent part $W_{div}[B^{\alpha}_3]$ of the effective action
takes the form
\begin{eqnarray}
&&W_{div}[B^{\alpha}_3]=-{1\over 2}{1\over (4\pi)^{3/2}}\left(V_{B_3^{\alpha}}
\int_{\epsilon^2}^{\infty}
{ds\over s^{5/2}}e^{-\mu s/ l^2}~+~A_+~(2\pi\alpha)~c_2(\alpha )
\int_{\epsilon^2}^{\infty}
{ds\over s^{3/2}}e^{-\mu s/ l^2}~ \right) \nonumber \\
&&=-{1\over (4\pi)^{3/2}}~[V_{B_3^{\alpha}}~ ({1\over 3\epsilon^3}-{
\mu\over l^2\epsilon}+{2\over 3}{\mu^{3/2}\over l^3}\sqrt{\pi}+O(\epsilon ))
\nonumber \\
&& +A_+~(2\pi\alpha)~c_2(\alpha )({1\over \epsilon}-{\sqrt{\mu\pi}\over l}+O(\epsilon ))]~~.
\label{3.11'}
\end{eqnarray}
Recall that Eqs.(\ref{3.14}), (\ref{3.11'}) must be analytically continued
by means of (\ref{?}) and (\ref{??}) to deal with the characteristics of the
Lorentzian black hole.
Note that the rotation parameter $J$ enters the
UV-infinite part (\ref{3.14}) only via $A_+$.
The form of (\ref{3.14}) is therefore the same for rotating
and non-rotating holes.
Similar behavior for an uncharged Kerr black hole
was previously observed in four dimensions \cite{MS}.
The classical gravitational action
$$
W=-{1\over 16\pi G_B}\int_M (R+{2\over l^2})=-{1\over 16\pi G_B}\int_M R
-\lambda_B \int_M~1~~,
$$
where $\lambda_B={1\over 8\pi G_B} {1\over l^2}$.
In the presence of a conical singularity with angular
deficit $\delta=2\pi (1-\alpha )$ on a surface of area $A_+$
this has the form
\begin{equation}
W=-{1\over 16\pi G_B}\int_{M^\alpha} R-{1\over 4 G_B}A_+(1-\alpha )
-\lambda_B \int_{M^\alpha}~1~~.
\label{3.21}
\end{equation}
The ${1\over \epsilon}$ and ${1\over \epsilon^3}$ UV-divergences of the
effective action (\ref{3.14})-(\ref{3.11'}) for $\alpha=1$ (regular manifold
without conical singularities) are known to be absorbed in the
renormalization of respectively the bare Newton
constant $G_B$ and cosmological constant $\lambda_B$ of the
classical action. As was pointed out in \cite{SS} and \cite{FS} the
divergences of the effective action that are of first order with respect
to $(1-\alpha )$ are automatically removed by the same
renormalization of Newton's constant $G_B$ in the classical action
(\ref{3.21}). This statement is important in the context of the
renormalization of UV-divergences of the black hole entropy. Its validity
in the case under consideration can be easily demonstrated if we note
that $(2\pi\alpha )c_2 (\alpha )={2\over 3}\pi (1-\alpha )
+O((1-\alpha )^2)$ and define the renormalized quantities $G_{ren}$
and $\lambda_{ren}$ as follows:
\begin{equation}
{1\over 16\pi G_{ren}}={1\over 16\pi G_B}+{1\over 12}~
{1\over (4\pi)^{3/2}}~\int_{\epsilon^2}^{\infty}{ds\over s^{3/2}}e^{-\mu s/ l^2}
\label{3.22}
\end{equation}
and
\begin{equation}
\lambda_{ren}=\lambda_B+{1\over 2}~
{1\over (4\pi)^{3/2}}~\left(\int_{\epsilon^2}^{\infty} {ds\over s^{5/2}}e^{-\mu s/ l^2}+
l^2\int_{\epsilon^2}^{\infty} {ds\over s^{3/2}}e^{-\mu s/ l^2} \right)~~.
\label{3.23}
\end{equation}
Then all the divergences in (\ref{3.11'}) which are up to order
$(1-\alpha )$ are renormalized by (\ref{3.22}), (\ref{3.23}). The
renormalization of terms $\sim O((1-\alpha )^2)$ requires in principle
the introduction of some new
counterterms. However they are irrelevant for black
hole entropy.
The prescription (\ref{3.22}), (\ref{3.23}) includes in part some
UV-finite renormalization. This is in order that
$G_{ren}$ and $\lambda_{ren}$ be treated as macroscopically measurable
constants. Note also that the relation between the bare
constants: $\lambda_B G_B={1\over 8\pi}{1\over l^2}$ is no longer valid
for the renormalized quantities (\ref{3.22})-(\ref{3.23}).
\bigskip
\section{Entropy}
\setcounter{equation}0
A consideration of the conical singularity at the horizon for the
Euclidean black hole is a convenient way to obtain the thermodynamic
quantities of the hole. Geometrically, the angular deficit
$\delta =2\pi (1-\alpha ),~\alpha={\beta\over \beta_H}$ appears when
we close the Euclidean time coordinate with an arbitrary period
$2\pi\beta$. Physically it means that we consider the statistical
ensemble containing a black hole at a temperature
$T=(2\pi\beta )^{-1}$ different from the Hawking value
$T_H=(2\pi\beta_H )^{-1}$. The state of the system at the
Hawking temperature is the equilibrium state corresponding to the
extremum of the free energy \cite{FWS}. The entropy of the black
hole appears in
this approach as the result of a small deviation from equilibrium.
Therefore in some sense the entropy is an off-shell quantity.
If $W[\alpha ]$ is the action calculated for arbitrary angular
deficit $\delta$ at the horizon we get
\begin{equation}
S=(\alpha\partial_\alpha-1)W[\alpha ]|_{\alpha=1}~~.
\label{5.1}
\end{equation}
for the black hole entropy.
Applying this formula to the classical gravitational action (\ref{3.21})
we obtain the classical Bekenstein-Hawking entropy:
\begin{equation}
S_{BH}={A_+\over 4G_B}~~.
\label{5.2}
\end{equation}
Applying (\ref{5.1}) to the (renormalized) quantum action $W+W_{eff}$
(\ref{3.14}), (\ref{3.21})
we obtain the (renormalized) quantum entropy of black hole:
\begin{eqnarray}
&&S={A_+\over 4G}+\sum_{n=1}^\infty
s_n~~, \nonumber \\
&&s_n={1\over 2n}{e^{-\sqrt{\mu}\bar{A}_+ n}\over (\cosh \bar{A}_+n-\cosh \bar{A}_-n)}
(1+\bar{A}_+ n \coth \bar{A}_+ n \nonumber \\
&&- {(\bar{A}_+n \sinh \bar{A}_+ n -
\bar{A}_-n \sinh \bar{A}_-n )\over (\cosh \bar{A}_+n-\cosh \bar{A}_-n)}
)~~,
\label{5.3}
\end{eqnarray}
where $G\equiv G_{ren}$ is the renormalized Newton constant. We already
have done the analytic continuation (\ref{??}) in (\ref{5.3}) in
order to deal with the characteristics of the Lorentzian black hole.
The second term in the right hand side of (\ref{5.3}) can be
considered to be the one-loop
quantum (UV-finite) correction to the classical entropy of black hole.
Since ${A_-\over A_+}=k<1$, $s_n$ is a non-negative quantity which
monotonically decreases with $n$ and
has asymptotes:
\begin{equation}
s_n\rightarrow {1\over 4n}e^{-(1+\sqrt{\mu})\bar{A}_+n}~~if~~A_+\rightarrow \infty
\label{5.4}
\end{equation}
and
\begin{equation}
s_n\rightarrow {1\over 6n}-{\mu\over 6}\bar{A}_+ ~~if~~A_+\rightarrow 0~~.
\label{5.5}
\end{equation}
Note that both asymptotes (\ref{5.4}), (\ref{5.5})
are independent of the parameter $A_-$ characterizing
the rotation of the hole.
The infinite sum in (\ref{5.3}) can be approximated by integral. We
find that
\begin{equation}
S={A_+\over 4G}+\int_{\bar{A}_+}^\infty
~s(x)~dx~~,
\label{5.6}
\end{equation}
where
\begin{equation}
s(x)={1\over 2x}{e^{-\sqrt{\mu}x}\over (\cosh x-\cosh kx)}
\left(1+x \coth x - {(x\sinh x -
kx \sinh kx )\over (\cosh x-\cosh kx)}
\right)~~.
\label{5.7}
\end{equation}
For large enough $\bar{A}_+\equiv {A_+\over l}>>1$ the integral in (\ref{5.6}) exponentially
goes to zero and we have the classical Bekenstein-Hawking formula for entropy.
On the other hand, for small $\bar{A}_+$ the integral in (\ref{5.6})
is logarithmically divergent
so that we have
\begin{equation}
S={A_+\over 4G}+{\sqrt{\mu}\over 6}{A_+\over l}-{1\over 6}\ln {A_+\over l}+
O(({A_+\over l})^2)~~.
\label{5.8}
\end{equation}
This logarithmic divergence can also be understood by
examining the expression (\ref{5.3}). {}From (\ref{5.5}) it follows that
every $n$-mode gives a finite contribution $s_n={1\over 6n}$ at
zero $A_+$. Their sum $\sum_{n=1}^\infty {1\over 6n}$, however,
is not convergent since $s_n$ does not decrease fast enough.
This divergence appears as the logarithmic one in (\ref{5.8}). This
logarithmic behavior for small $A_+$ is universal,
independent of the constant $\xi$
(or $\mu$) in the field operator
and the area of the inner horizon ($A_-$) of the black hole. Hence
the rotation parameter $J$ enters (\ref{5.8}) only via the area $A_+$
of the larger horizon. It should be note that similar
logarithmic behavior was previously observed in various models both
in two \cite{SS}, \cite{FWS}
and four \cite{F1}, \cite{Zas} dimensions. Remarkably, it appears in
the three dimensional model as the result of an
explicit one-loop calculation.
The first quantum correction to the
Bekenstein-Hawking entropy due to quantization of the three-dimensional
gravity itself was calculated in \cite{rev2} and was shown to be proportional
to area $A_+$ of outer horizon.
\section{Concluding Remarks}
\setcounter{equation}0
Our computation of the quantum-corrected entropy (\ref{5.6})
of the BTZ black hole
has yielded the interesting result that the entropy is not proportional to
the outer horizon area ({\it i.e.} circumference) $A_+$, but instead
develops a minimum for sufficiently small $A_+$.
(The plot of the entropy as function of area $A_+$ for non-rotating case
is represented in Fig.2.) This minimum is
a solution to the equation
\begin{equation}
{l\over 4G}=s({A_{+ min}\over l})~~.
\label{5.9}
\end{equation}
The constants $G$ and $l$ determine two different scales in the theory.
The former determines the strength of the gravitational interaction.
The distance $l_{pl}\sim G$ can be interpreted as the Planck scale in this
theory. It determines the microscopic behavior of quantum
gravitational fluctuations.
On the other hand, the constant $l$ (related to curvature via $R=-6/l^2$)
can be interpreted as radius of the Universe that contains the black hole.
So $l$ is a large distance (cosmological) scale.
Regardless of the relative sizes of $G$ and $l$, the entropy is always
minimized for $A_+ \leq G$.
If we assume $G<<l$ then (\ref{5.9}) is solved as $A_{+min}={2\over 3}G$,
However if $G >> l$, then (\ref{5.9}) becomes (for $\mu=0$, say)
$A_+/G \simeq e^{-A_+/l} < 1$. In either case,
the minimum of the entropy occurs for a hole whose horizon
area is of the order of the
Planck length $r_{+}\sim l_{pl}$. In the process of evaporation the
horizon area of a hole
typically shrinks. The evaporation is expected to stop when the
black hole takes the minimum entropy configuration. In our case it is the
configuration with horizon area $A_+=A_{+ min}$. Presumably it has zero
temperature and its geometry is a reminscent of an extremal black hole.
However at present we cannot definitively conclude this
since our considerations do not take into account quantum back reaction
effects. These effects are supposed to drastically change the geometry at
a distance $r\sim l_{pl}$. Therefore the minimum entropy configuration is
likely to have little in common with the classical black hole
configuration described in Section 2.
Further investigation of this issue will necessitate
taking the back reaction into account.
\section*{Acknowledgements}
This work was supported by the Natural Sciences and Engineering Research
Council of Canada and by a NATO Science Fellowship.
\newpage
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proofpile-arXiv_065-629
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
The Southern sky was surveyed at 2.7{\rm\thinspace GHz}\ by the Parkes radio
telescope between 1968 and 1979 (see Bolton, Savage \& Wright 1979,
and references therein), resulting in a catalogue of more than 10,000
radio sources. Over this period, an extensive programme of optical
identifications was undertaken. In its early stages, this programme
was frustrated by lack of a Southern radio calibrator grid, poor radio
positions (the original Parkes positions were only accurate to
10--15~arcsec) and a lack of optical sky survey plates. Modern
methods, using accurate (better than 1 arcsec) radio positions and
complete catalogues of digitised optical sky survey data, with the
radio and optical reference frames tied to an accuracy of better than
100 milliarcsec (Johnston et al.\ 1995) now allow unambiguous
optical identification of most of the radio sources, supplemented by
CCD imaging for the remainder. In this paper we present
new identifications of a sample of Parkes flat-spectrum radio sources
using these techniques. The Parkes Catalogue contains both steep- and
flat-spectrum sources. Radio samples are biased towards
core-dominated quasars if the flat-spectrum sources are selected and
towards lobe-dominated quasars and galaxies for steep-spectrum
sources. Since the scientific questions of primary interest to us are
related to core-dominated quasars, we concentrate on flat-spectrum
sources in this study.
Other workers have compiled a number of complete samples of radio
sources. Each has been selected on different criteria, leading to the
inclusion of different objects. Low frequency samples contain more
radio galaxies than quasars, high frequency (i.e. flat-spectrum)
samples reverse that bias, and lower flux limits increase the mean
redshift of the objects in the sample. Four notable samples are the
3CR Sample (Spinrad et al.\ 1985, Laing, Riley \& Longair 1983), the 2
Jansky Sample (Wall \& Peacock 1985), the 1 Jansky Sample (K\"uhr et
al.\ 1981; Stickel, Meisenheimer \& K\"uhr 1994), and the Parkes
Selected Regions (Dunlop et al.\ 1989).
The 3CR sample
comprises 173 sources selected with
$S_{178{\rm\thinspace MHz}} > 10${\rm\thinspace Jy}\ over an area of 4.23{\rm\thinspace Sr}.
The high flux limit biases this sample towards lower
redshift objects (18\% have $z>1$), and the use of a low frequency
biases the sample towards steep-spectrum radio galaxies.
The 2 Jansky
sample, which was selected at 2.7{\rm\thinspace GHz}\ over an area of 9.81{\rm\thinspace Sr},
contains 233 objects which are
mainly steep-spectrum, again biasing the sample towards low-redshift
radio galaxies.
The 1 Jansky
sample was selected at 5{\rm\thinspace GHz}, also over 9.81{\rm\thinspace Sr}\ of sky and
comprises 518 sources. 55\% are flat-spectrum sources,
and of those $\sim 90\%$ are
quasars or BL Lacs; many are Parkes sources.
Finally, the Parkes Selected Regions (a total of 0.075{\rm\thinspace Sr})
contain 178
sources with $S_{2.7{\rm\thinspace GHz}} > 0.1{\rm\thinspace Jy}$, most of which are
steep-spectrum extended sources identified as galaxies. Only 23\% are
flat-spectrum sources, but these objects have a distribution of
properties similar to our sample.
Our primary interest in this paper is the compilation of a large
unbiased sample of radio-selected quasars. Note that we use the standard
definitions of ``quasar'' for radio-loud sources and quasi-stellar-object
(QSO) for optically-selected sources.
There were several motivations for
defining the sample. First, we were interested in using quasars for
gravitational lensing studies. A proper determination of lensing
statistics requires the identification of complete quasar samples, as
well as an understanding of any selection effects which might bias
selection of gravitationally lensed quasars. Secondly, the recent
completion of the Large Bright QSO Survey (LBQS: Hewett, Foltz \&
Chaffee 1995) has meant that there is a well-defined sample of
optically-selected QSOs, allowing the determination of global
spectroscopic properties. The completion of a comparable sample of
radio-selected quasars will allow a detailed phenomenological
comparison of the optical spectra of these two classes of object,
perhaps allowing the determination of differences in underlying
physical conditions. Finally, quasars are one of the most effective
probes of the universe to high redshift, providing a measure of
evolution as well as the formation of large-scale structure. Of course
the complete identification of a sample of radio sources can also
provide some surprises, if unexpected objects, such as very high
redshift quasars, are found.
The Parkes Half-Jansky Flat-spectrum Sample we define here contains
323 sources selected in an area of 3.90{\rm\thinspace Sr}\ and is similar to the
earlier compilation by Savage et al.\ (1990). We have made
significant progress in the optical identification of the sources
which were previously termed ``Empty Fields'', particularly by using
near infrared \hbox{$Kn$}\ band (2.0--2.3 microns) imaging to detect the
optically faint sources. The extremely red optical to near infrared
colours of these sources imply that most are heavily reddened, viewed
through dust either in the line-of-sight to the quasar or within the
immediate quasar environment (see Webster et al.\ 1995). In this
paper we present optical identifications for 321 sources (99\%
of the sample), and redshifts for 277 sources (86\%).
The outline of the paper is as follows. The selection criteria for
the radio sources are described in Section~\ref{sec_sample}. In
Section~\ref{sec_positions} we explain how the accurate radio positions
were obtained, and present radio images of the resolved sources in the
sample. Section~\ref{sec_optical} describes the mapping of the
accurate radio positions onto the optical catalogues. We present a
full discussion of the accuracy of this procedure, locate the likely
optical counterparts, classify these images as either stellar or
non-stellar and provide the optical magnitudes. Where there is no
optical survey image at the location of the radio source, we use
R-band CCD frames and \hbox{$Kn$}\ near infrared images to determine the source
identification and
morphology. In Section~\ref{sec_spec} we present spectroscopic
classifications and redshifts of these sources; for those sources
which do not have a published spectrum, we also include the spectra.
All these results are summarised in a master catalogue of the sample in
Section~\ref{sec_cat}. Finally,
Section~\ref{sec_corr} presents a summary of the most important
features of our sample. An electronic version of our catalogue is
available from the Centre de Donn\'ees astronomiques de Strasbourg
in Section VIII (``Radio Data'') of the catalogue archive.
\section{Selection of the Sample}
\label{sec_sample}
\subsection{Radio Surveys}
Our basic selection criteria are very similar to those used by Savage et
al.\ (1990). We started with the machine-readable version of the Parkes
catalogue (PKSCAT90, Wright \& Otrupcek 1990) and applied the following
criteria:
\begin{enumerate}
\item 2.7{\rm\thinspace GHz}\ ($S_{2.7}$) and 5.0{\rm\thinspace GHz}\ ($S_{5.0}$) fluxes defined
\item $S_{2.7} > 0.5${\rm\thinspace Jy}\
\item spectral index $\alpha_{2.7/5.0} > -0.5$, where $S(\nu) \propto \nu^{\alpha}$
\item Galactic latitude $|b|>20\hbox{$^\circ$}$
\item $-45\hbox{$^\circ$}<{\rm Declination~(B1950)}<+10\hbox{$^\circ$} $
\end{enumerate}
In our search of PKSCAT90, three objects did not have a 5.0{\rm\thinspace GHz}\ flux
defined but satisfied all the other criteria. One of these was later
included from our search of the discovery papers, but the other
objects were not measured at 5.0{\rm\thinspace GHz}\ in the discovery papers,
presumably because they are very bright radio sources associated with
bright optical galaxies: PKS~0131$-$367 (5.6{\rm\thinspace Jy}, 15mag) and PKS~0320$-$374
(98{\rm\thinspace Jy}, 10mag) and were too extended to measure properly. This
search resulted in an initial sample of 325 objects.
We then carefully checked all the radio fluxes in the original
discovery papers of the radio survey, as listed in
Table~\ref{radio_surveys}. Many objects have more recent (but
unreferenced) flux measurements listed in PKSCAT90, but we replaced
these with the original fluxes in order to quantify the time
difference between the 2.7 and 5.0{\rm\thinspace GHz}\ measurements and thus estimate
the effects of variability. After the original fluxes were adopted, 15
sources no longer satisfied the flux and spectral criteria and so were
excluded. We also found 17 sources whose original fluxes in the
discovery papers satisfied the selection criteria so these were added
to the sample.
In two regions (samples A and F; see Table~\ref{radio_surveys}) our
search of PKSCAT90 produced several sources not listed in the original
papers. These two regions of the original survey were not complete
because the flux limit was not well-defined; subsequent unpublished
observations detected additional sources satisfying our selection
criteria that were included in PKSCAT90. We retained these additional
objects (12 in each region), but flagged them with a minus sign in
front of the reference code (Rf) in the final catalogue
(Table~\ref{tab_master}).
Finally we removed four planetary nebulae from the sample on the basis
that we are interested in extragalactic sources. This gave a final
sample of 323 sources which are listed in Table~\ref{tab_master} in
Section~\ref{sec_cat}. Our new sample is complete in 6 of the 8
sub-regions listed in Table~\ref{radio_surveys} but in two of the
regions (A and F) the original surveys are incomplete and we have
added additional sources from PKSCAT90. The distributions of the
fluxes and spectral indices are given in Figs. \ref{fig_flux} and
\ref{fig_alpha} respectively and a diagram showing the regions
surveyed and the distribution of our sample across the sky is shown in
Fig.~\ref{fig_sky}.
\begin{table*}
\caption{The Parkes Survey Regions}
\label{radio_surveys}
\noindent
\begin{tabular}{rrrrrlr}
Region &Dec range & RA range & $\Delta T^1$ & Flux Limit & Reference (code) & $N^2$ \\
& & & month & $S_{2.7}$(Jy) \\
\\
A&$+10\hbox{$^\circ$},+04\hbox{$^\circ$}$ & 7\hbox{$^{\rm h}$}-18\hbox{$^{\rm h}$},20\h30-5\h30 & 2,9 & $\ax0.5^3$ & Shimmins, Bolton \& Wall (1975) (79) & 46 \\
B&$+04\hbox{$^\circ$},-04\hbox{$^\circ$}$ & 7\h20-17\h50,19\h40-6\hbox{$^{\rm h}$} & 2-9 & 0.35 & Wall, Shimmins \& Merkelijn (1971)$^4$ (102) & 63 \\
C&$-04\hbox{$^\circ$},-15\hbox{$^\circ$}$ & 10\hbox{$^{\rm h}$}-15\hbox{$^{\rm h}$} & 2-14 & 0.25 & Bolton, Savage \& Wright (1979) (8) & 21 \\
D&$-15\hbox{$^\circ$},-30\hbox{$^\circ$}$ & 10\hbox{$^{\rm h}$}-15\hbox{$^{\rm h}$} & 3 & 0.25 & Savage, Wright \& Bolton (1977) (70) & 34 \\
E&$-04\hbox{$^\circ$},-30\hbox{$^\circ$}$ & 22\hbox{$^{\rm h}$}-5\hbox{$^{\rm h}$} & 1-12 & 0.22 & Wall, Wright \& Bolton (1976) (103) & 65 \\
F&$-04\hbox{$^\circ$},-30\hbox{$^\circ$}$ & 5\hbox{$^{\rm h}$}-6\h30,8\hbox{$^{\rm h}$}-10\hbox{$^{\rm h}$},15\hbox{$^{\rm h}$}-17\hbox{$^{\rm h}$},19\hbox{$^{\rm h}$}-22\hbox{$^{\rm h}$}
& 2-9 &$\ax0.6^3$& Bolton, Shimmins \& Wall (1975) (7) &39 \\
G&$-30\hbox{$^\circ$},-35\hbox{$^\circ$}$ & 9\hbox{$^{\rm h}$}-16\h30,18\h30-7\h15 & 1-10 & 0.18 & Shimmins \& Bolton (1974) (78)& 25 \\
H&$-35\hbox{$^\circ$},-45\hbox{$^\circ$}$ & 10\hbox{$^{\rm h}$}-15\hbox{$^{\rm h}$},19\hbox{$^{\rm h}$}-7\hbox{$^{\rm h}$} & 9 & 0.22 & Bolton \& Shimmins (1973) (6) & 30 \\
&& &&&total & 323 \\
\\
\end{tabular}
Notes: 1. $\Delta T$ is the time delay between the 2.7 and 5.0{\rm\thinspace GHz}\
measurements. 2. $N$ is the number of sources each region contributes
to our sample. 3. No completeness analysis was made for regions A and
F so extra objects from PKSCAT90 not in the original papers were
included (12 in each case) and the flux limits are only indicative.
4. The 5{\rm\thinspace GHz}\ fluxes for region B were published separately by Wall
(1972).
\end{table*}
\begin{figure}
\epsfxsize=\one_wide \epsffile{fig1flux.eps}
\centering
\caption{Histogram of the 2.7{\rm\thinspace GHz}\ fluxes of the sources in the sample.}
\label{fig_flux}
\end{figure}
\begin{figure}
\epsfxsize=\one_wide \epsffile{fig2alph.eps}
\centering
\caption{Histogram of the (2.7 to 5.0{\rm\thinspace GHz}) radio spectral
indices of the sample sources($S(\nu) \propto \nu^{\alpha}$).}
\label{fig_alpha}
\end{figure}
\begin{figure*}
\epsfxsize=16.5cm \epsffile{fig3skyd.eps}
\centering
\caption{Distribution on the sky of The Parkes Half-Jansky Flat-Spectrum Sample
(equal-area projection). The solid lines indicate the the survey regions
and the limits of Galactic latitude ($|b|>20\hbox{$^\circ$}$).}
\label{fig_sky}
\end{figure*}
\subsection{Variability}
Flat-spectrum radio sources are well-known to be variable, which
introduces two biases in our sample. First, our sample was selected
to have a 2.7{\rm\thinspace GHz}\ flux above 0.5{\rm\thinspace Jy}\ {\em at the observation epoch}.
Some of the sample may have been in a particularly bright state; their
average fluxes may be below our limit. Likewise some flat-spectrum
sources with average fluxes above 0.5{\rm\thinspace Jy}\ may have been excluded from
the sample because they were in a particularly faint state when the
sample was defined.
Secondly, the 5{\rm\thinspace GHz}\ observations of the sample
sources were not obtained simultaneously with the 2.7{\rm\thinspace GHz}\ observations
(see Table~\ref{radio_surveys}). The 5{\rm\thinspace GHz}\ observations were usually
taken after the 2.7{\rm\thinspace GHz}\ observations; the time interval being more than
6 months in $\sim 40$\% of cases; six months is a typical
variability timescale for compact radio sources (Fiedler et al.
1987). If a source varied between the two observations, its spectral
index could be in error, and the object might be wrongly included in,
or excluded from, the flat-spectrum sample.
Stannard \& Bentley (1977) investigated the variability of 50 Parkes
flat-spectrum radio sources, substantially overlapping our sample. They
compared 2.7{\rm\thinspace GHz}\ fluxes taken two years apart, and found that $\sim
50$\% of sources had varied by 15\% or more. The number of sources
included in the flux limited sample because they were brighter than
average at the time of observation will exceed the number of sources
missed because they were fainter than average. This is because there
are more sources with mean fluxes just below 0.5{\rm\thinspace Jy}\ that there are with
fluxes just above 0.5{\rm\thinspace Jy}, due to the steepness of the number/flux
relation. Using Stannard \& Bentley's numbers, we estimate that $\sim$
30--40 of our sources have mean fluxes below 0.5{\rm\thinspace Jy}, and that we missed
$\sim$ 20--30 sources with mean fluxes above 0.5{\rm\thinspace Jy}.
Allowing for the time delay between the 2.7{\rm\thinspace GHz}\ and 5{\rm\thinspace GHz}\ measurements,
we can also estimate that $\sim 10$ flat-spectrum sources with $-0.5 <
\alpha < -0.3$ will have been mistakenly classified as steep-spectrum
and excluded from our sample, while another $\sim 10$ with $-0.7 <
\alpha < -0.5$ will have been wrongly included. This calculation
ignores the dependence of variability on spectral index. Fiedler et
al.\ (1987) showed that most bright compact radio sources with
relatively steep-spectra ($\alpha < -0.2$) vary by only $\sim 5$\% on
timescales of two years. This implies that we will only misclassify
$\sim 5$ objects with $-0.5 < \alpha < -0.3$ as being steep-spectrum.
However, they also find that a small fraction of very flat-spectrum
sources ($\alpha > -0.2$) can vary by 50\% or more on timescales of two
years. Applying their numbers to our sample, we estimate that $\sim 2$
sources with $\alpha > -0.2$ may have varied by enough to have been
misclassified as steep-spectrum. These numbers may be an overestimate;
Fiedler et al.\ only considered compact sources, whereas several of our
objects, particularly those with steeper spectra, are extended and may
be less variable. We plan to address this uncertainty by remeasuring
the sample making simultaneous flux measurements at both frequencies.
In summary, variability imposes an uncertainty on our 0.5{\rm\thinspace Jy}\
completeness limit at 2.7{\rm\thinspace GHz}: some 30--40 ($\sim$11\%) sources in our
sample have mean fluxes below the limit, and we missed some 20--30
($\sim$8\%) sources with mean fluxes above the limit. This bias is
inherent to any single-epoch flux-limited sample.
On the other hand we find that $\sim$5--10 objects in our sample
actually have $\alpha <-0.5$ (steep-spectrum) and have been wrongly
included because they varied between the epochs of the 2.7{\rm\thinspace GHz}\ and
5.0{\rm\thinspace GHz}\ measurements, but that another $\sim$5--10 flat-spectrum
objects were missed for the same reason.
\section{Radio Positions}
\label{sec_positions}
We had to improve on the poor (10--20 arcsec) accuracy of the original
Parkes radio positions before being able to make optical
identifications of the radio sources by positional coincidence. To
this end we have obtained more accurate radio positions for all
sources in the sample using published data, The VLA Calibrator Manual
(as compiled by Perley \& Taylor, 1996)
and our own Very Large Array (VLA) and Australia
Telescope Compact Array (ATCA) observations. The sources of these
positions and the associated errors are listed in
Table~\ref{tab_Rradio}. The source positions are listed in
Table~\ref{tab_master}; note that we use the original naming scheme
for the sources based on B1950 coordinates but we include the J2000
coordinates for all the sources in Table~\ref{tab_master} for
reference.
\begin{table}
\centering
\caption{Sources of accurate radio positions}
\label{tab_Rradio}
\begin{tabular}{lr}
Reference (code) & uncertainty \\
& (arcsec) \\
\\
Jauncey et al.\ (1989) (39) & 0.15 \\
Johnston et al.\ (1995) (40) & 0.01 \\
Lister et al.\ (1994) (43) & $\approx$0.3 \\
Ma et al.\ (1990) (45) & 0.01 \\
Morabito et al.\ (1982) (50) & 0.6 \\
Patnaik (1996) (55) & $\approx$0.02 \\
Perley (1982) (56) & 0.15 \\
Perley \& Taylor (1996) (57) & $\approx$0.15 \\
Preston et al.\ (1985) (60) & 0.6 \\
Ulvestad et al.\ (1981) (97) & 0.40 \\
This paper: ATCA (120) & 0.0-0.3 \\
This paper: VLA (121) & 0.2-0.5 \\
\end{tabular}
\end{table}
\subsection{VLA Observations and Data Reduction}
On 1986 October 1 and 4 we observed the majority of the sources that
lacked accurate published positions with the VLA. The observations were
made at 4.86{\rm\thinspace GHz}\ with the VLA in its ``CnB'' configuration to yield
nearly circular synthesised beams with approximately 6 arcsec FWHM
resolution. Each programme source was covered with a single ``snapshot''
scan of about 3 minutes duration, and each group of snapshots was
preceded and followed by scans on a phase calibrator whose rms absolute
position uncertainty is not more than 0.1 arcsec in each coordinate.
The phase calibrator flux densities were bootstrapped to the Baars et
al.\ (1977) scale via observations of 3C 48 and 3C 286.
The (u,v) data recorded from both circular polarizations in two 50 {\rm\thinspace MHz}\
bands centered on 4.835 and 4.885{\rm\thinspace GHz}\ were edited, calibrated, and
mapped with AIPS. The images were cleaned, and the clean components
were used to self-calibrate the antenna phases, yielding images with
dynamic ranges typically $> 200:1$. Nearly every programme source
contains a dominant compact component that should coincide in position
with any possible optical identification. The positions of these
compact components were determined by Gaussian fitting on the images.
The formal fitting residuals are $< 0.1$ arcsec because the synthesised
beam is small and the signal-to-noise ratios are high. Thus the radio
position uncertainties are dominated by atmospheric phase drifts and
gradients not removed by the calibration. They range from about 0.2
arcsec at Dec $+10\hbox{$^\circ$}$ to about 0.5 arcsec at Dec $-45\hbox{$^\circ$}$.
\subsection{ATCA Observations and Data Reduction}
Several remaining sources in the sample were observed with the ATCA
during 1993 March and November using all 6 antennas with a maximum
baseline of 6~km. Observations were made at 4.80 and 8.64{\rm\thinspace GHz}\ in ``cuts''
mode with orthogonal linear polarisations at a bandwidth of
128{\rm\thinspace MHz}. The synthesised beam at 4.80{\rm\thinspace GHz}\ has a constant East-West
resolution of 2 arcsec FWHM and a North-South resolution varying
from 3 arcsec (at Dec $-45\hbox{$^\circ$}$) to 8 arcsec (Dec $-21\hbox{$^\circ$}$).
``Cuts'' mode involves observing each object for a period of one
minute on at least 6 occasions spread evenly over 12 hours. In this
way, it is possible to obtain imaging data on approximately 40 sources
within a 12 hour observation. Secondary calibrators with accurate,
milliarcsec positions close to the programme sources were observed
at least once every 2 hours. The flux density scale was determined
from observations of the primary calibrator at the ATCA, PKS~1934$-$638.
The data were edited and calibrated within AIPS and images made using
the Caltech Difmap software (Shepherd, Pearson \& Taylor, 1995). The
final self-calibrated images have typical dynamic ranges in excess of
400:1 for strong and relatively compact sources, decreasing to
approximately 100:1 for objects with weak or extended emission.
Source positions were calculated by fitting a Gaussian to the peak in
the brightness distribution of a cleaned (but not self-calibrated)
8.64{\rm\thinspace GHz}\ image. The uncertainty in source positions measured from these
ATCA images comprises a component due to thermal noise, which scales
inversely with S/N ($\sim$beamwidth/(S/N) ) and a component due to
systematic effects arising from the phase-referencing. The latter term
dominates for strong sources and scales linearly with angular distance
between the source and the phase-reference used to calibrate its
position. The error is approximately 0.1 arcsec for an angular
separation of 5\hbox{$^\circ$}\ (Reynolds et al.\ 1995).
\subsection{The Radio Positions and Morphology}
The new radio positions are presented in Table~\ref{tab_master}. As
shown in Table~\ref{tab_Rradio} these are all accurate to 0.6
arcsec or better for unresolved sources. Any radio sources we know
to be resolved are noted in the comments column of
Table~\ref{tab_master} and we present radio images of these sources in
Fig.~A1. We indicate five different categories of
resolved source in the Table using the terminology of Downes et al.
(1986):
\begin{enumerate}
\item ``P'' signifies partially resolved sources: the position is
well-defined by a peak.
\item ``Do'' indicates double sources with no central component or dominant
peak. There is no clear maximum, so the centroid of the image was used to
define the position.
\item ``Do+CC'' indicates a double-lobed source with a central component
or peak that gives a well-defined position.
\item ``H'' indicates a diffuse halo around a central source which
gives a well-defined position.
\item ``HT'' indicates a complex head-tail structure with no
well-defined position.
\end{enumerate}
\subsection{Notes on Individual Radio Positions}
\label{sec_notes}
In this section we describe any sources with extended structure
making the position difficult to define. We also note any sources
for which our final accurate positions differ by more than 24 arcsec
from the original Parkes catalogue positions.
\begin{enumerate}
\item PKS~0114+074: there are 3 components to the VLA radio image in
Fig.~A1. We have adopted the centroid of the stronger
double source to the South, although the Northern source also has an
optical counterpart. The PKSCAT90 position corresponds to the Northern
source; our position is therefore some 30 arcsec different. Our
spectroscopic observations show that the Northern source (at
01:14:49.51 $+$07:26:30.0 B1950) is a broad-lined quasar at $z=0.858$
consistent with previous publications. The correct identification (at
01:14:50.48 $+$07:26:00.3 B1950) is a narrow-line galaxy at $z=0.342$.
\item PKS~0130$-$447: this position is some 30 arcsec from the
original value.
\item PKS~0349$-$278: the VLA image in Fig.~A1 is confused
with a compact source some 2.5 arcmin from the PKSCAT90 position
and a marginal detection at the PKSCAT90 position. We made an
independent check of the radio centroid position for this source by
measuring it on the 4.85{\rm\thinspace GHz}\ survey images made with the NRAO 140-foot
telescope (Condon, Broderick \& Seielstad, 1991). A Gaussian fit gave
a position of 03:49:31.5 $-$27:53:41 (B1950), consistent with the
original position (and coincident with an optical galaxy) but not with
the stronger VLA source at 03:49:41.17 $-$27:52:07.0 (B1950).
Furthermore, the fit is clearly extended (source size 280 arcsec by 109
arcsec with position angle 50\hbox{$^\circ$}\ after the beam has been deconvolved).
The 4.85{\rm\thinspace GHz}\ flux of PKS~0349$-$278 is just over 2{\rm\thinspace Jy}, but the strong
source in the VLA image is only 0.3{\rm\thinspace Jy}. The VLA has resolved out most
of the flux, leaving only two components plus some residuals visible in
the contour plot. The strong VLA component is probably only a hotspot
in the northeastern lobe of the radio source. We adopt the fainter VLA
position (03:49:31.81 $-$27:53:31.5 B1950) which is consistent with the
single-dish positions.
\item PKS~0406$-$311: the VLA image in Fig.~A1 shows a
complex head-tail source with no clear centre. The Northern limit of
the source is close to a bright galaxy. We tentatively claim this as
the identification, although the separation is 7.25 arcsec from
the poorly defined ``head'' of the radio source and about 35
arcsec from the original position.
\item PKS~0511$-$220: we find a very large difference between our
position for this source (05:11:41.81 $-$22:02:41.2, B1950) and that
quoted in Hewitt \& Burbidge (1993) (05:11:49.94 $-$22:02:44.8). We
attribute this difference to a typographical error made with respect
to the position (05:11:41.94 $-$22:02:44.8) given by Condon, Hicks \&
Jauncey (1977). We are concerned that any published redshifts of this
object may correspond to an object near the wrong position so we do
not quote a redshift for this source pending our own observations.
\item PKS~1008$-$017: (see Fig.~A1) our new position is
about 40 arcsec from the original value.
\item PKS~1118$-$056: this is 60 arcsec away from the original survey
position; we suspect a typographical error in the discovery paper
(Bolton et al.\ 1979).
\item PKS~2335$-$181: in the case of this double source (see
Fig.~A1) with no central component the centroid of the
image was not used to define the position; the North-East component
was adopted instead. This was chosen because of the very good
positional correspondence with a quasar at redshift z=1.45 and also
the fact that no optical counterpart for the South-West component was
detected in the Hubble Space Telescope Snapshot Survey (Maoz et al.\
1993).
\end{enumerate}
\section{Optical Identifications}
\label{sec_optical}
\subsection{Matching to Sky Survey Positions}
A major advance that we present in this paper is the matching of our
accurate radio source positions to the accurate optical data now
available in large digitised sky catalogues based on the U.K. Schmidt
Telescope (UKST) and Palomar sky surveys. A factor contributing to
our successful identifications is the greatly improved agreement
between the radio and optical reference frames in the South (e.g.
Johnston et al.\ 1995).
Our major source of optical data is the COSMOS/UKST Southern Sky
Catalogue. This lists image parameters derived from automated
measurements of the ESO/SERC Southern Sky Survey plates, taken on
IIIa-J emulsion with the GG395 filter to give the photographic blue
passband \Bj\ (3950--5400\AA). The catalogue is described further by
Yentis et al.\ (1992). There are systematic errors in the astrometry
of the COSMOS catalogue: we made a first-order correction as described
by Drinkwater, Barnes \& Ellison (1995) by using the PPM star catalogue
(R\"oser, Bastian \& Kuzmin 1994) to calculate a mean shift in the
positions for each Schmidt field used.
For sources North of +3 degrees we used data from the Automated Plate
Measuring facility (APM; see Irwin, Maddox \& McMahon 1994) at
Cambridge based on blue (unfiltered 103a-O emulsion; 3550--4650\AA)
and red (red plexiglass 2444 filter plus 103a-E emulsion;
6250--6750\AA) plates from the first Palomar Observatory Sky Survey
(POSS~I).
The sky catalogues were used to generate finding charts for all the
sources which we present in Appendix~\ref{sec_charts}. These charts
are a good approximation to the photographic data, but we stress that
there can be problems with image merging in crowded fields: close
objects (e.g. two stars) can be misclassified as a ``merged'' object
or galaxy. The ``Field'' code at the bottom of each
chart indicates the UKST field number (or the plate number for POSS~I)
with a prefix describing the type of plate. The prefix ``J'' indicates
UKST \Bj\ plates measured by COSMOS. For APM data ``j'' indicates UKST
\Bj\ plates, ``O'' blue POSS~I plates and ``E'' red POSS~I plates.
The procedure to find the optical counterpart to each radio source
started with the selection of the nearest optical image in the
catalogues to each radio position. The relative positions of these
nearest-neighbours are shown in Fig.~\ref{fig_scatter}. There is a
clear concentration at small separations (less than 3 arcsec), but
we note that in some cases the nearest-neighbours are at larger
separations (greater than 5 arcsec). We made a preliminary
estimate of the spread in the position offsets by fitting Gaussians to
the distributions in RA and Dec; the rms scatter was found to be
about 0.9 arcsec in each direction. A preliminary cutoff separation of
4 arcsec (about 4$\sigma$) was then imposed.
\begin{figure}
\epsfxsize=\one_wide \epsffile{fig4scat.eps}
\centering
\caption{Distribution of the Position Offsets between each radio
source position and the nearest detected image in the sky catalogues.}
\label{fig_scatter}
\end{figure}
We removed the outliers more distant than 4 arcsec and then
recalculated the distributions of position offsets: these are shown
plotted in Fig.~\ref{fig_histo} as histograms of the offsets between
the two positions in RA and Dec. We estimated the statistical range of
this distribution by measuring the Gaussian dispersions in RA and Dec.
These results are given in Table~\ref{tab_offsets}. This shows that
the core of the distribution has dispersions of only about 0.8
arcsec in each direction. (The same table also shows the final
results with fainter objects matched on CCD frames included.) The
mean differences are small (about 0.2 arcsec) but significant (4 sigma
formally) in both RA and Dec: these indicate that some residual
systematic effects remain, mostly likely due to remaining second-order
errors in the COSMOS astrometry. The important point is that the small
dispersion in both measurements allows us to place very strong limits
on the identification of our sources.
We adopted a maximum difference of $\pm2.5$ arcsec in RA and $\pm2.5$
arcsec in Dec between the radio source position and nearest optical
image, corresponding to a $3\sigma$ confidence level in each
coordinate. We did not remove the small systematic mean offsets before
applying these limits. The maximum total separation among the
objects satisfying these criteria was 2.7 arcsec. In all cases where
the matching criteria were satisfied the image parameters from the
automated catalogues are given in Table~\ref{tab_master}: the
optical$-$radio position offsets in arcsec, the morphological
classification and the catalogue \hbox{$B_J$}\ magnitude.
\begin{figure*}
\epsfxsize=15cm \epsffile{fig5hist.eps}
\centering
\caption{Histograms of the Position Offsets between the Radio and the
Optical sources.}
\label{fig_histo}
\end{figure*}
\begin{table}
\caption{Mean Optical--Radio Position Offsets}
\label{tab_offsets}
\begin{tabular}{lrrrrr}
sample & N & $\overline{\Delta{\rm{RA}}}$ & $\sigma_{RA}$
& $\overline{\Delta{\rm{Dec}}}$ & $\sigma_{Dec}$ \\
& & arcsec & arcsec & arcsec & arcsec \\
\\
sky survey matches & 290 & -0.17 & 0.82 & -0.21 & 0.81 \\
all matches & 320 & -0.16 & 0.83 & -0.18 & 0.82 \\
\\
\end{tabular}
Note: each offset is measured in the sense optical$-$radio and
PKS~0406$-$311 is not included.
\end{table}
The morphological classifications are based on how extended the
optical images are and define the images as galaxies (g), stellar (s),
or too-faint-to-classify (f). In the case of the APM data there is a
further category of merged images (m) where 2 or more images are too
close to separate. We intentionally do not include in
Table~\ref{tab_master} the object classifications from PKSCAT90
because there is evidence that the distinction between ``galaxies'' and
``quasars'' was not applied uniformly over the whole survey (see
Drinkwater \& Schmidt 1996).
The calibration accuracy of the \hbox{$B_J$}\ photographic magnitudes from the
COSMOS catalogue is quoted as being about $\pm0.5$ magnitudes (H.
MacGillivray, private communication). We have found that some fields
lack any calibration data and some seem to be incorrect by more than
one magnitude, so the catalogue magnitudes should be treated with
caution. We specifically checked the calibration of any fields in
which the COSMOS magnitude differed by more than 2 mag from a value
published in the literature by comparison with data from adjacent
COSMOS fields and corrected any large errors. A histogram of the
magnitudes is given in Fig.~\ref{fig_mags}.
\begin{figure}
\epsfxsize=\one_wide \epsffile{fig6mags.eps}
\centering
\caption{Histogram of the optical \hbox{$B_J$}\ magnitudes of all sources identified
on the photographic sky surveys.}
\label{fig_mags}
\end{figure}
There is a further problem of objects where 2 or 3 close optical
images have been merged into a single catalogue object whose centroid
position is still within our $\pm2.5$ arcsec matching criteria.
These are easy to find because the resulting ``merged'' image is very
extended and thus mis-classified as a galaxy. This is a problem
inherent in automated catalogues for which reason the classifications
should always be checked. We visually inspected all the objects
classified as ``galaxies'' to check for merging. A total of 10 such
objects were found in the matched list; they are noted in
Table~\ref{tab_master} as ``(merge)''. We derived corrected image
parameters for these merged objects by analysing images from the
Digitised Sky Survey or CCD images (at other wavelengths, see next
section). If the image data was obtained from CCD data, no
\hbox{$B_J$}\ magnitude is given for the object in the table.
One additional object was included in the matched list although its
position difference was greater than the $\pm2.5$ arcsec
limits. This was PKS~0406$-$311 which we identified with a galaxy 7
arcsec from the nominal radio position because the head-tail radio
structure did not give an accurate position but is very indicative of
this type of galaxy (see above). With this galaxy and the merged
objects included, a total of 291 sources from our sample of 323 have
confirmed matches to objects in the optical catalogues.
This leaves a total of 32
sources with no matching image in the optical catalogues. We
undertook the identification of these sources using CCD imaging at
other wavelengths as described in the next section.
\subsection{Identifications at Other Wavelengths}
This section describes the methods we used to find optical
counterparts for the 32 sources not matched to images listed in the
optical catalogues. We first inspected all the fields visually on the
optical sky survey plates. Most of the sources (25) were found to be
genuine ``empty fields'' in the sense that no optical counterpart was
visible on the survey plates. In the remaining cases (7) however a
counterpart was clearly visible on the plate, but it was too faint to
be included in the automated catalogue or it had been merged with a
neighbouring object.
We note that six of these unmatched sources were assigned optical
identifications in PKSCAT90; our new accurate positions show that these
need to be revised. In three cases (PKS~1349$-$145, PKS~1450$-$338,
PKS~2127$-$096) there is a faint matching object but it is merged with a
brighter image and in the other cases there is no optical counterpart
at all at the correct position (PKS~0005$-$262, PKS~1601$-$222,
PKS~2056$-$369).
To identify the sources unmatched on the sky surveys we turned to longer
wavelengths, using optical \hbox{$R$}-band, \hbox{$I$}-band, and
infrared \hbox{$Kn$}-band (2.0--2.3
microns) imaging on the
3.9m Anglo-Australian Telescope
(AAT) and the
Australian National University (ANU) 2.3m Telescope. These data were
analysed using the {\footnotesize IRAF}\footnote{IRAF is distributed
by the National Optical Astronomy Observatories, which are operated by
the Association of Universities for Research in Astronomy, Inc. (AURA)
under cooperative agreement with the National Science Foundation.}
analysis software. The observations resulted in identifications of 30
of the remaining sources including the merged objects, one of which
was separated using a \hbox{$B$}-band image. These sources are listed in
Table~\ref{tab_master} in the same way as the sources identified from
the digitised survey data, except that no \Bj\ magnitude is given and
the position offsets are estimated from the CCD frames. The source of
the identifications is indicated in the comment column as ``(R)'' or
``(K)''. We will present a full analysis of the \hbox{$R$}- and \hbox{$Kn$}-band
data in later papers.
The 2 sources we did not identify include PKS~1213$-$172 which lies
too close to a bright star to be identified in our data but Stickel et
al.\ (1994) report having identified it with a ``$m=21.4$ mag resolved
galaxy''. The remaining source, PKS~0320$+$015 was not detected in a
\hbox{$Kn$}\ image (approximate limit of \hbox{$Kn$}=18) but we anticipate identifying
it when a deeper exposure is available.
We note that PKS~2149+056 which we detected in our \hbox{$Kn$}\ image
was previously detected and identified as a
quasar with a measured redshift by Stickel \& K\"uhr (1993).
\subsection{Reliability of Identifications}
For the majority of the matched sources for which spectroscopic redshifts
have been measured we are confident of having made the correct optical
identification. For the remaining sources for which we have not yet obtained
redshifts, the identifications must be made on positional coincidence alone.
A very detailed analysis of the statistics of source identifications was
made by Sutherland \& Saunders (1992) in the context of matching IRAS sources
with poor positions to the optical sky survey data. Our problem is much simpler
because both our source (radio) and survey (optical) positions are accurate.
Furthermore, we do not wish to include the image magnitudes in the analysis
because we do not know the true distribution of optical magnitudes---a large
fraction of the sources without spectroscopic confirmation are at the faint
limit of the magnitude distribution.
We made an estimate of the number of ``identifications'' in our sample
that might just be coincidences by calculating the mean surface density
of images in the sky survey catalogues at the plate limit and finding
how many of these would lie within the match criteria. For the 46
fields without spectroscopic confirmation we would expect 1 random
matches within a radius of 3 arcsec. In fact most objects lie
within 2 arcsec: at this separation we would only get 0.4 random
matches. It is therefore possible that one of the identifications we
claim without spectroscopic confirmation is wrong: ideally only the
sources with spectroscopic identifications should be used for analysis
purposes.
\section{Spectroscopic Identifications}
\label{sec_spec}
\subsection{Previous Results}
Earlier versions of the flat-spectrum sample have been the subject of
extensive campaigns of spectroscopic follow-up observations. Some two
thirds of the sample were identified in the summary made by Savage
et al.\ (1990) and we have drawn on this work for the current sample.
We carried out a very detailed literature review to find published
redshifts for as much of the sample as possible. We based our search
on the quasar catalogue compiled by Hewitt \& Burbidge (1993) with
additional material from the V\'eron-Cetty \& V\'eron (1993) quasar
catalogue, the Center for Astrophysics Redshift Catalog (Version of
May 28, 1994; see Huchra et al.\ 1992), the NASA/IPAC Extragalactic
Database (NED, Helou et al.\ 1991), and the Lyon-Meudon Extragalactic
Database (LEDA). There are occasional errors in some of these large
compilations, so for every redshift found in the catalogues we checked
the reference cited and only accepted values for which we found a
measured redshift in the original reference. We present these
redshifts in Table~\ref{tab_master} along with a code that specifies
the source of the measurement. For some objects the source reference
indicated that the redshift was uncertain (e.g. due to a single line
or a lower limit derived from the redshift of an absorption system);
in these cases the reference code is prefaced by a ``$-$'' sign. For
some additional objects we found no published original reference (some
were given by private communications): these are assigned a reference
code of zero and we have not listed the redshift in our table.
After our critical search of the literature we accepted published
redshifts for 206 sources in our sample of 323 sources. At the same
time we searched for published spectra of any sources in our sample;
references to these are also given in Table~\ref{tab_master}. Again,
we only include those spectra we have checked in the original
references.
\subsection{New Measurements}
As a result of the identifications presented in this paper we started
a campaign of new spectroscopic identifications. This has resulted in
114 new spectra and 90 new redshift measurements which we present
here. The journal of observations and the new redshifts are given in
Table~\ref{spectral_id} and we present the spectra in
Appendix~\ref{sec_spectra}. Notes on some individual spectra are given
in Section~\ref{sec_snotes} below. Note that three sources are
presented (``EXTRAS'' in Table~\ref{spectral_id}) that are not in our
final sample. These were part of an earlier version of the sample and
are included here to provide a published reference to their
redshifts. Details of our observations are as follows.
\begin{table*}
\caption{New Spectral Identifications}
\label{spectral_id}
\begin{tabular}{lrrrllrrrl}
name & tel & date & $z_{em}$ & comment &name & tel & date & $z_{em}$ & comment \\
\\
PKS~0036$-$216&AAT&1995 Sep 22&none & & PKS~1143$-$245&ANU&1995 May 25&1.940 & \\
PKS~0048$-$097&AAT&1994 Dec 02&none & & PKS~1144$-$379&AAT&1996 Apr 21&1.047 &(87) \\
PKS~0104$-$408&AAT&1984 Jun 30&none &(105)& PKS~1156$-$094&AAT&1996 Apr 20&none & \\
PKS~0114$+$074&AAT&1995 Sep 22&0.343 ¬e & PKS~1228$-$113&AAT&1996 Apr 21&3.528 & \\
PKS~0118$-$272&AAT&1994 Dec 04&$>$0.556& & PKS~1237$-$101&ANU&1995 May 25&0.751 & \\
PKS~0131$-$001&AAT&1994 Dec 03&0.879 & & PKS~1250$-$330&AAT&1996 Apr 20&none & \\
PKS~0138$-$097&ANU&1995 Sep 28&none &(90) & PKS~1256$-$229&AAT&1995 Mar 05&1.365 & \\
PKS~0153$-$410&AAT&1994 Dec 04&0.226 & & PKS~1258$-$321&AAT&1988 May 10&0.017 &(18) \\
PKS~0213$-$026&AAT&1994 Dec 04&1.178 & & PKS~1317$+$019&AAT&1996 Apr 21&1.232 & \\
PKS~0216$+$011&AAT&1994 Dec 03&1.61 & & PKS~1318$-$263&AAT&1995 Mar 05&2.027 & \\
PKS~0220$-$349&AAT&1994 Dec 04&1.49 & & PKS~1333$-$082&AAT&1988 May 10&0.023 &(26) \\
PKS~0221$+$067&AAT&1986 Aug 09&0.510 & & PKS~1336$-$260&AAT&1995 Mar 05&1.51 ¬e \\
PKS~0229$-$398&AAT&1994 Dec 04&1.646? & & PKS~1340$-$175&AAT&1996 Apr 20&1.50? &1 line \\
PKS~0256$+$075&AAT&1994 Dec 03&0.895 & & PKS~1354$-$174&AAT&1995 Mar 06&3.137 & \\
PKS~0301$-$243&AAT&1995 Sep 22&none & & PKS~1359$-$281&AAT&1984 May 01&0.803 & \\
PKS~0327$-$241&AAT&1994 Dec 04&0.888 & & PKS~1404$-$267&AAT&1988 May 10&0.022 &(21) \\
PKS~0332$-$403&ANU&1995 Sep 27&none & & PKS~1406$-$267&AAT&1996 Apr 20&2.43 & \\
PKS~0336$-$017&AAT&1987 Sep 17&3.202 & & PKS~1430$-$155&AAT&1996 Apr 21&1.573 & \\
PKS~0346$-$163&ANU&1995 Sep 28&none & & PKS~1435$-$218&ANU&1996 Feb 25&1.187 & \\
PKS~0346$-$279&AAT&1986 Aug 09&0.987 & & PKS~1445$-$161&AAT&1984 May 01&2.417 & \\
PKS~0357$-$264&AAT&1995 Sep 22&1.47? & & PKS~1450$-$338&AAT&1996 Apr 20&0.368 & \\
PKS~0400$-$319&AAT&1994 Dec 03&1.288 & & PKS~1456$+$044&AAT&1988 May 11&0.394 & \\
PKS~0405$-$331&AAT&1987 Sep 17&2.562 & & PKS~1511$-$210&AAT&1994 Apr 30&1.179 & \\
PKS~0406$-$311&ANU&1995 Sep 27&0.0565 & & PKS~1518$+$045&ANU&1995 May 25&0.052 & \\
PKS~0422$+$004&ANU&1995 Sep 28&none & & PKS~1519$-$273&ANU&1996 Apr 11&none & \\
PKS~0423$+$051&AAT&1994 Dec 02&1.333 & & PKS~1535$+$004&AAT&1996 Apr 21&3.497 & \\
PKS~0454$+$066&ANU&1995 Sep 28&0.4050 & & PKS~1615$+$029&AAT&1996 Apr 21&1.341 &(110) \\
PKS~0456$+$060&AAT&1995 Mar 05&none & & PKS~1616$+$063&AAT&1996 Apr 21&2.088 &(3) \\
PKS~0459$+$060&AAT&1994 Dec 03&1.106 & & PKS~1635$-$035&AAT&1988 May 11&2.856? & \\
PKS~0502$+$049&AAT&1995 Mar 05&0.954 & & PKS~1648$+$015&AAT&1996 Apr 20&none ¬e \\
PKS~0508$-$220&ANU&1995 Nov 29&0.1715 & & PKS~1654$-$020&AAT&1996 Apr 20&2.00 & \\
PKS~0532$-$378&AAT&1995 Mar 05&1.668 & & PKS~1706$+$006&AAT&1994 Sep 09&0.449 & \\
PKS~0829$+$046&AAT&1994 Dec 02&none & & PKS~1933$-$400&ANU&1995 May 25&0.965 & \\
PKS~0837$+$035&AAT&1995 Mar 05&1.57 & & PKS~1958$-$179&AAT&1996 Apr 21&0.652 &(10) \\
PKS~0859$-$140&AAT&1996 Apr 21&1.337 &(84) & PKS~2004$-$447&AAT&1984 May 02&0.240 & \\
PKS~0907$-$023&AAT&1995 Mar 05&0.957 &(110)& PKS~2021$-$330&AAT&1996 Apr 21&1.471 &(98) note \\
PKS~0912$+$029&AAT&1988 May 11&0.427 & & PKS~2022$-$077&AAT&1988 May 10&1.388 & \\
PKS~0922$+$005&AAT&1995 Mar 05&1.717 & & PKS~2056$-$369&AAT&1995 Jul 06&none & \\
PKS~1008$-$017&ANU&1996 Apr 10&0.887 ¬e & PKS~2058$-$135&AAT&1988 May 10&0.0291 &(21) \\
PKS~1016$-$311&AAT&1988 May 10&0.794 & & PKS~2058$-$297&AAT&1984 May 02&1.492 & \\
PKS~1020$-$103&ANU&1996 Apr 26&0.1966 &(112)& PKS~2059$+$034&AAT&1996 Apr 21&1.012 &(110) \\
PKS~1021$-$006&ANU&1996 Apr 26&2.549 &(110)& PKS~2120$+$099&AAT&1987 Sep 17&0.932 & \\
PKS~1036$-$154&AAT&1995 Mar 05&0.525 & & PKS~2127$-$096&AAT&1995 Jul 06&$>$0.780&$>$0.733 \\
PKS~1038$+$064&ANU&1996 Apr 26&1.264 &(84) & PKS~2128$-$123&AAT&1996 Apr 21&0.499 &(95) \\
PKS~1048$-$313&AAT&1995 May 31&1.429 & & PKS~2131$-$021&ANU&1995 Jun 01&1.285 ¬e \\
PKS~1055$-$243&AAT&1995 Mar 05&1.086 & & PKS~2143$-$156&ANU&1995 May 25&0.698 & \\
PKS~1102$-$242&AAT&1984 May 01&1.666 & & PKS~2145$-$176&AAT&1987 Sep 17&2.130 & \\
PKS~1106$+$023&AAT&1988 May 10&0.157 & & PKS~2215$+$020&AAT&1986 Aug 10&3.572 & \\
PKS~1107$-$187&AAT&1995 Mar 05&0.497 & & PKS~2229$-$172&AAT&1995 Jul 06&1.780 & \\
PKS~1110$-$217&AAT&1996 Apr 20&none & & PKS~2233$-$148&AAT&1995 Jul 06&$>$0.609& \\
PKS~1115$-$122&AAT&1988 May 10&1.739 & & PKS~2252$-$090&AAT&1996 Jul 19&0.6064 & \\
PKS~1118$-$056&AAT&1988 May 11&1.297? & & PKS~2254$-$367&AAT&1988 May 11&0.0055 &(21) \\
PKS~1124$-$186&ANU&1996 Apr 26&1.048 ¬e & PKS~2312$-$319&ANU&1995 Sep 28&1.323 &$>$1.0453 \\
PKS~1127$-$145&AAT&1996 Apr 21&1.187 &(107)& PKS~2329$-$415&AAT&1987 Sep 17&0.671 & \\
PKS~1128$-$047&AAT&1984 May 01&0.266 & & PKS~2335$-$181&AAT&1987 Sep 17&1.450 & \\
PKS~1133$-$172&AAT&1994 Apr 30&1.024 & & \multicolumn{4}{c}{EXTRAS} \\
PKS~1136$-$135&ANU&1996 Apr 26&0.5566 &(107)& PKS~0114$+$074b&AAT&1995 Sep 22&0.858 ¬e \\
PKS~1142$+$052&AAT&1986 Apr 11&1.342 &(105)& PKS~0215$+$015&AAT&1994 Dec 02&1.718 ¬e \\
PKS~1142$-$225&AAT&1996 Apr 21&1.141 & & PKS~1557$+$032&AAT&1995 Mar 5 &3.88 ¬e \\
& \\
\end{tabular}
\\
Notes: 1. Specific notes on individual spectra are given in
Section~\ref{sec_snotes}. 2. Redshifts of any absorption systems
identified in the spectra are prefaced by ``$>$'' as these give a
lower limit to the source redshift. 3. Reference numbers in
parentheses refer to previous published redshift estimates. 4. The
final three sources observed are not in our sample but are included
here in order to provide a published reference to their redshifts.
\end{table*}
Our identification of most of the ``Empty Field'' sources in our \hbox{$Kn$}\
and \hbox{$R$}\ band imaging enabled us to attempt spectroscopic
identifications of these very faint sources. We made these observations
using the AAT equipped with the RGO
spectrograph (grating 250B: a resolution of 5\AA\ in the blue) and the
faint object red spectrograph (FORS: a resolution of 20\AA\ in the
red). We also used the AAT to observe a number of brighter objects
with unconfirmed redshifts.
We made an extensive search of the AAT archive for observations of
sources in our sample with no published redshifts: this provided
27 measurements.
A number of the brighter objects were observed with the ANU 2.3m
Telescope using the double beam spectrograph (with a resolution of
8\AA\ in both the blue and red arms).
All these spectra were analysed with the IRAF package and any new
redshifts we obtained are included in Table~\ref{tab_master} with the
reference code ``121''.
Combining this new data with the published redshifts we now have
confirmed redshifts for 277 or 86\% of the sample and possible
redshifts for a further 10. This represents a significant improvement
over the last major compilation of this sample (Savage et al.\ 1990)
when only 67\% of the redshifts were measured, and not all of them
published. A histogram of all the redshifts is given in
Fig.~\ref{fig_reds}.
\begin{figure}
\epsfxsize=\one_wide \epsffile{fig7reds.eps}
\centering
\caption{Redshift histogram for the sample: the dotted line indicates
just the optically resolved sources (galaxies). The redshift histogram
of a large optically selected sample, the Large Bright QSO Survey is
shown in the lower panel for comparison.}
\label{fig_reds}
\end{figure}
\subsection{Notes on Individual Spectra}
\label{sec_snotes}
\begin{enumerate}
\item PKS~0114$+$074b: this is not part of the sample, but is close to
PKS~0114$+$074 and was the source of the previously quoted redshift
(see Section~\ref{sec_notes}).
\item PKS~0215$+$015: this is not part of the sample,
but was measured as part of a preliminary version of the sample and is
included here for reference.
\item PKS~1008$-$017: also observed with the AAT, 1988 May 11;
the combined spectrum was used.
\item PKS~1124$-$186: has weak lines but they were also observed on
the AAT 1984 May 01.
\item PKS~1336$-$260: also observed with the AAT, 1996 Apr 20;
combined spectrum used.
\item PKS~1557$+$032: this is not part of the sample,
but was measured as part of a preliminary version of the sample and is
included here for reference.
\item PKS~1648$+$015: also observed with the AAT, 1995 Jun 01; combined
spectrum used.
\item PKS~2021$-$330: possible broad absorption line structure near CIV.
\item PKS~2131$-$021: also observed with the ANU 2.3m Telescope, 1995 Sep
28; combined spectrum used. The redshift is based on OII and MgII in
our spectra and a reported ``definite'' line at 3541\AA\ (Baldwin et
al.\ 1989) which we identify with CIV.
\end{enumerate}
\section{The Catalogue}
\label{sec_cat}
We present all the data we have collected on our sample in
Table~\ref{tab_master}. We indicate the source of all published data
in the table by a reference number; the references are listed in
numerical order at the end of the paper. In all cases a minus sign in
front of the reference number indicates an uncertain value. The
specific reference numbers 120 and 121 refer to new data we present in
this paper: 120 to accurate radio positions measured with the ATCA and
121 to all our other data including the VLA radio positions.
The columns in the table are as follows:
\begin{enumerate}
\item name: the Parkes source name.
\item $S_{2.7}, S_{5.0}, \alpha$, Rf: the 2.7 and 5.0{\rm\thinspace GHz}\ source fluxes and
corresponding spectral index as published in reference Rf
(see Table~\ref{radio_surveys}).
\item RA(B1950), Dec(B1950), Rc: the accurate B1950
(i.e. equinox B1950 and epoch B1950)
radio source positions from reference Rc.
\item comment: (1) a brief description of the radio morphology if the
source is resolved using the terminology of Downes et al.\ (1986):
``P'' for partially resolved sources, ``Do'' for double sources with
no central component with the position defined by the centroid of the
source, ``Do+CC'' for double sources with a central component or peak
giving a well-defined position, ``H'' for a diffuse halo around a
central source, and ``HT'' for a complex head-tail morphology. (2)
comments in parentheses refer to the optical identification. In cases
where there was no match to the sky catalogues but the source was
identified using CCD data, these are indicated as ``(B)'', ``(R)'',
``(I)'' and ''(K)'' for the respective wavebands. If the CCD imaging
did not identify the source, the comment ``null'' is made and
``STR'' indicates a source too near a bright star. If the source was
confused with a close neighbour in the sky catalogues, but separated
by a CCD image the comment ``merge'' is made followed by the waveband
used; in some cases the Digitized Sky Survey data was used to separate
the object (``DSS'').
\item $\Delta RA, \Delta Dec, \Delta r$, cl, \hbox{$B_J$}: the position offsets
(arcsec, in the sense optical$-$radio) of the corresponding optical
image (if any), the total separation, the image classification (``g''
galaxy, ``s'' stellar, ``f'' too faint to classify, ``m'' merged) and
the apparent \hbox{$B_J$}\ magnitude if the counterpart was found in the sky
survey data.
\item z, Rz, Rsp: the emission redshift of the source obtained from
reference code Rz. If no emission redshift has been measured, but
absorption lines have been identified these are used to place a lower
limit on the source redshift indicated in the form ``$>$0.500''. If a
spectrum has been published it can be found in reference Rsp.
\item RA(J2000), Dec(J2000): the corresponding
J2000 positions.
\end{enumerate}
\section{Overview of the Sample}
\label{sec_corr}
We defer a detailed analysis of the sample to other papers, but we take
this opportunity to make a brief overview of the sample.
The sources with measured redshifts span the redshift range 0.05--3.78
with a median redshift of 1.07 (see Fig.~\ref{fig_reds}). The
redshift histogram is smooth, and broadly similar to that of optical
surveys such as the Large Bright QSO Survey (LBQS; see Hewett et al.\
1995, and references therein). The distribution of our sample is
compared to that of the LBQS in Fig.~\ref{fig_reds}. The lack of LBQS
quasars in the lowest redshift bin is due to the absolute magnitude
and redshift cut-off of that survey. The 2-sample Kolmogorov-Smirnov
test (comparing sources with redshift $z>0.22$ in both samples: 235
Parkes and 1018 LBQS) gives results consistent with the two samples
having the same redshift distribution at the 35\% probability level.
This makes the two samples ideal for comparing the properties of
radio- and optically-selected quasars.
The distribution of optical \hbox{$B_J$}\ magnitudes of our sources (see Figs.
\ref{fig_mags} and \ref{fig_mags2}) shows that despite the fact that
our sample was not selected on the basis of
optical magnitude, the sources occupy
a restricted range of magnitudes. The majority have $\hbox{$B_J$} = 18 \pm
3$. The well-defined mode in the distribution of \hbox{$B_J$}\ magnitudes is
not an artifact of the plate limit of $\hbox{$B_J$} \approx 22.5$;
the number of sources with $22 > \hbox{$B_J$} > 20$ is clearly below that
with $20 > \hbox{$B_J$} > 18$. However a small fraction of sources are clearly
very faint in \hbox{$B_J$}.
In common with Browne \& Wright (1985) we find that the modal \hbox{$B_J$}\
magnitude of our flat-spectrum sample is a function of radio flux; the
most radio-bright sources have slightly brighter typical \hbox{$B_J$}\
magnitudes (Fig.~\ref{fig_mags2}). This is also shown in
Fig.~\ref{fig_opt_rad}, a plot of the \hbox{$B_J$}\ magnitudes against the
2.7{\rm\thinspace GHz}\ radio fluxes.
\begin{figure}
\epsfxsize=\one_wide \epsffile{fig8mag2.eps}
\centering
\caption{The distribution of \hbox{$B_J$}\ magnitudes in the sample as a
function of 2.7{\rm\thinspace GHz}\ radio fluxes.}
\label{fig_mags2}
\end{figure}
There is some suggestion from our data that the radio-to-optical ratio
may be a physically meaningful parameter, as originally suggested by
Schmidt (1970). In Fig.~\ref{fig_ratio} we plot radio-to-optical
ratios $R$ as a function of radio luminosity, for different classes of
source. A clear correlation can be seen, with the lowest
radio-luminosity sources having low values of $R$. If, however, we
exclude galaxies, the correlation disappears, and no sources remain
with $R<100$.
The sources without redshifts have radio-to-optical ratios above those
of the detected sources, which may reflect dust
obscuration of the \hbox{$B_J$}\ emission (Webster et al.\ 1995).
\begin{table*}
\vspace{-0.6cm}
\rotate{
\begin{tabular}{rrrrrrrrrlrrrrrlrrrr}
N& name& $S_{2.7}$& $S_{5.0}$& $\alpha$& Rf& RA(B1950)& Dec(B1950)& Rc& comment& $\Delta$RA& $\Delta$Dec& $\Delta$r& cl& $B_J$& z& Rz& Rsp& RA(J2000)& Dec(J2000)\\
\\
1& PKS0003$-$066& 1.46& 1.58& 0.13& 103
& 0:03:40.29& $-$6:40:17.3& 56& & 1.15
& $-$0.22& 1.17& s& 18.47& ~~0.347& 87& 87
& 0:06:13.89& $-$6:23:35.2\\
2& PKS0005$-$239& 0.58& 0.53& $-$0.15& 103
& 0:05:27.47& $-$23:56:00.0& 121& & 0.04
& $-$0.31& 0.31& s& 16.61& ~~1.407& 114& 108
& 0:08:00.37& $-$23:39:18.0\\
3& PKS0005$-$262& 0.58& 0.58& 0.00& 103
& 0:05:53.51& $-$26:15:53.4& 121& (K)& 0.39
& $-$0.02& 0.39& s& 0.00& ~~0.0& 0& 0
& 0:08:26.25& $-$25:59:11.5\\
4& PKS0008$-$264& 0.67& 0.81& 0.31& 103
& 0:08:28.89& $-$26:29:14.8& 57& & 1.17
& 0.42& 1.24& s& 19.46& ~~1.096& 117& 0
& 0:11:01.25& $-$26:12:33.3\\
5& PKS0013$-$005& 0.89& 0.79& $-$0.19& 102
& 0:13:37.35& $-$0:31:52.5& 57& & $-$0.75
& $-$0.21& 0.78& s& 19.41& ~~1.574& 105& 25
& 0:16:11.08& $-$0:15:12.3\\
6& PKS0036$-$216& 0.53& 0.60& 0.20& 103
& 0:36:00.44& $-$21:36:33.1& 57& & 0.28
& $-$1.29& 1.31& g& 21.05& ~~0.0& 0& 121
& 0:38:29.95& $-$21:20:03.9\\
7& PKS0038$-$020& 0.61& 0.79& 0.42& 102
& 0:38:24.23& $-$2:02:59.3& 56& & $-$0.03
& $-$1.21& 1.21& s& 18.80& ~~1.178& 110& 0
& 0:40:57.61& $-$1:46:32.0\\
8& PKS0048$-$097& 1.44& 1.92& 0.47& 103
& 0:48:09.98& $-$9:45:24.3& 56& & 0.48
& 0.44& 0.65& s& 16.75& ~~0.0& 0& 121
& 0:50:41.32& $-$9:29:05.2\\
9& PKS0048$-$071& 0.70& 0.67& $-$0.07& 103
& 0:48:36.20& $-$7:06:20.5& 57& & $-$0.87
& $-$1.92& 2.11& f& 22.06& ~~1.974& 107& 108
& 0:51:08.20& $-$6:50:01.8\\
10& PKS0048$-$427& 0.68& 0.58& $-$0.26& 6
& 0:48:49.02& $-$42:42:51.8& 121& & 0.35
& 1.16& 1.21& s& 19.98& ~~1.749& 105& 0
& 0:51:09.49& $-$42:26:33.0\\
\\
11& PKS0056$-$001& 1.80& 1.38& $-$0.43& 102
& 0:56:31.76& $-$0:09:18.8& 56& & $-$0.59
& $-$0.60& 0.84& s& 17.79& ~~0.717& 44& 4
& 0:59:05.51& 0:06:51.8\\
12& PKS0104$-$408& 0.57& 0.85& 0.65& 6
& 1:04:27.58& $-$40:50:21.2& 56& & $-$1.59
& $-$0.61& 1.70& s& 18.92& ~~0.584& 105& 121
& 1:06:45.11& $-$40:34:19.5\\
13& PKS0106+013& 1.88& 2.82& 0.66& 102
& 1:06:04.52& 1:19:01.1& 56& & 0.08
& 0.06& 0.10& s& 18.82& ~~2.094& 110& 4
& 1:08:38.77& 1:35:00.4\\
14& PKS0108$-$079& 1.02& 0.89& $-$0.22& 103
& 1:08:19.00& $-$7:57:37.6& 57& & 1.11
& 0.15& 1.12& s& 18.47& ~~1.773& 112& 108
& 1:10:50.01& $-$7:41:41.1\\
15& PKS0111+021& 0.61& 0.67& 0.15& 102
& 1:11:08.57& 2:06:24.8& 56& & 0.05
& 1.26& 1.26& g& 16.42& ~~0.047& 112& 0
& 1:13:43.14& 2:22:17.4\\
16& PKS0112$-$017& 1.38& 1.60& 0.24& 102
& 1:12:43.92& $-$1:42:55.0& 56& & $-$1.15
& $-$1.06& 1.57& s& 17.85& ~~1.381& 4& 4
& 1:15:17.09& $-$1:27:04.5\\
17& PKS0113$-$118& 1.78& 1.88& 0.09& 103
& 1:13:43.22& $-$11:52:04.5& 56& & $-$0.28
& $-$1.85& 1.87& s& 19.42& ~~0.672& 91& 108
& 1:16:12.52& $-$11:36:15.3\\
18& PKS0114+074& 0.90& 0.67& $-$0.48& $-$79
& 1:14:50.48& 7:26:00.3& 121& Do& $-$0.45
& $-$1.50& 1.57& g& 22.14& ~~0.343& 121& 121
& 1:17:27.13& 7:41:47.7\\
19& PKS0116+082& 1.50& 1.11& $-$0.49& $-$79
& 1:16:24.24& 8:14:10.0& 97& P& 0.00
& $-$1.30& 1.30& s& 21.85& ~~0.594& 81& 81
& 1:19:01.27& 8:29:55.2\\
20& PKS0116$-$219& 0.57& 0.51& $-$0.18& 103
& 1:16:32.40& $-$21:57:15.2& 56& & $-$1.20
& $-$2.01& 2.34& s& 19.64& ~~1.161& 107& 108
& 1:18:57.25& $-$21:41:29.9\\
\\
21& PKS0118$-$272& 0.96& 1.18& 0.33& 103
& 1:18:09.53& $-$27:17:07.4& 56& & $-$0.03
& $-$0.93& 0.93& s& 17.47& $>$0.556& $-$121& 121
& 1:20:31.66& $-$27:01:24.4\\
22& PKS0119+041& 1.83& 2.01& 0.15& 79
& 1:19:21.39& 4:06:44.0& 56& & $-$0.15
& 0.20& 0.25& s& 19.18& ~~0.637& 63& 0
& 1:21:56.86& 4:22:24.8\\
23& PKS0122$-$003& 1.43& 1.24& $-$0.23& 102
& 1:22:55.18& $-$0:21:31.3& 56& & $-$0.39
& $-$0.15& 0.42& s& 16.49& ~~1.08& 10& 0
& 1:25:28.84& $-$0:05:55.9\\
24& PKS0130$-$171& 0.99& 0.97& $-$0.03& 103
& 1:30:17.66& $-$17:10:11.3& 120& & 0.24
& $-$1.53& 1.55& s& 17.65& ~~1.022& 107& 108
& 1:32:43.45& $-$16:54:47.8\\
25& PKS0130$-$447& 0.59& 0.49& $-$0.30& 6
& 1:30:52.91& $-$44:46:05.1& 120& (K)& 0.05
& $-$1.31& 1.31& f& 0.00& ~~0.0& 0& 0
& 1:33:00.33& $-$44:30:50.9\\
26& PKS0131$-$001& 0.68& 0.50& $-$0.50& 102
& 1:31:38.98& $-$0:11:35.9& 55& (K)& $-$1.38
& 0.31& 1.41& f& 0.00& ~~0.879& 121& 121
& 1:34:12.71& 0:03:45.1\\
27& PKS0133$-$204& 0.68& 0.63& $-$0.12& 103
& 1:33:13.59& $-$20:24:04.6& 39& & $-$0.81
& $-$0.55& 0.98& s& 18.18& ~~1.141& 107& 108
& 1:35:37.46& $-$20:08:46.1\\
28& PKS0135$-$247& 1.37& 1.65& 0.30& 103
& 1:35:17.12& $-$24:46:08.2& 120& & 0.26
& $-$0.96& 0.99& s& 18.93& ~~0.829& 107& 108
& 1:37:38.35& $-$24:30:53.2\\
29& PKS0137+012& 1.07& 0.82& $-$0.43& 102
& 1:37:22.87& 1:16:35.4& 43& & $-$0.36
& $-$0.07& 0.37& g& 19.44& ~~0.260& 44& 2
& 1:39:57.34& 1:31:45.8\\
30& PKS0138$-$097& 0.71& 1.19& 0.84& 103
& 1:38:56.86& $-$9:43:51.8& 45& & $-$0.36
& $-$0.20& 0.42& s& 18.50& $>$0.501& $-$90& 121
& 1:41:25.83& $-$9:28:43.7\\
\\
31& PKS0142$-$278& 0.82& 0.90& 0.15& 103
& 1:42:45.01& $-$27:48:34.7& 120& & $-$0.84
& 1.35& 1.59& s& 17.47& ~~1.153& 107& 108
& 1:45:03.41& $-$27:33:33.5\\
32& PKS0146+056& 0.72& 0.73& 0.02& 79
& 1:46:45.53& 5:41:00.8& 56& & $-$0.60
& 0.50& 0.78& s& 20.67& ~~2.345& 75& 0
& 1:49:22.37& 5:55:53.7\\
33& PKS0150$-$334& 0.92& 0.86& $-$0.11& 78
& 1:50:56.99& $-$33:25:10.7& 56& & $-$0.50
& $-$0.39& 0.64& s& 17.38& ~~0.610& 114& 108
& 1:53:10.13& $-$33:10:25.7\\
34& PKS0153$-$410& 1.22& 0.94& $-$0.42& 6
& 1:53:31.07& $-$41:03:22.2& 120& & 2.40
& 0.50& 2.45& g& 19.41& ~~0.226& 121& 121
& 1:55:37.04& $-$40:48:42.3\\
35& PKS0202$-$172& 1.40& 1.38& $-$0.02& 103
& 2:02:34.52& $-$17:15:39.4& 56& & 1.48
& $-$1.69& 2.25& s& 18.21& ~~1.74& 117& 0
& 2:04:57.68& $-$17:01:19.7\\
36& PKS0213$-$026& 0.50& 0.57& 0.21& 102
& 2:13:09.87& $-$2:36:51.5& 57& (R)& 0.33
& 1.87& 1.90& f& 0.00& ~~1.178& 121& 121
& 2:15:42.01& $-$2:22:56.7\\
37& PKS0216+011& 0.50& 0.64& 0.40& 102
& 2:16:32.46& 1:07:13.4& 56& & $-$0.60
& 0.47& 0.77& f& 21.82& ~~1.61& 121& 121
& 2:19:07.03& 1:20:59.8\\
38& PKS0220$-$349& 0.60& 0.61& 0.03& 78
& 2:20:49.61& $-$34:55:05.2& 40& & $-$1.07
& $-$0.13& 1.08& f& 21.50& ~~1.49& 121& 121
& 2:22:56.40& $-$34:41:28.7\\
39& PKS0221+067& 0.79& 0.77& $-$0.04& 79
& 2:21:49.96& 6:45:50.4& 56& & $-$0.58
& 0.10& 0.59& s& 20.76& ~~0.510& 121& 121
& 2:24:28.42& 6:59:23.4\\
40& PKS0226$-$038& 0.66& 0.55& $-$0.30& 102
& 2:26:21.96& $-$3:50:58.6& 120& & 2.12
& 0.30& 2.14& s& 17.59& ~~2.0660& 94& 84
& 2:28:53.09& $-$3:37:37.1\\
\\
\end{tabular}}
\caption{Master Source Catalogue}
\label{tab_master}
\end{table*}
\clearpage
\begin{table*}
\vspace{-0.6cm}
\rotate{
\begin{tabular}{rrrrrrrrrlrrrrrlrrrr}
N& name& $S_{2.7}$& $S_{5.0}$& $\alpha$& Rf& RA(B1950)& Dec(B1950)& Rc& comment& $\Delta$RA& $\Delta$Dec& $\Delta$r& cl& $B_J$& z& Rz& Rsp& RA(J2000)& Dec(J2000)\\
\\
41& PKS0229$-$398& 0.64& 0.68& 0.10& 6
& 2:29:51.99& $-$39:49:00.2& 39& (mrg R) & $-$0.69
& 0.86& 1.10& m& 22.28& ~~1.646& $-$121& 121
& 2:31:51.79& $-$39:35:47.1\\
42& PKS0232$-$042& 0.84& 0.62& $-$0.49& 103
& 2:32:36.51& $-$4:15:08.9& 39& & 0.62
& $-$1.15& 1.31& s& 16.21& ~~1.437& 84& 84
& 2:35:07.25& $-$4:02:04.1\\
43& PKS0237+040& 0.73& 0.77& 0.09& 79
& 2:37:14.41& 4:03:29.7& 56& & $-$0.74
& 0.20& 0.77& s& 18.48& ~~0.978& 75& 0
& 2:39:51.26& 4:16:21.6\\
44& PKS0238$-$084& 0.58& 1.40& 1.43& 103
& 2:38:37.36& $-$8:28:09.0& 56& & $-$0.34
& 0.53& 0.63& g& 11.73& ~~0.005& 67& 0
& 2:41:04.80& $-$8:15:20.7\\
45& PKS0240$-$217& 0.97& 0.82& $-$0.27& 103
& 2:40:19.23& $-$21:45:11.6& 120& & 0.99
& 1.01& 1.42& g& 19.05& ~~0.314& 117& 108
& 2:42:35.80& $-$21:32:27.8\\
46& PKS0240$-$060& 0.53& 0.52& $-$0.03& 103
& 2:40:43.24& $-$6:03:37.5& 121& & 0.18
& $-$0.86& 0.88& s& 18.20& ~~1.800& 3& 0
& 2:43:12.46& $-$5:50:55.2\\
47& PKS0256+075& 0.69& 0.98& 0.57& 79
& 2:56:46.99& 7:35:45.2& 57& & $-$0.44
& $-$0.40& 0.60& s& 18.89& ~~0.895& 121& 121
& 2:59:27.07& 7:47:39.7\\
48& PKS0301$-$243& 0.52& 0.39& $-$0.47& 103
& 3:01:14.22& $-$24:18:52.6& 121& H& 0.67
& $-$0.50& 0.84& s& 16.43& ~~0.0& 0& 121
& 3:03:26.50& $-$24:07:11.0\\
49& PKS0316$-$444& 0.82& 0.62& $-$0.45& 6
& 3:16:13.40& $-$44:25:09.2& 121& & 0.61
& $-$2.09& 2.18& g& 14.87& ~~0.076& 48& 0
& 3:17:57.68& $-$44:14:15.1\\
50& PKS0320+015& 0.52& 0.42& $-$0.35& 102
& 3:20:34.61& 1:35:12.7& 121& (K null) & 0.00
& 0.00& 0.00& $-$& 0.00& ~~0.0& 0& 0
& 3:23:09.86& 1:45:50.7\\
\\
51& PKS0327$-$241& 0.63& 0.73& 0.24& 103
& 3:27:43.87& $-$24:07:22.9& 57& & $-$1.56
& 0.07& 1.56& s& 19.39& ~~0.888& 121& 121
& 3:29:54.06& $-$23:57:08.5\\
52& PKS0332+078& 0.74& 0.81& 0.15& 79
& 3:32:12.10& 7:50:16.7& 56& (R) & 0.13
& $-$0.42& 0.44& f& 0.00& ~~0.0& 0& 0
& 3:34:53.31& 8:00:14.5\\
53& PKS0332$-$403& 1.96& 2.60& 0.46& 6
& 3:32:25.23& $-$40:18:24.0& 57& & 0.28
& $-$0.29& 0.40& s& 16.80& ~~0.0& 0& 121
& 3:34:13.64& $-$40:08:25.3\\
54& PKS0336$-$017& 0.58& 0.46& $-$0.38& 102
& 3:36:28.79& $-$1:43:00.4& 121& & $-$0.81
& $-$1.09& 1.36& s& 20.14& ~~3.202& 121& 121
& 3:39:00.98& $-$1:33:17.5\\
55& PKS0336$-$019& 2.23& 2.30& 0.05& 102
& 3:36:58.95& $-$1:56:16.9& 56& & $-$0.74
& $-$1.01& 1.25& s& 18.43& ~~0.850& 110& 0
& 3:39:30.93& $-$1:46:35.7\\
56& PKS0338$-$214& 0.82& 0.94& 0.22& 103
& 3:38:23.28& $-$21:29:07.9& 56& & 0.74
& $-$0.17& 0.76& s& 16.03& ~~0.048& 114& 108
& 3:40:35.60& $-$21:19:31.1\\
57& PKS0346$-$163& 0.54& 0.55& 0.03& 103
& 3:46:21.83& $-$16:19:25.5& 121& & 0.61
& $-$0.74& 0.95& s& 17.41& ~~0.0& 0& 121
& 3:48:39.27& $-$16:10:17.7\\
58& PKS0346$-$279& 1.10& 0.96& $-$0.22& 103
& 3:46:34.03& $-$27:58:20.7& 57& & 0.48
& 0.96& 1.07& s& 20.49& ~~0.987& 121& 121
& 3:48:38.14& $-$27:49:13.2\\
59& PKS0348+049& 0.54& 0.41& $-$0.45& 79
& 3:48:15.50& 4:57:21.2& 121& (R) & $-$1.37
& 0.93& 1.66& f& 0.00& ~~0.0& 0& 0
& 3:50:54.20& 5:06:21.3\\
60& PKS0348$-$120& 0.50& 0.54& 0.12& 103
& 3:48:49.16& $-$12:02:21.1& 121& & $-$1.16
& $-$1.91& 2.23& s& 17.87& ~~1.520& 117& 0
& 3:51:10.96& $-$11:53:22.5\\
\\
61& PKS0349$-$278& 2.89& 2.21& $-$0.44& 103
& 3:49:31.81& $-$27:53:31.5& 121& H& $-$0.47
& 1.44& 1.51& g& 16.77& ~~0.0662& 49& 0
& 3:51:35.77& $-$27:44:34.9\\
62& PKS0357$-$264& 0.58& 0.47& $-$0.34& 103
& 3:57:28.46& $-$26:23:57.9& 121& & $-$0.58
& $-$0.56& 0.81& s& 21.76& ~~1.47& $-$121& 121
& 3:59:33.67& $-$26:15:31.0\\
63& PKS0400$-$319& 1.14& 1.03& $-$0.16& 78
& 4:00:23.61& $-$31:55:41.9& 121& & $-$0.66
& $-$1.63& 1.75& s& 20.21& ~~1.288& 121& 121
& 4:02:21.27& $-$31:47:25.8\\
64& PKS0402$-$362& 1.04& 1.39& 0.47& 6
& 4:02:02.60& $-$36:13:11.8& 56& & $-$0.01
& $-$0.61& 0.61& s& 17.03& ~~1.417& 58& 108
& 4:03:53.75& $-$36:05:01.8\\
65& PKS0403$-$132& 3.15& 3.24& 0.05& 103
& 4:03:13.98& $-$13:16:18.1& 57& & 0.63
& $-$0.17& 0.65& s& 16.78& ~~0.571& 117& 0
& 4:05:34.00& $-$13:08:13.6\\
66& PKS0405$-$385& 1.02& 1.06& 0.06& 6
& 4:05:12.07& $-$38:34:24.7& 57& & $-$0.83
& $-$0.16& 0.84& g& 19.76& ~~1.285& 98& 98
& 4:06:59.08& $-$38:26:26.7\\
67& PKS0405$-$123& 2.35& 1.81& $-$0.42& 103
& 4:05:27.46& $-$12:19:32.5& 57& & 1.84
& $-$1.24& 2.22& s& 14.45& ~~0.574& 93& 93
& 4:07:48.42& $-$12:11:36.6\\
68& PKS0405$-$331& 0.70& 0.63& $-$0.17& 78
& 4:05:38.55& $-$33:11:42.0& 57& & 0.60
& $-$0.03& 0.61& s& 19.41& ~~2.562& 121& 121
& 4:07:33.91& $-$33:03:45.9\\
69& PKS0406$-$311& 0.55& 0.55& 0.00& 78
& 4:06:28.10& $-$31:08:00.0& $-$121& HT& $-$5.59
& $-$4.52& 7.19& g& 15.99& ~~0.0565& 121& 121
& 4:08:26.34& $-$31:00:07.2\\
70& PKS0406$-$127& 0.59& 0.61& 0.05& 103
& 4:06:45.33& $-$12:46:39.0& 57& & 1.03
& 0.01& 1.03& s& 17.99& ~~1.563& 116& 108
& 4:09:05.77& $-$12:38:48.1\\
\\
71& PKS0407$-$170& 0.55& 0.41& $-$0.48& 103
& 4:07:21.64& $-$17:03:24.1& 121& (K) & 0.11
& $-$1.18& 1.19& f& 0.00& ~~0.0& 0& 0
& 4:09:37.33& $-$16:55:35.4\\
72& PKS0413$-$210& 1.79& 1.36& $-$0.45& 103
& 4:13:53.62& $-$21:03:51.1& 57& & $-$0.27
& 0.16& 0.31& s& 18.64& ~~0.808& 107& 108
& 4:16:04.36& $-$20:56:27.6\\
73& PKS0414$-$189& 1.18& 1.31& 0.17& 103
& 4:14:23.35& $-$18:58:29.7& 56& & $-$0.04
& 0.07& 0.09& s& 19.35& ~~1.536& 34& 108
& 4:16:36.54& $-$18:51:08.3\\
74& PKS0420$-$014& 1.92& 2.14& 0.18& 102
& 4:20:43.54& $-$1:27:28.7& 121& & $-$0.56
& $-$1.56& 1.65& s& 17.38& ~~0.914& 110& 109
& 4:23:15.80& $-$1:20:33.0\\
75& PKS0421+019& 0.76& 0.72& $-$0.09& 102
& 4:21:32.67& 1:57:32.7& 56& & 0.37
& 0.25& 0.45& s& 17.10& ~~2.0548& 94& 84
& 4:24:08.56& 2:04:25.0\\
76& PKS0422+004& 1.25& 1.55& 0.35& 102
& 4:22:12.52& 0:29:16.7& 57& & $-$0.26
& $-$0.49& 0.56& s& 16.19& ~~0.0& 0& 121
& 4:24:46.84& 0:36:06.4\\
77& PKS0423$-$163& 0.55& 0.44& $-$0.36& 103
& 4:23:37.66& $-$16:19:25.1& 55& (K) & 0.62
& $-$0.23& 0.66& f& 0.00& ~~0.0& 0& 0
& 4:25:53.56& $-$16:12:40.4\\
78& PKS0423+051& 0.61& 0.68& 0.18& 79
& 4:23:57.23& 5:11:37.3& 57& & $-$0.29
& $-$0.40& 0.49& g& 19.23& ~~1.333& 121& 121
& 4:26:36.59& 5:18:19.8\\
79& PKS0426$-$380& 1.04& 1.14& 0.15& 6
& 4:26:54.71& $-$38:02:52.1& 56& & 1.00
& $-$0.13& 1.01& s& 18.37& $>$1.030& $-$90& 90
& 4:28:40.42& $-$37:56:19.5\\
80& PKS0430+052& 3.30& 3.78& 0.22& $-$79
& 4:30:31.60& 5:14:59.5& 56& & 0.74
& $-$0.60& 0.95& g& 12.85& ~~0.033& 66& 0
& 4:33:11.09& 5:21:15.5\\
\\
\end{tabular}
}
\contcaption{Master Source Catalogue}
\end{table*}
\clearpage
\begin{table*}
\vspace{-0.6cm}
\rotate{
\begin{tabular}{rrrrrrrrrlrrrrrlrrrr}
N& name& $S_{2.7}$& $S_{5.0}$& $\alpha$& Rf& RA(B1950)& Dec(B1950)& Rc& comment& $\Delta$RA& $\Delta$Dec& $\Delta$r& cl& $B_J$& z& Rz& Rsp& RA(J2000)& Dec(J2000)\\
\\
81& PKS0434$-$188& 1.05& 1.19& 0.20& 103
& 4:34:48.97& $-$18:50:48.2& 56& & $-$0.46
& 0.06& 0.46& s& 18.70& ~~2.705& 107& 108
& 4:37:01.48& $-$18:44:48.6\\
82& PKS0438$-$436& 6.50& 7.00& 0.12& 6
& 4:38:43.18& $-$43:38:53.1& 56& & $-$0.05
& $-$0.21& 0.22& s& 19.08& ~~2.852& 52& 52
& 4:40:17.17& $-$43:33:08.1\\
83& PKS0440$-$003& 3.53& 3.13& $-$0.20& 102
& 4:40:05.29& $-$0:23:20.6& 56& & $-$0.31
& $-$0.60& 0.67& s& 18.21& ~~0.844& 75& 4
& 4:42:38.66& $-$0:17:43.4\\
84& PKS0445+097& 0.68& 0.56& $-$0.32& 79
& 4:45:36.99& 9:45:37.2& 121& & 1.02
& 0.40& 1.10& g& 20.22& ~~2.115& 94& 5
& 4:48:21.70& 9:50:51.1\\
85& PKS0448$-$392& 0.89& 0.89& 0.00& 6
& 4:48:00.45& $-$39:16:15.7& 39& & 0.51
& $-$0.18& 0.54& s& 16.76& ~~1.288& 114& 108
& 4:49:42.24& $-$39:11:09.4\\
86& PKS0451$-$282& 2.38& 2.50& 0.08& 103
& 4:51:15.13& $-$28:12:29.3& 56& & 0.84
& 0.05& 0.84& s& 17.75& ~~2.5637& 8& 8
& 4:53:14.64& $-$28:07:37.2\\
87& PKS0454+066& 0.50& 0.44& $-$0.21& $-$79
& 4:54:26.41& 6:40:30.1& 56& & $-$0.91
& $-$0.90& 1.28& s& 19.79& ~~0.4050& 121& 121
& 4:57:07.71& 6:45:07.3\\
88& PKS0454$-$234& 1.76& 2.00& 0.21& 103
& 4:54:57.29& $-$23:29:28.7& 56& & $-$0.11
& 0.55& 0.56& s& 18.16& ~~1.003& 87& 87
& 4:57:03.16& $-$23:24:52.4\\
89& PKS0456+060& 0.78& 0.58& $-$0.48& 79
& 4:56:08.15& 6:03:33.9& 121& (K) & $-$1.25
& $-$1.26& 1.78& f& 0.00& ~~0.0& 0& 121
& 4:58:48.76& 6:08:04.0\\
90& PKS0457+024& 1.63& 1.47& $-$0.17& 102
& 4:57:15.54& 2:25:05.6& 56& & $-$1.92
& 1.08& 2.20& s& 18.24& ~~2.382& 110& 4
& 4:59:52.04& 2:29:31.1\\
\\
91& PKS0458$-$020& 1.99& 1.76& $-$0.20& 102
& 4:58:41.35& $-$2:03:33.9& 57& & 1.15
& $-$2.23& 2.51& s& 19.12& ~~2.310& 3& 4
& 5:01:12.81& $-$1:59:14.3\\
92& PKS0459+060& 0.99& 0.78& $-$0.39& 79
& 4:59:34.78& 6:04:52.0& 121& & $-$0.16
& $-$1.10& 1.11& s& 19.68& ~~1.106& 121& 121
& 5:02:15.44& 6:09:07.5\\
93& PKS0500+019& 2.47& 1.85& $-$0.47& 102
& 5:00:45.18& 1:58:53.8& 56& (K) & $-$1.45
& 0.52& 1.54& f& 0.00& ~~0.0& 0& 0
& 5:03:21.20& 2:03:04.5\\
94& PKS0502+049& 0.59& 0.82& 0.53& 79
& 5:02:43.81& 4:55:40.6& 57& & $-$0.31
& 0.10& 0.32& s& 18.70& ~~0.954& 121& 121
& 5:05:23.18& 4:59:42.8\\
95& PKS0508$-$220& 0.90& 0.68& $-$0.45& $-$7
& 5:08:53.20& $-$22:05:32.5& 120& & $-$0.38
& $-$0.10& 0.39& g& 16.89& ~~0.1715& 121& 121
& 5:11:00.50& $-$22:01:55.3\\
96& PKS0511$-$220& 1.21& 1.27& 0.08& 7
& 5:11:41.82& $-$22:02:41.2& 56& & $-$1.37
& $-$0.69& 1.54& g& 20.24& ~~0.0& 0& 108
& 5:13:49.11& $-$21:59:16.0\\
97& PKS0514$-$161& 0.80& 0.76& $-$0.08& 7
& 5:14:01.08& $-$16:06:22.6& 56& & $-$1.25
& $-$1.39& 1.87& s& 16.85& ~~1.278& 114& 84
& 5:16:15.93& $-$16:03:07.6\\
98& PKS0521$-$365& 12.50& 9.23& $-$0.49& 6
& 5:21:12.99& $-$36:30:16.0& 121& Do+CC& 1.09
& $-$0.18& 1.10& g& 16.74& ~~0.0552& 96& 96
& 5:22:57.99& $-$36:27:30.9\\
99& PKS0528$-$250& 1.32& 1.13& $-$0.25& 7
& 5:28:05.21& $-$25:05:44.6& 57& & $-$0.18
& $-$0.21& 0.28& s& 17.73& ~~2.765& 80& 80
& 5:30:07.96& $-$25:03:29.8\\
100& PKS0532$-$378& 0.70& 0.59& $-$0.28& 6
& 5:32:35.25& $-$37:49:21.8& 121& & $-$0.14
& 1.22& 1.23& s& 21.37& ~~1.668& 121& 121
& 5:34:17.49& $-$37:47:25.8\\
\\
101& PKS0533$-$120& 0.80& 0.64& $-$0.36& $-$7
& 5:33:13.75& $-$12:04:14.2& 55& & $-$0.77
& $-$1.56& 1.74& g& 18.64& ~~0.1573& 15& 0
& 5:35:33.31& $-$12:02:22.4\\
102& PKS0537$-$158& 0.63& 0.61& $-$0.05& 7
& 5:37:17.18& $-$15:52:05.1& 39& & $-$1.08
& $-$0.27& 1.11& s& 16.54& ~~0.947& 116& 108
& 5:39:32.03& $-$15:50:30.8\\
103& PKS0537$-$441& 3.84& 3.80& $-$0.02& 6
& 5:37:21.00& $-$44:06:46.8& 56& & 0.66
& 1.55& 1.68& s& 15.45& ~~0.893& 107& 108
& 5:38:50.28& $-$44:05:11.1\\
104& PKS0537$-$286& 0.74& 0.99& 0.47& 7
& 5:37:56.93& $-$28:41:28.0& 56& & $-$0.53
& 0.11& 0.54& s& 19.29& ~~3.11& 115& 115
& 5:39:54.27& $-$28:39:55.9\\
105& PKS0622$-$441& 0.77& 0.89& 0.24& 6
& 6:22:02.68& $-$44:11:23.0& 39& (mrgDSS& 0.00
& $-$0.15& 0.15& m& 18.59& ~~0.688& 114& 108
& 6:23:31.74& $-$44:13:02.4\\
106& PKS0629$-$418& 0.53& 0.74& 0.54& 6
& 6:29:37.72& $-$41:52:15.7& 121& & $-$1.11
& 0.21& 1.13& s& 18.07& ~~1.416& 38& 0
& 6:31:12.05& $-$41:54:28.3\\
107& PKS0823+033& 0.87& 1.13& 0.42& 102
& 8:23:13.54& 3:19:15.3& 56& (mrg R)& 0.02
& 0.86& 0.86& m& 0.00& ~~0.506& 90& 90
& 8:25:50.33& 3:09:24.3\\
108& PKS0829+046& 0.62& 0.70& 0.20& 79
& 8:29:10.89& 4:39:50.8& 57& & $-$0.17
& $-$0.50& 0.53& s& 16.03& ~~0.0& 0& 121
& 8:31:48.87& 4:29:39.0\\
109& PKS0837+035& 0.69& 0.59& $-$0.25& 102
& 8:37:12.37& 3:30:32.8& 57& & $-$0.73
& 0.60& 0.95& s& 20.40& ~~1.57& 121& 121
& 8:39:49.19& 3:19:53.6\\
110& PKS0859$-$140& 2.93& 2.29& $-$0.40& $-$7
& 8:59:54.95& $-$14:03:38.9& 56& & $-$0.52
& 0.30& 0.60& s& 16.33& ~~1.337& 121& 121
& 9:02:16.83& $-$14:15:31.0\\
\\
111& PKS0906+015& 1.20& 1.04& $-$0.23& 102
& 9:06:35.19& 1:33:48.0& 56& & 0.81
& $-$0.43& 0.92& s& 17.17& ~~1.018& 14& 108
& 9:09:10.10& 1:21:35.4\\
112& PKS0907$-$023& 0.57& 0.42& $-$0.50& 102
& 9:07:13.13& $-$2:19:16.4& 39& (mrgDSS& 0.21
& 0.38& 0.43& m& 19.11& ~~0.957& 110& 121
& 9:09:44.95& $-$2:31:30.8\\
113& PKS0912+029& 0.54& 0.46& $-$0.26& 102
& 9:12:01.95& 2:58:27.7& 121& & $-$0.01
& $-$0.20& 0.20& s& 19.56& ~~0.427& 121& 121
& 9:14:37.92& 2:45:59.0\\
114& PKS0921$-$213& 0.53& 0.42& $-$0.38& 7
& 9:21:21.82& $-$21:22:52.3& 121& Do+CC& $-$0.64
& $-$0.32& 0.72& g& 16.40& ~~0.052& 59& 0
& 9:23:38.87& $-$21:35:47.1\\
115& PKS0922+005& 0.74& 0.72& $-$0.04& 102
& 9:22:33.76& 0:32:12.2& 56& & $-$0.03
& $-$0.05& 0.06& s& 17.26& ~~1.717& 121& 121
& 9:25:07.82& 0:19:13.7\\
116& PKS0925$-$203& 0.81& 0.70& $-$0.24& 7
& 9:25:33.52& $-$20:21:44.7& 39& Do+CC & $-$0.21
& $-$0.42& 0.47& s& 16.35& ~~0.348& 59& 108
& 9:27:51.79& $-$20:34:51.1\\
117& PKS1004$-$018& 0.56& 0.60& 0.11& 102
& 10:04:31.71& $-$1:52:30.9& 56& & $-$0.03
& 0.77& 0.77& s& 20.33& ~~1.212& 110& 4
& 10:07:04.34& $-$2:07:11.3\\
118& PKS1008$-$017& 0.80& 0.61& $-$0.44& 102
& 10:08:18.94& $-$1:45:31.6& 121& P& $-$0.34
& 0.53& 0.63& s& 19.63& ~~0.887& 121& 121
& 10:10:51.67& $-$2:00:19.8\\
119& PKS1016$-$311& 0.62& 0.65& 0.08& 78
& 10:16:12.60& $-$31:08:51.0& 121& & $-$0.71
& 0.09& 0.72& s& 17.58& ~~0.794& 121& 121
& 10:18:28.77& $-$31:23:54.5\\
120& PKS1020$-$103& 0.64& 0.49& $-$0.43& 8
& 10:20:04.18& $-$10:22:33.4& 39& & 0.22
& $-$0.38& 0.44& s& 15.07& ~~0.1966& 121& 121
& 10:22:32.76& $-$10:37:44.2\\
\\
\end{tabular}
}
\contcaption{Master Source Catalogue}
\end{table*}
\clearpage
\begin{table*}
\vspace{-0.6cm}
\rotate{
\begin{tabular}{rrrrrrrrrlrrrrrlrrrr}
N& name& $S_{2.7}$& $S_{5.0}$& $\alpha$& Rf& RA(B1950)& Dec(B1950)& Rc& comment& $\Delta$RA& $\Delta$Dec& $\Delta$r& cl& $B_J$& z& Rz& Rsp& RA(J2000)& Dec(J2000)\\
\\
121& PKS1021$-$006& 0.95& 0.75& $-$0.38& 102
& 10:21:56.19& $-$0:37:41.6& 121& & $-$0.09
& 0.23& 0.25& s& 17.90& ~~2.549& 121& 121
& 10:24:29.58& $-$0:52:55.8\\
122& PKS1032$-$199& 1.10& 1.15& 0.07& 70
& 10:32:37.37& $-$19:56:02.2& 39& & $-$0.45
& 0.15& 0.47& s& 18.33& ~~2.189& 107& 108
& 10:35:02.16& $-$20:11:34.5\\
123& PKS1034$-$293& 1.33& 1.51& 0.21& 70
& 10:34:55.83& $-$29:18:27.0& 56& & $-$0.62
& 0.76& 0.99& s& 15.94& ~~0.312& 87& 108
& 10:37:16.08& $-$29:34:03.0\\
124& PKS1036$-$154& 0.75& 0.78& 0.06& 70
& 10:36:39.48& $-$15:25:28.1& 56& & $-$0.54
& 0.42& 0.69& s& 21.80& ~~0.525& 121& 121
& 10:39:06.71& $-$15:41:06.8\\
125& PKS1038+064& 1.74& 1.40& $-$0.35& $-$79
& 10:38:40.88& 6:25:58.3& 121& (mrgDSS& $-$0.15
& $-$0.22& 0.27& m& 16.10& ~~1.264& 121& 121
& 10:41:17.15& 6:10:16.6\\
126& PKS1042+071& 0.50& 0.50& 0.00& 79
& 10:42:19.46& 7:11:25.0& 121& & $-$1.03
& 0.00& 1.03& s& 18.55& ~~0.698& 105& 0
& 10:44:55.92& 6:55:37.9\\
127& PKS1045$-$188& 0.94& 1.11& 0.27& 70
& 10:45:40.09& $-$18:53:44.1& 57& & $-$1.02
& 0.00& 1.02& s& 18.42& ~~0.595& 53& 0
& 10:48:06.61& $-$19:09:35.9\\
128& PKS1048$-$313& 0.80& 0.73& $-$0.15& 78
& 10:48:43.38& $-$31:22:18.5& 121& & $-$0.32
& $-$0.15& 0.35& s& 18.49& ~~1.429& 121& 121
& 10:51:04.80& $-$31:38:14.4\\
129& PKS1055$-$243& 0.77& 0.61& $-$0.38& 70
& 10:55:29.94& $-$24:17:44.6& 56& & $-$0.66
& $-$0.02& 0.66& s& 19.90& ~~1.086& 121& 121
& 10:57:55.41& $-$24:33:49.0\\
130& PKS1055+018& 3.02& 3.07& 0.03& 102
& 10:55:55.32& 1:50:03.5& 56& & 0.04
& 0.27& 0.27& s& 18.47& ~~0.888& 110& 4
& 10:58:29.61& 1:33:58.7\\
\\
131& PKS1101$-$325& 0.93& 0.73& $-$0.39& 78
& 11:01:08.51& $-$32:35:06.2& 121& Do+CC& $-$0.99
& 1.53& 1.82& s& 16.45& ~~0.3554& 47& 108
& 11:03:31.57& $-$32:51:17.0\\
132& PKS1102$-$242& 0.50& 0.57& 0.21& 70
& 11:02:19.82& $-$24:15:13.6& 39& & $-$0.11
& $-$0.15& 0.18& s& 20.61& ~~1.666& 121& 121
& 11:04:46.18& $-$24:31:25.7\\
133& PKS1106+023& 0.64& 0.50& $-$0.40& 102
& 11:06:11.19& 2:18:56.2& 39& Do& 0.58
& $-$0.64& 0.87& g& 18.01& ~~0.157& 121& 121
& 11:08:45.52& 2:02:40.2\\
134& PKS1107$-$187& 0.65& 0.50& $-$0.43& 70
& 11:07:31.75& $-$18:42:31.8& 120& (K) & $-$1.07
& 1.18& 1.59& f& 0.00& ~~0.497& 121& 121
& 11:10:00.45& $-$18:58:49.2\\
135& PKS1110$-$217& 0.94& 0.76& $-$0.34& 70
& 11:10:21.67& $-$21:42:08.7& 39& (I) & 1.09
& 0.19& 1.11& f& 0.00& ~~0.0& 0& 121
& 11:12:49.81& $-$21:58:28.8\\
136& PKS1115$-$122& 0.67& 0.63& $-$0.10& 8
& 11:15:46.13& $-$12:16:29.5& 121& & $-$0.45
& $-$1.67& 1.73& s& 18.15& ~~1.739& 121& 121
& 11:18:17.14& $-$12:32:54.2\\
137& PKS1118$-$056& 0.66& 0.57& $-$0.24& 8
& 11:18:52.51& $-$5:37:29.1& 121& & $-$0.60
& $-$0.31& 0.67& s& 18.99& ~~1.297& $-$121& 121
& 11:21:25.10& $-$5:53:56.2\\
138& PKS1124$-$186& 0.61& 0.84& 0.52& 70
& 11:24:34.02& $-$18:40:46.4& 57& & $-$0.82
& $-$0.06& 0.82& s& 18.65& ~~1.048& 121& 121
& 11:27:04.39& $-$18:57:17.6\\
139& PKS1127$-$145& 5.97& 5.46& $-$0.14& 8
& 11:27:35.67& $-$14:32:54.4& 56& & 0.72
& $-$0.01& 0.72& s& 16.95& ~~1.187& 121& 121
& 11:30:07.05& $-$14:49:27.5\\
140& PKS1128$-$047& 0.74& 0.90& 0.32& 8
& 11:28:57.50& $-$4:43:46.1& 56& & $-$2.35
& 0.65& 2.43& f& 21.41& ~~0.266& 121& 121
& 11:31:30.52& $-$5:00:19.9\\
\\
141& PKS1133$-$172& 0.65& 0.52& $-$0.36& 70
& 11:33:31.60& $-$17:16:36.6& 121& & 0.12
& $-$1.89& 1.89& f& 22.43& ~~1.024& 121& 121
& 11:36:03.05& $-$17:33:12.9\\
142& PKS1136$-$135& 2.76& 2.22& $-$0.35& 8
& 11:36:38.43& $-$13:34:06.0& 121& Do& 1.00
& 0.86& 1.31& s& 16.30& ~~0.5566& 121& 121
& 11:39:10.62& $-$13:50:43.7\\
143& PKS1142+052& 0.60& 0.46& $-$0.43& 79
& 11:42:47.16& 5:12:06.7& 121& & $-$0.88
& $-$0.60& 1.06& s& 19.79& ~~1.342& 105& 121
& 11:45:21.33& 4:55:26.9\\
144& PKS1142$-$225& 0.54& 0.63& 0.25& 70
& 11:42:50.23& $-$22:33:51.8& 39& (mrg R)& 0.80
& 1.42& 1.63& f& 0.00& ~~1.141& 121& 121
& 11:45:22.05& $-$22:50:31.8\\
145& PKS1143$-$245& 1.32& 1.18& $-$0.18& 70
& 11:43:36.37& $-$24:30:52.9& 56& & $-$0.08
& 0.54& 0.54& s& 17.66& ~~1.940& 121& 121
& 11:46:08.10& $-$24:47:33.1\\
146& PKS1144$-$379& 1.07& 2.22& 1.18& 6
& 11:44:30.87& $-$37:55:30.6& 56& & 0.19
& 1.25& 1.27& s& 18.43& ~~1.047& 121& 121
& 11:47:01.38& $-$38:12:11.1\\
147& PKS1145$-$071& 1.09& 1.21& 0.17& 8
& 11:45:18.29& $-$7:08:00.7& 121& (mrgDSS& 1.44
& $-$0.91& 1.70& m& 19.03& ~~1.342& 107& 108
& 11:47:51.55& $-$7:24:41.3\\
148& PKS1148$-$001& 2.56& 1.95& $-$0.44& 102
& 11:48:10.13& $-$0:07:13.2& 57& & $-$0.11
& $-$0.82& 0.83& s& 17.13& ~~1.9803& 94& 119
& 11:50:43.87& $-$0:23:54.4\\
149& PKS1148$-$171& 0.60& 0.50& $-$0.30& 70
& 11:48:30.38& $-$17:07:18.7& 121& & 0.21
& $-$2.35& 2.36& s& 17.91& ~~1.751& 116& 108
& 11:51:03.21& $-$17:24:00.0\\
150& PKS1156$-$221& 0.71& 0.78& 0.15& 70
& 11:56:37.79& $-$22:11:54.9& 57& & $-$1.34
& $-$0.04& 1.35& s& 18.63& ~~0.565& 116& 108
& 11:59:11.29& $-$22:28:37.2\\
\\
151& PKS1156$-$094& 0.75& 0.66& $-$0.21& 8
& 11:56:39.06& $-$9:24:10.0& 121& & $-$0.85
& 0.44& 0.96& f& 22.57& ~~0.0& 0& 121
& 11:59:12.71& $-$9:40:52.2\\
152& PKS1200$-$051& 0.50& 0.46& $-$0.14& 8
& 12:00:00.44& $-$5:11:20.6& 121& & $-$0.43
& 0.53& 0.68& s& 16.42& ~~0.381& 116& 108
& 12:02:34.23& $-$5:28:02.8\\
153& PKS1202$-$262& 1.34& 0.99& $-$0.49& 70
& 12:02:58.82& $-$26:17:22.6& 39& Do& $-$0.23
& 0.14& 0.27& s& 19.76& ~~0.789& 107& 108
& 12:05:33.19& $-$26:34:04.8\\
154& PKS1206$-$399& 0.59& 0.53& $-$0.17& 6
& 12:06:59.46& $-$39:59:31.3& 39& & $-$0.67
& $-$0.18& 0.69& s& 17.20& ~~0.966& 36& 108
& 12:09:35.25& $-$40:16:13.1\\
155& PKS1213$-$172& 1.33& 1.28& $-$0.06& 70
& 12:13:11.67& $-$17:15:05.3& 56& (K STR)& 0.00
& 0.00& 0.00& $-$& 0.00& ~~0.0& 0& 0
& 12:15:46.75& $-$17:31:45.6\\
156& PKS1218$-$024& 0.54& 0.47& $-$0.23& 102
& 12:18:49.90& $-$2:25:12.0& 121& H& 0.14
& $-$0.11& 0.18& s& 20.25& ~~0.665& 105& 0
& 12:21:23.92& $-$2:41:50.4\\
157& PKS1222+037& 0.81& 0.86& 0.10& 102
& 12:22:19.10& 3:47:27.1& 56& & $-$0.17
& 0.00& 0.17& s& 19.28& ~~0.957& 110& 4
& 12:24:52.42& 3:30:50.2\\
158& PKS1226+023& 43.40& 40.00& $-$0.13& 102
& 12:26:33.25& 2:19:43.3& 56& & 1.41
& $-$0.41& 1.47& s& 12.93& ~~0.158& 2& 2
& 12:29:06.70& 2:03:08.5\\
159& PKS1228$-$113& 0.55& 0.46& $-$0.29& 8
& 12:28:20.06& $-$11:22:36.0& 121& & $-$0.31
& 0.45& 0.54& f& 22.01& ~~3.528& 121& 121
& 12:30:55.57& $-$11:39:09.9\\
160& PKS1229$-$021& 1.33& 1.05& $-$0.38& 102
& 12:29:25.88& $-$2:07:32.1& 121& Do+CC& 1.48
& $-$1.49& 2.10& s& 16.62& ~~1.045& 30& 108
& 12:31:59.99& $-$2:24:05.4\\
\\
\end{tabular}
}
\contcaption{Master Source Catalogue}
\end{table*}
\clearpage
\begin{table*}
\vspace{-0.6cm}
\rotate{
\begin{tabular}{rrrrrrrrrlrrrrrlrrrr}
N& name& $S_{2.7}$& $S_{5.0}$& $\alpha$& Rf& RA(B1950)& Dec(B1950)& Rc& comment& $\Delta$RA& $\Delta$Dec& $\Delta$r& cl& $B_J$& z& Rz& Rsp& RA(J2000)& Dec(J2000)\\
\\
161& PKS1236+077& 0.59& 0.67& 0.21& 79
& 12:36:52.31& 7:46:45.4& 56& & $-$0.46
& $-$0.80& 0.92& s& 19.14& ~~0.40& 105& 108
& 12:39:24.58& 7:30:17.1\\
162& PKS1237$-$101& 1.35& 1.13& $-$0.29& 8
& 12:37:07.29& $-$10:07:00.7& 57& & $-$0.47
& 0.34& 0.58& s& 17.46& ~~0.751& 121& 121
& 12:39:43.07& $-$10:23:28.9\\
163& PKS1243$-$072& 0.79& 1.11& 0.55& 8
& 12:43:28.79& $-$7:14:23.5& 57& & 0.07
& 0.30& 0.31& s& 17.64& ~~1.286& 107& 108
& 12:46:04.23& $-$7:30:46.7\\
164& PKS1244$-$255& 1.34& 1.55& 0.24& 70
& 12:44:06.71& $-$25:31:26.7& 57& & $-$0.80
& $-$0.12& 0.81& s& 16.17& ~~0.638& 107& 108
& 12:46:46.79& $-$25:47:49.4\\
165& PKS1250$-$330& 0.52& 0.49& $-$0.10& 78
& 12:50:14.95& $-$33:03:41.9& 121& P& $-$0.99
& $-$0.03& 0.99& s& 21.42& ~~0.0& 0& 121
& 12:52:58.47& $-$33:19:59.0\\
166& PKS1253$-$055& 12.00& 13.00& 0.13& 8
& 12:53:35.84& $-$5:31:08.0& 56& & $-$0.32
& $-$0.07& 0.33& s& 17.67& ~~0.540& 13& 0
& 12:56:11.17& $-$5:47:21.7\\
167& PKS1254$-$333& 0.72& 0.54& $-$0.47& 78
& 12:54:36.28& $-$33:18:33.6& 120& Do& $-$0.62
& 2.21& 2.29& s& 17.05& ~~0.190& 107& 108
& 12:57:20.71& $-$33:34:46.3\\
168& PKS1255$-$316& 1.49& 1.68& 0.19& 78
& 12:55:15.18& $-$31:39:05.0& 56& & 0.01
& $-$0.87& 0.87& s& 18.49& ~~1.924& 38& 0
& 12:57:59.07& $-$31:55:17.0\\
169& PKS1256$-$220& 0.65& 0.79& 0.32& 70
& 12:56:13.94& $-$22:03:20.4& 57& & 0.15
& $-$0.76& 0.78& s& 19.60& ~~1.306& 20& 20
& 12:58:54.48& $-$22:19:31.3\\
170& PKS1256$-$229& 0.50& 0.54& 0.12& 70
& 12:56:27.60& $-$22:54:27.7& 121& & 1.51
& 0.28& 1.54& s& 16.72& ~~1.365& 121& 121
& 12:59:08.45& $-$23:10:38.4\\
\\
171& PKS1258$-$321& 0.92& 0.79& $-$0.25& 78
& 12:58:16.17& $-$32:10:20.6& 120& H& 0.15
& $-$1.37& 1.38& g& 13.11& ~~0.017& 18& 121
& 13:01:00.80& $-$32:26:29.3\\
172& PKS1302$-$102& 0.89& 1.00& 0.19& 8
& 13:02:55.85& $-$10:17:16.5& 56& & $-$0.08
& 0.19& 0.21& s& 15.71& ~~0.286& 76& 0
& 13:05:33.01& $-$10:33:19.7\\
173& PKS1313$-$333& 1.00& 1.32& 0.45& 78
& 13:13:20.05& $-$33:23:09.7& 56& & 0.58
& 1.16& 1.29& s& 16.81& ~~1.21& 37& 0
& 13:16:07.99& $-$33:38:59.3\\
174& PKS1317+019& 0.55& 0.63& 0.22& 102
& 13:17:53.73& 1:56:19.7& 121& & 0.66
& 0.27& 0.71& s& 20.81& ~~1.232& 121& 121
& 13:20:26.78& 1:40:36.7\\
175& PKS1318$-$263& 0.65& 0.64& $-$0.03& 70
& 13:18:28.86& $-$26:20:28.7& 121& & 1.05
& 0.24& 1.08& s& 20.37& ~~2.027& 121& 121
& 13:21:13.99& $-$26:36:10.9\\
176& PKS1327$-$311& 0.52& 0.56& 0.12& 78
& 13:27:29.98& $-$31:07:30.8& 121& & 0.29
& $-$1.05& 1.09& s& 18.46& ~~1.335& 107& 108
& 13:30:19.09& $-$31:22:58.7\\
177& PKS1330+022& 1.91& 1.47& $-$0.42& 102
& 13:30:20.46& 2:16:08.8& 121& Do+CC& 0.37
& 1.02& 1.08& g& 19.40& ~~0.2159& 65& 0
& 13:32:53.25& 2:00:45.6\\
178& PKS1333$-$082& 0.50& 0.59& 0.27& 8
& 13:33:30.54& $-$8:14:34.4& 121& & $-$0.24
& 0.71& 0.75& g& 13.55& ~~0.023& 26& 121
& 13:36:08.26& $-$8:29:52.1\\
179& PKS1334$-$127& 2.01& 2.18& 0.13& 8
& 13:34:59.80& $-$12:42:09.7& 57& & 1.18
& 0.78& 1.41& s& 15.70& ~~0.5390& 89& 89
& 13:37:39.77& $-$12:57:24.8\\
180& PKS1336$-$260& 0.71& 0.77& 0.13& 70
& 13:36:32.48& $-$26:05:18.2& 121& & 0.35
& $-$0.30& 0.46& s& 20.13& ~~1.51& 121& 121
& 13:39:19.88& $-$26:20:30.5\\
\\
181& PKS1340$-$175& 0.76& 0.56& $-$0.50& 70
& 13:40:54.45& $-$17:32:51.7& 121& (K) & 1.55
& 1.33& 2.04& f& 0.00& ~~1.50& $-$121& 121
& 13:43:37.40& $-$17:47:55.9\\
182& PKS1349$-$145& 1.04& 0.93& $-$0.18& 8
& 13:49:10.75& $-$14:34:27.0& 57& (mrg K)& 2.15
& $-$1.03& 2.38& f& 0.00& ~~0.0& 0& 0
& 13:51:52.65& $-$14:49:15.0\\
183& PKS1351$-$018& 0.98& 0.94& $-$0.07& 102
& 13:51:32.03& $-$1:51:20.1& 121& & $-$0.98
& $-$1.11& 1.48& s& 21.30& ~~3.709& 25& 25
& 13:54:06.89& $-$2:06:03.3\\
184& PKS1352$-$104& 0.79& 0.98& 0.35& 8
& 13:52:06.85& $-$10:26:21.1& 121& Do+CC& $-$1.34
& $-$0.02& 1.34& s& 17.60& ~~0.332& 10& 108
& 13:54:46.54& $-$10:41:03.1\\
185& PKS1353$-$341& 0.64& 0.67& 0.07& 78
& 13:53:09.82& $-$34:06:31.3& 57& & 0.78
& 0.78& 1.10& g& 18.56& ~~0.223& 105& 0
& 13:56:05.39& $-$34:21:11.0\\
186& PKS1354$-$174& 1.28& 0.97& $-$0.45& 70
& 13:54:22.05& $-$17:29:24.7& 57& & $-$0.82
& $-$2.20& 2.35& s& 17.85& ~~3.137& 121& 121
& 13:57:06.08& $-$17:44:01.9\\
187& PKS1359$-$281& 0.82& 0.67& $-$0.33& 70
& 13:59:10.69& $-$28:07:59.7& 121& & $-$1.58
& 2.00& 2.55& s& 18.71& ~~0.803& 121& 121
& 14:02:02.50& $-$28:22:26.5\\
188& PKS1402$-$012& 0.71& 0.81& 0.21& 102
& 14:02:11.29& $-$1:16:01.8& 56& & 0.15
& $-$0.75& 0.77& s& 16.75& ~~2.5216& 94& 5
& 14:04:45.89& $-$1:30:22.1\\
189& PKS1402+044& 0.58& 0.71& 0.33& 79
& 14:02:29.97& 4:29:55.1& 57& & 0.45
& 0.80& 0.92& s& 21.29& ~~3.2109& 94& 108
& 14:05:01.11& 4:15:35.5\\
190& PKS1403$-$085& 0.71& 0.58& $-$0.33& 8
& 14:03:21.67& $-$8:33:49.7& 121& & 0.13
& 1.11& 1.11& s& 18.60& ~~1.758& 107& 108
& 14:06:00.72& $-$8:48:07.3\\
\\
191& PKS1404$-$267& 0.50& 0.40& $-$0.36& 70
& 14:04:38.30& $-$26:46:50.6& 121& & 0.69
& $-$0.21& 0.72& g& 13.56& ~~0.022& 121& 121
& 14:07:29.79& $-$27:01:05.1\\
192& PKS1404$-$342& 0.67& 0.62& $-$0.13& 78
& 14:04:57.20& $-$34:17:14.2& 121& & $-$0.52
& $-$0.20& 0.56& s& 17.66& ~~1.122& 105& 0
& 14:07:54.95& $-$34:31:27.9\\
193& PKS1406$-$076& 0.96& 1.05& 0.15& 8
& 14:06:17.90& $-$7:38:15.9& 57& & 1.49
& 0.39& 1.54& s& 20.30& ~~1.494& 107& 108
& 14:08:56.48& $-$7:52:26.8\\
194& PKS1406$-$267& 0.57& 0.90& 0.74& 70
& 14:06:58.43& $-$26:43:27.2& 39& & $-$0.72
& $-$2.21& 2.33& s& 21.75& ~~2.43& 121& 121
& 14:09:50.17& $-$26:57:36.3\\
195& PKS1411+094& 0.60& 0.45& $-$0.47& $-$79
& 14:11:32.40& 9:29:03.7& 121& Do& 0.85
& $-$1.40& 1.64& g& 19.73& ~~0.162& 105& 0
& 14:14:00.13& 9:15:05.1\\
196& PKS1417$-$192& 1.10& 0.83& $-$0.46& 70
& 14:17:02.63& $-$19:14:40.9& 121& Do+CC& $-$0.80
& 0.76& 1.10& g& 17.82& ~~0.1195& 11& 0
& 14:19:49.73& $-$19:28:25.9\\
197& PKS1425$-$274& 0.55& 0.60& 0.14& 70
& 14:25:33.56& $-$27:28:29.0& 121& & 0.48
& 0.29& 0.56& s& 18.14& ~~1.082& 107& 108
& 14:28:28.22& $-$27:41:52.1\\
198& PKS1430$-$178& 1.00& 0.93& $-$0.12& 70
& 14:30:10.65& $-$17:48:24.3& 56& & $-$1.29
& 1.82& 2.23& s& 17.82& ~~2.326& 107& 108
& 14:32:57.69& $-$18:01:35.3\\
199& PKS1430$-$155& 0.55& 0.66& 0.30& 70
& 14:30:36.13& $-$15:35:34.8& 57& (K) & $-$0.81
& $-$1.67& 1.86& f& 0.00& ~~1.573& 121& 121
& 14:33:21.46& $-$15:48:44.7\\
200& PKS1435$-$218& 0.79& 0.81& 0.04& 70
& 14:35:18.66& $-$21:51:57.9& 57& & $-$0.29
& $-$1.23& 1.27& s& 17.41& ~~1.187& 121& 121
& 14:38:09.47& $-$22:04:54.9\\
\\
\end{tabular}
}
\contcaption{Master Source Catalogue}
\end{table*}
\clearpage
\begin{table*}
\vspace{-0.6cm}
\rotate{
\begin{tabular}{rrrrrrrrrlrrrrrlrrrr}
N& name& $S_{2.7}$& $S_{5.0}$& $\alpha$& Rf& RA(B1950)& Dec(B1950)& Rc& comment& $\Delta$RA& $\Delta$Dec& $\Delta$r& cl& $B_J$& z& Rz& Rsp& RA(J2000)& Dec(J2000)\\
\\
201& PKS1437$-$153& 0.72& 0.64& $-$0.19& 70
& 14:37:11.35& $-$15:18:58.8& 57& & $-$1.24
& $-$0.02& 1.24& s& 19.87& ~~0.0& 0& 0
& 14:39:56.88& $-$15:31:50.6\\
202& PKS1438$-$347& 0.50& 0.45& $-$0.17& 78
& 14:38:20.36& $-$34:43:57.5& 121& & $-$2.30
& $-$0.14& 2.30& s& 17.58& ~~1.159& 37& 0
& 14:41:24.01& $-$34:56:45.8\\
203& PKS1443$-$162& 0.78& 0.65& $-$0.30& 70
& 14:43:06.68& $-$16:16:26.7& 57& & $-$0.70
& 0.04& 0.70& s& 20.53& ~~0.0& 0& 0
& 14:45:53.37& $-$16:29:01.7\\
204& PKS1445$-$161& 1.06& 0.80& $-$0.46& 70
& 14:45:28.34& $-$16:07:56.5& 120& & $-$0.68
& 0.49& 0.84& s& 20.40& ~~2.417& 121& 121
& 14:48:15.06& $-$16:20:24.7\\
205& PKS1450$-$338& 0.72& 0.54& $-$0.47& 78
& 14:50:58.10& $-$33:48:46.0& 121& (mrg K)& $-$1.01
& 0.07& 1.01& f& 0.00& ~~0.368& 121& 121
& 14:54:02.59& $-$34:00:57.6\\
206& PKS1454$-$060& 0.83& 0.62& $-$0.47& 8
& 14:54:02.68& $-$6:05:39.1& 121& & $-$1.84
& $-$1.68& 2.49& s& 18.27& ~~1.249& 12& 0
& 14:56:41.48& $-$6:17:42.0\\
207& PKS1456+044& 0.68& 0.72& 0.09& $-$79
& 14:56:29.16& 4:28:09.8& 121& H& $-$0.01
& $-$0.10& 0.10& s& 20.15& ~~0.394& 121& 121
& 14:58:59.36& 4:16:14.2\\
208& PKS1504$-$166& 2.30& 1.96& $-$0.26& $-$7
& 15:04:16.42& $-$16:40:59.3& 56& (mrgDSS& $-$0.04
& $-$0.70& 0.70& m& 19.05& ~~0.876& 34& 0
& 15:07:04.79& $-$16:52:30.3\\
209& PKS1508$-$055& 2.90& 2.33& $-$0.36& $-$7
& 15:08:14.98& $-$5:31:49.0& 56& & 0.51
& 0.40& 0.65& s& 17.12& ~~1.185& 107& 108
& 15:10:53.60& $-$5:43:07.5\\
210& PKS1509+022& 0.69& 0.54& $-$0.40& 102
& 15:09:43.83& 2:14:30.3& 121& H& $-$1.06
& 1.30& 1.68& g& 19.83& ~~0.219& 69& 108
& 15:12:15.75& 2:03:16.4\\
\\
211& PKS1510$-$089& 2.80& 3.25& 0.24& $-$7
& 15:10:08.90& $-$8:54:47.6& 56& & $-$0.56
& 0.10& 0.57& s& 16.21& ~~0.362& 107& 108
& 15:12:50.53& $-$9:05:59.9\\
212& PKS1511$-$100& 0.56& 0.70& 0.36& 7
& 15:11:02.25& $-$10:00:51.0& 57& & $-$0.94
& $-$0.36& 1.01& s& 17.61& ~~1.513& 107& 108
& 15:13:44.89& $-$10:12:00.4\\
213& PKS1511$-$210& 0.55& 0.77& 0.55& 7
& 15:11:03.95& $-$21:03:48.4& 57& & 0.62
& $-$0.58& 0.84& s& 21.88& ~~1.179& 121& 121
& 15:13:56.98& $-$21:14:57.5\\
214& PKS1514$-$241& 2.00& 1.94& $-$0.05& $-$7
& 15:14:45.28& $-$24:11:22.6& 56& & $-$1.34
& 0.33& 1.38& g& 16.44& ~~0.0486& 51& 108
& 15:17:41.82& $-$24:22:19.5\\
215& PKS1518+045& 0.50& 0.37& $-$0.49& $-$79
& 15:18:52.71& 4:31:14.2& 121& H& 0.31
& $-$0.20& 0.37& g& 12.82& ~~0.052& 121& 121
& 15:21:22.52& 4:20:30.4\\
216& PKS1519$-$273& 1.99& 2.28& 0.22& 7
& 15:19:37.24& $-$27:19:30.2& 121& & $-$0.22
& $-$1.21& 1.23& s& 18.02& ~~0.0& 0& 121
& 15:22:37.67& $-$27:30:10.8\\
217& PKS1532+016& 1.08& 0.94& $-$0.23& 102
& 15:32:20.16& 1:41:01.7& 121& & 0.42
& 0.53& 0.67& s& 19.04& ~~1.435& 105& 4
& 15:34:52.44& 1:31:04.2\\
218& PKS1535+004& 1.01& 0.87& $-$0.24& 102
& 15:35:42.56& 0:28:50.8& 57& (K) & $-$0.30
& 0.00& 0.30& f& 0.00& ~~3.497& 121& 121
& 15:38:15.96& 0:19:05.2\\
219& PKS1542+042& 0.53& 0.47& $-$0.19& 79
& 15:42:29.69& 4:17:07.6& 121& & 0.90
& 0.20& 0.93& s& 18.59& ~~2.184& 107& 108
& 15:44:59.39& 4:07:46.3\\
220& PKS1546+027& 1.27& 1.42& 0.18& 102
& 15:46:58.29& 2:46:06.1& 56& & 0.11
& 0.70& 0.71& s& 18.54& ~~0.415& 110& 4
& 15:49:29.43& 2:37:01.1\\
\\
221& PKS1548+056& 1.83& 2.18& 0.28& 79
& 15:48:06.93& 5:36:11.3& 56& & $-$0.01
& $-$0.20& 0.20& s& 18.45& ~~1.422& 105& 0
& 15:50:35.26& 5:27:10.4\\
222& PKS1550$-$269& 1.35& 1.14& $-$0.27& 7
& 15:50:59.64& $-$26:55:50.0& 97& P& 2.21
& $-$1.31& 2.56& s& 19.44& ~~2.145& 38& 0
& 15:54:02.34& $-$27:04:39.3\\
223& PKS1555+001& 2.01& 2.29& 0.21& 102
& 15:55:17.69& 0:06:43.5& 57& & $-$0.95
& $-$0.48& 1.06& f& 22.12& ~~1.77& 3& 0
& 15:57:51.43& $-$0:01:50.5\\
224& PKS1555$-$140& 0.73& 0.83& 0.21& 7
& 15:55:33.72& $-$14:01:26.4& 50& (mrgDSS& 0.00
& 0.28& 0.28& m& 16.99& ~~0.097& 59& 108
& 15:58:21.92& $-$14:09:59.0\\
225& PKS1556$-$245& 0.69& 0.54& $-$0.40& 7
& 15:56:41.20& $-$24:34:11.0& 60& & $-$1.75
& 0.04& 1.75& s& 17.79& ~~2.8179& 94& 5
& 15:59:41.40& $-$24:42:39.0\\
226& PKS1601$-$222& 0.57& 0.44& $-$0.42& 7
& 16:01:03.96& $-$22:15:30.2& 121& & $-$2.34
& $-$1.40& 2.73& s& 20.95& ~~0.0& 0& 0
& 16:04:01.65& $-$22:23:41.7\\
227& PKS1602$-$001& 0.53& 0.42& $-$0.38& 102
& 16:02:22.11& $-$0:11:00.6& 121& Do+CC& $-$0.15
& 0.24& 0.28& s& 17.71& ~~1.6241& 94& 119
& 16:04:56.14& $-$0:19:07.8\\
228& PKS1614+051& 0.67& 0.85& 0.39& 79
& 16:14:09.08& 5:06:54.4& 57& & $-$0.10
& $-$0.50& 0.51& s& 21.07& ~~3.2167& 94& 5
& 16:16:37.55& 4:59:32.7\\
229& PKS1615+029& 0.74& 0.66& $-$0.19& 102
& 16:15:19.11& 2:54:00.1& 57& & $-$1.09
& 1.10& 1.55& s& 18.20& ~~1.341& 121& 121
& 16:17:49.91& 2:46:43.0\\
230& PKS1616+063& 0.93& 0.89& $-$0.07& 79
& 16:16:36.54& 6:20:14.3& 56& & $-$0.45
& $-$0.30& 0.54& s& 19.63& ~~2.088& 121& 121
& 16:19:03.69& 6:13:02.2\\
\\
231& PKS1635$-$035& 0.51& 0.48& $-$0.10& 102
& 16:35:41.56& $-$3:34:10.0& 121& & $-$1.22
& 0.47& 1.31& s& 21.78& ~~2.856& $-$121& 121
& 16:38:19.29& $-$3:40:05.0\\
232& PKS1648+015& 0.72& 0.69& $-$0.07& 102
& 16:48:31.58& 1:34:25.7& 56& & 0.71
& $-$0.37& 0.80& f& 22.69& ~~0.0& 0& 121
& 16:51:03.66& 1:29:23.5\\
233& PKS1649$-$062& 0.70& 0.63& $-$0.17& 7
& 16:49:00.30& $-$6:13:16.0& 121& Do& $-$1.40
& 0.79& 1.61& g& 23.03& ~~0.0& 0& 0
& 16:51:41.08& $-$6:18:15.9\\
234& PKS1654$-$020& 0.64& 0.51& $-$0.37& 102
& 16:54:19.98& $-$2:02:12.0& 121& (K) & 0.17
& 0.06& 0.18& f& 0.00& ~~2.00& 121& 121
& 16:56:56.09& $-$2:06:49.8\\
235& PKS1655+077& 1.26& 1.60& 0.39& 79
& 16:55:43.95& 7:45:59.8& 57& & 0.40
& $-$0.20& 0.44& s& 21.66& ~~0.621& 107& 108
& 16:58:09.00& 7:41:27.6\\
236& PKS1656+053& 1.60& 2.10& 0.44& 79
& 16:56:05.62& 5:19:47.0& 121& & $-$0.02
& $-$0.30& 0.30& s& 17.11& ~~0.8873& 84& 85
& 16:58:33.44& 5:15:16.4\\
237& PKS1705+018& 0.53& 0.58& 0.15& 102
& 17:05:02.74& 1:52:38.4& 121& & $-$0.33
& $-$0.91& 0.97& s& 18.49& ~~2.5765& 94& 5
& 17:07:34.43& 1:48:45.8\\
238& PKS1706+006& 0.50& 0.38& $-$0.45& 102
& 17:06:11.54& 0:38:57.1& 121& & $-$0.23
& 0.22& 0.32& f& 22.80& ~~0.449& 121& 121
& 17:08:44.62& 0:35:09.4\\
239& PKS1725+044& 0.78& 1.21& 0.71& 79
& 17:25:56.34& 4:29:27.9& 56& (mrg K)& $-$0.52
& 0.09& 0.53& m& 18.20& ~~0.296& 107& 108
& 17:28:24.95& 4:27:04.9\\
240& PKS1732+094& 1.08& 0.82& $-$0.45& 79
& 17:32:35.66& 9:28:52.6& 57& (K) & 0.04
& 0.63& 0.63& f& 0.00& ~~0.0& 0& 0
& 17:34:58.38& 9:26:58.2\\
\\
\end{tabular}
}
\contcaption{Master Source Catalogue}
\end{table*}
\clearpage
\begin{table*}
\vspace{-0.6cm}
\rotate{
\begin{tabular}{rrrrrrrrrlrrrrrlrrrr}
N& name& $S_{2.7}$& $S_{5.0}$& $\alpha$& Rf& RA(B1950)& Dec(B1950)& Rc& comment& $\Delta$RA& $\Delta$Dec& $\Delta$r& cl& $B_J$& z& Rz& Rsp& RA(J2000)& Dec(J2000)\\
\\
241& PKS1933$-$400& 1.20& 1.44& 0.30& 6
& 19:33:51.12& $-$40:04:46.8& 56& & 0.41
& $-$1.28& 1.34& s& 17.73& ~~0.965& 121& 121
& 19:37:16.21& $-$39:58:00.8\\
242& PKS1953$-$325& 0.51& 0.63& 0.34& 78
& 19:53:48.42& $-$32:33:49.9& 57& & 0.36
& 1.71& 1.74& s& 19.94& ~~1.242& 37& 108
& 19:56:59.49& $-$32:25:46.2\\
243& PKS1954$-$388& 2.00& 2.00& 0.00& 6
& 19:54:39.06& $-$38:53:13.3& 56& & $-$0.54
& 0.20& 0.58& s& 17.82& ~~0.626& 107& 108
& 19:57:59.83& $-$38:45:06.1\\
244& PKS1958$-$179& 1.11& 1.17& 0.09& 7
& 19:58:04.61& $-$17:57:16.9& 56& & 0.30
& $-$0.02& 0.30& s& 17.05& ~~0.652& 121& 121
& 20:00:57.09& $-$17:48:57.5\\
245& PKS2000$-$330& 0.71& 1.20& 0.85& 78
& 20:00:13.02& $-$33:00:12.5& 56& (mrgDSS& 0.13
& 0.03& 0.13& m& 19.56& ~~3.7832& 94& 5
& 20:03:24.12& $-$32:51:44.4\\
246& PKS2002$-$185& 0.64& 0.48& $-$0.47& 7
& 20:02:24.43& $-$18:30:39.0& 121& & $-$0.10
& $-$0.45& 0.46& s& 17.44& ~~0.859& 107& 108
& 20:05:17.32& $-$18:22:03.3\\
247& PKS2004$-$447& 0.81& 0.65& $-$0.36& 6
& 20:04:25.13& $-$44:43:28.4& 39& & $-$0.83
& 0.36& 0.90& s& 18.09& ~~0.240& 121& 121
& 20:07:55.18& $-$44:34:44.0\\
248& PKS2008$-$159& 0.74& 1.35& 0.98& 7
& 20:08:25.91& $-$15:55:38.3& 56& & 0.76
& 0.26& 0.80& s& 15.95& ~~1.178& 107& 108
& 20:11:15.70& $-$15:46:40.3\\
249& PKS2021$-$330& 0.79& 0.90& 0.21& 78
& 20:21:26.60& $-$33:03:22.0& 121& & $-$0.45
& 0.72& 0.84& s& 17.60& ~~1.471& 121& 121
& 20:24:35.56& $-$32:53:36.3\\
250& PKS2022$-$077& 1.12& 0.89& $-$0.37& $-$7
& 20:22:59.59& $-$7:45:42.4& 121& & $-$0.20
& $-$0.86& 0.89& s& 18.47& ~~1.388& 121& 121
& 20:25:40.65& $-$7:35:52.0\\
\\
251& PKS2037$-$253& 0.93& 1.17& 0.37& 7
& 20:37:10.76& $-$25:18:26.4& 56& & 0.06
& $-$0.66& 0.66& s& 17.80& ~~1.574& 107& 108
& 20:40:08.77& $-$25:07:46.6\\
252& PKS2044$-$168& 0.77& 0.80& 0.06& 7
& 20:44:30.83& $-$16:50:09.5& 121& Do+CC& 0.15
& $-$1.04& 1.05& s& 17.49& ~~1.937& 106& 106
& 20:47:19.67& $-$16:39:05.6\\
253& PKS2047+098& 0.71& 0.85& 0.29& 79
& 20:47:20.78& 9:52:02.0& 56& (K) & $-$0.13
& 2.01& 2.01& f& 0.00& ~~0.0& 0& 0
& 20:49:45.86& 10:03:14.4\\
254& PKS2053$-$044& 0.55& 0.42& $-$0.44& 7
& 20:53:12.76& $-$4:28:18.2& 121& & $-$0.61
& $-$0.85& 1.04& s& 19.05& ~~1.177& 107& 108
& 20:55:50.24& $-$4:16:46.6\\
255& PKS2056$-$369& 0.51& 0.39& $-$0.44& 6
& 20:56:32.12& $-$36:57:37.5& 121& (K) & 0.72
& 0.00& 0.72& f& 0.00& ~~0.0& 0& 121
& 20:59:41.62& $-$36:45:54.5\\
256& PKS2058$-$297& 0.65& 0.87& 0.47& 7
& 20:58:00.91& $-$29:45:15.0& 56& & $-$0.91
& $-$0.35& 0.98& s& 16.21& ~~1.492& 121& 121
& 21:01:01.65& $-$29:33:27.7\\
257& PKS2058$-$135& 0.60& 0.52& $-$0.23& $-$7
& 20:58:59.35& $-$13:30:37.1& 121& & 1.84
& $-$0.60& 1.94& g& 10.60& ~~0.0291& 121& 121
& 21:01:44.38& $-$13:18:47.2\\
258& PKS2059+034& 0.59& 0.75& 0.39& 102
& 20:59:08.01& 3:29:41.5& 56& & $-$1.03
& $-$0.20& 1.05& s& 17.64& ~~1.012& 121& 121
& 21:01:38.83& 3:41:31.4\\
259& PKS2106$-$413& 2.11& 2.28& 0.13& 6
& 21:06:19.39& $-$41:22:33.4& 56& & $-$0.92
& $-$1.59& 1.83& s& 19.50& ~~1.055& 105& 0
& 21:09:33.18& $-$41:10:20.4\\
260& PKS2120+099& 0.65& 0.50& $-$0.43& $-$79
& 21:20:47.07& 9:55:02.2& 121& & 0.02
& $-$0.10& 0.10& s& 20.16& ~~0.932& 121& 121
& 21:23:13.34& 10:07:55.6\\
\\
261& PKS2121+053& 1.62& 3.16& 1.08& 79
& 21:21:14.80& 5:22:27.5& 56& & $-$0.63
& 0.20& 0.66& s& 18.31& ~~1.941& 84& 84
& 21:23:44.51& 5:35:22.3\\
262& PKS2126$-$158& 1.17& 1.24& 0.09& 7
& 21:26:26.78& $-$15:51:50.4& 56& & 0.22
& $-$0.79& 0.82& s& 16.60& ~~3.2663& 94& 68
& 21:29:12.18& $-$15:38:41.0\\
263& PKS2127$-$096& 0.51& 0.45& $-$0.20& 7
& 21:27:38.40& $-$9:40:49.2& 121& (mrg B)& 1.46
& $-$0.57& 1.57& f& 0.00& $>$0.780& $-$121& 121
& 21:30:19.08& $-$9:27:36.7\\
264& PKS2128$-$123& 1.90& 2.00& 0.08& $-$7
& 21:28:52.67& $-$12:20:20.6& 56& & 0.01
& $-$1.88& 1.88& s& 15.97& ~~0.499& 121& 121
& 21:31:35.25& $-$12:07:04.8\\
265& PKS2131$-$021& 1.91& 1.99& 0.07& 102
& 21:31:35.13& $-$2:06:40.0& 56& & $-$0.44
& $-$0.75& 0.86& s& 18.63& ~~1.285& 121& 121
& 21:34:10.30& $-$1:53:17.2\\
266& PKS2134+004& 7.59& 12.38& 0.79& 102
& 21:34:05.21& 0:28:25.1& 56& & $-$0.58
& 0.28& 0.65& s& 16.53& ~~1.937& 110& 4
& 21:36:38.58& 0:41:54.3\\
267& PKS2135$-$248& 0.77& 0.69& $-$0.18& 7
& 21:35:45.40& $-$24:53:28.5& 57& & 0.10
& $-$0.39& 0.40& s& 17.29& ~~0.821& 107& 108
& 21:38:37.18& $-$24:39:54.5\\
268& PKS2140$-$048& 0.77& 0.60& $-$0.40& 7
& 21:39:59.96& $-$4:51:27.8& 57& & 0.18
& $-$0.35& 0.39& s& 17.13& ~~0.344& 112& 0
& 21:42:36.89& $-$4:37:43.5\\
269& PKS2143$-$156& 1.11& 0.82& $-$0.49& 7
& 21:43:38.87& $-$15:39:37.3& 56& & 0.34
& $-$0.31& 0.46& s& 17.24& ~~0.698& 121& 121
& 21:46:22.97& $-$15:25:43.8\\
270& PKS2144+092& 0.95& 1.01& 0.10& 79
& 21:44:42.47& 9:15:51.2& 56& & $-$0.34
& 0.20& 0.39& s& 18.66& ~~1.113& 105& 0
& 21:47:10.15& 9:29:46.8\\
\\
271& PKS2145+067& 3.30& 4.50& 0.50& $-$79
& 21:45:36.08& 6:43:40.9& 56& & $-$0.60
& $-$0.40& 0.72& s& 16.75& ~~0.999& 84& 84
& 21:48:05.45& 6:57:38.7\\
272& PKS2145$-$176& 0.82& 0.79& $-$0.06& $-$7
& 21:45:51.48& $-$17:37:42.3& 121& & 1.86
& $-$0.41& 1.91& s& 20.24& ~~2.130& 121& 121
& 21:48:36.80& $-$17:23:43.5\\
273& PKS2149$-$307& 1.32& 1.15& $-$0.22& 78
& 21:49:00.59& $-$30:42:00.2& 56& & $-$0.54
& $-$0.83& 0.99& s& 17.69& ~~2.345& 107& 108
& 21:51:55.52& $-$30:27:53.7\\
274& PKS2149+069& 0.89& 0.94& 0.09& 79
& 21:49:02.10& 6:55:21.0& 60& & $-$0.59
& $-$0.30& 0.66& s& 18.64& ~~1.364& 112& 0
& 21:51:31.45& 7:09:27.0\\
275& PKS2149+056& 1.01& 1.19& 0.27& 79
& 21:49:07.70& 5:38:06.9& 57& (K) & 0.24
& 0.52& 0.57& f& 0.00& ~~0.740& 86& 86
& 21:51:37.87& 5:52:13.1\\
276& PKS2155$-$152& 1.67& 1.58& $-$0.09& $-$7
& 21:55:23.24& $-$15:15:30.2& 56& & $-$0.39
& $-$0.14& 0.41& s& 18.15& ~~0.672& 87& 87
& 21:58:06.28& $-$15:01:09.3\\
277& PKS2200$-$238& 0.53& 0.46& $-$0.23& 103
& 22:00:07.71& $-$23:49:41.1& 121& & $-$0.52
& $-$0.99& 1.12& s& 17.73& ~~2.120& 107& 108
& 22:02:55.99& $-$23:35:09.7\\
278& PKS2203$-$188& 5.25& 4.24& $-$0.35& 103
& 22:03:25.73& $-$18:50:17.1& 56& & 0.05
& $-$0.12& 0.13& s& 18.55& ~~0.619& 107& 108
& 22:06:10.41& $-$18:35:38.7\\
279& PKS2206$-$237& 1.33& 0.98& $-$0.50& 103
& 22:06:32.63& $-$23:46:38.2& 121& & 0.77
& $-$1.35& 1.56& g& 17.65& ~~0.0863& 15& 108
& 22:09:20.15& $-$23:31:53.2\\
280& PKS2208$-$137& 0.72& 0.53& $-$0.50& 103
& 22:08:42.93& $-$13:43:00.0& 121& & $-$1.92
& 1.50& 2.43& s& 16.79& ~~0.392& 64& 64
& 22:11:24.16& $-$13:28:10.8\\
\\
\end{tabular}
}
\contcaption{Master Source Catalogue}
\end{table*}
\clearpage
\begin{table*}
\vspace{-0.6cm}
\rotate{
\begin{tabular}{rrrrrrrrrlrrrrrlrrrr}
N& name& $S_{2.7}$& $S_{5.0}$& $\alpha$& Rf& RA(B1950)& Dec(B1950)& Rc& comment& $\Delta$RA& $\Delta$Dec& $\Delta$r& cl& $B_J$& z& Rz& Rsp& RA(J2000)& Dec(J2000)\\
\\
281& PKS2210$-$257& 0.96& 1.02& 0.10& 103
& 22:10:14.13& $-$25:44:22.5& 56& & $-$1.70
& $-$1.12& 2.03& s& 17.86& ~~1.833& 107& 108
& 22:13:02.50& $-$25:29:30.1\\
282& PKS2212$-$299& 0.54& 0.44& $-$0.33& 103
& 22:12:25.11& $-$29:59:19.8& 121& & $-$0.44
& $-$0.24& 0.50& s& 17.24& ~~2.703& 5& 5
& 22:15:16.03& $-$29:44:23.0\\
283& PKS2215+020& 0.70& 0.64& $-$0.15& 102
& 22:15:15.59& 2:05:09.0& 57& & $-$0.74
& $-$0.50& 0.89& s& 21.97& ~~3.572& 121& 121
& 22:17:48.24& 2:20:10.9\\
284& PKS2216$-$038& 1.04& 1.30& 0.36& 102
& 22:16:16.38& $-$3:50:40.7& 56& & $-$0.23
& $-$0.38& 0.45& s& 16.65& ~~0.901& 109& 109
& 22:18:52.03& $-$3:35:36.8\\
285& PKS2223$-$052& 4.70& 4.31& $-$0.14& 103
& 22:23:11.08& $-$5:12:17.8& 57& & 0.64
& $-$0.30& 0.71& s& 17.12& ~~1.404& 73& 0
& 22:25:47.26& $-$4:57:01.3\\
286& PKS2227$-$088& 1.49& 1.41& $-$0.09& 103
& 22:27:02.34& $-$8:48:17.6& 56& & 1.20
& 0.02& 1.20& s& 18.05& ~~1.561& 107& 108
& 22:29:40.08& $-$8:32:54.3\\
287& PKS2227$-$399& 1.02& 1.02& 0.00& 6
& 22:27:44.98& $-$39:58:16.8& 56& & $-$0.71
& $-$0.62& 0.94& s& 17.41& ~~0.323& 71& $-$71
& 22:30:40.27& $-$39:42:52.0\\
288& PKS2229$-$172& 0.52& 0.58& 0.18& 103
& 22:29:41.00& $-$17:14:29.6& 121& & 0.24
& $-$1.26& 1.28& f& 21.28& ~~1.780& 121& 121
& 22:32:22.56& $-$16:59:01.8\\
289& PKS2233$-$148& 0.50& 0.61& 0.32& 103
& 22:33:53.98& $-$14:48:56.7& 57& & $-$0.78
& 0.07& 0.78& s& 20.85& $>$0.609& $-$121& 121
& 22:36:34.08& $-$14:33:22.1\\
290& PKS2239+096& 0.65& 0.70& 0.12& 79
& 22:39:19.85& 9:38:09.9& 56& & $-$0.49
& 1.20& 1.29& s& 19.73& ~~1.707& 105& 0
& 22:41:49.72& 9:53:52.6\\
\\
291& PKS2240$-$260& 1.08& 1.00& $-$0.12& 103
& 22:40:41.84& $-$26:00:15.9& 120& & $-$0.77
& $-$0.72& 1.05& s& 17.87& ~~0.774& 88& 108
& 22:43:26.42& $-$25:44:29.0\\
292& PKS2243$-$123& 2.74& 2.38& $-$0.23& 103
& 22:43:39.80& $-$12:22:40.3& 56& & 0.40
& $-$0.46& 0.61& s& 16.50& ~~0.63& 10& 54
& 22:46:18.23& $-$12:06:51.2\\
293& PKS2245+029& 0.66& 0.58& $-$0.21& 102
& 22:45:26.02& 2:54:51.2& 121& & 0.84
& $-$0.40& 0.93& s& 19.52& ~~0.0& 0& 0
& 22:47:58.67& 3:10:42.7\\
294& PKS2245$-$328& 2.01& 1.80& $-$0.18& 78
& 22:45:51.50& $-$32:51:44.3& 121& & 0.11
& 0.63& 0.64& s& 18.24& ~~2.268& 59& 108
& 22:48:38.68& $-$32:35:51.9\\
295& PKS2252$-$090& 0.63& 0.69& 0.15& 103
& 22:52:27.49& $-$9:00:04.6& 57& (K) & $-$0.34
& $-$0.40& 0.53& s& 0.00& ~~0.6064& 121& 121
& 22:55:04.23& $-$8:44:03.8\\
296& PKS2254$-$367& 0.82& 0.72& $-$0.21& 6
& 22:54:23.11& $-$36:43:47.4& 121& & $-$0.18
& 0.24& 0.30& g& 11.41& ~~0.0055& 121& 121
& 22:57:10.61& $-$36:27:44.0\\
297& PKS2255$-$282& 1.38& 1.73& 0.37& 103
& 22:55:22.46& $-$28:14:25.6& 57& & 0.44
& 0.64& 0.77& s& 16.61& ~~0.925& 107& 108
& 22:58:05.96& $-$27:58:21.1\\
298& PKS2300$-$189& 0.98& 0.89& $-$0.16& 103
& 23:00:23.48& $-$18:57:35.8& 121& (mrgDSS& $-$0.61
& 0.09& 0.62& m& 18.46& ~~0.129& 19& 19
& 23:03:02.98& $-$18:41:25.6\\
299& PKS2301+060& 0.52& 0.54& 0.06& 79
& 23:01:56.29& 6:03:56.9& 121& & $-$0.01
& $-$0.20& 0.20& s& 17.69& ~~1.268& 105& 0
& 23:04:28.28& 6:20:08.5\\
300& PKS2303$-$052& 0.54& 0.45& $-$0.30& 103
& 23:03:40.15& $-$5:16:01.8& 121& & 0.18
& $-$0.08& 0.20& s& 18.32& ~~1.136& 107& 108
& 23:06:15.35& $-$4:59:48.2\\
\\
301& PKS2304$-$230& 0.59& 0.51& $-$0.24& 103
& 23:04:58.32& $-$23:04:08.0& 121& (K) & $-$0.28
& 0.09& 0.30& f& 0.00& ~~0.0& 0& 0
& 23:07:38.65& $-$22:47:53.0\\
302& PKS2312$-$319& 0.71& 0.58& $-$0.33& 78
& 23:12:06.37& $-$31:55:00.6& 121& & $-$0.75
& $-$1.31& 1.51& s& 17.60& ~~1.323& 121& 121
& 23:14:48.49& $-$31:38:38.7\\
303& PKS2313$-$438& 0.86& 0.69& $-$0.36& 6
& 23:13:34.82& $-$43:54:10.2& 121& & $-$0.50
& 0.66& 0.83& s& 19.01& ~~1.847& 105& 0
& 23:16:21.09& $-$43:37:47.0\\
304& PKS2314$-$409& 0.50& 0.42& $-$0.28& 6
& 23:14:02.01& $-$40:57:44.4& 39& & $-$1.85
& $-$1.46& 2.36& s& 18.20& ~~2.448& 38& 0
& 23:16:46.94& $-$40:41:20.8\\
305& PKS2318+049& 1.23& 1.17& $-$0.08& 79
& 23:18:12.13& 4:57:23.5& 56& & $-$0.16
& 0.00& 0.16& s& 18.65& ~~0.622& 75& 0
& 23:20:44.85& 5:13:50.1\\
306& PKS2320+079& 0.70& 0.68& $-$0.05& $-$79
& 23:20:03.91& 7:55:33.6& 55& & $-$0.23
& 0.10& 0.25& s& 17.65& ~~2.090& 111& 0
& 23:22:36.09& 8:12:01.6\\
307& PKS2325$-$150& 0.63& 0.71& 0.19& 103
& 23:25:11.60& $-$15:04:27.3& 57& & 0.97
& $-$0.62& 1.15& s& 19.50& ~~2.465& 107& 108
& 23:27:47.96& $-$14:47:55.6\\
308& PKS2329$-$162& 0.98& 1.03& 0.08& 103
& 23:29:02.40& $-$16:13:30.8& 121& & $-$2.05
& $-$1.22& 2.39& s& 20.87& ~~1.155& 107& 108
& 23:31:38.65& $-$15:56:56.8\\
309& PKS2329$-$384& 0.77& 0.67& $-$0.23& 6
& 23:29:18.94& $-$38:28:21.7& 121& & 0.63
& 0.45& 0.78& s& 17.08& ~~1.202& 36& 108
& 23:31:59.46& $-$38:11:47.4\\
310& PKS2329$-$415& 0.51& 0.47& $-$0.13& 6
& 23:29:37.82& $-$41:35:12.6& 121& & 0.67
& $-$0.69& 0.96& s& 18.20& ~~0.671& 121& 121
& 23:32:19.04& $-$41:18:38.1\\
\\
311& PKS2330+083& 0.52& 0.57& 0.15& 79
& 23:30:25.06& 8:21:36.1& 121& (K) & 0.33
& $-$0.42& 0.53& f& 0.00& ~~0.0& 0& 0
& 23:32:57.60& 8:38:10.7\\
312& PKS2331$-$240& 1.04& 1.06& 0.03& 103
& 23:31:17.98& $-$24:00:15.6& 56& & $-$0.83
& $-$0.42& 0.93& g& 16.47& ~~0.0477& 112& 108
& 23:33:55.27& $-$23:43:40.3\\
313& PKS2332$-$017& 0.64& 0.53& $-$0.31& 102
& 23:32:46.42& $-$1:47:45.3& 121& & $-$0.41
& $-$0.45& 0.61& s& 18.41& ~~1.184& 110& 4
& 23:35:20.41& $-$1:31:09.4\\
314& PKS2335$-$181& 0.69& 0.59& $-$0.25& 103
& 23:35:20.65& $-$18:08:57.6& 121& Do -note& 0.30
& $-$0.55& 0.63& s& 16.76& ~~1.450& 121& 121
& 23:37:56.63& $-$17:52:20.4\\
315& PKS2335$-$027& 0.60& 0.65& 0.13& 102
& 23:35:23.25& $-$2:47:34.5& 57& & 0.51
& 0.29& 0.59& s& 18.06& ~~1.072& 110& 4
& 23:37:57.33& $-$2:30:57.4\\
316& PKS2337$-$334& 1.36& 1.17& $-$0.24& 78
& 23:37:16.67& $-$33:26:54.8& 57& (R) & 0.06
& $-$0.04& 0.07& f& 0.00& ~~0.0& 0& 0
& 23:39:54.53& $-$33:10:16.7\\
317& PKS2344+092& 1.60& 1.42& $-$0.19& $-$79
& 23:44:03.77& 9:14:05.5& 56& & $-$0.42
& $-$0.50& 0.65& s& 16.15& ~~0.6726& 95& 95
& 23:46:36.83& 9:30:45.7\\
318& PKS2344$-$192& 0.54& 0.43& $-$0.37& 103
& 23:44:33.44& $-$19:12:59.1& 121& (R) & $-$0.61
& $-$0.58& 0.84& f& 0.00& ~~0.0& 0& 0
& 23:47:08.63& $-$18:56:18.6\\
319& PKS2345$-$167& 4.08& 3.47& $-$0.26& 103
& 23:45:27.69& $-$16:47:52.6& 56& & 0.96
& $-$0.62& 1.14& s& 17.32& ~~0.576& 93& 93
& 23:48:02.61& $-$16:31:11.9\\
320& PKS2351$-$006& 0.51& 0.47& $-$0.13& 102
& 23:51:35.39& $-$0:36:29.5& 121& & $-$2.05
& $-$0.75& 2.18& s& 18.05& ~~0.464& 110& 28
& 23:54:09.17& $-$0:19:47.7\\
\\
321& PKS2351$-$154& 1.08& 0.93& $-$0.24& 103
& 23:51:55.88& $-$15:29:53.0& 57& & $-$1.98
& $-$0.65& 2.09& s& 18.65& ~~2.6750& 94& 5
& 23:54:30.19& $-$15:13:11.1\\
322& PKS2354$-$117& 1.57& 1.39& $-$0.20& 103
& 23:54:57.20& $-$11:42:21.1& 121& Do+CC& $-$1.28
& $-$1.42& 1.91& s& 17.80& ~~0.960& 105& 0
& 23:57:31.19& $-$11:25:38.9\\
323& PKS2358$-$161& 0.50& 0.37& $-$0.49& 103
& 23:58:31.56& $-$16:07:49.1& 121& & $-$1.31
& $-$0.77& 1.52& s& 18.28& ~~2.033& 107& 108
& 0:01:05.34& $-$15:51:06.7\\
\end{tabular}}
\contcaption{Master Source Catalogue}
\end{table*}
\clearpage
\begin{figure}
\epsfxsize=\one_wide \epsffile{fig9optr.eps}
\centering
\caption{The distribution of \hbox{$B_J$}\ magnitudes as a function of 2.7{\rm\thinspace GHz}\
radio flux for all sources in the survey.}
\label{fig_opt_rad}
\end{figure}
\begin{figure*}
\epsfxsize=\two_wide \epsffile{fig10rat.eps}
\centering
\caption{Radio-to-optical ratios $R$ as a function of 2.7{\rm\thinspace GHz}\
luminosity. Absolute magnitudes and luminosities were computed
assuming $H_{\circ} = 75 {\thinspace\rm km\thinspace s}^{-1}{\rm
Mpc}^{-1}$ and $q_{\circ} = 0.5$; \hbox{$B_J$}\ magnitudes were $K$-corrected
assuming a continuum slope of $f_{\nu} \propto \nu^{-1}$, and radio
slope $f_{\nu} \propto \nu^{-0.09}$ which is the median slope of our
sources. Morphologically extended sources (as classified
automatically) are marked as circles and unresolved sources as crosses;
faint and merged sources are indicated by
triangles. Radio-to-optical ratios of sources without measured
redshifts are presented as a histogram in the top panel; dotted lines
show lower limits on radio-to-optical ratios for sources not detected
on the sky surveys.
}
\label{fig_ratio}
\end{figure*}
This correlation can be modelled by assuming a strict proportionality
between the \hbox{$B_J$}\ and 2.7{\rm\thinspace GHz}\ luminosities of our quasars, but adding
the light of a host galaxy. If we assume that all host galaxies have
absolute \hbox{$B_J$}\ magnitudes of $\sim -20.5$, the quasar light will
dominate over the host galaxy light for 2.7{\rm\thinspace GHz}\ luminosities $>10^{26}
{\rm W \ Hz}^{-1}$ (for assumed cosmology see the caption to
Fig.~\ref{fig_ratio}). Above this luminosity there will be no
correlation between $R$ and the radio flux, as observed, and below this
luminosity $R$ will be proportional to the radio flux, which
is consistent with the observed correlation.
The radio spectral indices do not correlate with redshift, apparent
\hbox{$B_J$}\ magnitude, radio flux or radio luminosity---but the range of
spectral index in the sample is of course limited.
\section*{Acknowledgements}
The compilation of this paper would not have been possible without the
efforts of many people who have worked on the Parkes radio samples over
the past 20 years or more. We particularly acknowledge the work done by
Graeme White on the optical identifications, Saul Caganoff on the
early stages of our image analysis, and Alan Wright and Robina
Otrupcek in compiling the machine-readable version of the Parkes Catalogue.
Our new radio observations were made with the VLA and the ATCA. John
Reynolds and Lucyna Kedziora-Chudczer assisted with the ATCA
observations and we would also like to thank John Reynolds for helpful
comments about astrometry.
We would like to thank Raylee Stathakis for making our service
observations on the AAT, Russell Cannon, Director of the AAO for
awarding us additional discretionary observing time and Roy Antaw for
extracting large amounts of data from the AAT archive for us. We are
also grateful to Katrina Sealey for obtaining some additional data for
us with the ANU 2.3m Telescope and Mike Bessell for giving technical
advice about the 2.3m out-of-hours.
We have made substantial use of the following on-line databases in the
compilation of this paper: The APM Catalogues (we thank Mike Irwin for
scanning additional fields for us); the Center for Astrophysics
Redshift Survey (with kind assistance from Cathy Clemens); the
COSMOS/UKST Southern Sky Catalogue supplied by the Anglo-Australian
Observatory; the Lyon-Meudon Extragalactic Database (LEDA) supplied by
the LEDA team at the CRAL-Observatoire de Lyon (France); the NASA/IPAC
Extragalactic Database (NED) which is operated by the Jet Propulsion
Laboratory, Caltech, under contract with the National Aeronautics and
Space Administration.
The Digitized Sky Survey was produced at the Space Telescope Science
Institute under U.S. Government grant
NAG W-2166. The images are based on photographic data
obtained using the Oschin Schmidt
Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were
processed into
compressed digital form with the permission of these institutions.
The Palomar Observatory Sky Survey was funded by the National Geographic
Society. The Oschin Schmidt
Telescope is operated by the California Institute of Technology and Palomar
Observatory.
The UK Schmidt Telescope was operated by the Royal Observatory Edinburgh,
with funding from the UK Science
and Engineering Research Council, until 1988 June, and thereafter by the
Anglo-Australian Observatory. Original
plate material is copyright the Royal Observatory Edinburgh and the
Anglo-Australian Observatory.
Finally we wish to thank the referee for many helpful suggestions.
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\newpage
\noindent Please note the following sections are not included in this
preprint but may be obtained from my
preprint page
http://www.aao.gov.au/local/www/mjd/papers/.
|
proofpile-arXiv_065-630
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{
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
The need to implement Gauss's law in QCD and Yang-Mills theory,
and the technical problems that complicate the implementation
of Gauss's law in non-Abelian theories have been discussed by a
number of authors\cite{goldjack,jackiw,khymtemp}. Strategies
for implementing Gauss's law have also been developed\cite{lands}.
In earlier work\cite{bellchenhall}, we
constructed states that implement
Gauss's law for Yang-Mills theory
and QCD --- in fact, for any `pure glue' gauge theory,
in a temporal gauge formulation that has
a non-Abelian $SU (N)$ gauge symmetry.
In that work, a state vector ${\Psi}\,|{\phi}\rangle$
was defined for which
\begin{equation}
\{\,b_{Q}^{a}({\bf{k}}) + J_{0}^{a}({\bf{k}})\,\}
{\Psi}\,|{\phi}\rangle =0\;,
\label{eq:subcon}
\end{equation}
where $b_{Q}^{a}({\bf{k}})$ and $J_{0}^{a}({\bf{k}})$
are the Fourier transforms of
$\partial_{i}{\Pi}^a_{i}({\bf r})$ (${\Pi}^a_{i}({\bf r})$ is
the momentum conjugate
to the gauge field) and of the gluon color charge density
\begin{equation}
J_{0}^{a}({\bf{r}})=g\,f^{abc}A_{i}^{b}({\bf{r}})\,\Pi_{i}^{c}({\bf{r}})
\label{eq:charge}
\end{equation}
respectively. Since the chromoelectric field
$E^a_{i}({\bf{r}})=-{\Pi}^a_{i}({\bf{r}})$, Eq.~(\ref{eq:subcon})
expresses
the momentum space representation of the non-Abelian `pure glue' Gauss's
law, and $\{\,b_{Q}^{a}({\bf{k}}) + J_{0}^{a}({\bf{k}})\,\}$ is
referred to as the `Gauss's law operator' for the `pure glue' case;
$|{\phi}\rangle$ is a perturbative state annihilated by
$\partial_{i}\Pi^a_{i}({\bf{r}})$. In Ref.~\cite{bellchenhall},
we exhibited an explicit form for the operator ${\Psi}$, namely
\begin{equation}
{\Psi}={\|}\,\exp({\cal{A}})\,{\|}\;,
\label {eq:Apsi}
\end{equation}
where bracketing between double bars denotes a normal
order in which all gauge fields
and functionals of gauge fields appear
to the left of all momenta conjugate to gauge fields. ${\cal{A}}$
was exhibited as an operator-valued series in
Ref.~\cite{bellchenhall}.
Its form was conjectured to all orders, and verified
for the first six orders.
\bigskip
In the work presented here we will
extend our previously published results in the
following ways: we will prove our earlier conjecture that the
state ${\Psi}|{\phi}\rangle$ implements the
`pure glue' form of Gauss's Law; we will extend our work
from the `pure glue' form of the theory to
include quarks as well as gluons; we will
construct gauge-invariant operator-valued
spinor (quark) and gauge (gluon) fields;
and we will adapt the QCD formulation to apply to the
$SU(2)$ Yang-Mills theory.
\section{Implementing the `pure glue' form of Gauss's law}
\label{sec-Implementing}
Our construction of ${\Psi}$ in Ref.~\cite{bellchenhall} was
informed by the realization that the operator ${\Psi}$ had
to implement $\{\,b_{Q}^{a}({\bf{k}}) + J_{0}^{a}({\bf{k}})\,\}\,
{\Psi}\,|{\phi}\rangle=
{\Psi}\,b_{Q}^{a}({\bf{k}})\,|{\phi}\rangle\,,$ or equivalently that
\begin{equation}
[\,b_{Q}^{a}({\bf{k}}),\,{\Psi}\,]=-J_{0}^{a}({\bf{k}})\,
{\Psi}\,+\,B_Q^{a}({\bf{k}})\;,
\label{eq:psicomm}
\end{equation}
where $B_{Q}^{a}({\bf{k}})$ is an operator product that has
$\partial_{i}\Pi^a_{i}({\bf{r}})$ on its extreme right and therefore
annihilates the same states as
$b_{Q}^{a}({\bf{k}})$, so that
$B_{Q}^{a}({\bf{k}})\,|{\phi}\rangle=0$ as well as
$b_{Q}^{a}({\bf{k}})\,|{\phi}\rangle=0$.
To facilitate the discussion of the structure of ${\Psi}$,
the following definitions are useful:
\begin{equation}
a_{i}^{\alpha} ({\bf{r}}) = A_{Ti}^{\alpha}({\bf{r}})\;
\label{eq:bookai}
\end{equation}
denotes the transverse part of the gauge field, and
\begin{equation}
x_i^\alpha ({\bf{r}}) = A_{Li}^{\alpha}({\bf{r}})\;
\label{eq:bookxi}
\end{equation}
denotes the longitudinal part, so that
$[\,a_i^\alpha({\bf{r}})+x_i^\alpha({\bf{r}})\,]=
A_{i}^{\alpha}({\bf{r}})$. We also will make use of the combinations
\begin{equation}
{\cal{X}}^\alpha({\bf{r}}) =
[\,{\textstyle\frac{\partial_i}{\partial^2}}A_i^\alpha({\bf{r}})\,]\;,
\end{equation}
and
\begin{equation}
{\cal{Q}}_{(\eta)i}^{\beta}({\bf{r}}) =
[\,a_i^\beta ({\bf{r}})+
{\textstyle\frac{\eta}{\eta+1}}x_i^\beta({\bf{r}})\,]\;,
\label{eq:bookaiQ}
\end{equation}
where $\eta$ is an integer-valued index.
\bigskip
We will furthermore refer to the composite operators
\begin{equation}
\psi^{\gamma}_{(\eta)i}({\bf{r}})= \,(-1)^{\eta-1}\,
f^{\vec{\alpha}\beta\gamma}_{(\eta)}\,
{\cal{R}}^{\vec{\alpha}}_{(\eta)}({\bf{r}})\;
{\cal{Q}}_{(\eta)i}^{\beta}({\bf{r}})\;,
\label{eq:psindef2}
\end{equation}
in which ${\cal{R}}^{\vec{\alpha}}_{(\eta)}({\bf{r}})$ is given by
\begin{equation}
{\cal{R}}^{\vec{\alpha}}_{(\eta)}({\bf{r}})=
\prod_{m=1}^\eta{\cal{X}}^{\alpha[m]}({\bf{r}})\;,
\label{eq:XproductN}
\end{equation}
and $f^{\vec{\alpha}\beta\gamma}_{(\eta)}$ is
the chain of $SU(3)$ structure functions
\begin{equation}
f^{\vec{\alpha}\beta\gamma}_{(\eta)}=f^{\alpha[1]\beta b[1]}\,
\,f^{b[1]\alpha[2]b[2]}\,f^{b[2]\alpha[3]b[3]}\,\cdots\,
\,f^{b[\eta-2]\alpha[\eta-1]b[\eta-1]}f^{b[\eta-
1]\alpha[\eta]\gamma}\;,
\label{eq:fproductN}
\end{equation}
where repeated indices are to be summed. For $\eta =1$,
the chain reduces to
$f^{\vec{\alpha}\beta\gamma}_{(1)}\equiv f^{\alpha\beta\gamma}$;
and for $\eta =0$,
$f^{\vec{\alpha}\beta\gamma}_{(0)}\equiv -\delta_{\beta ,\gamma}$.
Since the only properties of the structure functions that we will use
is their antisymmetry and the Jacobi identity, the
formalism we develop will be applicable to
$SU(2)$ as well as to
other models with an $SU(N)$ gauge symmetry.
\bigskip
The composite operators introduced so far
can help us to understand qualitatively how ${\Psi}$
can implement Eq.~(\ref{eq:subcon}). We observe, for example,
the product
\begin{equation}
\psi^{\gamma}_{(1)i}({\bf{r}})= \,f^{\alpha\beta\gamma}\,
{\cal{X}}^\alpha({\bf{r}})\;{\cal{Q}}_{(1)i}^{\beta}({\bf{r}})
=\,f^{\alpha\beta\gamma}\,
{\cal{X}}^\alpha({\bf{r}})\,[ a_i^\beta({\bf{r}}) +
{\textstyle\frac{1}{2}}x_i^\beta({\bf{r}}) ]\;,
\end{equation}
which as part of the expression
\begin{equation}
{\cal{A}}_1=ig{\int}d{\bf{r}}\,\psi^{\gamma}_{(1)i}({\bf{r}})\,
\Pi^{\gamma}_{i}({\bf{r}})\;,
\end{equation}
has the property that its commutator with $b_{Q}^{a}({\bf{k}})$,
\begin{eqnarray}
[\,b_Q^a({\bf{k}}),\,ig{\int}d{\bf{r}}\,\psi^{\gamma}_{(1)i}({\bf{r}})\,
\Pi^{\gamma}_{i}({\bf{r}})\,]\,=&&
-g\,f^{a\beta\gamma}{\int}d{\bf{r}}\;e^{-i{\bf{k\cdot r}}}\;
A_i^\beta({\bf{r}})\;\Pi_i^\gamma({\bf{r}})
\nonumber\\
&&-{\textstyle\frac{g}{2}}\,f^{a\beta\gamma}\,{\int}d{\bf{r}}\,
e^{-i{\bf{k\cdot r}}}\,
{\cal{X}}^\beta\;[\,\partial_i\Pi_i^\gamma({\bf{r}})\,]\;,
\label{eq:thetabqcom}
\end{eqnarray}
generates $-J_{0}^{a}({\bf{k}})$ when it acts on a state annihilated
by $b_Q^a({\bf{k}})\,.$
The expression $\exp({\cal{A}}_1)$
would therefore have been an appropriate choice for $\Psi$,
were it not for the fact that the commutator
$[\,b_Q^a({\bf{k}}),\,{\cal{A}}_1\,]$ fails
to commute with ${\cal{A}}_1$.
When Eq.~(\ref{eq:subcon}) is applied to a candidate
${\Psi}_{cand}=\exp({\cal{A}}_1)$, the commutator
$[\,b_Q^a({\bf{k}}),\,{\cal{A}}_1\,]$
is often produced within a polynomial consisting of ${\cal{A}}_1$
factors --- for example
${\cal{A}}_1^{(n-s)}\,[\,b_Q^a({\bf{k}}),\,
{\cal{A}}_1\,]\,{\cal{A}}_1^s\,$.
$[\,b_Q^a({\bf{k}}),\,{\cal{A}}_1\,]$ does not commute with
${\cal{A}}_{1},$ and can not move freely to annihilate the
state at the right of ${\Psi}_{cand}\,$, thereby excluding
$\exp({\cal{A}}_1)$ as a viable choice for $\Psi$.
\bigskip
The normal ordering denoted by bracketing between
double bars eliminates this problem, but only at the expense
of introducing another problem in its place ---
one that is more benign, but that nevertheless must be addressed.
When normal ordering is imposed,
the result of commuting $\exp({\cal{A}}_1)$ with $b_Q^a({\bf{k}})$
is not the formation of $J_{0}^{a}({\bf{k}})$
to the left of ${\Psi}_{cand}$, but the formation of only
$f^{a\beta\gamma}\,{\int}\,d{\bf{r}}\,e^{-i{\bf{k\cdot r}}}\,
A_i^\beta({\bf{r}})$ to the {\em left} of it, and of
$\Pi_i^\gamma({\bf{r}})$
to the extreme
{\em right} of all the gauge fields in the series
representation of the exponential.
Unwanted terms will be generated as
$\Pi_i^\gamma({\bf{r}})$ is commuted,
term by term, from the extreme right of
${\Psi}_{cand}$ to the extreme left to form the
desired $J_{0}^{a}({\bf{k}})$. To compensate for
these further terms, we modify
${\Psi}_{cand}$ by adding additional expressions to ${\cal{A}}_1$ to
eliminate the unwanted commutators
generated as $\Pi_i^\gamma({\bf{r}})$
is commuted from the right to the left hand sides of
operator-valued polynomials. The question
naturally arises whether the process of adding terms
to remove the unwanted contributions from
earlier ones, comes to closure --- whether
an operator-valued series ${\cal A}$,
that leads to a $\Psi$ for which Eq.~(\ref{eq:subcon})
is satisfied, can be specified to all orders.
In Ref.~\cite{bellchenhall}
we conjectured that this question can be answered affirmatively, by
formulating a recursive equation for ${\cal A}$, which we
verified to sixth order.
\bigskip
In Ref.~\cite{bellchenhall} we represented ${\cal A}$ as
the series ${\cal A}=\sum_{\,n=1}^\infty{\cal A}_{n}$;
we also showed that the requirement that ${\cal A}$ must satisfy
to implement Eq.~(\ref{eq:subcon}),
can be formulated as
\begin{equation}
{\|}\,[\,b_{Q}^{a}({\bf{k}}),\,
\sum_{n=2}^\infty{\cal A}_n\,]\exp({\cal A})\,{\|}\,
-\,{\|}\,g\,f^{a\beta\gamma}\int d{\bf{r}}\,e^{-i{\bf{k\cdot r}}}
A^{\beta}_{i}({\bf{r}})\,
[\,\exp({\cal A}),\,\Pi_{i}^{\gamma}({\bf r})\,]\,{\|}\approx 0\;,
\label{eq:psicom1f}
\end{equation}
where $\approx$ indicates a `soft' equality,
that only holds when the equation acts on a
state $|{\phi}\rangle$ annihilated by $b_Q^a({\bf k})$.
The commutator
$[\,\exp({\cal A}),\,\Pi_{i}^{\gamma}({\bf r})\,]$ in
Eq.~(\ref{eq:psicom1f}) reflects the fact that when
the gluonic `color' charge density is assembled to
the left of the candidate $\Psi$, the momentum conjugate
to the gauge field must be moved from the extreme
right to the extreme left of ${\|}\,\exp({\cal{A}})\,{\|}$.
Since ${\cal A}$ is a complicated multi-linear
functional of the gauge fields, but has a simple linear
dependence on $\Pi_i^{\gamma}({\bf{r}})$,
it is useful to represent it as
\begin{equation}
{\cal{A}}=
i{\int}d{\bf{r}}\;
\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})\;
\Pi_i^{\gamma}({\bf{r}})\;,
\label{eq:Awhole}
\end{equation}
where
\begin{equation}
\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})=
\sum_{n=1}^\infty g^n
{\cal{A}}_{(n)i}^{\gamma}({\bf{r}})\;,
\label{eq:calAbar}
\end{equation}
and the ${\cal{A}}_{(n)i}^{\gamma}({\bf{r}})$ are
elements in a series whose initial term is
${\cal{A}}^{\gamma}_{(1)i}({\bf{r}})=\psi^{\gamma}_{(1)i}({\bf{r}})$.
All the ${\cal{A}}_{(n)i}^{\gamma}({\bf{r}})$ consist of gauge fields
and functionals of gauge fields only; there are no
conjugate momenta, $\Pi_i^{\gamma}({\bf{r}})$, in
any of the ${\cal{A}}_{(n)i}^{\gamma}({\bf{r}})$.
We also showed in Ref.~\cite{bellchenhall}, that
Eq.~(\ref{eq:psicom1f})
is equivalent to
\begin{equation}
[\,b_Q^a({\bf{k}}),\,{\cal{A}}_n\,]\approx g\,f^{a\beta\gamma}
{\int}d{\bf r}\,e^{-i{\bf{k\cdot r}}}\,
A^\beta_i({\bf{r}})\,
[\,{\cal{A}}_{n-1},\,
\Pi^\gamma_i({\bf{r}})\,]\;,
\label{eq:recrel}
\end{equation}
for ${\cal{A}}_{n}$ with $n>1\,,$ where the ${\cal A}_{n}$
form the series ${\cal A}=\sum_{\,n=1}^\infty{\cal A}_{n},$
and each ${\cal A}_{n}$ can be represented as
\begin{equation}
{\cal A}_{n}=ig^n{\int}d{\bf r}\;{\cal{A}}_{(n)i}^{\gamma}({\bf{r}})
\Pi_{i}^{\gamma}({\bf{r}})\;.
\label{eq:asubn}
\end{equation}
If ${\cal A}_{n}$ satisfies
Eq.~(\ref{eq:recrel}), then the ${\Psi}$ defined in
Eq.~(\ref{eq:Apsi}) will also necessarily satisfy
Eq.~(\ref{eq:subcon}), and the state ${\Psi}\,|{\phi}\rangle$
will implement the non-Abelian `pure glue' Gauss's law.
\bigskip
In Ref.~\cite{bellchenhall} we gave the form of
${\cal A}$ as a functional of the auxiliary operator-valued
constituents
\begin{equation}
{\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})
=\prod_{m=1}^\eta{\textstyle \frac{\partial_{j}}{\partial^{2}}
\overline{{\cal A}_{j}^{\alpha [m]}}({\bf r})}=\prod_{m=1}^\eta
\overline{{\cal Y}^{\alpha[m]}}({\bf{r}})
=\overline{{\cal Y}^{\alpha[1]}}({\bf{r}})\,
\overline{{\cal Y}^{\alpha[2]}}({\bf{r}})\,\cdots
\overline{{\cal Y}^{\alpha[\eta]}}({\bf{r}})\;,
\label{eq:defM}
\end{equation}
and
\begin{equation}
\overline{{\cal B}_{(\eta) i}^{\beta}}({\bf r})=
a_i^{\beta}({\bf r})+\,
(\,\delta_{ij}-{\textstyle\frac{\eta}{\eta+1}}
{\textstyle\frac{\partial_{i}\partial_{j}}{\partial^{2}}}\,)
\overline{{\cal A}_{i}^{\beta}}({\bf r})\;,
\label{eq:calB1b}
\end{equation}
where
\begin{equation}
\overline{{\cal Y}^{\alpha}}({\bf r})=
{\textstyle \frac{\partial_{j}}{\partial^{2}}
\overline{{\cal A}_{j}^{\alpha}}({\bf r})}\;\;\;
\mbox{\small and}\;\;\;
{\cal Y}^{\alpha}_{(s)}({\bf r})=
{\textstyle \frac{\partial_{j}}{\partial^{2}}
{\cal A}_{(s)j}^{\alpha}({\bf r})}\;.
\label{eq:defY}
\end{equation}
The defining equation for ${\cal A}$ is the recursive
\begin{equation}
{\cal{A}}=\sum_{\eta=1}^\infty
{\textstyle\frac{ig^\eta}{\eta!}}{\int}d{\bf r}\;
\{\,\psi^{\gamma}_{(\eta)i}({\bf{r}})\,+
\,f^{\vec{\alpha}\beta\gamma}_{(\eta)}\,
{\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})\,
\overline{{\cal{B}}_{(\eta) i}^{\beta}}({\bf{r}})\,\}\;
\Pi^\gamma_i({\bf{r}})\;.
\label{eq:inteq2}
\end{equation}
In Ref.~\cite{bellchenhall}, we presented
this form as a conjecture that we had verified to sixth order only.
In this work, we will prove that ${\Psi}\,|{\phi}\rangle$
satisfies the `pure glue' Gauss's law by showing
that the ${\cal A}$ given in Eq.~(\ref{eq:inteq2}) satisfies
Eq.~(\ref{eq:recrel}).
\bigskip
The form of ${\cal A}$ suggests that the proposition that
it satisfies Eq.~(\ref{eq:recrel}) is
well suited to an inductive proof. We observe that
two kinds of terms appear on
the right hand side of Eq.~(\ref{eq:inteq2}). One is the
inhomogeneous term
$\psi^{\gamma}_{(\eta)i}({\bf{r}})$;
the other is the product of
$\overline{{\cal B}_{(\eta) i}^{\beta}}({\bf r})$
and ${\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})$.
$\overline{{\cal B}_{(\eta) i}^{\beta}}({\bf r})$ is a functional of
$\overline{{\textstyle{\cal A}_{i}^{\beta}}}({\bf r})$, and
${\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})$
is a multilinear functional of
$\overline{{\cal Y}^{\beta}}({\bf{r}})$, which is
given as a functional of
$\overline{{\textstyle{\cal A}_{i}^{\beta}}}({\bf r})$
in Eq.~(\ref{eq:defY}). It is useful to
examine the $r^{th}$ order components of
${\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})$ and
$\overline{{\cal B}_{(\eta) i}^{\beta}}({\bf r})$. These
are given, respectively, by
\begin{equation}
{\cal M}_{(\eta,r)}^{\vec{\alpha}}({\bf r})=
\Theta (r-\eta)\sum_{r[1],\cdots, r[\eta]}
\delta_{r[1]+\cdots +r[\eta]-r}\prod_{m=1}^{\eta}
{\cal Y}_{(r[m])}^{\alpha [m]}({\bf r})\;,
\label{eq:orderM}
\end{equation}
and
\begin{equation}
{\cal B}_{(\eta,r)i}^{\beta}({\bf r})=
\delta_{r}\,a_{i}^{\beta}({\bf r})
+(\,\delta_{ij}-{\textstyle\frac{\eta}{\eta+1}}\,
{\textstyle\frac{\partial_{i}\partial_{j}}{\partial^{2}}}\,)\,
{\cal A}_{(r)j}^{\beta}({\bf r})\;,
\label{eq:orderB}
\end{equation}
where the subscript $r$ is an integer-valued index that labels the
order in the expansion of
$\overline{{\textstyle{\cal A}_{i}^{\gamma}}}({\bf r}),$
and $\delta_{r}$ is the Kronecker `delta' that
vanishes unless $r=0$.
In Eqs.~(\ref{eq:defM}) and (\ref{eq:orderM}),
$\eta$ is a `multiplicity index' that defines
the multilinearity of
${\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})\,$ in
$\overline{{\cal Y}^{\beta}}({\bf{r}}).$
Eqs.~(\ref{eq:inteq2})-(\ref{eq:orderB}) demonstrate
that an ${\cal A}_{r}$ that appears on the l.h.s. of
Eq.~(\ref{eq:inteq2}) is given in terms of the $r^{th}$ order
inhomogeneous term $\psi_{(r)j}^{\gamma}({\bf r})\,
\Pi_{j}^{\gamma}({\bf r})$,
and ${\cal A}_{(r^{\prime }) j}^{\beta}$
terms on the r.h.s. of this equation in which $r^\prime < r$.
To emphasize this very crucial observation,
we note that in addition to the $g^\eta$ that appears as an
overall factor in Eq.~(\ref{eq:inteq2}), each
$\overline{{\cal A}_{j}^{\beta}}({\bf r})$ in
${\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})$
and $\overline{{\cal B}_{(\eta) i}^{\beta}}$
carries its own complement of coupling
constants --- $g^r$ for each
order $r$. The $r^{th}$ order term on the
l.h.s. of Eq.~(\ref{eq:inteq2}) therefore
depends on r.h.s.
contributions from ${\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})$ and
$\overline{{\cal B}_{(\eta) i}^{\beta}}\,({\bf{r}})$
whose orders do not add up to $r$,
but only to $r-\eta$. Since the summation in
Eq.~(\ref{eq:inteq2}) begins with $\eta=1,$ the
highest possible order of ${\cal A}^\gamma_{{(r^\prime)}j}$ that can
appear on the r.h.s. of Eq.~(\ref{eq:inteq2}),
when ${\cal A}_{r}$ is on the l.h.s., is
${\cal A}^\gamma_{{(r-1)}j}$ --- and that must stem from the
${\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})$ with
the multiplicity index $\eta=1$. Contributions
from ${\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})$
with higher multiplicity
indices are restricted to ${\cal A}^\gamma_{{(r^\prime)}j}$ with even
lower order $r^\prime$. This feature of Eq.~(\ref{eq:inteq2})
naturally leads us to consider an
inductive proof --- one in which we assume Eq.~(\ref{eq:inteq2})
for all ${\cal A}_{r}$
with $r\leq N,$ and then use that assumption to
prove it for ${\cal A}_{r}$ with $r=N+1.$
\bigskip
The fact that Eq.~(\ref{eq:recrel}) is a `soft' equation, is
an impediment to an inductive proof of the proposition
that ${\cal A}_{n},$ defined by Eq.~(\ref{eq:inteq2}),
satisfies it. In order to carry out the
needed inductive proof, we must infer correct `hard'
generalizations of both these
equations, in which ${\cal A}$ is replaced
by $i{\int}d{\bf{r}}\;
\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})\;
V_i^{\gamma}({\bf{r}}),$
where $V_i^{\gamma}({\bf{r}})$ is
{\em any} field that transforms appropriately, and
$\partial_{i}V_i^{\gamma}({\bf{r}})$ is
not required to annihilate any states.
The generalization we seek is an exact equality between
operator-valued quantities --- one that is true in general, and
not only when both sides of the equation project on a specified
subset of states. Such a generalization would, in
particular, allow us to use many different spatial vectors in
the role of $V_i^{\gamma}({\bf{r}})$ in the
course of the inductive proof.
\bigskip
We have made the necessary generalization,
and have arrived at the defining equation for
the $n^{th}$ order term of
$i{\int}d{\bf{r}}\;
\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})\;
V_i^{\gamma}({\bf{r}})$,
that generalizes Eq.~(\ref{eq:inteq2}):
\begin{eqnarray}
& & ig^{n}\int d{\bf r}
{\cal A}_{(n)i}^{\gamma}({\bf r})\,V_{i}^{\gamma}({\bf r})
={\textstyle \frac{ig^{n}}{n!}}\int d{\bf r}\,
\psi_{(n)i}^{\gamma}({\bf r})\,V_{i}^{\gamma}({\bf r})
\nonumber \\
& & \;\;\;\;+\sum_{\eta=1}{\textstyle\frac{ig^{\eta}}{\eta!} }\,
f^{\vec{\alpha}\beta\gamma}_{(\eta)} \,\sum_{u=0}\sum_{r=\eta}
\delta_{r+u+\eta-n}\int d{\bf r}\,
{\cal M}_{(\eta,r)}^{\vec{\alpha}}({\bf r})
\,{\cal B}_{(\eta,u)i}^{\beta}({\bf r})\,
V^{\gamma}_{i}({\bf r})\;.
\label{eq:LCA1}
\end{eqnarray}
The generalization of Eq.~(\ref{eq:recrel}) --- we make
use of the configuration-space representation
of the Gauss's law operator in this case, instead of its
Fourier transform --- is
\begin{eqnarray}
&& i \int d{\bf r}^\prime\left[\,\partial_{i}\Pi_{i}^{a}({\bf r}),
{\cal A}_{(n)j}^{\gamma}({\bf r}^\prime)\,\right]\,
V_{j}^{\gamma}({\bf r}^\prime)
+ \delta_{n-1}f^{a\mu \gamma}\,A_{i}^{\mu}({\bf r})\,
V_{i}^{\gamma}({\bf r})
\nonumber \\
& &\;\;\;\; - \sum_{\eta=1}\sum_{r=\eta}\delta_{r+\eta-(n-1)}
{\textstyle\frac{B(\eta)}{\eta!}}\,f^{a\mu c}
f^{\vec{\alpha}c\gamma}_{(\eta)}
A_{i}^{\mu}({\bf r})\,{\textstyle \frac{\partial_{i}}{\partial^{2}}}
\left({\cal M}_{(\eta,r)}^{\vec{\alpha}}({\bf r})\,\partial_{j}
V_{j}^{\gamma}({\bf r})\right)
\nonumber \\
& &\;\;\;\; +
\sum_{\eta=0}\sum_{t=1}\sum_{r=\eta}\delta_{r+t+\eta-n}
(-1)^{t-1}{\textstyle \frac{B(\eta)}{\eta!(t-1)!(t+1)}}\,
f^{\vec{\mu} a
\lambda}_{(t)}f^{\vec{\alpha}\lambda\gamma}_{(\eta)}\,
{\cal R}_{(t)}^{\vec{\mu}}({\bf r})\,
{\cal M}_{(\eta,r)}^{\vec{\alpha}}({\bf r})\,
\partial_{i}V_{i}^{\gamma}({\bf r})
\nonumber \\
& &\;\;\;\; + f^{a\mu d}A_{i}^{\mu}({\bf r})
\sum_{\eta=0}\sum_{t=1}\sum_{r=\eta}\delta_{r+t+\eta-(n-1)}
(-1)^{t}{\textstyle\frac{B(\eta)}{\eta!(t+1)!}}\,
f^{\vec{\nu}d\lambda}_{(t)}f^{\vec{\alpha}\lambda\gamma}_{(\eta)}
{\textstyle\frac{\partial_{i}}{\partial^{2}}}
\left({\cal R}_{(t)}^{\vec{\nu}}({\bf r})\,
{\cal M}_{(\eta,r)}^{\vec{\alpha}}({\bf r})\,
\partial_{j}V_{j}^{\gamma}({\bf r})\right)
\nonumber \\
&&\;\;\;\;= -if^{a\mu\sigma}A_{i}^{\mu}({\bf r})\;
\int d{\bf r}^\prime\left[\Pi_{i}^{\sigma}({\bf r}),\,
{\cal A}_{(n-1)j}^{\gamma}({\bf r}^\prime)\right]\,
V_{j}^{\gamma}({\bf r}^\prime)\;,
\label{eq:LCA2}
\end{eqnarray}
where $B(\eta)$ denotes the $\eta^{th}$ Bernoulli number.
Eq.~(\ref{eq:LCA2})
relates ${\cal A}_{(n)j}^{\gamma}({\bf r})$ with $n\geq 1$,
on the l.h.s. of the equation, to
${\cal A}_{(n-1)j^\prime}^{\gamma^{\,\prime}}({\bf r}^\prime)$ on
the r.h.s.; ${\cal A}_{(n)j}^{\gamma}({\bf r})$
with $n=0$ is not required for the representation
of $\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})$
given in Eq.~(\ref{eq:calAbar}),
and therefore does not have to be considered.
${\cal A}_{(n)j}^{\gamma}({\bf r})$ with
$n=1$ {\em is} required, but $\left[\partial_{i}\Pi_{i}^{a}({\bf r}),
{\cal A}_{(1)j}^{\gamma}({\bf r}^\prime)\right]$ can not be described
properly by Eq. (\ref{eq:LCA2}),
unless ${\cal A}_{(0)j}^{\gamma}({\bf r})$ on the r.h.s.
of Eq. (\ref{eq:LCA2}) is given an
appropriate definition.
The only equation like Eq. (\ref{eq:LCA2}), but with
$\int d{\bf r}^\prime\left[\partial_{i}\Pi_{i}^{a}({\bf r})
{\cal A}_{(1)j}^{\gamma}({\bf r}^\prime)\right]
V_{j}^{\gamma}({\bf r}^\prime)$ appearing on its l.h.s.,
is Eq.~(\ref{eq:thetabqcom}) with $\Pi_{i}^{\gamma}({\bf r})$
replaced by
$V_{i}^{\gamma}({\bf r})$. We have
formulated Eq. (\ref{eq:LCA2}) so that it includes the case of
$\int d{\bf r}^\prime\left[\partial_{i}\Pi_{i}^{a}({\bf r}),
{\cal A}_{(1)j}^{\gamma}({\bf r}^\prime)\right]
V_{j}^{\gamma}({\bf r}^\prime)$
on the l.h.s., by including the r.h.s. of Eq.~(\ref{eq:thetabqcom})
for the $n=1$ case. To include that case
correctly, we define the degenerate
${\cal M}_{(\eta,r)}^{\vec{\alpha}}({\bf r})$ with $\eta=r=0$
as ${\cal M}_{(0,0)}^{\vec{\alpha}}({\bf r})=1$, and the degenerate
${\cal A}_{(0)j}^{\gamma}({\bf r})=0$. We will refer to
Eq.~(\ref{eq:LCA2}) as the
`fundamental theorem' for this construction of ${\Psi}$.
\bigskip
The general plan for the inductive proof of
Eq.~(\ref{eq:LCA2}) is as follows: We {\em assume}
Eq.~(\ref{eq:LCA2})
for all values of $n\leq N.$ We then observe that, in the $n=N+1$
case to be proven, the r.h.s. of Eq.~(\ref{eq:LCA2}) becomes
${\sf RHS}_{(N+1)}=-if^{a\mu\sigma}A_{i}^{\mu}({\bf r})\;
\int d{\bf r}^\prime\left[\Pi_{i}^{\sigma}({\bf r}),\,
{\cal A}_{(N)\,j}^{\gamma}({\bf r}^\prime)\right]\,
V_{j}^{\gamma}({\bf r}^\prime).$
We use Eq.~(\ref{eq:LCA1}) to
substitute for the ${\cal A}_{(N)j}^{\gamma}\,V_{j}^{\gamma}\,$
in ${\sf RHS}_{(N+1)}$, and
evaluate the resulting commutators
$\left[\,\Pi_{i}^{\sigma}({\bf r}),\,
\psi_{(N)j}^{\gamma}({\bf r}^\prime)\,\right]$,
$\left[\,\Pi_{i}^{\sigma}({\bf r}),\,
{\cal M}_{(\eta,r)}^{\vec{\alpha}}({\bf r}^\prime)\,\right]$, and
$\left[\,\Pi_{i}^{\sigma}({\bf r}),\,
{\cal B}_{(\eta,u)j}^{\beta}({\bf r}^\prime)\,\right]$. Since
$\psi_{(N)j}^{\gamma}({\bf r}^\prime)$ is a
known inhomogeneity in Eq.~(\ref{eq:LCA1}),
$\left[\,\Pi_{i}^{\sigma}({\bf r}),\,
\psi_{(N)j}^{\gamma}({\bf r}^\prime)\,\right]$
can be explicitly evaluated. In expanding the
$f^{\vec{\alpha}\beta\gamma}_{(\eta)}
\left[\,\Pi_{i}^{\sigma}({\bf r}),\,
{\cal M}_{(\eta,r)}^{\vec{\alpha}}({\bf r}^\prime)\,\right]$ that
result from the substitution of Eq.~(\ref{eq:LCA1}) into
${\sf RHS}_{(N+1)}$, we make use
of the identity
\begin{eqnarray}
& & f^{\vec{\alpha}\delta\gamma}_{(\eta)}\sum_{r=\eta}
\delta_{r+\eta+u-N}\left[{\sf Q}({\bf r}),
{\cal M}_{(\eta,r)}^{\vec{\alpha}}({\bf r}^\prime)\right]
\nonumber \\
&&\;\;\;\;=- \left[{\cal P}^{(0)}_{(\alpha ,\beta [\eta-1])}
f^{\alpha\delta e}f^{\vec{\beta} e\gamma}_{(\eta-1)}\right]
\sum_{p=\eta-1}\sum_{r[\eta]=1}\delta_{p+r[\eta]+u+\eta-N}
\left[\,{\sf Q}({\bf r}),\,
{\cal Y}_{(r[\eta])}^{\alpha}({\bf r}^\prime)\,\right]
{\cal M}_{(\eta-1,p)}^{\vec{\beta}}({\bf r}^\prime)\;,
\label{eq:perm}
\end{eqnarray}
where ${\sf Q}({\bf r})$ is any arbitrary
operator; at times, the commutator $[{\sf Q}({\bf r})\,,
{\cal M}_{(\eta,r)}^{\vec{\alpha}}({\bf r}^\prime)\,]$ will represent
a partial derivative $\partial_{j}
{\cal M}_{(\eta,r)}^{\vec{\alpha}}({\bf r}^\prime)\,.$
${\cal P}^{(0)}_{(\alpha ,\beta [\eta-1])}$ represents
a sum over permutations over the indices labeling
the ${\cal Y}_{(r[\eta ])}^{\alpha[\eta]}({\bf r}^\prime)$ factors that
constitute ${\cal M}_{(\eta,r)}^{\vec{\alpha}}({\bf r}^\prime)$,
as shown in Eq.~(\ref{eq:defM}).
${\cal P}^{(0)}_{(\alpha ,\beta [\eta-1])}$
is defined by
\begin{equation}
\left[{\cal P}_{(e ,\beta [\eta-1])}^{(0)}f^{e\delta f}
f^{\vec{\beta} f\gamma}_{(\eta-1)}\right]
{\cal M}_{(\eta-1)}^{\vec{\beta}}=\sum_{s=0}^{\eta-1}
f^{\vec{\beta}\delta u}_{(s)}f^{ue v}
f^{\vec{\sigma}v\gamma}_{(\eta-s-1)}
{\cal M}_{(s)}^{\vec{\beta}}{\cal M}_{(\eta-s-1)}^{\vec{\sigma}}\;.
\label{eq:defperm}
\end{equation}
Eqs.~(\ref{eq:perm}) and (\ref{eq:defperm}) apply not only to
those specific cases, but
also to all other operators --- such as
${\cal R}_{(\eta)}^{\vec{\alpha}}({\bf r}^\prime)$ ---
that similarly are products of factors, identical
except for their Lie group indices
contracted over chains of structure functions.
\bigskip
With the substitution of of Eq.~(\ref{eq:LCA1}) into
${\sf RHS}_{(N+1)}$, and extensive integration by parts,
we have replaced the commutator
$[\,\Pi_{i}^{\sigma}({\bf r}),\,
{\cal A}_{(N)j}^{\gamma}({\bf r}^\prime)\,]$ which appears
in ${\sf RHS}_{(N+1)}$, with products of
chains of ${\cal A}_{(n^\prime)j}^{\beta^\prime}({\bf r}^\prime)\,$
and one commutator $[\,\Pi_{i}^{\sigma}({\bf r}),\,
{\cal A}_{(l)j}^{\gamma}({\bf r}^\prime)\,]$ with $l\leq N-1$.
Although the $[\,\Pi_{i}^{\sigma}({\bf r}),\,
{\cal A}_{(N)j}^{\gamma}({\bf r}^\prime)\,]$ in ${\sf RHS}_{(N+1)}$
is {\em not} covered by the inductive axiom --- it is
the r.h.s. of the equation
for the $n=N+1$ case --- the $[\,\Pi_{i}^{\sigma}({\bf r}),\,
{\cal A}_{(l)j}^{\gamma}({\bf r}^\prime)\,]$ with
$l\leq N-1$, which have been substituted into ${\sf RHS}_{(N+1)}$,
{\em are} covered by this axiom.
We can therefore use the inductive axiom to
replace all these latter commutators by their corresponding
left hand side equivalents from Eq.~(\ref{eq:LCA2}). After
extensive algebraic manipulations, we can demonstrate that
${\sf RHS}_{(N+1)}$ has been transformed into
the {\em left hand side}
of Eq.~(\ref{eq:LCA2}) for the case in which all $n$ have
been replaced by $n=N+1$. This, then, completes the
inductive proof of Eq.~(\ref{eq:LCA2}).
The details of the argument are given in two appendices. Appendix
A proves some necessary lemmas; Appendix B proves the
fundamental theorem.
\section{The inclusion of quarks}
In Eq.~(\ref{eq:subcon}), we have implemented
the `pure glue' form of Gauss's law. The complete Gauss's law
operator, when the quarks are included as sources
for the chromoelectric field, takes the form
\begin{equation}
{\hat {\cal G}}^{a}({\bf r})=\partial_{i}
\Pi_{i}^{a}({\bf r})+gf^{abc}A_{i}^{b}({\bf r})
\Pi_{i}^{c}({\bf r})+j_{0}^{a}({\bf r})\;,
\label{eq:Ghat}
\end{equation}
where
\begin{equation}
j^a_{0}({\bf{r}})=
g\,\,\psi^\dagger({\bf{r}})\,
{\textstyle\frac{\lambda^a}{2}}\,\psi({\bf{r}})\;,
\label{eq:quark}
\end{equation}
and where the ${\lambda^a}$ represent the Gell-Mann
matrices. To implement the `complete' Gauss's law --- a form that
incorporates quark as well as gluon color
--- we must solve the equation
\begin{equation}
{\hat {\cal G}}^{a}({\bf r})\,{\hat {\Psi}}\,|{\phi}\rangle =0\;.
\label{eq:Ggqlaw}
\end{equation}
Our approach to this problem will be based
on the fact that ${\hat {\cal G}}^{a}({\bf r})$ and
${\cal G}^{a}({\bf r})$ are unitarily equivalent, so that
\begin{equation}
\hat{\cal{G}}^a({\bf{r}})={\cal{U}}_{\cal{C}}\,
{\cal{G}}^a({\bf{r}})\,{\cal{U}}^{-1}_{\cal{C}}\;,
\label{eq:GgqGg}
\end{equation}
where ${\cal{U}}_{\cal{C}}=e^{{\cal C}_{0}}
e^{\bar {\cal C}}$ and where
${{\cal C}_{0}}$ and ${\bar {\cal C}}$ are given by
\begin{equation}
{\cal C}_{0}=i\,\int d{\bf{r}}\,
{\textstyle {\cal X}^{\alpha}}({\bf r})\,j_{0}^{\alpha}({\bf r})\;,
\;\;\;\;\;\mbox{and}\;\;\;\;\;\;\;
{\bar {\cal C}}=i\,\int d{\bf{r}}\,
\overline{{\cal Y}^{\alpha}}({\bf r})\,j_{0}^{\alpha}({\bf r})\;.
\label{eq:CCbar}
\end{equation}
We can demonstrate this unitary equivalence by noting
that Eq~(\ref{eq:GgqGg}) can be rewritten as
\begin{equation}
e^{-{\cal C}_{0}}\,{\hat {\cal G}}^{a}({\bf{r}})\,e^{{\cal C}_{0}}=
e^{\bar {\cal C}}\,{\cal G}^{a}({\bf{r}})\,e^{-\bar {\cal C}}\;.
\label{eq:gghatp}
\end{equation}
In this form, the unitary equivalence can be shown
to be a direct consequence of the
fundamental theorem --- $i.\,e.$ Eq.~(\ref{eq:LCA2}).
We observe that the l.h.s. of
Eq.~(\ref{eq:gghatp}) can be expanded, using the
Baker-Hausdorff-Campbell (BHC) theorem, as
\begin{equation}
e^{-{\cal C}_{0}}\,{\hat {\cal G}}^{a}({\bf{r}})\,e^{{\cal C}_{0}}=
{\hat {\cal G}}^{a}({\bf{r}})+{\cal S}_{(1)}^a+
\cdots +{\cal S}_{(n)}^a+\cdots\;,
\label{eq:uniser}
\end{equation}
where ${\cal S}_{(1)}^a=-[\,{\cal C}_{0},\,\hat{\cal{G}}^a({\bf{r}})\,]$
and
${\cal S}_{(n)}^a=-(1/n)[\,{\cal C}_{0},\,{\cal S}_{(n-1)}^a\,]$.
We observe that
\begin{equation}
{\cal S}_{(1)}^a=-\left[\,\delta_{a,c}+gf^{abc}{\cal X}^b({\bf{r}})
+gf^{abc}A_i^b({\bf{r}})
{\textstyle\frac{\partial_{i}}{\partial^2}}\,\right]\,j_0^c({\bf{r}})\;,
\label{eq:Sone}
\end{equation}
and that
\begin{eqnarray}
{\cal S}_{(n)}^a={\textstyle\frac{{(-1)}^{n+1}}{n\,!}}
&&\left[\left(\,g^{n -1}f^{\vec{\alpha}a\gamma}_{(n-1)}\,
{\cal R}^{\vec{\alpha}}_{(n-1)}({\bf r})\,+
g^{n}\, f^{\vec{\alpha}a\gamma}_{(n)}\,
{\cal R}^{\vec{\alpha}}_{(n)}({\bf r})\,\right)\,
j_0^{\gamma}({\bf{r}}\,)\right.
\nonumber\\
&&+ \left .g^n f^{abc}\,f^{\vec{\alpha}c\gamma}_{(n-1)}\,
A_i^b({\bf{r}})\,
{\textstyle\frac{\partial_{i}}{\partial^2}}
\left(\,{\cal R}^{\vec{\alpha}}_{(n-1)}({\bf r})\,
j_0^{\gamma}({\bf{r}})\,\right)\, \right]\;.
\label{eq:Snth}
\end{eqnarray}
Eq~(\ref{eq:Snth}) shows that two $g^{n}
f^{\vec{\alpha}a\gamma}_{(n)}\,
{\cal R}^{\vec{\alpha}}_{(n)}({\bf r})\,j_0^{\gamma}({\bf{r}})$ terms
will appear in this series:
one in ${\cal S}_{(n)}^a$, and one in ${\cal S}_{(n+1)}^a$. The sum of
these terms will have the
coefficient $\left[\frac{1}{n!}-\frac{1}{(n+1)!}\right]=
\frac{1}{(n+1)(n-1)!}$. When
the BHC series is summed, we find that
\begin{eqnarray}
e^{-{\cal C}_{0}}\,{\hat {\cal G}}^{a}({\bf r})\,e^{{\cal C}_{0}}
&&={\hat {\cal G}}^{a}({\bf r})-j_{0}^{a}({\bf r})-gf^{abc}
A_{i}^{b}({\bf r})\,{\textstyle\frac{\partial_{i}}{\partial^{2}}}
j_{0}^{c}({\bf r})
\nonumber \\
&&-\sum_{n=1} (-1)^{n} g^{n}{\textstyle \frac{1}{(n-1)!(n+1)}}
f^{\vec{\alpha}a\gamma}_{(n)}\,
{\cal R}_{(n)}^{\vec{\alpha}}({\bf r})\,
j_{0}^{\gamma}({\bf r})
\nonumber \\
&&+gf^{abc}A_{i}^{b}({\bf r})\sum_{n=1} (-1)^{n}g^{n}
{\textstyle \frac{1}{(n+1)!}}f^{\vec{\alpha}c\gamma}_{(n)}\,
{\textstyle \frac{\partial_{i}}{\partial^{2}}}
\left(\,{\cal R}_{(n)}^{\vec{\alpha}}({\bf r})\,
j_{0}^{\gamma}({\bf r})\,\right)\;.
\label{eq:GGhatleft}
\end{eqnarray}
\bigskip
To prepare for the evaluation of
$e^{\bar {\cal C}}\,{\cal G}^{a}({\bf r})\,e^{-\bar {\cal C}}$,
the r.h.s. of Eq.~(\ref{eq:gghatp}), we multiply both sides of
Eq~(\ref{eq:LCA2}) for the
$n^{th}$ order term, ${\cal A}_{(n)i}^{\gamma}({\bf r})$, by $g^n$,
and then sum over the integer-valued indices
$r$ and $n$ (in that order). The result --- a formulation
of the fundamental theorem that
no longer applies to the individual orders,
${\cal A}_{(n)j}^{\gamma}({\bf r}),$ but to their sum,
$\overline{{\cal A}_{j}^{\gamma}}({\bf r})$ --- is
\begin{eqnarray}
i\int d{\bf r}^\prime&&[\,\partial_{i}\Pi_{i}^{a}({\bf r}),\,
\overline{{\cal A}_{j}^{\gamma}}({\bf r}^\prime)\,]\,
V_{j}^{\gamma}({\bf r}^\prime)
+ igf^{a\beta d}A_{i}^{\beta}({\bf r})\int d{\bf r}^\prime
[\,\Pi_{i}^{d}({\bf r}),\,
\overline{{\cal A}_{j}^{\gamma}}({\bf r}^\prime)\,]\,
V_{j}^{\gamma}({\bf r}^\prime)
\nonumber \\
&&= -gf^{a\mu d}\,A_{i}^{\mu}({\bf r})\,V_{i}^{d}({\bf r})
\nonumber \\
&&+\sum_{\eta=1}{\textstyle\frac{g^{\eta +1}B(\eta)}{\eta!}}\,
f^{a\beta c}f^{\vec{\alpha}c\gamma}_{(\eta)}\,A_{i}^{\beta}({\bf r})\,
{\textstyle \frac{\partial_{i}}{\partial^{2}}}\left(\,
{\cal M}_{(\eta)}^{\vec{\alpha}}({\bf r})\,
\partial_{j}V_{j}^{\gamma}({\bf r})\,\right)
\nonumber \\
&&-\sum_{\eta=0}\sum_{t=1} (-1)^{t-1}g^{t+\eta}\,
{\textstyle \frac{B(\eta)}{\eta!(t-1)!(t+1)}}\,
f^{\vec{\mu}a\lambda}_{(t)}f^{\vec{\alpha}\lambda\gamma}_{(\eta)}\,
{\cal R}_{(t)}^{\vec{\mu}}({\bf r})\,
{\cal M}_{(\eta)}^{\vec{\alpha}}({\bf r})\,
\partial_{i}V_{i}^{\gamma}({\bf r})
\nonumber \\
&&-gf^{a\beta d}A_{i}^{\beta}({\bf r})\,
\sum_{\eta=0}\sum_{t=1} (-1)^{t}g^{t+\eta}\,
{\textstyle\frac{B(\eta)}{\eta!(t+1)!}}
f^{\vec{\mu}d\lambda}_{(t)}
f^{\vec{\alpha}\lambda\gamma}_{(\eta)}
{\textstyle\frac{\partial_{i}}{\partial^{2}}}\,
\left(\,{\cal R}_{(t)}^{\vec{\mu}}({\bf r})\,
{\cal M}_{(\eta)}^{\vec{\alpha}}({\bf r})\,
\partial_{j}V_{j}^{\gamma}({\bf r})\,\right)\;.
\label{eq:Asum}
\end{eqnarray}
If we again use the BHC expansion, as in Eq~(\ref{eq:uniser}),
but this time to represent
\begin{equation}
e^{\bar {\cal C}}\,{\cal G}^{a}({\bf r})\,e^{-\bar {\cal C}}=
{\cal G}^{a}({\bf r})+{\bar{\cal S}}_{(1)}^a+\cdots
+{\bar{\cal S}}_{(n)}^a+\cdots,
\end{equation}
we find that the first order term, ${\bar{\cal S}}_{(1)}^a$,
can be obtained directly from Eq.~(\ref{eq:Asum}) and is
\begin{eqnarray}
{\bar{\cal S}}_{(1)}^a &&=
-gf^{a\mu\gamma}\,A_{i}^{\mu}({\bf r})\,
{\textstyle\frac{\partial_{i}}{\partial^2}}
j_0^{\gamma}({\bf{r}})
\nonumber \\
&&+\sum_{s=1}{\textstyle\frac{g^{s+1}B(s)}{s!}}\,
f^{a\beta c}f^{\vec{\alpha}c\gamma}_{(s)}\,A_{i}^{\beta}({\bf r})\,
{\textstyle \frac{\partial_{i}}{\partial^{2}}}\left(\,
{\cal M}_{(s)}^{\vec{\alpha}}({\bf r})\,
j_0^{\gamma}({\bf{r}})\,\right)
\nonumber \\
&&-\sum_{s=0}\sum_{t=1} (-1)^{t-1}g^{t+s}\,
{\textstyle \frac{B(s)}{s!(t-1)!(t+1)}}\,
f^{\vec{\mu}a\lambda}_{(t)}f^{\vec{\alpha}\lambda\gamma}_{(s)}\,
{\cal R}_{(t)}^{\vec{\mu}}({\bf r})\,
{\cal M}_{(s)}^{\vec{\alpha}}({\bf r})\,
j_0^{\gamma}({\bf{r}})
\nonumber \\
&&-gf^{a\beta d}\,A_{i}^{\beta}({\bf r})\,
\sum_{s=0}\sum_{t=1} (-1)^{t}g^{t+s}
{\textstyle\frac{B(s)}{s!(t+1)!}}\,
f^{\vec{\mu}d\lambda}_{(t)}
f^{\vec{\alpha}\lambda\gamma}_{(s)}\,
{\textstyle\frac{\partial_{i}}{\partial^{2}}}
\left(\,{\cal R}_{(t)}^{\vec{\mu}}({\bf r})\,
{\cal M}_{(s)}^{\vec{\alpha}}({\bf r})\,
j_0^{\gamma}({\bf{r}})\,\right)\;;
\label{eq:SRone}
\end{eqnarray}
the $k^{th}$ order term is
\begin{eqnarray}
{\bar{\cal S}}_{(k)}^a & = & {\textstyle\frac{g^{k}}{k\, !}}\,
f^{a\mu d}f^{\vec{\alpha}d\gamma}_{(k-1)}\,A_{i}^{\mu}({\bf r})\,
{\textstyle \frac{\partial_{i}}{\partial^{2}}}
\left(\,{\cal M}_{(k-1)}^{\vec{\alpha}}({\bf r})\,
j_0^{\gamma}({\bf{r}})\,\right)
\nonumber \\
&+& \sum_{s=1}{\textstyle\frac{g^{s+k}B(s)}{s!\,k!}}\,
f^{a\beta c}f^{\vec{\alpha}c\gamma}_{(s+k-1)}\,
A_{i}^{\beta}({\bf r})\,
{\textstyle \frac{\partial_{i}}{\partial^{2}}}
\left(\,{\cal M}_{(s+k-1)}^{\vec{\alpha}}({\bf r})\,
j_0^{\gamma}({\bf{r}})\,\right)
\nonumber \\
&-& \sum_{s=0}\sum_{t=1} (-1)^{t-1}g^{t+s+k-1}\,
{\textstyle \frac{B(s)}{s!k!(t-1)!(t+1)}}\,
f^{\vec{\mu}a\lambda}_{(t)}
f^{\vec{\alpha}\lambda\gamma}_{(s+k-1)}\,
{\cal R}_{(t)}^{\vec{\mu}}({\bf r})\,
{\cal M}_{(s+k-1)}^{\vec{\alpha}}({\bf r})\,
j_0^{\gamma}({\bf{r}})
\nonumber \\
&-& gf^{a\beta d}\,A_{i}^{\beta}({\bf r})\,
\sum_{s=0}\sum_{t=1} (-1)^{t}g^{t+s+k-1}\,
{\textstyle\frac{B(s)}{s!k!(t+1)!}}\,
f^{\vec{\mu}d\lambda}_{(t)}
f^{\vec{\alpha}\lambda\gamma}_{(s+k-1)}\,
{\textstyle\frac{\partial_{i}}{\partial^{2}}}
\left(\,{\cal R}_{(t)}^{\vec{\mu}}({\bf r})\,
{\cal M}_{(s+k-1)}^{\vec{\alpha}}({\bf r})\,
j_0^{\gamma}({\bf{r}})\,\right)\;.
\label{eq:SRkth}
\end{eqnarray}
When we sum over the entire series, we can change
variables in the integer-valued indices to $\eta=k+s-1$,
and perform the summation over
$\eta$ and $s$, with $k=\eta-s+1$. The summation over $s$ then
involves nothing but the Bernoulli numbers
and fractional coefficients, so that we obtain
\begin{eqnarray}
e^{\bar {\cal C}}\,{\cal G}^{a}({\bf{r}})\,e^{-\bar {\cal C}}& =
& {\cal G}^{a}({\bf r})-gf^{a\beta\gamma}\,
A_{i}^{\beta}({\bf r})
{\textstyle\frac{\partial_{i}}{\partial^{2}}}\,j_{0}^{\gamma}({\bf r})
\nonumber \\
&+& \sum_{\eta=1}g^{\eta +1}D_0^\eta (\eta)f^{a\beta c}
f^{\vec{\alpha}c\gamma}_{(\eta)}\,A_{i}^{\beta}({\bf r})\,
{\textstyle \frac{\partial_{i}}{\partial^{2}}}\left(\,
{\cal M}_{(\eta)}^{\vec{\alpha}}({\bf r})\,
\,j_{0}^{\gamma}({\bf r})\,\right)
\nonumber \\
&+& \sum_{\eta=0}
\sum_{t=1} (-1)^{t}g^{t+\eta}\,
{\textstyle \frac{D_0^\eta (\eta)}{(t-1)!(t+1)}}\,
f^{\vec{\mu}a\lambda}_{(t)}
f^{\vec{\alpha}\lambda\gamma}_{(\eta)}\,
{\cal R}_{(t)}^{\vec{\mu}}({\bf r})\,
{\cal M}_{(\eta )}^{\vec{\alpha}}({\bf r})\,
j_{0}^{\gamma}({\bf r})
\nonumber \\
&-& gf^{a\mu d}\,A_{i}^{\mu}({\bf r})\,\sum_{\eta=0}\sum_{t=1}
(-1)^{t}g^{t+\eta}
{\textstyle\frac{D_0^\eta (\eta)}{(t+1)!}}\,
f^{\vec{\mu}d\lambda}_{(t)}
f^{\vec{\alpha}\lambda\gamma}_{(\eta)}\,
{\textstyle\frac{\partial_{i}}{\partial^{2}}}
\left(\,{\cal R}_{(t)}^{\vec{\mu}}({\bf r})\,
{\cal M}_{(\eta )}^{\vec{\alpha}}({\bf r})\,
j_{0}^{\alpha}({\bf r})\,\right)\;,
\label{eq:SRsum}
\end{eqnarray}
where $D_0^\eta (\eta)$ is the sum over Bernoulli numbers
defined in Eq.~(\ref{bernoulli}). $D_0^\eta (\eta)$ has the
values $D_0^\eta (\eta)=0$ for $\eta\neq 0$, and $D_0^0 (0)=1$.
Since $f^{\vec{\alpha}\lambda\gamma}_{(0)}=
-\delta_{\lambda ,\gamma}$,
we find that substitution of these values into
Eq.~(\ref{eq:SRsum}) reduces it identically to
Eq.~(\ref{eq:GGhatleft}) and thereby proves
Eqs.~(\ref{eq:GgqGg}) and (\ref{eq:gghatp}),
demonstrating the unitary equivalence of
${\hat {\cal G}}^{a}({\bf r})$ and ${\cal G}^{a}({\bf r})$.
\bigskip
The demonstration of unitary equivalence of
${\hat {\cal G}}^{a}({\bf r})$ and
${\cal G}^{a}({\bf r})$ enables us to assign two different roles to
${\cal G}^{a}({\bf r})$. On the one
hand, ${\cal G}^{a}({\bf r})$ can be viewed as the Gauss's law
operator for `pure glue' QCD and ${\hat {\cal G}}^{a}({\bf r})$ as the
Gauss's law operator for the theory that includes quarks as
well as gluons. But ${\cal G}^{a}({\bf r})$
can also be viewed as the Gauss's law operator for QCD {\em with}
interacting quarks and gluons, in a
representation in which all operators and states have been
transformed with a similarity transformation
that transforms ${\hat {\cal G}}^{a}({\bf r})$ into
${\cal G}^{a}({\bf r})\,$ and that similarly transforms all other
operators and states as well,
but that leaves matrix elements unchanged.
We will designate the representation in which
${\hat {\cal G}}^{a}({\bf r})$ represents the
Gauss's law operator for QCD with quarks as well
as gluons, and in which ${\cal G}^{a}({\bf r})$
represents the `pure glue' Gauss's law operator,
as the `common' or ${\cal C}$ representation.
The unitarily transformed representation,
in which ${\cal G}^{a}({\bf r})$ represents the Gauss's
law operator for QCD with interacting
quarks and gluons, will be designated the ${\cal N}$ representation.
We can use the relationship between
these two representations to construct states
that implement the `complete' Gauss's law ---
Eq.~(\ref{eq:Ggqlaw}) --- from
\begin{equation}
{\cal G}^{a}({\bf r})\,{\Psi}\,|{\phi}\rangle =0,
\label{eq:Gglaw}
\end{equation}
which is the `pure glue' form of Gauss's law in the ${\cal C}$
representation.
We can simply view Eq.~(\ref{eq:Gglaw}) as the statement of
the complete Gauss's law --- the version
that includes interacting quarks and gluons ---
but in the ${\cal N}$ representation.
In order to transform Eq.~(\ref{eq:Gglaw}) ---
now representing Gauss's law with interacting quarks and gluons ---
from the
${\cal N}$ to the ${\cal C}$ representation, we
make use of the fact that
\begin{equation}
{\hat {\cal G}}^{a}({\bf r})\,{\hat {\Psi}}\,
|{\phi}\rangle={\cal{U}}_{\cal{C}}\,
{\cal G}^{a}({\bf r})\,{\cal{U}}^{-1}_{\cal{C}}\,
{\cal{U}}_{\cal{C}}\,{\Psi}\,|{\phi}\rangle =0\;,
\label{eq:Gtrans}
\end{equation}
identifying ${\hat {\Psi}}\,|{\phi}\rangle =
{\cal{U}}_{\cal{C}}\,{\Psi}\,|{\phi}\rangle$ as a state
that implements Gauss's law for a theory with
quarks and gluons, in the ${\cal C}$ representation.
\section{Gauge-invariant spinor and gauge fields}
We can apply the unitary equivalence
demonstrated in the preceding section to
the construction of gauge-invariant spinor
and gauge field operators. We observe that
Gauss's Law has a central role in generating
local gauge transformations, in which
the operator-valued gauge and spinor fields in a
gauge theory --- QCD in this case ---
are gauge-transformed by an arbitrary c-number
field $\omega^a({\bf{r}})$ consistent with the gauge condition that
underlies the canonical theory. In this, the temporal gauge, such
gauge transformations are implemented by
\begin{equation}
{\cal{O}}({\bf{r}})\,\rightarrow\,
{\cal{O}}^\prime({\bf{r}})
=\,\exp\left(-\frac{i}{g}{\int}\hat{\cal{G}}^a({\bf{r}}^\prime)\,
\omega^a({\bf{r}}^\prime)\,d{\bf{r}}^\prime\,\right)\,
{\cal{O}}({\bf{r}})\,
\exp\left(\frac{i}{g}{\int}\hat{\cal{G}}^a({\bf{r}}^\prime)\,
\omega^a({\bf{r}}^\prime)\,d{\bf{r}}^\prime\,\right)\;,
\label{eq:gaugetrans}
\end{equation}
where $\omega^a({\bf{r}})$ is time-independent, and where
${\cal{O}}({\bf{r}})$ represents any of the
operator-valued fields of the gauge theory and
${\cal{O}}^\prime({\bf{r}})$ its gauge-transformed
form\cite{jacktop}. Eq.~(\ref{eq:gaugetrans}) applies
to QCD with quarks and gluons, and is expressed
in the ${\cal C}$ representation. It is obvious
that any operator-valued
field that commutes with ${\hat{\cal{G}}}^a({\bf{r}})$
is gauge-invariant.
\bigskip
We can also formulate the same gauge transformations
in the ${\cal N}$ representation, in which case they
take the form
\begin{equation}
{\cal{O}}_{\cal N}({\bf{r}})\,\rightarrow\,
{\cal{O}}_{\cal N}^\prime({\bf{r}})
=\,\exp\left(-\frac{i}{g}{\int}{\cal{G}}^a({\bf{r}}^\prime)\,
\omega^a({\bf{r}}^\prime)\,d{\bf{r}}^\prime\,\right)\,
{\cal{O}}_{\cal N}({\bf{r}})\,
\exp\left(\frac{i}{g}{\int}{\cal{G}}^a({\bf{r}}^\prime)\,
\omega^a({\bf{r}}^\prime)\,d{\bf{r}}^\prime\,\right)\;,
\label{eq:gaugetransN}
\end{equation}
where ${\cal{O}}_{\cal N}({\bf{r}})$ now represents a spinor
or gauge field in the ${\cal N}$ representation.
Eq.~(\ref{eq:gaugetransN}) has the same form as the equation that
implements gauge-transformations for `pure glue' QCD in the ${\cal C}$
representation, but it has
a very different meaning. In Eq.~(\ref{eq:gaugetransN}),
the operator-valued field ${\cal{O}}_{\cal N}({\bf{r}})$, and
${\cal{G}}^a({\bf{r}})$ which here
represents the {\em entire} Gauss's law --- quarks and
gluons included --- both are in the ${\cal N}$ representation.
\bigskip
It is easy to see that the spinor field $\psi({\bf{r}})$ is
a gauge-invariant spinor in the ${\cal N}$ representation, because
$\psi({\bf{r}})$ and ${\cal{G}}^a({\bf{r^\prime}})$ trivially
commute. To produce ${\psi}_{\sf GI}({\bf{r}}),$ this
gauge-invariant spinor
transposed into the ${\cal C}$ representation, we make use of
\begin{equation}
{\psi}_{\sf GI}({\bf{r}})={\cal{U}}_{\cal C}\,
\psi({\bf{r}})\,{\cal{U}}^{-1}_{\cal C}\;
\;\;\;\mbox{and}\;\;\;
{\psi}_{\sf GI}^\dagger({\bf{r}})={\cal{U}}_{\cal C}\,
\psi^\dagger({\bf{r}})\,{\cal{U}}^{-1}_{\cal C}\;.
\label{eq:psiqcd}
\end{equation}
We can easily carry out the unitary transformations in
Eq.~(\ref{eq:psiqcd})
to give
\begin{equation}
{\psi}_{\sf GI}({\bf{r}})=V_{\cal{C}}({\bf{r}})\,\psi ({\bf{r}})
\;\;\;\mbox{\small and}\;\;\;
{\psi}_{\sf GI}^\dagger({\bf{r}})=
\psi^\dagger({\bf{r}})\,V_{\cal{C}}^{-1}({\bf{r}})\;,
\label{eq:psiqcdg1}
\end{equation}
where
\begin{equation}
V_{\cal{C}}({\bf{r}})=
\exp\left(\,-ig{\overline{{\cal{Y}}^\alpha}}({\bf{r}})
{\textstyle\frac{\lambda^\alpha}{2}}\,\right)\,
\exp\left(-ig{\cal X}^\alpha({\bf{r}})
{\textstyle\frac{\lambda^\alpha}{2}}\right)\;,
\label{eq:el1}
\end{equation}
and
\begin{equation}
V_{\cal{C}}^{-1}({\bf{r}})=
\exp\left(ig{\cal X}^\alpha({\bf{r}})
{\textstyle\frac{\lambda^\alpha}{2}}\right)\,
\exp\left(\,ig{\overline{{\cal{Y}}^\alpha}}({\bf{r}})
{\textstyle\frac{\lambda^\alpha}{2}}\,\right)\;.
\label{eq:eldagq1}
\end{equation}
Because we have given an explicit expression for
$\overline{{\cal Y}^{\alpha}}({\bf{r}})$ in
Eqs.~(\ref{eq:defY}) and (\ref{eq:inteq2}), Eq.~(\ref{eq:psiqcdg1})
represents complete, non-perturbative expressions for
gauge-invariant spinors in the
${\cal C}$ representation. We can, if we choose,
expand Eqs.~(\ref{eq:psiqcdg1}) to arbitrary order. We then
find that to $O(g^3)$, we agree with Refs.~\cite{lavelle2,lavelle5} in
which a perturbative construction of a
gauge-invariant spinor is carried out to $O(g^3).$ Furthermore, in the
${\cal C}$ representation, ${\psi}({\bf{r}})$ gauge-transforms as
\begin{equation}
{\psi}({\bf{r}})\,\rightarrow\,
\psi^\prime({\bf{r}})=\,
\exp\left(i\omega^\alpha({\bf{r}})\,
{\textstyle\frac{\lambda^\alpha}{2}}\,\right)\,\psi({\bf{r}})\;.
\label{eq:psitransf}
\end{equation}
Since ${\psi}_{\sf GI}({\bf{r}})$ has been shown to be
gauge-invariant, it
immediately follows that $V_{\cal{C}}({\bf{r}})$
gauge-transforms as
\begin{equation}
V_{\cal{C}}({\bf{r}})\rightarrow
V_{\cal{C}}({\bf{r}})\exp\left(-i\omega^\alpha({\bf{r}})\,
{\textstyle\frac{\lambda^\alpha}{2}}\,\right)\;\;\;\;
\mbox{and}\;\;\;
V^{-1}_{\cal{C}}({\bf{r}})\rightarrow
\exp\left(i\omega^\alpha({\bf{r}})\,
{\textstyle\frac{\lambda^\alpha}{2}}\,\right)
V_{\cal{C}}^{-1}({\bf{r}})\;.
\end{equation}
The procedure we have used to construct gauge-invariant spinors
is not applicable to the construction of gauge-invariant gauge
fields, because we do not have ready access to a form of the gauge
field that is trivially gauge invariant in either the ${\cal C}$
or the ${\cal N}$ representation. We will, however, discuss two
methods for constructing gauge-invariant gauge fields.
One method is based on the observation that the states $|{\phi}\rangle$
for which
\begin{equation}
{\hat {\cal G}}^a{\hat {\Psi}}\,|{\phi}\rangle=
{\hat {\cal G}}^a{\cal{U}}_{\cal{C}}\,{\Psi}\,|{\phi}\rangle=0
\label{eq:phiset}
\end{equation}
include any state $|\phi_{A_{T\,i}^{b}({\bf{r}})}\rangle$ in which
the transverse gauge field $A_{T\,i}^{b}({\bf{r}})$
acts on another $|{\phi}\rangle$ state. This is an immediate
consequence of the fact that
${\hat {\cal G}}^a\,{\hat {\Psi}}={\hat {\Psi}}\,b_{Q}^{a}({\bf{k}})
+B_{Q}^{a}({\bf{k}}),$ and that $A_{T\,i}^{b}({\bf{r}})$
trivially commutes with $\partial_i\Pi_i^{a}({\bf{r}}^\prime\,).$
We use the commutator algebra
for the operator-valued fields to maneuver the transverse gauge field,
along with all further gauge field functionals generated in this
process, to the left of ${\cal{U}}_{\cal{C}}\,{\Psi}$ in
${\cal{U}}_{\cal{C}}\,{\Psi}\,A_{T\,i}^{b}({\bf{r}})\,
|{\phi}\rangle.$ We then obtain the result that
\begin{equation}
{\hat {\Psi}}\,A_{T\,i}^{b}({\bf{r}})\,|{\phi}\rangle=
A_{{\sf GI}\,i}^b({\bf{r}})\,{\hat {\Psi}}\,|{\phi}\rangle,
\label{eq:Agi}
\end{equation}
where $A_{{\sf GI}\,i}^b({\bf{r}})$ is a gauge-invariant gauge field
created in the process of commuting $A_{T\,i}^{b}({\bf{r}})$ past the
$\Psi$ to its left.
The gauge-invariance of $A_{{\sf GI}\,i}^b({\bf{r}})$ follows from
the fact that the Gauss's law operator ${\hat {\cal G}}^a$ annihilates
both sides of Eq.~(\ref{eq:Agi}).
Eqs.~(\ref{eq:phiset}) and (\ref{eq:Agi}) require that the
commutator $\left[{\hat {\cal G}}^a,\,
A_{{\sf GI}\,i}^b({\bf{r}})\, \right]=0,$ and it then follows
directly from Eq.~(\ref{eq:gaugetrans}) that
$A_{{\sf GI}\,i}^b({\bf{r}})$ is gauge-invariant.
It only remains for us to
find an explicit expression for $A_{{\sf GI}\,i}^b({\bf{r}}).$
We first observe from Eqs.~(\ref{eq:GgqGg}) and (\ref{eq:CCbar})
that the gauge field and all functionals of gauge fields commute with
${\cal{U}}_{\cal{C}}.$ We further see that
\begin{equation}
A_{{\sf GI}\,i}^b({\bf{r}})\,{\Psi}=\left[\Psi,\,
A_{T\,i}^{b}({\bf{r}})\, \right]
+A_{T\,i}^{b}({\bf{r}})\,\Psi.
\label{eq:Agieq}
\end{equation}
When we expand $\Psi$ as
\begin{eqnarray}
\Psi&&={\|}\,\exp({\cal{A}})\,{\|}={\|}\,
\exp\left(i{\int}\,d{\bf{r}}\,
\overline{{\cal{A}}^\gamma_{k}}({\bf{r}})\,
\Pi^\gamma_{k}({\bf{r}})\right){\|}
\nonumber\\
&&=1+i{\int}\,d{\bf{r}}_1)\,
\overline{{\cal{A}}^{\gamma}_{k}}({\bf{r}}_1)\,
\Pi^\gamma_{k}({\bf{r}}_1)+{\textstyle\frac{(i)^2}{2}}
{\int}\,d{\bf{r}}_1\,d{\bf{r}}_2\,
\overline{{\cal{A}}^{\gamma_1}_{k_1}}({\bf{r}}_1)\,
\overline{{\cal{A}}^{\gamma_2}_{k_2}}({\bf{r}}_2)\,
\Pi^{\gamma_1}_{k_1}({\bf{r}}_1)\,
\Pi^{\gamma_2}_{k_2}({\bf{r}}_2)+\cdots
\nonumber\\
&&+{\textstyle\frac{(i)^n}{n\,!}}{\int}\,
d{\bf{r}}_1\,d{\bf{r}}_2\,\cdots\,d{\bf{r}}_n\,
\overline{{\cal{A}}^{\gamma_1}_{k_1}}({\bf{r}}_1)\,
\overline{{\cal{A}}^{\gamma_2}_{k_2}}({\bf{r}}_2)\,\cdots\,
\overline{{\cal{A}}^{\gamma_n}_{k_n}}({\bf{r}}_n)\,
\Pi^{\gamma_1}_{k_1}({\bf{r}}_1)\,
\Pi^{\gamma_2}_{k_2}({\bf{r}}_2)\,\cdots\,
\Pi^{\gamma_n}_{k_n}({\bf{r}}_n)
\nonumber \\
&&+\,\cdots
\label{eq:Psiexp}
\end{eqnarray}
it becomes evident that
\begin{eqnarray}
[\,{\Psi},\,A_{T\,i}^b({\bf{r}})\,]&=&
\,(\delta_{ij}-{\textstyle\frac{\partial_{i}\partial_j}
{\partial^2}})\overline{{\cal{A}}^b_{j}}({\bf{r}})\,
+(\delta_{ij}-{\textstyle\frac{\partial_{i}\partial_j}
{\partial^2}})\overline{{\cal{A}}^b_{j}}({\bf{r}})\,
i{\int}\,d{\bf{r}}_1\,
\overline{{\cal{A}}^{\gamma}_{k}}({\bf{r}}_1)\,
\Pi^{\gamma}_{k}({\bf{r}}_1)+\cdots
\nonumber\\
&&+(\delta_{ij}-{\textstyle\frac{\partial_{i}\partial_j}
{\partial^2}})\overline{{\cal{A}}^b_{j}}({\bf{r}})\,
{\textstyle\frac{(i)^{n-1}}{(n-1)\,!}}{\int}\,
d{\bf{r}}_1\,d{\bf{r}}_2\,\cdots\,d{\bf{r}}_{n-1}\,
\overline{{\cal{A}}^{\gamma_1}_{k_1}}({\bf{r}}_1)\,
\overline{{\cal{A}}^{\gamma_2}_{k_2}}({\bf{r}}_2)\,\cdots\,
\overline{{\cal{A}}^{\gamma_{n-1}}_{k_{n-1}}}({\bf{r}}_{n-1})\,
\nonumber\\
&&\;\;\;\;\;\;\;\;\;\;\times
\Pi^{\gamma_1}_{k_1}({\bf{r}}_1)\,
\Pi^{\gamma_2}_{k_2}({\bf{r}}_2)\,\cdots\,
\Pi^{\gamma_{n-1}}_{k_{n-1}}({\bf{r}}_{n-1})
+\cdots \nonumber \\
&=&(\delta_{ij}-{\textstyle\frac{\partial_{i}\partial_j}
{\partial^2}})\overline{{\cal{A}}^b_{j}}({\bf{r}})\,\Psi\;,
\label{eq:bob4a}
\end{eqnarray}
and therefore that the gauge-invariant gauge field is
\begin{equation}
A_{{\sf GI}\,i}^b({\bf{r}})=
A_{T\,i}^b({\bf{r}}) +
[\delta_{ij}-{\textstyle\frac{\partial_{i}\partial_j}
{\partial^2}}]\overline{{\cal{A}}^b_{j}}({\bf{r}})=
a_{i}^{b} ({\bf{r}})+\overline{{\cal{A}}^b_{i}}({\bf{r}})
-\partial_{i}\overline{{\cal Y}^{b}}({\bf{r}})\;.
\label{eq:Adressedthree1b}
\end{equation}
Confirmation of this result can be obtained from the fact that
$A_{{\sf GI}\,i}^b({\bf{r}})$ commutes with ${\cal G}^a$ --- and
therefore also with
${\hat {\cal G}}^a.$ We observe that
\begin{eqnarray}
\left[\,{\cal{G}}^a({\bf{r}}),\,
A_{{\sf GI}\,i}^b({\bf{r}}^\prime)\right]=
\left[\,{\cal{G}}^a({\bf{r}}),\,
\left(A_{i\,T}^{b}({\bf{r}}^\prime)
+(\delta_{ij}-{\textstyle\frac{\partial_{i}\partial_j}{\partial^2}})
\overline{{\cal{A}}^b_{j}}({\bf{r}}^\prime)\,\right)\,\right]&=&
\nonumber \\
{\int}\,d{\bf{y}}\,
\left\{\left[\,{\cal{G}}^a({\bf{r}}),\,
{A}^b_{j}({\bf{y}})\,\right]+\left[\,{\cal{G}}^a({\bf{r}}),\,
\overline{{\cal{A}}^b_{j}}({\bf{y}})\,\right]\,\right\}
V_{ij}({\bf{y}}-{\bf{r}}^\prime)&=&0\;,
\label{eq:dirgi}
\end{eqnarray}
where
\begin{equation}
V_{ij}({\bf{y}}-{\bf{r}}^\prime)=
(\delta_{ij}-{\textstyle\frac{\partial_{i}\partial_j}{\partial^2}})\,
\delta({\bf{y}}-{\bf{r}}^\prime)\;.
\label{eq:vjay}
\end{equation}
Eq.~(\ref{eq:dirgi}) follows directly from Eq.~(\ref{eq:Asum});
$\int d{\bf y}\,\left[\,{\cal{G}}^a({\bf{r}}),\,
\overline{{\cal{A}}^b_{j}}({\bf{y}})\,\right]\,
V_{ij}({\bf{y}}-{\bf{r}}^\prime)$ can be identified as the
first line of that equation, when the integration
over ${\bf{y}}$ in Eq.~(\ref{eq:dirgi})
is identified with
the integration over ${\bf{r}}^\prime$ in Eq.~(\ref{eq:Asum}), and
when the tensor element
$V_{ij}({\bf{y}}-{\bf{r}}^\prime),$ with ${\bf r}^\prime$ and
$i$ fixed, is substituted for the
vector component $V^\gamma_{j}$ in Eq.~(\ref{eq:Asum}). Similarly,
$\int d{\bf y}\left[\,{\cal{G}}^a({\bf{r}}),\,
{A}^b_{i}({\bf{y}})\,\right]\,
V_{ij}({\bf{y}}-{\bf{r}}^\prime)$ can be identified as the second
line of Eq.~(\ref{eq:Asum}). The remaining three lines of
Eq.~(\ref{eq:Asum}) vanish because
$\partial_{j}V_{ij}({\bf{y}}-{\bf{r}}^\prime)=0$ is
an identity. In this way, Eq.~(\ref{eq:Asum}) accounts for the
gauge-invariance of $A_{{\sf GI}\,i}^b({\bf{r}}).$
\bigskip
Another method for constructing a gauge-invariant gauge field is based
on the observation that
$V_{\cal{C}}({\bf{r}})$ can be written as an exponential function.
We can make use of the BHC theorem that
$e^{\sf u}e^{\sf v}=e^{\sf w}\,,$
where ${\sf w}$ is a series whose initial term is ${\sf u}+{\sf v},$
and whose higher order terms are multiples of
successive commutators of ${\sf u}$ and ${\sf v}$ with earlier
terms in that series.
Since the commutator algebra of the Gell-Mann
matrices ${\lambda}^\alpha$ is closed,
$V_{\cal{C}}({\bf{r}})$ must be of the form
$\exp[-ig{\cal Z}^\alpha({\lambda}^\alpha/2)],$ where
\begin{equation}
\exp\left[-ig{\cal Z}^\alpha
{\textstyle\frac{{\lambda}^\alpha}{2}}\right]=
\exp\left[-ig{\overline {\cal Y}^\alpha}
{\textstyle\frac{{\lambda}^\alpha}{2}}\right]\,
\exp\left[-ig{\cal X}^\alpha
{\textstyle\frac{{\lambda}^\alpha}{2}}\right]
\label{eq:Zxy}
\end{equation}
and ${\cal Z}^\alpha$ is a functional of gauge fields
(but not of their canonical momenta).
$V_{\cal{C}}({\bf{r}})$ therefore can be viewed as a
particular case of the operator $\exp\left[i\omega^\alpha({\bf{r}})
\left(\lambda^{\alpha}/2\right)\right]$ that
gauge-transforms the spinor field ${\psi}({\bf r})\,;$
$\omega^\alpha$ in this
case is ${\cal Z}^\alpha$ and therefore a functional of gauge fields
that commutes with all other functionals
of gauge and spinor fields.
Moreover, we can refer to the Euler-Lagrange
equation (in the $A_0=0$ gauge)
for the spinor field ${\psi}({\bf r}),$
\begin{equation}
\left[im+\gamma_j\left(\partial_{j}-ig\,A_{j}^\alpha({\bf{r}})
{\textstyle\frac{\lambda^\alpha}{2}}\right)+
\gamma_{0}\partial_{0}\right]\,{\psi}({\bf r})=0,
\label{eq:Diracspin}
\end{equation}
where we have used the same non-covariant notation for the
gauge fields as in
Ref.~\cite{khymtemp} ($i.e.$ $A_{j}^\alpha({\bf{r}})$
designates contravariant and
$\partial_{j}$ covariant quantities), and where $\gamma_0=\beta$ and
$\gamma_j={\beta}{\alpha_j}.$ Although the gauge fields are
operator-valued, they commute with all other operators in
Eq.~(\ref{eq:Diracspin}) --- with the exception of the
derivatives $\partial_j\,$ --- so that, when only time-independent
gauge-transformations are considered, $V_{\cal{C}}({\bf{r}}),$ acting
as an operator that gauge-transforms $\psi,$
behaves as though ${\cal Z}^\alpha$ were a c-number.
The gauge-transformed
gauge field, that corresponds to the gauge-transformed spinor
${\psi}_{{\sf GI}}({\bf{r}})=V_{\cal{C}}({\bf{r}})\,\psi ({\bf{r}})$,
therefore also is gauge-invariant; it is given by
\begin{equation}
[\,A_{{\sf GI}\,i}^{b}({\bf{r}})\,{\textstyle\frac{\lambda^b}{2}}\,]
=V_{\cal{C}}({\bf{r}})\,[\,A_{i}^b({\bf{r}})\,
{\textstyle\frac{\lambda^b}{2}}\,]\,
V_{\cal{C}}^{-1}({\bf{r}})
+{\textstyle\frac{i}{g}}\,V_{\cal{C}}({\bf{r}})\,
\partial_{i}V_{\cal{C}}^{-1}({\bf{r}})\;.
\label{eq:AdressedAxz}
\end{equation}
Since {\em further} gauge transformations must be carried
out simultaneously
on $\psi ({\bf{r}})$ and $V_{\cal{C}}({\bf{r}}),$ and must leave
${\psi}_{{\sf GI}}({\bf{r}})$ untransformed,
$A_{{\sf GI}\,i}^{b}({\bf{r}})$ must also
therefore remain untransformed by further gauge transformations.
$A_{{\sf GI}\,i}^{b}({\bf{r}})$ thus is identified as a
gauge-invariant gauge field.
\bigskip
To find an explicit form for $[\,A_{{\sf GI}\,i}^{b}({\bf{r}})\,
{\textstyle\frac{\lambda^b}{2}}\,]$ from the r.h.s. of
Eq.~(\ref{eq:AdressedAxz}),
we use Eq.~(\ref{eq:LCA1}), with $V_{j}^{\gamma}({\bf r})=
\delta_{ij}({\lambda}^{\gamma}
/2)\,,$ to obtain
\begin{equation}
\left[a_{i}^{\gamma} ({\bf{r}})+
{\overline{{\cal{A}}^{\gamma}_i}}({\bf{r}})-\sum_{\eta=1}^\infty
{\textstyle\frac{g^\eta}{\eta!}}\,f^{\vec{\alpha}\beta\gamma}_{(\eta)}\,
{\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})\,
\overline{{\cal{B}}_{(\eta) i}^{\beta}}({\bf{r}})\,\right]
{\textstyle\frac{\lambda^\gamma}{2}}=
\left[a_{i}^{\gamma} ({\bf{r}})+\sum_{\eta=1}^\infty
{\textstyle\frac{g^\eta}{\eta!}}
\,\psi^{\gamma}_{(\eta)i}({\bf{r}})\,\right]
{\textstyle\frac{{\lambda}^\gamma}{2}}.
\label{eq:inteq3}
\end{equation}
It is straightforward but tedious to show that
\begin{equation}
\left[a_{i}^{\gamma} ({\bf{r}})+\sum_{\eta=1}^\infty
{\textstyle\frac{g^\eta}{\eta!}}
\,\psi^{\gamma}_{(\eta)i}({\bf{r}})\,\right]
{\textstyle\frac{{\lambda}^\gamma}{2}}=
\exp\left(\,-ig\,{\cal X}^\alpha ({\bf{r}})
{\textstyle\frac{\lambda^\alpha}{2}}\,\right)\,
\left[A_{i}^{\gamma} ({\bf{r}})
{\textstyle\frac{{\lambda}^\gamma}{2}\,+\frac{i}{g}}\partial_i\,\right]
\exp\left(\,ig\,{\cal X}^\alpha ({\bf{r}})
{\textstyle\frac{\lambda^\alpha}{2}}\,\right)\,,
\label{eq:psident}
\end{equation}
\begin{equation}
\left[a_{i}^{\gamma} ({\bf{r}})-\sum_{\eta=1}^\infty
{\textstyle\frac{g^\eta}{\eta!}}\,
f^{\vec{\alpha}\beta\gamma}_{(\eta)}\,
{\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})\,
a_{i}^{\gamma} ({\bf{r}})\right]
{\textstyle\frac{{\lambda}^\gamma}{2}}=
\exp\left(\,ig\,{\overline{{\cal Y}^\alpha}}({\bf{r}})
{\textstyle\frac{\lambda^\alpha}{2}}\,\right)\,
\left[a_{i}^{\gamma} ({\bf{r}})
{\textstyle\frac{{\lambda}^\gamma}{2}}\right]
\exp\left(\,-ig\,{\overline{{\cal Y}^\alpha}}({\bf{r}})
{\textstyle\frac{\lambda^\alpha}{2}}\,\right)\,,
\label{eq:aident}
\end{equation}
\begin{equation}
\left[\partial_{i}{\overline{{\cal Y}^{\gamma}}} ({\bf{r}})
-\sum_{\eta=1}^\infty
{\textstyle\frac{g^\eta}{\eta!}}\,
f^{\vec{\alpha}\beta\gamma}_{(\eta)}\,
{\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})\,
\partial_{i}{\overline{{\cal Y}^{\gamma}}} ({\bf{r}})\right]
{\textstyle\frac{{\lambda}^\gamma}{2}}=
\exp\left(\,ig\,{\overline{{\cal Y}^\alpha}}({\bf{r}})
{\textstyle\frac{\lambda^\alpha}{2}}\,\right)\,
\left[\partial_{i}{\overline{{\cal Y}^\gamma}} ({\bf{r}})
{\textstyle\frac{{\lambda}^\gamma}{2}}\right]
\exp\left(\,-ig\,{\overline{{\cal Y}^\alpha}}({\bf{r}})
{\textstyle\frac{\lambda^\alpha}{2}}\,\right)\,,
\label{eq:dyident}
\end{equation}
and
\begin{eqnarray}
&&\left[\overline{{\cal A}_{i}^{\gamma}}+\partial_{i}
{\overline{{\cal Y}^{\gamma}}} ({\bf{r}})-\sum_{\eta=1}^\infty
{\textstyle\frac{g^\eta}{\eta!}}\,
f^{\vec{\alpha}\beta\gamma}_{(\eta)}\,
{\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})\,
\left(\overline{{\cal A}_{i}^{\gamma}} ({\bf{r}})
+{\textstyle\frac{1}{\eta +1}}\partial_{i}
{\overline{{\cal Y}^\gamma}} ({\bf{r}})\right)\right]
{\textstyle\frac{{\lambda}^\gamma}{2}}
\nonumber \\
&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=
\exp\left(\,ig\,{\overline{{\cal Y}^\alpha}}({\bf{r}})
{\textstyle\frac{\lambda^\alpha}{2}}\,\right)\,
\left[\overline{{\cal A}_{i}^{\gamma}} ({\bf{r}})
{\textstyle\frac{{\lambda}^\gamma}{2}+\frac{i}{g}}\partial_i\,\right]
\exp\left(\,-ig\,{\overline{{\cal Y}^\alpha}}({\bf{r}})
{\textstyle\frac{\lambda^\alpha}{2}}\,\right)\,.
\label{eq:calaident}
\end{eqnarray}
Eqs. (\ref{eq:inteq3})-(\ref{eq:calaident}) leads to
\begin{equation}
V_{\cal{C}}({\bf{r}})\,[\,A_{i}^b({\bf{r}})\,
{\textstyle\frac{\lambda^b}{2}}\,]\,
V_{\cal{C}}^{-1}({\bf{r}})
+{\textstyle\frac{i}{g}}\,V_{\cal{C}}({\bf{r}})\,
\partial_{i}V_{\cal{C}}^{-1}({\bf{r}})\,=
A_{T\,i}^b ({\bf{r}}){\textstyle\frac{{\lambda}^b}{2}} +
[\delta_{ij}-{\textstyle\frac{\partial_{i}\partial_j}
{\partial^2}}]\overline{{\cal A}_{i}^b} ({\bf{r}})
{\textstyle\frac{{\lambda}^b}{2}}\;,
\end{equation}
so that the identical gauge-invariant gauge field is given in
Eqs.~(\ref{eq:Adressedthree1b}) and (\ref{eq:AdressedAxz}).
In the gauge-invariant gauge field, as in the earlier case
of the gauge-invariant spinor, we find that when we
expand Eq.~(\ref{eq:Adressedthree1b}) --- this time to $O(g^2)$ ---
we agree with Refs.~\cite{lavelle2,lavelle5} in which a perturbative
construction of a gauge-invariant gauge field is
carried out to that order.
\section{The case of Yang-Mills Theory}
Because of the simplicity of the $SU(2)$ structure constants,
it is instructive to examine
$\overline{{\cal{A}}^a_{j}}({\bf{r}})$ --- its defining equation and
its role in the `fundamental theorem' --- for the
case of Yang-Mills theory. For that purpose, we
substitute $\epsilon^{abc}$ --- the structure constants of
$SU(2)$ --- for the $f^{abc}$ required for $SU(3),$ in the
equations that pertain to
$\overline{{\cal{A}}^a_{j}}({\bf{r}})$.
$\epsilon^{\vec{\alpha}\beta\gamma}_{(\eta)},$
the $SU(2)$ equivalent of the $f^{\vec{\alpha}\beta\gamma}_{(\eta)}$
that are important in the definition of
$\overline{{\cal{A}}^a_{j}}({\bf{r}}),$ is given by
\begin{equation}
\epsilon^{\vec{\alpha}\beta\gamma}_{(\eta)}=(-1)^{\frac{\eta}{2}-1}
\delta_{\alpha[1]\alpha[2]}\,\delta_{\alpha[3]\alpha[4]}\,
\cdots\,\delta_{\alpha[\eta-3]\alpha[\eta-2]}\,
\epsilon^{\alpha[\eta-1]\beta b}\,
\epsilon^{b\alpha[\eta]\gamma}\;
\label{eq:fproductN2}
\end{equation}
and
\begin{equation}
\epsilon^{\vec{\alpha}\beta\gamma}_{(\eta)}=(-1)^{\frac{\eta-1}{2}}
\delta_{\alpha[1]\alpha[2]}\,\delta_{\alpha[3]\alpha[4]}\,
\cdots\,\delta_{\alpha[\eta-2]\alpha[\eta-1]}\,
\epsilon^{\alpha[\eta]\beta \gamma}\;
\label{eq:fproductN3}
\end{equation}
for even and odd $\eta$ respectively.
We can use Eqs.~(\ref{eq:fproductN2}) and (\ref{eq:fproductN3}) to
write the $SU(2)$ version of Eq.~(\ref{eq:inteq2}) for
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}}),$ which
appears (implicitly) as the coefficient of the
$\Pi_i^{\gamma}({\bf r})$ on the l.h.s. of that equation.
In doing so, we separate
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})$ into two
parts
\begin{equation}
\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})=
\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\cal X}+
\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\overline{\cal Y}}\;,
\label{eq:azero}
\end{equation}
where $\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\cal X}$
represents the part of
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})$ that depends only
on `known' quantities
that stem from the $\psi^{\gamma}_{(n)i}({\bf{r}})$ and are
functionals of gauge fields;
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\overline{\cal Y}}$
represents the
part that implicitly contains the
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})$ itself.
In Section~\ref{sec-Implementing}, we showed how the perturbative
expansion of
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})$ proceeds with the
construction of the
$n^{th}$ order term, ${\cal{A}}_{(n)i}^{\gamma}({\bf{r}}),$ from the
$\psi^{\gamma}_{(n)i}({\bf{r}})$ of
the same order, and from
${\cal{A}}_{(n^\prime)i}^{\gamma}({\bf{r}})$ of lower
orders --- in the $SU(2)$ case, the latter originating from
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\overline{\cal Y}}.$
The explicit forms of
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\cal X}$ and
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\overline{\cal Y}}$ are
\begin{eqnarray}
\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\cal X}
&&\,=g\epsilon^{\alpha\beta\gamma}\,
{\cal{X}}^\alpha({\bf{r}})\,A_i^\beta({\bf{r}})\,
{\textstyle\frac{\sin({\cal{N}})}{{\cal{N}}}}
\nonumber\\
&&-g\epsilon^{\alpha\beta\gamma}\,
{\cal{X}}^\alpha({\bf{r}})\,\partial_i{\cal{X}}^\beta({\bf{r}})\,
{\textstyle\frac{1-\cos({\cal{N}})}{{\cal{N}}^2}}
\nonumber\\
&&-g^2\epsilon^{\alpha\beta b}
\epsilon^{b\mu\gamma}\,
{\cal{X}}^\mu({\bf{r}})\,{\cal{X}}^\alpha({\bf{r}})
\,A_i^\beta({\bf{r}})\,
{\textstyle\frac{1-\cos({\cal{N}})}{{\cal{N}}^2}}
\nonumber\\
&&+g^2\epsilon^{\alpha\beta b}
\epsilon^{b\mu\gamma}\,
{\cal{X}}^\mu({\bf{r}})\,{\cal{X}}^\alpha({\bf{r}})\,
\partial_i{\cal{X}}^\beta({\bf{r}})\,
[{\textstyle\frac{1}{{\cal{N}}^2}}
-{\textstyle\frac{\sin({\cal{N}})}{{\cal{N}}^3}}]
\label{eq:a2X}
\end{eqnarray}
and
\begin{eqnarray}
\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\overline{\cal Y}}
&&\;=g\epsilon^{\alpha\beta\gamma}\,
\overline{{\cal{Y}}^\alpha}({\bf{r}})
\,\left(\,A_{T\,i}^\beta({\bf{r}}) +
(\delta_{ij}-{\textstyle\frac{\partial_{i}\partial_j}
{\partial^2}})\overline{{\cal{A}}^\beta_{j}}({\bf{r}})\,\right)\,
{\textstyle\frac{\sin(\overline{\cal{N}})}{\overline{\cal{N}}}}
\nonumber\\
&&+g\epsilon^{\alpha\beta\gamma}\,
\overline{{\cal{Y}}^\alpha}({\bf{r}})
\,\partial_i\overline{{\cal{Y}}^\beta}({\bf{r}})\,
{\textstyle\frac{1-\cos(\overline{\cal{N}})}
{\overline{\cal{N}}^2}}
\nonumber\\
&&+g^2\epsilon^{\alpha\beta b}
\epsilon^{b\mu\gamma}\,
\overline{{\cal{Y}}^\mu}({\bf{r}})\,
\overline{{\cal{Y}}^\alpha}({\bf{r}})
\,\left(\,A_{T\,i}^\beta({\bf{r}}) +
(\delta_{ij}-{\textstyle\frac{\partial_{i}\partial_j}
{\partial^2}})\overline{{\cal{A}}^\beta_{j}}({\bf{r}})\,\right)\,
{\textstyle\frac{1-\cos(\overline{\cal{N}})}{\overline{\cal{N}}^2}}
\nonumber\\
&&+g^2\epsilon^{\alpha\beta b}
\epsilon^{b\mu\gamma}\,
\overline{{\cal{Y}}^\mu}({\bf{r}})\,
\overline{{\cal{Y}}^\alpha}({\bf{r}})\,
\partial_i\overline{{\cal{Y}}^\beta}({\bf{r}})\,
[{\textstyle\frac{1}{\overline{\cal{N}}^2}}-
{\textstyle\frac{\sin(\overline{\cal{N}})}
{\overline{\cal{N}}^3}}]\;,
\label{eq:a2Y}
\end{eqnarray}
where
\begin{equation}
{\cal{N}}({\bf{r}})\equiv{\cal{N}}=\left[g^2\,
{\cal{X}}^\delta({\bf{r}})\,
{\cal{X}}^\delta({\bf{r}})\,\right]^{\frac{1}{2}}\;,
\end{equation}
and
\begin{equation}
\overline{\cal{N}}({\bf{r}})\equiv\overline{\cal{N}}=
\left[g^2\,\overline{{\cal{Y}}^\delta}({\bf{r}})\,
\overline{{\cal{Y}}^\delta}({\bf{r}})\,\right]^{\frac{1}{2}}\;.
\end{equation}
There is a striking resemblance in the structure of
Eqs.~(\ref{eq:a2X}) and (\ref{eq:a2Y}) on the one
hand, and $\left(A^{\gamma}\,\right)^{\prime}_{i},$ the
gauge-transformed gauge field $A^{\gamma}_{i},$
where the gauge transformation is by a finite gauge
function $\omega^\gamma.$
$\left(A^{\gamma}\,\right)^{\prime}_{i}$ is given by
\begin{eqnarray}
&&\left(A^{\gamma}\,\right)^{\prime}_{i}=\,
(\,A^\gamma_{i}+{\textstyle\frac{1}{g}}\,
\partial_i\omega^\gamma\,)
\nonumber\\
&&-\,\epsilon^{\alpha\beta\gamma}
\left(\,\omega^\alpha\,A_{i}^\beta\,
{\textstyle\frac{\sin(|\omega|)}{|\omega|}}\,
+\,{\textstyle\frac{1}{g}}\,
\omega^\alpha\,\partial_i\omega^\beta\,
{\textstyle\frac{1-\cos(|\omega|)}{|\omega|^2}}\,\right)
\nonumber\\
&&-\,\epsilon^{\alpha\beta b}\epsilon^{b\mu\gamma}\,
\left(\,\omega^\mu\omega^\alpha\,A^\beta_{i}\,
\,{\textstyle\frac{1-\cos(|\omega|)}{|\omega|^2}}\,
+{\textstyle\frac{\omega^\mu
\omega^\alpha\partial_i\omega^\beta}{g}}\,\,
(\,{\textstyle\frac{1}{|\omega|^2}}
-{\textstyle\frac{\sin(|\omega|)}{|\omega|^3}})\,\right)\;.
\label{eq:atranssu2}
\end{eqnarray}
The $SU(2)$ version of Eq.~(\ref{eq:Asum}) --- our
so-called `fundamental theorem' --- can similarly be given.
In that case, the summations over order and multiplicity
indices can be absorbed into trigonometric functions, and
we obtain the much
simpler equation
\begin{eqnarray}
i\int d{\bf r}^\prime&&[\,\partial_{i}\Pi_{i}^{a}({\bf r}),\,
\overline{{\cal A}_{j}^{\gamma}}({\bf r}^\prime)\,]\,
V_{j}^{\gamma}({\bf r}^\prime)
+ ig\epsilon^{a\beta d}A_{i}^{\beta}({\bf r})\int d{\bf r}^\prime
[\,\Pi_{i}^{d}({\bf r}),\,\overline{{\cal A}_{j}^{\gamma}}({\bf
r}^\prime)\,]\,
V_{j}^{\gamma}({\bf r}^\prime)
\nonumber \\
&&= -g\epsilon^{a\mu d}\,A_{i}^{\mu}({\bf r})\,V_{i}^{d}({\bf r})
\nonumber \\
&&-{\textstyle\frac{g^{2}}{2}}\,
\epsilon^{a\beta c}\epsilon^{{\alpha}c\gamma}\,A_{i}^{\beta}({\bf r})\,
{\textstyle \frac{\partial_{i}}{\partial^{2}}}\left(\,
\overline{{\cal Y}^{{\alpha}}}({\bf r})\,
\partial_{j}V_{j}^{\gamma}({\bf r})\,\right)
\nonumber \\
&&-g^{3}
\epsilon^{a\beta c}\epsilon^{\vec{\alpha}c\gamma}_{(2)}\,
A_{i}^{\beta}({\bf r})\,
{\textstyle \frac{\partial_{i}}{\partial^{2}}}
\left({\cal M}_{(2)}^{\vec{\alpha}}({\bf r})
\left[{\textstyle\frac{1}{2{\overline{\cal{N}}}}}
\cot{\left(\textstyle\frac{{\overline{\cal{N}}}}{2}\right)}
-{\textstyle\frac{1}{{\overline{\cal{N}}^2}}}\right]\,
\partial_{j}V_{j}^{\gamma}({\bf r})\,\right)
\nonumber \\
&&+g\epsilon^{{\mu}a\gamma}
\,{\cal X}^{{\mu}}({\bf r})
\left[{\textstyle\frac{\sin({\cal{N}})}{{\cal{N}}}}
-{\textstyle\frac{1-\cos({\cal{N}})}{{\cal{N}}^2}}
\right]\partial_{i}V_{i}^{\gamma}({\bf r})
\nonumber \\
&&+g^{2}\,
\epsilon^{\vec{\mu}a\gamma}_{(2)}\,
{\cal R}_{(2)}^{\vec{\mu}}({\bf r})\,
\left[{\textstyle\frac{\cos({\cal{N}})}{{\cal{N}}^2}}
-{\textstyle\frac{\sin({\cal{N}})}{{\cal{N}}^{3}}}\right]
\partial_{i}V_{i}^{\gamma}({\bf r})
\nonumber \\
&&+{\textstyle\frac{g^2}{2}}
\epsilon^{{\mu}a\lambda}\epsilon^{\alpha\lambda\gamma}
\,{\cal X}^{{\mu}}({\bf r})\overline{{\cal Y}^{{\alpha}}}({\bf r})\,
\left[{\textstyle\frac{\sin({\cal{N}})}{{\cal{N}}}}
-{\textstyle\frac{1-\cos({\cal{N}})}{{\cal{N}}^2}}
\right]\partial_{i}V_{i}^{\gamma}({\bf r})
\nonumber \\
&&+{\textstyle\frac{g^3}{2}}
\epsilon^{\vec{\mu}a\gamma}_{(2)}\,\epsilon^{\alpha\lambda\gamma}
{\cal R}_{(2)}^{\vec{\mu}}({\bf r})\,
\overline{{\cal Y}^{{\alpha}}}({\bf r})\,
\left[{\textstyle\frac{\cos({\cal{N}})}{{\cal{N}}^2}}
-{\textstyle\frac{\sin({\cal{N}})}{{\cal{N}}^{3}}}\right]
\partial_{i}V_{i}^{\gamma}({\bf r})
\nonumber \\
&&+g^3
\epsilon^{{\mu}a\lambda}
\,\epsilon^{\vec{\alpha}\lambda\gamma}_{(2)}
{\cal X}^{{\mu}}({\bf r}){\cal M}_{(2)}^{\vec{\alpha}}({\bf r})\,
\left[{\textstyle\frac{\sin({\cal{N}})}{{\cal{N}}}}
-{\textstyle\frac{1-\cos({\cal{N}})}{{\cal{N}}^2}}
\right]\left({\textstyle\frac{1}{2{\overline{\cal{N}}}}}
\cot{\left(\textstyle\frac{{\overline{\cal{N}}}}{2}\right)}
-{\textstyle\frac{1}{{\overline{\cal{N}}^2}}}\right)
\partial_{i}V_{i}^{\gamma}({\bf r})
\nonumber \\
&&+g^{4}\,
\epsilon^{\vec{\mu}a\lambda}_{(2)}\,
\epsilon^{\vec{\alpha}\lambda\gamma}_{(2)}
{\cal R}_{(2)}^{\vec{\mu}}({\bf r})\,
{\cal M}_{(2)}^{\vec{\alpha}}({\bf r})\,
\left[{\textstyle\frac{\cos({\cal{N}})}{{\cal{N}}^2}}
-{\textstyle\frac{\sin({\cal{N}})}{{\cal{N}}^{3}}}\right]
\left({\textstyle\frac{1}{2{\overline{\cal{N}}}}}
\cot{\left(\textstyle\frac{{\overline{\cal{N}}}}{2}\right)}
-{\textstyle\frac{1}{{\overline{\cal{N}}^2}}}\right)
\partial_{i}V_{i}^{\gamma}({\bf r})
\nonumber \\
&&-g^2 \epsilon^{a\beta d}
\epsilon^{{\mu}d\gamma}
A_{i}^{\beta}({\bf r})\,{\textstyle\frac{\partial_{i}}{\partial^{2}}}\,
\left(\,{\cal X}^{{\mu}}({\bf r})\,
{\textstyle\frac{1-\cos({\cal{N}})}{{\cal{N}}^2}}
\partial_{j}V_{j}^{\gamma}({\bf r})\,\right)
\nonumber \\
&&-g^3 \epsilon^{a\beta d}
\epsilon^{\vec{\mu}d\gamma}_{(2)}
A_{i}^{\beta}({\bf r})\,{\textstyle\frac{\partial_{i}}{\partial^{2}}}\,
\left(\,{\cal R}_{(2)}^{\vec{\mu}}({\bf r})\,
{\textstyle\frac{\sin({\cal{N}})-{\cal{N}}}{{\cal{N}}^3}}
\partial_{j}V_{j}^{\gamma}({\bf r})\,\right)
\nonumber \\
&&-{\textstyle\frac{g^3}{2}} \epsilon^{a\beta d}
\epsilon^{{\mu}d\lambda}\epsilon^{\alpha\lambda\gamma}
A_{i}^{\beta}({\bf r})\,
{\textstyle\frac{\partial_{i}}{\partial^{2}}}\,
\left(\,{\cal X}^{{\mu}}({\bf r})\,
\overline{{\cal Y}^{{\alpha}}}({\bf r})\,
{\textstyle\frac{1-\cos({\cal{N}})}{{\cal{N}}^2}}
\partial_{j}V_{j}^{\gamma}({\bf r})\,\right)
\nonumber \\
&&-{\textstyle\frac{g^4}{2}} \epsilon^{a\beta d}
\epsilon^{\vec{\mu}d\lambda}_{(2)}
\epsilon^{\alpha\lambda\gamma}
A_{i}^{\beta}({\bf r})\,
{\textstyle\frac{\partial_{i}}{\partial^{2}}}\,
\left(\,{\cal R}_{(2)}^{\vec{\mu}}({\bf r})\,
\overline{{\cal Y}^{{\alpha}}}({\bf r})\,
{\textstyle\frac{\sin({\cal{N}})-{\cal{N}}}{{\cal{N}}^3}}
\partial_{j}V_{j}^{\gamma}({\bf r})\,\right)
\nonumber \\
&&-g^4 \epsilon^{a\beta d}
\epsilon^{{\mu}d\lambda}\epsilon^{\vec{\alpha}\lambda\gamma}_{(2)}
A_{i}^{\beta}({\bf r})\,
{\textstyle\frac{\partial_{i}}{\partial^{2}}}\,
\left(\,{\cal X}^{{\mu}}({\bf r})\,
{\cal M}_{(2)}^{\vec{\alpha}}({\bf r})\,
{\textstyle\frac{1-\cos({\cal{N}})}{{\cal{N}}^2}}
\left[{\textstyle\frac{1}{2{\overline{\cal{N}}}}}
\cot{\left(\textstyle\frac{{\overline{\cal{N}}}}{2}\right)}
-{\textstyle\frac{1}{{\overline{\cal{N}}^2}}}\right]
\partial_{j}V_{j}^{\gamma}({\bf r})\,\right)
\nonumber \\
&&-g^5 \epsilon^{a\beta d}
\epsilon^{\vec{\mu}d\lambda}_{(2)}
\epsilon^{\vec{\alpha}\lambda\gamma}_{(2)}
A_{i}^{\beta}({\bf r})\,
{\textstyle\frac{\partial_{i}}{\partial^{2}}}\,
\left(\,{\cal R}_{(2)}^{\vec{\mu}}({\bf r})\,
{\cal M}_{(2)}^{\vec{\alpha}}({\bf r})\,
{\textstyle\frac{\sin({\cal{N}})-{\cal{N}}}{{\cal{N}}^3}}
\left[{\textstyle\frac{1}{2{\overline{\cal{N}}}}}
\cot{\left(\textstyle\frac{{\overline{\cal{N}}}}{2}\right)}
-{\textstyle\frac{1}{{\overline{\cal{N}}^2}}}\right]
\partial_{j}V_{j}^{\gamma}({\bf r})\,\right)\;.
\label{eq:fundthesu2}
\end{eqnarray}
To account for the general structure of
Eqs.~(\ref{eq:a2X}) and (\ref{eq:a2Y}),
we observe from Eqs.~(\ref{eq:Zxy}) and (\ref{eq:Diracspin}) that the
unitary transformation that transforms the spinor field
to its gauge-invariant
form {\em is itself a gauge transformation}.
$V_{\cal{C}}({\bf{r}})$ therefore
is an operator that gauge-transforms the spinor ${\psi}({\bf r})$ to a
form {\em that is then invariant to any
further gauge transformations}.
And $A_{{\sf GI}\,i}^{b}({\bf{r}}),$ which
is the corresponding gauge transform of the
gauge field $A_{i}^{b}({\bf{r}}),$ is similarly
invariant to any further gauge
transformations. Eq.~(\ref{eq:Adressedthree1b}) identifies
$\overline{{\cal{A}}_{i}^{b}}({\bf{r}})$ as an essential constituent of
$A_{{\sf GI}\,i}^{b}({\bf{r}}),$ and Eqs.~(\ref{eq:a2X})
and (\ref{eq:a2Y})
specialize $\overline{{\cal{A}}_{i}^{b}}({\bf{r}})$
to its $SU(2)$ structure.
It is therefore not surprising to find that the relation between
$\overline{{\cal{A}}_{i}^{b}}({\bf{r}})$ and
$A_{i}^{b}({\bf{r}})$ anticipates the
relation between $A_{{\sf GI}\,i}^{b}({\bf{r}})$ and
$A_{i}^{b}({\bf{r}})$ --- $i.e.$ that
$A_{{\sf GI}\,i}^{b}({\bf{r}})$ is the gauge-transform of
$A_{i}^{b}({\bf{r}})$ by the finite gauge function
${\cal Z}^{b}({\bf{r}}),$ defined in Eq.~(\ref{eq:Zxy}).
\section{Discussion}
This paper has addressed four main topics: The first
has been a proof of a previously published conjecture that states,
constructed in an earlier work\cite{bellchenhall} and
given in Eqs.~(\ref{eq:subcon}), (\ref{eq:Apsi}), and (\ref{eq:inteq2}),
implement the `pure glue' form of Gauss's law for QCD. Another
has been the construction of a unitary transformation
that extends these states so that they implement Gauss's
law for QCD with quarks as well as gluons. The third topic
is the construction of gauge-invariant spinor
and gauge field operators. And the last topic is
the application of the formalism
to the $SU(2)$ Yang-Mills case.
\bigskip
Implementation of Gauss's law is always required in a gauge theory,
but in earlier work it
was shown that in QED and other Abelian gauge theories, the failure to
implement Gauss's law does not affect the theory's physical
consequences\cite{khqedtemp,khelqed}.
And, in fact, it is known that the renormalized S-matrix in
perturbative QED
is correct, in spite of the fact that incident and
scattered charged particles are detached from all
fields, including the ones required to implement Gauss's law. In contrast,
the validity of perturbative QCD is more limited. It is not
applicable to low energy phenomena. And, it is likely
that all perturbative results in QCD are obscured, in some measure,
by long-range effects, so that the implications of QCD for
even high-energy phenomenology are still not fully known.
In particular, color confinement is not
well understood. One possible avenue for exploring QCD dynamics
beyond the perturbative
regime is the use of gauge-invariant operators and states in formulating
QCD dynamics. Although dynamical equations for gauge-invariant
operator-valued
fields have not yet been developed, we believe that
the mathematical apparatus we have constructed in this
paper can serve as a basis
for reaching such an objective.
\bigskip
We also note a feature of this work that is most clearly evident in
the $SU(2)$ example. The recursive equation for
$\overline{{\cal{A}}_{i}^{b}}({\bf{r}})$ --- Eq.~(\ref{eq:inteq2})
in the $SU(3)$ case,
with an arbitrary $V_i^\gamma({\bf r})$ replacing
the $\Pi_i^\gamma({\bf r})$,
and Eqs.~(\ref{eq:azero})--(\ref{eq:a2Y}) in the
$SU(2)$ Yang-Mills theory --- have many of the features that
we associate with finite gauge transformations applied to a
gauge field. This is particularly conspicuous for the parts of
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\cal X}$ and
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\overline{\cal Y}}$ that
correspond to the `pure gauge' components of
$\left(A^{\gamma}\,\right)^{\prime}_{i}$ displayed in
Eq.~(\ref{eq:atranssu2}).
These `pure gauge' parts are
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\cal X}^{(pg)}$ and
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\overline{\cal Y}}^{(pg)}$
respectively, and are given by
\begin{eqnarray}
\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\cal X}^{(pg)}\,
&&=-g\epsilon^{\alpha\beta\gamma}\,
{\cal{X}}^\alpha({\bf{r}})\,\partial_i{\cal{X}}^\beta({\bf{r}})\,
{\textstyle\frac{1-\cos({\cal{N}})}{{\cal{N}}^2}}
\nonumber\\
&&+g^2\epsilon^{\alpha\beta b}
\epsilon^{b\mu\gamma}\,
{\cal{X}}^\mu({\bf{r}})\,{\cal{X}}^\alpha({\bf{r}})\,
\partial_i{\cal{X}}^\beta({\bf{r}})\,
[{\textstyle\frac{1}{{\cal{N}}^2}}
-{\textstyle\frac{\sin({\cal{N}})}{{\cal{N}}^3}}]
\label{eq:a2Xpg}
\end{eqnarray}
and
\begin{eqnarray}
\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\overline{\cal Y}}^{(pg)}\,
&&=g\epsilon^{\alpha\beta\gamma}\,
\overline{{\cal{Y}}^\alpha}({\bf{r}})
\,\partial_i\overline{{\cal{Y}}^\beta}({\bf{r}})\,
{\textstyle\frac{1-\cos(\overline{\cal{N}})}
{\overline{\cal{N}}^2}}
\nonumber\\
&&+g^2\epsilon^{\alpha\beta b}
\epsilon^{b\mu\gamma}\,
\overline{{\cal{Y}}^\mu}({\bf{r}})\,
\overline{{\cal{Y}}^\alpha}({\bf{r}})\,
\partial_i\overline{{\cal{Y}}^\beta}({\bf{r}})\,
[{\textstyle\frac{1}{\overline{\cal{N}}^2}}-
{\textstyle\frac{\sin(\overline{\cal{N}})}
{\overline{\cal{N}}^3}}]\;.
\label{eq:a2Ypg}
\end{eqnarray}
The `pure gauge' parts of
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\cal X}$ and
$\overline{{\cal{A}}_{i}^{\gamma}}({\bf{r}})_{\overline{\cal Y}}$
correspond to the pure gauge part of
$\left(A^{\gamma}\,\right)^{\prime}_{i}$, with
$-g{\cal{X}}^\gamma({\bf{r}})$ and
$g\overline{{\cal{Y}}^\gamma}({\bf{r}})$
corresponding to the gauge function
$\omega^\gamma({\bf{r}}), $ and ${\cal{N}}$ and
$\overline{\cal{N}}$ corresponding to $|\omega|$ respectively.
This correspondence suggests that, in addition to the iterative
solution of Eq.~(\ref{eq:inteq2}), which we have discussed
extensively in this work, there may be non-perturbative solutions
that can not be represented as an iterated
series and that are related to the non-trivial topological sectors of
non-Abelian gauge fields\cite{topsect}.
\section{acknowledgements}
This research was supported by the Department of Energy under Grant
No.~DE-FG02-92ER40716.00.
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proofpile-arXiv_065-631
|
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\section*{\bf I.~~Preliminaries not involving group structure\\}
Before turning to a discussion of what is an appropriate definition of a
quaternionic projective group representation, we first address several
issues that do not involve the notion of a {\it group} of symmetries.
We follow throughout the Dirac notation used in our recent book [1],
in which linear operators in Hilbert space act on ket states from the
left and on bra states from the right, as in ${\cal O} |f \rangle$
and $\langle f|\cal O$, while quaternionic scalars in Hilbert
space act on ket states from the right and on bra states from the left, as in
$|f \rangle \omega$ and $\omega \langle f|$.
We begin by recalling the statement (see Sec. 2.3 of Ref.~[1])
of the quaternionic extension of
Wigner's theorem, which gives the Hilbert space representation of an
individual
symmetry in quantum mechanics. Physical states in quaternionic quantum
mechanics are in one-to-one correspondence with unit rays of the form
$|{\bf f} \rangle=\{ |f\rangle \omega \}$, with $|f\rangle$ a unit normalized
Hilbert space vector and $\omega$ a quaternionic phase of unit magnitude. A
symmetry operation $\cal S$ is a mapping of the unit rays $|{\bf f}\rangle$
onto images $|{\bf f}^{\prime} \rangle$, which preserves all transition
probabilities,
\begin{eqnarray}
{\cal S} |{\bf f}\rangle &=& |{\bf f}^{\prime} \rangle \nonumber\\
|\langle {\bf f}^{\prime}|{\bf g}^{\prime} \rangle|&=&
|\langle {\bf f} | {\bf g} \rangle |.
\label{one}
\end{eqnarray}
Wigner's theorem, as extended to quaternionic Hilbert space, asserts that
by an appropriate $\cal S$-dependent choice of ray representatives for the
states, the mapping $\cal S$ can always be represented
(in Hilbert spaces of dimension greater
than 2) by a unitary transformation $U_{\cal S}$ on the state vectors,
so that
\begin{equation}
|f^{\prime}\rangle = U_{\cal S}|f \rangle~.
\label{two}
\end{equation}
Conversely, any unitary transformation of the form of Eq.~(2) clearly implies
the preservation of transition probabilities, as in Eq.~(1). When only one
symmetry transformation is involved, the issue of projective
representations does not enter, since Wigner's theorem asserts that this
transformation can be given a unitary representation on appropriate
ray representative states in Hilbert space. The issue of projective
representations arises only when we are dealing with two (or more) symmetry
transformations, in which case the ray representative choices which reduce
the first symmetry transformation
to unitary form may not be compatible with the ray representative
choices which reduces a second symmetry transformation to unitary form.
Thus we disagree with Emch's statement, in the semifinal paragraph of his
Comment, that Wigner's theorem (which he notes is a form of the first
fundamental theorem of projective geometry) may be dependent on the
definition adopted for quaternionic projective group representations.
In the first section of his Comment, Emch proves a Proposition stating that
if an operator $\cal O$ commutes with all of the projectors
$|f\rangle \langle f| $ of a quaternionic Hilbert space of dimension 2
or greater, then $\cal O$ must be
a real multiple of the unit operator $1$ in Hilbert space. When $\cal O$
is further restricted to be a unitary operator (as obtained from a symmetry
transformation via the Wigner theorem), the real multiple is further
restricted to be $\pm 1$. Since we will refer to this result in the next
section, let us give an alternative proof, based on the
spectral representation of a general unitary operator $U$ in quaternionic
Hilbert space,
\begin{equation}
U=\sum_{\ell}|u_{\ell}\rangle e^{i \theta_{\ell}} \langle u_{\ell}|~,
~~0 \le \theta_{\ell} \le \pi~,
\label{three}
\end{equation}
in which the sum over $\ell$ spans a complete set of orthonormal
eigenstates of $U$.
Let us focus on a two state subspace spanned by $|u_1\rangle$ and
$|u_2\rangle$, and construct the projector $P=|\Phi\rangle \langle\Phi|$,
with
\begin{eqnarray}
|\Phi\rangle&=&|u_1\rangle + |u_2\rangle \omega~,\nonumber\\
\overline{\omega}&=&-\omega~,~~\omega=\omega_{\alpha} +j \omega_{\beta}~,~~
\omega_{\alpha}\omega_{\beta} \ne 0 ~,
\label{four}
\end{eqnarray}
where $\omega_{\alpha,\beta}$ are symplectic components lying in the complex
subalgebra of the quaternions spanned by $1$ and $i$. Then the
projector $P$ is given by
$$
P=|u_1\rangle \langle u_1|+|u_2 \rangle \langle u_2|
+|u_2 \rangle \omega \langle u_1|-|u_1 \rangle \omega \langle
u_2|~,\eqno(5a)
$$
and the part of $U$ lying in the $|u_{1,2}\rangle$ subspace is
$$
U_{1,2}=|u_1\rangle e^{i \theta_1} \langle u_1|
+|u_2\rangle e^{i \theta_2} \langle u_2| ~.\eqno(5b)
$$
The commutator of $U$ and $P$ is then given by
\setcounter{equation}{5}
\begin{equation}
[U,P]=[U_{1,2},P]=
|u_2\rangle (e^{i \theta_2} \omega -\omega e^{i \theta_1})\langle u_1|
-|u_1\rangle (e^{i \theta_1} \omega -\omega e^{i \theta_2})\langle u_2|~,
\label{six}
\end{equation}
which vanishes only if $e^{i \theta_1}=e^{i \theta_2}$ (from equating to
zero the coefficient of $\omega_{\alpha}$) and $e^{i \theta_1} =
e^{-i \theta_2}$ (from equating to zero the coefficient of $\omega_{\beta}$).
Since $0 \le \theta_{1,2} \le \pi$, this requires either $\theta_1=
\theta_2=0$ or $\theta_1=\theta_2=\pi$. Repeating the argument for each
dimension 2 subspace in turn, we learn that $U=\pm 1$. Note that in a
complex Hilbert space, the analogous argument shows only that
$e^{i \theta_1}=e^{i \theta_2}$, from which we conclude
(again by repeating the
argument for each dimension 2 subspace in turn) that $U=e^{i\theta}$,
which commutes with all projectors because any complex number is
a $c$-number in complex Hilbert space.
Clearly, the argument just given involves only elementary properties of
the projectors in Hilbert space, and makes no reference to the notion of
a group of symmetries. The same is true of the proposition given in Sec. I
of Emch's Comment. Since Schur's Lemma ordinarily describes the restrictions
on an operator that commutes with the representation matrices of an
irreducible group representation,
and since the projectors in Hilbert space do not form a group (they
are not invertible and the product of two different projectors is not
a projector),
it is a misnomer to describe Emch's
Proposition, or the corollary given here, as a ``quaternionic Schur's
lemma''.
In addition to disagreeing with Emch's terminology, we also
disagree with his statement, in the second paragraph of Sec. III of
his Comment,
that the analysis leading to his Proposition is
dependent on the definition adopted for quaternionic projective group
representations; in fact, the notion of a group of symmetries does not enter
into either his analysis, or the corollary for unitary matrices proved here.
\section*{II.~~How should one define quaternionic projective
group representations?}
Let us now address the central question of how one should generalize to
quaternionic Hilbert space the notion of a projective group representation.
We begin by reviewing how projective group representations arise in complex
Hilbert space. Let $\cal G$ be a symmetry group composed of abstract
elements $a$ with group multiplication $ab$. By Wigner's theorem, each
group element is represented, after an $a$-dependent choice of ray
representatives, by a unitary operator $U_a$ acting on the states of
Hilbert space. In the simplest case, in which the $U_a$ are said to
form a vector representation, the $U$'s obey a multiplication law
isomorphic to that of the corresponding abstract group elements,
\begin{equation}
U_aU_b=U_{ab}~.
\label{seven}
\end{equation}
However, when the complex rephasings of the states used in Wigner's
theorem are taken into account, there exists the more general possibility
that for any state $|f\rangle$, the states $U_aU_b|f\rangle $ and
$U_{ab}|f \rangle$ are not equal, but rather differ from one another by a
change of ray representative, i.e.,
\begin{equation}
U_aU_b|f\rangle = U_{ab}|f \rangle e^{i \phi(a,b;f)} ~.
\label{eight}
\end{equation}
Corresponding to Eq.~(8), there are two possible definitions of a projective
representation in complex Hilbert space:\hfill\break
{\it Definition (1)} In a {\it weak} projective representation, the
multiplication
law of the $U$'s obeys Eq.~(8) on one complete set of states $\{ |f\rangle\}$.
This suffices, by superposition, to determine the multiplication law of the
$U$'s on all states. \hfill \break
{\it Definiton (2)} In a {\it strong} projective representation, the
multiplication law of the $U$'s obeys Eq.~(8) on all states in Hilbert
space. In this case, we can easily prove that the phases $\phi(a,b;f)$ are
independent of the state label $f$. To see this, let us define
$V_{ab}=U_{ab}^{-1}U_aU_b$; then Eq.~(8) implies that
\begin{equation}
V_{ab}|f\rangle=|f \rangle e^{i \phi(a,b;f)}~,
\label{nine}
\end{equation}
which immediately implies that $V_{ab}$ commutes with the projector
$|f\rangle \langle f|$, for all states $|f\rangle$ in Hilbert space.
But invoking the complex Hilbert space specialization of the result of
the preceding section, we learn that $V_{ab}$ must be a $c$-number,
$V_{ab}=e^{i\phi(a,b)}$. This is the customary definition of a projective
representation in complex Hilbert space, and is well known to have nontrivial
realizations.
Let us now turn to the question of how to define projective representations
in quaternionic Hilbert space. Emch choses as his generalization the
strong definition given above, which by the reasoning following Eq.(9),
and the quaternionic result of Sec. 1, implies that $V_{ab}=(-1)^{n_{a,b}}$,
with $n_{a,b}$ an integer that can depend in general on $a$ and $b$.
In other words, {\it the only strong quaternionic projective representations
are real projective representations}.
The problem with adopting the strong definition, however, is that it excludes
from consideration as a quaternionic projective representation the
embedding into quaternionic Hilbert space of a nontrivial complex projective
representation realized on a complex Hilbert space. Thus, potentially
interesting structure is lost. To avoid this problem, Ref. [1] adopts as
the quaternionic generalization of the notion of a projective representation
the weak definition given above, which in quaternionic Hilbert space
states that
\begin{equation}
U_aU_b|f\rangle=U_{ab}|f\rangle \omega_{a,b}~,~~|\omega_{a,b}|=1~
\label{ten}
\end{equation}
for one particular complete set of states $\{|f\rangle \}$. As discussed
in Ref.~1, Eq.~(10) can also be rewritten in the operator form
$$
U_aU_b=U_{ab}\Omega(a,b) ~,
\eqno(11a)
$$
with
$$
\Omega(a,b)=\sum_f |f\rangle \omega(a,b;f)\langle f|~.
\eqno(11b)
$$
Since the operator $\Omega$ depends on the particular complete set
of states on which the projective phases are given, a more complete notation
(not employed in Ref.~1) would in fact be $\Omega(a,b;\{ |f\rangle\})$.
Using the result of an analysis [2] of the associativity condition for
weak quaternionic projective representations, Tao and Millard [3] have
recently given a beautiful complete structural classification theorem for
weak quaternionic projective representations. The complex
specialization of their Corollary 2, incidentally,
states that in a complex Hilbert space,
the weak definition of a projective representation implies the strong one.
Can the weak definition of a quaternionic projective representation
be weakened even further,
by using a {\it different} complete set of states $\{ |f\rangle\}$ to
specify the projective phases for each pair of group elements $a$ and $b$ [4]?
In this case, the operator $\Omega$ takes the form $\Omega(a,b; \{ |f\rangle \}
_{a,b})$. However, since any unitary operator is diagonalizable on
some complete set of states, this further weakening allows an
arbitrary specification
of $\Omega$ for each $a,b$, and any relationship of the unitary
representation to the underlying group structure is lost.
\section*{III.~~ Discussion}
We conclude that the difference between our analysis and that of Emch
is traceable to what I have here termed the difference between a {\it strong}
and a {\it weak} definition of projective representation. The strong
definition
is the customary one in complex Hilbert space, but it
excludes potentially interesting structure when applied to quaternionic
Hilbert space. Since the weak definition leads to a detailed theory
[1, 2, 3] of projective group representations in quaternionic Hilbert
space, and since it implies [3] the strong definition in complex Hilbert
space, the weak definition is in fact the more appropriate one in both
complex and quaternionic Hilbert spaces.
\acknowledgments
This work was supported in part by the Department of Energy under
Grant \#DE--FG02--90ER40542. I wish to thank A.C. Millard and T. Tao for
informative conversations, and to acknowledge the hospitality of the
Aspen Center for Physics, where this work was done.
|
proofpile-arXiv_065-632
|
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\section{Introduction}
The investigation of the transport properties of highly
correlated fermionic systems has attracted much
attention in recent years. A thorough understanding
of the conductivity in particular is essential
for the technical application of materials such
as metallic oxides in electronic devices.
The development of a new analytic
approach, the limit of infinite dimension for fermionic
systems \cite{metzn89a,vollh93},
allowed the numerical description of the metal-insulator
occuring in the half-filled Hubbard model in $d=\infty$ for higher
values of the interaction $U$ assuming a homogeneous phase
\cite{georg96,prusc96}.
The latter assumption means that one deliberately ignores
the possible occurence of symmetry breaking for the sake
of simplicity. It is argued that on frustrated lattices
symmetry breaking is suppressed so that the metal-insulator
transition occurs at higher temperatures than those at which
symmetry breaking sets in.
With this background in mind, it is the aim of this work
to extend and to complement the results known so far
into two directions.
First, the finite dimensionality of realistic systems, i.e.\
mostly $d=3$, shall be included at least to lowest non-trivial
order in an expansion in $1/d$.
Much care is used in including these correction without
physical and/or analytic inconsistencies.
It is shown
that it is {\em not} sufficient to use a conserving, $\Phi$-derivable
approximation in the sense of Baym/Kadanoff.
Furthermore, the true
three-dimensional DOS will be used.
Second, the influence of symmetry breaking on the conductivity,
especially the question of possible metal-insulator transitions
induced by symmetry breaking shall be investigated.
To this end, the model of spinless fermions with repulsive
interaction for particles on adjacent sites is considered on
a generic bipartite lattice, namely the simple cubic lattice.
Its Hamiltonian at half-filling $n=1/2$ reads
\begin{equation}\label{hamil4}
\hat{H} = -\frac{t}{\sqrt{Z}}\sum_{<i,j>} {\hat c}^+_i{\hat
c}^{\phantom{+}}_j + \frac{U}{2Z}\sum_{<i,j>} {\hat n}_i {\hat n}_j
-\frac{U}{2} \sum_i {\hat n}_i\ .
\end{equation}
where $ {\hat c}^+_i \; ({\hat
c}^{\phantom{+}}_i)$ creates (annihilates) a fermion at site
$i$. The sum $\sum_{<i,j>}$ runs over all sites $i$ and $j$
which are nearest neighbors. The coordination number
$Z=2d=6$ appears for the proper scaling of
the kinetic energy \cite{metzn89a} and for the proper scaling of
the potential energy \cite{mulle89a}. The interaction constant
is $U$.
In this model the symmetry is broken yielding an AB-CDW at
half-filling \cite{halvo94} for infinitesimal values of the
interaction at $T=0$ and for sufficiently large interaction
at all finite temperatures. The AB-CDW consists of alternating
sites with a particle density above (below) average. The order
parameter $b$ is the absolute
deviation of the particle density from its
average \cite{halvo94}.
As far as the occurence of a symmetry broken phase is concerned,
the model of spinless fermions at half-filling is similar
to the Hubbard model at half-filling which displays
antiferromagnetic behavior. The main differences are that the
broken symmetry for spinless fermions is discrete whereas it
is continuous in the Hubbard model, and the fact that a local
interaction like the one in the Hubbard model does not
favor a spatial order by itself. The latter fact leads to a value
of $T_c \propto 1/U$ for large $U$ in the Hubbard model whereas
one has $T_c \propto U$ in the spinless fermions model.
The article is organized as follows. Succeeding this introduction
it is discussed how a thermodynamically and analytically consistent
extension of the limit $Z\to \infty$ can be performed.
Next the basic equations for the extension to linear order
$1/Z$ are derived and their
numerical evaluation is sketched.
This third section contains also
results for the DOS and the corresponding proper self-energy.
In sect.\ 4 the Bethe-Salpeter equation is set up
and solved for the conductivity $\sigma(\omega)$. The preservation
of the f-sum rule is discussed.
Numerical results for the dc- and the ac-conductivity
are presented in sect.\ 5.
The findings are
summarized and dicussed in the final section.
All energies (temperatures, respectively)
throughout this article will be given in units of the
root-mean-square of the ``free'', i.e.\ non-interacting,
density-of-states of the lattice model concerned.
All conductivities will be given in units
of $e^2/(\hbar a^{d-2})$ where $a$ is the lattice constant.
The constants $a$, $\hbar$, and $k_{\scriptstyle\rm B}$
(Boltzmann's constant) are set to unity.
\section{Proper self-consistent extension of $Z=\infty$}
In the case $Z=\infty$, the evaluation of diagrams and the treatment
of quantities like the DOS is conceptually simple. It is always
the leading contribution in $1/Z$ and only this which must be kept.
There is no dependence on the sequence in which certain quantities
and the equations relating them are considered. All sum rules
which hold in any dimension also hold at $Z=\infty$, continuity
provided for the limit $Z\to\infty$. This simplicity is lost
as soon as corrections in $1/Z$ are to be included. For
concreteness, let us consider the linear corrections $1/Z$; the
problems are illustrated for the free DOS, the Dyson equation
and the free energy $F$ as function of the order parameter $b$.
The DOS is a non-negative function of which the zeroth moment is
unity. This holds in any dimension, hence in $Z=\infty$. On including
the linear corrections \cite{mulle89a} one realizes that the
approximate expression becomes negative at large values of $\omega$.
This is a disadvantage of the otherwise systematic expansion.
Another inconvenience catches the eye in fig.\ \ref{fi:1}.
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(16,7)(0,0.7)
\put(8.3,0){\psfig{file=fig1.ps,height=7.2cm,width=8cm,angle=270}}
\end{picture}
\caption{Non-interacting DOS in $d=\infty$
(short-dashed curve), in $d=3$
(solid curve) and the DOS expanded in $1/d$ evaluated in $d=3$
(long-dashed curve). These densities of states are symmetric
about the y-axis.}
\label{fi:1}
\end{figure}
The
expanded DOS does not improve considerably the agreement with the true
finite dimensional DOS (here $d=3$). A finite expansion in $1/Z$ cannot
produce the van-Hove-singularities.
To circumvent the problem of the DOS expansion, we decide to use
the exact finite dimensional DOS, i.e. the $d=3$ DOS.
This procedure provides often even in $d=1$ a remarkable
agreement \cite{halvo94,strac91}. In $d=3$, this approximation yields
qualitatively agreement for the local DOS as compared to finite
dimensional perturbation results \cite{schwe91a}.
Presently, the approach of using a finite dimensional DOS in an otherwise
infinite dimensional calculation as approximation for the finite
dimensional problem is employed as so-called ``dynamical mean-field
theory'' \cite{prusc96} or
``local impurity self-consistent approximation'' \cite{georg96}.
Next the problem of a systematic $1/Z$-expansion is discussed for
the Dyson equation. It is stated in a simple case when the
self-energy is strictly local in real space, i.e. constant in
momentum space
\begin{equation}\label{dysgl}
g(\omega)= g_0(\omega-\Sigma(\omega)) \ .
\end{equation}
This case is realized, for instance, in the
Hubbard model in $d=\infty$ \cite{mulle89a,janis92a}.
No lattice site or spin index appears since the phase is assumed to be
homogeneous and non-magnetic. The quantity $g(\omega)$ stands for the
full local Green function $G_{i,i}(\omega)$ and $g_0(\omega)$
stands for the free Green function $G_{0;\, i,i}(\omega)$. The
expansion of the Green function corresponds to the expansion of the
thermodynamic potential since they depend linearly on each other
\cite{ricka80}. An expansion of the self-energy, however,
yields a {\em different} expression for $g(\omega)$ since $g_0(\omega)$
is not a linear function.
The expansion of the self-energy seems more
promising since it preserves the Dyson equation by construction.
Moreover, it is able to describe the shift of singularities, e.g. the
shifts of the band edges. (Note that we discuss here finite expansions of
the quantities considered).
In spite of the choice to expand the self-energy some ambiguity
persists.
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,7)(0,0.7)
\put(8.3,0){\psfig{file=fig2.ps,height=7.2cm,width=8cm,angle=270}}
\end{picture}
\caption{Externally applied field $h_{\scriptstyle \rm EXT}$
as function of the order
parameter $b$ for $U=9$ and $T=0$ in $d=3$. The short-dashed
curve depicts the $1/d$ self-consistent result, the long-dashed
curve the result of a systematic expansion of the self-energy.
The zeros of the curves correspond to thermodynamic equilibrium.
But only zeros with positive slope are locally stable
($b\approx 0.48$).}
\label{fi:2}
\end{figure}
In fig.\ \ref{fi:2}, this problem is illustrated.
It arises in the description of spontaneous symmetry breaking.
Two results for the
dependence of the conjugated field on the order parameter
are opposed. The data refers to the AB-CDW occuring in the spinless
fermion problem at half-filling. The dotted curve results from
a fully self-consistent calculation whereas the dashed curve results
from a systematic expansion of the self-energy. Note that
the self-consistent approach generates higher order contributions.
The argument results now from the strange behavior of the dashed curve
in the vicinity of the origin. The free energy belonging to
the dotted curve can be found by integration; it has an unstable maximum
($\partial h_{\scriptstyle\rm EXT}/\partial b <0$) at $b=0$ and
two stable minima ($\partial h_{\scriptstyle\rm EXT}/\partial b >0$)
at $b\approx \pm 0.48$. But there is no free energy belonging to the
dashed curve since it would have three maxima in sequence around
$b=0$ which is mathematically impossible (theorem of Rolle).
This is a very strong argument in favor
of a self-consistent calculation.
For completeness, it shall be mentioned that one may argue that
in the vicinity of the physical solutions, i.e. the minima, the
difference of both approaches is negligible. There are also cases
known where the systematic, non self-consistent approach yields
better results \cite{schwe90b}. But there is still another advantage of
the self-consistent treatment
which will be crucial for what follows.
In the sense of Baym/Kadanoff \cite{baym61,baym62}
it covers also the calculations of two-particle properties and
ensures the preservation of sum rules.
So, Schweitzer and Czycholl resorted in their calulation of
resistance and thermopower for the periodic Anderson model to
the self-consistent treatment \cite{schwe91b}
although their results for the local
DOS did not necessarily favor this approach \cite{schwe90b}.
As result of the above discussion the starting point for the
inclusion of $1/Z$ correction is the generating functional
$\Phi$ according to Baym/Kadanoff \cite{baym61,baym62}. This
is the quantity which is expanded in a $1/Z$ series. Then the truncation
of this series yields an approximation to the corresponding order.
The power counting for the diagrams of $\Phi$ has been explained
previously \cite{halvo94,uhrig95d}.
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,4.3)(0,0)
\put(2,0.6){\psfig{file=fig3.ps,height=4.3cm,width=14.5cm}}
\end{picture}
\caption{Diagrams contained in
$\Phi_{\scriptstyle \rm A}[G]$. The first generates
the Hartree term, the second the Fock term, and the third the
local correlation term. The solid lines represent dressed
propagators, the wavy lines the interactions. The sum runs over
the lattice sites $i,j$.}
\label{fi:3}
\end{figure}
Here it shall just be stated
that the first diagram in fig.\ \ref{fi:3} is of order ${\cal O}(1)$
and the two other diagrams in fig.\ \ref{fi:3} produce the linear
corrections ${\cal O}(1/Z)$ whereas the diagrams in
fig.\ \ref{fi:4}
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,4.2)(0,0)
\put(3,0.6){\psfig{file=fig4.ps,height=4.2cm,width=13.5cm}}
\end{picture}
\caption{Two examples of diagrams
in higher order (here: quadratic) in $1/Z$.
The sites $i$ and $j$ are adjacent as are the sites $i'$ and $j'$.
Additionally, $i\neq i'$ and $j\neq j'$ holds.}
\label{fi:4}
\end{figure}
are examples for ${\cal O}(1/Z^2)$ contributions.
Thus fig.\ \ref{fi:3} visualizes the approximate
$\Phi_{\scriptstyle\rm A}$ potential
which will be used in this work.
By functional derivation the self-energy shown in fig.\ \ref{fi:5}
is obtained.
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,3.6)(0,0)
\put(4,0.6){\psfig{file=fig5.ps,height=3.6cm,width=12.5cm}}
\end{picture}
\caption{The self-energy diagrams
derived from fig.\ \protect\ref{fi:3} by taking
out one propagator line. The diagrams shown contribute in order
$1/Z$.}
\label{fi:5}
\end{figure}
Note that the Fock diagram is seemingly of another
order, namely ${\cal O}(1/Z^{3/2})$, than the third diagram,
${\cal O}(1/Z)$, which is called the local correlation
diagram henceforth. What matters, however,
is the order relative to the free Green
function which is ${\cal O}(1/Z^{1/2})$ for adjacent sites.
It is another advantage of the Baym/Kadanoff formalism that
one does not need to bother about these questions once the
approximate $\Phi$-potential is chosen.
Now a point shall be highlightened which
has not been mentioned before to our knowledge. In spite of the many
arguments in favor of the Baym/Kadanoff formalism its naive
application does not guarantee the absence of unphysical results.
A counter example serves as illustration. Consider an approximate
$\Phi$ consisting only of the diagram in fig.\ \ref{fi:4}(a),
summed over all sites $i,j,i',j'$, such that $i$ and $j$
($i'$ and $j'$, respectively) are adjacent to one another and
fulfill $j\ne j'$ and $i\ne i'$. The resulting nearest-neighbor
self-energy $\Sigma_{i,j}=t(\omega)=t'(\omega)+i t''(\omega)$ has a finite
imaginary part $t''(\omega)$. Using the Dyson equation, one obtains
in the homogeneous phase
\begin{eqnarray}\nonumber
G_{{\bf k},{\bf k}}(\omega+0i) &=&
\frac{1}{\omega+0i-(1+t(\omega+0i))\varepsilon({\bf k})}
\\ \nonumber
&=& \frac{\omega-(1+t')\varepsilon({\bf k})+i t''\varepsilon({\bf k})}
{\left(\omega+0i-(1+t')\varepsilon({\bf k})\right)^2+
\left(t''\varepsilon({\bf k})\right)^2}\ .
\end{eqnarray}
By choosing an appropriate wave vector ${\bf k}$ at fixed $\omega$
one can have the sign of $\varepsilon({\bf k})$ such that the imaginary
part of $G_{{\bf k},{\bf k}}(\omega+0i)$ is positive \cite{note1}. This is
a contradiction to the exact result \cite{ricka80}. Note that the
details of $t(\omega)$ are not essential as long as the imaginary
part is finite.
The counter example above is not only of academic interest.
Schweitzer and Czycholl observed as well that the inclusion
of a nearest-neighbor self-energy leads to wrong signs of the
imaginary parts. They considered the $1/d$ expansion of a $U^2$
perturbation theory around Hartree-Fock for the Hubbard model
and the periodic Anderson model \cite{schwe91a,schwe90a}.
They reached consistency by
including higher $1/d$ corrections (for $d=1$ up to 50 terms)
\cite{schwe91a}.
Problems with the analyticity (uniqueness) of the solution
occurred also in the first investigations
of $1/d$ corrections in the Hubbard model \cite{georg96}
(Falicov-Kimball model \cite{schil95}).
To the author's knowledge there is no necessary or sufficient
theory so far, which predicts under which circumstances
such problems have to be expected or can be excluded.
A sufficient argument excluding wrong signs of the imaginary
part of the approximate self-energy is given by the theorem:
If the approximation considered can be interpreted as an
expansion of the self-energy in a parameter $\lambda>=0$
and if $m$ is the leading order, in which the imaginary
part of the self-energy does {\em not} vanish, then
the self-energy approximated in the $m$-th order has the
right sign.
The proof relies on the continuity of limits if the expansion exists.
According to the precondition holds
\begin{equation}\label{gegen2}
0 \geq {\mathop{\rm Im}\nolimits} \Sigma_{\lambda}(\omega,{\bf k})
\, = \, \lambda^m {\mathop{\rm Im}\nolimits} \Sigma^{(m)}(\omega,{\bf k})
+{\cal O}(\lambda^{(m+1)}) \ ,
\end{equation}
which is equivalent to
\begin{equation}\label{gegen3}
0 \geq \lim_{\lambda\to 0+} \lambda^{-m}
{\mathop{\rm Im}\nolimits}\Sigma_{\lambda}(\omega,{\bf k})
\, = \, {\mathop{\rm Im}\nolimits} \Sigma^{(m)}(\omega,{\bf k})\ .
\end{equation}
The index ${\bf k}$ is the wave vector in a homogeneous,
translationally invariant phase. The derivation for
general phases, for instance
the AB-CDW, is given in appendix A.
The derivation in (\ref{gegen2}) and in (\ref{gegen3}) holds strictly
only for the non self-consistent treatment. In the generic situation,
however, the leading order of the self-energy with non-vanishing
imaginary part results from a certain diagram class and the analytic
properties do not depend on the specific form of the Green function
entering. If this is the case, the statement of the theorem extends
also to the self-consistent treatment where the quantitative form of the
Green functions are not known a priori.
The theorem helps one to understand the observations made
by Schweitzer and Czycholl. In the $1/d$ expansion of the
$d$-dimensional Hubbard model and of the periodic Anderson model
one has $\lambda=1/d$ and $m=0$ since the self-energy is imaginary
already in the first order. For the perturbation theory in $U$
one has $\lambda=U$ and $m=2$ since the self-energy stays real
in Hartree-Fock. Applying the rationale of the theorem
twice one understands that the self-energy in $U^2$ of the infinite
dimensional model has the right analytic behavior. If further
$1/d$ corrections are included this does not need to be true.
The result of Schweitzer and Czycholl, that the linear $1/d$ correction
leads to wrong signs, proves that the theorem is sharp: If the
precondition fails, the implication fails, too. The second obervation,
that the inclusion of {\em very many} $1/d$ correction terms remedies
the failure, can also be understood easily. In this case the
calculations approximates the $U^2$ perturbation theory of the
{\em finite} dimensional models very well. According to the theorem,
this perturbation theory displays the right sign, too.
The above observations indicate that also the analyticity problems
encountered for $1/d$ corrections in the Hubbard model \cite{georg96}
are not due to the approximations used to solve the effective
impurity problems. Rather each time that the theorem does not
apply one has to expect that analyticity problems arise for certain
parameters.
Considering eq.\ (6a) in ref.\ 18 or equivalently eq.\ (370) in ref.\ 3
one realizes that the spectral density of the local self energy might
change sign. This cannot obviously be excluded from the way how the
impurity self energies are computed.
Turning to the $1/d$ expansion of
the present model of spinless fermions ($\lambda=1/d$), one notes that
the theorem applies with $m=1$. Therefore, the equations
including linear $1/d$ corrections display the right analyticity.
These equation will be set up in the following.
\subsection{Resulting equations and one-particle results}
This section is kept very concise since it contains
material which is partly published elsewhere \cite{halvo94}.
For two reasons, however, it cannot be omitted. Firstly,
a different notation using different intermediate quantities
shall be introduced. Secondly, the one-particle results
are necessary requisites to understand the conductivity
results in the subsequent section.
The treatment of a self-energy of the type depicted in
fig.\ \ref{fi:5} is commonly known (see e.\ g.\ refs.\
11, 19, 20). Dealing with the symmetry
broken phase, however, requires some extension. In a previous work
\cite{halvo94} local Green function and the self-energy are distinguished
according to the sublattice to which they belong. In the present work,
sum and difference of the quantities on the two sublattices
will be used. The local quantities on site $i$ belonging to
sublattice $\tau \in \{A,B\}$ are
\begin{mathletters}
\label{sigdef1}
\begin{eqnarray}\label{sigdef1a}
g_\tau &:=& G_{i,i}(\omega)
\\ \label{sigdef1b}
\Sigma_\tau(\omega)&:=&\Sigma^{\scriptstyle\rm H}_{i,i}(\omega)
+\Sigma^{\scriptstyle\rm C}_{i,i}(\omega) \ ,
\end{eqnarray}
\end{mathletters}
where $G_{i,i}(\omega)$ is the full local Green function
and $\Sigma$ is the local self-energy.
The Fock part will be treated subsequently. The index $^{\scriptstyle\rm H}$
stands for the Hartree term (first diagram in fig.\ \ref{fi:5});
the index $^{\scriptstyle\rm C}$ stands for the local correlation
(third diagram in fig.\ \ref{fi:5}). Let us define
\begin{mathletters}
\label{sumdif}
\begin{eqnarray}\label{sumdifa}
g_{\scriptstyle\rm S}(\omega)&:=&
(g_{\scriptstyle\rm A}(\omega)+g_{\scriptstyle\rm B}(\omega))/2
\\ \label{sumdifb}
g_{\scriptstyle\rm D}(\omega)&:=&
(g_{\scriptstyle\rm A}(\omega)-g_{\scriptstyle\rm B}(\omega))/2
\\ \label{sumdifc}
\Sigma(\omega)&:=&(\Sigma_{\scriptstyle\rm A}(\omega)+
\Sigma_{\scriptstyle\rm B}(\omega))/2
\\ \label{sumdifd}
\Delta(\omega)&:=&(\Sigma_{\scriptstyle\rm A}(\omega)-
\Sigma_{\scriptstyle\rm B}(\omega))/2 \ .
\end{eqnarray}
\end{mathletters}
The spectral functions of the Green function
are called $N_{\scriptstyle\rm S}$ and $N_{\scriptstyle\rm D}$, respectively;
the spectral functions of the self-energy $\Sigma$ and $\Delta$
are called $N_\Sigma$ and $N_\Delta$, respectively.
The non local Fock term is $\Sigma^{\scriptstyle\rm F}:=\Sigma_{i,j}$, where
$i$ and $j$ are adjacent sites. It turns out, that
$\Sigma^{\scriptstyle\rm F}$
is negative (for repulsive interaction),
real, and that it does not depend on whether the fermion hops
from $A$ to $B$ or vice versa. Hence, it renormalizes the hopping
\begin{equation}\label{tren}
t \to \gamma t \quad \mbox{with} \quad \gamma:=1-\sqrt{Z}
\Sigma^{\scriptstyle\rm F}/t\ .
\end{equation}
Note that for attractive interaction $\gamma$ could become 0 which
would lead to a breakdown of the theory. Such a singularity is
absent in the repulsive case.
In the AB-CDW, the modes at ${\bf k}$ couple to those at
${\bf k}+{\bf Q}$. Hence one has
\begin{equation}\label{block2}
\left( \begin{array}{lr}
G_{{\bf k},{\bf k}}& G_{{\bf k},{\bf k}+{\bf Q}}\\
G_{{\bf k+{\bf Q}},{\bf k}} & G_{{\bf k+{\bf Q}},{\bf k}+{\bf Q}}
\end{array} \right)
= \left( \begin{array}{lr}
\omega-\Sigma(\omega)-\gamma\varepsilon & -\Delta(\omega) \\
-\Delta(\omega) & \omega-\Sigma(\omega)+\gamma
\varepsilon \end{array} \right)^{-1} \ .
\end{equation}
From this equation one obtains
\begin{mathletters}
\label{green3}
\begin{eqnarray}\nonumber
g_{\scriptstyle\rm S}(\omega)&=&
\frac{w}{\gamma\sqrt{w^2-\Delta^2(\omega)}}
g_0(\sqrt{w^2-\Delta^2(\omega)}/\gamma)
\\ \label{green3a}
&=&\int\limits_{-\infty}^\infty \frac{w}{w^2-(\gamma\varepsilon)^2-\Delta^2}
N_0(\varepsilon)d\varepsilon
\end{eqnarray}
\begin{eqnarray}\nonumber
g_{\scriptstyle\rm D}(\omega)&=&
\frac{\Delta(\omega)}{\gamma\sqrt{w^2-\Delta^2(\omega)}}
g_0(\sqrt{w^2-\Delta^2(\omega)}/\gamma)
\\ \label{green3b}
&=&\int\limits_{-\infty}^\infty
\frac{\Delta}{w^2-(\gamma\varepsilon)^2-\Delta^2}
N_0(\varepsilon)d\varepsilon
\ , \end{eqnarray}
\end{mathletters}
where $w$ is short hand for $\omega-\Sigma(\omega)$.
The averaged Hartree term $U(n_{\scriptstyle\rm A}+n_{\scriptstyle\rm B})/2$
renormalizes the chemical potential \cite{uhrig95d}. The Hartree
contribution to $\Delta$ is $Ub$ where
$b:=(n_{\scriptstyle\rm B}-n_{\scriptstyle\rm A})/2$
is the order parameter, i.e.\ the particle density difference.
It is given by $b=-\int\limits_{-\infty}^\infty
N_{\scriptstyle\rm D}(\omega) f_{\scriptstyle\rm F}(\omega) d\omega $,
where $f_{\scriptstyle\rm F}(\omega)$ is the Fermi function.
The Fock term can be calculated from the nearest-neighbor
Green function $G_{j+a,j}$
\begin{equation}\label{kinen2}
\Sigma^{\scriptstyle\rm F} = \frac{U}{\pi Z}
\int\limits_{-\infty}^\infty
{\mathop{\rm Im}\nolimits}\left( G_{j+a,j}(\omega+0i)
\right) f_{\scriptstyle\rm F}(\omega)d\omega \ ,
\end{equation}
which is given by
\begin{eqnarray}\nonumber
G_{j+a,j}(\omega)&=& -\frac{1}{\sqrt{Z}} \int\limits_{\scriptstyle\rm BZ}
\varepsilon({\bf k})G_{{\bf k},{\bf k}} \frac{dk^d}{(2\pi)^d}
\\ \label{kinen3}
&=& -\frac{1}{\gamma\sqrt{Z}}
\left[(\omega-\Sigma) g_{\scriptstyle\rm S}(\omega)-
\Delta g_{\scriptstyle\rm
D}(\omega)\right]
d\varepsilon \ .
\end{eqnarray}
The Fock term is related to the kinetic energy
$\Sigma^{\scriptstyle\rm F} =(U/Z^{3/2})\langle \hat T \rangle$.
Thus, (\ref{kinen2}) can be evaluated using (\ref{kinen3}) and (\ref{green3}).
The local correlation term is given in terms of the Matsubara frequencies
$\omega_\lambda$ (fermionic) and $\omega_l$ (bosonic) by
\begin{equation}\label{korr0}
\Sigma_\tau^{\scriptstyle\rm C}(i\omega_\nu) = -\frac{U^2T^2}{Z}
\sum\limits_{l,\lambda} g_{\overline{\tau}}(i\omega_\lambda+i\omega_l)
g_{\overline{\tau}}(i\omega_\lambda)g_\tau(i\omega_\l+i\omega_\nu)
\ .
\end{equation}
Here, the index $\overline{\tau}$ stands for the {\em other}
sublattice, i.e.\ for $A$ if $\tau=B$ and vice versa.
By performing the Matsubara sum one obtains the convolution
\begin{eqnarray}\nonumber
N_{\Sigma_\tau}(\omega)
& = &\frac{U^2}{Z}\int\limits_{-\infty}^\infty \int\limits_{-\infty}^\infty
N_{\overline{\tau}}(\omega'') N_{\overline{\tau}}(\omega''-\omega')
N_\tau(\omega-\omega') \cdot \hspace{0.5cm}
\\ && \hspace{0.5cm} \label{korr4}
\left[ f_{\scriptstyle\rm F}(\omega'-\omega)
f_{\scriptstyle\rm F}(-\omega'')f_{\scriptstyle\rm F}(\omega''-\omega')+
f_{\scriptstyle\rm F}(\omega-\omega')
f_{\scriptstyle\rm F}(\omega'')
f_{\scriptstyle\rm F}(\omega'-\omega'') \right]
d\omega' d\omega''
\end{eqnarray}
for the spectral function $N_{\Sigma_\tau}(\omega)$
belonging to $\Sigma_\tau^{\scriptstyle\rm C}(\omega)$.
The convolution can be expressed most conveniently in the
Fourier transforms
\begin{mathletters}
\label{four1}
\begin{eqnarray}\label{four1a}
\widetilde N^\pm(t)&:=&\int\limits_{-\infty}^\infty
\exp{(-i\omega t)} N(\pm\omega) f_{\scriptstyle\rm F}(-\omega)
\\ \label{four1b}
\widetilde N(t)&:=&\int\limits_{-\infty}^\infty \exp{(-i\omega t)} N(\omega)
\ .
\end{eqnarray}
\end{mathletters}
Eq.\ (\ref{korr4}) becomes as simple as $\widetilde N_{\Sigma_\tau}(t) =
\frac{U^2}{Z} \left[\left.\widetilde N_{\overline{\tau}}^+
\widetilde N_\tau^+\widetilde N_{\overline{\tau}}^-\right|_t
+ \left.\widetilde N_{\overline{\tau}}^-\widetilde N_\tau^-
\widetilde N_{\overline{\tau}}^+
\right|_{-t}\right]$. In sums and differences one obtains
\begin{mathletters}\label{four4}
\begin{eqnarray}
\label{four4a}
\widetilde N_{\Sigma}(t) &=&
\frac{U^2}{Z} \left[\left. \{(\widetilde N_{\scriptstyle\rm S}^+)^2-
(\widetilde N_{\scriptstyle\rm D}^+)^2\}
\widetilde N_{\scriptstyle\rm S}^-\right|_t
+ \left. \{(\widetilde N_{\scriptstyle\rm S}^+)^2-
(\widetilde N_{\scriptstyle\rm D}^+)^2\}
\widetilde N_{\scriptstyle\rm S}^+\right|_{-t}\right]
\\ \label{four4b}
\widetilde N_{\Delta}(t) &=&
-\frac{U^2}{Z} \left[\left. \{(\widetilde N_{\scriptstyle\rm S}^+)^2-
(\widetilde N_{\scriptstyle\rm D}^+)^2\}
\widetilde N_{\scriptstyle\rm D}^-\right|_t
+\left. \{(\widetilde N_{\scriptstyle\rm S}^+)^2-
(\widetilde N_{\scriptstyle\rm D}^+)^2\}
\widetilde N_{\scriptstyle\rm D}^+\right|_{-t}\right]
\ .\end{eqnarray}
\end{mathletters}
The complete self-energy $\Sigma$ and $\Delta$
are given by the following inverse Fourier transforms
\begin{mathletters}\label{four5}
\begin{eqnarray} \label{four5a}
\Sigma(\omega+0i)&=& -i\int\limits_0^\infty \exp{(i\omega t-0t)}
\widetilde N_{\Sigma}(t) dt
\\ \label{four5b}
\Delta(\omega+0i)&=& Ub -i\int\limits_0^\infty \exp{(i\omega t-0t)}
\widetilde N_{\Delta}(t) dt\ .
\end{eqnarray}
\end{mathletters}
In (\ref{four5b}) the Hartree part has been added.
So far, no assumptions concerning the DOS entered. The
formulae hold for all fillings. At the particular value
of half-filling the additional symmetries
$N_{\scriptstyle\rm S}(\omega)=N_{\scriptstyle\rm S}(-\omega)$,
$N_{\scriptstyle\rm D}(\omega)=-N_{\scriptstyle\rm D}(-\omega)$,
$N_{\Sigma}(\omega)=N_{\Sigma}(-\omega)$ and
$N_\Delta(\omega)=-N_{\Delta}(-\omega)$ can be
exploited.
The fact that the spectral densities
are real tells us that $\widetilde N(-t)$ is the
complex conjugate (c.c.) of $\widetilde N(t)$.
Thus (\ref{four4}) simplifies at half-filling to
\begin{mathletters}\label{four7}
\begin{eqnarray}\label{four7a}
\widetilde N_{\Sigma}(t) &=&
\frac{U^2}{Z} \left[\left. \{(\widetilde N_{\scriptstyle\rm S}^+)^2-
(\widetilde N_{\scriptstyle\rm D}^+)^2\}
\widetilde N_{\scriptstyle\rm S}^+\right|_t
+{}\mbox{c.c.} \right]
\\ \label{four7b}
\widetilde N_{\Delta}(t) &=&
\frac{U^2}{Z} \left[\left. \{(\widetilde N_{\scriptstyle\rm S}^+)^2-
(\widetilde N_{\scriptstyle\rm D}^+)^2\}
\widetilde N_{\scriptstyle\rm D}^+\right|_t
-{}\mbox{c.c.} \right]
\ .\end{eqnarray}
\end{mathletters}
This terminates the set up of the equations which have to be
solved self-consistently on the one-particle level.
For those who intend to implement these equations or similar ones
some remarks on the numerical realization are in order.
As usual, the self-consistent set of equations is solved by
iteration. At $T=0$ it is favorable to use a relaxed iteration.
This means that the self-energy $\Sigma$ and $\Delta$ from the
$n$-th and from the $n+1$ iteration are averaged and used
for the subsequent calculation instead of using only the $n+1$ iteration.
This procedure damps oscillatory deviations from the fixed point
more rapidly. It is even more advantageous to let the programme
decide whether relaxed or non relaxed iteration converges faster.
The Fourier transformation is the most time consuming step.
The best algorithm for this task is the so called Fast Fourier
Transformation (FFT). The extremely large number of points,
which can be used with the FFT, overcompensates the disadvantage
of an equidistant mesh which cannot be adapted to regions where
the DOS changes
rapidly \cite{halvo94}.
In the AB-CDW $2^{19}$ points were used. The vectorization
on a IBM3090 still permitted to do one iteration step comprising
four FFT in 19 seconds. A very good precision could be achieved.
The sum rules
\begin{mathletters}\label{sum}
\begin{eqnarray}\label{suma}
\int\limits_0^\infty N_{\Sigma}(\omega)d\omega &=& \frac{U^2}{Z}\frac{1}{2}
\left(\frac{1}{4} -b^2\right)
\\ \label{sumb}
\int\limits_0^\infty N_{\Delta}(\omega)d\omega &=& \frac{U^2}{Z} b
\left(\frac{1}{4} -b^2\right)
\end{eqnarray}
\end{mathletters}
are preserved up to $10^{-6}$. Note that (\ref{sumb}) holds
only at $T=0$ whereas (\ref{suma}) holds for all temperatures.
In order to achieve the high precision also at $T=0$, it is
necessary to discretize the DOS carefully. At the gap edges
the DOS displays inverse square root divergences
$a/\sqrt{\omega-\omega_{\Delta}}$. The parameters $a$ and $\omega_{\Delta}$
are determined directly from the self-energy using (\ref{green3}).
The diverging part of the DOS is discretized by using the
average value in the interval
$[\omega_i-\delta\omega/2,\omega_i+\delta\omega/2]$ instead of the
DOS value at $\omega_i$.
Once the Fourier transforms are essentially linear one as to avoid
a non-linear time loss
in the calculation of the complex free Green function
$g_0(z)$. Therefore, the integration from the Hilbert representation
must be avoided. This is done by using the approximate expression
\begin{eqnarray}
N_3(\varepsilon) &\approx&
\frac{1}{\pi} \left[
\left\{ \frac{13033}{29088}+\frac{8675}{174528}\varepsilon^2 \right\}
\sqrt{6-\varepsilon^2} - {} \right.
\nonumber\\
&& \left\{ \frac{4167}{6464} +\frac{459}{6464\sqrt{6}}\varepsilon
+ \frac{729}{12928}\varepsilon^2 \right\}
\sqrt{2/3-(\varepsilon-2\sqrt{2/3})^2} - {}
\nonumber\\
&&\left. \left\{ \frac{4167}{6464} -\frac{459}{6464\sqrt{6}}\varepsilon
+ \frac{729}{12928}\varepsilon^2 \right\}
\sqrt{2/3-(\varepsilon+2\sqrt{2/3})^2}
\right] \label{rho3}
\ ,\end{eqnarray}
for the three dimensional DOS $N_3(\varepsilon)$. The identities
$h(z;a):= (1/\pi)\int_{-\sqrt{a}}^{\sqrt{a}}
\sqrt{a-\varepsilon^2}/(z-\varepsilon) d\varepsilon =z\pm\sqrt{z^2-a}$
and $(1/\pi)\int_{-\sqrt{a}}^{\sqrt{a}}
\varepsilon \sqrt{a-\varepsilon^2}/(z-\varepsilon)
d\varepsilon = -(a/2)+ z h(z;a)$ permit to
compute $g_0(z)$ for any $z$ quickly.
The r.h.s.\ of (\ref{rho3}) is chosen such that the van-Hove-singularities
are at the right places and such that the first moments (including the 8th)
are reproduced exactly. The relative accuracy achieved is $4\cdot10^{-4}$
for $N_3(0)$ and $10^{-5}$ for the 10th and the 12th moment.
The calculation of the Hartree and of the Fock parts are linear
in the number of discretization points. Concluding the remarks
on the numerical realization we state that all parts of an
iteration step are essentially linear in the number of points used.
This allows a reliable and efficient computation.
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,7)(0,0.7)
\put(-0.7,0){\psfig{file=fig6a.ps,height=7.2cm,width=8cm,angle=270}}
\put(8.3,0){\psfig{file=fig6b.ps,height=7.2cm,width=8cm,angle=270}}
\end{picture}
\caption{Density of states and
spectral function of the self-energy in the
homogeneous phase at $U=2$ and $T=0$ and $T=2$ in $d=3$. For
definitions see eqs.\ (\protect\ref{sumdif}).}
\label{fi:6}
\end{figure}
In fig.\ \ref{fi:6}, results for the DOS and the spectral density
of the self-energy in the homogeneous phase
are shown. The spontaneous symmetry breaking is deliberately
suppressed.
Only positive frequencies are displayed since the
functions are even. At $T=0$, one notes that the imaginary part
of the self-energy tends quadratically to zero for $\omega \to 0$.
From (\ref{korr4}) this follows for all free DOSes with finite
non-singular value at the Fermi edge. Thus the homogeneous low
temperature phase of interacting spinless fermions is a
Fermi liquid. But this phase is thermodynamically unstable (see
below). The DOS still bears signs of the van-Hove-singularities
which are smeared out only a little due to the interaction.
Note that the width is increased by the Fock term. In the free case
the half-width is $\sqrt{6} \approx 2.45$.
High temperatures smear out the minimum of $N_\Sigma$ at $\omega=0$
completely. The solution depicted is stable since at $T=2$
no AB-CDW is possible.
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,15)(0,0.7)
\put(-0.7,7.5){\psfig{file=fig7a.ps,height=7.2cm,width=8cm,angle=270}}
\put(8.3,7.5){\psfig{file=fig7b.ps,height=7.2cm,width=8cm,angle=270}}
\put(-0.7,0){\psfig{file=fig7c.ps,height=7.2cm,width=8cm,angle=270}}
\put(8.3,0){\psfig{file=fig7d.ps,height=7.2cm,width=8cm,angle=270}}
\end{picture}
\caption{Density of states and
spectral function of the self-energy in
the AB-charge density wave at $U=2$ and $T=0$ ($b=0.311005$),
$T=0.225658$ ($b=0.250000$) in $d=3$. For
definitions see eqs.\ (\protect\ref{sumdif}). The sum quantities in
(a) and (b) are even functions of frequency; the difference
quantities in (c) and (d) are odd functions.}
\label{fi:7}
\end{figure}
In fig.\ \ref{fi:7}, stable solutions with $b>0$ are shown. Note the
square root divergence in the DOSes (left column) in the vicinity
of the gap.
At $T=0$ the gap is at $2\omega_\Delta\approx0.6$
whereas the spectral density of the self-energy
becomes finite at about $1.8\approx 6\omega_\Delta$. This results
from the two convolutions involved \cite{halvo94}. They make the
gap in the density of the self-energy to be exactly three times
the gap in the DOS.
Put differently, the finite spectral
density of the self-energy corresponds to
the inelastic scattering of a particle or a hole involving
an additional particle-hole pair. Thus, the necessary minimum energy
is three times the elementary gap.
The physically important implication is the
existence of quasi-particles with energies
between $\omega_\Delta$ and $3\omega_\Delta$
with infinite life-time. Following the arguments of Luttinger
\cite{lutti61} by which he shows that the density of the
self-energy generically
goes like $\omega^2$ at the Fermi edge one comes to the conclusion
that this factor 3 is not an artifact of the approximation but
valid to all orders. Therefore, if the conditions are such that the
the homogeneous phase is a Fermi liquid, i.e.\ Luttinger's
argument holds, a gapped, spontaneously symmetry broken phase
has a factor 3 between the gap in the DOS and the gap in the
self-energy. This implies also the existence of undamped quasi-particles
which have interesting consequences on the transport properties
(see below). The exponent of the
power law with which the imaginary parts of the
self-energy rises at $\omega=3\omega_\Delta$ is $3/2$.
At finite temperatures the energy gap is smaller since the
order parameter has decreased. This effect is visible already
in the Hartree treatment. In addition, the energy gap is smeared out:
thermal fluctuations
represented by the local correlation term $\Sigma^{\scriptstyle\rm C}$
induce a certain spectral weight within
the ``gap'' which does no longer exist in the rigorous sense.
The occurrence of two maxima in $N_\Sigma$ and in
$N_\Delta$ should be noted.
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,15)(0,0.7)
\put(-0.7,7.5){\psfig{file=fig8a.ps,height=7.2cm,width=8cm,angle=270}}
\put(8.3,7.5){\psfig{file=fig8b.ps,height=7.2cm,width=8cm,angle=270}}
\put(-0.7,0){\psfig{file=fig8c.ps,height=7.2cm,width=8cm,angle=270}}
\put(8.3,0){\psfig{file=fig8d.ps,height=7.2cm,width=8cm,angle=270}}
\end{picture}
\caption{Density of states and
spectral function of the self-energy in
the AB-charge density wave at $U=8$ and $T=0$ ($b=0.479312$),
$T=1.509384$ ($b=0.260004$) in $d=3$. For
definitions see eqs.\ (\protect\ref{sumdif}).
At $\omega \approx 12.0$ hardly visible satellite
bands are present in $N_{\scriptstyle\rm S}$ and
$N_{\scriptstyle\rm D}$ for $T=0$. They result from
the imaginary parts of the self-energy around
this frequency.}
\label{fi:8}
\end{figure}
In fig.\ \ref{fi:8}, the generic results for large values
of the interaction are shown. At $T=0$ the factor $3$
between the gap in the DOSes and the gap of the spectral
densities of the self-energies is even more easily discernible.
At the finite temperature ($T\approx 1.5$), all the structures
are smeared out; the order parameter is considerably smaller
than at $T=0$: $b=0.260$ at finite $T$ to $b=0.479$ at $T=0$.
The comparison of the spectral weights of the self-energy at zero and at
finite temperature illustrates an important effect.
The correlation term is suppressed by the symmetry breaking.
The larger $b$ the smaller is the area under the curves
in fig.\ \ref{fi:8}(b) and (d). The effect can be understood
quantitatively with the help of the equations (\ref{sum})
which imply that the area under the curves vanishes for $b\to 1/2$.
This leads to the counter-intuitive
effect that the significance of the
correlation term decreases on increasing interaction at $T=0$
albeit it is quadratic in the interaction
In fig.\ \ref{fi:8}, hardly discernible
satellite bands exist at $\omega\approx12$. They are
engendered by the finite imaginary part of the self-energy
at these energies (see fig.\ \ref{fi:8}(b) and (d)).
To demonstrate that there are in fact infinitely many
satellite bands
with exponentially decreasing weights, the densities
$N_{\scriptstyle\rm S}$ and $N_\Sigma$ are
plotted logarithmically in fig.\ \ref{fi:9}.
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,7)(0,0.7)
\put(8.3,0){\psfig{file=fig9.ps,height=7.2cm,width=8cm,angle=270}}
\end{picture}
\caption{Density of states $N_{\scriptstyle \rm S}$
(short dashed curve)
and spectral function $N_\Sigma$ (long dashed curve)
in the AB-CDW at $U=8$
in logarithmic scale. The difference quantities are not shown
since their values lie only slightly under those of the sum
quantities.}
\label{fi:9}
\end{figure}
The
principal band of the DOS consists of quasi-particles with infinite
life-time at $\omega_\Delta\approx 4$. The satellite bands
correspond to peaks in the spectral density of the self-energy.
The satellite bands are located at $(2m+1)\omega_\Delta$ where $m$ is
an integer. The peaks in the spectral density of the self-energy
are located at $(2m+1)\omega_\Delta$ where $m$ is
an integer but {\em not} $0$ or $-1$. This phenomenon is
generic for the self-consistent solution of a system of
equation comprising convolutions of strongly peaked
functions. It appears only at large values of $U$
because it is necessary that $\omega_\Delta\approx U/2$ is larger
than the band width in order to resolve the peaks.
Note that according to (\ref{green3}), a large value
of $\Delta$ induces band narrowing. Whereas the principal
band is $\sqrt{6}$ wide at $U=0$, its width is shrunk to about unity
in fig.\ \ref{fi:8}(a).
For detailed numerical results on the order parameter
as function of interaction and of temperature as well as on
the critical temperature the reader is referred to
ref.\ 6. The asymptotic behavior at small
$U$ is discussed analytically by van Dongen
\cite{donge91,donge94b}. In a nutshell, the correlation term
renormalizes the Hartree results for $b$ and $T_{\scriptstyle\rm c}$
by a constant factor of order unity which tends to unity
for $d\to \infty$.
\section{Conductivity: Foundations}
Due to the point symmetry group of the hypercubic
lattices the conductivity $\sigma(\omega)$ can be
treated as a scalar. Previous one-particle results
showed that the treatment on the level of linear
$1/d$ corrections should yield reasonable results \cite{halvo94}
in $d=3$.
The conductivity
is calculated from a two-particle correlation function.
This will be done here from the current-current
correlation function $\chi^{\scriptstyle\rm JJ}$. The conductivity
comprises two contributions
$\sigma(\omega) = \sigma_1(\omega) +\sigma_2(\omega)$.
The first term depends on the occupation of the
momentum states $\langle {\hat n}_{\bf k}\rangle$
whereas the second term is proportional to
$\chi^{\scriptstyle\rm JJ}(\omega)$ \cite{mahan90}
\begin{mathletters}\label{a2.6}
\begin{eqnarray}\label{a2.6a}
\sigma_1(\omega) &=& \frac{i}{\omega} \int\limits_{\scriptstyle\rm BZ}
\frac{\partial^2\varepsilon({\bf k})}{\partial k_1^2}
\langle {\hat n}_{\bf k}\rangle
\frac{dk^d}{(2\pi)^d}
\\ \label{a2.6b}
\sigma_2(\omega) &=& \frac{i}{\omega}\chi^{\scriptstyle\rm JJ}(\omega)
\ .\end{eqnarray}
\end{mathletters}
The current-currrent correlation function will be
computed including $1/d$ corrections with the help
of the Baym/Kadanoff formalism \cite{baym61,baym62}.
Specific correlation functions are determined from
the general two-particle correlation function
$L(12,1'2')$ via
\begin{equation}\label{baym0}
\chi^{AB} = \int A(1,1')L(12,1'2')B(2,2')d11'22'\ .
\end{equation}
The numbers stand for composite space and time
coordinates (or momentum and frequency coordinates).
The measure $d11'22'$ tells which coordinates are
integrated. The quantities $A$ and $B$ represent the
operators for which the correlation function is computed.
The Bethe-Salpeter equation determines $L(12,1'2')$
implicitly using the kernel (or effective two-particle
interaction) $\Xi(35,46)$ and the Green function
$G(1,2)$
\begin{equation}\label{baym1}
L(12,1'2') = G(1,2')G(2,1') +\int
G(1,3) G(1',4) \Xi(35,46) L(62,52') d3456 \ .
\end{equation}
Like the kernel of the Dyson equation, namely the
self-energy, the kernel $\Xi(35,46)$ of the
Bethe-Salpeter equation is given as functional derivative
with respect to the Green function
\begin{equation}\label{funcder2}
\Xi(35,46) = \frac{\partial\Sigma(3,4)}{\partial G(6,5)}
= \frac{\partial^2\Phi}{\partial G(4,3)\partial G(6,5)}
\ .\end{equation}
Diagrammatically, the functional derivation is the omission
of a propagator line. Applying these steps to the
approximate generating functional $\Phi_{\scriptstyle\rm A}$ in
fig.\ \ref{fi:3} yields the diagrammatic representation
of the Bethe-Salpeter equation (\ref{baym1}) in fig.\
\ref{fi:19}.
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,5.8)(0,0)
\put(1,0.6){\psfig{file=fig10.ps,height=5.8cm,width=16.4cm,angle=270}}
\end{picture}
\caption{Diagrammatic representation
of the Bethe-Salpeter equation
resulting from $\Phi_{\scriptstyle \rm A}$
according to Baym/Kadanoff.
The wavy lines stand for interaction; the solid ones for
fermionic propagators. The direction of the lower propagators
is opposite to the one of the upper propagators.}
\label{fi:19}
\end{figure}
The first diagram with a wavy interaction
line in the upper row stems from the Hartree diagram,
the last diagram in the upper row results from the Fock
diagram. The diagrams in the lower row in fig.\ \ref{fi:19}
are generated by the different possibilities to take out
two propagator lines from the correlation diagram.
Fortunately, the summation in fig.\ \ref{fi:19}
simplifies considerably for the evaluation of the
current-current correlation function $\chi^{\scriptstyle\rm JJ}$. Fig.\
\ref{fi:20}
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,1.7)(0,0)
\put(9,0.6){\psfig{file=fig11.ps,height=1.7cm,width=7.4cm}}
\end{picture}
\caption{Diagrammatic representation
of eq.\ (\protect\ref{baym0}) for the
current-currrent correlation.}
\label{fi:20}
\end{figure}
displays equation (\ref{baym0}). The squares
represent the current vertices
\begin{equation}\label{strver0}
\mbox{J}(1,1') = \delta({\bf k}_1-{\bf k}_{1'})
\delta(\omega_1-\omega_{1'}-\omega)
\frac{\partial \varepsilon}{\partial k_{1,1}}
\ .\end{equation}
Due to symmetry it does not matter for which spatial direction
$\mbox{J}(1,1')$ is calculated; $k_{1,1}$ is one
arbitrarily chosen component.
The crucial property of the current vertex is its oddness
as function of $k_{1,1}$. All interaction terms
which are even in $k_{1,1}$ do not contribute. This is the
case for all the diagrams resulting from the local
correlation in the lower row and for the diagram coming
from the Hartree term since only {\em one} site appears
on either side. Hence, only the geometric series
depicted in fig.\ \ref{fi:21}
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,1.6)(0,0)
\put(0.8,0.6){\psfig{file=fig12.ps,height=1.6cm,width=15.7cm}}
\end{picture}
\caption{Current-currrent correlation with $1/d$ corrections.}
\label{fi:21}
\end{figure}
caused by the non local Fock
term is left.
For comparison: in the infinite dimensional Hubbard model the
simplifications are even more drastic. All vertex corrections
drop out and the current-current correlation function is just the
convolution of two Green functions \cite{khura90}.
Let us call the value of the first diagram in fig.\ \ref{fi:21}
the ``free'' current-current correlation function and let us use the
symbol $\chi^{\scriptstyle\rm JJ}_0$ for it.
In the homogeneous phase one obtains with the help of
(\ref{baym0}), (\ref{strver0}) and of the propagator in ${\bf k}$-space
$(\omega-\Sigma(\omega)-\gamma\varepsilon({\bf k}))^{-1}$
\begin{equation}\label{freihom}
\chi^{\scriptstyle\rm JJ}_0(i\omega_m) =
\frac{4T}{Z} \sum\limits_{\omega_\nu-\omega_\lambda=\omega_m}
\;\int\limits_{\scriptstyle\rm BZ} \frac{\sin^2(k_1)}
{\left(w_{\nu}-\gamma\varepsilon({\bf k})\right)
\left(w_\lambda-\gamma\varepsilon({\bf k})\right)}
\frac{dk^d}{(2\pi)^d}
\ ,\qquad\quad\end{equation}
where $w_{\nu/\lambda}:=i\omega_{\nu/\lambda}-
\Sigma(i\omega_{\nu/\lambda})$.
We focus now on the segments between two wavy lines in fig.\ \ref{fi:21}.
The conservation of energy and of momentum makes
it possible to carry out the sum over all momentums and energies
by considering independent momentums and energies circulating in
each segment. Then the momentum {\em in} a wavy line is the difference
of two adjacent wave vectors ${\bf k}$ and
${\bf k'}$. A second time,
the evenness and the oddness in the components of the wave vector is used
to write for the factor of an interaction line
\begin{eqnarray}
-\frac{U}{d}\sum\limits_{i=1}^d\cos(k_i- k_i')&=&
-\frac{2U}{Z}\sum\limits_{i=1}^d
\left[\sin(k_i)\sin(k_i')+\cos(k_i)\cos(k_i')\right]
\nonumber \\
&\to& -\frac{2U}{Z} \sin(k_1)\sin(k_1')
\ .
\label{well}
\end{eqnarray}
The argument is obvious for one of the border segments and follows
for those in the middle by induction.
At the end one realizes that each segment corresponds to a
factor of $-(U/2)\chi^{\scriptstyle\rm JJ}_0$ which justifies to call
the right side of fig.\ \ref{fi:21} a geometric series
which takes the value
\begin{equation}\label{vollchi}
\chi^{\scriptstyle\rm JJ}(\omega+0i) =
\frac{\chi^{\scriptstyle\rm JJ}_0
(\omega+0i)}{1+U\chi^{\scriptstyle\rm JJ}_0(\omega+0i)/2}
\end{equation}
after analytic continuation. The derivation of a similar
formulae in the AB-CDW is given in appendix B. The results are
cited below.
The momentum integration in (\ref{freihom}) requires a modified
DOS, to be called the conductivity DOS henceforth
\begin{equation}\label{ldich0}
N_{c,0}(\omega) :=
\int\limits_{\scriptstyle\rm BZ}\sin^2(k_1)
\delta(\omega-\varepsilon({\bf k}))\frac{dk^d}{(2\pi)^d}
\ ,\end{equation}
from which we define also the conductivity Green function
$g_{c,0}(z) := \int\limits_{-\infty}^\infty
N_{c,0}(\omega)/(z-\omega) d\omega$.
The conductivity DOS can be simply derived once the DOS
is known. These two functions are related via
\begin{equation}\label{zufo3}
N_0(\omega) = -\frac{2}{\omega}
\frac{\partial N_{c,0}}{\partial\omega}(\omega) \ .
\end{equation}
This relation stems from the fact that one has to replace
one of the $d$ factors $(1/\pi) 1/\sqrt{t^2-\omega^2}$ in the convolution
for the DOS by $(1/\pi) \sqrt{t^2-\omega^2}$ in order to calculate the
conductivity DOS. The derivation uses the representation of
convolutions as products in Fourier space.
Using the definition of the conductivity Green function and
partial fraction expansion it is straightforward to rewrite
(\ref{freihom})
\begin{equation}\label{hom0}
\chi^{\scriptstyle\rm JJ}_0(i\omega_m) = - \frac{4T}{\gamma Z}
\sum\limits_{\omega_\nu-\omega_\lambda=\omega_m}
\frac{g_c(i\omega_\nu/\gamma)-g_c(i\omega_\lambda/\gamma)}
{w_\nu-w_\lambda} \ .
\end{equation}
Analytic continuation of the latter gives the general formula
(eq.\ (14) in ref.\ 26)
for the current-current correlation function in the homogeneous phase.
In the AB-CDW, it is also possible to sum the series in
fig.\ \ref{fi:21} as geometric series. The main difference is
the fact that $2\times2$ matrices instead of scalars are
involved. The details are given in appendix B; the results
\cite{uhrig95c} are
\begin{equation}\label{vollchi2}
\chi^{\scriptstyle\rm JJ}(i\omega_m) = \frac{2}{U}-\frac{2}{U}\frac{1-A_2}{
(1-A_1)(1-A_2)-A^2_3}
\ ,\end{equation}
where the quantities $A_1, A_2$ and $A_3$ are defined by
\begin{mathletters}\label{basis0}
\begin{eqnarray}
A_1(i\omega_m) &=&
\frac{2UT}{Z}\sum\limits_{\omega_\nu-\omega_\lambda=\omega_m}
\nonumber \\
&&\hspace*{-0.8cm} \left[
\frac{(w_\lambda+w_\nu)(g_{c,\scriptstyle\rm S}(i\omega_\nu)-
g_{c,\scriptstyle\rm S}(i\omega_\lambda))-
(\Delta(i\omega_\lambda)+\Delta(i\omega_\nu))
(g_{c,\scriptstyle\rm D}(i\omega_\nu)-g_{c,\scriptstyle\rm D}
(i\omega_\lambda))}
{w_\nu^2-w_\lambda^2-(\Delta^2(i\omega_\nu)-\Delta^2(i\omega_\lambda))}
\right]
\label{basis0a} \\
A_2(i\omega_m) &=&
\frac{2UT}{Z}\sum\limits_{\omega_\nu-\omega_\lambda=\omega_m}
\nonumber \\
&&\hspace*{-0.8cm} \left[
\frac{(w_\lambda-w_\nu)(g_{c,\scriptstyle\rm S}(i\omega_\nu)+
g_{c,\scriptstyle\rm S}(i\omega_\lambda))-
(\Delta(i\omega_\lambda)-\Delta(i\omega_\nu))
(g_{c,\scriptstyle\rm D}(i\omega_\nu)+g_{c,\scriptstyle\rm D}
(i\omega_\lambda))}
{w_\nu^2-w_\lambda^2-(\Delta^2(i\omega_\nu)-\Delta^2(i\omega_\lambda))}
\right]
\label{basis0b} \\
A_3(i\omega_m) &=&
\frac{2UT}{Z}\sum\limits_{\omega_\nu-\omega_\lambda=\omega_m}
\nonumber \\
&&\hspace*{-0.8cm} \left[
\frac{\Delta(i\omega_\lambda)g_{c,\scriptstyle\rm S}(i\omega_\nu)-
w_\lambda g_{c,\scriptstyle\rm D}(i\omega_\nu)+
\Delta(i\omega_\nu)g_{c,\scriptstyle\rm S}(i\omega_\lambda)-
w_\nu g_{c,\scriptstyle\rm D}(i\omega_\lambda)}
{w_\nu^2-w_\lambda^2-(\Delta^2(i\omega_\nu)-\Delta^2(i\omega_\lambda))}
\right]\ . \label{basis0c}
\end{eqnarray}
\end{mathletters}
In complete analogy to the usual Green functions, the
conductivity Green functions are
$g_{c,\scriptstyle\rm S} := (g_{c,\scriptstyle\rm A}+
g_{c,\scriptstyle\rm B})/2$ and
$g_{c,\scriptstyle\rm D} := (g_{c,\scriptstyle\rm A}-
g_{c,\scriptstyle\rm B})/2$, hence
\begin{mathletters}\label{lgreen3}
\begin{eqnarray} \label{lgreen3a}
g_{c,\scriptstyle\rm S}(\omega)&=&
\frac{\omega-\Sigma(\omega)}{\gamma\sqrt{\left(\omega-
\Sigma(\omega)\right)^2 -\Delta^2(\omega)}}
g_{c,0}(\sqrt{w^2-\Delta^2(\omega)}/\gamma)
\\ \label{lgreen3b}
g_{c,\scriptstyle\rm D}(\omega)&=&
\frac{\Delta(\omega)}{\gamma
\sqrt{w^2-\Delta^2(\omega)}}
g_{c,0}(\sqrt{w^2-\Delta^2(\omega)}/\gamma)
\ ,
\end{eqnarray}
\end{mathletters}
which compares to (\ref{green3})
($w$ is short-hand for $\omega-\Sigma(\omega)$).
Now a relation for the dc-conductivity shall be derived.
In order that the limit
$\lim_{\omega\to0} \sigma(\omega)$ exists
\begin{equation}\label{wfsum0}
\chi^{\scriptstyle\rm JJ}(0) =
\int\limits_{\scriptstyle\rm BZ}
\frac{\partial^2\varepsilon}{\partial k_1^2}({\bf k})
\langle {\hat n}_{\bf k}\rangle \frac{dk^d}{(2\pi)^d}
\; =\; \frac{\langle \hat{T}\rangle}{d}
\end{equation}
must hold according to (\ref{a2.6}). The operator $\hat {T}$
stands for the kinetic energy.
Eq.\ (\ref{wfsum0}) implies also the $f$-sum rule
$\int\limits_{-\infty}^\infty (i\chi^{\scriptstyle\rm JJ}/\omega) d\omega =
-\pi \langle \hat{T}\rangle/d$. At the end of
appendix B, it is shown explicitly
that (\ref{wfsum0}) is valid since $A_3$ vanishes at $\omega=0$
and $A_1=-U\langle \hat{T}\rangle/(2 \gamma d)=1-1/\gamma$.
For the dc-conductivity one obtains
\begin{equation}\label{dcltf}
\sigma(0) = i \left.\frac{\partial\chi^{\scriptstyle\rm JJ}}{\partial\omega}
\right|_{\omega=0}
= -\frac{2i \gamma^2}{U}
\left.\frac{\partial A_1}{\partial\omega}\right|_{\omega=0}
\ .\end{equation}
For explicit evaluation it is useful to split
$\sigma_{\scriptstyle\rm dc}(0)$
into a term including retarded and advanced Green functions
$\sigma_{\scriptstyle\rm dc1}$ and
a term including only retarded or advanced Green functions
$\sigma_{\scriptstyle\rm dc2}$ after analytic continuation. This yields
\begin{equation}\label{dcltf1}
\sigma_{\scriptstyle\rm dc1} = \frac{(\gamma)^2}{\pi Z}
\int\limits_{-\infty}^{\infty} \frac{(1-{\mathop{\rm Re}\nolimits}\Sigma)
N_{c,\scriptstyle\rm S}-({\mathop{\rm
Re}\nolimits} \Delta) N_{c,\scriptstyle\rm D}}
{(1-{\mathop{\rm Re}\nolimits}\Sigma)N_\Sigma+
({\mathop{\rm Re}\nolimits} \Delta) N_\Delta}
(-f_{\scriptstyle\rm F}'(\omega)) d\omega
\ ,\end{equation}
where $f_{\scriptstyle\rm F}'(\omega)$ is
the derivative of the Fermi distribution, and
\begin{eqnarray}
\sigma_{\scriptstyle\rm dc2} &=& -\frac{\gamma^2}{\pi Z}
\int\limits_{-\infty}^{\infty} \left.\frac{(1-\Sigma)
\partial_\omega g_{c,\scriptstyle\rm S}-
\Delta \partial_\omega g_{c,\scriptstyle\rm D}}
{(1-\Sigma)(\partial_\omega\Sigma-1)+\Delta
\partial_\omega\Delta}\right|_{\omega+0i}
(-f_{\scriptstyle\rm F}'(\omega)) d\omega
\nonumber \\ \label{dcltf2}
&=& \frac{1}{\pi Z}
\left[1-{\mathop{\rm Re}\nolimits} \int\limits_{-\infty}^{\infty}
\left((\omega-\Sigma)g_{\scriptstyle\rm S}-
\Delta g_{\scriptstyle\rm D}\right)_{\omega+0i}
(-f_{\scriptstyle\rm F}'(\omega)) d\omega \right]
\ .
\end{eqnarray}
In the last expressions, all the Green functions are retarded.
In the homogeneous phase, the contribution (\ref{dcltf1})
is more important than the one in (\ref{dcltf2}). The former diverges
for $T\to 0$ and $\omega\to 0$, the latter does not. In the symmetry
broken AB-CDW, however, both terms turn out to be essential.
Eqs.\ (\ref{vollchi2}), (\ref{basis0}), (\ref{dcltf1}) and (\ref{dcltf2})
are the foundation for the calculation of the conductivity for
zero and for non-zero order parameter. The focus of the present
work is on the AB-CDW. The properties of the conductivity
in the homogeneous phase (e.g.\ Fermi liquid behavior) are presented
in detail in ref.\ 26
where also the influence of the
truncation of the $1/d$ expansion is discussed.
\section{Conductivity: Results}
In this section we present and discuss results which follow
from the general equations derived in the previous section.
All results are calculated at half-filling and for
$d=3$.
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,7)(0,0.7)
\put(8.3,0){\psfig{file=fig14.ps,height=7.2cm,width=8cm,angle=270}}
\end{picture}
\caption{Scaled real part of the dynamic
conductivity Re$\ \sigma(\omega)$ in the non symmetry broken
phase at $U=4.243$ for $T=0.393$ (solid lines),
$T=0.196$ (dashed lines), $T=0.049$ (dashed-dotted lines),
and $T=0.025$ (dotted lines). Main figure:
spinless fermions in $d=3$; inset: Hubbard model in the
non-crossing approximation
(data from Th.~Pruschke).}
\label{fi:res1}
\end{figure}
In fig.\ \ref{fi:res1}, the real part of the
dynamic conductivity is depicted in the non symmetry broken
phase for different temperatures, i.e.\ the occurrence of
a symmetry broken phase at low temperature is discarded
deliberately for the moment.
They are compared with results of Pruschke, Cox, and Jarrell
\cite{prusc93a,prusc93b}
for the half-filled Hubbard model in $d=\infty$, obtained in the
non-crossing approximation. In both cases the interaction
value is $U=4.243$ (in our units)
which is just below the value where the
Mott-Hubbard transition occurs in the Hubbard model \cite{prusc93b}.
For spinless fermions the Drude peak is absolutely dominant.
Its weight is very large. Its width is given by the
imaginary part of the self-energy at the Fermi level
$N_\Sigma(0)$ (see (\ref{basis0}) with $\Delta=0$ or
eq.\ (14) in ref.\ 26)
, i.e.\ the width
is proportional to $T^2$. The shape of the Drude peak
corresponds very well to a lorenzian.
Only at low temperatures a
shoulder emerges. This shoulder is the effect of
interaction induced scattering. The fluctuations are not
particularly strong. It was already shown previously
\cite{uhrig95a} that the average over the $Z$ interaction
partners reduces the relative fluctuations. There is no Mott-Hubbard
transition without symmetry breaking in the spinless fermion model
because an increasing interaction enhances not only the
fluctuations but also the Fock term (absent in the Hubbard
model) which stabilizes the Fermi liquid phase.
These features are particularly obvious in the comparison with
the Hubbard model data. In this model, the Drude peak
is very reduced at all displayed temperatures since much of the
weight is shifted to the peaks induced by the strong
local particle density fluctuations.
Besides the difference
shoulder vs.\ peak it is interesting to note the difference in
energy scales. In the Hubbard model, it is more or less
$U$ which sets the energy at which the peak occurs. This can be
understood as the energetic effect of whether or not an electron
with a different spin is present.
The typical energy for the shoulder is obviously much smaller.
This in turn can be understood in the same way as before but it
has to be taken into account that the number of possible interaction
partners $Z$ leads to a reduction of the relative
fluctuations of the order of
$1/\sqrt{Z}$. This yields an energy
of roughly $1.7$ in the particular example
which is in good agreement with the numerical result.
Due to the nesting at half-filling, the system of spinless fermions
undergoes a transition to a spontaneously broken translation
symmetry for all (positive) values of the interaction
on lowering the temperature. This spontaneously broken
discrete symmetry implies the occurrence of a gap which
grows exponentially $\omega_\Delta \propto \exp(-c/U)$
for low values of the interaction at $T=0$ (see ref.\ 6
and refs.\ therein).
It is visible in the dynamic conductivity\cite{uhrig95c}.
In fig.\ \ref{fi:23},
its growth on decreasing temperature is shown in four snap-shots.
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,15)(0,0.7)
\put(-0.7,7.5){\psfig{file=fig15a.ps,height=7.2cm,width=8cm,angle=270}}
\put(8.3,7.5){\psfig{file=fig15b.ps,height=7.2cm,width=8cm,angle=270}}
\put(-0.7,0){\psfig{file=fig15c.ps,height=7.2cm,width=8cm,angle=270}}
\put(8.3,0){\psfig{file=fig15d.ps,height=7.2cm,width=8cm,angle=270}}
\end{picture}
\caption{Real part of the dynamic
conductivity Re$\ \sigma(\omega)$
in $d=3$ at $U=2.0$ in logarithmic scale. Fig.\ (a)
$T=0.300000$ and $b=0$; fig.\ (b) $T=0.225658$ and $b=0.250000$;
fig.\ (c) $T=0.155286$ and $b=0.299801$; fig.\ (d)
$T=0$ and $b=0.311005$. In fig.\ (d) the $\delta$-peak
at $\omega=0.13312$ is not shown, its weight is 0.062336.}
\label{fi:23}
\end{figure}
In fig.\ \ref{fi:23}(a), $T$ is still above its critical value. No
structure is visible except for the dominant Drude peak already
discussed in fig.\ \ref{fi:res1}.
In figs.\ \ref{fi:23}(b)-(d) the gap is present and
discernible. Its value is
approximately $2\omega_\Delta$ if $\omega_\Delta$ is the value of the
energy gap in the DOS, see figs.\ \ref{fi:7} and \ref{fi:8}.
But there is also
some weight within the gap for $T>0$ since the correlation
contribution blurred already the gap in the DOS.
Note in passing that the f-sum rule can be verified numerically on
the results shown in fig.\ \ref{fi:23} very accurately
(to the fraction of a percent at $T=0$;
to the fraction of a permille in the homogeneous phase).
The Drude peak does not vanish immediately in the AB-CDW.
It becomes smaller and narrower on decreasing temperature.
Its maximum value does not vanish for $T\to 0$ (see below) but
its weight does. In fig.\ \ref{fi:24},
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,7)(0,0.7)
\put(-0.7,0){\psfig{file=fig16a.ps,height=7.2cm,width=8cm,angle=270}}
\put(8.3,0){\psfig{file=fig16b.ps,height=7.2cm,width=8cm,angle=270}}
\end{picture}
\caption{Enlargements
of two frequency intervals for $T=0.0833833$ and
$b=0.310773$. Fig.\ (a) shows details of the Drude peak;
fig.\ (b) the excitonic resonance.}
\label{fi:24}
\end{figure}
two frequency intervals
are shown in detail for a fairly low temperature. Fig.\ \ref{fi:24}(a)
displays the Drude peak again. The interesting feature
is its small width (compared with the width of the Drude
peaks in figs.\ \ref{fi:23}(b) and (c)). It cannot be explained
by a factor of $T^2$ but corresponds to an exponential shrinking
$\exp(-\omega_\Delta/T)$. As already observed in the
one-particle properties, an increasing gap reduces the influence
of the fluctuations.
Fig.\ \ref{fi:24}(b) shows a very interesting feature below the
proper band edge at $\omega\approx 2\omega_\Delta$. This resonance is also
visible in fig.\ \ref{fi:23}(c) whereas the resonance and the
band edge are not resolved at a higher temperature, fig.\
\ref{fi:23}(b). The resonance can very well be approximated by
a lorenzian. At $T=0$, it is also present as a $\delta$-peak
(not shown in fig.\ \ref{fi:23}(d)). It originates from
a zero of the denominator in (\ref{vollchi2}). At $T>0$, only
the real part of the denominator vanishes and its imaginary
part leads to the observed broadening which depends
strongly, namely exponentially, on the temperature.
Physically the resonance can be interpreted as a bound state,
an exciton, between a particle in the upper band and a hole
in the lower band in the reduced Brillouin zone of the AB-CDW.
The energy difference between the position of the exciton and the
band edge is its binding energy. The type of diagrams which
yield the denominator in (\ref{vollchi2}) corroborates the interpretation
as an exciton. The vertical interaction lines stand for the
repeated interaction between particle and hole in the two
propagators involved in the calculation of $\chi^{\scriptstyle\rm JJ}$.
It should be noted that, for instance, for the parameters of
fig.\ \ref{fi:23}(d) about 70\% of the weight of the conductivity
are found in the excitonic resonance (one may not be misled
by the logarithmic scale). This means that the excitonic
effect is not at all a small side effect.
Concluding the part on the dynamic conductivity, we
discuss fig.\ \ref{fi:25}
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,7)(0,0.7)
\put(8.3,0){\psfig{file=fig17.ps,height=7.2cm,width=8cm,angle=270}}
\end{picture}
\caption{Real part of the dynamic conductivity Re$\
\sigma(\omega)$
in $d=3$ at $U=8.0$ for $T=0$ in logarithmic scale. The
$\delta$-distribution is not displayed.}
\label{fi:25}
\end{figure}
which shows results for
a large interaction value $U$. Due to the induced large gap
and due to the narrow
effective band width several
frequency intervals of absorption are well separated.
The peaks are caused by the convolution of the satellite
band presented for the one-particle properties.
Note, however, that the weight of these satellites
decreases rapidly by a factor of 100 from peak to peak.
These small amplitudes render an experimental verification
certainly extremely difficult if not impossible.
Nevertheless, it would be interesting to know whether such satellites
exist. Their existence would support the application
of a self-consistent approximation since the non self-consistent
calculation yields only two peaks besides the $\delta$-peak
which is not shown.
Since the dc-conductivity in absence of symmetry breaking
has been extensively discussed
in ref.\ 26
we will treat here exclusively the case with symmetry breaking.
The result of (\ref{dcltf1}) and (\ref{dcltf2}) is depicted
in fig.\ \ref{fi:26}
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,7)(0,0.7)
\put(-0.7,0){\psfig{file=fig18a.ps,height=7.2cm,width=8cm,angle=270}}
\put(8.3,0){\psfig{file=fig18b.ps,height=7.2cm,width=8cm,angle=270}}
\end{picture}
\caption{Temperature dependence of the dc-conductivity
at $U=1.0$ (fig.\ (a))
and at $U=8.0$ (fig.\ (b)). Below $T=0.026$ in fig.\ (a) and below
$T=0.6$ in fig.\ (b) a fit was used (see main text).}
\label{fi:26}
\end{figure}
for weak and strong interaction \cite{uhrig95c}.
To the right of the cusp the system is in the
non symmetry broken phase. The conductivity is
essentially proportional \cite{uhrig95a} to $T^2$.
On entering the symmetry broken phase with gap, the
conductivity falls drastically since the energy gap
reduces the DOS at the Fermi level.
Surprisingly, however, the conductivity does {\em not} vanish
for $T\to 0$ although the DOS vanishes in this limit.
There is even a very slight uprise of $\sigma_{\scriptstyle\rm dc}$ close
to $T=0$.
This phenomenon is again a manifestation of the suppression
of correlation effects by the energy gap. The DOS is
reduced by a factor of $\exp(-\omega_\Delta/T)$ but so is the
imaginary part of the self-energy in (\ref{basis0}) which
is responsible for the quasi-particle life-time.
These two effects cancel exactly.
Put differently, an exponentially small number of quasi-particles
of exponentially large life-time
carries a constant current (but see discussion below).
It remains an algebraic dependence on $T$
of the dc-conductivity. The constant term and the
linear one can be computed analytically and where
used to complete the curves in fig.\ \ref{fi:26} for
small values of $T$ where the numerical calculation
is no longer precise enough due to extinction.
The limit value $\lim_{T\to 0} \sigma(\omega=0)$ is given
in fig.\ \ref{fi:27}
\begin{figure}[hbt]
\setlength{\unitlength}{1cm}
\begin{picture}(8.2,7)(0,0.7)
\put(8.3,0){\psfig{file=fig19.ps,height=7.2cm,width=8cm,angle=270}}
\end{picture}
\caption{dc-conductivity $\sigma(\omega=0)$ in the
limit $T\to 0$ in logarithmic scale.}
\label{fi:27}
\end{figure}
as function of $U$. As expected
it decreases rapidly for $U\to \infty$. Note the logarithmic
scale.
What do the above findings for $\sigma_{\scriptstyle\rm dc}$ imply
for the existence of a metal-insulator transition?
Seemingly, even spontaneous symmetry breaking
does not suffice to render the system insulating.
But it must be noted that the ``residual''
conductivity $\lim_{T\to 0} \sigma(\omega=0)$
is infinitely fragile: any other
arbitrarily weak scattering mechanism
which does not die out on $T\to 0$
e.g.\ disorder or scattering at the borders of the sample, will take
over. The exponentially vanishing DOS will yield an
exponentially vanishing dc-conductivity.
This is reflected in the exponentially decreasing
width of the Drude peak which, at constant height,
implies an exponentially decreasing weight.
Experimentally, very pure samples might allow to see
the beginning of the plateaus in fig.\ \ref{fi:26}
before the above cited other scattering mechanism
reduce the conductivity. This behavior is in complete analogy
to the one observed for the shear viscosity $\eta(T)$ of
Helium 3 in the B phase \cite{vollh90b}.
In this system like in the system of spinless fermions
in the AB-CDW one observes an exponentially diverging
mean free path since the collision between (quasi-)particles is
suppressed by a gap. In the so-called ``Knudsen regime''
collisions of quasi-particles with the wall
of the container dominate the collisions {\em between}
the quasi-particles.
In Helium 3, one observes a sharp drop below $T_c$
and then the beginning of a plateau
before finally $\eta(T)$ vanishes rapidly.
The theoretical result for the infinite
system predicts a gentle uprise just like the one
we predict in fig.\ \ref{fi:26}. In both cases, a
factor $\exp(-\omega_\Delta/T)$ in the DOS cancels
with the same factor in the scattering rate \cite{vollh90b}.
This interesting analogy underlines the validity of the
results of our $1/d$ approach.
\section{Discussion}
Two main questions are addressed in the present paper:
(i) How one can an infinite
dimensional result be improved by including $1/d$ corrections
in a systematic way? (ii) Which influence does spontaneous
symmetry breaking have on the conductivity?
It turned out that it is highly non trivial to
construct systematic and reasonable approximations to arbitrary
order. This is true already on the conceptual level.
It was argued in detail
that the self-consistent calculation has certain advantages
since it yields thermodynamically consistent and conserving
approximations. The Baym/Kadanoff formalism, however, is
{\em not sufficient} to guarantee an approximation
which is free from obvious
contradictions. It was shown that an inappropriate
approximation may lead to the wrong analytic behavior
of Green functions and self-energies even though the
approximation was derived from a generating functional.
A general theorem was presented which allows to judge
whether wrong analyticity may occur. If
the conditions of the theorem are fulfilled
the appearance of the wrong analyticity is excluded.
This theorem explains a couple of observations
which were made in the last years on the application of
perturbation expansions and/or $1/d$ expansions.
It is used to show that the self-consistent
treatment of $1/d$ corrections for spinless fermions
is a good approximation: it possesses the necessary
analytic behavior.
For $1/d$ corrections in the Hubbard model the presented
theorem makes no statement since the self-energy has already
an imaginary part for $d=\infty$. This does not imply that the
systematic inclusion of $1/d$ corrections for the Hubbard model
is impossible, but one may expect further
difficulties. As a matter of fact, analyticity problems have been
encountered in the first calculations of $1/d$ corrections in the
Hubbard model \cite{georg96}.
It should be stated that the self-consistent
treatment of $1/d$ corrections to any finite order in $1/d$
remains a mean-field theory. As in the $d=\infty$ treatment of
the Hubbard model \cite{janis92a} the mean field is dynamic, i.e.\
it retains a dependence on frequency.
But in the skeleton diagrams, which are considered
in any finite order in $1/d$, only lattice sites of
{\em finite} distance occur. This means that critical
fluctuations are always cut off. In $d=1$, for instance, the inclusion
of $1/d$ corrections reduces the order parameter considerably
\cite{halvo94} but does not destroy the order completely.
In the self-consistent $1/d$ treatment of spinless
fermions, two-particle properties can be reached, too.
In this work, the Bethe-Salpeter equation was set up in general and
solved in the particular case of the conductivity
$\sigma(\omega)$. This was possible for the non symmetry
broken phase as well as for the charge density wave.
The equations were evaluated in $d=3$ since the approximation
should yield the best results for this value of all
experimentally accessible dimensions \cite{halvo94}.
A number of phenomena were described in the $1/d$
expansion which can be compared with other theoretical
predictions or experiments:\\
-- the dynamic conductivity $\sigma(\omega)$ in the
homogeneous phase has a Drude peak. Its width
decreases quadratically in $T$ for small values of $T$.
The dc-conductivity is always finite \cite{uhrig95a}.\\
-- The Drude peak persists in the CDW but its weight
vanishes exponentially $\propto \exp(-\omega_\Delta/T)$, where
$\omega_\Delta$ is the gap in the one-particle spectra.
The height of the Drude peak, however, does {\em not} vanish
since the diverging quasi-particle life-time cancels the
vanishing density of states.\\
-- The real part of $\sigma(\omega)$ displays a band edge
at $\approx 2\omega_\Delta$. The singularity at the edge
is a square root. Just below the edge an excitonic
resonance is situated which is the bound state between a
particle and a hole in the empty and in the full band, respectively.
These bands are created by the spontaneous symmetry breaking.\\
-- For strong interactions the real part of $\sigma(\omega)$
shows exponentially decreasing peaks at
$\omega \approx 2m \omega_\Delta ; m\in \{1,2,3,\ldots \}$,
which reflect the peaks in the one-particle DOS
at $\omega \approx (2m -1) \omega_\Delta$.\\
-- Strictly speaking, there is no metal-insulator transition.
But the Drude weight decays rapidly on $T\to 0$. Finally,
other scattering mechanisms will dominate over
quasi-particle--quasi-particle collisions.
In summary, we conclude that the self-consistent treatment
of $1/d$ corrections describes successfully a large
variety of phenomena since it includes the leading frequency
dependence of the self-energy. It is a generalized and improved
mean-field theory.
\section*{Acknowledgements:}
The author is grateful to D.~Vollhardt
and E.~M\"uller-Hartmann for valuable hints and to Th.\ Pruschke
for the data shown in the inset of fig.\ \ref{fi:res1}.
The author would like to
thank H.~J.~Schulz, V.~Jani\v{s}, P.~G.~J.~van Dongen,
and R.~Vlaming for helpful discussions and the
Laboratoire de Physique des Solides for its
hospitality. Furthermore,
the author acknowledges the financial support of the
Deutsche Forschungsgemeinschaft (SFB 341) and of the
European Community (grant ERBCHRXCT 940438).
|
proofpile-arXiv_065-633
|
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\begin{document}
\title{Anomalous dimensions of operators in polarized deep inelastic
scattering at $O(1/N_{\! f})$}
\author{J.A. Gracey, \\ Department of Mathematical Sciences, \\ University of
Liverpool, \\ P.O. Box 147, \\ Liverpool, \\ L69 3BX, \\ United Kingdom.}
\date{}
\maketitle
\vspace{5cm}
\noindent
{\bf Abstract.} Critical exponents are computed for a variety of twist-$2$
composite operators, which occur in polarized and unpolarized deep inelastic
scattering, at leading order in the $1/\Nf$ expansion. The resulting
$d$-dimensional expressions, which depend on the moment of the operator, are in
agreement with recent explicit two and three loop perturbative calculations. An
interesting aspect of the critical point approach which is used, is that the
anomalous dimensions of the flavour singlet eigenoperators, which diagonalize
the perturbative mixing matrix, are computed directly. We also elucidate the
treatment of $\gamma^5$ at the fixed point which is important in simplifying
the calculation for polarized operators. Finally, the anomalous dimension of
the singlet axial current is determined at $O(1/\Nf)$ by considering the
renormalization of the anomaly in operator form.
\vspace{-19cm}
\hspace{13.5cm}
{\bf LTH 384}
\newpage
\sect{Introduction.}
Our understanding of the structure of nucleons is derived primarily from
experiments where they are bombarded by other nucleons or electrons at high
energies. These deeply inelastic processes are, in general, well understood in
most instances. Current activity, however, centres on examining polarized
reactions due, for example, to the discrepancy observed in results for the spin
of the proton and theoretical predictions by the EMC collaboration, \cite{1}.
Consequently in order to make accurate statements about the data current
theoretical interest has focussed on carrying out higher order perturbative
calculations in the underlying field theory, quantum chromodynamics, (QCD). As
this theory is asymptotically free at high energies, \cite{2}, the coupling
constant is sufficiently small so that perturbative calculations give a good
description of the deep inelastic phenomenology. Indeed unpolarized scattering
is well understood with one and two loop results available for the anomalous
dimensions of the twist-$2$ flavour non-singlet and singlet operators which
arise in the operator product expansion, [3-5]. Moreover, the Dokshitzer,
Gribov, Lipatov, Altarelli, Parisi, (DGLAP), splitting functions, \cite{6},
which are a measure of the probability that a constituent parton fragments into
other partons are known to the same accuracy, \cite{3,5}. The moments of the
scattering amplitudes have also been studied. More recently the three loop
structure has been obtained exactly, for low moments, through huge impressive
analytic computations, both for non-singlet and singlet cases, \cite{7}.
The situation for polarized scattering is less well established. The one loop
anomalous dimensions for the corresponding twist-$2$ (and $3$) operators were
computed by Ahmed and Ross in \cite{8}. However, only recently has the two
loop structure been determined for the flavour singlet operators, \cite{9}.
(The non-singlet polarized dimensions are equivalent to the non-singlet
unpolarized case.) This was checked by Vogelsang in \cite{10,11} by
calculating the splitting functions themselves and then comparing with \cite{9}
by taking the inverse Mellin transform. In this the moment $n$ of the operator
has the conjugate variable $x$ which is the momentum fraction of the parton in
the nucleon. These results have been important for the next to leading order
evolution of the structure functions to low $x$ and $Q^2$ regions, \cite{12}.
(For review articles see \cite{13}.) Crucial in this exercise is the
dependence of the results on the moment of the operators.
To go beyond this two loop picture would require a great amount of
computation based, for example, on the unpolarized results of \cite{7}. One
way of improving our knowledge would be to compute the appropriate quantities
using another approximation. For example, the large $\Nf$ expansion, where
$\Nf$ is the number of quark flavours, has been used to obtain the leading
order coefficients of the anomalous dimensions of the twist-$2$ unpolarized
non-singlet operators to all orders in the perturbative coupling constant,
\cite{14}. The resulting analytic function of $n$ provided a useful check on
the exact low moment non-singlet three loop calculation of \cite{7}. Briefly
the method involved studying the scaling behaviour of the operator at the
non-trivial fixed point in $d$-dimensional QCD. With that success and the
current interest in polarized physics required for exploring new $x$
r\'{e}gimes it is appropriate to apply the large $\Nf$ analysis to study the
dimensions of the underlying (twist-$2$) operators. Aside from providing
$n$-dependent results and getting a flavour of the structure beyond two loops,
it will give at least another partial check on the recent results of [9-11]. In
particular we will determine the critical exponents at $O(1/\Nf)$ which encode
the all orders coefficients of the twist-$2$ polarized singlet operators. As a
prelude we need to study the unpolarized singlet case which will extend the
result of \cite{14}.
The paper is organised as follows. The basic formalism and notation is
introduced in section 2. We review previous work in section 3, including the
technical details of the computation of singlet unpolarized operators anomalous
dimensions. As the treatment of four dimensional objects, such as $\gamma^5$,
is needed we review previous $1/\Nf$ work involving this in section 4. The
remaining sections 5 and 6 are devoted to the application of the results of
earlier sections to polarized operators. In particular the latter section
centers on the treatment of the singlet axial current which is not conserved
due to the chiral anomaly. Future work and perspectives are discussed in
section 7. An appendix gives details of the relation of the $1/\Nf$ exponent
results to the DGLAP splitting functions.
\sect{Background.}
To begin with we review several of the more field theoretic aspects of the
formalism including the role of the critical renormalization group. First, we
recall that critical exponents are fundamental quantities. In experiments and
condensed matter problems dealing with phase transitions they completely
characterize the physics. In order to describe such phenomena one determines
estimates of the exponents by calculating, for example, in the underlying
quantum field theory describing the transition. In practice this means
carrying out a perturbative calculation of the renormalization constants of
the theory, in some renormalization scheme. This information is then encoded
in the corresponding renormalization group functions such as the anomalous
dimensions of the field or the mass. The critical coupling, $g_c$, is
subsequently determined from the $\beta$-function of the theory. It is defined
to be a non-trivial zero of $\beta(g)$. The appropriate critical exponents are
then found by evaluating the anomalous dimensions at the critical coupling. In
practice provided enough terms of the series have been computed, relatively
accurate numerical estimates can be obtained. (Useful background material can
be found in, for example, \cite{15}.)
For the present problem we will examine a fixed point in QCD in $d$-dimensions
and obtain the critical exponents as a function of $d$ and $\Nf$ which
characterize the transition. As indicated these correspond to the
renormalization group equation, (RGE), functions evaluated at $g_c$. Therefore
provided the location of $g_c$ is known in some approximation like $1/\Nf$ one
can decode the information contained in the exponent and determine the
coefficients of the RGE functions for non-critical values of the coupling. In
this large $\Nf$ method it turns out that the structure of $\beta(g)$ is such
that for leading order calculations in $1/\Nf$ only knowledge of the one loop
coefficient is required, \cite{16}. We illustrate these remarks with a general
example, which will set notation for later sections. As we are interested in
the coefficients in the series of an RGE function we denote such a function by
$\gamma(g)$ and define its expansion to be, with explicit $\Nf$ dependence,
\begin{equation}
\gamma(g) ~=~ a_1g ~+~ (a_2\Nf + b_1)g^2 ~+~ (a_3{\Nf}^2 + b_2\Nf + c_1)g^3
{}~+~ O(g^4)
\end{equation}
where the obvious definition for the $O(g^4)$ term is understood. The
coefficients $\{ a_i$, $b_i$, $c_i$, $\ldots \}$ can of course be functions of
other parameters such as colour group Casimirs or $n$. To evaluate (2.1) at a
fixed point we take the general structure of the $\beta$-function to be, in
$d$-dimensions,
\begin{equation}
\beta(g) ~=~ (d-4)g ~+~ A\Nf g^2 ~+~ (B\Nf + C)g^3 ~+~ O(g^4)
\end{equation}
Setting $d$ $=$ $4$ $-$ $2\epsilon$, then there is a non-trivial fixed point,
$g_c$, at
\begin{equation}
g_c ~=~ \frac{2\epsilon}{A\Nf} ~+~ O \left( \frac{1}{\Nf^2} \right)
\end{equation}
So at $O(1/\Nf)$
\begin{equation}
\eta ~\equiv~ \gamma(g_c) ~=~ \frac{1}{\Nf} \sum_{n=1}^\infty \frac{2^na_n
\epsilon^n}{A^n} ~+~ O\left( \frac{1}{\Nf^2} \right)
\end{equation}
where $\eta$ is the corresponding critical exponent. Clearly at leading order
in $1/\Nf$, when $\Nf$ is large, the coefficients $\{ a_i \}$ are accessed. To
determine them one needs to compute $\eta$ directly in the $O(1/\Nf)$ expansion
in $d$-dimensions. This is the aim of the paper for the operators discussed
earlier.
Before reviewing that we make several parenthetical remarks. In assumimg a
$\beta$-function of the form (2.2) we are restricting ourselves to a particular
class of theories which includes QED and QCD. If the two loop term of (2.2)
had been quadratic and not linear in $\Nf$ and likewise the three loop term
cubic in $\Nf$ and so on, then it would not be possible to determine a simple
form for $g_c$ at leading order in $1/\Nf$. Instead an infinite number of
terms of $\beta(g)$ would be required. This would imply a large $\Nf$
expansion would not be possible in that case. This is similar to the large
$N_{\! c}$ expansion of QCD. Then the structure of the $\beta$-function has
this nasty form and it is not easy to study QCD in a critical $1/N_{\! c}$
approach.
The method to compute critical exponents corresponding to the anomalous
dimensions of operators in powers of $1/\Nf$ is based on an impressive series
of papers, [17-19]. In \cite{16,17} the $O(N)$ $\sigma$ model was considered
and the technique has been developed for fermion and gauge theories more
recently, \cite{20,21}. Essentially one studies the theory precisely at the
fixed point $g_c$ where there are several simplifications. First, at $g_c$
there is no mass in the problem so all propagators are massless. Second, the
structure of the (full) propagators can be written down, in the approach to
criticality. Therefore in momentum space a fermion and gauge field will have
the respective propagators $\psi$ and $A_{\mu\nu}$ of the form, in the limit
$k^2$ $\rightarrow$ $\infty$ \cite{16},
\begin{equation}
\psi(k) ~ \sim ~ \frac{Ak \! \! \! /}{(k^2)^{\mu-\alpha}} ~~~,~~~
A_{\mu\nu}(k) ~ \sim ~ \frac{B}{(k^2)^{\mu-\beta}} \left[ \eta_{\mu\nu} ~-~
(1-b) \frac{k_\mu k_\nu}{k^2} \right]
\end{equation}
where $d$ $=$ $2\mu$, $b$ is the covariant gauge parameter and $A$ and $B$ are
amplitudes. The dimensions $\alpha$ and $\beta$ of the fields (in coordinate
space) comprise two pieces. For example,
\begin{equation}
\alpha ~=~ \mu ~-~ 1 ~+~ \half \eta
\end{equation}
where the first term is the canonical dimension of the fermion determined by
ensuring that the kinetic term in the action is dimensionless. The second term
is the critical exponent corresponding to the anomalous dimension of $\psi$ or
the wave function renormalization evaluated at $g_c$. It reflects the effect
radiative corrections have on the dimension of $\psi$.
With (2.5) one can analyse any set of Feynman diagrams in the neighbourhood of
$g_c$ and determine the scaling behaviour of the integral. In particular one
can examine the $2$-point Schwinger Dyson equation at criticality to obtain a
representation of those equations. It turns out that one obtains a set of
self-consistent equations which can be solved to determine $\eta$ analytically
as a function of $d$. Furthermore the approach is systematic in that
$O(1/\Nf^2)$ corrections can be studied too. In \cite{19} this was extended
to $n$-point Green's function. If, for example, one considers the $3$-point
interaction then the exponent or the vertex anomalous dimension is found by
computing the (regularized) set of leading order integrals with (2.5). The
residue of the simple pole of each graph contributes to the anomalous
dimension. We will illustrate these remarks explicitly in the next section.
However, we note that the regularization that is used is obtained by replacing
$\beta$ of (2.5) by $\beta$ $-$ $\Delta$. Here $\Delta$ is assumed to be small
like the $\epsilon$ used in dimensional regularization, \cite{19}.
We conclude this section by recalling another feature of critical theory which
is important in analysing QCD in large $\Nf$. So far the above remarks have
been completely general and summarize the approach taken in other models.
Another common feature is that the theory that underlies a fixed point is not
necessarily the unique model describing the physics. More than one model can
be used to determine the (measured) critical exponents. In this case such
theories are said to be in the same universality class. From a field theoretic
point of view one can use this to simplify large $\Nf$ calculations. For
example, the $O(N)$ $\sigma$ model and $\phi^4$ theory with an $O(N)$
symmetry are equivalent at the $d$-dimensional fixed point where the former is
defined in $2$ $+$ $\epsilon$ dimensions and the latter in $4$ $-$ $\epsilon$,
\cite{15}. The critical exponents computed in either are the same. For QCD
there is also a similar equivalence which has been demonstrated by Hasenfratz
and Hasenfratz in \cite{12}. They showed that as $\Nf$ $\rightarrow$ $\infty$
QCD and a non-abelian version of the Thirring model are equivalent. The
lagrangians of each are, for QCD,
\begin{equation}
L ~=~ i \bar{\psi}^{iI} ( D \! \! \! \! / \psi)^{iI} ~-~ \frac{(G^a_{\mu\nu})^2}{4e^2}
\end{equation}
and
\begin{equation}
L ~=~ i \bar{\psi}^{iI} ( D \! \! \! \! / \psi)^{iI} ~-~ \frac{(A^a_\mu)^2}{2\lambda^2}
\end{equation}
for the non-abelian Thirring model, (NATM), where $1$ $\leq$ $i$ $\leq$ $\Nf$,
$1$ $\leq$ $I$ $\leq$ $N_{\! c}$, $1$ $\leq$ $a$ $\leq$ $N^2_{\! c}$ $-$ $1$,
$D_{\mu \, IJ}$ $=$ $\partial_\mu \delta_{IJ}$ $+$ $T^a_{IJ}A^a_\mu$,
$G^a_{\mu\nu}$ $=$ $\partial_\mu A^a_\nu$ $-$ $\partial_\nu A^a_\mu$ $+$
$f^{abc}A^b_\mu A^c_\nu$, $\psi^{iI}$ is the quark field and $A^a_\mu$ is the
gluon field. The coupling constants of each lagrangian are $e$ and $\lambda$
and are dimensionless in $4$ and $2$ dimensions respectively. The auxiliary
spin-$1$ field of the NATM can be eliminated to produce a $4$-fermi interaction
which is renormalizable in strictly two dimensions. Essentially at a fixed
point it is the interactions which are important and which suggest (2.7) and
(2.8) are equivalent. The quadratic terms serve only to define the canonical
dimensions of the fields and coupling constants as well as establishing various
scaling laws for the exponents. Although the NATM does not appear to contain
the triple and quartic vertices typical of a Yang Mills theory, it was
demonstrated in \cite{21} that as $\Nf$ $\rightarrow$ $\infty$ that they
correctly emerge from the three and four point functions of (2.8) involving a
quark loop. Therefore to compute fundamental critical exponents in QCD at
$O(1/\Nf)$ it is sufficient to use the much simpler theory (2.8) to compute
these as only one interaction arises. We will point out later where the
contributions from $3$-gluon vertices would arise in the exponent calculation
with respect to perturbation theory. Having defined the NATM in which we will
calculate, we define $\beta$ as
\begin{equation}
\beta ~=~ 1 ~-~ \eta ~-~ \chi
\end{equation}
where $\chi$ is the quark gluon vertex anomalous dimension. Also in using a
gauge field in a covariant gauge the ghost sector needs to be added to both
(2.7) and (2.8). However it turns out that at $O(1/\Nf)$ there are no
contributions to any exponent we compute and we therefore omit them here,
\cite{16}.
Finally we state several earlier results which will be required. The
$\beta$-function of QCD is \cite{2,23},
\begin{eqnarray}
\beta(g) &=& (d-4)g + \left[ \frac{2}{3}T(R)\Nf - \frac{11}{6}C_2(G) \right]
g^2 \nonumber \\
&+& \left[ \frac{1}{2}C_2(R)T(R)\Nf + \frac{5}{6}C_2(G)T(R)\Nf
- \frac{17}{12}C^2_2(G) \right] g^3 \nonumber \\
&-& \left[ \frac{11}{72} C_2(R)T^2(R)\Nf^2 + \frac{79}{432} C_2(G) T^2(R)
\Nf^2 ~-~ \frac{205}{288}C_2(R)C_2(G)T(R)\Nf \right. \nonumber \\
&&+~ \left. \frac{1}{16} C^2_2(R) T(R) \Nf
- \frac{1415}{864} C^2_2(G)T(R)\Nf + \frac{2857}{1728}C^3_2(G)
\right] g^4 + O(g^5)
\end{eqnarray}
where the three loop term was given in \cite{24} and the colour group Casimirs
are defined as $\mbox{tr}(T^a T^b)$ $=$ $T(R)\delta^{ab}$, $T^a T^a$ $=$
$C_2(R)$ and $f^{acd} f^{bcd}$ $=$ $\delta^{ab}C_2(G)$. Although in our earlier
ansatz we omitted a constant term in the one loop coefficient its contribution
to $g_c$ does not appear until $O(1/\Nf^2)$ and so
\begin{equation}
g_c ~=~ \frac{3\epsilon}{T(R)\Nf}
+ O \left( \frac{1}{\Nf^2} \right)
\end{equation}
Various basic exponents are known to $O(1/\Nf)$ and we note \cite{15}
\begin{equation}
\eta_1 ~=~ \frac{[(2\mu-1)(\mu-2)+\mu b] C_2(R)\eta^{\mbox{o}}_1}
{(2\mu-1)(\mu-2)T(R)}
\end{equation}
where $\eta$ $=$ $\sum_{i=1}^\infty \eta_i(\epsilon)/\Nf^i$,
$\eta^{\mbox{o}}_1$ $=$
$(2\mu-1)(\mu-2)\Gamma(2\mu)/[4\Gamma^2(\mu)\Gamma(\mu+1)\Gamma(2-\mu)]$ and
\cite{16}
\begin{equation}
\chi_1 ~=~ -~ \frac{[(2\mu-1)(\mu-2)+\mu b] C_2(R)\eta^{\mbox{o}}_1}
{(2\mu-1)(\mu-2)T(R)} ~-~ \frac{[(2\mu-1)+ b(\mu-1)] C_2(G)\eta^{\mbox{o}}_1}
{2(2\mu-1)(\mu-2)T(R)}
\end{equation}
Throughout that paper we will work in an arbitrary covariant gauge. We note
that the physical operators which occur in the operator product expansion have
gauge independent anomalous dimensions and by including a non-zero $b$ this
will give us a minor check on the corresponding exponent calculations. The
combination $z$ $=$ $A^2B$ arises too and
\begin{equation}
z_1 ~=~ \frac{\Gamma(\mu+1)\eta^{\mbox{o}}_1}{2(2\mu-1)(\mu-2)T(R)}
\end{equation}
\sect{Unpolarized operators.}
We illustrate the large $\Nf$ technique by computing the critical exponent of
the simplest operator which arises in the operator product expansion. This is
the twist-$2$ flavour non-singlet operator, \cite{25,3},
\begin{equation}
{\cal O}^{\mu_1 \ldots \mu_n}_{\mbox{\footnotesize{ns}},a} ~=~
i^{n-1} {\cal S} \bar{\psi}^I \gamma^{\mu_1} D^{\mu_2} \ldots D^{\mu_n}
T^a_{IJ} \psi^J - \mbox{trace terms}
\end{equation}
where ${\cal S}$ denotes symmetrization on the Lorentz indices. Although this
has already been treated in $1/\Nf$ in \cite{14} its value forms part of the
flavour singlet calculation detailed later. The full critical exponent
associated with (3.1) is $\eta_{\footnotesize{\mbox{ns}}}^{(n)}$,
\begin{equation}
\eta_{\footnotesize{\mbox{ns}}}^{(n)} ~=~ \eta ~+~ \eta_{\cal O}
\end{equation}
The first piece corresponds in exponent language to the wave function
renormalization of the constituent fields of (3.1). The second part reflects
the renormalization of the operator itself. Although each term of (3.2) is
gauge dependent the combination is gauge independent. In perturbation theory
the renormalization is carried out by inserting (3.1) in some Green's function
and examining its divergence structure. Here we determine the scaling
behaviour of the integrals where ${\cal O}_{\mbox{\footnotesize{ns}}}$ is
inserted in a quark $2$-point function. The two leading order Feynman diagrams
are given in fig 1. With the regularization each graph is evaluated with the
critical propagators (2.5) in $d$-dimensions. As in perturbative calculations
\cite{3} we project the Lorentz indices of the operator into a basis using a
null vector $\Delta_\mu$, with $\Delta^2$ $=$ $0$. (This is not to be confused
with the regularizing parameter $\Delta$ which is a scalar object.) They have
the general form, omitting the external momentum dependence,
\begin{equation}
\frac{X}{\Delta} ~+~ Y ~+~ O(\Delta)
\end{equation}
where $X$ and $Y$ are functions of $d$. The integrals are straightforward to
compute using standard rules for massless integrals. To obtain the leading
order large $\Nf$ contribution $\alpha$ and $\beta$ are replaced by $\mu$ and
$1$ respectively. Following \cite{19} the residue $X$ of each graph contributes
to $\eta_{{\cal O},1}^{(n)}$. In this instance we have for the respective
graphs
\[
\frac{2\mu C_2(R)\eta^{\mbox{o}}_1}{(\mu-2)(2\mu-1)T(R)} \left[ 1 ~-~ b ~-~
\frac{(\mu-1)^3}{(\mu+n-1)(\mu+n-2)} \right]
\]
and
\begin{equation}
\frac{4\mu(\mu-1)C_2(R)\eta^{\mbox{o}}_1}{(\mu-2)(2\mu-1)T(R)}
\sum_{l=2}^n \frac{1}{(\mu+l-2)}
\end{equation}
where we have included a factor of $2$ in the second to account for the
contibution of the mirror image and used the value of $z$, (2.14). Summing the
contributions yields, \cite{14},
\begin{equation}
\eta^{(n)}_{{\footnotesize{\mbox{ns}}},1} ~=~
\frac{2C_2(R)(\mu-1)^2\eta^{\mbox{o}}_1}{(2\mu-1)(\mu-2)T(R)}
\left[ \frac{(n-1)(2\mu+n-2)}{(\mu+n-1)(\mu+n-2)} ~+~ \frac{2\mu}{(\mu-1)}
[\psi(\mu+n-1) \, - \, \psi(\mu)] \right]
\end{equation}
where $\psi(x)$ is the logarithmic derivative of the $\Gamma$-function. We
recall that this result is in agreement with all known perturbative $\MSbar$
results to three loops, \cite{3,7}. In concentrating on the detail for this
operator we will follow the same procedure in the remainder of the paper with
minimal comment.
We now turn to the treatment of the flavour singlet twist-$2$ operators. Before
analysing at the fixed point we need to recall several features of their
perturbative renormalization. First, the operators are, \cite{25,4},
\begin{eqnarray}
{\cal O}^{\mu_1 \ldots \mu_n}_{\mbox{\footnotesize{F}}} &=&
i^{n-1} {\cal S} \bar{\psi}^I \gamma^{\mu_1} D^{\mu_2} \ldots D^{\mu_n}
\psi^J - \mbox{trace terms} \\
{\cal O}^{\mu_1 \ldots \mu_n}_{\mbox{\footnotesize{G}}} &=&
\half i^{n-2} {\cal S} \, \mbox{tr} \, G^{a \, \mu_1\nu} D^{\mu_2}
\ldots D^{\mu_{n-1}} G^{a \, ~ \mu_n}_{~~\nu} - \mbox{trace terms}
\end{eqnarray}
As each operator has the same dimension in four dimensions and quantum numbers
they mix under renormalization. In other words defining the vector ${\cal O}_i$
$=$ $\{ {\cal O}_F, {\cal O}_G \}$ then the bare and renormalized operators are
related by
\begin{equation}
{\cal O}^{\footnotesize{\mbox{ren}}}_i ~=~ Z_{ij}
{\cal O}^{\footnotesize{\mbox{bare}}}_j
\end{equation}
where $Z_{ij}$ is a $2$ $\times$ $2$ matrix of renormalization constants.
Consequently the associated anomalous dimension is a $2$ $\times$ $2$ matrix
$\gamma_{ij}(g)$. It has the following structure, with the $\Nf$ dependence
explicit,
\begin{equation}
\gamma_{ij}(g) ~=~ \left(
\begin{array}{ll}
\gamma^{qq} & \gamma^{gq} \\
\gamma^{qg} & \gamma^{gg} \\
\end{array}
\right)
{}~=~ \left(
\begin{array}{ll}
a_1g + (a_2\Nf + a_3)g^2 & b_1g + (b_2\Nf + b_3)g^2 \\
c_1\Nf g + c_2\Nf g^2 & (d_1\Nf + d_2)g + (d_3\Nf + d_4)g^2 \\
\end{array}
\right)
\end{equation}
where, for example,
\begin{eqnarray}
a_1 &=& 2C_2(R) \left[ 4 S_1(n) ~-~ 3 ~-~ \frac{2}{n(n+1)} \right] ~~~,~~~
b_1 ~=~ -~ \frac{4(n^2+n+2)C_2(R)}{n(n^2-1)} \nonumber \\
c_1 &=& -~ \frac{8(n^2+n+2)T(R)}{n(n+1)(n+2)} ~~~,~~~
d_1 ~=~ \frac{8}{3} T(R) \nonumber \\
a_2 &=& T(R)C_2(R) \left[ \frac{4}{3} ~-~ \frac{160}{9}S_1(n) ~+~
\frac{32}{3}S_2(n) \right. \nonumber \\
&&+~ \left. \frac{16[11n^7+49n^6+5n^5-329n^4-514n^3-350n^2-240n-72]}
{9n^3(n+1)^3(n+2)^2(n-1)} \right] \nonumber \\
b_2 &=& \frac{32C_2(R)T(R)}{3} \left[ \frac{1}{(n+1)^2} ~+~ \frac{(n^2+n+2)}
{n(n^2-1)} \left( S_1(n) ~-~ \frac{8}{3} \right) \right]
\end{eqnarray}
The remaining entries of (3.9) can be found in, for instance, \cite{4,12} and
are not important for the present situation. To compute $\gamma_{ij}(g)$ the
operators are inserted in both quark and gluon $2$-point functions. Various
one loop graphs which occur are illustrated in figs 1 and 2. Clearly from (3.9)
the $\Nf$ dependence is not the same in each term. For example, at $g_c$ the
$\Nf$ dependence of each entry is respectively, $O(1/\Nf)$, $O(1/\Nf)$, $O(1)$
and $O(1)$. Further, in practical applications it is sometimes useful to
compute with the operator eigenbasis of (3.9) which simplifies the RGE
involving $\gamma_{ij}(g)$ and therefore the evolution of the Wilson
coefficients of the operator product expansion. From (3.9) this leads to the
eigenvalues
\begin{eqnarray}
\lambda_\pm &=& \frac{1}{2} ( d_1\Nf ~+~ a_1 ~+~ d_2 ~ \pm ~ \sqrt{A_1}) g
\nonumber \\
&&+~ \frac{1}{2} \left( (a_2+d_3)\Nf ~+~ a_3 ~+~ d_4 ~ \pm ~
\frac{A_2}{2\sqrt{A_1}} \right) g^2 ~+~ O(g^3)
\end{eqnarray}
where
\begin{eqnarray}
A_1 &=& d_1^2 \Nf^2\left[ 1 ~+~ \frac{2(d_4-a_1)}{d_1\Nf} ~+~ \frac{4b_1c_1}
{d^2_1\Nf^2} \right] \nonumber \\
A_2 &=& 2\Nf [ (d_1(d_3-a_2) + 2c_1b_2)\Nf \\
&&+~ (d_2-a_1)(d_3-a_2) + d_1(d_4-a_3) + 2(c_1b_3+c_2b_1) ] \nonumber
\end{eqnarray}
Or evaluating at $g_c$ the related eigenexponents are at leading order in large
$\Nf$
\begin{eqnarray}
\lambda_+ &=& d_1 \Nf g \nonumber \\
\lambda_- &=& \left( a_1 - \frac{b_1c_1}{d_1}\right) g ~+~
\left( a_2 - \frac{b_2c_1}{d_1}\right) g^2\Nf ~+~ O(\Nf^2 g^3)
\end{eqnarray}
Clearly the $\Nf$ dependence in each eigenexponent differs. The eigenoperators
associated with each eigenvalue, $\lambda_\pm$, are a combination of the
original operators. For example, that associated with $\lambda_-$ has
predominant contributions from the fermionic operator (3.6). Likewise
$\lambda_+$ is associated primarily with (3.7).
For the critical point analysis there will be a $2$ $\times$ $2$ matrix of
critical exponents analogous to (3.9) which are computed by inserting the
critical propagators into the diagrams of figs 1 and 2. In addition the graphs
of fig 3 are also of the same order in $1/\Nf$. However in determining the
contribution to $X$ of each of the graphs it turns out that several are trivial
due to the imbalance of the $\Nf$ dependence already mentioned. For instance
the leading order term for $\lambda_+$ arises purely from the tree graph of fig
2. Therefore we take as its entry in $\eta_{ij}$ $\equiv$ $\gamma_{ij}(g_c)$ as
\cite{24},
\begin{equation}
\eta_{\mbox{\footnotesize{GG}},1} ~=~ 2 \epsilon
\end{equation}
Also $\eta_{\mbox{\footnotesize{FG}},1}$ does not need to be evaluated as its
leading order value is given purely by the one loop perturbative result. Next
the contribution from the final graph of fig 2 is identically zero. That is,
with (2.5) the graph is $\Delta$-finite. Therefore the only non-trivial entry
to compute is $\eta_{\mbox{\footnotesize{FF}},1}$. As the non-singlet part has
already been determined this reduces to evaluating the two loop graphs of fig
3. Each is $b$-independent and respectively contribute, for even $n$,
\begin{eqnarray}
&&-~ \frac{\mu(\mu-1)\Gamma(n)\Gamma(2\mu)\eta^{\mbox{o}}_1}
{(\mu-2)(2\mu-1)(\mu+n-1)(\mu+n-2)\Gamma(2\mu-1+n)} \nonumber \\
&&~~~ \times [n(n(n-2) + 2(\mu-2+n)(2\mu-3)+(2\mu-2+n)) + 2(n-2)(\mu+n-1)]
\nonumber
\end{eqnarray}
and
\begin{equation}
\frac{8\mu(\mu-1)\Gamma(n-1)\Gamma(2\mu)C_2(R)\eta^{\mbox{o}}_1}
{(\mu-2)(2\mu-1)\Gamma(2\mu-1+n)T(R)}
\end{equation}
Hence,
\begin{eqnarray}
\eta_{{\footnotesize{\mbox{FF}}},1}^{(n)}
&=& \frac{(\mu-1)C_2(R)\eta^{\mbox{o}}_1}{(2\mu-1)
(\mu-2)T(R)\Nf} \left[ \frac{2(\mu-1)(n-1)(2\mu+n-2)}{(\mu+n-1)(\mu+n-2)}
{}~+~ 4\mu[\psi(\mu-1+n) - \psi(\mu)] \right. \nonumber \\
&&-~ \left. \frac{\mu\Gamma(n-1)\Gamma(2\mu)}{(\mu+n-1)(\mu+n-2)
\Gamma(2\mu-1+n)} \right. \nonumber \\
&&~~~~ \times \left. [(n^2+n+2\mu-2)^2 + 2(\mu-2)(n(n-1)(2\mu-3+2n)
+ 2(\mu-1+n))] \frac{}{} \right]
\end{eqnarray}
We now discuss the structure of $\eta_{ij}$. Unlike the perturbative mixing
matrix $\gamma_{ij}(g)$, $\eta_{ij}$ is triangular. At first sight this would
appear to be inconsistent with perturbation theory. However, at leading order
in $1/\Nf$ the calculation of $\eta_{ij}$ in fact determines the critical
anomalous dimensions of the eigenoperators {\em directly}. This is not
unexpected if one studies the dimensions of (3.6) and (3.7) at $g_c$. There
clearly the canonical dimensions of each operator is different and therefore
there is no mixing. The vanishing of certain graphs of fig 2 is merely a
reflection of this in the large $\Nf$ calculation. This indirect relation
between the exponents of the eigenoperators (3.6) and (3.7) is the reason why
we distinguish the perturbative entries of (3.9) by $q$ and $g$ in contrast to
$F$ and $G$ for the eigenoperators. A further justification of this point of
view comes from the comparison of the coefficients of the $O(\epsilon)$ and
$O(\epsilon^2)$ terms in the expansion of (3.16) with $\lambda_-$ evaluated to
the same order at $g_c$. We have checked that they are in total agreement with
(3.10) for all $n$. A further check is that the anomalous dimension must vanish
at $n$ $=$ $2$. Then the original operator corresponds to a conserved physical
quantity, the energy momentum tensor which has zero anomalous dimension. From
(3.16) it is easy to check that $\eta_{{\footnotesize{\mbox{FF}}},1}^{(2)}$ $=$
$0$.
It is worth commenting on this calculation in relation to the NATM and QCD
equivalence noted earlier, \cite{22}. Clearly $\lambda_-$ and
$\eta_{{\footnotesize{\mbox{FF}}}}$ contain contributions from the insertion of
gluonic operators in a Green's function. However the graphs we evaluate to
obtain $\eta_{{\footnotesize{\mbox{FF}}}}$ involve only (3.6). The resolution
of this apparent inconsistency is obtained by studying the integration of each
quark loop in fig 3 with (2.5) and the $\epsilon$ expansion of the individual
graphs. Clearly in perturbation theory the graphs of fig 1 will contribute to
the one loop renormalization whilst those of fig 3 will give part of the two
loop value of the anomalous dimension. So one would expect the large $\Nf$
graphs to be $O(\epsilon)$ and $O(\epsilon^2)$ respectively. This is not the
case. Studying (3.15) each graph of fig 3 is $O(\epsilon)$ and from (3.16)
their sum is also of this order. The point is that after performing the quark
loop integral and examining the resulting one loop integral, it contains a part
which would correspond to the ordinary perturbation theory two loop value as
well as a piece that corresponds to the final graph of fig 2 which is a
{\em one} loop integral. In other words an effective gluonic operator like
(3.7) emerges naturally in the exponent calculation. In effect we are
confirming in our calculation the equivalence observed in \cite{22} where we
recall that the three and four point gluon interactions were similarly
reproduced by integrating out quark loops.
We conclude this section by giving an indication of the $n$-dependence of at
least the leading order $1/\Nf$ coefficients of higher loop terms in the series
for $\gamma_-(g)$. Having established the correctness of our expansion at two
loops the higher order coefficients are
\begin{eqnarray}
a_3 ~-~ \frac{b_3c_1}{d_1} &=& \frac{2}{9}S_3(n) ~-~ \frac{10}{27}S_2(n) ~-~
\frac{2}{27}S_1(n) ~+~ \frac{17}{72} ~-~
\frac{2(n^2+n+2)^2[S_2(n)+S^2_1(n)]}{3n^2(n+2)(n+1)^2(n-1)} \nonumber \\
&&-~ \frac{2S_1(n)(16n^7+74n^6+181n^5+266n^4+269n^3+230n^2+44n-24)}
{9(n+2)^2(n+1)^3(n-1)n^3} \nonumber \\
&&-~ [100n^{10}+682n^9+2079n^8+3377n^7+3389n^6+3545n^5+3130n^4
\nonumber \\
&&~~~~~ + \, 118n^3-940n^2-72n+144]/[27(n+2)^3(n+1)^4n^4(n-1)]
\end{eqnarray}
and
\begin{eqnarray}
a_4 ~-~ \frac{b_4c_1}{d_1} &=&
\frac{2}{27}S_4(n) ~-~ \frac{10}{81}S_3(n) ~-~ \frac{2}{81}S_2(n) ~-~
\frac{2}{81}S_1(n) ~+~ \frac{131}{1296} \nonumber \\
&&+~ \zeta(3) \left[ \frac{4}{27}S_1(n) - \frac{2}{27n(n+1)} - \frac{1}{9}
- \frac{2(n^2+n+2)^2}{9n^2(n+2)(n+1)^2(n-1)} \right] \nonumber \\
&&-~ \frac{4(n^2+n+2)^2[2S_3(n) + 3S_2(n)S_1(n) + S_1^3(n)]}
{27(n+2)n^2(n-1)(n+1)^2} \nonumber \\
&&+\, 2[S_2(n) + S_1^2(n)] \frac{(16n^7 \! + 74n^6 + 181n^5 + 266n^4
+ 269n^3 + 230n^2 \! + 44n - 24)}{27n^3(n+2)^2(n+1)^3(n-1)}
\nonumber \\
&&-~ 2S_1(n)[88n^{10} + 608n^9 + 1947n^8 + 3405n^7 + 3670n^6 + 3693n^5
\nonumber \\
&&~~~~~~~~~~~~~ + 2973n^4 - 8n^3 - 920n^2 - 48n + 144]
/[81(n+2)^3(n+1)^4n^4(n-1)] \nonumber \\
&&+~ [68n^{13} + 548n^{12} + 1861n^{11} + 2474n^{10} - 817n^9 - 4143n^8
- 1712n^7 \nonumber \\
&&~~~~ - 2871n^6 - 7702n^5 - 2586n^4 + 3136n^3 + 1952n^2 \nonumber \\
&&~~~~ - 288n - 288]/[81(n+2)^4(n+1)^5(n-1)n^5]
\end{eqnarray}
For future reference we list the values of (3.17) calculated for low moments in
table 1. A similar table was produced for the analogous coefficient in the
non-singlet case. It is important to note that all the fractions up to $n$ $=$
$8$ are in {\em exact} agreement with the recent explicit three loop singlet
results of \cite{7}, when allowance is made for different coupling constant
definitions.
\sect{$\gamma^5$.}
To apply the large $\Nf$ method to polarized operators we need to review the
treatment of $\gamma^5$ in perturbation theory and earlier $1/\Nf$
calculations. As is well known one must be careful in arbitrary spacetime
dimensions when $\gamma^5$ or the pseudotensor $\epsilon_{\mu\nu\sigma\rho}$
are present, \cite{26}. The simple reason is that both are purely four
dimensional objects unlike, say, $\gamma^\mu$ and $\eta^{\mu\nu}$ and do not
generalize in the arbitrary dimensional case. Therefore problems will arise in
perturbation theory when one uses dimensional regularization. With this
regularization calculations are performed in $d$ $=$ $4$ $-$ $2\epsilon$
dimensions where the infinities are removed before taking the $\epsilon$
$\rightarrow$ $0$ limit. There are, however, various ways of incorporating
$\gamma^5$ in such calculations, [26-28]. (A review is, for example,
\cite{29}.) The original approach of \cite{26} was to split the $d$-dimensional
spacetime into physical and unphysical complements. In the former subspace
Lorentz indices run from $1$ to $4$ whilst they range over the remaining
dimensions in the latter. So, for example, the $\gamma$-matrices are split into
two components
\begin{equation}
\gamma^\mu ~=~ \bar{\gamma}^\mu ~+~ \hat{\gamma}^\mu
\end{equation}
where the bar, $\bar{}~$, denotes the physical four dimensional spacetime and
the hat, $\hat{}~$, the remaining $(d-4)$-dimensional subspace. Then the
Clifford algebra reduces to
\begin{equation}
\{ \bar{\gamma}^\mu , \bar{\gamma}^\nu \} ~=~ 2 \bar{\eta}^{\mu\nu} ~~,~~
\{ \bar{\gamma}^\mu , \hat{\gamma}^\nu \} ~=~ 0 ~~,~~
\{ \hat{\gamma}^\mu , \hat{\gamma}^\nu \} ~=~ 2 \hat{\eta}^{\mu\nu}
\end{equation}
The anti-commutativity of $\gamma^5$ is not preserved in the full spacetime.
Instead the following relations are used
\begin{equation}
\{ \bar{\gamma}^\mu , \gamma^5 \} ~=~0 ~~,~~
[ \hat{\gamma}^\mu , \gamma^5 ] ~=~ 0
\end{equation}
It is known that these definitions give a consistent method for treating
$\gamma^5$, \cite{28}. Traces involving an odd number of $\gamma^5$'s are
performed via, in our conventions,
\begin{equation}
\mbox{tr} ( \gamma^5 \gamma^\mu \gamma^\nu \gamma^\sigma \gamma^\rho ) ~=~
4 \bar{\epsilon}^{\mu\nu\sigma\rho}
\end{equation}
which acts like a projection into the physical dimensions. Further if (4.4)
occurs in a loop integral where the $\gamma$-matrices are contracted with loop
momenta the integral is performed first and then the Lorentz index contractions
carried out, with the caveat that external momenta are physical, $\hat{p}_\mu$
$=$ $0$.
For high order perturbative calculations this splitting of the algebra is not
always practical, \cite{30}. It would be easier if a $d$-dimensional
calculation could be performed. Such an approach has been introduced in
\cite{30,31} and carried out successfully for $3$-loop calculations in
\cite{30}. The first step there is to replace $\gamma^5$ by
\begin{equation}
\gamma^5 ~=~ \frac{1}{4!} \epsilon_{\mu\nu\sigma\rho} \gamma^\mu \gamma^\nu
\gamma^\sigma \gamma^\rho
\end{equation}
and remove the $\epsilon$-tensor from the renormalization procedure. The
$\gamma$-matrices of (4.5) are treated as $d$-dimensional in the calculation
before projecting to the physical dimension. If two such $\epsilon$-tensors
are present then they can be replaced by a sum of products of $\eta$-tensors
which is treated as $d$-dimensional. One performs the renormalization in a
minimal way as usual to determine the renormalization constants. To complete
the calculation, in relation to the $\MSbar$ scheme, one must introduce a
finite renormalization constant $Z_5$ in addition to the first renormalization
constant in order to restore the Ward identity, \cite{31}.
For the treatment of $\gamma^5$ in the $1/\Nf$ expansion we recall the simple
example of the flavour non-singlet axial current. In \cite{33} the method
outlined above was followed to correctly determine the anomalous dimension of
${\cal O}^{\mu 5}_{\mbox{\footnotesize{ns}}}$ $=$ $\bar{\psi} \gamma^\mu
\gamma^5 \psi$ at $O(1/\Nf)$. First, if one wishes to find the critical
exponent associated with the non-singlet vector current
${\cal O}^\mu_{\mbox{\footnotesize{ns}}}$ $=$ $\bar{\psi} \gamma^\mu \psi$
then it is inserted in a $2$-point function and the residue with respect to
$\Delta$ is determined. The only relevant graph at $O(1/\Nf)$ is the first
graph of fig 1. If we insert the more general non-singlet operator $\bar{\psi}
\Gamma \psi$ then the contribution to the critical exponent from the graph is
\begin{equation}
-~ \frac{ [ \gamma^\nu\gamma^\sigma \Gamma \gamma_\sigma \gamma_\nu ~-~
2\mu(1-b) \Gamma ] \eta^{\mbox{o}}_1}{2(2\mu-1)(\mu-2)T(R)}
\end{equation}
where the square brackets are understood to mean the coefficient of the matrix
$\Gamma$ after all $\gamma$-matrix manipulations have been performed for an
explicit form of $\Gamma$. Therefore for $\Gamma$ $=$ $\gamma^\mu$, (4.6)
gives
\begin{equation}
-~ \frac{[(2\mu-1)(\mu-2) + b \mu] \eta^{\mbox{o}}_1 }{(2\mu-1)(\mu-2)}
\end{equation}
and so with (2.12) and (2.13)
\begin{equation}
\eta_{{\cal O}^\mu_{\mbox{\footnotesize{ns}}}} ~=~ 0
\end{equation}
consistent with the Ward identity in exponent language, \cite{20}. For
${\cal O}^{\mu 5}_{\mbox{\footnotesize{ns}}}$ one performs the $\gamma$-algebra
of (4.6) using (4.2), to give
\begin{equation}
-~ \frac{[(2\mu-9)(\mu-2) + b\mu]\eta^{\mbox{o}}_1}{(2\mu-1)(\mu-2)T(R)}
\end{equation}
Thus
\begin{equation}
\tilde{\eta}_{{\cal O}^{\mu 5}_{\mbox{\footnotesize{ns}}}} ~=~
\frac{8\eta^{\mbox{o}}_1}{(2\mu-1)T(R)}
\end{equation}
where $\tilde{}$ denotes that the object still has to be augmented by the
finite renormalization. As discussed in \cite{33} this does not preserve four
dimensional chiral symmetry and is not consistent with the Ward identity. To
proceed correctly we need to include a finite renormalization constant. In
\cite{33} this was computed to be
\begin{equation}
Z_5 ~=~ 1 ~+~ \frac{C_2(R)\epsilon}{6T(R)\Nf} \hat{\mbox{L}} \left\{
\frac{\ln [1 - 4T(R)\Nf a_{\mbox{\footnotesize{S}}}/(3\epsilon)]}
{B(2-\epsilon,2-\epsilon) B(3-\epsilon,1+\epsilon)} \right\} ~+~
O \left( \frac{1}{\Nf^2} \right)
\end{equation}
where $\hat{\mbox{L}}$ is the Laurent operator which removes non-singular terms
from the expansion of the braces and $B(x,y)$ is the Euler $\beta$-function.
The constant $Z^{\mbox{\footnotesize{ns}}}_5$ is defined from the requirement
that, \cite{30},
\begin{equation}
Z^{\mbox{\footnotesize{ns}}}_5 ~ {\cal R}_{\footnotesize{\MSbar}} ~ \langle
\bar{\psi} \, {\cal O}^{\mu 5}_{\mbox{\footnotesize{ns}}} \, \psi \rangle ~=~
\gamma^5 \, {\cal R}_{\footnotesize{\MSbar}} ~ \langle \bar{\psi} \,
{\cal O}^\mu_{\mbox{\footnotesize{ns}}} \, \psi \rangle
\end{equation}
where ${\cal R}_{\footnotesize{\MSbar}}$ denotes the $R$-operator or
renormalization procedure. In other words the anti-commutativity of $\gamma^5$
is restored by this condition. Using the information in this finite
renormalization together with (4.10) the correct $\MSbar$ anomalous dimension
does emerge to all orders in the coupling at $O(1/\Nf)$.
There are several disadvantages, however, with the form of (4.11). First, it is
not as compact as the $O(1/\Nf)$ exponents that have been produced in earlier
work, \cite{16}. Second by examining (4.11) the result can be simplified since
the construction of $Z^{\mbox{\footnotesize{ns}}}_5$ is in effect equivalent to
the difference of the exponents (4.7) and (4.9) at $O(1/\Nf)$. In other words
the contribution from the finite renormalization to the final $\MSbar$ exponent
is equal to
\begin{equation}
-~ \frac{8\eta^{\mbox{o}}_1}{(2\mu-1)T(R)}
\end{equation}
Thus the sum of (4.11) and (4.13) correctly gives in $\MSbar$
\begin{equation}
\eta_{{\cal O}^{\mu 5}_{\mbox{\footnotesize{ns}}}} ~=~ 0
\end{equation}
Another difficulty with this procedure is that there is a quicker derivation
based on features of the fixed point approach. In perturbation theory the
regularization used is dimensional in contrast to the critical point method.
There the spacetime dimension is fixed and the regularization is analytic as it
is the gluon dimension which is adjusted. The upshot is that, at least for
non-singlet currects, one can use the anti-commutativity of $\gamma^5$ in
$d$-dimensions. Therefore with $\Gamma$ $=$ $\gamma^\mu\gamma^5$ in (4.6)
anti-commuting $\gamma^5$ twice immediately gives the same contribution as
$\Gamma$ $=$ $\gamma^\mu$. Hence the $\MSbar$ result (4.14) follows directly.
We have checked this procedure explicitly for other non-singlet operators such
as $\bar{\psi} \gamma^5 \psi$ and ${\cal S} \bar{\psi} \gamma^5 \gamma^{\mu_1}
D^{\mu_2} \ldots D^{\mu_n} \psi$ by calculating the analogous finite
renormalization constant from a condition similar to (4.12) and observing that
the result agrees with the direct anti-commuting $\gamma^5$ calculation. So,
for example, the unpolarized and polarized non-singlet twist-$2$ operators have
the same anomalous dimensions, (3.5). In other words we have justified the use
of an anti-commuting $\gamma^5$ in non-singlet sectors of calculations.
Although much of the content of this section may appear straightforward, there
is an important lesson in the result (4.11) from \cite{33} for singlet
operators. Then closed quark loops with an odd number of $\gamma^5$ matices
will occur which means quantities like (4.11) will need to be computed. As we
have demonstrated that this is equivalent to the difference in the anomalous
dimensions of the operators of the renormalization condition (4.12) defining
the finite renormalization constant, flavour singlet operators can be handled
in an efficient way. We will come back to this point in a later section.
\sect{Polarized singlet operators.}
We now extend the unpolarized singlet calculation of section 3 to the polarized
case as it is important to compare with recent perturbative calculations
[9-11]. The twist-$2$ operators are \cite{8},
\begin{eqnarray}
{\cal O}_F^{\mbox{\footnotesize{pol}}} &=& i^{n-1} {\cal S} \bar{\psi} \gamma^5
\gamma^{\mu_1} D^{\mu_2} \ldots D^{\mu_n} \psi - \mbox{trace terms} \\
{\cal O}_G^{\mbox{\footnotesize{pol}}} &=& \half i^{n-2} {\cal S}
\epsilon^{\mu_1\alpha\beta\gamma} \, \mbox{tr} \, G_{\beta\gamma} D^{\mu_2}
\ldots D^{\mu_{n-1}} G^{\mu_n}_{~~\, \alpha} - \mbox{trace terms}
\end{eqnarray}
Several features of the computation of the critical exponents will parallel
section 3 such as the triangularity of the mixing matrix and the $\Nf$
dependence of $\gamma^{\mbox{\footnotesize{pol}}}_{ij}(g)$. The essential
difference is the effect $\gamma^5$ has in the two two loop graphs of fig 3
which we focus on here. The contribution from the graphs of fig 1 is the same
as (3.15).
With (4.4) the second graph of fig 3 is $\Delta$-finite and gives no
contribution to
$\eta^{\mbox{\footnotesize{pol}}}_{{\mbox{\footnotesize{FF}}},1}$. For the
other graph one can compute the quark loop in $d$-dimensions before carrying
out the second loop integral, also in arbitrary dimensions. The projection to
four dimensions is made at the end. Adding all pieces we have,
\begin{eqnarray}
\eta^{\mbox{\footnotesize{pol}}}_{{\mbox{\footnotesize{FF}}},1} &=&
\frac{2C_2(R)\eta^{\mbox{o}}_1}{(2\mu-1)(\mu-2)T(R)} \left[
\frac{(n-1)(2\mu+n-1)(\mu-1)^2}{(\mu+n-1)(\mu+n-2)} \right. \\
&&\left. +~ 2\mu(\mu-1) [\psi(\mu-1+n) - \psi(\mu)] ~-~
\frac{\mu(2\mu+n-5)(n+2)\Gamma(n)\Gamma(2\mu)}
{2(\mu+n-1)(\mu+n-2)\Gamma(2\mu+n-2)} \right] \nonumber
\end{eqnarray}
As in section 3, due to the $\Nf$ dependence we have
\begin{equation}
\eta^{\mbox{\footnotesize{pol}}}_{{\mbox{\footnotesize{GG}}},1} ~=~
2\epsilon
\end{equation}
We have checked that the $\epsilon$-expansion of (5.3) agrees with the
anomalous dimension of the predominantly fermionic eigenoperator of the mixing
matrix at two loops, [9-11]. For completeness we note in the notation of (3.9),
\begin{eqnarray}
a_1^{\mbox{\footnotesize{pol}}} &=& 2C_2(R) \left[ 4 S_1(n) ~-~
3 ~-~ \frac{2}{n(n+1)} \right] ~~~,~~~
b_1^{\mbox{\footnotesize{pol}}} ~=~ -~ \frac{4(n+2)C_2(R)}{n(n+1)}
\nonumber \\
c_1^{\mbox{\footnotesize{pol}}} &=& -~ \frac{8(n-1)T(R)}{n(n+1)} ~~~,~~~
d_1^{\mbox{\footnotesize{pol}}} ~=~ \frac{8}{3} T(R) \nonumber \\
a_2^{\mbox{\footnotesize{pol}}} &=& T(R)C_2(R) \left[ \frac{4}{3} ~-~
\frac{160}{9}S_1(n) ~+~ \frac{32}{3}S_2(n) ~+~
\frac{32[10n^4+17n^3+10n^2+21n+9]}{9n^3(n+1)^3} \right] \nonumber \\
b_2^{\mbox{\footnotesize{pol}}} &=& - \, \frac{32(n+2)C_2(R)T(R)}{3n(n+1)}
\left[ S_1(n) ~-~ \frac{8}{3} ~+~ \frac{(n+2)}{(n+1)} \right]
\end{eqnarray}
This agreement, moreover, justifies our treatment of $\gamma^5$ at the fixed
point to be an anti-commuting object whose appearance in closed loops is
treated with (4.4). Finally we deduce
\begin{eqnarray}
\left[ a_3 ~-~ \frac{b_3c_1}{d_1} \right]^{\mbox{\footnotesize{pol}}}
&=& \frac{2}{9}S_3(n) ~-~ \frac{10}{27}S_2(n) ~-~ \frac{2}{27}S_1(n) \,+\,
\frac{17}{72} - \frac{2(n+2)(n-1)[S_2(n)+S^2_1(n)]}{3n^2(n+1)^2} \nonumber \\
&&+~ \frac{2S_1(n)(13n^3-6n^2+2n+3)(n+2)}{9(n+1)^3n^3} \\
&&-~ \frac{[61n^6+83n^5+27n^4+217n^3+68n^2-36n-18]}{27(n+1)^4n^4} \nonumber
\end{eqnarray}
and
\begin{eqnarray}
\left[ a_4 ~-~ \frac{b_4c_1}{d_1} \right]^{\mbox{\footnotesize{pol}}}
&=& \frac{2}{27}S_4(n) ~-~ \frac{10}{81}S_3(n) ~-~ \frac{2}{81}S_2(n) ~-~
\frac{2}{81}S_1(n) ~+~ \frac{131}{1296} \nonumber \\
&&+~ \zeta(3) \left[ \frac{4}{27}S_1(n) - \frac{2}{27n(n+1)} - \frac{1}{9}
- \frac{2(n+2)(n-1)}{9n^2(n+1)^2} \right] \nonumber \\
&&-~ \frac{4(n+2)(n-1)[2S_3(n) + 3S_2(n)S_1(n) + S_1^3(n)]}{27n^2(n+1)^2}
\nonumber \\
&&+~ \frac{2(13n^3 - 6n^2 + 2n + 3)(n+2)[S_2(n) + S_1^2(n)]}{27n^3(n+1)^3} \\
&&-~ \frac{2S_1(n)(49n^5 - 29n^4 + 95n^3 + 41n^2 - 15n - 9)(n+2)}
{81(n+1)^4n^4} \nonumber \\
&&+~ \frac{(19n^8 - 46n^7 + 3n^6 + 195n^5 - 392n^4 - 317n^3 - 32n^2 + 54n
+ 18)}{81(n+1)^5n^5} \nonumber
\end{eqnarray}
In the second column of table 1 we have evaluated (5.6) for low moments as a
check for future three loop calculations. Although we have given exact
fractions for the coefficients at three loops, the numerical values of the
polarized and unpolarized entries do not differ significantly as $n$ increases.
\sect{Singlet axial current.}
Having considered a variety of fermionic operators which are both flavour
non-singlet and singlet we turn to the remaining current. The renormalization
of the singlet axial current ${\cal O}^{\mu 5}_{\mbox{\footnotesize{s}}}$ $=$
$\bar{\psi} \gamma^\mu \gamma^5 \psi$ is somewhat special. Unlike the singlet
vector current the conservation of ${\cal O}^{\mu 5}_{\mbox{\footnotesize{s}}}$
is spoiled at the {\em quantum} level by the chiral anomaly, [34-36].
Consequently under renormalization the composite operator can develop a
non-zero anomalous dimension. By contrast the conservation of the vector
current ensures it has a zero anomalous dimension at all orders in the coupling
constant. Before attacking the problem of computing the $O(1/\Nf)$ exponent
for ${\cal O}^{\mu 5}_{\mbox{\footnotesize{s}}}$ in the $\MSbar$ scheme, it is
worthwhile reviewing the perturbative approach \cite{32} and in particular
\cite{30}. (Other related contributions to the renormalization of the axial
anomaly are [37-39].) In the three loop analysis, \cite{30}, two
renormalization constants are determined in a manner described earlier for
other currents. One is the renormalization constant which removes the
infinities in the usual way but using the standard $\gamma$-algebra and the
definition of $\gamma^5$, (4.5). This renormalization does not preserve the
axial anomaly, in operator form, in four dimensions. To remedy this a second
finite renormalization constant $Z_5^{\mbox{\footnotesize{anom}}}$ is
required. The relevant constraint in the present instance is determined by
ensuring that the operator form of the anomaly, [34-36],
\begin{equation}
\partial_\mu {\cal O}^{\mu 5}_{\mbox{\footnotesize{s}}} ~=~
\frac{T(R)\Nf}{4g} ~ \epsilon^{\mu\nu\sigma\rho} G^a_{\mu\nu} G^a_{\sigma\rho}
\end{equation}
is preserved, leading to, \cite{36,29},
\begin{equation}
Z_5^{\mbox{\footnotesize{anom}}} ~ {\cal R}_{\footnotesize{\MSbar}} ~
\langle A \, \partial_\mu {\cal O}^{\mu 5}_{\mbox{\footnotesize{s}}} \,
A \rangle ~=~ \frac{T(R)\Nf}{4g} \, {\cal R}_{\footnotesize{\MSbar}} ~
\langle A \, \epsilon^{\mu\nu\sigma\rho} G^a_{\mu\nu} G^a_{\sigma\rho} \,
A \rangle
\end{equation}
The large $\Nf$ calculation follows this two stage approach. In other words the
exponent corresponding to the $\MSbar$ anomalous dimension of
${\cal O}^{\mu 5}_{\mbox{\footnotesize{s}}}$ is given by
\begin{equation}
\eta_{\mbox{\footnotesize{s}}} ~=~ \eta ~+~ \eta_{5,\mbox{\footnotesize{s}}}
{}~+~ \eta_5^{\mbox{\footnotesize{fin}}}
\end{equation}
It is straightforward to compute the first graph of fig 3 in $d$-dimensions
with the rules given previously. With (4.9)
\begin{equation}
\eta_1 ~+~ \eta_{5,\mbox{\footnotesize{s}},1} ~=~
\frac{C_2(R)\eta^{\mbox{o}}_1}{T(R)} \left[ \frac{8}{(2\mu-1)} ~-~
\frac{6}{(\mu-1)} \right]
\end{equation}
where the $b$-dependence has cancelled. We have used a split $\gamma$-algebra
here to be consistent with the treatment of closed fermion loops in determining
$\eta_5^{\mbox{\footnotesize{fin}}}$. There the first graphs of fig 1 and 3
will occur as subgraphs. We have checked that the $\epsilon$-expansion of (6.4)
agrees with the three loop result for the same quantity in \cite{30}.
To compute $\eta_5^{\mbox{\footnotesize{fin}}}$ we use the result of section 4.
There with a split $\gamma$-algebra the finite renormalization exponent was
determined from the difference in the anomalous dimensions of the operators
arising in the defining relation. In that case the restoration of the Ward
identity was simple in that the result obtained was equivalent to using a fully
anti-commuting $\gamma^5$ initially and the operators themselves were similar
in nature. For $\eta_5^{\mbox{\footnotesize{fin}}}$ the exponents of
$\partial_\mu{\cal O}^{\mu 5}_{\mbox{\footnotesize{s}}}$ and $G$ $=$
$\epsilon^{\mu\nu\sigma\rho} G^a_{\mu\nu} G^a_{\sigma\rho}$, which are total
derivatives must be determined separately in $d$-dimensions. In detailing that
calculation we focus on
$\partial_\mu{\cal O}^{\mu 5}_{\mbox{\footnotesize{s}}}$ first.
We insert $\partial_\mu{\cal O}^{\mu 5}_{\mbox{\footnotesize{s}}}$ into a gluon
$2$-point function as illustrated in fig 4. For the moment we take the
momentum flow to be $p$ into the left gluon leg and $(p-q)$ out through the
other. This leaves a net flow of $q$ through the operator insertion which is
needed since a non-zero momentum must contract with $\gamma^\mu\gamma^5$ in
momentum space. To simplify the calculation of each integral we differentiate
with respect to $q_\phi$ and contract with $\epsilon_{\lambda\psi\theta\phi}
p^\theta$ where $\lambda$ and $\psi$ are the Lorentz indices of the gluon legs.
Then $q$ is set to zero, \cite{30}. This procedure ensures that part of the
integrals contributing to the renormalization of the operator is projected out.
We have given the $O(1/\Nf)$ diagrams in figs 5 and 6. The former is the one
loop anomaly and with the critical propagators it is $\Delta$-finite.
However, as the remaining graphs represent the higher order corrections the
value of the first graph of fig 5 must be factored off each to leave a formal
sum of terms
\begin{equation}
-~ \frac{6T(R)\Nf (2\mu-1)(\mu-2)}{(\mu-1)} \left[ 1 ~+~ \frac{1}{\Nf}
\left( \frac{X}{\Delta} ~+~ O(1) \right) \right]
\end{equation}
The overall factor is the $d$-dimensional value of the anomaly which is
non-zero in four dimensions. In (6.5) we have included $z_1$ from the
amplitudes of the quark fields which explains the origin of the factor
$(\mu-2)$. The residue $X$ is the value of the $O(1/\Nf)$ part of the
dimension of $\partial_\mu{\cal O}^{\mu 5}_{\mbox{\footnotesize{s}}}$ we
require.
With the momentum flow as indicated we have computed the value of each graph
of fig 6. No graphs have been included where the vertex with the external
gluon is dressed. These graphs together with the vertex counterterm do not
contribute to $X$ as they are $\Delta$-finite in sum. With the critical
propagators only the first two graphs are non-zero and give
\begin{equation}
X ~=~ -~ \frac{C_2(R)\eta^{\mbox{o}}_1}{T(R)} \left[
\frac{[(2\mu-9)(\mu-2) + b\mu]}{(2\mu-1)(\mu-2)} ~+~ \frac{3}{(\mu-1)} \right]
\end{equation}
The remaining graphs are each $\Delta$-finite and we note the colour factors
of the last two graphs are each $C_2(G)$. Recalling the field content of
$\partial_\mu{\cal O}^{\mu 5}_{\mbox{\footnotesize{s}}}$ we have
\begin{equation}
\eta_{\partial {\cal O},1} ~=~ -~ \frac{C_2(R)\eta^{\mbox{o}}_1}{T(R)} \left[
\frac{8}{(2\mu-1)} ~-~ \frac{3}{(\mu-1)} \right]
\end{equation}
The treatment of $G$ is parallel to that just outlined. With the same
projection of momenta the tree graph of fig 5 gives the normalization value of
$(-$ $6)$ analogous to that of (6.5). The relevant graphs are given in fig 7
and we list their respective contributions to $X$ as
\begin{eqnarray}
&-& \frac{C_2(R)\eta^{\mbox{o}}_1}{T(R)} ~~,~~
-~ \frac{[2C_2(R) - C_2(G)]\eta^{\mbox{o}}_1}{T(R)} ~~,~~
\frac{C_2(G)[4\mu^2-6\mu+1+b]\eta^{\mbox{o}}_1}{2(2\mu-1)(\mu-2)T(R)}
\nonumber \\
&-& \frac{C_2(G) [8\mu^2-13\mu+4-\mu (1-b)] \eta^{\mbox{o}}_1 }
{2(2\mu-1)(\mu-2)T(R)}
\end{eqnarray}
Useful in carrying out this calculation was the symbolic manipulation
programme {\sc Form}, \cite{40}. The value of the three loop graph accounted
for the most tedious part of the calculation. However, we made use in part of
results of integrals which arose in the computation of the QCD
$\beta$-function, \cite{41}. This was achieved by computing the dimension of
the composite operator $(G^a_{\mu\nu})^2$ associated with the coupling
constant in a gluon $2$-point function. We have included a non-zero $b$ to
observe its cancellation as a minor calculational check. Although the graphs
involved in computing the dimension of $G$ in fig 7 are similar in topology to
those for $\partial {\cal O}^{\mu 5}_{\mbox{\footnotesize{s}}}$ in fig 6, the
values obtained are somewhat different. For example, the last graphs of each
figure are similar once the loop integral with the singlet current insertion
is performed which leaves a Feynman integral with an effective $G$ insertion.
The difference in the values arises due to the critical propagators used and
the fact that this loop integral changes the dimension of the gluon lines
contracted with it and therefore the nature of the remaining loop integrations.
One check on this is that the leading terms in the $\epsilon$ expansion of
each graph ought to agree. It is easy to observe that the first two values of
(6.8) give the same leading coefficient as the second term of (6.6). Likewise
the remaining two terms of (6.8) are $O(\epsilon)$.
With the field content dimension (2.12) and (2.13), we find
\begin{equation}
\eta_{\mbox{\footnotesize{G}},1} ~=~ -~ \frac{3C_2(R)\eta^{\mbox{o}}_1}{T(R)}
\end{equation}
It is reassuring to note the cancellation of the terms involving $C_2(G)$ again
as the overall $\MSbar$ renormalization of
${\cal O}^{\mu 5}_{\mbox{\footnotesize{s}}}$ at $O(1/\Nf)$ is expected to be
proportional to $C_2(R)$ only.
With (6.7) and (6.9) the finite renormalization exponent is
\begin{equation}
\eta_{5,1}^{\mbox{\footnotesize{fin}}} ~=~ -~ \frac{C_2(R)\eta^{\mbox{o}}_1}
{T(R)} \left[ \frac{8}{(2\mu-1)} ~+~ \frac{3(\mu-2)}{(\mu-1)} \right]
\end{equation}
Therefore
\begin{equation}
\eta_{\mbox{\footnotesize{s}},1} ~=~ -~ \frac{3\mu C_2(R)\eta^{\mbox{o}}_1}
{(\mu-1)T(R)}
\end{equation}
where the cancellation of the terms proportional to $8/(2\mu-1)$ reflects the
non-singlet calculation of section 4. A final check on this relatively simple
result is that it correctly reproduces the large $\Nf$ leading order two and
three loop $\MSbar$ coefficients of \cite{30,32}. This agreement, moreover,
again strengthens the validity of our treatment of $\gamma^5$. Consequently
we deduce, in the notation of (2.1) and our coupling constant conventions,
\begin{equation}
a_4 ~=~ -~ \frac{4C_2(R)}{27} ~~~,~~~
a_5 ~=~ \frac{[9\zeta(3)-7]C_2(R)}{81}
\end{equation}
\sect{Discussion.}
We conclude our study by remarking on possible future calculations in this
area. The natural task to be performed next will be the $O(1/\Nf^2)$
corrections to the non-singlet twist-$2$ operators. Such a calculation would
mimic the determination of the mass operator dimension but would require the
quark dimension $\eta_2$ first. Only the abelian values are available for both
these quantities, \cite{21}. On another front the corrections to (3.14) and
(5.4) are needed. This would parallel the calculation of the QCD
$\beta$-function in $1/\Nf$, \cite{41}. Both these results for the gluonic
operators would give important insight into the $n$-dependence of the higher
order anomalous dimensions and the $x$-behaviour of the DGLAP splitting
functions.
From a more mathematical physics point of view such analyses may become
important for studying the operator content of strictly four dimensional gauge
theories which have (infrared) fixed points, \cite{42,43}. Evaluating the
perturbative anomalous dimension of the composite operator at these points
would be necessary to gain information on the (conformal) field content of the
underlying theory in the perturbatively accessible region. Moreover the
existence of fixed points such as that of Banks and Zaks in QCD, \cite{42}, for
a range of $\Nf$ values have been the subject of recent interest in
supersymmetric theories with various gauge groups and matter content,
\cite{43}. Therefore any information that can be determined from traditional
field theory methods and which sum perturbation theory beyond present low
orders such as $1/\Nf$, could be used to compare estimates of, for example,
critical exponents deduced from exact non-perturbative arguments.
\vspace{1cm}
\noindent
{\bf Acknowledgements.} The author acknowledges support for this work through
a PPARC Advanced Fellowship, thanks Dr D.J. Broadhurst for encouragement and
Drs A. Vogt and J. Bl\"umlein for useful discussion on their work. The tedious
algebra was performed in part through use of {\sc Form}, \cite{39}, and
{\sc Reduce}, \cite{44}.
|
proofpile-arXiv_065-634
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\section*{Introduction}
The role of chaos in the classical Yang-Mills fields has been
examined by several authors, the studies typically being divided into two
r\'{e}gimes. In the first, one studies the full field theory \cite{fft}
and tries to determine such global measures of chaos as the spectrum of
Lyapunov exponents and spatial-temporal correlations. In the second, one
studies the homogeneous or zero-dimensional limit of the problem
\cite{zerod} which admits a more microscopic analysis. Following this approach,
one is led to consider the three dimensional potential $V=x^2y^2+y^2z^2+z^2x^2$
and its simpler two dimensional cousin $V=x^2y^2$. The two dimensional
problem has been independently studied since it is an interesting
dynamical system in its own right. Until Dahlqvist and Russberg showed
otherwise \cite{dahlruss}, it was commonly believed that the classical
motion in this potential was completely chaotic. Although this is not
true, it remains one of the most chaotic potential systems known. It
also serves as a useful example of intermittency \cite{dahl1,dahl2}. Far from
the origin, the motion is confined within one of four channels within
which the problem is adiabatic so that a trajectory behaves in a smooth,
regular manner. Upon exiting the channel, the trajectory undergoes a
burst of strongly irregular motion before re-entering one of the
channels. This form of dynamics, regular behaviour with episodes of
irregularity, is called intermittency and is found in
various physical systems including the classical helium atom
\cite{helium} and the hydrogen atom in a strong magnetic field
\cite{hydrogen}. The first of these is governed by a potential very similar
in form to $x^2y^2$ \cite{eckwin}. The
three dimensional problem shares the properties of strong chaos
and intermittency although this has been less extensively studied.
We will be interested in the requantisation of these potentials,
particularly in their densities of states. As proved by Simon
\cite{simon} and later analysed in greater detail by Tomsovic
\cite{tom}, the two dimensional potential has a discrete quantum
spectrum in spite of having an energetically accessible phase space of
infinite volume. This potential therefore violates
the semiclassical relation that the average number of
quantum energy levels below energy $E$ is proportional to the
volume of energetically accessible classical phase space.
In this paper we discuss a related property of this potential -
the manner in which the average density of states decomposes among the
various irreducible representations (irreps). Normally, the ratio of the
number of states belonging to a given irrep $R$
of dimension $d_R$ is roughly $d_R^2/|G|$ \cite{pavloff,us},
where $|G|$ is the order of the group. There are then
small $\hbar$ corrections depending on the symmetry properties of the irreps
\cite{us}. We will show here that for the potentials mentioned above, the
symmetry ``corrections'' can be anomalously large and in two
dimensions are essentially leading order in their effect.
The relevant symmetry groups for
the two and three dimensional potentials are $C_{4v}$ and the extended
octahedral group respectively (``extended'' because we allow for
inversions as well as rotations.) These groups have 5 and 10 conjugacy
classes of group elements, respectively, and we need to analyse them
all in order to calculate the average density of each irrep. The
method for doing this when there are no channels was discussed in
Ref.~\cite{btu} for reflection operations and \cite{sw} in the context
of the permutation group of symmetric groups. It was then developed in
a more general context in Ref.~\cite{us}. For some of the classes
which appear here, the analysis is a straight-forward application of
this theory. For other classes, however, the channel
effects make it inapplicable and we use a different analysis
based on the adiabatic nature of the Hamiltonian deep in the channels
as introduced in \cite{tom}. In both two and three dimensions, each
channel calculation involves an analysis of the subgroup
which leaves that channel invariant.
The structure of the paper is as follows. In the next section, we
review the formalism used in constructing the average density of states
from approximations of the heat kernels. The approximations are based
on Wigner transforms of the Hamiltonian and of unitary transformations
which correspond to the group elements. This formalism will be used in
the central region of the potential but will be adapted for
application to the channels. In section II we apply this to the two
dimensional potential and show that there are
very strong effects arising from this decomposition
- much stronger than what one would expect for a normal bound potential.
In section III, we verify these results numerically and also point out
the existence of a subtle numerical effect which is only apparent on
doing the symmetry decomposition. In section IV we introduce the three
dimensional generalisation and discuss the Wigner transforms
corresponding to the various group elements. In section V we do the
channel analysis of the three dimensional problem and use this to get
the final results for all classes. In three dimensions,
the channel effects are less dramatic but still introduce
modifications to what one expects for a generic potential.
\section{Formalism}
We will interest ourselves in the smooth average part $\bar{\rho}(E)$
of the density of states, often called the Thomas-Fermi term. There is
also an oscillating part $\rho_{\mbox{osc}}(E)$ given by periodic
orbits \cite{gutz} but we will not discuss this in great detail so in
what follows we suppress the bar on the smooth functions. The
specification of only concerning ourselves with the Thomas-Fermi
term in the density of states is made by invoking $\hbar$ expansions
rather than expansions involving oscillatory
functions of $1/\hbar$. One way to find the Thomas-Fermi density of
states is to work with the partition function (often called the heat
kernel), which is the Laplace transform of the density of states,
\begin{equation} \label{hk}
Z(\beta) = \mbox{Tr}\left(e^{-\beta\hat{H}}\right) = {\cal
L}\left(\rho(E)\right).
\end{equation}
In the presence of a symmetry group, each quantum state will belong to
one specific irreducible representation of that group so we will
consider the heat kernels of each irrep separately,
\begin{equation} \label{phk}
Z_R(\beta) = \mbox{Tr}\left(\hat{P}_Re^{-\beta\hat{H}}\right).
\end{equation}
$\hat{P}_R$ is the projection operator onto the irrep $R$ and
for a discrete group is given by \cite{hamermesh}
\begin{equation} \label{proj}
\hat{P}_R = {d_R \over |G|}\sum_g\chi^*_R(g)\hat{U}(g).
\end{equation}
The sum is over the elements of the group, $|G|$ in number,
$\chi_R(g)$ is the character of group element $g$ in irrep $R$, $d_R$
is the dimension of irrep $R$ and
$\hat{U}(g)$ is the unitary operator corresponding to the element $g$,
\begin{equation} \label{unit}
\langle{\bf r}|\hat{U}(g)|\psi\rangle = \langle
g^{-1}{\bf r}|\psi\rangle = \psi(g^{-1}{\bf r}).
\end{equation}
One standard way to proceed \cite{jbb} is to find the Wigner transform of the
operators $e^{-\beta\hat{H}}$ and $\hat{P}_R$ and integrate them to
evaluate the trace. The Wigner transform $A_W({\bf q,p})$ of a quantum
operator
$\hat{A}$ is a representation of it in classical phase space and is defined as
\begin{equation} \label{wt}
A_W({\bf q,p}) = \int d{\bf x} \left\langle{\bf q} + {{\bf x} \over
2}\right|\hat{A}\left|{\bf q} - {{\bf x} \over
2}\right\rangle e^{-i{\bf p\cdot x}/\hbar}.
\end{equation}
To leading order in $\hbar$, it is valid to replace
$\left(e^{-\beta\hat{H}}\right)_W$ by $e^{-\beta H_W}$ where the
Wigner transform of the quantum Hamiltonian is just the
corresponding classical Hamiltonian.
Traces are simply evaluated in this representation since
\begin{eqnarray}
\mbox{Tr}(\hat{A})
& = & {1\over (2\pi\hbar)^n}\int d{\bf q}d{\bf p}
A_W({\bf q,p}) \nonumber \\
\mbox{Tr}(\hat{A}\hat{B}) & = & {1\over (2\pi\hbar)^n}\int d{\bf q}d{\bf p}
A_W({\bf q,p})B_W({\bf q,p}), \label{trwt}
\end{eqnarray}
where $n$ is the dimension of the system. As we will see below,
na\"{\i}ve application of these formulas may diverge in the channels;
nevertheless, the formalism can be adapted.
In the evaluation of the Wigner transform of the projection operators
(\ref{proj}), we need the Wigner transforms of the unitary
operators $\hat{U}(g)$. This is discussed in detail in Ref.~\cite{us};
the results for all possible group elements in two dimensions are,
\begin{eqnarray}
\left(\hat{U}(I)\right)_W({\bf q,p}) & = & 1 \nonumber \\
\left(\hat{U}(\sigma_i)\right)_W({\bf q,p}) & = &
\pi\hbar\delta(q_i)\delta(p_i) \nonumber \\
\left(\hat{U}(R_\theta)\right)_W({\bf q,p}) & \approx &
{\pi^2\hbar^2\over \sin^2({\theta\over 2})}
\delta(q_1)\delta(q_2)\delta(p_1)\delta(p_2).
\label{res}
\end{eqnarray}
The Wigner transform of the identity operator gives unity; the
transform of a reflection operator gives the delta functions of the
position and momentum corresponding to the symmetry plane; and, the
transform of a rotation gives the delta functions evaluated at the
symmetry axis. (The third result is exact for
$\theta=\pi$, otherwise it has higher order $\hbar$ contributions.)
An additional useful property is that the Wigner transform of the
product of two commuting operators is simply the product of their
respective transforms. Using this, we obtain from (\ref{res}) the
following relations for the three dimensional operators
\begin{eqnarray}
\left(\hat{U}(\sigma_1\sigma_2\sigma_3)\right)_W({\bf q,p}) & = &
\pi^3\hbar^3 \delta({\bf q})\delta({\bf p})\nonumber\\
\left(\hat{U}(\sigma R_\theta)\right)_W({\bf q,p}) & \approx &
{\pi^3\hbar^3\over \sin^2({\theta\over 2})}
\delta({\bf q})\delta({\bf p}).
\label{3dres}
\end{eqnarray}
The first of these says that the transform of the product of three
perpendicular reflections gives delta functions in all coordinates and momenta.
The second says that the transform of a reflection through a plane
times a rotation about the perpendicular axis gives the same delta functions.
In both (\ref{res}) and (\ref{3dres}), the relative power of
$\hbar$ equals the co-dimension of the set of points left invariant
by the group element. We follow Ref.~\cite{us} in constructing
``class heat kernels''
\begin{equation} \label{chk}
Z(g;\beta) = \mbox{Tr}\left(\hat{U}(g)e^{-\beta\hat{H}}\right)
\end{equation}
so that
\begin{equation} \label{dodo}
Z_R(\beta) = {d_R \over |G|} \sum_g\chi^*_R(g)Z(g;\beta).
\end{equation}
The functions defined in Eq.~(\ref{chk}) are ``class
functions''; they do not depend explicitly on the group
element $g$ but only on the class to which it belongs.
\section{The potential $V=\lowercase{x}^2\lowercase{y}^2$}
The equipotential curves of this potential are shown as the light
curves in Fig.~\ref{system}. The symmetry group is $C_{4v}$, the same
as that of the square. It consists of 8 elements: the identity
$\{I\}$; reflections through the channel axes $\{\sigma_x,\sigma_y\}$;
reflections through the diagonal axes $\{\sigma_1,\sigma_2\}$;
rotations by angle $\pi/2$, $\{R_{\pi/2},R_{-\pi/2}\}$; and, rotation
by angle $\pi$, $\{R_\pi\}$. These five sets of elements comprise the
five conjugacy classes. It follows that there are five irreps, four
are one dimensional and one is two dimensional. The character table is
given in Table 1.
We set out to calculate the five heat kernels corresponding to the
five classes. The integral corresponding to the identity is
\begin{equation} \label{Id}
Z(I;\beta) = {1\over(2\pi\hbar)^2} \int dxdydp_xdp_ye^{-\beta H},
\end{equation}
where $H=(p_x^2+p_y^2)/2 + x^2y^2$ is the classical Hamiltonian (and
the Wigner transform of the quantum Hamiltonian). This integral is
extensively discussed in Ref.~\cite{tom} where it is shown that it has
a logarithmic divergence. We return to this point below. According to
Eqs.~(\ref{trwt}) and (\ref{res}), the integral
corresponding to $\sigma_y$ is given by
\begin{eqnarray}
Z(\sigma_y;\beta) & = & {1\over(2\pi\hbar)^2} \int
dxdydp_xdp_ye^{-\beta H}\pi\hbar\delta(y)\delta(p_y) \nonumber\\
& = & \sqrt{1\over{8\pi\beta\hbar^2}}\int dx. \label{refy}
\end{eqnarray}
Since the $x$ integral runs from $-\infty$ to $\infty$, this
integral diverges even more violently than (\ref{Id}). We will also
return to consider this more carefully below.
The remaining three class heat kernels are well behaved. For the
reflection through the diagonal axis
$\sigma_1$, we change variables to $\xi=(x+y)/\sqrt{2}$ and
$\eta=(x-y)/\sqrt{2}$ so that
\begin{eqnarray}
Z(\sigma_1;\beta) & = & {1\over(2\pi\hbar)^2} \int
d\xi d\eta dp_\xi dp_\eta e^{-\beta
H}\pi\hbar\delta(\eta)\delta(p_\eta)
\nonumber\\
& = &
{\Gamma\left({1\over 4}\right)\over 4\sqrt{\pi}}
{1\over\beta^{3/4}\hbar}.
\label{ref1}
\end{eqnarray}
The kernels corresponding to rotations by $\pi/2$ and $\pi$ are
trivial since all integrals are done by delta functions leaving
\begin{equation} \label{rot1}
Z(R_{\pi/2};\beta) = {1\over 2} \hspace{8ex}
Z(R_\pi;\beta) = {1\over 4}.
\end{equation}
We now go back and analyse in greater detail the first two integrals.
The first was studied by Tomsovic \cite{tom} but for completeness and
consistency of notation, we review the calculation.
Deep in one of the channels, $x\gg 1$ for example,
the approximation using the Wigner transforms breaks down. This can be
understood as follows.
This approximation assumes that for short times one is free to
ignore the dynamics so that the calculation involves only the local
value of the Hamiltonian. Usually this is not problematic, however
one finds here that the channel effects violate this assumption. This
is because in one of the channels, $x\gg 1$ for example, we can treat the
problem adiabatically so that in the y
direction there is harmonic motion with a frequency $\omega_x=\sqrt{2}x$.
This frequency becomes arbitrarily large in the channel and there is no
time scale over which the dynamics can be ignored. We overcome this
problem by using an alternate representation of the heat kernels based
on the approximate separation of the problem as introduced in
Ref.~\cite{tom} and which we discuss below. This is a
complementary representation which is valid deep in the channels but fails
near the origin. To proceed, we assume that there is a domain of $x$
where both representations are valid. Let $Q$ be a value of
$x$ in this domain. The condition for the adiabatic representation to
be valid is that we be deep in one of the channels,
in terms of dimensionless quantities this is $\beta^{1/4}Q\gg 1$.
The condition for the Wigner function
representation to be valid is $\beta\hbar Q\ll 1$. These conditions
are compatible if $\beta^{3/4}\hbar\ll 1$. (If we determine all quantities
in units of energy $[E]$, then $Q$ has units of $[E^{1/4}]$, $\beta$ has
units of $[E^{-1}]$ and $\hbar$ has units of $[E^{3/4}]$ so that all the
conditions mentioned above are in terms of dimensionless combinations.)
We will use the Wigner
representation in the square $|x|\leq Q$, $|y|\leq Q$ and the adiabatic
representation elsewhere. We then replace (\ref{Id}) by
\begin{equation} \label{Id_1}
Z_0(I;\beta) = {1\over(2\pi\hbar)^2}{2\pi\over\beta}\int_{-Q}^Qdxdy
e^{-\beta x^2y^2}.
\end{equation}
(We have introduced the subscript 0 to denote that this is the
contribution from the central region around the origin.)
This integral can be done by the change of variables
$u=\sqrt{\beta}xy$ and
$v=x$, so that the integrand is proportional to $\exp(-u^2)/v$. Doing
the $v$ integral first and using $\beta^{1/4}Q\gg 1$, one finds
\begin{equation} \label{Id_1_res}
Z_0(I;\beta) =
\sqrt{{1\over\pi\beta^3\hbar^4}}\left(\log(2\sqrt{\beta}Q^2) +
{\gamma\over 2} \right),
\end{equation}
where $\gamma=0.5772...$ is Euler's constant.
Similarly, for the reflection operator
$\sigma_y$, integration of (\ref{refy}) between the limits $-Q$ and $Q$
leads to
\begin{equation} \label{refy_1}
Z_0(\sigma_y;\beta) = \sqrt{{1\over 2\pi\beta\hbar^2}}Q.
\end{equation}
To do the integrals in the channels we assume a local separation of the
Hamiltonian into a free particle in the $x$ direction and a
harmonic oscillator in the $y$ direction, with a
frequency which depends parametrically on $x$
\begin{equation} \label{separate}
h_x = {1 \over 2}p_y^2 + {\omega_x^2\over 2}y^2.
\end{equation}
Henceforth we will use small letters to denote objects related to the
local Hamiltonian $h_x$. It has eigenenergies $e_n=(n+1/2)\omega_x\hbar$
and eigenstates $|\phi_n\rangle$ which depend parametrically on $x$.
All the symmetry information to do with the channel calculation
is encoded in these local
eigenenergies and eigenstates. In particular, we are interested in the
subgroup of $C_{4v}$ which leaves $x$ invariant and so maps the local
eigenstates onto one another. This subgroup is just the parity group
with group elements $\{I,\sigma_y\}$.
This group has a trivial character table but we include it
for completeness as Table. 2. For fixed $x$, we proceed in analogy to
(\ref{chk}) by defining heat kernels based on the local eigenvalues
and corresponding to these two group operations,
\begin{eqnarray}
z_x(g,\beta) & = &
\mbox{tr}\left(\hat{U}^\dagger(g)e^{-\beta\hat{h}}\right) \nonumber \\
& = & \sum_n \eta_n(g) e^{-\beta e_n} \label{lhk}
\end{eqnarray}
where $g$ is either the identity or the reflection element. The trace
operator ``tr'' denotes the local integral over the $y$ degree of freedom
and can be found by summing over the index $n$. It is clear that the
operator $\hat{U}^\dagger(g)$ is unity when $g=I$ and changes the sign
of the odd states when $g=\sigma_y$, so that $\eta_n(I)=1$ and
$\eta_n(\sigma_y)=(-1)^n$.
To evaluate the full trace, we note that the integrals in $p_y$ and $y$ have
already been done implicitly in (\ref{lhk}) so we only need to do the
$x$ and $p_x$ integrals. Since this is only one dimensional,
the prefactor of the integral has only one power of
$2\pi\hbar$ and we conclude
\begin{eqnarray}
Z_c(g;\beta) & = & f_g{1\over 2\pi\hbar}\int_{-\infty}^\infty dp_x
e^{-\beta p_x^2/2}\int_Q^\infty dx z_x(g,\beta) \nonumber \\
& = & f_g \sqrt{{1\over\pi\beta^3\hbar^4}}
\sum_n \eta_n(g){\xi^{2n+1}\over 2n+1}, \label{stuff}
\end{eqnarray}
where we have defined the factor $\xi=\exp(-\beta\hbar Q/\sqrt{2})$.
(We include a subscript $c$ to denote that this is the channel contribution.)
We have also introduced a factor $f_g$ which represents the number of channels
which map to themselves under the action of the group element $g$.
When working with the identity element, all the channels map onto themselves
and $f_I=4$; when working with the reflection operator the two
such channels along the $x$ axis map onto themselves and $f_{\sigma_y}=2$.
We now make use of the series identities
\begin{eqnarray}
\sum_n{\xi^{2n+1}\over 2n+1} & = & {1\over 2}\log\left({1+\xi\over
1-\xi}\right)
\nonumber\\
\sum_n(-1)^n{\xi^{2n+1}\over 2n+1} & = & \arctan\xi
\label{serid}
\end{eqnarray}
and the fact that $\beta\hbar Q\ll 1$ to conclude
\begin{eqnarray}
Z_c(I;\beta) & = &
-\sqrt{{1\over\pi\beta^3\hbar^4}}\log\left({\hbar^2\beta^2Q^2\over
8}\right)\nonumber\\
Z_c(\sigma_y;\beta) & = & \sqrt{{\pi\over 4\beta^3\hbar^4}} -
\sqrt{{1\over2\pi\beta\hbar^2}}Q.
\label{chann_both_res}
\end{eqnarray}
We add the results from inside the square
(\ref{Id_1_res}) and (\ref{refy_1}) to the channel results
(\ref{chann_both_res}) to get
\begin{eqnarray}
Z(I;\beta) & = &
\sqrt{{1\over 4\pi\beta^3\hbar^4}}
\left(\log\left({1\over\beta^3\hbar^4}\right)+\gamma+8\log2\right)
\nonumber\\
Z(\sigma_y;\beta) & = & \sqrt{{\pi \over 4\beta^3\hbar^4}}. \label{total}
\end{eqnarray}
Note that the $Q$ dependence has cancelled from both results
leaving a finite answer. (This prescription actually overcounts some
regions of phase space but the errors so introduced are
exponentially small in $\beta^{1/4}Q$.)
We have now calculated the five class heat kernels which we need. All
that remains is to compute their inverse Laplace transforms. In fact, we
will not be interested in the densities $\rho(g;E)$ themselves but rather
in their integrals $N(g;E)$ which are given by
\begin{equation} \label{ilt}
N(g;E) = {\cal L}^{-1}\left({Z(g;\beta)\over\beta}\right).
\end{equation}
The inverse Laplace transforms are
\begin{eqnarray}
N(I;E) &\ =\ & {2 \over 3\pi}y^2\left(4\log y + 4\gamma + 14\log2 -
8\right) \nonumber \\[1ex]
N(\sigma_y;E) &\ =\ & {2 \over 3}y^2 \nonumber \\[1ex]
N(\sigma_1;E) &\ =\ & {\Gamma^2({1\over 4}) \over \sqrt{18\pi^3}}y\nonumber\\[1ex]
N(R_{\pi/2};E) &\ =\ & {1 \over 2} \nonumber \\[1ex]
N(R_\pi;E) &\ =\ & {1 \over 4}. \label{hwg}
\end{eqnarray}
We have defined the dimensionless scaled energy $y=E^{3/4}/\hbar$,
which is a semiclassically large quantity. If we explicitly include
the mass $m$ in the kinetic energy of the Hamiltonian and a parameter
$\alpha$ in front of the potential energy then Eq.~(\ref{hwg}) still
applies but with $y=(m^{1/2}E^{3/4})/(\alpha^{1/4}\hbar)$. We further
note that the inverse Laplace transforms imply that all the functions
are zero for negative energies. The first of these relations is the
average integrated density of states summed over all irreps and was
already found by Tomsovic \cite{tom}. To construct the integrated
densities of states for each of the five irreps, we use
Eq.~(\ref{dodo}) with the symbols $Z$ replaced by the symbols $N$. It
should also be mentioned that these are just the leading order results
in an asymptotic semiclassical expansion. The terms in this series
will eventually diverge in a manner controlled by the shortest
periodic orbit \cite{berhowl}.
For typical two dimensional potentials with finite phase space
volumes, the term $N(I;E)$ scales as $1/\hbar^2$. The prefactor of
that term in (\ref{hwg}) has this scaling but there is a further
logarithmic dependence on $\hbar$ which causes it to grow somewhat
faster. This logarithmic factor arises from the fact that the integral
in (\ref{Id_1}) diverges logarithmically with $Q$. One must be careful
in discussing ``orders'' when expressions involve logarithms of large
quantities and for practical purposes, the non-logarithmic term
$4\gamma+14\log 2-8$ represents an essential correction, as
discussed in Ref.~\cite{tom}. Based on Eq.~(\ref{res}), we expect
terms involving reflection operators to be weaker by a relative power
of $\hbar$ and therefore to scale as $1/\hbar$. This is not true for
$N(\sigma_y;E)$ which is amplified by a
factor of $1/\hbar$ so that it is of the same order as the
non-logarithmic term in $N(I;E)$. The fact that it has been amplified
by a full power of $1/\hbar$ can be traced to the fact that the
integral (\ref{refy}) diverges linearly with $Q$. Therefore, rather
than being a relatively weak correction, this reflection operator is
almost leading order in its effect. In particular, the approximate
relation that the fraction of states in irrep $R$ is
approximately $d_R^2/|G|$ fails in general, since it comes from
considering just the identity operator. (However it is valid for the $E$
irrep which is independent of that reflection class.) A similar behaviour
is also apparent in the related problem of the hyperbola billiard
\cite{hyp,dahl1}.
The other reflection
class function $N(\sigma_1;E)$ does scale as $1/\hbar$ as we expect for normal
reflection operations. The two rotation classes also
behave normally \cite{us}, being constants independent of $\hbar$.
\section{Numerical Comparison of Two Dimensional Results}
We have numerically diagonalised the quantum Hamiltonian and found the
first few hundred eigenvalues of the problem. We used appropriately
symmetrised bases involving harmonic oscillator wave functions in the
$x$ and $y$ directions to separately find the eigenvalues belonging to
each irrep. All results are for bases of 200 oscillators in each direction.
To make the comparison more explicit, we convolved the numerically
obtained density of states by a Gaussian of width $w$,
\begin{equation} \label{smooth}
\tilde{\rho}_R(E) = {1\over\sqrt{2\pi w^2}}\sum_n\exp{\left
(-{(E-E_n)^2\over 2w^2}\right)}.
\end{equation}
The integrated density of states is then obtained by replacing the
sharp steps at the quantum eigenvalues by the corresponding error
functions. For large $w$, this convolution washes out all oscillations
leaving just the average behaviour.
In Fig.~\ref{irreps} we show the results for all five irreps with a
smoothing width $w=3$. The solid curves are the numerics and the
dashed curves are the analytical forms. The first thing which is
apparent is that there is a great distinction between the $A_1$ and
$B_1$ states compared to the $A_2$ and $B_2$ states, resulting from
the large contribution of $N(\sigma_y;E)$. Between each of these pairs
there is a much smaller splitting due to $N(\sigma_1;E)$. The
deviations between the solid and dashed curves are completely
numerical in origin and arise from the finite basis used in
determining the quantum eigenvalues. Due to the
channels, the eigenvalues converges very slowly with increased basis
size. It is interesting to note that the irreps which are odd with
respect to reflections through the channels are better converged.
Being odd, they are less sensitive to the effects of
the channels and are therefore less error prone. Nevertheless, their error is
still dominated by channel effects as we will demonstrate.
The other three irreps are not odd with respect to both
channels ($A_1$ and $B_1$ are even with respect to both channels
and the $E$ states can be chosen as even with
respect to one and odd with respect to the other.) All three of
them fail at approximately the same energy of $E\approx 18$. The
number of accurate eigenvalues is roughly 35 for $A_1$ and $B_1$
and 45 for $E$ (recall that $E$ is doubly degenerate so the number of
independent eigenvalues obtained is half the number of states plotted.)
This is rather dismal considering the 40,000 oscillator states used.
The irreps $A_2$ and $B_2$ are accurate up to energies near $E\approx
60$ representing roughly 115 states each.
It is also interesting to numerically isolate the contributions from
the various classes and compare them to (\ref{hwg}) directly as done in
Ref.~\cite{us}. This is a simple exercise since the entries in the
character table are components of a unitary matrix which is readily
inverted. The result is
\begin{equation} \label{invert}
\left(\begin{array}{c}
N(I;E)\\N(\sigma_y;E)\\N(\sigma_1;E)\\N(R_{\pi/2};E)\\N(R_\pi;E)
\end{array}\right) =
\left(\begin{array}{rrrrr}
1& 1& 1& 1& 1\\
1&-1& 1&-1& 0\\
1& 1&-1&-1& 0\\
1&-1&-1& 1& 0\\
1& 1& 1& 1&-1
\end{array}\right)
\left(\begin{array}{c}
N_{A1}(E)\\N_{A2}(E)\\N_{B1}(E)\\N_{B2}(E)\\N_E(E)
\end{array}\right)
\end{equation}
This can be written compactly as
\begin{equation} \label{compact}
N(g;E) = \sum_R \eta_R(g) N_R(E)
\end{equation}
where the factors $\eta_R(g)$ are defined in (\ref{invert}) and
can be thought of as the inverse of the group characters.
In Fig.~\ref{I_sx_r2} we plot $N(I;E)$, $N(\sigma_y;E)$ and
$N(R_\pi;E)$ from the theory and with the numerical eigenvalues
combined according to (\ref{invert}).
As mentioned, the first is just the total number of states.
The third is shown in its own panel since its value is of
a very different scale than the other two. They all fail around
$E\approx 18$ which is consistent with the previous
figure. $N(R_\pi;E)$ depends on very fine cancellations
and is more sensitive to small errors so it is consistent that it
produces noticeable deviations at a slightly smaller energy than the other
two. Eq.~(\ref{hwg}) predicts a flat line for $N(R_\pi;E)$, the structure
at smaller $E$ comes from the convolution (\ref{smooth}) which is applied
to the analytical forms as well as to the numerical data.
The other two conjugacy classes behave very differently. We plot these
results in Fig.~\ref{s1_r1}. The upper panel shows $N(\sigma_1;E)$ and
the lower panel shows $N(R_{\pi/2};E)$. For the lower panel, we choose
two different smoothing widths, the relevance of which we discuss
below. For now, consider the comparison between the smooth solid curve
and the dashed curve in each case. The results are now accurate up to
energies of $E\approx 800$ or more than forty times the range
observed in the previous figure. This indicates that the numerics are,
in some sense, better than a quick study of Fig.~\ref{irreps} would
indicate. Although the various irreps are individually error prone
even at relatively modest energies, these errors are very correlated
so that appropriate combinations cause them to cancel. In fact, this is
apparent in Fig.~\ref{irreps} since the pairs $A_1$ and $B_1$ and also
$A_2$ and $B_2$ deviate from their expected behaviour
in very correlated manners. From (\ref{invert}) we see that both
$N(\sigma_1;E)$ and $N(R_{\pi/2};E)$ involve the differences
$N_{A1}(E)-N_{B1}(E)$ and $N_{A2}(E)-N_{B2}(E)$ and the systematic
effects cancel for these two classes. Since these functions agree with the
numerics up to $E\approx 800$, it is reasonable to associate all the problems
in the numerics with $N(I;E)$ and $N(\sigma_y;E)$, i.e. with the channels.
This is obviously true for the irreps $A_1$, $B_1$ and $E$, however it is
also true for the odd irres $A_2$ and $B_2$. Their staircase functions fail
at $E\approx 60$ which is better than the other irreps but still very
much smaller than the classes $N(\sigma_1;E)$ and $N(R_{\pi/2};E)$.
We now briefly discuss the oscillatory structure visible in the bottom
panel of Fig.~\ref{s1_r1}. This type of structure was also observed in
Ref.~\cite{us} where it was explained in terms of fractions of
periodic orbits \cite{robbins}. In this example, the structure arises
from the square-like periodic orbit shown in Fig.~\ref{system}. After
completing, one quarter of a cycle, the trajectory is related to its
initial point by a rotation of angle $\pi/2$. This quarter-orbit then
contributes an oscillatory contribution to the function
$N(R_{\pi/2};E)$. This is a scaling system whose classical mechanics
is independent of energy, after appropriate scalings. In particular,
the period of an orbit scales as $T\propto E^{-1/4}$ which
explains the growing wavelength with energy. Additionally, the smoothing
suppresses the oscillatory contribution by a factor proportional to
$\exp{(-w^2 T^2/2)}$ which explains why the amplitude of oscillation
increases with energy. At the highest end of the energy range, one sees the
contributions of higher repetitions - for example three quarters of
the square orbit will also contribute to $N(R_{\pi/2};E)$. The
function $N(R_{\sigma_1};E)$ receives contributions from fractional
orbits which map to themselves under reflection through the diagonal.
Examples of this include the diagonal orbit after a half period and after a
full period. Such structure is visible at
the upper end of the energy range but is less apparent than in the
bottom panel because of the different vertical scale.
Similar structure exists for the other classes as well but is not
visible due to the short energy range available. $N(R_{\pi};E)$
receives contributions from one half the diagonal orbit and one half
the square orbit. $N(\sigma_y;E)$ receives a strong contribution from
the almost periodic family of orbits corresponding to the adiabatic
oscillation deep in the channels (actually, from the fractional
periodic family which has one half the period.) This is a non-standard
contribution due to the intermittency, such effects are discussed in
Refs.\cite{dahl1,dahl2,helium,hydrogen}. The function $N(I;E)$ receives
contributions from all the complete orbits but not from any fraction
of them. The periodic orbit theory of this system has been discussed
in detail in Ref.~\cite{dahl2} and the references therein, so we
forego a more detailed discussion.
\section{The three dimensional generalisation}
In this section we discuss the three dimensional potential
$V=x^2y^2+y^2z^2+z^2x^2$. This is the potential which actually appears
in the zero dimensional limit of the $SU(2)$ Yang-Mills equations.
The symmetry group is that of the octahedral group in which we allow
spatial inversions --- the extended octahedral group. In Fig.~\ref{3dpot}
we show a three dimensional constant energy contour of the potential
and also an octahedron whose vertices are aligned along the channel
directions. In total there are 48 group elements organised into 10
conjugacy classes. This group is the direct product of the
inversionless octahedral group and the inversion parity group. The
first of these is composed of 24 group elements organised into 5
classes \cite{lomont} and we start by enumerating these. First, there
is the identity $I$, which is in a class by itself. There is a class
of six elements involving rotations by $\pm\pi/2$ about any of the
three axes, such as $R_{x,\pi/2}$. Similarly, there is a class of
three elements involving rotations by $\pi$ about these axes, for
example $R_{x,\pi}$. There is a class of 8 elements involving rotation
by $\pm 2\pi/3$ about any of the face-face axes, such as $R_{a,2\pi/3}$.
Finally, there is a class of six elements involving rotations by $\pi$
about any of the the six edge-edge axes, such as $R_{1,\pi}$. We refer
to these classes as $C_1$ to $C_5$ respectively. This group has 5
irreps and the character table is the top left quarter of Table~3.
To construct the full group, we multiply representative members of
each class by the the inversion operation $\Sigma=\sigma_x\sigma_y\sigma_z$.
The effect of this is to map the identity to the inversion element
$\Sigma$ and to map each rotation into either a single reflection or into a
rotation times a reflection. This induces five additional classes. The
element $\Sigma$ is in a class by itself. Composition of the second
class with $\Sigma$ gives a class of six elements which are rotations
by $\pm\pi/2$ through an axis times reflection through that axis, such
as $R_{x,\pi/2}\sigma_x$. Composition of the third class with
$\Sigma$ gives the reflection elements about the three planes, such as
$\sigma_x$. The fourth class becomes a product of a rotation about a
face-face axis times a reflection through the perpendicular plane,
such as $R_{a,2\pi/3}\sigma_a$. Finally, the fifth class becomes
reflections through planes defined by the edges and vertices. An
example is the plane defined by the point $1$ together with the vertices at
positive and minus $z$. We call reflections through this plane $\sigma_1$.
We denote these five additional classes $C_1'$ to $C_5'$ respectively.
The addition of these classes doubles the number of irreps and the
full character table is shown in Table~3.
We proceed by analogy with the two dimensional problem. There we found
that to analyse the contribution of a single channel, it was necessary
to consider the subgroup which mapped that channel onto itself --- in
that case it was the parity group. We do the same here. The eight group
elements which map the channel $x\gg 1$ (for example) onto itself are
$I$, $\sigma_{y,z}$, $\sigma_{2,3}$, $R_{x,\pm\pi/2}$ and $R_{x,\pi}$
and these belong to classes $C_1$, $C_3'$, $C_5'$, $C_2$ and $C_3$
respectively. ($\sigma_{2,3}$ are defined in analogy to $\sigma_1$;
they are reflections through the two planes defined by the vertices
at plus and minus $x$ and the midpoints of the two edges connecting the
$z$ vertex to the positive and negative $y$ vertices.) We can expect the
integrals associated with these elements to be problematic and to
possibly require the adiabatic matching used in section III.
Together these eight elements comprise the subgroup $C_{4v}$ which is,
of course, the group we studied in the two dimensional problem.
This will prove useful in the subsequent analysis.
We start by studying the five classes which do not require an
adiabatic analysis. The class $C_1'$ involves three orthogonal
reflections while the classes $C_2'$ and $C_4'$ involve rotations and
perpendicular reflections. Their Wigner transforms are given by
(\ref{3dres}) and are trivial to integrate since they involve delta
functions of all the quantities. Their constributions are $1/8$, $1/4$
and $1/6$ respectively. The class $C_4$ involves rotations through the
face axes. For rotation by $2\pi/3$ through the point $a$, we define a
change of variables
\begin{equation}
\xi = {1\over\sqrt{3}}( x - y + z) \hspace{1cm}
\eta = {1\over\sqrt{6}}(2x + y - z) \hspace{1cm}
\zeta = {1\over\sqrt{2}}(y+z), \label{cov_4}
\end{equation}
so that the potential along the $\xi$ axis is $V=\xi^4/3$. We then use
the third equation of (\ref{res}) with this choice of variables to find
\begin{equation} \label{c_4}
Z(C_4;\beta) = { \Gamma({1\over 4}) \over \sqrt{24\sqrt{3}\pi} }
{1\over\beta^{3/4}\hbar }.
\end{equation}
For rotation by $\pi$ through the point $1$, we define a change of
variables
\begin{equation}
\xi = {1\over\sqrt{2}}(x + y) \hspace{1cm}
\eta = {1\over\sqrt{2}}(x - y) \hspace{1cm}
\zeta = z \label{cov_5}
\end{equation}
so that the potential along the $\xi$ axis is $V=\xi^4/4$. We then
find
\begin{equation} \label{c_5}
Z(C_5;\beta) = {\Gamma({1\over 4})\over 8\sqrt{\pi}}
{1\over\beta^{3/4}\hbar}.
\end{equation}
We now consider the more interesting classes which map at least one
channel onto itself. We earlier suggested that the integrals
corresponding to them might be problematic. In fact, this is true for
all of them except the identity whose integral converges without
such an analysis. Therefore, we do it first,
\begin{eqnarray}
Z(I;\beta) & = & {1\over(2\pi\hbar)^3} \int dxdydzdp_xdp_ydp_ze^{-\beta H}
\nonumber\\
& = & {\Gamma^3({1\over 4})\over \sqrt{32\pi^3}}
{1\over\beta^{9/4}\hbar^3}.
\label{c_1}
\end{eqnarray}
(The $p$ integrals are done trivially and the spatial integrals can be done
by a change to cylindrical coordinates.)
The convergence of this integral is due to the fact that deep in one
of the channels, the energetically accessible area pinches off as $1/x^2$,
which is integrable. The analogous integral in two dimensions pinches
off as $1/x$ and is not integrable. The remaining four classes follow
from using (\ref{trwt}), (\ref{res}) and (\ref{chk}) inside a cube
$|x|\leq Q$, $|y|\leq Q$ and $|z|\leq Q$.
For reflection in $z$, which is a member of the $C_3'$ class,
we use Eq.~(\ref{res}) and so arrive at the following integral,
\begin{equation} \label{c_3'_0a}
Z_0(\sigma_z;\beta) = {1\over 2} {1\over (2\pi\hbar)^2}
\int_{-\infty}^\infty dp_xdp_y e^{-\beta(p_x^2+p_y^2)/2}
\int_{-Q}^Q dxdy e^{-\beta x^2y^2}.
\end{equation}
Other than the factor of one half, this is the same integral
we evaluated to get the total density of states in the two dimensional
problem. The result is given by (\ref{Id_1_res}) and so we conclude
\begin{equation} \label{c_3'_0}
Z_0(\sigma_z;\beta) = \sqrt{{1\over\pi\beta^3\hbar^4}}
\left(\log Q + \log\beta^{1/4} + {\gamma\over 4} + {1\over 2}\log 2\right).
\end{equation}
Reflection in $\sigma_3$, which is a member of the $C_5'$ class,
requires a more complicated calculation. We define a change of
coordinates so that $\eta = (z+y)/\sqrt{2}$ and $\zeta =
(z-y)/\sqrt{2}$ and then use Eq.~(\ref{res}) with the delta functions
acting on $\zeta$ and $p_\zeta$ so that the integral to be evaluated is
\begin{equation} \label{c_5'_0a}
Z_0(\sigma_3;\beta) = {1\over \pi\beta\hbar^2}
\int_0^Qdx\int_0^{\sqrt{2}Q}d\eta
e^{-\beta(x^2\eta^2+\eta^4/4)}.
\end{equation}
We have done the trivial momentum integrals and have noted that by its
definition, $\eta$ has a different integration range than $x$. This
integral can be done in a manner analogous to (\ref{Id_1}), we define
integration variables $u=x\eta$ and $v=\eta$. Doing the $v$
integration first and using $\beta^{1/4}Q\gg 1$ one
arrives at
\begin{equation} \label{c_5'_0}
Z_0(\sigma_3;\beta) = \sqrt{{1\over 4\pi\beta^3\hbar^4}}
\left(\log Q + \log\beta^{1/4} +{\gamma\over 4} + {3\over 2}\log 2\right).
\end{equation}
Rotation by $\pi/2$ about the $x$ axis is a member of the $C_2$ class
and implies delta functions in the other two variables so that the
integral to be done is
\begin{eqnarray}
Z_0(R_{\pi/2};\beta) & = & {1\over 4\pi\hbar}
\int_{-\infty}^\infty dp_x e^{-\beta p_x^2/2}\int_{-Q}^Q dx \nonumber \\
& = & \sqrt{{1\over 2\pi\beta\hbar^2}}Q.
\label{c_2_0}
\end{eqnarray}
Rotation by $\pi$ about the $x$ axis, which is a member of the $C_3$
class, involves an integral which is identical except for a factor
of two from the $\sin^2(\theta/2)$ factor in (\ref{res}). Therefore
\begin{equation} \label{c_3_0}
Z_0(R_\pi;\beta) = \sqrt{{1\over 8\pi\beta\hbar^2}}Q.
\end{equation}
\section {Channel Calculations in Three Dimensions}
In this section we evaluate the contribution of the channels in three
dimensions. As discussed before, this is is only necessary for some of
the group elements. In analogy with
(\ref{separate}) we define a local two-dimensional Hamiltonian as
\begin{equation} \label{separate_3d}
h_x = {1 \over 2}(p_y^2 + p_z^2) + {\omega_x^2\over 2}(y^2+z^2) + y^2z^2,
\end{equation}
where again $\omega_x=\sqrt{2}x$ and $x$ is assumed large.
Deep in the channel, the
final term can be thought of as a small perturbation which has
virtually no effect on the eigenenergies. If that term were completely
absent, the local Hamiltonian would have an $SU(2)$ symmetry
corresponding to a two-dimensional harmonic oscillator. The
eigenvalues of the Hamiltonian would then be $e_n=(n+1)\hbar\omega_x$,
each with a degeneracy of $(n+1)$. The degenerate states can be
labelled by the rotational quantum number $m$ which runs from $-n$ to
$n$ in even increments. The perturbation $y^2z^2$ will not affect the
energies in a significant manner but will act to break up the
degenerate collections of states into specific irreps of $C_{4v}$ as
follows. All states with odd $m$ correspond to the $E$ irrep.
The $m=0$ states are all $A_1$. For $m$ non-zero and divisible by 4, the
states are either $A_1$ or $B_2$ (corresponding to $\cos(m\theta)$ and
$\sin(m\theta)$ respectively). Otherwise, if $m$ is even but not
divisible by $4$, the states are either $A_2$ or $B_1$ (corresponding
to $\sin(m\theta)$ and $\cos(m\theta)$ respectively.) We then define
local heat kernels corresponding to the five irreps by adding
the contributions of all values of $n$ with the appropriate degeneracy
factor for each irrep so that
\begin{eqnarray}
z_{A_1}(\beta) & = & \sum_{n=\mbox{even}} \left[{n+4 \over 4}\right]
e^{-\beta\hbar\omega_x(n+1)} \nonumber\\[1ex]
z_{B_2}(\beta) & = & \sum_{n=\mbox{even}} \left[{n\over 4}\right]
e^{-\beta\hbar\omega_x(n+1)} \nonumber\\[1ex]
z_{B_1}(\beta) = z_{A_2}(\beta) & = & \sum_{n=\mbox{even}}
\left[{n+2\over 4}\right] e^{-\beta\hbar\omega_x(n+1)} \nonumber\\[1ex]
z_{E}(\beta) & = & \sum_{n=\mbox{odd}}
(n+1) e^{-\beta\hbar\omega_x(n+1)},
\label{lhk_3d}
\end{eqnarray}
where $[x]$ is the largest integer less than or equal to $x$. We will
refer to these relations collectively as
\begin{equation} \label{compacter}
z_R(\beta) = \sum_{n=0}^\infty c_R(n)e^{-\beta\hbar\omega_x(n+1)},
\end{equation}
where $c_R(n)$ are the degeneracy factors defined in (\ref{lhk_3d}).
To evaluate the traces, we integrate over the remaining $x$ dependence
\begin{eqnarray}
Z_R(\beta) & = & {1\over 2\pi\hbar}\sum_n c_R(n)
\int_{-\infty}^\infty dp_x e^{-\beta p_x^2/2}
\int_Q^\infty dx e^{-\beta\hbar\sqrt{2}(n+1)x}\nonumber\\
& = & \sqrt{{1\over 4\pi\beta^3\hbar^4}}\sum_n c_R(n)
{\xi^{n+1} \over n+1},
\label{Z_R_gen}
\end{eqnarray}
where we have defined $\xi=\exp{(-\sqrt{2}\beta\hbar Q)}
\approx 1-\sqrt{2}\beta\hbar Q$. (Note that this is different by a
factor of two from the analogous variable in two dimensions.)
All of this discussion is in terms of the local irreps; what we really
want, however, are the local class heat kernels. These we can get by
appropriate combinations of the irreps as in (\ref{compact}) to arrive
at the class sums
\begin{equation} \label{classsum}
S(g;\beta) = \sum_n c(g,n){\xi^{n+1}\over n+1}.
\end{equation}
For the moment we omit the prefactor of (\ref{Z_R_gen}), this will be
reintroduced later. The degeneracy factor $c(g,n)$ corresponding to a
group element $g$ is found by adding together the degeneracy factors
$c_R(n)$ with the appropriate weightings as given by (\ref{invert}), i.e.
\begin{equation} \label{dgn}
c(g,n) = \sum_R \eta_R(g)c_R(n).
\end{equation}
We start with the identity element. Earlier it was argued that we do
not need a channel calculation since the central integration
converges. However, it is of interest to see how this is also apparent
in the channel calculation. Comparing (\ref{invert}) and (\ref{lhk_3d})
it is apparent that $c(I,n)=n+1$ so that
\begin{eqnarray}
S(I;\beta) = \sum_{n=0}^\infty \xi^{n+1} & = & {\xi\over 1-\xi}\nonumber\\
& = & {1\over \sqrt{2}\beta\hbar Q}.
\label{sum_Id_chann}
\end{eqnarray}
We now need to reinsert the prefactor of (\ref{Z_R_gen}) and must also
include an integral
factor representing the number of channels left invariant by the
corresponding element $f_g$, as in two dimensions. We
trivially have $f_I=6$ so that the channel result for the identity
element is
\begin{equation} \label{Id_chann}
Z_c(I;\beta) = {3\over\sqrt{2\pi}} {1\over \beta^{5/2}\hbar^3Q}.
\end{equation}
We now compare this result to (\ref{c_1}), the present contribution is
very much smaller if $\beta^{1/4}Q\gg 1$ which is precisely the
limit we are considering. Therefore, we again observe that no channel
calculation is necessary for the identity element.
We next consider the reflection element $\sigma_z$. It is in the same class
as $\sigma_y$ so comparing (\ref{invert}) and (\ref{lhk_3d}) we conclude
$c(\sigma_z,n)=1$ when $n$ is even and $0$ when $n$ is odd. The sum which
must be done is
\begin{eqnarray}
S(\sigma_z;\beta) = \sum_{n=\mbox{even}}{\xi^{n+1}\over n+1}
& = & \sum_{m=0}^\infty {\xi^{2m+1}\over 2m+1}\nonumber\\
& = & {1\over 2} \log\left({\sqrt{2}\over \beta\hbar Q}\right),
\label{sum_refz_chann}
\end{eqnarray}
where we have used (\ref{serid}) and the approximation immediately
below (\ref{Z_R_gen}). Note that $f_{\sigma_z}=4$
since $\sigma_z$ leaves four channels invariant, so that
\begin{equation} \label{C_3'_chann}
Z_c(\sigma_z;\beta) = \sqrt{{1\over \pi\beta^3\hbar^4}}
\left(\log{\sqrt{2}\over\beta\hbar} - \log Q\right).
\end{equation}
Recalling now the corresponding result for the central region
(\ref{c_3'_0}), we conclude that for the class $C_3'$,
\begin{equation} \label{C_3'_finally}
Z(C_3';\beta) = \sqrt{{1\over 16\pi\beta^3\hbar^4}}
\left(\log\left({1\over\beta^3\hbar^4}\right) + \gamma + 4\log 2\right).
\end{equation}
This is independent of $Q$ as we expect.
The equality of $z_{B_1}$ and $z_{A_2}$ in (\ref{c_3'_0}) implies that
$S(\sigma_3;\beta)=S(\sigma_z;\beta)$ (since they both equal
$z_{A_1}-z_{B_2}$ from (\ref{invert}).) The only difference is the
subsequent calculation is that $f_{\sigma_3}=2$ so that the channel
calculation is one half of that for $\sigma_z$ (\ref{C_3'_chann}). We
combine this result with the result from the central region (\ref{c_5'_0})
to determine
\begin{equation} \label{C_5'_finally}
Z(C_5';\beta) = \sqrt{{1\over 64\pi\beta^3\hbar^4}}
\left(\log\left({1\over\beta^3\hbar^4}\right) + \gamma +8\log 2\right).
\end{equation}
For rotations by $\pi/2$ about the $x$ axis we note that
$c(R_{\pi/2},n)=(-1)^{n/2}$ for $n$ even and is $0$ for $n$ odd
so that
\begin{eqnarray}
S(R_{\pi/2};\beta) = \sum_{n=\mbox{even}}(-1)^{n/2}{\xi^{n+1}\over n+1}
& = & \sum_{m=0}^\infty (-1)^m{\xi^{2m+1}\over 2m+1}\nonumber\\
& = & \arctan\xi,
\label{sum_rot1_chann}
\end{eqnarray}
where we have again used (\ref{serid}). We now note that
$\arctan\xi\approx \pi/4 -\beta\hbar Q/\sqrt{2}$ and also that only
two channels are left invariant implying $f_{R_{\pi/2}}=2$ so that
\begin{equation} \label{C_2_chann}
Z_c(R_{\pi/2};\beta) = \sqrt{\pi\over 16\beta^3\hbar^4}
- \sqrt{{1\over 2\pi\beta\hbar^2}}Q.
\end{equation}
We now combine this with the calculation from the central region
(\ref{c_2_0}) to arrive at
\begin{equation} \label{C_2_finally}
Z(C_2;\beta) = \sqrt{{\pi\over 16\beta^3\hbar^4}}.
\end{equation}
The final class to be analysed is $C_3$ of which rotations about the
$x$ axis by $\pi$ is a representative member. We now have
$c(R_\pi,n)=(-1)^n$ so that the relevant sum is
\begin{eqnarray}
S(R_\pi;\beta) & = & \sum_{n=0}^\infty (-1)^n\xi^{n+1} = {\xi\over
1+\xi} \nonumber \\
& = & {1\over 2} \left( 1-{\beta\hbar \over \sqrt{2}}Q\right).
\label{sum_rot2_chann}
\end{eqnarray}
As for the previous rotation class, we have $f_{R_\pi}=2$ so that
the result of the channel calculation is
\begin{equation} \label{C_3_chann}
Z_c(R_\pi;\beta) = \sqrt{{1\over 4\pi\beta^3\hbar^4}} -
\sqrt{{1\over 8\pi\beta\hbar^2}}Q.
\end{equation}
Combining this with the calculation from the central region
(\ref{c_3_0}) we conclude
\begin{equation} \label{C_3_finally}
Z(C_3;\beta) = \sqrt{{1\over 4\pi\beta^3\hbar^4}}.
\end{equation}
The final analysis we will do is to find the inverse Laplace transform
of the various relations and thereby express them in the energy domain.
The ten results as a function of $\beta$ are scattered over the
previous two sections. As in the two dimensions, we go directly
to the integrated densities of states by use of (\ref{ilt}). The result is
\begin{eqnarray}
N(C_1;E) & \ =\ & {16\Gamma^2({1\over 4})\over45\sqrt{2\pi^3}}y^3 \nonumber\\[1ex]
N(C_2;E) & \ =\ & {1\over 3} y^2 \nonumber\\[1ex]
N(C_3;E) & \ =\ & {2\over 3\pi} y^2 \nonumber\\[1ex]
N(C_4;E) & \ =\ & {\Gamma^2({1\over 4})\over\sqrt{27\sqrt{3}\pi^3}} y \nonumber\\[1ex]
N(C_5;E) & \ =\ & {\Gamma^2({1\over 4})\over 6\sqrt{2\pi^3}} y \nonumber\\[1ex]
N(C_1';E) & \ =\ & {1\over 8} \nonumber\\[1ex]
N(C_2';E) & \ =\ & {1\over 4} \nonumber\\[1ex]
N(C_3';E) & \ =\ & {1\over 3\pi} y^2 (4\log y+4\gamma+10\log 2-8) \nonumber\\[1ex]
N(C_4';E) & \ =\ & {1\over 6} \nonumber\\[1ex]
N(C_5';E) & \ =\ & {1\over 6\pi} y^2 (4\log y+4\gamma+14\log 2-8),
\label{heretheyare}
\end{eqnarray}
where again we use the semiclassically large quantity $y=E^{3/4}/\hbar$.
For comparison, we remark that for generic potentials, use of
(\ref{res}), would imply that the first term scales as $y^3$, the
following four as $y$, the set $\{C_1',C_2',C_4'\}$ as $y^0$, and the
set $\{C_3',C_5'\}$ as $y^2$.
The leading order behaviour, as given by the first expression, scales
generically with $\hbar$. There are no other terms
which are competitive with it so the relation that
the fraction of states in irrep $R$ is approximately $d_R^2/|G|$ is
valid. As discussed in the text the reflection classes $C_3'$ and
$C_5'$ are amplified somewhat, having an additional logarithmic
dependence on $\hbar$ in addition to the $1/\hbar^2$ prefactor. This
is in analogy to the total density of states of the two dimensional
problem. In fact, the class $C_5'$ is, within a factor of four, the same
as the total density of states in two dimensions.
Two of the rotation classes are amplified by $1/\hbar$ so that they
scale as $1/\hbar^2$. This makes them competitive with the reflection
classes (since, as argued in the two dimensional problem, the logarithmic
term is a rather weak amplification). This is analogous to the
behaviour of one of the reflection operators in the two dimensional
case.
In Fig.~\ref{3dresults} we show the integrated densities of states
found from using the results of (\ref{heretheyare}) combined according
to the characters of Table~III.
(It should be remarked that this is not entirely consistent since the
leading order terms have semiclassical corrections which are almost
certainly of the same order or larger than the smallest terms we are
considering. However, the point of this paper is not a systematic
semiclassical expansion but rather a study of the symmetry effects.)
The structure now looks more typical; irreps of the same
dimensionality have roughly similar numbers of states with slight
differences arising from the contributions of the other group
elements. In particular, the largest four curves are the four three
dimensional irreps and the differences among them arise from the terms
of order $y^2\log y$ and $y^2$; the largest of these curves belongs
to $\Gamma_5'$. The middle two curves belong to the two dimensional
irreps and the smallest four curves belong to the one dimensional
irreps. The largest of these is the trivial irrep $\Gamma_1$; this is
reasonable since it receives positive contributions from all the classes.
In Fig.~\ref{more3dresults} we show the same data but on a smaller
energy scale. At the right edge of the figure ($E=35$), the curves are
ordered the same as in Fig.~\ref{3dresults} (i.e. their asymptotic ordering).
However, it is clear that there is a lot of crossing of these curves
at lower energies. This is because for moderate energies the
contribution corresponding to identity in (\ref{heretheyare}) does not
dominate the others. Additionally, in calculating the functions for
each irrep via (\ref{dodo}) (with the symbol $Z$ replaced by $N$), we
must sum over all the group elements and so the contribution of any
given class is amplified by the number of elements in that class. The
identity class only has one element but the classes which contribute
to next order, $\{C_2,C_3,C_3',C_5'\}$, have six, three, three and six
elements respectively. As mentioned, it is difficult to calculate many
accurate eigenvalues when a potential has channels and this is especially
true in three dimensions. Therefore, the non-asymptotic behaviour in
Fig.~\ref{more3dresults} is relevant to any numerical study since
the results will probably all be in that energy domain.
\section*{Conclusion}
We have shown that the symmetry reduction of the
Thomas-Fermi density of states discussed in Ref.~\cite{us} is easily
generalised to more perverse systems where the Wigner representation fails.
In two dimensions, the symmetry decomposition introduces
essentially leading order contributions to the densities of states of
the one dimensional irreps. The results were verified numerically and
seen to work well. However, the problem studied is numerically very
difficult and only a handful of states of each irrep are reliably
calculated. Nevertheless, certain combinations of the densities of
states are found to be accurate to very high energies even though the
density of states of each individual irrep is not. This effect is
noticeable only by studying the class functions derived here and
would not otherwise have been apparent, thus underlining the
importance of symmetry decompositions.
In three dimensions, we find that the symmetry decomposition
does not introduce terms which are essentially leading order. However,
there are still interesting effects; two of the reflection classes
have a logarithmic dependence on $\hbar$ beyond what one might have
expected and two of the rotation classes have an additional power of
$1/\hbar$ thus making them of essentially the same order as the
reflection elements. Furthermore, we observed that even in this case
one must consider rather high energies before the ordering of the
functions $N_R(E)$ achieves its final form. This is in spite of the
fact that the leading behaviour is not affected by the decomposition.
Rather it arises from the fact that the classes which contribute at
next to leading order have several group elements and their
contributions are correspondingly amplified. This is an effect which
we can expect to become even more important in higher dimensions if we
consider potentials of the form $V(\{x_i\}) = \sum_i\sum_{j>i}x_i^2x_j^2$.
In higher dimensions, more and more of the terms will behave with the normal
$\hbar$ dependence. The only terms with anamalous dependences are
those for which one would initially expect a dependence of $1/\hbar^2$
or $1/\hbar$. If the corresponding group element leaves at least one
channel invariant, they will be amplified by factors of
$\log(1/\hbar)$ and $1/\hbar$ respectively.
\begin{acknowledgements}
The author would like to thank Stephen Creagh and Bent Lauritzen for
useful discussions and the National Sciences and Engineering Research
Council of Canada for support.
\end{acknowledgements}
|
proofpile-arXiv_065-635
|
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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|
\section{Carnot--Carath\'eodory Metric and Sub-Riemannian Manifolds}
A Carnot--Carath\'eodory (C-C) metric on a manifold appears when one is
allowed to travel not along arbitrary paths but along {\em
distinguished} ones on the manifold.
The main example of this kind is {\em polarized Riemannian
manifolds}.
A {\em polarization} of a manifold $M$ is a subbundle $H$ of the
tangent bundle $TM$.
The polarization $H$ distinguishes a set of directions (tangent
vectors) in $M$ which are usually called {\em horizontal}.
A piecewise smooth curve in $M$ is called {\em distinguished}, {\em
admissible}, or {\em horizontal} with respect to $H$ if the tangent
vectors to this curve are horizontal, i.e., belong to $H$.
The Carnot--Carath\'eodory (C-C) metric $d_{H,\/g}(x_1 , x_2 )$ associated
to the polarization $H$ of the Riemannian manifold $(M, g)$ is defined
as the infimum of the $g$-lengths of the distinguished curves joining
the points $x_1$ and $x_2$.
Thus
$$
d_{H,g} (x_1, x_2) := \operatorname{inf} \,(g\text{-length of } H\text{-horizontal curves
between } x_1 \text{ and } x_2).
$$
This distance obviously satisfies the axioms of a metric, provided
that every two points in $M$ can be connected by a distinguished
curve.
This connectivity property takes place if the Lie
brackets $[[H,H] \ldots]$ of $H$ span the tangent bundle $TM$.
This is the case, e.g., for contact subbundles and for general
non-holonomic (completely non-integrable) distributions.
(About Carnot--Carath\'eodory spaces we refer to \cite{Gro}, where one
can find also a broad bibliography.)
Notice that to define the Carnot--Carath\'eodory metric on $(M,g)$ one
needs a Riemannian structure only on the polarization $H \subset TM$
(but not on the whole tangent bundle $TM$).
A smooth manifold $M$ with a polarization $H \subset TM$ and a
Riemannian structure $g$ on $H$ is refered to as the {\em
sub-Riemannian manifold} $(M, H, g)$.
We assume that the polarization $H$ on $M$ is {\em generic} in the
sense that $H$ is a smooth distribution of subspaces tangent to $M$,
of equal dimension, and such that Lie brackets (commutators of all
degrees) of $H$ span the tangent bundle $TM$.
We also assume that the polarization $H$ is {\em equiregular}, i.e.,
that the dimension of the subspace in $T_xM$ generated by the
commutators of fixed degree does not depend on $x \in M$. Then the
tangent bundle $TM$ is filtered by smooth subbundles
\begin{equation}
\eqlabel{1}
H = H_1 \subset H_2 \subset \ldots \subset H_j \subset \ldots \subset
H_k = TM
\end{equation}
such that $H_j$ is spanned by the $j$-th degree commutators of the
fields in $H$.
For example, this is the case if $M$ is a Lie group and $H$ is left
(or right) translation-invariant non-integrable field of tangent
spaces, obtained by translation of the corresponding subspace $t_0M
\subset T_0 M$ tangent to $M$ at the neutral element $0$ of the group.
Here $t_0 M$ is supposed to generate whole the Lie algebra of the
group.
The Heisenberg group $H^n$ with natural (but very unusual)
translation-invariant metric structure is, probably, the most
important example of this kind.
As the simplest example (`flat' in an approrpiate sense), $H^n$ plays
the same role relative to general strictly pseudoconvex
$CR$-manifolds, as $\R^n$ relative to Riemannian manifolds.
(The structure of the Heisenberg group $H^n$ of dimension $n = 2l + 1$
is the same as the
structure of the unit sphere $S^{n}$ in $\C^{l + 1}$. Mostow
observed the close relation between rigidity properties of homogeneous
spaces and quasiconformal mappings of the corresponding sphere at
infinity \cite{Mos}. Complete framework for a theory of quasiconformal
mappings on the Heisenberg group is presented in \cite{KR2}.)
\section{Hausdorff Measure on Carnot--Carath\'eodory Space}
Consider a sub-Riemannian manifold $(M^n, H, g)$ i.e., the smooth
$n$-dimensional manifold $M^n$ with `horizontal' subbundle $H \subset
T M^n$ and a Riemannian metric $g$ on $H$.
We suppose the subbundle $H$ is regular in the sense defined above.
Then $H$ and $g$ induce the {\em Carnot--Carath\'eodory metric}
$d_{H,g}$ on $M^n$ and turn $M^n$ into {\em Carnot--Carath\'eodory
manifold} (with horizontal curves as distinguished ones).
Now one can consider the Hausdorff measure of any degree with respect
to this Carnot--Carath\'eodory metric.
It is important to notice that the {\em metric} (or the {\em
Hausdorff}) {\em dimension} $m$ of the Carnot--Carath\'eodory space
$(M^n, d_{H,g})$ is usually greater than the topological dimension $n$
of the manifold $M^n$.
Namely,
$$
m = \sum^k_{j=1} j \cdot \text{ rank } (H_j/H_{j-1}),
$$
where $H = H_1 \subset \ldots \subset H_j \subset \ldots \subset H_k =
TM$ is the commutator filtration defined in (1) (we assume $H_0 =
\emptyset$).
The inequality $n < m$ is related to the shape of the
Carnot--Carath\'eodory ball. Carnot--Carath\'eodory metric is highly
non-isotropic and non-homogeneous
in contrast to the usual Euclidean one.
Nevertheless, the Hausdorff measure (volume) induced by
Carnot--Carath\'eodory metric may coincide with the usual Lebesque
measure on a Riemannian manifold or with the invariant Haar measure on
a Lie group.
For instance, the Haar measure on the Heisenberg group $H^n$ coincides
with the Lebesque measure on it $(H^n \approx \R^n)$, as well as with
the Hausdorff measure induced by the above Carnot--Carath\'eodory metric
on $H^n$.
At the same time the metric (Hausdorff) dimension $m$ of the
Heisenberg group (or any domain in it) with respect to the
Carnot--Carath\'eodory metric is equal to $n + 1$.
The corresponding Carnot--Carath\'eodory Hausdorff dimension of a smooth
hypersurface that bounds a domain in $H^n$ is equal to $n$.
A change of the Riemannian structure $g$ on the horizontal bunde $H$
of the sub-Riemannian manifold $(M^n, H,g)$ results in locally
quasiisometric change of the Carnot--Carath\'eodory metric and
thus it does not change the Hausdorff dimension and many other
characteristics of the Carnot--Carath\'eodory space. This is why one
often uses a shorter notation $(M^n, H)$ for the Carnot--Carath\'eodory
structure on the manifold $M^n$.
\section{Horizontal Gradient}
The Riemannian structure $g$ on the horizontal subbundle $H$ is,
nevertheless, very useful for the following definition.
The {\em horizontal gradient} $\nabla f$ (or $\operatorname{grad}_{H,g} f$)
of a function $f$ on $M^n$ is defined as the unique horizontal vector
such that
$$
\< \nabla f, X \>_g = Xf
$$
for all horizontal vectors $X$.
Here $\<, \>g$ is the inner product with respect to $g$ and $Xf$ is
the Lie derivative of $f$ along $X$ (that is supposed to exist).
The {\em horizontal normal unit vector} to a hypersurface $\{f = 0\}$
is defined by
$$
{\bf n} := \frac{1}{\| \nabla f \|} \nabla f.
$$
This is the horizontal normal pointing outward for the domain
$\{ f < 0 \}$.
The vector $ {\bf n} (x)$ is undefined at points $x \in M^n$ where
$\nabla f = 0$. However, such points form a submanifold of essentially
lower dimension since the horizontal subbundle $H$ is completely
non-integrable. We shall use this remark below without additional comments.
The sub-Riemannian structure $(M^n, H,g)$ on the manifold may be
considered (see e.g., \cite{KR1}) as a limit of the following
Riemannian structures $(M^n, g_\tau)$.
Consider the decomposition $T M^n = H \oplus H'$. Fix $n$ independent
vector fields $X_1, \ldots, X_d, \ldots X_n$ such that $X_1, \ldots,
X_d$
span $H$ and are orthonormal with respect to the Riemannian
structure $g$ on $H$, and $X_{d + 1}, \ldots , X_n$ span $H'$.
We introduce the Riemannian structure $g_\tau$ on $M^n$ by the condition
that the vectors $\tau X_{d + 1}, \ldots, \tau X_n \ (\tau > 0)$ also
form an orthonormal system.
The one parameter family of Riemannian metrics $g_\tau$ tends to the
singular metric $g$ as $\tau \rightarrow 0$ in the sense that the
length $l_\tau$ of any curve $\gamma \subset M^n$ measured with
respect to $g_\tau$ tends to the Carnot--Carath\'eodory length $l$ of
$\gamma$ with respect to the initial sub-Riemannian structure $(H,g)$
on $M^n$.
This remark allows one to clarify the Carnot--Carath\'eodory counterparts
of some clasical notions and relations that we use below.
For instance, the horizontal gradient is the limit of Riemannian
$g_\tau$-gradients as $\tau \rightarrow 0$.
The classical Fubini-type integral formula %
\begin{equation}
\eqlabel{2}
\int f \, dv = \int_\R \ dt \int_{\{ u = t \} } f \ \frac{d
\sigma}{|\nabla u|}
\end{equation}
(where the domain of integration is foliated by the level surfaces
$\{u = t \}$ of a function $u$) remains valid for sub-Riemannian
manifolds with respect to the induced Hausdorff volume and area measures
$dv, d\sigma$ and the horizontal grandient $\nabla u$, respectively.
In the special case when $f \equiv 1$ and $u$ is the distance to the
fixed point $O \in M^n$, we obtain the standard relation
\begin{equation}
\eqlabel{3}
V'(r) = S(r)
\end{equation}
between the volume of the r-ball and the area of its boundary.
The formula \eqref{2} also imply the general relation
\begin{equation}
\eqlabel{4}
dV = \sigma \, dl
\end{equation}
between the volume variation of a regular domain and the area
$\sigma$ of its boundary under the
horizontally-normal variation of the boundary.
The formula \eqref{4} is essentially local: it remains valid for the
local variation of the boundary as well. In this case $\sigma$
should be replaced by the area $d\sigma$
of the variated part of the boundary.
Thus,
\begin{equation}
\eqlabel{5}
d v = d \sigma \ dl
\end{equation}
where $dl$ is the oriented Carnot--Carath\'eodory length of the local
infinitesimal displacement in the direction horizontally normal to the
variated germ of the hypersurface. (One may consider \eqref{5} as the
volume formula for the cylinder.)
Notice that the volume and area integrals in the left- and right-hand
sides of the formula \eqref{2} are meaningful
even when the function under the integration is not
well-defined on some set of volume zero or area zero respectively.
For instance, the Carnot--Carath\'eodory sphere is not a smooth
hypersurface. Thus the horizontal normal
is not well-defined somewhere. But the singularities form a set of area
zero.
\section{Capacity}
In order to characterize conformal types of Riemannian manifolds we
used in \cite{ZK1} some conformal invariants. Namely, the capacity of
an annulus (ring) and the modulus (extremal length) of a family of
manifolds (curves). We need similar special conformal invariants for
the sub-Riemannian case. However, we start with describing the general
notion of capacity (in the spirit of \cite{Maz}) adopted for
sub-Riemannian manifolds.
Consider a smooth $n$-dimensional manifold $M^n$ with a regular
horizontal subbundle $H \subset T M^n$ and a Riemannian metric $g$ on
$H$. Thus, $(M^n, H, g)$ is a sub-Riemannian manifold.
Let $m$ be its Hausdorff dimension with respect to the induced
Carnot--Carath\'eodory metric, and $dv, d\sigma$
are the volume and area elements, i.e., elements of Hausdorff $m$-measure
for domains and $(m{-}1)$-measure for hypersurfaces in $M^n$.
Let $\Phi(x, \xi)$ be a nonnegative continuous function on the tangent
bundle $T M^n$ which is positive homogeneous of the first degree with
respect to $\xi \in T_x M^n$.
Let $C$ be a compact set in a domain $D \subset M^n$ and $A = A(C,D)$
be the set $\{ u \}$ of smooth nonnegative functions with compact
support in $D$ such that $u \equiv 1$ in a neighborhood of $C$. We
refer to them as {\em admissible functions} for the pair $(C,D)$.
The $(p, \Phi)$-{\em capacity} of the set $C$ relative to the domain
$D$ is defined as follows
\begin{equation}
\eqlabel{6}
(p ,\Phi)-\operatorname{cap}(C,D) := \operatorname{inf}_{u \in A} \int_D \Phi^p (x, \nabla u)
\ dv (x),
\end{equation}
where the infimum is taken over all admissible functions; $\nabla u$
is the horizontal gradient; $dv(x)$ is the volume element of Hausdorff
$m$-measure induced by the Carnot--Carath\'eodory metric of $(M^n, H,g)$.
We need this general notion of capacity only for a special
case of $C,D, p $ and for $\Phi(x, \nabla u) = |\nabla u|$.
Nevertheless, it is reasonable to mention some useful capacity
relations in their general form.
One can rewrite the definition \eqref{6} of the capacity in the
following form for $p > 1$:
\begin{equation}
\eqlabel{7}
(p, \Phi)-\operatorname{cap} (C,D) = \operatorname{inf}_{u \in A} \left( \int^1_0 \
\frac{dt}{(\int_{\{u = t \}} \Phi^p (x, \nabla u) \frac{d
\sigma}{|\nabla u|})^{1/p-1}}\right)^{1-p},
\end{equation}
(see \cite[section 2.2.2]{Maz}). Here the inner integration is over the
level surface $\{ u = t \}$ of the function $u$ and $d\sigma$ is the
element of area, i.e., of Hausdorff
$(m{-}1)$-measure induced on the level hypersurface by the
Carnot--Carath\'eodory metric.
This new representation \eqref{7} leads to the estimates of capacity
we need below.
Set
\begin{equation}
\eqlabel{8}
P(v) := \operatorname{inf}_{G: |G| \geq v} \int_{\partial G} \Phi(x, {\bf n}(x))\
d\sigma,
\end{equation}
where the infimum is taken over all domains $G \subset M^n$ with
regular boundary $\partial G$ and such that the volume $|G|$ of $G$ is not
less than $v$; ${\bf n} (x)$ is the unit vector horizontally normal
to $\partial G$ at the point $x \in \partial G$ directed towards the interior
of $G$.
(The integral in \eqref{8} is well-defined even if $\partial G$ fails to
be smooth on some subset of Hausdorff $(m{-}1)$-measure zero.)
Thus $P(v)$ in \eqref{8} may be considered as a generalized
isoperimetric function. For $\Phi(x, \xi) := |\xi|$ the function
$P(v)$ is the ordinary isoperimetric function of the manifold.
Indeed, in this case for any regular domain $G$ the following relation
is fulfilled
\begin{equation}
\eqlabel{9}
P(v) \leq S,
\end{equation}
where $v$ is the volume (Hausdorff $m$-measure) of $G$ and $S$ is
the area (Hausdorff $(m{-}1)$-measure) of the boundary $\partial G$.
Recall that a function $P: \R_+ \rightarrow \R_+ $ is said to be
the {\em isoperimetric function} on a manifold (equipped with volume
and area measures) if for every domain $G$ with regular boundary $\partial
G$ the relation \eqref{9} holds.
If the relation \eqref{9} holds for a special family of domains they
say that $P$ is the isoperimetric function for this family.
For example, we need below such an isoperimetric function for the
large Carnot--Carath\'eodory balls that form an exhaustion of a
sub-Riemannian manifold.
At the moment we need function $P$ in \eqref{8}
only for domains $G$ bounded by level surfaces of a function $u$
admissible for the pair $(C,D)$.
By means of this function and of representation \eqref{7} of the
capacity one obtains the following inequality
\begin{equation}
\eqlabel{10}
(p, \Phi)- \operatorname{cap} (C,D) \geq \left (\int^{|D|}_0 \
\frac{dv}{P^{p/p-1}(v)}\right)^{1-p},
\end{equation}
where $|D|$ is the volume of the domain $D$ and $P$ is defined by
the formula \eqref{8} applied to $G = \{ u \leq t \}$.
The proof of this inequality for a sub-Riemannian manifold is the same
as for the case of a Riemannian one (see \cite[section 2.2.2]{Maz}).
Now turn to the special cases we need.
Let $(M^n, H, g)$ be the sub-Riemannian manifold which is not compact,
but complete with respect to the natural Carnot--Carath\'eodory metric.
Fix a point $0 \in M^n$ that will play the role of
the origin. Let $B(r)$ be the ball of Carnot--Carath\'eodory radius $r$
centered at $0$, $v(r)$ the volume (Hausdorff $m$-measure) of
$B(r)$, $S(r)$ the area (Hausdorff $(m{-}1)$-measure) of the sphere
$\partial B(r)$, $R^b_a := B(b)\backslash \bar{B}(a)$ the annulus
(or ring) domain whose boundary components are spheres of radius $a$
and $b$ $(a < b)$ respectively.
For $\Phi(x, \xi) := |\xi|$ define
\begin{equation}
\eqlabel{11}
\operatorname{cap}_p R^b_a := (p, \Phi) - \operatorname{cap} (\bar{B}(a), B(b)).
\end{equation}
For this function $\Phi(x, \xi) = |\xi|$ and for the special choice
$u(x) = (b-a)^{-1}(r(x) - a)$ of the admissible function, where $r(x)$
is the Carnot--Carath\'eodory distance of the point $x$ to the origin
$0$, by means of representation \eqref{7} of the capacity we obtain
the following estimate for the $p$-capacity of the annulus $R^b_a$:
\begin{equation}
\eqlabel{12}
\operatorname{cap}_p R^b_a \leq \left (\int_a^b \
\frac{dr}{S^{\frac{1}{p-1}}(r)}\right )^{1- p} .
\end{equation}
Along with \eqref{10} applied to the isoperimetric function $P(v)$
of geodesic balls $B(r)$, we finally obtain the following estimates
\begin{equation}
\eqlabel{13}
\left (\int^{v(r_2)}_{v(r_1)} P^{\frac{p}{p - 1}}\right )^{1-p} \leq
\operatorname{cap}_p R^{r_2}_{r_1} \leq \left (\int_{r_1}^{r_2} S^{-\frac{1}{p
-1}}\right )^{1-p},
\end{equation}
where $v(r_1), v(r_2)$ are volumes of $B(r_1)$ and $B(r_2)$
respectively and $p > 1$ as in \eqref{10} and \eqref{12}.
Remind that in our special case
\begin{equation}
\eqlabel{14}
\operatorname{cap}_p R^b_a := \operatorname{inf} \int_{M^n} |\nabla u |^{p}(x)\, \, dv(x),
\end{equation}
where the infimum is taken over the smooth functions such that
$u \equiv 0$ in the neighborhood of $B(a)$ and $u \equiv 1$ in the
neighborhood of $M^n \backslash B(b)$, while $dv(x)$ is the element
of the volume (Hausdorff $m$-measure) generated by the
Carnot--Carath\'eodory metric of the sub-Riemannian manifold $(M^n, H, g)$.
Conformal change $\lambda^2 g$ of the Riemannian tensor $g$ on the
bundle $H$ produces local rescaling of the
Carnot--Carath\'eodory lengths element by factor $\lambda$, of horizontal
gradient by factor $\lambda^{-1}$, and of Hausdorff $m$-measure
element by factor $\lambda^m$.
Thus, for $p = m$ the integral in \eqref{14} is invariant with
respect to conformal changes of the sub-Riemannian structure and the
Carnot--Carath\'eodory metric.
We get the following conformally invariant capacity of the annulus
(ring, or condenser) $R^b_a$ in the Carnot--Carath\'eodory space $(M^n, H, g)$:
\begin{equation}
\eqlabel{15}
\operatorname{cap}_m R^b_a = \operatorname{inf} \int_{M^n} |\nabla u|^m (x) \, d v (x),
\end{equation}
where $m$ is the Hausdorff dimension of the Carnot--Carath\'eodory space
$(M^n, H,g)$, $dv(x)$ is the element of the induced Hausdorff
$m$-measure on $M^n$, $\nabla u$ is the horizontal gradient of the
function $u$, and infimum is taken over all admissible functions $u$
mentioned above.
It is well known (see e.g., \cite{Vai}) that the conformal
capacity of a spherical condenser $R^b_a$ in $\R^n$ (thus $m = n$) is
equal to
\begin{equation}
\eqlabel{16}
\operatorname{cap}_n R^b_a = \omega_{n-1} \left (\operatorname{ln} \frac{b}{a}\right )^{-n+1},
\end{equation}
where $\omega_{n-1}$ is the area of the unit sphere in $\R^n$.
For the case of the Heisenberg group $H^n$ (instead of $\R^n$),
equipped with the natural $n{-1}$-dimensional polarization $H$ and the
Carnot-Carath\'eodory metric, the conformal capacity $(m = n + 1)$ of
the spherical (with respect to Carnot-Carath\'eodory metric) condenser
$R^b_a$ is equal (see \cite{KR1}) to
\begin{equation}
\eqlabel{17}
\operatorname{cap}_{n + 1} R^b_a = \omega_{n -1} \left (\operatorname{ln} \frac{b}{a}\right )^{-n},
\end{equation}
where $\omega_{n-1}$ is as above.
After these preparations one can carry over to sub-Riemannian
manifolds the main notions and results described in \cite{ZK1} for
Riemannian manifolds.
Below we formulate the notions and results in full detail, sometimes
with comments, but we omit the proofs parallel to those in \cite{ZK1}.
\section{Ahlfors--Gromov Lemma}
Most of standard geometric characteristics of a space are not
conformally invariant. For instance, the unit ball $B^n \subset \R^n$
is flat with respect to Euclidean metric, but it admits a conformally
Euclidean metric of constant negative curvature (the Poincar\'e
metric) and at the same time admits a conformal (stereographic)
projection to the sphere $S^n \subset \R^{n + 1}$ of positive constant
curvature.
Nevertheless, some of geometric relations (that are actually
homogeneous-like or so) preserve or change in a controllable way under
conformal changes of the initial metric.
Recall the following useful Ahlfors--Gromov lemma that we formulated
now for sub-Riemannian manifolds.
\begin{lemma}
If two sub-Riemannian manifolds $(M^n, H, g)$ and $(M^n, H,
\tilde{g})$ are conformally equivalent, then
$$
\int^{\tilde{v}(r_1)}_{\tilde{v}(r_0)} \tilde{P}^{\frac{m}{1-m}} \geq
\int^{r_1}_{r_0} S^{\frac{1}{1-m}}
$$
where $m$ is the common Hausdorff dimension of the Carnot--Carath\'eodory
spaces under consideration; $\tilde{P}$ is any isoperimetric function
of $(M^n, H, \tilde{g})$; $\tilde{v}(r_0), \tilde{v}(r_1)$ are volumes
in $(M^n, H, \tilde{g})$ of any two concentric Carnot--Carath\'eodory
balls $B(r_0), B(r_1)$ of the Carnot--Carath\'eodory space $(M^n, H, g)$;
and $S = S(r)$ is the area (Hausdorff $(m{-}1)$-measure) of the sphere
$\partial B(r)$ in $(M^n, H, g)$.
\end{lemma}
The proof of the lemma for a sub-Riemannian manifold is parallel to
that for the Riemannian case (see e.g., \cite{ZK1}).
The proof shows that Ahlfors--Gromov lemma remains valid even if
$\tilde{P}$ is not a universal $\tilde{g}$-isoperimetric function but
is a $\tilde{g}$-isoperimetric function for $g$-balls only.
\section{Conformal Types of Sub-Riemannian Manifolds}
Consider a noncompact manifold $M^n$ endowed with a horizontal
subbundle $H \subset T M^n $ and a Riemannian metric $g$ on $H$ that
induces the Carnot--Carath\'eodory metric on $M^n$. Let $m$ be the
Hausdorff dimension of a sub-Riemannian manifold $(M^n, H, g)$ with
respect to the induced Carnot--Carath\'eodory metric.
Let $C$ be a nondegenerate compact set in $M^n$, say, a ball. We are
interested in the conformal capacity $\operatorname{cap}_m(C, M^n)$.
The manifold $M^n$, or more precisely, the sub-Riemannian manifold
$(M^n, H, g)$, is called {\em conformally parabolic} or {\em
conformally hyperbolic} if $\operatorname{cap}_m (C, M^n) = 0$ or $\operatorname{cap}_m
(C, M^n) > 0$ respectively.
The capacity relations described in the definition do not depend on
the choice of the compact set $C$ and reflect some conformally
invariant properties of the manifold `at infinity'.
In other words, for the manifold of conformally parabolic type
one has
$$
\lim_{b \rightarrow + \infty} \operatorname{cap}_m R^b_a = 0,
$$
and for conformally hyperbolic manifold
$$
\lim_{b \rightarrow + \infty} \operatorname{cap}_m R^b_a > 0
$$
independently of $a > 0$.
The formula \eqref{16} shows that the standard Euclidean space $\R^n$
is of conformally parabolic type.
If we consider the ordinary hyperbolic space modeled on the Euclidean
unit ball, we conclude by means of \eqref{16} that the Lobachevsky
space of constant negative curvature is of conformally hyperbolic
type.
The formula \eqref{17}
shows that the Heisenberg group as a
sub-Riemannian manifold is of conformally parabolic type.
The situation changes if we consider the same group equipped with a
translation-invariant Riemannian structure.
Notice, that the in both these cases the Heisenberg group $H^n$
by itself, endowed with the word-length metric, induces isoperimetric
inequalities with the same isoperimetric function $P(v) = v^{\frac{n +
1}{n}}$.
But the Hausdorff dimension $m$ of the manifold (or rather
Carnot--Carath\'eodory space) $H^n$ is different in the two cases under
consideration.
In the former (sub-Riemannian) case $m = n + 1$ while in the latter
(Riemannian) one $m = n$.
The left-hand-side of \eqref{13} with $p = n$ and $P(v) =
v^{{\frac{n + 1}{n}}}$ shows now that the Heisenberg group $H^n$
equipped with the translation-invariant Riemannian metric is indeed a
manifold of conformally hyperbolic type.
Note that the Carnot--Carath\'eodory metric in many respects is more
adequate for the Heisenberg group than the translation-invariant
Riemannian one. The Carnot--Carath\'eodory metric on $H^n$ is also
translation-invariant but, in addition, this metric follows the
similitudes of the Heisenberg group. It changes by a factor under such
homogeneous transformations.
These two metrics on the Heisenberg group are not conformal or
quasiconformal equivalent even locally.
By the way, every metric space that admits self similitudes (such as
$\R^n$ with Euclidean metric or $H^n$ with Carnot--Carath\'eodory metric)
must be of conformally parabolic type.
\section{Asymptotic Geometry and Conformal Types of Sub-Riemannian
Manifolds}
As it was mentioned above the conformal type of a manifold depends on
the behavior of the manifold at infinity.
We present now an explicit geometric version of this general claim.
We say that a certain property or relation is {\em realizable in a
class of metrics} if it is valid for some metric of the class.
For instance, any sub-Riemannian manifold $(M^n, H, g)$ can be
realized as a complete one in the conformal class
of its metric.
Below under conformal change of the metric on sub-Riemannian manifold
$(M^n, H, g)$ we mean a conformal change $\tilde{g} = \lambda^2 g$ of
the Riemannian structure on the horizontal bundle $H$. It results in
the conformal change of the corresponding Carnot--Carath\'eodory metric.
Now we are in a position to formulate the following theorem.
\begin{theorem}
Let $(M^n, H, g)$ be a noncompact sub-Riemannian manifold of the
Hausdorff dimension $m$ with respect to the induced
Carnot--Carath\'eodory metric.
The manifold is of conformally parabolic type if and only if any of
the following equivalent conditions is realizable in the class of
complete metrics conformally equivalent to the initial one
\begin{quote}
\begin{itemize}
\item[(i)] $\operatorname{vol}_m (M^n) < \infty$ ,
\item[(ii)] $\int^\infty \ S^{\frac{1}{1-m}} (r) \ dr = \infty$,
\item[(iii)] $\int^\infty \ (\frac{r}{v(r)})^{\frac{1}{m{-}1}} \ dr =
\infty$,
\item[(iv)] $\operatorname{liminf}_{r \rightarrow \infty} \ \frac{v(r)}{r^m} <
\infty$.
\end{itemize}
\end{quote}
\end{theorem}
Here, as above, $v(r)$ and $S(r)$ are the volume and area (i.e., the
Hausdorff $m$-measure and $(m{-}1)$-measure) respectively of the
Carnot--Carath\'eodory ball $B(r)$ and its boundary sphere $\partial B(r)$;
$\operatorname{vol}_m (M^n)$ is the Hausdorff $m$-measure of the whole manifold $M^n$.
The proof is similar to that for the Riemannian manifolds and we omit
it (see \cite{ZK1}).
\section{Canonical Forms of Isoperimetric Function}
In this section we supplement two statements related to possible
variations of the isoperimetric function under conformal changes of
the Carnot--Carath\'eodory metric.
Consider a $g$-spherical exhaustion of a sub-Riemannian manifold
$(M^n, H, g)$, i.e., a system of Carnot--Carath\'eodory balls $B(r)$ of
varying radius $r$ and fixed center $0 \in M^n$. The system is
invariant under the {\em spherically conformal change} $\tilde{g} =
\lambda^2 (r)g$ of the Riemannian metric $g$ on $H$. Let $m$ be the
Hausdorff dimension of $(M^n, H, g)$ with respect to the induced
Carnot--Carath\'eodory metric.
We denote by $v(r)$, $S(r)$ and $\tilde{v}(r)$, $\tilde{S}(r)$ the
volume (Hausdorff $m$-measure) of the ball $B(r)$ and the area
(Hausdorff $(m{-}1)$-measure) of the sphere $\partial B(r)$ considered in
spaces $(M^n, H, g)$ and $(M^n, H, \tilde{g})$ respectively.
Let $P$ be a nonnegative function on a sub-Riemannian manifold $(M^n,
H, g)$. Consider the class of metrics spherically conformally
equivalent to the given one along with the corresponding spherical
exhaustion of the manifold.
\begin{proposition}
\label{1}
The function $P$ is an isoperimetric function and, moreover, the
maximal one (i.e., $P(\tilde{v}) = \tilde{S}$) for sufficiently large
balls of the spherical exhaustion in a certain metric from this class
if and only if the integrals
$$
\text{ a) } \int^{\bullet} P^{\frac{m}{1-m}} \quad, \quad \text{ b) }
\int^{\bullet} S^{\frac{1}{1-m}}
$$
converge or diverge simultaneously at the upper bound of the domains
of integrands.
Moreover, the new Carnot--Carath\'eodory metric
(induced by $\tilde{g} = \lambda^2 (r) g$) is complete if and only if
$\int^{\bullet} P^{-1} = \infty$.
\end{proposition}
\begin{proposition}
\label{2}
Let $(M^n, H, g)$ be a sub-Riemannian manifold of the Hausdorff
dimension $m$ with respect to the Carnot--Carath\'eodory metric on
$M^n$ induced by the Riemannian structure $g$ on the horizontal
subbundle $H \subset T M^n$.
Consider the class of metrics conformally
equivalent to the given one on $(M^n, H, g)$ along with the geodesic
exhaustion of $M^n$, i.e., the exhaustion by balls with respect to the
corresponding Carnot--Carath\'eodory metric.
Then in this class of conformal metrics and
of the corresponding geodesic exhaustion the maximal isoperimetric
function (for large balls) can be reduced to the following canonical
forms
$$
P(\tilde{v}) = \tilde{v}^{\frac{m-1}{m}} \quad \text{ or } \quad
P(\tilde{v}) = \tilde{v}
$$
according to whether the sub-Riemannian manifold $(M^n, H, g)$ is of
conformally parabolic or conformally hyperbolic type respectively.
\end{proposition}
The Proposition \ref{2} is in a sense incomplete. Roughly speaking,
any function $P$ may occur on the conformally parabolic manifold as
the maximal isoperimetric function for large balls of the geodesic
exhaustion corresponding to a certain metric conformally equivalent to
the initial one.
Besides, it is sometimes possible to get the canonical form of the
isoperimetric function even by the spherically conformal change of the
initial metric.
At last, there is no indication to what extent the Proposition \ref{2}
is invertible.
We omit proofs of Proposition \ref{1} and Proposition \ref{2}. They
are similar to the ones for Riemannian manifolds (cf. \cite{ZK1}).
Notice, however, that Proposition \ref{2} can be essentially developed
in spirit of the paper \cite{ZK2}.
\section{Concluding Remarks}
Asymptotic geometry or, in other words, behavior of the manifold
(space) at infinity sometimes plays a decisive role in global
problems. For instance, it is responsible for the existence of special
solutions of operators
on the manifold. In the case of the Laplace operator the problem is
often closely related to (and for two dimensional manifolds coincides
with) the problem of the manifold type discussed above.
Conformal type of the manifold arises as we consider global
transformations of the manifold or the questions related to the
mappings of one manifold into another.
Liouville-type theorems are examples of this kind. In the classical
form the Liouville theorem claims that there is no bounded
non-constant entire function. In other words, there is no holomorphic
mapping of the plane into a disk. This phenomenon of nonexistence is
of rather general nature and it holds for a much broader class of
mappings. It is related to different conformal types of the Euclidean
plane and the disk (or the hyperbolic space). The former is
conformally
parabolic, while the latter is conformally hyperbolic.
We complete the paper with one more example.
The global homeomorphism theorem (GHT) is the following specifically
multidimensional phenomenon: {\em any locally invertible
quasiconformal mapping} $f: \R^n
\rightarrow \R^n$ {\em is globally invertible provided} $n \geq 3$.
The theorem essentially remains valid for mapings $f: M^n \rightarrow
N^n$ of Riemannian manifolds $(M^n, g_M)$, $(N^n, g_N)$ provided $n
\geq 3$, $\pi_1 (N^n) = 0$, and $(M^n, g_M)$ is of conformally
parabolic type.
There are serious evidence to expect the validity of the GHT for
sub-Riemannian manifolds as well, and with the same condition of
conformal parabolicity on $(M^n, H, g)$.
The initial idea of the proof \cite{Zo1}, which was used in further
generalizations and developments of the GHT
(see \cite{Zo2}), seems to be working for the sub-Riemannian case.
Indeed, the topological part of the proof holds for sub-Riemannian
manifolds as well. Thus, it remains to prove some capacity estimates
equivalent to lemmas 1 and 2 of \cite{Zo1}.
For instance, it is sufficient to prove the following.
\begin{slemma}
Let $D$ be a domain in the Carnot--Carath\'eodory space $(M^n, H, g)$
star-like relative to a point $0 \in D$, i.e., $D$ is formed by rays
or by their intervals originating from $0$ and horizontally normal to
Carnot--Carath\'eodory spheres centered at $0$. The finite intervals end
at points of the part $\triangle$ of the boundary $\partial D$ `visible'
from $0$.
Let $\Gamma_\epsilon $ be the family of all curves in $D$ which join
$\triangle $ with $(\epsilon > 0)$-neighborhood $B(\epsilon)$ of $0$.
If $\operatorname{mod}_m \Gamma_\epsilon = 0$ then
\begin{itemize}
\item[(i)] projection of $\triangle$ along these rays to the sphere
$\partial B(\epsilon) $ does not contain connected components different
from a point;
\item[(ii)] $\partial D$ does not locally divide the manifold $M^n$
provided $n \geq 3$.
\end{itemize}
\end{slemma}
Here $m$ is the Hausdorff dimension of the sub-Riemannian manifold
$(M^n, H, g)$ considered.
For our purposes we may suppose the manifold to be of conformally
parabolic type.
Conformal modulus $\operatorname{mod}_m \Gamma$ of the family $\Gamma$ of curves in
$(M^n, H, g) $ is defined as follows
$$
\operatorname{mod}_m \Gamma := \operatorname{inf} \int_{M^n} \rho^m (x) \ d v(x),
$$
where the infimum is taken over all Borel measurable nonnegative functions
such that $\int_\gamma \rho \geq 1$ for each curve $\gamma \in \Gamma$.
One readily shows (cf. the arguments above
concerning the conformal capacity) that $\operatorname{mod}_m \Gamma$ is a conformal
invariant.
It is closely related to the conformal capacity. For instance,
$$
\operatorname{cap}_m R^b_a = \operatorname{mod}_m \Gamma^b_a,
$$
where $\Gamma^b_a$ is the family of all curves that join boundary
components of the spherical condenser $R^b_a$.
Notice that the Star lemma formulated above for the sub-Riemannian
case is true for Riemannian manifolds. In the latter case it can be
proved by means of the slightly modified method that was used in the
proof of lemmas 1 and 2 in \cite{Zo1}.
\subsection*{Acknowledgements.} I wish to thank M.~Gromov, G.~Margulis, and
A.~Schwarz for invitations, fruitful discussions and hospitality during
my visit to University of Maryland, Yale University and University of
California in Spring 1996. I am grateful to B.~Khesin who helped me to
correct and improve the initial text of the manuscript.
This work was done in the stimulating atmosphere of
MSRI.
|
proofpile-arXiv_065-636
|
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\section{Introduction}
In the course of molecular dynamics calculations of the frequency
spectrum of commensurate monolayer solids of nitrogen adsorbed on
graphite [\ref{ref:HB95}, \ref{ref:HBT95}], we became aware of
paradoxical phenomena in what was expected to be the
most ideal and simple regime. At low temperatures the center-of-mass
one-phonon approximation to the intermediate scattering function has
a nearly pure sinusoidal oscillation over periods of 20 to 50 $ps$; however
the amplitude generally is quite different from that anticipated from
equipartition theory for harmonic oscillators and from a sum-rule
of Hansen and Klein [\ref{ref:HK}].
There is some indication in the simulation data that
the mean-square oscillation
amplitude, averaged over hundreds of picoseconds, is on the scale
expected for oscillator coordinates.
The only suggestions of such phenomena which we have found in the literature
are a comment by Hansen and Klein [\ref{ref:HK}] that their sum-rule
was satisfied to 10\% except at small wave numbers where spectral peaks were
quite sharp and a comment by Shrimpton and Steele [\ref{ref:SS91}]
that simulation times much longer than 400 $ps$
would be needed to ensure thermal equilibration of the long-lived
Brillouin-zone-center phonons of commensurate Krypton/graphite. A
nanosecond time scale is inferred from experiments and modeling of
the sliding friction of incommensurate inert gas monolayers
on metal surfaces [\ref{ref:DK}], but the relative
importance of processes determining such long lifetimes is in
dispute [\ref{ref:PN}].
We have used perturbation theory for the effect of cubic and quartic
anharmonicity in the adatom--substrate interaction on the lifetime of
the adlayer phonons[\ref{ref:KK61}, \ref{ref:MF62}]. With parameters
appropriate to commensurate Krypton/graphite and to a spherical molecule
version of commensurate Nitrogen/graphite, the estimated zone-center
phonon lifetimes again are on the scale of nanoseconds. Such processes are
not likely to be the dominant ones in determining the lifetimes,
as has been appreciated by Mills and his co-workers [\ref{ref:HMB},
\ref{ref:HM89}].
The principal mechanism which sets the lifetimes of zone-center phonons
in a commensurate monolayer solid is the radiative damping arising
because the adlayer normal mode is actually a surface resonance
that overlaps a continuum of substrate normal modes
[\ref{ref:HMB}]. In previous modeling of this process, the substrate was
treated as an isotropic elastic medium. There was good success in explaining
the phonon line-widths (for motions primarily polarized perpendicular
to the substrate surface) of inert gases adsorbed on metals [\ref{ref:HM89}].
The time scale for such damping is of the order of picoseconds and the
damping is expected to be larger for a low-density substrate such as graphite
than for high-density metal substrates. Inelastic helium atomic scattering
experiments for xenon adsorbed on graphite [\ref{ref:TV89}] and for
commensurate and incommensurate krypton monolayers on
graphite [\ref{ref:CJD92}] give evidence for a strong mixing of the adlayer
perpendicular vibration with substrate modes over a range of wave vectors and
for strong damping of the normal modes at small wavenumbers.
Here, we develop the elastic substrate theory for
the case of adsorption on the (0001) ($c$-axis) surface of a hexagonal
substrate. This nominally includes the case of the basal plane surface
of graphite. However, to mimic the strong anomalous dispersion of
the TA$_{\perp}$ branch of the graphite spectrum [\ref{ref:NWS},
\ref{ref:IV94}], the continuum approximation of
Komatsu [\ref{ref:KOM55}] and Yoshimori and Kitano [\ref{ref:YK56}] for
the bond-bending energies is adopted.
There is a nonzero frequency at small wave numbers for perpendicular motions
of an incommensurate monolayer, mainly determined by the curvature of
the adsorbate--substrate potential well, and such modes experience both
hybridization with substrate modes and damping [\ref{ref:HMB},\ref{ref:HG91}].
For a commensurate monolayer solid,
there is a Brillouin zone-center gap for motions both parallel and
perpendicular to the substrate surface; the radiative damping mechanism
acts for both polarizations.
The long wavelength motions of graphite
parallel to the surface plane are governed
by isotropic elasticity theory, a simplification relative to the
situation for the (111)
surface of face-centered-cubic metals. The large elastic anisotropy
of graphite between motions parallel and perpendicular to the $c$-axis
has the consequence that the strong radiative damping is
confined to a much smaller fraction of the adlayer Brillouin zone for
the branch with parallel polarization than for that with perpendicular
polarization.
The organization of this paper is: Section II describes the models of the
interactions and the intrinsic dynamics of the decoupled adlayer and
substrate. Section III contains the formulation of the coupled adlayer and
substrate dynamics. Section IV contains the formal solution
for the adlayer response functions. Some special cases are treated in
Section V and the results for commensurate monolayers on graphite are
presented in Section VI. Section VII contains concluding remarks.
A summary of our experience with
Molecular Dynamics simulations for the zone-center modes of the nearly
harmonic solid on a static substrate is contained in Appendix A.
\section{Interaction model and intrinsic dynamics}
The required components are models for the substrate dynamics,
the adsorbate--adsorbate interactions, and the adsorbate--substrate coupling.
The substrate dynamics are modeled with an elastic continuum approximation
which enables a quite explicit treatment at small wave vectors. Apart from
an approximation for bond-bending energy terms in the graphite substrate
[\ref{ref:KOM55},\ref{ref:YK56}], the dispersion of the substrate
normal modes is omitted.
The adsorbate--adsorbate interactions are taken to be central pair potentials.
For the small wave number modes, near the Brillouin zone-center gap,
the form of the pair potential is not crucial to the treatment.
Finally, Steele's Fourier decomposition of the
adatom--substrate interaction is used to make a simple parameterization of the
adsorbate--substrate coupling. As discussed in Sec.III.D, these
choices affect the calculated wave vector dependence of the dispersion
and damping of the vibrational spectra, but we believe the qualitative
features of the results are reliable.
In this paper, the $z-$axis of the Cartesian coordinate system is taken
parallel to the $c-$axis of the hexagonal substrate, the equilibrium
configuration of the adsorbed monolayer
is in the $x-y$ plane, and the wave vector
{\bf q} of the adlayer normal modes is a 2D vector lying in this $x-y$ plane.
\subsection{Intrinsic substrate dynamics}
The hexagonal solid substrate is approximated
as an elastic continuum with 5 independent elastic constants. The equations
of motion for substrate displacements with components u$_i$ are, with the
summation convention for repeated indices,
\begin{equation}
\rho \ddot{u}_i = c_{iklm} \frac{\partial^2 u_l}{\partial x_k \partial x_m}\, .
\label{eq:fyh1}
\end{equation}
With hexagonal symmetry, there are only a few nonzero elements in
the fourth rank tensor $c_{iklm}$. In the Voigt notation [\ref{ref:BH}], the
subscripts for Cartesian axes are denoted 1 = $xx$, 2 = $yy$, 3 = $zz$,
4 = $yz$, 5 = $xz$, and 6 = $xy$.
Then the independent elastic constants are $C_{11} = C_{22}$,
$C_{33}$, $C_{44} = C_{55}$, $C_{66}$, and $C_{13} = C_{23}$. A further
relation derived from the planar isotropy is
$C_{12} = C_{11} - 2 C_{66}$.
For graphite, the mass density is $\rho$ = 2.267 gm/cm$^3$ and the
elastic constants are, all in $10^{11}$ dyn/cm$^2$, $C_{11} = 106$,
$C_{33} = 3.65$, $C_{13} = 1.50$, $C_{44} = 0.40$, and $C_{66} = 44$,
from reference [\ref{ref:RAW}]. Values for $C_{44}$
in the recent literature range from 0.325 to 0.47, derived from
measurements on highly oriented pyrolytic graphite
with Brillouin scattering [\ref{ref:LL90}] and inelastic neutron
scattering [\ref{ref:IV94}], respectively. We adopt the
value $C_{44} = 0.40$ used in [\ref{ref:RAW}], since it is at the middle of
the range and this choice facilitates comparison with
lattice dynamical calculations [\ref{ref:RAWX},\ref{ref:RAWK}].
The speed of the Rayleigh wave [\ref{ref:DM76}] is nearly equal
to that of
the TA$_{\perp}$ mode for wavevector in the $x-y$ plane and polarization
parallel to the $c$-axis, $\sqrt{C_{44}/\rho}$,
because of the large elastic anisotropy of the graphite.
For the graphite substrate, it is
essential to include the strong anomalous dispersion
[\ref{ref:NWS},\ref{ref:IV94}] of the TA$_{\perp}$ mode. This can be
accomplished with a continuum approximation to the bond-bending
energy [\ref{ref:KOM55},\ref{ref:YK56}] by adding the following term to the
total substrate energy:
\begin{equation}
\Delta E = (\lambda / 2) \int d^3 r [ \nabla_2^2 u_z
+ \partial_z \nabla_2 \cdot \text{{\bf u}} ]^2 \, \, ,
\label{eq:KKb}
\end{equation}
where the subscript `2' on the gradient denotes the $x-y$ components.
Eq.(\ref{eq:KKb}) has been constructed so that (i) the associated stress
tensor is symmetric and (ii) isotropy in the graphite basal plane is
retained. Then for displacements with spatial dependence
\begin{equation}
\text{{\bf u}({\bf R},t)} \propto \exp (\imath \text{{\bf q}}
\cdot \text{{\bf R}} ) \, ,
\label{eq:fyh1a}
\end{equation}
and wavevectors {\bf q} in the
$x-y$ plane, the generalized elasticity theory for motions in the
sagittal plane ($SP$) defined by $\hat z$ and {\bf q} has the
replacement
\begin{equation}
C_{44} \rightarrow C_{44}(\text{eff}) = C_{44} + \lambda q^2 \, .
\label{eq:KKc}
\end{equation}
For motions with shear horizontal polarization ($SH$), the `bare'
$C_{44}$ is retained and is denoted $C_{44}^{(0)}$ in this paper.
We set $\lambda = \rho K^2$ with $K = 5.04 \times 10^{-3}$
cm$^2$/sec fitted to the curvature of the TA$_{\perp}$ branch observed by
inelastic neutron scattering experiments [\ref{ref:IV94}].
\subsection{Adatom-adatom interaction and intrinsic adlayer dynamics}
The adlayer consists of atoms or ``spherical molecules" of mass $m$
in a 2D Bravais lattice; for commensurate monolayers of
krypton or xenon on graphite, it is a triangular lattice. We assume
that the adatoms interact via a central potential $\psi$
and denote by {\bf r} the projections of the equilibrium positions
{\bf R} onto the $x-y$ plane. The analysis of this subsection is for modes
polarized in the $x-y$ plane; the dominant interaction for the out-of-plane
motions arises from the adatom-substrate potential treated in Sec.II.C.
The normal modes with wavevector {\bf q} have the form
\begin{equation}
\text{{\bf w}}_{j_a} = \text{{\bf w(q)}}
\exp (\imath [\text{{\bf q}}\cdot\text{{\bf r}}_{j_a} -
\omega(\text{{\bf q}})t]) \, .
\label{eq:mode1}
\end{equation}
The amplitude and angular frequency are obtained from
the solutions of the eigenvalue problem
\begin{equation}
m \, \omega (\text{{\bf q}})^2 \, \text{{\bf w}({\bf q})}
= \text{{\bf D}({\bf q})} \cdot \text{{\bf w}({\bf q}})
\label{eq:mode2}
\end{equation}
for the dynamical matrix {\bf D}({\bf q}) defined by
\begin{equation}
\text{{\bf D}({\bf q})} = \sum_{j_a \neq 0} \nabla \nabla \psi
\, [1 - \cos (\text{{\bf q}} \cdot \text{{\bf r}}_{j_a} )] \, .
\label{eq:mode3}
\end{equation}
The small $\vert \text{{\bf q}} \vert $ solutions are transverse and longitudinal acoustic
waves with frequency proportional to $q$.
The commensurate
monolayer on a static substrate has in-plane motions with angular
frequencies given by
\begin{equation}
\Omega (\text{{\bf q}})^2 = \omega_{0 \parallel }^2
+ \omega (\text{{\bf q}})^2 \, ,
\label{eq:mode4}
\end{equation}
where $ \omega_{0 \parallel }$ is the zone-center gap defined in
Sec.(2.3). Eq.(\ref{eq:mode4}) provides
the basis for the remark at the beginning of this section that the
analysis near the Brillouin zone center only depends weakly on the
form of the pair potential $\psi$. Thus, we adopt the primitive
Lennard-Jones (12,6) interaction for $\psi$;
parameters for krypton and xenon are listed in Table I.
\subsection{Adatom-substrate interaction}
In Sec.III.A, we assume that the interaction of the adlayer atoms or ``spherical"
molecules j$_a$ with substrate atoms J$_s$ is given by a sum of central pair
potentials:
\begin{equation}
\Phi_{as} = \sum_{j_a, J_s} \phi (\vert \text{{\bf R}}_{j_a} -
\text{{\bf R}}_{J_s} \vert) \, .
\label{eq:Steele1}
\end{equation}
For the case of a static substrate lattice with planar surface, $\Phi_{as}$
may be transformed following Steele [\ref{ref:Steele}]:
\begin{equation}
\Phi\vert_{static} = \sum_{j_a} [ V_o (z_{j_a}) + \sum_{\text{\bf g}}
V_g (z_{j_a}) \exp (\imath \text{{\bf g}} \cdot \text{{\bf r}}_{j_a})] \, .
\label{eq:Steele2}
\end{equation}
The notation in Eq.(\ref{eq:Steele2}) follows that of
Sec.II.B: the $z$-axis is
perpendicular to the surface and {\bf r}$_{j_a}$ is the component
of {\bf R}$_{j_a}$
parallel to the surface. The {\bf g} are the 2D reciprocal lattice
vectors of the substrate surface. This representation of the static
interaction is more general than Eq.(\ref{eq:Steele1}) since it may include
effects of noncentral forces and many-body forces. For the following,
we truncate the {\bf g}-sum at the first shell of reciprocal lattice
vectors.
In static substrate models of a commensurate adlayer such as Krypton/graphite
with one atom of mass $m$ per unit
cell, the zone-center modes polarized perpendicular and parallel to the
surface have angular frequencies given by [\ref{ref:LB88}]
\begin{eqnarray}
\omega_{0 \perp }^2 &=& (1/m) {d^2 \over dz^2}
[V_0 (z) + 6 V_g (z) ] \vert_{z = z_{eq}} \nonumber \\
\omega_{0 \parallel }^2 &=& - 3 g^2 V_g (z_{eq}) /m \, .
\label{eq:Steele3}
\end{eqnarray}
where the equilibrium overlayer height is denoted $z_{eq}$.
We use these frequencies to parameterize the dynamic coupling of the
adlayer to the substrate, with a further assumption
which is made explicit in Sec.III. Some values of the frequencies,
based on a combination of experimental data and modeling, are listed
in Table I.
\section{Dynamic coupling of adlayer and substrate}
In principle, one might solve the coupled dynamics of the adlayer and the
substrate using atomistic interaction models for all the constituents.
Such calculations have been performed
[\ref{ref:RAWX},\ref{ref:RAWK}] for the normal modes of coupled
inert gas--graphite slab systems, but not for the
effective damping in an adlayer response function.
Further, there is only limited knowledge of the adatom--substrate
corrugation energy. Therefore, we develop a formalism sufficiently detailed to
show the damping phenomenon and yet one in which the substrate dynamics
and the adatom-substrate coupling are treated with a few empirical
parameters.
\subsection{Parametric forms}
We must first examine the relation
between the descriptions in Secs.II.A and II.C.
In the former the substrate was treated as a continuous medium with
a displacement function {\bf u}({\bf r}, z, t); in the latter
the atomic discreteness of the substrate was basic to the lateral
periodicity of the adatom--substrate potential,
the amplitudes $V_g$ in Eq.(\ref{eq:Steele2}).
The formulation for the dynamic coupling of the adlayer and the continuum
substrate requires a specification of where
the stress from the adlayer is applied in the substrate.
We follow Hall et al. [\ref{ref:HMB}] and assume it to be concentrated
on the surface layer of substrate atoms at height $z_0$, displaced slightly
inward from the boundary $z=0$ of the elastic continuum. In the
final results, $z_0$ is taken to be vanishingly small.
However, an initial distinction between $z = 0$ and $z = z_0$ is
made to bypass complications of $\delta$-function
stresses applied precisely at the edge of the continuum.
We specialize immediately to the case where the oscillatory displacements
of the substrate and the adlayer are represented by
\begin{eqnarray}
\text{{\bf u}}_{J_s} &=& \text{{\bf u}}(\text{{\bf q}}, z)
\exp (\imath [ \text{{\bf q}}\cdot \text{{\bf r}}_{J_s} -
\omega t]) \nonumber \\
\text{{\bf w}}_{j_a} &=& \text{{\bf w}}(\text{{\bf q}})
\exp (\imath [ \text{{\bf q}} \cdot \text{{\bf r}}_{j_a} - \omega t]) \, .
\label{eq:mode5}
\end{eqnarray}
The corresponding interaction energy is derived from the
second order Taylor series expansion for the potential $\Phi_{as}$ of
Eq.(\ref{eq:Steele1})
\begin{eqnarray}
\Delta \Phi_{as} &=& \frac{1}{2} \sum_{j_a,J_s} \nabla \nabla \phi :
\, (\text{{\bf w}}_{j_a} -\text{{\bf u}}_{J_s}) \,
(\text{{\bf w}}_{j_a} - \text{{\bf u}}_{J_s})^* \nonumber \\
&=& \frac{N}{2} \sum_{J_s} \nabla \nabla \phi_{j_a J_s} :
\, [\text{{\bf w(q)}} -\exp(\imath \text{{\bf q}}\cdot [\text{{\bf r}}_{J_s}
-\text{{\bf r}}_{j_a}]) \text{{\bf u(q}}, z)] \nonumber \\
& \times &[\text{{\bf w(q)}} -\exp(\imath \text{{\bf q}}\cdot
[\text{{\bf r}}_{J_s} -\text{{\bf r}}_{j_a}]) \text{{\bf u(q}}, z)]^* \, ,
\label{eq:mode6}
\end{eqnarray}
where $N$ is the total number of adatoms, the $J$-sum is assumed
to be for atoms in the $z_0$-layer, and Umklapp processes involving
reciprocal lattice vectors of the adlayer are neglected.
Although conclusions about the dispersion based on
an atom--atom model for $\Phi_{as}$ have limited
generality, such a model was used[\ref{ref:RAWX},\ref{ref:RAWK}]
for commensurate inert gas / graphite cases treated in Sec.VI, so that
it is useful to specify the differences in the approach used here.
Eq.(\ref{eq:mode6}) may be reduced using a tensor generalization
of the analysis which gives Eq.(\ref{eq:Steele2}). However, in view of the
several approximations already made which limit the quantitative accuracy
with which the dispersion may be treated, we make one further simplification
and drop the phase factor[\ref{ref:PF3}] so that
\begin{equation}
\Delta \Phi_{as} \approx \frac{N}{2} \text{{\bf K}}_0 : \int dz \, \delta(z - z_0)
[\text{{\bf w(q)}} -\text{{\bf u}}(\text{{\bf q}},z)]
[\text{{\bf w(q)}} -\text{{\bf u}}(\text{{\bf q}},z)]^* \, ,
\label{eq:mode7}
\end{equation}
with the tensor coupling constant {\bf K}$_0$ given in dyadic form by
\begin{equation}
\text{{\bf K}}_0 = m ( \omega_{0 \perp}^2 {\hat z}{\hat z}
+ \omega_{0 \parallel}^2 [{\hat x}{\hat x} + {\hat y}{\hat y}]) \, .
\label{eq:mode8}
\end{equation}
Eq.(\ref{eq:mode8}) is a parameterized representation
for the effect of the adatom-substrate interaction
in terms of the Brillouin zone-center gap frequencies
discussed in Sec.II.C. It may be
used to represent the coupling for cases
where $\Phi_{as}$ is not determined as a sum of pair potentials,
such as commensurate layers on metals. It is also a way to bypass the
incomplete understanding of the origin
of realistic corrugation amplitudes $V_g$ for adsorption on graphite.
Finally, Eq.(\ref{eq:mode8}) enables a technical simplification
in the calculation. When combined
with the planar elastic isotropy of the hexagonal surface, the
problem of coupled adlayer and substrate separates into
$SP$ and $SH$ motions.
\subsection{Equations of motion}
The equation of motion for the adlayer normal coordinate becomes
\begin{equation}
m \, \omega^2 \, \text{{\bf w(q)}} = \text{{\bf D(q)}}
\cdot \text{{\bf w(q)}}
+ \text{{\bf K}}_0 \cdot [\text{{\bf w(q)}} -
\text{{\bf u}}(\text{{\bf q}},z_0)] \, .
\label{eq:mode9}
\end{equation}
The differential equations for the components of the substrate amplitude
{\bf u}({\bf q},z$_0$)
are, for {\bf q} parallel to the $x$-axis and
$A_{tot}/N$ equal to the area per adatom in the commensurate adlayer,
\begin{eqnarray}
\rho \, \omega^2 \, u_x (q) &=& (C_{11} q^2 - C_{44} \partial_z^2)
u_x (q) \nonumber \\
&-& \imath \, q \, (C_{13} + C_{44}) \partial_z u_z (q) \nonumber \\
&+& (N/A_{tot}) \, \delta(z - z_0) K_{0xx} [u_x (q) - w_x (q)] \nonumber \\
\rho \, \omega^2 \, u_z (q) &=& (C_{44} q^2 - C_{33} \partial_z^2) u_z (q) \nonumber \\
&-& \imath \, q \, (C_{13} + C_{44}) \partial_z u_x (q) \nonumber \\
&+& (N/A_{tot}) \, \delta(z - z_0) K_{0zz} [u_z (q) - w_z (q)] \, .
\label{eq:tmode1}
\end{eqnarray}
and
\begin{eqnarray}
\rho \, \omega^2 \, u_y (q) &=& (C_{66} q^2 -
C_{44}^{(0)} \partial_z^2) u_y (q) \nonumber \\
&+& (N/A_{tot}) \, \delta(z - z_0) K_{0yy} [u_y (q) - w_y (q)]
\label{eq:mode11}
\end{eqnarray}
\subsection{Boundary conditions}
As the boundary conditions on the substrate displacement
function {\bf u(q}, z), we take [\ref{ref:HMB}] the
$z = 0$ boundary to be a free surface where the following components
of the stress tensor vanish:
\begin{equation}
T_{z \beta} \vert_{z = 0} = 0, \, \, \beta = x, y, z \, .
\label{eq:boun1}
\end{equation}
The components of the stress tensor are given by
\begin{equation}
T_{\alpha, \beta} = c_{\alpha \beta k l} \, \partial_k u_l \, ,
\label{eq:boun2}
\end{equation}
using the 4-index form of the elastic constants. For the
displacement function of Eq.(\ref{eq:mode5}), with {\bf q} parallel
to the $x$-axis and returning to Voigt notation,
Eqs.(\ref{eq:boun1}) become (with $z = 0$):
\begin{eqnarray}
\partial_z u_x (q,z) + \imath \, q u_z (q,z) &=& 0 \nonumber \\
\partial_z u_y (q,z) &=& 0 \nonumber \\
\partial_z u_z (q,z) + \imath q (C_{13} / C_{33} ) u_x (q,z) &=& 0 \, .
\label{eq:boun3z}
\end{eqnarray}
Finally, the theory of the adlayer response involves substrate motions
driven by an
initial displacement of adlayer atoms. Then, deep in the substrate,
$z \to - \infty$, the
disturbance created by the adlayer must decay exponentially or take
the form of an `outgoing' wave. This
becomes a requirement that the solutions
of Eqs.(\ref{eq:tmode1}) and (\ref{eq:mode11}) for $z < z_0$
have the form $\exp(- K \vert z \vert)$ or
$\exp(\imath K \vert z \vert)$ with $K > 0$ -- see Sec.IV.B.
\subsection{Comments}
Eqs.(\ref{eq:mode9}) to (\ref{eq:mode11}) generalize the treatment of
Hall et al. [\ref{ref:HMB}] in two ways: the substrate is an anisotropic
elastic continuum and there are driving terms which
arise from the coherent addition of lateral force terms for the commensurate
adlayer. There is an increase in complexity beyond their treatment,
but, as shown in Sec.V.C, quite simple results are again
obtained at the Brillouin zone center.
We summarize the approximations that have been
made which have serious consequences for the treatment of the
dependence of the mode damping on the wave vector:
\begin{enumerate}
\item A distinction is made between the edge of the elastic
continuum at $z = 0$ and the height $z_0$ where the adlayer stress
is applied. However the limit $z_0 \to 0$ is taken in the
analysis.
\item The anomalous dispersion of the TA$_{\perp}$ branch
of the graphite substrate is approximated in the elastic continuum
description by using the effective elastic constant
C$_{44}$(eff) defined in Eq.(\ref{eq:KKc}). This replaces C$_{44}$ in
Eqs.(\ref{eq:tmode1}).
It leads to a large shift in the wave number where the
Rayleigh wave of the graphite hybridizes with the $\omega_{\perp}$ adlayer
mode and improves the agreement with the $HAS$ experiments
[\ref{ref:TV89},\ref{ref:CJD92}].
\item If the dynamical matrix {\bf D(q)} is dropped from the adlayer
equation of motion, there is only a small effect for small wave numbers.
\item The approximation in dropping certain phase factors to obtain
Eq.(\ref{eq:mode7}) is accurate at small wave numbers; however it
omits a $q-$dependence of the dynamic coupling of the
adlayer and substrate [\ref{ref:PF3}].
\end{enumerate}
\section{Correlation functions}
The response of the adlayer in the presence of the substrate is
characterized using the time Fourier transform of correlation
functions of displacement amplitudes defined by
\begin{equation}
S_{\alpha \alpha}(q,t) = \langle W_{\alpha} (q,t)
W_{\alpha} (q,0) \rangle \, ,
\label{eq:corrf1}
\end{equation}
where $\alpha = x, y, z$. The initial conditions on the displacements are
\begin{eqnarray}
W_{\alpha} (q,t=0) &=& W_{\alpha 0} \nonumber \\
{\dot W}_{\alpha} (q,t=0) &=& 0 \, ,
\label{eq:corrf4}
\end{eqnarray}
with zero for $t < 0$, and the substrate is initially unperturbed and
static. Then the Fourier transform for Eq.(\ref{eq:mode9}) is
generalized to
\begin{equation}
\int_0^{\infty} \exp(\imath \omega t) \ddot{W}_{\alpha}(q,t) \, dt =
- \omega^2 w_{\alpha}(q,\omega) + \imath \omega W_{\alpha 0} \, ,
\label{eq:corrf5}
\end{equation}
using
\begin{equation}
W_{\alpha} (q,t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty}
w_{\alpha}(q,\omega) \exp(- \imath \omega t) \, d\omega \, .
\label{eq:corrf3}
\end{equation}
In this and the following sections, the
dependence on wave number $q$ has been omitted from the notation, to reduce
the complexity of the formulae.
\subsection{Green's function solution}
The solution to the set of coupled dynamical equations posed in
Secs.III.B and III.C is conveniently
stated in terms of the values at $z = z' = z_0$ of
a set of Green's functions
g$_{\alpha \beta}$(\text{{\bf q}}, $\omega$, z $\vert z'$) satisfying
the following set of equations [\ref{ref:DM76}]
\begin{equation}
(\text{{\bf L}} \cdot \text{{\bf g}})_{\alpha \beta} =
\delta_{\alpha \beta} \delta(z - z') \, ,
\label{eq:corrf6}
\end{equation}
where the $3 \times 3$ differential tensor {\bf L} has the following
nonzero elements
\begin{eqnarray}
L_{xx} &=& \rho \omega^2 - C_{11} q^2 + C_{44} \partial_z^2 \nonumber \\
L_{zz} &=& \rho \omega^2 - C_{44} q^2 + C_{33} \partial_z^2 \nonumber \\
L_{yy} &=& \rho \omega^2 - C_{66} q^2 + C_{44} \partial_z^2 \nonumber \\
L_{zx} &=& L_{xz} = \imath q (C_{13} + C_{44}) \partial_z \, .
\label{eq:corrf6a}
\end{eqnarray}
The boundary conditions at $z = 0$ based on Eqs.(\ref{eq:boun3z}) are, for
$\alpha = x,y,z$,
\begin{eqnarray}
\partial_z g_{x \alpha} + \imath q g_{z \alpha} &=& 0 \nonumber \\
\partial_z g_{y \alpha} &=& 0 \nonumber \\
\partial_z g_{z \alpha} + \imath q { {C_{13}} \over {C_{33}} }
g_{x \alpha} &=& 0 \, .
\label{eq:corrf7}
\end{eqnarray}
The problem separates so that the functions g$_{xy}$, g$_{yz}$,
g$_{yx}$, and g$_{zy}$ vanish [\ref{ref:ggr}].
Then with the definitions
\begin{eqnarray}
\lambda_x &=& (N/A_{tot}) m \omega_{0 \parallel }^2 \nonumber \\
\lambda_z &=& (N/A_{tot}) m \omega_{0 \perp }^2 \, ,
\label{eq:corrf9}
\end{eqnarray}
the functions u$_{\alpha}$(z$_0$) are given in terms of the
functions $g_{\alpha \beta} \equiv g_{\alpha \beta}(z_0 \vert z_0)$ by
\begin{eqnarray}
u_x &=& g_{xx} \lambda_x (u_x - w_x) + g_{xz}
\lambda_z (u_z - w_z) \nonumber \\
u_z &=& g_{zx} \lambda_x (u_x - w_x) + g_{zz}
\lambda_z (u_z - w_z) \nonumber \\
u_y &=& g_{yy} \lambda_x (u_y - w_y) \, .
\label{eq:corrf12}
\end{eqnarray}
A formal solution for the driving terms in the
adlayer equations of motion is
\begin{eqnarray}
u_x - w_x &=& b_{11} w_x + b_{12} w_z \nonumber \\
u_z - w_z &=& b_{21} w_x + b_{22} w_z \nonumber \\
u_y - w_y &=& w_y / (g_{yy} \lambda_x - 1) \, ,
\label{eq:corrf13b}
\end{eqnarray}
where the coefficients $b_{ij}$ are given by
\begin{eqnarray}
b_{11} &=& [g_{zz} \lambda_z - 1] / W_b \nonumber \\
b_{12} &=& - g_{xz} \lambda_z / W_b \nonumber \\
b_{21} &=& - g_{zx} \lambda_x / W_b \nonumber \\
b_{22} &=& [g_{xx} \lambda_x -1] / W_b
\label{eq:corrf13a}
\end{eqnarray}
and
\begin{equation}
W_b = [g_{xx} \lambda_x -1] [g_{zz} \lambda_z - 1] - g_{xz}
\lambda_z g_{zx} \lambda_x \, .
\label{eq:corrf13}
\end{equation}
The adlayer equations of motion then become
\begin{eqnarray}
[\omega^2 - \omega_{\ell}^2 (q) + \omega_{0 \parallel}^2 b_{11}] w_x
+ \omega_{0 \parallel}^2 b_{12} w_z &=& \imath \omega W_{x0} \nonumber \\
\omega_{0 \perp}^2 b_{21} w_x + [\omega^2 + \omega_{0 \perp}^2 b_{22}] w_z
&=& \imath \omega W_{z0}
\label{eq:gfn2}
\end{eqnarray}
and
\begin{equation}
[\omega^2 - \omega_t^2 (q) +
(\omega_{0 \parallel}^2 /[g_{yy} \lambda_x - 1]) ] w_y =
\imath \omega W_{y0} \, .
\label{eq:gfn3}
\end{equation}
The $\omega_{\ell}$ and $\omega_t$ are the frequencies of
longitudinal and transverse polarization, respectively, in the intrinsic
adlayer dynamics, Eq.(\ref{eq:mode2}), and the $x-$axis is taken
to be a high symmetry direction, $\Gamma M$ or $\Gamma K$, of
the adlayer Brillouin zone.
The solutions are used to form
\begin{equation}
S_{\alpha \alpha}(q, \omega) = \vert w_{\alpha} (\omega, q) \vert^2
\label{eq:gfn3a}
\end{equation}
with initial condition
\begin{equation}
W_{\beta 0} = \delta_{\alpha \beta} \, .
\label{eq:gfn4}
\end{equation}
The damping of the adlayer normal modes manifests itself
as broadened peaks in $S(q,\omega)$ as a function of $\omega$
at fixed $q$. Generally, peaks in $S(q,\omega)$ may be assigned as
derived from the intrinsic adlayer frequencies or from the Rayleigh
wave of the substrate. As shown in Sec.IV.B, the radiative damping
mechanism operates for sufficiently small $q$.
\subsection{Evaluation of the Green's functions}
The problem separates into analysis of the $SP$ and $SH$ motions with
the coupled $x-z$ equations and $y$-equation, respectively.
\subsubsection{Sagittal plane}
The solution is very similar
to one given by Dobrzynski and Maradudin [\ref{ref:DM76}] for the Green's
function of a hexagonal elastic half space.
We seek solutions of the homogeneous versions of Eqs.(\ref{eq:corrf6})
which have exponential dependences on $z$:
\begin{equation}
g_{\beta \delta} \sim \exp (\alpha z) \, .
\label{eq:boun4}
\end{equation}
With the definitions
\begin{eqnarray}
\gamma_1 &=& [\rho \, \omega^2 - C_{11} q^2 ]/C_{44} \nonumber \\
\gamma_4 &=& [\rho \, \omega^2 - C_{44} q^2 ]/C_{33} \nonumber \\
\sigma_1 &=& \gamma_1 + \gamma_4 + q^2 [(C_{13} + C_{44})^2 /C_{33}
C_{44}] \nonumber \\
\sigma_2 &=& \sqrt{ \sigma_1^2 - 4 \gamma_1 \gamma_4} \, ,
\label{eq:boun7}
\end{eqnarray}
there are two inverse length scales $\alpha_j$ given by:
\begin{eqnarray}
\alpha_1^2 &=& [- \sigma_1 + \sigma_2 ]/2 \nonumber \\
\alpha_2^2 &=& [- \sigma_1 - \sigma_2 ]/2 \, .
\label{eq:boun8}
\end{eqnarray}
For $z < z'$, the roots of Eq.(\ref{eq:boun8}) are chosen to give
damped or outgoing waves according to whether $\alpha_i$ is real or
imaginary[\ref{ref:zroot}]. Denote
the longitudinal acoustic and transverse acoustic modes for wave vector
in the $x-y$ plane by LA (SP$_{\parallel}$) and TA$_{\perp}$,
respectively, and the LA mode for wave vector along the $z-$axis
by LA$_z$. The corresponding speeds in the long wavelength limit are
\begin{eqnarray}
c_{LA} &=& \sqrt{C_{11}/\rho} \nonumber \\
c_{TA} &=& \sqrt{C_{44}/\rho} \nonumber \\
c_{LAz} &=& \sqrt{C_{33}/\rho} \, .
\label{eq:gfn20}
\end{eqnarray}
The choice of roots for Eqs.(\ref{eq:boun8}) is then
\begin{eqnarray}
\alpha_1 &=& - \imath \vert \alpha_1 \vert \,, \, \omega > c_{LA}\, q \, ,
\nonumber \\
\alpha_1 &=& \vert \alpha_1 \vert \, , \, \, \, \, \, \omega < c_{LA} \, q \, ,
\label{eq:gfn21}
\end{eqnarray}
and
\begin{eqnarray}
\alpha_2 &=& - \imath \vert \alpha_2 \vert \, , \, \omega > c_{TA} \, q \, ,
\nonumber \\
\alpha_2 &=& \vert \alpha_2 \vert \, , \, \, \, \, \, \omega < c_{TA} \, q \, .
\label{eq:gfn22}
\end{eqnarray}
According to Eqs.(\ref{eq:corrf6}), the Green's functions are coupled
in pairs (g$_{xx}$ , g$_{zx}$) and (g$_{xz}$, g$_{zz}$). Then, for
Eq.(\ref{eq:boun4}) we have
\begin{equation}
g_{x \beta} (\alpha_j) = \imath f_j \, g_{z \beta}(\alpha_j) \, ,
\label{eq:boun8a}
\end{equation}
with the proportionality factor $f_j$ defined by
\begin{equation}
f_j = - q \alpha_j (C_{13} + C_{44})/[C_{44} (\alpha_j^2 + \gamma_1)] \, .
\label{eq:boun9}
\end{equation}
The solution of the homogeneous form of Eqs.(\ref{eq:corrf6})
in the range $z < z'$ is
\begin{eqnarray}
g_{z \beta} &=& a \exp (\alpha_1 z) + b \exp (\alpha_2 z) \nonumber \\
g_{x \beta} &=& \imath a f_1 \exp (\alpha_1 z) + \imath
b f_2 \exp (\alpha_2 z) \, ,
\label{eq:boun10}
\end{eqnarray}
and in the range $z' < z < 0$ is
\begin{eqnarray}
g_{z \beta} &=& [A_{+} \exp (\alpha_1 z)+ A_{-} \exp (-\alpha_1 z)] \nonumber \\
&+& [B_{+} \exp (\alpha_2 z) + B_{-} \exp (-\alpha_2 z)] \nonumber \\
g_{x \beta} &=& \imath f_1 \, [A_{+} \exp (\alpha_1 z)- \imath
A_{-} \exp (-\alpha_1 z)] \nonumber \\
&+& \imath f_2 \,[B_{+} \exp (\alpha_2 z)
- \imath B_{-} \exp (-\alpha_2 z)] \, .
\label{eq:boun12}
\end{eqnarray}
The six coefficients $a, b, A_{+}, A_{-}, B_{+}$ and $B_{-}$ are
determined from six equations:
the $z = 0$ boundary condition Eqs.(\ref{eq:corrf7}),
the continuity of $g_{x \beta}$ and $g_{z \beta}$ at $z = z'$,
and the matching of the discontinuities in the
first derivatives at $z = z'$ to the strengths of the $\delta$-functions.
The latter equations are
\begin{eqnarray}
C_{44} \, [\partial_z g_{xx} \vert_{z=z' +} -
\partial_z g_{xx} \vert_{z=z' -} ] &=& 1 \nonumber \\
C_{33} \, [\partial_z g_{zz} \vert_{z=z'+} -
\partial_z g_{zz} \vert_{z=z' -} ] &=& 1 \, .
\label{eq:boun12b}
\end{eqnarray}
The $z-$derivatives of g$_{xz}$ and g$_{zx}$ are continuous at
$z=z'$. Completion of the explicit solution for the Green's functions then
is an exercise in linear algebra.
The solutions for g$_{\alpha \beta}(z'\vert z')$ with $z' \to 0$ can be given
in compact form using the definitions:
\begin{eqnarray}
a_{11} &=& q + \alpha_1 f_1 \nonumber \\
a_{12} &=& q + \alpha_2 f_2 \nonumber \\
a_{21} &=& \alpha_1 - q (C_{13} /C_{33} ) f_1 \nonumber \\
a_{22} &=& \alpha_2 - q (C_{13} /C_{33} ) f_2
\label{eq:coup4}
\end{eqnarray}
and the Wronskian
\begin{equation}
W_a = a_{11} a_{22} - a_{12} a_{21} \, .
\label{eq:coup4a}
\end{equation}
The Green's function components are
\begin{eqnarray}
g_{zx} &=& - \imath (a_{21} - a_{22})/W_a C_{44} \nonumber \\
g_{xx} &=& (f_2 a_{21} - f_1 a_{22})/W_a C_{44} \nonumber \\
g_{xz} &=& -\imath (f_2 a_{11} - f_1 a_{12})/W_a C_{33} \nonumber \\
g_{zz} &=& (a_{12} - a_{11})/W_a C_{33} \, .
\label{eq:gfn6d}
\end{eqnarray}
There are three $q-$ranges: I, $q < \omega /c_{LA}$, where both
transverse and longitudinal substrate waves are involved in the damping;
II, $\omega / c_{LA} < q < \omega / c_{TA}$, where only the
transverse waves are involved; and III, $\omega / c_{TA} < q$,
where there is no radiative damping. Characteristic values for
$\omega$ are $\omega_{0 \parallel}$ for the parallel
polarization mode and $\omega_{0 \perp}$ for the perpendicular polarization.
\subsubsection{$SH$ mode}
Define
\begin{equation}
\alpha_3^2 = (C_{66} q^2 - \rho \omega^2 )/C_{44}^{(0)} \, .
\label{eq:boun13}
\end{equation}
The speed of the transverse elastic waves in the $x-y$ plane, denoted the
$SH$ mode, is
\begin{equation}
c_{SH} = \sqrt{C_{66}/\rho} \, ,
\label{eq:gfn14}
\end{equation}
and the choice of root of Eq.(\ref{eq:boun13}) is
\begin{eqnarray}
\alpha_3 &=& - \imath \vert \alpha_3 \vert \, , \, \omega > c_{SH}\, q \, ,
\nonumber \\
\alpha_3 &=& \vert \alpha_3 \vert \,, \, \, \, \, \,
\omega < c_{SH}\, q \, .
\label{eq:gfn15}
\end{eqnarray}
Then the solution for $g_{yy}$ has the form[\ref{ref:string}]
\begin{eqnarray}
g_{yy} &=& Y_1 \exp (\alpha_3 z), \, \, \, \, z < z' \nonumber \\
&=& Y_2 \cosh(\alpha_3 z), \, \, z' < z < 0 \, ,
\label{eq:boun14}
\end{eqnarray}
where the $z = 0$ boundary condition, Eq.(\ref{eq:corrf7}), has been used.
The coefficients $Y_1$ and $Y_2$ are obtained from the continuity
conditions at $z = z'$
\begin{eqnarray}
g_{yy}(z=z'+) - g_{yy}(z=z'-) &=& 0 \nonumber \\
C_{44}^{(0)} \, [\partial_z g_{yy} \vert_{z=z'+} -
\partial_z g_{yy} \vert_{z=z' -} ] &=& 1 \, .
\label{eq:gfn10}
\end{eqnarray}
The solution for $Y_2$ is
\begin{equation}
Y_2 = - \exp(\alpha_3 z') /(\alpha_3 C_{44}^{(0)}) .
\label{eq:gfn11}
\end{equation}
Then, with $z_0 \to 0$, the value of $g_{yy}$ entering in Eq.(\ref{eq:gfn3})
is
\begin{equation}
g_{yy} = -1/(\alpha_3 C_{44}^{(0)}) \, .
\label{eq:gfn12}
\end{equation}
\section{Special cases}
We discuss here three special cases where the present
formalism overlaps with other work.
\subsection{Rayleigh wave}
The frequency (speed) of the Rayleigh wave of wave number $q$ is the
root of $W_a = 0$, for the Wronskian defined in Eq.(\ref{eq:coup4a}).
In the limit
$q \to 0$, this reproduces the result of Dobrzynski and
Maradudin [\ref{ref:DM76}]. As noted by others [\ref{ref:RAW}],
in the small$-q$ limit the speed of the Rayleigh wave of the
graphite basal plane surface is only 0.02\% smaller than $c_{TA}$. The
solution for the Rayleigh wave frequency at finite $q$ with the
effective elastic constant Eq.(\ref{eq:KKc}) is formally the same,
but the quantitative results change somewhat. With the parameters
used here, the Rayleigh frequency is 0.1\% smaller than the
TA$_{\perp}$ frequency at $q=0.3$ {\AA}$^{-1}$ and 0.8\% smaller
at $q=0.6$ {\AA}$^{-1}$. Even so, the difference between the Rayleigh
frequency and the TA$_{\perp}$ frequency remains much smaller than
the 8\% difference found for the case of a Cauchy isotropic elastic
solid.
\subsection{Isotropic elastic medium}
The formalism reduces to the case treated by Hall et al. [\ref{ref:HMB}]
by choosing
\begin{eqnarray}
\omega_{0 \parallel} &=& 0 \nonumber \\
C_{11} &=& C_{33} \nonumber \\
C_{44} &=& C_{66} \nonumber \\
C_{13} &=& C_{12} \, ,
\label{eq:isot1}
\end{eqnarray}
and examining the structure of the response function
$S_{ZZ}(q,\omega)$ for
fixed $q$.
Results for the peak frequencies and full-widths at half-maximum for
the damped peaks of $S_{ZZ} (q,\omega)$ are shown
for a model of Xe/Ag(111) in Figure 1.
For Figure 1, we extended slightly the original calculation of Hall
et al. [\ref{ref:HMB},\ref{ref:fac15}] using
the parameters $\omega_{0 \perp} = 2.8$ meV (0.67$_6$ THz) and
$\rho = 10.635$ gm/cm$^3$ and omitting adatom -- adatom interactions
($\psi =0$). The effective elastic
constants C$_{11} = 17.7$ and C$_{66} = 2.86$
(10$^{11}$ dyn/cm$^2$) were fitted to the calculated speeds of longitudinal
and transverse sound for the Ag(111) surface [\ref{ref:HTW}].
Qualitatively [\ref{ref:HMB}, \ref{ref:HM89}], the peak frequencies
follow trajectories characteristic of an avoided level crossing of the
substrate Rayleigh wave and the $\omega_{\perp}$ adlayer
mode at $q \approx 0.3$ {\AA}$^{-1}$.
For $q < \omega_{\perp} /c_{TA}$, the $\omega_{\perp}-$mode is damped
and there is a sharp resonance at a
frequency somewhat reduced (the avoided crossing phenomenon) from that of the
bare Rayleigh wave. At $q \approx \omega_{\perp} /c_{LA}$, near
0.1 {\AA}$^{-1}$,
there is an additional contribution to the damping and a perturbation
to the peak frequency derived from $\omega_{\perp}$, a
phenomenon that has been termed a van Hove anomaly [\ref{ref:Zepp90}].
The branch which is the Rayleigh mode at small $q$
approaches $\omega_{0\perp}$ at large $q$, but is still 7.5\% below
that limit at 0.4 {\AA}$^{-1}$.
A novel feature occurs for the present choice of parameters:
there is only one sharp resonance at small $q$, but at
sufficiently large $q$ there are two sharp resonances. One is derived from
the Rayleigh mode and one from $\omega_{\perp}$. The
second sharp resonance arises because the upper `repelled' frequency
lies between the bare substrate Rayleigh frequency $c_R \, q$ and the
continuum of substrate frequencies that begins at $c_{TA} \, q$.
There is a 6\% difference between $c_R$ and $c_{TA}$ in this model.
That there is a signature of the substrate Rayleigh wave at wave numbers
both above and below the avoided crossing has been considered a notable
phenomenon in helium atom scattering from adsorbed
monolayers [\ref{ref:water}]. We do not find the
corresponding effect in the calculations for adsorbates on graphite,
Sec.VI, apparently because there the increment between the Rayleigh
frequency and the bulk continuum is quite small.
\subsection{Small q-limit}
In the $q \to 0$ limit, the results of Sec.IV have simple
explicit forms. The coefficients $g_{zx}$, $g_{xz}$, $b_{12}$, and
$b_{21}$ vanish, so that the
$w_x$, $w_y$, and $w_z$ motions are decoupled. The remaining
Green's function components become (for $\omega > 0$)
\begin{eqnarray}
g_{xx} &=& -1/[C_{44} \alpha_2]
= - \imath/ [\rho c_{TA} \omega] \nonumber \\
g_{zz} &=& -1/[C_{33} \alpha_1]
= - \imath /[\rho c_{LAz} \omega] \nonumber \\
g_{yy} &=& g_{xx} \, .
\label{eq:smq1}
\end{eqnarray}
Then, defining
\begin{eqnarray}
\Gamma_x &=& \lambda_x /[\rho c_{TA} ] \nonumber \\
\Gamma_z &=& \lambda_z / [\rho c_{LAz}] \, .
\label{eq:smq2}
\end{eqnarray}
the spectral functions are
\begin{eqnarray}
S_{XX}(0, \omega) &=& { {(\omega^2 + \Gamma_x^2)^2} \over
{\omega^2 (\omega^2 + \Gamma_x^2 - \omega_{0\parallel}^2)^2 +
\omega_{0\parallel}^4 \Gamma_x^2}} \nonumber \\
S_{ZZ}(0, \omega) &=& { {(\omega^2 + \Gamma_z^2)^2} \over
{\omega^2 (\omega^2 + \Gamma_z^2 - \omega_{0\perp}^2)^2 +
\omega_{0\perp}^4 \Gamma_z^2}} \, ,
\label{eq:smq3}
\end{eqnarray}
and $S_{YY}(0, \omega) = S_{XX}(0, \omega)$.
Approximate expressions for the full-widths at half-maximum
for $S_{XX}$ and $S_{ZZ}$, respectively, are
\begin{eqnarray}
\delta \omega_x &\simeq& \Gamma_x \nonumber \\
\delta \omega_z &\simeq& \Gamma_z \, .
\label{eq:smq5}
\end{eqnarray}
Eqs.(\ref{eq:smq2}) show that the damping
is enhanced for lower density substrates if the other parameters remain
similar. This indeed is the trend found in comparing the damping of the
$\omega_{\perp}-$modes of Xe/Ag(111) and Xe/graphite, see Sec.VI.
Eqs.(\ref{eq:smq5}) are accurate for weak damping; the
results reported in Sec.VI are obtained with the full formalism of
Sec.IV and include self-consistent solutions for cases with strong damping.
Using the $N_2$/graphite parameters in Table I,
we find $\Gamma_x /\omega_{\parallel} \simeq 0.25$ and
an estimate of 3 $ps$ for the decay time. This supports the
assertion in Sec.I that the radiative damping mechanism is the
dominant process determining the lifetime of the zone-center mode.
\section{Commensurate monolayers on graphite}
We present applications of the elastic substrate theory of radiative
damping to commensurate monolayers of Xe/graphite and Kr/graphite
and also compare to the inert gas /
graphite slab frequency spectra calculated by
DeWette et al.[\ref{ref:RAWX},\ref{ref:RAWK}].
Although the lateral interactions are the same as in that work, we
have adjusted the frequencies $\omega_{0\parallel}$ and $\omega_{0\perp}$
to incorporate more recent information, so that there are quantitative
differences which arise from differences in the interaction models.
Figure 2 shows the results for the Kr/graphite $\sqrt{3}-$commensurate
monolayer and Figure 3 shows the results for the corresponding Xe/graphite
case. The direction of $q$ is along the $\Gamma K$ axis of the
adlayer Brillouin zone.
The $\omega_{0\perp}-$frequency is marked as a solid horizontal line in
both graphs and dotted and dot-dash lines denote the thresholds of
bulk graphite continua based on the SP$_{\parallel}$ and TA$_{\perp}$
modes, respectively. Lateral interactions, with the parameters of
Table I,
are included following the discussion of Secs.II.B and IV.A.
The plotted points are the derived peak frequencies of $S_{XX}$ and
$S_{ZZ}$, as noted, with widths of damped peaks indicated by error bars.
Cases where an error bar coincides with a substrate threshold denote
a local minimum for the response function, without a full
decrease to half the peak height.
Experiments with Helium Atomic Scattering ($HAS$) [\ref{ref:TV89},
\ref{ref:CJD92}] indicate there is a strong damping of the
$\omega_{\perp}-$mode at small $q$ and show a strong perturbation for
$q \approx 0.25 - 0.3$ {\AA}$^{-1}$ where the TA$_{\perp}-$mode of the
bare graphite crosses $\omega_{0 \perp}$. The
extrapolated crossing using the initial slope of the TA$_{\perp}-$mode
is $q \sim 0.4-0.5$ {\AA}$^{-1}$. However, including the strong anomalous
dispersion of the TA$_{\perp}-$branch with the prescription in
Eq.(\ref{eq:KKc}), leads to a semi-quantitative
account of the crossing.
Second, we
examine the radiative damping for the in-plane zone-center gap
$\omega_{0 \parallel}$. The region of strong damping for peaks
of $S_{XX}(q, \omega)$ is
confined to region I defined in Sec.IV.B.1, i.e., to the
left of the substrate SP$_{\parallel}$ threshold shown in the
Figures. In region II, between the SP$_{\parallel}$ and TA$_{\perp}$
thresholds, the widths of the peaks in $S_{XX}$ are small but finite and
correspond to lifetimes on the scale of $ns$.
The large elastic anisotropy
of the graphite makes region I much smaller than for the
isotropic elastic medium: using the parameters for Ag(111) in Sec.V.B
the ratio $c_{TA}/c_{LA}$ is 0.38, but for graphite it is less than 0.1 for
$q < 0.3$ {\AA}$^{-1}$. Another
manifestation of the large elastic anisotropy is that the elliptical
polarization of the Rayleigh wave nearly degenerates to a transverse
($z$) polarization, with only weak coupling to in-plane motions of the
adlayer.
Third, the damping of the $\omega_{\perp}$ branch is very strong
in both regions I and II. In contrast to the model for Xe/Ag(111),
Figure 1, we obtain only one sharp resonance (denoted by $+$)
in $S_{ZZ}$ for region
III. The Figures show a large shift of the resonant
frequency in region III relative to the value for the static substrate
used as input to the calculation.
Finally, we compare the resonant frequencies themselves
with the atomistic calculations of DeWette and coworkers [\ref{ref:RAWX},
\ref{ref:RAWK}] at large $q$ for a test of the size of
effects of the neglected spatial dispersion. The
most significant discrepancy is
that `deflection' of the trajectories in the region of the avoided level
crossing is much larger for the continuum calculation than in the
atomistic calculation.
In the elastic continuum theory, the
shift remains on the order of 10\% to the largest $q$ of the
calculation; this is a 50\% larger shift than in the atomistic calculations.
The dispersion with $q$ of the peak `$\omega_x$' ($x$ and $\circ$)
of $S_{XX}$, from
effects of adatom--adatom interactions, is similar to that in
the atomistic calculations. The present
calculations show the $\omega_x$ branch crossing the TA$_{\perp}$
branch (actually, the Rayleigh mode), as do the atomistic calculations.
The main previous test of the elastic continuum theory of radiative
damping was for the damping of incommensurate
inert gas adlayers on Pt(111) [\ref{ref:HM89}], where
the formalism
tended to underestimate the zone-center damping. The factor of
ten in the mass density of the substrate between graphite and platinum
has the effect of making the damping much larger
for Xe/graphite. This was anticipated by Toennies
and Vollmer [\ref{ref:TV89}] in their discussion of the rather broad peaks
for the $\omega_{\perp}-$mode in the $HAS$ inelastic
scattering experiments.
\section{Prospects}
These calculations show that the radiative damping mechanism proposed
by Hall et al. has a major effect on line-widths which may be observed
in inelastic scattering experiments from commensurate adlayers on graphite.
The understanding of the coupling of the commensurate layer to the
substrate is more advanced for substrates such as graphite than for
metallic substrates. Thus, adsorbates on graphite
may be good subjects for detailed further study.
It would be of interest to determine whether there are related
effects of substrate dynamics on the monolayer fluid which would disrupt the
{\it long-time} tails seen in molecular dynamics simulations of the
N$_2$/graphite fluid for 1 to 10 ps.
Another question is how to relate the size of the avoided level
crossing of an adlayer mode and the substrate Rayleigh mode
to adlayer--substrate coupling constants. Comparison of our
elastic continuum results to the model calculations of DeWette and
co-workers indicates that there are significant effects of
spatial dispersion to be included. This might be explored in
future work based on a technique such as lattice Green's
functions[\ref{ref:HM89}],
now that the elastic continuum theory is in place. For the damping at
intermediate and large wave numbers, where the radiative damping
mechanism becomes small, a treatment of the more conventional
anharmonic damping will be needed [\ref{ref:HM89}].
The large differences in the damping of parallel and perpendicular
motions for the commensurate monolayer on graphite seem well-based
and may have ironic consequences. The $HAS$ experiments for such monolayers
could have more prominent inelastic peaks for
the parallel than for the perpendicular motions, in spite of
the role of polarization considerations in the coupling to
the helium atom to the adlayer.
\section*{Acknowledgments}
This work has been partially supported by the National Science
Foundation under Grant No. DMR-9423307 (LWB) and by The Danish
Natural Science Foundation (FYH).
L. W. B. thanks the Fysisk-Kemisk Institut and the Technical University
of Denmark for hospitality during the period this work was begun.
We thank for Professor C. J. Goebel and
Professor H. Taub for several helpful comments and suggestions.
|
proofpile-arXiv_065-637
|
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\pubnumber{ DTP 96-37}
\pubdate{September 1996}
\title{Classical backgrounds and scattering for affine Toda theory on a half-line}
\author{
P. BOWCOCK\thanks{ Email \tt
[email protected]}
\address{Dept. of Mathematical Sciences,
University of Durham,
Durham, DH1 3LE, U.K.}
}
\abstract{We find classical solutions to the simply-laced affine Toda equations which satisfy integrable boundary conditions using solitons which are
analytically continued from imaginary coupling theories. Both static
`vacuum' configurations and time-dependent perturbations about them
which correspond respectively to classical vacua and particle
scattering solutions are considered.
A large class of classical scattering matrices are calculated and
found to satisfy the reflection bootstrap equation. }
\maketitle
\section{Introduction}
Integrable models in two-dimensions possess remarkable features. In particular
the S-matrix for such a theory factorises into products of
two-particle scattering matrices $S^{ab}_{cd}(\theta)$ (which gives the matrix
element for the process $a+b \to c+d$) \cite{ZZ}. Under reasonable physical assumptions,
it is possible to show that $S$ must satisfy a number of rather simple
algebraic equations. Solutions have been found to these equations for all
affine Toda theories and these
are postulated to give a non-perturbative expression for the scattering
matrix of the theory \cite{BCDS,DGZ,CM}.
These ideas have been extended to include
integrable theories on a half-line \cite{C,FK1,GZ}.
In this case it is necessary to introduce
another matrix $K$ which describes the reflection of particles off the boundary.
Again it is generally believed that all scattering amplitudes factorise
into products of $S$ and $K$. In addition the algebraic equations which were
solved by $S$ alone previously can be modified to include $K$. Recently
a number of solutions to these equations for $S$ and $K$ have been found
in the context of affine Toda theories \cite{CDR,CDRS,FK1,FK2,Ki,S}.
However, many of these solutions make no reference to the
boundary conditions which presumably need to be imposed to make sense of
the theory. (The exceptions to this are \cite{Ki} whose perturbative analysis
is tied to Neumann boundary conditions and \cite{CDR,CDRS}).
In fact, integrability places
severe constraints on the boundary conditions one can impose; for
simply-laced affine Toda theories there are only a finite number of
possibilities \cite{CDR,CDRS,BCDR}.
Fortunately, there seems to be a straightforward way of analysing which
solutions for $S$ and $K$ correspond to which boundary conditions . Whilst
$S$ is known to tend to unity in the classical limit, it seems that in
general the reflection factor $K$ does not. Thus if we know the
classical reflection factors for a particular boundary condition,
it should be possible to identify the corresponding quantum reflection
factor by considering its classical limit \cite{CDR}.
The aim of this paper is to make some progress towards calculating
classical scattering for Toda theories on a half-line. This problem
naturally divides into two parts.
{}First it is necessary to find the `vacuum configuration' or lowest
energy solution (which is presumably static) which satisfies a particular
boundary condition. Then one solves the linear equations for infinitesimal
perturbations about this solution still ensuring that the boundary
conditions are satisfied. Far from the wall the solution consists
of a superposition of incoming and outgoing waves. The relative phase
between the waves is interpreted as the classical limit of the reflection
factor $K$. Some examples of calculations along these lines are to be found
in \cite{CDR} where particular cases within $a_2^{(1)},d_5^{(1)}$ and
$a_1^{(1)}$ were considered.
The paper will be divided into five sections. In the next section we review
affine Toda theories, and in particular the boundary conditions which can
be imposed which are consistent with the (classical) integrability of the
theory \cite{BCDR,BCR}.
It is then shown that the boundary conditions combine in a
particularly neat way with the equations of motion to give linear
equations that the tau functions must satisfy at the boundary.
This is the key result of the paper which enables us to solve for the
classical scattering solutions.
In the third section we consider static solutions to the boundary conditions
for the special case of $a_n^{(1)}$ Toda theories.
For these theories a large number of tau functions can be constructed
explicitly by analytically continuing the multi-soliton solutions of the
imaginary coupling theory \cite{H}.
Whilst it can be shown that these solutions are
singular on the whole line, it is possible that all the singularities can
be placed `behind' the boundary for a theory on the half-line. By substituting
this family of tau-functions into the equation derived in the previous section
we obtain a large number of possible solutions. The correspondence with
particular boundary conditions is discussed.
In section four, whilst remaining within the context of $a_n^{(1)}$ Toda
theories, the analysis is extended
to cope with the scattering perturbations
around vacuum solutions. In this way we are able to derive an expression for
the classical reflection matrix. It is shown that this satisfies the
classical reflection bootstrap equation.
We conclude with comments on our results and directions for future research.
\section{Classical affine Toda theory on a half-line}
In this section we briefly review some of the features of affine Toda theory
on a half line that we shall need later on to establish notation.
To each affine Kac-Moody algebra $\hat g$ we can associate an affine Toda
theory \cite{MOP}. For simplicity in this paper we shall restrict ourselves to
simply-laced algebras, and for the most part to $a_n^{(1)}$ which are in some
senses the simplest cases of all.
The equations of motion for the affine Toda theory associated to $\hat g$ are
given by
\begin{equation}
\partial_{\mu} \partial^{\mu}\phi+{m\over \beta}\sum_{i=0}^r n_i \alpha^i
e^{\beta \alpha^i \cdot \phi}=0.
\label{eqm}
\end{equation}
Here we have used the notation that $\alpha^i$ for $i=1$ to $r$ are the
simple roots of $g$, the finite Lie algebra associated to $\hat g$, and
$\alpha^0=-\psi$ where $\psi$ is the highest root of $g$
(for untwisted algebras). Also we have
defined $r$ to be the rank of $g$ and introduced $m$ the mass parameter, and
$\beta$ the coupling constant of the theory. In this paper we shall consider
theories for which $\beta$ is real, and henceforth we set $\beta=m=1$.
The constants $n_i$ are given by the equation
\begin{equation}
\alpha_0+\sum_{i=1}^r n_i \alpha^i=0
\end{equation}
and are sometimes referred to as `marks'.
When the classical theory is considered
on the half line $-\infty\le x\leq 0$, we need to supplement the equations
of motion by boundary conditions. In this paper we shall restrict ourselves
to conditions of the form
\begin{equation}
\partial_x \phi=F(\phi)
\end{equation}
although more general possibilities can be considered \cite{NW,BCR}.
An arbitrary choice of $F(\phi)$ will generically break the integrability of
the system. It turns out that for
simply-laced theories (with the exception of sinh-Gordon or $A^{(1)}_1$
affine Toda theory \cite{M,T})
there are only a finite number of choices for $F$ which preserve
integrability \cite{CDR,CDRS,BCDR}.
These can be summarised as follows:
\noindent
(1) Either
\begin{equation}
\partial_x \phi= 0;
\label{eq.Neumann}
\end{equation}
these are `free' or Neumann boundary conditions
\noindent
(2) Or
\begin{equation}
\partial_x \phi= \sum_{i=0}^r A_i \alpha^i \sqrt{ n_i} e^{\alpha^i \cdot \phi/2}
\label{eq.non.Neumann}
\end{equation}
where the $A_i=\pm 1$.
The first of these two possibilities is fairly easy to analyse completely at
a classical level. The classical energy functional on the half-line is
\begin{equation}
E=\int_{-\infty}^0 dx \left ({1\over 2}(\partial_x \phi)^2+
{1\over 2}(\partial_t \phi)^2+\sum_{i=0}^r n_i (e^{\alpha^i \cdot \phi} -1)
\right ).
\label{eq.bulk.energy}
\end{equation}
This is non-negative and the lowest energy configuration is clearly $\phi=0$
which also satisfies the Neumann boundary conditions. If we now take the field
$\phi(x,t)=\epsilon(x,t)$ to be some infinitesimal perturbation to this
vacuum configuration we see that in a linear approximation we have
\begin{equation}
\partial_{\mu} \partial^{\mu} \epsilon +M \epsilon =0
\label{eq.wave}
\end{equation}
where $M=\sum_{i=0}^r n_i \alpha^i \otimes \alpha^i$ is the mass matrix. The
eigenvectors $\rho_a$, $a=1$ to $r$, of $M$ are in one-to-one
correspondence with the
fundamental representations of the Lie algebra $g$ and are interpreted as
the basic particle-like excitations of the theory. The eigenvalue of $\rho_a$
is given by $\lambda_a=m_a^2$
where $m_a$ is the mass of the
corresponding particle. It is a remarkable fact that the set of masses
form the lowest eigenvalue eigenvector of the Cartan
matrix of $g$ \cite{BCDS,F,FLO}.
In terms of $\rho$ the basic solutions of \ref{eq.wave} are given by
\begin{equation}
\phi(x,t)=\epsilon(x,t)=\rho_a e^{iEt}(e^{ipx}+K_a e^{-ipx})
\label{eq.sol}
\end{equation}
where $E^2-p^2=m_a^2$. This is a superposition of left- and right-moving
waves. The classical reflection factor is given by the phase factor $K_a$
which can be determined by substituting the solution (\ref{eq.sol})
into the boundary conditions (\ref{eq.Neumann}). This yields the solution
$K_a=1$. Classically, at any rate, this boundary condition seems rather
uninteresting. There has been considerable progress in understanding
the quantum case in \cite{Ki}, where the semi-classical reflection
factor has been calculated using perturbative techniques.
In this paper we shall be interested in
boundary conditions of the form (\ref{eq.non.Neumann}). It turns out to
be particularly convenient to analyse this case in the language of {\it
tau-functions} \cite{H}.
These are introduced as a particular parametrisation of
the function $\phi$;
\begin{equation}
\phi=-\sum_{i=0}^r \alpha^i ln(\tau_i)
\label{eq.def.tau}
\end{equation}
Note that we have introduced $r+1$ functions $\tau_i$ to describe the $r$-component
field $\phi$ and this is reflected in the freedom to scale
all the tau-functions $\tau_i\to f(x,t)\tau_i$
without affecting the value of $\phi$. This unphysical degree of freedom is
fixed by demanding that $\tau_i$ satisfies the following particular
form of the equations of motion
\begin{equation}
\ddot{\tau_i} \tau_i -\dot{\tau_i}^2-\tau_i''\tau_i +(\tau_i')^2
=\left (\prod_{j=0}^r \tau_j^{I_{ij}}-\tau_i^2\right )n_i,
\label{eq.tau.eqm}
\end{equation}
where $I_{ij}$ is the incidence matrix of $g$ given by
\begin{equation}
I_{ij}=2\delta_{ij}-\alpha^i \cdot \alpha^j
\end{equation}
Essentially, $I_{ij}$ takes the value one if the nodes corresponding to $i$ and
$j$ are connected on the extended Dynkin diagram, and vanishes otherwise.
Substituting the form (\ref{eq.def.tau}) into the boundary conditions
(\ref{eq.non.Neumann}), and taking the inner product with the fundamental
weight $\lambda_i$, we discover that the boundary conditions can be
written in terms of the tau-functions as
\begin{equation}
{\tau_i' \over \tau_i}+\sqrt{n_i} A_i e^{\alpha^i \cdot \phi /2}=
n_i C
\end{equation}
where
\begin{equation}
C=({\tau_0'\over \tau_0} +A_0 e^{\alpha^0 \cdot \phi /2}).
\end{equation}
From this we see that
\begin{equation}
n_i \prod_j \tau_j^{I_{ij}}=\tau_i^2 (n_i C- {\tau_i' \over \tau_i})^2
\end{equation}
Substituting this into the equations of motion (\ref{eq.tau.eqm}) we find
\begin{equation}
\ddot{\tau}_i-{(\dot{\tau_i})^2 \over \tau_i}-\tau_i''+2 n_i C \tau_i'
-(n_i^2 C^2 -n_i)\tau_i =0.
\label{eq.master}
\end{equation}
This equation is the main result of this section. Immediately we see that
it has two attractive features. Firstly the equations for $\tau_i$ have
decoupled so that each equation only involves $\tau_i$ for some $i$.
Secondly, if we assume that the solution is static then the first
two terms vanish and we are left with the linear equation
\begin{equation}
\tau_i''-2 n_i C \tau_i' +(n_i^2 C^2-n_i)\tau_i=0
\label{eq.static.master}
\end{equation}
At this point we perhaps should remind the reader that this equation
is deduced from the equations of motion and the boundary conditions, and
that the latter are only valid at the boundary $x=0$. Thus the above
equation cannot be solved as a differential equation in $x$ since it is
only valid at the boundary, but it will prove a valuable tool in
determining the subset of solutions of the equations of motion which
satisfy one of the boundary conditions (\ref{eq.non.Neumann}). One of
the drawbacks of the equation (\ref{eq.static.master}) is that we
have essentially squared the boundary conditions, so that we have lost the
information contained in the $A_i$. This we will have to recover by
explicitly examining the proposed solutions at the boundary
Perhaps even more surprisingly we can use a similar equation in the
case of time-dependent perturbations to a static vacuum solution, i.e. in
the situation where
\begin{equation}
\tau_i(x,t)=(\tau_i(x))_{vac}+\epsilon_i(x,t)
\end{equation}
where $\epsilon_i(x,t)$ is some infinitesimal perturbation to the vacuum
solution $(\tau_i)_{vac}$. In this case, the important point is that
$\dot{\tau}_i=O(\epsilon)$, so discarding terms of $O(\epsilon^2)$ in
(\ref{eq.master}) we again arrive at the linear equation
\begin{equation}
\ddot{\tau}_i-\tau_i''+2 n_i C \tau_i'
-(n_i^2 C^2 -n_i)\tau_i =0
\label{eq.td.master}
\end{equation}
We shall use this equation in section four to solve for classical scattering
about the vacuum.
One may be concerned that one has introduced a spurious constant $C$ in
equation (\ref{eq.static.master}). Remarkably $C$ has a
physical interpretation as proportional to the
energy of the solution corresponding to
$\tau_i(x,t)$ on the half-line. The energy on the half-line for boundary
conditions of type (\ref{eq.non.Neumann}) is given by
\begin{equation}
E=E_{bulk}-\sum_{i=0}^r 2 A_i \sqrt{n_i} e^{\alpha^i \cdot \phi/2}|_{x=0}
\end{equation}
where $E_{bulk}$ is the bulk energy given by (\ref{eq.bulk.energy}) and
the second term is the contribution to the energy from the boundary
compatible with the boundary conditions (\ref{eq.non.Neumann}).
It has been shown in \cite{OTU} that for soliton solutions
the energy density is a total derivative, so that the bulk energy can
be written as a boundary contribution which in terms of tau-functions is
\begin{equation}
E_{bulk}=\left [ -2 \sum_{i=0}^r {\tau_i'\over \tau_i} \right ]^0_{x=-\infty}.
\label{eq.der.energy}
\end{equation}
In the sections that follow we shall be using tau-functions that tend
to a constant as $x\to -\infty$ so the contribution to (\ref{eq.der.energy})
come only from $x=0$. Adding the contribution from the bulk and the boundary
we find that
\begin{eqnarray}
E&=&-2\sum_{i=0}^r \left ( {\tau_i' \over \tau_i} + A_i \sqrt{n_i} e^{\alpha^i
\cdot \phi /2} \right )\\
&=&-2 \sum_{i=0}^r n_i C= -2 h C
\label{eq.total.energy}
\end{eqnarray}
where $h$ is the Coxeter number associated to $g$. Thus, although
initially it may have seemed that the appearance of $C$ in the equation (\ref{eq.static.master}) was a drawback, it turns out that it is an added
bonus, giving the energy of the solution that we are
considering. This is important in trying to determine
the `vacuum' for a given boundary condition, since the vacuum
is defined to be the static solution
with the lowest energy which satisfies the boundary condition.
\section{Static vacuum configurations for $a_n^{(1)}$ Toda theories from
analytically continued solitons}
In this section we shall confine ourselves to affine Toda theories
based on the algebra $a_n^{(1)}$. We have seen in the previous section that
the requirement that a solution satisfies one of the integrable boundary
conditions of the form (\ref{eq.non.Neumann}) can be neatly expressed in
terms of tau-functions. For the $a_n^{(1)}$ we have a particularly
rich source of such tau-functions which can be constructed by analytically
continuing multi-soliton solutions of the imaginary coupling theory \cite{H}.
At this point let us briefly review the nature of these solutions in the
imaginary coupling theory.
Let us reintroduce the coupling constant, so that we write
\begin{equation}
\beta \phi=-\sum_{i=0}^r \alpha^i ln(\tau_i)
\label{eq.def.tau2}
\end{equation}
The tau-functions for an $N$-soliton solutions can be compactly written as
\begin{equation}
\tau_j(x,t)=\sum_{\mu_1=0}^1\dots\sum_{\mu_N=0}^1 {\rm exp}
\left ( \sum_{p=1}^N \mu_p \omega^{a_p j} \Phi_p +\sum_{1\leq p\le q\leq N}
\mu_p\mu_q ln A^{(a_p a_q)}\right ).
\label{eq.multi.sol}
\end{equation}
Here we have introduced the notation $\Phi_p=\sigma_p(x-v_p t)+\xi_p$
where $v_p$ is the velocity of the $p$-th soliton, $\xi_p$ is a complex
parameter whose real part and imaginary part are related respectively to
the position and the topological charge of the $p$-th soliton. Also
$\sigma_p$ and $v_p$ are related by the mass-shell condition
\begin{equation}
\sigma_p^2(1-v_p^2)=m_{a_p}^2
\label{eq.mass.shell}
\end{equation}
where $a_p$ labels the species of soliton and $m_{a_p}=2\sin({{\pi a_p}\over {n+1}})$. We define $\omega=exp(2\pi i /(n+1))$.
The variables $\sigma_p$ and $v_p$ are often conveniently
parametrised in terms of the two-dimensional rapidity $\theta_p$ by putting
\begin{eqnarray}
\sigma_p &=& m_{a_p} \cosh(\theta_p)\\
\sigma_p v_p &=& m_{a_p} \sinh(\theta_p)
\label{eq.rap.one}
\end{eqnarray}
The constants $A^{(a_p a_q)}$ describe the interaction between the $p$-th and
$q$-th solitons and are given as
\begin{eqnarray}
A^{(a_p a_q)}(\Theta)
&=&-{{(\sigma_p-\sigma_q)^2-(\sigma_p v_p-\sigma_q v_q)^2-4 \sin^2
{\pi \over {n+1}}(a_p-a_q)}\over {(\sigma_p+\sigma_q)^2-
(\sigma_p v_p+\sigma_q v_q)^2-4 \sin^2
{\pi \over {n+1}}(a_p+a_q)}}\\
&=& {{\sin({\Theta \over 2i}+{\pi(a_p-a_q)\over {2(n+1)}})
\sin({\Theta \over 2i}-{\pi(a_p-a_q)\over {2(n+1)}})}\over
{\sin({\Theta \over 2i}+{\pi(a_p+a_q)\over {2(n+1)}})
\sin({\Theta \over 2i}-{\pi(a_p+a_q)\over {2(n+1)}})}}
\label{eq.inter.def}
\end{eqnarray}
where $\Theta=\theta_p-\theta_q$. In this section we shall only be
interested in static solitons so we shall take $\theta_p=0$ or equivalently
$v_p=0$, $\sigma_p=m_{a_p}$.
For single solitons the expression (\ref{eq.multi.sol})
reduces to
\begin{equation}
\tau_j=1+\omega^{j a} e^{\Phi_p}
\end{equation}
The topological charge of such a soliton of species $a$ which is defined
by
\begin{equation}
\lim_{x\to \infty} (\phi(x,t)-\phi(-x,t))
\end{equation}
can be shown to lie in the fundamental representation with highest
weight $\lambda_{a}$ where $\lambda_a\cdot \alpha^b= \delta_{a}^b$.
It is therefore natural to associate each species of soliton with nodes on the
Dynkin diagram of $a_n$. By analogy with the representation theory of
$a_n$ we refer to solitons of type $a$ and type $\bar{a}=n+1-a$ as conjugate.
We shall (ab)use the notation $\bar{p}$ to denote a soliton whose type is
congugate to that of soliton $p$.
Now let us specialise to the case of interest, namely static solitons.
Examining the expression (\ref{eq.multi.sol}) carefully
we see that solitons obey a kind of Pauli-exclusion principle. We can
only construct multi-soliton solutions whose constituent solitons have
either different velocity and/or different species. If we attempt to
consider two solitons of the same species and velocity, the interaction
constant $A^{(aa)}(0)$ vanishes and we simply end
up with one constituent soliton of that velocity and species at some
different position. This places severe constraints on the possibilities
for static soliton configurations; since the velocities of each constituent
is identically zero, it follows that each constituent
soliton must be of different species. Thus we can consider at most
an $n$-soliton solution where each constituent soliton corresponds to
a different node on the Dynkin diagram of $a_n$. Indeed we believe that
this is the most general static solution whose energy-density tends to zero
at spatial infinity.
The energy of a stationary soliton of type $a$ on the whole line
is simply given by
\begin{equation}
E=-{2\over \beta^2}(n+1) m_a
\label{eq.single.energy}
\end{equation}
Note that for imaginary $\beta$ this is positive, as we might expect. The
energy of $N$ stationary solitons is simply given as the sum of energies
of the constituent solitons $E=\sum_p E_p$.
The above discussion has all been in the context of imaginary coupling
Toda theory, where it is accepted that the field $\phi$ is allowed to
be complex. We are interested in real coupling Toda theory where
the field is $\phi$ is taken to be real. A little thought shows that for
tau-functions of the type given in (\ref{eq.multi.sol}), reality of $\phi$
implies the reality of $\tau_j$. This can be ensured by insisting that
each constituent soliton $p$ is paired with a congugate soliton $\bar{p}$
with the position/topological charge variables related by
$\xi_p=(\xi_{\bar{p}})^*$. An exception to this rule occurs for $n$ odd,
where the soliton corresponding to the middle node of the dynkin
diagram, i.e. $a_p=(n+1)/2$, is unpaired (since it is its own conjugate),
so we must take$\xi_p$ to be real in this case. Thus the reality
of the tau-functions closely corresponds to the representation theory
of $a_n$, where only the middle node corresponds to a real representation
and the other fundamental representations must be taken in conjugate
pairs if we wish to restrict ourselves to real representations. Similar
remarks apply equally well to tau-functions associated with other
simply-laced algebras
Finally, let us note that the energy $E$ for such solutions on the whole
line given in (\ref{eq.single.energy}) is negative for real $\beta$. This
may seem surprising in view of the manifestly positive energy density of the
theory (\ref{eq.bulk.energy}). This reflects the fact that all such
solutions become singular somewhere on the real line, and this is perhaps
why they are not generally considered in the
real coupling theories. However, in the present context the possibility
exists that all the singularities in $\phi$ lie in the `unphysical' region
$x>0$, so that they are perfectly acceptable physically.
\subsection{Two-soliton solutions}
As a pedagogical introduction to a more general solution, we begin by
considering the possible static two-soliton solutions that satisfy the
boundary conditions (\ref{eq.non.Neumann}). This solution and some of its
features have already been discussed in \cite{FU}. A two-soliton solution
consisting of a soliton of type $a$ and its anti-soliton of type $\bar{a}$ have
tau-functions of the form
\begin{equation}
\tau_i=1+2 d \cos(\chi+{2\pi i a\over n+1})e^{m_a x}+A^{(a\bar{a})} d^2
e^{2 m_a x}
\label{eq.two.soliton}
\end{equation}
Here we have set $\chi= Im(\xi)$ and $d={\rm exp}(Re(\xi))$ in the
previous notation. Also
we calculate $A^{(a\bar{a})}$ from setting $\Theta$ to zero in
(\ref{eq.inter.def}) as
\begin{equation}
A^{(a\bar{a})}=\cos^2({\pi a\over n+1})=1-{m_a^2 \over 4}.
\label{eq.appbar}
\end{equation}
In general multi-soliton tau-functions (\ref{eq.multi.sol}) can be split up
into a sum of `charge' sectors by writing
\begin{equation}
\tau_j= \sum_k T_k \omega^{k j}.
\label{eq.charge.tau}
\end{equation}
The linearity of
equation (\ref{eq.static.master}) and the fact that for $a_n$, we have that
$n_i=1$ for all $i$ implies that
each of the $T^k$ separately satisfy the equation;
\begin{equation}
(T_k)''-2 C (T_k)'+(C^2-1)T_k=0\;\;\;{\rm at}\;\;x=0
\label{eq.mod.stat}
\end{equation}
In the two-soliton case there are three charge sectors:$k=0,k=\pm a$. Obviously $T_a=
\bar{T}_{-a}$, so there are essentially two independent equations;
\begin{eqnarray}
(C^2-1)+((C-2 m_a)^2-1)A^{(a\bar{a})}d^2&=&0\\
((C-m_a)^2-1)d &=&0
\end{eqnarray}
The second of these equations implies that for a non-trivial solution
\begin{equation}
C_{\pm}=m_a\pm 1,
\end{equation}
and, substituting this into the first we find that
\begin{equation}
d={2\over (2\mp m_a)}.
\end{equation}
The energy of the two solutions
are given by $E_{\pm}=-2 h C_{\pm}=-2(n+1)(m_p\pm 1)$. Actually, it is clear that solutions should come in pairs. The reason is
that if $\phi(x,t)$ is a solution to the equations of motion and the
boundary conditions, then so is the parity reversed solution $\phi(-x,t)$,
since it satisfies boundary conditions of the form (\ref{eq.non.Neumann}) but
with $A_i\to -A_i$. Note that if we sum the energies of the two solutions,
the boundary contributions of each solution cancel, and we are left with
a bulk contribution of $\phi(x,t)$ and $\phi(-x,t)$ from $-\infty<x<0$ which
can be rewritten as simply the bulk energy of $\phi(x,t)$ on the whole line,
and indeed we find that
\begin{equation}
E_+ +E_- =-2(n+1)m_a
\end{equation}
which is the energy of two static solitons of type $a$.
Before we pick the lower energy
solution as the vacuum we should be careful to consider whether the
singularities of the corresponding solution for $\phi$ lie in the physical
region $x<0$ or the unphysical region $x>0$. Since we have established that
the two solutions are parity conjugate, and we know that the solution is
singular at some point on the whole line, it follows that at least one
solution must be singular somewhere in the physical region. (It is possible
that the singularity lies at $x=0$ which is energetically allowed. We discuss
this case later). It turns out that the `good' solution whose singularities
lie in the region $x>0$ corresponds to the higher
energy solution $C_-$. For this solution the
tau-function can be written
\begin{equation}
\tau_j={{(2+m_a)+4\cos(\chi+{{2\pi j a}\over n+1})e^{m_a x}+(2-m_a)e^{2 m_a x}}
\over 2+m_a}
\label{eq.tau.two}
\end{equation}
Singularities in $\phi$ occur when $\tau_j$ vanishes for some $j$. Solving the
equation $\tau_j=0$ yields
\begin{equation}
e^{m_a x}={{-\cos(\chi+{{2\pi j a}\over n+1})\pm\sqrt{\cos^2(\chi+{{2\pi j a}\over n+1})-\cos^2({\pi a\over n+1})}}\over {1-\sin({\pi a\over n+1})}}
\label{eq.solv.sing}
\end{equation}
For a real solution for $x$ this implies that
\begin{eqnarray}
\cos^2(\chi+{{2\pi j a}\over n+1})&\geq&\cos^2({\pi a\over n+1})\\
\cos(\chi+{{2\pi j a}\over n+1})&\leq &0.
\end{eqnarray}
Combining these two conditions gives
\begin{equation}
-1\leq \cos(\chi+{{2\pi j a}\over n+1}) \leq -\cos({\pi a\over n+1})
\end{equation}
It is easy to check that for any choice of $\chi$ there will be some
$j$ for which this condition is satisfied, so that there will always be
a singularity at some real value of $x$ for some $j$.
To see where these singularities lie consider the right hand side of
(\ref{eq.solv.sing}) where the square-root is taken with the minus sign,
since this is the smaller of the two solutions, and hence the more likely
to yield negative values of $x$. A simple calculation shows that
as a function of $\cos(\chi+{{2\pi j a}\over n+1})$ the right hand side is
a monotonically increasing function, so its lowest value is attained when
$\cos(\chi+{{2\pi j a}\over n+1})=-1$. At this point the right hand side is
equal to one, corresponding to a singularity at $x=0$. For other values
of $\cos(\chi+{{2\pi j a}\over n+1})$, the right hand side will be greater than
one, and the singularities will be in the unphysical region $x>0$.
Having found a class of two soliton solutions which correspond to non-singular fields $\phi$, it only remains to ascertain precisely which boundary conditions
each solution satisfies. Substituting $x=0$ into the expression (\ref{eq.tau.two}), we find that the tau-function at this point can be
written as
\begin{equation}
\tau_j={\cos^2({\chi\over 2}+{\pi j a\over n+1}) \over {1+\sin({\pi a\over n+1})}}
\end{equation}
Thus we can write
\begin{equation}
e^{\alpha^j \cdot \phi} ={ {\cos^2({\chi\over 2}+{\pi (j+1) a)\over n+1})\cos^2({\chi\over 2}+{\pi (j-1) a \over n+1})}\over
\cos^4({\chi\over 2}+{\pi j a\over n+1})}
\end{equation}
In the boundary conditions (\ref{eq.non.Neumann}), we have terms involving
${\rm exp}(\alpha^j \cdot \phi/2)$ which should be interpreted as the
positive square root of the above quantity. The values of $A_i$ are
found to be given by
\begin{equation}
A_j=-{\rm sign}\left ({\cos({\chi\over 2}+{\pi (j+1) a)\over n+1})
\cos({\chi\over 2}+{\pi (j-1) a)\over n+1})}\right )
\label{eq.two.aj}
\end{equation}
\begin{figure}
\hspace{1.0cm}
\epsfxsize=12truecm
\epsfysize=12truecm
\epsfbox{bt1.eps}
\caption{ Boundary conditions for $N=1$, $n=8$ and $a=2$}
\end{figure}
In Figure 1. we have plotted $\cos({\chi\over 2}+{\pi j a\over n+1})$,
tabulated its sign, and tabulated the sign of the right hand side of
(\ref{eq.two.aj}). From this we can see that the $A_j$ is generically
negative except at near a point where $\cos({\chi\over 2}+{\pi j a\over n+1})$
has a zero, where a pair of positive $A_j$ are produced. The number of
positive pairs is related to the number of such zeroes which are in turn
related to the value of $a$. One also observes that the expression
(\ref{eq.two.aj}) is invariant under
\begin{eqnarray}
\chi &\to& \chi-{\pi a \over n+1}\\
j &\to& j+1
\end{eqnarray}
This implements the $Z_n$ symmetry of $A^{(1)}$ Toda theories, and
cyclically permutes the boundary conditions $A_j\to A_{j+1\;{\rm mod}\;n+1}$.
\subsection{Multi-soliton solutions}
In this section we shall generalise the two-soliton solution given above
to multi-soliton solutions with arbitrarily many constituent solitons.
Whilst this case is much more complicated, and we cannot guarantee finding
a complete set of solutions to the basic equation (\ref{eq.static.master}),
we shall give a wide class of explicit solutions.
To begin with we shall restrict our attention to multi-soliton solutions
with an even number of constituent solitons; that is, we shall assume that
the soliton corresponding to the middle node of the Dynkin diagram for $n$
odd does not feature. Thus our ansatz consists of $N$ pairs of conjugate
solitons which we shall label $p$ and $\bar{p}$ respectively.
The real tau-function can be written in the form
\begin{equation}
1+2 \sum_{p=1}^N d_p \cos(\chi_p+{2 \pi a_p \over n+1})e^{m_{a_p} x}+\ldots
\label{eq.tau.many}
\end{equation}
where the dots compactly represent the (many) interaction terms. As in the
two-soliton case we can decompose the tau-function into sectors as in
(\ref{eq.charge.tau}). As a further simplification we shall assume that $n$ is
large with respect to the charges of the solitons, or more particularly
that the largest charge $Q_{max}=\sum_p a_p \leq (n+1)/2$.
With this restriction, the charges in (\ref{eq.charge.tau}) take values between $Q_{max}$ and $-Q_{max}$, and only the term
\begin{equation}
\prod_{p=1}^N \left (d_p e^{i(\chi_p+{2 \pi a_p \over n+1})}\right )\left (\prod_{1\leq p <q\leq N}A^{(pq)} \right ){\rm exp}\left (\sum_{p=0}^N m_{a_p} x
\right )
\label{eq.largest.term}
\end{equation}
contributes to the charge $Q_{max}$. Substituting this into the equation
(\ref{eq.mod.stat}), we immediately find
\begin{equation}
C_\pm=\sum_{p=0}^N m_{a_p}\pm 1
\label{eq.many.energy}
\end{equation}
and the energy is as usual $-2(n+1)C_{\pm}$. Once more this pair of
solutions are the parity inverses of one another. To determine the `positions'
of the constituent solitons we consider the terms with charge $Q_p=Q_{max}-a_p$.
Again we make the assumption the charges are chosen in such a way that only
two terms in the $\tau_j$ contribute to $T_{Q_p}$: namely the
term containing all the solitons except the $p$-th, and the term containing
all the solitons and the $p$-th conjugate soliton. More explicitly we have
\begin{eqnarray}
T_{Q_p}&=&
\prod_{r\neq p} \left (D_r e^{i(\chi_r+{2 \pi a_r \over n+1})}\right )\left (\prod_{r <s:r,s\neq p}A^{(a_ra_s)} \right ){\rm exp}\left (\sum_{r\neq p} m_{a_r} x
\right )\nonumber\\*&&\mbox{}
\times \left ( 1+d_p^2 A^{p\bar{p}}\prod_{r\neq p} \left( A^{(rp)}A^{(r\bar{p})}
\right)e^{2 m_{a_p} x}\right )
\label{eq.nargest.term}
\end{eqnarray}
Substituting this into (\ref{eq.mod.stat}) we discover that
\begin{equation}
(d_p)^2 A^{p\bar{p}}\prod_{r\neq p} \left( A^{(rp)}A^{(r\bar{p})}
\right)={2\pm m_{a_p} \over 2\mp m_{a_p}}
\label{eq.temp1}
\end{equation}
One can prove that
\begin{equation}
A^{(rp)}A^{(r\bar{p})}={(m_{a_r}-m_{a_p})^2\over (m_{a_r}+m_{a_p})^2}
\end{equation}
so that
\begin{equation}
d_p=\prod_{r\neq p} {(m_{a_r}+m_{a_p})\over |m_{a_r}-m_{a_p}|}
{2\over {2\mp m_{a_p}}}
\label{eq.expression.D}
\end{equation}
Here we have made a choice for the sign of $d_p$, since we can absorb this
sign into an $i\pi $ shift in $\chi_p$.
Strictly speaking we have only given necessary conditions that some of
the charge
sectors in the tau-function satisfy (\ref{eq.mod.stat}), but extensive
numerical investigation supports the conjecture that (\ref{eq.many.energy})
and (\ref{eq.expression.D}) provide a solution to (\ref{eq.static.master}).
Moreover
whilst we enforced fairly stringent conditions on $a_p$ to derive the necessary
constraints on $d_p$ and $C$, many examples seem to confirm the idea
that these values give a solution even when the conditions are not met.
However, in those cases one generally might
expect other solutions not of the form (\ref{eq.many.energy}), (\ref{eq.expression.D}).
By analogy with the two-soliton case, we should proceed by
discussing the position of the singularites
in $\phi$. Since the solutions corresponding to $C_{\pm}$ are parity
conjugate, and we know from the negative energy that $\phi$ is singular
somewhere on the real line $-\infty<x<\infty$, we know that one of the
two solutions must be singular in the physical region $x<0$. Unfortunately,
finding the zeroes
of the tau-function in general is very difficult.
Numerical work suggests that it is more difficult
to find regular solutions as more solitons are introduced. As alluded to above,
solitons in the real coupling theory have negative energy, so that it seems
that the lowest energy configuration consists of adding in as many solitons
as possible without making the solution singular for $x<0$. One possibility
is that the best we can do is to
place the singularity at the wall. In this case the analysis leading
to the expression for the total energy (\ref{eq.total.energy}) is still
correct and yields a finite answer. Physically the infinities in the
bulk and boundary energies exactly cancel.
In parallel with the two-soliton case, one can ask
which boundary condition does the solution given above correspond to; that
is for what $A_i$ does the above solution satisfy the boundary condition.
Again the answer seems to be a lot more complicated than for
two-solitons, but nonetheless some of the features of that case do
seem to persist. For the two-soliton case it was seen that one could
write $\tau_j(x=0)$ in the form $L W_j^2$ where all the $j$-dependence is in
$W$. From this we deduced that
\begin{equation}
e^{\alpha^j\cdot \phi/2}|_{x=0}={|W_{j-1}W_{j+1}|\over W_j^2}
\end{equation}
and that
\begin{equation}
A_j=\pm{\rm sign}(W_{j-1}W_{j+1})
\label{eq.A.def}
\end{equation}
where the plus/minus sign corresponds to the two solutions $C_{\pm}$.
We postulate that we can write the tau-function (\ref{eq.tau.many})
evaluated at $x=0$ in the form $L W_j^2$ where we can expand $W$ as
\begin{equation}
W_j=\sum_{\sigma_1=1,\sigma_p=\pm1} c_{\sigma_1\sigma_2\ldots\sigma_N}
\cos(\sum_p \sigma_p \{{\chi_p\over 2}+{\pi a_p j\over n+1}\})
\label{eq.W.def}
\end{equation}
By comparing this with the tau-function with $d_p$ given in
(\ref{eq.expression.D}), we find
\begin{equation}
c_{\sigma_1\sigma_2\ldots\sigma_N}=\prod_{r<s} \sin({\pi\over n+1}
(|\sigma_r a_r-\sigma_s a_s|))
\label{eq.c.def}
\end{equation}
and
\begin{equation}
L=4\prod_r {2\over(2\mp m_{a_r})} \prod_{r<s} {4\over {(m_{a_r}-m_{a_s})^2}}.
\end{equation}
We have checked that this ansatz for $\tau_j(0)$ is true up to six solitons.
By combining (\ref{eq.c.def},\ref{eq.W.def},\ref{eq.A.def}) we can deduce
$A_j$ for any given values of $\chi_p$, and use this to deduce a `phase'
diagram for different boundary conditions as a function of the $\chi_p$.
One such example is given in Figure 2, where we have taken $n=8, N=2, a_1=1,
a_2=3$. In this figure we have plotted the lines in the parameter space
$(\chi_1,\chi_2)$ along which $\tau_j(0)=0$ for $j=0,..,8$.
\begin{figure}
\hspace{1.0cm}
\epsfxsize=12.5truecm
\epsfysize=10truecm
\epsfbox{bt2.eps}
\caption{ Boundary conditions for $N=2$, $n=8$, $a_1=1$,
$a_2=3$}
\end{figure}
In the region marked $I$ the boundary conditions are found to be
\begin{equation}
(A_0,A_1,..,A_8)=(-1,1,1,-1,-1,-1,-1,1,1).
\end{equation}
The boundary conditions for the other regions
can be deduced by noting that $W_j$ changes sign as the line $\tau_j=0$ is
crossed. So by (\ref{eq.A.def}), $A_{j-1}$ and $A_{j+1}$ change sign when this
line is crossed. One can check that the resulting boundary conditions are
compatible with the $Z_n$ symmetry which is realised as
\begin{eqnarray}
\chi_p &\to & \chi_p -{2\pi a_p\over (n+1)}\\
j &\to& j+1
\label{eq.zn.many}
\end{eqnarray}
Suppose that we choose $\chi_p$ so that we lie on one of the lines $\tau_j=0$.
Then the corresponding solution for $\phi$ is singular, and in fact
\begin{eqnarray}
{e}^{\alpha^j \cdot \phi}&=& +\infty,\\
{e}^{\alpha^{j+1} \cdot \phi}&=& 0,\\
{e}^{\alpha^{j-1} \cdot \phi}&=& 0.
\end{eqnarray}
Thus along this line the coefficients $A_{j-1}$ and $A_{j+1}$ are undetermined
as we may have expected. It follows that a particularly rich variety of
boundary conditions are allowed at the points where many such lines intersect.
If $n$ is odd we can consider static soliton solutions containing a single
constituent soliton of type $a_{N+1}=(n+1)/2$. Reality of the tau-function
constrains the associated parameter $\xi_{N+1}$ to be real, or equivalently
$\chi_p=0,\pi$. Following much the same argument as in the previous section
we initially assume that the $a_p=n+1-a_{\bar{p}}$, $p=1,..,N$ are chosen in
such a way that only one term contributes to the sector of maximal charge
$Q_{max}=\sum_{p=1}^N a_p +a_{N+1}$ and two terms contribute to sectors of
charge $Q_p=Q_{max}-a_p$, $p=1,..,N$. This results in similar expressions
for $C_{\pm}$ and $d_p$, $p=1,..,N$ except that the sums in (\ref{eq.many.energy}), (\ref{eq.expression.D}) run from $1,..,N+1$. It
only remains to determine the value of $d_{N+1}$. Only one term contributes
to the charge $Q_{N+1}$ sector since as there is no soliton conjugate to
the middle soliton $p=N+1$, we cannot add a soliton-conjugate
soliton pair to the term involving solitons of type $p=1,..,N$. Instead the
resulting equation is
\begin{equation}
0=(C_{\pm}-\sum_{p=1}^N m_{a_p})^2-1=(m_{a_{N+1}}\pm 1)^2-1
\end{equation}
Since $m_{a_{N+1}}=2\sin(\pi/2)=2$, we see that only the $C_-$ solution is
allowed. The value of $d_{N+1}$ is unconstrained by this or any other
equation, so we are free to place the middle soliton anywhere! Under
a parity transformation, the energy and the values of $d_p$, $p=1,..,N$
remain unchanged. This is in agreement with the idea that the energy of
the solution on the whole line is formally given by the sum of the half-line
energies of the solution and its parity conjugate; that is in this case
\begin{equation}
-2(n+1)(C_-+C_-)=-2(n+1)(2\sum_{p=1}^{N+1}m_{a_p}-2)=-2(n+1)(2\sum_{p=1}^N m_{a_p}+m_{a_{N+1}})
\end{equation}
as we should have expected from the masses of the constituent solitons.
\section{Scattering solutions for $a_n^{(1)}$ Toda theories}
In the previous section we showed how one can construct static background
configurations. In this section we show
how to
solve the classical linearised perturbation equations around this background,
and explicitly calculate the classical scattering matrix. Once more we
shall rely on the tau-functions given by multi-soliton solutions.
Let us assume that the field is infinitesimally perturbed about the vacuum
configuration
\begin{equation}
\phi(x,t)=\phi(x)_{vac}+\epsilon(x,t).
\end{equation}
Substituting this into the equations of motion and the boundary conditions
(\ref{eq.non.Neumann}), and keeping terms linear in $\epsilon(x,t)$ yields
\begin{equation}
\partial_{\mu}\partial^{\mu}\epsilon(x,t) +\sum_{i=0}^r n_i \alpha^i e^{\alpha^i
\cdot \phi_{vac(x)}} (\alpha^i \cdot \epsilon(x,t)) = 0
\label{eq.linear.eqm}
\end{equation}
and
\begin{equation}
\partial_x \epsilon(x,t)={1\over 2}\sum_{i=0}^r A_i \alpha^i \sqrt{n_i} e^{\alpha^i
\cdot \phi_{vac}(x)/2}(\alpha^i \cdot \epsilon(x,t)).
\label{eq.linear.bc}
\end{equation}
Far away from the boundary $x=0$, the field $\phi_{vac}(x,t)\to 0$ for finite
energy configurations, and in this limit the equation (\ref{eq.linear.eqm})
reduces to (\ref{eq.wave}) and the solution for $\epsilon(x,t)$ tends
to
\begin{equation}
\epsilon(x,t)\to \rho_a e^{-iEt}(e^{ipx}+K_a e^{-ipx})
\label{eq.sol2}
\end{equation}
where $E^2-p^2=m_{a}^2$. This solution
consists of the superposition of two oscillatory solutions
corresponding to incoming and outgoing `waves' or quantum
mechanically `particles'. On the other hand, the field $\phi(x,t)$ corresponding to a single soliton of type $a$
has the asymptotic form
\begin{equation}
\phi(x,t)\sim \rho_a e^{\Phi(x,t)}
\end{equation}
where $\Phi(x,t)=\sigma(x-vt)$, and $\sigma^2-\sigma^2 v^2=m_{a}^2$. Comparison of the two asymptotic behaviours suggests
that we can obtain appropriate oscillatory solutions from the soliton tau-functions if we make the identification
\begin{eqnarray}
\sigma&=&\pm i p\\
\sigma v &=& i E.
\label{eq.identification}
\end{eqnarray}
From here it is clear how to proceed. We take the tau-function $\tau_{vac}$ corresponding
to the static vacuum solution that we found in the previous section and add
in two further time-dependent solitons with the identifications (\ref{eq.identification}). If we label the incoming and outgoing solitons by
indices $I$, and $O$ respectively, then we take
\begin{eqnarray}
&\sigma_I&=-\sigma_O=i p\;\;,\;\;
\sigma_I v_I =\sigma_O v_O =i E\\
&d_I&=d_O=\epsilon\;\;,\;\;
\chi_I=-\chi_O=\psi\;\;,\;\;
a_I=a_O=b
\end{eqnarray}
Note that these two solitons are {\it not} a conjugate pair but are both
of species $b$. The relative phase between the two waves is given by
\begin{equation}
K^b=e^{-2i\psi}
\end{equation}
This tau-function describes the scattering of a particle
of species $b$ on the background given by $\tau_{vac}$. As described in
section two, requiring that this tau-function satisfies one of the boundary
conditions (\ref{eq.non.Neumann}) up to $O(\epsilon)$ amounts
to the equation
\begin{equation}
\ddot{\tau}_i-\tau_i''+2 C \tau_i'
-(C^2 -1)\tau_i|_{x=0} =0.
\label{eq.more}
\end{equation}
Considering the cases where the vacuum solution has even/odd number
of constituent solitons in turn, let us assume first that it has $2N$ solitons.
The linearity of equation (\ref{eq.more}
implies that we can divide $\tau_j$ into
charge sectors as in the previous equation,
and use the equation (\ref{eq.mod.stat}) with the addition of a time-derivative
term as in(\ref{eq.more}), i.e.
\begin{equation}
\ddot{T_k}-T_k''+2 C T_k'
-(C^2 -1)T_k|_{x=0} =0.
\label{eq.more2}
\end{equation}
There are at
least two highest charge terms in the tau-function
with charge $Q_{max}=\sum_{p=1}^N a_p +b$ of the form
\begin{eqnarray}
T_{Q_{max}} &=& \prod_{j=1}^N d_p \prod_{1\leq j< k\leq N} A^{(jk)}
e^{i\sum_{j=1}^N \chi_j} e^{(\sum_{j=1}^N m_{a_j})x}\nonumber\\*&&\mbox{}
\times \left ( e^{i\psi}e^{iEt+ipx} \prod_{j=1}^N A^{(a_j b)}(p)+
e^{-i\psi}e^{iEt-ipx} \prod_{j=1}^N A^{(a_j b)}(-p)\right )
\end{eqnarray}
where $A^{(a_j b)}(p)$ are calculated using the definition (\ref{eq.inter.def}),
(\ref{eq.identification}). Substituting this expression into (\ref{eq.more2})
we find that the scattering matrices on the two solutions corresponding to
$C_{\pm}$ are given by
\begin{equation}
K^b_{\pm}=e^{-2i\psi}={{2ip\mp m_b^2}\over {2ip\pm m_b^2}}
{{\prod_{j=1}^N A^{(a_j b)}(p)}\over {\prod_{j=1}^N A^{(a_j b)}(-p)}}.
\label{eq.scattering}
\end{equation}
If the vacuum solution has an odd number of solitons, then we find that the
scattering matrix is given essentially by the above expression for $K_-$, but
where the product runs from $1,..,N+1$. Note that the reflection factor
given by (\ref{eq.scattering}) only depends on the number and species
of solitons in the background solution, not on the topological charge
parameters $\chi_p$. It follows that boundary conditions related by
the $Z_n$ symmetry (\ref{eq.zn.many}) will have identical scattering
matrices.
\subsection{Classical reflection bootstrap equations}
In the introduction, it was pointed out that in two dimensions, integrability
placed strong constraints on the $S$-matrix of the theory, and that $S$ and $K$
satisfied various algebraic constraints. In the case at hand both $S$ and $K$
are diagonal and in the classical limit $S$ tends to unity, and these two
facts ensure that most of the algebraic constraints are satisfied automatically.
However, one non-trivial check is the reflection bootstrap equation, which
in the classical limit becomes
\begin{equation}
K^c(\theta_c)=K^a(\theta_c+i\bar{\theta}^b_{ac})
K^b(\theta_c-i\bar{\theta}^a_{bc})
\label{eq.classical.rbp}
\end{equation}
where the fusion angles are given by
\begin{eqnarray}
{\theta}^b_{ac}&=&{\pi(a+b)\over 2(n+1)}\\
\bar{\theta}&=&{\pi/2}-\theta
\end{eqnarray}
and $a+b+c=n+1$. (For simplicity we ignore the case that $a+b+c=2(n+1)$.
For more details about fusion angles see for instance \cite{BCDS} and about
the classical reflection equation see \cite{FK1,CDR}.)
In the equation (\ref{eq.classical.rbp}), the argument of $K^c$ is the usual
rapidity variable $\phi$ which is related to the momentum $p$ and
energy $E$ of the incoming particle of type $a$ by
the formulae
\begin{eqnarray}
p&=&m_{a} \sinh(\phi)\\
E&=&m_{a} \cosh(\phi).
\end{eqnarray}
Note that using the identification (\ref{eq.identification}), we see that
$\theta_p=\phi+i{\pi\over 2}$ where $\theta_p$ appears in(\ref{eq.rap.one}).
If, as is usual, one defines a bracket notation
\begin{equation}
(x)={\sinh({\phi\over 2}+{{x i \pi}\over{2(n+1)}})\over
\sinh({\phi\over 2}-{{x i \pi}\over{2(n+1)}})}
\label{eq.bracket}
\end{equation}
then one can write in this notation
\begin{equation}
{A^{(a_j b)}(p)\over A^{(a_j b)}(-p)} = {{({n+1\over 2}+a_j-b)({n+1\over 2}-a_j+b)}\over {({n+1\over 2}+a_j+b)({n+1\over 2}-a_j-b)}}
\label{eq.inter.bracket}
\end{equation}
where the left hand side of this equation is one of the factors appearing
in the scattering matrix $K^b_{\pm}$ in equation (\ref{eq.scattering}).
A straightforward calculation shows that each of these factors individually
satisfy the reflection bootstrap equation. To prove that $K^b$ satisfies
(\ref{eq.classical.rbp}) it remains only to show that the remaining factor
\begin{equation}
{{2ip-m_b^2}\over {2ip+m_b^2}}=-{1\over {(n+1-a_b)(a_b)}}
\end{equation}
satisfies the bootstrap equation, and this is indeed the case.
\section{Other simply-laced algebras}
In this section we shall make a few remarks about extending the results found
in the previous two sections to affine Toda theories based on the $D$ and $E$
series of algebras. The major technical difficulty in extending the results
to these algebras is that there is no explicit form for tau-functions which
correspond to more than two-soliton
solutions, and even those solutions that are known are considerably
more complicated than those of the $a_n^{(1)}$ theories \cite{RH}.
Because of this, we
shall restrict our search to finding which single soliton solutions
\cite{ACFGZ} can be
found which satisfy the boundary conditions (\ref{eq.non.Neumann}).
The general form for the corresponding tau-functions is
\begin{equation}
\tau_j = 1 + \delta^{(p)}_1 d_p e^{m_p x} + .... +\delta^{(p)}_{n_j} (d_p)^{n_j}
e^{n_j m_p x}
\label{eq.single.soliton}
\end{equation}
where $p$ labels the species of soliton as before.
The coefficients $\delta^{(p)}_i$ have been explicitly
calculated for all affine algebras on
a case by case basis, and they are found to be real if and only if the
minimal representation $\lambda_p$ of the corresponding finite Lie algebra
is real. Restricting ourselves to such real tau-functions, we substitute
the tau-functions (\ref{eq.single.soliton})
case by case into the equation (\ref{eq.static.master})
and solve for $C$ and $d_p$. The results are
presented below. The labelling of nodes on the Dynkin diagram is given in
Figure 3.
\begin{figure}
\hspace{3.0cm}
\epsfxsize=12truecm
\epsfysize=18truecm
\epsfbox{bt3.eps}
\caption{ Root Labels for simply-laced algebras }
\end{figure}
\subsection{$d_4^{(1)}$}
The only solution is associated with the triality invariant soliton of
species $p=2$. The corresponding tau-functions are given by
\begin{equation}
\tau_0=\tau_1=\tau_3=\tau_4=1+d_2 e^{\sqrt{6} x},\;\;{\rm and}\;\;
\tau_2=1-4 d_2 e^{\sqrt{6} x} + (d_2)^2 e^{ 2\sqrt{6} x}
\end{equation}
and we find the solutions $C=\sqrt{6}/2,\sqrt{6}/2\pm 3\sqrt{2}/2,\sqrt{6}/2\pm
\sqrt{2}/2$ and correspondingly $d_2=-1,(5\pm 3\sqrt{3})/(-5\pm 3\sqrt{3}),
2\pm \sqrt{3}$. Only the solutions corresponding to
$-1,(5- 3\sqrt{3})/(-5+ 3\sqrt{3}),2-\sqrt{3}$ are
singularity free in the region $x<0$.
For $d_2=-1$,$\tau_0\to 0$ as $x\to 0$, so that the only non vanishing
component of ${\lambda_2\cdot \phi}\to -\infty$ in this limit. Nonetheless
the solution has finite energy, and obeys the boundary condition
$A_0=A_1=A_3=A_4=-1$ and $A_2$ is unspecified since $e^{\alpha_2 \cdot \phi}$
vanishes at the wall.
For $d_2=2-\sqrt{3}$ we find that $\tau_2$ vanishes at the wall so that
${\lambda_2\cdot \phi}\to \infty$. We find that $A_0,A_1,A_3,A_4$ are
unconstrained
and $A_2=1$. Finally the solution with $d_2=(5- 3\sqrt{3})/(-5+ 3\sqrt{3})$ is
regular at the wall, and obeys boundary conditions with $A_i=-1$ for all $i$.
\subsection{$d_5^{(1)}$}
Again the only solution is associated with the species $p=2$. The corresponding
tau-functions are given by
\begin{equation}
\tau_0=\tau_1=\tau_4=\tau_5=1+d_2 e^{2 x},\;\;{\rm and}\;\;
\tau_2=\tau_3=1-2 d_2 e^{2 x} + (d_2)^2 e^{4 x}.
\end{equation}
We find the solutions $C=(1\pm\sqrt{2\pm \sqrt{2}})$. The corresponding
values of $d_2$ are $(2\pm \sqrt{2\pm \sqrt{2}})/(2\mp \sqrt{2\pm \sqrt{2}})$. The
solutions which are regular in the region $x<0$ are those with
$C=(1-\sqrt{2\pm \sqrt{2}})$. These
solutions satisfies the boundary condition with $A_0=A_1=A_4=A_5=-1$ and
$A_2=A_3=\mp 1$.
\subsection{$d_n^{(1)}$, $n>5$}
The solitons corresponding to the tip nodes do not yield any solutions.
The other solitons correspond to roots $\alpha_p$ with $n_p=2$ and their
tau-functions are given by the formulae
\begin{eqnarray}
\tau_0=-\tau_1&=&1+d_p e^{m_p x}\\
\tau_i&=& 1+2 {\cos({(2i-1) \pi p \over 2(n-1)})\over
\cos({ \pi p \over 2(n-1)})} d_p e^{m_p x} + (d_p)^2 e^{2 m_p x}
\;\;,2\leq i
\leq n-2 \\
\tau_{n-1}=\tau_n &=& 1+(-1)^p d_p e^{m_p x}
\label{eq.tau.dn}
\end{eqnarray}
The value of the coefficient involving cosine takes at least two different
values for different $i$, so inserting the difference of two
such distinct tau-functions into (\ref{eq.static.master}) immediately
yields that $2C=m_p\pm \sqrt{2}$. Now substituting $\tau_0$ and $\tau_2$ into
the same equation we find that $m_p=\sqrt{6}$ and that $d_p=2\pm \sqrt{3}$.
Given that
\begin{equation}
m_p=2\sqrt{2} \sin ({ \pi p \over 2(n-1)})
\end{equation}
we find that we must have $n=3x+1$, $p=2x$. With these values (\ref{eq.tau.dn})
reduces to
\begin{eqnarray}
\tau_0=-\tau_1=\tau_{n-1}=\tau_n&=&1+d_p e^{\sqrt{6} x}\\
\tau_{3i}=\tau_{3i+1}&=&1+2 d_p e^{\sqrt{6} x}+(d_p)^2 e^{2 \sqrt{6} x}\\
\tau_{3i+2}=1-4 d_p e^{\sqrt{6} x}+(d_p)^2 e^{2\sqrt{6} x}.
\end{eqnarray}
Only the solution with $d_p=2-\sqrt{3}$ is non-singular in the region $x<0$.
The solution is compatible with the boundary conditions $A_{3i+2}=-1$, and the
other $A_i$ are unconstrained.
\subsection{$e_6^{(1)}$}
In this case only the solitons of species $p=2,4$ have real tau-functions.
Solutions which satisfy the boundary conditions can be found in both cases.
If $p=2$, then $m_2=\sqrt{6+2\sqrt{3}}$ and solutions can be found with
$C=m_2/2,m_2/2\pm \sqrt{6-2\sqrt{3}}/2$ and $d_2=-1,5\pm 2\sqrt{6}$.
Only the solution with $d_2=5-2\sqrt{6}$ is non-singular for $x<0$, and
satisfies boundary conditions (\ref{eq.non.Neumann}) with $A_4=-1$ and the
other $A_i=1$.
If $p=4$, then $m_2=\sqrt{6-2\sqrt{3}}$ and solutions can be found with
$C=m_2/2,m_2/2\pm \sqrt{6+2\sqrt{3}}/2$ and $d_4=-1,5\pm 2\sqrt{6}$. The
solutions with $d_4=-1,5-\sqrt{6}$ are regular for $x<0$ and satisfy
the boundary conditions $A_0=A_1=A_6=A_4=1$ and the other $A$'s unspecified,
and $A_i=1$ for all $i$ respectively.
\subsection{$e_7^{(1)}$}
Only solitons of species $p=3$ and $m_3=\sqrt{6}$ can satisfy the boundary conditions with $C=m_3/2,m_3\pm \sqrt{2}/2$. The corresponding values of
$d_3$ are $-1,2\pm \sqrt{3}$ and only the solutions with $d_3=-1,2-\sqrt{3}$
are free of singularities for $x<0$.
The solutions with $d_3=-1$ does not seem to correspond to any sensible
boundary condition of the type (\ref{eq.non.Neumann}) but for
$d_3=2-\sqrt{3}$ we find that $A_1=A_2=A_6=-1$ with the other $A_i$
unconstrained.
\subsection{$e_8^{(1)}$}
By using the two tau-functions corresponding to $n_i=2$ and also $\tau_0$, one
can use similar arguments to those used for $d_n^{(1)},\; n>5$ to
deduce that $m_p=\sqrt(6)$. Since there are no single solitons with such
a `mass' we find a contradiction so that there are no single-soliton solutions.
\vskip 1cm
\subsection{Scattering}
In section four we showed how by adding in two further solitons with
imaginary momentum, we could describe classical scattering solutions on
the static `vacuum' configurations. The tau-functions had to obey the equation
\begin{equation}
\ddot{\tau}_i-\tau_i''+2 C n_i\tau_i'
-(C^2 n_i^2 -n_i)\tau_i|_{x=0} =0.
\label{eq.more.again}
\end{equation}
As explained above, multi-soliton tau-functions for algebras other than
$a_n^{(1)}$ are very complicated in general. However, the exception to this
rule is $\tau_0$, which has exactly the same form as for the $a_n^{(1)}$ case,
except that the interaction coefficient $A^{(pq)}(\theta)$ defined in
(\ref{eq.inter.def}) must be generalised. For a definition of the generalisation
$X^{(pq)}(\theta)$ and a discussion of the properties it enjoys please see
reference \cite{OK}. Restricting ourselves to single soliton backgrounds,
and to linear order in the perturbation parameter $\epsilon$ we find that
\begin{eqnarray}
\tau_0&=&1+d_r e^{m_r x}+\epsilon e^{i \psi} e^{-iEt+ipx}
+\epsilon e^{-i \psi} e^{-iEt-ipx}+d_r\epsilon e^{i \psi} e^{-iEt+ipx+m_r x}X^{(rb)}(p)
\nonumber\\*&&\mbox{}
+d_r \epsilon e^{-i \psi} e^{-iEt-ipx+m_r x}X^{(rb)}(-p).
\end{eqnarray}
Substituting this into (\ref{eq.more.again}) and looking at the term linear
in $\epsilon$ one recovers
\begin{equation}
e^{-2 i \psi}= _{{{\left ( (C-m_r)^2-1)((C-m_r)^2+2ip(C-mr)+m_b^2-1)
X^{(rb)}(p)-(C^2-1)(C^2+2ipC+m_b^2-1)\right )}\over {\left ( (C-m_r)^2-1)((C-m_r)^2+2ip(C-mr)+m_b^2-1)
X^{(rb)}(p)-(C^2-1)(C^2+2ipC+m_b^2-1)\right )}}}
\end{equation}
\section{Conclusions}
We have seen that integrable boundary
conditions (\ref{eq.non.Neumann}) and the equations of motion combine neatly
to yield a simple equation satisfied by
the tau-functions of affine Toda theory.
In the case of $A^{(1)}_n$ a large class of static
solutions were found, and moreover it proved relatively straightforward to
extract the scattering data on these backgrounds. As a consistency check we
showed that the scattering matrices
satisfy the classical reflection boostrap equation.
Still, many open
questions remain even in this simple case.
We only provided a plausible argument
that the conjectured form of tau-function did indeed satisfy the boundary
condition, and we did not rule out the possibility of other solutions, amongst
which the true vacuum solution may lie . The solutions consisting
of $2N$ solitons contained $N$ unspecified parameters $\chi_p$
(those with $2N+1$
solitons were specified by $N+1$ parameters). These can be thought of as
moduli for zero-modes, since the energy of the solutions is independent of
$\chi_p$. We can change the boundary conditions satisfied by the solutions
by varying these parameters, in essentially the same way that one varies
the topological charge of solitons in the imaginary coupling theory.
The `phase' diagram which specifies which boundary conditions
can be obtained from a particular family of solutions by varying $\chi_p$
seems in general very complicated and this makes it difficult to isolate
the true vacuum solution. Nonetheless, the scattering data is found to be
independent of the topological charge, so all the resulting boundary conditions
will share the same scattering matrix $K$.
A related difficulty is to find which solutions
are singularity free in the region $x<0$. The general rule seems to be that as
we add in more solitons the energy of the solution decreases rather than
increases because of the reality of the coupling, but also the solution
develops more singularities. The trick to finding the true vacuum is
therefore to add in as many solitons as one can without introducing
singularities into the physical region. It may be that the best one can do
is to have a singularity at the wall itself which is still physically
acceptable. This may also provide a mechanism for removing the zero-modes of
the vacuum solution, since it is conceivable that varying the parameters
$\chi_p$ would inevitably move one or more of the singularities in $\phi$
into the physical region.
Although the formalism extends to other simply-laced algebras, the application
to these algebras is hampered by the relatively complicated
technology for tau-functions other than those of the $a_n^{(1)}$ series.
Whilst some partial results were obtained in these cases, it is clear that
more powerful methods are required. Clearly one way forward is to try and
adopt the methods of \cite{OTU} which provide exact (if not explicit)
formulae for multi-soliton solutions which are equally neat for any algebra.
In their notation the tau-functions can be expressed as
\begin{equation}
\tau_j = \langle \Lambda_j | e^{-{1\over 2} E_1(t+x)} g e^{-{1\over 2} E_{-1}(t-x)}| \Lambda_j \rangle e^{-{n_j\over 4} (t^2-x^2)}
\end{equation}
and the condition (\ref{eq.static.master}) can be written neatly as
\begin{equation}
\langle \Lambda_j | (C n_j +E_1) g (C n_j - E_{-1})| \Lambda_j \rangle=0
\end{equation}
where $g$ is an element of the group associated with the affine algebra.
It would be interesting to see if this yields a more complete solution
to the problem.
Finally, it is worth re-emphasising that analytically continued solitons
seem to play an important role in real coupling Toda theory on the
half-line and that this provides a respectable home for them. The
model seems to be on a firmer footing than imaginary coupling theories
with their manifest unitarity problems, and the singularities that inevitably
occur for the solitons in the real-coupling theories can be placed behind
the boundary out of harms way.
|
proofpile-arXiv_065-638
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\section{Introduction}
The notion of {\em sphaleron} refers to the special type
of static classical solution in a gauge field theory
with periodic vacuum structure and broken scale invariance \cite{Manton}.
Specifically, sphaleron relates to the top of the potential barrier
between distinct topological vacua, such that
its energy determines the barrier height. Sphaleron can
play the important role in the transition processes when
the system interpolates between distinct topological sectors.
Consider a thermal ensemble over one of the
perturbative topological vacua.
Such a system is metastable since the field modes are able to reach the
neighboring topological sectors both via the quantum tunneling
and due to the thermal overbarrier fluctuations.
If temperature is high enough, the latter effect is dominant,
in which case the sphaleron, `sitting' on the top of the barrier,
controls the transition rate.
To evaluate the rate of such sphaleron-mediated thermal transitions,
the Langer-Affleck formula is often used \cite{langer}:
\begin{equation} \label{1}
\Gamma=-\frac{|\omega_{-}|}{\pi}\frac{{\rm Im}Z_{1}}{Z_{0}},
\end{equation}
which relates the probability of the decay of the unstable phase
with the imaginary part of the free energy.
Here $Z_{0}$ and $Z_{1}$ are the
partition functions for the small fluctuations around the vacuum
and the sphaleron, respectively. Since the sphaleron has one unstable mode
whose eigenvalue $\omega_{-}^{2}<0$, the quantity $Z_{1}$ is
purely imaginary. To compute $\Gamma$ at the one-loop level
is usually rather difficult. The problem becomes especially hard
in the standard model case, where the sphaleron solution itself is
known only numerically. That is why the other sphaleron models for which
$\Gamma$ can be evaluated exactly have been investigated
\cite{shap}, \cite{mottola}, however these models
exist only in $1+1$ spacetime dimensions.
\section{The sphaleron on $S^{3}$}
To find an exactly solvable sphaleron model in $3+1$
dimensions \cite{main}, we consider the theory of a {\em pure}
non-Abelian $SU(2)$ gauge field in
the static Einstein universe $(M,{\bf g})$,
where $M=R^{1}\times S^{3}$, and the metric is
\begin{equation} \label{2}
ds^{2}={\bf a}^{2}\{-d\eta^{2}+ d\xi^{2}+sin^{2}\xi
(d\vartheta^{2}+sin^{2}\vartheta d\varphi^{2})\};
\end{equation}
here ${\bf a}$ is a constant scale factor. Consider the following
$SU(2)$-valued function on the $S^{3}$:
\begin{equation} \label{3}
U=U(\xi,\vartheta,\varphi)=
\exp\left\{- i \xi\ n^{a}\tau^{a}\right\},
\end{equation}
where $n^{a}=(\sin\vartheta\cos\varphi, \sin\vartheta\sin\varphi,
\cos\vartheta)$ and $\tau^{a}$ are the Pauli matrices.
This function defines the mapping
$S^{3}\rightarrow SU(2)$ with the unit winding number. Using
$U$, we construct the following sequence of the static gauge field
potentials:
\begin{equation} \label{4}
A[h]=i\frac{1+h}{2}UdU^{-1},
\end{equation}
where the parameter $h\in [-1,1]$. When $h=-1$ this field vanishes,
whereas for $h=1$ it is a pure gauge whose winding number is one,
by construction. Thus fields (\ref{4}) interpolate between
the two distinct topological vacua, and the energy
\begin{equation} \label{5}
E[h]=\int T^{0}_{0}\sqrt{^{3}{\bf g}}d^{3}x=
\frac{3\pi^{2}}{g^{2}{\bf a}}(h^{2}-1)^{2}
\end{equation}
has the typical barrier shape --- it vanishes at the vacuum
values of $h$, $h=\pm 1$, and reaches its maximum in between,
at $h=0$; ($g$ in (\ref{5}) stands for the gauge coupling constant).
The top of the barrier relates to the field configuration
\begin{equation} \label{6}
A^{(sp)}=
\frac{i}{2}\ UdU^{-1},
\end{equation}
which obeys the Yang-Mills equations and therefore
can be naturally called sphaleron. It is worth noting that the sphaleron
configuration consists of the gauge field alone. The violation
of the scale invariance in this case is provided by the background
curvature. Since the spacetime geometry is $SO(4)$-symmetric,
the sphaleron inherits the same symmetries, such that, for instance, the
energy-momentum tensor for the field (\ref{6}) has the manifest
$SO(4)$-symmetric structure.
\section{The sphaleron transition rate}
Our main task is to compute the transition rate (\ref{1}) for the
sphaleron solution (\ref{6}). We pass to the imaginary time $\tau$
in the metric (\ref{2}) and impose the periodicity condition,
$\tau\in [0,\beta]$. Let us introduce $A^{\{j\}}_{\mu}=j A^{(sp)}_{\mu}$,
which corresponds to the vacuum of the gauge field for $j=0$ and
to the sphaleron field for $j=1$. Next we consider small fluctuations
around the background gauge field:
$A^{\{j\}}_{\mu}\rightarrow A^{\{j\}}_{\mu}+\phi_{\mu}$.
Notice that we assume the spacetime metric (\ref{2}) to be fixed
and therefore do not take into account the
gravitational degrees of freedom.
The partition functions $Z_{j}$ are then given
by the Euclidean path integral over $\phi_{\mu}$. To compute the
integral, we impose the background gauge condition and use the
Faddeev-Popov procedure. The result is \cite{main}:
\begin{equation} \label{7}
Z_{j}=\exp (-S[A^{\{j\}}])\ {\cal N}
\frac{Det'(\hat{M}^{FP}_{j}/\mu_{0}^{2})}
{\sqrt{Det'(\hat{M}_{j}/\mu_{0}^{2}})},
\end{equation}
where $S$ is the Euclidean action, the factor ${\cal N}$ is due to
the zero and negative modes whereas $Det'$ has all such modes omitted,
$\mu_{0}$ is an arbitrary normalization
scale, and the fluctuation operators are
\begin{equation} \label{8}
\hat{{\bf M}_{j}}\phi^{\nu}=-D_{\sigma}D^{\sigma}\phi^{\nu}
+R^{\nu}_{\sigma}\phi^{\sigma}+
2i[F^{\nu}_{\ \sigma},\phi^{\sigma}],\ \ \ \ \
\hat{{\bf M}_{j}}^{FP}\alpha=-D_{\sigma}D^{\sigma}\alpha.
\end{equation}
Here $D_{\mu}=\nabla_{\mu}-i[A^{\{j\}}_{\mu},\ \ ]$ is the covariant
derivative,
$R^{\nu}_{\sigma}$ is the Ricci tensor for the geometry (\ref{2}),
$F^{\nu}_{\ \sigma}$ is the gauge field tensor for $A^{\{j\}}_{\mu}$,
and $\alpha$ is a Lie algebra valued scalar field.
To find spectra of these operators, we introduce
the 1-form basis $\{\omega^{0},\omega^{a}\}$ on the spacetime manifold,
where $\omega^{0}=d\tau$, and $\omega^{a}$ are the
left invariant 1-forms on $S^{3}$. It is convenient to expand
the fluctuations as
$\phi=(\phi^{0}_{p}\omega^{0}+\phi^{a}_{p}\omega^{a})\tau^{p}/2$.
Let $e_{a}$ be the
left-invariant vector fields dual to $\omega^{a}$, such that
${\bf L}_{a}=\frac{i}{2}e_{a}$ are the $SO(4)$ angular momentum
operators. We introduce also spin and isospin operators as follows:
${\bf S}_{a}\phi^{b}_{p}=\frac{1}{i}\varepsilon_{abc}\phi^{c}_{p}$ and
${\bf T}_{p}\phi^{a}_{r}=\frac{1}{i}\varepsilon_{prs}\phi^{a}_{s}$.
As a result, the fluctuation operators (\ref{8}) can be expressed
entirely in terms of the operators ${\bf L}_{a}$, ${\bf S}_{a}$ and ${\bf T}_{p}$,
such that the spectra can be explicitly obtained by the purely
algebraic methods \cite{main}.
All of the eigenvalues are
positive except for the following ones:
the sphaleron fluctuation operator $\hat{{\bf M}_{1}}$ has one
negative mode, whereas the vacuum operators $\hat{{\bf M}_{0}}$ and
$\hat{{\bf M}^{FP}_{0}}$ have three zero modes each. It is worth noting
that, since the sphaleron field configuration is $SO(4)$ invariant,
the sphaleron itself does not have zero modes at all (in the
background gauge imposed).
The next step is to compute the products of the eigenvalues
to evaluate the determinants in (\ref{7}). For this, zeta
function regularization scheme has been used. Omitting all
technical details given in \cite{main}, the resulting
expression for the transition rate can be represented in the
following form:
\begin{equation} \label{9}
\Gamma=
\frac{1}{8\sqrt{2}\pi^{2}\sin(\beta/\sqrt{2})}
\exp\left\{-\frac{3\pi^{2}}{g^{2}({\bf a})}\beta-{\cal E}_{0}\beta
-\beta(F_{1}-F_{0}) \right\}.
\end{equation}
In this expression, the prefactor in the right hand side is the
overall contribution of zero and negative modes.
$3\pi^{2}\beta/g^{2}({\bf a})$ is the
Euclidean action of the sphaleron, where the gauge coupling
constant receives the quantum correction due to the
scaling behavior of the functional determinants:
\begin{equation} \label{10}
\frac{1}{g^{2}({\bf a})}=\frac{1}{g^{2}({\bf a}_{0})}
-\frac{11}{12\pi^{2}}\ln\left(\frac{{\bf a}}{{\bf a}_{0}}\right).
\end{equation}
Here we have replaced $g$ by $g({\bf a}_{0})$, where ${\bf a}_{0}=1/\mu_{0}$.
This expression agrees with the renormalization group flow,
such that it does not depend on the scale ${\bf a}_{0}$ if $g({\bf a}_{0})$ is
chosen to obey the Gell-Mann-Low equation. To fix the scale, we assume that
the value of $g({\bf a}_{0})$ is determined by the physical
temperature, $T({\bf a}_{0})=1/\beta{\bf a}_{0}$, and use the QCD data:
\begin{equation} \label{11}
T({\bf a}_{0})=100\ {\rm GeV},
\ \ \ \ \ \ \frac{g^{2}({\bf a}_{0})}{4\pi}=0.12.
\end{equation}
One can assume that the weak coupling region extends up to some
${\bf a}_{max}\sim 10\div 100{\bf a}_{0}$.
The next term in (\ref{9}), ${\cal E}_{0}$, is the
contribution of the zero field
oscillations, that is, the Casimir energy. This quantity can be
computed exactly \cite{main}, the numerical value is
${\cal E}_{0}=-1.084$.
The contribution of the thermal degrees of freedom in
(\ref{9}) is
$$
\beta (F_{1}-F_{0})=
4\ln(1-e^{-\beta})
+2\sum_{\sigma=0,1,2}\ \sum_{n=3}^{\infty}(n^{2}-\sigma^{2})
\ln(1-e^{-\beta\sqrt{n^{2}+\sigma^{2}-3}})-
$$
\begin{equation} \label{12}
-6\sum_{n=2}^{\infty}(n^{2}-1)
\ln(1-e^{-\beta n}).
\end{equation}
Altogether Eqs.(\ref{9})-(\ref{12}) provide the
desired solution
of the one-loop sphaleron transition problem. The numerical
curves of $\Gamma(\beta)$ evaluated according to these formulas
for several values of ${\bf a}$ are presented in \cite{main}.
This solution makes sense under the following assumptions:
\begin{equation} \label{13}
{\bf a}\leq{\bf a}_{max},\ \ \ \ \ \
\frac{1}{\sqrt{2}\pi}<\frac{1}{\beta}\ll\frac{3\pi^{2}}{g^{2}({\bf a})}.
\end{equation}
The first condition is the the weak coupling requirement.
When the scale factor ${\bf a}$ is too large, the running coupling
constant (\ref{10}) becomes big (confinement phase), and
the effects of the strong coupling can completely change the
semiclassical picture. That is why our solution can be trusted only
for the small values of the size of ${\bf a}$. The other condition
in (\ref{13}) requires that the thermal fluctuations are
small compared to the classical sphaleron energy, such that
the perturbation theory is valid.
|
proofpile-arXiv_065-639
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}
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proofpile-arXiv_065-640
|
{
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
}
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\section{introduction}
The Kauffman bracket skein module is a deceptively simple
construction, occurring naturally in several fields of mathematics and
physics. This paper is a survey of the various ways in which it is a
quantization of a classical object.
Przytycki \cite{P1} and Turaev \cite{T1} introduced skein modules.
Shortly thereafter, Turaev \cite{T2} discovered that they formed
quantizations of loop algebras; further work in this direction was
done by Hoste and Przytycki \cite{HPquant}. We will look at some of
the heuristic reasons for treating skein modules as deformations, and
then realize the Kauffman bracket module as a precise quantization in
two different ways.
Traditionally, this is done by locating a non-commutative algebra that
deforms a commutative algebra in a manner coherent with a Poisson
structure. The importance of the Kauffman bracket skein module began
to emerge from its relationship with $\g$ invariant theory. It is
well known that the $\g$-characters of a surface group form a Poisson
algebra \cite{BG,Go}. The skein module is the appropriate deformation.
The idea of a lattice gauge field theory quantization of surface
group characters is due to Fock and Rosly \cite{FR}. It was developed
by Alekseev, Grosse and Schomerus \cite{AGS} and by Buffenoir and
Roche \cite {BR1,Bu}. We tie the approaches together by showing that
the skein module coincides with the lattice quantization.
\section{The Kauffman Bracket Skein Module}
Quantum topology began with the discovery of several new link
polynomials, the first and most well known being the Jones polynomial
\cite{jones1}, \cite{jones2}. Many subsequent invariants arose from
alternative proofs of its existence. The state sum approach
\cite{kauffman1} yielded an invariant known as the Kauffman bracket,
on which we will focus. The Kauffman bracket is a function on the set
of framed links in ${\hspace{-0.3pt}\mathbb R}\hspace{0.3pt}^3$. Since we will take a combinatorial view
throughout, one may as well think of a link as represented by a
diagram in ${\hspace{-0.3pt}\mathbb R}\hspace{0.3pt}^2$ (see Figure \ref{link-diagram}).
\begin{figure}[t]
\centering
\epsfig{file=link-diagram.eps}
\caption{Diagram of a two-component link.}
\label{link-diagram}
\end{figure}
A link is an embedding of circles, of which a diagram is a
particularly convenient picture. Two diagrams represent the same link
if one can be deformed into the other.\footnote{These deformations may include some Reidemeister moves.}
In a framed link each circle
is actually the centerline of an embedded annulus. Since we always
work with diagrams, it makes sense to assume the annulus lies flat in
${\hspace{-0.3pt}\mathbb R}\hspace{0.3pt}^2$ as illustrated in Figure \ref{framed-hopf}.
\begin{figure}[b]
\centering
\epsfig{file=hopf-diagram,width=1in,angle=-45}
represents
\epsfig{file=framed-hopf,width=1in,angle=-45}
\caption{Diagram for a framed Hopf link}
\label{framed-hopf}
\end{figure}
The Kauffman bracket, $\langle\; \rangle$, takes values in the ring
${\hspace{-0.3pt}\mathbb Z}\hspace{0.3pt}[A^{\pm1}]$ and is uniquely determined by the
rules:\footnote{Actually, Kauffman did not include the empty diagram
$\emptyset$, and his normalization was $\langle\text{unknot}\rangle =
1$. Later, the quantum group approach \cite{KM,RT1} to knot
polynomials---and to some extent skein modules---indicated that
normalization at $\emptyset$ was preferable.}
\begin{enumerate}
\item $\langle\emptyset\rangle=1$,
\vspace{5pt}
\item $\displaystyle{\left\langle\raisebox{-5pt}{\mbox{}\hspace{1pt\right\rangle =
A \left\langle\raisebox{-5pt}{\mbox{}\hspace{1pt\right\rangle +
A^{-1} \left\langle\raisebox{-5pt}{\mbox{}\hspace{1pt\right\rangle}$, and
\vspace{5pt}
\item $\displaystyle{\left\langle\bigcirc\right\rangle
= (-A^2-A^{-2})\langle\emptyset\rangle}$.
\end{enumerate}
The arguments of $\langle\; \rangle$ in (2) and (3) represent diagrams
which are identical except in a neighborhood where they differ as
shown in the formulas. One evaluates the function on a diagram by
first applying (2) until no crossings remain, and then reducing each
diagram to a polynomial via (3) and (1).
\begin{theorem}[Kauffman]
The function $\langle\; \rangle$ is well defined. If $D_1$ and $D_2$
represent the same framed link, then $\langle D_1\rangle =\langle
D_2\rangle$.
\end{theorem}
Kauffman's construction is an example of a link invariant defined by
skein relations on the set of all diagrams. Since skein relations are
defined only in small neighborhoods, the idea generalizes naturally to
spaces locally modeled on ${\hspace{-0.3pt}\mathbb R}\hspace{0.3pt}^3$.
The notion of a skein module of a $3$-manifold\footnote{A suggestion
appeared in Conway's treatment of the Alexander polynomial
\cite{conway}.} was introduced independently by Przytycki in \cite{P1}
and Turaev in \cite{T1}. Roughly speaking, the construction consists
of dividing the linear space of all links by an appropriate set of
skein relations, usually the same as those known to define a
polynomial invariant in ${\hspace{-0.3pt}\mathbb R}\hspace{0.3pt}^3$. We will give the explicit definition
for the Kauffman bracket skein module.
Let ${\mathcal L}_M$ be the set of framed links (including $\emptyset$) in a
$3$-manifold $M$. Denote by ${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}{\mathcal L}_M$ the vector space consisting of
all linear combinations of framed links. Take ${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}{\mathcal L}_M[[h]]$ to be
formal power series with coefficients in ${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}{\mathcal L}_M$, and give it the
$h$-adic topology.\footnote{Skein modules were originally less
technical \cite{Lick2,P1,T1}. Power series first appeared in
\cite{T2}. Topological considerations were first addressed in
\cite{quant}.} This is an example of a topological module (see
\cite{kassel} for a nice introduction), however, one may think of
${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}{\mathcal L}_M[[h]]$ as just the completion of a vector space with basis
${\mathcal L}_M$ and scalars ${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}[[h]]$.
Let $t$ denote the formal series $e^{h/4}$ in ${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}[[h]]$. We define
the module of skein relations, $S(M)$, to be the the smallest subset
of ${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}{\mathcal L}_M[[h]]$ that is closed under addition, multiplication by
scalars and the $h$-adic topology, and which contains all expressions
of the form
\begin{enumerate}
\item $\displaystyle{\raisebox{-5pt}{\mbox{}\hspace{1pt+t\raisebox{-5pt}{\mbox{}\hspace{1pt+t^{-1}\raisebox{-5pt}{\mbox{}\hspace{1pt}$, \quad and
\vspace{5pt}
\item $\bigcirc+t^2+t^{-2}$.
\end{enumerate}
As before, (1) and (2) indicate relations that hold among links which
can be isotoped in $M$ so that they are identical except in the
neighborhood shown. The Kauffman bracket skein module is the quotient
\[K(M) = {\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}{\mathcal L}_M[[h]]/S(M).\]
This process can be mimicked for any choice of basis (oriented links,
links up to homotopy, etc.), any choice of scalars, any set of skein
relations, and with or without requiring topological completion. The
resulting quotient is generically called a skein module. For
instance, an older version of the Kauffman bracket skein module is
\[K_A(M)={\hspace{-0.3pt}\mathbb Z}\hspace{0.3pt}[A^{\pm 1}]{\mathcal L}_M / S(M),\]
with $t=-A$ in the skein relations, and without topology. If
$M={\hspace{-0.3pt}\mathbb R}\hspace{0.3pt}^3$ (or $B^3$ or $S^3$)
the new version is just an outrageous way of expanding the Kauffman
bracket into a power series.
\begin{theorem}[Kauffman--Przytycki--B--F--K] $K({\hspace{-0.3pt}\mathbb R}\hspace{0.3pt}^3) \cong {\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}[[h]]$
via $L \mapsto \langle L\rangle_{A=-t}$.
\end{theorem}
On one level, $K(M)$ is a generalization of the Kauffman bracket
polynomial. If $K(M)$ is topologically free (i.e. isomorphic to
$V[[h]]$ for some vector space $V$) then the isomorphism gives a power
series link invariant for each vector in a basis of $V$. The
coefficients behave like finite type link invariants \cite{quant},
generalizing a well known property of the Jones polynomial expanded as
a power series \cite{birman-lin}. In order to utilize the module in
this fashion, one would like to know that it is free, and what the
basis is; information that is decidedly difficult to come by.
The survey article \cite{HP3} contains a nearly complete list of those
manifolds for which the explicit computation has been done, the
exception being \cite{skein}. These computations predate the
topological version of the module, but whenever $K_A(M)$ is free,
$K(M)$ is just its completion after substituting $A=-t$.
There is, however, a deeper understanding of skein modules,
$K(M)$ in particular. Przytycki often refers to skein theory as
``algebraic topology based on knots,'' alluding strongly to
skein modules as a sort of non-commutative alternative to homology.
This is also reflected in the principle that loops up to homotopy
carry classical information, whereas knots up to isotopy carry quantum
information. The notion that a skein module is a quantization or a
deformation of some kind can be made very explicit for $M=F \times I$,
$F$ being a compact oriented surface.
In this case, $K(F\times I)$ is an
algebra. Multiplication of links in ${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}{\mathcal L}_{F\times I}$ is by
stacking one atop the other; it extends obviously to ${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}{\mathcal L}_{F\times
I}[[h]]$, and it is a simple matter to check that $S(F \times I)$ is
an ideal. Crossings form a barrier to commutativity in
${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}{\mathcal L}_{F\times I}$, and, for most surfaces, the obstruction survives
in the quotient.\footnote{ The exceptions are planar surfaces with
$\chi(F) \geq -1$.}
It is possible for non-homeomorphic surfaces $F_1$ and $F_2$ to have
homeomorphic cylinders, $F_1\times I$ and $F_2\times I$. The
homeomorphism does not preserve the algebra
structure.\footnote{$F\times I$ in Theorem \ref{BP} is homeomorphic to
$\raisebox{-2pt}{\epsfig{file=pants.eps,height=8pt}} \times I$, whose
skein algebra is commutative.} For this reason, it makes sense to
compress to notation into $K(F) = {\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}{\mathcal L}_F[[h]]/S(F)$.
For an example of $K(F)$ as a ``deformation'', recall that the
commutative algebra
of polynomials in three variables (over your favorite scalars) is
presented by
\[ \langle x,y,z\ |\ xy-yx=0, yz-zy=0, zx-xz=0 \rangle.\]
\begin{theorem}[B--Przytycki]\label{BP} If $F$ is a once punctured
torus, then $K_A(F)$ is presented by\footnote{The variables $x$, $y$
and $z$ are a meridian, a longitude and a slope one curve on $F$.}
\begin{align*}
\langle x,y,z\ |\ &Axy - A^{-1}yx=(A^2-A^{-2})z \\
&Ayz - A^{-1}zy=(A^2-A^{-2})x \\
&Azx - A^{-1}xz=(A^2-A^{-2})y \rangle.
\end{align*}
\end{theorem}
Theorem \ref{BP} can be thought of as a $1$-parameter family of
presentations,\footnote{Barrett \cite{barrett} showed that a spin
structure on $M$ induces an isomorphism $K_A(M)\cong K_{-A}(M)$.}
which reduces to the commutative polynomials if $A=\pm 1$. Other
examples can be found in \cite{bptorus}.
Yet another way to see the module as a deformation is to let the
parameter $h$ go to $0$ (or $t$ to $1$, or $A$ to $-1$). Formally,
this is achieved by passing to the quotient $K_0(F)=K(F)/hK(F)$. In
this case the skein relations would become
\begin{enumerate}
\item $\displaystyle{\raisebox{-5pt}{\mbox{}\hspace{1pt+\raisebox{-5pt}{\mbox{}\hspace{1pt+\raisebox{-5pt}{\mbox{}\hspace{1pt}$, \quad and
\vspace{5pt}
\item $\bigcirc+2$.
\end{enumerate}
Taken together these allow crossings to be changed at will and make
framing irrelevant. Hence, the ``undeformed'' module is a commutative
algebra spanned by free homotopy classes of collections of loops.
The multiplication in this algebra is commutative.
This is all quite heuristic for we lack a precise definition of
quantization or deformation. The next section will address this. Even
so, if one can only understand the underlying commutative algebra as
an obvious quotient of the deformation, there is little here but
tautology. We will close this section with an interpretation of
$K_0(F) = K(F)/hK(F)$ in terms of group characters.
Suppose that $G$ is a finitely presented group with generators
$\{a_i\}_{i=1}^m$ and relations $\{r_j\}_{j=1}^n$. The space of $\g$
representations of $G$ is a closed affine algebraic
set.\footnote{Shafarevich \cite{shaf} is a good reference for the
algebraic geometry.} You can view the representations as lying in
$\prod_{i=1}^m \g \subset {\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}^{4m}$. Each of the relations $r_j$
induces four equations from the coefficients of $r_j(A_1, \ldots
,A_m)=I$. The zero set of these polynomials restricted to the variety
$\prod_{i=1}^m \g$ is the representation space.
We might naively try to construct the coordinate ring of the
representations as follows. Let ${\mathcal I}$ be the ideal generated by the
equations $r_j(A_1, \ldots ,A_m)=I$ in the coordinate ring $
C[\prod_{i=1}^m\g]$. Let $R(G)= C[\prod_{i=1}^m\g]/{\mathcal I}$. The
problem is that $R(G)$ might have nilpotents (equivalently,
$\sqrt{{\mathcal I}} \neq {\mathcal I}$). However, it was proved in \cite{LM} that
$R(G)$ is an isomorphism invariant of $G$. There is an action of $\g$
on $R(G)$ induced by conjugation in the factors of $\prod_{i=1}^m
\g$. The part of the ring fixed by this action is the affine
$\g$-characters of $G$, denoted $R(G)^{\g}$. It, too, is an isomorphism
invariant of $G$ \cite{BH}. For our purposes, it suffices to define
the ring of $\g$-characters of $G$ to be\footnote{It is a deep result
of Culler and Shalen \cite{CS} that the set of characters of $\g$
representations of $G$ is a closed affine algebraic set. Its
coordinate ring is $\Xi(G)$.}
\[ \Xi(G) = R(G)^{\g}/\sqrt{0}.\]
If $X$ is a manifold, we will write $\Xi(X)$ rather than
$\Xi(\pi_1(X))$.
The connection with 3-manifolds is quite simple (see
\cite{reps,estimate,isomorphism,PS1} for details). Suppose that
$\rho:\pi_1(M) \rightarrow \g$ is a representation and $\chi_\rho$ is its
character. Let $K$ be a loop, thought of as a conjugacy class in
$\pi_1(M)$. Since the trace of a matrix in $\g$ is invariant under
inversion and conjugation, it makes sense to speak of $\chi_\rho(K)$
regardless of the choice of a starting point or orientation. The loop
$K$ determines an element of $R(G)^{\g}$ by $K(\rho) = -\chi_\rho(K)$. The
function extends to
\[ \Phi :K_0(M) \rightarrow R(G)^{\g}\]
by requiring it to be a map of algebras. It is well defined because
the relations in $K_0(M)$ are sent to the fundamental $\g$ trace
identities:
\begin{enumerate}
\item ${\rm tr}(AB)+{\rm tr}(AB^{-1})={\rm tr}(A){\rm tr}(B)$, and
\item ${\rm tr}(I)=2$.
\end{enumerate}
It is shown in \cite{isomorphism} that the image of $\Phi$ is a
particular presentation of the affine characters \cite{GM}. Sikora
\cite{sikora} has achieved this by directly identifying a version of
$K_0(M)$ with $R(\pi_1(M))^{\g}$ as defined by Brumfiel and Hilden.
Przytycki and Sikora \cite{PS1,PS2} have computed $K_0(M)$ for a large
number of manifolds, including $M=F\times I$, for which they can prove
it has no nilpotents.\footnote{They have also worked with various
scalars. Certainly if the scalar ring has nilpotents then $K_0(M)$
does as well, and they have even located a nilpotent with scalar field
$Z_2$. However, no nilpotents have ever been found with scalar ring
${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}[[h]]$.} Summarizing, we have a good idea of what $K_0(M)$ is in
general, and we know exactly what $K_0(F)$ is.
\begin{theorem}[B-Przytycki-Sikora]\label{BPS}
$\Phi :K_0(F) \rightarrow \Xi(F)$ is an isomorphism.
\end{theorem}
\section{Poisson Quantization of Surface Group Characters}
In the previous section we saw how a non-commutative algebra shrinks
to a commutative specialization for some particular value of a
deformation parameter. The formal definition of quantization reverses
this process. Beginning with a commutative algebra, one introduces a
parameter $h$, and a ``direction'' of deformation. The direction is a
Poisson bracket.
To make this precise, a commutative algebra $A$ is
called a Poisson algebra if it is equipped with a bilinear, antisymmetric map
$ \{\ ,\ \} : A \otimes A \rightarrow A$ which
satisfies the Jacobi identity:
\[ \{a,\{b,c\}\} +\{b,\{c,a\}\} + \{c,\{a,b\}\} = 0 , \]
and is a derivation:
\[ \{ab,c\} = a\{b,c\}+ b\{a,c\}, \]
for any $a,b,c \in A$.
A quantization of a complex Poisson algebra $A$ is a
${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}[[h]]$-algebra, $A_h$, together with a ${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}$-algebra isomorphism,
$\Phi: A_h/hA_h \rightarrow A$, satisfying the following properties:
\begin{itemize}
\item as a ${\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}[[h]]$-module $A_h$ is topologically free
(i.e. $A_h\equiv V[[h]]$);
\item if
$a,b \in A$ and $a',b'$ are any elements of $A_h$ with $\Phi(a')=a$
and $\Phi(b')=b$, then
\[\Phi\left( \frac{a'b'-b'a'}{h}\right) = \{a,b\}. \]
\end{itemize}
Hoste and Przytycki \cite{HPquant}, and Turaev \cite{T2} knew that
certain skein modules gave Poisson quantizations of various algebras
based on loops in a surface.\footnote{Their modules have a slightly
different flavor than the one defined here, both because topology is
not considered and because the scalars are not necessarily power
series. Their definitions of Poisson quantization are analogously
distinct. It is also interesting to note that their work predates the
appearance of quantum groups in low-dimensional topology.} Since
$K(F)$ is topologically free (\cite{P1}, \cite{quant}), one can easily
see it as a Poisson quantization of $K_0(F)$ with the obvious bracket:
\[\{a,b\}=\text{lead coefficient of $a'b'-b'a'$ in $K(M)$.}\]
As noted in Section 3, however, understanding the Poisson algebra $K_0(F)$
as a formal quotient of $K(M)$ yields no new insight.
This is where character theory reenters.
Since $\Xi(F)$ is the complexification of the $SU(2)$-characters of
$\pi_1(F)$, it has a Poisson structure given by complexifying the
standard one on $SU(2)$-characters \cite{Go,BG}. Recall
(Theorem \ref{BPS}) that the algebra $\Xi(F)$ is generated by the
functions corresponding to loops. The Poisson bracket is given by an
intersection pairing on oriented loops, and extended to all of
$\Xi(F)$. In \cite{quant} this is reformulated as a state sum using
unoriented loops, proving
\begin{theorem}[B-F-K]
$K(F)$ and the map $\Phi : K_0(F \times I) \rightarrow \Xi(F)$ form a
Poisson quantization of the standard Poisson algebra $\Xi(F)$.
\end{theorem}
\section{Lattice Gauge Field Theory}
Lattice gauge field theory gives an alternative quantization of
$\Xi(F)$. To see this, we first sketch how an $SU(2)$ gauge theory on
$F$ recovers the $SU(2)$-characters of $\pi_1(F)$. We then pass to a
lattice model of the theory, in which a Lie group may be replaced
with its universal enveloping algebra. Finally, the enveloping
algebra may be deformed to a quantum group. Along the way, of course,
we will complexify to return to the $\g$ setting.
An $SU(2)$ gauge theory over $F$ consists of connections, gauge
transformations (also called the gauge group) and gauge fields.
These objects have technical definitions involving the geometry of an
$SU(2)$-bundle over $F$, but for our purposes only a few consequences
are relevant.
First of all, a connection determines a notion of parallel transport
along a path, $\gamma$, which assigns to it an element $hol(\gamma)$
of $SU(2)$. This element is called the holonomy of the connection
along $\gamma$. Notice that if you traverse the path in the opposite
direction then the holonomy is the inverse. A connection is flat if
holonomy only depends on the homotopy class of a path relative to its
endpoints.
Second, the gauge group acts on connections. A gauge
transformation can be thought of as an element of $SU(2)$ assigned to
each point of $F$. Its effect on a connection is irrelevant; its
effect on holonomy is $hol(\gamma) \mapsto g\;hol(\gamma)\;h^{-1}$,
where $g$ and $h$ correspond to the beginning and
end points of $\gamma$.
Finally, gauge fields are (real analytic) functions on connections.
There is an adjoint action of the gauge group on gauge
fields;\footnote{For a gauge transformation $g$, a gauge field $f$,
and a connection $x$, $(g\bullet f)(x) = f(g\bullet x)$.}
invariant gauge fields are called observables.
Flat connections give rise to representations of $\pi_1(F)$ into $\g$
via holonomy of loops. There are actually more flat connections than
representations. However, two connections are gauge equivalent if and
only if their holonomy representations are conjugate.
The observables, restricted to flat connections, are a space of (real
analytic) functions on $SU(2)$ representations, which are invariant
under conjugation. The ``polynomials'' in this space---a dense
set---are the $SU(2)$-characters of $F$.
Much of the technical detail glossed over in the last few paragraphs
vanishes if we pass to a lattice model; a combinatorial setting in
which geometry is disposed of and the behavior of holonomy is
axiomatized. As a bonus, one need not base the theory on a compact Lie
group. What follows works for any affine algebraic group, but we will
stick to $\g$ for continuity.
Suppose that $F$ is triangulated. The 1-skeleton of the triangulation
of $F$ is a graph. Let $V$ denote the set of vertices and $E$ the set
of edges, each with an orientation. The objects of a lattice gauge
field theory over $F$ are:
\begin{enumerate}
\item the connections, $\displaystyle{{\hspace{-0.3pt}\mathbb A}\hspace{0.3pt}=\prod_{e \in E} \g}$,
\item the gauge group, $\displaystyle{{\mathcal G}=\prod_{v\in V} \g}$, and
\item the gauge fields, $\displaystyle{C[{\hspace{-0.3pt}\mathbb A}\hspace{0.3pt}]= \bigotimes_{e\in E}
C[\g]}$.
\end{enumerate}
In the formula above, $C[\g]$ is the coordinate ring of $\g$.
One thinks of a connection as assigning an element of $\g$ to each
edge. A path is a string of edges. Holonomy of $(x_1,x_2,x_3)$ along
the path $\{e_1,e_2,e_3\}$ is depicted in Figure \ref{holonomy}. Note
that holonomy is clearly inverted if the path is reversed.
\begin{figure}
\centering
\makebox[117pt]{\mbox{}\hfill $x_1$ \hfill $x_2$ \hfill $x_3$ \hfill\mbox{}}\makebox[160pt]{}\\
\raisebox{-3pt}{\mbox{\epsfig{file=holonomy.eps,width=117pt}}}
\makebox[160pt]{\mbox{}\hfill $\Longrightarrow$ \hfill
$hol(x_1,x_2,x_3)=x_1x_2^{-1}x_3$ \hfill\mbox{}}\\
\makebox[117pt]{\mbox{}\hfill $e_1$ \hfill $e_2$ \hfill $e_3$ \hfill\mbox{}}
\makebox[160pt]{}
\caption{Example of holonomy in a lattice.}
\label{holonomy}
\end{figure}
One thinks of a gauge transformation as an element of $\g$ at each
vertex. The action of the gauge group on a connection is illustrated
near a vertex in Figure \ref{action}. Note that the action is by
$y^{-1}$ on the right if an edge points in and by $y$ on the left if
it points out, a convention we adhere to through this and the next two
sections.
\begin{figure}
\centering
\makebox[30pt]{}\makebox[.15in][r]{\raisebox{-14pt}{$y$}}
\makebox[.35in][l]{$x_2$}\makebox[90pt]{}
\makebox[.5in]{$yx_2$}
\makebox[30pt]{}\\
\vspace{-12pt}
\makebox[30pt][r]{\raisebox{.25in}{$x_3$}}
\epsfig{file=vertex.eps,width=.5in}
\makebox[90pt]{\raisebox{.25in}{$x_1$}\hfill
\raisebox{.25in}{$\Longrightarrow$}\hfill
\raisebox{.25in}{$x_3y^{-1}$}}
\epsfig{file=vertex.eps,width=.5in}
\makebox[30pt][l]{\raisebox{.25in}{$yx_1$}}\\
\makebox[30pt]{}\makebox[.5in]{$x_4$}\makebox[90pt]{}
\makebox[.5in]{$yx_4$}\makebox[30pt]{}\\
\caption{Gauge group action at a vertex.}
\label{action}
\end{figure}
The gauge fields can be evaluated on connections in the obvious way.
By taking adjoints we get an action of the gauge group on the gauge
fields. The fixed subring of this action is the ring of
$\g$-characters of the one skeleton. If $G$ is the fundamental group
of the 1-skeleton this ring is isomorphic to $\Xi(G)$.
Flatness should amount to holonomy being independent of path, but in a
lattice model we prefer the following equivalent definition. A
connection is flat on a face of the triangulation if it is gauge
equivalent to one which has 1 on each edge of the face. A flat
connection is flat on each face. Invariant gauge fields evaluated on
flat connections form a ring of observables which, regardless of the
choice of triangulation, is isomorphic to
$\Xi(F)$.\footnote{Technically, divide the gauge field algebra by the
annihilator of all flat connections and then restrict to the gauge
invariant part of the quotient.}
This is an easily manipulated model of a gauge theory, but groups do
not quantize; algebras do. So, replace $\g$ with the universal
enveloping algebra $U(sl_2)$. This is a cocommutative Hopf algebra. The
interested reader may find a full explanation in \cite{abe} for
example, but we can get by with less. There is an involution
$S:U(sl_2)\rightarrowU(sl_2)$ that corresponds to inversion in the group, a
counit $\epsilon :U(sl_2)\rightarrow{\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}$, and a comultiplication
$\Delta:U(sl_2)\rightarrowU(sl_2)\otimesU(sl_2)$. One may regard $\Delta^n$ as an
operation that breaks an element of $U(sl_2)$ into states residing in
$U(sl_2)^{\otimes(n+1)}$. The notation for this is due to Sweedler
\cite{sweedler}. For example,
\[ \Delta^3(y) = \sum_{(y)} y^{(1)}\otimes y^{(2)}\otimes
y^{(3)}\otimes y^{(4)}. \]
Since $C[\g]$ lies in the dual of $U(sl_2)$, we can almost repeat the
entire process with
\begin{enumerate}
\item connections $\displaystyle{{\hspace{-0.3pt}\mathbb A}\hspace{0.3pt}=\bigotimes_{e \in E} U(sl_2)}$,
\item gauge algebra $\displaystyle{{\mathcal G}=\bigotimes_{v\in V} U(sl_2)}$, and
\item gauge fields $\displaystyle{C[{\hspace{-0.3pt}\mathbb A}\hspace{0.3pt}]= \bigotimes_{e\in E}C[\g]}$.
\end{enumerate}
The catch is the gauge action. In order to make sense of it, we need
to assign an ordering to the edges at each vertex. This is done by
marking the vertex with a cilium (see Figure \ref{ciliated-vertex})
after which the orientation on $F$ gives a counter-clockwise ordering
of the edges.
\begin{figure}[b]
\centering
\makebox[30pt]{}\makebox[.5in]{\raisebox{2pt}{$e_2$}}\makebox[30pt]{}\\
\makebox[30pt][r]{\raisebox{.25in}{$e_3$}}
\epsfig{file=ciliated-vertex.eps,width=.5in}
\makebox[30pt][l]{\raisebox{.25in}{$e_1$}} \\
\makebox[30pt]{}\makebox[.5in]{$e_4$}\makebox[30pt]{}
\caption{Ciliated vertex with edges ordered $e_1<e_2<e_3<e_4$.}
\label{ciliated-vertex}
\end{figure}
It is best to think of connections and gauge transformations as pure
tensors, remembering always to extend linearly. We thus view a
connection as an assignment of an element of $U(sl_2)$ to each edge of the
triangulation. Holonomy is apparent; for the path in Figure
\ref{holonomy} it would be $x_1S(x_2)x_3$, where $S$ is the antipode
of $U(sl_2)$. We continue to think of a gauge transformation as an element
of $U(sl_2)$ at each vertex, with the action at a vertex illustrated in
Figure \ref{q-action}.
\begin{figure}
\centering
\makebox[30pt]{}\makebox[.15in]{\raisebox{-14pt}{$y$}}
\makebox[.34in][l]{$x_2$}\makebox[90pt]{}
\makebox[.5in]{$y^{(2)}x_2$}
\makebox[30pt]{}\\
\vspace{-12pt}
\makebox[30pt][r]{\raisebox{.25in}{$x_3$}}
\epsfig{file=ciliated-vertex.eps,width=.5in}
\makebox[90pt]{\raisebox{.25in}{$x_1$}\hfill
\raisebox{.25in}{$\Longrightarrow$}\hfill
\raisebox{.25in}{$x_3S(y^{(3)})$}}
\epsfig{file=ciliated-vertex.eps,width=.5in}
\makebox[30pt][l]{\raisebox{.25in}{$y^{(1)}x_1$}}\\
\makebox[30pt]{}\makebox[.5in]{$x_4$}\makebox[90pt]{}
\makebox[.5in]{$y^{(4)}x_4$}\makebox[30pt]{}\\
\caption{Gauge algebra action at a vertex.}
\label{q-action}
\end{figure}
A further problem with the action is that gauge ``equivalence'' is not
an equivalence relation anymore, necessitating a slight technical
modification of flatness which we will not address here. Also, the
word ``invariant'' means $y\bullet x = \epsilon(y) x.$ However, the
passage to gauge fields on flat connections modulo the gauge algebra
proceeds as before, giving exactly the same ring.
Finally we pass to $U_h(sl_2)$. This is a quasi-triangular ribbon Hopf
algebra \cite{kassel}. It is non-cocommutative in a fashion
constrained by an element of $U_h(sl_2)\otimesU_h(sl_2)$ called the universal
$R$-matrix. The antipode $S$ is no longer an involution; rather $S^2$
acts as conjugation by the so-called charmed element, $k$. The
definition of flat connection is further altered, preserving
independence of path but deforming the holonomy of a trivial loop to
$k^{\pm 1}$. The dual of $U_h(sl_2)$ contains a deformation, $\mbox{}_qSL_2$, of
$C[\g]$. Thus one hopes to obtain a quantized ring of observables by
replacing each object with its quantum analogue.
There is one small problem. The natural multiplication on
$C[{\hspace{-0.3pt}\mathbb A}\hspace{0.3pt}]= \otimes_{e\in E}\mbox{}_qSL_2$ (i.e. the one dual to
the natural comultiplication on ${\hspace{-0.3pt}\mathbb A}\hspace{0.3pt}=\otimes_{e \in
E} U_h(sl_2)$) is not gauge invariant. This is a major obstruction, and
the solution is notable enough to occupy the next section. However,
once it has been addressed, we will have
\begin{theorem}
Quantum observables exist. They form a ring,
$\Xi_h(F)$, which is independent of triangulation and ciliation, and
which quantizes $\Xi(F)$.
\end{theorem}
\section{Nabla}
\begin{figure}[b]
\centering
\makebox[30pt]{}\makebox[.5in]{\raisebox{2pt}{$e_2$}}\makebox[30pt]{}\\
\makebox[30pt][r]{$e_3$}
\epsfig{file=trivalent.eps,width=.5in}
\makebox[30pt][l]{$e_1$}
\caption{Trivalent ciliated vertex.}
\label{trivalent}
\end{figure}
The natural comultiplication on the coalgebra of quantum connections
is a tensor power of $\Delta$ composed with a permutation. For
instance, it would send
\[x_1\otimes x_2 \mapsto x_1^{(1)}\otimes x_2^{(1)}\otimes
x_1^{(2)}\otimes x_2^{(2)}.\]
Expanding on a theme of quantum topology, we denote this morphism by a
tangle built from branches for each application of $\Delta$ and a
braid corresponding to the permutation. We then obtain a quantized
comultiplication
\[\nabla : {\hspace{-0.3pt}\mathbb A}\hspace{0.3pt} \rightarrow {\hspace{-0.3pt}\mathbb A}\hspace{0.3pt}\otimes {\hspace{-0.3pt}\mathbb A}\hspace{0.3pt} \]
by allowing crossings to encode actions of the $R$-matrix.
There is a fundamental tangle associated to any vertex---the one in
Figure \ref{trivalent}, for example---whose construction proceeds in
stages. First, assign a coupon to each edge as in Figure
\ref{level-one}.
\begin{figure}
\centering
\epsfig{file=level-one.eps,width=1.5in,height=.5in}\\
$e_1$\hspace{.5in}$e_2$\hspace{.5in}$e_3$
\caption{Coupons for each edge.}
\label{level-one}
\end{figure}
There are two types of these, depending on whether the edge points in
or out, and they must be ordered left to right matching the cilial
order of the edges. Next, we construct a $2n$-braid ($n=\text{valence
of the vertex}$) by dragging odd numbered strands left and even
numbered strands right.\footnote{The inherent ambiguity evaporates
when we construct the morphism because the $R$-matrix solves the
Yang-Baxter equation \cite{kassel}. It is an elegant feature of
quantum topology that isotopies of tangles correspond to identities in
a quantum group.} Evens lie over odds. Our example is the the 6-braid
in Figure \ref{level-two}.
\begin{figure}
\centering
\epsfig{file=level-two.eps,width=1.5in,height=.5in}
\caption{Six-braid encoding the permutation $(1)(2453)(6)$.}
\label{level-two}
\end{figure}
The fundamental tangle is formed by stacking the braid atop the
coupons. Orientation of the coupons carries over to the strands of
the braid.
Now imagine $x_1\otimes x_2\otimes x_3$ entering the tangle from the
bottom and traveling upward. Each branch indicates comultiplication
with the output ordered as in Figure \ref{branches}.
\begin{figure}[b]
\centering
\makebox[.4in]{$x''$}\makebox[.4in]{$x'$}\hspace{.4in}
\makebox[.4in]{$x'$}\makebox[.4in]{$x''$}\\
\vspace{2pt}
\epsfig{file=comult.eps,width=1.6in,height=.5in}\\
\makebox[.8in]{$x$}\hspace{.4in}\makebox[.8in]{$x$}
\caption{Comultiplication acting at a branch.}
\label{branches}
\end{figure}
Note that we are suppressing the summation symbols.
Each crossing corresponds to an action of the $R$-matrix, which we
write as $R = \sum_i \alpha_i\otimes \beta_i$. The four possibilities
are shown in Figure \ref{crossings}, again suppressing summation.
\begin{figure}
\centering
\makebox[.4in]{$\beta_iy$}\makebox[.4in]{$\alpha_ix$}\hspace{.4in}
\makebox[.4in]{$\beta_iy$}\makebox[.4in]{$xS(\alpha_i)$}\hspace{.4in}
\makebox[.4in]{$yS(\beta_i)$}\makebox[.4in]{$\alpha_ix$}\hspace{.2in}
\makebox[.5in]{$yS(\beta_i)$}\makebox[.5in]{$xS(\alpha_i)$} \\
\vspace{2pt}
\epsfig{file=R-mat.eps,width=4in,height=.5in}\\
\makebox[.4in]{$x$}\makebox[.4in]{$y$}\hspace{.4in}
\makebox[.4in]{$x$}\makebox[.4in]{$y$}\hspace{.4in}
\makebox[.4in]{$x$}\makebox[.4in]{$y$}\hspace{.2in}
\makebox[.5in]{$x$}\makebox[.5in]{$y$}
\caption{Actions of the $R$-matrix at crossings.}
\label{crossings}
\end{figure}
Note that, as usual, left and right multiplication correspond
respectively to outward and inward pointing edges, and that right
multiplication is preceded by an application of $S$.
Sweeping out, we get a morphism
\[ {\hspace{-0.3pt}\mathbb F}\hspace{0.3pt}_v : \bigotimes_{\text{edges at $v$}} U_h(sl_2) \rightarrow
\left(\bigotimes_{\text{edges at $v$}} U_h(sl_2)\right)^{\otimes 2}.\]
In our example,
\begin{multline*}
x_1\otimes x_2 \otimes x_3 \mapsto \\
x_1'\otimes x_2'S(\beta_1)S(\beta_3) \otimes
x_3'S(\beta_2)S(\beta_4)S(\beta_5) \otimes \alpha_5\alpha_3 x_1''
\otimes x_2''S(\alpha_2)S(\alpha_4) \otimes x_3''S(\alpha_2).
\end{multline*}
Eight summations are suppressed, and the subscripts on $\alpha_j$ and
$\beta_j$ are shorthand for summation over the $j$-th application of
the $R$-matrix.
The morphism ${\hspace{-0.3pt}\mathbb F}\hspace{0.3pt}_v$ is coassociative in the sense that $(Id \otimes
{\hspace{-0.3pt}\mathbb F}\hspace{0.3pt}_v)\circ {\hspace{-0.3pt}\mathbb F}\hspace{0.3pt}_v = ( {\hspace{-0.3pt}\mathbb F}\hspace{0.3pt}_v\otimes Id)\circ {\hspace{-0.3pt}\mathbb F}\hspace{0.3pt}_v$. Furthermore, its
effect in a given factor of $(\bigotimes_{\text{edges at $v$}}
U_h(sl_2))^{\otimes 2}$ is either entirely by right multiplication or
entirely by left multiplication. This allows us to combine the
effects of $\{{\hspace{-0.3pt}\mathbb F}\hspace{0.3pt}_v\;|\;v\in V\}$ into a single morphism
\[\nabla : {\hspace{-0.3pt}\mathbb A}\hspace{0.3pt} \rightarrow {\hspace{-0.3pt}\mathbb A}\hspace{0.3pt} \otimes {\hspace{-0.3pt}\mathbb A}\hspace{0.3pt}.\]
\begin{theorem}
$\nabla$ is coassociative and gauge invariant.
\end{theorem}
\section{Quantum Observables for $\g$}
At the end of Section 2 we saw how loops became functions generating
$\Xi(F)$. In this section we will describe the quantum analogue of
that fact. All definitions are given in terms of a running example,
so assume that $\Gamma$ is the oriented, ciliated graph in Figure
\ref{bowtie}.
\begin{figure}[b]
\centering
\makebox[2.5in]{
\mbox{}\hfill $v_4$ \hfill \raisebox{-.2in}{$e_4$} \hfill
\raisebox{-.4in}{$v_3$} \hfill \raisebox{-.2in}{$e_1$} \hfill $v_1$
\hfill\mbox{}} \\ \vspace{-.3in}
\makebox[2.5in]{
\raisebox{.5in}{$e_5$} \hfill \epsfig{file=bowtie.eps,width=2in}
\hfill \raisebox{.5in}{$e_2$}} \\ \vspace{-.1in}
\makebox[2.5in]{
\mbox{}\hfill $v_5$ \hfill \raisebox{.2in}{$e_6$} \hfill \mbox{}
\hfill \raisebox{.2in}{$e_3$} \hfill $v_2$ \hfill\mbox{}}
\caption{Oriented ciliated graph $\Gamma$.}
\label{bowtie}
\end{figure}
Following Section 6 we have
\begin{enumerate}
\item the connection coalgebra,
$\displaystyle{{\hspace{-0.3pt}\mathbb A}\hspace{0.3pt} =U_h(sl_2)^{\otimes 6}}$,
\item the gauge algebra,
$\displaystyle{{\mathcal G}=U_h(sl_2)^{\otimes 5}}$, and
\item the algebra of gauge fields,
$\displaystyle{C[{\hspace{-0.3pt}\mathbb A}\hspace{0.3pt}] = (\mbox{}_qSL_2)^{\otimes 6}}$.
\end{enumerate}
Bowing to technicalities, a loop in $\Gamma$ will be allowed to meet
each edge at most once, and each vertex at most twice. In accordance
with the theme of quantization by crossings, we say a $q$-loop is a
loop with a choice of under or over crossing whenever it intersects
itself transversely. We express this as a sequence of edges with $+$
and $-$ signs interspersed. For example,
\[l = \{e_1,e_2,e_3,+,e_4,e_5,e_6,-\} \]
is a $q$-loop. It defines an element of $C[{\hspace{-0.3pt}\mathbb A}\hspace{0.3pt}]$ via the following
graphical recipe.
\begin{enumerate}
\item Choose a pure tensor $x = x_1\otimes\cdots\otimes x_6 \in {\hspace{-0.3pt}\mathbb A}\hspace{0.3pt}$.
Draw a picture of $\Gamma$ with $x_i$ labeling each corresponding edge.
\item Apply $\epsilon$ to any edge not appearing in the loop. (No effect
on this example.)
\item At the crossing, act by an $R$-matrix, either
$\sum\alpha_i\otimes\beta_i$ or $\sum\beta_i\otimes S(\alpha_i)$. The
action will take place on the first two edges in the ciliation; the
$R$-matrix is chosen so that $\beta_i$ acts on the bottom strand
($\sum\alpha_i\otimes\beta_i$ in this example); and the action follows
left/right rules as in Section 5. Write this on the appropriate
edges, suppressing summations.
\item If an edge is oriented against the direction of the loop,
multiply on the right by $k$, and then apply $S$.
\item Each time the loop passes through a vertex check to see if the
incoming edge goes before the outgoing edge in the ciliation. If not,
then right multiply by $k$ on the incoming edge. The picture should
now look like Figure \ref{sample-loop}.
\begin{figure}
\centering
\makebox[1in]{}
\makebox[1in][l]{\hspace{.5in}$x_4k$}
\makebox[1in][r]{$\beta_ix_1$\hspace{.5in}}
\makebox[1in]{} \\ \vspace{-.2in}
\makebox[1in][r]{\raisebox{.5in}{$S(x_5k)k$}}\makebox[2in]{
\epsfig{file=bowtie.eps,width=2in}}
\makebox[1in][l]{\raisebox{.5in}{$x_2$}} \\ \vspace{-.2in}
\makebox[1in]{}
\makebox[1in][l]{\hspace{.5in}$x_6k$}
\makebox[1in][r]{$x_3S(\alpha_i)$\hspace{.5in}}
\makebox[1in]{}
\caption{Action of the $q$-loop $l$ on $x$.}
\label{sample-loop}
\end{figure}
\item Multiply everything together as you traverse the loop. Take the
image of this ``quantum holonomy'' in the fundamental representation
(see \cite{KM} or \cite{RT1}) of $U_h(sl_2)$. Finally, take the trace
to get a complex number:
\[x \mapsto \sum_i{\rm tr}(\beta_ix_1x_2x_3S(\alpha_i)x_4S(x_5)kx_6k).\]
\item Extend linearly over all of ${\hspace{-0.3pt}\mathbb A}\hspace{0.3pt}$ to obtain a function
$W_l$.
\end{enumerate}
The function we have defined is usually called a Wilson loop in the
literature. It should be clear that a $q$-loop is just a knot diagram
with a base point and an orientation. Our goal is to assign a quantum
observable to each equivalence class of link diagrams.
The first step is to note that the rules we gave for acting on edges
prior to computing holonomy are local. One could just as well apply
them to a link of loops, compute holonomy along each, and take the
product of the resulting traces. Clearly the individual traces are
independent of base points. Reversing orientation is less trivial, as
it involves $S$ rather than inversion. But ${\rm tr}(S(x))={\rm tr}(x)$ in the
fundamental representation, so orientations don't matter. Gauge
invariance is easily checked at individual vertices. Finally, suppose
that $\Gamma$ is the 1-skeleton of a triangulated surface. Let $l$ and
$l'$ be equivalent link diagrams and $x$ a flat connection. Since
flatness implies independence of path, $W_l = W_{l'}$.
At this point we see that a link $L$ determines an observable, $W_L$,
provided one has a fine enough triangulation of $F$. Since $\Xi_h(F)$
is independent of triangulation, we can make the assignment
\[ L \mapsto (-1)^{|L|}W_L,\]
where $|L|$ is the number of components of $L$. Linearity and
continuity extend it uniquely to a map
\[\Phi_h : {\hspace{-0.3pt}\mathbb C}\hspace{0.3pt}{\mathcal L}_{F\times I}[[h]] \rightarrow \Xi_h(F).\]
This is the quantum analogue of $\Phi$ from Section 2, taking loops to
the character ring. That map took (adjoints of) the skein relations
in $K_0$ to the fundamental $\g$-trace identity and to ${\rm tr}(I)=2$.
The corresponding quantized identities in $U_h(sl_2)$ are
\begin{align*}
t\;{\rm tr}(ZY)+t^{-1}\;{\rm tr}(S(Z)W) &= \sum_i{\rm tr}(\alpha_iz){\rm tr}(\beta_ix),
\quad\text{and}\\
{\rm tr}(k^{\pm 1})& = t^2+t^{-2}.
\end{align*}
It is clear from the definition of flat connection that the (adjoint
of the) skein relation $\bigcirc+(t^2+t^{-2})$ maps to ${\rm tr}(k) =
t^2+t^{-2}$, but
\[ \raisebox{-5pt}{\mbox{}\hspace{1pt+t\raisebox{-5pt}{\mbox{}\hspace{1pt+t^{-1}\raisebox{-5pt}{\mbox{}\hspace{1pt\]
is more complicated because it has less symmetry than the quantum
trace identity. In some cases, the (adjoint) skein relation is
obviously mapped to an identity, while others require more manipulation.
\begin{theorem}
$\Phi_h(S(F)) = 0$. Furthermore, the quotient map
\[\widetilde{\Phi}_h:K(F) \rightarrow \Xi_h(F)\]
is an isomorphism.
\end{theorem}
\section{The Future}
The results described in this survey place skein theory at the
confluence of ideas from topology, representation theory,
noncommutative algebra and mathematical physics. Standard
techniques from skein theory \cite{HP3,HP4} extend our lattice
construction of $K(F)$ to a description of
the Kauffman bracket skein module of an arbitrary compact
$3$-manifold. Consequently, $K(M)$ has an intensive definition in
terms of links and skein relations, and an extensive definition in
terms of quantized invariant theory. This nexus suggests some avenues
for further research.
It is proved in \cite{quant} that the affine $\g$-characters induce
topological generators of $K(M)$. In particular, the Kauffman bracket
skein module of a small $3$-manifold (i.e. containing no
incompressible surface) is finitely generated, and thus can be used as
a classification tool.
If $K(M)$ is topologically free then there is a meaningful
pairing between it and the set of equivalence classes of
$\g$-representations of $\pi_1(M)$. For nilpotent free $K_0(M)$ this
is a duality pairing. In the case of $F\times I$ the pairing has an
especially easy form because the basis is a canonical set of links
\cite{quant}. The Yang-Mills measure on the algebra of observables
can be computed along the same lines \cite{Bu}. This holds out the
promise of producing integral formulas for the
Witten-Reshetikhin-Turaev invariants of a $3$-manifold that will admit
to asymptotic analysis.
The focus in this paper has been $\g$, but the lattice gauge field
theory works for any algebraic group \cite{lattice}. There should be
skein modules corresponding to the other groups, just as $K(M)$
corresponds to $\g$. We will need two kinds of skein relations:
fundamental relations in the Hecke algebra associated to the group,
and the quantized Cayley-Hamilton identity. There has been some study
of these ideas due to Kuperberg \cite{spiders} and Anderson, Mattes
and Reshetikhin \cite{AMR}.
In another direction, it should be possible to commence the study of
the syzygies of skein modules. A syzygy is a relationship between
relationships. For instance, we can define a homology theory for the
Kauffman bracket skein module. The 0-chains are spanned by all links;
the 1-chains by all ``Kauffman bracket skein triples''; the 2-chains
by all ``triples of triples'', etc. The 0-th homology of this complex
is $K(M)$, and have examples to show that the theory is not always
trivial. Notice that the $n$-th homology is measuring relations among
relations.
The opacity of the structure of $K(M)$ poses many questions. Is it
possible for $K(M)$ of a compact manifold to have torsion and still be
topologically finitely generated? What is the relationship between
$K_A(M)$ and $K(M)$? Przytycki has an example of a noncompact manifold
where $K_A(M)$ is infinitely generated yet $K(M)$ is trivial. There is
a grading of $K(M)$ by cables. The top term in the grading is
everything; after that you take the span of all $2$-fold cables, then
$3$-fold cables, etc. How is torsion in $K(M)$ reflected in this grading?
Are there nilpotents in any $K_0(M)$, and if so, how do they affect
the geometry of the representation space? Finally, in the interest of
computability, what is a relative skein module and is there a gluing
theorem?
The Kauffman bracket skein module is organic to many fields. We hope
it, and other skein modules, will act as catalysts for the
synergistic mixing of ideas from these fields.
|
proofpile-arXiv_065-641
|
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\section{Introduction}
There has lately been some interest in the problem of how to
accommodate an extra gauge singlet field into the minimal
supersymmetry standard model (MSSM). This is the simplest
extension which is consistent with a lightest higgs boson whose mass
exceeds the upper bound found in the MSSM~\cite{mssm}. Previously it
was thought that, by acquiring a
vacuum expectation value of ${\cal O} (M_W)$, such a singlet could also
provide a simple solution to a fine-tuning problem in the MSSM,
the so-called `$\mu$--problem'~\cite{muprob,gm}. Because of
difficulties with cosmology (specifically the appearance of domain
walls) this now no longer appears to be the case~\cite{aw,us}. In
fact, it was shown in ref.\cite{us} that models with singlets are likely to
require symmetries in addition to those in the MSSM if they are to
avoid problems with either domain walls or fine-tuning. In this
respect models with gauge singlets are singularly {\em less} efficient
at solving fine-tuning problems. However since they allow for more
complicated higgs phenomenology, it is still worth pursuing them.
This paper concentrates on the task of building an MSSM extended by
a singlet, which avoids reintroducing the hierarchy problem,
fine-tuning, {\em and} domain walls.
Let us take as our starting point a low-energy effective theory
which includes all the fields of the MSSM, plus one additional
singlet $N$. The superpotential is assumed to be the standard MSSM
Yukawa couplings plus the higgs interaction
\begin{equation}
\label{superpot}
W_{\rm higgs}=\mu H_1 H_2 + \mu' N^2 +
\lambda{N}H_{1}H_{2}-\frac{k}{3}N^3,
\end{equation}
and the soft supersymmetry breaking terms are taken to be
of the form
\begin{eqnarray}
V_{\rm soft higgs} &=
&B \mu h_1 h_2 + B' \mu' n^2 +
\lambda A_{\lambda}nh_1h_2-
\frac{k}{3} A_k n^3 + {\rm h.c.} \nonumber\\
&&+ m^2_1 |h_1|^2
+ m^2_2 |h_2|^2
+ m^2_N |n|^2,
\end{eqnarray}
where throughout scalar components will be denoted by lower case
letters. For the moment let us put aside the question of how the
$\mu$ and $\mu'$ terms get to be so small (i.e. ${\cal O}(M_W)$ instead of
${\cal O}(M_{\rm Pl})$), and return to it later. From a low-energy point of view
the only requirement is that the additional singlet should
significantly alter the higgs mass spectrum. This means that
$\lambda\neq 0$. There are four possibilities which can arise:
If all the other operators are absent, then in the low energy
phenomenology there is an apparent (anomalous) global ${\tilde{U}}(1)$
symmetry (orthogonal to the hypercharge), which leads to a
massless goldstone boson. Generally one expects significant
complication to be required in
order that axion bounds are satisfied.
There are two cases which lead to a discrete symmetry. These are $\mu
=0$, $k=0$ which leads to a $Z_2$ symmetry, and $\mu =0$, $\mu' =0$
which leads to a $Z_3$ symmetry. The latter is usually
referred to as the next-to-minimal supersymmetric standard model
(NMSSM)~\cite{nmssm,ellis}, and has been the main focus of work on
singlet extensions of the MSSM.
Thus the second possibility is that there is an {\em exact} discrete
symmetry, and thus a domain wall problem associated with the existence
of degenerate vacua after the electroweak phase transition. Weak scale
walls cause severe cosmological problems (for example their density
falls as $T^2$ whereas that of radiation falls as $T^4$ so they
eventually dominate and cause power law inflation)~\cite{us}. This is
not true however, if the discrete symmetry is embedded in a broken
gauge symmetry. In this case the degenerate vacua are connected by a
gauge transformation in the full theory~\cite{ls}. After the
electroweak phase transition, one expects a network of domain walls
bounded by cosmic strings to form and then collapse \cite{ls}.
As discussed in ref.\cite{meme} bounds
from primordial
nucleosynthesis (essentially on the reheat temperature after
inflation) require that the potential be very flat.
In addition this mechanism depends rather strongly on
the cosmology, and so models with discrete symmetry (such as
the NMSSM) remain questionable.
The third possibility is that the discrete symmetry is
broken~\cite{zko} by gravitationally suppressed
interactions~\cite{ellis,rai}. This was the case considered and
rejected in ref.\cite{us}. Here the very slight non-degeneracy in the
vacua, causes the true vacuum to dominate once the typical curvature
scale of the domain wall structure becomes large enough. However one
must ensure that the domain walls disappear before the onset of
nucleosynthesis and this means that the gravitationally suppressed
terms must be of order five. It was shown in ref.\cite{us} that, no
matter how complicated the full theory (i.e. including gravity), there
is {\em no} symmetry which can allow one of these terms, whilst
forbidding the operator $\nu N$, where $\nu$ is an effective
coupling. Furthermore, any such operator large enough to make the
domain walls disappear before nucleosynthesis generates these terms at
one loop anyway (with magnitude $\sim M_W^2 M_{\rm Pl} N$), even if they are
set to zero initially. This constitutes a reintroduction of the
hierarchy problem as emphasised in ref.\cite{destab} and as will be
clarified in the following section.
The final case which is the subject of this paper, is when there is no
discrete symmetry at the weak scale (exact or apparent). This is true
when either $\mu\neq 0$ or both $\mu'\neq 0$ and $k\neq 0$.
It is well known that (as in the previous case) this type of model can
lead to dangerous divergences
due to the existence of tadpole diagrams. Such divergences have the potential to
destroy the gauge hierarchy unless they are either fine-tuned away, or
removed by some higher symmetry.
In the next section the problem is quantified for the model in eq.(\ref{superpot}),
and the dangerous diagrams identified.
It is also shown that normal gauge symmetries are not able to forbid
these diagrams, and that they are therefore not a good candidate for the
higher symmetry in question. Then in sections 2 and 3, it is shown that
models which possess gauged-$R$ symmetry and target space duality respectively,
can avoid such problems. (For the reasons discussed in
ref.\cite{herbi}, gauged $R$-symmetry~\cite{herbi,gaugedr} might be
favoured over global, although the arguments presented will apply to
either case.)
\section{The Dangerous Diagrams}
In order to demonstrate which are the dangerous diagrams associated with the
model of eq.(\ref{superpot}), it is convenient to use the formalism of $N=1$
supergravity~\cite{sriv}. In this section the formalism will be described,
and some specific examples given. Using standard power counting rules, some general
observations will then be made about the divergent diagrams.
For completeness, let us first summarize the pertubation theory
calculation of the offending, divergent diagrams~\cite{sriv,destab}.
The lagrangian of $N=1$ supergravity depends only on the K\"ahler function,
\begin{equation}
{\cal G} = K(z^i,z^{\overline{i}}) + \ln |\hat{W}(z^i)|^2
\end{equation}
where $z^i$ is used to denote a generic chiral superfield (visible or
hidden), and $z^{\overline{i}}=\overline{z}^i$. Although the holomorphic
function $\hat{W}$ is referred to
as the superpotential, it does not necessarily correspond to the
superpotential in the low energy (i.e. softly-broken, global
superymmetry) approximation. This point will be important later; hence
the hat on this superpotential. The function $K = K^\dagger $ is the
K\"ahler potential. When supersymmetry is spontaneously broken,
divergent diagrams are most efficiently calculated using the augmented
perturbation theory rules described in ref.\cite{destab}
which are as follows.
The breaking of supersymmetry is embodied in $\theta $ and $\overline{\theta}$ dependent,
classical VEVs for the chiral compensator, $\phi $, and K\"ahler
potential which take the form
\begin{eqnarray}
\label{obvious}
\phi &\sim & 1 + \frac{M_S^2}{M_{\rm Pl}} \theta^2 \nonumber\\
e^{-K/3 M^2_{\rm Pl}} &\sim & 1 + \frac{M_S^2}{M_{\rm Pl}} \theta^2 +
\frac{M_S^2}{M_{\rm Pl}} \overline{\theta}^2
+ \frac{M_S^4}{M_{\rm Pl}^2} \theta^2 \overline{\theta}^2,
\end{eqnarray}
where $M_S$ is the scale of supersymmetry breaking in the hidden sector, of
order $M_S^2 \sim M_W M_{\rm Pl}$. (The precise forms, which are not important here, may
be found in ref.\cite{destab}.)
Generally, in addition to renormalisable terms, the K\"ahler potential and superpotential
are expected to contain an infinite number of non-renormalisable terms
suppressed by powers of $M_{\rm Pl}$.
There are therefore two types of vertex which can appear in diagrams; those coming from
the dimension-3, $\hat{W}$ operators of the form
\begin{equation}
\phi^3 \hat{W}_{ij...},
\end{equation}
and those coming from dimension-2, $K$ operators, of the form
\begin{equation}
\phi \overline{\phi} \left( -3 e^{-K/3 M^2_{\rm Pl}}\right)_{ij\overline{k}\overline{l}...},
\end{equation}
for a vertex with $z^i,z^j,z^{\overline{k}},z^{\overline{l}}...$
exiting. Here the indices $ij\overline{k}\overline{l}...$ denote covariant
differentiation (with respect to K\"ahler transformations), so that
\begin{eqnarray}
D_i \hat{W} & =& e^{-K/M_{\rm Pl}^2} \partial _i e^{K/M_{\rm Pl}^2} \nonumber\\
\hat{W}_{ij}& =&D_j \hat{W}_i - \Gamma^k_{ij} \hat{W}_k
\end{eqnarray}
where $\Gamma^k_{ij}$ is the connection of the K\"ahler manifold
described by the metric $\partial_i \partial_{\overline{j}}K$.
In order to calculate the divergent diagrams, one may now use
global superspace perturbation rules. In particular,
using the standard definitions for $D_\alpha$ and
$\overline{D}^{\dot{\alpha}}$ operators~\cite{sriv},
a $K$-vertex with $m$ chiral legs and $n$ antichiral legs throws
$m$ of the $-\overline{D}^2 /4 $ and $n$ of the $-D^2 /4 $ operators onto the
surrounding propagators. On the other hand
a chiral vertex with $n$ chiral legs throws only $n-1$ of the $-\overline{D}^2 /4 $ operators
onto the surrounding propagators and similarly for antichiral with $-D^2 /4 $ operators
(the difference being due to the conversion of integrations to full
superspace ones).
The propagators are as follows~\cite{destab},
\begin{eqnarray}
\langle z^i z^{\overline{j}} \rangle & = &
K^{i\overline{j}} P_1 \frac{e^{K(\theta, \overline{\theta '})/3}}{\phi(\theta) \overline{\phi}
(\overline{\theta '})}
\frac{\delta^4 (x-x') \delta^4 (\theta -\theta ')}{\Box } \nonumber\\
\langle z^{\overline{i}} z^j \rangle & = &
K^{\overline{i}j} P_2 \frac{e^{K(\theta ',
\overline{\theta })/3}}{\phi(\theta ') \overline{\phi} (\overline{\theta })}
\frac{\delta^4 (x-x') \delta^4 (\theta -\theta ')}{\Box },
\end{eqnarray}
where $P_1$ and $P_2$ are the chiral and anti-chiral projection operators
\begin{eqnarray}
P_1 & = & \frac{ D^2 \overline{D}^2 }{16\Box} \nonumber\\
P_2 & = & \frac{ \overline{D}^2 D^2 }{16\Box} ,
\end{eqnarray}
and where
\begin{equation}
\delta^4 (\theta -\theta ')=(\theta -\theta ')^2 (\overline{\theta} -\overline{\theta '})^2 .
\end{equation}
Since we are only interested in determining the leading divergences,
it is quite sufficient to use the massless approximation here.
This completes our review of the perturbation theory rules. Now let us
consider the NMSSM, in which the renormalisable part of k\"ahler
potential has the canonical form,
\begin{equation}
K= z^i z^{\overline{j}}\delta_{i\overline{j}} + K_{\rm non-renorm}
\end{equation}
and the superpotential is of the following form;
\begin{equation}
\label{superpot2}
\hat{W}_{\rm higgs}=
\lambda{N}H_{1}H_{2}-\frac{k}{3}N^3 + \hat{W}_{\rm non-renorm} .
\end{equation}
The extra terms, which represent possible higher order, non-renormalisable
operators, are the terms which we are going to examine. As a
warm-up exercise, consider the case where there are no
non-renormalisable operators in $K$, and only a single non-renormalisable
coupling in the superpotential of the form
\begin{equation}
\label{superpot3}
\hat{W}_{\rm non-renorm}=
\frac{\lambda'}{M_{\rm Pl}} (H_{1}H_{2})^2 .
\end{equation}
One may hope that by adding such a coupling it is possible to remove
the domain walls which would otherwise form due to the global
$Z_3$ symmetry apparent in the renormalisable part of
eq.(\ref{superpot2}). However, as discussed
in ref.\cite{us}, {\em there is no sufficiently large,
non-renormalisable operator that can be added to the superpotential,
which does not destabilise the gauged hierarchy}. Here `sufficiently
large' means that the cosmological walls must disappear before the onset of
primordial nucleosynthesis for which one requires $\lambda'\,\raisebox{-0.6ex}{$\buildrel > \over \sim$}\, 10^{-7}$.
For the operator in question, this is due to the 3-loop diagram in
fig.(1), which gives rise to a contribution to the effective action of the form,
\begin{eqnarray}
\label{3-loop}
\delta S &=& \frac{-k\lambda' \lambda^2}{M_{\rm Pl}}
\int {\rm d}^4 x_1 \ldots {\rm d}^4 x_4 {\rm d}^4\theta_1\ldots {\rm d}^4\theta_4
N(x_1,\theta_1) \frac{\phi(\theta_1)}{\phi(\theta_4)}
{ e^{K_{(12)}/3}e^{K_{(13)}/3}e^{2 K_{(42)}/3}e^{2 K_{(43)}/3}}
\nonumber\\
& & \hspace{1cm}\times
\left( \frac{\overline{D}^2_1 \delta_{12}}{4\Box_1}\right)
\left( \frac{ D ^2_2 \delta_{24}}{4\Box_2}\right)
\left( \frac{\overline{D}^2_4 \delta_{43}}{4\Box_4}\right)
\left( \frac{ D ^2_3 \delta_{31}}{4\Box_3}\right)
\left( \frac{D_2^2 \overline{D}^2_2 \delta_{24}}{16\Box_2}\right)
\left( \frac{\overline{D}^2_4 D^2_4 \delta_{43}}{16\Box_4}\right),
\end{eqnarray}
where $\delta_{ij}=\delta^4 (x_i-x_j) \delta^4 (\theta_i -\theta_j) $,
and here $K_{(ij)}=K(\theta_i,\overline{\theta}_j )$.\\
\begin{picture}(375,250)(0,40)
\Line(200,250)(238,167)
\Line(200,250)(162,167)
\Line(200,150)(200,100)
\Vertex(200,150){3}
\Vertex(200,250){3}
\Vertex(238,167){3}
\Vertex(162,167){3}
\CArc(200,200)(50,270,90)
\CArc(200,200)(50,90,270)
\Text(200,160)[]{\scriptsize $1$}
\Text(200,260)[]{\scriptsize $4$}
\Text(238,180)[]{\scriptsize $3$}
\Text(162,180)[]{\scriptsize $2$}
\Text(195,100)[]{\scriptsize $N$}
\Text(142,208)[]{\scriptsize $H_2$}
\Text(172,208)[]{\scriptsize $H_1$}
\Text(212,208)[]{\scriptsize $H_2$}
\Text(242,208)[]{\scriptsize $H_1$}
\Text(170,153)[]{\scriptsize $N$}
\Text(230,153)[]{\scriptsize $N$}
\Text(200,75)[]{ figure 1: Divergent tadpole diagram from $(H_1 H_2)^2$ operator.}
\end{picture}\\
\noindent
One can evaluate this expression by integrating by parts to expose
factors of $\delta^4 (\theta_i-\theta_j)$ and thus eliminating
$\theta$ integrals in the standard manner. Acting on the $\phi $ or
$e^{K/3}$ factors always reduces the degree of divergence as is
obvious from eqn.(\ref{obvious}). Factors of $D^2 \overline{D}^2 $
may be removed using the identities,
\begin{eqnarray}
\label{didents}
D^2 \overline{D}^2 D^2 &=& 16 \Box D^2 \nonumber \\
\overline{D}^2 D^2 \overline{D}^2 &=& 16 \Box \overline{D}^2 \nonumber \\
16 &=&\int {\rm d}^4\theta_2 \delta^4(\theta_2-\theta_1) D^2 \overline{D}^2
\delta^4(\theta_2-\theta_1) \nonumber\\
16 &=&\int {\rm d}^4\theta_2 \delta^4(\theta_2-\theta_1) D^2 \overline{D}^2
\delta^4(\theta_2-\theta_1).
\end{eqnarray}
The integral is reduced to a single
integral over $\theta_1$ of the form,
\begin{equation}
\delta S = \frac{-2 k\lambda' \lambda^2}{M_{\rm Pl}}\int {\rm d}^4 x_1 \ldots {\rm d}^4 x_4 {\rm d}^4\theta_1
N(x_1,\theta_1)
e^{2 K_{(11)}}
\left( \frac{\delta^4 x_{31}}{\Box_3}\right)
\left( \frac{\delta^4 x_{43}}{\Box_4}\right)^2
\left( \frac{\delta^4 x_{24}}{\Box_2}\right)^2
\delta^4 x_{12} ,
\end{equation}
where $\delta^4 x_{ij}=\delta^4 (x_i-x_j) $.
Converting the delta functions to momentum space, one finds a
contribution to the effective action of
\begin{equation}
\delta S = -2 k\lambda' \lambda^2\int {\rm d}^4 x_1 {\rm d}^4\theta_1
N(x_1,\theta_1)
e^{2 K_{(11)}} I_3,
\end{equation}
in which $I_3$ is the quadratically divergent 3-loop integral,
\begin{equation}
I_3=\int \dk{1}\dk{2}\dk{3} \frac{1}{k_1^2 k_2^2 k_3^2 (k_1-k_2)^2
(k_1-k_3)^2 }= {\cal O} (M_{\rm Pl}^2 /(16\pi^2 )^{3}) ,
\end{equation}
where the integral has been regularised with a cut-off of order $M_P$.
Inserting the $\theta$ dependent VEVs of eqn.(\ref{obvious}) into the
above, results in terms in the effective potential of the form
\begin{equation}
\delta V\approx \frac{2 k\lambda' \lambda^2}{(16 \pi^2)^3}
\left( (n+n^*) M_{\rm Pl} M_W^2 + (F_N + F_N^*) M_{\rm Pl} M_W\right)
\end{equation}
which clearly destabilises the hierarchy unless $\lambda'$ is
sufficiently small, so small in fact that it is unable to remove the
cosmological domain walls before the onset of nucleosynthesis~\cite{us}.
The non-renormalisable term in eq.(\ref{superpot3}), is
(to leading order in $M_{\rm Pl}^{-1}$) equivalent to adding instead the term
\begin{equation}
K_{\rm non-renorm} =
- \frac{\lambda'}{\lambda} \left(\frac{N^{\dagger}H_{1}H_{2} +
{\rm h.c.}}{M_{{\rm Pl}}}\right)
- \frac{k \lambda'}{\lambda^2} \left(\frac{N^{\dagger}H_{1}H_{1}^\dagger +
{\rm h.c.}}{M_{{\rm Pl}}}\right) ,
\end{equation}
in the K\"ahler potential. This may be seen by making the redefinitions
\begin{eqnarray}
N &\rightarrow & N - \frac{\lambda' H_1 H_2}{\lambda M_{\rm Pl}} \nonumber\\
H_1 &\rightarrow & H_1 - \frac{\lambda' k N H_1}{\lambda^2 M_{\rm Pl}}.
\end{eqnarray}
This provides a useful check of the perturbation theory rules. The
divergent diagrams in the redefined model are of the form shown in
fig.(2), where black vertices are chiral and white ones come from
the $K_{\rm non-renorm}$ terms in the K\"ahler potential. \\
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\Text(312,100)[]{+}
\Text(175,50)[]{figure 2: Equivalent diagrams to fig.(1) when the fields
are redefined. }
\end{picture}\\
The 1-loop divergent contributions were shown by
Jain in ref.\cite{destab} to cancel unless the
trilinear terms couple directly to hidden sector fields. This result
can easily be recovered here, since the diagram gives
\begin{equation}
\delta S = \frac{M_{\rm Pl}}{2 (16 \pi^2)} \int {\rm d}^4 x_1 {\rm d}^4\theta_1
K_{N H_1 \overline{H}_1} K^{H_1 \overline{H}_1} N(x_1,\theta_1) + {\rm h.c.}
\end{equation}
where we have approximated
\begin{equation}
\int \dk{1} \frac{1}{k_1^2}= {\cal O} ( M_{\rm Pl}^2 / (16 \pi^2) ) .
\end{equation}
Without any direct coupling between $H_1$ and a hidden sector field,
the VEVs of eq.(\ref{obvious}) do not appear, and the diagram does not
give dangerous terms.
The 2-loop contributions are easily found to cancel amongst
themselves. With a little effort the remaining divergences can
also be shown to cancel except the single (Mercedes) diagram
of fig.(3).\\
\begin{picture}(375,250)(0,40)
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\Text(238,180)[]{\scriptsize $3$}
\Text(162,180)[]{\scriptsize $2$}
\Text(195,100)[]{\scriptsize $N$}
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\Text(195,225)[]{\scriptsize $N$}
\Text(190,75)[]{ figure 3 }
\end{picture}\\
The contribution of this diagram to the effective action is,
\begin{eqnarray}
\delta S &=& \frac{-k\lambda' \lambda^2}{M_{\rm Pl}}
\int {\rm d}^4 x_1 \ldots {\rm d}^4 x_5 {\rm d}^4\theta_1\ldots {\rm d}^4\theta_5
N(x_1,\theta_1) \frac{\phi(\theta_1)}{\phi(\theta_5)}
\nonumber\\
&& \times
e^{K_{(12)}/3}e^{K_{(13)}/3}e^{K_{(42)}/3}e^{K_{(43)}/3}
e^{K_{(45)}/3}e^{K_{(52)}/3}e^{K_{(53)}/3}
\left( \frac{\overline{D}^2_1 \delta_{12}}{4\Box_1}\right)
\left( \frac{D ^2_2 \overline{D}^2_2 \delta_{25}}{16\Box_2}\right)
\nonumber\\
& & \times
\left( \frac{\overline{D}^2_5 \delta_{53}}{4\Box_4}\right)
\left( \frac{ D ^2_3 \delta_{31}}{4\Box_3}\right)
\left( \frac{D_2^2 \overline{D}^2_2 \delta_{24}}{16\Box_2}\right)
\left( \frac{\overline{D}^2_4 D^2_4 \delta_{43}}{16\Box_4}\right)
\left( \frac{ D^2_5 \delta_{54}}{4\Box_5}\right) .
\end{eqnarray}
By integrating by parts with $\overline{D}_4^2$, $\overline{D}_5^2$
and $D_5^2$, and using the rules in eqn.(\ref{didents}),
the last factor becomes simply $ \delta_{54}$. The $\langle 4 5 \rangle$
propagator effectively collapses and the integral over
$(x_5,\theta_5)$ results in eqn.(\ref{3-loop}) as required. (Again,
when evaluating the leading divergences, one
may ignore $D^2$ operators acting on $\phi $ and $e^{K/3}$.)
Having gained some confidence in calculation of divergences, we can
now go
on to systematically consider the other operators which may appear in
$\hat{W}$ or $K$. In order to determine
exactly which ones are dangerous, let us first restrict our attention
to operators in $\hat{W}_{\rm non-renorm}$.
Obviously the degree of fine-tuning
decreases with higher order since each loop gives a factor $\Lambda^2
/(16 \pi^2 )$ where $\Lambda $ is a cut-off, and involves more Yukawa
couplings. It therefore seems reasonable to disregard contributions
which are higher than six-loop since they
are unable to destabilise the hierarchy. Upto and including six loop,
the following operators are potentially dangerous if they
appear in the superpotential (multiplied by any function of
hidden sector fields), since one can write down a tadpole diagram using
them (together with the trilinear operators of the NMSSM);
\vspace{0.5cm}
\begin{center}
\begin{tabular}{||l|l|r||} \hline
\mbox{Operator} & \mbox{resp. diagram} & \mbox{Loop-order} \\ \hline\hline
$N^2$, $H_1 H_2$ & 3a,3a & 1 \\ \hline
$N^4$, $N^2 H_1 H_2$ & 3b,3b & 2 \\ \hline
$(H_1 H_2)^2 $, $N (H_1 H_2)^2$,
$N^3 (H_1 H_2)$, $N^5$ & 3c,3d,3d,3d &3 \\ \hline
$N^3 (H_1 H_2)^2$, $N^5 (H_1 H_2)$, $N^7$ & 3e,3e,3e,3e & 4 \\ \hline
$N (H_1 H_2)^3$, $N^2 (H_1 H_2)^3$, $N^4 (H_1 H_2)^2$,
$N^6 (H_1 H_2)$, $N^8$ & 3f,3g,3g,3g,3g & 5 \\ \hline
$N^4 (H_1 H_2)^3$, $N^6 (H_1 H_2)^2$,
$N^8 (H_1 H_2)$, $N^{10}$ & 3h,3h,3h,3h & 6 \\ \hline
\end{tabular}
\end{center}
\vspace{0.5cm}
The corresponding tadpole diagrams for each operator are shown in
fig.(4a-h). (Figure (4c) is the diagram which was evaluated above.)
Notice that, since the leading divergences involve chiral or antichiral
vertices only, an operator must break the $Z_3$ symmetry in
$\hat{W}$ in order for it to be dangerous (so that for example
$N^2 (H_1 H_2)^2 $ does not destabilise the hierarchy).
The first two operators are the exception in this list,
since one cannot say with certainty whether or not their contributions to the
effective potential will be dangerous. This depends on how the couplings $\mu $
or $\mu'$ are generated. Specifically, the diagram in fig.(4a)
generates logarithmically divergent terms of the form
\begin{equation}
\delta V = \frac{\log \Lambda^2}{32 \pi^2}
\int \mbox{d}^4\theta e^{2 K/3 M_{\rm Pl}^2} \varphi\overline{\varphi}
\hat{W}_{ij} \overline{\hat{W}}^{ij} +\ldots
\end{equation}
These are the divergent terms which lead to logarithmic
running of the soft-breaking scalar masses. However, if there is a
$\mu$-term produced directly in the superpotential from some product
of hidden sector fields ($\mu = \Phi^m/M_{\rm Pl}^{m-1}$ for
example), the contribution above includes
\begin{equation}
\frac{\log \Lambda^2}{32 \pi^2}
\int {\rm d}^4\theta \mu (\Phi) \lambda^\dagger N^\dagger =
\frac{\log \Lambda^2}{32 \pi^2} \lambda^\dagger F_N^\dagger
\frac{m \phi^{m-1} F_\Phi}{M^{m-1}_{\rm Pl}}
\sim \left( \frac{M_{\rm Pl}}{M_W}\right)^{1/m}M_W^{2} F_N^\dagger .
\end{equation}
where since $\Phi$ is a hidden sector field, one can assume that
$F_\Phi \sim M_W M_{\rm Pl} $, and that also
$\langle |\phi |^m \rangle \sim M_W M_{\rm Pl}^{m-1}$ in
order to get $\mu \sim M_W $. This leads to a value of $F_N \gg M_W$
unless $m$ is extremely large, destabilising the gauge hierarchy.
If $\mu $ is generated in the visible sector on the other hand, it may be
possible to avoid this conclusion\footnote{I would like to thank G.~G.~Ross
for pointing this out.}. In this sense such terms have the same status as
the trilinear couplings in the K\"ahler potential which were discussed above.
It has already been demonstrated that the next three operators will
lead to dangerous divergences and must be forbidden. Not all of the
remaining operators are dangerous however. Consider for instance
adding a dimension-7 operator to the superpotential;
\begin{equation}
\hat{W}_{\rm non-renorm} = \frac{\lambda'}{M_{\rm Pl}^4} N^7 .
\end{equation}
In this case the (Garfield) diagram of fig.(4e) looks potentially
dangerous, since it also appears to be a divergent tadpole
contribution. Its contribution to the effective action is
\begin{eqnarray}
\label{garfield}
\delta S &=& \frac{k^2\lambda'}{18 M^4_{\rm Pl}}
\int {\rm d}^4 x_1 {\rm d}^4 x_2 {\rm d}^4 x_3 {\rm d}^4\theta_1{\rm d}^4\theta_2{\rm d}^4\theta_3
N(x_1,\theta_1) \frac{1}{\phi(\theta_1)^3}
e^{K_{(12)}}e^{K_{(13)}}
\nonumber\\
& & \hspace{1cm}\times
\left( \frac{D^2_2\overline{D}^2_2 \delta_{21}}{16\Box_2}\right)^2
\left( \frac{\overline{D}^2_2 D^2_2 \delta_{21}}{16\Box^2_2}\right)
\left( \frac{D^2_3\overline{D}^2_3 \delta_{31}}{16\Box_3}\right)^2
\left( \frac{-\overline{D}^2_3 \delta_{31}}{4\Box_3}\right).
\end{eqnarray}
\newpage
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\Text(200,100)[]{figure 4: Tadpole diagrams for non-renormalisable
operators in $\hat{W}$ upto 6-loop. }
\end{picture}\\
Again by integrating by parts with $\overline{D}_2^2$
and $\overline{D}_3^2$ one can extract the leading term, but this
time, one is forced to act at least once upon the $e^K$ factors,
because in total there is an odd number of $D^2$ and $\overline{D}^2$
operators.
The result is
\begin{equation}
\delta S = \frac{k^2\lambda'}{18 M^4_{\rm Pl}}\int {\rm d}^4 x_1 {\rm d}^4\theta_1
N(x_1,\theta_1) \frac{1}{\phi(\theta_1)^3}
\left(-\frac{\overline{D}^2}{4}e^{ 2 K_{(11)}}\right) I_4,
\end{equation}
in which $I_4$ is the quartically divergent 4-loop integral,
\begin{equation}
I_4=\int \dk{1}\dk{2}\dk{3}\dk{4} \frac{1}{k_1^2 k_2^2 k_3^2 k_4^2
(k_1-k_2)^2 (k_3-k_4)^2 }= {\cal O} (M_{\rm Pl}^4 /(16\pi^2 )^{4}) .
\end{equation}
The final contribution to the effective potential is not harmful to
the gauge hierarchy;
\begin{equation}
\delta V \approx \frac{-k^2\lambda'}{9 (16 \pi^2)^4}
\left( (F_N+F_N^*) M_W^2 + (n+n^*) M_W^3 \right).
\end{equation}
This is clearly the case whenever the total number of $D^2$ and
$\overline{D}^2$ operators is odd. This fact leads one quite easily
to the chief result of this section, which is that, for the
model of eqn.(\ref{superpot2}), {\em any extra
odd-dimension operators in $\hat{W}$ or even-dimension operators in
$K$ are not harmful to the gauge hierarchy.}
This may be deduced by first generalising the supergraph, power counting
rules. Let there be $V_d$ superpotential vertices of dimension $d+3$
(that is of the form $z^{d+3}/M_{\rm Pl}^d $), and $U_d$ K\"ahler potential
vertices of dimension $d+2$ (of the form $z^{d+2}/M_{\rm Pl}^d $). To the
divergence, a propagator counts as $ 1/p^2 $, a $V_d$
vertex as $p^{d+2}$ (from the $D^2$ factors on its legs),
a $U_d$ vertex as $p^{d+2}$, and each loop variable as $p^2$.
In addition each external chiral leg removes a $D^2$ operator of
the vertex, effectively contributing $1/p$.
Hence the total degree of divergence is~\cite{sriv},
\begin{equation}
D= 2 L -2 P - E_c + \sum_d V_d (d+2) + \sum_d U_d (d+2),
\end{equation}
where $L$ is the number of loops, $P$ is the number of propagators,
and $E_c$ is the number of external chiral legs. There are two useful
relations; the first is
\begin{equation}
\label{pident}
2 P + E_c = \sum_d V_d (d+3) + \sum_d U_d (d+2),
\end{equation}
the right hand side being simply the number of external legs when
there are no propagators; the second arises from counting the internal
momentum variables, one of which is removed by each vertex delta function,
\begin{equation}
P - L = \sum_d V_d + \sum_d U_d -1 .
\end{equation}
Substituting these gives the following value for the divergence
\begin{equation}
D= 2 - E_c + \sum_d V_d + \sum_d U_d.
\end{equation}
The actual contribution to the effective potential is therefore of the
form
\begin{equation}
\delta V \sim \frac{\Lambda^{2-E_c + \sum_d V_d + \sum_d U_d}}
{M_{\rm Pl}^{ \sum_d V_d + \sum_d U_d}} \sim M_{\rm Pl}^{2-E_c}.
\end{equation}
This is the result of ref.\cite{sriv,destab}, which says that in
$N=1$ supergravity, apart from a quadratic vacuum term,
the only divergent contribution to the effective potential is linear in fields ($E_c=1$).
Now consider the total number, $N_{D^2}$, of $D^2$ and $\overline{D}^2$ operators.
There are $d+2$ from every vertex, $-1$ from every external chiral
line, and 2 on every propagator, giving
\begin{equation}
N_{D^2} = 2 P - E_c + \sum_d V_d (d+2) + \sum_d U_d (d+2)
\end{equation}
in total. In order for a diagram to be harmful, this number must be
even, and hence when $E_c=1$,
\begin{equation}
\sum_d V_d d + \sum_d U_d d = \mbox{odd}.
\end{equation}
This can only be satisfied if there is at least one vertex which has
an odd $d$, thus proving the statement above.
(Substituting eq.(\ref{pident}) shows that this also means the
total number of chiral and antichiral vertices is even.)
The relatively restrictive constraint that the superpotential be a
holomorphic function means that there are now only 13
dangerous operators in $\hat{W}$. The K\"ahler potential is restricted
only by the condition, $K=K^\dagger$ however. Apart from the trilinear
operators (which as we
have seen above only destabilise the gauge hierarchy if they directly
couple visible and hidden sector fields), there is a much larger number
of higher dimension operators which must be forbidden here. For example
the operator,
\begin{equation}
K_{\rm non-renorm}=\lambda' N^{\dagger 2} N (H_1 H_2)
\end{equation}
leads to the diagram in Fig.(5), whose contribution to the effective
action is
\begin{equation}
\delta S \approx -\frac{M_{\rm Pl} k \lambda \lambda'}{18 (16 \pi^2)^4}
\int {\rm d}^4 x_1 {\rm d}^4\theta_1
N(x_1,\theta_1) \frac{\phi(\theta_1)}{\overline{\phi}(\overline{\theta}_1)}
e^{5 K_{(11)}/3},
\end{equation}
which again gives $n$ a VEV of ${\cal O} (10^{11} \,{\rm GeV} )$. Clearly {\em any}
odd-dimension operator which breaks the $Z_3$ symmetry of
eq.(\ref{superpot3}) may appear in $K$ and will destroy the gauge
hierarchy if it does so.\\
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\Text(200,155)[]{\scriptsize $N$}
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\Text(285,170)[]{\scriptsize $N$}
\Text(340,205)[]{\scriptsize $N$}
\Text(250,100)[]{ figure 5 }
\end{picture}\\
Hence a particularly attractive way to ensure a model with
singlets which is natural, is to devise a symmetry which forbids
odd-dimension terms in $K$, and even-dimension
terms in $\hat{W}$.
This is the approach taken in the next two sections. (A
possible alternative which will not be considered here is to
include an extra symmetry in
the visible sector, which ensures these couplings are always
suppressed by some field whose VEV is extremely small.)
To finish this section, let us recapitulate the arguments of
ref.\cite{us} which make it clear that such a symmetry cannot
be a normal gauge symmetry.
For simplicity, take this to be a $U(1)_X$
symmetry (the extension to non-abelian cases is trivial), and let the
$Z_3$ symmetry be broken by a $H_1 H_2$ or $N^2$ term in $K$. Such couplings
provide naturally small $\mu\sim M_W$ or $\mu' \sim M_W$ in the
effective low energy global superpotential $W$~\cite{gm}.
The other effective couplings at the weak scale are in general
arbitrary functions of hidden sector fields which carry charge under
the new $U(1)_X$ which shall be referred to collectively as $\Phi$ (with
$\xi =\Phi/M_{\rm Pl}$). It is simple to see that one cannot use this
symmetry to forbid terms linear in $N$.
If $\mu (\xi)\neq 0$ then $\mu (\xi)$ must have the same charge as
$\lambda (\xi) N$ and therefore $(\mu (\xi))^\dagger \lambda (\xi) N $
is uncharged. If both $\mu'\neq 0$ and $k\neq 0$ then $\mu' (\xi)$
must have the same charge as $k (\xi) N$ and therefore $(\mu'
(\xi))^\dagger k (\xi) N $ is uncharged. Once such a linear operator
has been constructed, it is of course trivial to construct all the
other dangerous operators.
One should bear in mind that if one sets these couplings to zero by
hand in the first place, they remain small to higher order in
perturbation theory. So this is merely a fine-tuning problem.
One might also argue that the nature of this fine-tuning problem is
different from that of the $\mu$-problem, since in the latter
the coupling has to be very small, whereas here the couplings may
just happen to be absent (as for example are superpotential mass
terms in string theory).
However, the extremely large number of dangerous operators
makes this fine tuning problem a particularly serious one.
In the next two sections, two examples are presented which are able to
avoid this problem.
\section{Models with $R$-symmetry}
The reason that it has not been possible to forbid divergent tadpole
diagrams
in the models that have been discussed here and in ref.\cite{us}, is
that the K\"ahler potential and superpotential have the same charges
(i.e. zero). There are however two available symmetries in which the
K\"ahler and superpotentials transform differently. These may
accommodate singlet extensions to the MSSM simply and without fine-tuning.
The first is gauged $U(1)_R$-symmetry~\cite{herbi,gaugedr}. In this case the
K\"ahler potential has zero $R$-charge, but the superpotential has $R$-charge
2. This means that the standard renormalisable NMSSM higgs superpotential,
\begin{equation}
\label{rwhiggs}
\hat{W}_{\rm higgs}=\lambda N H_{1}H_{2}-
\frac{k}{3}N^3,
\end{equation}
has the correct $R$-charge if $R(N)= 2/3$ and $R(H_1)+R(H_2) =4/3 $.
So consider the K\"ahler potential
\begin{equation}
\label{quad}
{\cal G} = y_i y^i + \Phi \overline{\Phi}
+ \left( \frac{\alpha}{M^2_{\rm Pl}}\Phi H_1 H_2 +
\frac{\alpha '}{M^2_{\rm Pl}}\Phi N^2 +{\rm h.c.} \right)
+ \log |\hat{W} + g(\Phi )|^2 ,
\end{equation}
where $y_i$ are the visible sector fields and
where $\Phi$ represents a hidden sector field with superpotential
$g(\Phi )$ which aquires a VEV of ${\cal O} (M_{\rm Pl})$.
(It may represent arbitrary functions of hidden
sector fields in what follows). This next-to-minimal choice
of K\"ahler potential is the one proposed in ref.\cite{gm} which leads
to naturally small $\mu$ and $\mu'$ couplings in the low energy
(global supersymmetry) approximation $W$. Specifically, the terms which
arise in the scalar potential are~\cite{gm,us}
\begin{equation}
V_{\rm scalar} = W_{i} W^{i} + m^2 y_{i}y^{i}
+ m \left[y^{i}W_{i}+(A-3)\tilde{W}
+ (B-2)m \mu H_{1} H_{2} + (B-2)m \mu^\prime N^2
+ {\rm h.c.}\right],
\end{equation}
where $\tilde{W}$ are the trilinear terms of the superpotential $\hat{W}$,
rescaled according to
\begin{equation}
\tilde{W} = \langle\exp{(\Phi \overline{\Phi}/2 M_{{\rm Pl}}^2)}
\rangle \hat{W}.
\end{equation}
Here $W$ is the new low energy superpotential including the
$\mu$ and $\mu'$ terms,
\begin{equation}
W = \hat{W} + \mu H_{1} H_{2} + \mu' N^2,
\end{equation}
and $m$ is the gravitino mass
\begin{equation}
m = \langle \exp{(\Phi \overline{\Phi} / 2 M_{{\rm Pl}}^2)} g^{(2)}\rangle,
\end{equation}
where $g^{(2)}$ are the quadratic terms in $g$, and
where the VEV of $g^{(2)} = M_S^2/M_{{\rm Pl}}$ is set by hand such
that $M_{S}\sim 10^{11}$ GeV.
Applying the constraint of vanishing cosmological constant, one finds
that the universal trilinear scalar coupling, $A=\sqrt{3}
\langle \Phi/M_{{\rm Pl}}\rangle $, and that the bilinear couplings are given
by,
\begin{eqnarray}
B &= &(2 A-3)/(A-3) \nonumber\\
|\mu| &= &\left|\frac{m\alpha(A-3)}{\sqrt{3}}\right| \nonumber\\
|\mu^\prime| &= &\left|\frac{m\alpha^\prime(A-3)}{\sqrt{3}}\right|.
\end{eqnarray}
All dimensionful parameters at low energy are of order $M_W$.
Invariance of the K\"ahler potential requires that $R(\Phi) =- 4/3 $.
It is easy to see that with this set of $R$-charges there
can never be odd-dimension operators in $K$, or even-dimension ones in
$\hat{W}$. Indeed the operators which can appear in the superpotential
can be written as,
\begin{equation}
\hat{O}_{c}=
\frac{\Phi^c}{M_{\rm Pl}^c} \frac{y^{(d+3)}}{M_{\rm Pl}^d},
\end{equation}
where $y$ stands for any of the visible sector fields. In order to
have $R$-charge 2, they must satisfy
\begin{equation}
\frac{2 (d+3)}{3} - \frac{4 c}{3}=2
\end{equation}
or $d=2c$. Hence only odd-dimension operators are allowed in
$\hat{W}$. The operators which can appear in the K\"ahler potential
are of the form
\begin{equation}
\hat{O}_{abc}=
\frac{(\Phi \overline{\Phi})^b}{M_{\rm Pl}^{2b}}
\frac{\Phi^c}{M_{\rm Pl}^c}
(y y^\dagger)^a \frac{y^{(d+2-2a)}}{M_{\rm Pl}^d},
\end{equation}
where negative $c$ can be taken to represent powers of $\overline{\Phi} $.
The condition $R=0$ becomes,
\begin{equation}
d=2 (a+c-1),
\end{equation}
so that only even-dimension operators may appear in $K$ as required.
In a fully viable model, one would also have to take account of
anomalies in the $R$ symmetry which can usually be cancelled if
there are enough hidden sector singlets~\cite{herbi}. This will
not be considered here.
\section{Models with Duality Symmetry}
The second symmetry one can use to forbid terms linear in $N$ is
target space duality in a string effective action.
Generally, these have flat directions,
some of which correspond to moduli determining the size and shape of the
compactified space. Furthermore these moduli have discrete
duality symmetries, which at certain points of enhanced symmetry become
continuous gauge symmetries~\cite{duality}.
In Calabi-Yau models, abelian orbifolds and fermionic strings the
moduli include three K\"ahler class moduli ($T$-type) which are always
present, plus the possible deformations of the complex structure
($U$-type), all of which are gauge singlets. Additionally there
will generally be complex Wilson line fields~\cite{moduli,moduli2}.
When the latter acquire a vacuum expectation value they result in
the breaking of gauge symmetries. There has been continued interest
in string effective actions since they may induce the higgs
$\mu$-term~\cite{gm,moduli2,ant1,brignole}, be able to explain the
Yukawa structure~\cite{kpz,binetruy}, and be able to explain the
smallness of the cosmological constant in a {\em no-scale}
fashion~\cite{kpz,noscale}. Since the main objective here is
simply to find a route to a viable low energy model with visible
higgs singlets, these questions will only be partially addressed.
Typically the moduli and matter fields describe a space whose local
structure is given by a
direct product of $SU(n,m)/SU(n)\times SU(m)$ and $SO(n,m)/SO(n)\times
SO(m)$ factors~\cite{moduli,moduli2}. As an example consider
the K\"ahler potential derived in refs.\cite{moduli2}, which
at the tree level is of the form
\begin{equation}
\label{stringkahler}
K=-\log (S+\overline{S}) -\log
[(T+\overline{T})(U+\overline{U})-\frac{1}{2}
(\Phi_1+\overline{\Phi}_2)(\Phi_2+\overline{\Phi}_1)]+\ldots
\end{equation}
The $S$ superfield is the dilaton/axion chiral multiplet, and the
ellipsis stands for terms involving the matter fields.
The fields $\Phi_1$ and $\Phi_2$ are two Wilson line moduli. As in
ref.\cite{gm,moduli2,ant1,brignole}, let us identify these fields with the neutral
components of the higgs doublets in order to provide a $\mu$-term.
Problems such as how the dilaton acquires a VEV, or the eventual
mechanism which seeds supersymmetry breaking will not be addressed here.
The moduli space is given locally by
\begin{equation}
{\cal{K}}_0 = \frac{SU(1,1)}{U(1)}\times \frac{SO(2,4)}{SO(2)\times
SO(4)},
\end{equation}
which ensures the vanishing of the scalar potential at least at the
tree level, provided that the $S$, $T$ and $U$ fields all participate
in supersymmetry breaking (i.e. $G_S$, $G_T$, $G_U\neq 0$).
In fact writing the K\"ahler function as
\begin{equation}
G=K(z_i,z^i)+\ln \left| \hat{W}(z_i) \right|^2 ,
\end{equation}
the scalar potential becomes
\begin{equation}
\hat{V}_s = - e^{G} \left(3- G_i G^{i\overline{j}} G_{\overline{j}} \right)
+ \frac{g^2}{2} {\rm Re} (G^i T^{Aj}_iz_j)(G^k T^{Al}_kz_l),
\end{equation}
where $G_i=\partial G/\partial z^i$, and $G^{i{\overline{j}}}=
(G_{{\overline{j}}i})^{-1}$. The dilaton contribution separates, and
gives $G_S G^{S\overline{S}} G_{\overline{S}}=1$. To show that the
remaining contribution is $2$, it is simplest to define the vector
\begin{equation}
A^\alpha= a (t,u,h,\overline{h})
\end{equation}
where the components are defined as $\alpha=(1\ldots 4) \equiv
(T,U,\Phi_1,\Phi_2)$, and $u=U+\overline{U}$, $t=T+\overline{T}$,
$h=\Phi_1+\overline{\Phi}_2$. It is easy to show that
\begin{equation}
G_\alpha A^\alpha =-2 a.
\end{equation}
The vector $A^\alpha$ is designed so that $G_{\overline{\beta}\alpha}
A^\alpha $ is proportional to $G_{\overline{\beta}}$; viz,
\begin{equation}
G_{\overline{\beta}\alpha}A^\alpha = - a G_{\overline{\beta}}.
\end{equation}
Multiplying both sides by $G_\alpha G^{\alpha\overline{\beta}}$ gives
the desired result, i.e. that $G_\alpha G^{\alpha\overline{\beta}}
G_{\overline{\beta}}=2$. Thus, if the VEVs of the matter fields are
zero, the potential vanishes and is flat for all values of the moduli
$T$ and $U$, along the direction $\langle |\Phi_1|\rangle = \langle
|\Phi_2|\rangle=\rho_{\phi}$ (since this is the direction in which the
$D$-terms vanish). The gravitino mass is therefore undetermined
at tree level, being given by
\begin{equation}
m^2 = \langle e^G \rangle = \frac{|\hat{W}|^2}{s(ut-2 \rho^2_{\phi})}.
\end{equation}
In addition to the properties described above, there is an
$O(2,4,Z)$ duality corresponding to automorphisms of the compactification
lattice~\cite{duality,moduli2}. This constrains the possible form of the
superpotential. The $PSL(2,Z)_T$ subgroup implies
invariance under the transformations~\cite{duality,moduli2},
\begin{eqnarray}
\label{dualtrans}
T &\rightarrow& \frac{aT-ib}{icT+d} \nonumber\\
U &\rightarrow& U-\frac{ic}{2}\frac{\Phi_1\Phi_2}{icT+d} \nonumber\\
z_i &\rightarrow& z_i (icT +d)^{n_i},
\end{eqnarray}
where $a,b,c,d~\epsilon~Z$, $ad-bc=1$, and
where $z_i$ stands for general matter superfields with
weight $n_i$ under the modular transformation above.
The $\Phi_1$ and $\Phi_2$ fields have modular weight $-1$.
It is easy to verify the invariance of the K\"ahler function under
this transformation provided that
\begin{equation}
\label{wdual}
\hat{W}\rightarrow (ic T + d)^{-1} \hat{W}.
\end{equation}
The superpotential should be defined to be consistent with this
requirement in addition to charge invariance, and this leads to a
constraint on the modular weights of the Yukawa couplings and matter
fields. (Anomalies occur here also, and must be cancelled in addition
to the gauge anomalies. Again this is considered to be beyond the
scope of the present paper.)
One may now easily find examples where this symmetry is able
by itself, to forbid dangerous operators.
Consider the NMSSM superpotential of
eqn.(\ref{superpot2}). Identifying $\Phi_1$ and $\Phi_2$ with
the higgs superfields $H_1$ and $H_2$ (in order to generate a $\mu H_1
H_2 $ term in the low energy superpotential $W$)
means that both of these fields have weight $-1$.
Since the superpotential must transform as in eq.(\ref{wdual}), the
other weights must obey the following;
\begin{eqnarray}
3 n_N + n_k &=& -1 \nonumber\\
n_N + n_\lambda &=& +1.
\end{eqnarray}
Since the Yukawa couplings are functions of the moduli, they too can
carry weight under the transformation in eqn.(\ref{dualtrans}).
One simple solution which forbids dangerous divergences
is $n_N=-1$ and $n_k=n_\lambda=+2$. In this case it is obvious that
(since the visible fields all have weight $-1$) even operators may be
avoided in $\hat{W}$. As for the K\"ahler potential, one
expects the terms in $K_{\rm non-renorm}$ to be
multiplied by powers of $(T+\overline{T})$. Thus terms in which the
holomorphic and anti-holomorphic weights are the same may be allowed.
Since all the weights are $-1$, this can obviously only be achieved for
operators which have an even number of fields.
There are clearly many ways in which one could devise similar models.
A perhaps more obvious example would be models in which the
superpotential transforms with weight $-3$. There all the physical
fields could be given weight $-1$, with the couplings having weight
$0$. It is then clear that only trilinear couplings can exist in
the superpotential, and only even-dimension terms can appear in the
K\"ahler potential.
\section{Conclusions}
The problem of destablising divergences in models which extend the
MSSM with a singlet field has been discussed.
In this paper the case where there is no discrete or global symmetry
at the weak scale has been examined, and the dangerously divergent
tadpole diagrams have been identified. In particular it was shown
that half of the possible operators (i.e. those with odd-dimension in the
superpotential $\hat{W}$, or even-dimension in the K\"ahler potential)
are perfectly harmless in the sense that they do not destroy the
gauged hierarchy. Thus an attractive possibility for
extending the higgs sector with a singlet is to generate the $\mu $ term from
couplings in the K\"ahler potential. Two examples were demonstrated in which
all operators which are dangerous to the gauge hierarchy are
forbidden. In order to achieve this, they had to incorporate
either a gauged $R$-symmetry or a target space duality symmetry in
the full theory including gravity.
These models clearly satisfy all constraints from fine-tuning,
primordial nucleosynthesis and cosmological domain walls.
Since they have no discrete or continuous
global symmetries in the weak scale effective theories, one
expects all possible couplings
(i.e. $\mu H_1 H_2 $, $\mu N^2 $, $\lambda N H_1 H_2 $ and $k N^3
$) to be present. The phenomenological
implications of these more general cases, have been discussed recently
in ref.\cite{moorehouse}.
\vspace{1cm}
\noindent
{\bf \Large Acknowledgements:} I would like to thank H.~Dreiner,
J.-M.~Fr\`ere, M.~Hindmarsh, S.~King, D.~Lyth, G.~Ross, S.~Sarkar
and P.~Van Driel for valuable discussions. This work was
supported in part by the European Flavourdynamics Network
(ref.chrx-ct93-0132) and by INTAS project 94/2352.
\newpage
\small
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\section{Introduction}
\noindent
The longitudinal structure function in deep inelastic scattering,
$F_L(x,Q^2)$, is one of the observables from
which the
gluon density can be unfolded.
In leading order (LO)~\cite{R1} it is given by
\begin{equation}
\label{fl1}
F_L^{ep}(x,Q^2) = \frac{\alpha_s(Q^2)}{\pi}
\left \{ \frac{4}{3} c_{L,1}^q(x) \otimes F_2^{ep}(x,Q^2) + 2 \sum_q
c_{L,1}^g(x,Q^2) \otimes [
xG(x,Q^2)] \right \}
\end{equation}
with
\begin{equation}
\label{fl2}
c_{L,1}^q(x) = x^2~~~~~~~c_{L,1}^g(x) = x^2 (1-x),
\end{equation}
and $\otimes$ denoting the Mellin convolution.
Eq.~(\ref{fl1}) applies for light quark flavours. Due to the power
behaviour of the coefficient functions $c_{L,1}^{q,g}(x)$,
an approximate relation for the gluon density at small $x$
\begin{equation}
\label{fl3}
xG(x, Q^2) \simeq \frac{3}{5} \times 5.85 \left \{ \frac{3\pi}
{4 \alpha_s(Q^2)} F_L(0.4x, Q^2) - \frac{1}{2} F_2(0.8x, Q^2) \right \},
\end{equation}
has been used to derive a simple estimate for $xG(x,Q^2)$ in the
past~\cite{R2}.
Heavy quark contributions and the next-to-leading order (NLO)
QCD corrections
complicate the unfolding of the gluon density using $F_L(x,Q^2)$
and have to be
accounted for in terms of $K$-factors.
In the present note, these contributions are studied numerically for the
HERA energy range.
The NLO corrections for the case of light quark flavours were calculated
in ref.~\cite{R3} and the LO and NLO contributions for the heavy
flavour terms were derived in refs.~\cite{R4} and \cite{R5},
respectively. While in LO the heavy flavour part of $F_L(x,M^2)$
is only due to $\gamma^* g$ fusion, in NLO also light quark terms
contribute. Moreover, the choice of the factorization scale $M^2$
happens to affect
$F_L^{Q\overline{Q}}(x,M^2)$ substantially.
\vspace{5mm}
\noindent
{\large\bf Light flavour contributions }\\
\vspace{1mm}
\noindent
The leading order contributions to $F_L(x,Q^2)$ are shown in Figure~1
for $x \geq 10^{-4}$ and $10 \leq Q^2 \leq 500~{\rm GeV}^2$.
Here and in the following we refer to the CTEQ
parametrizations~\cite{R6} and assume $N_f = 4$. We also show the
quarkonic contributions which are suppressed by one order of magnitude
against the gluonic ones in the small $x$ range. The ratio of the
NLO/LO contributions is depicted in figure~2. Under the above
conditions, it exhibits a fixed point at $x \sim 0.03$. Below, the
correction grows for rising $Q^2$ from $K = 0.9$ to 1 for $x = 10^{-4}$,
$Q^2 \epsilon [10, 500]~{\rm GeV^2}$. Above, its behaviour is reversed.
The correction factor $K$ rises for large values of $x$. For $x \sim
0.3$ it reaches e.g. $1.4$ for $Q^2 = 10~{\rm GeV}^2$.
In NLO the quarkonic contributions are suppressed similarly as in the
LO case at small $x$ and contribute to $F_L$ by $15 \%$ if only
light flavours are assumed.
\vspace{5mm}
\noindent
{\large\bf Heavy flavour contributions }\\
\vspace{1mm}
\noindent
The heavy flavour contributions to $F_L$ are shown in figures~3 and 4,
comparing the results for the choices of the factorization scale
$M^2 = 4 m_c^2$ and $M^2 = 4 m_c^2 +Q^2$, with $m_c = 1.5~{\rm GeV}$.
Here we used again parametrization~\cite{R6} for the description of
the parton densities but referred to three light flavours only unlike
the case in the previous section. The comparison of Figures~3a and 4a
shows that the NLO corrections are by far less sensitive to the
choice of the factorization scale than the LO results.
Correspondingly the $K_{c\overline{c}}-$factors
$F_L^{c\overline{c}}(NLO)
/F_L^{c\overline{c}}(LO)$ are strongly scale dependent. Note that the
ratios $K_{c\overline{c}}$ and $K$ behave different and compensate
each other
partially. Thus the overall correction depends on the heavy-to-light
flavour composition of $F_L(x,Q^2)$.
\vspace{5mm}
In summary we note that the NLO corrections to $F_L$ are large. Partial
compensation between different contributions can emerge. For
an unfolding of the gluon density from $F_L(x,Q^2)$ the NLO corrections
are indispensable.
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\section{Motivation}
There is a recent increased interest in $QED_2$. This concerns
the continuum as well as the lattice version of the model
(c.f. \cite{Fr92}
\nocite{Fr93,GaSe94,GaLa94b,AzDiGa94,Fr95,GaLa95,Ga95,Ga95a,AzDiGa96b,IrSe96}-
\cite{IrSe96}).
The one flavor massless continuum model \cite{Sc62a,Sc62b} is
analytically solvable and has been studied extensively. The
reason for the increased interest is that $QED_2$ shows $QCD_4$
like behavior. This applies especially to the multi flavor
situation \cite{GaSe94}. The Maxwell equations for two
dimensional electrodynamics also have topologically non trivial
$C^\infty$ solutions with finite action which can be classified
by their Chern number. These topological objects called
instantons are considerably simpler to imagine for $\U1$ in
$D=2$ than for $SU(2)$ in $D=4$ which is an additional appeal
to study $QED_2$. Therefore one finds in $QED_2$ three
closely related problems. There is the problem of the
$\theta$-vacua, which naively speaking are superpositions of
all topological sectors corresponding to different Chern
numbers. Also observed in both models is the occurrence of the
${\rm U}_A(1)$ problem \cite{Ad69,BeJa69}. $QED_2$ further
allows for a Witten-Veneziano type formula
\cite{Wi79,Ve79,GaSe94}.
It is not clear how important these topological nontrivial
configurations are indeed for quantum physics. Naively such
$C^\infty \in \cal{S}^\ast$ solutions should not contribute in
the functional integration since the subset of such smooth
solutions is of measure zero for the measure over
$\cal{S}^\ast$. Nevertheless the topological susceptibility,
which is the first Chern character for $QED_2$ vice versa the
second Chern character for $QCD_4$, appears in the ${\rm
U}_A(1)$ anomaly.
The lattice situation is quite different. First of all the
lattice regularized version is analytically not solvable.
Further assuming that the lattice model approximates in a
certain limit the continuum model and thus also contributions
from topology it is a priori not clear what differential
geometry means for a set of points. Any straightforward bundle
reconstruction will only lead to trivial bundles with Chern
number zero. One way out is to provide the lattice with a very
special topology and construct partially ordered sets which
allow for non trivial bundles \cite{BaAl96}. $QED_2$ can be
also defined on a fuzzy sphere which allows a topological
classification in a surprisingly intuitive way via the Hopf
fibration \cite{GrMa92,GrKl96}. A third possibility is to
regard the lattice as a directed complex with a certain
realization like $\torus^2$. This idea was pioneered by
L\"uscher \cite{Lu82a} for $SU(2)$ in $D=4$ and put on a more
axiomatic approach in \cite{Ph84} for $\U1$ in $D=2$.
Without the explicit construction of bundles the
$\theta$-vacuum problem and the topological charge problem on
the lattice could also be addressed by possible remnants of the
Atiyah Singer index theorem \cite{AtSi68}. For the numerical
simulation of these models it turns out that {\it lattice
topological charge} \cite{SeSt82} leads to an unpleasant
problem. As observed by \cite{SmVi87, SmVi87a, Vi88} the
lattice Dirac operator indeed shows (approximate) zero modes
depending on the {\it lattice topological charge} of the
configuration. The lattice Dirac operator thus cannot be
inverted and the numerical procedure breaks down for such
configurations, although the measure of the configuration is
almost zero.
In this paper we follow the strategy pioneered by L\"uscher
\cite{Lu82a} and assume that the lattice is a directed
two-complex with $\torus^2$ as realization. We further assume that
the topological structure is trivial below a certain scale
(i.e. within a region which is about of the size of a
plaquette). This means, that any local pullback connection
one-form is a pure gauge. This assumption is physically
justified, since in the continuum limit it is assumed that any
local lattice structure does not contribute. Formally it
shrinks to a point and thus has no structure.
\section{Classical Lattice Gauge Theory}
Let us introduce the concept of a classical lattice model which
is used to approximate classical gauge theory.
\begin{Definition}
Let $\Lambda$ be a $2$-d complex and $\BB$ be a realization of
$\Lambda$, i.e. the space underlying the complex $\Lambda$.
The complex $\Lambda$
is called \EM{lattice on $\BB$}.
A $0$-cell $x$ of $\Lambda$ is called \EM{site} and a
directed $1$-cell $\link{xy}$ of $\Lambda$ is called \EM{link} or
\EM{bond}.
\end{Definition}
\begin{Definition}
Let $\xi=(\EE,\pi,\BB)$ be a principal $G$ bundle,
$\omega:\TT\EE\to \GG$ be a connection one-form and
$\Lambda$ be a lattice on $\BB$.
The bundle $\xi_\Lambda=(\EE_\Lambda,\pi_\Lambda,\Lambda)$
is called
\EM{lattice-bundle} and
the tuple $(\xi_\Lambda,\omega)$ is called
\EM{classical lattice model}.
\end{Definition}
Let $j:\Lambda\to \BB$ be the inclusion map.
Then the lattice bundle $\xi_\Lambda$ could be identified with
the restriction $\xi\restrict{\Lambda}$.
The induced bundle $j^\ast(\xi)$ of $j$ is the bundle
$(\EE',\pi',\Lambda)$ with the total-space
$$\EE'=\{(x,e)\in\Lambda\times\EE \restr j(x)=\pi(e)\}$$
and the projection
$\pi'=\pr_1$.
On the other hand we have
an isomorphism $(u, \id_\Lambda)$ to the induced bundle
$j^\ast(\xi)$, i.e. the following diagram commutes
\diagramquad[\EE_\Lambda-\pi_\Lambda->\Lambda-\id_\Lambda->\BB'<-\pi'-\EE'<-u-]
with $u:\EE_\Lambda\to \EE':x\mapsto (\pi_\Lambda(e),e)$ and
$e\in\EE$.
Finally one obtains the following commutative diagram:
$$ \begin{array}{rcccl}
\EE_\Lambda & \maprightt{u} & \EE' & \maprightt{\hat j} & \EE\\
\mapdownr{\pi_\Lambda} & & \mapdownr{\pi'} & & \mapdownr{\pi}\\
\Lambda & \maprightt{\id_\Lambda} & \BB' & \maprightt{j} & \BB
\end{array} $$
where $\hat j$ is defined as usual by $\hat j:\EE'\to \EE:(x,e)\mapsto e$.
We also know that each fiber of the pullback $j^\ast(\xi)$ is
homeomorphic to the fiber $G$ of $\xi$.
Therefore our lattice bundle $\xi_\Lambda$ has typical fiber $G$
and is also a principal $G$-bundle.
\begin{Definition}
Let
$\Lambda$ be a lattice on $\BB$ and $x_0, x_1$ two neighboring
$0$-cells.
$\gamma:[0,1] \to\BB:0\mapsto x_0:1\mapsto x_1$ be a path in $\BB$.
The corresponding image in $\Lambda$ is the
directed $1$-cell $\link{x_0 x_1}$,
and called \EM{path in $\Lambda$}.
\end{Definition}
If the path $\gamma$ is a \EM{loop} then the corresponding path in
$\Lambda$ is a $1$-cycle.
\begin{figure}
\begin{center}
\psfig{figure=f-t2parallel.eps}
\end{center}
\caption{Path of the horizontal lift $\tilde\gamma$
of $\gamma$ in $\EE_\Lambda$}
\protect\label{f-lat-trans}
\end{figure}
\begin{Definition}\label{d-lat-parallel}
Let $(\xi_\Lambda,\omega)$ be a
classical lattice model $\gamma:[0,1] \to\BB$ be a path and $\link{x_0 x_1}$ be
the corresponding path in $\Lambda$.
The \EM{lattice parallel translation} along the path $\gamma$ is
a map
$$\transport_{\link{x_0 x_1}}:\pi_\Lambda\inv(x_0)\to
\pi_\Lambda\inv(x_1):h_0\mapsto h_1$$
where $h_1$ denotes the parallel transport of $h_0$ along
the horizontal lift $\tilde\gamma$ of $\gamma$, i.e.
$\tilde\gamma(0)=h_0$ and $h_1:= \tilde\gamma(1)$.
\end{Definition}
Let $(\xi_\Lambda,\omega)$ be a classical lattice model,
$\sigma:U\to\EE_\Lambda$ be a local section. One obtains the
lattice parallel translation $\transport_{\link{x_0 x_1}}$ in
terms of the local connection one-form $\ga = \sigma^\ast
\omega$ \begin{equation}\label{e-local-lat-trans}
\transport_{\link{x_0 x_1}}:h_0\mapsto h_1=h_0 \circ {\bf P}
\exp\left( -\int_{x_0}^{x_1} j^\ast\ga \right), \end{equation}
where the boundary condition of the horizontal lift function
$$g(t)={\bf P} \exp\left(
-\int_{x(0)}^{x(t)} j^\ast\ga \right), $$ has been set to
$g(0):=e$.
\begin{Definition}
Let $(\xi_\Lambda,\omega)$ be a
classical lattice model.
To each $1$-cell one can assign a lattice parallel translation
which leads to a map
$$\link{xy}\mapsto \transport_{\link{xy}}$$
which is called a \EM{gauge field on $\Lambda$}.
The collection $\{\transport_{\link{xy}}\}$ of all this lattice parallel
translations
is called \EM{configuration on $\Lambda$}.
\end{Definition}
In general one cannot define a global gauge field
on $\Lambda$ except the bundle $\xi_\Lambda$ is a trivial bundle.
Therefore a configuration contains elements which belong to
different local trivialisations.
\begin{Definition}
Let
$\Lambda$ be a complex such that the realization of $\Lambda$ is the
2-Torus $\torus^2$.
A directed complex $\Lambda$ with
\begin{enumerate}
\item $0$-cells $\lpoint{i}{j}$,
\item $1$-cells $(i,i+1)\times \{j\}$ and
$\{i\}\times (j,j+1)$
\item $2$-cells $(i,i+1)\times (j,j+1)$
\end{enumerate}
for all $i\in\ZZ_M$ and $j\in\ZZ_N$
is called a \EM{cubic lattice on $\torus^2$} and
is denoted by $\Lambda(N,M)$.
The closure of a $2$-cell $(i,i+1)\times (j,j+1)$ is called \EM{plaquette}
and is denoted $\Lambda_{(i,j)}$.
\end{Definition}
Since the 2-Torus $\torus^2$ cannot be covered
by a single chart
we choose an atlas
$${\cal A}(\torus^2):=\{
(U_{(i,j)},\phi_{(i,j)})\restr 0\leq i\leq M-1, 0\leq j\leq
N-1\}$$
where the charts be
all the open subsets
$U_{(i,j)}\subset \torus^2$ which
cover the corresponding $2$-cells $(i,i+1)\times (j,j+1)$.
\begin{figure}
\begin{center}
\psfig{figure=f-lattice.eps}
\end{center}
\caption{Cubic lattice on $\torus^2$}
\protect\label{f-lattice}
\end{figure}
Let $U_{(i,j)}$ be a chart on $\torus^2$.
We denote the corresponding local section/trivialisation
by $\sigma^{(i,j)}(x)=\vphi^{(i,j)}(x,g^{(i,j)})$
and $\vphi^{(i,j)}$, respectively.
The local connection one-form is denoted by $\ga^{(i,j)}$.
Since we denote an open interval by $(i,j)$
a site is denoted by $\lpoint{i}{j}\in\torus^2$.
To make the lattice bundle $\xi_\Lambda$ unique
one has to fix the collection
of all transition functions
$\{ t_{(i,j)(k,l)}(x) \}$.
Our goal is to reconstruct the transition functions, i.e. lattice bundle, from
a given configuration of the lattice model.
In order to define our $\U1$ gauge theory over $\torus^2$ one
needs to specify a {\em global} connection one-form
$$\omega:\TT\EE\to\I\RR.$$
Since we are interested in a connection form which has a
trivial topological structure in a local trivialisation
$\vphi^{(i,j)}$ (no topological structure below a certain scale)
we define the {\em local} connection one-forms
to be
\begin{equation}\label{e-localconnection}
\ga^{(i,j)}\restrict{U_{(i,j)}}=\sigma^{(i,j)\ast}
\omega:={t^{(i,j)}}\inv (p) \circ\baseCTB{t^{(i,j)}}(p),
\end{equation}
for all $p\in U_{(i,j)}$,
i.e. the local connection one-form restricted to the chart $U_{(i,j)}$
has to be a pure gauge in the local trivialisation
$\vphi^{(i,j)}$.
This connection together with the lattice bundle
$\xi_{\Lambda(M,N)}=(\EE_\Lambda,\pi_\Lambda,\Lambda(M,N))$
defines our model $(\xi_{\Lambda(M,N)}, \omega)$.
Since the choice of all the $g^{(i,j)}$ is arbitrary this leads
to $N\cdot M$ degrees of freedom.
The choice of the $g^{(i,j)}$ is equivalent to the
choice of the local trivialisations $\vphi^{(i,j)}$,
but due to left invariance of our connection one-form (Cartan Maurer form)
the final result does not depend on these degrees.
\section{Reconstruction of the Bundle}
This property of the connection one-form $\omega$ leads to some
restrictions in the choice of local trivialisations. In
general, the only information one has are the 'transporters'
which are assigned to each link of the lattice, i.e. the
configuration of the lattice model. Since we have an atlas
${\cal A}(\torus^2)$ of the torus one has to be careful how to
assign the 'transporters' to the given charts.
\begin{Lemma}\label{l-charts-choice}
Let $(\xi_{\Lambda(M,N)}, \omega)$ be our lattice model.
Let $U_{(i,j)}$ be a chart on $\torus^2$ and $\Lambda_{(i,j)}$
the corresponding plaquette. Let
${\cal A}(\torus^2)$
be our atlas of $\torus^2$ and
$$\omega^{(i,j)}\restrict{U_{(i,j)}}=\sigma^{(i,j)\ast}
\omega:={t^{(i,j)}}\inv(p)
\circ\baseCTB{t^{(i,j)}}(p)$$
our local connection
one-form.
Let $\{\transport\}$ be a configuration.
Only three of the four lattice parallel translations
$$\transport^{(i,j)}_{\link{x_1 x_2}},\, \transport^{(i,j)}_{\link{x_2
x_3}},\,\transport^{(i,j)}_{\link{x_3 x_4}}\,\mbox{and}\,\,\,
\transport^{(i,j)}_{\link{x_4 x_1}}$$
which belong to the plaquette $\Lambda_{(i,j)}$ can be assigned to
the corresponding local trivialisation
$\vphi^{(i,j)}$, i.e. belong to the same local representation.
\end{Lemma}
\begin{proof}
Since the local connection one-form $\omega^{(i,j)}\restrict{U_{(i,j)}}$
is a pure gauge the lattice parallel translations around the plaquette
must be closed.
Therefore the lattice parallel
translation $\transport^{(i,j)}_{\gamma}$ has to be the group
identity $e$, thus
three of the four lattice parallel translations
have to be given in the local trivialisation and the fourth
has to be the inverse of the composition of the given three.
\end{proof}
The next step is to reconstruct the transition functions $\{
t_{(i,j)(k,l)}(x) \}$ from a given configuration of the lattice
model.
\begin{figure}
\begin{center}
\psfig{figure=f-t2charts.eps}
\end{center}
\caption{Choice of the charts}
\protect\label{f-t2charts}
\end{figure}
Take a local section $\sigma^{(i,j)}$ together with the four
neighboring local sections $\sigma^{(i-1,j)}$,
$\sigma^{(i+1,j)}$, $\sigma^{(i,j-1)}$ and $\sigma^{(i,j+1)}$.
Denote the transition function which maps from the fiber $\U1$
in the local trivialisation $\vphi^{(i-1,j)}$ to the same fiber
in the local trivialisation $\vphi^{(i,j)}$ at $\lpoint{k}{l}$
by
$$t_{(i,j)(i-1,j)}(\lpoint{k}{l}),$$
we obtain the following relation for the elements
$h^{(i-1,j)}(\lpoint{k}{l})$ and $h^{(i,j)}(\lpoint{k}{l})$ of
$\U1$:
\begin{equation}\label{e-sect-trans}
h^{(i-1,j)}(\lpoint{k}{l})=h^{(i,j)}(\lpoint{k}{l})
\circ t_{(i,j)(i-1,j)}(\lpoint{k}{l}).
\end{equation}
Since we want to calculate the transition function from the
local sections we rewrite (\ref{e-sect-trans}) to obtain
\begin{equation}\label{e-trans-sect}
t_{(i,j)(i-1,j)}(\lpoint{k}{l})=
{h^{(i,j)}}\inv(\lpoint{k}{l}) \circ
h^{(i-1,j)}(\lpoint{k}{l}).
\end{equation}
In each local
trivialisation $\vphi^{(i,j)}$ the local connection one-form
$\omega^{(i,j)}\restrict{U_{(i,j)}}$ has to be a pure gauge.
We choose our charts according to Fig.~\ref{f-t2charts} where
the three links which correspond to the three
lattice parallel translation which are assigned to
the corresponding local trivialisation
$\vphi^{(k,l)}$ are marked as bold lines.
In a trivialisation $\vphi^{(i,j)}$ we can
express the lattice parallel translation
in terms of the local connection one-form $\ga^{(i,j)}$ by
\begin{equation}\label{e-t2local-lat-trans}
\transport^{(i,j)}_{\link{x_1 x_2}}:h_1^{(i,j)}\mapsto
h_2^{(i,j)} = h_1^{(i,j)}\circ {\bf P}\exp\left(
-\int_{x_0}^{x_1} j^\ast \ga^{(i,j)} \right).
\end{equation}
Since we have one degree of freedom per local
trivialisation we choose
$$h_1^{(i,j)}:=g^{(i,j)}$$
where $g^{(i,j)}$ is an arbitrary $\U1$-element.
Denote the three lattice parallel translations along the links
$\link{x_1 x_2}$, $\link{x_2 x_3}$ and $\link{x_1 x_4}$ by
$\transport^{(i,j)}_{\link{x_1 x_2}}, \transport^{(i,j)}_{\link{x_2
x_3}}$ and
$\transport^{(i,j)}_{\link{x_1 x_4}}$, respectively.
The fourth
lattice parallel translation is nothing but
$$\transport^{(i,j)}_{\link{x_3 x_4}}:=\transport^{(i,j)}_{\link{x_3
x_2}}\circ
\transport^{(i,j)}_{\link{x_2 x_1}}
\circ\transport^{(i,j)}_{\link{x_1 x_4}},$$
since our local connection one-form has to be a pure gauge.
\begin{table}
\caption{Notation of local coordinates $x$ and fiber elements $h$ in different charts}
\begin{tabular}{llllll}
\noalign{\smallskip\hrule\smallskip}
point of $\torus^2$ & $U_{(i,j)}$ & $U_{(i-1,j)}$ & $U_{(i+1,j)}$
& $U_{(i,j-1)}$ & $U_{(i,j+1)}$\\
\noalign{\smallskip\hrule\smallskip}
$\lpoint{i}{j}$ & $x_1^{(i,j)}$ & $x_2^{(i-1,j)}$ & - & $x_4^{(i,j-1)}$ & - \\
$\lpoint{i+1}{j}$ & $x_2^{(i,j)}$& - & $x_1^{(i+1,j)}$ & $x_3^{(i,j-1)}$ & - \\
$\lpoint{i+1}{j+1}$ & $x_3^{(i,j)}$& - & $x_4^{(i+1,j)}$ & - & $x_2^{(i,j+1)}$ \\
$\lpoint{i}{j+1}$ & $x_4^{(i,j)}$& $x_3^{(i-1,j)}$ & - & - & $x_1^{(i,j+1)}$ \\[3ex]
$\lpoint{i}{j}$ & $h_1^{(i,j)}$ & $h_2^{(i-1,j)}$ & - & $h_4^{(i,j-1)}$ & - \\
$\lpoint{i+1}{j}$ & $h_2^{(i,j)}$& - & $h_1^{(i+1,j)}$ & $h_3^{(i,j-1)}$ & - \\
$\lpoint{i+1}{j+1}$ & $h_3^{(i,j)}$& - & $h_4^{(i+1,j)}$ & - & $h_2^{(i,j+1)}$ \\
$\lpoint{i}{j+1}$ & $h_4^{(i,j)}$& $h_3^{(i-1,j)}$ & - & - & $h_1^{(i,j+1)}$ \\
\noalign{\smallskip\hrule\smallskip}
\end{tabular}
\protect\label{t-t2notat}
\end{table}
We 'transport' the element $g^{(i,j)}$ at $x_1^{(i,j)}$ via
these lattice parallel translations to obtain the fiber elements
at all sites (c.f. Fig.~\ref{f-lat-trans}) of this plaquette:
$$
\begin{array}{l}
h_1^{(i,j)}:=g^{(i,j)},\\
h_2^{(i,j)}=g^{(i,j)} \circ\transport^{(i,j)}_{\link{x_1 x_2}},\\
h_3^{(i,j)}=g^{(i,j)} \circ\transport^{(i,j)}_{\link{x_1 x_2}}
\circ\transport^{(i,j)}_{\link{x_2 x_3}},\\
h_4^{(i,j)}=g^{(i,j)} \circ\transport^{(i,j)}_{\link{x_1 x_4}}.\\
\end{array}
$$
Now we calculate the transition functions from the
local trivialisations $\vphi^{(i,j)}$.
\begin{figure}
\begin{center}
\psfig{figure=f-t2transition.eps}
\end{center}
\caption{Notation of the local sections}
\protect\label{f-t2notation}
\end{figure}
Each site is covered by four charts.
The first step is to recognize that only three
of the four transition functions have to be calculated since
the cocycle conditions give
some additional relations.
We use the charts according to Fig.~\ref{f-t2charts} and
summarize the notation of the local coordinates in Table~\ref{t-t2notat}.
Our choice of charts gives the two relations
\begin{equation}\label{e-simpl}
\transport^{(i,j)}_{\link{x_3 x_2}}=\transport^{(i+1,j)}_{\link{x_4 x_1}}
\quad\mbox{and}\quad \transport^{(i,j)}_{\link{x_4 x_1}}=
\transport^{(i-1,j)}_{\link{x_3 x_2}},
\end{equation}
which can be used to simplify the results.
Also in the non-Abelian case they are useful because if one
calculates Chern classes one takes the trace over the
transition functions.
For the Abelian case together with
the two relations of (\ref{e-simpl}) and
with the use of
(\ref{e-trans-sect})
we obtain:
\begin{itemize}
\item Site $\lpoint{i}{j}$
\begin{eqnarray}\label{trans1}
t_{(i,j)(i-1,j)}(\lpoint{i}{j}) & = & {g^{(i,j)}}\inv \circ
g^{(i-1,j)} \circ\transport^{(i-1,j)}_{\link{x_1x_2}} \nonumber \\
t_{(i,j)(i,j-1)}(\lpoint{i}{j}) & = & {g^{(i,j)}}\inv \circ
g^{(i,j-1)} \circ\transport^{(i,j-1)}_{\link{x_1 x_4}}
\end{eqnarray}
\item Site $\lpoint{i+1}{j}$
\begin{eqnarray}\label{trans2}
t_{(i,j)(i,j-1)}(\lpoint{i+1}{j}) & = & \transport^{(i,j)}_{\link{x_2 x_1}}\circ
{g^{(i,j)}}\inv \circ
g^{(i,j-1)} \circ\transport^{(i,j-1)}_{\link{x_1 x_2}}
\circ\transport^{(i,j-1)}_{\link{x_2 x_3}} \nonumber \\
t_{(i,j)(i+1,j)}(\lpoint{i+1}{j}) & = & \transport^{(i,j)}_{\link{x_2 x_1}}\circ
{g^{(i,j)}}\inv \circ g^{(i+1,j)}
\end{eqnarray}
\item Site $\lpoint{i+1}{j+1}$
\begin{eqnarray}\label{trans3}
t_{(i,j)(i+1,j)}(\lpoint{i+1}{j+1}) & = &
\transport^{(i,j)}_{\link{x_2 x_1}} \circ
{g^{(i,j)}}\inv \circ
g^{(i+1,j)} \\ \nonumber
t_{(i,j)(i,j+1)}(\lpoint{i+1}{j+1}) & = &\transport^{(i,j)}_{\link{x_3 x_2}} \circ
\transport^{(i,j)}_{\link{x_2 x_1}} \circ
{g^{(i,j)}}\inv \circ
g^{(i,j+1)} \circ\transport^{(i,j+1)}_{\link{x_1 x_2}}
\end{eqnarray}
\item Site $\lpoint{i}{j+1}$
\begin{eqnarray}\label{trans4}
t_{(i,j)(i,j+1)}(\lpoint{i}{j+1}) & = & \transport^{(i,j)}_{\link{x_4 x_1}}\circ
{g^{(i,j)}}\inv \circ g^{(i,j+1)} \nonumber \\
t_{(i,j)(i-1,j)}(\lpoint{i}{j+1}) & = &
{g^{(i,j)}}\inv \circ
g^{(i-1,j)} \circ\transport^{(i-1,j)}_{\link{x_1 x_2}}
\end{eqnarray}
\end{itemize}
\section{Topological Invariants}
The Chern character is used to measure the twist of
a bundle.
Integrating the first Chern character $\chch_1(\gf)$
over the whole lattice gives an integer called
\EM{Chern number}
$$\Ch(\xi_{\Lambda(M,N)}):=\int_{\torus^2}\chch_1(\gf)=
\frac{\I}{2\pi}\int_{\torus^2}\gf,$$
which is a topological invariant and which can be used
to classify the $\U1$-bundles over $\Lambda(M,N)$.
One has to be careful if integrating over $\Lambda(M,N)$ since
our bundle is constructed by patching together local pieces
via the transition functions.
One also should remember that integration of a $n$-form over
a manifold is done via integration over
$n$-cells in the corresponding complex.
Let $\ga$ be a $2$-form and $j:\Lambda(M,N) \to \torus^2$.
Then one writes simply
$$\int_{\torus^2}\ga$$
for
$$\int_{\torus^2}\ga:=\int_{\Lambda(M,N)}j^\ast\ga,$$
because the integral is independent of the cellular subdivision.
Let $\{\lambda_{(i,j)}\}$ be a partition of unity subordinate to
the covering $\{U_{(i,j)}\}$.
Then our pullback {\em global} connection one-form
can be written as
\begin{equation}\label{e-pb-conn}
\ga:=\sum_{(i,j)\in\ZZ_M\times\ZZ_N}
\lambda_{(i,j)}\,\ga=\sum_{(i,j)\in\ZZ_M\times\ZZ_N} \ga_{(i,j)}.
\end{equation}
Therefore we get
$$\gf=\baseCTB{}\,\ga=\sum_{(i,j)\in\ZZ_M\times\ZZ_N}
\baseCTB{}\,\ga_{(i,j)}.$$
Integration is now be done via partition of unity by
$$\int_{\torus^2}\gf:=\sum_{(i,j)\in\ZZ_M\times\ZZ_N}
\int_{U_{(i,j)}}\baseCTB{}\,\ga_{(i,j)}.$$
Since our lattice model $(\xi_{\Lambda(M,N)},\omega)$
is designed in such a way that
there is
no topological structure below a certain scale
we have
$$
\ga^{(i,j)}:=\ga^{(i,j)}\restrict{U_{(i,j)}}=\sigma^{(i,j)\ast}
\omega:={g^{(i,j)}}\inv (x) \circ\baseCTB{g^{(i,j)}}(x),
$$
for all $x\in U_{(i,j)}$.
We notice that the part of our pullback {\em global} connection one-form
with compact support on $U_{(i,j)}$ denoted by $\ga_{(i,j)}$
is obtained by
rewriting
$$
\ga\restrict{U_{(i,j)}}=\ga_{(i,j)}+
\sum_{\mbox{\scriptsize neighbors}} \ga_{(k,l)}\restrict{U_{(i,j)}
\cap U_{(k,l)}}.
$$
to get
$$
\ga_{(i,j)}=\ga\restrict{U_{(i,j)}}-
\sum_{\mbox{\scriptsize neighbors}} \ga_{(k,l)}\restrict{U_{(i,j)}
\cap U_{(k,l)}}
.$$
\begin{figure}
\begin{center}
\psfig{figure=f-t2ovl.eps}
\end{center}
\caption{Partition of the connection one-form}
\protect\label{f-t2ovl}
\end{figure}
Let $(\xi_{\Lambda(M,N)},\omega)$ be
our lattice model. Take overlapping charts
$U_1$ and $U_2$ on $\torus^2$ and let $\ga^1$ and $\ga^2$ be
the local connection one-form on $U_1$ and $U_2$, respectively.
Let $\{\lambda_{(i)}\}$ be a partition of unity subordinate to
the covering $\{U_i\}$.
The corresponding pullback connection one-form is
$\ga\restrict{U_1\cup U_2}=\ga_{(1)}+\ga_{(2)}$. With
the two relations
$$\ga_{(1)}=\ga\restrict{U_1}-\ga_{(2)}\restrict{U_1\cap U_2}$$
and
$$\ga_{(2)}=\ga\restrict{U_2}-\ga_{(1)}\restrict{U_1\cap U_2}$$
the integral
\begin{eqnarray*}
\int_{U_1\cup U_2}\baseCTB{}\,\ga & = &
\int_{U_1}\baseCTB{}\,\ga_{(1)}+\int_{U_2}\baseCTB{}\,\ga_{(2)}\\
& = & \int_{U_1}\baseCTB{}\,\ga\restrict{U_1} -
\int_{U_1}\baseCTB{}\,\ga_{(2)}\restrict{U_1\cap U_2}\\
& & + \int_{U_2}\baseCTB{}\,\ga\restrict{U_2} -
\int_{U_2}\baseCTB{}\,\ga_{(1)}\restrict{U_1\cap U_2}
\end{eqnarray*}
expands to
$$\int_{U_1\cup U_2}\baseCTB{}\,\ga=-\int_{U_1}\baseCTB{}\,\ga_{(2)}\restrict{U_1\cap U_2}
-\int_{U_2}\baseCTB{}\,\ga_{(1)}\restrict{U_1\cap U_2},$$
where we had assumed that the local connection forms
have to be pure gauges, i.e.
$\baseCTB{}\,\ga^1\restrict{U_1}\equiv 0$ and
$\baseCTB{}\,\ga^2\restrict{U_2}\equiv 0$.
Applying Stokes' theorem gives
$$\int_{U_1\cup U_2}\baseCTB{}\,\ga=-\int_{\partial U_1}
\ga_{(2)}\restrict{U_1\cap U_2}
-\int_{\partial U_2}\ga_{(1)}\restrict{U_1\cap U_2}.$$
Finally we realize (c.f. Fig~\ref{f-t2ovl}) that at the boundaries
of $U_1$ and $U_2$
only the local connections $\ga^2$ and $\ga^1$, respectively, count.
Note that due to the left invariance of our local connection one-form
we have with $\tilde t(x) = g \circ t(x)$ and $g$ constant
\begin{equation}\label{constg}
t\inv(x)\circ\baseCTB{}\,t(x) = \tilde t\inv(x)\circ\baseCTB{}\,\tilde t(x).
\end{equation}
We further notice that due to the definition of the integral
over a cell-complex
our map $j$ is an inclusion and can
be omitted.
Therefore we get
$$\int_{U_1\cup U_2}\baseCTB{}\,\ga=-\int_{\link{x_1 x_2}}\ga^2
-\int_{\link{x_2 x_1}}\ga^1,$$
and together with
$$\ga^1=\ga^2+
t_{21}\inv \circ \,\baseCTB{}\,t_{21},$$
the result
\begin{eqnarray*}\label{e-int-trans}
\int_{U_1\cup U_2}\baseCTB{}\,\ga &=& -\int_{\link{x_1 x_2}}\ga^2
-\int_{\link{x_2 x_1}}\ga^2-\int_{\link{x_2 x_1}}
t_{21}(x)\inv \circ \,\baseCTB{}\,t_{21}(x)\\
& = & -\int_{\link{x_2 x_1}}
t_{21}(x)\inv \circ \,\baseCTB{}\,t_{21}(x)\\
& = & \log t_{21}(x_1)-\log t_{21}(x_2).
\end{eqnarray*}
If we further assume that
\begin{equation}\label{lesspi}
| \int_{U_1\cup U_2}\baseCTB{}\,\ga | < \pi
\end{equation}
then the above equation can be written as
\begin{equation}\label{e-chern2}
\int_{U_1\cup U_2}\baseCTB{}\,\ga = \log \left( t_{21}(x_1) \circ
t_{21}\inv(x_2) \right) ,
\end{equation}
where $\log ( t_{21}(x_1) \circ t_{21}\inv(x_2))$ is defined as
the principal value with range $[-\pi,\pi)$.
From (\ref{lesspi}) follows that $ t_{21}(x_1) \circ
t_{21}\inv(x_2)\not=-1$.
As we will see later there can be
\EM{configurations on $\Lambda$} which violate
assumption (\ref{lesspi}). Since the values
of each transition function $ t_{(i,j)(k,l)}(x) $ are only known
on the two
end points of the region of integration,
a parameterization of $\U1$, such that at least
$$
| \int_{U_1\cup U_2}\baseCTB{}\,\ga | \leq \pi
$$
holds, can always be assumed. Note that this assumption is an addition
to (\ref{e-localconnection}).
Due to the fact that
on $U_{(i,j)}\cap U_{(k,l)}$ the local connection one-forms
are related as
$$\ga^{(k,l)}(x)=\ga^{(i,j)}(x) +
t_{(i,j)(k,l)}\inv(x)\circ\baseCTB{}\,t_{(i,j)(k,l)}(x) ,$$
we obtain:
\begin{eqnarray*}
\int_{\torus^2}\gf & = & - \sum_{\link{x_a x_b}}
\int_{\link{x_a x_b}} t_{(i,j)(k,l)}(x)\inv \circ\baseCTB{}\,t_{(i,j)(k,l)}(x)\\
& = & - \sum_{\link{x_a x_b}}\left[
\log t_{(i,j)(k,l)}(x_b)- \log t_{(i,j)(k,l)}(x_a)\right],
\end{eqnarray*}
where the sum is over all directed links $\link{x_a x_b}$
according to Fig.~\ref{f-t2orient}.
Thus the Chern number is
\begin{equation}\label{e-chernnumber}
\Ch(\xi_{\Lambda(M,N)})=\frac{\I}{2\pi}
\sum_{\link{x_a x_b}}\left[
\log t_{(i,j)(k,l)}(x_a)- \log t_{(i,j)(k,l)}(x_b)\right].
\end{equation}
\begin{figure}
\begin{center}
\psfig{figure=f-t2orient.eps}
\end{center}
\caption{Orientation of the plaquettes}
\protect\label{f-t2orient}
\end{figure}
When integrating over all links one should remember that our
lattice is a directed complex, i.e. we have
an orientation (c.f. Fig.~\ref{f-t2orient}).
Let $M$ and $N$ be even
integers,
then the Chern number (c.f. (\ref{e-chernnumber})) gives
\begin{eqnarray*}
\Ch(\xi_{\Lambda}) & = &\frac{\I}{2\pi}
\sum_{\{\bar\imath, \bar \jmath\}}\left[
\log t_{(i,j)(i,j-1)}(\lpoint{i}{j})- \log
t_{(i,j)(i,j-1)}(\lpoint{i+1}{j})\right]\\
& + & \frac{\I}{2\pi}
\sum_{\{\bar\imath, \bar \jmath\}}\left[
\log t_{(i,j)(i,j+1)}(\lpoint{i+1}{j+1})- \log
t_{(i,j)(i,j+1)}(\lpoint{i}{j+1})\right]\\
& + & \frac{\I}{2\pi}
\sum_{\{\bar\imath, \bar \jmath\}}\left[
\log t_{(i,j)(i-1,j)}(\lpoint{i}{j+1}) - \log
t_{(i,j)(i-1,j)}(\lpoint{i}{j})\right]\\
& + & \frac{\I}{2\pi}
\sum_{\{\bar\imath, \bar \jmath\}}\left[
\log t_{(i,j)(i+1,j)}(\lpoint{i+1}{j+1}) - \log
t_{(i,j)(i+1,j)}(\lpoint{i+1}{j})\right],
\end{eqnarray*}
where the sum is over all even or odd sites $\{\bar\imath, \bar \jmath\}$.
The last two sums give zero because
we have
$$t_{(i,j)(i-1,j)}(\lpoint{i}{j})=t_{(i,j)(i-1,j)}(\lpoint{i}{j+1})$$
and
$$t_{(i,j)(i+1,j)}(\lpoint{i+1}{j})=t_{(i,j)(i+1,j)}(\lpoint{i+1}{j+1}).$$
If we straightforwardly insert the transition functions then this
gives with the use of (\ref{constg})
\begin{eqnarray}\label{e-ch-sum}
\Ch(\xi_{\Lambda}) & = & \frac{\I}{2\pi}
\sum_{\{\bar\imath, \bar \jmath\}}
\log \transport^{(i,j-1)}_{\link{x_1 x_4}}
-
\log \transport^{(i,j)}_{\link{x_2 x_1}}\circ
\transport^{(i,j-1)}_{\link{x_1 x_2}}
\circ\transport^{(i,j-1)}_{\link{x_2
x_3}}\nonumber\\
& + & \frac{\I}{2\pi}
\sum_{\{\bar\imath, \bar \jmath\}}
\log
\transport^{(i,j)}_{\link{x_3 x_2}} \circ
\transport^{(i,j)}_{\link{x_2 x_1}} \circ
\transport^{(i,j+1)}_{\link{x_1 x_2}}
-\log\transport^{(i,j)}_{\link{x_4 x_1}}
\end{eqnarray}
Note that this definition of the Chern number
is not lattice gauge invariant in the usual
sense. This means that for a general
\EM{configuration on $\Lambda$}
different lattice gauges might
lead to different results for the Chern number. We also note that
reversing all transporters, which should lead to
$-\Ch(\xi_{\Lambda})$, does in general not hold for the above result.
To derive a unique result we must apply assumption (\ref{lesspi})
and obtain
\begin{eqnarray}\label{e-ch-sum1}
\Ch(\xi_{\Lambda}) & = & \frac{\I}{2\pi}
\sum_{\{\bar\imath, \bar \jmath\}}
\log \Bigl
( \transport^{(i,j-1)}_{\link{x_1 x_4}}
\circ (\transport^{(i,j-1)}_{\link{x_1 x_2}}
\circ\transport^{(i,j-1)}_{\link{x_2 x_3}}
\circ \transport^{(i,j)}_{\link{x_2 x_1}})\inv \Bigr) \nonumber\\
& + & \frac{\I}{2\pi}
\sum_{\{\bar\imath, \bar \jmath\}}
\log \Bigl(
\transport^{(i,j+1)}_{\link{x_1 x_2}} \circ
\transport^{(i,j)}_{\link{x_3 x_2}} \circ
\transport^{(i,j)}_{\link{x_2 x_1}} \circ
(\transport^{(i,j)}_{\link{x_4 x_1}})\inv \Bigr)
\end{eqnarray}
In (\ref{e-ch-sum}) as well as in (\ref{e-ch-sum1}) the sum
over all even sites can be replaced by the sum over
all odd sites replacing
$(i,j)$ by $(i,j-1)$ and
$\log u$ by $-\log u\inv$.
Finally, we rewrite the second sum such that we can take the sum
instead of all even sites over all sites $\lpoint{i}{j}$ and obtain
the following theorem.
\begin{Theorem}\label{th-chern}
Let $(\xi_{\Lambda(M,N)},\omega)$ be our lattice model and
choose the charts according to Fig.~\ref{f-t2charts}. The
local connection one-form is a pure gauge and defined as in
(\ref{e-localconnection}). Let the transition functions be as
in (\ref{trans1}) to (\ref{trans4}). Assume that for each
1-cell (link)
$$|\int_{\link{x_a x_b}} t_{(i,j)(k,l)}(x)\inv
\circ\baseCTB{}\,t_{(i,j)(k,l)}(x)| < \pi $$
holds; i.e. for each 0-cell (site) $\{i,j\}$ we must have
$$\transport^{(i,j)}_{\link{x_2 x_1}}
\circ\transport^{(i,j-1)}_{\link{x_4 x_1}}
\circ\transport^{(i,j-1)}_{\link{x_1 x_2}}
\circ\transport^{(i,j-1)}_{\link{x_2 x_3}} \neq -1$$
Choose $M$ and $N$ to be even integers.
The Chern number of the lattice bundle $\xi_{\Lambda(M,N)}$ is
then given by
\begin{equation}\label{e-t2plaqangle}
\Ch(\xi_{\Lambda}) = -\frac{\I}{2\pi}
\sum_{\lpoint{i}{j}}
\log \left(
\transport^{(i,j)}_{\link{x_2 x_1}}
\circ\transport^{(i,j-1)}_{\link{x_4 x_1}}
\circ\transport^{(i,j-1)}_{\link{x_1 x_2}}
\circ\transport^{(i,j-1)}_{\link{x_2 x_3}}
\right).
\end{equation}
\end{Theorem}
\begin{proof}
Previous calculation.
\end{proof}
Note that such configurations for which the above Theorem
holds are often called \EM{continuous configurations} and the excluded
ones are called \EM{exceptional configurations}.
\begin{figure}
\begin{center}
\psfig{figure=f-t2gij.eps}
\end{center}
\caption{Two neighboring local trivialisations}
\protect\label{f-t2gij}
\end{figure}
If we denote the lattice parallel translations
according to the standard notation
in lattice field theories,
i.e.
\begin{eqnarray*}
U_{\lpoint{i}{j-1},\hat 1} & := &\transport^{(i,j-1)}_{\link{x_1 x_2}},\\
U_{\lpoint{i}{j-1},\hat 2} & := &\transport^{(i,j-1)}_{\link{x_1 x_4}},\\
U_{\lpoint{i+1}{j-1},\hat 2} & := &\transport^{(i,j-1)}_{\link{x_2 x_3}},\\
U_{\lpoint{i}{j},\hat 1} & := &\transport^{(i,j)}_{\link{x_1 x_2}},
\end{eqnarray*}
we obtain for (\ref{e-t2plaqangle})
$$
\Ch(\xi_{\Lambda}) = -\frac{\I}{2\pi}
\sum_{\lpoint{i}{j}}
\log \left({U_{\lpoint{i}{j},\hat 1}}\inv
\circ {U_{\lpoint{i}{j-1},\hat 2}}\inv
\circ U_{\lpoint{i}{j-1},\hat 1}
\circ U_{\lpoint{i+1}{j-1},\hat 2}\right),
$$
where the logarithm
$$
K_{(i,j-1)}:= \frac{\I}{2\pi}
\log \left({U_{\lpoint{i}{j},\hat 1}}\inv
\circ {U_{\lpoint{i}{j-1},\hat 2}}\inv
\circ U_{\lpoint{i}{j-1},\hat 1}
\circ U_{\lpoint{i+1}{j-1},\hat 2}\right),
$$
is called the \EM{plaquette angle} of the plaquette
$\Lambda_{(i,j-1)}$ and corresponds to the result obtained in \cite{Ph84}.
\section{Summary}
Starting with the physically reasonable assumption of a connection
which is locally represented by pure gauges, we were basically able to
calculate or better to assign a Chern number to each \EM{configuration
on $\Lambda$}. This so obtained result is unfortunately not consistent
with the usual understanding of lattice gauge invariance. However even
more problematic is the fact that the general result for
$\Ch(\xi_{\Lambda})$ does not lead to $-\Ch(\xi_{\Lambda})$ for all
\EM{configurations on $\Lambda$} when
inverting all parallel translations $\transport_{\link{x y}}$. These two
problems can be resolved with one additional assumption on the
connection which is expressed in an assumption on the parameterization
of the transition functions such that the integrals over the overlap
areas are less than $\pi$. This can always be assumed as far as
$\transport^{(i,j)}_{\link{x_2 x_1}}
\circ\transport^{(i,j-1)}_{\link{x_4 x_1}}
\circ\transport^{(i,j-1)}_{\link{x_1 x_2}}
\circ\transport^{(i,j-1)}_{\link{x_2 x_3}} \neq -1$ for all $\{i,j\}$.
As already observed in \cite{Ph84} without such a condition or at
least some restricting assumption there is no unique result. Depending
on the parameterization of $\U1$ there is always one group element which,
to put it crudely, allows for \EM{two results} thus a tie breaker is
needed.
\section*{Acknowledgments}
\markboth{Acknowledgments}{}
We would like to thank Ch. Gattringer, H. Grosse, C.B. Lang and L. Pittner
for many discussions.
\cleardoublepage
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\section{\bf Introduction}
\vspace*{-0.5pt}
\noindent
The number of exactly solvable eigenproblems in non-relativistic quantum
mechanics is small, and most of them can be dealt with the factorization
method. This technique, introduced long ago by Schr\"odinger,${}^1$ was
analysed in depth by Infeld and Hull,${}^2$ who made an exhaustive
classification of factorizable potentials. Later on, Witten noticed the
possibility of arranging the Schr\"odinger's Hamiltonians into isospectral
pairs (supersymmetric (SUSY) partners).${}^3$ The resulting {\it
supersymmetric quantum mechanics} catalysed the study of hierarchies of
`exactly solvable Hamiltonians'. An additional step was Mielnik's `atypical'
factorization${}^4$ through which the general SUSY partner for the oscillator
was found; this technique was immediately applied to the hydrogen
potential.${}^5$ Meanwhile, Nieto${}^6$ and Andrianov {\it et. al.}${}^7$
put the method on its natural background discovering the links between SUSY,
factorization and Darboux algorithm. These developments caused the renaissance
of factorization and related algebraic methods, with particular attention
focused on the first order differential shift operators.${}^{8-16}$
As can be noticed, however, the scheme is still narrow. An obvious
generalization arises when higher order differential shift operators are used
to connect the Hamiltonian pair. The idea of HSUSY (higher order SUSY),
recently put forward by Andrianov {\it et. al.}${}^{17-18}$ (see also
${}^{19}$), incubated since 70-tieth.${}^{20-21}$ In this paper we will
restrict ourselves to the case when the shift operator is of second order, and
we name it SUSUSY.
\pagebreak
\textheight=7.8truein
\setcounter{footnote}{0}
\renewcommand{\fnsymbol{footnote}}{\alph{footnote}}
\section{\bf Second order shift operator technique}
\noindent
We postulate the existence of a second order differential operator
interconnecting two different Hamiltonians $H, \ {\widetilde H}$:
\begin{equation}
{\widetilde H} A^\dagger = A^\dagger H, \label{(1)}
\end{equation}
\begin{equation}
H = - {d^2\over dx^2} + V(x), \quad {\widetilde H} = - {d^2\over dx^2} +
{\widetilde V}(x), \label{(2)}
\end{equation}
\begin{equation}
A^\dagger = {d^2\over dx^2} + \beta(x) {d \over dx} + \gamma(x).
\label{(3)}
\end{equation}
Equality (1) imposes some restrictions to the functions $\{V(x),{\widetilde
V}(x),\beta(x),\gamma(x)\}$:
\begin{equation}
{\widetilde V} = V + 2\beta', \quad
2V+\delta = \beta^2 - 2\gamma -\beta' , \quad
V'' + \beta V' = 2 \gamma \beta' -\gamma'', \label{(4)}
\end{equation}
where $\delta$ is an integration constant. We shall suppose that $\beta(x)$ is
given and we shall express the other functions $\{ V(x), \ {\widetilde V}(x),
\ \gamma(x)\}$ in terms of it. If we solve $\gamma(x)$ from second equation
(4) and substitute the result in the third equation (4), we get:
\begin{equation}
{\beta''' \over 2} -2 \beta'^2 + \beta'\beta^2 - \beta \beta'' -\delta\beta' =
\beta V' + 2 \beta' V . \label{(5)}
\end{equation}
Multiplying by $\beta$, it can be immediately integrated, yielding:
\begin{equation}
V(x)= {\beta''\over 2\beta}-\left({\beta'\over 2\beta}\right)^2 - \beta' +
{\beta^2\over 4} + {c\over \beta^2} - {\delta\over 2}, \label{(6)}
\end{equation}
where $c$ is a new integration constant. The other two unknown functions
become:
\begin{equation}
{\widetilde V}(x)= {\beta''\over 2\beta}-\left({\beta'\over 2\beta}\right)^2 +
\beta' + {\beta^2\over 4} + {c\over \beta^2} - {\delta\over 2}, \quad
\gamma(x) = -{\beta''\over 2\beta}+\left({\beta'\over 2\beta}\right)^2 +
{\beta' \over 2} + {\beta^2\over 4} - {c\over \beta^2} . \label{(7)}
\end{equation}
Before going to the particular cases, let us notice a curious relation between
the second order shift operator and Witten idea of the SUSY quantum mechanics.
\section{\bf Second order SUSY (SUSUSY)}
\noindent
According to Witten,${}^3$ SUSY arises by defining a set of operators $Q_i$
that commute with the (supersymmetric) Hamiltonian $H_s$,
\begin{equation}
[Q_i,H_s]=0, \quad i=1\cdots N, \label{(8)}
\end{equation}
and satisfy the algebra
\begin{equation}
\{Q_i,Q_j\} = \delta_{ij} H_s, \label{(9)}
\end{equation}
where $[\cdot,\cdot]$ represents the commutator and $\{\cdot,\cdot\}$ the
anticommutator. Now, with the aid of the operators $A^\dagger, \ A$ of the
previous section, one can construct a case of the supersymmetric algebra
(8-9) with $N=2$.
With this aim, define the supercharges:${}^{17-18}$
\begin{equation}
Q=\left(\matrix{0 & 0 \cr A & 0} \right), \qquad Q^\dagger = \left(\matrix{0
& A^\dagger \cr 0 & 0} \right), \label{(10)}
\end{equation}
where $A^\dagger$ is given in (3) and $A$ is its adjoint. Notice that $Q^2
= Q^{\dagger 2} = 0$. Let us construct an operator, which we cannot abstain to
call the SUSUSY `Hamiltonian':
\begin{equation}
H_{ss} = \{ Q, Q^\dagger \} = \left( \matrix{ A^\dagger A & 0 \cr 0 &
AA^\dagger} \right) = \left( \matrix{H^+ & 0 \cr 0 & H^-}\right). \label{(11)}
\end{equation}
Using the SUSY languaje, $H^+=A^\dagger A$ and $H^-=AA^\dagger$ should be
called the SUSY partners. Notice that the SUSUSY `Hamiltonian' $H_{ss}$
commutes with the two supercharges $Q$ and $Q^\dagger$. The SUSY generators
$Q_1= (Q^\dagger + Q)/\sqrt{2}$, $Q_2=(Q^\dagger - Q)/i\sqrt{2}$, and $H_{ss}$
satisfy the supersymmetric algebra (8-9).
Notice that the SUSY partners $H^+,H^-$ are now the fourth order differential
operators. It can be shown that $H^+$ commutes with ${\widetilde H}$ and $H^-$
commutes with $H$. Hence, $H^+$ can be a certain function of ${\widetilde H}$
and $H^-$ a function of $H$. Indeed:
\begin{equation}
H^+ = \left({\widetilde H} + {\delta \over 2}\right)^2 -c, \quad
H^- = \left(H + {\delta \over 2}\right)^2 -c. \label{(12)}
\end{equation}
A physical Hamiltonian $H_s$ can be defined (compare with the recent ideas of
${}^{17-18}$),
\begin{equation}
H_s = \left( \matrix{ {\widetilde H} & 0 \cr 0 & H}\right), \label{(13)}
\end{equation}
and the SUSUSY `Hamiltonian' $H_{ss}$ is related to $H_s$ by means of:
\begin{equation}
H_{ss} = \left( H_s + {\delta \over 2}\right)^2 - c. \label{(14)}
\end{equation}
Thus, the SUSUSY `Hamiltonian' $H_{ss}$ is a quadratic form of a physical
Hamiltonian $H_s$. The diagonal elements of $H_s$ are the two Hamiltonians $H
, \ {\widetilde H}$ of the previous section, which are related by the second
order differential operators $A, \ A^\dagger$ (compare ${}^{17-18}$).
\newpage
\section{\bf Example: the SUSUSY oscillator}
\noindent
We shall now look for the SUSUSY analogue of the oscillator Hamiltonian
\begin{equation}
H = -{d^2\over dx^2} + x^2 \label{(15)}
\end{equation}
We will try to show the existence of a 2-parametric family of potentials
isospectral to $V(x) = x^2$. This has to do with the general solution
$\beta(x)$ of equation (6). This solution of course should include the ladder
operator $A^\dagger = (a^\dagger)^2$, where $a^\dagger = - d/dx +x$ is the
standard ladder operator of the harmonic oscillator. This means that for $V(x)
= x^2$ and $\beta_p(x) = -2 x$, equation (6) should become an identity, which
fixes the constants to $c=1, \delta=4$. Substituting these results again in
(6) and multiplying by $2\beta^2$, we get:
\begin{equation}
\beta\beta'' - {\beta'^2\over 2} - 2\beta^2\beta' + {\beta^4\over 2} -
4\beta^2 -2 x^2 \beta^2 + 2 = 0. \label{(16)}
\end{equation}
Let us notice the existence of an explicit solution more general than $\beta_p
= -2 x$. It arises after multiplying the standard raising operator $a^\dagger$
by the operator $b^\dagger$ of atypical factorizations,${}^4$ i.e., $A^\dagger
= b^\dagger a^\dagger$, leading to:
\begin{equation}
\beta_p(x) = -2x - {e^{-x^2}\over \lambda + \int_0^x e^{-y^2} dy}, \quad
\vert\lambda\vert > {\sqrt{\pi}\over 2}. \label{(17)}
\end{equation}
The general solution of (16), which depends on two constants, should reduce
itself to (17) as one of them takes a particular value (or one of them becomes
a function of the other one).
Here, I would like to present some partial numeric results obtained when (16)
is integrated to provide $\beta(x)$ which arises as continuous deformations of
the particular solutions (17). With this aim we choose the initial point
$(\beta(0),\beta'(0))$ on the Poincar\'e plane close to
$(\beta_p(0),\beta_p'(0))$ and use then a standard numeric integration
package\fnm{a}\fnt{a}{We have employed the routine `NDSolve' of `Mathematica'}
to find $\beta(x)$ and to look for singularities in the corresponding SUSUSY
potential ${\widetilde V}(x) = x^2 + 2\beta'(x)$. If there is no singularity,
we increase slightly $\beta'(0)$ maintaining $\beta(0)$ fixed, and repeat the
integration until finding the upper threshold of $\beta'(0)$: above this
threshold a singularity arises while below it disappears. A similar procedure
is used to find the low threshold. After that, we make a small change of
$\beta(0)$ along $(\beta_p(0),\beta_p'(0))$ and start again the whole
process. In this way we can split the $\beta\beta'$-plane into the region
where the SUSUSY potential ${\widetilde V}(x)$ is free of singularities and
the rest. Notice that the points $(\beta_p(0),\beta_p'(0))$ provide a curve on
$\beta\beta'$-plane:
\begin{equation}
\beta_p'(0) = -2 + \beta_p^2(0), \quad \vert\beta_p(0)\vert <
{2\over\sqrt\pi}. \label{(18)}
\end{equation}
Departing from (18) we made the classification on figure 1.
\begin{figure}[htbp]
\vspace*{13pt}
\begin{minipage}{10truecm}
\hspace*{3truecm}
\epsfxsize=10truecm
\epsfbox{sususyf1.eps}
\end{minipage}
\vspace*{13pt}
\fcaption{Classification of the $\beta\beta'$-plane into regions where the
SUSUSY potential ${\widetilde V}(x)$ has no singularities (the shadowed
regions) and those with singularity (the white regions).}
\end{figure}
As we can see, there is a non-trivial region, the shadowed one, where the
SUSUSY potential ${\widetilde V}(x)$ has no singularity; it comprises the
curve in (18). For the purposes of this paper, to show that the general
family of SUSUSY oscillator potentials is two parametric, our calculation
brings already some information.
The SUSUSY potentials ${\widetilde V}(x)$ for various values of the pair
$(\beta(0), \beta'(0))$ lying at the shadowed region (no singularity, we have
fixed $\beta(0) = -0.7$) are shown in figure 2\fnm{b}\fnt{b}{Indeed,
${\widetilde V}(x)$ is displaced with respect to $V(x)=x^2$ a quantity $\delta
E = -4$. This can be seen after realizing that ${\widetilde V}(x) + 4
\rightarrow x^2$ when $(\beta(0),\beta'(0))\rightarrow (0,-2)$. Hence, we
decided to represent on the vertical axis of figure 2 the potentials
${\widetilde V}(x)+4$ rather that ${\widetilde V}(x)$.}. This family is richer
than the Abraham-Moses (AM) SUSY potentials:${}^{4,22}$
\begin{equation}
V_\lambda(x) = x^2 - 2 {d\over dx}\left({e^{-x^2}\over \lambda + \int_0^x e^{-
y^2}dy}\right). \label{(19)}
\end{equation}
This is so because ${\widetilde V}(x)$ is essentially two-parametric while
(19) is just one-parame\-tric. Indeed, the $V_\lambda(x)$ of (19) can be
numerically reconstructed by solving (16) with the points of (18) as the
initial conditions. The SUSUSY family obtained by this procedure coincides
with a plot of the analytic results (19).
\begin{figure}[htbp]
\vspace*{13pt}
\begin{minipage}{10truecm}
\hspace*{3truecm}
\epsfxsize=10truecm
\epsfbox{sususyf2.eps}
\end{minipage}
\vspace*{13pt}
\fcaption{The SUSUSY potentials ${\widetilde V}(x)+4$ for some values of
$\beta'(0)$ and $\beta(0) = -0.7$. All pairs $(\beta(0), \beta'(0))$ lie in
the region where there is no singularity for ${\widetilde V}(x)$.}
\end{figure}
Interesting that the SUSUSY family ${\widetilde V}(x)$ embraces some cases of
the widely discussed {\it double well potentials} (DWP). The dynamics of a
system in these potentials is of some relevance, because it illustrates the
differences between the classical and quantum regimes. In particular, it
clearly shows one of the most intriguing quantum effects, the tunneling of the
system from one well to the other as a result of the evolution. In most of the
situations where a DWP is a SUSY pair of the oscillator potential, the DWP
spectrum has one level more below the ground state energy of the oscillator.
Moreover, it is usually symmetric with respect to $x=0$ (see e.g.
${}^{9,23}$). For our SUSUSY DWP, apparently, it is unneccessary to add any
level below the ground state energy of the oscillator to generate the double
well: the spectra of ${\widetilde V}(x) + 4$ and $V(x) = x^2$ are
equal.\fnm{c}\fnt{c}{This is at the moment a hypothesis supported by our
numerical plots of $\beta(x)$ and ${\widetilde V}(x)$. (The continuity
argument might be important.)} The price to pay is that ${\widetilde V}(x)$
and $V(x) = x^2$ are not precisely the SUSY partners, as shown in section 3.
Moreover, although ${\widetilde V}(x)$ is a double well, it turns out that it
is not symmetric with respect to any point $x=x_0$. We hope that the SUSUSY
treatment here presented can be implemented to other physically interesting
potentials.
\nonumsection{\bf Acknowledgements}
\noindent
The support of CONACYT is acknowledged.
\newpage
\nonumsection{\bf References}
\noindent
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\section{Introduction}
The supergravity theories that arise as the low-energy limits of string
theory or M-theory admit a multitude of $p$-brane solutions. In general,
these solutions are characterised by their mass per unit $p$ volume, and the
charge or charges carried by the field strengths that support the solutions.
Solutions can be extremal, in the case where the charges and the mass per
unit $p$-volume saturate a Bogomol'nyi\ bound, or non-extremal if the mass per unit
$p$-volume exceeds the bound. There are two basic types of solution, namely
elementary $p$-branes, supported by field strengths of rank $n=p+2$, and
solitonic $p$-branes, supported by field strengths of degree $n=D-p-2$,
where $D$ is the spacetime dimension. Typically, we are interested in
considering solutions in a ``fundamental'' maximal theory such as $D=11$
supergravity, which is the low-energy limit of M-theory, and its various
toroidal dimensional reductions. A classification of extremal
supersymmetric $p$-branes in M-theory compactified on a torus can be found
in \cite{lpsol}.
The various $p$-brane solutions in $D\le 11$ can be represented as
points on a ``brane scan'' whose vertical and horizontal axes are the
spacetime dimension $D$ and the spatial dimension $p$ of the $p$-brane
world-volume. The same process of Kaluza-Klein dimensional reduction that
is used in order to construct the lower-dimensional toroidally-compactified
supergravities can also be used to perform dimensional reductions of the
$p$-brane solutions themselves: Since the Kaluza-Klein procedure
corresponds to performing a {\it consistent} truncation of the
higher-dimensional theory, it is necessarily the case that the
lower-dimensional solutions will also be solutions of the higher-dimensional
theory. There are two types of dimensional reduction that can be carried
out on the $p$-brane solutions. The more straightforward one involves a
simultaneous reduction of the spacetime dimension $D$ and the spatial
$p$-volume, from $(D,p)$ to $(D-1,p-1)$; this is known as ``diagonal
dimensional reduction'' \cite{dhis,lpss1}. It is achieved by choosing
one of the spatial world-volume coordinates as the compactification
coordinate. The second type of dimensional reduction corresponds to a
vertical descent on the brane scan, from $(D,p)$ to $(D-1,p)$, implying that
one of the directions in the space transverse to the $p$-brane world-volume
is chosen as the compactification coordinate. This requires that one first
construct an appropriate configuration of $p$-branes in $D$ dimensions that
has the necessary $U(1)$ isometry along the chosen direction. It is not
{\it a priori} obvious that this should be possible, in general. However,
it is straightforward to construct such configurations in the case of
extremal $p$-branes, since these satisfy a no-force condition which means
that two or more $p$-branes can sit in neutral equilibrium, and thus multi
$p$-brane solutions exist \cite{cg}. By taking a limit corresponding to an
infinite continuum of $p$-branes arrayed along a line, the required
$U(1)$-invariant configuration can be constructed \cite{k,ghl,ht,lps}
In this paper, we shall investigate the dimensional-reduction
procedures for non-extremal $p$-branes. In fact the process of diagonal
reduction is the same as in the extremal case, since the non-extremal
$p$-branes also have translational invariance in the spatial world-volume
directions. The more interesting problem is to see whether one can also
describe an analogue of vertical dimensional reduction for non-extremal
$p$-branes. There certainly exists an algorithm for constructing a
non-extremal $p$-brane at the point $(D-1,p)$ from one at $(D,p)$ on the
brane scan. It has been shown that there is a universal prescription for
``blackening'' any extremal $p$-brane, to give an associated non-extremal
one \cite{dlp}. Thus an algorithm, albeit inelegant, for performing the
vertical reduction is to start with the general non-extremal $p$-brane in
$D$ dimensions, and then take its extremal limit, from which an extremal
$p$-brane in $D-1$ dimensions can be obtained by the standard
vertical-reduction procedure described above. Finally, one can then invoke
the blackening prescription to construct the non-extremal $p$-brane in $D-1$
dimensions. However, unlike the usual vertical dimensional reduction for
extremal $p$-branes, this procedure does not give any physical
interpretation of the $(D-1)$-dimensional $p$-brane as a superposition of
$D$-dimensional solutions.
At first sight, one might think that there is no possibility of
superposing non-extremal $p$-branes, owing to the fact that they do not
satisfy a no-force condition. Indeed, it is clearly true that one cannot
find well-behaved static solutions describing a finite number of black
$p$-branes located at different points in the transverse space. However, we
do not require such general kinds of multi $p$-brane solutions for the
purposes of constructing a configuration with a $U(1)$ invariance in the
transverse space. Rather, we require only that there should exist static
solutions corresponding to an infinite number of $p$-branes, periodically
arrayed along a line. In such an array, the fact that there is a
non-vanishing force between any pair of $p$-branes is immaterial, since the
net force on each $p$-brane will still be zero. The configuration is in
equilibrium, although of course it is highly unstable. For example, one can
have an infinite static periodic array of $D=4$ Schwarzschild black holes
aligned along an axis. In fact the instability problem is overcome in the
Kaluza-Klein reduction, since the extra coordinate $z$ is compactified on a
circle. Thus there is a stable configuration in which the $p$-branes are
separated by precisely the circumference of the compactified dimension.
Viewed from distances for which the coordinates orthogonal to $z$ are large
compared with this circumference, the fields will be effectively independent
of $z$, and hence $z$ can be used as the compactification coordinate for the
Kaluza-Klein reduction, giving rise to a non-extremal $p$-brane in $D-1$
dimensions.
In section 2, we obtain the equations of motion for axially symmetric
$p$-branes in an arbitrary dimension $D$. We then construct multi-center
non-extremal $D=4$ black hole solutions in section 3, and show how they may
be used for vertical dimensional reduction of non-extremal black holes.
In section 4, we generalise the construction to non-extremal $(D-4)$-branes
in arbitrary dimension $D$.
\section{Equations of motion for axially symmetric $p$-branes}
We are interested in describing multi-center non-extremal $p$-branes
in which the centers lie along a single axis in the transverse space.
Metrics with the required axial symmetry can be parameterised in the
following way:
\be
ds^2=-e^{2U}dt^2+e^{2A}dx^idx^i+e^{2V}(dz^2+dr^2)+e^{2B}r^2d\Omega^2
\ ,\label{metricans}
\ee
where $(t,x^i),\,\, i=1,\ldots,p$, are the coordinates of the $p$-brane
world-volume. The remaining coordinates of the $D$ dimensional spacetime,
\ie those in the transverse space, are $r$, $z$ and the coordinates on a
$\tilde d$-dimensional unit sphere, whose metric is $d\Omega^2$, with $\tilde
d=D-p-3$. The functions $U$, $A$, $V$ and $B$ depend on the coordinates
$r$ and $z$ only. We find that the Ricci tensor for the metric
(\ref{metricans}) is given by
\bea
R_{00}&=&e^{2U-2V}\Big(U''+\ddot U+U'^2 + \dot U^2 + p(U'A'+\dot U \dot A)+
\tilde d(U'B' +\dot U \dot B) + \fft{\tilde d}{r}U'\Big)\ ,\nonumber\\
R_{rr}&=&-\Big(U'' - U'V' + U'^2 +\dot U \dot V +\ddot V + V'' +
p(A'' - V' A' + A'^2 + \dot V \dot A) \nonumber\\
&&+ \tilde d (B''-V'B'-\fft{1}{r}V' +\dot V \dot B + B'^2 +\fft{2}{r}B')\Big)\ ,
\nonumber\\
R_{zz}&=&-\Big(\ddot U -\dot U \dot V + {\dot U}^2 + U' V' + \ddot V
+V'' +p(\ddot A - \dot V \dot A +{\dot A}^2 + V' A' ) \nonumber\\
&&+ \tilde d (\ddot B -\dot V \dot B + V' B' + {\dot B}^2 +\fft{1}{r} V')\Big)\ ,
\nonumber\\
R_{rz}&=&\Big( -\dot U' +\dot V U' -\dot U U' +\dot U V' +
p(-\dot A' +\dot V A' -\dot A A' + V' \dot A ) \label{ricci}\\
&&+ \tilde d (-\dot B' +\dot V B' +\fft{1}{r} \dot V -\fft{1}{r} \dot B -\dot B B'
+V' \dot B )\Big)\ ,\nonumber\\
R_{ab}&=&-e^{2B-2V}\Big(B'' +\ddot B +U' B'+\dot U \dot B +\fft{1}{r} U' +
p(A'B' +\dot A \dot B + \fft{1}{r} A') \nonumber\\
&&+ \tilde d (B'^2 +{\dot B}^2 +\fft{2}{r} B')\Big)\, r^2\, \bar g_{ab} +
(\tilde d-1)(1-e^{2B-2V})\bar g_{ab}\ ,\nonumber\\
R_{ij}&=&-e^{2A-2V}\Big(A'' +\ddot A +U' A' +\dot U \dot A +p(A'^2 +{\dot A}^2)
+\tilde d (A'B' +\dot A \dot B +\fft{1}{r} A')\Big)\, \delta_{ij}\ ,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}
\eea
where the primes and dots denote derivatives with respect to $r$ and $z$
respectively, $\bar g_{ab}$ is the metric on the unit $\tilde d$-sphere, and
the components are referred to a coordinate frame.
Let us consider axially-symmetric solutions to the theory described by
the Lagrangian
\be
e^{-1} {\cal L} = R -\ft12 (\partial \phi)^2 -\fft1{2 n!}\, e^{-a\phi}\, F_n^2\ ,
\label{boslag}
\ee
where $F_n$ is an $n$-rank field strength. The constant $a$ in the dilaton
prefactor can be parameterised as
\be
a^2 = \Delta - \fft{2(n-1)(D-n-1)}{D-2}\ ,\label{avalue}
\ee
where the constant $\Delta$ is preserved under dimensional reduction
\cite{lpss1}. (For supersymmetric $p$-branes in M-theory compactified on a
torus, the values of $\Delta$ are $4/N$ where $N$ is an integer $1\le N \le
N_c$, and $N_c$ depends on $D$ and $p$ \cite{lpsol}. Non-supersymmetric
$(D-3)$-branes with $\Delta = 24/(N(N+1)(N+2))$ involving $N$ 1-form field
strengths were constructed in \cite{lptoda}. Their equations of motion
reduce to the $SL(N+1,R)$ Toda equations. Further non-supersymmetric
$p$-branes with other values of $\Delta$ constructed in \cite{lpsol} however
cannot be embedded into M-theory owing to the complications of the
Chern-Simons modifications to the field strengths.) We shall concentrate on
the case where $F_n$ carries an electric charge, and thus the solutions will
describe elementary $p$-branes with $p=n-2$. (The generalisation to
solitonic $p$-branes that carry magnetic charges is straightforward.) The
potential for $F_n$ takes the form $A_{0i_1\cdots i_p}= \gamma
\epsilon_{i_1\cdots i_p}$, where $\gamma$ is a function of $r$ and $z$. Thus
the equations of motion will be
\be
\Box\phi = -\ft12 a \, S^2\ ,\qquad R_{MN} = \ft12\partial_M\phi\, \partial_N\phi
+S_{MN}\ ,\qquad \partial_{M_1}(\sqrt{-g}\, e^{-a\phi}\, F^{M_1\cdots M_n}) =0
\ ,\label{eom}
\ee
where
\bea
S_{00} = \fft{\tilde d}{2(D-2)} S^2 e^{2U}({\dot\gamma}^2+{\gamma'}^2)\ , &&
S_{rr} = \fft{1}{2(D-2)} S^2 e^{2V} ( d {\dot\gamma}^2 -{\tilde d} {\gamma'}^2
)\ , \nonumber \\
S_{zz} = \fft{1}{2(D-2)} S^2 e^{2V} ( -{\tilde d} {\dot\gamma}^2 + d
{\gamma'}^2)\ , &&
S_{rz} = -\ft{1}{2} S^2 e^{2V} {\dot\gamma} \gamma' \ , \label{sab} \\
S_{ab} = \fft{d}{2(D-2)} S^2 ( {\dot\gamma}^2 + {\gamma'}^2 ) e^{2B} r^2
\bar g_{ab} \ , &&
S_{ij} = -\fft{{\tilde d}}{2(D-2)} S^2 e^{2A}(\dot\gamma^2 + \gamma'^2)
{\delta}_{ij} \ , \nonumber} \def\bd{\begin{document}} \def\ed{\end{document}
\eea
$S^2 = e^{-2pA -2V -a\phi-2U} $ and $d=p+1$.
\section{$D=4$ black holes and their dimensional reduction}
\subsection{Single-center black holes}
Let us first consider black hole solutions in $D=4$, whose charge is
carried by a 2-form field strength. By appropriate choice of coordinates,
and by making use of the field equations, we may set $B=-U$. Defining also
$V=K-U$, we find that equations of motion (\ref{eom}) for the metric
\be
ds^2 = -e^{2U}\, dt^2 + e^{2K-2U}\, (dr^2 + dz^2) + e^{-2U}\, r^2\,
d\theta^2\label{d4metric}
\ee
can be reduced to
\bea
{\nabla}^2 U = \ft{1}{4} e^{-a\phi -2 U} ({\dot\gamma}^2 +{\gamma'}^2) \ ,
&& {\nabla}^2 K - \ft{2}{r} K' = \ft{1}{2} e^{-a\phi - 2 U} {\gamma'}^2 -
2 U'^2 -\ft{1}{2} {\phi'}^2 \ , \nonumber \\
{\nabla}^2 K = \ft{1}{2} e^{-a\phi -2 U} {\dot\gamma}^2 - 2 {\dot U}^2 -
\ft{1}{2} {\dot\phi}^2 \ ,
&& \ft{1}{r} {\dot K} = -\ft{1}{2} e^{-a\phi -2 U} {\dot\gamma} {\gamma'}
+ 2{\dot U} U' + \ft{1}{2} {\dot\phi} {\phi}' \ , \label{d4eom} \\
{\nabla}^2 \phi = \ft{1}{2} a e^{-a\phi - 2 U} ( {\dot\gamma}^2
+{\gamma'}^2 ) \ ,
&& {\nabla}^2 {\gamma} = (a \phi' + 2 U') \gamma' + (a {\dot\phi} +
2 {\dot U} ) {\dot\gamma} \, \nonumber} \def\bd{\begin{document}} \def\ed{\end{document}
\eea
where ${\nabla}^2 = {d^2 \over dr^2} + \ft{1}{r} {d \over dr} +
{d^2 \over dz^2} $ is the Laplacian for axially-symmetric functions in
cylindrical polar coordinates.
We shall now discuss three cases, with increasing generality, beginning
with the pure Einstein equation, with $\phi=0$ and $\gamma=0$. The
equations (\ref{d4eom}) then reduce to
\be
\nabla^2U=0\ ,\qquad K'=r({U'}^2 -\dot U^2)\ ,\qquad \dot K =2r U'\dot U\ ,
\label{ricflat}
\ee
thus giving a Ricci-flat axially-symmetric metric for any harmonic function
$U$. The solution for $K$ then follows by quadratures.
A single Schwarzschild black hole is given by taking $U$ to be the
Newtonian potential for a rod of mass $M$ and length $2M$ \cite{ik}, \ie
\be
U = \ft12 \log\fft{\sigma +\tilde\sigma -2M}{\sigma+\tilde\sigma +2M} \ ,
\label{rod}
\ee
where $\sigma =\sqrt{r^2+(z-M)^2}$ and $\tilde \sigma = \sqrt{r^2 +
(z+M)^2}$. The solution for $K$ is
\be
K= \ft12 \log\fft{(\sigma+\tilde\sigma -2M)(\sigma+\tilde\sigma +2M)}{4\sigma
\tilde\sigma}\ .\label{ksol1}
\ee
Now we shall show that this is related to the standard Schwarzschild
solution in isotropic coordinates, \ie
\be
ds^2= - \fft{(1-\fft{M}{2R})^2}{(1+\fft{M}{2R})^2}\, dt^2 +
(1+\fft{M}{2R})^4 (d\rho^2 + dy^2 + \rho^2 d \theta^2) \ ,\label{sch}
\ee
where $R\equiv \sqrt{\rho^2 + y^2}$. To do this, we note that (\ref{sch})
is of the general form (\ref{metricans}), but with $B\ne -U$. As discussed
in \cite{s}, setting $B=-U$ depends firstly upon having a field
equation for which the $R_{00}$ and $R_{\theta\theta}$ components of the
Ricci tensor are proportional, and secondly upon performing a holomorphic
coordinate transformation from $\eta\equiv r + {\rm i} z$ to new variables
$\xi\equiv \rho+ {\rm i} y$. Comparing the coefficients of $d\theta^2$ in
(\ref{d4metric}) and (\ref{sch}), we see that we must have $\Re(\eta)=
\Re(\xi)(1-m^2/(4\bar\xi\xi))$, and hence we deduce that the required
holomorphic transformation is given by
\be
\eta=\xi -\fft{m^2}{4\xi}\ .\label{trans}
\ee
It is now straightforward to verify that this indeed transforms the metric
(\ref{d4metric}), with $U$ and $K$ given by (\ref{rod}) and (\ref{ksol1}),
into the standard isotropic Schwarzschild form (\ref{sch}).
Now let us consider the pure Einstein-Maxwell case, where $\gamma$ is
non-zero, but $a=0$ and hence $\phi=0$. We find that the equations of
motion (\ref{d4eom}) can be solved by making the ansatz
\be
e^{-U} = e^{- {\widetilde U} } - c^2 e^{\widetilde U} \ , \qquad
\gamma = 2 c e^{ 2 {\widetilde U} } \Big( 1 - c^2 e^{ 2 {\widetilde U} } \Big)^{-1}
\ , \label{einmax}
\ee
where $c$ is an arbitrary constant and $\widetilde U$ satisfies $\nabla^2 \widetilde
U=0$. Substituting into (\ref{d4eom}), we find that all the equations are
then satisfied if
\be
K' = r\, ( {\widetilde U}'^2 - {\dot{\widetilde U}}^2 ) \ , \qquad
{\dot K} = 2r\, {\widetilde U}'\, {\dot{\widetilde U}} \ . \label{ksol2}
\ee
(Our solutions in this case are in agreement with \cite{g}, after
correcting some coefficients and exponents.) The solution for a single
Reissner-Nordstr{\o}m black hole is given by taking the harmonic function
$\widetilde U$ to be the Newtonian potential for a rod of mass $\ft12 k$ and
length $k$, implying that $\widetilde U$ and $K$ are given by
\bea
\widetilde U &=& \ft12 \log\fft{\sigma +\tilde\sigma -k}{\sigma+\tilde\sigma +k}
\ ,\nonumber\\
K&=& \ft12 \log\fft{(\sigma+\tilde\sigma -k)(\sigma+\tilde\sigma +k)}{4\sigma
\tilde\sigma}\ .\label{uk}
\eea
where $\sigma = \sqrt{r^2 + (z-k/2)^2}$ and $\tilde \sigma =
\sqrt{r^2 + (z+ k/2)^2}$. The metric can be re-expressed in terms of the
standard isotropic coordinates $(\hat t,\rho,y,\theta)$ by performing the
transformations
\be
\eta=\fft{1}{1-c^2}\, (\xi -\fft{\hat k^2}{16\xi}) \ ,\qquad
t=(1-c^2)\, \hat t\ ,\label{redef}
\ee
where $\xi=\rho+{\rm i} y$ and $\hat k = (1-c^2) k$, giving
\bea
ds^2&=& - \Big(1 + \fft{\hat k R}{(R+\ft14 \hat k)^2} \, \sinh^2\mu\Big)^{-2}\,
\Big(\fft{R-\ft14 \hat k}{R+\ft14 \hat k}\Big)^2\, d\hat t^2 \nonumber\\
&& + \Big(1 + \fft{\hat k R}{(R+\ft14 \hat k)^2} \, \sinh^2\mu\Big)^2 \,
(1+\fft{\hat k}{4R})^4\, (d\rho^2 + dy^2 + \rho^2 d\theta^2)\ ,\label{rn}
\eea
where $c=\tanh\mu$, and again $R\equiv\sqrt{\rho^2+y^2}$. (It is necessary
to rescale the time coordinate, as in (\ref{redef}), because the function
$e^{-U}$ given in (\ref{einmax}) tends to $(1-c^2)$ rather than 1 at
infinity.) Equation (\ref{rn}) is the standard Reissner-Nordstr{\o}m
metric in isotropic coordinates, with mass $M$ and charge $Q$ given in terms
of the parameters $\hat k$ and $\mu$ by
\be
M= \hat k\sinh^2\mu +\ft12 \hat k\ ,\qquad Q= \ft14 \hat k
\sinh 2\mu \ . \label{mc}
\ee
The extremal limit is obtained by taking $\hat k\rightarrow 0$ at the same time
as sending $\mu\rightarrow\infty$, while keeping $Q$ finite, implying that
$Q=\ft12 M$. This corresponds to setting $c\rightarrow 1$ in
(\ref{einmax}). The description in the form (\ref{d4metric}) becomes
degenerate in this limit, since the length and mass of the Newtonian rod
become zero. However, the rescalings (\ref{redef}) also become singular,
and the net result is that the metric (\ref{rn}) remains well-behaved in the
extremal limit. The previous pure Einstein case is recovered if $\mu$ is
instead sent to zero, implying that $Q=0$ and $c=0$.
Finally, let us consider the case of Einstein-Maxwell-Dilaton black
holes. We find that the equations of motion (\ref{d4eom}) can be solved by
making the ans\"atze
\bea
\phi &=& 2a(U -\widetilde U)\ ,
\qquad e^{-\Delta U} = (e^{-\widetilde U} - c^2\, e^{\widetilde U}) e^{-a^2 \widetilde U}\ ,
\nonumber\\
\gamma &=& 2c\, e^{2\widetilde U} \Big(1 - c^2\, e^{2\widetilde U} \Big)^{-1}
\ ,\label{phians}
\eea
where, as in the pure Einstein-Maxwell case, $c$ is an arbitrary constant
and $\widetilde U$ satisfies $\nabla^2 \widetilde U=0$. Substituting the ans\"atze into
(\ref{d4eom}), we find that all the equations are satisfied provided that
the function $K$ satisfies (\ref{ksol2}). The solution for a single
dilatonic black hole for generic coupling $a$ is given by again taking the
harmonic function $\widetilde U$ to be the Newtonian potential for a rod of mass
$\ft12k$ and length $k$. After performing the coordinate transformations
\be
\eta=(1-c^2)^{-\ft1{\Delta}}\, (\xi-\fft{\hat k^2}{16\xi})\ ,\qquad
t=(1-c^2)^{\ft{1}{\Delta}}\, \hat t\ ,
\ee
where $\hat k=(1-c^2)^{1/\Delta} k$, and writing $c=\tanh \mu$, we find that
the metric takes the standard isotropic form
\bea
ds^2&=& - \Big(1 + \fft{\hat k R}{(R+\ft14 \hat k)^2} \,
\sinh^2\mu\Big)^{-\ft2{\Delta}}\,
\Big(\fft{R-\ft14 \hat k}{R+\ft14 \hat k}\Big)^2\, d\hat t^2 \nonumber\\
&& + \Big(1 + \fft{\hat k R}{(R+\ft14 \hat k)^2} \,
\sinh^2\mu\Big)^{\ft2{\Delta}} \, (1+\fft{\hat k}{4R})^4\, (d\rho^2 + dy^2 +
\rho^2 d\theta^2)\ .\label{dbh}
\eea
The mass $M$ and charge $Q$ are given by
\be
M = \fft{\hat k}{\Delta} \sinh^2 \mu + \ft12 \hat k\ ,\qquad
Q= \fft{\hat k}{4\sqrt\Delta} \sinh 2\mu\ .
\ee
Again, the extremal limit is obtained by taking $\hat k \rightarrow 0$,
$\mu\rightarrow \infty$, while keeping $Q$ finite, implying that
$Q=\sqrt{\Delta} M/2$.
\subsection{Vertical dimensional reduction of black holes}
The vertical dimensional reduction of a $p$-brane solution
requires that the Kaluza-Klein compactification coordinate should lie in the
space transverse to the world-volume of the extended object. In order to
carry out the reduction, it is necessary that the higher-dimensional
solution be independent of the chosen compactification coordinate. In the
case of extremal $p$-branes, this can be achieved by exploiting the fact
that there is a zero-force condition between such objects, allowing
arbitrary multi-center solutions to be constructed. Mathematically,
this can be done because the equations of motion reduce to a Laplace
equation in the transverse space, whose harmonic-function solutions can be
superposed. Thus one can choose a configuration with an infinite line of
$p$-branes along an axis, which implies in the continuum limit that the the
solution is independent of the coordinate along the axis.
As we saw in the previous section, the equations of motion for an
axially-symmetric non-extremal black-hole configuration can also be cast in
a form where the solutions are given in terms of an arbitrary solution of
Laplace's equation. Thus again we can superpose solutions, to describe
multi-black-hole configurations. We shall discuss the general case of
dilatonic black holes, since the $a=0$ black holes and the uncharged black
holes are merely special cases. Specifically, a solution in which $\widetilde U$
is taken to be the Newtonian potential for a set of rods of mass $\ft12 k_n$
and length $k_n$ aligned along the $z$ axis will describe a line of charged,
dilatonic black holes:
\bea
\tilde U&=& \ft12 \sum_{n=1}^N \log
\fft{\sigma_n +\tilde \sigma_n -k_n}{\sigma_n +\tilde \sigma_n +k_n}\ ,
\label{multirod}\\
K &=& \ft14 \sum_{m,n=1}^N \log\fft{[\sigma_m \tilde\sigma_n +
(z-z_m -\ft12 k_m)(z-z_n+ \ft12 k_n) + r^2]}{[\sigma_m \sigma_n +
(z-z_m -\ft12 k_m)(z-z_n- \ft12 k_n) + r^2]}\label{multisol}\\
&&+\ft14 \sum_{m,n=1}^N \log\fft{[\tilde\sigma_m \sigma_n +
(z-z_m +\ft12 k_m)(z-z_n- \ft12 k_n) + r^2]}{[\tilde\sigma_m \tilde\sigma_n +
(z-z_m +\ft12 k_m)(z-z_n+ \ft12 k_n) + r^2]}\ ,\nonumber
\eea
where $\sigma_n^2 = r^2 + (z-z_n-\ft12 k_n)^2$ and $\tilde \sigma_n^2 =
r^2 +(z-z_n+\ft12 k_n)^2$. (Multi-center Schwarzschild solutions were
obtained in \cite{ik}, corresponding to (\ref{multirod}) and (\ref{multisol})
with $k_n=2M_n$, and $U$ equal to $\widetilde U$ rather than the expression given
in (\ref{phians}).) This describes a system of $N$ non-extremal
black holes, which remain in equilibrium because of the occurrence of
conical singularities along the $z$ axis. These singularities correspond to
the existence of (unphysical) ``struts'' that hold the black holes in place
\cite{g2,hs}. If, however, we take all the constants $k_n$ to be equal, and
take an infinite sum over equally-spaced black holes lying at points $z_n=n
b$ along the entire $z$ axis, the conical singularities disappear \cite{m}.
In the limit when the separation goes to zero, the resulting solution
(\ref{multisol}) becomes independent of $z$. For small $k=k_n$, we have
$U\sim -\ft12 k (r^2 + (z-nb)^2)^{-1/2}+ O(k^3/r^3)$, and thus in the limit
of small $b$, the sum giving $\widetilde U$ in (\ref{multirod}) can be replaced by
an integral:
\be
\widetilde U \sim -\fft{k}{2b} \int_L^L \fft{dz'}{\sqrt{r^2 + z'^2}}\ ,
\ee
in the limit $L\rightarrow\infty$. Subtracting out the divergent constant
$-k(\log2 + \log L)/b$, this gives the $z$-independent result \cite{m}
\be
\widetilde U = \fft{k}{b} \log r\ .\label{zindep}
\ee
Similarly, one finds that $K$ is given by
\be
K = \fft{k^2}{b^2}\, \log r \ .\label{kzindep}
\ee
One can of course directly verify that these expressions for $\widetilde U$ and
$K$ satisfy the equations of motion (\ref{d4eom}). Since the associated
metric and fields are all $z$-independent, we can now perform a dimensional
reduction with $z$ as the compactification coordinate, giving rise to a
solution in $D=3$ of the dimensionally reduced theory, which is
obtained from (\ref{boslag}), with $D=4$ and $n=2$, by the standard
Kaluza-Klein reduction procedure. A detailed discussion of this procedure
may be found, for example, in \cite{lpss1}. From the formulae given there,
we find that the relevant part of the $D=3$ Lagrangian, namely the part
involving the fields that participate in our solution, is given by
\be
e^{-1}{\cal L} = R - \ft12 (\partial\phi)^2 -\ft12(\partial\varphi)^2
-\ft14 e^{-\varphi-a\phi}\, F_2^2\ ,\label{d3lag}
\ee
where $\varphi$ is the Kaluza-Klein scalar coming from the dimensional
reduction of the metric, \ie $ds_4^2 = e^{\varphi}\, ds_3^2 + e^{-\varphi}
dz^2$. A ``standard'' black hole solution in $D=3$ would be one where only
the combination of scalars $(-\varphi-a\phi)$, occurring in the exponential
prefactor of the field strength $F_2$ that supports the solution, is
non-zero. In other words, the orthogonal combination should vanish, \ie
$a\varphi-\phi=0$. Since our solution in $D=4$ has $\phi=2a(U-\widetilde U)$, it
follows that $\varphi=2U-2\widetilde U$, and hence we should have
\be
ds_4^2 = e^{2U-2\widetilde U}\, ds_3^2 + e^{2\widetilde U-2U}\, dz^2 \ .
\ee
Comparing this with the $D=4$ solution, whose metric takes the form
(\ref{d4metric}), we see that the $D=3$ solution will have the above
single-scalar structure if $K=\widetilde U$. From (\ref{zindep}) and
(\ref{kzindep}), this will be the case if the parameter $k$ setting the
scale size of the rods, and the parameter $b$ determining the spacing
between the rods, satisfy $k=b$.
It is interesting to note that since $k$ is the length of
each rod, and $b$ is the period of the array, the condition $k=b$ implies that
the rods are joined end to end, effectively describing a single rod of
length $L$ and mass $\ft12 L$ in the limit $L\rightarrow \infty$. In
other words, the $D=4$ multi-black-hole solution becomes a single black hole
with $k=L\rightarrow \infty$ in this case. If $r$ is large compared with
$z$, the solution is effectively independent of $z$, and thus one can reduce
to $D=3$ with $z$ as the Kaluza-Klein compactification coordinate. (This is
rather different from the situation in the extremal limit; in that case, the
lengths and masses of the individual rods are zero, and the sum over an
infinite array does not degenerate to a single rod of infinite length.)
If $k$ and $b$ are not equal, the dimensional reduction of the $D=4$
array of black holes will of course still yield a 3-dimensional solution of
the equations following from (\ref{d3lag}), but now with the orthogonal
combination $a\varphi -\phi$ of scalar fields active also. Such a solution
lies outside the class of $p$-brane solitons that are normally discussed;
we shall examine such solutions in more detail in the next section.
\section{Reductions of higher-dimensional black $(D-4)$-branes}
The equations of motion (\ref{eom}) for general black $p$-branes in
$D$ dimensions become rather difficult to solve in the axially symmetric
coordinates, owing to the presence final term involving $(\tilde
d-1)(1-e^{2B-2V})$ in $R_{ab}$ given in (\ref{ricci}). This term vanishes
if $\tilde d=1$, as it did in the case of 4-dimensional black holes discussed in
section 3. The simplest generalisation of these 4-dimensional results is
therefore to consider $(D-4)$-branes, which have $\tilde d =1$ also. They will
arise as solutions of the equations of motion following from (\ref{boslag})
with $n=D-2$. The required solutions can be obtained by directly solving
the equations of motion (\ref{eom}), with Ricci tensor given by
(\ref{ricci}). However, in practice it is easier to obtain the solutions by
diagonal Kaluza-Klein oxidation of the $D=4$ black hole solutions. The
ascent to $D$ dimensions can be achieved by recursively applying the inverse
of the one-step Kaluza-Klein reduction procedure.
The one step reduction of the metric from $(\ell + 1)$ to $\ell$
dimensions takes the form
\be
ds^2_{\ell +1} = e^{2 \a_\ell \varphi_\ell} ds_{\ell}^2 +
e^{-2(\ell -2) \a_\ell\varphi_{\ell}} dx_{5-\ell}^2
\ ,\label{kk} \ee
where $\a_{\ell}^{-2} = 2 (\ell -1) (\ell -2)$. (We have omitted the
Kaluza-Klein vector potential since it is not involved in the solutions that
we are discussing.) The kinetic term for the field strength $F_{\ell-1}$
in $(\ell+1)$ dimensions, \ie $e^{-a_{\ell
+1} \phi_{\ell +1} } F_{\ell-1}^2$, reduces to the kinetic term
$e^{-\a_{\ell +1} \phi_{\ell +1} + 2 \a_\ell \varphi_\ell} F_{\ell -2}^2$ in
$\ell$ dimensions for the relevant field strength $F_{\ell-2}$. We may
define $-a_{\ell+1}\, \phi_{\ell+1} +2\a_\ell\, \varphi_\ell \equiv -a_\ell
\phi_\ell$, where $a_\ell^2 = a_{\ell+1}^2 + 4\a_\ell^2$. In fact although the
dilaton coupling constant $a_\ell$ is different in different dimensions
$\ell$, the related quantity $\Delta$, defined in (\ref{avalue}), is
preserved under dimensional reduction \cite{lpss1}. The solutions that we
are considering have the feature that the combination of scalar fields
orthogonal to $\phi_\ell$ in $\ell$ dimensions vanishes, \ie $2\a_\ell\,
\phi_{\ell+1} + a_{\ell+1}\, \varphi_{\ell}=0$. This ensures that a
single-scalar solution in $D$ dimensions remains a single-scalar solution
in all the reduction steps. Thus we have the following recursive relations
\bea
&&\fft{\phi_{\ell+1}}{a_{\ell+1}} = \fft{\phi_\ell}{a_\ell} = \cdots =
\fft{\phi_4}{a_4} = 2 (U-\widetilde U)\ ,\nonumber\\
&& \varphi_{\ell} = - 2 \a_{\ell} \fft{\phi_\ell}{a_\ell} =
-\fft{4}{(\ell-1) (\ell-2)} (U - \widetilde U)\ ,
\eea
where $U$ and $\tilde U$ are the functions for the four-dimensional
dilatonic black holes given in section 3. We find that the metric for the
$(D-4)$-brane in $D$ dimensions is then given by
\bea
ds^2_{\sst D}\!\!\! &=&\!\!\!e^{-\ft{2(D-4)}{D-2}(U-\widetilde U)}\, ds_4^2 +
e^{\ft{4}{D-2} (U-\widetilde U)}\, (dx_1^2 + \cdots + dx_p^2)
\label{dmetric}\\
\!\!\!&=&\!\!\! e^{\ft{4}{D-2} (U-\widetilde U)} (-e^{2\widetilde U} dt^2 + dx^idx^i)
+ e^{\ft{2(D-4)}{D-2} \widetilde U- \ft{4(D-3)}{D-2} U}
\Big(e^{2K} (dr^2 + dz^2) + r^2 d\theta^2\Big)\ ,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}
\eea
and the dilaton is given by $\phi_{\sst D} = 2a_{\sst D} (U-\widetilde U)$. If
the functions $\widetilde U$ and $K$ are those for a Newtonian rod, given by
(\ref{uk}), and the function $U$ is given by (\ref{phians}), the metric
describes a single black $(D-4)$-brane. The coordinate transformations
\be
\eta = (\xi - \fft{\hat k^2}{16\xi}) (\cosh\mu)^{\ft{4(D-3)}{\Delta(D-2)}}\ ,
\qquad
x^\mu = \hat x^\mu (\cosh\mu)^{-\ft{4}{\Delta(D-2)}}\ ,
\ee
where $\hat k =(\cosh\mu)^{-4/(\Delta(D-2))}$, put the metric into the
standard isotropic form for a black $(D-4)$-brane, where $c=\tanh \mu$. The
further transformation $\hat r = (R+ \ft14 \hat k)^2/R$ puts the metric into
the form
\bea
ds_{\sst D}^2 &=& \Big( 1 + \fft{\hat k}{\hat r}
\sinh^2\mu\Big)^{-\ft{4}{\Delta(D-2)}}\, (-e^{2f} d\hat t^2 +
d\hat x^id\hat x^i)\nonumber\\
&& \Big( 1 + \fft{\hat k}{\hat r} \sinh^2\mu \Big)^{\ft{4(D-3)}{\Delta(D-2)}}\,
(e^{-2f} d\hat r^2 + \hat r^2 d\theta^2)\ ,
\eea
where $e^{2f} = 1 -\hat k/\hat r$. This is the standard form for black
$(D-4)$-branes discussed in \cite{dlp}.
Since we again have general solutions given in terms of the harmonic
function $\widetilde U$, we may superpose a set of Newtonian rod potentials, by
taking $\widetilde U$ and $K$ to have the forms (\ref{multirod}) and
(\ref{multisol}). Equilibrium can again be achieved, without conical
singularities on the $z$ axis, by taking an infinite line of such rods,
with equal masses $\ft12 k$, lengths $k$, and spacings $b$. As discussed
in section 3, the resulting functions $\widetilde U$ and $K$ become
$z$-independent, and are given by (\ref{zindep}) and (\ref{kzindep}). Thus
we can perform a vertical dimensional reduction of the black
$(D-4)$-brane metric (\ref{dmetric}) in $D$ dimensions to a solution in
$(D-1)$ dimensions. For generic values of $k$ and $b$, this solution will
involve two scalar fields. However, as discussed in section 3, it will
describe a single-scalar solution if $k=b$. In this case, we have $K=\widetilde U
=\log r$, and hence we find that the $(D-1)$-dimensional metric $d\tilde s_{\sst
D-1}^2$ , obtained by taking $z$ as the compactification coordinate, so that
$ds^2_{\sst D}= e^{2\a\varphi} d\tilde s_{\sst D-1}^2 + e^{-2(D-3)\a\varphi}
dz^2$, is given by
\bea
d\tilde s_{\sst D-1}^2 &=& - e^{2\widetilde U}\, dt^2 + dx^i dx^i + e^{4\widetilde U-4 U}
\, dr^2 + r^2 e^{2 \widetilde U-4 U}\, d\theta^2 \ ,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&=&-r^2 \, dt^2 + dx^i dx^i+ (1-c^2 r^2)^{\ft4{\Delta}}\,
(dr^2 +d\theta^2)\ .\label{vertox}
\eea
Although the condition that the length $k$ of the rods and their
spacing $b$ be equal is desirable from the point of view that it gives rise
to a single-scalar solution in the lower dimension, it is clearly
undesirable in the sense that the individual single $p$-brane solutions are
being placed so close together that their horizons are touching. This
reflects itself in the fact that the sum over the single-rod potentials is
just yielding the potential for one rod, of infinite length and infinite
mass, and accordingly, the higher-dimensional solution just describes a
single infinitely-massive $p$-brane. The corresponding vertically-reduced
solution (\ref{vertox}), which one might have expected to describe a black
$((D-1)-3)$-brane in $(D-1)$ dimensions,\footnote{The somewhat clumsy
notation is forced upon us by the lack of a generic $D$-independent name for
a $(D-3)$-brane in $D$ dimensions.} thus does not have an extremal limit.
This can be understood from another point of view: A vertically-reduced
extremal solution is in fact a line of uniformly distributed extremal
$p$-branes in one dimension higher. In order to obtain a black
$((D-1)-3)$-brane in $(D-1)$-dimension that has an extremal limit, we should
be able to take a limit in the higher dimension in which the configuration
becomes a line of extremal $p$-branes. Thus a more appropriate superposition
of black $p$-branes in the higher dimension would be one where the spacing
$b$ between the rods was significantly larger than the lengths of the rods.
In particular, we should be able to pass to the extremal limit, where the
lengths $k$ tend to zero, while keeping the spacing $b$ fixed. In this
case, the functions $\widetilde U$ and $K$ will take the form (\ref{zindep}) and
(\ref{kzindep}) with $k<b$. Defining $\beta = k/b$, we then find that the
lower-dimensional metric, after compactifying the $z$ coordinate, becomes
\be
d\hat s_{\sst D-1}^2 = r^{\ft{2\beta(\beta-1)}{D-3}} \,
\Big( -r^{2\beta}\, dt^2 + dx^i dx^i \Big) +
r^{-2\beta +\ft{2\beta(\beta-1)}{D-3} } (1-c^2 r^{2\beta} )^{\ft{4}{\Delta}}
\, \Big( r^{2\beta^2}\, dr^2 + r^2\, d\theta^2 \Big) \ .\label{met2}
\ee
This can be interpreted as a black $((D-1)-3)$-brane in $(D-1)$ dimensions.
(In other words, what is normally called a $(D-3)$-brane in $D$ dimensions.)
The extremal limit is obtained by sending $k=b\beta$ to zero and $\mu$ to
infinity, keeping $b$ and the charge parameter $Q=(\hat k \sinh
2\mu)/(4\sqrt\Delta)$ finite. At the same time, we must rescale the $r$
coordinate so that $r\rightarrow r (\cosh\mu)^{4/\Delta}$, leading to the
extremal metric
\be
ds^2 = -dt^2 + dx^i dx^i + \Big( 1- \fft{4\sqrt\Delta Q}{b} \log
r\Big)^{\ft{4}{\Delta}} (dr^2 + r^2 d\theta^2 )\ .\label{extreme}
\ee
Thus the solution (\ref{met2}) seems to be the natural non-extremal
generalisation of the extremal $((D-1)-3)$-brane (\ref{extreme}). Note that
the black solutions (\ref{met2}) involve two scalar fields, as we discussed
previously, although in the extremal limit the additional scalar decouples.
In fact the above proposal for the non-extremal generalisation of
$(D-3)$-branes in $D$ dimensions receives support from a general analysis
of non-extremal $p$-brane solutions. The usual prescription for
constructing black $p$-branes, involving a single scalar field, as described
for example in \cite{dlp}, breaks down in the case of $(D-3)$-branes in $D$
dimensions, owing to the fact that the transverse space has dimension 2, and
hence $\tilde d=0$. Specifically, one can show in general that there is a
universal procedure for ``blackening'' the extremal single-scalar $p$-brane
$ds^2= e^{2A} (-dt^2 + dx^i dx^i) + e^{2B}(dr^2 + r^2 d\Omega^2)$, by
writing \cite{dlp}
\be
ds^2 = e^{2A}(-e^{2f} dt^2 + dx^i dx^i) + e^{2B}( e^{-2f} dr^2 + r^2
d\Omega^2) \ ,\label{dlp1}
\ee
where $e^{2f}=1-\hat k r^{-\tilde d}$, and the functions $A$ and $B$ take the same
form as in the extremal solution, but with rescaled charges:
\be
e^{-\ft{\Delta(D-2)}{2\tilde d} A} = e^{\ft{\Delta(D-2)}{2d} B} = 1+
\fft{\hat k}{r^{\tilde d}} \sinh^2\mu\ .\label{dlp2}
\ee
However, the case where $\tilde d=0$ must be treated separately, and we find
that the black solutions then take the form
\be
ds^2= -(1-\hat k\log r)dt^2 + dx^i dx^i + \fft{1}{r^2}\, \Big(1 + \hat k
\sinh^2\mu
\log r\Big)^{\ft{4}{\Delta}} \Big((1-\hat k\log r)^{-1}\, dr^2 +
d\theta^2 \Big)
\ .\label{dlp3}
\ee
In the extremal limit, \ie $\hat k\rightarrow 0$ and
$\mu\rightarrow\infty$, the metric becomes
\be
ds^2= -dt^2 + dx^i dx^i + (1 + Q R)^{\ft{4}{\Delta}} (dR^2 + d\theta^2) \ ,
\label{ext3}
\ee
where $R=\log r$. Unlike the situation for non-zero values of $\tilde d$,
where the analogous limit of the black $p$-branes gives a normal isotropic
extremal $p$-brane, in this $\tilde d=0$ case the extremal limit describes a
line of $(D-3)$-branes in $D$ dimensions, lying along the $\theta$
direction, rather than a single $(D-3)$-brane. (In fact this line of
$(D-3)$-branes can be further reduced, by compactifying the $\theta$
coordinate, to give a domain-wall solution in one lower dimension
\cite{clpst,bdgpt}.) Thus it seems that there is no appropriate
single-scalar non-extremal generalisation of an extremal $(D-3)$-brane in
$D$ dimensions, and the two-scalar solution (\ref{met2}) that we obtained by
vertical reduction of a black $(D-3)$-brane in one higher dimension is the
natural non-extremal generalisation.
\section{Conclusions}
In this paper, we raised the question as to whether one can generalise
the procedure of vertical dimensional reduction to the case of non-extremal
$p$-branes. It is of interest to do this, since, combined with the more
straightforward procedure of diagonal dimensional reduction, it would provide
a powerful way of relating the multitude of black $p$-brane solutions of
toroidally-compactified M-theory, analogous to the already well-established
procedures for extremal $p$-branes. Vertical dimensional reduction
involves compactifying one of the directions transverse to the $p$-brane
world-volume. In order to achieve the necessary translational invariance
along this direction, one needs to construct multi-center $p$-brane
solutions in the higher dimension, which allow a periodic array of
single-center solutions to be superposed. This is straightforward for
extremal $p$-branes, since the no-force condition permits the construction
of arbitrary multi-center configurations that remain in neutral equilibrium.
No analogous well-behaved multi-center solutions exist in general in the
non-extremal case, since there will be net forces between the various
$p$-branes. However, an infinite periodic array along a line will still be
in equilibrium, albeit an unstable one. This is sufficient for the
purposes of vertical dimensional reduction.
The equations of motion for general axially-symmetric $p$-brane
configurations are rather complicated, and in this paper we concentrated on
the simpler case where the transverse space is 3-dimensional. This leads to
simplifications in the equations of motion, and we were able to obtain the
general axially-symmetric solutions for charged dilatonic non-extremal
$(D-4)$-branes in $D$ dimensions. These solutions are determined by a
single function $\widetilde U$ that satisfies a linear equation, namely the
Laplace equation on a flat cylindrically-symmetric 3-space, and thus
multi-center solutions can be constructed as superpositions of basic
single-center solutions. The single-center $p$-brane solutions correspond
to the case where $\widetilde U$ is the Newtonian potential for a rod of mass $k$
and length $\ft12 k$.
The rather special features that allowed us to construct general
multi-center black solutions when the transverse space is 3-dimensional also
have a counterpart in special features of the lower-dimensional solutions
that we could obtain from them by vertical dimensional reduction. The
reduced solutions are expected to describe non-extremal $(D-3)$-branes in
$D$ dimensions. Although a general prescription for constructing
single-scalar black $p$-branes from extremal ones for arbitrary $p$ and $D$
was given in \cite{dlp}, we found that an exceptional case arises when
$p=D-3$. In this case the general analysis in \cite{dlp} degenerates, and
the single-scalar black solutions take the form (\ref{dlp3}), rather than
the naive $\tilde d\rightarrow 0$ limit of (\ref{dlp1}) and (\ref{dlp2}) where
one would simply replace $r^{-\tilde d}$ by $\log r$. The extremal limit of
(\ref{dlp3}) in fact fails to give the expected extremal $(D-3)$-brane, but
instead gives the solution (\ref{ext3}), which describes a line of
$(D-3)$-branes. Interestingly enough, we found that the vertical reduction
of the non-extremal $p$-branes obtained in this paper gives a class of
$(D-3)$-branes which are much more natural non-extremal generalisations of
extremal $(D-3)$-branes. In particular, their non-extremal limits {\it do}
reduce to the standard extremal $(D-3)$-branes. The price that one pays for
this, however, is that the non-extremal solutions involve two scalar fields
(\ie the original dilaton of the higher dimension and also the Kaluza-Klein
scalar), rather than just one linear combination of them. Thus we see that
a number of special features arise in the cases we have considered. It
would be interesting to see what happens in the more generic situation when
$\tilde d>0$.
\section*{Acknowledgement}
We are grateful to Gary Gibbons for useful discussions, and to Tuan
Tran for drawing our attention to some errors in an earlier version of the
paper. H.L.\ and C.N.P.\ are grateful to SISSA for hospitality in the early
stages of this work. K.-W.X.\ is grateful to TAMU for hospitality in the
late stages of this work.
|
proofpile-arXiv_065-646
|
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\section{Introduction}
\setcounter{equation}{0}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
Modifications of hadron properties in
nuclear medium is of great interest
in connection with the ongoing experimental plans
at CEBAF and RHIC etc.
Especially, the mass shift of vector mesons
is directly accessible by inspecting the change
of the lepton pair spectra
in the electro- or photo- production
experiments of the vector mesons from the
nuclear targets.
To study this issue,
Hatsuda-Lee (HL) applied the
QCD sum rule (QSR) method to the
vector mesons
in the nuclear medium, and got 10-20 \% {\it decrease}
of the masses of the $\rho$ and $\omega$ mesons at
the nuclear matter
density\,\cite{HL}.
Later one of the present authors\,\cite{Koike}
reexamined the analysis of \cite{HL}
based on the observation that their density effect
in the vector current
correlator
comes from
the current-nucleon forward
scattering amplitude,
and accordingly the effect should be interpretable
in terms of the
physical effect in the forward amplitude\,\cite{KM}.
This analysis showed
slight {\it increase} of the $\rho$, $\omega$ meson masses
in contradiction to \cite{HL}.
Subsequently,
the analysis in \cite{Koike} was criticized
by Hatsuda-Lee-Shiomi\cite{HLS}.
This paper is prepared as a reexamination
and a more expanded discussion of \cite{Koike}.
We present a new analysis on the $\rho$, $\omega$ and $\phi$-
nucleon scattering lengths. By introducing a constraint relation
among the parameters in the spectral function, we eventually got
a decreasing mass similar
to \cite{HL}, although the interpretation presented in \cite{Koike}
essentially persists.
We also provide informative comments and replies to \cite{HLS},
and clarify the misunderstanding in the
literature on the interpretation of the mass shift
\,\cite{HL,HLS,Hatsuda}.
We first wish to give a brief sketch
of the debate.
The information about the spectrum of a vector meson
in the nuclear medium
with the nucleon density $\rho_N$
can be extracted from the correlation function
\begin{eqnarray}
\Pi_{\mu\nu}^{\rm NM}(q) = i\int d^{4}x
e^{iq \cdot x}\langle TJ_{\mu}(x)
J_{\nu}^{\dag}(0) \rangle _{\rho_N},
\label{eq1.1}
\end{eqnarray}
where $q=(\omega,\mbox{\boldmath $q$})$ is the four momentum and
$J_\mu$ denotes the vector current for the vector
mesons in our interest:
\begin{eqnarray}
J_{\mu}^{\rho}(x) = \frac{1}{2}
(\overline{u}\gamma_{\mu}u-\overline{d}\gamma_{\mu}d)(x),\
J_{\mu}^{\omega}(x) = \frac{1}{2}
(\overline{u}\gamma_{\mu}u+\overline{d}\gamma_{\mu}d)(x),\
J_{\mu}^{\phi}(x) = \overline{s}\gamma_{\mu}s(x).
\label{eq1.2}
\end{eqnarray}
Following a common wisdom in the QSR method\,\cite{SVZ},
Hatsuda-Lee
applied an operator product expansion (OPE)
to this correlator at large $Q^2 = -q^2 > 0$.
The basic assumption employed in this procedure
is that the $\rho_N$-dependence of the correlator
is wholely ascribed to the $\rho_N$ dependence in the
condensates\,\cite{DL}:
\begin{eqnarray}
\Pi^{\rm NM}(q^{2}\rightarrow -\infty) \stackrel{\rm OPE}{=}
\sum_{i}C_{i}(q^{2},\mu^{2})
\langle {\cal O}_{i}(\mu^{2})\rangle _{\rho_N},
\label{eq1.3}
\end{eqnarray}
where $C_i$ is the Wilson coefficient
for the operator ${\cal O}_i$ and we suppressed
all the Lorentz indices
for simplicity.
A new feature in the finite density sum rule
is that both Lorentz scalar and nonscalar operators
survive as the condensates $\langle {\cal O}_i \rangle_{\rho_N}$.
An assumption of the Fermi gas model for the nuclear medium
was introduced to estimate the $\rho_N$-dependence of
$\langle {\cal O}_i \rangle_{\rho_N}$, which is
expected to be valid at
relatively low density\,\cite{DL}:
\begin{eqnarray}
\langle {\cal O}_i\rangle_{\rho_N}
&=& \langle {\cal O}_i \rangle_0
+ \sum_{\rm spin, isospin}\int^{p_f}
{ d^3p \over (2\pi)^3 2p^0 }
\langle ps|{\cal O}_i | ps \rangle \nonumber\\
&=& \langle {\cal O}_i\rangle _{0} +
\frac{\rho_N}{2M_{N}}\langle {\cal O}_i\rangle _{N} + o(\rho_N),
\label{eq1.4}
\end{eqnarray}
where $\langle\cdot\rangle_0$ represents the
vacuum expectation value,
$|ps\rangle$ denotes the nucleon state with momentum $p$
and the spin $s$
normalized covariantly
as $\langle ps|p's'\rangle = (2\pi)^3 2p^0 \delta_{ss'}
\delta^{(3)}(\mbox{\boldmath $p$}
-\mbox{\boldmath $p$}')$, and $\langle \cdot\rangle_N$
denotes the expectation value with respect to the nucleon state
with $\mbox{\boldmath $p$}=0$.
The effect of $\mbox{\boldmath $p$}
\neq 0$ introduces $O(\rho_N^{5/3})$ correction to
(\ref{eq1.4}).
This way the $\rho_N$-dependence of the condensates
can be incorporated through the nucleon matrix elements
in the linear density approximation.
By inserting (\ref{eq1.4}) in (\ref{eq1.3}), one can easily see that
the approximation to the condensate,
(\ref{eq1.4}), is equivalent to the following approximation to
the correlation function itself:
\begin{eqnarray}
\Pi^{\rm NM}_{\mu\nu}(q) = \Pi^0_{\mu\nu}(q)
+ \sum_{\rm spin, isospin}\int^{p_f} { d^3p \over (2\pi)^3 2p^0 }
T_{\mu\nu}(p,q),
\label{eq1.5}
\end{eqnarray}
where $\Pi^0_{\mu\nu}(q)$ is the vector current correlator
in the vacuum,
\begin{eqnarray}
\Pi^0_{\mu\nu}(q)=i\int d^{4}x e^{iq \cdot x}\langle
\mbox{T}J_{\mu}(x)
J_{\nu}^{\dag}(0) \rangle_0,
\label{eq1.6}
\end{eqnarray}
and
$T_{\mu\nu}(p,q)$ is the current-nucleon forward amplitude
defined as
\begin{eqnarray}
T_{\mu\nu}(p,q)= i \int d^4x e^{iq\cdot x}\langle ps |
T J_\mu(x)J^{\dag}_\nu (0)|ps \rangle.
\label{eq1.7}
\end{eqnarray}
Since \cite{HL} adopted (\ref{eq1.4}),
one should be able to interpret the result in \cite{HL}
from the point of view of the current-nucleon forward amplitude.
What was the essential ingredient
in $T_{\mu\nu}$ which lead to the decreasing
mass in \cite{HL}? What kind of approximation
in the analysis of $T_{\mu\nu}(p,q)$ corresponds to
the analysis of $\Pi_{\mu\nu}^{\rm NM}$ in \cite{HL}?
To answer these questions
we first note that the linear density approximation
(\ref{eq1.4}) to the condensates becomes better at smaller $\rho_N$
or equivalently smaller $p_f$. As long as the OPE side
is concerned,
the effect of the nucleon Fermi motion can be
included in $\langle{\it O}\rangle_{\rho_N}$
as is discussed in \cite{HLS}.
It turned out, however, that its effect is
negligible.
Therefore what is relevant in the mass shift in the QSR approach
is the structure of $T_{\mu\nu}$ in the
$\mbox{\boldmath $p$}=0$ limit.
We observe that in this limit,
$T_{\mu\nu}$
is reduced to the vector meson-nucleon scattering length $a_V$
at $q=(\omega=m_V,\mbox{\boldmath $q$}=0)$
($m_V$ is the mass of the vector meson).
If one knows
$a_V$, the mass shift of the vector meson becomes
\begin{eqnarray}
\delta m_{V} = 2\pi \frac{M_{N}+m_{V}}{M_{N}m_{V}}a_{V }\rho_N
\label{eq1.8}
\end{eqnarray}
in the linear density approximation. In the following discussion
we argue that what
was observed in \cite{HL} as a decreasing mass shift is essentially
the one in (\ref{eq1.8}).
Of course, whether the approximation (\ref{eq1.4}), (\ref{eq1.5})
to $\Pi^{\rm NM}_{\mu\nu}$
is a good one or not
at the nuclear matter density is a different issue.
What we wish to stress is that the approximation
adopted in \cite{HL} is certainly interpretable in terms of the
vector meson-nucleon ($V-N$)
scattering lengths unlike the argument in \cite{HLS}.
To motivate our idea from a purely mathematical point of view,
let's forget about the $V-N$ scattering lengths
for the moment, and translate what was observed in \cite{HL}
into the language of $T_{\mu\nu}$.
HL analyzed $\Pi^{\rm NM}_1(\omega^2)\equiv
\Pi_\mu^{{\rm NM}\mu}(q)/(-3\omega^2)$
at $\mbox{\boldmath $q$}=0$ in QSR. At $\rho_N=0$,
namely in the vacuum,
$\Pi^{\rm NM}_1(\omega^2)$ is reduced to $\Pi_1(q^2)$
defined by the relation
$\Pi^0_{\mu\nu}(q)=(q_\mu q_\nu -g_{\mu\nu}q^2)\Pi_1(q^2)$.
HL obtained a QSR relation for $\Pi_1^{\rm NM}$ as
\begin{eqnarray}
{1 \over 8\pi^2}{\rm ln}\left( {s_0^* - q^2 \over -q^2 } \right)
+{ A^* \over q^4} + { B^* \over q^6} = { F'^* \over m_V^{2*}-q^2}
+ { \rho_{sc} \over q^2},
\label{eq1.9}
\end{eqnarray}
where $A^*$ and $B^*$ are
the in-medium condensates with dim.=4 and dim.=6, respectively,
and $m_V^{*2}$, $F^*$ and $s_0^*$ are the in-medium values
of the (squared) vector meson mass,
pole residue and the continuum threshold,
which are to be determined by fitting the above equation.
$\rho_{sc}$ is the so called Landau damping term
which is purely a medium effect and is thus
$O(\rho_N)$. Actual values are
$\rho_{sc}=-{\rho_N \over 4M_N}$ for the $\rho$,
$\omega$ mesons and $\rho_{sc}=0$ for $\phi$ meson\,\cite{HL,BS}.
At $\rho_N=0$, (\ref{eq1.9}) is simply the well known sum rule in the
vacuum\,\cite{SVZ}:
\begin{eqnarray}
{1 \over 8\pi^2}{\rm ln}\left( {s_0 - q^2 \over -q^2 } \right)
+{ A \over q^4} + { B \over q^6} = { F' \over m_V^2-q^2}.
\label{eq1.10}
\end{eqnarray}
Since HL
included the linear density correction (\ref{eq1.4})
in $A^*$ and $B^*$,
they got the change in $m_V^{*2}$, $F^*$ and $s_0^*$
to $O(\rho_N)$ accuracy.
Indeed, HL got a clear linear change in these quantities.
We write $A^*=A+{\rho_N \over 2M_N}\delta A$ and
similarly for $B^*$ corresponding to (\ref{eq1.4}), where
$\delta A$ and $\delta B$ are the nucleon matrix elements
of the same operators as $A$ and $B$ respectively.
Correspondingly it is legitimate to write
$m_V^{2*}=m_V^2+{\rho_N \over 2M_N}\delta m_V^2 $,
$F'^*=F'+{\rho_N \over 2M_N}\delta F'$ and
$s_0^*=s_0+{\rho_N \over 2M_N}\delta s_0$.
Expand (\ref{eq1.9}) to $O(\rho_N)$ and subtract
(\ref{eq1.10}) from it. Then one gets
\begin{eqnarray}
{ \delta A \over q^4} + {\delta B \over q^6} =
{-F'\delta m_V^{2} \over (m_V^2 - q^2)^2} +{\delta F'
\over m_V^2 -q^2}
-{\delta s_0/(8\pi^2) \over s_0 -q^2 } +
{\delta\rho_{sc} \over q^2}.
\label{eq1.11}
\end{eqnarray}
The left hand side of this equation is precisely the
OPE expression for
$T_\mu^\mu (p,q)/(-3\omega^2)$
at $\mbox{\boldmath $p$}=\mbox{\boldmath $q$}=0$, and thus
(\ref{eq1.11}) is the QSR for the same quantity which
is equivalent to
the QSR for $\Pi_1^{\rm NM}(\omega^2)$ assumed in \cite{HL}.
Regardless of what HL intended in their
sum rule analysis for the the vector mesons in the medium,
(\ref{eq1.11}) is the equivalent sum rule relation for
$T_{\mu\nu}$ in their analysis.
What is the physical content of this
sum rule for $T_{\mu\nu}$?
In this paper
we shall show that our analysis on the vector meson
nucleon scattering
lengths precisely leads to the sum rule (\ref{eq1.11}).
This paper is organized as follows.
In section 2, we present
a new analysis for the $\rho$, $\omega$ and $\phi$
meson-nucleon spin-isospin averaged scattering lengths
in the framework of QSR. The difference from
the previous analysis\,\cite{Koike} is emphasized.
The contents of this section
should be taken as independent from the issue of the mass shift
of these vector mesons in the nuclear medium.
In section 3, we discuss the relation between the scattering lengths
obtained in section 2 and the mass shift of \cite{HL}. In section 4,
we shall
give detailed answers and comments to the criticisms raised in
\cite{HLS}.
Section 5 is devoted to summary and conclusion. Some of the formula
will be discussed in the appendix.
\section{$\rho$, $\omega$, $\phi$-nucleon scattering lengths}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
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In this section we analyze
the vector current-nucleon forward scattering
amplitude (\ref{eq1.7})
at $\mbox{\boldmath $p$}=0$ in the framework of the QCD sum rule,
and present a new estimate for the $\rho$, $\omega$ and $\phi$-meson
nucleon scattering lengths.
We first write
\begin{eqnarray}
T_{\mu\nu}(\omega,\mbox{\boldmath $q$})
& = & i\int d^{4}x e^{iq \cdot x}
\langle ps| \mbox{T}J_{\mu}(x)J_{\nu}^{\dag}(0) |ps\rangle,
\label{eq2.1}
\end{eqnarray}
suppressing the explicit dependence on
the four momentum of the nucleon $p=(M_N,0)$.
As was noticed in the introduction,
we are interested in the structure
of $T_{\mu\nu}(\omega,\mbox{\boldmath $q$}=0)$
around $\omega=m_V$ which
affects the pole structure of the vector current
correlator in the medium.
Near the pole position of the vector meson,
$T_{\mu\nu}$ can be associated
with the $T$ matrix for the forward $V-N$ ($V=\rho,\omega, \phi$)
scattering amplitude
${\cal T}_{hH,h'H'}$
by the following relation
\begin{eqnarray}
\epsilon^{*\mu}_{(h)}(q)T_{\mu\nu}(\omega,\mbox{\boldmath $q$})
\epsilon^{\nu}_{(h')}(q) \simeq
\frac{-f_{V}^{2}m_{V}^{4}}{(q^{2}-m_{V}^{2}+i\varepsilon)^{2}}
{\cal T}_{hH,h'H'}(\omega,\mbox{\boldmath $q$}),
\label{eq2.2}
\end{eqnarray}
where $h$($h'$) denotes the helicities for the initial (final)
vector meson, and similarly
$H$($H'$) for the nucleon. In (\ref{eq2.2}) the coupling $f_V$
is introduced
by the relation
$\langle 0|J_{\mu}^{V}|V^{(h)}(q)\rangle =
f_{V}m_{V}^{2}\epsilon_{\mu}^{(h)}(q)$
with the
polarization vector $\epsilon_\mu^{(h)}$ normalized as
$\sum_{h}\epsilon_{\mu}^{(h)^{*}}(q)\epsilon_{\nu}^{(h)}(q)
=-g_{\mu\nu}+q_{\mu}q_{\nu}/q^{2}$.
$T_{\mu\nu}$ can be decomposed into the four scalar functions
respecting the invariance under parity and time reversal and the
current conservation.
Taking the spin average on both sides of
(\ref{eq2.2}) (see appendix A),
$T_{\mu\nu}(\omega,\mbox{\boldmath $q$})$ is projected onto
$T(\omega,\mbox{\boldmath $q$})=
T_\mu^\mu/(-3)$\,
\cite{err} and
${\cal T}_{hH,h'H'}$ is projected onto
the spin averaged $V-N$ $T$ matrix, ${\cal T}(\omega,
\mbox{\boldmath $q$})$.
At $q=(m_V,0)$ and $p=(M_N,0)$, ${\cal T}$ is connected to
the spin averaged $V-N$ scattering length $a_V$
as ${\cal T}(m_V, 0)=
8\pi (M_N+m_V)a_V$\,\cite{err}
with $a_V={1 \over 3}(2a_{3/2}+a_{1/2})$
where $a_{3/2}$ and $a_{1/2}$
are the $V-N$ scattering lengths in the spin-3/2 and 1/2
channels respectively. We also remind that the $\rho^0-N$
scattering
length corresponds to the
isospin-averaged scattering length owing to the
isospin symmetry.
The retarded correlation function defined by
\begin{eqnarray}
T_{\mu\nu}^{R}(\omega,\mbox{\boldmath $q$})
= i\int d^{4}x e^{iq \cdot x}
\langle N|\theta(x^{0}) [J_{\mu}(x),J_{\nu}^{\dag}(0)]
|N\rangle
\label{eq2.2p}
\end{eqnarray}
satisfies the following dispersion relation
\begin{eqnarray}
T_{\mu\nu}^{R}(\omega,\mbox{\boldmath $q$})
= \frac{1}{\pi}\int_{-\infty}^{\infty}du
\frac{\mbox{Im}\ T^{R}_{\mu\nu}
(u,\mbox{\boldmath $q$})}{u-\omega-i\varepsilon}.
\label{eq2.3}
\end{eqnarray}
We recall that
for nonreal values of $\omega$,
$T^R_{\mu\nu}(\omega, \mbox{\boldmath $q$})$
becomes identical to
$T_{\mu\nu}(\omega,\mbox{\boldmath $q$})$.
Applying the same spin-averaging
procedure to both sides of (\ref{eq2.3}) as above,
we get the following dispersion relation for $\omega^2\neq$
positive real number:
\begin{eqnarray}
T(\omega,0)= \int_{-\infty}^\infty\, d\,u{ \rho(u, 0)
\over u-\omega -i\epsilon} = \int_0^\infty\,d\,u^2 {\rho(u,0)
\over u^2 -\omega^2 },
\label{eq2.4}
\end{eqnarray}
where we introduced the
spin-averaged spectral
function $\rho(\omega,\mbox{\boldmath $q$})$
constructed from ${1\over \pi}{\rm Im}
T_{\mu\nu}^R(\omega,\mbox{\boldmath $q$})$.
The second equality in (\ref{eq2.4}) comes from the relation
$\rho(-\omega,-\mbox{\boldmath $q$})=-\rho(\omega,
\mbox{\boldmath $q$})$.
Using (\ref{eq2.2}),
$\rho(u, 0)$
can be expressed
in terms of the spin-averaged $V-N$ forward $T$-matrix
${\cal T}$ as
\begin{eqnarray}
\lefteqn{\rho(u>0,\mbox{\boldmath $q$}=0)} \nonumber \\
&=& \frac{1}{\pi}\mbox{Im} \left[
\frac{-f_{V}^{2}m_{V}^{4}}{(u^{2}-m_{V}^{2}
+i\varepsilon)^{2}}{\cal T}(u,0)
\right]+\cdots\nonumber \\
&=& \frac{-f_{V}^{2}m_{V}^{4}}{\pi}\left[
\mbox{Im} \frac{1}{(u^{2}-m_{V}^{2}+i\varepsilon)^{2}} \mbox{Re}
{\cal T}(u,0)
+ \mbox{Re} \frac{1}{(u^{2}-m_{V}^{2}+i\varepsilon)^{2}}
\mbox{Im} {\cal T}(u,0)
\right] +\cdots\label{eq2.5} \\
&\equiv & a\delta'(u^{2}-m_{V}^{2}) + b\delta(u^{2}-m_{V}^{2})
+ c\delta(u^{2}-s_{0}),
\label{eq2.6}
\end{eqnarray}
where
\begin{eqnarray}
a &=& -f_V^2 m_V^4 {\rm Re}{\cal T}(u,0)|_{u=m_V} = -8\pi f_V^2
m_V^4(M_N +m_V)a_V,\label{eq2.6p}\\
b &=& -f_V^2 m_V^4 { d \over du^2}{\rm Re}{\cal T}(u,0)|_{u=m_V},
\label{eq2.6pp}
\end{eqnarray}
and $\cdots$ in (\ref{eq2.5}) represents the
continuum contribution which is not associated with
the $\rho-N$ scattering.\footnote{In (\ref{eq2.2}),
${\cal T}$ is defined only around $\omega=m_V$ and thus we
introduced the contribution $\cdots$ in (\ref{eq2.5}).}
The first two terms in (\ref{eq2.6}) come from the first term in
(\ref{eq2.5}) when (\ref{eq2.5}) is substituted into the dispersion
integral (\ref{eq2.4}).
The $b$-term (simple pole term) in (\ref{eq2.6}) represents
the off-shell effect in the ${\cal T}$ matrix of the forward
$VN\to VN$ scattering.
We note that no other higher
derivatives of ${\rm Re}{\cal T}(u,0)$ appear here.
The third term in (\ref{eq2.6})
corresponds to $\cdots$ in (\ref{eq2.5}) and represents the
scattering contribution in the continuum part of $J_V$ which starts at
the threshold $s_0$. The value of $s_0$ is fixed as $s_0=1.75$ GeV$^2$
for the $\rho$ and $\omega$ mesons
and $s_0= 2.0$ GeV$^2$ for the $\phi$ meson,
since these values are known to reproduce the masses of these
mesons\,\cite{SVZ}.
What is not included in the ansatz (\ref{eq2.6})
is the second term in $[\cdots ]$ of (\ref{eq2.5}) which represents
inelastic (continuum)
contribution such as $\rho N\to \pi N, \pi \Delta$ for
the $\rho$ meson and $\phi N\to K\Lambda,
K\Sigma$ for the $\phi$ meson etc.
The strength of
these contributions could be sizable, so we should take the following
analysis with caution. (See discussion below.)
The OPE expression for $T(q^2=\omega^2)=T(\omega,0)$ in (\ref{eq2.4})
is given in Eq. (6) of \cite{Koike}
(and Eq. (2.13) of \cite{HKL}) for the $\rho$ and $\omega$-mesons,
and it is not repeated here. It takes the following form
including the operators with dimension up to 6:
\begin{eqnarray}
T^{\rm OPE}(q^2) = { \alpha \over q^2} + {\beta \over q^4},
\label{eq2.7}
\end{eqnarray}
where $\alpha$ is
the sum of the nucleon matrix elements of the dim.=4 operators
and $\beta$ for the
dim.=6 operators. In our analysis, we adopt the same values
for these matrix elements as \cite{Koike}: $\alpha= 0.39$ GeV$^2$
for the $\rho$ and $\omega$
mesons and $\beta = -0.23 \pm 0.07$ ($-0.16 \pm
0.10$) GeV$^4$
for the $\rho$ ($\omega$) mesons. The difference in $\beta$
between $\rho$ and $\omega$
originates from the twist-4 matrix elements for which we adopted the
parameterization used in \cite{CHKL}.
For the $\phi$ meson,
$\alpha=0.24$ GeV$^2$ and $\beta=-0.12$ GeV$^4$. See \cite{Koike}
for the detail.
Up to now our procedure
for analyzing $T_{\mu\nu}$ is completely the same
as \cite{Koike}. Here we start to deviate from \cite{Koike}
and introduce a constraint relation
among $a$, $b$ and $c$ which is imposed by the low energy theorem
for the vector current-nucleon forward scattering amplitude.
In the low energy limit,
$p\rightarrow (M_N, 0)$ and $q=(\omega, \mbox{\boldmath $q$})
\rightarrow (0,0)$, $T_{\mu\nu}(\omega, \mbox{\boldmath $q$})$
is determined
by the Born diagram contribution (Fig.1)
as in the case of the Compton scattering\,\cite{Bj}.
Since we are considering the
case $\mbox{\boldmath $q$}=0$,
we first put $\mbox{\boldmath $q$}=0$
and then take the limit $\omega\to 0$ (See appendix B):
\begin{equation}
T^{\rm Born}(\omega^2)\equiv T^{\rm Born}(\omega,0) = \left\{
\begin{array}{ll} \frac{-2M_{N}^{2}}{4M_{N}^2-\omega^2}
\stackrel{\omega\rightarrow 0}{\longrightarrow} -\frac{1}{2}
& (\rho^{0},\omega) \\
\hspace*{3mm}0 & (\phi)
\end{array}
\right.
\label{eq2.9}
\end{equation}
At $q_{\mu} \neq 0$, the Born term is not the total contribution
and there remains an ambiguity
in dealing with $T^{\rm Born}$.
We thus assume two forms of the parameterization
for the phenomenological side of
the sum rules for $\rho$ and $\omega$ mesons:
\begin{enumerate}
\item[(i)] With explicit Born term:
\begin{eqnarray}
T^{\rm ph}(q^{2}) = T^{\rm Born}(q^{2}) + \frac{a}
{(m_{V}^{2}-q^2)^{2}}
+ \frac{b}{m_{V}^{2}-q^2}
+ \frac{c}{s_{0}-q^2}
\label{eq2.10}
\end{eqnarray}
with the condition
\begin{eqnarray}
\frac{a}{m_{V}^{4}}
+ \frac{b}{m_{V}^{2}}
+ \frac{c}{s_{0}} = 0.
\label{eq2.11}
\end{eqnarray}
\item[(ii)] Without explicit Born term:
\begin{eqnarray}
T^{\rm ph}(q^{2}) = \frac{a}{(m_{V}^{2}-q^2)^{2}}
+ \frac{b}{m_{V}^{2}-q^2}
+ \frac{c}{s_{0}-q^2}
\label{eq2.12}
\end{eqnarray}
with the condition
\begin{eqnarray}
\frac{a}{m_{V}^{4}}
+ \frac{b}{m_{V}^{2}}
+ \frac{c}{s_{0}} = T^{\rm Born}(0).
\label{eq2.13}
\end{eqnarray}
\end{enumerate}
With the phenomenological sides of the sum rules
((\ref{eq2.10}) or (\ref{eq2.12})) and the OPE side (\ref{eq2.7}),
the QSR is given by the relation
\begin{eqnarray}
T^{\rm OPE}(q^2) = T^{\rm ph}(q^2).
\label{QSR}
\end{eqnarray}
Several comments are in order here.
\begin{enumerate}
\item
Because of the conditions
(\ref{eq2.11}) and (\ref{eq2.13}), $T^{\rm ph}(q^2)$
satisfies $T^{\rm ph}(0)=T^{\rm Born}(0)$ and
has two independent parameters to be determined in either case.
This part is the essential difference from the previous study
in \cite{Koike}. In \cite{Koike}, $a$, $b$ and $c$ were treated as
independent parameters which were determined
in the Borel sum rule (BSR).
In the following, we eliminate $c$ by these relations and regard
$T^{\rm ph}$ as a functions of $a$ and $b$.
\item
The leading behavior of $T^{\rm ph}(q^2)$
at large $-q^2 >0$ is consistent with $T^{\rm OPE}(q^2)$:
Both sides start with the ${1\over q^2}$ term, which supports
the form of the spectral function in (\ref{eq2.6}).
\item
Inclusion of $T^{\rm Born}(q^2)$ in (\ref{eq2.10}) has a
similar effect
as the inclusion of the ``second continuum'' contribution
with the threshold $4M_N^2$. In the QSR analysis for
the lowest resonance
contribution, the result is more reliable if it does not depend
on the details of the higher energy part.
We shall see this is indeed the case in the
following Borel sum rule method.
\end{enumerate}
By expanding $T^{\rm ph}(q^2)$ with respect to
$1/(-q^2)$ and comparing the coefficients
of $1/q^2$ and $1/q^4$ in $T^{\rm ph}(q^2)$
with those in $T^{\rm OPE}(q^2)$, one gets the
finite energy sum rules (FESR). These relations are solved to give
\begin{eqnarray}
a &=& { 1 \over 1- { s_0 \over m_V^2}} \left[ m_V^2 \left(
1 + {s_0 \over m_V^2}\right)\left(-\alpha + 2M_N^2\right)
+ \left( \beta - 8 M_N^4 \right) \right],
\label{eq2.14}\\
b &=& { 1 \over \left( 1 - {s_0 \over m_V^2} \right)^2 }
\left[ \left(1 + { s_0^2 \over m_V^4 } \right)
\left( -\alpha + 2 M_N^2
\right)
+ { s_0 \over m_V^4 } \left( \beta - 8M_N^4 \right) \right],
\label{eq2.15}
\end{eqnarray}
for the case (i) and
\begin{eqnarray}
a &=& { 1 \over 1- { s_0 \over m_V^2}} \left[ m_V^2 \left(
1 + {s_0 \over m_V^2}\right)\left(-\alpha + { 1 \over 2}s_0 \right)
+ \left( \beta - {1 \over 2}s_0^2 \right) \right],
\label{eq2.16}\\
b &=& { 1 \over \left( 1 - {s_0 \over m_V^2} \right)^2 }
\left[ \left(1 + { s_0^2 \over m_V^4 } \right)
\left(-\alpha + { 1 \over 2}s_0
\right)
+ { s_0 \over m_V^4 } \left( \beta -
{ 1 \over 2}s_0^2 \right) \right],
\label{eq2.17}
\end{eqnarray}
for the case (ii).
These FESR relations give $a_{\rho}=-0.68$ fm,
$a_{\omega}=-0.66$ fm
for the case (i) and $a_{\rho}=-0.13$ fm, $a_{\omega}=-0.11$ fm
for the case (ii).
For the $\phi$ meson, $a_{\phi}=-0.06$ fm.
Two ways of dealing with the Born term
give quite different results. This is not surprising.
Since the leading order contribution in $T^{\rm ph}$ comes from
the continuum contribution, the results in FESR strongly depends on
the treatment of this part.
These small negative numbers,
however, suggest that the $V-N$ interaction
is weakly attractive.
In order to give more
quantitative prediction, we proceed to the Borel
sum rule (BSR) analysis.
In this method, the higher energy contribution
in the spectral function is suppressed compared
to the $V-N$ scattering
contribution. We thus have an advantage that the ambiguity in
dealing with the Born term becomes less important in BSR.
We shall try the following two methods in BSR:
\begin{enumerate}
\item[(1)] Derivative Borel Sum Rule (DBSR):
After the Borel transform of (\ref{QSR})
with respect to $Q^2=-q^2>0$, take the derivative of both sides
with respect to the Borel mass $M^2$, and use those two equations
to get $a$ and $b$ by taking the average in a Borel window,
$M_{min}^2<M^2<M_{max}^2$.
\item[(2)] Fitting Borel Sum Rule (FBSR):
Determine $a$ and $b$ in order to make the
following quantity minimum
in a Borel window $M_{\rm min}^2 < M^2 < M^2_{\rm max}$:
\begin{equation}
F(a,b)=\int_{M_{min}^{2}}^{M_{max}^{2}}dM^{2}[T^{\rm OPE}(M^{2})
- T^{\rm ph}(M^{2};a,b)]^{2}
\label{eq2.18}
\end{equation}
where $T^{\rm ph}(M^2; a, b)$ is the Borel transform
of $T^{\rm ph}(q^2)$ which is a functional of $a$ and $b$.
\end{enumerate}
After getting $a$ and $b$ by these methods, we determine
$a_V$ from the relation (\ref{eq2.6p}) using the
experimental values
of $M_N$, $m_V$ and $f_V$. The numbers we adopted are
$M_N=940$ MeV, $m_{\rho,\omega}=770$ MeV, $f_{\rho,\omega}=0.18$,
$m_{\phi}=1020$ MeV and $f_{\phi}=0.25$.
Borel curves for $a_V$ ($V$=$\rho$, $\omega$, $\phi$) in the DBSR
are shown in Figs. 2 and 3.
Stability of these curves is reasonably good
around $M^2=1$ GeV$^2$ for the $\rho$ and $\omega$ mesons
and around $M^2=1.5$ GeV$^2$ for the $\phi$ meson.
We take the average over the window $0.8\ {\rm GeV}^2\ < M^2 <
1.3\ {\rm GeV}^2$ for $\rho$, $\omega$ mesons and
$1.3\ {\rm GeV}^2\ < M^2 < 1.8\ {\rm GeV}^2$ for the $\phi$ meson.
These windows are typical for the analysis of these vector meson
masses, and the experimental values are well reproduced
with the continuum threshold $s_0=1.75$ GeV$^2$
for $\rho$, $\omega$
and $s_0=2.0$ GeV$^2$ for $\phi$\,\cite{SVZ}.
The obtained values are
$a_{\rho}=-0.5$ ($-0.4$) fm
and $a_{\omega}=-0.45$ ($-0.35$) fm for the case (i) ((ii)),
and
$a_{\phi}=-0.15$ fm.
In the FBSR method with the same window, we get close numbers
$a_{\rho}=-0.52$ ($-0.42$) fm
and $a_{\omega}=-0.46$ ($-0.36$) fm for (i) ((ii)),
and $a_{\phi}=-0.15$ fm.
We tried FBSR for various Borel windows within
$0.6$ GeV$^2$ $<M^2<1.8$ GeV$^2$
($0.9$ GeV$^2$ $<M^2<2.0$ GeV$^2$)
for the $\rho$, $\omega$ ($\phi$) mesons and found that
the results change within 20 \% level.
From these analyses, we get
\begin{eqnarray}
a_{\rho} &=& -0.47\pm 0.05\ \mbox{fm}, \nonumber \\
a_{\omega} &=& -0.41 \pm 0.05\ \mbox{fm}, \nonumber \\
a_{\phi} &=& -0.15\pm 0.02\ \mbox{fm},
\label{eq2.19}
\end{eqnarray}
where the assigned error bars are due to the uncertainty in
the Borel analysis.
We first note that the magnitudes of these scattering lengths are
quite small, i.e., smaller than the typical hadronic size
of 1 fm. For $\pi N$ and $K N$ systems,
the scattering lengths
are known to be small due to the chiral symmetry.
The above numbers
are not so different from $a_{\pi N}$ and $a_{K N}$.
Small negative values suggest that these $V-N$ interactions
are weakly attractive. The ansatz (\ref{eq2.6})
for the spectral function
ignores various inelastic contributions as was noted
below (\ref{eq2.6pp}). So we should take the above numbers as
a rough estimate of the order of magnitude.
Recently Kondo-Morimatsu-Nishino calculated the $\pi N$ and $KN$
scattering lengths by
applying the same QSR method to the correlator of
the axial vector current
\,\cite{KMN}.
The results with the lowest dimensional operators
in the OPE side is the same as the current algebra
calculation. QSR supplies the correction due to the
nucleon matrix elements of the higher
dimensional operator. Since there is no
algebraic technique (such as current algebra)
to calculate the scattering lengths
in the vector channels,
it is interesting to see that OPE provides
a possibility to estimate the strengths of the $VN$ interactions.
\section{Mass shift of the vector mesons in the nuclear medium}
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\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
In the previous section, we have identified the pole
structure of $T_{\mu\nu}(\omega,0)$ around $\omega^2 = m_V^2$ as
\begin{eqnarray}
T_{\mu\nu}(\omega,0)=
\left( {q_\mu q_\nu \over \omega^2}-g_{\mu\nu}\right)
\left({ a \over (m_V^2 - \omega^2)^2 }
+ { b \over m_V^2-\omega^2 }
+ \ldots \right).
\label{eq3.0}
\end{eqnarray}
By combining this piece with the vacuum
piece $\Pi_{\mu\nu}^0(\omega,0)$
in (\ref{eq1.6}),
the vector current correlation function in the nuclear medium
take the following form around $\omega^2=m_V^2$:
\begin{eqnarray}
\Pi_{\mu\nu}^{\rm NM}(\omega,0) &\simeq &
\left( {q_\mu q_\nu \over \omega^2}-g_{\mu\nu}\right)
\left(\Pi(\omega^2)
+ { \rho_N \over 2M_N}T(\omega,0)\right)\nonumber\\
&\propto & \frac{F}{m_{V}^{2}-\omega^2}+
\frac{\rho_N}{2M_{N}}\left\{\frac{a}{(m_{V}^{2}-\omega^2)^{2}} +
\frac{b}{m_{V}^{2}-\omega^2} \right\}\cdot\cdot\cdot \nonumber\\
&\simeq & \frac{F+\delta F}{(m_{V}^{2}+\Delta m_{V}^{2})-\omega^2}
+ \cdot\cdot\cdot,
\label{eq3.1}
\end{eqnarray}
where $\Pi(q^2)$ is defined as $\Pi_{\mu\nu}^0(q)=
\left( {q_\mu q_\nu \over q^2}- g_{\mu\nu}\right)\Pi(q^2)$ and
the pole residue $F$ in $\Pi^0_{\mu\nu}$
is related to $f_V$ and $m_V$ by the relation
$F=f_V^2 m_V^4$ and $\delta F={\rho_N \over 2M_N}b$.
The quantity
\begin{eqnarray}
\Delta m_{V}^{2} = \frac{-\rho_N}{2M_{N}}\frac{a}{F}
= \frac{\rho_N}{2M_{N}}8\pi(M_{N}+m_{V})a_{V}
\label{eq3.2}
\end{eqnarray}
is regarded as the shift of the squared vector
meson mass in nuclear matter.
We thus have the mass shift
$\delta m_V$ as shown in (\ref{eq1.8})
from the relation
\begin{eqnarray}
m_V^*=m_{V}+\delta m_V = \sqrt{m_{V}^{2}+\Delta m_{V}^{2}}.
\label{eq3.3}
\end{eqnarray}
Using the scattering lengths obtained in the previous section,
we plotted the vector meson masses in Fig. 4 as a function
of the density $\rho_N$ based on the linear density
approximation.
At the nuclear matter density $\rho_N=0.17$ fm$^{-3}$ as
\begin{eqnarray}
\delta m_{\rho} &=& -45 \sim -55\ \mbox{MeV}\
(6 \sim 7\%), \nonumber \\
\delta m_{\omega} &=& -40 \sim -50\ \mbox{MeV}\
(5 \sim 6\%), \nonumber \\
\delta m_{\phi} &=& -10 \sim -20\ \mbox{MeV}\ (1 \sim 2\%).
\label{table2}
\end{eqnarray}
In order to clarify the
relation between the above mass shifts and the
approach by Hatsuda-Lee,
we briefly recall QSR for the vector meson mass in the vacuum.
The correlation function in the vacuum defined in (\ref{eq1.6})
has the structure
\begin{eqnarray}
\Pi_{\mu\nu}^0(q)= (q_\mu q_\nu -g_{\mu\nu}q^2)\Pi_1(q^2).
\label{eq3.4}
\end{eqnarray}
In QSR one starts with the dispersion relation for $\Pi_1(q)$
(See Appendix C):
\begin{eqnarray}
\Pi_1(q^2) = {q^2 \over \pi}\int_{0+}^\infty\,d\,s {
{\rm Im}\Pi_1(s)
\over s(s -q^2)} + \Pi_1(0),
\label{eq3.5}
\end{eqnarray}
where we introduced
one subtraction to avoid the logarithmic divergence.
In the deep Euclidean region $q^2\to -\infty$,
$\Pi_1(q^2)$ in the the left hand side of
(\ref{eq3.5})
has the OPE expression including the operators up to
dim.=6 as
\begin{eqnarray}
\Pi_1^{\rm OPE}(q^2) = -{ 1 \over 8\pi^2 }
{\rm ln}(-q^2) + { A \over q^4}
+ {B \over q^6},
\label{eq3.6}
\end{eqnarray}
where $A$ and $B$ are respectively the
sums of dim.=4 and dim.=6
condensates, and the
perturbative correction factor $1 + {\alpha_s \over \pi}$
to the first term is omitted for simplicity.
We also suppressed the scale dependence in each
term in (\ref{eq3.6}).
The spectral function in
(\ref{eq3.5}) is often modeled by the sum of the pole
contribution from the vector meson and the continuum
contribution:
\begin{eqnarray}
{1 \over \pi}{\rm Im}\Pi_1(s)
=F'\delta(s-m_V^2)+{1\over 8\pi^2}\theta(s-s_0),
\label{eq3.7}
\end{eqnarray}
where $F'=f_V^2 m_V^2$.
With this form in (\ref{eq3.5}) together with (\ref{eq3.6}),
one gets the
sum rule relation (See Appendix C) as
\begin{eqnarray}
{1 \over 8\pi^2}{\rm ln}\left( {s_0-q^2 \over -q^2}\right)
+{ A \over q^4} + { B \over q^6} = {F' \over m_V^2 - q^2}.
\label{eq3.8}
\end{eqnarray}
Hatsuda-Lee considered the sum rule
for $\Pi_1^{\rm NM}(\omega^2)=
\Pi^{{\rm NM}\mu}_\mu (\omega,\mbox{\boldmath $q$}=0)/(-3\omega^2)$.
The QSR for $\Pi_1^{\rm NM}(q^2)$ is reduced to
(\ref{eq3.8}) at $\rho_N\to 0$
limit. At $\mbox{\boldmath $q$}=0$, $\Pi_1^{\rm NM}(q^2)$ becomes
\begin{eqnarray}
\Pi_1^{\rm NM}(q^2)=\Pi_1(q^2)+{\rho_N\over 2M_N}{ T(q^2)\over q^2}
+O(\rho_N^{5/3}).
\label{eq3.9}
\end{eqnarray}
Thus one has to analyze $T(q^2)/q^2$ to understand the density
dependence in $\Pi_1^{\rm NM}(\omega^2)$.
We write the
dispersion relation for $T(q^2)/q^2$:
\begin{eqnarray}
{T(q^2) \over q^2} = \int_{0+}^\infty\,d\,s { \rho(s) \over
s(s-q^2)} + {T(0) \over q^2},
\label{eq3.10}
\end{eqnarray}
where the pole contribution
at $q^2=0$ is explicitly taken care of by $T(0)$.
Substituting the spectral function (\ref{eq2.6})
in this equation and equating it to the OPE side, one gets
the QSR relation for the case (ii) as
\begin{eqnarray}
{ \alpha \over q^4} + {\beta \over q^6}
={ a' \over (m_V^2 - q^2)^2} + { b' \over m_V^2 -q^2}
+{(T^{\rm Born}(0)-b') \over s_0 -q^2} + { T^{\rm Born}(0) \over q^2},
\label{eq3.11}
\end{eqnarray}
where
\begin{eqnarray}
a'= {a \over m_V^2},\ \ \ \ \ b'={a \over m_V^4}+{b \over m_V^2},
\label{eq3.12}
\end{eqnarray}
and the relation $T(0)=T^{\rm Born}(0)$
is used in the last term of
(\ref{eq3.11}).
We note that
(\ref{eq3.11}) is nothing but the relation obtained by
dividing both sides of (\ref{QSR}) by $q^2$ for the case (ii),
which guarantees the absence of the $1\over q^2$ term
in the right hand side.
(Note that the condition $T(0)=T^{\rm Born}(0)$ itself is
not required
to guarantee this consistency condition.)
Using (\ref{eq3.8}) and (\ref{eq3.11}) in (\ref{eq3.9}),
we can construct
the QSR for $\Pi_1^{\rm NM}(q)$ in the linear
density approximation:
\begin{eqnarray}
{1 \over 8\pi^2}{\rm ln}\left( {s_0 - q^2 \over -q^2} \right)
+{ A+\widetilde{\alpha} \over q^4}+{ B+ \widetilde{\beta}
\over q^6}
={ F' + \widetilde{b'} \over m_V^2 - q^2} +
{ \widetilde{a'} \over (m_V^2-q^2)^2}+
{ \widetilde{T}^{\rm Born}(0) -\widetilde{b'}
\over s_0 -q^2} + { \widetilde{T}^{\rm Born}(0) \over q^2},
\nonumber\\
\label{eq3.13}
\end{eqnarray}
where $\widetilde{\alpha}={\rho_N \over 2M_N}\alpha$,
$\widetilde{a'}={\rho_N \over 2M_N}a'$, etc.
To $O(\rho_N)$ accuracy
(\ref{eq3.13}) can be rewritten as
\begin{eqnarray}
\frac{1}{8\pi^2}\mbox{ln}\left(\frac{s_0^*-q^2}{-q^2}\right)
+ \frac{A^*}{q^4} + \frac{B^*}{q^6} =
\frac{F'^{*}}{m_{V}^{*2}-q^{2}}
+\frac{\widetilde{T}^{\rm Born}(0)}{q^{2}},
\label{eq3.14}
\end{eqnarray}
with
\begin{eqnarray}
A^{*} = A + \widetilde{\alpha}, \ \ \ \
B^{*} = B +\widetilde{\beta},
\label{eq3.15a}
\end{eqnarray}
\begin{eqnarray}
F'^* =F'+\widetilde{b'},\ \ \ \
m_V^{*2}=m_V^2 - { \widetilde{a'} \over F'},
\ \ \ \
s_{0}^{*} = s_{0} - 8\pi^2( \widetilde{T}^{\rm Born}(0)
- \widetilde{b'}).
\label{eq3.15b}
\end{eqnarray}
From the above demonstration, it is now clear that
our analysis of $T_{\mu\nu}$ in the previous section
(case (ii)) leads to (\ref{eq1.9}) for $\Pi_1^{\rm NM}$
by the identification $\rho_{sc}=\widetilde{T}^{\rm Born}(0)$.
In fact $\rho_{sc}={-\rho_N \over 2M_N}$ for
the $\rho$, $\omega$ mesons
and $\rho_{sc}=0$ for the $\phi$ meson in \cite{HL}, which is
consistent with (\ref{eq2.9}).
The mass shift in (\ref{eq3.15b}) is obviously the same as
given in (\ref{eq3.2}).
We should emphasize that it is our constraint relation
$T^{\rm ph}(0)=T^{\rm Born}(0)$ in the analysis of the
scattering lengths
which leads to the same sum rule for $\Pi_1^{\rm NM}(q^2)$
as in \cite{HL}.
Our use of low energy theorem is in parallel with
the calculation of the Landau damping term $\rho_{sc}$
from the Born diagram in \cite{HL}.
If one did not have such information
on $T(0)$, one would have to use the approach in \cite{Koike}
with the matrix elements of the dim.=8 or higher operators.
We point out, however, a small difference from \cite{HL}.
From the first and the third relation in (\ref{eq3.15b}), one obtains
\begin{eqnarray}
F'^* - F' = {1 \over 8\pi^2}\left( s_0^* - s_0 \right) +
\widetilde{T}^{\rm Born}(0),
\label{eq3.15p}
\end{eqnarray}
which is the same as the first FESR relation obtained
from $\Pi_1^{\rm NM}$. (In FESR our
present analysis is completely equivalent
to \cite{HL}.)
Namely the shift of $F'$ is determined by that of $s_0$.
In the Borel sum rule in \cite{HL}, $F'^*$ and $s_0^*$
are regarded as independent fitting parameters.
But if one recognizes that the QSR
for $T_{\mu\nu}$ is independent from
that for $\Pi_{\mu\nu}$,
it is easy to see that this condition has to be also satisfied
in the approach of \cite{HL}. In fact, in
(\ref{eq1.11}) which was derived purely mathematically
from the sum rule
in \cite{HL},
absence of $1/q^2$ term in the left hand side
of (\ref{eq1.11}) imposes the consistency requirement
in the right hand side of (\ref{eq1.11}), which is exactly
(\ref{eq3.15p}).
Since HL took the view that $\rho_{sc}$ is calculable
(owing to the low energy theorem),
they could have eliminated $\delta F'$ or $\delta s_0$
from the outset.
In our BSR for $T_{\mu\nu}$, we were lead to use the condition
(\ref{eq3.15p}) explicitly, which is imposed by the low energy theorem
$T(0)= T^{\rm Born}(0)$.
In our opinion, this is more natural because
the QSR for $T_{\mu\nu}$ is completely independent from
the one for $\Pi^0_{\mu\nu}$, i.e.,
the density $\rho_N$ is simply
an external parameter
which connects these
quantities in the sum rule for $\Pi^{\rm NM}_1$.
Although the mass shifts discussed in this section
are essentially the same
as those in \cite{HL}, the numerical values in
(\ref{table2}) are approximately factor two
smaller
than those in \cite{HL}, especially for the $\rho$ and $\omega$ mesons.
This is mainly because their calculation is done
at the chiral limit
(they ignored a correction due to the condensate
$m_q\langle \bar{\psi}\psi\rangle$), and correspondingly their
value for the continuum
threshold $s_0$
is different from ours. They used $s_0=1.43$ GeV$^2$
for $\rho$, $\omega$ in the vacuum.
Another reason is that their QSR
was for the total sum of $\Pi^0_{\mu\nu}$ and $T_{\mu\nu}$,
the latter being small ($O(\rho_N)$) correction to the
former as noted above, while our QSR is for the latter.
These differences eventually leads to
factor-two difference in the mass shifts
at around nuclear matter density.
Hatsuda claims\,\cite{Hatsuda} that, although
$m_V^*/m_V$ at $\rho_N=0$ in \cite{HL,JL}
is consistent with our scattering lengths,
the mass shift in \cite{HL,JL} at higher $\rho_N$
becomes bigger than expected from the scattering length,
with the reasoning that the scattering length
can be used only at very close to zero density
and the prediction in \cite{HL} contains more than that.
This deviation, however, should not be regarded as
a meanigful one, since
the OPE side includes only $O(\rho_N)$ density effect
and therefore only the $O(\rho_N)$ effect
represented by the scattering length
is a valid physical prediction.
It is probably useful to add a brief comment on the
calculation
of $\rho_{sc}$ in \cite{HL}.
Using the general relation
\begin{eqnarray}
\lim_{\mbox{\boldmath $q$}\to 0}\Pi_1^{\rm NM}(\omega,
\mbox{\boldmath $q$})
=\lim_{\mbox{\boldmath $q$}\to 0}
{\Pi_{00}^{\rm NM}(\omega,\mbox{\boldmath $q$})
\over|\mbox{\boldmath $q$}|^2},
\label{eq3.16}
\end{eqnarray}
they calculated $\rho_{sc}$ from the spectral function
of $\Pi_{00}^{\rm NM}(\omega,
\mbox{\boldmath $q$})/|\mbox{\boldmath $q$}|^2$
which corresponds to
$T_{00}(\omega,
\mbox{\boldmath $q$})/|\mbox{\boldmath $q$}|^2$ in our method.
They included
the pole contributions which appear at $\omega=\pm 0$ and
ignored the contributions from $\omega=\pm 2M_N$.
But this treatment suffices as long as one needs the value of
$T^{\rm Born}(0)$.
The residue at $\omega=\pm 0$ (=$-1/2$ for $\rho$, $\omega$ meson)
of $\lim_{\mbox{\boldmath $q$}\to 0}T_{00}
(\omega,\mbox{\boldmath $q$})/|\mbox{\boldmath $q$}|^2$
precisely gives $T^{\rm Born}(0)$. (See Appendix B.)
As was noticed below (\ref{QSR}), their neglection
of the poles at $\omega=\pm 2M_N$ in
$\lim_{\mbox{\boldmath $q$}\to 0}
\Pi_{00}^{\rm NM}(\omega,
\mbox{\boldmath $q$})/|\mbox{\boldmath $q$}|^2$
corresponds to
the assumption
that those contributions are taken care of
in the continuum part of (\ref{eq2.10})
( $1/(s_0-q^2)$ term) in our language.
\section{Comments and Replies to Hatsuda-Lee-Shiomi (HLS)}
\setcounter{equation}{0}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
It is by now clear that our present QSR analysis
on the current-nucleon
forward scattering amplitude is essentially equivalent
to the medium QSR for the vector mesons in \cite{HL}.
Namely their result is certainly interpretable in terms
of the $V-N$ scattering lengths.
Although this already resolves the essential controversy
between \cite{Koike} and \cite{HL}, we summarize in the following
our replies and comments to HLS\,\cite{HLS}.
\vspace{1cm}
\noindent
(1) HLS claims that $V-N$ scattering lengths
are not calculable in QSR without including dim.=8 matrix elements.
This is because phenomenological side contains
three unknown parameters
but finite energy sum rules (FESR) provide only two relations.
(Sec. III.B of \cite{HLS})
\vspace{0.5cm}
\noindent
Reply:
In our present analysis, we eliminated one parameter
by the constraint relation
at $q^\mu =0$ and thus have two unknown parameters to be determined
by QSR.
This constraint relation due to the low energy theorem
renders our analysis
equivalent to HL in the FESR.
\vspace{1.0cm}
\noindent
(2) HLS claims $\Pi^{\rm NM}
(\omega^2)=\omega^2\Pi_1^{\rm NM}(\omega^2)$
is not usable to predict the mass of the vector mesons
either in medium or in the vacuum.
(Sec. III.C of \cite{HLS})
\vspace{0.5cm}
\noindent
Reply:
This argumentation is based on the number of
available FESRs' and the
Borel stability of
the sum rules. As is shown in Appendix C, the QSR itself (before
Borel transform) is the same for
$\Pi^{\rm NM}(\omega^2)$ and $\Pi_1^{\rm NM}(\omega^2)$
as long as one starts with the same consistent assumption
for these quantities.
The QSR for $\Pi^{\rm NM}$ is simply the one obtained
by multiplying $\omega^2$ to $\Pi_1^{\rm NM}$.
Accordingly the FESRs' are the same.
Whether one applies Borel transform before or
after multiplying $\omega^2$
to $\Pi_1^{\rm NM}(\omega^2)$ causes numerical difference,
especially because one loses information from the
polynomial terms in the BSR method.
We agree that applying Borel transform
to $\Pi_1^{\rm NM}$ leads to more
stable Borel curve than applying to $\Pi^{\rm NM}$.
We, however, note that the reason HL obtained
the stable Borel curve for $\Pi_1^{\rm NM}$ is that
the Borel curve for $\Pi_1$ in the vacuum is very stable
and the curve for ${\rho_N\over 2 M_N} {T(q^2)\over q^2}$
(see (\ref{eq3.9})) is only an $O(\rho_N)$ correction to the former.
In our case, what is plotted in Figs. 2 and 3 are the
Borel curves for $T(q^2)$ itself.
The authors of \cite{JL} raised a similar criticism against
\cite{Koike} and claimed that they have clarified the
origin of discrepancy
between \cite{HL} and \cite{Koike}.
But this does not solve the problem.
In \cite{Hatsuda}, it was advertised that
the Borel curves for $m_V^*$ in \cite{HL}
are more stable than
those for our scattering lengths shown in Figs. 2 and 3.
But the reason for this is obvious. The stability of the Borel curve
for $m_V$ is excellent in the vacuum, and the density
effect based on the scattering length is simply a small
$O(\rho_N)$ correction to it.
\vspace{1.0cm}
\noindent
(3) HLS claims that the $V-N$ scattering lengths and the mass
shift of the vector mesons in the nuclear matter have no direct
relation due to the momentum dependence of the $V-N$ forward
scattering amplitude.
They also claim that
the analysis in \cite{HL} did not use this relation.
(Sec. III.A of \cite{HLS})
\vspace{0.5cm}
\noindent
Reply:
As is noted in the introduction, the analysis in \cite{HL} is
{\it mathematically}
equivalent to the QSR for $T_{\mu\nu}$ shown in
(\ref{eq1.11}).
The right hand side of (\ref{eq1.11}) is precisely
reproduced by the spectral function
shown in (\ref{eq2.6}) and the Born
contribution to $T_{\mu\nu}$. Thus the physical effect
which caused the mass shift in \cite{HL} is essentially
the same as the one based on the $V-N$ scattering lengths.
HLS stressed the
importance of the momentum dependence of $T_{\mu\nu}$. However,
it is not conspicuous in the OPE side.
So one can not claim its importance
in the phenomenological side from the QSR analysis itself.
In fact the effect of the fermi motion of
the nucleon can be included
in the OPE side, but they are at least $O(\rho_N^{5/3})$ and
they can be neglected as was shown in sec. IV of \cite{HLS}.
How come one can claim the importance
of the effect which is negligible
in the OPE side?
Since the common starting point of our analysis was
the linear density approximation
to the OPE side shown in (\ref{eq1.4})
the negligible effect in (\ref{eq1.4}) should be taken
as the effect which is either
negligible in the phenomenological side or
beyond the resolution of the analysis.
The phenomenological basis on
which HLS emphasize the effect of Fermi motion
of the nucleons is as follows:
Nucleon's fermi momentum is
$p_f=270$ MeV in Nuclear matter and thus one should take into
account the $\rho-N$
scattering from $\sqrt{s}=m_{\rho}+M_N=1709$ MeV
through $\sqrt{s}=[(m_\rho
+\sqrt{M_N^2+p_f^2})^2-p_f^2]^{1/2}=1726$
MeV. In this interval there are
some $s$-channel resonances such as $N(1710)$ and $N(1720)$
which couple to $\rho-N$ channel, thus
$T_{\mu\nu}$ should change rapidly in this
interval. However, these resonances
together with the other near resonances
($N(1700)$, $\Delta(1700)$) have broad widths of over $100$ MeV
and that
the whole interval
$0<|\mbox{\boldmath $p$}| <p_f$ is buried under these
broad resonance regions.
In this situation,
it is unlikely that the $V-N$ phase shift changes rapidly
in this interval.
It might be a good approximation to take the $T$-matrix
at $\mbox{\boldmath $p$}=0$ as a representative value of it.
How about $\phi$ meson? The $\phi-N$ scattering occurs from
$\sqrt{s}=m_{\phi}+M_N=1960$ MeV
through $\sqrt{s}
=[(m_\phi+\sqrt{M_N^2+p_f^2})^2-p_f^2]^{1/2}=1980$
MeV. In this interval there is no resonance which could couple to
$\phi-N$ system. The situation is better.
In \cite{FH,KM2}, there is a debate on the interpretation
of the nucleon sum rule in the nuclear medium.
We agree with the interpretation of \cite{KM2}.
A difference between the sum rules for the $V-N$ and $N-N$
interactions is the smallness of
the obtained $V-N$ scattering lengths, which, together with
the argument above, may
justify the use of (\ref{eq1.8}) to predict the mass
shift in the linear density approximation.
One can organize the finite temperature ($T$) QSR in a similar way,
replacing the Fermi gas of nucleons by the ideal gas of pions
\,\cite{HKL,Koi}. In this formalism, the $T$-dependence
of correlation functions
comes from the current-pion forward amplitude.
Since the pion-hadron
scattering lengths are zero in the chiral limit,
there is no $O(T^2)$ mass shift\,\cite{Koi,EI}.
This is in parallel with
our present analysis that $O(\rho_N)$-dependence of
the mass is determined by the scattering length.
\section{Summary and Conclusions}
\setcounter{equation}{0}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
In this paper,
we have presented a new analysis on the $\rho$, $\omega$
and $\phi$ meson-nucleon spin-isospin
averaged scattering lengths
$a_V$ ($V=\rho,\omega,\phi$)
in the framework of the QCD sum rule. Essential difference from
the previous calculation in \cite{Koike} is that
the parameters in the spectral
function of the vector current-nucleon
forward amplitude is constrained by the relation at $q^\mu =0$
(low energy theorem for the vector-current nucleon scattering amplitude).
We obtained small negative values for $a_V$ as
\begin{eqnarray}
a_{\rho} &=& -0.47\pm 0.05\ \mbox{fm}, \nonumber \\
a_{\omega} &=& -0.41 \pm 0.05\ \mbox{fm}, \nonumber \\
a_{\phi} &=& -0.15\pm 0.02\ \mbox{fm}.
\end{eqnarray}
This suggests that these $V-N$ interactions are weakly attractive
in contrast to the previous study \cite{Koike}.
Since the form of the spectral function is greatly simplified,
these numbers should be taken as a rough estimate of the
order of magnitude.
In the axial vector channel, the method works as a tool to
introduce a correction to the current algebra calculation
due to the higher dimensional operators. Present application
to the vector channel in which current algebra technique
does not work is suggestive in that the QSR provides us with a
possibility to express the $V-N$ scattering lengths in terms of
various nucleon matrix elements.
If one applies above $a_V$s' to the
vector meson masses in the nuclear medium in the linear
density approximation, one gets for the mass shifts as
\begin{eqnarray}
\delta m_{\rho} &=& -45 \sim -55\ \mbox{MeV}\
(6 \sim 7\%), \nonumber \\
\delta m_{\omega} &=& -40 \sim -50\ \mbox{MeV}\
(5 \sim 6\%), \nonumber \\
\delta m_{\phi} &=& -10 \sim -20\ \mbox{MeV}\ (1 \sim 2\%),
\end{eqnarray}
at the nuclear matter density.
We have shown that the physical
content of the mass shifts
discussed in \cite{HL}
are essentially the one due to the scattering
lengths shown above and have resolved the discrepancy
between \cite{HL} and \cite{Koike}.
One might naturally ask whether the previous QSR\,\cite{Koike}
for the scattering lengths is wrong or not.
Compared with \cite{Koike}, the present
analysis utilizes more available information, i.e. the constraint
from the low energy theorem. In this sence,
one may say that the present analysis is a more
sound one.
If one did not have such information
on $T(0)$, one would have to use the approach in \cite{Koike}
with the inclusion of the
matrix elements of the dim.=8 or higher operators.
In this sence, the way of constructing sum rule itself
in \cite{Koike} is also correct.
Another point we wish to emphasize is that
regardless of the availabilty of the information
on $T(0)$ (such as the low energy theorem),
the sum rule for $m_V^*$ in (\ref{eq1.9})\,\cite{HL}
in the linear density approximation is automatically
equivalent to the mass shift due to the scattering lengths
as is shown in (\ref{eq1.11}).
Finally, we wish to make some comments on
the interpretation in the literature
about the mass shifts of the vector mesons in the nuclear medium.
Several effective theories for the vector mesons
($\rho$, $\omega$)\,\cite{SMS}
predicts decreasing masses in the nuclear medium, and
the magnitude of the mass shifts is quite similar to the
QSR analysis in \cite{HL}.
Accordingly, the ``similarity'' and ``consistency''
between QSR in medium and
the effective theories has been erroneously advertized
in the literature\,\cite{Hatsuda, HLS}.
The essential ingredient of the mass shifts
predicted by those effective theories
is the polarization in the Dirac sea of the nuclear
medium, which leads to a smaller effective mass of the
nucleon in the nuclear medium.
If one switch off this effect, the vector meson propagators
receives only the effects of the Fermi sea of the nucleons,
which leads to small positive mass shifts of those vector
mesons\,\cite{Chin}. One has to recognize that
the physical effect which QSR for the vector mesons in medium
is enjoying is simply the scattering with this Fermi sea
of the nucleons (through the forward scattering amplitude
with the nucleon) which has the same mass as in the vacuum,
and accordingly the QSR in medium does not pick up any effect
of the polarization of the Dirac sea of the nucleons.
Similarity in prediction on the mass shift between the medium
QSR\,\cite{HL} and
the effective theories\,\cite{SMS} looks fortuitous and
rather causes new problems.
As has been clarified in this work, the medium QSR
presented by \cite{HL} should be interpreted as
a QCD sum rule analysis on the vector current-nucleon forward amplitude,
and should not be interpreted as a method which picks up
an effect of the vacuum polarization due the finite baryon number
density. It is misleading to celebrate the medium QSR
in \cite{HL} as
a tool to incorporate the
effect of ``change of QCD vacuum''
due to the finite baryon density.
\vskip 0.5cm
\centerline{\bf Acknowledgement}
We thank O. Morimatsu for useful comments on the manuscript.
\newpage
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\section{Introduction}
\label{sec:intro}
An interacting electron gas in one dimension has many unusual
properties, such as the spin-charge separation, the power law of
correlation functions, and the linear dependence of the electron
relaxation rate on temperature and frequency (see Ref.\
\cite{Firsov85} for a review). These one-dimensional (1D) results are
well established, in many cases exactly, by applying a variety of
mathematical methods including the Bethe Ansatz, the bosonization, and
the parquet, or the renormalization group. To distinguish the exotic
behavior of the 1D electron gas from a conventional Fermi-liquid
behavior, Haldane introduced a concept of the so-called Luttinger
liquid \cite{Haldane81}.
The discovery of high-$T_c$ superconductivity renewed interest in
the Luttinger-liquid concept. Anderson suggested that a
two-dimensional (2D) electron gas behaves like the 1D Luttinger
liquid, rather than a conventional Fermi liquid \cite{Anderson92}. It
is difficult to verify this claim rigorously, because the methods that
prove the existence of the Luttinger liquid in 1D cannot be applied
directly to higher dimensions. The Bethe Ansatz construction does not
work in higher dimensions. The bosonization in higher dimensions
\cite{Haldane92,Khveshchenko93a,Khveshchenko94b,Marston93,Marston,Fradkin,LiYM95,Kopietz95}
converts a system of interacting electrons into a set of harmonic
oscillators representing the electron density modes. This procedure
replaces the exact $W_\infty$ commutation relations
\cite{Khveshchenko94b} with approximate boson commutators, which is a
questionable, uncontrolled approximation. On the other hand, the
parquet method, although not being as exact as the two other methods,
has the advantage of being formulated as a certain selection rule
within a standard many-body diagram technique; thus, it can be applied
to higher dimensions rather straightforwardly. The parquet method has
much in common with the renormalization-group treatment of Fermi
liquids \cite{Shankar94}.
The 1D electron gas has two types of potential instabilities: the
superconducting and the density-wave, which manifest themselves
through logarithmic divergences of the corresponding one-loop
susceptibilities with decreasing temperature. Within the parquet
approach, a sum of an infinite series of diagrams, obtained by adding
and inserting the two basic one-loop diagrams into each other, is
calculated by solving a system of nonlinear differential equations,
which are nothing but the renormalization-group equations
\cite{Solyom79}. This procedure was developed for the first time for
meson scattering \cite{Diatlov57} and later was successfully applied
to the 1D electron gas \cite{Bychkov66,Dzyaloshinskii72a}, as well as
to the Kondo problem \cite{Abrikosov} and the X-ray absorption edge
problem \cite{Nozieres69a}. By considering both the superconducting
and the density-wave instabilities on equal footing and adequately
treating their competition, the parquet approximation differs from a
conventional ladder (or mean-field) approximation, commonly applied in
higher dimensions, where only one instability is taken into account.
Under certain conditions in the 1D case, the superconducting and
density-wave instabilities may cancel each other, giving rise to a
non-trivial metallic ground state at zero temperature, namely the
Luttinger liquid. In this case, the parquet derivation shows that the
electron correlation functions have a power-law structure, which is
one of the characteristic properties of the Luttinger liquid
\cite{Dzyaloshinskii72a,Larkin73}. One may conclude that the
competition between the superconducting and density-wave instabilities
is an important ingredient of the Luttinger liquid theory.
In a generic higher-dimensional case, where density-wave
instability does not exist or does not couple to superconducting
instability because of corrugation of the Fermi surface, the parquet
approach is not relevant. Nevertheless, there are a number of
higher-dimensional models where the parquet is applicable and produces
nontrivial results. These include the models of multiple chains
without single-electron hopping \cite{Gorkov74} and with
single-electron hopping but in a magnetic field \cite{Yakovenko87}, as
well as the model of an isotropic electron gas in a strong magnetic
field \cite{Brazovskii71,Yakovenko93a}. In all of these models, the
electron dispersion law is 1D, which permits to apply the parquet
method; at the same time, the interaction between electrons is
higher-dimensional, which makes a nontrivial difference from the
purely 1D case. The particular version of the parquet method used in
these cases is sometimes called the ``fast'' parquet, because, in
addition to a ``slow'', renormalization-group variable, the parquet
equations acquire supplementary, ``fast'' variables, which label
multiple electron states of the same energy.
Taking into account these considerations, it seems natural to start
exploring a possibility of the Luttinger liquid behavior in higher
dimensions by considering a model that combines 1D and
higher-dimensional features. This is the model of an electron gas
whose Fermi surface has flat regions on its opposite sides. The
flatness means that within these regions the electron dispersion law
is 1D: The electron energy depends only on the one component of
momentum that is normal to the flat section. On the other hand, the
size of the flat regions is finite, and that property differentiates
the model from a purely 1D model, where the size is infinite, since
nothing depends on the momenta perpendicular to the direction of a 1D
chain. A particular case of the considered model is one where the 2D
Fermi surface has a square shape. This model describes 2D electrons
on a square lattice with the nearest-neighbor hopping at the half
filling. It is a simplest model of the high-$T_c$ superconductors.
The model has already attracted the attention of
theorists. Virosztek and Ruvalds studied the ``nested Fermi liquid''
problem within a ladder or mean-field approximation
\cite{Ruvalds90,Ruvalds95}. Taking into account the 1D experience,
this approach may be considered questionable, because it does not
treat properly the competition between the superconducting and the
density-wave channels. Houghton and Marston \cite{Marston93} mapped
the flat parts of the Fermi surface onto discrete points. Such an
oversimplification makes all scattering processes within the flat
portion equivalent and artificially enhances the electron interaction.
Mattis \cite{Mattis87} and Hlubina \cite{Hlubina94} used the
bosonization to treat the interaction between the electron density
modes and claimed to solve the model exactly. However, mapping of the
flat Fermi surface onto quantum chains and subsequent bosonization by
Luther \cite{Luther94} indicated that the treatment of Mattis and
Hlubina is insufficient, because the operators of backward and umklapp
scattering on different quantum chains require a consistent
renormalization-group treatment. Luther did not give solution to this
problems, as well as he missed the interaction between the electrons
located on four different quantum chains.
In the present paper, we solve the model consistently, using the
fast parquet approach, where all possible instabilities occurring in
the electron system with the flat regions on the Fermi surface are
treated simultaneously. This approach was applied to the problem
earlier \cite{Dzyaloshinskii72b} in order to explain the
antiferromagnetism of chromium. In the present paper, we advance the
study further by including the order parameters of the odd symmetry,
missed in \cite{Dzyaloshinskii72b}, performing detailed numerical
calculations, and investigating the effect of a curvature of the Fermi
surface. To simplify numerical calculations and to relate to the
high-$T_c$ superconductors, we consider the 2D case, although the
method can be straightforwardly generalized to higher dimensions as
well.
We find that the presence of the boundaries of the flat portions of
the Fermi surface has a dramatic effect on the solutions of the
parquet equations. Even if the initial vertex of interaction between
electrons does not depend on the momenta along the Fermi surface
(which are the ``fast'' variables), the vertex acquires a strong
dependence on these variables upon renormalization, which greatly
reduces the feedback coupling between the superconducting and
density-wave channels relative to the 1D case. Instead of the two
channels canceling each other, the leading channel, which is the
spin-density-wave (SDW) in the case of the repulsive Hubbard
interaction, develops its own phase transition, inducing on the way a
considerable growth of the superconducting $d$-wave susceptibility.
At the same time, the feedback from the superconducting to the SDW
channel, very essential in the 1D case, is found negligible in the 2D
case. These results are in qualitative agreement with the picture of
the antiferromagnetically-induced $d$-wave superconductivity, which
was developed within a ladder approximation for the flat Fermi surface
in \cite{Ruvalds95} and for a generic nested Hubbard model in
\cite{Scalapino}. Recent experiments strongly suggest that the
high-$T_c$ superconductivity is indeed of the $d$-wave type
\cite{d-wave}. On the other hand, our results disagree with Refs.\
\cite{Mattis87,Hlubina94}. The origin of the discrepancy is that the
bosonization arbitrarily replaces the exact $W_\infty$ commutation
relations \cite{Khveshchenko94b} by approximate boson commutators;
thus, the renormalization of the electron-electron interaction, which
is an important part of the problem, becomes neglected.
In addition to having the flat sides, the square Fermi surface also
has sharp corners, where the saddle points of the electron dispersion
law, which produce the van Hove singularity in the density of states,
are located. The presence of the van Hove singularity at the Fermi
level enhances the divergence of the superconducting and density-wave
loops to the square of the temperature logarithm. The fast parquet
problem was formulated in this case in Ref.\ \cite{Dzyaloshinskii87a},
where the contribution from the flat sides, being less divergent than
the contribution from the saddle points, was neglected. The present
paper completes the study by considering a Fermi surface with the flat
sides and rounded corners, that is, without saddle points at the Fermi
level. Our physical conclusions for both models are in qualitative
agreement.
As photoemission experiments \cite{ZXShen93} demonstrate (see also
\cite{Ruvalds95}), many of the high-$T_c$ superconductors indeed have
flat regions on their Fermi surfaces. Hence, some of the results of
this paper may be applicable to these materials. However, the primary
goal of our study is to elucidate general theoretical concepts rather
than to achieve detailed description of real materials.
In order to distinguish the new features brought into the problems
by introducing higher dimensions, we present material in an inductive
manner. In Sec.\ \ref{sec:spinless}, we recall the derivation of the
parquet equations in the simplest case of 1D spinless electrons. In
Sec.\ \ref{sec:spin1D}, we generalize the procedure to the case of 1D
electrons with spin \cite{Bychkov66,Dzyaloshinskii72a}. Then, we
derive the parquet equations in the 2D case in Sec.\ \ref{sec:2D} and
solve them numerically in Sec.\ \ref{sec:numerical}. The paper ends
with conclusions in Sec.\ \ref{sec:conclusion}.
\section{Parquet Equations for One-Dimensional Spinless Fermions}
\label{sec:spinless}
Let us consider a 1D electron gas with a Fermi energy $\mu$ and a
generic dispersion law $\varepsilon(k_x)$, where $\varepsilon$ is the
energy and $k_x$ is the momentum of the electrons. As shown in Fig.\
\ref{fig:1D}, the Fermi surface of this system consists of two points
located at $k_x=\pm k_F$, where $k_F$ is the Fermi momentum. Assuming
that the two points are well separated, let us treat the electrons
whose momenta are close to $\pm k_F$ as two independent species and
label them with the index $\pm$. In the vicinity of the Fermi energy,
the dispersion laws of these electrons can be linearized:
\begin{equation}
\varepsilon_{\pm}(k_x) = \pm v_F k_x ,
\label{eps}
\end{equation}
where the momenta $k_x$ are counted from the respective Fermi points
$\pm k_F$ for the two species of the electrons, $\pm v_F$ are the
corresponding Fermi velocities, and the energy $\varepsilon$ is
counted from the chemical potential $\mu$.
First, let us consider the simplest case of electrons without spin.
The bare Hamiltonian of the interaction between the $\pm$ electrons,
$\hat{H}_{\rm int}$, can be written as
\begin{equation}
\hat{H}_{\rm int}= g \int\frac{dk_x^{(1)}}{2\pi}
\frac{dk_x^{(2)}}{2\pi}\frac{dk_x^{(3)}}{2\pi}
\hat{\psi}^+_+(k_x^{(1)}+k_x^{(2)}-k_x^{(3)})
\hat{\psi}^+_-(k_x^{(3)}) \hat{\psi}_-(k_x^{(2)})
\hat{\psi}_+(k_x^{(1)}),
\label{Interaction:spinless}
\end{equation}
where $g$ is the bare vertex of interaction, and the operators
$\hat{\psi}^+_\pm$ and $\hat{\psi}_\pm$ create and destroy the $\pm$
electrons.
The tendencies toward the superconducting or density-wave ($2k_F$)
instabilities in the system are reflected by the logarithmic
divergences of the two one-loop diagrams shown in Fig.\
\ref{fig:loops}, where the solid and dashed lines represent the Green
functions $G_+$ and $G_-$ of the $+$ and $-$ electrons, respectively.
The two diagrams in Fig.\ \ref{fig:loops} differ in the mutual
orientation of the arrows in the loops. In the Matsubara technique,
the integration of the Green functions over the internal momentum
$k_x$ and energy $\omega_n$ produces the following expressions for the
two diagrams:
\begin{eqnarray}
&&\pm T\sum_n\int\frac{dk_x}{2\pi}
G_-(\mp\omega_n,\mp k_x)G_+(\omega_n+\Omega_n,k_x+q_x)
\nonumber \\
&&=-T\sum_n\int\frac{dk_x}{2\pi}\frac{1}
{(i\omega_n+v_Fk_x)(i\omega_n+i\Omega_m -v_F(k_x+q_x))}
\nonumber \\
&&\approx\frac{1}{2\pi v_F}\ln\left(\frac{\mu}
{ \max\{T,|v_Fq_x|,|\Omega_m|\} } \right) \equiv \xi,
\label{1loop}
\end{eqnarray}
where the upper sign corresponds to the superconducting and the lower
to the density-wave susceptibility. In Eq.\ (\ref{1loop}), $T$ is the
temperature, $\Omega_m$ is the external energy passing through the
loop, and $q_x$ is the external momentum for the superconducting loop
and the deviation from $2k_F$ for the density-wave loop. With
logarithmic accuracy, the value of the integral (\ref{1loop}) is
determined by the upper and lower cutoffs of the logarithmic
divergence. In Eq.\ (\ref{1loop}), the upper and lower cutoffs are
written approximately, up to numerical coefficients of the order of
unity, whose logarithms are small compared to $\xi\gg1$. The variable
$\xi$, introduced by Eq.\ (\ref{1loop}), plays a very important role
in the paper. Since $\xi$ is the logarithm of the infrared cutoff,
the increase of $\xi$ represents renormalization toward low
temperature and energy.
The two primary diagrams of Fig.\ \ref{fig:loops} generate
higher-order corrections to the vertex of interaction between
electrons, $\gamma$, as illustrated in Fig.\ \ref{fig:sample}. In
this Figure, the dots represent the bare interaction vertex $g$,
whereas the renormalized vertex $\gamma$ is shown as a circle. The
one-loop diagrams in Fig.\ \ref{fig:sample} are the same as in Fig.\
\ref{fig:loops}. The first two two-loop diagrams in Fig.\
\ref{fig:sample} are obtained by repeating the same loop twice in a
ladder manner. The last two diagrams are obtained by inserting one
loop into the other and represent coupling between the two channels.
The diagrams obtained by repeatedly adding and inserting the two basic
diagrams of Fig.\ \ref{fig:loops} in all possible ways are called the
parquet diagrams. The ladder diagrams, where only the addition, but
not the insertion of the loops is allowed, represent a subset of the
more general set of the parquet diagrams. Selection of the parquet
diagrams is justified, because, as one can check calculating the
diagrams in Fig.\ \ref{fig:sample}, they form a series with the
expansion parameter $g\xi$: $\gamma=g\sum_{n=0}^\infty a_n(g\xi)^n$.
If the bare interaction vertex $g$ is small and the temperature is
sufficiently low, so that $\xi(T)$ is big, one can argue
\cite{Diatlov57,Bychkov66,Dzyaloshinskii72a} that nonparquet diagrams
may be neglected, because their expansion parameter $g$ is small
compared to the parquet expansion parameter $g\xi$.
Every diagram in Fig.\ \ref{fig:sample}, except the bare vertex $g$,
can be divided into two disconnected pieces by cutting one solid and one
dashed line, the arrows of the cut lines being either parallel or
antiparallel. The sum of those diagrams where the arrows of the cut
lines are parallel (antiparallel) is called the superconducting
(density-wave) ``brick''. Thus, the vertex $\gamma$ can be decomposed
into the bare vertex $g$, the superconducting brick $C$, and the
density-wave brick $Z$:
\begin{equation}
\gamma=g+C+Z.
\label{vertex:spinless}
\end{equation}
Eq.\ (\ref{vertex:spinless}) is illustrated in Fig.\
\ref{fig:SpinlessVertex}, where the bricks are represented as
rectangles whose long sides, one being a solid and another a dashed
line, represent the lines to be cut.
In a general case, the vertices and the bricks depend on the
energies and momenta
($\omega_1,\:\omega_2,\:\omega_3,\:v_Fk_x^{(1)},\:v_Fk_x^{(2)}$, and
$v_Fk_x^{(3)}$) of all incoming and outgoing electrons. Equations for
the bricks can be found in closed form in the case where all their
arguments are approximately equal within the logarithmic accuracy,
that is, the ratios of the arguments and of their linear combinations
are of the order of unity
\cite{Diatlov57,Bychkov66,Dzyaloshinskii72a}. Practically, this means
that all vertices and bricks are considered to be functions of the
single renormalization-group variable $\xi$, defined in Eq.\
(\ref{1loop}). It was proved in \cite{Diatlov57} that the two pieces
obtained by cutting a brick are the full vertices of interaction, as
illustrated graphically in Fig.\ \ref{fig:SpinlessBricks}.
Analytically, the equations for the bricks are
\begin{mathletters}%
\label{integral}
\begin{eqnarray}
C(\xi)&=&-\int_0^\xi d\zeta\,\gamma(\zeta)\gamma(\zeta),
\label{C}\\
Z(\xi)&=&\int_0^\xi d\zeta\,\gamma(\zeta)\gamma(\zeta).
\label{Z}
\end{eqnarray}
\end{mathletters}%
The two vertices $\gamma$ in the r.h.s.\ of Eqs.\ (\ref{integral})
represent the two pieces obtained from a brick by cutting, whereas the
integrals over $\zeta$ represent the two connecting Green functions
being integrated over the internal momentum and energy of the loop.
The value of the renormalized vertex $\gamma(\zeta)$ changes as the
integration over $\zeta$ progresses in Eqs.\ (\ref{integral}). In
agreement with the standard rules of the diagram technique \cite{AGD},
a pair of the parallel (antiparallel) lines in Fig.\
\ref{fig:SpinlessBricks} produces a negative (positive) sign in the
r.h.s.\ of Eq.\ (\ref{C}) [(\ref{Z})].
Eqs.\ (\ref{integral}) can be rewritten in differential,
renormalization-group form:
\begin{mathletters}%
\label{differential}
\begin{eqnarray}
&& \frac{dC(\xi)}{d\xi}=-\gamma(\xi)\gamma(\xi),
\quad\quad C(\xi\!\!=\!\!0)=0; \\
&& \frac{dZ(\xi)}{d\xi}=\gamma(\xi)\gamma(\xi),
\quad\quad Z(\xi\!\!=\!\!0)=0.
\end{eqnarray}
\end{mathletters}%
Combining Eqs.\ (\ref{differential}) with Eq.\
(\ref{vertex:spinless}), we find the renormalization equation for the
full vertex $\gamma$:
\begin{mathletters}%
\label{RG:spinless}
\begin{eqnarray}
&& \frac{d\gamma(\xi)}{d\xi}=\gamma(\xi)\gamma(\xi)
-\gamma(\xi)\gamma(\xi)=0,
\label{cancellation} \\
&& \gamma(\xi\!\!=\!\!0)=g.
\end{eqnarray}
\end{mathletters}%
We see that the two terms in the r.h.s.\ of Eq.\ (\ref{cancellation}),
representing the tendencies toward density-wave and superconducting
instabilities, exactly cancel each other. In a ladder approximation,
where only one term is kept in the r.h.s., the result would be quite
different, because $\gamma(\xi)$ would diverge at a finite $\xi$
indicating an instability or generation of a pseudogap in the system.
In order to study possible instabilities in the system, we need to
calculate corresponding generalized susceptibilities. For that
purpose, let us add to the Hamiltonian of the system two fictitious
infinitesimal external fields $h_{\rm SC}$ and $h_{\rm DW}$ that
create the electron-electron and electron-hole pairs:
\begin{eqnarray}
\hat{H}_{\rm ext}=\int\frac{dq_x}{2\pi}\frac{dk_x}{2\pi}&&\left[
h_{\rm SC}(q_x)\,\hat{\psi}^+_-\left(\frac{q_x}{2}-k_x\right)
\hat{\psi}^+_+\left(\frac{q_x}{2}+k_x\right) \right.
\nonumber \\
&&\left.{}+h_{\rm DW}(q_x)\,\hat{\psi}^+_-\left(k_x+\frac{q_x}{2}\right)
\hat{\psi}_+\left(k_x-\frac{q_x}{2}\right) + {\rm H.c.} \right].
\label{Hext}
\end{eqnarray}
Now we need to introduce triangular vertices ${\cal T}_{\rm SC}$
and ${\cal T}_{\rm DW}$ that represent the response of the system to
the fields $h_{\rm SC}$ and $h_{\rm DW}$. Following the same
procedure as in the derivation of the parquet equations for the bricks
\cite{Bychkov66,Dzyaloshinskii72a,Brazovskii71,Dzyaloshinskii72b}, we
find the parquet equations for the triangular vertices in graphic
form, as shown in Fig.\ \ref{fig:SpinlessTriangle}. In that Figure,
the filled triangles represent the vertices ${\cal T}_{\rm SC}$ and
${\cal T}_{\rm DW}$, whereas the dots represent the fields $h_{\rm
SC}$ and $h_{\rm DW}$. The circles, as in the other Figures,
represent the interaction vertex $\gamma$. Analytically, these
equations can be written as differential equations with given initial
conditions:
\begin{mathletters}%
\label{triangular}
\begin{eqnarray}
\frac{d{\cal T}_{\rm SC}(\xi)}{d\xi}=-\gamma(\xi)
{\cal T}_{\rm SC}(\xi),
&\quad\quad\quad& {\cal T}_{\rm SC}(0)=h_{\rm SC}; \\
\frac{d{\cal T}_{\rm DW}(\xi)}{d\xi}=\gamma(\xi)
{\cal T}_{\rm DW}(\xi),
&\quad\quad\quad& {\cal T}_{\rm DW}(0)=h_{\rm DW}.
\end{eqnarray}
\end{mathletters}%
We will often refer to the triangular vertices ${\cal T}$ as the
``order parameters''. Indeed, they are the superconducting and
density-wave order parameters induced in the system by the external
fields $h_{\rm SC}$ and $h_{\rm DW}$. If, for a finite $h_i$ ($i$=SC,
DW), a vertex ${\cal T}_i(\xi)$, which is proportional to $h_i$,
diverges when $\xi\rightarrow\xi_c$, this indicates that a {\em
spontaneous} order parameter appears in the system, that is, the order
parameter may have a finite value even when the external field $h_i$
is zero. The external fields are introduced here only as auxiliary
tools and are equal to zero in real systems. We also note that the
two terms in the r.h.s.\ of Eq.\ (\ref{Hext}) are not Hermitially
self-conjugate; thus, the fields $h_i$ are the complex fields.
Consequently, the order parameters ${\cal T}_i(\xi)$ are also complex,
so, generally speaking, ${\cal T}$ and ${\cal T}^*$ do not coincide.
According to Eqs.\ (\ref{RG:spinless}), $\gamma(\xi)=g$, so Eqs.\
(\ref{triangular}) have the following solution:
\begin{mathletters}%
\label{triangular:solutions}
\begin{eqnarray}
{\cal T}_{\rm SC}(\xi)&=&h_{\rm SC}\exp(-g\xi), \\
{\cal T}_{\rm DW}(\xi)&=&h_{\rm DW}\exp(g\xi).
\end{eqnarray}
\end{mathletters}%
Now we can calculate the susceptibilities. The lowest order
corrections to the free energy of the system due to the introduction
of the fields $h_{\rm SC}$ and $h_{\rm DW}$, $F_{\rm SC}$ and $F_{\rm
DW}$, obey the parquet equations shown graphically in Fig.\
\ref{fig:SpinlessSusceptibility} and analytically below:
\begin{mathletters}%
\label{FreeEnergy}
\begin{eqnarray}
F_{\rm SC}(\xi)&=&\int_0^\xi d\zeta\;{\cal T}_{\rm SC}(\zeta)
{\cal T}_{\rm SC}^*(\zeta), \\
F_{\rm DW}(\xi)&=&\int_0^\xi d\zeta\;{\cal T}_{\rm DW}(\zeta)
{\cal T}_{\rm DW}^*(\zeta).
\end{eqnarray}
\end{mathletters}%
Substituting expressions (\ref{triangular:solutions}) into Eqs.\
(\ref{FreeEnergy}) and dropping the squares of $h_{\rm SC}$ and
$h_{\rm DW}$, we find the susceptibilities:
\begin{mathletters}%
\label{susceptibilities}
\begin{eqnarray}
\chi_{\rm SC}(\xi)&=&-\bigm[\exp(-2g\xi)-1\bigm]/2g, \\
\chi_{\rm DW}(\xi)&=&\bigm[\exp(2g\xi)-1\bigm]/2g.
\end{eqnarray}
\end{mathletters}%
According to Eqs.\ (\ref{susceptibilities}), when the interaction
between electrons is repulsive (attractive), that is, $g$ is positive
(negative), the density-wave (superconducting) susceptibility increases
as temperature decreases ($T\rightarrow0$ and $\xi\rightarrow\infty$):
\begin{equation}
\chi_{\rm DW(SC)}(\xi)\propto\exp(\pm 2g\xi)
=\left(\frac{\mu}{ \max\{T,|v_Fq_x|,|\Omega_m|\} } \right)^{\pm2g}.
\label{PowerLaw}
\end{equation}
Susceptibilities (\ref{PowerLaw}) have power dependence on the
temperature and energy, which is one of the characteristic properties
of the Luttinger liquid. The susceptibilities are finite at finite
temperatures and diverge only at zero temperature, in agreement with
the general theorem \cite{Landau-V} that phase transitions are
impossible at finite temperatures in 1D systems. Mathematically, the
absence of divergence at finite $\xi$ is due to the cancellation of
the two terms in the r.h.s.\ of Eq.\ (\ref{cancellation}) and
subsequent nonrenormalization of $\gamma(\xi)$. This nontrivial 1D
result can be obtained only within the parquet, but not the ladder
approximation.
\section{Parquet Equations for One-Dimensional Fermions with Spin}
\label{sec:spin1D}
Now let us consider 1D electrons with spin. In this case, there
are three vertices of interaction, conventionally denoted as
$\gamma_1$, $\gamma_2$, and $\gamma_3$, which represent backward,
forward, and umklapp scattering, respectively
\cite{Bychkov66,Dzyaloshinskii72a}. Umklapp scattering should be
considered only when the change of the total momentum of the electrons
in the interaction process, $4k_F$, is equal to the crystal lattice
wave vector, which may or may not be the case in a particular model.
In this paper, we do not consider the vertex $\gamma_4$, which
describes the interaction between the electrons of the same type ($+$
or $-$), because this vertex does not have logarithmic corrections.
The bare Hamiltonian of the interaction, $\hat{H}_{\rm int}$, can be
written as
\begin{eqnarray}
\hat{H}_{\rm int}&=&\sum_{\sigma,\tau,\rho,\nu=\uparrow\downarrow}
\int\frac{dk_x^{(1)}}{2\pi}
\frac{dk_x^{(2)}}{2\pi}\frac{dk_x^{(3)}}{2\pi}
\nonumber \\
&& \times\biggm\{
(-g_1\delta_{\rho\tau}\delta_{\sigma\nu} +
g_2\delta_{\rho\nu}\delta_{\sigma\tau} )
\hat{\psi}^+_{\nu+}(k_x^{(1)}+k_x^{(2)}-k_x^{(3)})
\hat{\psi}^+_{\tau-}(k_x^{(3)})
\hat{\psi}_{\sigma-}(k_x^{(2)}) \hat{\psi}_{\rho+}(k_x^{(1)})
\nonumber \\
&& +\left[ g_3\delta_{\rho\nu}\delta_{\sigma\tau}
\hat{\psi}^+_{\nu-}(k_x^{(1)}+k_x^{(2)}-k_x^{(3)})
\hat{\psi}^+_{\tau-}(k_x^{(3)})
\hat{\psi}_{\sigma+}(k_x^{(2)})
\hat{\psi}_{\rho+}(k_x^{(1)}) + {\rm H.c.} \right]
\biggm\},
\label{Interaction}
\end{eqnarray}
where the coefficients $g_{1-3}$ denote the bare (unrenormalized)
values of the interaction vertices $\gamma_{1-3}$. The operators
$\hat{\psi}^+_{\sigma s}$ and $\hat{\psi}_{\sigma s}$ create and
destroy electrons of the type $s=\pm$ and the spin
$\sigma={\uparrow\downarrow}$. The spin structure of the interaction
Hamiltonian is dictated by conservation of spin. We picture the
interaction vertices in Fig.\ \ref{fig:interaction}, where the solid
and dashed lines represent the $+$ and $-$ electrons. The thin solid
lines inside the circles indicate how spin is conserved: The spins of
the incoming and outgoing electrons connected by a thin line are the
same. According to the structure of Hamiltonian (\ref{Interaction}),
the umklapp vertex $\gamma_3$ describes the process where two +
electrons come in and two -- electrons come out, whereas the complex
conjugate vertex $\gamma_3^*$ describes the reversed process.
The three vertices of interaction contain six bricks, as shown
schematically in Fig.\ \ref{fig:vertices}:
\begin{mathletters}%
\label{vertices}
\begin{eqnarray}
\gamma_1 &=& g_1+C_1+Z_1, \\
\gamma_2 &=& g_2+C_2+Z_2, \\
\gamma_3 &=& g_3+Z_I+Z_{II},
\end{eqnarray}
\end{mathletters}%
where $C_1$ and $C_2$ are the superconducting bricks, and $Z_1$,
$Z_2$, $Z_I$, and $Z_{II}$ are the density-wave bricks. In Fig.\
\ref{fig:vertices}, the thin solid lines inside the bricks represent
spin conservation. The umklapp vertex has two density-wave bricks
$Z_I$ and $Z_{II}$, which differ in their spin structure.
Parquet equations for the bricks are derived in the same manner as
in Sec.\ \ref{sec:spinless} by adding appropriate spin structure
dictated by spin conservation. It is convenient to derive the
equations graphically by demanding that the thin spin lines are
continuous, as shown in Fig.\ \ref{fig:bricks}. Corresponding
analytic equations can be written using the following rules. A pair
of parallel (antiparallel) lines connecting two vertices in Fig.\
\ref{fig:bricks} produces the negative (positive) sign. A closed loop
of the two connecting lines produces an additional factor $-2$ due to
summation over the two spin orientations of the electrons.
\begin{mathletters}%
\label{bricks}
\begin{eqnarray}
\frac{dC_1(\xi)}{d\xi} &=& -2\gamma_1(\xi)\:\gamma_2(\xi), \\
\frac{dC_2(\xi)}{d\xi} &=& -\gamma_1^2(\xi)-\gamma_2^2(\xi), \\
\frac{dZ_1(\xi)}{d\xi} &=& 2\gamma_1(\xi)\:\gamma_2(\xi)
-2\gamma_1^2(\xi), \\
\frac{dZ_2(\xi)}{d\xi} &=& \gamma_2^2(\xi)
+\gamma_3(\xi)\gamma_3^*(\xi), \\
\frac{dZ_I(\xi)}{d\xi} &=& 2\gamma_3(\xi)
[\gamma_2(\xi)-\gamma_1(\xi)], \\
\frac{dZ_{II}(\xi)}{d\xi} &=& 2\gamma_3(\xi)\:\gamma_2(\xi).
\end{eqnarray}
\end{mathletters}%
Combining Eqs.\ (\ref{vertices}) and (\ref{bricks}), we obtain the
well-known closed equations for renormalization of the vertices
\cite{Dzyaloshinskii72a}:
\begin{mathletters}%
\label{RG1D}
\begin{eqnarray}
\frac{d\gamma_1(\xi)}{d\xi} &=& -2\gamma^2_1(\xi), \\
\frac{d\gamma_2(\xi)}{d\xi} &=& -\gamma_1^2(\xi)
+\gamma_3(\xi)\gamma_3^*(\xi), \\
\frac{d\gamma_3(\xi)}{d\xi} &=& 2\gamma_3(\xi)
[2\gamma_2(\xi)-\gamma_1(\xi)].
\end{eqnarray}
\end{mathletters}%
In the presence of spin, the electron operators in Eq.\
(\ref{Hext}) and, correspondingly, the fields $h_i$ and the triangular
vertices ${\cal T}_i(\xi)$ acquire the spin indices. Thus, the
superconducting triangular vertex ${\cal T}_{\rm SC}(\xi)$ becomes a
vector:
\begin{equation}
{\cal T}_{\rm SC}(\xi) = \left( \begin{array}{c}
{\cal T}_{\rm SC}^{\uparrow \uparrow}(\xi) \\
{\cal T}_{\rm SC}^{\uparrow \downarrow}(\xi) \\
{\cal T}_{\rm SC}^{\downarrow \uparrow}(\xi) \\
{\cal T}_{\rm SC}^{\downarrow \downarrow}(\xi)
\end{array} \right).
\label{TSC}
\end{equation}
Parquet equations for the triangular vertices are given by the
diagrams shown in Fig.\ \ref{fig:SpinlessTriangle}, where the spin
lines should be added in the same manner as in Fig.\ \ref{fig:bricks}.
The superconducting vertex obeys the following equation:
\begin{equation}
\frac{d{\cal T}_{\rm SC}(\xi)}{d\xi} =
\Gamma_{\rm SC}(\xi)\;{\cal T}_{\rm SC}(\xi),
\label{MTRX}
\end{equation}
where the matrix $\Gamma_{\rm SC}(\xi)$ is
\begin{equation}
\Gamma_{\rm SC}(\xi) = \left( \begin{array}{cccc}
-\gamma_2 + \gamma_1 & 0 & 0 &0 \\
0 & -\gamma_2 & \gamma_1 & 0 \\
0 & \gamma_1 & -\gamma_2 & 0 \\
0 & 0 & 0 & -\gamma_2 + \gamma_1 \end{array} \right).
\label{GSC}
\end{equation}
Linear equation (\ref{MTRX}) is diagonalized by introducing the
singlet, ${\cal T}_{\rm SSC}$, and the triplet, ${\cal T}_{\rm TSC}$,
superconducting triangular vertices:
\begin{mathletters}%
\label{SC}
\begin{eqnarray}
{\cal T}_{\rm SSC}(\xi) &=&
{\cal T}_{\rm SC}^{\uparrow \downarrow}(\xi) -
{\cal T}_{\rm SC}^{\downarrow \uparrow}(\xi) ,
\label{SCS} \\
{\cal T}_{\rm TSC}(\xi) &=& \left( \begin{array}{c}
{\cal T}_{\rm SC}^{\uparrow \uparrow}(\xi) \\
{\cal T}_{\rm SC}^{\uparrow \downarrow}(\xi) +
{\cal T}_{\rm SC}^{\downarrow \uparrow}(\xi) \\
{\cal T}_{\rm SC}^{\downarrow \downarrow}(\xi)
\end{array} \right),
\label{SCT}
\end{eqnarray}
\end{mathletters}%
which obey the following equations:
\begin{equation}
\frac{d{\cal T}_{\rm SSC(TSC)}(\xi)}{d\xi} =
[\mp\gamma_1(\xi)-\gamma_2(\xi)]
\;{\cal T}_{\rm SSC(TSC)}(\xi).
\label{SCOP}
\end{equation}
In Eq.\ (\ref{SCOP}) the sign $-$ and the index SSC correspond to the
singlet superconductivity, whereas the sign $+$ and the index TSC
correspond to the triplet one. In the rest of the paper, we use the
index SC where discussion applies to both SSC and TSC.
Now let us consider the density-wave triangular vertices, first in
the absence of umklapp. They form a vector
\begin{equation}
{\cal T}_{\rm DW}(\xi) = \left( \begin{array}{c}
{\cal T}_{\rm DW}^{\uparrow \uparrow}(\xi) \\
{\cal T}_{\rm DW}^{\uparrow \downarrow}(\xi) \\
{\cal T}_{\rm DW}^{\downarrow \uparrow}(\xi) \\
{\cal T}_{\rm DW}^{\downarrow \downarrow}(\xi)
\end{array} \right),
\label{TDW}
\end{equation}
which obeys the equation
\begin{equation}
\frac{d{\cal T}_{\rm DW}(\xi)}{d\xi} =
\Gamma_{\rm DW}(\xi)\;{\cal T}_{\rm DW}(\xi)
\label{DWMTRX}
\end{equation}
with the matrix
\begin{equation}
\Gamma_{\rm DW}(\xi) = \left( \begin{array}{cccc}
-\gamma_1 + \gamma_2 & 0 & 0 &-\gamma_1 \\
0 & \gamma_2 & 0 & 0 \\
0 & 0 & \gamma_2 & 0 \\
-\gamma_1 & 0 & 0 & -\gamma_1 + \gamma_2 \end{array} \right).
\label{GDW}
\end{equation}
Eq.\ (\ref{DWMTRX}) is diagonalized by introducing the charge-,
${\cal T}_{\rm CDW}$, and the spin-, ${\cal T}_{\rm SDW}$, density-wave
triangular vertices:
\begin{mathletters}%
\label{DW}
\begin{eqnarray}
{\cal T}_{\rm CDW}(\xi) &=&
{\cal T}_{\rm DW}^{\uparrow \uparrow}(\xi) +
{\cal T}_{\rm DW}^{\downarrow \downarrow}(\xi) ,
\label{DWS} \\
{\cal T}_{\rm SDW}(\xi) &=& \left( \begin{array}{c}
{\cal T}_{\rm DW}^{\uparrow \downarrow}(\xi) \\
{\cal T}_{\rm DW}^{\downarrow \uparrow}(\xi) \\
{\cal T}_{\rm DW}^{\uparrow \uparrow}(\xi) -
{\cal T}_{\rm DW}^{\downarrow \downarrow}(\xi)
\end{array} \right),
\label{DWT}
\end{eqnarray}
\end{mathletters}%
which obey the following equations:
\begin{mathletters}%
\label{DWOP}
\begin{eqnarray}
\frac{d{\cal T}_{\rm CDW}(\xi)}{d\xi} &=&
[-2\gamma_1(\xi)+\gamma_2(\xi)]\;{\cal T}_{\rm CDW}(\xi), \\
\frac{d{\cal T}_{\rm SDW}(\xi)}{d\xi} &=&
\gamma_2(\xi)\;{\cal T}_{\rm SDW}(\xi).
\end{eqnarray}
\end{mathletters}%
When the umklapp vertices $\gamma_3$ and $\gamma_3^*$ are introduced,
they become offdiagonal matrix elements in Eqs.\ (\ref{DWOP}), mixing
${\cal T}_{\rm CDW}$ and ${\cal T}_{\rm SDW}$ with their complex
conjugates. Assuming for simplicity that $\gamma_3$ is real, we find
that the following linear combinations diagonalize the equations:
\begin{equation}
{\cal T}_{{\rm CDW(SDW)}\pm}={\cal T}_{\rm CDW(SDW)}
\pm {\cal T}^*_{\rm CDW(SDW)},
\label{DW+-}
\end{equation}
and the equations become:
\begin{mathletters}%
\label{DWOP+-}
\begin{eqnarray}
\frac{d{\cal T}_{{\rm CDW}\pm}(\xi)}{d\xi} &=&
[-2\gamma_1(\xi)+\gamma_2(\xi)\mp\gamma_3(\xi)]
\;{\cal T}_{{\rm CDW}\pm}(\xi), \\
\frac{d{\cal T}_{{\rm SDW}\pm}(\xi)}{d\xi} &=&
[\gamma_2(\xi)\pm\gamma_3(\xi)]\;{\cal T}_{{\rm SDW}\pm}(\xi).
\end{eqnarray}
\end{mathletters}%
If the external fields $h_i$ are set to unity in the initial
conditions of the type (\ref{triangular}) for all triangular vertices
$i$ = SSC, TSC, CDW$\pm$, and SDW$\pm$, then the corresponding
susceptibilities are equal numerically to the free energy corrections of
the type (\ref{FreeEnergy}):
\begin{equation}
\chi_i(\xi)= \int_0^\xi d\zeta\;
{\cal T}_i(\zeta){\cal T}_i^*(\zeta).
\label{chii}
\end{equation}
Eqs.\ (\ref{RG1D}), (\ref{SCOP}), (\ref{DWOP+-}), and (\ref{chii})
were solved analytically in Ref.\ \cite{Dzyaloshinskii72a}, where a
complete phase diagram of the 1D electron gas with spin was obtained.
\section{Parquet Equations for Two-Dimensional Electrons}
\label{sec:2D}
Now let us consider a 2D electron gas with the Fermi surface shown
schematically in Fig.\ \ref{fig:2DFS}. It contains two pairs of flat
regions, shown as the thick lines and labeled by the letters $a$ and
$b$. Such a Fermi surface resembles the Fermi surfaces of some
high-$T_c$ superconductors \cite{ZXShen93}. In our consideration, we
restrict the momenta of electrons to the flat sections only. In this
way, we effectively neglect the rounded portions of the Fermi surface,
which are not relevant for the parquet consideration, because the
density-wave loop is not divergent there. One can check also that the
contributions of the portions $a$ and $b$ do not mix with each other
in the parquet manner, so they may be treated separately. For this
reason, we will consider only the region $a$, where the 2D electron
states are labeled by the two momenta $k_x$ and $k_y$, the latter
momentum being restricted to the interval $[-k_y^{(0)},k_y^{(0)}]$.
In our model, the energy of electrons depends only on the momentum
$k_x$ according to Eq.\ (\ref{eps}) and does not depend on the
momentum $k_y$. We neglect possible dependence of the Fermi velocity
$v_F$ on $k_y$; it was argued in Ref.\ \cite{Luther94} that this
dependence is irrelevant in the renormalization-group sense.
In the 2D case, each brick or vertex of interaction between
electrons acquires extra variables $k_y^{(1)}$, $k_y^{(2)}$, and
$k_y^{(3)}$ in addition to the 1D variables
$\omega_1,\:\omega_2,\:\omega_3,\:v_Fk_x^{(1)},\:v_Fk_x^{(2)}$, and
$v_Fk_x^{(3)}$. These two sets of variables play very different
roles. The Green functions, which connect the vertices and produce
the logarithms $\xi$, depend only on the second set of variables.
Thus, following the parquet approach outlined in the previous
Sections, we dump all the $\omega$ and $v_Fk_x$ variables of a vertex
or a brick into a single variable $\xi$. At the same time, the
$k_y^{(1)}$, $k_y^{(2)}$, and $k_y^{(3)}$ variables remain independent
and play the role of indices labeling the vertices, somewhat similar
to the spin indices. Thus, each vertex and brick is a function of
several variables, which we will always write in the following order:
$\gamma(k_y^{(1)},k_y^{(2)};\:k_y^{(3)},k_y^{(4)};\:\xi)$. It is
implied that the first four variables satisfy the momentum
conservation law $k_y^{(1)}+k_y^{(2)}=k_y^{(3)}+k_y^{(4)}$, and each
of them belongs to the interval $[-k_y^{(0)},k_y^{(0)}]$. The
assignment of the variables $k_y^{(1)}$, $k_y^{(2)}$, $k_y^{(3)}$, and
$k_y^{(4)}$ to the ends of the vertices and bricks is shown in Fig.\
\ref{fig:vertices}, where the labels $k_j$ ($j=1-4$) should be
considered now as the variables $k_y^{(j)}$. To shorten notation, it
is convenient to combine these variable into a single four-component
vector
\begin{equation}
{\cal K}=(k_y^{(1)},k_y^{(2)};\:k_y^{(3)},k_y^{(4)}),
\label{K}
\end{equation}
so that the relation between the vertices and the bricks can be
written as
\begin{mathletters}%
\label{2Dgammas}
\begin{eqnarray}
\gamma_1({\cal K},\xi) &=&
g_1 + C_1({\cal K},\xi) + Z_1({\cal K},\xi),\\
\gamma_2({\cal K},\xi) &=&
g_2 + C_2({\cal K},\xi) + Z_2({\cal K},\xi),\\
\gamma_3({\cal K},\xi) &=&
g_3 + Z_I({\cal K},\xi) + Z_{II}({\cal K},\xi).
\end{eqnarray}
\end{mathletters}%
After this introduction, we are in a position to write the parquet
equations for the bricks. These equations are shown graphically in
Fig.\ \ref{fig:bricks}, where again the momenta $k_j$ should be
understood as $k_y^{(j)}$. Analytically, the equations are written
below, with the terms in the same order as in Fig.\ \ref{fig:bricks}:
\begin{mathletters}%
\label{2Dbricks}
\begin{eqnarray}
\frac{\partial C_1({\cal K},\xi)}{\partial \xi} &=&
-\gamma_1({\cal K}_1,\xi)\circ\gamma_2({\cal K}_1^{\prime},\xi) -
\gamma_2({\cal K}_1,\xi)\circ\gamma_1({\cal K}_1^{\prime},\xi),
\label{C1} \\
\frac{\partial C_2({\cal K},\xi)}{\partial \xi} &=&
-\gamma_1({\cal K}_1,\xi)\circ\gamma_1({\cal K}_1^{\prime},\xi)
-\gamma_2({\cal K}_1,\xi)\circ\gamma_2({\cal K}_1^{\prime},\xi),
\label{C2} \\
\frac{\partial Z_1({\cal K},\xi)}{\partial \xi} &=&
\gamma_1({\cal K}_2,\xi)\circ\gamma_2({\cal K}_2^{\prime},\xi) +
\gamma_2({\cal K}_2,\xi)\circ\gamma_1({\cal K}_2^{\prime},\xi)
- 2 \gamma_1({\cal K}_2,\xi)\circ\gamma_1({\cal K}_2^{\prime},\xi)
\nonumber \\
&& - 2 \tilde{\gamma}_3({\cal K}_2,\xi)
\circ\tilde{\bar{\gamma}}_3({\cal K}_2^{\prime},\xi)
+ \tilde{\gamma}_3({\cal K}_2,\xi)
\circ\bar{\gamma}_3({\cal K}_2^{\prime},\xi)
+ \gamma_3({\cal K}_2,\xi)
\circ\tilde{\bar{\gamma}}_3({\cal K}_2^{\prime},\xi),
\label{Z1} \\
\frac{\partial Z_2({\cal K},\xi)}{\partial \xi} &=&
\gamma_2({\cal K}_2,\xi)\circ\gamma_2({\cal K}_2^{\prime},\xi)
+\gamma_3({\cal K}_2,\xi)\circ\bar{\gamma}_3({\cal K}_2^{\prime},\xi),
\label{Z2} \\
\frac{\partial Z_I({\cal K},\xi)}{\partial \xi} &=&
\tilde{\gamma}_3({\cal K}_3,\xi)
\circ\gamma_2({\cal K}_3^{\prime},\xi)
+ \gamma_2({\cal K}_3,\xi)
\circ\tilde{\gamma}_3({\cal K}_3^{\prime},\xi) +
\gamma_1({\cal K}_3,\xi)\circ\gamma_3({\cal K}_3^{\prime},\xi)
\nonumber \\
&& + \gamma_3({\cal K}_3,\xi)\circ\gamma_1({\cal K}_3^{\prime},\xi)
- 2\tilde{\gamma}_3({\cal K}_3,\xi)
\circ\gamma_1({\cal K}_3^{\prime},\xi) -
2\gamma_1({\cal K}_3,\xi)
\circ\tilde{\gamma}_3({\cal K}_3^{\prime},\xi),
\label{ZI} \\
\frac{\partial Z_{II}({\cal K},\xi)}{\partial \xi} &=&
\gamma_3({\cal K}_2,\xi)\circ\gamma_2({\cal K}_2'',\xi)
+ \gamma_2({\cal K}_2,\xi)\circ\gamma_3({\cal K}_2'',\xi),
\label{ZII}
\end{eqnarray}
\end{mathletters}%
where
\begin{mathletters}%
\label{2DK}
\begin{eqnarray}
&& {\cal K}_1=(k_y^{(1)},k_y^{(2)};\:k_y^{(A)},k_y^{(B)}),\quad
{\cal K}_1'=(k_y^{(B)},k_y^{(A)};\:k_y^{(3)},k_y^{(4)}),\\
&& {\cal K}_2=(k_y^{(1)},k_y^{(B)};\:k_y^{(3)},k_y^{(A)}),\quad
{\cal K}_2'=(k_y^{(A)},k_y^{(2)};\:k_y^{(B)},k_y^{(4)}),\quad
{\cal K}_2''=(k_y^{(2)},k_y^{(A)};\:k_y^{(4)},k_y^{(B)}),\\
&& {\cal K}_3=(k_y^{(1)},k_y^{(B)};\:k_y^{(4)},k_y^{(A)}),\quad
{\cal K}_3'=(k_y^{(2)},k_y^{(A)};\:k_y^{(3)},k_y^{(B)}),
\end{eqnarray}
\end{mathletters}%
and the tilde and the bar operations are defined as
\begin{mathletters}%
\label{TildeBar}
\begin{eqnarray}
\tilde{\gamma}_j(k_y^{(1)},k_y^{(2)};\:k_y^{(3)},k_y^{(4)};\:\xi)
&\equiv&\gamma_j(k_y^{(1)},k_y^{(2)};\:k_y^{(4)},k_y^{(3)};\:\xi),
\label{tilde}\\
\bar{\gamma}_3(k_y^{(1)},k_y^{(2)};\:k_y^{(3)},k_y^{(4)};\:\xi)
&\equiv&\gamma_3^*(k_y^{(4)},k_y^{(3)};\:k_y^{(2)},k_y^{(1)};\:\xi).
\end{eqnarray}
\end{mathletters}%
In Eqs.\ (\ref{2Dbricks}), we introduced the operation $\circ$ that
represents the integration over the internal momenta of the loops in
Fig.\ \ref{fig:bricks}. It denotes the integration over the
intermediate momentum $k_y^{(A)}$ with the restriction that both
$k_y^{(A)}$ and $k_y^{(B)}$, another intermediate momentum determined
by conservation of momentum, belong to the interval
$[-k_y^{(0)},k_y^{(0)}]$. For example, the explicit form of the first
term in the r.h.s.\ of Eq.\ (\ref{C1}) is:
\begin{eqnarray}
&&\gamma_1({\cal K}_1,\xi)\circ\gamma_2({\cal K}_1^{\prime},\xi)=
\displaystyle
\int_{ -k_y^{(0)} \leq k_y^{(A)} \leq k_y^{(0)};\;\;
-k_y^{(0)} \leq k_y^{(1)}+k_y^{(2)}-k_y^{(A)} \leq k_y^{(0)} }
\frac{\displaystyle dk_y^{(A)}}{\displaystyle 2\pi}\,
\nonumber \\ && \times
\gamma_1(k_y^{(1)},k_y^{(2)};\:k_y^{(A)},k_y^{(1)}+k_y^{(2)}-k_y^{(A)};\:\xi)
\,
\gamma_2(k_y^{(1)}+k_y^{(2)}-k_y^{(A)},k_y^{(A)};\:k_y^{(3)},k_y^{(4)};\:\xi).
\label{o}
\end{eqnarray}
Eqs.\ (\ref{2Dbricks}) and (\ref{2Dgammas}) with definitions
(\ref{K}), (\ref{2DK}), and (\ref{TildeBar}) form a closed system of
integrodifferential equations, which will be solved numerically in
Sec.\ \ref{sec:numerical}. The initial conditions for Eqs.\
(\ref{2Dbricks}) and (\ref{2Dgammas}) are that all the $C$ and $Z$
bricks are equal to zero at $\xi=0$.
Parquet equations for the superconducting triangular vertices can
be found in the 2D case by adding the $k_y$ momenta to the 1D
equations (\ref{SCOP}). The equations are shown graphically in Fig.\
\ref{fig:SpinlessTriangle}, where the momenta $k$ and $q$ should be
interpreted as $k_y$ and $q_y$:
\begin{equation}
\frac{\partial{\cal T}_{\rm SSC(TSC)}(k_y,q_y,\xi)}{\partial\xi}
= f_{\rm SSC(TSC)}({\cal K}_{\rm SC},\xi)\circ
{\cal T}_{\rm SSC(TSC)}(k'_y,q_y,\xi),
\label{2DSCOP}
\end{equation}
where
\begin{eqnarray}
&& f_{\rm SSC(TSC)}({\cal K}_{\rm SC},\xi)=
\mp\gamma_1({\cal K}_{\rm SC},\xi)-
\gamma_2({\cal K}_{\rm SC},\xi),
\label{fSC} \\
&& {\cal K}_{\rm SC}=(k'_y+q_y/2,-k'_y+q_y/2; -k_y+q_y/2,k_y+q_y/2),
\end{eqnarray}
and the operator $\circ$ denotes the integration over $k'_y$ with the
restriction that both $k'_y+q_y/2$ and $-k'_y+q_y/2$ belong to the
interval $[-k_y^{(0)},k_y^{(0)}]$. The $\mp$ signs in front of
$\gamma_1$ in Eq.\ (\ref{fSC}) correspond to the singlet and triplet
superconductivity. As discussed in Sec.\ \ref{sec:spinless}, the
triangular vertex ${\cal T}_{\rm SC}(k_y,q_y,\xi)$ is the
superconducting order parameter, $q_y$ and $k_y$ being the
$y$-components of the total and the relative momenta of the electrons
in a Cooper pair. Indeed, the vertex ${\cal T}_{\rm SC}(k_y,q_y,\xi)$
obeys the linear equation shown in Fig.\ \ref{fig:SpinlessTriangle},
which is the linearized Gorkov equation for the superconducting order
parameter. As the system approaches a phase transition, the vertex
${\cal T}_{\rm SC}(k_y,q_y,\xi)$ diverges in overall magnitude, but
its dependence on $k_y$ for a fixed $q_y$ remains the same, up to a
singular, $\xi$-dependent factor. The dependence of ${\cal T}_{\rm
SC}(k_y,q_y,\xi)$ on $k_y$ describes the distribution of the emerging
order parameter over the Fermi surface. The numerically found
behavior of ${\cal T}_{\rm SC}(k_y,q_y,\xi)$ is discussed in Sec.\
\ref{sec:numerical}.
Due to the particular shape of the Fermi surface, the vertices of
interaction in our 2D model have two special symmetries: with respect to
the sign change of all momenta $k_y$ and with respect to the exchange of
the $+$ and $-$ electrons:
\begin{mathletters}%
\label{symmetry}
\begin{eqnarray}
\gamma_i(k_y^{(1)},k_y^{(2)};\:k_y^{(3)},k_y^{(4)};\:\xi) &=&
\gamma_i(-k_y^{(1)},-k_y^{(2)};\:-k_y^{(3)},-k_y^{(4)};\:\xi),
\quad i=1,2,3; \\
\gamma_i(k_y^{(1)},k_y^{(2)};\:k_y^{(3)},k_y^{(4)};\:\xi) &=&
\gamma_i(k_y^{(2)},k_y^{(1)};\:k_y^{(4)},k_y^{(3)};\:\xi),
\quad i=1,2,3; \\
\gamma_3(k_y^{(1)},k_y^{(2)};\:k_y^{(3)},k_y^{(4)};\:\xi) &=&
\gamma_3(k_y^{(4)},k_y^{(3)};\:k_y^{(2)},k_y^{(1)};\:\xi),
\label{*}
\end{eqnarray}
\end{mathletters}%
where in Eq.\ (\ref{*}) we assume for simplicity that $\gamma_3$ is
real. As a consequence of (\ref{symmetry}), Eqs.\ (\ref{2DSCOP}) are
invariant with respect to the sign reversal of $k_y$ in ${\cal T}_{\rm
SC}(k_y,q_y,\xi)$ at a fixed $q_y$. The following combinations of the
triangular vertices form two irreducible representations of this
symmetry, that is, they are independent and do not mix in Eqs.\
(\ref{2DSCOP}):
\begin{equation}
{\cal T}^\pm_{\rm SSC(TSC)}(k_y,q_y,\xi)=
{\cal T}_{\rm SSC(TSC)}(k_y,q_y,\xi)
\pm {\cal T}_{\rm SSC(TSC)}(-k_y,q_y,\xi).
\label{SASC}
\end{equation}
The triangular vertices ${\cal T}^\pm_{\rm SSC(TSC)}(k_y,q_y,\xi)$
describe the superconducting order parameters that are either
symmetric or antisymmetric with respect to the sign change of $k_y$.
When ${\cal T}^+_{\rm SSC}$ is extended over the whole 2D Fermi
surface (see Fig.\ \ref{fig:2DFS}), it acquires the $s$-wave symmetry,
whereas ${\cal T}^-_{\rm SSC}$ the $d$-wave symmetry. The symmetrized
vertices ${\cal T}^\pm_{\rm SSC(TSC)}(k_y,q_y,\xi)$ obey the same
Eqs.\ (\ref{2DSCOP}) as the unsymmetrized ones.
The equations for the density-wave triangular vertices are obtained
in a similar manner:
\begin{mathletters}%
\label{CSDWOPA}
\begin{eqnarray}
\frac{\partial{\cal T}_{{\rm CDW}\pm}^{\pm}(k_y,q_y,\xi)}
{\partial\xi} &=&
f_{{\rm CDW}\pm}({\cal K}_{\rm DW},\xi)\circ
{\cal T}_{{\rm CDW}\pm}^{\pm}(k'_y,q_y,\xi),
\label{CDWOPA} \\
\frac{\partial{\cal T}_{{\rm SDW}\pm}^{\pm}(k_y,q_y,\xi)}
{\partial\xi} &=&
f_{{\rm SDW}\pm}({\cal K}_{\rm DW},\xi)\circ
{\cal T}_{{\rm SDW}\pm}^{\pm}(k'_y,q_y,\xi),
\label{SDWOPA}
\end{eqnarray}
\end{mathletters}%
where
\begin{eqnarray}
&& f_{{\rm CDW}\pm}({\cal K}_{\rm DW},\xi)=
-2\gamma_1({\cal K}_{\rm DW},\xi)
\mp 2\tilde{\gamma}_3({\cal K}_{\rm DW},\xi)
+\gamma_2({\cal K}_{\rm DW},\xi)
\pm \gamma_3({\cal K}_{\rm DW},\xi),
\label{fCDW} \\
&& f_{{\rm SDW}\pm}({\cal K}_{\rm DW},\xi)=
\gamma_2({\cal K}_{\rm DW},\xi) \pm \gamma_3({\cal K}_{\rm DW},\xi),
\label{fSDW} \\
&& {\cal K}_{\rm DW} = (k'_y+q_y/2,k_y-q_y/2; k'_y-q_y/2,k_y+q_y/2).
\end{eqnarray}
The $\pm$ signs in the subscripts of ${\cal T}$ in Eqs.\
(\ref{CSDWOPA}) and in front of $\gamma_3$ in Eqs.\
(\ref{fCDW})--(\ref{fSDW}) refer to the umklapp symmetry discussed in
Sec.\ \ref {sec:spin1D}, whereas the $\pm$ signs in the superscripts
of ${\cal T}$ refer to the symmetry with respect to sign reversal of
$k_y$, discussed above in the superconducting case. The
$k_y$-antisymmetric density waves are actually the waves of charge
current and spin current \cite{Halperin68,Dzyaloshinskii87a}, also
known in the so-called flux phases \cite{FluxPhases}.
Once the triangular vertices ${\cal T}_i$ are found, the
corresponding susceptibilities $\chi_i$ are calculated according to
the following equation, similar to Eq.\ (\ref{chii}):
\begin{equation}
\chi_i(q_y,\xi)=\int_0^\xi d\zeta \int\frac{dk_y}{2\pi}
{\cal T}_i(k_y,q_y,\zeta){\cal T}_i^*(k_y,q_y,\zeta),
\label{2Dchii}
\end{equation}
where the integration over $k_y$ is restricted so that both
$k_y\pm q_y/2$ belong to the interval $[-k_y^{(0)},k_y^{(0)}]$.
Using functions (\ref{fSC}), (\ref{fCDW}), and (\ref{fSDW}) and
symmetries (\ref{symmetry}), we can rewrite Eqs.\ (\ref{2Dbricks}) in
a more compact form. For that purpose, we introduce the SSC, TSC,
CDW, and SDW bricks that are the linear combinations of the original
bricks:
\begin{mathletters}%
\label{NEWbricks}
\begin{eqnarray}
C_{\rm SSC(TSC)} &=& C_2 \pm C_1,
\label{CST} \\
Z_{{\rm CDW}\pm} &=& \tilde{Z}_2 - 2 \tilde{Z}_1
\pm (\tilde{Z}_{II} - 2 Z_I),
\label{ZCW} \\
Z_{{\rm SDW}\pm} &=& Z_2 \pm Z_{II},
\label{ZSW}
\end{eqnarray}
\end{mathletters}%
where the tilde operation is defined in Eq.\ (\ref{tilde}). Then,
Eqs.\ (\ref{2Dbricks}) become:
\begin{mathletters}%
\label{NEWRG}
\begin{eqnarray}
\frac{\partial C_{\rm SSC(TSC)}({\cal K},\xi)}{\partial \xi}
&=& -f_{\rm SSC(TSC)}({\cal K}_1,\xi)
\circ f_{\rm SSC(TSC)}({\cal K}_1^{\prime},\xi),
\label{SC1} \\
\frac{\partial Z_{{\rm CDW}\pm}({\cal K},\xi)}{\partial \xi}
&=& f_{{\rm CDW}\pm}({\cal K}_3,\xi)
\circ f_{{\rm CDW}\pm}({\cal K}_3^{\prime},\xi),
\label{CW1} \\
\frac{\partial Z_{{\rm SDW}\pm}({\cal K},\xi)}{\partial \xi}
&=& f_{{\rm SDW}\pm}({\cal K}_2,\xi)
\circ f_{{\rm SDW}\pm}({\cal K}_2^{\prime},\xi).
\label{SW1}
\end{eqnarray}
\end{mathletters}%
The parquet equations in the form (\ref{NEWRG}) were obtained in
Ref.\ \cite{Dzyaloshinskii72b}.
It is instructive to trace the difference between the parquet
equations (\ref{NEWRG}) and the corresponding ladder equations. Suppose
that, for some reason, only one brick, say $C_{\rm SSC}$, among the six
bricks (\ref{NEWbricks}) is appreciable, whereas the other bricks may be
neglected. Using definitions (\ref{2Dgammas}) and (\ref{fSC}), we find
that Eq.\ (\ref{SC1}) becomes a closed equation:
\begin{equation}
\frac{\partial f_{\rm SSC}({\cal K},\xi)}{\partial \xi} =
f_{\rm SSC}({\cal K}_1,\xi)\circ f_{\rm SSC}({\cal K}_1',\xi),
\label{fSCRG}
\end{equation}
where
\begin{equation}
f_{\rm SSC}({\cal K}_1,\xi)=-g_1-g_2-C_{\rm SSC}({\cal K},\xi).
\label{fSSC}
\end{equation}
Eq.\ (\ref{fSCRG}) is the ladder equation for the singlet
superconductivity. When the initial value $-(g_1+g_2)$ of the vertex
$f_{\rm SSC}$ is positive, Eq.\ (\ref{fSCRG}) has a singular solution
($f_{\rm SSC}\rightarrow\infty$ at $\xi\rightarrow\xi_c$), which
describes a phase transition into the singlet superconducting state at
a finite temperature. Repeating this consideration for every channel,
we construct the phase diagram of the system in the ladder
approximation as a list of necessary conditions for the corresponding
instabilities:
\begin{mathletters}%
\label{LadderPhaseDiagram}
\begin{eqnarray}
{\rm SSC:} & \quad & g_1+g_2<0, \\
{\rm TSC:} & \quad & -g_1+g_2<0, \\
{\rm CDW+:} & \quad & -2g_1+g_2-g_3>0, \\
{\rm CDW-:} & \quad & -2g_1+g_2+g_3>0, \\
{\rm SDW+:} & \quad & g_2+g_3>0, \\
{\rm SDW-:} & \quad & g_2-g_3>0.
\end{eqnarray}
\end{mathletters}%
The difference between the ladder and the parquet approximations
shows up when there are more than one appreciable bricks in the
problem. Then, the vertex $f_{\rm SSC}$ contains not only the brick
$C_{\rm SSC}$, but other bricks as well, so Eqs.\ (\ref{NEWRG}) get
coupled. This is the case, for example, for the 1D spinless
electrons, where the bricks $C$ and $Z$ are equally big, so they
cancel each other in $\gamma$ (see Sec.\ \ref{sec:spinless}).
\section{Results of Numerical Calculations}
\label{sec:numerical}
The numerical procedure consists of three consecutive steps; each of
them involves solving differential equations by the fourth-order
Runge--Kutta method. First, we solve parquet equations
(\ref{2Dgammas}) and (\ref{2Dbricks}) for the interaction vertices,
which are closed equations. Then, we find the triangular vertices
${\cal T}_i$, whose equations (\ref{2DSCOP}) and (\ref{CSDWOPA})
involve the interaction vertices $\gamma_i$ through Eqs.\ (\ref{fSC}),
(\ref{fCDW}), and (\ref{fSDW}). Finally, we calculate the
susceptibilities $\chi_i$ from Eqs.\ (\ref{2Dchii}), which depend on
the triangular vertices ${\cal T}_i$.
We select the initial conditions for the interaction vertices to be
independent of the transverse momenta ${\cal K}$: $\gamma_i({\cal
K},\,\xi\!\!=\!\!0) = g_i$. The momentum-independent interaction
naturally appears in the Hubbard model, where the interaction is local
in real space. In this Chapter, the results are shown mostly for the
repulsive Hubbard model without umklapp: $g_1=g_2=g,\;g_3=0$ (Figs.\
\ref{fig:GammaData}--\ref{fig:PhaseDiagram110}), or with umklapp:
$g_1=g_2=g_3=g$ (Figs.\ \ref{fig:chi111}--\ref{fig:PhaseDiagram111}),
where $g$ is proportional the Hubbard interaction constant $U$. The
absolute value of $g$ (but not the sign of $g$) is not essential in
our calculations, because it can be removed from the equations by
redefining $\xi$ to $\xi'=|g|\xi$. After the redefinition, we
effectively have $|g|=1$ in the initial conditions. The actual value
of $|g|$ matters only when the logarithmic variable $\xi'$ is
converted into the temperature according to the formula
$T=\mu\exp(-2\pi v_F\xi'/|g|)$.
The initial independence of $\gamma_i({\cal K},\,\xi\!\!=\!\!0)$ on
${\cal K}$ does not imply that this property is preserved upon
renormalization. On the contrary, during renormalization,
$\gamma_i({\cal K},\xi)$ develops a very strong dependence on ${\cal
K}$ and may even change sign in certain regions of the ${\cal
K}$-space. We illustrate this statement in Fig.\ \ref{fig:GammaData}
by showing typical dependences of $\gamma_1({\cal K},\xi)$ and
$\gamma_2({\cal K},\xi)$ on the average momentum
$p_y=(k_y^{(1)}+k_y^{(2)})/2$ of the incoming electrons at $k_1=k_3$
and $k_2=k_4$ after some renormalization ($\xi = 1.4$). In Figs.\
\ref{fig:GammaData}--\ref{fig:TDWData}, the upper and lower limits on
the horizontal axes are the boundaries $\pm k_y^{(0)}$ of the flat
region on the Fermi surface, which are set to $\pm1$ without loss of
generality. One can observe in Fig.\ \ref{fig:GammaData} that the
electron-electron interaction becomes negative (attractive) at large
$p_y$, even though initially it was repulsive everywhere.
Mathematically, the dependence of $\gamma_i({\cal K},\xi)$ on
${\cal K}$ arises because of the finite limits of integration,
$[-k_y^{(0)},k_y^{(0)}]$, imposed on the variables $k_y^{(A)}$ and
$k_y^{(B)}$ in Eqs.\ (\ref{2Dbricks}). For example, in Eq.\
(\ref{C1}), when $p_y=(k_y^{(1)}+k_y^{(2)})/2$ equals zero,
$k_y^{(A)}$ may change from $-k_y^{(0)}$ to $k_y^{(0)}$ while
$k_y^{(B)}$ stays in the same interval. However, when $p_y>0$,
$k_y^{(A)}$ has to be confined to a narrower interval
$[-k_y^{(0)}+2p_y,k_y^{(0)}]$ to ensure that
$k_y^{(B)}=2p_y-k_y^{(A)}$ stays within $[-k_y^{(0)},k_y^{(0)}]$.
This difference in the integration range subsequently generates the
dependence of $\gamma_i({\cal K},\xi)$ on $p_y$ and, more generally,
on ${\cal K}$. Since many channels with different geometrical
restrictions contribute to $\partial\gamma_i({\cal
K},\xi)/\partial\xi$ in Eqs.\ (\ref{2Dbricks}), the resultant
dependence of $\gamma_i({\cal K},\xi)$ on the four-dimensional vector
${\cal K}$ is complicated and hard to visualize. Because of the
strong dependence of $\gamma_i({\cal K},\xi)$ on ${\cal K}$, it is not
possible to describe the 2D system by only three renormalizing charges
$\gamma_1(\xi)$, $\gamma_2(\xi)$, and $\gamma_3(\xi)$, as in the 1D
case. Instead, it is absolutely necessary to consider an infinite
number of the renormalizing charges $\gamma_i({\cal K},\xi)$ labeled
by the continuous variable ${\cal K}$. This important difference was
neglected in Ref.\ \cite{Marston93}, where the continuous variable
${\cal K}$ was omitted.
Having calculated $\gamma_i({\cal K},\xi)$, we solve Eqs.\
(\ref{2DSCOP}) and (\ref{CSDWOPA}) for the triangular vertices (the
order parameters) ${\cal T}(k_y,q_y,\xi)$, which depend on both the
relative ($k_y$) and the total ($q_y$) transverse momenta. We find
numerically that the order parameters with $q_y=0$ diverge faster than
those with $q_y\neq0$. This is a natural consequence of the
integration range restrictions discussed above. For this reason, we
discuss below only the order parameters with zero total momentum
$q_y=0$. We select the initial conditions for the symmetric and
antisymmetric order parameters in the form:
\begin{equation}
{\cal T}_i^+(k_y,\,\xi\!\!=\!\!0)=1,\quad
{\cal T}_i^-(k_y,\,\xi\!\!=\!\!0)=k_y.
\label{SA}
\end{equation}
In Figs.\ \ref{fig:TSCData} and \ref{fig:TDWData}, we present typical
dependences of the superconducting and density-wave order parameters
on the relative momentum $k_y$ at the same renormalization ``time''
$\xi = 1.4$ as in Fig.\ \ref{fig:GammaData}. The singlet
antisymmetric component (${\cal T}_{\rm SSC}^{-}$) dominates among the
superconducting order parameters (Fig.\ \ref{fig:TSCData}), whereas
the symmetric SDW order parameter (${\cal T}_{SDW}^+$) is the highest
in the density-wave channel (Fig.\ \ref{fig:TDWData}).
Having calculated the triangular vertices ${\cal T}$, we find the
susceptibilities from Eq.\ (\ref{2Dchii}). The results are shown in
Fig.\ \ref{fig:chi110}. The symmetric SDW has the fastest growing
susceptibility $\chi^+_{\rm SDW}$, which diverges at $\xi_{\rm
SDW}=1.76$. This divergence indicates that a phase transition from
the metallic to the antiferromagnetic state takes place at the
transition temperature $T_{\rm SDW}=\mu\exp(-2\pi v_F\xi_{\rm
SDW}/g)$. A similar result was obtained in Ref.\
\cite{Dzyaloshinskii72b} by analyzing the convergence radius of the
parquet series in powers of $g\xi$. In the ladder approximation, the
SDW instability would take place at $\xi_{\rm SDW}^{\rm lad}=1/g_2=1$,
as follows from Eqs.\ (\ref{fSDW}) and (\ref{SW1}). Since $\xi_{\rm
SDW}>\xi_{\rm SDW}^{\rm lad}$, the transition temperature $T_{\rm
SDW}$, calculated in the parquet approximation, is lower than the
temperature $T_{\rm SDW}^{\rm lad}$, calculated in the ladder
approximation: $T_{\rm SDW}<T_{\rm SDW}^{\rm lad}$. The parquet
temperature is lower, because competing superconducting and
density-wave instabilities partially suppress each other.
Thus far, we considered the model with ideally flat regions on the
Fermi surface. Suppose now that these regions are only approximately
flat. That is, they can be treated as being flat for the energies
higher than a certain value $E_{\rm cutoff}$, but a curvature or a
corrugation of the Fermi surface becomes appreciable at the smaller
energies $E<E_{\rm cutoff}$. Because of the curvature, the Fermi
surface does not have nesting for $E<E_{\rm cutoff}$; thus, the
density-wave bricks in the parquet equations (\ref{2Dbricks}) stop to
renormalize. Formally, this effect can be taken into account by
introducing a cutoff $\xi_{\rm cutoff}=(1/2\pi v_F)\ln(\mu/E_{\rm
cutoff})$, so that the r.h.s.\ of Eqs.\ (\ref{Z1})--(\ref{ZII}) for
the density-wave bricks are replaced by zeros at $\xi>\xi_{\rm
cutoff}$. At the same time, Eqs.\ (\ref{C1}) and (\ref{C2}) for the
superconducting bricks remain unchanged, because the curvature of the
Fermi surface does not affect the superconducting instability with
$q_y=0$. The change of the renormalization equations at $\xi_{\rm
cutoff}$ is not a completely rigorous way \cite{Luther88} to take into
account the Fermi surface curvature; however, this procedure permits
obtaining explicit results and has a certain qualitative appeal. For
a more rigorous treatment of the corrugated Fermi surface problem see
Ref.\ \cite{Firsov}.
In Fig.\ \ref{fig:chiCutoff}, we show the susceptibilities
calculated using the cutoff procedure with $\xi_{\rm cutoff}=1.4$.
The density-wave susceptibilities remain constant at $\xi>\xi_{\rm
cutoff}$. At the same time, $\chi_{\rm SSC}^-(\xi)$ diverges at
$\xi_{\rm SSC}^-=2.44$ indicating a transition into the singlet
superconducting state of the $d$-wave type. Thus, if the SDW
instability is suppressed, the system is unstable against formation of
the $d$-wave superconductivity. This result is in agreement with the
conclusions of Refs.\ \cite{Ruvalds95,Scalapino,Dzyaloshinskii87a}.
From our numerical results, we deduce that the dependence of
$\xi_{\rm SSC}^-$ on $\xi_{\rm cutoff}$ is linear: $\xi_{\rm
SSC}^-=a-b\,\xi_{\rm cutoff}$ with $b=2.06$, as shown in the inset to
Fig.\ \ref{fig:PhaseDiagram110}. Converting $\xi$ into energy in this
relation, we find a power law dependence:
\begin{equation}
T_{\rm SSC}^- \propto \frac{1}{E_{\rm cutoff}^b}.
\label{TCR1}
\end{equation}
Eq.\ (\ref{TCR1}) demonstrates that increasing the cutoff energy
$E_{\rm cutoff}$ decreases the temperature of the superconducting
transition, $T_{\rm SSC}^-$. Such a relation can be qualitatively
understood in the following way. There is no bare interaction in the
superconducting $d$-wave channel in the Hubbard model, so the
transition is impossible in the ladder approximation. The growth of
the superconducting $d$-wave correlations is induced by the growth of
the SDW correlations, because the two channels are coupled in the
parquet equations (\ref{NEWRG}). If $E_{\rm cutoff}$ is high, the SDW
correlations do not have enough renormalization-group ``time'' $\xi$
to develop themselves because of the early cutoff of the density-wave
channels; thus, $T_{\rm SSC}^-$ is low. Hence, decreasing $E_{\rm
cutoff}$ increases $T_{\rm SSC}^-$. However, when $E_{\rm cutoff}$
becomes lower than $T_{\rm SDW}$, the SDW instability overtakes the
superconducting one. Corresponding phase diagram is shown in Fig.\
\ref{fig:PhaseDiagram110}. Generally speaking, the phase diagram
plotted in the energy variables, as opposed to the logarithmic
variables $\xi$, may depend on the absolute value of the bare
interaction constant $|g|$. In Fig.\ \ref{fig:PhaseDiagram110}, we
placed the points for the several values of $g$ = 0.3, 0.4, and 0.5;
the phase boundary does not depend much on the choice of $g$. The
phase diagram of Fig.\ \ref{fig:PhaseDiagram110} qualitatively
resembles the experimental one for the high-$T_c$ superconductors,
where transitions between the metallic, antiferromagnetic, and
superconducting states are observed. The value of $E_{\rm cutoff}$
may be related to the doping level, which controls the shape of the
Fermi surface. Taking into account the crudeness of our
approximations, detailed agreement with the experiment should not be
expected.
We perform the same calculations also for the Hubbard model with
umklapp scattering ($g_1 = g_2 = g_3 =1$). As one can see in Fig.\
\ref{fig:chi111}, where the susceptibilities are shown, the umklapp
process does not modify the qualitative picture. The leading
instability remains the SDW of the symmetric type, which is now also
symmetric with respect to the umklapp scattering, whereas the next
leading instability is the singlet $d$-wave superconductivity. The
SDW has a phase transition at $\xi_{\rm SDW+}^+=0.54$, which is close
to the ladder result $\xi_{\rm SDW+}^{\rm lad}=1/(g_2+g_3)=0.5$. Some
of the susceptibilities in Fig.\ \ref{fig:chi111} coincide exactly,
which is a consequence of a special SU(2)$\times$SU(2) symmetry of the
Hubbard model at the half filling \cite{SO(4)}. The phase diagram
with the energy cutoff (Fig.\ \ref{fig:PhaseDiagram111}) is similar to
the one without umklapp (Fig.\ \ref{fig:PhaseDiagram110}), but the
presence of the umklapp scattering decreases the transition
temperature of the $d$-wave superconductivity.
An important issue in the study of the 1D electron gas is the
so-called $g$-ology phase diagram, which was constructed for the first
time by Dzyaloshinskii and Larkin \cite{Dzyaloshinskii72a}. They
found that, in some regions of the $(g_1,g_2,g_3)$ space, the 1D
electron system develops a charge or spin gap, which is indicated by
divergence of $\gamma_i(\xi)$ with increasing $\xi$. In the region
where none of the gaps develops, the Luttinger liquid exists. It is
interesting whether such a region may exist in our 2D model. To study
the phase diagram of the 2D system, we repeat the calculations,
systematically changing relative values of $g_1$, $g_2$, and $g_3$.
From the physical point of view, the relative difference of $g_1$,
$g_2$, and $g_3$ roughly mimics dependence of the interaction vertex
on the momentum transfer. As an example, we show the susceptibilities
in the case where $g_1=2$, $g_2=1$, and $g_3=0$ in Fig.\
\ref{fig:chi210}. In this case, the leading instabilities are
simultaneously the triplet superconductivity of the symmetric type
(TSC+) and the spin-density wave.
For all studied sets of $g_i$, we find that the leading
instabilities calculated in the parquet and the ladder approximations
always coincide. (We do not introduce the energy cutoff here.) Thus,
the parquet effects do not modify the $g$-ology phase diagram of the
2D model derived in the ladder approximation, even though the
transition temperatures in the parquet approximation are always lower
than those obtained in the ladder approximation. In that sense, the
parquet corrections are much less important in the 2D case than in the
1D case. From the mathematical point of view, this happens because a
leading divergent brick develops a strong dependence on the transverse
momenta ${\cal K}$ and acquires the so-called mobile pole structure
\cite{Gorkov74,Brazovskii71,Dzyaloshinskii72b}:
\begin{equation}
Z({\cal K},\xi)\propto\frac{1}{\xi_c({\cal K})-\xi}.
\label{MovingPole}
\end{equation}
The name ``mobile pole'' is given, because the position of the pole in
$\xi$ in Eq.\ (\ref{MovingPole}), $\xi_c({\cal K})$, strongly depends
on the momenta ${\cal K}$. It was shown in Refs.\
\cite{Brazovskii71,Gorkov74,Dzyaloshinskii72b} that, because of the
mobility of the pole, the leading channel decouples from the other
channels, and the parquet description effectively reduces to the
ladder one, as described at the end of Sec.\ \ref{sec:2D}. The phase
diagram of the 2D system in the ladder approximation is given by Eqs.\
(\ref{LadderPhaseDiagram}). It follows from Eqs.\
(\ref{LadderPhaseDiagram}) that every point in the $(g_1,g_2,g_3)$
space has some sort of instability. Thus, the Luttinger liquid,
defined as a nontrivial metallic ground state where different
instabilities mutually cancel each other, does not exist in the 2D
model.
Generally speaking, other models may have different types of
solutions of the fast parquet equations, such as immobile poles
\cite{Gorkov74} or a self-similar solution \cite{Yakovenko93a}, the
latter indeed describing some sort of a Luttinger liquid. In our
study of a 2D model with the van Hove singularities
\cite{Dzyaloshinskii87a}, we found a region in the $g$-space without
instabilities, where the Luttinger liquid may exist
\cite{Dzyaloshinskii}. However, we find only the mobile-pole
solutions in the present 2D model.
\section{Conclusions}
\label{sec:conclusion}
In this paper we derive and numerically solve the parquet equations
for the 2D electron gas whose Fermi surface contains flat regions.
The model is a natural generalization of the 1D electron gas model,
where the Luttinger liquid is known to exist. We find that, because
of the finite size of the flat regions, the 2D parquet equations
always develop the mobile pole solutions, where the leading
instability effectively decouples from the other channels. Thus, a
ladder approximation is qualitatively (but not necessarily
quantitatively) correct for the 2D model, in contrast to the 1D case.
Whatever the values of the bare interaction constants are, the 2D
system always develops some sort of instability. Thus, the Luttinger
liquid, defined as a nontrivial metallic ground state where different
instabilities mutually cancel each other, does not exist in the 2D
model, contrary to the conclusions of Refs.\
\cite{Mattis87,Hlubina94}.
In the case of the repulsive Hubbard model, the leading instability
is the SDW, i.e., antiferromagnetism \cite{Dzyaloshinskii72b}. If the
nesting of the Fermi surface is not perfect, the SDW correlations do
not develop into a phase transition, and the singlet superconductivity
of the $d$-wave type appears in the system instead. These results may
be relevant for the high-$T_c$ superconductors and are in qualitative
agreement with the findings of Refs.\
\cite{Ruvalds95,Scalapino,Dzyaloshinskii87a}.
In the bosonization procedure
\cite{Haldane92,Khveshchenko93a,Khveshchenko94b,Marston93,Marston,Fradkin,LiYM95,Kopietz95},
a higher-dimensional Fermi surface is treated as a collection of flat
patches. Since the results of our paper do not depend qualitatively on
the size of the flat regions on the Fermi surface, the results may be
applicable, to some extent, to the patches as well. Precise relation
is hard to establish because of the infinitesimal size of the patches,
their different orientations, and uncertainties of connections between
them. On the other hand, the bosonization procedure seems to be even
better applicable to a flat Fermi surface, which consists of a few big
patches. Mattis \cite{Mattis87} and Hlubina \cite{Hlubina94} followed
that logic and claimed that the flat Fermi surface model is exactly
solvable by the bosonization and represents a Luttinger liquid. The
discrepancy between this claim and the results our paper indicates
that some conditions must restrict the validity of the bosonization
approximations. Luther gave a more sophisticated treatment to the flat
Fermi surface problem by mapping it onto multiple quantum chains
\cite{Luther94}. He found that the bosonization converts the
interaction between electrons into the two types of terms, roughly
corresponding to the two terms of the sine-Gordon model: the
``harmonic'' terms $(\partial \varphi/\partial x)^2$ and the
``exponential'' terms $\exp(i\varphi)$, where $\varphi$ is a
bosonization phase. The harmonic terms can be readily diagonalized,
but the exponential terms require a consistent renormalization-group
treatment. If the renormalization-group equations were derived in the
bosonization scheme of \cite{Luther94}, they would be the same as the
parquet equations written in our paper, because the
renormalization-group equations do not depend on whether the boson or
fermion representation is used in their derivation \cite{Wiegmann78}.
Long time ago, Luther bosonized noninteracting electrons on a
curved Fermi surface \cite{Luther79}; however, the interaction between
the electrons remained intractable because of the exponential
terms. The recent bosonization in higher dimensions
\cite{Haldane92,Khveshchenko93a,Khveshchenko94b,Marston93,Marston,Fradkin,LiYM95,Kopietz95}
managed to reformulate the problem in the harmonic terms only. This is
certainly sufficient to reproduces the Landau description of sound
excitations in a Fermi liquid \cite{Landau-IX}; however, it may be not
sufficient to derive the electron correlation functions. The validity
of the harmonic approximation is hard to trace for a curved Fermi
surface, but considerable experience has been accumulated for the flat
Fermi surface models.
In the model of multiple 1D chains without single-electron
tunneling between the chains and with forward scattering between
different chains, the bosonization produces the harmonic terms only,
thus the model can be solved exactly
\cite{Larkin73,Gutfreund76b}. However, a slight modification of the
model by introducing backward scattering between different chains
\cite{Gorkov74,PALee77} or interaction between four different chains
\cite{Yakovenko87} adds the exponential terms, which destroy the exact
solvability and typically lead to a CDW or SDW instability. Even if no
instability occurs, as in the model of electrons in a high magnetic
field \cite{Yakovenko93a}, the fast parquet method shows that the
electron correlation functions have a complicated, nonpower structure,
which is impossible to obtain within the harmonic bosonization.
Further comparison of the fast parquet method and the bosonization in
higher dimensions might help to establish the conditions of
applicability of the two complementary methods.
The work at Maryland was partially supported by the NSF under Grant
DMR--9417451, by the Alfred P.~Sloan Foundation, and by the David and
Lucile Packard Foundation.
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proofpile-arXiv_065-648
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\section{introduction}
Recently, the importance of the parametric resonance is recognized in
the reheating phase of the inflationary model\cite{sht,kof}. Due to the coherent
oscillation of the background inflaton field, the fluctuation of the
non-linearly coupled boson field or the fluctuation of
self interacting inflaton field itself are amplified and the
catastrophic particle creation occurs. This effect drastically
changes the scenario of the reheating considered so far. When one
pay attention to the evolution of the metric
perturbation during the coherent oscillatory phase of the scalar
field dominated universe, a naive question arises:does the metric
perturbation undergo a influence of the parametric resonance by
the background oscillation of the scalar field?
It is well known that the cosmological perturbation of the scalar field
in the oscillatory stage has the problematic aspect\cite{ks}. If one try to
write down the evolution equation for the density contrast or the
Newtonian potential, the coefficient of the equation becomes singular
periodically in time due to the background oscillation of the scalar
field. This behavior is not real. It appears through the reduction of
the constrained system to the second order differential equation. But
it makes difficult to understand the behavior of the metric
perturbation in the oscillatory phase.
In this paper, we consider the evolution of the metric perturbation
using the gauge invariant variable introduced by Mukhanov. The
evolution equation for this variable has no singular behavior and is
suitable to apply the oscillatory phase of the scalar field.
We treat a spatially flat FRW universe with a minimally
coupled scalar field with a potential
\begin{equation}
V(\phi)=\frac{\lambda}{4}\phi_{0}^{4}
\left(\frac{\phi}{\phi_{0}}\right)^{2n},~~(n=1,2,\ldots)
\end{equation}
The background equations are
\begin{eqnarray}
H^{2}&=&\frac{\kappa}{3}(\frac{1}{2}\dot\phi^{2}+V(\phi)), \\
\dot H&=&-\frac{\kappa}{2}\dot\phi^{2}, \\
\ddot\phi&+&3H\dot\phi+V_{\phi}=0,
\end{eqnarray}
where $\kappa=8\pi G$.
The scalar field oscillates if the condition $\phi\llm_{pl}$ is
satisfied. In such a situation, we cannot have the exact solution, but
using the time averaged potential energy and
the kinetic energy have the relation
\begin{equation}
<\dot\phi^{2}>=2n<V(\phi)>,
\end{equation}
the scale factor and the Hubble parameter can be approximately expressed as
\begin{equation}
a(t)\approx\left(\frac{t}{t_{0}}\right)^{\frac{n+1}{3n}},
~~~H\approx\frac{n+1}{3n}\frac{1}{t}.
\end{equation}
We define new variables for the scalar field and the time:
\begin{eqnarray}
\eta&=&nt_{0}a^{\frac{3}{n+1}}=nt_{0}\left(\frac{t}{t_{0}}\right)^{1/n}, \\
\phi(t)&=&\phi_{0}a^{-\frac{3}{n+1}}\tilde\phi,
\end{eqnarray}
where $\tilde\phi=1$ at $t=t_{0}$ and $\phi_{0}$ is a initial value of the scalar
field($\phi_{0}\llm_{pl}$). Using the new variables, the evolution
equation of the scalar field becomes
\begin{equation}
\tilde\phi_{\eta\eta}+m^{2}n\tilde\phi^{2n-1}=0,~~~
m^{2}=\frac{\lambda}{2}\phi_{0}^{2}. \label{sc}
\end{equation}
For $n=1$(massive scalar field), $\tilde\phi=\cos(m\eta)$. For $n=2$,
$\tilde\phi=cn(\sqrt{\lambda}\phi_{0}\eta;\frac{1}{\sqrt{2}})$ where $cn$
is a elliptic function.
We use the gauge invariant variables to treat the perturbation whose
wavelength is larger than the horizon scale. We found that the most
convenient variable is
\begin{equation}
Q=\delta\phi-\frac{\dot\phi}{H}\psi=\delta\phi^{(g)}-\frac{\dot\phi}{H}\Psi,
\end{equation}
where $\psi$ is the perturbation of the three curvature, $\delta\phi^{(g)}$
is the gauge invariant variable for the scalar
field perturbation and $\Psi$ is the gauge invariant Newtonian
potential. For the zero curvature slice, $Q$ represents the fluctuation of the
scalar field. This variable $Q$ was first introduced by
Mukhanov\cite{muk}. As already mentioned, the coefficient of
the evolution equation for the Newtonian
potential or the gauge invariant density contrast becomes singular
because of the background oscillation of the scalar field.
But the evolution equation for $Q$
does not have such a singular behavior:
\begin{equation}
\ddot Q+3H\dot Q+\left[V_{\phi\phi}+\left(\frac{k}{a}\right)^{2}+
2\left(\frac{\dot H}{H}+3H\right)^{\cdot}\right]Q=0. \label{muk}
\end{equation}
The Newtonian potential and the variable $Q$ is connected by the
relation
\begin{equation}
-\frac{k^{2}}{a^{2}}\Psi=\frac{\kappa}{2}\frac{\dot\phi^{2}}{H}
\left(\frac{H}{\dot\phi}Q\right)^{\cdot}, \label{pot}
\end{equation}
and the gauge invariant density contrast $\Delta$ which is equal to
$\left(\frac{\delta\rho}{\rho}\right)$ on the co-moving time slice is
\begin{equation}
\Delta= \frac{\kappa}{3}
\frac{\dot\phi^{2}}{H^{3}}\left(\frac{H}{\dot\phi}Q\right)^{\cdot}.
\label{delta}
\end{equation}
To treat the eq.(\ref{muk}) more tractable, we change the variable
\begin{equation}
Q=a^{-\frac{3}{n+1}}\tilde Q,~~~
\eta=nt_{0}\left(\frac{t}{t_{0}}\right)^{1/n},
\end{equation}
Then
\begin{equation}
\tilde Q_{\eta\eta}+a^{\frac{6(n-1)}{n+1}}\left[
V_{\phi\phi}+\left(\frac{k}{a}\right)^{2}+2\left(\frac{\dot
H}{H}+3H\right)^{\cdot}-\frac{9n}{(n+1)^{2}}H^{2}-\frac{3}{n+1}\dot
H\right]\tilde Q=0. \label{muk2}
\end{equation}
Using the background equation, we can estimate the time dependence of
the each terms in this equation:
\begin{eqnarray}
&&V_{\phi\phi}=n(2n-1)m^{2}a^{-\frac{6(n-1)}{n+1}}\tilde\phi^{2n-2}\sim
O\left(\left(\frac{\eta}{t_{0}}\right)^{2-2n}\right), \nonumber \\
&&\left(\frac{\dot H}{H}+3H\right)^{\cdot}=
6Hn^{n}\left(\frac{\eta}{t_{0}}\right)^{1-n}\tilde\phi^{2n-1}\tilde\phi_{\eta}
\sim O\left(\left(\frac{\eta}{t_{0}}\right)^{1-2n}\right), \nonumber \\
&&H^{2},~\dot H\sim O\left(\left(\frac{\eta}{t_{0}}\right)^{-2n}\right). \nonumber
\end{eqnarray}
As we are considering the situation $\eta\geq t_{0}$, we can
neglect the $H^{2}, \dot H$ terms in eq.(\ref{muk2}) because they are
higher order in powers of $\left(1/\eta\right)$ compared to other terms. Our basic
equation for the gauge invariant variable becomes
\begin{equation}
\tilde Q_{\eta\eta}+\left[n(2n-1)m^{2}\tilde\phi^{2n-2}
+k^{2}a^{\frac{4(n-2)}{n+1}}
+4n(n+1)\frac{1}{\eta}\tilde\phi^{2n-1}\tilde\phi_{\eta}\right]\tilde Q=0.
\label{muk3}
\end{equation}
$\tilde\phi$ is the solution of eq.(\ref{sc}).
We first consider $n=1$ case(massive scalar). The background scalar
field is sinusoidal and given by
\begin{equation}
\tilde\phi=\cos(m\eta).
\end{equation}
Eq.(\ref{muk3}) becomes
\begin{equation}
\tilde Q_{\eta\eta}+\left[m^{2}+\left(\frac{k}{a}\right)^{2}-
\frac{8}{\eta}\sin(m\eta)\cos(m\eta)\right]\tilde Q=0.
\end{equation}
We introduce a dimensionless time variable $\tau=m\eta$,
\begin{equation}
\tilde Q_{\tau\tau}+\left[1+\left(\frac{k}{ma}\right)^{2}
-\frac{4}{\tau}\sin(2\tau)\right]\tilde Q=0, \label{ma1}
\end{equation}
where $a\propto\tau^{2/3}$. This equation has the same form as
the Mathieu equation:
\begin{equation}
Y_{\tau\tau}+\left[A-2q\sin(2\tau)\right]Y=0. \label{math}
\end{equation}
In our case the coefficient $A, q$ are time dependent functions
and the relation between $A$ and $q$ is
\begin{equation}
A=1+\left(\frac{mt_{0}}{2}\right)^{4/3}
\left(\frac{k}{m}\right)^{2}q^{4/3}.
\end{equation}
Using the stability/instability chart of the Mathieu equation, we can know
that the perturbation will have the effect of parametric resonance of
the first unstable band of the Mathieu function and
grows in time(see figure). We can derive its time evolution by solving
eq.(\ref{ma1}) using multi-time scale method\cite{nay}. We introduce a parameter
\begin{equation}
\epsilon=\frac{4}{\tau_{0}}.
\end{equation}
As the condition $\tau_{0}\gg 1$ is equivalent to the condition of coherent
oscillation, $\epsilon$ is small parameter. Rewrite the
eq.(\ref{ma1}) as
\begin{equation}
\tilde Q_{\tau\tau}+\left[1+2\epsilon\omega_{1}-
\epsilon\frac{\tau_{0}}{\tau}\sin(2\tau)\right]\tilde Q=0,
~~(\tau\ge\tau_{0})
\end{equation}
where $\omega_{1}=\frac{1}{2\epsilon}\left(\frac{k}{ma}\right)^{2}$.
We assume the condition $\left(\frac{k}{ma}\right)^{2}<1$ to be the
term $2\epsilon\omega_{1}$ small. This means we consider the
wavelength larger than the Compton length. We define
slow time scale $\tau_{n}=\epsilon^{n}\tau$. The time derivative with
respect to $\tau$ is replaced by
\begin{equation}
\frac{d}{d\tau}=D_{0}+\epsilon D_{1}+\cdots,
\end{equation}
where $D_{n}=\frac{\partial}{\partial\tau_{n}}$. We expand
\begin{equation}
\tilde Q=Q^{(0)}+\epsilon Q^{(1)}+\cdots.
\end{equation}
By substituting these expression to eq.(\ref{ma1}) and collect the
terms of each order of $\epsilon$. From the $O(\epsilon^{0})$ and
$O(\epsilon^{1})$, we have
\begin{eqnarray}
O(\epsilon^{0})&:&~~~D_{0}^{2}Q^{(0)}+Q^{(0)}=0, \label{mul1} \\
O(\epsilon^{1})&:&~~~D_{0}^{2}Q^{(1)}+Q^{(1)}=-\left(2D_{0}D_{1}Q^{(0)}+
2\omega_{1}Q^{(0)}-\frac{\tau_{0}}{\tau}\sin(2\tau)Q^{(0)}\right).
\label{mul2}
\end{eqnarray}
The solution of eq.(\ref{mul1}) is
\begin{equation}
Q^{(0)}={\cal A}(\tau_{1})e^{i\tau}+{\cal A}^{*}(\tau_{1})e^{-i\tau}. \label{e0}
\end{equation}
We substitute this to the right hand side of eq.(\ref{mul2}) and
demand that the secular term which is proportional to $e^{i\tau}$
vanishes. This gives the equation for the amplitude ${\cal A}$:
\begin{equation}
i\frac{\partial
{\cal A}}{\partial\tau_{1}}+\omega_{1}{\cal A}+i\frac{\tau_{0}}{4\tau}{\cal A}^{*}=0.
\end{equation}
Using the definition of $\epsilon$ and $\tau_{1}$, we have
\begin{equation}
i\frac{\partial
{\cal A}}{\partial\tau}+\frac{1}{2}\left(\frac{k}{ma}\right)^{2}{\cal A}
+\frac{i}{\tau}{\cal A}^{*}=0.
\end{equation}
Writing ${\cal A}=u+iv$($u,v$ are real), $u$ satisfies the following second
order differential equation:
\begin{equation}
u_{\tau\tau}+\frac{4}{3\tau}u_{\tau}+\left[
\frac{1}{4}\left(\frac{k}{ma}\right)^{4}-\frac{2}{3\tau}\right]u=0.
\label{u}
\end{equation}
Using the eq.(\ref{delta}) and (\ref{e0}),
\begin{equation}
\Delta\propto\tilde Q_{\eta}\tilde\phi_{\eta}-\tilde Q\tilde\phi_{\eta\eta}-
\frac{1}{\eta}\tilde Q_{\eta}\tilde\phi=u+O(\frac{u}{\eta}).
\end{equation}
Therefore the function $u$ is equal to the
gauge invariant density contrast $\Delta$ within the approximation we
are using here. The solution of eq.(\ref{u}) is
\begin{equation}
u=\tau^{-1/6}Z_{\pm
5/2}\left(\left(\frac{k}{a}\right)^{2}\frac{1}{mH}\right),
\end{equation}
where $Z_{\nu}$ is a Bessel function of order $\nu$.
We have the critical wavelength $\lambda_{J}=(mH)^{-1/2}$. The mode
whose wavelength is larger than $\lambda_{J}$ can grow. If the
wavelength is shorter than $\lambda_{J}$ initially, the wavelength is
stretched by the cosmic expansion and its exceeds the critical length.
We can see this behavior by using the chart of Mathieu function. The
trajectory which started from the stable region moves to the unstable
region. For $k\rightarrow 0$ limit,
\begin{eqnarray}
\Delta&\propto&\tau^{2/3}=a,~~\tau^{-1}=a^{-3/2}, \nonumber \\
\Psi&\propto&\hbox{constant},~~\tau^{-5/3}=\frac{H}{a}.
\end{eqnarray}
This behavior is the same as the perturbation in the dust dominated universe.
For $n\geq 2$, the scalar filed oscillation is not sinusoidal.
We start searching the solution of the equation for $y=\tilde\phi_{\eta}$:
\begin{equation}
y_{\eta\eta}+n(2n-1)m^{2}\tilde\phi^{2n-2}y=0. \label{y}
\end{equation}
Eq.(\ref{muk3}) reduces to this equation if the third and the forth
term do not exist. We approximate the solution of eq.(\ref{y}) by
sinusoidal function:$y=m\sin(cm\eta)$. $c$ is a some numerical
constant which defines the period of scalar field oscillation. Using
the equation for $\tilde\phi$, we have
\begin{eqnarray}
&&n(2n-1)m^{2}\tilde\phi^{2n-2}=-\frac{y_{\eta\eta}}{y}, \nonumber \\
&&2n(n+1)m^{2}\tilde\phi^{2n-1}\tilde\phi_{\eta}
=(\tilde\phi_{\eta}\tilde\phi)_{\eta\eta}, \nonumber
\end{eqnarray}
and using $y=\tilde\phi_{\eta}=m\sin(cm\eta)$, the equation of $\tilde Q$ becomes
\begin{equation}
\tilde Q_{\eta\eta}+\left[c^{2}m^{2}-\frac{4cm}{\eta}\sin(2cm\eta)
+k^{2}a^{\frac{4(n-2)}{n+1}}\right]\tilde Q=0.
\end{equation}
By introducing the dimensionless time variable $\tau=cm\eta$,
\begin{equation}
\tilde Q_{\tau\tau}+\left[1-\frac{4}{\tau}\sin(2\tau)
+\left(\frac{k}{cm}\right)^{2}a^{\frac{4(n-2)}{n+1}}\right]\tilde Q=0.
\label{ma2}
\end{equation}
This is also Mathieu equation. It is surprising that this equation contains
the $n=1$ case if we set $c=1$. The function $A$ and $q$ are
\begin{equation}
A=1+\alpha\tau^{\frac{4(2-n)}{3}},~~q=\frac{2}{\tau},
\end{equation}
where $\alpha=\left(\frac{k}{cm}\right)^{2}(cnt_{0}m)^{4(n-2)/3}$ and
we have the relation
\begin{equation}
A(q)=1+\alpha\left(\frac{q}{2}\right)^{\frac{4(2-n)}{3}}.
\end{equation}
Using the chart of Mathieu function, we find that the perturbation
also get the effect of the parametric resonance of the first unstable
band and grows. But as the time goes on, the trajectory moves from the
unstable region to the stable region and the perturbation will oscillate
with a constant amplitude.
To investigate these behavior, we introduce slow time variable and
derive the equation for slowly changing amplitude of $\tilde Q$. The
procedure is the completely same as $n=1$ case. The result is
\begin{eqnarray}
&&\tilde Q={\cal A}e^{i\tau}+{\cal A}^{*}e^{-i\tau}, \\
&&i\frac{\partial {\cal A}}{\partial\tau}+
\frac{1}{2}\left(\frac{k}{cm}\right)^{2}a^{\frac{4(n-2)}{n+1}}{\cal
A}
+\frac{i}{\tau}{\cal A}^{*}=0.
\end{eqnarray}
The real part of ${\cal A}$ obeys
\begin{equation}
u_{\tau\tau}-\frac{4}{3}(n-2)\frac{1}{\tau}u_{\tau}
+\left[\frac{\alpha^{2}}{4}\tau^{\frac{8}{3}(n-2)}
-\left(\frac{4}{3}n-\frac{2}{3}\right)\frac{1}{\tau^{2}}\right]u=0.
\end{equation}
The solution of this equation is
\begin{equation}
u=\tau^{\frac{1}{6}(4n-5)}Z_{\nu}\left(\frac{3\alpha}{2(4n-5)}
\tau^{\frac{1}{3}(4n-5)}\right),~~\nu=\pm\frac{4n+1}{2(4n-5)}.
\end{equation}
For $k\rightarrow 0$ limit, we have
\begin{eqnarray}
\Delta&\propto&\tau^{\frac{2}{3}(2n-1)}=a^{\frac{2(2n-1)}{n+1}},
~~\tau^{-1}=a^{-\frac{3}{n+1}}, \\
\Psi&\propto&\hbox{constant},~~\tau^{-\frac{1}{3}(4n+1)}=\frac{H}{a}.
\end{eqnarray}
In summary, we found that the evolution equation of the Mukhanov's
gauge invariant variable in the oscillatory phase of the scalar
field can be reduced to the Mathieu equation and
the evolution of this variable undergoes the effect of the parametric
resonance. We can interpret the growth of the density perturbation in
this phase is caused by the parametric resonance.
Now we comment on previous works. In paper \cite{ns}, the analysis is
done by using the Newtonian approximation which means the wavelength
of the perturbation is smaller than the horizon length. But the
obtained equation for $\left(\delta\rho/\rho\right)$ coincides with
the result of this paper(eq.(\ref{u})). In paper \cite{kh}, the long
wave approximation is used. As the eq.(\ref{muk}) has the exact
solution for $k=0$, they take in the effect of small $k$ perturbatively
and derive the evolution of the gauge invariant variables whose
wavelength is larger than the horizon length.
The assumption we used in this paper is the wavelength is larger than
the Compton length which is smaller than the horizon scale in the
oscillatory phase of the scalar field.
So our treatment is
more general. Extension to the non-linearly
interacting two scalar field system which is a realistic model of the
reheating is straightforward and the analysis is now going on.
We will show the result in a separate publication.
\newpage
\begin{center}
{\Large FIGURE}
\end{center}
The stability/instability chart of the Mathieu equation
(\ref{math}). The grey is stable and the white is unstable
region. Three curves shows the time evolution of the parameter $A, q$
for the power index $n$ of the scalar field potential
$V=\frac{\lambda}{4}\phi_{0}^{4}\left(\frac{\phi}{\phi_{0}}\right)^{2n}$.
$A=1$ line corresponds to $k=0$.
\newpage
\thispagestyle{empty}
\vspace*{22 cm}
\special{epsf=fig_reheat.eps}
|
proofpile-arXiv_065-649
|
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
}
|
\section{Introduction}
Many ``connection-dynamic'' theories of gravity with propagating torsion
have been proposed in the last decades. Contrary to the usual
Einstein-Cartan (EC) gravity\cite{hehl}, in such theories one could
in principle
have long-range torsion mediated interactions.
In the same period, we have also witnessed a spectacular progress in the
experimental description of the solar system\cite{will}.
Many important tests using
the parameterized post-Newtonian (PPN) formalism have been performed. Tight
limits for the PPN parameters have been establishing and
several alternatives
theories to General Relativity (GR)
have been ruled out. Indeed, such solar system experiments
and also observations of the binary pulsar $1913+16$ offer strong evidence
that the metric tensor must not deviate too far from the predictions of
GR\cite{will}.
Unfortunately, the situation with respect to the torsion tensor is
much more obscure. The interest in experimental consequences of propagating
torsion models has been revived recently\cite{CF,hamm}.
Carroll and Field\cite{CF} have examined the
observational consequences of propagating torsion in a wide class of
models involving scalar fields. They conclude that for reasonable models
the torsion must decay quickly outside matter distribution, leading to
no long-range interaction which could be detected experimentally.
Nevertheless, as also stressed by them, this does not mean that torsion
has not relevance in Gravitational Physics.
Typically, in
propagating torsion models the Einstein-Hilbert action is modified in
order to induce a differential equation for the torsion tensor, allowing for
non-vanishing torsion configurations to the vacuum. In almost all cases
a dynamical scalar field is involved, usually related to the torsion
trace or pseudo-trace. Such modifications are introduced in a
rather arbitrary way; terms are added to the Lagrangian in
order to produce previously desired differential
equations for the torsion tensor.
The goal of this paper is to present a propagating torsion model
obtained from first principles of EC theory. By exploring some basic
features of the Einstein-Hilbert action in spacetimes with torsion we
get a model with a new and a rather intriguing type of propagating torsion
involving a non-minimally coupled scalar field.
We write and discuss the metric and torsion equations for the vacuum
and in the presence of different matter fields.
Our model does not belong to
the large class of models studied in \cite{CF}.
The work is organized as follows. Section II is a brief revision of
Riemann-Cartan (RC)
geometry, with special emphasis to the concept of parallel volume
element. In the Section III, we show how a propagating torsion model
arises from elementary considerations on the compatibility between
minimal action principle and minimal coupling procedure. The Section
IV is devoted to study of the proposed model in the vacuum and in presence of
various type of matter. Section V is left to some concluding
remarks.
\section{RC manifolds and parallel volume elements}
A RC spacetime is a differentiable four dimensional
manifold endowed with a metric tensor $g_{\alpha\beta}(x)$ and with a
metric-compatible connection $\Gamma_{\alpha\beta}^\mu$, which is
non-symmetrical in its lower indices. We adopt in this work
${\rm sign}(g_{\mu\nu})=(+,-,-,-)$.
The anti-symmetric part of the
connection defines a new tensor, the torsion tensor,
\begin{equation}
S_{\alpha\beta}^{\ \ \gamma} = \frac{1}{2}
\left(\Gamma_{\alpha\beta}^\gamma-\Gamma_{\beta\alpha}^\gamma \right).
\label{torsion}
\end{equation}
The metric-compatible connection can be written as
\begin{equation}
\Gamma_{\alpha\beta}^\gamma = \left\{_{\alpha\beta}^\gamma \right\}
- K_{\alpha\beta}^{\ \ \gamma},
\label{connection}
\end{equation}
where $\left\{_{\alpha\beta}^\gamma \right\}$ are
the usual Christoffel symbols
and $K_{\alpha\beta}^{\ \ \gamma}$ is the
contorsion tensor, which is given in terms of the torsion tensor by
\begin{equation}
K_{\alpha\beta}^{\ \ \gamma} = - S_{\alpha\beta}^{\ \ \gamma}
+ S_{\beta\ \alpha}^{\ \gamma\ } - S_{\ \alpha\beta}^{\gamma\ \ }.
\label{contorsion}
\end{equation}
The connection (\ref{connection}) is used to define the covariant derivative of
vectors,
\begin{equation}
D_\nu A^\mu = \partial_\nu A^\mu + \Gamma_{\nu\rho}^\mu A^\rho,
\label{covariant}
\end{equation}
and
it is also important to our purposes to introduce the covariant derivative of
a density $f(x)$,
\begin{equation}
D_\mu f(x) = \partial_\mu f(x) - \Gamma^\rho_{\rho\mu}f(x).
\end{equation}
The contorsion tensor (\ref{contorsion}) can be covariantly
split in a traceless part and in a trace,
\begin{equation}
K_{\alpha\beta\gamma} = \tilde{K}_{\alpha\beta\gamma} -
\frac{2}{3}\left(
g_{\alpha\gamma} S_\beta - g_{\alpha\beta} S_\gamma
\right),
\label{decomposit}
\end{equation}
where $\tilde{K}_{\alpha\beta\gamma}$ is the traceless part and $S_\beta$ is
the trace of the torsion tensor, $S_\beta = S^{\ \ \alpha}_{\alpha\beta}$.
In four dimensions the traceless part $\tilde{K}_{\alpha\beta\gamma}$
can be also decomposed in a pseudo-trace and a part with vanishing
pseudo-trace, but for our purposes (\ref{decomposit}) is sufficient.
The curvature tensor is given by:
\begin{equation}
\label{curva}
R_{\alpha\nu\mu}^{\ \ \ \ \beta} = \partial_\alpha \Gamma_{\nu\mu}^\beta
- \partial_\nu \Gamma_{\alpha\mu}^\beta
+ \Gamma_{\alpha\rho}^\beta \Gamma_{\nu\mu}^\rho
- \Gamma_{\nu\rho}^\beta \Gamma_{\alpha\mu}^\rho .
\end{equation}
After some algebraic manipulations we get the following expression for the
scalar of curvature $R$, obtained from suitable contractions of (\ref{curva}),
\begin{equation}
R\left(g_{\mu\nu},\Gamma^\gamma_{\alpha\beta}\right) =
g^{\mu\nu} R_{\alpha\mu\nu}^{\ \ \ \ \alpha} =
{\cal R} - 4D_\mu S^\mu + \frac{16}{3}S_\mu S^\mu -
\tilde{K}_{\nu\rho\alpha} \tilde{K}^{\alpha\nu\rho},
\label{scurv}
\end{equation}
where ${\cal R}\left(g_{\mu\nu},\left\{_{\alpha\beta}^\gamma \right\}\right)$
is the Riemannian scalar of curvature, calculated from the
Christoffel symbols.
In order to define a general covariant volume element in a manifold, it is
necessary to introduce a density quantity $f(x)$ which will compensate
the Jacobian that arises from the transformation law
of the usual volume element
$d^4x$ under a coordinate transformation,
\begin{equation}
d^4x \rightarrow f(x) d^4x = d{\rm vol}.
\end{equation}
Usually, the density $f(x) = \sqrt{-g}$ is took to this purpose. However,
there are
natural properties that a volume element shall exhibit.
In a Riemannian manifold, the usual covariant volume element
\begin{equation}
d{\rm vol} = \sqrt{-g}\, d^4x,
\label{vele}
\end{equation}
is parallel, in the sense that the scalar density $\sqrt{-g}$ obeys
\begin{equation}
{\cal D}_\mu\sqrt{-g} = 0,
\end{equation}
where ${\cal D}_\mu$ is the covariant derivative defined using the
Christoffel symbols.
One can infer that the volume element (\ref{vele}) is not parallel
when the spacetime is not torsionless, since
\begin{equation}
D_\mu\sqrt{-g}= \partial_\mu\sqrt{-g} - \Gamma^\rho_{\rho\mu}\sqrt{-g} =
-2 S_\mu\sqrt{-g},
\end{equation}
as it can be checked using Christoffel symbols properties. This is the
main point that we wish to stress, it will be the basic argument to our claim
that the usual volume element (\ref{vele}) is not the most appropriate one
in the presence of torsion, as it will be discussed in the next section.
The question
that arises now is if it is possible to define a
parallel volume element in RC manifolds.
In order to do it,
one needs to find a density $f(x)$ such that
$D_\mu f(x)=0$. Such density exists only if the trace
of the torsion tensor, $S_\mu$, can be obtained
from a scalar potential\cite{saa1}
\begin{equation}
S_\beta(x) = \partial_\beta \Theta(x),
\label{pot}
\end{equation}
and in this case we have $f(x)=e^{2\Theta}\sqrt{-g}$, and
\begin{equation}
d{\rm vol} = e^{2\Theta}\sqrt{-g} \,d^4x,
\label{u4volume}
\end{equation}
that is the parallel RC volume element,
or in another words, the volume element (\ref{u4volume}) is compatible
with the connection in RC manifolds obeying (\ref{pot}).
It is not usual to find in the literature applications where volume
elements different from the canonical one are used.
Non-standard
volume elements have been used
in the characterization of half-flats solutions of
Einstein equations\cite{volu},
in the description of field theory on Riemann-Cartan
spacetimes\cite{saa1,saa2} and of dilatonic gravity\cite{saa4},
and in the study of some aspects of BRST symmetry\cite{AD}.
In our case the
new volume element appears naturally; in the same way that we require
compatibility conditions between the metric tensor and the linear
connection we can do it for the connection and volume element.
With
the volume element (\ref{u4volume}), we have the following generalized
Gauss' formula
\begin{equation}
\int d{\rm vol}\, D_\mu V^\mu =
\int d^4x \partial_\mu e^{2\Theta}\sqrt{-g} V^\mu =\
{\rm surface\ term},
\label{gauss}
\end{equation}
where we used
that
\begin{equation}
\label{gammacontr}
\Gamma^\rho_{\rho\mu}=\partial_\mu\ln e^{2\Theta}\sqrt{-g}
\end{equation}
under the hypothesis (\ref{pot}). It is easy to see that one cannot have a
generalized Gauss' formula of the type (\ref{gauss}) if the torsion does not
obey (\ref{pot}). We will return to discuss the actual role of the condition
(\ref{pot}) in the last section.
\section{Minimal coupling procedure and minimal action principle}
As it was already said, our model arises from elementary considerations
on the minimal coupling procedure and minimal action principle.
Minimal coupling procedure (MCP) provides us with an useful rule to get
the equations for any physical field on non-Minkowskian manifolds starting
from their versions of Special Relativity (SR). When studying classical
fields on a non-Minkowskian manifold $\cal X$ we usually require that the
equations of motion for such fields have an appropriate SR limit. There are,
of course, infinitely many covariant equations on $\cal X$ with the same
SR limit, and MCP solves this arbitrariness by saying that the relevant
equations should be the ``simplest'' ones. MCP can be heuristically formulated
as follows. Considering the equations of motion for a classical field in
the SR, one can get their version for a non-Minkowskian spacetime $\cal X$
by changing the partial derivatives by the $\cal X$ covariant ones and the
Minkowski metric tensor by the $\cal X$ one. MCP is also
used for the classical and quantum
analysis of gauge fields, where the gauge field is to be interpreted
as a connection, and it is in spectacular agreement with
experience for QED an QCD.
Suppose now that the SR equations of motion for a classical field follow
from an action functional via minimal action principle (MAP). It is natural to
expect that the equations obtained by using MCP to the SR equations
coincide with the Euler-Lagrange equations of the action obtained
via MCP of the SR one. This can be better visualized with the help of the
following diagram\cite{saa5}
\setlength{\unitlength}{1mm}
$$
\addtocounter{equation}{1}
\newlabel{diagr}{{3.1}{3.1}}
\hspace{106pt}
\begin{picture}(52,28)
\put(3,20) {$ {\cal C}_{ {\cal L}_{\rm SR} }$}
\put(7,18){\vector(0,-1){9}}
\put(3,5){$ E({\cal L}_{\rm SR}) $}
\put(45,20){${ \cal C_{L_X} }$ }
\put(40,5){$ E({\cal L}_{\cal X})$}
\put(47,18){\vector(0,-1){9}}
\put(12,22){\vector(1,0){30}}
\put(17,7){\vector(1,0){22}}
\put(24,24){${\scriptstyle \rm MCP}$}
\put(27,9){${\scriptstyle \rm MCP}$}
\put(8,13){${\scriptstyle \rm MAP}$}
\put(48,13){${\scriptstyle \rm MAP}$}
\end{picture}
\hspace{116pt}\raise 7ex \hbox{(\theequation)}
$$
where $E({\cal L})$ stands to the Euler-Lagrange equations for the
Lagrangian $\cal L$, and ${\cal C}_{\cal L}$ is the equivalence class
of Lagrangians, ${\cal L}'$ being equivalent to $\cal L$ if
$E({\cal L}')=E({\cal L})$.
We restrict ourselves to the case of non-singular Lagrangians.
The diagram (\ref{diagr}) is verified in GR. We say that MCP is
compatible with MAP if (\ref{diagr}) holds. We stress that if (\ref{diagr})
does not hold we have another arbitrariness to solve, one needs to choose
one between two equations, as we will shown with a simple example.
It is not difficulty to check that in general MCP is not compatible with MAP
when spacetime is assumed to be non-Riemannian.
Let us examine for simplicity
the case of a massless scalar field $\varphi$ in the frame of Einstein-Cartan
gravity\cite{saa1}. The equation for $\varphi$ in SR is
\begin{equation}
\partial_\mu\partial^\mu\varphi=0,
\label{e2}
\end{equation}
which follows from the extremals of the action
\begin{equation}
\label{act}
S_{\rm SR} =
\int d{\rm vol}\, \eta^{\mu\nu}\partial_\mu\varphi\partial_\nu\varphi.
\end{equation}
Using MCP to (\ref{act}) one gets
\begin{equation}
\label{act1}
S_{\cal X} = \int d{\rm vol}\, g^{\mu\nu}
\partial_\mu\varphi\partial_\nu\varphi,
\end{equation}
and using the Riemannian volume element for $\cal X$, $
d{\rm vol} = \sqrt{g}d^nx$, we get the following equation from the
extremals of (\ref{act1})
\begin{equation}
\label{aa22}
\frac{1}{\sqrt{g}}\partial_\mu \sqrt{g}\partial^\mu\varphi = 0.
\end{equation}
It is clear that (\ref{aa22}) does not coincide in general with the
equation obtained via MCP of (\ref{e2})
\begin{equation}
\label{e3}
\partial_\mu\partial^\mu\varphi + \Gamma^\mu_{\mu\alpha}
\partial^\alpha\varphi =
\frac{1}{\sqrt{g}}\partial_\mu \sqrt{g}\partial^\mu\varphi
+ 2 \Gamma^\mu_{[\mu\alpha]} \partial^\alpha\varphi = 0.
\end{equation}
We have here an ambiguity, the equations (\ref{aa22}) and (\ref{e3}) are in
principle equally acceptable ones, to choose one of them corresponds to choose
as more fundamental the equations of motion or the action formulation from
MCP point of view. As it was already said, we do not have such ambiguity
when spacetime is assumed to be
a Riemannian manifold. This
is not a feature of massless scalar fields, all matter fields have the
same behaviour in the frame of Einstein-Cartan gravity.
An accurate analysis of the diagram (\ref{diagr}) reveals that the source
of the problems of compatibility between MCP and MAP is the volume element
of $\cal X$. The necessary and sufficient condition to the validity of
(\ref{diagr}) is that the equivalence class of Lagrangians
${\cal C}_{\cal L}$ be preserved under MCP. With our definition of
equivalence we have that
\begin{equation}
\label{class}
{\cal C}_{ {\cal L}_{\rm SR} } \equiv \left\{ {\cal L}'_{\rm SR}|
{\cal L}'_{\rm SR} - {\cal L}_{\rm SR} = \partial_\mu V^\mu \right\},
\end{equation}
where $V^{\mu}$ is a vector field. The application of MCP to the
divergence $\partial_\mu V^\mu$ in (\ref{class}) gives $D_\mu V^\mu$,
and in order to the set
\begin{equation}
\left\{ {\cal L}'_{\cal X}|
{\cal L}'_{\cal X} - {\cal L}_{\cal X} = D_\mu V^\mu \right\}
\end{equation}
be an equivalence class one needs to have a Gauss-like law like
(\ref{gauss}) associated to
the divergence $D_\mu V^\mu$.
As it was already said in Section II, the necessary and sufficient
condition to have such a Gauss law is that the trace of the torsion
tensor obeys (\ref{pot}).
With the use of the parallel volume element in the
action formulation for EC gravity we can have qualitatively
different predictions. The scalar of curvature (\ref{scurv})
involves terms quadratic in the torsion.
Due to (\ref{pot}) such quadratic terms will provide a differential
equation for $\Theta$, what will allow for non-vanishing torsion
solutions for the vacuum.
As to the matter fields, the
use of the parallel volume element, besides of guarantee
that the diagram (\ref{diagr}) holds, brings also qualitative changes.
For example, it is possible to have a minimal interaction between
Maxwell fields and torsion preserving gauge symmetry. The next section
is devoted to the study of EC equations obtained by using the
parallel volume element (\ref{u4volume}).
\section{The model}
Now, EC gravity will be reconstructed by
using the results of the previous sections. Spacetime will be
assumed to be a Riemann-Cartan
manifold with the parallel volume element (\ref{u4volume}), and of course,
it is implicit the restriction that the trace of the torsion tensor is
derived from a scalar potential, condition (\ref{pot}).
With this hypothesis, EC theory of gravity will predict new effects, and they
will be pointed out in the following subsections.
\subsection{Vacuum equations}
According to our hypothesis,
in order to get the EC gravity equations we will assume that they
can be obtained from an Einstein-Hilbert action using the scalar of
curvature (\ref{scurv}), the condition (\ref{pot}), and the
volume element (\ref{u4volume}),
\begin{eqnarray}
\label{vaction}
S_{\rm grav} &=& -\int d^4x e^{2\Theta} \sqrt{-g} \, R \\
&=&-\int d^4x e^{2\Theta} \sqrt{-g} \left(
{\cal R} + \frac{16}{3} \partial_\mu\Theta \partial^\mu \Theta
- \tilde{K}_{\nu\rho\alpha} \tilde{K}^{\alpha\nu\rho}
\right) + {\rm surf. \ terms}, \nonumber
\end{eqnarray}
where the generalized Gauss' formula (\ref{gauss}) was used.
The equations for the $g^{\mu\nu}$, $\Theta$, and
$\tilde{K}_{\nu\rho\alpha}$ fields follow from the extremals of the action
(\ref{vaction}).
The variations of $g^{\mu\nu}$ and $S_{\mu\nu}^{\ \ \rho}$ are assumed to
vanish in the boundary.
The equation $\frac{\delta S_{\rm grav}}{\delta\tilde{K}_{\nu\rho\alpha}} =0$
implies that $\tilde{K}^{\nu\rho\alpha} = 0$,
$\frac{\delta S_{\rm grav}}{\delta\tilde{K}_{\nu\rho\alpha}}$ standing for the
Euler-Lagrange equations for
${\delta\tilde{K}_{\nu\rho\alpha}}$.
For the other equations we have
\begin{eqnarray}
\label{1st}
-\frac{e^{-2\Theta}}{\sqrt{-g}}
\left.\frac{\delta }{\delta g^{\mu\nu}}S_{\rm grav}
\right|_{\tilde{K}=0} &=& {\cal R}_{\mu\nu}
-2D_\mu \partial_\nu\Theta \nonumber \\
&&-\frac{1}{2}g_{\mu\nu}
\left(
{\cal R} + \frac{8}{3}\partial_\rho\Theta \partial^\rho \Theta
-4 \Box \Theta
\right) = 0, \\
-\frac{e^{-2\Theta}}{2\sqrt{-g}}
\left.\frac{\delta }{\delta \Theta}S_{\rm grav}
\right|_{\tilde{K}=0} &=&
{\cal R} + \frac{16}{3}\left(
\partial_\mu\Theta \partial^\mu \Theta -
\Box \Theta \right) =0, \nonumber
\end{eqnarray}
where
${\cal R}_{\mu\nu}
\left(g_{\mu\nu},\left\{_{\alpha\beta}^\gamma \right\}\right)$
is the usual Ricci tensor, calculated using the
Christoffel symbols, and $\Box = D_\mu D^\mu$.
Taking the trace of the first equation of (\ref{1st}),
\begin{equation}
{\cal R} + \frac{16}{3}\partial_\mu\Theta \partial^\mu \Theta =
6\Box\Theta,
\end{equation}
and using it, one finally obtains
the equations for the vacuum,
\begin{eqnarray}
\label{vacum0}
{\cal R}_{\mu\nu} &=& 2D_\mu\partial_\nu \Theta
- \frac{4}{3} g_{\mu\nu}\partial_\rho\Theta \partial^\rho \Theta
= 2D_\mu S_\nu - \frac{4}{3}g_{\mu\nu}S_\rho S^\rho, \nonumber \\
\Box \Theta &=& \frac{e^{-2\Theta}}{\sqrt{-g}}
\partial_\mu e^{2\Theta}\sqrt{-g}\partial^\mu\Theta = D_\mu S^\mu = 0, \\
\tilde{K}_{\alpha\beta\gamma} &=& 0. \nonumber
\end{eqnarray}
The vacuum equations (\ref{vacum0})
point out new features of our model. It is
clear that torsion, described by the last two equations,
propagates.
The torsion mediated interactions are not of
contact type anymore. The traceless tensor $\tilde{K}_{\alpha\beta\gamma}$
is zero for the vacuum, and only the trace $S_\mu$ can be non-vanishing
outside matter distributions. As it is expected, the gravity field
configuration for the vacuum is determined only
by boundary conditions, and if
due to such conditions we have that $S_\mu=0$, our equations reduce to the
usual vacuum equations, $S_{\alpha\gamma\beta}=0$, and
${\cal R}_{\alpha\beta}=0$. Note that this is the case if one considers
particle-like solutions (solutions that go to zero asymptotically).
Equations (\ref{vacum0}) are valid only to the exterior region of the
sources. For a discussion to the case with sources see \cite{H1}.
The first term in the right-handed side of the first equation
of (\ref{vacum0}) appears
to be non-symmetrical under the change $(\mu\leftrightarrow\nu)$,
but in fact it is symmetrical as one can see using (\ref{pot}) and
the last equation of (\ref{vacum0}). Of course that if
$\tilde{K}_{\alpha\beta\gamma}\neq 0$ such term will be non-symmetrical,
and this is the case when fermionic fields are present, as we will see.
It is not difficult to generate solutions for (\ref{vacum0})
starting from the well-known solutions of the minimally coupled
scalar-tensor gravity\cite{saa6}.
\subsection{Scalar fields}
The first step to introduce matter fields in our discussion
will be the description of
scalar fields on RC manifolds.
In order to do it, we will use MCP according to Section II.
For a massless scalar field one gets
\begin{eqnarray}
\label{scala}
S &=& S_{\rm grav} + S_{\rm scal} = -\int \,d^4xe^{2\Theta}\sqrt{-g}
\left(R -\frac{g^{\mu\nu}}{2} \partial_\mu\varphi \partial_\nu \varphi
\right)\\
&=&-\int d^4x e^{2\Theta} \sqrt{-g} \left(
{\cal R} + \frac{16}{3} \partial_\mu\Theta \partial^\mu \Theta
- \tilde{K}_{\nu\rho\alpha} \tilde{K}^{\alpha\nu\rho}
-\frac{g^{\mu\nu}}{2} \partial_\mu\varphi \partial_\nu \varphi
\right), \nonumber
\end{eqnarray}
where surface terms were discarded.
The equations for this case are obtained by varying (\ref{scala}) with
respect to $\varphi$, $g^{\mu\nu}$, $\Theta$, and
$\tilde{K}_{\alpha\beta\gamma}$. As in the vacuum case, the equation
$\frac{\delta S}{\delta \tilde{K}}=0$
implies $\tilde{K}=0$. Taking it into
account we have
\begin{eqnarray}
\label{e1}
-\frac{e^{-2\Theta}}{\sqrt{-g}} \left.
\frac{\delta S}{\delta\varphi}
\right|_{\tilde{K}=0} &=& \frac{e^{-2\Theta}}{\sqrt{-g}}\partial_\mu
e^{2\Theta}\sqrt{-g}\partial^\mu\varphi
=\Box \varphi = 0, \nonumber \\
-\frac{e^{-2\Theta}}{\sqrt{-g}} \left.
\frac{\delta S}{\delta g^{\mu\nu}}
\right|_{\tilde{K}=0} &=& {\cal R}_{\mu\nu}
- 2 D_\mu S_\nu - \frac{1}{2} g_{\mu\nu}
\left(
{\cal R} + \frac{8}{3}S_\rho S^\rho - 4 D_\rho S^\rho
\right) \nonumber \\
&&-\frac{1}{2} \partial_\mu \varphi \partial_\nu\varphi
+ \frac{1}{4} g_{\mu\nu}\partial_\rho \varphi \partial^\rho \varphi = 0, \\
-\frac{e^{-2\Theta}}{2\sqrt{-g}} \left.
\frac{\delta S}{\delta \Theta}
\right|_{\tilde{K}=0} &=& {\cal R} +
\frac{16}{3}\left( S_\mu S^\mu - D_\mu S^\mu\right)
-\frac{1}{2} \partial_\mu\varphi \partial^\mu\varphi = 0. \nonumber
\end{eqnarray}
Taking the trace of the second equation of (\ref{e1}),
\begin{equation}
{\cal R} + \frac{16}{3} S_\mu S^\mu = 6 D_\mu S^\mu +
\frac{1}{2} \partial_\mu\varphi \partial^\mu \varphi,
\end{equation}
and using it, we get the following
set of equations for the massless scalar case
\begin{eqnarray}
\label{aa}
\Box \varphi &=& 0, \nonumber \\
{\cal R}_{\mu\nu} &=& 2D_\mu S_\nu - \frac{4}{3}g_{\mu\nu} S_\rho S^\rho
+\frac{1}{2} \partial_\mu\varphi \partial_\nu\varphi, \\
D_\mu S^\mu &=& 0, \nonumber \\
\tilde{K}_{\alpha\beta\gamma} &=& 0. \nonumber
\end{eqnarray}
As one can see, the torsion equations have the same form than the ones
of the vacuum case (\ref{vacum0}). Any
contribution to the torsion will be due to boundary conditions, and not due
to the scalar field itself.
It means that if such boundary conditions imply that $S_\mu=0$, the
equations for the fields $\varphi$ and $g_{\mu\nu}$ will be the same ones
of the GR.
One can interpret this by saying that,
even feeling the torsion (see the second equation of (\ref{aa})),
massless scalar fields do not produce it. Such behavior is
compatible with the idea that torsion must be governed by spin distributions.
However, considering massive scalar fields,
\begin{eqnarray}
S_{\rm scal} = \int \,d^4xe^{2\Theta}\sqrt{-g}
\left(\frac{g^{\mu\nu}}{2} \partial_\mu\varphi \partial_\nu \varphi
-\frac{m^2}{2}\varphi^2 \right),
\end{eqnarray}
we have the
following set of equations instead of (\ref{aa})
\begin{eqnarray}
\label{aa1}
(\Box+m^2) \varphi &=& 0, \nonumber \\
{\cal R}_{\mu\nu} &=& 2D_\mu S_\nu - \frac{4}{3}g_{\mu\nu} S_\rho S^\rho
+\frac{1}{2} \partial_\mu\varphi \partial_\nu\varphi
-\frac{1}{2} g_{\mu\nu} m^2\varphi^2, \\
D_\mu S^\mu &=& \frac{3}{4}m^2\varphi^2, \nonumber \\
\tilde{K}_{\alpha\beta\gamma} &=& 0. \nonumber
\end{eqnarray}
The equation for the trace of the torsion tensor is different than the one of
the vacuum case, we have that massive scalar field
couples to torsion in a different way than the massless one.
In contrast to the massless case, the equations (\ref{aa1}) do not admit as
solution $S_\mu=0$ for non-vanishing $\varphi$ (Again for particle-like
solutions we have $\phi=0$ and $S_\mu=0$).
This is in disagreement with the traditional belief that torsion must be
governed by spin distributions. We will return to this point in the last
section.
\subsection{Gauge fields}
We need to be careful with the use of MCP to gauge fields. We will restrict
ourselves to the abelian case in this work,
non-abelian gauge fields will bring some
technical difficulties that will not contribute to the understanding
of the basic problems of gauge fields on Riemann-Cartan spacetimes.
Maxwell field can be described by the differential
$2$-form
\begin{equation}
F = dA = d(A_\alpha dx^\alpha) = \frac{1}{2}F_{\alpha\beta}dx^\alpha
\label{form}
\wedge dx^\beta,
\end{equation}
where $A$ is the (local) potential $1$-form, and
$F_{\alpha\beta}=\partial_\alpha A_\beta- \partial_\beta A_\alpha$ is the
usual electromagnetic tensor. It is important to stress that the
forms $F$ and
$A$ are covariant objects in any differentiable manifolds. Maxwell equations
can be written in Minkowski spacetime in terms of exterior calculus as
\begin{eqnarray}
\label{maxeq}
dF&=&0, \\
d {}^*\!F &=& 4\pi {}^*\! J, \nonumber
\end{eqnarray}
where ${}^*$ stands for the Hodge star operator and $J$ is the current
$1$-form, $J=J_\alpha dx^\alpha$. The first equation in (\ref{maxeq}) is
a consequence of the definition (\ref{form}) and of Poincar\'e's lemma.
In terms of components, one has the familiar homogeneous and non-homogeneous
Maxwell's equations,
\begin{eqnarray}
\label{maxeq1}
\partial_{[\gamma} F_{\alpha\beta]} &=& 0, \\
\partial_\mu F^{\nu\mu} &=& 4\pi J^\nu, \nonumber
\end{eqnarray}
where ${}_{[\ \ \ ]}$ means antisymmetrization. We know also that the
non-ho\-mo\-ge\-nous equation follows from the extremals
of the following action
\begin{equation}
S = -\int \left(4\pi{}^*\!J\wedge A +\frac{1}{2} F \wedge {}^*\!F\right) =
\int d^4x\left(4\pi J^\alpha A_\alpha - \frac{1}{4}
F_{\alpha\beta}F^{\alpha\beta} \right).
\label{actmink}
\end{equation}
If one tries to cast (\ref{actmink}) in a covariant way by using MCP in the
tensorial quantities, we have that Maxwell tensor will be given by
\begin{equation}
\label{tilda}
F_{\alpha\beta}\rightarrow
\tilde{F}_{\alpha\beta} =
F_{\alpha\beta} - 2 S_{\alpha\beta}^{\ \ \rho}A_\rho,
\end{equation}
which explicitly breaks gauge invariance. With this analysis, one usually
arises the conclusion that gauge fields cannot interact minimally with
Einstein-Cartan gravity. We would stress another undesired
consequence, also related to the breaking of gauge symmetry, of the use of MCP
in the tensorial quantities. The homogeneous Maxwell equation, the
first of (\ref{maxeq1}), does not come from a Lagrangian, and of course,
if we choose to use
MCP in the tensorial quantities we need also apply MCP to it. We get
\begin{equation}
\partial_{[\alpha} \tilde{F}_{\beta\gamma]} +
2 S_{[\alpha\beta}^{\ \ \rho} \tilde{F}_{\gamma]\rho} = 0 ,
\label{falac}
\end{equation}
where $\tilde{F}_{\alpha\beta}$ is given by (\ref{tilda}). One can see that
(\ref{falac}) has no general solution for arbitrary
$S_{\alpha\beta}^{\ \ \rho}$. Besides the breaking of gauge symmetry,
the use of MCP in the tensorial quantities also leads to a non consistent
homogeneous equation.
However, MCP can be successfully applied for general gauge fields
(abelian or not) in the differential form quantities \cite{saa2}. As
consequence, one has that the homogeneous equation is already in a
covariant form in any differentiable manifold, and that the covariant
non-homogeneous equations can be gotten from a Lagrangian obtained only by
changing the metric tensor and by
introducing the parallel volume element in the Minkowskian action
(\ref{actmink}). Considering the case where $J^\mu=0$, we have the
following action to describe the interaction of Maxwell fields and
Einstein-Cartan gravity
\begin{equation}
\label{actmax}
S = S_{\rm grav} + S_{\rm Maxw} = -\int \,d^4x e^{2\Theta} \sqrt{-g}
\left(
R + \frac{1}{4}F_{\mu\nu}F^{\mu\nu}
\right).
\end{equation}
As in the previous cases, the equation $\tilde{K}_{\alpha\beta\gamma}=0$
follows from the extremals of (\ref{actmax}).
The other equations will be
\begin{eqnarray}
\label{ee1}
&&\frac{e^{-2\Theta}}{\sqrt{-g}}\partial_\mu e^{2\Theta}\sqrt{-g} F^{\nu\mu}
=0, \nonumber \\
&& {\cal R}_{\mu\nu} = 2D_\mu S_\nu - \frac{4}{3}g_{\mu\nu}S_\rho S^\rho
-\frac{1}{2} \left(F_{\mu\alpha}F^{\ \alpha}_\nu
+\frac{1}{2}g_{\mu\nu} F_{\omega\rho}F^{\omega\rho} \right), \\
&& D_\mu S^\mu = -\frac{3}{8}F_{\mu\nu}F^{\mu\nu}. \nonumber
\end{eqnarray}
One can see that the equations (\ref{ee1}) are invariant under the usual
$U(1)$ gauge transformations. It is also clear
from the equations (\ref{ee1}) that Maxwell fields can interact with the
non-Riemannian structure of spacetime. Also, as in the massive
scalar case, the equations do not admit as solution $S_\mu=0$ for arbitrary
$F_{\alpha\beta}$, Maxwell fields are also sources to the spacetime torsion.
Similar results can be obtained also for non-abelian gauge fields\cite{saa2}.
\subsection{Fermion fields}
The Lagrangian for a (Dirac)
fermion field with mass $m$ in the Minkowski spacetime
is given by
\begin{equation}
{\cal L}_{\rm F}=\frac{i}{2}\left(\overline{\psi}\gamma^a\partial_a\psi
- \left(\partial_a\overline{\psi} \right)\gamma^a\psi \right)
- m\overline{\psi}\psi,
\label{fermion}
\end{equation}
where $\gamma^a$ are the Dirac matrices and
$\overline{\psi}=\psi^\dagger\gamma^0$. Greek indices denote spacetime
coordinates (holonomic), and roman ones locally flat coordinates
(non-holonomic). It is well known\cite{hehl}
that in order to cast (\ref{fermion}) in a covariant way, one needs to
introduce the vierbein field, $e^\mu_a(x)$, and
to generalize the Dirac matrices,
$\gamma^\mu(x) = e^\mu_a(x)\gamma^a$. The partial derivatives also must be
generalized with the introduction of the spinorial connection $\omega_\mu$,
\begin{eqnarray}
\partial_\mu\psi \rightarrow
\nabla_\mu\psi &=& \partial_\mu\psi+ \omega_\mu \psi, \nonumber \\
\partial_\mu\overline{\psi} \rightarrow
\nabla_\mu\overline{\psi} &=& \partial_\mu\overline{\psi} -
\overline{\psi}\omega_\mu,
\end{eqnarray}
where the spinorial connection is given by
\begin{eqnarray}
\label{spincon}
\omega_\mu &=& \frac{1}{8}[\gamma^a,\gamma^b]e^\nu_a\left(
\partial_\mu e_{\nu b} -\Gamma^\rho_{\mu\nu}e_{\rho b}\right) \\
&=& \frac{1}{8}\left(
\gamma^\nu\partial_\mu\gamma_\nu - \left(\partial_\mu\gamma_\nu \right)
\gamma^\nu - \left[\gamma^\nu,\gamma_\rho \right] \Gamma^\rho_{\mu\nu}
\right). \nonumber
\end{eqnarray}
The last
step, according to our hypothesis, shall be
the introduction of the parallel
volume element, and after that one
gets the following action for fermion fields on RC manifolds
\begin{equation}
S_{\rm F} = \int d^4x e^{2\Theta}\sqrt{-g}\left\{
\frac{i}{2}\left(\overline{\psi}\gamma^\mu(x)\nabla_\mu\psi -
\left(\nabla_\mu\overline{\psi}\right)\gamma^\mu(x)\psi \right)
-m\overline{\psi}\psi
\right\}.
\label{fermioncov}
\end{equation}
Varying the action (\ref{fermioncov}) with respect to $\overline{\psi}$ one
obtains:
\begin{equation}
\frac{e^{-2\Theta}}{\sqrt{-g}}\frac{\delta S_{\rm F}}{\delta\overline{\psi}} =
\frac{i}{2}\left(\gamma^\mu\nabla_\mu\psi + \omega_\mu\gamma^\mu\psi \right)
-m \psi + \frac{i}{2}\frac{e^{-2\Theta}}{\sqrt{-g}} \partial_\mu
e^{2\Theta}\sqrt{-g}\gamma^\mu\psi = 0.
\end{equation}
Using the result
\begin{equation}
[\omega_\mu,\gamma^\mu]\psi = - \left(
\frac{e^{-2\Theta}}{\sqrt{-g}}\partial_\mu e^{2\Theta}\sqrt{-g}\gamma^\mu
\right)\psi,
\end{equation}
that can be check using (\ref{spincon}),
(\ref{gammacontr}), and
properties of ordinary Dirac matrices and of the vierbein field,
we get the following equation for $\psi$ on a RC spacetime:
\begin{equation}
\label{psi}
i\gamma^\mu(x)\nabla_\mu\psi - m\psi =0.
\end{equation}
The equation for $\overline{\psi}$ can be obtained in a similar way,
\begin{equation}
\label{psibar}
i \left( \nabla_\mu\overline{\psi}\right) \gamma^\mu(x)
+ m\overline{\psi} = 0.
\end{equation}
We can see that the equations (\ref{psi}) and (\ref{psibar}) are the same
ones that arise from MCP used in the minkowskian equations of motion. In the
usual EC theory, the equations obtained from the action principle do not
coincide with the equations gotten by generalizing the minkowskian
ones. This is another new feature of the proposed model.
The Lagrangian that describes the interaction of fermion fields with the
Einstein-Cartan gravity is
\begin{eqnarray}
\label{actferm}
S &=& S_{\rm grav} +
S_{\rm F} \\ &=& - \int d^4x e^{2\Theta}\sqrt{-g} \left\{
R - \frac{i}{2}\left(\overline{\psi}\gamma^\mu\partial_\mu\psi -
\left(\partial_\mu\overline{\psi}\right)\gamma^\mu\psi
\right.\right. \nonumber \\
&& \ \ \ \ \ \ \ \ \ \ \
+ \left.\left.
\overline{\psi}\left[\gamma^\mu,\omega_\mu\right] \psi\right)
+ m\overline{\psi}\psi \right\} \nonumber \\
&=& - \int d^4x e^{2\Theta}\sqrt{-g} \left\{
R - \frac{i}{2}\left(\overline{\psi}\gamma^\mu\partial_\mu\psi -
\left(\partial_\mu\overline{\psi}\right)\gamma^\mu\psi
\right.\right. \nonumber \\
&& \ \ \ \ \ \ \ \ \ \ \
+ \left.\left.
\overline{\psi}\left[\gamma^\mu,\tilde{\omega}_\mu\right] \psi\right)
-\frac{i}{8}\overline{\psi}\tilde{K}_{\mu\nu\omega}
\gamma^{[\mu}\gamma^\nu\gamma^{\omega]} \psi
+ m\overline{\psi}\psi \right\},\nonumber
\end{eqnarray}
where it was used that
$\gamma^a\left[\gamma^b,\gamma^c\right]+
\left[\gamma^b,\gamma^c\right]\gamma^a=
2\gamma^{[a}\gamma^b\gamma^{c]}$, and that
\begin{equation}
\omega_\mu = \tilde{\omega}_\mu +\frac{1}{8}K_{\mu\nu\rho}
\left[\gamma^\nu,\gamma^\rho\right],
\end{equation}
where $\tilde{\omega}_\mu$ is the
Riemannian spinorial connection, calculated by
using the Christoffel symbols instead of the full connection in
(\ref{spincon}).
The peculiarity of fermion fields is that one has a non-trivial equation
for $\tilde{K}$ from (\ref{actferm}).
The Euler-Lagrange equations for $\tilde{K}$ is given by
\begin{eqnarray}
\frac{e^{-2\Theta}}{\sqrt{-g}} \frac{\delta S}{\delta\tilde{K}} =
\tilde{K}^{\mu\nu\omega} + \frac{i}{8}\overline{\psi}
\gamma^{[\mu}\gamma^\nu\gamma^{\omega]}\psi = 0.
\label{ka}
\end{eqnarray}
Differently from the previous cases, we have that the traceless part of
the contorsion tensor,
$\tilde{K}_{\alpha\beta\gamma}$, is proportional to the spin
distribution. It is still zero outside
matter distribution, since its equation is an algebraic one, it does not
allow propagation. The other equations follow from the extremals of
(\ref{actferm}). The main difference between these equations and the usual
ones obtained from standard EC gravity, is that in the present case one
has non-trivial solution for the trace of the torsion tensor, that is
derived from $\Theta$. In the standard EC gravity, the torsion tensor is
a totally anti-symmetrical tensor and thus it has a vanishing trace.
\section{Final remarks}
In this section, we are going to discuss the role of the
condition (\ref{pot}) and the source for torsion in the proposed model.
The condition (\ref{pot}) is the necessary
condition in order to be possible the definition of a parallel
volume element on a manifold. Therefore, we have that our approach is
restrict to spacetimes which admits such volume elements.
We automatic have this restriction if we wish to use
MAP in the sense discussed in Section II.
Although it is not clear how to get EC gravity equations without
using a minimal action principle, we can speculate about matter fields
on spacetimes not obeying (\ref{pot}). Since it is not equivalent
to use MCP in the equations of motion or in the action formulation, we
can forget the last and to cast the equations of motion for matter
fields in a covariant way directly. It can be done easily, as example,
for scalar fields\cite{saa1}. We get the equation (\ref{e3}),
which is, apparently, a consistent equation. However, we need to define
a inner product for the space of the solutions of (\ref{e3})
\cite{dewitt}, and we are able to do it only if (\ref{pot}) holds.
We have that the dynamics of matter fields requires some restrictions
to the non-riemannian structure of spacetime, namely, the condition
(\ref{pot}). This is more evident for gauge fields, where
(\ref{pot}) arises directly as an integrability condition for the
equations of motion \cite{saa2}. It seems that condition (\ref{pot}) cannot
be avoided.
We could realize from the matter fields studied that the trace of the
torsion tensor is not directly related to spin distributions. This is a
new feature of the proposed model, and we naturally arrive to the
following question: What is the source of torsion? The situation for the
traceless part of the torsion tensor is the same that one has in the
standard EC theory, only fermion fields can be sources to it. As to the
trace part, it is quite different.
Take for example $\tilde{K}_{\alpha\beta\gamma}=0$, that corresponds to
scalar and gauge fields.
In this case, using the definition of the energy-momentum tensor
\begin{equation}
\frac{e^{-2\Theta}}{\sqrt{-g}}
\frac{\delta S_{\rm mat}}{\delta g^{\mu\nu}} = -\frac{1}{2}T_{\mu\nu},
\end{equation}
and that for scalar and gauge fields we have
\begin{equation}
\frac{e^{-2\Theta}}{\sqrt{-g}}
\frac{\delta S_{\rm mat}}{\delta \Theta} = 2 {\cal L}_{\rm mat},
\end{equation}
one gets
\begin{equation}
D_\mu S^\mu = \frac{3}{2}
\left( {\cal L}_{\rm mat} - \frac{1}{2}T
\right),
\end{equation}
where $T$ is the trace of the energy-momentum tensor.
The quantity between parenthesis, in general, has nothing to do with spin, and
it is the source for a part of the torsion, confirming that in
our model part of torsion is not determined by spin distributions. See
also \cite{H1} for a discussion on possible source terms to the torsion.
This work was supported by FAPESP. The author wishes to thank an anonymous
referee for pointing out the reference \cite{H1}.
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{
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
The freezing transition of heteropolymers, in which the number of
thermodynamically relevant states goes from an exponentially large
value (${\cal O}(e^{N})$) in the random globule state, to only a
few (${\cal O}(1)$) conformations in the frozen state, has
attracted a great deal of interest. In addition to providing an
interesting problem in the statistical mechanics of disordered
materials\cite{Previous}, this system is
potentially relevant to the biologically important question of
protein folding.
Most previous investigations have focused on heteropolymers
with short-range interactions. Recently, however, there has
been renewed theoretical\cite{Polyamph,KLK,DobRub} and
experimental\cite{Copart,Tanaka} interest in polyampholytes (PAs),
which are heteropolymers with charged monomers of both signs.
It has been shown that, due to screening effects, PAs
collapse to compact globules if their net charge is below a
critical value\cite{KK}. There is also some evidence from exact
enumeration studies of short chains\cite{KKenum} that
dense globules of neutral PAs may have a freezing transition.
However, it is unclear how long range (LR) interactions
affect freezing, or whether the formalism developed for globular
polymers with short range (SR) interactions remains
applicable to the LR case.
The freezing transition of SR heteropolymers is most commonly
described by the Random Energy Model (REM)\cite{Derrida},
although it is not always applicable even in this
case\cite{HowGoodREM}).
As the principle underlying assumption of REM is the statistical
independence of energies of states (polymer conformations) over
disorder (sequence of charges along the chain), we first examine
correlation of the energies and then discuss the
resulting freezing transition.
Our starting point is the Hamiltonian
\begin{equation} {\cal H} = \sum_{I \neq J}^N B s_I s_J f({\bf
r}_I - {\bf r}_J) ,
\end{equation}
where $B$ is a constant, $I$ labels monomers along the chain,
and $s(I) \in \pm 1$ is the charge of monomer $I$. The range of
interactions is indicated through $f(r)$, such that $f(r) = \Delta (r)$
for SR interactions, and $f(r)=1/r^{d-2}$ for Coulomb forces in $d$
dimensional space.
Finally, we only consider the case of
maximally compact polymers, assuming that maximal density is
maintained independently of Coulomb interactions, i.e. by an external
box, poor solvent, or internal attractions, such that $R \sim N^{1/d}$.
The simplest characteristics of statistical dependence of energies
is the pair correlation between two arbitrary conformations
$\alpha$ and $\beta$, given by
\begin{equation}
\left< E_{\alpha}E_{\beta} \right>_c \equiv
\left< E_\alpha E_\beta \right> - \left< E_\alpha \right>
\left<E_\beta \right>
= B^2 {\cal Q}_{\alpha \beta} \ ,
\label{eq:E1E2}
\end{equation}
with ${\cal Q}_{\alpha \beta} \equiv
\sum_{I \neq J} f({\bf r}_I^\alpha - {\bf r}_J^\alpha) f({\bf
r}_I^\beta - {\bf r}_J^\beta)$.
In the familiar case of SR interactions, ${\cal Q}_{\alpha
\beta}^{\rm SR} = \sum_{I \neq J}
\Delta({\bf r}_I^\alpha - {\bf r}_J^\alpha)
\Delta({\bf r}_I^\beta - {\bf r}_J^\beta)$
is just the number of bonds in common between configurations
$\alpha$ and $\beta$.
Numerical simulations\cite{HowGoodREM} indicate that in many
cases the probability distribution for ${\cal Q}_{\alpha
\beta}^{\rm SR}$, i.e. $P_{\rm SR}({\cal Q}) \equiv \sum_{\alpha
\beta} \delta({\cal Q} - {\cal Q}^{\rm SR}_{\alpha \beta})$ is sharply
peaked at small ${\cal Q}$. This happens because one can easily
``hide'' monomers by moving them only a small distance and
decreasing their contribution to ${\cal Q}^{\rm SR}$. Large
statistical dependence is thus achieved only for conformations
that are closely related. The validity of REM rests on the statistical
rarity of such closely related conformations. REM is valid when
configurations that are statistically dependent can be ignored in
a large $N$ limit.
By contrast, with long range interactions, the relevant
parameter for judging statistical dependence is
${\cal Q}_{\alpha \beta}^{\rm LR} = \sum_{I \neq J} [|{\bf
r}_I^\alpha-{\bf r}_J^\alpha| \cdot |{\bf r}_I^\beta-{\bf
r}_J^\beta|]^{-(d-2)}$. While the geometric interpretation of
${\cal Q}_{\alpha \beta}^{\rm LR}$ is not as clear as ${\cal
Q}_{\alpha \beta}^{\rm SR}$, it measures the similarity in
contributions from monomer pairs $(I,J)$ in conformations
$\alpha$ and $\beta$ to the overall energy.
Unlike the SR case, polymeric bonds always keep monomers
within the scale of LR interactions. Thus, for two conformations
chosen at random, the overlap ${\cal Q}^{\rm LR}_{\rm rand}$
may not be negligible (even if ${\cal Q}^{\rm SR}_{\rm rand}$ is).
The following scaling argument provides an estimate of the width of
the probability distribution $P_{\rm LR}({\cal Q}) \equiv
\sum_{\alpha \beta} \delta({\cal Q} - {\cal Q}^{\rm LR}_{\alpha
\beta})$.
\begin{figure}
\epsfxsize=3.3in \centerline{ \epsplace{RandScalingQvsN.eps.art} }
\caption{
Scaling of ${\cal Q}_{\rm rand}$ and
${\cal Q}_{\rm max}$ with $N$ for LR and SR interactions ($d=3$).
Power law scaling of the form ${\cal Q} \sim N^\gamma$ indicates
that
${\cal Q}^{\rm LR}_{\rm rand}/{\cal Q}^{\rm LR}_{\rm max}$ does not
vanish in the thermodynamic limit, whereas
${\cal Q}^{\rm SR}_{\rm rand}/{\cal Q}^{\rm SR}_{\rm max}$ does.
}\end{figure}
\begin{figure}
\epsfxsize=3.3in \centerline{\epsplace{PQSRLR.eps}}
\caption{
Probability distributions $P({\cal Q}^{\rm LR})$ and $P({\cal Q}^{\rm
SR})$,
obtained from 64-mers on a cubic lattice. Due to finite size effects,
there is
some
residual overlap in the SR case (here peaked at 0.1). However, we
expect that the SR residual overlap vanishes in the thermodynamic
limit, while the LR overlap does not.
}\end{figure}
First, consider the maximum overlap which occurs (for both LR
and SR) when {\it all} elements are correlated (i.e. ${\cal Q}_{\rm
max}=
{\cal Q}_{\alpha \alpha}$ is the correlation of a configuration with
itself).
To compute this, we note that for each of the $N$ monomers, there is a
contribution from ${\cal O}(r^{d-1})$ monomers at a distance $r$
(for {\it compact} states in $d$ dimensions), resulting in
${\cal Q}_{\rm max} \sim N \int dr r^{d-1} f(r)^{2}$. For SR
interactions, this integral is dominated by contributions at a
microscopic length scale (set by the interaction range)
and we get ${\cal Q}^{\rm SR}_{\rm max} \sim N $. For LR
interactions, while contributions from monomers far
away are smaller, there are more of them. For Coulomb interactions
in $d\leq 4$, the integral is dominated by the longest distance, and
for a polymer of size $R$, we get
${\cal Q}^{\rm LR}_{\rm max} \sim N R^d/R^{2(d-2)} \sim
NR^{4-d}$.
We can use similar arguments for the
overlap between two conformations chosen at random
(${\cal Q}^{\rm LR}_{\rm rand}$).
In fact, for the LR problem,
${\cal Q}^{\rm LR}_{\rm max}$ and ${\cal Q}^{\rm LR}_{\rm rand}$ scale
identically,
as both cases involve ${\cal O}(N^2)$ pairs of monomers
each giving a contribution ${\cal O}(1/R^{2(d-2)})$, for a
total of ${\cal Q}^{\rm LR}_{\rm max} \sim
{\cal Q}^{\rm LR}_{\rm rand} \sim N^{2}R^{2(2-d)}$.
Moreover, as
the main contribution to ${\cal Q}^{\rm LR}_{\rm rand}$ comes from
far away sites, this residual overlap is only weakly conformation
dependent.
The existence of a
residual overlap changes the problem fundamentally from the
SR case: REM is not valid as there is always a statistical
dependence in $d < 4$ \cite{Note1}.
Computer simulations support the above arguments. To examine a large
range in $N$, we generated random conformations on a lattice by
first choosing a radius $R$, and then enumerating random paths
\cite{Enum}
on the set of lattice sites which are within $R$.
$R$ was varied from 3 to 10 lattice sites, and the following results
represent averages over 20 conformations for each $R$ value.
Fig.~1 shows that the scaling exponents $\gamma$ defined by
${\cal Q} \sim N^\gamma$ appear to be the same within error for random
pairs of conformations, as well as the overlap of any conformation with
itself. Furthermore, the fits agree well with the predictions
$\gamma^{\rm LR}_{\rm max} = \gamma^{\rm LR}_{\rm rand} = 4/3$.
By contrast, with SR interactions $\gamma^{\rm SR}_{\rm max} = 1$,
while $\gamma^{\rm SR}_{\rm rand} \approx 0.75$ is distinctly smaller.
We also calculated SR and LR overlaps ${\cal Q}^{\rm SR}$ and
${\cal Q}^{\rm LR}$ for 1000 pairs of 64-mer conformations ($d=3$,
cubic lattice). The resulting histograms, with overlaps normalized
by the maximal value, are shown in Fig.~2. SR overlaps are peaked
at small values whereas the LR overlaps are peaked closer to
unity. Furthermore, the sharpness of the distribution suggests that
${\cal Q}^{\rm LR}$ is approximately independent of the chosen
pairs of conformations.
Having demonstrated the residual overlap between energies
of conformations with LR interactions, and hence the breakdown
of REM, we go on to better characterize the density of states.
This will take us a step closer to understanding the freezing of PAs.
To describe the density of states, we use the following three
characteristics: the annealed energy variance $\sigma_{\rm ann}$ (the
width of the density of states for annealed disorder), the average
quenched
energy variance $\sigma_{\rm quen}$ (the width of the density of states
for quenched disorder), and the quenched energy correlation function
$g$ (the statistical dependence between states). These quantities are
given by the formul\ae
\begin{eqnarray}\label{define}
\sigma^2_{\rm ann} & \equiv & \left< \overline{(E^2)} \right>_c =
\left< \overline{(E^2)} \right> - \left<\, \overline{E}\, \right>^2,
\nonumber \\
\sigma^2_{\rm quen} & \equiv &\left< \overline{(E^2)}_c \right> =
\left< \overline{(E^2)} \right> - \left< (\overline{E})^2
\right>,
\\ g & \equiv & \left< {(\overline{E} )}^2 \right>_c = \left<
{(\overline{E} )}^2 \right> -
\left<\, \overline{E} \,\right>^2,
\nonumber
\end{eqnarray}
where $\overline{ {}^{ } \ldots }$ and $\left< \ldots
\right>$ denote averaging over conformations and sequences
respectively. Note that these quantities are related by a
mathematical identity
$\sigma^2_{\rm ann} = \sigma^2_{\rm quen} + g$.
\begin{figure}
\epsfxsize=3.3in \centerline{ \epsplace{meanE.eps.art} }
\caption{
Mean and width of the energy spectra for
80 sequences of 36-mers, determined by full enumeration over all
maximally compact conformations (see text for details).
}\end{figure}
\bigskip
In the annealed case, the energy variance is $\sigma^2_{\rm ann} =
B^2 {\cal Q}_{\rm max}$, since, in this case, all possible states can be
accessed and thus the width of the energy spectrum must be maximal.
This result is also easily extracted from equation (\ref{eq:E1E2}) by
averaging over conformations with $\alpha = \beta$. Averaging the same
equation over {\it all pairs} of states $\alpha$ and $\beta$, we can
find $g$:
for ${\cal M}$ conformations, there are ${\cal M}$ pairs $\alpha=\beta$
which completely overlap ${\cal Q}_{\alpha \beta} = {\cal Q}_{\rm
max}$,
but this is overshadowed by the remaining ${\cal M}({\cal M}-1)$ pairs
with overlap ${\cal Q}_{\alpha \beta}= {\cal Q}_{\rm rand}$, resulting
in $g \approx B^2 {\cal Q}_{\rm rand}$.
In addition to measuring the statistical dependence between states,
$g=\left< (\overline{ E })^2 \right>_c$ also describes how the mean of
the
energy spectrum for a given sequence varies between sequences.
Finally the width of the energy spectrum for a typical sequence is
$\sigma^2_{\rm quen} \equiv \sigma^2_{\rm ann}-g = B^2 ({\cal
Q}_{\rm max} - {\cal Q}_{\rm rand})$. This makes sense physically
as correlation (anticorrelation) in the energies should narrow
(broaden) the width of the energy spectra. Also, we see that when
there is no correlation ($g=0$),
$\sigma_{\rm ann} = \sigma_{\rm quen}$, as in the REM.
The following picture emerges from the above results.
As ${\cal Q}^{\rm SR}_{\rm rand}=0$, we have $g=0$ for
the SR case above the freezing temperature, and the
mean of the energy spectrum does not vary significantly
between sequences. Also, the width of the spectrum for a
given sequence is large (the maximum possible value, as
in the annealed case). The variation of the means of the energy
spectra between sequences $g$, is much smaller
than the typical width of each spectrum $\sigma^2_{\rm quen}$;
thus disorder is not important for SR interactions above freezing.
Of course, below the freezing temperature, self averaging breaks
down, and disorder is relevant. By contrast, for LR interactions,
${\cal Q}^{\rm LR}_{\rm rand}$ does not vanish and is significant.
We thus expect the widths of the energy spectra to be small
and the means to vary widely from sequence to sequence.
The results of a computational test of the above scenario,
obtained from the exact enumeration of all globular states of
36-mers on a cubic lattice ($d=3$) are presented in Fig.~3.
We see that for SR interactions, the means of the spectra are
indeed well defined and their width (gray region) is large.
For LR interactions, the means are poorly defined, with a
variance between sequences which is greater than the
widths of individual spectra (error bars).
Is the insight gained above sufficient to analyze the freezing
transition in
PAs?
In general, freezing is governed by the low energy tail of the density
of
states $\rho (E) = {\cal M} P(E)$, where ${\cal M}$ is the total number
of
conformations, and $P(E)$ is the single level energy distribution.
In the standard REM entropy crisis scenario, the system freezes in a
microstate, much like a snapshot, at a
temperature $T_f$ at which $\rho _T \sim 1$, where $\rho _T =\rho(E_T)$
is the density of states at the equilibrium energy $E_T$ at the
temperature
$T$.
The density of states in the high temperature regime is governed by
$\sigma _{\rm ann}$, as can be seen by a high temperature expansion:
The partition function $Z={\rm tr}\left[ \exp\left( -\beta{\cal H}
\right)
\right]$
is first expanded in powers of $\beta=1/T$, resulting in (after
averaging over
sequences) $-\beta F=\left\langle\ln Z\right\rangle=\ln{\cal M}-
\beta\left\langle \overline{E} \right\rangle+
\beta^2\langle \overline{(E^2)} \rangle_c/2+\cdots$.
{}From this expression (and using Eq.(\ref{define})), the entropy is
calculated as $S(T)=\ln {\cal M}-\beta^2\sigma^2_{\rm quen}/2+\cdots$,
where (as demonstrated earlier) for Coulomb interactions in $d=3$,
$\sigma^2_{\rm quen}\sim e^2N^2/R$,
yielding
\begin{equation}
\rho_T \sim
{\cal M}
\exp \left[- {1\over 2}\left({e^2 N \over TR} \right)^2 \right] .
\end{equation}
>From the structure of the series\cite{KLK}, we expect the high
temperature
expansion to break down for temperatures $T<T_D \equiv e^2N/R$. This
temperature can also be obtained by regarding the polymer globule as a
(non-polymeric) plasma of the same $N$ charges confined within the
volume
$R^3$. As the Debye screening length for this plasma is of the order
$r_D \sim (TR^3/N e^2)^{1/2}$, there are two regimes: For $T<T_D$, the
plasma
is fully screened as $r_D < R$. However, for $T>T_D$, $r_D > R$ and the
charges are not screened. The latter regime is meaningless for a regular
plasma, but describes the high temperature behavior of the polymer
globule.
It is not clear that, with the constraints of polymeric bonds, the
scaling
for a PA should be the same as that for a screened plasma at low
temperatures. However, assuming that this is the case, the entropy can
be
estimated by noting that the plasma is composed of roughly
${\cal N} \sim R^3/r_D^3\sim (Ne^2/RT)^{3/2}$ independent Debye volumes.
Assuming that the entropy is proportional to ${\cal N}$, we finally
conclude
\begin{equation}
\rho_T \sim
{\cal M}
\exp \left[-c \left( \frac{e^2 N}{TR} \right)^{3/2} \right]
\label{eq:renormdos}
\end{equation}
where $c$ is a numerical constant. Note that
Eq.~(\ref{eq:renormdos}) indicates a very sharp decrease of the density
of
states in the low energy tail, proportional to $\exp [ - c^{\prime} ( E
-
\overline{E} )^3 ]$, which reflects the fine tuning of configurations
necessary
for screening.
Typically the number of conformations of a polymer scales as
${\cal M} \sim e^{\omega N}$, with $\omega$ of the order of unity.
In the limit where the polymer is kept maximally compact by an external
box, poor solvent, or internal attractions, such that $R \sim a
N^{1/3}$,
where $a$ is a monomeric length scale, $\omega$ is approximately the
entropy of Hamiltonian walks.
Freezing, which is signaled by $\rho \sim 1$,
can take place in the unscreened regime only for short chains with
$N < 1/\omega$. (The ``apparent'' freezing temperature for unscreened
polymers grows as $N^{1/6}$.) In this case, a further decrease of
temperature will not lead to screening, of course. For longer chains,
we predict freezing at an $N$-independent temperature of
$T_f\sim e^2/(a\omega^{2/3})$ in the screened regime. In this sense, the
compact PA freezes in a phase transition that is similar to REM. We
stress that this happens despite the unusual scaling of the width of the
density of states, $\sigma \sim N^{2/3}$. The distinction between the
two
behaviors is important for understanding the results of lattice
simulations, as it appears that 36-mers are in the short chain regime.
We expect that the nature of the frozen state also depends on $T_f/T_D$.
For freezing in the screened regime ($T_f<T_D$), the system looks much
like
that of the SR case, i.e. like a disordered version of a salt crystal.
For freezing in the unscreened regime ($T_f>T_D$), we expect a smaller
degree of antiferrogamnetic ordering; consistent with the
idea
that freezing at a higher temperature leads to a state which is less
energetically optimized.
An important class of PAs are {\it proteins}. In the light of our
findings
in this work, we make here some concluding remarks about protein
folding and evolution.
Of the 20 natural amino-acids, three are positively charged (Lys, Arg,
His),
two are negatively charged (Asp, Glu), and the rest are neutral.
Nevertheless, it is often assumed that LR interactions are not
essential to proteins, as the screening length in biological solvents is
often quite small. It is less clear that screening is also effective in
compact globular configurations with little or no solvent in their
interiors. Furthermore, secondary structural elements such as
$\alpha$-helices effectively reduce the conformational flexibility of
proteins. Indeed, the conformation space of small proteins (i.e. 70-90
amino--acids) perhaps corresponds to that of lattice 27-mers
\cite{CoresStates},
and small proteins are likely to be in the short chain regime with
respect to LR interactions. Thus, while the total charge on a given
protein
may be small, in solvents with few counter ions, this may be sufficient
to
lead to a REM-violating correlated energy landscape, making the
results
obtained here relevant. Moreover, for the
typical separation of charges in a globular protein (roughly
20 \AA), and given a dielectric constant of order 5-10, and
$\omega\approx 2$, the characteristic freezing temperature $T_f$ is of
the order of (biologically relevant) room temperatures.
We have discussed how the mean of the density of states can vary greatly
from
sequence to sequence. It appears that a large contribution to this mean
comes from the interaction between monomers that are not far apart along
the
sequence. For example, while next nearest neighbors along the chain can
somewhat vary their spatial distance from each other, this will still
not
break their great contribution to the mean energy. This is why the
conformational average energy depends strongly on the correlations
between
charges quenched along the sequence. For Coulomb interactions,
chains
with anti-correlated sequences have low mean energies. This is
intriguing, considering the recent finding that protein sequences are
indeed
anti-correlated with respect to their charge \cite{ProtCor}. This
indicates
that perhaps protein evolution was not just dictated solely by the
degree of
hydrophobicity of monomers (which depends on the degree of charge, not
the
sign), but by Coulomb effects as well.
\bigskip
The work was supported by NSF (DMR 94-00334).
AYG acknowledges the support of Kao Fellowship. Computations were
performed on Project SCOUT (ARPA contract MDA972-92-J-1032).
\vspace{-0.3in}
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}\label{intro}
In systems at or close to a critical point the asymptotic power laws
governing the behavior of thermodynamic
quantities are modified in the vicinity
of surfaces or
other imhomogenities\cite{binder,diehl}.
The characteristic distance within which changes
occur is given by the bulk correlation length $\xi$.
In general each bulk
universality class splits up in several {\it surface
universality classes}, depending upon whether the tendency to order
in the surface is smaller or larger than or the same as in the bulk.
In the case of the
two-dimensional (2-$d$) Ising model in a semi-infinite geometry there exist
two surface universality classes. Since the boundary is one-dimensional
and, thus cannot become critical itself, the surface generally reduces the
tendency to order. More precisely, for any positive starting value
the exchange coupling between surface spins $J_1$ is driven to zero by
successive renormalization-group transformations.
The relevant field pertaining to
the surface is the magnetic field $h_1$,
which acts on surface spins only and which
for instance may take into account the influence of an
adjacent noncritical medium. The two universality classes then are
labelled by $h_1=0$, the ``ordinary'' transition, and $h_1=\infty$, the
``normal'' transition, where the former is a unstable fixed point and
the latter is a stable fixed point of the
renormalization-group flow\cite{binder,diehl}.
In this work
we are mainly concerned with order-parameter profiles
for {\it finite} $h_1$ in the crossover region between the fixed points.
Nonetheless, let us first recapitulate the situation {\it at} the fixed points.
For the sake of simplicity, in the Introduction our
considerations remain restricted to bulk criticality $T=T_c$, but
the extension to the critical region is straightforward and
will be done below.
At the critical point and $h_1=0$, the order-parameter
(or magnetization) profile
$m(z)$ is zero
for any distance $z\ge 0$ from the surface. In the other extreme,
$h_1=\infty$,
it is well known that at macroscopic distances from the surface the
magnetization decays as $\sim z^{-x_{\phi}}$,
where $x_{\phi}=\beta/\nu$ is the scaling dimension of the bulk order-parameter
field, with the exact value $1/8$ in
the 2-$d$ Ising model.
What do we expect, when $h_1$ takes some intermediate
value, i.e., in the crossover region between the fixed points?
Now, $h_1$ will certainly generate a surface magnetization
$m_1$. As far as the profile $m(z)$ is concerned,
a first guess would perhaps be that the magnetization should
decay from that value
as $z$ increases away from the surface. This guess is supported
by mean-field theory, where one can calculate $m(z)$
and indeed finds a monotonously decreasing function of $z$
\cite{binder,lubrub,bray}.
In Ref.\,\cite{czeri} the present authors have shown that,
contrary to the naive (mean-field)
expectation, fluctuations may cause the order parameter to steeply
{\it increase} to values $m(z)\gg m_1$
in a surface-near regime. The range
within which this growth occurs (at bulk criticality) is determined
by $h_1$, the characteristic
length scale being $l_{1}\sim h_1^{-1/y_1}$, where
$y_1=\Delta_1/\nu$ is the scaling dimension of $h_1$ \cite{diehl}.
As further demonstrated by the present authors in Ref.\,\cite{czeri},
the growth of order in the near-surface regime $z\ll l_1$
is described by a {\it universal} power law
\begin{equation}\label{power}
m \sim z^{\kappa}\quad \mbox{with} \quad \kappa=y_1-x_{\phi}\>,
\end{equation}
i.e., the growth exponent $\kappa$ is governed by the difference
between the scaling dimensions $y_1$ and $x_{\phi}$.
For $z\simeq l_1$ the profile has a maximum and
farther away from the surface, at distances much
larger than $l_{1}$, the magnetization decays as $z^{-x_{\phi}}$.
Largely analogous results---monotonous behavior at the fixed points
and profiles with one extremum in the crossover regime---where
previously found
by Mikheev and Fisher \cite{mifi} for the {\it energy density}
of the 2-$d$ Ising model. The authors also suggested to calculate
the order parameter in the crossover region.
Below we focus our attention exactly on this problem.
The questions posed in this work are the following:
\begin{itemize}
\item Does the short-distance
growth of $m(z)$, found for instance in the three-dimensional Ising
model, also occur in two dimensions?
\item Does the simple power
law (\ref{power}) quantitatively describe the magnetization profile
in $d=2$, or are modifications to be expected?
\item Is the scenario for the crossover between ``ordinary'' and
``normal'' transition, developed for the three-dimensional
Ising model, also valid in $d=2$?
\end{itemize}
As we will demonstrate below, the answer to the first
and third question is ``yes",
but the simple power law (\ref{power}) is modified by a {\it logarithm}.
The rest of this paper is organized as follows: In Sec.\,\ref{two}
the theoretical framework is expounded.
We first summarize the results of Ref.\,\cite{czeri} and
then generalize the scaling analysis
by taking into account the available
exact results on the dependence of $m_1$ on $h_1$ and
on the magnetization profile.
In Sec.\,\ref{three}, in order to corroborate our analytical findings,
we present Monte Carlo (MC) data for $m(z)$ and
compare with exact results for the order-parameter profile.
In two Appendices exact literature results on the dependence
of the surface magnetization on $h_1$ and on the order-parameter
profile are briefly reviewed.
\section{Theory}\label{two}
We consider the semi-infinite Ising system with a free boundary
on a plane square lattice.
The exchange coupling between
neighboring spins is $J$.
A surface magnetic field $H_1$ is
imposed on the boundary spins and bulk magnetic fields are set to
zero such that the Hamiltonian of the model reads
\begin{equation}\label{ising}
{\cal H}= -J\!\sum_{<ij>\in V}\,s_is_j-H_1\sum_{i\in\partial V}\,s_i\>,
\end{equation}
where
$\partial V$ and $V$ stand for the boundary and the
whole system (including the boundary), respectively.
As usual, we work with the dimensionless variables
\begin{equation}\label{Kandh1}
K=J/k_BT\quad \mbox{and}\quad h_1=H_1/k_BT\>.
\end{equation}
The bulk critical point corresponds to $K_c=\frac12\, \mbox{ln}
(1+\sqrt{2})$.
\subsection{Scaling analysis for Ising system in $2<d<4$}\label{twoone}
In the critical regime, where $|\tau|\equiv |(T-T_c)/T_c|\ll 1$,
thermodynamic quantities are described
by homogeneous functions of the scaling fields.
As a consequence,
the behavior of the local magnetization under
rescaling of distances should be described by
\begin{equation}\label{scal}
m(z,\tau,{h}_1)\sim b^{-x_{\phi}}\,m(zb^{-1},\,\tau b^{1/\nu},\,
{h}_1\,b^{y_1}),
\end{equation}
where $x_{\phi}=\beta/\nu$ and
$y_1=\Delta_1/\nu$ is the scaling dimension of $h_1$ \cite{diehl}.
In terms of other surface exponents we have
$y_1=(d-\eta_{\parallel})/2$\cite{diehl,foot}.
In Eq.\,(\ref{scal}) it was further assumed
that the distance $z$ from the surface is much larger than
the lattice spacing or any other {\it microscopic} length scale.
One is interested in the behavior at {\it macroscopic} scales, and,
for the present,
$z$ may be considered as a continuous variable ranging
from zero to infinity.
Removing the arbitrary rescaling parameter $b$ in
Eq.\,(\ref{scal}) by setting it
$\sim z$, one obtains the scaling form of the magnetization
\begin{mathletters}\label{scal2}
\begin{equation}\label{scalm}
m(z,\tau,h_1)\sim z^{-x_{\phi}}\,{\cal M}(z/\xi, z/l_1)\>,
\end{equation}
where, as already stated,
\begin{equation}\label{length}
l_1\sim h_1^{-1/y_1}
\end{equation}
\end{mathletters}
\noindent
is the length scale set by the surface field. The second length
scale pertinent to the semi-infinite system and occurring in
(\ref{scalm}) is the bulk correlation length $\xi=\tau^{-\nu}$.
Regarding the interpretation of MC data, which are normally obtained
from finite lattices, one has to take into account a third
length scale, the characteristic dimension $L$ of the system,
and a finite-size scaling analysis
has to be performed. The latter will be briefly described
in Sec.\,\ref{twothree}.
Going back to the semi-infinite case and setting
$\tau=0$,
the only remaining length scale is $l_1$, and
the order-parameter profile can be written in the
critical-point scaling form
\begin{equation}\label{h1}
m(z,{h}_1)\sim z^{-x_{\phi}}\,{\cal M}_c(z/l_1)\>.
\end{equation}
As said above, for $z\to \infty$ the magnetization decays as $\sim
z^{-x_{\phi}}$
and, thus, ${\cal M}_c(\zeta)$ should approach a constant for
$\zeta\to \infty$.
In order to work out the {\it short-distance}
behavior of the scaling function ${\cal M}_c(\zeta)$,
we demand that $m(z)\sim m_1$ as $z\to 0$.
This means that in general, in terms of macroscopic quantities,
the boundary value of $m(z)$ is {\it not} $m_1$. If
the $z$-dependence of $m(z)$ is described
by a power law, it cannot approach any value
different from zero or infinity as $z$ goes to zero.
However, the somewhat weaker relation symbolized by ``$\sim$'' should hold,
stating that the respective
quantity asymptotically (up to constants) ``behaves as" or
``is proportional to".
This is in accord with and actually motivated by the
field-theoretic short-distance expansion \cite{syma,diehl}, where
operators near a boundary are represented in terms of
boundary operators multiplied by $c$-number functions.
In the case of the three-dimensional Ising model the foregoing
discussion leads to the conclusion
that $m(z)\sim h_1$ because the ``ordinary''
surface---the universality class to which also a free surface
belongs---is paramagnetic
and responds linearly to a small magnetic field\cite{bray}.
The consequence for the scaling function in (\ref{h1})
is that ${\cal M}_c(\zeta)\sim \zeta^{y_1}$, and, inserting
this in (\ref{h1}),
we obtain that the exponent governing the short-distance
behavior of $m(z)$ is given by the difference between
$y_1$ and $x_{\phi}$ (as already stated in
Eq.\,(\ref{power})).
Using the scaling relation $\eta_{\perp}=(\eta+\eta_{\parallel})/2$
\cite{diehl} among anomalous dimensions, one can reexpress
the exponent $\kappa$ as \cite{foot}
\begin{equation}\label{kappa}
\kappa=1-\eta_{\perp}\>.
\end{equation}
In the mean field approximation the value of $\kappa$ is zero, and
one really has $m(z\to 0)=m_1$.
However, a positive value is obtained when fluctuations are taken into account
below the upper critical dimensionality $d^*$.
For instance, the result for the $n$-vector model with $n=1$
(belonging to the Ising universality class)
in the framework of the $\epsilon$-expansion is
$\kappa=\epsilon/6$ \cite{czeri}. Thus, the
magnetization indeed grows as $z$ increases away from the surface.
For the 2-$d$ Ising model the exponent $\eta_{\perp}$
is known {\it exactly} \cite{cardy} and one obtains
$\kappa=3/8$. However, as will be discussed in Sec.\,\ref{twotwo},
the pure
power law found in Ref.\,\cite{czeri} for $d=3$ is modified in two
dimensions by a {\it logarithmic} term, and the exponent $3/8$ cannot
directly be seen in the profile.
The above phenomenological analysis is straightforwardly extended
to the case $\tau>0$. In $d>2$, we may assume that the
behavior near the surface for $z<<\xi$ is unchanged compared
to (\ref{power}), and, thus,
the increasing profiles are also expected slightly above the
critical temperature. The behavior farther away from the surface depends
on the ratio $l_1/\xi$.
In the case of $l_1>\xi$ a crossover
to an exponential decay will take place for $z\simeq \xi$
and the regime of nonlinear decay does not occur.
For $l_1< \xi$ a crossover
to the power-law decay $\sim z^{-\beta/\nu}$ takes place
and finally at $z\simeq \xi$ the exponential behavior sets in.
Below the critical temperature, the short-distance
behavior of the order parameter is also described by a power
law, this time governed by a different exponent, however. The essential
point is that below $T_c$ the surface orders even for $h_1=0$.
Hence, in the scaling analysis the scaling dimension of $h_1$
has to be replaced by the scaling dimension of $m_1$, the
conjugate density to $h_1$,
given by $x_1=\beta_1/\nu$\cite{diehl}. The exponent that describes
the increase of the profile is thus
$x_1-x_{\phi}$ \cite{gompper}, a number that even in
mean-field theory is different from zero ($=1/2$).
Phenomena to some extent analogous to the ones discussed above were
reported for the crossover between {\it special} and
normal transition\cite{brezin,ciach}.
Also near the special transition the
surface field $h_1$ gives rise to a length scale. However, the
respective exponent, the analogy to $\kappa$ in (\ref{power}),
is negative, and, thus, one
finds a profile that monotonously decays for all
(macroscopic) $z$, with different power laws
in the short-distance and the long-distance regime and
a crossover at distances comparable to the length scale
set by $h_1$. However,
{\it non-monotonous} behavior in the crossover
region is a common feature in the case
of the energy density in $d=2$ \cite{mifi} and as well as in
higher dimensionality\cite{eisenriegler}.
The {\it spatial} variation of the
magnetization discussed so far
strongly resembles the {\it time} dependence of the
magnetization in relaxational processes at the critical point.
If a system with nonconserved order parameter (model A) is quenched from a
high-temperature initial state to the critical point, with a small
initial magnetization $m^{(i)}$, the order parameter behaves as $m \sim
m^{(i)}\,t^{\theta}$ \cite{jans}, where the short-time exponent $\theta$
is governed
by the difference between the scaling dimensions of initial
and equilibrium magnetization divided by the
dynamic (equilibrium) exponent\cite{own}. Like the exponent $\kappa$
in (\ref{power}), the exponent
$\theta$ vanishes in MF theory, but becomes positive
below $d^*$.
\subsection{Scaling analysis in $d=2$}\label{twotwo}
During the years,
initiated by the work of McCoy and Wu\cite{mccoy,other},
the 2-$d$ Ising model with a surface magnetic field
received a great deal of attention,
because many aspects can be treated
exactly and it is a simple special
version of the 2-$d$ Ising model in a inhomogeneous bulk
field, a problem to which an exact
solution would be highly desirable.
Some of these exact results, namely those considering the
vicinity of the critical point \cite{fishau,bariev},
will be used in the following as a guiding
line for our phenomenological scaling analysis and
to compare numerical data with.
The dependence of $m_1$ on the surface magnetic field (bulk field $h=0$)
in the 2-$d$ Ising model was calculated exactly by Au-Yang
and Fisher \cite{fishau} in a $n\times \infty$ (strip) geometry.
The limit $n\to \infty$, yielding results for the semi-infinite
geometry, was also considered. Whereas above two dimensions
at the ordinary transition the surface, in a sense, is
paramagnetic, i.e., the response
of $m_1$ to a small $h_1$ is linear, in two dimensions
the function $m_1(h_1)$ has a more complicated form. As summarized
in Appendix A (see Eq.\,(\ref{m1h1})),
there is a logarithmic correction to the
linear term in $d=2$; for $h_1\to 0$ the surface magnetization
behaves as $\sim h_1\,\mbox{ln}\,h_1$.
Further, as shown by Bariev \cite{bariev} and summarized in Appendix B,
the length scale $l_1$ determined by $h_1$ behaves as
$\sim \left[\tanh (h_1)\right]^{-2}$.
For small $h_1$, where the scaling analysis is expected to be
correct,
this is consistent with (\ref{length}) as
$y_1=1/2$ in the 2-$d$ Ising model. Thus the characteristic length
scale that enters the scaling analysis
depends in the same way upon $h_1$ as in higher dimensions.
The foregoing discussion allows us to generalize our scaling analysis,
especially Eqs.\,(\ref{h1}) and the near-surface law (\ref{power}),
such that the speacial features of the 2-$d$ Ising model are taken into
account.
Again, the only available length scale at
$\tau=0$ is $l_1$, and
the magnetization can be represented in the form given
in Eq.\,(\ref{h1}).
For $z\to \infty$ we expect that $m\sim z^{-1/8}$
and, thus, ${\cal M}_c(\zeta)$ should approach a constant for
$\zeta\to \infty$.
In order to find the short-distance behavior
we assume again that $m(z)\sim m_1$ as $z\to 0$.
Taking into account the logarithmic correction
mentioned above and discussed in more detail in Appendix A
(see Eq.\,(\ref{m1h1}), we find that
${\cal M}_c(\zeta)\sim \zeta^{1/2}\,\mbox{ln}\,\zeta$
for $\zeta\to 0$. Hence, for the short-distance behavior of $m(z)$
in the semi-infinite system we obtain
\begin{equation}\label{sdbe}
m(z,h_1)\sim h_1\, z^{\kappa}\,\mbox{ln}(h_1\,z^{y_1})\>,
\end{equation}
where the exact values of the exponents are
$\kappa=1-\eta_{\perp}=3/8$ and $y_1=1/2$.
Thus, for $z < l_1$ the magnetization $m(z)$
for a given value of $h_1$ behaves as $\sim z^{3/8}\,\mbox{ln}\,z$.
The result (\ref{sdbe}) should hold for any value of the exchange
coupling $J_1$ in the surface. In our MC analyses to be presented below
we implemented free boundary conditions with
$J=J_1$, but we expect (\ref{sdbe}) to hold for any value of $J_1$ with
possible $J_1$-dependent nonuniversal
constants leaving the qualitative behavior of the profiles unchanged.
Eq.\,(\ref{sdbe}) is the main analytic result of this work.
As discussed in the following,
it is consistent with Bariev's exact solution\cite{bariev}
(see Appendix B) and with MC data for the profile.
It tells us that the short-distance power law behavior is
modified by a logarithmic term. This logarithm
can be traced back to the logarithmic singularity
of the surface
susceptibility\cite{binder}, which, in turn,
causes the logarithmic dependence
of $m_1$ on $h_1$, and eventually leaves its fingerprint
also on the near-surface behavior of the magnetization.
The result (\ref{sdbe}) provides
a thorough understanding of the near-surface
behavior of the order parameter and allows to relate special
features of the two-dimensional system, which were (as
we will discuss in more detail below)
previously known from the exact analyses \cite{bariev}, to the
somewhat simpler short-distance law in higher dimensions.
\subsection{Finite Size scaling}\label{twothree}
In order to assess the finite size effects to be
expected in the MC simulations, we have to take into account the
finite-size length scale $L$, which is proportional to the linear
extension $N$ of the lattice (compare Sec.\,\ref{threeone} below).
The generalization of (\ref{scal}) reads\cite{fisi}
\begin{equation}\label{scalfs}
m(z,\tau,{\sf h}_1,L)\sim b^{-x_{\phi}}\,m(zb^{-1},\,\tau b^{1/\nu},\,
{h}_1\,b^{y_1}, Lb^{-1})\>,
\end{equation}
and proceeding as before, we obtain as the generalization
of (\ref{scalm}) to a system of finite size:
\begin{equation}\label{scalmfs}
m(z,\tau,h_1,L)\sim z^{-x_{\phi}}\,{\cal M}(z/\xi, z/l_1,z/L)\>.
\end{equation}
Thus even at $T_c$ there are two pertinent length scales,
on the one hand $L$ (imposed by
the geometry that limits the wavelength of fluctuations)
and on the other hand $l_1$ (the scale set by $h_1$).
It is well known that for large $z\gtrsim L$ we have to expect
an exponential decay of $m(z)$ on the
scale $L$. In the opposite limit,
when $z$ is smaller than both $L$ and $l_1$, we expect
the short-distance
behavior (\ref{sdbe}) to occur.
However, for $d=2$ it can be concluded from the
finite-size result (\ref{m1h1fin})
that there will be an $L$-dependent amplitude, a prefactor
to the function given in (\ref{sdbe}). But otherwise
the logarithmically modified power law (\ref{sdbe}) should occur.
Farther away from the surface, the form of the profile
depends on the ratio between $l_1$ and $L$.
For $z$ smaller than both $L$ and $l_1$, the behavior
is described
by (\ref{sdbe}). In the case of $l_1>L$ a crossover
to an exponential decay will take place for $z\simeq L$. In the
opposite case, a crossover
to the power-law decay $\sim z^{-x_{\phi}}$ takes place,
followed by the crossover to the exponential
behavior at $z\simeq L$. Thus, qualitatively,
the discussion is
completely analogous to the one in Sec.\,\ref{twoone},
where the behavior at finite $\xi$ was described.
\section{Monte Carlo simulation}\label{three}
\subsection{Method}\label{threeone}
The results of the scaling analysis, especially the
short-distance law (\ref{sdbe}), were checked by MC simulations.
To this end, we calculated order-parameter profiles for the 2-$d$ Ising model
with uniform exchange coupling $J$.
The geometry of our systems was that of
a rectangular (square)
lattice with two free boundaries (opposite to each other)
and the other boundaries periodically coupled, such that the
effective geometry was that of a cylinder of finite length.
The linear dimension perpendicular to the free surfaces
was taken to be four times larger than the lateral extension
in order to keep corrections due to the second surface
small\cite{fidege}. Hence,
when we talk about a lattice of size $N$ in the
following, we refer to a rectangular
$N\times 4\,N$ system.
In order to generate an equilibrium sample of spin
configurations, we used
the Swendson-Wang algorithm\cite{swewa}.
It effectively avoids critical slowing
down by generating new spin configurations
via clusters of bonds, whereby
the law of detailed balance is obeyed.
For a given spin configuration, a bond between two neighboring
spins of {\it equal} sign exits with probability $1-e^{-2K}$.
There are no bonds between {\it opposite} spins.
Then, clusters are defined
as any connected configuration of bonds.
Also isolated spins define
a cluster, such that eventually
each spin belongs to one of the clusters.
After having identified the clusters,
the new configuration is generated by assigning to
each cluster of spins a new orientation, with equal
probability for each spin value as long as the cluster does {\it not}
extend to a surface.
In order to take into account $h_1$, we introduced,
in the same way as
suggested by Wang for taking into account bulk fields
\cite{wang},
two ``ghost'' layers of spins next to
each surface that couple to the surface spins with
coupling strength $h_1$ and that all point in the direction of $h_1$.
If at least one bond between
a surface and a ghost spin exists the
cluster has to keep its old spin
when the system is updated. This preserves
detailed balance.
In the practical calculation this
rule was realized by a modified (reduced) spin-flip probability
\begin{equation}
p(k)=1-\frac{1}{2} \,\exp(-2\,h_1\, k)
\end{equation}
for clusters pointing in the direction of $h_1$ (and $1/2$
for clusters pointing in opposite direction),
$k$ being the number of {\it surface}
spins contained in the cluster.
In order to obtain an equilibrium distribution of configurations
we discharged several hundred (depending on system size)
configurations after the start of the simulation.
To keep memory consume low, we used multispin-coding techniques,
i.e., groups of 64 spins were coded in one long integer.
\subsection{Comparison with exact results}\label{threetwo}
A crucial test for the MC program is the comparison with
known exact results. On the other hand, if both MC data
an exact results agree, the former can be regarded also
as a confirmation of the exact results.
Having calculated order-parameter profiles
for different values of $h_1$, in particular
the magnetization at the boundaries
(the ends of the cylinder) can be obtained.
In Fig.\,1 we show the results for $m_1(h_1)$. The squares
respresent our MC data for a system of size $N=512$.
Also depicted is the exact result for the semi-infinite system
taken from Ref. \cite{fishau} (see
also Appendix\,\ref{appa}). It is clear that the MC
values approach the exact curve for large $h_1$.
Below $h_1\simeq 0.03$,
the MC results show a linear dependence on $h_1$,
significantly deviating from the exact curve.
The linear behavior can be regarded as a finite-size effect,
and it is qualitatively consistent with the result (\ref{m1h1fin})
of Au-Yang and Fisher\cite{fishau}. The latter was
derived in a strip of finite width with infinite lateral extension,
however,
so that a quantitative comparison with our data is not possible.\\[2mm]
\def\epsfsize#1#2{0.6#1}
\hspace*{2.5cm}\epsfbox{fig1.eps}\\[0mm]
{\small {\bf Fig.\,1}: Monte Carlo results for $m_1$ as a function of $h_1$
for $N=512$ (represented by full squares)
compared to the exact result of
Ref.\,\cite{fishau} (see Appendix A). The statistical errors of the
Monte Carlo data are about the same size as the symbols. A detailed
discussion and comparison of the data is in the text.} \\[0.1cm]
Next we compare with the exact solution obtained
by Bariev \cite{bariev} for the order-parameter profile.
The explicit result (\ref{asympt}) holds in the limit
$h_1\to 0$. This limit is hard to access in the MC simulation
since the signal $m(z)$ becomes small and is
eventually lost in the noise.
To obtain a concrete result to compare with, we have calculated
the profile numerically from (\ref{bariev}) terminating the
series in (\ref{exact2})
after the third term. It turned out that
the series converges rather quickly as long as
the distance from the surface is not too small.
Only very close to the surface
higher orders need to be taken into account.
Concretely, we took $K=0.999 K_c$ (i.e. $\tau \simeq 0.001$) and $h_1=0.01$.
Then, employing (\ref{correl}) and
(\ref{bariev}) one obtains
$\xi=567$ and $l_1=2069$, respectively. The result
of the numerical evaluation of (\ref{bariev}) is shown in Fig.\,2
(dashed curve), where $m(z)$ versus the distance $z\equiv n-1$
is plotted.
Then, with exactly
the same parameters, the MC profiles were calculated,
with the size $N$ varying between 128 and
1024. The results are depicted in double-logarithmic representation
in Fig.\,2 (solid curves). It is obvious that the MC data approach
the (approximated, in principle) exact profile
with increasing lattice size. Most importantly, both
results show the short-distance behavior anticipated from the
scaling analysis above and expressed in (\ref{sdbe}).
This is demonstrated by the dotted line, which depicts
the function $0.016\, n^{3/8}\,(6.15-\mbox{ln}\,n)$ (the
constants were fitted). As expected, it only describes
the profile for short distances from the boundary and becomes
wrong for large distances, completely analogous to the asymptotic
form (\ref{m1h1}) of $m_1(h_1)$.
\\[2mm]
\def\epsfsize#1#2{0.6#1}
\hspace*{2.5cm}\epsfbox{fig2.eps}\\[0mm]
{\small {\bf Fig.\,2}: Monte Carlo profiles for $K/K_c=0.001$ and $h_1=0.01$
for lattices of size 128$\times$512, 256$\times$1024, 512$\times$2048, and 1024$\times$4096 (solid lines from below to above) compared with
the numerical evaluation of the exact result (\ref{bariev}). The latter
holds for the semi-infinite system. It is clearly visible that the
Monte Carlo data approach the exact result for increasibng system
size. The dotted line represents the
asymptotic ($z\to 0$)
behavior expressed in Eq.\,(\ref{sdbe}) $\sim z\,\mbox{ln}\,z$
that becomes wrong for
large distances. }\\[-.8cm]
\subsection{Monte Carlo results}
\label{threethree}
First we discuss a set of profiles which were obtained with
$N=512$
by setting $h_1=0.01$ and $K/K_c$ varying between 0.996 to
1.004, in steps of 0.001. The data are depicted in Fig.\,3.
Depending on the temperature, we averaged over 10\,000 to
30\,000 configurations. Especially the shape
of the critical profile, marked
in Fig.\,3, is consistent with
the scaling analysis of Sec.\,\ref{twotwo}.
It increases up to $z\simeq 60$, then has a maximum
and farther away from the surface it decays.
With increasing distance the influence of the second surface
(here at $z=2047$) becomes stronger, such that the profile
has a minimum about halfway between the boundaries. (The data
of Fig.\,3 were not symmetrized after the average over configurations
was taken.)
For $T$ above $T_c$ (curves below the critical profile), the
maximum moves towards the surface and the regime of growing
magnetization becomes
smaller. This is consistent with the scaling analysis
of Sec.\,\ref{twoone}, in this case the growth is limited
by the correlation length $\xi$.
On the other hand, for $T<T_c$, the tendency to decay
in between the surfaces becomes weaker, as the the bulk
value of the magnetization in the ordered phase grows.
As said in Sec.\,\ref{twoone}, the short-distance
growth below $T_c$ is described by $z^{x_1-x_{\phi}}$, where the
difference $x_1-x_{\phi}$ takes also the exact
value 3/8 \cite{foot5}. This time, there is no logarithm
however, and the growth is steeper than above $T_c$.
\\[2mm]
\def\epsfsize#1#2{0.6#1}
\hspace*{2.5cm}\epsfbox{fig3.eps}\\[0mm]
{\small {\bf Fig.\,3}: Order-parameter profiles for various temperatures
below and above $T_c$ for fixed $h_1=0.01$ and $N=512$
compared with the critical profile.
The three curves
above the critical profile
were obtained with $K/K_c = 1.001$, 1.002, 1.003.
The four profiles below correspond to
$K/K_c=0.999 \ldots 0.996$. The data are not symmetrized.}\\
Fig.\,4 shows the MC results for the critical point, for
different values of $h_1$ in double-linear representation.
In Fig.\,5 the same data are plotted double-logarithmically.
The lower dashed line shows the short-distance behavior $\sim z^{3/8}\,
\mbox{ln}\,z$ according to
Eq.\,(\ref{sdbe}) and (as already discussed in connection with
Fig.\,2) the MC profiles
confirm the scaling analysis of Sec.\,\ref{twotwo}.
The upper dashed line is the pure power law $z^{-1/8}$ that
describes the decay in the regime where $z$ is
larger than $l_1$ but still smaller
than $L$ or $\xi$. In our simulation this regime is only reached for
relatively large values of $h_1\gtrsim 0.5$. The uppermost profile ($h_1=1.0$)
obviously goes through
this regime for $10\lesssim z\lesssim 60$. The finite-size
exponential behavior (see Sec.\ref{twothree}) can be observed in all
curves for $z\gtrsim 100$. The data of Figs.\,4 and 5 are symmetrized, i.e.,
after averaging over configurations we computed the mean value
of the left and right halfs of the system. Hence, the profiles
are only displayed up to halfway between the
boundaries (here $z=1023$).
The location of the maximum of $m(z)$,
$z_{\rm max}$, as a function of $h_1$
is depicted in Fig.\,6. The maximum $z_{\rm max}$ was determined from the
profiles by a graphical method. Error bars are estimated.
For small values of $h_1$ the near-surface growth
is limited by finite-size effects.
Up to about $h_1=0.03$, the value of
$z_{\rm max}$
is roughly independent of $h_1$.
For larger values of $h_1$, $z_{\rm max}$
moves towards the surface. As indicated by the dashed line, the
dependence of $z_{\rm max}$ on $h_1$ is completely consistent with
$l_1 \sim h_1^{-2}$ obtained from the scaling analysis (see (\ref{length})).
For $h_1=1.0$ (upper curve) the maximum value
of $m(z)$ is at the boundary, and the
magnetization monotonously decays for $z>0$.\\[2mm]
\def\epsfsize#1#2{0.6#1}
\hspace*{2.5cm}\epsfbox{fig4.eps}\\[0mm]
{\small {\bf Fig.\,4}: Monte Carlo profiles for $N=512$ at $T=T_c$ for
$h_1=0.005$, 0.01, 0.02, 0.03, 0.07, 0.1, and 1.0
(from bottom to top).
The data are symmetrized and are
only shown up to $z=1023$.} \\[0.1cm]
\def\epsfsize#1#2{0.6#1}
\hspace*{2.5cm}\epsfbox{fig5.eps}\\[0mm]
{\small {\bf Fig.\,5}: The same profiles as in Fig.\,4 in double-logarithmic
representation, pronouncing the short-distance behavior.
For small $h_1$ (lower curves) the growth of $m(z)$ described
by Eq.\,(\ref{sdbe}) is clearly visible
For $h_1=1.0$ our Monte Carlo
result (upper solid line) is in accord with $m(z)\sim z^{-1/8}$
(dashed line). The behavior of the Monte Carlo data for $z\gtrsim 100$
is described by the (finite-size) exponential decay and corrections
to the semi-infinite profiles due to the second boundary become stronger.} \\[0.1cm]
\def\epsfsize#1#2{0.6#1}
\hspace*{2.5cm}\epsfbox{fig6.eps}\\[0mm]
{\small {\bf Fig.\,6}: The location of the maximum $z_{\rm max}$ of $m(z)$
in dependence of $h_1$ as obtained from the data of Fig.\,4 and
other profiles for the same system
(not displayed). The dashed line represents
$l_1\sim h_1^{-2}$. For larger values of $h_1$ we have
$z_{\rm max}\sim l_1$. For small $h_1$ where $l_1$
becomes larger than the finite-size scale $L$, $z_{\rm max}$
is determined by $L$.} \\[0.1cm]
\def\epsfsize#1#2{0.6#1}
\hspace*{2.5cm}\epsfbox{fig7.eps}\\[0mm]
{\small {\bf Fig.\,7}: Order-parameter profiles for $K=K_c$ and $h_1=0.01$
for system size $N=128$, 256, 512, 1024, and 2048.
} \\[0.1cm]
In Fig.\,7 profiles for fixed $h_1$ and $T=T_c$ for
various system sizes (ranging between $N=128$ and 2048) are displayed.
With increasing $N$, the maximum keeps moving away from the
surface, $z_{\rm max}\simeq$ in the largest system.
This means that with these parameters we are still in the regime
where $L<l_1$. When $h_1=0.03$ is taken instead, the situation
is different. The respective MC data are depicted in
Fig.\,8. For the small system we have again increasing $z_{\rm max}$.
For the two largest systems $N=1024$ and 2048, however, the
maximum is roughly located at the same
distance from the surface, signaling that now $l_1<L$.\\[3mm]
\def\epsfsize#1#2{0.6#1}
\hspace*{2.5cm}\epsfbox{fig8.eps}\\[0mm]
{\small {\bf Fig.\,8}: Profiles for $h_1=0.03$ and otherwise
the same parameters as in Fig.\,7.
} \\[0.1cm]
\section{Discussion} \label{disc}
We studied the short-distance behavior of order-parameter profiles
in the two-dimensional semi-infinite Ising system at or above
the bulk critical temperature. Our main goals
were a detailed understanding of the near-surface
behavior of the order parameter and
a complete scenario for the crossover between
``ordinary'' and ``normal'' transition.
With regard to the short-distance behavior, especially for
small surface fields $h_1$, the Ising model in $d=2$
turned out to be quite special. The
functional form is
quantitatively not captured by
the analysis of Ref.\,\cite{czeri},
where the main emphasis was put on the situation in $d=3$.
Here we demonstrated by means of a scaling analysis
that a small $h_1$ induces a magnetization
in a surface-near regime that grows as $z^{3/8}\,\mbox{ln}\,z$
as the distance $z$ increases away from the surface
(see Eq.\,(\ref{sdbe})).
Our findings are not only consistent
with available exact results (to which we made a detailed comparison),
but they also allow a detailed understanding of the
physical reasons for the growth and its quantitative
form. The surface magnetization $m_1$ generates in the region that is
(on macroscopic scales) close to the surface and that
is much more suceptible than the surface itself, a magnetization
$m(z)$ much larger than $m_1$.
The exponent $3/8$ that governs the power-law part of the growth
is the difference between the scaling dimensions $y_1$ (of $h_1$)
and $x_{\phi}$ (of the bulk magnetization).
Eventually, as was demonstrated Sec.\,\ref{twotwo},
the logarithmic factor can be traced back to the logarithmic
dependence of $m_1$ on $h_1$.
The scenario for the crossover
between ``ordinary'' and ``normal'' developed
in Ref.\,\cite{czeri} and generalized
in this work to include the 2-$d$ Ising model is
the following: At $T=T_c$ a small $h_1$ causes the increasing
near-surface behavior described above. The magnetization
grows up to distances $l_1\sim h_1^{-2}$ and then
the crossover to the power-law decay $\sim z^{-1/8}$ takes place.
With increasing $h_1$ the surface-near regime becomes smaller, and
eventually for $h_1\to\infty$ the length scale $l_1$ goes to zero, such
that the region with increasing magnetization vanishes completely
and the situation of the ``normal'' transition is reached.
For $T$ slightly above $T_c$ our scenario essentially
remains valid as long as
$\xi >l_1$. Only when $z$ is of the order of $\xi$ the
crossover to the exponential decay takes place. For $\xi < l_1$,
on the other hand, the
growth is limited to the region $z<\xi$.
Concerning three-dimensional systems
several experiments were pointed out in Ref.\,\cite{czeri},
whose results are possibly related
related to the anomalous short-distance behavior \cite{mail,franck}.
It would be an interesting question for the future, whether
similar, surface-sensitive
experiments in two-dimensional systems are feasible.
{\small {\it Acknowledgements}: We thank R. Z. Bariev, E. Eisenriegler,
and M. E. Fisher for helpful comments.
This work was supported in part by the Deutsche Forschungsgemeinschaft
through Sonderforschungsbereich 237.}
|
proofpile-arXiv_065-652
|
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
There has been recent activity in the area of
weak-scale supersymmetry, spurred on by a number of suggestive
experimental results.
First, there is the single
$ee\gamma\gamma +
\thinspace\thinspace{\not{\negthinspace\negthinspace E}}_T$ event
observed by CDF\cite{eegamgam}. This particular event does
not seem to have a Standard Model interpretation.
Also, in supersymmetry, the $Z\rightarrow b\bar{b}$ rate
($R_b = \Gamma(Z \rightarrow b \bar b) /
\Gamma(Z \rightarrow {\rm hadrons})$),
the value of $\alpha_s$ extracted from $\Gamma_Z$, and
the branching ratio for $b\rightarrow s\gamma$ are all
affected by loop diagrams containing charginos and stop squarks.
At present \cite{Warsaw}, all three of these quantities are
1.5--2.0$\sigma$ from their Standard Model predictions,
each in precisely the directions expected from supersymmetry
\cite{Warsaw,Shifman,RbA,KaneKoldaWells,CDM,Global}
if there is a light stop squark.
Remarkably, these
experimentally independent ``mysteries'' can all be
explained by a single reasonably well-determined set of parameters
within the framework of weak scale supersymmetry.
The $ee\gamma\gamma +
\thinspace\thinspace{\not{\negthinspace\negthinspace E}}_T$ event
has a natural interpretation
in terms of selectron pair production.
Two different scenarios are possible, depending upon whether
the lightest supersymmetric particle (LSP) is a gravitino
\cite{selectronA,selectronB}
or Higgsino-like neutralino\cite{selectronA}.
The neutralino and chargino parameters suggested by the
second scenario overlap with the values required to
account for the $R_b$ difference\cite{RbA},
provided that one of the stop squark eigenstates is
light ($\widetilde{M}_{t} \alt m_W$).
One might be concerned that such a low stop mass
would have undesirable side-effects. Indeed, an immediate
consequence\cite{KaneKoldaWells} is that the decays
\begin{equation}
t \rightarrow \tilde{t}\neut{i}
\label{TopToStopDecay}
\end{equation}
where $\tilde{t}$ is the lighter of the two stop mass eigenstates
and $N_i$ is a kinematically accessible
neutralino mass eigenstate should
occur with a total branching ratio in the
neighborhood of 50\%.
The consequences of this depend on how the stop
decays.
When at least one chargino is light enough, the decay
\begin{equation}
\tilde{t} \rightarrow C_i b
\label{OtherDecay}
\end{equation}
dominates~\cite{StopPhenom}.
In this case, the subsequent chargino decay
to a fermion-antifermion pair plus neutralino leads to
a $\neut{1}\neut{1}f \bar{f}' b$ final state whenever
a top undergoes the decay~(\ref{TopToStopDecay}). Since the
final state for the SM decay is identical except for the (invisible)
neutralinos, there is potential for both direct stop decays as
well as top to stop decays to mimic ordinary top decays.
This possibility has been investigated
by several authors\cite{StopPhenom,BST,Lopez,Abraham,Sender}.
In particular, we note that
Sender~\cite{Sender} finds that
models with a ``large'' ${\cal B}(t \rightarrow \tilde{t}\neut{1})$
have not been ruled out by Tevatron data, provided the stop
decays according to~(\ref{OtherDecay}).
This scenario, although interesting, is not the focus of this
paper. Instead, we wish to examine the situation where
the one-loop decay
\begin{equation}
\tilde{t} \rightarrow \neut{1} c
\label{OneLoop}
\end{equation}
dominates, which happens when the decay~(\ref{OtherDecay}) is
kinematically forbidden~\cite{FCNC}. In this case,
a top undergoing the decay~(\ref{TopToStopDecay}) would
produce a $c\neut{1}\neut{1}$ final state and
be effectively invisible to standard searches.
Two independent analyses appropriate to this case
have been performed~\cite{Sender,Yuan}
which conclude that ${\cal B}(t \rightarrow X)$
where $X\ne Wb$ is at most 20--25\%.
However, neither analysis accounts for the possibility
that supersymmetry can lead to additional sources of
top quarks, without the need for stop decays masquerading as
top decays~\cite{XtraTops}. For example, if the gluino
is lighter than the other (non-stop) squark flavors, but
heavier than ${m}_{t} + \widetilde{M}_{t}$, then it decays
exclusively via~\cite{StopPhenom}
\begin{equation}
\tilde{g} \rightarrow t \tilde{t}^{-}, \bar{t}\tilde{t}^{+},
\label{GluinosToTheRescue}
\end{equation}
making the production of gluinos a source
of top quarks~\cite{XtraTops,Kon}.
In fact, the authors of Ref.~\cite{XtraTops} argue that there is
indirect evidence for the decays~(\ref{TopToStopDecay}),
(\ref{OneLoop}), and (\ref{GluinosToTheRescue}) in the
Fermilab data on top rates and distributions.
In light of the indirect hints at weak scale supersymmetry, it
is important to take every opportunity to
obtain some direct evidence that nature is indeed supersymmetric,
or, to show that it is not.
A discussion of search strategies for the direct production
of stop pairs at the Tevatron already exists
in the literature\cite{BST}.
The authors of Ref.~\cite{BST} claim that a stop
squark with a mass of up to about 100 GeV
should be visible at the Tevatron in the $\ge2$ jets plus
missing transverse energy channel given 100 pb$^{-1}$ of data.
Nevertheless,
we feel that it is beneficial to augment the direct search
with a search for stops coming from top decay:
observation of a signal in both channels would greatly
boost the case for SUSY.
To this end, we present a method that may facilitate the
search for top to stop decay at the Fermilab Tevatron.
Our method consists of defining a
superweight $\widetilde{\cal X}$ whose
function is to discriminate between signal and background events.
The superweight is constructed from various observables
in the events so that it is ``large'' for the signal events
and ``small'' for the background.
We will illustrate the superweight method using the case
of the stops coming from top quark decays; it can also
be applied to stop pair production. Although the required
analysis is not easy, it should be possible to determine
directly whether about half of all tops indeed decay to stops.
Our goal in this paper is to help this process.
The authors of Ref.~\cite{Yuan} have also looked at the
problem of searching for stops from top decay at the Tevatron.
However, they employ the traditional method of cutting
only on the kinematic observables in the event.
As a result, their signal efficiencies are rather low
(6\%--8\%). Also, they do not include the possibility of
SUSY-induced $t \bar t$ production. Consequently, they conclude
the prospects for observing a signal, if present, are
most promising at an {\it upgraded}\ Tevatron.
As we shall see, the superweight method allows us to reduce
the backgrounds to a level similar to that of Ref.~\cite{Yuan},
but with efficiencies as high as 16\%, providing for
the possibility of finding a signal in the {\it current}\ data set.
The D0 experiment at Fermilab as well as the various LEP
experiments have reported limits based upon searches
for the pair production of stop squarks~\cite{D0stop,LEPstop}
(see Fig.~\ref{exclusion}).
In the event that the current
run at LEP finds a stop signal, the confirmation process
could be greatly aided by the Tevatron data, depending
upon the stop and LSP masses. On the other hand, even if
LEP sees nothing, there is still a significant region in the
stop-LSP mass plane to which the Tevatron is sensitive and
which will not have been excluded by LEP.
The remainder of the paper is organized as follows:
in Sec.~\ref{Features} we
briefly examine the generic features of SUSY models
hinted at by the data,
and determine the experimental signature we will concentrate on.
Sec.~\ref{SUwgt} contains a general discussion of the superweight
and the methods by which it is constructed.
We discuss the detection of the decay $t \rightarrow \tilde{t}\neut1$
in Sec.~\ref{BasicSignal}, within the framework of a simplified
model where no other neutralinos are light enough to be
produced, and where only SM $t \bar t$ production mechanisms
are considered. Such an analysis is appropriate for any
SUSY model which contains the decays~(\ref{TopToStopDecay})
and~(\ref{OneLoop}), whether or not there are extra sources
of top quarks.
We expand our discussion to include the other neutralinos
and SUSY $t \bar t$ production mechanisms in Sec.~\ref{FullSignal}.
Finally, Sec.~\ref{CONCLUSIONS}
contains our conclusions; Fig.~\ref{Nevents} summarizes our
results.
\section{Stops from Top Decay} \label{Features}
\subsection{A SUSY Model}
In this section we flesh-out the
supersymmetric scenario described in the introduction.
Specifically, the picture implied by
Refs.~\cite{RbA,selectronA,XtraTops} contains a Higgsino-like
neutralino $\neut1$ with a mass in the 30 to 55 GeV range,
a light stop squark with a mass in the 45 to 90 GeV range,
a gluino with a mass in the 210 to 250 GeV range,
and $\tilde{u},\tilde{d},\tilde{s},\tilde{c}$ squarks
in the 225 to 275 GeV range. The ``heavy'' stop eigenstate
as well as both $\tilde{b}$ squark eigenstates
may be heavier.
We further assume that the ``light'' stop eigenstate
is lighter than the charginos, and that the gluino is lighter
than the squarks (except for $\tilde{t}$).
The stop and the lighter chargino could be
approximately degenerate; we ignore such a complication here.
We take the top quark mass
to be 163 GeV.
With these masses and couplings, the decays and branching ratios
relevant to our study are
\begin{equation}
{\cal B}(\tilde{q} \rightarrow q\tilde{g}) \sim 25\% \hbox{--} 75\%
\label{SquarkDecay}
\end{equation}
\begin{equation}
{\cal B}(\tilde{g} \rightarrow t\tilde{t}^{-}) =
{\cal B}(\tilde{g}\rightarrow\bar{t}\thinspace\tilde{t}^{+}) = 50\%
\end{equation}
\begin{equation}
{\cal B}(t \rightarrow Wb) \sim
{\cal B}(t \rightarrow \tilde{t}\neut{i}) \sim 50\%
\label{ThreeThree}
\end{equation}
\begin{equation}
{\cal B}(\tilde{t} \rightarrow c\neut{1}) \sim 100 \%.
\end{equation}
The large variation in the branching ratio for squark decays
is a consequence of the relatively small phase space available for
producing gluinos; hence, 2-body decays to the electroweak
superpartners are able to compete effectively with the strong decays.
The gluinos, however, have no other 2-body decay modes: if
top-stop is open, it dominates.
As indicated by~(\ref{ThreeThree}), the total branching
ratio for top to all of the kinematically accessible neutralinos
is about 50\%.
In these models, $N_1$ is the LSP and
assumed stable. The interpretation of the $ee\gamma\gamma$
which inspired our closer examination of this particular
region of parameter space requires
\begin{equation}
{\cal B}(\neut{2} \rightarrow \neut{1}\gamma) > 50\%.
\end{equation}
In principle, one could look for this photon as an aid
in selecting events with $t \rightarrow\tilde{t}\neut{2}$.
However,
because of the large photino content of the $N_2$,
its production in top decays is suppressed compared to
$N_1$ or $N_3$. So rather than concentrate on
a small fraction of events, we make no attempt to
identify the photon in our study, and instead allow it
to mimic a jet.
Quite often, $N_3$ is also light enough to be produced
by decaying tops.
Its decays are
more complicated: if the sneutrinos happen to be light
enough to provide a 2-body channel, then $\tilde{\nu}\nu$ is favored;
otherwise, the 3-body
decays $\neut{1}f\bar{f}$ where $f$ is a light fermion
dominate. The net result is of all of this is that
allowing the top to decay to SUSY states other than $N_1$
simply adds additional (relatively) soft jets to the final
state.
\subsection{The Supersymmetric Signal}
Within the supersymmetric scenario proposed in Ref.~\cite{XtraTops},
there are several different production mechanisms for top
quarks, and hence many different final states which must be
considered.
For the top pairs produced by the usual SM processes,
we end up with mainly three different final states, depending upon
the way in which they decay.
Firstly, both tops could decay to $Wb$, according to the Standard Model.
In this case, the final state consists of
2 leptons, 2 jets, and missing $p_T$ (dilepton);
1 lepton, 4 jets, and missing $p_T$ ($W$ + 4 jets); or
6 jets (all jets).
Secondly, one top could decay to $Wb$, and the other
to $\tilde{t}\neut1$.
In this case, the final state consists of
1 lepton, 2 jets, and missing $p_T$ ($W$ + 2 jets);
or 4 jets and missing $p_T$ (missing + 4 jets).
Finally, both tops could decay to $\tilde{t}\neut1$. In this
case the final state consists entirely of 2 jets and missing
$p_T$ (missing + 2 jets). This last final state is identical
to that of direct stop pair production, which is considerably
more difficult because of the large
Standard Model multijets background.
Of the remaining final states coming
from supersymmetric sources, the $W$ + 2 jets mode is
the most promising, and the one we will discuss in detail.
In our illustrations, we will describe the situation where
the top decays to a supersymmetric final state, and
the antitop decays according to the Standard Model:
\begin{equation}
p \bar p \rightarrow t \bar t \rightarrow c \neut1 \neut1 \bar b
\ell^{+} \bar\nu_\ell.
\label{TopToStopSignal}
\end{equation}
The presence of the charge-conjugated process is always implicitly
assumed, and is included in all of the rates reported below.
In addition to~(\ref{TopToStopSignal}),
we must consider the effect of
top quarks arising from gluino and squark decays, which,
as argued in \cite{XtraTops}, must be present in significant
numbers if the non-$Wb$ top quark branching ratio is to be
as large as is typical for a light stop. Top quarks produced
in this manner are accompanied by extra jets. Consider
first the pair production of gluinos. Both gluinos will decay
to a top and a stop. The tops then decay
as described above, and the stops each yield a charm jet
and a neutralino.
Thus,
gluino pair production leads to the same final states as top
pair production, but with two additional charm jets and
additional missing energy.
Likewise, for the chain beginning with a squark,
we pick up an additional jet from the decay~(\ref{SquarkDecay}).
For a summary of conventional gluino physics at FNAL, see
Ref.~\cite{Haber}.
It is useful to examine some kinematical consequences of the
scenario we have proposed.
Consider first purely SM $t \bar t$ production at the Tevatron,
which takes place
relatively close to threshold.
We would expect
the ordering in $E_T$ of the $\bar b$ and $c$ jets
to reflect fairly accurately
the relative sizes of the $t$-$W$
and $\tilde{t}$-$\neut1$ mass splittings. For the range of masses
we consider here,
$ m_t - m_W > \widetilde{M}_{t} - \mneut1. $
Hence, the highest $E_T$ jet should come
from the $\bar b$ quark most of the time.
Our simulations confirm this, with the $\bar b$ quark
becoming the leading jet more than 70\% of the time over most
of the range of SUSY masses examined.
Fig.~\ref{BfractLEGO} shows the results for the kinematically
allowed masses in the ranges
$30 {\rm \enspace GeV} < \mneut{1} < 70 {\rm \enspace GeV}$,
$45 {\rm \enspace GeV} < \widetilde{M}_{t} < 100 {\rm \enspace GeV}$.
The situation is only slightly worse when we add squark and gluino
production. The additional jets from the cascade decays down
to top are rather soft, given the relatively small mass splittings
involved. Thus, in the all of the cases we examine here,
the identification of the $\bar b$ parton with the leading
jet is a reasonably good one.
Because the jets coming from gluino and squark decays are
relatively soft,
we will organize our results around the premise that the
process~(\ref{TopToStopSignal})
is the framework about which
the complications from such decays are relatively small perturbations.
That is, we will first describe the situation as if the
only processes going on are the SM backgrounds plus decays
of top to stop, parameterized by the stop mass, the LSP mass,
and the branching ratio ${\cal B}(t\rightarrow\tilde{t}\neut1)$
(Sec.~\ref{BasicSignal}).
Then, we will expand our consideration to include a full-blown
SUSY model where
additional top quarks are being produced by decaying gluinos
(Sec.~\ref{FullSignal}). As we shall see,
the same superweight derived under the simplifying assumptions
works well in the more realistic environment.
\section{The Superweight} \label{SUwgt}
We now describe a procedure which may be employed to
construct a quantity we call the ``superweight'' out
of the various observables associated with a given process.
In principle, this procedure may be used to differentiate
between signal and background in a wide range of processes,
although we will concentrate on the detection of the
decay~(\ref{TopToStopDecay}).
For each event in the data sample passing our selection criteria
(correct number and stiffness of jets, sufficient missing
energy, correct number of leptons, {\it etc.})
we define a superweight $\widetilde{\cal X}$ by a sum of the form
\begin{equation}
\widetilde{\cal X} = \sum_{i=1}^{N} {\cal C}_i
\label{SUwgtGeneric}
\end{equation}
where the ${\cal C}_i$'s evaluate to 0 or 1 depending upon whether
or not some given criterion is satisfied.
The number of terms $N$ in the sum defining $\widetilde{\cal X}$ is
arbitrary: one should use as many terms as there
are ``good'' criteria.
One could
consider a more general form including separate weighting factors
for the components and continuous values for the ${\cal C}_i$'s
(as in a full-blown neural net analysis). However, our intent
is to search for new particles over some range of masses
and couplings. In such a situation, too much refinement
could narrow the range of parameters to which the superweight
is a good discriminant between signal and background.
Furthermore,
the components appearing in (\ref{SUwgtGeneric})
are easily given a physical interpretation, which guides
us in the optimization of the ${\cal C}$'s.
Let us consider a criterion of the form
\begin{equation}
{\cal C} = \cases{1, & if $ {\cal Q} > {\cal Q}_0$; \cr
0, & otherwise, \cr}
\label{Crit}
\end{equation}
where ${\cal Q}$ is some measurable quantity associated with
the event, and ${\cal Q}_0$ is the cut point\cite{OtherWay}.
Although we will refer to ${\cal Q}_0$ as a cut point, we
don't actually cut events from the sample which
have ${\cal C}=0$.
Note that~(\ref{Crit}) implies that the value of ${\cal C}$
averaged over the entire sample is exactly the fraction of
the cross section satisfying the constraint ${\cal Q} > {\cal Q}_0$.
A ``good'' superweight component should have the property that
its average for Standard Model events is much less than its average
for SUSY events. That is, we want
\begin{equation}
\Delta{\cal C} \equiv \langle{\cal C}\rangle_{\rm SUSY}
-\langle{\cal C}\rangle_{\rm SM}
\label{IFdiff}
\end{equation}
to be as large as possible.
So, to develop a new superweight, one should first devise a set
of cuts to produce a data set
where the number of background events versus the number of
signal events is reasonable (S:B of order 1:4, say).
Next, separate Monte Carlos of
both the signal and main backgrounds should be run, in order
to generate plots of $\Delta{\cal C}$ as a function of ${\cal Q}_0$.
The physical interpretation of $\langle{\cal C}\rangle$ as the
fraction of events satisfying ${\cal Q} > {\cal Q}_0$ may be used as a
guide when deciding which ${\cal Q}_0$'s are worth investigating.
For each value of the new physics parameters, there will be
an ideal value of ${\cal Q}_0$ for which $\vert\Delta{\cal C}\vert$
is maximal. A good superweight component should not only
have a ``large'' value of $\vert\Delta{\cal C}\vert$, but the
corresponding value of ${\cal Q}_0$ at that point should be reasonably
stable over the entire parameter space to be investigated.
An issue that arises concerns the question of correlations
among the ${\cal C}_i$'s. Our philosophy in this respect is
to evaluate the effectiveness of the superweight in terms of
how well it separates the signal from the background, {\it i.e.}\ what
is the purity of an event sample with a certain minimum
superweight? Thus, while we avoid using two ${\cal C}_i$'s whose
values are 100\% correlated (on the grounds that doing
so is no more beneficial than using only one of the two),
we don't worry about using partially correlated ${\cal C}_i$'s.
The main effect of correlations among the ${\cal C}_i$'s is
that the overall performance of the sum of the ${\cal C}_i$'s will
be less than what is implied by
considering the ${\cal C}_i$'s individually.
Thus, to evaluate the effectiveness of a given superweight definition,
one should compare the predicted distributions
in $\widetilde{\cal X}$ for the signal and background.
We now give an example of the steps used to determine
one of the superweight elements for the
$t \rightarrow\tilde{t}\neut1$ search method described in
detail in Sec.~\ref{BasicSignal}. To begin, we take a moment to
recall the definition of the transverse mass.
Given particles
of momenta $P$ and $Q$, the transverse mass of the pair
is defined by
\begin{equation}
m_T^2(P,Q) = 2 P_T Q_T [ 1 - \cos {\it\Phi}_{PQ}],
\label{transX}
\end{equation}
where $P_T \equiv \sqrt{P_x^2+P_y^2}$ and ${\it\Phi}_{PQ}$
is the azimuthal opening angle between $P$ and $Q$.
An important feature of the transverse mass is that if the
particles $P$ and $Q$ were produced in the decay of some
parent particle $X$, then the maximum value of $m_T(P,Q)$ is
precisely the mass of $X$.
As already discussed in Sec.~\ref{Features},
the signal~(\ref{TopToStopSignal})
for top to stop
appears in the detector as a charged lepton,
2 jets, and missing energy.
The largest background turns out to be
the Standard Model production of a $W$ plus 2 jets, so we determine our
superweight criteria using that background.
Furthermore, we know that for signal events, the
leading jet is usually from the $\bar{b}$ quark in the
$\bar{t}\rightarrow W^{-} \bar{b}$ decay.
Consequently, most of the time the leading jet and
the charged lepton
should reconstruct to no more than the top quark mass
(some energy and momentum is carried away by the unseen neutrino).
This suggests
an upper limit on the value of $m_T(j_1,\ell)$, which may
violated at least some of the time by ordinary $W$ plus 2 jet
events.
In Fig.~\ref{SuperPlot}, we show the differential cross section
in $m_T(j_1,\ell)$
for both the signal and the background, as determined
from VECBOS\cite{Vecbos} (relevant details of our simulations
will be discussed in Sec.~\ref{BasicSignal}).
Note that for the signal there is the expected sharp drop-off
for large values of $m_T(j_1,\ell)$.
In Fig.~\ref{AAPlot} we show the
fraction of events with a $j_1\ell$ transverse
mass above $m_T(j_1,\ell)$,
as a function of $m_T(j_1,\ell)$.
The individual magnitudes of these two
curves are not critical in making a good superweight component,
but rather the difference in these two curves, which is plotted
in Fig.~\ref{DiffPlot} not only for the masses used in
Figs.~\ref{SuperPlot} and~\ref{AAPlot}, but also for two additional
values as well. The presence of a dip ranging in depth
from about $-0.4$ to $-0.5$ in the vicinity
of $m_T(j_1,\ell) = 125 {\rm \enspace GeV}$ for each of the
masses used suggests that
this is indeed a worthwhile superweight element, and that
the criterion should read
\begin{equation}
{\cal C} = \cases{1, & if $ m_T(j_1,\ell) < 125 {\rm \enspace GeV}$; \cr
0, & otherwise. \cr}
\end{equation}
The key quantities to look
for in this evaluation were approximate stability in peak (dip)
position and ``large'' magnitude for the peak for the range of
parameters to be investigated. Narrowness of the peak is not
a requirement. In fact, a broad peak is better, since then the
exact placement of the cut point is unimportant.
Note also that since we are exploiting the difference in the
{\it shapes}\ of the signal and background distributions,
there is no reason we can't use an observable both for cutting
and in the superweight. For example, even after requiring
a minimum missing transverse momentum, we can (and do) still use a
superweight criterion based on the
shapes of the missing transverse momentum
distributions for the surviving events.
\section{The Process $\noexpand\lowercase{p}
\bar{\noexpand\lowercase{p}} \rightarrow
\noexpand\lowercase{t} \bar{\noexpand\lowercase{t}}
\rightarrow \noexpand\lowercase{c}
\neut1 \neut1 \bar{\noexpand\lowercase{b}}
\ell^{+} \bar\nu_\ell$} \label{BasicSignal}
Our simulations of the signal and backgrounds in this section
are based upon tree level matrix elements, with the hard-scattering
scale for the structure functions and first-order running
$\alpha_s$ set to the partonic center of mass energy.
For vector-boson plus jet production, we employ
VECBOS~\cite{Vecbos} running with the structure
functions of Martin, {\it et. al.}~\cite{BCDMS} (the ``BCDMS fit'').
For the processes containing top pairs, we
perform a Monte Carlo integration of the matrix element
folded with the HMRS(B) structure functions~\cite{MRSEB}.
Under these conditions, the tree-level SM $t \bar t$ production
cross section is 5.1 pb for 163 GeV top quarks,
while two recent computations of the NLO rate including
the effects of multiple soft gluon emission give
$6.95^{+1.07}_{-0.91}$ pb \cite{CERNnlo}
and $8.12^{+0.12}_{-0.66}$ pb \cite{ANLnlo}
for this mass, implying a $K$ factor in the 1.4 to 1.6 range.
We refrain from applying any
$K$ factor to the rates we report below, although
the reader may wish to do so. On the the other hand, we do
use a somewhat light value of $m_t$ (163 GeV).
Hadronization and detector effects are mocked up by applying
gaussian smearing with a width of
$125\%/\sqrt{E} \oplus 2.5\%$. When the simulation of merging jets
is called for, we combine final state partons which lie within
0.4 units of each other in $(\eta,\phi)$ space.
Since our intent is to demonstrate that the superweight method
is viable, we have avoided detailed simulation of the CDF
or D0 detectors. Instead, we have tried to capture enough
of the general features in order to demonstrate the
viability of the method. Of course, the superweight criteria
used in an actual analysis should be determined by the
experimenters from a complete detector simulation.
\subsection{Discussion of Backgrounds}
There are several ways to mimic our signal of a hard lepton,
missing $E_T$, and two (or more) jets within the Standard Model.
The most obvious background process, and the one with the
largest raw cross section is the direct production of a $W$
plus 2 jets.
However, we can also have
contributions from $Z$ plus 2 jets should one of the leptons
be missed by the detector.
Furthermore, we must beware of Standard Model sources of
top quarks. In the context of $t \bar t$ production,
the dilepton mode can fake the signal if one of the two leptons
is lost, which is particularly likely if one of the $W$'s decays
to $\tau\nu$.
Since $\tau$ leptons,
can appear as either a jet of hadrons plus missing momentum
($\tau \rightarrow j \nu_\tau$) or as a lepton plus missing
momentum ($\tau \rightarrow \ell\bar\nu_\ell\nu_\tau$),
we have been careful to study these backgrounds separately.
The $W$ + 4 jets mode is also a potential troublespot,
since jets can merge or simply be too soft to be detected.
Finally,
single top production followed by SM top decay leads to a
final state of a $W$, two $b$ jets, and missing energy
(plus possibly an extra jet if $W$-gluon fusion is the production
mechanism). Fortunately, the small rate for single tops is effectively
dealt with by the cuts described below.
The cuts we impose on the data before embarking on our superweight
analysis are listed in Table~\ref{TopToStopCuts}.
The entries above the dividing line are our ``basic'' cuts.
They were inspired by the CDF top analysis\cite{CDFtop}, in
order to automatically incorporate some of the
coverage and sensitivity limitations
imposed by the detector, and to produce a ``clean'' sample
of events. Thus we require the lepton to have a minimum $p_T$
of 20 GeV, be centrally located ($|\eta|<1$)
and to lie at least 0.4 units in $\Delta R$ from the jets
($\Delta R \equiv \sqrt{ (\Delta\eta)^2 + (\Delta\varphi)^2}$).
The $p_T$ cut on the lepton aids in the rejection of taus
which decay leptonically.
Some discrimination against events with fake missing $E_T$ is
obtained by setting a minimum
$\thinspace\thinspace{\not{\negthinspace\negthinspace E}}_T$ of 20 GeV.
The leading two jets should each have a $p_T$ of at least 15 GeV,
and a pseudorapidity $|\eta|<2$.
All jets must have a minimum separation of 0.4 units in $\Delta R$.
To reject Standard Model
$t \bar{t} \rightarrow W + 4 \enspace{\rm jets}$ events,
we require that the third hardest jet have a {\it maximum}
$p_T$ of 10 GeV.
While effective in this task, such a cut does have the
unwanted side-effect of suppressing signal events containing
extra jets, such as those containing squarks
and gluinos~\cite{XtraTops}.
In addition, some signal events will contain extra jets because of
QCD radiation. Inclusion of either class of events in the
data sample requires the relaxation of this cut, as is
done in Sec.~\ref{FullSignal}. Here we note that the data
in Table~\ref{NevtFULL} imply that no more than 25\% of the
signal events contain extra jets above 10 GeV in $p_T$,
so the ultra-conservative reader may wish to reduce the signals
we report in this section by that amount. However, since we have
neglected a $K$ factor of 1.4--1.6 in our figures, we feel that
our values are indeed reasonable.
Table~\ref{TopToStopBk} lists the sources of background
discussed above along with the estimated cross section
surviving the cuts for each mode.
Note that we report the $t\bar{t}$ backgrounds as if
${\cal B}(t\rightarrow Wb)$ were unity: the actual
contributions to the background in the presence of a
signal are smaller by a factor of this branching ratio squared.
While the basic cuts are nearly adequate for most of the
backgrounds, the contribution
from $W+2$ jets is still an overwhelming 39.1 pb,
necessitating an additional cut.
Given an ideal detector, the
only source of missing momentum in a background $W$ + 2 jet event
is the neutrino from the decaying $W$. Hence,
the transverse mass of the charged lepton and missing momentum
(energy) must be less than or equal to the $W$ mass.
Allowing for the finite width of the $W$ as well as
detector resolution effects, a number of
events spill over into higher $m_T$ values.
In contrast, for
SUSY events given by~(\ref{TopToStopSignal}),
the presence of the two neutralinos in addition to the neutrino
frequently produces events with a transverse mass well above
$m_W$. Thus, we require that
$m_T(\ell,\thinspace{\not{\negthinspace p}}_T)>100 {\rm \enspace GeV}$.
This cut is highly effective against the $W$ + 2 jets background,
while preserving about half of the remaining signal.
It also removes the small contribution from single top production.
However, it is
less effective against the $t \bar t$ backgrounds,
especially those containing $\tau$'s. Fortunately,
those backgrounds are already under control.
When all of our cuts are imposed, the surviving background
is about 0.42 pb, nearly 90\% of which comes from
Standard Model production of a $W$ plus 2 jets.
Hence, we consider only that background in developing the
${\cal C}$'s that make up the superweight.
We plot the efficiency for retaining the signal in
Fig.~\ref{EffHi} as a function of the
stop and LSP masses, and supply numerical values for
several representative pairings in Table~\ref{TopToStopEff}.
\subsection{Construction of the Superweight}
In Table~\ref{TopToStopSUwgt} we list the 10 criteria used to
build the superweight for the process~(\ref{TopToStopSignal}),
in approximate order of decreasing usefulness.
We now provide intuitive explanations for our selections:
although
the exact placement of the cut points is determined from
the Monte Carlo, we should still be able to understand
from a physical point of view why criteria of the forms
listed are sensible.
We begin our discussion with the three criteria (${\cal C}_5$,
${\cal C}_8$, and ${\cal C}_9$) which depend on joint properties
of the charged lepton ($\ell$) and leading jet ($j_1$).
As already discussed in Sec.~\ref{Features}, the $\bar b$ quark
frequently becomes the leading jet. Since the $\bar b$ quark
and the charged lepton come from the same parent top quark,
not only would we expect an upper limit on the mass of the
pair (${\cal C}_9$, discussed previously), but there should be some
tendency for the lepton and jet 1 to align. On the
other hand, in Standard Model $W$ + 2 jet events, the $W$ is recoiling
against the two jets, leading to a tendency for the lepton
and jet 1 to {\it anti-}align. Hence, we adopt ${\cal C}_5$,
which contributes when the $j_1$-$\ell$ azimuthal angle
is less than 2.4 radians, and ${\cal C}_8$, which contributes
when the cosine of the $j_1$-$\ell$ opening angle is greater
than $-0.15$.
The next group of criteria (${\cal C}_1$, ${\cal C}_2$,
${\cal C}_6$, ${\cal C}_7$) are various combinations of the
transverse momenta in the event. Naturally, we make use of
the ``classic'' supersymmetric signature: the missing
transverse momentum (${\cal C}_1$), which we require to be
at least 65 GeV to add one unit to the superweight,
that being the point where the two integrated fractions
differ the most.
In addition, we make use
of the fact that Standard Model $W$ + 2 jets production
falls off rapidly
with increasing $p_T$; that is, we expect the lepton and jets
from the signal process to be somewhat harder on average.
Instead of the
individual $p_T$'s, however, we use their scalar sum with the
missing $p_T$. Admittedly, there are some correlations introduced
by this choice; however, as discussed in Sec.~\ref{SUwgt},
that is not important for our purposes.
The remaining criteria (${\cal C}_3$, ${\cal C}_4$ and ${\cal C}_{10}$)
may be described as ``miscellaneous.''
The first of these is tied to the difference between the
missing $p_T$ and charged lepton $p_T$,
\begin{equation}
\Delta{\cal P}_T \equiv p_T(miss) - p_T(\ell).
\end{equation}
In Standard Model events,
the neutrino from the decaying $W$-boson
is the only source of missing momentum. Even though the 2-body
decay of a polarized $W$-boson is not isotropic in its rest frame,
we expect little or no net polarization in the $W$ bosons produced
at the Tevatron.
Consequently, the distribution in $\Delta{\cal P}_T$
ought to be symmetric
about zero: there is no preferred direction for the charged lepton
relative to the $W$ boost direction.
On the other hand, for events with a supersymmetric origin,
there are a pair of $\neut1$'s in the final state. On average,
these neutralinos will tend to increase the mean value of
the missing transverse momentum. Hence, we expect that the
distribution in $\Delta{\cal P}_T$ will be
asymmetric, with a peak for some positive value.
We find that a criterion reading
$\Delta{\cal P}_T > 0{\rm \enspace GeV}$ is useful.
Earlier, we commented on the use of
the transverse mass of the charged lepton and missing $p_T$
for the purpose of reducing the $W$ + 2 jets background.
Among the events satisfying this cut, the distributions
{\it still} differ enough to produce a useful superweight criterion:
the spectrum of Standard Model events falls more rapidly than for the
SUSY events. Thus, we select a criterion of
the form
$m_T(\ell,\thinspace{\not{\negthinspace p}}_T) >
125 {\rm \enspace GeV}$ (${\cal C}_4$).
The final criterion we employ is the ``visible'' mass, defined
by summing the observed 4-momenta of the charged lepton and
the leading two jets, and forming an invariant mass-squared.
If all of the final state particles were represented by these
three objects, then this quantity would be equal to the
center of mass energy squared of the hard scattering,
that is $\agt 2{m}_{t}$ for the signal, and $\agt 2M_W$ for
the background.
However,
not all of the particles are detected: some go down the beampipe,
some are too soft, and some are weakly interacting.
We expect the first two
kinds of losses to be comparable across signal and background.
In contrast, since the signal events contain two extra
weakly-interacting particles (the $\neut1$'s), an even larger
proportion of the total mass is invisible.
Although it is not immediately obvious which
way the net effect will go, it is clear that
that distributions in this variable should be different.
From a study like the one described
in Sec.~\ref{SUwgt}, we find that we should set ${\cal C}_{10}=1$ when
$m(\ell,j_1,j_2) < 200 {\rm \enspace GeV}$.
\subsection{Results}
\label{BasicResults}
The procedure we have in mind for the detection of top to stop
decays is a simple counting experiment. We apply all of the cuts in
Table~\ref{TopToStopCuts} to the data, and evaluate the
superweight for each of the surviving events. Our signal
consists of an excess of events which have a superweight greater
than some value determined by comparing the expected superweight
distributions for the signal and background.
We now consider
various pieces of data relevant to evaluating the effectiveness
of the superweight we have just defined.
Fig.~\ref{SUwgtSigDist} shows the distribution of signal events
according to their superweight, for the specific masses
$\widetilde{M}_{t} = 65 {\rm \enspace GeV}$,
$\mneut1 = 45 {\rm \enspace GeV}$. A significant
tendency for signal events to have a high superweight is
readily apparent.
Fig.~\ref{SUwgtLegoFig} presents the mean value of $\widetilde{\cal X}$
as a function of the stop and neutralino masses for kinematically
allowed points in the range
$45 {\rm \enspace GeV} \le \widetilde{M}_{t} \le
100 {\rm \enspace GeV}$, $30 {\rm \enspace GeV} \le
\mneut1 \le 70 {\rm \enspace GeV}$.
Note the flatness of this distribution: this implies
that our superweight has roughly the same effectiveness
over the entire range.
Numerical results are presented in
Table~\ref{TopToStopSUwgtTable} for a few selected points.
Over the entire range the mean superweight is in excess of 7, and
typically 75\% or more of the events have a superweight of
6 or greater.
Of course, the significance of these results depends upon the
behavior of the backgrounds. We plot the superweight distributions
for all backgrounds which were estimated to be 1 fb or greater
in Fig.~\ref{SUwgtBkDist}, and supply the mean values and
fraction of events of each type with superweights of 6 or greater
in Table~\ref{TopToStopBkSUwgtTable}. It is readily apparent
that our criteria were tailored to reject $W$ + 2 jets events:
they do that very well. On the other hand, the backgrounds
from Standard Model $t \bar t$ production
do not typically have low superweights.
In fact, their superweight distributions resemble that of the
signal. Fortunately, the cross section times branching ratio
surviving our cuts for such events is only 0.023 pb
(0.006 pb if we include the effect of
${\cal B}(t\rightarrow \tilde{t} \neut1)=50\%$),
while for most (but not all)
values of the SUSY masses, the signal has a cross section 3 to 5
times greater than this particular background.
To get a feeling for the range of masses to which we are
sensitive, we present Fig.~\ref{Nevents}, which shows the
predicted number of signal events in 100 pb$^{-1}$ of data,
the approximate size of the present CDF and D0 data sets.
To guide the eye, we have included the contour where $S/\sqrt{B}=3$.
We must caution the reader, however, that the exact area in which
we can exclude or discover the top squark depends upon a more
complete analysis involving full detector simulations and
Poisson statistics where appropriate. Note that
the numbers in Fig.~\ref{Nevents} assume a 50\% branching ratio of
top to stop (which is the most favorable case). However,
we have omitted the expected increase in rate
from the 1--loop radiative corrections and summation
of multiple soft gluon emission. Furthermore, we have
reported $t\bar{t}$ backgrounds that do not include the
effects of the reduced branching ratio to $Wb$.
So overall, we believe our numbers
to be reasonably conservative. One might hope to increase the
signal somewhat by a careful tuning of the cut choices in
Table~\ref{TopToStopCuts} and the superweight definition in
Table~\ref{TopToStopSUwgt}. Also, in the event that a
signal is found, it would be useful to vary the final cut
on the superweight, as a check on systematics.
It is interesting to compare our results to those of
Mrenna and Yuan~\cite{Yuan},
who consider the same search, but only employ cuts on the
``traditional'' event observables. They obtain a background
of 1.8 events for 100 pb$^{-1}$ of data~\cite{YuanRemark},
compared to our 4.9 events in the $\widetilde{\cal X}\ge6$ sample.
However, the efficiencies they report for retaining the signal are
only in the 6\%--8\% range: our efficiencies are as high as 16\%.
The net result is that we have a larger $S/\sqrt{B}$:
indeed, their $S/\sqrt{B}=3$
contour would lie somewhere in the vicinity of the $N=8$ contour on
our Fig.~\ref{Nevents}.
Conspicuously absent from our discussion to this point has
been the issue of $b$-tagging. We have avoided using
such information so far for two reasons.
First, the efficiency for $b$-tagging reported by CDF is
currently about 30\% per $b$ jet\cite{TeVMM}.
Hence, the rejection of events without a $b$-tag lowers
the efficiency significantly. Furthermore, this tagging
efficiency implies that a superweight criterion reading
\begin{equation}
{\cal C} = \cases{1, & \hbox{if there is a $b$-tag} \cr
0, & otherwise, \cr}
\end{equation}
only adds about 0.3 units of separation in the mean superweights
of the signal and background. Compared to the criteria already
in use, this is only a modest separation. Therefore, we would
prefer to use $b$-tagging to verify that the high superweight
events do indeed contain top quarks in the event that a signal
is observed.
Note that this assessment would
change should the tagging algorithms improve:
we urge the experimentalists to vary the parameters
and criteria in Tables~\ref{TopToStopCuts} and~\ref{TopToStopSUwgt}
to obtain the optimum balance.
Finally, given that the SUSY signal contains both a
$b$ jet and a $c$ jet, we remark that the development
of a specific charm-tagging algorithm would be useful
in this connection.
\section{Inclusion of Squarks and Gluinos} \label{FullSignal}
In this section we consider our superweight analysis in
the context of a ``complete'' SUSY model. Our aim is to
demonstrate that the addition of other sources of top quarks
can only help in the observability of a signal, if present.
At the same time, we will show that it is indeed
sufficient to tune the superweight criteria using the
simplified assumptions of Sec.~\ref{BasicSignal}.
To illustrate these points, we have chosen a specific
model which has a stop mass of 65 GeV and a LSP mass of 45 GeV:
we believe this model to be representative of the
types of models described in Sec.~\ref{Features}.
We list a few other features of this model in Table~\ref{ModelT}.
The data in this section were generated using
{\tt PYTHIA 5.7}~\cite{PYTHIA}
with supersymmetric extensions~\cite{SPYTHIA}.
Tree level matrix elements are used,
along with the CTEQ2L structure functions~\cite{CTEQ}.
The square of the hard-scattering scale
for the structure functions and first-order running
$\alpha_s$ is set to the average of the squares of the
transverse masses of the two outgoing particles participating
in the hard scattering (the program default).
For this choice of calculational parameters, the raw SM
$t \bar t$ production cross section is reported as 6.8 pb,
which is rather close to the NLO estimates.
Although we do not do so, the reader may wish to apply
a $K$ factor of 1.0--1.2 to the signals we report in this
section.
Jets are constructed using a cone algorithm ($R=0.7$) inside a
toy calorimeter using the routine supplied by {\tt PYTHIA}.
No attempt is made to simulate the out-of-cone corrections
required to ensure that the jet energy accurately reflects
the parton energy. Thus the output systematically underestimates
the jet energies. As a result, it is not possible to
directly compare the results appearing in this section
with the results from the previous section. In particular,
the efficiencies implied by the data in this section will be
lower than what should be expected under actual conditions.
This merely underscores the importance of having each experiment
do the analysis with their full detector simulations in place.
Our goal in this section is to document the effect of
adding SUSY-induced $t \bar t$ production mechanisms
to the analysis, and so the only direct comparisons we
need to make to this end are self-contained within this set
of Monte Carlos.
In order to take advantage of the SUSY-produced top events,
we must relax our cut on the $p_T$ of the third jet.
However, we must beware of the background represented
by $W$ + 4 jet SM decays of the top. In Fig.~\ref{ThirdJet}
we compare the $p_T$ distributions of the third jet for
signal ($\tilde{g}\tilde{g}$, $\tilde{q}\tilde{q}$,
and $\tilde{g}\tilde{q}$) and $t\bar{t} \rightarrow W + 4$ jets
background, employing the cuts in Table~\ref{TopToStopCuts}
{\it except}\ for the requirement on the third jet.
It is apparent from Fig.~\ref{ThirdJet} that it is possible to
raise the cut on the maximum allowed $p_T$ of the third jet
without totally swamping the signal in SM $t \bar t$
background. We will present our results for the cases
$p_T(j_3)<10, 20, 30 {\rm \enspace GeV}$.
Table~\ref{NevtFULL} lists the number of events
in 100 pb$^{-1}$ predicted to pass the
cuts in Table~\ref{TopToStopCuts}, as a function of the maximum
allowed $p_T$ of jet 3. The entries above the dividing line
are within the context of our SUSY model.
For the purposes of this study, we define as ``signal'' any
event which contains a pair of $\neut{1}$'s in the final state,
whether or not it contains a $t\rightarrow\tilde{t} c$ decay.
Thus, we list separate entries for $t \bar t$ events which contain
at least one SUSY decay ($t \bar t$ signal) and those which
don't ($t \bar t$ background), but do not distinguish between
squark and gluino events which do or do not contain tops
in the intermediate states.
Should a SUSY scenario of this type prove to be correct
and one wanted to study
only $t\rightarrow\tilde{t} c$ events,
additional work would be required to purify the sample to remove
these non-$t \bar t$ SUSY ``backgrounds.''
Note that, as expected,
for the tightest $p_T(j_3)$ cut (10 GeV), the squark and gluino
channels have little effect on the expected number of events.
However, by relaxing this cut to 30 GeV, we allow nearly 2/3
of the $\tilde{g}\tilde{g}$, $\tilde{q}\tilde{q}$, and
$\tilde{g}\tilde{q}$ events into the sample, with only
a modest increase in the background from SM $t\bar{t}$ decays.
The entries below the line give the number of counts assuming
purely SM $t \bar t$ production and decay.
For good discriminating power, the cuts on $j_3$ and the
superweight should be chosen so that the total number of
counts expected with SUSY is greatly different from the
total number of counts expected without SUSY.
Since the background from $W/Z + {\rm jets}$ in the absence
of a superweight cut (nearly 40 events)
is significantly larger than the
entries in Table~\ref{NevtFULL}, it is necessary to impose such a cut.
Fig.~\ref{SUwgtFULL} shows the
superweight distribution for the signal
($\neut{1}$-containing) events.
Compared to Fig.~\ref{SUwgtSigDist}, we see a somewhat
broader distribution. However, there is still a significant
peaking at high superweight, and the cut $\widetilde{\cal X}\ge6$
still retains the majority of the signal (73\% in this case).
Hence, we present Table~\ref{NevtHI}, which is the same
as Table~\ref{NevtFULL}, but with the additional requirement
$\widetilde{\cal X}\ge6$. Now the $W/Z + {\rm jets}$ background
is reduced to the point where, for example, taking
the $p_T(j_3)$ cut to be at 30 GeV
yields a factor of 2 difference in the number of
counts with and without SUSY.
Thus, the prospects for observing or excluding this type of
model are quite good.
\section{Conclusions} \label{CONCLUSIONS}
We have investigated the possibility of detecting a light
stop squark in the decays of top quarks using the present
Fermilab Tevatron data set (approximately 100 pb$^{-1}$).
Instead of a traditional analysis which relies on cutting
on the kinematic observables individually with a low
resultant efficiency, we have defined a composite
observable, the superweight. The superweight is
assigned event-by-event depending upon how many of the criteria
from a predetermined list are true. By construction,
events with a large superweight are likely to be signal,
while those with a small superweight are likely to be
background. Since we do not require {\it all}\ of the
criterion to be true to accept an event, our efficiency
is significantly better; for example, compared to the analysis
of Mrenna and Yuan~\cite{Yuan}, our signal efficiencies are typically
twice as large.
For the given set of cuts and superweight
criteria, we have shown that the prospects for finding a top-to-stop
signal are good. Fig.~\ref{Nevents} can be viewed as a summary
of the results.
The collaborations are urged
to view this work as a starting point, since a proper
analysis must be based upon the actual event reconstruction
program used by each experiment. Furthermore, by adjusting
the parameters in Tables~\ref{TopToStopCuts} and~\ref{TopToStopSUwgt}
it may be possible to do even better.
Finally, we remark that
although we have applied the superweight concept to the specific
case of a light stop squark in supersymmetric models,
the method is applicable in any situation where the
individual kinematic cuts
required to reduce the background result in a low signal
efficiency. Thus, for example, one could consider developing
a superweight suited for the direct search for stop pair production.
\acknowledgements
High energy physics research at the University of Michigan
is supported in part by the U.S. Department of Energy,
under contract DE-FG02-95ER40899.
GDM would like to thank Soo--Bong Kim, Graham Kribs,
Steve Martin, Steve Mrenna, and Stephen Parke for useful discussions.
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\section{Introduction}
Many experiments in nuclear physics require targets with a precise
characterization. In particular, one needs a precise determination
of quantities such as the target thickness, the homogeneity, and the
amount and kind of impurities, in order to investigate
rare processes or perform high accuracy measurements.
Recently, we have investigated $^3$He- and $^4$He-induced
nuclear fission of several compound nuclei at bombarding
energies between 20 and 145 MeV measured at the 88-Inch
Cyclotron of the Lawrence Berkeley National Laboratory
\cite{Mor95,Rub96a,Rub96b,Rub96c}.
To study the excitation energy dependence of the first chance
fission probability, which is determined by subtracting similar
cross sections of two neighboring isotopes \cite{Rub96c},
it is essential to
measure the cumulative fission cross sections with high precision.
While statistical errors can be minimized by measuring a sufficiently
large number of fission events, systematic errors, as for example
caused by uncertainties in the target thickness or uniformity, are of
particular concern and must be evaluated.
\begin{figure}[htb]
\centerline{\psfig{file=fig1.eps,height=12cm}}
\caption{
Schematic excitation functions for the cumulative fission of the
compound nuclei $^{209}$Po, $^{210}$Po, $^{211}$Po. The
first chance fission probability can be determined by
subtracting similar cross sections of the mother
(triangle) and daughter nucleus (circle).
}
\label{f1}
\end{figure}
To qualitatively illustrate the accuracy needed for our measurements,
we schematically show in Fig.~\ref{f1} the fission excitation
functions of three neighboring lead isotopes \cite{Rub96c}.
The first chance
fission probability is determined by the difference in the cross
sections of the mother (triangle) and daughter nucleus (circle)
separated by the kinetic and the binding energies of
the evaporated neutron.
Due to the flattening of the curves
at large excitation energies, the cross sections
become more similar and thus precise cross sections
measurements are required.
The presence of contaminations from heavier elements
represents another systematic
uncertainty in fission cross sections measured from targets made
of lighter elements in the rare earth region: Due to their
substantially lower fission barriers, even a small contamination from
heavy elements ($<$ 1 ppm) can significantly increase the measured
fission cross sections \cite{Rai67}. This effect is most
prominent at low excitation energies near the fission barrier of the
lighter element.
In this paper, we report on the use of Rutherford backscattering
and particle induced x-ray emission for a precise off-line
characterization of targets used in nuclear physics experiments.
Furthermore, we introduce a sensitive method to check the
relative accuracy of cross section measurements.
\section{The Rutherford backscattering technique}
Rutherford scattering was studied at the beginning of the century
by Rutherford \cite{Rut11}, Geiger and Marsden \cite{Gei13}.
Their experiments were purely of nuclear physics interest, i.e.
they were designed to confirm the atomic model proposed by
Rutherford. The analytical nature of the Rutherford backscattering
method (RBS), however,
was not fully realized until the late 1950s \cite{Rub57}.
For several decades, RBS has been used as a technique to
characterize the surface and near surface properties
of thin films of thicknesses between $\sim$100{\AA}
and 1$\mu$m (see e.g. Ref.~\cite{Wil78}).
The major push to use this method has come from the need to analyze
electronic materials like semiconductors \cite{Nav83,Wit83}.
The technique has also been used to investigate ion
implantations into solids \cite{Wil84}.
As in the original experiments by Rutherford, Geiger and Marsden,
the RBS technique analyzes the Coulomb interaction between a projectile
of charge $Z_1 e$ and a target nucleus of charge $Z_{2} e$.
As we will briefly discuss in this section, the energy and scattering
angle of the scattered particle provide information on the thickness,
the nature of constituents, and the profile of the target.
A typical experimental setup requires a beam generating device
(providing a collimated monoenergetic beam of charged particles),
a scattering chamber where the beam interacts with the target, and a
detector for the backscattered particles. As mentioned before, the
measured quantities are the backscattering angle $\theta$ and the energy
$E$ of the detected particle. Good energy resolution is obviously an
essential quantity for the accuracy of the analysis.
In the following, we give a brief description of the method.
More detailed information can be found,
e.g., in the book by Chu, Mayer and Nicolet \cite{Chu78}.
The most important quantity determined in RBS is the kinematic
scattering factor $k$, defined by the ratio of the energy of the
backscattered particle $E$ and the incident energy of the projectile
$E_0$:
\begin{equation}
k = \frac{E}{E_0} = \left( \frac{\sqrt{M_2^2-M_1^2 \sin^2\theta}
+ M_1 \cos\theta}{M_1 + M_2} \right)^2.
\end{equation}
Here, $M_1$ and $M_2$ are the masses of the projectile and the target,
respectively. The knowledge of the mass and energy of the projectile
and the measurement of the energy $E$ and the angle $\theta$ of the
backscattered particle allows the identification of the elementary
constituents of the sample.
The thickness $t$ of the sample can be derived from the energy
loss $dE/dx$, i.e. by determining the energy of the backscattered
particles $E_1$ and $E_2$ at both edges of the sample \cite{Chu78}.
\begin{figure}[htb]
\centerline{\psfig{file=fig2.eps,height=8.5cm}}
\caption{Schematic RBS spectrum of a sample
which contains three different constituents (A, B, and C).
The individual contributions are shown as a dashed-dotted
(A), a dotted (B), and a dashed line (C). The sum spectrum
is displayed with a full line.
}
\label{f2}
\end{figure}
Due to the specific energy loss in different materials,
contaminations in the sample show up as distortions of
the RBS spectrum. This is schematically shown in Fig.~\ref{f2}
for a sample which contains three different
constituents.
Since the amount of backscattered particles from any
given element is proportional to its concentration,
RBS can be used to investigate quantitatively the depth
profile of individual elements in the sample.
We note that due to the strong $Z$ dependence of
the scattering cross section, the
RBS technique shows a lack of sensitivity for low $Z$ contaminants
imbedded in high Z materials.
RBS spectrometers using heavy ions as projectiles have been designed
and utilized to improve the sensitivity to low $Z$ constituents
\cite{Yu84}.
The advantages of the RBS method are many. It provides precise
information about the sample without employing physical or chemical
sectioning techniques and gives a quantitative analysis without references
or standards. Furthermore, this technique is fast and non destructive.
\section{Target thickness and homogeneity}
We have utilized an RBS spectrometer at Lawrence Berkeley National
Laboratory using monoenergetic $^4$He$^+$ particles of $E_0$ =
1.95 MeV generated by a 2.5 MeV van der Graaf accelerator.
The diameter of the beam size was 0.75~mm.
A silicon surface barrier detector was positioned at 165$^{\circ}$
with respect to the ion beam
to collect and analyze the scattered helium particles
\cite{Yu96}.
Four different targets made of natural and isotopic lead
($^{{\rm nat},206,207,208}$Pb) have been investigated. The
free standing targets were mounted on a thin aluminum target frame
with a circular opening of 19~mm. The target thicknesses were
$\sim$0.5 mg/cm$^2$. The commercially made targets were
manufactured using an evaporation method \cite{Mic}.
\begin{figure}[htb]
\centerline{\psfig{file=fig3.eps,height=12cm}}
\caption{RBS energy spectra for four lead targets
($^{\rm{nat}}$Pb, $^{206}$Pb, $^{207}$Pb, $^{208}$Pb).
The different symbols correspond to different positions
on the target: center (full circles), upper (open squares)
and lower edge (open triangles).}
\label{f3}
\end{figure}
In Fig.~\ref{f3}, we show the measured energy spectra from the RBS
analysis for four lead targets.
The thicknesses of the foils are deduced from the
widths of the RBS spectra using the energy loss data of the
ions in Pb. The high energy edge reflects the
front and the low energy edge the back of the sample.
Small inhomogeneities in the target thickness can clearly be seen
in the figure.
In general, the spectral edges are sharply defined indicating
well defined surfaces.
In Table \ref{t1}, we compare the thicknesses determined
by direct weighing, using a geometric correction factor
to account for evaporation nonuniformity \cite{Mic},
with thicknesses determined by the RBS method.
Note that the thickness measured by RBS is given in areal density
(atoms/cm$^2$), i.e. the amount of materials present to scatter the
incident He ions. This areal density can be directly compared to the
data obtained by the direct weighing method using Avogadro's number and
the known isotopic weight of the Pb isotope.
To determine the overall homogeneity of the target,
we have measured the thickness at 3 different points
(center, lower left and upper right edge). The distance
between the different points was 6~mm.
The standard weighing technique provides only an
average thickness and does not provide any information
on the homogeneity of the target foils.
We have also calculated an average thickness using
the results from the RBS measurements according to
$ <t^{RBS}> = (t^{RBS}_{center} + t^{RBS}_{low} + t^{RBS}_{up})/3$.
The observed agreement between the average thicknesses
determined by the two methods is good.
\begin{figure}[htb]
\centerline{\psfig{file=fig4.eps,height=8.5cm}}
\caption{Target thickness determined by RBS as a function
of position on the target surface for $^{207}$Pb.
The error bars show the absolute uncertainty of
the measurement.
}
\label{f4}
\end{figure}
In Fig.~\ref{f4}, we show the thickness as a function
of the distance from the center
on the surface for the $^{207}$Pb target.
Measurements were made in 2~mm steps to determine the homogeneity.
Within the central 8-10~mm, the thickness fluctuation
is small. However, the sides are not symmetric.
A systematic decrease of the target thickness
from the center to the edges is found which is due to
the evaporation process used to produce the target.
In our fission experiments, the diameter of the beam spot
on the target was less than 5~mm and the accuracy of
the center focus was $\sim$1~mm.
Therefore, the differences in the target thickness given in Table \ref{t1}
represent an upper limit for the uncertainty in the
homogeneity.
\begin{table}[tb]
\caption{Target thicknesses $t$ determined from weighing in
comparison to the results of the RBS technique. The thickness
has been measured at three different points on the target
(center, upper edge, lower edge). Furthermore, an average
thickness $<t^{RBS}>$has been calculated from these values.}
\begin{tabular}{lccccc}
\hline
\hline
Target &
$t^{weighing}$ &
$<t^{RBS}>$ &
$t^{RBS}_{center}$ &
$t^{RBS}_{low}$ &
$t^{RBS}_{up}$ \\
&
($\mu$g/cm$^{2})$ &
($\mu$g/cm$^{2})$ &
($\mu$g/cm$^{2})$ &
($\mu$g/cm$^{2})$ &
($\mu$g/cm$^{2})$ \\
\hline
$^{nat}$Pb & 544 & 553 & 582 & 538 & 538 \\
$^{206}$Pb & 555 & 543 & 558 & 531 & 541 \\
$^{207}$Pb & 560 & 548 & 550 & 534 & 561 \\
$^{208}$Pb & 500 & 490 & 503 & 496 & 472 \\
\hline
\hline
\end{tabular}
\label{t1}
\end{table}
We note that the relative uncertainty of the RBS target
thickness measurement is below 1\% and thus
provides the necessary accuracy to minimize
systematic errors in cross section measurements and associated quantities
like, in our experiment, the first chance fission probability.
\begin{figure}[htb]
\centerline{\psfig{file=fig5.eps,height=8.5cm}}
\caption{RBS energy spectra for $^{206}$Pb (center of target).
The dashed line shows the results of the simulated
spectrum. The deviations indicate surface contaminations
and/or surface inhomogeneities.
}
\label{f5}
\end{figure}
The sharpness of the low energy edges of the RBS energy spectra shown in
Fig.~\ref{f3} provides information on the
surface condition, surface contaminations, and foil roughness.
Non uniformities of the surface or significant contaminations
will broaden the energy of the backscattered $^4$He particle.
For all targets the front edges are very sharp and for
three targets the back edges are also quite sharp.
However, for the
$^{206}$Pb case, the back edge is significantly
washed out compared to other targets.
In Fig.~\ref{f5}, we show $^{206}$Pb data in comparison to
the simulated sum spectrum obtained from the RBS
analysis \cite{Saa92}.
While the agreement is good for the front edge and
inside the target material,
a large deviation is observed at the back edge.
This is most likely caused by either
a significant surface inhomogeneity or the presence of
large particles in the foil.
\section{Target impurities}
In order to determine whether any significant
target impurities were present, particle induced x-ray
emission (PIXE) has been measured simultaneously during
the RBS experiments.
PIXE is an analytical method which relies on the spectrometry
of characteristic x-rays emitted by the target atoms due to
the irradiation with a high energy ion beam. The method can
identify various constituents in a compound target
via their characteristic x-rays.
To measure the x-rays, we have used a lithium drifted silicon
detector which was located at 30$^{\circ}$ with respect to the
incident beam. Under the most favorable conditions, a detection limit
of $\sim$ 1~ppm for thin foils can be achieved \cite{Yu96}.
Compared to RBS, this method is significantly more sensitive
to determine target impurities \cite{Yu96}.
\begin{figure}[htb]
\centerline{\psfig{file=fig6.eps,height=8.5cm}}
\caption{Particle induced x-ray emission (PIXE) spectrum
for $^{207}$Pb (center of target).
}
\label{f6}
\end{figure}
In Fig.~\ref{f6}, we show the accumulated x-ray spectrum for one
of the targets ($^{207}$Pb). The spectrum is dominated by
the various M and L x-ray peaks of Pb confirming that
Pb is the major constituent.
In addition, a small peak from the carbon backing of the target
is seen. No sizable contribution of other contaminations has
been detected. We note that for the present experimental
conditions the detectable limit for most transition metals
is 10-50~ppm.
\section{Relative cross sections}
A good relative accuracy of the measured fission cross sections of
the neighboring compound nuclei is very important to minimize
the associated error in measurements of first chance fission cross
sections. To check this quantity for several separated isotopic targets,
we have applied an independent method based on the measurement
of the cross section of the corresponding natural target.
In our experiment, we have measured the fission cross sections
of four different lead targets ($^{206,207,208}$Pb and $^{\rm nat}$Pb).
The composition of natural lead is: 52.4\% of $^{208}$Pb,
22.1\% of $^{207}$Pb, 24.1\% of $^{206}$Pb, and 1.4\% of $^{204}$Pb.
Unfortunately, we have not measured the fission cross section
of the latter isotope and had to estimate it from the
ratio of the cross sections for $^{208}$Pb and $^{206}$Pb.
This estimate is in agreement with measured fission cross sections
for all three isotopes \cite{Kho66}.
We have calculated the ``natural'' cross section by
adding up the relative isotopic cross sections using the target
thicknesses determined by RBS:
\begin{equation}
\sigma_{\rm nat}^{\rm calc} = \sum_{i=204}^{208} p_{i} \sigma_{i}.
\label{nat_xs}
\end{equation}
Here, $p$ represents the contribution of the isotope $i$ to
the natural composition.
\begin{figure}[htb]
\centerline{\psfig{file=fig7.eps,height=8.5cm}}
\caption{Ratio of the calculated fission cross section
for natural lead using the individually measured
cross sections of the lead isotopes and the measured cross section
using a natural lead target. The projectile is $^3$He.
}
\label{f7}
\end{figure}
In Fig.~\ref{f7}, we show the results of this analysis; the
calculated cross sections from Eq.~\ref{nat_xs} have been
normalized by the cross section measured for the natural lead
target.
A rather constant value close to unity has been found.
This good agreement
allows us to conclude that the relative cross sections
are known to $\pm$2\%.
This accuracy is a substantial improvement over
previous experiments \cite{Kho66} and is sufficiently
good to allow extraction of first chance fission
probabilities from our data \cite{Rub96c}.
\section{Summary}
In this paper, we have presented results of a
method that allows precise characterization
of thin target foils used in nuclear physics experiments.
The applied Rutherford backscattering and particle induced
x-ray emission techniques provide
information on the thickness, homogeneity,
and constituents of a target
material. Furthermore, this method is fast and -- more
importantly -- non-destructive.
The information allows one to minimize systematic errors due to
uncertainties in the target thickness and homogeneity.
The technique described in this paper thus
provides a powerful tool to determine
the purity of a target and is especially useful if it is applied
in advance of an experiment.
\bigskip
This work was supported by the Director, Office of Energy Research,
Office of High Energy and Nuclear Physics, Nuclear Physics Division
of the US Department of Energy, under contract DE-AC03-76SF00098.
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\section{Introduction}
Extended objects, known as `branes', currently play an essential role in
our understanding of the non-perturbative dynamics underlying
ten-dimensional (D=10) superstring theories and the 11-dimensional (D=11)
M-theory (see \cite{schwarz} for a recent review). In the context of the
effective D=10 or D=11 supergravity theory a `p-brane' is a solution of the
field equations representing a p-dimensional extended source for an abelian
$(p+1)$-form gauge potential $A_{p+1}$ with $(p+2)$-form field strength
$F_{p+2}$. As such, the $p$-brane carries a charge
\begin{equation}
Q_p = \int_{S^{D-p-2}}\!\!\! \star F_{p+2}\ ,
\label{eq:introa}
\end{equation}
where $\star$ is the Hodge dual in the D-dimensional spacetime and the integral
is over a $(D-p-2)$-sphere encircling the brane, as shown schematically in the
figure below:
\vskip 0.5cm
\epsfbox{figa.eps}
\vskip 0.5cm
\noindent
In the case of a static infinite planar p-brane this formula is readily
understood as a direct generalization of the $p=0$ case, i.e. a point particle
in electrodynamics, with the $(D-p-1)$-dimensional `transverse space' (spanned
by vectors orthogonal to the $(p+1)$-dimensional worldvolume) taking the place
of space. In the case of a closed p-brane, static or otherwise, the charge $Q_p$
can be understood (after suitable normalization) as the linking number of the
p-brane with the $(D-p-2)$-sphere in the $(D-1)$-dimensional space.
Examples for $p=1$ are provided by the D=10 heterotic strings, for which
\begin{equation}
Q_1 = \int_{S^7}\! \star H\ ,
\label{eq:introb}
\end{equation}
where $H$ is the 3-form field strength for the 2-form gauge potential $B$ from
the massless Neveu-Schwarz (NS) sector of the string spectrum. Further
D=10 examples are provided by the type II superstrings, with the
difference that $B$ is now the 2-form of ${\rm NS}\otimes{\rm NS}\ $ origin in type II superstring
theory. A D=11 example is the charge
\begin{equation}
Q_2 = \int_{S^7}\! \star F
\label{eq:introc}
\end{equation}
carried by a supermembrane, where $F=dA$ is the 4-form field strength for the
3-form potential $A$ of D=11 supergravity.
The statement that a $p$-brane carries a charge of the above type can be
rephrased as a statement about interaction terms in the effective
worldvolume action governing the low-energy dynamics of the object. Consider,
for example, type II and heterotic strings. Let
$\sigma^i$ be the worldsheet coordinates and let $X^\mu(\sigma)$ describe the
immersion of the worldsheet in the D=10 spacetime. Then the worldsheet
action in a background with a non-vanishing 2-form $B$ will include the
term
\begin{equation}
I_B= \int\! d^2\sigma\; \varepsilon^{ij} \partial_i X^\mu \partial_j X^\nu\
B_{\mu\nu}\big(X(\sigma)\big)\ ,
\label{eq:introd}
\end{equation}
where $\varepsilon^{ij}$ is the alternating tensor density on the worldsheet.
Thus the string is a source for $B$ and, since the coupling is `minimal', it will
contribute to the charge $Q_1$ defined above. Similarly, in a D=11 background
with non-vanishing 3-form potential $A$, the membrane action includes the
term \cite{bst}
\begin{equation}
I_A= \int\! d^3\xi\; \varepsilon^{ijk} \partial_i X^M \partial_j X^N
\partial_k X^P A_{MNP}\ ,
\label{eq:introe}
\end{equation}
where $X^M(\xi)$ describes the immersion of the supermembrane's worldvolume in
the D=11 spacetime, and $\xi^i$ are the worldvolume coordinates. This
minimal interaction implies that the membrane is a source for $A$ with
non-vanishing charge $Q_2$.
The actions $I_A$ and $I_B$ are actually related by double-dimensional reduction,
as are the full supermembrane and IIA superstring actions \cite{DHIS}. The
dimensional reduction to D=10 involves setting $X^M=(X^\mu, y)$ where $y$ is
the coordinate of the compact 11th dimension, and taking all fields to be
independent of $y$. From the worldvolume perspective this amounts to a special
choice of background for which $k=\partial/\partial y$ is a Killing vector
field. Double-dimensional reduction is then achieved by setting $\xi=
(\sigma,\rho)$ where $\rho$ is the coordinate of a compact direction of the
membrane, and then setting $\partial_\rho X^\mu=0$ and $dy=d\rho$, which is the
ansatz appropriate to a membrane that wraps around the 11th dimension. The
action $I_A$ then becomes $I_B$ after the identification $B={\it i}_kA$, where ${\it i}_k$ indicates contraction with the vector field $k$.
A coupling to $B$ of the form (\ref{eq:introd}) is possible only for oriented
strings. Of the five D=10 superstring theories all are theories of oriented
strings except the type I theory. Thus, the type I string does {\it not} couple
minimally to $B$. Instead, it couples {\it non-minimally}. In the
Lorentz-covariant GS formalism in which the worldsheet fermions,
$\theta$, are in a spinor representation of the D=10 Lorentz group, the
worldsheet interaction Lagrangian is
\begin{equation}
L_B = \bar \theta \Gamma^{\mu\nu\rho} \theta H_{\mu\nu\rho}\ .
\label{eq:introf}
\end{equation}
Because of the `derivative' coupling of the string to $B$ through its field
strength $H$, the $Q_1$ charge carried by the type I string vanishes. As the
above interaction shows, the type I string theory origin of $B$ is in the ${\rm R}\otimes{\rm R}\ $
sector rather than the ${\rm NS}\otimes{\rm NS}\ $ sector. This example illustrates a general feature of
string theory: ${\rm R}\otimes{\rm R}\ $ charges are {\it not} carried by the fundamental string.
If there is anything that carries the charge $Q_1$ in type I string theory it
must be non-perturbative. It is now known that there is such a
non-perturbative object in type I string theory \cite{dab,hull,polwit}; it
is just the $SO(32)$ heterotic string! This is one of the key pieces of evidence
in favour of the proposed `duality', i.e. non-perturbative equivalence, of the
type I and $SO(32)$ heterotic string theories. Another is the fact that the two
effective supergravity theories are equivalent, being related to each other by a
field redefinition that takes $\phi\rightarrow -\phi$, where
$\phi$ is the dilaton \cite{wita,ark}. Since the vacuum expectation value
$\langle e^\phi\rangle$ is the string coupling constant $g_s$ this means that
the weak coupling limit of one theory is the strong coupling limit of the other.
An important consequence of the charge $Q_p$ carried by a $p$-brane is that it
leads to a BPS-type bound on the p-volume tension, $T_p$ of the form
$T_p\ge c_p|Q_p|$, where $c_p$ is some constant characteristic of the particular
supergravity theory, the choice of vacuum solution of this theory, and the value
of $p$. If one considers the class of static solutions with $p$-fold
translational symmetry then a bound of the above form follows from the
requirement that there be no naked singularities. This bound is saturated by the
solution that is `extreme' in the sense of General Relativity, i.e. for which
the event horizon is a degenerate Killing horizon\footnote{When a dilaton is
present this is true only for an appropriate definition of the metric.}.
However, these considerations are clearly insufficient to show that the p-brane
tension actually {\it is} bounded in this way because the physically relevant
class of solutions is the much larger one for which only an appropriate {\it
asymptotic} behaviour is imposed. Remarkably, the attempt to establish a
BPS-type bound succeeds if and only if the theory is either a supergravity
theory, or a consistent truncation of one \cite{ght}\footnote{In contrast, the
proof of positivity of the ADM mass of asymptotically-flat spacetimes is not
subject to this restriction since, for example, it is valid for arbitrary
D.}. In particular, {\it the presence of various Chern-Simons terms in the
Lagrangians of D=10 and D=11 supergravity theories is crucial to the existence of
a BPS-type bound on the tensions of the $p$-brane solutions of these theories}.
This is so even when, as is usually the case, these Chern-Simons (CS) terms play
no role in the $p$-brane solutions themselves in the sense that they are equally
solutions of the (non-supersymmetric) truncated theory in which the CS terms are
omitted. These facts hint at a more important role for the supergravity CS terms
in determining the properties of $p$-branes than has hitherto been appreciated.
This observation provided the principal motivation for this article, as will
become clear.
Although the charge $Q_p$ has only a magnitude, it is associated with an object
whose spatial orientation is determined by a $p$-form of fixed magnitude. Thus,
a $p$-brane is naturally associated with a $p$-form charge of magnitude $Q_p$.
Indeed, the supersymmetrization of terms of the form (\ref{eq:introd}) or
(\ref{eq:introe}) leads to a type of super-Wess-Zumino term that implies a
modification of the standard supersymmetry algebra to one of the (schematic)
form \cite{azc,pktc}
\begin{equation}
\{Q,Q\} = \Gamma\cdot P + \Gamma^{(p)}\cdot Z_p \ ,
\label{eq:extraone}
\end{equation}
where $\Gamma^{(p)}$ is an antisymmetrized product of $p$ Dirac matrices and
$Z_p$ is a $p$-form charge whose magnitude is given by the coefficient of the
Wess-Zumino term. For $p=0$ this is the well-known modification that
includes $Z_0=Q_0$ as a central charge. More generally, $Q_p$ may be identified
as the magnitude of $Z_p$, and an extension of the arguments used in the $p=0$
case \cite{witol,gh} shows that the supersymmetry algebra (\ref{eq:extraone})
implies the BPS-type bound on the p-brane tension $T_p$. It also shows that the
`extreme' $p$-brane solutions of supergravity theories which saturate the bound
must preserve some of the supersymmetry, and the fraction preserved is always
1/2 for $p$-brane solutions in D=10 and D=11\footnote{There are other solutions
which preserve less than half the supersymmetry, and which have an
interpretation as $p$-branes in $D<10$, but these can always be viewed as
composites (e.g. intersections) of $p$-branes in D=10 or D=11. We shall not need
to consider such solutions here.}. The heterotic and type II superstrings and
the D=11 supermembrane are examples not only of charged $p$-branes but also of
{\it extreme} charged $p$-branes. This follows from the `$\kappa$-symmetry' of
their Lorentz covariant and spacetime supersymmetric worldsheet/worldvolume
actions (see \cite{pktb} for a review). The BPS-saturated $p$-branes are
important in the context of the non-perturbative dynamics of superstring
theories or M-theory for essentially the same reasons that BPS-saturated
solitons are important in D=4 field theories. In fact, most of the the latter
can be understood as originating in D=10 or D=11 BPS-saturated $p$-branes. For
these reasons, the BPS-saturated p-branes are the ones of most interest and will
be the only ones considered here. It should therefore be understood in what
follows that by `brane' we mean `BPS-saturated brane'.
One of the lessons of recent years has been that much can be learned about the
non-perturbative dynamics of superstring theories from the effective
D=10 supergravity theories. One example of this is the fact that there exist
p-brane solutions of type II supergravity theories which are charged, in the
sense explained above, with respect to the $(p+1)$-form gauge fields from the ${\rm R}\otimes{\rm R}\ $
sector of the corresponding string theory. By supposing these ${\rm R}\otimes{\rm R}\ $ branes to be
present in the non-perturbative string theory one can understand how otherwise
distinct superstring theories might be dual versions of the same underlying
theory. The basic idea is that branes can `improve' string theory in
the same way that strings `improve' Kaluza-Klein (KK) theory. For example, the
$S^1$-compactified IIA and IIB supergravity theories have an identical massless
D=9 spectrum but are different as Kaluza-Klein theories because
their massive modes differ. The corresponding string theories are the same,
however, because the inclusion of the string winding
modes restores the equivalence of the massive spectra. Similarly, the
$K_3$-compactified type IIA superstring theory and the $T^4$-compactified
heterotic string theory have an identical massless D=6 spectrum, but since
they differ in their perturbative massive spectra they are inequivalent as
perturbative string theories. However, the non-perturbative massive spectrum of
the IIA superstring includes `wrapping' modes of 2-branes around
2-cycles of $K_3$ \cite{townhull}. The inclusion of these leads to the same
massive BPS spectrum in the two theories, and there is now strong evidence of
a complete equivalence \cite{wita,vafwit,vafa,sen}. This evidence rests, in
part, on the fact that ${\rm R}\otimes{\rm R}\ $ branes now have a remarkably simple description
\cite{pol} in string theory as D-branes, or D-$p$-branes if we wish to specify
the value of
$p$; the worldvolume of a D-$p$-brane is simply a (p+1)-dimensional hyperplane
defined by imposing $(D-p-1)$ Dirichlet boundary conditions at the boundaries of
open string worldsheets.
The distinctions between the various kinds of type II p-brane, such as whether
they are of ${\rm NS}\otimes{\rm NS}\ $ or ${\rm R}\otimes{\rm R}\ $ type, are not intrinsic but are rather artefacts (albeit
very useful ones) of perturbation theory. Non-perturbatively, all are on an
equal footing since any one can be found from any other one by a combination of
`dualities'. This feature is apparent in the IIA or IIB effective supergravity
theories which treat all p-form gauge fields `democratically'. Their string
theory origin is nevertheless apparent from the supergravity solutions if the
latter are expressed in terms of the {\it string metric}, instead of the
canonical, or `Einstein', metric. One then finds that there are three categories
of p-brane, `fundamental' (F), `Dirichlet' (D) and `solitonic' (S) according to
the dependence of the p-volume tension on the string coupling constant
$g_s\equiv\langle e^\phi\rangle$. Specifically,
\begin{equation}
T\sim \cases{1 & {\rm for a fundamental string}\cr
1/g_s & {\rm for a Dirichlet p-brane}\cr
1/g_s^2 & {\rm for a solitonic 5-brane}\ . }
\label{eq:introg}
\end{equation}
Note that according to this classification only strings can be `fundamental'.
This is hardly surprising in view of the fact that we are discussing the
dependence of the tension in terms of the string metric, but it seems to be a
reflection of a more general observation \cite{wita,hullb} that a sensible
perturbation theory can be found only for particles or strings. Similarly only
5-branes can be `solitonic'. This is a reflection of the electric/magnetic
duality between strings and 5-branes in D=10 and the fact that the magnetic dual
of a fundamental object is a solitonic one. It does not follow from this that all
strings are fundamental and all 5-branes solitonic. This is nicely illustrated
by the string solution of N=1 supergravity representing the heterotic string.
This solution is `fundamental' as a solution of the effective supergravity
theory of the heterotic string, as it must be of course, but it is a D-string
when viewed as a solution of the effective supergravity theory of the type I
string. Thus, a single supergravity solution can have two quite different string
theory interpretations.
The M-theory branes, or `M-branes', consist of only the D=11 membrane and its
magnetic dual, a fivebrane. We saw earlier that the classical IIA superstring
action is related to that of the D=11 supermembrane by double-dimensional
reduction. This was widely considered to be merely a `coincidence', somewhat
analogous to the fact that IIA supergravity happens to be the dimensional
reduction of D=11 supergravity; after all, the {\it quantum} superstring theory
has D=10 as its critical dimension. However, the critical dimension
emerges from a calculation in {\it perturbative} string theory. It is still
possible that the non-perturbative theory really is 11-dimensional, but if this
is so the KK spectrum of the $S^1$-compactified D=11 supergravity must appear in
the non-perturbative IIA superstring spectrum. It was pointed out in
\cite{tow,wita} that the extreme black holes of IIA supergravity, now regarded
as the effective field theory realization of D-0-branes, are candidates for this
non-perturbative KK spectrum. This means that the IIA superstring really {\it
is} an $S^1$-wrapped D=11 supermembrane, but it does not then follow that the
supermembrane is also `fundamental' because this adjective is meaningful only in
the context of a specific perturbation theory. For example, the $SO(32)$
heterotic string is `fundamental' at weak coupling but as the coupling increases
it transmutes into the D-string of the type I theory. Another example is the IIB
string which is `fundamental' at weak coupling but which transmutes into the
D-string of a dual IIB theory at strong coupling \cite{jhs,wita}. In the IIA
case the strong coupling limit is a decompactification limit in which the D=11
Lorentz invariance is restored and the effective D=10 IIA supergravity is
replaced by D=11 supergravity \cite{wita}. The `fundamental' IIA superstring
transmutes, in this limit, into the unwrapped D=11 membrane of M-theory but,
because of the absence of a dilaton, there is no analogue of string
perturbation theory in D=11 and so there is no analogous basis for deciding
whether or not the membrane is `fundamental'. Nevertheless, as we shall shortly
see, there is an intrinsic asymmetry between M-theory membranes and fivebranes
which suggests a fundamental role for the membrane in some as yet unknown
sense\footnote{This is also suggested by the `n=2 heterotic string' approach to
M-theory \cite{KM}.}.
Given that the heterotic string appears as a D-brane in type I string theory one
might wonder whether the type I string should make an appearance somewhere in
the non-perturbative SO(32) heterotic string theory. As we have seen, however,
the type I string carries no $Q_1$ charge, so its description in the effective
supergravity theory would have to be as a non-extreme, or `black', string.
Infinite uncharged black strings have been shown to be
unstable against perturbations that have the tendency to break the string into
small segments \cite{greg} (whereas extreme strings are stable because they
saturate a BPS-type bound). This is exactly what one expects from string theory
since a closed type I string can break, i.e. type I string theory is a theory of
both closed and open strings. The reason that this is possible for type I
strings, but not for heterotic or type II strings, is precisely that the type I
string carries no $Q_1$ charge. To see this, suppose that a string carrying a
non-zero $Q_1$ charge were to have an endpoint. One could then `slide off' the
7-sphere encircling the string and contract it to a point. Provided that the
integral defining $Q_1$ is homotopy invariant, which it will be if $d\star H=0$,
the charge $Q_1$ must then vanish, in contradiction to the initial assumption. We
conclude that the only breakable strings are those for which $Q_1=0$. Thus type
II and heterotic strings cannot break. Clearly, similar arguments applied to
$p$-branes carrying non-zero $Q_p$ charge lead to the conclusion that they too
cannot break.
By `break' we mean to imply that the $(p-1)$-brane boundary created in this
process is `free' in the sense that its dynamics is determined entirely by the
$p$-brane of which it is the boundary. An `unbreakable' p-brane may nevertheless
be open if its boundary is tethered to some other object because there may then
be an obstruction to sliding the $(D-p-2)$-sphere off the end of the p-brane.
Examples of such obstructions are the D-branes on which type II superstrings can
end. One way to understand how this is consistent with
conservation of the charge $Q_1$ is to consider the D-brane's effective
worldvolume action, which governs its low-energy dynamics. The field content of
this action is found from the massless sector of an open type II superstring
with mixed Neumann/Dirichlet boundary conditions at the ends. These fields are
essentially the same as those of the open type I string without Chan-Paton
factors with the difference that they depend only on the D-brane's worldvolume
coordinates (see e.g. \cite{polb}). In particular, these worldvolume fields
include an abelian 1-form potential $V$. The bosonic sector of the effective
worldvolume action, in a general ${\rm NS}\otimes{\rm NS}\ $ background, can be deduced from the
requirement of conformal invariance of the type II string action for a
worldsheet with a boundary \cite{leigh}. This effective worldvolume action is
found to contain the term
\begin{equation}
-{1\over4} \int d^{p+1}\xi\; |dV-B|^2\ ,
\label{eq:introh}
\end{equation}
where the integral is over the $(p+1)$-dimensional worldvolume $W$ and it is to
be understood that the spacetime 2-form $B$ is pulled back to $W$.
This shows that the D-brane is a source of $B$. If we modify the equation
$d\star H=0$ in order to include this source we find, by integration, that
\begin{equation}
Q_1 = \int_{S^{p-2}}\!\! * dV\ ,
\label{eq:introi}
\end{equation}
where $Q_1$ is defined as before in (\ref{eq:introa}). The integral on the right
hand side of (\ref{eq:introi}) is over a $(p-2)$-sphere in the D-brane
surrounding the string's endpoint and $*$ is the {\it worldvolume} Hodge dual.
This result can be interpreted as the statement that the charge of the string
can be `transferred' to an electric charge of a particle on the D-brane, so
charge conservation is compatible with the existence of an open string provided
that its endpoints are identified with charged particles living on a D-brane.
A similar analysis can be applied to the D=11 membrane which, we recall, is an
electric-type source for the 3-form gauge potential $A$ of D=11 supergravity.
In this case, the D=11 fivebrane has a worldvolume action containing the terms
\cite{pkt,aha}
\begin{equation}
-{1\over12}\int d^6\xi\big\{ |{\cal F}_3|^2 -\varepsilon^{ijklmn}
A_{ijk}\partial_l V_{mn}\big\}\ ,
\label{eq:introj}
\end{equation}
where ${\cal F}_3= (dV_2 -A)$ is the 3-form field strength for a worldvolume
2-form potential $V_2$, and it is again to be understood that $A$ is the
pullback of the spacetime field to the worldvolume. Actually, the
worldvolume 3-form
${\cal F}_3$ is {\it self-dual}, but this condition must be imposed after
variation of the action (\ref{eq:introj}). The second term in this action is
needed for consistency of the self-duality condition with the $V_2$ field
equation. Apart from this subtlety, we see from its worldvolume action that the
fivebrane is a source for
$A$. Its inclusion in the field equation for $A$ leads, after integration, to the
equation
\begin{equation}
Q_2 = \int_{S^3}\! * dV_2\ ,
\label{eq:introl}
\end{equation}
which can be interpreted as the statement that the membrane charge can be
transferred to a charge carried by a self-dual string within the fivebrane.
This string is just the boundary of an open membrane. Thus, the D=11 fivebrane is
the M-theory equivalent of a D-brane \cite{pkt,strom}.
The above analysis can be generalized \cite{strom} to determine whether a
$p$-brane can end on a $q$-brane, as follows. One first determines the
worldvolume field content of the $q$-brane. If this includes a $p$-form gauge
field $V_p$, and if the spacetime fields include a $(p+1)$-form gauge potential
$A_{p+1}$, then one can postulate a coupling of the form $\int |dV_p
-A_{p+1}|^2$ in the $q$-brane's effective worldvolume action. This leads to the
$q$-brane appearing as a source for $A_{p+1}$ such that
\begin{equation}
Q_p= \int_{S^{q-p}}\!\! * dV_p\ ,
\label{eq:introm}
\end{equation}
where the integral in the $q$-brane is over a $(q-p)$-sphere surrounding the
$(p-1)$-brane boundary of the p-brane. Thus, the $p$-brane charge can be
transferred to the electric charge of the $(p-1)$-brane boundary living in the
$q$-brane. That is, charge conservation now permits the $p$-brane to be open
provided its boundary lies in a $q$-brane. The cases discussed above clearly fit
this pattern, but there are drawbacks to this approach. Firstly, it is indirect
because one must first determine the worldvolume field content of all relevant
branes. Secondly, it is {\sl ad hoc} because, in general, the worldvolume
coupling is postulated rather than derived. The subtleties alluded to above in
the construction of the fivebrane action show that this is not a trivial matter.
In fact, even the bosonic fivebrane action is not yet fully known and until it
is one cannot be completely certain that the wanted terms in this action
really are present.
In this contribution I will present a new, and extremely simple, method
for the determination of when $p$-branes may have boundaries on $q$-branes.
Essentially, {\it one can read off from the Chern-Simons terms in the
supergravity action whether any given $p$-brane can have a boundary and, if so,
in what $q$-brane the boundary must lie}. As such, the method provides a further
example of how much can be learned about the non-perturbative dynamics of
superstring theories, or M-theory, from nothing more than the
effective supergravity theory. I have called the method `brane surgery' because
of a notional similarity to the way in which manifolds can be `glued' together
by the mathematical procedure known as `surgery', but it is not intended that
the term should be understood here in its technical sense.
It is pleasure to
dedicate this contribution to the memory of Claude Itzykson, who would surely
have apreciated the remarkable confluence of ideas that has marked
recent advances in the theory that is still, misleadingly, called `string
theory'.
\section{IIB brane boundaries}
I shall explain the `brane surgery' method initially in the context of the IIB
theory. Both IIA and IIB supergravity have in common the bosonic fields
($g_{\mu\nu}, \phi, B_{\mu\nu}$) from the ${\rm NS}\otimes{\rm NS}\ $ sector, all of which have already
made an appearance above. The remaining bosonic fields come from the ${\rm R}\otimes{\rm R}\ $ sector.
The (massless) ${\rm R}\otimes{\rm R}\ $ fields of the IIB theory are
\begin{equation}
(\ell, B'_{\mu\nu}, C^+_{\mu\nu\rho\sigma})\ ,
\label{eq:onea}
\end{equation}
i.e. a pseudoscalar $\ell$, another 2-form gauge potential $B'$ and a 4-form
gauge potential $C^+$ with a {\it self-dual} 5-form field strength $D^+$. The
self-duality condition makes the construction of an action problematic but, as
with the self-duality condition on the D=11 fivebrane's worldvolume field
strength ${\cal F}_3$, one can choose to impose this condition {\it after}
varying the action. When the IIB action is understood in this way it contains
the CS term\footnote{The conventions can be chosen such that the coefficient is
as given.}
\begin{equation}
C^+ \wedge H\wedge H'\ ,
\label{eq:oneb}
\end{equation}
where $H=dB$, as before, and $H'=dB'$. This CS term modifies the $B$, $B'$ and
$C^+$ field equations.
Consider first the $B$ equation. This becomes
\begin{equation}
d\star H = - D^+ \wedge H'\ ,
\label{eq:onec}
\end{equation}
where $D^+=dC^+$ is the self-dual 5-form field strength for $C^+$.
This can be rewritten as
\begin{equation}
d(\star H - D^+ \wedge B') =0 \ .
\label{eq:oned}
\end{equation}
Since $\star H$ is no longer a closed form its integral over a 7-sphere will no
longer be homotopy invariant. Clearly, the well-defined, homotopy invariant,
charge associated with the fundamental IIB string is {\it not} $Q_1$ as defined
in (\ref{eq:introa}) but rather
\begin{equation}
\hat Q_1 = \int_{S^7}[\star H - D^+ \wedge B']\ .
\label{eq:onee}
\end{equation}
Let us again suppose that the IIB string has an endpoint. Far away from this
endpoint we can ignore all fields other than $H$, to a good approximation, so
that $\hat Q_1\approx Q_1$. How good this aproximation is actually depends on
the ratio of the radius $R$ of the 7-sphere to the its distance $L$ from the
end of the string, and it can be made arbitrarily good by increasing $L$ for
fixed $R$. This shows, in particular, that $\hat Q_1\ne0$ for the fundamental
IIB string. Let us now `slide' the 7-sphere along the string towards the
endpoint. If the $D^+\wedge B'$ term could be entirely ignored we would be back
in the situation described previously in which we arrived at a contradiction,
so we are forced to suppose that an endpoint is associated with a non-vanishing
value of $D^+ \wedge B'$. Nevertheless, the approximate equality of $\hat Q_1$
to $Q_1$, which ignores the $D^+ \wedge B'$ term, can be maintained, to the
same precision, if the 7-sphere surrounding the string is contracted as we
approach the endpoint so as to keep the ratio $R/L$ constant. Then, as
$L\rightarrow 0$ so also $R\rightarrow 0$, until at $L=0$ the 7-sphere is
contracted to the endpoint itself. We can then deform the 7-sphere into the
product $S^5\times S^2$ so that $\hat Q_1$ now receives its entire contribution
from the $D^+ \wedge B'$ term as follows:
\begin{equation}
\hat Q_1 = -\int_{S^5}D^+\ \times\,\int_{S^2} B'\ .
\label{eq:onef}
\end{equation}
The $S^5\times S^2$ integration region is illustrated schematically by the
figure below:
\vskip 0.5cm
\epsfbox{figb.eps}
\vskip 0.5cm
\noindent
Observe that the $S^5$ integral is just the definition of the charge $Q_3$
carried by a 3-brane, so the IIB string has its endpoint on a 3-brane; the
$S^2$ integration surface lies within the 3-brane and surrounds the string
endpoint. Let us choose $Q_3=1$. If we further suppose that $H'\equiv dB' =0$
within the 3-brane, which is reasonable in the absence of any D-string source
for this field, then $B'$ is a closed 2-form which we may write, locally, as
$B'=dV'$ for some 1-form $V'$. Then
\begin{equation}
\hat Q_1 = -\int_{S^2}\! dV'\ .
\label{eq:onefa}
\end{equation}
Effectively, $V'$ is a field living on the worldvolume of the 3-brane.
Clearly, it cannot be globally defined because the right hand side of
(\ref{eq:onefa}) is
a magnetic charge on the 3-brane associated with the vector potential $V'$.
Now consider the $B'$ equation. Taking the CS term (\ref{eq:oneb}) into account
we have
\begin{equation}
d\star H = D^+ \wedge H\ .
\label{eq:onefb}
\end{equation}
By the same reasoning as before we deduce that the D-string can end on a
3-brane. Charge conservation is satisfied because the D-string charge can be
expressed as
\begin{equation}
\hat Q'_1 = Q_3 \times \int_{S^2}\! B\ .
\label{eq:onefc}
\end{equation}
Since there is no fundamental string source in the problem we may suppose that
$H=0$, so that now $B$ is a closed 2-form which we may write, locally, as
$B=dV$. For $Q_3=1$ we now have
\begin{equation}
\hat Q'_1 = \int_{S^2}\! dV\ ,
\label{eq:oneg}
\end{equation}
so the D-string charge has been transferred to a magnetic charge of the 1-form
potential $V$ on the 3-brane's worldvolume.
It must be regarded as a weakness of the above analysis that it does not
supply the relation between $V$ and $V'$, although we know that there must be
one because both supersymmetry and an analysis of the small fluctuations about
the 3-brane solution show that there is only {\it one} worldvolume 1-form
potential. In fact, $V$ and $V'$ are dual in the sense that
\begin{equation}
dV' = *dV\ ,
\label{eq:onega}
\end{equation}
where we recall that $*$ indicates the {\it worldvolume} Hodge dual.
Using this relation, (\ref{eq:oneg}) becomes
\begin{equation}
\hat Q_1 = \int_{S^2}* dV\ ;
\label{eq:onegb}
\end{equation}
i.e. the endpoint of the IIB string on the 3-brane is an {\it electric}
charge associated with $V$. We thereby recover the D-brane picture for the IIB
3-brane; the fact that the D-string can end on the magnetic charge associated
with $V$ is then a consequence of the strong/weak coupling duality in IIB
superstring theory interchanging the fundamental string with the D-string.
It will be seen from the examples to follow that the need to impose a
condition of the type (\ref{eq:onega}) is a general feature, which is not
explained by the `brane surgery' method. However, the method does determine
whether a given p-brane can have a boundary and, if so, the possible q-branes in
which the boundary must lie.
As a further illustration we now observe that whereas (\ref{eq:onec}) was
previously rewritten as (\ref{eq:oned}), we could instead rewrite it as
\begin{equation}
d(\star H + H'\wedge C^+)=0\ .
\label{eq:onegc}
\end{equation}
Thus an equivalent definition of $\hat Q_1$ is
\begin{equation}
\hat Q_1 = \int_{S^7} [\star H + H'\wedge C^+]\ .
\label{eq:onegd}
\end{equation}
Proceeding as before, but now deforming the $S^7$ into the product $S^3\times
S^4$, we can express $\hat Q_1$ as
\begin{equation}
\hat Q_1 = \int_{S^3} H' \times \int_{S^4} C^+\ .
\label{eq:oneh}
\end{equation}
We recognise the first integral as the D-5-brane charge $Q'_5$. Setting
$Q'_5=1$ and $D^+=0$, we conclude that
\begin{equation}
\hat Q_1 = \int_{S^4} dV_3\ ,
\label{eq:oneha}
\end{equation}
where $V_3$ is a locally-defined 3-form field on the 5-brane worldvolume, which
can be traded for a 1-form potential $V$ via the relation
\begin{equation}
dV_3 = * dV\ .
\label{eq:onehb}
\end{equation}
We conclude that the CS term allows the fundamental IIB string to end on a
5-brane as well as on a 3-brane, and that the end of the string is
electrically charged with respect to a 1-form potential $V$ living on the
5-brane's worldvolume. This is just the usual picture of the D-5-brane.
Interchanging the roles of $B$ and $B'$ leads to the further possibility of the
D-string ending on the solitonic 5-brane.
We have not yet exhausted the implications of the CS term (\ref{eq:oneb}) because
we have still to consider how it affects the $C^+$ equation of motion. We find
that\footnote{The same equation follows, given the self-duality of $D^+$, from
the `modified' Bianchi identity for $D^+$.}
\begin{equation}
d\star D^+ = -H \wedge H'
\label{eq:onei}
\end{equation}
or
\begin{equation}
d(\star D^+ + H' \wedge B) =0\ .
\label{eq:onej}
\end{equation}
This means that the 3-brane charge should be modified to
\begin{equation}
\hat Q_3 = \int_{S^5} [ \star D^+ + H' \wedge B]\ .
\label{eq:onek}
\end{equation}
This reduces to the previously-defined 3-brane charge $Q_3$ if the 5-sphere
surrounds a 3-brane sufficiently far from the boundary. As before the 5-sphere
can be slid towards, and contracted onto, the boundary, after which it emerges
as the product $S^3\times S^2$. Setting $B=dV$ again we arrive at the expression
\begin{equation}
\hat Q_3 = \int_{S^3}H' \times \int_{S^2} dV
\label{eq:onel}
\end{equation}
for the 3-brane charge. The singularity involved in this deformation of the
7-sphere is now the 2-brane boundary of the 3-brane within a D-5-brane, since we
recognise the first integral on the right hand side of (\ref{eq:onel}) as $Q'_5$.
Setting
$Q'_5=1$ we learn that the 3-brane charge can be transferred to a magnetic
charge of a D-5-brane worldvolume 1-form potential $V$, defined by a 2-sphere in
the D-5-brane surrounding the 2-brane boundary. The main point in all this is
that a 3-brane can have a boundary in a D-5-brane, as pointed out in
\cite{strom}. In fact, this possibility follows by T-duality from the previous
results: the configuration of a D-string ending on a D-3-brane is mapped to a
D-3-brane ending on a D-5-brane by T-duality in two directions orthogonal to
both the D-string and the D-3-brane. By interchanging the roles of $B$ and $B'$
in the above analysis one sees that a 3-brane can also end on a solitonic
5-brane.
We have seen that the CS term (\ref{eq:oneb}) allows a IIB string to end on a
D-3-brane or a D-5-brane, but we know from string theory that it can also end on
a D-string or a D-7-brane. As we shall see shortly, these possibilities are
consequences of the fact that the kinetic term for $H'$ actually has the form
\begin{equation}
-{1\over6}|H'- \ell H|^2\ .
\label{eq:onem}
\end{equation}
There is no obvious relation to CS terms yet, but if we perform a duality
transformation to replace the 2-form $B'$ by its 6-form dual $\tilde B'$ with
7-form field strength $\tilde H'$, so that {\it on shell}
\begin{equation}
\tilde H' =\star H'\ ,
\label{eq:onema}
\end{equation}
then one finds that the dualized action contains the CS term
\begin{equation}
\ell \tilde H' \wedge H\ .
\label{eq:onen}
\end{equation}
Clearly, this modifies the $B$ equation so that, following the steps explained
previously, we end up with an expression
\begin{equation}
\hat Q_1 = \int_{S^7}\tilde H' \times\int_{S^{0}} \ell\ .
\label{eq:oneo}
\end{equation}
The first integral can be identified, using (\ref{eq:onema}), as the D-string
charge. The final `integral' over $S^0\equiv Z_2$ is just the difference between
the value of $\ell$ on either side of the string boundary on the D-string; by the
same logic as before we may assume that $d\ell=0$, {\it locally}, but allow the
constant $\ell$ to be different on either side. Thus, the charge $Q_1$ on the
fundamental IIB string is transformed into the topological charge of a type of
`kink' on the D-string.
Alternatively, we can deform $S^7$ to $S^1\times S^6$, so that
\begin{equation}
\hat Q_1 = -\int_{S^1} d\ell \times \int_{S^{6}} \tilde B'\ .
\label{eq:onep}
\end{equation}
The first integral is the charge $Q_7$ associated with the D-7-brane. This
charge can be non-zero because of the periodic identification of $\ell$
implied by the conjectured $Sl(2;\bb{Z})$ invariance of IIB superstring theory
\cite{townhull}. For $Q_7=1$, and setting $\tilde B'=d\tilde V'_5$ for 5-form
potential $\tilde V'_5$ (since we may assume that $\tilde H'=0$), we have
\begin{equation}
\hat Q_1 = -\int_{S^{6}}\! d\tilde V'_5 \ .
\label{eq:oneq}
\end{equation}
Defining the 1-form $V$ on the 7-brane's worldvolume by
\begin{equation}
dV = * d{\tilde V}'_5 \ ,
\label{eq:oner}
\end{equation}
we can rewrite (\ref{eq:oneq}) as
\begin{equation}
\hat Q_1 = \int_{S^{6}}\! * dV \ .
\label{eq:ones}
\end{equation}
We conclude that the IIB string may end on an electric charge in a
7-brane. This is just the description of the D-7-brane.
\section{IIA boundaries}
The `brane surgery' method should now be clear. We shall now apply it to
IIA supergravity, for which the ${\rm R}\otimes{\rm R}\ $ gauge potentials are
\begin{equation}
(C_\mu, A_{\mu\nu\rho})\ ,
\label{eq:twoa}
\end{equation}
i.e. a 1-form $C$ and a 3-form $A$. We might start by considering the CS term
\begin{equation}
F\wedge F\wedge B\ ,
\label{eq:twob}
\end{equation}
where $F$ is the 4-form field strength of $A$. Consideration of this term leads
to the conclusion that (i) a IIA string can end on a 4-brane, and (ii) a
2-brane can end on either a 4-brane or a 5-brane. Since the CS term
(\ref{eq:twob}) is so obviously related to the similar one in D=11 to be
considered below we shall pass over the details. The fact that the IIA string can
also end on either a 2-brane or a 6-brane follows from the fact that the field
strength $F$ has a `modified' Bianchi identity
\begin{equation}
dF = H\wedge K\ ,
\label{eq:twoc}
\end{equation}
where $K=dC$ is the field strength of $C$ (this has a Kaluza-Klein origin in
D=11). We can dualize $A$ to convert this modified Bianchi into a CS term of
the form\footnote{This dualization is inessential to the result, but it allows a
convenient uniformity in the description of the method.}
\begin{equation}
\tilde F \wedge K \wedge B\ ,
\label{eq:twod}
\end{equation}
where the 6-form $\tilde F$ is, on-shell, the Hodge dual of $F$. This modifies
the $B$ equation to
\begin{equation}
d\star H = -\tilde F \wedge K\ .
\label{eq:twoe}
\end{equation}
We may therefore take the modified charge $\hat Q_1$ to be
\begin{equation}
\hat Q_1 =\int_{S^7} \! [\star H + \tilde F\wedge C]\ .
\label{eq:twof}
\end{equation}
Now, by the identical reasoning used in the IIB case, we first deform the
7-sphere so as to arrive at the formula
\begin{equation}
\hat Q_1 =\int_{S^6}\! \tilde F\times \int_{S^1}\! C\ .
\label{eq:twog}
\end{equation}
We then identify the first integral as the charge $Q_2$ of a membrane. We then
set $Q_2=1$ and $C=dy$ for some scalar $y$ defined locally on the
worldvolume of the membrane to conclude that the IIA string can end on a
membrane, with the string's charge now being transferred to the
magnetic-type charge
\begin{equation}
\int_{S^1}\! dy
\label{eq:twoh}
\end{equation}
of a particle on the membrane \cite{asy}. This charge can be non-zero if $y$ is
periodically identified. Clearly, from the KK origin of $C$, we should interpret
$y$ as the coordinate of a hidden 11th dimension. Defining the worldvolume
1-form $V$ by
\begin{equation}
dV= * dy\ ,
\label{eq:twoi}
\end{equation}
we recover \cite{duff,pkt,schm} the usual description of the IIA D-2-brane, in
which the end of the string on the membrane carries the electric charge
\begin{equation}
\int_{S^1}\! *dV\ .
\label{eq:twoj}
\end{equation}
Returning to (\ref{eq:twoe}) we can alternatively define the modified string
charge to be
\begin{equation}
\hat Q_1 = =\int_{S^7} \! [\star H + K\wedge \tilde A]\ ,
\label{eq:twok}
\end{equation}
where $\tilde A$ is the 5-form potential associated with $\tilde F$, i.e.
$\tilde F= d\tilde A$. Since
\begin{equation}
Q_6=\int_{S^2}\! K
\label{eq:twol}
\end{equation}
is the 6-brane charge, similar reasoning to that above, but now setting $\tilde
A=d\tilde V_4$, leads to the conclusion that a IIA string can also end on a
6-brane and that the string charge is transferred to the 6-brane magnetic charge
\begin{equation}
\int_{S^5} \!d\tilde V_4\ ,
\label{eq:twom}
\end{equation}
which can be rewritten in the expected electric charge form
\begin{equation}
\int_{S^5} \! * dV
\label{eq:twon}
\end{equation}
by introducing the worldvolume 1-form potential $V$ dual to $\tilde V_4$.
The remaining IIA D-branes are the 0-brane and the 8-brane. The
possibility of a IIA string ending on a 0-brane is {\it not} found by the
`brane surgery' method for the good reason that it is actually forbidden by
charge conservation unless the 0-brane is the endpoint of two or more
strings. Thus, a modification of the method will be needed to deal with this
case. Neither is it it clear how the method can cope with the IIA 8-brane,
because of the non-generic peculiarities of this case.
Leaving aside these limitations of the method, there are further consequences to
be deduced from the CS term (\ref{eq:twod}). We have still to consider its
effect on the $\tilde A$ equation of motion. Actually it is easier to return to
the modified Bianchi identity (\ref{eq:twoc}), which we can rewrite as
\begin{equation}
d(F- K\wedge B)=0\ .
\label{eq:twoo}
\end{equation}
This shows that the homotopy-invariant magnetic 4-brane charge is actually
\begin{equation}
\hat Q_4 = \int_{S^4} [F - K\wedge B]\ .
\label{eq:twop}
\end{equation}
By the now familiar reasoning we deform the 4-sphere and set
$H=0$ to arrive at
\begin{equation}
\hat Q_4 = -\int_{S^2}\! K \times \int_{S^2}\! dV\ .
\label{eq:twoq}
\end{equation}
We recognise the first integral as the charge $Q_6$ of a 6-brane. The second
integral is the magnetic charge associated with a 3-brane within the 6-brane.
The 3-brane is of course the 4-brane's boundary. Thus a 4-brane can end on a
6-brane. This is not unexpected because it follows by T-duality from the
fact that a IIB 3-brane can end on a D-5-brane.
We could as well have rewritten the modified Bianchi identity (\ref{eq:twoc}) as
\begin{equation}
d(F + H\wedge C)=0\ ,
\label{eq:twor}
\end{equation}
in which case a similar line of reasoning, but setting $K=0$, and so $C=dy$,
leads to the expression
\begin{equation}
\hat Q_4 = \int_{S^3}\! H \times \int_{S^1}\! dy\ .
\label{eq:twos}
\end{equation}
The first integral is the magnetic 5-brane charge $Q_5$, so we deduce that a
4-brane can also end on a (solitonic) 5-brane. The 3-brane boundary in the
5-brane is a magnetic source for the scalar field $y$. The KK origin of $y$
suggests a D=11 interpretation of this possibility. It is surely closely
related to the fact that two D=11 fivebranes can intersect on a 3-brane
\cite{paptown}, since by wrapping one of the 5-branes (but not the other one)
around the 11th dimension we arrive at a D-4-brane intersecting a solitonic
5-brane in a 3-brane. This is not quite the same as a D-4-brane {\it ending} on
a 5-brane, but the intersection could be viewed as two 4-branes which happen to
end on a common 3-brane boundary in the 5-brane. This illustrates a close
connection between the `brane boundary' rules and the `brane
intersection rules', which will not be discussed here.
\section{M-brane boundaries}
Finally, we turn to M-theory, or rather D=11 supergravity and its p-brane
solutions. The bosonic fields of D=11 supergravity are the 11-metric and
a 3-form gauge potential $A$ with 4-form field strength $F=dA$. The Bianchi
identity for $F$ is
\begin{equation}
dF=0\ ,
\label{eq:threea}
\end{equation}
from which we may immediately conclude that the D=11 fivebrane must be closed.
The same is not true of the D=11 membrane, however, because there is a CS
term in the action of the form
\begin{equation}
{1\over 3} F\wedge F\wedge A
\label{eq:exa}
\end{equation}
which leads to the following field equation\footnote{An additional
singular 5-brane source term was included in \cite{wit} leading to a rather
different interpretation of the significance of the $F\wedge F$ term. We note
that since the 5-brane is actually a completely non-singular solution of the
D=11 field equations \cite{ght} it should not be necessary to include it as a
source.}
\begin{equation}
d\star F = - F\wedge F\ .
\label{eq:threeb}
\end{equation}
We see that the well-defined membrane charge is actually
\begin{equation}
\hat Q_2 = \int_{S^7}\! [\star F + F\wedge A]\ .
\label{eq:threec}
\end{equation}
Now consider a membrane with a boundary. Contract the 7-sphere to the boundary
and deform it to the product $S^4\times S^3$ so that the entire contibution to
$\hat Q_2$ is given by
\begin{equation}
\hat Q_2 = \int_{S^4}\! F\times \int_{S^3}\! A \ .
\label{eq:threed}
\end{equation}
The first integral is the charge $Q_5$ associated with a fivebrane. Set
$Q_5=1$. We may also set to zero the components of $F$ `parallel' to the
fivebrane, so that $A=dV_2$ in the second integral. We then have
\begin{equation}
\hat Q_2 = \int_{S^3}\! dV_2\ ,
\label{eq:threee}
\end{equation}
which is the magnetic charge of the string boundary of the membrane in the
fivebrane.
In fact, the 3-form field-strength $F_3=dV_2$ (or rather ${\cal F}_3 =dV_2-A$
in a general background) is {\it self-dual} but we do not learn this fact from
the `brane surgery' method. As for the IIB 3-brane, where we saw that the
worldvolume 1-forms $V$ and $\tilde V$ are related by Hodge duality of their
2-form field strengths, this information must be gleaned from a different
analysis. The similarity between these constraints on the worldvolume gauge
fields suggests that a deeper understanding of the phenomenon should be
possible.
In this contribution I have discussed the rules governing `brane boundaries'
in superstring and M-theory and shown that they follow from consideration of
interactions in the corresponding effective supergravity theories. It should be
appreciated that brane boundaries constitute a subset of possible `brane
interactions', which include intersecting branes and branes of varying
topologies. A reasonably complete picture is now emerging of the static aspects
of brane interactions, but little is known at present about the dynamic aspects,
i.e. the analogue of the splitting and joining interaction in string theory.
This problem is presumably bound up with the problem of finding an intrinsic
definition of M-theory, which may well require a substantially new
conceptual framework. Hopefully, the current focus on branes will prove to be of
some help in this daunting task.
\vskip 0.5cm
\noindent
{\it Acknowledgements}: I am grateful to George Papadopoulos for helpful
discussions.
|
proofpile-arXiv_065-655
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\chapter{Introduction}
There is now ample evidence that the IIA superstring theory is an $S^1$
compactification of an 11-dimensional supersymmetric quantum theory called
M-theory. It was pointed out in [\pkta,\wita] that this interpretation requires
the presence in the non-perturbative IIA superstring theory of
BPS-saturated particle states carrying Ramond-Ramond (RR) charge,
corresponding to the Kaluza-Klein (KK) modes of D=11 supergravity, and it was
argued that these should be identified with the IIA 0-branes. At the time, the
only evidence for the required 0-branes was the existence of extreme electric
`black hole' solutions of the effective IIA supergravity theory [\hs],
but their
presence in the IIA superstring theory was subsequently confirmed by the
interpretation of D-branes as the carriers of RR charge [\pol].
Actually, one needs not just the D-0-brane, for which the effective field
theory
realization is the extreme black hole of lowest charge associated with the
first
KK harmonic, but also a bound state at threshold in the system of $n$
D-0-branes
for each $n>1$, a prediction that has still to be confirmed although there is
good evidence that it is true [\sen]. Assuming that these bound states exist,
M-theory provides a KK interpretation of the D-0-branes of IIA superstring
theory. However, as emphasized in [\pkta], {\it all} the IIA p-branes must have
a D=11 interpretation. Indeed, many of them can be interpreted as reductions,
either `direct' or `double', of D=11 branes, i.e. M-branes. The cases of
interest to us here are those IIA p-branes that that have a D=11 interpretation
as (p+1)-branes wrapped around the compact 11th dimension. The massless
worldvolume action for the D=10 p-brane is then a dimensional reduction
on $S^1$
of the worldvolume action of the (p+1)-brane of M-theory. Thus, the D=11
interpretation of these D=10 p-branes requires the existence of massive
particle-like excitations `on the brane' that can be identified with the KK
harmonics of the `hidden' $S^1$. From the D=10 string theory perspective these
excitations can only be BPS-saturated 0-brane/p-brane bound states [\doug].
Moreover, since the worldvolume KK states preserve 1/2 of the (p+1)-dimensional
worldvolume supersymmetry and the p-brane preserves 1/2 of the spacetime
supersymmetry, these `brane within brane' states must preserve 1/4 of the
spacetime supersymmetry [\douglas]. The IIA $p$-branes for which we
should expect to find such bound states are (i) the 1-brane, i.e. the
fundamental IIA
superstring, since this is a wrapped D=11 membrane, (ii) the D-4-brane, since
this is a wrapped D=11 fivebrane, and possibly (iii) the D-8-brane, since
it has been suggested [\bgprt] that the D-8-brane might be a wrapped
D=11 ninebrane.
The required bound states are not difficult to identify in case (i).
A fundamental string can end on a 0-brane; actually, charge conservation
requires a 0-brane to be the end of at least two fundamental strings. Two
such strings can be joined at their other ends to produce a closed string
loop with a 0-brane `bead'. One could replace the 0-brane by a bound state
of several 0-branes. Thus, the bound states needed for the KK interpretation of the IIA superstring as a wrapped D=11 membrane are an immediate consequence
of the 0-brane bound states needed for the KK interpretation of the
effective IIA supergravity theory. This is not so in cases (ii) and
(iii) for which we need
to find bound states of D-0-branes with D-4-branes or D-8-branes. The
existence of such bound states is consistent with the `D-brane
intersection rules' [\polch,\gg] which allow, in particular, the possibility of a p-brane within a q-brane preserving 1/4 of the supersymmetry for $p=q$ mod 4.
The issue of 0-brane/4-brane bound states has been discussed recently in the
context of the D-brane effective action [\doug]. Here we investigate this
question in the context of solutions of the effective IIA supergravity theory.
We shall show that solutions representing 0-branes within p-branes preserving
precisely 1/4 of the supersymmetry exist for $p=1,4$ but not otherwise
(completing previous partial constructions by other methods [\BBJ]).
This result is consistent with the standard D=11 interpretation of all the type
II p-branes for $p\le6$ but {\it not} with the interpretation of the
IIA 8-brane as an $S^1$-wrapped M-theory ninebrane. A further argument
against the M-theory ninebrane interpretation of the IIA 8-brane comes from
consideration of a solution of IIA supergravity preserving 1/4 of the spacetime
supersymmetry that represents a D-4-brane within a D-8-brane (the metric for
this solution is already known [\BBJ]; here we present the complete solution).
If there were an M-theory ninebrane it would be natural to interpret this
4-brane within 8-brane solution as a fivebrane within a ninebrane wrapped on
$S^1$ along a fivebrane direction. However, if such an M-theory configuration
were to exist it could also be reduced to a IIA 5-brane within an 8-brane
but there does not exist any such solution of IIA supergravity preserving
precisely 1/4 of the spacetime supersymmetry.
In what follows we shall use the notation $(q|q,p)$ to represent a q-brane
within a p-brane preserving 1/4 of the supersymmetry. This is the special
case of $(r|p,q)$, which we use to denote a solution representing an r-brane
intersection of a p-brane with a q-brane. Thus, in this notation the
supersymmetric solutions representing a 0-brane within a p-brane for $p=1$
and $p=4$ are $(0|0,1)$ and $(0|0,4)$. These solutions have magnetic duals,
$(5|5,6)$ and $(2|2,6)$, respectively, whose existence is required by
M-theory. To see this recall that the D=11 interpretation of the 6-brane is
simply as a D=11 spacetime of the form $H_4\times \bb{M}_7$ where $H_4$ is a
particular (non-compact) hyper-K{\" a}hler manifold and $\bb{M}_7$ is
7-dimensional Minkowski spacetime [\pkta]. Clearly, there is nothing to prevent
the worldvolumes of either the D=11 membrane or fivebrane from lying within the
$\bb{M}_7$ factor, and from the D=10 perspective this is a membrane or a 5-brane
within a 6-brane. We shall show how the $(5|5,6)$ and $(2|2,6)$ can also be
deduced from known intersecting M-brane solutions.
\chapter{Branes within branes in IIA supergravity}
As just explained, M-theory predicts the existence of a variety of IIA
supergravity solutions preserving precisely 1/4 of the N=2 spacetime
supersymmetry that represent `branes within branes' (by `precisely` we
mean to exclude solutions preserving more than 1/4 of the supersymmetry).
A summary of these predictions is as follows: we expect $(0|0,p)$ solutions for p=1,4 and possibly p=8, {\it but not otherwise}. We also expect the magnetic
duals of $(0|0,p)$ for $p=1,4$, and a $(4|4,8)$ solution.
That there are no $(0|0,2)$, $(0|0,5)$ or $(0|0,6)$ solutions of IIA
supergravity preserving 1/4 (as against any other fraction) of the
supersymmetry
follows from consideration of the projection operators associated with
Killing spinors. A single p-brane solution is associated with a projection
operator $P_p$, of which precisely half the eigenvalues vanish, such that only
spinors $\kappa$ satisfying $P_p\kappa=\kappa$ can be Killing. This accounts
for
the fact that such solutions preserve half the supersymmetry. Configurations
representing a p-brane within a q-brane for $p\ne q$ can also preserve some
supersymmetry since $P_p$ and $P_q$ must {\it either} commute {\it or}
anticommute. If $P_p$ and $P_q$ commute then the product $P_pP_q$ is also a
projector. In such cases one may find a supersymmetric solution preserving 1/4
of the supersymmetry, representing either two intersecting branes or a `brane
within a brane'. If $P_p$ and $P_q$ anticommute then the matrix
$$
\alpha P_p +\beta P_q \qquad (\alpha^2 +\beta^2 =1)
\eqn\newa
$$
is another projector with precisely half of its eigenvalues vanishing. In this
case one can hope to find `brane within brane' solutions preserving 1/2 the
supersymmetry. An example of such a solution is the D=11 membrane within a
fivebrane solution [\memdy]; as shown in [\gp,\glpt], this reduces to a
$(2|2,4)$ solution of IIA supergravity preserving 1/2 the supersymmetry.
Consideration of T-duality then implies the existence of $(0|0,2)$ solutions
preserving 1/2 the supersymmetry \foot{It has been pointed out to us
independently by J. Maldacena and J. Polchinski that such a solution could be
interpreted as a D-2-brane boosted in the 11th dimension. The solution has
since been constructed [\tsey].}. Here we are interested in solutions
preserving precisely 1/4 of the supersymmetry, so only those cases for which
$P_p$ and
$P_q$ {\it commute} are relevant. When both branes are D-branes one can show
that $P_p$ and
$P_q$ commute if and only if $q=p\ $ mod 4, so that $(0|0,2)$ and $(0|0,6)$
solutions preserving 1/4 of the supersymmetry are immediately excluded, whereas
$(0|0,4)$, $(2|2,6)$ and
$(4|4,8)$ are allowed, as is $(0|0,8)$. This D-brane rule says nothing about
$(0|0,1)$ or $(0|0,5)$ since neither the IIA string nor the IIA
5-brane is a D-brane. It happens that $P_1$ commutes with $P_0$ whereas $P_5$
does not, so a $(0|0,1)$ solution preserving 1/4 of the supersymmetry is
allowed
whereas a $(0|0,5)$ solution is not. A putative $(5|5,8)$ solution
preserving 1/4 of the supersymmetry is similarly ruled out. For the
reason given earlier, this fact is evidence against the existence of
a $(0|0,8)$ solution. Thus,
the projection operator analysis provides arguments both for and
against the possibility of a $(0|0,8)$ solution.
Leaving aside $(0|0,8)$, we have now seen that the solutions not expected from
M-theory considerations are indeed absent, while the solutions that
M-theory requires to exist are permitted. We shall now show that all of
the latter, among those mentioned above, not only exist but can be
constructed from known intersecting M-brane solutions preserving
1/4 of the supersymmetry [\guv,\paptown,\ark,\kleb,\jer] by means of
the various dualities connecting M-theory with the IIA and IIB
superstring theories. The relevant M-theory solutions can be obtained from the
`M-theory intersection rules' determining the allowed M-brane intersections
together with the `harmonic function rule' that allows one to write down the
general solution. For example, the $(0|0,1)$ solution of IIA
supergravity can be
deduced from the solution of D=11 supergravity associated with the
intersection
of two membranes at a point, i.e. $(0|2,2)_M$. This is achieved by
consideration of the `duality chain'
$$
(0|2,2)_M\rightarrow (0|1,2){\buildrel T \over\rightarrow}
(0|1,1_D)_B{\buildrel T \over\rightarrow} (0|0,1)\ ,
\eqn\branea
$$
where the subscript $B$ indicates a solution of IIB supergravity and $1_D$
denotes the IIB D-string. In the first step one of the two D=11
membranes is wrapped around the 11th dimension; the corresponding D=10 solution
being obtained by double-dimensional reduction. In the second step we T-dualize
along a direction parallel to the IIA 2-brane to arrive at the IIB solution.
A further T-dualization along one of the two directions determined by the
D-strings leads to the required IIA solution.
To make clear the unambiguous nature of the derivation we shall give all the
intermediate solutions for this example, while giving just the final result
for the examples to follow. Thus, we begin with the
$(0|2,2)$ solution of D=11 supergravity
$$
\eqalign{ds^2&=U^{1/3} V^{1/3}\big[ -U^{-1} V^{-1} dt^2+ U^{-1}
ds^2(\bb{E}^2)+V^{-1} ds^2(\bb{E}^2)+ds^2(\bb{E}^6)\big]
\cr
G_4&=-3 dt \wedge d\big( U^{-1}J_1+V^{-1} J_2\big)\ ,}
\eqn\bone
$$
where (in the terminology of [\paptown]) $U,V$ are harmonic functions of the
overall transverse space $\bb{E}^6$ and $J_1\oplus J_2$ is a complex structure on
the relative transverse space $\bb{E}^2\oplus \bb{E}^2$ . Double-dimensional
reduction
along one of the relative transverse directions results in the following
$(0|1,2)$ solution of IIA supergravity:
$$
\eqalign{
ds_{(10)}^2&= V^{1/2}\big[ -U^{-1} V^{-1} dt^2+ U^{-1}
dx^2+V^{-1} ds^2(\bb{E}^2)+ds^2(\bb{E}^6)\big]
\cr
e^{{4\over3}\phi}&= U^{-{2\over3}} V^{1\over3}
\cr
F_4&=-3 dt \wedge d\big(V^{-1} J_2\big)
\cr
F_3&=-3 dt\wedge dx\wedge d U^{-1}\ , }
\eqn\btwo
$$
where $x$ is the string coordinate. Next, using the T-duality rules of [\bho]
(adapted to our conventions) to T-dualize along one of the directions of the
2-brane, we get the $(0|1,1_D)_B$ solution
$$
\eqalign{
ds_{(10)}^2&= V^{1/2}\big[ -U^{-1} V^{-1} dt^2+ U^{-1}
dx^2+V^{-1} du^2 + ds^2(\bb{E}^7)\big]
\cr
e^{{2\over3}\varphi}&=U^{-{1\over3}} V^{1\over3}
\cr
F^{(2)}_3&= -3 dt\wedge du\wedge dV^{-1}
\cr
F^{(1)}_3&= -3 dt\wedge dx\wedge d U^{-1}\ , }
\eqn\bthree
$$
where $\varphi$ is the IIB dilaton. Finally, we transform \bthree\ using
T-duality along the $u$ direction to get the following $(0|1,0)\equiv (0|0,1)$
solution of IIA supergravity:
$$
\eqalign{
ds_{(10)}^2&= V^{1/2}\big[ -U^{-1} V^{-1} dt^2+ U^{-1}
dx^2+ds^2(\bb{E}^8)\big]
\cr
e^{{2\over3}\phi}&=U^{-{1\over3}} V^{1\over2}
\cr
F_3&=-3 dt\wedge dx\wedge d U^{-1}
\cr
F_2&= -{9\over2} dt\wedge dV^{-1}\ . }
\eqn\bfour
$$
The above solutions, as others given below, depend on two
{\it independent} harmonic functions, each of which is associated with a single
p-brane. For simplicity, let us suppose that each harmonic function has
just one singularity (at the position of the brane). Clearly, one must
further suppose that
both harmonic functions have their singularities at the {\it same} location in
order to be able to interpret the configuration as a `brane within brane'
solution associated with the long range fields of a bound state. The same
solution could equally well represent the simple coincidence of two branes; the
fact that solutions exist with two independent harmonic functions indicates
that any bound state would be a bound state at threshold. It is a
weakness of the effective field theory approach that it cannot distinguish
between a bound state at threshold of two branes or their simple coincidence
because both have the same long range fields. The evidence for bound states
provided by the effective field theory is, therefore, not particularly strong,
Nevertheless, when the harmonic functions in \bfour\ are restricted in the
way just described these solutions do give the long range fields of the
KK modes
that arise from the wrapping of the D=11 membrane on $S^1$ to give a D=10
string.
Before proceeding we pause to remark that the magnetic dual of the $(0|0,1)$
solution can be found from the $(3|5,5)_M$ solution of M-theory by the
following
duality chain:
$$
(3|5,5)_M\rightarrow (3|5,4){\buildrel T \over\rightarrow}
(4|5,5_D)_B{\buildrel T \over\rightarrow} (5|5,6)\ ,
\eqn\bseven
$$
where $5_D$ denotes the D-5-brane of the IIB theory ($5$ denoting the NS-NS
5-brane). In the second step we have T-dualized in a direction parallel to the
IIA 5-brane, which is mapped to the IIB NS-NS 5-brane under this operation.
The final $(5|5,6)$ solution dual to $(0|0,1)$ (which has been found
previously
by other means [\ark]), is
$$
\eqalign{ds^2&= U V^{{1\over2}}\big[U^{-1} V^{-1} ds^2(\bb{M}^6)+V^{-1}
dv^2+ds^2(\bb{E}^3)\big]
\cr
e^{{2\over3}\phi}&=U^{1\over3} V^{-{1\over2}}
\cr
F_3&=9 dv \wedge \star dU
\cr
F_2&=27 \star dV\ ,}
\eqn\beight
$$
where $U,V$ are harmonic functions on the Euclidean transverse space
$\bb{E}^3$ and
$\star$ is the Hodge dual for $\bb{E}^3$.
We turn next to the $(0|0,4)$ case. This can be found from the following
duality chain,
$$
(0|2,2)_M\rightarrow (0|2,2){\buildrel T \over\rightarrow}
(0|1_D,3)_B{\buildrel T \over\rightarrow} (0|0,4)\ ,
\eqn\bafive
$$
where the first step is the direct reduction to D=10 of the D=11 solution.
The resulting $(0|0,4)$ solution is
$$
\eqalign{ds^2&=U^{{1\over2}} V^{{1\over2}} \big[ -U^{-1} V^{-1} dt^2+V^{-1}
ds^2(\bb{E}^4)+ds^2(\bb{E}^5)\big]
\cr
e^{{2\over3}\phi}&=V^{1\over2} U^{-{1\over6}}
\cr
F_4&=3\star dU
\cr
F_2&=-{9\over2} dt\wedge dV^{-1}\ ,}
\eqn\bfive
$$
where $U,V$ are harmonic functions on $\bb{E}^5$ and $\star$ is now the Hodge dual
for $\bb{E}^5$. The magnetic dual of this solution is $(2|2,6)$, which can
be found
from the duality chain
$$
(3|5,5)_M\rightarrow (2|4,4){\buildrel T \over\rightarrow}
(2|3,5_D)_B{\buildrel T \over\rightarrow} (2|2,6)\ .
\eqn\bten
$$
The final $(2|2,6)$ solution is
$$
\eqalign{ds^2&=U^{{1\over2}}V^{{1\over2}}\big[U^{-1} V^{-1}
ds^2(\bb{M}^3)+U^{-1}ds^2(\bb{E}^4)+ds^2(\bb{E}^3)\big]
\cr
e^{{2\over3}\phi}&=V^{1\over6} U^{-{1\over2}}
\cr
F_4&=-{3\over2}\epsilon(\bb{M}^3)dV^{-1}
\cr
F_2&=27\star dU \ ,}
\eqn\bnine
$$
where $U,V$ are harmonic functions on $\bb{E}^3$ and $\star$ is the Hodge dual for
$\bb{E}^3$.
We remark that both $(0|0,4)$ and its magnetic dual $(2|2,6)$ can also be found
from $(1|2,5)_M$ as follows:
$$
(1|2,5)_M\rightarrow (1|2,4){\buildrel T \over\rightarrow}
(0|1_D,3)_B{\buildrel T \over\rightarrow} (0|0,4)\ ,
\eqn\bsix
$$
and
$$
(1|2,5)_M\rightarrow (1|2,4){\buildrel T \over\rightarrow}
(1|1_D,5_D)_B{\buildrel T \over\rightarrow} (2|2,6)\ .
\eqn\beleven
$$
The above solutions confirm the current D=11 interpretations of all IIA
p-branes for $p\le6$. In addition, the duality chain of \beleven\ can be
continued as follows:
$$
(2|2,6) {\buildrel T \over\rightarrow} (3|3,7)_B {\buildrel T
\over\rightarrow} (4|4,8) \ .
$$
In principle, the 7-brane appearing in the penultimate solution is the
D-7-brane. However, the 7-brane solution needed for this construction is the
`circularly-symmetric' 7-brane of IIB supergravity since, as shown in [\bgprt],
it is this solution that is mapped to either the 6-brane or the 8-brane
solution of $S^1$ compactified IIA supergravity. A further point is that the
T-duality transformations to be used in the last link of the duality chain are
the `massive' ones of [\bgprt] connecting solutions of IIB supergravity with
those of the {\it massive} IIA supergravity theory. Apart from these subtleties
the construction proceeds as before, with the final result
$$
\eqalign{ds^2&=U^{1\over2}V^{1\over2}\big(U^{-1}V^{-1}ds^2(\bb{M}^5)+
U^{-1}ds^2(\bb{E}^4)+ dy^2\big)
\cr
e^{{2\over3}\phi}&=V^{-{1\over 6}} U^{-{5\over6}}
\cr
M&=\partial_yU
\cr
F_4&=3 \epsilon(\bb{E}^4)\partial_yV\ ,}
\eqn\ctwo
$$
where $U,V$ are harmonic functions of $y$.
Finally, we return to the question of whether there exists a $(0|0,8)$ solution
which might represent KK modes in a possible M-theory ninebrane
interpretation of the IIA 8-brane. If it exists we should be able
to deduce it from
M-theory. It cannot be so deduced from the intersecting M-brane solutions
considered so far, but there exists a solution of D=11 supergravity preserving
1/4 of the supersymmetry that has been interpreted as the intersection of two
M-theory fivebranes on a string, i.e. as a $(1|5,5)$ solution [\jer]. Taking
this solution as the starting point of the following duality chain
$$
\eqalign{
(1|5,5)_M &\rightarrow (0|4,4) {\buildrel T \over\rightarrow} (0|3,5_D)_B
{\buildrel T \over\rightarrow} (0|2,6) \cr
& {\buildrel T \over\rightarrow} (0|1_D,7)_B {\buildrel T \over\rightarrow}
(0|0,8)\ , }
\eqn\branetwo
$$
we could apparently deduce the existence of the sought $(0|0,8)$ solution.
However, the starting $(1|5,5)$ solution has a quite different form from the
other intersecting M-brane solutions. In particular, the two harmonic functions
associated with each fivebrane are independent of the `overall transverse'
coordinate. On the other hand, consistency with the KK ansatz needed for the
various T-duality steps in the above chain requires that both harmonic
functions be independent of all the other coordinates. Therefore, the only
acceptable starting solution for the above duality chain is the special case
of the $(1|5,5)$ solution for which both harmonic functions are constant;
this is just the Minkowski vacuum which obviously preserves all the
supersymmetry rather than just 1/4 of it. Of course, this shows only that a
IIA $(0|0,8)$ solution preserving precisely 1/4 of the supersymmetry
cannot be obtained from a particular
starting point. However, supposing such a solution to exist we could reverse
the steps in the duality chain \branetwo\ to deduce the existence of a
$(1|5,5)$ solution of `conventional' form for intersecting M-branes, i.e. with
both harmonic functions depending only on the overall transverse coordinate.
It is not difficult to see that there is no such solution because the
associated 4-form field strength does not satisfy the field equation
$d*G=G\wedge G$. Thus, there is no $(0|0,8)$ solution of the required type.
\chapter{Conclusions}
The interpretation of certain p-brane solutions of IIA supergravity
as wrapped (p+1)-branes of M-theory requires the
existence of massive KK modes `on the brane'. In turn, this requires the
existence (and in other cases, absence) of `brane within brane' solutions of
IIA supergravity preserving 1/4 of the supersymmetry. We have shown that the
list of such solutions is compatible both with the current M-theory
interpretations of the IIA p-branes with $p\le6$, but not with an
interpretation of the IIA 8-brane as an M-theory 9-brane.
\vskip 0.5cm
\noindent{\bf Acknowledgments:} GP thanks The Royal Society for a University
Research Fellowship. We thank the organisers of the Benasque Centre for Physics
in Spain, where part of this work was done, Michael Douglas,
Michael Green and Christopher Hull for discussions. We also thank Juan
Maldacena and Joseph Polchinski for comments on an earlier version of this
paper and especially Eric Bergshoeff, who pointed out a serious error in it.
\refout
\end
|
proofpile-arXiv_065-656
|
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|
\subsection{Right Tail}
Let us first describe the essentially inviscid
instantons producing the right tails of the PDFs
for gradients and differences \cite{Pol95,95GM,GK96}.
At $t=0$, the
field $p$ is localized near the origin. At moving backwards in time the
viscosity will spread the field $p$. Nevertheless,
a positive velocity slope
``compresses'' the field $p$ so that one can expect that the width of $p$ remains
much smaller than $L$. Then, it is possible to formulate the closed
system of equations for the quantities $a(t)$ and
$c(t)=-i \int dx\,x\,p(t,x)$ since for narrow $p$ and small $x$ we can put
$\int dx'\chi(x-x')p(t,x')\to-i\partial_x\chi(x)c(t)\approx2i\omega^3xc(t)$:
\begin{equation}
\partial_tc=2ac, \quad
\partial_ta=-a^2+2\omega^3c.
\label{vca} \end{equation}
The instanton is a separatrix
solution of (\ref{vca}).
The initial condition
$a(0)c(0)=n$ by virtue of the energy conservation
gives $a(0)=\omega^3c^2(0)/n=\omega n^{1/3}$.
For differences, $w=2a(0)\rho$.
One can check that
$ {\cal I}_{\rm extr}=i\int dt\, c\partial_t a\sim a(0)c(0)=n $ which
is negligible in comparison with $n\ln[a(0)]$ so that
$\langle(u')^n\rangle\sim[a(0)]^n \sim \omega^n n^{n/3}$
which gives the right cubic tails of the PDFs
$\ln{\cal P}(u')\sim-(u'/\omega)^3$ \cite{GK96} and
$\ln{\cal P}(w)\sim-[w/(\rho\omega)]^3$
\cite{Pol95,95GM}.
One can show that the width of $p$ is much less than $L$ through the
time of evolution $T\sim n^{-1/3}\omega^{-1}$
giving the main contribution into the action \cite{95GM}.
The right tails of ${\cal P}(u')$ and ${\cal P}(w)$ are thus
universal i.e. independent of the large-scale properties of the pumping.
Above consideration does not imply that the instanton is completely inviscid,
it may well have viscous shock at $x\sim L$, this has no influence
on the instanton answer (since $p$ is narrow) while may influence the fluctuation
contribution i.e. predexponent in the PDF.
The main subject of this paper is the analysis of the instantons that give
the tails of ${\cal P}(u)$ and the left tails of ${\cal P}(u')$
and ${\cal P}(w)$ corresponding to negative $a$, $w$.
Even though the field $p$ is narrow at $t=0$, we cannot use the simple
system (\ref{vca}) to describe those instantons.
The reason is that sweeping by a negative velocity slope provides for
stretching (rather than compression) of the field $p$ at moving backwards
in time. As a result, the support of $p(x)$ stretches up to $L$ so that
one has to account for the given form of the pumping correlation function
$\chi(x)$ at $x\simeq L$. This leads to a nonuniversality of ${\cal P}(u)$ and of
the left tails of ${\cal P}(u')$ and ${\cal P}(w)$ which depend
on the large-scale properties of the pumping. As we shall see, the form of
the tails is universal, nonuniversality is related to a single constant in PDF.
Additional complication in analytical description is due to
the shock forming from negative slope near the
origin. The shock cannot be described in terms of the inviscid
equations so that we should use the complete system
(\ref{va2},\ref{vam}) to describe what can be called
viscous instantons.
Apart from a narrow front near $x=0$, the velocity field
has $L$ as the only characteristic scale of change. The life time $T$ of
the instanton is then determined by the moment when the position of $p$
maximum reaches $L$ due to sweeping by the velocity
$u_0$: $T\sim L/u_0$. Such a velocity $u_0$ itself has been created during
the time $T$ by the forcing so that $u_0\sim|c|_{max}TL\omega^3$.
To estimate the maximal value of $|c(t)|$, let us consider the backward
evolution from $t=0$. We first notice that the width of $p$ (which was zero
at $t=0$) is getting larger than the width of the velocity front $\simeq u_0/a$
already after the short time $\simeq a^{-1}$. After that time, the
values of $c$ and $a$ are of order of their values at $t=0$.
Then, one may consider that $p(t,x)$ propagates (backwards in time)
in the almost homogeneous velocity field $u_0$ so that
$$\partial_t c=-i\int_{-\infty}^{\infty} dx\, xup_x\approx 2iu_0\int_0^\infty dx\, p
\ .$$ The (approximate) integral
of motion $i\int dx\, p$ can be estimated by it's value at $t=0$ which
is $n/2u_0$. Therefore, we get $c_{max}\simeq nT$ so that
$T\simeq n^{-1/3}\omega^{-1}$ and $u_0\simeq L\omega n^{1/3}$.
At the viscosity-balanced shock, the velocity $u_0$ and the gradient $a$
are related by $u_0^2\simeq\nu a$ so that $a(0)\simeq \omega{\rm Re}\,n^{2/3}$.
Let us briefly describe now the consistent analytic procedure of the derivation of
the function $c(t)$ that confirms above estimates. We use the
Cole-Hopf substitution \cite{Burg} for the velocity $\partial_x\Psi=-{u}\Psi/{2\nu}$
and introduce $P=2\nu\partial_xp/\Psi$.
The saddle-point equations for $\Psi$ and $P$
\begin{eqnarray} &&
\partial_t\Psi-\nu\partial_x^2\Psi+\nu F\Psi=0,
\label{ha7} \\ &&
\partial_t P+\nu\partial_x^2P-\nu FP
-{2\nu}\lambda'(x)\delta(t)\Psi^{-1}=0
\label{ha3} \end{eqnarray}
contain $F$ determined by $\partial_xF(t,x)=-{i}
\int dx'\chi(x-x')p(t,x')/{2\nu^2}$ and fixed by the condition $F(t,0)=0$.
We introduce the evolution operator $\hat U(t)$
which satisfies the equation $\partial_t\hat U=\hat H\hat U$ with
$\hat H(t)=\nu(\partial_x^2-F)$. It is remarkable that one
can develop the closed description
in terms of two operators $\hat A=\hat U^{-1} x\hat U$ and $\hat B=
\hat U^{-1}\partial_x\hat U$:
$$\partial_t\hat A=-2\nu\hat B\,,\quad \partial_t\hat B=-\nu F_x(t,\hat A)\ .$$
Since we are
interesting in the time interval when $p(t,x)$ is narrow, it is enough for our
purpose to consider $x\ll L$ where $F(t,x)=c(t)x^2\omega^3/2\nu^2$. Further
simplification can be achieved in this case and the closed ODE for $c(t)$ can
be derived after some manipulations:
$$(\partial_t c)^2=4\omega^3c^3+16\xi_2^2+4\omega^3\xi_1^3\ ,$$
where $\xi_1\!=i\int\! dx\lambda(x) x$ and $4\xi_2\!=-{i}\int\! dx\lambda(x)
\partial_x[xu(0,x)]$. Integrating we get
\begin{eqnarray}&&
t=\frac{1}{2}
\int_{c(0)}^{c}\frac{dx}{\sqrt{\omega^3x^3+4\xi_2^2+\xi_1^3}}\ ,
\label{b7}\end{eqnarray}
which describes $c(t)$ in an implicit form. Further analysis depends on the
case considered. For the gradients, we substitute $\xi_1=n/a_0$ and $\xi_2=-n/2$
and see that, as time goes backwards, negative $c(t)$ initially decreases by
the law $c(t)=c(0)+2nt$ until $T= \omega^{-1}(n/2)^{-1/3}$ then it grows
and the approximation looses validity when $c(t)$ approaches zero and the account
of the pumping form $\chi(x)$ at $x\simeq L$ is necessary. Requiring the width
of $p(x)$ at this time to be of order $L$ we
get the estimate $a(0)\simeq \omega{\rm Re}\,n^{2/3}$ and thus confirm the above
picture.
The main contribution to the saddle-point value (\ref{vaa})
is again provided by the term $[\partial_xu(0,0)]^n$
and we find $\langle(u')^n\rangle\simeq[a(0)]^n\simeq
(\omega{\rm Re})^n n^{2n/3}$, which corresponds to the following left
tail of PDF at $u'\gg \omega{\rm Re}$
\begin{equation}
{\cal P}(u')\propto
\exp[-C(-u'/\omega{\rm Re})^{3/2}]\ .
\label{an2} \end{equation}
For higher derivatives $u^{(k)}$, by using (\ref{b7}) we get initial growth
$c(t)=c(0)+n(k+1)t$ which gives
$u^{(k)}(0,0)\sim N^{k+1}L^{1-k}\omega {\rm Re}^{k}$ leading to
$\langle[u^{(k)}]^n\rangle\sim\omega {\rm Re}^{k}
L^{1-k} n^{(k+1)/3}$ which can be rewritten in terms of PDF:
\begin{eqnarray} &&
{\cal P}\left(|u^{(k)}|\right)\!\propto\!
\exp\left[-C_k\left({|u^{(k)}|L^{k-1}}/
{\omega{\rm Re}^k}\right)^{3/(k+1)}\right].
\label{hd4}\end{eqnarray}
Note that the non-Gaussianity increases with increasing $k$. On the other hand,
the higher $k$ the more distant is the validity region of (\ref{hd4}):
$u^{(k)}\gg u^{(k)}_{\rm rms}\sim L^{1-k}\omega{\rm Re}^k$.
For the differences, $\xi_1={2n\rho_0}/{w}$ and
$4\xi_2=-{n}[1+{2\rho_0u_x(0,\rho_0)}/{w}]$ and we get
$\langle w^n\rangle\simeq (L\omega)^nn^{n/3}$
which corresponds to the cubic left tail
\begin{equation} {\cal P}(w)\propto \exp\{-B[w/(L\omega)]^3\}
\label{an3} \end{equation}
valid at $w\gg L\omega$. In the intermediate region $L\omega\gg w\gg\rho\omega$,
there should be a power asymptotics which is the subject of current debate
\cite{Pol95,GK96,KS96}.
It is natural that $\rho$-dependence of ${\cal P}(w)$ cannot be found in a
saddle-point approximation; as a predexponent, it can be obtained only at the
next step by calculating the contribution of fluctuations around the instanton
solution. This is consistent with the known fact that the scaling exponent
is $n$-independent for $n>1$: $\langle w^n(\rho)\rangle\propto\rho$.
For the velocity, $\lambda(x)=-{in}\delta(x)/u(0,0)$ is an even function
so that $F$ is a linear (rather than quadratic)
function of $x$ for narrow $p$: $F(x)={\chi(0)bx}/{2\nu^2}$ with
$b=-{i}\int dx p(x)$. Direct calculation shows that energy and momentum
conservation makes $b$ time independent: $b=n/u(0,0)$.
It is easy then to get the $n$-dependence of $u(0,0)$:
Velocity stretches the field $p$ so that the width of $p$
reaches $L$ at $T\simeq L/u(0,0)$ while the velocity itself is produced by
the pumping during the same time: $u(0,0)\simeq \chi(0)bT=\chi(0)nT/2u(0,0)
\simeq n\chi(0)L/u(0,0)$. That gives $u(0,0)\simeq L\omega n^{1/3}$ and
$$ {\cal P}(u)\propto \exp\{-D[u/(L\omega)]^3\}\ .$$
The product $L\omega$ plays the role of the root-mean square velocity
$u_{\rm rms}$. The numerical factors $C$, $B$ and $D$
are determined by the evolution at $t\simeq T$ i.e. by the behavior of pumping
correlation function $\chi(x)$ at $x\simeq L$.
We thus found the main exponential factors in the PDF tails.
Complete description of the tails requires the analysis
of the fluctuations around the instanton which will be the subject of
future detailed publications. Here, we briefly outline some important
steps of this analysis. The account of the fluctuations in the Gaussian
approximation is straightforward and leads to the shift of ${\cal I}_{\rm extr}$
insignificant at $n\gg1$. However, the terms of the perturbation theory with
respect to the interaction of fluctuations are infrared divergent (proportional
to the total observation time). That means that there is a soft mode which
is to be taken into account exactly. Such an approach has been already developed
in \cite{95FKLM} for the simpler problem of the PDF tails for a passive scalar
advected by a large-scale velocity where the comparison with the exactly
solvable limits was possible.
A soft mode usually corresponds to a global symmetry
with a continuous group: if one allows the slow spatio-temporal
variations of the parameters of the transformation then small variations of
the action appears.
Our instantons break Galilean invariance so that the respective
Goldstone mode has to be taken into account. Namely, under the transformation
\begin{equation}x\rightarrow x-r,\ u(x)\rightarrow u(x-r)+v,\
r=\int_t^0v(\tau)d\tau\ ,\label{sym}\end{equation}
the action is transformed as ${\cal I}\rightarrow{\cal I}-i\int dxdtp\partial_tv$.
The source term $\int dxdt\lambda u$ is invariant with respect to (\ref{sym}) for
antisymmetric $\lambda(x)$. To integrate exactly along the direction specified by
(\ref{sym}) in the functional space we use Faddeev-Popov trick
inserting
the additional factor
\begin{equation}
1=\int{\cal D}v(t)\delta\left[u\biggl(t,\int_t^0v(\tau)d\tau\biggr)-
v(t)\right] {\cal J}\ .
\label{unity}\end{equation}
into the integrand in (\ref{si1},\ref{sio}).
Jacobian ${\cal J}$ is determined by a regularization of (\ref{sym})
according to our choice of the retarded regularization for the initial
integral: at
discretizing time we put $\partial_tu+u\partial_xu
\rightarrow ({u_n-u_{n-1}})/{\epsilon}+u_{n-1} u'_{n-1}$ (otherwise,
some additional $u$-dependent term appears \cite{DP78}).
The discrete version of (\ref{sym}),
$p_n(x)\rightarrow p_n(x-\epsilon\sum_{j=n}^{N-1}v_j)$,
$u_n(x)\rightarrow u_n(x-\epsilon\sum_{j=n}^{N-1}v_j)-v_n$, $u_N(x)\rightarrow
u_N(x)-v_N$ gives
$${\cal J}=\exp\left[\int_{-T}^0dtu'\biggl(t,\int_t^0 v(\tau)d\tau\biggr)\right]\ .$$
Substituting (\ref{unity}) into (\ref{si1},\ref{sio})
and making (\ref{sym}) we calculate $\int{\cal D}v$ as a Fourier integral
(the saddle-point method is evidently
inapplicable to such an integration) and conclude that after the integration
over the mode (\ref{sym}) the measure ${\cal D}u{\cal D}pe^{i{\cal I}}$
acquires the additional factor
$$\prod\limits_t\!\delta\biggl[\int\partial_t^2 p(t,x)dx\biggr]
\delta[u(t,0)]\exp\left[
\int_{-T}^0\!u'(t,0)dt\right].$$
The last (jacobian) term here exactly corresponds
to the term $u'{\cal P}(u')$ in the equation for ${\cal P}(u')$
derived in \cite{GK93,GK96}. This term
makes the perturbation theory for the fluctuations around the instanton to be
free from infrared
divergences, the details will be published elsewhere.
Let us summarize.
At smooth almost inviscid ramps, velocity differences and gradients are
positive and linearly related $w(\rho)\approx
2\rho u'$ so that the right tails of PDFs have the same cubic form
\cite{Pol95,95GM,GK96}.
Those tails are universal i.e. they are determined by a single characteristics
of the pumping correlation function $\chi(r)$, namely, by it's second
derivative at zero $\omega=[-(1/2)\chi''(0)]^{1/3}$.
Contrary, the left tails found here contain nonuniversal constant which
depends on a large-scale behavior of the pumping. The left tails
come from shock fronts where $w^2\simeq -\nu u'$ so that cubic tail
for velocity differences (\ref{an3}) corresponds to semi-cubic tail for gradients
(\ref{an2}).
The formula (\ref{an3}) is valid for $w\gg u_{\rm rms}\simeq L\omega$
where ${\cal P}(w)$ should coincide with
a single-point ${\cal P}(u)$ since the
probability is small for both $u(\rho)$ and $u(-\rho)$ being large
simultaneously. Indeed, we saw that the tails of
$\ln{\cal P}(u)$ at $u\gg u_{\rm rms}$ are cubic as well.
Note that
(\ref{hd4}) is the same as obtained for decaying turbulence with white (in space)
initial conditions
by a similar method employing the saddle-point approximation in the
path integral with time as large parameter \cite{Avel}.
That, probably, means that white-in-time forcing
corresponds to white-in-space initial conditions. Note that
if the pumping has a finite correlation time $\tau$ then our
results, strictly speaking, are valid for
$u,w\ll L/\tau$ and $u'\ll1/\tau$.
We are grateful to M. Chertkov, V. Gurarie, D. Khmelnitskii, R. Kraichnan
and A. Polyakov for useful discussions.
This work was supported
by the Minerva Center for Nonlinear Physics (I. K. and V. L.),
by the Minerva Einstein Center (V. L.), by the Israel Science Foundation
founded by the Israel Academy (E.B.) and by
the Cemach and Anna Oiserman Research Fund (G.F.).
|
proofpile-arXiv_065-657
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
The origin of CP violation has remained an unsolved problem since the
discovery of CP violation in K meson system a quarter ago\cite{Christ}.
Although the observed CP violation in K meson system can be accommodated in
the standard model (SM) of electroweak interactions by virtue of a physical
complex phase in the three by three Cabibbo-Kobayashi-Maskawa matrix (CKM)
\cite{Kob}, it is not clear if CKM mechanism is really correct or the only
source for CP/T violation \cite{Lee}. To verify CKM mechanism one needs not
only the information on K meson mixing and decay but also that from the B
meson system or other systems. The main physical purpose of B factory is to
test the CKM mechanism. However even if CKM is the correct mechanism to
describe the CP violation in K and B meson mixing and decay, it is not
necessary that the CKM matrix is the only source of CP/T violation in the
nature \cite{Jar}. As pointed out by Weinberg \cite{Wei}, unless the Higgs
sector is extremely simple, it would be unnatural for Higgs-boson exchange
not to contribute to CP/T non-conservation. CKM matrix may explain the
observed CP violation in K meson system and possibly the CP violation in B
meson system, while other new sources of CP/T violation may occur everywhere
it can. In fact there are some physical motivations for people to seek the
new sources of CP/T violation. One motivation is from strong CP problem in
the SM \cite{Pec1}. For most of the scenarios to solve this problem they
need more complex vacuum structure and therefore new CP non-conservation
origin. Another motivation is from cosmology, most astrophysical
investigation shows that the additional sources of CP violation are needed
to account for the baryon asymmetry of universe at present \cite{Sha}. The
third motivation is from supersymmetry. Even in the minimal supersymmetrical
standard model (MSSM), there are some additional CP non-conservation sources
beyond the CKM matrix \cite{Hab}. Now the question is at what places the
possible new CP/T violation effects may show up and what is the potential to
search for those effects. In this work we are going to study systematically
on the possibility to find new CP/T violation effects at Tau-Charm factory
(TCF).
TCF is a very good place to test the SM and search new physics phenomena
because of its high luminosity and precision \cite{TCF}. Especially the $%
\tau $ sector is a good place to seek for non-SM CP/T violation effects
because in the SM CP violation in lepton sector occurs only at multi loop
level and is way below any measurable level in high energy experiments, only
non-SM sources of CP/T non-conservation may contribute and another reason is
that $\tau $ has abundant decay channels with sizable branching ratio, which
can be used to measure CP/T violation. Furthermore, the production-decay
sequences of $\tau $ pair by electron-positron annihilation is also favored.
The reason is as the following: (i) $\tau $ pair production by
electron-positron annihilation is a purely electroweak process and can be
perturbatively calculated; (ii) For the unpolarized electron-positron
collision, its initial state is CP invariant in the c.m. frame; (iii) when
the electron and/or positron beams are longitudinally polarized, the initial
state is still effectively CP even, which presents extra chances to detect
possible CP violation. To detect the possible CP/T violation, one can either
compare certain decay properties of $\tau ^{-}$ with corresponding CP/T
conjugations, or measure some CP/T-odd correlation of momentum or spin of
the final state particles from $\tau $ pair decay. These CP/T violating
observables can and should be constructed model independently, since
normally in non-SM these observables are not well predicted due to the
complexity and many free parameters. The sensitivity of the experimental
measurement on the possible CP/T violation is determined by the sensitivity
of the measurement on momentum, spin or other physical quantities of the
final state particles, from them the physical CP/T violating observables are
constructed. The better one can measure these quantities, the momenta, for
example, the smaller the CP violation phase can be reached. In TCF, it is to
expect about $10^7$ $\tau $ pair in one year, and the precision of
measurement on kinematic parameters at $10^{-3}$. The statistical and
systematic error can be around or below this level. Therefore generally a
CP/T violation phase as small as order of $10^{-3}$ can be reached at TCF
\cite{TCF}. In a non-SM the CP/T violating phase may appear in various
stages of the process of production-decay chain, $e^{+}e^{-}\to \tau
^{+}\tau ^{-}\to final~~particles$. We sort them in three cases; (i) CP/T
violation is generated in the tree level production process, $e^{+}e^{-}\to
\gamma ,Z,X\to \tau ^{+}\tau ^{-}$, where X is some new Higgs or gauge
bosons, CP/T violating phase appears either in the propagator of X or in the
coupling to lepton pairs, and the simplest possibility is X being a neutral
Higgs in two or multi-Higgs doublets model. In this case the size of CP/T
violation is proportional to the interference between the X exchange and $%
\gamma $, Z exchange processes. Unfortunately for X being Higgs doublet this
interference term is proportional to the initial and final states fermion
masses $m_em_\tau $ as a result of chirality conservation. This factor along
contributes a suppression factor $m_e/m_\tau \sim 3\times 10^{-4}$ to all
CP/T violating observables in this kind of processes at TCF besides other
possible suppression factor, like the large mass of X, small coupling
between X and leptons. We conclude that it is hopeless to search for CP
non-conservation from the tree-level production process at TCF. (ii) CP/T
violation is also generated at production stage, but through loop level. The
most hopeful cases are that there may exist large electric or weak dipole
moment (EDM or WDM) for $\tau $ lepton, i.e. there are sizable CP/T
violation phase at the vertex $\tau ^{-}-\gamma ,Z-\tau ^{+}$. For this
situation the new physical particles beyond SM only appear as virtual
particles through loops and the size of CP violation is proportional to EDM
or WDM and is not suppressed by other factors, so the point is just whether
EDM or WDM of $\tau $ is large enough to be observed. Generally the
Lagrangian describing the CP/T violation in $\tau $ pair production related
to EDM and WDM is
\begin{eqnarray}
L_{CP}=-1/2i{\bar \tau }\sigma ^{\mu \nu }\gamma _5\tau [d_\tau
^E(q^2)F_{\mu \nu }+d_\tau ^W(q^2)Z_{\mu \nu }]
\end{eqnarray}
$F_{\mu \nu }$ and $Z_{\mu \nu }$ are the electromagnetic and weak field
tensors. The momentum transfer at TCF is around 4 GeV, and in LEP
experiments it is around the mass of Z boson. Therefore at TCF we expect the
contribution from WDM is a factor of $\frac{4m_\tau ^2}{M_Z^2}\simeq 2\times
10^{-3}$ smaller than the contribution from EDM, if EDM and WDM at the same
order of the magnitude. On the other hand the EDM term is less important at
LEP energy. That is the reason why the LEP data constrain more strictly on
WDM than EDM of $\tau $ \cite{ALE1,ALE}. We will neglect the WDM
contribution from now on in this work. (iii) It is possible that the CP/T
violation phase is small in the production process but it is relatively
large in the $\tau $ pair decay processes. The processes like $\tau $ to
neutrino plus light leptons or hadrons through some new bosons exchange at
tree level can contribute significantly to CP/T violation observables.
Obviously in this situation any CP/T violation effect from loop level is
negligible, since any loop effect is at least suppressed by a factor $\frac
1{16\pi ^2}\frac{m_\tau ^2}{M^2}$, where M is the mass of some new heavy
particles appearing in loops. This factor is smaller than $10^{-4}$ if M is
heavier than about 20 GeV.
Now let us recall that how one detects CP violation in $K$ meson decays: One
measures the partial widths for a decay channel and compares it with that
for the corresponding CP-conjugate decay process. Underlying such a
philosophy is the interference between a CP violating phase and a CP
conserving strong interaction phase, i.e. CP violation effect is only
manifested in the process with strong final state interaction. To observe
possible non-CKM CP violation effects in tau decays, however, one has to
invoke new methodology in the most cases. The basic reason is that both in
production vertex of $\tau $ pair (EDM of $\tau $) and in some tau decay
channels (like pure leptonic decay, $\pi \nu $, $\rho \nu $ decay channels
etc. ), there is no strong interaction phase, caused by hadronic final state
interaction, to interfere with possible CP violating phase. So far some
efforts have been made to investigate the CP/T violation effects in TCF.
Mainly those work are trying to find various ways to measure possible CP/T
violation. The simple and very useful method is to construct observables
which are CP/T-odd operators being made from momenta of final state
particles coming from $\tau $ pair decay or polarization vector of the
initial electron (or both electron and positron) beam \cite{Ber}. These
operators can be used very conveniently to test any CP/T violation from
either EDM of $\tau $ lepton or from the decay of the $\tau $ pair without
much model dependence. Some of the operators are constructed by considering
the reactions
\begin{eqnarray}
e^{+}(p)+e^{-}(-p)\to \tau ^{+}+\tau ^{-}\to A(q_{-})+\bar B(q_{+})+X
\end{eqnarray}
in the laboratory system, where $A(\bar B)$ can be identified as a charged
particle coming from the $\tau ^{-}(\tau ^{+})$ decay. Some CP/T-odd
operators (so CPT even, we will not consider CPT-odd operator in this work
since it is certainly much smaller violation effect) can be expressed as
following \cite{Ber}
\begin{eqnarray}
&O_1=
{\hat p}\cdot \frac{{\hat q}_{+}\times {\hat q}_{-}}{|{\hat q}_{+}\times {\
\hat q}_{-}|}, \nonumber \\
&T^{ij}=({\hat q}_{+}-{\hat q}_{-})^i\cdot \frac{({\hat q}%
_{+}\times {\hat q}_{-})^j}{|{\hat q}_{+}\times {\hat q}_{-}|}%
+(i\leftrightarrow j),
\end{eqnarray}
where $\hat p,\hat q$ denote the unit momenta. If the initial electron
and/or positron beams are polarized, one can construct some more observables
making use of the initial polarization vector. For example a T violating
operator
\begin{eqnarray}
O_2=\vec \sigma \cdot \frac{{\hat q}_{+}\times {\hat q}_{-}}{|{\hat q}%
_{+}\times {\hat q}_{-}|}
\end{eqnarray}
can be constructed from the electron polarization vector $\vec \sigma $ and
momenta of final state particles. If there exists any sizable CP/T violation
from EDM of $\tau $ or in $\tau $ pair decay vertex, in principle the
experimental expectation values of these operators are nonzero. For EDM of $%
\tau $ lepton, $d_\tau $, the theoretical expectation values of these
operators are worked out and expressed only as a function of $d_\tau $ \cite
{Ana}. Since at TCF the precision of measurement for these operators are at $%
10^{-3}$ level, one expects to probe $d_\tau $ as small as $\displaystyle
\frac{10^{-3}}{2m_\tau }\simeq 10^{-17}$ e-cm. An example is the measurement
of $d_\tau $ or $d_\tau ^W$ in LEP experiment . Expectation value of $T^{ij}$
operator is directly related to $d_\tau $ \cite{ALE1},
\begin{eqnarray}
<T_{AB}^{ij}>=\frac{E_{cm}}ed_\tau C_{AB}diag(-1/6,-1/6,1/3).
\end{eqnarray}
By the term diag means a diagonal matrix with diagonal elements given above,
$E_{cm}$ is the energy at c.m. frame. The proportional constants $C_{AB}$
depend on the $\tau $ decay modes, but generally this constant is order of
one for all the decay models \cite{Ber}. The decay channels, which can be
measured in experiments, may be classified as $l-l$, $l-h$ and $h-h$
classes, here $l$ is the lighter leptons, $h$ is charged hadron like $\pi $,
$\rho $ and $a_1$. Very impressively, if the initial electron (or both
electron and positron) is polarized, one may use the polarization
asymmetrized distribution. The distribution is defined as the differential
cross section difference between two different polarizations. With this
method, a $d_\tau $ as small as $10^{-19}$ e-cm can be reached at TCF \cite
{Ana}, this corresponds to a sensitivity of $10^{-5}$ of CP/T violation. Up
to now the best experimental bound on $d_\tau $ is from LEP experimental
data, which is used to exclude indirectly the $d_\tau $ as large as $%
10^{-17} $ e-cm \cite{ALE}, so two order of magnitudes improvement on $%
d_\tau $ measurement can be achieved at TCF.
Besides the CP/T-odd operator method, several other useful strategy were
proposed to test these violation in $\tau $ decay. 1) C. A. Nelson and
collaborators \cite{Nel} investigated systematically the feasibility of
using the so-called stage-two spin-correlation functions to detect possible
non-CKM CP violation in the tau-pair production-decay sequence and the
corresponding CP-conjugate sequence. The two-variable energy-correlation
distribution $I(E_A,E_B,\Psi )$, where $\Psi $ is the opening angle between
the final $A$ and $B$ particles, is essentially a kinematic consequence of
the tau-pair spin correlation which depends on the dynamics of $Z^0$ or $%
\gamma ^{*}\to \tau ^{-}\tau ^{+}$ amplitude, and of the $\tau ^{-}\to
A^{-}X_A$ and $\tau ^{+}\to B^{+}X_B$ amplitudes. By including $\theta _e$
and $\phi _e$ which specify the initial electron beam direction relative to
the final-state $A$ and $B$ momentum directions in the c.m. frame of $%
e^{-}e^{+}$ system, one obtains the so-called beam-referenced stage-two
spin-correlation function $I(\theta _e,\phi _e,E_A,E_B,\Psi )$. For the $%
\gamma ^{*}\to \tau ^{-}\tau ^{+}$ vertex, there are four complex helicity
amplitudes. Hence, the beam-referenced stage-two spin-correlation function
constructs four distinct tests for possible CP violation in $e^{-}e^{+}\to
\tau ^{-}\tau ^{+}$. To illustrate the discovery limit in using the
beam-referenced stage-two spin-correlation function, Goozovat and Nelson
\cite{Nel2} calculated the ideal statistical errors corresponding to the
four tests. An advantage of detecting CP violation by use of the stage-two
spin-correlation function is that the model independence and amplitude
significance of the results is manifest. It is complementary to the greater
dynamical information that can be obtained through other approaches, such as
from higher-order diagrammatic calculations in the multi-Higgs extensions of
the SM. 2) Another strategy to test CP violation in the two-pion channels of
tau decay is due to Y.S. Tsai \cite{Tsa}, the basic ingredient of which is
to invoke a highly polarized tau-pair. Consider the tau-pair production by
electron-positron annihilation near threshold. If the initial electron and
positron beams are polarized longitudinally (along the same direction), the
tau-pair will be produced mainly in the $S$-wave, resulting in polarizations
of $\tau ^{\pm }$ both pointing in the same direction as that of the initial
beams. Such a polarization is independent of the production angle and the
corresponding polarization vector supplies us with an important block to
form products with the final particle momenta. By comparing such
polarization-vector-momentum products for a specific tau decay channel with
those for the corresponding CP-conjugate process, one can perform a series
of tests for possible CP violation effects in the tau decay. However, it is
impossible to detect a CP violation in the $\tau \to \pi \nu _\tau $ decay
without violating CPT symmetry. As for the two-pion channel, the existence
of a complex phase due to the hadronic final-state interactions, given by
the Breit-Wigner formula for the $P$-wave resonance $\rho $, enables
detecting possible non-CKM violation by measuring asymmetry of $({\bf w}%
\times {\bf q}_1)\cdot {\bf q}_2$ without violating the CPT symmetry, where (%
${\bf w}$ is the tau polarization vector and ${\bf q}_i$ (i=1,2) are the
final pion momenta). By limiting the weak interaction to be transmitted only
by exchange of spin-one and spin-0 particles, one can know that only the $S$%
-wave part of the amplitude for the exchange of the extra spin-1 particle
make contributions to CP violating observables. A very generic conclusion is
that unless two diagrams have different strong interactions phases, one
cannot observe the existence of weak phase using terms involving ${\bf %
w\cdot q_1}$. Tsai \cite{Tsai2} also points out that T violation can not be
detected in the pure leptonic decay without detecting the polarization of
the decay lepton. Because it is impossible to construct T-odd operator by
the momenta of the initial and final state particles in pure leptonic three
body decays. This also implies that with CPT symmetry, one can not detect CP
violation in $\tau $ decay processes with unpolarized $\tau $. On the other
hand, however, with polarized initial electron and positron beams, one can
construct T-odd operators using the momenta and polarization vector of $\tau
$ and the decay lepton. Therefore polarization of initial electron and
positron is very desirable for detecting of CP/T violation at TCF. 3) As for
the $\tau \to (3\pi )\nu _\tau $ decay, it can proceed either via $J^P=1^{+}$
resonance $a_1$ and the $J^P=0^{-}$ resonance $\pi ^{\prime }$. Choi,
Hagiwara and Tanabashi \cite{Cho} investigated the possibility that the
large width-mass ratios of these resonances enhance CP-violation effects in
the multi-Higgs extensions of the SM. To detect possible CP-violation
effects, these authors compare the differential decay width for the $\tau
^{-}\to \pi ^{+}\pi ^{-}\pi ^{-}\nu _\tau $ with that for the corresponding
CP-conjugate decay process. To optimize the experimental limit, they
suggested considering several CP-violating forward-backward asymmetry of
differential decay widths, with appropriate real weight functions. 4) To
probe possible CP-violating effects in the tau decay with $K^{-}\pi ^{-}\pi
^{+}$ or $K^{-}\pi ^{-}K^{+}$ final states, Kilian, K\"orner, Schilcher and
Wu \cite{Wu} partitioned the final-state phase space into several sectors
and constructed some asymmetries of the differential decay widths. As a
result, they showed that T-odd triple momentum correlations are connected to
certain asymmetries, so their non-vanishing would indicate a possible
non-CKM CP violation in the exclusive semileptonic $\tau \to $three
pseudoscalar-meson decays.
With these knowledge and results obtained in the previous papers in mind,
now the crucial question, which is also the motivation of this work, is
whether for CP/T violation appearing in EDM close to $d_\tau \sim 10^{-19}$
e-cm and CP/T violation effects in $\tau $ decay as $10^{-3}$ are possible
values theoretically. If for all possible extensions of the SM, which people
can visualize now, with natural parameter choice, these values are much
smaller than the theoretically predicted ones, then the effort to search for
such small CP/T violation signal at TCF would be not much meaningful, at
least from the theoretical point of view. In this paper we are trying to
answer this question by investigating various possible mechanisms for
generating large EDM of $\tau $, CP/T violation in $\tau $ decay. This paper
is organized as the following. In section 2 we review the generation of EDM
of $\tau $ lepton in various popular beyond standard models and stress on
what models can produce possible large EDM of $\tau $. Following the
discussion of EDM, in the section 3 we concentrate on CP/T violation effects
from $\tau $ decay in the beyond standard models. The last section is
reserved for some further discussion, and the conclusion on the possibility
of finding CP/T violation at TCF is given.
\section{EDM of $\tau$ lepton}
EDM of the lepton $d_l$ is a dimension-5 operator. It can only be generated
from the loop level. Because this operator changes the chirality of the
lepton, it must be proportional to a fermion mass. In the SM EDM of lepton
is generated from three loop diagrams and is proportional to lepton mass
itself, so $d_l$ is very small \cite{Khr}. However generally $d_l$ can be
produced from one loop diagrams in beyond standard model. At one loop level,
the $d_l$ can be expressed as
\begin{eqnarray}
d_l\sim \frac{e\lambda }{16\pi ^2}\frac{ M_F}{V^2}\sin\phi\sim 10^{-18}(
\frac{\lambda}{1} )(\frac {M_F}{100GeV})(\frac{100GeV}V)^2\sin\phi ~~\rm{e-cm}
\end{eqnarray}
where $M_F$ is some fermion mass, $V$ is a large scale from intermediate
states in the loops and $\lambda $ denotes other couplings. $\phi$ is a CP/T
violation phase. In the following part we assume maximal CP/T violation
phase, i.e. $\sin\phi\simeq 1$. From this equation one sees that $d_l$ can
be at most as large as $10^{-18}-10^{-19}$ e-cm if $\lambda$ is between $%
1.0-0.1 $. Since $V$ is a scale around or larger than weak scale, in order
to obtain large $d_l$, $M_F$ must be a large fermion mass such as t quark
mass or new heavy fermion masses. For example if M is the $\tau $ mass then $%
d_\tau $ is smaller than $10^{-20}$ e-cm which is not detectable at TCF.
Same is true for the scale $V$ . If $V$ is at TeV scale $d_l$ is smaller
than $10^{-20}$ e-cm. Although, in principle, $d_\tau $ is possibly as large
as $10^{-19}$e-cm , one has to avoid too large EDM of electron $d_e$ at the
same time. Current experimental upper limit on $d_e$ is about $10^{-26}$%
e-cm. This is a very strong constraint especially when one is expecting
large $d_\tau $. So in any beyond standard model, two requirements must be
satisfied in order to obtain measurable $d_\tau $. The first one is that the
model must provide $d_\tau $ at one loop level and $d_\tau $ is not
suppressed by a small fermion mass term, the fermion mass term should be a
top quark mass, supersymmetric partner of bosons or other exotic fermion
masses. The second one is that the predicted $d_e$ associated with large $%
d_\tau $ is below its current experimental bound. These two conditions
altogether exclude most of beyond standard models which can provide large
enough $d_\tau $ observable for TCF. We will see from the following
discussion that many beyond standard models do not satisfy the two
requirements.
Usually EDM of lepton is generated from one loop diagrams in extension of
the SM. Fig. 1 is a typical one loop diagram for the lepton EDM. The virtual
particles are scalar or vector boson $S$ and fermion $F$ in the loop. Photon
is attached to the charged intermediate particles. The $d_l$ from this
diagram is approximately proportional to the fermion mass $M_F$ and it is
divided by a scale $V$, which is larger or equal to $M_F$. Besides, there
are two more couplings at the vertex $l-S-F$. In a practical model there
could be many possible virtual bosons and fermions in the loop, but we only
consider the dominant contribution here as an order of magnitude estimation.
The diagram in Fig. 1 is evaluated as
\begin{eqnarray}
d_i/Q\simeq \frac{|\lambda _i\lambda ^{\prime *}_i|}{16\pi ^2}\frac{M_F}{V^2}
\xi\sin\phi
\end{eqnarray}
where $i=e,~\mu ,~\tau $ denotes three generation leptons and $Q$ is the
electric charge of the virtual particles. $\xi$ is an order of one factor
from the loop integral. Eq. (7) is true up to a factor of order one. And
there should be a logarithmic dependence on $\frac{M_F}V$ in $\xi$, which is
slowly varying.
In order to obtain measurable $d_\tau $ and avoid too large $d_e$, one needs
a large $M_F$ as discussed before and $\lambda $, $\lambda ^{\prime }$ must
be around order of one for $\tau $ but much small (smaller than about $%
10^{-3}$) for electron. We systematically investigate and review most of the
popular extensions of the standard model and point out that the following
type of models can fulfill the requirements.
{\bf {Scalar leptoquark models}}\cite{Dav} \hskip 0.5 cm CP violation effect
in $\tau$ sector for the models are recently discussed extensively by some
authors \cite{Cho,Bar}. It is particularly interesting for generating a
large $d_l$. These are the models which do not need to introduce additional
fermion. Because the top quark mass is large, it is possible to generate a
large $d_{\tau}$ through coupling of $\tau$, top quark and the corresponding
leptoquark. $d_e$ could be small enough due to the coupling of electron, top
quark and leptoquark is independent of that for $d_{\tau}$. So long as there
is a relative large hierarchy for the couplings for different generations,
the two requirements can be satisfied.
There are five types of scalar leptoquarks which can couple to leptons and
quarks. We denote them by $S_1$, $S_2$, $S_3$, $S_4$ and $\vec{S_5}$. Their
quantum numbers under standard gauge group transformation are $(3, 2, \frac{%
7 }{3})$, $(3, 1, -\frac{2}{3})$, $(3,2,\frac{1}{3})$, $(3,1,-\frac{7}{3})$
and $(3, 3, -\frac{2}{3})$ respectively. The Yukawa coupling terms are
therefore given by
\begin{eqnarray*}
&L_1=(\lambda_1^{ij}
{\bar Q}_{Li}i\tau_2E_{Rj}+\lambda^{\prime ij}_1{\bar U}_{Ri}l_{Lj})S_1+h.c. \nonumber
\\ &L_2=(\lambda_2^{ij}
{\bar Q}_{Li}i\tau_2l^c_{Lj}+\lambda^{\prime ij}_2{\bar U}%
_{Ri}E^c_{Rj})S_2+h.c. \nonumber \\& L_3=\lambda_3^{ij}
{\bar D}_{Ri}l_{Lj}S_3+h.c. \nonumber \\ &L_4=\lambda_4^{ij}
{\bar D}_{Ri}E^C_{Rj}S_4+h.c. \nonumber \\ &L_5=(\lambda_5^{ij} {\bar Q}_{Li}i\tau_2
\vec{\tau} l^c_{Lj})\cdot\vec{ S_5}+h.c.
\end{eqnarray*}
Here $l_L$ and $Q_L$ are lepton and quark doublets respectively, $U_R$, $D_R$
and $E_R$ are singlet quark and lepton respectively. Individually only $S_1$
and $S_2$ contribute to the EDM of lepton.
$\xi $ factor in Eq. (7) is evaluated as $\xi =\frac 23ln\frac{M_F^2}{V^2}+
\frac{11}6$ \cite{Bar}. Currently the constraints on mass and coupling of
leptoquark are relatively weak \cite{DAT}. For leptoquark coupled only to
third generation, its lower mass bound is about 45 GeV with order of unit
coupling \cite{DAT}. This bound is from a leptoquark pair production from
LEP experiments. On the other hand with the leptoquark mass at weak scale,
the coupling is very weakly bounded too. In fact the coupling could be as
large as order of one. If we take $\lambda ^{33}$ , $\lambda ^{\prime
}{}^{33}$ as 0.5 and the mass of leptoquark as 200GeV and assume maximal
CP/T violation phase, we estimate that $d_\tau \simeq 2\times 10^{-19}$
e-cm, while $d_e$ is determined by other coupling components, so a small $%
d_e $ is not necessary in conflict with a large $d_\tau $ in this model.
{\bf Models with the fourth generation or other exotic lepton}\hskip 0.5cm
The SM with fourth generation is another possible model to generate a large $%
d_{\tau}$. The heavy fourth generation leptons may play a role of the heavy
fermion F in the loop. However it is well known that if the fourth
generation exists, it must satisfy the constraints from LEP experiments \cite
{LEP}. Here we propose a realistic model for this purpose.
Besides the fourth generation fermions, we also introduce a right-handed
neutrino $\nu_R$ and a singlet scalar $\eta^-$ with one unit electric charge
\cite{Zee}. The new interaction terms are
\begin{eqnarray}
L=\lambda_{ij}l_i^Ti\tau_2l_j\eta^-+\lambda^{\prime}_{i}E_{Ri}^T\nu_{R}%
\eta^-+ M_R\nu_R^T\nu_R+M_i^D{\bar\nu}_{Li}\nu_R+h.c.
\end{eqnarray}
where $\lambda_{ij}$ is antisymmetric due to the Fermi statistics. $M^D$ is
the Dirac neutrino mass from standard Higgs vacuum expectation value. In
this model three light neutrinos remain massless and the fourth neutrino is
massive \cite{Li}. The constraints from LEP experiments and other low energy
data can be satisfied so long as $M_R$ is at weak scale or up and $M_i^D$ is
not much smaller than $M_R$. In the one loop diagram contribution to $d_\tau
$, $\eta ^{-} $ appears as the scalar S. The fermion line is two massive
neutrinos $\nu _4$ and $\nu _H$ in the mass basis and they are related to
each other,
\begin{eqnarray}
\nu _{L4}=\cos \theta \nu _4-\sin \theta \nu _H \nonumber\\
\nu _R=\sin \theta \nu _4+\cos \theta \nu _H
\end{eqnarray}
We assume $\nu _4$ is the lighter neutrino and the dominant contribution is
from either $\nu_H$ or $\nu _4$ depending on whether $\nu _H$ is heavier
than the mass of $\eta $, $M_\eta $ . $d_\tau $ is evaluated as in (7) with $%
M_F=M_H\cos \theta \sin \theta $ and $V\simeq M_\eta $ if $M_\eta \geq M_H$;
with $M_F=M_{\nu _4}\cos \theta \sin \theta $ and $V\simeq M_H$ if $M_\eta
\leq M_H$. Choosing $\lambda _{34}=\lambda _3^{\prime }=1.0$ and $M_F=50$
GeV, $V=200$ GeV, we have the numerical result $d_\tau \simeq 10^{-19}$
e-cm. Also in this model a hierarchy on the coupling $\lambda $ and $\lambda
^{\prime }$ for different generation is needed to keep small enough $d_e$,
i.e. $\lambda _{34}>>\lambda _{14}$ and $\lambda _3^{\prime }>>\lambda
_1^{\prime }$.
Existence of exotic leptons provide another possibility to generate a
measurable $d_{\tau}$. It can be realized in horizontal models \cite{Bar1}.
With only three standard leptons, it is impossible to obtain large enough $%
d_{\tau}$, because the largest fermion mass in the loop is $m_{\tau}$.
However, with some new heavy leptons this model can provide a large $%
d_{\tau} $. The constraints from low energy data can be avoided if one
assumes that the horizontal interaction is strong between $\tau$ and the
exotic lepton, but it is much weaker in other sectors. Similar result on $%
d_{\tau}$ as for the case with the fourth generation can be obtained.
Finally, we should point out that for our purpose it is clear that some new
exotic heavy leptons are needed in the new physics models, however even
though there exists some kind of models with some new heavy leptons, they
are able to generate $d_l$ only from two loop diagrams \cite{Fab}, so they
may result in interesting $d_e$ but not $d_\tau $.
{\bf Generic MSSM } \hskip 0.5pc Generic MSSM contains 63 parameters not
including the parameters in the non-SUSY SM. Ferminic superpartners of the
ordinary bosons can be the heavy fermions in the loop diagrams for $d_l$. It
provides some new sources for CP/T violation. It is well known that the
electron and neutron can acquire large EDM \cite{Pol} in this model. In
fact, in order to obey the experimental bounds on $d_n$ and $d_e$, some
parameters in the model are strongly restricted \cite{Bab}. For $d_l$
generation, it is dominated by photino mediated one loop diagram. Both left-
and right-handed sleptons also appear in the loop. The contribution to $d_l$
from this diagram is proportional to left- and right-handed slepton mixing
matrices $M_{LR}=(A_l-\mu \tan \beta )M_l$. $A_l$ is the matrix of
soft-SUSY-breaking parameters that appears in the SUSY Yukawa terms of
slepton coupling to Higgs doublet. Here $M_l$ is diagonal mass matrix of
lepton mass. Usually it is assumed that $A_l$ is diagonal and the diagonal
elements are not much different for different generation, for example in
supergravity inspired model $A_l$ is universal for three generation \cite
{Hab}, therefore one can get $d_\tau /d_e\simeq m_\tau /m_e$. Using the
experimental limit $d_e\leq 10^{-26}$ e-cm, one concludes that $d_\tau \leq
4\times 10^{-23}$ e-cm \cite{Mah}. However in the generic MSSM all the
elements of $A_l$ are free parameters, so the above constraint is not
necessarily true. For example if for some unknown reason the 33 component of
$A_l$ is much larger than other elements, and $\mu $ term is much smaller
than SUSY breaking scale, then $d_\tau $ still can be larger than $10^{-22}$
e-cm and $d_e$ is in the allowed region. In this case $d_\tau $ can also be
expressed as Eq. (7), but with $M_F={\tilde m}_\gamma $, $V={\tilde m}_\tau
^2/M_{LR}$, $\lambda _{33}=\lambda _{33}^{\prime }=e$ and $\phi =arg(M_{LR}^2%
{\tilde m}_\gamma )$. The loop integral $\xi $ was four times the function
calculated some years ago in dealing with $d_e$ in MSSM known as
Polchinski-Wise function \cite{Wis}. Here ${\tilde m}_\gamma $ and ${\tilde m%
}_\tau $ are photino and the third slepton masses respectively. We estimate
that $d_\tau \simeq 10^{-19}$ e-cm with ${\tilde m}_\gamma =100$ GeV and $%
V=200$ GeV.
As for other popular extensions of the SM, we would like to point out here,
though they have some new sources of CP/T violation, they can not offer a
observable $d_\tau $ at TCF. These include multi-Higgs doublet model (
including two Higgs doublet model) \cite{Lee,Wei2}, Left-Right symmetric
model \cite{Pat}, mirror fermion model \cite{Don} and universal soft
breaking SUSY model \cite{Hab} etc. In multi-Higgs doublet model electron
\cite{Barr} and neutron \cite{Wei} may obtain a large EDM close to current
experimental bounds through two loop diagrams, but $d_\tau $ generated in
the model is quite below the TCF observable value. The reason is that $%
d_\tau $ is proportional $m_\tau $, but not a large fermion mass. We
estimate $d_\tau \leq 4\times 10^{-21}$e-cm \cite{Son} that in this model.
For Left-Right symmetric model, Nieves, Chang and Pal \cite{Nie} find that
the upper bound for $d_\tau $ is $2.4\times 10^{-22}$e-cm. It is the right-
or left-handed gauge boson in the loop as the role of $S$ particle, while
right-handed neutrino is the virtual fermion particle in the loop. $d_\tau $
in this model is proportional to left- and right-handed gauge boson mixing
angle. Though it is not suppressed by the small fermion mass ( $M_F$ is a
large right-handed neutrino mass), the mixing angle is constrained to be
smaller than $0.004$ \cite{Don2} from purely non-leptonic strange decays. It
leads to about three order of magnitude suppression. In the mirror fermion
model, standard gauge bosons couple to ordinary lepton and the mirror lepton
with a mixing angle. It is $Z$ and $W$ bosons in the one loop diagrams, the
heavy fermion line is the mirror lepton. However the mixing angle in this
model is constrained by various experiments \cite{Lan}, and most stringently
by LEP data on $Z\to \tau ^{+}\tau ^{-}$ \cite{Bha}. The constraint from LEP
data on the mixing angle is less than about 0.3. The resulting bound is $%
d_\tau \leq 2.1\times 10^{-20}$e-cm, which is a few times smaller than TCF
measurable value. As we have mentioned above in the universal soft breaking
SUSY model, $d_\tau \leq 4\times 10^{-23}$e-cm due to the constraint on $d_e$%
. The only alternative situation is discussed above on Generic MSSM in this
section.
\section{CP/T violated $\tau$ decays}
As we have pointed out in the introduction, CP/T violation effects in $\tau $
decays, if observed, must occur at tree level diagrams. That is the
interference between the SM $\tau $ decay processes and new tree level
processes of $\tau $ decays, in which CP/T violation phases appear at the
interaction vertexes, provides the information of CP/T violation in the $%
\tau $ sector. Feynman diagrams of these processes can be shown as in the
Fig. 2, where $f_i$, $f_j$ and $f_k$ are light
fermions. X is a new particle ( scalar or vector boson) which mediates CP/T
violating interaction. The size of CP/T violation is always proportional to
the interference of the tree level diagrams. We denote the amplitudes for
these diagrams as $A_1$ for W boson exchange diagram, $A_2$ for other X
boson exchange diagrams. The size of CP/T violation in the $\tau $ decay can
be characterized by a dimensionless quantity
\begin{eqnarray}
\epsilon =\frac{Im(A_1^{*}A_2)}{|A_1|^2+|A_2|^2}
\end{eqnarray}
Practically physical quantity expectation values which are used to reflect
CP/T violation, like the expectation values of CP/T-odd operators,
difference of a partial decay widths of a $\tau ^{-}$ decay channel and its
conjugate $\tau ^{+}$ decay channel, are model dependent and generally quite
complicate. It needs the detailed information of the new physics model and a
lot of parameters enter into the expression. This makes it a very much
involved work to write down these quantities in a specific model beyond the
SM. And the exact CP/T violation quantity expression written down from a
model should be different from the $\epsilon $ defined above. However as a
simple and reasonable estimation, the quantity $\epsilon $ in Eq. (11) can
be used as an indication of how large of CP/T violation may happen at
various $\tau $ decays. Moreover, the amplitude $A_2$ is usually much
smaller than $A_1$ because so far all the experimental data agree with the
SM prediction very well. So $A_2$ term in the denominator can be neglected.
Using $A_1$ as the amplitude from W boson exchange and $A_2$ from the new
boson X exchange, we estimate its size,
\begin{eqnarray}
\epsilon\sim (4\sqrt{2}G_F)^{-1}\frac{Im(\lambda\lambda^{\prime}{}^*)}{M_X^2}
\end{eqnarray}
Here $G_F$ is Fermi constant and $\lambda$, $\lambda^{\prime}$ are couplings
in $A_2$. From Eq. (12) one sees that the size of CP/T violation is
determined by the parameter $\frac{Im(\lambda\lambda^{\prime}{}^*)}{M_X^2}$.
For different models, this parameter is constrained by some other physical
processes. So the possible size of CP/T violation depends on the parameter
region which is restricted in a specific model.
In Fig. 2 the final state fermions can be a pair of leptons and quarks
besides $\nu _\tau $. It corresponds to pure leptonic and hadronic decays
respectively. At the quark level, the diagrams with a pair of quarks in the
final states denote an inclusive process, it includes all possible hadronic
channels originated from quark pair hadronization. Some of the useful
hadronic final states like $2\pi $, $3\pi $, $K\pi $, $K\pi \pi $, $KK\pi $
and $\rho $, $a_1$ can be used to measure the properties of $\tau $.
However, it is often difficult to make a reliable quantitative prediction
for CP/T violation in exclusive hadronic decay modes, because of the
uncertainty in the hadronic matrix elements. On the other hand, for the
inclusive cases, one may make a more reliable quantitative estimation due to
the fact that one has no need to deal with the hadronization of quarks in
this case. In addition, QCD correction should not change the order of the
tree level diagram evaluation as the energy scale for $\tau $ decay
processes is around 1GeV. In this section we only deal with the diagrams
containing quark pair inclusively, So the CP/T violation size we estimate
below is for all the possible hadronic decay channels. In the last section
we will comment on our results in exclusive processes. Because of the scale
of $\tau $ mass, its decay products can only be neutrino, electron, muon and
hadrons containing only light u,d, s quarks as other heavy quarks are
kinematically forbidden. Therefore there are not many possibilities for X
particle being the candidate for mediating CP/T violation in the Fig. 2. In
fact all the possible choices are the following: X being leptoquark, charged
Higgs singlet, doublet and triplet, and double charged singlet. Now we come
to discuss these different cases separately.
{\bf Scalar leptoquark models} \hskip 1cm At tree level it is obvious that
only $S_1$, $S_2$ and $\vec S_5$ contribute to $\tau $ decays. There are two
type of decay processes at quark level, $\tau \to \nu _\tau {\bar u}d$ and $%
\tau \to \nu _\tau {\bar u}s$. The $\epsilon $ parameter is determined by $%
\lambda ^{31}{\lambda ^{\prime }}^{31}{}^*$ and $\lambda ^{32}{\lambda
^{\prime }}^{31}{}^{*}$ for these two type of decays respectively in model 1
and 2 in Eq. (8). For model 5 there is CP/T violation effect only in the
second type process, which is determined by $\lambda ^{32}{\lambda ^{\prime }%
}^{31}{}^{*}$. A direct constraint on these parameters can be obtained
through comparing the theoretical value $\Gamma ^{th}(\tau \to \pi \nu _\tau
)=(2.480\pm 0.025)\times 10^{-13}$ GeV and the measurement value of $\Gamma
^{exp}(\tau \to \pi \nu _\tau )=(2.605\pm 0.093)\times 10^{-13}$ GeV \cite
{Mar}. Assuming that real and imaginary part of the coupling $\lambda {%
\lambda ^{\prime }}{}^{*}$ are approximately equal, one has from $\tau\to
\pi \nu_{\tau}$ \cite{Cho}
\begin{eqnarray}
\frac{|Im(\lambda ^{31}{\lambda ^{\prime }}^{31}{}^{*})|}{M_X^2}\sim \frac{%
|Re(\lambda ^{31}{\lambda ^{\prime }}^{31}{}^{*})|}{M_X^2}<3\times
10^{-6}GeV
\end{eqnarray}
at $2\sigma $ level for model one and two. And from $\tau \to \ K\nu _\tau $
a similar result can be obtained for all the three models. Using the
theoretical value $\Gamma ^{th}(\tau \to K\nu_{\tau} )=(0.164\pm
0.036)\times 10^{-13}$ GeV \cite{Mar,Mar1} and the measurement value $\Gamma
^{exp}(\tau \to K\nu _\tau )=(0.149\pm 0.051)\times 10^{-13}$ GeV for the $%
\tau \to K\nu _\tau $ decay width we obtain
\begin{eqnarray}
\frac{|Im(\lambda ^{32}{\lambda ^{\prime }}^{31}{}^{*})|}{M_X^2}\sim \frac{%
|Re(\lambda ^{32}{\lambda ^{\prime }}^{31}{}^{*})|}{M_X^2}<7\times
10^{-6}GeV
\end{eqnarray}
at $2\sigma $ level. This constraint is less stringent due to the large
uncertainties in $\Gamma ^{exp}(\tau \to K\nu _\tau )$. With these
constraints, one estimates the upper bound of $\epsilon $ value for the two
type of processes as
\begin{eqnarray}
\epsilon (\tau^- \to \nu _\tau {\bar u}d)\simeq (4\sqrt{2}G_F)^{-1}\frac{%
Im(\lambda ^{31}{\lambda ^{\prime }}^{31}{}^{*})}{M_X^2}\leq 4\times 10^{-2}
\end{eqnarray}
and
\begin{eqnarray}
\epsilon (\tau^- \to \nu _\tau {\bar u}s)\simeq (4\sqrt{2}G_F)^{-1}\sin
\theta _C\frac{Im(\lambda ^{32}{\lambda ^{\prime }}^{31}{}^{*})}{M_X^2}\leq
2\times 10^{-2}
\end{eqnarray}
where $\theta _C$ is Cabibbo angle. $\epsilon (\tau^- \to \nu _\tau {\bar u}%
s)$ is proportional to $\sin \theta _C$ and is smaller than $\epsilon (\tau
\to \nu _\tau {\bar u}d)$ because this process is Cabibbo suppressed, even
though the coupling is less constrained than that of Cabibbo unsuppressed
process. From this estimation we expect CP/T violation in these models could
be large enough for TCF or in the other words TCF data can put stronger
direct restriction on the parameters of the model. However, if one assumes
that all the couplings $\lambda $ and $\lambda ^{\prime }$ are at the same
size irrespective of the generation indexes, then much more stringent bounds
exist. These bounds are obtained from experimental bounds of $Br(K_L\to\mu
e) $, $Br(\pi\to e\nu_e(\gamma))$, $Br(\pi\to \mu\nu_{\mu}(\gamma))$ and $%
\Gamma(\mu Ti\to e Ti)/ \Gamma(\mu Ti\to capture)$ \cite{Cho}. They are
generally about five order of magnitude smaller than the direct bounds.
Therefore the size of CP/T violation is $\epsilon \le 4\times 10^{-7}$ which
is far below the capability of TCF.
{\bf Multi-Higgs doublet models (MHD)} \hskip 1cm With the natural
suppression of flavor changing neutral current, it is necessary to have more
than two Higgs doublets, so that there are at least two physical charged
Higgs particles. CP/T violation may generally happen through the mixing of
these charged Higgs particles. We consider a multi-Higgs doublet model, say,
n Higgs doublets. In this model there are 2(n-1) charged and (2n-1) neutral
physical scalars. Since only the Yukawa interactions of the charged scalars
with fermions are relevant for our purpose. Following Grossman \cite{Gro} we
write down the Yukawa interactions in fermion mass eigenstates as
\begin{eqnarray}
L_{MHD}=\sqrt{2\sqrt{2}G_F}\Sigma _{i=2}^n[X_i({\bar U}_LVM_DD_R)+Y_i({\bar U%
}_RM_UVD_L+Z_i({\bar l}_LM_EE_R)]H_i^{+}+h.c.
\end{eqnarray}
Here $M_U$, $M_D$ and $M_E$ denote the diagonal mass matrices of up-type
quarks, down type quarks and charged leptons respectively. $V$ is KM matrix.
$X$, $Y$ and $Z$ are complex couplings which arise from the mixing of the
charged scalars and CP/T violation in $\tau $ decay processes is due to
these couplings. How large is the $\epsilon $ for various $\tau $ decay
channels depends on the values of these parameters. More precisely, in the
pure leptonic decays the size of CP/T violation is determined by $%
Im(Z_iZ_j^{*})$ with $i\not =j$ and in hadronic decays it is determined by $%
Im(X_iZ_j^{*})$ and $Im(Y_iZ_j^{*})$. The three combinations of parameters
are constrained by various experiments \cite{Gro}. The strongest constraint
on $Z$ is from $e-\mu $ universality in $\tau $ decay, which gives $|Z|\le
1.93M_H$GeV$^{-1}$ for Higgs mass $M_H$ around 100GeV. $Im(XZ^{*})$ is
bounded from above from the measurement of the branching ratio $Br(B\to
X\tau \nu _\tau )$, $Im(XZ^{*})\le |XZ|\le 0.23M_H^2$GeV$^{-2}$ if $M_H\le
440$ GeV. Finally a upper bound is given as $Im(YZ^{*})\le |YZ|\le 110$ from
the experimental data of the process $K^{+}\to \pi ^{+}\nu {\bar \nu }
This bound is obtained for t quark mass at 140 GeV \cite{Gro} and $M_H=45$%
GeV, however for a different $M_H$, say 100GeV, this bound is expected not
to change much. With these bounds we can estimate CP/T violation size of $%
\tau $ leptonic and hadronic decays. For the leptonic decay $\tau \to \mu
\nu {\bar \nu }$, we have the quantity
\begin{eqnarray}
\epsilon \simeq \frac 12\frac{Im(ZZ^{*})m_\mu m_\tau }{M_H^2}\cdot \frac{%
m_\mu }{m_\tau }=\frac 12\frac{m_\mu ^2}{M_H^2}Im(ZZ^{*})\leq 2\times
10^{-2}.
\end{eqnarray}
Here the additional factor $\frac{m_\mu }{m_\tau }$ comes from the
interference of left- and right-handed muon lines in the final states. So we
expect that CP/T violation effect in the process $\tau \to e\nu {\bar \nu }$
is suppressed by a factor $m_e/m_\mu $ and is negligible. For the hadronic
decay $\tau \to {\bar u}d\nu $ we have
\begin{eqnarray}
\epsilon \simeq \frac 12\frac{m_d{\bar m}_d}{M_H^2}Im(XZ^{*})\leq 3\times
10^{-4},
\end{eqnarray}
With the current $d$ quark mass $m_d=7$ MeV and the dynamical $d$ quark mass
${\bar m}_d=300$ MeV . For hadronic decay $\tau \to {\bar u}s\nu $ similar
result is obtained
\begin{eqnarray}
\epsilon \simeq \frac 12\frac{m_s{\bar m}_s}{M_H^2}Im(XZ^{*})\leq 1.5\times
10^{-3}
\end{eqnarray}
Here we use current and dynamical $s$ quark masses as 150 MeV and 400 MeV
respectively. In summary, in multi-Higgs doublet model CP/T violation effect
is possibly as large as order of $10^{-3}$ for exclusive hadronic decays and
It could be even close to $10^{-2}$ in pure leptonic decay to $\mu $ and
neutrinos.
{\bf Other extensions of the SM for pure leptonic decays}\hskip 1cm Besides
leptoquark and Higgs doublet, there are three other kind of scalars which
can couple to leptons. We denote $l$ as a lepton doublet and $E$ as a
singlet lepton. Two $l$ can combine to a charged singlet or a triplet. Two $%
E $ can combine to a double charged singlet. Corresponding to these three
cases one can introduce a charged singlet scalar $h^{-}$, triplet scalar $%
\Delta $ and double charged scalar $K^{--}$. However $K^{--}$ only induce a
lepton family- number-violating process $\tau \to 3l$. There is no diagram
corresponding SM contribution, so there is no CP/T violation mediated by
this particle. Also the branching ratio ($\leq 10^{-5}$) for this decay is
much smaller than TCF reachable CP/T violation precision $10^{-3}$. In
principle if there exists more than one $h$ or $\Delta $, CP/T violation can
be induced by the interference of the W exchange diagram and $h$ or $\Delta $
exchange diagram in the process $\tau \to l{\bar \nu }\nu $ with $l=e,\mu $.
Now let us discuss these two possibilities in details. We can write down the
new interaction terms which couple the new scalar particles to leptons as
the following
\begin{eqnarray}
L_h=\frac 12f_{ij}l^T{}_iCi\tau _2l_jh+h.c.
\end{eqnarray}
\begin{eqnarray}
L_\Delta =\frac 12g_{ij}l^T{}iCi\tau _2{\vec \tau }l_j{\vec \Delta }+h.c.,
\end{eqnarray}
where $C$ is the Dirac charge conjugation matrix and $f_{ij}$ is
antisymmetric, $g_{ij}$ is symmetric due to Fermi statistics. $\epsilon $
parameter for these singlet and triplet models are given by,
\begin{eqnarray}
\epsilon _h\simeq (4\sqrt{2}G_F)^{-1}\frac{Im(f_{\tau l}f_{l\tau }^{*})}{%
M_h^2}
\end{eqnarray}
in singlet model and
\begin{eqnarray}
\epsilon _\Delta \simeq (4\sqrt{2}G_F)^{-1}\frac{Im(g_{\tau l}g_{l\tau
}^{*}) }{M_\Delta ^2}
\end{eqnarray}
in triplet model respectively.
For the singlet model we assume that $f_{e\mu }$ is considerably smaller
than $f_{\tau l}$, so that one does not need to readjust the Fermi constant $%
G_F$. This assumption is also consistent with the constraint set by
universality between $\beta $ and $\mu $ decay \cite{Zee,Bri}. The parameter
$\frac{Im(f_{\tau l}f_{l\tau }^{*})}{M_h^2}$ is constrained only by the
measurement of $\tau $ leptonic decays. At $2\sigma $ level (which is about $%
2\sim 3$\% precision) we estimate approximately $\frac{Im(f_{\tau l}f_{l\tau
}^{*})}{M_h^2}\leq 10^{-6}$ GeV$^{-2}$ \cite{Tao}. It implies that
\begin{eqnarray}
\epsilon _h\simeq (4\sqrt{2}G_F)^{-1}\frac{Im(f_{\tau l}f_{l\tau }^{*})}{%
M_h^2}\leq 1.4\times 10^{-2}
\end{eqnarray}
with $M_h=100$ GeV. Therefore in this model there is a possibility that CP/T
violation effect may show up with a size reachable at TCF in pure leptonic
decay channels.
For the triplet model the direct constraint is also from the measurement of
pure leptonic decays. The same result is obtained as that in the singlet
model , i.e. $\frac{Im(g_{\tau l}g_{l\tau }^{*})}{M_h^2}\leq 10^{-6}$ GeV$%
^{-2}$. As the result of this constraint one has
\begin{eqnarray}
\epsilon _h\simeq (4\sqrt{2}G_F)^{-1}\frac{Im(g_{\tau l}g_{l\tau }^{*})}{%
M_\Delta ^2}\leq 1.4\times 10^{-2}
\end{eqnarray}
with $M_\Delta =100$ GeV. However, in this model the new interactions will
induce lepton family number violating decay $\tau \to 3l$ and $\mu \to 3e$
through exchange of the double charged scalar particle $\Delta ^{--}$.
Without seeing any signal, one obtains some approximate bounds on the
coupling constants as the following \cite{Dat1}
\begin{eqnarray}
\frac{|g_{\mu e}g_{ee}^{*}|}{M_\Delta ^2}\leq 5\times 10^{-12}
\end{eqnarray}
and
\begin{eqnarray}
\frac{|g_{\tau l}g_{ll}^{*}|}{M_\Delta ^2}\leq 10^{-8}
\end{eqnarray}
for $M_\Delta =100$ GeV. If one assumes that all the couplings $g_{ij}$ are
at the same order of magnitude, then these bounds will restrict the CP/T
violation size far below the ability of TCF. Again we see some hierarchies
on the couplings are needed for this model to give rise observable CP/T
violation effects. Additionally in the triplet model one has to avoid the
restriction from neutrino mass generation \cite{Gel}. If neutrino develops a
mass at tree level, either the couplings or the vacuum expectation value of
the neutral component of the triplet $\Delta ^0$ are extremely small. The
natural way to deal with this problem is to impose some symmetry on this
model. An example is to introduce a discrete symmetry:
\begin{eqnarray}
l\to il;\hskip 2cmE\to iE\hskip 2cm\Delta \to -\Delta
\end{eqnarray}
With this symmetry, $\Delta ^0$ will never develop a nonzero vacuum
expectation value, therefore the couplings are not constrained by the
neutrino mass generation.
\section{Discussion and conclusion}
In this work we systematically investigate the possibility of finding CP/T
violation in the $\tau $ sector with TCF. The origin of CP/T violation is
from the extensions of the SM. We discuss most of the popular beyond the SM
and present the models which may give rise large CP/T violation in $\tau $
sector through either EDM or decay of $\tau $ lepton. Before making our
conclusion, some interesting points should be further discussed or
emphasized. (1) Polarization of initial electron and/or positron is very
desired for our purpose. First with polarization the precision of
measurement of EDM will be increased by about two order of magnitude, as $%
10^{-19}$e-cm, which is used through this work. Without polarization, from
our above discussion one sees that we have no hope to expect a detectable
EDM of $\tau $ at TCF. Secondly in some decay channels without final state
interaction, like pure leptonic decays and two body decays $\pi \nu _\tau $
etc., polarization is needed to search for CP/T violation occurring at $\tau
$ decay vertex. With unpolarized electron and positron beams the CP/T
violation could only be detected using channels with final state interaction
phase, like $2\pi \nu _\tau $ etc. (2) For the hadronic decay we only
consider inclusive processes. The advantage of inclusive process is that one
does not need not to consider the hadronization of quarks, which may bring
in large uncertainties in the estimation. And the event number in inclusive
process is larger than that in certain exclusive processes. However we
should mention that for certain exclusive decays the CP/T violation
parameter $\epsilon $ can be larger than that in inclusive decay. One
example is from the multi-Higgs double model. We estimate that $\epsilon
\leq 3\times 10^{-4}$ for the decay $\tau \to {\bar u}d\nu _\tau $. Here we
may also consider the exclusive decay $\tau \to 3\pi \nu _\tau $ contributed
by $a_1$ and $\pi ^{\prime }$ resonances. Compared to inclusive decay, the $%
\epsilon $ parameter is larger by a factor of (using current algebra
relation)
\begin{eqnarray}
\frac{<o|{\bar u}_Ld_R|\pi ^{\prime }>}{<o|{\bar u}_L\gamma _0d_L|\pi
^{\prime }>}\simeq \frac{m_{\pi ^{\prime }}}{m_u+m_d}\simeq 100.
\end{eqnarray}
So $\epsilon \leq 3\times 10^{-2}$ is obtained. However on the other hand
the event number decreases by a factor of
\begin{eqnarray}
\frac{f_\pi ^{\prime }}{f_\pi }\frac{Br(\tau \to \pi \nu _\tau )}{Br(\tau
\to hadron+\nu _\tau )}\simeq 10^{-2}
\end{eqnarray}
Here $f_{\pi ^{\prime }}=5\times 10^3$ GeV is used. Therefore statistical
error increases by about 10 times. In the other words the measurement
precision at TCF for this channel is about $10^{-2}$. As the result, at $%
2\sigma $ level $\epsilon \simeq 3\times 10^{-2}$ is observable. This
estimation agrees with the exact result of reference \cite{Cho}. (3)
Obviously the numerical result we obtained above is quite crude. More
accurate estimation is necessary in the future. For instance through this
paper we assume that EDM as large as $10^{-19}$e-cm and $\epsilon $ as large
as $10^{-3}$ can be observed. This of course is a rough estimation. To be
more precise, Monte Carlo simulation is needed, which will tell us more
confidently how large CP/T violation is able to be observed at TCF.
Especially the Monte Carlo simulation on EDM of $\tau $ will give us a quite
clear result , because in this case the $d_\tau $ is the only parameter we
should take care. All the model dependence is included in it. Recently a
group of people analyzed the data from BEPC experiments to
set bound on the T-violating effect for $\tau$ system \cite{Qi}.
Following the suggestion
by T.D. Lee, they considered the pure leptonic $\tau^{\pm}$ decays to
$e^{\pm}\mu^{\mp}$ plus neutrinos in the final states. The T-violating
amplitude
\begin{eqnarray}
A=<\hat{p}_e\cdot(\hat{p}_1\times \hat{p}_2)>_{average}
\end{eqnarray}
is measured, where $\hat{p}_e$ is the unit momentum vector of the initial
electron beam, $\hat{p}_1$ and $\hat{p}_2$ are the unit momenta of the final
state electron and muon respectively. Totally 251 events are analyzed and it
results in
\begin{eqnarray}
A=-0.097\pm 0.039\pm 0.135
\end{eqnarray}
This result agrees with no T-violation as expected from our previous
discussion on pure leptonic $\tau$ decays.
(4) In order to generate
detectable large CP/T violation effects, we know from our investigation that
there must exist new physics and the new physics scale is not far above the
weak scale. Therefore if there is a observable CP/T violation effect in $%
\tau $ sector at TCF, the associated new physics phenomena should be
observed at high energy experiments, like LHC and LEP II experiments. It is
interesting to see if the new particles predicted by the various models we
have discussed in this paper are indeed detectable in these high energy
experiments. (5) Precise measurement of the pure leptonic decay is another
way to test the new physics responsible for CP/T violation. Since if there
is CP/T violation effect at level of $10^{-3}$, the $\tau $ leptonic decay
width must deviate from the SM prediction at the same level. So we expect to
observe the deviation by measuring the branching ratio of the pure leptonic
decay. However it is not true {\it vice versa}, since a deviation of
leptonic branching ratio from that of the SM does not necessarily indicate
CP/T violation.
Finally we come to our conclusion. There exists the possibility that CP/T
violation in $\tau $ sector is large enough to be discovered at TCF,
although for this large violation effect some specific new physics phenomena
beyond the SM are needed and the parameter spaces of the models are strongly
restricted.
Z. J. Tao is supported by the National Science Foundation of China (NSFC).
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\section{Introduction.}
The $q$-calculus, in the generic case in which the fixed complex number $q$ is
not a root of unity but is otherwise arbitrary, involves a single
non-commuting variable $\theta$
and its left derivative operator ${{\cal D}} = {{\cal D}}_L$ and is
governed by the commutation relation
\begin{equation}
{{\cal D}} \theta -q \theta {{\cal D}} =1 \quad .
\end{equation}
It is closely related to the $q$-deformed oscillator \cite{AC} \cite{AJM}
\cite{LCB}, as is shown below.
The context in which $q$ is a root of unity, $q=\exp {{2\pi i} \over n}$,
is also of great interest. It involves $\theta$ such that $\theta^n=0$ and
and can be discussed by truncating the generic case so as to exclude powers of
$\theta$ higher than the $(n-1)$-th.
However, if
we pass with care from the generic case to the
limit in which $q$ is a root of unity much more structure can be exposed.
The algebraic structure in question is the full algebraic structure of
fractional supersymmetry (FSUSY), not only the generalised
Grassmann sector of this
${{\cal Z}}_n$-graded theory which is the part that where
$\theta$ enters but also its bosonic sector.
The paper shows how both these `sectors' emerge and
discusses the representation of the theory in a product Hilbert space. This has
an ordinary oscillator factor for the bosonic degree of freedom, and
relates the generalised Grassmann sector to the $q$-deformed
oscillator with deformation parameter
$q^{1/2}$, which is exactly what is needed to ensure proper hermiticity
properties.
We do not here make any extensive discussion of the interplay between
the sectors. But some idea of the
insights regarding this interplay can be obtained from \cite{DMdAPBplb}
which is devoted to the case of $q=-1$, which is that of ordinary
({\it i.e.}, ${{\cal Z}}_2$-graded) supersymmetry in zero space dimension.
It seems worthwhile emphasising that the
$q$-deformed oscillators at deformation parameter $q^{1/2}$ emerge as
those generalisations from $n=2$ to higher $n$ of the fermions
of supersymmetry which are best suited to the development of FSUSY.
References to FSUSY, including many to the extensive work of others, can be
found in our published \cite{dAM} and forthcoming \cite{DMdAPB} work.
\section{The $q$-calculus.}
For any graded algebra, we define a graded bracket, initially for elements
$A$ and $B$ of pure grade $g(A)$ and $g(B)$, by
\begin{equation}
[A,B]_{\gamma (A,B)}=AB-\gamma (A,B) \, BA, \quad
\gamma (A,B):=q^{-g(A) g(B)} \quad .
\label{gb}
\end{equation}
\nit This satisfies
\begin{equation}
[AB ,C]_{\gamma (AB,C)} = A[B,C]_{\gamma (B,C)}+\gamma (B,C)[A,C]_
{\gamma (A,C)} B \quad , \nonumber
\end{equation}
\begin{equation}
[A ,BC]_{\gamma (A,BC)} = [A,B]_{\gamma (A,B)} C +\gamma (A,B) B
[A,C]_{\gamma (A,C)} \quad ,
\end{equation}
\nit wherein $g(AB) =g(A)+g(B)$ is implicit.
The definition (\ref{gb}) extends by linearity to elements
of the algebra not of pure grade.
In (\ref{gb}) and until section four,
$q$ is `generic' {\it i.e.}, it is a fixed but arbitrary complex
quantity that is not a root of $1$. To define the $q$-calculus
algebra, we employ a single non-commuting variable $\theta$
of grade $1$, together with left and right
derivatives ${{\cal D}}_L$ and ${{\cal D}}_R$ of grade $-1$.
Since we shall not refer to ${{\cal D}}_R$ here (cf. \cite{dAM,NEW}),
we shall write ${{\cal D}}_L \equiv {{\cal D}} $.
The action
of ${{\cal D}}$ upon powers
and hence functions of $\theta$ is defined algebraically with the help
of the graded bracket
\begin{equation}
1=[{{\cal D}} , \theta ]_q \, := \, {{\cal D}} \theta -q \theta {{\cal D}} \quad ,
\label{deriv}
\end{equation}
\nit so that, for any positive integer $m$, we have
\begin{equation}
[{{\cal D}} ,\theta^{(m)}]_{q^m}=\theta^{(m-1)} \quad ,
\end{equation}
\nit
where the bracketed exponent is defined by
\begin{equation}
B^{(m)}:=B^m/[m]_q! \quad , \quad [m]_q=(1-q^m)/(1-q) \quad .
\label{notn}
\end{equation}
\nit The action extends obviously to $f(\theta)=\sum_{m=0}^\infty C_m
\theta^{(m)}$, where the $C_m$ are complex numbers
\begin{equation}
{{df} \over {d\theta}}
\equiv [{{\cal D}} ,f(\theta)]_\gamma \quad
= \sum_{m=0}^\infty C_m [{{\cal D}} ,\theta^{(m)}]_\gamma \quad = \sum_{m=1}^\infty
C_m \theta^{(m-1)} \quad .
\end{equation}
\nit It extends further also to $f(\theta) =\sum_{m=0}^\infty \theta^{(m)}
A_m$, where
the $A_m$ are quantities independent of $\theta$ and of pure grade $g(A_m)$,
so that $[\theta ,A_m]_\gamma$ with $\gamma = q^{-g(A_m)}$, giving
\begin{eqnarray}
{{df} \over {d\theta}}
& \equiv & [{{\cal D}} ,f(\theta)]_\gamma \quad
= \sum_{m=0}^\infty [{{\cal D}} ,\theta^{(m)} A_m]_\gamma \quad , \nonumber \\
& = & \sum_{m=0}^\infty ( [{{\cal D}} ,\theta^{(m)}]_{q^m} \, A_m + q^m A_m
[{{\cal D}} ,A_m]_\gamma ) \quad = \sum_{m=1}^\infty \theta^{(m-1)} A_m \quad ,
\end{eqnarray}
\nit provided that $[{{\cal D}} , A_m]_\gamma =0$
where $\gamma =q^{g(A_m)}$ (cf. \ref{gb}), which
can be seen to be compatible with the corresponding result assumed for
$\theta$ and $A_m$. An illustration, featuring $\exp _q(\theta A)
=\sum_{m=0}^\infty (\theta A)^{(m)}$ and wherein $A$ is of pure grade,
gives rise to
\begin{equation}
{{d\exp _q(\theta A)} \over {d\theta}}
=A \, \exp _q(\theta A) \quad .
\end{equation}
\nit Another notable result, involving a parameter $\varepsilon$ of grade $1$
with $[{{\cal D}} ,\varepsilon ]_{q^{-1}}=0$ and
$[\varepsilon , \theta ]_{q^{-1}}=0 $,
\nit shows that the quantity $G_L(\varepsilon)=\sum_{m=0}^\infty
\varepsilon^{(m)} {{\cal D}}^m $ generates the translation $\theta \mapsto \theta+
\varepsilon$, {\it i.e.}
$G_L(\varepsilon) \, f(\theta) \, G_L(\varepsilon)^{-1} =f(\theta +
\varepsilon) \quad . $
\section{Number, creation and destruction operators.}
We begin by constructing a number operator $N$ of grade zero
with the properties
\begin{eqnarray}
{[}N,\theta {]}=\theta \quad , & & q^N \theta q^{-N}=q\theta \quad ,
\nonumber \\
{[}N,{{\cal D}} {]}=-{{\cal D}} \quad , & & q^N {{\cal D}} q^{-N}=q^{-1}{{\cal D}} \quad .
\label{numb}
\end{eqnarray}
\nit Since $N$ is of grade zero, an expression for it in terms of $\theta$
and ${{\cal D}}$ may be expected to be of the form
\begin{equation}
N=\sum_{m=0}^\infty C_m \theta^m {{\cal D}}^m \quad ,
\end{equation}
and
\begin{equation}
N=\sum_{m=1}^\infty {{(1-q)^m} \over {1-q^m}} \theta^m {{\cal D}}^m \quad ,
\end{equation}
\nit satisfies both lines of (\ref{numb}). Likewise the right entries of
(\ref{numb}) may be shown to be satisfied by
\begin{equation}
q^N={{\cal D}} \theta -\theta {{\cal D}} \quad = 1-(1-q)\theta {{\cal D}} \quad ,
\label{qton}
\end{equation}
\nit where (\ref{deriv}) has been used. Useful consequences of these results
include
\begin{equation}
\theta {{\cal D}} = [N]_q \quad , \quad
\theta^m {{\cal D}}^m = {{[N]_q!} \over {[N-m]_q!}}
\quad , \quad
N=\sum_{m=1}^\infty {{(1-q)^m} \over {1-q^m}} {{[N]_q!} \over {[N-m]_q!}}
\quad .
\label{props}
\end{equation}
\nit The last result in (\ref{props}) is of interest as it gives
an expression for $N$ in terms of $q^N$.
Acting on an eigenstate of $N$ whose eigenvalue is a positive
integer $r$, this yields the identity
\begin{equation}
r=\sum_{m=1}^r {{(1-q)^m} \over {1-q^m}} {{[r]_q!} \over {[r-m]_q!}} \quad .
\end{equation}
If we now make the identification (to within a similarity transformation,
discussion of which may be sought in \cite{DMdAPB} )
\begin{equation}
\theta =a^{\dagger}, \quad {{\cal D}}=q^{N/2} a \quad ,
\label{repaadag}
\end{equation}
\nit then (\ref{deriv}) and (\ref{qton}) imply
\begin{equation}
a a^{\dagger}-q^{\mp 1/2} a^{\dagger} a = q^{\pm N/2} \quad.
\label{defcr}
\end{equation}
\nit This important result indicates how the $q$-calculus is related to the
$q$-deformed harmonic oscillator \cite{AC}, \cite{AJM} and \cite{LCB}. If $q$
is real, (\ref{defcr}) admits representations in which
$a^{\dagger}$ is indeed the adjoint of $a$ in a positive definite Hilbert
space. Further, for $q=\exp {2\pi i/n}$ when $n$ is an odd integer, the
situation to be concentrated upon below, a similar statement also holds true
because of the fact that
the deformation parameter in (\ref{defcr}) is $q^{1/2}$.
For simplicity the remaining sections of the Colloquium
talk confined discussion to the indicated set of roots of unity.
But the case $q=-1$ can also be treated in a similar
spirit. It is of interest because
it underlies an instructive view \cite{DMdAPBplb} of ordinary supersymmetry
in much the same way as the present work does for fractional supersymmetry.
\section{Lemmas for use at $q$ a root of 1.}
We now confine attention
--as mentioned at the end of the previous section--
to the $q$-values $q=\exp {{2\pi i} \over n}$
for odd integer $n$. We use the shorthand ${{\cal L}}$
to indicate the passage to the limit in which $q$ takes on such $q$-values,
${{\cal L}}:=\lim_{q\to\exp(2\pi i/n)}$.
We here
deduce a sequence of lemmas to be used in subsequent sections to effect the
systematic passage to the limit in question in the work of previous sections.
The following results can be proved in the order given:
\begin{equation}
{{\cal L}} {{[rn]_q} \over {[n]_q}}=r \; , \;
{\rm for} \; {\rm integer} \; r \quad ,
\quad
{{\cal L}} {{[rn]_q!} \over {[n]_q! [(r-1)n]_q!}}=r \quad , \quad
{{\cal L}} {{[rn]_q!} \over {([n]_q!)^r}} =r! \quad ,
\end{equation}
In the next section, we shall retain $\theta^{(m)}$, in the notation
(\ref{notn}), for $m=1,2, \dots ,(n-1)$ as ${{\cal Z}}_n$- graded variables,
and explain the use of the case $m=n$ to define a variable $z$ of zero grade
by setting $z={{\cal L}} \theta^{(n)}$.
To handle powers $m$ greater than $n$, we require further lemmas to be deduced
in order. Set $m=rn+p$ for integer $r$ and $p=1,2, \dots,(n-1)$. Then we have
\begin{equation}
[rn+p]_q=[p]_q \quad , \quad
{{\cal L}} {{[rn+p]_q!} \over {[rn]_q!}}=[p]_q! \quad , \quad p=1,\ldots,
(n-1)\quad,
\end{equation}
\begin{equation}
{{\cal L}} \theta^{(rn+p)}={{\cal L}} {{\theta^{rn+p}} \over {([n]_q!)^r r!}}
\Big / {{\cal L}} {{[rn+p]_q!} \over {[rn]_q!}} \, = \, {{\theta^p} \over {[p]_q!}} \,
{1 \over {r!}} \, {{\cal L}}
\Bigl( {{\theta^n} \over {[n]_q!}} \Bigr)^r \, = \,
{{z^r} \over {r!}} \theta^{(p)}
\quad .
\label{lem6}
\end{equation}
An illustration of the use of these lemmas indicates what happens to
$\exp _q (C \theta )$, $C$ a complex number,
in the limit under study. We find
\begin{equation}
\exp _q (C \theta )=\sum_{m=0}^\infty C^m \theta^{(m)}
= \bigl( \sum_{r=0}^\infty {{(zC^n)^r} \over {r!}} \bigr)
\bigl( \sum_{p=0}^{n-1} C^p \theta^{(p)} \bigr)\quad,
\end{equation}
\nit In other words
\begin{equation}
\exp _q (C \theta )=\exp (zC^n) \times {\rm truncated} \;
{\rm series} \quad .
\end{equation}
\section{The $q$-calculus for $q=\exp {{2\pi i} \over n}$
for odd integer $n$.}
Now we consider what happens to the $q$-calculus for those values of $q$.
We look first at an identity valid for generic complex values of $q$ and
any positive integer $m$, namely
\begin{equation}
[{{\cal D}} , \theta^{(m)} ]=\theta^{(m-1)} \quad ,
\label{idy}
\end{equation}
\nit where the notation (\ref{notn}) is used.
This makes sense for $m=n$ and $[n]_q=0$ only if $\theta^n=0$ at this $q$, and
if ${{\cal L}} \theta^{(n)}$
attains a finite non-zero value.
Indeed, we hereby define a new variable
$z = {{\cal L}} \theta^{(n)}$ of grade zero, so that (\ref{idy}) assumes the form
\begin{equation}
[{{\cal D}} , z]=\theta^{(n-1)} \quad .
\end{equation}
\nit Also we see that the $q$-calculus involves the variables
\begin{equation}
1,\theta , \theta^{(2)} ,\dots , \theta^{(n-1)} \quad
{\rm of} \; {\rm grades} \quad
0, 1, 2, \dots ,n-1 \quad .
\label{vbles}
\end{equation}
\nit It is natural at this point to ask what happens to powers of
the generalised Grassmann variable $\theta$
higher than the $n$-th. If they are simply discarded much insight into the
nature of fractional supersymmetry \cite{DMdAPB} (and likewise of ordinary
supersymmetry \cite{DMdAPBplb}) is lost. Actually lemma (\ref{lem6}) of the
previous section gives us directly an explicit non-trivial answer to
the question. It
follows that the generalised superfields of the context are linear
combinations of the variables (\ref{vbles})
with coefficients that are functions of $z$
Thus $z$ plays the role for the present (${{\cal Z}}_n$-graded
fractional supersymmetry) context that $t$ plays in ordinary
(${{\cal Z}}_2$-graded) supersymmetric mechanics.
Next it is natural to ask about ${{\cal D}}^n$ and to ask how ${{\partial} \over
{\partial z}}$ enters the picture, plainly not unrelated matters. By looking at
a suitable $n$-fold graded bracket involving $\theta$ and
${{\cal D}}$ each $n$ times,
it is not hard to show that ${{\cal D}}^n$ must be a well defined
quantity such that
\begin{equation}
[{{\cal D}}^n , z]=1 \quad .
\end{equation}
\nit Thus we make the identification ${{\cal D}}^n={{\partial} \over
{\partial z}}$.
It is clear that we must adjust somewhat our view of the nature of the
derivative operator ${{\cal D}}$. Presenting (\ref{idy}) in the form
\begin{equation}
[{{\cal D}}, z]= \theta^{(n-1)}= \Bigl( {{dz} \over {d\theta}} \Bigr)\quad ,
\end{equation}
\nit suggests that we now must view ${{\cal D}}$ as a total derivative with
respect to $\theta$ and write
\begin{equation}
{{\cal D}} = \partial_{\theta} +\theta^{(n-1)} \partial_z \quad ,
\label{supe}
\end{equation}
\nit which corresponds to the result
\begin{equation}
\Bigl( {{df} \over {d\theta}} \Bigr) =\Bigl( {{\partial f} \over
{\partial \theta}} \Bigr)+ \Bigl( {{dz} \over {d\theta}} \Bigr)
{{\partial f} \over {\partial z}} \quad .
\end{equation}
\nit It follows from (\ref{supe}) that
\begin{equation}
1=[\partial_{\theta}, \theta]_q \quad, \quad
(\partial_{\theta})^n=0 \quad .
\end{equation}
It might be judged from the form of (\ref{supe}) that ${{\cal D}}$ is closely related
to the full supercharge of the ${{\cal Z}}_n$-graded fractional
supersymmetry (FSUSY),
and it can be seen in \cite{dAM} \cite{DMdAPB} (see \cite{NEW}
for ${\cal Z}_3$) that this is
exactly correct. That ${{\cal D}}$ should therefore generate the full translational
invariance of the theory is one aspect of this. We wish to exhibit
how this emerges
from the results at the end of section two where ${{\cal D}}$ is seen to generate
translation of $\theta$ at generic $q$. First we note that $\theta \mapsto
\theta + \varepsilon$ is compatible with $\theta^n=0$ only if $\varepsilon^n
=0,$ holds in addition, of course, to $\varepsilon \theta = q^{-1} \theta
\varepsilon$. Next, using lemmas from section four, we deduce
\begin{equation}
G_L = {{\cal L}} \, \sum_{m=0}^\infty \varepsilon^{(m)} {{\cal D}}^m
= {{\cal L}} \, \sum_{r=0}^\infty \sum_{p=0}^{n-1}
{{\varepsilon^{p} {{\cal D}}^p} \over {[p]_q!}} \times
{{\varepsilon^{rn} {{\cal D}}^{rn}} \over {([n]_q!)^r r!}} \quad .
\label{gen}
\end{equation}
\nit This makes it clear that we should define a grade zero parameter
to associate with a
translation of $z$ by means of
\begin{equation}
{{\cal L}} \varepsilon^{(n)}= z_{\varepsilon} \quad.
\end{equation}
\nit For then it follows that we may write (\ref{gen}) in the form
\begin{equation}
G_L(z_{\varepsilon}, \epsilon )= \sum_{r=0}^\infty \sum_{p=0}^{n-1}
{{z_{\varepsilon}^r \partial_z^r} \over {r!}} \varepsilon^{(p)} {{\cal D}}^p
=\exp (z_{\varepsilon} \partial_z) \sum_{p=0}^{n-1}
\varepsilon^{(p)} {{\cal D}}^p \quad .
\end{equation}
\nit The first factor --an ordinary exponential of zero grade quantities--
generates $z \mapsto z+z_\varepsilon$ and the second factor is exactly
the one obtained in \cite{dAM} as the generator of translations of $\theta$
in the FSUSY context. However, the key result, showing that the full
non-trivial FSUSY transformation of $z$ is generated by
$G_L(z_{\varepsilon}, \epsilon )$, is
\begin{equation}
z \mapsto G_L z {G_L}^{-1} =
z+z_{\varepsilon}+ \sum_{p=1}^{n-1} \varepsilon^{(p)} \theta^{(n-p)}
\quad ,
\end{equation}
in agreement with \cite{dAM}.
\section{Reduction of the Representation space.}
It is rather obvious how we are to represent the algebra of $z, \partial_z,
\theta$ and $\partial_{\theta}$. The first two describe a bosonic degree of
freedom that commutes with the latter pair,
one that describes in Bargmann
style a harmonic oscillator Hilbert space ${{\cal V}}_{HO}$.
Also, with the evident
analogue
\begin{equation}
\theta=a^{\dagger} \quad , \quad \partial_{\theta} =q^{N/2} \, a \quad ,
\end{equation}
\nit of (\ref{repaadag}), we see that (\ref{defcr}) still follows.
Also
$\theta^n=0$ and $\partial_{\theta}^n=0$ imply $a^n=0 \, , \, a^{\dagger n}=0$
so that the variables of non-zero grade are represented in a vector space
${{\cal V}}^n$ of $n$ degrees of freedom. Crucially, since (\ref{defcr})
involves the deformation parameter $q^{1/2}$,
in the natural representation of $a$ and
$a^{\dagger}$ in ${{\cal V}}^n$ of positive definite metric,
the latter operator is indeed the true adjoint of the former.
It is our purpose now to demonstrate how the structure just described emerges
from the work of section three when one passes from the case of generic $q$ to
$q=\exp (2\pi i/n)$ for odd integer $n$. A representation of
${{\cal D}}$ and $\theta$
at generic $q$ in a space spanned by eigenkets of $N$, namely $|m\rangle$ for
$m=0,1,2, \dots $, can be taken to within equivalence in the form
\begin{equation}
{{\cal D}} |m\rangle =|m-1\rangle \quad , \quad {{\cal D}} |0\rangle =0 \quad ,
\quad \theta |m\rangle =[m+1]_q|m+1\rangle \quad .
\end{equation}
\nit This implies
\begin{equation}
\theta^{(n)} |m\rangle= ([m+n]_q! \, /([m]_q! \, [n]_q!) \; |m+n\rangle
\end{equation}
\nit is valid for generic $q$.
Setting $m=rn+p$ as in section four and passing to the limit for $q$
a root of 1 with the aid of lemmas from section four gives
$z|rn+p\rangle =(r+1)|(r+1)n+p\rangle$. Also $\partial_z={{\cal D}}^n$ leads to
$\partial_z |rn+p\rangle =|(r-1)n+p\rangle $. Indeed we can see that the
representation space at generic $q$ in the limit acquires a
product structure.
Setting $|rn+p\rangle \equiv |r\, , \, p \rangle \in
{{\cal V}}_{HO} \otimes {{\cal V}}^n $, we may view $z, \partial_z$
as $z \otimes 1, \partial_z \otimes 1$ in the product space, so that
\begin{equation}
z|r\rangle =(r+1)|r+1\rangle \quad ,
\quad \partial_z |r\rangle =|r-1\rangle \quad .
\end{equation}
\nit Likewise we may view $\theta$, etc., as $1 \otimes \theta$ and use
\begin{equation}
{{\cal D}} =1 \otimes \partial_\theta +\partial_z \otimes \theta^{(n-1)} \quad .
\end{equation}
\nit to express ${{\cal D}}$ in terms of creation and destruction operators.
There is of course a similarity transformation involved in placing
the representations considered here
explicitly in equivalence with those in which
\begin{equation}
a|p\rangle= \Bigl( {{q^{p/2}-q^{-p/2}} \over {q^{1/2}-q^{-1/2}}} \Bigr)^{1/2}
\; |p-1\rangle \quad ,
\end{equation}
and in which the correct adjoint properties of $a^{\dagger}$ are evident.
This is discussed in \cite{DMdAPB} .
\section*{Acknowledgements}
This paper describes research supported in part by E.P.S.R.C and P.P.A.R.C.
(UK) and by the C.I.C.Y.T (Spain).
J.C.P.B. wishes to acknowledge an FPI grant
from the CSIC and the Spanish Ministry of Education and Science.
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\section{Introduction}
\footnotetext {e--mail: [email protected]}
In the last two decades, redshift surveys provided a wealth of informations
about the spatial distribution of local galaxies,
revealing the existence of large--scale structures.
The most widely used statistical tool to quantify
the degree of clustering has been the galaxy two--point correlation function,
both in its angular $w_g(\theta)$ and spatial
$\xi_g(r)$ versions
(see, e.g., Peebles 1980).
As previous studies were confined to the nearby universe, nowadays
new observational resources permit to extend the correlation
analysis to deeper samples.
In fact,
the Canada--France Redshift Survey has recently provided the new
opportunity
to investigate the clustering properties of galaxies out to redshifts
$z \sim 1$ (Le F\`evre et al. 1996).
Moreover, only lately
it has been possible to analyse the angular distribution
of faint galaxies by using
the Hawaii Keck K--band survey (Cowie et al. 1996) and the Hubble Deep
Field data (Villumsen, Freudling \& da Costa 1997).
Therefore, we are now confident of
detecting a direct signature of redshift dependence in the
observed correlation function.
For this reason, the theoretical analysis of the
evolution of the mass two--point correlation function, $\xi(r)$,
is becoming a fundamental topic of modern cosmology.
However,
it is worth stressing that
the interpretation of the observational data is not
immediate:
before
obtaining the ``real'' change of the large--scale structure one
has to consider the possible evolution of the galaxy population
(as well as the related selection effects)
and of the
bias factor that formally relates $\xi_g$ to $\xi$
(see, e.g., Matarrese et al. 1997).
The observational results should then be compared
to the predictions of the existing models for structure
formation. One of the several issues involved in this comparison
is represented by the lack of a standard description of clustering
evolution: analytical
treatments are generally unable to manage this fully non--linear problem
while numerical simulations are limited in resolution.
However, new light has been recently shed on this argument.
Hamilton et al. (1991) suggested that the correlation function obtained
through N-body simulations of an Einstein--de Sitter universe,
in which the structure develops hierarchically,
can be easily reproduced by applying a
non--local and non--linear transformation to the linear $\xi(r)$.
This ansatz has been refined and extended
to more general cosmological scenarios by a number of authors
(Peacock \& Dodds 1994, Jain, Mo \& White 1995, Peacock \& Dodds 1996).
Moreover it is possible to give theoretical arguments that account
for the scaling hypothesis (Nityananda \& Padmanabhan 1994).
The main purpose of this paper is to compare the predictions of the Zel'dovich
approximation (Zel'dovich 1970) with the scaling ansatz
formulated in the
version of
Jain, Mo \& White (1995, hereafter JMW).
Actually, it would be very interesting to
obtain all the details of the semi--empirical
scaling relationship
in the framework of the gravitational
instability scenario. However, in the absence of a model for the
advanced phases
of clustering evolution, we are forced to analyse only
the onset of non--linear dynamics.
From the theoretical point of view, the evolution of the two--point
correlation function is strictly related to the dynamical development
of the density field
$\varrho({\bf x},t)$. When the dimensionless density contrast
$ \delta ({\bf x},t)=[\varrho({\bf x},t)-\bar \varrho(t)]/\bar \varrho(t)$
is much smaller than unity, the growth of the fluctuations can be
followed performing a perturbative approach (see, e.g., Peebles 1980,
Fry 1984, Scoccimarro \& Frieman 1996a). At the lowest order (linear theory),
the different Fourier modes
of $\delta({\bf x},t)$ evolve independently provided
that their power spectrum is less steep than
$k^4$ at small $k$ (Zel'dovich 1965, Peebles 1974).
As the fluctuations grow, however, the interactions
between different modes become more and more important.
The effect of this mode--coupling on the two--point statistics has been
studied by many authors using higher--order than linear terms
in the perturbation expansion.
Juskiewicz, Sonoda \& Barrow (1984) computed the second--order contribution to
$\xi(r)$ for an exponentially smoothed linear spectrum $P(k)\propto k^2$,
finding that non--linear interactions among long wavelength modes
act as a source for short $\lambda$ perturbations.
As a matter of fact, they found a substantial decrease of the characteristic
scale of clustering with the evolution.
However Suto \& Sasaki (1991) and Makino, Sasaki \& Suto (1992), analysing
exponentially filtered scale--free spectra, found that second--order effects
can either suppress or enhance the growth of perturbations on large scales,
depending on the shape and the amplitude of the fluctuation spectrum.
In particular Makino, Sasaki \& Suto (1992), modelling a CDM spectrum with
two different power laws, concluded that the effects of
mode--coupling are generally very small and completely negligible on
scales $r \magcir 20 \,h^{-1}{\rm Mpc}$
(where $h$ denotes the Hubble constant in units of $100
\,{\rm km\,s^{-1}\,Mpc^{-1}}$).
The second--order correction to the ``true'' linear CDM spectrum
has been calculated
by Coles (1990) who computed also the respective correlation function.
The results show that, for moderate evolution,
the large--scale distortions are of no importance,
while later (for $\sigma_8 \magcir 1$,
where $\sigma_8$ represents the {\em rms} linear mass fluctuation in spheres
of radius $8 \,h^{-1}{\rm Mpc}$)
non--linear effects can increase the clustering strength on scales
$r > 35 \,h^{-1}{\rm Mpc}$;
for example, the first zero crossing of $\xi(r)$ can be significantly
shifted with
respect to linear predictions.
Similar results were obtained by Baugh \& Efstathiou (1994) who also found
good agreement with the output of numerical simulations.
However,
Jain \& Bertschinger (1994)
pointed out that the perturbative approach
is able to reproduce the N--body outcomes only at early times
($\sigma_8 \mincir 0.5-1$).
Moreover, the recent analysis
applied to scale--free spectra by Scoccimarro \& Friemann
(1996b) showed
that the validity of perturbation theory
is restricted to a small range of spectral indices.
In this paper, we want to study the
non--linear evolution
of the mass autocorrelation function by describing the growth of
density fluctuations through the Zel'dovich approximation
(hereafter ZA).
In effect, Eulerian second--order perturbation theory may break down once
the mass variance becomes sufficiently large. On the other hand, we know that
ZA, especially in its ``truncated'' form,
is able to reproduce fairly well the outcomes of
N--body simulations even in the mildly non--linear regime
(Melott, Pellman \& Shandarin 1994).
The main advantage of ZA over other dynamical approximations
(for a recent review see, e.g., Sahni \& Coles 1995)
is that it permits analytical investigations ensuring at the same time
good accuracy, at least for quasi--linear scales.
The pioneering analysis by Bond \& Couchmann (1988) showed
that ZA is able to predict
the shifting of the first zero crossing of the correlation function.
In Section 3 we will give a detailed
quantitative description of this effect.
Other features of the mass two--point correlation function in ZA have
been discussed by Mann, Heavens \& Peacock (1993, hereafter MHP).
Moreover,
the related evolution of the power spectrum has been studied by
Taylor (1993),
Schneider \& Bartelmann (1995) and Taylor \& Hamilton (1996).
These authors
showed that ZA is able to describe the generation of small--scale
power through mode coupling, at least at early times.
Besides Fisher \& Nusser (1996) and Taylor \& Hamilton (1996)
succeeded in computing
the power spectrum
also in redshift space.
This paper is organized as follows. In Section 2 we briefly introduce the
Zel'dovich approximation while
in Section 3 we compute
the cross correlation function between the mass density field
evaluated at two different times. The usual two--point correlation function
is obtained as a particular case of this more general quantity.
The redshift evolution of $\xi(r)$ in a
CDM model is the last subject of Section 3.
In Section 4 we compare the predictions of ZA with the
scaling ansatz of JMW.
In Section 5 we use our results to evaluate the correlation
function of a collection of objects sampled by an observer in a
wide redshift interval of his past light cone.
We then propose a simplified scheme to compute this quantity so as
to improve another approximation presented in the literature.
A brief summary is given in Section 6.
\section{The Zel'dovich approximation}
Let us consider a set of collisionless, self--gravitating particles
in an expanding universe with scale factor $a(t)$.
We can describe the motion of each point--like particle
writing its actual (Eulerian) comoving position, ${\bf x}$, at time $t$
as the sum of
its initial (Lagrangian) comoving position, ${\bf q}$,
plus a displacement:
\begin{equation}
{\bf x}({\bf q},t)={\bf q}+{\bf S}({\bf q},t).
\label{Eq:lagrange}
\end{equation}
The displacement vector field ${\bf S}({\bf q},t)$ represents
the effect of density perturbations on the trajectories.
The Zel'dovich approximation is obtained by assuming the separability
of the temporal and spatial parts of ${\bf S}({\bf q},t)$
and by requiring equation (\ref{Eq:lagrange}) to give
the correct evolution of $\delta({\bf x},t)$
in the linear
regime.
Considering only the growing mode for
a pressureless fluid, one gets
(Zel'dovich 1970):
\begin{equation}
{\bf S}({\bf q},t)= - b(t) {\bf \nabla}
\phi \big|_{\mathbf q}
\label {Eq:zeld}
\end{equation}
where
$b(t)$ is the linear growth factor and
$\phi({\bf q})$ represents the initial peculiar velocity potential
that at the linear stage is proportional to the gravitational potential
$\Phi_0({\bf q})$.
The Zel'dovich approximation can be also extracted from a fully Lagrangian
approach to the evolution of density fluctuations
(Buchert 1989, Moutarde et al. 1991, Bouchet et al. 1992, Buchert 1993,
Catelan 1995).
In this case, ZA
corresponds to the first order solution provided that
the initial velocity field is irrotational and the
initial peculiar velocity
and acceleration fields are everywhere parallel.
Equations (\ref{Eq:lagrange}) and (\ref{Eq:zeld}) define a mapping from
Lagrangian
to Eulerian space that develops caustics as time goes on
(Shandarin \& Zel'dovich 1989). However, the
``Zel'dovich fluid'' is a system with infinite memory: even after the
intersection of two trajectories, the motion of the particles is determined
by their initial conditions according to equation
(\ref{Eq:zeld}).
The lack of self--gravity between intersecting streams causes the
forming structure
to be rapidly washed out. This is a severe problem especially in hierarchical
models of structure formation,
where caustics appear early on small scales causing ZA
to become soon inaccurate.
Nevertheless Coles, Melott \& Shandarin (1993)
showed that a modified version of ZA, the ``truncated'' ZA, obtained
by smoothing the initial conditions, is able to reproduce with good accuracy
the density distributions obtained from numerical simulations.
Melott, Pellman \& Shandarin (1994) found that the optimal
version of the truncation procedure is accomplished by using a Gaussian window
to smooth the linearly extrapolated power spectrum of the
density fluctuation field $b^2(t) P(k)$:
\footnote{We set $b=1$ at
the present epoch.}
\begin{equation}
P_{T}(k,t)=b^2(t) P(k)\exp{\left[ -k^2 R_f^2(t)\right]}
\label{Eq:trunc}
\end{equation}
where the filtering radius $R_f(t)$ increases with time being
related to the
typical scale going non--linear.
The success of this approximation can be justified by noticing that
the non--linearly evolved gravitational potential resembles
its smoothed linear counterpart
(Pauls \& Melott 1995).
In the following we will adopt the filtering prescription given in equation
(\ref{Eq:trunc}).
\section{The two-point correlation function in the Zel'dovich approximation}
Assuming that initially the mass is evenly distributed in
Lagrangian space, implies that the Eulerian density field
is related to the Lagrangian displacement field via the relation:
\begin{equation} \varrho({\bf x},t)= \bar \varrho(t) \int
d^3q \, \delta _D \left[{\bf x}-{\bf q}-{\bf S}({\bf q},t)\right],
\label {Eq:rho-S}
\end{equation}
where $ \delta_D({\bf x})$ denotes the three--dimensional Dirac delta function.
For purposes that will be clarified in Section 5, we are
interested in computing the cross correlation function between the
density contrast field evaluated at two different times:
\begin{equation}
\langle \delta({\mathbf x}_1,t_1) \delta({\mathbf x}_2,t_2) \rangle =
\langle \int d^3q_1 d^3q_2
\, \, \delta_D \left[{\mathbf x}_1-{\mathbf q}_1-{\mathbf S}({\mathbf q}_1,
t_1)\right]
\delta_D \left[{\mathbf x}_2-{\mathbf q}_2-{\mathbf S}({\mathbf q}_2,t_2)
\right]
\rangle -1
\label{Eq:deldel}
\end{equation}
where $\langle \cdot \rangle$ represents the average over an ensemble of
realizations.
Before going any further,
it is convenient to
Fourier transform the Dirac delta functions in equation (\ref{Eq:deldel})
obtaining:
\begin{equation}
1+\langle \delta({\mathbf x}_1, t_1) \delta({\mathbf x}_2,t_2) \rangle=
\int d^3q_1 d^3q_2 \, \, {d^3 w_1 \over (2 \pi)^3}\, {d^3 w_2
\over (2 \pi)^3} \exp \left[ i \sum _{j=1}^2 {{\mathbf w}_j \cdot
({\mathbf x}_j-{\mathbf q}_j)} \right]
\langle \exp \left[ -i \sum_{\ell=1}^2 {\mathbf w}_\ell\cdot {\mathbf S}
({\mathbf q}_\ell,t_\ell)
\right] \rangle.
\label{Eq:deldel2}
\end{equation}
We then use equation (\ref{Eq:zeld}) to introduce ZA
into equation (\ref{Eq:deldel2}).
In such a way, by assuming, as usual, that $\phi({\bf q})$ is
a statistically homogeneous and isotropic Gaussian field,
uniquely specified by its power spectrum $P_\phi(k)\propto P(k)/k^4$,
the ensemble average contained in equation
(\ref{Eq:deldel2})
can be written as a functional integral:
\begin{equation}
\langle \exp \left[ -i \sum_{\ell=1}^2 {\mathbf w}_\ell\cdot {\bf S}
({\bf q}_\ell,t_\ell)
\right] \rangle
= \left( \det {K}\right)^{1/2}
\int {\em D}[\phi] \exp{ \left[ -{1\over 2} \int \phi({\mathbf q})
K({\mathbf q},
{\mathbf q}^\prime) \phi({\mathbf q}^\prime) d^3q d^3q^\prime +
i \sum _{\ell=1}^2 b(t_\ell){\mathbf w}_\ell\cdot \nabla
\phi
\big|_{{\mathbf q_\ell}} \right]}
\label{Eq:path}
\end{equation}
where the kernel $K({\bf q},{\bf q}^\prime)$ represents the functional
inverse of the two--point correlation function of the field $\phi({\bf q})$.
By defining a six--dimensional vector ${\mathbf c^t}=({\mathbf w}_1, {\mathbf w}_2)$
and choosing the $z$-axis of our reference frame in the direction of the
vector ${\mathbf q}={\mathbf q}_1-{\mathbf q}_2$, we can reduce
equation (\ref{Eq:path}) to the form:
\begin{equation}
\langle \exp \left[ -i \sum_{\ell=1}^2 {\mathbf w}_\ell\cdot {\mathbf S}
({\mathbf q}_\ell,t_\ell)
\right] \rangle
= \exp \left[-{1\over 2} \mathbf c^t M c \right]
\label{Eq:pathsolve}
\end{equation}
where the matrix $\mathbf M$ has the structure
$$
\mathbf {M} =\gamma
\left( \begin{array} {cccccc}
b_1^2 & 0 & 0 & b_1 b_2 \psi _\perp & 0 & 0 \nonumber \\
0 & b_1^2 & 0 & 0 & b_1 b_2 \psi _\perp & 0 \nonumber \\
0 & 0 & b_1^2 & 0 & 0 & b_1 b_2 \psi _\parallel \nonumber\\
b_1 b_2 \psi _\perp & 0 & 0 & b_2^2 & 0 & 0 \\
0 & b_1 b_2 \psi _\perp & 0 & 0 & b_2^2 & 0 \nonumber \\
0 & 0 & b_1 b_2 \psi _\parallel & 0 & 0 & b_2^2 \nonumber
\end{array}
\right)\eqno (9)
\setcounter{equation}{9}
$$
with $b_i=b(t_i)$ and
\begin{equation}
\gamma= {1\over 6 \pi^2} \int _0 ^\infty \!\!\! P(k) dk\;, \ \ \
\gamma \psi_{\parallel} (q)= {1\over 2 \pi^2} \int _0 ^\infty \!\!\! P(k)
\left[ j_0(kq)-{2\over kq} j_1(kq)\right] dk \;, \ \ \
\gamma \psi_{\perp} (q)= {1\over 2 \pi^2} \int _0 ^\infty \!\!\! P(k)
{1\over kq} j_1(kq) dk\;,
\end{equation}
having denoted by $j_\ell (x)$
the spherical Bessel function of order $\ell$.
By substituting this result into equation (\ref{Eq:deldel2}) we can
easily solve the Gaussian integration over the ${\mathbf w}_i$. In order
to perform the remaining integrations, it is convenient to introduce the new
variables ${\bf q}$ and ${\bf Q}={\bf q}_1+{\bf q}_2$. In this way, after some
algebra, we finally obtain:
\begin {eqnarray}
\lefteqn {1+\xi(r,t_1,t_2) \equiv
1+\langle \delta({\mathbf x}_1, t_1) \delta({\mathbf x}_2,t_2) \rangle=
{1\over (2\pi )^{1/2}r}
\int _0 ^\infty {q^2 dq\over (b_1 b_2)^{1/2} \gamma
(\psi _\perp -\psi _\parallel)^{1/2} (b_1^2+b_2^2-2 b_1 b_2
\psi _\perp)^{1/2}}\times} \nonumber\\
& &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\times \left\{ D(u_+) \exp\left[ -{(q-r)^2\over 2 \gamma
(b_1^2+b_2^2-2 b_1 b_2
\psi _\parallel)}\right] -D(u_-) \exp \left[
-{(q+r)^2\over 2 \gamma (b_1^2+b_2^2-2 b_1 b_2
\psi _\parallel)}\right]
\right\}
\label {Eq:result}
\end{eqnarray}
where $r=|{\mathbf x}_1-{\mathbf x}_2|$,
\begin{equation}
u_{\pm}=\left[ {b_1 b_2 (\psi _\perp -\psi _\parallel) \over
\gamma (b_1^2+b_2^2-2b_1b_2\psi_\perp)(b_1^2+b_2^2-2b_1b_2\psi_\parallel)
}\right] ^{1/2}
\left[{b_1^2+b_2^2-2b_1b_2\psi_\perp\over 2 b_1 b_2 (\psi _\perp -
\psi_ \parallel)}q\pm r \right]
\end{equation}
and $D(x)$ represents the Dawson's integral
\footnote{It is worth stressing that when $\psi_\perp < \psi_\parallel$,
in order to avoid a complex argument for the Dawson's integral,
it is convenient
to express
the integrand in equation (\ref{Eq:result}) in terms of exponentials and
error functions
(see also the discussion in
Schneider \& Bartelmann, 1995).
However, since for the CDM spectrum (the only one considered in our analysis)
$\psi_\parallel$ is never larger than $ \psi_\perp$, we preferred to write
the solution using $D(x)$.}
(see, e.g., Abramowitz \& Stegun,
1968).
It is straightforward to show that for $t_1=t_2$ the previous formula
reduces to
the usual expression for the mass two--point correlation function in ZA
(Bond \& Couchman 1988, Mann, Heavens \& Peacock 1993, Schneider \&
Bartelmann 1995).
\begin{figure*}
\vskip -8truecm
\epsfxsize=16cm
\centerline{\epsfbox{copenh1.ps}}
\caption[]{
Left panel: the mass autocorrelation
function, obtained using ZA, for a
{\sl COBE} normalized CDM linear spectrum
is plotted for different values of the truncation
radius $R_f$ (in $\,h^{-1}{\rm Mpc}$).
Right panel: dependence of the correlation function evaluated at
$r= 1 \,h^{-1}{\rm Mpc}$ on $R_f$.}
\label{Fig:Rf}
\end{figure*}
We numerically evaluated the two--point correlation function
$\xi(r,t)\equiv \xi(r,t,t)$
employing a
{\sl COBE} normalized
standard CDM linear power spectrum
(with density parameter $\Omega=1$ and $h=0.5$).
We used the transfer function of Bardeen et al. (1986) while the
normalization to the four--year {\sl COBE} DMR data is given in Bunn \& White
(1997) and corresponds to $\sigma_8=1.22$.
As already noted by MHP, the small scale behaviour of the resulting correlation
function depends on the value assigned to
the truncation radius, $R_f$,
defined in equation (\ref{Eq:trunc}) (see Fig. \ref{Fig:Rf}).
If $R_f$ is very small, then shell crossing will not be suppressed
and $\xi(r)$ will show an unusually flat behaviour.
On the contrary, if $R_f$ is too large, the smoothing procedure will
remove an important contribution to the power spectrum,
causing again too low a correlation.
Therefore we need a criterion to select $R_f$. Since our main purpose is
to compare
the clustering amplitudes predicted by ZA with those extracted from
the scaling ansatz of JMW,
we can choose
$R_f$ so as to optimize the agreement between the
respective correlation functions.
Anyway, we find that this method conforms quite well
to a simpler one already used by MHP: the best $R_f$ is the one that
maximizes $\xi(r,R_f)$ on small scales.
Strictly speaking,
the optimal smoothing radius depends on the scale selected for
maximizing the correlation: the smaller is $r$ the larger comes out $R_f$
(we find that the difference between the smoothing lengths
obtained by maximizing
$\xi$ at
$r= 0.1 \,h^{-1}{\rm Mpc}$ and at $ r=1 \,h^{-1}{\rm Mpc}$ roughly amounts to $0.2 \,h^{-1}{\rm Mpc}$ and remains
nearly constant by varying $\sigma_8$). However, the effect of this discrepancy
on the correlation evaluated on larger scales is indeed minimal.
Following Schneider \& Bartelmann (1995), we select $r=1 \,h^{-1}{\rm Mpc}$ as the
scale at which we require $\xi(R_f)$ to be maximal.
As previously stated,
the optimum filtering length increases as the field evolves;
the dependence of the best $R_f$ on $\sigma_8$
is almost linear and for $\sigma_8>0.3$
(that in our model corresponds to $z\sim 3$)
it can
be approximated by:
\begin{equation}
R_f(\sigma_8)=(3.16\, \sigma_8 - 0.65) \,h^{-1}{\rm Mpc} \;.
\end{equation}
\begin{figure*}
\epsfxsize=8cm
\centerline{\epsfbox{def5.ps}}
\caption[]{
Redshift evolution
of the mass two--point correlation function obtained
using ZA to evolve a linear CDM spectrum.}
\label{Fig:zeldevo}
\end{figure*}
The redshift evolution of the correlation
function is shown in Fig. \ref{Fig:zeldevo}.
As expected, on scales that are not affected by shell crossing ($r>R_f$),
$\xi(r,z)$ steepens with decreasing $z$.
Moreover, we note that the
first zero crossing radius of $\xi(r,z)$
increases as time goes on (see also Bond \& Couchmann 1988).
A similar pattern has been noticed by Coles (1990) and by Baugh \&
Efstathiou (1994) in the context of second--order Eulerian perturbation theory.
The displacement of the first zero crossing of $\xi$ as a function of time is
plotted in Fig. \ref{Fig:0cross}.
Measuring the degree of dynamical evolution of the density field through
$\sigma_8$,
this shifting can be described with good approximation
by the function:
\begin{equation}
r_{\rm 0C}(\sigma _8)-r_{\rm 0C}^{\rm lin} \simeq
5.3 \, \sigma_8 ^{(1.5+0.1/\sigma_8)} \,h^{-1}{\rm Mpc}
\end{equation}
where we denoted by $r_{\rm 0C}$ the scale at which the correlation function
crosses for the first time the zero--level and by $r_{\rm 0C}^{\rm lin}$
its linear
counterpart.
It would be interesting to compare this result with the predictions of
second--order Eulerian (and Lagrangian) perturbation theory and of other
dynamical approximations.
\begin{figure*}
\epsfxsize=8cm
\centerline{\epsfbox{pisa4c.ps}}
\caption[]{
The first zero crossing radius of the correlation function is plotted
against $1/\sigma_8$. The circles represent the results obtained using ZA,
the dashed line
is the fitting function given in the text while the dotted line shows
the prediction of Eulerian linear theory.}
\label{Fig:0cross}
\end{figure*}
\section {Comparison with the scaling hypothesis}
The analysis of a large set of numerical simulations suggests that,
in hierarchical models,
the non--linear two--point correlation function, $\xi(r,z)$,
can be related to the linear one, $\xi_{\rm L}(r,z)$, through a simple scaling
relation
(Hamilton et al. 1991,
Peacock \& Dodds 1994, Jain, Mo \& White 1995,
Peacock \& Dodds 1996).
The main idea is that the action of gravity can be represented
as a continuous change of scale or, better, that the `flow of information'
about clustering propagates along the curves of equation:
\begin{equation}
r_0=[1+\bar \xi(r,z)]^{1/3} r \;,
\end{equation}
where $\bar \xi(r,z)$ represents the average correlation function within a
sphere of radius $r$
\begin{equation}
\bar \xi(r,z) = {3 \over r^3} \int_0^r y^2 \xi(y,z) dy
\end{equation}
and $r_0$ is a sort of Lagrangian coordinate determining a
`conserved pair surface'
(Hamilton et al. 1991, Nityananda \& Padmanabhan 1994).
In fact, by definition, the average number of
neighbours of a particle
contained within a spherical volume of radius $r_0$ at the linear stage
(when $\bar \xi \ll 1$) equals the average number
of neighbours inside a sphere of radius
$r$ in the evolved field.
Here, we want to compare the results obtained in the previous
section, using ZA, with the predictions of the scaling ansatz
(hereafter SA) formulated in the
version of JMW:
\begin{equation}
\bar \xi(r,z) = B(n_{\rm eff})
F\left[{\bar \xi_{\rm L}(r_0,z) \over B(n_{\rm eff})}\right]\;
\end{equation}
with
\begin{equation}
F(x) = { x + 0.45 x^2 - 0.02 x^5 + 0.05 x^6 \over 1 + 0.02 x^3 +
0.003 x^{9/2}} \;, \ \ \ \ \ \
B(n_{\rm eff}) = \left({3+n_{\rm eff}\over 3}\right)^{0.8}\;,
\ \ \ \ \ \
n_{\rm eff}(z) = \left.{ d \ln P(k)\over d \ln k} \right| _{k_{\rm NL}(z)}
\end{equation}
where $k_{\rm NL}^{-1}$ denotes the radius of the top--hat window function in
which the {\em rms} linear mass fluctuation is unity.
However, it would be useless to perform the comparison between the
spherically
averaged correlation functions since, on
small scales, $\xi(r)$
obtained using ZA is seriously affected by
shell crossing
and the computation of $\bar \xi$ requires an integration starting from $r=0$.
For this reason we prefer to use directly $\xi(r)$. The
two--point correlation function deriving from the ansatz of JMW can
be obtained performing a simple differentiation:
\begin{equation}
\xi(r,z) = { \bigl [1 + B(n_{\rm eff}) F(X) \bigr] F'(X) \Delta \xi_{\rm L}
(r_0,z)
\over 1 + B(n_{\rm eff}) F(X) - F'(X) \Delta \xi_{\rm L}(r_0,z)}
+ B(n_{\rm eff}) F(X) \;, \ \ \ \ \ \ X={\bar \xi_{\rm L}(r_0,z)
\over B(n_{\rm eff})}\; ,
\label{Eq:jmwdiff}
\end{equation}
with $F'(x) = dF/dx$ and
\begin{equation}
\Delta \xi_{\rm L}(r_0,z) \equiv \xi_{\rm L}(r_0,z) - \bar \xi_{\rm L}(r_0,z)
= {b^2(z) \over 2 \pi^2} \int_0^\infty \!\!\! k^2 P(k)
\bigl[ j_0(kr_0) - {3 \over kr_0} j_1(kr_0)\bigr] dk \;.
\end{equation}
\begin{figure*}
\epsfxsize=8cm
\centerline{\epsfbox{def2.ps}}
\caption[]{
Comparison between the mass autocorrelation functions computed for a CDM
model by using: the Zel'dovich approximation (ZA), the scaling ansatz of
Jain, Mo \& White (JMW) and Eulerian linear theory (ELT).
The linear power spectrum extrapolated to the present epoch ($z=0$) is
normalized to match the {\sl COBE} DMR data ($\sigma_8=1.22$).}
\label{Fig:JMW1}
\end{figure*}
We evaluated the correlation function
given in equation (\ref{Eq:jmwdiff}) using a {\sl COBE} normalized,
linear CDM spectrum.
In Fig. \ref{Fig:JMW1} we plot the result obtained at $z=0$ with
the corresponding one achieved by using ZA.
For comparison we also show the prediction of Eulerian linear
theory.
The agreement between ZA and SA
is remarkable on mildly non--linear scales ($4\,h^{-1}{\rm Mpc} \mincir r \mincir 20 \,h^{-1}{\rm Mpc}$)
and on completely
linear scales ($r > 50 \,h^{-1}{\rm Mpc}$).
For example,
at $r = 5 \,h^{-1}{\rm Mpc}$, linear theory overestimates
the correlation of JMW by $82 \%$, ZA underestimates it by $2 \%$ while
the accuracy of the JMW fit is about $15-20 \%$.
However, we find that in the interval
$20 \,h^{-1}{\rm Mpc} \mincir r \mincir 50 \,h^{-1}{\rm Mpc}$ ZA predicts more non--linear
evolution than SA (for example the $r_{\rm 0C}$ obtained by using ZA
is larger than the one determined through SA).
In order to consider a less evolved field, in Fig. \ref{Fig:JMW2}
we repeat the comparison using the correlation functions evaluated
at $z=1$. Now,
the main item to note is
that the JMW result
matches the linear solution
on scales ($r \sim 10 \,h^{-1}{\rm Mpc}$)
that, according to ZA, are already involved in non--linear phenomena.
In any case, we do not know the accuracy of the scaling hypothesis
on large scales. In fact, the function $F(x)$ is obtained by requiring
the resulting $\xi(r)$
to reproduce the linear behaviour where $\bar \xi_{\rm L} \to 0$ and,
simultaneously, to approximate properly the correlation function
extracted from
N--body simulations. However, in order to achieve a detailed description
of non--linear scales, JMW used a relatively small box to perform their
simulations.
Therefore, imposing the match to linear theory on large scales,
without having any constraint from numerical data on quasi--linear scales,
could seriously alter
the accuracy of $F(x)$.
This probably implies that the JMW fitting function could be
improved on large scales.
Our conclusion is shared by
Baugh \& Gazta\~{n}aga (1996, hereafter BG), who tested the scaling ansatz
for the evolution of the power spectrum
against the results
of 5 N--body simulations performed within a $378 \,h^{-1}{\rm Mpc}$ box.
Indeed, they found
that the JMW formula gives a relatively poor description
of the large--scale behaviour
even though the agreement between the spectra
remains always within the quoted 20 \% accuracy.
By using the output of their simulations,
BG proposed a new scaling formula calibrated on large scales.
As initially suggested by Peacock \& Dodds (1994), the analytic
expression of this SA concerns the dimensionless power spectrum,
$\Delta^2(k,z)=k^3 P(k,z)/ 2 \pi^2$
(i.e. the contribution to the the variance of the density contrast
per bin of $\ln k$),
while,
following JMW, it takes account of a spectral dependence of the
transformation:
\begin{equation} \Delta^2(k,z)= \beta(n_{\rm eff}) f\left[
{\Delta^2_{\rm L}(k_{\rm L},z)\over \beta(n_{\rm eff})} \right]\;,
\ \ \ \ \ \
k_{\rm L}=\left[ 1+\Delta^2(k,z)\right]^{-1/3} k
\end{equation}
where
\begin{equation}
f(x)=x \left( {1+0.598x-2.39x^2+8.36x^3-9.01x^{3.5}+2.895x^4
\over 1-0.424x+[2.895/(11.68)^2]x^3}\right)^{1/2}\;, \ \ \ \ \ \
\beta(n_{\rm eff})=1.16\left( {3+n_{\rm eff}\over 3}\right) ^{1/2} \end{equation}
and the subscript {\rm L} marks linear quantities.
The function $f(x)$ has been obtained by matching the power spectrum in
the simulations at $\sigma_8=1$,
with an accuracy of $5 \%$, over the range $0.02 \, h\,{\rm Mpc}^{-1}< k <
1.0 \, h \,
{\rm Mpc}^{-1}$ and by forcing the fit to have the asymptotic form
$f(x) \to 11.68 \,x^{3/2}$ when $x \to \infty$ (Hamilton {\it et al.}
1991).
The two--point correlation function is related to
$\Delta^2(k,z)$ through
the Fourier relation:
\begin{equation} \xi(r,z)=\int_0^\infty \Delta^2(k,z) j_0(kr)
{dk\over k}.
\end{equation}
In Fig. \ref{Fig:BAU} we compare the correlations obtained by
using ZA and the JMW formula with the results of the scaling ansatz by BG:
we are considering a standard CDM linear spectrum at the epoch in which
$\sigma_8=1.22$.
We immediately note that using larger simulation boxes to calibrate the SA
allows a better
determination of the correlation function for
$r \magcir 20 \,h^{-1}{\rm Mpc}$.
In fact, we find that the correlations obtained with ZA and with the BG formula
agree by better than $ 20 \%$ for $r > 4.6 \,h^{-1}{\rm Mpc}$ (with
the exception of a very small $r$-interval centred in the first zero crossing
of $\xi$) while
the discrepancy between ZA and the JMW ansatz is less
than $ 20 \%$ over the ranges
$4.1 \,h^{-1}{\rm Mpc} < r < 18.3 \,h^{-1}{\rm Mpc} $ and $ r > 49.6 \,h^{-1}{\rm Mpc}$.
Similar patterns are obtained considering different values of $\sigma_8$.
This shows that the BG fit, that has been calibrated against
large box CDM simulations, gives also a very good description
of the mass clustering predicted by ZA on intermediate scales.
In any case, as expected, the JMW formula is sensibly more accurate for
$5 \,h^{-1}{\rm Mpc} \mincir r \mincir 15 \,h^{-1}{\rm Mpc}$ where the BG predictions
grow worse as $\sigma_8$ assumes values significantly larger than 1.
\begin{figure*}
\epsfxsize=8cm
\centerline{\epsfbox{defjmw1.ps}}
\caption[]{
As in Fig. 4, but at $z=1$ ($\sigma_8=0.61$).}
\label{Fig:JMW2}
\end{figure*}
On the other hand, it would be interesting
to check the reliability of ZA and second--order Eulerian perturbation
theory by directly comparing their predictions on these scales.
Bond \& Couchmann (1988), studying the weakly non--linear evolution of
the CDM
power spectrum, found
remarkable agreement between the two approximations.
Moreover, Baugh \& Efstathiou (1994)
showed that second--order Eulerian
perturbation theory can reproduce, at least qualitatively,
the evolution of the power spectrum predicted by numerical simulations.
However,
Jain \& Bertschinger (1994)
found that the agreement between perturbation theory and N--body outcomes
gets worse as the density field evolves. Besides, their results
are inconsistent with the low--$k$ behaviour
of the second--order Eulerian correction to the
CDM power spectrum computed by Bond \& Couchmann (1988),
raising again the issue about the compatibility between ZA and
perturbation theory.
In a recent work concerning the evolution of scale invariant spectra,
Scoccimarro \& Friemann (1996b) showed that, if the spectral index $n$
satisfies $-3<n<-1$,
Eulerian perturbation theory is able to reproduce fairly well the
power spectrum obtained though the scaling ansatz, while the one--loop
perturbative version of ZA gives worse results.
Anyway, Bharadwaj (1996a,b) pointed out that the effects of multistreaming
on the correlation function
cannot be studied perturbatively.
This fact implies that our result, obtained considering the full Zel'dovich
approximation,
should be more reliable than any other achieved by adopting a
perturbative version of ZA.
In any case,
it would be interesting to clarify to which extent ZA and
Eulerian perturbation theory agree on large scales.
\begin{figure*}
\epsfxsize=8cm
\centerline{\epsfbox{tac122sm.ps}}
\caption[]{
Mass two--point correlation functions at the epoch in which $\sigma_8=1$
obtained from a linear CDM spectrum evolved through the Zel'dovich
approximation (ZA) and through the scaling ans\"atze by Jain, Mo
\& White (JMW) and by Baugh \& Gazta\~naga (BG).}
\label{Fig:BAU}
\end{figure*}
\section {The correlation of high redshift objects}
In this section, we study the evolution of the
cross correlation function of the mass density contrast evaluated
at two different times as defined in equation (\ref{Eq:result}).
This quantity could play an important role in comparing
the clustering properties extracted from deep redshift surveys
to the predictions of theoretical models for structure formation.
In practice, one always collects data on correlations in a finite
redshift strip of his past light cone
while the quantity $\xi(r,t)$, normally used in theoretical works, refers
to objects
selected on an hypersurface of constant cosmic time.
Therefore, as far as one is considering a deep sample of cosmic objects,
it is not correct to relate the observed clustering properties to
$\xi(r,t)$. This issue is addressed in detail
by Matarrese et al. (1997, hereafter MCLM)
who build
a theoretical quantity that allows a direct comparison
of model predictions to the observed correlations.
Their approach can be
divided into three steps:
first of all they compute the redshift evolution of mass correlations,
then they relate the clustering properties of cosmic objects to
the matter distribution by means of a linear bias relationship
and finally they convolve the result with the observed
redshift distribution of the class of objects under analysis.
By assuming that the effects of redshift distortions and of the
magnification bias due to weak gravitational lensing are negligible and
by considering isotropic selection functions,
MCLM showed that
the theoretical estimate for the observed two--point correlation
function can be formally expressed as an integral over
$z_1$ and $z_2$
of the
function $\xi(r,z_1,z_2)$ weighted by geometrical factors and effective
bias parameters (all dependent on $z_1$ and $z_2$).
Different classes of objects are selected by changing the
amplitude and the redshift dependence of the effective bias.
However, in the absence of a model for the evolution of
the cross--correlation,
only assuming that the above mentioned integral is dominated by
the contribution of
objects whose redshifts are nearly the same,
can one
estimate the observed correlation function deriving from
a particular scenario of structure formation.
In this way, one is
allowed to replace
$\xi (r,z_1,z_2)$ with
$\xi(r,\bar z)$, where
$\bar z$ is a suitably defined average between $z_1$ and $z_2$
that, for simplicity, MCLM
identify with
$\bar z=(z_1+z_2)/2$.
This is a crucial approximation, as it allows MCLM
to use the JMW ansatz to compute the non linear mass correlation
function
(there is no known scaling ansatz for $\xi(r,z_1,z_2)$).
However, as shown in the previous paragraphs, ZA allows the computation
of $\xi(r,z_1,z_2)$ so that
we are able to compute the theoretical estimate for the observed
correlation function by using both the complete and the approximated formulae
given by MCLM (respectively their equations 15 and 18).
Therefore we can check here, within the validity of ZA, the reliability of the
approximation introduced by MCLM.
Large discrepancies between the exact and the approximated correlations
would obviously invalidate their whole analysis and consequently
also their complete formula for $\xi_{\rm obs}$ would be unutilizable.
On the other hand, if the approximated correlation function turns out
to reproduce accurately the complete one, MCLM formulae could represent
an important tool to disprove cosmological models
in the light of present and future observations.
\begin{figure*}
\epsfxsize=8cm
\centerline{\epsfbox{def6.ps}}
\caption[]{
Cross correlation between the density contrast field
evaluated at two different redshifts vs. comoving separation.}
\label{Fig:duet}
\end{figure*}
In order to compute $\xi(r,z_1,z_2)$ using equation (\ref{Eq:result}), we
truncated the linearly extrapolated power spectrum
$b(z_1) b(z_2) P(k)$
according to the prescription:
\begin{equation}
P_{T}(k,z_1,z_2)=b(z_1) b(z_2) P(k)\exp{\left[ -k^2 R_{f}(z_1) R_{f}(z_2)\right]}
\label{Eq:t2D}
\end{equation}
where $R_{f}(z)$ represents the optimum filtering length for the
density field at redshift $z$, determined by following the method described
in Section 3.
On small scales,
the correlation functions that we obtain
opting for this truncation procedure appear
much more
flattened than those computed at a single time.
The evolution of $\xi(r,z_1,z_2)$ as $z_2$ changes is shown, for a CDM model, in
Fig. \ref{Fig:duet}.
It is evident that even though the correlation decreases
as $z_2$ grows, its decay is very slow.
Actually, the ratios between the correlations computed at the same $r$,
for different
pairs of redshifts, are very similar to the predictions of linear theory.
We find that the redshift evolution of the cross correlation function can
be approximately described by the relation:
\begin{equation}
\xi(s,z_1,z_2) \simeq \left[ \xi(s,z_1) \xi(s,z_2)\right]^{1/2}
\left[ 1-2\Theta(s-1) \right]
\label {Eq:approx2t}
\end{equation}
where the quantity $s=r/r_{\rm 0C}(z)$ is introduced in order to
take into account the shifting of the first zero crossing
of $\xi(r,z)$
and $\Theta(x)$ is the Heaviside step function.
Moreover, the first zero crossing radius of $\xi(r,z_1,z_2)$
is nearly given by the geometric average of
$r_{\rm 0C}(z_1)$ and $r_{\rm 0C}(z_2)$.
For $s>0.1$
equation (\ref{Eq:approx2t}),
which is meaningful up to the scale at which the first of the two $\xi(s,z)$
reaches its second zero crossing,
reproduces $\xi(s,z_1,z_2)$ with an accuracy of
$\sim 5 \%$.
Anyway, for $s \magcir 2$, the usual relation
$ \xi(r,z_1,z_2) \simeq [\xi(r,z_1) \xi(r,z_2)]^{1/2}{\rm sign}[\xi(r,z_1)]$
deriving from linear
theory is preferable.
We can now check the accuracy of the approximation introduced by MCLM
that consists in computing the
theoretical estimate for the
observed correlation function
by replacing $\xi(r,z_1,z_2)$ with $\xi(r,\bar z)$, where $\bar z=(z_1+z_2)/2$,
in the appropriate formula.
For simplicity
(and in order to isolate the phenomenon of clustering evolution)
we will assume no bias, no selection effects and a constant comoving
number density in an Einstein--de Sitter universe.
In this case, equation 15 of MCLM reduces to:
\begin{equation}
\xi_{\rm obs}(r,z_{\rm min},z_{\rm max})=
{\displaystyle \int_{z_{\rm min}}^{z_{\rm max}}
{2+z_1-2(1+z_1)^{1/2} \over (1+z_1)^{5/2}} \,
{2+z_2-2(1+z_2)^{1/2} \over (1+z_2)^{5/2}} \,
\xi(r,z_1,z_2) \, dz_1 dz_2 \over
\displaystyle \left[ \int_{z_{\rm min}}^{z_{\rm max}}
{2+z-2(1+z)^{1/2} \over (1+z)^{5/2}}\, dz
\right] ^2}
\label{Eq:mat}
\end{equation}
where we denoted by $\xi_{\rm obs}(r,z_{\rm min},z_{\rm max})$ the
(ensemble averaged) theoretical estimate for the two--point
correlation function measured by an observer that acquires data from
the region of his past light cone corresponding to
the redshift interval $ [z_{\rm min},z_{\rm max}]$.
Considering only the linear evolution of density fluctuations,
$\xi(r,z_1,z_2)=\xi(r,0,0)/[(1+z_1)(1+z_2)]$,
the integrals contained in equation (\ref{Eq:mat}) can
be analytically performed.
In this case, the quantity $\xi_{\rm obs}(r,z_1,z_2)/\xi(r,0,0)$ does not
depend on $r$; for example we obtain
$\xi_{\rm obs}(r,0,2)/\xi(r,0,0)\simeq 0.224$ and
$\xi_{\rm obs}(r,0,1)/\xi(r,0,0)\simeq 0.375$.
In this regime, we find that the approximation for $\xi_{\rm obs}$ introduced
by MCLM is accurate to $2-3 \%$.
In order to extend our analysis also to the mildly non--linear evolution,
we numerically computed $\xi_{\rm obs}$ by using the cross correlation
given in equation (\ref{Eq:result}). The result obtained
for $ [z_{\rm min},z_{\rm max}]=[0,2]$
is shown in Fig. \ref{Fig:integrata}: also in this case
$\xi_{\rm obs}$
looks like the usual correlation function evaluated at some
intermediate redshift.
We then tested the accuracy of the above mentioned
simplified scheme for the computation of
$\xi_{\rm obs}$,
finding good agreement between the exact and the rough estimates
(excluding a small neighbourhood of the
zero--crossing radius of $\xi(r,z_1,z_2)$,
where the approximated method breaks down,
we find a maximum discrepancy of $6 \%$
for $[z_{\rm min},z_{\rm max}]=[0,2]$ and of $3 \%$ for
$[z_{\rm min},z_{\rm max}]=[0,1]$).
Anyway, the simplified procedure to compute $\xi_{\rm obs}$
can be further improved:
adopting
a different way of performing the average between redshifts, namely
$1+\bar z=[(1+z_1)(1+z_2)]^{1/2}$, ensures more accurate predictions
(in this case the maximum error is always of the order of $1\%$).
Probably this higher precision is due to the fact that we are considering mildly non--linear
scales and the latter approximation gives exact results for linear
evolution.
\begin{figure*}
\epsfxsize=8cm
\centerline{\epsfbox{def9.ps}}
\caption[]{
The observed two--point correlation function computed using equation (26)
for a CDM
model with
$[z_{\rm min},z_{\rm max}]=[0,2]$. For comparison, the corresponding $\xi(r,z)$
evaluated for $z=0,1,2$
are plotted.}
\label{Fig:integrata}
\end{figure*}
\section{Summary}
In this paper,
we have studied in detail the evolution of the mass two--point correlation
function
by describing the growth of density perturbations through ZA.
Our motivations were originated by the well known ability
of ZA to reproduce the weakly non--linear regime of
gravitational dynamics.
On scales that are not affected by shell--crossing, we found that
the correlation function steepens as the clustering amplitude
increases. Moreover,
we showed that non--linear interactions are able to move the first
zero crossing
of $\xi(r)$ and we gave a quantitative description of this
shifting for a CDM linear spectrum.
We then compared our results with the predictions of the scaling ansatz
for clustering evolution formulated by JMW, obtaining remarkable
agreement between the correlations on mildly non--linear scales
and on completely linear scales.
However, between these two regimes, the
JMW prescription, which has been obtained requiring the resulting correlation
to reproduce the linear behaviour on large scales, predicts
smaller clustering amplitudes than ZA.
We think that this disagreement is caused by the smallness of the box
used by JMW to perform their N--body simulations.
Actually, imposing to match the linear solution where
$\bar \xi_{\rm L} \to 0$, without
having any constraint from numerical data on quasi--linear scales, could alter
the accuracy of the fitting function that embodies the scaling ansatz.
In connection with this hypothesis, we compared ZA predictions on
correlations with the output of a different scaling ansatz calibrated
against large box simulations by BG.
In effect, on large scales, the BG formula agrees better with
ZA, keeping the same accuracy of the JMW fit on intermediate scales.
On the other hand, the reliability of ZA on these scales and for dynamically
evolved fields ($\sigma_8 \magcir 1$) should be verified by directly comparing
its predictions with the results of other approximations and numerical
simulations.
Finally,
we studied the evolution of the cross correlation between the density
field evaluated at two different epochs and,
adopting the method introduced by MCLM,
we used our results to
compute the theoretical prediction for the observed correlation function
deriving from a deep catalogue of objects.
In this context,
we proposed a simplified procedure for
the computation of $\xi_{\rm obs}$ that, at least for
quasi--linear scales, significantly improves another approximation
previously introduced by MCLM.
This result confirms that the MCLM method can be used to make
quantitative predictions about clustering evolution that find
a direct observative counterpart in the analysis of deep surveys.
\section*{Acknowledgments.}
I would like to thank Sabino Matarrese for the encouragement and the useful
suggestions.
I am grateful to the referee, Carlton Baugh, for helpful comments on
the manuscript.
Francesco Lucchin, Lauro Moscardini and Pierluigi Monaco are also thanked
for discussions.
Italian MURST is acknowledged for financial support.
\vspace{1.5cm}
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\section{Introduction}
\label{I}
The relativistic approach to nuclear physics has attracted much
attention. From a theoretical point of view, it allows to implement,
in principle, the important requirements of relativity, unitarity,
causality and renormalizability~\cite{Wa74}. From the phenomenological
side, it has also been successful in reproducing a large body of
experimental data~\cite{Wa74,Ho81,Se86,Re89,Se92}. In the context of
finite nuclei a large amount of work has been done at the Hartree
level but considering only the positive energy single particle nucleon
states. The Dirac sea has also been studied since it is required to
preserve the unitarity of the theory. Actually, Dirac sea corrections
have been found to be non negligible using a semiclassical expansion
which, if computed to fourth order, seems to be quickly
convergent~\cite{Ca96a}. Therefore, it would appear that the overall
theoretical and phenomenological picture suggested by the relativistic
approach is rather reliable.
However, it has been known since ten years that such a description is
internally inconsistent. The vacuum of the theory is unstable due to
the existence of tachyonic poles in the meson propagators at high
Euclidean momenta~\cite{Pe87}. Alternatively, a translationally
invariant mean field vacuum does not correspond to a minimum; the
Dirac sea vacuum energy can be lowered by allowing small size mean
field solutions~\cite{Co87}. Being a short distance instability it
does not show up for finite nuclei at the one fermion loop level and
within a semiclassical expansion (which is an asymptotic large size
expansion). For the same reason, it does not appear either in the
study of nuclear matter if translational invariance is imposed as a
constraint. However, the instability sets in either in an exact mean
field valence plus sea (i.e., one fermion loop) calculation for finite
nuclei or in the determination of the correlation energy for nuclear
matter (i.e., one fermion loop plus a boson loop). Unlike quantum
electrodynamics, where the instability takes place far beyond its
domain of applicability, in quantum hadrodynamics it occurs at the
length scale of 0.2~fm that is comparable to the nucleon size and
mass. Therefore, the existence of the instability contradicts the
original motivation that lead to the introduction of the field
theoretical model itself. In such a situation several possibilities
arise. Firstly, one may argue that the model is defined only as an
effective theory, subjected to inherent limitations regarding the
Dirac sea. Namely, the sea may at best be handled semiclassically,
hence reducing the scope of applicability of the model. This
interpretation is intellectually unsatisfactory since the
semiclassical treatment would be an approximation to an inexistent
mean field description. Alternatively, and taking into account the
phenomenological success of the model, one may take more seriously the
spirit of the original proposal~\cite{Wa74}, namely, to use specific
renormalizable Lagrangians where the basic degrees of freedom are
represented by nucleon and meson fields. Such a path has been
explored in a series of papers~\cite{Ta90,Ta91,TB92} inspired by the
early work of Redmond and Bogolyubov on non asymptotically free
theories~\cite{Re58,Bo61}. The key feature of this kind of theories is
that they are only defined in a perturbative sense. According to the
latter authors, it is possible to supplement the theory with a
prescription based on an exact fulfillment of the K\"all\'en-Lehmann
representation of the two point Green's functions. The interesting
aspect of this proposal is that the Landau poles are removed in such a
way that the perturbative content of the theory remains unchanged. In
particular, this guarantees that the perturbative renormalizability is
preserved. It is, however, not clear whether this result can be
generalized to three and higher point Green's functions in order to
end up with a completely well-behaved field theory. Although the
prescription to eliminate the ghosts may seem to be ad hoc, it
certainly agrees more with the original proposal and provides a
workable calculational scheme.
The above mentioned prescription has already been used in the context
of nuclear physics. In ref.~\cite{Lo80}, it was applied to ghost
removal in the $\sigma$ exchange in the $NN$ potential. More recently, it
has been explored to study the correlation energy in nuclear matter in
the $\sigma$-$\omega$ model~\cite{TB92} and also in the evaluation of
response functions within a local density approximation~\cite{Ta90}.
Although this model is rather simple, it embodies the essential field
theoretical aspects of the problem while still providing a reasonable
phenomenological description. We will use the $\sigma$-$\omega$ model
in the present work, to estimate the binding energy of finite nuclei
within a self-consistent mean field description, including the effects
due to the Dirac sea, after explicit elimination of the ghosts. An
exact mean field calculation, both for the valence and sea, does make
sense in the absence of a vacuum instability but in practice it
becomes a technically cumbersome problem. This is due to the presence
of a considerable number of negative energy bound states in addition
to the continuum states\cite{Se92}. Therefore, it seems advisable to
use a simpler computational scheme to obtain a numerical estimate.
This will allow us to see whether or not the elimination of the ghosts
induces dramatic changes in the already satisfactory description of
nuclear properties. In this work we choose to keep the full Hartree
equations for the valence part but employ a semiclassical
approximation for the Dirac sea. This is in fact the standard
procedure~\cite{Se86,Re89,Se92}. As already mentioned, and discussed
in previous work~\cite{Ca96a}, this expansion converges rather quickly
and therefore might be reliably used to estimate the sea energy up to
possible corrections due to shell effects.
The paper is organized as follows. In section~\ref{II} we present the
$\sigma$-$\omega$ model of nuclei in the $1/N$ leading approximation,
the semiclassical treatment of the Dirac sea, the renormalization
prescriptions and the different parameter fixing schemes that we will
consider. In section~\ref{III} we discuss the vacuum instability
problem of the model and Redmond's proposal. We also study the
implications of the ghost subtraction on the low momentum effective
parameters. In section~\ref{IV} we present our numerical results for
the parameters as well as binding energies and mean quadratic charge
radii of some closed-shell nuclei. Our conclusions are presented in
section~\ref{V}. Explicit expressions for the zero momentum
renormalized meson self energies and related formulas are given in the
appendix.
\section{$\sigma$-$\omega$ model of nuclei}
\label{II}
In this section we revise the $\sigma$-$\omega$ model description of
finite nuclei disregarding throughout the instability problem; this
will be considered in the next section. The Dirac sea corrections are
included at the semiclassical level and renormalization issues as well
as the various ways of fixing the parameters of the model are also
discussed here.
\subsection{Field theoretical model}
Our starting point is the Lagrangian density of the $\sigma$-$\omega$
model~\cite{Wa74,Se86,Re89,Se92} given by
\begin{eqnarray}
{\cal L}(x) &=& \overline\Psi(x) \left[ \gamma_\mu ( i \partial^\mu - g_v
V^\mu(x)) - (M - g_s \phi(x)) \right] \Psi(x)
+ {1\over 2}\, (\partial_\mu \phi(x)
\partial^\mu \phi(x) - m_s^2 \, \phi^2(x)) \nonumber \\
& & - {1\over 4} \, F_{\mu \nu}(x) F^{\mu \nu}(x)
+ {1\over 2} \, m_v^2 V_\mu(x) V^\mu(x) + \delta{\cal L}(x)\,.
\label{lagrangian}
\end{eqnarray}
$\Psi(x)$ is the isospinor nucleon field, $\phi(x)$ the scalar field,
$V_\mu(x)$ the $\omega$-meson field and $F_{\mu\nu} =\partial_\mu
V_\nu-\partial_\nu V_\mu$. In the former expression the necessary
counterterms required by renormalization are accounted for by the
extra Lagrangian term $\delta{\cal L}(x)$ (including meson
self-couplings).
Including Dirac sea corrections requires to take care of
renormalization issues. The best way of doing this in the present
context is to use an effective action formalism. Further we have to
specify the approximation scheme. The effective action will be
computed at lowest order in the $1/N$ expansion, $N$ being the number
of nucleon species (with $g_s$ and $g_v$ of order $1/\sqrt{N}$), that
is, up to one fermion loop and tree level for bosons~\cite{TB92}. This
corresponds to the Hartree approximation for fermions including the
Dirac sea~\cite{NO88}.
In principle, the full effective action would have to be computed by
introducing bosonic and fermionic sources. However,
since we will consider only stationary situations, we do not
need to introduce fermionic sources. Instead, we will
proceed as usual by integrating out exactly the fermionic degrees of
freedom. This gives directly the bosonic effective action at leading
order in the $1/N$ expansion:
\begin{equation}
\Gamma[\phi,V]= \Gamma_B[\phi,V]+\Gamma_F[\phi,V] \,,
\end{equation}
where
\begin{equation}
\Gamma_B[\phi,V]=\int\left({1\over 2}\, (\partial_\mu \phi
\partial^\mu \phi - m_s^2 \, \phi^2) - {1\over 4} \, F_{\mu \nu} F^{\mu \nu}
+ {1\over 2} \, m_v^2 V_\mu V^\mu \right) d^4x \,,
\label{GammaB}
\end{equation}
and
\begin{equation}
\Gamma_F[\phi,V]= -i\log {\rm Det}\left[ \gamma_\mu ( i \partial^\mu - g_v
V^\mu) - (M - g_s \phi) \right] +\int\delta{\cal L}(x)d^4x \,.
\end{equation}
The fermionic determinant can be computed perturbatively, by adding up
the one-fermion loop amputated graphs with any number of bosonic legs,
using a gradient expansion or by any other technique. The ultraviolet
divergences are to be canceled with the counterterms by using any
renormalization scheme; all of them give the same result after fitting
to physical observables.
The effective action so obtained is uniquely defined and completely
finite. However, there still remains the freedom to choose different
variables to express it. We will work with fields renormalized at zero
momentum. That is, the bosonic fields $\phi(x)$ and $V_\mu(x)$ are
normalized so that their kinetic energy term is the canonical
one. This is the choice shown above in $\Gamma_B[\phi,V]$. Other usual
choice is the on-shell one, namely, to rescale the fields so that the
residue of the propagator at the meson pole is unity. Note that the
Lagrangian mass parameters $m_s$ and $m_v$ do not correspond to the
physical masses (which will be denoted $m_\sigma$ and $m_\omega$ in
what follows) since the latter are defined as the position of the
poles in the corresponding propagators. The difference comes from the
fermion loop self energy in $\Gamma_F[\phi,V]$ that contains terms
quadratic in the boson fields with higher order gradients.
Let us turn now to the fermionic contribution, $\Gamma_F[\phi,V]$. We
will consider nuclear ground states of spherical nuclei, therefore the
space-like components of the $\omega$-meson field vanish \cite{Se92}
and the remaining fields, $\phi(x)$ and $V_0(x)$ are stationary. As
it is well-known, for stationary fields the fermionic energy, i.e.,
minus the action $\Gamma_F[\phi,V]$ per unit time, can be formally written
as the sum of single particle energies of the fermion moving in the
bosonic background~\cite{NO88},
\begin{equation}
E_F[\phi,V_0] = \sum_n E_n \,,
\end{equation}
and
\begin{equation}
\left[ -i {\bf \alpha} \cdot \nabla + g_vV_0(x) + \beta (
M-g_s\phi(x)) \right] \psi_n(x) = E_n \,\psi_n(x)\,.
\label{Dirac-eq}
\end{equation}
Note that what we have called the fermionic energy contains not only
the fermionic kinetic energy, but also the potential energy coming
from the interaction with the bosons.
The orbitals, and thus the fermionic energy, can be divided into
valence and sea, i.e., positive and negative energy orbitals. In
realistic cases there is a gap in the spectrum which makes such a
separation a natural one. The valence energy is therefore given by
\begin{equation}
E_F^{\rm val}[\phi,V] = \sum_n E_n^{\rm val}\,.
\end{equation}
On the other hand, the sea energy is ultraviolet divergent and
requires the renormalization mentioned above~\cite{Se86}. The (at zero
momentum) renormalized sea energy is known in a gradient or
semiclassical expansion up to fourth order and is given by~\cite{Ca96a}
\begin{eqnarray}
E^{\rm sea}_0 & = & -{\gamma\over 16\pi^2} M^4 \int d^3x \,
\Biggl\{ \Biggr.
\left({\Phi\over M}\right)^4 \log {\Phi\over M}
+ {g_s\phi\over M} - {7\over 2} \left({g_s \phi\over M}\right)^2
+ {13\over 3} \left({g_s \phi\over M}\right)^3 - {25\over 12}
\left({g_s \phi\over M}\right)^4 \Biggl. \Biggr\} \nonumber\\
E^{\rm sea}_2 & = & {\gamma \over 16 \pi^2} \int d^3 x \,
\Biggl\{ {2 \over 3} \log{\Phi\over M} (\nabla V)^2
- \log{\Phi\over M} (\nabla \Phi)^2 \Biggr\} \nonumber\\
E^{\rm sea}_4 & = & {\gamma\over 5760 \pi^2} \int d^3 x \,
\Biggl\{ \Biggr. -11\,\Phi^{-4} (\nabla \Phi)^4
- 22\,\Phi^{-4} (\nabla V)^2(\nabla \Phi)^2
+ 44 \, \Phi^{-4} \bigl( (\nabla_i \Phi) (\nabla_i V) \bigr)^2
\nonumber\\
& & \quad - 44 \, \Phi^{-3} \bigl( (\nabla_i \Phi) (\nabla_i V)
\bigr) (\nabla^2 V)
- 8 \, \Phi^{-4} (\nabla V)^4 + 22 \, \Phi^{-3} (\nabla^2 \Phi)
(\nabla \Phi)^2
\nonumber\\
& & \quad
+ 14 \, \Phi^{-3} (\nabla V)^2 (\nabla^2 \Phi)
- 18 \, \Phi^{-2} (\nabla^2 \Phi)^2 + 24 \, \Phi^{-2} (\nabla^2 V)^2
\Biggl. \Biggr\}\,.
\label{Esea}
\end{eqnarray}
Here, $V=g_vV_0$, $\Phi=M-g_s\phi$ and $\gamma$ is the spin and
isospin degeneracy of the nucleon, i.e., $2N$ if there are $N$ nucleon
species (in the real world $N=2$). The sea energy is obtained by
adding up the terms above. The fourth and higher order terms are
ultraviolet finite as follows from dimensional counting. The first two
terms, being renormalized at zero momentum, do not contain operators
with dimension four or less, such as $\phi^2$, $\phi^4$, or $(\nabla
V)^2$, since they are already accounted for in the bosonic term
$\Gamma_B[\phi,V]$. Note that the theory has been renormalized so that
there are no three- or four-point bosonic interactions in the
effective action at zero momentum~\cite{Se86}.
By definition, the true value of the classical fields
(i.e., the value in the absence of external
sources) is to be found by minimization of the
effective action or, in the stationary case, of the energy
\begin{equation}
E[\phi,V] = E_B[\phi,V]+E_F^{\rm val}[\phi,V] + E_F^{\rm sea}[\phi,V] \,.
\end{equation}
Such minimization yields the equations of motion for the bosonic
fields:
\begin{eqnarray}
(\nabla^2-m_s^2)\phi(x) &=& -g_s\left[\rho_s^{\rm val}(x)+
\rho_s^{\rm sea}(x)\right] \,, \nonumber \\
(\nabla^2-m_v^2)V_0(x) &=& -g_v\left[\rho^{\rm val}(x)+
\rho^{\rm sea}(x) \right] \,.
\label{Poisson-eq}
\end{eqnarray}
Here, $\rho_s(x)=\langle\overline\Psi(x)\Psi(x)\rangle$
is the scalar density and $\rho(x)=\langle\Psi^\dagger(x)
\Psi(x)\rangle$ the baryonic one:
\begin{eqnarray}
\rho_s^{\rm val~(sea)}(x) &=& -\frac{1}{g_s}\frac{\delta E_F^{\rm
val~(sea)}}{\delta \phi(x)}\,, \nonumber \\
\rho^{\rm val~(sea)}(x) &=& +\frac{1}{g_v}\frac{\delta E_F^{\rm
val~(sea)}}{\delta V_0(x)} \,.
\label{densities}
\end{eqnarray}
The set of bosonic and fermionic equations, eqs.~(\ref{Poisson-eq})
and (\ref{Dirac-eq}) respectively, are to be solved self-consistently.
Let us remark that treating the fermionic sea energy using a gradient
or semiclassical expansion is a further approximation on top of the
mean field approximation since it neglects possible shell effects in
the Dirac sea. However, a direct solution of the mean field equations
including renormalization of the sum of single-particle energies would
not give a physically acceptable solution due to the presence of
Landau ghosts. They will be considered in the next section.
At this point it is appropriate to make some comments on
renormalization. As we have said, one can choose different
normalizations for the mesonic fields and there are also several sets
of mesonic masses, namely, on-shell and at zero momentum. If one were
to write the mesonic equations of motion directly, by similarity with
a classical treatment, there would be an ambiguity as to which set
should be used. The effective action treatment makes it clear that the
mesonic field and masses are those at zero momentum. On the other
hand, since we have not included bosonic loops, the fermionic
operators in the Lagrangian are not renormalized and there are no
proper vertex corrections. Thus the nucleon mass $M$, the nuclear
densities $\langle\Psi\overline\Psi\rangle$ and the combinations
$g_s\phi(x)$ and $g_v V_\mu(x)$ are fixed unambiguously in the
renormalized theory. The fermionic energy $E_F[\phi,V]$, the
potentials $\Phi(x)$ and $V(x)$ and the nucleon single particle
orbitals are all free from renormalization ambiguities at leading
order in $1/N$.
\subsection{Fixing of the parameters}
The $\sigma$-$\omega$ and related theories are effective models of
nuclear interaction, and hence their parameters are to be fixed to
experimental observables within the considered approximation. Several
procedures to perform the fixing can be found in the literature
\cite{Ho81,Re89,Se92}; the more sophisticated versions try to adjust,
by minimizing the appropriate $\chi^2$ function, as many experimental
values as possible through the whole nuclear table \cite{Re89}. These
methods are useful when the theory implements enough physical elements
to provide a good description of atomic nuclei. The particular model
we are dealing with can reproduce the main features of nuclear force,
such as saturation and the correct magic numbers; however it lacks
many of the important ingredients of nuclear interaction, namely
Coulomb interaction and $\rho$ and $\pi$ mesons. Therefore, we will
use the simple fixing scheme proposed in ref.~\cite{Ho81} for
this model.
Initially there are five free parameters: the nucleon mass ($M$), two
boson Lagrangian masses ($m_s$ and $m_v$) and the corresponding
coupling constants ($g_s$ and $g_v$). The five physical observables to
be reproduced are taken to be the physical nucleon mass, the physical
$\omega$-meson mass $m_\omega$, the saturation properties of nuclear
matter (binding energy per nucleon $B/A$ and Fermi momentum $k_F$) and
the mean quadratic charge radius of $^{40}$Ca. In our approximation,
the equation of state of nuclear matter at zero temperature, and hence
its saturation properties, depends only on the nucleon mass and on
$m_{s,v}$ and $g_{s,v}$ through the combinations~\cite{Se86}
\begin{equation}
C_s^2 = g_s^2 \frac{M^2}{m_s^2}\,, \qquad
C_v^2 = g_v^2 \frac{M^2}{m_v^2}\,.
\end{equation}
At this point, there still remain two parameters to be fixed, e.g.,
$m_v$ and $g_s$. Now we implement the physical $\omega$-meson mass
constraint. From the expression of the $\omega$ propagator at the
leading $1/N$ approximation, we can obtain the value of the physical
$\omega$ pole as a function of the Lagrangian parameters $M$, $g_v$
and $m_v$ or more conveniently as a function of $M$, $C_v$ and $m_v$
(see appendix). Identifying the $\omega$ pole and the physical
$\omega$ mass, and given that $M$ and $C_v$ have already been fixed,
we obtain the value of $m_v$. Finally, the value of $g_s$ is adjusted
to fit the mean quadratic charge radius of $^{40}$Ca. We will refer to
this fixing procedure as the {\em $\omega$-shell scheme}: the name
stresses the correct association between the pole of the
$\omega$-meson propagator and the physical $\omega$ mass. The above
fixing procedure gives different values of $m_s$ and $g_s$ depending
on the order at which the Dirac sea energy is included in the
semiclassical expansion (see section~\ref{IV}).
Throughout the literature the standard fixing procedure when the Dirac
sea is included has been to give to the Lagrangian mass $m_v$ the
value of the physical $\omega$ mass \cite{Re89,Se92} (see, however,
refs.~\cite{Ta91,Ca96a}). Of course, this yields a wrong value for the
position of the $\omega$-meson propagator pole, which is
underestimated. We will refer to this procedure as the {\em naive
scheme}. Note that when the Dirac sea is not included at all, the
right viewpoint is to consider the theory at tree level, and the
$\omega$-shell and the naive schemes coincide.
\section{Landau instability subtraction}
\label{III}
As already mentioned, the $\sigma$-$\omega$ model, and more generally
any Lagrangian which couples bosons with fermions by means of a
Yukawa-like coupling, exhibits a vacuum instability~\cite{Pe87,Co87}.
This instability prevents the actual calculation of physical
quantities beyond the mean field valence approximation in a systematic
way. Recently, however, a proposal by Redmond~\cite{Re58} that
explicitly eliminates the Landau ghost has been implemented to
describe relativistic nuclear matter in a series of
papers~\cite{Ta90,Ta91,TB92}. The main features of such method are
contained already in the original papers and many details have also
been discussed. For the sake of clarity, we outline here the method as
applies to the calculation of Dirac sea effects for closed-shell
finite nuclei.
\subsection{Landau instability}
Since the Landau instability shows up already at zero nuclear density,
we will begin by considering the vacuum of the $\sigma$-$\omega$
theory. On a very general basis, namely, Poincar\'e invariance,
unitarity, causality and uniqueness of the vacuum state, one can show
that the two point Green's function (time ordered product) for a
scalar field admits the K\"all\'en-Lehmann representation~\cite{BD65}
\begin{eqnarray}
D(x'-x) = \int\,d\mu^2\rho(\mu^2)\,D_0(x'-x\, ;\mu^2)\,,
\label{KL}
\end{eqnarray}
where the full propagator in the vacuum is
\begin{eqnarray}
D(x'-x) = -i \langle
0|T\phi(x')\phi(x)|0\rangle\,,
\end{eqnarray}
and the free propagator reads
\begin{eqnarray}
D_0(x'-x\,;\mu^2) = \int\,{d^4p\over (2\pi)^4}{ e^{-ip(x'-x)}\over
p^2-\mu^2+i\epsilon }\,.
\end{eqnarray}
The spectral density $\rho(\mu^2)$ is
defined as
\begin{eqnarray}
\rho(q^2) = (2\pi)^3\sum_n\delta^4(p_n-q)|\langle
0|\phi(0)|n\rangle|^2\,.
\end{eqnarray}
It is non negative, Lorentz invariant and vanishes for space-like four
momentum $q$.
The K\"all\'en-Lehmann representability is a necessary condition for
any acceptable theory, yet it is violated by the $\sigma$-$\omega$
model when the meson propagators are approximated by their leading
$1/N$ term. It is not clear whether this failure is tied to the theory
itself or it is an artifact of the approximation ---it is well-known
that approximations to the full propagator do not necessarily preserve
the K\"all\'en-Lehmann representability---. The former possibility
would suppose a serious obstacle for the theory to be a reliable one.
In the above mentioned approximation, eq.~(\ref{KL}) still holds both for the
$\sigma$ and the $\omega$ cases (in the latter case with obvious
modification to account for the Lorentz structure)
but the spectral density gets modified to be
\begin{eqnarray}
\rho(\mu^2) = \rho^{\rm KL}(\mu^2) - R_G\delta(\mu^2+M_G^2)
\end{eqnarray}
where $\rho^{\rm KL}(\mu^2)$ is a physically acceptable spectral
density, satisfying the general requirements of a quantum field
theory. On the other hand, however, the extra term spoils these
general principles. The residue $-R_G$ is negative, thus indicating
the appearance of a Landau ghost state which contradicts the usual
quantum mechanical probabilistic interpretation. Moreover, the delta
function is located at the space-like squared four momentum $-M_G^2$
indicating the occurrence of a tachyonic instability. As a
perturbative analysis shows, the dependence of $R_G$ and $M_G$ with
the fermion-meson coupling constant $g$ in the weak coupling regime is
$R_G\sim g^{-2}$ and $M_G^2 \sim 4M^2\exp(4\pi^2/g^2)$, with $M$ the
nucleon mass. Therefore the perturbative content of $\rho(\mu^2)$ and
$\rho^{\rm KL}(\mu^2)$ is the same, i.e., both quantities coincide
order by order in a power series expansion of $g$ keeping $\mu^2$
fixed. This can also be seen in the propagator form of the previous
equation
\begin{eqnarray}
D(p) = D^{\rm KL}(p) - {R_G\over p^2+M_G^2}\,.
\label{Delta}
\end{eqnarray}
For fixed four momentum, the ghost term vanishes as
$\exp(-4\pi^2/g^2)$ when the coupling constant goes to zero. As noted
by Redmond~\cite{Re58}, it is therefore possible to modify the theory
by adding a suitable counterterm to the action that exactly cancels
the ghost term in the meson propagator without changing the
perturbative content of the theory. In this way the full meson
propagator becomes $D^{\rm KL}(p)$ which is physically acceptable and
free from vacuum instability at leading order in the $1/N$ expansion.
It is not presently known whether the stability problems of the
original $\sigma$-$\omega$ theory are intrinsic or due to the
approximation used, thus Redmond's procedure can be interpreted either
as a fundamental change of the theory or as a modification of the
approximation scheme. Although both interpretations use the
perturbative expansion as a constraint, it is not possible, at the
present stage, to decide between them. It should be made quite clear
that in spite of the seemingly arbitrariness of the no-ghost
prescription, the original theory itself was ambiguous regarding its
non perturbative regime. In fact, being a non asymptotically free
theory, it is not obvious how to define it beyond finite order
perturbation theory. For the same reason, it is not Borel summable and
hence additional prescriptions are required to reconstruct the Green's
functions from perturbation theory to all orders. As an example, if
the nucleon self energy is computed at leading order in a $1/N$
expansion, the existence of the Landau ghost in the meson propagator
gives rise to a pole ambiguity. This is unlike physical time-like
poles, which can be properly handled by the customary $+i\epsilon$
rule, and thus an additional ad hoc prescription is needed. This
ambiguity reflects in turn in the Borel transform of the perturbative
series; the Borel transform presents a pole, known as renormalon in
the literature~\cite{Zi79}. In recovering the sum of the perturbative
series through inverse Borel transformation a prescription is then
needed, and Redmond's proposal provides a particular suitable way of
fixing such ambiguity. Nevertheless, it should be noted that even if
Redmond's prescription turns out to be justified, there still remains
the problem of how to extend it to the case of three- and more point
Green's functions, since the corresponding K\"all\'en-Lehmann
representations has been less studied.
\subsection{Instability subtraction}
To implement Redmond's prescription in detail we start with the
zero-momentum renormalized propagator in terms of the proper
self-energy for the scalar field (a similar construction can be
carried out for the vector field as well),
\begin{eqnarray}
D_s(p^2) = (p^2-m_s^2 - \Pi_s(p^2))^{-1}\,,
\end{eqnarray}
where the $m_s$ is the zero-momentum meson mass and the corresponding
renormalization conditions are $\Pi_s(0)= \Pi_s^\prime(0)=0$. The
explicit formulas for the scalar and vector meson self energies are
given in the appendix. Of course, $D_s(p^2)$ is just the inverse of
the quadratic part of the effective action $K_s(p^2)$. According to
the previous section, the propagator presents a tachyonic pole. Since
the ghost subtraction is performed at the level of the two-point
Green's function, it is clear that the corresponding Lagrangian
counterterm must involve a quadratic operator in the mesonic fields.
The counterterm kernel $\Delta K_s(p^2)$ must be such that cancels the
ghost term in the propagator $D_s(p^2)$ in eq.~(\ref{Delta}). The
subtraction does not modify the position of the physical meson pole
nor its residue, but it will change the zero-momentum parameters and
also the off-shell behavior. Both features are relevant to nuclear
properties. This will be discussed further in the next section.
Straightforward calculation yields
\begin{eqnarray}
\Delta K_s(p^2) =
-{1\over D_s(p^2)}{R_G^s\over R_G^s+(p^2+{M^s_G}^2)D_s(p^2)} \,.
\label{straightforward}
\end{eqnarray}
As stated, this expression vanishes as $\exp(-4\pi^2/g_s^2)$ for small
$g_s$ at fixed momentum. Therefore it is a genuine non perturbative
counterterm. It is also non local as it depends in a non polynomial
way on the momentum. In any case, it does not introduce new
ultraviolet divergences at the one fermion loop level. However, it is
not known whether the presence of this term spoils any general
principle of quantum field theory.
Proceeding in a similar way with the $\omega$-field $V_\mu(x)$, the
following change in the total original action is induced
\begin{eqnarray}
\Delta S = {1\over 2}\int{d^4p\over (2\pi)^4}\phi(-p)
\Delta K_s(p^2)\phi(p) - {1\over 2}\int{d^4p\over
(2\pi)^4}V_\mu(-p)\Delta K^{\mu\nu}_v(p^2)V_\nu(p) \,,
\label{Delta S}
\end{eqnarray}
where $\phi(p)$ and $V_\mu(p)$ are the Fourier transform of the scalar
and vector fields in coordinate space, $\phi(x)$ and $V_\mu(x)$
respectively. Note that at tree-level for bosons, as we are
considering throughout, this modification of the action is to be added
directly to the effective action ---in fact, this is the simplest way
to derive eq.~(\ref{straightforward})---.
Therefore, in the case of static fields, the total mean field energy
after ghost elimination reads
\begin{equation}
E= E_F^{\rm val} + E_F^{\rm sea} + E_B + \Delta E \,,
\end{equation}
where $E_F^{\rm val}$, $E_F^{\rm sea}$ and $E_B$ were given in
section~\ref{II} and
\begin{equation}
\Delta E[\phi,V] = {1\over 2}\int\,d^3x\phi(x)
\Delta K_s(\nabla^2)\phi(x) - {1\over 2}\int\,d^3x
V_0(x)\Delta K^{00}_v(\nabla^2)V_0(x) \,.
\end{equation}
One can proceed by minimizing the mean field total energy as a
functional of the bosonic and fermionic fields. This yields the usual
set of Dirac equations for the fermions, eqs.~(\ref{Dirac-eq}) and
modifies the left-hand side of the bosonic eqs.~(\ref{Poisson-eq}) by
adding a linear non-local term. This will be our starting point to
study the effect of eliminating the ghosts in the description of
finite nuclei. We note that the instability is removed at the
Lagrangian level, i.e., the non-local counterterms are taken to be new
terms of the starting Lagrangian which is then used to describe the
vacuum, nuclear matter and finite nuclei. Therefore no new
prescriptions are needed in addition to Redmond's to specify how the
vacuum and the medium parts of the effective action are modified by
the removal of the ghosts.
So far, the new counterterms, although induced through the Yukawa
coupling with fermions, have been treated as purely bosonic terms.
Therefore, they do not contribute directly to bilinear fermionic
operators such as baryonic and scalar densities. An alternative
viewpoint would be to take them rather as fermionic terms, i.e., as a
(non-local and non-perturbative) redefinition of the fermionic
determinant. The energy functional, and thus the mean field equations
and their solutions, coincide in the bosonic and fermionic
interpretations of the new term, but the baryonic densities and
related observables would differ, since they pick up a new
contribution given the corresponding formulas similar to
eqs.~(\ref{densities}). Ambiguities and redefinitions are ubiquitous
in quantum field theories, due to the well-known ultraviolet
divergences. However, in well-behaved theories the only freedom
allowed in the definition of the fermionic determinant comes from
adding counterterms which are local and polynomial in the
fields. Since the new counterterms induced by Redmond's method are not
of this form, we will not pursue such alternative point of view in
what follows. Nevertheless, a more compelling argument would be needed
to make a reliable choice between the two possibilities.
\subsection{Application to finite nuclei}
In this section we will take advantage of the smooth behavior of the
mesonic mean fields in coordinate space which allows us to apply a
derivative or low momentum expansion. The quality of the gradient
expansion can be tested a posteriori by a direct computation. The
practical implementation of this idea consists of treating the term
$\Delta S$ by expanding each of the kernels $\Delta K(p^2)$ in a power
series of the momentum squared around zero
\begin{equation}
\Delta K(p^2) = \sum_{n\ge 0}\Delta K_{2n}\, p^{2n}\,.
\end{equation}
The first two terms are given explicitly by
\begin{eqnarray}
\Delta K_0 & = & -\frac{m^4R_G}{M_G^2-m^2R_G} ,\nonumber\\
\Delta K_2 & = & \frac{m^2 R_G(m^2-m^2R_G+2M_G^2)}{(M_G^2-m^2R_G)^2}.
\end{eqnarray}
The explicit expressions of the tachyonic pole
parameters $M_G$ and $R_G$ for each meson can be found below.
Numerically, we have found that the fourth and higher orders in this
gradient expansion are negligible as compared to zeroth- and
second orders. In fact, in ref.~\cite{Ca96a} the same behavior was
found for the correction to the Dirac sea contribution to the binding
energy of a nucleus. As a result, even for light nuclei, $E_4^{\rm
sea}$ in eq.~(\ref{Esea}) can be safely neglected. Furthermore, it has
been shown~\cite{Ca96} that the fourth order term in the gradient
expansion of the valence energy, if treated semiclassically, is less
important than shell effects. So, it seems to be a general rule that,
for the purpose of describing static nuclear properties, only the two
lowest order terms of a gradient expansion need to be considered. We
warn, however, that the convergence of the gradient or semiclassical
expansion is not the same as converging to the exact mean field
result, since there could be shell effects not accounted for by this
expansion at any finite order. Such effects, certainly exist in the
valence part~\cite{Ca96}. Even in a seemingly safe case as infinite
nuclear matter, where only the zeroth order has a non vanishing
contribution, something is left out by the gradient expansion since
the exact mean field solution does not exist due to the Landau ghost
instability (of course, the situation may change if the Landau pole is
removed). In other words, although a gradient expansion might appear
to be exact in the nuclear matter case, it hides the very existence of
the vacuum instability.
From the previous discussion it follows that the whole effect of the
ghost subtraction is represented by adding a term $\Delta S$ to the
effective action with same form as the bosonic part of the original
theory, $\Gamma_B[\phi,V]$ in eq.~(\ref{GammaB}). This amounts to a
modification of the zero-momentum parameters of the effective
action. The new zero-momentum scalar field (i.e., with canonical
kinetic energy), mass and coupling constant in terms of those of the
original theory are given by
\begin{eqnarray}
{\wh\phi}(x) &=& (1+\Delta K^s_2)^{1/2}\phi(x)\,, \nonumber \\
\wh{m}_s &=& \left(\frac{m_s^2-\Delta K^s_0}{1+\Delta
K^s_2}\right)^{1/2} \,, \\
\wh{g}_s &=& (1+\Delta K^s_2)^{-1/2}g_s \,. \nonumber
\end{eqnarray}
The new coupling constant is obtained recalling that $g_s\phi(x)$
should be invariant. Similar formulas hold for the vector meson. With
these definitions (and keeping only $\Delta K_{s,v}(p^2)$ till second
order in $p^2$) one finds\footnote{Note that $E_{B,F}[~]$ refer to the
functionals (the same at both sides of the equations) and not to their
value as is also usual in physics literature.}
\begin{eqnarray}
E_B[\wh{\phi},\wh{V};\wh{m}_s,\wh{m}_v] &=&
E_B[\phi,V; m_s,m_v] + \Delta E[\phi,V; m_s,m_v] \,, \\
E_F[\wh{\phi},\wh{V};\wh{g}_s,\wh{g}_v] &=&
E_F[\phi,V;g_s,g_v] \,. \nonumber
\end{eqnarray}
The bosonic equations for the new meson fields after ghost removal are
hence identical to those of the original theory using
\begin{eqnarray}
\wh{m}^2 &=& m^2 \, M_G^2 \, \frac{M_G^2 - m^2\, R_G}{M_G^4 + m^4 \, R_G }
\,,\nonumber \\
\wh{g}^2 &=& g^2 \, \frac{(M_G^2 - m^2\, R_G)^2}{M_G^4 + m^4 \, R_G } \,,
\label{mg}
\end{eqnarray}
as zero-momentum masses and coupling constants respectively. In the
limit of large ghost masses or vanishing ghost residues, the
reparameterization becomes trivial, as it should be. Let us note that
although the zero-momentum parameters of the effective action
$\wh{m}_{s,v}$ and $\wh{g}_{s,v}$ are the relevant ones for nuclear
structure properties, the parameters $m_{s,v}$ and $g_{s,v}$ are the
(zero-momentum renormalized) Lagrangian parameters and they are also
needed, since they are those appearing in the Feynman rules in a
perturbative treatment of the model. Of course, both sets of parameters
coincide when the ghosts are not removed or if there were no ghosts in
the theory.
To finish this section we give explicitly the fourth order
coefficient in the gradient expansion of $\Delta E$, taking into account
the rescaling of the mesonic fields, namely,
\begin{equation}
\frac{\Delta K_4}{1+\Delta K_2}=
-\frac{R_G (M_G^2 + m^2)^2}{
(M_G^4 + m^4 \, R_G)\, (M_G^2 - m^2 \, R_G)}
- \frac{\gamma g^2}{\alpha \, \pi^2 \, M^2} \,
\frac{m^4 \, R_G^2 - 2 \, m^2 \, M_G^2 \, R_G }
{M_G^4 + m^4 \, R_G } \,,
\end{equation}
where $\alpha$ is $160$ for the scalar meson and $120$ for the vector
meson. As already stated, for typical mesonic profiles the
contribution of these fourth order terms are found to be numerically
negligible. Simple order of magnitude estimates show that squared
gradients are suppressed by a factor $(RM_G)^{-2}$, $R$ being the
nuclear radius, and therefore higher orders can also be
neglected. That the low momentum region is the one relevant to nuclear
physics can also be seen from the kernel $K_s(p^2)$, shown in
fig.~\ref{f-real}. From eq.~(\ref{Delta S}), this kernel is to be
compared with the function $\phi(p)$ that has a width of the order of
$R^{-1}$. It is clear from the figure that at this scale all the
structure of the kernel at higher momenta is irrelevant to $\Delta E$.
\subsection{Fixing of the parameters after ghost subtraction}
As noted in section \ref{II}, the equation of state at zero
temperature for nuclear matter depends only on the dimensionless
quantities $C^2_s$ and $C_v^2$, that now become
\begin{equation}
C_s^2 = \wh{g}_s^2 \, \frac{M^2}{\wh{m}_s^2}\,,\qquad
C_v^2 = \wh{g}_v^2 \, \frac{M^2}{\wh{m}_v^2}\,.
\label{CsCv}
\end{equation}
Fixing the saturation density and binding energy to their observed
values yields, of course, the same numerical values for $C_s^2$ and
$C_v^2$ as in the original theory. After this is done, all static
properties of nuclear matter are determined and thus they are
insensitive to the ghost subtraction. Therefore, at leading order in
the $1/N$ expansion, to see any effect one should study either
dynamical nuclear matter properties as done in ref.~\cite{Ta91} or
finite nuclei as we do here.
It is remarkable that if all the parameters of the model were to be
fixed exclusively by a set of nuclear structure properties, the ghost
subtracted and the original theories would be indistinguishable
regarding any other static nuclear prediction, because bosonic and
fermionic equations of motion have the same form in both
theories. They would differ however far from the zero four momentum
region where the truncation of the ghost kernels $\Delta K(p^2)$ at
order $p^2$ is no longer justified. In practice, the predictions will
change after ghost removal because the $\omega$-meson mass is quite large
and is one of the observables to be used in the fixing of the
parameters.
To fix the parameters of the theory we choose the same observables as
in section \ref{II}. Let us consider first the vector meson
parameters $\wh{m}_v$ and $\wh{g}_v$. We proceed as follows:
1. We choose a trial value for $g_v$ (the zero-momentum coupling
constant of the original theory). This value and the known physical
values of the $\omega$-meson and nucleon masses, $m_\omega$ and $M$
respectively, determines $m_v$ (the zero-momentum mass of the original
theory), namely
\begin{equation}
m_v^2 = m_\omega^2 + \frac{\gamma \, g_v^2}{8 \, \pi^2}\, M^2\,
\left\{ \frac{4}{3} + \frac{5}{9}\, \frac{m_\omega^2}{M^2}
- \frac{2}{3} \left(2 + \frac{m_\omega^2}{M^2}\right)
\sqrt{\frac{4 \, M^2}{m_\omega^2} - 1 } \,
\arcsin\left(\frac{m_\omega}{2 \, M}\right)\right\} \,.
\end{equation}
(This, as well as the formulas given below, can be deduced from those
in the appendix.)
2. $g_v$ and $m_v$ provide the values of the tachyonic
parameters $R_G^v$ and $M_G^v$. They are given by
\begin{eqnarray}
M_G^v &=& \frac{2M}{\sqrt{\kappa_v^2-1}} \nonumber \\
\frac{1}{R_G^v} &=& -1 + \frac{\gamma \, g_v^2}{24\,\pi^2}
\left\{ \left(\frac{\kappa_v^3}{4} + \frac{3}{4\,\kappa_v}\right)
\log \frac{\kappa_v + 1}{\kappa_v - 1} - \frac{\kappa_v^2}{2} -
\frac{1}{6}\right\}\,,
\label{RGv}
\end{eqnarray}
where the quantity $\kappa_v$ is the real solution of the following
equation (there is an imaginary solution which corresponds to the
$\omega$-meson pole)
\begin{equation}
1 + \frac{m_v^2}{4 \, M^2} \, (\kappa_v^2-1)
+\frac{\gamma \, g_v^2}{24\,\pi^2}
\left\{ \left(\frac{\kappa_v^3}{2} - \frac{3\, \kappa_v}{2}\right)
\log \frac{\kappa_v + 1}{\kappa_v - 1} - \kappa_v^2 +
\frac{8}{3}\right\} =0 \,.
\label{kappav}
\end{equation}
3. Known $g_v$, $m_v$, $M_G^v$ and $R_G^v$, the values of $\wh{m}_v$ and
$\wh{g}_v$ are obtained from eqs.~(\ref{mg}). They are then inserted
in eqs.~(\ref{CsCv}) to yield $C^2_v$. If necessary, the initial trial
value of $g_v$ should be readjusted so that the value of $C^2_v$ so
obtained coincides with that determined by the saturation properties
of nuclear matter.
The procedure to fix the parameters $m_s$ and $g_s$ is similar but
slightly simpler since the physical mass of the scalar meson
$m_\sigma$ is not used in the fit. Some trial values for $m_s$ and
$g_s$ are proposed. This allows to compute $M_G^s$ and $R_G^s$ by
means of the formulas
\begin{eqnarray}
M_G^s &=& \frac{2M}{\sqrt{\kappa_s^2-1}} \nonumber \\
\frac{1}{R_G^s} &=& -1 -
\frac{\gamma \, g_s^2}{16\,\pi^2}
\left\{ \left(\frac{\kappa_s^3}{2} \, - \frac{3\,\kappa_s}{2}\right)
\log \frac{\kappa_s + 1}{\kappa_s - 1}
- \kappa_s^2 + \frac{8}{3}\right\}\,,
\label{RGs}
\end{eqnarray}
where $\kappa_s$ is the real solution of
\begin{equation}
1 + \frac{m_s^2}{4 \, M^2} \, (\kappa_s^2-1)
- \frac{\gamma \, g_s^2}{16\,\pi^2}
\left\{ \kappa_s^3 \, \log \frac{\kappa_s + 1}{\kappa_s - 1}
- 2 \, \kappa_s^2 - \frac{2}{3}\right\} = 0\,.
\label{kappas}
\end{equation}
One can then compute $\wh{m}_s$ and $\wh{g}_s$ and thus $C_s^2$ and
the mean quadratic charge radius of $^{40}$Ca. The initial values of
$m_s$ and $g_s$ should be adjusted to reproduce these two quantities.
We will refer to the set of masses and coupling constants so obtained
as the {\em no-ghost scheme} parameters.
\section{Numerical results and discussion}
\label{IV}
As explained in section~\ref{II}, the parameters of the theory are
fitted to five observables. For the latter we take the following
numerical values: $M=939$~MeV, $m_\omega=783$~MeV, $B/A=15.75$~MeV,
$k_F=1.3$~fm$^{-1}$ and $3.82$~fm for the mean quadratic charge radius
of $^{40}$Ca.
If the Dirac sea is not included at all, the numerical values that we
find for the nuclear matter combinations $C_s^2$ and $C_v^2$ are
\begin{equation}
C_s^2 = 357.7\,, \qquad C_v^2 = 274.1
\end{equation}
The corresponding Lagrangian parameters are shown in
table~\ref{t-par-1}. There we also show $m_\sigma$ and $m_\omega$ that
correspond to the position of the poles in the propagators after
including the one-loop meson self energy. They are an output of the
calculation and are given for illustration purposes.
When the Dirac sea is included, nuclear matter properties fix the
following values
\begin{equation}
C_s^2 = 227.8\,, \qquad C_v^2 = 147.5
\end{equation}
Note that in nuclear matter only the zeroth order $E_0^{\rm sea}$ is
needed in the gradient expansion of the sea energy, since the meson
fields are constant. The (zero momentum renormalized) Lagrangian meson
masses $m_{s,v}$ and coupling constants $g_{s,v}$ are shown in
table~\ref{t-par-1} in various schemes, namely, $\omega$-shell,
no-ghost and naive schemes, previously defined. The scalar meson
parameters differ if the Dirac sea energy is included at zeroth order
or at all orders (in practice zeroth plus second order) in the
gradient expansion. For the sake of completeness, both possibilities
are shown in the table. The numbers in brackets in the no-ghost scheme
are the zero-momentum parameters of the effective action,
$\wh{m}_{s,v}$ and $\wh{g}_{s,v}$ (in the other schemes they coincide
with the Lagrangian parameters). Again $m_\sigma$ and $m_\omega$ refer
to the scalar and vector propagator-pole masses after including the
one fermion loop self energy for each set of Lagrangian parameters.
Table~\ref{t-par-2} shows the ghost masses and residues corresponding
to the zero-momentum renormalized propagators. The no-ghost scheme
parameters have been used.
The binding energies per nucleon (without center of mass corrections)
and mean quadratic charge radii (without convolution with the nucleon
form factor) of several closed-shell nuclei are shown in
tables~\ref{t-dat-1} and \ref{t-dat-2} for the $\omega$-shell and for
the naive and no-ghost schemes (these two schemes give the same
numbers), as well as for the case of not including the Dirac sea.
The experimental data are taken from refs.~\cite{Ja74,Va72,Wa88a}.
From table~\ref{t-par-1} it follows that the zero-momentum vector meson
mass $m_v$ in the $\omega$-shell scheme is considerably larger than the
physical mass. This is somewhat unexpected. Let us recall that the
naive treatment, which neglects the meson self energy, is the most
used in practice. It has been known for a long time~\cite{Pe86,FH88}
that the $\omega$-shell scheme is, as a matter of principle, the
correct procedure but on the basis of rough estimates it was assumed
that neglecting the meson self energy would be a good approximation
for the meson mass. We find here that this is not so.
Regarding the consequences of removing the ghost, we find in
table~\ref{t-par-1} that the effective parameters $\wh{m}_{s,v}$ and
$\wh{g}_{s,v}$ in the no-ghost scheme are similar, within a few per
thousand, to those of the naive scheme. This similarity reflects in
turn on the predicted nuclear properties: the results shown in
tables~\ref{t-dat-1} and \ref{t-dat-2} for the no-ghost scheme
coincide, within the indicated precision, with those of the naive
scheme (not shown in the table). It is amazing that the outcoming
parameters from such a sophisticated fitting procedure, namely the
no-ghost scheme, resemble so much the parameters corresponding to the
naive treatment. We believe this result to be rather remarkable for it
justifies a posteriori the nowadays traditional calculations made with
the naive scheme.
The above observation is equivalent to the fact that the zero-momentum
masses, $\wh{m}_{s,v}$, and the propagator-pole masses
$m_{\sigma,\omega}$ are very similar in the no-ghost scheme. This
implies that the effect of removing the ghosts cancels to a large
extent with that introduced by the meson self energies. Note that
separately the two effects are not small; as was noted above $m_v$ is
much larger than $m_\omega$ in the $\omega$-shell scheme. To interpret
this result, it will be convenient to recall the structure of the
meson propagators. In the leading $1/N$ approximation, there are
three kinds of states that can be created on the vacuum by the meson
fields. Correspondingly, the spectral density functions $\rho(q^2)$
have support in three clearly separated regions, namely, at the ghost
mass squared (in the Euclidean region), at the physical meson mass
squared, and above the $N\overline{N}$ pair production threshold
$(2M)^2$ (in the time-like region). The full meson propagator is
obtained by convolution of the spectral density function with the
massless propagator $(q^2+i\epsilon)^{-1}$ as follows from the
K\"alle\'en-Lehmann representation, eq.~(\ref{KL}). The large
cancelation found after removing the ghosts leads to the conclusion
that, in the zero-momentum region, most of the correction induced by
the fermion loop on the meson propagators, and thereby on the
quadratic kernels $K(p^2)$, is spurious since it is due to unphysical
ghost states rather than to virtual $N\overline{N}$ pairs. This can
also be seen from figs.~\ref{f-real} and \ref{f-imaginary}. There, we
represent the real and imaginary parts of $K_s(p^2)$ respectively, in
three cases, namely, before ghost elimination, after ghost elimination
and the free inverse propagator. In all three cases the slope of the
real part at zero momentum is equal to one and the no-ghost (sea 2nd)
set of parameters from table~\ref{t-par-1} has been used. We note the
strong resemblance of the free propagator and the ghost-free
propagator below threshold. A similar result is obtained for the
vector meson.
One may wonder how these conclusions reflect on the sea energy. Given
that we have found that most of the fermion loop is spurious in the
meson self energy it seems necessary to revise the sea energy as well
since it has the same origin. Technically, no such problem appears in
our treatment. Indeed the ghost is found in the fermion loop attached
to two meson external legs, i.e., terms quadratic in the fields.
However, the sea energy used, namely, $E_0^{\rm sea}+E_2^{\rm sea}$,
does not contain such terms. Quadratic terms would correspond to a
mass term in $E_0^{\rm sea}$ and a kinetic energy term in $E_2^{\rm
sea}$, but they are absent from the sea energy due to the
zero-momentum renormalization prescription used. On the other hand,
terms with more than two gradients were found to be
negligible~\cite{Ca96a}. Nevertheless, there still exists the
possibility of ghost-like contributions in vertex functions
corresponding to three or more mesons, similar to the spurious
contributions existing in the two-point function. In this case the
total sea energy would have to be reconsidered. The physically
acceptable dispersion relations for three or more fields have been
much less studied in the literature hence no answer can be given to
this possibility at present.
\section{Summary and conclusions}
\label{V}
We summarize our points. In the present paper, we have studied the
consequences of eliminating the vacuum instabilities which take place
in the $\sigma$-$\omega$ model. This has been done using Redmond's
prescription which imposes the validity of the K\"all\'en-Lehmann
representation for the two-point Green's functions. We have discussed
possible interpretations to such method and have given plausibility
arguments to regard Redmond's method as a non perturbative and non
local modification of the starting Lagrangian.
Numerically we have found that, contrary to the naive expectation, the
effect of including fermionic loop corrections to the mesonic
propagators ($\omega$-shell scheme) is not small. However, it largely
cancels with that of removing the unphysical Landau poles. A priori,
this is a rather unexpected result which in fact seems to justify
previous calculations carried out in the literature using a naive
scheme. Actually, as compared to that scheme and after proper
readjustment of the parameters to selected nuclear matter and finite
nuclei properties, the numerical effect becomes rather moderate on
nuclear observables. The two schemes, naive and no-ghost, are
completely different beyond the zero four momentum region, however,
and for instance predict different values for the vector meson mass.
Therefore it seems that in this model most of the fermionic loop
contribution to the meson self energy is spurious. The inclusion of
the fermionic loop in the meson propagator can only be regarded as an
improvement if the Landau ghost problem is dealt with simultaneously.
We have seen that the presence of Landau ghosts does not reflect on
the sea energy but it is not known whether there are other
spurious ghost-like contributions coming from three or higher point
vertex functions induced by the fermionic loop.
\section*{Acknowledgments}
This work is supported in part by funds provided by the U.S.
Department of Energy (D.O.E.) under cooperative research agreement
\#DF-FC02-94ER40818, Spanish DGICYT grant no. PB92-0927 and Junta de
Andaluc\'{i}a grant no. FQM0225. One of us (J.C.) acknowledges the
Spanish Ministerio de Educaci\'on y Ciencia for a research grant.
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\section{Introduction}
The inclusion of gravity into a physical theory fundamentally
alters its basic assumptions and structure.
Perhaps the most general and universal of all theories is
thermodynamics. It is therefore particularly interesting to understand
the changes induced by gravity in its underlying principles and
to organize them into a general framework that serves as the foundation
of a thermodynamic theory that incorporates strongly self-gravitating
systems. Although this is relevant in its own right, it is also
important in a somewhat different context: gravitational
thermodynamics is expected to arise as one of the macroscopic limits of
quantum gravity. Despite the fact that progress continues to be made
into the way gravity alters the structure of quantum field theories
\cite{AsLe}, there does not exist yet a complete theory of quantum
gravity. We believe that the characteristic features and principles of a
theory of gravitational thermodynamics have to be understood
properly before they can be fully justified by means of statistical
methods derived from one or several candidate theories of quantum gravity.
It is our purpose in this paper to formulate the general
principles of gravitational thermodynamics.
We shall discuss in detail the basic definitions and concepts of this
formalism, its overall structure, its fundamental problem, the minimal
set of assumptions (the postulates) that lead to the formal resolution of
this problem as well as their mathematical and physical consequences.
This formulation clarifies not only the differences and similarities
between gravitational and nongravitational thermodynamics
within a coherent framework, but also a number of existing misconceptions
concerning its logical foundations and results.
It generalizes the overall structure of thermodynamic theory and may
provide a basic framework to incorporate current and future progress
\cite{AsLe,qg} in a statistical mechanical description of
self-gravitating systems.
A clear and elegant formulation of the foundations and
structure of ordinary (that is, nongravitational) thermodynamics has
been introduced by Callen \cite{Ca}. However, this formalism cannot
describe the thermodynamics of gravitational systems in its present form
and needs to be modified. To recognize this, we shall follow the
following strategy. We believe that the most effective way to gain
insight into the limitations of a set of physical principles
is by studying a model system that illustrates them without unnecessary
complications. We introduce such a model problem in Sec. II. It is
the simplest example of a composite self-gravitating system at finite
temperature and resembles as closely as possible the textbook example of
a composite system in ordinary thermodynamics. The model is
general enough to capture all the non-trivial behavior of gravity, but
simple enough to allow an exact evaluation of all quantities, and
provides insight into the places where the thermodynamic formalism has to
be refined. As a preparatory step in the study of the postulatory basis
of thermodynamics, we show that it provides
a counterexample to a basic assertion \cite{Ca,GrCa} of ordinary
thermodynamics: in gravitational thermodynamics, additivity of entropies
applied to spatially separate subsystems does {\it not} depend on or
require the entropies of the latter to be homogeneous first-order
functions of their extensive parameters.
The general principles of gravitational thermodynamics are developed
from the start in Sec. III, which constitutes the core of the
paper. It is shown that a general and rigorous characterization of the
defining properties of thermodynamical self-gravitating systems
originates from two factors, namely (1) the particular characteristics of
their extensive variables, and (2) the homogeneous properties of their
intensive parameters as functions of extensive variables. The correct
definitions of these variables are discussed in detail.
Extensive variables include quasilocally defined quantities like
quasilocal energy and provide the background for the postulates.
We discuss the composition of thermodynamic systems, the existence of
fundamental equations, and formulate the fundamental problem of
gravitational thermodynamics. Our object is to follow
as closely as possible the logic of Callen's formalism and generalize
its basic definitions and postulates wherever it is required.
The definitions and postulates arrived at are a natural extension of
the ones of ordinary thermodynamics with appropriate modifications
necessary to accommodate the global aspects and nonlinearities
characteristic of gravity. The postulates are the basic principles of
thermodynamics when strongly self-gravitating systems are considered.
We revise completely the logic of Sec. II and show that additivity of
entropies (and, in general, additivity of all Massieu functions) as well
as the generalized first and second laws of thermodynamics are part of
these postulates and, as such, are not to be proven within the
thermodynamic formalism. We demonstrate that the fundamental problem is
formally solved by these postulates in spite of strong
interaction between constituent subsystems and the associated
nonadditivity of extensive variables. The preeminence of Massieu
functions over thermodynamic potentials is discussed. The conditions of
thermodynamical equilibrium and its (spatially) inhomogeneous nature
(equivalence principle) are a direct result of the postulates.
Although the fundamental equation and the intensive variables maintain
their mutual relationships, the former is no longer a homogeneous
first-order function of extensive parameters.
This property is not a consequence of
the inhomogeneity of equilibrium configurations but of
the functional form of the gravitational equations of state. These are
in general no longer homogeneous zeroth-order functions under rescaling
of extensive parameters. We show that these central
differences with nongravitational thermodynamics are not forbidden by
the logical structure of the formalism and can be incorporated easily
into it by relaxing several assumptions in the postulates.
However, the mathematical consequences of the new
postulates are quite different from the ones familiar in
regular thermodynamics. Formal relationships (like the Euler
equation) must be reformulated and there is no direct analogue of the
ordinary Gibbs-Duhem relation among intensive
variables. Nonetheless, this does not influence significantly the
applicability of the thermodynamic formalism.
The principles of the framework and its logic apply to general
systems at finite temperature.
They incorporate the so-called gravothermal catastrophe and other
peculiar thermodynamical behavior observed in self-gravitating objects.
As such, {\it all} standard equilibrium thermodynamics of gravitational
configurations is a consequence of the general postulates presented here.
The approach generalizes ordinary thermodynamics and provides a
consistent treatment of composite self-gravitating systems (with or
without matter fields). Moreover, it becomes evident that the
modifications in the thermodynamic formalism necessary to incorporate
gravity liberates it from assumptions that appeared (and are not)
fundamental, and as such, allows us to regard its general character and
the extent of its logic in their full measure.
\section{Homogeneity and additivity}
We motivate our presentation of the postulates of
gravitational thermodynamics and their consequences with a model
problem. It consists of a spherically symmetric uncharged black hole
surrounded by matter (represented by a thin shell).
We shall find in this section the
relationship among the entropy functions $S_0$, $S_B$, and $S_S$ whenever
thermodynamic equilibrium is achieved and its connection with the
functional dependence of those functions.
In what follows, subscripts $B$ and $S$ refer to quantities for black
hole and shell constituents respectively, whereas the subscript $0$ refers
to quantities for the total system.
The composite system is characterized by the surface area
$A_0 = 4\pi r_0^2$ of a two-dimensional spherical boundary
surface $B_0$ (located at $r=r_0$) that encloses the components, and
the quasilocal energy $E_0$ contained within \cite{MaYo,Yo}.
The Arnowitt-Deser-Misner (ADM) mass $m_+$ of the system as a
function of these variables is
\begin{equation}
m_+ (E_0, A_0) = E_0 \, \Bigg( 1 - {{E_0}\over{2r_0}} \Bigg)\ .
\label{m+}
\end{equation}
Throughout the paper, units are chosen so that
$G =\hbar = c = k_{Boltzmann} =1$. Due to spherical symmetry of the
model problem, we use areas and radii interchangeably.
The pressure $p_0$ associated to the surface $B_0$ is obtained as
(minus) the partial derivative of the energy $E_0$ with respect to $A_0$
at constant $m_+$. It corresponds to a negative surface tension and reads
\begin{equation}
p_0 (E_0, A_0) = {{{E_0}^2}\over{16 \pi {r_0}^3}}
\Bigg(1 - {{E_0}\over{r_0}} \Bigg)^{-1} = \,
{{1}\over{16\pi \, r_0 \, k_0}} (1 - k_0)^2 \ , \label{p0}
\end{equation}
where $k_0 \equiv (1 - 2m_+/r_0)^{1/2}$.
Let $\beta_0$ denote the inverse temperature
at the surface $B_0$. The first law of thermodynamics for the system
reads
\begin{equation}
dS_0 = \beta_0 \, (dE_0 + p_0 \, dA_0) \ ,
\end{equation}
which can be written as a total differential by using Eqs. (\ref{m+})
and (\ref{p0}) as
\begin{equation}
dS_0 = \beta_0 \, {k_0}^{-1} \, dm_+ \ . \label{ds0}
\end{equation}
Consider now the constituent systems. The black hole is characterized
by the surface area $A_B = 4\pi R^2$ of a spherical boundary
surface $B_R$ located at $r=R \leq r_0$, and by its
quasilocal energy $E_B$.
The ADM mass $m_-$ of the black hole as a function of
these variables is \cite{Yo}
\begin{equation}
m_- (E_B, A_B) = E_B \, \Bigg( 1 - {{E_B}\over{2R}} \Bigg) \ . \label{m-}
\end{equation}
The horizon radius of the black hole is $2m_-$. The
different radii satisfy the inequalities $2m_- \leq 2m_+ \leq R \leq r_0$,
where $2m_+$ represents the horizon radius
of the total system.
The pressure associated to the gravitational field of the black hole at
the surface $B_R$ is
\begin{equation}
p_B (E_B, A_B) = {{{E_B}^2}\over{16 \pi {R}^3}}
\Bigg(1 - {{E_B}\over{R}} \Bigg)^{-1} = \,
{{1}\over{16 \pi R\, k_-}} (1 - k_-)^2 \ , \label{pb}
\end{equation}
where $k_- \equiv (1 - 2m_- /R)^{1/2}$.
As it is well known,
the first law of thermodynamics for the black hole can be
expressed as a total differential by using Eqs. (\ref{m-}) and
(\ref{pb}), namely
\begin{eqnarray}
dS_B &=& \beta_B \, (dE_B + \, p_B \, dA_B) \nonumber \\
&=& \beta_B \, {k_-}^{-1} \, dm_- \ , \label{dsb}
\end{eqnarray}
where $\beta_B$ is the inverse temperature of the
black hole at the surface $B_R$. This equation must be contrasted
with the entropy differential (\ref{ds0}) for the total system.
For thermodynamical purposes, the shell is considered (effectively) at
rest. For simplicity, we assume that the shell surface
coincides with the surface $B_R$. (This does not
imply loss of generality \cite{MaYo}.)
Thus, the surface areas for
the black hole and shell coincide: $A_B = A_S \equiv A_R = 4\pi R^2$.
The gravitational junction conditions \cite{Is,La} at the shell position
determine its surface energy density $\sigma$ and surface pressure $p_S$
to be
\begin{equation}
E_S \equiv 4\pi R^2 \sigma = R \, (k_- - k_+) \ , \label{m}
\end{equation}
and
\begin{equation}
p_S = {{1}\over{16 \pi R \, k_- \, k_+}} \,
\Big[ k_- (1-k_+)^2 - k_+
(1-k_-)^2 \Big] \ , \label{ps}
\end{equation}
respectively. The symbol $E_S$ denotes the local mass-energy of matter
and $k_+ \equiv (1 - 2m_+ /R)^{1/2}$. To simplify the analysis further,
we consider only the case when the total number of particles $N_S$
in the shell is constant. The condition $m_+ \geq m_-$ guarantees that
both $E_S$ and $p_S$ are non-negative. Let $\beta_S$ denote the inverse
local temperature at the shell. Its entropy differential reads
\begin{equation}
dS_S = \beta_S \, (dE_S + p_S \, dA_R) \ . \label{dss}
\end{equation}
Use of Eqs. (\ref{m}) and (\ref{ps}) allows us to write the matter
entropy as
\begin{equation}
dS_S = \beta_S \, ({k_+}^{-1} \, dm_+ - {k_-}^{-1} \, dm_-) \ .
\end{equation}
How are the three entropies related when the system is in
equilibrium?
The emergence of equilibrium conditions from general principles within
thermodynamics is the subject of the following section. However,
intuitively
the system is in a state of thermal equilibrium provided
(1) $\beta_B = \beta_S \equiv \beta_R$, and
(2) $\beta_R = \beta_0 \, {k_0}^{-1} \, {k_+}$ at the surface $B_R$.
The first condition constrains the temperature at the shell
surface to coincide with the black hole temperature there
(black hole and shell in mutual thermal equilibrium),
whereas the second guarantees that the total
system is in thermal equilibrium with its components \cite{To}.
Mechanical equilibrium of the matter shell with the black
hole is guaranteed by the shell pressure equation (\ref{ps}).
Under these conditions, Eqs. (\ref{ds0}), (\ref{dsb}), and
(\ref{dss}) jointly imply that
\begin{equation}
dS_0 = dS_B + dS_S \ .
\end{equation}
The entropy of the composite system is, therefore, additive
with respect to its constituent subsystems. Since the entropy is a
function of energy and size (its ``extensive variables" discussed below)
this means that, up to a global constant,
\begin{equation}
S_0 (E_0, A_0) = S_B (E_B, A_R) + S_S (E_S, A_R) \ .
\label{additivity}
\end{equation}
In the preceding analysis, additivity is a direct consequence of the
conditions of thermal and mechanical equilibrium. Observe that it is
independent of the functional form of the parameters $\beta_S (E_S,
A_R)$ and $\beta_B (E_B, A_R)$. This is as expected, since inverse
temperature appears in the first law of thermodynamics as an
integrating factor. In particular, the foregoing derivation of
additivity does not depend on the special choice of boundary
conditions or phenomenological matter action employed in Ref.
\cite{MaYo} or on spherical symmetry \cite{Za1}.
It depends only on the adopted definition of stress-energy tensor
\cite{BrYo1} in terms of quasilocal quantities.
As discussed in the following section, the entropies $S_0 (E_0, A_0)$,
$S_B (E_B, A_R)$, and $S_S (E_S, A_R)$ can be determined from Eqs.
(\ref{ds0}), (\ref{dsb}) and (\ref{dss}) {\it only if} the
precise forms of all the thermodynamical equations
of state are known. The latter express intensive parameters as functions
of the appropriate extensive parameters. For example, it is well
known that if the thermal equation of state for a black hole is given
by Hawking's semiclassical expression\cite{Ha}
\begin{equation}
\beta_B (E_B, A_R) = 8 \pi E_B \,
\Bigg( 1 - {{E_B}\over{2R}} \Bigg)
\Bigg( 1 - {{E_B}\over{R}} \Bigg) \ , \label{betah}
\end{equation}
then Eq. (\ref{dsb}) yields the Bekenstein-Hawking entropy \cite{Yo}
\begin{equation}
S_B (E_B, A_R) = 4 \pi {E_B}^2
\Bigg( 1 - {{E_B}\over{2R}} \Bigg)^2
= 4 \pi {m_-}^2 \ . \label{sbh}
\end{equation}
These expressions are sufficient to demonstrate that
additivity of entropies for spatially separate subsystems does {\it not}
require the entropy of each constituent system to be a homogeneous
first-order
function of its extensive parameters. This is in contrast to ordinary
thermodynamics \cite{Ca,GrCa}. Recall that a function $f(x_1,...,x_n)$ is
said to be a homogeneous $m$-th order function of the variables
$(x_1,...,x_n)$ if it satisfies the identity
\begin{equation}
f(\lambda x_1,...,\lambda x_n) = {\lambda}^{m} \, f(x_1,...,x_n) \ ,
\end{equation}
where $\lambda$ is a constant. Upon the
rescaling $E_B \to \lambda E_B$, $A_R \to {\lambda}^2 A_R$
($R \to \lambda R $) the entropy $S_B (E_B, A_R)$ in Eq. (\ref{sbh})
behaves as a homogeneous second-order function of $E_B$ and
as a first-order function of $A_R$, namely \cite{Yo,MaYo}
\begin{equation}
S_B (\lambda E_B, {\lambda}^2 A_R) =
{\lambda}^2 \, S_B (E_B, A_R) \ . \label{sb}
\end{equation}
Equations (\ref{betah}) and (\ref{pb}) also illustrate a central
characteristic of gravitational systems: the inverse temperature and
pressure are not homogeneous zeroth-order functions. Under rescaling
they behave as \cite{Yo} $\beta_B (\lambda E_B, {\lambda}^2 A_R) =
{\lambda} \, \beta_B (E_B, A_R) $
and
$p_B (\lambda E_B, {\lambda}^2 A_R) =
{\lambda}^{-1} \, p_B (E_B, A_R) $.
The functional form of $S_S (E_S, A_R)$ and its homogeneous properties
depend on the explicit form of the matter equations of state.
These arise from either a phenomenological or a field theoretical
description of the matter fields involved, and their precise
form does not concern gravity. Examples of equations of
state for a self-gravitating matter shell (in the absence of a black
hole) in thermal equilibrium with itself have been studied
in Ref. \cite{Ma}. Observe that
Eq. (\ref{p0}) for $p_0$ and Eq. (\ref{pb}) for
$p_B$ are indeed equations of state while
Eq. (\ref{ps}) for $p_S$ is not. If the equations of state were known for both
components, the total entropy $S_0$ could be obtained by
Eq. ({\ref{additivity}) as a function of the extensive parameters of
the subsystems. A discussion of this point and of further properties of
this model are delayed to the following section.
\section{The fundamental problem and the postulates}
The preceding analysis motivates the search for principles
that are independent of model problems and that incorporate the
characteristics of gravity into a logical framework more general than
ordinary thermodynamics. We must start, therefore, by reviewing the basic
assumptions.
Gravitational thermodynamics describes (effectively) static states of
macroscopic finite-size self-gravitating systems. As in regular
thermodynamics, it is expected that very few variables survive the
statistical average involved in a macroscopic measurement.
What are these macroscopic parameters?
The relationship between thermodynamical and dynamical variables in
gravity has been amply discussed and we refer
the reader to the literature \cite{BrMaYo,ensembles,BrYoRev,LoWh}. For
our purposes, it is enough to recall the following points.
Firstly, it has been shown in a wide variety of problems (involving
black holes at finite temperature in interaction with matter fields)
that the
thermodynamical energy coincides with the quasilocal energy $E$ that
naturally follows from the action of a spatially bounded region.
If ${^3\!{B}}$ denotes the three-dimensional boundary of the system
and ${^2\!{B}}$ the two-surface resulting from its intersection with a
spacelike hypersurface $\Sigma$, the quasilocal
energy is the value of the Hamiltonian that generates unit time
translations on ${^3\!{B}}$ in the
direction orthogonal to the surface ${^2\!{B}}$
\cite{BrMaYo,BrYo1}.
We {\it postulate} that this is the appropriate energy variable in
(gravitational) thermodynamics for {\it all} self-gravitating systems.
Secondly, the analog of the thermodynamical ``size" of the system
is the induced two-metric ${\sigma}_{ab}$ of the two-dimensional
boundary surface ${^2\!{B}}$ \cite{Yo,WhYo,BrMaYo}.
This property reflects the ``surface character" of gravitational
thermodynamics and is in part a consequence of the lack of an
operational definition of ``volume" in the presence of black holes.
The size reduces to the surface area $A$ of the two-surface only in the
case of spherical symmetry \cite{BrMaYo}. For composite systems,
quantities that measure size for internal matter components have to be
found. Finally, the remaining macroscopic variables are a
finite number of conserved charges. These may include, for example,
angular momentum \cite{BrMaYo}, suitable combinations of electric
\cite{RN} or magnetic \cite{Za2} charges, cosmological constant
\cite{BrCrMa}, other types of hair \cite{NuQuSu}, and number of
particles for matter systems \cite{Ma}. (The thermodynamic conjugate
quantities to these parameters are chemical potentials defined at the
boundary of the system \cite{RN,BrMaYo}.)
The existence of these macroscopic parameters motivates the first
postulate:
\noindent {\bf Postulate I:}\, There exist particular states
(called equilibrium states) of self-gravitating systems that are
completely characterized macroscopically by the specification of a
finite set of variables. These variables are the quasilocal energy, size,
and a small number of conserved quantities (denoted generically by the
symbol $N$).
In ordinary thermodynamics a similar postulate is usually applied to
so-called ``simple" systems \cite{Ca}.
These systems do not include gravitational or electromagnetic fields
and are by definition macroscopically homogeneous. The previous
postulate incorporates strongly self-gravitating
configurations (with or without matter fields). As shown
below, these systems may be spatially inhomogeneous.
Systems describable by these parameters may be termed ``simple" in
gravitational thermodynamics.
Observe that the preceding postulate does not imply that every
gravitational system has equilibrium configurations. Very often a
system does not possess an equilibrium state even though it has
definite values of energy and other parameters. Rather, the postulate
maintains that equilibrium states, in case they exist, are completely
described by the foregoing finite set of parameters.
The variables $(E, \sigma_{ab}, N)$ that describe a gravitational
equilibrium state are to be called {\it extensive} parameters.
Extensive quantities are constructed entirely from the dynamical phase
space variables.
Another essential difference with usual thermodynamics appears
here: in the latter, the extensive parameters of a
composite system equal the sum of their values in each of the subsystems.
As we illustrate below, this is {\it not} the case in gravitational
thermodynamics.
Some extensive variables of a self-gravitating system cannot
be constrained in the conventional thermodynamic sense.
For example, there exist no walls restrictive with respect to angular
momentum of a stationary black hole system. However, this is not unusual
or particular to gravity. For instance, it also occurs in the treatment
of magnetic systems: there exist no walls restrictive with respect to
magnetic moment. However, one can maintain always the value of magnetic
moment constant at a boundary surface that delimits the system by a
feedback mechanism that continually adjusts the magnetic field
\cite{Ca}. The same happens in gravitational thermodynamics:
unconstrainable quantities can be kept constant at a given boundary
surface by continually monitoring the value of their respective
conjugate chemical potentials at this surface. The unavailability of
walls that restrict certain extensive variables is only an idiosyncrasy
that does not affect the applicability of thermodynamics. As in ordinary
thermodynamics, a boundary that does (does not) allow the flux of heat
can be called diathermal (adiabatic). Observe that the quasilocal energy
adopted here has a very important property for thermodynamics.
It is ``macroscopically controllable" in the usual thermodynamic sense:
it can be ``trapped" by restrictive boundaries and ``manipulated" by
diathermal ones.
A boundary that does not allow the flow of heat and work can be
called ``restrictive with respect to quasilocal energy." A closed
system is defined as one whose extensive variables (quasilocal energy,
size, etc.) remain effectively constant at its boundary surface.
It is of course difficult to split a self-gravitating system into
independent ``component" systems in the manner familiar in
ordinary thermodynamics. Although one can speak of a ``composite"
system formed by two or more subsystems, the latter
interact strongly among themselves. Clearly, if a
composite system is closed, the simple systems are not necessarily so.
However, internal constraints may exist among the component systems.
These are constraints that prevent the flow of energy or any other
extensive parameter among subsystems. For example, in our
model problem internal constraints can restrict the flow of energy
between the two subsystems (for instance, by fixing a particular
value of $E_B$) or area (by fixing $A_R$). The relaxation of internal
constraints in an equilibrium composite system will create processes
that will tend to bring the system to a new equilibrium state.
The central problem of thermodynamics of strongly self-gravitating
systems is, therefore, a reflection of the central problem of ordinary
thermodynamics \cite{Ca}, namely: The determination of the equilibrium
states that will result when internal constraints are removed in a
closed, composite system.
What assumptions are needed in order to solve this problem? Equilibrium
states in gravitational thermodynamics must be characterized by a
simple extremum principle. As any other thermodynamic system, a
gravitational system will select, in the absence of constraints, any one
of a number of states, each of which can also be realized in the presence
of a suitable constraint. Each of these constrained equilibrium
states corresponds to particular values of the extensive parameters
of each constituent system and has a definite entropy. Therefore, the
extremum postulate states that if constraints are lifted, the system
will select the state with the largest entropy. Paraphrasing
Callen:
\vfill\eject
\noindent {\bf Postulate II:} \, There exists a single-valued
function (the entropy $S$) of the extensive variables of any composite
system, defined for all equilibrium states, and possessing the following
property. In the absence of internal constraints, the values assumed
by the extensive parameters are those that maximize the entropy over the
manifold of constrained equilibrium states.
This postulate has to be interpreted carefully.
Physical equilibrium states correspond to states that extremize the total
entropy over the manifold of constrained equilibrium states.
Equilibrium states are, therefore, either maxima, minima, or inflection
points of the entropy. However, in the absence of constraints, the
extensive parameters of the components in the final equilibrium state
will be those that maximize the entropy.
Postulates I and II not only predict equilibrium states, but
also determine their stability properties.
Equilibrium states corresponding to maxima of entropy are {\it stable}
whereas {\it unstable} equilibrium states correspond to extrema other
than maxima. It is important to emphasize that Postulate II
makes no reference to nonequilibrium states. Furthermore, it
implies neither that all equilibrium states of a gravitational system
must have maximum entropy nor that stable states do exist.
After all, it is common to find systems possessing
equilibrium states that are local minima of entropy. Simple examples
include a nonrelativistic self-gravitating gas in a spherical box or
isothermal stellar systems \cite{BiTr}.
The entropy as a function of its extensive variables constitutes the
``fundamental equation" of a self-gravitating system.
The first differentials of the fundamental equation
define its {\it intensive} variables.
For systems with a vanishing shift vector, the intensive variables
in the entropy representation are $(\beta, \beta p, \beta \mu)$, where
$\beta$ denotes inverse temperature, $p$ pressure, and $\mu$ chemical
potentials. Systems possessing a nonvanishing shift require
functional differentials in the
definitions of their intensive parameters.
(This happens, for example, in the thermodynamic description of
stationary geometries \cite{BrMaYo}.)
For general spacetimes the conjugate quantities to the size $\sigma_{ab}$
are proportional to (minus) the spatial stresses introduced
in Ref. \cite{BrYo1}. The intensive parameters are always functions of the
extensive parameters. The set of functional relationships
expressing intensive in terms of extensive parameters are
the thermodynamical equations of state of a self-gravitating system.
For example, for static (as opposed to stationary) systems these are
\begin{eqnarray}
\beta &=& \beta (E, A, N) \ , \nonumber \\
\beta p &=& \beta p \,(E, A, N) \ ,\nonumber \\
\beta \mu &=& \beta \mu \,(E, A, N) \ .\label{eqst}
\end{eqnarray}
As in ordinary thermodynamics, once the fundamental equation
$S(E, {\sigma_{ab}}, N)$ of a system (or, alternatively, its complete
set of equations of state) is known, {\it all} its thermodynamical
information can be obtained from it.
The criteria for global and local stability of equilibrium states in
terms of the entropy function are identical to the ones familiar in
ordinary thermodynamics \cite{Ca,Ma}. In particular, global stability
requires that the entropy hypersurface $S(E, {\sigma_{ab}}, N)$ lies
everywhere below its family of tangent hyperplanes.
It is possible to express the
fundamental equation in terms of different sets of independent variables
by performing Legendre transformations on the entropy.
These are the so-called Massieu functions \cite{Massieu,Ca}.
They play a more fundamental role in gravitational than in
ordinary thermodynamics because they are in a
one-to-one correspondence with actions \cite{ensembles}.
Their preeminence over ``thermodynamic potentials" has not been
emphasized sufficiently. (The latter are Legendre transforms of energy
and include the Helmholtz potential $F$ and Gibbs potential $G$.)
For static systems, Massieu functions include, for example,
${\cal S}_1 (\beta, A, N) = S - \beta E = - \beta F$, in which
quasilocal energy has been replaced by its conjugate entropic intensive
parameter (inverse temperature) as independent variable, and
${\cal S}_2 (\beta, \beta p, N) = S - \beta E - \beta p A = - \beta G $
in which, in addition, the size of the system is replaced by its
entropic intensive parameter.
The above equations can be generalized to stationary geometries
if one recalls that in general it is not possible to choose
all intensive variables constant at a given choice of two-dimensional
boundary surface \cite{BrMaYo,BrYo2}.
The basic extremum postulate is very general and can be reformulated in
these alternative representations: each Legendre transform of the
entropy is a maximum for constant values of the transformed (intensive)
parameters \cite{Ca}.
It is important to emphasize that the second postulate incorporates
not only the so-called first law, {\it but also} the
generalized second law \cite{Be,FrPa,Za3} into a thermodynamic formalism.
Finally, how is the entropy of a composite system
related to the entropies of the subsystems?
In the previous section we illustrated that entropies are additive
despite strong interaction between subsystems. On the one hand,
additivity of entropies does not seem to depend critically on the
particular functional form of the intensive parameters.
On the other hand, it seems to be a natural consequence of the
additivity of actions in a path integral approach to statistical
thermodynamics \cite{MaYo,Za1}. These reasons motivate us to assume
the additivity postulate:
\noindent {\bf Postulate III:}\, The entropy of a composite system
is additive over the constituent subsystems. Furthermore, it is a
continuous, differentiable, and monotonically increasing function of
the quasilocal energy $E$.
We emphasize three important points. First, we shall {\it not} assume
in this postulate that the entropy of each subsystem is a
homogeneous first-order function of the extensive parameters. The
postulate is, therefore, more general than the corresponding one in
regular thermodynamics \cite{Ca,GrCa}. Second, the preceding postulate
implies that all Massieu functions are additive over component Massieu
functions. As we illustrate below, this is not the case for
thermodynamic potentials \cite{MaYo}. Third, the monotonic property
implies that the temperature is postulated to be non-negative as in
ordinary thermodynamics.
The logic of the previous section must be contrasted with the present
one: additivity is neither the result of equilibrium conditions among
intensive variables, nor of the functional form of intensive or
extensive parameters, but a fundamental assumption. Additivity is valid
{\it even} when the subsystems cannot be considered independent and
strongly interact among themselves. As we show in the following
paragraphs, equilibrium conditions are indeed a consequence of the
postulates.
The preceding postulates are the natural extension of the postulates of
nongravitational thermodynamics necessary to accommodate the extensive
parameters characteristic of gravitational systems. Are these postulates
sufficient to solve the fundamental problem despite strong
interactions among subsystems? The answer is on the affirmative.
To illustrate this consider again our model problem in the light of the
logic resulting from the postulates. We shall determine the equilibrium
state of the closed, composite system, namely, the relationships that
must exist among extensive variables of the subsystems for the total
system to be in thermal and mechanical (and in general, chemical)
equilibrium. We also shall indicate how far one can proceed in the
explicit solution of this problem without assuming particular
expressions for the equations of state of the subsystems.
By Postulate I, the black hole and matter subsystems are simple systems
characterized by the extensive variables $(E_B, A_B, N_B)$
and $(E_S, A_S, N_S)$, respectively. The composite system is itself a
simple system and is characterized by the variables $(E_0, A_0, N_0)$.
The size of all systems reduces to the area
of their respective surfaces.
By Postulate II, the
fundamental thermodynamical equations in the entropy representation are
the functions $S_0 = S_0(E_0, A_0, N_0)$, $S_B = S_B(E_B, A_B, N_B)$, and
$S_S = S_S (E_S, A_S, N_S)$.
Postulate III states that $ S_0(E_0, A_0, N_0) = S_B(E_B, A_B, N_B) +
S_S (E_S, A_S, N_S)$.
For simplicity and with no loss of generality, we assume $A_B = A_S
\equiv A_R$ and the quantities $N_0$, $N_B$ and $N_S$ to be constant.
The system is considered closed if its quasilocal energy and area are
kept effectively constant at the boundary $B_0$,
namely
\begin{equation}
E_0 = {\rm const.}; \,\,\, A_0 = {\rm const.} \label{closure}
\end{equation}
The fundamental problem is to determine the extensive variables
$(E_B, E_S, A_R)$ as functions of these constant quantities whenever
equilibrium is attained as a result of relaxing internal constraints.
Postulate II establishes that the total entropy of a
composite system in a state of equilibrium
is an extremum, namely, it does not change as a result of an infinitesimal
virtual transfer of heat or work from one subsystem to the other.
Therefore, in equilibrium
\begin{eqnarray}
dS_0 = 0 &=& dS_B + dS_S \nonumber \\
&=& \beta_B \, (dE_B + p_B \, dA_R) +
\beta_S \, (dE_S + p_S \, dA_R) \ ,\label{ds0a}
\end{eqnarray}
where the second equality is a consequence of Postulates I and
II. The entropic intensive variables are defined in the conventional
way:
\noindent $\beta_S (E_S, A_R) \equiv (\partial S_S /
\partial E_S)_{A_R} $,
$\beta_S p_S (E_S, A_R) \equiv (\partial S_S / \partial A_R)_{E_S} $;
$\beta_B (E_B, A_R) \equiv (\partial S_B / \partial E_B)_{A_R} $, and
$\beta_B p_B (E_B, A_R) \equiv (\partial S_B / \partial A_R)_{E_B} $.
Since the quasilocal energy can be expressed as
$E_0 = r_0 (1-k_0)$, the closure equations are
equivalent to the condition
$m_+ = {\rm const.}$ Because the energy $E_B$ refers to the surface
$B_R$ which coincides with the shell surface, it is easy to see that
the total quasilocal energy at $B_R$ is
\begin{eqnarray}
E_R &\equiv & E_B + E_S \nonumber \\
&=& R \, (1 - k_+) \ . \label{er}
\end{eqnarray}
This equation is a consequence of the additivity of quasilocal
energy discussed in \cite{MaYo,BrYo1}.
(If the black hole energy $E_B$ is defined at a surface which does not
coincide with the shell surface, the energies $E_B$ and $E_S$ are not
simply additive as in Eq. (\ref{er}) \cite{MaYo}.)
The closure conditions and Eq. (\ref{er}) allow the total entropy
(\ref{ds0a}) to be written as
\begin{equation}
dS_0 = 0 = (\beta_S - \beta_B)\, dE_S +
(\beta_B \, p_B + \beta_S \, p_S - \beta_B \, p_R) \, dA_R
\label{ds0b} \ ,
\end{equation}
where the pressure $p_R$ is defined as
\begin{equation}
p_R (E_R, A_R) \equiv {{{E_R} ^2}\over{16 \pi {R}^3}}
\Bigg(1 - {{E_R}\over{R}} \Bigg)^{-1} = \,
{{1}\over{16\pi \, R \, k_+}} (1 - k_+)^2 \ . \label{pr}
\end{equation}
Since the equality in Eq. (\ref{ds0b}) must be satisfied by
independent and arbitrary
variations of $E_S$ and $A_R$, we must necessarily have
\begin{equation}
\beta_S = \beta_B \equiv \beta_R \ , \label{equilb}
\end{equation}
and
\begin{equation}
p_S + p_B = p_R \ . \label{equilp}
\end{equation}
The preceding equations are the sought equilibrium
conditions. They state the relationship among intensive variables of the
subsystems for the composite system to be in thermal and mechanical
equilibrium. As in nongravitational thermodynamics, they yield
a formal solution to the fundamental problem {\it provided}
the equations of state
\begin{eqnarray}
\beta_B &=& \beta_B (E_B, A_R, N_B) \ ,
\,\, p_B = p_B (E_B, A_R, N_B) \ ;
\nonumber \\
\beta_S &=& \beta_S (E_S, A_R, N_S) \ ,
\,\, p_S = p_S (E_S, A_R, N_S)
\end{eqnarray}
for the subsystems are known. If this is so, Eqs. (\ref{equilb}) and
(\ref{equilp}) are two formal relationships among $E_B$, $E_S$, $A_R$
and $m_+$ (with $N_S$ and $N_B$ each held fixed). Equations
(\ref{closure}), (\ref{er}), (\ref{equilb}), and (\ref{equilp}) are,
therefore, the four desired equations that determine the four sought
variables $(E_B, E_S, A_R, m_+)$.
Naturally, the variable $m_+$ in Eqs. (\ref{closure}) and
(\ref{er}) does not play an important role in the formalism:
the relationship among the three energies
$E_0$, $E_B$, and $E_S$ can be written explicitly as
\begin{equation}
E_0 = r_0 \, \Bigg\{ 1 - \Bigg[ 1 - {{2(E_B + E_S)}\over{r_0}}
\bigg(1-{{E_B + E_S}\over{2R}} \bigg) \Bigg]^{1/2} \Bigg\}
\ . \label{e02}
\end{equation}
For a closed system, Eqs. (\ref{equilb}), (\ref{equilp}),
and (\ref{e02}) (with Eq. (\ref{closure})) provide three desired equations
to determine the three sought variables $(E_B, E_S, A_R)$.
The fundamental problem is formally solved by the postulates {\it
despite} the quasilocal energy $E_0$ not being the simple sum of the
component energies $E_B$ and $E_S$ due to binding and self-energy
interactions characteristic of gravitational systems. (It is easy to
see, by using Eq. (\ref{e02}) that all thermodynamic potentials
obtained from $E_0$ by Legendre transforms are {\it not} the
simple sum of the component potentials.)
This indicates not only that the postulates
form a complete set of assumptions for a more general class of
thermodynamic systems than previously considered, but also the
appropriateness of the adopted definitions of extensive parameters.
Consider some further consequences of the postulates.
Firstly, the mechanical equilibrium condition (\ref{equilp}) represents
the spatial stress component of the junction conditions at the surface
$B_R$. (It reduces to Eq. (\ref{ps}) if the pressure equation of state
for the black hole is given by ({\ref{pb})). Secondly, additivity of
entropies and the equilibrium conditions (\ref{equilb}) and
(\ref{equilp}) allow the differential of the total entropy to be written
as
\begin{eqnarray}
dS_0 &=& \beta_0 \, (dE_0 + p_0 \, dA_0) \nonumber \\
&=& \beta_R \, (d(E_B + E_S) + (p_B + p_S) \, dA_R)
\nonumber \\
&=& \beta_R \, (dE_R + p_R \, dA_R) \ . \label{dst2}
\end{eqnarray}
This expression confirms that there is no
``gravitational" entropy associated to the shell \cite{DaFoPa,MaYo}
and illustrates that $E_R$ is the total quasilocal energy and $p_R$
the associated pressure associated to the surface $B_R$.
In turn, Eq. (\ref{dst2}) implies that
\begin{equation}
dS_0 = \beta_0 \, {k_0}^{-1} \, dm_+ = \beta_R \, {k_+}^{-1} \, dm_+ \ .
\end{equation}
Therefore, the inverse temperature at the surface
$B_R$ is given in terms of the inverse temperature $\beta_0$ at the
boundary $B_0$ by \begin{equation}
\beta_0 \, {k_0}^{-1}= \beta_R \, {k_+}^{-1} \ .
\end{equation}
The (spatially) inhomogeneous character \cite{To}
of thermodynamic equilibrium (equivalence principle) is a
consequence of the postulates of thermodynamics and the definition of
quasilocal stress-energy. Thus, the postulates do incorporate
equilibrium states in inhomogeneous systems in contrast to the ordinary
postulates of thermodynamics \cite{Carr}, where a system that is not
homogeneous is not in thermodynamic equilibrium even if its properties
remain constant in time.
The preceding treatment of a composite self-gravitating system must be
contrasted with the equivalent one of a composite nongravitational
system presented in Appendix A. Although the systems are physically
different, their similarities and differences are readily apparent.
In particular, the gravitational equations (\ref{e02}) and
(\ref{closure}) substitute the relations (\ref{etapp}) and (\ref{vtapp})
of flat spacetime thermodynamics.
The formalism provides the methodology to solve the fundamental
problem for self-gravitating systems. In the spirit of
thermodynamics, it yields explicit answers for explicit functional forms
of the fundamental equations (or equivalently, the associated
equations of state) of each of the subsystems \cite{Ca}.
These are outside the realm of thermodynamics and are the result
of either phenomenological or statistical mechanical descriptions of the
constituent systems. We reiterate its formal structure:
for a composite self-gravitating system one must assume the
fundamental equation of the components to be known in principle.
If the total system is in a constrained
equilibrium state (characterized by particular values of the extensive
parameters for each constituent system), the total entropy is obtained
by adding the individual entropies of the components and
is, therefore, a known function of their extensive parameters.
The extrema of the total
entropy define the equilibrium states. Stable states correspond to
maxima of entropy. As an illustration, if explicit
equations of state are known for both black hole and matter in our
model problem, their entropy values can be found (up to an overall
constant) by integrating Eqs. (\ref{dsb}) and (\ref{dss}), and
substituting the values of $(E_B, E_S, A_R)$ at equilibrium. The total
entropy is then given by Eq. (\ref{additivity}).
Can this logical framework accommodate
runaway instabilities (the so-called gravothermal catastrophe) observed
in bounded self-gravitating systems? The answer is on the affirmative.
This behavior is a consequence of the postulates and the particular form
of the fundamental equations characteristic of gravity.
Typically, the latter are such that there exist, besides equilibrium
states that locally maximize the entropy, equilibrium states that
locally minimize it. (The existence of these state is well-known in
stellar dynamics \cite{BiTr} and black hole physics \cite{Yo,WhYo}.)
Consider as illustration an isothermal self-gravitating gas in a closed
spherical container \cite{BiTr}.
The system might be thought of as formed by two components,
the `core' and the `halo.' The formalism states that if the entropies
for the components are known as functions of their extensive parameters,
the total entropy is $S = S_c + S_h$. Equilibrium states are
obtained by extremizing this function and are characterized by
particular values of the extensive parameters of the components. For
simplicity, consider only the energies (or equivalently, the density
contrast between components). The entropy functions for the gas
components are such that there exists an equilibrium state (described by
a particular critical value of the density contrast) that is a local
entropy minimum over the set of all possible equilibrium states
\cite{BiTr}. But Postulate II predicts that this state is unstable. The
onset of instability in the gas obeys the postulates: if the system finds
itself in that state and the density contrast between core and halo is
allowed to change, the system will try to reach equilibrium states of
higher entropy. The system finds it advantageous to transfer energy (or
work) from one region to another, developing more internal
inhomogeneities. Local stability conditions \cite{Ca,Ma} imply that a
negative heat capacity is associated to a local entropy minimum: if the
core gets hotter than the halo, heat flows from the core to the halo and
the core temperature raises. The end result depends on the form of the
entropy function and on the direction of the fluctuation that started the
instability. It might be that a local entropy maximum exists in which
the system settles down (as discussed in Ref. \cite{BiTr}, this occurs if
the entropy is such that, for example, $C_h < |C_c|$). In this case the
halo temperature rises more than the core's and the system shuts off in a
stable state. A runaway instability happens if there does not exist a
local maximum for the system to settle down (this happens if the
fundamental equation is such that $C_h > |C_c|$). In this case the
temperature difference between halo and core keeps growing. Whether a
black hole is created or the system runs out of equilibrium before that
occurs, the important point is that, as long as the system can be
described by equilibrium physics, the postulates {\it predict} its
behavior if the fundamental equations of the components are known. The
above argument applies equally to a collection of stars or other
astronomical systems.
We have studied so far the impact of gravitational extensive variables
in the thermodynamic formalism.
But the latter is also characterized by the
functional forms of its intensive variables (equations of state). These
arise from a dynamical theory but their main characteristic is that, in
general, they are no longer homogeneous zeroth-order functions of
extensive parameters (for instance, the inverse temperature $\beta_B$ in
Eq. (\ref{betah}) is homogeneous first-order in energy and half-order
in area; although the intensive variable $\beta p$ in the entropy
representation of a static black hole is homogeneous zeroth-order,
this is not the case for other systems \cite{Ma}.) This implies that the
consequences of the postulates are different than in ordinary
thermodynamics, particularly the mathematical properties of fundamental
equations. Fundamental equations are in general
no longer homogeneous first-order
functions of their extensive variables. (Alternatively, the
homogeneous properties of fundamental equations in gravitational
thermodynamics imply that intensive variables are no longer
homogeneous zeroth-order functions.)
This does not affect the formalism itself, but has direct
implications for at least two formal relationships among thermodynamic
quantities. Firstly, the so-called Euler relation is necessarily
different from the one familiar in ordinary thermodynamics \cite{Yo}.
An Euler
relation is a consequence of Euler theorem stating that a homogeneous
function $f(x_1, ..., x_n)$ of $m$-th order satisfies the equality
\begin{equation}
m\, f(x_1, ..., x_n) = x_1 \Bigg( {{\partial f}
\over{\partial x_1}} \Bigg) + \dots +
x_n \Bigg( {{\partial f}\over{\partial x_n}}\Bigg)\ .
\end{equation}
In standard thermodynamics entropy is a homogeneous first-order function
and the Euler relation is therefore
$S = \beta E + \beta p V - \beta \mu N$.
In contrast and as an example, the Euler relation for a static
charged black hole reads
\begin{equation}
S = {{1}\over{2}} \, \beta \, E + \beta \, p \, A -
{{1}\over{2}} \, \beta \, \mu \, N \ . \label{euler}
\end{equation}
Euler relations for hollow self-gravitating thin shells with
power law thermal equations of state have been presented in
Ref. \cite{Ma}.
Secondly, there does not exist a Gibbs-Duhem relation in gravitational
thermodynamics \cite{Ma}. The Gibbs-Duhem relation in ordinary
thermodynamics is a direct consequence of the homogeneous
first-order properties of the fundamental equation and relates the
intensive parameters of a system. It states that the sum of products of
extensive parameters and the differentials of the (conjugate) intensive
parameters vanishes. In the entropy representation it reads
$E d\beta + V d(\beta p) - N d(\beta \mu) = 0$. In contrast, if one
combines the first law with the
Euler relation (\ref{euler}) for a charged black hole one obtains
\begin{equation}
E \, d \beta - \beta \, dE + 2 A \, d(\beta p)
- N \, d(\beta \mu) + \beta \, \mu \, dN = 0 \ .
\end{equation}
The reformulation of an Euler relation and the absence of a
Gibbs-Duhem relation set gravity apart from other interactions: even for
magnetic systems the Euler and the Gibbs-Duhem expressions maintain their
usual relationship.
There are no obstacles in applying the preceding formalism
to any self-gravitating system. These may include not only
nonrelativistic astronomical objects, but also highly relativistic
systems involving general black holes in interaction with matter.
The (spherically symmetric) model problem was
used {\it only} as an illustration because of its simplicity and
transparency. For general situations,
a larger state space is required to incorporate a larger number of
extensive variables and thermodynamic equilibrium includes equilibrium
under ``interchange of number $N$." It is possible to
employ a condensed notation where the symbols $X_i$ and $P_j$ denote
generically all extensive and intensive parameters, respectively
(excluding energy and inverse temperature), as in Ref. \cite{Ca}. In
this way the equilibrium conditions (\ref{equilb}) and (\ref{equilp})
are easily generalized to include all chemical potentials for the
system. Although the treatment of these and other
systems may be technically difficult, the resolution of the
fundamental problem of thermodynamics obeys the same
logical structure as the one presented here.
We emphasize again that the characteristics of gravitational
thermodynamics are the result of its extensive parameters (which must
include quasilocal quantities like quasilocal energy) and the particular
homogeneity properties of its intensive variables (equations of state) as
functions of extensive parameters. This is a more general and rigorous
characterization of the defining properties of gravitational
thermodynamics than, for example, the one presented in Ref. \cite{Fr}
in the context of black holes.
The definitions of extensive and intensive variables as well as the
changes introduced in Callen's postulates are the minimal changes
necessary to incorporate the global aspects and nonlinearities
characteristic of the gravitational interaction into a postulatory
formulation of thermodynamics.
To summarize, we have presented the overall structure and principles
of a thermodynamic framework that incorporates self-gravitating
systems. All the results of standard (gravitational) thermodynamics are
a consequence of the generalized postulates and the solution of the
fundamental problem and can be extracted from them by following the
standard procedures described in Refs. \cite{Ca,Gu}. Possible
applications of the formalism include, for example, the description of
quasi-static reversible and irreversible
processes, alternative representations, phase transitions, etc.
Finally, a further postulate is usually introduced in standard
thermodynamics: the so-called third law. However, the main body of
thermodynamics does not require this postulate since in the latter there
is no meaning for the absolute value (and therefore for the zero) of
entropy. The role and interpretation of this kind of assumption in the
statistical mechanics of gravitation is the subject of a future
publication \cite{Maip}.
\acknowledgments
It is a pleasure to thank Abhay Ashtekar, Valeri Frolov, Gerald Horwitz,
Werner Israel, Lee Smolin and especially James York for
stimulating conversations and critical remarks.
Research support was received from the National
Science Foundation Grants No. PHY 93-96246 and No. PHY 95-14240,
and from the Eberly Research Funds of The Pennsylvania State University.
|
proofpile-arXiv_065-663
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\section{Introduction}
\noindent
Has the quark gluon plasma been discovered at the CERN SPS?
Experiment NA50 has reported an abrupt decrease in
$\psi$ production in Pb+Pb collisions at 158 GeV per nucleon
\cite{na50}. Specifically, the collaboration presented a striking
`threshold effect' in the $\psi$--to--continuum ratio by plotting it
as a function of a calculated quantity, the mean path length of the
$\psi$ through the nuclear medium, $L$, as shown in fig.~1a. This apparent
threshold has sparked considerable excitement as it may signal
deconfinement in the heavy Pb+Pb system \cite{bo}.
\begin{figure}
\vskip -1.0in
\epsfxsize=4.5in
\leftline{\epsffile{fig1.ps}}
\vskip -2.4in
\caption[]{(a) The NA50 \cite{na50} comparison of $\psi$ production in
Pb+Pb and S+U collisions as a function of the average path length $L$,
see eq.\ (3). $B$ is the $\psi\rightarrow \mu^+\mu^-$ branching
ratio. (b) Transverse energy dependence of Pb+Pb data. Curves in (a)
and (b) are computed using eqs.\ (4--6).}
\end{figure}
In this talk I report on work with Ramona Vogt in ref.~\cite{gv2}
comparing Pb results to predictions \cite{gv,gstv} using a hadronic
model of charmonium suppression. We first demonstrate
that the behavior in the NA50 plot, fig.~1a, is not a threshold effect
but, rather, reflects the approach to the geometrical limit of $L$ as
the collisions become increasingly central. When plotted as a
function of the {\it measured} neutral transverse energy $E_{T}$ as in
fig.~1b, the data varies smoothly as in S+U measurements in fig.~3b
below \cite{na50,na38,na38c,na38d,na38e}. The difference between S+U
and Pb+Pb data lies strictly in the relative magnitude. To assess
this magnitude, we compare $\psi$ and $\psi^\prime$ data to
expectations based on the hadronic comover model \cite{gv,gstv}. The
curves in fig.~1 represent our calculations using parameters fixed
earlier in Ref.\ \cite{gstv}. Our result is essentially the same as
the Pb+Pb prediction in \cite{gv}.
Our primary intention is to demonstrate that there is no evidence for
a strong discontinuity between $p$A, S+U and Pb+Pb data. However, to
quote Maurice Goldhaber, ``$\ldots$ absence of evidence is {\it not}
evidence of absence.'' Our secondary goal is to show that our model
predictions agree with the new Pb+Pb data. The consistency of these
predictions is evident from the agreement of our old $p$A and S+U
calculations with more recent NA38 and NA51 data. Nevertheless, the
significance of this result must be weighted by the fact that all
$p$A and AB data are preliminary and at different beam energies.
In this work, we do not attempt to show that our comover
interpretation of the data is unambiguous -- this is certainly
impossible at present.
\section{Nucleons and Comovers}
The hadronic contribution to charmonium suppression arises from scattering of
the nascent $\psi$ with produced particles -- the comovers -- and
nucleons \cite{gv,gstv}. To determine the suppression from nucleon
absorption of the $\psi$, we calculate the probability that a
$c{\overline c}$ pair produced at a point $(b, z)$ in a nucleus
survives scattering with nucleons to form a $\psi$. The standard
\cite{gstv,gh} result is
\begin{equation}
S_{A} = {\rm exp}\{-\int_z^\infty\! dz\, \rho_{A}(b, z) \sigma_{\psi N}\}
\end{equation}
where $\rho_{A}$ is the nuclear density, $b$ the impact parameter and
$\sigma_{\psi N}$ the absorption cross section for $\psi$--nucleon
interactions. One can estimate $S_{A}\sim \exp\{-
\sigma_{\psi N} \rho_0 L_{A}\}$, where $L_{A}$ is the path length
traversed by the $c\overline{c}$ pair.
Suppression can also be caused by scattering with mesons that happen
to travel along with the $c\overline{c}$ pair (see refs.\ in
\cite{gv}). The density of such comovers scales roughly as $E_{T}$.
The corresponding survival probability is
\begin{equation}
S_{\rm co} = {\rm exp}\{- \int\! d\tau n\,
\sigma_{\rm co} v_{\rm rel}\},
\end{equation}
where $n$ is the comover density and $\tau$ is the time in the $\psi$
rest frame. We write $S_{\rm co}\sim {\rm exp}\{-\beta E_{T}\}$,
where $\beta$ depends on the scattering frequency,
the formation time of the comovers and the transverse size of the
central region, $R_{T}$, {\it cf.} eq.\ (8).
To understand the saturation of the Pb data with $L$ in fig.~1a, we apply
the schematic approximation of Ref.~\cite{gh} for the moment to write
\begin{equation}
{{\sigma^{AB}_\psi(E_{T})}\over{\sigma^{AB}_{\mu^+\mu^-}(E_{T})}}
\propto \langle S_{A}S_{B}S_{\rm co}\rangle
\sim
{\rm e}^{-\sigma_{\psi N}\rho_{0}L}{\rm e}^{-\beta E_{T}},
\end{equation}
where the brackets imply an average over the collision geometry for
fixed $E_{T}$ and $\sigma(E_T) \equiv d\sigma/dE_T$. The path length
$L\equiv \langle L_{A}+L_{B}\rangle$ and transverse size $R_T$ depend
on the collision geometry. The path length grows with $E_{T}$,
asymptotically approaching the geometric limit $R_A + R_B$. Explicit
calculations show that nucleon absorption begins to {\it saturate} for
$b < R_A$, where $R_A$ is the smaller of the two nuclei, see fig.~4
below. On the other hand, $E_{T}$ continues to
grow for $b < R_A$ due, {\it e.g.}, to fluctuations in the number of
$NN$ collisions. Equation (2) falls exponentially in this regime
because $\beta$, like $L$, saturates.
In fig.~1b, we compare the Pb data to calculations of the
$\psi$--to--continuum ratio that incorporate nucleon and comover
scattering. The contribution due to nucleon absorption indeed levels
off for small values of $b$, as expected from eq.\ (3). Comover
scattering accounts for the remaining suppression.
These results are {\it predictions} obtained using the computer code
of Ref.~\cite{gv} with parameters determined in Ref.~\cite{gstv}.
However, to confront the present NA50 analysis \cite{na50}, we
account for changes in the experimental coverege as follows:
\begin{itemize}
\item Calculate the continuum dimuon yield in the new mass range $2.9
< M < 4.5$~GeV.
\item Adjust the $E_T$ scale to the pseudorapidity
acceptance of the NA50 calorimeter, $1.1 < \eta < 2.3$.
\end{itemize}
The agreement in fig.~1 depends on these updates.
\section{$J/\psi$ Suppression}
We now review the details of our calculations, highlighting the
adjustments as we go. For collisions at a fixed $b$, the
$\psi$--production cross section is
\begin{equation}
\sigma_\psi^{AB}(b)
=
\sigma^{NN}_{\psi}\!\int\! d^2s dz dz^\prime\,\rho_A(s,z)
\rho_B(b-s,z^\prime)\, S,
\end{equation}
where $S\equiv S_AS_BS_{\rm co}$ is the product of the survival
probabilities in the projectile $A$, target $B$ and comover matter.
The continuum cross section is
\begin{equation}
\sigma_{\mu^{+}\mu^{-}}^{AB}(b) =
\sigma^{NN}_{\mu^+\mu^-}\!\int\! d^2s dz dz^\prime\,\rho_A(s,z)
\rho_B(b-s,z^\prime).
\end{equation}
The magnitude of (4,5) and their ratio are fixed by the elementary
cross sections $\sigma^{NN}_{\psi}$ and
$\sigma^{NN}_{\mu^{+}\mu^{-}}$. We calculate $\sigma^{NN}_{\psi}$
using the phenomenologically--successful color evaporation model
\cite{hpc-psi}. The continuum in the mass range used by NA50, $2.9 <
M < 4.5$~GeV, is described by the Drell--Yan process. To confront
NA50 and NA38 data in the appropriate kinematic regime, we compute
these cross sections at leading order following \cite{hpc-psi,hpc-dy}
using GRV LO parton distributions with a charm $K$--factor $K_c= 2.7$
and a color evaporation coefficient $F_\psi =2.54\%$ and a Drell--Yan
$K$--factor $K_{DY}=2.4$. Observe that these choices were fixed by
fitting $pp$ data at all available energies \cite{hpc-psi}. Computing
$\sigma^{NN}_{\mu^{+}\mu^{-}}$ for $2.9<M<4.5$~GeV corresponds to the
first update.
To obtain $E_T$ dependent cross sections from eqs.\ (4) and (5), we
write
\begin{equation}
\sigma^{AB}(E_{T}) =
\int\! d^2b\, P(E_T,b) \sigma^{AB}(b).
\end{equation}
The probability $P(E_T,b)$ that a collision at impact parameter $b$
produces transverse energy $E_T$ is related to the minimum--bias
distribution by
\begin{equation}
\sigma_{\rm min}(E_{T}) = \int\! d^{2}b\; P(E_{T}, b).
\end{equation}
We parametrize $P(E_{T}, b) = C\exp\{- (E_{T}- {\overline
E}_{T})^2/2\Delta\}$, where ${\overline E}_{T}(b) = \epsilon {\cal
N}(b)$, $\Delta(b) = \omega \epsilon {\overline E}_{T}(b)$,
$C(b)=(2\pi\Delta(b))^{-1}$ and ${\cal N}(b)$ is the number of
participants (see, {\it e.g.}, Ref.~\cite{gv}).
We take $\epsilon$ and $\omega$ to be phenomenological
calorimeter--dependent constants.
We compare the minimum bias distributions for total hadronic $E_T$
calculated using eq.\ (7) for $\epsilon = 1.3$~GeV and $\omega = 2.0$
to NA35 S+S and NA49 Pb+Pb data \cite{na49}. The agreement in fig.~2a
builds our confidence that eq.\ (7) applies to the heavy Pb+Pb system.
\begin{figure}
\vskip -1.5in
\epsfxsize=4.0in
\centerline{\epsffile{fig2.ps}}
\vskip -1.0in
\caption{Transverse energy distributions from eq.\ (7).
The S--Pb comparison (a) employs the same parameters.}
\end{figure}
Figure 2b shows the distribution of neutral transverse energy
calculated using eqs.\ (5) and (6) to simulate the NA50 dimuon
trigger. We take $\epsilon = 0.35$~GeV, $\omega = 3.2$, and
$\sigma^{NN}_{\mu^+\mu^-}\approx 37.2$~pb as appropriate for the
dimuon--mass range $2.9 < M < 4.5$~GeV. The $E_T$ distribution for
S+U~$\rightarrow \mu^+\mu^- + X$ from NA38 was described \cite{gstv}
using $\epsilon = 0.64$~GeV and $\omega = 3.2$ -- the change in
$\epsilon$ corresponds roughly to the shift in particle production
when the pseudorapidity coverage is changed from $1.7 < \eta < 4.1$
(NA38) to $1.1 < \eta < 2.3$ (NA50). Taking $\epsilon = 0.35$~GeV for
the NA50 acceptance is the second update listed earlier.
We now apply eqs.\ (1,2,4) and (5) to charmonium suppression in Pb+Pb
collisions. To determine nucleon absorption, we used $p$A data to fix
$\sigma_{\psi N}\approx 4.8$~mb in Ref.~\cite{gstv}. This choice is in
accord with the latest NA38 and NA51 $pA$ data, see fig.~3a.
To specify comover scattering \cite{gstv}, we assumed that the
dominant contribution to $\psi$ dissociation comes from exothermic
hadronic reactions such as $\rho + \psi \rightarrow D+ \overline{D}$.
We further took the comovers to evolve from a formation time
$\tau_{0}\sim 2$~fm to a freezeout time $\tau_{F}\sim R_{T}/v_{\rm
rel}$ following Bjorken scaling, where $v_{\rm rel}\sim 0.6$ is
roughly the average $\psi-\rho$ relative velocity. The
survival probability, eq.\ (2), is then
\begin{equation}
S_{\rm co} = \exp\{ - \sigma_{\rm co}v_{\rm rel}n_{0}\tau_{0}
\ln(R_{T}/v_{\rm rel}\tau_{0})\}
\end{equation}
where $\sigma_{\rm co} \approx 2\sigma_{\psi N}/3$, $R_{T}\approx
R_{A}$ and $n_{0}$ is the initial density of sufficiently massive
$\rho, \omega$ and $\eta$ mesons. To account for the variation of
density with $E_{T}$, we take $n_{0} = {\overline
n}_{0}E_{T}/{\overline E}_{T}(0)$ \cite{gv}. A value $\overline{n}_{0}
= 0.8$~fm$^{-3}$ was chosen to fit the central S+U datum. Since we
fix the density in central collisions, this simple {\it ansatz} for
$S_{\rm co}$ may be inaccurate for peripheral collisions. [Densities
$\sim 1$~fm$^{-3}$ typically arise in hadronic models of ion
collisions, e.g., refs.~\cite{cascade}. The internal consistency of
hadronic models at such densities demands further study.]
We expect the comover contribution to the suppression to increase in
Pb+Pb relative to S+U for central collisions because both the
initial density and lifetime of the system can increase. To be
conservative, we assumed that Pb and S beams achieve the same mean initial
density. Even so, the lifetime of the system essentially doubles in
Pb+Pb because $R_T \sim R_{A}$ increases to 6.6~fm from 3.6~fm in S+U.
The increase in the comover contribution evident in comparing figs.~1b
and 3b is described by the seemingly innocuous logarithm in eq.\ (8),
which increases by $\approx 60\%$ in the larger Pb system.
\begin{figure}
\vskip -2.8in
\epsfxsize=4.5in
\rightline{\epsffile{fig3.ps}}
\vskip -0.5in
\caption[]{(a) $p$A cross sections \cite{na50} in the NA50 acceptance
and (b) S+U ratios from '91 \cite{na38c} and '92 \cite{na50}
runs. The '92 data are scaled to the '91 continuum. The dashed line
indicates the suppression from nucleons alone. The $pp$ cross section
in (a) is constrained by the global fit to $pp$ data in
ref.~\cite{hpc-psi}.}
\end{figure}
In Ref.~\cite{gstv}, we pointed out that comovers were necessary to
explain S+U data from the NA38 1991 run \cite{na38}. Data just
released \cite{na50} from their 1992 run support this conclusion. The
'91 $\psi$ data were presented as a ratio to the dimuon continuum in
the low mass range $1.7 < M < 2.7$~GeV, where charm decays are an
important source of dileptons. On the other hand, the '92 $\psi$ data
\cite{na50,na38e} are given as ratios to the Drell--Yan cross section
in the range $1.5< M < 5.0$~GeV. That cross section is extracted from
the continuum by fixing the $K$--factor in the high mass region
\cite{na38f}. To compare our result from Ref.~\cite{gstv} to these
data, we scale the '92 data by an empirical factor. This factor is
$\approx 10\%$ larger than our calculated factor
$\sigma^{NN}_{DY}(92)/\sigma^{NN}_{\rm cont.}(91) \approx 0.4$; these
values agree within the NA38 systematic errors. [NA50 similarly
scaled the '92 data to the high--mass continuum to produce fig.~1a.]
Because our fit is driven by the highest $E_T$ datum, we see from
fig.~3b that a fit to the '92 data would not appreciably change our
result. Note that a uniform decrease of the ratio would increase the
comover contribution needed to explain S+U collisions.
NA50 and NA38 have also measured the total $\psi$--production cross
section in Pb+Pb \cite{na50} and S+U reactions \cite{na38c}. To
compare to that data, we integrate eqs.\ (4, 6) to obtain the total
$(\sigma/AB)_{\psi} = 0.95$~nb in S+U at 200~GeV and 0.54~nb for Pb+Pb
at 158~GeV in the NA50 spectrometer acceptance, $0.4 > x_{F}> 0$ and
$-0.5 < \cos\theta < 0.5$ (to correct to the full angular range and $1
> x_{F} > 0$, multiply these cross sections by $\approx 2.07$). The
experimental results in this range are $1.03 \pm 0.04 \pm 0.10$~nb for
S+U collisions \cite{na38} and $0.44 \pm 0.005 \pm 0.032$ nb for Pb+Pb
reactions \cite{na50}. Interestingly, in the Pb system we find a
Drell--Yan cross section $(\sigma/AB)_{{}_{DY}} = 37.2$~pb while NA50
finds $(\sigma/AB)_{{}_{DY}} = 32.8\pm 0.9\pm 2.3$~pb. Both the
$\psi$ and Drell--Yan cross sections in Pb+Pb collisions are somewhat
above the data, suggesting that the calculated rates at the $NN$ level
may be $\sim 20-30\%$ too large at 158~GeV. This discrepancy is
within ambiguities in current $pp$ data near that low energy
\cite{hpc-psi}. Moreover, nuclear effects on the parton densities
omitted in eqs.\ (4,5) can affect the total S and Pb cross sections at
this level.
We remark that if one were to neglect comovers and take $\sigma_{\psi
N} = 6.2$~mb, one would find $(\sigma/AB)_{\psi} = 1.03$~nb in S+U at
200~GeV and 0.62~nb for Pb+Pb at 158~GeV. The agreement with S+U data
is possible because comovers only contribute to the total cross
section at the $\sim 18\%$ level in the light system. This is
expected, since the impact--parameter integrated cross section is
dominated by large $b$ and the distinction between central and
peripheral interactions is more striking for the asymmetric S+U
system. As in Ref.~\cite{gstv}, the need for comovers is evident for
the $E_{T}$--dependent ratios, where central collisions are singled
out.
\section{Saturation and the Definition of $L$}
To see why saturation occurs in Pb+Pb collisions but not in S+U, we
compare the NA50 $L(E_T)$ \cite{na50} to the average impact parameter
$\langle b\rangle (E_T)$ in fig.~4. To best understand fig.~1a, we
show the values of $L(E_T)$ computed by NA50 for this figure. We use
our model to compute $\langle b\rangle = \langle b
T_{AB}\rangle/\langle T_{AB}\rangle$, where $\langle f(b)\rangle
\equiv \int\!d^2b\; P(E_T,b)f(b)$ and $T_{AB} =
\int\!d^{2}sdzdz^\prime \rho_{A}(s,z)\rho_{B}(b-s,z^\prime)$. [Note
that NA50 reports similar values of $\langle b\rangle (E_T)$
\cite{na50}.] In the $E_T$ range covered by the S experiments, we see
that $\langle b\rangle$ is near $\sim R_{\rm S} = 3.6$~fm or larger.
In this range, increasing $b$ dramatically reduces the collision
volume and, consequently, $L$. In contrast, in Pb+Pb collisions
$\langle b\rangle \ll R_{\rm Pb} =$~6.6~fm for all but the lowest
$E_T$ bin, so that $L$ does not vary appreciably.
\begin{figure}
\vskip -2.8in
\epsfxsize=4.5in
\rightline{\epsffile{fig4.ps}}
\vskip -0.5in
\caption[]{$E_T$ dependence of $L$ (solid) used by NA50 \cite{na50}
(see fig.~1a) and the average impact parameter $\langle b\rangle$
(dot--dashed). The solid line covers the measured $E_T$ range.}
\end{figure}
\begin{figure}
\vskip -2.8in
\epsfxsize=4.5in
\rightline{\epsffile{l_et_all.ps}}
\vskip -0.5in
\caption[]{NA50 $L(E_T)$ [1] (points) compared to calculations
for realistic nuclear densities (solid), as used here, and for
a sharp--surface approximation (dot-dashed).}
\end{figure}
\begin{figure}
\vskip -2.0in
\epsfxsize=4.5in
\centerline{\epsffile{fig1_L.ps}}
\vskip -2.0in
\caption{NA50 data replotted with a realistic $L(E_T)$ from (9).}
\end{figure}
To understand the sensitivity of fig.~1a to the definition of the path
length, we now estimate $L(E_T)$ \cite{gv3}. We identify (3) with the
exact expression formed from the ratio of (4) and (5). Expanding in
$\sigma_{\psi N}$ and neglecting comovers, we find:
\begin{equation}
L(E_T) = \{2\rho_0\langle T_{AB}\rangle\}^{-1}
\left\langle\int\! d^2s\; [T_A(s)]^2T_B(b-s) +
[T_B(b-s)]^2T_A(s)\right\rangle,
\end{equation}
where $T_A(s) = \int \rho_A(s,z) dz$.
In fig.~5 we compare the NA50 $L(E_T)$ to the path length calculated
using two assumptions for the nuclear density profile: our realistic
three--parameter Fermi distribution and the sharp--surface
approximation $\rho = \rho_0\Theta(R_A -r)$. NA38~\cite{borhani}
obtained $L$ for S+U using the empirical prescription of
ref.~\cite{gh}, while NA50 calculated $L$ assuming the sharp-surface
approximation~\cite{claudie}. Indeed, we see that the NA50 Pb+Pb
values agree with our sharp--surface result, while the NA38 S+U values
are nearer to the realistic distribution.
To see how the value of the path length can affect the appearance of
fig.~1a, we replot in fig.~6 the NA50 data using $L(E_T)$ from (9)
with the realistic density. We learn that the appearance of fig.~1a
is very sensitive to the definition of $L$. Furthermore, with a
realistic $L$, one no longer gets the impression given by the NA50
figure \cite{na50} of Pb+Pb data ``departing from a universal curve.''
Nevertheless, the saturation phenomena evident in fig.~1a does not
vanish. Saturation is a real effect of geometry.
\section{$\psi^\prime$ Suppression}
\begin{figure}
\vskip -3.2in
\epsfxsize=4.5in
\rightline{\epsffile{fig5.ps}}
\vskip -0.3in
\caption[]{Comover suppression of $\psi^\prime$ compared to (a) NA38 and
NA51 $p$A data \cite{na50,na38e} and (b) NA38 S+U data \cite{na38d}
(filled points) and preliminary data \cite{na50}.}
\end{figure}
\begin{figure}
\vskip -2.0in
\epsfxsize=4.5in
\centerline{\epsffile{fig6.ps}}
\vskip -2.0in
\caption{Comover suppression in Pb+Pb~$\rightarrow \psi^\prime +X$.}
\end{figure}
To apply eqs.\ (4-6) to calculate the $\psi^{\prime}$--to--$\psi$
ratio as a function of $E_{T}$, we must specify
$\sigma_{\psi^{\prime}}^{NN}$, $\sigma_{\psi^{\prime} N}$, and
$\sigma_{\psi^{\prime} {\rm co}}$. Following Ref.~\cite{hpc-psi}, we
use $pp$ data to fix $B\sigma_{\psi^{\prime}}^{NN}/B\sigma_{\psi}^{NN}
= 0.02$ (this determines $F_{\psi^\prime}$). The value of
$\sigma_{\psi^{\prime} N}$ depends on whether the nascent
$\psi^{\prime}$ is a color singlet hadron or color octet
$c\overline{c}$ as it traverses the nucleus. In the singlet case, one
expects the absorption cross sections to scale with the square of the
charmonium radius. Taking this {\it ansatz} and assuming that the
$\psi^\prime$ forms directly while radiative $\chi$ decays account for
40\% of $\psi$ production, one expects $\sigma_{\psi'}\sim
2.1\sigma_{\psi}$ for interactions with either nucleons or comovers
\cite{gstv}. For the octet case, we take $\sigma_{\psi^{\prime} N}
\approx \sigma_{\psi N}$ and fix $\sigma_{\psi^{\prime} {\rm
co}}\approx 12$~mb to fit the S+U data. In fig.~7a, we show that the
singlet and octet extrapolations describe $p$A data equally well.
Our predictions for Pb+Pb collisions are shown in fig.~8. In the
octet model, the entire suppression of the $\psi^{\prime}$--to--$\psi$
ratio is due to comover interactions. In view of the schematic nature
of our approximation to $S_{\rm co}$ in eq.\ (8), we regard the
agreement with data of singlet and octet extrapolations as equivalent.
\section{Summary}
In summary, the Pb data \cite{na50} cannot be described by nucleon
absorption alone. This is seen in the NA50 plot, fig.~1a, and
confirmed by our results. The saturation with $L$ but not $E_T$
suggests an additional density--dependent suppression mechanism.
Earlier studies pointed out that additional suppression was already
needed to describe the S+U results \cite{gstv}; recent data
\cite{na50} support that conclusion (see, however, \cite{bo}).
Comover scattering explains the additional suppression. Nevertheless,
it is unlikely that this explanation is unique. SPS
inverse--kinematics experiments ($B < A$) and AGS $p$A studies near
the $\psi$ threshold can help pin down model uncertainties.
After the completion of \cite{gv2}, several cascade calculations
\cite{cascade} have essentially confirmed our conclusions. This
confirmation is important, because such calculations do not employ the
simplifications ({\it e.g.\ } $n_0\propto E_T$) needed to derive (8).
In particular, these models calculate $E_T$ and the comover density
consistently. Some of these authors took $\sigma_{\psi N} \sim 6$~mb
(instead of $\sim 5$~mb) to fit the NA51 data in fig.~3a somewhat
better.
I am grateful to Ramona Vogt for her collaboration in this work. I
also thank C.~Gerschel and M.~Gonin for discussions of the NA50 data,
and M.~Gyulassy, R.~Pisarski and M.~Tytgat for insightful comments.
\nonumsection{References}
|
proofpile-arXiv_065-664
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\section{Introduction}
\label{sec:introduction}
Turbulence in two dimensional fluids is simple enough to allow some
analytically tractable models
\cite{Onsager49,KM80,Kuzmin82,Polyakov92} and high-resolution
computation \cite{BW89,MSMOM91,CMcWPWY91,CMcWPWY92,MMSMO92}, but still
sufficiently complex to exhibit the general challenging nature of
turbulence. In addition, many features of plasma and geophysical
fluid dynamics are essentially two dimensional.
The principal difficulty encountered by any analytical approach to
fluid turbulence is that the underlying equations of motion are
nonintegrable, a quality independent of the abilities of the
researcher. To overcome this difficulty, two major analytical
approaches have been attempted so far, namely, (a)~statistical
closures dealing with simplifying, but usually uncontrollable,
modifications of the dynamical equations and (b)~considerations of
{\em a priori\/} equally probable states constrained by the known
integrals of motion. In this work we take the second approach, where
the consistency requirement is to allow for {\em all\/} constants of
the motion.
The conventional Boltzmann-Gibbs statistical mechanics, when applied to
partial differential equations \cite {Tasso87,LRS88,LRS89,Pomeau92a},
encounters a fundamental obstacle because of the underlying infinite
number of degrees of freedom and the necessity to use a finite ($N$)
dimensional approximation. The continuum limit $N\to\infty$, in common
to all classical fields, results in the so-called ultraviolet
catastrophe (the divergence of energy at finite temperature), a problem
which goes back to Jeans. Unlike the equilibrium electromagnetic
radiation, the ultraviolet catastrophe in fluid turbulence cannot be
remedied by quantization and should be resolved within the classical
framework. The tendency toward the equipartition of energy between the
degrees of freedom (by $T/2$ for each degree, where $T$ is the energy
temperature) in a closed continuum system can only be satisfied by
letting the temperature to zero. In fact, the way the temperature goes
to zero as the number of degrees of freedom goes to infinity is the
heart of the problem. Nontrivial equilibrium states are obtained when
there are more than one integrals of motion, which diverge at different
rates as $N\to\infty$.
The absence of a well-defined concept of measure in a functional
(infinite dimensional) space requires an $N$ dimensional
discretization, even though the final results are obtained by letting
$N\to\infty$.
\label{loc2}
Lee \cite {Lee52} was the first to use the truncated Fourier series
to show the validity of an infinite dimensional Liouville theorem. The
choice of the discrete variables is not unique, and one ought to make sure
that the results of the statistical theory of turbulence be invariant
with respect to the way this choice is made.
A less formal, physical motivation for the discretization procedure can
be found in the finiteness of the number $N(t)$ of the ``effectively
excited'' collective modes at any finite time $t$. Under ``effectively
excited'' we mean modes with amplitudes not yet exponentially small (a
smooth field, for instance, has exponentially small high Fourier
harmonics). There are examples of $N(t)$ becoming infinite in finite
time \cite {FS91}, even when a real-space collapse \cite{Zakharov72}
does not occur. It appears, however, that in two dimensional ideal
hydrodynamics and magnetohydrodynamics $N(t)$ behaves algebraically,
$N(t)\simeq(t/{\tau}_A)^p$, so that the time of the doubling of $N(t)$ is
of order $t$ and the time of the increment of $N(t)$ by one is of order
${\delta} t={\tau}_A(t/{\tau}_A)^{1-p}$, if $0<p<1$. Hence, on time scale longer
than the eddy turnover (nonlinear mode interaction) time ${\tau}_A$, one
may expect an equilibrium statistical distribution among the
effectively excited $N(t)$ modes. By letting $N(t)\gg1$ in this
distribution, we shall infer the most probable direction of the
long-time system evolution. The probability of a deviation from the
most probable state goes to zero as the number of degrees of freedom
goes to infinity. As for a continuum conservative system we really
mean $N(t)\to\infty$ as $t\to\infty$, the probabilistic nature of the
statistical prediction assumes a rather deterministic quality.
Artificial finite dimensional approximations are notorious for
destroying an infinity of topological invariants (also known as
freezing-in integrals or Casimirs), which impose important constraints
on the evolution. An interesting alternative proposed by Zeitlin \cite
{Zeitlin91}, whereby an $N$ dimensional hydrodynamic-type system
conserves $\sim\sqrt{N}$ invariants, is not quite suitable for
continuum fields because of the implied periodicity in the Fourier
space corresponding to modulated point vortices in real space. It
would be very interesting to construct other ``meaningful'' (that is,
having many invariants) finite-mode hydrodynamics with well-behaved
real-space velocity fields.
So far, most statistical theories of continuum hydrodynamics,
\label{loc3}
most notably the absolute equilibrium ensemble (AEE) theory
\cite {Kraichnan67,Kraichnan75,FM76}, simply ignored all topological
invariants except quadratic ones, such as enstrophy and helicity in
hydrodynamics or magnetic helicity, cross helicity, and square vector
potential (in two dimensions) in magnetohydrodynamic turbulence. These
integrals were honored the special attention in part because of their
ruggedness (survivability under the approximation of a finite number of
Fourier modes), but mostly because of the convenience to handle
quadratic integrals.
Despite the dissatisfaction with such a reasoning (cf.~\cite{CF87}),
the attempts to incorporate all topological invariants in Gibbs
statistics have been less frequent, the examples including
Vlasov-Poisson system \cite{Lynden-Bell67} and 2D Euler turbulence
\cite {Kuzmin82,Miller90,RS91,MWC92}. For the reasons discussed in
Sec.~\ref{sec:conclusion} and Appendix~D, these attempts appear not
quite successful, because the non-Gaussianity of turbulent
fluctuations in these systems poses a fundamental difficulty in making
quantitative predictions about the equilibrium turbulent states.
\label{loc4}
The key problem is that the averaging with respect to the given
probability functional of a turbulence involves integration in a
functional (infinite dimensional) space. These averages are well
defined---that is, independent of the discretization procedure
involved, if and only if the probability functional is Wienerian, or
Gaussian in the highest-order derivative \cite{Isichenko95}.
Otherwise, the result of the averaging is sensitive to the arbitrary
choice of the sequence of discrete representations, which makes such
probability functionals ambiguous, to say the least.
In the present work we point out that these difficulties are absent
from a certain class of systems; namely, those where all integrals of
motion are not higher than quadratic in the highest-order-derivative
variable. Important examples of such systems, allowing a valid
Gibbs-ensemble description of turbulence, include two dimensional and
reduced magnetohydrodynamics.
The problem of accounting for all invariants is circumvented by the
representation of turbulence in the form of a gas of point vortices,
which is a very singular, and very special topologically, although an
asymptotically exact solution of the hydrodynamic equations. The
localization of vorticity in point vortices makes the topological
constraints trivially fulfilled for any motions. The conservation of
only energy and the Liouville theorem expressed in the convenient form
of the spatial variables of the vortices yield nicely to the
statistical mechanical description, although the thermodynamic limit
of infinitely many point vortices has long been a controversial issue
\cite {Onsager49,MJ73,ET74,FR82,BKH91}. It remains unclear to what
extent the gas of many point vortices represents a continuum two
dimensional turbulence. The frustrating dependence of the statistics
of point vortices on the arbitrary choice of their strengths was noted
by Onsager and reflects the above-mentioned fundamental difficulty in
the 2D Euler turbulence.
We pursue an analytical approach to continuum two dimensional ideal MHD
turbulence where all topological invariants are respected. We use the
Gibbs ensemble analysis to predict the following evolution of the
turbulence. An initial state evolves into (a)~a stationary, stable
coherent structure, which appears as the ``most probable state'' and
(b)~small-scale turbulence (fluctuations) with Gaussian statistics.
\label{loc5}
The Gaussianity of the MHD fluctuations was recently numerically
confirmed by Biskamp and Bremer \cite {BB93}. At large time the
fluctuations of the magnetic flux and the fluid stream function assume
vanishing amplitude and length scale (while containing finite energy
and dominating phase volume), and become essentially invisible on the
background of the coherent structure. In this sense, the coherent
structure can be regarded as an attractor or a relaxed state, although
the underlying dynamics is perfectly Hamiltonian.
The concept of ``statistical attractor,'' introduced by Vladimir Yankov
\cite {KY80,PY89,DZPSY89}, emphasizes the method of analysis and describes
this kind of Hamiltonian relaxation, when the excess of phase-space
volume and energy get hidden in obscure (small-scale) corners of the
infinite dimensional phase space. The fundamental difference between
the statistical attractors in nonintegrable wave-type systems
\cite {KY80,PY89,DZPSY89,Gruzinov93a} (where there are only a finite number
of integrals of motion) and the hydrodynamic-type systems (where the
number of integrals is infinite) is the universal shape of the
attractor---soliton---in the first case, and the nonuniversal shape of
the coherent structure---vortex---in the second case. The appearance
of the coherent structure depends on the initial condition, but is the
same for all initial conditions with identical topological invariants
[Eqs.~({6}) and ({7})]. In this sense, the asymptotically emerging
coherent structures (relaxed states) in hydrodynamic systems are
attractors only within a certain subclass (basin) of initial
conditions. These relaxed states can be called {\em topological
attractors}.
The specific of two dimensional MHD is that the coherent structure
inherits from the initial state all magnetic topology invariants, but
only a fraction of energy, the rest of which goes to the Gaussian
fluctuations. These invariant sharing properties can be interpreted in
terms of the well-known reasoning of turbulent cascades. In the case of
2D MHD the energy cascade is direct (i.e., toward the small-scale
fluctuations), while the magnetic topology cascades inversely (toward a
large-scale magnetic structure). However, compared to the cascade
description, the invariant sharing properties appear to present a
clearer physics of what happens to the conserved quantities in a closed
turbulent system. In fact, this allows us to predict the appearance of
the relaxed state, which should minimize energy subject to certain
topological constraints. One of the novelties of our analysis is using
a second functional set of ``cross'' topology invariants \cite
{MH84,IM87}, which was not used in MHD turbulence theories so far,
including the previous note \cite {Gruzinov93b}. We find that these
invariants have an important effect on the statistical description of
turbulence; specifically, the shape of the coherent structure is
sensitive to the cross topology invariants. In many features our
development is analogous to the Gibbs ensemble treatment of the
truncated Fourier representation of 2D MHD which was investigated by
Fyfe and Montgomery \cite{FM76}. The reason for this resemblance is
partial consistency of the Gibbs statistics with partial invariants
accounting. In fact, our theory shows that truncated 2D MHD equations
partially represent the true statistics of ideal 2D MHD.
Unlike neutral fluid dynamics, magnetohydrodynamics in two dimensions
are known to produce energetic small scales. This makes the difference
between 2D and 3D for MHD turbulence less drastic than that for fluid
turbulence. In our model, we observe two types of small-scale
behavior: (a)~for a generic (that is, topologically nontrivial) initial
condition, the coherent structure must have discontinuities in the form
of current sheets and (b)~the Gaussian fluctuations in the long evolved
state have both vanishing length scale and amplitude so that the
gradients and the energy are finite. The numerically observed
current-sheet-type structures \cite{BW89} are explained by our theory
as the singular coherent structures.
The paper is organized as follows. In Sec.~\ref{sec:equations} we
present the equations and the constants of the motion. In
Sec.~\ref{sec:stationary} we discuss the stationary solutions of the
MHD equations and formulate the Arnold variational principle in the
form suitable for 2D MHD. In Sec.~\ref{sec:gibbs} the canonical
ensemble approach to MHD turbulence is set forth
(Sec.~\ref{sec:ensemble}), and the most probable state
(Sec.~\ref{sec:coherent}) and the fluctuations about this state
(Sec.~\ref{sec:fluctuations}) are analyzed.
The key issue of how the integrals of motion are divided between the
coherent structure and the fluctuations is addressed in
Sec.~\ref{sec:partition}. In Sec.~\ref{sec:relaxation} we reformulate
the properties of the coherent structure using a variational principle
of iso-topological relaxation, which allows us to predict the
appearance of the structure. This prediction is then compared with
numerical results (Sec.~\ref{sec:numerical}). In
Sec.~\ref{sec:dissipation} we speculate on the role of small
dissipation and the relation between the resistive and the ideal MHD
relaxation in the kink tearing mode in tokamaks.
Section~\ref{sec:conclusion} restates the principal steps of our
statistical method and summarizes our work. Some technical details and
results not directly related to MHD turbulence are set in Appendices.
In Appendix~A we discuss the Lyapunov stability of MHD and Euler fluid
equilibria and point out the relation of minimum- and maximum-energy
stability to positive- and negative-temperature Gibbs states,
respectively. Appendix B addresses the Liouvillianity of the
eigenmodes that we use in the Gibbs statistics. In Appendix~C the
spectrum of the eigenmodes is studied. In Appendix~D we discuss the
application of the Gibbs-ensemble formalism to the turbulence of two
dimensional Euler fluid.
\section{Equations and constants of the motion}
\label{sec:equations}
We consider the set of equations of two dimensional incompressible
ideal magnetohydrodynamics (cf.~\cite {TMM86})
\begin{eqnarray}
{\partial}_t a &=& \{{\psi},a\}\ ,
\label{1} \\
{\partial}_t{\omega} &=& \{{\psi},{\omega}\} + \{j,a\}\ ,
\label{2}
\end{eqnarray}
where $\{A,B\}\equiv{\mbox{\boldmath $\bf\nabla$}} A\times{\mbox{\boldmath $\bf\nabla$}} B\cdot\wh{\bf z}$ denotes
the Poisson bracket, $\wh{\bf z}$ the unit vector in the $z$ direction,
${\psi}(x,y,t)$ the stream function of the fluid velocity field
${\bf v}={\mbox{\boldmath $\bf\nabla$}}\times({\psi}\wh{\bf z}),\; {\omega}=-{\mbox{\boldmath $\bf\nabla$}}^2{\psi}$ the fluid vorticity,
$a(x,y,t)$ the normalized vector potential of the magnetic field
${\bf B}=(4\pi{\rho})^{1/2}{\mbox{\boldmath $\bf\nabla$}}\times(a\wh{\bf z})$ (${\rho}$ being the constant fluid
density), and $j=-{\mbox{\boldmath $\bf\nabla$}}^2a$ the normalized electric current flowing
perpendicular to the $(x,y)$ plane. The boundary conditions $a={\psi}=0$ are
assumed at a rigid boundary encompassing the finite domain
with the area ${\cal S}$.
The incompressibility of the fluid motion is a reasonable approximation
in tokamaks where the strong toroidal magnetic field $B_z$ makes plasma
compression energetically expensive. In the reduced MHD approximation
\cite {KP73,RMSW76,Strauss76}, where the fast \Alfven\ and magnetosonic
waves due to $B_z$ are ignored, the strong uniform field $B_z$ drops
out of the equations of the motion.
The system ({1})---({2}) conserves the following quantities: the energy
\begin{equation}
E = E_m + E_f =
\frac{1}{2}\int\left[ ({\mbox{\boldmath $\bf\nabla$}} a)^2 + ({\mbox{\boldmath $\bf\nabla$}}{\psi})^2 \right]\,d^2{\bf x} =
\frac{1}{2}\int\left( aj + {\psi}{\omega} \right)\,d^2{\bf x}\ ,
\label{3}
\end{equation}
consisting of the magnetic part $E_m$ and the fluid part $E_f$,
the momentum (with translationally invariant, or in the absence of,
boundaries)
\begin{equation}
{\bf P} = \int{\bf v}\,d^2{\bf x}=
\frac{1}{2}\int{\bf x}\times\wh{\bf z}{\omega}\,d^2{\bf x}\ ,
\label{4}
\end{equation}
the angular momentum (with circular or no boundaries)
\begin{equation}
M\wh{\bf z} = \int{\bf x}\times{\bf v}\,d^2{\bf x}=
\frac{1}{3}\int{\bf x}\times({\bf x}\times\wh{\bf z}{\omega})\,d^2{\bf x}\ ,
\label{5}
\end{equation}
the magnetic topology invariants
\begin{equation}
I_F = \int F(a)\,d^2{\bf x}\ ,
\label{6}
\end{equation}
and the ``cross'' topology invariants
\begin{equation}
J_G = \int {\omega} G(a)\,d^2{\bf x} = \int G'(a){\mbox{\boldmath $\bf\nabla$}}{\psi}\cdot{\mbox{\boldmath $\bf\nabla$}} a\,d^2{\bf x}\ ,
\label{7}
\end{equation}
where $F$ and $G$ are arbitrary functions. Along with the continuum
set of integrals ({6}) and ({7}), we will also use their discretized
analogues,
\renewcommand{\theequation}{\arabic{equation}a}
\addtocounter{equation}{-2}
\begin{eqnarray}
I_n &=& \int a^n\,d^2{\bf x}\ ,
\label{6a}
\\
J_n &=& \int {\omega} a^n\,d^2{\bf x}\ ,
\label{7a}
\end{eqnarray}
\renewcommand{\theequation}{\arabic{equation}}
through which the continuum invariants can be Taylor expanded.
Strictly speaking, invariants ({6}) and ({7}) do not yet imply the
conservation of topology. Equation ({6}) only means that allowed
motions are incompressible interchanges of fluid elements together with
their ``frozen'' values of the magnetic flux $a$. The topology of the
contours of $a$ will be conserved only if these interchanges are
performed by continuous movements, a constraint which is not built into
Eq.~({6}) but follows from the equations of motion for smooth initial
conditions. Then the conservation of magnetic topology expressed by
Eq.~({6}) means that if the contour $a(x,y)=a_1$ initially lies inside
the contour $a(x,y)=a_2$, then this topological relation is preserved
by the motion; if a contour of $a$ has a hyperbolic (saddle, $x$)
point, this quality will also persist. In addition to the topological
constraints, the integrals ({6}) also specify the incompressibility of
the fluid, so that the area inside a given contour of $a$ remains
constant. The geometrical meaning of the integrals ({7}) is that the
amount of the fluid vorticity ${\omega}$ on a given contour of $a$ (to be
more precise, the integral of ${\omega}$ over the anulus between two
infinitesimally close contours) is conserved. Although stating
nothing of the contours of ${\omega}$ or ${\psi}$, the conservation of the
integrals ({7}) also bears certain topological relation between the
magnetic field and the vorticity field, which motivates our notation of
the ``cross topology invariants.''
The invariants $J_G$ appear to be poorly known, although the particular
member of the family ({7a})---the cross helicity
\begin{equation}
J_1 = \int {\omega} a\,d^2{\bf x} =
\frac{1}{\sqrt{4\pi{\rho}}}\int {\bf v}\cdot{\bf B}\,d^2{\bf x}
\label{8}
\end{equation}
---has been extensively discussed in the literature. The invariants
({7}) were first noted by Morrison and Hazeltine \cite {MH84}.
Independently, a similar set of integrals was used to study the vortex
stability in the framework of two dimensional electron MHD \cite
{IM87}. The idea towards the existence of a second set of topological
invariants is suggested by the observation that there is another
frozen-in quantity, namely the vorticity ${\omega}$, in the Euler limit
$a\equiv0$, which must have a counterpart in the MHD case $a\ne0$.
Once the existence of a second functional set of invariants is
suspected, it is not hard to guess the form of the topological
invariants ({7}). Although this is not straightforward, one can trace
the transition, as $a\to0$, from Eqs.~({{6}) and ({7}) to the Euler
invariants $\int F({\omega})\,d^2{\bf x}$.
Another way to find the topological invariants is to identify the
Hamiltonian structure using a noncanonical Poisson bracket \cite
{MH84}, whereby the topological invariants appear as Casimirs.
\section{MHD equilibria and Arnold's variational principle}
\label{sec:stationary}
The system of equations ({1})---({2}) has stationary solutions
satisfying
\begin{eqnarray}
\{{\psi},a\} &=& 0\ ,
\label{9}
\\
\{{\psi},{\mbox{\boldmath $\bf\nabla$}}^2{\psi}\} &=& \{a,{\mbox{\boldmath $\bf\nabla$}}^2a\}\ .
\label{10}
\end{eqnarray}
The equilibrium condition can be rewritten in a more convenient form by
substituting the functional dependence ${\psi}={\psi}(a)$, which is implied by
Eq.~({9}), into Eq.~({10}). Then, after simple manipulations, we find
\begin{equation}
\{{\Psi}(a),{\mbox{\boldmath $\bf\nabla$}}^2{\Psi}(a)\}=0\ ,
\label{9a}
\end{equation}
where
\begin{equation}
{\Psi}(a)=\int_0^a da'\sqrt{\pm\left(1-[d{\psi}(a')/da']^2\right)\,}\ ,
\label{9b}
\end{equation}
and the sign is chosen to make the square root real. As equation
(11) shows, any two dimensional MHD equilibrium with fluid flow
(${\psi}\ne0$) is reduced to a purely magnetic (${\psi}=0$) equilibrium for a
modified magnetic vector potential $a'={\Psi}(a)$. Note that the magnetic
field lines (the contour lines of the vector potential) are identical
for both the true field $a$ and the modified field $a'$, although the
values of $a$ and $a'$ on these lines are different.
The most evident stationary solution is given by an arbitrary circular
distribution ${\psi}={\psi}(r),\;a=a(r)$, where $r$ is the distance from the
origin. Another solution to ({9})---({10}) corresponds to the
identical zero in one of the Els\"asser variables, where
${\psi}(x,y)\equiv\pm a(x,y)$ is an arbitrary function of $x$ and $y$, so
that ${\Psi}(a)\equiv0$. For the case when $|{\psi}|$ and $|a|$ are not
identical, there exist many periodic and quasiperiodic solutions in the
form
\begin{equation}
C{\psi}({\bf x})=a({\bf x})=\sum_{m=1}^{N}A_m\cos({\bf k}_m{\bf x}+\theta_m)\ ,
\label{11}
\end{equation}
where the moduli of the wavevectors ${\bf k}_m$ are the same. In addition
to the smooth solutions one can devise a wide class of singular
solutions with appropriate boundary conditions at the lines of
discontinuity. Without additional physical constraints, such as
stability or topology, the class of all equilibria is too wide to be
useful. In Secs.~\ref{sec:gibbs} and \ref{sec:relaxation} we provide
such constraints, which specify the physically interesting (attracting)
equilibria. In many cases these equilibria must be singular.
There exists a profound relation between the stationary solutions and
the constants of the motion. For finite dimensional conservative
systems, the D'Alembert variational principle says that the energy
variation be zero at an equilibrium. For a hydrodynamic-type
conservative system, the counterpart of the D'Alembert theorem is the
Arnold variational principle. Originally formulated for the Euler
equation, but carried over without difficulty to other hydrodynamics,
the principle states that at a stationary (and only stationary)
solution the variation of energy, {\em subject to the conservation of
all topological invariants}, be zero:
\begin{equation}
{\delta} E\bigg|_{I_F,J_G={\rm const},\ \forall F {\ \rm and\ } G}=0\ .
\label{arnold}
\end{equation}
If, in addition, the second variation is definite---that is, the energy
assumes a nondegenerate conditional extremum, then the equilibrium is
Lyapunov stable. In fact, this was the search for stable fluid flows
which motivated Arnold's work. The power of the method lies in the
possibility to write the general iso-topological (iso-vortical in
Arnold's notation) variation in a closed form. When all the integrals
({6}) and ({7}) are to be conserved, such a variation is
\begin{equation}
{\Delta}\left(\begin{array}{l} a\\
{\omega}
\end{array}\right) =
{\delta}\left(\begin{array}{l} a\\
{\omega}
\end{array}\right) +
\frac{1}{2!}
{\delta}^2\left(\begin{array}{l} a\\
{\omega}
\end{array}\right) + \ldots\ ,
\label{12}
\end{equation}
where the infinitesimal iso-topological variation is given by
\cite {IM87}
\begin{equation}
{\delta}\left(\begin{array}{c} a\\
{\omega}
\end{array}\right) =
\left(\begin{array}{c} \{\mu,a\}\\
\{\mu,{\omega}\} + \{\nu,a\}
\end{array}\right)\ ,
\label{13}
\end{equation}
$\mu(x,y)$ and $\nu(x,y)$ being arbitrary functions. The operator of
finite variation ${\Delta}$ can be symbolically expressed through the
infinitesimal variation ${\delta}$ as
\begin{equation}
{\Delta} = \exp{\delta} - 1\ .
\label{14}
\end{equation}
It is easy to verify that the finite variation (15) conserves both sets
of integrals $I_F$ and $J_G$ to all orders in $\mu$ and $\nu$. The
form of the variation is suggested by the form of Eqs.~({1})---({2}),
where one can substitute the quantities ${\psi}$ and $j$ by arbitrary
${\partial}\mu/{\partial} t$ and ${\partial}\nu/{\partial} t$, respectively, to preserve only the
Poisson-bracket structure of the equations and thereby to
iso-topologically (that is, at constant $I_F$ and $J_G$) drag the
fields $a$ and ${\omega}$ to a new state, where the values of all other
integrals, if any, are generally different from those of the initial
state.
Let us see what happens to the energy ({3}) under the variation
(15)---(16). Writing the total change in the energy in the form
${\delta} E+{\delta}^2E/2!+\ldots$, we obtain after integrating by parts
\begin{equation}
{\delta} E = -\int[\mu\left(\{{\psi},{\omega}\}+\{j,a\}\right)+
\nu\{{\psi},a\}]\,d^2{\bf x}\ .
\label{15}
\end{equation}
By requiring that the first energy variation ({18}) be zero for all
$\mu$ and $\nu$ we arrive exactly at the system of equations
({9})---({10}) specifying the equilibrium solution. This strongly
suggests that the two sets of the topological invariants ({6}) and
({7}) are indeed complete, a condition necessary to apply statistics
(Sec.~\ref{sec:gibbs}) in a meaningful way.
The second iso-topological variation of energy can be used to
investigate the stability of MHD equilibria, as discussed in
Appendix~A.
\section{Gibbs statistics of two dimensional MHD turbulence}
\label{sec:gibbs}
We now wish to analyze the long-time evolution of the system subject to
Eqs.~({1}) and ({2}) for a given initial condition
$a_0({\bf x}),\;{\psi}_0({\bf x})$. The probability distribution functional
$P[a({\bf x}),{\psi}({\bf x})]$ can serve this purpose. This functional
specifies the relative probability, with respect to time measure, of
the spatial behaviors of various states $a({\bf x},t)$ and ${\psi}({\bf x},t)$.
As $P$ is invariant under the evolutional change of the fields $a$ and
${\psi}$, it must be a function of the constants of motion ({3})---({7}).
\subsection{Choice of statistical ensemble}
\label{sec:ensemble}
For a conservative Hamiltonian system, $P$ is given by the microcanonical
ensemble,
\begin{equation}
P_{MC}[a,{\psi}]=
{\delta}(E[a,{\psi}]-E_0)\,\prod_n{\delta}(I_n[a,{\psi}]-I_{n0})\,{\delta}(J_n[a,{\psi}]-J_{n0})\ ,
\label{MC}
\end{equation}
specifying a uniform distribution on the manifold of the specified
(initial) integrals of motion. It must be emphasized that the validity
of the microcanonical ensemble requires at least four assumptions.
\begin{enumerate}
\item
The phase space must be finite dimensional; that is, the fields $a$ and
${\psi}$ are parameterized by discrete dynamical variables
$f_m,\ m=1,2,\ldots,N$, so that the concept of measure in the space of
states is meaningful. This is a tricky issue as to how many variables
are needed (see discussion in Sec.~\ref{sec:introduction}).
\item
The motion on the manifold of conserved invariants is nonintegrable
(chaotic). The fundamental phenomenon lying behind the ergodic
behavior (uniform distribution over the manifold) is Hamiltonian chaos,
whose principal manifestation is the exponential divergence of nearby
trajectories. The chaotic motion is possible only if the
dimension of the manifold ($N-N_I$, where $N_I$ is the number of
invariants) is three or more (four or more to allow the Arnold
diffusion, so that all of the manifold might be visited by each
trajectory). The property of ergodicity was proved for very special
cases \cite {Sinai70} but is believed to be generally valid if the
dimension of the manifold is sufficiently large: $N-N_I\gg1$. The
remarkable accuracy of the classical thermodynamics is connected with
the macroscopic numbers of degrees of freedom ($N\sim10^{23}$) and only
a few invariants. It is also natural to expect that the microcanonical
statistics will work in turbulence (which can be loosely defined as
chaos in PDE), where the limit $N\to\infty$ should be carefully taken.
\item
The manifold of conserved integrals specified by the delta functions in
({19}) must be connected. The problem of connectivity of complicated
iso-surfaces is related to the percolation problem \cite {Isichenko92}.
The percolation threshold, above which the connectivity takes place, is
inversely proportional to the dimension of the iso-surface. This
suggests that in the continuum (infinite dimensional) limit the
connectivity of the manifold of conserved integrals should not be a
problem.
\item
The dynamical variables $f_m(t)$ must satisfy the Liouville theorem:
$\sum_m{\partial}\dot f_m/{\partial} f_m=0$. For non-Liouvillian variables a weighting
factor (the Jacobian of change to Liouvillian variables) should be
included in Eq.~({19}).
\end{enumerate}
In a hydrodynamic-type system, where the number of dynamical
constraints is infinite, we encounter another difficulty. Namely, any
attempt to restrict the dimension of the phase space without
restriction on the number of conserved quantities immediately drives
the manifold of conserved integrals into an empty set, where no mixing
may occur. Motivated by the experimental/numerical observation that
turbulence does exist, as well as by the functional arbitrariness of
the iso-topological variation ({16}), we adopt a hypothesis that there
exists a ``meaningful'' $N$ dimensional MHD approximation with at most
$N_I(N)$ conserved integrals, where both $N_I$ and $N-N_I$ go to
infinity as $N\to\infty$. (In Zeitlin's example \cite {Zeitlin91} for
the Euler fluid $N_I\simeq N^{1/2}$.) The specific form of this
approximation is unimportant for our arguments.
The microcanonical ensemble ({19}) is inconvenient to handle and is
commonly transformed into the more convenient canonical (Gibbs)
ensemble by integrating $P_{MC}$ over most of the dynamical variables
in the amount of $N_{th}\gg N-N_{th}\gg1$. These $N_{th}$ degrees of
freedom can referred to as ``thermal bath.'' The integration over the
thermal bath variables leads to an exponential dependence of the
resulting distribution on the integrals of motion expressed through the
remaining $N-N_{th}$ variables, the rest of the information being
stored in arbitrary constants called temperatures.
In our problem, the dimension $N'=N-N_{th}$ of the subsystem can
be also taken large, which amounts to another finite dimensional ($N'$)
MHD approximation. However, now we have the canonical distribution
over the remaining $N'$ variables,
\begin{equation}
P[a,{\psi}] =
\exp\left[-{\alpha}\left(
E[a,{\psi}] + \sum_n{\beta}_n I_n[a] + \sum_n{\gamma}_n J_n[a,{\psi}]
\right)\right]\ ,
\label{canonical}
\end{equation}
instead of the microcanonical one ({19}). A drastic simplification
achieved by the change of the ensemble is that in the
finite dimensional Gibbs distribution ({20}), where the fields $a$ and
${\psi}$ are parameterized by $N'$ modes and the summation over invariants
runs up to $N_I(N')$, we may extend the summation up to infinity
without significant change in the result, which was impossible for the
product of delta functions in the microcanonical ensemble ({19}).
In Eq.~({20}), the constants ${\alpha},\;{\alpha}{\beta}_n$ and ${\alpha}{\gamma}_n$ appear
as the reciprocal temperatures corresponding to each invariant. These
constants are to be determined from the initial state by solving
the infinite system of equations:
\begin{equation}
\left<E\right>_P=E_0\ ,
\quad
\left<I_n\right>_P=I_{n0}\ ,
\quad
\left<J_n\right>_P=J_{n0}\ ,
\quad n=0,1,2,\ldots\ ,
\label{conservation}
\end{equation}
expressing the conservation of the integrals of motion. Here the
subscript ``0'' refers to the initial state. As a result of solving
Eqs.~({21}), each parameter ${\alpha},\;{\beta}_n,$ or ${\gamma}_n$
($n=0,1,2,\ldots$) is a function of the infinity of the initial
invariants $E_0,\;I_{n0}$ and $J_{n0}$. It is emphasized that there is
no arbitrariness in the temperatures characterizing the Gibbs
distribution of a closed system. In fact, this is the central problem
in our theory how to determine these temperatures in order to predict
the final state from the given initial state.
The angular brackets in Eqs.~({21}) denote the ensemble averaging,
\begin{equation}
\left<A\right>_P=
\frac{\int A\, P[a,{\psi}]{\cal D} a{\cal D}{\psi}}{\int P[a,{\psi}]{\cal D} a{\cal D}{\psi}}\ ,
\label{averaging}
\end{equation}
which involves functional integrals over the space of the system
states. This kind of integrals do not always exist.
\label{loc6}
However, when the probability functional $P$ is Gaussian, the
functional integrals belong to the important class of Wiener integrals
\cite {Wiener58} (their complex counterparts are known as path integrals
\cite {FH65}), which are soluble and well-behaved. This is exactly
what we use in order to resolve the ultraviolet catastrophe. In fact,
we seek the long evolved state in the form of a coherent structure plus
small-amplitude fluctuations. This allows us to expand the integrals
in the exponential ({20}) about the coherent structure up to quadratic
terms, which will result in a Gaussian probability functional.
The uniform and additive (with respect to the eigenmodes) invariants is
another assumption lying behind the transition from the microcanonical
({19}) to the canonical ({20}) distribution functionals. The
additivity of the invariants can be achieved by the procedure of
diagonalization, which only works for quadratic forms.
In the spirit of the conventional statistical mechanics we call the
state specified by the detailed list of variables $(f_1,\ldots,f_N)$
{\em the microstate}, whereas the union of all microstates with the
same $(f_1,\ldots,f_{N'}),\ N'\ll N$ {\em the macrostate}. The entropy
$S$---a functional of the macrostate---is then introduced as the logarithm
of the number of various microstates corresponding to the given
macrostate. Up to an additive constant, $S$ is the logarithm of the
microstate phase volume on the manifold ({19}):
\begin{equation}
S[f_1,\ldots,f_{N'}]=
\ln\int P_{MC}[f_1,\ldots,f_N]\,df_{N'+1}\ldots df_N\ .
\label{S1}
\end{equation}
In other words, the entropy is simply the logarithm of the canonical
distribution functional ({20}),
\begin{equation}
S[a,{\psi}]=\ln P[a,{\psi}] = -{\alpha}(E[a,{\psi}]+I_{\beta}[a]+J_{\gamma}[a,{\psi}])\ ,
\label{S2}
\end{equation}
if $N'\gg1$. In Eq.~({24}), the integrals $I_{\beta}$ and $J_{\gamma}$ are defined
by Eqs.~({6}) and ({7}), respectively, and the functions
\begin{equation}
{\beta}(a)=\sum_n{\beta}_na^n \quad{\rm and}\quad
{\gamma}(a)=\sum_n{\gamma}_na^n
\label{be_and_ga}
\end{equation}
can be regarded as ``topological temperature functions.''
It is emphasized that the macrostate specified by $N'$ degrees of
freedom can be arbitrarily detailed, as we may let $N'\to\infty$ (while
preserving the requirement $N'\ll N$), so that formally there is little
difference between macrostates and microstates in a continuum system,
although the apparatus of the canonical distribution ({20}) and the
entropy ({24}) is much more convenient than that of the microcanonical
ensemble ({19}).
\subsection{Coherent structure: the most probable state}
\label{sec:coherent}
Maximizing the probability ({20}) or, equivalently, the entropy
({24}) yields ``the most probable state'' of the system.
Upon varying $S$ with no restriction on the field variations ${\delta} a$ and
${\delta}{\psi}$ we obtain
\begin{equation}
{\delta}(E[a,{\psi}]+I_{\beta}[a,{\psi}]+J_{\gamma}[a,{\psi}])=0\ ,
\label{MPS}
\end{equation}
which results in a stationary solution $\left(a_s({\bf x}),{\psi}_s({\bf x})\right)$
satisfying
\begin{eqnarray}
{\mbox{\boldmath $\bf\nabla$}}^2a_s &=& {\beta}'(a_s) + {\gamma}'(a_s){\mbox{\boldmath $\bf\nabla$}}^2{\gamma}(a_s)\ ,
\label{25}
\\
{\psi}_s &=& -{\gamma}(a_s)
\label{26}
\end{eqnarray}
[compare with ({9})---({10})]. There is nothing surprising in that the
most probable state is stationary, because varying a linear combination
of the energy and the topological integrals ({26}) amounts to the
Arnold (iso-topological) variation written with the help of Lagrange
multipliers. The relation (28) stating that the fluid flow is along
the magnetic field lines is characteristic of the ``dynamic alignment''
developing in course of turbulent MHD relaxation \cite {TMM86}.
Similarly to the transformation (11)---(12), we can rewrite
Eqs.~(27) and (28) in the form
\begin{equation}
{\mbox{\boldmath $\bf\nabla$}}^2{\Gamma}(a_s)={\beta}'(a_s)/{\Gamma}'(a_s)\ ,
\label{26a}
\end{equation}
where
\begin{equation}
{\Gamma}(a_s)=\int_0^{a_s} da\sqrt{1-[{\gamma}'(a)]^2\,}\ .
\label{26b}
\end{equation}
This representation of the coherent structure will be used in
Sec.~\ref{sec:relaxation}.
The quantity
\begin{equation}
{\gamma}'(a_s({\bf x}))=-\frac{{\mbox{\boldmath $\bf\nabla$}}{\psi}_s}{{\mbox{\boldmath $\bf\nabla$}} a_s}=
-\frac{{\bf v}_s}{{\bf B}_s/\sqrt{4\pi{\rho}}}
\label{Mach}
\end{equation}
is the local Mach number of the fluid flow. Although some interesting
phenomena may occur near the lines where $|{\gamma}'|=1$, we will restrict
our attention to the sub-\Alfvenic\ case $|{\gamma}'|<1$. A sound
motivation for this is found in the absence of maximum-energy states in
2D magnetohydrodynamics (see Appendix~A) and the relaxation of
turbulence to minimum-energy states where $|{\gamma}'|$ is necessarily less
than one [see Eq.~({44}) and Appendix~C]. Even though the initial
condition is highly super-\Alfvenic, $|{\mbox{\boldmath $\bf\nabla$}}{\psi}_0|\gg|{\mbox{\boldmath $\bf\nabla$}} a_0|$, the
necessary magnetic field will be generated by means of turbulent
dynamo.
In addition to equations (27) and (28), we must require that the
equilibrium state $(a_s,{\psi}_s)$ be actually the maximum of the entropy
$S$. This requirement, which is pursued in the next subsection, means
that the coherent structure must be Lyapunov stable, which is natural
to expect of a relaxed state. Indeed, the ``fine-grained entropy''
$S$, as defined by Eq.~({24}) is an integral of motion playing the role
of a Lyapunov functional.
\subsection{Fluctuations: the Gaussian turbulence}
\label{sec:fluctuations}
Now that we have identified (or, rather, assumed the presence of) the
coherent structure $(a_s,{\psi}_s)$, we seek solution to the problem
({1})---({2}) in the form
\begin{equation}
a({\bf x},t)=a_s({\bf x})+\wt a({\bf x},t)\ ,\quad
{\psi}({\bf x},t)={\psi}_s({\bf x})+\wt{\psi}({\bf x},t)\ ,
\label{decomposition}
\end{equation}
where the amplitude of the fluctuations ${\bf f}=(\wt a,\wt{\psi})$ is
expected (and below confirmed) to be small in the long-time limit.
With this in mind, we calculate the second variation of the entropy:
\begin{equation}
{\delta}^2S = -{\alpha}\int\Big[
({\mbox{\boldmath $\bf\nabla$}}\wt a)^2 + ({\mbox{\boldmath $\bf\nabla$}}\wt{\psi})^2 + {\beta}^*\wt a^2 + 2{\gamma}'(a_s)\wt a\wt{\omega}
\Big]\,d^2{\bf x}\ ,
\label{27}
\end{equation}
where we denote ${\delta} a=\wt a,\;{\delta}{\psi}=\wt{\psi}$, and
${\beta}^*\equiv{\beta}''(a_s({\bf x}))+{\omega}_s({\bf x}){\gamma}''(a_s({\bf x}))$. In order
for the fluctuations to be finite, ${\delta}^2S$ must be negative definite.
The integral quadratic form on the right hand side (RHS) of Eq.~(33)
can be represented as the matrix element $\left<{\bf f}|W|{\bf f}\right>$ of
the linear self-adjoint tensor operator,
\begin{equation}
W\left(\begin{array}{l}
\wt a \\
\wt{\psi}
\end{array}\right)=
\left(\begin{array}{rr}
({\beta}^*-{\mbox{\boldmath $\bf\nabla$}}^2) & \quad-{\gamma}'{\mbox{\boldmath $\bf\nabla$}}^2 \\
-{\mbox{\boldmath $\bf\nabla$}}^2({\gamma}'\ldots) & \quad-{\mbox{\boldmath $\bf\nabla$}}^2
\end{array}\right)
\left(\begin{array}{l}
\wt a \\
\wt{\psi}
\end{array}\right)
\equiv
\left(\begin{array}{c}
{\beta}^*\wt a+\wt j+{\gamma}'\wt{\omega} \\
-{\mbox{\boldmath $\bf\nabla$}}^2({\gamma}'\wt a) + \wt{\omega}
\end{array}\right)\ ,
\label{W}
\end{equation}
acting on a pair of functions ${\bf f}=(\wt a,\wt{\psi})$. The boundary
conditions are $\wt a=0$ (tangential magnetic field) and $\wt{\psi}=0$
(tangential velocity) at the boundary of the finite domain.
The orthonormal set of the eigenfunctions $(a_m,{\psi}_m)$ of $W$ provides a
natural representation of the fluctuations:
\begin{equation}
W\left(\begin{array}{l}
a_m\\
{\psi}_m
\end{array}\right)={\lambda}_m
\left(\begin{array}{l}
a_m\\
{\psi}_m
\end{array}\right)\ ,
\label{eigen}
\end{equation}
The standard definition of the orthonormality implies
\begin{equation}
\int(a_ma_n+{\psi}_m{\psi}_n)d^2{\bf x}={\delta}_{mn}\ .
\label{orthonormality}
\end{equation}
Upon expanding the fluctuation field
\begin{equation}
{\bf f}({\bf x},t)=\sum_m
f_m(t)\left(\begin{array}{l}
a_m({\bf x}) \\
{\psi}_m({\bf x})
\end{array}\right)\ ,
\label{28b}
\end{equation}
in a series over the complete set of the eigenfunctions, the probability
distribution of the fluctuations is conveniently written as
\begin{equation}
P[{\bf f}]\equiv\exp\left(\frac{{\delta}^2S}{2}\right) =
\exp\left(-\frac{{\alpha}}{2}\sum_m{\lambda}_m f_m^2\right)\ ,
\label{Gaussian}
\end{equation}
which is a Gaussian distribution.
In Appendix B we discuss the Liouvillianity of the variables $f_m$ and
show that the averages over the distribution ({38}) are done by
replacing ${\cal D} a{\cal D}{\psi}$ by $\prod_mdf_m$ in Eq.~({22}). Then the
fundamental averages are
\begin{equation}
\left<f_m\right>=0\ ,\quad \left<f_mf_n\right>={\delta}_{mn}/({\alpha}{\lambda}_m)\ .
\label{averages}
\end{equation}
The eigenvalues ${\lambda}_m$ depend on the temperature parameters
${\alpha},\;{\beta}_n$, and ${\gamma}_n$ and, through those, on the initial state.
However, the behavior of the eigenvalues becomes universal in the
ultraviolet ($m\gg1$) limit. In Appendix~C we show that the spectrum
of the matrix operator $W$ is similar to that of the standard scalar
Schr\"odinger operator $U({\bf x})-{\mbox{\boldmath $\bf\nabla$}}^2$, whose quasiclassical
eigenvalues are determined by the Bohr-Sommerfeld quantization rule:
\begin{equation}
{\lambda}_m^\pm\simeq C^\pm\frac{4\pi m}{{\cal S}}\ ,\quad m\gg k_s^2{\cal S}\ ,
\label{spectrum}
\end{equation}
where ${\cal S}$ is the area of the domain and $k_s$ the characteristic
wavenumber of the smooth part of the coherent structure. (In
Sec.~\ref{sec:relaxation} we show that the coherent structure can also
have singularities---current sheets---which are not important in this
context.) In the Schr\"odinger case the constant $C$ in Eq.~({40}) is
unity, whereas for the operator ({34}) there are two branches of
eigenmodes with
\begin{equation}
1-|{\gamma}'|_{\mathop{\scriptstyle\max}}<C^-<C^+<1+|{\gamma}'|_{\mathop{\scriptstyle\max}}\ .
\label{Cpm}
\end{equation}
As also shown in Appendix~C, the eigenfunctions behave in the
ultraviolet [Wentzel-Kramers-Brillouin (WKB)] limit as
\begin{equation}
{\psi}_m^\pm({\bf x})\simeq\pm a_m^\pm({\bf x})\ ,\quad m\gg k_s^2{\cal S}\ ,
\label{eigenfunctions}
\end{equation}
and the WKB wavenumber of the $m$th mode is
\begin{equation}
{\bf k}_m^2\simeq4\pi m/{\cal S}\ ,\quad m\gg k_s^2{\cal S}\ .
\label{wavenumber}
\end{equation}
The maximum of entropy (${\delta}^2S<0$) is equivalent to the
non-negativeness of all eigenvalues ${\lambda}_m$ of operator ({34}). It
appears difficult to formulate the exact criterion of the positive
definiteness of $W$ in the general case; however, a sufficient
condition can be derived by applying the Silvester criterion to the
integrand in (33) considered as a plain quadratic form of fifth order
expressed through the variables $\wt a,\ {\mbox{\boldmath $\bf\nabla$}}\wt a$, and
${\mbox{\boldmath $\bf\nabla$}}\wt{\psi}$. Then the result is
\begin{equation}
{\alpha}>0\ ,
\quad
|{\gamma}'|<1\ ,
\quad
{\beta}^*(1-{\gamma}'^2)>({\mbox{\boldmath $\bf\nabla$}}{\gamma}')^2\ .
\label{44}
\end{equation}
Under these (or perhaps milder) constraints, the coherent structure is
Lyapunov stable as realizing minimum of the conserved quantity
$-S/{\alpha}=E+I_{\beta}+J_{\gamma}$.
\subsection{Partition of conserved quantities between the coherent
structure and the fluctuations}
\label{sec:partition}
So the long evolved state of the 2D MHD turbulence involves two constituents,
namely the stationary, stable coherent structure $(a_s({\bf x}),{\psi}_s({\bf x}))$ and
the fluctuations $(\wt a({\bf x},t),\wt{\psi}({\bf x},t))$ distributed according to the
Gaussian law ({38}). The initial state's invariants are shared
between the structure and the fluctuations,
\begin{equation}
E_0=E_s+\wt E\ ,\quad I_{F0}=I_{Fs}+\wt I_F\ ,\quad J_{G0}=J_{Gs}+\wt J_G\ ,
\label{sharing}
\end{equation}
where the subscripts $0$ and $s$ refer to the initial state and the coherent
structure, respectively, and tilde to the fluctuations. The Gaussianity and
the integral sharing properties ({45}) follow from our assumption of
the small amplitude of the fluctuations, which we confirm below. In addition,
we establish that $\wt I_F=0$, which bears a useful topological corollary.
We start with the fluctuation energy
\begin{equation}
\wt E=\left<\frac{1}{2}
\int(\wt j\wt a+\wt{\psi}\wt{\omega}) d^2{\bf x}\right>=
\frac{1}{2}
\sum_{mn}\left<f_mf_n\right>\int(j_ma_n+{\psi}_m{\omega}_n) d^2{\bf x}\ .
\label{E1}
\end{equation}
Using formula ({39}) and eliminating $j_m$ and ${\omega}_m$ with the help
of Eqs.~({34}) and ({35}), Eq.~({46}) can be rewritten in terms of
only $a_m$ and ${\psi}_m$. At $m\gg k_s^2{\cal S}$ the principal term in the
fluctuation energy is
\begin{equation}
\wt E=\frac{1}{2{\alpha}}\sum_m
\int\frac{a_m^2-2{\gamma}'a_m{\psi}_m+{\psi}_m^2}{1-{\gamma}'^2}d^2{\bf x}\ .
\label{E2}
\end{equation}
The integrand in (47) is greater than $(a_m^2+{\psi}_m^2)/(1+|{\gamma}'|)$ and
less than $(a_m^2+{\psi}_m^2)/(1-|{\gamma}'|)$. As the orthonormality condition
({36}) then implies, the sum ({47}) diverges with the number
of eigenmodes $N\gg1$ {\em linearly:}
\begin{equation}
\wt E=\frac{C_NN}{2{\alpha}}\ ,\quad N\gg1, \quad
\frac{1}{1+|{\gamma}'|_{\mathop{\scriptstyle\max}}}\le C_N\le\frac{1}{1-|{\gamma}'|_{\mathop{\scriptstyle\max}}}\ .
\label{E3}
\end{equation}
Equation ({48}) is a remnant of the equipartition of energy between the
degrees of freedom (eigenmodes). At finite temperature the energy would
diverge as $N\to\infty$, which constitutes the well-known ``ultraviolet
catastrophe.'' However, $\wt E$ is bounded from above by the initial energy
$E_0$. Therefore the energy temperature $1/{\alpha}$ of the fluctuations should
decrease with the number $N$ of the effectively excited modes. From the
conservation of energy [Eq.~({45})] we infer
\begin{equation}
{\alpha}=\frac{C_NN}{2(E_0-E_s)}\ ,\quad N\gg k_s^2{\cal S}\ .
\label{al}
\end{equation}
Analogously to the fluctuation energy, $\wt I_F$ and $\wt J_G$ also diverge at
constant ${\alpha}$, as $N\to\infty$. However, the divergence of $\wt I_F$ is
only {\em logarithmic} in $N$, because the role of small scales is less
pronounced in the magnetic topology invariants ({6}), which involve no
derivatives of $a$ and ${\psi}$. [The linear divergence of $\wt E$ and $\wt J_G$
is due to the terms $({\mbox{\boldmath $\bf\nabla$}} a)^2$ and ${\omega}=-{\mbox{\boldmath $\bf\nabla$}}^2{\psi}$ in ({3}) and
({7}), respectively.] Similarly to ({46})---({48}), we have
\begin{equation}
\wt I_F=\frac{1}{2}\int F''(a_s)\left<\wt a^2\right>d^2{\bf x}=
\sum_m\frac{1}{2{\alpha}{\lambda}_m}\int F''(a_s)a_m^2\,d^2{\bf x}\ ,
\label{I1}
\end{equation}
and, in accordance with Eqs.~({36}) and ({40}),
\begin{equation}
|\wt I_F|\le
\frac{|F''|_{\mathop{\scriptstyle\max}}}{2{\alpha}}\sum_m\frac{1}{{\lambda}_m}\le
\frac{|F''|_{\mathop{\scriptstyle\max}}{\cal S}}{8\pi{\alpha}(1-|{\gamma}'|_{\mathop{\scriptstyle\max}})}\,\ln N\ .
\label{I2}
\end{equation}
Upon substituting expression ({49}) into Eq.~({51}) we
obtain
\begin{equation}
|\wt I_F|\le
\frac{|F''|_{\mathop{\scriptstyle\max}} E_0{\cal S}}{4\pi}\,
\frac{1+|{\gamma}'|_{\mathop{\scriptstyle\max}}}{1-|{\gamma}'|_{\mathop{\scriptstyle\max}}}\,
\frac{\ln N}{N}\ .
\label{I3}
\end{equation}
Thus we conclude that
\begin{equation}
\wt I_F\,\to\,0
\quad{\rm as}\quad
N\,\to\,\infty\ .
\label{I=0}
\end{equation}
That is, in the long evolved state, the invariants $I_F$ are exclusively
contained in the coherent structure $(a_s,{\psi}_s)$, which therefore inherits the
exact magnetic topology of the initial state. On the contrary, the energy and
the cross topology invariants, due to their linear divergence at $N\to\infty$,
are shared between the coherent structure and the fluctuations.
Analogously to conserved quantities, we can estimate the mean square
norm of the fluctuations:
\begin{equation}
\left<{\bf f}|{\bf f}\right>\equiv
\int(\wt a^2 + \wt{\psi}^2)d^2{\bf x}=
\sum_m\frac{1}{{\lambda}_m}\le
\frac{E_0{\cal S}}{2\pi(1-|{\gamma}'|_{\mathop{\scriptstyle\max}})}\,\frac{\ln N}{N}
\,\to\,0\ ,\quad{\rm as}\quad N\to\infty\ .
\label{NORM=0}
\end{equation}
Thus the mean square amplitude of the fluctuations, measured in the magnetic
flux and the stream function, goes to zero in the continuum, or the long-time
limit $N\to\infty$. It is emphasized that the amplitude of {\em all}, not only
higher, fluctuation modes goes to zero.
The assumption of $\wt a\ll a_s,\;\wt{\psi}\ll{\psi}_s$ was indeed necessary for the
quadratic expansion of the probability functional $P[a,{\psi}]$ near the
equilibrium leading to the well-behaved Gaussian distribution. In fact, the
small amplitude of $\wt a$ and $\wt{\psi}$ is not sufficient to apply the Gibbs
formalism, because the fluctuations in ${\mbox{\boldmath $\bf\nabla$}}\wt a$ and ${\mbox{\boldmath $\bf\nabla$}}\wt{\psi}$, which
enter the integrals of energy ({3}) and cross topology ({7}), are found
to be not small. Fortunately, the quadratic expansion of Eqs.~({3}) and
({7}) is also valid, because the energy and the cross topology invariants
are {\em themselves quadratic (bilinear) with respect to the derivatives}
$\nabla a$ and $\nabla \psi$.
This fortune is not extended to many other systems, most notably the
two dimensional Euler equation, where the resulting non-Gaussianity of the
fluctuations makes it difficult to draw any quantitative conclusions based on
the Gibbs ensemble (see Appendix~D).
\section{Iso-topological relaxation and topological attractors}
\label{sec:relaxation}
Now the formal procedure of varying the integral of entropy ({26})
[which leads to the equilibrium (28)---(29)] can be rendered more
physical sense. Namely, the coherent structure minimizes the energy subject to
the conservation of all magnetic topology invariants ($I_F,\ \forall F$) and one
of the cross topology invariants ($J_{\gamma}$) of the initial condition:
\begin{equation}
\min\,E[a,{\psi}]\bigg|_{
I_F[a]={\rm const},\ \forall F,\ {\rm and}\ J_{\gamma}[a,{\psi}]={\rm const}}\ .
\label{isotop}
\end{equation}
The term ``iso-topological relaxation'' describes a process whereby
this minimization may be achieved. As we are now interested in the coherent
(coarse-grained) part of the MHD system, and the fluctuations are set aside,
the discussed relaxation is no longer Hamiltonian and resembles that
occurring in dissipative systems. In fact, the usual dissipation in
macroscopic systems also originates from the purely Hamiltonian
molecular dynamics, where one is not interested in the microscopic
degrees of freedom.
Due to the seemingly dissipative nature of the iso-topological
relaxation, the relaxed state may be considered as an attractor. The
qualitative arguments developed in this section predict the appearance
of the relaxed state without solving the complicated nonlinear problem
of the reconstruction of the functions ${\gamma}(a)$ and ${\beta}(a)$ from the
initial state, a task which must be complete for a quantitative
prediction and appears to be feasible only numerically.
Examine the equation of the coherent structure (29), or its variational
form (55). Introduce the modified magnetic flux function by the
ansatz
\begin{equation}
a'={\Gamma}(a)\ ,
\label{ansatz}
\end{equation}
where the function ${\Gamma}(a)$ is defined by Eq.~(30) and is in principle
known from the initial condition. We note that the conservation of the
magnetic topology invariants $I_F$ for the field $a$ is equivalent to the
conservation of those for the modified field $a'$. Then Eq.~(29) can
be interpreted as an equilibrium condition for the modified magnetic field $a'$
with no fluid flow. In other words, the problem reduces to the incompressible,
iso-topological minimization of the modified magnetic energy
\begin{equation}
E'_m=\frac{1}{2}\int({\mbox{\boldmath $\bf\nabla$}} a')^2\,d^2{\bf x}\ ,
\label{45}
\end{equation}
starting from the specified initial condition $a'_0({\bf x})={\Gamma}(a_0({\bf x}))$.
Upon minimizing Eq.~(57) subject to the iso-topological variation ${\delta}
a'=\{\mu,a'\}$ with arbitrary $\mu(x,y)$ results in $\{{\mbox{\boldmath $\bf\nabla$}}^2a',a'\}=0$,
implying a functional dependence between the modified magnetic flux and the
modified current.
Once the relaxed state $a'_\infty({\bf x})$ is found, the coherent structure of
the original magnetic field is recovered by inverting Eq.~(56):
\begin{equation}
a_s({\bf x})={\Gamma}^{-1}(a'_\infty({\bf x}))\ .
\label{relaxed}
\end{equation}
Then the stream function of the fluid flow in the coherent structure is given by
\begin{equation}
{\psi}_s({\bf x})=-{\gamma}(a_s({\bf x}))\ .
\label{relaxed_psi}
\end{equation}
The kind of relaxation undergone by the modified field $a'$ will take
place in an incompressible, viscous fluid with an ideal conductivity,
where the viscosity damps down the fluid motion. It is well known that
such an iso-topological relaxation may not be attainable in the class
of smooth magnetic fields \cite {Syrovatskii71,Arnold74,Moffatt90}. In
two dimensions, these are the saddle ($x$) points of the initial
magnetic field that lead to singularities---current sheets---in the
relaxed field. It is important to note that the location and the shape
of the current sheet is not locally determined by the $x$ point alone,
but rather depends on the shape of the separatrix coming through the
$x$ point. The qualitative arguments of Ref.~\cite {Gruzinov93a} show
that each initial magnetic separatrix, in course of the iso-topological
relaxation, turns into a characteristic structure with a current
sheet---the {\em asymptotic separatrix structure\/} shown in Fig.~1.
\begin{figure}
\centerline{
\psfig{figure=ass.eps,width=6.5in}
}
\caption{
Asymptotic separatrix structures resulting from iso-topological
relaxation. Topologically nontrivial initial state (a) leads to the
formation of a relaxed state (b) with current sheets (shown bold). The
arrows indicate the direction of the magnetic field.
}
\end{figure}
The orientation of the current sheet is such as to lie within a ``figure
eight'' separatrix and to border the outside of an ``inside-out figure
eight.''
It appears that the final state $a'_\infty$ of the iso-topological
relaxation is uniquely determined by the initial state $a'_0$. The
same is true of the corresponding magnetic fluxes $a_s$ and $a_0$, once
the function ${\gamma}(a)$ [and thereby ${\Gamma}(a)$] is known. Even without
any information about ${\gamma}(a)$ the appearance of the relaxed state is
well understood qualitatively through the above construct, because
applying a function to $a$ does not change the geometry of magnetic
field lines.
\subsection{Comparison with numerical data}
\label{sec:numerical}
The computation of the long-time evolution of nearly ideal 2D MHD
turbulence reported by Biskamp and Welter \cite {BW89} clearly shows
current sheets terminating at $Y$ points, which are characteristic of
the asymptotic separatrix structures, although the reconnection due to
finite magnetic (hyper)diffusivity smears out the individual topology
of separatrices.
The spatial distribution of the fluctuations $(\wt a,\wt{\psi})$ is
determined by the eigenfunctions of operator ({34}). The potential of
this operator involves second spatial derivatives of $a_s$ (through the
term ${\beta}^*$). The singularities of $j_s=-{\mbox{\boldmath $\bf\nabla$}}^2a_s$ are
delta-function singularities at current sheets. This must lead to the
localization of the wavefunctions (the fluctuations) near the potential
wells (the current sheets) where ${\gamma}''(a_s){\mbox{\boldmath $\bf\nabla$}}^2{\gamma}(a_s)<0$. This
kind of localization of the microscopic turbulence near the current
sheets is indeed observed in the computation of Ref.~\cite {BW89}.
Earlier simulations of turbulent magnetic reconnection \cite
{ML86,MM81} also confirm this picture.
\subsection{Iso-topological relaxation and magnetic reconnection}
\label{sec:dissipation}
\label{loc7}
So far we were mostly concerned with the ideal model of two-dimensional
MHD, and the question is in order as to the evolution of a more
realistic dissipative system involving finite electrical resistivity
and fluid viscosity.
In general, this is a very difficult problem, because no
straightforward perturbation theory can be built for small coefficients
appearing in front of higher derivatives in the equations. We
therefore restrict ourselves to the qualitative analysis of the role of
small dissipation.
If the dissipation is small, the system behavior clearly must resemble,
up to a certain point, the prediction of the ideal MHD theory. The
deviation of a weakly dissipative evolution from the ideal behavior is
always a matter of time of the evolution. In order to neglect the
effects of dissipation in MHD turbulence relaxation, not only must the
resistivity ${\eta}$ and the viscosity $\nu$ be small but also the length
scales should be sufficiently large. The ideal MHD evolution discussed
above does lead to the formation of small scale structures which
trigger, in the long run, the strong effects of the weak dissipation.
If the initial state is smooth, the small-scale structures do not
appear at ones; it takes several nonlinear (eddy turnover) times for
the small scales to show up. In the meantime, the system evolves
towards, however not quite attains, the ideal statistical equilibrium.
In fact, the principal manifestation of approaching the statistical
equilibrium is the separation of scales into long-wavelength coherent
structures and short-wavelength fluctuations. It is reasonable to
assume that this separation of scales is not only necessary, but also
{\em sufficient\/} for the statistical equilibrium to set in. Then, by
the time when the initially small dissipation becomes important, the
coherent part of the turbulent field is essentially built by the
statistical mechanics of ideal MHD turbulence. The smaller the
dissipation, the shorter scales are allowed to evolve in the
Hamiltonian fashion, and therefore the closer the attained shape of the
coherent structures to the exact predictions of the Gibbs-ensemble
theory.
The time scale ${\tau}^*$ specifying the crossover from the ideal regime to
the dissipative regime is certainly much shorter than the diffusive
time ${\tau}_{\eta}\sim(k_s^2c^2{\eta})^{-1}$ and may not be very long compared
to the characteristic nonlinear time ${\tau}_A$. Numerical
results \cite{MSMOM91,MMSMO92} indicating the enstrophy decay in 2D
fluid in just a few eddy turnover times suggest that ${\tau}^*/{\tau}_A$ is a
small power or even logarithm of the large Reynolds number. The fast
crossover to the dissipative regime directly indicates the fast
production of small scales and, therefore, the equally fast approaching
to the statistical equilibrium.
After the approximate equilibrium is set in, the dissipation takes over
and the small-scale fluctuations are significantly damped over several
times ${\tau}^*$, whereas the coherent structures remain little affected,
at least in the case when these structures involve no singularities.
If the initial magnetic field has $x$ points, the coherent structure
will develop current sheets. The coherent structure will then undergo
fast magnetic reconnection. The reconnection occurs in a
characteristic time ${\tau}_r$ much longer than the \Alfvenic\ time
${\tau}_A$, if the magnetic Reynolds number $R_m={\tau}_{\eta}/{\tau}_A$ is large.
By different models, ${\tau}_r/{\tau}_A$ ranges from $R_m^{1/2}$
\cite {Sweet58,Parker57} to $(\ln R_m)^{p},\ p>0$ \cite
{Petschek64,Sonnerup70}, although the former (Sweet-Parker) model
appears to be more typical \cite {Biskamp85}.
So the ideal MHD turbulence theory describes the early,
$t<\min({\tau}^*,{\tau}_r)$, iso-topological stage of the turbulent MHD
relaxation and predicts the appearance of the coherent structures
entering the later stages where magnetic reconnection and/or viscosity
play the dominant role. Even then, some topological invariants survive
better than others, also providing useful variational tools for the
prediction of fully relaxed \cite
{Taylor74,Kadomtsev75,Taylor86,MS87,AT91}
\label{loc8}
or selective-decay \cite {MSMOM91,MMSMO92} states.
Our theory can be used to qualitatively describe the relatively early
stage, ${\tau}_A\ll t\ll({\tau}_A{\tau}_{\eta})^{1/2}$, of the nonlinear kink
tearing mode in a tokamak, where two dimensional MHD models are
commonly used for helically symmetric magnetic perturbations \cite
{KP73,Kadomtsev75,RMSW76,Waelbroeck89}. The kink tearing is
accompanied by changes in magnetic topology. First, an $x$ point in
the ``auxiliary magnetic field'' ${\bf B}_*={\bf B}-q{\bf B}_\th$ is created
near the linearly unstable $q=1$ surface. Then the resulting
``magnetic bubble'' is pushed to the exterior of the plasma column by
essentially ideal MHD motions. This process is likely to be of
turbulent nature and, until very small scales are generated, the ideal
turbulent relaxation will proceed in the direction of forming a
coherent structure with a current sheet corresponding to the initial
$x$ point, as suggested by the Gibbs statistics. This stage of
evolution may be pretty long, as the magnetic Reynolds number in
tokamaks can be of order $10^6$ and more. Later on, magnetic
reconnection via the current sheet \cite {Biskamp85} will occur at a
characteristic time of order $({\tau}_A{\tau}_{\eta})^{1/2}$. Dynamically, the
reconnection develops through a sequence of singular MHD equilibria
with the same local helicity
\cite {Kadomtsev75}, as analytically described by Waelbroeck
\cite {Waelbroeck89}. The first of the sequence of these current-sheet
equilibria can be interpreted as the asymptotic separatrix structure
arising from the initial state via the iso-topological turbulent
relaxation.
\section{Summary and conclusion}
\label{sec:conclusion}
The main result of this paper lies in working out the Gibbs statistics
for a Hamiltonian PDE system with an infinity of constants of the
motion. This formalism was demonstrated in the example of two
dimensional magnetohydrodynamics but can be carried over to other
systems. We review again the principal steps of our approach in terms
of a general nonlinear Hamiltonian system describing the fields
${\mbox{\boldmath $\bf\psi$}}({\bf x},t)$ and having a finite or an infinite number of invariants
${\bf I}[{\mbox{\boldmath $\bf\psi$}}]=I_1[{\mbox{\boldmath $\bf\psi$}}],I_2[{\mbox{\boldmath $\bf\psi$}}],\ldots$. Here it does not matter
what these invariants are; one can think of $I_1$ as the energy and of
the rest as topological invariants.
\renewcommand{\labelenumi}{(\alph{enumi})}
\begin{enumerate}
\item
The solution to the underlying nonlinear system is sought in the form
${\mbox{\boldmath $\bf\psi$}}({\bf x},t)={\mbox{\boldmath $\bf\psi$}}_s({\bf x})+\wt{\mbox{\boldmath $\bf\psi$}}({\bf x},t)$, where ${\mbox{\boldmath $\bf\psi$}}_s({\bf x})$
is yet unspecified stationary, Lyapunov stable solution (coherent
structure). We then anticipate that the amplitude of the fluctuation
field $\wt{\mbox{\boldmath $\bf\psi$}}({\bf x},t)$ is going to be small and hence the exact
integrals of motion can be expanded about the coherent structure
${\mbox{\boldmath $\bf\psi$}}_s$ up to quadratic terms: ${\bf I}={\bf I}_s+\wt{\bf I}$.
\item
The Gibbs ensemble is introduced in the fluctuation space in the
standard form of the exponential of a linear combination of all
invariants, $P[\wt{\mbox{\boldmath $\bf\psi$}}]=\exp(-{\mbox{\boldmath $\bf\alpha$}}\cdot\wt{\bf I})$, where
${\mbox{\boldmath $\bf\alpha$}}={\alpha}_1,{\alpha}_2,\ldots$ are the reciprocal temperatures to be
determined from the initial state. Having an infinity of Casimirs,
which depend on an arbitrary function, is not an obstacle, because any
linear combination of the Casimirs is again one of them. In order to
have non-diverging fluctuations, we exercise our right to suitably
choose the coherent structure ${\mbox{\boldmath $\bf\psi$}}_s$. Namely, ${\mbox{\boldmath $\bf\psi$}}_s({\bf x})$ is
required to minimize the linear combination ${\mbox{\boldmath $\bf\alpha$}}\cdot{\bf I}[{\mbox{\boldmath $\bf\psi$}}]$ of
the invariants, where the stationarity of the resulting state is
ensured by the Arnold variational principle. Then ${\mbox{\boldmath $\bf\alpha$}}\cdot\wt{\bf I}$
is a positive definite quadratic form, and the Gibbs distribution of
the fluctuations is a Gaussian distribution. From now on, the standard
Boltzmann-Gibbs statistics is applied in a straightforward way, at
least for a finite dimensional approximation using $N$ eigenmode
amplitudes $f_m$ satisfying the Liouville theorem.
\item
The eigenmodes are introduced such as to diagonalize the Gibbs
exponential, ${\mbox{\boldmath $\bf\alpha$}}\cdot\wt{\bf I}=({\alpha}_1/2)\sum_m{\lambda}_mf_m^2$. Then
averages can be computed in the conservation laws,
${\bf I}_0={\bf I}_s+\left<\wt{\bf I}[\wt{\mbox{\boldmath $\bf\psi$}}]\right>$, in order to infer the
equations for the temperatures.
\item
The fluctuations' share of the invariants,
$\left<\wt{\bf I}[\wt{\mbox{\boldmath $\bf\psi$}}]\right>=\sum_m1/(2{\alpha}_1{\lambda}_m){\partial}^2\wt{\bf I}/{\partial}
f_m^2$, when expanded in the eigenmodes, turns out to diverge as
$N\to\infty$, unless the temperatures $1/{\alpha}_m$ are let to zero (even
then the temperature ratios remain finite and keep useful information).
This is the ``ultraviolet catastrophe.'' The regularization of this
divergence requires the reciprocal temperatures to also diverge, e.g.,
${\alpha}_1(N)\propto N$.
\item
If the square norm of the fluctuations
$\left<\wt{\mbox{\boldmath $\bf\psi$}}|\wt{\mbox{\boldmath $\bf\psi$}}\right>=\sum_m({\alpha}_1{\lambda}_m)^{-1}$ diverges at
constant temperatures slower (e.g., logarithmically) than the fastest
diverging invariant (say, the energy $I_1$), then the average norm goes
to zero as $N\to\infty$. This is the crucial point, which justifies
the assumption of the small amplitude necessary for the Gaussianity of
the fluctuations in the given representation. If this condition is not
fulfilled, one can always pick other variables involving lower order of
derivatives, such as ${\mbox{\boldmath $\bf\psi$}}'={\mbox{\boldmath $\bf\nabla$}}^{-2}{\mbox{\boldmath $\bf\psi$}}$, and repeat the above
steps. However, the exact Gaussianity of the fluctuations requires
another important property of the integrals of motion, which is
independent of the variables used. Namely, {\em each invariant must be
not more than quadratic in the highest-order-derivative variables}.
Then the quadratic expansion will be also valid even for those (fastest
diverging) invariants, whose fluctuations are finite. This property
holds for 2D MHD, but it does not for 2D Euler turbulence or
Vlasov-Poisson system (Appendix~D).
\item
If, in addition, there are invariants (such as magnetic topology
invariants) diverging slower than the fastest diverging integral of
motion, then the average fluctuation's share of those invariants
vanishes as $N\to\infty$. The presence of such invariants simplifies
the analysis of the coherent structure.
\end{enumerate}
\renewcommand{\labelenumi}{\arabic{enumi}.}
We use the above steps to study the relaxation of ideal two dimensional
MHD turbulence, where both infinite sets of topological invariants,
magnetic ({6}) and cross ({7}), are incorporated. We show that
accounting for all topological invariants leads to the prediction that
the long evolved MHD turbulent state consist of a coherent structure
(the most probable state) and a small-amplitude, small-scale Gaussian
turbulence (the fluctuations). The fluctuations are small if measured
in terms of the magnetic vector potential $\wt a$ and the flow stream
function $\wt{\psi}$. The fluctuations in the magnetic field $\wt{\bf B}$
and the fluid velocity $\wt{\bf v}$ are of the same order as in the
coherent structure. The fluctuation current $\wt{\bf j}$ and vorticity
$\wt{\omega}$ are infinite in the long time limit.
We find that in 2D ideal MHD turbulence the coherent structure has the
same magnetic topology as the initial state, while energy and cross
topology are shared between the coherent structure and the
fluctuations. Therefore, for a sufficiently wide class of initial
conditions having the same topological invariants, the final coherent
state is the same, whereas the fluctuations, when measured by the
standard norm (54), become asymptotically ``invisible.'' In this sense,
the coherent structures emerging from the turbulent MHD relaxation can
be regarded as ``topological attractors,'' even though the underlying
dynamics is perfectly Hamiltonian. (The theorem of the absence of
attractors in Hamiltonian systems is not valid for infinite dimensional
PDE systems.)
The presence of the fluctuations on the top of the coherent structure
is conceptually important even though the amplitude of these
fluctuations goes to zero in the long time limit: the fluctuations
appear as the storage of the ``lost'' integrals of motion, if only the
most probable state is compared with the initial state. This explains
the well-known result (cf.~\cite {Miller90}), that the topological
invariants of the coherent vortex emerging from 2D Euler turbulence,
are different from those of the initial state. In 2D
magnetohydrodynamics the role of the initial topology is more
important. In Appendix~D we discuss the application of the
Gibbs-ensemble formalism to the two dimensional Euler equation.
We formulate the variational principle of iso-topological relaxation,
which allows us to predict the shape of the coherent structure for the
given initial state. We show how the problem of the ideal MHD
relaxation with plasma flow is reduced to the viscous relaxation of
magnetic field with no flow in the final state. The numerical results
suggest that the asymptotic separatrix structures with current sheets
are indeed observed during the turbulent relaxation. It appears that
these structures are the route to reconnection in the nonlinear kink
tearing mode in tokamaks.
Many problems of MHD turbulence remain, most notably the role of small
dissipation. As discussed in Sec.~\ref{sec:dissipation}, this is the
dynamics of producing small scales which determines when and how the
dissipative processes become important. In order to study the
phenomena of crossover from the ideal to the dissipative turbulent
relaxation, the nonequilibrium dynamics of the ideal relaxation must be
worked out.
It appears that the formalism of the weak turbulence theory
\cite{FS91,ZLF92,Pomeau92b} can be appropriately suited for
Eq.~(\ref{f-dynamics}) in order to study the nonequilibrium
statistics of 2D MHD turbulence.
\label{loc9}
However, the important role of the
ideal Gibbs turbulence for weakly dissipative systems is found in that
the ideal turbulence forms predictable coherent structures, which enter
the later, dissipative stages of the turbulent evolution.
The comparison of the MHD and the Euler turbulence prompts us to
distinguish between three kinds of advected fields. The first kind is
passive field, such as the concentration of a dye or the temperature
that do not affect the advecting velocity field. Passive fields tend
to become spatially uniform due to turbulent diffusion. The second
kind is active field, such as the vorticity in Euler fluid, which does
affect the velocity field but whose lines or contours can be
indefinitely stretched at no significant energy price. The active
fields therefore tend to self-organize assuming topologically simple
structures, like monopole vortices, whose topology is different from
that of the initial state. The third kind can be referred to as
``reactive field,'' such as the magnetic flux frozen into an ideally
conducting fluid. Stretching of magnetic field lines is energetically
expensive and cannot last indefinitely. The topology of the reactive
field is therefore much more robust than that of passive or active
fields, and the self-organization can lead to nontrivial coherent
structures with singularities (current sheets).
\subsection*{Acknowledgments}
We wish to thank V.~V. Yankov, P.~J. Morrison, F.~L. Waelbroeck, F.
Porcelli, P.~H. Diamond, G.~E. Falkovich, and J.~B. Taylor for
stimulating discussions. This work was partially supported by the
U.S.~Department of Energy under Contracts No.~DE-FGO3-88ER53275 and
DE-FG05-80ET53088.
\clearpage
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proofpile-arXiv_065-665
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\section{Introduction}
The study of exactly solvable potentials, for which the
quantum mechanical eigenfunctions may be expressed in terms of
hypergeometric functions, has a long and varied history. One approach
is an algebraic solution of the problem. Early work by Infeld and
Hull classified factorizations of the Schroedinger operator for solvable
potentials which then allow one to generate other solutions to the
problem. \cite{Infeld_Hull} A related technique, supersymmetric
quantum mechanics, discovered
as a limiting case ($d=1$) of
supersymmetric field theory, was introduced by Witten and later
developed by other authors. \cite{SUSY_QM} In
particular,
Gendenshtein gave a
criteria, shape invariance, which when satified insures that the
complete spectrum of the supersymmetric Hamiltonian may be found.
\cite{Gendenshtein} Finally, spectrum generating algebras, whose use
dates back to Pauli's work on the hydrogen atom, have been studied
more recently as a method to find the spectrum and eigenstates of
solvable potentials. \cite{Pauli,SGA}
Another method to find the energy eigenvalues and wavefunctions of a
solvable potential is to use an operator transformation, essentially a
change of independent and dependent variables, to relate it to a
Schroedinger equation for a potential whose solutions are known.
Duru and Kleinert described such a method for transforming the resolvant
operator, whose matrix element is the propagator. \cite{Duru_Kleinert}
They used this
technique to transform the time-sliced form of a path integral into a
known path integral, such as that for the harmonic oscillator, by
transforming both the space and time variables in the path integral
expression. We will discuss these transformations, outside the
context of path integrals, in the next section.
We will show that the operator transformations not only allow one to find
algebraic relations between the Fourier transform of the propagators
for two different quantum systems but also, in the case of a real
transformation function, provide a mapping between the group
generators for the spectrum generating algebra. Thus quantum systems which may
be mapped to one another by real Duru-Kleinert transformations have
the same formulation in terms of the enveloping algebra of the same
Lie group.
In the first section we describe the operator transformations with
special attention to how the measure for the normalization of states
transforms. We next illustrate
the method with a derivation of the relation between the propagators
for the trigonometric Poschl-Teller and Rosen-Morse potentials and
give the relations for the propagators for some other exactly solvable
potentials. Finally we examine the corresponding
transformation of the Lie group generators.
\section{Operator Transformations and Causal Green's Functions}
We will consider transformations of the Fourier transform of the
causal propagator for a quantum mechanical system. Hereafter
operators
will be denoted by a caret. The propagator is
given by
\beq
K(x_{0},x_{f},t) \equiv \theta(t) \: \langle x_{f} |
e^{-{\imath \over \hbar} {\hat{\cal H}} t} | x_{0} \rangle
\eeq
and its Fourier transform is defined by
\begin{eqnarray}
\label{E_prop}
G(x_{0},x_{f},E) &\equiv& \imath \int_{-\infty}^{\infty}dt\:
e^{{\imath \over \hbar}E t} \: K(x_{0},x_{f},t) \\ \label{propagator}
&=& \imath \int_{0}^{\infty}dt \: \langle x_{f} |
e^{-{\imath \over \hbar} ({\hat{\cal H}}-E) t} | x_{0} \rangle \nonumber\\
&=& \langle x_{f} | {\hbar \over {{\hat{\cal H}} - E - \imath \epsilon}}
| x_{0} \rangle \nonumber
\end{eqnarray}
where the infinitesimal imaginary constant in the last line gives the causal
propagator.
Duru and Kleinert realized that Eqn. \ref{E_prop} is invariant
under two types of operator transformations.
One type is simply a point canonical transformation, which for a
one-dimensional system is
\begin{eqnarray}
\hat{x} \rightarrow f(\hat{x}) \\ \label{point_CT}
\hat{p} \rightarrow {1 \over f'(\hat{x})}\hat{p} \nonumber
\end{eqnarray}
with $\hat{x}$, $\hat{p}$ the canonical position and momentum respectively.
This point canonical transformation may be implemented by a similarity
transformation on the operators, which is also called a quantum canonical
transformation, since if it is applied to all operators it preserves
the canonical commutation
relations.\cite{Anderson} Under such a similarity transformation
\begin{eqnarray}
{\hat{\cal H}} - E &\rightarrow& {\hat{\cal O}} ({\hat{\cal H}} - E) {\hat{\cal O}}^{-1} \\ \label{similarity}
\langle x | &\rightarrow& \langle x | {\hat{\cal O}}^{-1}.
\end{eqnarray}
The operator ${\hat{\cal O}}$ which implements the transformation is composed of
the canonical position and momentum operators. We will assume that
${\hat{\cal O}}$ is invertible although, with proper care, operators with a
nonzero kernel may also be considered. \cite{Anderson}
Clearly, this type of transformation leaves invariant any matrix
element of an operator.
Another type of transformation which leaves Eqn. \ref{propagator}
invariant is what Duru and Kleinert denoted as an f-transformation.
We will distinguish between two types of f-transformations, since the
normalization measure transforms differently in each case. The first
type of f-transformation is a similarity transformation with ${\hat{\cal O}} =
f(\hat{x})$, where $f(q)$ is some function of q. The other type of
transformation, which we will call conjugation, is
\begin{eqnarray}
{\hat{\cal H}} - E &\rightarrow& f(\hat{x}) ({\hat{\cal H}} - E) f(\hat{x}) \\ \label{conjugation}
\langle x | &\rightarrow& \langle x | f(\hat{x}).
\end{eqnarray}
Eqn. \ref{propagator} is invariant under this transformation, however
a general matrix element of an operator is not invariant.
We next examine the change in the measure factor for these
transformations. First consider a similarity transformation,
Eqn. \ref{similarity}. The original wavefunction $\psi(r)$ and the
transformed wavefunction $\psi'(r)$ are defined as
\begin{eqnarray}
\psi(r) = \langle r | \psi \rangle \\
\psi'(r) = \langle r | {\hat{\cal O}} | \psi \rangle
\end{eqnarray}
with $\langle r |$ an eigenstate of the position operator with
eigenvalue $r$. We then may find the transformation of the (in
general operator valued) measure factor $\hat{\mu}$.
\begin{eqnarray}
\langle \psi | \psi \rangle_{\hat{\mu}} &=& \int dr \langle \psi | \hat{\mu} | r \rangle
\langle r | \psi \rangle \\
&=& \int dr \langle \psi | \hat{\mu} {\hat{\cal O}}^{-1}| r \rangle
\langle r | {\hat{\cal O}} | \psi \rangle \nonumber\\
&=& \int dr \langle \psi | {\hat{\cal O}}^{\dagger} ({\hat{\cal O}}^{-1})^{\dagger} \hat{\mu}
{\hat{\cal O}}^{-1}| r \rangle \langle r | {\hat{\cal O}} | \psi \rangle \nonumber\\
&=& \int dr \langle \psi | {\hat{\cal O}}^{\dagger} \hat{\mu}' | r \rangle \langle r | {\hat{\cal O}} |
\psi \rangle \nonumber\\
&=& \langle \psi' | \psi' \rangle _{\hat{\mu}'}.\nonumber
\end{eqnarray}
Therefore the measure factor for the transformed wavefunctions is
$\hat{\mu}' = ({\hat{\cal O}}^{-1})^{\dagger} \hat{\mu} {\hat{\cal O}}^{-1}$.
We next assume that the measure factor contains only the position operator,
{\it i.e.}, $\hat{\mu} = g(\hat{x})$. Without ambiguity we may then use the
notation $g(r)$ for
the measure factor. For a point canonical transformation,
Eqn. \ref{point_CT}, the measure transforms as a differential
\beq
g(r) \rightarrow g(f(r)){df(r) \over dr}. \label{PCT_measure}
\eeq
For the similarity transformation with ${\hat{\cal O}} = f(\hat{x})$ the measure
factor transforms multiplicatively as
\beq
g(r) \rightarrow f^{-2}(r)g(r).
\eeq
Finally the measure factor remains unchanged for the conjugation
transformation of Eqn. \ref{conjugation}.
\section{Example: Rosen-Morse to Poschl-Teller Potential}
The transformation from a Hamiltonian with potential $V_{0}(r)$
\beq
{\hat{\cal H}} = {\hat{p}^{2}\over {2\mu}} + V_{0}(\hat{x})
\eeq
to another with potential $V_{f}(r)$ is specified by a single function $f(r)$.
We will illustrate the general sequence of transformations along with
the specific example with $V_{0}(r)$ the Rosen-Morse I potential and
$V_{f}(r)$ the hyperbolic Poschl-Teller potential.
First a point canonical transformed is performed as in
Eqn. \ref{point_CT}. For the example, the function is $f(r) = {1\over
a}\mathrm{\ arctanh} \cos 2ar$, giving the operator transformation
\begin{eqnarray}
\hat{x} &\rightarrow& {\hat{\cal O}}_{0}\hat{x}{\hat{\cal O}}_{0}^{-1}= {1\over a}\mathrm{\ arctanh} \cos 2a\hat{x} \\
\hat{p} &\rightarrow& {\hat{\cal O}}_{0}\hat{p}{\hat{\cal O}}_{0}^{-1}= -{1\over 2} \left(\sin
2a\hat{x}\right) \hat{p}, \nonumber
\end{eqnarray}
which transforms the original operator, ${\hat{\cal S}}_{0} \equiv {\hat{\cal H}}_{0} - E$
\beq
{\hat{\cal S}}_{0} = {1\over 2\mu}\hat{p}^{2} + A\mathrm{\ csch}^{2} a\hat{x} + B\coth ax \mathrm{\ csch} ax
- E
\eeq
into
\beq
{\hat{\cal S}}_{1} \equiv {\hat{\cal O}}_{0}{\hat{\cal S}}_{0}{\hat{\cal O}}_{0}^{-1} = {1\over 8\mu}
\left( \sin^{2} 2a\hat{x} \hat{p}^{2}- 2a\imath \hbar \sin 2a\hat{x}
\cos 2ax \hat{p} \right) + A \cos 2a\hat{x} - B \sin^{2} 2a\hat{x} -E.
\eeq
According to Eqn. \ref{PCT_measure} the measure transforms as
\beq
dx \rightarrow {-2\over \sin 2a\hat{x}} dx.
\eeq
The propagator becomes
\begin{eqnarray}
G_{\mathrm{R-M}}(x_{f},x_{0},E) &=& \imath \int dT \langle x_{f} |\:
e^{-{\imath\over \hbar}{\hat{\cal S}}_{0}T}\: | x_{0}\rangle \\
&=& \imath \int dT \langle x_{f} |\: {\hat{\cal O}}_{0}^{-1} e^{-{\imath\over
\hbar}{\hat{\cal S}}_{0}T} \left({\hat{\cal O}}^{-1}\right)^{\dagger}\: | x_{0}\rangle \\ \nonumber
&=& \imath \int dT \langle {1\over 2a}\arccos(\tanh ax_{f}) |\:
e^{-{\imath\over \hbar}{\hat{\cal S}}_{1}T} \:| {1\over 2a} \arccos(\tanh ax_{0})
\rangle. \nonumber
\end{eqnarray}
Next one performs the similarity transformation with ${\hat{\cal O}}_{1} =
\left({df(r)\over dr}\right)^{{1\over 2}}=\sin^{-{1\over 2}} 2a\hat{x}$ to get
\begin{eqnarray}
{\hat{\cal S}}_{2} \equiv {\hat{\cal O}}_{1}{\hat{\cal S}}_{1}{\hat{\cal O}}_{1}^{-1} &=& {1\over 8\mu}\sin^{2}
2a\hat{x} \hat{p} - {\imath\hbar a\over 2\mu}\sin 2a\hat{x} \cos 2a\hat{x} \hat{p}
+ {2\hbar^{2}a^{2}\over {8\mu}} \sin^{2} 2a\hat{x} \\
&+& A \cos 2a\hat{x} - B \sin^{2}
2a\hat{x} - {\hbar^{2}a^{2}\over {8\mu}} - E. \nonumber
\end{eqnarray}
The measure transforms as
\beq
{-2\over \sin 2a\hat{x}} dx \rightarrow -2 dx
\eeq
and the propagator is then
\begin{eqnarray}
G_{\mathrm{R-M}}&(&x_{f},x_{0},E) = \imath \int dT \langle {1\over 2a}\arccos(\tanh
ax_{f}) |\: {\hat{\cal O}}^{-1}
e^{-{\imath\over\hbar}{\hat{\cal S}}_{2}T}\left({\hat{\cal O}}^{-1}\right)^{\dagger}\: |
{1\over 2a}\arccos(\tanh ax_{0})\rangle \\
&=&\imath (\mathrm{\ sech}^{{1\over 2}} ax_{f})(\mathrm{\ sech}^{{1\over 2}} ax_{0}) \int dT
\langle {1\over 2a}\arccos(\tanh
ax_{f}) |\: e^{-{\imath\over\hbar}{\hat{\cal S}}_{2}T}\:
|{1\over 2a}\arccos(\tanh ax_{0})\rangle. \nonumber
\end{eqnarray}
Next a conjugation transformation follows, Eqn. \ref{conjugation}, with the
function $C{df(r)\over dr}$. The constant $C$ is chosen to give the
correct kinetic energy factor in the Hamiltonian.
\begin{eqnarray}
{\hat{\cal S}}_{3} \equiv {2\over \sin 2a\hat{x}}{\hat{\cal S}}_{2}{2\over \sin 2a\hat{x}} &=&
{1\over 2\mu}\hat{p}^{2} + \left(A - E - {{\hbar^{2}a^{2}}\over
{8\mu}}\right) \csc^{2} 2a\hat{x} \\
&+& \left(-A - E - {{\hbar^{2}a^{2}}\over
{8\mu}}\right) \sec^{2} 2a\hat{x} -{1\over 2}\hbar^{2}a^{2} - 4B. \nonumber
\end{eqnarray}
The transformed propagator is
\begin{eqnarray}
G_{\mathrm{R-M}}&(&x_{f},x_{0},E) = \imath (\mathrm{\ sech}^{{1\over 2}} ax_{f})
(\mathrm{\ sech}^{{1\over 2}} ax_{0}) \\ \label{RM_propagator}
&\times& \int dT \langle {1\over 2a}\arccos(\tanh
ax_{f}) |\:\left({2\over \sin 2a\hat{x}}\right)
e^{-{\imath\over\hbar}{\hat{\cal S}}_{3}T} \left({2\over \sin 2a\hat{x}}\right)
\:|{1\over 2a}\arccos(\tanh ax_{0})\rangle \nonumber\\
&=& 4 \imath(\cosh^{{1\over 2}} ax_{f}(\cosh^{{1\over 2}} ax_{0}) \nonumber\\
&\times&\int dT \langle {1\over 2a}\arccos(\tanh
ax_{f}) |\:
e^{-{\imath\over\hbar}{\hat{\cal S}}_{3}T}\:| {1\over 2a}\arccos(\tanh
ax_{f})\rangle. \nonumber
\end{eqnarray}
Finally the the Hilbert space is rescaled so that the measure becomes
the usual one, $\mu = dx$,
\beq
^{\mathrm{norm}}\langle x | \equiv \sqrt{2} \langle x |.
\eeq
This introduces a factor of ${1\over 2}$ in the propagator,
Eqn. \ref{RM_propagator}. The final result is
then obtained
from Eqn. \ref{RM_propagator} by matching parameters in the operator
${\hat{\cal S}}_{3}$ with those for the Poschl-Teller potential.
The algebraic relations between the Fourier transform of the
propagator for several solvable potentials are shown in the table
along with
the function $f(r)$ used for the operator transformations.
\footnote{The transformation functions given in the table are also listed in
Ref. \cite{point_CT_map}, however we correct them for the Rosen-Morse
II and Eckart potentials.}
Although all of the potentials for which we give explicit results in
the table are shape invariant, the operator transformations are valid
for a general potential. It is interesting to note that although not
all one dimensional
solvable potentials, classified by Natanzon, are shape invariant, they
are related to a shape invariant potential by an operator
transformation. \cite{Natanzon,solvable_SUSY}
\section{Operator Transformations for Lie Group Generators}
The operator transformations from ${\hat{\cal S}}_{0} \equiv {\hat{\cal H}}_{0} - E_{0}$ to
${\hat{\cal S}}_{f} \equiv {\hat{\cal H}}_{f} - E_{f}$ may be summarized by
\beq
\label{S_trans}
{\hat{\cal S}}_{f} = C\:(f')^{3/2}\:{\hat{\cal O}}_{0}\:{\hat{\cal S}}_{0}\:{\hat{\cal O}}_{0}^{-1}\:(f')^{{1\over 2}}.
\eeq
${\hat{\cal O}}_{0}$ is the operator implementing the point canonical
transformation, Eqn. \ref{point_CT}, with function $f(q)$ and C is a
constant. Since the
eigenvalue equation, ${\hat{\cal S}}_{f}=0$, is homogeneous one may multiply
Eqn. \ref{S_trans} by $C^{-1}(f')^{-2}$ on the left to obtain the following
equation, valid for
an interval in which $f'\neq 0$ and finite,
\beq
\label{L_trans}
(f')^{-{1\over 2}}\:{\hat{\cal O}}_{0}\:{\hat{\cal S}}_{0}\:{\hat{\cal O}}_{0}^{-1}\:(f')^{{1\over 2}}=0.
\eeq
The operator transformation between the eigenvalue equation for the
Hamiltonian ${\hat{\cal H}}_{0}$ and ${\hat{\cal H}}_{f}$ now preserves the commutators
of operators on the
two Hilbert spaces, {\it e.g.}, it is a Lie algebra isomorphism.
The new generators $\hat{T}_{f}^{i}$ are related to the Lie
algebra generators for the original potential, $\hat{T}_{0}^{i}$ as
\beq
\label{gen_trans}
\hat{T}_{f}^{i}=(f')^{-{1\over 2}}\:{\hat{\cal O}}_{0}\:\hat{T}_{0}^{i}\:{\hat{\cal O}}_{0}^{-1}\:(f')^{{1\over 2}}
\eeq
Therefore, in the cases where the
eigenvalue equation for ${\hat{\cal H}}_{0}$ may be written as an element of the
enveloping
algebra of a particular Lie algebra, the transformed eigenvalue
equation, Eqn. \ref{L_trans}, has the same formulation in terms of Lie
group generators, however in a different representation. The
eigenvalue equation for the
potentials listed in the table
then have the same Lie algebraic form as either the radial harmonic
oscillator, the trigonometric Poschl-Teller, or the hyperbolic
Poschl-Teller potential. $SU(1,1)$ generators for the radial
harmonic oscillator Schroedinger operator and those related to it by
Eqn. \ref{gen_trans} are well known and given in Ref. \cite{so21_algebra}.
As an example, we consider the Lie algebraic form for the
trigonometric Poschl-Teller potential and then find the transformed
generators for the Rosen-Morse I potential.
The Poschl-Teller potential is known to have an algebraic formulation
in terms of the Lie group $SU(2) \otimes SU(2)$.
One may find the generators for $SO(4)=SU(2) \otimes SU(2)$ by
considering the generators of rotations in $\Re^{4}$
\begin{eqnarray}
J_{1} &=& {\imath\over 2}\left(-x_{1}\partial_{4} + x_{2}\partial_{3} -
x_{3}\partial_{2} + x_{4}\partial_{1}\right), \\
J_{2} &=& {\imath\over 2}\left(-x_{1}\partial_{3} - x_{2}\partial_{4} +
x_{3}\partial_{1} + x_{4}\partial_{2}\right), \nonumber \\
J_{3} &=& {\imath\over 2}\left(-x_{1}\partial_{2} + x_{2}\partial_{1} +
x_{3}\partial_{4} - x_{4}\partial_{3}\right), \nonumber\\
K_{1} &=& {\imath\over 2}\left(-x_{1}\partial_{2} + x_{2}\partial_{1} -
x_{3}\partial_{4} + x_{4}\partial_{3}\right), \nonumber\\
K_{2} &=& {\imath\over 2}\left(x_{1}\partial_{3} - x_{2}\partial_{4} -
x_{3}\partial_{1} + x_{4}\partial_{2}\right), \nonumber\\
K_{3} &=& {\imath\over 2}\left(x_{1}\partial_{4} + x_{2}\partial_{3} -
x_{3}\partial_{2} - x_{4}\partial_{1}\right). \nonumber
\end{eqnarray}
Changing to Euler angle coordinates for the double cover of $S^{3}$
\begin{eqnarray}
x_{1} &=& \cos\left({\theta\over 2}\right)
\cos\left({{\phi+\psi}\over 2}\right), \\
x_{2} &=& \cos\left({\theta\over 2}\right)
\sin\left({{\phi+\psi}\over 2}\right), \nonumber\\
x_{3} &=& \sin\left({\theta\over 2}\right)
\cos\left({{\phi-\psi}\over 2}\right), \nonumber\\
x_{4} &=& \sin\left({\theta\over 2}\right)
\sin\left({{\phi-\psi}\over 2}\right), \nonumber
\end{eqnarray}
and scaling $\theta \rightarrow 2a\theta$ we obtain the generators
\begin{eqnarray}
\label{PT_generators}
J_{1} &=& \imath\left({1\over{2a}}\sin{\psi}\partial_{\theta}
- \csc2a\theta \cos\psi \partial_{\phi}
+ \cot2a\theta \cos\psi\partial_{\psi}\right), \\
J_{2} &=&
\imath\left(-{1\over{2a}}\cos{\psi}\partial_{\theta}
-\csc 2a\theta \sin\psi \partial_{\phi}
+ \cot 2a\theta \sin \psi \partial_{\psi}\right), \nonumber\\
J_{3} &=& -\imath\partial_{\psi}, \nonumber\\
K_{1} &=& \imath\left({1\over{2a}}\sin{\phi}\partial_{\theta}
+ \cot2a\theta \cos\phi\partial_{\phi}
- \csc2a\theta \cos\phi \partial_{\psi}\right), \nonumber\\
K_{2} &=&
\imath\left(-{1\over{2a}}\cos{\phi}\partial_{\theta}
+ \cot 2a\theta \sin \phi \partial_{\phi}
-\csc 2a\theta \sin\phi \partial_{\psi}\right), \nonumber\\
K_{3} &=& -\imath\partial_{\phi}. \nonumber
\end{eqnarray}
These obey the commutation relations
\begin{eqnarray}
\left[J_{l},J_{m}\right] &=& \imath\:\epsilon_{lmn}\:J_{n}, \\
\left[K_{l},K_{m}\right] &=& \imath\:\epsilon_{lmn}\:K_{n}, \nonumber\\
\left[J_{l},K_{m}\right] &=& 0, \nonumber
\end{eqnarray}
and $J_{i}$ is obtained from $K_{i}$ by interchanging $\phi
\leftrightarrow \psi$. These operators are similar to those found in
Ref. \cite{Barut}, which were deduced from the corresponding
Infeld-Hull factorization. The Casimir operator $J^{2}$ is
\begin{eqnarray}
4a^{2}J^{2} &=& -\partial^{2}_{\theta} + a^{2}\left(-\partial^{2}_{\phi} -
\partial^{2}_{\psi} + 2\partial_{\phi}\partial_{\psi} - {1\over 4}\right)
\csc^{2} a\theta \\
&+& a^{2} \left(-\partial^{2}_{\phi} -
\partial^{2}_{\psi} - 2\partial_{\phi}\partial_{\psi} -
{1\over 4}\right)\sec^{2}a\theta - a^{2}. \nonumber
\end{eqnarray}
The other Casimir operator $K^{2}$ is identical.
One may express the eigenfunction equation for a unitary
representation of the group $SU(2)$ as
\begin{eqnarray}
\label{su2_rep}
J^{2}|klm\rangle &=& k(k+1)|klm\rangle, \ k=0,{1\over 2},1,{3\over 2},\ldots \\
J_{3}|klm\rangle &=& n|klm\rangle, \ n = -k,\ldots,0,\ldots,k. \nonumber
\end{eqnarray}
If one chooses the eigenfunction $|klm\rangle=u_{mn}^{k}(\theta)
\mathrm{e}^{\imath(l\phi+m\psi)}$ then Eqn. \ref{su2_rep} becomes
\begin{eqnarray}
{{2a^{2}}\over\mu}&J&^{2}u_{lm}^{k}(\theta) \\
&=& \left[-{1\over{2\mu}}
\partial^{2}_{\theta}
+ {a^{2}\over{2\mu}}\left((l-m)^{2}-{1\over 4}\right)\csc^{2}a\theta
+ {a^2\over{2\mu}}\left((l+m)^{2}-{1\over 4}\right)\sec^{2}a\theta
- {a^{2}\over{2\mu}}\right]
u_{lm}^{k}(a\theta) \nonumber \\
&=& {{2a^{2}}\over \mu}k(k+1)u_{lm}^{k}(\theta). \nonumber
\end{eqnarray}
This is the Schroedinger equation for the Poschl-Teller potential,
which if we define the coefficients in the potential
$A\equiv \hbar^{2}\gamma(\gamma-1)$ and
$B\equiv \hbar^{2}\delta(\delta-1)$, gives $\gamma = l-m+{1\over 2}$,
$\delta=l+m+{1\over 2}$ and $E_{k} = {{2a^{2}\hbar^{2}}\over
\mu}(k+{1\over 2})^{2}$. Since $l = k - j,\ j = 0,1,\ldots,2k$ the
energy eigenvalues are
\beq
E_{k} = {{a^{2}\hbar^{2}}\over{2\mu}}(\gamma+\delta+2j)^{2}
\eeq
with $j\geq {1\over 2}(1-\gamma-\delta)$.
The same procedure for the $K_{i}$ operators gives the same energy
eigenvalues.
If one transforms the $SU(2)$ generators $J_{i}$, in
Eqn. \ref{PT_generators}, into the corresponding ones for the
Rosen-Morse I potential, using Eqn. \ref{gen_trans}, one obtains
\begin{eqnarray}
J^{\mathrm{RM}}_{1} &=& \imath\left({-1\over a}\cosh a\theta
\sin \psi \partial_{\theta} - \cosh a\theta \cos \psi
\partial_{\phi} + \sinh a\theta \cos \psi \partial_{\psi}\right), \\
J^{\mathrm{RM}}_{2} &=& \imath\left({1\over a}\cosh a\theta
\cos \psi \partial_{\theta} -\cosh a\theta \sin \psi \partial_{\phi}
+\sinh a\theta \sin \psi \partial_{\psi}\right), \nonumber \\
J^{\mathrm{RM}}_{3} &=& -\imath\partial_{\psi}. \nonumber
\end{eqnarray}
The Casimir operator acting on the state $|klm\rangle \equiv
u_{lm}^{k}(\theta)\mathrm{e}^{\imath(l\phi-m\psi)}$ gives
\begin{eqnarray}
J^{2}u_{lm}^{k}(\theta) &=& \left[{{-\cosh^{2}
a\theta}\over a}
\partial_{\theta}^{2} + (l^{2}+m^{2})\cosh^{2} a\theta
+2lm \sinh a\theta \cosh a\theta \right]u_{lm}^{k}(\theta) \\
&=& k(k+1)u_{lm}^{k}(\theta) \nonumber
\end{eqnarray}
and $J^{\mathrm{RM}}_{3}|klm\rangle = -n|klm\rangle$.
Multiplying by $-a^{2}\hbar^{2}\mathrm{\ sech}^{2} a\theta/2\mu$ leads to
the Schroedinger
equation for the Rosen-Morse potential
\beq
\hbar^{2}\left[-{1\over 2\mu}\partial_{\theta}^{2} +
{{a^{2}lm}\over{\mu}}\tanh a\theta
- {{a^{2}k(k+1)}\over{2\mu}}\mathrm{\ sech}^{2} a\theta \right]u_{lm}^{k}(\theta)
= -{{a^{2}\hbar^{2}(l^{2}+m^{2})}\over{2\mu}}u_{lm}^{k}(\theta)
\eeq
with parameters $A=a^{2}lm/\mu$ and $B=a^{2}k(k+1)/2\mu$ and energy eigenvalue
$E = -a^{2}\hbar^{2}(l^{2}+m^{2})/2\mu$. Since the energy eigenvalues
are non-positive only the bound states energies may be found. Again
for a unitary representation of
$SU(2)$ we have $-m = -k + j,\ j=0,1,\ldots,2k$. Substituting this
in the equation for the energy eigenvalue and expressing the result in
terms of the potential coefficients
\begin{eqnarray}
E_{j} &=& -\hbar^{2}\left[{\mu A^{2}\over{2a^{2}}}\left({1\over n^{2}}\right)
+{a^{2}\over{2\mu}}n^{2}\right] \\
n &=& -{1\over 2} + {1\over 2}\sqrt{1 + {{8\mu B}\over a^{2}}}-j,\ j = 0,1,\ldots,
\left(-1+\sqrt{1+{{8\mu B}\over a^{2}}}\right). \nonumber
\end{eqnarray}
Furthermore we may assume that $A\geq0$, since under the change of
variables $\theta \rightarrow -\theta$, $A \rightarrow -A$. Similar
to the Poschl-Teller case, the other $SU(2)$ operators $K_{i}$ may be
found from $J_{i}$ by exchanging $\phi \leftrightarrow \psi$
and furthermore the
Casimirs are equal,
$K^{2}=J^{2}$. Therefore, with $K_{3}|klm\rangle = l |klm\rangle$
the range of the eigenvalue is $l = -k,-k+1,\ldots,k-1,k$ and one
finds the following bound on the coefficients in the potential in order for the
existence of a bound state
\beq
\left({\mu A\over{a^{2}}}\right)^{{1\over 2}} = lm \leq k^{2} = \left(-{1\over 2} +
{1\over 2}\sqrt{1+{{8\mu B}\over a^{2}}}\right)^{2}.
\eeq
\section{Conclusion}
We have shown that if a particular type of operator transformation,
which is not
necessarily unitary, exists between two Schroedinger operators there
is a procedure for finding an algebraic relation between the
respective propagators and that the two eigenvalue problems have the
same formulation in terms of Lie group generators. Also a knowledge of
the Fourier transform
of the propagator for the new potential allows one, in principle, to
find the energy eigenvalues and wavefunctions for both the bound and
scattering states.
One interesting generalization of this procedure would be to find such
operator transformations between multiparticle exactly solvable systems,
such as those of the Calogero-Sutherland type.
\section*{Acknowledgements}
This work was supported by the Japanese Society for the Promotion of Science.
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\section{Introduction}
The non-relativistic interacting Bose gas is certainly among the most
extensively investigated problems of many-body physics. Early interest
in this problem has been strongly motivated by the low temperature
properties of helium \cite{LP}. It has been sustained, on the
theoretical side, by a persisting elusiveness of the deeper nature of
the condensation process \cite{NA} and, on the experimental side, by
developments which led to therecent observation of the more artificial
Bose condensates \cite{Rb}-\cite{Li}.
Much of the persisting uneasiness about the condensation process of a
nonideal bose gas hinges on the fact that the body of available
results relies mainly on perturbative methods, a circumstance
which tends to be considered as a liability. Non perturbative
approximation schemes have been developed, on the other hand, in
connection with the problem of self-interacting, relativistic Bose
fields which became relevant e.g. for inflationary models of the
universe \cite{UI}. This problem has also been used as a testing
ground for non perturbative methods which could then be applied to
more complicated systems of interacting fields. In this context, the
gaussian variational approach \cite{JI} has received considerable
attention, also in view of its relation to extended mean field methods
traditionally employed in non relativistic many-body physics
\cite{LinTh,LinP,TT,KeLin}.
The purpose of this paper is to describe an application of the
gaussian variational approximation to the much studied problem of
interacting non relativistic bosons in order to bring about a direct
confrontation of this approach with at least part of the available
results for this problem. We use a formulation \cite{LinP} which is
very close to the standard language of non relativistic, extended mean
field approaches of the Hartree-Fock-Bogolyubov type, which can on the
other hand be directly related to the methods adopted in connection
with the Schr\"odinger representation of quantum field theory
\cite{TT}. Although for simplicity we restrict ourselves to the case
of uniform systems, extending the formulation to finite, inhomogeneous
systems such as those actually realized in the recent alkeli atom
experiments is completely straightforward \cite{LinTh} using well
known many-body techniques \cite{KTr}. The inclusion of finite
temperature effects is also straightforward in the formulation we use,
so that thermodynamic properties can be studied rather easily. We show
that several of the old results can be retrieved from {\it truncated}
versions of the full gaussian approximation, including ground state
energy and phonon spectrum. This last feature is lost when one adheres
to the full gaussian approximation, notwithstanding the fact that it
is supported theoretically by the Hugenholtz-Pines theorem
\cite{HP,GN} to all perturbative orders, a feature which has been
noted long ago by Girardeau and Arnowitz \cite{GA}. The phonon
spectrum can be recoverd in a generalized RPA treatment \cite{KP} or
by using the functional derivative method developed by Hohenberg and
Martin \cite{HM} and recently re-discussed by Griffin \cite{GG}.
Problems of thermodynamic instability develop for contact forces when
one uses a renormalization procedure in which the coupling constant is
made to approach zero from {it negative} values, which has been
considered in a relativistic context e.g. by Bardeen and Moshe
\cite{BM} and which has also been extensively discussed by Stevenson
as the ``precarious'' renormalized $\lambda\phi^4$ theory \cite{SF}
and more recently by T\"urk\"oz \cite{TT} and by Kerman and Lin
\cite{KeLin}.
The formulation which we adopt is reviewed in Section 2, where the
formal, finite temperature equilibrium solutions for a contact
repulsive interaction are obtained. A truncated version of the
gaussian variational equations is analyzed in Section 3. In Section 4,
we deal with the full gaussian approximetion. Two different
prescriptions for dealing with the divergences are considered. The
first one involves a renormalization scheme related to that proposed
by Stevenson. As a second, alternative scheme we treat the
Hamiltonian with contact interactions as an effective theory to be
implemented with a fixed cut-off in momentum space. Numerical results
are given in this case for properties of the different phases and
phase equilibria. Section 5 contains our conclusions.
\section{Thermal gaussian approximation}
We consider an extended, uniform and isotropic system of
non-relativistic, interacting, spinless bosons described by
the Hamiltonian (in momentum representation with periodic
boundary conditions in volume $V$)
\begin{equation}
H=\sum_{\vec{k}}^{} e(k) a_{\vec{k}}^{\dag} a_{\vec{k}} +
\frac{\lambda}{2 V} \sum_{\vec{k_{1}} \vec{k_{2}}
\vec{q}}^{} a_{\vec{k_{1}}+\vec{q}}^{\dag} a_{\vec{k_{2}}-
\vec{q}}^{\dag} a_{\vec{k_{2}}} a_{\vec{k_{1}}}
\end{equation
\noindent where \( e(k) = \frac{\hbar^{2} k^{2}}{2 m}\) is the free
particle kinetic energy, and the contact repulsive ($\lambda > 0$)
interaction between a pair of particles is $\lambda \delta( \vec{r} -
\vec{r'})$. The field operators satisfy standard boson commutation
relations. Within a grand canonical description the state of the
system is described by the density operator
\[
{\cal{F}}=\frac{1}{Z}e^{-\beta{\cal{H}}}
\]
\noindent where ${\cal{H}}=H-\mu N$, $N$ being the boson number
operator, and $Z$ the grand canonical partition function
\[
Z=Tr e^{-\beta{\cal{H}}}.
\]
\noindent Following the variational approach of Balian and
V\'en\'eroni \cite{BV}, we look for extrema of the object
\[
f(M)=Tr e^{-M}
\]
\noindent under the constraint $M-\beta{\cal{H}}=0$, which is taken
into account through the introduction of a Lagrange multiplier matrix
$B$. This leads to the variational problem
\begin{equation}
\label{02}
\delta\Phi(M,B)=\delta Tr\left[e^{-M}+B(M-\beta {\cal{H}})\right]=0.
\end{equation}
\noindent Variation of $M$ gives $B=e^{-M}$, and elimination of $B$
from Eq. \ref{02} leads to the $M$-dependent object
\begin{equation}
\label{03}
\Phi(M,B)\rightarrow\Psi(M)=Tr\left[e^{-M}(1+M-\beta{\cal{H}})\right].
\end{equation}
Balian and V\'en\'eroni show that $Z\geq\Psi(M)$ for {\it any}
$M$. Defining $z=Tr e^{-M}$ we can write
\[
e^{-M}=z{\cal{F}}_0
\]
\noindent where now $Tr{\cal{F}}_0=1$. Substituting this in
Eq.\ref{03} and varying $z$ we find the variational expression for the
grand potential $\Omega$
\begin{equation}
\label{04}
\Omega=-\frac{1}{\beta}\ln Z\leq Tr[({\cal{H}}+KT\ln {\cal{F}}_0)
{\cal{F}}_0)]
\end{equation}
\noindent where ${\cal{F}}_0$ is an arbitrary density with unit trace,
and we used the notation $1/\beta=KT$. In Eq. \ref{04} we can in
particular identify an entropy factor as $S_0=-K\;\;Tr[{\cal{F}}_0\ln
{\cal{F}}_0]$.
The most general gaussian approximation consists in adopting for $M$
an ansatz of the form \cite{DC}
\[
M\rightarrow\sum_{\vec{k}\vec{k}'}[A_{\vec{k}\vec{k}'}
a^\dagger_{\vec{k}} a_{\vec{k}'} + (B_{\vec{k}\vec{k}'}
a^\dagger_{\vec{k}} a^\dagger_{\vec{k}'} + h.c.)+(C_{\vec{k}}
a^\dagger_{\vec{k}} +h.c.)]
\]
\noindent where the matrix $A_{\vec{k}\vec{k}'}$ is hermitean. The
quadratic form appearing in $M$ can be diagonalized by a general
canonical transformation of the Bogolyubov type, which amounts to
changing to the natural orbital representation of the extended one
boson density corresponding to the gaussian density ${\cal{F}}_0$
\cite{LinTh,LinP}. The uniformity and isotropy assumptions we make
allow us to restrict this general ansatz so that $A_{\vec{k}\vec{k}'}$
is diagonal, $B_{\vec{k}\vec{k}'}$ vanishes unless $\vec{k}=
-\vec{k}'$ and both of these matrices and the $C_{\vec{h}}$ depend
only on the magnitudes of the momentum vectors. The diagonalization of
the quadratic form is achieved in this case by defining transformed
boson operators as
\[
\eta_{\vec{k}} = x_{k}^{\ast} b_{\vec{k}} + y_{k}^{\ast}
b_{-\vec{k}}^{\dag}
\]
\[
\eta_{\vec{k}}^{\dag} = x_{k} b_{\vec{k}}^{\dag} + y_{k}
b_{-\vec{k}}
\]
\noindent where
\[
b_{\vec{k}} = a_{\vec{k}} - \Gamma_{k}
\]
\noindent and we have used the isotropy of the uniform system to
make the c-number transformation parameters $x_{k}$, $y_{k}$ and
$\Gamma_{k}$ dependent only on the magnitude of $\vec{k}$. In order
for this transformation to be canonical we have still to impose on
the $x_{k}$ and $y_{k}$ the usual normalization condition
\begin{equation}
\label{05}
|x_{k}|^2 - |y_{k}|^2 = 1.
\end{equation}
\noindent The trace-normalized gaussian density operator is now
written explicitly as
\begin{equation}
\label{06}
{\cal F}_{0} = \prod_{\vec{k}} \frac{1}{1 + \nu_{k}}
\left(\frac{\nu_{k}}{1 + \nu_{k}} \right)
^{\eta_{\vec{k}}^{\dag} \eta_{\vec{k}}}.
\end{equation}
\noindent Straightforward calculation shows that
\[
Tr(\eta_{\vec{k}}^{\dag} \eta_{\vec{k'}} {\cal F}_{0})=
\nu_{k} \delta_{\vec{k} \vec{k'}}
\]
\noindent so that the $\nu_{k}$ are positive quantities corresponding
to mean occupation numbers of the $\eta$-bosons. One also finds that
\[
Tr[(x^{*}_{k}a_{\vec{k}}+y^{*}_{k}a_{-\vec{k}}^{\dag})^n {\cal F}_{0}]=
Tr[(\eta_{\vec{k}}+A_{k})^n {\cal F}_{0}]= A_{k}^n
\]
\noindent with $A_{k}=x_{k}^{*} \Gamma_{k}+y_{k}^{*} \Gamma_{k}^{*}$
so that non vanishing values of the $\Gamma_{k}$ correspond to
coherent condensates of unshifted, Bogolyubov transformed bosons. We
again invoke the system uniformity to impose
\[
\Gamma_{k}=\delta_{k,0} \Gamma_{0}
\]
\noindent in the calculations to follow.
It is important to note that the truncated density ${\cal F}_{0}$ in
general breaks the global gauge symmetry of $H$ which is responsible
for the conservation of the number of $a$-bosons. The quadratic
dispersion of the number operator $N=\sum_{\vec{k}}a_{\vec{k}}^{\dag}
a_{\vec{k}}$ in this state can in fact be obtained explicitly as
\begin{eqnarray}
\langle N^{2} \rangle - \langle N \rangle^{2} &=& 2 | \Gamma_{0} |^{2}
[| x_{0} |^{2} \nu_{0} + |y_{0}|^{2} (1 + \nu_{0})] -
2\Gamma_{0}^{\ast^{2}} x_{0} y_{0}^{\ast} (1 + 2 \nu_{0}) - \nonumber
\\ &&\nonumber \\ &&-2\Gamma_{0}^{2} y_{0} x_{0}^{\ast} (1 + 2
\nu_{0}) + |\Gamma_{0}|^{2} +\sum_{\vec{k}}[|x_{k}|^{2} \nu_{k} + (1 +
\nu_{k}) |y_{k}|^{2} ] + \nonumber \\ &&+\sum_{\vec{k}} \{
[|x_{k}|^{2} \nu_{k} + (1 + \nu_{k}) |y_{k}|^2]^{2} + |x_{k}|^{2}
|y_{k}|^{2} (1+ 2 \nu_{k})^{2} \}. \nonumber \\ \nonumber
\end{eqnarray}
\noindent Furthermore, mean values of many-boson operators taken with
respect to ${\cal F}_{0}$ will contain no irreducible many-body parts,
so that the replacement of $\cal F$ by ${\cal F}_{0}$ amounts to a
mean field approximation. The states described by ${\cal F}_{0}$ have
therefore to be interpreted as ``intrinsic'' mean field states.
An important simplification which occurs in the case of stationary
states such as we consider here is that the transformation parameters
$x_{k}$, $y_{k}$ and $\Gamma_{0}$ can be taken to be real and we can
use a simple parametric representation that automatically satisfies
the canonicity condition. It reads
\begin{equation}
\label{07}
x_{k} = \cosh \sigma_{k}, \; \; \; y_{k} = \sinh \sigma_{k}.
\end{equation}
\noindent It is then straightforward to evaluate the traces involved
in Eq.\ref{04} to obtain
\begin{eqnarray}
\label{08}
\Omega &\leq&\sum_{\vec{k}} {\left(e(k) - \mu + \frac{2 \lambda
\Gamma_{0}^{2}}{V}\right)\left[\frac{(1 + 2 \nu_{k}) \cosh 2\sigma_{k}
-1}{2} \right]}-\mu \Gamma_{0}^{2}+ \frac{\lambda \Gamma_{0}^{4}}{2 V}
\nonumber \\ &&-\frac{\lambda \Gamma_{0}^{2}}{2 V} \sum_{\vec{k}}^{}
{(1 + 2 \nu_{k})\sinh 2 \sigma_{k} }+ \frac{\lambda}{V}\left
\{\sum_{\vec{k}}\left[ \frac{(1+ 2 \nu_{k})\cosh 2\sigma_{k} -1
}{2}\right] \right\}^{2} \\ &&+\frac{\lambda}{8 V}\{ \sum_{\vec{k}}^{}
{(1 + 2 \nu_{k}) \sinh 2 \sigma_{k} }\}^{2} \nonumber \\ &&-KT
\sum_{\vec{k}}^{} [(1+\nu_{k})\ln(1+\nu_{k})-
\nu_{k}\ln\nu_{k}]. \nonumber
\end{eqnarray}
\noindent In a similar way the number fixing condition $Tr[{\cal
F}_{0}N]=\langle N \rangle$ evaluates to
\begin{equation}
\label{09}
\langle N \rangle = \Gamma_{0}^{2} + \sum_{\vec{k}}^{}
\left[\frac{(1 + 2 \nu_{k}) \cosh2 \sigma_{k} -1}{2} \right].
\end{equation}
\subsection{Formal equilibrium solutions}
Equations determining the form of the truncated density
${\cal F}_{0}$ appropriate for thermal equilibrium are in
general derived by requiring that $\Omega$, eq.\ref{08}, is
stationary under arbitrary variations of $\Gamma_{0}$,
$\sigma_{k}$ and $\nu_{k}$. Variation with respect to
$\Gamma_{0}$ gives the gap equation
\begin{equation}
\Gamma_{0} \left\{\frac{2 \lambda}{V} \Gamma_{0}^{2} - 2 \mu -
\frac{\lambda}{V} \sum_{\vec{k}}\left[(1+2 \nu_{k})(\sinh 2
\sigma_{k}- 2 \cosh 2\sigma_{k}) +2\right] \right\} = 0
\end{equation}
\noindent
which, besides the trivial solution $\Gamma_{0}=0$, may also admit a
solution with a non vanishing value of $\Gamma_{0}$ obtained by
requiring that the expression in curly brackets vanishes. This
solution involves the number constraint, Eq.\ref{09}, in addition to
the values of $\nu_{k}$ and $\sigma_{k}$, which are determined by the
remaining variational conditions on $\Omega$. In order to simplify the
algebraic work involved in the study of this class of solutions it is
convenient to use the number constraint Eq.\ref{09} to eliminate
$\Gamma_{0}$ from the right hand side of Eq.\ref{08} which then
assumes the form
\[
\Omega \leq F - \mu \langle N \rangle
\]
\noindent
with $\langle N \rangle$ given by Eq.\ref{09}. This identifies a
free energy $F$ as
\begin{eqnarray}
\label{11}
F &=&\sum_{\vec{k}}^{} {\left(e(k) + \lambda \rho\right)\left[\frac{(1
+ 2 \nu_{k})\cosh 2\sigma_{k} -1}{2}\right]}-\frac{\lambda \rho}{2 }
\sum_{\vec{k}}{(1 + 2\nu_{k})\sinh 2 \sigma_{k}} \nonumber\\ &
&-\frac{\lambda}{2 V}\left\{\sum_{\vec{k}}^{}\left[\frac{ (1 + 2
\nu_{k}) \cosh 2\sigma_{k} -1 }{2}\right] \right\}^{2} +
\frac{\lambda}{8 V} \{ \sum_{\vec{k}}^{} {(1 + 2 \nu_{k})\sinh 2
\sigma_{k}}\}^{2} \nonumber \\ & &+\frac{\lambda}{2 V}
\sum_{\vec{k},\vec{k'}}^{} (1 + 2 \nu_{k'})\sinh 2 \sigma_{k'}
\left[\frac{(1 + 2 \nu_{k}) \cosh 2\sigma_{k}-1}{2}\right]
+\frac{\lambda \rho^2 V}{2}\nonumber\\ & &-KT \sum_{\vec{k}}^{}
[(1+\nu_{k})\ln(1+\nu_{k})- \nu_{k}\ln\nu_{k}]
\end{eqnarray}
\noindent Extremizing $F$ by setting derivatives with respect to
$\sigma_{k}$ and $\nu_{k} $ equal to zero one gets
\begin{eqnarray}
\label{12}
& &\tanh 2 \sigma_{k} = \\
& & \frac{\lambda \rho-\frac{\lambda}{2 V}\sum_{\vec{k}}^{}
[ (1 + 2 \nu_{k})\cosh 2\sigma_{k} -1] - \frac{\lambda}{2 V}
\sum_{\vec{k}}^{} {(1 + 2 \nu_{k}) \sinh 2 \sigma_{k}}}
{e(k) + \lambda \rho -\frac{\lambda}{2 V}\sum_{\vec{k}}^{}
[(1 + 2 \nu_{k}) \cosh 2\sigma_{k} -1] + \frac{\lambda}{2 V}
\sum_{\vec{k}} {(1 + 2 \nu_{k})\sinh 2 \sigma_{k}}} \nonumber
\end{eqnarray}
\noindent and
\begin{equation}
\label{13}
\nu_{k} =\frac{1}{\{exp[\sqrt{\Delta}/{KT}]-1\}}
\end{equation}
\noindent where
\begin{eqnarray}
\label{14}
\Delta &=& e(k)^{2} + 2 \lambda e(k)\times\nonumber\\
& &\times\{\rho -\frac{\lambda}{2 V}\sum_{\vec{k}}[(1 + 2 \nu_{k})
\cosh 2\sigma_{k} -1] + \frac{\lambda}{2 V} \sum_{\vec{k}}
{(1 + 2 \nu_{k})\sinh 2 \sigma_{k}}\} \nonumber \\
& &+4 \lambda^{2} \{\rho - \frac{\lambda}{2 V}\sum_{\vec{k}}
[(1 + 2 \nu_{k}) \cosh 2\sigma_{k} -1] \} \times\nonumber\\
& & \times \frac{\lambda}{2 V}
\sum_{\vec{k}} {(1 + 2 \nu_{k})\sinh 2 \sigma_{k}}
\end{eqnarray}
As for the trivial solution $\Gamma_{0} = 0$, we differentiate
$\Omega$ as written in Eq.\ref{08} and get
\begin{equation}
\label{15}
\tanh 2 \sigma_{k} = \frac{-\frac{\lambda}{2 V} \sum_{\vec{k}} (1 + 2
\nu_{k}) \sinh 2 \sigma_{k}}{e(k) - \mu + \frac{\lambda}{V}
\sum_{\vec{k}}[(1 + 2 \nu_{k}) \cosh 2 \sigma_{k} - 1 ] }.
\end{equation}
Finally, we stress that the results obtained in the present section
are largely formal, as they involve divergent sums. In order to allow
for the derivation of thermodynamic properties of the different phases
they must therefore be supplemented by suitable regularization and
renormalization procedures. These will be discussed in sections 3 and
4 below.
\section{Independent $\eta$-bosons and dilute \\ system limit}
In this section we consider two different truncation schemes of the
gaussian variational equations which lead to well known results for
the low temperature properties of the interacting boson system. The
first and most drastic truncation of the gaussian approximation
consists in neglecting all terms representing interactions between
$\eta$-bosons. The result is entirely trivial in the case
$\Gamma_{0}=0$ since this implies $\sigma_{k}=0$. All effects of the
interaction are thus discarded, giving us just the ideal gas results
\begin{equation}
\nu_{k} =\frac{1}{\{exp[\frac{e(k) -\mu}{KT}]-1\}}
\end{equation
\noindent where $\mu$ is determined by the number constraint
\begin{equation}
\rho = \frac{1}{4 \pi^2} \int_{0}^{\infty}
\frac{k^2 dk}{ \{ \exp[\frac{e(k)-\mu}{KT} ] -1\}}.
\end{equation
As for the solution corresponding to $\Gamma_{0} \ne 0$, discarding
interaction between $\eta$-bosons amounts to dropping all double sums
in Eq.~(9). The variational conditions on $\sigma_{k}$ and $\nu_{k}$
appear then as
\begin{equation}
\tanh 2 \sigma_{k} = \frac{\lambda \rho}{e(k) + \lambda \rho}
\end{equation
\noindent and
\begin{equation}
\nu_{k} = \frac{1}{e^{\frac{1}{KT} \sqrt{e(k)^{2}
+ 2 \lambda \rho e(k)}}-1}.
\end{equation
\noindent When using these results for calculating $F$ we replace sums
by integrals and introduce a cut-off $\Lambda$ in the range of
integration over momenta. Ignoring terms that vanish in the limit
$\Lambda \rightarrow \infty$ we get
\begin{eqnarray}
\frac{F}{V} & = &\frac{\lambda\rho^{2}}{2}
-\frac{\lambda^{2} m \Lambda \rho^{2}}{4 \pi^{2}\hbar^{2}}
+\frac{8 m^{3/2} \lambda^{5/2} \rho^{5/2}}{15 \pi^{2}\hbar^{3}}
\nonumber \\
& &+\frac{KT}{2 \pi^{2}} \int_{0}^{\Lambda}k^2
\ln\left\{1-\exp\left[-\frac{\sqrt{e(k)^{2}+2e(k)\lambda\rho}}
{KT}\right]\right\}dk
\end{eqnarray
\noindent which shows that we get a linearly divergent term
proportional to $\lambda^{2}$. This term is a leftover of the
non-normal ordered terms of $H$ which involve two $\eta$-operators.
It is therefore a direct consequence of the non-trivial nature of the
Bogolyubov transformation, and can be compensated by introducing the
Fermi pseudo-potential, where the contact interaction is replaced by
\begin{eqnarray}
V_{p}(\vec{r}-\vec{r\prime})&=&\lambda\frac{\partial}{\partial(|\vec{r}
-\vec{r\prime})|}[(|\vec{r}-\vec{r\prime}|)\delta(\vec{r}-\vec{r\prime})]
\nonumber \\ &=&\lambda\delta(\vec{r}-\vec{r\prime})+
(|\vec{r}-\vec{r\prime}|)\lambda\delta'(\vec{r}-\vec{r\prime})
\nonumber
\end{eqnarray}
\noindent which can be related, in the case of dilute, cold systems to
the two boson scattering length $a$ through
\[
\lambda = \frac{4 \pi \hbar^{2} a}{m}.
\]
\noindent The regularized ground state energy becomes
\[
\frac{E_{0}}{V}=\frac{\lambda\rho^{2}}{2}\left(1+\frac{128}{15}
\frac{(a^{3}\rho)^{1/2}}{\pi^{1/2}}\right)
\]
\noindent and the chemical potential appears as
\begin{eqnarray}
\mu & = &\lambda\rho+\frac{4 \rho^{3/2}m^{3/2}
\lambda^{5/2}}{3 \pi^{2}\hbar^{3}}\nonumber \\
& &+\frac{1}{2 \pi^{2}} \int_{0}^{\infty}
\frac{\lambda e(k) k^{2} dk}
{ \left\{\exp\left[ \frac{\sqrt{e(k)^{2}+2 e(k) \lambda \rho}}
{KT}\right]-1\right\} \sqrt{e(k)^{2}+2 e(k) \lambda \rho } }.
\end{eqnarray
These are just the results obtained by Lee, Huang and Yang \cite{KY},
by Beliaev \cite{SB} and by Hugenholtz and Pines \cite{HP} under the
assumption of a macroscopic (c-number) occupation of the zero momentum
mode. Note however that in the present formulation this is replaced by
the coherent condensate associated with $\Gamma_{0}$. In addition to
this we still have non-coherent occupation of the zero momentum mode
as given by the limit of $\nu_{k}$, Eq.~(17), for $k \rightarrow 0$.
For small, non-zero temperatures this diverges as $1 / k$ and
therefore does not contribute to the density of the system. Finally,
the values obtained for the chemical potential in the condensate
($\Gamma_{0} \ne 0$) and non-condensate ($\Gamma_{0}=0$) phases
indicate the instability of the former. This may be seen very easily
from the fact that $\mu$ is always positive in Eq.~(19) and always
negative or zero in Eq.~(15).
A possible way to circumvent this drawback in the framework of the
gaussian approximation is to perform a somewhat less drastic if more
delicate truncation of the complete variational expressions, which
relies on the fact that we are working with dilute systems. There is
little change in the case of the $\Gamma_{0}\neq 0$ phase. Eqs.~(7)
and ~(16) give the depletion
\[
\rho = \rho_{0}+\frac{8}{3}\frac{(a^{3}\rho)^{1/2}}{\pi^{1/2}}.
\]
\noindent In the limit of a dilute system one has
$(a^{3}\rho_{0})^{1/2} \simeq(a^{3}\rho)^{1/2}\ll 1$ so that one may
replace $\rho$ by $\rho_{0}$ in Eq.~(16). With this replacement the
excitation spectrum of the truncated Hamiltonian is related to the
density of the condensate as
\[
w(k)=\sqrt{e(k)^{2}+2\lambda\rho_{0}e(k)}
\]
As for the phase with $\Gamma_{0}=0$, the occupation numbers $\nu_{k}$
are obtained by keeping $\sigma_{k}=0$ but with no truncation. The
chemical potential $\mu$ is determined using the constraint condition
(7) that in this case reads
\[
\rho=\frac{1}{4\pi^{2}}\int_{0}^{\infty}\frac{k^{2}dk}{\exp{\frac{e(k)
-\mu+2\lambda\rho}{KT}}-1}.
\]
\noindent Since the truncation has been avoided in the case when
$\Gamma_0=0$, this phase is no longer treated as a free bose gas, and
calculation shows that at $T=0$ the condensate phase now appears as
the stable one, since its chemical potential $\mu=\lambda\rho$ is
lower than that of the non condensed phase, for which
$\mu=2\lambda\rho$. It is apparent that in this derivation the two
phases are not treated on the same footing as far as the truncation is
concerned. However the truncation of the two body terms has a quite
different significance in each of the two cases. The rationale for
this procedure rests in fact on the expectation that the most relevant
effects of the two body interaction are incorporated to the result via
the symmetry breaking processes which take place in the treatment of
the $\Gamma_{0}\neq 0$ phase but not in that of the $\Gamma_{0}=0$
phase.
\section{Handling the full gaussian approximation }
In this section we examine two different prescriptions for dealing
with the divergences of the complete gaussian approximation developed
in section 2. The first prescription involves a renormalization scheme
similar to the one proposed by Stevenson \cite{SF} in the context of
the relativistic $\phi^{4}$ theory under the name of ``precarious
theory'', in which the bare coupling constant is made to approach zero
by negative values. It represents an attempt at sticking as much as
possible to the dynamics of contact interactions as such in the
present context. Although successful in removing the divergences in a
consistent way, this scheme will be shown to lead to a
thermodynamically unstable system. We therefore consider also, as a
second prescription, the simple alternate ``effective theory'' scheme
in which a fixed cut-off is introduced in momentum space in the spirit
of the work of Amelino-Camelia and Pi \cite{GP}.
\subsection{Contact forces: precarious theory}
In order to reduce unessential complications to a minimum we restrict
ourselves in the following development to the properties of the system
at $T= 0$ in the phase corresponding to $\Gamma_{0}\ne 0$, since the
extension to $T \ne 0$ involves no additional divergences. The gap
equation, Eq.~(8), combined with the number constraint, Eq.~(7), gives
\begin{equation}
\mu = \lambda \rho +\frac{\lambda}{2 V}\sum_{\vec{k}}
[\cosh 2\sigma_{k} -1] -\frac{\lambda}{2 V} \sum_{\vec{k}}^{}
{\sinh 2 \sigma_{k} }.
\end{equation
\noindent and from Eqs.~(10) to ~(12) we get
\begin{equation}
\tanh 2 \sigma_{k} = \frac{\lambda \rho -\frac{\lambda}{2 V}
\sum_{\vec{k}}[\cosh 2 \sigma_{k} -1] - \frac{\lambda}{2 V}
\sum_{\vec{k}} {\sinh 2 \sigma_{k}}}{e(k) + \lambda \rho
-\frac{\lambda}{2 V} \sum_{\vec{k}}[\cosh 2 \sigma_{k} -1]
+ \frac{\lambda}{2 V} \sum_{\vec{k}} {\sinh 2 \sigma_{k}}}.
\end{equation
\noindent This is an implicit equation for the transformation
parameters $\sigma_{k}$ which appear in the sums of Eqs.~(20) and
cut-off $\Lambda$ and, again neglecting contributions that vanish in
the limit $\Lambda \rightarrow \infty $, make the ansatze
\begin{equation}
\frac{1}{2 V} \sum_{\vec{k}}^{}{ \sinh 2 \sigma_{k}} =
\alpha + \beta \Lambda
\end{equation
\noindent and
\begin{equation}
\frac{1}{2 V} \sum_{\vec{k}}^{}{(\cosh 2\sigma_{k}-1)} = \gamma
\end{equation
\noindent where $\alpha$, $\beta$ and $\gamma$ are assumed to approach
finite values in the limit $\Lambda \rightarrow \infty$. We next
introduce a renormalized coupling constant $\lambda_{r}$ as
\begin{equation}
\lambda = \frac{\lambda_{r}}{1 - \frac{\lambda_{r} m \Lambda}
{2 \pi^{2} \hbar^{2}}}
\end{equation
\noindent so that when $\Lambda \rightarrow \infty $ the bare coupling
constant $\lambda$ approaches zero from negative values
\cite{KeLin,TT,SF}. Eq.~(21) becomes
\[
\tanh 2 \sigma_{k}=\frac{\frac{\lambda_{r} \rho-\lambda_{r}
\gamma-\lambda_{r}\alpha-\lambda_{r}\beta\Lambda} {1-\frac{\lambda_{r}
m \Lambda}{2 \pi^{2}\hbar^{2}}}} {e(k) + \frac{\lambda_{r} \rho
-\lambda_{r} \gamma + \lambda_{r} \alpha +\lambda_{r} \beta \Lambda}
{1-\frac{\lambda_{r} m \Lambda}{2 \pi^{2}\hbar^{2}}}}\equiv
-\frac{M+\frac{Q}{\Lambda}+{\cal{O}}(\Lambda^{-2})}
{e(k)+M+\frac{R}{\Lambda}+{\cal{O}}(\Lambda^{-2})}
\]
\noindent with
\begin{eqnarray}
M&=&-\frac{2\pi^{2} \hbar^{2} \beta}{m},\nonumber \\
Q&=&\frac{2\pi^{2} \hbar^{2}}{m}(\rho-\gamma-\alpha-\frac{2\pi^{2}
\hbar^{2} \beta}{\lambda_{r} m}),\nonumber \\
R&=&-\frac{2\pi^{2} \hbar^{2}}{m}(\rho-\gamma+\alpha+\frac{2\pi^{2}
\hbar^{2} \beta}{\lambda_{r} m}).
\end{eqnarray
\noindent This makes good sense for all $\vec{k}$ and for any
$\Lambda$ provided $M$ is positive. We can also obtain $\sinh 2
\sigma_{k}$ and $\cosh 2 \sigma_{k}$ in terms of $M$, $Q$ and $R$ to
evaluate the sums of Eqs.~(22) and ~(23) up to terms
${\cal{O}}(\Lambda^{-1})$ with the results
\begin{eqnarray}
\alpha + \beta \Lambda &=& -\frac{1}{4 \pi^{2}}
\int_{0}^{\Lambda}{\frac{k^{2}(M+\frac{Q}{\Lambda})} {\sqrt{e(k)^{2} +
2 (M+\frac{Q}{\Lambda}) e(k)+G}} dk} \nonumber \\ &=&\frac{m^{3/2}
M^{3/2}}{\pi^2 \hbar^{3}}- \frac{mQ}{2 \pi^{2} \hbar^{2}} -\frac{m M
\Lambda}{2 \pi^{2} \hbar^{2}}, \nonumber
\end{eqnarray}
\noindent which is consistent with the expression for $M$ in Eq.(25),
and
\[
\gamma = \frac{1}{4 \pi^{2}} \int_{0}^{\Lambda}
{k^2\left\{\frac{[e(k)+M+\frac{R} {\Lambda}]}{\sqrt{e(k)^{2} +
2(M+\frac{Q}{\Lambda}) e(k)+G}}-1\right\}dk} = \frac{ m^{3/2}
M^{3/2}}{3 \pi^{2} \hbar^{3}}
\]
\noindent where we defined
\[
G=\frac{2 M (R-Q)}{\Lambda}+\frac{R^{2}-Q^{2}}{\Lambda^{2}}.
\]
\noindent It is easy to see that for $M=0$ we obtain the
ideal Bose gas results (free theory). For $M>0$, in order to evaluate
$F$ (which for $T=0$ reduces to the ground state energy $E_{0}$), we
also need
\[
\frac{1}{2 V} \sum_{\vec{k}}^{}{e(k)(\cosh 2\sigma_{k}-1)} =
\frac{m M^{2} \Lambda}{4 \pi^{2} \hbar^{2}}+
\frac{mMQ}{2 \pi^{2} \hbar^{2}}-
\frac{4 m^{3/2} M^{5/2}}{5 \pi^{2} \hbar^{3}}.
\]
\noindent Taking these results to Eq.~(9) (with $\nu_{k}=0$ for $T=0$)
we see that the remaining linearly divergent terms cancel and we are
left with
\begin{equation}
\frac{F(T=0)}{V}=\frac{E_{0}}{V}=
\frac{8 m^{3/2} M^{5/2}}{15 \pi^{2}
\hbar^{3}} - M \rho - \frac{M^{2}}{2 \lambda_{r}}
\end{equation
\noindent which identifies the chemical potential as $\mu=-M$. Note
that $M>0$ now implies $\mu<0$. Finally, in order to relate $\mu$ (or
$M$) to the density of the system we evaluate the grand potential,
Eq.~(6), with the result
\begin{equation}
\Omega(T=0)=\frac{8m^{3/2}(-\mu)^{5/2}V}
{15\pi^{2}\hbar^{3}}-\frac{\mu^{2}V}{2\lambda_{r}}
\end{equation
\noindent and use the relation $N=-(\frac{\partial\Omega}
{\partial\mu})_{T,V}$ to obtain
\begin{equation}
\rho=\frac{\mu}{\lambda_{r}}+\frac{4m^{3/2}
(-\mu)^{3/2}}{3\pi^{2}\hbar^{3}}.
\end{equation
\noindent With the condition $\mu<0$ when $\Lambda \rightarrow
\infty$, we get a phonon-like spectrum as in Sec. 3,
\[
E(k) = \sqrt{e(k)^{2}-2 \mu e(k)}.
\]
\noindent We can also take the appropriate derivative of $F$ (or,
equivalently, evaluate $-\Omega / V$ using Eq.~(27)) to get the
pressure at $T=0$ as
\begin{equation}
P = -\left(\frac{\partial F}{\partial V}\right)_{T,N} =
-\frac{\Omega}{V}=
-\frac{8 m^{3/2} (-\mu^{5/2})}{15 \pi^{2} \hbar^{3}}
+\frac{\mu^{2}}{2 \lambda_{r}}
\end{equation
\noindent and it is easy to show, using Eqs.~(26),~(28),~(29) and
$\rho>0$, that $\frac{d P}{d \rho}$ and $E_{0}/V$ are always negative.
This shows that the renormalized theory is thermodynamically unstable.
As a matter of fact, this instability should be seen as the
non-relativistic counterpart of the ``intrinsic instability'' pointed
quite some time ago by Bardeen and Moshe \cite{BM} in their analysis
of the phase-structure of the relativistic $\lambda\phi^4$ theory in
four dimensions, which has also been raised recently in the context of
multi-field theories \cite{GAC}. It underlies also the
``precariousness'' of this theory in Stevenson's treatment \cite{SF}.
Another possible phase to be studied corresponds to a solution where
$\Gamma_{0}=0$. In order to account for the fact that in this case a
macroscopic occupation $\nu_{0}$ may develop, we separate the
$\vec{k}=0$ contribution in the sums
\begin{equation}
\frac{1}{2 V} \sum_{\vec{k}}\left(\cosh 2 \sigma_{k} - 1 \right)
\rightarrow c + \frac{1}{2 V} \sum_{\vec{k}\neq 0}\left( \cosh 2
\sigma_{k} - 1 \right) \nonumber
\end{equation}
\noindent and
\begin{equation}
\frac{1}{2 V} \sum_{\vec{k}}\sinh 2 \sigma_{k} \rightarrow d +
\frac{1}{2 V} \sum_{\vec{k}\neq 0}\sinh 2 \sigma_{k} \nonumber
\end{equation}
\noindent so that Eq. (13) becomes
\begin{equation}
\tan 2 \sigma_{k} = \frac{-\lambda d - \frac{\lambda}{2 V}
\sum_{\vec{k}\neq 0} \sinh 2 \sigma_{k}}{e(k) - \mu + 2 \lambda \rho }.
\nonumber
\end{equation}
\noindent Using the ansatze (22) and (23) we obtain
\begin{equation}
\tanh 2 \sigma_{k}=\frac{\frac{-\lambda_{r}
d-\lambda_{r}\alpha-\lambda_{r}\beta\Lambda} {1-\frac{\lambda_{r} m
\Lambda}{2 \pi^{2}\hbar^{2}}}} {e(k) - \mu + \frac{2 \lambda_{r} \rho}
{1-\frac{\lambda_{r} m \Lambda}{2 \pi^{2}\hbar^{2}}}}\equiv
-\frac{M+\frac{Q}{\Lambda}+{\cal{O}}\left(\Lambda^{-2}\right)}
{e(k)-\mu+\frac{R}{\Lambda}+{\cal{O}}\left(\Lambda^{-2}\right)}
\end{equation}
\noindent with
\begin{eqnarray}
M&=&-\frac{2\pi^{2} \hbar^{2} \beta}{m},
\nonumber \\
Q&=&+\frac{2\pi^{2} \hbar^{2}}{m} d,
\nonumber \\
R&=&-\frac{4\pi^{2} \hbar^{2}}{m} \rho.
\end{eqnarray}
\noindent If we examine
\begin{equation}
\gamma = \frac{1}{4 \pi^{2}} \int_{0}^{\Lambda} k^{2} \left[\frac{e(k)
-\mu}{\sqrt{[e(k)-\mu]^{2}-M^{2}}} -1 \right] dk \nonumber
\end{equation}
\noindent we see that the only acceptable solution is $M=\mu$. If
$M=0$ it follows that $\sigma_{k}=0$, leading to a possible solution
$c=d=\nu_{0}/V$ which corresponds essentially a free Bose gas. If, on
the other hand, $M\ne0$ we are again led to a thermodynamically
unstable situation.
\subsection{Effective theory}
The renormalization prescription of the preceding subsection tells us
that if $\lambda_{r}$ is positive and held fixed when $\Lambda
\rightarrow \infty$ we must have $\lambda \rightarrow 0_{-}$ which
results in a thermodynamically unstable theory that we want to
avoid. Since a positive value of $\lambda$ leads to a trivial theory
with $\lambda_{r}=0$ when the limit $\Lambda \rightarrow \infty$ is
taken, we next consider the results that one obtains for the
non-trivial effective theory in which $\Lambda$ is kept fixed at a
finite value allowing for both $\lambda_{r} > 0$ and $\lambda >
0$. This requires that one must have
\[
\frac{\lambda_{r} m \Lambda}{2 \pi^{2} \hbar^{2}} < 1
\]
\noindent and implies a finite resolution ${\cal{O}}(\Lambda^{-1})$ in
configuration space. The restricted momentum space also implies that
the validity of the results will be restricted to temperatures which
are low in the scale
\[
\frac{\hbar^{2} \Lambda^{2}}{2 K m}.
\]
\noindent Because in the derivations given in the preceding subsection
all terms that vanish in the limit $\Lambda \rightarrow \infty$ were
neglected, we use here the general expressions given in Section 2.
Eq.~(24) will be used with a fixed value of $\Lambda$ to relate the
bare and effective coupling constants $\lambda$ and $\lambda_{r}$
respectively.
Consider first the condensed phase, $\Gamma_{0}\neq 0$. The sums
appearing in Eqs.~(9),~(10) and ~(11) are now finite, and we therefore
define
\begin{eqnarray}
A &=& \frac{1}{2 V} {\sum_{\vec{k}}^{}}' {(1 + 2 \nu_{k})\sinh 2
\sigma_{k}} \nonumber \\
B &=& \frac{1}{2 V} {\sum_{\vec{k}}^{}}' {[(1 + 2 \nu_{k})
\cosh 2\sigma_{k} -1]} \nonumber \\
C &=& \frac{1}{2 V} {\sum_{\vec{k}}^{}}' {e(k)[(1 + 2 \nu_{k})
\cosh 2\sigma_{k} -1]}. \nonumber
\end{eqnarray}
\noindent where the primed sums are restricted to $|\vec{k}|\leq
\Lambda$. Eqs.~(8), ~(10) and ~(12) appear then as
\begin{equation}
\mu = \lambda \rho -\lambda A + \lambda B
\end{equation
\[
\tanh 2 \sigma_{\vec{k}} = \frac{\lambda [\rho -B-A]}{e(k) +
\lambda[\rho -B+A]}
\]
\noindent and
\[
\Delta = e(k)^{2} + 2 \lambda e(k) [\rho - B + A] +
4 \lambda^{2} [\rho -B] A.
\]
\noindent This last quantity determines $\nu_{k}$ through Eq.~(11) and
shows explicitly the presence of an energy gap in the quasiparticle
spectrum which is related to the quasiparticle interaction term $2
\lambda \rho_{0} A $ (we use $\rho-B=\rho_{0}$, the density of the
coherent condensate). The free energy $F$ and the pressure $P$ are
given by
\begin{eqnarray}
F&=&CV-\lambda \rho (A-B)V+\frac{\lambda}{2}(\rho^{2}+A^{2}-B^{2})V
+\lambda ABV \nonumber \\
& &-KT {\sum_{\vec{k}}}'[(1+\nu_{k})\ln(1+\nu_{k})-\nu_{k}
\ln\nu_{k}]
\end{eqnarray
\noindent and
\begin{eqnarray}
P&=&\frac{\lambda B^{2}}{2} + \frac{\lambda \rho^{2}}{2} - C -\lambda
A B -\frac{\lambda A^{2}}{2} + \frac{1}{V}
{\sum_{\vec{k}}}'{\nu_{\vec{k}} \sqrt{\Delta}} \nonumber\\ & &-
\frac{KT}{V} {\sum_{\vec{k}}}'{\ln\{ 1-\exp[-\sqrt{\Delta}/KT]\}}
\end{eqnarray
After going to the continuum limit, we can calculate $A$ and $B$ for
given values of $\rho$ and $T$ by solving numerically the system of
equations
\[
\left\{ \begin{array}{lll}
A &=&\frac{1}{4 \pi^{2}} \int_{0}^{\Lambda}
{k^2 \{ \frac{\lambda [\rho -B-A]}{\sqrt{\Delta}}[1+\frac{2}
{\exp\frac{\sqrt{\Delta}}{KT}-1}]}\}dk \\
B &=&\frac{1}{4 \pi^{2}} \int_{0}^{\Lambda}
{k^2 \{ \frac{e(k)+\lambda [\rho -B+A]}{\sqrt{\Delta}}[1+\frac{2}
{\exp\frac{\sqrt{\Delta}}{KT}-1}]-1\}dk}.
\end{array}
\right.
\]
\noindent Finally, in terms of the values of $A$ and $B$ we can
calculate $C$ and the thermodynamic functions also numerically.
As for the case $\Gamma_{0} = 0$ with the sums restricted to the
cut-off, it's easy to see that the only solution is $\sigma_{k}=0$. In
the continuum limit,
\begin{equation}
\rho = \frac{1}{2 \pi^{2}} \int_{0}^{\Lambda} \frac{k^{2} dk}
{\exp[(e(k)-\mu+2 \lambda \rho)/KT] -1}
\end{equation
\noindent which serves to determine $\mu$. The pressure is in this
case given by
\begin{equation}
P=\lambda \rho^{2} - \frac{K T}{2 \pi^{2}} \int_{0}^{\Lambda}
{k^{2} \ln\{ 1-\exp[-(e(k)-\mu+2 \lambda \rho)/KT]\}}.
\end{equation
\subsection{Numerical results}
\subsubsection{Cut-off dependence}
In order to perform numerical calculations we used a system of units
such that the Boltzmann constant $K=1$ and $c=1$, so that energies are
expressed in degrees Kelvin. We use $m=4\times10^{13}\;^oK$ and
$\lambda_{r} = 50 \;^oK \mbox{\AA}^{3}$. In Fig.~(1), we show the
dependence of the ground state energy, for a density $\rho=0.01
\;\mbox{\AA}^{-3}$ as the cut-off is varied in the interval $1.0
\;\mbox{\AA}^{-1} < \Lambda < 5.0\; \mbox{\AA}^{-1} $. For the
calculations described below we take $\Lambda = 3.0\; \mbox{\AA}^{-1}
$. According to Eq.~(24), this corresponds to a bare coupling constant
$\lambda=119.124^oK\mbox{\AA}^3$. For densities larger than the quoted
value one starts having stronger cut-off dependence, so that the
present scheme is, in this sense, also limited to rather dilute
systems.
\subsubsection{Phase transition}
Results of some sample calculations using the various expressions of
the preceding section are shown in Figs.~(2) to ~(5). For the range of
temperatures and densities considered here, the adopted value of the
cut-off momentum $\Lambda$ essentially saturates the values of the
finite momentum integrals involved. The density $\rho$ is plotted in
the figures as the number density in units of $\mbox{\AA}^{-3}$.
In order to study the properties of the condensed ($\rho_{0}> 0$)
and of the non-condensed ($\rho_{0}=0$) phases, as well as their
coexistence, we show in Fig.~2 the chemical potential $\mu$ calculated
as a function of the density $\rho$ in each of these two cases, for a
constant temperature fixed as $T=2^{o} K$, using Eqs.~(38) and ~(41)
respectively. The condensed phase has two different branches with
different chemical potentials in the range $\rho > \sim .7 \times
10^{-2}\mbox{\AA}^{-3}$ and ceases to exist for lower values of the
density. The chemical potential of the non-condensed phase increases
monotonically approaching the upper branch of the condensed phase
asymptotically from above. It is interesting to note that these two
solutions essentially merge at a density where the non-condensed
solution develops a macroscopic occupation $\nu_{0}$ of the zero
momentum state, and that the solution involving instead a coherent
condensate ($\rho_{0}> 0$) leads to a lower value of $\mu$. In
order to decide about the thermodynamic stability of the various
solutions we give in Fig.~3 the corresponding plot of $\mu$ against
the pressure $P$, which displays the pattern of a first order
transition. The densities of the two stable phases are indicated by
the light dashed lines in Fig.~2. Fig.~4 shows the $T=2^{o}K$ and the
$T=0^{o}K$ isotherms in a standard $P \times \rho^{-1}$ diagram
together with the appropriate Maxwell construction for $T=2^{o}K$. The
condensed solution is the only one stable at $T=0^{o}K$. The pattern
found for $T=2^{o}K$ repeats itself for higher temperatures with no
evidence of a critical temperature within the range allowed by the
limitations of the effective theory. The stable isotherms for
temperatures of 0, 2, 3 and 4 $^oK$ are shown in Fig.~5.
\section{Concluding remarks}
In the gaussian approximation it is possible to renormalize a
repulsive contact interaction using a procedure akin to the so called
precarious renormalization scheme of Stevenson\cite{SF}. Despite being
finite and exhibiting some desirable properties, such as an excitation
spectrum of the phonon type, the condensate phase is thermodynamically
unstable for all system densities. This is a manifestation of
a corresponding instability poited out in a relativistic context by
Bardeen and Moshe \cite{BM}. Solutions with no pairing
($\sigma_{k}=0$), on the other hand, can be verified to correspond
just to a free bose gas. We stress that thethermodynamic instability
of the paired, condensed phase cannot be recognized just from the
expression for the ground state energy, Eq.~(27), but requires the use
of the equation of state.
As an alternative to such an unstable, renormalized theory we
considered a cut-off dependent effective theory for a dilute system.
Among the results of this theory we have a gap in the excitation
spectrum in disagreement with the Hugenholz-Pines theorem\cite{HP},
and a first order transition between a condensed phase and a
non-condensed phase. The ocurence of the gap can be associated to the
drastic reordering and infinite partial ressumation of the
perturbative series implied in the Gaussian variational
approximation. It should be noted in this connection that the
variational calculation involving the truncated density $F_{0}$ is
designed to optimize the determination of the grand-potential, and
that this does not imply the variational optimization of the thermal
occupation probabilities from which the excitation spectrum is derived
\cite{BV}. On the other hand, the sensitivity of the excitation
spectrum to the dynamical ingredients included in the calculation can
be illustrated by the fact that, using the free quasi-boson truncation
described in section 3 (which breaks the variational self consistency
of the calculation), the results obtained agree with the
Hugenholtz-Pines theorem.
We also remark that it is a well known tendency of mean field
approximations to overrepresent discontinuities in situations
involving phase transitions, e.g. by predicting discontinuous
behaviors in the case of finite systems. This recommends, as already
observed in ref. \cite{GP} on the basis of a reliability analysis of
power-counting type, that caution should be exerted in the
interpretation of the first order character obtained for the phase
transition in this calculation.
In conclusion, the introduction of quantum effects at the level of a
paired mean field calculation for a non-ideal bose system gives
results which differ even qualitatively from usual perturbative
results. Comparison with experiment is still hampered by the fact
that a very dilute though sufficiently non-ideal system has not yet
been realized in practice.
|
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\section{Acknowledgments}
This research was supported in part by the Comisi\'on Interministerial de
Ciencia y Tecnolog\'{\i}a of Spain under contract MAT 94-0982-C02-01 and by
the Commission of European Communities under contract Ultrafast
CHRX-CT93-0133.
|
proofpile-arXiv_065-668
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section*{Resumen}
\else \small
\begin{center}
{\bf Resumen\vspace{-.5em}\vspace{0pt}}
\end{center}
\quotation
\fi}
\def\thebibliography#1{\section*{Referencias\markboth
{REFERENCIAS}{REFERENCIAS}}
\list
{[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}
\leftmargin\labelwidth
\advance\leftmargin\labelsep
\usecounter{enumi}}
\def\hskip .11em plus .33em minus -.07em{\hskip .11em plus .33em minus -.07em}
\sloppy
\sfcode`\.=1000\relax}
\begin{document}
\preprint{EFUAZ FT-96-26}
\title{Interacci\'on `Oscilador' de Part\'{\i}culas
Relativistas\thanks{Ciertas partes de este art\'{\i}culo
han sido presentados en el Seminario del IFUNAM, 19 de noviembre de 1993,
el Seminario de la EFUAZ, 25 de mayo de 1994 y en el Simposio de
Osciladores Arm\'onicos, Cocoyoc, M\'exico, 23-25 de marzo de 1994.
Enviado a ``Investigaci\'on Cientifica".}}
\author{{\bf Valeri V. Dvoeglazov}}
\address{
Escuela de F\'{\i}sica, Universidad Aut\'onoma de Zacatecas \\
Antonio Doval\'{\i} Jaime\, s/n, Zacatecas 98068, ZAC., M\'exico\\
Correo electronico: [email protected]}
\date{25 de junio de 1996}
\maketitle
\abstract{
Es una introducci\'on en el nivel accesible a las recientes ideas
en mec\'anica relativista de part\'{\i}culas con diferentes
espines, interacci\'on de cuales es del tipo oscilador. Esta
construcci\'on matem\'atica propuesta por M. Moshinsky pudiera proveer
aplicaciones en la descripci\'on de los processos mediados por los
campos tensoriales y en la teor\'{\i}a de los estados ligados.
}
\pacs{PACS: 12.90}
\newpage
\setlength{\baselineskip}{24pt}
\section{Introducci\'on}
Con el oscilador arm\'onico ha trabajado gran parte de su vida
el doctor Marcos Moshinsky~\cite{Mosh4}, alumno de Eugene Wigner y
el primer {\it Ph. D.} en f\'{\i}sica de M\'exico, este tipo de
interacci\'on se ha sido aplcado a muchos problemas en F\'{\i}sica
Matem\'atica, F\'{\i}sica At\'omica y Molecular, \'Optica, F\'{\i}sica
Nuclear y Part\'{\i}culas Fundamentales. Desde 1992 se selebran
Simposios Internacionales de los problemas relacionados con el oscilador
\'armonico. Otros f\'{\i}sicos conocidos de M\'exico, Rusia y de los
EE.UU., tales como N. Atakishiyev (IIMAS, Cuernavaca y Baku, Azerbaijan),
L. C. Biedenharn (Duke, EE. UU.), O. Casta\~nos (ICN-UNAM), J. P.
Draayer (Louisiana), A. Frank (ICN-UNAM), F. Iachello (Yale, EE. UU.), Y.
S. Kim (Maryland, EE. UU.), V. I. Man'ko (Lebedev, Mosc\'u) , M. M.
Nieto (LANL, EE. UU.), L. de la Pe\~na (IF-UNAM), Yu. F. Smirnov
(IF-UNAM y MSU, Mosc\'u), K. B. Wolf (CIC, Cuernavaca) trabajan en esta
\'area. Entonces, el objetivo de dar a conocer una parte de estos
problemas a los estudiantes de la UAZ y otras instituciones de la
provincia mexicana tiene suficientes razones.
En esta Secci\'on me permito echar una breve mirada al desarollo de estas
materias hasta los a\~nos noventas. Tanto en mec\'anica cl\'asica como en
mec\'anica cu\'antica no relativista los problemas de movimiento del
una part\'{\i}cula con masa en un punto en el campo potencial
`oscilador' $V(x) = {1\over 2} Kx^2$ pueden ser resueltos en forma
exacta. La ecuaci\'on de Schr\"odinger de la mec\'anica cu\'antica
\begin{equation}
-{\hbar^2 \over 2m} \frac{d^2 \Psi (x)}{dx^2} +{1\over 2} Kx^2 \Psi (x)
=E\Psi (x)
\end{equation}
nos da valores propios de energia que son discretos ($\omega=\sqrt{K/m}$):
\begin{equation}
E_n = \hbar \omega (n+{1\over 2})\quad,\quad n=0,1,2\ldots
\end{equation}
Este tipo del espectro se llama espectro {\tt equidistante}.
En este problema tenemos la energ\'{\i}a {\tt zero-point} (o, {\tt
del punto cero}, $n=0$). Como se menciona en muchos libros de
mec\'anica cu\'antica esta energ\'{\i}a est\'a relacionada con el
principio de incertidumbre --- no podemos medir exactamente la coordenada
y el momento lineal al mismo tiempo, lo que resulta en la existencia de
la energ\'{\i}a minima $E_0 \sim \hbar \omega/2$ del sistema
part\'{\i}cula -- campo potencial~\cite{Yariv}. Desde mi punto de vista
este tipo de problemas demuestra la complicada estructura de vac\'{\i}o en
las teor\'{\i}as cu\'anticas e importancia del concepto de paridad.
Ademas, como todos los problemas cu\'anticos el sistema manifiesta el
principio de correspondencia, probablemente, el principio m\'as grande en
f\'{\i}sica y filosof\'{\i}a (vease, por ejemplo, ref.~\cite[\S
13]{Shiff}).\footnote{Estoy impresionado por su expresion por M.M. Nieto
en Memorias del II Simposio ``Oscilador Arm\'onico": ``\ldots if you can
solve (or not solve) something classically the same is true quantum
mechanically, and {\it vice versa}".}
Entre las importantes aplicaciones del concepto `interacci\'on
oscilador' (incluyendo los modelos para el problema de muchos cuerpos y los
modelos con amortiguaci\'on) quisiera mensionar:
\begin{itemize}
\item
La representaci\'on de un n\'umero infinito de los osciladores del campo
y cuantizaci\'on secund\'aria en teor\'{\i}a de campos cuantizados
({\it e.g.}, el libro de N. N. Bogoliubov, 1973);
\item
Los espectros de las vibraciones de moleculas en el tratamiento
algebr\'aico ({\it e.g.}, los art\'{\i}culos de F. Iachello, A. Frank);
\item
En los modelos de c\'ascaras ({\it e.g.}, el libro de M. Mayer y J. Jensen,
1955) y modelos del movimiento colectivo, tales como el modelo
simpl\'ectico nuclear (D. Rowe, 1977-85);
\item
La representaci\'on de los estados `squeeze' y los estados coherentes en
\'optica cu\'antica ({\it e.g.}, en las Mem\'orias de la ELAF'95, V. I.
Man'ko);
\item
Finalmente, el oscilador arm\'onico entr\'o en mec\'anica cu\'antica
relativista~\cite{Mosh}.
\end{itemize}
\section{Ecuaci\'on de Dirac y ecuaciones en
las representaciones m\'as altas}
De la mecanica cu\'antica relativista sabemos la relaci\'on entre
la energ\'{\i}a, la masa y el momento lineal:\footnote{El contenido de
esa Secci\'on tambien fue considerado desde diferentos puntos de
vista en el articulo antecedente de Dvoeglazov~\cite{DV-IC}. Ser\'{\i}a
\'util leer antes el art\'{\i}culo citado.}
\begin{equation} E^2 = {\bf
p}^{\,2} c^2 +m^2 c^4\quad. \label{rd} \end{equation}
La relaci\'on es
conectada \'{\i}ntimamente con las transformaciones de Lorentz de la
teor\'{\i}a de la relatividad~\cite[\S 2.3]{Ryder}. Pero esa ecuaci\'on
no contiene informaci\'on acerca del espin, la variable adicional sin
fase~\cite{Wigner} y puede describir la evoluci\'on del campo escalar, o
part\'{\i}cula escalar, \'unicamente. Despu\'es de la aplicaci\'on de la
transformaci\'on de Fourier obtenemos la ecuaci\'on de Klein-Gordon:
\begin{equation} \left ( {1\over c^2} {\partial^2 \over
\partial t^2} - \bbox{\nabla}^2 + {m^2 c^2 \over \hbar^2} \right ) \Psi
({\bf x}, t) = 0\quad.\label{kg}
\end{equation}
La funci\'on $\Psi ({\bf
x}, t)$ tiene una sola componente. La energ\'{\i}a que corresponde a las
soluciones de la ec. (\ref{kg}) puede tener valores tanto negativos como
positivos. Aunque la densidad $\rho = {i\hbar \over 2mc^2} \left
(\Psi^\ast {\partial \Psi \over \partial t} - {\partial \Psi^\ast \over
\partial t} \Psi \right )$ satisface la ecuaci\'on de continuidad, hay
dificultades con su interpretaci\'on como densidad de
probabilidad, de acuerdo con la ecuaci\'on antecedente, $\rho$ puede ser
positiva o negativa (como la energ\'{\i}a) y no podemos ignorar las
soluciones con $E<0$ porque en este caso las soluciones con $E>0$ no
forman el sistema completo en sentido matem\'atico. Por esas razones en
los veintes y treintas buscaron las ecuaciones para la funci\'on con
m\'as componentes y que sean lineales en la primer derivada respecto al
tiempo (como la ecuaci\'on de Schr\"odinger). Aunque la ecuaci\'on lineal
de primer orden fue encontrada por P. Dirac en 1928, sin cierta
reinterpretaci\'on ella ten\'{\i}a los mismos defectos. M\'as tarde W.
Pauli, V. Weisskopf y M. Markov~\cite{Markov} argumentaron que $\rho$
se tiene que considerar como la densidad de carga y quitaron una de
las objeciones contra la teor\'{\i}a con las ecuaciones del segundo orden.
?` Como manejar
el grado de libertad de esp\'{\i}n en las ecuaciones de
segundo orden? Sabemos de la necesidad de introducirlo por los
experimentos como la separaci\'on de las lineas espectrosc\'opicas de la
part\'{\i}cula cargada en el campo magn\'etico, el efecto de Zeeman. La
propuesta obvia para introducir el esp\'{\i}n es representar el operador
$(E^2/c^2) -{\bf p}^2$ como
\begin{equation}
\left ( {E^{(op)}\over c}
-{\bbox \sigma} \cdot {\bf p} \right ) \left ( {E^{(op)}\over c} + {\bbox
\sigma} \cdot {\bf p} \right ) =(mc)^2\label{eq-wdw}
\end{equation}
y considerar la funci\'on como la con dos componentes. $E^{(op)} \equiv
i\hbar {\partial \over \partial t}$ y $\bbox{\sigma}$ son las matrices
de Pauli de la dimensi\'on $2\times 2$ . La forma estandar de ellas es
\begin{eqnarray} \sigma_x=\pmatrix{0&1\cr 1&0\cr}\quad,\quad
\sigma_y=\pmatrix{0&-i\cr i&0\cr}\quad,\quad
\sigma_z=\pmatrix{1&0\cr 0&-1\cr}\quad. \label{mp}
\end{eqnarray}
Sin embargo, las diferentes representaciones de aquellos tambi\'en son
posibles~\cite[p.84]{Sakurai}.
La forma (\ref{eq-wdw}) es compatible con la relaci\'on dispercional
relativista (\ref{rd}) gracias a las propiedades de las matrices de Pauli:
\begin{equation}
\bbox{\sigma}_i \bbox{\sigma}_j + \bbox{\sigma}_j \bbox{\sigma}_i =
2\delta_{ij} \quad,
\end{equation}
donde $\delta_{ij}$ es el simbolo de Kronecker.
En el espacio de coordenadas la ecuaci\'on se lee
\begin{equation}
\left ( i\hbar {\partial \over \partial x_0} + i\hbar \bbox{\sigma}\cdot
\bbox{\nabla} \right ) \left (i\hbar {\partial \over \partial x_0} -
i\hbar \bbox{\sigma}\cdot \bbox{\nabla} \right )\phi = (mc)^2
\phi\quad,\label{ww}
\end{equation}
donde $x_0 =ct$. Como mension\'o
Sakurai~\cite[p.91]{Sakurai} R. P. Feynman y L. M. Brown usaron esa
ecuaci\'on del segundo orden en derivadas en tiempo y fue perfectamente
v\'alido para un electr\'on. Pero, como sabemos de la teor\'{\i}a de las
ecuaciones diferenciales parciales, cuando tratamos de resolver una
ecuaci\'on de segundo orden (para predecir la conducta futura) necesitamos
especificar las condiciones iniciales para la funci\'on $\phi$ y su
primera derivada en el tiempo (la condici\'on adicional). En este punto hay
diferencia con lo que hizo Dirac en 1928 cuando propuse una ecuaci\'on
para electr\'on y positr\'on de primer orden en las derivadas. Me permito
recordar que el bispinor de Dirac $j=1/2$ es con cuatro componentes y el
problema puede ser resuelta si sabemos s\'olo la funci\'on en el instante
$t=0$.\footnote{Otras ecuaciones en la representaci\'on $(1/2,0)\oplus
(0,1/2)$ del grupo de Poincar\`e (para part\'{\i}culas del tipo Majorana,
estados auto/contr-auto conjugados de carga) fueron propuestas
recientamente por D. V. Ahluwalia, G. Ziino, A. O. Barut y por
Dvoeglazov (1993-96), pero esas materias est\'an fuera de las metas del
presente art\'{\i}culo.} Concluyendo, podemos decir que el n\'umero
de componentes independentes que tenemos que especificar para la
descripci\'on de una part\'{\i}cula cargada es cuatro, no importa si
usamos la ecuaci\'on de Dirac o la ecuaci\'on de Waerden (\ref{ww}).
Sin embargo, es posible reconstruir la ecuaci\'on de Dirac empezando desde
la forma (\ref{ww}). Para este objetivo vamos a definir \begin{equation}
\phi_{_R} \equiv {1\over mc} \left (i\hbar {\partial \over \partial x_0} -
i\hbar \bbox{\sigma}\cdot \bbox{\nabla} \right ) \phi\quad,\quad \phi_{_L}
\equiv \phi\quad.
\end{equation}
Entonces tenemos la equivalencia entre la ecuaci\'on (\ref{ww}) y
el conjunto
\begin{mathletters}
\begin{eqnarray}
\label{d1} \left [i\hbar (\partial/\partial x_0) - i\hbar
\bbox{\sigma}\cdot \bbox{\nabla} \right ] \phi_{_L} &=& mc \phi_{_R}\quad,\\
\label{d2} \left [i\hbar (\partial/\partial x_0) + i\hbar
\bbox{\sigma}\cdot \bbox{\nabla} \right ] \phi_{_R} &=& mc \phi_{_L}\quad.
\end{eqnarray} \end{mathletters}
Tomando la suma y la diferencia de las
ecuaciones (\ref{d1},\ref{d2}) llegamos a la famosa ecuaci\'on
presentada por Dirac ($\psi = (\phi_{_R} +\phi_{_L})/\sqrt{2}$,\,\, $\chi
= (\phi_{_R} -\phi_{_L})/\sqrt{2}$): \begin{equation} \pmatrix{i\hbar
(\partial /\partial x_0) & i\hbar \bbox{\sigma}\cdot \bbox{\nabla}\cr
-i\hbar \bbox{\sigma}\cdot \bbox{\nabla}& -i\hbar (\partial
/\partial x_0)\cr} \pmatrix{\psi \cr \chi} = mc \pmatrix{\psi \cr
\chi}\quad,
\end{equation}
o bien,
\begin{equation} \left [ i\gamma^\mu
\partial_\mu - m \right ] \Psi (x^\mu) = 0\quad,\quad \hbar=c=1\quad,
\label{ed}
\end{equation}
lo que coincide con~\cite[ec.(10)]{DV-IC}, \, la ecuaci\'on para los
fermiones -- part\'{\i}culas con el espin $j=1/2$.
Las matrices de Dirac tienen la siguiente forma en esta representaci\'on
que se llama la representaci\'on estandar (o bien,
can\'onica):\footnote{Existe el n\'umero infinito de las representaciones
de las matrices $\gamma$. Por eso se dice que esas matrices se
definen con la precisi\'on de la transformaci\'on unitaria.}
\begin{equation} \gamma^0 =\pmatrix{\openone & 0\cr 0
&-\openone\cr}\quad,\quad \bbox{\gamma}^i =\pmatrix{0&\bbox{\sigma}^i\cr
\bbox{-\sigma}^i &0\cr}\quad, \label{md} \end{equation} donde $\openone$ y
$0$ se deben entender como matrices de $2\times 2$. La
ecuaci\'on de Klein-Gordon pudiera sido reconstruida de la ecuaci\'on
(\ref{ed}) despu\'es de multiplicaci\'on por el operador $\left [
i\gamma^\mu \partial_\mu + m\right ]$ que es otra `raiz cuadrada' de la
ecuaci\'on (\ref{kg}). Este es otro
camino para deducir la ecuaci\'on de
Dirac\footnote{V\'ease Dvoeglazov~\cite{DV-IC} para otros dos.}, propuesto
por B. L. van der Waerden y J. J. Sakurai~\cite{Sakurai}.
Como ya sabemos la funci\'on de Dirac tiene cuatro componentes complejos.
Pero, es posible saber la dimensi\'on de las matrices de algebra de Dirac
(que entran en la ecuaci\'on (\ref{ed})) en base de la simple deducci\'on
matem\'atica. De la definici\'on (\ref{md}) podemos concluir que
cuatro matrices $\gamma^\mu$ (o, bi\'en, $\alpha^\mu$, v\'ease la forma
hamiltoniana~\cite[ec.(8)]{DV-IC}) son anticomutados,
\begin{equation}
\gamma^\mu \gamma^\nu
+\gamma^\nu \gamma^\mu = 2g^{\mu\nu}\label{rc}
\end{equation}
con $g^{\mu\nu} = diag\,(1,-1,-1,-1)$
es el tensor de m\'etrica en el espacio de Minkowski.
Si $i\neq j$ tenemos
$\gamma_i \gamma_j +\gamma_j \gamma_i =0$. Entonces, de acuerdo con las
reglas del \'algebra lineal:
\begin{equation} Det (\gamma_i \gamma_j) = Det
(-\gamma_j \gamma_i) = (-1)^d \, Det (\gamma_i \gamma_j )\quad,
\end{equation}
que nos da informaci\'on que la dimesi\'on tiene que ser
{\bf par}. En el caso $d=2$ se tienen s\'olo tres matrices que
anticonmutan una con otra, son matrices de Pauli (\ref{mp}) que forman el
sistema completo en sentido matem\'atico. Pero necesitamos cuatro
matrices $\gamma^1, \gamma^2, \gamma^3$ y $\gamma^0$. Conclu\'{\i}mos que
la dimensi\'on tiene que ser mayor o igual a cuatro $d \geq 4$.
La representaci\'on m\'as simple es $d=4$, v\'ease (\ref{md}).
En otras representaciones del grupo de Poincar\`e tambi\'en se pueden
proponer ecuaciones del primer orden, como hizieron de
Broglie, Duffin, Kemmer y Bhabha (por
ejemplo,~\cite{Fish}).\footnote{V\'ease acerca de las relaciones entre las
ecuaciones de primer y de segundo orden para las part\'{\i}culas con el
esp\'{\i}n $j=1$ en ref.~\cite[Secci\'on \# 4]{DV-IC}.} Ellos tienen la
forma de la ecuaci\'on de Dirac pero en los casos del esp\'{\i}n alto la
funci\'on del campo ya no es la funci\'on con cuatro componentes y las
matrices no satisfacen la relaci\'on de anticonmutaci\'on (\ref{rc}).
Pero en cada representaci\'on existen relaciones m\'as complicadas entre
esas matrices. Para el algebra de Duffin, Kemmer y Petiau, la
representaci\'on $(1,0)\oplus (0,1)\oplus (1/2,1/2)\oplus (1/2,1/2) \oplus
(0,0)\oplus (0,0)$, ellas se denotan como matrices $\beta$ (en lugar de
matrices $\gamma$) y satisfacen
\begin{equation} \beta_\mu \beta_\nu
\beta_\lambda + \beta_\lambda \beta_\nu \beta_\mu =\beta_\mu
g_{\nu\lambda} +\beta_\lambda g_{\mu\nu}\quad.
\end{equation}
Adem\'as, se
puede presentar la ecuaci\'on para las part\'{\i}culas escalares (el campo
escalar) en la forma de las derivadas de primer orden.
Introduciendo para la funci\'on de Klein-Gordon la siguiente notaci\'on
\begin{equation} \psi_0 \equiv {\partial \psi \over \partial t}\quad,\quad
\psi_i \equiv {\partial \psi \over \partial x^i}\quad,\quad
\psi_4 \equiv m\Psi
\end{equation}
el 'vector' con cinco componentes satisface la ecuaci\'on
de primer orden, que ponen en la forma hamiltoniana~\cite{It}:
\begin{equation}
i{\partial \psi \over \partial t} = \left ({1\over i} \bbox{\alpha}
\cdot \bbox{\nabla} +m\beta \right) \psi\quad.
\end{equation}
La ecuaci\'on de Klein-Gordon se presenta tambi\'en en la
forma del conjunto de dos ecuaciones~\cite{Fish,Kos}
\begin{equation}
{\partial \Psi \over \partial x^\alpha}
=\kappa \Xi_\alpha\quad,\quad {\partial \Xi^\alpha \over \partial
x^\alpha} = -\kappa \Psi\quad, \label{kg10}
\end{equation}
con $\kappa \equiv mc/\hbar$,
que toma la forma de matrices siguiente:
\begin{equation} i{\partial \over \partial t}
\pmatrix{\phi\cr\chi_1\cr\chi_2\cr\chi_3\cr}
= \left [ \pmatrix{0&p_1&p_2&p_3\cr
p_1&0&0&0\cr
p_2&0&0&0\cr
p_3&0&0&0\cr} +m \pmatrix{1&0&0&0\cr
0&-1&0&0\cr
0&0&-1&0\cr
0&0&0&-1\cr}\right ]\pmatrix{\phi\cr\chi_1\cr\chi_2\cr\chi_3\cr}\quad,
\label{kg1}
\end{equation}
donde
\begin{eqnarray}
\cases{\phi = i\partial_t \Psi +m\Psi &\cr
\chi_i = -i\bbox{\nabla}_i \Psi = {\bf p}_i \Psi &\cr}\quad.
\end{eqnarray}
Finalmente, gracias a Dowker~\cite{Dowker} sabemos que una part\'{\i}cula
de cualquier esp\'{\i}n puede ser descrita por el sistema de
ecuaciones de primer orden:
\begin{mathletters} \begin{eqnarray}
\alpha^\mu \partial_\mu \Phi &=& m\Upsilon\quad,\label{dow1}\\
\overline{\alpha}^\mu \partial_\mu \Upsilon &=& -m\Phi \label{dow2} \quad.
\end{eqnarray} \end{mathletters}
$\Phi$ se transforma de acuerdo con la representaci\'on
$(j,0)\oplus (j-1,0)$ y $\Upsilon$ de acuerdo con
$(j-1/2,1/2)$. Las matrices $\alpha^\mu$ en este caso generalizado tienen
dimensi\'on $4j \times 4j$.
Muchas caracter\'{\i}sticas de las part\'{\i}culas pueden ser obtenidas
por el an\'alisis de la teor\'{\i}a de campos libres, por la ecuaciones
de la mec\'anica cu\'antica relativista que presentamos en esa Secci\'on.
Lo importante es prestar atenci\'on al formalismo matem\'atico
del grupo de Poincar\`e, desarollado basicamente por Wigner. Como
dir\'{\i}a el profesor A. Barut esa materia es muy viva hasta ahora. Pero,
la f\'{\i}sica siempre tiene muchos caminos: gracias al desarollo de los
aceleradores de altas energ\'{\i}as la tarea de los f\'{\i}sicos en
los \'ultimos cincuenta a\~nos era explicar los procesos con el cambio del
n\'umero de part\'{\i}culas, para este objetivo era necessario desarollar
la met\'odica de calculaciones, tales como la met\'odica de diagramos de
Feynman, la teor\'{\i}a de la matriz $S$, modelos potenciales etc.
Gran parte de esos c\'alculos se basan en el concepto de la
interacci\'on, principalmente el concepto de la interacci\'on minimal,
$\partial_\mu \rightarrow \partial_\mu -ieA_\mu$, donde $A_\mu$ es el
potencial 4-vector. Pero, matem\'aticamente, es posible introducir otros
tipos de interacci\'on como lo hizo el doctor Moshinsky.
\section{Oscilador de Dirac de Moshinsky}
El concepto del oscilador arm\'onico relativista fue propuesto
por primera vez hace mucho tiempo~\cite{Ito} pero fue olvidado
y redescubierto en 1989 por el doctor Marcos Moshinsky~\cite{Mosh}.
En el caso del problema de un cuerpo \'el caracteriza por la
siguiente substituci\'on de interacci\'on {\bf no-minimal}
en la ecuaci\'on de Dirac:
\begin{equation}
{\bf p}
\rightarrow {\bf p}-im\omega {\bf r} \beta\quad,
\end{equation}
donde $m$ es la masa del fermi\'on, $\omega$ es la frequencia del
oscilador, ${\bf r}$ es la coordenada 3-dimensional, y $\beta \equiv
\gamma^0$ es una de matrices de algebra de Dirac (que tambi\'en es la
matriz del operador de paridad). Existen muy pocos problemas
de interacci\'on de una part\'{\i}cula, tales como a) potencial de
Coulomb; b) campo magn\'etico uniforme; c) la onda electromagn\'etica
plana, que se puede resolver en forma exacta en mec\'anica cu\'antica. El
oscilador de Dirac de Moshinsky es un de ellos y un poco parecida al
problema b).\footnote{En el problema de la part\'{\i}cula en un campo
magn\'etico uniforme tenemos los terminos adicionales $\sim ({\bf A}\cdot
{\bf p})$. Es tarea para el lector comprobar los c\'alculos
en~\cite[p.67]{It} y compararlos con el problema que consideramos en el
texto de este art\'{\i}culo.} Las soluciones del conjunto para
2-espinores~\cite{Mosh}
\begin{mathletters}
\begin{eqnarray}
\label{do1}
(E-mc^2) \psi &=& c \bbox{\sigma} \cdot ({\bf p} +im\omega {\bf r})
\chi\quad,\\
\label{do2} (E+mc^2) \chi &=& c \bbox{\sigma} \cdot ({\bf
p} -im\omega {\bf r}) \psi
\end{eqnarray} \end{mathletters}
se han dados por el {\tt ket}
\begin{equation}
\vert N (l {1\over 2}) jm> =
\sum_{\mu\sigma} <l\mu, {1\over 2} \sigma\vert jm> R_{_{Nl}} (r)
Y_{_{lm}} (\theta,\phi) \chi_\sigma\quad.
\end{equation}
El espectro de energ\'{\i}a es entonces
\begin{equation}
(mc^2)^{-1} (E_{_{Nlj}}^2 - m^2 c^4) =
\cases{\hbar \omega \left [ 2(N-j)+1 \right ]\quad,\quad
\mbox{if}\,\,\,\,\, l=j-{1\over 2}&\cr \hbar \omega \left [ 2(N+j)+3
\right ]\quad,\quad \mbox{if}\,\,\,\,\, l=j+{1\over 2}&\quad.\cr}
\end{equation}
Fue demostrado en los art\'{\i}culos~\cite{Moreno1} que la ecuaci\'ones
(\ref{do1},\ref{do2}) pueden ponerse en la forma covariante:
\begin{equation}
\left ( \hat p - mc +\kappa {e\over 4m} \sigma^{\mu\nu}
F_{\mu\nu} \right ) \Psi = 0\quad, \quad\kappa =2m^2\omega/e\quad,
\label{fc}
\end{equation}
que significa que la interacci\'on `oscilador' en el problema de un
cuerpo es esencialmente la interacci\'on tensorial con el campo el\'ectrico
(!` {\bf no} con el vector potencial!). Aunque en esa consideraci\'on
$F^{\mu\nu} = u_\mu x_\nu -u_\nu x_\mu$ con $u_\mu = (1, {\bf 0})$ {\it
i.e.} es dependiente del sistema de referencia, no es dif\'{\i}cil
aplicar las transformaciones de Lorentz ({\tt boost} y rotaciones) para
reconstruir todos los resultados para los observables f\'{\i}sicos de
cualquier sistema inercial. Dos notas que pudiera ser \'utiles para las
investigaciones futuras: 1) El vector $u^\mu$ pudiera ser utilizado para
la definici\'on de la parte transversal de $x^\mu$, a saber $x^\mu_{\perp}
\equiv x^\mu + (x^\nu u_\nu) u^\mu$; 2) La necesidad de la interacci\'on
tensorial ya fue aprobada en base del analisis de los datos experimentales
de los decaimientos de $\pi^-$ y $K^+$ mesones (V. N. Bolotov {\it et
al.}, S. A. Akimenko {\it et al.}, 1990-96).
Como se ha demostrado en unos art\'{\i}culos {\it
e.g.}~\cite{Moreno2}, ese tipo de interacci\'on preserva la {\tt
supersimetr\'{i}a} de Dirac, el caso particular de {\tt
supersimetr\'{\i}a}. Generalmente, el concepto de {\tt
supersimetr\'{\i}a} se define en el sentido de teor\'{\i}a de grupos como
una algebra:
\begin{mathletters} \begin{eqnarray}
\left \{ \hat Q\,\,, \hat
Q\,\,\right \}_+ &=& \left \{ \hat Q^{\,\dagger}, \hat Q^{\,\dagger}\right
\}_+ =0\quad,\label{ss1}\\
\left [ \hat Q, \hat {\cal H}\right ]_- &=& \left [\hat
Q^{\,\dagger}, \hat {\cal H}\right ]_- =0\quad, \label{ss2}\\
\left \{
\hat Q, \hat Q^{\,\dagger} \right \}_+ &=& \hat{\cal H}\quad.\label{ss3}
\end{eqnarray} \end{mathletters}
$\hat Q^{\,\dagger}$ y $\hat Q$ se llama
{\tt supercargas}. En el caso de la mec\'anica cu\'antica relativista de
particulas cargadas con espin $j=1/2$, el hamiltoniano se ha dado
por~\cite{Moreno2}
\begin{equation} \hat {\cal H} = Q +Q^{\,\dagger}
+\lambda\quad, \label{dh} \end{equation}
$\lambda$ es hermitiana. Entonces, si
\begin{equation}
\left \{ Q,\,\lambda \right \}
=\left \{ Q^{\,\dagger},\,\lambda \right \} =0 = Q^{\,2} =
Q^{\,\dagger^{\,2}}\label{dss1}
\end{equation}
tenemos
\begin{equation}
\left \{ Q, \,Q^{\,\dagger} \right \} = \hat{\cal H}^2 - \lambda^2
\quad.\label{dss2}
\end{equation}
Por ejemplo, si
\begin{equation}
Q =\pmatrix{0&0\cr \bbox{\sigma}\cdot ({\bf p} -im\omega {\bf
r})&0\cr}\quad,
\quad \mbox{y}\quad,
\quad Q^{\,\dagger} =
\pmatrix{0&\bbox{\sigma}\cdot ({\bf p} +im\omega{\bf r})\cr 0&0\cr}
\end{equation}
todas las condiciones (\ref{dh},\ref{dss1},\ref{dss2}) se satisfacen y de
(\ref{dss2}) obtenemos la ecuaci\'on del oscilador de Dirac.
Otros tipos del oscilador arm\'onico relativista para el problema de un
cuerpo han sido propuestas en~\cite{DV-HJ}, es interesante que \'estos
est\'an relacionados con la interacci\'on con la carga quiral o con
coplamiento pseudoescalar, $m \rightarrow m \left [1+(w/c) r\gamma_5
\right ]$.
El profesor Moshinsky dijo en muchos seminarios que sus objetivos al
inventar ese tipo de interacci\'on eran aplicarlo al problema cu\'antico
relativista de muchos cuerpos. Aunque existe el formalismo de Bethe y
Salpeter~\cite{BS} y los m\'etodos para manejar este
formalismo~\cite{DV-PPN} con el tiempo relativo,\footnote{Recuerda las
palabras de Eddington: ``Un electron ayer y un proton hoy no forman el
atomo de hidrogeno".} fueron
aprobados en base a la comparaci\'on de los resultados te\'oricos y del
experimento, no todos f\'{\i}sicos quieren usarlo, principalmente, por su
complejidad. Unos problemas de descripci\'on alternativa han sido
considerados~\cite{Barut,Mosh2,Mosh3,Mosh4} desde diferentes puntos de
vista. Las interacci\'ones del tipo `oscilador' han sido propuestas para
la ``ecuaci\'on de Dirac" para dos cuerpos. En este caso podemos
considerar la ecuaci\'on \begin{eqnarray} \left [ (\bbox{\alpha}_1 -
\bbox{\alpha}_2)\cdot ({\bf p} -i{m\omega \over 2} {\bf r} {\cal B}) +mc
(\beta_1 +\beta_2) \right ] \psi ={E\over c}\psi\label{dodc} \quad.
\end{eqnarray}
Los \'{\i}ndices $1$ y $2$ indican el
espacio de representaci\'on de primera o segunda part\'{\i}cula. En lugar
de la matriz ${\cal B}$ pueden ser substituidos $B=\beta_1 \beta_2$ o
$B\Gamma_5=\beta_1 \beta_2 \gamma^5_1 \gamma^5_2$. Entonces, tenemos dos
tipos de oscilador para dos cuerpos. Los espectros son
parecidos~\cite{Mosh5}. Adem\'as, ambos dan los valores propios de
energ\'{\i}a $E=0$, {\tt relativistic cockroach nest
(RCN)}~\cite{Mosh2,Mosh5}, como les nombr\'o el doctor
Moshinsky.\footnote{Pienso que al problema del RCN se requiere m\'as
atenci\'on. \'El puede ser relacionado con las soluciones con $E=0$ que
eran descubiertas en otros sitemas f\'{\i}sicos (por ejemplo, en las
conocidas ecuaciones para el campo tensorial antisim\'etrico de segundo
rango, as\'{\i} como en las ecuaciones de primer orden para $j=3/2$ y
$j=2$, el \'ultimo es el campo gravitacional en la $(2,0)\oplus (0,2)$
representaci\'on) por D. V. Ahluwalia, A. E. Chubykalo, M. W. Evans
y J.-P. Vigier, y por V. V. Dvoeglazov. Pero esa materia tiene que ser
discutida en un art\'{\i}culo aparte.}
Las contribuciones de otros grupos cient\'{\i}ficos
ser\'an consideradas en la siguiente Secci\'on.
\section{Interacci\'on `oscilador' para
las part\'{\i}culas con espin alto.}
Las ecuaciones con la interacci\'on `oscilador' relativista han sido
tratados en los
art\'{\i}culos~\cite{Deb,Bruce,Ned,DV-NASA,DV-NC1,DV-NC2,DV-RMF1,DV-RMF2}
desde diversos puntos de vista. Ellos son para los espines diferentes del
espin $j=1/2$.
El operador de coordenada y el operador de
momento~\cite{Bruce} lineal han sido escogidos como $n\times n$ matrices:
$\widehat {\bf Q} = \hat \eta {\bf q}$ y $\widehat {\bf P} = \hat \eta
{\bf p}$. La interacci\'on introducida en la ecuaci\'on de
Klein-Gordon fue entonces
$\widehat{\bf P} \rightarrow \widehat {\bf P} -im\hat
\gamma \hat \Omega \cdot \widehat{\bf Q}$. Las matrices satisfacen las
condiciones \begin{equation} \hat \eta^2 =\openone\quad,\quad \hat
\gamma^2 = \openone\quad,\quad \mbox{y}\quad \left \{ \hat \gamma, \hat
\eta \right \}_+ =0\quad.
\end{equation}
$\Omega$ es la matriz de las
frecuencias , de dimensi\'on $3\times 3$. Como resultado tenemos el
oscilador anisotr\'opico en tres dimensiones: \begin{equation}
-{\partial^2 \over \partial t^2} \Psi ({\bf q}, t) = \left ( {\bf p}^{\,2}
+m^2 {\bf q}\cdot \hat \Omega^2 \cdot {\bf q} +m\hat \gamma \,\mbox{tr}
\Omega +m^2 \right ) \Psi ({\bf q},t)\quad, \end{equation} donde la forma
explicita de las matrices constituyentes es
\begin{equation} \hat \eta
=\pmatrix{0&1\cr 1&0\cr}\quad,\quad \hat \gamma =\pmatrix{-1&0\cr
0&1\cr}\quad.
\end{equation}
El espectro en el l\'{\i}mite no relativista llega a ser el espectro del
oscilador anisotr\'opico.
Pero las razones de introducir la forma matricial
en la ecuaci\'on de Klein-Gordon no han sido claros en este
art\'{\i}culo. En otros trabajos~\cite{DV-NASA,DV-NC1} otro
formalismo para la desripci\'on de part\'{\i}cula con $j=0$ y $j=1$ es
presentado. Como mencionamos, la ecuaci\'on de Klein-Gordon puede
ser presentada en la forma (\ref{kg10}). Entonces, la interacci\'on
`oscilador' se introduce a la ecuaci\'on (\ref{kg1}) en la misma manera:
${\bf p} \rightarrow {\bf p} -im\omega \beta {\bf r}$ con $\beta$,
la matriz ante el t\'ermino de masa en (\ref{kg1}). En caso $\omega_1
=\omega_2 =\omega_3 \equiv \omega$ la ecuaci\'on resultante coincide con
(10a) de ref.~\cite{Bruce}.
Las ecuaci\'ones para la interacci\'on `oscilador' de
part\'{\i}culas con $j=0$ y $j=1$
tambi\'en se han discutido
en~\cite{Deb,Ned,DV-RMF1,DV-RMF2}. La ecuaci\'on hamiltoniana en el
formalismo de Duffin, Kemmer y Petiau toma la forma\footnote{Recuerda que
las matrices $\beta$ no anticonmutan, entonces, la reducci\'on de la
ecuaci\'on covariante a forma hamiltoniana es m\'as complicada. Adem\'as,
es necesario mencionar que en el proceso de deducci\'on de la forma
hamiltoniana~\cite{Kemmer} los autores hicieron un procedimiento
matem\'aticamente dudoso cuando se multiplic\'o la ecuaci\'on por la
matriz singular $\beta_0$. Sin embargo, depende del lector si ser de
acuerdo con la discuci\'on en p. 110 de ref.~[32a].}
\begin{equation} i{\partial
\Phi \over \partial t} = \left ( {\bf B}\cdot {\bf p} +m\beta_0 \right )
\Phi\quad,\quad B_\mu = \left [\beta_0, \beta_\mu \right ]_- \quad,
\end{equation} donde $\Phi$ es la
funci\'on con 5 componentes en el caso $j=0$ y con 10 componentes en el
caso $j=1$. La interaci\'on que introducen N. Debergh {\it et
al.}~\cite{Deb} es ${\bf p} \rightarrow {\bf p}-im\omega \eta_0 {\bf r}$,
\,\,$\eta_0 =2\beta_0^2 -1$. Como el resultado, en el limite no
relativista se encuentra el mismo t\'ermino ${1\over 2} m\omega^2 {\bf
r}^{\,2}$ que en el caso de la representaci\'on $(1/2,0)\oplus (0,1/2)$
anterior. Adem\'as, tambi\'en tenemos el coplamiento espin-orbita.
Este es el oscilador de Duffin y Kemmer. La misma
substituci\'on nominimal fue discutida por Nedjadi y Barrett~\cite{Ned}
pero fue introducida en la forma covariante de la ecuaci\'on de Duffin,
Kemmer y Petiau (v\'ease footnote \# 10).
En otro art\'{\i}culo se empes\'o desde el sistema de
ecuaciones de Bargmann-Wigner (ecs. (2,3) en ref.~\cite{DV-RMF1})
y se consider\'o la funci\'on de Bargmann-Wigner antisim\'etrica en los
indices espinoreales (ec. (4)en ref.~\cite{DV-RMF1}). Los resultados de
ese art\'{\i}culo llegan a una conclusi\'on acerca de la existencia de los
estados doble degenerados en $N$, el numero cu\'antico principal, en el
l\'{\i}mite $\hbar \omega << mc^2$ excepto el nivel de base.
Los t\'erminos de interacci\'on en tres art\'{\i}culos citados
pueden ser presentados en forma covariante como los t\'erminos de
interacci\'on de la forma $\kappa S^{\mu\nu} F_{\mu\nu}$, {\it cf.}
con~(\ref{fc}). Vamos a componer una tabla
en que se comparan las formas de interacci\'on de varios art\'{\i}culos:\\
\medskip
\begin{tabular}{cc}
\hline
Referencia & El termino de interacci\'on\\
\hline
&\\
\cite{Deb}& $S^{\mu\nu} = -2i \left \{ \beta^\mu,\, B^\nu\right \}_+$\\
&\\
\cite{Ned}& $S^{\mu\nu}=\beta^\mu \eta^\nu$\\
&\\
\cite{DV-RMF1}& $S^{\mu\nu} = \beta^\mu
\beta^\nu - \beta^\nu \beta^\mu$ \\
&\\
\hline
\end{tabular}
Tambi\'en han sido considerados:
\begin{itemize}
\item
La interacci\'on `oscilador' en el formalismo de Sakata y
Taketani~\cite{Deb,DV-NASA,DV-NC1};
\item
El oscilador de Dirac en $(1+1)$ dimensi\'on~\cite{Dom,DV-NC1};
\item
El oscilador de Dirac en la forma con cuaterniones~\cite{DV-NASA};
\item
El oscilador para el sistema de ecuaciones de Dowker
(\ref{dow1},\ref{dow2}), lo que significa que este tipo de
interacci\'on puede ser introducido para cualquier espin~\cite{DV-NC2};
\item
El oscilador en el $2(2j+1)$ formalismo~\cite{DV-NASA};
\item
El oscilador de Dirac para el sistema de dos cuerpos
y su conexi\'on con el formalismo de Proca, y de
Bargmann y Wigner~\cite{DV-NASA,DV-RMF2}.
\end{itemize}
Finalmente, del conjunto de las ecuaciones de Crater y van
Alstine~\cite{Crater} y Sazdjian~\cite{Sazdjian} para un problema
de dos cuerpos podemos deducir una ecuaci\'on muy parecida a la
ecuaci\'on de oscilador de Dirac para dos cuerpos con el t\'ermino
de potencial m\'as general~[22b]:
\begin{equation}
{\cal V}^{int} (r) = {1\over r} \frac{dV(r)/dr}{1- \left [ V(r) \right ]^2
} \, i \left ( \bbox{\alpha}_2 -\bbox{\alpha}_1 \right ) B \Gamma_5 {\bf
r} \quad.
\end{equation}
M. Moshinsky {\it et al.} escogieron $V(r) =
\tanh (\omega r^2/4)$ y obtuvieron el oscilador de Dirac con la
interacci\'on del segundo tipo $B \Gamma_5 {\bf r}$, v\'ease (\ref{dodc}).
Nosotros queremos conectar esa formulaci\'on con las antecedentes y
proveer alguna base para escoger el potencial. Para este objetivo
vamos recordar el art\'{\i}culo~\cite{Skachkov} donde el potencial en la
representaci\'on configuracional relativista
\begin{equation} V(r) = -g^2
\frac{\coth (rm\pi)}{4\pi r}
\label{ps}
\end{equation}
ha sido deducido en base del an\'alisis de la
serie principal y la serie complementaria del grupo de Lorentz y la
aplicaci\'on de las transformaciones de Shapiro en lugar de las
transformaciones de Fourier al potencial de Coulomb en el espacio del
momento lineal; la coordenada en el espacio configuracional se considera
en la forma m\'as general y puede ser imaginario $r \rightarrow i\rho$.
Esa forma de potencial encontr\'o algunas aplicaciones en los modelos
potenciales de cromodin\'amica cu\'antica y electrodin\'amica cu\'antica.
Entonces, el uso del potencial de Skachkov tiene razones definitivas. En
caso del uso del potencial (\ref{ps}) tenemos una conducta
asintotica del termino ${\cal V}^{int}$ diferente en tres regiones. En
la regi\'on $r>> {1\over m\pi}$ \begin{eqnarray} {\cal V}^{int} (r) &\sim&
{1\over r \left [ r^2 -(g/4\pi)^2 \right ]} ( \bbox{\alpha}_1
-\bbox{\alpha}_2 ) B\Gamma_5 {\bf r}\approx\\ & & \approx \cases{(1/r^3) i
( \bbox{\alpha}_1 -\bbox{\alpha}_2 ) B\Gamma_5 {\bf r} \quad, \quad
\mbox{si}\quad r>>{1\over m\pi}\quad \mbox{y}\quad r> ({g\over 4\pi})^2
&\cr -(1/r) i(\bbox{\alpha}_1 -\bbox{\alpha}_2) B\Gamma_5 {\bf
r}\quad,\quad\mbox{si}\quad {1\over m\pi} << r < ({g\over
4\pi})^2\quad;&\cr}\nonumber
\end{eqnarray}
y en la regi\'on
$r<<{1\over \kappa}$
\begin{equation}
{\cal V}^{int} (r) \sim -im
(\bbox{\alpha}_1 -\bbox{\alpha}_2 ) B\Gamma_5
{\bf r}\quad,\quad\mbox{si}\quad r<<{1\over m\pi}\quad.
\end{equation}
Entonces, podemos ver que en la regi\'on de las distancias peque\~nas
tenemos precisamente la conducta del potencial del oscilador de Dirac. En
la regi\'on de las distancias grandes tenemos la conducta del potencial de
inversos grados en $r$.
Como conclusi\'on: el concepto del `oscilador de Dirac' aunque
ha sido propuesto recientemente se ha desarrollado
mucho en los \'ultimos a\~nos pues nos permite describir bien diferentes
sistemas f\'{\i}sicos relativistas (incluyendo espectros de mesones y
bariones) desde un punto de vista diferente al punto de vista com\'un.
Esas ideas se publican en las revistas de mayor nivel internacional
como {\it Physical Review Letters}, {\it Physics Letters}, {\it Journal of
Physics} y {\it Nuovo Cimento}. Por eso yo llamo a los
jovenes f\'{\i}sicos mexicanos aplicar sus talentos a esa area de
investigaciones.
Quiero indicar que en esa nota \'unicamente delineemos unos
rasgos de ese problema de la mec\'anica cu\'antica relativista
y no toquemos muchas ideas (por ejemplo, las teor\'{\i}as
del electr\'on extendido con adicionales grados de libertad
intr\'{\i}nsecos~\cite{Dirac}) que pueden tener cierta relaci\'on con el
`oscilador de Dirac' pero que todav\'{\i}a no han sido desarrollados
suficientemente y no han sido aceptados por mucha gente.
\bigskip
Agradezco mucho al doctor D. Armando Contreras Solorio por su invitaci\'on
a trabajar en la Escuela de F\'{\i}sica de la Universidad Aut\'onoma de
Zacatecas y los doctores M. Moshinsky, Yu. F. Smirnov y A. Del Sol Mesa por
sus valiosas discusiones. Reconozco la ayuda en la ortograf\'{\i}a
espa\~nola del Sr. Jes\'us Alberto C\'azares.
\smallskip
|
proofpile-arXiv_065-669
|
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
}
|
\section{Introduction}
The Drell-Yan (DY) process, i.e. the production of massive lepton pairs in
hadronic collisions, has remained, together with deep inelastic scattering
(DIS), one of the most prominent processes in strong interaction physics.
In recent years, in connection with the availability of high energy
machines like HERA and the Tevatron, much attention has been devoted to the
small-$x$ region of QCD, where parton densities become high and
perturbative methods reach their limits.
Although extensive work exists in the field of small-$x$ DIS and related
processes, the small-$x$ limit of lepton pair production has received only
limited theoretical attention. In our opinion the small-$x$ or high energy
region of the DY process, i.e. the region where the lepton pair mass $M$ is
much smaller than the available energy $\sqrt{s}$, deserves study for at
least two reasons:
First, it is of general theoretical interest to understand the
interrelations between the high energy limits of DIS and DY pair
production on both nucleon and nuclear targets. Though general
factorization theorems are established (see e.g. \cite{css}), it is still
worthwhile to develop an intuition for the way they are realized
specifically in the small-$x$ region.
Second, the DY process may provide new tools for the experimental
investigation of the small-$x$ dynamics in QCD. In particular, lepton pair
production in the region $M^2\ll s$ may be one of the cleanest processes
for the study of new phenomena in heavy ion collisions at future colliders.
Recently, a new approach to the DY process has been suggested by
Kopeliovich \cite{kop}, with the aim to understand the observed nuclear
shadowing at small $x_{\mbox{\scriptsize{target}}}$ \cite{ald}. It has been
observed, that in analogy to DIS, the DY cross section at high energies can
be expressed in terms of the scattering cross section of a color-neutral
$q\bar{q}$-pair.
In the present investigation, we derive the high energy DY cross section
in the target rest frame. The dominant underlying process is the scattering
of a parton from the projectile structure function off the target color
field. This parton radiates a massive photon, which subsequently decays into
a lepton pair. Our treatment of the interaction of the projectile parton
with the target hadron makes use of the high energy limit, but it is not
restricted to the exchange of a finite number of gluons.
Using the non-perturbative $q\bar{q}$-cross section $\sigma(\rho)$, where
$\rho$ is the transverse separation of the pair, a parallel description of
DIS and DY pair production is presented in the rather general framework
given above. The cross section $\sigma(\rho)$ appears in DIS since the
incoming photon splits into a $q\bar{q}$-pair, testing the target field at
two transverse positions \cite{nz}. Similarly, $\sigma(\rho)$ appears in
the DY process due to the interference of amplitudes in which the fast
quark of the projectile hits the target at different impact parameters.
Our main focus is on the role of the photon polarization. The interplay of
small and large transverse distances, characterized by different values of
the parameter $\rho$, is compared in DIS and the DY process for transverse
and longitudinal photons. In addition, the azimuthal angular correlations
in the DY process provide a new tool for the investigation of
$\sigma(\rho)$, which is not available in inclusive DIS.
The paper is organized as follows: After reviewing the impact parameter
description of DIS in Sect.~\ref{dis}, an analogous calculation of the
cross section for DY pair production is presented in Sect.~\ref{dy}. In
Sect.~\ref{dyad} the angular distribution of the produced lepton is given
in terms of integrals of the $q\bar{q}$-cross section $\sigma(\rho)$.
Concluding remarks in Sect.~\ref{con} are followed by an appendix, which
describes the technical details of the calculation.
\section{Deep inelastic scattering in the target rest frame}\label{dis}
A detailed discussion of small $x$ DIS in the target rest frame and in
impact parameter space has been given in \cite{nz}. The main
non-perturbative input is the scattering cross section $\sigma(\rho)$ of a
quark-antiquark pair with fixed transverse separation $\rho$. In the
present section this approach is briefly reviewed and reformulated in a way
allowing straightforward generalization to the DY process.
Consider first the scattering of a single energetic quark off an external
color field, e.g. the field of a proton (Fig.~\ref{quark}). The
complications associated with the color of the quark in the initial and
final state can be neglected at this point, since the quark amplitude is
only needed as a building block for the scattering of a color-neutral
$q\bar{q}$-pair.
\begin{figure}[ht]
\begin{center}
\parbox[b]{8cm}{\psfig{width=6cm,file=fig1.eps}}\\
\end{center}
\refstepcounter{figure}
\label{quark}
{\bf Fig.\ref{quark}} Scattering of a quark off the proton field.
\end{figure}
In the high energy limit the soft hadronic field cannot change the energy
of the quark significantly. Furthermore, we assume helicity conservation
and linear growth of the amplitude with energy. Therefore, introducing an
effective vertex $V(k',k)$, the amplitude can be given in the form
\begin{equation}
i2\pi\delta(k_0'-k_0)T_{fi}=\bar{u}_{s'}(k')V(k',k)u_s(k)=i2\pi\delta(k_0'
-k_0)\,2k_0\,\delta_{s's}\,\tilde{t}_q(k_\perp'-k_\perp)\, .\label{tdef}
\end{equation}
Here $\tilde{t}_q(p_\perp)$ can be interpreted as the Fourier transform of
an impact parameter space amplitude,
\begin{equation}
\tilde{t}_q(p_\perp)=\int d^2x_\perp\,t_q(x_\perp)\,e^{-ip_\perp x_\perp}
\, .
\end{equation}
Note that $t$ is a matrix in color space.
If the interaction of the quark with the color field is treated in the
non-Abelian eikonal approximation, $t_q$ is given explicitly by \cite{nac}
\begin{equation}
1+it_q(x_\perp)=F(x_\perp)=P\exp\left(-\frac{i}{2}\int_{-\infty}^{\infty}A_-
(x_+,x_\perp)dx_+\right)\, .\label{eik}
\end{equation}
Here $x_\pm=x_0\pm x_3$ are the light-cone components of $x$,
$A(x_+,x_\perp)$ is the gauge field, and the path ordering $P$ sets the
field at smallest $x_+$ to the rightmost position. The $x_-$-dependence of
$A$ is irrelevant as long as it is sufficiently smooth.
However, our analysis in the following does not rely on the specific form
of $t_q$ provided by the eikonal approximation Eq.~(\ref{eik}).
Consider now the forward elastic scattering of a photon with virtuality
$Q^2$ off an external field which is related to the total cross section
via the optical theorem. In the limit of very high photon energy,
corresponding to the small-$x$ region, the dominant process is the
fluctuation of the photon into a $q\bar{q}$-pair long before the target
(see Fig.~\ref{vp}). The quark and antiquark then scatter independently off
the external field and recombine far behind the target. The virtualities of
the quarks, which are small compared to their energies, do not affect their
effective scattering vertices. They enter the calculation only via the
explicit quark propagators connected to the photon.
\begin{figure}[ht]
\begin{center}
\parbox[b]{10cm}{\psfig{width=9.5cm,file=fig2.eps}}\\
\end{center}
\refstepcounter{figure}
\label{vp}
{\bf Fig.\ref{vp}} Elastic forward scattering of a virtual photon off the
proton field.
\end{figure}
The necessary calculations have been performed many years ago for the
Abelian case in light-cone quantization \cite{bks} and, more recently, in
a covariant approach, treating two gluon exchange in the high energy
limit \cite{nz}.
In the notation of \cite{nz} the transverse and longitudinal photon cross
sections read
\begin{equation}
\sigma_{T,L}=\int_0^1d\alpha\int d^2\rho_\perp\sigma(\rho)W_{T,L}
(\alpha,\rho)\, ,
\end{equation}
where $\rho=|\rho_\perp|$ is the transverse separation of quark and
antiquark when they hit the target proton, and $\alpha$ is the longitudinal
momentum fraction of the photon carried by the quark. The cross section
$\sigma(\rho)$ for the scattering of the $q\bar{q}$-pair is given by
\begin{equation}
\sigma(\rho)=\frac{2}{3}\,\mbox{Im}\int d^2x_\perp\,\mbox{tr}\left[it_q
(x_\perp)t_{\bar{q}}(x_\perp+\rho_\perp)+t_q(x_\perp)+t_{\bar{q}}(x_\perp+
\rho_\perp)\right].\label{sigmaqq}
\end{equation}
Here $t_q(x_\perp)$ is the quark scattering amplitude in impact parameter
space introduced above and $t_{\bar{q}}(x_\perp+\rho_\perp)$ is its
antiquark analogue. The last two terms in Eq.~(\ref{sigmaqq}) correspond to
diagrams where only the quark or only the antiquark is scattered.
We denote by $W_{T,L}$ the squares of the light-cone wave functions of a
transverse photon and a longitudinal photon with virtuality $Q^2$. In the
case of one massless quark generation with one unit of electric charge they
are given by
\begin{eqnarray}
W_T(\alpha,\rho)&=&\frac{6\alpha_{\mbox{\scriptsize em}}}{(2\pi)^2}N^2
[\alpha^2+(1-\alpha)^2]K_1^2(N\rho)\label{wt}\\ \nonumber\\
W_L(\alpha,\rho)&=&\frac{24\alpha_{\mbox{\scriptsize em}}}{(2\pi)^2}N^2
[\alpha(1-\alpha)]K_0^2(N\rho)\, ,\label{wl}
\end{eqnarray}
where $N^2=N^2(\alpha,Q^2)\equiv\alpha(1-\alpha)Q^2$ and $K_{0,1}$ are
modified Bessel functions. As is illustrated in Fig.~\ref{lcwf} the
variables $\alpha$ and $1\!-\!\alpha$ denote the longitudinal momentum
fractions of the photon carried by quark and antiquark.
\begin{figure}[ht]
\begin{center}
\parbox[b]{7cm}{\psfig{width=6.5cm,file=fig3.eps}}\\
\end{center}
\refstepcounter{figure}
\label{lcwf}
{\bf Fig.\ref{lcwf}} Light-cone wave function of the virtual photon in the
mixed representation.
\end{figure}
Notice that the color factor $1/3$ in Eq.~(\ref{sigmaqq}) is compensated
by a factor 3 in Eqs. (\ref{wt}),(\ref{wl}), so that $\sigma(\rho)$ is the
cross section for one color-neutral $q\bar{q}$-pair and the color summation
is included in the definition of $W_{T,L}$.
Consider now the region of a relatively soft quark, $\alpha<\Lambda^2/Q^2$,
with a hadronic scale $\Lambda\ll Q$. This region, where $N^2\simeq\alpha
Q^2=a^2$, corresponding to Bjorken's aligned jet model \cite{bj} (see also
\cite {lps}), gives a higher-twist contribution to $\sigma_L$ and a
leading-twist contribution to $\sigma_T$,
\begin{equation}
\sigma_{T,\bar{q}}=\frac{6\alpha_{\mbox{\scriptsize em}}}{(2\pi)^2Q^2}
\int_0^{\Lambda^2}da^2\int d^2\rho_\perp a^2K_1^2(a\rho)\sigma(\rho)\, .
\end{equation}
A possible interpretation of DIS in this kinematical region is the
splitting of the photon into a fast, on-shell antiquark and a soft quark,
which is not far off-shell. In a frame where the proton is fast, the latter
one corresponds to an incoming antiquark described by a scaling antiquark
distribution $\bar{q}(x)$. We thus denote the cross section by
$\sigma_{T,\bar{q}}$. Using the standard formula for the contribution of
the antiquark structure to the transverse cross section,
\begin{equation}
\sigma_{T,\bar{q}}=\frac{(2\pi)^2\alpha_{\mbox{\scriptsize em}}}{Q^2}x
\bar{q}(x)\, ,
\end{equation}
the antiquark distribution can then be given in terms of the
$q\bar{q}$-cross section,
\begin{equation}
x\bar{q}(x)=\frac{6}{(2\pi)^4}\int_0^{\Lambda^2}da^2\int d^2\rho_\perp
a^2K_1^2(a\rho)\sigma(\rho)\, .\label{qdis}
\end{equation}
This formula will be reproduced below from the impact parameter space
description of DY pair production at small
$x_{\mbox{\scriptsize{target}}}$, in agreement with factorization.
The above discussion in terms of scatterings off an external field can be
generalized to the case of a realistic hadron target by summing
appropriately over all contributing field configurations. Such an approach
has already been used for the treatment of DIS in \cite{bal} and for the
treatment of diffraction in \cite{bdh}.
\section{Drell-Yan Process in the target rest frame}\label{dy}
In this section the DY pair production cross section at small
$x_{\mbox{\scriptsize{target}}}$ will be calculated in the target rest
frame. Such an approach has recently been suggested by Kopeliovich in the
context of nuclear shadowing \cite{kop}.
Consider the kinematical region where the mass of the produced lepton pair
is large compared to the hadronic scale, but much smaller than the hadronic
center of mass energy, $\Lambda^2\ll M^2\ll s$. Furthermore, let the
longitudinal momentum fraction $x_F$ of the projectile hadron carried by
the DY pair be large, but not too close to 1. We assume here that the last
condition allows us to neglect higher-twist contributions from spectator
partons in the projectile \cite{bb}.
In the parton model the above process is described as the fusion of a
projectile quark with momentum fraction $x\approx x_F$ and a target
antiquark with momentum fraction $x_{\mbox{\scriptsize{target}}}\approx
M^2/sx_F\ll1$. (Here and below we neglect the antiquark distribution of the
projectile at the relevant values of $x_F$.)
However, a different physical picture of this process is appropriate in the
target rest frame: A large-$x$ quark of the projectile scatters off
the gluonic field of the target and radiates a massive photon, which
subsequently decays into leptons (compare \cite{bln}). The two relevant
diagrams, corresponding to the photon being radiated before or after the
interaction with the target, are shown in Fig.~\ref{mp}. Diagrams where the
quark interacts with the target both before and after the photon vertex are
suppressed in the high energy limit \cite{bh}. Note that in the above
approach no antiquark distribution of the target has to be introduced.
Instead, its effect is produced by the target color field.
\begin{figure}[ht]
\begin{center}
\parbox[b]{12cm}{\psfig{width=11.5cm,file=fig4.eps}}\\
\end{center}
\refstepcounter{figure}
\label{mp}
{\bf Fig.\ref{mp}} Production of a massive photon by a quark scattering off
the target field. A quark with momentum $k$ interacts with an external
field producing a photon with momentum $q$ and an outgoing quark with
momentum $k'$.
\end{figure}
In the high energy limit, i.e. $q_0,k_0,k_0'\gg M^2$, the corresponding
cross section, including the decay of the photon into the lepton pair,
reads ($e^2=4\pi\alpha_{\mbox{\scriptsize em}}$)
\begin{equation}
\frac{d\hat{\sigma}}{dx_FdM^2}=\frac{e^2}{72(2\pi)^3}\cdot\frac{1}
{x_Fk_0k_0'M^2}\int\frac{d^2q_\perp}{(2\pi)^2}\frac{d^2k_\perp'}{(2\pi)^2}
|T|^2\, .\label{dycs}
\end{equation}
Here $T$ is the amplitude for the production of the virtual photon, given
by the sum of the two diagrams in Fig.~\ref{mp},
\begin{equation}
i2\pi\delta(q_0\!+\!k_0'\!-\!k_0)T_\lambda=e\bar{u}_{s'}(k')\left[V(k',k\!-
\!q)\frac{i}{k\!\!\!/-q\!\!\!/}\epsilon\!\!/_\lambda(q)+\epsilon\!\!/_\lambda(q)\frac{i}
{k\!\!\!/'+q\!\!\!/}V(k'\!+\!q,k)\right]u_s(k)\, .\label{dyt}
\end{equation}
The matrix $V$ is the effective quark scattering vertex introduced in the
previous section, and $\epsilon(q)$ is the polarization vector of the
produced photon, accessible via the lepton angular distribution. Averaging
over $s$ and summation over $s'$ and $\lambda$ is understood in
Eq.~(\ref{dycs}).
When the cross section is explicitly calculated, the quark scattering
amplitudes $t_q(x_\perp)$ implicit in $V$ combine in a way very similar to
the case of DIS. Therefore, the final result can be expressed in terms of
the $q\bar{q}$-cross section introduced in the previous section \cite{kop}.
This cross section arises from the interference of the two diagrams in
Fig.~\ref{mp}. To understand the parallelism of the DY process and DIS,
observe that in the DY cross section the product of two quark amplitudes
tests the external field at two different transverse positions. In DIS this
corresponds to the quark-antiquark pair wave function of the virtual
photon, which tests the external field at two transverse positions as well.
The details of this calculation are presented in the appendix.
To make the analogy to DIS more apparent, the cross sections for the
production of the lepton pair via transversely and longitudinally polarized
photons are given separately:
\begin{equation}
\frac{d\sigma_{T,L}}{dx_FdM^2}=\frac{\alpha_{\mbox{\scriptsize em}}}{9(2\pi)
M^2}\!\!\int\limits_0^{(1-x_F)/x_F}\!\!\!\!\!d\alpha\int d^2\rho_\perp
\frac{q\Big(x_F(1+\alpha)\Big)}{(1+\alpha)^2}
\sigma(\rho)W^{DY}_{T,L}(\alpha,\rho)\, .\label{dyex}
\end{equation}
Here $q(x)$ is the quark distribution of the projectile, $\alpha=k_0'/q_0$
is the ratio of energies or longitudinal momenta of outgoing quark and
photon, and $W^{DY}_{T,L}$ are the analogues of the squares of the photon
wave functions defined for DIS in the previous section,\footnote{
Our formula for the transverse polarization is similar but not identical to
the result given in \cite{kop}.}
\begin{eqnarray}
W^{DY}_T(\alpha,\rho)&=&\frac{12\alpha_{\mbox{\scriptsize em}}}{(2\pi)^2}
N^2[\alpha^2+(1+\alpha)^2]K_1^2(N\rho)\label{wdyt}\\ \nonumber\\
W^{DY}_L(\alpha,\rho)&=&\frac{24\alpha_{\mbox{\scriptsize em}}}{(2\pi)^2}N^2
[\alpha(1+\alpha)]K_0^2(N\rho)\, .\label{wdyl}
\end{eqnarray}
As in the DIS case a subsidiary variable $N^2=\alpha(1+\alpha)M^2$ has
been introduced.
We have defined the polarization of the massive photon in the $u$-channel
frame. In this frame the photon is at rest and the $z$-axis, defining the
longitudinal polarization vector, is antiparallel to the momentum of the
target hadron. Since the polarizations are invariant with respect to boosts
along the $z$-axis, one could also say that the longitudinal polarization is
defined by the direction of the photon momentum, in a frame where photon
and target hadron momenta are antiparallel. This last definition makes it
obvious that the polarizations in the $u$-channel frame of DY pair
production are analogous to the standard polarization choice in DIS,
defining $\sigma_T$ and $\sigma_L$.
Note, that Eqs. (\ref{wdyt}),(\ref{wdyl}) can be obtained from their
analogues in DIS, Eqs. (\ref{wt}),(\ref{wl}), by the substitutions
$Q^2\to M^2$ and $1\!-\!\alpha\,\to\,1\!+\!\alpha$. The last substitution
reflects the fact that the longitudinal parton momenta in units of the
photon momentum are $\alpha$ and $1-\alpha$ in DIS, as opposed to $\alpha$
and $1+\alpha$ in DY pair production. Since the transverse polarizations
are summed rather than averaged in the DY process, an additional factor of
$2$ appears in Eq.~(\ref{wdyt}) as compared to Eq.~(\ref{wt}).
Consider now the region of a relatively soft outgoing quark,
$\alpha<\Lambda^2/M^2$, with a hadronic scale $\Lambda\ll M$. In analogy to
the DIS case, this region gives a higher-twist contribution for
longitudinal polarization and a leading-twist contribution for transverse
polarization:
\begin{equation}
\frac{d\sigma_{T,\bar{q}}}{dx_FdM^2}=\frac{\alpha_{\mbox{\scriptsize em}}^2
q(x_F)}{6\pi^3M^4}\int_0^{\Lambda^2}da^2\int d^2\rho_\perp a^2K_1^2(a\rho)
\sigma(\rho)\, .\label{sdy}
\end{equation}
Here, assuming a sufficiently smooth behavior of $q(x)$, terms suppressed
by powers of $\Lambda/M$ have been dropped.
The above kinematical region corresponds to the contribution from the
antiquark distribution of the target as calculated in the parton model at
leading order,
\begin{equation}
\frac{d\sigma_{T,\bar{q}}}{dx_FdM^2}=\frac{4\pi
\alpha_{\mbox{\scriptsize em}}^2}{9M^4}q(x_F)\cdot x_t\bar{q}(x_t)\, .
\end{equation}
By comparing this formula with Eq.~(\ref{sdy}), an expression for
$x\bar{q}(x)$ can be derived which is identical to Eq.~(\ref{qdis})
obtained in the case of DIS. This, of course, was to be expected in view of
the factorization theorems (see e.g. \cite{css}) relating DIS and the DY
process.
So far the target hadron has been treated simply as a given external color
field. As already pointed out in the last section, a more realistic model
has to include an appropriate summation over all contributing field
configurations. These field configurations, together with the produced
lepton pair and the projectile remnant, form the final state of the
scattering process. If we assume that at some stage before hadronization
the target field is separated from the rest of the final state, the
inclusiveness of the process translates into a summation over all field
configurations in the cross section. This corresponds exactly to the
discussion of the previous section, where a summation over all field
configurations of the target had to be performed for the cross section of
DIS.
\section{Angular distributions}\label{dyad}
In DY pair production the transverse and longitudinal photon polarizations
can be distinguished by measuring the angle between the direction of the
decay lepton and the $z$-axis. However, more information can be obtained by
considering the azimuthal angle as well. In particular, additional
integrals involving the $q\bar{q}$-cross section $\sigma(\rho)$ are
provided by the angular correlations.
As explained in the last section the $u$-channel frame is most suitable for
an analysis along the lines of small-$x$ DIS. We work with a right-handed
coordinate system, the $z$-axis being antiparallel to $\vec{p}_t$ and the
$y$-axis parallel to $\vec{p}_p\times\vec{p}_t$, where $\vec{p}_p$ and
$\vec{p}_t$ are the projectile and target momenta in the photon rest frame
(see e.g. \cite{fal}).
The direction of the produced lepton is characterized by the standard
polar and azimuthal angles $\theta$ and $\phi$. To obtain the complete
angular dependence of the cross section, interference terms between
different photon polarizations have to be considered in equations analogous
to (\ref{dycs}) and (\ref{dyt}). The obtained contributions are multiplied
by typical angle dependent functions obtained from the leptonic tensor (for
details see e.g. \cite{mir}).
In general, the angular dependence can be given in the form
\begin{equation}
\frac{1}{\sigma}\frac{d\sigma}{d\Omega}\sim1+\lambda\cos^2\theta+\mu\sin2
\theta\cos\phi+\frac{\nu}{2}\sin^2\theta\cos2\phi\,.
\end{equation}
The cross section will be presented after integration over the transverse
momentum of the pair. The dependence on $q_\perp^2$ can be recovered from
the formulae in the appendix, where some details of the calculation are
given.
In compact notation the results of our impact parameter space calculation
of the DY cross section read
\begin{equation}
\frac{d\sigma}{dx_FdM^2d\Omega}=\frac{\alpha_{\mbox{\scriptsize em}}^2}
{2(2\pi)^4M^2}\!\!\!\!\!\!\!\int\limits_0^{(1-x_F)/x_F}\!\!\!\!\!\!\!d
\alpha\int d^2r_\perp\frac{q\Big(x_F(1+\alpha)\Big)}{(1+\alpha)^2}
\sigma(r/N)\sum_if_i(\alpha,r)h_i(\theta,\phi)\, ,\label{dya}
\end{equation}
where $i\in \{T,\,L,\,TT,\,LT\}$ labels the contributions of transverse and
longitudinal polarizations and of the transverse-transverse and
longitudinal-transverse interference terms. Note, that in contrast to
Eq.~(\ref{dyex}) here the integration is over the dimensionless variable
$r_\perp=N\rho_\perp$. The angular dependence is given by the functions
\begin{eqnarray}
h_T(\theta,\phi)=1+\cos^2\theta&,&\qquad h_{TT}(\theta,\phi)=
\sin^2\theta\cos2\phi\, ,\label{h1}\\
h_L(\theta,\phi)=1-\cos^2\theta&,&\qquad h_{LT}(\theta,\phi)=\sin2\theta
\cos\phi\,.\label{h2}
\end{eqnarray}
Finally, the $\alpha$- and $r$-dependent coefficients read
\begin{eqnarray}
f_T(\alpha,r)&=&[\alpha^2+(1+\alpha)^2]K_1^2(r)\\
f_L(\alpha,r)&=&4\alpha(1+\alpha)K_0^2(r)\\
f_{TT}(\alpha,r)&=&\alpha(1+\alpha)\Big[r^{-1}K_1''(r)+r^{-2}K_1'(r)-r^{-3}
K_1(r)-2K_1^2(r)\Big]\\
f_{LT}(\alpha,r)&=&r^{-1}(1+2\alpha)\sqrt{\alpha(1+\alpha)}\Big[K_0(r)\Big(
rA(r)-1\Big)\label{flt}\\
&&-K_1(r)\Big((2r)^{-1}-rA'(r)\Big)-K_1'(r)/2\Big]\, .\nonumber
\end{eqnarray}
Here the first two functions give the transverse and longitudinal
contributions of the last section. The function $A$ is defined by the
following definite integral, that can be expressed through the difference
of the modified Bessel function $I_0$ and the modified Struve function
$\mbox{\bf L}_0$ \cite{gr},
\begin{equation}
A(r)=\int_0^\infty\frac{dt\sin rt}{\sqrt{1+t^2}}=\frac{\pi}{2}\Big(I_0(r)-
\mbox{\bf L}_0(r)\Big)\, .
\end{equation}
As discussed in the previous section the integral involving $f_T$ receives
a contribution from large $\rho$. In Eq.~(\ref{dya}) this is most easily
seen by recalling that $\sigma(\rho)\sim\rho^2$ at small $\rho$. Replacing
$\sigma(r/N)$ with the model form $r^2/N^2$ results in a divergent
$\alpha$-integration. This shows the sensitivity to the large
$\rho$-behavior of $\sigma(\rho)$. In DY pair production on nuclei this
sensitivity will show up as leading-twist shadowing, since configurations
with large cross section are absorbed at the surface. This is analogous to
the leading-twist shadowing in DIS \cite{fs}.
In contrast to the integral of $f_T$, the integrals involving $f_L,\,
f_{TT}$ and $f_{LT}$ are dominated by the region of small $\rho$ at leading
twist. To see this, notice that replacing $\sigma(r/N)$ with $r^2/N^2$ in
Eq.~(\ref{dya}) results in finite $\alpha$-integrations for $f_L,\, f_{TT}$
and $f_{LT}$. This leading-twist contribution corresponds to the effect of
the gluon distribution of the target. Integrations involving higher powers
of $r$ are sensitive to large transverse distances, but they are suppressed
by powers of $M$. This corresponds to the fact that in the leading order
(and leading-twist) parton model these angular coefficients vanish.
The above discussion shows that in the longitudinal contribution and in the
interference terms, shadowing appears only at higher twist or at higher
order in $\alpha_S$. While higher-twist terms are suppressed by
$\Lambda^2/M^2$ in the longitudinal cross section and in the
transverse-transverse interference term, they are only suppressed by
$\Lambda/M$ in the longitudinal-transverse interference. This results from
the weaker suppression of $f_{LT}$ at small $\alpha$ (see Eq.~(\ref{flt})).
The presented formulae contain contributions from all transverse sizes of
the effective $q\bar{q}$-pair interacting with the target gluonic field,
thus including all higher-twist corrections from this particular source.
Our analysis also gives a simple and intuitive derivation of the
dominant QCD-corrections at small $x$, associated with the gluon
distribution of the target.
\section{Conclusions}\label{con}
A detailed calculation of the DY cross section, including its angular
dependence, has been performed in the target rest frame in the limit of
high energies and small $x_{\mbox{\scriptsize{target}}}$. The close
similarity with the impact parameter description of DIS has been
established for transverse and longitudinal photon polarizations and the
availability of additional angular observables in the DY process has been
demonstrated.
As is well known, in the small-$x$ limit DIS can be calculated from the
elastic scattering of the quark-antiquark component of the virtual photon
wave function off the hadronic target. The DIS cross section is given by a
convolution of the photon wave function with the $q\bar{q}$-cross section
$\sigma(\rho)$. This picture holds even when the interaction of each of the
quarks with the target is completely non-perturbative.
The cross section for DY pair production via transversely and
longitudinally polarized massive photons can be given as a convolution of
the above $q\bar{q}$-cross section with analogues of the transverse and
longitudinal photon wave functions. These functions depend on the photon
momentum fractions carried by the quarks and on the photon virtuality in
exactly the same way as in DIS. For this analogy to hold polarization by
polarization the DY process has to be analyzed in the $u$-channel frame.
This frame corresponds to the $\gamma^*p$-frame of small-$x$ DIS, since it
uses photon and target hadron momenta for the definition of the $z$-axis.
As in the DIS case, the transverse photon contribution is sensitive to
large distances in impact parameter space. It receives a leading-twist
contribution from large $\rho$, which corresponds to the effect of a
non-perturbative antiquark distribution in the target. Our approach
includes, beyond this leading-twist contribution and the
$\alpha_S$-correction from small $\rho$, all higher-twist terms associated
with different transverse distances inside the target. The universal
function $\sigma(\rho)$ relates these contributions directly to the
analogous terms in DIS.
In addition to the transverse-longitudinal analysis, which can be performed
using the polar angle of the produced lepton, the azimuthal angle allows the
investigation of interference terms of different polarizations. These terms
involve convolutions of $\sigma(\rho)$ with new functions, not available in
DIS.
Our analysis shows that rather detailed information about the function
$\sigma(\rho)$ can be obtained from a sufficiently precise measurement of
angular correlations in the DY process at small
$x_{\mbox{\scriptsize{target}}}$. Even more could be learned from a
measurement of the nuclear dependence of these angular correlations. Using
the Glauber approach to nuclear shadowing, this type of measurement would
provide additional information about the functional dependence of the
$q\bar{q}$-cross section on $\rho$. We expect that future measurements of
the DY process will help to disentangle the interplay of small and large
transverse distances in small-$x$ physics.
Several aspects of the presented approach require further study:
The leading-twist part of our calculation combines the standard
$q\bar{q}$-annihilation cross section with the $\alpha_S$-corrections
associated with the target gluon density. It is certainly necessary to
include other $\alpha_S$-corrections systematically into our approach. For
example, corrections associated with the radiation of a gluon off the
projectile quark can be treated in the impact parameter space by methods
developed in \cite{bdh}.
Furthermore, higher-twist contributions from sources not considered here
should be carefully analyzed. At small $x_{\mbox{\scriptsize{target}}}$,
corresponding to large $x_F$, higher-twist corrections from comoving
projectile partons are potentially important \cite{bb}.
Finally, to go beyond the classical field model, we have argued that the
summation over all field configurations of the target is identical in DIS
and the DY process. It would be highly desirable, to derive this statement
in the framework of QCD and to specify the type of expected corrections.
\\*[0cm]
We would like to thank M.~Beneke, W.~Buchm\"uller, L.~Frankfurt, P.~Hoyer,
B.~Kopeliovich, A.H.~Mueller, M.~Strikman, and R.~Venugopalan for valuable
discussions and comments. A.H. and E.Q. have been supported by the Feodor
Lynen Program of the Alexander von Humboldt Foundation.
\section*{Appendix}
Some details of the calculations leading to the results of Sections
\ref{dy} and \ref{dyad} are presented below.
In analogy to Eq.~(\ref{dycs}) the angular distribution of the lepton in
the DY process is given by
\begin{equation}
\frac{d\hat{\sigma}}{dx_FdM^2d\Omega}=\frac{e^2}{192(2\pi)^4}\cdot\frac{1}
{x_Fk_0k_0'M^4}\int\frac{d^2q_\perp}{(2\pi)^2}\frac{d^2k_\perp'}{(2\pi)^2}
\Big(T_\lambda T^*_{\lambda'}\Big)\,L^{\mu\nu}\epsilon^\lambda_\mu
\epsilon^{\lambda'*}_\nu\, ,\label{adycs}
\end{equation}
where the polarization sum is understood. The leptonic tensor $L^{\mu\nu}$
is contracted with the photon polarization vectors
\begin{equation}
\epsilon_\pm=(0,1,\pm i,0)\, ,\quad \epsilon_0=(0,0,0,1)\, ,\label{defe}
\end{equation}
defined in the $u$-channel frame, which has been specified in
Sect.~\ref{dyad}. This expression gives the functions $h_i(\theta,\phi)$,
introduced in Eqs.~(\ref{h1}),(\ref{h2}).
The amplitudes $T_\lambda$ are most conveniently calculated in the target
rest frame, in a system where $q_\perp=0$. This corresponds to the
$u$-channel frame, boosted appropriately along its $z$-axis. Of course now
the amplitude is a function of $k_\perp$ $(k_\perp\neq0)$, and the
$q_\perp$-integration in Eq.~(\ref{adycs}) has to be replaced by a
$k_\perp$-integration,
\begin{equation}
\int\frac{d^2q_\perp}{q_0^2} \rightarrow \int\frac{d^2k_\perp^2}{k_0^2}\, ,
\end{equation}
leading to
\begin{eqnarray}
\!\!\!\!\!
\frac{d\hat{\sigma}}{dx_FdM^2d\Omega}&=&\frac{e^2}{96(2\pi)^4}\cdot
\frac{q_0^2}{x_Fk_0^3k_0'M^4}\int\frac{d^2k_\perp d^2k_\perp'}{(2\pi)^4}
\Bigg[\frac{h_T}{2}\Big(|T_+|^2+|T_-|^2\Big)+h_L|T_0|^2\nonumber\\
\!\!\!\!\!\label{csa}\\
\!\!\!\!\!
&&-\frac{h_{TT}}{2}\Big(T_+T_-^*+T_-T_+^*\Big)-\frac{h_{LT}}{2\sqrt{2}}
\Big(T_0T_+^*+T_0T_-^*+T_+T_0^*+T_-T_0^*\Big)\Bigg]\, .\nonumber
\end{eqnarray}
Now the amplitudes have to be calculated explicitly. In the high energy
approximation the fermion propagators appearing in Eq.~(\ref{dyt}) can be
treated as follows,
\begin{equation}
\frac{1}{p\!\!/}\approx\frac{\sum_r u_r(p)\bar{u}_r(p)}{p^2}\, .
\end{equation}
Here $u(p)\equiv u(p_+,\bar{p}_-,p_\perp)$, with $\bar{p}_-\equiv
p_\perp^2/p_+$, for off-shell momentum $p$. This approximation, which has
been used in the above form in \cite{bdh}, corresponds to dropping the
instantaneous terms in light-cone quantization \cite{bks}.
In our frame with $q_\perp=0$ the resulting spinor products in the high
energy approximation are given by
\begin{equation}
\bar{u}(k-q)\epsilon\!\!/_\lambda u(k)\equiv g_\lambda(k_\perp,q_0,\alpha)\,
,\quad\bar{u}(k')\epsilon\!\!/_\lambda u(k'+q)\equiv g_\lambda(k_\perp',q_0,
\alpha)\, ,\label{gg}
\end{equation}
where the helicity dependence has been suppressed. By choosing some spinor
representation explicit formulae are easily obtained.
Using the definition of $\tilde{t}_q$ in Eq.~(\ref{tdef}), the following
expression for the product of two amplitudes can now be given,
\begin{equation}
T_\lambda T_{\lambda'}^*=\Big[2eq_0\alpha(1\!+\!\alpha)\Big]^2\,\Big|
\tilde{t}_q(k_\perp'\!-\!k_\perp)\Big|^2\,\left(\frac{g_\lambda(k_\perp)}
{k_\perp^2\!+\!N^2}-\frac{g_\lambda(k_\perp')}{k_\perp'^2\!+\!N^2}\right)
\left(\frac{g_{\lambda'}(k_\perp)}{k_\perp^2\!+\!N^2}-\frac{g_{\lambda'}
(k_\perp')}{k_\perp'^2\!+\!N^2}\right)^*.\label{tt}
\end{equation}
Here the $q_0$- and $\alpha$-dependence of the function $g$ has been
suppressed. To relate this formula to the $q\bar{q}$-cross section of
Sect.~\ref{dis}, observe that in the high energy limit the quark and
antiquark scattering amplitudes are dominated by gluon exchange. This
implies the relation $t_{\bar{q}}(x_\perp)=-t^\dagger_q(x_\perp)$. Note in
particular, that this relation is respected by the eikonal approximation
Eq.~(\ref{eik}). From the definition of $\sigma(\rho)$ in Eq.~(5) one now
derives
\begin{equation}
|\tilde{t}_q(k_\perp'\!-\!k_\perp)|^2=-\frac{3}{2}\int d^2\rho_\perp
\sigma(\rho)\,e^{i\rho_\perp(k'-k)_\perp}+2(2\pi)^2\delta^2(k_\perp'\!-\!
k_\perp)\,\mbox{Im}\,\tilde{t}_q(0)\, .\label{ts}
\end{equation}
When inserted into Eq.~(\ref{tt}) the second term on the r.h.s. of
Eq.~(\ref{ts}) does not contribute, so that the product $T_\lambda
T_{\lambda'}^*$ can indeed be expressed through the the $q\bar{q}$-cross
section $\sigma(\rho)$.
Two remarks have to be made concerning the treatment of the photon
polarization vectors in Eqs.~(\ref{gg}),(\ref{tt}):
First, recall that Eq.~(\ref{tt}) holds in the target rest frame with
$z$-axis parallel to $\vec{q}$. Therefore, the transverse polarizations are
the same as in the $u$-channel frame (see Eq.~(\ref{defe})). However,
before the $k_\perp$-integration in Eq.~(\ref{csa}) can be performed, the
$k_\perp$-dependence of the orientation of $x$- and $y$-axis has to be
explicitly introduced. This is most easily done by assuming $k$ to be
exactly parallel to the projectile momentum and writing $\hat{e}_x=k_\perp/
|k_\perp|$ (see the definition of the $u$-channel frame in
Sect.~\ref{dyad}).
Second, the boost from the $u$-channel frame to the target rest frame
transforms the longitudinal polarization vector to
\begin{equation}
\epsilon_0=\frac{q}{M}-\frac{M}{q_0}(1,\vec{0})\, .
\end{equation}
However, taking advantage of gauge invariance, the first term can be
dropped. This significantly simplifies the evaluation of the corresponding
spinor products in Eq.~(\ref{gg}).
It is now straightforward to choose an explicit spinor representation, to
evaluate Eq.~(\ref{gg}), and to combine Eqs.~(\ref{ts}),(\ref{tt}) and
(\ref{csa}). Performing the $k_\perp$ and $k_\perp'$-integrations, the
result stated in Eq.~(\ref{dya}) is obtained.
|
proofpile-arXiv_065-670
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\section{Introduction}
\label{introduction}
\subsection{General motivation for the $1/N$ expansion}
\label{intro-general}
The approach to quantum field theory and statistical mechanics based
on the identification of the large-$N$ limit and the perturbative
expansion in powers of $1/N$, where $N$ is a quantity related to the
number of field components, is by now almost thirty years old. It
goes back to the original work by Stanley \cite{Stanley-II} on the
large-$N$ limit of spin systems with ${\rm O}(N)$ symmetry, soon
followed by Wilson's suggestion that the $1/N$ expansion may be a
valuable alternative in the context of renormalization-group
evaluation of critical exponents, and by 't Hooft's extension
\cite{THooft-planar} to gauge theories and, more generally, to fields
belonging to the adjoint representation of ${\rm SU}(N)$ groups. More
recently, the large-$N$ limit of random-matrix models was put into a
deep correspondence with the theory of random surfaces, and therefore
it became relevant to the domain of quantum gravity.
In order to understand why the $1/N$ expansion should be viewed as a
fundamental tool in the study of quantum and statistical field theory,
it is worth emphasizing a number of relevant features:
1) $N$ is an intrinsically dimensionless parameter, representing a
dependence whose origin is basically group-theoretical, and leading to
well-defined field representations for all integer values, hence it is
not subject to any kind of renormalization;
2) $N$ does not depend on any physical scale of the theory, hence we
may expect that physical quantities should not show any critical
dependence on $N$ (with the possible exception of finite-$N$ scaling
effects in the double-scaling limit);
3) the large-$N$ limit is a thermodynamical limit, in which we observe
the suppression of fluctuations in the space of internal degrees of
freedom; hence we may expect notable simplifications in the algebraic
and analytical properties of the model, and even explicit
integrability in many instances.
Since integrability does not necessarily imply triviality, the
large-$N$ solution to a model may be a starting point for finite-$N$
computations, because it shares with interesting finite values of $N$
many physical properties. (This is typically not the case for the
standard free-field solution which forms the starting point for the
usual perturbative expansions.) Moreover, for reasons which are
clearly, if not obviously, related to the three points above, the
physical variables which are naturally employed to parameterize
large-$N$ results and $1/N$ expansions are usually more directly
related to the observables of the models than the fields appearing in
the original local Lagrangian formulation.
More reasons for a deep interest in the study of the large-$N$
expansion will emerge from the detailed discussion we shall present in
the rest of this introductory section. We must however anticipate
that many interesting review papers have been devoted to specific
issues in the context of the large-$N$ limit, starting from Coleman's
lectures \cite{Coleman-erice}, going through Yaffe's review on the
reinterpretation of the large-$N$ limit as classical mechanics
\cite{Yaffe}, Migdal's review on loop equations
\cite{Migdal-equations}, and Das' review on reduced models
\cite{Das-review}, down to Polyakov's notes \cite{Polyakov-book} and
to the recent large commented collection of original papers by Brezin
and Wadia \cite{Brezin-Wadia}, not to mention Sakita's booklet
\cite{Sakita-book} and Ma's contributions
\cite{Ma-introduction,Ma-largeN}. Moreover, the $1/N$ expansion of
two-dimensional spin models has been reviewed by two of the present
authors a few years ago \cite{Campostrini-Rossi-review}. As a
consequence, we decided to devote only a bird's eye overview to the
general issues, without pretension of offering a self-contained
presentation of all the many conceptual and technical developments
that have appeared in an enormous and ever-growing literature; we even
dismissed the purpose of offering a complete reference list grouped by
arguments, because the task appeared to be beyond our forces.
We preferred to focus on a subset of all large-$N$ topics, which has
never been completely and systematically reviewed: the issue of
unitary-matrix models. Our self-imposed limitation should not appear
too restrictive, when considering that it still involves such topics
as ${\rm U}(N) \times {\rm U}(N)$ principal chiral models, virtually
all that concerns large-$N$ lattice gauge theories, and an important
subset of random-matrix models with their double-scaling limit
properties, related to two-dimensional conformal field theory.
The present paper is organized on a logical basis, which will neither
necessarily respect the sequence of chronological developments, nor
it will keep the same emphasis that was devoted by the authors of the
original papers to the discussion of the different issues.
Sect.\ \ref{unitary-matrices} is devoted to a presentation of the
general and common properties of unitary-matrix models, and to an
analysis of the different approaches to their large-$N$ solution that
have been discussed in the literature.
Sect.\ \ref{single-link} is a long and quite detailed discussion of the
most elementary of all unitary-matrix systems. Since all essential
features of unitary-matrix models seem to emerge already in the
simplest example, we thought it worthwhile to make this discussion as
complete and as illuminating as possible.
Sect.\ \ref{2d-YM} is an application of results obtained by studying
the single-link problem, which exploits the equivalence of this model
with lattice YM$_2$ and principal chiral models in one dimension.
Sect.\ \ref{chiral-chains} is devoted to a class of reasonably simple
systems, whose physical interpretation is that of closed chiral chains
as well as of gauge theories on polyhedra.
Sect.\ \ref{simplicial-chiral} presents another class of integrable
systems, corresponding to chiral models defined on a $d$-dimensional
simplex, whose properties are relevant both in the discussion of the
strong-coupling phase of more general unitary-matrix models and in the
context of random-matrix models.
Sect.\ \ref{principal-chiral} deals with the physically more
interesting applications of unitary-matrix models: two-dimensional
principal chiral models and four-dimensional lattice gauge theories,
sharing the properties of asymptotic freedom and ``confinement'' of
the Lagrangian degrees of freedom. Special issues, like numerical
results and reduced models, are considered.
\subsection{Large $N$ as a thermodynamical limit: factorization}
\label{intro-factorization}
As we already mentioned briefly in the introduction, one of the
peculiar features of the large-$N$ limit is the occurrence of notable
simplifications, that become apparent at the level of the quantum
equations of motion, and tend to increase the degree of integrability
of the systems. These simplifications are usually related to a
significant reduction of the number of algebraically-independent
correlation functions, which in turn is originated by the property of
factorization.
This property is usually stated as follows: connected Green's
functions of quantities that are invariant under the full symmetry
group of the system are suppressed with respect to the corresponding
disconnected parts by powers of $1/N$. Hence when $N\to\infty$ one
may replace expectation values of products of invariant quantities
with products of expectation values.
One must however be careful, since factorization is not a property
shared by all invariant operators without further qualifications. In
particular, experience shows that operators associated with very high
rank representations of the symmetry group, when the rank is $O(N)$,
do not possess the factorization property. A very precise
characterization has been given by Yaffe \cite{Yaffe}, who showed that
factorization is a property of ``classical'' operators, i.e., those
operators whose coherent state matrix elements have a finite
$N\to\infty$ limit.
It is quite interesting to investigate the physical origin of
factorization. The property
\begin{equation}
\lim_{N\to\infty} \left<AB\right> = \left<A\right>\left<B\right>
\end{equation}
implies in particular that
\begin{equation}
\lim_{N\to\infty} \langle A^2\rangle = \left<A\right>^2,
\end{equation}
i.e., the vacuum state of the model, seen as a statistical ensemble,
seems to possess no fluctuations. To be more precise, all the field
configurations that correspond to a nonvanishing vacuum wavefunction
can be related to each other by a symmetry transformation. This
residual infinite degeneracy of the vacuum configurations makes the
difference between the large-$N$ limit and a strictly classical limit
$\hbar\to0$, and allows the possibility of violations of factorization
when infinite products of operators are considered; this is in a sense
the case with representations whose rank is $O(N)$.
More properly, we may view large $N$ as a thermodynamical limit
\cite{Haan}, since the number of degrees of freedom goes to
infinity faster than any other physical parameter, and as a
consequence the ``macroscopic'' properties of the system, i.e., the
invariant expectation values, are fixed in spite of the great number
of different ``microscopic'' realizations. This realization does not
rule out the possibility of searching for the so-called ``master
field'', that is a representative of the equivalence class of the
field configurations corresponding to the large-$N$ vacuum, such that
all invariant expectation values of the factorized operators can be
obtained by direct substitution of the master field value into the
definition of the operators themselves \cite{Coleman-erice}.
There has been an upsurge of interest on master fields in recent years
\cite{Gopakumar-Gross,Douglas-master}, triggered by new results in
non-commutative probability theory applied to the stochastic master
field introduced in Ref.\ \cite{Greensite-Halpern-master}.
\subsection{$1/N$ expansion of vector models in
statistical mechanics and quantum field theory}
\label{intro-vector}
The first and most successful application of the approach based on the
large-$N$ limit and the $1/N$ expansion to field theories is the
analysis of vector models enjoying ${\rm O}(N)$ or ${\rm SU}(N)$
symmetry. Actually, ``vector models'' is a nickname for a wide class
of different field theories, characterized by bosonic or fermionic
Lagrangian degrees of freedom lying in the fundamental representation
of the symmetry group (cfr.\ Ref.\ \cite{Campostrini-Rossi-review} and
references therein).
A quite general feature of these models is the possibility of
expressing all self-interactions of the fundamental degrees of freedom
by the introduction of a Lagrange multiplier field, a boson and a
singlet of the symmetry group, properly coupled to the Lagrangian
fields, such that the resulting effective Lagrangian is quadratic in
the $N$-component fields. One may therefore formally perform the
Gaussian integration over these fields, obtaining a form of the
effective action which is nonlocal, but depends only on the singlet
multiplier, acting as a collective field; in this action $N$ appears
only as a parameter.
The considerations developed in Subs.\ \ref{intro-factorization} make
it apparent that all fluctuations of the singlet field must be
suppressed in the large-$N$ limit (no residual degeneracy is left in
the trivial representation). As a consequence, solving the models in
this limit simply amounts to finding the singlet field configuration
minimizing the effective action. The problem of nonlocality is easily
bypassed by the consideration that translation invariance of the
physical expectation values requires the action-minimizing field
configuration to be invariant in space-time; hence the saddle-point
equations of motion become coordinate-independent and all nonlocality
disappears.
As one may easily argue from the above considerations, the large-$N$
solution of vector models describes some kind of Gaussian field
theory. Nevertheless, this result is not as trivial as one might
imagine, since the free theory realization one is faced with usually
enjoys quite interesting properties, in comparison with the na{\"\i}ve
Lagrangian free fields. Typical phenomena appearing in the large-$N$
limit are an extension of the symmetry and spontaneous mass
generation. Moreover, when the fundamental fields possess some kind
of gauge symmetry, one may also observe dynamical generation of
propagating gauge degrees of freedom; this is the case with
two-dimensional ${\rm CP}^{N-1}$ models and their generalizations
\cite{Witten-CPN,DAdda-Luscher-DiVecchia-1/N}.
The existence of an explicit form of the effective action offers the
possibility of a systematic expansion in powers of $1/N$. The
effective vertices of the theory turn out to be Feynman integrals over
a single loop of the free massive propagator of the fundamental field.
In two dimensions, where the physical properties of many vector models
are especially interesting (e.g., asymptotic freedom), these one-loop
integrals can all be computed analytically in the continuum version,
and even on the lattice many analytical results have been obtained.
The $1/N$ expansion is the starting point for a systematic computation
of critical exponents, which are nontrivial in the range $2<d<4$, for
the study of renormalizability of superficially nonrenormalizable
theories in the same dimensionality range, and for the computation of
physical amplitudes. Notable is the case of the computation of
amplitude ratios, which are independent of the coupling in the scaling
region, and therefore are functions of $1/N$ alone; hopefully, their
$1/N$ expansion possesses a nonvanishing convergence radius. The
$1/N$ expansion was also useful to explore the double-scaling limit
properties of vector models
\cite{Nishigaki-Yoneya,DiVecchia-Kato-Ohta,Damgaard-Heller}.
The properties of the large-$N$ limit and of the $1/N$ expansion of
continuum and lattice vector models were already reviewed by many
authors. We therefore shall not discuss this topic further. We only
want to stress that this kind of studies can be very instructive,
given the physical interest of vector models as realistic prototypes
of critical phenomena in two and three dimensions and as models for
dynamical Higgs mechanism in four dimensions. Moreover, some of the
dynamical properties emerging mainly from the large-$N$ studies of
asymptotically free models (in two dimensions) may be used to mimic
some of the features of gauge theories in four dimensions; however,
at least one of the essential aspects of gauge theories, the presence
of matrix degrees of freedom (fields in the adjoint representation),
cannot be captured by any vector model.
\subsection{$1/N$ expansion of matrix models: planar diagrams}
\label{intro-planar-diagrams}
The first major result concerning the large-$N$ limit of matrix-valued
field theories was due to G. 't Hooft, who made the crucial
observation that, in the $1/N$ expansion of continuum gauge theories,
the set of Feynman diagrams contributing to any given order admits a
simple topological interpretation. More precisely, by drawing the
${\rm U}(N)$ fundamental fields (``quarks'') as single lines and the
${\rm U}(N)$ adjoint fields (``gluons'') as double lines, each line
carrying one color index, a graph corresponding to a $n$th-order
contribution can be drawn on a genus $n$ surface (i.e., a surface
possessing $n$ ``holes''). In particular, the zeroth-order
contribution, i.e., the large-$N$ limit, corresponds to the sum of all
planar diagrams. The extension of this topological expansion to gauge
models enjoying ${\rm O}(N)$ and ${\rm Sp}(2N)$ symmetry has been
described by Cicuta \cite{Cicuta}. Large-$N$ universality among ${\rm
O}(N)$, ${\rm U}(N)$, and ${\rm Sp}(2N)$ lattice gauge theories has
been discussed by Lovelace \cite{Lovelace-universality}.
This property has far-reaching consequences: it allows for
reinterpretations of gauge theories as effective string theories, and
it offers the possibility of establishing a connection between matrix
models and the theory of random surfaces, which will be exploited in
the study of the double-scaling limit.
As a byproduct of this analysis, 't Hooft performed a summation of all
planar diagrams in two-dimensional continuum Yang-Mills theories, and
solved QCD$_2$ to leading nontrivial order in $1/N$,
finding the meson spectrum \cite{THooft-mesons,Callan-Coote-Gross}.
Momentum-space planarity has a coordinate-space counterpart in lattice
gauge theories. It is actually possible to show that, within the
strong-coupling expansion approach, the planar diagrams surviving in
the large-$N$ limit can be identified with planar surfaces built up of
plaquettes by gluing them along half-bonds
\cite{Kazakov-pl,OBrien-Zuber-expansion,Kostov-sc}. This construction
however leads quite far away from the simplest model of planar random
surfaces on the lattice originally proposed by Weingarten
\cite{Weingarten-pathological,Weingarten-nonplanar}, and hints at some
underlying structure that makes a trivial free-string interpretation
impossible.
\subsection{The physical interpretation: QCD phenomenology}
\label{intro-QCD}
The sum of the planar diagrams has not till now been performed in the
physically most interesting case of four-dimensional ${\rm SU}(N)$
gauge theories. It is therefore strictly speaking impossible to make
statements about the relevance of the large-$N$ limit for the
description of the physically relevant case $N=3$. However, it is
possible to extract from the large-$N$ analysis a number of
qualitative and semi-quantitative considerations leading to a very
appealing picture of the phenomenology predicted by the $1/N$
expansion of gauge theories. These predictions can be improved
further by adopting Veneziano's form of the large-$N$ limit
\cite{Veneziano-unified}, in which not only the number of colors $N$
but also the number of flavors $N_f$ is set to infinity, while their
ratio $N/N_f$ is kept finite. We shall not enter a detailed
discussion of large-$N$ QCD phenomenology, but it is certainly useful
to quote the relevant results.
\subsubsection{The large-$N$ property of mesons}
Mesons are stable and noninteracting; their decay amplitudes are
$O(N^{-1/2})$, and their scattering amplitudes are $O(N^{-1})$.
Meson masses are finite.
The number of mesons is infinite.
Exotics are absent and Zweig's rule holds.
\subsubsection{The large-$N$ property of glueballs}
Glueballs are stable and noninteracting, and they do not mix with
mesons; a vertex involving $k$ glueballs and $n$ mesons is
$O(N^{1-k-n/2})$.
The number of glueballs is infinite.
\subsubsection{The large-$N$ property of baryons}
A large-$N$ baryon is made out of $N$ quarks, and therefore it
possesses peculiar properties, similar of those of solitons
\cite{Witten-baryons}.
Baryon masses are $O(N)$.
The splitting of excited states is $O(1)$.
Baryons interact strongly with each other; typical vertices are
$O(N)$.
Baryons interact with mesons with $O(1)$ couplings.
\subsubsection{The $\eta'$ mass formula}
The spontaneous breaking of the ${\rm SU}(N_f)$ axial symmetry in QCD
gives rise to the appearance of a multiplet of light pseudoscalar
mesons. This symmetry-breaking pattern was explicitly demonstrated in
the context of large-$N$ QCD by Coleman and Witten
\cite{Coleman-Witten}. However, the singlet pseudoscalar is not
light, due to the anomaly of the ${\rm U}(1)$ axial current. Since
the anomaly equation
\begin{equation}
\partial_\mu J_\mu^5 = {g^2 N_f \over 16\pi^2}
\mathop{\operator@font Tr}\nolimits \widetilde F_{\mu\nu} F^{\mu\nu}
\end{equation}
has a vanishing right-hand side in the limit $N_c\to\infty$ with $N_f$
and $g^2N_c$ fixed (the standard large-$N$ limit of non-Abelian gauge
theories), the leading-order contribution to the mass of the $\eta'$
should be $O(1/N_c)$. The proportionality constant should be related
to the symmetry-breaking term, which in turn is related to the
so-called topological susceptibility, i.e., the vacuum expectation
value of the square of the topological charge. The resulting
relationship shows a rather satisfactory quantitative agreement with
experimental and numerical results \cite{Witten-algebra,%
DiVecchia-Veneziano,Rosenzweig-Schechter-Trahern,Nath-Arnowitt,%
Teper-topol-SU3}.
\subsection{The physical interpretation: two-dimensional quantum
gravity}
\label{intro-2dqg}
In the last ten years, a new interpretation of the $1/N$ expansion of
matrix models has been put forward. Starting from the relationship
between the order of the expansion and the topology of two-dimensional
surfaces on which the corresponding diagrams can be drawn, several
authors \cite{Ambjorn-Durhuus-Frohlich,David-planar,David-random,%
Kazakov-bilocal,Kazakov-Kostov-Migdal} proposed that large-$N$ matrix
models could provide a representation of random lattice
two-dimensional surfaces, and in turn this should correspond to a
realization of two-dimensional quantum gravity. These results were
found consistent with independent approaches, and proper modifications
of the matrix self-couplings could account for the incorporation of
matter.
The functional integrals over two-dimensional closed Riemann manifolds
can be replaced by the discrete sum over all (piecewise flat)
manifolds associated with triangulations. It is then possible to
identify the resulting partition function with the vacuum energy
\begin{equation}
E_0 = - \log Z_N,
\end{equation}
obtained from a properly defined $N\times N$ matrix model, and the
topological expansion of two-dimensional quantum gravity is nothing
but the $1/N$ expansion of the matrix model.
The partition function of two-dimensional quantum gravity is expected
to possess well-defined scaling properties
\cite{Knizhnik-Polyakov-Zamolodchikov}. These may be recovered in the
matrix model by performing the so-called ``double-scaling limit''
\cite{Brezin-Kazakov,Douglas-Shenker,Gross-Migdal}. This limit is
characterized by the simultaneous conditions
\begin{equation}
N \to \infty, \qquad g \to g_c \, ,
\label{double-scaling}
\end{equation}
where $g$ is a typical self-coupling and $g_c$ is the location of some
large-$N$ phase transition. The limits are however not independent.
In order to get nontrivial results, one is bound to tune the two
conditions (\ref{double-scaling}) in such a way that the combination
\begin{equation}
x = (g-g_c) N^{2/\gamma_1}
\end{equation}
is kept finite and fixed. $\gamma_1$ is a computable critical
exponent, usually called ``string susceptibility''. According to
Ref.\ \cite{Knizhnik-Polyakov-Zamolodchikov}, it is related to the
central charge $c$ of the model by
\begin{equation}
\gamma_1 = {1\over12}\left[25 - c + \sqrt{(1-c)(25-c)}\right].
\end{equation}
An interesting reinterpretation of the double-scaling limit relates it
to some kind of finite-size scaling in a space where $N$ plays the
r\^ole of the physical dimension $L$
\cite{Damgaard-Heller,Carlson,Brezin-ZinnJustin-RG}. Research in this
field has exploded in many directions. A wide review reflecting the
state of the art as of the year 1993 appeared in the already-mentioned
volume by Brezin and Wadia \cite{Brezin-Wadia}. Here we shall only
consider those results that are relevant to our more restricted
subject.
\section{Unitary matrices}
\label{unitary-matrices}
\subsection{General features of unitary-matrix models}
\label{general-unitary-matrices}
Under the header of unitary-matrix models we class all the systems
characterized by dynamical degrees of freedom that may be expressed in
terms of the matrix representations of the unitary groups ${\rm U}(N)$
or special unitary groups ${\rm SU}(N)$ and by interactions enjoying a
global or local ${\rm U}(N)_L \times {\rm U}(N)_R$ symmetry.
Typically we shall consider lattice models, with no restriction on the
lattice structure and on the number of lattice points, ranging from 1
(single-matrix problems) to infinity (infinite-volume limit) in an
arbitrary number of dimensions.
In the field-theoretical interpretation, i.e., when considering models
in infinite volume and in proximity of a fixed point of some (properly
defined) renormalization group transformation, such models will have a
continuum counterpart, which in turn shall involve unitary-matrix
valued fields in the case of spin models, while for gauge models the
natural continuum representation will be in terms of hermitian matrix
(gauge) fields.
A common feature of all unitary-matrix models will be the
group-theoretical properties of the functional integration measure:
for each dynamical variable the natural integration procedure is based
on the left- and right-invariant Haar measure
\begin{equation}
{\mathrm{d}}\mu(U) = {\mathrm{d}}\mu(UV) = {\mathrm{d}}\mu(VU), \qquad
\int {\mathrm{d}}\mu(U) = 1.
\label{haar}
\end{equation}
An explicit use of the invariance properties of the measure and of the
interactions (gauge fixing) can sometimes lead to formulations of the
models where some of the symmetries are not apparent. Global ${\rm
U}(N)$ invariance is however always assumed, and the interactions,
as well as all physically interesting observables, may be expressed in
terms of invariant functions.
It is convenient to introduce some definitions and notations. An
arbitrary matrix representation of the unitary group ${\rm U}(N)$ is
denoted by ${\cal D}_{ab}^{(r)}(U)$. The characters and dimensions of
irreducible representations are $\chi_{(r)}(U) = {\cal
D}_{aa}^{(r)}(U)$ and $d_{(r)}$ respectively. $(r)$ is characterized
by two set of decreasing positive integers $\{l\} = l_1,...l_s$ and
$\{m\} = m_1,...,m_t$. We may define the ordered set of integers
$\{\lambda\} = \lambda_1,...,\lambda_N$ by the relationships
\begin{eqnarray}
\lambda_k &=& l_k,\ (k=1,...,s), \quad
\lambda_k = 0,\ (k=s+1,...,N-t), \nonumber \\
\lambda_k &=& -m_{N-k+1},\ (k=N-t+1,...,N).
\label{lambda}
\end{eqnarray}
It is then possible to write down explicit expressions for all
characters and dimensions, once the eigenvalues $\exp{\mathrm{i}}\phi_i$ of the
matrix $U$ are known:
\begin{eqnarray}
\chi_{(\lambda)}(U) &=& {\det\Vert\exp\{{\mathrm{i}}\phi_i(\lambda_j+N-j)\}\Vert
\over \det\Vert\exp\{{\mathrm{i}}\phi_i(N-j)\}\Vert}, \\
d_{(\lambda)} &=& {\prod_{i<j}(\lambda_i-\lambda_j+j-i) \over
\prod_{i<j} (j-i)} = \chi_{(\lambda)}(1).
\label{chi-d}
\end{eqnarray}
The general form of the orthogonality relations is
\begin{equation}
\int {\mathrm{d}}\mu(U) \, {\cal D}_{ab}^{(r)}(U) \, {\cal D}_{cd}^{(s)\,*}(U) =
{1\over d_{(r)}} \, \delta_{r,s}\,\delta_{a,c}\,\delta_{b,d} \,.
\label{ortho}
\end{equation}
Further relations can be found in Ref.\ \cite{Itzykson-Zuber}.
The matrix $U_{ab}$ itself coincides with the fundamental
representation $(1)$ of the group, and enjoys the properties
\begin{equation}
\chi_{(1)}(U) = \mathop{\operator@font Tr}\nolimits U, \qquad
d_{(1)} = N, \qquad
\sum_a U_{ab}U_{ac}^* = \delta_{bc} \, .
\end{equation}
The measure ${\mathrm{d}}\mu(U)$ (which we shall also denote simply by ${\mathrm{d}} U$),
when the integrand depends only on invariant combinations, may be
expressed in terms of the eigenvalues \cite{Mehta-book}.
\subsection{Chiral models and lattice gauge theories}
\label{chiral-and-lattice}
Unitary matrix models defined on a lattice can be divided into two
major groups, according to the geometric and algebraic properties of
the dynamical variables: when the fields are defined in association
with lattice sites, and the symmetry group is global, i.e., a single
${\rm U}(N)_L \times {\rm U}(N)_R$ transformation is applied to all
fields, we are considering a spin model (principal chiral
model); in turn, when the dynamical variables are defined on the links
of the lattice and the symmetry is local, i.e., a different
transformation for each site of the lattice may be performed, we are
dealing with a gauge model (lattice gauge theory).
As we shall see, these two classes are not unrelated to each other: an
analogy between $d$-dimensional chiral models and $2d$-dimensional
gauge theories can be found according to the following correspondence
table \cite{Green-Samuel-chiral}:
\begin{eqnarray*}
\begin{tabular}{c@{\quad}c}
{\bf spin} & {\bf gauge} \\
site, link & link, plaquette \\
loop & surface \\
length & area \\
mass & string tension \\
two-point correlation & Wilson loop \\
\end{tabular}
\end{eqnarray*}
While this correspondence in arbitrary dimensions is by no means
rigorous, there is some evidence supporting the analogy.
In the case $d=1$, which we shall carefully discuss later, one can
prove an identity between the partition function (and appropriate
correlation functions) of the two-dimensional lattice gauge theory and
the corresponding quantities of the one-dimensional principal chiral
model. Both theories are exactly solvable, both on the lattice and in
the continuum limit, and the correspondence can be explicitly shown.
Approximate real-space renormalization recursion relations obtained by
Migdal \cite{Migdal} are identical for $d$-dimensional chiral models
and $2d$-dimensional gauge models.
The two-dimensional chiral model and the (phenomenologically
interesting) four-dimensional non-Abelian gauge theory share the
property of asymptotic freedom and dynamical generation of a mass
scale. In both models these properties are absent in the Abelian case
(${\rm XY}$ model and ${\rm U}(1)$ gauge theory respectively), which
shows no coupling-constant renormalization in perturbation theory.
The structure of the high-temperature expansion and of the
Schwinger-Dyson equations is quite similar in the two models.
It will be especially interesting for our purposes to investigate the
Schwinger-Dyson equations of unitary-matrix models and discuss the
peculiar properties of their large-$N$ limit.
\subsection{Schwinger-Dyson equations in the large-$N$ limit}
\label{SD-largeN}
In order to make our analysis more concrete, we must at this stage
consider specific forms of interactions among unitary matrices, both
in the spin and in the gauge models. The most dramatic restriction
that we are going to impose on the lattice action is the condition of
considering only nearest-neighbor interactions. The origin of this
restriction is mainly practical, because non nearest-neighbor
interactions lead to less tractable problems. We assume that, for the
systems we are interested in, it will always be possible to find a
lattice representation in terms of nearest-neighbor interactions
within the universality class.
Let us denote by $x$ an arbitrary lattice site, and by $x,\mu$ an
arbitrary lattice link originating in the site $x$ and ending in the
site $x+\mu$: $\mu$ is one of the $d$ positive directions in a
$d$-dimensional hypercubic lattice.
A plaquette is identified by the label $x,\mu,\nu$, where the
directions $\mu$ and $\nu$ ($\mu\ne\nu$) specify the plane where the
plaquette lies. The dynamical variables (which we label by $U$ in the
general case) are site variables $U_x$ in spin models and link
variables $U_{x,\mu}$ in gauge models.
The general expression for the partition function is
\begin{equation}
Z = \int\prod{\mathrm{d}}\mu(U) \exp[-\beta S(U)],
\end{equation}
where $\beta$ is the inverse temperature (inverse coupling) and the
integration is extended to all dynamical variables. The action $S(U)$
must be a function enjoying the property of extensivity and of (global
and local) group invariance, and respect the symmetry of the lattice.
Adding the requisite that the interactions involve only nearest
neighbors, we find that a generic contribution to the action of spin
models must be proportional to
\begin{equation}
\sum_{x,\mu} \chi_{(r)}(U_x U^\dagger_{x+\mu}) + \hbox{h.c.} \, ,
\label{S-spin}
\end{equation}
and for gauge models to
\begin{equation}
\sum_{x,\mu,\nu} \chi_{(r)}(U_{x,\mu} U_{x+\mu,\nu}
U^\dagger_{x+\nu,\mu} U^\dagger_{x,\nu}) + \hbox{h.c.} \, ,
\label{S-gauge}
\end{equation}
where $(r)$ is in principle arbitrary, and the summation is extended
to all oriented links of the lattice in the spin case, to all the
oriented plaquettes in the gauge case.
In practice we shall mostly focus on the simplest possible choice,
corresponding to the fundamental representation. In order to reflect
the extensivity of the action, i.e., the proportionality to the number
of space and internal degrees of freedom, it will be convenient to
adopt the normalizations
\begin{eqnarray}
S(U) &=& -\sum_{x,\mu} N(\mathop{\operator@font Tr}\nolimits U_x U^\dagger_{x+\mu} + \hbox{h.c.})
\qquad{\rm(spin)},
\label{action-spin} \\
S(U) &=& -\sum_{x,\mu,\nu} N(\mathop{\operator@font Tr}\nolimits U_{x,\mu} U_{x+\mu,\nu}
U^\dagger_{x+\nu,\mu} U^\dagger_{x,\nu} + \hbox{h.c.})
\qquad{\rm(gauge)}.
\label{action-gauge}
\end{eqnarray}
Once the lattice action is fixed, it is easy to obtain sets of
Schwinger-Dyson equations relating the correlation functions of the
models. These are the quantum field equations and solving them
corresponds to finding a complete solution of a model. It is
extremely important to notice the simplifications occurring in the
Schwinger-Dyson equations when the large-$N$ limit is considered.
These simplifications are such to allow, in selected cases, explicit
solutions to the equations.
Before proceeding to a derivation of the equations, we must
preliminarily identify the sets of correlation functions we are
interested in. For obvious reasons, these correlations must involve
the dynamical fields at arbitrary space distances, and must be
invariant under the symmetry group of the model. Without pretending
to achieve full generality, we may restrict our attention to such
typical objects as the invariant correlation functions of a spin model
\begin{equation}
G^{(n)}(x_1,y_1,...,x_n,y_n) = {1\over N}\left<\mathop{\operator@font Tr}\nolimits\prod_{i=1}^n
U_{x_i} U^\dagger_{y_i}\right>
\label{corr-spin}
\end{equation}
and to the so-called Wilson loops of a gauge model
\begin{equation}
W({\cal C}) = {1\over N}\left<\mathop{\operator@font Tr}\nolimits\prod_{l\in{\cal C}} U_l\right>,
\label{corr-gauge}
\end{equation}
where ${\cal C}$ is a closed arbitrary walk on the lattice, and
$\prod_{l\in{\cal C}}$ is the ordered product over all the links along
the walk. It is worth stressing that the action itself is a sum of
elementary Green's functions (elementary Wilson loops).
More general invariant correlation functions may involve expectation
values of products of invariant operators similar to those appearing
in the r.h.s.\ of Eqs.\ (\ref{corr-spin}) and (\ref{corr-gauge}). The
already mentioned property of factorization allows us to express the
large-$N$ limit expectation value of such products as a product of
expectation values of the individual operators. As a consequence, the
large-$N$ form of the Schwinger-Dyson equations is a (generally
infinite) set of equations involving only the above-defined
quantities.
For sake of clarity and completeness, we present the explicit
large-$N$ form of the Schwinger-Dyson equations for the models
described by the standard actions (\ref{action-spin}) and
(\ref{action-gauge}). For principal chiral models
\cite{GonzalezArroyo-Okawa-reduced},
\begin{eqnarray}
0 &=& G^{(n)}(x_1,y_1,...,x_n,y_n) \nonumber \\
&+& \beta\sum_\mu\Bigl[G^{(n+1)}(x_1,x_1+\mu,x_1,y_1,...,x_n,y_n)
- G^{(n)}(x_1+\mu,y_1,...,x_n,y_n)\Bigr] \nonumber \\
&+& \sum_{s=2}^n \Bigl[\delta_{x_1,x_s} \,
G^{(s-1)}(x_1,y_1,...,x_{s-1},y_{s-1}) \,
G^{(n-s+1)}(x_s,y_s,...,x_n,y_n) \nonumber \\
&&\quad -\,
\delta_{x_1,y_s} \,
G^{(s)}(x_1,y_1,...,x_s,y_s) \,
G^{(n-s)}(x_{s+1},y_{s+1},...,x_n,y_n)\Bigr].
\label{princ-SD}
\end{eqnarray}
For lattice gauge theories \cite{Makeenko-Migdal-exact,Wadia-study},
\begin{eqnarray}
\beta\Bigl[\sum_\mu W({\cal C}_{x,\mu\nu}) -
W({\cal C}_{x-\mu,\mu\nu})\Bigr] =
\sum_{y\in{\cal C}} \delta_{x,y}\,W({\cal C}_{x,y})\,W({\cal C}_{y,x}),
\label{Migdal-Makeenko}
\end{eqnarray}
where $W({\cal C}_{x,\mu\nu})$ is obtained by replacing $U_{x,\nu}$
with $U_{x,\mu} U_{x+\mu,\nu} U^\dagger_{x+\nu,\mu}$ in the loop
${\cal C}$, and ${\cal C}_{x,y}$, ${\cal C}_{y,x}$ are the sub-loops
obtained by splitting ${\cal C}$ at the intersection point, including
the ``trivial'' splitting. Eqs.\ (\ref{Migdal-Makeenko}) are commonly
known as the lattice Migdal-Makeenko equations. The derivation of the
Schwinger-Dyson equations is obtained by performing infinitesimal
variations of the integrand in the functional integral representation
of expectation values and exploiting invariance of the measure.
\subsection{Survey of different approaches}
\label{approach-survey}
Schwinger-Dyson equations are the starting point for most techniques
aiming at the explicit evaluation of large-$N$ vacuum expectation
values for nontrivial unitary-matrix models. The form exhibited in
Eqs.\ (\ref{princ-SD}) and (\ref{Migdal-Makeenko}) involves in
principle an infinite set of variables, and it is therefore not
immediately useful to the purpose of finding explicit solutions.
Successful attempts to solve large-$N$ matrix systems have in general
been based on finding reformulations of Schwinger-Dyson equations
involving more restricted sets of variables and more compact
representations (collective fields). As a matter of fact, in most
cases it turned out to be convenient to define generating functions,
whose moments are the correlations we are interested in, and whose
properties are usually related to those of the eigenvalue
distributions for properly chosen covariant combinations of matrix
fields.
By ``covariant combination'' we mean a matrix-valued variable whose
eigenvalues are left invariant under a general ${\rm SU}(N) \times
{\rm SU}(N)$ transformation of the Lagrangian fields. Such objects
are typically those appearing in the r.h.s.\ of Eqs.\ (\ref{corr-spin})
and (\ref{corr-gauge}) {\em before} the trace operation is performed.
Under the ${\rm SU}(N) \times {\rm SU}(N)$ transformation $U \to
VUW^\dagger$, these operators transform accordingly to ${\cal O} \to
V{\cal O}V^\dagger$, and therefore their eigenvalue spectrum is left
unchanged.
Without belaboring on the details (some of which will however be
exhibited in the discussion of the single-link integral presented in
Sect.\ \ref{single-link}), we only want to mention that the approach
based on extracting appropriate Schwinger-Dyson equations for the
generating functions is essentially algebraic in nature, involving
weighted sums of infinite sets of equations in the form
(\ref{princ-SD}) or (\ref{Migdal-Makeenko}), identification of the
relevant functions, and resolution of the resulting algebraic
equations, where usually a number of free parameters appear, whose
values are fixed by boundary and/or asymptotic conditions and
analyticity constraints.
The approach based on direct replacement of the eigenvalue
distributions in the functional integral and the minimization of the
resulting effective action leads in turn to integral equations which
may be solved by more or less straightforward techniques.
These two approaches are however intimately related, since the
eigenvalue density is usually connected with the discontinuity along
some cut in the complex-plane extension of the generating function,
and one may easily establish a step-by-step correspondence between the
algebraic and functional approach.
Let us finally mention that the procedure based on introducing
invariant degrees of freedom and eigenvalue density operators has been
formalized by Jevicki and Sakita
\cite{Jevicki-Sakita-collective,Jevicki-Sakita-euclidean} in terms of
a ``quantum collective field theory'', whose equations of motion are
the Schwinger-Dyson equations relevant to the problem at hand.
A quite different application of the Schwinger-Dyson equations is
based on the strong-coupling properties of the correlation functions.
In the strong-coupling domain, expectation values are usually analytic
in the coupling $\beta$ within some positive convergence radius, and
their boundary value at $\beta=0$ can easily be evaluated. As a
consequence, it is formally possible to solve Eqs.\ (\ref{princ-SD})
and (\ref{Migdal-Makeenko}) in terms of strong-coupling series by
sheer iteration of the equations. This procedure may in practice turn
out to be too cumbersome for practical purposes; however, in some
circumstances, it may lead to rather good approximations
\cite{Marchesini-loop,Marchesini-Onofri-convergence} and even to a
complete strong-coupling solution. Continuation to the weak-coupling
domain is however a rather nontrivial task.
As a special application of the strong-coupling approach, we must
mention the attempt (pioneered by Kazakov, Kozhamkulov and Migdal
\cite{Kazakov-Kozhamkulov-Migdal}) to construct an effective action
for the invariant degrees of freedom by means of a modified
strong-coupling expansion, and explore the weak-coupling regime by
solving the saddle-point equations of the resulting action. This
technique might be successful at least in predicting the location and
features of the large-$N$ phase transition which is relevant to many
physical problems, as mentioned in Sect.\ \ref{introduction}.
A numerical approach to large-$N$ lattice Schwinger-Dyson equations
based on the minimization of an effective large-$N$ Fokker-Plank
potential and suited for the weak-coupling regime was proposed by
Rodrigues \cite{Rodrigues-numerical}.
Another relevant application of the Schwinger-Dyson equations is found
in the realm of the so-called ``reduced'' models. These models, whose
prototype is the Eguchi-Kawai formulation of strong-coupling large-$N$
lattice gauge theories \cite{Eguchi-Kawai-reduction}, are based on the
physical intuition that, in the absence of fluctuations, due to
translation invariance, the space extension of the lattice must be
essentially irrelevant in the large-$N$ limit, since all invariant
physics must be already contained in the expectation values of
(properly chosen) purely local variables. More precisely, one might
say that, when $N\to\infty$, the ${\rm SU}(N)$ group becomes so large
that it accommodates the full Poincar\`e group as a subgroup, and in
particular it should be possible to find representations of the
translation and rotation operators among the elements of ${\rm
SU}(N)$. As a consequence, one must be able to reformulate the full
theory in terms of a finite number of matrix field variables defined
at a single space-time site (or on the $d$ links emerging from the
site in the case of a lattice gauge theory) and of the above-mentioned
representations of the translation group. This reformulation is
called ``twisted Eguchi-Kawai'' reduced version of the theory
\cite{Eguchi-Nakayama-simplification,GonzalezArroyo-Okawa-EK}.
We shall spend a few more words on the reduced models in Sect.\
\ref{principal-chiral}. Moreover, a very good review of their
properties has already appeared many years ago \cite{Das-review}. In
this context, we must only mention that the actual check of validity
of the reduction procedure is based on deriving the Schwinger-Dyson
equations of the reduced model and comparing them with the
Schwinger-Dyson equations of the original model. Usually the
equivalence is apparent already at a superficial level when
na{\"\i}vely applying to correlation functions of the reduced model
the symmetry properties of the action itself. This procedure however
requires some attention, since the limit of infinitely many degrees of
freedom within the group itself allows the possibility of spontaneous
breakdown of some of the symmetries which would be preserved for any
finite value of $N$. In this context, we recall once more that large
$N$ is a thermodynamical limit: $N$ must go to infinity before any
other limit is considered, and sometimes the limiting procedures do
not commute. It is trivial to recognize that, when the
strong-coupling phase is considered, symmetries are unbroken, and the
equivalence between original and reduced model may be established
without further ado. Problems may occur in the weak-coupling side of
a large-$N$ phase transition.
An unrelated and essentially numeric approach to solving the large-$N$
limit of lattice matrix models is the coherent state variational
algorithm introduced by Yaffe and coworkers
\cite{Brown-Yaffe,Dickens-Lindqwister-Somsky-Yaffe}. We refer to the
original papers for a presentation of the results that may be obtained
by this approach.
\section{The single-link integral}
\label{single-link}
\subsection{The single-link integral in external field: finite-$N$
solution}
\label{single-link-finite-N}
All exact and approximate methods of evaluation of the functional
integrals related to unitary-matrix models must in principle face the
problem of performing the simplest of all relevant integrations: the
single-link integral. The utmost importance of such an evaluation
makes it proper to devote to it an extended discussion, which will
also give us the opportunity of discussing in a prototype example the
different techniques that may be applied to the models we are
interested in.
A quite general class of single-link integrals may be introduced by
defining
\begin{equation}
Z(A^\dagger A) = \int{\mathrm{d}} U
\exp\bigl[N\mathop{\operator@font Tr}\nolimits(A^\dagger U + U^\dagger A)\bigr],
\label{one-link}
\end{equation}
where as usual $U$ is an element of the group ${\rm U}(N)$ and $A$ is
now an arbitrary $N\times N$ matrix. The ${\rm U}(N)$ invariance of
the Haar measure implies that the one link integral (\ref{one-link})
must depend only on the eigenvalues of the Hermitian matrix $A^\dagger
A$, which we shall denote by $x_1,...,x_N$. The function
$Z(x_1,...,x_N)$ must satisfy a Schwinger-Dyson equation: restricting
the variables to the ${\rm U}(N)$ singlet subspace, the
Schwinger-Dyson equation was shown to be equivalent to the partial
differential equation \cite{Brower-Nauenberg,Brezin-Gross}
\begin{eqnarray}
{1\over N^2}\, x_k\,{\partial^2 Z\over\partial x_k^2} +
{1\over N}\,{\partial Z\over\partial x_k} +
{1\over N^2} \sum_{s \ne k} {x_s\over x_k-x_s}
\left({\partial Z\over\partial x_k} -
{\partial Z\over\partial x_s}\right)
= Z, \nonumber \\
\label{SD-Z}
\end{eqnarray}
with the boundary condition $Z(0,...,0) = 1$ and the request that $Z$
be completely symmetric under exchange of the $x_i$.
It is convenient to reformulate the equation in terms of the new
variables $z_k = 2N\sqrt{x_k}$, and to parameterize the solution in
terms of the completely antisymmetric function
${\hat Z}(z_1,...,z_N)$ by defining
\begin{equation}
Z(z) = {{\hat Z}(z)\over\prod_{i<j}(z_i^2-z_j^2)}\,.
\end{equation}
The equation satisfied by $\hat Z$ can be shown to reduce to
\begin{eqnarray}
&&\Biggl[ \sum_k z_k^2\,{\partial^2\over\partial z_k^2} +
(3-2N) \sum_k z_k\,{\partial\over\partial z_k} - \sum_k z_k^2 +
{2\over3}\,N(N-1)(N-2)\Biggr] {\hat Z} = 0. \nonumber \\
\label{Zhat}
\end{eqnarray}
Eq.\ (\ref{Zhat}) has the structure of a fermionic many-body
Schr\"odinger equation. With some ingenuity it may be solved in the
form of a Slater determinant of fermion wavefunctions. In conclusion,
we obtain, after proper renormalization
\cite{Brower-Rossi-Tan-chains} (see also \cite{Gaudin-Mello}),
\begin{equation}
Z(z_1,...,z_N) = 2^{N(N-1)/2} \Biggl(\prod_{k=0}^{N-1} k!\Biggr)
{\det\Vert z_j^{i-1} I_{i-1}(z_j)\Vert \over
\det\Vert z_j^{2(i-1)}\Vert},
\label{Z}
\end{equation}
where $I_i(z)$ is the modified Bessel function. Eq.\ (\ref{Z}) is
therefore a representation of the single-link integral in external
field for arbitrary ${\rm U}(N)$ groups. By taking proper derivatives
with respect to its arguments one may in principle reconstruct all the
cumulants for the group integration of an arbitrary string of
(uncontracted) matrices \cite{Samuel-integrals,Bars}.
Some special limits of the general expression (\ref{Z}) may prove
useful. Let us first of all consider the case when $A$ is
proportional to the identity matrix: $A = a 1$ and therefore
$z_i = 2Na$ and
\begin{equation}
Z(2Na,...,2Na) = \det\Vert I_{i-j}(2Na)\Vert.
\end{equation}
As we shall see, this is exactly Bars' and Green's solution for ${\rm
U}(N)$ lattice gauge theory in two dimensions
\cite{Bars-Green-largeN}.
When only one eigenvalue of $A$ is different from zero the result is
\begin{equation}
Z(2Na,0,...,0) = (N-1)!\,(Na)^{1-N}\,I_{N-1}(2Na).
\end{equation}
The large-$N$ limit will be discussed in the next subsection.
\subsection{The external field problem: large-$N$ limit}
\label{external-field-large-N}
For our purposes it is extremely important to extract the limiting
form of Eq.\ (\ref{Z}) when $N\to\infty$. In principle, it is a very
involved problem, since the dependence on $N$ comes not only through
the $z_i$ but also from the dimension of the matrices whose
determinant we must evaluate. It is however possible to obtain the
limit, either by solving separately the large-$N$ version of
Eq.\ (\ref{SD-Z}), or by directly manipulating Eq.\ (\ref{Z}).
In the first approach, we introduce the large-$N$ parameterization
\begin{equation}
Z = \exp NW,
\end{equation}
where $W$ is now proportional to $N$; we then obtain from
Eq.\ (\ref{SD-Z}), dropping second-derivative terms that are manifestly
depressed in the large-$N$ limit \cite{Brezin-Gross},
\begin{equation}
x_k\left(\partial W\over\partial x_k\right)^{\!\!2} +
{\partial W\over\partial x_k} +
{1\over N}\sum_{s\ne k}{x_s\over x_s-x_k}
\left({\partial W\over\partial x_s} -
{\partial W\over\partial x_k}\right) = 1.
\label{SD-Z-largeN}
\end{equation}
It is possible to show that in the large-$N$ limit
Eq.\ (\ref{SD-Z-largeN}) admits solutions, which can be parameterized
by the expression
\begin{equation}
{\partial W\over\partial x_k} = {1\over\sqrt{x_k+c}}
\left[1 - {1\over2N}\sum_s{1\over \sqrt{x_k+c} + \sqrt{x_s+c}}\right],
\qquad c \ge 0.
\label{SD-Z-largeN-sol}
\end{equation}
Substitution of Eq.\ (\ref{SD-Z-largeN-sol}) into
Eq.\ (\ref{SD-Z-largeN}) and some algebraic manipulation lead to the
consistency condition
\begin{equation}
c\left[{1\over2N}\sum_s{1\over\sqrt{x_s+c}} - 1\right] = 0,
\end{equation}
which in turn admits two possible solutions:
{\it a}) $c$ is determined by the condition
\begin{equation}
{1\over2N}\sum_s{1\over\sqrt{x_s+c}} = 1,
\label{SD-Z-strong-cond}
\end{equation}
implying $c\le{1\over4}$; this is a ``strong coupling'' phase,
requiring that the eigenvalues satisfy the bound
\begin{equation}
{1\over2N}\sum_s{1\over \sqrt{x_s}} \ge 1,
\label{SD-Z-strong-reg}
\end{equation}
i.e., at least some of the $x_s$ are sufficiently small;
{\it b}) when
\begin{equation}
{1\over2N}\sum_s{1\over\sqrt{x_s}} \le 1,
\label{SD-Z-weak-cond}
\end{equation}
then the solution corresponds to the choice $c=0$; this is a ``weak
coupling'' phase, and all eigenvalues are large enough.
Direct integration of Eq.\ (\ref{SD-Z-largeN-sol}) with proper boundary
conditions leads to the large-$N$ result \cite{Brezin-Gross}
\begin{eqnarray}
W(x) &=& 2\sum_k\sqrt{x_k+c} - {1\over2N}\sum_{k,s}\log
\bigl(\sqrt{x_k+c} + \sqrt{x_s+c}\bigr)
- Nc - {3\over4}N, \nonumber \\
\label{SD-W}
\end{eqnarray}
which must be supplemented with Eq.\ (\ref{SD-Z-strong-cond}) in the
strong-coupling regime (\ref{SD-Z-strong-reg}), while $c=0$ reproduces
the weak-coupling result by Brower and Nauenberg. Amazingly enough,
setting $c=0$ in Eq.\ (\ref{SD-W}) one obtains the n{a\"\i}ve
one-loop estimate of the functional integral, which turns out to be
exact in this specific instance.
It is possible to check that Eq.\ (\ref{SD-W}) is reproduced
by carefully taking the large-$N$ limit of Eq.\ (\ref{Z}), which
requires use of the following asymptotic limits of Bessel functions
\cite{Brower-Rossi-Tan-chains}
\begin{eqnarray}
k! \left(2\over z\right)^{\!\!k} I_k(z) &&\mathop{\;\longrightarrow\;}_{z\to\infty}
\left[{1\over2}\left(1+\sqrt{1+{z^2\over k^2}}\right)\right]^{1-k}
\left(1+{z^2\over k^2}\right)^{\!\!-1/4} \nonumber \\
&&\quad\times\,\exp\left(\sqrt{k^2+z^2} - k\right) \qquad
\hbox{(strong coupling)}, \\
I_k(z)\, &&\approx {1\over\sqrt{2\pi z}} \exp z \qquad
\hbox{(weak coupling)}.
\end{eqnarray}
An essential feature of Eq.\ (\ref{SD-W}) is the appearance of two
different phases in the large-$N$ limit of the single-link integral.
Such a transition would be mathematically impossible for any finite
value of $N$; however it affects the large-$N$ behavior of all
unitary-matrix models and gives rise to a number of interesting
phenomena. A straightforward analysis of Eq.\ (\ref{SD-W}) shows that
the transition point corresponds to the condition
\begin{equation}
t \equiv {1\over2N}\sum_s{1\over\sqrt{x_s}} = 1.
\end{equation}
It is also possible to evaluate the difference between the strong- and
weak-coupling phases of $W$ in the neighborhood of $t=1$, finding the
relationship \cite{Brezin-Gross}
\begin{equation}
W_{\rm strong} - W_{\rm weak} \sim (t-1)^3.
\end{equation}
As a consequence, we may classify this phenomenon as a ``third order
phase transition''.
\subsection{The properties of the determinant}
\label{properties-determinant}
The large-$N$ factorization of invariant amplitudes is a
well-estab\-lished property of products of operators defined starting
from the fundamental representation of the symmetry group. Operators
corresponding to highly nontrivial representations may show a more
involved pattern of behavior in the large-$N$ limit. Especially
relevant from this point of view are the properties of determinants of
covariant combinations of fields
\cite{Green-Samuel-chiral,Green-Samuel-un2}; we will consider the
quantities
\begin{equation}
\Delta(x) = \det\left[U_0 U^\dagger_x\right]
\end{equation}
for lattice chiral models and
\begin{equation}
\Delta({\cal C}) = \det\prod_{l\in{\cal C}} U_l
\end{equation}
for lattice gauge theories.
The expectation values of these operators may act as an order
parameter for the large-$N$ phase transition characterizing the class
of models we are taking into consideration. Indeed the determinant
picks up the phase characterizing the ${\rm U}(1)$ subgroup that
constitutes the center of ${\rm U}(N)$. Moreover, since
\[
{\rm U}(N) \approx {\rm U}(1) \times {{\rm SU}(N)\over Z_N}\,,
\]
${\rm SU}(N) \to {\rm U}(N)$ as $N\to\infty$ because $Z_N\to{\rm
U}(1)$; therefore the determinant of the ${\rm U}(N)$ theory in the
large-$N$ limit reflects properties of the center of ${\rm SU}(N)$.
In lattice models this Abelian ${\rm U}(1)$ subgroup is not decoupled,
as it happens in the continuum theory, and therefore
$\left<\Delta\right>$ does not in general have on the lattice the
free-theory behavior it has in the continuum.
The basic properties of the determinant may be explored by focusing
once more on the external field problem we discussed above. Let us
introduce a class of determinant operators, and define their
expectation values as \cite{Aneva-Brihaye-Rossi-pseudoAbelian}
\begin{equation}
\Delta^{(l)} = \left<\det U^l\right> =
{\int{\mathrm{d}} U \det U^l \exp[N\mathop{\operator@font Tr}\nolimits(U^\dagger A + A^\dagger U)] \over
\int{\mathrm{d}} U \exp[N\mathop{\operator@font Tr}\nolimits(U^\dagger A + A^\dagger U)]} \,.
\end{equation}
In order to parameterize the ${\rm SU}(N)$ external-source integral,
besides the eigenvalues $x_i$ of $AA^\dagger$, a new external
parameter must be introduced, that couples to the determinant:
\begin{equation}
\theta = {{\mathrm{i}}\over2N}(\log\det A^\dagger - \log\det A).
\label{theta-def}
\end{equation}
Because of the symmetry properties, $\Delta^{(l)}$ may only depend on
the eigenvalues $z$ and on $\theta$.
It was found that, when $U$ enjoys ${\rm U}(N)$ symmetry (with finite
$N$),
\begin{equation}
\Delta^{(l)} = \exp({\mathrm{i}} Nl\theta) {\hat Z_l\over\hat Z_0} \,,
\end{equation}
where $\hat Z_l$ is the solution of the following Schwinger-Dyson
equation, generalizing Eq.\ (\ref{Zhat}):
\begin{eqnarray}
&&{1\over N}\Biggl[\sum_k z^2_k\,{\partial^2\over\partial z_k^2} +
(3-2N)\sum_k z_k\,{\partial\over\partial z_k} - \sum_k z^2_k
\nonumber \\ && \qquad+\,
{2\over3}N(N-1)(N-2)\Biggr]\hat Z_l = l^2 \hat Z_l;
\label{SD-hatZl}
\end{eqnarray}
$\hat Z_l$ satisfy the property
\begin{equation}
\hat Z_l = \Biggl(\prod_k z_k\Biggr)^{\!\!|l|}
\Biggl(\prod_k {1\over z_k}\,
{\partial\over\partial z_k}\Biggr)^{\!\!|l|}
\hat Z_0 = \det\Vert z_i^{j-1} I_{j-1-l}(z_i)\Vert.
\label{Zhatl-def}
\end{equation}
When the weak-coupling condition $t\equiv\sum_k 1/z_k\le1$ is
satisfied, the leading contribution to the large-$N$ limit of all
$\hat Z_l$ is the same:
\begin{equation}
\hat Z_l \to \hat Z^{(\infty)} =
\exp\left[\sum_k z_k - {1\over2}\sum_k\log 2\pi z_k +
\sum_{i<k}\log(z_i-z_k)\right].
\label{hatZl}
\end{equation}
In order to determine the large-$N$ limit of $\Delta^{(l)}$, one
therefore needs to compute the $O(1)$ factor in front of the
exponentially growing term (\ref{hatZl}). It is convenient to define
\begin{equation}
X_l = {\hat Z_l\over\hat Z^{(\infty)}} \,,
\label{Xl-def}
\end{equation}
whose Schwinger-Dyson equation may be extracted from
Eq.\ (\ref{SD-hatZl}) and takes the form
\begin{eqnarray}
&&{1\over N}\left[\sum_k z^2_k\,{\partial^2 X_l\over\partial z_k^2} +
2 \sum_k z^2_k\,{\partial X_l\over\partial z_k} +
\sum_{k\ne i} {z_k z_i\over z_k-z_i}
\left({\partial X_l\over\partial z_k} -
{\partial X_l\over\partial z_i}\right)\right]
\nonumber \\ =\,&&
\left(l^2-{1\over4}\right) X_l \,.
\label{SD-Xl}
\end{eqnarray}
Let us introduce the large-$N$ Ansatz
\begin{equation}
X_l = X_l(t),
\end{equation}
reducing Eq.\ (\ref{SD-Xl}) to
\begin{equation}
{1\over N} \sum_k{1\over z^2_k}\,{{\mathrm{d}}^2X_l\over{\mathrm{d}} t^2} +
2(t-1){{\mathrm{d}} X_l\over{\mathrm{d}} t} = \left(l^2-{1\over4}\right) X_l \,.
\label{SD-Xl-reduced}
\end{equation}
Removing terms that are depressed by two powers of $1/N$, we are left
with a consistent equation whose solution is
\begin{equation}
X_l = (1-t)^{{1\over2}(l^2-{1\over4})}.
\label{SD-Xl-sol}
\end{equation}
Finally we can compute the weak-coupling large-$N$ limit of
$\Delta^{(l)}$:
\begin{equation}
\Delta^{(l)} \mathop{\;\longrightarrow\;}_{N\to\infty} \exp({\mathrm{i}} Nl\theta) \,
(1-t)^{{1\over2}l^2}, \qquad t \le 1.
\end{equation}
From the standard strong-coupling expansion we may show that
\begin{equation}
\Delta^{(l)} \mathop{\;\longrightarrow\;}_{N\to\infty} 0
\qquad \hbox{when} \quad t \ge 1.
\end{equation}
An explicit evaluation, starting from the exact expression
(\ref{Zhatl-def}), expanded in powers of $1/z_k$ for arbitrary $N$,
allows us to show that the quantities $\hat Z_l$ may be obtained from
Eqs.\ (\ref{hatZl}) and (\ref{Xl-def}) by expanding
Eq.\ (\ref{SD-Xl-sol}) up to 2nd order in $t$ with no $O(1/N^2)$
corrections. $\Delta^{(l)}$ according to this result violate
factorization; in turn, they take the value which would be predicted
by an effective Gaussian theory governing the ${\rm U}(1)$ phase of
the field $U$.
\subsection{Applications to mean field and strong coupling}
\label{mean-field+strong-coupling}
The single-link external-field integral has a natural domain of
application in two important methods of investigation of lattice field
theories: mean-field and strong-coupling expansion. Extended papers
and review articles have been devoted in the past to these topics
(cfr.\ Ref.\ \cite{Drouffe-Zuber} and references therein), and we
shall therefore focus only on those results that are specific to the
large-$N$ limit and to the $1/N$ expansion.
Let us first address the issue of the mean-field analysis, considering
for sake of definiteness the case of $d$-dimensional chiral models,
but keeping in mind that most results can be generalized in an
essentially straightforward manner to lattice gauge theories. The
starting point of the mean-field technique is the application of the
random field transform to the functional integral:
\begin{eqnarray}
Z_N &=& \int{\mathrm{d}} U_n \exp\Biggl\{N\beta\sum_{n,\mu}
\mathop{\operator@font Tr}\nolimits\left(U_n U^\dagger_{n+\mu}
+ U_{n+\mu} U^\dagger_n\right)\Biggr\}
\nonumber \\
&=& \int{\mathrm{d}} V_n {\mathrm{d}} A_n \exp\Biggl\{N\beta\sum_{n,\mu}
\mathop{\operator@font Tr}\nolimits\left(V_n V^\dagger_{n+\mu}
+ V_{n+\mu} V^\dagger_n\right)
\nonumber \\
&&\qquad\quad-\, N\sum_n \mathop{\operator@font Tr}\nolimits\left(A_n V_n^\dagger
+ V_n A^\dagger_n\right) \Biggr\}
\nonumber \\
&&\qquad\times\,
\int{\mathrm{d}} U_n \exp\left\{N \sum_{n,\mu}
\mathop{\operator@font Tr}\nolimits\left(A_n U_n^\dagger+ U_n A^\dagger_n\right)\right\},
\end{eqnarray}
where $V_n$ and $A_n$ are arbitrary complex $N{\times}N$ matrices.
Therefore the integration over $U_n$ is just the single-link integral
we discussed above. As a consequence, the original chiral model is
formally equivalent to a theory of complex matrices with effective
action
\begin{eqnarray}
-{1\over N} \, S_{\rm eff}(A,V) &=& \beta\sum_{n,\mu}
\mathop{\operator@font Tr}\nolimits\left(V_n V_{n+\mu}^\dagger+ V_{n+\mu} V^\dagger_n\right)
- \sum_n \mathop{\operator@font Tr}\nolimits\left(A_n V_n^\dagger+ V_n A^\dagger_n\right)
\nonumber \\ &+&
\sum_n W(A_n A^\dagger_n).
\label{chiral-Seff}
\end{eqnarray}
The leading order in the mean-field approximation is obtained by
applying saddle-point techniques to the effective action, assuming
saddle-point values of the fields $A_n$ and $V_n$ that are
translation-invariant and proportional to the identity.
We mention that, in the case at hand, the large-$N$ saddle-point
equations in the weak-coupling phase are:
\begin{equation}
A_n = a = 2 \beta d v, \qquad V_n = v = 1 - {1 \over 4a},
\end{equation}
and they are solved by the saddle-point values
\begin{equation}
\overline a = \beta d \left(1 + \sqrt{1 - {1 \over 2 \beta d}}\right),
\qquad \overline v = {1\over2} +
{1\over2} \sqrt{1 - {1 \over 2 \beta d}},
\end{equation}
leading to a value of the free and internal energy
\begin{equation}
{Fd \over N^2 L} =
\overline a - {1\over2}\log 2\overline a - {1\over2}, \qquad
{1\over2d}\,{\partial\over\partial\beta}\,{Fd \over N^2 L} =
\overline v^2.
\end{equation}
The strong-coupling solution is trivial: $v = a = 0$, and there is a
first-order transition point at
\begin{equation}
\beta_c d = \casefr{1}{2}, \qquad \overline v_c = \casefr{1}{2},
\qquad \overline a_c = \casefr{1}{2}.
\end{equation}
One may also compute the quadratic fluctuations around the mean-field
saddle point by performing a Gaussian integral, whose quadratic form
is related to the matrix of the second derivatives of $W$ with respect
to the fields, and generate a systematic loop expansion in the
effective action (\ref{chiral-Seff}), which in turns appears to be
ordered in powers of $1/d$. Therefore mean-field methods are
especially appropriate for the discussion of models in large space
dimensions, and not very powerful in the analysis of $d=2$ models.
The very nature of the transition cannot be taken for granted,
especially at large $N$. However, when $d\ge3$ there is independent
evidence of a first-order phase transition for $N\ge3$. We mention
that a detailed mean-field study of ${\rm SU}(N)$ chiral models in $d$
dimensions appeared in Refs.\
\cite{Kogut-Snow-Stone-mean,Brihaye-Rossi-weak}.
When willing to extend the mean-field approach, it is in general
necessary to find a systematic expansion of the functional
$W(AA^\dagger)$ in the powers of the fluctuations around the
saddle-point configurations. Moreover, one may choose to consider not
only the large-$N$ value of the functional, but also its expansion in
powers if $1/N^2$, in order to make predictions for large but finite
values of $N$. The expansion of $W_0$ up to fourth order in the
fluctuations was performed in Ref.\ \cite{Brihaye-Taormina-mean},
where explicit analytic results can be found. A technique for the
weak-coupling $1/N^2$ expansion of $W$ can be found in Ref.\
\cite{Brihaye-Rossi-integrals}. We quote the complete $O(1/N^4)$
result:
\begin{eqnarray}
{W \over N} = {1 \over N^2} &&\Biggl[\sum_a z_a
- {1\over2} \sum_{a,b}\log{z_a+z_b \over 2N} - {3\over4}\,N^2
+ \log(1-t)^{-1/8}
\nonumber \\ &&\quad+\,
{3\over2^7}(1-t)^{-3} \sum_a {1 \over z_a^3}\Biggr]
+ O\left(1 \over N^6\right),
\label{W-1/N2}
\end{eqnarray}
where $t = \sum_a 1/z_a$. Eq.\ (\ref{W-1/N2}) can also be expanded in
the fluctuations around a saddle-point configuration. Extension to
${\rm SU}(N)$ with large $N$ was also considered. A discussion of
large-$N$ mean field for lattice gauge theories can be found in
Refs.\ \cite{Brihaye-Rossi-weak,Muller-Ruhl-mean-1,Muller-Ruhl-mean-2,%
Hasegawa-Yang-mean-1,Hasegawa-Yang-mean-2}.
Let us now turn to a discussion of the main features of the large-$N$
strong-coupling expansion. A preliminary consideration concerns the
fact that it is most convenient to reformulate the strong-coupling
expansion (i.e., the expansion in powers of $\beta$) into a character
expansion, which is ordered in the number of lattice steps involved in
the effective path that can be associated with each nontrivial
contribution to the functional integral. The large-$N$ character
expansion will be discussed in greater detail in Subs.\
\ref{character-expansion}. Here we only want to discuss those
features that are common to any attempt aimed at evaluating
strong-coupling series for expectation values of invariant operators
in the context of ${\rm U}(N)$ and ${\rm SU}(N)$ matrix models, with
special focus on the large-$N$ behavior of such series.
The basic ingredient of strong-coupling computations is the knowledge
of the cumulants, i.e., the connected contributions obtained
performing the invariant group integration of a string of uncontracted
$U$ and $U^\dagger$ matrices. ${\rm U}(N)$ group invariance insures
us that these group integrals can be non-zero only if the same number
of $U$ and $U^\dagger$ matrices appear in the integrand. ${\rm
SU}(N)$ is slightly different in this respect, and its peculiarities
will be discussed later and are not relevant to the present analysis.
It was observed a long time ago that the cumulants, whose group
structure is that of invariant tensors with the proper number of
indices, involve $N$-dependent numerical coefficients. The asymptotic
behavior of these coefficients in the large-$N$ limit was studied
first by Weingarten \cite{Weingarten-asymptotic}. However, for finite
$N$, the coefficients written as function of $N$ are formally plagued
by the so-called DeWit-'t Hooft poles \cite{DeWit-tHooft}, that are
singularities occurring for integer values of $N$. The highest
singular value of $N$ grows with the number $n$ of $U$ matrices
involved in the integration, and therefore for sufficiently high
orders of the series it will reach any given finite value. A complete
description of the pole structure was presented in Ref.\
\cite{Samuel-integrals}; not only single poles, but also arbitrary
high-order poles appear for large enough $n$, and analyticity is
restricted to $N \ge n$. Obviously, since group integrals are well
defined for all $n$ and $N$, this is only a pathology of the $1/N$
expansion. Finite-$N$ results are finite, but they cannot be obtained
as a continuation of a large-$N$ strong-coupling expansion. However,
it is possible to show that the strict $N\to\infty$ limit of the
series exists, and moreover, for sufficiently small $\beta$ and
sufficiently large $N$, the limiting series is a reasonable
approximation to the true result, all nonanalytic effects being
$O(\beta^{2N})$ in ${\rm U}(N)$ models and $O(\beta^N)$ in ${\rm
SU}(N)$ models. As a consequence, computing the large-$N$ limit of
the strong-coupling series is meaningful and useful in order to
achieve a picture of the large-$N$ strong-coupling behavior of matrix
models, but the evaluation of $O(1/N^2)$ or higher-order corrections
in the strong-coupling phase is essentially pointless.
The large-$N$ limit of the external-field single-link integral has
been considered in detail from the point of view of the
strong-coupling expansion. In particular, one may obtain expressions
for the coefficients of the expansion of $W$ in powers of the moments
of $AA^\dagger$: setting
\begin{equation}
\rho_n = {1 \over N}\mathop{\operator@font Tr}\nolimits(AA^\dagger)^n, \qquad
W = \sum_{n=1}^\infty
\sum_{\stackrel{\scriptstyle \alpha_1,...,\alpha_n}%
{\sum_k k \alpha_k = n}}
W_{\alpha_1,...,\alpha_n}
\rho_1^{\alpha_1} ... \rho_n^{\alpha_n} \, ,
\end{equation}
one gets
\begin{equation}
W_{\alpha_1,...,\alpha_n} =
(-1)^n {(2 n + \sum_k \alpha_k - 3)! \over (2 n)!}
\prod_k \left[-{(2 k)! \over (k!)^2}\right]^{\alpha_k}
{1 \over \alpha_k!} .
\end{equation}
Further properties of this expansion can be found in the original
reference \cite{OBrien-Zuber-note}.
A character-expansion representation of the single-link integral was
also produced for arbitrary ${\rm U}(N)$ integrals in Ref.\
\cite{Bars}. Strong-coupling expansions for large-$N$ lattice gauge
theories have been analyzed in detail by Kazakov
\cite{Kazakov-pl,Kazakov-jetp}, O'Brien and Zuber
\cite{OBrien-Zuber-expansion}, and Kostov \cite{Kostov-sc}, who
proposed reinterpretations in terms of special string theories.
\subsection{The single-link integral in the adjoint representation}
\label{single-link-adjoint}
The integral introduced at the beginning of Sect.\ \ref{single-link} is
by no means the most general single-link integral one can meet in
unitary-matrix models. As mentioned in Sect.\ \ref{unitary-matrices},
any invariant function of the $U$'s is in principle a candidate for a
lattice action. In practice, the only case that has been considered
till now that cannot be reduced to Eq.\ (\ref{one-link}) is the
integral introduced by Itzykson and Zuber \cite{Itzykson-Zuber}
\begin{equation}
I(M_1,M_2) = \int{\mathrm{d}} U \exp\mathop{\operator@font Tr}\nolimits(M_1 U M_2 U^\dagger),
\label{I-def}
\end{equation}
where $M_1$ and $M_2$ are arbitrary Hermitian matrices. This is a
special instance of the single-link integral for the coupling of the
adjoint representation of $U$ to an external field.
The result, because of ${\rm U}(N)$ invariance, can only depend on the
eigenvalues $m_{1i}$ and $m_{2i}$ of the Hermitian matrices. Several
authors \cite{Itzykson-Zuber,Mehta-integration,HarishChandra} have
independently shown that
\begin{equation}
I(M_1,M_2) = \Biggl(\prod_{p=1}^{N-1} p!\Biggr)
{\det\left\Vert\exp(m_{1i} m_{2j})\right\Vert \over
\Delta(m_{11},...,m_{1N})\,\Delta(m_{21},...,m_{2N})},
\label{I-value-det}
\end{equation}
where $\Delta(m_1,...,m_N) = \prod_{i>j}(m_i-m_j)$ is the Vandemonde
determinant. A series expansion for $I(M_1,M_2)$ in terms of the
characters of the unitary group takes the form
\begin{equation}
I(M_1,M_2) = \sum_{(r)} {1\over|n|!}\,
{\sigma_{(r)}\over d_{(r)}}\,\chi_{(r)}(M_1)\,\chi_{(r)}(M_2),
\end{equation}
where $\sigma_{(r)}$ is the dimension of the representation $(r)$ of
the permutation group; we will present an explicit evaluation of
$\sigma_{(r)}$ in Eq.\ (\ref{sigma-def}). Eq.\ (\ref{I-value-det})
plays a fundamental r\^ole in the decoupling of the ``angular''
degrees of freedom when models involving complex Hermitian matrices
are considered.
An interesting development based on the use of
Eq.\ (\ref{I-value-det}) is the so-called ``induced QCD'' program,
aimed at recovering continuum large-$N$ QCD by taking proper limits in
the parameter space of the lattice Kazakov-Migdal
model \cite{Kazakov-Migdal-induced}
\begin{equation}
S = N\sum_x \mathop{\operator@font Tr}\nolimits V(\Phi_x) - N \sum_{x,\mu}
\mathop{\operator@font Tr}\nolimits(\Phi_x U_{x,\mu} \Phi_{x+\mu} U^\dagger_{x,\mu}),
\end{equation}
where $U_{x,\mu}$ is the non-Abelian gauge field and $\Phi_x$ is a
Hermitian $N \times N$ (matrix-valued) Lorentz-scalar field. The
Itzykson-Zuber integration (\ref{I-def}) allows the elimination of the
gauge degrees of freedom and reduces the problem to studying the
interactions of Hermitian matrix fields (with self-interactions
governed by the potential $V$). Discussion of the various related
developments is beyond the scope of the present report. It will be
enough to say that, while one may come to the conclusion that this
model does {\em not\/} induce QCD, it is certainly related to some
very interesting (and sometimes solvable) matrix models (cfr.\ Ref.\
\cite{Weiss} for a review).
\section{Two-dimensional lattice Yang-Mills theory}
\label{2d-YM}
\subsection{Two-dimensional Yang-Mills theory as a single-link
integral}
\label{2d-YM-single-link}
The results presented in the previous section allow us to analyze
the simplest physical system described by a unitary-matrix model. As
we shall see, one of the avatars of this system is a Yang-Mills theory
in two dimensions (YM$_2$), in the lattice Wilson formulation.
Notwithstanding the enormous simplifications occurring in this model
with respect to full QCD, still some nontrivial features are retained,
and even in the large-$N$ limit some interesting physical properties
emerge. It is therefore worth presenting a detailed discussion of
this system, which also offers the possibility of comparing the
different technical approaches to the large-$N$ solution in a
completely controlled situation.
The lattice formulation of the two-dimensional ${\rm U}(N)$ gauge
theory is based on dynamical variables $U_{x,\mu}$ which are defined on
links; however, because of gauge invariance, in two dimensions there
are no transverse gauge degrees of freedom, and a one-to-one
correspondence can be established between link variables and
plaquettes. A convenient way of exploiting this fact consists in
fixing the gauge \cite{Gross-Witten}
\begin{equation}
U_{x,0} = 1
\label{temporal-gauge}
\end{equation}
(the lattice version of the temporal gauge $A_0 = 0$). An extremely
important consequence of the gauge choice (\ref{temporal-gauge})
emerges from considering the gauge-fixed form of the single-plaquette
contribution to the lattice action:
\begin{equation}
\mathop{\operator@font Tr}\nolimits\left(U_{x,0} U_{x+0,1} U_{x+1,0}^\dagger U_{x,1}^\dagger\right)
\to \mathop{\operator@font Tr}\nolimits U_{x+0,1} U_{x,1}^\dagger .
\end{equation}
This is nothing but the single-link contribution to the
one-dimen\-sional lattice action of a principal chiral model whose
links lie along the 0 direction. When considering invariant
expectation values (Wilson loops), we then recognize that they can be
reduced to contracted products of tensor correlations of variables
defined on decoupled one-dimensional models. As a consequence, YM$_2$
factorizes completely into a product of independent chiral models
labeled by their 1 coordinate. Not only the partition function, but
also all invariant correlations can be systematically mapped into
those of the corresponding chiral models. The area law for non
self-interacting Wilson loops in YM$_2$ and the exponential decay of
the two-point correlations in one-dimensional chiral models are
trivial corollaries of these results \cite{Gross-Witten}.
The above considerations allow us to focus on the prototype model
defined by the action
\begin{equation}
S = - N \sum_i \mathop{\operator@font Tr}\nolimits(U_i U_{i+1}^\dagger + U_i^\dagger U_{i+1}),
\label{S-proto}
\end{equation}
where $i$ is the site label of the one-dimensional lattice. By
straightforward manipulations we may show that the most general
nontrivial correlation one really needs to compute involves product of
invariant operators of the form
\begin{equation}
\mathop{\operator@font Tr}\nolimits(U_0 U_l^\dagger)^k,
\end{equation}
where $l$ plays the r\^ole of the space distance, and $k$ is a sort
of ``winding number''.
An almost trivial corollary of the above analysis is the observation
that YM$_2$ and principal chiral models in one dimension enjoy a
property of ``geometrization'', i.e., the only variables that can turn
out to be relevant for the complete determination of expectation
values are the single-plaquette (single-link) averages of products of
powers of moments \cite{Rossi-qcd2}
\begin{equation}
\prod_k \left[\mathop{\operator@font Tr}\nolimits(U_0 U_1^\dagger)^k\right]^{m_k}
\label{mega-product}
\end{equation}
and the geometrical features of the correlations (in YM$_2$, areas of
Wilson loops and subloops; in chiral models, distances of correlated
points), such that all coupling dependence is incorporated in the
expectation values of the quantities (\ref{mega-product}). This
result is sufficiently general to apply not only to the Wilson action
formulation, but also to all ``local'' actions such that the
interaction depends only on invariant functions of the
single-plaquette (single-link) variable, i.e., any linear combination
of the expressions appearing in Eqs.\ (\ref{S-spin}), (\ref{S-gauge})
\cite{Rossi-qcd2,Jurkiewicz-Zalewski,Chen-Tan-Zheng-universality}.
In order to proceed to the actual computation, it is convenient to
perform a change of variables, allowed by the invariance of the Haar
measure, parameterizing the fields by
\begin{equation}
V_l = U_{l-1} U_l^\dagger;
\label{chiral-variable-change}
\end{equation}
the action (\ref{S-proto}) explicitly factorizes into
\begin{equation}
S = -N \sum_l \mathop{\operator@font Tr}\nolimits(V_l + V_l^\dagger).
\end{equation}
It is now easy to get convinced that in the most general case a Wilson
loop expectation value (correlation function) can be represented as a
finite product of invariant tensors, each of which is originated by a
single-link integration of the form
\begin{equation}
{\int{\mathrm{d}} V_l \, f(V_l) \exp\left[N \beta \mathop{\operator@font Tr}\nolimits(V_l + V_l^\dagger)\right]
\over \int{\mathrm{d}} V_l \exp\left[N \beta\mathop{\operator@font Tr}\nolimits(V_l + V_l^\dagger)\right]}
\equiv \left<f(V_l)\right>,
\end{equation}
where $f(V_l)$ is any (tensor) product of $V_l$'s and $V_l^\dagger$'s,
and the only nontrivial contributions to the full expectation value
come from integrations extended to plaquettes belonging to the area
enclosed by the loop itself (in chiral models, links comprised between
the extremal points of the space correlation).
For sake of definiteness, we may focus on the correlators
\cite{Rossi-Vicari-QCD2}
\begin{equation}
W_{l,k} \equiv {1 \over N}\left<\mathop{\operator@font Tr}\nolimits(U_0 U_l^\dagger)^k\right>,
\end{equation}
and find that
\begin{eqnarray}
W_{l,k} = {\int{\mathrm{d}} V_1 ...{\mathrm{d}} V_l\,(1/N) \mathop{\operator@font Tr}\nolimits(V_1 ... V_l)^k
\exp\left[N \beta \sum_{i=1}^l \mathop{\operator@font Tr}\nolimits(V_i + V_i^\dagger)\right]
\over \prod_i
\int{\mathrm{d}} V_i \exp\left[N \beta\mathop{\operator@font Tr}\nolimits(V_i + V_i^\dagger)\right]} \, .
\nonumber \\
\end{eqnarray}
This problem can be formally solved for arbitrary $N$ by a character
expansion, which we shall discuss in Subs.\ \ref{character-expansion}.
It is however immediate to recognize that we are ultimately led to
computing the general class of group integrals whose form is
\begin{equation}
\int{\mathrm{d}} V \prod_k\left(\mathop{\operator@font Tr}\nolimits V^k\right)^{\!\!m_k}
\exp\left[N \beta \mathop{\operator@font Tr}\nolimits(V + V^\dagger)\right]
\label{general-group-integral}
\end{equation}
(where the product runs over positive and negative values of $k$), and
in turn it is in principle an exercise based on the exploitation of
the result for the external fiend single-link integral introduced in
Eq.\ (\ref{one-link}).
By the way, integrals of the form (\ref{general-group-integral}) can
easily be expressed as linear combinations of integrals belonging to
the class
\begin{equation}
\int{\mathrm{d}} V \, \chi_{(\lambda)}(V)
\exp\left[N \beta \mathop{\operator@font Tr}\nolimits(V + V^\dagger)\right],
\label{character-goup-integral}
\end{equation}
where $\lambda$ labels properly chosen representations of ${\rm U}(N)$.
Eq.\ (\ref{character-goup-integral}) is in turn related to the
definition of the character coefficients in the character expansion of
$\exp[N \beta \mathop{\operator@font Tr}\nolimits(V + V^\dagger)]$. For arbitrary $N$, as a matter of
principle, $\chi_{(\lambda)}(V)$ has a representation in terms of the
eigenvalues $\phi_i$ of the matrix $V$, while $\mathop{\operator@font Tr}\nolimits(V + V^\dagger) =
2\sum_i \cos\phi_i$ and the measure itself can in this case be
expressed in terms of the eigenvalues as
\begin{equation}
{\mathrm{d}}\mu(V) \sim \prod_i {\mathrm{d}}\phi_i\,\Delta^2(\phi_1,...,\phi_N),
\end{equation}
where
\begin{eqnarray}
\Delta(\phi_1,...,\phi_N) &\equiv& \det\exp\Vert{\mathrm{i}}(i\phi_j)\Vert,
\nonumber \\
\Delta^2(\phi_1,...,\phi_N) &=&
\prod_{i<j} 4\sin^2{\phi_i-\phi_j \over 2} \,.
\end{eqnarray}
As a consequence, it is always possible to express all ${\rm U}(N)$
integrals in the class (\ref{character-goup-integral}) in terms of
linear combinations of products of modified Bessel functions
$I_k(2N\beta)$, with $k<N$.
Let us now come to the specific issue of evaluating the relevant
physical quantities in the large-$N$ limit of ${\rm U}(N)$ models, and
comparing the procedures corresponding to different possible
approaches. Basic to most subsequent developments is the observation
that the large-$N$ factorization property allows us to focus on a very
restricted class of interesting correlations, which we label by
\begin{equation}
w_k \equiv \left<{1 \over N} \mathop{\operator@font Tr}\nolimits V^k\right> \equiv W_{1,k} \,.
\end{equation}
The first explicit solution to the problem of evaluating $w_k$ in the
large-$N$ limit was offered by Gross and Witten \cite{Gross-Witten}.
To this purpose, they introduced the eigenvalue density
\begin{equation}
\rho(\phi) = {1\over N} \sum_i \delta(\phi-\phi_i),
\end{equation}
and considered the group integral defining the partition function of
the single-link model
\begin{equation}
Z(\beta) \sim \int\prod_i{\mathrm{d}}\phi_i \, \Delta^2(\phi_1,...,\phi_N)
\exp\!\left(2 N \beta \sum_i\cos\phi_i\right).
\label{single-link-Z}
\end{equation}
The integral (\ref{single-link-Z}) can be evaluated in the
$N\to\infty$ limit by a saddle-point technique
\cite{Brezin-Itzykson-ZinnJustin-Zuber} applied to the
effective action
\begin{equation}
2\beta \int \rho(\phi) \cos(\phi)\,{\mathrm{d}}\phi +
\princint \rho(\phi)\,\rho(\phi')
\log\sin{\phi-\phi'\over2}\,{\mathrm{d}}\phi\,{\mathrm{d}}\phi' ,
\end{equation}
with the constraint $\int \rho(\phi)\,{\mathrm{d}}\phi = 1$. The support of the
function $\rho(\phi)$ is dynamically determined. The saddle-point
integral equation is
\begin{equation}
2\beta \sin\phi = \int_{-\phi_c}^{\phi_c}{\mathrm{d}}\phi' \,
\rho(\phi') \cot{\phi-\phi'\over2} \,,
\end{equation}
and it is possible to identify two distinct solutions, corresponding
to weak and strong coupling. When $\beta$ is small, it is easy to
find out that
\begin{equation}
\rho(\phi) = {1\over2\pi} (1 + 2\beta\cos\phi), \qquad
-\pi \le \phi \le \pi ;
\label{easy-rho-strong-coupling-UN}
\end{equation}
$\rho(\phi)$ is positive definite whenever $\beta\le\casefr{1}{2}$.
When $\beta$ is large, $\phi_c<\pi$ and
\begin{equation}
\rho(\phi) = {2\beta\over\pi} \cos{\phi\over2}
\sqrt{{1\over2\beta} - \sin^2{\phi\over2}}, \qquad
\sin^2{\phi_c\over2} = {1\over2\beta} \,,
\label{easy-rho-weak-coupling-UN}
\end{equation}
submitted to the condition $\beta\ge\casefr{1}{2}$. Therefore it is
possible to identify the location of the third-order phase transition
\cite{Gross-Witten}:
\begin{equation}
\beta_c = \casefr{1}{2} \,.
\label{beta_c-UN}
\end{equation}
By direct substitution, one finds the values of the free and internal
energy (per unit link or unit plaquette):
\begin{equation}
{F \over N^2} =
\left\{
\renewcommand\arraystretch{1.3}
\begin{array}{l@{\quad}l}
\beta^2 \,, & \beta\le\casefr{1}{2} \,, \\
2 \beta - \casefr{1}{2} \log 2\beta - \casefr{3}{4} \,, &
\beta\ge\casefr{1}{2} \,,
\end{array}
\right.
\label{F-UN}
\end{equation}
\begin{equation}
w_1 = {1\over2}\,{\partial\over\partial\beta}\,{F \over N^2} =
\left\{
\renewcommand\arraystretch{1.3}
\begin{array}{l@{\quad}l}
\beta \,, & \beta\le\casefr{1}{2} \,, \\
\displaystyle 1 - {1\over4\beta} \,, & \beta\ge\casefr{1}{2} \,.
\end{array}
\right.
\label{w1-UN}
\end{equation}
More generally, one may evaluate $w_k$ from $\rho(\phi)$, thanks to
the relationship
\begin{eqnarray}
w_k &=& \int_{-\phi_c}^{\phi_c}{\mathrm{d}}\phi \cos k\phi\,\rho(\phi)
\nonumber \\ &=&
\left\{
\renewcommand\arraystretch{1.3}
\begin{array}{l@{\quad}l}
0 \,, & \beta\le\casefr{1}{2},\ k \ge 2 \,, \\
\displaystyle \left(1 - {1\over2\beta}\right)^{\!\!2}
{1\over k-1}\,P^{(1,2)}_{k-2}\!\left(1 - {1\over\beta}\right), &
\beta\ge\casefr{1}{2} \,,
\end{array}
\right.
\end{eqnarray}
where $P^{(\alpha,\beta)}_k$ are the Jacobi polynomials. All $w_k$
are differentiable once in $\beta=\beta_c$, but their second
derivatives are discontinuous. Let us notice that
Eqs.\ (\ref{beta_c-UN}), (\ref{F-UN}), and (\ref{w1-UN}) are an
immediate consequence of Eqs.\ (\ref{SD-Z-strong-cond}) and
(\ref{SD-W}) for the special choice
\begin{equation}
x_s = \beta^2 \,.
\end{equation}
\subsection{The Schwinger-Dyson equations of the two-dimensional
Yang-Mills theory}
\label{sd-YM}
It is interesting to obtain the above results from the algebraic
approach to the Schwinger-Dyson equations of the model. We can
restrict Eqs.\ (\ref{Migdal-Makeenko}) to the set of Wilson loops
${\cal C}_k$ consisting of $k$ turns around a single plaquette, in
which case by definition $W({\cal C}_k) = w_k$. Formally, the
Schwinger-Dyson equations do not close on this set of expectation
values; however, one may check by inspection, using the factorization
property of two-dimensional functional integral for the Yang-Mills
theory, that contributions from other Wilson loops cancel in the
equations for $w_k$ (this is strictly a two-dimensional property). As
a consequence, we obtain the large-$N$ relationships
\cite{Paffuti-Rossi-solution}
\begin{equation}
\beta(w_{n-1} - w_{n+1}) = \sum_{k=1}^n w_k w_{n-k} \,,
\label{MM-reduced}
\end{equation}
with a boundary condition $w_0 = 1$. The solution is found by
defining a generating function
\begin{equation}
\Phi(t) \equiv \sum_{k=0}^\infty w_k t^k
\end{equation}
and noticing that Eq.\ (\ref{MM-reduced}) corresponds to
\begin{equation}
\Phi t^2 - (\Phi - 1 - w_1 t) =
{t\over\beta} (\Phi^2 - \Phi),
\end{equation}
which is solved by
\begin{eqnarray}
\Phi(t) = {\beta\over2t}
\sqrt{\left(1 + {t\over\beta} + t^2\right)^{\!\!2}
- 4 t^2 \left(1 - {w_1\over\beta}\right)}
- {\beta\over2t} \left(1 - {t\over\beta} - t^2\right).
\nonumber \\
\label{Phi-strong-coupling}
\end{eqnarray}
The condition $|w_k|\le1$ implies that $\Phi(t)$ is holomorphic within
the unitary circle. On the boundary of the analyticity domain, $t =
\e^{{\mathrm{i}}\phi}$ and
\begin{equation}
w_k = {1\over\pi} \int_{-\pi}^\pi
[\mathop{\operator@font Re}\nolimits\Phi(\phi) - \casefr{1}{2}] \cos k\phi \, {\mathrm{d}}\phi,
\end{equation}
and as a consequence we may identify
\begin{equation}
\mathop{\operator@font Re}\nolimits\Phi(\phi) - \casefr{1}{2} = \rho(\phi).
\end{equation}
The positivity condition on $\rho(\phi)$ leads to a complete
determination of the solution, implying either
\begin{equation}
w_1 = \beta, \qquad w_k = 0 \ \ (k\ge2), \qquad \beta\le\casefr{1}{2}
\end{equation}
or $\rho(\pi) = 0$, which in turn leads to
\begin{equation}
w_1 = 1 - {1\over4\beta}, \qquad -\phi_c\le\phi\ge\phi_c,
\qquad \beta\le\casefr{1}{2} \, ,
\end{equation}
and $\phi_c$ is given by Eq.\ (\ref{easy-rho-weak-coupling-UN}).
It is immediate to check that the resulting eigenvalue densities are
the same as Eqs.\ (\ref{easy-rho-strong-coupling-UN}) and
(\ref{easy-rho-weak-coupling-UN}).
Let us mention that these methods may in principle be applied to more
general formulation of the theory based on ``local'' actions, and in
particular Wilson loop expectation values can be computed for the
fixed-point version of the model, corresponding to the continuum
action \cite{Rossi-qcd2}. The fixed-point action in YM$_2$ in turn is
nothing but the ``heat kernel'' action \cite{Drouffe-heat}, discussed
in the large-$N$ context in Ref.\ \cite{Menotti-Onofri}. Large-$N$
continuum YM$_2$ is slightly beyond the purpose of the present review.
We must however mention that in recent years a number of interesting
results have appeared in a string theory context. It is worth quoting
Refs.\ \cite{Rusakov,Douglas-Kazakov,Gross-Taylor} and references
therein.
While the problem of evaluating the more general expectation values
$W_{l,k}$ is solved in principle, in practice it is not always simple
to obtain compact closed-form expressions whose general features can
be easily understood. In the strong-coupling regime
$\beta<\casefr{1}{2}$, it is not too difficult to determine from
finite-$N$ results the large-$N$ limit in the form
\cite{Rossi-Vicari-QCD2}
\begin{equation}
\lim_{N\to\infty} W_{l,k} = {(-1)^{k-1}\over k}
\left(
\renewcommand\arraystretch{1.3}
\begin{array}{c}
lk - 2 \\
k - 1
\end{array}
\right)
\beta^{kl},
\label{w_lk-SU}
\end{equation}
and one may show that the corresponding Schwinger-Dyson equations
close on the set $W_{l,k}$ for any fixed $l$ and are solved by Eq.\
(\ref{w_lk-SU}). As a matter of fact, by defining
\begin{equation}
\Phi_l(t) \equiv \sum_{k=0}^\infty W_{l,k} t^k,
\end{equation}
one may show that the strong-coupling Schwinger-Dyson equations reduce
to
\begin{equation}
[\Phi_l(t) - 1] [\Phi_l(t)]^{l-1} = \beta^l t .
\end{equation}
For the interesting values $l=1$ and $l=2$, Eq.\ (\ref{w_lk-SU})
reduces to
\begin{equation}
\Phi_1(t) = 1 + \beta t,
\end{equation}
consistent with the strong-coupling solution
(\ref{Phi-strong-coupling}), and
\begin{equation}
\Phi_2(t) = \casefr{1}{2}\left(\sqrt{1 + 4 \beta^2 t^2} + 1\right),
\end{equation}
related to the generating function for the moments of the energy
density
\begin{eqnarray}
&&{1 \over N}
\left<\mathop{\operator@font Tr}\nolimits{1 \over 1 - \beta t (V_n + V_{n+1}^\dagger)}\right>
= 1 + 2 t \beta^2 + 2 \sum_{k=1}^\infty (\beta t)^{2k} W_{2,k}
\nonumber \\
&&\qquad=\, 2 \beta^2 t + \sqrt{1 + 4 \beta^4 t^2} \,.
\label{moments-generating-function-UN}
\end{eqnarray}
Eq.\ (\ref{moments-generating-function-UN}) is related to a different
approach for solving large-$N$ unitary-matrix models, based on an
integration of the matrix angular degrees of freedom to be performed
in strong coupling \cite{Kazakov-Kozhamkulov-Migdal,Barsanti-Rossi}.
The corresponding weak-coupling problem is definitely more difficult.
As far as we can see, the Schwinger-Dyson equations close only on a
larger set of correlation functions, defined by the generating
function \cite{Rossi-unpublished}
\begin{eqnarray}
&&D_{k,n}^{(l)}(t) =
{1 \over N} \mathop{\operator@font Tr}\nolimits \left[(V_k)^{n+1} V_{k+1} ... V_l \,
{1 \over 1 - t V_1 ... V_l}\right], \nonumber \\
&&0 < k \le l,\ n \ge 0,
\end{eqnarray}
such that
\begin{equation}
\Phi_l(t) = 1 + t D_{1,0}^{(l)}(t).
\end{equation}
The explicit form of the equations is
\begin{eqnarray}
&&\sum_{j=0}^{n-1} w_j D_{k,n-j}^{(l)}(t) +
D_{l,n-1}^{(l)}(t)\, D_{k,0}^{(l)}(t) \nonumber \\
&&\quad+\,
\beta\left[D_{k,n+1}^{(l)}(t) - D_{k,n-1}^{(l)}(t)\right] = 0,
\qquad 1 \le k \le l .
\end{eqnarray}
When $l=1,2$ it is possible to find explicit weak-coupling solutions,
but the general case $l>2$ has not been solved so far.
More about the calculability of Wilson loops with arbitrary contour in
two-dimensional ${\rm U}(\infty)$ lattice gauge theory can be found in
Ref.\ \cite{Kazakov-Kostov-wilson}. The corresponding continuum
calculations are presented for arbitrary ${\rm U}(N)$ groups in
Ref.\ \cite{Kazakov-wilson}.
\subsection{Large-$N$ properties of the determinant}
\label{determinant}
It is quite interesting to apply the results of Subs.\
\ref{properties-determinant}, concerning the properties of the
determinant, to YM$_2$ and principal chiral models in one dimension.
Exploiting the factorization of the functional integration and the
possibility of performing the variable change
(\ref{chiral-variable-change}) in the operators as well as in the
action, we can easily obtain the relationship
\begin{equation}
\Delta_l \equiv \det\left[U_0 U^\dagger_l\right] =
\det\left[V_1 ... V_l\right] = \det V_1 ... \det V_l,
\end{equation}
and, as a consequence,
\begin{equation}
\left<\Delta_l\right> = \left<\det V_1\right>^l.
\end{equation}
The problem is therefore reduced to that of evaluating $\left<\det
V\right>$ in the single-plaquette model. It is immediate to recognize
from Eqs.\ (\ref{SD-Xl-reduced}) and (\ref{SD-Xl-sol}) that
\begin{eqnarray}
&\displaystyle \left<\det V\right> \to \sqrt{1 - {1\over2\beta}},
&\qquad \beta \ge \casefr{1}{2},
\label{detV-weak-coupling} \\
&\displaystyle \left<\det V\right> \to 0,
&\qquad \beta \le \casefr{1}{2}.
\end{eqnarray}
Apparently, this expectation value acts as an order parameter for the
phase transition between the weak- and strong-coupling phases. More
precisely, according to Green and Samuel
\cite{Green-Samuel-un1,Green-Samuel-largeN}, one must identify the
order parameter with the quantity
\begin{equation}
\left<\Delta_l\right>^{1/N}
\end{equation}
and notice that
\begin{eqnarray}
&\displaystyle \left<\Delta_l\right>^{1/N} \to 1
&\qquad \hbox{in weak coupling,}
\label{Deltal-wc} \\
&\displaystyle \left<\Delta_l\right>^{1/N} \to \exp(-\sigma l)
&\qquad \hbox{in strong coupling,}
\label{Deltal-sc}
\end{eqnarray}
where $\sigma$ acts as a ${\rm U}(1)$ ``string tension''. Eqs.\
(\ref{Deltal-wc}) and (\ref{Deltal-sc}) generalize to higher
dimensions, when replacing $l$ with the (large) area of the
corresponding Wilson loop. Notice that the weak-coupling result is
consistent with the decoupling of the ${\rm U}(1)$ degrees of freedom
from the ${\rm SU}(N)$ degrees of freedom, and with the interpretation
of ${\rm U}(1)$ as a free massless field.
It is therefore interesting to compute
\begin{equation}
\sigma = -{1\over N} \log\left<\det V\right>
\end{equation}
in the case of the single-matrix model; this requires taking the
large-$N$ limit only after the strong-coupling calculation of
$\left<\det V\right>$ has been performed. Since the technique of
evaluation of $\sigma$ has some relevance for subsequent developments,
we shall briefly sketch its essential steps. Standard manipulations
of the single-link integrals for finite $N$ allow to evaluate
\begin{eqnarray}
A_{m,N}(\beta) &=& \int{\mathrm{d}} V \exp\left[N \beta\mathop{\operator@font Tr}\nolimits(V+V^\dagger)\right]
(\det V)^m = \det\Vert I_{k-l-m}(2N\beta)\Vert. \nonumber \\
\end{eqnarray}
These quantities can be shown to satisfy the recurrence relations
\cite{Guha-Lee-chiral}
\begin{equation}
A^2_{m,N} - A_{m+1,N} A_{m-1,N} = A_{m,N-1} A_{m,N+1} .
\label{A-recurrence}
\end{equation}
Willing to compute expectation values, we define
\begin{equation}
\Delta_{m,N}(\beta) = \left<(\det V)^m\right> = {A_{m,N}\over A_{0,N}}
\,.
\end{equation}
Eq.\ (\ref{A-recurrence}) implies that
\begin{equation}
\Delta^2_{m,N} - \Delta_{m+1,N} \Delta_{m-1,N} =
\Delta_{m,N-1} \Delta_{m,N+1} (1 - \Delta^2_{1,N}).
\label{Delta-recurrence}
\end{equation}
Since all $\Delta_{m,1}$ are known, it is possible to reconstruct all
$\Delta_{m,N}$ from Eq.\ (\ref{Delta-recurrence}) once $\Delta_{1,N}$
is determined. Now $\Delta_{1,N}$ is exactly $\left<\det V\right>$,
and it is possible to show that it obeys the following second-order
differential equation \cite{Rossi-exact}
\begin{eqnarray}
&&{1\over s}\,{{\mathrm{d}}\over{\mathrm{d}} s}\,s\,{{\mathrm{d}}\over{\mathrm{d}} s}\,\Delta_{1,N} +
{1\over1-\Delta^2_{1,N}}\left[\left({{\mathrm{d}}\over{\mathrm{d}} s}
\,\Delta_{1,N}\right)^{\!\!2}
- {N^2\over s^2}\right]\Delta_{1,N} \nonumber \\
&&\quad+\,
(1-\Delta^2_{1,N})\Delta_{1,N} = 0,
\label{Delta1N-differential}
\end{eqnarray}
where $s = 2N\beta$. Eq.\ (\ref{Delta1N-differential}) can be analyzed
in weak and strong coupling and in the large-$N$ limit. In particular
the weak-coupling $1/N$ expansion leads to
\begin{equation}
\Delta_{1,N} \to \sqrt{1 - {1\over2\beta}} - {1\over N^2}\,
{1\over128\beta^3}\left(1 - {1\over2\beta}\right)^{\!\!5/2}
+ O\left(1\over N^4\right),
\end{equation}
thus confirming Eq.\ (\ref{detV-weak-coupling}), while in strong
coupling one may show that
\begin{equation}
\Delta_{1,N} = J_N(2N\beta) + O(\beta^{3N+2})
\mathop{\;\longrightarrow\;}_{N\to\infty} J_N(2N\beta),
\end{equation}
where $J_N$ is the standard Bessel function, whose asymptotic behavior
is well known. As an immediate consequence, we find
\begin{equation}
-\sigma = \sqrt{1-4\beta^2} - \log{1 + \sqrt{1-4\beta^2}
\over 2\beta}, \qquad \beta<\casefr{1}{2}.
\end{equation}
This result was first guessed by Green and Samuel
\cite{Green-Samuel-largeN}, and then explicitly demonstrated in
Ref.\ \cite{Rossi-exact}.
\subsection{Local symmetry breaking in the large-$N$ limit}
\label{local-SB}
Another interesting application of the external-field single-link
integral to the large-$N$ limit of two-dimensional Yang-Mills theories
is the study of the possibility of breaking a local symmetry, as a
consequence of the thermodynamical nature of the limit. If we
introduce an infinitesimal explicit ${\rm U}(N)$ symmetry breaking
term in the action \cite{Celmaster-Green}
\begin{equation}
S = - \beta N \left[\mathop{\operator@font Tr}\nolimits V + J N V^{ij} + \hbox{h.c.}\right],
\end{equation}
corresponding to replacing
\begin{equation}
A_{lm} \to \beta\left[\delta_{lm} + N J \delta_{lj}\delta_{mi}\right]
\end{equation}
in Eq.\ (\ref{one-link}), we find that the eigenvalues of $AA^\dagger$
are
\begin{eqnarray}
x_{1,2} &=& \beta^2 \left[1 + \casefr{1}{2} N^2 J^2 \pm \casefr{1}{2}
\sqrt{J^4 N^4 + 4 J^2 N^2}\right], \nonumber \\
x_l &=& \beta^2, \qquad l>2 .
\end{eqnarray}
When taking the large-$N$ limit of the free energy, we find
\begin{equation}
\lim_{N\to\infty} {\log Z\over N^2} = F_0(\beta) + 2 \beta|J|,
\end{equation}
and in the limit $J\to0^\pm$ we then find
\begin{equation}
\left<\mathop{\operator@font Re}\nolimits V^{ij}\right> = \pm 1.
\end{equation}
We therefore expect that, for finite $k$, the ${\rm U}(k)$ global
symmetries of large-$N$ chiral models and ${\rm U}(k)$ gauge
symmetries are broken in any number of dimensions
\cite{Celmaster-Green}. This phenomenon cannot occur for any finite
value of $N$ in two dimensions.
\subsection{Evaluation of higher-order corrections}
\label{higher-order-corrections}
In the context of large-$N$ two-dimensional Yang-Mills theory, it is
worth mentioning that it is possible to compute systematically
higher-order corrections to physical quantities in the powers of
$1/N^2$. It is interesting to notice that the weak-coupling
corrections to the free energy \cite{Goldschmidt} (see also
\cite{Muller-Ruhl-mean-1})
\begin{eqnarray}
F &=& F_0 + {1\over N^2}\left[{1\over12} - A - {1\over12} \log N
- {1\over8} \log\left(1 - {1\over2\beta}\right)\right]
\nonumber \\ &+&\,
{1\over N^4}\left[{3\over1024\beta^3}
\left(1 - {1\over2\beta}\right)^{\!\!-3}
- {1\over240}\right] + ... \,, \\
U &=& 1 - {1\over4\beta} -
{1\over N^2}\,{1\over32\beta^2}
\left(1 - {1\over2\beta}\right)^{\!\!-1}
\nonumber \\ &-&\,
{1\over N^2}\,{1\over1024\beta^4}
\left(1 - {1\over2\beta}\right)^{\!\!-4} +
O\left(1\over N^4\right),
\end{eqnarray}
where $A = 0.24875...$, are well defined, but become singular when
$\beta\to\casefr{1}{2}$. In turn, when evaluating higher-order
corrections in the strong-coupling phase, one finds out that there are
no corrections proportional to powers of $1/N$, while there are
contributions that fall off exponentially with large $N$, as expected
from the general arguments discussed in
Subs.\ \ref{mean-field+strong-coupling} in connection with the
appearance of the DeWit-'t Hooft poles.
Let us however mention that Eqs.\ (\ref{Delta-recurrence}) and
(\ref{Delta1N-differential}) are also the starting point for a
systematic $1/N$ expansion of the free energy in the weak-coupling
regime, alternative to Goldschmidt's procedure. The basic ingredient
is the observation that, defining the free energy at finite $N$ by
\begin{equation}
F_N(\beta) = \log A_{0,N}(\beta),
\end{equation}
one may show that
\begin{equation}
{{\mathrm{d}}\over{\mathrm{d}} s}(\log F_N - \log F_{N-1}) =
{\Delta_{1,N}\over1-\Delta_{1,N}^2}
\left({{\mathrm{d}}\over{\mathrm{d}} s}\,\Delta_{1,N} +
{N\over s}\,\Delta_{1,N}\right),
\end{equation}
and this allows for a systematic reconstruction of $F_N$, whose
strong-coupling form is \cite{Guha-Lee-chiral}
\begin{equation}
F_N(\beta) = N^2\beta^2 - \sum_{k=1}^\infty k J_{N+k}^2(2N\beta)
+ O(\beta^{4N+4}).
\end{equation}
\subsection{Mixed-action models for lattice YM$_2$}
\label{mixed-ation}
Another instance of the problem of the single-link integration for
matrix fields in the adjoint representation of the full symmetry group
occurs in the discussion of the so-called ``mixed action'' models.
Consider the following single-link integral
\cite{Chen-Tan-Zheng-phase}, resulting from a different formulation of
lattice YM$_2$,
\begin{equation}
Z(\beta_{\rm f},\beta_{\rm a}) = \int{\mathrm{d}} U
\exp\left\{N\beta_{\rm f}\mathop{\operator@font Tr}\nolimits(U+U^\dagger)
+ \beta_{\rm a} |\mathop{\operator@font Tr}\nolimits U|^2\right\}.
\label{Z-fa}
\end{equation}
It is possible to show that, in the large-$N$ limit, the corresponding
free energy can be obtained by the same saddle-point technique
presented in Subs.\ \ref{2d-YM-single-link}, i.e., by introducing a
spectral density $\rho(\theta)$ for the eigenvalues of $U$. This
spectral density turns out to be precisely the same as the one
obtained when $\beta_{\rm a} = 0$, if one simply replaces $\beta_{\rm
f}$ by an effective coupling
\begin{equation}
\beta_{\rm eff} = \beta_{\rm f} + \beta_{\rm a} w_1(\beta_{\rm eff}),
\label{beff-def}
\end{equation}
where $w_1$ can be evaluated in terms of $\rho(\theta)$ as
\begin{equation}
w_1(\beta_{\rm eff}) = \int{\mathrm{d}}\theta \cos\theta\,\rho(\theta).
\label{w1-eff}
\end{equation}
Eq.\ (\ref{w1-eff}) is a self-consistency condition for $w_1$, which
allows a determination of
$\beta_{\rm eff}(\beta_{\rm f},\beta_{\rm a})$.
Finally, by substitution into the effective action, one finds the
relationship
\begin{equation}
F(\beta_{\rm f},\beta_{\rm a}) =
F(\beta_{\rm eff}(\beta_{\rm f},\beta_{\rm a}),0) -
\beta_{\rm a} w_1^2(\beta_{\rm eff}(\beta_{\rm f},\beta_{\rm a})),
\label{F-fa}
\end{equation}
where $F(\beta,0)$ is nothing but the free energy obtained in
Subs.\ \ref{2d-YM-single-link}.
The strong- and weak-coupling solutions are separated by the line
$2\beta_{\rm f} + \beta_{\rm a} = 1.$ In strong coupling one obtains
\begin{eqnarray}
\beta_{\rm eff} &=& w_1 = {\beta_{\rm f}\over1-\beta_{\rm a}}\,,
\nonumber \\
F &=& {\beta_{\rm f}^2\over1-\beta_{\rm a}}\,,
\end{eqnarray}
while in weak coupling
\begin{eqnarray}
\beta_{\rm eff} &=& {1\over2} \left[\beta_{\rm f} + \beta_{\rm a} +
\sqrt{(\beta_{\rm f} + \beta_{\rm a})^2 - \beta_{\rm a}}\right],
\nonumber \\
w_1 &=& {1\over2\beta_{\rm a}} \left[\beta_{\rm a} - \beta_{\rm f} +
\sqrt{(\beta_{\rm f} + \beta_{\rm a})^2 - \beta_{\rm a}}\right],
\nonumber \\
F &=& \beta_{\rm f} + {\beta_{\rm a}\over2}
- {\beta_{\rm f}^2\over2\beta_{\rm a}} - {1\over2}
- {1\over2}\log\left[\beta_{\rm f} + \beta_{\rm a} +
\sqrt{(\beta_{\rm f} + \beta_{\rm a})^2 - \beta_{\rm a}}\right]
\nonumber \\
&&\quad+\,
{1\over2}\left(1 + {\beta_{\rm f}\over\beta_{\rm a}}\right)
\sqrt{(\beta_{\rm f} + \beta_{\rm a})^2 - \beta_{\rm a}}.
\end{eqnarray}
It may be interesting to quote explicitly the limiting case
$\beta_{\rm f} = 0$, where \cite{Brihaye-Rossi-weak}
\begin{eqnarray}
Z(0,\beta_{\rm a}) &\equiv& \int{\mathrm{d}} U \exp\beta_{\rm a}|\mathop{\operator@font Tr}\nolimits U|^2
\nonumber \\
&=& \left\{
\renewcommand\arraystretch{1.3}
\begin{array}{l@{\quad}c}
0, & \beta_{\rm a} < 1, \\
\displaystyle {1\over2}\beta_{\rm a} +
{1\over2}\beta_{\rm a}\sqrt{1 - {1\over\beta_{\rm a}}} -
{1\over2}\log\beta_{\rm a}
\left(1 + \sqrt{1 - {1\over\beta_{\rm a}}}\right), &
\beta_{\rm a} > 1.
\end{array}
\right. \nonumber \\
\end{eqnarray}
One may actually show that, in any number of dimensions, a lattice
gauge theory with mixed action
\cite{Samuel-adjoint,Makeenko-Polikarpov,Samuel-phase} (a trivial
generalization of Eq.\ (\ref{Z-fa})) is solved in the large-$N$ limit
in terms of the solution of the corresponding theory with pure Wilson
action; Eqs.\ (\ref{beff-def}) and (\ref{F-fa}) hold as they stand, and
\begin{equation}
w_1(\beta_{\rm eff}) = \left.{1\over N} \left<\mathop{\operator@font Tr}\nolimits U_p\right>
\right|_{\beta_{\rm f} = \beta_{\rm eff},\ \beta_{\rm a} = 0} .
\end{equation}
More about the large-$N$ behavior of variant actions can be found in
Refs.\ \cite{Ogilvie-Horowitz,Jurkiewicz-KorthalsAltes,%
Jurkiewicz-KorthalsAltes-Dash}. Different kinds of variant actions
have been studied in the large-$N$ limit in Refs.\
\cite{Rodrigues-variant,Lang-Salomonson-Skagerstam-third,Samuel-heat}.
\subsection{Double-scaling limit of the single-link integral}
\label{double-scaling-single-link}
In the Introduction, we mentioned that one of the most interesting
phenomena related to the large-$N$ limit of matrix models is the
appearance of the so-called ``double-scaling limit''
\begin{equation}
\left\{
\renewcommand\arraystretch{1.3}
\begin{array}{l}
N\to\infty, \\
g\to g_c,
\end{array}
\right.
\qquad N^{2/\gamma_1}(g_c-g) = \hbox{const},
\end{equation}
where $g$ is a (weak) coupling related to the inverse of $\beta$. We
already discussed the general physical interpretation of this limit as
an alternative description of two-dimensional quantum gravity and its
relationship to the theory of random surfaces. Here we only want to
consider the double-scaling limit properties for those simple models
of unitary matrices that can be reformulated as a single-link model
(cfr.\ Ref.\ \cite{Demeterfi-Tan}).
This specific subject was pioneered by Periwal and Shevitz
\cite{Periwal-Shevitz}, who discussed the double-scaling limit in
models belonging to the class
\begin{equation}
Z_N = \int{\mathrm{d}} U \exp\left[N\beta\mathop{\operator@font Tr}\nolimits {\cal V}(U+U^\dagger)\right],
\label{ZN-ds}
\end{equation}
where ${\cal V}(U)$ is a polynomial in $U$. Because of the invariance
of the measure, Eq.\ (\ref{ZN-ds}) can be reduced to
\begin{equation}
Z_N \sim \int{\mathrm{d}}\phi_i |\Delta(\e^{{\mathrm{i}}\phi_1},...,\e^{{\mathrm{i}}\phi_N})|^2
\exp\left[N\beta \textstyle\sum_i {\cal V}(2\cos\phi_i)\right],
\end{equation}
and solved by the method of orthogonal polynomials. One starts by
defining polynomials
\begin{equation}
P_n(z) = z^n + \sum_{k=0}^{n-1} a_{k,n} z^k,
\end{equation}
that satisfy
\begin{equation}
\oint {{\mathrm{d}} z\over2\pi{\mathrm{i}} z}\,P_n(z)\,P_m\!\left(1\over z\right)
\exp\left[N\beta {\cal V}\!\left(z + {1\over z}\right)\right] =
h_n\,\delta_{mn}\,,
\end{equation}
where the integration runs over the unit circle, and moreover obey the
recursion relation
\begin{equation}
P_{n+1}(z) = zP_n(z) + R_nz^nP_n\left(1\over z\right), \qquad
{h_{n+1}\over h_n} = 1 - R^2_n \,.
\end{equation}
where $R_n \equiv a_{0,n+1}$. As a corollary,
\begin{equation}
Z_N \propto N! \prod_i\left(1-R^2_{i-1}\right)^{N-i},
\end{equation}
and one may show that
\begin{eqnarray}
(n+1)(h_{n+1}-h_n) &=& \oint {{\mathrm{d}} z\over2\pi{\mathrm{i}} z}
\exp\left[N\beta {\cal V}\!\left(z + {1\over z}\right)\right]
N\beta {\cal V}'\!\left(z + {1\over z}\right)
\nonumber \\ &\times&\,
\left(1 - {1\over z^2}\right)P_{n+1}(z)\,P_n\!\left(1\over z\right),
\end{eqnarray}
which in turn leads to a nonlinear functional equation for $R_n$.
The simplest example, corresponding to YM$_2$, amounts to choosing
${\cal V}' = 1$, obtaining
\begin{equation}
(n+1)R^2_n = N\beta R_n(R_{n+1}+R_{n-1})(1-R_n^2),
\label{YM2-ds}
\end{equation}
and in the large-$N$ limit, setting $n=N$ and $R_N=R$, we obtain the
limiting form
\begin{equation}
R^2 = 2\beta R^2(1-R^2),
\end{equation}
showing that $\beta_c=\casefr{1}{2}$ (degeneracy of solution $R_c=0$).
One may now look for the scaling solution to Eq.\ (\ref{YM2-ds}) in the
form
\begin{equation}
R_N-R_c = R_N = N^{-\mu} f\left[N^\rho(g_c-g)\right], \qquad
g = {1\over\beta},
\end{equation}
where $f^2$ is related to the second derivative of the free energy.
This is a consistent Ansatz when
\begin{equation}
\mu=\casefr{1}{3}, \qquad
\rho=\casefr{2}{3},
\label{k=1-exponents}
\end{equation}
leading to the equation
\begin{equation}
-2xf + 2f^3 = f'', \qquad x=N^\rho(g_c-g).
\end{equation}
In the case ${\cal V}' = 1 + \lambda u$, one finds the equation
\begin{equation}
{1\over\beta} = -2(1-R^2)(-1-\lambda+3\lambda R^2),
\end{equation}
which reduces to $1/\beta = \casefr{3}{2}(1-R^4)$ when
$\lambda=\casefr{1}{4}$. A scaling solution to the corresponding
difference equation requires $\mu=\casefr{1}{5}$ and
$\rho=\casefr{4}{5}$.
When ${\cal V}' = 1 + \lambda_1 u + \lambda_2 u^2$, multicriticality
sets at $\lambda_1=-\casefr{3}{7}$ and $\lambda_2=\casefr{1}{14}$, and
$1/\beta = \casefr{10}{7}(1-R^6)$, leading to the exponents
$\mu=\casefr{1}{7}$ and $\rho=\casefr{6}{7}$.
Rather general results can be obtained for an arbitrary order $k$ of
the polynomial ${\cal V}$: $\mu = 1/(2k+1)$, $\rho = 2k/(2k+1)$, and
$c = 1 - 6/(k(k+1))$.
The double-scaling limit can also be studied in the case of the
external-field single-link integral \cite{Gross-Newman}, and it was
found that its critical behavior is simple enough to be identified
with that of the $k=1$ unitary-matrix model. In the language of
quantum gravity, the only effect of introducing $N^2$ real parameters
$A_{ij}$ is that of renormalizing the cosmological constant, without
changing the universality class of the critical point.
A few interesting features of the double-scaling limit for the $k=1$
model are worth a more detailed discussion \cite{Damgaard-Heller}. In
particular let us recall that, according to Eq.\ (\ref{k=1-exponents}),
\begin{equation}
\rho = {2\over\gamma_1} = {2\over3},
\end{equation}
and therefore $\gamma_1=3$, implying $c=-2$. We may now reinterpret
the double-scaling limit of matrix models as a finite-size scaling
with respect to the ``volume'' parameter $N$ in a two-dimensional
$N{\times}N$ space. As a consequence, we obtain relationships with
more conventional critical exponents through the identification
$\gamma_1=2\nu$, which in turn by hyperscaling leads to a
determination of the specific heat exponent $\alpha = 2(1-\nu)$.
Numerically we obtain $\nu=\casefr{3}{2}$ and $\alpha=-1$. The result
$\alpha=-1$ can be easily tested on the solution of the model
\begin{equation}
C(\beta) = {1\over2}\,\beta^2\,{{\mathrm{d}}^2F\over{\mathrm{d}}\beta^2} =
\left\{
\renewcommand\arraystretch{1.3}
\begin{array}{l@{\quad}l}
\beta^2, & \beta\le\beta_c, \\
\casefr{1}{4}, & \beta\ge\beta_c,
\end{array}
\right.
\end{equation}
with $\beta_c=\casefr{1}{2}$,
consistent with a negative critical exponent $\alpha=-1$.
It is also interesting to find tests for the exponent $\nu$,
especially in view of the fact that the most direct checks are not
possible in absence of a proper definition for the relevant
correlation length. Numerical studies have been performed by
considering the partition function zero $\beta_0$ closest to the
transition point $\beta_c=\casefr{1}{2}$, finding that the relationship
\begin{equation}
\mathop{\operator@font Im}\nolimits\beta_0 \propto N^{-1/\nu}
\end{equation}
is rather well satisfied even for very low values of $N$; at $N\ge5$,
it is valid within one per mille. Another test concerns the location
of the peak in the specific heat in ${\rm U}(N)$ models, whose
position $\beta_{\rm peak}(N)$ should approach $\beta_c$ with
increasing $N$. Finite-size scaling arguments predict
\begin{equation}
\beta_{\rm peak}(N) \cong \beta_c + a N^{-1/\nu} ,
\end{equation}
and large-$N$ results are very well fitted by the choice
$\nu=\casefr{3}{2}$, $a\cong0.60$
\cite{Campostrini-Rossi-Vicari-chiral-3}.
\subsection{The character expansion and its large-$N$ limit:
${\rm SU}(N)$ vs.\ ${\rm U}(N)$}
\label{character-expansion}
The general features of the character expansion for lattice spin and
gauge models have been extensively discussed by different authors. In
particular, Ref.\ \cite{Drouffe-Zuber}, besides offering a general
presentation of the issues, presents tables of character coefficients
for many interesting groups, including ${\rm U}(\infty) \cong {\rm
SU}(\infty)$, for the Wilson action. Let us therefore only briefly
recall the fundamental points of this approach, which is relevant
especially in the analysis of the strong-coupling phase and of the
phase transition.
In Sect.\ \ref{unitary-matrices} we classified the representations and
characters of ${\rm U}(N)$ groups. Because of the orthogonality and
completeness relations, every invariant function of $V$ can be
decomposed in a generalized Fourier series in the characters of $V$.
Let us now consider for sake of definiteness chiral models with action
given by Eq.\ (\ref{action-spin}); extension to lattice gauge theories
is essentially straightforward, at least on a formal level.
We can replace the Boltzmann factor corresponding to each lattice link
by its character expansion:
\begin{eqnarray}
&& \exp\left\{\beta N\mathop{\operator@font Tr}\nolimits\left[U_x U^\dagger_{x+\mu} +
U_{x+\mu} U^\dagger_{x}\right]\right\} \nonumber \\
&=&
\exp \Biggl\{ N^2 F(\beta) \sum_{(r)} d_{(r)} \tilde z_{(r)}(\beta)
\, \chi_{(r)}\bigl(U_x U^\dagger_{x+\mu}\bigr)\Biggr\},
\label{link-char-exp}
\end{eqnarray}
where the sum runs over all the irreducible representations of ${\rm
U}(N)$, $F(\beta)$ is the free energy of the single-link model
\begin{eqnarray}
F(\beta) = {1\over N^2}
\log\int {\mathrm{d}} V \exp\left[N\beta\mathop{\operator@font Tr}\nolimits(V+V^\dagger)\right] =
{1\over N^2}\,\log\det\Vert I_{j-i}(2N\beta)\Vert,
\nonumber \\
\label{single-link-F}
\end{eqnarray}
and $\tilde z_{(r)}(\beta)$ are the character coefficients, defined by
orthogonality and representable in terms of single-link integrals as
\begin{equation}
d_{(r)} \tilde z_{(r)}(\beta) = \left<\chi_{(r)}(V)\right> =
{\det\Vert I_{\lambda_i+j-i}(2N\beta)\Vert
\over \det\Vert I_{j-i}(2N\beta)\Vert} \,,
\end{equation}
with $\lambda$ defined by Eq.\ (\ref{lambda}). We may notice that, for
any finite $N$, $\tilde z_{(r)}(\beta)$ are meromorphic functions of
$\beta$, with no poles on the real axis, which is relevant to the
series analysis. However, singularities may develop, as usual, in the
large-$N$ limit. Eqs.\ (\ref{link-char-exp}) and (\ref{single-link-F})
become rapidly useless with growing $N$. However, an extreme
simplification occurs in the large-$N$ limit, owing to the property
\begin{eqnarray}
d_{(l,m)} \tilde z_{(l,m)}(\beta) =
{1\over n_+!}\,{1\over n_-!}\,\sigma_{(l)}\sigma_{(m)}\,
(N\beta)^{n_++n_-} \left[1 + O(\beta^{2N})\right],
\nonumber \\
\label{dz}
\end{eqnarray}
where $n_+ = \sum_i l_i$, $n_- = \sum_i m_i$, and $\sigma_{(l)}$ is
the dimension of the representation $(l)$ of the permutation group,
which in turn can be computed explicitly as
\begin{equation}
{1\over n_+!}\,\sigma_{(l_1,...,l_s)} = {\prod_{1 \le j \le k \le s}
(l_j-l_k+k-j)!\over\prod_{i=1}^s(l_i+s-i)!} \,;
\label{sigma-def}
\end{equation}
$d_{(l,m)}$ can be parameterized by
\begin{equation}
d_{(l,m)} = {1\over n_+!}\,{1\over n_-!}\,\sigma_{(l)}\sigma_{(m)}
\,C_{(l,m)},
\end{equation}
where $C_{(l,m)}$ can be expressed as a finite product:
\begin{eqnarray}
C_{(l,m)} &=& \prod_{i=1}^s {(N-t-i+l_i)!\over(N-t-i)!}
\prod_{j=1}^t {(N-s-j+m_j)!\over(N-s-j)!}
\nonumber \\ &\times&\,
\prod_{i=1}^s \prod_{j=1}^t {(N+1-i-j+l_i+m_j)!\over(N+1-i-j)!}
\, ,
\end{eqnarray}
allowing for a conceptually simple $1/N$ expansion.
These results are complemented with the result
\begin{equation}
F(\beta) = \beta^2 + O(\beta^{2N+2})
\label{F-beta}
\end{equation}
and with the unavoidable large-$N$ constraint $\beta \le
\casefr{1}{2}$.
The character expansion now proceeds as follows.
We notice that, thanks to Eq.\ (\ref{dz}), only a finite number of
nontrivial representations contributes to any definite order in the
strong-coupling series expansion in powers of $\beta$, and each
lattice integration variable can appear only once for each link where
a nontrivial representation in chosen. A systematic treatment leads
to a classification of contributions in terms of paths (surfaces in a
gauge theory) along whose non self-interacting sections a particular
representation is assigned. Self-intersection points are submitted to
constraints deriving from the orthogonality of representations and
their composition rules.
In the case of chiral models, all relevant assignments can be
generated by considering the class of the lattice random paths
satisfying a non-backtracking condition
\cite{Campostrini-Rossi-Vicari-chiral-1}.
Once all nontrivial configurations are classified and counted, one is
left with the task of computing the corresponding group integrals.
Only integrations at intersection points are nontrivial, since other
integrations follow immediately from the orthogonality
relationships. Unfortunately, no special computational
simplifications occur in the large-$N$ limit of group integrals.
Apparently, the character expansion is the most efficient way of
computing the strong-coupling expansion of lattice models. In
particular, very long strong-coupling series have been obtained in the
large-$N$ limit for the free energy, the mass gap, and the two-point
Green's functions of chiral models in two and three dimensions (for
the free energy, 18 orders on the square lattice, 26 orders on the
honeycomb lattice, and 16 orders on the cubic lattice; for the Green's
functions, 15 orders on the square lattice, 20 orders on the honeycomb
lattice, and 14 orders on the cubic lattice). The analysis of these
series will be discussed in Sect.\ \ref{principal-chiral}.
Before leaving the present subsection, we must make a few comments
concerning the relationship between ${\rm SU}(N)$ and ${\rm U}(N)$
groups. We already made the observation that when $N\to\infty$ there
is essentially no difference between ${\rm SU}(N)$ and ${\rm U}(N)$
models, at least when considering operators not involving the
determinant. In order to explore this relationship more carefully, we
may start as usual from the expression of the single-link integral
(\ref{one-link}).
Representations of $Z(A^\dagger A) $ in the ${\rm SU}(N)$ case can be
obtained \cite{Brower-Rossi-Tan-SUN} in terms of the eigenvalues $x_i$
of $A^\dagger A$ and of $\theta$, defined in Eq.\ (\ref{theta-def}).
Introducing the Vandemonde determinant
\begin{equation}
\Delta(\lambda_1,...,\lambda_N) = \prod_{j>i} (\lambda_j-\lambda_i) =
\det\Vert\lambda_j^{i-1}\Vert,
\end{equation}
one obtains
\begin{eqnarray}
Z(A^\dagger A) &=& {1\over N!}
\Biggl(\prod_{k=1}^{N-1} {k!\over2\pi}\Biggr)
\int\prod_i{\mathrm{d}}\phi_i\,\delta\!\left(\sum_i\phi_i + N\theta\right)
\nonumber \\ &\times&\,
{|\Delta(\e^{{\mathrm{i}}\phi_1},...,\e^{{\mathrm{i}}\phi_N})|^2 \over
\Delta(2\sqrt{x_1},...,2\sqrt{x_N})\,
\Delta(\cos\phi_1,...,\cos\phi_N)}
\exp\Biggl[2\sum_k\sqrt{x_k}\cos\phi_k\Biggr],
\nonumber \\
\end{eqnarray}
or alternatively
\begin{eqnarray}
Z(A^\dagger A) &=& \prod_{k=1}^{N-1} {k!\over2\pi}
\int\prod_i{\mathrm{d}}\phi_i\,\delta\!\left(\sum_i\phi_i + N\theta\right)
\nonumber \\ &\times&\,
{\Delta(\sqrt{x_1}\e^{{\mathrm{i}}\phi_1},...,\sqrt{x_N}\e^{{\mathrm{i}}\phi_N}) \over
\Delta(x_1,...,x_N)} \exp\Biggl[2\sum_k\sqrt{x_k}\cos\phi_k\Biggr].
\end{eqnarray}
The only difference between ${\rm SU}(N)$ and ${\rm U}(N)$ is due to
the presence of the (periodic) delta function
$\delta\left(\sum_i\phi_i + N\theta\right)$, introducing the
dependence on $\theta$ corresponding to the constraint $\det U = 1$.
A formal solution is obtained by expanding in powers of $\e^{{\mathrm{i}}
N\theta}$:
\begin{eqnarray}
Z(A^\dagger A) &=& \sum_{m=-\infty}^\infty\e^{{\mathrm{i}} Nm\theta}
\det\Vert z_i^{j-1} I_{j-1-|m|}(2z_i)\Vert
\Biggl(\prod_{k=1}^{N-1} k!\Biggr)
{1\over\Delta(z_1^2,...,z_N^2)} \,,
\nonumber \\
\label{Z-sum-m}
\end{eqnarray}
where $z_i = \sqrt{x_i}$. Eq.\ (\ref{Z-sum-m}) in turn leads to the
following representation of the free energy for the ${\rm SU}(N)$
single-link model:
\begin{equation}
F_N(\beta,\theta) = \log \sum_{m=-\infty}^\infty
A_{m,N}(\beta) \, \e^{{\mathrm{i}} Nm\theta},
\label{FN-theta}
\end{equation}
where for convenience we have redefined the coupling:
$\beta\to\beta\e^{{\mathrm{i}}\theta}$. Eq.\ (\ref{FN-theta}) is useful for a
large-$N$ mean-field study \cite{Guha-Lee-chiral}, but it is certainly
inconvenient at small $N$, where more specific integration techniques
may be applied.
We mention that a large-$N$ analysis of Eq.\ (\ref{FN-theta}) for
$\theta=0$ leads to
\begin{eqnarray}
F_N(\beta,0) &=& N^2\beta^2 + 2 J_N(2N\beta) -
2 J_{N-1}(2N\beta)\,J_{N+1}(2N\beta)
\nonumber \\ &-&\,
\sum_{k=1}^\infty k J^2_{N+k}(2N\beta) + O(\beta^{3N}).
\end{eqnarray}
It is also possible to establish a relationship between ${\rm SU}(N)$
and ${\rm U}(N)$ groups at the level of character coefficients.
Thanks to the basic relationships
\begin{equation}
\chi_{\lambda_1+s,...,\lambda_N+s}(U) =
(\det U)^s \chi_{\lambda_1,...,\lambda_N}(U),
\end{equation}
holding in ${\rm U}(N)$, one may impose the condition $\det U = 1$ in
the integral representation of the character coefficients and obtain
\begin{equation}
z_{(r)} = {\sum_{s=-\infty}^\infty \tilde z(r,s) \over
\sum_{s=-\infty}^\infty \tilde z(0,s)},
\end{equation}
where, by definition, for ${\rm U}(N)$ groups
\begin{equation}
\tilde z(0,s) = \left<\det U^s\right>, \qquad
d_{(r)}\,\tilde z(r,s) =
\left<\det U^s \chi_{(r)}(U)\right>.
\end{equation}
These relationships are the starting point for a systematic
implementation of the corrections due to the ${\rm SU}(N)$ condition
in the $1/N$ expansion of ${\rm U}(N)$ models
\cite{Green-Samuel-chiral,Rossi-Vicari-chiral2}. A peculiarity of the
${\rm SU}(N)$ condition can be observed in the finite-$N$ behavior of
the eigenvalue density function $\rho(\phi,N)$, which shows a
non-monotonic dependence on $\phi$, characterized by the presence of
$N$ peaks. This is already apparent in the $\beta\to0$ limit of the
single-link integral, where \cite{Campostrini-Rossi-Vicari-chiral-3}
\begin{eqnarray}
\rho_{{\rm U}(N)}(\phi) \mathop{\;\longrightarrow\;}_{\beta\to0} {1\over2\pi}\,, &\qquad&
\rho_{{\rm SU}(N)}(\phi) \mathop{\;\longrightarrow\;}_{\beta\to0}
{1\over2\pi}\left(1 + (-1)^{N+1}\,{2\over N}\cos N\phi\right).
\nonumber \\
\end{eqnarray}
\section{Chiral chain models and gauge theories on polyhedra}
\label{chiral-chains}
\subsection{Introduction}
\label{sec4intr}
The use of the steepest-descent techniques allows to extend the number
of the unitary-matrix models solved in the large-$N$ limit to some few
unitary-matrix systems. The interest for few-matrix models may arise
for various reasons. Their large-$N$ solutions may represent
non-trivial benchmarks for new methods meant to investigate the
large-$N$ limit of more complex matrix models, such as QCD. Every
matrix system may have a r\^ole in the context of two-dimensional
quantum gravity; indeed, via the double scaling limit, its critical
behavior is connected to two-dimensional models of matter coupled to
gravity. Furthermore, every unitary-matrix model can be reinterpreted
as the generating functional of a class of integrals over unitary
groups, whose knowledge would be very useful for the strong-coupling
expansion of many interesting models.
This section is dedicated to a class of finite-lattice chiral
models termed chain models and defined by the partition function
\begin{equation}
Z_L=\int \prod_{i=1}^L{\mathrm{d}} U_i \exp\left[
N\beta \sum_{i=1}^L \mathop{\operator@font Tr}\nolimits \left(U_iU^\dagger_{i+1}
+U^\dagger_iU_{i+1}\right)\right],
\label{Zf}
\end{equation}
where periodic boundary conditions are imposed: $U_{L+1}=U_1$.
Chiral chain models have interesting connections with gauge models.
Fixing the gauge $A_0 = 0$, YM$_2$ on a $K\times L$ lattice (with
free boundary conditions in the direction of size $K$) becomes
equivalent to $K$ decoupled chiral chains of length $L$.
Chiral chains with periodic boundary conditions enjoy another
interesting equivalence with lattice gauge theories defined on the
surface of polyhedra, where a link variable is assigned to each edge
and a plaquette to each face. By choosing an appropriate gauge,
lattice gauge theories on regular polyhedra like tetrahedron, cube,
octahedron, etc., are equivalent respectively to periodic chiral
chains with $L=4,6,8$, etc.\ \cite{Brower-Rossi-Tan-chains}.
The thermodynamic properties of chiral chains can be derived by
evaluating their partition functions. Free-energy density, internal
energy, and specific heat are given respectively by
\begin{equation}
F_L={1\over LN^2} \log Z_L,
\label{FL}
\end{equation}
\begin{equation}
U_L= {1\over 2} {\partial F_L\over \partial\beta},
\label{UL}
\end{equation}
\begin{equation}
C_L=\beta^2 {\partial U_L\over \partial \beta}.
\label{CL}
\end{equation}
When $L\rightarrow\infty$, $Z_L$ can be reduced to the partition
function of the Gross-Witten single-link model, and therefore shares
the same thermodynamic properties. In particular, the free energy
density at $N=\infty$ is piecewise analytic with a third-order
transition at $\beta_c=\casefr{1}{2}$ between the strong-coupling and
weak-coupling domains. Furthermore, the behavior of $C_\infty$ around
$\beta_c$ can be characterized by a specific heat critical exponent
$\alpha=-1$. It is easy to see that the $L=2$ chiral chain is also
equivalent to the Gross-Witten model, but with $\beta$ replaced by
$2\beta$; therefore $\beta_c=\casefr{1}{4}$ and the critical
properties are the same, e.g., $\alpha=-1$.
\subsection{Saddle-point equation for chiral $L$-chains}
\label{exactres}
The strategy used in Refs.\
\cite{Brower-Rossi-Tan-chains,Brower-Rossi-Tan-qcd} to compute the
$N=\infty$ solutions for chiral chains with $L\leq 4$ begins with
group integrations in the partition function (\ref{Zf}), with the help
of the single-link integral, for all $U_i$ except two. This leads to
a representation for $Z_L$ in the form
\begin{equation}
Z_L = \int{\mathrm{d}} U \, {\mathrm{d}} V
\exp \left[N^2 S_{\rm eff}^{(L)}(UV^\dagger)\right]
\label{str1}
\end{equation}
suitable for a large-$N$ steepest-descent analysis. Since the
integral depends only on the combination $UV^\dagger$, changing
variable to $\theta_j$, $\e^{{\mathrm{i}}\theta_j}$ being the eigenvalues of
$UV^\dagger$, leads to
\begin{equation}
Z_L \sim \int\prod_i {\mathrm{d}}\theta_i |\Delta(\theta_1,...,\theta_N)|^2
\exp \left[N^2 S_{\rm eff}^{(L)}(\theta_k)\right]
\label{str2}
\end{equation}
where $-\pi\leq \theta_j \leq \pi$, $\Delta(\theta_1,...,\theta_N) =
\det \Vert\Delta_{jk}\Vert$, $\Delta_{jk}=\e^{{\mathrm{i}} j\theta_k}$. In the
large-$N$ limit, $Z_L$ is determined by its stationary configuration,
and the distribution of $\theta_j$ is specified by a density function
$\rho_L(\theta)$, which is the solution of the equation
\begin{equation}
\princint{\mathrm{d}}\phi\,\rho_L(\phi)
\cot{\theta-\phi\over 2}
+ {\delta\over\delta\theta} S_{\rm eff}^{(L)}(\theta,\rho_L)=0,
\label{eqrho}
\end{equation}
with the normalization condition
\begin{equation}
\int^{\pi}_{-\pi} \rho_L(\theta)\,{\mathrm{d}}\theta = 1.
\label{normco}
\end{equation}
For $L=2$, $Z_2$ is already in the desired form with
\begin{equation}
S_{\rm eff}^{(2)} =
2\beta {1\over N} \mathop{\operator@font Tr}\nolimits \left(U_1U_2^\dagger + U_1^\dagger U_2\right),
\label{l2eq1}
\end{equation}
and the large-$N$ eigenvalue density $\rho_2(\theta)$ of the matrix
$U_1U_2^\dagger$ satisfies the Gross-Witten equation
\begin{equation}
\princint{\mathrm{d}}\phi\,\rho_2(\phi)
\cot{\theta-\phi\over 2}
-4\beta\sin \theta = 0,
\label{l2eq}
\end{equation}
which differs from that of the infinite-chain model only in replacing
$\beta$ by $2\beta$.
\subsection{The large-$N$ limit of the three-link chiral chain}
\label{sol3}
In the $L=3$ chain model, setting $U=U_1$ and $V=U_2$,
$S_{\rm eff}^{(3)}$ is given by
\begin{equation}
\exp\left[ N^2 S_{\rm eff}^{(3)}\right] =
\exp\left[2N\beta \mathop{\operator@font Re}\nolimits \mathop{\operator@font Tr}\nolimits UV^\dagger\right]
\int{\mathrm{d}} U_3 \exp\left[2N\beta \mathop{\operator@font Re}\nolimits \mathop{\operator@font Tr}\nolimits A U_3^\dagger\right],
\label{l3eq}
\end{equation}
where $A=U+V$. Recognizing in the r.h.s.\ of (\ref{l3eq}) a
single-link integral, one can deduce that the large-$N$ limit of the
spectral density $\rho_3(\theta)$ of the matrix $UV^\dagger$ satisfies
the equation
\begin{eqnarray}
2\beta\left(\sin\theta+\sin\half\theta\right)
- \princint{\mathrm{d}}\phi\,\rho_3(\phi)
\left[\cot{\theta-\phi\over 2} + {1\over 2}
{\sin\half\theta\over \cos\half\theta + \cos\half\phi}
\right] = 0, \nonumber \\
\label{l3eq2}
\end{eqnarray}
with the normalization condition $\int\rho_3(\theta)\,{\mathrm{d}}\theta=1$. In
order to find a solution for the above equation, one must distinguish
between strong-coupling and weak-coupling regions.
In the weak-coupling region the solution of Eq.\ (\ref{l3eq2}) is
\begin{equation}
\rho_3(\theta) = {\beta\over \pi} \cos {\theta\over 4}
\left[2\cos {\theta\over 2} + \sqrt{1 - {1\over 3\beta}}\,\right]
\left[2\cos {\theta\over 2} - 2\sqrt{1 -{1\over 3\beta}}\,\right]^{1/2}
\label{wesol3}
\end{equation}
for
\begin{equation}
|\theta| \leq \theta_c = 2\arccos \sqrt{1-{1\over 3\beta}}
\label{thetac}
\end{equation}
and $\rho_3(\theta)=0$ for $\theta_c\leq|\theta|\leq\pi$. This
solution is valid for $\beta\geq\beta_c=\casefr{1}{3}$, indicating
that a critical point exists at $\beta_c=\casefr{1}{3}$. Similarly
one can calculate $\rho_3(\theta)$ in the strong-coupling domain
$\beta\leq\beta_c$
\cite{Brower-Rossi-Tan-chains,Brower-Rossi-Tan-qcd,Friedan} finding:
\begin{eqnarray}
\rho_3(\theta) &=& {\beta\over 2\pi}
\left(y(\theta)+1 - {\sqrt{c} + \sqrt{4+c}\over 2}\right)
\nonumber \\ &\times&
\left[\left(y(\theta)+\sqrt{c}\right)
\left(y(\theta) + \sqrt{4+c}\right)\right]^{1/2},
\label{scsol3}
\end{eqnarray}
where
\begin{equation}
y(\theta) = \sqrt{4\cos^2{\theta\over 2} + c},
\end{equation}
and the parameter $c$ is related to $\beta$ by the equation
\begin{equation}
1 + \sqrt{c} + \casefr{1}{2} c +
\left(1-\casefr{1}{2}\sqrt{c}\right) \sqrt{4 + c}
= {1\over\beta} \, .
\label{constr3}
\end{equation}
At $\beta=\beta_c$, $c=0$ and therefore
\begin{equation}
\rho_3(\theta)_{\rm crit}=
{1\over 3\pi} \left(2 \cos{\theta\over 2}\right)^{3/2}
\cos{\theta\over 4},
\label{rho3cr}
\end{equation}
in agreement with the critical limit of the weak-coupling solution
(\ref{wesol3}).
Since $\rho_3(\pi) > 0$ for $\beta < \beta_c$ and $\rho_3(\pi)=0$ for
$\beta\geq \beta_c$, the critical point $\beta_c$ can be also seen as
the compactification point for the spectral density $\rho_3(\theta)$,
similarly to what is observed in the Gross-Witten model.
\subsection{The large-$N$ limit of the four-link chiral chain}
\label{sol4}
For $L=4$, setting $U=U_1$ and $V=U_3$, $S_{\rm eff}^{(4)}$
is given by
\begin{eqnarray}
\exp\left(N^2 S_{\rm eff}^{(4)}\right) &=&
\int{\mathrm{d}} U_2\exp\left(2N\beta \mathop{\operator@font Re}\nolimits \mathop{\operator@font Tr}\nolimits A U_2^\dagger\right) \nonumber \\
&\times&
\int{\mathrm{d}} U_4\exp\left(2N\beta \mathop{\operator@font Re}\nolimits \mathop{\operator@font Tr}\nolimits A U_4^\dagger\right),
\label{l4eq}
\end{eqnarray}
where again $A=U+V$. The large-$N$ limit of the spectral density
$\rho_4(\theta)$ of the matrix $UV^\dagger$ must be solution of the
equation
\begin{equation}
4\beta\sin\casefr{1}{2}\theta
-\princint{\mathrm{d}}\phi \rho_4(\phi)
\left[\cot{\theta-\phi\over 2} +
{\sin\casefr{1}{2}\theta \over
\cos\casefr{1}{2}\theta + \cos\casefr{1}{2}\phi}
\right] = 0,
\label{l4eq2}
\end{equation}
satisfying the normalization condition $\int \rho_4(\theta){\mathrm{d}}\theta=1$.
In order to solve Eq.\ (\ref{l4eq2}) one must again separate weak- and
strong-coupling domains. In the weak-coupling region the solution is
\begin{equation}
\renewcommand\arraystretch{1.3}
\begin{array}{l@{\qquad}l}
\displaystyle \rho_4(\theta) = {2\beta\over\pi}
\sqrt{\sin^2 {\theta_c\over 2} - \sin^2 {\theta\over 2}} &
{\rm for}\ 0\leq \theta\leq\theta_c\leq\pi, \\
\rho_4(\theta) = 0 &
{\rm for}\ \theta_c\leq\theta\leq\pi,
\end{array}
\label{rho4w}
\end{equation}
with $\theta_c$ implicitly determined by the normalization condition
$\int_{-\theta_c}^{\theta_c}\rho_4(\theta){\mathrm{d}}\theta=1$. The solution
(\ref{rho4w}) is valid for $\beta\geq\beta_c=\casefr{1}{8}\pi$, since the
normalization condition can be satisfied only in this region.
$\casefr{1}{8}\pi$ is then a point of non-analyticity representing the
critical point for the transition from the weak to the strong-coupling
domain.
In the strong-coupling domain $\beta < \beta_c=\casefr{1}{8}\pi$ one
finds
\begin{equation}
\rho_4(\theta)={\beta\over 2}
\sqrt{\lambda - \sin^2 {\theta\over 2}}
\label{rho4s}
\end{equation}
where $\lambda$ is determined by the normalization condition
$\int^\pi_{-\pi}\rho_4(\theta){\mathrm{d}}\theta=1$. The strong- and
weak-coupling expressions of $\rho_4(\theta)$ coincide at $\beta_c$:
\begin{equation}
\rho_4(\theta)_{\rm crit}=
{\beta\over 2} \sqrt{1 - \sin^2 {\theta\over 2}} \, .
\label{rho4cr}
\end{equation}
Notice that again the critical point $\beta_c=\casefr{1}{8}\pi$
represents the compactification point of the spectral density
$\rho_4(\theta)$; indeed $\rho_4(\pi)> 0$ for $\beta< \beta_c$, and
$\rho_4(\pi)=0$ for $\beta\geq \beta_c$.
\subsection{Critical properties of chiral chain models with $L\leq 4$}
\label{crit}
In the following we derive
the $N=\infty$ critical behavior of the specific heat in the
models with $L=3,4$, using the exact results of
Subs.\ \ref{sol3} and \ref{sol4}.
From the spectral density $\rho_3(\theta)$, the internal energy can be
easily derived by $U_3=\int{\mathrm{d}}\theta\,\rho_3(\theta) \cos \theta$. One
finds that $U_3$ is continuous at $\beta_c$. In the weak-coupling
region $\beta\geq \beta_c=\casefr{1}{3}$,
\begin{eqnarray}
U_3&=& \beta + {1\over 2} - {1\over 8\beta} -
\beta\left(1 - {1\over 3\beta}\right)^{\!\!3/2},\nonumber \\
C_3&=& \beta^2 + {1\over 8} - \beta^2\left(1 + {1\over 6\beta}\right)
\sqrt{1 - {1\over 3\beta}} \, .
\label{ec3w}
\end{eqnarray}
Close to criticality, i.e., for $0\leq \beta/\beta_c-1 \ll 1$,
\begin{equation}
C_3={17\over 72}
- {1\over 2\sqrt{3}}\left(\beta-\beta_c\right)^{\!\!1/2}
+ O(\beta-\beta_c).
\label{c3cr}
\end{equation}
In the strong-coupling region, one finds
\begin{equation}
C_3={17\over 72}
- {1\over 2\sqrt{3}}\left(\beta_c-\beta\right)^{\!\!1/2}
+ O(\beta_c-\beta).
\label{c3crs}
\end{equation}
for $0\leq 1-\beta/\beta_c\ll 1$. Then the weak- and strong-coupling
expressions of $C_3$ show that the critical point
$\beta_c=\casefr{1}{3}$ is of the third order, and the critical
exponent associated with the specific heat is $\alpha=-\casefr{1}{2}$.
In the $L=4$ case, recalling that $\rho_4(\theta)$ is the spectral
distribution of $U_1 U_3^\dagger$, one writes
\begin{eqnarray}
F_4 &=& {1\over4}\left[
8\beta \int{\mathrm{d}}\theta\,\rho_4(\theta) \cos{\theta\over 2}
- \int{\mathrm{d}}\theta\,{\mathrm{d}}\phi\,\rho_4(\theta)\,\rho_4(\phi)
\log \left(\cos {\theta\over 2} +\cos {\phi\over 2}\right)\right.
\nonumber \\ &&\quad - \,
\left. {3\over 2} - \log 2\beta
+\princint{\mathrm{d}}\theta\,{\mathrm{d}}\phi\,\rho_4(\theta)\,\rho_4(\phi)
\log \sin^2 {\theta-\phi\over 2}\right] \,.
\label{freen}
\end{eqnarray}
Observing that, since $\rho_4(\theta)$ is a solution of the
variational equation $\delta F_4 / \delta \rho_4 = 0$, the following
relation holds
\begin{equation}
{{\mathrm{d}} F_4\over{\mathrm{d}}\beta} = {\partial F_4\over \partial \beta},
\label{dergfreen}
\end{equation}
one can easily find that
\begin{equation}
U_4 = - {1\over 8\beta} +
\int{\mathrm{d}}\theta\,\rho_4(\theta)\cos{\theta\over 2} \, .
\label{en}
\end{equation}
In this case, the study of the critical behavior around
$\beta_c=\casefr{1}{8}\pi$ is slightly subtler, since it requires the
expansion of elliptic integrals $F(k)$ and $E(k)$ around $k=1$.
Approaching criticality from the weak-coupling region, i.e., when
$\beta\rightarrow\beta_c^+$, one obtains
\begin{equation}
C_4={\pi^2\over 32}+{1\over 8} -{\pi^2\over 16\log (4/\delta_w)}
+ O(\delta^2_w) ,
\label{C4w}
\end{equation}
where $\delta^2_w\sim \beta-\beta_c$, apart from logarithms.
For $\beta\rightarrow\beta_c^-$
\begin{equation}
C_4={\pi^2\over 32}+{1\over 8} -{\pi^2\over 16\log (4/\delta_s)}
+ O(\delta_s^2) ,
\label{C4s}
\end{equation}
where $\delta_s^2\sim \beta_c-\beta$, apart from logarithms. A
comparison of Eqs.\ (\ref{C4w}) and (\ref{C4s}) leads to the conclusion
that the phase transition is again of the third order, with a specific
heat critical exponent $\alpha=0^-$.
In conclusion we have seen that chain models with $L=2,3,4,\infty$
have a third-order phase transition at increasing values of the
critical coupling, $\beta_c={1\over4}$, ${1\over3}$,
$\casefr{1}{8}\pi$, ${1\over2}$ respectively, with specific heat
critical exponents $\alpha=-1$, $-\casefr{1}{2}$, $0^-$, $-1$
respectively. It is worth noticing that $\alpha$ increases when $L$
goes from 2 to 4, reaching the limit of a third order critical
behavior, but in the large-$L$ limit it returns to $\alpha=-1$.
The critical exponent $\nu$, describing the double-scaling behavior
for $N\rightarrow\infty$ and $\beta\rightarrow\beta_c$, can then be
determined by the two-dimensional hyperscaling relationship
$2\nu=2-\alpha$. This relation has been proved to hold for the
Gross-Witten problem, and therefore for the $L=2$ and $L=\infty$ chain
models, where it is related to the equivalence of the corresponding
double scaling limit with the continuum limit of a two-dimensional
gravity model with central charge $c=-2$. It is then expected to hold
in general for all values of $L$. At $L=4$, the value $\nu=1$ has
been numerically verified, within a few per cent of uncertainty, by
studying the scaling of the specific heat peak position at finite $N$.
Notice that the exponents $\alpha=0^-$, $\nu=1$ found for $L=4$
correspond to a central charge $c=1$.
\subsection{Strong-coupling expansion of chiral chain models}
\label{SC}
Strong-coupling series of the free energy density of chiral chain
models can be generated by means of the character expansion, which
leads to the result
\begin{equation}
F_L(\beta) = F(\beta) + \widetilde{F}_L(\beta),
\label{be1}
\end{equation}
where $F(\beta)$ is the free energy of the single unitary-matrix model,
\begin{equation}
\widetilde{F}_L = {1\over LN^2}\log \sum_{(r)}d_{(r)}^2 z_{(r)}^L,
\label{be1b}
\end{equation}
$\sum_{(r)}$ denotes the sum over all irreducible representations of
${\rm U}(N)$, and $d_{(r)}$ and $z_{(r)}(\beta)$ are the corresponding
dimensions and character coefficients. The calculation of the
strong-coupling series of $F_L(\beta)$ is considerably simplified in
the large-$N$ limit, due to the relationships (\ref{F-beta}) and
\begin{equation}
z_{(r)}(\beta) =
\bar{z}_{(r)} \beta^n + O\left(\beta^{2N}\right),
\label{be4}
\end{equation}
where $\bar{z}_{(r)}$ is independent of $\beta$ and $n$ is the order
of the representation $(r)$. Explicit expressions for $d_{(r)}$ and
$\bar{z}_{(r)}$ were reported in Subs.\ \ref{character-expansion}.
The large-$N$ strong-coupling expansion of $\widetilde{F}_L(\beta)$ is
actually a series in $\beta^L$, i.e.,
\begin{equation}
\widetilde{F}_L = \sum_n c(n,L)\beta^{nL}.
\label{be5}
\end{equation}
It is important to recall that the large-$N$ character coefficients
have jumps and singularities at $\beta={1\over 2}$
\cite{Green-Samuel-chiral}, and therefore the relevant region for a
strong-coupling character expansion is $\beta<{1\over 2}$.
Another interesting aspect of the large-$N$ limit of chain models,
studied by Green and Samuel using the strong-coupling character
expansion \cite{Green-Samuel-un2}, concerns the determinant channel,
which should provide an order parameter for the phase transition. The
quantity
\begin{equation}
\sigma = -{1\over N} \log \langle \det U_i U_{i+1}^\dagger \rangle
\label{detch}
\end{equation}
is non-zero in the strong-coupling domain and zero in weak coupling at
$N=\infty$. $\beta_c$ may then be evaluated by determining where the
strong-coupling evaluation of the order parameter $\sigma$ vanishes.
Like the free-energy, $\sigma$ is calculable via a character
expansion. Indeed
\begin{equation}
\langle \det U_i U_{i+1}^\dagger \rangle =
{ \sum_{(r)} d_{(r)} z_{(r)}^{L-1} d_{(r,-1)} z_{(r,-1)} \over
\sum_{(r)} d_{(r)}^2 z_{(r)}^L}
\label{scdet}
\end{equation}
Green and Samuel evaluated a few orders of the above character
expansion, obtaining estimates of $\beta_c$ from the vanishing point
of $\sigma$. Such estimates compare well with the exact results for
$L=3,4$. In the cases where $\beta_c$ is unknown, they found
$\beta_c\simeq 0.44$ for $L=5$, $\beta_c\simeq 0.47$ for $L=6$, etc.,
with $\beta_c$ monotonically approaching the value $\casefr{1}{2}$
with increasing $L$.
In order to study the critical behavior of chain models for $L\geq 5$,
one can also analyze the corresponding strong-coupling series of the
free energy (\ref{be1})
\cite{Brower-Campostrini-Orginos-Rossi-Tan-Vicari}. An integral
approximant analysis of the strong-coupling series of the specific
heat led to the estimates $\beta_c\simeq 0.438$ for $L=5$ and
$\beta_c\simeq 0.474$ for $L=6$, with small negative $\alpha$, which
could mimic an exponent $\alpha=0^-$. For $L\geq 7$ a such
strong-coupling analysis would lead to $\beta_c$ larger than
$\casefr{1}{2}$, that is out of the region where a strong-coupling
analysis can be predictive. Therefore something else must occur
earlier, breaking the validity of the strong-coupling expansion. An
example of this phenomenon is found in the Gross-Witten single-link
model (recovered when $L\rightarrow\infty$), where the strong-coupling
expansion of the $N=\infty$ free energy is just $F(\beta) = \beta^2$,
an analytical function without any singularity; therefore, in this
model, $\beta_c={1\over 2}$ cannot be determined from a
strong-coupling analysis of the free energy.
From such analysis one may hint at the following possible scenario: as
for $L\leq4$, for $L=5,6$, that is when the estimate of $\beta_c$
coming from the above strong-coupling analysis is smaller than
${1\over 2}$ and therefore acceptable. The term
$\widetilde{F}(\beta)$ in Eq.\ (\ref{be1}) should be the one relevant
for the critical properties, determining the critical points and
giving $\alpha\neq -1$ (maybe $\alpha=0^-$ as in the $L=4$ case). For
$L\geq 7$ the critical point need not be a singular point of the free
energy in strong or weak coupling, but just the point where
weak-coupling and strong-coupling curves meet each other. This would
cause a softer phase transition with $\alpha=-1$, as for the
Gross-Witten single-link problem. We expect $\beta_c<{1\over 2}$ also
for $L\geq 7$. This scenario is consistent with the results of the
analysis of the character expansion of $\sigma$, defined in Eq.\
(\ref{detch}).
\section{Simplicial chiral models}
\label{simplicial-chiral}
\subsection{Definition of the models}
\label{simplicial-def}
Another interesting class of finite-lattice chiral models is obtained
by considering the possibility that each of a finite number of unitary
matrices may interact in a fully symmetric way with all other
matrices, while preserving global chiral invariance; the resulting
systems can be described as chiral models on $(d-1)$-dimensional
simplexes, and thus termed ``simplicial chiral models''
\cite{Brower-Campostrini-Orginos-Rossi-Tan-Vicari,Rossi-Tan}.
The partition function for such a system is:
\begin{equation}
Z_d = \int\prod_{i=1}^d {\mathrm{d}} U_i \,
\exp\Biggl[N\beta\sum_{i=1}^d\sum_{j=i+1}^d
\mathop{\operator@font Tr}\nolimits\left(U_i U_j^\dagger + U_j U_i^\dagger\right)\Biggr].
\label{Z-simplicial}
\end{equation}
Eq.~(\ref{Z-simplicial}) encompasses as special cases a number of
models that we have already introduced and solved; in particular, the
chiral chains with $L\le3$ correspond to the simplicial chiral models
with $d\le3$.
One of the most attractive features of these models is their
relationship with higher-dimensional systems, with which they share
the possibility of high coordination numbers. This relationship
becomes exact in the large-$d$ limit, where mean-field results are
exact.
In the large-$N$ limit and for arbitrary $d$ a saddle-point equation
can be derived, whose solution allows the evaluation of the large-$N$
free energy
\begin{equation}
F_d = {1\over N^2}\,\log Z_d
\end{equation}
and of related thermodynamical quantities.
\subsection{Saddle-point equation for simplicial chiral models}
\label{simplicial-saddle}
The strategy for the determination of the large-$N$ saddle-point
equation is based on the introduction of a single auxiliary variable
$A$ (a complex matrix), allowing for the decoupling of the unitary
matrix interaction:
\begin{equation}
Z_d = {\widetilde Z_d\over \widetilde Z_0} \,,
\end{equation}
where
\begin{eqnarray}
\widetilde Z_d = \int\prod_{i=1}^d {\mathrm{d}} U_i\,{\mathrm{d}} A \,
\exp\Biggl[- N\beta && \mathop{\operator@font Tr}\nolimits AA^\dagger + N\beta\mathop{\operator@font Tr}\nolimits A\sum_i U_i^\dagger
\nonumber \\ + N\beta && \mathop{\operator@font Tr}\nolimits A^\dagger\sum_i U_i - N^2\beta d\Biggr].
\end{eqnarray}
We are now back to the single-link problem and, since we have solved
it in Sect.\ \ref{single-link} in terms of the function $W$, whose
large-$N$ limit is expressed by Eq.~(\ref{SD-W}), we obtain
\begin{equation}
\widetilde Z_d = \int {\mathrm{d}} A \,
\exp\left[-N\beta \mathop{\operator@font Tr}\nolimits A A^\dagger + N d W(\beta^2 A A^\dagger)
- N^2 \beta d\right].
\end{equation}
It is now convenient to express the result in terms of the eigenvalues
$x_i$ of the Hermitian semipositive-definite matrix $4 \beta A
A^\dagger$, obtaining
\begin{equation}
\widetilde Z_d = \int {\mathrm{d}}\mu(x_i) \,
\exp\Biggl[-{N\over4\beta} \sum_i x_i + N d W\left(x_i\over4\right)
- N^2 \beta d\Biggr].
\end{equation}
The angular integration can be performed, leading to
\begin{equation}
{\mathrm{d}}\mu(x_i) = \prod_i {\mathrm{d}} x_i \prod_{i>j} (x_i-x_j)^2.
\end{equation}
The saddle-point equation is therefore
\begin{equation}
{\sqrt{r+x_i}\over2\beta} - d =
{1\over N}\sum_{i\ne j}
{(4-d)\sqrt{r+x_i} + d \sqrt{r+x_j} \over x_i-x_j} \, ,
\label{simpl-SP}
\end{equation}
subject to the constraint (needed to define $r$)
\begin{equation}
\left\{
\renewcommand\arraystretch{1.3}
\begin{array}{l@{\quad}l}
\displaystyle {1\over N}\sum_i {1\over\sqrt{r+x_i}} = 1 &
\hbox{(strong coupling)}; \\
r = 0 & \hbox{(weak coupling)}.
\end{array}
\right.
\end{equation}
The energy
\begin{equation}
U_d = {1\over2}\,{\partial F_d\over\partial\beta}
\end{equation}
is easily expressed in terms of the eigenvalues:
\begin{equation}
d(d-1)U_d = {1\over4\beta^2} \sum_i x_i - d - {1\over\beta} \,.
\label{simpl-Ud}
\end{equation}
In the large-$N$ limit, after a change of variables to
$z_i=\sqrt{r+x_i}$, we introduce as usual an eigenvalue density
function $\rho(z)$, and turn Eq.~(\ref{simpl-SP}) into the integral
equation
\begin{equation}
{z\over2\beta} - d = \princint_a^b {\mathrm{d}} z' \, \rho(z')
\left[{2\over z-z'} - {d-2\over z+z'}\right],
\label{simpl-IE}
\end{equation}
subject to the constraints
\begin{equation}
\int_a^b \rho(z') \,{\mathrm{d}} z' = 1
\end{equation}
and
\begin{equation}
\int_a^b \rho(z') \, {{\mathrm{d}} z'\over z'} \le 1,
\end{equation}
with equality holding in strong coupling, where $a=\sqrt{r}$.
The easiest way of evaluating the free energy $F_d$ is the integration
of the large-$N$ version of Eq.\ (\ref{simpl-Ud}) with respect to
$\beta$.
Very simple solutions are obtained for a few special values of $d$.
When $d=0$, the problem reduces to a Gaussian integration, and one
easily finds that Eq.\ (\ref{simpl-IE}) is solved by
\begin{equation}
\rho(z) = {z\over4\pi\beta}\,
{\sqrt{16\beta - (z^2-a^2)}\over\sqrt{z^2-a^2}}
\end{equation}
and $\widetilde Z_0 = \exp(N^2\log\beta)$, independent of $a$ as
expected.
When $d=2$ we obtain
\begin{eqnarray}
\rho_w(z) &=& {1\over4\pi\beta}\sqrt{8\beta-(z-4\beta)^2}, \qquad
\beta\ge\casefr{1}{2}, \\
\rho_s(z) &=& {1\over4\pi\beta} \, z\sqrt{1+6\beta-z\over z-(1-2\beta)} \,,
\qquad r(\beta) = (1-2\beta)^2,
\qquad \beta\le\casefr{1}{2},
\end{eqnarray}
and these results are consistent with the reinterpretation of the
model as a Gross-Witten one-plaquette system. Notice however that the
matrix whose eigenvalue distribution has been evaluated is not the
original unitary matrix, and corresponds to a different choice of
physical degrees of freedom. This is the reason why, while knowing
the solution for the free energy of the $d=1$ system (trivial,
non-interacting) and of the $d=3$ system (three-link chiral chain), we
cannot find easily explicit analytic forms for the corresponding
eigenvalue densities.
The saddle-point equation (\ref{simpl-IE}) has been the subject of
much study in recent times, because it is related to many different
physical problems in the context of double-scaling limit
investigations. In particular, in the range of values $0\le d\le4$,
the same equation describes the behavior of ${\rm O}(n)$ spin models
on random surfaces in the range $-2\le n\le2$, with the very simple
mapping $n=d-2$ \cite{Gaudin-Kostov}. In this range, the equation has
been solved analytically in Refs.\ \cite{Eynard-Kristjansen-1} and
especially \cite{Eynard-Kristjansen-2} in terms of $\theta$-functions.
\subsection{The large-$N$ $d=4$ simplicial chiral model}
\label{simpl-d=4}
The chiral model on a tetrahedron is the first example within the
family of simplicial chiral models which turns out to be really
different from all the systems discussed in the previous sections.
Explicit solutions were found for both the weak and the strong
coupling phases, and they are best expressed in terms of a rescaled
variable
\begin{equation}
\zeta = \sqrt{1 - {z^2\over b^2}}
\end{equation}
and of a dynamically determined parameter
\begin{equation}
k = \sqrt{1 - {a^2\over b^2}} \, .
\end{equation}
The resulting expressions, after defining
$\beta\bar{\rho}(\zeta)\,{\mathrm{d}}\zeta \equiv \rho(z)\,{\mathrm{d}} z$, are
\begin{equation}
\bar{\rho}_w(\zeta) = {8\over E(k)^2}
\left[{\sqrt{k^2-\zeta^2}\over\sqrt{1-\zeta^2}}\,K(k) -
\sqrt{k^2-\zeta^2}\sqrt{1-\zeta^2}\,\Pi(\zeta^2,k)\right]
\end{equation}
and
\begin{eqnarray}
\bar{\rho}_s(\zeta) &=& {8\over[E(k)-(1-k^2)K(k)]^2} \nonumber \\
&\times&
\left[k^2\,{\sqrt{1-\zeta^2}\over\sqrt{k^2-\zeta^2}}\,K(k) -
\sqrt{k^2-\zeta^2}\sqrt{1-\zeta^2}\,\Pi(\zeta^2,k)\right],
\end{eqnarray}
where $K$, $E$ and $\Pi$ are the standard elliptic integrals, and
$0\le\zeta\le k$.
The complete solution is obtained by enforcing the normalization
condition, which leads to a relationship between $\beta$ and $k$, best
expressed by the equation
\begin{equation}
{1\over\beta} = \int_0^k {\mathrm{d}}\xi\,\bar{\rho}(\zeta,k).
\end{equation}
Criticality corresponds to the limit $k\to1$, and it is easy to
recognize that both weak and strong coupling results lead in this
limit to $\beta_c=\casefr{1}{4}$ and
\begin{equation}
\beta\bar{\rho}_c(\zeta) = \zeta\log{1+\zeta\over1-\zeta}\,.
\end{equation}
Many interesting features of this model in the region around
criticality can be studied analytically, and one may recognize that
the critical behavior around $\beta_c=\casefr{1}{4}$ corresponds to a
limiting case of a third-order phase transition with critical exponent
of the specific heat $\alpha=0^-$. In the double-scaling limit
language this would correspond to a model with central charge $c=1$
and logarithmic deviations from scaling. The critical behavior of the
specific heat on both sides of criticality is described by
\begin{equation}
C \equiv \beta^2\,{\partial U\over\partial\beta}
\mathop{\;\longrightarrow\;}_{k'\to0}
{\pi^2 + 3\over36} - {\pi^2\over12\log(4/k')}
+ O\left(1\over\log^2k'\right),
\end{equation}
where $k'\equiv\sqrt{1-k^2}$.
\subsection{The large-$d$ limit}
\label{simpl-large-d}
By introducing a function defined by
\begin{equation}
f(z) = \int_a^b {\rho(z')\over z-z'}\,{\mathrm{d}} z', \qquad
f(z) \mathop{\;\longrightarrow\;}_{|z|\to\infty} {1\over z}\,,
\label{simpl-f}
\end{equation}
analytic in the complex $z$ plane with the exception of a cut on the
positive real axis in the interval $[a,b]$, we can turn the
saddle-point equation (\ref{simpl-IE}) into the functional equation
\begin{equation}
{z\over2\beta} - d = 2 \mathop{\operator@font Re}\nolimits f(z) + (d-2) f(-z).
\label{simpl-FE}
\end{equation}
This equation can be the starting point of a systematic $1/d$
expansion, on whose details we shall not belabor, especially because
its convergence for small values of $d$ is very slow.
It is however interesting to solve the large-$d$ limit of
Eq.\ (\ref{simpl-FE}) by the Ansatz
\begin{equation}
\rho(z) = \delta(z-\bar z),
\end{equation}
whose substitution into Eq.\ (\ref{simpl-f}) leads to the solution
\begin{equation}
\renewcommand\arraystretch{1.3}
\begin{array}{l@{\qquad}l}
\displaystyle \bar z = \beta d
\left(1 + \sqrt{1 - {1\over\beta d}}\right), &
\beta d\ge1, \\
\bar z = 1, &
\beta d\le1.
\end{array}
\end{equation}
The large-$d$ limit predicts the location of the critical point
$\beta_c = 1/d$, and shows complete equivalence with the mean-field
solution of infinite-volume principal chiral models on a
$d/2$-dimensional hypercubic lattice. The large-$d$ prediction for
the nature of criticality is that of a first-order phase transition,
with
\begin{equation}
U = {1\over2} + {1\over2}\sqrt{1 - {1\over\beta d}} -
{1\over4\beta d} \,, \qquad \beta d \ge 1.
\end{equation}
\subsection{The large-$N$ criticality of simplicial models}
\label{simpl-crit}
The connection with the double-scaling limit problem naturally leads
to the study of the finite-$\beta$ critical behavior. In the regime
$0\le d\le4$ one is helped by the equivalence with the solved problem
of ${\rm O}(n)$ spin models on a random surface, which allows not only
a determination of the critical value (found to satisfy the
relationship $\beta_c d=1$), but also an evaluation of the eigenvalue
distribution at criticality \cite{Gaudin-Kostov}:
\begin{equation}
\rho_c(z) = {2\over\pi\theta}\,\cos{\pi\theta\over2}\,
{\sinh\theta u\over\cosh u}\,,
\end{equation}
and
\begin{equation}
a_c = 0, \qquad b_c = {2\over\theta}\,\tan{\pi\theta\over2} \,,
\end{equation}
where $\theta$ and $u$ are defined by the parametrizations
\begin{equation}
4\cos^2{\pi\theta\over2} \equiv d = {1\over\beta_c} \,, \qquad
\cosh u \equiv {b_c\over z} \,.
\end{equation}
Unfortunately, the technique that was adopted in order to find the
above solution does not apply to the regime $d>4$, in which case one
cannot choose $a_c=0$. The saddle-point equation at criticality can
however be solved numerically with very high accuracy, and one finds
that the relationship
\begin{equation}
\beta_c d = 1
\end{equation}
is satisfied for all $d$, thus also matching the large-$d$
predictions. The combinations $(a_c+b_c)/2$ and $a_cb_c$ admit a
$1/d$ expansion, and the coefficients of the expansion are found
numerically to be integer numbers up to order $d^{-8}$.
An analysis of criticality for $d>4$ shows that its description is
fully consistent with the existence of a first-order phase transition,
with a discontinuity of the internal energy measured by
$d a_c^2/(4(d-1))$, again matching with the large-$d$ (mean-field)
predictions.
\subsection{The strong-coupling expansion of simplicial models}
\label{simpl-sc}
There is nothing peculiar in performing the strong-coupling expansion
of Eq.\ (\ref{Z-simplicial}). There is however a substantial
difference with respect to the case of chiral chains discussed in the
previous section: because of the topology of simplexes, the
strong-coupling configurations entering the calculation are no longer
restricted to simple graphs whose vertices are joined by at most one
link, and the full complexity of group integration on arbitrary graphs
is now involved \cite{Campostrini-Rossi-Vicari-chiral-1}.
As a consequence, as far as the simplicial models can be solved by
different techniques, they may also be used as generating functionals
for these more involved group integrals, that enter in a essential way
in all strong-coupling calculations in higher-dimensional standard
chiral models and lattice gauge theories.
\section{Asymptotically free matrix models}
\label{principal-chiral}
\subsection{Two-dimensional principal chiral models}
\label{sec6intr}
Two dimensional ${\rm SU}(N)\times {\rm SU}(N)$ principal chiral
models, defined by the action
\begin{equation}
S={1\over T} \int{\mathrm{d}}^2x \mathop{\operator@font Tr}\nolimits\partial_\mu U(x) \, \partial_\mu
U^\dagger(x),
\label{caction}
\end{equation}
are the simplest asymptotically free field theories whose large-$N$
limit is a sum over planar diagrams, like four dimensional ${\rm
SU}(N)$ gauge theories.
Using the existence of an infinite number of conservation laws and
Bethe-Ansatz methods, the on-shell solution of the ${\rm SU}(N)\times
{\rm SU}(N)$ chiral models has been proposed in terms of a factorized
$S$-matrix \cite{Abdalla-Abdalla-LimaSantos,Wiegmann}. The analysis
of the corresponding bound states leads to the mass spectrum
\begin{equation}
M_r=M{\sin(r\pi/N) \over \sin(\pi/N)},
\qquad 1\leq r\leq N-1,
\label{masses}
\end{equation}
where $M_r$ is the mass of the $r$-particle bound state transforming
as totally antisymmetric tensors of rank $r$. $M\equiv M_1$ is the
mass of the fundamental state determining the Euclidean long-distance
exponential behavior of the two-point Green's function
\begin{equation}
G(x)= {1\over N}\langle {\rm Tr} \,U(0) U(x)^\dagger \rangle.
\label{fgf}
\end{equation}
The mass-spectrum (\ref{masses}) has been verified numerically at
$N=6$ by Monte Carlo simulations
\cite{Rossi-Vicari-chiral1,Drummond-Horgan}: Monte Carlo data of the
mass ratios $M_2/M$ and $M_3/M$ agree with formula (\ref{masses})
within statistical errors of about one per cent. Concerning the
large-$N$ limit of these models, it is important to notice that the
$S$-matrix has a convergent expansion in powers of $1/N$, and becomes
trivial, i.e., the $S$-matrix of free particles, in the large-$N$
limit.
By using Bethe-Ansatz techniques, the mass/$\Lambda$-parameter
ratio has also been computed, and the result is
\cite{Balog-Naik-Niedermayer-Weisz}
\begin{equation}
{M\over\Lambda_{\overline {MS}}}=\sqrt{{8\pi\over\e}}
\, {\sin (\pi/N)\over \pi/N},
\label{mass-lambda}
\end{equation}
which again enjoys a $1/N$ expansion with a finite radius of
convergence. This exact but non-rigorous result has been substantially
confirmed by Monte Carlo simulations at several values of $N$
\cite{Rossi-Vicari-chiral2,Manna-Guttmann-Hughes}, and its large-$N$
limit also by $N=\infty$ strong-coupling
calculations \cite{Campostrini-Rossi-Vicari-chiral-brief,%
Campostrini-Rossi-Vicari-chiral-2}.
While the on-shell physics of principal chiral models has been
substantially solved, exact results of the off-shell physics are still
missing, even in the large-$N$ limit. When $N\rightarrow\infty$,
principal chiral models should just reproduce a free-field theory in
disguise. In other words, a local nonlinear mapping should exist
between the Lagrangian fields $U$ and some Gaussian variables
\cite{Polyakov-book}. However, the behavior of the two-point Green's
function $G(x)$ of the Lagrangian field shows that such realization of
a free-field theory is nontrivial. While at small Euclidean momenta,
and therefore at large distance, there is a substantial numerical
evidence for an essentially Gaussian behavior of $G(x)$
\cite{Rossi-Vicari-chiral2}, at short distance renormalization group
considerations lead to the asymptotic behavior
\begin{equation}
G(x) \sim
\left[\log\left({1\over x \Lambda}\right)\right]^{\gamma_1/b_0} ,
\label{n14}
\end{equation}
where $\Lambda$ is a mass scale, and
\begin{equation}
{\gamma_1 \over b_0} = 2\left(1 -{2\over N^2}\right)
\mathop{\;\longrightarrow\;}_{N\rightarrow\infty} 2.
\end{equation}
$b_0$ and $\gamma_1$ are the first coefficients respectively of the
$\beta$-function and of the anomalous dimension of the fundamental
field. We recall that a free Gaussian Green's function behaves like
$\log\left(1/x \right)$. Then at small distance $G(x)$ seems to
describe the propagation of a composite object formed by two
elementary Gaussian excitations, suggesting an interesting
hadronization picture: in the large-$N$ limit, the Lagrangian fields
$U$, playing the r\^ole of non-interacting hadrons, are constituted by
two confined particles, which appear free in the large momentum limit,
due to asymptotic freedom.
Numerical investigations by Monte Carlo simulations of lattice chiral
models in the continuum limit show that the large-$N$ limit is rapidly
approached, which confirms that the $1/N$ expansion, were it
available, would be an effective predictive tool in the analysis of
these models.
\subsection{Principal chiral models on the lattice}
\label{sec6s2}
In the persistent absence of an explicit solution, the large-$N$ limit
of two-dimensional chiral models has been investigated by applying
analytical and numerical methods of lattice field theory, such as
strong-coupling expansion and Monte Carlo simulations. In the
following we describe the main results achieved by these studies.
A standard lattice version of the continuum action (\ref{caction}) is
obtained by introducing a nearest-neighbor interaction, according to
Eq.\ (\ref{action-spin}):
\begin{equation}
S_L=-2N\beta\sum_{x,\mu}
{\rm Re}{\rm Tr}\left[ U_x U^\dagger_{x+\mu}\right],
\qquad \beta={1\over NT}\,.
\label{laction}
\end{equation}
${\rm SU}(N)$ and ${\rm U}(N)$ lattice chiral models, obtained by
constraining respectively $U_x\in {\rm SU}(N)$ and $U_x\in {\rm
U}(N)$, are expected to have the same large-$N$ limit at fixed
$\beta$. In the continuum limit $\beta\rightarrow\infty$, ${\rm
SU}(N)$ and ${\rm U}(N)$ lattice actions should describe the same
theory even at finite $N$, since the additional ${\rm U}(1)$ degrees
of freedom of ${\rm U}(N)$ models should decouple. In other words,
the ${\rm U}(N)$ lattice theory represents a regularization of the
${\rm SU}(N)\times {\rm SU}(N)$ chiral field theory when restricting
ourselves to its ${\rm SU}(N)$ degrees of freedom, i.e. when
considering Green's functions of the field
\begin{equation}
\hat{U}_x = {U_x\over(\det U_x)^{1/N}} \, ,
\label{Uhat}
\end{equation}
e.g.,
\begin{equation}
G(x)\equiv {1\over N}
\langle {\rm Tr} \hat{U}_0 \hat{U}_x^\dagger\rangle,
\end{equation}
whose large-distance behavior allows to define the fundamental mass
$M$.
At finite $N$, while ${\rm SU}(N)$ lattice models should not have any
singularity at finite $\beta$, ${\rm U}(N)$ lattice models should
undergo a phase transition, driven by the ${\rm U}(1)$ degrees of
freedom corresponding to the determinant of $U(x)$. The determinant
two-point function
\begin{equation}
G_d(x)\equiv \langle \det [ U^\dagger (x) U(0) ]\rangle^{1/N}
\end{equation}
behaves like $x^{-f(\beta,N)}$ at large $x$ in the weak-coupling
region, with $f(\beta,N)\sim O(1/N)$, but drops off exponentially in
strong-coupling region, where $G_d(x)\sim\e^{-m_d x}$ with
\cite{Green-Samuel-un2}
\begin{equation}
m_d=-\log\beta + {1\over N}\log{N!\over N^N} + O(\beta^2).
\label{scmd}
\end{equation}
This would indicate the existence of a phase transition at a finite
$\beta_d$ in ${\rm U}(N)$ lattice models. Such a transition, being
driven by ${\rm U}(1)$ degrees of freedom, should be of the
Kosterlitz-Thouless type: the mass propagating in the determinant
channel $m_d$ should vanish at the critical point $\beta_d$ and stay
zero for larger $\beta$. Hence for $\beta > \beta_d$ this ${\rm
U}(1)$ sector of the theory would decouple from the ${\rm SU}(N)$
degrees of freedom, which alone determine the continuum limit
($\beta\to\infty$) of principal chiral models.
The large-$N$ limit of principal chiral models has been investigated
by Monte Carlo simulations of ${\rm SU}(N)$ and ${\rm U}(N)$ models
for several large values of $N$, studying their approach to the
$N=\infty$
limit \cite{Rossi-Vicari-chiral2,Campostrini-Rossi-Vicari-chiral-3}.
Many large-$N$ strong-coupling calculations have been performed which
allow a direct study of the $N=\infty$ limit. Within the
nearest-neighbor formulation (\ref{laction}), the large-$N$
strong-coupling expansion of the free energy has been calculated up to
18th order, and that of the fundamental Green's function $G(x)$
(defined in Eq.\ (\ref{fgf})) up to 15th order
\cite{Green-Samuel-un2,Campostrini-Rossi-Vicari-chiral-1}. Large-$N$
strong-coupling calculations have been performed also on the honeycomb
lattice, within the corresponding nearest-neighbor formulation, which
is expected to belong to the same class of universality with respect
to the critical point $\beta=\infty$. On the honeycomb lattice the
free energy has been computed up to $O\left(\beta^{26}\right)$, and
$G(x)$ up to $O\left(\beta^{20}\right)$
\cite{Campostrini-Rossi-Vicari-chiral-1}.
Monte Carlo simulations show that ${\rm SU}(N)$ and ${\rm U}(N)$
lattice chiral models have a peak in the specific heat
\begin{equation}
C= {1\over N} {{\mathrm{d}} E\over{\mathrm{d}} T}
\end{equation}
which becomes sharper and sharper with increasing $N$, suggesting the
presence of a critical phenomenon for $N=\infty$ at a finite
$\beta_c$. In ${\rm U}(N)$ models the peak of $C$ is observed in the
region where the determinant degrees of freedom are massive, i.e., for
$\beta < \beta_d$ (this feature characterizes also
two-dimensional ${\rm XY}$ lattice models \cite{Gupta-Baillie}). An
estimate of the critical coupling $\beta_c$ has been obtained by
extrapolating the position $\beta_{\rm peak}(N)$ of the peak of the
specific heat (at infinite volume) to $N\rightarrow\infty$ using a
finite-$N$ scaling Ansatz \cite{Campostrini-Rossi-Vicari-chiral-3}
\begin{equation}
\beta_{\rm peak}(N) \simeq \beta_c+ cN^{-\epsilon},
\label{FNS}
\end{equation}
mimicking a finite-size scaling relationship. The above Ansatz arises
from the idea that the parameter $N$ may play a r\^ole quite analogous
to the volume in the ordinary systems close to the criticality. This
idea was already exploited in the study of one-matrix models
\cite{Damgaard-Heller,Carlson,Brezin-ZinnJustin-RG}, where the double
scaling limit turns out to be very similar to finite-size scaling in a
two-dimensional critical phenomenon. The finite-$N$ scaling Ansatz
(\ref{FNS}) has been verified in the similar context of the large-$N$
Gross-Witten phase transition, as mentioned in Subs.\
\ref{double-scaling-single-link}. Since $\epsilon$ is supposed to be
a critical exponent associated with the $N=\infty$ phase transition,
it should be the same in the ${\rm U}(N)$ and ${\rm SU}(N)$ models.
The available ${\rm U}(N)$ and ${\rm SU}(N)$ Monte Carlo data (at
$N=9,15,21$ for ${\rm U}(N)$ and $N=9,15,21,30$ for ${\rm SU}(N)$) fit
very well the Ansatz (\ref{FNS}), and their extrapolation leads to the
estimates $\beta_c= 0.3057(3)$ and $\epsilon=1.5(1)$. The
interpretation of the exponent $\epsilon$ in this context is still an
open problem. It is worth noticing that the value of the correlation
length describing the propagation in the fundamental channel is finite
at the phase transition: $\xi^{(c)}\simeq 2.8$.
The existence of this large-$N$ phase transition is confirmed by an
analysis of the $N=\infty$ 18th-order strong-coupling series of the
free energy
\begin{eqnarray}
F=&&2\beta^2+2\beta^4+4\beta^6+19\beta^8+96\beta^{10}+
604\beta^{12}\nonumber \\ &&+4036\beta^{14}
+ {58471\over 2}\beta^{16}+{663184\over 3}\beta^{18}+
O\left(\beta^{20}\right),
\label{Fseries}
\end{eqnarray}
which shows a second-order critical behavior:
\begin{equation}
C ={1\over 4} \beta^2 {\partial^2 F\over \partial\beta^2}
\sim |\beta - \beta_c |^{-\alpha},
\label{Ccrit}
\end{equation}
with $\beta_c = 0.3060(4)$ and $\alpha = 0.27(3)$, in agreement with
the extrapolation of Monte Carlo data. The above estimates of
$\beta_c$ and $\alpha$ are slightly different from those given in
Ref.\ \cite{Campostrini-Rossi-Vicari-chiral-2}; they are obtained by a
more refined analysis based on integral approximant techniques
\cite{Guttmann-Joyce,Hunter-Baker-AI,Fisher-Yang} and by the so-called
critical point renormalization method \cite{Hunter-Baker-CPRM}.
Green and Samuel argued that the large-$N$ phase transition of
principal chiral models on the lattice is nothing but the large-$N$
limit of the determinant phase transition present in ${\rm U}(N)$
lattice models \cite{Green-Samuel-chiral,Green-Samuel-largeN}.
According to this conjecture, $\beta_d$ and $\beta_{\rm peak}$
should both converge to $\beta_c$ in the large-$N$ limit, and the
order of the determinant phase transition would change from the
infinite order of the Kosterlitz-Thouless mechanism to a second order
with divergent specific heat. The available Monte Carlo data of ${\rm
U}(N)$ lattice models at large $N$ provide only a partial confirmation
of this scenario; one can just get a hint that $\beta_d(N)$ is
also approaching $\beta_c$ with increasing $N$. The large-$N$ phase
transition of the ${\rm SU}(N)$ models could then be explained by the
fact that the large-$N$ limit of the ${\rm SU}(N)$ theory is the same
as the large-$N$ limit of the ${\rm U}(N)$ theory.
The large-$N$ character expansion of the mass $m_d$ propagating
in the determinant channel has been calculated up to 6th order in the
strong-coupling region, indicating a critical point (determined by the
zero of the $m_d$ series) slightly larger than our determination
of $\beta_c$: $\beta_d(N{=}\infty)\simeq 0.324$
\cite{Green-Samuel-chiral}. This discrepancy might be explained
either by the shortness of the available character expansion of
$m_d$ or by the fact that such a determination of $\beta_c$
relies on the absence of singular points before the strong-coupling
series of $m_d$ vanishes, and therefore a non-analyticity at
$\beta_c\simeq 0.306$ would invalidate all strong-coupling predictions
for $\beta > \beta_c$.
It is worth mentioning another feature of this large-$N$ critical
behavior which emerges from a numerical analysis of the phase
distribution of the eigenvalues of the link operator
\begin{equation}
L = U_x \, U^\dagger_{x+\mu}:
\end{equation}
the $N=\infty$ phase transition should be related to the
compactification of the eigenvalues of $L$
\cite{Campostrini-Rossi-Vicari-chiral-3}, like the Gross-Witten phase
transition.
The existence of such a phase transition does not represent an
obstruction to the use of strong-coupling expansion for the
investigation of the continuum limit. Indeed large-$N$ Monte Carlo
data show scaling and asymptotic scaling (in the energy scheme) even
for $\beta$ smaller then the peak of the specific heat, suggesting an
effective decoupling of the modes responsible for the large-$N$ phase
transition from those determining the physical continuum limit. This
fact opens the road to tests of scaling and asymptotic scaling at
$N=\infty$ based only on strong-coupling computations, given that the
strong-coupling expansion should converge for $\beta < \beta_c$. (The
strong-coupling analysis does not show evidence of singularities in
the complex $\beta$-plane closer to the origin than $\beta_c$.)
In the continuum limit the dimensionless renormalization-group
invariant function
\begin{equation}
A(p;\beta)\equiv{\widetilde{G}(0;\beta)\over\widetilde{G}(p;\beta)}
\label{ldef}
\end{equation}
turns into a function $A(y)$ of the ratio $y\equiv p^2/M_G^2$ only,
where $M_G^2\equiv 1/\xi_G^2$ and $\xi_G$ is the second moment
correlation length
\begin{equation}
\xi_G^2\equiv {1\over 4}\,{\sum_x x^2G(x)\over \sum_x G(x)}.
\label{xig}
\end{equation}
$A(y)$ can be expanded in powers of $y$ around $y=0$:
\begin{equation}
A(y)=1 + y + \sum_{i=2}^\infty c_i y^i,
\label{lexp}
\end{equation}
and the coefficients $c_i$ parameterize the difference from a
generalized Gaussian propagator. The zero $y_0$ of $A(y)$ closest to
the origin is related to the ratio $M^2/M_G^2$, where $M$ is the
fundamental mass; indeed $y_0=-M^2/M_G^2$. $M^2/M_G^2$ is in general
different from one; it is one in Gaussian models (i.e. when
$A(y)=1 + y $).
Numerical simulations at large $N$, which allow an investigation of
the region $y\geq 0$, have shown that the large-$N$ limit of the
function $A(y)$ is approached rapidly and that its behavior is
essentially Gaussian for $y \mathrel{\mathpalette\vereq<} 1$, indicating that $c_i\ll 1$ in
Eq.\ (\ref{lexp}) \cite{Rossi-Vicari-chiral1}. Important logarithmic
corrections to the Gaussian behavior must eventually appear at
sufficiently large momenta, as predicted by simple weak-coupling
calculations supplemented by a renormalization group resummation:
\begin{equation}
\widetilde{G}(p)\sim {\log p^2 \over p^2}
\label{gpert}
\end{equation}
for $p^2/M_G^2\gg 1$ and in the large-$N$ limit.
The approximate Gaussian behavior at small momentum is also confirmed
by the direct estimate of the ratio $M^2/M_G^2$ obtained by
extrapolating Monte Carlo data to $N=\infty$. The large-$N$ limit of
the ratio $M^2/M_G^2$ is rapidly approached, already at $N=6$ within
few per mille, leading to the estimate $M^2/M_G^2=0.982(2)$, which is
very close to one \cite{Rossi-Vicari-chiral2}. Large-$N$
strong-coupling computations of $M^2/M_G^2$ provide a quite stable
curve for a large region of values of the correlation length, which
agrees (within about one per cent) with the continuum large-$N$ value
extrapolated by Monte Carlo data
\cite{Campostrini-Rossi-Vicari-chiral-2}.
Monte Carlo simulations at large values of $N$ ($N\geq 6$) also show
that asymptotic scaling predictions applied to the fundamental mass
are verified within a few per cent at relatively small values of the
correlation length ($\xi\gtrsim2$) and even before the peak of the
specific heat in the so-called ``energy scheme'' \cite{Parisi-betaE};
the energy scheme is obtained by replacing $T$ with a new temperature
variable $T_E \propto E$, where $E$ is the internal energy density.
At $N=\infty$ a test of asymptotic scaling may be performed by using
the large-$N$ strong-coupling series of the fundamental mass. The
two-loop renormalization group and a Bethe Ansatz evaluation of the
mass/$\Lambda$-parameter ratio \cite{Balog-Naik-Niedermayer-Weisz}
lead to the following large-$N$ asymptotic scaling prediction in the
$\beta_E$ scheme:
\begin{eqnarray}
&&M \cong 16\,\sqrt{\pi\over\e} \exp\!\left(\pi\over4\right)
\Lambda_{E,2l}(\beta_E),\nonumber \\
&&\Lambda_{E,2l}(\beta_E) =
\sqrt{8\pi\beta_E}\exp(-8\pi\beta_E) ,\nonumber \\
&&\beta_E = {1\over 8E}\,.
\label{mass-lambdaE}
\end{eqnarray}
Strong-coupling calculations, where the new coupling $\beta_E$ is
extracted from the strong-coupling series of $E$, show asymptotic
scaling within about 5\% in a relatively large region of values of the
correlation length ($1.5\mathrel{\mathpalette\vereq<}\xi\lesssim3$)
\cite{Campostrini-Rossi-Vicari-chiral-brief,%
Campostrini-Rossi-Vicari-chiral-2}.
The good behavior of the large-$N$ $\beta$-function in the $\beta_E$
scheme, and therefore the fact that physical quantities appear to be
smooth functions of the energy, together with the critical behavior
(\ref{Ccrit}), can be explained by the existence of a non-analytical
zero at $\beta_c$ of the $\beta$-function in the standard scheme:
\begin{equation}
\beta_L(T)\equiv a{{\mathrm{d}} T\over{\mathrm{d}} a}\sim |\beta-\beta_c|^\alpha
\label{betasing}
\end{equation}
around $\beta_c$, where $\alpha$ is the critical exponent of the
specific heat. This is also confirmed by an analysis of the
strong-coupling series of the magnetic susceptibility $\chi$ and
$M^2_G$, which supports the relations
\begin{equation}
{{\mathrm{d}}\log\chi\over{\mathrm{d}}\beta} \sim
{{\mathrm{d}}\log M^2_G\over{\mathrm{d}}\beta} \sim
|\beta-\beta_c|^{-\alpha}
\label{chi_crit}
\end{equation}
in the neighborhood of $\beta_c$, which are consequences of
Eq.\ (\ref{betasing}) \cite{Campostrini-Rossi-Vicari-chiral-2}.
We finally mention that similar results have been obtained for
two-dimensional chiral models on the honeycomb lattice by a large-$N$
strong-coupling analysis. In fact an analysis of the 26th-order
strong-coupling series of the free energy indicates the presence of a
large-$N$ phase transition, with specific heat exponent $\alpha \cong
0.17$, not far from that found on the square lattice (we have no
reasons to expect that the large-$N$ phase transition on the square
and honeycomb lattices are in the same universality class).
Furthermore the mass-gap extracted from the 20th-order strong-coupling
expansion of $G(x)$ allows to check the corresponding asymptotic
scaling predictions in the energy scheme within about 10\%
\cite{Campostrini-Rossi-Vicari-chiral-2}.
\subsection{The large-$N$ limit of ${\rm SU}(N)$
lattice gauge theories}
\label{secQCD}
An overview of the large-$N$ limit of the continuum formulation of QCD
has been already presented in Sect.\ \ref{unitary-matrices}. In the
following we report some results concerning the lattice approach.
Gauge models on the lattice have been mostly studied in their Wilson
formulation
\begin{eqnarray}
S_{\rm W} &=& N\beta \sum_{x,\mu>\nu} {\rm Tr} \left[
U_\mu(x) U_\nu(x+\mu) U_\mu^\dagger(x+\nu) U_\nu^\dagger(x)
+ {\rm h.c.}\right]. \nonumber \\
\label{wilsonac}
\end{eqnarray}
In view of a large-$N$ analysis one may consider both ${\rm SU}(N)$
and ${\rm U}(N)$ models, since they are expected to reproduce the same
statistical theory in the limit $N\rightarrow\infty$ (at fixed
$\beta$). As for two-dimensional chiral models, ${\rm SU}(N)$ and
${\rm U}(N)$ models should have the same continuum limit for any
finite $N\geq 2$.
The phase diagram of statistical models defined by the Wilson action
has been investigated by standard techniques, i.e., strong-coupling
expansion, mean field \cite{Drouffe-Zuber}, and Monte Carlo
simulations
\cite{Creutz-SU5,Creutz-Moriarty,Moriarty-Samuel-comparison}. These
studies show the presence of a first-order phase transition in ${\rm
SU}(N)$ models for $N\geq 4$, and in ${\rm U}(N)$ models for any
finite $N$. A first-order phase transition is then expected also in
the large-$N$ limit at a finite value of $\beta$, which is estimated
to be $\beta_c \approx 0.38$ by mean-field calculations and by
extrapolation of Monte Carlo results. A review of these results can
be found in Ref.\ \cite{Itzykson-Drouffe}. Some speculations on the
large-$N$ phase diagram can be also found in Refs.\
\cite{Kostov-sc,Green-Samuel-largeN}. The r\^ole of the determinant
of Wilson loops in the phase transition of ${\rm U}(N)$ gauge models
has been investigated in Ref.\ \cite{Green-Samuel-largeN} by
strong-coupling character expansion, and in Ref.\
\cite{Moriarty-Samuel} by Monte Carlo simulations.
Large-$N$ mean-field calculations suggest the persistence of a
first-order phase transition when an adjoint-representation coupling
is added to the Wilson action
\cite{Chen-Tan-Zheng-phase,Ogilvie-Horowitz}.
The first-order phase transition of ${\rm SU}(N)$ lattice models at
$N>3$ can probably be avoided by choosing appropriate lattice actions
closer to the renormalization group trajectory of the continuum limit,
as shown in Ref.\ \cite{Itoh-Iwasaki-Yoshie-absence} for ${\rm
SU}(5)$. In ${\rm U}(N)$ models the use of such improved actions
should leave a residual transition, due to the extra ${\rm U}(1)$
degrees of freedom which should decouple at large $\beta$ in order to
reproduce the physical continuum limit of ${\rm SU}(N)$ gauge models.
It is worth mentioning two studies of confinement properties at large
$N$, obtained essentially by strong-coupling arguments. In Ref.\
\cite{Greensite-Halpern}, the authors argue that deconfinement of
heavy adjoint quarks by color screening is suppressed in the large-$N$
limit. At $N=\infty$, the adjoint string tension is expected to be
twice the fundamental string tension, as implied by factorization.
In Ref.\ \cite{Lovelace-universality}, strong-coupling based arguments
point out that Wilson loops in ${\rm O}(N)$, ${\rm U}(N)$, and ${\rm
Sp}(N)$ lattice gauge theories should have the same large-$N$ limit,
and therefore these theories should share the same confinement
mechanism. Such results should be taken into account when studying
confinement mechanisms.
Studies based on Monte Carlo simulations for $N>3$ have not gone
beyond an investigation of the phase diagram, so no results concerning
the continuum limit of ${\rm SU}(N)$ lattice gauge theories with $N>3$
have been produced. Estimates of the mass of the lightest glueball,
obtained by a variational approach within a Hamiltonian lattice
formulation, seem to indicate a rapid convergence of the $1/N$
expansion \cite{Chin-Karliner}.
An important breakthrough for the study of the large-$N$ limit of
${\rm SU}(N)$ gauge theories has been the introduction of the
so-called reduced models. A quite complete review on this subject can
be found in Ref.\ \cite{Das-review}.
Eguchi and Kawai \cite{Eguchi-Kawai-reduction} pointed out that, as a
consequence of the large-$N$ factorization, one can construct
one-site theories equivalent to lattice YM in the limit
$N\rightarrow\infty$. The simplest example is given by the one-site
matrix model obtained by replacing all link variables of the standard
Wilson formulation with four ${\rm SU}(N)$ matrices according to the
simple rule
\begin{equation}
U_\mu(x) \to U_\mu.
\label{EK}
\end{equation}
This leads to the reduced action
\begin{equation}
S_{\rm EK}= N\beta \sum_{\mu>\nu}{\rm Tr} \left[
U_\mu U_\nu U_\mu^\dagger U_\nu^\dagger + {\rm h.c.}\right].
\label{EKaction}
\end{equation}
Reduced operators, and in particular reduced Wilson loops, can be
constructed using the correspondence (\ref{EK}). In the large-$N$
limit one can prove that expectation values of reduced Wilson loop
operators satisfy the same Schwinger-Dyson equations as those in the
Wilson formulation. Assuming that all features of the $N=\infty$
theory are captured by the Schwinger-Dyson equations of Wilson loops,
the reduced model may provide a model equivalent to the standard
Wilson theory at $N=\infty$. In the proof of this equivalence the
residual symmetry of the reduced model
\begin{equation}
U_\mu \to Z_\mu U_\mu, \qquad Z_\mu\in Z_N,
\label{Zn}
\end{equation}
where $Z_N$ is the center of the ${\rm SU}(N)$ group, plays a crucial
r\^ole. Therefore, the equivalence in the large-$N$ limit of the
Wilson formulation and the reduced model (\ref{EKaction}) is actually
valid if the symmetry (\ref{Zn}) is unbroken. This is verified only in
the strong-coupling region; indeed in the weak-coupling region the
$Z_N^4$ symmetry gets spontaneously broken and therefore the
equivalence cannot be extended to weak coupling
\cite{Bhanot-Heller-Neuberger}.
In order to avoid this unwanted phenomenon of symmetry breaking and to
extend the equivalence to the most interesting region of the continuum
limit, modifications of the original Eguchi-Kawai model have been
proposed \cite{Eguchi-Nakayama-simplification,Bhanot-Heller-Neuberger,%
GonzalezArroyo-Okawa-twisted}. The most promising one for numerical
simulation is the so-called twisted Eguchi-Kawai (TEK)
model \cite{Eguchi-Nakayama-simplification,%
GonzalezArroyo-Okawa-twisted}. Instead of the correspondence
(\ref{EK}), the twisted reduction prescription consists in replacing
\begin{equation}
U_\mu(x) \to T(x)U_\mu T(x)^\dagger,
\label{TEK}
\end{equation}
where
\begin{equation}
T(x)= \prod_\mu (\Gamma_\mu)^{x_\mu}
\label{trasl}
\end{equation}
and $\Gamma_\mu$ are traceless ${\rm SU}(N)$ matrices obeying the
't Hooft algebra
\begin{equation}
\Gamma_\nu \Gamma_\mu = Z_{\mu\nu}\Gamma_\mu \Gamma_\nu ;
\label{twist}
\end{equation}
$Z_{\mu\nu}$ is an element of the center of the group $Z_N$,
\begin{equation}
Z_{\mu\nu} = \exp\!\left({\mathrm{i}}{2\pi\over N} n_{\mu\nu}\right),
\end{equation}
where $n_{\mu\nu}$ is an antisymmetric tensor with $n_{\mu\nu}=1$ for
$\mu<\nu$. $\Gamma_\mu$ are the matrices implementing the
translations by one lattice spacing in the $\mu$ direction (here it is
crucial that the fields $U_\mu$ are in the adjoint representation).
The twisted reduction applied to the Wilson action leads to the
reduced action
\begin{equation}
S_{\rm TEK}= N\beta \sum_{\mu>\nu}
{\rm Tr}\left[ Z_{\mu\nu}U_\mu U_\nu U_\mu^\dagger U_\nu^\dagger
+ {\rm h.c.}\right].
\label{TEKaction}
\end{equation}
The correspondence between correlation functions of the large-$N$ pure
gauge theory and those of the reduced twisted model is obtained as
follows. Let ${\cal A}[U_\mu(x)]$ be any gauge invariant functional
of the field $U_\mu(x)$, then
\begin{equation}
\langle {\cal A}[U_\mu(x)]\rangle_{[N=\infty,\ {\rm YM}]}=
\langle {\cal A}[ T(x)U_\mu T(x)^\dagger]
\rangle_{[N=\infty,\ {\rm TEK}]}
\label{TEKfunc}
\end{equation}
Once again the Schwinger-Dyson equations for the reduced Wilson loops,
constructed using the correspondence (\ref{TEK}), are identical to the
loop equations in the Wilson formulation when $N\rightarrow\infty$.
The residual symmetry (\ref{Zn}), which is again crucial in the proof
of the equivalence, should not be broken in the weak-coupling region,
and therefore the equivalence should be complete in this case.
One can also show that:
(i) the reduced TEK model is equivalent to the corresponding field
theory on a periodic box of size $L=\sqrt{N}$ \cite{Das-review};
(ii) in the large-$N$ limit finite-$N$ corrections are $O(1/N^2)$,
just as in the ${\rm SU}(N)$ lattice gauge theory.
Moreover, since $N^2=L^4$, finite-$N$ corrections can be seen as
finite-volume corrections. Therefore in twisted reduced models the
large-$N$ and thermodynamic limits are connected and approached
simultaneously.
Monte Carlo studies of twisted reduced models at large $N$ confirm the
existence of a first-order phase transition at $N=\infty$ located at
$\beta_c = 0.36(2)$ \cite{GonzalezArroyo-Okawa-string}, which is
consistent with the mean-field prediction $\beta_c\simeq 0.38$
\cite{Itzykson-Drouffe}. This transition is a bulk transition, and it
does not spoil confinement. The few and relatively old existing Monte
Carlo results obtained in the weak-coupling region (cfr.\ e.g.\ Refs.\
\cite{GonzalezArroyo-Okawa-string,Fabricius-Haan,Haan-Meier}) seem to
support a rapid approach to the $N\rightarrow\infty$ limit of the
physical quantities, and are relatively close to the corresponding
results for ${\rm SU}(3)$ obtained by performing simulations within
the Wilson formulation. This would indicate that $N=3$ is sufficiently
large to consider the large-$N$ limit a good approximation of the
theory.
We mention that hot twisted models can be constructed, which should be
equivalent to QCD at finite temperature in the large-$N$ limit (cfr.\
Ref.\ \cite{Das-review} for details on this subject).
|
proofpile-arXiv_065-671
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{
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section*{References}%
\begin{quotation}\mbox{}\par}
\def\refer#1\par{{\setlength{\parindent}{-\leftmargin}\indent#1\par}}
\def\end{quotation}{\end{quotation}}
{\noindent\small{\bf Abstract:}
We present the first results of a new set of population synthesis models,
which utilize the latest stellar evolutionary tracks, recent non-LTE
atmosphere models which include stellar winds, and observed line strengths
in WR spectra to predict the relative strengths of various WN and
WC/WO emission features in the spectra of starburst galaxies.
Our results will be used to derive accurate numbers of WN and WC stars
in starburst galaxies.
We also analyze the frequency and the WN and WC content of WR-rich galaxies
in low metallicity samples; the theoretical predictions are found to be in
good agreement with the observed frequencies.
We also discuss the possible role of massive close binaries in starburst regions.
If the starburst regions are formed in relatively instantaneous bursts
we argue that, given their young age as derived from emission lines equivalent widths,
{\em (1)} in the majority of the observed WR galaxies massive close binaries
have not contributed significantly to the WR population, and
{\em (2)} nebular He~{\sc ii}\ 4686 emission is very unlikely due to massive X-ray
binaries.
}
\section{Introduction}
The presence of large numbers of Wolf-Rayet (WR) stars in extragalactic star-forming
objects (hereafter called WR-galaxies)
of quite heterogeneous types is well established (see e.g.~the compilation of
Conti 1991). New serendipitous discoveries of WR galaxies have resulted from studies
covering a wide range of topics, from the primordial He abundance determination
(cf.~Izotov et al.\ 1996a) to the nature of Seyfert galaxies (Heckman et al.\ 1996),
and a considerable number of new observations can be expected with the new generation
of 8-10 m class telescopes.
In most cases the presence of WR stars can be used as a powerful constraint
on the age of the starburst episode (typically $3 - 8$ Myr). The luminosity
in the broad ``WR-bump'' (centered at $\lambda$ 4650) can be used to derive the
total number of WR stars present in the burst.
From the strength of the nebular emission lines one can also determine the
numbers of OB stars, which are the dominant contributors to the Lyman continuum flux.
Additional information on the slope of the Initial Mass Function (IMF)
can also be obtained (Meynet 1995; Contini et al.~1995; Schaerer 1996).
When compared with evolutionary models, the derived WR/O number ratios indicate
that star formation occurs in ``bursts'' short compared to the lifetime
of massive stars (Arnault et al.~1989; Vacca \& Conti 1992; Meynet 1995).
We refer the reader to the contribution by Vacca in these proceedings for a
more detailed discussion of the properties and analysis of WR galaxies.
Many aspects of WR galaxies remain to be explored and there are several
questions regarding the effect of large numbers of WR stars on their host
galaxies that remain unanswered. Among these are the following, which we hope to
address in this contribution:
What fraction of starbursts have gone through or are currently in
a WR-rich phase (Kunth \& Joubert 1985; Meynet 1995) ?
How frequent are WC stars in WR-galaxies and what is their importance
(Meynet 1995; Schaerer 1996) ? Are WR stars responsible for the high excitation
nebular lines observed in the optical spectra of some young starbursts (Garnett et
al.~1991; Motch et al.~1994; Schaerer 1996) ? How important is the formation of WR stars
in binary systems for WR-galaxies (Cervi\~{n}o \& Mas-Hesse 1996; Vanbeveren et al.~1996) ?
Answers to these questions require a detailed knowledge of the stellar populations in
the host galaxies. The aim of our work is to provide new predictions for the WR populations
in young starbursts, by explicitly taking into account the two main W-R subtypes,
WN and WC stars. Our synthesis approach, based on well-tested evolutionary models,
recent atmosphere models for O and WR stars, and observed line-strengths in WR stars,
provides a number of relevant observable quantities. (A similar, but less comprehensive
attempt, was carried out by Kr\"uger et al.\ 1992.)
Here we present preliminary results from this on-going work, which will be used
for the future analysis of a large sample of WR-galaxies.
It is our hope to shed some light on some of the aforementioned questions.
\section{Evolutionary synthesis models}
In this Section we briefly describe the adopted model ingredients for
our evolutionary synthesis models, the most important input parameters,
and the synthesized quantities.
{\em Stellar evolution:} We adopt the recent tracks of the the Geneva
group, which cover the metallicity range from $Z$=0.001 (1/20 \ifmmode Z_{\odot} \else $Z_{\odot}$\fi) to
$Z$=0.04 (2 \ifmmode Z_{\odot} \else $Z_{\odot}$\fi) (see Meynet et al.~ 1994 and references therein).
As shown by Maeder \& Meynet (1994) the models with enhanced mass loss
rates reproduce a large number of observations regarding massive star populations,
including WR/O star ratios in various nearby galaxies.
These models are preferred over the earlier models of Schaller et al.~
(1992) adopted in the calculations of Vanbeveren (1995) and other
population synthesis models (e.g.~Cervi\~{n}o \& Mas-Hesse 1994, 1996).
{\em Evolution of massive close binaries:}
In order to explore the effects of forming WR stars via
mass transfer in massive close binaries on the total population of massive stars,
we adopted the following simplified treatment of binaries.
We used the recent calculations for various metallicities of de Loore
\& Vanbeveren (1994), who assume an initial mass ratio of 0.6, Case B
mass transfer, and neglect the possibility of subsequent WR formation by
the secondary.
For our synthesis models one free parameter $f$ determines the binary
population; $f$ is defined as the fraction of stars, which are primaries
in close binary systems and which
will therefore experience Roche lobe overflow during their evolution.
The total WR population formed through the binary channel and the
distribution among the different subtypes can be derived directly
from the stellar lifetimes and the duration of the respective phases
(see de Loore \& Vanbeveren 1994).
To determine the impact of binaries on observational properties (nebular
lines and broad WR emission lines) we adopted an average Lyman continuum
flux of $Q_0 = 10^{49} \, {\rm photons \, s^{-1}}$ which roughly corresponds
to the average contribution
from single WR stars at the time during the evolution of a burst population
when binary stars are first expected to be formed.
The broad WR line emission is treated in the same manner as for single stars (see below).
{\em Continuum spectral energy distribution:}
To determine the stellar continuum spectral energy distribution at each time
during a burst, we relied on three different sets of theoretical models:
{\em 1)}
For massive stars we used the spectra from the combined stellar structure and atmosphere
({\em CoStar}) models of Schaerer et al.~ (1996ab), which
include non--LTE\ effects, line blanketing, and stellar winds.
These models cover the entire parameter space of O stars during their main sequence
evolution.
{\em 2)}
For later spectral types we use the line-blanketed plane-parallel
LTE models of Kurucz (1992).
{\em 3)}
For W-R stars, we used the spherically expanding
non--LTE\ models of Schmutz, Leitherer \& Gruenwald (1992).
In addition to the stellar continuum one also needs to account for
the nebular continuous spectrum. Its emission is calculated assuming
$T_e=$ 10 kK, $N_e=100 \, {\rm cm^{-3}}$, and solar H/He abundances.
{\em Nebular and WR emission lines:}
The strengths of the nebular recombination lines (primarily \ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi, \ifmmode {\rm H{\alpha}} \else $\rm H{\alpha}$\fi,
He~{\sc ii}\ $\lambda$ 4686) are calculated with the same values of the electron
temperature and density as used for the nebular continuum. We have compiled
average {\em stellar} line fluxes of the
strongest WR emission lines for WN, WC, and WO stars.
We distinguish 5 WC subtypes as well as the WO subtype, as these objects show
considerable differences in their line fluxes.
We also include possible emission from OfI stars.
The line fluxes have been taken from the following sources:
Crowther (1996, private communication) and Smith et al.~ (1996)
for WN stars, and Smith et al.~ (1990ab) for WC/WO stars.
More details are given in Schaerer \& Vacca (1996).
The WR stage, including the WC subtype (see Smith \& Maeder 1991),
is determined by the surface abundances predicted from the evolutionary models.
{\em Input parameters:}
In the present work we consider the time evolution of an instantaneous
burst of star-formation. The basic parameters of our models are therefore
the metallicity, the binary fraction, and the slope and upper mass cut-off of the
initial mass function. Here, we adopt a Salpeter IMF with
an upper mass cut-off of 120 \ifmmode M_{\odot} \else $M_{\odot}$\fi. The results do not depend
on the lower mass cut-off, as long as it is less than about $5$ \ifmmode M_{\odot} \else $M_{\odot}$\fi.
Variations of the IMF slope are considered in Schaerer \& Vacca (1996).
{\em Synthesized quantities:}
The major predictions from our models include:
{\em (1)} the relative populations of O stars (where an O stars is defined by
\ifmmode T_{\rm eff} \else $T_{\rm eff}$\fi\ $>$ 30 kK), WN stars, and WC/WO stars,
{\em (2)} emission line fluxes and equivalent widths of the following
broad WR lines: He~{\sc ii}\ $\lambda$ 1640, N~{\sc iii} $\lambda$ 4640,
C~{\sc iii/iv} $\lambda$ 4650, He~{\sc ii}\ $\lambda$ 4686, total $\lambda$
4650 WR-bump, C~{\sc iv} $\lambda$ 5696, and C~{\sc iv} $\lambda$ 5808,
{\em (3)} the WR contributions to \ifmmode {\rm H{\alpha}} \else $\rm H{\alpha}$\fi\ and \ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi,
{\em (4)} the ionizing photon fluxes in the H, He~{\sc i}, and He~{\sc ii}\ continua, and
{\em (5)} the emission line fluxes and equivalent widths of nebular lines of
\ifmmode {\rm H{\alpha}} \else $\rm H{\alpha}$\fi, \ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi, and He~{\sc ii}\ $\lambda$ 4686.
\begin{figure}[htb]
\centerline{\psfig{figure=e004_4.eps,height=8cm}
\psfig{figure=plot_binaries_004.eps,height=8cm}}
\caption{{\em Left panels:} Time evolution of the equivalent width
of the strongest WR lines (upper left) and WR/O star ratios including
subtypes (lower left) at Z=0.004 for standard evolutionary models.
{\em Right panels:} Models including massive close binaries for $f$=0.2.
Upper right: same as upper left.
Lower right: evolution of the relative He~{\sc ii}/\ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi line intensities as
a function of the \ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi\ equivalent width (solid line). The dashed line
shows the contribution from single stars. The two ``epochs'' where WR
stars are formed from single stars and by the binary channel are well
separated in this diagram}
\label{fig_e004}
\end{figure}
\section{Probing WN and WC populations in starbursts}
We will illustrate the model predictions for a burst with a metallicity of $Z=0.004$,
a typical value for the WR-galaxies analyzed by Vacca \& Conti
(1992). [The entire set of results, which depend strongly on metallicity,
will be discussed in Schaerer \& Vacca (1996).]
The left panels in Figure 1 present the results from our standard models
(Salpeter IMF, instantaneous burst, single star evolution), while the right
panels include massive close binary stars (cf.~Sect.~4).
The lower left Figure shows the relative WR and populations as a function
of the age of the burst. The WR-rich phase lasts from $\sim$ 2.5 to
5.5 Myr. WC stars evolving from the most massive stars
dominate the WR population from about $3 - 4$ Myr, while
WNL stars are more numerous from $4 - 5.5$ Myr. Thus, the models predict
a WC-rich phase shortly after the first appearance of
WR stars. For an instantaneous burst the last period of the
WR-rich phase is always dominated by WNL stars, as these objects represent
the descendents of the least massive stars which barely manage to peel
off their outer layers revealing the processed material resulting from H-burning.
The upper left panel shows the corresponding evolution of the equivalent
widths of the most important WR lines.
The He~{\sc ii}\ and N~{\sc iv} 4640 emission predicted {\em before} the WR rich phase
is due to the (relatively large) contribution adopted for OIf stars.
Broad He~{\sc ii}\ 4686 emission usually dominates the optical spectrum except during
the short ($\sim$ 1 Myr) WC-rich phase, during which C~{\sc iii/iv} 4650 dominates
the broad classical ``WR bump'' and the presence of WC stars can be unambiguously
deduced from the strong C~{\sc iv} 5808 feature.
Although the predicted strength of N~{\sc iv} 4640 is relatively uncertain, its
is always lower than that of He~{\sc ii}\ 4686 except at solar or higher
metallicities. This is an immediate consequence of the abundance effect
pointed out by Smith et al.~ (1996).
As expected, C~{\sc iii} 5696 is very weak at $Z=0.004$; this feature is strong
only in late WC stars, which are not found at low metallicities.
\section{The frequency of WR-rich starbursts}
The predictions illustrated above can be used to determine the WR content
in individual starbursts and allow us to determine separately the WN and WC
populations. In addition to the study of individual objects, however,
a statistical analysis of a set of starburst galaxies also provides
a test of the models, as recently stressed by Meynet (1995).
Although large samples adequate for such statistical
studies are not yet available, we would like to point out briefly some
interesting results from the low metallicity samples of Izotov et al.~ (1994, 1996a)
and Pagel et al.~ (1992), which have been obtained as part of a systematic determination
of the primordial He abundance.
Since the major goal of these studies is to obtain as many low metallicity
objects as possible these samples are suited for statistical studies of
starbursts over a low, and clearly specified metallicity interval.
The Izotov sample contains 33 objects with $Z$ between $\sim 0.001$
and 0.004, of which 14 exhibit WR features, including 4 WC with signatures.
Thus $\sim$ 40 \% of the objects show evidence of WR stars, and $\sim$ 30
\% of those include WC stars.
Similar, or even larger, percentages of WR detections are found in the
Pagel et al.~ sample over a similar metallicity range.
Interestingly these numbers are fairly close to the percentages
of starbursts containing WR stars
predicted from evolutionary models (Meynet 1995)\footnote{The values
for the high mass loss models at $Z=0.004$ in Table 1 of Meynet (1995)
are erroneous. Furthermore the duration of the WC-rich phase given by
Meynet is overestimated.
This accounts for the difference between our
results in Fig. 1, and those in Fig. 3 of Meynet (1995).}.
The expected percentage is between $\sim$ 18 and 40 \% for $Z$ between 0.001 and
0.004; the duration of WC-rich phase is predicted to be $\sim$ 1/3 of the WR phase
(see Fig.~\ref{fig_e004}).
An observational bias is introduced by the requirement that the
[O~{\sc iii}] 4363 line can be detected and reliably measured. This requirement
favours inclusion of objects with the youngest bursts and could therefore lead
to an overestimate of the percentage of WR-rich objects
as compared to the definition used by Meynet (1995).
This might be responsible for the apparent difference with
the model predictions at low $Z$ .
The approximate agreement between models and observations regarding
the statistical number of WR-rich objects is very encouraging
although admittedly the present samples are fairly small.
In particular the detection of a significant fraction of WC stars at low
metallicities gives strong support to the adopted high-mass-loss
evolutionary models.
In this context it is also interesting to note that to date no
WR features have been detected in objects with metallicities below O/H
$\le$ 7.7--7.8 (Pagel et al.~ 1992; Izotov et al.~ 1996), corresponding to an
absolute metal abundance of $Z \le$ 0.0012--0.0015, or about $0.06 Z_\odot$.
Although no formal low metallicity cut-off for the presence of WR stars is expected
from evolutionary models, this observed limit seems to be in fair agreement
with the predicted sharp decrease in the duration of the WR phase
between $Z=0.004$ and $0.001$ (cf.~Meynet 1995).
\section{The role of massive close binaries in young starbursts}
Recent studies have begun to explore the importance of the formation of WR
stars in massive close binaries (MCB's) on massive star populations in
starbursts (Cervi\~{n}o \& Mas-Hesse 1996; Vanbeveren et al.~ 1996).
Here we briefly discuss some basic considerations, which are useful to estimate
those circumstances in which binary stars may be of relevance for the WR
populations in starbursts.
An important property of binary models is that, because the high mass loss
rate prevents a large increase in the stellar radius, primaries with initial
masses $M_1 >$ 40-50 \ifmmode M_{\odot} \else $M_{\odot}$\fi\ should, in general, avoid Roche lobe overflow
(cf.~Vanbeveren 1995); for those stars that do experience Roche lobe overflow,
their evolution is nearly indistinguishable from that of single stars (Langer 1995).
{\em Therefore, in instantaneous bursts with ages
$\leq 5$ Myr the stellar population is unaltered by the formation of
WR stars through the binary channel.}
{\em Do WR galaxies contain a significant population of WR stars formed
through the binary channel ?}
The observed \ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi\ equivalent width in the spectrum of an H II region exhibits
a monotonic decrease with time can be used as a good indicator of the age of
a starburst (e.g., Leitherer \& Heckman 1995). Bursts with ages $\tau \ge 5$
Myr are predicted to have $W(\ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi) <$ 60 \ang\ for $Z \sim 0.001$, while at larger
metallicity the upper limit for $W(\ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi)$ is even lower.
An inspection of the compilation of WR galaxies given Conti (1991)
reveals that most objects have large \ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi\ equivalent widths:
12 out of 37 objects have $W(\ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi) <$ 60 \ang, and only 3 show
$W(\ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi) <$ 30 \ang. In fact, because of various physical effects which serve to
artifically reduce the observed $W(\ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi)$, these fractions are actually
{\em upper} limits to the true number of WR galaxies with low equivalent widths.
Therefore, most WR galaxies experienced bursts of star formation
less than 5 Myr ago.
If star-formation has taken place on such a short timescale compared to the
lifetime of massive stars (``instantaneous burst'') roughly 70 to 90 \%
of the burst populations in WR galaxies are too young to be affected
by WR formation through the binary channel and therefore they should
be well described by single star models.
{\em The link between population synthesis models and observable
quantities.}
In recent studies Cervi\~{n}o \& Mas-Hesse (1996) and Vanbeveren et al.~
(1996) have included massive close binaries (MCB's) in population synthesis models.
They find that
{\em (1)} the WR-rich phase of a starburst lasts much longer (up to 12-20 Myr)
when MCBs are taken into account,
{\em (2)} WR/O number ratios can be larger than those predicted by synthesis models
including only single stars, and
{\em (3)} even with a ``standard'' IMF the observed WNL/O ratios are well
reproduced by their models (Vanbeveren et al.\ 1996).
These findings require some remarks.
As shown above, in the vast majority of the observed WR galaxies the
bursts are very young and therefore, in general, their WR population has probably not
been formed through the binary channel. Older objects with a possibly large WR population
remain to be found; however most searches are biased against finding such objects.
Given the young age of the known WR galaxies, the ``observed'' WNL/O star ratios
of Vacca \& Conti (1992) cannot be compared to the large values obtained by
Vanbeveren et al.~ (1996) in the ``binary rich'' WR phase.
Moreover, as shown by Schaerer (1996) the observed WR/O population can be
explained with single star models and a ``standard'' Salpeter IMF.
To allow for a direct comparison between synthesized stellar populations and
observations the relevant observable quantities (line fluxes, equivalent widths
etc.) need to be modeled (see Sects.~2 and 3).
Predictions from exploratory calculations which also include binary stars are
shown in Fig.~\ref{fig_e004} (right panel).
The behaviour of the equivalent widths of the most important broad WR lines
(upper right) nicely illustrates the prolonged WR phase.
The lower right panel shows that, compared to the flux in \ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi, a relatively
large flux in the broad He~{\sc ii}\ 4686 line can be obtained if binaries are included.
However, as mentioned before, such behaviour can be obtained only at ages
$\tau >$ 5 Myr corresponding roughly to $W(\ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi) <$ 30--60 \ang.
{\em Massive X-ray binaries as the origin of nebular HeII emission ?}
Based on the same age considerations we would like to mention several arguments
regarding the role of high-mass X-ray binaries (HMXRB) in the origin of
{\em nebular} He~{\sc ii}\ emission in extragalactic H~{\sc ii}\ regions (see Garnett et al.~ 1991;
Schaerer 1996). There are several lines of evidence that indicate that HMXRBs are
{\em not} the source of the nebular He~{\sc ii}\ emission:
{\em (1)}
All the objects from the samples of Campbell et al.~ (1986) and Izotov et al.~
(1994, 1996ab) have large \ifmmode {\rm H{\beta}} \else $\rm H{\beta}$\fi\ equivalent widths, corresponding
to burst ages of less than $\sim$ 5 Myr.
If nebular He~{\sc ii}\ emission is due to HMXRB these systems must have had
primaries with very large masses ($M_1 \ge$ 40-50 \ifmmode M_{\odot} \else $M_{\odot}$\fi) necessary
to form neutron star remnants. Such a scenario for the formation of HMXRB
seems to be very unlikely (van den Heuvel 1994).
The age argument was also put forward by Motch et al.~ (1994).
{\em (2)} Given the short duration of the X-ray emitting phase
($\sim 5 \times 10^4$ yr, van den Heuvel 1994), it is very difficult to produce HMXRB
in large numbers (e.g.~comparable to the number of equivalent O7 stars
in SBS 0335-052 according to Izotov et al.~ 1996b).
{\em (3)} It is not clear why the spatial distribution of nebular He~{\sc ii}\ should
preferentially follow the continuum instead of the remaining emission lines
as found by (Izotov et al.~ 1996b).
{\em (4)} Motch et al.\ (1994) find that the He~{\sc ii}\ emission and the X-ray
emission are not spatially coincident, as would be expected if HMXRB are the source of
the He~{\sc ii}\ emission.
The above results render the MXRB hypothesis rather unlikely.
In many objects WR stars appear to be a very likely source of the high energy photons
needed to ionize He II (Motch et al.\ 1994; Schaerer 1996) although peculiar O stars
close to the Eddington limit (Gabler et al.~ 1992) cannot be excluded.
We also note that out of the 38 objects
from Campbell et al.~ and Izotov et al.~ (1994, 1996a) which have a definite measurement of
He~{\sc ii}, only 7 objects are found at very low metallicities (O/H $<$ 7.72),
for which WR features have never been detected.
{\small
\section*{Acknowledgements}
We thank Paul Crowther for providing us with observational data.
The work of DS is supported by a grant of the Swiss National Foundation
of Scientific Research. Additional support from the Directors
Discretionary Research Fund of the STScI is also acknowledged.
WDV acknowledges support in the form of a fellowship from the
Beatrice Watson Parrent Foundation.
}
\vspace*{-0.5cm}
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\end{document}
|
proofpile-arXiv_065-672
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{
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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|
\section{Introduction}
X-ray observations of clusters of galaxies show that in the
central regions of most clusters the cooling time of the IntraCluster Medium
(ICM) is significantly less than the Hubble time ({\it e.g.\ } Edge, Stewart \& Fabian
1992). The observed cooling, which takes place primarily through the
emission of X-rays, leads to a slow net inflow of material towards the cluster
centre; a process known as a {\it cooling flow} (Fabian 1994).
Within the idealized model of an {\it homogeneous} cooling flow, all of the cooling
gas flows to the centre of the cluster where it is deposited, having
radiated away its thermal energy.
However, observations of cooling
flows show that the simple homogeneous model is not correct.
Although the X-ray surface brightness profiles of clusters with cooling flows
are substantially more sharply-peaked than those of non cooling-flow
systems, the emission is not as sharply-peaked as it would be in the case of homogeneous
flows. Rather than all of the cooling gas flowing to the very
centres of the clusters, the X-ray data show that
gas is `cooling out' throughout the central few tens to hundreds of
kpc. Typically the cooled material is deposited with ${\dot M}(r) \propto
r$, where ${\dot M}(r)$ is the integrated mass deposition rate within radius
$r$. The X-ray data firmly require that cooling flows
are {\it inhomogeneous} with a range of density and temperature phases at all
radii.
Spatially resolved X-ray spectroscopy of clusters also confirms the the presence of
distributed cool (and rapidly cooling) gas in cooling flows. The spatial distribution
and luminosity of the cool components determined from the spectral data are well-matched
to values inferred from the X-ray images ({\it e.g.\ } Allen {\it et al.\ } 1993, Allen, Fabian \& Kneib
1996; Fabian {\it et al.\ } 1996; Allen \& Fabian 1996).
For a detailed review of the theory and observations of cooling flows see
Fabian (1994).
In this paper we present observations, made with the ASCA and ROSAT X-ray
astronomy satellites, of three exceptionally luminous cluster cooling
flows. Two of the systems, Zwicky 3146 ($z=0.291$) and Abell 1835 ($z=0.252$) were identified as
X-ray luminous clusters during optical-follow up studies to
the ROSAT All-Sky Survey (Allen {\it et al.\ } 1992a). The third system,
E1455+223 (or Zwicky 7160; $z=0.258$) was identified
by Mason
{\it et al.\ } (1981) in a follow-up to X-ray observations made with
Einstein Observatory. All three clusters are included in the ROSAT
Brightest Cluster Sample (Ebeling {\it et al.\ } 1996a).
Combining the high spectral resolution ASCA data with ROSAT images
we present consistent determinations of the temperatures, metallicities, luminosities
and cooling rates in the clusters.
The data for Zwicky 3146, Abell 1835 and E1455+223 identify them as
the three largest cluster cooling flows known to date. We constrain the level
of intrinsic X-ray absorption in the cooling flows and relate the results to
measurements of intrinsic reddening in
the Central Cluster Galaxies (CCGs) of the systems.
Results on the distributions of X-ray gas, galaxies, and the total gravitating
matter in the clusters are reported.
The structure of this paper is as follows. In Section 2 we summarize the
observations. In Section 3 we present the X-ray and optical imaging data
and discuss the morphological relationships between the
clusters and their CCGs. In Section 4 we discuss
the spectral analysis of the X-ray data. In Section 5 present the results from the
deprojection analyses of the ROSAT images. Section 6 discusses the
optical properties of the clusters. In Section 7 we discuss some of the more
important results in detail, and in Section 8 summarize our conclusions.
Throughout this paper, we assume $H_0$=50
\hbox{$\kmps\Mpc^{-1}$}, $\Omega = 1$ and $\Lambda = 0$.
\section{Observations}
\begin{table*}
\vskip 0.2truein
\begin{center}
\caption{Observation summary}
\vskip 0.2truein
\begin{tabular}{ c c c c c c c }
\hline
Cluster & ~ & Instrument & ~ & Observation Date & ~ & Exposure (ks) \\
&&&&&& \\
Zwicky 3146 & ~ & ASCA SIS0 & ~ & 1993 May 18 & ~ & 26.3 \\
& ~ & ASCA SIS1 & ~ & " " & ~ & 31.6 \\
& ~ & ASCA GIS2 & ~ & " " & ~ & 30.2 \\
& ~ & ASCA GIS3 & ~ & " " & ~ & 30.2 \\
& ~ & ROSAT HRI \#1 & ~ & 1992 Nov 27 & ~ & 15.2 \\
& ~ & ROSAT HRI \#2 & ~ & 1993 May 17 & ~ & 10.8 \\
& ~ & ROSAT PSPC & ~ & 1993 Nov 13 & ~ & 8.62 \\
& ~ & ESO 3.6m (R) & ~ & 1992 Nov 11 & ~ & 0.12 \\
&&&&&& \\
Abell 1835 & ~ & ASCA SIS0 \#1 & ~ & 1994 Jul 20 & ~ & 18.2 \\
& ~ & ASCA SIS1 \#1 & ~ & " " & ~ & 17.2 \\
& ~ & ASCA GIS2 \#1 & ~ & " " & ~ & 13.0 \\
& ~ & ASCA GIS3 \#1 & ~ & " " & ~ & 13.0 \\
& ~ & ASCA SIS0 \#2 & ~ & 1994 Jul 21 & ~ & 8.51 \\
& ~ & ASCA SIS1 \#2 & ~ & " " & ~ & 8.22 \\
& ~ & ASCA GIS2 \#2 & ~ & " " & ~ & 6.55 \\
& ~ & ASCA GIS3 \#2 & ~ & " " & ~ & 6.56 \\
& ~ & ROSAT HRI & ~ & 1993 Jan 22 & ~ & 2.85 \\
& ~ & ROSAT PSPC & ~ & 1993 Jul 03 & ~ & 6.18 \\
& ~ & Hale 5m (Gunn i) & ~ & 1994 Jun 09 & ~ & 1.00 \\
& ~ & Hale 5m (KC B) & ~ & 1994 Jun 10 & ~ & 0.50 \\
& ~ & Hale 5m (KC U) & ~ & 1994 Jun 09 & ~ & 3.00 \\
&&&&&& \\
E1455+223 & ~ & ASCA SIS0 & ~ & 1994 Jul 18 & ~ & 30.5 \\
& ~ & ASCA SIS1 & ~ & " " & ~ & 28.3 \\
& ~ & ASCA GIS2 & ~ & " " & ~ & 18.3 \\
& ~ & ASCA GIS3 & ~ & " " & ~ & 18.3 \\
& ~ & ROSAT HRI \#1 & ~ & 1992 Jan 11 & ~ & 4.09 \\
& ~ & ROSAT HRI \#2 & ~ & 1993 Jan 20 & ~ & 4.23 \\
& ~ & ROSAT HRI \#3 & ~ & 1994 Jul 07 & ~ & 6.57 \\
& ~ & Hale 5m (Gunn i) & ~ & 1994 Jun 10 & ~ & 0.50 \\
& ~ & Hale 5m (KC B) & ~ & 1994 Jun 10 & ~ & 0.60 \\
& ~ & Hale 5m (KC U) & ~ & 1994 Jun 10 & ~ & 3.00 \\
\hline
&&&&&& \\
\end{tabular}
\end{center}
\parbox {7in}
{Notes: Exposure times are effective exposures after all
cleaning and correction procedures have been carried out. For Abell 1835 the
ASCA observations were carried out in 2 parts, yielding total exposures of 26.7,
25.4, 19.6 and 19.6 ks for the S0, S1, G2 and G3 detectors, respectively. For Zwicky 3146,
two separate ROSAT HRI observations were made, yielding a total exposure of 26.0
ks. For E1455+223 three HRI observations were carried out providing a total
exposure of 14.9 ks. }
\end{table*}
\begin{table*}
\vskip 0.2truein
\begin{center}
\caption{Target summary}
\vskip 0.2truein
\begin{tabular}{ c c c c c c c c c c c c c c c c }
\hline
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{} &
\multicolumn{2}{c}{OPTICAL (J2000.)} &
\multicolumn{2}{c}{X-RAY (J2000.)} &
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{} \\
Cluster & ~ & $z$ & R.A. & Dec. & R.A. & Dec. & $F_X$ & $L_X$ \\
&&&&&&&& \\
Zwicky 3146 & ~ & 0.2906 & $10^{\rm h}23^{\rm m}39.6^{\rm s}$ & $04^{\circ}11'10''$ & $10^{\rm h}23^{\rm m}39.8^{\rm s}$ & $04^{\circ}11'11''$ & 6.6 & 2.8 \\
Abell 1835 & ~ & 0.2523 & $14^{\rm h}01^{\rm m}02.0^{\rm s}$ & $02^{\circ}52'42''$ & $14^{\rm h}01^{\rm m}01.9^{\rm s}$ & $02^{\circ}52'43''$ & 12.3 & 3.8 \\
E1455+223 & ~ & 0.2578 & $14^{\rm h}57^{\rm m}15.1^{\rm s}$ & $22^{\circ}20'31''$ & $14^{\rm h}57^{\rm m}15.0^{\rm s}$ & $22^{\circ}20'36''$ & 3.7 & 1.3 \\
\hline
&&&&&&&& \\
\end{tabular}
\end{center}
\parbox {7in}
{Notes: Redshifts and CCG coordinates (J2000) from
Allen {\it et al.\ } (1992). X-ray coordinates denote the position of the X-ray peak determined from the
HRI data. X-ray fluxes ($F_X$) in units of $10^{-12}$ \hbox{$\erg\cm^{-2}\s^{-1}\,$} and luminosities ($L_X$)
in $10^{45}$ \hbox{$\erg\s^{-1}\,$}~are determined from the ASCA (S0) data. Fluxes are quoted in the 2-10 keV band of the observer. Luminosities
are absorption-corrected and are quoted in the 2-10 keV rest-frame of the object.}
\end{table*}
The details of the observations are summarised in Table 1. Exposure times for
the ASCA data sets are effective exposures after standard data screening and
cleaning procedures have been applied (Day {\it et al.\ } 1995). Hashed numbers
following an instrument name indicate that those observations were carried out
on more than one date. The ASCA observations of Abell 1835 were made in two
parts, on consecutive days in 1994 July.
The ROSAT HRI observations of Zwicky 3146 were carried out on 2 dates, in 1992
November and 1993
May, giving a
total exposure of 26.0 ks. The HRI observations of E1455+223 were carried out
in 3 parts, in 1992 January -- 1994 July, with a total exposure time of 14.9 ks.
The ASCA observation of Zwicky 3146 was carried out in 1993 May during the PV
stage of the mission. The SIS detectors were used in 2-CCD mode with
the target positioned approximately at the boundary between chips 1 and 2 in
SIS0 (the nominal pointing position in 2-CCD mode during the PV phase). The
observations of Abell 1835 and E1455+223 were carried out during the AO-1 stage
of the ASCA program. These clusters were also observed in 2-CCD mode, but with
the targets positioned more centrally in chip 1 of SIS0. For a detailed
discussion of ASCA observing modes and instrument configurations see Day {\it et al.\ }
(1995). Reduction of the ASCA data was carried out using the FTOOLS package.
Standard selection and screening criteria were applied (Day {\it et al.\ } 1995).
The ROSAT data were analysed using the STARLINK ASTERIX package.
The optical observations of Zwicky 3146 was carried out with the 3.6m
telescope at the European Southern Observatory (ESO), La Silla, Chile. The ESO
Faint Object Spectrograph Camera was used with the TEK $512 \times 512$ CCD
(pixel scale 0.61 arcsec). An exposure of 120 sec in the R band was made in
seeing of $\sim 1.5$ arcsec. The optical observations of Abell 1835 and
E1455+223 were carried out with the 5m Hale Telescope, Palomar,
as part of a follow up study (Edge {\it et al.\ } 1996) of
the most X-ray luminous clusters in the ROSAT Brightest Cluster Sample.
The COSMIC instrument and TEK $2048 \times 2048$ chip (pixel scale 0.28
arcsec) were used.
Exposures of 1000 and 500 s were made in Gunn i, 500s and 600s in KC B, and 3000s in KC U,
for Abell 1835 and E1455+223 respectively. The seeing was $\sim 1.1$ arcsec.
The optical data were reduced and analysed in IRAF.
Figs. 1-3 show the optical images of the clusters, with the
ROSAT HRI X-ray contours overlaid. (Details of the smoothing algorithms are
given in the figure captions.)
\begin{figure*}
\vskip12.5cm
\caption{ The ESO 3.6m R band image of Zwicky 3146 with the ROSAT HRI X-ray
contours overlaid. The optical data have a pixel scale of $0.61$ arcsec and
were taken in $\sim 1.5$ arcsec seeing. The X-ray image has a pixel size of $2
\times 2$ arcsec$^2$ and has been adaptively smoothed (Ebeling, White \& Rangarajan 1996)
to give $\geq 36$ count smoothing element$^{-1}$. Contours are drawn at eight evenly-spaced
logarithmic intervals between 0.89 and 22.38 ct pixel$^{-1}$.
}
\end{figure*}
\begin{figure*}
\vskip12.5cm
\caption{ The Hale 5m Gunn i image of Abell 1835 with the ROSAT HRI X-ray
contours overlaid. The optical data have a pixel scale of $0.28$ arcsec and
were taken in $\sim 1.1$ arcsec seeing. The X-ray image has a pixel size
of $4 \times 4$ arcsec$^2$ and has been adaptively smoothed to give
$\geq 16$ count smoothing element$^{-1}$. Contours are drawn at six evenly-spaced logarithmic
intervals between 0.79 and 7.94 ct pixel$^{-1}$.
}
\end{figure*}
\begin{figure*}
\vskip12.5cm
\caption{ The Hale 5m Gunn i image of E1455+223 with the ROSAT HRI X-ray
contours overlaid. The optical data have a pixel scale of $0.28$ arcsec and
were taken in $\sim 1.1$ arcsec seeing. The X-ray image has a pixel size of $2
\times 2$ arcsec$^2$ and has been adaptively smoothed to give $\geq 25$ count
smoothing element$^{-1}$. Contours are drawn at nine evenly-spaced logarithmic
intervals between 0.21 and 8.32 ct pixel$^{-1}$.
}
\end{figure*}
\section{Morphology Analysis}
\begin{table*}
\vskip 0.2truein
\begin{center}
\caption{Isophote Analysis}
\vskip 0.2truein
\begin{tabular}{ c c c c c c c c c c c c c }
\hline
\multicolumn{1}{c}{Cluster} &
\multicolumn{1}{c}{} &
\multicolumn{4}{c}{OPTICAL} &
\multicolumn{1}{c}{} &
\multicolumn{4}{c}{X-RAY} \\
& ~ & pixel & range & ellipticity & P.A. & ~~ & pixel & range & ellipticity & P.A. \\
&&&&&&&&& \\
Zwicky 3146 & ~ & $0.61 \times 0.61$ & $1.2-3.7$ & $0.32 \pm 0.01$ & $125 \pm 1$ & ~~ & $8.0 \times 8.0$ & $16-24$ & $0.16 \pm 0.04$ & $127 \pm 8$ \\
Abell 1835 & ~ & $0.57 \times 0.57$ & $1.1-4.5$ & $0.21 \pm 0.03$ & $145 \pm 4$ & ~~ & $8.0 \times 8.0$ & $16-56$ & $0.20 \pm 0.07$ & $163 \pm 16$ \\
E1455+223 & ~ & $0.57 \times 0.57$ & $1.1-4.5$ & $0.16 \pm 0.01$ & $35 \pm 2$ & ~~ & $8.0 \times 8.0$ & $16-48$ & $0.20 \pm 0.05$ & $40 \pm 7$ \\
\hline
&&&&&&&&& \\
\end{tabular}
\end{center}
\parbox {7in}
{Notes: A summary of the results from the isophote analysis of the optical
(CCG) and X-ray (cluster) data. Columns (2) and (6) give the pixel sizes in
arcsec of the re-binned optical and X-ray images used in the analyses. Columns
(3) and (7) list the range (in arcsec) of semi-major axes analysed. Columns
(4) and (8)
give the mean ellipticities (defined as $1-b/a$ where $b$ and $a$ are the
semi-minor and semi-major axes respectively) of the optical and X-ray data in
the regions analysed. Columns (5) and (9) list the mean position angles (PA)
in degrees in these same regions.}
\end{table*}
The images presented in Figs. 1-3, and the optical and X-ray co-ordinates listed in Table 2,
demonstrate excellent agreement between the positions of the CCGs and the
positions of the peaks of the
X-ray emission from the clusters. [Errors of $\mathrel{\spose{\lower 3pt\hbox{$\sim$} 5$ arcsec may be
associated with the aspect solutions of the HRI data. The astrometry of the
optical data is accurate to within 1 arcsec.]
The ellipticities and position angles of the X-ray emission from the clusters
and the optical emission from the CCGs have been examined using the ELLIPSE
isophote-analysis routines in IRAF. The images were re-binned to a suitable
pixel size ($8 \times 8$ arcsec$^2$ for the X-ray data and $0.57 \times 0.57$
arcsec$^2$ for the optical images) and were modelled with elliptical isophotes
(Jedrzejewski 1987). The ellipticities, position angles and centroids of the
isophotes were free parameters in the fits. The results are summarized in
Table 3 where we list the mean ellipticities and position angles over the
range of semi-major axes studied. We find excellent agreement between the
position angles of the optical (CCG) and X-ray (cluster) isophotes.
The agreement between the position angles of the CCG and cluster isophotes,
and the coincidence of the CCG coordinates and the peaks of the cluster X-ray
emission, are similar to the results from studies of other large cooling-flow
clusters at lower redshifts (White {\it et al.\ } 1994; Allen {\it et al.\ } 1995; Allen {\it et al.\ } 1996).
\section{Spectral Analysis of the ASCA data}
\subsection{Method of Analysis}
\begin{table}
\vskip 0.2truein
\begin{center}
\caption{Regions included in the spectral analysis}
\vskip 0.2truein
\begin{tabular}{ c c c c c }
&&&& \\
\hline
Cluster & ~ & Detector & ~ & Radius (arcmin/kpc) \\
&&&& \\
Zwicky 3146 & ~ & SIS0 & ~ & 5.0/1620 \\
& ~ & SIS1 & ~ & 4.0/1230 \\
& ~ & SIS2 & ~ & 6.0/1940 \\
& ~ & SIS3 & ~ & 6.0/1940 \\
&&&& \\
Abell 1835 & ~ & SIS0 & ~ & 4.0/1190 \\
& ~ & SIS1 & ~ & 3.0/890 \\
& ~ & SIS2 & ~ & 6.0/1780 \\
& ~ & SIS3 & ~ & 6.0/1780 \\
&&&& \\
E1455+223 & ~ & SIS0 & ~ & 3.8/1140 \\
& ~ & SIS1 & ~ & 2.8/840 \\
& ~ & SIS2 & ~ & 6.0/1800 \\
& ~ & SIS3 & ~ & 6.0/1800 \\
\hline
&&&& \\
\end{tabular}
\end{center}
\end{table}
SIS spectra were extracted from circular regions centred on the positions of
the X-ray peaks. The radii of these regions were selected to minimize the
number of chip boundaries crossed (and thereby minimize systematic
uncertainties introduced into the data by such crossings) whilst covering as
large a region of the clusters as possible. The compromise of these
considerations lead to the choice of regions for spectral analysis listed in
Table 4. For Abell 1835 and E1455+223 the spectra were extracted from a single
chip in each SIS (chip 1 in SIS0 and chip 3 in SIS1). For Zwicky 3146, which
is centred on the boundary between chips 1 and 2 in SIS0 (chips 3 and 0 in
SIS1), the data were extracted across the two chips in circular regions
bounded by the outer chip edges. The GIS spectra used for the analysis of the
cluster properties were extracted from circular regions of radius 6 arcmin
(corresponding to $\sim 2$ Mpc at the redshifts of the clusters) again centred
on the peak of the X-ray emission from the clusters.
Background subtraction was carried out using the `blank sky'
observations compiled during the performance verification stage of the ASCA
mission. (The blank sky observations are compiled from observations of high
Galactic latitude fields free of bright X-ray sources). For X-ray sources
lying in directions of relatively low
Galactic column density, like the targets discussed in this paper, the blank sky
observations provide a reasonable
representation of the cosmic and instrumental backgrounds in the detectors over
the energy ranges of interest.
The modelling of the X-ray spectra has been carried out using the XSPEC
spectral fitting package (version 8.50; Shafer {\it et al.\ } 1991). For the SIS
analysis, the 1994 November 9 release of the response matrices from GSFC was
used. Only those counts in pulse height analyser (PHA) channels corresponding
to energies between 0.5 and 10 {\rm\thinspace keV}~ were included in the fits (the energies
between which the calibration of the SIS is best-understood). For the GIS
analysis, the 1995 March 6 release of the GSFC response matrices was used and
only data in the energy range $1 - 10$ {\rm\thinspace keV}~were included in the fits. All
spectra were grouped before fitting to ensure a minimum of 20 counts per PHA
channel, thereby allowing $\chi^2$ statistics to be used.
The X-ray emission from the clusters has been modelled using the plasma codes
of Raymond \& Smith (1977; with updates incorporated into XSPEC version 8.50)
and Kaastra \& Mewe (1993). The results for the two plasma codes show good
agreement. For clarity, only the results for the Raymond \& Smith (hereafter
RS) code will be presented in detail in this paper although the conclusions
drawn may equally be applied to the analysis with the Kaastra \& Mewe code.
We have modelled the ASCA spectra both by fitting the data from the individual detectors
independently and by combining the data from all 4 detectors. The results from the
fits to the individual detectors are summarised in Tables 5-7. When combining the data for the different detectors
the temperature, metallicity and column density values were linked together. However, the
normalizations of both the ambient cluster emission and the cooling flow components
were allowed to vary independently, due to
the different source extraction areas used and residual uncertainties in the flux calibration
of the instruments. The results from the fits to the combined data sets are summarized in Table 8.
\subsection{The spectral models}
\begin{table*}
\vskip 0.2truein
\begin{center}
\caption{Spectral Analysis of the ASCA data for Zwicky 3146}
\vskip 0.2truein
\begin{tabular}{ c c c c c c c c c c c }
&&&&&&&&&& \\
\hline
& ~ & Parameters & ~ & S0 & ~~~ & S1 & ~~~ & G2 & ~~~ & G3 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $5.6^{+0.3}_{-0.3}$ & ~~~ & $5.6^{+0.3}_{-0.4}$ & ~~~ & $6.2^{+0.6}_{-0.6}$ & ~~~ & $6.2^{+0.6}_{-0.6}$ \\
& ~ & $Z$ & ~ & $0.22^{+0.07}_{-0.06}$ & ~~~ & $0.30^{+0.09}_{-0.08}$ & ~~~ & $0.19^{+0.11}_{-0.11}$ & ~~~ & $0.31^{+0.13}_{-0.11}$ \\
MODEL A & ~ & $N_{\rm H}$ & ~ & $0.34$ & ~~~ & $0.34$ & ~~~ & $0.34$ & ~~~ & $ 0.34$ \\
& ~ & $\chi^2$/DOF & ~ & 243.7/235 & ~~~ & 256.1/207 & ~~~ & 182.3/209 & ~~~ & 173.9/214 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $5.9^{+0.7}_{-0.4}$ & ~~~ & $5.6^{+0.6}_{-0.5}$ & ~~~ & $6.5^{+0.6}_{-0.7}$ & ~~~ & $6.3^{+0.8}_{-0.8}$ \\
& ~ & $Z$ & ~ & $0.22^{+0.07}_{-0.07}$ & ~~~ & $0.30^{+0.10}_{-0.08}$ & ~~~ & $0.19^{+0.11}_{-0.11}$ & ~~~ & $0.32^{+0.12}_{-0.12}$ \\
MODEL B & ~ & $N_{\rm H}$ & ~ & $0.16^{+0.15}_{-0.15}$ & ~~~ & $0.34^{+0.18}_{-0.17}$ & ~~~ & $<0.39$ & ~~~ & $< 0.97$ \\
& ~ & $\chi^2$/DOF & ~ & 239.8/234 & ~~~ & 256.1/206 & ~~~ & 180.1/208 & ~~~ & 173.8/213 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $6.6^{+4.4}_{-1.0}$ & ~~~ & $5.6^{+2.3}_{-0.5}$ & ~~~ & $11.2^{+8.2}_{-5.0}$ & ~~~ & $12.3^{+4.4}_{-5.8}$ \\
& ~ & $Z$ & ~ & $0.23^{+0.08}_{-0.07}$ & ~~~ & $0.30^{+0.10}_{-0.09}$ & ~~~ & $0.30^{+0.17}_{-0.19}$ & ~~~ & $0.48^{+0.17}_{-0.19}$ \\
MODEL C & ~ & $N_{\rm H}$ & ~ & $0.38^{+0.43}_{-0.30}$ & ~~~ & $0.42^{+0.58}_{-0.24}$ & ~~~ & $<1.29$ & ~~~ & $1.26^{+0.82}_{-0.96}$ \\
& ~ & ${\dot M}$ & ~ & $<2900$ & ~~~ & $<2800$ & ~~~ & $<2290$ & ~~~ & $2240^{+260}_{-1760}$ \\
& ~ & $\chi^2$/DOF & ~ & 238.5/233 & ~~~ & 256.0/205 & ~~~ & 178.0/207 & ~~~ & 170.2/212 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $6.6^{+2.4}_{-0.9}$ & ~~~ & $5.4^{+1.3}_{-0.5}$ & ~~~ & $11.5^{+6.9}_{-5.1}$ & ~~~ & $12.7^{+3.9}_{-6.1}$ \\
& ~ & $Z$ & ~ & $0.23^{+0.08}_{-0.08}$ & ~~~ & $0.31^{+0.08}_{-0.09}$ & ~~~ & $0.30^{+0.17}_{-0.17}$ & ~~~ & $0.49^{+0.15}_{-0.20}$ \\
MODEL D & ~ & $N_{\rm H}$ & ~ & $0.34$ & ~~~ & $0.34$ & ~~~ & $0.34$ & ~~~ & $0.34$ \\
& ~ & ${\dot M}$ & ~ & $1150^{+1400}_{-850}$ & ~~~ & $<1100$ & ~~~ & $1740^{+570}_{-1340}$ & ~~~ & $2240^{+250}_{-1740}$ \\
& ~ & $\Delta N_{\rm H}$ & ~ & $<1.1$ & ~~~ & $U.C.$ & ~~~ & $<2.44$ & ~~~ & $<5.43$ \\
& ~ & $\chi^2$/DOF & ~ & 238.6/233 & ~~~ & 255.5/205 & ~~~ & 177.9/207 & ~~~ & 170.2/212 \\
\hline
&&&&&&&&&& \\
\end{tabular}
\end{center}
\parbox {7in}
{ Notes: The best-fit parameter values and 90 per cent ($\Delta \chi^2 =
2.71$) confidence limits from the spectral analysis of the ASCA data for Zwicky
3146. Temperatures ($kT$) are in keV and metallicities ($Z$) are quoted as a
fraction of the Solar value (Anders \& Grevesse 1989). Column densities
($N_{\rm H}$) are in units of $10^{21}$ atom cm$^{-2}$ and mass deposition
rates (\hbox{$\dot M$}) in \hbox{$\Msun\yr^{-1}\,$}. $\chi^2$ values and the number of degrees of
freedom (DOF) in the fits are given for the four spectral models discussed in
Section 4.2. }
\end{table*}
\begin{table*}
\vskip 0.2truein
\begin{center}
\caption{Spectral Analysis of the ASCA data for Abell 1835}
\vskip 0.2truein
\begin{tabular}{ c c c c c c c c c c c }
&&&&&&&&&& \\
\hline
& ~ & Parameters & ~ & S0 & ~~~ & S1 & ~~~ & G2 & ~~~ & G3 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $9.4^{+0.7}_{-0.6}$ & ~~~ & $9.4^{+1.1}_{-0.7}$ & ~~~ & $6.5^{+0.7}_{-0.6}$ & ~~~ & $6.8^{+0.6}_{-0.7}$ \\
& ~ & $Z$ & ~ & $0.24^{+0.10}_{-0.09}$ & ~~~ & $0.26^{+0.12}_{-0.12}$ & ~~~ & $0.26^{+0.13}_{-0.12}$ & ~~~ & $0.23^{+0.12}_{-0.11}$ \\
MODEL A & ~ & $N_{\rm H}$ & ~ & $0.22$ & ~~~ & $0.22$ & ~~~ & $0.22$ & ~~~ & $0.22$ \\
& ~ & $\chi^2$/DOF & ~ & 416.7/369 & ~~~ & 363.5/298 & ~~~ & 200.0/216 & ~~~ & 199.5/261 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $7.4^{+0.8}_{-0.6}$ & ~~~ & $7.1^{+0.9}_{-0.7}$ & ~~~ & $6.7^{+0.8}_{-1.0}$ & ~~~ & $6.0^{+0.9}_{-0.7}$ \\
& ~ & $Z$ & ~ & $0.23^{+0.08}_{-0.07}$ & ~~~ & $0.26^{+0.09}_{-0.09}$ & ~~~ & $0.26^{+0.14}_{-0.12}$ & ~~~ & $0.23^{+0.11}_{-0.10}$ \\
MODEL B & ~ & $N_{\rm H}$ & ~ & $0.73^{+0.15}_{-0.14}$ & ~~~ & $0.81^{+0.19}_{-0.19}$ & ~~~ & $<0.82$ & ~~~ & $1.04^{+0.75}_{-0.69}$ \\
& ~ & $\chi^2$/DOF & ~ & 379.2/368 & ~~~ & 334.4/297 & ~~~ & 199.9/215 & ~~~ & 195.7/260 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $8.8^{+6.6}_{-1.8}$ & ~~~& $7.2^{+5.0}_{-0.7}$ & ~~~ & $12.1^{+3.3}_{-6.1}$ & ~~~ & $6.0^{+6.1}_{-0.7}$ \\
& ~ & $Z$ & ~ & $0.26^{+0.09}_{-0.08}$ & ~~~& $0.26^{+0.08}_{-0.09}$ & ~~~ & $0.34^{+0.17}_{-0.17}$ & ~~~ & $0.23^{+0.13}_{-0.10}$ \\
MODEL C & ~ & $N_{\rm H}$ & ~ & $1.04^{+0.32}_{-0.40}$ & ~~~& $0.81^{+0.33}_{-0.19}$ & ~~~ & $1.03^{+0.83}_{-0.64}$ & ~~~ & $1.05^{+1.07}_{-0.52}$ \\
& ~ & ${\dot M}$ & ~ & $<2600$ &~~~ & $<2600$ & ~~~ & $<3500$ & ~~~ & $<4400$ \\
& ~ & $\chi^2$/DOF & ~ & 377.4/367 & ~~~& 334.4/296 & ~~~ & 198.3/214 & ~~~ & 195.7/259 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $9.7^{+3.5}_{-1.2}$ & ~~~ & $8.0^{+1.6}_{-1.2}$ & ~~~ & $9.5^{+7.5}_{-3.6}$ & ~~~ & $6.2^{+5.9}_{-0.9}$ \\
& ~ & $Z$ & ~ & $0.28^{+0.06}_{-0.08}$ & ~~~ & $0.28^{+0.11}_{-0.10}$ & ~~~ & $0.32^{+0.16}_{-0.16}$ & ~~~ & $0.24^{+0.13}_{-0.11}$ \\
MODEL D & ~ & $N_{\rm H}$ & ~ & $0.22$ & ~~~ & $0.22$ & ~~~ & $0.22$ & ~~~ & $0.22$ \\
& ~ & ${\dot M}$ & ~ & $2000^{+550}_{-450}$ & ~~~ & $2000^{+1000}_{-600}$ & ~~~ & $<4300$ & ~~~ & $<4300$ \\
& ~ & $\Delta N_{\rm H}$ & ~ & $3.25^{+2.25}_{-0.85}$ & ~~~ & $6.68^{+5.50}_{-2.59}$ & ~~~ & U.C. & ~~~ & U.C. \\
& ~ & $\chi^2$/DOF & ~ & 372.9/367 & ~~~ & 327.2/296 & ~~~ & 198.9/214 & ~~~ & 195.5/259 \\
\hline
&&&&&&&&&& \\
\end{tabular}
\end{center}
\parbox {7in}
{ Notes: The best-fit parameter values and 90 per cent ($\Delta \chi^2 = 2.71$)
confidence limits from the spectral analysis of the ASCA data for Abell 1835.
Details as for Table 5.}
\end{table*}
\begin{table*}
\vskip 0.2truein
\begin{center}
\caption{Spectral Analysis of the ASCA data for E1455+223}
\vskip 0.2truein
\begin{tabular}{ c c c c c c c c c c c }
&&&&&&&&&& \\
\hline
& ~ & Parameters & ~ & S0 & ~~~ & S1 & ~~~ & G2 & ~~~ & G3 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $5.0^{+0.4}_{-0.3}$ & ~~~ & $5.4^{+0.5}_{-0.5}$ & ~~~ & $4.5^{+0.8}_{-0.6}$ & ~~~ & $4.5^{+0.6}_{-0.5}$ \\
& ~ & $Z$ & ~ & $0.29^{+0.10}_{-0.10}$ & ~~~ & $0.14^{+0.11}_{-0.11}$ & ~~~ & $0.20^{+0.23}_{-0.16}$ & ~~~ & $0.39^{+0.24}_{-0.20}$ \\
MODEL A & ~ & $N_{\rm H}$ & ~ & $0.31$ & ~~~ & $0.31$ & ~~~ & $0.31$ & ~~~ & $0.31$ \\
& ~ & $\chi^2$/DOF & ~ & 178.0/170 & ~~~ & 138.1/138 & ~~~ & 94.0/83 & ~~~ & 123.2/104 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $4.2^{+0.4}_{-0.3}$ & ~~~ & $4.5^{+0.6}_{-0.5}$ & ~~~ & $4.0^{+1.0}_{-0.8}$ & ~~~ & $4.2^{+0.9}_{-0.7}$ \\
& ~ & $Z$ & ~ & $0.30^{+0.10}_{-0.09}$ & ~~~ & $0.16^{+0.10}_{-0.10}$ & ~~~ & $0.24^{+0.26}_{-0.21}$ & ~~~ & $0.41^{+0.25}_{-0.20}$ \\
MODEL B & ~ & $N_{\rm H}$ & ~ & $0.89^{+0.23}_{-0.22}$ & ~~~ & $0.89^{+0.29}_{-0.28}$ & ~~~ & $< 3.0$ & ~~~ & $< 2.2$ \\
& ~ & $\chi^2$/DOF & ~ & 157.7/169 & ~~~ & 125.1/137 & ~~~ & 92.5/82 & ~~~ & 122.4/103 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $5.3^{+2.3}_{-1.3}$ & ~~~ & $5.5^{+4.1}_{-1.4}$ & ~~~ & $7.1^{+3.1}_{-3.8}$ & ~~~ & $4.6^{+5.5}_{-1.0}$ \\
& ~ & $Z$ & ~ & $0.32^{+0.10}_{-0.10}$ & ~~~ & $0.16^{+0.11}_{-0.10}$ & ~~~ & $0.34^{+0.25}_{-0.27}$ & ~~~ & $0.43^{+0.35}_{-0.22}$ \\
MODEL C & ~ & $N_{\rm H}$ & ~ & $1.63^{+0.25}_{-0.25}$ & ~~~ & $1.44^{+0.52}_{-0.75}$ & ~~~ & $2.5^{+1.2}_{-2.2}$ & ~~~ & $<3.3$ \\
& ~ & ${\dot M}$ & ~ & $<3550$ & ~~~ & $<3500$ & ~~~ & $<2000$ & ~~~ & $<2390$ \\
& ~ & $\chi^2$/DOF & ~ & 155.6/168 & ~~~ & 123.8/136 & ~~~ & 91.6/81 & ~~~ & 122.3/102 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $5.2^{+2.3}_{-0.8}$ & ~~~ & $6.6^{+3.1}_{-1.9}$ & ~~~ & $6.9^{+3.2}_{-3.7}$ & ~~~ & $5.1^{+5.0}_{-1.6}$ \\
& ~ & $Z$ & ~ & $0.33^{+0.10}_{-0.10}$ & ~~~ & $0.17^{+0.11}_{-0.11}$ & ~~~ & $0.34^{+0.26}_{-0.26}$ & ~~~ & $0.46^{+0.33}_{-0.24}$ \\
MODEL D & ~ & $N_{\rm H}$ & ~ & $0.31$ & ~~~ & $0.31$ & ~~~ & $0.31$ & ~~~ & $0.31$ \\
& ~ & ${\dot M}$ & ~ & $2290^{+1040}_{-960}$ & ~~~ & $2500^{+750}_{-1100}$ & ~~~ & $<2320$ & ~~~ & $<2670$ \\
& ~ & $\Delta N_{\rm H}$ & ~ & $3.8^{+4.2}_{-1.5}$ & ~~~ & $2.6^{+4.0}_{-1.0}$ & ~~~ & U.C. & ~~~ & U.C. \\
& ~ & $\chi^2$/DOF & ~ & 154.1/168 & ~~~ & 124.0/136 & ~~~ & 91.4/81 & ~~~ & 122.3/102 \\
\hline
&&&&&&&&&& \\
\end{tabular}
\end{center}
\parbox {7in}
{Notes: The best-fit parameter values and 90 per cent ($\Delta \chi^2 = 2.71$)
confidence limits from the spectral analysis of the ASCA data for E1455+223.
Details as for Table 5.}
\end{table*}
\begin{figure}
\centerline{\hspace{2.3cm}\psfig{figure=spectrum_z3146_s0.ps,width=0.75\textwidth,angle=270}}
\caption{ (Upper Panel) The S0 spectrum for Zwicky 3146 with the best-fit
solution for Model B overlaid. The data have been binned-up by a factor 5
along the energy axis for display purposes.
(Lower Panel) Residuals to the fit. The positive residuals at $E \sim 0.55$ keV are
due to small systematic uncertainties in response matrix around the oxygen edge in the detector.
}
\end{figure}
\begin{figure}
\centerline{\hspace{2.3cm}\psfig{figure=spectrum_a1835_s0.ps,width=0.75\textwidth,angle=270}}
\caption{ (Upper Panel) The S0 spectrum for Abell 1835 with the best-fit
solution for Model B overlaid. (Lower Panel) Residuals to the fit. Details as in Fig. 4.}
\end{figure}
\begin{figure}
\centerline{\hspace{2.3cm}\psfig{figure=spectrum_e1455_s0.ps,width=0.75\textwidth,angle=270}}
\caption{ (Upper Panel) The S0 spectrum for E1455+223 with the best-fit
solution for Model B overlaid. (Lower Panel) Residuals to the fit. Details as in Fig. 4.}
\end{figure}
The ASCA spectra were first examined with a simple single-phase model
consisting of an RS component, to account for the X-ray emission from the
cluster, and a photoelectric absorption component (Morrison \& McCammon 1983)
normalized to the equivalent Galactic hydrogen column density
along the line-of-sight to the cluster. The
free parameters in this model (hereafter Model A) were the temperature,
metallicity and emission measure of the X-ray gas.
The redshift
of the X-ray emission from the cluster was fixed at the optically-determined
values for the CCGs (Table 2).
We then examined a second model (Model B) in which the absorbing
column density was also allowed to be a free parameter in the fits.
The fits to Abell 1835 and E1455+223 in particular showed
highly significant improvements with the introduction of this single
extra fit parameter.
(Note that absorbing material was assumed to lie at zero redshift in this model).
The best-fit parameter values and 90 per cent
($\Delta \chi^2 = 2.71$) confidence limits from the analyses
with the single-phase models (A and B) are summarized in Tables 5--8.
Although the single-phase modelling can provide a useful parameterization of
the properties of the cluster gas, the results obtained with such a model
should be interpreted with caution. The deprojection analyses presented
in Section 5
show that all three of the clusters discussed in this paper contain large cooling
flows. The gas in the central regions of these clusters must therefore be
highly multiphase {\it i.e.\ } contain a wide range of densities and temperatures at all
radii.
We therefore next examined the data with more sophisticated spectral models in which
the spectrum of the cooling flow was accounted for explicitly. The first of
these models
(Model C) consists of an RS component (to model the emission from the ambient
ICM in the region of interest) and a cooling-flow component (following the
models of Johnstone {\it et al.\ } 1992) modelling the X-ray spectrum of gas cooling
from the ambient cluster temperature, to temperatures below the X-ray waveband,
at constant pressure. Note that Model C introduces only one
extra free parameter into the fits relative to the single-phase Model B; the
mass deposition rate of cooling gas. The upper temperature of the cooling gas,
the metallicity, and the absorbing column density acting on the cooling flow were
tied to those of the ambient cluster emission modelled by the RS component.
Fourthly, we examined a further cooling-flow model (D; which we expect to be
the most physically-appropriate model) in which an
intrinsic X-ray absorbing column density was associated with the cooling-flow.
The excess absorption is modelled as a uniform absorbing screen in front of
the cooling flow, at the redshift of the cluster, with the column
density a free parameter in the fits. The
column density acting on the ambient cluster emission was fixed at the Galactic
value. The best-fit parameter values and confidence limits obtained with the
multiphase, cooling-flow models (C,D) are also summarized in Tables 5--8.
Finally, a fifth model in which the X-ray emission from the cluster was
parameterized by a combination of two RS components was studied. However, the
statistical significance of including the second RS component in the fits (2
extra free parameters) is low and the results on the temperatures and emission
measures of the two components, which are poorly constrained by the
data, are not presented here.
\subsection{Results from the spectral analysis}
\begin{table*}
\vskip 0.2truein
\begin{center}
\caption{All instruments combined together }
\vskip 0.2truein
\begin{tabular}{ c c c c c c c c c c c }
&&&&&&&&&& \\
\hline
& ~ & Parameters & ~ & Model A & ~~~ & Model B & ~~~ & Model C & ~~~ & Model D \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $5.74^{+0.19}_{-0.18}$ & ~~~ & $6.07^{+0.33}_{-0.33}$ & ~~~ & $6.9^{+2.7}_{-0.9}$ & ~~~ & $6.6^{+1.1}_{-0.7}$ \\
& ~ & $Z$ & ~ & $0.26^{+0.05}_{-0.04}$ & ~~~ & $0.26^{+0.04}_{-0.05}$ & ~~~ & $0.27^{+.05}_{-0.05}$ & ~~~ & $0.27^{+0.05}_{-0.05}$ \\
Zwicky 3146 & ~ & $N_{\rm H}$ & ~ & $0.34$ & ~~~ & $0.17^{+0.12}_{-0.09}$ & ~~~ & $0.54^{+0.34}_{-0.29}$ & ~~~ & $<1.02$ \\
& ~ & ${\dot M}$ & ~ & --- & ~~~ & --- & ~~~ & $1870^{+1270}_{-1350}$ & ~~~ & $1330^{+1220}_{-820}$ \\
& ~ & $\chi^2$/DOF & ~ & 864.4/871 & ~~~ & 858.6/870 & ~~~ & 852.3/866 & ~~~ & 853.6/866 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $8.41^{+0.38}_{-0.39}$ & ~~~ & $7.03^{+0.34}_{-0.33}$ & ~~~ & $9.1^{+5.3}_{-1.6}$ & ~~~ & $9.5^{+1.3}_{-1.7}$ \\
& ~ & $Z$ & ~ & $0.26^{+0.05}_{-0.05}$ & ~~~ & $0.26^{+0.06}_{-0.05}$ & ~~~ & $0.30^{+0.06}_{-0.06}$ & ~~~ & $0.31^{+0.06}_{-0.05}$ \\
Abell 1835 & ~ & $N_{\rm H}$ & ~ & $0.22$ & ~~~ & $0.72^{+0.11}_{-0.10}$ & ~~~ & $1.12^{+0.24}_{-0.34}$ & ~~~ & $3.8^{+1.6}_{-0.4}$ \\
& ~ & ${\dot M}$ & ~ & --- & ~~~ & --- & ~~~ & $2050^{+1280}_{-1440}$ & ~~~ & $2090^{+630}_{-700}$ \\
& ~ & $\chi^2$/DOF & ~ & 976.4/958 & ~~~ & 909.2/957 & ~~~ & 898.7/953 & ~~~ & 891.8/953 \\
&&&&&&&&&& \\
& ~ & $kT$ & ~ & $5.01^{+0.26}_{-0.26}$ & ~~~ & $4.29^{+0.25}_{-0.24}$ & ~~~ & $5.0^{+2.6}_{-0.7}$ & ~~~ & $5.4^{+1.9}_{-0.7}$ \\
& ~ & $Z$ & ~ & $0.23^{+0.06}_{-0.07}$ & ~~~ & $0.25^{+0.07}_{-0.06}$ & ~~~ & $0.26^{+0.07}_{-0.06}$ & ~~~ & $0.27^{+0.07}_{-0.07}$ \\
E1455+223 & ~ & $N_{\rm H}$ & ~ & $0.31$ & ~~~ & $0.90^{+0.17}_{-0.16}$ & ~~~ & $1.48^{+0.45}_{-0.49}$ & ~~~ & $3.8^{+2.0}_{-0.8}$ \\
& ~ & ${\dot M}$ & ~ & --- & ~~~ & --- & ~~~ & $1890^{+1300}_{-1490}$ & ~~~ & $2030^{+720}_{-880}$ \\
& ~ & $\chi^2$/DOF & ~ & 541.8/501 & ~~~ & 503.5/500 & ~~~ & 498.8/496 & ~~~ & 498.2/496 \\
\hline
&&&&&&&&&& \\
\end{tabular}
\end{center}
\parbox {7in}
{ Notes: The best-fit parameter values and 90 per cent ($\Delta \chi^2 =
2.71$) confidence limits from the spectral analyses with data from all 4 detectors
combined. Temperatures ($kT$) are in keV and metallicities ($Z$) are quoted as a
fraction of the solar Value (Anders \& Grevesse 1989). Column densities
($N_{\rm H}$) are in units of $10^{21}$ atom cm$^{-2}$. $kT$, $Z$,
and $N_{\rm H}$ are linked in the fits. However, the mass deposition
rates for each detector (\hbox{$\dot M$}; quoted in \hbox{$\Msun\yr^{-1}\,$}) are included as independent fit
parameters, due to variations in source extraction area and uncertainties in the flux
calibration of the instruments.
The \hbox{$\dot M$}~value quoted for models C and D is for the S0 detector.
}
\end{table*}
The results from the spectral analysis, presented in Tables 5--8, provide a
consistent description of the X-ray properties of the clusters. We find
good agreement in the results from the different detectors. Interestingly,
in a reduced $\chi^2$ sense, all four
spectral models provide a statistically adequate description of the ASCA spectra.
Even with the single-phase models, however, the statistical
improvement obtained by allowing the X-ray absorption to fit freely ({\it i.e.\ } the
improvement obtained with Model B over Model A) is very high -- particularly for
Abell 1835 and E1455+223 where a simple $F$-test (Bevington
1969) indicates the improvement to be significant at $ >> 99.9$ per cent
confidence. Model B indicates excess column densities
(assumed to lie at zero redshift)
in Abell 1835 and E1455+223 of $5.0^{+1.1}_{-1.1}
\times 10^{20}$ \rm atom cm$^{-2}$ and $5.9^{+1.7}_{-1.6} \times 10^{20}$ \rm atom cm$^{-2}$, respectively. The data for
Zwicky 3146 prefer a column density marginally less than the nominal Galactic value
(Stark {\it et al.\ } 1992).
The introduction of the cooling flow component into the fits with Model C
results in a further reduction in $\chi^2$. However, the statistical
significance of this improvement, relative to the model B results, is marginal, being required
at $> 95$ per cent confidence
only with the Abell 1835 data. (Note, however, that the improvement obtained with model C with respect to
model A is very high.)
In general, the lowest $\chi^2$ values are obtained with Model D.
However, it is difficult to interpret the improvement in $\chi^2$ obtained
with Model D with respect to Models A--C
in terms of a statistical significance since Model D
includes fit parameters and constraints not present in the other models.
Within the context of Model D the cooling flow component is required at high significance (Table 8). The data for
Abell 1835 and E1455+223 also require significant amounts of intrinsic absorption associated with their cooling flows
($N_{\rm H} = 3.8^{+1.6}_{-0.4} \times 10^{21}$ \rm atom cm$^{-2}$ for Abell 1835 and $3.8^{+2.0}_{-0.8} \times 10^{21}$
\rm atom cm$^{-2}$ for E1455+223) whereas the data for Zwicky 3146 are consistent with Galactic absorption.
Note that only the SIS data have the spectral resolution and sensitivity at lower
energies ($E \mathrel{\spose{\lower 3pt\hbox{$\sim$} 1$ keV) to detect the presence of cooling flows in the clusters. The GIS
data do not provide firm constraints on the emission from cooling gas.
We conclude that the ASCA spectra alone are unable to
discriminate at high significance between the multiphase cooling flow
models (C, D) and the
single-phase model (B) for the clusters. Although adopting Model D as intuitively the most
reasonable description of the X-ray emission from the clusters leads to a strong spectral requirement
for large cooling flows in all three systems, it is only through the
combination of the spectral results with the results from the deprojection analyses
discussed in Section 5, that the presence of massive cooling flows in these clusters is
firmly established.
Finally in this Section, we note the possible effects of uncertainties in
the low-energy calibration of the SIS data on our results. Our own analyses
of ASCA observations of bright, nearby X-ray sources indicate that the current
GSFC response matrices (released 1994 November 9) may slightly
overestimate the low-energy response of the SIS instruments. This
systematic effect can lead to overestimates of Galactic column densities
by $1-3 \times 10^{20}$ \rm atom cm$^{-2}$ (see also the discussion of calibration
uncertainties associated with the ASCA instruments on the ASCA Guest
Observer Facility World Wide Web pages at
${http://heasarc.gsfc.nasa.gov/docs/asca/cal\_probs.html}$). Fixing the
Galactic column densities in our analyses with spectral Model D at values
$1.7 \times 10^{20}$ \rm atom cm$^{-2}$ in excess of the nominal Galactic values for the
clusters [$1.7 \times 10^{20}$ \rm atom cm$^{-2}$ being the best-fit systematic excess
column density determined from our analysis of ASCA observations of the
Coma cluster, which we expect to contain little or no intrinsic absorbing
material (White {\it et al.\ } 1991, Allen \& Fabian 1996), we determine best-fit
intrinsic column densities for Zwicky 3146, Abell 1835 and E1455+223 of
$N_{\rm H} < 0.6 \times 10^{21}$ \rm atom cm$^{-2}$, $N_{\rm H} = 3.2^{+1.4}_{-0.6}
\times 10^{21}$ \rm atom cm$^{-2}$, and $N_{\rm H} = 3.1^{+1.9}_{-0.6} \times 10^{21}$ \rm atom cm$^{-2}$,
respectively. Hence, our conclusions on the presence of
intrinsic absorbing material in these clusters are
essentially unaffected by uncertainties in the calibration of the
SIS instruments. Note that the temperature constraints are also
little-affected by the calibration uncertainties, with the best-fit
temperatures and 90 per cent confidence limits from the column
density-adjusted fits being $kT = 6.9^{+1.2}_{-0.7}$, $9.1^{+2.1}_{-1.3}$ and
$5.2^{+2.2}_{-0.7}$ keV, respectively.
\subsection{Single-phase and Multiphase temperature results}
The most notable differences between the results obtained with the single-phase (Model B)
and multiphase (Models C,D) models are in the temperature determinations for the clusters.
The cooling flow models imply significantly higher ambient cluster temperatures, and therefore
larger integrated cluster masses.
The presence of a cooling flow will naturally lead to differences
between the mean emission-weighted and mass-weighted temperatures for a
cluster. The X-ray emissivity of the cooler, denser material in the cooling flow
will be significantly higher than that of the surrounding hotter gas. The mean
emission-weighted temperature will therefore be biased to
temperatures below the mass-weighted value for the system. The
effects of emission-weighting become particularly important for
observations made with the comparatively low spectral resolution and limited ($0.1-2.4$ keV)
bandpass of the ROSAT PSPC. In Table 9 we present
the results from a spectral analysis of the PSPC data for the central 6 arcmin
radius regions of Zwicky 3146 and Abell 1835. The single-phase models again provide a
statistically adequate description of the spectra ($\chi^2_\nu \sim 1.0$) but the measured emission weighted temperatures
are only $3.2^{+1.4}_{-0.7}$ keV and $3.8^{+1.6}_{-0.9}$ keV respectively,
much less than $6.6^{+1.1}_{-0.7}$ keV and $9.5^{+1.3}_{-1.7}$ keV determined from the multiphase analysis of the ASCA spectra
(Model D). The importance
of distinguishing between single-phase and multiphase models in the analysis of X-ray data for clusters
is discussed in more detail in Section 7.1.
\begin{table}
\vskip 0.2truein
\begin{center}
\caption{PSPC spectra for Zwicky 3146 and Abell 1835}
\vskip 0.2truein
\begin{tabular}{ c c c c c }
&& \\
\hline
Parameter & ~ & Zwicky 3146 & ~ & Abell 1835 \\
&& \\
$kT$ & ~ & $3.2^{+1.4}_{-0.7}$ & ~ & $3.8^{+1.6}_{-0.9}$ \\
$Z$ & ~ & $0.52^{+1.68}_{-0.48}$ & ~ & $0.0^{+0.22}_{-0.0}$ \\
$N_{\rm H}$ & ~ & $0.23^{+0.05}_{-0.06}$ & ~ & $0.18^{+0.03}_{-0.02}$ \\
$\chi^2$/DOF & ~ & 26.3/22 & ~ & 17.0/20 \\
\hline
&& \\
\end{tabular}
\end{center}
\parbox {3.3in}
{ Notes: The best-fit parameter values and 90 per cent ($\Delta \chi^2 = 2.71$)
confidence limits from the spectral analysis of the PSPC data. }
\end{table}
\section{Deprojection Analysis}
\subsection{General results}
\begin{table}
\vskip 0.2truein
\begin{center}
\caption{Deprojection analyses of the clusters}
\vskip 0.2truein
\begin{tabular}{ c c c c c c c }
&&&&&& \\
\hline
& ~ & $t_{\rm cool}$ & ~ & $r_{\rm cool}$ & ~ & ${\dot M}$ \\
&&&&&& \\
Zwicky 3146 & ~ & $1.01^{+0.06}_{-0.06}$ & ~ & $231^{+50}_{-37}$ & ~ & $1355^{+408}_{-129}$ \\
Abell 1835 & ~ & $1.36^{+0.31}_{-0.31}$ & ~ & $231^{+26}_{-13}$ & ~ & $1106^{+455}_{-425}$ \\
E1455+223 & ~ & $1.15^{+0.17}_{-0.15}$ & ~ & $213^{+47}_{-33}$ & ~ & $732^{+162}_{-64}$ \\
\hline
&&&&&& \\
\end{tabular}
\end{center}
\parbox {3.3in}
{ Notes: A summary of the results from the deprojection analyses of the ROSAT
HRI data. Cooling times ($t_{\rm cool}$) are mean values for the central (8
arcsec) bin and are in units of $10^9$ {\rm\thinspace yr}. Cooling radii ($r_{\rm cool}$), the radii at
which the cooling time exceeds the Hubble time ($1.3 \times 10^{10}$ yr),
are in {\rm\thinspace kpc}. Integrated mass deposition rates within the cooling radii
(${\dot M}$ ) are in units of \hbox{$\Msun\yr^{-1}\,$}. Errors on the cooling times are the 10 and 90 percentile values from
100 Monte Carlo simulations. The upper and lower confidence limits on the
cooling radii are the points where the 10 and 90 percentiles exceed, and become
less than, the Hubble time, respectively. Errors on the mass deposition rates
are the 90 and 10 percentile values at the upper and lower limits for the
cooling radius. Galactic column densities as listed in Table 1 are
assumed.
}
\end{table}
\begin{figure*}
\centerline{\hspace{3.2cm}\psfig{figure=z3146_deproj_new.ps,width=1.35\textwidth,angle=270}}
\caption{ A summary of the results from the deprojection analysis of
the HRI data for Zwicky 3146. From left to right, top to bottom, we plot; (a)
surface brightness, (b) pressure, (c) integrated luminosity, (d) temperature,
(e) electron density, (f) cooling time, (g) integrated gas and gravitational
mass and (h) integrated mass deposition rate. Data points are mean values and
1$\sigma$ errors (in each radial bin) from 100 Monte Carlo simulations, except
for (d), (f) and (h) where the median and 10 and 90 percentile values have
been plotted.}
\end{figure*}
\begin{figure*}
\centerline{\hspace{3.2cm}\psfig{figure=a1835_deproj.ps,width=1.35\textwidth,angle=270}}
\caption{ A summary of the results from the deprojection analysis of
the HRI data for Abell 1835. Details as for Fig. 7}
\end{figure*}
\begin{figure*}
\centerline{\hspace{3.2cm}\psfig{figure=e1455_deproj_new.ps,width=1.35\textwidth,angle=270}}
\caption{ A summary of the results from the deprojection analysis of
the HRI data for E1455+225. Details as for Fig. 7}
\end{figure*}
We have carried out a deprojection analysis of the ROSAT images
using an updated version of the code of Fabian {\it et al.\ } (1981). Using assumptions of spherical
symmetry and hydrostatic equilibrium in the ICM, the deprojection technique can
be used to study the properties of the intracluster gas ({\it e.g.\ } density,
pressure, temperature, cooling rate) as a function of radius. The deprojection
method requires that either the total mass profile (which defines the pressure
profile) or the gas temperature profile be specified.
Following ASCA observations of nearby cooling flow clusters (Fabian
{\it et al.\ } 1996), and the results from the combined X-ray and
gravitational lensing study of the cooling-flow cluster PKS0745-191 (Allen {\it et al.\ }
1996), we assume that the mass-weighted temperature profiles
in the clusters remain constant, at the temperatures determined from the fits with
the multiphase spectral models to the combined detector data sets (Table 8).
Spectral model D is intuitively the preferred model, but using
the full temperature range allowed by models C and D probably provides a more
realistic estimate of the true uncertainty on the mass-weighted temperatures
in the highly complex, multiphase environments of the cooling flows.
Column densities were are fixed at the Galactic values from Stark {\it et al.\ } (1992),
but see also Section 5.3.
The clusters discussed in this paper are remarkably similar in their X-ray
properties to PKS0745-191 and the analogy to that system is a reasonable one.
It should be noted that although the
deprojection method of Fabian {\it et al.\ } (1981) is essentially a single-phase
technique, it produces results in good agreement with the more detailed
multi-phase treatment of Thomas, Fabian \& Nulsen (1987) and, due to its
simple applicability at large radii, is better-suited to the present project.
The azimuthally averaged X-ray surface brightness profiles of the clusters
determined from the HRI data (background-subtracted and corrected for telescope
vignetting) and the results from the deprojection analyses are summarized in
Figs. $7-9$. The primary results on the cooling flows in the clusters; the
central cooling times, the cooling radii and the integrated mass deposition
rates within the cooling radii are listed in Table 10. The results on the
cooling flow in Zwicky 3146 are in good agreement with those reported by Edge
{\it et al.\ } (1994) from an earlier analysis of the HRI data. The mass deposition from the
cooling flows is distributed throughout the inner $\sim 200$ kpc of the clusters with ${\dot M}
\mathrel{\spose{\lower 3pt\hbox{$\sim$} r$. As noted in Section 1, such distributed mass deposition
profiles requires that the central ICM is inhomogeneous (Nulsen
1986; Thomas, Fabian \& Nulsen 1987; Fabian 1994). Note also that the mass
deposition profiles shown in Figs. $7-9$(h) flatten at radii $< 200$ kpc,
where the cooling time is $\mathrel{\spose{\lower 3pt\hbox{$\sim$} 10^{10}$ yr. Thus accounting for the
look-back time to the clusters ($\sim 4 \times 10^{9}$ yr) does not
significantly alter the integrated mass deposition rates.
\subsection{Parameterization of the cluster masses}
\begin{figure}
\centerline{\hspace{3.2cm}\psfig{figure=baryon_all_new.ps,width=0.7\textwidth,angle=270}}
\caption{The ratio of the gas mass to the total mass as a function of radius.}
\end{figure}
\begin{table}
\vskip 0.2truein
\begin{center}
\caption{Cluster mass distributions}
\vskip 0.2truein
\begin{tabular}{ c c r l r l c }
\hline
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{} &
\multicolumn{2}{c}{$kT$} &
\multicolumn{2}{c}{$\sigma$} &
\multicolumn{1}{c}{$r_c$} \\
&&&& \\
Zwicky 3146 & ~ & $6.6^{+3.0}_{-0.7}$ & $(^{+1.1}_{-0.7})$ & $850^{+175}_{-50}$ & $(^{+60}_{-50})$ & 45 \\
Abell 1835 & ~ & $9.5^{+4.9}_{-2.0}$ & $(^{+1.3}_{-1.7})$ & $1000^{+300}_{-100}$ & $(^{+150}_{-70})$ & 50 \\
E1455+223 & ~ & $5.4^{+2.2}_{-1.1}$ & $(^{+1.9}_{-0.7})$ & $720^{+180}_{-70}$ & $(^{+150}_{-40})$ & 45 \\
\hline
&&&& \\
\end{tabular}
\end{center}
\parbox {3.3in}
{ Notes: A summary of the temperature constraints (in keV) and the
velocity dispersions (in \hbox{$\km\s^{-1}\,$}) and core radii (in kpc) of the
isothermal mass distributions (Binney \& Tremaine 1989) required to produce the
flat temperature profiles shown in Figs. $7-9$(d). Errors on the temperatures give the
the full range allowed by spectral models C and D. (The tighter
constraints determined with model D alone are given in parentheses.)
Errors on the velocity dispersions show the range of values required
to match the temperature results.}
\end{table}
In Table 11 we summarize the mass distributions required to produce
the flat temperature profiles
described in Section 5.1 We have parameterized
the mass distributions as isothermal spheres
(Equation 4-125 of Binney \& Tremaine 1987) with core radii, $r_c$, and
velocity dispersions, $\sigma$. In Fig. 11 we
plot the X-ray gas mass/total mass ratios for the central 500 kpc of the
clusters (where the total mass is assumed to be described by the
best-fit parameters listed in Table 11).
\subsection{The correction for intrinsic absorption}
As discussed in Section 5.1, for the purposes of deprojection
we have assumed that the column densities to the clusters are given by the
Galactic values of Stark {\it et al.\ } (1992). However, the spectral analysis of Section 4 clearly
shows that both Abell 1835 and E1455+223 exhibit significant excess absorption.
This can have a significant effect on the mass deposition rates determined from the deprojection
analyses.
To correct the deprojection results for intrinsic absorption, we
adopt spectral model D as the most reasonable description for the clusters.
The intrinsic column densities in Abell 1835 and E1455+223 ($N_{\rm H} = 3.8 \times 10^{21}$
\rm atom cm$^{-2}$; Table 8) are assumed to be in uniform screens in front of the cooling flows. The effects of
absorption on the observed HRI count rates from the cooling flows have been calculated from
XSPEC simulations, using the ROSAT HRI response matrix issued by GSFC. For
both Abell 1835 and E1455+223, the intrinsic absorption acts to reduce the count rates
from the cooling flows by a factor of two (in detail, factors 2.07 and 2.04, respectively). The true mass deposition rates from
the cooling flows can therefore be assumed to be a factor 2 larger than the values
listed in Table 10. [For Zwicky 3146, the maximum allowed intrinsic column density of $10^{21}$
\rm atom cm$^{-2}$ implies a maximum correction factor to the mass deposition rate of 1.28.]
The absorption-corrected mass deposition rates for the clusters are summarized in Table 12.
The agreement between these values and the spectrally-determined mass
deposition rates (Model D) is excellent.
\section{Optical properties of the clusters}
\subsection{The galaxy populations}
The galaxy populations in Abell 1835 and E1455+223 have been studied using the
Palomar B and I images of the clusters. The SExtractor software of Bertin
(1995) was used to identify galaxy candidates from their total (Kron)
I magnitudes and $(B-I)$ colours (measured in 3.0 arcsec diameter apertures from
seeing-matched images). For Abell 1835, 283 objects were selected with
$15.0<I<21.0$ and $2.5<(B-I)<3.2$. The CCG (which is bluer than allowed by this range)
was also included, making 284 galaxy identifications in total. For E1455+223,
188 objects with $15.0<I<21.0$ and $1.8<(B-I)<2.7$ were identified.
Absolute V magnitudes were
calculated from the observed I magnitude, and applying the appropriate
K-corrections.
In Figs. 11 (a), (b) we plot the (projected) galaxy profiles and the
ratio of the galaxy mass to the total mass
for Abell 1835 and E1455+223. A value for (M/L)$_V$ of 10 has been assumed.
Note the large central peaks in the ratio profiles [Fig. 12(b)] due to the
luminous central galaxies, and the near-flatness of the ratios at large radii
($r \mathrel{\spose{\lower 3pt\hbox{$\sim$} 0.5$ Mpc) indicating that the galaxies follow an approximately isothermal
distribution.
\begin{figure}
\centerline{\hspace{3.2cm}\psfig{figure=galaxies_total.ps,width=0.7\textwidth,angle=270}}
\vskip -0.2truein
\centerline{\hspace{3.2cm}\psfig{figure=galaxies.ps,width=0.7\textwidth,angle=270}}
\caption{(a) The projected galaxy mass as a function of radius in Abell 1835
and E1455+223. (b) The ratio of the galaxy mass to the total mass (derived using
the temperature constraints from spectral Model D).}
\end{figure}
\subsection{Spectra of the central cluster galaxies}
\begin{figure*}
\centerline{\hspace{0.0cm}\psfig{figure=e1455_spectrum_optical.ps,width=0.8\textwidth,angle=270}}
\caption{ Optical spectrum of the CCG of E1455+223 obtained with the Faint
Object Spectrograph on the INT in June 1991.}
\end{figure*}
Optical spectra for the CCGs of Zwicky 3146 and Abell 1835 are presented and
discussed by Allen (1995). These CCGs are two of the most
optically line-luminous central galaxies known, with luminosities in
H$\alpha\lambda$6563 emission alone of $\sim 10^{43}$ \hbox{$\erg\s^{-1}\,$}.
In Fig. 13 we present the spectrum of the CCG in
E1455+223. The data were obtained with the Faint Object Spectrograph on the
Isaac Newton Telescope (INT) in June 1991. A slit width of 1.5 arcsec was used,
providing a resolution of $\sim 16$ \AA~in the first spectral order.
The CCG of E1455+223 exhibits strong, narrow emission
lines (FWHM $\sim 500$ \hbox{$\km\s^{-1}\,$}) and an enhanced blue continuum with respect to a typical elliptical
galaxy spectrum. Both features are characteristic properties of CCGs in large
cooling flows (Johnstone, Fabian \& Nulsen
1987; Heckman {\it et al.\ } 1992; McNamara \& O'Connell 1989; Crawford \& Fabian 1993; Allen 1995).
The observed flux in H$\alpha\lambda$6563 of $1.9 \pm 0.1 \times
10^{-15}$ \hbox{$\erg\cm^{-2}\s^{-1}\,$} implies a slit luminosity of $6.1 \pm 0.3 \times
10^{41}$ \hbox{$\erg\s^{-1}\,$}. However, the H$\beta\lambda$4861 emission line is only
marginally detected and, after correcting for
absorption by the underlying stellar continuum (a factor $\sim 2$ correction),
we derive an H$\alpha\lambda$6563/H$\beta\lambda$4861
flux ratio of $\mathrel{\spose{\lower 3pt\hbox{$\sim$} 4.0$. This implies significant intrinsic reddening
at the source [$E(B-V) \mathrel{\spose{\lower 3pt\hbox{$\sim$} 0.3$] and an intrinsic
H$\alpha\lambda$6563 (slit) luminosity of $\mathrel{\spose{\lower 3pt\hbox{$\sim$} 10^{42}$ \hbox{$\erg\s^{-1}\,$}.
[The reddening laws of Seaton (1979) and Howarth (1983) have been
used.] Donahue, Stocke and Gioia (1992) present results from a narrow-band
H$\alpha$+[NII] imaging study of E1455+223. These authors show that the
line emission from the CCG is highly extended and that the total
line flux exceeds the slit flux reported here by a factor $\sim 7-8$.
The Donahue {\it et al.\ } (1992) results, together with the spectral results
reported here, thus imply an H$\alpha\lambda$6563 luminosity for
the CCG of E1455+223 of $\mathrel{\spose{\lower 3pt\hbox{$\sim$} 7 \times 10^{42}$ \hbox{$\erg\s^{-1}\,$},
comparable to that of Zwicky 3146 and Abell 1835.
\begin{table*}
\vskip 0.2truein
\begin{center}
\caption{Reddening and Excess absorption}
\vskip 0.2truein
\begin{tabular}{ c c c c c c c }
\hline
& ~ & $N_{{\rm H, X-ray}}$ & Spectral ${\dot M}$ & Corrected deproj. ${\dot M}$ & $E(B-V)$ & $N_{{\rm H, OPT}}$ \\
&&&&&& \\
Zwicky 3146 & ~ & $<1.0$ & $1330^{+1220}_{-820}$ & $1355^{+637}_{-161}$ & $0.22^{+0.15}_{-0.14}$ & $1.3^{+0.9}_{-0.8}$ \\
Abell 1835 & ~ & $3.8^{+1.6}_{-0.4}$ & $2090^{+630}_{-700}$ & $2291^{+943}_{-881}$ & $0.49^{+0.17}_{-0.15}$ & $2.8^{+1.0}_{-0.9}$ \\
E1455+223 & ~ & $3.8^{+2.0}_{-0.8}$ & $2040^{+720}_{-880}$ & $1491^{+330}_{-130}$ & $\mathrel{\spose{\lower 3pt\hbox{$\sim$} 0.3$ & $\mathrel{\spose{\lower 3pt\hbox{$\sim$} 1.7 $ \\
\hline
&&&&&& \\
\end{tabular}
\end{center}
\parbox {7in}
{Notes: Columns 2 and 3 list the intrinsic X-ray column densities
(without account for systematic uncertainties in the SIS calibration;
Section 4.3) and mass deposition rates determined from the spectral
analysis of the combined instrument data sets (Table 8). Column 4 lists the
mass deposition rates, determined by deprojection, corrected a posteriori for
the effects of X-ray absorption (Section 5.3). For Zwicky 3146, zero intrinsic
absorption has been assumed. Applying the maximum allowed correction factor
for this cluster (1.28) gives a mass deposition rate of $1734^{+815}_{-206}$.
$E(B-V)$ estimates in column 5 are determined from the
H$\alpha$6563/H$\beta$4861 line ratios in the CCGs. The data for Zwicky 3146
and Abell 1835 are from Allen (1995). $N_{\rm H, OPT}$ values are the column
densities of X-ray absorbing material implied by the $E(B-V)$ estimates,
following the relation of Bohlin, Savage \& Drake (1978). }
\end{table*}
The CCGs of Zwicky 3146 and Abell 1835 also exhibit
significant intrinsic reddening.
In Table 12 we summarize the results on intrinsic X-ray absorption and optical
reddening for the clusters. These results, together with the optical/X-ray/UV results
discussed by Allen {\it et al.\ } (1995) for a larger sample of cooling flows at
intermediate redshifts ($z \sim 0.15$),
indicate an interesting tendency for clusters with large column densities of
intrinsic X-ray absorbing material to
exhibit significant intrinsic reddening. The similarity of the column densities
inferred from the X-ray, optical and UV data, across a
variety of aperture sizes, suggests a dust-to-gas ratio in these galaxies similar to that
in our own Galaxy. The results also suggest that much of the dust
may be associated with, or entrained within, the X-ray absorbing gas.
\section{Discussion}
We have discussed, in detail, the X-ray properties of Zwicky 3146, Abell 1835
and E1455+223. We have shown that all three of these clusters contain
exceptionally large cooling flows.
Zwicky 3146 and Abell 1835 are amongst the most X-ray luminous clusters
known (Table 2). The cooling flows in these systems account for $\sim 15-20$ per cent
of the total intrinsic luminosity in the $2-10$ keV band, and as much as $\sim 40$ per cent in the
$0.1 - 2.4$ keV ROSAT band.
With E1455+223, which is a factor $2-3$ less luminous than the other clusters,
the cooling flow accounts for $\sim 35$ per cent of the luminosity in 2-10 keV band
and $\sim 60$ per cent of the emission between 0.1 and 2.4 keV.
Both Abell 1835 and E1455+223 exhibit significant intrinsic absorption in their ASCA spectra.
The need for excess absorption is found
in both the single-phase and multiphase (cooling flow) spectral analyses and cannot
reasonably be attributed to uncertainties in the Galactic column densities (Stark {\it et al.\ } 1992).
The most plausible interpretation of the excess absorption is that it is due
to material associated with the cooling flows (spectral Model D).
The mass of absorbing gas implied by an intrinsic column density of $\sim 3.8 \times 10^{21}$ \rm atom cm$^{-2}$,
distributed in a uniform (circular) screen across the central 220 kpc (radius $\sim r_{\rm cool}$) of the
clusters, is $\sim 4.6 \times 10^{12}$ \hbox{$\rm\thinspace M_{\odot}$}. Such a mass could plausibly be accumulated by the
cooling flows in Abell 1835 and E1455+223 in $\sim 2-3 \times 10^{9}$ yr. [See also White {\it et al.\ } (1991) and
Allen {\it et al.\ } (1993).] Note also the excellent
agreement in the mass deposition rates for the cooling flows
determined with the spectral and deprojection methods (Section 5.3) under this assumption for the
distribution of absorbing gas.
With spectral models B and C, wherein the excess absorption is assumed to cover
the whole cluster, the mass of absorbing gas is implausibly high.
The intrinsic column densities inferred for Abell 1835 and E1455+223 are similar to those observed in nearby
cooling flows (White {\it et al.\ } 1991; Allen {\it et al.\ } 1993; Fabian {\it et al.\ } 1994; Fabian {\it et al.\ } 1996).
Note also that the metallicities of $Z \sim 0.25 -0.30 Z_\odot$ measured for these clusters
are similar to those observed in
nearby systems, and imply that the the bulk of the enrichment of the ICM in
these clusters occurred before redshifts of $\sim 0.3$.
\subsection{Cooling flows and multiphase models}
\begin{table*}
\vskip 0.2truein
\begin{center}
\caption{Comparison of multiphase and single-phase results}
\vskip 0.2truein
\begin{tabular}{ c c c c c c c c }
\hline
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{} &
\multicolumn{2}{c}{PSPC (SP)} &
\multicolumn{2}{c}{ASCA (SP)} &
\multicolumn{2}{c}{ASCA (MP)} \\
Cluster & ~ & $kT$ & $\chi^2_\nu$ ($\nu$) & $kT$ & $\chi^2_\nu$ ($\nu$) & $kT$ & $\chi^2_\nu$ ($\nu$) \\
&&&&&&& \\
Zwicky 3146 & ~ & $3.2^{+1.4}_{-0.7}$ & 1.20 (22) & $6.1^{+0.3}_{-0.3}$ & 0.987 (870) & $6.6^{+1.1}_{-0.7}$ & 0.986 (866) \\
Abell 1835 & ~ & $3.8^{+1.6}_{-0.9}$ & 0.85 (20) & $7.0^{+0.3}_{-0.3}$ & 0.950 (957) & $9.5^{+1.3}_{-1.7}$ & 0.936 (953) \\
\hline
&&&&&&& \\
\end{tabular}
\end{center}
\parbox {7in}
{ Notes: A comparison of the temperature results obtained with the Single-phase (SP) and
MultiPhase (MP) models. SP results from ASCA are for spectral Model B. MP results are for spectral Model
D. }
\end{table*}
One of the most important results from this paper is the marked
difference in the temperatures of the clusters determined from the single-phase and multiphase
spectral models. These results are summarized in Table 13. The single-phase temperatures
consistently (and significantly) underestimate the multiphase results.
For Zwicky 3146 and Abell 1835, the ASCA single-phase results underestimate the
multiphase temperatures by $\sim 10$ and 25 per cent, respectively.
With the ROSAT data, the discrepancy in much more severe, with the PSPC values
underestimating the ASCA multiphase results by $\sim 3.4 $ keV (50 per cent) for Zwicky 3146, and
$\sim 6$ keV (60 per cent) for Abell 1835. Note, however, that in all cases the
reduced $\chi^2$ values indicate statistically acceptable fits.
The multiphase $kT$ results (Models C,D) should approximate the
true mass-weighted temperatures in the clusters
(Thomas {\it et al.\ } 1987; Waxman \& Miralda-Escude 1995; Allen {\it et al.\ }
1996). The single-phase results, however, are simply emission/detector-weighted average
values for the integrated cluster emission.
Since the X-ray emissivity of cluster gas rises with increasing density (decreasing temperature),
the presence of a large cooling flow
naturally leads to a decrease in the emission-weighted temperature of a cluster.
The effects on the emission-weighted $kT$ are most dramatic
in the $0.1-2.4$ keV band of the PSPC, where
the emission from cooler gas phases dominates the
detected flux. However, the (comparatively) poor spectral resolution and limited band-pass
of the PSPC, mean that the single-phase models can still provide a statistically adequate description of
the data.
The PSPC data are unable to discern the need for multi-temperature components
(although the imaging data clearly require them).
These results imply that caution should be applied in the interpretation of temperatures
determined with simple, single-phase models and, in particular, those determined
from ROSAT data.
With ASCA data the single-phase results should be more reliable, although significant
discrepancies can still arise (as in the case of Abell 1835).
The presence of a range of density and temperature phases is clearly established by the
data for cooling flow clusters. However, it should not assumed that the absence of a cooling flow
implies that a single-phase modelling of the ICM is appropriate.
The existence of cooling flows with distributed mass deposition requires significant
inhomogeneity (a density/temperature spread of $\sim$ a factor 2) in the ambient cluster
gas before the cooling flow forms (Nulsen 1986; Thomas, Fabian \& Nulsen 1987). The best data for clusters
are consistent with such a range of inhomogeneity (Allen {\it et al.\ } 1992b). The absence of cooling flows
in some nearby, luminous clusters such as the Coma cluster is usually attributed to merger events
having disrupted the cluster cores and having re-heated and redistributed the cooling gas throughout the
cluster. In such circumstances it seems unlikely that
the merger will completely homogenize the gas and, therefore, that a single-phase
model will provide an exact measure of the mass-weighted cluster temperature.
\subsection{A comparison with lensing masses}
\begin{figure*}
\vskip 12.5cm
\caption{ The Hale 5m U band image of Abell 1835.
Arc `A' is indicated to the South East of the CCG.
}
\end{figure*}
\begin{figure}
\centerline{\hspace{3.2cm}\psfig{figure=lensmass.ps,width=0.7\textwidth,angle=270}}
\caption{A comparison of the X-ray and lensing mass estimates for arc `A'. The bold
curve shows the mass within the arc determined from the standard lensing formula
(for a spherical mass distribution) as a function of the redshift of the arc.
The solid horizontal lines show the X-ray constraints on the mass within this
radius determined with spectral Model D ($7.8 < kT < 10.8$ keV). The
dashed lines show the constraints for spectral Model C
($7.5 < kT < 14.4$ keV). The lensing and multiphase X-ray results together
imply $z_{\rm arc} > 1.6$ for Model D, and $z_{\rm arc} > 0.7$ for Model C.
The lower dotted line shows the mass within arc `A'
implied by the single-phase spectral results ($M_{\rm proj} \sim 9.1 \times 10^{13}$ \hbox{$\rm\thinspace M_{\odot}$} for
$kT = 7.0$ keV). The single-phase X-ray results are inconsistent with the lensing data.}
\end{figure}
In Fig. 13 we show the
Palomar U band image of the central 2.5 arcmin$^2$ of Abell 1835. The CCG is the bright
source in the centre of the field. The image shows a number of
distorted features
(arcs, arclets and image pairs) attributable to gravitational lensing by
the cluster.
In particular, we observe a bright, elongated arc
(`A' in Fig. 13) at a radius of 30.3 arcsec from the centre of the CCG,
along a PA of 133 degree. The arc has a length of $\sim
16$ arcsec, is extended along a PA of 221 degree, and exhibits reflection
symmetry about the point $14^{\rm h}01^{\rm m}03.7{\rm s}$,
$02^{\circ}52'21''$.
For a simple, circular mass distribution the projected mass within the
tangential critical radius, $r_{\rm ct}$, is given by
\begin{equation}
M_{\rm proj}(r_{\rm ct}) ~ =
\frac{c^2 }{4 G} \left( \frac{D_{\rm arc}} {D_{\rm clus} D_{{\rm
arc-clus}}}
\right) ~ r_{\rm ct}^2
\end{equation}
where $r_{\rm ct}$ can be approximated by the arc radius
($r_{\rm arc} = 150$ kpc) and
$D_{\rm clus}$, $D_{\rm arc}$ and $D_{\rm arc-clus}$ are respectively
the angular
diameter distances from the observer to the cluster, the observer to the
lensed object, and the cluster to the lensed object.
In Fig. 14 we show the mass within arc `A' as a function of the redshift of
the arc, calculated with the above formula.
Also shown are the X-ray constraints on
the projected mass within this radius [$1.0 \times
10^{14}$ \hbox{$\rm\thinspace M_{\odot}$} $< M_{\rm proj} < 2.1 \times 10^{14}$ \hbox{$\rm\thinspace M_{\odot}$}~from the
multiphase analysis using spectral Model
C, and $1.1 \times 10^{14}$ \hbox{$\rm\thinspace M_{\odot}$} $ < M_{\rm proj} < 1.6 \times 10^{14}$
\hbox{$\rm\thinspace M_{\odot}$}~using spectral Model D].
Combining the X-ray and lensing mass results we are able
constrain the redshift of arc `A' to be $> 0.7$.
It is also important to note that if the single-phase
(Model B) X-ray temperature results for Abell 1835 were (wrongly) used in
place of the multiphase values,
no consistent solution for the X-ray and lensing masses would be possible.
The intrinsic ellipticity of the lensing potential may lead to a
slight ($\mathrel{\spose{\lower 3pt\hbox{$\sim$} 20$ per cent) overestimate of the lensing mass
determined with the circular mass model (Bartlemann 1995).
However, the effects of ellipticity also lead to a slight overestimate
of the X-ray mass within this aperture and, to first order, the conclusions
on the redshift of the arc should not be dramatically affected.
[The lensing properties of Abell 1835, and
those of a larger sample of X-ray luminous clusters observed with the Palomar
5m telescope, are discussed further by Edge {\it et al.\ } (1996). ]
Smail {\it et al.\ } (1995) report results from a study of (weakly) gravitationally
distorted images in the field of E1455+223, from which they derive a projected
mass within 450 kpc of the cluster centre of $\sim 3.6 \times 10^{14}$ \hbox{$\rm\thinspace M_{\odot}$}.
This mass exceeds the X-ray determination of the projected mass within this
radius, $1.6 \times 10^{14}$ \hbox{$\rm\thinspace M_{\odot}$}~(for an isothermal mass distribution
corresponding to a temperature of 5.4 keV),
by a factor $\sim 2$. [Note that the
determination of the X-ray mass assumes that the cluster remains isothermal
and extends to 3Mpc. However, extrapolating the mass profile to 5 Mpc
increases the projected mass within 450 kpc of the cluster centre by only $\sim
2$ per cent.] Using a potential consistent with the upper-limit to the
X-ray temperature of 7.6 keV ({\it i.e.\ } $\sigma = 900$
\hbox{$\km\s^{-1}\,$}, $r_c = 40$ kpc) we still determine a projected mass within 450 kpc of
only $2.5 \times 10^{14}$ \hbox{$\rm\thinspace M_{\odot}$}.
The lensing result on the cluster mass for E1455+223 appears high.
E1455+223 is a regular, relaxed cluster with a large cooling
flow, and a $2-10$ keV X-ray luminosity of $1.3 \times 10^{45}$ \hbox{$\erg\s^{-1}\,$}. The
ASCA constraints on the X-ray temperature ($4.3-7.6$ keV) are
consistent with results for other nearby, cooling-flow clusters of similar
X-ray luminosity ({\it e.g.\ } Abell 1795; Edge {\it et al.\ } 1990, Fabian {\it et al.\ } 1996), which
lends support to the X-ray mass determination. The velocity dispersion
of $\sigma = 660-900$ \hbox{$\km\s^{-1}\,$} (Table 11) implied by the X-ray
data is also in good agreement with optical observations
($\sigma \sim 700$ \hbox{$\km\s^{-1}\,$}; Mason {\it et al.\ } 1981, Smail {\it et al.\ } 1995).
A cluster of exceptional X-ray luminosity
and temperature is required to provide a projected mass
within 450 kpc consistent with the Smail {\it et al.\ } (1995) result for E1455+223.
Abell 1835, discussed in this paper, provides a mass
of $2.5-5.3 \times 10^{14}$ \hbox{$\rm\thinspace M_{\odot}$}. Similarly, the
exceptionally X-ray
luminous cooling-flow cluster
PKS0745-191 ($L_X = 2.8 \times 10^{45}$ \hbox{$\erg\s^{-1}\,$}), for
which Allen {\it et al.\ } (1996) present a self-consistent determination of the
mass distribution from X-ray and gravitational lensing data, provides
a projected mass within 450 kpc of only $\sim
3.7 \times 10^{14}$ \hbox{$\rm\thinspace M_{\odot}$}. Given the X-ray luminosity of E1455+223 (which is
a factor 2-3 less than PKS0745-191 or Abell 1835), the
X-ray mass measurement for the cluster seems reasonable
and the lensing mass high. The result of Smail {\it et al.\ } (1995) may imply an unusual redshift
distribution for the weakly distorted sources, or a projected mass
distribution that deviates significantly from the simple isothermal mass
model used (perhaps due to
some line of sight mass enhancement from material external to the X-ray
luminous part of the cluster).
\subsection{Optical and X-ray properties }
CCGs in cooling flows frequently exhibit characteristic
low-ionization, optical emission-line
spectra ({\it e.g.\ } Johnstone, Fabian \& Nulsen 1987; Heckman {\it et al.\ }
1989; Crawford \& Fabian 1992; Allen {\it et al.\ } 1995). The optical
(H$\alpha\lambda6563$) line luminosity correlates with
the excess UV/blue continuum luminosity
(the excess with respect to the UV/blue emission
expected from a normal gE/cD galaxy;
Johnstone, Fabian \& Nulsen 1987; McNamara \& O'Connell 1989; Allen {\it et al.\ } 1992a;
Crawford {\it et al.\ } 1995; Allen 1995).
Typically, both the emission lines and the excess UV/blue continua
are extended across the
central $10-20$ kpc of the clusters.
The clusters discussed in this paper contain three of the largest cooling
flows known. They also host three of the most optically line-luminous
(and UV/blue luminous) CCGs. Only the CCG of the massive cooling flow cluster
PKS0745-191 exhibits a
comparable optical line luminosity (Allen {\it et al.\ } 1996).
Although no simple correlation
between ${\dot M}$, $t_{\rm cool}$ and the optical line
luminosity (and therefore
UV/blue continuum luminosity) exists, the data presented here
confirm a tendency for the most optically-line-luminous CCGs
to be found in the largest cooling flows ({\it e.g.\ } Allen {\it et al.\ } 1995).
The UV/blue continua in Abell 1835 and Zwicky 3146 appear dominated by
emission from
hot, massive stars (Allen 1995). These young stellar populations may also
provide the bulk of the ionizing continuum emission responsible for
the observed optical emission lines.
The exceptionally high mass deposition rates from the cooling flows in
the clusters can naturally provide the
large reservoirs of cooled material necessary to fuel the
observed (very high) star formation rates (Allen 1995).
The star formation may also account for some of
the dust in the CCGs.
The excellent alignments between the (optical) CCG and (X-ray) cluster isophotes
are consistent with the results for other massive cooling flows at
intermediate ($z \sim 0.15$) redshifts (Allen {\it et al.\ } 1995). These results again
reveal the unique and intimate link between CCGs and their host clusters.
\section{CONCLUSIONS}
We have presented detailed results on the X-ray properties of
Zwicky 3146, Abell 1835
and E1455+223, three of the most distant, X-ray luminous clusters known.
We have shown that these clusters contain the
three largest cooling flows known, with mass deposition rates of $\sim 1400, 2300$
and 1500 \hbox{$\Msun\yr^{-1}\,$}, respectively. We have presented mass models for the clusters and
have highlighted the need for multiphase analyses to consistently explain
the spectral and imaging X-ray data for these systems.
The inappropriate use of single-phase models leads to significant underestimates
of the cluster temperatures and masses.
For Abell 1835 it was shown that a mass distribution
that can consistently explain both the X-ray and gravitational lensing data
for the cluster can only be formed
when multiphase X-ray spectral models are used.
We have discussed the relationship between intrinsic X-ray absorption
and optical reddening in the clusters. These results suggest that the X-ray
absorbing material frequently observed in the X-ray spectra of cooling
flows is dusty.
\section*{Acknowledgments}
SWA, ACF and ACE thank the Royal Society for support.
We thank I. Smail for communicating the results on the galaxy
photometry in Abell 1835 and E1455+223.
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\section{Introduction}
It is now a common belief among cosmologists and relativists that although
spacetime appears smooth, nearly flat, and four dimensional on large scales, at
sufficiently small distances and early times, it is highly curved with all
possible topologies and of arbitrary dimensions. The initial idea of
Kaluza-Klein has been extensively used in unified theories of fundamental
interactions; e.g. Green, Shwarz, \& Witten \shortcite{gsw}. There the extra dimensions are assumed to be compactified to
Planckian size, and therefore do not display themselves in macroscopic
processes. The multidimensional cosmologies based on these ideas have been
extensively studied in the last years [see e.g. Cho \shortcite{cho}, Kirillov \& Melnikov \shortcite{km}, Wesson \shortcite{wess}]. \\
Usually in cosmological models based on higher dimensions the problem of
the dimensionality of the gravitational coupling constant is not tackled
on, being tacitly assumed to be
\begin{equation}
\kappa = 8 \pi G, \label{kappa} \end{equation}
where $G$ is the Newtonian gravitational constant. This is ofcourse a relation
being derived in four dimension. Even in some textbooks this has not been
differentiated[see e.g. Kolb \& Turner \shortcite{kt}].
The effective gravitational constant in the
mutidimensional cosmologies is defined through the multiplication of this
four-dimensional value with some volume of the internal space which could
even be infinite[Rainer \& Zhuk \shortcite{rz}]. \\
There are however cosmological models in higher dimension
where this picture do not work and one should be cusious about the value of
the coupling constant in dimensions higher than four[see e.g. Chatterjee \shortcite{ca}, Chaterjee \& Bhui \shortcite{cb}, Khorrami, {\it et. al} \shortcite{kmme}, Khorrami, Mansouri \& Mohazzab \shortcite{kmm}]. In fact, Chatterjee \shortcite{ca}, Chaterjee \& Bhui \shortcite{cb} calculate a homogeneous model in a higher dimensional
gravity using the same 4-dimensional relation as in (1).\\
We propose to show some deficiencies of this misuse and propose a
generalization for the gravitational coupling constant for any arbitrary
dimension. Any definition of a coupling constant in higher dimension will
influence the time dependence of $G$ and therfore could have direct
observational consequences; e.g. Barrow \shortcite{barr}, Degl'Innocenti
, {\it et.al} \shortcite{df},
Ramero \& Melnikov \shortcite{rm}. \\
In this note we confine ourselves to a mere generalization of the $\kappa$,
using discrepancies in comparing relativistic and Newtonian cosmologies
in higher dimensions.
It belongs to the folklore of the theories of gravitation that their weak
limit must be the Newtonian theory of Gravitation. Moreover, on account of the
relation (1), there is a Newtonian derivation of the FRW cosmologies. We will
show that this derivation, using the coupling constant (1) is just valid in
$(3+1)$-dimension, and has to be changed for higher diemensional theories.
On account of the Gauss theorem we give a generalization of it, which makes
the Newtonian derivation valid in arbitrary dimensions.
\section{Friedmann models and Newtonian cosmology in D dimension}
Consider the following Hilbert-Einstein action in ${\bf D+1}$ dimensional
space-time with ${\bf D}$ as the fixed dimension of space:
\begin{equation}
S_g = -\frac{1}{2\kappa} \int \ ^{(D+1)}R \sqrt {-g} \ d^{D+1}x
+\frac{1}{2} \int T \sqrt{-g} \ d^{D+1}x, \label{h-e action}
\end{equation}
\noindent where $\kappa$ is the gravitational constant, again. For simplicity, we consider the $k=0$ FRW model. In an arbitrary fix dimension we ontain
the following Friedmann equation, \cite{kmme}
\begin{equation}
(\frac{\dot a}{a})^2=\frac{2 \kappa}{D(D-1)}\rho, \label{grfr}
\end{equation}
which leads to the familiar Friedmann equation in ${\bf D=}3$:
\begin{equation}
(\frac{\dot a}{a})^2=\frac{\kappa}{3} \rho = \frac{8 \pi G }{3} \rho.
\label{grfr=3}
\end{equation}
Now, it is well known that the Friedmann equation (\ref{grfr=3}) can be derived
from a Newtonian point of view. Taking the Newtonian equation of
gravitation in ${\bf D}$ dimension in the usual form
\begin{equation}
\nabla^2_d \;\varphi = 4\pi G \rho, \label{dnabla} \end{equation}
with $\varphi$ the gravitational constant and $\nabla_d$ the ${\bf D}$
dimensional $\nabla$ operator, then the corresponding Friedmann equation
can easily be obtained to be
\begin{equation}
(\frac{\dot a}{a})^2=\frac{8 \pi G }{D(D-2)} \rho. \label{nfr}
\end{equation}
Now, as expected, for ${\bf D}=3$, the familiar relation (\ref{grfr=3})
is obtained,
assuming the relation (\ref{kappa}). But, how if ${\bf D}\neq 3$. Then one will
realise that the agreement between (\ref{grfr=3}) and (\ref{nfr}) will fail.
\section{Modification}
Looking for the roots of the factor $8\pi$ in (1) we come across the relation
\begin{equation}
R_{00}=\nabla^2 \varphi. \label{ric-po}
\end{equation}
Now, the coefficient in the Poisson equation, i.e. $4 \pi$ has been obtained
, using Gauss law, for three dimensional space. Thus we should first derive
the correct coefficient for a {\bf D} dimensinal space. Applying
Guass's law for a {\bf D} dimensional volume, we find the Poisson equation
for arbitrary fixed dimension,
\begin{equation}
\nabla^2 \varphi = \frac{2 \pi ^{D/2} G}{(D/2-1)!} \rho. \label{modpoiss}
\end{equation}
On the other hand we get for arbitrary $D$
\begin{equation}
R_{00}=(\frac{D-2}{D-1}) \kappa \rho. \label{ricc-kappa}
\end{equation}
A comparison of (\ref{ric-po}), (\ref{modpoiss}), and (\ref{ricc-kappa}) will
give us the following modified Einstein gravitational constant,
\begin{equation}
\kappa_D = \frac{2(D-1)\pi^{D/2} G}{(D-2)(D/2-1)!}. \label{modkappa}
\end{equation}
The correct form of Friedmann equation in any
arbitrary fixed dimension is now derived to be
\begin{equation}
(\frac{\dot a}{a})^2 = \frac{4 \pi^{D/2} G}{D(D-2)(D/2-1)!}\rho.
\end{equation}
As it is easily seen, the above relation is in complete agreement with its
Newtonian counterpart in all dimensions.
\section{conclusion}
The dimensional dependence of the gravitational constant $\kappa$ may have
very different and serious field theoretic and astrophysical consequences
hitherto unnoticed.
It would be interesting, e.g., to see which changes are to be expected if
the results of this note are combined with Kaluza-Klein paradigm. The
consequences on time variation of G within various models is another
very interesting cosmological issue which is under investigation.
\section*{ACKNOWLEDGEMENT}
A.N. wishes to thank Prof.Padmanabhan for many useful discussions. A.N. was financially supported by the Council of Scientific and Industerial Research, India.
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\section{WHY DRESSINGS?}
Many aspects of hadronic physics can be well described in terms
of constituent quarks. The role played by such objects in our
discovery of colour and QCD is well known. However, we do not yet
have a good understanding of how such objects can emerge from QCD.
This talk describes a new approach to this problem (for a review see~\cite{LaMcMu96a}).
Any description
of a (colour) charged particle in a field theory has to fulfill certain
requirements:
({\em i}) we need to include a
chromo-(electro-) magnetic cloud around the charge. This is known to
underlie the infra-red problem and we recall the long postulated link
between the infra-red structure of QCD and confinement;
({\em ii}) since constituent quarks have, in the realm where the quark model is valid,
a physical meaning, it is essential to describe them in a gauge invariant
way. This, in turn, ensures that the constituent quarks have a well defined colour charge as
required by the standard quark model\cite{LaMcMu96a}.
\subsection{QED}
In QED, an approach which incorporates the above requirements
was originally proposed by Dirac\cite{Di55}. If some function $f_\mu(z,x)$ satisfies
$\partial_\mu^z f^\mu(z,x)=
\delta^{(4)}(z-x)$, then
a charged particle at $x$ with an electromagnetic cloud around it
may be written in a gauge invariant manner as
\begin{equation}
\psi_{\rm f}(x)=\exp\left\{ ie\int d^4zf_\mu(z,x)A^\mu(z)\right\}
\psi(x)\,.
\label{eq:fcond}
\end{equation}
The phase factor in~(\ref{eq:fcond}) is usually called
the {\em dressing}. There are as many dressings as there are possible choices of $f_\mu$
in~(\ref{eq:fcond}).
We will study two such choices. Our first example (which we will refer to as ({\em i}) below)
is
\begin{eqnarray}
&&f_0=f_1=f_2=0,\nonumber\\
&&f_3(z,x)=-\theta(x^3-z^3)\prod_{j=0}^2\delta(z^j-x^j),
\end{eqnarray}
which can be easily seen to correspond to a string attached to the charge at $x$
and going along a straight line in the $x^3$ direction out to infinity. One could equally
well choose a more complicated path $\Gamma$ and in general
we call such a dressed field, $\psi_\Gamma$.
Example ({\em ii}), which is the case we are primarily concerned with, is
\begin{eqnarray}
&&f_0=0,\nonumber\\
&&\vec f ={1\over4\pi}\; \delta(z^0-x^0)\vec\nabla_z{1\over|\vec z-\vec x|}
,
\end{eqnarray}
from which we find the dressed field
\begin{equation}\psi_c(x)=
\exp\left\{
ie{\partial_iA_i\over \nabla^2}(x)
\right\}\psi(x) .
\label{eq:Coul}
\end{equation}
Example~({\em i}) is not well suited to describe physical charges. First of all
it corresponds to a very singular field configuration. Second,
its path dependence is difficult
to interpret and, finally, it has been shown that this field configuration is unstable~\cite{Shab}.
Essentially what happens is that the charge generates a Coulombic field and the string radiates away
to infinity. Fig.~\ref{fig:nicolas} shows the time evolution of a similar situation: the
electric field is initially concentrated in the straight line
joining two charges in such a way that Gau\ss's law holds. Again the string radiates away and
only the dipole field generated by the charges remains in the far future\cite{www}.
\begin{figure}[t]
\vspace{9pt}
\hbox{\epsfxsize=6.49cm
\epsfbox{fig2b.ps}
}
\caption{Six frames\protect\cite{www} of the time evolution of a electric field initially
located in the straight
line joining two opposite (static) charges. The dashed lines show the extension of the
region where the radiation field from the decay of the string is present.
Only the dipole field survives if we wait long enough.}
\label{fig:nicolas}
\end{figure}
In contrast, example~({\em ii}) has very nice properties: ({\em a}) it is stable;
({\em b}) it is possible to factor out the path dependence in $\psi_\Gamma$ giving
$\psi_c$, i.e.,
$\psi_\Gamma=N_\Gamma \psi_c $ where $N_\Gamma$ is a gauge invariant, but
path dependent, factor; ({\em c}) using the fundamental commutation relations
one has
\begin{equation}
[\vec E (\vec x),\psi_c(\vec y)]=
{e\over4\pi}{\vec x-\vec y\over|\vec x-\vec y|^3}
\psi_c(\vec y)
\,,
\end{equation}
where we recognize the factor before $\psi_c$ as the electric Coulomb field.
This immediately suggests that example~({\em ii}) is the right dressing for a static charge.
One can generalize this dressing to the case of a charge moving with arbitrary velocity
$\vec v$
\begin{equation}
\psi_v=\exp\left\{
ie{g^{\mu\nu}\!-(\eta+ v )^\mu(\eta- v)^\nu\over\partial^2\!-
(\eta\cdot\partial)^2+(
v\cdot\partial)^2}\partial_\nu A_\mu
\right\}
\psi ,
\label{eq:boos}
\end{equation}
where $v=(0,\vec v)$, and $\eta=(1,\vec 0)$; ({\em d}) One can perform perturbative calculations
using these dressings. We shall do this in the next section where we also show that the
(one loop)
mass shell renormalized dressed electron
propagator,
\begin{equation}
i S_v(p)=\int d^4x\, \exp\{i p\cdot x\}\;
\langle0|\psi_v(x)\bar\psi_v(0)|0\rangle,
\end{equation}
is infra-red finite
provided $p=m\gamma(1,\vec v)$\cite{slow,fast},
as was already predicted in Ref.\cite{LaMcMu96a}. As far as we know,
there are only two other gauges with an infra-red finite charge propagator in the mass shell scheme:
the Yennie gauge and the Coulomb gauge. The latter is a particular case of our approach
($\vec v\to \vec 0$). The infra-red finiteness of $S_v(p)$ can be understood as a consequence
of having the charge dressed with the asymptotic electromagnetic
field of a classical charge moving with velocity $\vec v$. This is a
boosted Coulomb field. Since the
infra-red behaviour is related to its slow fall-off, one would expect that the same dressing
should lead to infra-red finite results also for scalar QED.
This has been verified explicitly. The lack of infra-red divergences in the propagator
of the dressed charge is a necessary and highly non-trivial
requirement for the construction of an asymptotic
state with sharp momentum that can be interpreted as a (single) physical charge.
Before closing this section, we would like to comment on some subtleties
associated with charged states.
Note that $\psi_v$ is both non local and non covariant, which one might regard as a problem.
However, it can be proved that these are {\em unavoidable} features of {\em any}
operator that creates
charged {\em physical} states out of the vacuum\cite{LaMcMu96a,non-loc,non-cov}.
Note also that Eq.(\ref{eq:boos})
is not just a Lorentz boost of Eq.(\ref{eq:Coul}). This is a consequence of the lack of covariance
and locality necessarily associated with charged states\cite{LaMcMu96a}.
\subsection{QCD}
All the properties that have been discussed in the previous section go through to QCD
in perturbation theory. It is also possible to define dressed gluon fields
perturbatively. A new reason for introducing dressings in non-abelian gauge theories is that
the colour charge is only well-defined on gauge invariant states such as a dressed
quark\cite{LaMcMu96a}.
However, it has been shown that beyond perturbation theory the Gribov ambiguity
obstructs the construction of dressings\cite{LaMcMu96a}. As a result, one cannot obtain any
true observable out of a single lagrangian (quark) field. Therefore, our approach explains
confinement in the sense that one cannot construct an asymptotic quark field.
\section{THE DRESSED ELECTRON/QUARK PROPAGATOR}
The Feynman diagrams for the one loop dressed propagator, $S_v(p)$, are shown in
Fig.\ref{fig:diagrams}.
\begin{figure}[htb]
\vspace{9pt}
\leftline{
\hbox{\epsfxsize=6.50cm
\epsfbox{fig3.ps}
}
}
\caption{The diagrams which yield the one loop dressed propagator, $S_v(p)$. The vertices
coming from the perturbative expansion of the dressing are denoted by a black box}
\label{fig:diagrams}
\end{figure}
The diagram
of Fig.\ref{fig:diagrams}.a gives the standard self~energy. The other three diagrams,
Figs.\ref{fig:diagrams}.b---d, contain a new vertex (the black box). The corresponding
Feynman rule, which can be easily obtained
from the perturbative expansion of Eq.(\ref{eq:boos}), reads
\begin{equation}
e{(\eta+v)_\mu(\eta-v)_\rho-g_{\mu\rho}\over
k^2-(k\cdot\eta)^2+(k\cdot v)^2}k^\rho,
\end{equation}
where $k$ ($\mu$) is the momentum (Lorentz index) of the incoming photon.
We can now proceed in two ways: ({\em i}) compute the diagrams of Figs.\ref{fig:diagrams}.a---d
in a Feynman gauge and check that the result (before integrating the loop momentum)
is independent of the gauge parameter; ({\em ii}) use the so called {\em dressing gauge},
in which the dressing phase is 1, i.e., $\psi_v=\psi$ and compute only the diagram of
Fig.\ref{fig:diagrams}.a. In the dressing gauge the photon propagator is
\begin{eqnarray}
&&
{1\over k^2}
\left\{-g_{\mu\nu}+
{k^2-[k\cdot(\eta-v)]^2\gamma^{-2}\over
[k^2-(k\cdot\eta)^2+(k\cdot v)^2]^2}\;k_\mu k_\nu\right.\nonumber\\
&&\left.
-{k\cdot(\eta-v)\over k^2-(k\cdot\eta)^2+(k\cdot v)^2}\;
k_{(\mu}\; (\eta+v)_{\nu)}\right\}.
\label{eq:prop}
\end{eqnarray}
Note that in the limit $\vec v\to\vec0$ this reduces to the propagator in Coulomb gauge.
We have explicitly checked that the two procedures give the same loop momentum integral.
To integrate the loop momentum we have chosen to work in dimensional regularization
with a space-time dimension $D=4-2\epsilon$.
This regularizes both the ultra-violet and infra-red divergences. In particular, the
latter show up as $\int_0^1 du\, u^{D-5}\sim 1/\epsilon$, where $u$ is a Feynman parameter.
\eject
\subsection{Ultra-violet divergences}
The ultra-violet divergent part of the electron self-energy has the following structure
\begin{eqnarray}
\Sigma_{UV}\sim &{\displaystyle{1\over\epsilon}}&\Bigl\{
-3m+(p\hspace{-0.45em}/-m){\cal F}_1(\vec v) \nonumber \\ &+&
2 [p\cdot v\eta\hspace{-0.45em}/
-p\cdot\eta v\hspace{-0.45em}/] {\cal F}_2(\vec v)
\Bigr\} ,
\label{eq:UVpiece}
\end{eqnarray}
where ${\cal F}_1$ and ${\cal F}_2$ do not depend on the external momentum, $p$. The last term
in Eq.(\ref{eq:UVpiece}) seems to endanger the multiplicative renormalization of the propagator.
However, one can check that Eq.(\ref{eq:UVpiece}) can be cancelled by introducing the standard
mass shift, $m\to m-\delta m$, and the
following multiplicative
{\em matrix} renormalization for the electron
\begin{equation}
\psi^{({\rm bare})}_v=\sqrt{Z_2} \exp\left\{
-i { Z'\over Z_2}\sigma^{\mu\nu}\eta_\mu v_\nu \right\}\psi_v,
\end{equation}
which is reminiscent of a naive Lorentz boost upon a fermion.
\subsection{Renormalization}
To actually compute $\delta m$, $Z'$ and $Z_2=1+\delta Z_2$ it is convenient to write
the renormalized self energy as
\begin{equation}
-i\Sigma=m\alpha+p\hspace{-0.45em}/\beta+p\cdot\eta \eta
\hspace{-0.45em}/\delta +m v\hspace{-0.45em}/ \varepsilon,
\end{equation}
where
$\alpha$, $\beta$, $\delta$ and $\varepsilon$ depend upon
$p^2$, $p\cdot\eta$,
$p\cdot v$ and $v$ and they also contain the counterterms $\delta m$, $Z'$ and $\delta Z_2$.
We recall that in the mass shell scheme one has to impose the following
two conditions: ({\em i}) The propagator has a simple pole at $m$, i.e.,
$m$ is the {\em physical} mass of the fermion;
({\em ii}) the residue of the propagator at $m$ is unity. From these two conditions
it must be possible to determine $\delta m$, $Z'$ and $\delta Z_2$.
Condition ({\em i}) implies that the renormalized $\Sigma$ must obey
\begin{equation}
\tilde\alpha +\tilde\beta+{(p\cdot\eta)^2\over m^2}\tilde\delta+
{p\cdot v\over m}\tilde\varepsilon=0,
\label{eq:cond1}
\end{equation}
where the tildes signify that we put the momentum $p^2$ on shell: $p^2=m^2$. Note
that $Z'$ and $\delta Z_2$ do not enter in~(\ref{eq:cond1}) so just $\delta m$ is determined.
We find
\begin{equation}
\delta m={e^2\over(4\pi)^2}\left(
{3\over\epsilon}+4\right).
\end{equation}
It is important to emphasize that this is the standard result for the mass shift and that
it solves Eq.(\ref{eq:cond1}) for arbitrary $p\cdot\eta$, $p\cdot v$ and $v$.
Condition ({\em ii}) can only be satisfied if $p=m\gamma(\eta+v)=m\gamma(1,\vec v)$,
which is precisely the momentum of the real electron moving with velocity $\vec v$. In
addition we need that
\begin{eqnarray}
0&=& 2 m^2 \bar\Delta +\bar\beta -\bar\delta\nonumber\\
0&=&\gamma\left(2m^2\bar\Delta+\bar\beta\right)-\bar\varepsilon\nonumber\\
0&=& 2 m^2 \bar\Delta +\bar\alpha +2\bar\beta,
\label{eq:cond2}
\end{eqnarray}
where
\begin{equation}
\Delta={\partial\alpha\over\partial p^2}+
{\partial\beta\over\partial p^2}+{(p\cdot\eta)^2\over m^2}
{\partial\delta\over\partial p^2}+{p\cdot v\over m}
{\partial\varepsilon\over\partial p^2}
\end{equation}
and the bar upon functions denotes that they must be computed at $p=m\gamma(1,\vec v)$.
If we now explicitly separate out the contributions of $Z'$ and $\delta Z_2$ from the
rest (to which we give a subscript R) then we find that (\ref{eq:cond2}) can be rewritten
as
\begin{equation}
\begin{array}{lcrcl}
-i\delta Z_2\negsp&+&\negsp2\vec v^2 i Z'\negsp&=&\negsp 2 m^2 \bar\Delta +
\bar\beta_R -\bar\delta_R\\
-i\delta Z_2\negsp&+&\negsp2 i Z'\negsp&=&\negsp\gamma\left(2m^2\bar\Delta+
\bar\beta_R\right)-\bar\varepsilon_R\\
-i \delta Z_2\negsp&&\negsp\negsp&=&\negsp 2 m^2 \bar\Delta +
\bar\alpha_R +2\bar\beta_R.
\end{array}
\end{equation}
Since we have three equations and two unknowns ($Z'$ and $\delta Z_2$) one might worry
that perhaps no solution exists. However, a unique solution exists for our choice of mass shell.
It reads
\begin{eqnarray}
Z'&=&{1\over2i}[\gamma^2\bar\delta_R-\gamma\bar\epsilon_R]\nonumber\\
\delta Z_2&=&-{1\over i}[\bar\alpha_R+2\bar\beta_R+2m^2 \bar\Delta].
\end{eqnarray}
As is the case for the standard fermion propagator, the infra-red singularities can only enter
through derivatives with respect to $p^2$. Hence, $Z'$ is infra-red finite and
infra-red divergences can only arise in
$\delta Z_2$ through $\bar\Delta$. The total infra-red divergent contribution to $\bar\Delta$ is
\begin{eqnarray}
\bar\Delta_{{\rm IR}}\negsp&\sim&\negsp\!\!\int_0^1 du\, u^{D-5}
\left\{
-2+2\int_0^1 \!{dx \over\sqrt{1-x}\sqrt{1-\vec v^2 x}}\right.\nonumber\\
\negsp&\times&\negsp(1+\vec v^2-2\vec v^2 x)
\nonumber\\
\negsp&-&\negsp\left.
\gamma^{-2}\int_0^1 \! {dx\, x \over\sqrt{1-x}\sqrt{1-\vec v^2 x}}
{3+\vec v^2-2\vec v^2 x\over2(1-\vec v^2 x )}
\right\}\nonumber\\
\negsp&=&\negsp0,
\end{eqnarray}
and no infra-red divergence arises in the mass shell renormalize
propagator of the dressed electron. The full expressions for $Z'$ and $Z_2$ can
be found in Ref.\cite{fast}.
As we have already mentioned, one can also repeat the calculation in scalar QED
where the algebra is not so heavy. Again one can renormalize the propagator of
the dressed scalar electron, defined as in~(\ref{eq:boos}) replacing $\psi$ by the
scalar field. In this case, no $Z'$ is needed and we can get rid of the ultra-violet
divergences through the usual counterterms $Z_2$ and $\delta m$. Again,
no infra-red divergence arise. The consistency of these calculations with our
expectations is compelling evidence for the validity of
this approach.
\section{SUMMARY}
To conclude we note that
\begin{itemize}
\item Any description of a physical charge must be gauge invariant. Gau\ss's law implies
an intimate link between charges and a chromo-(electro-)~magnetic cloud.
\item Not all gauge invariant descriptions are physically relevant. One needs to find
the right ones.
\item In QCD there is no such description of a single quark outside of perturbation
theory. This sets the limits of the constituent quark model and fixes when jets start to
hadronise.
\item The perturbative tests reported here all worked. They are now being extended to vertex
studies.
\item Phenomenologically, the main question is: at what scale does the Gribov ambiguity
prevent any description of a quark from being stable?
\end{itemize}
\section*{Acknowledgments}
EB \& BF were supported by
CICYT research project AEN95-0815 and
ML by AEN95-0882. NR was supported
by a grant from the region Rh\^one-Alpes.
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
The Red Rectangle nebula that surrounds the star HD\,44179 (Cohen et al.
1975) is famous for the molecular and dusty emission it displays in the red
and infrared parts of the spectrum. The central star is a peculiar
A-supergiant. A major puzzle is that the infrared-to-optical luminosity of
the central object in the Red Rectangle is very high (about 33, Leinert and
Haas 1989), despite the fact that the extinction of HD\,44179 is not larger
than \(E(B-V)\,=\,0.4\). This led Rowan-Robinson and Harris (1983) to invoke
the presence of an embedded M-giant companion; Leinert and Haas went even
further, and argued that HD\,44179 is a foreground object. The solution of
this puzzle is contained in the observations by Roddier et al. (1995), who
showed that the optical flux observed from HD~44179 is entirely
scattered light from
two lobes located above and below a dusty disk, the star itself being hidden
by the disk. There is then no need to invoke another source than HD~44179
to power the luminosity of the nebula.
Waelkens et al. (1992) have shown that HD\,44179 is severely iron-deficient,
with \( [Fe/H]\,=\,-3.3\), that also other metals such as Mg, Si, and Ca are
severely underabundant, but that the CNO and S abundances of this star are
nearly solar. This star is then clearly of the same nature as the other
extremely iron-poor supergiants HR~4049, HD~52961 and BD +39$^{\circ}$4926
(Lambert et al. 1988; Waelkens et al. 1991a; Kodaira 1973; Bond 1991), which
show the same most peculiar abundance pattern, and two of which (HR~4049 and
HD~52961) are also surrounded by circumstellar dust. The location of these
other low-gravity stars rather far from the galactic plane strongly suggests
that they are not massive supergiants, but evolved low-mass stars, and thus
that the central star of the Red Rectangle also is an evolved low-mass star.
Moreover, the dust features show clearly that the nebula is carbon-rich, which
argues that the star has undergone the third dredge-up
typical for late AGB stars. Still, the large amount of circumstellar matter
and the low galactic latitude \(b\,=\,-12^{\circ}\) may indicate that HD\,44179
is somewhat more massive than the other stars of the group. The carbon
richness of the circumstellar environment, as well as the high luminosity
combined with the high or intermediate galactic latitudes, suggest that
these stars are low- or intermediate-mass objects in a post-AGB stage of evolution.
While there is no ground any more to consider HD~44179 as a component of a
{\it wide} binary, Van Winckel et al. (1995) have shown that HD\,44179 is a
{\it spectroscopic} binary with an orbital period of about 300 days. Also the
other mentioned extremely iron-poor post-AGB stars are binaries with periods
of the order of one year.
\begin{table*}
\caption{Photospheric abundances of the extremely iron-poor post-AGB stars
(the zinc abundance for HD~44179 is from the present study, the other values
are taken from Van Winckel 1995)}
\begin{tabular}{l|lrrrrrrrr}
\hline
Object & Iras & [Fe/H] & [C/H] & [N/H] & [O/H] & [S/H] & [Mg/H] & [Si/H] & [Zn/H] \\
\hline
HR 4049 & 10158$-$2844 & -4.8 & -0.2 & 0.0 & -0.5 & -0.4 & ? & ? & ? \\
HD 52961 & 07008$+$1050 & -4.8 & -0.4 & -0.4 & -0.6 & -1.0 & ? & ? & -1.5 \\
BD +39$^{\circ}$4926 & & -3.3 & -0.3 & -0.4 & -0.1 & 0.1 & -1.5 & -1.9 & ? \\
HD 44179 & 06176$-$1036 & -3.3 & 0.0 & 0.0 & -0.4 & -0.3 & -2.1 & -1.8 & -0.6 \\
\hline
\end{tabular}
\end{table*}
In this paper we discuss new observations of HD~44179 that were triggered by
the similarity of this object with the other peculiar binaries. In Section~2
we report on the determination of the zinc abundance in HD~44179, which
confirms that the photospheric peculiarity is due to accretion of circumstellar
gas. In Section~3 we discuss the photometric variability of HD~44179; for
HR~4049 the brightness and colours vary with orbital phase, as a result of
variable circumstellar absorption; in HD~44179 it appears more likely that the
observed variability is due to periodically variable scattering. In Section~4
we discuss the constraints the close-binary nature of HD~44179 imposes on the
previous and future evolution of this object.
\section{The zinc abundance of HD~44179}
In Table\,1 we summarize the photospheric composition of the four objects
that are known to belong to the group of extremely iron-poor post-AGB stars.
These stars are characterized by very low abundances of refractory elements
such as Fe, Mg and Si, and about normal abundances of CNO and S. Following
a suggestion by Venn and Lambert (1990), Bond (1991) first suggested that
the low iron abundances are not primordial, but are due to fractionation
onto dust. Indeed, the abundance pattern of these stars follows that of the
interstellar gas rather closely.
Convincing evidence for this scenario came from the detection of a rather
solar Zn abundance in HD\,52961 by Van Winckel et al. (1992): in the
photosphere of this star, there is more zinc than iron in absolute numbers!
The unusual zinc-to-iron ratio cannot be understood in terms of
nucleosynthetic processes, but must be due to the different {\it chemical}
characteristics of both elements, zinc having a much higher condensation
temperature than iron (Bond 1992).
From the study of a spectrum of HD~44179 obtained with the Utrecht Echelle
Spectrograph at the William Herschel Telescope at La Palma, we can confirm
that zinc follows CNO and S also in this star. In Figure~1 we show a spectrum
with the 4810 $\AA$\ zinc line for HD~44179. The zinc abundance derived from
this line and the 4722 $\AA$\ line is [Zn/H] = -0.6, while [Fe/H] = -3.3.
Also in the photosphere of HD\,44179 the amount of zinc is near that of iron
in absolute numbers. This result further underscores that the central star of
the Red Rectangle belongs to the same group of objects as HR\,4049, HD\,52961
and BD+39$^{\circ}$4926. The fact that all four such objects known occur in
binaries with similar periods then strengthens the suggestion by Waters et al.
(1992) that the chemical separation process occurs in a
stationary {\it circumbinary} disk.
\begin{figure}
\mbox{\epsfxsize=3.2in\epsfysize=2.2in\epsfbox[55 40 550
780]{art13fig1.ps}}
\caption[]{The observation of a zinc line in the spectrum of HD\,44179}
\end{figure}
\section{Photometric variability}
Seen at a high inclination, a circumbinary disk can cause variable
circumstellar extinction, because the amount of dust along the line of sight
varies with the orbital motion of the star. Such an effect has indeed been
observed for HR\,4049 (Waelkens et al. 1991b). We have therefore obtained
68 photometric measurements in the Geneva photometric system, with the Geneva
photometer attached to the 0.70 Swiss Telescope at La Silla Observatory in
Chile, between 1992 and 1996, covering now six orbital cycles. In order to
improve on the orbital elements, we have also obtained new radial-velocity
measurements with the CES spectograph fed by the CAT telescope at ESO; the
data now cover more than five cycles. In Figure~2 we fold the observed visual
magnitudes and [U-B] colors with the orbital phase. The orbital period of
318\,$\pm$\,3\,days is somewhat longer as the one determined previously
(Van Winckel et al. 1995). The vertical lines on the figure indicate the
epochs of inferior and superior conjunction.
\begin{figure}
\begin{flushleft}
\mbox{\epsfxsize=3.2in\epsfysize=4in\epsfbox[117 345 470
710]{art13fig2.ps}}
\caption[]{The phase diagram for the photometric and radial-velocity
variations of HD\,44179. The phase of inferior(superior) conjunction is
marked by a full (dashed) vertical line. Phase 0 corresponds arbitrarily to JD
2448300. A typical error-bar for an individual measurement is shown in the
upper-right corner of each panel.
}
\end{flushleft}
\end{figure}
It is apparent from Figure~2 that photometric variability occurs with the
orbital
period. As in the case of HR~4049, minimum brightness occurs at
inferior conjunction and maximum brightness at superior conjunction. In the
case of HR~4049 the photometric variations are caused by variable obscuration
by the circumbinary disk during the orbital motion. This interpretation was
confirmed by the colour variations, which are consistent with extinction.
However, in the case of HD~44179, {\it no color variations are observed, not in
[U-B], nor in any other color in the optical range}.
If variable extinction along the line of sight
is responsible for the variability, then the grains causing
it must be larger than in HR~4049.
On the other hand, many spectral features
attest the prominent presence of small grains in the Red Rectangle nebula.
We propose that the photometric variability of HD~44179 is not caused
by variable extinction, but by the
variability around the orbit of the scattering angle of the light that is
observed.
The two scattering clouds that are observed cannot be located on the orbital
axis of the system, since then no orbital motion would be observed at all!
It is much more likely that what we observe is light scattered from the
transition region between the optically thick disk and the optical nebula.
Roddier et al. (1995) found a smaller opening angle of the inner source
(40$^{\circ}$) than for the nebula (70$^{\circ}$); this can be understood in
our model: the scattering angle at the edge of the cone, as seen in projection,
is 90$^{\circ}$ and so probably too large for a significant flux in our
direction; the light we observe, must be reflected by that part of the cone
that is directed to us, i.e. where the scattering angle is smallest. In the
following, we therefore assume that the inclination at which the orbital motion
is observed, is equal to half the opening angle of the cone, i.e. 35$^{\circ}$.
\begin{figure}
\begin{flushleft}
\mbox{\epsfxsize=3.4in\epsfysize=2.in\epsfbox{art13fig3.ps}}
\caption[]{A geometric model for the inner part of the Red Rectangle.
Note that the thickness of the disk is two orders of magnitude
larger than the size of the binary system. During the orbit, the scattering
angle toward the observer varies, being largest at inferior conjunction,
when minimum brightness is observed.}
\end{flushleft}
\end{figure}
Our model is schematically presented in Figure~3. The orbital plane of the
binary is nearly edge-on, as is assumed commonly, since the star is hidden and
the nebula is remarkably symmetric. The variable scattering angle can be
estimated from the size of the orbit
and the geometry of the nebula. Roddier et al. (1995) determined that the
scattering clouds are located some 0.07" above and under the orbital plane.
Assuming an absolute magnitude -4.0 for the star we then find from the observed
bolometric luminosity and reddening (Leinert \& Haas, 1989) a distance
of 360\,pc, close to the value originally estimated by Cohen et al. (1975)
on the basis of different assumptions. This distance then implies that the
scattering occurs at a vertical distance of 25\,AU from the orbital plane;
with an opening angle of the nebula of 70$^{\circ}$, it follows that the
distance of this region to the orbital axis is some 17.5\,AU. From the
orbital elements we derive that the radial distance of HD~44179 from the
center of mass of the system amounts to 0.53\,A.U. at both inferior and
superior conjunction, so that the scattering angle of the light we observe
varies between 54.2$^{\circ}$ and 55.8$^{\circ}$.
It is customary to parametrize the scattering function S($\theta$) of an
astronomical source by a Heyney-Greenstein function of the form
$$
S(\theta) = (1\,-\,g^{2})\,(1\,+\,g^{2}\,-2\,g\,cos\,\theta)^{-3/2}
$$
Isotropic scattering corresponds to \(g=0\) and $g$ approaches unity for strong
forward scattering. In typical sources, $g$ ranges between 0.6 and 0.8.
For $g$ values of 0.6, 0.7, and 0.8, the ratio of forward scattered light to
that scattered at an angle of 55$^{\circ}$ varies by factors 8.6, 21.1, and
76.8, respectively; in the latter case, the brightness would be much lower than
is observed, so that it appears likely that $g$ falls in the range 0.6-0.7.
The variable scattering angle then induces photometric variations with an
amplitude between 0.067 and 0.076 mag. The observed amplitude of some 0.12 mag
is slightly larger; nevertheless, the agreement with a model in which the
scattering surface was assumed constant and no additional extinction variations
were taken into account, is encouraging.
\section{Evolutionary history of the system}
The mass function of the spectroscopic binary is 0.049\,$M_{\odot}$. Assuming
an `effective' inclination of 35$^{\circ}$, the mass of the unseen companion
can be derived for various masses of the primary. For primary masses in
the range between 0.56 and 0.80 $M_{\odot}$, typical for post-AGB stars,
the mass of the secondary falls in the range betwen 0.77 and 0.91 $M_{\odot}$,
i.e. masses well below the initial mass of the primary. It is then most
natural to assume that the secondary is a low-mass main-sequence star.
The present orbital parameters of the system are such that no AGB star with
the same luminosity can fit into the orbit. On the other hand, if Roche-lobe
overflow had occurred on the AGB, it is dubious whether the system could have
survived as a relatively wide binary. This problem is already encountered for
HR\,4049, whose orbital period is 429 days. It is even more severe in the
case of HD\,44179, because its orbit is shorter, and moreover the initial mass
of the star was probably larger than for HR\,4049. Nevertheless, the carbon
richness of the nebula does suggest that the star has gone through the thermal
pulses which normally occur near the end of AGB evolution.
The present characteristics of the Red Rectangle therefore suggest that
filling of the Roche lobe was not necessary for mass transfer to occur.
Probably, then, mass loss started on the AGB before the Roche lobe
has been filled, altering substantially the evolution of HD~44179.
Apparently, this mass-loss process prevented the star from ever filling
its Roche lobe, even during the thermally pulsing AGB. The luminosity may
have increased at a normal rate, but the radius would be lower than for single
AGB stars, i.e. the AGB evolution takes place on a track which is much bluer
than for single stars.
It is then not at all clear whether the present envelope mass is as low as
the 0.05 solar mass that is usually assumed at the start of a post-AGB
evolutionary track. Indeed, in the scenario we propose, one may argue whether
these stars can be called {\it post}-AGB stars. The present evolutionary
timescale of HD~44179 could then be longer than for typical post-AGB stars.
We note that a longer timescale is indeed more consistent with the huge extent
and small outflow velocity of the nebula: from a coronograhic picture taken at
the ESO NTT, the extent of the nebula is some 40" on both sides; the expansion
velocity, deduced from the CO lines, amounts to some 6\,km/s (Jura et al.
1995); hence, an age of more than 10\,000 years follows for the nebula, much
longer than the typical post-AGB timescale. A very short evolutionary
timescale seems unlikely also in view of the fact that not less than four
such objects brighter than nineth magnitude are known.
Similar problems with orbital sizes are well known for the barium stars, which
are binaries containing a white dwarf, which thus formerly was an AGB star.
The overabundances of the s-process elements in the barium stars are
interpreted as due to wind accretion from this AGB star. Also barium stars
occur with orbital periods that are too short for normal AGB evolution.
It has been suggested that the progenitors of these close barium stars have
always remained detached, and that mass transfer occurred via wind accretion
(Boffin \& Jorissen 1988; Jorissen \& Boffin 1992; Theuns \& Jorissen 1993).
We suggest that HR~4049 and HD~44179 are progenitors of barium stars.
Indeed, they are binaries with similar periods, the primary of which is
presently finalizing its evolution before it becomes a planetary nebula and
then a white dwarf. In our systems, the secondaries, that will later be
seen as barium stars, are still on the main sequence. Important mass loss
is presently observed, and it is likely that a fraction of it is captured by
the companion.
Unfortunately, this suggestion cannot easily be checked directly.
The companion is much too faint to detect the s-process elements it accreted
after they had been produced during the thermal pulses of the primary. Only
for one star of the group, the coolest member HD\,52961, could the s-process
elements Ba and Sr be detected in the spectrum of the primary: they follow
the iron abundance rather closely, because also these elements have been
absorbed by the dust and were not reaccreted.
Another peculiar characteristic which is shared by our systems and the
short-period barium stars, is the fact that the orbits are eccentric,
while one would expect that tidal interactions are very effective in
circularizing the orbits. Here again, the substantial circumbinary disks
these objects develop may yield the answer. Recent studies show that
binaries may acquire their eccentricity through tidal interactions with the
disks in which they are formed (Artymovicz et al. 1991). It is then natural
to conjecture that the mass lost by HD~44179 and HR~4049, which appears to
accumulate preferentially in a disk, finally increases the eccentricity rather
than circularizing the orbit. If this conjecture proves true, it would
strengthen the link with the barium stars, because it would imply that a
previous circumbinary disk must be invoked to explain the present
eccentricities of barium stars in close binary systems.
\acknowledgements{The authors thank Drs. Henny Lamers, Alain Jorissen, and
Ren\'e Oudmaijer fruitful discussions. We thank the staff of
Geneva Observatory for their generous awarding of telescope time at the Swiss
Telescope at La Silla Observatory. We also thank Dr. Hugo Schwarz for his
help for obtaining the coronographic image and Hans Plets for the fitting
of the velocity curve. This work has been sponsored by the Belgian National
Fund for Scientific Research, under grant No. 2.0145.94. LBFMW gratefully
acknowledges support from the Royal Netherlands Academy of Arts and Sciences.
EJB was supported by grant No. 782-371-040 by ASTRON, which receives funds
from the Netherlands Organisation for the Advancement of Pure Research (NWO).}
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
The massive Schwinger model is two-dimensional QED with one massive fermion.
In this model there are instanton-like gauge field configurations present, and,
therefore, a $\theta$ vacuum has to be introduced as a new, physical vacuum
(\cite{CJS,Co1}). Further, confinement is realized in this model in the sense
that there are no fermions in the physical spectrum (\cite{AAR,GKMS}). The
fermions form charge neutral bosons, and only the latter ones exist as physical
particles.
The fundamental particle of the theory is a massive, interacting boson with
mass $\mu =M_1$ (Schwinger boson).
In addition, there exist $n$-boson bound states. The two-boson
bound state is stable (mass $M_2$), whereas the higher bound states may decay
into $M_1$ and $M_2$ particles (\cite{Co1}).
All these features have been discussed in
\cite{GBOUND} within mass perturbation theory (\cite{Co1},
\cite{FS1} -- \cite{SMASS}),
which uses the exacly soluble massless Schwinger model (\cite{AAR},
\cite{Sc1} -- \cite{Adam}) as a starting point.
Here we will find that another type of unstable bound states has to be included
into the theory, namely hybrid bound states composed of $M_1$ and $M_2$
particles. In addition, we will compute the decay widths of the $M_3$ bound
state and of the lightest hybrid bound state (which consists of one $M_1$ and
one $M_2$ and has mass $M_{1,1}$).
\section{Bound states}
For later convenience we define the functions
\begin{equation}
E_\pm (x)=e^{\pm 4\pi D_\mu (x)} -1
\end{equation}
and their Fourier transforms $\widetilde E_\pm (p)$, where $D_\mu (x)$ is the massive
scalar propagator. As was discussed in \cite{GBOUND}, all the $n$-boson bound
state masses $M_n$ may be inferred from the two-point function ($P=\bar\Psi
\gamma_{5} \Psi$, $S=\bar\Psi \Psi$, $S_\pm =\bar\Psi \frac{1}{2}(1\pm \gamma_{5})\Psi $)
\begin{equation}
\Pi (x):=\delta (x)+g\langle P(x)P(0)\rangle
\end{equation}
(or $g\langle S(x)S(0)\rangle$ for even bound states) where $g=m\Sigma +o(m^2)$
is the coupling constant of the mass perturbation theory for vanishing vacuum
angle $\theta=0$ (the general $\theta$ case we discuss in a moment); $\Sigma$
is the fermion condensate of the massless model. $\Pi (x)$ is related to the
bosonic $n$-point functions of the theory via the Dyson-Schwinger equations
\cite{GBOUND}. In momentum space $\widetilde \Pi (p)$ may be resummed,
\begin{equation}
\widetilde \Pi (p)=\frac{1}{1-g\widetilde{\langle PP\rangle}_{\rm n.f.}(p)}
\end{equation}
where n.f. means non-factorizable and denotes all Feynman graphs that may not
be factorized in momentum space. In lowest order $\widetilde{\langle PP\rangle}_{\rm
n.f.}$ is
\begin{equation}
\widetilde{\langle PP\rangle}_{\rm n.f.} (p)=\frac{1}{2}(\widetilde E_+ (p)-\widetilde E_- (p))
\end{equation}
(and with a + for $\widetilde{\langle SS\rangle}_{\rm n.f.}$). Expanding the
exponential one finds $1-g\sum_{n=1}^\infty \frac{(4\pi)^n}{n!}
\widetilde{D_\mu^n}(p)$ in the denominator of (3) (more precisely, the odd powers for
$\widetilde{\langle PP\rangle}_{\rm n.f.}$, the even powers for $\widetilde{\langle SS
\rangle}_{\rm n.f.}$). At $p^2 =(n\mu)^2$,
$\widetilde{D_\mu^n}(p)$ is singular, therefore there are mass poles $p^2 =M_n^2$
slightly below the $n$-boson thresholds. Further $\widetilde{D_\mu^m}(p)$ have
imaginary parts at $p^2 =M_n^2$ for $m<n$, therefore decays into $mM_1$ are
possible (more precisely, for the parity conserving case $\theta =0$, only
odd $\rightarrow$ odd or even $\rightarrow$ even decays are possible).
Up to now we did not mention the $M_2$ particle, although decays into some
$M_2$ are perfectly possible. So where is it?
The boson bound states are found by a resummation, so a further resummation is
a reasonable idea. Let us look at the $M_1 +M_2$ final state for definiteness.
Within $g\widetilde{\langle PP\rangle}_{\rm n.f.}(p)$ we may find the following term
\begin{equation}
H(p):= \int\frac{d^2 q}{(2\pi)^2}g\widetilde{\langle PP\rangle}_{\rm n.f.}(q)
\widetilde \Pi (q) g\widetilde{\langle PP\rangle}_{\rm n.f.}(q) 4\pi\widetilde D_\mu (p-q).
\end{equation}
It is simply a loop where we have selected one boson to run along the first
line, whereas the $g\widetilde{\langle PP\rangle}_{\rm n.f.}(q)$
and $\widetilde\Pi (q)$ run along
the other line. $H(p)$ is a loop, therefore it is non-factorizable.
The additional $(g\widetilde{\langle PP\rangle}_{\rm n.f.}(q))^2$ factor is necessary,
because $\widetilde\Pi (q)$ starts at zeroth order ($\widetilde\Pi (q) =1+g\widetilde{\langle
PP\rangle}_{\rm n.f.}(q)+\ldots$), and without this factor we would
include some
diagrams into $H(p)$ that were already used for the $n$-boson bound-state
formation (double counting).
The claim is that $H(p)$ has a threshold singularity precisely at $p^2 =(M_1
+M_2)^2$, and therefore an imaginary part for $p^2 >(M_1 +M_2)^2$. But this is
easy to see. At $q^2 =M_2^2$ $\widetilde\Pi(q)$ has the $M_2$ one-particle singularity
and $g\widetilde{\langle PP\rangle}_{\rm n.f.}(M_2)\equiv 1$. Therefore, near $p^2
=(M_1 +M_2)^2$, $H(p)$ is just the $M_1 ,M_2$- two-boson loop (up to a
normalization constant).
Observe that this line of reasoning is not true for higher bound states, $p^2
\simeq (M_n +M_1)^2 ,n>2$. $\widetilde\Pi (M_n)$ contains imaginary parts and is not
singular for $n>2$ (because the $M_n$ are unstable), and therefore $H(p)$ has
no thresholds at higher $p^2$.
A further consequence is that $H(p)$ gives rise to a further mass pole slightly
below $p^2 =(M_1 +M_2)^2$ in (3).
These considerations may be generalized, and we find $n_1 M_1 +n_2 M_2$
particle-production thresholds at $p^2 =(n_1 M_1 +n_2 M_2)^2$ and (unstable)
$n_1 M_1 +n_2 M_2$-bound states slightly below.
After all, this is not so surprizing. The $M_2$ are stable particles and
interacting via an attractive force. In two dimensions this {\em must} give
rise to a bound state formation. (Similar conclusions may be drawn from
unitarity when $M_2$-scattering is considered, \cite{SCAT}.)
Before starting the actual computations, we should generalize to arbitrary
$\theta \ne 0$. There the coupling constant is complex, $g\rightarrow g_\theta
,g_\theta^*$, and, because of parity violation, the Feynman rules acquire a
matrix structure (the propagators are $2\times 2$ matrices, the vertices
tensors, etc.). The exact propagator may be inverted, analogously to (3), and
leads to (see \cite{GBOUND})
\begin{equation}
\frac{{\cal M}_{ij}}{1-\alpha -\alpha^* +\alpha \alpha^* -\beta \beta^* }
\end{equation}
\begin{equation}
\alpha (p)=g_\theta \widetilde{\langle S_+ S_+ \rangle}_{\rm n.f.}(p) \quad ,\quad
\beta (p)=g_\theta \widetilde{\langle S_+ S_- \rangle}_{\rm n.f.}(p)
\end{equation}
and ${\cal M}_{ij}$ ($i,j=+,-$) gives the $\widetilde{\langle S_i S_j \rangle}$
component of the propagator. For our considerations only the denominator in
(6) is important. In leading order
\begin{equation}
\alpha (p) =g_\theta \widetilde E_+ (p)\quad ,\quad \beta (p)=g_\theta \widetilde E_- (p)
\quad ,\quad g_\theta =\frac{m\Sigma}{2} e^{i\theta}
\end{equation}
and the denominator reads
\begin{equation}
1-m\Sigma\cos\theta \widetilde E_+ (p) +\frac{m^2 \Sigma^2}{4}(\widetilde E_+^2 (p)-\widetilde E_-^2
(p)).
\end{equation}
Inserting the $n$-boson functions ($d_n
(p):=\frac{(4\pi)^n}{n!}\widetilde{D_\mu^n}(p)$) results in
\begin{equation}
1-m\Sigma\cos\theta (d_1 +d_2 +\ldots )+m^2 \Sigma^2
\Bigl( d_1 (d_2 +d_4 +\ldots) +d_3 (d_2 +d_4 +\ldots)+\ldots \Bigr)
\end{equation}
Now suppose we are e.g. at the $M_3$ bound state mass. Then the real part of
(10) vanishes by definition and $m\Sigma\cos\theta d_3 (M_3) =1+o(m)$, and we
get
\begin{equation}
-im\Sigma\cos\theta {\rm Im\,} d_2 (M_3) +im^2\Sigma^2 d_3 (M_3){\rm Im\,} d_2 (M_3)=
-im\Sigma (\cos\theta -\frac{1}{\cos\theta}){\rm Im\,} d_2 (M_3).
\end{equation}
This computation may be generalized easily, and we find that each parity
allowed decay acquires a $\cos\theta$, whereas a parity forbidden decay
acquires a $(\cos\theta - \frac{1}{\cos\theta})$ factor.
To include the decays into $M_2$ we have to perform a further resummation
analogous to above, however, the resummed contributions enter into the
functions $\alpha$, $\beta$ in a way that is perfectly consistent with our
parity considerations (a $n_1 M_1 +n_2 M_2$-state has parity $P=(-1)^{n_1}$).
\section{Bound state masses}
We are now prepared for explicit computations, but before computing decay
widths we need the masses and residues of the propagator at the various mass
poles. The masses $M_1 ,M_2 ,M_3$ have already been computed (\cite{GBOUND};
there is, however, a numerical error in the $M_2$ mass formula in
\cite{GBOUND}),
\begin{equation}
M_1^2 \equiv \mu^2 =\mu_0^2 +\Delta_1 +o(m^2) \quad ,\quad \Delta_1 =4\pi
m\Sigma\cos\theta
\end{equation}
\begin{equation}
M_2^2 =4\mu^2 -\Delta_2 \quad ,\quad \Delta_2 =\frac{4\pi^4 m^2 \Sigma^2 \cos^2
\theta}{\mu^2}
\end{equation}
\begin{equation}
M_3^2 =9\mu^2 -\Delta_3 \quad ,\quad \Delta_3 \simeq
6.993 \mu^2 \exp (-0.263\frac{\mu^2}{m\Sigma \cos\theta})
\end{equation}
and the three-boson binding energy is smaller than polynomial in the coupling
constant $m$ (or $g$).
In leading order the $n$-th mass pole is the zero of the function
\begin{equation}
f_n (p^2)=1-m\Sigma\cos\theta d_n (p^2),
\end{equation}
therefore the residue may be inferred from the first
Taylor coefficient around $(p^2 -M_n^2 )$,
\begin{equation}
f_n (p^2) \simeq c_n (p^2 -M_n^2).
\end{equation}
The $c_n$ may be inferred from the computation of the mass poles
(\cite{GBOUND}) and are
related to the binding energies. Explicitly they read
\begin{equation}
c_1 =\frac{1}{4\pi m\Sigma\cos\theta}=\frac{1}{\Delta_1}
\end{equation}
\begin{equation}
c_2 =\frac{\mu^2}{8\pi^4 (m\Sigma\cos\theta)^2}=\frac{1}{2\Delta_2}
\end{equation}
\begin{equation}
c_3 =\frac{m\Sigma\cos\theta}{0.263 \mu^2 \Delta_3}
\end{equation}
The mass $M_{1,1}$ is the solution of $1=(g_\theta +g_\theta^* )H(p)$, which
looks difficult to solve. However, there is an approximation. At threshold
$\widetilde \Pi
(q)$ equals the $M_2$ propagator, so this may be a reasonable approximation
provided that the binding energy is sufficiently small, $\Delta_{1,1}\equiv
(M_1 +M_2)^2 -M_{1,1}^2 <\Delta_2$. In this approximation we have for $M_{1,1}$
\begin{eqnarray}
1 &=& m\Sigma\cos\theta\int\frac{d^2 q}{(2\pi)^2}\frac{8\pi^4
m\Sigma\cos\theta}{\mu^2 (q^2 -M_2^2)}\frac{4\pi}{(p-q)^2 -M_1^2} \nonumber \\
&=& \frac{32\pi^5 m^2 \Sigma^2 \cos^2 \theta}{2\pi \mu^2\bar w(p^2 ,M_2^2
,M_1^2 )}\Bigl( \pi +\nonumber \\
&& \arctan \frac{2p^2}{\bar w(p^2 ,M_2^2 ,M_1^2 )
-\frac{1}{\bar w(p^2 ,M_2^2 ,M_1^2 )}(p^2 +M_1^2 -M_2^2)(p^2 -M_1^2 +M_2^2 )}
\Bigr)
\end{eqnarray}
\begin{equation}
\bar w(x,y,z):=(-x^2 -y^2 -z^2 +2xy+2xz+2yz)^{\frac{1}{2}}
\end{equation}
where we inserted the residues that may be derived from the Taylor coefficients
$c_1 ,c_2$ (17,18) (${\rm Res}_i =\frac{1}{c_i m\Sigma\cos\theta}$).
The solution is
\begin{equation}
M_{1,1}^2 =(M_1 +M_2)^2 -\Delta_{1,1}\quad ,\quad \Delta_{1,1}=
\frac{32\pi^{10}(m\Sigma\cos\theta)^4}{\mu^6}
\end{equation}
which shows that our approximation is justified for sufficiently small $m$.
$M_{1,1}$ was computed in a way analogous to $M_2$ (see \cite{GBOUND}),
therefore it leads to an analogous Taylor coefficient
\begin{equation}
c_{1,1} =\frac{1}{2\Delta_{1,1}}=\frac{\mu^6}{64\pi^{10}(m\Sigma\cos\theta)^4}.
\end{equation}
\section{Decay width computation}
The decay widths may be inferred in a simple way from the imaginary parts of
the propagator. Generally
\begin{equation}
G(p)\sim \frac{{\rm const.}}{p^2 -M^2 -i\Gamma M}
\end{equation}
and $\Gamma$ is the decay width. In our case the poles have their Taylor
coefficients,
\begin{equation}
\widetilde \Pi (p) \sim \frac{{\rm const.}}{c_i (p^2 -M_i^2) -i{\rm Im\,} (\cdots)}
\sim \frac{{\rm const'.}}{p^2 -M_i^2 -i\frac{{\rm Im\,} (\cdots)}{c_i}}
\end{equation}
and therefore
\begin{equation}
\Gamma_i \sim \frac{{\rm Im\,} (\cdots)}{c_i M_i}.
\end{equation}
Before performing the explicit computations let us add a short remark.
The $c_i$ are related to the binding energies, $c_i \sim \frac{1}{\Delta_i}$.
Therefore, all the decay widths are restricted by the binding energies,
$\Gamma_i\sim \Delta_i$. But this is a very reasonable result. The denominator
of the propagator (25) has zero real part at $M_i^2$ and infinite real part
at the real particle production threshold. Suppose $\widetilde \Pi (p)$ contributes to
a scattering process (to be discussed in detail in a further publication,
\cite{SCAT}). It will give rise to a local maximum (resonance) at $p^2 =M_i^2$,
and to a local minimum at the production threshold $p^2 =M_i^2 +\Delta_i$.
Therefore the resonance width (decay width) {\em must} be bounded by
$\Delta_i$.
Now let us perform the explicit calculations. At $M_{1,1}^2$ the propagator is
\begin{equation}
\widetilde \Pi (p)\sim\frac{1}{c_{1,1} (p^2 -M_{1,1}^2) -im\Sigma (\cos\theta
-\frac{1}{\cos\theta}) {\rm Im\,} d_2 (p)}
\end{equation}
\begin{displaymath}
{\rm Im\,} d_2 (p)=\frac{8\pi^2}{2w(p^2 ,M_1^2 ,M_1^2 )}
\end{displaymath}
\begin{equation}
w(x,y,z)=(x^2 +y^2 +z^2 -2xy-2xz-2yz)^{\frac{1}{2}}
\end{equation}
leading to the decay width ($M_1 \equiv \mu$)
\begin{equation}
\Gamma_{M_{1,1}}=\frac{2^8 \pi^{12} (m\Sigma\cos\theta)^5}{9\sqrt{5}\mu^9}
(\frac{1}{\cos^2 \theta} -1) \simeq 21340 \mu(\frac{m\cos\theta}{\mu})^5
(\frac{1}{\cos^2 \theta} -1)
\end{equation}
($\Sigma =\frac{e^\gamma \mu}{2\pi}=0.283 \mu$)
for the decay $M_{1,1}\rightarrow 2M_1$. This decay is parity forbidden, and therefore
$M_{1,1}$ is stable for $\theta =0$.
For the $M_3$ decay there exist two channels, $M_3 \rightarrow M_2 +M_1 ,M_3 \rightarrow 2M_1$,
\begin{equation}
\widetilde \Pi (p)\sim\frac{1}{c_3 (p^2 -M_3^2) -im\Sigma (\cos\theta
-\frac{1}{\cos\theta})\frac{4\pi^2}{w(p^2 ,M_1^2 ,M_1^2 )}
-i(m\Sigma\cos\theta)^2\frac{16\pi^5}{\mu^2 w(p^2 ,M_2^2 ,M_1^2)}}
\end{equation}
leading to the partial decay widths
\begin{equation}
\Gamma_{M_3 \rightarrow 2M_1}=0.263 \frac{4\pi^2
\Delta_3}{9\sqrt{5}\mu}(\frac{1}{\cos^2 \theta}-1)
\simeq 3.608 \mu (\frac{1}{\cos^2 \theta}-1)
\exp (-0.929\frac{\mu}{m\cos\theta})
\end{equation}
\begin{equation}
\Gamma_{M_3 \rightarrow M_2 +M_1}=0.263\frac{4\pi^3 \Delta_3}{3\sqrt{3}\mu}
\simeq 43.9 \mu \exp (-0.929\frac{\mu}{m\cos\theta})
\end{equation}
and to the ratio
\begin{equation}
\frac{\Gamma_{M_3 \rightarrow 2M_1}}{\Gamma_{M_3 \rightarrow M_2 +M_1}}=\frac{\frac{1}{\cos^2
\theta}-1}{\sqrt{15}\pi}.
\end{equation}
The latter is independent of the approximations that were used in the
computation of $M_3$ and $c_3$. Observe that $\Gamma_{M_3 \rightarrow M_2 +M_1}$ is
larger than $\Gamma_{M_3 \rightarrow 2M_1}$, although $M_1 +M_2 \sim M_3$. This is so
because the phase space "volume" does not rise with increasing momentum in
$d=1+1$.
Remark: there seems to be a cheating concerning the sign of $\Gamma_{M_3 \rightarrow
2M_1}$ (see (30), (31)). This is a remnant of the Euclidean conventions that
are implizit in our computations (see e.g. \cite{GBOUND}). There the
conventions are such that $\theta$ is imaginary and, consequently, $\cos\theta
-\frac{1}{\cos\theta}\ge 0$. In a really Minkowskian computation, roughly
speaking, the roles of $E_+$ and $E_-$ in (6) are exchanged, leading to a
relative sign between odd and even states. The final results (29), (31) and
(32) are expressed for Minkowski space and for real $\theta$
($\frac{1}{\cos^2 \theta} -1\ge 0$), which explains the sign.
\section{Summary}
By a closer inspection of the massive Schwinger model we have found that its
spectrum is richer than expected earlier. In addition to the $n$-boson bound
states there exist hybrid bound states that are composed of fundamental bosons
and stable two-boson bound states. A posteriori their existence is not too
surprizing and may be traced back to the fact that particles
that attract each other form at
least one bound state in $d=2$; or it may be
understood by some unitarity arguments. For
the special case of vanishing vacuum angle, $\theta =0$, the lowest of these
hybrid bound states is even stable and must be added to the physical particles
of the theory.
Further we computed the decay widths of some unstable bound states
and found that our results are consistent with an interpretation of the
bound states as resonances. Even more insight
into these features would be possible by a discussion of scattering, which will
be done in a forthcoming publication (\cite{SCAT}).
Of course, it would be interesting to compare our results to other approaches,
like e.g. lattice calculations.
\section*{Acknowledgement}
The author thanks the members of the Institute of Theoretical Physics of the
Friedrich-Schiller-Universit\"at Jena, where this work was done, for their
hospitality. Further thanks are due to Jan Pawlowski for helpful discussions.
This work was supported by a research stipendium of the Vienna University.
|
proofpile-arXiv_065-677
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\section{Introduction}
\setcounter{equation}{0}
The discovery of the top quark at Fermilab fulfilled the three-family
quark picture in the standard electroweak model.
Up to now, some knowledge on the mass spectra of $(u, c, t)$ and
$(d, s, b)$ quarks has been accumulated through both experimental and
theoretical (or phenomenological) attempts \cite{Gasser}.
The ratios of quark mass
eigenvalues are obtainable after one renormalizes them
to a common reference scale, e.g., $\mu = 1$ GeV or $M_Z$. There exists
a clear mass hierarchy in each quark sector:
\begin{equation}
m_u \; \ll \; m_c \; \ll \; m_t \; ; ~~~~~~~~
m_d \; \ll \; m_s \; \ll \; m_b \; .
\end{equation}
In comparison, the masses of three charged leptons manifest a similar
hierarchical pattern \cite{PDG}.
\vspace{0.3cm}
Quark mass eigenstates are related to quark weak (flavor)
eigenstates by the Kobayashi-Maskawa (KM) matrix $V$ \cite{KM}, which
provides a quite natural description of flavor mixings and $CP$ violation
in the standard model. To date, many experimental constraints on
the magnitudes of the KM matrix elements have been achieved.
The unitarity of $V$ together with current data requires
a unique hierarchy among the nine matrix elements \cite{XingV}:
\begin{eqnarray}
|V_{tb}| \; > \; |V_{ud}| \; > \; |V_{cs}| & \gg & |V_{us}| \; > \; |V_{cd}| \;
\nonumber \\
& \gg & |V_{cb}| \; > \; |V_{ts}| \; \nonumber \\
& \gg & |V_{td}| \; > \; |V_{ub}| \; > \; 0 \; .
\end{eqnarray}
Here $|V_{ub}|\neq 0$ is a necessary condition for the presence of $CP$
violation in the KM matrix.
\vspace{0.3cm}
How to understand the hierarchies of quark masses and flavor mixings is an important
but unsolved problem in particle physics. A natural approach to the final solution
of this problem is to look for the most favorable pattern of quark mass matrices
(see, e.g., Refs. \cite{Weinberg,Fritzsch77}),
which can account for all low-energy phenomena of quark mixings and $CP$ violation. The
relevant symmetries hidden in such phenomenological schemes are possible to provide
useful hints toward the dynamical details of fermion mass generation.
\vspace{0.3cm}
It has been speculated by some authors that the realistic fermion mass matrices
could arise from the flavor permutation symmetry and its spontaneous or
explicit breaking \cite{Democracy,Fritzsch90,Meshkov}.
Under exact $S(3)_{\rm L}\times S(3)_{\rm R}$ symmetry the
mass spectrum for either up or down quark sector consists of only two levels:
one is of 2-fold degeneracy with vanishing mass eigenvalues, and the other
is nondegenerate (massive). An appropriate breakdown of the above symmetry may
lead to the observed mass hierarchy and flavor mixings.
Although the way to introduce the minimum number
of free parameters for permutation symmetry breaking is technically
trivial, its consequences on quark mixings and $CP$ violation
may be physically instructive and may even shed some light on the proper
relations between the KM matrix elements and quark mass ratios.
Indeed there has not been a satisfactory symmetry breaking pattern with enough
predictive power in the literature.
\vspace{0.3cm}
In this work we first stress that some observed properties of the KM
matrix can be interpreted by the quark mass hierarchy without the
assumption of specific mass matrices. In the quark mass limits such as
$m_u=m_d=0$, $m_t\rightarrow \infty$ or $m_b\rightarrow \infty$,
we find that simple but instructive relations between the KM matrix
elements and quark mass ratios are suggestible from current experimental data.
Then we present a new quark mass {\it Ansatz} through the explicit
breakdown of flavor permutation symmetry at the weak scale ($M_Z = 91.187$ GeV).
This {\it Ansatz} contains seven free parameters, thus it can give rise to three
predictions for the phenomena of quark mixings and $CP$ violation. The typical
results are $|V_{cb}| \approx |V_{ts}| \approx \sqrt{2} ~ (m_s/m_b - m_c/m_t)$,
$|V_{ub}/V_{cb}|\approx \sqrt{m_u/m_c}$ and $|V_{td}/V_{ts}|\approx
\sqrt{m_d/m_s}$ in the leading order approximation.
Prescribing the same {\it Ansatz} at the supersymmetric grand unified theory
(GUT) scale ($M_X = 10^{16}$ GeV),
we derive the renormalized quark mass matrices at $M_Z$ for
small $\tan\beta_{\rm susy}$ (the ratio of Higgs vacuum expectation values
in the minimal supersymmetric model). We also
renormalize some relations between the KM matrix elements and quark mass ratios
at $M_Z$ for arbitrary $\tan\beta_{\rm susy}$, and find that the relevant
results are in good agreement with experimental data.
The scale-independent predictions of our {\it Ansatz} for the characteristic
measurables of $CP$ asymmetries in weak $B$ decays, i.e.,
$0.18 \leq \sin(2\alpha) \leq 0.58$, $0.5\leq \sin(2\beta) \leq 0.78$
and $-0.08 \leq \sin(2\gamma) \leq 0.5$, can be tested at the forthcoming
KEK and SLAC $B$-meson factories.
\vspace{0.3cm}
The remaining part of this paper is organized as follows.
Some qualitative implications of the quark mass hierarchy on the KM matrix
elements, which are almost independent of the specific forms of quark mass
matrices, are discussed in section 2. In section 3 we suggest
a new quark mass {\it Ansatz} from the flavor permutation symmetry breaking
at the weak scale, and study its various consequences on flavor mixings
and $CP$ violation. The same {\it Ansatz} is prescribed at the supersymmetric
GUT scale in section 4. By use of the one-loop renormalization group
equations, we run the mass matrices from $M_X$ to $M_Z$
and then discuss the renormalized relations between the KM matrix elements and
quark mass ratios. Section 5 is devoted to a brief summary of this work.
\section{Flavor mixings in quark mass limits}
\setcounter{equation}{0}
Without loss of any generality, the up and down quark mass matrices (denoted
by $M_{\rm u}$ and $M_{\rm d}$, respectively) can be chosen to be Hermitian.
After the diagonalization of $M_{\rm u}$ and $M_{\rm d}$ through the unitary
transformations
\begin{eqnarray}
O^{\dagger}_{\rm u} M_{\rm u} O_{\rm u} & = & {\rm Diag}\{ m_u, ~ m_c, ~ m_t
\} \; , \nonumber \\
O^{\dagger}_{\rm d} M_{\rm d} O_{\rm d} & = & {\rm Diag}\{ m_d, ~ m_s, ~ m_b
\} \; ,
\end{eqnarray}
one obtains the KM matrix $V\equiv O^{\dagger}_{\rm u} O_{\rm d}$, which describes
quark flavor mixings in the charged current. Explicitly, the KM matrix elements
read
\begin{equation}
V_{ij} \; = \; \sum^3_{k=1} \left ( O^{{\rm u}^*}_{ki} ~ O^{\rm d}_{kj} \right ) \; ,
\end{equation}
depending upon the quark mass ratios $m_u/m_c$, $m_c/m_t$ (from $O_{\rm u}$) and
$m_d/m_s$, $m_s/m_b$ (from $O_{\rm d}$) as well as other parameters of $M_{\rm u}$
and $M_{\rm d}$ (e.g., the non-trivial phase shifts between $M_{\rm u}$ and $M_{\rm d}$).
In view of the distinctive mass hierarchy in Eq. (1.1), we find that some
interesting properties of $V$ can be interpreted without the assumption of specific
forms of $M_{\rm u}$ and $M_{\rm d}$.
\begin{center}
{\large\bf A. ~ $|V_{us}|$ and $|V_{cd}|$ in the limits $m_t\rightarrow \infty$
and $m_b\rightarrow \infty$}
\end{center}
Since the mass spectra of up and down quarks are absolutely dominated by
$m_t$ and $m_b$ respectively, the limits $m_t\rightarrow \infty$ and
$m_b\rightarrow \infty$ are expected to be very reliable when we discuss flavor mixings
between $(u,d)$ and $(c,s)$. In this case, the effective mass matrices
turn out to be two $2\times 2$ matrices and the resultant
flavor mixing matrix (i.e., the Cabibbo matrix \cite{Cabibbo})
cannot accommodate $CP$ violation. The magnitudes of $V_{us}$ and $V_{cd}$ can
be obtained from Eq. (2.2), since $O_{i3}=O_{3i}=\delta_{i3}$ holds for both
sectors in the above-mentioned mass limits. We find that
$|V_{us}| = |V_{cd}|$
is a straightforward result guaranteed by the unitarity of $O_{\rm u}$ and $O_{\rm d}$.
The current experimental data, together with unitary conditions of the
$3\times 3$ KM matrix, have implied \cite{XingV,XingW}
\begin{equation}
|V_{us}| ~ - ~ |V_{cd}| \; \approx \; A^2\lambda^5 \left (\frac{1}{2} - \rho
\right ) \; < \; 10^{-3} \; ,
\end{equation}
which is insensitive to allowed errors of the Wolfenstein parameters
$A$, $\lambda$ and $\rho$ \cite{Wolfenstein}. From the discussions above we realize
that the near equality of $|V_{us}|$ and $|V_{cd}|$ is in fact
a natural consequence of $m_t\gg m_c, m_u$ and $m_b \gg m_s, m_d$.
\vspace{0.3cm}
The magnitude of $V_{us}$ (or $V_{cd}$) must be a function of the mass ratios
$m_u/m_c$ and $m_d/m_s$ in the limits $m_t\rightarrow \infty$ and
$m_b\rightarrow \infty$, if $M_{\rm u}$ and $M_{\rm d}$ have
parallel or quasi-parallel structures. Considering the experimentally allowed
regions of $m_u/m_c$ ($\sim 5\times 10^{-3}$ \cite{PDG}), $m_s/m_d$ ($= 18.9\pm 0.8$
\cite{Leutwyler}) and $|V_{us}|$ ($=0.2205\pm 0.0018$ \cite{PDG}),
one may guess that $|V_{us}|$ is dominated by
$\sqrt{m_d/m_s}$ but receives small correction from $\sqrt{m_u/m_c}$.
Indeed such an instructive result for $|V_{us}|$ or $|V_{cd}|$
can be derived from $2\times 2$ Hermitian mass matrices of the form \cite{Weinberg}
\begin{equation}
\left ( \matrix{
0 & A \cr
A^* & B } \right ) \; ,
\end{equation}
where $|B|\gg |A|$. Denoting the phase difference between $A_{\rm u}$ and
$A_{\rm d}$ as $\Delta\phi$, we obtain
\begin{equation}
|V_{us}| \; = \; |V_{cd}| \; = \;
\left | ~ \sqrt{\frac{m_c}{m_u + m_c}} \sqrt{\frac{m_d}{m_d + m_s}}
~ - ~ \exp({\rm i} \Delta\phi) \sqrt{\frac{m_u}{m_u + m_c}} \sqrt{\frac{m_s}{m_d + m_s}}
~ \right | \; .
\end{equation}
Although the $2\times 2$ flavor mixing matrix cannot accommodate $CP$ violation,
the phase shift $\Delta\phi$ is non-trivial on the point that it sensitively determines
the value of $|V_{us}|$. For illustration, we calculate the allowed region of $\Delta\phi$
as a function of $m_u/m_c$ in Fig. 1. It is clear that the possibilities
$\Delta\phi=0^0$, $90^0$ and $180^0$ have all been ruled out by current data
on $V_{us}$ and $m_s/m_d$, since $m_u/m_c \geq 10^{-3}$ is expected to be true.
We conclude that the presence of $\Delta\phi$ in the quark mass {\it Ansatz} above
is crucial for correct reproduction of $|V_{us}|$ and $|V_{cd}|$.
Such a non-trivial phase shift will definitely lead to $CP$ violation,
when the limits $m_t\rightarrow \infty$ and $m_b\rightarrow \infty$ are discarded.
\begin{center}
{\large\bf B. ~ $|V_{cb}|$ and $|V_{ts}|$ in the limit $m_u=m_d=0$}
\end{center}
Considering the fact that $m_u$ and $m_d$ are negligibly small in
the mass spectra of up and down quarks, one can take the reasonable
limit $m_u=m_d=0$ to discuss flavor mixings between the second and
third families. In this case, there is no mixing between $(u,d)$ and $(c,s)$
or between $(u,d)$ and $(t,b)$. Thus $M_{1i}=M_{i1}=0$ holds for both
up and down mass matrices, and then we get $O_{1i}=O_{i1}=\delta_{1i}$.
The relation $|V_{cb}| = |V_{ts}|$ is straightforwardly obtainable
from Eq. (2.2) by use of the unitary conditions of $O_{\rm u}$ and $O_{\rm d}$.
In contrast, the present data and unitarity of the KM matrix requires \cite{XingW}
\begin{equation}
|V_{cb}| ~ - ~ |V_{ts}| \; \approx \; A \lambda^4 \left (\frac{1}{2} - \rho
\right ) \; < \; 10^{-2} \; .
\end{equation}
We see that the near equality between $|V_{cb}|$ and $|V_{ts}|$
can be well understood, because the quark mass limit $m_u=m_d=0$ is
a good approximation for $M_{\rm u}$ and $M_{\rm d}$.
\vspace{0.3cm}
We expect that $|V_{cb}|$ and $|V_{ts}|$ are functions of the mass ratios
$m_c/m_t$ and $m_s/m_b$ in the limit $m_u=m_d=0$. Current experimental
data give $|V_{cb}|=0.0388\pm 0.0032$ \cite{Neubert}, while
$m_c/m_t \sim 10^{-3}$ \cite{PDG} and $m_b/m_s = 34\pm 4$ \cite{Narison}
are allowed. Thus $|V_{cb}|$ (or $|V_{ts}|$) should be dominated by
$m_s/m_b$, and it may get a little correction from $m_c/m_t$. To obtain
a linear relation among $V_{cb}$, $m_s/m_b$ and $m_c/m_t$ in the leading order
approximation, one can investigate mass matrices of the following Hermitian form:
\begin{equation}
\left (\matrix{
0 & 0 & 0 \cr
0 & A & B \cr
0 & B^* & C } \right ) \; ,
\end{equation}
where $A\neq 0$ and $|C|\gg |B| \sim |A|$ for both quark sectors.
This generic pattern can also be regarded as a trivial generalization of the
Fritzsch {\it Ansatz}, in which $A=0$ is assumed \cite{Fritzsch77},
but they have rather different consequences on
the magnitudes of $V_{cb}$ and $V_{ts}$.
Denoting $\Delta\varphi = \arg(B_{\rm u}/B_{\rm d})$,
$R_{\rm u} = |B_{\rm u}/A_{\rm u}|$ and $R_{\rm d} = |B_{\rm d}/
A_{\rm d}|$, we find the approximate result
\begin{equation}
|V_{cb}| \; = \; |V_{ts}| \; \approx \;
\left | ~ R_{\rm d} \frac{m_s}{m_b} ~ - ~ \exp({\rm i} \Delta\varphi)
R_{\rm u} \frac{m_c}{m_t} ~ \right | \; .
\end{equation}
One can see that $|V_{cb}|\propto m_s/m_b$ holds approximately, if
$R_{\rm u}$ is comparable in magnitude with $R_{\rm d}$.
Here the phase shift $\Delta\varphi$ plays an insignificant
(negligible) role in confronting Eq. (2.8) with the experimental
data on $|V_{cb}|$, since the term proportional to $m_c/m_t$ is
significantly suppressed. To determine the values of $R_{\rm u}$
and $R_{\rm d}$, however, one has to rely on a more specific
{\it Ansatz} of quark mass matrices.
\begin{center}
{\large\bf C. ~ $|V_{ub}/V_{cb}|$ in $m_b\rightarrow \infty$
and $|V_{td}/V_{ts}|$ in $m_t\rightarrow \infty$}
\end{center}
Now let us take a look at the two smallest matrix elements of $V$,
$|V_{ub}|$ and $|V_{td}|$, in the quark mass limits.
Taking $m_b\rightarrow \infty$, we have $O^{\rm d}_{i3}=
O^{\rm d}_{3i}=\delta_{i3}$, because $M_{\rm d}$ turns out to be
an effective $2\times 2$ matrix in this limit. Then the ratio of
$|V_{ub}|$ to $|V_{cb}|$ reads
\begin{equation}
\lim_{m_b\rightarrow \infty} \left | \frac{V_{ub}}{V_{cb}} \right |
\; = \; \left | \frac{O^{\rm u}_{31}}{O^{\rm u}_{32}} \right | \; ,
\end{equation}
obtained from Eq. (2.2). Contrary to common belief, $|V_{ub}/V_{cb}|$ is absolutely
independent of the mass ratio $m_d/m_s$ in the limit $m_b\rightarrow
\infty$! Therefore one expects that the left-handed side of Eq. (2.9) is
dominated by a simple function of the mass ratio $m_u/m_c$, while the
contribution from $m_c/m_t$ should be insignificant in most cases.
The present numerical knowledge of $|V_{ub}/V_{cb}|$ ($=0.08\pm 0.02$
\cite{PDG}) and $m_u/m_c$ ($\sim 5\times 10^{-3}$ \cite{PDG}) implies
that $|V_{ub}/V_{cb}|\approx \sqrt{m_u/m_c}$ is likely to be true.
Indeed such an approximate result can be reproduced from
the Fritzsch {\it Ansatz} and a variety of its modified versions
\cite{Hall}.
\vspace{0.3cm}
In the mass limit $m_t\rightarrow \infty$, $M_{\rm u}$ becomes an effective
$2\times 2$ matrix, and then $O^{\rm u}_{i3}=O^{\rm u}_{3i}=\delta_{i3}$
holds. The ratio of $|V_{td}|$ to $|V_{ts}|$ is obtainable from Eq. (2.2)
as follows:
\begin{equation}
\lim_{m_t\rightarrow \infty} \left | \frac{V_{td}}{V_{ts}} \right |
\; = \; \left | \frac{O^{\rm d}_{31}}{O^{\rm d}_{32}} \right | \; .
\end{equation}
Here again we find that $|V_{td}/V_{ts}|$ is independent of both
$m_u/m_c$ and $m_c/m_t$ in the limit $m_t\rightarrow \infty$, thus it
may be a simple function of the mass ratios $m_d/m_s$ and $m_s/m_b$.
The current data give $0.15 \leq |V_{td}/V_{ts}| \leq 0.34$ \cite{Ali},
$m_s/m_d = 18.9\pm 0.8$ \cite{Leutwyler} and $m_b/m_s =34\pm 4$ \cite{Narison}.
We expect that $|V_{td}/V_{ts}|\approx \sqrt{m_d/m_s}$ has a
large chance to be true in the leading order approximation.
Note that this approximate relation can also be derived from the
Fritzsch {\it Ansatz} or some of its revised versions \cite{Hall}.
\vspace{0.3cm}
The qualitative discussions above have shown that
some properties of the KM matrix $V$ can be well understood just
from the quark mass hierarchy. For example, $|V_{us}|\approx |V_{cd}|$
and $|V_{cb}|\approx |V_{ts}|$ are natural consequences of arbitrary
(Hermitian) mass matrices with $m_3 \gg m_2, m_1$ and $m_1 \ll m_2, m_3$
respectively, where $m_i$ denote the mass eigenvalues of each quark
sector. To a good degree of accuracy, $|V_{us}|$ and $|V_{cd}|$ are
expected to be
independent of the mass ratios $m_c/m_t$ and $m_s/m_b$, while
$|V_{cb}|$ and $|V_{ts}|$ are independent of $m_u/m_c$ and $m_d/m_s$.
The ratios $|V_{ub}/V_{cb}|$ and $|V_{td}/V_{ts}|$ may be simple functions
of $m_u/m_c$ and $m_d/m_s$, respectively, in the leading order approximations.
These qualitative results should hold, in most cases and without fine
tuning effects, for generic (Hermitian) forms of $M_{\rm u}$ and $M_{\rm d}$.
They can be used as an enlightening clue for the construction of specific
and predictive $Ans\ddot{a}tze$ of quark mass matrices.
\section{A quark mass {\it Ansatz} at the weak scale}
\setcounter{equation}{0}
We are now in a position to consider the realistic $3\times 3$
mass matrices in no assumption of the quark mass limits. Such
an {\it Ansatz} should be able to yield the definite
values of $R_{\rm u}$ and $R_{\rm d}$ in Eq. (2.8), and account for
current experimental data on flavor mixings and $CP$ violation
at low-energy scales.
\begin{center}
{\large\bf A. ~ Flavor permutation symmetry breaking}
\end{center}
We start from the flavor permutation symmetry to construct
quark mass matrices at the weak scale, so that the resultant
KM matrix can be directly confronted with the experimental data.
The mass matrix with the $S(3)_{\rm L}\times S(3)_{\rm R}$ symmetry
reads
\begin{equation}
M_0 \; = \; \frac{c}{3} \left ( \matrix{
1 & 1 & 1 \cr
1 & 1 & 1 \cr
1 & 1 & 1 } \right ) \; ,
\end{equation}
where $c=m_3$ denotes the mass eigenvalue of the third-family
quark ($t$ or $b$). Note that $M_0$ is obtainable from another
rank-one matrix
\begin{equation}
M_{\rm H} \; = \; c \left ( \matrix{
0 & 0 & 0 \cr
0 & 0 & 0 \cr
0 & 0 & 1 } \right ) \;
\end{equation}
through the unitary transformation $M_0 = U^{\dagger} M_{\rm H} U$,
where
\begin{equation}
U \; = \; \frac{1}{\sqrt{6}} \left ( \matrix{
\sqrt{3} & -\sqrt{3} & 0 \cr
1 & 1 & -2 \cr
\sqrt{2} & \sqrt{2} & \sqrt{2} } \right ) \; .
\end{equation}
To generate masses for the second- and first-family quarks, one has to
break the permutation symmetry of $M_0$ to
the $S(2)_{\rm L}\times S(2)_{\rm R}$ and $S(1)_{\rm L}\times S(1)_{\rm R}$
symmetries, respectively.
Here we assume that the up and down mass
matrices have the parallel symmetry breaking patterns, corresponding to
the parallel dynamical details of quark mass generation. We further assume
that each symmetry breaking chain (i.e., $S(3)_{\rm L}\times S(3)_{\rm R}
\rightarrow S(2)_{\rm L}\times S(2)_{\rm R}$ or
$S(2)_{\rm L}\times S(2)_{\rm R} \rightarrow S(1)_{\rm L}\times S(1)_{\rm R}$)
is induced by a single real parameter,
and the possible phase shift between two quark sectors arises from
an unknown dynamical mechanism.
\vspace{0.3cm}
With the assumptions made above, a new {\it Ansatz} for the up and down
mass matrices can be given as follows:
\begin{equation}
M^{\prime}_0 \; = \; \frac{c}{3} \left [
\left ( \matrix{
1 & 1 & 1 \cr
1 & 1 & 1 \cr
1 & 1 & 1 } \right )
+ \epsilon \left ( \matrix{
0 & 0 & 1 \cr
0 & 0 & 1 \cr
1 & 1 & 1 } \right )
+ \sigma \left ( \matrix{
1 & 0 & -1 \cr
0 & -1 & 1 \cr
-1 & 1 & 0 } \right ) \right ] \; ,
\end{equation}
where $\epsilon$ and $\sigma$ are real (dimensionless) perturbation
parameters responsible for the breakdowns of $S(3)_{\rm L}\times
S(3)_{\rm R}$ and $S(2)_{\rm L}\times S(2)_{\rm R}$ symmetries of
$M_0$, respectively. In the basis of $M_{\rm H}$, the mass matrix
$M^{\prime}_0$ takes the form
\begin{equation}
M^{\prime}_{\rm H} \; = \; c
\left (\matrix{
0 & \displaystyle\frac{\sqrt{3}}{3} \sigma & 0 \cr \cr
\displaystyle\frac{\sqrt{3}}{3} \sigma
& \displaystyle -\frac{2}{9} \epsilon
& \displaystyle -\frac{2\sqrt{2}}{9} \epsilon \cr \cr
0 & \displaystyle -\frac{2\sqrt{2}}{9} \epsilon
& \displaystyle 1+\frac{5}{9} \epsilon \cr } \right ) \; ,
\end{equation}
which has three free parameters and three texture zeros.
Diagonalizing $M^{\prime}_{\rm H}$ through the unitary
transformation $O^{{\prime}^{\dagger}} M^{\prime}_{\rm H} O^{\prime}
= {\rm Diag} \{ m_1, ~ m_2, ~ m_3 \}$,
one can determine $c$, $\epsilon$
and $\sigma$ in terms of the quark mass eigenvalues. In the
next-to-leading order approximations, we get
\begin{eqnarray}
c & \approx & m_3 \left ( 1 + \frac{5}{2} \frac{m_2}{m_3}
\right ) \; , \nonumber \\
\epsilon & \approx & -\frac{9}{2} \frac{m_2}{m_3}
\left ( 1- \frac{1}{2} \frac{m_2}{m_3} \right ) \; , \nonumber \\
\sigma & \approx & \frac{\sqrt{3 m_1 m_2}}{m_3}
\left ( 1 - \frac{5}{2} \frac{m_2}{m_3} \right ) \; .
\end{eqnarray}
Then the elements of $O^{\prime}$ are expressible in terms of
the mass ratios $m_1/m_2$ and $m_2/m_3$.
\vspace{0.3cm}
The flavor mixing matrix can be given as $V= O^{{\prime}^{\dagger}}_{\rm u}
P O^{\prime}_{\rm d}$, where $P$ is a diagonal phase matrix taking
the form $P={\rm Diag} \{ 1, ~ \exp({\rm i} \Delta \phi),
~ \exp({\rm i} \Delta \phi) \}$.
Here $\Delta \phi$ denotes the phase shift between up and
down mass matrices, and its presence is necessary for the {\it Ansatz}
to correctly reproduce both $|V_{us}|$ (or $|V_{cd}|$) and $CP$ violation.
\begin{center}
{\large\bf B. ~ Flavor mixings and $CP$ violation}
\end{center}
Calculating the KM matrix elements $|V_{us}|$ and $|V_{cd}|$
in the next-to-leading order approximation, we obtain
\begin{equation}
|V_{us}| \; \approx \; |V_{cd}| \; \approx \;
\sqrt{\left (\frac{m_u}{m_c} +\frac{m_d}{m_s} - 2 \sqrt{\frac{m_u m_d}{m_c m_s}}
~ \cos \Delta\phi \right )
\left ( 1-\frac{m_u}{m_c}-\frac{m_d}{m_s}\right )} \; .
\end{equation}
This result is clearly consistent with that in Eq. (2.5). The allowed
region of $\Delta\phi$ has been shown by Fig. 1 with the inputs of
$m_s/m_d$ and $|V_{us}|$. We find $73^0 \leq \Delta\phi \leq 82^0$
for reasonable values of $m_u/m_c$. In the leading order
approximation of Eq. (3.7) or Eq. (2.5), it is easy to check that
$|V_{cd}|$, $\sqrt{m_u/m_c}$ and $\sqrt{m_d/m_s}$ form a triangle in
the complex plane \cite{FritzschXing}.
\vspace{0.3cm}
In the next-to-leading order approximation, $|V_{cb}|$ and $|V_{ts}|$
can be given as
\begin{equation}
|V_{cb}| \; \approx \; |V_{ts}| \; \approx \; \sqrt{2}
\left (\frac{m_s}{m_b} -\frac{m_c}{m_t}\right )
\left [ 1+3 \left (\frac{m_s}{m_b} + \frac{m_c}{m_t}\right ) \right ] \; .
\end{equation}
Comparing between Eqs. (3.8) and (2.8), we get
$R_{\rm u} = R_{\rm d} = \sqrt{2}$, determined by the
quark mass {\it Ansatz} in Eq. (3.4).
By use of $m_b/m_s = 34\pm 4$ \cite{Narison}, we illustrate the allowed
region of $|V_{cb}|$ as a function of $m_c/m_t$ in Fig. 2, where
the experimental constraint on $|V_{cb}|$ ($=0.0388 \pm 0.0032$
\cite{Neubert})
has also been shown. We see that the result of $|V_{cb}|$
obtained in Eq. (3.8) is rather favored by current data. This implies
that the pattern of permutation symmetry breaking
(i.e., $S(3)_{\rm L}\times S(3)_{\rm R} \rightarrow S(2)_{\rm L}
\times S(2)_{\rm R}$) in Eq. (3.4) may have a large chance to be true.
\vspace{0.3cm}
The ratios $|V_{ub}/V_{cb}|$ and $|V_{td}/V_{ts}|$ are found to be
\begin{equation}
\left | \frac{V_{ub}}{V_{cb}} \right | \; \approx \; \sqrt{\frac{m_u}{m_c}} \; ,
~~~~~~~~
\left | \frac{V_{td}}{V_{ts}} \right | \; \approx \; \sqrt{\frac{m_d}{m_s}} \;
\end{equation}
to a good degree of accuracy
\footnote{More exactly, we obtain
$|V_{ub}/V_{cb}| \approx \sqrt{m_u/m_c} ~ (1 - \delta )$ with
$\delta = \sqrt{(m_c m_d)/(m_u m_s)} ~ (m_s/m_b) \cos\Delta\phi$.
The magnitude of $\delta$ may be as large as $10\%$ to $15\%$
for $\Delta\phi \sim 0^0$ or $180^0$, but it is only about $2\%$
for $73^0 \leq \Delta\phi \leq 82^0$ allowed by Eq. (3.7).}.
By use of Leutwyler's result $m_s/m_d = 18.9\pm 0.8$ \cite{Leutwyler}, we get
$0.225\leq |V_{td}/V_{ts}| \leq 0.235$.
In comparison, the current data together with unitarity of
the $3\times 3$ KM matrix yield $0.15\leq |V_{td}/V_{ts}|\leq 0.34$
\cite{Ali}. The allowed region of
$|V_{ub}/V_{cb}|$ is constrained by that of $m_u/m_c$, which has not been
reliably determined.
We find that $0.0036 \leq m_u/m_c \leq 0.01$ is necessary for the quark mass
{\it Ansatz} in Eq. (3.4) to accommodate the experimental
result $|V_{ub}/V_{cb}| = 0.08\pm 0.02$ \cite{PDG}.
\vspace{0.3cm}
In the leading order approximations, we have
$|V_{ud}|\approx |V_{cs}| \approx |V_{tb}| \approx 1$. Small corrections to
these diagonal elements are obtainable with the help of the unitary
conditions of $V$. If we rescale three sides of the unitarity triangle
$V^*_{ub}V_{ud} + V^*_{cb}V_{cd} + V^*_{tb}V_{td} =0$ by $V^*_{cb}$,
then the resultant triangle is approximately equivalent to that formed
by $V_{cd}$, $\sqrt{m_u/m_c}$ and $\sqrt{m_d/m_s}$ in the complex plane
\cite{FritzschXing}. This interesting result can be easily shown by use of
Eqs. (3.7), (3.8) and (3.9). Three inner angles of the unitarity triangle
turn out to be
\begin{eqnarray}
\alpha & = & \arg \left (- \frac{V^*_{ub}V_{ud}}{V^*_{tb}V_{td}} \right )
\; \approx \; \Delta\phi \; , \nonumber \\
\beta & = & \arg \left (- \frac{V^*_{tb}V_{td}}{V^*_{cb}V_{cd}} \right )
\; \approx \; \tan \left (\frac{\sin\Delta \phi}
{\displaystyle \sqrt{\frac{m_c m_d}{m_u m_s}} - \cos\Delta \phi} \right ) \; ,
\nonumber \\
\gamma & = & \arg \left (- \frac{V^*_{cb}V_{cd}}{V^*_{ub}V_{ud}} \right )
\; \approx \; 180^0 - \alpha - \beta \;
\end{eqnarray}
in the approximations made above. At the forthcoming $B$-meson factories,
these three angles will be determined from $CP$ asymmetries in
a variety of weak $B$ decays (e.g., $B_d \rightarrow J/\psi K_S$,
$B_d\rightarrow \pi^+\pi^-$ and $B_s\rightarrow \rho^0 K_S$).
For illustration, we calculate
$\sin (2\alpha)$, $\sin (2\beta)$ and $\sin (2\gamma)$ by use of Eq. (3.10)
and plot their allowed regions in Fig. 3.
Clearly the quark mass {\it Ansatz} under discussion favors
$0.18\leq \sin(2\alpha) \leq 0.58$, $0.5\leq \sin(2\beta) \leq 0.78$
and $-0.08 \leq \sin(2\gamma) \leq 0.5$. These results do not involve
large errors, and they can be confronted with the relevant experiments
of $B$ decays and $CP$ violation in the near future.
\vspace{0.3cm}
Finally we point out that $CP$ violation in the KM matrix, measured by the
Jarlskog parameter $J$ \cite{Jarlskog}, can also be estimated in terms of quark
mass ratios. It is easy to obtain
\begin{equation}
J \; \approx \; 2 \sqrt{\frac{m_u}{m_c}} \sqrt{\frac{m_d}{m_s}}
\left ( \frac{m_s}{m_b} - \frac{m_c}{m_t} \right )^2
\left [ 1 + 6 \left ( \frac{m_s}{m_b} + \frac{m_c}{m_t} \right ) \right ]
\sin\Delta\phi \; .
\end{equation}
Typically taking $m_u/m_c = 0.005$, $m_s/m_d = 19$, $m_c/m_t = 0.005$,
$m_b/m_s = 34$ and $\Delta\phi = 80^0$, we get $J\approx 2.3 \times 10^{-5}$.
This result is of course consistent with current data on $CP$ violation in the
$K^0-\bar{K}^0$ mixing system \cite{PDG}.
\section{A quark mass {\it Ansatz} at the GUT scale}
\setcounter{equation}{0}
It is interesting to speculate that the quark mass hierarchy and flavor
mixings may arise from a certain symmetry breaking pattern in the
context of supersymmetric GUTs \cite{SUSY,Froggatt}.
Starting from the flavor permutation
symmetry, here we prescribe the same {\it Ansatz} for quark mass matrices
as that proposed in Eq. (3.4) at the supersymmetric GUT scale $M_X$.
For simplicity we use $\hat{M}_0$ and $\hat{M}_{\rm H}$,
which correspond to $M_0^{\prime}$ in Eq. (3.4) and $M^{\prime}_{\rm H}$
in Eq. (3.5), to denote the mass matrices at $M_X$ in two
different bases. They are related to each other through the unitary
transformation $\hat{M}_0 = U^{\dagger} \hat{M}_{\rm H}
U$. The flavor mixing matrix derived from $\hat{M}_0$ (or
$\hat{M}_{\rm H}$) is denoted by $\hat{V}$. The subsequent
running effects of $\hat{M}_0$ and $\hat{V}$ from $M_X$ to $M_Z$ can be
calculated with the help of the renormalization group equations in the
minimal supersymmetric standard model.
\begin{center}
{\large\bf A. ~ Renormalized mass matrices at $M_Z$}
\end{center}
The simplicity of $\hat{M}_0$ (or $\hat{M}_{\rm H}$)
may be spoiled after it evolves from $M_X$ to $M_Z$. To illustrate
this point, here we derive the renormalized mass matrices
$\hat{M}^{\rm u}_0$ and $\hat{M}^{\rm d}_0$ at $M_Z$
by use of the one-loop renormalization group equations for the
Yukawa matrices and gauge couplings \cite{Babu}.
To get instructive analytical
results, we constrain the ratio of Higgs vacuum expectation values
$\tan\beta_{\rm susy}$ to be small enough ($\tan\beta_{\rm susy} < 10$),
so that all non-leading
terms in the Yukawa couplings different from that of the top quark
can be safely neglected \cite{Giudice}.
In this approximation, the evolution equations
of $\hat{M}^{\rm u}_0$ and $\hat{M}^{\rm d}_0$ read
\begin{eqnarray}
16 \pi^2 \frac{{\rm d} \hat{M}^{\rm u}_0}{{\rm d} \chi}
& = & \left [ \frac{3}{v^2} {\rm Tr} \left ( \hat{M}^{\rm u}_0
\hat{M}^{{\rm u}^{\dagger}}_0 \right ) + \frac{3}{v^2} \left (
\hat{M}^{\rm u}_0 \hat{M}^{{\rm u}^{\dagger}}_0
\right ) - G_{\rm u} \right ] \hat{M}^{\rm u}_0 \; , \nonumber \\
16 \pi^2 \frac{{\rm d} \hat{M}^{\rm d}_0}{{\rm d} \chi}
& = & \left [ \frac{1}{v^2} \left (
\hat{M}^{\rm u}_0 \hat{M}^{{\rm u}^{\dagger}}_0
\right ) - G_{\rm d} \right ] \hat{M}^{\rm d}_0 \; ,
\end{eqnarray}
where $\chi = \ln (\mu /M_Z)$,
$G_{\rm u}$ and $G_{\rm d}$ are functions of the gauge couplings
$g^{~}_i$ ($i=1,2,3$), and $v$ is the overall Higgs vacuum expectation
value normalized to 175 GeV. For the charged lepton mass matrix
$\hat{M}^{\rm e}_0$, its evolution equation is dominated only by
a linear term $G_{\rm e}$ in the case of small $\tan\beta_{\rm susy}$.
Thus the Hermitian structure of $\hat{M}^{\rm e}_0$ will be unchanged
through the running from $M_X$ to $M_Z$ (in our discussions the neutrinos
are assumed to be massless). The quantity $G_{\rm n}$
(n = u, d or e) obeys the following equation:
\begin{equation}
G_{\rm n} (\chi) \; = \; 8\pi^2 \sum^3_{i=1} \left [ \frac{c^{\rm n}_i ~
g^2_i (0)}{8\pi^2 - b_i ~ g^2_i (0) ~ \chi} \right ] \; ,
\end{equation}
where $c^{\rm n}_i$ and $b_i$ are coefficients in the context of
the minimal supersymmetric standard model. The values of $g^2_i (0)$,
$c^{\rm n}_i$ and $b_i$ are listed in Table 1.
\begin{table}
\begin{center}
\begin{tabular}{c|ccccc} \hline\hline
$i$ & ~~ $c^{\rm u}_i$ ~~ & ~ $c^{\rm d}_i$ ~
& ~~ $c^{\rm e}_i$ ~~ & ~~ $b_i$ ~~ & ~~ $g^2_i(0)$ \\ \hline \\
1 & 13/9 & 7/9 & 3 & 11 & 0.127 \\ \\
2 & 3 & 3 & 3 & 1 & 0.42 \\ \\
3 & 16/3 & 16/3 & 0 & $-$3 & 1.44
\\ \\ \hline\hline
\end{tabular}
\end{center}
\caption{The values of $c^{\rm n}_i$, $b_i$ and $g^2_i(0)$ in the
minimal supersymmetric standard model.}
\end{table}
In order to solve Eq. (4.1), we diagonalize $\hat{M}^{\rm u}_0$ through
the unitary transformation $\hat{O}^{\dagger} \hat{M}^{\rm u}_0 \hat{O}
= \hat{M}^{{\rm u}^{\prime}}_0$. Making the same transformation for
$\hat{M}^{\rm d}_0$, i.e., $\hat{O}^{\dagger} \hat{M}^{\rm d}_0 \hat{O}
= \hat{M}^{{\rm d}^{\prime}}_0$, we obtain the simplified evolution equations
as follows:
\begin{eqnarray}
16 \pi^2 \frac{{\rm d}\hat{M}^{{\rm u}^{\prime}}_0}{{\rm d} \chi}
& = & \left [ 3 f^2_t \left ( \matrix{
0 & 0 & 0 \cr
0 & 0 & 0 \cr
0 & 0 & 1 } \right ) +
\left ( 3 f^2_t - G_{\rm u} \right ) \left ( \matrix{
1 & 0 & 0 \cr
0 & 1 & 0 \cr
0 & 0 & 1 } \right ) \right ] \hat{M}^{{\rm u}^{\prime}}_0
\; , \nonumber \\
16 \pi^2 \frac{{\rm d}\hat{M}^{{\rm d}^{\prime}}_0}{{\rm d} \chi}
& = & \left [ f^2_t \left ( \matrix{
0 & 0 & 0 \cr
0 & 0 & 0 \cr
0 & 0 & 1 } \right ) -
G_{\rm u} \left ( \matrix{
1 & 0 & 0 \cr
0 & 1 & 0 \cr
0 & 0 & 1 } \right ) \right ] \hat{M}^{{\rm d}^{\prime}}_0 \; ,
\end{eqnarray}
where $f_t = m_t/v$ is the top quark Yukawa coupling eigenvalue.
For simplicity in presenting the results, we define
\begin{eqnarray}
\Omega_{\rm n} & = & \exp \left [ + \frac{1}{16\pi^2} \int^{\ln (M_X/M_Z)}_0
G_{\rm n}(\chi) ~ {\rm d}\chi \right ] \; , \nonumber \\
\xi_i & = & \exp \left [ - \frac{1}{16\pi^2} \int^{\ln (M_X/M_Z)}_0
f^2_i (\chi) ~ {\rm d}\chi \right ]
\end{eqnarray}
with $i=t$ (or $i=b$). By use of Eq. (4.2) and the inputs listed in Table 1, one can
explicitly calculate the magnitude of $\Omega_{\rm n}$. We find
$\Omega_{\rm u}=3.47$, $\Omega_{\rm d}=3.38$ and $\Omega_{\rm e}=1.49$
for $M_X=10^{16}$ GeV and $M_Z=91.187$ GeV.
The size of $\xi_t$ depends upon the value of $\tan\beta_{\rm susy}$
and will be estimated in the next subsection.
Solving Eq. (4.3) and transforming $\hat{M}^{{\rm n}^{\prime}}_0$
back to $\hat{M}^{\rm n}_0$, we get
\begin{eqnarray}
\hat{M}^{\rm u}_0 (M_Z) & = & \Omega_{\rm u} ~ \xi^3_t ~ \hat{O}
\left ( \matrix{
1 & 0 & 0 \cr
0 & 1 & 0 \cr
0 & 0 & \xi^3_t } \right )
\hat{O}^{\dagger} ~ \hat{M}^{\rm u}_0 (M_X) \; , \nonumber \\
\hat{M}^{\rm d}_0 (M_Z) & = & \Omega_{\rm d} ~ \hat{O}
\left ( \matrix{
1 & 0 & 0 \cr
0 & 1 & 0 \cr
0 & 0 & \xi_t } \right )
\hat{O}^{\dagger} ~ \hat{M}^{\rm d}_0 (M_X) \;
\end{eqnarray}
in the leading order approximation.
\vspace{0.3cm}
Since $\hat{O}$ can be easily determined from $\hat{M}^{\rm u}_0 (M_X)$ and
$\hat{M}^{{\rm u}^{\prime}}_0 (M_X)$ in the approximation of
$\hat{M}^{{\rm u}^{\prime}}_0 (M_X) \approx {\rm Diag} \{0, ~ 0, ~ m_t \}$
made above,
we explicitly express $\hat{M}^{\rm u}_0 (M_Z)$ and $\hat{M}^{\rm d}_0 (M_Z)$
as follows:
\begin{equation}
\hat{M}^{\rm u}_0 (M_Z) \; = \; \frac{c_{\rm u}}{3} \Omega_{\rm u} \xi_t^3
\left [ \xi^3_t
\left ( \matrix{
1 & 1 & 1 \cr
1 & 1 & 1 \cr
1 & 1 & 1 } \right )
+ \epsilon_{\rm u} \left ( \matrix{
x_{\rm u} & x_{\rm u} & y_{\rm u} \cr
x_{\rm u} & x_{\rm u} & y_{\rm u} \cr
y_{\rm u} & y_{\rm u} & z_{\rm u} } \right )
+ \sigma_{\rm u} \left ( \matrix{
1 & 0 & -1 \cr
0 & -1 & 1 \cr
-1 & 1 & 0 } \right ) \right ] \;
\end{equation}
with $x_{\rm u} = (\xi^3_t -1)/9$, $y_{\rm u} =(7 \xi^3_t +2)/9$
and $z_{\rm u} = (13 \xi^3_t -4)/9$; and
\begin{equation}
\hat{M}^{\rm d}_0 (M_Z) \; = \; \frac{c_{\rm d}}{3} \Omega_{\rm d}
\left [ \xi_t
\left ( \matrix{
1 & 1 & 1 \cr
1 & 1 & 1 \cr
1 & 1 & 1 } \right )
+ \epsilon_{\rm d} \left ( \matrix{
x_{\rm d} & x_{\rm d} & y_{\rm d} \cr
x_{\rm d} & x_{\rm d} & y_{\rm d} \cr
y_{\rm d} & y_{\rm d} & z_{\rm d} } \right )
+ D_{\epsilon}
+ \sigma_{\rm d} \left ( \matrix{
1 & 0 & -1 \cr
0 & -1 & 1 \cr
-1 & 1 & 0 } \right ) \right ] \;
\end{equation}
with $x_{\rm d} = (\xi_t -1)/9$, $y_{\rm d} =(7 \xi_t +2)/9$,
$z_{\rm d} = (13 \xi_t -4)/9$ and
\begin{equation}
D_{\epsilon} \; = \; 2 \left (\epsilon_{\rm d} - \epsilon_{\rm u} \right )
x_{\rm d} \left ( \matrix{
1 & 1 & 1 \cr
1 & 1 & 1 \cr
-2 & -2 & -2 } \right ) \; .
\end{equation}
If one takes $M_Z=M_X$, which leads to $\Omega_{\rm n}=1$,
$\xi_i =1$ and in turn $x_{\rm n}=0$, $y_{\rm n}=1$,
$z_{\rm n}=1$ and $D_{\epsilon}=0$,
then Eqs. (4.6) and (4.7) recover the form of
$\hat{M}_0 (M_X)$ as assumed in Eq. (3.4).
To a good degree of accuracy, $\hat{M}^{\rm u}_0 (M_Z)$
remains Hermitian. The Hermiticity of $\hat{M}^{\rm d}_0 (M_Z)$
is violated by $D_{\epsilon}$, which would vanish
if the top and bottom quark masses were identical
(i.e., $\epsilon_{\rm d} = \epsilon_{\rm u}$). The presence of
nonvanishing $D_{\epsilon}$ reflects the fact that $m_t$ dominates
the mass spectra of both quark sectors \cite{Albright}.
Of course, one can transform the mass matrices obtained in Eqs. (4.6)
and (4.7) into
the basis of $\hat{M}_{\rm H}$. In doing so, we will find the inequality
between (2,3) and (3,2) elements of $\hat{M}^{\rm d}_{\rm H} (M_Z)$,
arising from $D_{\epsilon}$.
\begin{center}
{\large\bf B. ~ Renormalized flavor mixings at $M_Z$}
\end{center}
Calculating the magnitudes of flavor mixings from $\hat{M}_0$ or
$\hat{M}_{\rm H}$ at
$M_X$, we can obtain the same asymptotic relations between the KM matrix
elements and quark mass ratios as those given in Eqs. (3.7), (3.8) and (3.9).
Now we renormalize such relations at the weak scale $M_Z$ by
means of the renormalization group equations. The quantities
$\xi_t$ and $\xi_b$ defined in Eq. (4.4) will be evaluated below for arbitrary
$\tan\beta_{\rm susy}$,
so that one can get some quantitative feeling about the running effects
of quark mass matrices and flavor mixings from $M_X$ to $M_Z$.
\vspace{0.3cm}
The one-loop renormalization group equations for quark mass ratios and
elements of the KM matrix $\hat{V}$ have been explicitly presented by
Babu and Shafi in Ref. \cite{Babu}.
In view of the hierarchy of Yukawa couplings and quark mixing angles, one can
make reliable analytical approximations for the relevant evolution equations
by keeping only the leading terms. It has been found that
(1) the running effects of $m_u/m_c$ and $m_d/m_s$ are negligibly small;
(2) the diagonal elements of the KM matrix have negligible evolutions with energy;
(3) the evolutions of $|\hat{V}_{us}|$ and $|\hat{V}_{cd}|$ involve
the second-family Yukawa couplings and thus they are negligible;
(4) the KM matrix elements $|\hat{V}_{ub}|$, $|\hat{V}_{cb}|$, $|\hat{V}_{td}|$
and $|\hat{V}_{ts}|$ have identical running behaviors.
Considering these points as well as the dominance of the third-family
Yukawa couplings (i.e., $f_t$ and $f_b$), we get three key evolution equations
in the minimal supersymmetric standard model:
\begin{eqnarray}
\left . \frac{m_s}{m_b} \right |_{M_Z} & = & \frac{1}{\xi_t ~ \xi^3_b} ~
\left . \frac{m_s}{m_b} \right |_{M_X} \; , \nonumber \\
\left . \frac{m_c}{m_t} \right |_{M_Z} & = & \frac{1}{\xi^3_t ~ \xi_b} ~
\left . \frac{m_c}{m_t} \right |_{M_X} \; , \nonumber \\
\left |\hat{V}_{ij} \right |_{M_Z} & = & \frac{1}{\xi_t ~ \xi_b} ~
\left |\hat{V}_{ij} \right |_{M_X} \;
\end{eqnarray}
with $(ij) = (ub)$, $(cb)$, $(td)$ or $(ts)$. In the same approximations,
the renormalization group equations for the Yukawa coupling eigenvalues
$f_t$, $f_b$ and $f_{\tau}$ read \cite{Babu}:
\begin{eqnarray}
16 \pi^2 \frac{{\rm d} f_t}{{\rm d} \chi} & = & f_t \left ( 6 f^2_t ~ + ~
f^2_b ~ - ~ G_{\rm u} \right ) \; , \nonumber \\
16 \pi^2 \frac{{\rm d} f_b}{{\rm d} \chi} & = & f_b \left ( f^2_t ~ + ~
6 f^2_b ~ + ~ f^2_{\tau} ~ - ~ G_{\rm d} \right ) \; , \nonumber \\
16 \pi^2 \frac{{\rm d} f_{\tau}}{{\rm d} \chi} & = & f_{\tau} \left (
3 f^2_b ~ + ~ 4 f^2_{\tau} ~ -~ G_{\rm e} \right ) \; ,
\end{eqnarray}
where the quantities $G_{\rm n}$ have been given in Eq. (4.2).
\vspace{0.3cm}
With the typical inputs $m_t (M_Z) \approx 180$ GeV, $m_b (M_Z) \approx 3.1$ GeV,
$m_{\tau} (M_Z) \approx 1.78$ GeV and those listed in Table 1,
we calculate $\xi_t$ and $\xi_b$ for arbitrary $\tan\beta_{\rm susy}$
by use of the above equations. Our result is illustrated in Fig. 4.
We see that $\xi_b \approx 1$ for $\tan\beta_{\rm susy} \leq 10$. This
justifies our approximation made previously in deriving
$\hat{M}^{\rm u}_0 (M_Z)$ and $\hat{M}^{\rm d}_0 (M_Z)$.
Within the perturbatively allowed region of
$\tan\beta_{\rm susy}$ \cite{Froggatt},
$\xi_b$ may be comparable in magnitude with $\xi_t$ when $\tan\beta_{\rm susy}
\geq 50$. In this case, the evolution effects of quark mass matrices and
flavor mixings are sensitive to both $f_t$ and $f_b$.
\vspace{0.3cm}
Clearly the analytical results of $|\hat{V}_{us}|$, $|\hat{V}_{cd}|$,
$|\hat{V}_{ub}/\hat{V}_{cb}|$ and $|\hat{V}_{td}/\hat{V}_{ts}|$
as those given in Eqs. (3.7) and (3.9) are almost scale-independent,
i.e., they hold at both $\mu=M_X$ and $\mu=M_Z$. Non-negligible
running effects can only appear in the expression of $|\hat{V}_{cb}|$ or
$|\hat{V}_{ts}|$, which is a function of the mass ratios $m_s/m_b$ and
$m_c/m_t$ (see Eq. (3.8) for illustration). With the help of Eq. (4.9),
we find the renormalized relation between $|\hat{V}_{cb}|$ (or $|\hat{V}_{ts}|$)
and the quark mass ratios at the weak scale $M_Z$:
\begin{equation}
|\hat{V}_{cb}| \; \approx \; |\hat{V}_{ts}| \; \approx \;
\sqrt{2} \left ( \xi^2_b \frac{m_s}{m_b} - \xi^2_t \frac{m_c}{m_t} \right )
\left [ 1 + 3 \xi_t \xi_b \left ( \xi^2_b \frac{m_s}{m_b} + \xi^2_t \frac{m_c}{m_t}
\right ) \right ] \; .
\end{equation}
This result will recover that in Eq. (3.8) if one takes $M_Z=M_X$ (i.e.,
$\xi_t=\xi_b=1$). Using $m_b/m_s=34\pm 4$ \cite{Narison}
and taking $m_c/m_t=0.005$ typically,
we confront Eq. (4.11) with the experimental data on $\hat{V}_{cb}$
(i.e., $|\hat{V}_{cb}|=0.0388\pm 0.0032$ \cite{Neubert}).
As shown in Fig. 5, our result is in good agreement with experiments
for $\tan\beta_{\rm susy} < 50$. This implies that
the quark mass pattern $\hat{M}_0$ or $\hat{M}_{\rm H}$,
proposed at the supersymmetric GUT scale $M_X$, may have a large chance to
survive for reasonable values of $\tan\beta_{\rm susy}$.
\vspace{0.3cm}
Note that evolution of the $CP$-violating parameter $J$ is dominated by
that of $|\hat{V}_{cb}|^2$. Note also that
three sides of the unitarity triangle $\hat{V}^*_{ub}\hat{V}_{ud} +
\hat{V}^*_{cb}\hat{V}_{cd} + \hat{V}^*_{tb}\hat{V}_{td}=0$ have
identical running effects from $M_X$ to $M_Z$, hence its three
inner angles are scale-independent and take the same values as those
given in Eq. (3.10) or Fig. 3. As a result, measurements of $\alpha$,
$\beta$ and $\gamma$ in the forthcoming experiments of $B$ physics
may check both the quark mass {\it Ansatz} at the
weak scale and that at the supersymmetric GUT scale.
\section{Summary}
\setcounter{equation}{0}
Without the assumption of specific mass matrices, we have pointed out
that part of the observed properties of flavor mixings can be well understood
just from the quark mass hierarchy. In the quark mass limits such as
$m_u=m_d=0$, $m_t\rightarrow \infty$ or $m_b\rightarrow \infty$,
a few instructive relations between the KM matrix
elements and quark mass ratios are suggestible from current experimental data.
We stress that such {\it Ansatz}-independent results may serve as
a useful guide in constructing the specific quark mass matrices at either
low-energy scales or superheavy scales.
\vspace{0.3cm}
Starting from the flavor permutation symmetry and assuming an explicit pattern
of symmetry breaking, we have proposed a new quark mass {\it Ansatz}
at the weak scale. We find that all experimental
data on quark mixings and $CP$ violation can be accounted for
by our {\it Ansatz}. In particular, we obtain an
instructive relation among $|V_{cb}|$, $m_s/m_b$ and $m_c/m_t$ in the
next-to-leading approximation (see Eq. (3.8)). The scale-independent
predictions of our quark mass pattern, such as
$0.18 \leq \sin(2\alpha) \leq 0.58$, $0.5\leq \sin(2\beta) \leq 0.78$
and $-0.08 \leq \sin(2\gamma) \leq 0.5$, can be confronted with
the forthcoming experiments at KEK and SLAC $B$-meson factories.
\vspace{0.3cm}
With the same {\it Ansatz} prescribed at the supersymmetric GUT scale
$M_X$, we have derived the renormalized quark mass matrices at the weak
scale $M_Z$ for small $\tan\beta_{\rm susy}$ and calculated the renormalized
flavor mixing matrix elements at $M_Z$ for arbitrary $\tan\beta_{\rm susy}$.
Except $|\hat{V}_{cb}|$ and $|\hat{V}_{ts}|$, the other asymptotic
relations between the KM matrix elements and quark mass ratios
are almost scale-independent. We find that the renormalized
result of $|\hat{V}_{cb}|$ (or $|\hat{V}_{ts}|$) is
in good agreement with the relevant experimental data for reasonable
values of $\tan\beta_{\rm susy}$.
\vspace{0.3cm}
In this work we neither assumed a specific form for the charged lepton
mass matrix nor supposed its relation with the down quark mass matrix within the
supersymmetric GUT framework. Of course, this can be done by following
the strategy proposed in Ref. \cite{Georgi}. Then one may obtain the
relations between $m_d$, $m_s$, $m_b$ and $m_e$, $m_{\mu}$, $m_{\tau}$.
Such an {\it Ansatz}, based on the specific GUT scheme and flavor
permutation symmetry breaking, will be discussed somewhere else.
\vspace{0.5cm}
\begin{flushleft}
{\Large\bf Acknowledgements}
\end{flushleft}
The author would like to thank A.I. Sanda for his warm hospitality and the Japan
Society for the Promotion of Science for its financial support. He is also
grateful to H. Fritzsch, A.I. Sanda and K. Yamawaki for their useful comments
on the topic of permutation symmetry breaking and on part of this work.
\newpage
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proofpile-arXiv_065-678
|
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\subsection*{1. Introduction}
The connection of positive knots with transcendental numbers, via the
counterterms of quantum field theory, proposed in~\cite{DK1} and
developed in~\cite {DK2}, and has been vigorously tested against
previous~\cite{GPX,DJB} and new~\cite{BKP} calculations,
entailing knots with up to 11 crossings, related by counterterms with
up to 7 loops to numbers that are irreducible
multiple zeta values (MZVs)~\cite{DZ,LM}.
Cancellations of transcendentals in gauge
theories have been illuminated by knot theory~\cite{BDK}. All-order results,
from large-$N$ analyses~\cite{BGK} and Dyson-Schwinger methods~\cite{DKT},
have further strengthened the connection of knot theory and number theory,
via field theory. A striking feature of this connection is that the
first irreducible MZV of depth 2 occurs at weight 8~\cite{DJB,BBG}, in
accord with the appearance of the first positive 3-braid knot at crossing
number 8. Likewise the first irreducible MZV of depth 3 occurs at weight
11~\cite{BG}, matching the appearance of the first positive 4-braid
at 11 crossings, obtained from skeining link diagrams that encode momentum
flow in 7-loop counterterms~\cite{BKP}. Moreover, the
investigations in~\cite{BGK} led to a new discovery at weight 12,
where it was found that the reduction of MZVs first entails alternating
Euler sums. The elucidation of this phenomenon
resulted in an enumeration~\cite{EUL} of irreducible Euler sums
and prompted intensive searches for evaluations of sums of
arbitrary depth~\cite{BBB}. A review of all these developments is
in preparation~\cite{DK}.
This paper pursues the connection to 8 and 9 loops, entailing knots
with up to 15 crossings.
In Section~2, we enumerate irreducible MZVs by weight.
Section~3 reports calculations of Feynman diagrams that yield
transcendental knot-numbers entailing MZVs up to weight 15.
In Section~4 we enumerate positive knots, up to 15 crossings,
and give the braid words and HOMFLY polynomials~\cite{VJ} for
all knots associated with irreducible MZVs of weight $n<17$.
Section~5 gives our conclusions.
\subsection*{2. Multiple zeta values}
We define $k$-fold Euler sums~\cite{BBG,BG} as in~\cite{EUL,BBB},
allowing for alternations of signs in
\begin{equation}
\zeta(s_1,\ldots,s_k;\sigma_1,\ldots,\sigma_k)=\sum_{n_j>n_{j+1}>0}
\quad\prod_{j=1}^{k}\frac{\sigma_j^{n_j}}{n_j^{s_j}}\,,\label{form}
\end{equation}
where $\sigma_j=\pm1$, and the exponents $s_j$ are positive integers,
with $\sum_j s_j$ referred to as the weight (or level) and $k$ as the depth.
We combine the strings of exponents and signs
into a single string, with $s_j$ in the $j$th position when $\sigma_j=+1$,
and $\overline{s}_j$ in the $j$th position when $\sigma_j=-1$.
Referring to non-alternating sums as MZVs~\cite{DZ},
we denote the numbers of irreducible Euler sums
and MZVs by $E_n$ and $M_n$, at weight $n$, and find that
\begin{equation}
1-x -x^2=\prod_{n>0}(1-x^n)^{E_n}\,;\quad
1-x^2-x^3=\prod_{n>0}(1-x^n)^{M_n}\,,\label{EM}
\end{equation}
whose solutions, developed in Table~1, are given in closed form by
\begin{eqnarray}
E_n=\frac{1}{n}\sum_{d|n}\mu(n/d)L_d\,;
&&L_n=L_{n-1}+L_{n-2}\,;\quad L_1=1\,;\quad L_2=3\,,\label{Es}\\
M_n=\frac{1}{n}\sum_{d|n}\mu(n/d)P_d\,;
&&P_n=P_{n-2}+P_{n-3}\,;\quad P_1=0\,;\quad P_2=2;\quad P_3=3\,,\label{Ms}
\end{eqnarray}
where $\mu$ is the M\"obius function, $L_n$ is a Lucas number~\cite{EUL} and
$P_n$ is a Perrin number~\cite{AS}.
\noindent{\bf Table~1}:
The integer sequences\Eqqq{Es}{Ms}{Kn} for $n\leq20$.
\[\begin{array}{|r|rrrrrrrrrrrrrrrrrrrr|}\hline
n&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20\\\hline
E_n&1&1&1&1&2&2&4&5&8&11&18&25&40&58&90&135&210&316&492&750\\
M_n&0&1&1&0&1&0&1&1&1&1&2&2&3&3&4&5&7&8&11&13\\
K_n&0&0&1&0&1&1&1&2&2&3&4&5&7&9&12&16&21&28&37&49\\\hline
\end{array}\]
In~\cite{EUL}, $E_n=\sum_k E_{n,k}$ was apportioned, according to
the minimum depth $k$ at which irreducibles of weight $n$ occur.
Similarly, we have apportioned $M_n=\sum_k M_{n,k}$.
The results are elements of Euler's triangle~\cite{EUL}
\begin{equation}
T(a,b)=\frac{1}{a+b}\sum_{d|a,b}\mu(d)\,P(a/d,b/d)\,,
\label{ET}
\end{equation}
which is a M\"obius transform of Pascal's triangle, $P(a,b)={a+b\choose a}
=P(b,a)$.
We find that
\begin{equation}
E_{n,k}=T(\df{n-k}{2},k)\,;\quad
M_{n,k}=T(\df{n-3k}{2},k)\,,
\label{EMb}
\end{equation}
for $n>2$ and $n+k$ even.
There is a remarkable feature of the result for $M_{n,k}$: it gives
the number of irreducible Euler sums of weight $n$ and depth $k$ that occur
in the reduction of MZVs, which is {\em not\/} the same as the
number of irreducible MZVs of this weight and depth. It was shown
in~\cite{BGK,EUL} that alternating multiple sums occur in the reduction of
non-alternating multiple sums. For example, $\zeta(4,4,2,2)$
cannot be reduced to MZVs of lesser depth, but it can~\cite{EUL} be reduced
to the alternating Euler sum $\zeta(\overline9,\overline3)$.
Subsequently we found
that an analogous ``pushdown'' occurs at weight 15, where depth-5 MZVs,
such as $\zeta(6,3,2,2,2)$, cannot be reduced to MZVs of lesser depth,
yet can be reduced to alternating Euler sums, with
$\zeta(9,\overline3,\overline3)-\frac{3}{14}\zeta(7,\overline5,\overline3)$
serving as the corresponding depth-3 irreducible. We conjecture
that the number, $D_{n,k}$, of MZVs of weight $n$ and depth $k$
that are not reducible to MZVs of lesser depth
is generated by
\begin{equation}
1-\frac{x^3 y}{1-x^2}+\frac{x^{12}y^2(1-y^2)}{(1-x^4)(1-x^6)}
=\prod_{n\ge3} \prod_{k\ge1} (1-x^n y^k)^{D_{n,k}},\label{Pd}
\end{equation}
which agrees with~\cite{BBG,BG} for $k<4$ and all weights $n$,
and with available data on MZVs, obtained from binary
reshuffles~\cite{LM} at weights $n\leq20$ for $k=4$, and $n\leq18$
for $k>4$. Further tests of\Eq{Pd} require very large scale
computations, which are in progress, with encouraging results.
However, the work reported here does not rely on this conjecture;
the values of $\{M_n\mid n\le15\}$ in Table~1 are sufficient for present
purposes, and these are amply verified by exhaustive use of
the integer-relation search routine MPPSLQ~\cite{DHB}.
Finally, in this section, we remark on the simplicity of the prediction
of\Eq{Ms} for the dimensionality, $K_n$, of the search space
for counterterms that evaluate to MZVs of weight $n$.
Since $\pi^2$, with weight $n=2$, does
not occur in such counterterms, it follows that they
must be expressible in terms of transcendentals that are enumerated by
$\{M_n\mid n\geq3\}$, and products of such knot-numbers~\cite{DK1,BGK,EUL}.
Thus $K_n$ is given by a Padovan sequence:
\begin{equation}
\sum_n x^n K_n=\frac{x^3}{1-x^2-x^3}\,;\quad
K_n=K_{n-2}+K_{n-3}\,;\quad K_1=0\,;\quad K_2=0\,;\quad K_3=1\,,\label{Kn}
\end{equation}
which is developed in Table~1. Note that the dimension of
the search space for a general MZV of weight $n$ is $K_{n+3}$~\cite{DZ},
which exceeds $K_n$ by a factor~\cite{AS} of $2.324717957$, as $n\to\infty$.
\subsection*{3. Knot-numbers from evaluations of Feynman diagrams}
The methods at our disposal~\cite{DK1,DK2,BKP} do not yet permit us to
predict, {\em a priori\/}, the transcendental knot-number assigned to
a positive knot by field-theory counterterms; instead we need
a concrete evaluation of at least one diagram
which skeins to produce that knot.
Neither do they allow us to predict the rational coefficients with which
such knot-numbers, and their products, corresponding to factor knots,
occur in a counterterm; instead we must, at present, determine these
coefficients by analytical calculation, or by applying a lattice method,
such as MPPSLQ~\cite{DHB}, to (very) high-precision numerical data.
Nonetheless, the consequences of~\cite{DK1,DK2} are highly predictive
and have survived intensive testing with amazing fidelity. The origin
of this predictive content is clear: once a knot-number is determined
by one diagram, it is then supposed, and indeed found, to occur in the
evaluation of all other diagrams which skein to produce that knot.
Moreover, the search
space for subdivergence-free counterterms that evaluate to MZVs
is impressively smaller than that for the MZVs themselves,
due to the absence of any knot associated with $\pi^2$,
and again the prediction is borne out by a wealth of data.
We exemplify these features by considering diagrams that
evaluate to MZVs of depths up to 5, which is the greatest depth
that can occur at weights up to 17, associated with knots up to
crossing-number 17, obtained from diagrams with up to 10 loops.
We follow the economical notation of~\cite{GPX,DJB,BKP}, referring
to a vacuum diagram by a so-called angular diagram~\cite{GPX}, which
results from choosing one vertex as origin, and indicating all vertices
that are connected to this origin by dots, after removing the origin
and the propagators connected to it.
{From} such an angular diagram one may uniquely reconstruct
the original Feynman diagram. The advantage of this notation is that
the Gegenbauer-polynomial $x$-space technique~\cite{GPX} ensures
that the maximum depth of sum which can result is the smallest
number of propagators in any angular diagram that characterizes
the Feynman diagram. Fig.~1 shows a naming convention for log-divergent
vacuum diagrams with angular diagrams that yield up to 5-fold sums.
To construct, for example, the 7-loop diagram $G(4,1,0)$ one places
four dots on line 1 and one dot on line 2.
Writing an origin at any point disjoint from the angular diagram,
and joining all 6 dots to that origin, one recovers the
Feynman diagram in question.
Using analytical techniques developed
in~\cite{GPX,DJB,BG,EUL}, we find that all subdivergence-free diagrams
of $G$-type, up to 13 loops (the highest number computable in
the time available), give counterterms that evaluate
to $\zeta(2n+1)$, their products, and depth-3 knot-numbers
chosen from the sets
\begin{eqnarray}
N_{2m+1,2n+1,2m+1}&=&
\zeta(2m+1,2n+1,2m+1)-\zeta(2m+1)\,\zeta(2m+1,2n+1)\nonumber\\&&{}
+\sum_{k=1}^{m-1}{2n+2k\choose2k}\zeta_P(2n+2k+1,2m-2k+1,2m+1)\nonumber\\&&{}
-\sum_{k=0}^{n-1}{2m+2k\choose2k}\zeta_P(2m+2k+1,2n-2k+1,2m+1)
\,,\label{K3o}\\
N_{2m,2n+1,2m}&=&
\zeta(2m,2n+1,2m)+\zeta(2m)\left\{\zeta(2m,2n+1)+\zeta(2m+2n+1)\right\}
\nonumber\\&&{}
+\sum_{k=1}^{m-1} {2n+2k\choose2k }\zeta_P(2n+2k+1,2m-2k,2m)\nonumber\\&&{}
+\sum_{k=0}^{n-1} {2m+2k\choose2k+1}\zeta_P(2m+2k+1,2n-2k,2m)
\,,\label{K3e}
\end{eqnarray}
where $\zeta_P(a,b,c)=\zeta(a)\left\{2\,\zeta(b,c)+\zeta(b+c)\right\}$.
The evaluation of a 9-loop non-planar example, $G(3,2,2)$,
is given in~\cite{EUL}: it evaluates in terms of MZVs of weights
ranging from 6 to 14. Choosing from\Eqq{K3o}{K3e}
one knot-number at 11 crossings and two at 13 crossings,
one arrives at an expression from which all powers of $\pi^2$ are banished,
which is a vastly more specific result than for a generic collection of MZVs
of these levels, and is in striking accord with what is required by the
knot-to-number connection entailed by field theory. Moreover, all planar
diagrams that evaluate to MZVs have been found to contain terms purely of
weight $2L-3$ at $L$ loops, matching the pattern of the zeta-reducible
crossed ladder diagrams~\cite{DK1,DK2}.
Subdivergence-free counterterms obtained from the $M$-type angular diagrams of
Fig.~1 evaluate to MZVs of weight $2L-4$, at $L$-loops, with depths up to 4.
Up to $L=8$ loops, corresponding to 12 crossings, the depth-4 MZVs can be
reduced to the depth-2 alternating sums~\cite{EUL}
$N_{a,b}\equiv\zeta(\overline{a},b)-\zeta(\overline{b},a)$. The knot-numbers
for the $(4,3)$ and $(5,3)$ torus knots may be taken as $N_{5,3}$
and $N_{7,3}$, thereby banishing $\pi^2$ from the associated 6-loop
and 7-loop counterterms. In general, $N_{2k+5,3}$ is a $(2k+8)$-crossing
knot-number at $(k+6)$ loops. Taking the second knot-number at 12 crossings
as~\cite{BGK,EUL} $N_{7,5}-\frac{\pi^{12}}{2^5 10!}$,
we express all 8-loop $M$-type counterterms in a $\pi$-free form.
At 9 loops, and hence 14 crossings, we encounter the first depth-4 MZV
that cannot be pushed down to alternating Euler sums of depth 2.
The three knot-numbers are again highly specific: to $N_{11,3}$
we adjoin
\begin{equation}
N_{9,5}+\df{5\pi^{14}}{7032946176}\,;\quad
\zeta(5,3,3,3)+\zeta(3,5,3,3)-\zeta(5,3,3)\zeta(3)
+\df{24785168\pi^{14}}{4331237155245}\,.\label{k14}
\end{equation}
Having determined these knot-numbers by applying MPPSLQ to
200 significant-figure evaluations of two counterterms, in a search space
of dimension $K_{17}=21$, requisite for generic MZVs of weight 14,
knot theory requires that we find the remaining five $M$-type counterterms
in a search space of dimension merely $K_{14}=9$. This prediction is totally
successful. The rational coefficients are too cumbersome to write here;
the conclusion is clear: when counterterms evaluate to MZVs they live
in a $\pi$-free domain, much smaller than that inhabited by a generic
MZV, because of the apparently trifling circumstance that a knot with
only two crossings is necessarily the unknot.
Such wonders continue, with subdivergence-free diagrams of types
$C$ and $D$ in Fig.~1
Up to 7 loops we have obtained {\em all\/} of them in terms of the
established knot-numbers $\{\zeta(3),\zeta(5),\zeta(7),N_{5,3},\zeta(9),
N_{7,3},\zeta(11),N_{3,5,3}\}$,
associated in~\cite{BKP,BGK} with the positive knots
$\{(3,2),(5,2),(7,2),8_{19}=(4,3),(9,2),10_{124}=(5,3),(11,2),11_{353}
=\sigma_1^{}\sigma_2^{3}\sigma_3^{2}\sigma_1^{2}\sigma_2^{2}\sigma_3^{}\}$,
and products of those knot-numbers, associated with the corresponding
factor knots.
A non-planar $L$-loop diagram may have terms of different weights,
not exceeding $2L-4$.
Invariably, a planar $L$-loop diagram evaluates purely
at weight $2L-3$.
Hence we expect the one undetermined MZV knot-number at
15 crossings to appear in, for example, the planar 9-loop diagram
$C(1,0,4,0,1)$. To find the precise combination of
$\zeta(9,\overline3,\overline3)-\frac{3}{14}\zeta(7,\overline5,\overline3)$
with other Euler sums would require a search in a space of dimension
$K_{18}=28$. Experience suggests that would require an evaluation
of the diagram to about 800 sf, which is rather ambitious,
compared with the 200 sf which yielded\Eq{k14}. Once the number is found,
the search space for further counterterms shrinks to dimension $K_{15}=12$.
\subsection*{4. Positive knots associated with irreducible MZVs}
Table~2 gives the braid words~\cite{VJ} of 5 classes of positive knot.
For each type of knot, ${\cal K}$,
we used the skein relation to compute the HOMFLY polynomial~\cite{VJ},
$X_{\cal K}(q,\lambda)$, in terms of
$p_n=(1-q^{2n})/(1-q^2)$, $r_n=(1+q^{2n-1})/(1+q)$,
$\Lambda_n=\lambda^n(1-\lambda)(1-\lambda q^2)$.
\noindent{\bf Table~2}:
Knots and HOMFLY polynomials associated with irreducibles MZVs.
\[\begin{array}{|l|l|l|}\hline{\cal K}&X_{\cal K}(q,\lambda)\\\hline
{\cal T}_{2k+1}=\sigma_1^{2k+1}&T_{2k+1}=\lambda^k(1+q^2(1-\lambda)p_k)\\
{\cal R}_{k,m}=\sigma_1^{}\sigma_2^{2k+1}\sigma_1^{}\sigma_2^{2m+1}&
R_{k,m}= T_{2k+2m+3}+q^3p_k p_m\Lambda_{k+m+1}\\
{\cal R}_{k,m,n}=
\sigma_1^{}\sigma_2^{2k}\sigma_1^{}\sigma_2^{2m}\sigma_1^{}\sigma_2^{2n+1}&
R_{k,m,n}=R_{1,k+m+n-1}+q^6p_{k-1}p_{m-1}r_n\Lambda_{k+m+n+1}\\
{\cal S}_{k}=
\sigma_1^{}\sigma_2^{3}\sigma_3^{2}\sigma_1^{2}\sigma_2^{2k}\sigma_3^{}&
S_{k}= T_3^2T_{2k+3}+q^2p_k r_2(q^2(\lambda-2)+q-2)\Lambda_{k+3}\\
{\cal S}_{k,m,n}=
\sigma_1^{}\sigma_2^{2k+1}\sigma_3^{}\sigma_1^{2m}\sigma_2^{2n+1}\sigma_3^{}
&S_{k,m,n}=T_{2k+2m+2n+3}+q^3(p_k p_m+p_m p_n+p_n p_k\\&\phantom{S_{k,m,n}=}
\quad{}+(q^2(3-\lambda)-2q)p_k p_m p_n)\Lambda_{k+m+n+1}\\\hline
\end{array}\]
Noting that ${\cal S}_{1,1,1}={\cal S}_{1}$ and
${\cal S}_{m,n,0}={\cal R}_{m,n,0}={\cal R}_{m,n}$,
one easily enumerates the knots of Table~2. The result is given,
up to 17 crossings, in the last row of Table~3,
where it is compared with the enumeration of all prime knots, which is known
only to 13 crossings, and with the enumeration of positive knots,
which we have
achieved to 15 crossings, on the assumption that the HOMFLY polynomial
has no degeneracies among positive knots. It is apparent
that positive knots are sparse, though they exceed the irreducible
MZVs at 10 crossings and at all crossing numbers greater than 11.
The knots of Table 2 are equal in number to the irreducible MZVs up to
16 crossings; thereafter they are deficient.
Table~4 records a finding that may be new: the Alexander
polynomial~\cite{VJ}, obtained by setting $\lambda=1/q$ in the HOMFLY polynomial,
is not faithful for positive knots. The Jones polynomial~\cite{VJ}, with
$\lambda=q$, was not found to suffer from this defect.
Moreover, by using REDUCE~\cite{RED}, and assuming the fidelity of the
HOMFLY polynomial in the positive sector, we were able to prove,
by exhaustion, that none of the $4^{14}\approx2.7\times10^8$ positive
5-braid words of length 14 gives a true 5-braid 14-crossing knot.
\noindent{\bf Table~3}:
Enumerations of classes of knots by crossing number, $n$,
compared with\Eq{Ms}.
\[\begin{array}{|r|rrrrrrrrrrrrrrr|}\hline
n&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17\\\hline
\mbox{prime knots}&1&1&2&3&7&21&49&165&552&2176&9988&?&?&?&?\\
\mbox{positive knots}&1&0&1&0&1&1&1&3&2&7&9&17&47&?&?\\
M_n&1&0&1&0&1&1&1&1&2&2&3&3&4&5&7\\
\mbox{Table~2 knots}&1&0&1&0&1&1&1&1&2&2&3&3&4&5&5\\\hline
\end{array}\]
\noindent{\bf Table~4}:
Pairs of positive knots with the same Alexander polynomial,
$X_{\cal K}(q,1/q)$.
\[\begin{array}{|l|l|l|}\hline{\cal K}_1&{\cal K}_2&
X_{{\cal K}_1}(q,\lambda)-X_{{\cal K}_2}(q,\lambda)\\\hline
{\cal S}_{2,1,2}=
\sigma_1^{}
\sigma_2^{5}
\sigma_3^{}
\sigma_1^{2}
\sigma_2^{5}
\sigma_3^{}&
\sigma_1^{3}
\sigma_2^{4}
\sigma_3^{}
\sigma_1^{2}
\sigma_2^{2}
\sigma_3^{2}
\sigma_2^{}&
q^4(1-\lambda q)p_2r_2\Lambda_6\\
(\sigma_1^{}
\sigma_2^{2}
\sigma_3^{})^2
\sigma_1^{}
\sigma_2^{5}
\sigma_3^{}
&
(\sigma_1^{}
\sigma_2^{2}
\sigma_3^{})^2
\sigma_1^{3}
\sigma_2^{}
\sigma_1^{2}
\sigma_3^{}&
q^5(1-\lambda q)p_2\Lambda_6\\
\sigma_1^{5}
\sigma_2^{}
\sigma_3^{}
\sigma_1^{2}
\sigma_2^{3}
\sigma_3^{2}
\sigma_2^{}&
\sigma_1^{2}
\sigma_2^{2}
\sigma_1^{3}
\sigma_2^{7}&
q^5(1-\lambda q)p_2\Lambda_6\\\hline
\end{array}\]
The association~\cite{DK1,DK2} of the 2-braid torus knots
$(2k+1,2)={\cal T}_{2k+1}$
with the transcendental numbers $\zeta(2k+1)$ lies at the heart of our work.
In~\cite{DK2,BKP}, we associated the 3-braid torus knot
$(4,3)=8_{19}={\cal R}_{1,1}$ with the unique irreducible MZV
at weight 8, and in~\cite{BKP} we associated $(5,3)=10_{124}={\cal R}_{2,1}$
with that at weight 10. The 7-loop counterterms of $\phi^4$-theory
indicate that the knot-numbers associated with
$10_{139}=\sigma_1^{}\sigma_2^{3}\sigma_1^{3}\sigma_2^{3}$
and
$10_{152}=\sigma_1^{2}\sigma_2^{2}\sigma_1^{3}\sigma_2^{3}$
are not~\cite{BKP} MZVs.
At 11 crossings, the association of the knot-number $N_{3,5,3}$
with ${\cal S}_1={\cal S}_{1,1,1}\equiv11_{353}$ is rock solid:
we have obtained
this number analytically from 2 diagrams and numerically from another 8,
in each case finding it with different combinations of $\zeta(11)$
and the factor-knot transcendental $\zeta^2(3)\zeta(5)$.
In~\cite{BGK} we associated the family of knots ${\cal R}_{k,m}$
with the knot-numbers $N_{2k+3,2m+1}$, modulo multiples of $\pi^{2k+2m+4}$
that have now been determined up to 14 crossings.
It therefore remains to explain here how:
(a) two families of 4-braids, ${\cal S}_{k}$ and ${\cal S}_{k,m,n}$,
diverge from their common origin at 11 crossings, to give two knots
at 13 crossings, and three at 15 crossings, associated with triple Euler sums;
(b) a new family, ${\cal R}_{k,m,n}$, begins at 14 crossings, giving
the $(7,3)$ torus knot, $(\sigma_1^{}\sigma_2^{})^7
=(\sigma_1^{}\sigma_2^4)^2\sigma_1^{}\sigma_2^3
={\cal R}_{2,2,1}$, associated with a truly irreducible four-fold sum.
To relate the positive knots of Table 2 to Feynman diagrams
that evaluate to MZVs we shall
dress their braid words with chords. In each of Figs.~2--8, we
proceed in two stages: first we extract, from a braid word,
a reduced Gauss code that defines a trivalent chord diagram;
then we indicate how to shrink propagators to obtain a scalar diagram that
is free of subdivergences and has an overall divergence
that evaluates to MZVs. Our criterion for reducibility to MZVs is
that there be an angular diagram, obtained~\cite{GPX,DJB}
by choosing one vertex as an origin, such that the angular
integrations may be performed without encountering
6--j symbols, since these appeared in all the diagrams involving the
non-MZV knots $10_{139}$ and $10_{152}$ at 7 loops~\cite{BKP}, whereas
all the MZV-reducible diagrams could be cast in a form free of 6--j
symbols.
The first step -- associating a chord diagram with a knot --
allows considerable freedom: each
chord is associated with a horizontal propagator connecting vertical
strands of the braid between crossings, and there are almost
twice as many crossings as there are chords in the corresponding
diagram. Moreover, there are several braid words representing the same knot.
Thus a knot can be associated with several chord diagrams. Figs.~3b and~4b
provide an example: each diagram obtained from the $(5,2)$ torus knot
yields a counterterm involving $\zeta(5)$,
in a trivalent theory such as QED or Yukawa theory.
In Table 2 we have five families of braid words: the 2-braid torus knots,
two families of 3-braids, and two families of 4-braids. We begin with
the easiest family, ${\cal T}_{2k+1}$.
Consider Fig.~2a. We see the braid $\sigma_1^3$, dressed with two
horizontal propagators. Such propagators will be called chords,
and we shall refer to Figs.~2a--8a as chorded braid diagrams.
In Fig.~2a the two chords are labelled 1 and 2.
Following the closed braid, starting from the upper end
of the left strand, we encounter each chord twice, at vertices
which we label $\{1,{1^\prime}\}$ and $\{2,{2^\prime}\}$. These
occur in the order
$1,2,{1^\prime},{2^\prime}$ in Fig.~2a. This is the same order as they
are encountered on traversing the circle of Fig.~2b, which
is hence the same diagram as the chorded braid of Fig.~2a.
As a Feynman diagram, Fig.~2b is indeed associated with
the trefoil knot, by the methods of~\cite{DK1}.
We shall refer to the Feynman diagrams of Figs.~2b--8b as
chord diagrams\footnote{The reader familiar with
recent developments in knot theory and topological field theory might
notice that our notation is somewhat motivated by the connection between
Kontsevich integrals~\cite{LM} and chord diagrams. In~\cite{DK}
this will be discussed in detail and related to the work in~\cite{DK1}.}.
Each chord diagram is merely
a rewriting of the chorded braid diagram that precedes it,
displaying the vertices
on a hamiltonian circuit that passes through all the vertices.
The final step is trivial in this example:
the scalar tetrahedron is already log-divergent in 4 dimensions, so no
shrinking of propagators is necessary. Fig.~2c records the trivial
angular diagram, obtained~\cite{GPX} by choosing ${2}$ as an origin
and removing the propagators connected to it:
this merely represents a wheel with 3 spokes. In general~\cite{DJB}
the wheel with $n+1$ spokes delivers $\zeta(2n-1)$.
In Fig.~3a we give a chording of the braid $\sigma_1^{2n-1}$,
which is the simplest representation of the $(2n-1,2)$ torus knot,
known from previous work~\cite{DK1,DK2} to be associated with
a $(n+1)$-loop diagram, and hence with a hamiltonian circuit that has $n$
chords. Thus each addition of $\sigma_1^2$ involves adding a
single chord, yielding the chord diagram of Fig.~3b. To obtain
a logarithmically divergent scalar diagram, we shrink the propagators
connecting vertex ${2^\prime}$ to vertex ${n^\prime}$, drawn with a thick
line on the hamiltonian circuit of Fig.~3b, and hence obtain
the wheel with $n+1$ spokes, represented by the angular diagram of Fig.~3c.
To show how different braid-word representations of the
same knot give different chord diagrams, yet the same transcendental,
we consider Fig.~4.
In Fig.~4a we again have a chorded braid with $n$ chords, which this time
is obtained by combining $\sigma_1\sigma_2\sigma_1\sigma_2$
with $n-2$ powers of $\sigma_2^2$.
The resultant braid word,
$\sigma_1^{}\sigma_2^{}\sigma_1^{}\sigma_2^{2n-3}$,
is the $(2n-1,2)$ torus knot written as a 3-braid.
Labelling the pairs of vertices of Fig.~4b, one sees that it is identical
to the closure of the braid of Fig.~4a.
Shrinking together the vertices
$\{{2^\prime},{n^\prime},\ldots,{3^\prime}\}$
gives the angular diagram of Fig.~4c, which is
the same as Fig.~3c and hence delivers $\zeta(2n-1)$.
This ends our consideration of the 2-braid torus knots. We now turn
to the class ${\cal R}_{k,m}$ in Fig.~5.
The first member ${\cal R}_{1,1}=8_{19}=(4,3)$ appears at 6 loops,
with five chords.
It was obtained from Feynman diagrams in \cite{DK2}, and found
in~\cite{BKP} to be associated with an MZV of depth 2.
In Fig.~5a we add singly-chorded powers of $\sigma_2^2$
to a chorded braid word that delivers a Feynman diagram for which the
procedures of~\cite{DK1} gave $8_{19}$ as one of its skeinings.
In general, we have $k+m+3$ chords and thus $k+m+4$ loops.
The resulting chord diagram
is Fig.~5b, whose 7-loop case was the basis for associating
$10_{124}$ with the MZV $\zeta(7,3)$~\cite{BKP}.
Shrinking the propagators indicated by thickened lines in Fig.~5b,
we obtain diagram $M(k,1,1,m)$, indicated by the angular diagram of
Fig.~5c. Explicit computation of all such diagrams, to 9
loops, shows that this family is indeed MZV-reducible, to 14 crossings.
In Fig.~6 we repeat the process of Fig.~5 for the knot class
${\cal R}_{k,m,n}$. Marked boxes, in Fig.~6a,
indicate where we increase the number of chords.
Fig.~6b shows the highly non-planar chord diagram
for this knot class. This non-planarity is maintained
in the log-divergent diagram obtained by shrinking the thickened lines
in Fig.~6b. The parameters $m$ and $k$ correspond to the
series of dots in the corresponding angular diagram of Fig.~6c.
Non-planarity is guaranteed by the two remaining dots,
which are always present.
For $n>1$, we see even more propagators in the angular diagram.
The absence of 6--j symbols from angular integrations leads us to believe
that the results are reducible to MZVs; the non-planarity entails
MZVs of even weight, according to experience up to 7 loops~\cite{BKP}.
We now turn to the last two classes of knots: the 4-braids of Table~2.
In Fig.~7a we give a chorded braid diagram for knots
of class ${\cal S}_k$. Again, the marked box indicates
how we add chords to a chorded braid diagram
that corresponds to a 7-loop Feynman diagram, already known~\cite{BKP}
to skein to produce ${\cal S}_1=11_{353}$.
Shrinking the thickened lines in Fig.~7b, we obtain a log-divergent
planar diagram containing: a six-point coupling,
a $(k+3)$-point coupling, and $k+5$ trivalent vertices.
This is depicted in Fig.~7c as an angular diagram obtained by
choosing the $(k+3)$-point coupling as an origin.
Choosing the 6-point coupling as an origin for
the case $k=1$ confirms that ${\cal S}_1=11_{353}$ is associated
with $\zeta(3,5,3)$ via the 7-loop diagram $G(4,1,0)$.
However, for $k=3$ there is no way of obtaining MZVs of depth
3 from either choice of 6-point origin. Hence we expect a depth-5
MZV to be associated with the 15-crossing knot ${\cal S}_3$, with the
possibility of depth-7 MZVs appearing at higher crossings.
Finally we show that the three-parameter class
${\cal S}_{k,m,n}$, also built on $11_{353}={\cal S}_{1,1,1}$,
is associated with depth-3 MZVs.
The chorded braid of Fig.~8a indicates
the three places where we can add further chords.
Fig.~8b gives the chord diagram associated with it,
and indicates how to shrink propagators to obtain a log-divergent
diagram, represented by the angular diagram $G(m+n+2,k,0)$
of Fig.~8c, which evaluates in terms of depth-3 MZVs
up to 13 loops, and presumably beyond.
\subsection*{5. Conclusions}
In summary, we have
\begin{enumerate}
\item enumerated in\Eqq{Es}{Ms} the irreducibles entailed by Euler sums
and multiple zeta values at weight $n$; apportioned them by depth in\Eq{EMb};
conjectured the generator\Eq{Pd} for the number, $D_{n,k}$, of MZVs of weight
$n$ and depth $k$ that are irreducible to MZVs of lesser depth;
\item determined all MZV knot-numbers to 15 crossings, save one,
associated with a 9-loop diagram that evaluates to MZVs of depth 5
and weight 15;
\item enumerated positive knots to 15 crossings, notwithstanding
degenerate Alexander polynomials at 14 and 15 crossings;
\item developed
a technique of chording braids so as to generate families of knots
founded by parent knots whose relationship to Feynman diagrams
was known at lower loop numbers;
\item combined all the above to identify,
in Table~2, knots whose enumeration, to 16 crossings, matches that of MZVs.
\end{enumerate}
Much remains to be clarified in this rapidly developing area.
Positive knots, and hence the transcendentals associated with them
by field theory, are richer in structure than MZVs: there are more of them
than MZVs; yet those whose knot-numbers are MZVs evaluate in search
spaces that are significantly smaller than those for the MZVs, due to
the absence of a two-crossing knot. After 18 months of intense collaboration,
entailing large scale computations in knot theory, number theory, and field
theory, we are probably close to the boundary of what can be discovered
by semi-empirical methods. The trawl, to date, is impressive, to our minds.
We hope that colleagues will help us to understand it better.
\noindent{\bf Acknowledgements}
We are most grateful to Don Zagier for his generous comments,
which encouraged us to believe in the correctness of our discoveries\Eq{EMb},
while counselling caution as to the validity of\Eq{Pd}
in so far uncharted territory with depth $k>4$.
David Bailey's MPPSLQ~\cite{DHB}, Tony Hearn's REDUCE~\cite{RED}
and Neil Sloane's superseeker~\cite{NJAS} were instrumental in this work.
We thank Bob Delbourgo for his constant encouragement.
\newpage
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proofpile-arXiv_065-679
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
\section{Introduction}
\noindent
The past twenty years have shown much progress in the field of
perturbative calculations in strong interaction physics [1]. This in
particular holds for the radiative corrections to the deep inelastic
structure functions. Sometimes these corrections could be even extended
up to third order in the strong coupling constant $\alpha_s$.
The structure
functions we would like to discuss are measured in deep inelastic
lepton-hadron scattering
\begin{equation}
l_1 (k_{1}) + H(p) \rightarrow l_{2} (k_{2}) + "X"
\end{equation}
\vspace*{3mm}
\noindent
where $l_1,l_2$
stand for the in- and outgoing leptons respectively. The
hadron is denoted by $H$ and $"X"$ stands for any inclusive hadronic state.
The relevant kinematical and scaling variables are defined by
\begin{equation}
q = k_1 - k_2 \hspace{1cm}
q^2 = - Q^2 > 0 \hspace{1cm}
x = \frac{Q^2}{2pq} \hspace{1cm}
y = \frac{pq}{pk_1}\\[3mm]
\end{equation}
with the boundaries
\begin{equation}
0 < y < 1 \hspace{3cm} 0 < x \le 1 \\[3mm]
\end{equation}
Reaction (1) proceeds via the exchange of one of the intermediate vector
bosons $V$ of the standard model which are represented by
$V=\gamma,Z,W$.
In the case of unpolarized scattering with
$V=\gamma$ one can measure the
structure functions $F_L(x,Q^2)$ (longitudinal) and
$F_1(x,Q^2)$ (transverse)
or the better known $F_2(x,Q^2)$ which is related to the former two via
\begin{equation}
F_2 \, (x,Q^2) \, = \, 2x \, F_1 \, (x,Q^2) \, + \,\,F_L
\,(x,Q^2) \\[3mm]
\end{equation}
When $V=W$ or $V=Z$ one can in addition to $F_1$, $F_2$
and $F_L$ also measure
the structure function $F_3(x,Q^2)$ which is due to parity violation of
the weak interactions. In the case the incoming lepton and hadron are
polarized one measures besides the structure functions $F_i$ $(i=1,2,3,L)$
also the spin structure functions denoted by $g_i(x,Q^2)$ $(i=1,\cdots 5)$.
At
this moment, because of the low $Q^2$ available, reaction (1) is only
dominated by the photon $(V=\gamma)$ so that one has data for $g_1(x,Q^2)$
(longitudinal spin) and $g_2(x,Q^2)$ (transverse spin) only.
The measurement of the structure functions at large $Q^2$ gives us insight
in the structure of the hadrons. According to the theory of quantum
chromodynamics (QCD) the hadrons consist out of quarks and gluons where
the latter are carriers of the strong force. When $Q^2$ gets large one can
probe the light cone behaviour of the strong interactions which can be
described by perturbation theory because the running coupling constant
denoted by $\alpha_s(Q^2)$ is small. In
particular perturbative QCD predicts the $Q^2$-evolution of the deep
inelastic structure functions mentioned above. Unfortunately the theory
is not at that stage that it enables us to predict the $x$-dependence so
that one has to rely on parametrizations which are fitted to the data.
A more detailed description of the structure functions is provided by
the parton model which can be applied if one can neglect power corrections
of the type $(1/Q^2)^p$ (higher twist effects). Here one asumes that in the
Bjorken limit $(Q^2 \rightarrow \infty$, $x$ is fixed)
the interaction between the hadron
and lepton in process (1) proceeds via the partons (here the quarks and
the gluons) of the hadron. If the scattering of the lepton with the partons
becomes incoherent the structure function can be written as
\begin{eqnarray}
F^{V,V'} (x,Q^2) & = & \int^1_x \,\frac{dz}{z} \, \left[
\sum^{n_f}_{k=1}\,
\left(v ^{(V)}_k \, v^{(V')}_k \,+\, a^{(V)}_k \,
a^{(V')}_k \right) \,
\left\{ \Sigma (\frac{x}{z}, \mu^2) \, C^S_{i,q} \,
(z, \frac{Q^2}{\mu^2}) \right. \right.
\nonumber \\
& + & \left. \left.
G\, \left(\frac{x}{z}, \mu^2 \right) \, C_{i,g} \,
\left( z, \frac{Q^2}{\mu^2} \right) \right\} \,
+ \sum^{n_f}_{k=1} \, \left( v^{(V)}_k \,
v^{(V')}_k \, + \,a^{(V)}_k
\, a^{(V')}_k \right) \right. \nonumber \\
& \Delta_k & \left. \left(\frac{x}{z}, \mu^2 \right) \,
C^{NS}_{i,q} \, \left( z, \,\frac{Q^2}{\mu^2} \right) \right]
\hspace{2cm} i = 1, 2, L
\end{eqnarray}
\begin{eqnarray}
F^{V,V'}_3 \left( x,Q^2 \right) & = & \int^1 _x \,
\frac{dz}{z} \, \left[ \sum^{n_f}_{k=1} \,
\left( v^{(V)}_k a^{(V')}_k \, + \, a^{(V)}_k \,
v^{(V')}_k \right) \right. \nonumber \\
& V_k & \left. \left( \frac{x}{z}, \mu^2 \right) \,
C ^{NS}_{3,q} \, \left( z,
\frac{Q^2}{\mu^2} \right) \right]
\end{eqnarray}
\noindent
with similar expressions for the twist two contributions to the spin
structure functions $g_1(x,Q^2)$ in which case we introduce the notations
$\Delta \Sigma, \Delta G , \Delta C_{i,l}$ etc.. The vector- and axial-vector
electroweak couplings of the standard model are given by $v_k^{(V)}$ and
$a_k^{(V)}$ respectively with $V=\gamma,Z,W$ and $k=1 (u), 2 (d), 3
(s)$
.... .
Further $n_f$ denotes the number of light flavours and $\mu$ stands for the
factorization/renormalization scale. The singlet $(\Sigma)$ and non-singlet
combinations of parton densities $(\Delta_k, V_k)$ are defined by
\begin{equation}
\Sigma \left( z, \mu^2 \right) = \frac{1}{n_f} \sum_{k=1}^{n_f}
\left( f_k \left( z, \mu^2 \right)
+ f_{\bar{k}} \left( z, \mu^2 \right) \right)
\end{equation}
\begin{equation}
\Delta_k \, (z, \mu^2)\,=\, f_k\,(z, \mu^2)\, +\, f_{\bar{k}}\, (z,
\mu^2) \,-\, \Sigma \,(z, \,\mu^2)
\end{equation}
\begin{equation}
V_k\,(z, \,\mu^2)\, =\, f_k \,(z, \,\mu^2) - f_{\bar{k}} \,(z,\,
\mu^2)
\end{equation}
where $f_k , f_{\bar k}$ denote the quark and anti-quark densities of species
$k$ respectively. The gluon density is defined by $G(z,\mu^2)$. The same
nomenclature holds for the coefficient functions $C_{i,l} (l=q,g)$ which can
also be distinguished in a singlet (S) and a non-singlet (NS) part. Like
in the case of the structure functions the $x$-dependence of the parton
densities cannot be determined by perturbative QCD and it has to be
obtained by fitting the parton densities to the data. Fortunately these
densities are process independent and they are therefore universal. This
property is not changed after including QCD radiative corrections. It
means that the same parton densities also show up in other so called hard
processes like jet production in hadron-hadron collisions,direct photon
production, heavy flavour production, Drell-Yan process etc. Another
firm prediction of QCD is that the scale $(\mu)$ evolution of the parton
densities is determined by the DGLAP [2] splitting functions $P_{ij}$
$(i,j=q,\bar q,g)$ which can be calculated order by order in the strong
coupling constant $\alpha_s$. The perturbation series of $P_{ij}$
gets the form
\begin{equation}
P_{kl} = a_s \, P^{(0)}_{kl} + a^2_s \,P^{(1)}_{kl} + a^3_s \,
P^{(2)}_{kl} + .\
.. \\[3mm]
\end{equation}
with $a_s=\alpha_s(\mu^2)/{4\pi}$. The splitting functions $P_{ij}$
are related
to the anomalous dimensions $\gamma_{ij}^{(n)}$
corresponding to twist two local
operators $O_i^{\mu_1 ...\mu_n} (x)$ of spin $n$ via the Mellin transform
\begin{equation}
\gamma^{(n)}_{ij} = - \int^1 _o dz z^{n-1} \, P_{ij} (z) \\[3mm]
\end{equation}
These operators appear in the light cone expansion of the product of two
electroweak currents which shows up in the calculation of the cross section
of process (1)
\begin{equation}
J(x)\, J(0) \sim \sum^{\infty}_{n=0} \sum_k \,\widetilde{C}^{(n)}_k
\,
(\mu^2x^2) \, x_{\mu{_1}} .. x_{\mu{_n}} \, O_k^{\mu_1 ...\mu_n} (0,
\mu^2) \\[3mm]
\end{equation}
where $\widetilde {C}_k^{(n)}$ (12) are the Fourier transforms of
the coefficient
functions $C_k^{(n)}$ (5),(6) $(k=q,g)$ in Minkowski space
$(x_\mu)$.
Like the
splitting functions they are calculable order by order in $\alpha_s$ and the
perturbation series takes the form
\begin{equation}
C_{i,k} = \delta_{kq} + a_s \, C^{(1)}_{i,k} + a_s^2 \,C^{(2)}_{i,k} +
a_s^3 \, C^{(3)}_{i,k} + ... \\[3mm]
\end{equation}
with $i=1,2,3,L$ and $k=q,g$.
We will now review the higher order QCD corrections to the splitting
functions and the coefficient functions which have been calculated
till now.
\section {Splitting Functions}
The splitting functions are calculated by
\begin{itemize}
\item[1.] $P_{ij}^{(0)}$\hspace*{1cm} Gross and Wilczek (1974) [3];
Altarelli and Parisi (1977) [2].
\item[2.] $\Delta P_{ij}^{(0)}$\hspace*{1cm} Sasaki (1975) [4];
Ahmed and Ross (1976) [5];
Altarelli and Parisi [2].
\item[3.] $P_{ij}^{(1)}$\hspace*{1cm} Floratos, Ross, Sachrajda (1977) [6];
Gonzales-Arroyo, Lopez, Yndurain (1979)
\hspace*{15mm} [7];
Floratos, kounnas, Lacaze (1981) [8];
Curci, Furmanski, Petronzio (1980) [9].
\item[4.] $\Delta P_{ij}^{(1)}$\hspace*{1cm} Zijlstra and van Neerven (1993)
[10];
Mertig and van Neerven (1995) [11];
\hspace*{2cm} Vogelsang (1995) [12].
\end{itemize}
Notice that till 1992 there was a discrepancy for $P_{gg}^{(1)}$ between
the covariant gauge [6--8] and the lightlike axial gauge calculation
[9] which was decided in favour of the latter by Hamberg and van Neerven
who repeated the covariant gauge calculation in [12]. The DGLAP splitting
functions satisfy some special relations. The most interesting one is
the so called supersymmetric relation which holds in ${\cal N} =1$
supersymmetry [13]. Here the colour factors, which in $SU(N)$ are given by
$C_F=(N^2-1)/{2N}$ , $C_A=N$, $T_f=1/2$ become $C_F=C_A=2T_f=N$. The
supersymmetric relation then reads
\begin{equation}
P^{S,(k)}_{qq} + P^{(k)}_{gq} - P^{(k)}_{qg} - P^{(k)}_{gg} = 0 \\[3mm]
\end{equation}
\begin{equation}
\Delta P^{S,(k)}_{qq} + \Delta P^{(k)}_{gq} - \Delta P^{(k)}_{qg}
- \Delta P^{(k)}_{gg} = 0 \\[3mm]
\end{equation}
which is now confirmed up to first $(k=0)$ and second $(k=1)$ order in
perturbation theory.
The third order splitting functions $P_{ij}^{(2)},\Delta
P_{ij}^{(2)}$
are not known
yet. However the first few moments $\gamma_{ij}^{(2),(n)}$ for $n=2,4,6,8,10$
have been calculated by Larin, van Ritbergen, Vermaseren (1994) [14].
Besides exact calculations one has also determined the splitting functions
and the anomalous dimensions in some special limits. Examples are the
large $n_f$ expansion carried out by Gracey (1994) [15].
Here one has computed the
coefficients $b_{21}$ and $b_{31}$ in the perturbations series of the
non-singlet
anomalous dimension
\begin{eqnarray}
\left. \gamma^{NS}_{qq} \right|_{n_f \rightarrow \infty }
& = & a_s^2 \, \left[ n_f \, C_F \, b_{21} \right]
+ a_s^3\, \left[ n^2_f \, C_F \, b_{31} \right. \nonumber \\
& + & \left. n_f \, C_A \, C_F \, b_{32}
+ n_f \, C_F^2 \, b_{33} \right] + ...
\end{eqnarray}\\[2ex]
Further Catani and Hautmann (1993) [16] calculated the splitting functions
$P_{ij}(x)$ in the limit $x \rightarrow 0$. The latter take the following
form
\vspace*{3mm} \noindent
\begin{equation}
\left. P^{(k)}_{ij} (x) \right|_{x \rightarrow 0}
\sim \, \frac{ln^kx}{x} \rightarrow
\left. \gamma^{(k),(n)}_{ij} \right|_{n \rightarrow 1}
\sim \, \frac{1}{(n-1)^{k+1}} \\[3mm]
\end{equation}
The above expressions follow from the BFKL equation [17] and
$k_T$-factorization [18]. Some results are listed below. The leading
terms in $\gamma_{gg}^{(n)}$ are given by
\begin{equation}
\left. \gamma^{(n)}_{gg}\right|_{n \rightarrow 1} = \left[ C_A \frac{a_s}{n-1}
\right] + 2\zeta(3) \,
\left[ C_A
\frac{a_s}{n-1} \right] ^4 + 2\zeta(5) \, \left[ C_A \frac{a_s}{n-1}
\right] ^6 \\[3mm]
\end{equation}
where $\zeta(n)$ denotes the Riemann zeta-function. Further we have in
leading order $1/(n-1)$
\begin{equation}
\left. \gamma^{(n)}_{gq}\right|_{n \rightarrow 1} = \frac{C_F}{C_A} \,\left.
\gamma^{(n)}_{gg}\right|_{n \rightarrow 1}
\end{equation}
\begin{eqnarray}
\left. \gamma^{(n)}_{qg} \right|_{n \rightarrow 1} & = & a_s T_f \frac{1}{3}\,
\left[ 1 + 1.67 \,
\left\{ \frac{a_s}{n-1} \right\} \, + \, 1.56 \,
\left\{ \frac{a_s}{n-1} \right\} ^2 \right. \nonumber \\
& + & 3.42 \left. \left\{ \frac{a_s}{n-1} \right\} ^3 \, + \, 5.51
\left\{ \frac{a_s}{n-1} \right\} ^4 \, + ...\right]
\end{eqnarray}
\begin{equation}
\left. \gamma^{S,(n)}_{qq}\right|_{n \rightarrow 1} = \frac{C_F}{C_A} \,
\left[\left. \gamma^{(n)}_{qg} \right|_{n \rightarrow 1} -
\frac{1}{3} \, a_s T_f \right] \\[3mm]
\end{equation}\\[3mm]
Kirschner and Lipatov (1983) and Bl\"umlein and Vogt (1996)
have also determined the subleading terms
in the splitting functions (anomalous dimensions). They behave like
\begin{equation}
\left. P^{(k)}_{ij} (z) \right|_{z \rightarrow 0}
\sim \, ln^{2k}\,z
\left. \hspace{2cm} \gamma^{(k),(n)}_{ij}
\right|_{n \rightarrow 0}
\sim \, \frac{1}{n^{2k+1}}
\end{equation}\\[3mm]
The same logarithmic behaviour also shows up in $\Delta P_{ij}$ and $\Delta$
$\gamma_{ij}^{(n)}$. In the latter case the expressions in (22) become the
leading ones since the most singular terms in (17) decouple in the spin
quantities. The expressions in (22) have been calculated for the spin
case by Bartels, Ermolaev, Ryskin (1995) [20] and by Bl\"umlein and
Vogt (1996) [21] who also investigated the effect of these type of
corrections on the spin structure function $g_1(x,Q^2)$.
Finally the three-loop anomalous dimension $\Delta \gamma_{qq}^{S,(1)}$
is also known (see Chetyrkin,K\"uhn (1993) [22] and Larin (1993) [23]).
It reads
\begin{equation}
\Delta \gamma^{S,(1)}_{qq} = a^2_s \,
\left[ -6n_f \, C_F \right] + a^3_s \,
\left[ \left( 18 C^2_F -
\frac{142}{3} C_AC_F \right) n_f + \frac{4}{3}\, n^2_f C_F \right] \\[3mm]
\end{equation}
Notice that the second order coefficient was already determined by Kodaira
(1980) [24].
\section {Coefficient Functions}
\noindent
The higher order corrections to the coefficient functions are calculated by
\begin{itemize}
\item[1.] $C_{i,q}^{(1)}$ , $C_{i,g}^{(1)}$~~~$i=1,2,3,L$ \hspace*{1cm}
Bardeen, Buras,
Muta, Duke (1978) [25],\\
\hspace*{6cm} see also Altarelli (1980) [26].
\item[2.] $\Delta C_q^{(1)}$ , $\Delta C_g^{(1)}$ \hspace*{2cm}
Kodaira et al. (1979) [27],\\
\hspace*{5cm} see also Anselmino, Efremov, Leader
(1995) [28]
\end{itemize}
Together with the splitting functions $P_{ij}^{(k)}$, $\Delta
P_{ij}^{(k)}\,(k=0,1)$
one is now able to make a complete next-to-leading (NLO) analysis of the
structure functions $F_i(x,Q^2)\, (i=1,2,3,L)$ and $g_1(x,Q^2)$. The second
order contributions to the coefficient functions are also known
\begin{itemize}
\item[1.] $C_{i,q}^{(2)} , C_{i,g}^{(2)}$~~~ $i=1,2,3,L$ \hspace*{1cm}
Zijlstra and van Neerven (1991) [29]
\item[2.] $\Delta C_q^{(2)} , \Delta C_g^{(2)}$ \hspace*{3cm} Zijlstra and
van Neerven (1993) [10]
\end{itemize}
The first few moments of $C_{i,k}^{(2)}$~~$(i=2,L ; k=q,g)$ were calculated
by
Larin and Vermaseren (1991) [30] and they agree with Zijlstra and van
Neerven [29]. The first moment of $\Delta C_q^{(2)}$ was checked by Larin
(1993) [31] and it agrees with the result of Zijlstra and van Neerven
[10]. The third order contributions to the coefficient functions are
not known except for some few moments. They are given by
\begin{itemize}
\item[1.] $C_{1,q}^{(3),(1)}$ (Bjorken sum rule) \hspace*{3cm} Larin,
Tkachov,
Vermaseren (1991) [32];
\item[2.] $C_{3,q}^{(3),(1)}$ (Gross-Llewellyn
Smith sum rule)\hspace*{1cm} Larin and Vermaseren (1991)
[33];
\item[3.] $\Delta C_q^{(3),(1)}$ (Bjorken sum
rule )\hspace*{3cm} Larin and Vermaseren (1991) [33];
\item[4.] $C_{i,q}^{(3),(n)}$~~ $(i=2,L)~~ n=2,4,6,8$ \hspace*{15mm}
Larin, van Ritbergen, Vermaseren (1994) [14],\\
\hspace*{8cm} (see also [34]).
\end{itemize}
Since the three-loop splitting functions $P_{ij}^{(2)}$, $\Delta
P_{ij}^{(2)}$ are
not known, except for a few moments, it is not possible to obtain a
full next-to-next-to-leading order (NNLO) expression for the structure
functions. However recently Kataev et al. (1996) [35] made a NNLO analysis
of the structure functions $F_2(x,Q^2)$, $F_3(x,Q^2)$ (neutrino scattering)
in the kinematical region $x > 0.1$ which is based on
$\gamma_{qq}^{NS,(2),(n)}$
for $n=2,4,6,8,10$ [14].
Like in the case of the DGLAP splitting functions Catani and Hautmann (1994)
[16]
also derived the small $x$-behaviour of the coefficient functions.
At small $x$ the latter behave like
\begin{equation}
\left. C^{(l)}_{i,k} \right|_{x \rightarrow 0}
\sim \, \frac{ln^{l-2}x}{x} \hspace{2cm}
\left. C^{(l),(n)}_{i,k} \right|_{n \rightarrow 1}
\sim \, \frac{1}{(n-1)^{l-1}} \hspace*{1cm} (l \ge 2)
\end{equation}[3mm]
The ingredients of the derivation are again the BFKL equation [17] and
$k_T$-factorizaton [18]. from [16] we infer the following Mellin-transformed
coefficient functions.
\begin{eqnarray}
\left. C^{(n)}_{L,g}\right|_{n \rightarrow 1} & = & a_s \, T_f \, n_f \,
\frac{2}{3} \, \left[ 1 - 0.33 \,
\left\{ \frac{a_s}{n-1} \right\} \,
+ \, 2.13 \, \left\{ \frac{a_s}{n-1} \right\}^2 \right.
\nonumber \\
& + & \left.
2.27 \left\{ \frac{a_s}{n-1} \right\} ^3 \, + \,
0.43 \, \left\{ \frac{a_s}{n-1} \right\} ^4 \, + \, ... \right]
\end{eqnarray}
\begin{eqnarray}
\left. C^{(n)}_{2,g}\right|_{n \rightarrow 1} & = & a_s \, T_f \, n_f \,
\frac{1}{3}
\, \left[ 1 + 1.49 \,
\left\{ \frac{a_s}{n-1} \right\}
\, + \, 9.71 \, \left\{ \frac{a_s}{n-1} \right\}^2 \right. \nonumber
\\
& + & \left. 16.43 \, \left\{ \frac{a_s}{n-1} \right\}^3 \, +
\, 39.11 \, \left\{ \frac{a_s}{n-1} \right\} ^4 \, + ... \right]
\end{eqnarray}
\begin{equation}
\left. C^{S,(n)}_{L,q}\right|_{n \rightarrow 1} = \frac{C_F}{C_A} \, \left[
\left. C^{(n)}_{L,g}\right|_{n \rightarrow 1} -
\frac{2}{3} \, a_s
\, n_f \, T_f \right] \\[3mm]
\end{equation}
\begin{equation}
\left. C^{S,(n)}_{2,q}\right|_{n \rightarrow 1} = \frac{C_F}{C_A} \, \left[
\left. C^{(n)}_{2,g} \, \right|_{n \rightarrow 1} - \,
\frac{1}{3} \, a_s \, n_f \, T_f \right] \\[3mm]
\end{equation}
The order $\alpha_s^2$ coefficients were already obtained via the exact
calculation performed by Zijlstra and van Neerven (1991) [29]. The
subleading terms given by
\begin{equation}
\left. C^{(l)}_{2,k} \right|_{x \rightarrow 0}
\sim \ln ^{2l-1} x \hspace{2cm} (l \ge 1) \\[3mm]
\end{equation}
were investigated by Bl\"umlein and Vogt (1996) [21]. The most singular
terms shown in (24) do not appear in the spin coefficient functions
$\Delta C_k^{(l)}$ because the Lipatov pomeron decouples in polarized
lepton-hadron scattering. Therefore the most singular behaviour near $x=0$
is given by (29) (see [10],[21]). Besides the logarithmical enhanced
terms which are characteristic of the low $x$-regime we also find similar
type of logarithms near $x=1$. Their origin however is completely different
from the one determining the small $x$-behaviour. The logarithmical
enhanced terms near $x=1$, which are actual distributions, originate from
soft gluon radiation. They dominate the structure functions $F_i$ and $g_i$
near $x=1$ because other production mechanisms are completely suppressed
due to limited phase space. Following the work in [36] and [37] the
DGLAP splitting functions and the coefficient functions behave near $x=1$
like
\begin{equation}
P^{NS,(k)}_{qq} = \Delta \, P^{NS,(k)}_{qq} \sim \left( \frac{1}{1-x}\right)_+
\hspace{1cm}
P^{(n)}_{gg} =
\Delta P^{(k)}_{gg} \sim \, \left( \frac{1}{1-x}\right)_+
\end{equation}
\begin{equation}
\Delta C^{NS,(k)}_q =
C^{NS,(k)}_{i,q} \sim \left( \frac{\ln^{2k-1} (1-x)}{1 - x} \right)_+
\hspace{1cm} (l = 1, 2, 3) \\[3mm]
\end{equation}
Notice that the above corrections cannot be observed in the kinematical
region $(x < 0.4)$ accessible at HERA. Furthermore the behaviour in (30)
is a conjecture (see [7]) which is confirmed by the existing calculations
carried out up to order $\alpha_s^2$.
\section {Heavy Quark Coefficient Functions}
\noindent
The heavy quark coefficient functions have been calculated by
\begin{itemize}
\item[1.] $C_{i,g}^{(1)}(x,Q^2,m^2)$~~$(i=2,L)$\hspace*{1cm} Witten (1976)
[38];
\item[2.] $\Delta C_g^{(1)}(x,Q^2,m^2)$ \hspace*{2cm} Vogelsang (1991)
[39];
\item[3.] $C_{i,g}^{(2)}(x,Q^2,m^2),$~~ $C_{i,q}^{(2)}(x,Q^2,m^2)$~~$(i=2,L)$
\hspace*{5mm} Laenen, Riemersma, Smith, van Neerven
\hspace*{9cm} (1992) [40].
\end{itemize}
where $m$ denotes the mass of the heavy quark. The second order heavy quark
spin coefficient functions $\Delta C_g^{(2)}(x,Q^2,m^2)$ and
$\Delta C_q^{(2)}(x,Q^2,m^2)$ are not known yet. Due to the presence of the
heavy quark mass one was not able to give explicit analytical expressions
for $C_{i,k} (i=2,L ; k=q,g)$. However for experimental and
phenomenological use
they were presented in the form of tables in a computer program [41].
Analytical expressions do exist when either $x \rightarrow 0$ or
$Q^2 \gg m^2$. In the
former case Catani, Ciafaloni and Hautmann [42] derived the general form
\begin{equation}
\left. C^{(l)}_{i,k} \right|_{x \rightarrow 0}
\sim \, \frac{1}{x} \, \ln^{l-2} (x) \, f(Q^2, m^2)
\hspace{1cm} (l \ge 2,\, i = 2,\, L; \, k=q,g) \\[3mm]
\end{equation}
Like for the light parton coefficient functions (see (24)) the above
expression is based on the BFKL equation [17] and $k_T$-factorization [18].
In second order Buza et al. (1996) [43] were able to present analytical
formulae for the heavy quark coefficient functions in the asymptotic
limit $Q^2 \gg m^2$. This derivation is based on the operator product
expansion and mass factorization.
\section {Phenomenology at low $x$}
\noindent
Since the calculation of the higher order corrections to the DGLAP
splitting functions $P_{ij}$ and the coefficient functions $C_{ik}$ is very
cumbersome various groups have tried to make an estimate of the NNLO
corrections to structure functions in particular to $F_2(x,Q^2)$. The
most of these estimates concerns the small $x$-behaviour. In [44]
Ellis, Kunszt and Levin Hautman made a detailed study
of the $Q^2$-evolution
of $F_2$ using the small $x$-approximation for $P_{ij}$ (17) and
$C_{2,k}^{(2)}$ (24).
Their results heavily depend on the set of parton densities used and
the non leading small $x$-contributions to $P_{ij}^{(2)}$. The latter are
e.g.
needed to satisfy the momentum conservation sum rule condition. Large
corrections appear when for $x \rightarrow 0$ the gluon density behaves like
$xG(x,\mu^2)\rightarrow$
const. whereas
they are small when the latter has the behaviour $xG(x,\mu^2) \rightarrow
x^{-\lambda} (\lambda
\sim 0.3 - 0.5$ ; Lipatov pomeron).\\
However other investigations reveal that
the singular terms at $x=0$, present in $P_{ij}$ and $C_{i,k}$,
do not dominate
the radiatve corrections to $F_2(x,Q^2)$ near low $x$. This became apparent
after the exact coefficient functions or DGLAP splitting functions were
calculated.\\
In [45] Gl\"uck, Reya and Stratmann (1994) investigated the
singular behaviour of the second order heavy quark coefficient functions
(32) in
electroproduction and they found that its effect on $F_2$ was small.\\
Similar work was done by Bl\"umlein and Vogt (1996) [21] on the effect of
the logarithmical terms (22),(29) on $g_1(x,Q^2)$ which contribution to
the latter turned out to be negligable.\\
Finally we would like to illustrate
the effect of the small $x$-terms, appearing in the coefficient functions
$C_{2,k}^{(2)}$
and $C_{L,k}^{(2)}$, on the structure functions $F_2(x,Q^2)$ and
$F_L(x,Q^2)$.
For that purpose we compute the order $\alpha_s^2$ contributions to $F_2$
and $F_L$. Let us introduce the following notations. When the exact
expressions for the coefficient functions $C_{i,k}^{(2)}$ are adopted
the order
$\alpha_s^2$ contributions to $F_i$ will be called $\delta F_i^{(2),exact}$.
If we replace the exact coefficient functions by their most singular
part which is proportional to $1/x$ (see (24)) the order $\alpha_s^2$
contributions to $F_i$ are denoted by $\delta F_i^{(2),app}$. The results
are listed in table 1 and 2 below. Further we have used the parton density
sets MRS(D0) $(xG(x,\mu^2) \rightarrow$ $const.$ for $x \rightarrow 0)$
and MRS(D-) $(xG(x,\mu^2) \rightarrow x^{-\lambda}$ for $x
\rightarrow 0)$ [46]
\vspace*{2cm}
\begin{tabular}{||l||r|r|r||r|r|r||}
\hline \hline
\multicolumn{1}{||c|}{ }&
\multicolumn{3}{ c|}{MRS(D0)}&
\multicolumn{3}{c||}{MRS(D-)}\\
\multicolumn{1}{||c|}{$x$}&
\multicolumn{1}{||c|}{$F_2^{\rm NLO}$}&
\multicolumn{1}{c|}{$ \delta F_2^{(2),exact}$}&
\multicolumn{1}{c|}{$ \delta F_2^{(2),app}$}&
\multicolumn{1}{|c|}{$F_2^{\rm NLO}$}&
\multicolumn{1}{c|}{$ \delta F_2^{(2),exact}$}&
\multicolumn{1}{c||}{$ \delta F_2^{(2),app}$}\\
\hline \hline
$10^{-3}$ & 0.67 & -0.069 & 0.088 & 0.99 & -0.084 & 0.116 \\
$10^{-4}$ & 0.82 & -0.088 & 0.158 & 2.29 & -0.226 & 0.349 \\
$10^{-5}$ & 1.00 & -0.092 & 0.251 & 5.99 & -0.665 & 1.059 \\
\hline\hline
\multicolumn{1}{||c|}{$x$}&
\multicolumn{1}{||c|}{$F_L^{\rm NLO}$}&
\multicolumn{1}{c|}{$ \delta F_L^{(2),exact}$}&
\multicolumn{1}{c|}{$ \delta F_L^{(2),app}$}&
\multicolumn{1}{|c|}{$F_L^{\rm NLO}$}&
\multicolumn{1}{c|}{$ \delta F_L^{(2),exact}$}&
\multicolumn{1}{c||}{$ \delta F_L^{(2),app}$}\\
\hline\hline
$10^{-3}$ & 0.149 & -0.029 & -0.040 & 0.263 & 0.008 & -0.052 \\
$10^{-4}$ & 0.210 & -0.062 & -0.071 & 0.780 & 0.031 & -0.156 \\
$10^{-5}$ & 0.281 & -0.102 & -0.113 & 2.370 & 0.105 & -0.475 \\
\hline\hline
\end{tabular}
\vspace*{2cm} \noindent
{}From the table above we infer that a steeply rising gluon density near
$x=0$ (MRS(D-)) leads to small corrections to $F_2$ and $F_L$.
On the other hand
if one has a flat gluon density (MRS(D0)) the corrections are much larger
in particular for $F_L$. A similar observation was made for $F_2$ in [44].
However the most important observation is that the most singular part
of the coefficient functions gives the wrong prediction for the order
$\alpha_s^2$ contributions to the structure functions except for $F_L$
provided the set MRS(D0) is chosen. This means that the subleading terms
are important and they cannot be neglected. Therefore our main conclusion
is that only exact calculations provide us with the correct NNLO analysis
of the structure functions. The asymptotic expressions obtained in the
limits $x \rightarrow 0 , x \rightarrow 1$ and $Q^2 \gg m^2$ can
only serve as a check on the
exact calculations of the DGLAP splitting functions and the coefficient
functions.
|
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{
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
}
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\section{Ultra-High Energy Neutrinos}
\vskip -0.2true in
Active Galactic Nuclei (AGNs) are the most powerful
sources of high-energy gamma rays. If these gamma rays originate in
the decay of $\pi^{0}$,
then AGNs may also be prodigious sources
of high-energy neutrinos.
Neutrinos are undeflected by magnetic fields and have
long interaction lengths, so they may
potentially provide valuable information
about astrophysical sources. Gammas,
on the other hand, are absorbed by a few hundred grams of material.
As underground neutrino telescopes
achieve larger instrumental areas, prospects for measuring fluxes from
AGNs become realistic.
The diffuse flux of AGN neutrinos,
summed over all sources, is isotropic, so
the event rate is $A \int dE_\nu P_\mu(E_\nu,E_\mu^{\rm min})
S(E_\nu){dN_\nu/dE_\nu}$,
given a neutrino spectrum $dN_\nu/dE_\nu$ and detector
area $A$.
Attenuation of neutrinos in the Earth,
described by a shadowing
factor $S(E_\nu)$, depends on the $\nu_\mu N$ cross section through the
neutrino interaction length, while
the probability
that the neutrino
converts to a muon that arrives at the detector
with $E_\mu$ larger than the threshold energy $E_\mu^{\rm min}$,
$P_\mu(E_\nu,E_\mu^{\rm min})$
is
directly proportional to the charged-current cross section.
Here we present predictions of
event rates for several models of the AGN
neutrino flux.\cite{stecker}
We also compare the predicted rates with the atmospheric neutrino
background (ATM).\cite{volkova}
These rates reflect a new calculation \cite{Gqrs} of the
neutrino-nucleon cross section that incorporates recent results from
the HERA $ep$ collider.\cite{He}
The classic signal for cosmic neutrinos is energetic muons produced in
charged-current interactions of neutrinos with nucleons.
To reduce the background from muons produced
in the atmosphere, we consider
upward-going muons produced in and below the detector in $\nu_\mu N$ and
$\bar\nu_\mu N$ interactions.
We also give predictions for downward-moving
(contained) muon event rates
due to $\bar\nu_e e$ interactions in the
PeV range and for neutrinos produced in the collapse of
topological defects.
In Table \ref{upward} we show the event rates for a detector with $A=0.1\hbox{
km}^2$ for $E_\mu^{\rm min}=1\hbox{ TeV}$ and $10\hbox{ TeV}$.
The CTEQ--DIS rates are representative of the new generation of
structure functions.\cite{CTEQ} The older rates derived from the EHLQ
structure functions are given for comparison.\cite{Rq} If the most
optimistic flux predictions are accurate, the observation of AGNs by
neutrino telescopes is imminent.
\begin{table}[t!]
\caption{
Number of upward $\mu+\bar{\mu}$ events
per year per steradian for $A=0.1$ km$^2$.}
\begin{center}
\begin{tabular}{ccccc}
\hline
\raisebox{-1.5ex}{Flux} & \multicolumn{2}{c} { $ E_\mu^{\rm min}=1\hbox{ TeV}$}
&
\multicolumn{2}{c}{ $ E_\mu^{\rm min}=10\hbox{ TeV}$} \\
& EHLQ & CTEQ--DIS & EHLQ & CTEQ--DIS \\ \hline
AGN--SS \cite{stecker} & 82 & 92 & 46 & 51 \\
AGN--NMB \cite{stecker} & 100 & 111 & 31 & 34 \\
AGN--SP \cite{stecker} & 2660 & 2960 & 760 & 843 \\
ATM \cite{volkova}& 126 & 141 & 3 & 3 \\ \hline
\end{tabular}
\end{center}\label{upward}
\vskip -0.1 true in
\end{table}
Only in the neighborhood of $E_\nu=6.3\hbox{ PeV}$, where the $W$-boson is
produced as a $\bar{\nu}_e e$ resonance, are electron targets
important.
The contained event rate for resonant $W$ production is
${(10/18)} V_{\rm eff} N_A
\int dE_{\bar{\nu}} \sigma_{\bar{\nu}e}(E_\nu)
S(E_{\bar{\nu}}){dN/dE_{\bar{\nu}}}$.
We show event rates for downward resonant $W$-boson production in Table
\ref{electron}. (The Earth is opaque to upward-going
$\bar{\nu}_{e}$s at resonance.)
\begin{table}[b!]
\caption{
$\bar\nu_e e\rightarrow W^-$ events per
year per steradian for a detector with effective volume
1~km$^3$ and the downward (upward) background
from $(\nu_\mu,\bar\nu_\mu) N$ interactions above 3 PeV.}
\begin{center}
\begin{tabular}{ccc}
\hline
Mode & AGN--SS & AGN--SP \\ \hline
$W\rightarrow \bar{\nu}_\mu \mu$ & 6 & 3 \\
$W\rightarrow {\rm hadrons}$ & 41 & 19 \\ \hline
$(\nu_\mu,\bar\nu_\mu)N$ CC & 33 (7) & 19 (4) \\
$(\nu_\mu,\bar\nu_\mu)N$ NC & 13 (3) & 7 (1) \\ \hline
\end{tabular}
\end{center}\label{electron}
\end{table}
We note that a 1-km$^3$ detector with energy threshold in the PeV range
would be suitable for
detecting resonant $\bar\nu_e e\rightarrow W$ events,
though the $\nu_\mu N$ background is not negligible.
Another possible source of UHE neutrinos is
topological defects such as monopoles, cosmic strings, and
domain walls, which might have been formed
in symmetry-breaking phase transitions in the early Universe. When
topological defects are destroyed by collapse or annihilation, the energy
stored in them is released in the form of massive $X$-quanta
of the fields that generated the defects. The $X$ particles can then
decay into quarks, gluons, leptons, and such, that eventually materialize
into energetic neutrinos and other particles.
Table \ref{TDrates} shows rates induced by the neutrino flux from the collapse
of cosmic-string loops, in a model \cite{hill} that survives the Fr\'{e}jus
bound \cite{frejus} at low energies.
We take this flux as a plausible example to consider the
sensitivity of a km$^{3}$ detector to fossil neutrinos from the
collapse of topological defects.
\begin{table}[t]
\caption{
Downward $\mu^{+}+\mu^{-}$ events per steradian per year
from $(\nu_{\mu},\bar{\nu}_{\mu})N$ interactions in
a detector with effective volume 1 km$^{3}$, for the
BHS$_{p= 1.0}$ flux from topological defects.}
\begin{center}
\begin{tabular}{ccc} \hline
\raisebox{-1.5ex}{Parton Distributions} & \multicolumn{2}{c}{$E_{\mu}^{\
mathrm{min}}$} \\
& $10^7\hbox{ GeV}$ & $10^8\hbox{ GeV}$ \\ \hline
CTEQ--DIS & 10 & 6 \\
CTEQ--DLA & 8 & 4 \\
MRS D\_ & 12 & 8 \\
EHLQ & 6 & 3 \\ \hline
\end{tabular}
\end{center}\label{TDrates}
\end{table}
For our nominal set (CTEQ-DIS) of parton distributions, the BHS$_{p =
1.0}$ flus leads to 10 events per steradian per year with
$E_{\mu}>10^{7}\hbox{ GeV}$, far larger than the rate expected from
``conventional'' pion photoproduction on the cosmic microwave
background. This is an attractive
target for a 1-km$^{3}$ detector, and raises the possibility
that even a 0.1-km$^{3}$ detector could see hints of the collapse of
topological defects.
\vskip -1.5true in
\frenchspacing
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{
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
}
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\section{Introduction}
\def4.\arabic{equation}{1.\arabic{equation}}
\setcounter{equation}{0}
~~
The Higgs boson is the last remaining ingredient of a complete
standard model. It's persistent elusiveness is perhaps not surprising. Within
the framework of the standard model, there are no symmetries which can be invoked
to make a fundamental scalar light. The existence of a light scalar
degree of freedom which remains fundamental above the weak-scale
would argue for supersymmetry since supersymmetry provides the only explicitly known solution
to the naturalness problem which accompanies fundamental scalars~\cite{Witten}.
Of course, the Higgs boson may not be fundamental at all, and
the only testament to its existence may be the eventual unitarization of the longitudinal
$W$ scattering cross section at TeV scale energies. However,
although no vestige of the
Higgs boson may be seen until the LHC, a failure to observe a Higgs boson in
pre-LHC experiments could significantly challenge the principle motivation for
weak-scale supersymmetry, at least in its minimal forms.
If nature is supersymmetric above the weak-scale, the allowable range of
Higgs boson masses is considerably restricted.
In the minimal supersymmetric extension of the standard model (MSSM),
the lightest Higgs boson lies below $m_Z$ at tree level,
\begin{equation}
m_{h} \leq | \cos2\beta | m_{Z},
\end{equation}
where $\tan\beta = v_u/v_d$ is the ratio of Higgs boson vacuum expectation values.
Quantum corrections can lift the
light Higgs boson mass above $m_Z$ \cite{mh1}, but the magnitude of these corrections
are restricted if supersymmetry provides a successful solution to the
naturalness problem. Radiative corrections to the light Higgs boson mass in
supersymmetry have been
calculated by many authors~\cite{mh1,mh2,mh3}.
From these corrections, upper bounds for the lightest Higgs boson mass have been computed either by choosing arbitrary heavy masses for superpartners or
by demanding the theory remains perturbative up to some high scale~\cite{mh1,mh2,mh3}.
While these upper bounds reasonably approximate an important, unexceedable
upper-limit on the Higgs boson mass, they do not
provide a complete picture of
our expectations for the mass of the
lightest Higgs boson in supersymmetric models. Realistically, we expect
the Higgs boson mass to be significantly lighter.
To achieve Higgs boson masses as heavy as these upper-bounds
requires some or all superpartner masses to be much heavier than the weak-scale.
The appearance of this heavy mass scale in turn requires demonstrably
large, unexplained cancellations among heavy masses
in order to maintain a light weak-scale.
However, avoiding this fine-tuning is
the principle reason that supersymmetry was
introduced at the weak-scale.
In this article, we observe that it would be quite unnatural for the lightest Higgs
boson mass to saturate the maximal upper bounds which have been previously computed.
We compute the natural upper bound on the Higgs boson masses in minimal,
low-energy supergravity (MLES), and we show
the extent to which naturalness is lost as the experimental lower bound on the
lightest Higgs boson mass increases.
Section two provides a brief review of naturalness and how it is reliably quantified.
An analysis of the natural upper bound on the Higgs boson mass follows in section three.
We find that for $m_t < 175$ GeV, if $m_h > 120 $ GeV, minimal low energy supergravity
does not accommodate the weak-scale naturally. Moreover,
in the {\it most} natural cases, $m_h < 108 $ GeV
when $m_t < 175$ GeV. For modest
$\tan\beta$, the natural upper-bound is even more restrictive.
In particular,
for $\tan\beta <2$ and $m_t < 175$ GeV, if $m_h > 100$ GeV large fine-tuning is
required, while the {\it most} natural values of the Higgs boson mass lie below $m_Z$.
This has important implications for challenging weak-scale supersymmetry
at collider experiments.
In particular, if the lightest supersymmetric Higgs boson is not observed at
CERN's $e^{+}e^{-}$ collider LEP-II, requiring
natural electroweak symmetry breaking
in MLES will progressively increase the lower bound on
$\tan\beta$ as LEP-II increases in energy.
In the {\it most} natural cases, if the energy of LEP-II is
extended to $\sqrt{s}= 205 $ GeV, a light Higgs boson would be observed
provided it decays appreciably to $b \bar{b}$, but it would
not be possible to argue that natural electroweak symmetry
breaking is untenable in the minimal supersymmetric standard model
if the Higgs boson lies above the kinematic reach of LEP-II. By contrast,
the proposed Run-III of Fermilab's Tevatron with
${\cal L} = 10^{33} {\rm cm}^{-2}{\rm s}^{-1}$ (TeV33) can pose a
very serious challenge to the minimal supersymmetric standard model.
The projected mass-reach for a standard model Higgs boson at
TeV33 is 100 (120) GeV with integrated luminosities of 10 (25)
${\rm fb}^{-1}$~\cite{TeV2000}.
If the possibility that the light Higgs boson decays primarily to neutralinos
can be excluded on the basis of combined searches for superpartners
at LEP-II and the Tevatron, natural electroweak symmetry breaking
in the minimal supersymmetric standard model will no longer be possible
if TeV33 fails to observe a light Higgs boson.
\section{Naturalness}
\def4.\arabic{equation}{2.\arabic{equation}}
\setcounter{equation}{0}
~~
The original and principle motivation
for weak-scale supersymmetry is naturalness.
Supersymmetry provides the only explicitly known mechanism which allows fundamental
scalars to be light without an unnatural fine-tuning of parameters.
Naturalness also implies
that superpartner masses can not lie much above the weak-scale if we are to avoid
the fine-tuning which would be needed to keep the weak-scale light.
In this section, we recall the principle of naturalness and briefly review how it
can be reliably quantified. A more complete discussion of naturalness
criteria can be found in
Ref.~\cite{Fine_Tuning}.
Although fine-tuning is an aesthetic criterion, once we adopt the prejudice that
large unexplained-cancellations are unnatural, a quantitative fine-tuning measure can
be constructed and placed on solid footing. For any effective
field theory, it is straightforward to identify whether
large cancellations occur, and when these fine-tunings are present
their severity can be reliably quantified.
In non-supersymmetric theories, light fundamental scalars are unnatural
because scalar particles receive quadratically divergent contributions to
their masses. Generically, at one-loop, a scalar mass is of the form
\begin{equation}
m_{S}^2(g) = g^2 \Lambda_{1}^2 - \Lambda_{2}^2,
\end{equation}
where $\Lambda_1 $ is the ultraviolet cutoff of the effective
theory, and $\Lambda_2$ is a bare term.
The divergence in Eq. (2.1)
must be almost completely cancelled against the counter term
or the fundamental scalar will have a
renormalized mass on the order of the cutoff.
In supersymmetry, additional loops involving super-partners conspire to
cancel these quadratic divergences,
but when supersymmetry is broken, the cancellation is no longer
complete, and the dimensionful
terms in Eq. (2.1) are replaced by the mass splitting between
standard particles and their super-partners.
In this toy example, the cancellation is self-evident, and no
abstract quantitative prescription is needed to determine when the
parameters of the theory must conspire to give a light scalar mass.
We are interested in a more complicated example, and this requires
a quantitative prescription for identifying instances of
fine-tuning. In the toy example, if
we examine the sensitivity of the scalar mass to variations
in the coupling $g$:
\begin{equation}
\frac{ \delta m_{S}^2}{m_{S}^2} = c(m_{S}^2,g) \frac{\delta g}{g},
\end{equation}
where
\begin{equation}
c(m_{S}^2;g) = 2\frac{g^2 \Lambda_{1}^2}{m_{S}^2(g)},
\end{equation}
the scalar mass will be unusually sensitive to minute changes in
$g$ when we arrange for large unexplained-cancellations~\cite{Wilson}:
\begin{equation}
c(m_{S}^2 \ll \Lambda^2) \gg c(m_{S}^2 \sim \Lambda^2 ).
\end{equation}
However, the bare sensitivity parameter $c$,
by itself is not a measure of naturalness.
Although physical quantities depend sensitively on minute variations of
the fundamental parameters when there is fine-tuning,
fine-tuning is not necessarily implied by $c\gg 1$.
Large sensitivities can occur in a theory even
when there are no large cancellations
\footnote{For example the mass of the proton depends very sensitively on
minute variations in the value of the strong coupling constant at high
energy, but the lightness of the proton is a consequence of asymptotic freedom
and the logarithmic running of the QCD gauge coupling and
not the result of unexplained cancellations.}.
In particular, this is true for supersymmetric
extensions of the standard model, where it is known that bare sensitivity
provides a poor measure of fine-tuning~\cite{Fine_Tuning}.
A reliable measure of fine-tuning must compare
the sensitivity of a particular choice of parameters
$c$ to a measure of the average, global sensitivity in
parameter space, $\bar{c}$. The naturalness measure
\begin{equation}
\gamma = c/\bar{c}
\end{equation}
will greatly exceed unity if and only if fine-tuning is
encountered~\cite{Fine_Tuning}
\footnote{Alternatively, we could define a measure of fine-tuning as the
ratio of the amount of parameter space in the theory supporting typical values of $m_S$ to
the amount of parameter space giving a unusually light value of $m_S$.
This criterion
is in fact equivalent to the ratio of sensitivity over typical sensitivity
\cite{Fine_Tuning}.}.
This definition is a quantitative implementation of a refined
version of Wilson's naturalness criterion: Observable properties
of a system should not be unusually unstable against minute variations
of the fundamental parameters.
In supersymmetric extensions of the standard model, as the masses of superpartners
become heavy, increasingly large fine-tuning
is required to keep the weak-scale light.
Naturalness places an upper bound on supersymmetry-breaking
parameters and superpartner masses. Because the radiative
corrections to the Higgs boson mass increase with heavier superpartner
masses, naturalness translates into an upper limit on the mass
of the lightest Higgs boson. This limit is computed in the following section.
\section{Analysis}
\def4.\arabic{equation}{3.\arabic{equation}}
\setcounter{equation}{0}
~~
Following the methods of Ref. 6,
we have computed the severity of fine-tuning in the minimal
supersymmetric standard model. For definiteness, we consider soft
supersymmetry breaking parameters with (universal)
minimal, low-energy supergravity (MLES) boundary conditions.
We quantify the severity of large cancellations, and
present our results as upper limits on the Higgs boson mass as a
function of the degree of fine-tuning. Although our quantitative results
were obtained in a framework with universal soft terms at a scale
near $10^{16}$ GeV, as motivated by MLES, we do not expect our bounds on the Higgs
boson mass to significantly increase in models with more general
soft supersymmetry breaking masses provided they have
minimal particle content at the weak-scale. Because there are enough
free parameters in MLES to independently adjust the parameters
in the minimal supersymmetric standard model (MSSM) which most
significantly increase the Higgs boson mass,
more general soft terms could allow one to increase the masses
of the squarks from the first two generations above their naturalness
limits in MLES, for example, but these new degrees of freedom
will not significantly increase the upper limit on the Higgs boson mass.
Qualitatively, our results are even more general, if we enlarge
the particle content beyond the MSSM, the upper-limit on the lightest
Higgs boson mass can be increased~\cite{nonminimal}, but natural values of the
lightest Higgs boson mass will lie significantly below any maximal
upper-bounds.
Our calculation evolves the dimensionless couplings of the theory at
two-loops and includes one-loop threshold contributions and one-loop
correction to the Higgs potential.
From the resulting weak-scale parameters, we calculate
the pole masses for the Higgs bosons at one-loop
following standard diagrammatic techniques~\cite{mh2}.
The remaining next-to-leading order corrections to the Higgs boson mass arising from the
two-loop evolution of dimensionful couplings
are small in the natural region of parameter space~\cite{mh2,mh3}.
Figures 1-3 show the naturalness of the Higgs boson mass as a function
of $\tan\beta$, $m_A$, and $m_t$, respectively.
In all three figures ideally natural solutions correspond to $\gamma =1$
and fine-tuning is implied by $\gamma \gg 1$.
Figure 1 shows contours where the severity of
fine-tuning - $\gamma$ exceeds $2.5$, $5$, $10$ and $20$
in the $\tan{\beta}$-$m_h$ plane for $m_t =175$ GeV.
From Fig. 1 we see that
the mass of the lightest Higgs boson can not exceed $120$ GeV without
very significant fine-tuning, while in the most natural cases it
lies below $108$ GeV. When $\tan\beta$ is small these limits are
even more restrictive.
Figure 2 shows naturalness contours for the lightest Higgs boson mass
in MLES as a function of the CP-odd Higgs mass,
$m_A$ for $m_t =175$ GeV and arbitrary $\tan\beta$.
If we restrict ourselves to modest or small values of $\tan\beta$
these curves will become more restrictive in the $m_h$ direction.
Figure 3 shows naturalness contours for the lightest Higgs boson mass
in MLES as a function of the top quark mass. The inset in Fig. 3
displays the current uncertainty in the top quark mass, and the projected
uncertainties after run-II of Fermilab's Tevatron and
after TeV33~\cite{TeV2000,topmass}.
Fine-tuning increases both with increasing superpartner masses and with
an increasing top quark Yukawa coupling. Therefore,
in contrast to the case of fixed superpartner masses where
the corrections to the mass squared
of the Higgs boson increases as $m^{4}_t$, for fixed
naturalness these corrections increases roughly as $m^{2}_t$.
We can assess the challenge to weak-scale
supersymmetry from Higgs boson searches at colliders
from the natural regions of parameter space identified in
Figs.1-3.
The dominant production mechanism for light CP-even Higgs boson
at LEP-II is Higgs-strahlung
\begin{equation}
e^{+} e^{-} \rightarrow Z^{*} \rightarrow Z + h
\end{equation}
If Higgs boson decays into light neutralino pairs,
$h \rightarrow \tilde{\chi}^{0}_1 \tilde{\chi}^{0}_1 $,
are kinematically forbidden, $h$ will decay primarily to
$b \bar{b}$.
An upper bound on the light Higgs mass reach in this mode
is set by kinematics
and scales as $m_h < \sqrt{s} - m_Z -$ (a few) GeV.
The combined 95\% CL exclusion reaches for a standard model
(SM) Higgs
boson at LEP-II are 83 (98) ((112)) GeV at $\sqrt{s} = $
175 (192) ((205)) GeV, with integrated Luminosities of
75 (150) ((200)) ${\rm pb}^{-1}$, per experiment~\cite{LEPII}.
However, it is well known that
the observability of the lightest supersymmetric scalar $h$
can be degraded with respect to the standard model in two respects.
First, the $ZZh$ vertex carries a suppression of $\sin(\alpha - \beta)$
relative to the standard model vertex, where $\alpha$ is the mixing
angle of the CP-even Higgs scalars.
The departure of this factor from unity can be appreciable for relatively
light values of the $CP$-odd mass $m_A$, but
it approaches one as the mass of the $CP$-odd Higgs increases.
For $m_A \mathrel{\raise.3ex\hbox {$>$}\mkern-14mu \lower0.6ex\hbox{$\sim$}}
200$ GeV, $\cos^{2}(\beta-\alpha) < .01$. If the $CP$-odd Higgs
mass is light it may be produced and seen through
associated production $ e^{+} e^{-} \rightarrow A \,h$,
but this mode provides
a less significant challenge to weak-scale supersymmetry because
the CP-odd scalar mass $m_A$ is much less constrained by naturalness
arguments (see Fig. 2). Second,
the mass reach for the lightest Higgs $h$ can also be reduced
if $h$ decays invisibly into a pair of lightest superpartners,
$\tilde{\chi}^{0}_1 \tilde{\chi}^{0}_1$.
This branching ratio
can approach 100\% when allowed~\cite{neutralinos},
and this mode becomes more probable as the mass of the lightest Higgs
boson increases.
In the relatively clean environment of an $e^{+}e^{-}$ collider, a Higgs
with such invisible decays could be seen from the
acoplanar jet or lepton pair topologies resulting from the decay of the
associated $Z$, but the Higgs mass reach in this case is reduced to roughly
half of the reach when $h$ decays visibly~\cite{LEPII}.
When
$\sin^{2}(\alpha - \beta) BR(h\rightarrow b\bar{b})$
is maximal, in the {\it most} natural cases, LEP-II
operating up to $\sqrt{s}= 205$ GeV would observe a light
Higgs, but this energy is not large enough
to argue that natural electroweak symmetry
breaking is untenable in minimal supersymmetry if the Higgs
boson lies above the kinematic reach of LEP-II.
Kinematically,
the proposed Run-III of Fermilab's Tevatron with
${\cal L} = 10^{33} {\rm cm}^{-2}{\rm s}^{-1}$ (TeV33)~\cite{TeV2000} can pose a
very serious challenge to weak-scale supersymmetry.
The best single mode for discovery of a light Higgs boson at
the Tevatron is $q'\bar{q} \rightarrow W h$, with
$h\rightarrow b \bar{b}$~\cite{SMW}. TeV33 can probe a SM Higgs up to
100 (120) GeV with integrated luminosities of 10 (25) ${\rm fb}^{-1}$.
A Higgs boson mass in excess of $120$ GeV would be extremely
unnatural in the MSSM. However,
the $Wb\bar{b}$ cross section from $W h$ production
is also reduced by the
factor $BR(h\rightarrow b \bar{b}) \sin^{2}(\alpha-\beta)$. So
the significance of the challenge to weak-scale supersymmetry
from light Higgs searches at TeV33 will depend strongly
on the ability of searches for neutralinos
and charginos at the Tevatron and LEP-II
to eliminate the possibility of
$h \rightarrow \tilde{\chi}^{0}_1 \tilde{\chi}^{0}_1$,
by raising the limits on the LSP mass.
If this is the case, natural
electroweak symmetry breaking in the minimal supersymmetric
standard model will no longer be tenable if TeV33 achieves
$\int {\cal L}dt = 25 {\rm fb}^{-1}$
and fails to observe any signal of a Higgs boson.
\section{Conclusions}
\def4.\arabic{equation}{4.\arabic{equation}}
\setcounter{equation}{0}
~~
Natural choices of parameters in supersymmetric models lead to Higgs
boson masses which lie significantly below
the maximal upper-bounds determined previously
in the literature.
We have computed the natural upper bound on the Higgs mass in MLES,
and we have quantified the extent to which naturalness is lost
as the lower bound on $m_h$ increases. A Higgs mass above $120$
GeV will require very large fine-tuning, while the most natural
values of the Higgs mass lie below $108$ GeV.
The natural values of the
lightest Higgs boson mass have important implications for
the challenge to weak-scale supersymmetry at colliders.
In particular, if the possibility that the Higgs decays predominantly
to neutralino pairs can be excluded from neutralino mass limits
inferred from other superpartner searches,
natural electroweak symmetry breaking will no longer be tenable in
the MSSM if TeV33 achieves the projected reach of $m_h = 120$ GeV
and fails to observe signals of a Higgs boson.
\section*{Acknowledgments}
GA acknowledges the support of the U.S. Department of Energy
under contract DE-AC02-76CH03000.
DC is supported by the U.S. Department of Energy under grant
number DE-FG-05-87ER40319.
AR is supported by
the DOE and NASA under Grant NAG5--2788.
Fermilab is operated by the Universities Research Association, Inc.,
under contract DE-AC02-76CH03000 with the U.S. Department of Energy.
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\section{Introduction}
The possibility of having observed quantum cavitation in superfluid
$^4$He has been first put
forward by Balibar and coworkers \cite{Bal1}. These authors have used
a
hemispherical transducer that focusses a sound wave in a small region
of a cell where cavitation is induced in liquid $^4$He at low
temperature. The analysis of their experimental data is
complicated by the fact that neither the pressure (P) nor the
temperature (T) at the focus can
be directly measured. This makes the determination of the
thermal-to-quantum
cavitation crossover temperature T$^*$ to depend on the theoretical
equation of state (EOS) near the spinodal point. Using the results of
Ref. \cite{Maris1}, they conclude that T$^* \sim$ 200 mK, in agreement
with the prediction of \cite{Maris1}. However, using for instance
the EOS of Ref. \cite{Guirao}, which reproduces the spinodal point
microscopically calculated by Boronat et al \cite{Boronat1,Boronat2},
the "experimental" result becomes 120 mK.
The first detailed description of the cavitation process in liquid
helium was provided by Lifshitz and Kagan \cite{Lif1}, who used the
classical capillarity model near the saturation line, and a density
functional-like description near the spinodal line. More recently,
the method has been further elaborated by Xiong and Maris \cite{Xiong}.
These authors conclude that there is no clear way to interpolate
between these two regimes, which makes quite uncertain the range of
pressures in which each of them is valid.
In this work, we determine T$^*$ for $^3$He and $^4$He using a
functional-integral approach (FIA) in conjunction with a density
functional description of liquid helium. The method overcomes the
conceptual limitations of previous works based on the application of
zero-temperature
multidimensional WKB methods \cite{Maris1}, and the technical ones
inherent to the use of parametrized bubble density profiles
\cite{Guilleumas1}, thus putting on firmer grounds the
theoretical results. Moreover, it gives T$^*$ in the whole pressure
range.
Thermally assisted quantum tunneling is nowadays well
understood (see for example Ref. \cite{Chud} and Refs. therein). Let
us simply recall that at high temperatures, the cavitation rate, i.e.,
the number of bubbles formed per unit time and volume, is given by
%
\begin{equation}
J_T = J_{0T}\, e^{-\Delta\Omega_{max}/T}\, ,
\label{eq1}
\end{equation}
where $\Delta\Omega_{max}$ is the barrier height for thermal
activation and $J_{0T}$ is a prefactor which depends on the dynamics
of the cavitation process. At low T, it becomes
%
\begin{equation}
J_Q = J_{0Q}\, e^{-S_{min}}\, ,
\label{eq2}
\end{equation}
where $S_{min}$ is the minimum of the imaginary-time action
%
\begin{equation}
S(T) = \oint {\rm d}\tau \int {\rm d}{\vec r}\,\, {\cal L}\, ,
\label{eq3}
\end{equation}
${\cal L}$ being the imaginary-time classical Lagrangian density of
the system and the time-integration is extended over a period in the
potential well obtained by inverting the potential barrier. These
equations hold provided the rate can be calculated in the
semiclassical limit, i.e., $S_{min} >> 1$, which is the present case.
For a given value of T, one has to obtain periodic solutions to the
variational problem embodied in Eq. (\ref{eq3}). Among these many
periodic solutions, called thermons in Ref. \cite{Chud}, those
relevant for the problem of finding T$^*$ are the ones corresponding
to small oscillations around the minimum of the potential, which has
an energy equal to $-\Delta\Omega_{max}$. If $\omega_p$ is the angular
frequency of this oscillation, T$^*=\hbar\omega_p/2\pi$. It is worth
realizing that contrarily to WKB, this procedure
permits to go continously from one regime to the other: at T$^*$, Eqs.
(\ref{eq1}) and (\ref{eq2}) coincide, whereas the WKB approach forces
to equal a zero-temperature barrier penetrability to a
finite-temperature Arrhenius factor
\cite{Maris1,Guilleumas1}. Whether this is justified or not, can only
be ascertained a posteriori comparing the WKB with FIA
results.
To obtain the Lagrangian density ${\cal L}$ we have resorted to a
zero-temperature density functional description of the system
\cite{Guirao,Guilleumas2}. This is justified in view of the low-T
that are expected to come into play ($\le$ 200 mK). The critical
cavity
density profile $\rho_0(r)$ is obtained solving the Euler-Lagrange
equation \cite{Xiong,Jezek}
%
\begin{equation}
\frac{\delta\omega} {\delta\rho} = 0\, ,
\label{eq4}
\end{equation}
where $\omega(\rho)$ is the grand potential density and $\rho$ is the
particle density. $\Delta\Omega_{max}$ is given by
%
\begin{equation}
\Delta\Omega_{max} =\int {\rm d}{\vec
r}\left[\omega(\rho_0)-\omega(\rho_m)\right]\, ,
\label{eq5}
\end{equation}
where $\rho_m$ is the density of the metastable homogeneous liquid. It
is now simple to describe the dynamics of the cavitation process in
the
inverted barrier well, whose equilibrium configuration corresponds to
$\rho_0(r)$ and has an energy $-\Delta\Omega_{max}$. We suppose that
the collective velocity of the fluid associated with the bubble growth
is irrotational. This is not a severe restriction since one expects
only radial displacements (spherically symmetric bubbles). Introducing
the velocity potential field s$({\vec r},t)$, we have
%
\begin{equation}
{\cal L} =m\dot{\rho}s - {\cal H}(\rho, s)\, ,
\label{eq6}
\end{equation}
where ${\cal H}(\rho,s)$ is the imaginary-time hamiltonian density.
Defining ${\vec u}({\vec r},t)\equiv \nabla s({\vec r},t)$,
%
\begin{equation}
{\cal H} =\frac{1}{2}m\rho{\vec u}^2
-\left[\omega(\rho)-\omega(\rho_m)\right]\,\, .
\label{eq7}
\end{equation}
Hamilton's equations yield
%
\begin{equation}
m\dot{\rho} = \frac{\delta {\cal H}} {\delta s} = -m \nabla(\rho
{\vec u})
\label{eq8}
\end{equation}
\begin{equation}
m\dot{s} = -\frac{\delta{\cal H}} {\delta\rho}\, .
\label{eq9}
\end{equation}
Eq. (\ref{eq8}) is the continuity equation. Taking the gradient of Eq.
(\ref{eq9}) we get
\begin{equation}
m\frac{{\rm d}{\vec u}}{{\rm dt}} = -\nabla
\left\{
\frac{1}{2}m{\vec u}\,^2-\frac{\delta\omega}{\delta\rho}
\right\}\, .
\label{eq10}
\end{equation}
Thermons $\rho({\vec r},t)$ are periodic solutions of Eqs. (\ref{eq8})
and (\ref{eq10}). From Eq. (\ref{eq3}) and using Eqs. (\ref{eq6}) and
(\ref{eq8}) we can write
\begin{equation}
S_{min}(T) = \oint {\rm d}\tau \int {\rm d}{\vec r}
\left\{
\frac{1}{2}
m\rho{\vec u}^2+\omega(\rho)-\omega(\rho_m)
\right\}\, .
\label{eq11}
\end{equation}
Within this model, to {\it exactly} obtain T$^*$ only a linearized
version
of Eqs. (\ref{eq8}) and (\ref{eq10}) around $\rho_0(r)$ is needed.
Defining the T$^*$-thermon as
\begin{equation}
\rho(r,t) \equiv \rho_0(r) + \rho_1(r)\, e^{i\omega_p t}\, ,
\label{eq12}
\end{equation}
where $\rho_1(r)$ is much smaller than $\rho_0(r)$, and keeping only
first order terms in ${\vec u}(r,t)$ and $\rho_1(r)$, we get:
\begin{equation}
m\omega_p^2 \rho_1(r)= \nabla\left[\rho_0(r) \nabla\left(
\frac{\delta^2\omega}{\delta\rho^2}\bullet\rho_1(r)\right)\right]\, .
\label{eq13}
\end{equation}
Here, $\frac{\delta^2\omega}{\delta\rho^2}\bullet\rho_1(r)$ means that
$\delta\omega / \delta\rho$ has to be linearized, keeping only
terms in $\rho_1(r)$ and its derivatives.
Eq. (\ref{eq13}) is a fourth-order linear differential, eigenvalue
equation. A careful analysis shows that its
physical solutions have to fulfill
$\rho_1'(0)=\rho_1'''(0)=0$, and fall
exponentially
to zero at large distances. The linearized continuity equation
$\rho_1(r)\propto -\nabla(\rho_0{\vec u})$ imposes the integral of
$\rho_1(r)$ to yield zero when taken over the whole space.
We have solved Eq. (\ref{eq13}) using seven point Lagrange
formulae to discretize the r-derivatives together with a standard
diagonalization
subroutine. The sensibility of the solution to the precise value of
the
r-step has been carefully checked, and in most cases a value $\Delta
r$ = 0.25 \AA\, has been used.
For all pressures, only one positive
$m\omega_p^2$ eigenvalue has been found. Fig. 1 (a) and (b) shows
T$^*$
(mK) as a function of P(bar) for $^4$He and $^3$He, respectively. In
the case of $^4$He, the maximum T$^*$ is 238 mK at -8.58 bar, and for
$^3$He it is 146 mK at -2.91 bar. It is worth noting that T$^*$ is
strongly dependent on P in the spinodal region, falling to zero at the
spinodal point (see also Ref. \cite{Xiong}).
We display in
Fig. 2 the $\rho_1(r)$-component of the thermon (\ref{eq12}) in the
case of $^4$He (a similar figure could be drawn for $^3$He). For large
bubbles, $\rho_1(r)$ is localized at the surface: the thermon is a
well defined surface excitation.
It justifies the use of the capillarity approximation
near saturation, or more elaborated approaches,
like that of Ref.
\cite{Guilleumas1}, that consists in a simplified one-dimensional
model in which the oscillations are just described by rigid
displacements of the critical bubble surface.
When the density inside the bubble becomes sizeable, a mixed
surface-volume thermon develops, which eventually becomes a pure
volume
mode in the spinodal region. This mode can no longer be described as a
rigid density displacement, and the above mentioned models fail:
the exact T$^*$ is higher than the prediction of the rigid surface
displacement model because volume modes involve higher frequencies.
To determine which of the T$^*$(P) shown in Fig. 1 corresponds to
the actual experimental conditions, we have calculated the homogeneous
cavitation pressure P$_h$ \cite{Xiong,Jezek}. It is the one
the system can sustain before bubbles nucleate at an appreciable
rate. We have solved the equation
\begin{equation}
1 = (Vt)_e\, J
\label{eq14}
\end{equation}
taking J$=$J$_T$ and
\begin{equation}
J_{0T} = \frac{k_B T}{h V_0}.
\label{eq14b}
\end{equation}
\noindent $V_0 = 4 \pi R^3_c/3$ represents the volume of the critical
bubble, for which we have taken R$_c = $10 \AA. For T $<$
T$^*$, J$_{T}$ has to be replaced by J$_{Q}$. Lacking of a
better choice, we have taken J$_{0Q} =$ J$_{0T}$(T$=$T$^*$), and
for the experimental factor
(Vt)$_e$ (experimental volume$\times$time), two values at the limits of
the experimental range
\cite{Bal1,Maris1}, namely 10$^{14}$ and 10$^4$ \AA$^3$ s. For
$^4$He it yields P$_h$=-8.57 bar and -8.99 bar, respectively. The
corresponding values for $^3$He are -2.97 and -3.06 bar. This means
that for both isotopes P$_h$ is close to the spinodal pressure. Table
1 displays the associated T$^*$-values.
The crossover temperatures are similar to those
given in Ref. \cite{Maris1}, although different functionals
have been used in both calculations.
As a matter of fact, this is irrelevant, since
both functionals reproduce equally well the experimental
quantities pertinent to the description of the cavitation process.
An explanation for the agreement between these calculations can be
found in Ref. \cite{Guilleumas1}. In that work, using a simplified
one-dimensional model in which the oscillations were modelled by rigid
displacements of the bubble surface,
the cavitation process was described within FIA from
T=0 to the thermal regime. It was shown that thermally assisted
quantum cavitation only adds small corrections to the T=0 "instanton"
solution (formally equivalent to WKB if S$_{min}>>1$) in the
quantum-to-thermal transition region.
Let us recall that the formalism used in Ref. \cite{Maris1} to
describe quantum cavitation is a multidimensional WKB one, appropiated
for a T=0, pure quantum state with a well defined energy value. This
approximation is well known to fail for energies close to the
top of the barrier. On the contrary, the FIA here adopted deals with
thermally mixed quantum states, making it possible to smoothly connect
quantum and thermal regimes \cite{Chud}.
Besides, it is
technically complicated to obtain the E=0 instanton
solution to Eqs. (\ref{eq8}) and (\ref{eq10}) without using some
numerical approximations
\cite{Maris1} that might be unworkable in more complex physical
situations, like that of a $^3$He-$^4$He liquid mixture.
We also want to stress again that, to determine the quantity of
experimental significance, namely T$^*$,
only the thermon solution of the much simpler
eigenvalue Eq. (\ref{eq13}) is required.
To conclude, within density functional theory,
we have performed a thorough description of the
quantum-to-thermal transition in the process of cavitation in liquid
helium based on the
functional-integral approach. Our quantitative results (see also Ref.
\cite{Maris1}) indicate that the crossover temperature is below 240 mK
for $^4$He, and below 150 mK for $^3$He. The experiments on
$^4$He yield results
which, depending on which equation of state is used, are in the
120-200 mK range. Given the present uncertainties in theoretical and
experimental results as well, we consider the agreement as
satisfactory.
We would like to thank Sebastien Balibar,
Eugene Chudnovsky and Jacques Treiner
for useful discussions. This work has been supported by DGICYT
(Spain) Grant No. PB92-0761, by the Generalitat de Catalunya Grant No.
GRQ94-1022, by the CONICET (Argentine) Grant No. PID 97/93 and by the
IN2P3-CICYT agreement.
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\section{Introduction}
The precise determination of weak mixing angles $V_{cb}$ and $V_{ub}$
is a demanding task. In spite of progress in this field they still
remain ones of the worse known parameters of the Standard Model. The
uncertainties which appear here are both of an experimental and
theoretical origin. The relatively large theoretical errors mainly
reflect the lack of quantitative knowledge about the structure of
hadrons and QCD higher order perturbative corrections to the
amplitudes of weak decays of $b$~quarks. The most valuable source of
information about the weak mixing angles are the semileptonic decays
of $B$ and $\bar{B}$ mesons. The leptons in the final state do not
interact strongly and the process is less affected by unknown QCD
effects than a hadronic decay. Furthermore pseudoscalar $B$~mesons are
the simplest bottom hadrons. The $b$~quark mass is about 5~GeV and
thus it exceeds roughly ten times typical energy scales which
characterize the infrared dynamics in the hadrons. Moreover the
presence of this mass justifies the perturbative treatment of most of
the processes involving the $b$~quark. The simple facts have given
rise to a quantitative description of dynamics of hadrons containing
heavy quarks (Heavy Quark Effective Theory
\cite{VS,PW,IsgW,EH,Grin,Geor}). In the framework of HQET many
observables describing heavy hadrons may be expressed as a power
series in $1/m_b$. In particular it was shown in Ref.~\cite{CGG},
that the inclusive lepton distributions from a bottom hadron decay may
be treated in such a way. It follows from the operator product
expansion (OPE) that a matrix elements which should be evaluated to
derive the distributions may be expanded into a series of local
operators characterizing the decaying bound state. The very advantage
of this approach is that subsequent unknown non-perturbative matrix
elements are suppressed by increasing powers of $m_b$. The leading
term corresponds to a parton contribution to the process. As argued in
Ref.~\cite{CGG} the next-to-leading term vanishes. The $1/m_b ^2$
corrections have been calculated by for a case of massless
\cite{IB,MWB} and \cite{nonp1,nonp2,nonp3} massive lepton. Recently
also the third order terms have become known \cite{GK}.
The first order perturbative QCD corrections to the inclusive lepton
distributions in a process of decay: $b \rightarrow ql\bar\nu$ are as
important as the HQET corrections for the corresponding $B$ decay.
They have been evaluated \cite{JK,CJ} for the vanishing lepton
mass. In the case of a non-zero lepton mass only a differential
distribution of the lepton pair invariant mass is known to the first
order in strong coupling constant \cite{CJK}.
In the present article we present our recent calculation
of the first order QCD correction to the double
differential inclusive lepton distribution from $b$~decay with a
massive lepton in the final state. The complete analytical
result and details of the calculation will be published elsewhere
\cite{JM}.
Here we give results for
the perturbative correction to the $\tau$ lepton
energy spectrum which has been
obtained by numerical integration of this double differential
distribution.
\section{Kinematical variables}
The purpose of this section is to define the kinematical variables which are
used in this paper. We describe also the constraints imposed on these variables
for three and four-body decays of the heavy quark.
The calculation is performed in the rest frame of the decaying $b$ quark. Since
the first order perturbative QCD corrections to the inclusive process are taken
into account, the final state can consist either of produced quark $c$, lepton
$\tau$ and $\tau$ anti-neutrino or of the three particles and a real gluon. The
four-momenta of the particles are denoted in the following way: $Q$ for
$b$~quark, $q$ for $c$~quark, $\tau$ for the charged lepton, $\nu$ for the
corresponding anti-neutrino and $G$ for the real gluon. By the assumption that
all the particles are on-shell, the squares of their four-momenta are equal to
the squares of masses:
\begin{equation}
Q^2 = m_b ^2, \qquad q^2 = m_c ^2 , \qquad \tau^2 = m_\tau ^2 , \qquad
\nu^2 = G^2 = 0.
\end{equation}
The four-vectors $P = q + G$ and $W = \tau + \nu$ characterize
the quark--gluon system and the virtual $W$ boson respectively. We define a set
of variables scaled in the units of mass of heavy quark $m_b$:
\begin{equation}
\varrho = {m_c ^2 \over m_b ^2} , \qquad \eta = {m_\tau ^2 \over m_b ^2},\qquad
x = {2 E_\tau \over m_b ^2}, \qquad t = {W^2 \over m_b ^2}, \qquad
z = {P^2 \over m_b ^2}.
\end{equation}
We introduce light-cone variables describing the charged lepton:
\begin{equation}
\tau_\pm = {1\over 2} \left( x \pm \sqrt{x^2 - 4\eta} \right)
\end{equation}
The system of $c$ quark and real gluon is characterized by the following
quantities:
\begin{eqnarray}
P_0 (z) &=& {1 \over 2}(1-t+z), \nonumber \\
P_3 (z) &=& \sqrt{P_0 ^2 - z} = {1\over 2} [1+t^2+z^2-2(t+z+tz)]^{1/2},\nonumber \\
P_\pm (z) &=& P_0 (z) \pm P_3 (z), \nonumber \\
{\cal Y}_p (z) &=& {1 \over 2} \ln {P_+ (z) \over P_- (z) } =
\ln { P_+ (z) \over \sqrt{z} } \nonumber \\
\end{eqnarray}
where $P_0 (z)$ and $P_3 (z)$ are the energy and length of the momentum vector
of the system in $b$ quark rest frame, ${\cal Y}_p (z)$ is the corresponding
rapidity.
Similarly for virtual $W$:
\begin{eqnarray}
W_0 (z) &=& {1 \over 2}(1+t-z), \nonumber \\
W_3 (z) &=& \sqrt{W_0 ^2 - t} = {1\over 2} [1+t^2+z^2-2(t+z+tz)]^{1/2}, \nonumber \\
W_\pm (z) &=& W_0 (z) \pm W_3 (z), \nonumber \\
{\cal Y}_w (z) &=& {1 \over 2} \ln {W_+ (z) \over W_- (z) } =
\ln { W_+ (z) \over \sqrt{t} }, \nonumber \\
\end{eqnarray}
>From kinematical point of view the three body decay is a special case of the
four body one with vanishing gluon four-momentum, what is equivalent to
$z=\varrho$. It is convenient to use in this case the following variables:
\[
p_0 = P_0 (\varrho) = {1\over 2} (1 - t + \varrho ), \qquad
p_3 = P_3 (\varrho) = \sqrt{p_0 ^2 - \varrho},
\]
\[
p_\pm = P_\pm (\varrho) = p_0 \pm p_3, \qquad
w_\pm = W_\pm (\varrho) = 1 - p_\mp .
\]
\begin{equation}
Y_p = {\cal Y}_p (\varrho ) = {1 \over 2} \ln {p_+ \over p_-}, \qquad
Y_w = {\cal Y}_w (\varrho ) = {1 \over 2} \ln {w_+ \over w_-}.
\end{equation}
We express also the scalar products which appear in the calculation by the
variables $x$, $t$ and $z$:
\begin{equation}
\begin{array}{ll}
Q \!\cdot\! P \,= {1\over 2} (1+z-t) & \tau\!\cdot\!\nu \, = {1\over 2}(t - \eta) \\
Q \!\cdot\! \nu \, = {1\over 2} (1-z-x+t)\hspace{3em} &
\tau\!\cdot\! P \, = {1\over 2}(x-t-\eta) \\
Q \!\cdot\! \tau \, = {1\over 2} x & \nu\!\cdot\!\tau \, = {1\over 2}(1-x-z+\eta) \\
\end{array}
\end{equation}
All of the written above products are scaled in the units of the mass of
$b$~quark.
The allowed ranges of $x$ and $t$ for the three-body decay
are given by following inequalities:
\begin{equation}
2\sqrt{\eta} \leq x \leq 1 + \eta - \varrho = x_{max}, \qquad
\label{xbound}
\end{equation}
\begin{equation}
t_1 = \tau_- \left( 1 - {\varrho \over 1 - \tau_-} \right)
\leq t \leq
\tau_+ \left( 1 - {\varrho \over 1 - \tau_+} \right) = t_2
\end{equation}
(a region A).
In the case of the four-body process the available region of the phase space
is larger than the region~A. The additional, specific for the four body decay area
of the phase space is denoted as a region~B. Its boundaries are given by the
formulae:
\begin{equation}
2\sqrt{\eta} \leq x \leq x_{max}, \qquad
\eta \leq t \leq t_1
\label{xyps}
\end{equation}
We remark, that if the charged lepton mass tends to zero than
the region B vanishes.
One can also parameterize the kinematical boundaries of $x$ as
functions of~$t$. In this case we obtain for the region A:
\begin{equation}
\eta \leq t \leq (1-\sqrt{\varrho})^2, \qquad
w_- + {\eta \over w_-} \leq x \leq w_+ + {\eta \over w_+},
\label{yxps}
\end{equation}
and for the region B:
\begin{equation}
\eta \leq t \leq \sqrt{\eta} \left( 1 - {\varrho\over 1 - \sqrt{\eta}}
\right), \qquad
2\sqrt{\eta} \leq x \leq w_- + {\eta \over w_-}.
\end{equation}
The upper limit of the mass squared of the $c$-quark --- gluon system is
in both regions given by
\begin{equation}
z_{max} = (1-\tau_+)(1-t/\tau_+),
\end{equation}
whereas the lower limit depends on a region:
\begin{equation}
z_{min} = \left\{
\begin{array}{ll}
\varrho & \mbox{\rm in the region A} \\
(1-\tau_-)(1-t/\tau_-) & \mbox{\rm in the region B.} \\
\end{array} \right.
\end{equation}
\section{Evaluation of the QCD corrections}
The QCD corrected differential rate for
$b \rightarrow c + \tau^- + \bar{\nu}$
reads:
\begin{equation}
d\Gamma = d\Gamma_0 + d\Gamma_{1,3} + d\Gamma_{1,4},
\end{equation}
where
\begin{equation}
d\Gamma_0 = G_F ^2 m_b ^5 |V_{CKM}| ^2 {\cal M}_{0,3} ^-
d{\cal R}_3 (Q;q,\tau,\nu) / \pi^5
\end{equation}
in Born approximation,
\begin{equation}
d\Gamma_{1,3} = {2 \over 3}\alpha_s G_F ^2 m_b ^5 |V_{CKM}|^2 {\cal M}_{1,3} ^-
d{\cal R}_3 (Q;q,\tau,\nu) / \pi^6
\end{equation}
comes from the virtual gluon contribution and
\begin{equation}
d\Gamma_{1,4} = {2 \over 3}\alpha_s G_F ^2 m_b ^5 |V_{CKM}|^2 {\cal M}_{1,4} ^-
d{\cal R}_4 (Q;q,\tau,\nu) / \pi^7
\end{equation}
describes a real gluon emission. $V_{CKM}$ is the Cabbibo--Kobayashi--Maskawa
matrix element associated the $b$ to $c$ or $u$ quark weak transition.
Lorentz invariant $n$-body phase space is defined as
\begin{equation}
d{\cal R}_n(P;p_1, \ldots , p_n ) =
\delta^{(4)} (P - \sum p_i) \prod_i { d^3 {\bf p}_i \over 2 E_i}
\end{equation}
In Born approximation the rate for the decay into three body final state is
proportional to the expression
\begin{equation}
{\cal M}_{0,3} ^- = F_0 (x,t) =
4 q \!\cdot\! \tau \; Q \!\cdot\! \nu \;= (1 - \varrho - x + t )(x - t - \eta),
\end{equation}
where the quantities describing the $W$~boson propagator are neglected.
Interference between virtual gluon exchange and Born amplitude yields:
\begin{eqnarray}
{\cal M}_{1,3} ^- & = &
- [
\; q \!\cdot\!\tau\; Q \!\cdot\! \nu \; H_0 + \varrho\; Q\!\cdot\!\nu\; Q\!\cdot\!\tau\; H_+
+ \; q \!\cdot\! \nu\; q \!\cdot\! \tau \; H_- + \nonumber\\
& & + {1 \over 2} \varrho \; \nu \!\cdot\! \tau \; ( H_+ + H_-)
+ {1 \over 2} \eta \varrho \; Q \!\cdot\! \nu \; ( H_+ - H_- + H_L )
- {1 \over 2} \eta \; q \!\cdot\! \nu \; H_L ] , \nonumber \\
\end{eqnarray}
where
\begin{eqnarray}
H_0 & = & 4(1-Y_p p_0/p_3 ) \ln \lambda_G + (2p_0/p_3)
\left[ \mbox{Li}_2 \left( 1 - {p_- w_- \over p_+ w_+ } \right) \right. \nonumber\\
& & - \left. \mbox{Li}_2 \left( 1 - {w_- \over w_+} \right) - Y_p (Y_p+1) +
2(\ln\sqrt\varrho + Y_p)(Y_w + Y_p) \right] \nonumber\\
& & + [2p_3 Y_p + (1 - \varrho - 2t) \ln \sqrt \varrho ] / t + 4, \nonumber\\
H_\pm & = & {1 \over 2} [ 1 \pm (1-\varrho) / t ] Y_p / p_3 \pm
{1 \over t} \ln\sqrt\varrho , \nonumber \\
H_L & = & {1 \over t} ( 1 -\ln\sqrt\varrho) + {1- \varrho \over t^2}
\ln\sqrt\varrho + {2 \over t^2} Y_p p_3 + {\varrho\over t}
{Y_p \over p_3}. \nonumber\\
\end{eqnarray}
In ${\cal M}_{1,3} ^-$
infrared divergences are regularized by
a small mass of gluon denoted by $\lambda_G$.
According to Kinoshita--Lee--Naunberg theorem, the infrared divergent part
should cancel with the infrared contribution of the four-body decay amplitude
integrated over suitable part of the phase space.
The rate from real gluon emission is proportional to
\begin{equation}
{\cal M}^- _{1,4} =
{{\cal B}^- _1 \over (Q\!\cdot\! G)^2 } -
{{\cal B}^- _2 \over Q\!\cdot\! G \; P\!\cdot\! G} +
{{\cal B}^- _3 \over (P\!\cdot\! G)^2 } ,
\end{equation}
where
\begin{eqnarray}
{\cal B}_1 ^- & = & \,
q \!\cdot\! \tau \; [\, Q \!\cdot\! \nu\; (Q \!\cdot\! G \, - 1) + \, G \!\cdot\! \nu \, -
\, Q \!\cdot\! \nu\; Q \!\cdot\! G \, +\, G \!\cdot\! \nu\; Q \!\cdot\! G \, ],\nonumber\\
{\cal B}_2 ^- & = & \,
q \!\cdot\! \tau \; [\, G \!\cdot\! \nu \; Q \!\cdot\! q \, - \, q\!\cdot\!\nu \; Q\!\cdot\! G\, +
\, Q \!\cdot\! \nu \; (\, q \!\cdot\! G \, - \, Q \!\cdot\! G\, - 2\, q \!\cdot\! Q\,)]+ \nonumber\\
& & + \, Q \!\cdot\! \nu \; (\, Q \!\cdot\! \tau \; q \!\cdot\! G \,
- \, G \!\cdot\! \tau\; q \!\cdot\! Q\, ), \nonumber\\
{\cal B}_3 ^- & = & Q \!\cdot\! \nu \; (\, G \!\cdot\! \tau \; q \!\cdot\! G \, -
\varrho \; \tau \!\cdot\! P \, ). \nonumber\\
\end{eqnarray}
Integrating and adding all the contributions one arrives at the
following double differential distribution of leptons:
\begin{equation}
{d\Gamma \over dx\, dt} =
\left\{
\begin{array}{ll}
12 \Gamma_0 \left[ F_0 (x,t) - {2\alpha_s \over 3\pi } F_1 ^A (x,t) \right] &
\mbox{for $(x,t)$ in A}, \\
12 \Gamma_0 {2\alpha_s \over 3\pi} F_1 ^B (x,t) & \mbox{for $(x,t)$ in B} \\
\end{array}
\right.
\label{main}
\end{equation}
where
\begin{equation}
\Gamma_{0} = {G_F ^2 m_b ^5 \over 192\pi^3} |V_{CKM}|^2,
\end{equation}
\begin{equation}
F_0 (x,t) = (1 - \varrho - x + t )(x - t - \eta),
\end{equation}
and the functions $F_1 ^A (x,t)$, $F_1 ^B (x,t)$ describe the
perturbative correction in the regions $A$ and $B$.
Explicite formulae for $F_1 ^A (x,t)$ and $F_1 ^B (x,t)$
will be given in \cite{JM}.
The factor of 12 in the formula (\ref{main}) is introduced to meet widely used
\cite{nonp1,CJK,MV} convention for $F_0 (x)$ and $\Gamma_0$.
The obtained results were tested by comparison with
earlier calculations.
One of the cross checks was arranged by fixing the
mass of the produced lepton to zero.
Our results are in this limit algebraically identical
with those for the massless charged lepton \cite{JK,CJ}.
On the other hand one can numerically integrate the calculated
double differential distribution over $x$, with the limits given by the
kinematical boundaries:
\begin{equation}
\int_{2\sqrt{\eta}} ^{w_+ + \eta / w_+ } {d\Gamma\over dx\, dt}
(x,t;\varrho,\eta) = {d\Gamma \over dt} (t;\varrho,\eta)
\label{testint}
\end{equation}
Obtained in such a way differential
distribution of $t$ agrees with recently published \cite{CJK}
analytical formula describing this distribution. This test is particularly
stringent because one requires two functions of three variables
($t,\varrho$ and $\eta$) to be numerically equal for any values of the arguments.
We remark, that for higher values of $t$
only the region A contributes to the integral (\ref{testint}) and for lower
values of $t$ both regions~A and B contribute.
This feature of the test is very helpful ---
the formulae for $F_1 (x,t)$, which are different for the
regions A and B can be checked separately.
\begin{figure}
\hbox{
\epsfxsize = 200pt
\epsfysize = 200pt
\epsfbox[36 366 453 765]{fig1a.ps}
\epsfxsize = 200pt
\epsfysize = 200pt
\epsfbox[36 366 453 765]{fig1b.ps} \vspace{1em} }
\caption{(a) The distributions $f_0 (x)$, $f_1 (x)$ and (b) the
ratio $f_1 (x) / f_0 (x)$ for the pole mass of the $b$ quark $m_b =
4.5$~GeV (dotted), $m_b = 4.75$~GeV (solid) and $m_b = 5.0$~GeV (dashed).}
\end{figure}
\section{Differential distribution of $\tau$ energy}
The point of interest to check how the QCD corrections change
energy spectrum of the charged lepton. This aim may be reached by integration
of the double differential lepton distribution over the lepton pair invariant
mass:
\begin{equation}
{d\Gamma \over dx} = \int_{\eta} ^{t_2} {d\Gamma \over dx\,dt} \, dt,
\end{equation}
where $t_2$ is the upper kinematical boundary for $t$ given the formula
(\ref{yxps}). The decomposition of the resulting distribution into the Born
term and the perturbative QCD correction yields in a natural way definitions of
functions $f_0 (x)$ and $f_1 (x)$:
\begin{equation}
{d\Gamma \over dx} =
12 \Gamma_0 \left[ f_0 (x) - {2\alpha_s \over 3\pi } f_1 (x) \right].
\end{equation}
The analytical formula for $f_0 (x)$ reads
\begin{eqnarray}
f_0 (x) &=& 2\sqrt{x^2 - 4\eta}
\left\{ x_0 ^3 [x^2 - 3x(1 + \eta) + 8\eta] + \right. \nonumber \\
& & \hspace{100pt}\left. + x_0 ^2 [-3x^2 + 6x(1+\eta) - 12\eta ] \right\} \nonumber ,\\
\end{eqnarray}
where following \cite{nonp1} we introduced
\begin{equation}
x_0 = 1 - \varrho /(1+\eta-x) .
\end{equation}
An equivalent expression for $f_0 (x)$ is
\begin{eqnarray}
f_0 (x) &=& {1\over 6} x \sqrt{ x^2 - 4\eta }
\left( \frac{ x_{max} - x }{1+\eta -x} \right) ^2
[ 3 (1+\eta ) - 2x + \varrho - 4\eta / x + \nonumber \\
& & \hspace{105pt} + 2\varrho (1+\eta -4\eta / x) / (1+\eta -x)], \nonumber \\
\end{eqnarray}
where $x_{max}$ is given by (\ref{xbound}). The latter formula
clearly exhibits the behavior of $f_0 (x)$ for $x$ close to
the upper kinematical limit.
The integration of $F^{A,B} _1 (x,t)$ was performed numerically
for different masses of $b$-quark with fixed $m_b - m_c = 3.4$~GeV and
$m_\tau = 1.777$~GeV. The functions $f_0 (x)$ and
$f_1 (x)$ for $m_b = 4.75$~GeV are plotted on Fig.~1a and the ratios
$f_1 (x) / f_0 (x)$ for three different realistic values of
$m_b$ are plotted on Fig.~1b. As can be easily seen the ratios have logarithmic
singularities at the upper end of the spectra. Such a behavior would lead to a
inconsistence. The standard solution to problems of this kind is an
exponentiation which yields well known Sudakov form factor \cite{FJMW}.
Far from the end point the ratio of the correction term to the leading one
is almost constant and close to~2. It means that the perturbative correction
changes rather the normalization than the shape of lepton energy distribution.
\begin{figure}[hbpt]
\epsfxsize = 350pt
\epsfysize = 290pt
\hspace{2cm}
\epsfbox[85 366 517 775]{fig2.ps}
\caption{The QCD corrected $\tau$ lepton spectrum from the $b$ quark
decay for different values of $\alpha_s$. The mass of $b$ quark is chosen as
4.75~GeV.}
\end{figure}
\
The obtained distributions of the scaled charged lepton energy for
$m_b=4.75$~GeV with and without perturbative QCD corrections are
shown in
Fig.~2. The strong coupling constant was chosen as $0.2$ and $0.4$ since the
energy scale for this process in not known until the second order QCD
corrections are evaluated. The value of $\alpha_s$ for this decay is expected
to lay between the two numbers.
The knowledge of perturbative corrections to lepton energy is essential for
fixing HQET parameters, especially $\lambda_1$ and $\bar\Lambda$
\cite{IB,MWB}. Especially analysis of moments of the lepton energy spectrum
and other quantities involving integration over the energy distribution
appeared particularly valuable for this purpose \cite{GKLW} as was earlier
suggested in Refs.~\cite{CJK,MV}.
\section{Conclusions}
The first order QCD corrections to the double differential inclusive lepton
distributions from $b$~quark semileptonic decay have been
calculated for a massive fermion in the final state.
Non-trivial cross checks of the final the result have been performed.
We remark that including a real gluon radiation on the parton level yields a
increase of the phase space available in the decay process.
The QCD corrected $\tau$ energy spectrum has been obtained. The effect of the
correction may be estimated as about 10\% of the magnitude of uncorrected
distributions.
The presented above results can be utilized to improve an analysis of
semileptonic decays of beauty hadrons with a $\tau$ in the final state.
Thus the values of involved weak mixing angles may be fixed more exactly.
The decrease of theoretical uncertainty increases the sensitivity to hypothetic
deviations from the Standard Model \cite{Kal,GL,GHN} which should have to be
particularly distinct in the case of the heaviest family. The better
understanding of the perturbative QCD effects allows one to perform more
stringent tests of HQET predictions \cite{nonp1,nonp2,nonp3} and narrow the
error bars for HQET parameters. Moreover one can extract more precisely some
information about masses of quarks and strong coupling constant from the data.
Finally, the process that we considered may appear a background for other
processes so precise theoretical knowledge about the process is valuable.
At present however the statistics of measured
$b \rightarrow c(u)\tau\bar\nu_\tau$
transitions is rather low and ten-percent effects are not seen. Probably the
application of provided here formulae to the expected data from $B$-factories
will be really fruitful.
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\newcommand{
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\section{INTRODUCTION}
The underlying assumptions of the dual superconductivity\cite{tmp} of
gauge theories, and its appropriatenss for describing quark confinement,
are not rigorously founded, and it is necessary to perform precise numerical
or analytic tests of this conjecture whenever possible.
The internal structure of the color flux tube joining a quark pair provides an
important test of these ideas, because it should show, as the dual of
an Abrikosov vortex, a very peculiar property: it is expected to have
a core of normal, hot vacuum as contrasted with the surrounding
medium, which is in the dual superconducting phase. The location of the core
would be given by the vanishing of the disorder parameter $\langle\Phi_M(x)\rangle=0$,
where $\Phi_M$ is some effective magnetic Higgs field.
In a pure gauge theory, the formulation of this property from the
first principles poses some problems, because no local, gauge invariant,
disorder field $\Phi_M(x)$ is known. As a consequence, one cannot define in a
meaningful, precise way the notion of core of the dual vortex.
A possible way out is suggested by the fact that in a medium in which
$\langle\Phi_M\rangle=0$ the quarks should be deconfined, then it is expected that
the interquark potential inside the flux tube gets modified. As a consequence,
one may try to define a gauge-invariant notion of normal core of the flux
tube as the region where the interquark interaction mimics a deconfined
behavior.
Of course one cannot speak of a true deconfinement, as it would require
pulling infinitely apart the quarks, while the alleged core has a finite size.
A simple, practical way to study in a lattice gauge theory the influence
of the flux tube on the quark interaction is based on the study of the
system of four coplanar Polyakov loops $P_1,P_2,P_3$ and $P_4$
following two steps
\begin{description}
\item{~~} Modify the ordinary vacuum by
inserting in the action the pair $P_3,P_4^{\dagger}$
acting as sources at a fixed distance $R$.
\item{~~} Evaluate in this modified vacuum the correlator the other
pair $P_1,P_2^{\dagger}$ of Polyakov loops which are used as probes.
\end{description}
The correlators in the two vacua are related by
\begin{equation}
\langle P_1 P_2^{\dagger}\rangle_{q\bar{q}}=\frac{\langle P_1 P_2^{\dagger}P_3 P_4^{\dagger}\rangle}
{\langle P_3 P_4^{\dagger}\rangle}~~.
\end{equation}
In this note we study some general properties of these correlators for
$T\leq T_c$~. In particular, we point out that at $T=T_c$ the functional form
of these correlators is universal and in some $3D$ gauge theories
can be written explicitly, even in finite volumes.
\section{FOUR POLYAKOV LOOPS}
Consider the system of four parallel, coplanar Polyakov loops, symmetrically
disposed with respect the origin of a cubic lattice with periodic boundary
conditions in the direction of the imaginary time (which coincides with the
common direction of the loops). We study their correlator
\begin{equation}
\langle P_1 P_2^{\dagger}P_3 P_4^{\dagger}\rangle=
\langle P(\scriptsize{{-\frac{r}2}})P^{\dagger}(\scriptsize{{\frac{r}2}})P(\scriptsize{{-\frac{R}2}})P^{\dagger}(\scriptsize{{\frac{R}2}})\rangle
\label{four}
\end{equation}
as a function of $r\le R$.
For large $R$ and $r\sim R$ it obeys the asymptotic factorization
condition
\begin{equation}
\langle P_1 P_2^{\dagger} P_3 P_4^{\dagger}\rangle\sim
\langle P_1 P_3^{\dagger}\rangle\langle P_2 P_4^{\dagger}\rangle~.
\label{fact}
\end{equation}
When $T < T_c$, assuming the usual area law
$\langle P_1P_2^{\dagger}\rangle\propto\exp(-\sigma r/T)$,
where $\sigma$ is the string tension, yields
\begin{equation}
\langle P_1 P_2^{\dagger}\rangle_{q\bar{q}}\sim\exp(\sigma r/T)
\sim1/\langle P_1 P_2^{\dagger}\rangle~~,
\end{equation}
which gives an apparent repulsion between the two probes due to the
attraction of the two sources.
The other limit $r\ll R$ is more interesting, because the kinematics does not
force any factorization and different confinement models suggest
different behaviors. In particular in the naive string picture one is tempted
to assume the factorization (\ref{fact}) even in this limit, because within this
assumption the total area of the surfaces connecting the Polyakov loops is
minimal. On the contrary, in the dual superconductivity it is expected that
the test particles probe the short distance properties of the
hot core of the flux tube, thus the correlator in the modified vacuum would
approach to a constant ($\sim \langle P\rangle^2_{T>T_c}$)
from above and
\begin{equation}
\langle P_1 P_2^{\dagger}\rangle_{q\bar{q}}>\langle P_1 P_2^{\dagger}\rangle~~~(r\ll R\,, T<T_c)~.
\end{equation}
In the range $T\ge T_c$ the interior of the flux tube is in the same phase of
the surrounding region and the mutual interaction between the two near
probes should not depend on the presence of very far sources, then
\begin{equation}
\langle P_1 P_2^{\dagger}\rangle_{q\bar{q}}\sim\langle P_1 P_2^{\dagger}\rangle~~~
(r\ll R\,, T\ge T_c)~.
\label{faq}
\end{equation}
\subsection{ Critical Behavior}
According to the widely tested Svetitsky-Yaffe conjecture, any gauge theory
in $d+1$ dimensions with a continuous deconfining transition belongs to the
same universality class of a $d$-dimensional $C(G)$-symmetric spin model, where
$C(G)$ is the center of the gauge group. It follows that at the critical
point all the critical indices describing the two transitions and all the
adimensional ratios of correlation functions of corresponding observables
in the two theories should coincide.
In particular, since the order parameter the gauge theory is obviously
mapped in the corresponding one of the spin model, the correlation functions
among Polyakov loops should be proportional to the corresponding correlators
of spin operators:
\begin{equation}
\langle P_1\dots P_{2n}\rangle_{T=T_c}\propto \langle s_1\dots s_{2n}\rangle~~.
\end{equation}
Conformal field theory has been very successful in determining
the exact form of these universal functions for $d=2$ even in a finite box,
which is a precious information for a correct comparison with numerical
simulations.
In particular, using the known results of the $2D$ critical Ising model
in a rectangle $L_1\times L_2$ with periodic boundary conditions \cite{fsz}
we can write explicitly the correlator of any (even) number $2n$ of
Polyakov loops of any $2+1$ gauge theory with $C(G)=\hbox{{\rm Z{\hbox to 3pt{\hss\rm Z}}}}_2$. Let $x_j,y_j$
be the spatial coordinates of $P_j$ and define the complex variables
$z_j=\frac{x_j}{L_1}+i\frac{y_j}{L_2}$ and $\tau=iL_2/L_1$. Then
\begin{equation}
\langle P_1\dots P_{2n}\rangle^2=c_n\sum_{\nu=1}^{4}
\sum_{\varepsilon_i=\pm1}^{~}{\,}'
A_\nu(\varepsilon \cdot z)\prod_{i<j}B_{ij}
\label{crt}
\end{equation}
with $\varepsilon\cdot z=\sum_i \varepsilon_iz_i$ and the primed sum
is constrained by $\sum_i\varepsilon_i=0$~; $c_n$ is an overall constant
that can be expressed by factorization in terms of $c_1$.
The universal functions
$A_\nu$ and $B_{ij}$ can be written in terms of the four Jacobi theta
functions $\vartheta_\nu(z,\tau)$ as follows
\begin{equation}
B_{ij}=\left\vert\frac{\vartheta_1(z_i-z_j,\tau)}{\vartheta_1'(0,\tau)}
\right\vert^{\varepsilon_i\varepsilon_j/2},
\end{equation}
\begin{equation}
A_\nu(z)=\left\vert\frac{\vartheta_\nu(z,\tau)}{\vartheta_\nu(0,\tau)}
\right\vert^2,
\end{equation}
In the infinite box limit $L_1,L_2\to\infty$, using the Taylor expansion
\begin{equation}
\vartheta_\nu(z,\tau)=a_\nu(1-\delta_{1,\nu})+b_\nu\,z+O(z^2)~,
\end{equation}
the correlator (\ref{four}) becomes
\begin{equation}
\langle P_1P_2P_3P_4\rangle=\frac{4c_1^2}{(Rr)^\frac14}
\sqrt{\frac{R+r}{R-r}}~~,
\end{equation}
which satisfies both factorizations (\ref{fact},\ref{faq}).
\section {CLUSTER ALGORITHM}
In order to test the above formulae at criticality it is convenient to perform
the numerical simulations in the simplest model belonging to the
above-mentioned universality class, which is the the $3D$ $\hbox{{\rm Z{\hbox to 3pt{\hss\rm Z}}}}_2$ gauge model.
Using the duality transformation it is possible to build
up a one-to-one mapping of physical observables of the gauge system
into the corresponding spin quantities.
A great advantage of this method is that it can be used a non local cluster
updating algorithm \cite{sw}, which has been proven very successful in
fighting critical slowing down.
In this framework it is easily shown that the vacuum expectation value of
any set $\{C_1\dots C_n\}$ of Polyakov or Wilson loops of arbitrary
shapes is simply
encoded in the topology of Fortuin-Kasteleyn (FK) clusters: to each Montecarlo
configuration we assign a weight 1 whenever there is no FK cluster
topologically linked to any $C_i\in\{C_1\dots C_n\}$, otherwise we assign a
weight 0. Let $N_0$ and $N_1$ be the number
configurations of weight 0 and 1 respectively, then we have simply
\begin{equation}
\langle C_1\dots C_n\rangle=\frac{N_1}{N_0+N_1}~~.
\end{equation}
This method provides us with a handy, very powerful tool to estimate
the correlator of any set of Wilson or Polyakov
loops even at criticality.
\section{RESULTS}
In order to test the critical behavior of the multiloop correlator one has to
know with high precision the location of the critical temperature as a function
of the coupling $\beta$. We took advantage of ref.\cite{ch}, where these
critical values have been obtained with an extremely high statistical
accuracy. We report in Fig.1 some results at $\beta=0.746035$ corresponding
to $1/aT_c=N_{tc}=6 $
and to a string tension $\sigma a^2=0.0189(2)$.
The open circles are the data for the correlator
$\langle P(\scriptsize{{-\frac{r}2}})P(\scriptsize{{\frac{r}2}})\rangle$ in a $N_t\times N_x\times N_y$ lattice with
$N_t=3N_{tc},N_x=N_y=64$.
They are well fitted by the one-parameter formula
$c\exp(-\sigma N_t r)/\eta(i\frac{N_t}{2r})$, where the Dedekind $\eta$
function takes into account the quantum contribution of the flux tube
vibrations \cite{pv}.
The square symbols correspond to the correlator $\langle P(\scriptsize{{-\frac{r}2}})P(\scriptsize{{\frac{r}2}})\rangle_{q\bar{q}}$
at the same temperature, in presence of a pair of sources at a distance
$R=24a$. The data in the
central region are well fitted by a two-parameter formula
$c_{q\bar{q}}\exp(-\sigma N_t r)/\eta(i\frac{N_t}{2r})+b\to
c'_{q\bar{q}}\frac{e^{-mr}}{\sqrt{r}}+b$ which simulates a high temperature
behavior with a screening mass $m=\sigma N_t$ and an order parameter
$\langle P\rangle=\sqrt{b}$.
The black circles correspond to
$\langle P(\scriptsize{{-\frac{r}2}})P(\scriptsize{{\frac{r}2}})P(\scriptsize{{-\frac{R}2}})P(\scriptsize{{\frac{R}2}})\rangle$ evaluated at $T=T_c$ with $R=16a$.
They fit nicely eq.(\ref{crt}) (continuous line). Note that such a curve
does not contain any free parameters, being $c_2=\sqrt{2}c_1^2$ with
$c_1 N_x^{\frac14}=0.199(4)~$ as estimated by measuring $\langle P_1P_2\rangle$
on lattices of different sizes at $T=T_c$ and $N_{tc}=6$.
\vskip0.3cm
\vskip -2.3 cm
\hskip-2.cm\epsfig{file=figlat96.ps,height=10.5cm}
\vskip -1.9cm
Figure. 1. { Correlator of two Polyakov loops inside and outside
the flux tube}
\vskip0.2cm
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\section{Introduction}
A photon that covers a distance $L$ within a transverse,
homogeneous magnetic field of strength $B$
has a probability of converting into a graviton given by\cite{Gers,Zeldovich}
\begin{equation}
P\simeq 4\pi GB^2L^2\simeq 8\times 10^{-50}\left ({B\over {\rm Gauss}}
\right )^2\left ({L\over {\rm cm}}\right )^2,
\label{P}
\end{equation}
where $G$ is Newton's constant.
It has recently been suggested\cite{Chen,Magueijo}
that a primordial magnetic field may imprint observable consequences upon the
cosmic microwave background radiation through photon-graviton conversion.
According to eq. (\ref{P}), a primordial magnetic field
of present value around $10^{-8}$ Gauss, if it already existed
at the time of
decoupling of matter and radiation and was homogeneous over a Hubble radius,
would have induced
a degree-scale anisotropy of
the cosmic microwave background of about $10^{-5}$, of the order of the
observed value \cite{1deg}. Although current bounds suggest that a
cosmological magnetic field, if it exists, has present strength smaller than
around $10^{-9}$ Gauss\cite{Breviews}, photon-graviton conversion
could in principle provide an
independent method to constrain or eventually detect a primordial cosmological
magnetic field.
In this article we wish to point out that plasma effects due to the Universe
residual ionization make the photon-graviton oscillation length much shorter
than the Hubble radius, and the probability of photon-graviton conversion is
consequently much smaller than in the absence of free electrons. The effects
of a primordial magnetic field of present value around $10^{-9}$ Gauss or
smaller are consequently negligible.
\section{Photon-graviton conversion probability}
The interaction between a gravitational and an electromagnetic field
linearized in the small perturbation $h_{\mu\nu}$
around flat space-time is described, in General Relativity, by
the term in the action
\begin{equation}
S_{int}=%
{\displaystyle {1 \over 2}}
\displaystyle \int
h_{\mu \nu }T^{\mu \nu }d^4x \label{Sint}
\end{equation}
where $T_{\mu \nu }$ is the flat-space energy-momentum tensor
of the electromagnetic field.
In an external, homogeneous magnetic field $B$,
photons and gravitons can convert
into each other conserving energy and linear momentum. The
linearized interaction term between electromagnetic and gravitational
plane-waves with the same wave-vector can be written as
\begin{equation}
S_{int}=B\sin \theta
\displaystyle \int
\left[ h_{+}E_{\perp }+h_{\times }E_{\parallel }\right] d^4x\ . \label{Sint2}
\end{equation}
Here $E_\parallel$ and $ E_\perp $ denote respectively
the component of the electric field
in the electromagnetic wave that is either parallel or perpendicular
to the plane that contains the direction of propagation and
the external homogeneous magnetic field,
$h_{+}$ and $h_{\times }$ describe two independent polarization
modes of the gravitational wave, in the transverse-traceless gauge,
and $\theta$ is the angle between
the external magnetic field and the
common direction of propagation of the electromagnetic and gravitational
waves.
From eq. (\ref{Sint2}) the conversion probability
between photons and gravitons is easily read off.
Incoming photons with polarization either $\parallel$ or
$\perp$ convert into gravitons
with the same probability
\begin{equation}
P = 4\pi GB^2L^2\sin^2\theta\ ,
\label{P2}
\end{equation}
the only difference
being the polarization of the resulting graviton.
We wish to point out that precisely because these two independent
states of linear polarization have the same conversion probability,
unpolarized electromagnetic radiation does not become
linearly polarized
due to photon-graviton conversion
as it propagates through an homogeneous
magnetic field. In this respect, photon-graviton conversion differs
qualitatively from the conversion between photons and pseudoscalar
particles\cite{S,RS}. In the latter case, only $E_\parallel $
mixes with the pseudoscalar field. Photon-pseudoscalar conversion in a
cosmological magnetic field induces a small degree of linear
polarization in the cosmic microwave background \cite{HS}.
We conclude, however, and contrary to the claim in ref. \cite{Magueijo},
that photon-graviton conversion does not induce linear polarization in the
cosmic microwave background.
In the presence of a free electron density $n_e$, photons propagate
as if they had an
effective mass equal to the plasma frequency
$\omega^2_{pl}=4\pi \alpha n_e / m_e$,
where $m_e$ denotes the electron mass and
$\alpha =\frac{e^2}{4\pi }\sim \frac 1{137}$ is the fine
structure constant. We work in Heaviside-Lorentz natural units
(in which $1 {\rm Gauss} = 1.95 \times 10^{-2} {\rm eV}^2$).
If the external magnetic field and the electron density are
perfectly homogeneous, there are oscillations between
the electromagnetic and gravitational plane waves, over an
oscillation length given by\cite{RS}
\begin{equation}
\ell_{\rm osc}=%
{\displaystyle {4\pi \omega \over \omega _{pl}^2}}
\label{losc}
\end{equation}
where $\omega $ is the angular frequency of the electromagnetic wave.
Indeed, the photon-graviton conversion probability, for either
$\parallel$ or $\perp$ polarization, becomes\cite{RS}
\begin{equation}
P={4\over\pi} GB^2%
\ell_{\rm osc}^2
\sin ^2%
{\displaystyle {\pi L \overwithdelims() \ell_{\rm osc} }}
\sin ^2\theta \label{P3}
\end{equation}
Of course, if $L\ll l_{\rm osc}$ this expression reduces to
eq. (\ref{P2}), as if there were no free electrons.
Otherwise, the conversion probability does not accumulate
over distances larger then $\ell_{\rm osc}$.
The situation is different when there are processes, such as
inhomogeneities in the electron-density,
that affect the coherence of the photon-graviton oscillations.
In this case a fraction $f$ of the photons that mixed into gravitons
within one oscillation
does not oscillate back into photons.
Adding the effect over $N=L/l_{\rm osc}$ independent regions
the photon-graviton conversion probability
over a distance $L$ becomes
\begin{equation}
P\simeq f GB^2L\ell _{\rm osc} \sin ^2 \theta \label{PH}
\end{equation}
The precise value of the factor $f$ is model-dependent.
See for instance ref. \cite{CG} for an estimate of these effects
in the interstellar medium
in our galaxy. For our purposes it will be enough to consider
its largest possible value, $f\simeq 1$.
We shall see that even in this most favourable case, photon-graviton
conversion in a primordial magnetic field has negligible effects.
\section{CMB anisotropy induced by photon-graviton conversion}
Photon-graviton conversion in a cosmological magnetic field
induces anisotropies in the CMB due to the angular dependence of
the conversion probability \cite{Zeldovich,Chen,Magueijo}.
Ignoring plasma effects, the conversion probability is frequency-independent,
and thus preserves the black-body CMB spectrum.
Using eq. (\ref{P2}) we see that a
cosmological magnetic field of present value $B(t_\circ)$
assumed homogeneous over a scale of order the present Hubble
radius, $H_\circ^{-1}$, would induce (if plasma effects were negligible)
a large angular scale anisotropy of order
\begin{equation}
{\Delta T \over T}
\simeq 5\times 10^{-6}%
{\displaystyle {B(t_{\circ }) \overwithdelims() 1.3\times 10^{-6}
{\rm Gauss}}}^2%
{\displaystyle {h \overwithdelims() 0.5}}
^{-2}
\end{equation}
where $H_{\circ
}=100\ h\ {\rm km}\ {\rm seg}^{-1}{\rm Mpc}^{-1}.$
The anisotropy induced at present times by a cosmological magnetic field
of about $10^{-9}$ Gauss
would thus be negligible, about six orders of magnitude smaller
than the observed quadrupole CMB anisotropy \cite{cobe}, even
in the absence of plasma effects.
A cosmological magnetic field of present value $B(t_\circ)$
is expected to have been larger in the past, by a factor
$B(t)=B(t_\circ) a^2(t_\circ)/a^2(t)$, where $a$ is the Robertson-Walker
scale factor, due to flux conservation \cite{Breviews}.
Photon-graviton conversion would thus have had larger effects in the past,
if the magnetic field was always homogenous over a Hubble radius,
since the factor $(BH^{-1})^2$ scales with redshift as $1+z$
in a matter-dominated universe.
Anisotropies induced before decoupling, however, are quickly erased
by Thomson scattering during the period of tight coupling
between photons, electrons and baryons. The largest effect would thus
arise right around decoupling.
The anisotropy induced around
the time of decoupling of matter and radiation ($t=t_*$),
on angular scales of order the size of the horizon at
decoupling, which corresponds to about one degree on our sky is,
neglecting plasma effects
\begin{equation}
{\Delta T \over T} \approx 10^{-5}%
{\displaystyle {B(t_{*}) \overwithdelims() 0.04 {\rm Gauss}}}^2%
{\displaystyle {h \overwithdelims() 0.5}}
^{-2}%
{\displaystyle {1+z_{*} \overwithdelims() 1100}}
^{-3}\ .
\end{equation}
$10^{-5}$
is the order of the observed anisotropy on angular scales of
about one degree\cite{1deg}. The present value of
a primordial magnetic field which had a strength $B(t_*)\simeq 0.04$ Gauss
at decoupling is
$B(t_{\circ })\simeq 3\times 10^{-8}$ Gauss.
We thus conclude, as in refs. \cite{Chen,Magueijo},
that if plasma effects were negligible
the conversion between photons and gravitons in a primordial magnetic field
around the time of decoupling of matter and radiation
could have non-negligible effects upon the isotropy of the CMB.
Plasma effects, however, are not negligible. Even in the most favourable
case, with $f\approx 1$ in eq. (\ref{PH}),
the conversion probability drops precipitously.
Consider the Universe right after decoupling of
matter and radiation. The number density of free electrons is
\begin{equation}
n_e(t\approx t_*)=0.15%
{\displaystyle {\Omega _bh^2 \overwithdelims() 0.01}}
{\displaystyle {1+z_{*} \overwithdelims() 1100}}^3%
{\displaystyle {X \overwithdelims() 10^{-3}}} {\rm cm}^{-3}
\end{equation}
where $X$ is the fractional residual ionization and $\Omega_b$ is the baryon
energy-density in units of the critical density.
Notice that because the oscillation length depends on the photon
frequency, so does the conversion probability. Photon-graviton conversion does
not preserve the black body spectrum of the CMB. We still write, for
comparison purposes, the anisotropy in the CMB intensity induced by
photon-graviton conversion in terms of an effective temperature
anisotropy, at a given frequency. The anisotropy induced by a magnetic
field homogeneous over a Hubble radius at decoupling would be at most,
including plasma effects
\begin{eqnarray}
{\Delta T \over T}
&\lesssim &GB^2(t_{*})H^{-1}(t_{*})\ell _{\rm osc}(t_*) \nonumber
\\
\ &\simeq &10^{-5}%
{\displaystyle {B(t_{*}) \overwithdelims() 14 {\rm Gauss}}}^2%
{\displaystyle {h \overwithdelims() 0.5}}^{-1}%
{\displaystyle {\nu (t_{\circ }) \overwithdelims() 90 {\rm GHz}}}\
\nonumber \\
\end{eqnarray}
Here $\nu (t_\circ)$ is the present value of the CMB photons'
frequency. A magnetic field of strength 14 Gauss at decoupling
would have a strength of order $10^{-5}$ Gauss today.
A realistic value, smaller than $10^{-9}$ Gauss today, would
thus induce anisotropies through photon-graviton conversion
at least eight orders of magnitude smaller than those observed.
We have already seen that the large angular scale anisotropy in the CMB
induced at present times by a field of $10^{-9}$ Gauss would be negligible
even in the absence of free electrons. A small electron-density would
reduce the effect of photon-graviton conversion even further.
The present value of the free electron density in the intergalactic
medium is not known with certainty. The Gunn-Peterson limit on
the abundance of neutral Hydrogen \cite{Steidel}
however,
suggests that most of the intergalactic material is ionized.
A probably realistic figure for the present electron number density
is thus $n_e\approx 10^{-7} {\rm cm}^{-3}$.
The anisotropy induced today on large angular scales
by a cosmological magnetic field would thus be
\begin{eqnarray}
{\Delta T \over T}
&\lesssim &5\times 10^{-6}%
{\displaystyle {B(t_{\circ }) \overwithdelims() 0.006 {\rm Gauss}}}
^2 \nonumber \\
&&\ \ \times
{\displaystyle {\nu \overwithdelims() 90 {\rm GHz}}}
{\displaystyle {10^{-7}{\rm cm}^{-3} \overwithdelims() n_e}}
{\displaystyle {h \overwithdelims() 0.5}}
^{-1}
\end{eqnarray}
Clearly, the effect of a cosmological magnetic field of
current value around $10^{-9}$ Gauss would be completely negligible.
\section{Conclusions\label{conclus}}
Photon-graviton conversion induced by a cosmological magnetic field
of present strength $10^{-9}$ Gauss or smaller has negligible
effects upon the isotropy of the cosmic microwave background.
The effect would have been much larger in the absence of free electrons.
Plasma effects, however, make the characteristic length for photon-graviton
oscillations much smaller than the Hubble radius, preventing the conversion
probability to grow quadratically with distance over such large scales.
We have also seen that photon-graviton conversion does not induce linear
polarization upon the cosmic microwave background,
contrary to the case of photon-pseudoscalar
conversion \cite{HS,CG}.
The probability of photon-graviton conversion in a magnetic field
in the presence of free electrons depends on the photon frequency.
In principle, one could also attempt to detect the effects of a primordial
magnetic field through departures from the black body spectrum in the CMB.
Since thermalization processes are effective only at redshifts larger
than about $z\approx 10^6$, one could test in this way for the presence of
a primordial magnetic fields at very early times. One can show, however,
that the departure from the black body spectrum is also negligibly
small. For a present field of $10^{-9}$ Gauss at
CMB frequencies of order $10^3$ GHz,
the fractional departure from a black body spectrum is
at most of order $10^{-12}$, induced right after decoupling.
At earlier times, with matter fully ionized,
the large free electron density makes the effect
even smaller, of order $10^{-16}$ at the time of matter-radiation
equality, $z_{\rm eq}\approx 10^4$. At higher redshifts
the factor $B^2H^{-1}\ell_{\rm osc}$ remains constant.
We should mention that a primordial magnetic field may still have
significant effects upon the isotropy of the cosmic microwave background
by driving an anisotropic expansion of the Universe \cite{Zeldovich}.
Its direct effect through
photon-graviton conversion, however, is negligible due to plasma effects.
\section*{Acknowledgements}
This work was partially supported by grants from Universidad de Buenos
Aires and Fundaci\'on Antorchas. D.H. is also supported by CONICET.
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\section*{Introduction}
Two experimentally verified theories
describe at present the physical world: quantum
field theory and general relativity. Both have
been extremely successful in their tested
ranges of applicability.
Quantum field theory, particularly implemented
in the so-called standard model, describes the
types and behavior of elementary particles as
measured in accelerator experiments and as
experienced by everyday contact with matter.
General relativity is concerned with the
classical spacetime in which quantum field
theory takes place. This spacetime, a
four-dimensional Lorentzian manifold with events
being its points, can have a complicated
structure both locally and globally and can be
influenced by the presence of classically
understood matter.
Both quantum field theory as applied in the
standard model and general relativity indicate
intrinsically that they cannot be valid under
very extreme circumstances. But moreover they are
not fully compatible even under rather usual
conditions with the problem being that
matter is described by a quantum theory
whereas spacetime interacting with the
quantum matter is described classically.
For all these reasons it is believed that it
should be possible to find a more advanced theory
in which also gravity is quantized and in
which both the compatibility problem for
general relativity and quantum field theory and
their internal problems are resolved.
Such theories have already been proposed,
most notably string theory \cite{Witten96}.
While the internal consistency of such a theory
turns out to be a difficult problem, another
issue arises once the theory is formulated. How
can one relate it to the physical world? The
interpretational side of a physical theory has
at certain points of history not been trivial
but here the question stands with a new
urgency. Practically all measurements that are
performed in experimental physics use
implicitly the notion of classical spacetime.
The measurements of positions and times
play a dominant role, and there is a clear
practical understanding of them. But in a
theory where gravity is quantized, there is no
classical spacetime in its postulates. The
obvious conclusion is that unless one is able to
recover from such a theory classical spacetime,
at least in an approximative sense, the theory
may be a nice piece of mathematics but does
not make contact with the physical world and is
as a physical theory rather useless.
This work is concerned with providing a
tool for recovering classical spacetime from an
advanced theory and is thus aimed at the
interpretation of a quantum theory of gravity.
It is assumed here that such a theory can first
be simplified to an effective low energy
theory which will look like a usual quantum
field theory but without having specified
spacetime yet. In such a situation no
Lorentzian manifold is present, but there are
many structures that contain what one can call
spectral information. It comes from the structure
of the algebra of observables of the effective
theory and eventually from structures
like the decoherence functional of
generalized quantum mechanics. The problem is
thus to describe classical spacetime by spectral
data.
There is a theory doing just that for
Riemannian spaces: A. Connes'
noncommutative geometry
\cite{Connes94,Connes95}. Noncommutative
geometry describes classical spaces by
commutative algebras of functions on them
together with some additional structures on
them. It is actually more powerful than is
needed here: Noncommutative geometry is able
to deal even with noncommutative algebras not
corresponding to any classical space. In an
indirect way this fact is actually useful even
in the present situation where only a classical
space is wished for: The understanding of the
general noncommutative case is more direct
in separating out which concepts are of
fundamental importance and which are from a
broader perspective just particularities.
One structure recognized in this way as being
important, the spectral triple, will be
especially useful in the considerations presented.
So in a more specific view the problem
is to discuss how noncommutative geometry can
be used to describe spacetime in the
particular commutative case. Unfortunately,
the present mathematical framework is able
to deal only with spaces of Riemannian type,
having a nonnegative distance between any two
points. There it is very efficient in
using spectral data: Practically all the
geometric information is contained in just a few
relatively simple structures. The question is
whether the same is possible in the Lorentzian
case.
The answer to this question is the main topic
and result of this work. Compared to Riemannian
spectral geometry there is a new phenomenon
recognized: causal relationships. Inspired by
the thorough discussion of causality in quantum
field theory
\cite{Haag-Kastler,Streater-Wightman,Haag,Yurtsever1,Yurtsever2}, its place in the framework of
noncommutative geometry is found. With this
understanding it is possible to show that
again, as in Riemannian geometry, the
spectral data exhibit a remarkable efficiency
in the description of Lorentzian spaces, at least
if they are globally hyperbolic which will be
assumed throughout.
This gives hope that the adopted approach may
turn out to be actually useful in the way it
is wished to be useful from the physical
context. Several remarks and conjectures on
applications in physical interpretations are
put forward. Many technical questions are left
open for further considerations but have now a
clearer formulation and context and can
thus be attacked gradually.
The work is organized in the following way:
Section \ref{FieldSection} discusses a free Weyl
spinor field and the fermionic quantization of
its covariant phase space.
The local and causal structures of this field
theory are emphasized in Section
\ref{LocalObservables}.
Section \ref{ConnesSpectralTriple} reviews
briefly the spectral triple of Connes' spectral
geometry.
A na\"{\i}ve spectral description of
Lorentzian globally hyperbolic manifolds is
given in Section \ref{SNG}.
In Section \ref{LorentzSpectralData} the
information contained in causal relationships is
discussed and used to obtain a rather compact
description of spacetime. The view obtained is
the main result of this work.
This is summarized in the conclusion.
\section{Fermionic quantization of free Weyl
spinor fields} \label{FieldSection}
The field theory considered in this paper will be
that of a fermionic Weyl spinor field. This choice
is maybe not overly surprising in view of the
role played by such fields in the standard model
of particle physics. The primary motivation is,
however, the importance of spinor fields in
spectral geometry as will become clear in
Section \ref{ConnesSpectralTriple}.
The covariant phase space $\mathcal{S}$ of a
classical free Weyl spinor field $\psi$ is the
linear space of solutions of the equation of
motion following from the action $S$,
\begin{align}
S[ \psi ] &= Re \int_{\Omega }{\bar{\psi} D \psi
d\Omega},\label{Action}\\
\intertext{i.e., the Dirac equation}
D\psi &= 0.\label{DiracEquation}
\end{align}
Here $ D$ is the Dirac operator, $\bar{\psi }$ is
the Dirac adjoint of $\psi $
\cite{Budinich-Trautman} and $\Omega$ is an
arbitrarily chosen region of spacetime.
If the spacetime manifold $M$ is assumed to be
globally hyperbolic (i.e., is topologically
$\Sigma \times \mathbb{R} $, sliced by spacelike
Cauchy surfaces diffeomorphic to the
3-dimensional manifold $\Sigma $ \cite{Wald})
then there is a Hermitean inner product $s$ on
the space of solutions $\mathcal{S}$ of the Dirac
equation expressed as an integral over a
spacelike Cauchy surface $\Sigma $,
\begin{align}
\bar{\phi }\circ s \circ \psi = \int_{\Sigma
}{\bar{\phi }{\gamma }_{\mu}\psi
{d\Sigma }^{\mu }}\label{DiracProduct}
\end{align} Here ${d\Sigma }^{\mu }$ is the
future directed hypersurface element induced from
the spacetime volume element $d\Omega $. In order
for $s$ to be a Hermitean inner product on the
space of solutions $\mathcal{S}$, it has to be
independent of the choice of $\Sigma $. Indeed,
given two spacelike Cauchy hypersurfaces
${\Sigma}_{1}$ and ${\Sigma}_{2}$, the difference
in the corresponding Hermitean inner products can
be by Stokes' theorem expressed by a spacetime
integral over the region $\Omega $ enclosed by
${\Sigma}_{1}$ and ${\Sigma}_{2}$, vanishing in
consequence of the equation of motion
\ref{DiracEquation}:
\begin{align}
\int_{{\Sigma }_{1} }{\bar{\phi }{\gamma
}_{\mu}\psi {d\Sigma }^{\mu }}-\int_{{\Sigma
}_{1} }{\bar{\phi }{\gamma }_{\mu}\psi {d\Sigma
}^{\mu }}= \int_{\Omega }{\left( \bar{\phi} D
\psi - \overline{ D \phi}\psi\right) d\Omega }
\end{align}
The real part $\mu $ of the Hermitean inner
product $s$,
\begin{align}
\bar{\phi }\circ \mu \circ \psi = Re \int_{\Sigma }
{\bar{\phi }{\gamma }_{\mu}\psi
{d\Sigma }^{\mu }}
\end{align}
is a real bilinear symmetric inner product on the phase
space $\mathcal{S}$. Its inverse is the fermionic causal
Green's function ${\tilde{G}}_{F}$.
It can be shown \cite{DeWitt65} that the
fermionic causal Green's function
${\tilde{G}}_{F}$ has the meaning of the Poisson
bracket $\{\bullet , \bullet\} $ of classical
mechanics \cite{DeWitt83}:
\begin{align}
\{\bullet , \bullet\} = {\tilde{G}}_{F} =
{\mu}^{-1}
\end{align}
Once the classical description of a system (e.g.
a field) is known, one can make an educated guess
of what the correct quantum theory is, i.e., one
can quantize the field theory. In principle
there are two rather different ways to do that,
namely quantization by path integrals
\cite{Feynman-Hibbs} and canonical quantization
(see, e.g.\cite{DeWitt65,DeWitt83,Woodhouse}).
Here the latter is chosen, since it leads more
directly to an algebraic setting used in
noncommutative geometry.
In fermionic canonical quantization, chosen in
agreement with the spin-statistics
theorem\cite{Streater-Wightman}, one starts with
the classical phase space $\mathcal{S}$ equipped
with the symmetric inner product $\mu $. The
functions on the classical phase space
$\mathcal{S}$, the classical observables, are
then replaced by elements in a noncommutative
algebra, the algebra of observables following
some rules which turned out to be useful in
particular cases. The rules are as follows:
First, a special set $F(\mathcal{S})$ of function
on the phase space has to be selected. The set
$F(\mathcal{S})$ of chosen classical observables
should be closed under taking the Poisson bracket
$\{\bullet,\bullet\}$, i.e.,
\begin{align}
\{ a, b\} \in F(\mathcal{S}) &&\text{ for $a,
b\in F(\mathcal{S})$},
\end{align}
Second, a linear map $\hat{\psi}$ into a
complex associative algebra $\mathbf{A}$
should be given,
\begin{align}
\hat{\psi}:F(\mathcal{S}) &\rightarrow
\mathbf{A}.
\end{align}
The map $\hat{\psi}$ should satisfy a commutation
relation replacing the Poisson bracket by a
commutator:
\begin{align}
\hat{\psi}(a)\hat{\psi}(b) + \hat{\psi}(b)\hat{\psi}(a) =
i\hat{\psi}(\{ a, b \} ) \text{ for all $a, b\in F(\mathcal{S})$},\label{AnticommutationRelation}
\end{align}
and its image $\hat{\psi}(F(\mathcal{S}))$ should
generate the algebra $\mathbf{A}$.
\begin{note}
If $F(\mathcal{S})$ contains the constant
functions on $\mathcal{S}$ (which have vanishing
Poisson brackets with all other functions on
$\mathcal{S}$), then their image under the
mapping $\hat{\psi}$ must be in the centre of the
algebra $\mathbf{A}$, and if $\mathbf{A}$ is
central then the image of constant functions is
proportional to the unit $\mathbf{1}$ in the
algebra. A not very surprising addition to the
quantization rules then usually is the
requirement
\begin{align}
\hat{\psi}(k) = k\mathbf{1} &&\text{for all constant
functions $k$ on $\mathcal{S}$}.
\end{align}
\end{note}
In general, one of the difficulties of these
rules is the potentially complicated
anticommutation relation
(\ref{AnticommutationRelation}), and another is
the choice of $F(\mathcal{S})$. Obvious choices,
like the space of all continuous functions on
$\mathcal{S}$, are plagued by inconsistencies or
by giving an algebra that is far too big compared
with the one that gives a quantum theory in
agreement experiment. To deal with this
situation, additional information is usually
necessary (see e.g. \cite{Woodhouse}), and even
then it is a difficult problem. The situation
radically simplifies for a free system (i.e. one
with a linear phase space $\mathcal{S}$) as the
one considered here. The correct choice of
$F(\mathcal{S})$ is then the space of linear
observables.
One can define the field operator $\Psi (f)$ for
a classical solution $f \in \mathcal{S}$
\begin{align}
\Psi (f) =\hat{\psi}(\mu \circ
f),\label{SolutionField}
\end{align}
and write the anticommutation relation
(\ref{AnticommutationRelation}) in the form
\begin{align}\label{SolutionAnticommutationRelation}
\Psi (f)\Psi (g)-\Psi (g)\Psi (f)=i
(f\circ\mu\circ g) \mathbf{1} &&\text{for $f,g\in
\mathcal{S}$.}
\end{align}
The ${C}^{\ast}$-algebra of observables of the
quantum field generated from this anticommutation
relation is unique and independent of a
completion of $\mathcal{S}$
\cite{Plymen-Robinson,Bratteli-Robinson2}. It has
a unique minimal enveloping von Neumann algebra
\cite{Plymen-Robinson} having, up to unitary
isomorphism, a unique regular irreducible
representation by bounded operators in a Hilbert
space. There is no information whatsoever in this
algebra about the smooth structure of spacetime.
\section{Local algebras of
observables}\label{LocalObservables}
If the ${C}^{\ast}$-algebra of observables is
considered by itself, without reference to its
origin, then it is sufficient to express the
evolution of the field by automorphisms and the
space of states by normed positive linear
functionals (see \cite{Bratteli-Robinson1}), but
then the physical interpretation is completely
lost.
A somewhat similar loss of interpretation can be
observed if a classical system is judged on the
basis of its phase space only, where canonical
transformations can rather arbitrarily change the
meaning of coordinates and momenta. It is
possible to argue that, e.g., the topology of the
phase space is specific to the system, but this
is by no means sufficient to give a complete
description if there actually is a fundamental
distinction between coordinates and momenta.
As mentioned in Section \ref{FieldSection}, the
algebra of observables does in this case not
contain any information about spacetime.
Some structure has thus to be given to the
algebra of observables of a quantum field in
order to enable one to give its physical
interpretation. One could, of course, just
remember the whole construction of the algebra of
observables, starting with the classical field.
In a path integral approach this would not be so
bad, since classical histories are part of that
framework, but in an algebraic approach to
quantum field theory, where the classical field
has just the position of an effective
approximation, this is definitely not what one
would wish to do. The widely accepted solution
is to give the algebra of observables the
structure of a local algebra
\cite{Haag,Bratteli-Robinson1}. The idea is to
associate with each region of spacetime $\Omega$
a subalgebra $\mathbf{A}(\Omega)$ of the algebra
$\mathbf{A}$ of observables. Thus one obtains a
set of subalgebras indexed (not necessarily
unambiguously) by the set $I$ of open subsets of
spacetime.
For many technical purposes it is not necessary
to keep the reference to spacetime, and only some
properties of the index set $I$ are extracted and
required. This is the case of the definition of a
quasi-local algebra
\cite{Haag,Bratteli-Robinson1}. However, since
here interpretation is the main concern, the
full link to spacetime will be required
\cite{Yurtsever1,Yurtsever2}.
\begin{definition}
A ${C}^{\ast}$-algebra $\mathbf{A}$ together with
a spacetime manifold $M$ is local if the following
three conditions all hold:
\begin{enumerate}
\item{For each open subset $\Omega$ of $M$ there
is a central ${C}^{\ast}$-algebra
$\mathbf{A}(\Omega)$, with
$\mathbf{A}(\emptyset)=\mathbb{C}$, and
$\mathbf{A}(M)= \mathbf{A}$.}
\item{For any collection $\{ {\Omega}_{i}\}$ of
open subsets of $M$ one has
\begin{align}
\mathbf{A}\left( \cup_{i} {\Omega}_{i}\right) =
\overline{\langle\cup_{i} \mathbf{A}\left(
{\Omega}_{i}\right)\rangle}\notag
\end{align}
(On the right hand side of this equation is the
closure of the algebraic envelope
$\langle\cup_{i} \mathbf{A}\left(
{\Omega}_{i}\right)\rangle $ of $\cup_{i}
\mathbf{A}\left( {\Omega}_{i}\right) $.)
}
\item{If the regions ${\Omega}_{1}$,
${\Omega}_{2}$ are not in causal contact, then
the corresponding algebras $\mathbf{A}\left(
{\Omega}_{1}\right)$, $\mathbf{A}\left(
{\Omega}_{2}\right)$ commute in the Bose case and
graded-commute in the Fermi case.}
\end{enumerate}
\end{definition}
\begin{example}
The quantized Weyl spinor field can be given the
structure of a local algebra. The Green's
function ${\tilde{G}}_{F}$ of the field can be
used to produce from any smooth density $\nu $ on
the spacetime manifold $M$ a solution $f$:
\begin{align}
{f}^{p} = {({\tilde{G}}_{F})}^{pq}
{\nu}_{q}\label{MeasureFieldFermi}
\end{align}
and to each solution $f$ one can by
(\ref{SolutionField}) associate a quantum
observable $\Psi (f)$. Given a subset $\Omega$ of
spacetime, the algebra $\mathbf{A}(\Omega)$ can
be then generated by densities with support in
$\Omega$. If the supports of two measures ${\nu
}_{1}$, ${\nu }_{2}$ are not causally connected,
then the corresponding classical solutions
${f}_{1}$, ${f}_{2}$ can be checked to have a
vanishing product ${f}_{1}\circ\mu\circ
{f}_{2}$, and the corresponding quantum
observables $\Psi ({f}_{1})$, $\Psi ({f}_{2})$
thus anticommute.
\end{example}
A pleasant feature of the local algebra
structure is that the
${C}^{\ast}$-subalgebras
$\mathbf{A}(\Omega)$ (with $\Omega\in M$)
of $\mathbf{A}$ are actually sufficient
to reconstruct the spacetime $M$ as a
topological space and to determine its
causal structure, as observed by
U.Yurtsever \cite{Yurtsever1,Yurtsever2}.
\section{Connes' spectral triple}\label{ConnesSpectralTriple}
A geometric space may be described by its set of
points with some additional structures, or,
alternatively, by the algebra of functions on it,
again with some additional structures. The first
point of view is the one of classical geometry.
The second may be taken as a starting point for a
far more general and powerful theory, A. Connes'
noncommutative geometry \cite{Connes94}, and is
adopted here. In particular, a space can be
encoded in the form of a spectral triple
\cite{Connes95}.
\begin{definition}\label{SpectralTriple}
A {\em spectral triple} $(\mathbf{A},\mathcal{H},D)$ is given
by an involutive algebra of operators $\mathbf{A}$ in a Hilbert
space $\mathcal{H}$ and a selfadjoint operator
$D={D}_{\ast}$ in $\mathcal{H}$ such that
\begin{enumerate}
\item{The resolvent ${(D-\lambda)}^{-1}, \>
\lambda\not\in\mathbb{R}$, of $D$ is compact}
\item{The commutators $[D,a]=Da-aD$ are bounded,
for any $a\in\mathbf{A}$}
\end{enumerate}
The triple is said to be {\em even} if there is a
hermitean grading operator $\gamma $ on the
Hilbert space $\mathcal{H}$ (i.e.
${\gamma}^{\ast}=\gamma,\>
{\gamma}^{2}=\mathbf{1})$ such that
\begin{align}
\gamma a &= a \gamma &\text{ for all $a\in \mathbf{A}$}\\
\gamma D &= - D \gamma &
\end{align}
Otherwise the triple is called {\em odd}
\end{definition}
\begin{note}
This section is only concerned with introducing
the spectral triple and mentioning its properties
to be used in the applications. From that it is
not fully clear why one should be interested in
exactly this kind of structure, so some
motivation is clearly missing here. See however
\cite{Connes94,Connes95} for the deep and solid
structure of noncommutative geometry that is
supporting the spectral triple.
\end{note}
The following example is of great importance.
\begin{example}\label{DiracSpectralTriple}
On a compact Riemannian spin manifold $M$ there
is canonically the following spectral triple
$({C}^{\infty }(M) ,{L}^{2}(M,S),D)$, the Dirac
triple \cite{Connes94}, \cite{Connes95}. Here
${C}^{\infty }(M) $ is the commutative algebra of
smooth complex functions on $M$, ${L}^{2}(M,S)$
is the Hilbert space of square integrable
sections of the complex spinor bundle $S$ over
$M$ and D is the Dirac operator. The algebra of
functions ${C}^{\infty }(M) $ acts on the
Hilbert space ${L}^{2}(M,S)$ by pointwise
multiplication
\begin{align}
(f\psi ) (p) &= f(p)\psi (p) &\text{for all
$f\in {C}^{\infty }(M)
,\psi \in {L}^{2}(M,S), p\in M$}\\
\intertext{and the commutator with the Dirac
operator $D$ with a function $f$ is}
[D,f]&=\gamma df &\text{for $f\in {C}^{\infty
}(M) $.}
\end{align}
$\gamma $ is the Clifford map from the cotangent
bundle into operators on ${L}^{2}(M,S)$.
\end{example}
In Example \ref{DiracSpectralTriple} the algebra
was taken to be ${C}^{\infty }(M)$. Such a choice
contains a lot of information and is actually not
necessary. In the definition of the Dirac
spectral triple it is sufficient to take instead
of ${C}^{\infty }(M)$ any algebra $\mathbf{A}$
that has the same weak closure (double commutant)
${\mathbf{A}}^{''}$ as has ${C}^{\infty }(M)$.
Such an algebra does not necessarily contain any
information about the topology or differential
structure of $M$ whatsoever. From $\mathbf{A}$
alone only $M$ as a set of points can be obtained
as the spectrum of $\mathbf{A}$. The rest,
however, can then be recovered from the structure
of the spectral triple including the notion of
smooth functions and Lipschitz functions.
Lipschitz functions with Lipschitz constant $1$
can then be used to define a distance function
$d$ on $M$. This means that a Riemannian spin
manifold can be replaced by a spectral triple
without the loss of any information about it. The
facts are summarized in Proposition
\ref{DiracSpectralTripleProposition} (see
\cite{Connes95}).
\begin{proposition}\label{DiracSpectralTripleProposition}
Let $(\mathbf{A},{L}^{2}(M,S),D)$ be the Dirac
spectral triple associated to a compact
Riemannian spin manifold M. Then the compact
space $M$ is the spectrum of the commutative
${C}^{\ast}$-algebra norm closure of
\begin{align}
{\mathbf{A}}_{B}&= \{ a \in {\mathbf{A}}^{''}\mid [D,a]
\text{ bounded}\} &\\
\intertext{while the geodesic distance $d$ on $M$
is given by}
d(p,q) &=\sup{\{ \mid f(p)-f(q)\mid ;
f\in {\mathbf{A}}_{B}, \parallel [D,f]\parallel\leq 1 \}
}\label{DistanceFunction}
\end{align}
\end{proposition}
It is now in question whether one can reconstruct
from a spectral triple a manifold if one is not
assured that the spectral triple actually comes
from a manifold. With some additional conditions
it will certainly be possible to prove in the
future a theorem in this direction. One helpful
tool for this purpose is a real structure $J$ on
the spectral triple \cite{Connes95},
\cite{Connes96a}.
\begin{example}
In the case of the Dirac spectral triple of
Example \ref{DiracSpectralTriple} a real
structure is given by the charge conjugation
composed with complex conjugation (see
\cite{Budinich-Trautman}).
\end{example}
Before giving its general definition it should be
mentioned that for simply connected spaces the
real structure ensures that the spectrum of a
spectral triple will have the homotopy type of a
closed manifold \cite{Connes95},
\cite{Connes96a}. In addition to that, its
dimension is governed by the spectrum of the
Dirac operator \cite{Gilkey}. So a theorem
examining which commutative spectral triples are
classical Riemannian manifolds is not out of
sight. The considerations of the next sections
would be best motivated by such a theorem but
making use of it is at this
point probably premature.
\begin{definition}
A real structure $J$ on the spectral triple
$(\mathbf{A},\mathcal{H})$ is an antilinear
isometry $J$
\begin{align}
J:\mathcal{H}&\rightarrow \mathcal{H}&\\
\intertext{ such that}
Ja{J}^{-1} &= {a}^{\ast} &\text{for all
$a\in\mathbf{A}$}\\
{J}^{2} &= \epsilon &\\
JD &= {\epsilon}^{'}DJ &\\
J\Gamma &= {\epsilon }^{''}\Gamma J
\end{align}
where the signs $\epsilon, {\epsilon}^{'},
{\epsilon}^{''}\in \{ -1, +1\} $ are given by the following
table with $\nu$ being the dimension of the space $mod\> 8$:
\begin{equation}\label{RealStructureTable}
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
\nu & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
\epsilon & 1 & 1 &-1 &-1 &-1 &-1 & 1 & 1\\
\hline
{\epsilon}^{'} & 1 &-1 & 1 & 1 & 1 &-1 & 1 & 1\\
\hline
{\epsilon}^{''} & 1 & &-1 & & 1 & &-1 & \\
\hline
\end{array}
\end{equation}
\end{definition}
\begin{note}
The sign ${\epsilon}^{''}$ in Table
(\ref{RealStructureTable}) is shown for even
dimensions only, since for Riemannian spin
manifolds only in that case the grading
(helicity) operator ${\Gamma}$ preserves the
irreducible spin representation and has thus a
good meaning in it. In the odd case it is assumed
that only one of the two irreducible
representations is chosen and since $\Gamma$
switches between the two irreducible
representations it has no meaning just in one of
them. More details on spinors can be found in
\cite{Budinich-Trautman}. Also the periodicity
$mod\> 8$ of Table (\ref{RealStructureTable}), a
manifestations of the spinorial chessboard is
explained there.
\end{note}
\section{Spacetime in spectral
geometry}\label{SNG}
Here a Lorentzian globally hyperbolic spacetime
manifold will be characterized by spectral data.
This cannot be done directly by Connes' spectral
triple (see Definition \ref{SpectralTriple})
since it is well suited for the description of
generalized Riemannian spaces only. This is
obvious, e.g., from the distance function
(\ref{DistanceFunction}), which cannot be
negative. A simple idea to avoid this difficulty
is to foliate the spacetime $M$ by a family of
spacelike Cauchy slices ${\Sigma}_{t}$ with
$t\in\mathbb{R}$ a coordinate time (see Figure
\ref{Slicing}). Each hypersurface ${\Sigma}_{t}$
is then Riemannian and can be characterized by a
family of Dirac spectral triples $\left(
{L}^{\infty}({\Sigma}_{t}),
{L}^{2}({\Sigma}_{t},S), {D}_{t} \right)$ (see
Example \ref{DiracSpectralTriple} and Proposition
\ref{DiracSpectralTripleProposition}) together
with some additional information on how the
spacelike slices ${\Sigma}_{t}$ are related to
each other. In particular, the normal distance
between two infinitesimally close Cauchy surfaces
${\Sigma}_{t}$ is encoded by the lapse function
$N$ (see \cite{MTW} and Figure \ref{Slicing}).
The only further information needed is the
identification ${i}_{t}:{\Sigma}_{t}\rightarrow
{\Sigma}_{0}$ of points which lie on the same
curve normal to the hypersurfaces. This can be
established in the spectral data by specifying an
automorphism
${i}_{t}^{\ast}:{L}^{\infty}({\Sigma}_{0})\rightarrow
{L}^{\infty}({\Sigma}_{t})$
\begin{figure}\label{Slicing}
\epsfxsize= 5 in
\epsfbox{fig1.ps}
\caption{A Cauchy foliation. The globally
hyperbolic manifold $M$ can be sliced by
spacelike Cauchy surfaces ${\Sigma}_{t}$. Each of
them can be characterized by a Dirac spectral
triple $\left( {L}^{\infty}({\Sigma}_{t}),
{L}^{2}({\Sigma}_{t},S), {D}_{t} \right)$ with
${L}^{\infty}({\Sigma}_{t})$ being the algebra of
essentially bounded functions on ${\Sigma}_{t}$,
${L}^{2}({\Sigma}_{t},S)$ being the spinor bundle
over ${\Sigma}_{t}$ and ${D}_{t}$ being the Dirac
operator on ${\Sigma}_{t}$. The normal distance
between infinitesimally close Cauchy surfaces
${\Sigma}_{t}$, ${\Sigma}_{t+dt}$ is
characterized by the lapse function $N$ on
${\Sigma}_{t}$. $N$ can be thought of as an
element in the algebra
${L}^{\infty}({\Sigma}_{t}) = {\left(
{C}^{\infty}({\Sigma}_{t}) \right)}^{''}$, the
double commutant of the algebra of smooth
functions.}
\end{figure}
Since the square integrable sections of the spin
bundles over the Cauchy surfaces ${\Sigma}_{t}$,
$t\in\mathbb{R}$ are valid Cauchy data for weak
solutions of the equation of motion of a Weyl
spinor field on M, there is a preferred
isomorphism between the spin bundles
${L}^{2}({\Sigma}_{t},S)$ and the space of
solutions $\mathcal{S}$ of Weyl spinors. This
means that all spectral triples can be understood
to share the same Hilbert space $\mathcal{S}$.
Summarizing, a globally hyperbolic spacetime can
be described using spectral data by
\begin{itemize}
\item{a family of spectral triples
$({\mathbf{C}}_{t}, \mathcal{S}, {D}_{t})$ with
${\mathbf{C}}_{t}$ a commutative algebra of
bounded operators on $\mathcal{S}$ and ${D}_{t}$
Hermitean (possibly unbounded) on $\mathcal{S}$}
\item{a family of lapse functions
${N}_{t}\in{\mathbf{C}}_{t}$ }
\item{an automorphism ${i}^{\ast}$ between any
two of the commutative algebras
${\mathbf{C}}_{t}$}
\end{itemize}
\begin{note}
Usually it is not required that the
identification of Cauchy surfaces has to be done
along normal lines. Then the deviation of of the
direction of identification from the normal one
has to be characterized by a shift vector field
$\vec{N}$ on the Cauchy surfaces \cite{MTW}. The
restriction to the case $\vec{N}=0$ here avoids
the necessity of a replacement of vector fields
by spectral concepts.
\end{note}
The above description agrees with \cite{Hawkins}
except that there the automorphism ${i}^{\ast}$
is omitted. That omission seems to make the
spectral data appear incomplete from the point of
view presented there.
It is now possible to describe the quantum field
theory for Weyl spinors on the spacetime
specified by the spectral data. Since the Hilbert
space $\mathcal{S}$ in the spectral data is taken
to be the space of classical solutions equipped
with the canonical Hermitean inner product
(\ref{DiracProduct}), this is entirely trivial:
The quantum field algebra of observables is just
the Clifford algebra generated from $\mathcal{S}$
by the anticommutation relation
(\ref{SolutionAnticommutationRelation}).
This completes the discussion of quantum field
theory on spacetime using a spectral approach but
not taking in account the causal structure
information present in the problem. This is a
natural place to reflect on the above with a few
comments.
{}From the point of view of the motivations, one
would wish to start from an algebra of quantum
observables, to specify the spectral data, and
then to construct, if possible, classical
spacetime. Such an approach will however bring
rather difficult problems: At least in the cases
where one hopes to obtain a spacetime that is a
topological or smooth manifold, one would wish to
have the one-parameter family ${\mathbf{C}}_{t}$
in some sense continuous or smooth. (It may be
viewed as a continuous or smooth algebra bundle
over $\mathbb{R}$). This is an important, but on
the other hand technical, issue. Instead of
discussing it satisfactorily, the treatment will
rely on the case studied here starting with a
classical spacetime, producing the space of
solutions $\mathcal{S}$ of the Weyl spinor field
on it and obtaining by quantization the field
algebra $\mathbf{A}$. Then all the facts can be
viewed backwards, starting with the field algebra
$\mathbf{A}$. This is clearly dishonest to the
motivations in using as its input what should be
abandoned in the first place: classical
spacetime. On the other hand this allows one to
go through all the way from the quantum algebra
to spacetime avoiding some, in general difficult,
arguments bridged by the particular features of
this not-so-elegant example. The result is then
an understanding of what is important, and with
this, one can then gradually face the technically
difficult points. This approach has worked so far
extremely well in noncommutative geometry. In
this context, the aim here is to gain an
understanding only, thus considering the example
as a valid approach.
For a view starting from the quantum field
algebra according to the above motivations, it
would also be desirable to have a deeper
justification of the introduced structures,
particularly for the family of operator algebras
${\mathbf{C}}_{t}$ and the family of operators
${D}_{t}$ on the space $\mathcal{S}$ generating
the algebra of observables $\mathbf{A}$. It will
be suggested here in the form of two conjectures
that this may eventually be possible.
\begin{conjecture}\label{1stConjecture}
Another way to look at the family of commutative
algebras ${\mathbf{C}}_{t}$ will be offered now.
For a given value of the parameter $t={t}_{0}$,
the algebra ${\mathbf{C}}_{{t}_{0}}$ splits the
space $\mathcal{S}$ into orthogonal subspaces by
spectral projections. On the quantum level this
means that the field algebra $\mathbf{A}$ is
given preferred mutually commuting subspaces. In
the case in which the Hilbert space is finite
dimensional, these spaces are complex one
dimensional. It is conjectured that this
structure is sufficient to determine a preferred
complete set of commuting projectors in the
algebra of observables $\mathbf{A}$ or eventually
in its (unique) minimal enveloping von Neumann
algebra. If that is the case, then the choice of
${\mathbf{C}}_{{t}_{0}}$ may be understood as the
choice of a set of histories in generalized
quantum mechanics
\cite{Hartle,Isham,Isham-Linden,Isham-Linden-Schreckenberg}.
This would to a large degree justify the
introduced structures from a very fundamental
point of view.
\end{conjecture}
\begin{conjecture}\label{2ndConjecture}
If Conjecture \ref{1stConjecture} is in some way
correct, then the family ${D}_{t}$ of Hermitean
operators on $\mathcal{S}$ can be recovered from
the decoherence functional of generalized quantum
mechanics on histories of the quantum field
$\mathbf{A}$.
\end{conjecture}
These conjectures are a topic of future research.
They are stated here only to show that what was
reached so far is really following the call of
the motivations put forward in the Introduction,
which would not be so easy to see otherwise.
\section{Spectral data and the causal structure
of spacetime.} \label{LorentzSpectralData}
The spectral data describing spacetime as
presented in the previous section are sufficient.
But they do not take into account the fact that
causal structure information is also stored in
the family of spectral triples in a way that was
not yet exploited.
To understand that, consider two spacelike Cauchy
surfaces ${\Sigma}_{0}$, ${\Sigma}_{1}$ on the
spacetime manifold (see Fig\-ure
\ref{CausalityFigure}). They are de\-scri\-bed by
the spec\-tral triples $({\mathbf{C}}_{0},
\mathcal{S}, {D}_{0})$, $({\mathbf{C}}_{1},
\mathcal{S}, {D}_{1})$. Given two points
${p}_{0}$, ${p}_{1}$ on these Cauchy surfaces
(${p}_{0}\in {\Sigma}_{0}$, ${p}_{1}\in
{\Sigma}_{1}$) it is now possible just to decide
whether they are in causal contact or not. If and
only if the points ${p}_{0}$, ${p}_{1}$ are not
in causal contact, the value of the Weyl spinor
field at the point ${p}_{0}$ cannot influence the
value of the field at the point ${p}_{1}$. In
more precise terms on can say that there exist
open neighborhoods $\mathcal{U}({p}_{0})$,
$\mathcal{U}({p}_{1})$ of the points ${p}_{0}$,
${p}_{1}$ in ${\Sigma}_{0}$, ${\Sigma}_{1}$ such
that any solution $\psi$ of the equation of
motion of the Weyl spinor field with Cauchy data
on ${\Sigma}_{0}$ supported in
$\mathcal{U}({p}_{0})$ has a vanishing inner
product with any solution $\phi$ with Cauchy
data on ${\Sigma}_{1}$ supported in
$\mathcal{U}({p}_{1})$. To identify solutions in
$\mathcal{S}$ which have Cauchy data on
${\Sigma}_{i}$ supported in a certain region
$\mathcal{U}({p}_{i})\subset {\Sigma}_{i} $ from
the spectral data is easy: they are just given as
elements of the ranges of the spectral projection
corresponding to $\mathcal{U}({p}_{i})$.
\begin{figure}\label{CausalityFigure}
\epsfxsize= 5 in
\epsfbox{fig2.ps}
\caption{Causal contact. Any solution $\psi$ with
Cauchy data on ${\Sigma}_{0}$ supported in
$\mathcal{U}({p}_{0})$ has a vanishing inner
product with any solution $\phi$ with Cauchy
data on ${\Sigma}_{1}$ supported in
$\mathcal{U}({p}_{1})$. The points ${p}_{0}$,
${p}_{1}$ are not causally connected.}
\end{figure}
\begin{note}
If one is willing to use generalized eigenvectors
then causal contact can be expressed in the
following way. A (generalized) solution with
Cauchy data on ${\Sigma}_{0}$ supported in the
point ${p}_{0}$ is a generalized eigenvector of
the algebra ${\mathbf{C}}_{0}$ satisfying
\begin{align}
a\psi = a({p}_{0}) \psi &&\text{for
$a\in{\mathbf{C}}_{0}$,}
\end{align}
with $a({p}_{0})$ being the value of the function
$a$ at the point ${p}_{0}$. The vector $\psi$ can
then be for briefness called an eigenvector of
point ${p}_{0}$. Then two points are not in
causal contact if and only if all their
eigenvectors are orthogonal.
\end{note}
One can now summarize:
\begin{observation}\label{Observation}
Using the family ${\mathbf{C}}_{t}$ of
commutative algebras represented on the Hilbert
space $\mathcal{S}$ of solutions, one can recover
spacetime as a set of points and find by the
above procedure which points are in causal
contact, using the Hermitean inner product on
$\mathcal{S}$.
\end{observation}
This observation is of central importance. Before
using it to reduce the spectral data necessary to
describe a Lorentzian spacetime, a two
connections will be made.
First, from the point of view of differential
equations it is not surprising that the Hermitean
inner product on $\mathcal{S}$ contains
information on the causal structure, since as
mentioned in Section \ref{FieldSection} the real
part of it is the inverse of the causal Green's
function.
Second, from the point of view of quantum field
theory the orthogonality of classical solutions
with Cauchy data locally supported around two
points ${p}_{0}$, ${p}_{1}$ has as its
consequence (or, if one wishes, as its origin)
the graded commutativity of the corresponding
${C}^{\ast}$-subalgebras of the local algebra
$\mathbf{A}$ of observables generated from
$\mathcal{S}$. This is the point where the notion
of causality makes contact with Section
\ref{LocalObservables} and with some of the
motivations for this work given in the
Introduction.
Now the consequences of Observation
\ref{Observation} will be discussed. First of
all, the family of spectral triples
$({\mathbf{C}}_{t}, \mathcal{S}, {D}_{t})$ of
Section \ref{ConnesSpectralTriple} contains
already all necessary information about spacetime
and no automorphism ${i}^{\ast}$ between the
algebras ${\mathbf{C}}_{t}$ and no lapse function
$N$ need to be specified. Indeed, by knowing the
geometry of the Cauchy surfaces ${\Sigma}_{t}$
corresponding to the spectral triples
$({\mathbf{C}}_{t}, \mathcal{S}, {D}_{t})$ and
the causal structure one can find the normal
identifications of points and the normal
distances between infinitesimally close Cauchy
surfaces (see Figure \ref{LightCone}).
\begin{figure}\label{LightCone}
\epsfxsize= 5 in
\epsfbox{fig3.ps}
\caption{The geometry of Cauchy surfaces, causal
contact and the geometry of spacetime. The point
${p}_{t}$ on ${\Sigma}_{t}$ has as its region of
causal contact on ${\Sigma}_{t+dt}$ the disk
$\mathcal{U}({p}_{t+dt})$ (including its bounding
sphere). The square of the radius of the sphere
is the negative of the square of the normal
spacetime distance between the Cauchy surfaces
${\Sigma}_{t}$, ${\Sigma}_{t+dt}$, and the center
${p}_{t+dt}$ of the sphere
$\mathcal{U}({p}_{t+dt})$ is the point reached by
the normal vector $n$ based in ${p}_{t}$. }
\end{figure}
Thus a large part of the spectral data can be
just left out, and the remaining family of
spectral triples gives now a quite efficient
description. But it is still considerably
redundant. To see this is not difficult: If the
metric information contained in the operators
${D}_{t}$ is omitted, then the conformal
structure of spacetime is still rigidly fixed.
But not all metrics are conformally related, and
thus the ${D}_{t}$ determining the metric on the
Cauchy surfaces cannot be chosen at will but have
to agree with the conformal structure. This means
that the spectral data of spacetime can be
further reduced. How this has to be done in a
useful way will be left for consideration in the
future. But even without that a conceptual result
is appearing: The spectral data describing a
Lorentzian manifold do so in a very efficient
way. This result based on Observation
\ref{Observation} is the main claim of this work.
\begin{note}
There is a way of giving less redundant spectral
data, if one is willing to lose metric
information and keep just the conformal structure
of spacetime. It is shown in \cite{Connes94} that
for building just differential geometry without
metric information, it is sufficient to take,
instead of the spectral triple with an unbounded
operator $D$, the same spectral triple but with
$D$ replaced by $F=sgn \>D$, the sign of the
operator $D$. This is actually a grading operator
on $\mathcal{S}$ since ${F}^{2}=\mathbf{1}$. Thus
the spectral triple $({\mathbf{C}}_{t},
\mathcal{S}, {F}_{t})$ with a family of grading
operators contains the topological and causal as
well as differential geometric information on
spacetime.
\end{note}
\begin{note}
One may wonder where the efficiency of the
spectral data in the presented description comes
from. In the case of the spectral triple A.
Connes argued \cite{Connes94,Connes95} that most
of the information is not in the algebra of the
triple, giving basically just a set of points,
nor in the chosen Hermitean operator, fully
described by its spectrum, but in the
relationship between them. This explanation can
be used here again: Most of the information in
the spectral data is not in the commutative
algebras ${\mathbf{C}}_{t}$ represented on
$\mathcal{S}$ but in the relationships between
them. Indeed, the strong causal structure is
purely a result of this.
\end{note}
\section*{Conclusion}
Motivated by the need to recover classical
spacetime from a theory of quantum gravity in
order to achieve the theory's physical
interpretation, the thesis examines the
possibility of describing classical Lorentzian
spacetime manifolds by spectral data.
Following in Section \ref{SNG} a na\"{\i}ve
Hamiltonian approach, the spectral data for a
Lorentzian manifold are specified as a family of
A. Connes' spectral triples with a common Hilbert
space and additional structures known from
Hamiltonian general relativity: a family of lapse
functions and an identification of Cauchy
surfaces implemented by isomorphisms of the
algebras in the spectral triples. This gives a
complete description of spacetime, trivially
extended to a free quantum field theory on
spacetime.
However, in Section \ref{LorentzSpectralData} it
is realized that the spectral description of
spacetime automatically contains unused
information on causal relationships. The use of
this information leads to a significant reduction
of the spectral data. The family of lapse
functions and the identification of Cauchy
surfaces can be completely left out, and still
there is considerable redundancy in the data
present. The discovery of the place of causal
relationships in spectral geometry thus leads to
a very efficient spectral description of
spacetime. This is the main result of this
thesis.
With the result attained here, there are now two
well motivated problems of conceptual importance:
\begin{enumerate}
\item{The remaining redundancy in the spectral
data should be removed and the result put into a
useful form to be recognized as standard.}
\item{The way in which the result may fit into an
interpretation of quantum gravity should be
clarified, possibly along the lines of
Conjectures \ref{1stConjecture} and
\ref{2ndConjecture}}
\end{enumerate}
Moreover, there are also many further points of
technical nature, to be worked out. To suggest
just one of them as an example, it would be
desirable to have a usefully formulated
expression for spacetime distances.
With the insight obtained here, these questions
are now open to future investigations.
\section*{Acknowledgements}
The author would like to thank Pavel Krtou\v{s},
Don N. Page and Georg Peschke for a number of
invaluable discussions.
|
proofpile-arXiv_065-688
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
Quantum
groups
or
q-deformed
Lie
algebra
implies
some
specific
deformations
of classical Lie algebras.
From
a
mathematical
point
of
view,
it
is
a
non-commutative
associative
Hopf
algebra.
The
structure
and
representation
theory of quantum groups
have
been
developed
extensively
by
Jimbo
[1]
and
Drinfeld
[2].
The
q-deformation
of
Heisenberg
algebra
was
made
by
Arik and Coon [3], Macfarlane [4] and Biedenharn [5].
Recently
there
has
been
some
interest
in
more
general
deformations
involving
an
arbitrary
real
functions
of
weight
generators
and
including
q-deformed algebras as a special case [6-10].
\defa^{\dagger}{a^{\dagger}}
\defq^{-1}{q^{-1}}
\defa^{\dagger}{a^{\dagger}}
Recently Greenberg [11] has studied the following q-deformation of
multi mode boson
algebra:
\begin{displaymath}
a_i a^{\dagger}_j -q a^{\dagger}_j a_i=\delta_{ij},
\end{displaymath}
where the deformation parameter $q$ has to be real.
The main problem of Greenberg's approach is that we can not derive the relation
among $a_i$'s operators at all.
Moreover the above algebra is not covariant under $gl_q(n)$ algebra.
In order to solve this problem we should find the q-deformed multimode
oscillator
algebra which is covariant under $gl_q(n)$ algebra.
Recently the Fock space representation of $gl_q(n)$-covarinat multimode
oscillator
system was known by some authers [12].
In this paper we construct the correct form of coherent states for the above
mentioned oscillator system and obtain the q-symmetric states
generalizing the bosonic
states.
\section{Coherent states of $gl_q(n)$-covariant oscillator algebra}
$gl_q(n)$-covariant oscillator algebra is defined as [12]
\begin{displaymath}
a^{\dagger}_ia^{\dagger}_j =q a^{\dagger}_j a^{\dagger}_i,~~~(i<j)
\end{displaymath}
\begin{displaymath}
a_ia_j=\frac{1}{q}a_j a_i,~~~(i<j)
\end{displaymath}
\begin{displaymath}
a_ia^{\dagger}_j=q a^{\dagger}_ja_i,~~~(i \neq j)
\end{displaymath}
\begin{displaymath}
a_ia^{\dagger}_i =1+q^2 a^{\dagger}_ia_i +(q^2-1) \sum_{k=i+1}^na^{\dagger}_k a_k,~~~(i=1,2,\cdots,n-1)
\end{displaymath}
\begin{displaymath}
a_n a^{\dagger}_n =1+q^2 a^{\dagger}_n a_n,
\end{displaymath}
\begin{equation}
[N_i, a_j]=-\delta_{ij}a_j,~~~[N_i, a^{\dagger}_j]=\delta_{ij}a^{\dagger}_j,~~~(i,j=1,2,\cdots, n )
\end{equation}
where we restrict our concern to the case that $q$ is real and $0<q<1$.
Here $N_i$ plays a role of number operator and $a_i(a^{\dagger}_i)$ plays a role of
annihilation(creation) operator.
From the above algebra one can obtain the relation between the number operators
and mode opeartors as follows
\begin{equation}
a^{\dagger}_ia_i=q^{2\sum_{k=i+1}^nN_k}[N_i],
\end{equation}
where $[x]$ is called a q-number and is defined as
\begin{displaymath}
[x]=\frac{q^{2x}-1}{q^2-1}.
\end{displaymath}
\def|n_1,n_2,\cdots,n_n>{|n_1,n_2,\cdots,n_n>}
Let us introduce the Fock space basis $|n_1,n_2,\cdots,n_n>$ for the number operators
$N_1,N_2,\cdots, N_n$
satisfying
\begin{equation}
N_i|n_1,n_2,\cdots,n_n>=n_i|n_1,n_2,\cdots,n_n>,~~~(n_1,n_2,\cdots,n_n=0,1,2\cdots)
\end{equation}
Then we have the following representation
\begin{displaymath}
a_i|n_1,n_2,\cdots,n_n>=q^{\sum_{k=i+1}^nn_k}\sqrt{[n_i]}|n_1,\cdots, n_i-1,\cdots,n_n>
\end{displaymath}
\begin{equation}
a^{\dagger}_i|n_1,n_2,\cdots,n_n>=q^{\sum_{k=i+1}^nn_k}\sqrt{[n_i+1]}|n_1,\cdots, n_i+1,\cdots,n_n>.
\end{equation}
From the above representation we know that there exists the ground state
$|0,0,\cdots,0>$
satisfying
$a_i|0,0>=0$ for all $i=1,2,\cdots,n$. Thus the state $|n_1,n_2,\cdots,n_n>$ is obtatind
by applying the
creation operators
to the ground state $|0,0,\cdots,0>$
\begin{equation}
|n_1,n_2,\cdots,n_n>=\frac{(a^{\dagger}_n)^{n_n}\cdots(a^{\dagger}_1)^{n_1}}{\sqrt{[n_1]!\cdots
[n_n]!}}|0,0,\cdots,0>.
\end{equation}
If we introduce the scale operators as follows
\begin{equation}
Q_i=q^{2N_i},~~(i=1,2,\cdots,n),
\end{equation}
we have from the algebra (1)
\begin{equation}
[a_i,a^{\dagger}_i]=Q_iQ_{i+1}\cdots Q_n.
\end{equation}
Acting the operators $Q_i$'s on the basis $|n_1,n_2,\cdots,n_n>$ produces
\begin{equation}
Q_i|n_1,n_2,\cdots,n_n>=q^{2n_i}|n_1,n_2,\cdots,n_n> .
\end{equation}
From the relation $a_i a_j =\frac{1}{q}a_j a_i,~~(i<j)$, the coherent states
for $gl_q(n)$
algebra
is defined as
\begin{equation}
a_i|z_1,\cdots,z_i,\cdots,z_n>=z_i|z_1,\cdots, z_{i},qz_{i+1},\cdots,q z_n>.
\end{equation}
Solving the eq.(9) we obtain
\begin{equation}
|z_1,z_2,\cdots,z_n>=c(z_1,\cdots,z_n)\sum_{n_1,n_2,\cdots,n_n=0}^{\infty}
\frac{z_1^{n_1}z_2^{n_2}\cdots z_n^{n_n}}{\sqrt{[n_1]![n_2]!\cdots [n_n]!}}|n_1,n_2,\cdots,n_n> .
\end{equation}
Using eq.(5) we can rewrite eq.(10) as
\begin{equation}
|z_1,z_2,\cdots,z_n>=c(z_1,\cdots,z_n)
\exp_q(z_na^{\dagger}_n)\cdots\exp_q(z_2a^{\dagger}_2)\exp_q(z_1a^{\dagger}_1)|0,0,\cdots,0>.
\end{equation}
where q-exponential function is defined as
\begin{displaymath}
\exp_q(x)=\sum_{n=0}^{\infty}\frac{x^n}{[n]!}.
\end{displaymath}
The q-exponential function satisfies the following recurrence relation
\begin{equation}
\exp_q(q^2 x)=[1-(1-q^2)x]\exp_q(x)
\end{equation}
Using the above relation and the fact that $0<q<1$, we obtain the formula
\begin{equation}
\exp_q(x) =\Pi_{n=0}^{\infty}\frac{1}{1-(1-q^2)q^{2n}x}
\end{equation}
Using the normalization of the coherent state , we have
\begin{equation}
c(z_1,z_2,\cdots,z_n)=\exp_q(|z_1|^2)\exp_q(|z_2|^2)\cdots \exp_q(|z_n|^2).
\end{equation}
The coherent state satisfies the completeness relation
\begin{equation}
\int\cdots \int
|z_1,z_2,\cdots,z_n><z_1,z_2,\cdots,z_n|\mu(z_1,z_2,\cdots,z_n) d^2z_1
d^2z_2\cdots d^2 z_n=I,
\end{equation}
where the weighting function $\mu(z_1,z_2,\cdots,z_n)$ is defined as
\begin{equation}
\mu(z_1,z_2,\cdots,z_n)=\frac{1}{\pi^2}\Pi_{i=1}^n\frac{\exp_q(|z_i|^2)}
{\exp_q(q|z_i|^2)}.
\end{equation}
In deriving eq.(15) we used the formula
\begin{equation}
\int_0^{1/(1-q^2)}x^n \exp_q(q^2 x)^{-1} d_{q^2} x=[n]!
\end{equation}
\def\otimes{\otimes}
\section{q-symmetric states}
In this section we study the statistics of many particle state.
Let $N$ be the number of particles. Then the N-partcle state can be obtained
from
the tensor product of single particle state:
\begin{equation}
|i_1,\cdots,i_N>=|i_1>\otimes |i_2>\otimes \cdots \otimes |i_N>,
\end{equation}
where $i_1,\cdots, i_N$ take one value among $\{ 1,2,\cdots,n \}$ and the sigle
particle state is defined by $|i_k>=a^{\dagger}_{i_k}|0>$.
Consider the case that k appears $n_k$ times in the set $\{ i_1,\cdots,i_N\}$.
Then we have
\begin{equation}
n_1 + n_2 +\cdots + n_n =\sum_{k=1}^n n_k =N.
\end{equation}
Using these facts we can define the q-symmetric states as follows:
\begin{equation}
|i_1,\cdots, i_N>_q
=\sqrt{\frac{[n_1]!\cdots [n_n]!}{[N]!}}
\sum_{\sigma \in Perm}
\mbox{sgn}_q(\sigma)|i_{\sigma(1)}\cdots i_{\sigma(N)}>,
\end{equation}
where
\begin{displaymath}
\mbox{sgn}_q(\sigma)=
q^{R(i_1\cdots i_N)}q^{R(\sigma(1)\cdots \sigma(N))},
\end{displaymath}
\begin{displaymath}
R(i_1,\cdots,i_N)=\sum_{k=1}^N\sum_{l=k+1}^N R(i_k,i_l)
\end{displaymath}
and
\begin{displaymath}
R(i,j)=\cases{
1 & if $ i>j$ \cr
0 & if $ i \leq j $ \cr
}
\end{displaymath}
Then the q-symmetric states obeys
\begin{equation}
|\cdots, i_k,i_{k+1},\cdots>_q=
\cases{
q^{-1} |\cdots,i_{k+1},i_k,\cdots>_q& if $i_k<i_{k+1}$\cr
|\cdots,i_{k+1},i_k,\cdots>_q& if $i_k=i_{k+1}$\cr
q |\cdots,i_{k+1},i_k,\cdots>_q& if $i_k>i_{k+1}$\cr
}
\end{equation}
The above property can be rewritten by introducing the deformed transition
operator
$P_{k,k+1}$ obeying
\begin{equation}
P_{k,k+1}
|\cdots, i_k , i_{k+1},\cdots>_q =|\cdots, i_{k+1},i_k,\cdots>_q
\end{equation}
This operator satisfies
\begin{equation}
P_{k+1,k}P_{k,k+1}=Id,~~~\mbox{so}~~P_{k+1,k}=P^{-1}_{k,k+1}
\end{equation}
Then the equation (21) can be written as
\begin{equation}
P_{k,k+1}
|\cdots, i_k , i_{k+1},\cdots>_q
=q^{-\epsilon(i_k,i_{k+1})}
|\cdots, i_{k+1},i_k,\cdots>_q
\end{equation}
where $\epsilon(i,j)$ is defined as
\begin{displaymath}
\epsilon(i,j)=
\cases{
1 & if $ i>j$\cr
0 & if $ i=j$ \cr
-1 & if $ i<j$ \cr }
\end{displaymath}
The relation (24) goes to the symmetric relation for the ordinary bosons
when the deformation parameter $q$ goes to $1$.
If we define the fundamental q-symmetric state $|q>$ as
\begin{displaymath}
|q>=|i_1,i_2,\cdots,i_N>_q
\end{displaymath}
with $i_1 \leq i_2 \leq \cdots \leq i_N$, we have
\begin{displaymath}
||q>|^2 =1.
\end{displaymath}
In deriving the above relation we used following identity
\begin{equation}
\sum_{\sigma \in Perm } q^{2R(\sigma(1),\cdots, \sigma(N))}=
\frac{[N]!}{[n_1]!\cdots [n_n]!}
\end{equation}
The derivation of above formula will be given in Appendix.
\section{Concluding Remark}
In this paper the $gl_q(n)$-covariant oscillator algebra and its
coherent states are discussed. The q-symmetric states generalizing the
symmetric
(bosonic) states are obtained by using the $gl_q(n)$-covariant oscillators
and are shown to be orthonormal.
I think that the q-symmetric states will be important when we consider the new
statistical field theory generalizing the ordinary one.
\section*{Appendix}
In this appendix we prove the relation(25) by using the mathematical induction.
Let us assume that the relation (25) holds for $N$.
Now we should prove that eq.(25) still hold for $N+1$.
Let us consider the case that $i$ appears $n_i+1$ times.
Then we should show
\def\sigma{\sigma}
\begin{equation}
\sum_{\sigma \in Perm}q^{2R(\sigma(1), \cdots, \sigma(N+1))}=\frac{[N+1]!}{[n_1]!\cdots
[n_{i-1}]![n_i+1]![n_{i+1}]!\cdots [n_n]!}
\end{equation}
In this case the above sum can be written by three pieces:
\begin{equation}
\sum_{j=1}^{i-1} \sum_{ \sigma(1)=j}
+\sum_{\sigma(1)=i}
+\sum_{j=i+1}^{n} \sum_{ \sigma(1)=j}
\end{equation}
Thus the left hand side of eq.(26) is given by
\begin{eqnarray}
LHS&=& \sum_{j=1}^{i-1}\sum_{\sigma(1)=j} q^{2R(j,\sigma(2),\cdots,\sigma(n+1))}\cr
&+& \sum_{\sigma(1)=i} q^{2R(i,\sigma(2),\cdots,\sigma(n+1))} \cr
&+& \sum_{j=i+1}^{n}\sum_{\sigma(1)=j} q^{2R(j,\sigma(2),\cdots,\sigma(n+1))} \cr
\end{eqnarray}
Then we have
\begin{displaymath}
R(j,\sigma(2),\cdots,\sigma(n+1)) =\sum_{k=2}^{N+1} R(j,\sigma(k))
+R(\sigma(2),\cdots,\sigma(N+1))
\end{displaymath}
where
\begin{displaymath}
\sum_{k=2}^{N+1}R(j,\sigma(k))
=\cases{ n_1 + \cdots + n_{j-1} & if $ j \leq i$\cr
n_1 +\cdots + n_{j-1} +1 & if $ j >i$ \cr}
\end{displaymath}
Using the above relations the LHS of eq.(26) can be written as
\begin{eqnarray}
LHS
&=& \sum_{j=1}^{i-1}q^{2(n_1 + \cdots + n_{j-1})}
\frac{[N]!}{[n_i+1]![n_j-1]!\Pi_{k \neq
i,j} [n_k]!} \cr
&+& q^{2(n_1 + \cdots + n_{i-1})} \frac{[N]!}{\Pi_{k } [n_k]!} \cr
&+& \sum_{j=i+1}^{N+1}q^{2(n_1 + \cdots + n_{j-1}+1)}
\frac{[N]!}{[n_i+1]![n_j-1]!\Pi_{k
\neq i,j} [n_k]!} \cr
\end{eqnarray}
If we pick up the common factor of three terms of eq.(29), we have
\begin{displaymath}
I=J
\frac{[N]!}{[n_i+1]!\Pi_{k \neq i} [n_k]!}
\end{displaymath}
where
\begin{eqnarray}
J&=&\frac{1}{q^2-1}[
\sum_{j=1}^{i-1}q^{2(n_1 + \cdots + n_{j-1})} [n_j]
+q^{2(n_1 + \cdots + n_{i-1})}[n_1+1]
+\sum_{j=i+1}^{N+1}q^{2(n_1 + \cdots + n_{j-1}+1)}[n_j] \cr
&=&[N+1]
\end{eqnarray}
Thus we proved the relation (25).
\section*{Acknowledgement}
This paper was
supported by
the KOSEF (961-0201-004-2)
and the present studies were supported by Basic
Science
Research Program, Ministry of Education, 1995 (BSRI-95-2413).
\vfill\eject
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proofpile-arXiv_065-689
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\section*{Introduction}
In physics, tilings are used to model solids, in particular non-periodic ones.
Studying the possible types of long range ordered structures
(and their implication on physical quantities)
amounts therefore in parts to the study of
(a suitable class of) tilings. In fact, the investigation of certain
tilings as idealized models for quasicrystals began
more than a decade ago so that one can find by now a large amount of
articles many of them being collected in \cite{OsSt,JaMo,AxGr}.
Some elements of a theory of long range ordered structures
are based on tilings but require additional
information, for instance when it comes to
the calculation of diffraction patterns
(Fourier transforms).
Others depend only on the topological nature of the tiling, as e.g.\ the
$K$-theoretical gap labelling. Results of these are consequently more
qualitative in nature. The present article clearly belongs to the second
area. In particular, the specific shape or volume of the tiles which make
up the tiling will not be of importance for us. Furthermore, due to the
locality of the interactions in the solid, it is only the
local structure of the tiling that matters, i.e.\ the way the tiling looks on
finite patches.
One motivation to write this article is to illustrate that this local
structure can be described by an almost-groupoid\ resp.\ an
inverse semigroup. The groupoid associated to the tiling arises
together with its topology functorially from the almost-groupoid.
The algebraic structure
is defined on the most elementary level and therefore underlies the
construction of all topological invariants (including the group of possible
gap labels) of the tiling.
The main aim of this article is however to give an answer to the question
under which circumstances two tilings give rise to isomorphic groupoids.
For that we introduce the notion of topological equivalence of tilings.
This notion is closely related
to mutually locally derivability of the tilings,
a concept well known from physical considerations.\bigskip
The article is organized as follows.
We start with an informal description of the local structure of a tiling
as an example of an almost-groupoid\ resp.\ inverse semigroup.
After that we put this into a general context
and describe a functor which assigns to every almost-groupoid\
a groupoid with discrete orbits. We apply this to tilings,
obtaining the groupoid associated it, then emphasizing the particularities
of this case.
By that we mean the existence of a metric structure
which is well known for tilings and
with respect to which the functor looks like taking a closed
subspace of the metric completion.
We compare the groupoid, which we also call in distinction the discrete
groupoid associated to the tiling, to the continuous groupoid, which is
often considered in the literature.
In the next section we investigate the known concept
of local derivability of tilings which leads us to introduce
the notion of topological equivalence. Theorem~\ref{17093}
constitutes the main result of this article.
It shows that topological equivalence of tilings -- a purely "local"
notion -- is sufficient and necessary for them to have isomorphic groupoids.
Whereas the metric structure is not used to define topological
equivalence of tilings the proof of theorem relies on this structure.
Everything is restricted to tilings which are of finite type.
The finite type (or compactness) condition which they satisfy is
the hypotheses for many compactness arguments.
In the final section we give a selected overview on topological invariants for
tilings. We mention the invariants of the groupoid-$C^*$-algebra, the $K$-groups,
and groupoid cohomology. But we will neither discuss the construction
of groupoid-$C^*$-algebra s (see \cite{Ren})
nor of its $K$-groups (see \cite{Bla}) nor of groupoid cohomology
(see \cite{Ren,Kum}).
We will also not illustrate the $K$-theoretical gap labelling
but refer the reader to \cite{Be2,BBG,Ke2}.
\section{The local structure of a tiling}
In this article the following notion of tiling will be used.
A tile (in $\mathbb R^d$) is a connected bounded subset of $\mathbb R^d$ which is
the closure of its interior and may be decorated.\footnote{
The decoration, which may consist e.g.\ of arrows or colours,
may serve for the purpose to distinguish
translation classes of tiles which have
the same shape.}
A $d$ dimensional tiling is an infinite set of tiles which cover $\mathbb R^d$
overlapping at most at their boundaries.
A finite subset of a tiling is also called a pattern.
Although often formulated for a specific tiling, the relevant quantities
like the groupoid associated to it and the almost-groupoid\ of its local structure depend
only on the congruence class of the tiling.
A tile- resp.\ tiling-
resp.\ pattern-class shall here be an equivalence class under translation of a
tile resp.\ tiling resp.\ pattern, i.e.\ two such objects belong to the
same class if there is an $x\in\mathbb R^d$ such that translation by $x$
applied to the tile resp.\ the elements of one set yield the other tile resp.\
the elements of the other. Note that a pattern class does not consist
simply of tile classes.
The local structure of a tiling is a multiplicative structure
determined by its pattern classes.
On the set of patterns of a given tiling one can easily introduce an
associative binary operation (multiplication), the union.
But such an operation is not well defined on pattern classes.
In order to achieve this we need to
keep track of the relative position between patterns.
This can be done with the help of an additional choice of a tile in the
pattern, such a composed object is called a pointed pattern.
Calling two pointed
patterns composable if their choice of tile coincides (in the tiling),
one may define an associative binary operation from the set of
composable pairs
into the set of pointed patterns as follows: the
union of the patterns of the composable pair yields the new pattern
and their common choice of tile the new choice.
This multiplication being only partially defined, it appears at first
sight to be a draw back, but its advantage lies in the possibility to
extend it to a well defined
partial multiplication on translation classes:
Call two \mix es composable if they have representatives which are composable
in the above sense, multiply them in case
as above and take their translation class.
But this is not all we want. We want to be able to build arbitrary large
pattern classes from a finite set of small ones using a multiplication.
This can obviously not be achieved by the above. Instead, if we look at
pattern classes with a choice of an ordered pair of tiles in it, calling that a
\mixx, we can make larger pattern classes from smaller ones as follows:
Ignoring the first resp.\ second choice in the ordered pair of the
first resp.\ second pattern class we obtain two (simply) \mix es
which may be multiplied as above provided they are composable. Of the
resulting \mix\ we forget the choice of tile and take instead the
ordered pair which is given by the so far ignored tiles, namely the
first of the ordered pair of the first and the second
of the ordered pair of the second pattern class we started with.
As we will elaborate below, this is a useful
algebraic structure which we call almost-groupoid.
Equivalently one could work, after adding a zero element, with inverse
semigroups
\cite{Pet}.
We still keep the name almost-groupoid, because it is almost a groupoid and applying
a functor to it yields topological groupoids.
This functor is most natural in tiling theory since it furnishes tilings
from patterns.
\subsection{Almost-groupoids / inverse semigroups}
Let $\Gamma$ be a set. A partially defined associative multiplication
is given by a subset $\Gamma^\vdash\subset \Gamma\times\Gamma$ of composable pairs
(we write $x\vdash y$ for $(x,y)\in\Gamma^{\vdash}$) with a
map $m:\Gamma^\vdash\to\Gamma$
(we write $xy=m(x,y)$)
which is associative in the sense that, first,
$x\vdash y$ and $xy\vdash z$ is equivalent to $x\vdash yz$ and $y\vdash z$, and
second, if $x\vdash y$ and $xy\vdash z$ then $(xy)z = x(yz)$.
Hence we don't have to care about brackets.
Relations or equations like the above in a set with
partially defined associative multiplication make sense only if the
multiplications are defined, i.e.\
if the to be multiplied pairs are composable.
In order to avoid cumbersome notation we
shall agree from now on that a relation involving products is true
if all multiplications involved are defined and it is then true.
Given such a set $\Gamma$ with partially defined multiplication, suppose
that for some $a\in\Gamma$ the equations
$axa=a$ and $xax=x$ were true for some $x\in\Gamma$. Then $x$ is called an
inverse of $a$. $\Gamma$ has a unique inverse (map) if any $a$ has a unique
inverse. The inverse map is then denoted
by $a\mapsto a^{-1}$.
\begin{df}
An almost-groupoid\ is a set $\Gamma$ with partially defined associative multiplication
and unique inverse.
\end{df}
A set with fully defined associative multiplication
and unique inverse is an inverse semigroup, i.e.\
an inverse semigroup is an almost-groupoid\ for which
$\Gamma^\vdash=\Gamma\times\Gamma$.
In particular, adding a zero element $0$ to an almost groupoid $\Gamma$
and extending the multiplication by $xy=0$ if $x\not\vdash y$, and $0x=x0=0$,
yields an inverse semigroup with zero (which we write as $\Gru_0$).
Conversely, if $\Gru_0$ is an inverse semigroup with zero then
$\Gamma=\Gamma_0\backslash\{0\}$ with $\Gamma^\vdash=\{(x,y)|xy\neq 0\}$ is an
almost-groupoid. So we may apply the known results of inverse semigroup theory.
In fact, any statement below on almost-groupoid s may be reformulated as a statement
on inverse semigroups with zero element and vice versa. However, we find
the formulation in terms of almost-groupoid s more natural.
The elements of $\Gamma^0:=\{x x^{-1}|x\in\Gamma\}$ are called units. There are
the image of the frequently occurring maps
$r,d:\Gamma\rightarrow\Gamma^0$
given by $r(x) = xx^{-1}$ and $d(x)=r(x^{-1})$
Let us mention that the uniqueness of the inverse implies, first, that
units are the same as idempotents, i.e.\
$\Gamma^0:=\{x\in\Gamma| x^2=x\}$, and that they commute,
and second, that the inverse map is an involution,
in particular $(xy)^{-1}=y^{-1}x^{-1}$. A proof of that can be found in
\cite{Pet} formulated in the framework of inverse semigroups.
This has implications on which kind of elements are composable.
E.g.\ if $x\vdash y$ then $(xy)^{-1}=y^{-1}x^{-1}$ so that we must also have
$ y^{-1}\vdash x^{-1}$. Furthermore, under the same condition $x\vdash y$ we have
$xy = xy y^{-1} x^{-1} xy$ so that we must have composabilities like
$xy\vdash y^{-1}$ etc..
Similarly,
$d(x)=r(y)$ implies $x=xyy^{-1}$ so that we must have $x\vdash y$.
If $x\vdash y$ is even equivalent to $d(x)=r(y)$, then $\Gamma$ satisfies
cancellation, i.e.
$xy=xz$ implies $y= z$.
This is simply because for $xy=xz$ to be true we must have
$x\vdash y$ and $x\vdash z$. But then $y=r(y)y=d(x)y=d(x)z=z$.
Note that a groupoid -- for an explicit definition c.f.\ \cite{Ren} --
is the same as an almost-groupoid\ which satisfies cancellation.
The well known order relation
on inverse semigroups \cite{Pet} will be of great use here. One way of
formulating it here is:
\begin{df}
The order of an almost-groupoid\ is defined by\footnote{
We use here a direction of the order which coincides with the convention
used in semigroup theory. It is reversed to that in \cite{Ke5}.}
\begin{equation}\label{29052}
x\preceq y\quad\mbox{whenever}\quad
r(x)=xy^{-1}.
\end{equation}
\end{df}
Note that $x\succeq y$ is equivalent to $x^{-1}\succeq y^{-1}$, and, if
moreover $y\vdash z$, then $x\vdash z$ and $xz\succeq yz$. In other words
the order is compatible with multiplication. Note also that
a groupoid has trivial order.
\begin{lem}\label{20091}
The set of all minimal elements of an almost-groupoid\ is a (possibly empty) ideal
which is a groupoid.
\end{lem}
{\em Proof:}\
Let $\Gamma$ be an almost-groupoid\ and $x\vdash y$ for two of its elements.
Suppose that $x$ is
minimal and consider the
relation $xy\succeq z$, i.e.\ $z^{-1}z=z^{-1}xy$.
We want to show that $z=xy$ and hence it is minimal.
Since order is compatible with multiplication we have
$x\succeq z y^{-1}$ hence $x= z y^{-1}$ by minimality.
Since for units
$u$ holds $zu\preceq z$ we conclude
$xx^{-1}=zy^{-1}yz^{-1}\preceq zz^{-1}$, and
$xx^{-1}xy\preceq zz^{-1}xy=z$ showing that $xy=z$. Thus $xy$ is minimal.
In particular, all minimal elements form an almost-groupoid\ (which may be empty).
We want to show that it satisfies
cancellation, i.e.\ that $x\vdash y$ implies $d(x)=r(y)$.
If $x\vdash y$ then $d(x)\vdash r(y)$ and hence
$d(x), r(y)\succeq d(x)r(y)$. Minimality of
$x$ implying that of $d(x)$ and $r(x)$ shows that
$d(x)=r(y)$.\hfill q.e.d.\bigskip
Let $u\in\Gamma^0$ and $c\in\Gamma$. If $c\preceq u$ then
$c^{-1}=d(c)u$ and in particular $c\in\Gamma^0$. On the other hand
$u\preceq c$ does for $u\in\Gamma^0$ not have to imply that $c\in\Gamma^0$.
But this latter property is useful in the sequel so that we give it a name.
\begin{df}
An almost-groupoid\ is unit hereditary if, for $u\in\Gamma^0$ and $c\in\Gamma$,
$u\preceq c$ implies $c\in\Gamma^0$.
\end{df}
Either of the statements $xy^{-1}\in\Gamma^0$ or $yx^{-1}\in\Gamma^0$
implies that $x$ and $y$ have a lower bound (common smaller element).
E.g.\ if $xy^{-1}\in\Gamma^0$ then $xd(y)$ is such a lower bound.
For a unit hereditary almost-groupoid\ the converse holds as well, namely
if $x$ and $y$ have a lower bound $z$ then $r(z)$ is smaller
than both, $xy^{-1}$ and $yx^{-1}$. Moreover, in that case
$z=zd(z)\preceq xd(y)$. Therefore, if $\Gamma$ is unit hereditary and $x$ and
$y$ have a lower bound then
\begin{equation}\label{03061}
\max\{z\in\Gamma|z\preceq x,y\} = xd(y)
\end{equation}
and $r(x)y=r(y)x=yd(x)=xd(y)$.\bigskip
\noindent
{\bf Example 1.} Let $X$ be a topological space and $\beta_0(X)$ a
(not necessarily proper) subset of the topology of $X$
which has the property that
any open subset of $X$ is a union of sets of $\beta_0(X)$
(i.e.\ it is a base of the topology)
and that it is closed under intersection.
Then $UV=U\cap V$ defines a multiplication on $\beta_0(X)$.
Since the only solution of the equations $U\cap V\cap U=U$ and
$V\cap U\cap V=V$
is given by $U=V$ and $U\cap U=U$,
$\beta_0(X)$ is a commutative inverse semigroup which consists of units
(idempotents) only. The empty set is a zero element in it and consequently
$\beta(X):=\beta_0(X)\backslash\{\emptyset\}$ a
commutative almost-groupoid\ consisting of units only.
Its order is the inclusion of sets. Note that there are in general no
minimal elements in $\beta(X)$.\bigskip
\noindent
{\bf Example 2.} Let ${\mathcal T}$ be a tiling of $\mathbb R^d$.
We already have explained in
words that the set ${\mathcal M}_{\rm I\!I}$ of \mixx es
carries a partially defined multiplication.
Let us reformulate this in more technical terms.
We start with defining an order relation on ${\mathcal M}_{\rm I\!I}$, namely
$c\succeq c'$ if $c'$ can be obtained from $c$
by addition of tiles but keeping the ordered pair of chosen
tiles fixed. Let ${\mathcal M}_{\rm I\!I\!I}$ be the set of all pattern classes together with an
ordered triple of chosen tiles and denote for $\eta\in{\mathcal M}_{\rm I\!I\!I}$ by
$\eta_{\hat{i}}\in{\mathcal M}_{\rm I\!I}$
the \mixx\ which is obtained by forgetting the $i$th choice
in the triple.
Call two \mixx es $c,c'$ composable whenever there is
an $\eta\in{\mathcal M}_{\rm I\!I\!I}$ such that
$c\succeq \eta_{\hat{3}}$ and $c'\succeq \eta_{\hat{1}}$.
Then define the product of two composable elements
$$
cc' = \max\{\eta_{\hat{2}}|\eta\in{\mathcal M}_{\rm I\!I\!I},c\succeq \eta_{\hat{3}},
c'\succeq \eta_{\hat{1}}\}
$$
the maximum being taken with respect to the above order.
This defines an associative multiplication.
It turns out to have a unique inverse map $c\mapsto c^{-1}$ which is
given by
interchange of the elements of the ordered pair of chosen tiles.
Thus ${\mathcal M}_{\rm I\!I}$ forms an almost-groupoid\ which is in general
not commutative. The order of the almost-groupoid\ coincides with
the order used to define composability.
In particular, the almost-groupoid\ of a tiling is unit hereditary.
Note that there are no minimal elements in ${\mathcal M}_{\rm I\!I}$.
A well known equivalence relation among tilings is that of two tilings
being locally isomorphic \cite{SoSt}.
Thus are called two tilings which have the property that
every pattern class of either
tiling can also be found in the other.\footnote{
The notion is used here in a stronger sense than in \cite{SoSt}
in that pattern classes are considered as
equivalence classes under translations but not under rotations.}
This can here simply be expressed by saying that the
tilings lead to the same almost-groupoid.
Let ${\mathcal M}_{\rm I}$ be the set of \mix es which are pattern classes together
with one chosen tile. We may identify ${\mathcal M}_{\rm I}$ with the subset of ${\mathcal M}_{\rm I\!I}$
consisting of those elements which are invariant under the inverse map,
i.e.\ for which the chosen tiles in the ordered pair coincide.
Another specific property which holds for almost-groupoid s defined by tilings is that
elements which are equal to
their inverse have to be units, i.e.\ under the above
identification ${\mathcal M}_{\rm I}={\mathcal M}_{\rm I\!I}^0$.
We shall be interested in tilings
which satisfy the following finite type (or compactness) condition.
We call a pattern (and its class) connected if the subset it covers is
connected.
\begin{itemize}
\item
The set of connected \mixx es which consist of two tiles is finite.
\end{itemize}
Since tiles are bounded sets which have positive Lebesgue measure
this condition implies that, for any $r$, the maximal number of
tiles a pattern fitting inside an $r$-ball can have is finite.
From that one concludes that the above condition is
equivalent to the requirement that the number of
pattern classes fitting inside an $r$-ball is finite.
In particular ${\mathcal M}_{\rm I\!I}$ is countable.
\subsection{From almost-groupoid s to groupoids}
We now aim at a functorial construction to obtain a
topological groupoid from an almost-groupoid. For that we consider sequences
$(x_n)_{n\in\mathbb N}$ of elements $x_n\in\Gamma$ which are
decreasing in that for all $n$: $x_n\succeq x_{n+1}$.
The set of all decreasing sequences, which
is denoted by $\Gamma^{\mathbb N}_{\succeq}$,
carries a pre-order
\begin{equation}\label{18061}
\fol{x}\preceq\fol{y}\quad\mbox{\rm
whenever}\quad \forall n\exists m:x_m\preceq y_n.
\end{equation}
To turn this pre-order into an order one considers the
equivalence relation on $\Gamma^{\mathbb N}_{\succeq}$
\begin{equation}\label{18069}
\fol{x}\sim\fol{y}\quad\mbox{\rm
whenever}\quad \fol{x}\preceq\fol{y}\quad\mbox{\rm
and}\quad \fol{y}\preceq\fol{x}.
\end{equation}
On the set of equivalence classes, the elements of which we denote by
$[\fol{x}]$,
\begin{equation}\label{10091}
[\fol{x}]\preceq[\fol{y}] \quad\mbox{\rm
whenever}\quad \fol{x}\preceq\fol{y}
\end{equation}
is an order relation.
\begin{df}
For a given almost-groupoid\ $\Gamma$, $\fl{\Gamma}$ is the set
$\Gamma^{\mathbb N}_{\succeq}$ modulo relation (\ref{18069}) and
${\mathcal R}(\Gamma)$ the set of minimal elements of $\fl{\Gru}$ with respect to the
order (\ref{10091}).
\end{df}
We identify the elements of $\Gamma$ with constant sequences in $\fl{\Gru}$.
We use also the notation $\fl{x}$ for the elements of $\fl{\Gru}$.
\begin{lem}\label{21091}
If $\Gamma$ is a countable almost-groupoid\ then any $x\in\Gamma$ has a smaller
minimal element in $\fl{\Gru}$, in particular ${\mathcal R}(\Gamma)\neq\emptyset$.
\end{lem}
{\em Proof:}\
Given $x\in\Gamma$ there is a bijection
$\gamma:\mathbb N\to\Gamma$ such that $\gamma(1)=x$. Now define
$\hat{\gamma}(1)=\hat{\gamma}(1)$ and
$\hat{\gamma}(n)=\hat{\gamma}(n-1)d(\gamma(n))$ if
$\hat{\gamma}(n-1)\vdash d(\gamma(n))$ and else
$\hat{\gamma}(n)=\hat{\gamma}(n-1)$. Then $(\hat{\gamma}(n))_n\in
\Gamma^{\mathbb N}_{\succeq}$. Now suppose that $\fol{y}\preceq (\hat{\gamma}(n))_n$.
Then in particular $y_m$ and
$\hat{\gamma}(n)$ have for all $n,m\in\mathbb N$
a common smaller element. But this implies that
$\hat{\gamma}(\gamma^{-1}(y_m))\preceq y_m$ and hence
$\fol{y}\succeq (\hat{\gamma}(n))_n$. Thus $(\hat{\gamma}(n))_n$, which is
certainly smaller than the constant sequence $x$, is a minimal element.
\hfill q.e.d.\bigskip
Examples show that countability is not a necessary condition.
\begin{lem}
$\fl{\Gru}$ is an almost-groupoid\ under the operations induced by point-wise operations on
$\Gamma^{\mathbb N}_{\succeq}$, and its order coincides with the
order (\ref{10091}).
\end{lem}
{\em Proof:}\
$\Gamma^{\mathbb N}_{\succeq}$ is an almost-groupoid\ under point-wise operations, i.e.\
composability is given by
$\fol{x}\vdash\fol{y}$ if $\forall n:x_n\vdash y_n$ and then
$\fol{x}\fol{y}=(x_ny_n)_n$, $\fol{x}^{-1}=\fol{x^{-1}}$.
Since order is compatible with multiplication
$\fol{x'}\sim\fol{x}$ and $\fol{y'}\sim\fol{y}$ and $\fol{x}\vdash\fol{y}$
imply, first $\fol{x'}\vdash\fol{y'}$, and second $(x_ny_n)_n\sim (x'_ny'_n)_n$.
Furthermore $x\succeq y$ being equivalent to $x^{-1}\succeq y^{-1}$ implies that
$\fol{x}\sim\fol{y}$ is equivalent to $\fol{x^{-1}}\sim\fol{y^{-1}}$.
From this follows the uniqueness of inversion.
Hence also $\fl{\Gru}$ is an almost-groupoid. Its units are classes of sequences of decreasing
units of $\Gamma$.
It is straightforward to see that its order is given
by (\ref{10091}). \hfill q.e.d.
\subsubsection{Morphisms of almost-groupoid s}
It turns out that the natural morphisms to look at in the context of
tilings are not homomorphisms but certain prehomomorphisms.
An order ideal of an almost-groupoid\ $\Gamma$ is a subset ${\mathcal N}$ for which
$c\preceq c'\in{\mathcal N}$ implies $c\in{\mathcal N}$.
Any subset ${\mathcal N}$ generates an order ideal, namely
$I({\mathcal N})=\{x\in\Gamma|\exists y\in{\mathcal N}:x\preceq y\}$.
We call an element of $\Gamma^{\mathbb N}_{\succeq}$ approximating if its class is
minimal.
\begin{df}
A prehomomorphism $\varphi:\Gamma\to\Gamma'$ between two almost-groupoid s is a map
which preserves composability, commutes with the inversion map,
and satisfies for all
$x\vdash y$
\begin{equation}\label{07031}
\varphi(xy)\preceq\varphi(x)\varphi(y).
\end{equation}
A prehomomorphism is called approximating if it maps approximating
sequences onto approximating ones.
An approximating prehomomorphism\ $\varphi:D(\varphi)\subset\Gamma\to\Gamma'$ is called a
partial approximating prehomomorphism\ or
local morphism between $\Gamma$ and $\Gamma'$ if its domain $D(\varphi)$,
which is a sub-almost-groupoid\ of $\Gamma$,
is an order ideal.
\end{df}
\begin{lem}
Prehomomorphisms preserve the order.
\end{lem}
{\em Proof:}\
$x\preceq y$ is equivalent to $x=r(x)y$ and hence implies
$\varphi(x)\preceq r(\varphi(x))\varphi(y)\preceq\varphi(y)$.\hfill q.e.d.\bigskip
This lemma implies that prehomomorphisms are
composable, and since the domain of $\psi\circ\varphi$,
which is $D(\psi\circ\varphi)=\{x\in D(\varphi)|\varphi(x)\in D(\psi)\}$,
is an order ideal of $\Gamma$ local morphisms are composable as well.
A prehomomorphism $\varphi:\Gamma\to\Gamma'$ of almost-groupoid s can be extended to
a prehomomorphism $\varphi:\Gamma_0\to\Gamma'_0$ of inverse semigroups
by simply setting $\varphi(0)=0$.
The condition that $\varphi:\Gamma\to\Gamma'$
preserves composability implies then for the extension that it satisfies
$\varphi^{-1}(0)=0$. Conversely, any
prehomomorphism $\varphi:\Gamma_0\to\Gamma'_0$ of inverse semigroups with zero
which satisfies $\varphi^{-1}(0)=0$ restricts to a prehomomorphism
on the almost-groupoid s.
A homomorphisms between almost-groupoid s is a prehomomorphism for
which (\ref{07031}) is an equality.
By element wise application to sequences, a prehomomorphism maps
decreasing sequences onto decreasing sequences, and
moreover preserves equivalence classes.
Hence it extends to a
prehomomorphism $\tilde{\varphi}:\fl{\Gru}\to{\fl{\Gru}'}$
through
\begin{equation}\label{20061}
\fl{\varphi}[\fol{x}]:=[(\varphi(x_n))_n].
\end{equation}
If $\varphi$ is a local morphism then we
denote by ${\mathcal R}(\varphi):{\mathcal R}(D(\varphi))\to{\mathcal R}(\Gamma)$
the restriction of $\fl{\varphi}$ to the minimal elements ${\mathcal R}(D(\varphi))$.
\subsubsection{Topology}
A topological almost-groupoid\ is an almost-groupoid\ which carries a topology
such that the product and the inversion map are continuous,
$\Gamma^\vdash$ carrying the relative topology.
A (locally compact) groupoid is called $r$-discrete if
$r^{-1}(x)$ is discrete for any $x$,
or equivalently, if its set of
units $\Gamma^0$ is open \cite{Ren}.
If nothing else is said $\Gamma$ shall carry the discrete topology.
The topology of $\fl{\Gru}$ shall then be defined as the one which is generated by
$\beta_0(\fl{\Gru}):=\{\fl{U}_{x}|x\in\Gamma_0\}$,
\begin{equation}
\fl{U}_{x} = \{\fl{y}\in\fl{\Gru}|\fl{y}\preceq x\},
\end{equation}
$\fl{U}_0=\emptyset$, and
${\mathcal R}(\Gamma)$ shall carry the relative topology, i.e.\
the one generated by
$\beta_0({\mathcal R}(\Gamma))=\{{\mathcal U}_x,x\in\Gamma\}$,
${\mathcal U}_{x}=\fl{U}_{x}\cap{\mathcal R}(\Gamma)$.
Using set multiplication\footnote{For arbitrary subsets of
$\fl{\Gru}_0$ is $\fl{U}\fl{V}=\{\fl{x}\fl{y}|\fl{x}\in \fl{U},
\fl{y}\in \fl{V},\fl{x}\vdash \fl{y}\}$.} as multiplication on
$\beta_0(\fl{\Gru})$ resp.\ $\beta_0({\mathcal R}(\Gamma))$ we get:
\begin{lem}
The maps $x\mapsto \fl{U}_x$ resp.\ $x\mapsto {\mathcal U}_x$
furnish an isomorphisms between the
inverse semigroups $\Gamma_0$ and $\beta_0(\fl{\Gru})$ resp.\ $\beta_0({\mathcal R}(\Gamma))$.
\end{lem}
{\em Proof:}\
Let $x,y\in\Gamma$. $\fl{U}_{x}\fl{U}_{y}\subset\fl{U}_{xy}$ follows
directly from the compatibility between order and multiplication
and ${\mathcal U}_{x}{\mathcal U}_{y}\subset{\mathcal U}_{xy}$ is then a consequence of Lemma~\ref{20091}.
As for the converse, let
$\fl{z}\preceq xy$.
Then first $x^{-1}\fl{z}\preceq x^{-1}xy\preceq y$,
and second $\fl{z}=r(xy)\fl{z}\preceq r(x)\fl{z}$ hence $\fl{z}=x(x^{-1}\fl{z})$.
This shows that $\fl{z}\in\fl{U}_{x}\fl{U}_{y}$.
If moreover $\fl{z}$ is minimal then the factorization
$\fl{z}=(r(\fl{z}x))(x^{-1}\fl{z})$ shows $\fl{z}\in{\mathcal U}_{x}{\mathcal U}_{y}$
as both, $r(\fl{z}x)$ and $x^{-1}\fl{z}$ are minimal. Thus
\begin{equation}\label{28051}
\fl{U}_{x}\fl{U}_{y}=\fl{U}_{xy}\quad\mbox{and}\quad{\mathcal U}_x{\mathcal U}_{y}={\mathcal U}_{xy}.
\end{equation}
The considered maps are by definition surjective.
But either of $\fl{U}_{x}=\fl{U}_{y}$ or ${\mathcal U}_{x}={\mathcal U}_{y}$
implies that $x\preceq y$ and $y\preceq x$ so that
the maps are injective as well.\hfill q.e.d.\bigskip
If $\Gamma$ is unit hereditary
$\beta_0(\fl{\Gru})$ and $\beta_0({\mathcal R}(\Gamma))$ are closed under intersection.
In fact,
$\fl{U}_{x}\cap\fl{U}_{y}\neq\emptyset$ whenever $x$ and $y$ have a lower
bound in $\Gamma$ and therefore
$\fl{U}_{x}\cap\fl{U}_{y}=\fl{U}_{r(x)y}$ (which might be empty)
and hence also ${\mathcal U}_{x}\cap{\mathcal U}_{y}={\mathcal U}_{r(x)y}$.
\begin{thm}\label{12092}
${\mathcal R}(\Gamma)$ is an r-discrete topological groupoid whose topology is
$T_1$. If $\Gamma$ is unit hereditary then ${\mathcal R}(\Gamma)$ is even Hausdorff.
\end{thm}
{\em Proof:}
${\mathcal U}_x^{-1}={\mathcal U}_{x^{-1}}$ is open showing continuity of the inversion.
By (\ref{28051}),
$$m^{-1}({\mathcal U}_x)=\bigcup_{(x_1,x_2)\in\Gamma^\vdash:
x\succeq x_1x_2}({\mathcal U}_{x_1}\times{\mathcal U}_{x_2})\cap{\mathcal R}(\Gamma)^\vdash$$ is open as well
and hence multiplication continuous.
Moreover, since $d([\fol{x}])\preceq d(x_1)$ we have
${\mathcal R}(\Gamma)^0=\bigcup_{u\in\Gamma^0}{\mathcal U}_u$ which is open and hence the groupoid
$r$-discrete.
To show that ${\mathcal R}(\Gamma)$ is $T_1$, i.e.\
that for all $\fl{x},\fl{y}\in{\mathcal R}(\Gamma)$ with $\fl{x}\neq\fl{y}$
there is an open $U$ containing
$\fl{x}$ but not $\fl{y}$, let $\fol{x}$ resp.\ $\fol{y}$
be a representative for $\fl{x}$ resp.\ $\fl{y}$
observe that $\fl{x}\neq \fl{y}$ implies both, $\fl{x}\not\preceq \fl{y}$
and $\fl{x}\not\succeq \fl{y}$, and hence
the existence of an $n_0$ such that for all $n\geq n_0$:
$x_n\not\succeq\fl{y}$ and $y_n\not\succeq\fl{x}$.
Therefore any $U={\mathcal U}_{x_n}$, $n\geq n_0$, does the job.
Now suppose that $\Gamma$ is unit hereditary. We claim that
for some $m$, $x=x_{n_0}$ and $y_m$ do not have a smaller common element.
This then proves the Hausdorff property since
for that $m$ is ${\mathcal U}_x\cap{\mathcal U}_{y_m}=\emptyset$ and
$\fl{x}\in{\mathcal U}_x$ and $\fl{y}\in{\mathcal U}_{y_m}$.
To prove the claim suppose its contrary, i.e.\ $x$ and $y_m$ to have a
common smaller element for all $m$. Then $\fl{y}d(x)\preceq\fl{y}$ which by
minimality implies $\fl{y}\preceq\fl{y}d(x)$.
In particular $\exists l\exists m:y_m\preceq y_ld(x)$,
and since $y_lx^{-1}$ is a unit $y_ld(x)\preceq x$.
This contradicts the above.
\hfill q.e.d.
\begin{thm}\label{29051}
Let $\varphi:D(\varphi)\to\Gamma'$ be an
local morphism\ of almost-groupoid s and ${\mathcal R}(\varphi)$ be
the restriction of $\tilde{\varphi}$ to ${\mathcal R}(D(\varphi))$.
Then ${\mathcal R}(\varphi):{\mathcal R}(D(\varphi))\to {\mathcal R}(\Gamma')$ is a continuous homomorphism
between topological groupoids.
\end{thm}
{\em Proof:}
${\mathcal R}(\varphi)$ is a prehomomorphism by construction.
But cancellation implies that on groupoids the order is trivial and hence
prehomomorphisms are homomorphisms.
To show continuity of ${\mathcal R}(\varphi)$
let $x'\in\Gamma'$. Then
${\mathcal R}(\varphi)([\fol{x}])\in{\mathcal U}_{x'}$ is equivalent to
$\exists n:x'\succeq\varphi(x_n)$.
Hence ${\mathcal R}(\varphi)^{-1}({\mathcal U}_{x'})
\subset\bigcup_{y\in\Gamma:\varphi(y)\preceq x'}{\mathcal U}_y$.
Since also
${\mathcal R}(\varphi)({\mathcal U}_y)\subset {\mathcal U}_{\varphi(y)}$,
and $x\preceq y$ implies ${\mathcal U}_x\subset{\mathcal U}_y$, the above inclusion is in fact
an equality.
This shows continuity. \hfill q.e.d.\bigskip
In fact, it is easily checked that ${\mathcal R}$ is a covariant functor of the
category of almost-groupoid s with local morphism s
into the category of $r$-discrete groupoids with
partial continuous homomorphisms.
Since ${\mathcal R}(\Gamma)$ has trivial order decreasing
sequences are constant sequences. Therefore we may identify ${\mathcal R}\circ{\mathcal R}$
with ${\mathcal R}$.
In general an almost-groupoid\ is non commutative. But in Example~1 we
have seen that bases
of the topology of topological spaces
which are closed under intersection give rise to
almost-groupoid s which consist only of units so they are in particular commutative.
Theorem~\ref{12092} and (\ref{28051})
show that any almost-groupoid\ may be identified (after adding
a zero element) with a base of the topology of a $T_1$ space
but only in the case where the almost-groupoid\ consists only of units
its multiplication coincides with intersection.
In that case ${\mathcal R}(\Gamma)={\mathcal R}(\Gamma)^0$. Hence if $x\vdash y$,
which means for groupoids $d(x)=r(y)$, then $y=x$.
In other words the groupoid operations are trivial, i.e.\
$x$ is composable only with itself, $x^2=x$, and $x^{-1}=x$.
So a topological groupoid which consists only of units is an ordinary
topological space.
This indicates why one may call the field to which this
study of tilings belongs the non commutative topology\ of tilings.
\subsubsection{Inverse semigroups of groupoids}
It is instructive to compare the inverse semigroup $\Gamma_0$ from which we
obtained the groupoid with other
inverse semigroups which are often considered in connection with
groupoids. For instance in Renault's book \cite{Ren}
such inverse semigroups (with zero) are considered which consist of
$G$-sets. A $G$-set is a subset $s$ of the groupoid $G$
which has the property that the restrictions of $r$ and $d$ to $s$
are both injective.
Multiplication is then given by set multiplication (and inversion applies
element-wise). The order is inclusion of sets and the empty set is the
zero element. If no more restrictions on the $G$-sets are given this is
called the inverse semigroup of the groupoid, we denote it here by $\mathcal{ISG}(G)$.
In the context of $r$-discrete groupoids it is also interesting to look at
those $G$-sets which are compact and open. They form also a
sub-inverse semigroup,
the ample semigroup of $G$ denoted here by $\mathcal{ASG}(G)$.
Note that both, $\mathcal{ISG}(G)$ and $\mathcal{ASG}(G)$ are closed under intersection.
Since the assignment of an inverse semigroup to the groupoid is reverse
to the functor ${\mathcal R}$ the natural question is whether they are somehow inverse
(leaving aside the more subtle question of how to assign to a
groupoid homomorphism
a morphism of $\mathcal{ISG}(G)$ or $\mathcal{ASG}(G)$).
The answer is in general negative but we can say the following.
The relation between the inverse semigroup $\Gamma_0$ to start with and
the inverse semigroups of ${\mathcal R}(\Gamma)$-sets of ${\mathcal R}(\Gamma)$ is rather obvious:
${\mathcal U}_c$, $c\in \Gamma$, is a ${\mathcal R}(\Gamma)$-set so that
by (\ref{28051}) we may identify $\Gamma_0$ as a
sub-inverse semigroup of $\mathcal{ISG}({\mathcal R}(\Gamma))$.
Furthermore, if the ${\mathcal U}_c$ are compact then $\mathcal{ASG}({\mathcal R}(\Gamma))$
is given by (finite) unions of elements of $\Gamma_0$ under this identification.
But they are not equal (and not all finite unions are allowed).
A topological space is called first countable if any point has a
countable local base. If it is $T_1$ then for any point $x$
there exists a descending sequence $\fol{U}$ of neighbourhoods such that
$\bigcap_n U_n=\{x\}$. Such a sequence may be constructed as follows:
Let ${\mathcal U}(x)$ be a local base at $x$ and $\gamma:\mathbb N\to{\mathcal U}(x)$ be a bijection.
Define $\hat{\gamma}(1)=\gamma(1)$ and
$\hat{\gamma}(n)=\gamma(m)$ where $m$ is the smallest number such that
$\gamma(m)\subset\gamma(n)\cap\hat{\gamma}(n-1)$.
This is a descending sequence and $x\in\bigcap_n \hat{\gamma}(n)$.
Let $y\neq x$ and $V$
be an open set containing $x$ but not $y$. Then there is a $U\in{\mathcal U}(x)$
such that $U\subset V$ and hence $y\notin \hat{\gamma}(\gamma^{-1}(U))$.
Hence $y\notin \bigcap_n \hat{\gamma}(n)$.
Thus the $\hat{\gamma}(n)$ form the desired sequence.
\begin{thm} If $G$ is a first countable groupoid whose topology is $T_1$
and generated by $\mathcal{ASG}(G)$ then ${\mathcal R}(\mathcal{ASG}(G)\backslash\{\emptyset\})=G$.
\end{thm}
{\em Proof:}\
Let $\Gamma=\mathcal{ASG}(G)\backslash\{\emptyset\}$.
The map $p([\fol{U}]=\bigcap_n U_n$ is easily seen to be a well defined map
from $\fl{\Gamma}$ into the power set of $G$.
Since the elements of $\mathcal{ASG}(G)$ are compact $p([\fol{U}])$ is not empty.
Now suppose that $[\fol{U}]$ were minimal and $x\neq y$ both
in $\bigcap_n U_n$.
Then there is a $V\in\mathcal{ASG}(G):y\notin V,
x\in V$. It follows that $[\fol{V\cap U}]$ is strictly smaller
than $[\fol{U}]$ which yields a contradiction. We conclude that
$p$ maps ${\mathcal R}(\beta(X))$ onto singletons.
Hence $p$ defines a map $p':{\mathcal R}(\beta(X))\to X$.
Since $G$ is first countable and $T_1$ any point $x$
lies in the image of $p'$, namely according to the above remark
we can find a descending sequence of neighborhoods $(\hat{\gamma}(n))_n$
with $\{x\}=\bigcap_n\hat{\gamma}(n)$.
Now choose for any $n$ an $U'_n\in\mathcal{ASG}(G)$ with
$x\in U'_n\subset \hat{\gamma}(n)$,
and set $U_n=\bigcap_{i\leq n}U'_i$. Then $\fl{U}$ is a pre-image of $x$.
To show that $p'$
is injective suppose that $p([\fol{U^1}])=p([\fol{U^2}])$
so that $\fol{V}$ defined by $V_n=U^1_n\cap U^2_n$ is in $\fl{\Gamma}$.
Then $[\fol{V}]\preceq
[\fol{U^i}]$ and by minimality $[\fol{V}]=[\fol{U^i}]$.
It is clear that
${p'}^{-1}(U)={\mathcal U}_U$ for $U\in\Gamma$. Thus $p'$ is a homeomorphism.
So it remains to show that $p'$ preserves composability and
$p'([\fol{U}][\fol{V}])=p'([\fol{U}])p'([\fol{V}])$, in case
$[\fol{U}]\vdash[\fol{V}]$.
Let $[\fol{U}]\vdash[\fol{V}]$. This
is equivalent to $[(d(U)_n)_n]=[(r(U)_n)_n]$, and hence for
$\{x\}=\bigcap_n U_n$ and $\{y\}=\bigcap_n V_n$ it implies $x\vdash y$.
Moreover, in that case
$xy\in\bigcap_n U_n V_n=p([\fol{U}][\fol{V}])$, and
since $p([\fol{U}][\fol{V}])$ is a singleton the claim follows.\hfill q.e.d.
A topological space which has a base consisting
of closed (and open) sets is called zero dimensional. Hence the groupoid
of the last theorem is zero dimensional.
A zero dimensional $T_1$ space is totally disconnected,
i.e.\ the only connected set containing a point $x$
is the singleton (one point set) containing $x$.
\subsubsection{The universal groupoid of an inverse semigroup}
The question of how to assign a groupoid
$G(S)$ to an inverse semigroup $S$ in such
a way that $S$ may be identified with a sub inverse semigroup
of $\mathcal{ASG}(G(S))$ has been thoroughly addressed in \cite{Pat1,Pat2}.
In particular, a construction is presented which yields the universal
groupoid $G_u(S)$ of an inverse semigroup $S$.
We have not made use of Paterson's approach
but followed different lines and therefore include a brief comparison
for completion.
This is best done by first presenting ${\mathcal R}(\Gamma)$
in the manner it has been presented in \cite{Ke5} for tiling almost-groupoid s.
There is a right action of $\Gamma$ on the space of units $\Omega={\mathcal R}(\Gamma)^0$
by means of partial homeomorphisms. Let
\begin{equation} \nonumber
\Omega^\cp := \{(\fl{u},c)\in\Omega\times\Gamma|r(c)\succeq \fl{u}\}
\end{equation}
with relative topology, $\Omega\times\Gamma$ carrying the product topology. Let
$\gamma:\Omega^\cp\rightarrow \Omega:\,
(\fl{u},c)\mapsto d(\fl{u}c)$.
Then $\gamma(\cdot,c):{\mathcal U}_{r(c)}\to{\mathcal U}_{d(c)}$ is
a partial homeomorphism.
Now consider the equivalence relation on $\Omega^\cp$
\begin{equation}
(\fl{u},c)\sim (\fl{u},c')\quad\mbox{whenever}\quad \exists n:
u_n c = u_n c'.
\end{equation}
It is straightforward to see that this definition is independent of the
choice of the representative $\fol{u}$ of $\fl{u}$ and that the relation is
transitive. We denote the equivalence class of $(\fl{u},c)$ by
$[\fl{u},c]$.
\begin{lem}\label{21093}
Let ${\mathcal R}'(\Gamma)$ be quotient of $\Omega^\cp$ by the above equivalence relation
with quotient topology and consider the groupoid structure
defined by
$[\fl{u},c][\fl{u}',c']=[\fl{u},cc']$ provided $\fl{u}'=d(\fl{u} c)$ and
$[\fl{u},c]^{-1}=[d(\fl{u}c),c^{-1}]$.
Then ${\mathcal R}'(\Gamma)$ is a groupoid which is isomorphic to ${\mathcal R}(\Gamma)$.
\end{lem}
{\em Proof:}\
Let $f:\Omega^\cp\to {\mathcal R}(\Gamma)$, $f(\fl{u},c):=\fl{u}c$.
$f$ is surjective, since $\fl{c}=\fl{r(c)}c_1$ for some representative
$\fol{c}$ of $\fl{c}$. If $f(\fl{u},c)=f(\fl{u}',c')$ then first,
$\fl{u}=\fl{u}'$, and second $\exists n:u_n\preceq r(c),r(c')$ so that
$(\fl{u},c)$ and $(\fl{u},c')$ are equivalent in the above sense.
The topology of ${\mathcal R}'(\Gamma)$ is generated by sets of the form
$\left[{\mathcal U}_{u}\times\{c\}\cap\Omega^\cp\right]$. Such a set is equal
to $\left[{\mathcal U}_{r(uc)}\times\{uc\}\right]=\{[\fl{u},uc]|r(uc)\succeq\fl{u}\}$
in case $u\vdash c$ and otherwise empty.
Since
$f^{-1}({\mathcal U}_c)={\mathcal U}_{r(c)}\times\{c\}$ for any $c\in\Gamma$, $f$
induces a homeomorphism between ${\mathcal R}'(\Gamma)$ and ${\mathcal R}(\Gamma)$.
It is straightforward to check that this homeomorphism preserves
multiplication and inversion.\hfill q.e.d.
To compare this with the universal groupoid $G_u$ defined by $\Gamma_0$
\cite{Pat1,Pat2} we assume that $\Gamma$ is countable.
Paterson looks a the space $X$ of all nonzero semicharacters of
$\Gamma^0_0$, i.e.\ at nonzero (inverse semigroup) homomorphisms
$\alpha:\Gamma^0_0\to\{0,1\}$,
the latter being a group under multiplication.
Semicharacters yield an inverse semigroup under point-wise multiplication, but
$X$ not containing the zero map, it is an almost-groupoid\
under point-wise multiplication. We denote by $1$ the semicharacter which
is identically to $1$.
\begin{lem}
The map $\fl{\Gru}^0\to X\backslash\{1\}:\fl{u}\mapsto \alpha_{\fl{u}}$ where
$\alpha_{\fl{u}}(v)=1$ if and only if $v\succeq \fl{u}$ is an isomorphism
of almost-groupoid s (both containing only units).
\end{lem}
Let $\fl{u}\vdash\fl{u}'$ for two elements of $\fl{\Gru}^0$.
Then, for $v\in\Gamma^0_0$, $\fl{u}\fl{u}'\preceq v$ is equivalent to
$\fl{u}\preceq v$ and $\fl{u}'\preceq v$.
Hence $\alpha_{\fl{u}\fl{u}'}=\alpha_{\fl{u}}\alpha_{\fl{u}'}$. The above
map is therefore a homomorphism which is obviously injective.
Let $\alpha\in X$, $\alpha\neq 1$, and
${\Gamma^0}_\alpha=\{u\in\Gamma^0_0|\alpha(u)=1\}$ be the support of $\alpha$.
${\Gamma^0}_\alpha$ is a sub-inverse semigroup of $\Gamma^0_0$ which is lower
directed, i.e.\ any two of its elements have a lower bound in it.
From the countability condition and Lemma~\ref{21091} follows that
$\fl{{\Gamma^0}_\alpha}$ has a unique minimal element, call it $\fl{u}_\alpha$.
Then $\alpha=\alpha_{\fl{u}_\alpha}$. \hfill q.e.d.\bigskip
Identifying $X$ with $\fl{\Gru}^0\cup\{1\}$, where we consider $1$ as an
extra element of $\fl{\Gru}$ which satisfies
$\forall\fl{c}\in \fl{\Gru}:1\fl{c}=\fl{c}=\fl{c}1$ and $11=1$,
Paterson's topology can be described as the one which is
generated by sets of the form
\begin{equation}\label{25091}
A_{u;u_1,\cdots,u_k}:=A_u\cap A_{u_1}^{c}\cap\cdots
\cap A_{u_k}^{c}
\end{equation}
with $u,u_i\in\Gamma^0_0$, $u_i\preceq u$, $A_u=\fl{U}_u\cup\{1\}$, and
$A_{u_i}^c$ here denoting the complement of $A_{u_i}$.
In particular, the relative topology of this topology on $\fl{\Gru}^0$ is finer then
the one we consider.
The universal groupoid $G_u(\Gamma_0)$ is now obtained from a right action
of $\Gamma$ on $X$.
Define
\begin{equation}
X^\cp := \{(\fl{u},c)\in\fl{\Gru}^0\times\Gamma|r(c)\succeq \fl{u}\}\cup\{1\}\times\Gamma
\end{equation}
with relative topology, $X\times\Gamma$ carrying the product topology.
Let $\gamma:X^\cp\rightarrow X$ with $\gamma(x,c) = d(xc)$. Again,
$\gamma(\cdot,c):A_{r(c)}\to A_{d(c)}$ is a partial homeomorphism.
Consider the equivalence relation on $X^\cp$
\begin{equation}
(\fl{u},c)\sim (\fl{u},c')\quad\mbox{whenever}\quad \exists n:
u_n c = u_n c',
\end{equation}
for $\fl{u}\in\fl{\Gru}^0$,
whereas $(1,c)$ is only equivalent to itself.
Again, it is independent of the
choice of representative and
transitive. The universal groupoid
$G_u(\Gamma_0)$ is given by the quotient of $X^\cp$
w.r.t.\ the above equivalence relation
with quotient topology and groupoid structure
defined by $[x,c][x',c']=[x,cc']$ provided $x'=d(xc)$ and
$[x,c]^{-1}=[d(xc),c^{-1}]$, square brackets again denoting equivalence classes.
We will have more to say about the relation between ${\mathcal R}(\Gamma)$ and
$G_u(\Gamma_0)$ in the case where $\Gamma$ is a tiling almost-groupoid.
\subsection{Application to tilings}
Let us see what ${\mathcal R}$ yields applied to the almost-groupoid\ ${\mathcal M}_{\rm I\!I}$ of a tiling $T$.
For that
we consider a notion of radius of a \mixx.
Let $\mbox{\rm rad}:{\mathcal M}_{\rm I\!I}\to\mathbb R^+$ be defined by
the Euclidean distance between the two tiles of the ordered pair
and the boundary of a pattern class\footnote{Choosing a representative
for the pattern class it is the Euclidean distance between
the boundary of the subset it covers and the subset covered by
the two tiles of the ordered pair.}.
In particular $\mbox{\rm rad}(c)=\min\{\mbox{\rm rad}(r(c)),\mbox{\rm rad}(d(c))\}$
and $c\preceq c'$ implies $\mbox{\rm rad}(c)\geq\mbox{\rm rad}(c')$.
Furthermore, let $M_r(c)$, $r > 0$,
be the \mixx\ which is obtained from $c$ by eliminating
all tiles which have distance greater than or equal to $r$ from both pointed
tiles and $M_0(c)$ be the \mixx\ which is given by the pointed tiles only.
The finite type (compactness) condition takes then the form
\begin{itemize}
\item
The set $\{M_r(c)|c\in{\mathcal M}_{\rm I\!I}\}$ is finite for any $r$.
\end{itemize}
Of particular interest are \mixx es called $r$-patches which are those
which satisfy $c=M_r(c)$ and $\mbox{\rm rad}(c)\geq r$.
Consider the metric on $\Gamma$ defined by
\begin{equation}\label{12091}
d(c,c')=\inf(\{e^{-r}|M_r(c)=M_r(c')\}\cup\{e^{-1}\})).
\end{equation}
\begin{thm}\label{16091}
Let ${\mathcal M}_{\rm I\!I}$ be the almost-groupoid\ of a tiling which satisfies the finite type condition.
Then there is a continuous bijection between
$\fl{{\mathcal M}_{\rm I\!I}}$ and the metric completion
of ${\mathcal M}_{\rm I\!I}$ with respect to the above metric.
Furthermore, the sets $U_c$, $c\in{\mathcal M}_{\rm I\!I}$ are metric-compact.
\end{thm}
{\em Proof:}\
Let $\fol{c}$ be a decreasing sequence of \mixx es.
Since $c\succeq c'$ implies $M_r(c)\succeq M_r(c')$ the finite type condition implies
the existence of an $N$ such that for all $n\geq N:M_r(c_n)=M_r(c_N)$.
It follows that
$d(c_n,c_m)\leq e^{-r}$ for $n,m\geq N$, i.e.\ $\fol{c}$ is a
Cauchy sequence.
Moreover, if $\fol{c}$ and $\fol{c'}$ are two decreasing sequences which are
equivalent in the sense (\ref{18069}) a similar argument shows that
$d(c_n,c'_n)\to 0$, i.e.\ that they are equivalent as Cauchy sequences.
Now fix an increasing sequence $(r_k)_k$ of positive numbers which diverges.
If $\fol{c}$ is a Cauchy sequence then
$\forall k\exists N_k\forall n\geq N_k:M_{r_k}(c_{N_k})=M_{r_k}(c_n)$.
Defining $j_k=j(\fol{c})_k=M_{r_k}(c_{N_k})$ yields thus a decreasing
sequence for which $d(j_k,c_{N_k})\to 0$, i.e.\ which is equivalent to
$\fol{c}$ as Cauchy sequence.
Moreover, if $\fol{c}$ and $\fol{c'}$ are Cauchy equivalent sequences
then $j(\fol{c})=j(\fol{c'})$. If $\fol{c}$ is decreasing, then not only
$j(\fol{c})\succeq \fol{c}$, but
since $\forall n\exists k:r_k>\mbox{\rm rad}(c_n)$
also $j(\fol{c})\preceq \fol{c}$. So if $j(\fol{c})$ and $j(\fol{c'})$
are not equivalent
in the sense (\ref{18069}) they cannot belong to the same Cauchy class.
Therefore is the map
which sends $\fl{c}$ to its Cauchy class a well defined bijection
between $\fl{{\mathcal M}_{\rm I\!I}}$ and the metric completion of ${\mathcal M}_{\rm I\!I}$.
To compare the topologies
extend $M_r$ to $\fl{{\mathcal M}_{\rm I\!I}}$
through $M_r(\fl{c})=\lim_nM_r(c_n)$, $[\fol{c}]=\fl{c}$.
The limit exists and is independent of the chosen representative
by the same argument as above which in fact shows that
$\lim_nM_r(c_n)=M_r(c_N)$ for some $N$.
It is then straightforward to check that
the extension of the metric to the completion of ${\mathcal M}_{\rm I\!I}$ is given by formally
the same expression for $d$ as in (\ref{12091}).
Again using the finite type condition one sees that
the image of the (continuous) function $d(\fl{c},\cdot):\fl{{\mathcal M}_{\rm I\!I}}\to\mathbb R^+$
is discrete apart from a limit point at $0$. Therefore
$\epsilon$-neighbourhoods are closed and hence complete in the metric topology.
$\epsilon$-neighbourhoods are sets of the form
$$U_r(\fl{c})=\{\fl{c}'|M_r(\fl{c}')=M_r(\fl{c})\}$$
(the smaller $\epsilon$ the bigger $r$)
but since $r$ is finite $U_r(\fl{c})=U_r(c_n)$ for some $n$
and representative $\fol{c}$.
If $0<r_1<r_2$ then $U_{r_1}(c)=\bigcup_{c'|M_{r_1}(c')=M_{r_1}(c)}
U_{r_2}(c')$ but by the finite type condition only finitely many
sets in the union of the r.h.s.\ are mutually disjoint.
Thus for any $0<\epsilon_2<\epsilon_1$ holds that the
$\epsilon_1$-neighbourhood has a finite cover by $\epsilon_2$-neighbourhoods,
i.e.\ $\epsilon$-neighbourhoods are pre-compact and hence compact.
If $c$ is an $r$-patch then
$U_r(c)=U_c$. For arbitrary $c\in{\mathcal M}_{\rm I\!I}$ one has
$U_c=\bigcup_{c'\preceq c} U_r(c')$ where $r$ is some number bigger
than the diameter of $c$ (the diameter of the set covered by a
representative of the pattern class in $\mathbb R^d$).
In particular, the
metric topology is finer than the original topology on $\fl{{\mathcal M}_{\rm I\!I}}$.
But moreover, only finitely many
sets in the union of the r.h.s.\ are mutually disjoint
so that the $U_c$ are metric-compact.\hfill q.e.d.\bigskip
To proceed let us extend the radius function
$\mbox{\rm rad}:\fl{{\mathcal M}_{\rm I\!I}}\to\mathbb R^+\cup\{\infty\}$
through $\mbox{\rm rad}(\fl{x})=\lim_n \mbox{\rm rad}(x_n)$
the r.h.s.\ being independent of the representative.
\begin{lem} \label{18091}
Let ${\mathcal M}_{\rm I\!I}$ be the almost-groupoid\ of a tiling which satisfies the finite type condition.
$\fl{c}$ is minimal if and only if $\mbox{\rm rad}(\fl{c})=\infty$. Stated differently,
a sequence $\fol{c}\in{{\mathcal M}_{\rm I\!I}}^\mathbb N_\succeq$ is approximating
if and only if the sequence $(\mbox{\rm rad}(c_n))_n$ diverges.
\end{lem}
{\em Proof:}\
Suppose that $\mbox{\rm rad}(\fl{c})=R'<\infty$ and let $R>R'$.
There is at least one but at most finitely many $R$-patches $d_1,\dots,d_k$
for which $d_i\preceq M_R(\fl{c})$.
Now consider the sequence which is obtained from a representative $\fol{c}$
of $\fl{c}$ by replacing each $c_n$ by $k$ elements
$r(d_1)c_n,\dots,r(d_k)c_n$.
Since $U_{c_1}$ is metric-compact
the sequence has a metric-convergent subsequence, say $\fol{c'}$,
which we may assume to be decreasing (if not apply the map $j$ defined
in the proof of Theorem~\ref{16091}).
But then $\fol{c'}\preceq\fol{c}$ and since $\mbox{\rm rad}(\fol{c'})\geq R$,
$\fol{c'}$ cannot be equivalent to $\fol{c}$. Hence $\fl{c}$ is not minimal.
For the converse suppose that $\mbox{\rm rad}(c_n)$ diverges and $\fol{c'}\preceq\fol{c}$.
Since for all $n$ there is an $m$ such that $\mbox{\rm rad}(c_m)$ is larger
than the diameter of $c'_n$
this implies $c'_n\succeq c_m$ and thus
$\fol{c'}\succeq\fol{c}$. Note that we do not need the finite type condition
for this part.\hfill q.e.d.
\begin{lem} \label{20092}
Let ${\mathcal M}_{\rm I\!I}$ be the almost-groupoid\ of a tiling which satisfies the finite type condition.
Then the relative topologies on ${\mathcal R}({\mathcal M}_{\rm I\!I})$ coincide and
$\Gr(\mTxx)$ is metric-closed in $\fl{{\mathcal M}_{\rm I\!I}}$.
\end{lem}
{\em Proof:}\
The relative metric-topology on ${\mathcal R}({\mathcal M}_{\rm I\!I})$ is generated by sets
$U_r(\fl{c})\cap\Gr(\mTxx)$ where $\mbox{\rm rad}(\fl{c})=\infty$. Hence
$M_r(\fl{c})=M_r(c')$ for some $r$-patch $c'$ and thus $U_r(\fl{c})=U_{c'}$.
This shows that $U_r(\fl{c})\cap\Gr(\mTxx)$ is open with respect
to the original topology on $\Gr(\mTxx)$, i.e.\ the latter is finer
than the relative metric-topology. By Theorem~\ref{16091}
the topologies coincide.
Now suppose that $\fl{x}$ is not minimal, i.e.\ $\mbox{\rm rad}(\fl{x})=R'<\infty$.
Let $R>R'$ and $\fl{y}$ be an element of the $e^{-R}$-neighbourhood
of $\fl{x}$. Then $\mbox{\rm rad}(\fl{y})=R'$ as well,
and hence $\fl{y}$ is not minimal, i.e.\ $\fl{{\mathcal M}_{\rm I\!I}}\backslash\Gr(\mTxx)$
is metric open.\hfill q.e.d.
\begin{cor}
Under the requirements of Theorem~\ref{16091}
is ${\mathcal U}_c$ compact. In particular is
${\mathcal R}({\mathcal M}_{\rm I\!I})^0$ a compact zero dimensional metric space
and $\beta_o(\Gr(\mTxx))$ a sub-inverse semigroup of $\mathcal{ASG}(\Gr(\mTxx))$.
\end{cor}
The compactness of ${\mathcal U}_c$ follows from Theorem~\ref{16091} and
Lemma~\ref{20092}. Writing ${\mathcal R}({\mathcal M}_{\rm I\!I})^0=\bigcup_u{\mathcal U}_u$, the union being
taken over all $u\in{\mathcal M}_{\rm I}$ which consists only of one tile shows that
the finite type condition implies compactness for ${\mathcal R}({\mathcal M}_{\rm I\!I})^0$.
Roughly speaking, we have shown that the elements of $\Gr(\mTxx)$ can be seen
as limits of \mixx es whose radii eventually become infinite.
This can be formulated as follows:
To a given approximating sequence $\fol{c}$ construct a covering of $\mathbb R^d$
by first choosing a representative $\hat{c}_1$ for $c_1$ in $\mathbb R^d$.
Then there are unique representatives $\hat{c}_n$ for $c_n$ such that
$\hat{c}_n$ is obtained from $\hat{c}_1$ by addition of tiles
(but keeping the ordered pair fixed).
Since $\mbox{\rm rad}(c_n)$ diverges
$\bigcup_n\hat{c}_n$ is a covering of $\mathbb R^d$
(each $\hat{c}_n$ is a set of tiles) together with an ordered pair
of tiles. We call this a doubly pointed tiling. The elements of $\Gr(\mTxx)$
are the classes of doubly pointed tilings which are obtained in this way.
The set of units $\Omega={\mathcal R}({\mathcal M}_{\rm I\!I})^0={\mathcal R}({\mathcal M}_{\rm I})$
can than be identified with classes of
tilings together with one chosen tile. It is called the hull of the tiling.
The relative Paterson topology on $\Omega$, c.f.\ (\ref{25091}), coincides
with the topology on $\Omega$ considered above, since
the sets ${\mathcal U}_u$, $u\in{\mathcal M}_{\rm I\!I}^0$, which generate the latter are closed.
Moreover, since
$$\fl{U}_r(c)=\fl{U}_{M_r(c)}\backslash
\bigcup_{\mbox{\tiny $r$-patches }c'\neq c,c'\preceq c}\fl{U}_{M_r(c')}$$
the relative Paterson topology on $\fl{\Gru}^0$ is finer than the metric
topology and hence $\Omega$ is a Paterson-closed subset of $X$.
It follows that ${\mathcal R}'({\mathcal M}_{\rm I\!I})$ is a reduction of the
universal groupoid $G_u({\mathcal M}_{\rm I\!I}\cup\{0\})$ with respect to the
subset $\Omega$, which fits well into the general theory of \cite{Pat2}.
For later use we proof:
\begin{lem}\label{16092}
Let ${\mathcal M}_{\rm I\!I}$ be the almost-groupoid\ of a tiling which satisfies the finite type condition.
Then any $\fl{c}\in\fl{\Gru}$ has a smaller minimal element.
\end{lem}
{\em Proof:}\
Suppose that $\fl{c}$ is not minimal and therefore $\mbox{\rm rad}(\fl{c})=R<\infty$.
Fix an increasing diverging sequence of real numbers
$(r_k)_k$ which are greater than $R$.
As in the proof of Lemma~\ref{18091} we construct $\fl{c}'_k$ such that
$\fl{c}'_k\preceq \fl{c}$ and $\mbox{\rm rad}(\fl{c}'_k)\geq r_k$.
Hence $\fl{c}'_k\in U_r(\fl{c})$ and since the latter is metric-compact
the sequence $(\fl{c}'_k)_k$ has a
metric-convergent subsequence
converging to a class $\fl{c}'$ which is smaller than $\fl{c}$ and minimal.\hfill q.e.d.
\subsubsection{A continuous groupoid associated to the tiling}
There is another topological groupoid one can assign to a tiling, which we
want to mention for comparison. Here one starts with the local isomorphism
class ${\mathcal L}_{\mathcal T}$ of a tiling ${\mathcal T}$. This is the space of all tilings which
are locally isomorphic to ${\mathcal T}$.
${\mathcal L}_{\mathcal T}$ may be obtained as the closure
of the orbit of ${\mathcal T}$ under the action of the group $\mathbb R^d$ of translations
with respect to a metric. In fact, viewed as a geometrical object a tiling
may be translated, ${\mathcal T}-x$, $x\in\mathbb R^d$ is the covering given by
the sets $t-x:=\{y-x|y\in t\}$ where $t$ runs through all tiles of ${\mathcal T}$.
Then ${\mathcal L}_{\mathcal T}$ is the closure of $\{{\mathcal T}-x|x\in\mathbb R^d\}$ under the metric
$$ d(T,T')=\inf(\{\{\epsilon |\exists x,x'\in
\mathbb R^d: r(T-x,T'-x')\geq \frac{1}{\epsilon},
|x|,|x'|<\epsilon\}\cup\{\frac{1}{\sqrt{2}}\}) $$
where $r(T,T')$ is the largest $r$ such that
$T$ and $T'$ agree on the $r$-ball around $0$ \cite{AP}.
The other groupoid which may now be assigned to ${\mathcal T}$ is the transformation
group ${\mathcal C}_{\mathcal T}:={\mathcal L}_{\mathcal T}\times\mathbb R^d$.
Two of its elements $(T,x)$, $(T',x')$ are composable whenever
$T'=T-x$ and then $(T,x)(T',x')=(T,x+x')$. The topology is the
product topology.
For distinction we call it the continuous groupoid assigned to the
tiling as opposed to the discrete one. How is it
related to ${\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}))$?
Fix for each tile-class a point in its interior, we call it a puncture.
The punctures of the tiles of a tiling may be identified with a countable
subset of $\mathbb R^d$.
Let $\Omega_{\mathcal T}$ be the
subset of ${\mathcal L}_{\mathcal T}$ which consists of tilings with
the property that the puncture of one of its tiles
identifies with $0\in\mathbb R^d$.
The reduction of ${\mathcal C}_{\mathcal T}$ by $\Omega_{\mathcal T}$, which is the sub-groupoid
$\{(T,x)\in{\mathcal C}_{\mathcal T}|T,T-x\in\Omega\}$, is the groupoid which has been
associated to an aperiodic tiling in \cite{Ke2}
It is isomorphic to ${\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}))$,
an isomorphism is given by the map which assigns to $(T,x)$ the
doubly pointed tiling class which is given by the class of $T$ and the
pair of tiles given by, first, the one which covers $0$, and
second, the one which covers $x$.
Moreover, it has been proven by Anderson and Putnam \cite{AP}
that the above reduction of ${\mathcal C}_{\mathcal T}$ is an abstract transversal of ${\mathcal C}_{\mathcal T}$
in the sense of Muhly et al.\ so that by the work of the latter authors
\cite{MRW} the groupoid-$C^*$-algebra s of ${\mathcal C}_{\mathcal T}$ and ${\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}))$ are
stably isomorphic.
\section{Topological equivalence and mutual local derivability}
If we focus on the role tilings play in solid state physics
when describing spatial structures, then several properties of the tiling
are unimportant. First of all, only the congruence class of the tiling
matters, and second, due to the locality of the interactions
locally isomorphic tilings are equally well suited to describe that structure.
This can now all be taken into account by working with the almost-groupoid\ of the tiling.
However, investigating further the way how tilings model e.g.\ the
arrangement of atoms (or ions) in solids one may take the point of view that
this should only be understood in a topological way. In particular
details like the precise position and strength of the bondings are to be added,
i.e.\ are not to be derived from the tiling.
This led Baake et al.\ from the theoretical physics group in T\"ubingen
to introduce another
equivalence relation between tilings which is based on mutual local
derivability \cite{BSJ}, see also \cite{BaSc} for an overview.
Let $B_r(x)$ denote the closed
$r$-ball around $x$ and $B_r=B_r(0)$. Furthermore ${\mathcal T}\sqcap B_r(x)$
is the pattern consisting of all tiles of ${\mathcal T}$ which intersect with
$B_r(x)$.
\begin{df}
${\mathcal T}_2$ is locally derivable from ${\mathcal T}_1$ if there is an $r\geq 0$ such
that for all $x,y\in\mathbb R^d$
\begin{equation}\label{12061}
({\mathcal T}_1-x)\sqcap B_r = ({\mathcal T}_1-y)\sqcap B_r\quad\mbox{implies}\quad
({\mathcal T}_2-x)\sqcap \{0\} = ({\mathcal T}_2-y)\sqcap \{0\}.
\end{equation}
\end{df}
Restricting our interest to tilings which satisfy the finite type condition
the knowledge of the correspondence between
$({\mathcal T}_1-x)\sqcap B_r$ and $({\mathcal T}_2-x)\sqcap \{0\}$ for finitely many $x$
is enough to construct all tiles of ${\mathcal T}_2$ from ${\mathcal T}_1$. This obviously
defines a map $\ell:{\mathcal L}({\mathcal T}_1)\to{\mathcal L}({\mathcal T}_2)$, which is continuous, has
dense image and is therefore surjective. $\ell$ can be extended
to a surjective homomorphism of groupoids, $\ell:{\mathcal C}_{\mathcal T}\to{\mathcal C}_{{\mathcal T}'}$:
$(T,x)\mapsto (\ell(T),x)$.
We may call the replacement of
${\mathcal T}_1\sqcap B_r(x)$ by ${\mathcal T}_2\sqcap \{x\}$
a local derivation rule.
In particular the above definition is equivalent
to saying that for all $r'\geq 0$ there is an $r\geq 0$ such
that for all $x,y\in\mathbb R^d$
\begin{equation}\label{12062}
({\mathcal T}_1-x)\sqcap B_r = ({\mathcal T}_1-y)\sqcap B_r\quad\mbox{implies}\quad
({\mathcal T}_2-x)\sqcap B_{r'} = ({\mathcal T}_2-y)\sqcap B_{r'}.
\end{equation}
${\mathcal T}_1$ and ${\mathcal T}_2$ are called mutually locally derivable
if ${\mathcal T}_2$ is locally derivable from ${\mathcal T}_1$ and vice versa. This is an
equivalence relation which can be extended by saying that
${\mathcal T}_1$ and ${\mathcal T}_2$ belong to the same MLD-class if there is a ${\mathcal T}'_2$
which is locally isomorphic to
${\mathcal T}_2$ and mutually locally derivable from
${\mathcal T}_1$. That this extension is an equivalence relation
(in fact on LI-classes) follows from the observation that
if ${\mathcal T}_2$ is locally derivable from ${\mathcal T}_1$
and ${\mathcal T}'_1$ is locally isomorphic to ${\mathcal T}_1$ then the local derivation
rule can be used to locally derive a tiling
${\mathcal T}'_2$ from ${\mathcal T}'_1$. Then ${\mathcal T}'_2$
has to be locally isomorphic to ${\mathcal T}_2$.
Moreover, the local derivation of ${\mathcal T}_1$ from ${\mathcal T}_2'$ yields the inverse
of $\ell:{\mathcal C}_{\mathcal T}\to{\mathcal C}_{{\mathcal T}'}$ so that the latter becomes an isomorphism.
\begin{cor}\label{17092}
If ${\mathcal T}$ and ${\mathcal T}'$ are in the same MLD-class then the groupoids
${\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}))$ and ${\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}'))$
are reductions (in fact abstract transversals) of the same groupoid.
In particular they have stably isomorphic groupoid-$C^*$-algebra s.
\end{cor}
This follows directly from the fact that ${\mathcal C}_{\mathcal T}$ and ${\mathcal C}_{{\mathcal T}'}$
are isomorphic and the above mentioned theorem of \cite{AP}.
The above corollary indicates that the T\"ubingen
formulation of local derivability is a good starting point to answer
the question under which circumstances ${\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}))$ and ${\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}'))$
are isomorphic.
In order to cast it
in a form applicable to our framework, using almost-groupoid s and
the discrete groupoid,
we are naturally led to strengthen and at the
same time to generalize the concept of local derivation.
A strengthening comes along with the idea of preservation of the average
number of tiles per unit volume
whereas a generalization is necessary
as we want to work in a purely topological setting.
\subsection{Constructing local morphisms from local derivation rules}
Suppose that ${\mathcal N}$ is a sub-almost-groupoid\ of $\Gamma$
which is the order ideal generated by a finitely generated almost-groupoid, i.e.\
${\mathcal N}=I(\erz{{\mathcal C}})$ where ${\mathcal C}$ is a finite set and $\erz{{\mathcal C}}$ the
almost-groupoid\ generated by it.
Suppose furthermore that we have a map $\hat{\varphi}:{\mathcal C}\to\Gru'$
which satisfies conditions
which arise if it were the restriction of a prehomomorphism from ${\mathcal N}$
into an almost-groupoid\ $\Gamma'$. A question which is of prime importance for sequel
is whether we can construct
a local morphism $\varphi:{\mathcal N}\to\Gamma'$ from that map.
We call $n$ elements $c_1,\dots,c_n$ collatable if they may be composed,
i.e.\ if $\forall 1\leq k < n:c_1\dots c_k \vdash c_{k+1}$.
Let ${\mathcal C}^{-1}={\mathcal C}$ be a finite subset of an almost-groupoid\ and $\hat{\varphi}:{\mathcal C}\to\Gru'$
be a map into another almost-groupoid\ which satisfies for all $c,c_i\in{\mathcal C}$:
\begin{itemize}
\item[E1] $\hat{\varphi}(c^{-1})=\hat{\varphi}(c)^{-1}$,
\item[E2] if
$c_1,\dots ,c_n$ are collatable then $\hat{\varphi}(c_1),\dots ,\hat{\varphi}(c_n)$
are collatable,
\item[E3] if
$c_1\dots c_n$ is a unit then
$\hat{\varphi}(c_1)\dots \hat{\varphi}(c_n)$ is a unit.
\end{itemize}
Consider for $c\in\erz{{\mathcal C}}$
\begin{equation}
\Phi(c)=\{\hat{\varphi}(c_1)\dots \hat{\varphi}(c_n)|c_1\dots c_n=c,c_i\in{\mathcal C}\}.
\end{equation}
Since $c_1\dots c_n=c'_1\dots c'_{n'}$ implies that
$\hat{\varphi}(c_1)\dots \hat{\varphi}(c_n)(\hat{\varphi}(c'_1)\dots
\hat{\varphi}(c'_{n'}))^{-1}$
is a unit any two elements of $\Phi(c)$ have a common smaller element,
i.e.\ $\Phi(c)$ is a lower directed set.
Provided $\Phi(c)$ is finite we define
\begin{equation} \label{07034}
\varphi(c):=\min\Phi(c).
\end{equation}
Then $\varphi$
commutes with the inverse map, because of
$\Phi(c)^{-1}=\Phi(c^{-1})$, and it satisfies inequality (\ref{07031})
since $\Phi(c_1)\Phi(c_2)\subset \Phi(c_1c_2)$.
Thus $\varphi:\erz{{\mathcal C}}\to\Gru'$ is a prehomomorphism.
If $H_{\mathcal C}(c):=\{c'\in\erz{{\mathcal C}}|c'\succeq c\}$ has a unique minimal element
then $\pi:I(\erz{{\mathcal C}})\to \erz{{\mathcal C}}$: $\pi(c)=\min H_{\mathcal C}(c)$ is a
prehomomorphism as well, and we may extend $\varphi$ through
$\varphi\circ \pi$.
\begin{df}\label{18062}
We call a pair $(\varphi,{\mathcal C})$,
where ${\mathcal C}={\mathcal C}^{-1}\subset\Gamma$ is finite and
$\hat{\varphi}:{\mathcal C}\to\Gru'$ satisfies conditions E1-3,
a local derivation rule\ if it leads for all $c\in\Gamma$ to finite lower directed sets
$\Phi(c)$ and $H_{\mathcal C}(c)$ and $\varphi:I(\erz{{\mathcal C}})\to\Gamma'$,
\begin{equation}\label{17091}
\varphi(c):=\min\Phi(\min H_{\mathcal C}(c))
\end{equation}
is approximating.
\end{df}
\begin{lem}\label{13031}
Let ${\mathcal M}_{\rm I\!I}$ and ${\mathcal M}_{\rm I\!I}'$ be two tiling almost-groupoid s. Suppose that
there exist a finite ${\mathcal C}={\mathcal C}^{-1}\subset{\mathcal M}_{\rm I\!I}$ and
a map $\hat{\varphi}:{\mathcal C}\to{\mathcal M}_{\rm I\!I}'$ which
satisfies E1-3. Then $\Phi(c)$ and $H_{\mathcal C}(c)$ are finite lower directed sets.
\end{lem}
{\em Proof:}\
$H_{\mathcal C}(c)$ is finite since any \mixx\ has only finitely many tiles.
It is lower directed since ${\mathcal M}_{\rm I\!I}$ is unit hereditary.
As for $\Phi(c)$ we subdivide this set first into subsets
$c'\Phi_{c'c''}(c)c''$ where
$\Phi_{c'c''}(c):=\{
\hat{\varphi}(u_1)\dots\hat{\varphi}(u_n)|n\in\mathbb N,c=c' u_1\dots u_n c''\}$,
$u_i\in\erz{{\mathcal C}}^0$ and
$c'=c'_1\cdots c'_k,c'_i\in{\mathcal C}$ none of the
$c'_i\cdots c'_j$, $1\leq i\leq j\leq k$, being a unit, and
the same conditions for $c''$.
Since there are only finitely many different
units which satisfy $u \succeq {c'}^{-1}c{c''}^{-1}$, units commute, and
$\varphi(u) \varphi(u)=\varphi(u)$, $\Phi_{c'c''}(c)$ is finite.
Moreover, there are only finitely many different possibilities to choose
$c',c''$ so that $\Phi(c)$ is finite.\hfill q.e.d.\bigskip
There is no reason why $\varphi$ should be approximating.
To connect the T\"ubingen formulation of local derivability with
the above and justify double use of the
word local derivation rule\ we proof:
\begin{thm}
Let ${\mathcal T}'$ be locally derivable from ${\mathcal T}$. Then there exists a
local derivation rule\ in the sense of Definition~\ref{18062},
$\hat{\varphi}:{\mathcal C}\subset{\mathcal M}_{\rm I\!I}({\mathcal T})\to {\mathcal M}_{\rm I\!I}({\mathcal T}')$, such that
${\mathcal R}(I(\erz{{\mathcal C}}))={\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}))$ and
the induced homomorphism maps the class of ${\mathcal T}$ onto that of ${\mathcal T}'$.
\end{thm}
{\em Proof:}\
First introduce punctures for the tile classes of ${\mathcal T}$
which are chosen such
that none of the punctures of tiles of ${\mathcal T}$ lies on the boundary of
tiles of ${\mathcal T}'$.
For given $r'$ fix $r$ according to (\ref{12062}) and
let for any tile $t$ of ${\mathcal T}$, $\hat{\ell}(t)={\mathcal T}'\sqcap B_{r'}(t^{pct})$,
where $t^{pct}$ is the puncture of $t$.
We now define a local derivation rule\ on the set $\Cp{r}$ of all $r$-patches $c$ for which
$M_0(c)$ is connected.
Let $m$ be a doubly pointed
pattern in ${\mathcal T}$ of the class $\fl{m}\in\Cp{r}$. Denote the $i$th tile of its
ordered pair by $t_i(m)$.
Then $\hat{\varphi}(\fl{m})$ shall be the
class of the pattern
$\hat{\ell}(t_1(m))\cup\hat{\ell}(t_2(m))$ with the ordered pair
$({\mathcal T}'\sqcap B_0(t_1(m)^{pct}),{\mathcal T}'\sqcap B_0(t_2(m)^{pct}))$.
That $\hat{\varphi}(\fl{m})$ does not depend on the chosen representative
$m$ for $\fl{m}$
is precisely the definition of local derivability.
Defined in that geometrical way,
it is easy to see that $\hat{\varphi}$ satisfies the conditions E1-3.
If $r'$ is larger than twice the diameter of the largest tile in
${\mathcal T}_1$ then $\hat{\ell}(t_1(m))\cup\hat{\ell}(t_2(m))$ is connected and
$\varphi$ approximating. By construction it maps the class of
${\mathcal T}$ onto that of ${\mathcal T}'$.
\hfill q.e.d.\bigskip
Although the local derivation rule\ $\hat{\varphi}$ yields a homomorphism ${\mathcal R}(\varphi)$ which
is very similar to a restriction of the map
$\ell:{\mathcal C}_{\mathcal T}\to{\mathcal C}_{{\mathcal T}'}$
constructed from the local derivation rule\ in the T\"ubingen version
it is neither injective nor surjective, in general.
The geometrical picture of $\ell:{\mathcal C}_{\mathcal T}\to{\mathcal C}_{{\mathcal T}'}$ allows one to conclude
that ${\mathcal R}(\varphi)$
is surjective whenever the punctures for the tiles of ${\mathcal T}$ can be chosen
in such a way that
any tile of ${\mathcal T}'$ contains at least one puncture.
(First, doubly pointed tiling classes $\fl{c}\in{\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}'))$
for which $r(\fl{c})$ is in the same class then ${\mathcal T}'$ lie in the image
of ${\mathcal R}(\varphi)$, and then, by continuity, all of ${\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}'))$.)
Similarly, a necessary (but not sufficient)
condition for ${\mathcal R}(\varphi)$ to be injective is
that any tile of ${\mathcal T}'$ contains at most one puncture.
Hence the failure of ${\mathcal R}(\varphi)$
to be an isomorphism may have its cause in that
the average number of tiles per unit volume is not preserved.
The converse of the theorem is false.
If ${\mathcal T}'$ is obtained from ${\mathcal T}$ by a change of length scale
or an overall rotation there would (apart from symmetric cases) not be a
local derivation rule in the T\"ubingen sense but a local
derivation in the sense of Definition~\ref{18062} is
is given by applying the change of length scale
resp.\ rotation to the \mixx es.
\subsection{Topological equivalence}
An answer to the question under which circumstances two tilings lead to
isomorphic groupoids shall be given here
in purely "local" terms, i.e.\ in terms
of almost-groupoid s and local derivation rules.
For that let us start with a lemma.
Let us use the notation that for subsets of an ordered set $X\preceq Y$
if $\forall y\in Y \exists x\in X: x\preceq y$.
\begin{lem}
Let $\varphi$ be a local morphism from a countable unit hereditary almost-groupoid\
$\Gamma$ into itself.
Then ${\mathcal R}(\varphi)=\mbox{\rm id}$ if an only if
$D(\varphi)\preceq \Gamma$ and $\varphi(c)$ and $c$ have
for all $c\in D(\varphi)$
a lower bound.
\end{lem}
{\em Proof:}\
Suppose first that ${\mathcal R}(\varphi)=\mbox{\rm id}$ which in particular means
${\mathcal R}(D(\varphi))={\mathcal R}(\Gamma)$. Let $c\in\Gamma$, by Lemma~\ref{21091}
there is a smaller minimal element $[\fol{c}]$.
It has a representative $\fol{c}$, $c_n\in D(\varphi)$.
But then there exists already
some $c_n\in D(\varphi)$ for which $c_n\preceq c$.
Furthermore, $\varphi(c_n)$ and $c_n$ must have for any $n$ a lower bound
since they constitute equivalent sequences. Any such bound is also
a lower bound for $\varphi(c)$ and $c$.
As for the converse observe that under the assumption that
$\varphi(c)$ and $c$ have a lower bound for all $c\in D(\varphi)$ we have
$\fl{\varphi}(\fl{c})d(\fl{c})\preceq \fl{\varphi}(\fl{c}),\fl{c}$
and hence for minimal $\fl{c}$: $\fl{\varphi}(\fl{c})=\fl{c}$. Hence
${\mathcal R}(\varphi)=\mbox{\rm id}|_{{\mathcal R}(D(\varphi))}$.
But since $D(\varphi)$ is an order ideal,
$D(\varphi)\preceq \Gamma$ implies ${\mathcal R}(D(\varphi))={\mathcal R}(\Gamma)$.\hfill q.e.d.
\begin{df}\label{17094} Two countable unit hereditary almost-groupoid s
$\Gamma$ and $\Gamma'$ are called topologically equivalent
if there are local derivation rule s $\hat{\varphi}:{\mathcal C}\subset\Gamma \to\Gamma'$,
$\hat{\psi}:{\mathcal C}'\subset\Gamma'\to\Gamma$ such that for the induced
local morphisms $\varphi$ resp.\ $\psi$ holds
$D(\psi\circ\varphi)\preceq \Gamma$, $D(\varphi\circ\psi)\preceq \Gamma'$,
and $\psi(\varphi(c))$ and $c$ have
for all $c\in D(\psi\circ\varphi)$ resp.\
$\varphi(\psi(c'))$ and $c'$
for all $c'\in D(\varphi\circ\psi)$ a lower bound.
Two tilings of finite type are called
topologically equivalent if their corresponding almost-groupoid s are
topologically equivalent.
\end{df}
According to the above lemma the definition of topological equivalence may
equally well be formulated by saying that the local morphisms $\varphi$
and $\psi$ satisfy ${\mathcal R}(\psi\circ\varphi)=\mbox{\rm id}$ on ${\mathcal R}(\Gru)$ and
${\mathcal R}(\varphi\circ\psi)=\mbox{\rm id}$ on ${\mathcal R}(\Gru')$.
By the functorial properties of ${\mathcal R}$ it implies that ${\mathcal R}(\Gamma)$
and ${\mathcal R}(\Gamma')$ are isomorphic and
shows at once that topological equivalence is indeed
an equivalence relation.
According to Remark~1, being in the same MLD-class
is not sufficient to guarantee that the tilings are isomorphic.
It is sufficient only in case
any tile of ${\mathcal T}'$ contains exactly one of the punctures of ${\mathcal T}$.
\begin{thm}\label{17093}
Two almost-groupoid s of finite type tilings
are topologically equivalent whenever their associated
groupoids are isomorphic.
\end{thm}
{\em Proof:}\
We already mentioned above that topological equivalence implies
the existence of an isomorphism between the associated
groupoids.
For the converse let $f:{\mathcal R}({\mathcal M}_{\rm I\!I})\to{\mathcal R}({\mathcal M}_{\rm I\!I}')$ be an isomorphism,
${\mathcal R}({\mathcal M}_{\rm I\!I})={\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}))$, ${\mathcal R}({\mathcal M}_{\rm I\!I}')={\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}'))$.
Let $Y\subset {\mathcal R}({\mathcal M}_{\rm I\!I})$ resp.\ $Y'\subset {\mathcal R}({\mathcal M}_{\rm I\!I}')$
be the set of elements $y$ such that $M_0(y)$ is connected.
Furthermore, let $X=Y\cup f^{-1}(Y')$ and
${\mathcal C}(r)=\{M_r(\alpha)|\alpha\in X\}$.
Since
$f:{\mathcal R}({\mathcal M}_{\rm I\!I})\to {\mathcal R}({\mathcal M}_{\rm I\!I}')$ is continuous and $X$ compact,
\begin{equation} \label{07032}
\forall r'>0\exists r>0\forall \alpha\in X:
f({\mathcal U}_{M_r(\alpha)})\subset{\mathcal U}_{M_{r'}(f(\alpha))}.
\end{equation}
Choose $r>0$ and $r'>0$ satisfying (\ref{07032}), and define
$\hat{\varphi}:{\mathcal C}(r)\to{\mathcal M}_{\rm I\!I}'$ by
\begin{equation}
\hat{\varphi}(M_r(\alpha)) := M_{r'}(f(\alpha)).
\end{equation}
In particular (\ref{07032}) implies
\begin{equation} \label{07033}
f({\mathcal U}_{c})\subset{\mathcal U}_{\hat{\varphi}(c)}
\end{equation}
for all $c\in {\mathcal C}(r)$. To show that $\hat{\varphi}$ is a local derivation rule\ we first check
E1-3. E1 is clearly satisfied.
Using set multiplication and the convention that ${\mathcal U}_{cc'}={\mathcal U}_0=\emptyset$
if $c\not\vdash c'$ we obtain for collatable $c_1,\dots, c_n$
\begin{equation}\label{13032}
f ( {\mathcal U}_{c_1\dots c_n} ) = f ({\mathcal U}_{c_1})\dots f({\mathcal U}_{c_n})\subset
{\mathcal U}_{\varphi(c_1)\dots \varphi(x_n)}
\end{equation}
where we used (\ref{28051}) and that $f$ is a homomorphism of groupoids.
Therefore $ {\mathcal U}_{\varphi(c_1)\dots \varphi(x_n)}$
cannot be empty and hence $\varphi$ satisfies E2.
To show E3 let $c_1\dots c_n$ be a nonzero unit. Then $f({\mathcal U}_{c_1\dots c_n})
\subset{\mathcal R}({\mathcal M}_{\rm I\!I})^0$. Since, for tilings, either
${\mathcal U}_c\cap{\mathcal R}({\mathcal M}_{\rm I\!I})^0=\emptyset$ or ${\mathcal U}_c\subset{\mathcal R}({\mathcal M}_{\rm I\!I})^0$
(\ref{13032}) implies E3 for $\varphi$.
Therefore $\varphi$ extends to a prehomomorphism. Clearly
$D(\varphi)=I(\erz{{\mathcal C}})\preceq{\mathcal M}_{\rm I\!I}$.
Moreover, (\ref{13032}) implies that (\ref{07033}) holds even for all
$c\in I(\erz{{\mathcal C}})$.
Therefore, if $\fol{c}$ is an approximating sequence, then
$f([\fol{c}])\in\bigcap_n {\mathcal U}_{\varphi(c_n)}$ or, stated differently,
$f([\fol{c}])\preceq\fl{\varphi}([\fol{c}])$. Hence if
$\varphi$ is approximating then ${\mathcal R}(\varphi)=f$.
So far we have only used that $f$ is a homomorphism. To show that
$\varphi$ is approximating we need to use its invertibility.
Having nothing specific said about the choice of $r,r'$
we choose them now in a way that there exist
$0< r_2\leq r$ and $r_1'\geq r'$ such that apart from (\ref{07032}) also holds
$f^{-1}({\mathcal U}_{M_{r_1'}(\beta)})\subset{\mathcal U}_{M_{r}(f^{-1}(\beta))}$ and
$f^{-1}({\mathcal U}_{M_{r'}(\beta)})\subset{\mathcal U}_{M_{r_2}(f^{-1}(\beta))}$
for all $\beta\in f(X)$.
Since $f(X)$ is compact as well this is possible. We then define
${\mathcal C}'(r):=\{M_{r}(\beta)|\beta\in f(X)\}$, and
$\hat{\psi}_1:{\mathcal C}'(r'_1)\to\Gru$, $\hat{\psi}_2:{\mathcal C}'(r')\to\Gru$ by
\begin{equation}
\hat{\psi}_1(M_{r_1'}(\beta)) := M_{r}(f^{-1}(\beta))\:,
\quad
\hat{\psi}_2(M_{r'}(\beta)) := M_{r_2}(f^{-1}(\beta))
\end{equation}
for all $\beta\in f(X)$.
Alike $\varphi$, $\psi_i$, $i=1,2$, extend to a prehomomorphisms
and ${\mathcal R}(D(\psi_i))={\mathcal R}({\mathcal M}_{\rm I\!I}')$.
Moreover, $\hat{\varphi}\circ\hat{\psi}_1(M_{r_1'}(\beta))=M_{r'}(\beta)$
and $\hat{\psi}_2\circ\hat{\varphi}(M_{r}(\alpha))=M_{r_2}(\alpha)$
imply that $\psi_2\circ\varphi(c)\succeq c$ for all
$c\in D(\psi_2\circ\varphi)$ and
$\varphi\circ\psi_1(c')\succeq c'$ for all
$c'\in D(\varphi\circ\psi_1)$. In particular,
${\mathcal R}(\varphi\circ\psi_1)=\mbox{\rm id}$ on ${\mathcal R}({\mathcal M}_{\rm I\!I}')$
and ${\mathcal R}(\psi_2\circ\varphi)=\mbox{\rm id}$ on ${\mathcal R}({\mathcal M}_{\rm I\!I})$.
Therefore, if $\fol{c}$ is an approximating sequence then
$\fl{\psi}_2([\fol{c}])=[(\psi_2\circ\varphi\circ\psi_1(c_n))_n]
\succeq [\psi_1(c_n))_n]$.
In particular, if next to $\fl{c}$ also $\fl{\psi}_2(\fl{c})$
is minimal then
$\fl{\psi}_2(\fl{c})=\fl{\psi}_1(\fl{c})$.
Now let $\fl{c}\in{\mathcal R}({\mathcal M}_{\rm I\!I})$.
By Lemma~\ref{16092} there is a $\fl{c}'\in{\mathcal R}({\mathcal M}_{\rm I\!I}')$ with
$\fl{c}'\preceq\fl{\varphi}(\fl{c})$. Then
$\fl{\psi}_2(\fl{c}')\preceq\fl{\psi}_2\circ\fl{\varphi}(\fl{c})=\fl{c}$, i.e.\
$\fl{\psi}_2(\fl{c}')$ is minimal, and hence $\fl{c}=\fl{\psi}_1(\fl{c}')$,
and consequently
$\fl{\varphi}(\fl{c})=\fl{c}'$.
It follows that $\fl{\varphi}$ is approximating and hence ${\mathcal R}(\varphi)=f$.
But then the above implies that $\psi_i$, $i=1,2$, are approximating and
${\mathcal R}(\psi_i)=f^{-1}$. Hence $\hat{\varphi}$ and $\hat{\psi}_i$ for either of the
$i=1,2$ satisfy according to Lemma~\ref{21091} the requirements of
the definition of locally topological equivalence.\hfill q.e.d.\bigskip
In fact, we have proven a little more, namely that any isomorphism
between groupoids associated to finite type tilings is "locally defined",
i.e.\ it can be obtained by a local derivation rule.
One could also define a stronger form of topological equivalence between
two tilings ${\mathcal T}$, ${\mathcal T}'$ in that one requires in addition for the local
morphism of Definition~\ref{17094} that ${\mathcal R}(\varphi)$
maps the class of ${\mathcal T}$ onto that of ${\mathcal T}'$.
This is then equivalent to the existence of an isomorphism between
${\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}))$ and ${\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}'))$ which
maps the class of ${\mathcal T}$ onto that of ${\mathcal T}'$.
A simple example for which the construction
of a prehomomorphism of the theorem can be carried out, not yielding an
approximating one, is
the constant map $f:{\mathcal R}({\mathcal M}_{\rm I\!I})\to {\mathcal R}({\mathcal M}_{\rm I\!I}')$ given by
$f(\fl{c})=\fl{u}$, $\fl{u}\in{\mathcal R}({\mathcal M}_{\rm I\!I}')^0$ fixed.
The above construction yields $\varphi(c)=M_{r'}(\fl{u})$ for all
$c\in D(\varphi)$ which is not approximating.\bigskip
\section{A selected overview on topological invariants of tilings}
We have shown that the topological groupoid ${\mathcal R}({\mathcal M}_{\rm I\!I})$ is a
complete invariant for a topological equivalence class of tilings
which are of finite type.
This answers the question
under which circumstances two tilings of finite type lead to the same groupoid.
Furthermore it means that the groupoid
contains all physically interesting topological
information about a tiling, the
prime example of that being the $K$-theoretic gap labelling.
The question immediately following such a result is that after an
invariant for tiling-groupoids which is computable
and distinguishes between non-isomorphic groupoids (the term
invariant always referring to a quantity which depends on
isomorphism classes). In fact, the determination whether two
such groupoids are isomorphic or not can be rather difficult,
and what we have in mind here is something like Elliot's classification of
$AF$-algebras by means of their scaled ordered $K_0$-group \cite{Ell1}.
These groups may be in many cases easily computed \cite{Eff}.
So one might hope that the $K$-theory of the groupoid-$C^*$-algebra\ is a good
starting point to classify all groupoids coming from tilings.
And in fact, if one restricts its attention only to
the groupoid-$C^*$-algebra\ of the groupoid, then, for $1$-dimensional tilings -- which
may be viewed as topological dynamical systems --
one obtains a $C^*$-algebra\ which is the limit of circle algebras. Elliot's
classification extends to such algebras \cite{Ell2},
the scaled ordered $K_0$-group
of the groupoid-$C^*$-algebra\ is a complete invariant as well.
A full treatment of the one dimensional case including an interpretation in
dynamical terms can be found in \cite{HPS,GPS}.
In higher dimensions, it is not yet clear whether $K$-theory yields complete
invariants for the groupoid-$C^*$-algebra s of tilings but the ordered $K_0$-group is
still an interesting object to consider, after all it has physical
signification in the gap-labelling.
However, it should be said that there are non-isomorphic tiling
groupoids which give rise to isomorphic $C^*$-algebra s, so that the $K$-theory
of the latter cannot be a complete invariant for tiling groupoids.
It is known that groupoids are invariants for pairs of $C^*$-algebra s,
the groupoid-$C^*$-algebra\ and a Cartan subalgebra of it \cite{Ren}.
\subsection{K-theoretic invariants}
The definition of the (reduced or full) groupoid $C^*$-algebra\
of an $r$-discrete groupoid can be found in \cite{Ren} or,
in the special context of tilings, in \cite{Ke2,Ke5}.
In the latter case, it may be seen as the $C^*$-closure of a representation of
the inverse semigroup ${\mathcal M}_{\rm I\!I}\cup \{0\}$ by means of partial
symmetries of a Hilbert space and coincides with the algebra
of observables for particles moving in the tiling.
To be more precise, a priori on distinguishes two such closures, obtaining
the reduced or the full algebra. But since the (discrete) groupoid of
a tiling is the abstract transversal of a transformation group with
amenable group, its reduced and full groupoid-$C^*$-algebra\ coincide \cite{MRW,Muh}.
The $K$-theoretic invariants of the groupoid-$C^*$-algebra\
${\mathcal A}_{\mathcal T}$ of ${\mathcal R}({\mathcal M}_{\rm I\!I}({\mathcal T}))$ are topological invariants
of the tiling.
The results which could be obtained so far are, apart from
periodic tilings, all related to tilings which are invariant under
a primitive invertible substitution.
For one dimensional tilings the $K$-theory is computed
in \cite{For,Hos}. For higher dimensional tilings
the (integer) group of coinvariants (which is actually
a cohomology group) together with
a natural order could be obtained in \cite{Ke5}.
For tilings which allow for a locally defined $\mathbb Z^d$-action, $d\leq 3$, the
group of coinvariants embeds as ordered unital group into the $K_0$-group.
This is enough to solve the $K$-theoretical gap-labelling for these.
Explicit calculations include Penrose tilings \cite{Ke5} and
octagonal tilings \cite{Ke6}. Further results are obtained in terms of
cohomology groups, see below.
But before coming to that let us recall Corollary~\ref{17092} which has
as a consequence that $K_1$-groups and ordered $K_0$-groups alone (without
order unit) are invariants for MLD-classes of tilings.
That the order unit may distinguish elements of such a class may be seen
from the cases in which ${\mathcal R}(\varphi)$ is injective but not surjective.
In particular, any tile of ${\mathcal T}'$ contains at most one puncture of a tile
of ${\mathcal T}$ but some of them carry none. In this situation one can
identify ${\mathcal A}_{\mathcal T}$ with a full corner of ${\mathcal A}_{{\mathcal T}'}$ and the induced
order isomorphism between the ordered $K_0$-groups maps the order
unit of $K_0({\mathcal A}_{\mathcal T})$ onto an element which is strictly smaller than
the order unit of $K_0({\mathcal A}_{{\mathcal T}'})$ \cite{Ke5}.
\subsection{Cohomological invariants}
Another topological invariant of a groupoid is its cohomology.
If one considers cohomology groups of the discrete groupoid
with integer coefficients then, at least
for tilings which carry a local $\mathbb Z^d$-action, unordered $K$-groups are
isomorphic to cohomology groups \cite{FoHu}. E.g.\ the non-vanishing
cohomology group of highest degree, which is the group of coinvariants,
is a direct summand of the $K_0$-group. This was taken advantage of
already above.
On the other hand Anderson and Putnam showed that unordered $K$-theory of
two dimensional substitution\ tilings is isomorphic to the Czech-cohomology
of a certain CW-complex \cite{AP}.
They computed the latter for a number of tilings
including Penrose tilings. In particular they obtained as well
the $K_1$-group. The route they took is different from the one in
\cite{Ke5}, but the actual calculations, as far as they concern
the common part of the results, reduce
at the end in both cases to the computation of images and kernels of
combinatorial matrices, which are almost the same.
Comparing the types of invariants it can be said that
cohomology groups give a finer grading than $K$-groups
but a priori no order.
This is a severe draw back due to the vast possibilities
of orders on such groups. In particular, integer valued cohomology is
not a complete invariant for tilings either.\bigskip
Other cohomology groups of groupoids are also of interest for physics.
The second cohomology group of a groupoid with coefficients in the
circle group provides the twisting elements for the construction of
the twisted groupoid-$C^*$-algebra\ \cite{Ren}.
For the simpler case of the group $\mathbb Z^2$ (which is of course a groupoid)
the twisted group-$C^*$-algebra\ is very important.
It is an irrational rotation algebra which is the observable e.g.\ for
for particles which move on the lattice $\mathbb Z^2$ (a periodic tiling)
and which are subject to a constant perpendicular magnetic field
\cite{Zak,TNK,Be1}.
The flux through the unit cell (a tile)
may be interpreted as the cocycle which yields the twisting element.
It would therefore be rather interesting to compute the full second cohomology
group with coefficients in the circle group for non periodic tilings.
|
proofpile-arXiv_065-690
|
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\section{#1}}
\begin{document}
\begin{titlepage}
\begin{center}
\vskip .5 in
{\large \bf Non-Trivial Extensions of the $3D-$Poincar\'e Algebra}\\
{\large \bf and Fractional
Supersymmetry for Anyons}
\vskip .3 in
{
{\bf M. Rausch de Traubenberg}\footnote{[email protected]}
\vskip 0.3 cm
{\it Laboratoire de Physique Th\'eorique, Universit\'e Louis Pasteur}\\
{\it 3-5 rue de l'universit\'e, 67084 Strasbourg Cedex, France}\\
\vskip 0.3 cm
and \\
{\bf M. J. Slupinski\footnote{[email protected]}}\\
{\it Institut de Recherches en Math\'ematique Avanc\'ee}\\
{ \it Universit\'e Louis-Pasteur, and CNRS}\\
{\it 7 rue R. Descartes, 67084 Strasbourg Cedex, France}\\ }
\end{center}
\vskip .5 in
\begin{abstract}
Non-trivial extensions of the three dimensional Poincar\'e algebra, beyond
the supersymmetric one, are explicitly constructed. These algebraic
structures are the natural three dimensional generalizations of fractional
supersymmetry of order $F$ already considered in one and two dimensions.
Representations of these algebras are exhibited, and
unitarity is explicitly checked.
It is then shown that
these extensions generate symmetries which connect fractional spin states
or anyons. Finally, a natural classification arises according to the
decomposition of $F$ into its product of prime numbers leading to
sub-systems
with smaller symmetries.
\end{abstract}
\end{titlepage}
\renewcommand{\thepage}{\arabic{page}}
\setcounter{page}{1}
In $D-$dimensional spaces, particles are classified by irreducible
representations of the Poincar\'e algebra. This algebra generates the
space-time symmetries (Lorentz transformations and space-time
translations),
and after one has
gauged the space-time translations we naturally obtain a theory of
gravity.
Therefore, in order to understand the fundamental interactions and the
symmetries
in particle physics, it is interesting to study all the possible extensions
of
the Poincar\'e symmetry. Quantum Field Theory restricts considerably the
possible
generalizations. If one imposes the unitarity of the $S-$matrix with a
discrete
spectrum of massive one particle states, then
within the framework of Lie algebras,
the Coleman and Mandula theorem \cite{cm} allows only
internal symmetries, {\it i.e.} those
commuting with the generators of the Poincar\'e algebra\footnote{
In the massless case, the Poincar\'e group can be promoted to the conformal
one.}. However, if we go beyond Lie algebras, we can escape this
{\it no-go} theorem. The well-known supersymmetric extension is generated
by
fermionic charges which, by the Haag, Lopuszanski and Sohnius
theorem,
are in the spinorial representation of $SO(1,D-1)$ \cite{hls}.
So, it seems that there exists
a {\it unique} non-trivial extension of the Poincar\'e algebra, up
to the choice of the number $N$ of supercharges. Indeed, according to the
Noether theorem, all these symmetries correspond to conserved currents, and
are generated by charges which are expressed in terms of the fields.
By
the spin-statistics theorem we have two kinds of fields having integer or
half-integer spin. The former will close with commutators and the latter
with
anticommutators leading respectively to Lie and super-Lie algebras.
The consideration of algebraic extensions, beyond the Poincar\'e algebra,
is not new. Such a possibility was considered in \cite{ker, luis}.
In the second paper, Wills Toro showed that the generators of the
Poincar\'e
algebra might themselves have non-trivial indices. In this paper
we pursue a different possibility, namely the study of special
dimensions. Particular dimensions can reveal
exceptional behaviour. This opportunity to find ``particular'' dimensions
has already been exploited with success and has
led to generalizations of
supersymmetry. Fractional supersymmetry (FSUSY) which was introduced
in \cite{fsusy}, is one such generalization.
In one-dimensional spaces, where
no rotation is available, this symmetry is generated by one
generator which can be seen as the $F^{th}$ root of the time translation
$\left(Q_t\right)^F=\partial_t$. $F=2$ corresponds to the usual
supersymmetry.
A group theoretical justification was then given in \cite{am,fr}
and this symmetry was applied in the world-line formalism \cite{fr}.
The second peculiar cases, are the two-dimensional
spaces where, by use of conformal transformations, the
(anti)holomorphic part of the fields transforms independently \cite{bpz}.
In \cite{prs}, this situation
was exploited to build a Conformal Field Theory with fractional conformal
weight. The Virasoro algebra was extended by two generators satisfying
$\left(Q_z\right)^F=\partial_z$ and $\left(Q_{\bar z}\right)^F=
\partial_{\bar z}$ and besides the stress-energy tensor, a conserved
current of conformal weight ($1 + {1 \over F}$) was obtained.
Several groups have also studied this symmetry
in one \cite{fsusy1d} and two dimensions \cite{fsusy2d}.
Finally, in $1+2$ dimensions particles with arbitrary spin and statistics
exist. The so-called anyons were defined for the first time
in \cite{lm}. In fact, studying the representations of the $3D-$ Poincar\'e
algebra $P_{1,2}$ the unitary irreducible representations divide into two
classes: massive or massless.
For the massive particles, we can consider a one-dimensional wave function
with arbitrary real spin$-s$ ({\it i.e.} which picks up an arbitrary phase
factor $\exp(2i\pi s)$ when rotated through $2 \pi$).
In the massless case, only
two types of discrete spin exist \cite{b}. Then a relativistic wave
equation
for anyons was formulated following different approaches in \cite{jn,p}.
The purpose of this letter is to build non-trivial extensions of the
Poincar\'e
algebra which go beyond supersymmetry (SUSY).
We first give a fractional supersymmetric extension of
the Poincar\'e algebra of any order $F$. Then,
we study the representations of this algebra which turn out to contain
anyonic fields with spin ($\lambda,\lambda -{1 \over F}, \cdots,
\lambda -{F-1 \over F}$) (in the simplest case and with $\lambda$ an
arbitrary real number). We also explicitly check that the representations
we are considering are unitary.
It then appears that $3D-$FSUSY, like in $2D$, is a symmetry
which connects the fractional spin states previously obtained.
In this sense it is a natural
generalization of SUSY. We also prove that the algebras so-obtained can be
classified according to the decomposition of $F$ into its product
of prime numbers.
Introducing the generators of space-time translations $P^\alpha$ and the
generators of Lorentz
transformations \-$J^\alpha = {1 \over 2}
\eta^{\alpha \beta}$ $ \epsilon_{\beta \gamma \delta} J^{\gamma \delta}$,
we can rewrite the three dimensional Poincar\'e algebra as
follows
\beqa
\label{eq:P}
\left[ P^\alpha ,P^\beta \right] &=& 0 \nonumber \\
\left[ J^\alpha ,P^\beta \right] &=& i \eta^{\alpha \gamma} \eta^{\beta
\delta}
\epsilon_{\gamma \delta \eta} P^\eta \\
\left[ J^\alpha,J^\beta \right] &=& i \eta^{\alpha \gamma} \eta^{\beta
\delta}
\epsilon_{\gamma \delta \eta} J^\eta, \nonumber
\eeqa
\noindent
with $\eta_{\alpha \beta} = {\mathrm{diag}}(1,-1,-1)$ the Minkowski metric
and $\epsilon_{\beta \gamma \delta}$
the completely antisymmetric Levi-Civita tensor such that
$\epsilon_{012} =1$.
\noindent
Particles are then classified according to the values of the Casimir
operators of the Poincar\'e algebra. More precisely, for a mass $m$
particle
of positive/negative energy,
the unitary irreducible representations are obtained by studying the little
group leaving the rest-frame momentum $P^\alpha=(m,0,0)$ invariant. This
stability group in $\overline{SO(1,2)}$, the universal covering group of
${SO(1,2)}$,
is simply the universal covering group $\hbox{\it I\hskip -2.pt R }$ of
the abelian sub-group of rotation $SO(2)$ (generated by $J^0$).
As it is well-known, such a group is not quantized. This means that the
substitution $J^0 \to J^0 + s$ leaves the $SO(2)$ part invariant. But the
remarkable property of $\overline{SO(1,2)}$, is that the concomitant transformation
on the Lorentz boosts $J^i \to J^i + s { P^i \over P^0 + m}$ leaves the
algebraic structure (\ref{eq:P}) unchanged. Anyway, following the method
of induced representation for groups expressible as a semi-direct product
we
find that unitary irreducible representations for a massive particles are
one dimensional, and that the Lorentz generators are \cite{jn,b}
(for an arbitrary spin$-s$ representation)
\beqa
J^0_s &=& i \left(p^1 {\partial \over \partial p_2}
- p^2 {\partial \over \partial p_1}\right)
+ s \nonumber \\
J^1_s &=& - i \left(p^2 {\partial \over \partial p_0}-
p^0 {\partial \over \partial p_1}\right)
+ s { p^1 \over p^0+m} \\
J^2_s &=& - i \left(p^0 {\partial \over \partial p_1}-
p^1 {\partial \over \partial p_0}\right)
+ s { p^2 \over p^0+m}, \nonumber
\eeqa
\noindent
with $p^\alpha$ the eigenvalues of the operators $P^\alpha$. This
modification
of the $3D$ Lorentz generators was pointed out in \cite{sch} and is not
the most general one we can consider (see the last paper of \cite{p}).
\noindent
The main difference between $SO(1,2)$, or more precisely the
proper orthochronous Lorentz group, and $SO(3)$ is that $p^0+m$ never
vanishes with $SO(1,2)$ and $s$ does not need to be quantized.
In Ref.\cite{jn,p}, a relativistic wave equation for
massive anyons was given.
First, notice that the two Casimir operators are the
two scalars $P.P$ and $P.J$ and their eigenvalues for a spin$-s$ unitary
irreducible representation are respectively $m^2$ and $ms$. The equations
of
motion are then
\beqa
\label{eq:ms}
(P^2 - m^2) \Psi &=& 0 \\
(P.J - sm) \Psi &=& 0. \nonumber
\eeqa
However, to obtain manifestly covariant equations one has to go beyond the
mass-shell conditions (\ref{eq:ms}) given by the induced representation.
Therefore, we can start with a field which belongs to the appropriate
spin$-s$
representation of the {\it full} Lorentz group instead of the little group.
When $s$ is a negative integer, or a negative half-integer, this
representation is not unitary and is $2|s|+1$
dimensional, and the solution of the relativistic wave equations reduces to
the appropriate induced representation (see \cite{b,jn} for an
explicit calculation in the case $|s|=1,1/2$). When $s$ is an arbitrary
number,
the representation is infinite dimensional and belongs to the discrete
series of $\overline{SO(1,2)}$ \cite{wy}. A relativistic wave equation for an anyon
in the continuous series \cite{wy} was also considered in the
third paper of \cite{p}.
Noting $J_{s,\pm}=J^1_s \mp i~J^2_s$
($[J^0,J_\pm]=\pm J_\pm,~~[J_+,J_-]=-2J^0$)
the Lorentz generators of the spin$-s$
representation, and $|s,n \rangle$ the states ($n =0,\dots, \infty$)
we can build two spin$-s$ representations;
one bounded from below, noted ${\cal D}^+_s$
\beqa
\label{eq:sb}
J_s^0 |s_+,n \rangle &=& (s+n) |s_+,n \rangle \nonumber \\
J_{s,+} |s_+,n \rangle &=& \sqrt{(2s+n)(n+1)} |s_+,n+1 \rangle \\
J_{s,-} |s_+,n \rangle &=& \sqrt{(2s+n-1)n} |s_+,n-1 \rangle, \nonumber
\eeqa
\noindent
and one bounded from above (${\cal D}^-_s$)
\beqa
\label{eq:sa}
J_s^0 |s_-,n \rangle &=& -(s+n) |s_-,n \rangle \nonumber \\
J_{s,+} |s_-,n \rangle &=& - \sqrt{(2s+n-1)n}|s_-,n-1 \rangle \\
J_{s,-} |s_-,n \rangle &=& - \sqrt{(2s+n)(n+1)} |s_-,n+1 \rangle. \nonumber
\eeqa
\noindent
For both representations, the quadratic Casimir operator
of the Lorentz group equals $s(s-1)$.
For the first representation we have $J_{s,-} |s_+,0 \rangle =0$ and
for the second $J_{s,+} |s_-,0 \rangle =0$.
Jackiw and Nair \cite{jn} and Plyushchay \cite{p}
were able to define an equation of motion (plus some
subsidiary conditions) such that the solution of a spin$-s$ anyonic
equation
decomposes into a direct sum of a positive energy solution in the
representation bounded from
below and a negative energy in the one bounded from above. In
other
words, a solution of a spin$-s$ anyonic equation decomposes into
a positive energy state of helicity $h=s$ and a negative energy solution
with $h=-s$ : $|s \rangle = |h=s, +\rangle \oplus |h=-s,- \rangle$ and the
two states are $CP$ conjugate.
If $s$ is a negative integer or a negative half-integer number we get a
$2|s|+1$ dimensional representation,
but for a general $s$ we have an infinite number of states. Furthermore when
$s < 0$ the representation is non-unitary. Taking the spinorial representation
as a guidline, we choose the case $s=-1/F$ to build a non-trivial extension
of the Poincar\'e algebra. If we observe the relations (\ref{eq:sa}) and
(\ref{eq:sb}) with $s=-1/F$, we see an ambiguity in the square
root of $-2/F$. So a priori we have four different representations for
$s=-1/F$, (two bounded from below/above) with the two choices
$\sqrt{-1}=\pm i$.
We note ${\cal D}^\pm_{-1/F;\pm}$ (with obvious notations) these
representations. Next, we can
make the following identifications
\begin{itemize}
\item the dual representation of ${\cal D}^+_{-1/F;+}$ is obtained through
the substitution $J^a \longrightarrow -\left(J^a\right)^t$ and is given by
$\left[{\cal D}^+_{-1/F;+}\right]^*={\cal D}^-_{-1/F;+}$;
\item the complex conjugate representation of ${\cal D}^+_{-1/F;+}$ is
defined by $J^a \longrightarrow -\left(J^a\right)^\star$ \footnote{In the
mathematical literature because in the definition of Lie algebras there is no
$i$ factor --see equation (\ref{eq:P})-- we do not have a minus sign
in the definition of this representation.}
\footnote{Note that, for a complex matrix $X$, $X^\star$ denotes the
complex conjugate (and not the hermitian conjugate) matrix of $X$;
for a vector space $V$, $V^*$ is its dual.}
(we have to be careful when we do such
a transformation because we have {\it by definition $J^\pm = J^1 \mp i J^2$,
for any representation})
is given by $\overline{{\cal D}^+_{-1/F;+}}={\cal D}^-_{-1/F;-}$;
\item the dual of the complex conjugate representation of
${\cal D}^+_{-1/F;+}$ is given by
$\left[\overline{{\cal D}^+_{-1/F;+}}\right]^*={\cal D}^+_{-1/F;-}$.
\end{itemize}
If we note $\psi_a \in {\cal D}^+_{-1/F;+} ,\psi^a \in {\cal D}^-_{-1/F;+},
\bar{\psi}_{\dot a} \in {\cal D}^-_{-1/F;-}$ and
$\bar{\psi^{\dot a}} \in {\cal D}^+_{-1/F;-}$ then we have the following
transformation laws:
\beqa
\psi^\prime_a &=& S_a^{~~b} \psi_b \nonumber \\
\psi^{\prime a} &=& \left(S^{-1}\right)_{b}^{~~a} \psi^b \\
\bar{\psi}^\prime_{\dot a} &=& \left(S^\star\right)_{\dot a}^{~~ \dot b}
\bar {\psi}_{\dot b}
\nonumber \\
\bar{\psi}^{\prime {\dot a}}&=& \left((S^\star)^{-1} \right)_{\dot b}^
{~~ \dot a} \bar{\psi}^{\dot b}. \nonumber
\eeqa
\noindent
Furthermore, if we define
\beq
\psi^a = g^{a \dot a} \bar \psi_{\dot a},
\eeq
\noindent
we can write the following scalar product
\beq
\label{eq:ps}
\varphi^a \psi_a = -\bar \varphi_{\dot 0} \psi_0 + \sum \limits_{a>0}
\bar \varphi_{\dot a} \psi_a,
\eeq
\noindent
where the infinite matrix $g^{a \dot a}$ and its inverse $g_{\dot a a}$
are given by ${\mathrm{diag}} (-1,1,\cdots,1)$.
The reason why we have a pseudo-hermitian scalar product is because
we are dealing with a non-unitary representation of a
non-compact Lie group. The invariant scalar product gives an explicit
isomorphism between the two representations bounded from below (or above)
($\left(S^{-1}\right)_b^{~~a}= g^{a \dot a}
\left( S^\star\right)_{\dot a}^{~~\dot b} g_{\dot b b}$).
From now on, we choose $\sqrt{-2/F}=i\sqrt{2/F}$ for
representations bounded from below and $\sqrt{-2/F}=-i\sqrt{2/F}$ for
those bounded from above.
Using the representations (\ref{eq:sb}--\ref{eq:sa}),
and with the sign ambiguity resolved, we can define two
series
of operators, belonging to a non-trivial representation of the Poincar\'e
algebra. We denote now $\sqrt{-1}=i$.
Note $Q^+_{-1/F+n}$ those built from the representation bounded from below
(${\cal D}^+_{-1/F;+}$)
and $Q^-_{-1/F+n}$ the charges of the representation bounded from above
(${\cal D}^-_{-1/F;-}$).
Using (\ref{eq:sa}, \ref{eq:sb}) we get
\beqa
\label{eq:Q}
\left[J^0,Q^+_{-1/F+n} \right] &=& (n-1/F)~~ Q^+_{-1/F+n} \nonumber \\
\left[J_+,Q^+_{-1/F+n} \right] &=& \sqrt{(-2/F+n)(n+1)}~~ Q^+_{-1/F+n+1}
\nonumber
\\
\left[J_-,Q^+_{-1/F+n} \right] &=& \sqrt{(-2/F+n-1)n}~~ Q^+_{-1/F+n-1}
\nonumber \\
&& \\
\left[J^0,Q^-_{-1/F+n} \right] &=& - (n-1/F) ~~Q^-_{-1/F+n} \nonumber \\
\left[J_+,Q^-_{-1/F+n} \right] &=& - \left(\sqrt{(-2/F+n-1)n}\right)^\star
~~ Q^-_{-1/F+n-1}
\nonumber
\\
\left[J_-,Q^-_{-1/F+n} \right] &=& - \left(\sqrt{(-2/F+n)(n+1)}\right)^\star
~~Q^-_{-1/F+n+1}.
\nonumber
\eeqa
\noindent
We want to combine this algebra (\ref{eq:Q}) in a non-trivial way with the
Poincar\'e algebra (\ref{eq:P}). With such a choice, $Q^+_{-1 \over F}$ (resp.
$Q^-_{-1 \over F}$) has a helicity $h=-{1\over F}$ (${1 \over F}$ resp.).
With the above choices for the square roots of the negative numbers we know
that the representations are conjugate to each other {\it i.e.}
$\left(Q_{-1/F+n}^+\right)^\dag \equiv Q_{-1/F+n}^-$.
Having set the values of $s$, we have two reasons
to close the algebra with
the $Q$'s through a $F^{th}-$order product. First of all, we would like
the
algebra to be a direct generalization of the one built in two-dimensions.
Second,
the charges we have introduced are in the spin$-{1 \over F}$
representation
of the Poincar\'e algebra, and so the $Q$'s pick up an
$\exp{(-{2i\pi \over F})}$ phase factor when rotated through
$2\pi$. They have a non-trivial $\hbox{$Z$_F$ graduation, although
the generators of the Poincar\'e algebra are trivial with
respect to $\hbox{$Z$_F$. The algebra splits then into an
anyonic $\cal{A}$ and a bosonic $\cal{B}$ part. It can be written
\beqa
\label{eq:PQ0}
&&\left\{\cal{A},\cdots, \cal{A} \right\}_F \sim \cal{B} \nonumber \\
&&\left[\cal{B},\cal{A}\right] \sim \cal{A} \\
&&\left[\cal{B},\cal{B}\right] \sim \cal{B}, \nonumber
\eeqa
\noindent
with $\{{\cal A}_{s_1}, \cdots,{\cal A}_{s_F} \}_F={1 \over F !}
\sum\limits_{\sigma \in \Sigma_F}
{\cal A}_{i_{s_{\sigma(1)}}} \cdots {\cal A}_{i_{s_{\sigma(F)}}}$ and
$\Sigma_F$
the permutation group with $F$ elements.
Equations (\ref{eq:PQ0}) reveal the $\hbox{$Z$_F$ structure of the algebraic
extension of the Poincar\'e algebra we are considering. The bosonic part
of the algebra is generated by $J$ and $P$ and has a graduation zero.
The anyonic generators are the supercharges $Q^\pm$ and have graduation $\mp 1$
in $\hbox{$Z$_F$. To close the algebra, both sides of the equation have to have
the same graduation, justifying (\ref{eq:PQ0}). In the case of the
supersymmetric extension of the Poincar\'e algebra, (\ref{eq:PQ0})
corresponds to a $\hbox{$Z$_2-$graded Lie algebra or a superalgebra.
Now, we want to identify the whole algebraic extension of $P_{1,2}$.
Part of this algebra is known (see eqs.(\ref{eq:P}) and (\ref{eq:Q})).
Using adapted Jacobi identities, we calculate the remaining part of the
algebra, and justify the use of a completely symmetric product in
(\ref{eq:PQ0}). Those involving three bosonic fields
or two bosonic and one anyonic fields are the same as for
superalgebras. Using the Leibniz rule of ${\cal B}$ with $\{\dots\}_F$
we get the third Jacobi identity and the last one is obtained by a
direct calculation
\beqa
\label{eq:J}
&&\left[\left[{\cal B}_1,{\cal B}_2\right],{\cal B}_3\right] +
\left[\left[{\cal B}_2,{\cal B}_3\right],{\cal B}_1\right] +
\left[\left[{\cal B}_3,{\cal B}_1\right],{\cal B}_2\right] =0 \nonumber \\
&&\left[\left[{\cal B}_1,{\cal B}_2\right],{\cal A}_3\right] +
\left[\left[{\cal B}_2,{\cal A}_3\right],{\cal B}_1\right] +
\left[\left[{\cal A}_3,{\cal B}_1\right],{\cal B}_2\right] =0 \nonumber \\
&&\left[{\cal B},\left\{{\cal A}_1,\dots,{\cal A}_F\right\}_F\right] =
\left\{\left[{\cal B},{\cal A}_1 \right],\dots,{\cal A}_F\right\}_F +
\dots +
\left\{{\cal A}_1,\dots,\left[{\cal B},{\cal A}_F\right] \right\}_F \\
&&\sum\limits_{i=1}^{F+1} \left[ {\cal A}_i,\left\{{\cal A}_1,\dots,
{\cal A}_{i-1},
{\cal A}_{i+1},\dots,{\cal A}_{F+1}\right\}_F \right] =0. \nonumber
\eeqa
In order to identify the whole algebraic structure of the non-trivial
extension of the Poincar\'e algebra, assume, as a first step,
$\left[\-|\cal{A},\cdots, \cal{A}|\-\right]_F = \alpha. P + \beta. J,$
\noindent
with $[| \cdots |]$ a symmetric product of charges to be defined.
If we use the third Jacobi identity with ${\cal B} =P$, we obtain
$\beta =0$ ( $\left[ P, Q \right] =0$), the same Jacobi identity with
$ {\cal B} = J^0$ proves that both sides of the equation
have the same helicity.
In other words, this equation just tells us that we need to build
a mapping from a {\bf sub-space} of ${\cal S}^F({\cal D}_{-{1 \over F}}^\pm)$
(the $F-$fold symmetric product of
the representation ${\cal D}_{-{1 \over F}}^\pm$) to
the vectorial ($P$) representation of
$SO(1,2)$ which is equivariant for the action of $SO(1,2)$.
Now, we remark that there are primitive states in
${\cal S}^F({\cal D}_{-{1 \over F}}^\pm)$ from which we are
able to construct the vector representation of $SO(1,2)$:
\beqa
\label{eq:vrep}
\left[J^0,\left(Q^{\pm}_{-{1 \over F}}\right)^F\right]&=&
\mp \left(Q^{\pm}_{-{1 \over F}}\right)^F \\
\left[J_\mp,\left(Q^{\pm}_{-{1 \over F}}\right)^F\right]&=&0 \nonumber
\eeqa
\noindent
From these relations, it follows that the sub-space
$${\cal D}_{-1}=\left\{\left(Q^{\pm}_{-{1 \over F}}\right)^F,
\left[J_\pm,\left(Q^{\pm}_{-{1 \over F}}\right)^F\right],
\left[J_\pm,\left[J_\pm,\left(Q^{\pm}_{-{1 \over F}}\right)^F\right]\right] \right\}$$
\noindent
of ${\cal S}^F({\cal D}_{-{1 \over F}}^\pm)$ is
isomorphic to the vector representation of the Poincar\'e algebra.
Note that this relations also imply that
$\left[J_\pm,\left[J_\pm,\left[J_\pm, \left(Q^\pm_{-{1 \over F}}\right)^F
\right]\right]\right] = 0$. \\
So, we obtain the following algebra
(we have to be careful with the normalization appearing in the bracket
$\left\{\cdots \right\}$, for instance $\left(Q^\pm_{-{1 \over
F}}\right)^{F-1}
Q^\pm_{1-{1 \over F}} + \cdots Q^\pm_{1-{1 \over F}}
\left(Q^\pm_{-{1 \over F}}\right)^{F-1} = F \left\{Q^\pm_{-{1 \over F}},
\cdots,Q^\pm_{-{1 \over F}}, Q^\pm_{1-{1 \over F}}\right\}$).
\beqa
\label{eq:PQ}
&&\left\{Q^\pm_{-{1\over F}},\dots,Q^\pm_{-{1\over F}} \right\}_F = P_\mp
\nonumber \\
&&\left\{Q^\pm_{-{1\over F}},\dots,Q^\pm_{-{1\over F}},Q^\pm_{1-{1\over F}}
\right\}_F
=\pm i \sqrt{{2 \over F}} P^0 \\
&& -(F-1)
\left\{Q^\pm_{-{1\over F}},\dots,Q^\pm_{-{1\over F}},Q^\pm_{1-{1\over F}},
Q^\pm_{1-{1\over F}} \right\}_F
\pm i \sqrt{ F-2}
\left\{Q^\pm_{-{1\over F}},\dots,Q^\pm_{-{1\over F}},Q^\pm_{2-{1\over F}}
\right\}_F
= P_\pm \nonumber \\
&&\left[J_\pm,\left[J_\pm,\left[J_\pm, \left(Q^\pm_{-{1 \over F}}\right)^F
\right]\right]\right]=0 \nonumber \\
&& ~~~~~~~~~~~~ \vdots \nonumber
\eeqa
\noindent
with $P_\pm = P^1 \mp i P^2$. The normalization of the R.H.S. of
eq.(\ref{eq:PQ}) comes from the definition of the bracket
$\left\{\cdots\right\}_F$ and (\ref{eq:P},\ref{eq:Q}).
Now, we can address the question of the remaining brackets?
In fact, it is impossible to find a decomposition\footnote{We thank the
referee for pointing this tu us.}
\beq
{\cal S}^F\left({\cal D}^\pm_{-1/F}\right) = {\cal D}_{-1} \oplus V,
\eeq
\noindent
where $V$ is stable under $SO(1,2)$. Indeed, if there were such a
decomposition there would be a $SO(1,2)$ equivariant projection
\beq
\pi:~{\cal S}^F\left({\cal D}^\pm_{-1/F}\right) \longrightarrow {\cal D}_{-1}.
\eeq
\noindent
But then
$X^\pm=\pi\left( {\cal S}^F\left(Q^\pm_{-1/F},\cdots,Q^\pm_{-1/F},Q^\pm_{3-1/F}
\right) \right) \in {\cal D}_{-1}$ satisfies (see \ref{eq:Q})
$$\big[J_\mp,\big[J_\mp,\big[J_\mp,X^\pm \big]\big]\big]=
\pm i\sqrt{2/F}\sqrt{2(1-2/F)} \sqrt{3(2-2/F)} P_-\ne 0,$$
\noindent
and this is impossible because in the vector representation ${\cal D}_{-1}$,
$J_-^3$ acts as zero.
Finally, we can note that direct calculation easily shows that equations
(\ref{eq:PQ})
are stable under hermitian conjugation.
In this family of algebras, noted $FSP_{1,2}$ if
we take $F=2$ we are in an exceptional situation.
First, instead of having an infinite number of charges we have only two.
Secondly, the two representations $Q^\pm$ are equivalent
whereas the two series of charges are inequivalent
representations of $SO(1,2)$ when $F\neq 2$. In the case $F=2$, with
one series of supercharges $Q$ we obtain the well-known supersymmetric
extension of the Poincar\'e algebra, and (\ref{eq:Q}), (\ref{eq:PQ}) can
be
easily rewritten with the Pauli matrices. For more details on this algebra,
one can see, for example, the book of Wess and Bagger \cite{wb}.
The algebra we have obtained is then a direct generalization of the
super-Poincar\'e one.
It is remarkable that the supersymmetric algebra, which can
be generalized easily in one and two-dimensional spaces, can also
be considered in $1+2$ dimensions. This is a consequence of the special
feature of $SO(1,2)$ which allows to define states with
fractional statistics, {\it i.e.} anyons. If we try to go beyond, and
to build an extension of SUSY for higher dimensional spaces, one immediately
faces an obstruction. Indeed, when $D \ge 4$ one just has
bosonic or fermionic states and supersymmetry is the unique non-trivial
extension of the Poincar\'e algebra one can build.
Finally, let us mention that, the similarity of the algebra
(\ref{eq:PQ}) and the SUSY algebra
does not stop at this point.
If one considers now $N$ series of charges $Q^+$ and $Q^-$
we obtain, as in SUSY, algebraic extensions with central charges.
Before studying the representations of the algebra
(\ref{eq:PQ}) we can address
some general properties. First, $P^2$ commutes with all the generators so
that all states in an irreducible representation have the same mass.
Secondly, if
we define an anyonic-number operator $\exp({2i\pi{\cal N}_A})$ which gives
the
phase $e^{2i\pi s}$ on a spin$-s$ anyon we have ${\mathrm{tr}}
\exp({2i\pi{\cal N}_A}) =0$ showing that
in each irreducible representation there are $F$ possible statistics
($s,s-{1\over F},\dots, s-{ {F-1\over F}}$, where $s$ will be specified
later) and the dimension of the space with a given statistics is
always
the same. This can be checked proving by that ($\exp({2i\pi{\cal N}_A})
Q_s = e^{2i\pi s} Q_s \exp({2i\pi{\cal N}_A})$) and using cyclicity of
the trace
\beqa
&&{\mathrm{tr}} \left( \exp({2i\pi{\cal N}_A})
\left\{Q^+_{-{1\over F}},\dots, Q^+_{-{1\over F}},Q^+_{1-{1\over
F}}\right\}_F\right) \nonumber \\
&=&1/F \times {\mathrm{tr}}\left( \sum \limits_{a=0}^{F-1} e^{2i\pi{\cal N}_A}
\left(Q^+_{-{1\over F}}
\right)^a
\left(Q^+_{1-{1\over F}}\right)\left(Q^+_{-{1\over F}}\right)^{F-a-1}\right)
\nonumber \\
&=& 1/F \times \left(\sum \limits_{a=0}^{F-1} e^{-{2i \pi a \over F}}\right)
{\mathrm{tr}} \left(
\left(Q^+_{-{1\over F}}\right)^{F-1}e^{2i\pi{\cal N}_A}
\left(Q^+_{1-{1\over F}}\right)\right)=0.
\nonumber
\eeqa
\noindent
Of course because we are dealing with infinite dimensional algebras
the construction of the trace should be done with care.
However, we will explicitly see, by constructing the unitary representations,
that ${\mathrm{tr}} \exp({2i\pi{\cal N}_A}) =0$.
Having defined the anyonic extensions of the Poincar\'e algebra,
we now look at the massive representations of (\ref{eq:P}),
(\ref{eq:Q}) and (\ref{eq:PQ}).
Up to now we have written the algebra in such a way that there
is still one ambiguity: we do not know whether we can choose an
algebraic extension of the Poincar\'e algebra using only one series of
supercharges ($Q^+$ or $Q^-$) or whether we need both. In fact the unitarity
of the representation will force us to take {\it both simultaneously}.
Let us first concentrate on the case where one
series of supercharges is involved, say $Q^+$.
For the Poincar\'e as well as for its
supersymmetric extension, the irreducible representations are obtained,
using the Wigner method of induced representation. Then,
the massive representations $p^\alpha p_\alpha=m^2$ are constructed by
studying the sub-algebra leaving the rest-momentum $p^\alpha=(m,0,0)$
invariant. Similarly, within the framework of the FSUSY algebras,
the one particle-states are characterised by the eigenvalue
of the rotation in the $(x^1,x^2)$ plane {\it i.e.} by the helicity.
In other words, all the representations are obtained by studying the
sub-algebra where $P_\pm,J_\pm$ are set to zero.
On the level of the charges,
a similar assumption will be made (valid {\it only } on shell):
if we are looking at eqs.(\ref{eq:PQ}) only one fundamental bracket does
not vanish,
{\it i.e} the one involving $(F-1)$ times the charge $Q_{-{1 \over F}}$ and
the one involving $Q_{1-{1 \over F}}$ once. All brackets involving the
$Q_{n-{1 \over F}}$'s with $n>1$ acts trivially on the rest-frame states
(the R.H.S. always vanishes), so those charges can be represented by $0$
(this is not a new feature and this already appears in usual SUSY, and
for instance in four dimensions, in the massless case,
the surcharges $Q_2$ and $Q_{\dot 2}$ vanish)).
After an appropriate normalization (\ref{eq:PQ}) becomes
\beqa
\label{eq:PQr}
&&\left\{Q^+_{-{1 \over F}}, \dots,Q^+_{-{1 \over F}},Q^+_{1-{1 \over F}}
\right\}_F
= 1/F \\
&&\left\{Q^+_{s_1}, \dots,Q^+_{s_F} \right\}_F = 0,~~ ,
i_1,\cdots i_F=-1/F,1-1/F~~ \mathrm {and },~~i_1 + \dots + i_F \ne 0.
\nonumber
\eeqa
\noindent
Let us stress some properties of the algebras defined by (\ref{eq:PQr}).
This kind of algebra is known to mathematicians, and
is called the Clifford algebra of the polynomial $x^{F-1}y$ \cite{rr}.
Indeed, using (\ref{eq:PQr})
we obtain (developing explicitly the $F^{th}$ power) $\left(xQ_{-{1\over
F}} +
yQ_{1-{1\over F}}\right)^F=x^{F-1}y$. Hence, the algebra generated by the
two
charges $Q_{-{1\over F}}$ and $Q_{1-{1\over F}}$ is associated with the
linearization of the polynomial $x^{F-1}y$ and constitute a generalization
of
usual Clifford algebras. This procedure can be considered for any
polynomial.
However, this algebra does not
admit a finite dimensional faithful representation.
This means that, using a faithful
representation, we are able to build representations with an infinite
number
of states. It was shown in \cite{frr} that the Clifford algebra of a
polynomial of degree greater than $2$
admits a non-trivial, finite but not faithful representation.
For $F=2$, the situation is slightly different because Clifford algebras
admit a finite dimensional faithful representation in terms of the Dirac
$\gamma-$matrices.
Because we want to have a representation which contains a finite number
of states, we consider non-faithful representations.
An extensive study of the representations of Clifford algebras of cubic
polynomials was undertaken by Revoy \cite{re} and a family of
representations can be obtained. This result can be generalized for
$F \ge 4$. To obtain the irreducible representations for arbitrary $F$ we
first observe that $F$ is the first power of $Q_{-{1\over F}}$ which
is equal to zero (in other words the rank of $Q_{-{1\over F}}$ is $F-1$).
Indeed, if one assumes $Q^{F-b}_{-{1\over F}}=0$ (with $b > 1$), and
multiplies
the first equation of (\ref{eq:PQr}) by $Q_{-{1\over F}}$ on the left and
$Q^{F-b-2}_{-{1\over F}}$ on the right, one gets a contradiction.
Using the Jordan decomposition and the property that all eigenvalues of
$Q_{-{1\over F}}$ are zero, we can write
\beq
\label{eq:q}
Q^+_{-{1\over F}}=\pmatrix{0&0&0&\ldots&0&0& \cr
1&0&0&\ldots&0&0& \cr
0&1&0&\ldots&0&0& \cr
&\cr
\vdots&\vdots&&\ddots&\ddots&\vdots \cr
0&0&\ldots&0&1&0& }
~~\mathrm{and}~~ Q^+_{1-{1\over F}}=
\pmatrix{0&0&0&\ldots&0&1 \cr
0&0&0&\ldots&0&0& \cr
0&0&0&\ldots&0&0& \cr
&\cr\
\vdots&\vdots&&&\ddots&\vdots& \cr
0&0&0&\ldots&0&0&}.
\eeq
The matrix representation of $Q_{1-{1\over F}}$ has been obtained,
solving (\ref{eq:PQr}).
When $F=3$ the matrix given in (\ref{eq:q}) for $Q_{1-{1\over F}}$
is not the only possibility \cite{re}, and probably other
representations can be obtained when $F \ge 4$ \footnote{ This property
was pointed out to us by Ph. Revoy.}. However, the matrices given in
(\ref{eq:q}) are the only ones
consistent with the Poincar\'e
algebra: if some of the matrix elements which are
equal to zero in (\ref{eq:q}) are different from zero,
we obtain equations where both sides do
not have the same helicity (see below).
Finally, using the property
that the dimensions of the representations of Clifford algebras
are a multiple of the degree of the polynomial \cite{ht} ($F$ in this
case),
by similar arguments we can prove that
the other representations are reducible and are built with the two
matrices given in (\ref{eq:q}).
However, the matrices exhibited are not convenient to prove that the
representations of the FSUSY algebra are unitary. Indeed, we need quadratic
relations upon the matrices $Q^+$ and $Q^-=\left(Q^+\right)^\dag$. So, instead
of the two $Q$'s given on (\ref{eq:q})
(and their hermitien conjugate matrices) we would prefer
more suitable matrices obtained after a rescaling. At least two interesting
solutions have been found (the second was suggested by the referee)
{\tiny
\beqa
\label{eq:q1}
\begin{array}{ll}
Q^+_{-{1\over F}}=\left\{
\begin{array}{l}\pmatrix{0&0&0&\ldots&0&0& \cr
\sqrt{[1]}&0&0&\ldots&0&0& \cr
0&\sqrt{[2]} &0&\ldots&0&0& \cr
&\cr
\vdots&\vdots&&\ddots&\ddots&\vdots \cr
0&0&\ldots&0&\sqrt{[F-1]}&0& } \cr
~~~~~~ \cr
\pmatrix{0&0&0&\ldots&0&0& \cr
\sqrt{1(F-1)}&0&0&\ldots&0&0& \cr
0& \sqrt{2(F-2)}&0&\ldots&0&0& \cr
&\cr
\vdots&\vdots&&\ddots&\ddots&\vdots \cr
0&0&\ldots&0&\sqrt{(F-1)1}&0& }
\end{array}
\right.
Q^+_{1-{1\over F}}=\left\{
\begin{array}{ll}
\pmatrix{0&0&0&\ldots&0&\left\{\sqrt{[F-1]!}\right\}^{-1} \cr
0&0&0&\ldots&0&0& \cr
0&0&0&\ldots&0&0& \cr
&\cr\
\vdots&\vdots&&&\ddots&\vdots& \cr
0&0&0&\ldots&0&0&} \cr
~~~~~~ \cr
\pmatrix{0&0&0&\ldots&0&1/(F-1)!\cr
0&0&0&\ldots&0&0& \cr
0&0&0&\ldots&0&0& \cr
&\cr\
\vdots&\vdots&&&\ddots&\vdots& \cr
0&0&0&\ldots&0&0&}
\end{array}
\right.
\end{array}
\eeqa
}
\noindent
with $[a] = {q^{-a/2}-q^{a/2} \over q^{-1/2} - q^{1/2}}$,
$[F-1]!= [F-1] [F-2] \cdots [2] [1]$ and $q=\exp{(2i\pi /F)}$.
Of course the three sets of matrices given in (\ref{eq:q}) and
(\ref{eq:q1}) are related by a conjugation transformation (or a rescalling
of the vectors which belong to the representation --see after--).
From the basic conjugation we obtain immediately the associated representation
for the $Q^-$ charges
\beqa
\label{eq:qdag}
Q^-_{-{1\over F}}&=&\left(Q^+_{-{1\over F}}\right)^\dag \\
Q^-_{1-{1\over F}}&=&\left(Q^+_{1-{1\over F}}\right)^\dag \nonumber
\eeqa
\noindent
There are two consequences of the exhibited representations.
\begin{enumerate}
\item A direct calculation shows that the two charges $Q^+_{-1/F}$
and $Q^-_{-1/F}$ satisfy quadratic relations.
\begin{enumerate}
\item
In the case of the
first series we obtain the $q-$oscil\-lator algebra introduced by
Biedenharn and Macfarlane \cite{bm}
\beqa
\label{eq:qos}
&&Q^-_{-1/F} Q^+_{-1/F} - q^{\pm 1/2} Q^+_{-1/F} Q^-_{-1/F} = q^{\mp N/2} \\
\nonumber
&&[N,Q^+_{-1/F}]=Q^+_{-1/F} \\
&&[N,Q^-_{-1/F}]=-Q^-_{-1/F}, \nonumber
\eeqa
\noindent
with $N={\mathrm{diag}}(0,1,\cdots,F-1)$ the number operator
(which can be expressed with $Q_{-1/F}^\pm$).
\item
For the second choice we have
\beqa
\label{eq:sl}
&&[Q^-_{-1/F}, Q^+_{-1/F}] = N =\mathrm{diag} (F-1,F-3,\cdots,
1-F) \\
&&[ N,Q^\pm_{-1/F}]= \mp 2 Q^\pm_{-1/F}, \nonumber
\eeqa
showing that the $Q$ generate the $F-$dimensional representation of
$sl(2,\hbox{\it I\hskip -2.pt R })$.
\end{enumerate}
Among those two matrix representation of the FSUSY algebra (and eventually
others) we were not able to find arguments to select one rather the other
{\it i.e.} to obtain {\it naturally} and independently of
{\it any} matrix realization a quadratic relation among $Q^+_{-1/F}$ and
$Q^-_{-1/F}$ which characterizes the structure of the FSUSY algebra.
Some indications in this direction should be given. We can first notice
the property that the usual superspace construction of SUSY, by the
help of Grassmann variables, can be
generalized, and an adapted version has already been built
within the framework of FSUSY, at least when $D=1,2$
\cite{fr,prs,fsusy1d,fsusy2d}. Secondly, we can observe that
the quantization of the algebra generated by $Q^+_{-1/F}$
and its conjugate $Q^-_{-1/F}$ (variables fullfiling
$\theta^F=0$ and generalizing the well-known Grassmann variables)
is related with the $q-$deformed
Heisenberg algebra \cite{hq}. In other words we might have relations like
$Q^+_{-1/F} \sim \theta$, $Q^-_{-1/F} \sim \partial_\theta$ and
$\partial_\theta \theta -q \theta \partial_\theta \sim 1$.
Furthermore it is known that the algebra generated by $\theta$ and
$\partial_\theta$ is equivalent to the $q-$oscillators \cite{bm}.
These two remarks are surely related and
can be compared with the fact that the quantization of
the Grassmann algebra is the Clifford algebra.
As a consequence, the representation
built with the $Q$'s is unitary.
Indeed, the quadratic relations (\ref{eq:qos}) or (\ref{eq:sl}) enable us
to prove that the norm of the vector $\left(Q^+_{-1/F}\right)^n$ $ |0>$,
with $n=0,\cdots,F-1$ and $|0>$
the primitive vector on which the representation span by $Q^{\pm}_{-1/F}$
is built, is positive. This result can be obtained even more simply,
using the results of the $q-$oscillators for the first series \cite{bm},
or by proving that the matrices given in (\ref{eq:sl}) can be mapped to
the $F \times F$ hermician matrices of $SU(2)$, which generate unitary
representation (see after).
The deep reason for the emergence of a quadratic structure is
the non-faithfulness of the representation.
Indeed, relations (\ref{eq:PQr}) are
not strong enough to order the monomials in such a way that, say
$Q^+_{-1/F}$, is always on the left of $Q^+_{1-1/F}$, and the number of
monomials increase with their degree. If we have a finite-dimensional
representation then it means that we have obtained quadratic relations:
this allows us to order the monomials.
\item
We can observe directly that
$$Q^+_{1-1/F}= {1 \over [F-1]!} \left(Q^-_{-1/F}\right)^{F-1};$$
for the first choice, and
$$Q^+_{1-1/F}={1 \over ((F-1)!)^2} \left(Q^-_{-1/F}\right)^{F-1}$$
for the second.
Because of this constraint, the $Q_{-1/F}^\pm$ alone span the representation
of the FSUSY algebra.
\end{enumerate}
We note that the representations built with
the matrices $Q_{-{1 \over F}}$ and $Q_{1-{1 \over F}}$ can be obtained
in a way similar to the way one obtains representations of SUSY \cite{fs}.
We start with a vacuum $\Omega_\lambda$ in the spin$-\lambda$
representation
of $SO(1,2)$. On-shell, using the results established in \cite{jn,p},
we have the following decomposition
$$\Omega_\lambda = \Omega_{h=\lambda}^+ \oplus \Omega_{h=-\lambda}^-,$$
with two states of helicity $\pm \lambda$ and positive/negative energy.
These two vacua are $CP-$conj\-uga\-te and allow us to build a
$CP-$invariant
representation. This constraint of $CP$ invariance is very strong, because
as soon as we have chosen the representation built from
$\Omega_{h=\lambda,+}$,
the one built from $\Omega_{h=-\lambda,-}$ is not arbitrary.
Altogether, with (\ref{eq:q1}) and (\ref{eq:qdag}) we get the representation
($Q^-_{-1/F} \Omega_{h=\lambda}^+=0, Q^+_{-1/F} \Omega_{h=-\lambda}^-=0$,
and for our normalization we have chosen the first choice for the $Q$'s)
$$\vbox{\offinterlineskip \halign{
\tvi\vrule# & \cc{#} & \tvi\vrule# & \cc{#} & \tvi\vrule# &
\cc{#} & \tvi\vrule# & \cc{#} & \tvi\vrule# & \cc{#}& \tvi\vrule# \cr
\noalign{\hrule}
&\cc{states}&&\cc{helicity} &&\cc{states}&&\cc{helicity} & \cr
\noalign{\hrule}
&$\Omega_{\lambda}^+$&&$\lambda$&
&$\Omega_{-\lambda}^-$&&$-\lambda$& \cr
\noalign{\hrule}
&$Q^+_{-1/F}\Omega_{\lambda}^+$&&$\lambda-1/F$&
&$Q^-_{-1/F}\Omega_{-\lambda}^-$&
&$-\lambda+1/F$& \cr
\noalign{\hrule}
&$\vdots$&& && &&$\vdots$& \cr
\noalign{\hrule}
&${\left(Q^+_{-1/F}\right)^a \over \sqrt{[a]!}}\Omega_{\lambda}^+$&
&$\lambda-a/F$&
&${\left(Q^-_{-1/F}\right)^a \over \sqrt{[a]!}}\Omega_{-\lambda}^-$&
&$-\lambda+a/F$ &\cr
\noalign{\hrule}
&$\vdots$&& && &&$\vdots$& \cr
\noalign{\hrule}
&${\left(Q^+_{-1/F}\right)^{F-1} \over \sqrt{[F-1]!}}\Omega_{\lambda}^+$&
&$\lambda-(F-1)/F$&
&${\left(Q^-_{-1/F}\right)^{F-1}\over \sqrt{[F-1]!}}\Omega_{-\lambda}^-$&
&$-\lambda+(F-1)/F$ &\cr
\noalign{\hrule}
}}$$
\noindent
The states of positive energy and helicity ($\lambda,\lambda -{1 \over F},
\dots,\lambda -{F-1 \over F}$) are $CP-$ conjugate to the states
of negative negative energy and helicity ($-\lambda,-\lambda +{1 \over
F}, \dots,-\lambda +{F-1 \over F}$), and following the remarks given here above
it is known that the representation is unitary.
An interesting consequence of the second choice for the $Q-$matrices is the
fact that the representation of the FSUSY algebra belong to a $F-$dimensional
representation of $SU(2)$. Indeed, it is easy to check that the matrices
$K_1=1/2\left(Q^+_{-1/F}+Q^-_{-1/F}\right),
K_2=i/2\left(Q^+_{-1/F}-Q^-_{-1/F}\right)$ and $K_3=N/2$ are unitary
and generate the $SU(2)$ algebra.
Hence, FSUSY is a direct generalization of SUSY in the sense that these
fractional spin states or anyons are connected by FSUSY transformations.
The next step
would be to construct explicitly a Lagrangian invariant under a FSUSY
transformation which mixes these states, as has been done in one and
two dimensions \cite{fsusy,am,fr,prs,fsusy1d,fsusy2d}. As a starting point,
one could use the lagrangian formulation of anyonic fields given
in \cite{jn,p}.
To conclude this general study of the algebra, it is
of great interest to mention some properties when $F$ is not a prime
number.
Assuming $F=F_1 F_2$, we have $F_1SP_{1,2}
\subset FSP_{1,2}$. This property was already observed
in two dimensions in the second paper of \cite{prs}. So, this inclusion
(which can also be proven in one dimension) is a general property of FSUSY
and does not depend on the dimension.
To prove this statement, we focus on the case where we have only
the $Q^+$ charges and we omit the $+$ superscript.
If we define $\left(Q_{-{1\over F}} \right)^{F_2}=Q_{-{1\over F_1}} $,
using the algebra
we can build, from the spin$-{1 \over F}$ representation, a
spin$-{1 \over F_1}$ representation
of $SO(1,2)$ : $Q_{n-{1\over F_1}} \sim \left[J_+,\dots,
\left[J_+, Q_{-{1\over F_1}},\right],\dots\right]$ where $J_+$ has
been applied $n-$times. Using the Jacobi identities (\ref{eq:J}),
we can construct an algebraic generalization of (\ref{eq:PQ}) which
mixes the spin$-{1\over F}$ and spin$-{1\over F_1}$ anyonic operators.
The case where $F$ is an even number is special because the spin$-1/2$
representation is finite, so we have the same constraints as before for
(\ref{eq:PQ}).
From these inclusions of algebras, we are able to build sub-algebras with
smaller symmetries when $F$ is not a
prime number. In such a situation, the
$F-$multiplet of $FSP_{1,2}$ splits into
$F_2$~~$F_1-$multiplets of $F_1SP_{1,2}$
$$\Phi_\lambda^{(F)} = \bigoplus \limits_{a=0}^{F_2-1}
\Phi_{\lambda + {a\over F}}^{(F_1)}.$$
The $F_1-$multiplet $\Phi_{\lambda + {a\over F}}^{(F_1)}$ is built from the
vacuum $\Omega_{\lambda + {a\over F}}$. This can be checked directly from
the
definitions and using the representations ) or the matrices (\ref{eq:q1})
and (\ref{eq:q}).
In this letter, we have explicitly constructed non-trivial algebraic
extensions
of the $3D$ Poincar\'e algebra that go beyond the supersymmetric ones.
The study of their representations enables us to show that these symmetries
connect the fractional spin states given in (17-18).
We have pointed out an interesting
classification of these algebras by means of the decomposition of $F$
(the order of FSUSY) as a product of prime numbers.
This leads to sub-systems with
smaller symmetries. A first application of these algebras, would be to
build a Lagrangian formulation where FSUSY, among anyonic fields, is
manifest.
This could lead to some generalizations of the well known Wess-Zumino model
\cite{wz}. A further application would be to gauge FSUSY along the lines
given
in \cite{fr}, after having studied the massless representations of the
algebra (\ref{eq:P}),(\ref{eq:Q}) and (\ref{eq:PQ}).
Recently, a very interesting interpretation of supersymmetry and fractional
supersymmetry in one dimension was given as an appropriate limit of the
braided line \cite{bl}. Is it possible to understand, along these lines,
how
supersymmetry and fractional supersymmetry emerge in two and three
dimensions
and to prove that when the dimension is higher than three
only SUSY is allowed ?
Finally, it should be interesting to understand the consequences
of the FSUSY extensions of the Poincar\'e algebra, in relation with three
dimensional physics.
\vskip.5truecm
We would like to thank A. Comtet, E. Dudas, M. Plyushchay,
Ph. Revoy and C. A. Savoy
for critical remarks and useful discussions.
We would also like to thank the referee for his remarks and suggestions.\\
\vskip .3 in
\baselineskip=1.6pt
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{
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
The $B_K$ parameter serves to parameterize the weak hadronic
matrix element responsible for $K^0-\bar{K^0}$ mixing.
Since this mixing gives us the only CP violation
observed to date, $B_K$ is a crucial link between the measured
quantity $\epsilon$ and the parameters of the Standard Model.
Lattice calculations are well suited for the study of $B_K$
parameter, and it has by now received much attention.
After an early round of calculations\cite{GKPS1,GKPS2,GKPS3},
the statistics have now been raised to a level which
allows one to examine some of the fine points of the calculation,
such as checks on the reliability of one-loop lattice perturbation
theory~\cite{Ishizuka93},
the chiral behavior and nondegenerate quark masses~\cite{Lee,Aoki96},
the dependence of $B_K$ on the lattice spacing~\cite{GKPS3,Aoki95,Aoki96}
and the number of dynamical flavors\cite{Kilcup93}.
In this note we offer more information on these latter two points.
\section{Calculational Setup}
\begin{table}[hbt]
\setlength{\tabcolsep}{1.2pc}
\caption{Ensembles for $N_f$ Study}
\label{tab:dynamical}
\begin{tabular}{llll}
\hline
$N_f$ & $\beta$ & $N_{\rm config}$ & $m_\rho a$ \\
\hline
0 & 6.05 & 306 & 0.384(5) \\
2 & 5.7 & 83 & 0.384(4) \\
4 & 5.4 & 69 & 0.391(7) \\
\hline
\end{tabular}
\end{table}
For the dynamical fermion comparison we use lattices of
geometry $16^3\times 32$, with parameters as given in
table \ref{tab:dynamical}.
The quenched configurations were generated on
the Ohio Supercomputer Center T3D, while the
dynamical configurations with two and four flavors of
$m_qa = 0.01$ staggered fermions were generated on the 256-node
Columbia machine.
The parameters were chosen so as to make the scales of the
lattices exceedingly close (and equal to approximately
(2 GeV$)^{-1}$), as determined from the $\rho$-meson mass in
chiral limit (see Fig.~\ref{fig:rho} and Ref.~\cite{Chen}).
We employ 9 values of (degenerate) valence $d$ and $s$ quark masses
from $m_q=.01$ to $m_q=.05$.
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize 7.2cm
\epsfbox{osu-BK-fig1.eps}
\end{center}
\vskip-.2cm
\caption{Data and linear fit for $m_{\rho}a$ vs. quark mass, for
three sets of configurations with $N_f$=0,2 and 4.
(See the talk by D. Chen [9].)
\label{fig:rho} }
\end{figure}
For the continuum limit study we generated
7 ensembles of quenched configurations as listed
in table \ref{tab:quenched}, and used 7 to 9 values of $m_q$.
\begin{table}[htb]
\setlength{\tabcolsep}{0.7pc}
\caption{Quenched Ensembles for Continuum Extrapolation}
\label{tab:quenched}
\begin{tabular}{llll}
\hline
$\beta$ & Geometry & $N_{\rm config}$ & $m_q$ \\
\hline
5.70 & $16^3\times32$&259 & .01 to .08 \\
5.80 & $16^3\times32$&200 & .01 to .04 \\
5.90 & $16^3\times32$&200 & .01 to .04 \\
6.00 & $16^3\times32$&221 & .01 to .04 \\
6.05 & $16^3\times32$&306 & .01 to .05 \\
6.10 & $24^3\times32$& 60 & .01 to .04 \\
6.20 & $24^3\times48$&121 & .005 to .035 \\
\hline
\end{tabular}
\end{table}
For creating kaons (at rest) we use a wall of U(1) noise on timeslice $t=0$,
i.e. complex random numbers $\xi_{\vec{x}}$ at each space point such that
$\langle \xi_{\vec{x}}\xi^\dagger_{\vec{y}}\rangle = \delta_{\vec{x},\vec{y}}$.
This is statistically equivalent to computing a collection of delta-function
sources. In particular, our wall creates only pseudoscalars.
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize 7.2cm \epsfbox{osu-BK-fig2.eps}
\end{center}
\caption{We use periodic boundary conditions in space and time,
and the lattice is duplicated in time direction.}
\label{fig:period}
\end{figure}
We use a lattice duplicated in the time direction, with periodic
boundary conditions in space and time (see Fig.~\ref{fig:period}).
Computing propagators on the doubled lattice, we obtain forward-
and backward-going propagators which we use for computing $B_K$.
That is, if $G_\pi(t)$ is the $\pi$ propagator on the doubled
lattice, then our operator correlation functions are schematically
of the form $G_\pi(t)G_\pi(t+N_t)$, where $N_t=32$ or 48.
We employ three kinds of operators: Landau gauge, gauge invariant,
and tadpole improved. Landau gauge operators are defined by
fixing the gauge and omitting explicit links in non-local operators.
For gauge-invariant operators we supply the links, averaging over
all shortest paths. Tadpole-improved operators are gauge-invariant
operators, but with all links rescaled by $u_0^{-\Delta}$, where
$u_0=P^{1/4}$, $P$ is the average plaquette, and $\Delta$ is the
number of links needed to connect fermion fields.
We opted for tadpole-improved operators on all configurations,
using the others on a subset of configurations for checks.
The matching between continuum and lattice operators is of the form
$${\cal O}^{cont}_i = (\delta_{ij}+\frac{g^2}{16\pi^2}
(\gamma_{ij}\log{(\frac{\pi}{\mu a})} + C_{ij})) {\cal O}^{lat}_j, $$
where $\gamma_{ij}$ is the one-loop anomalous dimension matrix,
and $C_{ij}$ are finite coefficients, which can be sizable.
We take these from the calculations of \mbox{Refs.~\cite{IS,PS}.}
For the continuum scheme, we choose NDR, quoting results either
at scale $\mu=\pi/a$ or at $\mu= 2\mathop{\rm GeV}\nolimits$.
We use the $\overline{MS}$ coupling constant $g_{\overline{MS}}$,
defined as
$1/g^2_{\overline{MS}}(\pi /a) = P/g_{\rm bare}^2+0.02461-0.00704\, N_f.$
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize 7.2cm \epsfbox{osu-BK-fig3.eps}
\caption{$B_K$ with (lower points) and without (upper points)
one-loop perturbative matching.
The points are artificially displaced horizontally for clarity. }
\label{fig:pert2}
\end{center}
\end{figure}
To check how well the perturbation theory works, we computed all
three operators on a subset of the $N_f=2$ ensemble, finding that
after one-loop corrections are put in, the matrix elements
agree within our statistical error.
For the bulk of the calculation we used tadpole-improved operators
exclusively.
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize 7.2cm \epsfbox{osu-BK-fig4.eps}
\caption{Data and fit for $B_K$ vs. $m_K^2$ on the quenched ensemble.
The vertical line marks the physical kaon mass.}
\label{fig:BKQ}
\end{center}
\end{figure}
\section{Results for $N_f$ Dependence}
Figs.~\ref{fig:BKQ}~and~\ref{fig:BKD} show the results for $B_K$
on three ensembles of configurations. Values at 9 quark mass
points are fitted to the form expected from chiral perturbation
theory, $B_K=a+bm_K^2+cm_K^2\ln{m_K^2}$.
The \mbox{$N_f=4$} and \mbox{$N_f=2$} curves are similar in shape,
while the quenched curve crosses between the other two.
While this is perfectly allowed, we should also inject
a small note of caution---our ensembles have the same $\rho$-masses,
but these masses are presumably affected to some degree by the
finite volume. If this effect is sizable and depends significantly
on $N_f$, our curves could shift a little.
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize 7.2cm \epsfbox{osu-BK-fig5.eps}
\caption{Data and fit for $B_K$ vs. $m_K^2$ on two dynamical
ensembles. The dashed line shows the fit for the quenched
ensemble.}
\label{fig:BKD}
\end{center}
\end{figure}
Taking the results at face value, we note that the $N_f=2$
and $N_f=0$ results lie nearly on top of each other at the
kaon mass, consistent with our earlier results \cite{Kilcup93}.
Also, most of the $N_f=2$ data lie below $N_f=0$, consistent
with the observation by other groups that quenching seems to
increase $B_K$ slightly (see, e.g. ref. \cite{Soni}).
However, the $N_f=4$ data turn this picture upside down.
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize 7.2cm \epsfbox{osu-BK-fig6.eps}
\end{center}
\caption{Final results for $B_K$ at physical kaon mass and in
the chiral limit, vs. $N_f$.}
\label{fig:BKvsNf}
\end{figure}
Fig.~\ref{fig:BKvsNf} shows our final values for $B_K$,
obtained at the physical kaon mass and by extrapolation to
the chiral limit. We see that the interpolated $N_f=3$ result
is a few percent higher than quenched.
\section{Continuum Extrapolation}
Performing the same analysis on the quenched ensembles, we
obtain the result shown in figure \ref{fig:BKvsa}, where
we plot $B_K(NDR,\mu=2\mathop{\rm GeV}\nolimits)$ versus the scale as determined from $m_\rho$.
The data are well fit by the quadratic form
$B_K(a)= B_K(a=0) + (a\Lambda_2)^2 + (a\Lambda_4)^4$, where
the scale of the power correction parameters turns out to
be typical of QCD: $\Lambda_2\approx660\mathop{\rm MeV}\nolimits$, $\Lambda_4\approx650\mathop{\rm MeV}\nolimits$.
Alternatively, we note that we can avoid making reference to the
possibly problematic $m_\rho$ by using the scaling form
$$
a(\beta) = a_0 {\big({16 \pi^2\over 11 g^2}\big)}^{51\over121}
\exp({-8 \pi^2\over 11 g^2})
$$
where we take $g$ here to be the $\overline{MS}$ coupling.
This amounts to shuffling around the $a^4$ corrections,
and in practice tends to straighten the data out. That is
to say, much of the curvature in figure \ref{fig:BKvsa}
might be ascribed to scaling violations in $m_\rho$ itself.
To quote a final value we make the conservative choice
of a linear fit to the four points with $\beta\ge6.0$,
and obtain
$$B_K|_{a=0,N_f=0} = .573\pm.015.$$
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize 7.2cm \epsfbox{osu-BK-fig7.eps}
\end{center}
\caption{Linear (heavy line) and quadratic fits to $B_K(a)$.}
\label{fig:BKvsa}
\end{figure}
\section{Conclusions}
From the dynamical comparison, we find that $B_K(N_f=3)$
is ($5\pm2$)\% larger than $B_K(N_f=0)$.
Combining with the $a=0$ extrapolation we
we quote our current central value $B_K$ in
the real world:
$$B_K(NDR,\mu=2\mathop{\rm GeV}\nolimits,N_f=3,a=0) = .60\pm.02$$
Remaining uncertainties include possible finite-size effects
in the dynamical ensemble, higher order perturbative corrections
in the matching, and higher order chiral ($m_s-m_d$) effects.
A study of hadronic weak matrix elements relevant for $\epsilon^\prime
/\epsilon$ using the same techniques and ensembles is currently underway.
\bigskip
\noindent{\bf Acknowledgements.} We thank the Columbia collaboration
for access to the dynamical configurations. Cray T3D time was
supplied by the Ohio Supercomputer Center and the Los Alamos
Advanced Computing Laboratory.
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\section{INTRODUCTION}
Quasars emit most of their power in the UV to soft X-ray regime.
The PSPC detector aboard {\em ROSAT} allowed a significantly improved
study of the soft X-ray emission of quasars compared with earlier missions
(some of which were not sensitive below 2~keV),
such as {\em HEAO-1, EINSTEIN, EXOSAT}, and {\em GINGA}
(e.g. Mushotzky 1984; Wilkes \& Elvis 1987; Canizares \& White 1989;
Comastri {\it et al.} 1992; Lawson {\it et al.} 1992; Williams {\it et al.}
1992; and a recent review by Mushotzky, Done, \& Pounds 1993).
These earlier studies indicated that the X-ray emission above 1-2~keV
is well described by a power law with a spectral slope
$\alpha_x= d\ln f_{\nu}/d\ln \nu$ of about $-0.5$ for radio loud quasars and
about $-1.0$ for radio quiet quasars. Large heterogeneous samples of AGNs
were recently studied using the {\em ROSAT} PSPC by Walter \& Fink (1993) and
by Wang, Brinkmann, \& Bergeron (1996). However, the objects studied in
the papers mentioned above do not form a complete sample, and the available
results may be biased by various
selection effects which were not well defined a priori. In particular, most
of the studied objects are nearby, intrinsically X-ray bright AGNs.
To overcome the potential biases in existing studies we initiated
a {\em ROSAT} PSPC program to make an accurate determination of the soft
X-ray properties of a well defined and complete sample of quasars,
selected independent of their X-ray properties.
This program was designed
to address the following questions:
\begin{enumerate}
\item What are the soft X-ray spectral properties of
the low redshift quasar population?
\item Are simple thin accretion disk models (e.g. Laor 1990) able to fit the
observed
optical/UV/soft X-ray continuum? are other modifying mechanisms, such as a
hot corona (e.g. Czerny \& Elvis 1987) required? Are models invoking
optically thin free-free emission possible (e.g. Ferland, Korista \&
Peterson, 1990; Barvainis 1993)?
\item Do the observed soft X-ray properties display any significant correlations
with other properties of these quasars? Are these correlations compatible with
various models for the continuum and line emission mechanisms?
\end{enumerate}
Our sample includes all 23
quasars from the BQS sample
(Schmidt \& Green 1983) with $z\le 0.400$, and $N_{\rm H~I}^{\rm {\tiny Gal}}$$< 1.9\times10^{20}$
cm$^{-2}$, where $N_{\rm H~I}^{\rm {\tiny Gal}}$\ is the H~I Galactic
column density as measured at 21~cm. These selection criteria allow optimal
study of soft X-ray emission at the lowest possible energy.
The additional advantages of the BQS sample are that it has been extensively
explored at other wavelengths (see Paper I for further details), and
that it includes only bright quasars, thus allowing high S/N X-ray spectra
for most objects. The sample
selection criteria are independent of the quasar's X-ray properties, and
we thus expect our sample to be representative
of the low-redshift, optically-selected quasar population.
Preliminary results from the analysis of the first 10 quasars available
to us were described by Laor et al. (1994, hereafter Paper I).
Here we report the analysis of
the complete sample which allows us to address the three questions posed
above. The outline of the paper is as follows.
In \S 2 we describe the
observations and the analysis of the spectra. \S 3 describes the
analysis of correlations between the soft X-ray properties and other continuum
and emission line properties. In \S 4 we compare our results with other
soft X-ray observations and discuss some of the implications. We conclude
in \S 5 with answers to the questions raised above, and with some new
questions to be addressed in future studies.
\section {THE OBSERVATIONS AND ANALYSIS OF THE SPECTRA}
The complete sample of 23 quasars is listed
in Table 1 together with their redshifts, $m_B$ and
$M_B$ magnitudes (calculated for H$_0=50$~km~s$^{-1}$, q=0.5), $R$,
the radio to optical flux ratio, and
$N_{\rm H~I}^{\rm {\tiny Gal}}$. The redshifts, $m_B$, and $M_B$
magnitudes are taken from
Schmidt \& Green (1983), $R$ is taken from Kellermann {\it et al.} (1989).
Note that 4 of the 23 quasars in our sample are radio loud (defined here as
$R\ge 10$).
The Galaxy becomes optically thick below 0.2~keV for the typical high
Galactic column of $3\times 10^{20}$~cm$^{-2}$ (Dickey \& Lockman 1990;
Morrison \& McCammon 1983), and
accurate values of $N_{\rm H~I}^{\rm {\tiny Gal}}$\ are therefore crucial even for our low $N_{\rm H~I}^{\rm {\tiny Gal}}$\
sample. The $N_{\rm H~I}^{\rm {\tiny Gal}}$\ values given in column 7 of Table 1 are taken from
Elvis, Lockman \& Wilkes (1989), Savage et al.
(1993), Lockman \& Savage (1995), and the recent extensive
measurements by Murphy et al (1996). All these
measurements of $N_{\rm H~I}^{\rm {\tiny Gal}}$\ were made with the 140 foot telescope of the NRAO at
Green Bank, WV, using the ``bootstrapping'' stray
radiation correction method described by Lockman, Jahoda, \& McCammon
(1986), which provides an angular resolution of 21', and an uncertainty of
$\Delta$$N_{\rm H~I}^{\rm {\tiny Gal}}$=$1\times 10^{19}$ cm$^{-2}$ (and possibly lower for our
low $N_{\rm H}$\ quasars). This uncertainty introduces a flux
error of 10\% at 0.2~keV, 30\% at 0.15~keV, and nearly a factor of 2 at
0.1~keV. Thus, reasonably accurate fluxes can be obtained down to
$\sim 0.15$~keV. Note that Murphy et al. (1996) includes accurate $N_{\rm H~I}^{\rm {\tiny Gal}}$\
measurements for about 220 AGNs, including most AGNs observed by
{\em ROSAT}, which
would be very useful for eliminating the significant systematic
uncertainty in the PSPC spectral slope which must be present when
a low accuracy $N_{\rm H~I}^{\rm {\tiny Gal}}$\ is used.
Table 2 lists the PSPC observations of all the quasars. For the sake of
completeness
we include also the 10 quasars already reported in Paper I.
All sources were detected, and their net source counts range from 93 to
38,015, with a median value
of about 1900 counts. The PROS software package was used to extract
the source counts. Table 2 includes
the exposure times, the dates of the observations, the net number of counts
and their statistical error,
the count rate,
the radius of the circular aperture used to extract the source
counts, the offset of the X-ray position from the center of the PSPC field of
view, the {\it ROSAT} sequence identification number,
and the SASS version used for the calibration of the data.
All objects, except one, are
typically within $15''$ of the center of the PSPC field of view, so all the
identifications are secure. The one exception is PG~1440+356 where the
pointing was offset by $40'$ from the position of the quasar
(Gondhalekar et al. 1994).
Note that the exposure times are uncertain by about 4\% due to a number
of possible systematic errors, as described by Fiore {\it et al.} (1994).
The typically large number of counts for each object allows
an accurate determination of the spectral slope for most objects, as
described below.
Model fits to the extracted number of source counts per pulse
invariant (PI) channel, $N_{\rm ch}^{\rm ob}$, were carried out using the
XSPEC software package. PI channels 1-12 of the original 256 channels
spectra ($E<0.11$~keV), were ignored since
they are not well calibrated and are inherently uncertain due to the
large Galactic optical depth. The January 1993 PSPC calibration matrix was
used for observations made after 1991 Oct. 14, and earlier observations
were fit with the March 1992 calibration matrix. The best fit model parameters
are obtained by $\chi^2$ minimization. Nearby channels were merged when
$N_{\rm ch}^{\rm ob}< 10$. A 1\% error was added in quadrature to the
statistical
error in $N_{\rm ch}^{\rm ob}$, to take into account possible systematic
calibration errors (see Paper I for more complete details).
\subsection {A Single Power-Law}
As in Paper I, we fit each spectrum with a single power-law
of the form $f_E=e^{-N_{\rm H}\sigma_E}f_0E^{\alpha_x}$, where $f_E$
is the flux density,
$\sigma_E$ is the absorption cross section per H atom (Morrison
\& McCammon 1983), $f_0$ is the
flux density at 1~keV, and $E$ is in units of keV.
We make three different fits for each object, with:
1. $N_{\rm H}$\ a free parameter, 2. $N_{\rm H}$$=$$N_{\rm H~I}^{\rm {\tiny Gal}}$.
3. $N_{\rm H}$$=$$N_{\rm H~I}^{\rm {\tiny Gal}}$, and $0.47\le E \le 2.5$~keV, i.e. using only the hard
{\rm ROSAT} band (channels $12-34$ of the rebinned 34 channels
spectra).
A comparison of fits 1 and 2 allows us to determine
whether there is evidence for a significant intrinsic absorption excess or
emission excess relative to a single power-law fit with
$N_{\rm H}$$=$$N_{\rm H~I}^{\rm {\tiny Gal}}$.
This comparison also allows us,
as further shown in \S 5.3.3, to determine whether the 21 cm measurement of
$N_{\rm H~I}^{\rm {\tiny Gal}}$\ is a reliable measure of the Galactic soft X-ray opacity.
A comparison of fits 2 and 3 allows us to look for a dependence of
the power-law slope on energy.
Table 3 provides the results of fits 1-3 described above. The table
includes the 13 objects not reported in Paper I, and 6 objects from paper
I for which we only now have the accurate $N_{\rm H~I}^{\rm {\tiny Gal}}$\ values.
For each fit we give the best fitting spectral slope $\alpha_x$, the
normalization of the power-law flux at 1 keV ($f_0$), the best fitting
$N_{\rm H}$, the $\chi^2$ of the fit ($\chi^2_{\rm fit}$), the number of degrees
of freedom (dof), and the probability for $\chi^2\ge \chi^2_{\rm fit}$.
The errors $\Delta\alpha_x$ and $\Delta$$N_{\rm H}$\ in fit 1 were calculated
by making a grid search for models with $\Delta \chi^2=2.30$, as
appropriate for 1 $\sigma$ confidence level for two interesting
parameters (e.g. Press {\it et al.} 1989). The error on the slope
$\Delta\alpha_x$ in fits 2 and 3 is calculated by requiring $\Delta
\chi^2=1.0$ (i.e. 68\% for one interesting parameter). We neglect the
effect of $\Delta$$N_{\rm H~I}^{\rm {\tiny Gal}}$\ on $\Delta\alpha_x$ in fits 2 and 3 since we use
accurate $N_{\rm H~I}^{\rm {\tiny Gal}}$\ for all objects.
The observed and best fit spectra for the 13 quasars not reported in paper
I are displayed in Figure 1. There are 3 panels for each object.
The upper panel displays the observed
count rate per keV as a function of channel energy, the histogram
represents the expected
rate from the best fit power-law model, $N_{\rm ch}^{\rm ob}$,
with $N_{\rm H}$\ a free parameter (fit 1).
The middle panel displays $\Delta/\sigma$, where
$\Delta=N_{\rm ch}^{\rm ob}-N_{\rm ch}^{\rm mod}$, and $\sigma$ is the standard
error in $N_{\rm ch}^{\rm ob}$. This plot helps indicate what features in
the spectrum
are significant. The lower panel
displays the fractional deviations from the expected flux, or equivalently
$\Delta/N_{\rm ch}^{\rm mod}$, which indicates the fractional amplitude of
the observed features.
As shown in Table 3, in all 13 objects the simple power-law model
with $N_{\rm H~I}^{\rm {\tiny Gal}}$\ (fit 2) provides an acceptable fit (i.e.
Pr($\chi^2\ge\chi^2_{\rm fit})>0.01$). Note in particular the spectrum of
PG~1116+215, which despite the very high S/N available (24,272
net counts), shows no deviations from a simple power law above a level of
$\lesssim 10$\%. In Paper I a simple power-law model could not provide
an acceptable fit to three of the 10 quasars,
though in two of them the apparent features could not be fit with a
simple physical
model, and in one of them this may be due to calibration errors
(see \S 4.1).
As mentioned above, a comparison of the free $N_{\rm H}$\ fit (fit 1) with the
$N_{\rm H~I}^{\rm {\tiny Gal}}$\ fit (fit 2) allows us to look for evidence for an absorption
or an emission excess. We measure the statistical significance of the
reduction
in $\chi^2_{\rm fit}$ with the addition of $N_{\rm H}$\ as a free parameter using
the F test (Bevington 1969). In PG~1440+356 we find a significant
reduction with Pr$=7.5\times 10^{-4}$
(F=12.98 for 26 dof), where Pr is the probability that the
reduction in $\chi^2$ is not statistically significant
(calculated using the FTEST routine in Press {\it et al.} 1989). The $N_{\rm H}$\
obtained in fit 1 suggests intrinsic absorption of about
$5\times 10^{19}$~cm$^{-2}$ above the Galactic absorption. However,
unlike all other objects, PG~1440+356 was observed significantly off
axis (see Table 2), and some small systematic calibration
errors may be present there. Note also that the
$\chi^2_{\rm fit}$ of the fit with $N_{\rm H~I}^{\rm {\tiny Gal}}$\ (Pr=0.05)
is still acceptable. We therefore
cannot conclude that an extra absorber must be present in PG~1440+356.
Marshall et al. (1996) found a very steep slope ($\alpha=-4.7\pm 0.65$)
in PG~1440+35 at 0.1-0.15~keV (80-120\AA) using the Extreme UV Explorer.
There is no indication for such a component in the PSPC spectrum below
0.15~keV (Fig.1c). However,
given the very low sensitivity of the PSPC below 0.15~keV, such a steep
soft component may still be consistent with the PSPC spectrum.
In the other 12 objects the free $N_{\rm H}$\ fit does not provide a significant
improvement (i.e. Pr$>0.01$), and thus there is no clear evidence for either
intrinsic absorption, or low energy excess emission above a simple
power-law.
Figure 2 compares the Galactic $N_{\rm H}$\ deduced from the accurate 21~cm
measurements with the best fit X-ray column deduced using the free $N_{\rm H}$\
fit. The straight line represents equal columns. The $\chi^2$ of the
$N_{\rm H}$(21~cm)=$N_{\rm H}$(X-ray) model is 31.9 for 22 dof (PG~1001+054 was not
included because of the low S/N), which is acceptable at the 8\% level.
This result demonstrates that there is no significant excess
absorption over the Galactic value in any of our objects. It is interesting
to note that in our highest S/N spectra, PG~1116+215 and PG~1226+023,
$N_{\rm H}$(X-ray) is determined to a level of $0.8-1\times 10^{19}$~cm$^{2}$, and
is still consistent with $N_{\rm H}$(21~cm), indicating that both methods agree to
better than 10\%.
The average hard {\em ROSAT} band (0.5-2~keV) slope for the complete sample is
$-1.59\pm 0.08$ (excluding PG~1114+445 which is affected by a warm absorber,
and PG 1001+054 and PG 1425+267 where the S/N is very low). This slope is
not significantly different from the average slope for
the full {\em ROSAT} band, $-1.63\pm 0.07$.
Spectral fits to the PSPC data of some of the objects in our complete
sample have already been reported by Gondhalekar et al. (1994),
Ulrich \& Molendi (1996), Rachen, Mannheim, \& Biermann
(1996), and Wang, Brinkmann, \& Bergeron (1996). The results of the
single power-law fit with a free $N_{\rm H}$\ in these papers
are all consistent with our
results for the overlapping objects. The only discrepancy is with
PG~1444+407 where although both us and Wang et al. find a similar slope,
Wang et al. find evidence for absorption while we find no such
evidence. For a simple power-law fit to PG~1444+407 Wang et al.
find $\chi^2=26$ for 20 dof which is acceptable only at the 17\% level
($\sim 1.4\sigma$ level), while we find for such a fit
$\chi^2=14.7$ for 20 dof, which is acceptable at the 80\% level.
As discussed in paper I (\S 5.1.3), the difference in spectral slopes at hard
(2-10~keV) and soft X-rays raises the possibility that $\alpha_x$ may be
changing within the PSPC band itself.
The individual spectra are well fit by a simple power
law, and thus any spectral curvature must be consistent with zero.
Stronger constraints on the spectral curvature may be obtained by measuring
the average curvature parameter ($\beta$, defined in Paper I) for the
complete sample since the random error in the mean is smaller by
$\sqrt{N}\sim 5$ than the random error for individual objects.
Unfortunately, the PSPC calibration
uncertainty at low energy, discussed in Paper I, introduces a
systematic error in $\beta$ (which obviously does not cancel out as
$\sqrt{N}$), and as shown in paper I, does not allow a reliable
determination of the curvature parameter. We therefore did not try to
constrain the spectral curvature parameter in this paper.
\section {CORRELATION ANALYSIS}
Table 4 presents 8 of the 12 rest-frame continuum
parameters and 7 of the 18 emission
line parameters used for the correlation analysis.
The spectral slopes are defined as
(flux subscript indicates $\log\nu(\rm Hz)$):
$\alpha_o=\log(f_{15}/f_{14.5})/0.5$
(1-0.3~$\mu$m), $\alpha_{ox}=\log(f_{17.685}/f_{15})/2.685$
(3000~\AA-2~keV), and $\alpha_{os}=\log(f_{16.861}/f_{15})/1.861$
(3000~\AA-0.3~keV).
The X-ray continuum parameters are
from fit 2, and from Paper I. The near IR and optical continuum parameters are
taken from Neugebauer {\it et al.} (1987). The emission line parameters were
taken from Boroson \& Green (1992). Luminosities were calculated assuming
$H_0$=50~km~s$^{-1}$ and $q_0=0.5$.
The four additional continuum parameters not presented in Table 5
for the sake of brevity are
$\alpha_{\rm irx}=\log(f_{16.861}/f_{14.25})/2.611$ (1.69~$\mu$m-2~keV),
$\alpha_{\rm irs}=\log(f_{17.685}/f_{14.25})/3.435$ (1.69~$\mu$m-0.3~keV),
radio luminosity (Kellermann et al.
1989), and $1~\mu$m luminosity (Neugebauer et al. 1987). The 11
emission line parameters not detailed in Table 5 are: [O~III] EW,
Fe~II EW, H$\beta$\ EW, He~II EW, [O~III]/H$\beta$\ and He~II/H$\beta$\ flux ratio,
[O~III] peak flux to H$\beta$\ peak flux ratio, radio to optical flux ratio,
and the H$\beta$\ assymetry, shape, and shift parameters. All these 11
parameters are listed in Table 2 of Boroson \& Green (1992).
The significance of the correlations was tested
using the Spearman rank-order
correlation coefficient ($r_S$) which is
sensitive to any monotonic
relation between the two variables. A summary of
the main correlation coefficients and their two sided significance is given in
Table 5.
\subsection {The Significance Level}
In Paper I the correlation analysis was carried out using 10 objects,
and only relatively strong correlations ($r_S\ge 0.76$) could be detected
at the required significance level (Pr$\le 0.01$).
Here, with 23 objects, a Pr$\le 0.01$ corresponds to
$r_S\ge 0.52$, and we can thus test for the presence of weaker correlations,
and check whether the correlations
suggested in Paper I remain significant. We have searched for
correlations among the 12 continuum emission parameters, and between these 12
parameters and the 18 emission line parameters listed above, which gives a
total of
294 different correlations. One thus expects about 1 spurious correlation
with Pr$\le 3.4\times 10^{-3}$ in our analysis, and for a significance level
of 1\% one would now have to go to Pr$\le 3.4\times 10^{-5}$, rather
than Pr$\le 1\times 10^{-2}$. However, we
find that there are actually 42 correlations with
Pr$\le 3.4\times 10^{-3}$, rather than just one, in our sample.
Thus, the probability that any
one of them is the spurious one is only 2.4\%, and the significance
level of these correlations is reduced by a factor of 7 ($=0.024/0.0034$),
rather than a
factor of 300. Below we assume that correlations with
Pr$\le 1\times 10^{-3}$ are significant at the 1\% level
(there are 30 correlations with Pr$\le 1\times 10^{-3}$, versus an expected
number of 0.3). Thus, given the large number of correlations
we looked at, we can only test reliably for correlations with
$r_S\ge 0.64$ (which corresponds to Pr$\le 1\times 10^{-3}$ for 23 data
points).
\subsection {The Near IR to X-Ray Energy Distribution}
A comparison of the rest-frame spectral energy distributions of all 23 quasars
is shown in Figure 3. The 3-0.3~$\mu$m continuum is from Neugebauer et al.
(1987), and the 0.2-2~keV continuum is from paper~I and from this paper.
The upper panel shows the absolute luminosities, and the lower 2 panels the
luminosity
normalized to unity at $\log \nu=14.25$ for radio quiet quasars and for
radio loud quasars. Note the relatively small dispersion in the normalized
0.3~keV ($\log \nu=16.861$) luminosity.
The outlying objects are labeled. PG~1626+554 is
the only object where a steep $\alpha_x$ is clearly associated with a
strong soft
excess (relative to the near IR flux). In other objects a steep $\alpha_x$
tends to be associated with a low 2~keV flux. This trend is also suggested
by the
presence of a marginally
significant correlation between $\alpha_x$ and $\alpha_{ox}$
($r_S=0.533$, Pr$=0.0089$, see below), and the absence of a significant
correlation
between $\alpha_x$ and $\alpha_{os}$ ($r_S=-0.201$, Pr$=0.36$).
The X-ray luminosity distribution appears to be bimodal with two quasars,
PG~1001+054 and PG~1411+ 442, being a
factor of 30 weaker than the mean radio quiet quasar. These two quasars
appear to form a
distinct group of `X-ray weak quasars'. The statistics for the radio
loud quasars (RLQ) are
much poorer, and there is no well defined mean, but PG~1425+267 may be a
similar X-ray weak RLQ.
\subsection {Correlations with Emission Line Properties}
Figure 4 presents the correlations between the hard X-ray luminosity,
$L_{\rm 2\ keV}$ ($\log \nu=17.685$), or the soft X-ray luminosity,
$L_{\rm 0.3\ keV}$, with the luminosities of H$_{\beta}$, [O~III],
He~II, or Fe~II. The value of $r_S$ and the two sided significance
level (Pr) of $r_S$ are indicated above each panel. Upper limits were
not included in the correlations. Thus, the actual correlations for
He~II, where there are 5 upper limits, are likely to be smaller than found
here (there is only one upper limit for [O~III] and Fe~II, and none for
H$\beta$). Excluding He II, the X-ray luminosity is most strongly correlated
with $L_{\rm H\beta}$ ($r_S=0.734$, Pr$=6.6\times 10^{-5}$).
We note in passing that $L_{\rm H\beta}$ has
an even stronger correlation
with the luminosity at 3000~\AA ($r_S=0.866$, Pr$=9\times 10^{-8}$)
and with the near IR luminosity at 1~$\mu$m ($r_S=0.810$, Pr$=2.7\times
10^{-6}$).
The position of the X-ray weak quasars is marked in Fig.4. Both
PG~1001+054 and PG~1411+442 appear to have an X-ray luminosity weaker
by a factor of about 30 compared to other quasars with similar $L_{\rm H\beta}$.
PG~1425+267 is also weaker by a factor of $\sim 10$ compared with the other
RLQ. These ratios are the same as those found above in \S 3.2, based on the
spectral energy distribution.
Figures 5a-d displays various emission parameters which correlate
with $\alpha_x$ (as obtained with $N_{\rm H}$=$N_{\rm H~I}^{\rm {\tiny Gal}}$).
The FWHM of H$\beta$, $L_{\rm [O~III]}$, the Fe~II/H$\beta$ flux ratio,
and the ratio of [O III] peak flux to H$_{\beta}$ peak flux (as
defined by Boroson \& Green) are the emission line parameters which
correlate most
strongly with $\alpha_x$. As found in Paper I, all the $\alpha_x$ versus
emission line correlations become significantly weaker when we use
$\alpha_x$ obtained with the free $N_{\rm H}$\ fit.
The X-ray weak quasars are labeled in the
$\alpha_{\rm ox}$ vs. $\alpha_x$ correlation in Figure 5e. As expected
they have
a steeper than expected $\alpha_{\rm ox}$ for their $\alpha_x$. The
last parameter shown in Figure 5f is 1.5$L^{1/2}_{14.25}\Delta v^{-2}$,
where $\Delta v=$H$\beta$\ FWHM. This parameter is related, under some
assumptions, to the luminosity in Eddington units, as further discussed
in \S 4.7.
\section {DISCUSSION}
\subsection {The Soft X-Ray Spectral Shape}
We find an average spectral index $\langle\alpha_x\rangle=-1.62\pm 0.09$
for the complete sample of 23 quasars, where
the error here and below is the uncertainty in the mean.
This slope is consistent with
the mean slope $\langle\alpha_x\rangle=-1.57\pm 0.06$ which we found
for the subsample of 24 quasars out of the 58 AGNs analyzed by
Walter \& Fink (1993, the other 34 AGNs in their sample are Seyfert
galaxies as defined by V\'{e}ron-Cetty \& V\'{e}ron 1991).
A similar average slope of $-1.65\pm 0.07$ was found by Schartel et al.
(1996) for 72 quasars from the LBQS sample detected in the {\em ROSAT} all
sky survey (RASS). Puchnarewicz et al. (1996) find a significantly
flatter mean slope, $\langle\alpha_x\rangle=-1.07\pm 0.06$, in a large
sample of 108 soft X-ray selected (0.5-2~keV) AGNs.
Part of the difference is related to the exclusion of counts below 0.5~keV,
which selects against steep $\alpha_x$ quasars, but this bias cannot explain
the much flatter $\alpha_{ox}$ in their sample. As discussed by
Puchnarewicz et al., their sample appears to include a large proportion of
highly reddened quasars (see further discussion in \S 4.2).
RLQ are known to have a flatter $\alpha_x$ than radio quiet quasars (RQQ) at
energies above the PSPC band (e.g. Wilkes \& Elvis 1987, Lawson et al. 1992).
We find $\langle\alpha_x\rangle=-1.72\pm 0.09$ for the 19 RQQ, and
$\langle\alpha_x\rangle=-1.15\pm 0.14$ for the 4 RLQ in our sample. We find
a similar trend using the Walter \& Fink quasar data, where
$\langle\alpha_x\rangle=-1.61\pm 0.08$ for the RQQ
and $\langle\alpha_x\rangle=-1.36\pm 0.08$ for the RLQ. A similar
difference between RQQ and RLQ was found by Ciliegi \& Maccacaro (1996)
in PSPC spectra of a sample of 63 AGNs extracted from the {\em EINSTEIN}
extended medium sensitivity survey sample.
We therefore conclude that the trend observed at harder X-rays also extends
down to the 0.2-2~keV band. Schartel et al. (1995) stacked PSPC images
of 147 RQQ and 32 RLQ finding for the sum images
$\langle\alpha_x\rangle=-1.65\pm 0.18$ for RQQ
and $\langle\alpha_x\rangle=-1.00\pm 0.28$ for the RLQ. However, the mean
redshift of their objects is $\sim 1.3$ and thus their results apply to
the $\sim 0.45-4.5$~keV band.
As discussed in Paper I, the {\em ROSAT} PSPC indicates a significantly
different soft X-ray spectral shape for quasars compared with earlier results
obtained by the {\em EINSTEIN } IPC and {\em EXOSAT} LE+ME detectors
(e.g. Wilkes \& Elvis, 1987; Masnou et al. 1992;
Comastri et al. 1992; Saxton et al. 1993;
Turner \& Pounds 1989; Kruper, Urry \& Canizares 1990).
In particular, earlier missions suggested that the hard X-ray slope
(Lawson et al. 1992; Williams et al. 1992) extends down to
$\sim 0.5$~keV with a steep rise at lower energy. Here we find that the
0.2-2~keV spectrum is fit well by a single power-law with Galactic
absorption. This indicates that:
1). the break between the soft and hard X-ray slope must occur well above
0.5~keV, 2) the break must be gradual, and 3) there is no steep soft
component with significant flux down to $\sim 0.2$~keV.
ASCA observations of two of the quasars in our sample,
3C~273 by Yaqoob et al. (1994), and PG~1116+215 by Nandra et al. (1996)
find, as expected, significantly flatter spectra above 2~keV. However,
the exact break energy cannot be accurately determined from the ASCA
spectra due to likely calibration uncertainties below 1~keV.
As mentioned in Paper I, the different {\em EINSTEIN } IPC and
{\em EXOSAT} LE+ME results may be
traced back to the combined effect of the lower sensitivity of these
instruments below $\sim 0.5$~keV, and possibly
some calibration errors. Small systematic errors in the PSPC response function
appear to be present below 0.2~keV (Fiore {\it et al.} 1994), and this
instrument is thought to be significantly better calibrated at
low energy than earlier instruments.
No significant spectral features are present in the PSPC spectra of all 13
additional quasars reported here, indicating that intrinsic features must
have an
amplitude of less than 10-20\%. Note, in particular the high S/N spectrum
of PG~1116+215, where the number of counts is about 12 times the median
sample counts, yet this spectrum is still consistent with a simple
power-law. For the complete sample we find that only 1 quasar, PG 1114+445,
has a significant physical feature which is well described by a warm
absorber model. In two other quasars,
PG~1226+023 and PG~1512+370, there are significant features
below 0.5~keV (paper I). In the case of PG~1512+370 the features are at a
level of $\sim 30$\%, and in PG~1226+023 they are at a level $\lesssim$10\%
and may well
be due to small calibration errors. This result is consistent with the
result of Fiore et al. (1994) who found that a simple power-law
provides an acceptable fit to the individual spectra of six high S/N
PSPC quasar spectra.
A composite optical to hard X-ray spectral energy distribution for RLQ and
RQQ is displayed in Figure 6. To construct it we used the mean $L_{14.25}$
(Table 5), the mean $\alpha_o$, the mean $\alpha_x$,
and the mean $\alpha_{ox}$ in
our sample. We excluded from the mean the three X-ray weak quasars, and
PG~1114+445, where $\alpha_x$ is highly uncertain due to the presence
of a warm absorber. The mean spectra were extended above 2~keV assuming a slope of
$-1$ for RQQ and $-0.7$ for RLQ.
The Mathews \& Ferland (1987, hereafter MF) quasar
energy distribution is also displayed for the purpose of comparison. The
MF shape assumes a steep soft component with a break to the hard X-ray slope
above 0.3~keV, and it therefore significantly underestimates the soft X-ray
flux at $\sim 0.2-1$~keV.
RLQ tend to be somewhat stronger hard X-ray sources than RQQ.
This trend, together with the flatter X-ray slope of RLQ was interpreted by
Wilkes \& Elvis (1987) as possible evidence for a two component model. In this
interpretation RLQ have the same hard X-ray component with
$\alpha_x\sim -1$, as in RQQ, with an additional contribution from a
flatter
$\alpha_x\sim -0.5$ component, making their overall X-ray emission
flatter and brighter. The additional X-ray component in RLQ could be related
to the radio jet, e.g. through inverse Compton scattering. The composite
spectrum suggests that although RLQ
are brighter at 2~keV, they may actually be fainter at lower energy because
of their flatter $\alpha_x$. The RLQ composite is based only on four
objects and is therefore rather uncertain. In addition, the results of Sanders
et al. (1989 \S III.c) suggest that RLQ in the PG sample
are about twice as bright at 2~keV
compared with RQQ of similar optical luminosity, rather than
the $\sim 30$\% found for the composite, thus the difference in PSPC $\alpha_x$
would imply a smaller difference at 0.2~keV than shown in the composite.
If RLQ are indeed weaker than RQQ at 0.2~keV then the two component model
suggested above would not be valid, and RLQ need to have a different X-ray
emission process, rather than just an additional component.
The difference between RLQ and RQQ may actually be unrelated to the
radio emission properties, as discussed in \S 4.4.
Figure 6 also displays a simple cutoff power-law model of the form
$ L_{\nu}\propto \nu^{\alpha_o}e^{-h\nu/kT_{\rm cut}}$ with $\alpha_o=-0.3$
and $T_{\rm cut}=5.4\times 10^5$~K. This is an
alternative way to interpolate between the UV and soft X-ray emission,
and it is also
a reasonable approximation for an optically thick thermal component.
The lack of a very steep low energy component down to 0.2~keV
allows us to
set an upper limit on $T_{\rm cut}$. The upper limit is set using
$\alpha_o$ and $\alpha_{os'}$ the slope from 3000~\AA\ to rest frame
$0.15(1+z)$~keV
(the lowest energy where the Galactic absorption correction error$\le30$\%),
given by
\[ \alpha_{os'}=[2.685\alpha_{ox}-(17.685-\log\nu_{s'})\alpha_x]/
\log (\nu_{s'}/10^{15}), \]
\[ \ \ {\rm where}\ \
\log \nu_{s'}=16.560+\log(1+z). \]
The upper limit on the cutoff temperature $T_{\rm cut}^{\rm ul}$ is related to
the spectral slopes by
\[
T_{\rm cut}^{\rm ul}=\frac{
4.8\times 10^{-11}(\nu_{s'}-10^{15})\log e}{
(\alpha_{o}-\alpha_{os'})\log(\nu_{s'}/10^{15})}~{\rm K}.
\]
We find a rather small dispersion in $T_{\rm cut}^{\rm ul}$
with $\langle T_{\rm cut}^{\rm ul} \rangle=(5.5\pm 2.6)\times 10^5$~K,
averaged over
the complete sample (Table 4), which corresponds to a cutoff energy of
47~eV, or about
3.5 Ryd. This value of $T_{\rm cut}^{\rm ul}$ corresponds very closely to the
far UV continuum shape assumed by MF (see Fig.6).
Walter et al. (1994) fit such a cutoff model directly to
six quasars and Seyfert galaxies finding
$\langle E_{\rm cut}\rangle=63\pm 12$~eV, or
$\langle T_{\rm cut}\rangle=(7.3\pm 1.4)\times 10^5$~K, while
Rachen, Mannheim \& Biermann (1996) find using such a model
$\langle T_{\rm cut}\rangle=(6.3\pm 2.3)\times 10^5$~K for 7
quasars and Seyfert galaxies. These values are
consistent with our results. The small dispersion in $T_{\rm cut}$
reflects the small dispersion in $\alpha_{\rm os}$ in our sample, which
is in marked contradiction with the dispersion predicted by thin accretion
disk models, as further discussed in \S 4.5.
\subsubsection{The Far UV Continuum}
Zheng et al. (1996) have constructed a composite quasar spectrum
based on HST spectra of 101 quasars at $z>0.33$. They find a
far UV (FUV) slope (1050\AA-350\AA) of
$\langle \alpha_{\rm FUV}\rangle=-1.77\pm 0.03$ for RQQ and
$\langle \alpha_{\rm FUV}\rangle=-2.16\pm 0.03$ for RLQ, with slopes
of $\sim -1$ in the 2000\AA-1050\AA\ regime. The Zheng et al. mean
spectra, presented in Fig.6, together with the PSPC mean spectra,
suggest that the FUV power-law
continuum extends to the soft X-ray band. In the case of RQQ there
is remarkable agreement in both slope and normalization of the soft
X-ray and the FUV power-law
continua. RLQ are predicted to be weaker than RQQ at $\sim 100$~eV
by both the FUV and the PSPC composites. It thus appears that there
is no extreme UV sharp cutoff in quasars, and that the fraction
of bolometric luminosity in the FUV regime is significantly smaller
than assumed.
The UV to X-ray continuum suggested in Fig.6 is very different from the
one predicted by thin accretion disk models (\S 4.5), and suggested
by photoionization models. In particular, it implies about four times
weaker FUV ionizing continuum compared with the MF continuum which
was deduced based on the
He~II~$\lambda 1640$ recombination line.
One should note, however, that
the Zheng et al. sample is practically disjoint from our low $z$ sample,
so it may still be possible that low $z$ quasars have a different FUV
continuum.
\subsection {Intrinsic Absorption}
As shown in Fig.2, the H~I column deduced from our accurate 21~cm
measurements is consistent for all objects with the best fit X-ray column.
It is quite remarkable that even in our highest S/N spectra the 21~cm
and X-ray columns agree to a level of about $1\times 10^{19}$~cm$^{-2}$,
or 5-7\%.
This agreement is remarkable since the 21~cm line and the PSPC are actually
measuring the columns of different elements. Most of the soft X-ray
absorption is due to He~I or He~II, rather than H~I, and the H~I column
is indirectly inferred assuming the column ratio H~I/He~I$=10$.
The fact that the 21~cm line and the PSPC give the same H~I
column implies that the H~I/He column ratio at high Galactic latitudes
must indeed be close to 10. The dispersion in the H~I/He column ratio
is lower than 20\% (based on typical quasars in
our sample), and may even be lower than 5\% (based on our highest S/N
spectra). There is therefore no appreciable Galactic column at high
Galactic latitudes where the
ionized fraction of H differs significantly from the ionized fraction
of He, as found for example in H~II regions (e.g. Osterbrock 1989).
The consistency of the 21~cm and X-ray columns also indicates
that the typical column of cold gas intrinsic to the quasars in our sample
must be
smaller than the X-ray $N_{\rm H}$\ uncertainty, or about
$3\times 10^{19}(1+z)^3$~cm$^{-2}$.
An additional indication for a lack of an intrinsic cold column in quasars
comes from the fact the the strong correlations of $\alpha_x$
with the emission line parameters described above (\S 3.3) become weaker
when we use $\alpha_x$ from the free $N_{\rm H}$\ fit rather than $\alpha_x$ from
the fit with $N_{\rm H~I}^{\rm {\tiny Gal}}$. This indicates that $N_{\rm H~I}^{\rm {\tiny Gal}}$\
is closer to the true $N_{\rm H}$\ than the free $N_{\rm H}$\ (see discussion in Paper I).
In our highest S/N spectra we can set an upper limit of
$\sim2\times 10^{19}$~cm$^{-2}$ on any intrinsic absorption.
As discussed in Paper I, the lack of intrinsic X-ray column for most quasars
is consistent with more stringent
upper limits set by the lack of a Lyman limit edge, as well as the He~I and
the He~II bound-free edges in a few very high z quasars.
Puchnarewicz et al. (1996) suggest that the strong $\alpha_x$ vs.
$\alpha_{ox}$ correlation in their X-ray selected sample is due to absorption
of the optical
and soft X-ray emission by cold gas and dust. They show that the $\alpha_x$
vs. $\alpha_{ox}$ correlation for the 10 objects in Paper I can be explained
by a universal spectral shape absorbed by a gas with a column of up to
$N_{\rm H}$$=3\times 10^{20}$~cm$^{-2}$ (see their Figure 16). As described above,
such absorbing columns are clearly ruled out by our high S/N spectra.
The mean $\alpha_o$ in the Puchnarewicz et al. sample is $-0.92\pm 0.07$,
which is significantly steeper than the mean $\alpha_o$ for optically
selected quasars, e.g. a median of $-0.2$ for 105 PG quasars
(Neugebauer et al. 1987), a median of $-0.32$ for 718 LBQS quasars
(Francis et al. 1991), and $\langle\alpha_o\rangle=-0.36\pm 0.05$,
in our sample. Puchnarewicz et al. suggested that the much flatter
$\alpha_o$ of the PG quasars is a selection bias since these quasars were
selected by the strength of their UV excess. However, the PG sample was
selected on the basis of the color criterion $U-B<-0.44$, which using
the flux transformations of Allen (1973), corresponds to $\alpha_o\ge-1.8$.
Thus, most of the red quasars discovered by Puchnarewicz et al. fit into
the PG color criterion. The difference between the soft X-ray selected and
optically selected quasars must reflect the true tendency of
quasars selected above 0.5~keV
to be significantly redder than optically selected quasars.
These red quasars may very well be affected by a large
absorbing column ($N_{\rm H}$$>10^{21}$~cm$^{-2}$), as suggested by
Puchnarewicz et al.
Intrinsic absorption is common in Seyfert 1 galaxies. About half of the
primarily X-ray selected
Seyfert galaxies observed by Turner \& Pounds (1989) using the {\em EXOSAT}
LE+ME detectors,
by Turner, George \& Mushotzky (1993) using the {\em ROSAT} PSPC, and by
Nandra \& Pounds (1994) using the {\em GINGA} LAC for a largely overlapping
sample, show low energy
absorption, or spectral features inconsistent with the simple power-law
typically observed above 2~keV. Quasars are very different. Excess
absorption produces significant spectral features only in one object
(PG~1114+445, see paper~I), i.e. $\sim 5$\%
(1/23) of the objects, and the absorbing gas is partially ionized ("warm"),
rather than neutral. Given the typical S/N in our sample we estimate that
a partially ionized absorber which produces $\tau>0.3$ can be ruled out
in most of our objects.
We cannot rule out partial
absorption, or complete absorption and scattering,
by a very high column density ($N_{\rm H}$$>10^{24}$~cm$^{-2}$) gas since such
effects may only suppress the flux level without affecting
the spectral shape, and without inducing significant spectral features.
As described in \S 4.8, we suspect that such strong absorption may indeed be
present in about $\sim 10$\% (3/23) of the quasars in our sample (the X-ray
weak quasars).
\subsection {Implications of the Continuum-Continuum Correlations}
The continuum-continuum luminosity correlations found here are
all weaker than found in Paper I. This is mostly due to the three X-ray weak
quasars which were not present in Paper I. For example, in Paper I we
found that $f_{0.3~{\rm keV}}$ can be predicted to within a
factor of two, once $f_{1.69~\mu m}$ is given. This statement is still valid
if the 4 extreme objects labeled in Fig.3 middle panel are excluded.
The implications of the near IR versus X-ray luminosity correlation on the
X-ray variability power spectrum were discussed in Paper I.
In Paper I we noted the similarity
$\langle\alpha_{ox}\rangle=\langle\alpha_x\rangle=-1.50$, which was also
noted by Brunner {\it et al.} (1992) and Turner, George \& Mushotzky (1993).
However, we argued there that this similarity is only fortuitous, and
that it does not imply that the X-ray power law can be extrapolated into
the UV since the optical slope is significantly different. Here we find that
the relation
$\langle\alpha_{ox}\rangle\simeq \langle\alpha_x\rangle$ holds only
roughly for the complete sample where
$\langle\alpha_{ox}\rangle=-1.55\pm 0.24$, and
$\langle\alpha_x\rangle=-1.62\pm 0.45$. This relation
does not hold when the sample is broken to the RQQ where
$\langle\alpha_{ox}\rangle=-1.56\pm 0.26$, and
$\langle\alpha_x\rangle=-1.72\pm 0.41$,
and to the RLQ where
$\langle\alpha_{ox}\rangle=-1.51\pm 0.16$, and
$\langle\alpha_x\rangle=-1.15\pm 0.27$.
A significantly flatter $\langle\alpha_{ox}\rangle$ is obtained when the
three X-ray weak quasars, and the absorbed quasar PG~1114+445 are excluded.
Thus, ``normal'' RQQ quasars in our sample have
$\langle\alpha_{ox}\rangle=-1.48\pm 0.10$, $\langle\alpha_x\rangle=-1.69\pm
0.27$, while for the RLQ
$\langle\alpha_{ox}\rangle=-1.44\pm 0.12$, $\langle\alpha_x\rangle=-1.22\pm
0.28$, where the $\pm$ denotes here and above the dispersion about the mean,
rather than the error in the mean.
The $\alpha_x$ versus $\alpha_{ox}$ correlation found here is weaker than in
Paper I due to the presence of the X-ray weak quasars. However, the other
20 quasars appear to follow a trend of increasing $\alpha_x$ with
increasing $\alpha_{ox}$ (Fig.5e), indicating as discussed in Paper I
that a steep $\alpha_x$ is generally associated with a weak hard X-ray
component (at 2~keV), rather than a strong soft excess. The only object
which clearly violates this trend is PG~1626+554 (Fig.3), which has both
a steep $\alpha_x$ and a strong soft excess. Puchnarewicz, Mason \& Cordova
(1994) and Puchnarewicz et al. (1995a;
1995b) present PSPC spectra of three AGNs with extremely strong soft excess,
where $L_{\rm 0.2~keV}>L_{\rm 3000~\AA}$. Our sample suggests that such objects
are most likely rare, as can also be inferred from the selection criteria of
Puchnarewicz et al. who selected their three objects from
the {\em ROSAT} WFC all sky survey, in which only five AGNs were detected.
This selection criterion implies that these AGNs must
have a very high far UV flux.
The soft X-ray selected quasars in the Puchnarewicz et al. (1996) sample
have $\langle\alpha_{ox}\rangle=-1.14\pm 0.02$, and none of their quasars
is ``X-ray weak'', i.e with $\alpha_{ox}<-1.6$. The absence of X-ray
weak quasars in their sample is clearly a selection effect. The small
survey area implies that most quasars in their sample are optically rather
faint ($m_B\sim 18$-19). ``Normal'' $\alpha_{ox}$ quasars in their sample
produce a few hundred PSPC counts, but ``X-ray weak'' quasars are below
their detection threshold. The abundance of ``X-ray loud'' quasars
(i.e $\alpha_{ox}>-1$) in the Puchnarewicz et al. sample is consistent
with their rarity in optically selected samples. For example,
quasars with $\alpha_{ox}=-1$ are about 20 times fainter at 3000\AA\
than quasars with $\alpha_{ox}=-1.5$, for the same $L_x$. Since the
surface density of quasars increases as $\sim f_{\rm B}^{-2.2}$ (see
\S 2.2.2.1 and Figure 1 in Hartwick \& Schade), where
$f_{\rm B}$ is the B band flux,
there are about 700 times more of these fainter quasars per B magnitude
per square degree. Thus, even if only 0.3\% of quasars at a given
$f_{\rm B}$
have $\alpha_{ox}=-1$, there would still be twice as many quasars with
$\alpha_{ox}=-1$
than $\alpha_{ox}=-1.5$ per square degree in an X-ray flux limited sample,
such as the Puchnarewicz et al. sample.
\subsection {Implications of The Continuum-Line Correlations}
The presence of the strong correlations of $\alpha_x$ with the H$\beta$\ FWHM,
with $L_{\rm [O~III]}$, and with the Fe~II/H$\beta$\ ratio described in
Paper I is verified here. The correlation coefficients for the complete
sample are comparable or somewhat smaller than those found in Paper I, but
since the sample is larger the significance level is now much higher
(Fig.5). We also report here an additional strong correlation of
$\alpha_x$ with the ratio of [O~III] peak flux to the H$\beta$\ peak flux,
which is one of the emission line parameters measured by Boroson \& Green.
The $\alpha_x$ versus H$\beta$\ FWHM correlations is
the strongest correlation we find between any of the X-ray continuum
emission parameters and any of the emission line parameters reported by
Boroson \& Green (with the addition of line luminosities reported in Table 5).
The $\alpha_x$ versus H$\beta$\ FWHM correlation is much stronger than
the well known $\alpha_x$ correlation with radio loudness
($r_S=0.26$, Pr=0.23 in our sample, but see Wilkes \& Elvis 1987 and
Shastri et al. 1993 for stronger correlations). Thus, the fact that
the average $\alpha_x$ in RLQ is significantly flatter than in
RQQ (\S 4.1) may be completely unrelated to the presence of radio
emission, it may just reflect
the fact that RLQ tend to have broader lines than RQQ (e.g. Tables 3 and
5 in Boroson \& Green). This
appears to be the case in our sample, where the RLQ follow the same
$\alpha_x$ versus H$\beta$\ FWHM distribution defined by the RQQ (Fig.5a).
This intriguing suggestion can be clearly
tested by comparing $\alpha_x$ for RLQ and RQQ of similar H$\beta$\ FWHM.
Boller, Brandt \& Fink (1996) studied in detail narrow line Seyfert 1
galaxies (NLS1) and they also find an apparently significant trend of
increasing $\alpha_x$ with increasing H$\beta$\ FWHM. However, the scatter in
their sample is significantly larger than that found here. In particular
they find a large range of $\alpha_x$ for H$\beta$\ FWHM$<2000$~km~s$^{-1}$,
where only a few objects are available in our sample.
The overall larger scatter in the Boller, Brandt \& Fink data is probably
due in part to the generally larger statistical
errors in their $\alpha_x$ determinations. Large
systematic errors may also be induced by the use of H$\beta$\ FWHM from
a variety of sources. The measured H$\beta$\ FWHM can be sensitive to the
exact measuring procedure, such as continuum placement, subtraction
of Fe~II blends, and subtraction of the narrow component of the line
(produced in the narrow line region) which may not be well resolved
in low resolution spectra. For example, Shastri et al. 1993 and
Boroson \& Green
measured the H$\beta$\ FWHM independently for 13 overlapping objects, in 8
of which their values differ by more than 1000~km~s$^{-1}$.
Other than these technical reasons the
increased scatter may represent a real drop in the strength of the
correlation when the luminosity decreases from the quasar level studied here
to the Seyfert level studied by Boller, Brandt \& Fink. One should also
note that intrinsic absorption is probably common in Seyfert 1 galaxies
(Turner \& Pounds, 1989; Turner, George \& Mushotzky, 1993), and such an
absorption may lead to a large
systematic error in $\alpha_x$ unless a high S/N spectrum is available
indicating that features are present.
Wang, Brinkmann \& Bergeron (1996) analyzed PSPC spectra of 86 AGNs,
including 22 of the 23 quasars from our sample. Their sample is more
heterogeneous than ours and includes some high z quasars and a number
of AGNs selected by their strong Fe~II emission. The various correlations
found by Wang et al. are typically similar, or somewhat weaker than found
here. For example, their(our) values are $r_S=-0.73(-0.79)$ for
$\alpha_x$ versus H$\beta$\ FWHM, and $r_S=0.65(0.714)$ for $\alpha_x$ versus
Fe~II/H$\beta$. The somewhat smaller values found by Wang et al. may result
from their inclusion of $z>0.4$ quasars, where $\alpha_x$ measures a
higher energy slope than measured here, and from their use of a free $N_{\rm H}$\
fit (limited from below by $N_{\rm H~I}^{\rm {\tiny Gal}}$), which increases the random error
in $\alpha_x$ (see Paper I).
We verify the strong correlation between $L_{\rm H\beta}$ and
$L_{\rm 2\ keV}$ found in Paper I. The correlations of the other lines
with X-ray luminosity are significantly weaker than found in Paper I,
and they are only marginally significant.
Corbin (1993) found significant correlations of $L_{\rm 2\ keV}$ with
Fe~II/H$\beta$\ ($r_S=-0.474$), and of $L_{\rm 2\ keV}$ with the H$\beta$\ asymmetry
($r_S=-0.471$). We find that neither correlation is significant in our
sample ($r_S=-0.288$, Pr=0.19, and $r_S=-0.106$, Pr=0.63). Since
we can only test for correlation with $r_S>0.64$, we cannot securely
exclude the presence of the correlations reported by Corbin.
As discussed in Paper I, the $\alpha_x$ versus $L_{\rm [O~III]}$
correlation
can be used to place a limit on the $\alpha_x$ variability on timescales
shorter than a few years. Given the scatter in this correlation we estimate
that $\alpha_x$ should not vary by significantly more than 0.3 on these
timescales.
It is hard to interpret the $\alpha_x$ versus [O~III] to H$\beta$\ peak flux
ratio correlation since the physical meaning of the [O~III] to H$\beta$\ peak
flux ratio parameter defined by Boroson \& Green is rather obscure.
The [O~III] peak flux is related to the width of [O~III], and the
[O~III] to H$\beta$\ peak flux ratio may thus partly reflect the FWHM ratio of
these lines. Thus, this correlation may represent a correlation of the
[O~III] FWHM with $\alpha_x$. High spectral resolution measurements of the
[O~III] line profile are required to test this possibility.
\subsection {Inconsistency with Thin Accretion Disk Models}
Figure 7 presents the continuum emission from two thin accretion disk
models. The models are for a disk around a rotating
black hole, and viscous stress which scales like the
$\sqrt{P_{\rm gas}P_{\rm rad}}$, where $P_{\rm rad}$ is the radiation
pressure and $P_{\rm gas}$ is the gas pressure
(Laor \& Netzer 1989).
Significant soft X-ray emission is obtained for disks
with a high accretion rate and a small inclination. However, as discussed
by Fiore et al. (1995), the observed
soft X-ray spectral slope is always much flatter than the one produced
by a thin `bare' accretion disk model. As noted above there is no
indication in the 0.2-2~keV band for a very steep and soft ``accretion disk''
component.
Although thin disks cannot
reproduce the 0.2-2~keV spectral shape, they may still be able
to contribute a significant fraction of the flux at the lowest
observed energy, i.e. 0.2-0.3~keV, above which a non thermal power-law
component sets in. As noted by Walter et al. (1994) and in Paper I,
accretion disk models
predict a large dispersion in the optical/soft X-ray flux ratio, and
the strong correlation between these fluxes argues against
the idea that a thin disk produces both the optical and
soft X-ray emission. The arguments put by Walter et al. and
in Paper I were only qualitative, and were not based on actual disk
models. Furthermore, the objects in the small sample of Walter et al.
were selected from known optically bright AGNs, and they also had to
be bright soft X-ray sources since most spectra were obtained from the
{\em ROSAT} all sky survey. Thus, these objects were a priori selected
to be bright at both optical-UV and at soft X-rays, and the absence of
a large scatter in the UV/soft X-ray flux ratio may just reflect the sample
selection criteria. Such selection effects are not present in our sample
since the sample was defined independently of the X-ray properties, and
X-ray spectra were obtained for all objects.
Below we describe a detailed
calculation of the expected distribution of optical/soft X-ray flux
ratio for a complete optically selected sample
based on the thin disk models of Laor \& Netzer (1989), and
show that such models cannot be reconciled with the observed
distribution of optical/soft X-ray flux ratio in our complete sample.
The optical/soft X-ray flux ratio, $\alpha_{os}$, of a given disk
model depends on the black hole mass, accretion rate $\dot{m}$, and
inclination angle $\theta=\cos^{-1}\mu$. We now need to determine what
distributions of these parameters will be consistent with the
observed luminosity function in a complete optically selected sample.
The intrinsic distribution of disk inclinations must be random. However,
the observed distribution depends on the shape of the luminosity
function of quasars, and possible obscuration effects, as described below.
The luminosity function of quasars is parametrized using the number
density of quasars per unit volume per magnitude
$\Phi\equiv d^2N/dMdV$, and it is well fit by a power-law
over a restricted range of magnitude, M. Using Figure 2 in Hartwick \&
Schade (1990) we find $\log \Phi= 0.55M+c$ for $z<0.2$ and
$\log \Phi= 0.66M+c$ for $0.4<z<0.7$, where $c$ is a constant.
Since our sample is
restricted to $z<0.4$ we assume $\log \Phi= 0.6M+c$. Using the
relation $M=-2.5\log L+c$ we get that $dn/dL\propto L^{-2.5}$, where
$n\equiv dN/dV$.
The apparent luminosity of a flat disk $L_{\rm app}$ is related to its
intrinsic luminosity through $L_{\rm app}=2\mu L$, neglecting limb
darkening effects which steepen the $\mu$ dependence, and relativistic
effects which flatten the $\mu$ dependence. This provides a reasonable
approximation in the optical-near UV regime (see Laor, Netzer, \& Piran
1990).
Assuming $dN/d\mu=const$, i.e. a uniform distribution of inclination
angles for the intrinsic quasar population, we would like to find
$dn/d\mu$ for a given $L_{\rm app}$.
When $L_{\rm app}$ is fixed, $\mu \propto L^{-1}$, and substituting $\mu$
in the expression for $dn/dL$ we get $dn/d\mu\propto \mu^{0.5}$. Thus
although the disks are assumed to have a uniform distribution of
inclination angles the observed distribution at a given $L_{\rm app}$
is biased towards face on disks.
To reproduce the observed luminosity function we choose two
values of $\dot{m}=0.1, 0.3$, where $\dot{m}$ is measured in units of the
Eddington accretion rate. Since $L\propto m_9\dot{m}$, where
$m_9$ is the black hole mass in units of $10^9M_\odot$
the required mass distribution is $dn/dm_9\propto m_9^{-2.5}$.
The observed number of objects in a flux limited sample
is $dN_{\rm ob}/dL\propto dn/dL\times V(L)$, where
$V(L)\propto L^{3/2}$ is the observable volume for a flux limited
sample, such as the BQS sample. We therefore select a mass distribution
of $dN_{\rm ob}/dm_9\propto m_9^{-1}$.
Figure 8 compares the observed distribution of $\alpha_{os}$, as a
function of $\nu L_{\nu}$ at 3000\AA, with the one expected from thin
accretion disk
models with the parameter distribution described above.
Thin disk models cannot account for the very small scatter in
$\alpha_{os}$.
The range of observed disk inclinations may actually be smaller than
assumed here. For example, for a certain range of inclinations the
disk may be completely obscured by an optically thick torus, as
suggested in unification schemes for RQQ (e.g. Antonucci 1993). However,
even if $\mu$ is fixed at a given value for all AGNs (say $\mu=1$ which
corresponds to the points extending from $\alpha_{os}=-1.5$ on the
left axis to $\log \nu L_{\nu}=46.5$ on the bottom axis of Fig.8),
the range in $m_9$ and $\dot{m}$ will stil produce a range in $\alpha_{os}$
which is much larger than observed.
The X-ray power-law emission is most likely produced by Comptonization
of the thermal disk emission in a hot corona above the disk (e.g. Czerny
\& Elvis 1987). The slope and normalization of the power-law component
are determined by the temperature and electron scattering optical depth
in the corona (e.g. Sunyaev \& Titarchuk 1985; Titarchuk \& Lyubarskij 1995).
The small range in $\alpha_{os}$
implies that some physical mechanism which couples the optical and
soft X-ray emission processes must be operating, e.g. through a
feedback which regulates both the temperature (see Haardt \& Maraschi 1993)
and the optical depth of the corona.
As pointed out by various authors (Ross, Fabian \& Mineshige, 1992;
Shimura \& Takahara 1995;
Dorrer et al. 1996), and shown in Fig.7, simple thin accretion disks with no
corona can produce a significant flux below 1~keV. For various disk model
parameters $\alpha_{os}$ can in fact be significantly flatter than observed
(Fig.8), yet such extreme flat optical-soft X-ray spectra are only rarely
observed (e.g. Puchnarewicz 1995a). The flattest spectra are expected for disks
which are close to edge on (e.g. Laor, Netzer \& Piran 1990), and one therefore
needs to assume that such disks are not observable. This is indeed expected in
AGNs unification schemes which invoke obscuring material close to the disk
plane. Alternatively, the accreted material may form a geometrically thick,
rather than a thin,
configuration close to the center, which would display a smaller inclination
dependence.
\subsection {Inconsistency with Optically Thin Free-Free
Emission Models}
As was clearly demonstrated by Fiore et al. (1995) for 6 low redshift
quasars with a high S/N PSPC spectra, and by
Siemiginowska et al. (1995) using {\em EINSTEIN} data for 47 quasars
from Elvis et al. (1994), isothermal optically thin pure free-free emission
models (Barvainis 1993) cannot fit the
observed UV to soft X-ray energy distribution in AGNs. Furthermore, as was
pointed out by Kriss (1995), and Hamman et al. (1995), optically thin free-free
emission can also be ruled out based on the observed UV line emission.
\subsection {On the origin of the $\alpha_x$ versus H$\beta$\ FWHM
correlation}
What is the physical process behind the $\alpha_x$ versus H$\beta$\ FWHM
correlation? In Paper I we speculated that this may either be an
inclination effect, or that it could be an $L/L_{\rm Edd}$ effect.
Various authors raised the interesting suggestion that steep
$\alpha_x$ quasars may be analogous to `high'-state Galactic black
hole candidates (e.g. White, Fabian \& Mushotzky 1984;
Fiore \& Elvis 1995; Pounds, Done \& Osborne 1995), which display a
steep slope in the soft and the hard X-ray bands when
their brightness increases. The physical interpretation for this effect
described by Pounds, Done \& Osborne (1995) is that the hard X-ray
power-law is produced by Comptonization in a
hot corona and that as the object becomes brighter in the optical-UV,
Compton cooling of the corona increases, the corona becomes colder,
thus producing a steeper X-ray power-law. This is obviously far from
being a predictive model since the corona heating mechanism
is not specified, and it is implicitly assumed that the coronal heating
does not increase much as the quasar becomes brighter. However, the narrow
H$\beta$\ line profiles provide independent evidence that steep $\alpha_x$
quasars may indeed have a higher $L/L_{\rm Edd}$, as further
described below.
The $L/L_{\rm Edd}$ of quasars can be estimated under two
assumption: 1. The bulk motion of the gas in the broad line region is
virialized, i.e. $\Delta v\simeq \sqrt{GM/r}$,
where $\Delta v$=H$\beta$\ FWHM. This gives
\[ \Delta v_{3000}=2.19m_9^{1/2}R_{0.1}^{-1/2},\]
where
$\Delta v_{3000}=\Delta v/3000$~km~s$^{-1}$, $m_9=M/10^9~M_{\odot}$, and
$R_{0.1}=R/0.1$~pc. 2. The size of the broad line region is
determined uniquely by the luminosity, $R_{0.1}=L_{46}^{1/2}$~pc,
where $L_{46}=L_{\rm Bolometric}/10^{46}$. This
scaling is consistent with reverberation line mapping of AGNs
(Peterson 1993; Maoz 1995), and is theoretically expected if the
gas in quasars is dusty (Laor \& Draine 1993, Netzer \& Laor 1994).
Combining assumptions 1 and 2 gives
\[ \Delta v_{3000}=2.19m_9^{1/2}L_{46}^{-1/4}, \]
and thus the mass
of the central black hole is
\[ m_9=0.21\Delta v_{3000}^2L_{46}^{1/2}. \]
Using $L_{\rm Edd,46}=12.5m_9$ one gets
\[ L/L_{\rm Edd}=0.38\Delta v_{3000}^{-2}L_{46}^{1/2}. \]
Thus, given the two assumptions made above, narrow line quasars should
indeed have a high $L/L_{\rm Edd}$,
as previously suggested based only on their steep
$\alpha_x$, and analogy to Galactic black hole candidates.
To test whether $L/L_{\rm Edd}$ is indeed the underlying parameter
which determines $\alpha_x$, rather than just the H$\beta$\ FWHM, we
looked at the correlation of $\alpha_x$ versus
$\Delta v_{3000}^{-2}L_{46}^{1/2}$ displayed in Fig.5f, where we used the
1.7~$\mu$m luminosity and the relation $L_{\rm Bolometric}=15L_{14.25}$
(see Fig.7 in Laor \& Draine). This correlation is not as strong as the
$\alpha_x$ versus H$\beta$\ FWHM correlation, but it certainly appears
suggestive. Note that $L/L_{\rm Edd}>1$ for some of the objects in
the sample. These values are well above the thin accretion disk limit
($L/L_{\rm Edd}=0.3$, Laor \& Netzer 1989) and suggest a thick disk
configuration. However, the
assumptions used above to infer $L/L_{\rm Edd}$ are more qualitative
than quantitative since both the luminosity and the velocity field in the
broad line region may not
be isotropic and therefore the presence of $L/L_{\rm Edd}>1$
cannot be securely deduced.
The Pounds et al. mechanism implies that a steep $\alpha_x$ is
associated with a weak hard X-ray component, and as described in
\S 3.2, this indeed appears to be the trend in our
sample. If the Pounds et al. mechanism is true then steep $\alpha_x$
AGNs should also have a steep hard X-ray power-law. We are currently
pursuing this line of research using ASCA and SAX observations of
our sample.
An additional hint that a steep $\alpha_x$ may indeed be associated
with a high
$L/L_{\rm Edd}$ comes from the anecdotal evidence described by
Brandt, Pounds \& Fink (1995), Brandt et al. (1995), Grupe et al.
(1995), and Forster \& Halpern (1996) where a
number of Seyfert galaxies with a steep $\alpha_x$ display
rapid, large amplitude, soft X-ray variability, which as Boller, Brandt,
\& Fink discuss may imply a low mass black hole, and thus a high
$L/L_{\rm Edd}$. We are currently pursuing
a more systematic study of the soft X-ray variability properties of broad
versus narrow line quasars using the {\em ROSAT} HRI.
The Pounds et al. suggestion is very appealing since it allows
a physical explanation for the tight correlation of apparently
completely unrelated quantities. Although it is not clear a priori
that $\alpha_x$ must steepen with increasing $L/L_{\rm Edd}$,
it appears that this is indeed what happens in Galactic black hole
candidates.
Wang et al. also suggested that steep $\alpha_x$ objects have
a high $L/L_{\rm Edd}$ based on the fact that the fraction of luminosity
emitted in the X-ray regime in thin accretion disk models increases with
$L/L_{\rm Edd}$, as discussed above in \S 4.5. However, if this were
indeed the physical process behind the $\alpha_x$ versus H$\beta$\ FWHM
correlation then one would expect high $L/L_{\rm Edd}$ objects to
have a high soft X-ray to optical flux ratio, while we find no correlation
between H$\beta$\ FWHM and $\alpha_{os}$ ($r_S=-0.079$, Table 5).
We note in passing that one does not need to eliminate
the normal broad line region in narrow line AGNs, as one of the
options suggested by
Boller et al., and Pounds et al. The lines are narrow simply
because of the lower black hole mass. The broad line emitting gas
does not extend much closer to the center in narrow line AGNs,
as it does not extend much closer to the center in other AGNs,
simply because of the effects of a higher
ionization parameter, and a higher gas density, each of which
quenches line emission.
\subsection {The X-ray Weak Quasars}
Two of the quasars in our sample, the RQQ PG~1001 +054 and PG~1411+442,
and possibly also the RLQ PG~1425+267 appear to form a distinct group which
we term
here ``X-ray weak'' quasars, where the normalized X-ray luminosity is
a factor of 10-30 smaller than the sample median. The position of these
quasars as outliers can be noticed in the near IR normalized flux
distribution (Fig.3), in the $\alpha_x$ versus $\alpha_{ox}$ correlation
(Fig.5e), and in the H$\beta$\ versus 2~keV and 0.3~keV luminosity
correlations (Fig.4). The first two indicators are based on the spectral
shape, but the last one is independent of the spectral shape, and it also
suggests a deficiency of the X-ray luminosity by a factor of 10-30
relative to the one expected based on the H$\beta$\ luminosity.
An apparently bimodal distribution in $\alpha_{ox}$ can also be seen
in Figure 5b of Wang et al. where 6 of their 86 quasars appear to form
a distinct group with $\alpha_{ox}<-2$.
No bimodality of $\alpha_{ox}$ is seen in the Avni \& Tananbaum (1986)
{\em EINSTEIN} study of the PG quasars. All the ``X-ray weak'' quasars
found by the {\em ROSAT} PSPC have $\alpha_{ox}<-2$, but none of the quasars
detectd by Tananbaum et al. have $\alpha_{ox}<-2$ (see Fig.8 in Avni \&
Tananbaum). The lack of $\alpha_{ox}<-2$ and bimodality in the Tananbaum
et al. sample probably reflects its incompleteness, as only 86\% of the
quasars they observed were detected.
Although the three X-ray weak quasars in our sample
stand out in luminosity correlations, they
conform well to the $\alpha_x$ correlations (Figs.5a-d). They thus have
the ``right'' slope but the ''wrong'' flux level. Why are these quasars
different? A simple answer is that for some unknown reason the X-ray
emission mechanism, most likely Comptonization by $T\ge 10^8$~K electrons,
tends to be bimodal, and in about 10\% of quasars
(or in all quasars for $\sim 10$\% of the time) the X-ray flux
level is strongly suppressed, while the spectral slope is not affected.
Another option is that these are just normal quasars where the direct
X-ray flux happens to be obscured. In this case what we see is only the
scattered X-ray flux. Photoionization calculations indicate that a few
percent of the direct flux will be scattered, depending on the covering
factor of the absorber and the ionization parameter. If the
ionization parameter is large enough then the scattering will be
mostly by free electrons which preserves the spectral shape
(see Netzer 1993, and by Krolik \& Kriss 1995).
Such scattering will explain why the flux level is strongly reduced, while
the spectral shape is not affected. Note that the obscuring matter should
be transparent in the visible range, as is the case with the absorbing
matter in BALQSO.
Additional hints towards this
interpretation come from the fact that PG~1411+442 is a broad absorption
line quasar (BALQSO, Malkan Green \& Hutchings, 1987), and the UV absorbing gas
may also produce soft
X-ray absorption, as may also be the case in PHL~5200
(Mathur Elvis \& Singh, 1995).
In addition, Green \& Mathur (1996) find that BALQSO observed by
{\em ROSAT} have $\alpha_{ox}\lesssim -1.8$, i.e. as observed here in the
X-ray weak RQQ.
Another
hint is provided by the fact that PG~1114+445 is also somewhat
underluminous at 0.3~keV (Fig.3b), and this quasar is most likely seen
through a warm absorber. The X-ray weak quasars could therefore be more
extreme cases of PG~1114+445 and have an absorbing column which is large
enough to completely absorb the direct soft X-ray emission.
Note that PG~1425+267 is a RLQ, while all BALQSO are known to be RQQ
(Stocke et al.
1992). It would thus seem implausible to suggest that PG~1425+267 is a BAL.
However, PG~1425+267 has about the same relatively steep $\alpha_{ox}$,
compared to other RLQ, as in 3C~351 (Fiore et al. 1993),
where X-ray absorption by warm gas is
observed together with resonance UV absorption lines (Mathur et al. 1994)
which are narrower than
in `proper' BALQSO.
Forthcoming HST spectra of all 23 quasars in our sample will allow us to test if
there is a one to one correspondence between X-ray weakness and broad absorption
lines, i.e. if all X-ray weak quasars are BALQSO, and not just that
all BALQSO are X-ray weak, as strongly suggested by the
Green \& Mathur, and the Green et al. (1996) results.
A simple test of whether these are truly ``X-ray weak quasars'', or just
normal highly absorbed quasars, can be done by looking at their hard X-ray
emission. If the X-ray column is below $10^{24}$~cm$^{-2}$ then the
obscuring material would become transparent at $E<10$~keV, and the
observed hard X-ray emission will rise steeply above the cutoff
energy, as seen in various highly absorbed AGNs, such as Mkn~3 (Iwasawa et
al. 1994), NGC~5506 (Nandra \& Pounds 1994),
NGC~6552 (Fukazawa, et al. 1994; Reynolds
et al. 1994), and NGC~7582 (Schachter et al. 1996). One may also expect
significant spectral features produced by the obscuring material
(e.g. Matt et al. 1996), depending on the ionization state of this material.
Forthcoming ASCA observations of PG~1411+442 and
PG~1425+267 will allow us to test this scenario.
Another prediction is that the X-ray weak quasars should show lower
variability compared with other quasars of similar X-ray luminosity.
This is because: 1). they are intrinsically more X-ray luminous, and
variability amplitude tends to drop with increasing luminosity
(Barr \& Mushotzky 1986; also Fig.9 in
Boller, Brandt \& Fink). 2). the scattering medium must be significantly
larger than the X-ray source, and short time scale variability will be
averaged out. If the X-ray weak quasars are just due to large amplitude
intrinsic variability of the soft X-ray emission, as seen in some steep
narrow line Seyfert 1 galaxies (\S 4.7), then one may expect the exact
opposite behavior, i.e. these quasars may become significantly brighter
at soft X-rays at some stage in the future.
\section {SUMMARY}
We defined a complete sample of 23 optically selected quasars which
includes all
the PG quasars at $z\le 0.400$,
and $N_{\rm H~I}^{\rm {\tiny Gal}}$$< 1.9\times 10^{20}$~cm$^{-2}$. Pointed {\em ROSAT} PSPC
observations were made for all quasars, yielding high S/N spectra
for most objects. The high quality of the {\em ROSAT} spectra allows one
to determine the best fitting $\alpha_x$ with about an order of
magnitude higher precision
compared with previously available X-ray spectra. In this paper we report
the observations of 13 quasars not described in Paper I, analyze the
correlation of the X-ray properties of the complete sample with other
emission properties, determine the mean X-ray spectra of low z quasars,
discuss the possible origin of the $\alpha_x$ versus H$\beta$\ FWHM
correlation, the nature of X-ray weak quasars, and the physical origin
of the soft X-ray emission. Our major results are the following:
\begin{enumerate}
\item The spectra of 22 of the 23 quasars are consistent, to within
$\sim 10-30$\%, with a single power-law model over the rest frame range
$0.2-2$~keV.
There is no evidence for significant soft excess emission with
respect to the best fit power-law. We place a limit of
$\sim 5\times 10^{19}$~cm$^{-2}$ on the amount of
excess foreground absorption by cold gas in most of our quasars. The
limits are
$\sim 1\times 10^{19}$~cm$^{-2}$ in the two highest S/N spectra.
\item Significant X-ray absorption by partially ionized gas (``warm absorber'')
in quasars is rather rare, occurring for $\lesssim 5$\% of the population,
which is in sharp contrast to lower luminosity AGNs, where significant
absorption probably occurs for $\sim 50$\% of the population.
\item The average soft X-ray spectral slope for RQQ is
$\langle\alpha_x\rangle=-1.72\pm 0.09$, and it agrees remarkably
well with an extrapolation of the mean 1050\AA-350\AA\ continuum
recently determined by Zheng et al. (1996) for $z>0.33$ quasars.
For RLQ $\langle\alpha_x\rangle=-1.15\pm 0.16$, which suggests that
RLQ quasars are weaker than RQQ below 0.2~keV, as suggested also
by the Zheng et al. mean RLQ continuum. These results suggest that
there is no steep soft component below 0.2~keV.
\item Extensive correlation analysis of the X-ray continuum
emission parameters with optical emission line parameters indicates
that the strongest correlation
is between $\alpha_x$, and the H$\beta$\ FWHM. A possible explanation for
this remarkably strong correlation is a dependence of $\alpha_x$ on
$L/L_{\rm Edd}$, as observed in Galactic black hole candidates.
\item There appears to be a distinct class of ``X-ray weak'' quasars,
which form $\sim 10$\% of the population,
where the X-ray emission is smaller by a factor of 10-30 than expected
based on their luminosity at other bands, and on their H$\beta$\ luminosity.
\item Thin accretion disk models cannot reproduce the observed
0.2-2~keV spectral shape, and they also cannot reproduce the tight
correlation between the optical and soft X-ray emission.
\item The H~I/He~I ratio in the ISM at high Galactic latitudes must be
within 20\%, and possibly within 5\%, of the total H/He ratio.
\end{enumerate}
The main questions raised by this study are:
\begin{enumerate}
\item What is the true nature of X-ray quiet quasars? Are these quasars
indeed intrinsically X-ray weak, or are they just highly absorbed but
otherwise normal quasars?
\item What physical mechanism is maintaining the strong correlation between
the optical-UV and the soft X-ray continuum emission, or equivalently,
maintaining a very small dispersion in the maximum possible
far UV cutoff temperature?
\item What is the physical origin for the strong correlations
between $\alpha_x$, and $L_{\rm [O~III]}$, Fe~II/H$\beta$, and the peak
[O~III] to H$\beta$\ flux ratio?
\item Is the soft X-ray emission indeed related to the presence of radio
emission, or is it just a spurious relation and the primary effect is related
to the H$\beta$\ line width? Or, put differently,
do RLQ and RQQ of similar H$\beta$\ FWHM have similar $\alpha_x$?
\end{enumerate}
Extensions of the {\em ROSAT} PSPC survey described in this paper to the
hard X-ray regime with ASCA and SAX, to the UV with HST, and soft X-ray
variability monitoring with the {\em ROSAT} HRI, which are currently being
carried out, may provide answers to some of the questions raised above.
These studies
will also allow us to: 1) Test if steep $\alpha_x$ quasars have
a steep 2-10~keV slope, as expected based on the Pounds et al. $L/L_{\rm Edd}$
interpretations. 2) Test if soft X-ray variability is indeed strongly tied
to the H$\beta$\ FWHM, as expected if the H$\beta$\ FWHM is an indicator of
$L/L_{\rm Edd}$. 3) Explore the relation of the UV line emission properties
to the ionizing spectral shape.
\acknowledgments
We thank Niel Brandt, Hagai Netzer, Bev Wills and an anonymous
referee for many useful comments and suggestions.
This work was supported in part by NASA grants NAG 5-2087, NAG 5-1618,
NAG5-2496, NAG 5-30934, NAGW 2201 (LTSA), and NASA contract NAS 8-30751.
A. L. acknowledges support by LTSA grant NAGW-2144.
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{Introduction}
Theories attempting to unify the leptons and quarks in some common framework
often contain new states that couple to lepton-quark pairs, and hence are
called leptoquarks\cite{guts}. Necessarily leptoquarks are color triplets,
carry both baryon number and lepton number, and can be either spin-0
(scalar) or spin-1 (vector) particles.
Perhaps the most well-known examples of leptoquarks appear as gauge
bosons of grand unified theories\cite{gg}. To prevent rapid proton decay they
must be very heavy and unobservable, or their couplings must be
constrained by symmetries. Nonetheless, much work has been devoted
to signals for the detection of leptoquarks at present and future
colliders\cite{pp},\cite{ep},\cite{hr},\cite{ee},\cite{egam},\cite{gamgam}.
One potentially attractive source of light leptoquarks is in $E_6$ models
where the scalar leptoquark can arise as the supersymmetric partner to the
color-triplet quark that naturally resides in the fundamental representation
{\bf 27}. A recent review of the physics signals for leptoquarks can be found
in Ref.~\cite{review}.
At $e^+e^-$ and $\mu^+\mu^-$ colliders, pairs of leptoquarks can be produced
directly via the $s$-channel $\gamma $ and $Z$ exchange. The reach for the
leptoquark mass for this mode is essentially the kinematic limit, i.e.
$M_S< \sqrt{s}/2$. However even if a leptoquark is too massive to be produced
directly, it can contribute\cite{hr},\cite{dreiner},\cite{choudhury}
indirectly to the process
$\ell^+\ell^-\to q\bar{q}$ by interfering with the Standard Model diagrams
as shown in Fig.~1. By examining the overall rate and the angular distribution,
indirect evidence for leptoquarks can be obtained. In this note, we examine
the bounds which can be placed on the leptoquark mass in this way, paying
special attention to assessing the potential advantage that polarized electron
or muon beams might provide.
\begin{center}
\epsfxsize=4.0in
\hspace*{0in}
\epsffile{fig1.eps}
\vspace*{0in}
\parbox{5.5in}{\small Fig.1. The Feynman diagrams for the process
$\ell^+\ell^-\to q\overline{q}$ include the (a) Standard Model diagrams
involving $s$-channel $V=\gamma,Z$ exchange,
and (b) the hypothetical $t$-channel leptoquark $S$ exchange.}
\end{center}
The polarization of the beams of a lepton collider can serve two purposes
in indirect leptoquark searches:
(1) it can extend the reach of the indirect search by serving to enhance
the fraction of initial leptons to which the
leptoquark couples; (2) it can measure the left-handed and right-handed
couplings of the leptoquark separately.
Light leptoquarks (less than a few hundred GeV)
must also satisfy strong constraints from
flavor changing neutral current processes, so that leptoquarks must couple
to a single generation of quarks and leptons.
For the leptoquarks that might be detected at the
multi-TeV machines considered here, the constraints
from low energy processes do not
necessarily apply, since (as shown below)
the reach in leptoquark mass can exceed even
10~TeV, for which the FCNC effects should be very much suppressed.
\begin{center}
\epsfxsize=4.0in
\hspace*{0in}
\epsffile{lcos.eps}
\vspace*{-1.1in}
\parbox{5.5in}{\small Fig.2. The angular distribution
of $\ell^+\ell^-\to q\overline{q}$ in the Standard Model
and including the effects of a scalar leptoquark for $M_S=8$~TeV and
$\sqrt{s}=4$~TeV.}
\end{center}
The deviations from the Standard Model appear in the total cross section and
the forward-backward asymmetry, $A_{FB}$\cite{hr}.
In Fig.~2, the angular distribution
of the $q\overline{q}$ pair (the quark is taken to be $Q=2/3$)
is shown in the Standard Model and in the
presence of a scalar leptoquark. The total cross section and $A_{FB}$ amount
to integrating this distribution in one or two bins respectively. In order to
maximize the sensitivity and following Choudhury\cite{choudhury}, we bin
the cross section in 18 bins with $\Delta \cos \theta =0.1$
in the range $-0.9<\cos \theta < 0.9$ and perform
a $\chi ^2$-analysis to calculate the statistical significance of any
deviations from the Standard Model. Therefore this procedure is simply a
generalization of the measurement of the total cross section and $A_{FB}$.
The $\chi^2$ is determined in the usual way from the number of
events expected in each bin in the Standard Model, $n_j^{\rm SM}$, and the
number of events including the leptoquark, $n_j^{\rm LQ}$, expected in each
bin, as
\begin{eqnarray}
\chi ^2&=&\sum _{j=1}^{18}{{(n_j^{\rm LQ}-n_j^{\rm SM})^2}
\over {n_j^{\rm SM}}}\;.
\end{eqnarray}
The additional piece in the Lagrangian that is of relevance to us can be
parametrized in the form
\begin{eqnarray}
{\cal L}&=&gS\bar{q}(\lambda_L P_L + \lambda_R P_R)\ell \;,
\end{eqnarray}
where $g$ is the weak coupling constant (to set the overall magnitude of the
interaction) and $\lambda _{L,R}$ are dimensionless constants. $P_L$ and $P_R$ are the
left- and right-handed projectors. The size of the interference effect will be
determined by the three parameters $M_S$, $\lambda_L$ and $\lambda_R$.
Let us now concentrate on the interactions
$\ell^-(P^-) \ell^+(P^+) \rightarrow q\overline{q}$,
where the produced quark has $Q=2/3$.
$P^-$ and $P^+$ are the polarization of the colliding leptons, and can be
either left- or right-handed (we choose to define them such that they are
always positive).
The amplitudes for the diagrams presented in Fig.~1 have been presented for
the unpolarized case in Ref.~\cite{hr}, and is generalized to the case with
polarization in Ref.~\cite{choudhury}. So we do not repeat them here, and
proceed directly to the results.
\section{Electron-Positron Collider}
The possibility of a multi-TeV $e^+e^-$ collider has begun to be taken
seriously, and the
physics potential of such a machine has started to be assessed.
It is expected that substantial polarization in the electron beam can
be achieved, while the polarization of the positron beam might not be
possible.
Figure~3 shows the 95\% c.l. bounds that could be achieved on a leptoquark with
right-handed couplings ($\lambda _L=0$) at a
$\sqrt{s}=4$~TeV $e^+e^-$ collider, with nonpolarized beams and with
80\% and 100\% polarization of the electron beam. We have assumed
integrated luminosity $L_0$ and efficiency $\epsilon$ for detecting the final
state quarks so that $\epsilon L_0=70 {\rm fb}^{-1}$.
This reflects the luminosity benchmark of $L_0=100 {\rm fb}^{-1}$ and assumes
that the tagging efficiency for charm quarks might be as high as 70\% at
the machine.
Polarization from 80\% to 100\% roughly brackets
the range that might reasonably be achievable for the electron beam. The
option of polarizing the electron beam is clearly very useful, as it can lead
to an increase in the bound by as much as a factor of two. Figure~4 shows
the same bounds for the case where the leptoquark has left-handed couplings
($\lambda _R=0$). In this case the improvement is more modest but still
nonnegligible.
In general a leptoquark would have both left- and right-handed couplings.
The bounds that can be achieved are
substantially larger than the collider energy, provided the leptoquark
couplings are not too small compared to the weak coupling.
\begin{center}
\epsfxsize=4.0in
\hspace*{0in}
\epsffile{eer.eps}
\vspace*{-1.3in}
\parbox{5.5in}{\small Fig.3. The 95\% c.l. bounds on leptoquark mass and
couplings at
a $\sqrt{s}=4$~TeV $e^+e^-$ collider for a leptoquark with right-handed
couplings only ($\lambda _L=0$). The electron polarization $P$ is set to
0\%, 80\% and 100\%, and the positron is always unpolarized.
The area above each curve would be excluded.}
\end{center}
\begin{center}
\epsfxsize=4.0in
\hspace*{0in}
\epsffile{eel.eps}
\vspace*{-1.3in}
\parbox{5.5in}{\small Fig.4. The 95\% c.l. bounds on leptoquark mass and
couplings at
a $\sqrt{s}=4$~TeV $e^+e^-$ collider for a leptoquark with left-handed
couplings only ($\lambda _R=0$). The electron polarization $P$ is set to
0\%, 80\% and 100\%, and the positron is always unpolarized.
The area above each curve would be excluded.}
\end{center}
\section{Muon Collider}
There is increasing interest recently in the possible construction of a
$\mu^+\mu^-$ collider\cite{mupmumi},\cite{saus},\cite{montauk},\cite{sfproc}.
The expectation is that a muon collider
with multi-TeV energy and the high luminosity
can be achieved\cite{neuffersaus,npsaus}.
Initial surveys of the physics potential of muon colliders have been carried
out\cite{workgr},\cite{sf}.
Both $\mu^+$ and $\mu^-$ beams can be at least partially polarized, but perhaps
with some loss of
luminosity. At the Snowmass meeting
a first study of the tradeoff between polarization and luminosity at a muon
collider was
presented\cite{feas}. This analysis found that if one
tolerates a drop in luminosity of a factor two, then one can achieve
polarization of both beams at the level of $P^-=P^+=34\%$.
(One could extend the
polarization to 57\% with a reduction in the luminosity by a factor of eight.
This additional polarization does not prove useful if one must sacrifice so
much luminosity, at least for the leptoquark
searches studied here.)
It might be possible to maintain the luminosity at its full unpolarized value
if the proton source intensity (a proton beam is used to create pions that
decay into muons for the collider) could be increased\cite{feas}.
We have chosen to present results for each of these three possible scenarios
below.
\begin{center}
\epsfxsize=4.0in
\hspace*{0in}
\epsffile{mumur.eps}
\vspace*{-1.3in}
\parbox{5.5in}{\small Fig.5. The 95\% c.l. bounds on leptoquark mass and
couplings at
a $\sqrt{s}=4$~TeV $\mu^+\mu^-$ collider for a leptoquark with right-handed
couplings only ($\lambda _L=0$). The curves indicate the bounds for
nonpolarized beams, both $\mu^+$ and $\mu^-$ having
polarization $P$ is set to 34\% and no reduction in luminosity, and
both $\mu^+$ and $\mu^-$ having
polarization $P$ is set to 34\% and a reduction in luminosity of
a factor of two.
The area above each curve would be excluded.}
\end{center}
In Fig.~5 the 95\% c.l. bounds that can be obtained at a muon collider
for a leptoquark
with right-handed couplings are shown for
three cases: (1) unpolarized beams with integrated luminosity such that
$\epsilon L_0=70{\rm fb}^{-1}$; (2) both
the $\mu^+$ and $\mu^-$ beams with 34\% polarization with the same luminosity
$L_0$; and (3) both the $\mu^+$ and $\mu^-$ beams with 34\% polarization but
now including the expected reduction in luminosity
$L=L_0/2$. One sees that even with the reduction of luminosity one obtains
improved bounds with polarized $\mu $ beams.
In Fig.~6 the bounds that can be obtained at a muon collider for a leptoquark
with left-handed couplings are shown. In this case the expected luminosity
reduction associated with polarizing the muon beams does not result in an
improved bound.
\begin{center}
\epsfxsize=4.0in
\hspace*{0in}
\epsffile{mumul.eps}
\vspace*{-1.3in}
\parbox{5.5in}{\small Fig.6. The 95\% c.l. bounds on leptoquark mass and
couplings at
a $\sqrt{s}=4$~TeV $\mu^+\mu^-$ collider for a leptoquark with left-handed
couplings only ($\lambda _R=0$). The curves indicate the bounds for
nonpolarized beams, both $\mu^+$ and $\mu^-$ having
polarization $P$ is set to 34\% and no reduction in luminosity, and
both $\mu^+$ and $\mu^-$ having
polarization $P$ is set to 34\% and a reduction in luminosity of
a factor of two.
The area above each curve would be excluded.}
\end{center}
\section{Conclusions}
We have performed a first study of the indirect search for leptoquarks at
multi-TeV lepton colliders. It is known already that polarization can be
advantageous at the NLC\cite{review},\cite{choudhury},
and we have shown by how much polarization
is found to increase the lower bounds
on scalar leptoquark masses at both multi-TeV $e^+e^-$ machines and
$\mu^+\mu^-$ machines. Of particular interest is the utility of
polarization in the case of muon colliders, for which partial polarization of
both beams is possible but comes at the cost of loss in luminosity. If one
can achieve 34\% polarization in both muon beams, we find that this does
improve the reach for leptoquarks if they couple to the right-handed muon, but
does not either improve or disimprove substantially the reach for leptoquarks
that couple to the left-handed muon. One should keep in mind that the
expectations for the polarization and luminosity at a muon collider are very
preliminary, and it might be possible to achieve polarization without
significant reduction in the luminosity\cite{feas}. We find that
polarizing the electron beam at an $e^+e^-$ collider improves the reach in
scalar leptoquark mass, assuming no loss of luminosity.
Finally one can assess the utility of polarizing both beams as opposed to
polarizing just one beam. This can be done by comparing Figs.~3 and 5 for the
right-handed leptoquark case and Figs.~4 and 6 for the
left-handed leptoquark case. We summarize the bound for leptoquarks with
interactions of order the weak coupling strength in Table I, for both
left-handed couplings ($|\lambda _L|^2=0.5,|\lambda _R|^2=0$) and right-handed
couplings ($|\lambda _R|^2=0.5,|\lambda _L|^2=0$). The 95\% c.l. limits in the
two unknown parameter analysis, here translates into a 98.6\% c.l. when only
the leptoquark mass is unknown.
For both cases one sees that the 34\% polarization
of both beams gives roughly the same bounds as a collider with one beam
polarized at the 80-90\% level.
\begin{table}[h]
\begin{center}
\caption{Bounds on leptoquark masses at 98.6\%
confidence level, assuming either
left-handed couplings ($|\lambda _L|^2=0.5,|\lambda _R|^2=0$) or
right-handed couplings ($|\lambda _L|^2=0,|\lambda _R|^2=0.5$).}
\label{tab:sample}
\begin{tabular}{l|c|c}
\hline
\hline
Luminosity and & & \\
Polarization($\ell^-,\ell^+$) & Coupling & $M_S$-Bound (TeV)\\
\hline
$L_0$ (0\%,0\%) & Left & 14.3 \\
& Right & 10.8 \\
\hline
$L_0$ (80\%,0\%) & Left & 16.8 \\
& Right & 15.1 \\
\hline
$L_0$ (100\%,0\%) & Left & 17.7 \\
& Right & 16.7 \\
\hline
$L_0$ (34\%,34\%) & Left & 17.1 \\
& Right & 14.9 \\
\hline
$L_0/2$ (34\%,34\%) & Left & 14.4 \\
& Right & 12.5 \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
It should be emphasized that there are many uses for polarization at these
machines, and the leptoquark search is just one entry on a long list of
processes that should be studied to ascertain the full usefulness of including
of polarization. Even without polarization we find the reach of a 4~TeV
lepton collider is quite high: we find that leptoquarks
with couplings of roughly
electroweak strength can be ruled out well above 10~TeV, and discovered even
if they
have masses well above the collider energy.
Whether nature provides us with leptoquarks of about 10~TeV
is, however, another matter indeed.
\section*{Acknowledgement}
I would like to thank J.L.~Hewett for suggesting this topic.
This work was supported in part by the U.S. Department of
Energy under Grant No. DE-FG02-95ER40661.
\newpage
\begin{center}
{\large\bf REFERENCES}
\end{center}
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"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
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\section{{\it Effective Chiral Lagrangian}}
\indent
The effective chiral-quark Lagrangian\cite{mg} implemented with the QCD
conformal anomaly is\cite{Beane}
\begin{eqnarray}
L &=& \bar\psi i(\not\! \partial + \not\! V ) \psi + g_{A}
\bar \psi\not\! A \gamma_{5}\psi
- \frac{m}{f_d} \bar\psi \psi\chi +\frac{1}{4} \frac{f_\pi^2}{f_d^2}
tr(\partial_{\mu}U \partial^{\mu} U^{\dag})\chi^2\nonumber \\
&&+ \frac{1}{2} \partial_{\mu}\chi\partial^{\mu}\chi
- V(\chi)+... \label{sicq}
\end{eqnarray}
where
$V_\mu = \frac{1}{2}(\xi^\dagger
\partial_\mu\xi + \xi\partial_\mu\xi^\dagger)$,
$A_\mu = \frac{1}{2}i(\xi^\dagger
\partial_\mu\xi - \xi\partial_\mu\xi^\dagger)$ with $\xi^2 =U=
\exp(\frac{i2\pi_iT_i}{f_\pi})$ and $g_A=0.752$ is the
axial-vector coupling constant for the constituent quark
\footnote{Since the role of gluons in the chiral quark
model is negligible, we will ignore the terms containing gluons\cite{mg}
\cite{keaton}.}.
The potential term for the dilaton fields,
\footnote{The scalar
field that figures in the dilaton limit must be the quarkonium component
that enters in the trace anomaly, not the gluonium component that remains
``stiff" against the chiral phase transition. See \cite{Brown} for
a discussion on this point.}
\begin{eqnarray}
V(\chi) = -\frac{m_{\chi}^2}{8f_d^2 }[\frac{1}{2} \chi^4 -
\chi^4 \ln(\frac{\chi^2}{f_d^2})],
\end{eqnarray}
reproduces the trace anomaly of QCD at the effective
Lagrangian level.
We assume that this potential chooses the ``vacuum" of the broken chiral and
scale symmetries, $<0|\chi|0>=f_d$ and
the dilaton mass is determined by
$m_\chi^2 = \frac{\partial^2V(\chi)}{\partial\chi^2}|_{\chi= f_d}$.
After shifting the field, $\chi\rightarrow f_d +\chi^\prime$, eq.(\ref{sicq}) becomes
\begin{eqnarray}
L &=& \bar\psi i(\not\! \partial + \not\! V ) \psi + g_{A}
\bar \psi\not\! A \gamma_{5}\psi -m \bar\psi \psi
- \frac{m}{f_d} \bar\psi \psi\chi^\prime\nonumber \\
&+&\frac{1}{4}f_\pi^2
tr(\partial_{\mu}U \partial^{\mu} U^{\dag})
+\frac{1}{2} \frac{f_\pi^2}{f_d}
tr(\partial_{\mu}U \partial^{\mu} U^{\dag})\chi^\prime
+\frac{1}{4} \frac{f_\pi^2}{f_d^2}
tr(\partial_{\mu}U \partial^{\mu} U^{\dag})\chi^{\prime 2}\nonumber \\
&+& \frac{1}{2} \partial_{\mu}\chi^\prime\partial^{\mu}\chi^\prime
-V(f_d + \chi^\prime)+ ...~ .\label{asf}
\end{eqnarray}
Expanding $U$ in terms of pion fields and collecting the
terms relevant for one-loop corrections, the Lagrangian for the fluctuating
field can be written as
\begin{eqnarray}
L &=& \bar\psi i\not \partial \psi
-m \bar\psi \psi
- \frac{m}{f_d} \bar\psi \psi\chi^\prime
-\frac{g_A}{f_\pi}\bar \psi\not\! \partial\pi\gamma_{5}\psi
-\frac{1}{2f_\pi^2}\epsilon_{abc}T_c
\bar \psi\not\! \partial\pi^a\pi^b\psi \nonumber \\
&+&\frac{1}{2}\partial_{\mu}\pi\partial^{\mu}\pi
+\frac{f_\pi^2}{f_d}\partial_\mu\pi\partial^\mu\pi\chi^\prime
+\frac{1}{2f_d^2}\partial_{\mu}\pi\partial^{\mu}\pi \chi^{\prime 2}\nonumber \\
&+& \frac{1}{2} \partial_{\mu}\chi^\prime\partial^{\mu}\chi^\prime
-\frac{1}{2}m_\chi^2\chi^{\prime 2} -\frac{5}{6}\frac{m_\chi^2}{f_d^2}
\chi^{\prime 3} +...~.
\label{asb}
\end{eqnarray}
We analyze the Lagrangian using the large $N_c$
approximation at one-loop order.
From 't Hooft and Witten\cite{coleman}, we know the $N_c$ dependence of the
properties of mesons and baryons.
Some of them, which are relevant to us, are
summarized as follows:
Three meson vertiex is of order $\frac{1}{\sqrt{N_c}}$.
Baryon-meson-baryon vertex is of order $\sqrt{N_c}$.
But, since the constituent quarks in eq.(\ref{sicq}) have color indices,
there should be some modification to the meson-fermion vertex.
The mesons are created and annihilated by quark bilinears,
\begin{equation}
B=N_c^{-\frac{1}{2}}\bar\psi^a\psi^a\label{mes}
\end{equation}
If we consider matrix element of eq.(\ref{mes}) between two color-singlet
baryons, the order is
$N_c/N_c^{\frac{1}{2}}$ since $N_c$ quarks
are involved to be annihilated by $B$.
However in our case the ground-states are the
constituent quarks with definite color and
only one quark which has same color with ground-state quarks
can contribute the matrix element; thus the
quark-meson-quark vetrex is of order
$1/N_c^{\frac{1}{2}}$.
So quark-meson-quark vertex is of order
$\frac{1}{\sqrt{N_c}}$ while there are no changes in $N_c$ counting for
mesonic vertices.
Now we can determine $N_c$ dependences of one-loop diagrams.
The tadpole contribution to consitutent quark mass, Fig.2, is of order
$N_c(\frac{1}{\sqrt{N_c}})^2 = 1$ and that from Fig.3 is of order
$(\frac{1}{\sqrt{N_c}})^2 =\frac{1}{N_c}$.
The two diagrams, Fig. 4 and Fig. 5,
for dilaton mass corrections are of the same
$N_c$ order.\\
\section{{\it Perturbative calculations}}
\indent
We compute the diagrams representing mass corrections
at one-loop approximation.
At very low temperature, the Bose-Einstein distribution function
$n_B(k) =\frac{1}{e^{\beta k} -1}$ goes to zero but the Fermi-Dirac
distribution
function $n_F(p) =\frac{1}{e^{\beta (p-\mu)} +1}$ becomes $\theta (\mu-p)$.
Thus in our case, the finite-temperature and -density corrections come
from quark propagators.
In our calculations of Feynman integrals,
we shall follow the method of Niemi and Semenoff \cite{ns}.
In the case of renormalization of finite-temperature QED \cite{do}
one encounters a new
singularity like $ \frac{1}{k}\frac{1}{e^{\beta k} -1} $.
However in our case, because pion-quark vertices depend on momentum,
we do not have any additional infrared singularities at finite temperature
and density.
At finite temperature and density, the lack of explicit Lorentz
invariance causes some ambiguities in defining renormalized masses.
The standard practice is to define density-dependent (or $T$-dependent)
mass corrections as the energy of the particle at $\vec p =0$ \cite {bla}.
In our
case, this definition may not be the most suitable one because of the
Fermi blocking from the fermions inside the Fermi sphere. We find
it simplest and most convenient to define the mass at $\vec p=0$.\\
\leftline{{\bf \underbar{Pion mass}}}
\indent
We first consider pion mass in baryonic matter at low temperature.
There are three density-dependent diagrams, Fig. 1, that may contribute to
pion mass corrections.
The first one, Fig. 1a, vanishes identically due to isospin symmetry.
Explicit calculation of the two diagrams in Fig. 1b gives,
for $\beta\rightarrow\infty $,
\begin{eqnarray}
\Sigma_\pi(p^2) &=&i(\frac{g_A}{f_\pi})^2 tr\int\frac{d^4k}{(2\pi)^4}
\not \! p\gamma_5T^a(\not\! p+\not\! k +m)
\cdot\not \! p\gamma_5 T^a (\not\! k +m)\nonumber \\
& &[\frac{i}{(p+k)^2-m^{2}}(-2\pi)\delta (k^2-m^2)
\sin^2\theta_{k_0}\nonumber \\
& &+\frac{i}{k^2-m^{2}}(-2\pi)\delta ((k+p)^2-m^{2})
\sin^2\theta_{k_0+p_0}].\label{pm0}
\end{eqnarray}
With the change of variable on the second term,
$p+k\rightarrow k$, the above integral can be rewritten as
\begin{eqnarray}
\Sigma_\pi(p^2) &=&-2(\frac{g_A}{f_\pi})^2 \int\frac{d^4k}{(2\pi)^3}
[\frac{-p\cdot k(p^2+2k\cdot p )+2m^{2}p^2}{p^2+2k\cdot p} \nonumber \\
& &~~~~~~~~~~ + \frac{p\cdot k(p^2-2k\cdot p)+2m^{2}p^2}{p^2-2k\cdot p}]
\delta (k^2-m^{2})\sin^2\theta_{k_0}\label{pm1}\nonumber \\
&=&-2p^2(\frac{g_A}{f_\pi})^2 \int\frac{d^4k}{(2\pi)^3}[\frac{2m^2}{p^2+2k\cdot p}
+\frac{2m^2}{p^2-2k\cdot p}]\nonumber \\
&&~~~~~~\times\delta (k^2-m^{2})\sin^2\theta_{k_0}.\label{pm2}
\end{eqnarray}
Since the pion self-energy is proportional to $p^2$, the pole
of the pion propagator does not change. Hence the pion remains
massless as long as there is no explicit chiral symmetry breaking.
Naively we might expect that the pion may acquire a dynamical
mass due to dynamical screening from thermal particles or
particle density, which is the case for QED \cite{sa} even with
gauge invariance. The reason for this is the derivative pion coupling
to the quark field.
In eq.(\ref{pm2}), the terms with $p\cdot k$ may give rise to terms
which are not proportional to $p^2$, which lead to the pion
mass correction after $dk$ integration. However, those terms are
proportional to $p^2 + 2k\cdot p$ or $p^2 - 2k\cdot p$ which are
cancelled by the their denominators and contribute nothing after
the $dk$ integration. This feature has been explicitly
demonstrated in eq.(\ref{pm2}).
Since the pole of the pion propagator does not change, we can take
the renormalization point at $p^2=0$. Then the integrand itself
vanishes, so there is no wave-function
renormalization for the pion in dense medium.\\
\leftline{{\bf \underbar{Quark mass}}}
\indent
At one-loop order, the quark self-energy is given by the diagrams
of Fig. 2 and Fig. 3.
The diagram Fig. 2 gives
\begin{equation}
\Sigma^{(1)}_Q(p)=i(-i\frac{m}{f_d})^2\frac{i}{-m_\chi^2}\rho_s
\end{equation}
where $\rho_s$ is the scalar density obtained from the fermionic
loop with thermal propagator,
\begin{eqnarray}
\rho_s&=&-tr\int\frac{d^4p}{(2\pi)^4}(-2\pi)(\not\! p+m)\delta (p^2-m^2)
sin^2\theta_{p_0}\nonumber \\
&=&\frac{m}{\pi^2}\theta(\mu-m)[\mu\sqrt{\mu^2 - m^{2}}
- m^{2} ln ( \frac{\mu + \sqrt{\mu^2 - m^{2}} }{m})]
\end{eqnarray}
where $\vec p_F^2=\mu^2-m^2$.
With $I$ defined as
$\rho_s = 4mI$, we get
\begin{equation}
\Sigma^{(1)}_Q(p) = -(\frac{m}{f_d})^2\frac{4m}{m_\chi^2}I\label{qm1}
\end{equation}
The radiative correction from the pion field in Fig. 3a is
\begin{eqnarray}
\Sigma^{3a}_Q(p)=\frac{3}{8} (\frac{g_A}{f_\pi})^2
[(\not\! p +m)I].\label{fse}
\end{eqnarray}
Finally, the radiative correction due to the $\chi$-field in Fig. 3b is found
to be
\begin{eqnarray}
\Sigma^{3b}_Q(p)
=-(\frac{m}{f_d})^2[\not\! J - \frac{m}{2m_{\chi}^2}I]
\end{eqnarray}
where $\not\! J$ is defined
\begin{equation}
\not \! J\equiv \int\frac{d^4k}{(2\pi)^3}\frac{\not\! k}{2m^2-2p\cdot k
-m_\chi^2}\delta (k^2-m^2)\sin^2\theta_{k_0}.\label{jde1}
\end{equation}
In sum, the self-energy of the quark with four momentum $(E, \vec p)$ can be
written in the form
\begin{eqnarray}
\Sigma_Q(E, \vec p)=aE\gamma_0+b\vec\gamma\cdot\vec p +
c-(\frac{m}{f_d})^2\frac{4m}{m_\chi^2}I\label{gs}
\end{eqnarray}
where $a$, $b$, and $c$ are
\begin{eqnarray}
a&=& \frac{3}{8}(\frac{g_A}{f_\pi})^2I - \frac{1}{E}(\frac{m}{f_d})^2J^0\nonumber \\
b&=&-\frac{3}{8}(\frac{g_A}{f_\pi})^2I + (\frac{m}{f_d})^2
\frac{1}{\vec p^2}\vec J\cdot\vec p\nonumber \\
c&=&\frac{3}{8}(\frac{g_A}{f_\pi})^2mI + (\frac{m}{f_d})^2\frac{m}{2m_\chi^2}I
\label{abc}
\end{eqnarray}
where we have used the relation \cite{do},
$\vec J\cdot \vec \gamma =\frac{\vec J\cdot\vec p
~\vec p\cdot\vec\gamma}{\vec p^2}$.
The detailed calculations
of $J^0$ and $\vec J\cdot\vec p$ are given in appendix II.
From eq.(\ref{abc}), we can see that the scalar field radiative
corrections violate the
Lorentz symmetry, that is, $ a \neq -b$, unless $\frac{1}{E}J^0 =
\frac{1}{\vec p^2} \vec J\cdot \vec p$.
On the other hand, the pion radiative corrections in $a$ and $b$ preserve
Lorentz covariance.
Since we have only $O(3)$ symmetry in the medium for the quark
propagation, we will adopt the conventional definition of mass
as a zero of the inverse propagator with zero momentum.
The inverse propagator of the quark is given by
\begin{eqnarray}
G^{-1}(E,p) &=& \not\! p -m -\Sigma(E, \vec{p})) \\
&=& E(1-a)\gamma_0-(1+b)\vec\gamma\cdot\vec p - c-m
\end{eqnarray}
The mass is now defined as the energy which satisfies $det(G^{-1})
=0$ with $\vec p^2=0$,
\begin{equation}
(1-a)E =c+m\label{dr}.
\end{equation}
Since $a$ and $c$ are perturbative corrections, eq.(\ref{dr}) can be
approximated to the leading order as
\begin{equation}
E =m-(\frac{m}{f_d})^2\frac{4m}{m_\chi^2}I+c+am \label{dm}.
\end{equation}
If we define the mass as the energy of the particle at finite
density, the quark mass becomes
\begin{eqnarray}
m^*&=&m -(\frac{m}{f_d})^2\frac{4m}{m_\chi^2}I\nonumber \\
&&+\frac{3}{4}(\frac{g_A}{f_\pi})^2mI \nonumber \\
&&- (\frac{m}{f_d})^2J^0 + (\frac{m}{f_d})^2\frac{m}{2m_\chi^2}I.\label{qms}
\end{eqnarray}
The first line in eq.(\ref{qms}) is just the result of the mean-field
approximation, in
which only the tadpole diagram, Fig. 2, is taken into account, as
further elaborated on in the section 4. While the
dropping of the quark mass with density is obvious in the mean-field
approximation, it is not clear whether it is still true when the
radiative corrections, Fig.3, are included as in the second and
the third lines in eq.(\ref{qms}).
Thus the result obtained at the one-loop order does not indicate
in an unambiguous way that the quark mass is scaling in medium
according to BR scaling \cite{Rho}. The specific behavior depends on
the strength of the coupling constants involved in the theory, $g_A, m, f_\pi$
and $f_d$. This does not seem to be the correct physics for
BR scaling as evidenced in Nature.
However, as discussed in section 2,
the tadpole contribution to consitutent quark mass, Fig.2, is of order
$1$ and that from Fig.3 is of order
$\frac{1}{N_c}$.
So we can neglect the Fig.3 for large $N_c$ and obtain in-medium quark mass,
\begin{equation}
m^*=m -(\frac{m}{f_d})^2\frac{4m}{m_\chi^2}I.
\end{equation}
In the large $N_c$ limit, therefore, the quark propagator
retains Lorentz covariance and the mass does decrease according to
the mean-field approximation.\\
\leftline{{\bf \underbar{Scalar mass}}}
\indent
Now consider the mass shift of the dilaton field (see Fig. 4 and
Fig. 5).
The tadpole diagram, Fig. 4, gives
\begin{eqnarray}
\Sigma^{(2)}_\chi(p)&=&-\frac{5}{6}\frac{m_\chi^2}{f_d}
(-i\frac{m}{f_d})\rho_s(\frac{i}{-m_\chi^2})3 !\nonumber \\
&=&-20(\frac{m}{f_d})^2I \label{dlm}
\end{eqnarray}
where the factor $3!$ comes from the topology of the diagram. This corresponds
to the mean-field approximation at one-loop order.
The analytic expression for the contribution from Fig. 5 can be
obtained in the large scalar mass approximation, $m^2<<m_\chi^2$,
\begin{eqnarray}
\Sigma^{(5)}_\chi(p)&=&\frac{m^2}{m_\chi^2}
\frac{8}{\pi^2}(\frac{m}{f_d})^2[\frac{1}{2}\theta (\mu - m)
(\mu\sqrt{\mu^2 - m^2}
- m^2 \ln ( \frac{\mu + \sqrt{\mu^2 - m^2} }{m})) \nonumber \\
& &~~~~~~~~~~-\frac{E^2}{m_\chi^2}{\bf (}\frac{\mu(\mu^2-m^2)^{3/2}}{4m^2}
+\frac{\mu\sqrt{\mu^2-m^2}}{8}\nonumber \\
& &~~~~~~~~~~-\frac{m^2}{8}
\ln (\frac{\mu+\sqrt{\mu^2-m^2}}{m})~{\bf )}~].\label{scm}
\end{eqnarray}
Note that the terms on the second and the third lines contribute only to
the {\it energy} of the dilaton field and hence break Lorentz invariance.
This contribution, (\ref{scm}), however can be neglected since it is
suppressed compaired to that of tadpole, eq.(\ref{dlm}), by
$\frac{m^2}{m_\chi^2}$.
Hence the Lorentz invariance is maintained approximately in the
large scalar mass approximation.
Now the renormalized dilaton mass becomes
\begin{equation}
m_{\chi}^{*2}=m_\chi^2- 20(\frac{m}{f_d})^2 I.
\end{equation}
The mass of the dilaton drops as density increases. This is consistent
with the tendency for the scalar to become dilatonic at large density,
providing an answer to the question posed at the beginning.
\section{{\it Discussion}}
\indent
So far, we have not taken into account the fact that
introducing the thermal fluctuation can cause further
shifting of the vacuum expectation value of
the dilaton field.
The vacuum expectation value of $\chi^\prime$-field in eq.(\ref{asf})
is zero, $<0|\chi^\prime|0>$=0, at zero density.
The introduction of the new ground state, $|F>$, which
are already occupied by constituent
quarks with $E$$<$$ \it{E}_f$(fermi energy),
implies the necessity of shifting the vacuum.
Imposing the condition that all tadpole graphs
vanish on the physical vacuum,
in terms of Weinberg's notation\cite{web}, the condition is
\begin{equation}
(\frac{\partial P(\chi^\prime)}{\partial \chi^\prime})_{\chi^\prime
=<F|\chi^\prime|F>}
+\tilde T=0\label{tp}
\end{equation}
where $P(\chi^\prime )$ is a polynomial in $\chi^\prime$ and $\tilde T$ is the sum of
all tadpole graphs.
In our case, we are interested in the small change of vacuum
expectation value,
$\frac{<F|\chi^\prime|F>}{f_d} <1 $.
So we can drop higher powers of the $\chi^\prime$-field and retain
only the tadpoles that depend on density at one-loop order.
Then the eq.(\ref{tp}) reads
\begin{equation}
-m_\chi^2\chi_0 -\frac{m}{f_d}\rho_s=0\label{nt}
\end{equation}
where $\chi_0\equiv <F|\chi^\prime|F>$.
Then, we get the vacuum expectation value of the $\chi^\prime$-field.
\begin{equation}
\chi_0 =-\frac{m}{m_\chi^2 f_d} \rho_s
\end{equation}
This is equivalent to Fig. 1 without the external quark line.
With the shift of the $\chi^\prime$-field around $\chi_0$, i.e.
$\chi^\prime\rightarrow \chi_0+\tilde\chi$
in eq.(\ref{asf}) or
$\chi\rightarrow f_d + \chi_0+\tilde\chi$ in eq.(\ref{sicq}),
the Lagrangian is modified
effectively to
\begin{eqnarray}
L &=& \bar\psi i(\not\! \partial + \not\! V ) \psi + g_{A}
\bar \psi\not\! A \gamma_{5}\psi -\frac{m}{f_d}(f_d+ \chi_0) \bar\psi \psi
- \frac{m}{f_d} \bar\psi \psi\tilde\chi\nonumber \\
&&+\frac{1}{4} \frac{f_\pi^2}{f_d^2}(f_d + \chi_0)^2
tr(\partial_{\mu}U \partial^{\mu} U^{\dag})\nonumber \\
&&+ \frac{1}{2} \partial_{\mu}\tilde\chi\partial^{\mu}\tilde\chi
-V(f_d+\chi_0+\tilde\chi)+\cdots\label{lag}
\end{eqnarray}
where the ellipsis stands for the interaction terms including
pions and $\chi$-fields.
The Lagrangian, eq.(\ref{lag}), shows explicitly
the density dependence of $f_\pi^2$, $m_\chi^2$ and $m$:
\begin{eqnarray}
f_\pi^{*2}&=&\frac{f_\pi^2}{f_d^2}(f_d+\chi_0)^2
=f_\pi^2(1+\frac{\chi_0}{f_d})^2\nonumber \\
m_\chi^{*2} &=&\frac{\partial^2V(\chi)}
{\partial\chi^2 }\mid_{\chi =f_d+\chi_0}
=\frac{m_\chi^2}{f_d^2}(f_d+\chi_0)^2(1+3\ln \frac{f_d+\chi_0}{f_d})\nonumber \\
&\simeq&m_\chi^2(1+\frac{\chi_0}{f_d})^2(1+3\frac{\chi_0}{f_d})
\simeq m_\chi^2(1+5\frac{\chi_0}{f_d})\nonumber \\
m^*&=&\frac{m}{f_d}(f_d+\chi_0) =m(1+\frac{\chi_0}{f_d}).
\end{eqnarray}
This is the result obtained in the previous section.
Hence we can see that
the mean field approximation in the chiral quark model coupled to dilaton is
a good approximation in the large $N_c$ limit with
massive dilaton, $(\frac{m}{m_\chi})^2<<1$, and gives
\begin{equation}
\frac{m^{*}}{m}=\frac{f_\pi^*}{f_\pi}\cong\frac{m_\chi^*}{m_\chi}\label{sl}
\end{equation}
as predicted by BR scaling.
This is somewhat different from the observation of Nambu-Freund
model \cite{nf}
which consists of a matter field $\psi$ and a dilaton field $\phi$.
After spontaneous symmetry breaking the matter field and dilaton field
acquire masses.
There are universal dependences on the vacuum expectation values both for
the matter field and the dilaton field.
One can easily see that the universal dependence on the vacuum expectation
value is no longer valid
in our calculations due to the logarithmic potential from QCD trace anomaly.
In summary, we have found that the tadpole type corrections
lead to the decreasing
masses with increasing baryon density, while the radiative corrections induce
Lorentz-symmetry-breaking terms.
The pion remains massless at finite density in the chiral limit.
In the context of large $N_c$ approximation with large scalar mass,
tadpoles dominate and the mean-field approximation is reliable, giving
rise to a Lorentz-invariant Lagrangian with masses decreasing as the baryon
density increases according to eq.(\ref{sl}).
This analysis with large $N_c$ approximation gives a clue to construct
the Lorentz invariant lagrangian, {\it i.e}, dilated
chiral quark model which incoporate the mended symmetry
of Weinberg into chiral quark model, at finite density.
The dilated
chiral quark model can emerge in dense and hot medium -- if it does
at all -- only through
nonperturbative processes (e.g., large $N_c$ expansion) starting from
a chiral quark Lagrangian.\\
{\bf Acknowledgments}\\
We thank Mannque Rho for the useful discussions.
This work was supported in part by the Korea Ministry of Education
(BSRI-96-2441) and
in part by the Korea Science and Engineering Foundation under Grants No.
94-0702-04-01-3
|
proofpile-arXiv_065-695
|
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|
\section{Introduction}
Copious neutrino emission from collapse-driven supernovae
attracts significant attention because it provides rich
information not only on the mechanism of supernovae but also
on the neutrino physics through a number of events captured
in some underground neutrino detectors, such as the Super-Kamiokande
(SK)\cite{SK}. The most noteworthy subject on the nature of neutrinos
is the mass of neutrinos
and oscillations between different flavors induced by the mass difference.
However, the ordinary matter oscillation (well known as the MSW
effect\cite{MSW}) has its effect only on neutrinos but not on
antineutrinos under the direct mass hierarchy, and the vacuum
oscillation is not observable unless the mixing angle is unnaturally
large compared with that of the quark sector.
In this case, electron antineutrinos
($\bar\nu_e$'s), which is the most detectable in a water \v{C}erenkov
detector, do not undergo any oscillation.
One of some possibilities that $\bar\nu_e$'s would oscillate or
be converted into other species of neutrinos is the neutrino magnetic
moment. If the neutrinos have a nonvanishing magnetic moment, it
couples the left- and right-handed neutrinos, and interaction
with sufficiently strong magnetic fields induces the
precession between neutrinos with different chiralities
in the inner region of the collapse-driven supernova
\cite{Cisneros,Fujikawa-Shrock}.
In general, non-diagonal elements of the magnetic moment matrix
are possible, and neutrinos can be changed into different flavors
by this flavor changing moment\cite{Schechter-Valle}.
Furthermore, with the additional
effect of the coherent forward scattering by
dense matter in the collapsing star,
neutrinos can be resonantly converted\footnote{The
precession itself is suppressed by the matter potential.} into
neutrinos with different chiralities
\cite{Lim-Marciano,Akhmedov,Voloshin,Akhmedov-Berezhiani,Peltoniemi,%
Athar-etal} by the mechanism similar
to the MSW effect. This resonant
spin-flavor conversion induced by the neutrino magnetic moment
may drastically deform
the spectrum of electron antineutrinos ($\bar\nu_e$'s)
in the water \v{C}erenkov detectors.
The earlier publications have shown that in the future experiment
this effect will be
observable with inner magnetic fields of some reasonable strength,
if there is a magnetic
moment a little smaller than the current astrophysical upper
limits from the argument of the stellar cooling due to
the plasmon decay\footnote{This constraint refers to the norm of the
neutrino magnetic moment matrix,
$(\sum_{i,j}|\mu_{ij}|^2)^{1/2}$, i.e.,
this includes the flavor changing moment.}:
$\mu_\nu \alt 10^{-11}$--$10^{-10}
\mu_B$, where $\mu_B$ is the Bohr magneton \cite{Fukugita-Yazaki,%
Raffelt}. The magnetic moment
of neutrinos in the standard electro-weak theory with small neutrino
masses is very small due to
the chirality suppression; for example,
the standard SU(2)$_L \times$U(1) model with a singlet
right-handed neutrino gives $\mu_\nu \sim 3\times 10^{-19}
(m_\nu/1{\rm eV}) \mu_B$, far below the experimental/astrophysical
upper bounds\cite{Fujikawa-Shrock,Marciano-Sanda,Lee-Shrock,Petkov}.
However,
some particle-physics models\cite{Fukugita-Yanagida,Babu-Mathur}
have been proposed
in order to give a large magnetic moment of $\sim 10^{-11} \mu_B$
which would explain\cite{VVO,Akhmedov-b,Inoue-Dron}
the anticorrelation
between the time variability of the solar neutrino flux and the sun spot
numbers suggested in the $^{37}$Cl experiment\cite{Davis}.
(The anticorrelation in the Cl experiment, however,
has not yet been statistically settled.)
Therefore, the influence
of a large magnetic moment on various physical or astrophysical
phenomena including collapse-driven supernovae deserves more detailed
investigation. Discovery of a large magnetic moment
of neutrinos indicates that there exist interactions which
violate the chirality conservation beyond the standard theory.
In this paper, the resonant spin-flavor conversion between
right-handed
$\bar\nu_e$'s and left-handed mu or tau neutrinos ($\nu_\mu$
or $\nu_\tau$'s)
is studied assuming that the neutrino is the Majorana particle.
In general, the matter potential suppresses the interaction of
the magnetic moment and magnetic fields because of the generated
difference of the energy levels. However,
Athar et al.\cite{Athar-etal} pointed out that
the resonant conversion of this mode ($\bar\nu_e \leftrightarrow \nu_\mu$)
occurs quite effectively in the region above the iron core and below the
hydrogen envelope of collapsing stars, namely,
in the O+Si, O+Ne+Mg, O+C, and He layers (hereafter
referred to `the isotopically neutral region').
The reason is that the effective matter potential for the
$\bar\nu_e \leftrightarrow \nu_\mu$ mode
is given in the form proportional to the value of ($1-2Y_e$),
where $Y_e$ is the electron number fraction per nucleon,
and $Y_e$ is very close to 0.5 in this region (typically,
$(1-2Y_e) \sim 10^{-4}$--$10^{-3}$);
the matter effect is therefore strongly suppressed compared with the
magnetic interaction, and the adiabaticity condition
becomes considerably less stringent.
Athar et al.\cite{Athar-etal} have shown
that assuming $\mu_\nu (=\mu_{\bar\nu_e \nu_\mu})
\sim 10^{-12} \mu_B$, this resonant conversion
would occur with some reasonable assumptions about magnetic fields
in a star. We also consider $\mu_\nu$ around this value\footnote{%
There is a further stringent constraint on the transition magnetic
moment of massive neutrinos from observation of the 21-cm
(hyperfine) radiation from neutral hydrogen gas in external
galaxies: $\mu_\nu \leq
1.7 \times 10^{-15}$ for the neutrino masses above 30 eV
\cite{Raphaeli-Szalay,Nusseinov-Raphaeli}. However, we consider
only the range of $\Delta m^2 \alt 1$ [eV$^2$] and this upper bound
does not constrain our analysis.}.
In order to judge the deformation of an observed $\bar\nu_e$ spectrum
as the evidence of the existence of the neutrino magnetic
moment, it is necessary that the conversion probability is
calculated with high accuracy in a wide range of some parameters
such as the mass of neutrinos or the magnetic fields. However,
only rough estimates or demonstrations in some cases are given
in the earlier publications and the relation between the shape of
the deformed spectrum and the physical parameters has not yet been
clarified. Therefore we make the contour maps of
the conversion probability of $\bar\nu_e \leftrightarrow \nu_\mu$
for some used models of precollapse stars as a function of
the two parameters: $\Delta m^2 / E_\nu$ and $\mu_\nu B_0$,
where $\Delta m^2$ is the neutrino mass squared difference,
$E_\nu$ the neutrino energy,
and $B_0$ the magnetic field at the surface of the iron core.
The expected observational effects can be clearly understood by these maps.
Some examples of spectral deformation
are also calculated and qualitative features which will
be useful for the future observation are summarized by using
these maps.
It is apparent that the deviation of the value of $Y_e$ from 0.5
in the isotopically neutral region
is quite important, and this value is strongly dependent
on the isotopic composition. Since almost all nuclei in the
isotopically neutral region are symmetric
in the number of neutrons and protons,
this deviation is determined by rarely existent nuclei and the accurate
estimate of this deviation is quite difficult. Therefore,
the astrophysical uncertainty in $(1-2Y_e)$ should be discussed.
We use the latest 15 and 25 $M_\odot$ precollapse
models of Woosley and Weaver (hereafter WW) \cite{WW1995}
which include no less than 200 isotopes. Such a large number of isotopes
have never been used previously in the calculation of $(1-2Y_e)$.
It is also expected that this value strongly depends on the stellar
metallicity, and hence we use the WW models
with the two different metallicities: the solar and zero
metallicity, and the metallicity effect is investigated.
Also the models of 4 and 8 $M_\odot$ helium core of Nomoto \&
Hashimoto \cite{NH1988} (hereafter NH)
are used, which correspond approximately
to 15 and 25 $M_\odot$ main sequence stars, and the model dependence
of $(1-2Y_e)$ is discussed.
We consider $\Delta m^2$ smaller than about 1 [eV$^2$], therefore
the resonance occurs above the surface of the iron core.
The resonance in the iron core and its implications on the dynamics of
the supernova considered in a recent preprint\cite{Akhmedov-prep}
are not discussed here.
The global structure of magnetic field is assumed to be a dipole moment,
and the strength of the magnetic field is normalized at
the surface of the iron core with the values of $10^8$--$10^{10}$
[Gauss], which are inferred from the observation of the magnetic fields
on the surface of white dwarfs.
Throughout this paper, we consider the conversion between
two generations for simplicity. Because $\nu_\mu$ and $\nu_\tau$
can be regarded as identical particles
in the collapse-driven supernova, our results also apply to the
conversion of $\bar\nu_e \leftrightarrow \nu_\tau$.
Derivation of the equation which describes the propagation of
neutrinos and evolution of conversion probability is given in
section \ref{sec:osc-mech},
and the profile of the effective matter potential and
magnetic fields are given in section \ref{sec:astro} by using the
precollapse models of massive stars.
Qualitative features of the conversion
are also discussed in this section. Numerical results are given in section
\ref{sec:results},
and spectral deformation is also discussed. Discussion and conclusions
are given in sections \ref{sec:discussion} and \ref{sec:summary},
respectively.
\section{Formulations}
\label{sec:osc-mech}
The interaction of the magnetic moment of neutrinos
and magnetic fields is described as
\begin{equation}
<(\nu_i)_R|H_{\rm int}|(\nu_j)_L> = \mu_{ij} B_{\bot} \ ,
\end{equation}
where $\mu_{ij}$ is the magnetic moment matrix, $B_\bot$ the
magnetic field transverse to the direction of propagation,
$(\nu)_R$ and $(\nu)_L$ the right- and left-handed neutrinos,
respectively, and $i$ and
$j$ denote the flavor eigenstate of neutrinos, i.e., e, $\mu$, and
$\tau$. The magnetic moment interacts only with transverse
magnetic fields.
If neutrinos are the Dirac particles, right-handed neutrinos
and left-handed antineutrinos do not interact
with matter and therefore undetectable. The
conversion into these sterile neutrinos due to the magnetic moment
suffers strong constraints from the observation of neutrinos from
SN1987A by the Kamiokande II \cite{SN1987A-Kam} and
IMB \cite{SN1987A-IMB}, and also
from the argument on energy transportation in the collapse-driven
supernova\cite{Voloshin,Peltoniemi,Dar,Lattimer-Cooperstein,%
Barbieri-Mohapatra,Notzold}.
On the other hand, if neutrinos are the Majorana
particles, as assumed in this paper,
$\nu_R$'s are antineutrinos and interact with
matter, and the constraint becomes considerably weak.
The diagonal magnetic moment is forbidden for the Majorana
neutrinos, and therefore only the conversion between
different flavors is possible, e.g., $(\bar\nu_e)_R
\leftrightarrow (\nu_{\mu, \tau})_L$. As mentioned in introduction,
we investigate this mode because
the conversion of this mode occurs quite effectively in the
isotopically neutral region and also $\bar\nu_e$'s
are most easily detected in the water \v{C}erenkov detectors.
In dense matter of the collapsing stars, the coherent
forward scattering by matter leads to the effective potential
for neutrinos, and this potential for each type of neutrinos
is determined according to the
weak interaction theory.
The potential due to scattering with electrons is given as
(including both the charged- and neutral-current interactions)
\begin{equation}
V = \pm \sqrt{2} \ G_F \ (\pm \frac{1}{2} + 2 \sin^2 \theta_W) \ n_e \ ,
\end{equation}
where $n_e$ is the number density of electrons,
$G_F$ the Fermi coupling constant, and $\theta_W$ the
Weinberg angle. The $\pm$ sign in the parentheses refers to $\nu_e$ (+)
and $\nu_{\mu, \tau}$ ($-$), and that in front to $\nu$ (+) and
$\bar\nu$ ($-$). In the ordinary flavor oscillation
($\nu_e \leftrightarrow \nu_{\mu, \tau}$), the effective
potential is only due to the charged-current scattering by
electrons because the effect of
neutral-current interactions is the same for all flavors.
However, we have to consider the neutral-current interaction
in the conversion of $\nu$ and $\bar\nu$,
because of the opposite signs of the potential.
Therefore the neutral-current scattering by nucleons should also be
included, that is
\begin{equation}
V = \pm \sqrt{2} \ G_F \ (\frac{1}{2}-2 \sin^2 \theta_W) \ n_p
\mp \sqrt{2} \ G_F \ \frac{1}{2} \ n_n \ ,
\end{equation}
where $n_p$ is the number density of protons, $n_n$ that
of neutrons. The $\pm$ or $\mp$ signs refer to $\nu$ (upper)
and $\bar\nu$ (lower) for all three flavors of neutrinos.
We do not have to consider the form factor of nuclei
because the relevant
interaction is forward scattering and there is no
momentum transfer.
The isotopically neutral region is far beyond the neutrino sphere
and neutrinos go out freely in this region; hence
we do not have to consider the neutrino-neutrino scattering.
By using the charge neutrality,
the difference of the potentials for $\bar\nu_e$'s
and $\nu_\mu$'s (or $\nu_\tau$'s)
which we are interested in is as follows:
\begin{equation}
\Delta V \equiv V_{\bar\nu_e} - V_{\nu_\mu} =
\sqrt{2}G_F\rho/m_N (1 - 2 Y_e) \ ,
\end{equation}
where $\rho$ is the density,
$m_N$ the mass of nucleons,
and $Y_e = n_p / (n_p + n_n)$.
Now the time evolution of the mixed state of $\bar\nu_e$
and $\nu_\mu$ is described
by the following Schr\"{o}dinger equation:
\begin{equation}
i\frac{d}{dr}\left( \begin{array}{c} \bar\nu_e \\ \nu_\mu
\end{array}\right) = \left(
\begin{array}{cc} 0 & \mu_\nu B_\bot \\
\mu_\nu B_\bot & \Delta H \end{array} \right)
\left( \begin{array}{c} \bar\nu_e \\ \nu_\mu
\end{array}\right) \ ,
\label{eq:schrodinger}
\end{equation}
and $\Delta H$ is defined as:
\begin{equation}
\Delta H \equiv \frac{\Delta m^2}{2 E_\nu} \cos 2 \theta
- \Delta V \ ,
\end{equation}
where $E_\nu$ is the energy of neutrinos, $\Delta m^2 =
m^2_{2} - m^2_{1}$, $\theta$ the angle of the vacuum generation
mixing, and $r$ the radius from the center of the star.
Here we consider only $\bar\nu_e$ and $\nu_\mu$, but this equation
is actually a truncation of the original 4-component ($\nu_e$,
$\nu_\mu$, and antineutrinos) equation (see ref. \cite{Lim-Marciano}).
The neutrino masses, $m_1$ and $m_2$ are
those in the mass eigenstates ($m_2 > m_1$).
The direct mass hierarchy is assumed here and
therefore $\Delta m^2$ is positive. The other terms have
their standard meanings and the units of $ c = \hbar = 1 $ are used.
Also note that we can subtract an arbitrary constant times the unit
matrix from the Hamiltonian, which does not affect the probability
amplitudes.
In the MSW flavor oscillation, there appears
the term of generation mixing, $\Delta m^2 \sin 2 \theta
/ 4E_\nu$, in the off-diagonal elements of the Hamiltonian;
however, this term does not appear in this spin-flavor conversion
between neutrinos and antineutrinos.
In the following,
$\mu_\nu$ and $\cos 2 \theta$ are
set to be $10^{-12}\mu_B$ and 1, respectively, and
the scaling of $B$ or $\Delta m^2$
with respect to other values of $\mu_\nu$ or $\cos 2 \theta$
is obvious.
The resonant spin-flavor conversion occurs when the difference of
the diagonal elements
in the Hamiltonian vanishes, and hence the resonance condition is
given as $\Delta H = 0$.
By using this equation, the probability of conversion can be
calculated provided that $\rho(r), Y_e(r)$, and $B_\bot(r)$ are known.
\section{Astrophysical Aspects}
\label{sec:astro}
\subsection{Effective Matter Potential}
In this section, we consider the effective matter potential
in the isotopically neutral region.
The value of $(1-2Y_e)$ which we are interested in is easily
calculated as:
\begin{equation}
Y_e - \frac{1}{2} = \sum_i \left( \frac{Z_i}{A_i} - \frac{1}{2} \right) X_i
\ ,
\end{equation}
where $Z_i$, $A_i$, and
$X_i$ are the atomic number, mass number, and
the mass fraction of the $i$-th isotope, respectively, and
the subscript $i$ runs over all isotopes with $2Z \neq A$.
In order to get this value and the density profiles,
the precollapse models of massive stars of Woosley \& Weaver (WW)
\cite{WW1995} and Nomoto \& Hashimoto (NH) \cite{NH1988} are used.
We assume that the dynamical effect can be ignored within the
time scale of the neutrino emission, and hence use the above
static models. The mass and radius of the
helium core of a $15 M_\odot$ main sequence star is $\sim
4 M_\odot$ and $\sim 1 R_\odot$, respectively, and
its free-fall time scale, $(\sqrt{G \rho})^{-1}$ is $\sim
10^2$--$10^3$ [sec], which is longer than the neutrino emission
time scale (at most a few tens of seconds). It takes about
several tens of seconds for the shock wave generated at the core
bounce to reach the hydrogen envelope \cite{WW1995}, and
the inner region of the isotopically neutral region may be
disturbed by the shock wave. We will discuss about this in section
\ref{sec:discussion}.
The calculation of the WW models of 15 and 25 $M_\odot$ (hereafter
WW15 and WW25, respectively) includes 200 isotopes, up to $^{71}$Ge.
Although the network of 19 isotopes
is used for energy generation up to the end of oxygen burning,
the network of 200 isotopes is updated in each cycle and mixed using
the same diffusion coefficients.
The NH models are 4 and 8 $M_\odot$ helium cores (hereafter NH4 and NH8,
respectively) corresponding approximately to 15 and 25 $M_\odot$
main sequence stars. Their calculation includes 30 isotopes
up to the end of oxygen burning, which are also used for the energy
generation. The WW and NH models use the different reaction rates of
$^{12}{\rm C} (\alpha , \gamma) ^{16}{\rm O}$, and the treatment of convection
is also different. As for the WW models, we use
the models with two different metallicities: the solar and zero
metallicity, and the metal abundance of the NH models is that of the
Sun.
By using the data of composition as well as the density profile
of the solar metallicity
WW models (WW15S and WW25S, where `S' denotes the solar metallicity),
$|\Delta V|$ in the WW15S and WW25S models are
depicted in Figs. \ref{fig:WW15S-ham}
and \ref{fig:WW25S-ham}, respectively,
by the thick solid lines as a function of the radius from the center
of the star.
Also shown by the dashed lines is $|\Delta H|$ when
$\Delta m^2 / E_\nu$ is $10^{-4}$ and $10^{-6}$ [eV$^2$/MeV].
The dominantly existent nuclei are
also indicated for each layer in the top of these figures.
In the neutronized iron core,
$Y_e$ is smaller than 0.5, and $\Delta V$ is positive
and much larger than $\Delta m^2 / E_\nu$ unless $\Delta m^2 / E_\nu$
is larger than $10^{-1}$ [eV$^2$/MeV].
We consider the range of $\Delta m^2 / E_\nu$ below this
value, and hence the resonance does not occur in the iron core.
Above the iron core, i.e., in the isotopically
neutral region, $Y_e$ becomes quite close to 0.5 (still $Y_e < 0.5$)
and $\Delta V$
is strongly suppressed, typically by a factor of $\sim 10^{-3}$
in solar metallicity stars,
and the term $\mu_\nu B$ becomes
more effective. This suppression continues to the end of the
isotopically neutral region, namely, just below the hydrogen
envelope.
The isotopically neutral region is roughly divided into
the four layers: O+Si, O+Ne+Mg, O+C, and He layer,
from inner to outer region.
The values of $(1-2Y_e)$ and some nuclei which are relevant to
the deviation of $Y_e$ from 0.5 are tabulated in Table
\ref{table:ye} for each layer and for the six precollapse models
used in this paper.
For the solar metallicity models, $(1-2Y_e)$ is determined mainly by
the isotopes such as $^{22}$Ne, $^{25,26}$Mg, $^{27}$Al, $^{34}$S,
$^{38}$Ar, and so on.
The resonance occurs when $\Delta V$ becomes smaller than $\Delta m^2 /
2 E_\nu$, and after the resonance (above the resonance layer)
$\Delta H$ becomes constant with radius
because $\Delta V$ is negligibly small. If the strength of the magnetic
field is sufficiently strong for the satisfaction of the adiabaticity
condition at the resonance layer,
the neutrinos are resonantly converted into the other helicity state.
The magnetic fields and the adiabaticity condition are discussed in
the following subsection.
The resonance layer is in the isotopically neutral region
if $\Delta m^2 / E_\nu$ is in the range of roughly
$10^{-10}$--$10^{-1}$ [eV$^2$/MeV] (slightly dependent on the
stellar models), and the resonance layer moves inward with increasing
$\Delta m^2 / E_\nu$.
If $\Delta m^2 / E_\nu$ is smaller than $10^{-10}$ [eV$^2$/MeV],
the mass term has no effect on $\Delta H$ in this region of the
solar metallicity models,
and the resonance occurs at the boundary between the helium
layer and the hydrogen envelope due to the change of the sign of
($1-2Y_e$). In contrast with the flavor oscillation,
the matter potential changes its sign by itself and the resonance can
occur without the mass term, $\Delta m^2 / E_\nu$.
However, as explained in the next subsection,
if a dipole moment is assumed as the
global structure of the magnetic fields, it seems difficult
that $B$ is strong enough to satisfy the adiabaticity condition at
this boundary in the solar metallicity models.
In the hydrogen envelope, $Y_e$ is about 0.8 and
the suppression of $(1-2Y_e)$ does not work any more. We can see that
most of the qualitative features are the same for the two models:
WW15S and WW25S, and the dependence on the stellar masses is rather
small. Note that our result gives 1--2 orders of magnitude larger
$\Delta V$ than that in
the earlier calculation by Athar et al.\cite{Athar-etal}, in which
the older 15 $M_\odot$ Woosley \& Weaver model \cite{WW1986,%
Woosley-Langer-Weaver} is used.
It is probably because our calculation of $(1-2Y_e)$ includes
the larger network of isotopes used in the latest WW models.
In Figs. \ref{fig:NH4-ham} and \ref{fig:NH8-ham}, we show the
same with Figs. \ref{fig:WW15S-ham} and \ref{fig:WW25S-ham},
but for the Nomoto \& Hashimoto models. It can be seen that
the profiles of $\Delta V$ of the
WW and NH models are not so different, and the model dependence seems
rather small. However, the situation is drastically changed when
we consider the effect of different metallicities.
Figs. \ref{fig:WW15Z-ham} and \ref{fig:WW25Z-ham} are the same with
Figs. \ref{fig:WW15S-ham} and \ref{fig:WW25S-ham}, but for the
zero metallicity WW models: WW15Z and WW25Z. (`Z' denotes the
zero metallicity.) In the O+Si and O+Ne+Mg
layers, $(1-2Y_e)$ is smaller than that of the solar metallicity
models by about 1 order of magnitude,
and in the O+C and He layers, $(1-2Y_e)$ is
further strongly suppressed (4--6 orders of magnitudes)
because of the lack of the heavy nuclei which cause the deviation
of $Y_e$ from 0.5 (Table \ref{table:ye}).
In consequence, the metallicity effect becomes
especially important when $\Delta m^2 / E_\nu$ is smaller than $\sim
10^{-6}$ [eV$^2$/MeV].
How this effect changes the profile of
the conversion probability will be discussed in more detail in
section \ref{sec:results}.
\subsection{Magnetic Fields}
Let us consider the magnetic fields in the isotopically
neutral region.
In the earlier publication\cite{Athar-etal}, the strength of
magnetic fields was normalized at the surface of the newly born
neutron star ($r \sim$ 10 km), but it is unlikely that the
magnetic fields of a nascent neutron star have some effects on
the far outer region, such as the isotopically neutral region,
within the short time scale of the neutrino burst.
The magnetic fields should be normalized by the fields which
are static and
existent before the core collapse.
The strength of such magnetic fields above the
surface of the iron core may be inferred from that observed on the
surface of white dwarfs, because the iron core of giant stars
is similar to white dwarfs in the point that both are
sustained against the gravitational collapse by the degenerate
pressure of electrons. The observations of the magnetic fields
in white dwarfs show that the strength
spreads in a wide range of $10^7$--$10^9$ Gauss\cite{Chanmugam}.
Taking account of the possibility of the decay of magnetic fields
in white dwarfs, it is not unnatural to consider the magnetic fields
up to $10^{10}$ Gauss at the surface of the iron core.
As for the global structure of the fields, although the optimistic estimate
of $B \propto r^{-2}$ is sometimes discussed
from the argument of the flux freezing,
a magnetic dipole is natural as static and global fields;
we hence assume such fields in this paper. Therefore the off-diagonal
element of the Hamiltonian in Eq.(\ref{eq:schrodinger}),
$\mu_\nu B_\bot$ becomes $\mu_\nu B_0 (r_0/r)^3 \sin \Theta$,
where $B_0$ is the strength of the magnetic field at the equator
on the iron core surface, $r_0$ the radius of the iron core,
and $\Theta$ the angle between the pole of the magnetic dipole and
the direction of neutrino propagation.
If the magnetic field is normalized at the surface of the neutron
star, the radial dependence of a dipole ($\propto r^{-3}$) gives
too small field in the isotopically neutral region, but
the normalization at the surface of the
iron core inferred from the observations
of white dwarfs makes it possible that the magnetic field is
sufficiently strong in the isotopically neutral region
under the condition of a global dipole moment.
The lines of $\mu_\nu B$ [eV] are shown in Figs.
\ref{fig:WW15S-ham}--\ref{fig:WW25Z-ham} for $B_0 =
10^8$ and $10^{10}$ [Gauss], assuming $\mu_\nu = 10^{-12} \mu_B$.
The strength of magnetic fields is also discussed from the
argument of energetics. The energy density of
the maximum strength of magnetic
fields should at most be the same order of magnitudes
with that of the thermal plasma in the star.
Let us define the magnetic fields $B_{th}$,
whose energy density is the same
with that of the gas in the star:
\begin{equation}
\frac{1}{8\pi} B_{th}^2 = \frac{3}{2}\frac{\rho}
{\tilde{\mu} m_p}kT \ ,
\end{equation}
where $\rho$ is the density, $T$ temperature, $k$ the Bolzman
constant, and $\tilde{\mu}$ the mean molecular weight.
The line of $\mu_\nu B_{th}$ is depicted in Figures
\ref{fig:WW15S-ham}--\ref{fig:WW25Z-ham}, assuming $\tilde{\mu} = 1$
and $\mu_\nu = 10^{-12} \mu_B$,
and it can be seen that
the magnetic fields up to $B_0 \sim 10^{10}$ [Gauss] are
far below $B_{th}$ and therefore
natural from the view point of energetics.
If there is no matter potential, the complete precession of
$\nu_R \leftrightarrow \nu_L$ occurs; however, the precession
amplitude is suppressed by the matter potential.
The precession amplitude is given in the form\cite{Athar-etal,VVO}:
\begin{equation}
A_p = \frac{(2\mu_\nu B)^2}{(2\mu_\nu B)^2 + (\Delta H)^2} \ .
\label{eq:precession-amplitude}
\end{equation}
In the neutronized iron core, $\Delta H$ is much larger than
$\mu_\nu B$ even when $B_0 \sim 10^{10}$ [Gauss], and the precession
below the surface of the iron core can be completely neglected.
(In other words, we can start the calculation with the pure neutrino
states from the iron core surface, with $B_0 \alt 10^{10}$ [Gauss].)
Above the iron core, i.e., in the isotopically neutral region
in the solar metallicity models,
$\mu_\nu B$ is still much lower than $\Delta V$ (or $\Delta H$),
except at the resonance
layer or the boundary of the helium layer and the hydrogen envelope,
as shown in Figs. \ref{fig:WW15S-ham}--\ref{fig:NH8-ham}.
Therefore, the precession does not occur in
the solar metallicity stars.
However, if the strength of the magnetic fields is strong enough
to satisfy the adiabaticity condition at the resonance layer,
neutrinos are resonantly converted into other types of neutrinos.
The adiabaticity condition is satisfied
when the precession length at the resonance layer, $(\mu_\nu B)^{-1}$,
is shorter than the thickness of the resonance layer, i.e.,
\begin{equation}
\mu_\nu B \agt \left|\frac{d(\Delta H)}{dr}
\right|^{\frac{1}{2}} =
\left|\frac{d(\Delta V)}{dr}
\right|^{\frac{1}{2}} \ ({\rm at \ the \ resonance}),
\label{eq:adiabaticity-cond}
\end{equation}
since the thickness of the resonance layer, $\Delta r_{\rm res}$,
is given as
\begin{equation}
\Delta r_{\rm res} = \mu_\nu B \left( \left| \frac{d(\Delta H)}{dr}
\right| \right)^{-1} \ .
\end{equation}
Note that
the suppression of $(1-2Y_e)$ in the isotopically neutral region
makes the adiabaticity condition well satisfied because it reduces
the right hand side of Eq.(\ref{eq:adiabaticity-cond}) by a factor
of $(1-2Y_e)^{1/2}$.
In order to show how this condition is satisfied,
$|d(\Delta V)/dr|^{1/2}$ is shown by the thin solid lines
in Figs. \ref{fig:WW15S-ham}--\ref{fig:WW25Z-ham}.
If $\mu_\nu B$ is (roughly) larger than
$|d(\Delta V)/dr|^{1/2}$ at the resonance layer
($\Delta H = 0$), the adiabaticity
condition is satisfied and $\bar\nu_e$'s and $\nu_\mu$'s are
mutually converted.
In both the WW and NH models with the solar metallicity,
the region where this condition is
satisfied appears with $B_0 \agt 10^{10}$ [Gauss].
Because the slope of
$|d(\Delta V)/dr|^{1/2}$ is flatter than that of
$\mu_\nu B$, this condition is
satisfied better in the inner region of the star, in other words,
with large values of $\Delta m^2 / E_\nu$, in the solar metallicity
models. When $\Delta m^2 / E_\nu $ is smaller than
$\sim 10^{-10}$ [eV$^2$/MeV] and the resonance layer
lies at the boundary of the helium layer and the hydrogen envelope,
unnaturally strong
magnetic fields are necessary for satisfaction of the adiabaticity
condition. However, in the zero metallicity stars, because the value of
$(1-2Y_e)$ is very strongly suppressed in the O+C and He layers,
$\mu_\nu B$ becomes much larger than $\Delta H$ and hence
the strong precession between different chiralities occurs with
small $\Delta m^2 / E_\nu$
(Figs. \ref{fig:WW15Z-ham} and \ref{fig:WW25Z-ham}).
Since the adiabaticity may be broken at the quite large jump of
the matter potential at the boundary of the helium layer and the hydrogen
envelope, the detailed calculation is necessary for the conversion probability
when $\Delta m^2 / E_\nu \alt 10^{-8}$ [eV$^2$/MeV].
It is apparent that the conversion probability in zero metallicity stars
will be completely different from the solar metallicity stars.
Now all of the qualitative features of the conversion can be understood from
Figs. \ref{fig:WW15S-ham}--\ref{fig:WW25Z-ham}
and the results of final conversion probability obtained by
solving the evolution equation (Eq. \ref{eq:schrodinger}) numerically
are given in the following section.
\section{Results}
\label{sec:results}
\subsection{Conversion Probability Maps}
In this section the contour maps of the conversion
probability ($\bar\nu_e \leftrightarrow \nu_\mu$)
are given for all the models used in this paper
as a function of $\Delta m^2 / E_\nu$ and $B$
at the surface of the neutronized iron core ($B_0$).
Before we proceed to contour maps,
the evolution of conversion probability in the
isotopically neutral region along the trajectory of
neutrinos is shown for some cases as a demonstration.
Fig. \ref{fig:demo-1} shows the conversion probability as a
function of radius from the center of the star
using the model NH4 for some
values of $B_0$, with
$\Delta m^2 / E_\nu = 10^{-4}$[eV$^2$/MeV],
$\mu_\nu = 10^{-12}\mu_B$, and $\cos 2 \theta = 1$.
The resonance layer lies at $r \sim 5 \times 10^{-3} R_\odot$
in the O+Si layer and its location
is never changed by strength of the magnetic fields.
We can see in this figure that the conversion probability
becomes larger with increasing strength of magnetic fields, and
the complete conversion occurs with the magnetic fields strong
enough ($B_0 \agt 5 \times 10^9$ [Gauss], in this case)
to satisfy the adiabaticity condition (see also Figure
\ref{fig:NH4-ham}). Fig. \ref{fig:demo-2} is the same with Fig.
\ref{fig:demo-1},
but $\Delta m^2 / E_\nu$ is $10^{-5}$[eV$^2$/MeV] and
the resonance layer is hence in more outer region at
$r \sim 1.5 \times 10^{-2} R_\odot$ (O+Ne+Mg layer).
As shown in these figures, the necessary $B_0$ for the complete
conversion becomes larger with decreasing $\Delta m^2 / E_\nu$
in the solar metallicity models, because the adiabaticity condition
is well satisfied with larger $\Delta m^2 / E_\nu$,
as discussed in the previous section.
In both the figures, the conversion probability
jumps up a little at the radius of about 2.5 $\times 10^{-3} R_\odot$,
because this radius corresponds to the surface of the iron core and
the value of ($1-2Y_e$) drops quite suddenly here.
Now we calculate the contour maps of the conversion probability
for the solar metallicity models, in the region of
$\Delta m^2 / E_\nu = 10^{-8}$--$10^{-1}$[eV$^2$/MeV] and
$B_0 = 10^8$--$10^{10}$ [Gauss],
and the results are given in Figs.
\ref{fig:WW15S-contour}--\ref{fig:NH8-contour} (for WW15S, WW25S,
NH4, and NH8, respectively). In the region of
$\Delta m^2 / E_\nu < 10^{-8}$[eV$^2$/MeV] or $B_0 < 10^8$ [Gauss],
the conversion does not occur because of too weak magnetic fields.
Magnetic fields stronger than $10^{10}$ [Gauss] induce
the precession below the surface of
the iron core which cannot be ignored,
and $\Delta m^2 / E_\nu$ larger than
$10^{-1}$[eV$^2$/MeV] leads to the resonance below the
surface of the iron core. In this paper, we consider the parameter
region in which the conversion or precession
below the iron core surface can be neglected.
The contours are depicted with the probability intervals of 0.1, assuming
$\mu_\nu = 10^{-12} \mu_B$ and $\cos 2 \theta = 1$.
It can be seen that some observable effects on the
spectrum of the emitted $\bar\nu_e$'s are expected if $B_0$ is
stronger than $\sim$10$^9$ [Gauss] and $\Delta m^2$ is
larger than $\sim$10$^{-5}$ [eV$^2$]. Note that the typical energy
range of the neutrinos which are observed in a water \v{C}erenkov detector
is 10--70 MeV. The lower margins of the strong conversion region
($P >$ 0.9, where $P$ is the conversion probability) in
the contour maps are, in all the four models,
contours which runs from the upper left to the lower right
direction. This is due to the fact that the adiabaticity condition is
well satisfied with larger values of $\Delta m^2 / E_\nu$,
as discussed in the previous section.
We refer to this marginal region in the contour maps as ``the
continuous deformation region'', because the conversion probability
continuously decreases with increasing neutrino energy in this region
and the spectral deformation is expected to be continuous.
What is interesting about these maps
is that some band-like patterns can be
seen in the relation of the conversion
probability and the value of $\Delta m^2 /
E_\nu$. For example, the conversion probability in the region of
$\Delta m^2 / E_\nu = 5 \times 10^{-4}
$--$5 \times 10^{-3}$ [eV$^2$/MeV] in Fig. \ref{fig:NH4-contour}
is much lower than that in the other regions of the map. These patterns
come directly from the jumps in the matter potential due to
the onion-like structure of the isotopic composition
in giant stars\cite{Athar-etal}.
When $\Delta m^2 / E_\nu$ is in the above region,
the resonance in the model NH4
occurs at the surface of the iron core
and the interval of $\Delta m^2 / E_\nu$
corresponds to the jump of $\Delta V$
at the surface (see Fig. \ref{fig:NH4-ham}). Since the matter potential
changes suddenly here, very strong magnetic field is necessary
for the satisfaction of the adiabaticity condition, and consequently
the resonant conversion is significantly suppressed.
We refer to such bands
of $\Delta m^2 / E_\nu$ as ``the weak adiabaticity band'',
hereafter. At each boundaries of the onion-like structure of
massive stars,
this weak adiabaticity band appears due to the jump in the
matter potential.
It should also be noted that in the weak adiabaticity bands,
the conversion probability only weakly depends on $\Delta m^2 /
E_\nu$ because the location of the resonance layer is not changed
in a band. In Figs. \ref{fig:WW15S-contour}--\ref{fig:NH8-contour},
we can see the difference between the WW and NH models
as well as between 15 and 25 $M_\odot$ models.
Although there are some quantitative differences,
almost all qualitative features are the same
for these four models.
Next we show the contour maps of the conversion probability
for the zero metallicity models, in Figs. \ref{fig:WW15Z-contour}
and \ref{fig:WW25Z-contour} (for the models WW15Z and WW25Z,
respectively), with the region of $\Delta m^2 / E_\nu = 10^{-11}$%
--$10^{-1}$ [eV$^2$/MeV] and $B_0 = 10^8$--$10^{10}$.
When $\Delta m^2 / E_\nu$ is larger than $\sim 10^{-6}$
[eV$^2$/MeV], the profile of the contour maps is qualitatively similar
to that of the solar metallicity models. But the
adiabaticity condition can be satisfied with smaller strength
of magnetic fields and the region of complete conversion becomes
somewhat larger, because in the inner part of the isotopically neutral
region (O+Si and O+Ne+Mg layers) the value of $(1-2Y_e)$
is about 1 order of magnitude smaller than that in the solar
metallicity models. Especially, in the model WW25Z, $\Delta H$ and
$\mu_\nu B$ are comparable in this region (see Fig. \ref{fig:%
WW25Z-ham}) and the precession effect is no longer negligible,
leading to the more complicated feature of the contour map of WW25Z
than of WW15Z.
When $\Delta m^2 / E_\nu \alt
10^{-6}$ [eV$^2$/MeV], the strong precession occurs in the
outer part of the isotopically neutral region (O+C and He layers),
where $\mu_\nu B$ is much higher than $\Delta H$.
In contrast to
the solar metallicity models, the conversion still occurs with such a low
value of $\Delta m^2 / E_\nu$, even when $\Delta m^2 = 0$.
Further interesting is that with $\Delta m^2 / E_\nu$ lower than
$\sim$10$^{-6}$ [eV$^2$/MeV], the conversion probability
changes periodically with $\mu_\nu B_0$ (Fig. \ref{fig:%
WW15Z-contour}). This can be understood as follows.
The precession effect in the outer part of the isotopically neutral
region is very profound and then this precession is stopped almost suddenly
at the boundary of the helium layer and the hydrogen envelope
where $|\Delta V|$
increases by 5--10 orders of magnitude. The final phase of the precession
strongly depends on $\mu_\nu B_0$,
because the precession length is given as
\begin{equation}
L = \frac{\pi}{\sqrt{\left( \frac{\Delta m^2}{4 E_\nu}
\right)^2 + \left( \mu_\nu B \right)^2}} \ .
\label{eq:osc-length}
\end{equation}
Therefore the conversion probability
changes periodically with $\mu_\nu B_0$.
The examples of strong precession effect are shown in Fig.
\ref{fig:demo-3}, using the WW15Z model with $\Delta m^2 = 0$
and some values of $B_0$.
The conversion probability as a function of the radius is shown
in this figure.
The precession begins at $r = 0.025 R_\odot$ and ceases at $r =
0.28 R_\odot$ (see also Fig. \ref{fig:WW15Z-ham}). One can see
that the precession length becomes larger with propagation of
neutrinos, because $B$ decreases with $r$. It is also clear that
the change in $B_0$ leads to the change of the precession length,
and hence to the oscillation of the final phase of precession.
Below $\Delta m^2 / E_\nu \sim 10^{-11}$
[eV$^2$/MeV], the effect of the mass term can be completely neglected
and the conversion probability becomes constant (but never vanishes)
with neutrino mass or energy.
\subsection{Spectral Deformation}
All of the qualitative features of the spectral deformation
due to the resonant spin-flavor conversion of $\bar\nu_e \leftrightarrow
\nu_\mu$ are clearly understood by
the contour maps given in the previous section (Figs.
\ref{fig:WW15S-contour}--\ref{fig:WW25Z-contour}).
The most easily detectable flavor in a water \v{C}erenkov detector
is $\bar\nu_e$'s because of the large cross section of
the reaction $\bar\nu_e p \rightarrow n e^+$, and
they are detectable
above the positron energy of $\sim$5 MeV in the Super-Kamiokande
detector, which has the fiducial volume of 22,000 tons \cite{SK}.
If we consider the positron energy range of
10--70 MeV, which includes almost all of the events,
this range corresponds to a vertical bar in the contour maps
with fixed values of $\Delta m^2$ and $B_0$. The samples of such
a bar are shown in Figs.
\ref{fig:NH4-contour} and \ref{fig:NH8-contour} (NH models), and
the corresponding spectral deformation of the events at the SK
are shown in Fig. \ref{fig:spec-def-1}.
The distance of the supernova is set to 10 kpc and
the total energy of each type of neutrinos is assumed to be
$5 \times 10^{52}$ erg. We use 5 and 8 MeV as
the temperature of $\bar\nu_e$'s and $\nu_\mu$'s,
respectively, and the Fermi-Dirac distribution with zero chemical
potential is assumed for both $\bar\nu_e$'s and
$\nu_\mu$'s\cite{SNneu-property}.
The cross section of the dominant reaction of $\bar\nu_e
p \rightarrow n e^+$ is $9.72 \times 10^{-44}
E_e p_e$ cm$^2$\cite{SK},
where $E_e$ and $p_e$ is the energy
and momentum of recoil positrons. The appropriate detection efficiency
curve is also taken into account\cite{efficiency}. The thick
solid line in Fig. \ref{fig:spec-def-1}
is the expected differential event number
of $\bar\nu_e$'s without any oscillation or conversion.
As mentioned in the previous section, the three characteristic
regions appear in the contour maps for the solar metallicity
models: A) the complete conversion
region, B) the continuous deformation region, and C) the weak adiabaticity
band. (A, B, and C correspond to those in Figs.
\ref{fig:NH4-contour} and \ref{fig:NH8-contour}.)
When the conversion is complete, we can see the original
$\nu_\mu$ spectrum as $\bar\nu_e$'s, and the event number
is considerably enhanced because of the higher average energy
(thin solid line in Fig. \ref{fig:spec-def-1}).
When the vertical line in the contour map lies in the continuous
deformation region, conversion probability decreases with increasing
energy of neutrinos and consequently the original $\nu_\mu$'s
are dominant in the lower energy range, while the original $\bar\nu_e$'s
are dominant in the higher energy range (short-dashed line in Fig.
\ref{fig:spec-def-1},
also corresponding to the vertical line (B) in Fig.
\ref{fig:NH8-contour}).
Note that this feature is based upon the assumption
that the radial dependence of the magnetic fields is a dipole ($B
\propto r^{-3}$) and $\mu_\nu B$ drops faster than
$|d(\Delta V)/dr|^{1/2}$
with increasing radius. On the other hand, in the weak adiabaticity
band, the energy dependence of conversion probability is rather weak
(long-dashed line in Fig. \ref{fig:spec-def-1},
also corresponding to
the vertical line (C) in Fig. \ref{fig:NH4-contour}).
Because the resonance always
occurs in the same place (jumps in the matter potential),
this feature does not depend on the assumption of the radial
dependence of magnetic fields, in contrast to the case (B).
Finally, quite interesting deformation is expected if the vertical
line in the contour maps crosses the boundary of the weak adiabaticity
band (the vertical line (D) in Fig. \ref{fig:NH4-contour}). Because
the conversion probability changes almost suddenly at the boundary,
the spectrum of the event rate suffers drastical
deformation at a certain positron energy (dot-dashed line in Fig.
\ref{fig:spec-def-1}).
The used values of ($\Delta m^2$
[eV$^2$], $B_0$ [Gauss]) for
the vertical lines A, B, C, and D in Figs. \ref{fig:NH4-contour}
and \ref{fig:NH8-contour} are (5$\times 10^{-3},
7\times 10^9$), ($5\times 10^{-4}, 5\times 10^9$),
($3\times 10^{-2}, 5\times 10^9$), and ($1.5\times 10^{-2}, 2\times
10^9$), respectively.
In the zero metallicity models, the feature of the spectrum
deformation is similar to that of the solar metallictiy models
when $\Delta m^2 / E_\nu \agt 10^{-7}$ [eV$^2$/MeV].
However, if $\Delta m^2 \sim 10^{-7}$--$10^{-6}$ [eV$^2$],
the conversion probability increases with neutrino energy, because
the precession in the outer part of the isotopically neutral region
becomes effective (Fig. \ref{fig:WW15Z-contour}). With $\Delta m^2 /
E_\nu \alt 10^{-9}$ [eV$^2$/MeV], the conversion probability
becomes constant with neutrino energy, but changes
periodically with $B_0$. In the model WW25Z, the precession
in the inner part of the isotopically neutral region (O+Si and
O+Ne+Mg layers) is also effective, and the probability may change rapidly
and complicatedly with neutrino energy (Fig \ref{fig:WW25Z-contour}).
\section{Discussion}
\label{sec:discussion}
We found that the difference of the stellar metallicity
significantly affects the resonant spin-flavor
conversion of $\bar\nu_e \leftrightarrow \nu_\mu$,
and some implications from this fact are discussed in the following.
The lifetime of massive stars which end their life by the
gravitational collapses is very shorter than that of the Sun,
and the progenitors of observed supernovae are therefore younger.
Consequently, the metallicity of the Galactic
supernova is expected to be at least the solar abundance or more metal-rich.
If the metallicity is higher than that of the Sun,
the suppression of $(1-2Y_e)$ will be weaker and the $B_0$ which is required
to satisfy the adiabaticity condition becomes larger.
On the other hand, the Large and Small Magellanic Clouds are
known to be very metal-poor systems\cite{MC-metallicity}.
Therefore, the resonant conversion will occur with smaller
magnetic fields in supernovae in the Magellanic Clouds, and
also the precession effect may be observed. The another object which
has relation to the metallicity effect is the supernova relic neutrino
background (SRN)\cite{SRN}. Because the SRN is the accumulation
of neutrinos from supernovae which have ever occurred in the universe,
the SRN includes neutrinos from supernovae with quite low metallicity
in the early phase of galaxy formation. The conversion of
$\bar\nu_e \leftrightarrow \nu_\mu$ conpensates the energy
degradation due to the cosmological redshift effect and enhances
the expected event rate of the SRN.
However, because of the small expected event rate at the SK%
\cite{SRN}, it will be difficult to get some decisive information
on the spin-flavor conversion from the observation at the SK.
We did not consider the flavor conversion (the MSW effect)
in this paper although this can occur with appropriate
generation mixing and neutrino masses. With the same
$\Delta m^2 / E_\nu$, the resonance of the flavor conversion
occurs in more outer region than the resonance layer of the spin-flavor
conversion, because the matter potential for the flavor
conversion are not suppressed by $(1-2Y_e)$\cite{Athar-etal}.
Even if the flavor conversion occurs in more outer region,
the spectrum of $\bar\nu_e$ is not changed. There may be interesting
effect if we consider the conversion
in the iron core, or the mutual effect of spin-flavor and flavor
conversion among the three generations of neutrinos, as pointed out
by the earlier publication\cite{Athar-etal}.
The turbulence in the radial dependence of the magnetic fields
was ignored in this paper. In the solar metallicity models,
$\Delta H$ is much larger than $\mu_\nu B$ except at the
resonance, and the conversion probability is determined only from the
strength of the field at the resonance layer. Therefore the turbulence
does not affect the evolution of conversion probability of neutrinos.
However, the turbulence disturbs the relation
of $B_0$ and $B$ at the resonance
layer. When the neutrino energy changes, the location of the
resonance layer also changes, and hence the neutrino spectrum
can be disturbed by some strong turbulence in the magnetic fields.
The effect of dynamics in the collapse-driven supernova
was also not taken into consideration. Although it seems unlikely
that the shock wave is propagated
through the whole isotopically neutral region
in a few tens of seconds, the inner part of the isotopically
neutral region may be dynamically disturbed by the shock wave.
If $\Delta m^2 / E_\nu$
is large, the resonance occurs at the inner part of the isotopically
neutral region, and the dynamical effect may change the situation of
the resonant conversion of the neutrinos emitted in the later phase
of emission ($\agt$ 10 sec after the bounce).
We give here simple discussion on the effect of the change in density,
assuming that $Y_e$ is conserved during the shock propagation.
(The composition of matter is drastically changed
by the shock wave, but the change in $Y_e$ requires the weak
interaction.) The matter potential ($\Delta V$)
changes as $\propto \rho$. On the other hand,
if we assume the conservation of the magnetic flux in dynamical plasma,
the field strength changes as $\propto \rho^{2/3}$, and hence the precession
becomes more effective with decreasing matter density (see Eq.
(\ref{eq:precession-amplitude})).
However, the adiabaticity condition is the competition of $\mu_\nu B$ and
$|d(\Delta V)/dr|^{1/2}$, and if we assume
$|d(\Delta V)/dr|$ scales as (length)$^{-4}$
(homogeneous expansion or compression),
the scaling of $|d(\Delta V)/dr|^{1/2}$ is the same with that of magnetic
fields, $\propto \rho^{2/3}$. Therefore, the adiabaticity condition
is not strongly affected by the dynamics of the shock wave.
It should also be noted that the observed data of neutrinos from
SN1987A \cite{SN1987A-Kam,SN1987A-IMB} favor a softer neutrino
spectrum than theoretically plausible spectrum of electron
antineutrinos. If $\bar\nu_e$'s are exchanged with $\nu_\mu$-like
neutrinos ($\nu_\mu$, $\nu_\tau$, and their antiparticles) which
have higher average energy, this discrepancy becomes larger.
From this view point, an upper bound on the conversion probability
of $\bar\nu_e$'s and $\nu_\mu$-like neutrinos has been derived:
$P < 0.35$ at the 99 \% confidence level \cite{Smirnov-Spergel-%
Bahcall}. The earlier paper of the spin-flavor conversion
\cite{Athar-etal} used this constraint in order to
derive an upper bound on the neutrino magnetic moment.
However, the above constraint on $P$ and its confidence level
suffer considerable statistical uncertainty because of the small number
of events in Kamiokande and IMB. Therefore, we avoid
a decisive conclusion about the upper bound on $\mu_\nu$ from SN1987A,
although the strong conversion region ($P > 0.9$) in the contour maps
(Figs. \ref{fig:WW15S-contour}--\ref{fig:WW25Z-contour}) may be disfavored.
Also we point out here that the metallicity effect on the conversion
probability should be taken into account when one attempts to constrain
$\mu_\nu$ from the SN1987A data, because the Large Magellanic
Cloud is a low-metal system.
\section{Conclusions}
\label{sec:summary}
Neutrino spin-flavor conversion of $\bar\nu_e \leftrightarrow
\nu_\mu$ induced by the interaction of a flavor changing magnetic
moment of Majorana neutrinos and magnetic fields
above the iron core of collapsing stars was investigated
in detail. The effective matter potential of this conversion mode
($\bar\nu_e \leftrightarrow \nu_\mu$)
is proportional to $(1-2Y_e)$, and hence this value is quite
important to this resonant conversion. However, this
value is determined by isotopes which are quite rarely existent,
and in order to estimate the effect of the astrophysical uncertainties,
we used the six precollapse models, changing
the stellar masses, metallicities, and authors of the models.
The components of Hamiltonian in the propagation
equation (\ref{eq:schrodinger}) are shown for all the models
in Figs. \ref{fig:WW15S-ham}--\ref{fig:WW25Z-ham}, and
qualitative features of the conversion can be understood from
these figures. The results of the numerical calculation for all the
models are shown in Figs. \ref{fig:WW15S-contour}--\ref{fig:%
WW25Z-contour} as contour maps of conversion probability
as a function of the two parameters of
$\Delta m^2 / E_\nu$ and $B_0$, where $B_0$ is $B$ at the surface
of the iron core.
For the solar metallicity models,
observable effects are expected when $\Delta m^2 / E_\nu$
is in the range of
$10^{-5}$--$10^{-1}$ [eV$^2$/MeV] and $\mu_\nu \agt
10^{-12} (10^9{\rm G}/B_0)$ [$\mu_B$]
(Figs. \ref{fig:WW15S-contour}--\ref{fig:NH8-contour}).
The difference of the stellar masses leads to the different
thickness and location of the layers of the onion-like structure in
massive stars
and this effect appears in the contour maps,
although the effect is rather small.
The qualitative features of the contour maps
for the WW and NH models are also not so different, and the model
dependence of the conversion probability can be roughly estimated
by the comparison of these figures.
Although the dependence on the stellar models or stellar masses
is rather weak as shown in Figs.
\ref{fig:WW15S-contour}--\ref{fig:NH8-contour}, it was found that
the metal abundance of the precollapse star significantly affect the
value of $(1-2Y_e)$. The difference between the solar and
zero metallicity is prominent especially in the O+C and He layers,
and the strong precession between $\bar\nu_e \leftrightarrow \nu_\mu$
occurs with small $\Delta m^2 / E_\nu$,
because $\mu_\nu B$ is much larger than $\Delta m^2 / E_\nu$
in this region. In contrast to the solar metallicity models,
the conversion occurs even when $\Delta m^2 = 0$. The probability
changes periodically
with $B_0$ because of the precession effect. (See Figs.
\ref{fig:WW15Z-contour} and \ref{fig:WW25Z-contour}.)
Considering the above properties, the expected spectral
deformation of $\bar\nu_e$'s can be summarized as follows.
For the solar metallicity models,
there are roughly three types of the energy dependence of the
conversion probability: 1) complete conversion
in a range of neutrino energy, 2)
conversion probability decreases with increasing energy when
the energy range is in `the continuous deformation region'
in the contour maps, and 3) incomplete conversion and weak
energy dependence of conversion probability when the energy range
is in `the weak adiabaticity band' in the contour maps. Examples of
these types of spectral deformation are given in Fig. \ref{fig:spec-def-1}.
(For the explanation of `the continuous deformation region'
and `the weak adiabaticity band', see section \ref{sec:results}.)
Furthermore, there appear some interesting jumps in the spectrum if
the energy range of 10--70 MeV includes
boundaries of the complete conversion region and
the weak adiabaticity band. Irrespective of the stellar models or
stellar masses, such a boundary exists especially
at the surface of the iron core, where the matter potential suddenly
changes. For the zero metallicity models, although the feature
of the spectral deformation is
similar to that of the solar metallicity models when
$\Delta m^2 / E_\nu \agt 10^{-7}$ [eV$^2$/MeV],
energy-independent conversion is possible with quite small
$\Delta m^2 / E_\nu$ ($\alt 10^{-9}$ [eV$^2$/MeV]), which does not
occur in the solar metallicity models.
\section*{acknowledgments}
The authors would like to thank S.E. Woosley and M. Hashimoto,
for providing us the data of their precollapse models and useful comments.
They are also grateful to Y. Totsuka, for the information on
the detection efficiency of the SK detector. This work has been
supported in part by the Grant-in-Aid for COE Research (07CE2002) and
for Scientific Research Fund (05243103 and 07640386) of the Ministry
of Education, Science, and Culture in Japan.
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proofpile-arXiv_065-696
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\section{Introduction}
Instantons plays an important role in modern field theory and mathematics.
Till now the thorough studies of instantons were carried for gauge fields
and sigma models. Rigid string is another model of interest which was known to
posses instantons. The model was originally considered as a
string for gauge fields. Unfortunately, rigidity has quite complicated
structure when expressed in ordinary string variables. This prevents any
significant progress in quantization. Certain rigid string instantons were
derived and investigated in \cite{polrig,wheater}.
Despite this efforts little was known about the generality of the proposed
instanton
equations and its significance for physics of the model.
In recent paper \cite{inst} we
derived a new set of instanton equations for the 4d rigid string.
It was claimed
this set is rich enough to have representatives for all topological
sectors of the rigid string. Because the action of the model contains terms
with four derivative the relevant topological invariant is not only the genus of
the world-sheet surface but also the self-intersection number of the surface
immersed in a target 4d space-time. The instantons split not into
two families - instantons and anti-instantons but into three families.
We shall call them $J^{(P)}_1$-instantons, anti-$J^{(P)}_1$-instantons and minimal or
$J^{(P)}_2$-instantons.
Minimal instantons are just minimal
maps from the world-sheet to the target space-time.
In general, intersection of these
families is non-trivial even in $R^4$ what is also a novel feature.
Unfortunately the equations seemed to be very difficult and a
method (through the Gauss map) to solve them in full generality, failed.
In this paper we are going to study the rigid string instantons of \cite{inst}
in more
general setting. Thus we shall consider the rigid string moving in a
Riemannian 4-manifold $M$ with the metric $G^{(M)}_{\mu\nu}$. Using the twistor
method \cite{eells,ward}
we shall be able to show that in many cases one can give explicit formulas for
the instantons. Moreover the construction will reveal an interesting structure
of the equations, namely, the instantons will appear to be pseudo-holomorphic
curves in the twistor space of $M$. This unexpected result unfolds the
underlying simplicity of the equations and lies foundation of the successful
solution of the equations.
It is worth to note that the subject touches Yang-Mills
fields in two points. First of all, pseudo-holomorphic curves
were
used to build the string picture of YM$_2$ \cite{cmr}. Secondly the
dimension of the moduli space of $J^{(P)}_1$-instantons on $R^4$ and
$S^4$ is exactly the same as those of $SU(2)$ Yang-Mills instantons with the
appropriate identification of topological numbers.
Content of the paper is the following: in Sec.\ref{sec:rig} we introduce the
necessary notation and recall some results of \cite{inst}. In the next section
we show how to solve the $J^{(P)}_1$-instanton equations using the twistor method.
We also
calculate the dimension of the moduli space of the rigid string
instantons. In sec.\ref{sec:examples} we derive explicit formulas for the
cases of $M=R^4$ and $M=S^4$. In the final section we speculate on
new topological (smooth) invariants of 4-manifolds. We also discuss connection
of the rigid string instantons to string description of
Yang-Mills fields and shortly discuss the case of
3-dimensional target $M$.
\section{Rigid string instantons.}
\label{sec:rig}
In this section we introduce necessary notions and recall basic results of
\cite{inst}.
We start with some generalities concerning the problem.
We shall be interested in
maps $X:\Sigma\to M$ which are immersions i.e. $rank(dX)=2$ (
the tangent map is of maximal possible rank).
Roughly speaking it means that the image of $\Sigma$ in $M$ is smooth.
It means also that the induced metric $g_{ab}\equiv \p_a{\vec X}\p_b{\vec
X}$ is non-singular.
Any immersion
defines the Gauss map $t^{\mu\nu}:\Sigma\to G_{4,2}=S^2_+\times S^2_-$.
The appearance of product of two $S^2$ corresponds to the
fact that $t^{\mu\nu}$ can be decomposed into self-dual $t^{\mu\nu}_+$ and
anti-self-dual $t^{\mu\nu}_-$ part.
If $M$ has non-trivial
topology we can not expect the Gauss map to be defined globally. Thus we must
introduce the so-call Grassmann fiber bundle over $M$ with fibers $G_{4,2}$.
The map ${\tilde X}$ to this bundle is called the Gauss lift. Because the
fiber of this bundle splits into self-dual and anti-self-dual part we can
consider Gauss lifts to each of them independently i.e. we can define bundle
of tensors $\tp{mu}{nu}$ separately. This is a sphere bundle which shall play a crucial role in the next section.
The action of the rigid string (without the Nambu-Goto
term\footnote{The Nambu-Goto term breaks space-time scale invariance of the model
thus prevents existence of instantons.}) is
\begin{eqnarray}
\int_\Sigma\sqrt{g}g^{ab}\nabla_a t^{\mu\nu}\nabla_b
t_{\mu\nu}=
2\int_\Sigma\sqrt{g}(\D{ X^\mu})(\D{ X^\nu})G^{(M)}_{\mu\nu}-8\pi \chi.
\label{extc1}
\end{eqnarray}
where
$t^{\mu\nu}\equiv \epsilon^{ab}\p_a X^\mu \p_b X^\nu/\sqrt{g}$ are the element
of the Grassmann manifold $G_{4,2}$, $G^{(M)}_{\mu\nu}$ is the metric on $M$ and
$g_{ab}\equiv \p_aX^\mu \p_b X^\nu G^{(M)}_{\mu\nu}$ is the induced metric on a
Riemann surface of genus $h$.
Tensors $\p_a X^\mu$ are components of
$T^*\Sigma\otimes X^*TM$, where $X^*TM$ is the pull-back
bundle. The covariant derivatives are built with Levi-Civita connections on
$T^*\Sigma$ and $TM$. Explicitly $\nabla_b \p_aX^\mu
=\p_b\p_aX^\mu-\Gamma^{(\Sigma)c}_{\;\;ab}
\p_cX^\mu+\Gamma^{(M)\mu}_{\;\;\rho\sigma} \p_aX^\sigma \p_bX^\rho$.
The Euler
characteristic of the Riemann surface $\Sigma$ is given by the Gauss-Bonnet formula
$\chi=\frac{1}{4\pi}\int_\Sigma\sqrt{g}R$.
Immersions of Riemann surfaces in $R^4$ are classified by the
self-intersection number $I$ \cite{whitney}.
General arguments based on singularity theory showed that rigidity separates
topologically different string configurations.
The derivation of instanton equations was based on the knowledge of relevant
topological invariants. In our case these were the above mentioned
self-intersection number $I$ and
the Euler characteristic $\chi$. The equations were derived using
formulae for both invariants in terms of $t^{\mu\nu}$. Explicitly:
$\chi=I_+-I_-, \quad I={\mbox{\small $\frac{1}{2}$}}(I_++I_-)$,
where $I_{\pm}=\pm\frac{1}{32\pi}\int_\Sigma \epsilon^{ab}\p_a t^{\mu}_{\pm\;\nu}
\p_b t^{\nu}_{\pm\;\rho} t^{\rho}_{\pm\;\mu}$ and $t_{\pm}^{\mu\nu}\equiv
t^{\mu\nu}\pm {\tilde t}^{\mu\nu}$.
Standard reasoning yielded the following instanton equations
(denoted as $(+,\pm)$ with obvious sign convention)\footnote{The minus $-$ in
front
of the first term appeared in order to preserve notation of \cite{inst}.}:
\begin{eqnarray}
-\nabla_a \tp{\;\mu}{\nu}\pm \frac{\epsilon_a^{\;\;b}}{\sqrt{g}}
\tp{\;\rho}{\nu} \nabla_b \tp{\;\mu}{\rho}=0
\label{instpm}
\end{eqnarray}
Here the equations were adopted to the general manifold $M$.
Analogous equations hold for the anti-self-dual part of
$t^{\mu\nu}$. The $(+,+)$ equations are equivalent to
$\D X^\mu=0$ and their solutions will be called minimal
instantons or $J^{(P)}_2$-instantons. Appropriate equations for anti-self-dual
part of $t$ will give
only $J^{(P)}_1$-anti-instantons - the fourth possibility appeared to be
equivalent to the minimal instantons.
Thus instantons form 3 families.
The former two families behave as true instantons and anti-instantons in this
sense that they do not have continuation to the Minkowski space-time and their
role is interchanged under change of orientation of the space-time. Minimal
instantons have continuation to Minkowski space-time what is a novel feature
of this kind of solutions. It is also worth to note that change of orientation
of the world-sheet (change of sign of the world-sheet complex structure)
together with change of sign of $t^{\mu\nu}$
(change of sign of the space-time complex structure) do not change any of the
equations.
In $R^4$ the instanton families are not disjoint.
Intersection of $J^{(P)}_1$-instantons and minimal
instantons gives $\nabla_a \tp{\;\mu}{\nu}=0$ while intersection of
anti-$J^{(P)}_1$-instanton and minimal
instantons gives $ \nabla_a t^{-\;\,\mu}_{\,\nu}=0$.
These equations have solutions in $R^4$ \cite{wheater} and $S^4$
\cite{bryant,eells}.
For $R^4$ there is also one
nontrivial
intersection of $J^{(P)}_1$-instantons and anti-$J^{(P)}_1$-instantons at genus zero. The
solution was
found in \cite{inst} to be a sphere embedded in $R^3\subset R^4$. It
has 5-dimensional moduli space - four positions and one breathing
mode. The analogy with SU(2) Yang-Mills case is suggestive. In
sec.\ref{sec:twistor} we shall show that in fact the dimension of the
moduli space of $J^{(P)}_1$-instantons on $R^4$ and $S^4$ is exactly given by the
same formula as for
the $SU(2)$ Yang-Mills case with appropriate identification of topological numbers.
We want to stress here that these properties of the three family of instantons
were proven for $M=R^4$ and may be modified for other 4-manifolds.
The above mentioned spherical solution was found using properties of the Gauss
map of an immersion. Unfortunately we were not able to find other instantons
with this method. In the following section we shall use the twistors
\cite{eells,ward} in finding solutions to Eqs.\refeq{instpm}. The method
appears so powerful that
one can find closed formulae for all instantons for many interesting spaces $M$.
\section{Twistor construction of instantons}
\label{sec:twistor}
In this section we shall show that the rigid string instantons are
pseudo-holomorphic
curves in the twistor space of the space-time $M$. This will directly lead to
the explicit formulas on instantons for some manifolds $M$. Moreover, the
method will allow to calculate the dimension of the moduli space of
instantons. In the following we shall concentrate on the
self-dual part of $t^{\mu\nu}$ only, understanding that the behavior of the
anti-self-dual part is analogous.
Before we go to the main subject we recall some facts from complex
geometry and twistors.
We shall heavily use certain properties of almost complex
structures.\footnote{Several
different almost complex structure will appear in this paper. In order to
clarify the notation we
decided to denote by $\epsilon$ almost complex structures of 2d manifolds
and by $J$ almost complex structure of $M$ and the twistor space ${\cal P}_M$. These
will be supplemented with the
appropriate superscript of the manifold. An almost complex manifold $X$ with
given almost complex structure $J$ we shall denote as $(X,J)$.}
Thus, the space of all almost complex structures
on $R^4$ is $O(4)/U(2)=S^2\times {Z}_2$ i.e. the space of all orthonormal
frames up to unitary rotation which preserve choice of complex coordinates.
The $Z_2$ factor is responsible for change of orientation of $R^4$.
Hence the bundle of almost complex structures over $M$ (up to change of
orientation) is just the
sphere bundle ${\cal P}_M$ i.e. a bundle with $S^2$ as fibers.
In other words, any point $p\in {\cal P}_M$, with coordinates in a local
trivialization
$p=(u,x) \in S^2\times R^4$, fixes
an almost complex structure $J$ on $M$ at $x=\pi (p)\in M$. This almost complex
structure is given by the coordinate $u$ on the fiber $S^2$.
It appears that such sphere bundles have two
natural almost complex structures. The reason is that
the sphere $S^2$ has two canonical complex structures $\pm \epsilon^{(S)}$. In the
conformal metric on $S^2$ we have $(\epsilon^{(S)})^{ij}=\epsilon^{ij}$,
where $i,j=1,2$. Out of $\pm\ep^{(S)}$ we build two almost complex structures on ${\cal P}_M$.
With the help of Levi-Civita connection on $M$ we can decompose the tangent
space $T_p{\cal P}_M$ at $p\in {\cal P}_M$ into the horizontal part
$H_p$ and the vertical part $V_p$: $T_p{\cal P}_M=H_p\oplus V_p$.
The former is isomorphic to $T_{\pi (p)}M$. The isomorphism is given
by the lift defined with help of the Levi-Civita connection on $TM$. The lift
also defines the
action of the almost complex structure $J$ on $H_p$. The vertical space $V_p$
is tangent to the
fiber ($S^2$) and has the complex structure $\pm \epsilon^{(S)}$. It follows
that we can define two almost complex structure at $p\in {\cal P}_M$ given by the
formulas
\begin{eqnarray}
J^{(P)}_1&=&J\oplus \epsilon^{(S)}\nonumber\\
J^{(P)}_2&=&J\oplus -\epsilon^{(S)}.
\label{acs}
\end{eqnarray}
Both almost complex structures \refeq{acs} will appear in the subsequent
construction of the rigid string instantons. The sphere bundle ${\cal P}_M$ with given
almost complex structure $J^{(P)}_1$ or $J^{(P)}_2$ is sometimes called the twistor
space
\cite{eells} (see also \cite{ward}).
Now let us recall that a (complex) curve $y$ from a Riemann surface
$(\Sigma,\epsilon^{(\Sigma)})$ to a manifold $(N,J^{(N)})$ is said to be
pseudo-holomorphic if
\begin{eqnarray}
dy+J^{(N)}\circ dy\circ\ep^{(\Si)}=0
\label{jholo}
\end{eqnarray}
where $dy$ is the tangent map $dy:T\Sigma\to TN$. Sometimes, in order to
indicate the almost
complex structure of the target space we shall call \refeq{jholo}
$J$-holomorphic curve suppressing reference to $\ep^{(\Si)}$ \cite{gromov,dusa}.
In this paper we shall take
$\ep^{(\Si)}$ to be complex structure given by
$(\ep^{(\Si)})_a^{\;\;b}=g_{ac}\epsilon^{cb}/\sqrt{g}$ where $g$ is the
metric o n $\Sigma$. When we pull back the definition \refeq{jholo}
on $\Sigma$ we get
\begin{eqnarray}
\p_a\, y^m+ (J^{(N)})_n^{\;\;m}\; \p_b\, y^n\; \frac{ \epsilon_{a}^{\;\;b}}{\sqrt{g}}=0.
\label{jholoi}
\end{eqnarray}
$(a,b,c=1,2\quad m,n=1,...\dim{\cal P}_M)$. For the conformal metric and complex
coordinates on $\Sigma$ \refeq{jholoi} is $\pb y^m-i (J^{(N)})_n^{\;\;m}\;\pb y^n =0$. Thus
${\mbox{\small $\frac{1}{2}$}} (1-iJ^{(N)})$ is the projector on the holomorphic part, while
${\mbox{\small $\frac{1}{2}$}} (1+iJ^{(N)})$ on anti-holomorphic part of (complexified) $TN$.
As it was established in the previous section any immersions defines a sphere
bundle.
Explicitly we define ${\cal P}_M$ as the bundle of normalized, self-dual
tensors $t^{\mu\nu}_{+}$
over $M$. The fiber of this bundle is homeomorphic to $S^2$
(the normalization is $t_{+}^{\mu\nu}t_{+\mu\nu}=4$). The Gauss lift to this
bundle will be denoted by ${\tilde X}_+$.
\begin{eqnarray}
\begin{array}{ccc} & & {\cal P}_M \\
&\stackrel{{\tilde X}_+}{\nearrow} & \downarrow \pi \\
\Sigma & \stackrel{X}{\longrightarrow} & M
\end{array}
\end{eqnarray}
We see that
this bundle is isomorphic to the bundle of almost complex structures defined
previously. This is
the reason why we used the same notation in both cases.
After establishing this simple fact we go to the instanton equation
\refeq{instpm}.
We rewrite \refeq{instpm} and the equation which follows from definition of
$t_+$ in the conformal gauge for the induced metric
$g_{ab}\equiv \p_a X^\mu\p_b X^\nu G^{(M)}_{\mu\nu}\propto \d_{ab}$.
\begin{eqnarray}
(+,\pm)={\overline \na} \tp{\mu}{\nu}\pm i
\tp{\rho}{\nu}\,{\overline \na} \tp{\mu}{\rho}&=&0\nonumber\\
\pb X^\mu-i\tp{\rho}{\mu}\,\pb X^\rho&=&0
\label{insteq}
\end{eqnarray}
We have chosen complex
coordinates on $\Sigma$, thus ${\overline \na}$ is the anti-holomorphic part of the covariant
derivative.
Next we show that Eqs.\refeq{insteq} give pseudo-holomorphic curves on ${\cal P}_M$
with the
two almost complex structures \refeq{acs}.
Any Gauss lift defines $t_+^{\mu\nu}$ and hence with the
help of the metric $G^{(M)}_{\rho\nu}$ we can write down an expression for the almost
complex structure $J_{\mu}^{\;\;\nu}= t_{\;\mu}^{+\;\,\nu}$ on $X^* TM$ at $z\in\Sigma$.
We emphasize that $J$ depends on coordinates on the Grassmann
bundle ${\cal P}_M$.
This almost complex structure decomposes (the complexification of)
the tangent space $X^*T_{X(z)}M$
into holomorphic $T^{(1,0)}$ and anti-holomorphic $T^{(0,1)}$
part.\footnote{We shall suppress the index $X(z)$ of the tangent space at this
point.}
The former is defined as the
space of vectors of the form $T^{(1,0)}=\{(1-iJ)V;V\in X^*TM\}$ while the latter
are complex conjugate vectors.
We also choose locally almost
hermitian metric which
provides the following identification:
$T^{(0,1)}=T^{*(1,0)}$ and
$T^{*(0,1)}=T^{(1,0)}$. Thus from tautology
${\mbox{\small $\frac{1}{2}$}}(1-iJ){\mbox{\small $\frac{1}{2}$}}(1-iJ){\mbox{\small $\frac{1}{2}$}}(1-iJ)={\mbox{\small $\frac{1}{2}$}}(1-iJ)$ we get ${\mbox{\small $\frac{1}{2}$}}(1-iJ)\in
T^{(1,0)} \otimes T^{*(1,0)}$ so $J\in T^{(1,1)}$.
From $J({\overline \na} J)+({\overline \na} J) J=0$ we check that
${\mbox{\small $\frac{1}{2}$}}(1-iJ)[(1-iJ){\overline \na} J]{\mbox{\small $\frac{1}{2}$}}(1+iJ)=(1-iJ){\overline \na} J$ i.e. $(1-iJ){\overline \na} J\in
T^{(1,0)} \wedge T^{*(0,1)}\sim T^{(2,0)}$. Similarly $(1+iJ){\overline \na} J\in
T^{(0,2)}$. Any self-dual tensors
decomposes into
direct sum $T^{(2,0)}\oplus T^{(1,1)}\oplus T^{(0,2)}$ in the almost complex
structure defined by $J$. This can easily checked in particular orthonormal
basis of
$T^{(1,0)}\oplus T^{(0,1)}$, e.g. $\{e_1,e_2,{\bar e}_1,{\bar e}_2\}$.
In this basis $J=e_1\wedge {\bar e}_1 + e_2\wedge {\bar e}_2$ and the two
other self-dual tensors are $e_1\wedge e_2, {\bar e}_1\wedge {\bar e}_2$.
We note that ${\overline \na} J$ is also self-dual. Thus
${\overline \na} J \in T^{(2,0)}\oplus T^{(0,2)}$. As an immediate implication we infer
that ${\overline \na} J$ span the tangent space to the space of almost complex structures
at the point $J$.
Using the above we can built two almost complex structure on the fibers $S^2$.
We define $\ep^{(S)}$ to be such an almost complex structure that $T^{(2,0)}$ are
holomorphic
vectors while $T^{(0,2)}$ are anti-holomorphic
vectors. The choice $-\ep^{(S)}$ would reverse holomorphicity
properties. Thus, $(1-iJ){\overline \na} J$ is holomorphic, while $(1+iJ){\overline \na}
J$ is anti-holomorphic in the $\ep^{(S)}$
complex structure. One can easily find an explicit realization of $\ep^{(S)}$ for
$M=R^4$. For $J_0^{\;\;i}\equiv n^i$, ${\vec n}\in S^2$ and the following
coordinate system on $S^2$
\begin{eqnarray}
{\vec n}=(\frac{f{\bar f}-1}{1+|f|^2},\;-i\frac{f-{\bar
f}}{1+|f|^2},\;\frac{f+{\bar f}}{1+|f|^2})
\label{ffunction}
\end{eqnarray}
we get
\begin{eqnarray}
(1-iJ)\pb J=0\quad\Rightarrow \quad\pb f=0.
\end{eqnarray}
The above $\ep^{(S)}$ is just standard complex structure on $S^2$.
With the help of $\pm\ep^{(S)}$ we can define two
almost complex structures \refeq{acs} on the fiber bundle ${\cal P}_M$ just as we did
in the beginning of this section.
Now it is easy to see that rigid string instantons \refeq{insteq} are
pseudo-holomorphic
curves ${\tilde X}_+:\Sigma\to ({\cal P}_M,J^{(P)}_{1,2})$. Take $J^{(P)}$ given by that
of \refeq{acs}
and $J, \ep^{(S)}$ defined as above. Hence
if we split the map ${\tilde X}_+$
into vertical and horizontal components of $T{\cal P}_M$ then applying the notation
of \refeq{acs} we rewrite \refeq{jholo} as
\begin{eqnarray}
(1-iJ)dX=0, \quad (1\mp i\ep^{(S)})(d{\tilde X}_+)^v=0
\label{split}
\end{eqnarray}
In the first equation we have identified the horizontal component of the
pseudo-holomorphic equation with its counterpart on $M$. In the second equation
$(d{\tilde X}_+)^v$ denotes the vertical part of the map i.e. the space of
$T^{(2,0)}\oplus T^{(0,2)}$ vectors. Thus, accordingly
$(1\mp i\epsilon^{(S)})(d{\tilde X}_+)^v=(1\mp i J){\overline \na} J$. Recalling that $J=t_+$,
this implies that \refeq{split} is
equivalent to \refeq{insteq}.
We conclude that for conformal induced metric $g_{ab}\sim \d_{ab}$ on $\Sigma$
\begin{center}
{{\it
pseudo-holomorphic curves \refeq{jholo}
are solutions of the
instanton equations \refeq{insteq}.}}
\end{center}
The above considerations were applied in \cite{eells} in the context of
minimal and conformal harmonic maps
$X:\Sigma\to M$. In our present nomenclature these maps are $J^{(P)}_2$-holomorphic
curves in ${\cal P}_M$. The almost complex structure $J^{(P)}_2$ is non-integrable what
makes pseudo-holomorphic curves on the manifold (${\cal P}_M,J^{(P)}_2$) hard to explore.
We shall not dwell upon the case any more referring the reader to the
existing reviews \cite{eells2,osserman}.
On the other hand, the case of
$J^{(P)}_1$-instantons maybe relatively easy. The reason is that in
some cases the almost
complex structure $J^{(P)}_1$ is integrable thus defines a complex structure
\cite{ahs} on ${\cal P}_M$. There is a nice geometrical condition under which this
happens.
It states that $M$ must be a half-conformally flat manifold \cite{ahs,eells}.
A lot of classical 4-manifolds respect this condition. In this
work we shall concentrate on $M=R^4,\;S^4$. The other examples are
$T^4,\; S^1\times S^3,\; CP^2,\; K3$. Hence for the half-conformally flat $M$
there exists complex coordinates $\zeta_i$ on ${\cal P}_M$ and then \refeq{jholo} is
simply
\begin{eqnarray}
\pb \zeta_i=0
\label{holo}
\end{eqnarray}
Thus $J^{(P)}_1$-instantons are just holomorphic
maps $\Sigma\to {\cal P}_M$. Another important fact is that if $J^{(P)}_1$ is integrable
then it depends only on the conformal class of the metric $G^{(M)}$ on $M$.
This property gives $J^{(P)}_1$-instantons on $R^4$ if they are known on $S^4$
because
$R^4$ is conformally equivalent to $S^4$. The sphere bundle ${\cal P}_M$ for the
latter is $CP^3$ with unique complex structure being precisely $J^{(P)}_1$.
Following this facts we shall construct
all $J^{(P)}_1$-instantons for $\Sigma=S^2$ explicitly in the next section.
There is a remark necessary at this point. We have chosen
to work in the conformal metric $g_{ab}=e^\phi\d_{ab}$
on $\Sigma$ thus fixing the almost complex structure on $\Sigma$
from the very beginning. For higher genus surfaces Riemann surfaces $\Sigma$ this
is not possible globally unless one allows for some singularities of the
metric i.e. vanishing of the conformal factor. In such a case
solutions of the instanton equations will be so called branched
immersions \cite{eells}. One may try to avoid this working
with the most general complex structure $\ep^{(\Si)}$. This causes problems
with the definition of almost complex structures on ${\cal P}_M$. It is
because, for the rigid string, $\ep^{(\Si)}$
is determined by the induced metric from $X$, but not from ${\tilde X}$.
The problem can be resolved if both metrics are the same what happens for
intersection of $J^{(P)}_1$ and $J^{(P)}_2$ families.
It appears that if $M=S^4$ then
all minimal surfaces respect this condition \cite{bryant}.
\subsection{Moduli space}
We define the moduli space ${\cal M}$ of the problem \refeq{insteq}
as the space of solutions modulo automorphism group of solutions and
reparameterizations of $\Sigma$.
This moduli space is the same as the moduli space of \refeq{jholo}.
One of interesting quantities is the dimension of ${\cal M}$.
Unfortunately, fixing the metric on $\Sigma$ to
be conformal we have lost control (except the case when $\Sigma=S^2$)
over the space of reparameterizations.
Thus we first calculate the dimension of the space ${\tilde {\cal M}}$ of
solutions of
\refeq{jholo} with fixed metric and then we shall argue how to correct
formula in order to get $\dim({\cal M})$.
The (virtual) dimension of the moduli space
$\dim {\tilde {\cal M}}$ is expressed through an index of an operator \cite{index}
The latter is a deformation of \refeq{jholo}:
${\tilde X}_++\xi:\Sigma\to {\cal P}_M$.
After short calculations we get the deformation of \refeq{jholo}:
\begin{eqnarray}
[(1-i J^{(P)}){\overline \na}\xi
-i {\overline \na}_\xiJ^{(P)} \equiv (1-i J^{(P)}){\overline \na}\xi+O(\xi)=0.
\label{oper}
\end{eqnarray}
where $O(\xi)$ denotes terms linear in $\xi$ and not
containing derivatives of $\xi$.
The operator in \refeq{oper} acting on $\xi$ is the elliptic (twisted)
operator mapping
${\tilde X}_+^*T{\cal P}_M\to\Lambda^{(0,1)}\Sigma\otimes {\tilde X}_+^*T{\cal P}_M$.
Homotopic deformations of the
$O(\xi)$ part does not change its index
\cite{index,dusa}. Thus we can set it to zero and obtain the Dolbeault operator
$\pb_{J}=(1-i J^{(P)})\pb$. The index is given by general Atiyah-Singer theorem or by
Hirzerbruch-Riemann-Roch theorem.
\begin{eqnarray}
{\rm Index}(\pb_{J})&=&c_1({\tilde X}_+^*T{\cal P}_M)+{\mbox{\small $\frac{1}{2}$}} \dim_C({\cal P}_M) c_1(T\Sigma)\nonumber\\
&=&c_1({\tilde X}_+^*T{\cal P}_M)+3(1-h).
\label{index}
\end{eqnarray}
Thus $\dim_R({\tilde {\cal M}})=2c_1({\tilde X}_+^*T{\cal P}_M)+6(1-h)$.
For $g=0$ the moduli space ${\cal M}$ is ${\tilde {\cal M}}$ divided by the action
of the group of automorphisms of $S^2$ i.e. the M{\"o}bius group. Hence we
obtain $\dim_R({\cal M})=2c_1({\tilde X}_+^*T{\cal P}_M)$. For higher genus surfaces
$h>0$ if one assumes that the metric on $\Sigma_h$ is elementary or induced from
${\cal P}_M$ one would get
\begin{eqnarray}
\dim_R({\cal M})=
\dim_R({\tilde {\cal M}})-6(1-h)=2 c_1({\tilde X}_+^*T{\cal P}_M).
\label{dim}
\end{eqnarray}
The result agrees with \cite{gromov} where ${\cal M}$ denotes the space of
unparameterized pseudo-holomorphic curves $\Sigma\to {\cal P}_M$.
It is interesting to notice that the formal expression on $\dim_R({\cal M})$ is
independent on the almost complex structure on
${\cal P}_M$. Thus one can use the same formula for both families of instantons
\cite{gromov,cmr}.
It is known that for $M=S^4$ the sphere bundle is $CP^3$. In this case we can
easily find the dimension of ${\cal M}$ for maps from $\Sigma=S^2$.
If the map $S^2\to CP^3$ is given by the degree $k$ polynomials in the
variable $z$ we get $\dim_R({\cal M})=2c_1({\tilde X}_+^*CP^3)=2k c_1(CP^3)=8k$.
\section{Explicit formulae}
\label{sec:examples}
\subsection{ $M=S^4$}
From now on we shall discuss explicit solutions of the $J^{(P)}_1$-instanton equations.
There is vast literature for the minimal
instanton case \cite{eells} and we are not going to review it here.
It is known that for $S^4$ the
appropriate twistor space is $CP^3$ which has only one complex structure.
Complex projective space $CP^3$ is defined as
projective subspace
of $C^4$ i.e.
$CP^3=C^4/\sim$ where $\sim$ means that we identify $(Z_1,Z_2,Z_3,Z_4)$
and $\lambda (Z_1,Z_2,Z_3,Z_4)$ for all $0\neq\lambda\in C$. We can cover $CP^3$ with
four charts $k=1,...4$ for which $Z_k\neq 0$ respectively. In the $k$-th chart
we introduce (inhomogeneous) coordinates: $\zeta_i\equiv Z_i/Z_k$ ($i\neq k)$.
Eq. \refeq{holo} implies that $\zeta_i$ are meromorphic functions of $z$ on
$\Sigma$. This yields instantons on ${\cal P}_M$ which next must be projected on $S^4$.
We do this with help of a very convenient representation of $S^4$ as the
quaternionic projective space \cite{atiyah}.
We recall that quaternions are defined as $q=q^m\sigma^m$ (m=0,..3),
$\sigma^m=(1,i,j,k)\equiv (1,i{\vec \sigma})$\footnote{According to the
standard notation, $i$ on the l.h.s. of this definition denotes the matrix,
while on the r.h.s., the imaginary unit. This remark is applicable
whenever we use quaternions.}
The space of quaternions is denoted by $H$ and is
isomorphic to $C^2$. The isomorphism is such that $(Z_1,Z_2,Z_3,Z_4)
\leftrightarrow (Z_1+j Z_2,Z_3+j Z_4)\in H^2$.
Multiplication and conjugation of quaternions follows from
the above matrix representation. Now we have
\begin{eqnarray}
S^4=HP^1\equiv H^2/\sim
\end{eqnarray}
In the above $\sim$ means that we
identify $(q_1,q_2)$
and $(q_1q,q_2q)$ for all $0\neq q\in H$ i.e. $S^4$ is quaternionic projective
space (line).
Quaternionic representation of $S^4$ is so useful because $CP^3$ is complex
projective space in the same $C^4$.
Heaving a curve in $CP^3$ we can represent it in $H^2=C^4$ and then define two
maps $H^2\to R^4$ which cover $S^4$: $(q_1,q_2)\to (q_1,X_+q_1)$ for
$|q_1|\neq 0$,
and $(q_1,q_2)\to (X_-q_2,q_2)$ for $|q_2| \neq 0$.
The maps are stereographic projections of $S^4$ from the north and south poles
with the transition function $X_-=1/X_+$. The norm is $|X|^2=
(X^\dagger X)=X X^\dagger$ (the expression is proportional to the unit
matrix). Explicitly we have
\begin{eqnarray}
X_+=(Z_3+jZ_4)(Z_1+jZ_2)^{-1}=
\frac{({\bar Z}_1Z_3+Z_2{\bar Z}_4)+j({\bar Z}_1Z_4-Z_2{\bar Z}_3)}{|Z_1|^2+|Z_2|^2}
\label{quat}
\end{eqnarray}
Rotations $SO(4)=SU_L(2)\times
SU_R(2)/Z_2$ act as $X_+'=(\a_L+j \b_L)X_+(\a_R+j \b_R)$.
We see that the action of both $SU(2)$ groups (here unit quaternions) is
equivalent.
After these general remarks we go to the detailed description of the
$J^{(P)}_1$-instantons with topology of sphere $S^2$.
Let us first reproduce the only
compact $J^{(P)}_1$-instanton found in \cite{inst}.
We take $Z_i=a_i (z+b_i)$ (i=1,...4) i.e.
a complex line in $C^4$. For generic choice of $\{a_1,a_2,b_1,b_2\}$
the quaternion $q_1$ is not singular $q_1=Z_1+jZ_2\neq
0$.
By the conformal transformation (M\"{o}bius group),
$z\to \frac{\a z+\b}{\gamma z+ \d}$ ($\a,\b,\gamma,\d\in C,\;\a\d-\b\gamma=1$), we can
fix position of 3 point.
Thus we choose $b_1=\infty, b_2=0, a_1=a_2$. Going from $C^4$ to $CP^3$ fixes
$a_1=a_2=1$ so
$(Z_1+j Z_2)=(1+j z)$. Then we get
\begin{eqnarray}
X_+=\frac{(Z_3+jZ_4)(1-\zb j)}{1+|z|^2}=
X_0+ \frac{(Az+B)+j(-{\bar B}+{\bar A})}{1+|z|^2}
\label{s4}
\end{eqnarray}
for some constants $X_0\in H,\;A,B\in C$. Moding out by the rotation group
leaves only the scale $\lambda$ and the position $X_0$ as moduli . Hence
\begin{eqnarray}
X-X_0=\frac{\lambda}{1+|z|^2}(z+j),\quad \lambda\in R
\label{sphere}
\end{eqnarray}
what is exactly the result obtained in \cite{inst}. \refeq{sphere}
represents sphere of radius $\lambda/2$.
The above shows that \refeq{sphere} is the most general
$J^{(P)}_1$-instanton
with $\chi=2,I=0$.
We can easily generalize this to other topological sectors.
In order to get $J^{(P)}_1$-instantons of the $k$-th sector the
functions $Z_i$ which defines $\zeta_i$ must be polynomials of degree $k$
\begin{eqnarray}
Z_i=a_i\prod_{j=1}^k (z-a_{ij})\quad i=1,...4
\label{zes}
\end{eqnarray}
Thus $\zeta_i$'s are rational functions with poles at points where coordinates
are ill defined. We can calculate
dimension of the moduli space ${\cal M}$ directly from (\ref{quat},\ref{zes}).
The are $8k+6$ parameters involved in \refeq{quat}. Moding out by the
M{\"o}bius group subtract 6
parameters yielding $\dim({\cal M})=8k$. We can also divide by the rotation group
dropping additional 3 dimensions of the moduli space.
The instanton sectors are characterized by the self-intersection number of the
immersed surface in $S^4$: $I=k-1$.
We shall obtain this result by simple means in the next subsection.
The dimension of the moduli space is
quite remarkable result, because it is exactly the dimension of the moduli space
of $SU(2)$ instantons \cite{atiyah}.
Moreover we for $k=1$ topology of both spaces is exactly
the same. Topology of ${\cal M}$ for higher $k$ remains to be investigated.
\subsection{$M=R^4$}
It appeared that the rigid string instanton equations, which seemed so
complicated \cite{inst}, can be trivially solved in $R^4$. Using
the parameterization of
\refeq{ffunction} we can rewrite the second of Eqs.\refeq{insteq} as:
\begin{eqnarray}
\pb {\bar X}_+^1+ f\pb X_+^2&=&0\nonumber\\
- f \pb X_+^1+\pb {\bar X}_+^2&=&0
\label{flat}
\end{eqnarray}
where $X_+^1=X^0+iX^1,\;X_+^2=X^2+iX^3$. This is enormous and unexpected
simplification of the $(1-it_+)\pb X=0$ equation.
The first line of Eqs.\refeq{insteq} is
\begin{eqnarray}
\pb f&=&0\quad\mbox{ for $J^{(P)}_1$-instantons}\\
\pb {\bar f}&=&0\quad\mbox{ for minimal instantons}
\end{eqnarray}
Both system of equations are very simple and can be directly integrated.
$J^{(P)}_1$-instantons are identical with \refeq{s4}. Explicitly
\begin{equation}
X^1_+=\frac{{\bar w}_1(\zb)-{\bar f}(\zb) w_2(z)}{1+|f|^2},\quad
X^2_+=\frac{{\bar w}_2(\zb)+{\bar f}(\zb) w_1(z)}{1+|f|^2}
\end{equation}
Comparing with \refeq{quat} we see that $f= Z_2/ Z_1$. Because
$I_+$ is minus degree of the map:
$f:\Sigma\to S^2$ we get $I_+=k$. From the relation: $I_+=I+\chi/2$ follows that
$I=k-1$, what is the result quoted in the previous subsection.
We want to stress that results on $M=R^4$ and $M=S^4$ are almost identical
because $S^4$ is
conformally equivalent to $R^4$ and the integrable $J_1$ almost complex
structure is conformally
invariant \cite{ahs}. We also notice non-triviality of the complex structure
given by $f=f(z)$: holomorphic functions are ${\bar X}_+^1+f\,X_+^2$
and $- f\,X_+^1+{\bar X}_+^2$.
Minimal instantons can also be integrated and as one could expect they give
solutions of the equation $\D X^\mu=0$. Contrary to the previous case they do
not correspond to minimal surfaces on $S^4$. We shall not dwell upon this
subject referring to the rich existing literature
\cite{eells,bryant,osserman}.
\section{Speculations and final remarks}
In this section we allude on some possible applications of the presented
results to topology of 4-manifolds and indicate similarities with several
proposals for string picture of gauge fields. We also shortly discuss the case
of 3d target manifold.
\subsection{Topology of 4-manifolds}
Starting from works of Gromov \cite{gromov} and Witten \cite{witten}
pseudo-holomorphic curves were used to define certain topological invariants, so
called Gromov-Witten invariants \cite{dusa} of symplectic
manifolds (here denoted by $N$).
The invariants can be defined geometrically in descriptive way as follows:
take a set of homology cycles $\a_i\in H_{d_i}(N,Z)$ and
count (with an appropriate sign) those pseudo-holomorphic curves representing
2-cycle $A\in H_{2}(N,Z)$ which intersect all
classes ${\a_i}$ at some points. There is also ``physicist'' definition of the
invariants through a correlation function in a topological field theory
\cite{witten}. In
this case the invariants can be formally defined on any almost
complex manifold.
All of twistor spaces
are almost complex and some of them are K\"ahler (for $M=S^4,\;CP^2$) so also
symplectic.
Thus following these definitions one could define appropriate invariants for
the twistor spaces of 4-manifolds $M$ considered in this work.
The hypothesis is:
{\it the Gromov-Witten invariants of the twistor space ${\cal P}_M$ define some
invariants of the 4-manifold $M$}.
These new invariants are well
defined on $M$ if they are well defined on ${\cal P}_M$. Moreover we can define two
sets of invariants (if we require that ${\cal P}_M$ must be almost complex only) due
to two natural almost complex structures $J^{(P)}_{1,2}$ on ${\cal P}_M$.
The real problem is what kind of
topological information do they carry? Intersection of cycles in the twistor space
(say at $p\in{\cal P}_M$) corresponds to the situation when projection of the cycles
to $M$
have common tangents at common point $\pi(p)\in M$. This property is invariant
only under diffeomorphisms of $M$ (class $C^1(M)$) but not under
homeomorphisms of $M$!. It may be that the invariants carry some information
about smooth structures of $M$, so would be similar in nature to Donaldson or
Seiber-Witten invariants. The basic difference is that they are defined in
purely geometrical way avoiding any reference to gauge fields. Moreover the
invariants seems to be well defined on manifolds for which there are
no other invariants.
This includes very interesting cases
$M=R^4,\;S^4$ discussed in this paper. Both cases are, of course, different
because there are no compact $J^{(P)}_2$-instantons on $R^4$. Contrary,
the $J^{(P)}_1$-invariants should be the same due to one-to-one
correspondence between spaces of instantons in both cases.
This subject, if relevant, seems to be very exciting.
\subsection{Relation to gauge fields}
Going back to physics we want to discuss striking relations of
rigid string with gauge fields. Of course both theories uses
twistors in construction of instantons. Leaving this aside we go to more
quantitative
comparisons. First of all,
two-dimensional pseudo-holomorphic curves were
used to build the string picture of YM$_2$ \cite{cmr}. Rigid string
instantons provides natural generalization of these curves to 4-dimensions.
One can perform a
naive
dimensional reduction of 4d instantons to 2-dimensions by
suppressing two coordinates (say $X^2,\;X^3$).
This results in taking $|t^{01}|=1$ (there is no distinction between $t_-$
and $t_+$). Thus we get two families of pseudo-holomorphic curves
\begin{equation}
\frac{\epsilon_a^{\;\;b}}{\sqrt{\det (g)}}\,\p_b X^\mu\pm i J_\rho^{\;\;\mu}\, \p_a
X^\rho=0
\label{twodim}
\end{equation}
where now $J_\mu^{\;\;\nu}=G_{\mu\rho}\frac{\epsilon^{\rho\nu}}{\sqrt{\det (G)}}$ and
$G$ is
the metric on $M^2$. These are the maps of \cite{cmr} (here $g_{ab}$
is the elementary metric).
On this basis one can state a bold hypothesis that $YM_4$ is localized on the rigid
string instantons\footnote{This is a natural generalization of
\cite{horava}.}.
All these similarities suggest that rigid string instantons will play a
significant role in string description of YM fields. Some other ideas along this
line were posed in \cite{nfold}.
We also notice strange coincidence of the
dimensions of the moduli space of genus zero rigid string instantons on $R^4$ and
$S^4$ and the moduli space of $SU(2)$ Yang-Mills instantons (with the
appropriate identification of topological numbers). For $k=1$ both moduli
spaces are identical. We do not know what happens for other $k$.
\subsection{3d manifolds}
Finally we comment on 3d target manifolds. In this case the tensor
$t^{\mu\nu}$ has 3 components. Classification of immersions of
surfaces in $R^3$ is more complicated then for the $R^4$ case. There are $4^h$
distinct regular homotopy classes of immersions of a surface of genus $h$
into $R^3$ \cite{jamesthomas}.
One can easily derive appropriate instanton equations following \cite{inst} and
using $\chi$ only (the self-intersection number $I$ is strictly 4d notion!).
The equations are just Eqs.\refeq{insteq} with $t^{\mu\nu}$ in place of
$\tp{\mu}{\nu}$. One of the equation is equivalent to $\D X^\mu=0$ another
one represents so-called totally umbilic maps. In the case of $\Sigma=S^2$ we
have an immediate solution of the latter. This is just the
sphere embedded in $R^3$ given by Eq.\refeq{sphere}.
Unfortunately, because classification of immersions is so different and we do
not know the invariant which would distinguish all topological classes it is
hard to imagine that the instantons will represent all of them.
\vskip.5cm
{\bf Acknowledgment}. I would like to thank Erwin Schr\"odinger Institute
for kind hospitality where a part of this paper was prepared.
I also thank E.Corrigan, K.Gaw{\c e}dzki, H.Grosse, B.Jonson, C.Klimcik, A.Morozov,
R.Ward and J.Zakrzewski
for comments and interest in the work. Special thanks to P. Nurowski for many
illuminating discussions on twistor space.
|
proofpile-arXiv_065-697
|
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\section{Higgs bosons in the Standard Model}
In this lecture we will review some elementary and/or well established
features of the Higgs sector in the Standard Model (SM)~\cite{SM}.
Most of it should
be viewed as an introduction for beginners and/or students
in the field though we also have presented some recent results on
Higgs mass bounds obtained by this author in various collaborations.
The methods used to obtain the latter results are sometimes technical.
Therefore, we have simplified the analysis and presented only the relevant
results.
\subsection{\bf Why a Higgs boson?}
The Higgs mechanism~\cite{Higgs}
is the simplest mechanism to induce spontaneous
symmetry breaking of a gauge theory. In particular, in the Standard
Model of electroweak interactions it achieves the breaking
\begin{equation}
SU(2)_L \times U(1)_Y\longrightarrow U(1)_{em}
\label{rotura}
\end{equation}
in a renormalizable quantum field theory, and gives masses to the
gauge bosons $W^{\pm},Z$, the Higgs boson and the fermions.
The SM fermions are given by~\cite{Pepe}
\begin{eqnarray}
\label{fermiones}
q_L & = & \left(
\begin{array}{c}
u_L \\
d_L
\end{array}
\right)_{1/6},\
\left(u_R\right)_{2/3},\
\left(d_R\right)_{-1/3} \nonumber\\
\ell_L & = & \left(
\begin{array}{c}
\nu_L \\
\ell_L
\end{array}
\right)_{-1/2},\
\left(\ell_R\right)_{-1}
\end{eqnarray}
where the hypercharge $Y$ is related to the electric charge $Q$ by,
$Q=T_3+Y$, and we are using the notation $f=f_L+f_R$, with
\begin{eqnarray}
\label{fermquirales}
f_L &= & \frac{1}{2}(1-\gamma_5)f \nonumber \\
f_R &= & \frac{1}{2}(1+\gamma_5)f .
\end{eqnarray}
The Higgs boson is an $SU(2)_L$ doublet, as given by
\begin{equation}
\label{higgsSM}
H=\frac{1}{\sqrt{2}}\left(
\begin{array}{c}
\chi^+ \\
\Phi+i\chi^0
\end{array}
\right)_{1/2}
\end{equation}
The physical Higgs $\phi$ is related to $\Phi$ by, $\Phi=\phi+v$, where
$v=(\sqrt{2}G_F)^{-1/2}=246.22$ GeV
is the vacuum expectation value (VEV) of the Higgs.
The (massless) fields $\chi^\pm , \chi^0$ are the Goldstone
bosons.
A mass term for gauge bosons $V_{\mu}$, as
$\frac{1}{2}M_V^2 V_\mu V^\mu$
is not gauge invariant, and would spoil the renormalizability properties
of the theory. A mass term for fermions,
$m_u \bar{q}_L u_R+m_d \bar{q}_L d_R+m_\ell \bar{\ell}_L \ell_R$
does not even exist (it is not $SU(2)_L\times U(1)_Y$ invariant). Both goals
can be achieved through the Higgs mechanism~\cite{Higgs}.
One can write the part of the SM Lagrangian giving rise to
mass terms as
\begin{equation}
\label{lagrangiano}
{\cal L}=\left(D_\mu H\right)^\dagger \left(D_\mu H\right)
- (h_d \bar{q}_L H d_R + h_u \bar{q}_L H^c u_R+
h_\ell \bar{\ell}_LH\ell_R +h.c.) -V(H)
\end{equation}
where $H^c\equiv i \sigma_2 H^*$, the covariant derivative
$D_\mu$ of the Higgs field is defined by
\begin{equation}
\label{dercovariante}
D_\mu H\equiv \left(\partial_\mu+i g \frac{\vec{\sigma}}{2} \vec{W}_\mu
+i g' \frac{1}{2} B_\mu \right) H
\end{equation}
and the Higgs potential by
\begin{equation}
\label{Higgspot}
V(H)=-\mu^2 H^\dagger H +\frac{\lambda}{2}\left(H^\dagger H\right)^2
\end{equation}
Minimization of (\ref{Higgspot}) yields,
\begin{equation}
\label{minimo}
\langle 0|H|0 \rangle\equiv \frac{v}{\sqrt{2}}
\left(
\begin{array}{c}
0 \\
1
\end{array}
\right);\ v=\sqrt{\frac{2\mu^2}{\lambda}}
\end{equation}
Replacing now $\Phi=\phi+v$ into (\ref{lagrangiano}) yields:
\begin{eqnarray}
\label{lagmasas}
{\cal L}& = & -\frac{1}{4}g^2v^2 W_\mu^+ W^{\mu -}
-\frac{1}{8}v^2
\left(
\begin{array}{cc}
Z^\mu & A^\mu
\end{array}
\right)
\left(
\begin{array}{cc}
g^2+g'^2 & 0 \\
0 & 0
\end{array}
\right)
\left(
\begin{array}{c}
Z_\mu \\
A_\mu
\end{array}
\right) \nonumber \\
&&-\frac{vh_u}{\sqrt{2}}\bar{u}u
-\frac{vh_d}{\sqrt{2}}\bar{d}d
-\frac{vh_\ell}{\sqrt{2}}\bar{\ell}\ell
\end{eqnarray}
where
\begin{eqnarray}
W_\mu^{\pm}& = & \frac{1}{\sqrt{2}}
\left(W_{\mu 1}\pm i W_{\mu 2}\right) \nonumber\\
Z_\mu & = & \cos\theta_W W_{\mu 3}-\sin\theta_W B_\mu \\
A_\mu & = & \sin\theta_W W_{\mu 3}+\cos\theta_W B_\mu \nonumber
\end{eqnarray}
and the electroweak angle $\theta_W$ is defined by
$\tan\theta_W=g'/g$.
In this way the goal of giving masses to the gauge bosons and the
fermions has then been achieved as \footnote{In the following we will
use the notation $m_t,m_H$ for the top-quark and Higgs boson running
$\overline{\rm MS}$ on-shell masses
(defined at a scale equal to the corresponding mass), and $M_t,M_H$ for
the corresponding pole (physical) masses. They are related by a
contribution from self-energies. Thus for the Higgs boson, the running
and pole masses are related by~\cite{CEQR}
$M_H^2=m_H^2(M_H)+{\rm Re}\Pi_{\phi\phi}(M_H)-{\rm Re}\Pi_{\phi\phi}(0)$.}
\begin{eqnarray}
M_W^2 & = & \frac{1}{4} g^2 v^2 \nonumber \\
M_Z^2 & = & \frac{1}{4} \left(g^2+g'^2\right) v^2\\
m_f & = & \frac{1}{\sqrt{2}} h_f v \nonumber\\
m_H^2 & = & \lambda v^2\nonumber
\end{eqnarray}
\subsection{\bf What we know about the Higgs: its couplings}
The couplings $(g,g',v)$ are experimentally {\it traded} by
a set of three observables, as e.g. $(M_W,M_Z,G_F)$, or
$(\alpha_{em},M_Z,G_F)$, while the Yukawa couplings $h_f$ are
{\it measured} by the fermion masses, $m_f$. Only the quartic
coupling $\lambda$ in Eq.~(\ref{lagrangiano}), which should
be {\it measured} by the Higgs mass, is
at present {\bf unknown}.
All Higgs interactions (cross-sections, branching ratios,...)
are determined once the corresponding Feynman rules
are known~\cite{Japs}.
In Table~1 we summarize the main vertices involving the physical
Higgs boson in the SM along with the rest of particles in the
SM.
\begin{center}
\begin{tabular}{||c|c||}\hline
Vertex & Coupling \\ \hline
& \\
$\phi f\bar{f}$ & $-i\frac{g}{2M_W}m_f$ \\
& \\
$\phi W^{\pm}_\mu W^{\mp}_\nu$ & $i g M_W g_{\mu\nu}$ \\
&\\
$\phi Z_\mu Z_\nu$ & $ i\frac{g M_Z}{\cos\theta_W}g_{\mu\nu}$ \\
&\\
$\phi\phi\phi$ &$ -i\frac{3g}{2M_W}M^2_H $ \\
&\\
$\phi\phi W^{\pm}_\mu W^{\mp}_\nu $ & $i\frac{1}{2}g^2 g_{\mu\nu}$ \\
&\\
$\phi\phi Z_\mu Z_\nu$ & $i\frac{1}{2}\frac{g^2}{\cos^2\theta_W}g_{\mu\nu}$\\
&\\
$\phi\phi\phi\phi $ & $ -i\frac{3g^2M_H^2}{4M_W^2}$ \\
&\\
\hline
\end{tabular}
\end{center}
\begin{center}
Table 1
\end{center}
\subsubsection{Higgs production at LEP2}
The main mechanisms for production of Higgs particles at $e^+e^-$
colliders, at the LEP2 energies, are~\cite{LEP2}:
\begin{itemize}
\item
HIGGS-STRAHLUNG: $e^+e^-\rightarrow Z\phi$, where the Higgs boson
is radiated off the virtual $Z$-boson line exchanged in the s-channel.
[Fig.~\ref{diag1}, where the solid (fermion) lines are
electrons, the wavy line is a $Z$ boson and the dashed line a
Higgs $\phi$.]
\begin{figure}[hbt]
\centerline{
\psfig{figure=diag1.ps,width=5cm,bbllx=4.75cm,bblly=14.5cm,bburx=14.25cm,bbury=17.7cm}}
\caption{Higgs-strahlung process for Higgs production.}
\label{diag1}
\end{figure}
\item
WW-FUSION: $e^+e^-\rightarrow \phi\bar{\nu}_e\nu_e$, where the Higgs
boson is formed in the fusion of virtual $WW$
exchanged in the t-channel. The virtual $W$'s
are radiated off the electron and positron of the beam.
[Fig.~\ref{diag2}, where the incoming lower (upper)
fermion line is an electron
(positron) and the corresponding outcoming fermion a $\nu_e$
($\bar{\nu}_e$). Wavy lines are $W$ and the dashed line a
Higgs.]
\end{itemize}
\begin{figure}[hbt]
\centerline{
\psfig{figure=diag2.ps,width=5cm,bbllx=4.75cm,bblly=11.5cm,bburx=14.25cm,bbury=17.cm}}
\caption{Vector-Vector fusion process for Higgs production.}
\label{diag2}
\end{figure}
A detailed analysis of these processes for LEP2 can be found in
Ref.~\cite{LEP2}. There it is found that the Higgs-strahlung process
dominates the cross-section for low values of the Higgs mass
($M_H<105$ GeV), while the WW-fusion process dominates it for large
values of the Higgs mass ($M_H>105$ GeV).
\subsubsection{Higgs production at LHC}
The main mechanisms for production of Higgs bosons at $pp$ colliders,
at the LHC energies, are~\cite{Ferrando}:
\begin{figure}[hbt]
\centerline{
\psfig{figure=diag4.ps,width=5cm,bbllx=4.75cm,bblly=12.7cm,bburx=14.25cm,bbury=19.2cm}}
\caption{Gluon-gluon fusion process for Higgs production.}
\label{diag4}
\end{figure}
\begin{itemize}
\item
GLUON-GLUON FUSION: $gg\rightarrow \phi$, where two gluons in the sea
of the protons collide through a loop of top-quarks, which
subsequently emits a Higgs boson. [Fig.~\ref{diag4} where the
curly lines are gluons, the internal fermion line a top and the
dashed line a Higgs.]
\item
WW (ZZ)-FUSION: $W^\pm W^\mp (ZZ)\rightarrow \phi$, where the Higgs
boson is formed in the fusion of $WW (ZZ)$, the virtual $W (Z)$'s
being exchanged in the t-channel and radiated off
a quark in the proton beam. [Fig.~\ref{diag2}, where wavy lines
are $W$ ($Z$), the incoming fermions quarks $q$ and the
outcoming fermions quarks $q$ ($q'$). The dashed line is the
Higgs.]
\item
HIGGS STRAHLUNG: $q \bar{q}^{(')}\rightarrow Z (W) \phi$,
where the Higgs boson is radiated off the virtual $Z(W)$-boson line
exchanged in the s-channel. [Fig.~\ref{diag1}, where wavy lines
are $Z$ ($W$), the incoming fermion a quark $q$ and the
outcoming fermion a quark $q$ ($q'$).]
\item
ASSOCIATED PRODUCTION WITH $t\bar{t}$: $gg\rightarrow \phi t \bar{t}$,
where the gluons from the proton sea exchange a top quark in the
t-channel, which emits a Higgs boson. [Fig.~\ref{diag3}, where
curly lines are gluons and the fermion line corresponds to a
quark $t$. The dashed line is the Higgs boson.]
\end{itemize}
\begin{figure}[hbt]
\centerline{
\psfig{figure=diag3.ps,width=5cm,bbllx=4.75cm,bblly=13.cm,bburx=14.25cm,bbury=19.cm}}
\caption{Associated production of Higgs with $f\bar{f}$.}
\label{diag3}
\end{figure}
A complete analysis of the different production channels can be found,
e.g. in Ref.~\cite{LHC}. It is found that for a top mass in the
experimental range~\cite{top} the gluon-gluon fusion mechanism is
dominating the production cross-section for any value of the Higgs
mass. The subdominant process, WW(ZZ)-fusion is comparable in magnitude
to the gluon-gluon process only for very large values of the Higgs
mass $M_H\sim 1$ TeV. For low values of the Higgs mass, $M_H\sim 100$ GeV,
the gluon-gluon fusion process is still dominant over all other channels
by around one order of magnitude, while all the others are similar
in magnitude for these values of the Higgs mass.
\subsubsection{Higgs decays}
For values of the Higgs mass relevant at LEP2 energies, the main
decay modes of the Higgs boson are:
\begin{itemize}
\item
$\phi\rightarrow b\bar{b},c\bar{c},\tau^-\tau^+$, which is dominated by the
$b\bar{b}$ channel.
\item
$\phi\rightarrow gg$, where the gluons are produced by a top-quark loop
emitted by the Higgs. [The inverse diagram of Fig.~\ref{diag4}.]
\item
$\phi\rightarrow WW^*\rightarrow W f \bar{f}'$, which is relevant for values
of the Higgs mass, $M_H>M_W$.
\end{itemize}
A complete analysis of different Higgs decay channels reveals~\cite{LEP2}
that, for LEP2 range of Higgs masses, $M_H<110$ GeV, the $b\bar{b}$
channel dominates the Higgs branching ratio by $\sim$ one order
of magnitude.
For $M_H>110$ GeV, the main decay modes relevant for LHC energies and
$pp$ colliders are~\cite{LHC}:
\begin{itemize}
\item
$\phi\rightarrow \gamma\gamma$, where the photons are produced by a top-quark
loop emitted by the Higgs. [The inverse diagram as that of
Fig.~\ref{diag4}, where gluons are replaced by photons.]
\item
$\phi\rightarrow W^{\pm}W^{\mp}$, which requires $M_H>2M_W$.
\item
$\phi\rightarrow ZZ$, which requires $M_H>2M_Z$.
\item
$\phi\rightarrow t\bar{t}$, which requires $M_H>2M_t$.
\end{itemize}
For a heavy Higgs ($M_H>150$ GeV) the $WW (ZZ)$ decay channels
completely dominate the Higgs branching ratio, while the radiative
decay $\gamma\gamma$ dominates for low values of the Higgs mass and is
expected to close the LHC window for a light Higgs. The reader
is referred to Ref.~\cite{LHC} for more details.
\subsection{\bf What we do not know about the Higgs: its mass}
Being the Higgs boson the missing ingredient of the SM, the quartic
coupling $\lambda$, and so its mass, are unknown. However we can
have information on $M_h$ from experimental and theoretical input.
From experimental inputs we have direct and indirect information
on the Higgs mass.
Since direct
experimental searches at LEP have been negative up to now,
they translate into a lower bound on the Higgs mass~\cite{Higgsmass},
\begin{equation}
\label{expbound}
M_h>67\ {\rm GeV},
\end{equation}
Experimental searches also yield
indirect information, which is the influence the Higgs mass
has in radiative corrections and in precision measurements at
LEP~\cite{Higgsmass}.
However, unlike the top quark mass, on which the radiative
corrections are quadratically dependent, and so very
sensitive, the dependence of one-loop radiative corrections on
the Higgs mass is only logarithmic (the so-called Veltman's
screening theorem), which means that radiative corrections in
the SM have very little sensitivity to the Higgs mass, providing
only very loose bounds from precision measurements.
However, from the theoretical input the situation is rather
different. In fact the theory has a lot of information on $M_h$,
which can be used to put bounds on the Higgs mass. If these
bounds were evaded when the Higgs mass will be eventually
measured, this measurement might lead to the requirement of new physics,
just because the SM cannot accomodate such a value of the Higgs
(and the top-quark) mass.
For particular values of the Higgs boson and top quark masses, $M_H$ and $M_t$,
the effective potential of the Standard Model (SM) develops a deep non-standard
minimum for values of the field $\phi \gg G_F^{-1/2}$~\cite{L}.
In that case the
standard electroweak (EW) minimum becomes metastable and might decay into
the non-standard one. This means that the SM might have troubles
in certain regions
of the plane ($M_H$,$M_t$), a fact
which can be intrinsically interesting as evidence for
new physics. Of course, the mere existence of the non-standard minimum,
and also the decay rate
of the standard one into it, depends on the scale $\Lambda$ up to which
we believe the SM results. In fact, one can identify $\Lambda$
with the scale of new physics.
\subsubsection{Stability bounds}
The preliminary question one should ask is: When the standard EW
minimum becomes
metastable, due to the appearance of a deep non-standard
minimum? This question was
addressed in past years~\cite{L} taking into account leading-log (LL) and
part of next-to-leading-log (NTLL) corrections.
More recently, calculations have incorporated all
NTLL
corrections~\cite{AI,CEQ}
resummed to all-loop by the renormalization group equations (RGE),
and considered pole masses for the top-quark and
the Higgs-boson.
From the requirement of a stable (not metastable) standard EW minimum
we obtain a lower bound on
the Higgs mass, as a function of the top mass, labelled by the values of
the SM cutoff (stability bounds). Our
result~\cite{CEQ} is lower than previous estimates by ${\cal O}$(10) GeV.
The problem to attack is easily stated as follows:
The effective potential in the SM can be written as (\ref{Higgspot})
\begin{equation}
\label{poteff}
V=-\frac{1}{2}m^2\phi^2+\frac{1}{8}\lambda\phi^4+\cdots
\end{equation}
where the ellipsis refers to radiative corrections and all
parameters and fields in (\ref{poteff}) are running with the
renormalization group scale $\mu(t)=M_Z\exp(t)$. The condition for
having an extremal is
$V'(\phi(t))=0$, which has as solution
\begin{equation}
\label{vev}
\phi^2=\frac{2m^2}{\lambda-\frac{12}{32\pi^2}h_t^4
\left(\log\frac{h_t^2\phi^2}{2\mu^{2}}-1\right)}
\end{equation}
where $h_t$ refers to the top Yukawa coupling, and only the
leading radiative corrections have been kept for simplicity.
The curvature of the potential (\ref{poteff}) at the extreme
is given by
\begin{equation}
\label{curv}
V''(\phi)=2m^2+\frac{1}{2}\beta_\lambda \phi^2
\end{equation}
The condition $V'=0$ is obviously satisfied at the EW minimum where
$\langle\phi\rangle=v\sim 246$ GeV, $\lambda\sim(m_H/v)^2>1/16$,
$m^2\sim m_H^2/2$ and $V''(\langle\phi\rangle)>0$ (a minimum).
However, the condition $V'=0$ can also be satisfied for values
of the field $\phi\gg v$ and, since $m={\cal O}(100)$ GeV,
for those values
$$
\lambda\sim\left(\frac{m}{\phi}\right)^2\ll 1.
$$
Therefore, for the non-standard extremals we have
\begin{eqnarray}
\label{minmax}
\beta_\lambda < 0 & \Longrightarrow & V''<0\ {\rm
maximum}\nonumber \\
\beta_\lambda > 0 & \Longrightarrow & V''>0\ {\rm minimum}.
\end{eqnarray}
The one-loop effective potential of the SM improved by
two-loop RGE has been shown to
be highly scale independent~\cite{CEQR} and, therefore, very reliable for the
present study.
In Fig.~\ref{fval1} we show (thick solid line)
the shape of the effective potential for
$M_t=175$ GeV
and $M_H=121.7$ GeV. We see the appearance of the non-standard maximum,
$\phi_M$, while the global
non-standard minimum has been cutoff at $M_{P\ell}$.
We can see from Fig.~\ref{fval1} the
steep descent from the non-standard maximum. Hence,
even if the non-standard minimum is beyond
the SM cutoff, the
standard minimum becomes metastable and might be destabilized. So for fixed
values of $M_H$ and
$M_t$ the condition for the standard minimum not to become metastable is
\begin{equation}
\label{condstab}
\phi_M \stackrel{>}{{}_\sim} \Lambda
\end{equation}
Condition (\ref{condstab}) makes the stability condition $\Lambda$-dependent.
In fact we have plotted
in Fig.~\ref{fval2} the stability condition on $M_H$ versus $M_t$ for
$\Lambda=
10^{19}$ GeV and 10 TeV. The stability
region corresponds to the region above the dashed curves.
\begin{figure}[hbt]
\centerline{
\psfig{figure=fval1.ps,height=7.5cm,width=7cm,bbllx=4.75cm,bblly=3.cm,bburx=14.25cm,bbury=16cm}}
\caption{Plot of the effective potential for $M_t=175$ GeV, $M_H=121.7$
GeV at $T=0$ (thick solid line) and $T=T_t=2.5\times 10^{15}$ GeV
(thin solid line).}
\label{fval1}
\end{figure}
\subsubsection{Metastability bounds}
In the last subsection we have seen that
in the region of Fig.~\ref{fval2}
below the dashed line the standard EW minimum is
metastable. However we should not draw physical consequences
from this fact since we still do not
know at which minimum does the Higgs field sit. Thus, the real physical
constraint we have to impose is avoiding
the Higgs field sitting at its non-standard minimum.
In fact the Higgs field can be sitting at its
zero temperature non-standard minimum because:
\begin{enumerate}
\item
The Higgs field was driven from the origin to the non-standard minimum
at finite temperature
by thermal fluctuations in a non-standard EW phase transition at
high temperature.
This minimum evolves naturally to the non-standard minimum at zero
temperature. In this case
the standard EW phase transition, at $T\sim 10^2$ GeV, will not take place.
\item
The Higgs field was driven from the origin to the
standard minimum at $T\sim 10^2$ GeV, but decays,
at zero temperature, to the non-standard minimum by a quantum fluctuation.
\end{enumerate}
\begin{figure}[hbt]
\centerline{
\psfig{figure=fval2.ps,height=7.5cm,width=7cm,bbllx=5.cm,bblly=2.cm,bburx=14.5cm,bbury=15cm}}
\caption{Lower bounds on $M_H$ as a function of $M_t$, for
$\Lambda=10^{19}$ GeV (upper set) and $\Lambda=10$ TeV (lower set).
The dashed curves
correspond to the stability bounds and the solid (dotted)
ones to the metastability bounds at finite (zero) temperature.}
\label{fval2}
\end{figure}
In Fig.~\ref{fval1} we have depicted the
effective potential at $T=2.5\times
10^{15}$ GeV (thin solid line) which
is the corresponding
transition temperature. Our finite temperature
potential~\cite{EQtemp} incorporates plasma effects~\cite{Q}
by one-loop resummation of Debye masses~\cite{DJW}. The tunnelling
probability per unit time per
unit volume was computed long ago for thermal~\cite{Linde} and
quantum~\cite{Coleman} fluctuations.
At finite temperature it is given by $\Gamma/\nu\sim T^4 \exp(-S_3/T)$,
where $S_3$ is the euclidean action evaluated
at the bounce solution $\phi_B(0)$. The semiclassical
picture is that unstable bubbles are nucleated behind the
barrier at $\phi_B(0)$ with a probability given by $\Gamma/\nu$. Whether
or not they fill the Universe depends on
the relation between the probability rate and the expansion
rate of the Universe. By normalizing the former
with respect to the latter we obtain a normalized probability $P$,
and the condition for decay corresponds
to $P\sim 1$. Of course our results are trustable,
and the decay actually happens, only if
$\phi_B(0)<\Lambda$, so that the similar condition to (\ref{condstab}) is
\begin{equation}
\label{condmeta}
\Lambda< \phi_B(0)
\end{equation}
The condition of no-decay (metastability condition) has
been plotted in Fig.~\ref{fval2} (solid lines)
for $\Lambda=10^{19}$ GeV and 10 TeV. The region
between the dashed and the solid line corresponds to
a situation where the non-standard minimum exists
but there is no decay to it at finite temperature.
In the region below the solid lines the Higgs field is sitting
already at the non-standard minimum at $T\sim 10^2$ GeV, and the standard EW
phase transition does not happen.
We also have evaluated the tunnelling probability
at zero temperature from the standard EW minimum to the
non-standard one. The result of the calculation
should translate, as in the previous case, in lower bounds
on the Higgs mass for differentes
values of $\Lambda$. The corresponding bounds are shown
in Fig.~\ref{fval2} in
dotted lines. Since the dotted lines
lie always below the solid ones, the possibility of quantum tunnelling at
zero temperature does not impose any extra constraint.
As a consequence of all improvements in the
calculation, our bounds are lower than previous
estimates~\cite{AV}. To fix ideas, for $M_t=175$ GeV, the
bound reduces by $\sim 10 $ GeV for $\Lambda=10^4$ GeV,
and $\sim 30$ GeV for $\Lambda=10^{19}$ GeV.
\subsubsection{Perturbativity bounds}
Up to here we have described lower bounds on the Higgs mass
based on stability arguments. Another kind of bounds, which have
been used in the literature, are upper bounds based on the
requirement of perturbativity of the SM up to the high scale
(the scale of new physics) $\Lambda$.
Since the quartic coupling grows with the scale~\footnote{In fact
the value of the renormalization scale where the quartic
coupling starts growing depends on the value of the top-quark
mass.}, it will blow up to infinity at a given scale: the scale where
$\lambda$ has a Landau pole. The position of the Landau pole
$\Lambda$ is, by definition, the maximum scale up to which the
SM is perturbatively valid. In this way assuming the SM remains
valid up to a given scale $\Lambda$ amounts to requiring an
upper bound on the Higgs mass from the perturbativity
condition~\cite{LEP2}
\begin{equation}
\label{perturbcond}
\frac{\lambda(\Lambda)}{4\pi}\leq 1
\end{equation}
This upper bound depends on the scale $\Lambda$ and very mildly
on the top-quark mass $M_t$ through its influence on the
renormalization group equations of $\lambda$. We have plotted in
Fig.~\ref{lepp} this upper bound for different values of the high scale
$\Lambda$, along with the corresponding stability bounds.
\begin{figure}[hbt]
\centerline{
\psfig{figure=lepp.ps,height=10cm,width=17cm,angle=90}}
\caption{Perturbativity and stability bounds on the SM Higgs
boson. $\Lambda$ denotes the energy scale where the particles
become strongly interacting.}
\label{lepp}
\end{figure}
\subsection{\bf A light Higgs can {\it measure} the scale of New Physics}
From the bounds on $M_H(\Lambda)$ previously obtained
(see Fig.~\ref{fval6})
one can easily deduce that
a measurement of $M_H$ might provide an
{\bf upper bound} (below the Planck scale) on the
scale of new physics provided that
\begin{equation}
\label{final}
M_t>\frac{M_H}{2.25\; {\rm GeV}}+123\; {\rm GeV}
\end{equation}
Thus, the present
experimental bound from LEP, $M_H>67$ GeV, would imply, from
(\ref{final}), $M_t>153$ GeV, which is fulfilled
by experimental detection of the
top~\cite{top}. Even non-observation of the Higgs at
LEP2 (i.e. $M_H\stackrel{>}{{}_\sim} 95$ GeV), would
leave an open window ($M_t\stackrel{>}{{}_\sim} 165$ GeV)
to the possibility that a future Higgs detection
at LHC could lead to an upper bound on $\Lambda$. Moreover, Higgs
\begin{figure}[htb]
\centerline{
\psfig{figure=fval6.ps,height=7.5cm,width=7cm,bbllx=5.cm,bblly=2.5cm,bburx=14.5cm,bbury=15.5cm}}
\caption{SM lower bounds on $M_H$ from
metastability requirements as a function of
$\Lambda$ for different values of $M_t$.}
\label{fval6}
\end{figure}
detection at
LEP2 would put an upper bound on the scale of new physics. Taking,
for instance, $M_H\stackrel{<}{{}_\sim} 95$
GeV and 170 GeV $< M_t< $ 180 GeV, then $\Lambda\stackrel{<}{{}_\sim} 10^7$ GeV, while for
180 GeV $< M_t <$ 190 GeV, $\Lambda\stackrel{<}{{}_\sim} 10^4$
GeV, as can be deduced from Fig.~\ref{fval6}. Finally, using as upper
bound for the top-quark mass $M_t<180$ GeV [Ref.~\cite{top}] we obtain
from (\ref{final}) that only if the condition
\begin{equation}
M_h>128\ {\rm GeV}
\end{equation}
is fulfilled, the SM can be a consistent theory up to the Planck
scale, where gravitational effects can no longer be neglected.
\section{Higgs bosons in the Minimal Supersymmetric Standard Model}
The Minimal Supersymmetric Standard Model
(MSSM)~\cite{susy} is the best
motivated extension of the SM where some of their theoretical problems
(e.g. the hierarchy problem inherent with the fact that the
SM cannot be considered as a fundamental theory for energies
beyond the Planck scale) find at least a technical solution~\cite{Carlos}.
In this lecture we will concentrate on the Higgs
sector of the MSSM that is being the object of experimental
searches at present accelerators (LEP), and will equally be one of the
main goals at future colliders (LHC).
\subsection{\bf The Higgs sector in the Minimal Supersymmetric Standard
Model}
The Higgs sector of the MSSM~\cite{Hunter} requires two Higgs doublets, with
opposite hypercharges, as
\begin{equation}
\label{higgsmssm}
H_1 = \left(
\begin{array}{c}
H_1^0 \\
H_1^-
\end{array}
\right)_{-1/2}, \ \
H_2 = \left(
\begin{array}{c}
H_2^+ \\
H_2^0
\end{array}
\right)_{1/2}
\end{equation}
The reason for this duplicity is twofold. On the one hand it is
necessary to cancel the triangular anomalies generated by the
higgsinos. On the other hand it is required by the structure of
the supersymmetric theory to give masses to all fermions.
The most general gauge invariant scalar potential is given,
for a general two-Higgs doublet model, by:
\begin{eqnarray}
\label{higgs2}
V& = & m_1^2 |H_1|^2+m_2^2|H_2|^2+(m_3^2 H_1 H_2+h.c.)
+\frac{1}{2}\lambda_1(H_1^\dagger H_1)^2\nonumber\\
&&+\frac{1}{2}\lambda_2(H_2^\dagger
H_2)^2+\lambda_3(H_1^\dagger H_1)(H_2^\dagger H_2)
+\lambda_4(H_1 H_2)(H_1^\dagger H_2^\dagger) \\
&&+\left\{\frac{1}{2}\lambda_5(H_1 H_2)^2+
\left[\lambda_6(H_1^\dagger H_1)+\lambda_7(H_1^\dagger
H_2^\dagger) \right](H_1 H_2)+h.c.\right\} \nonumber
\end{eqnarray}
However, supersymmetry provides the following tree-level
relations between the previous couplings. The non-vanishing ones are:
\begin{equation}
\label{lambdastree}
\lambda_1 = \lambda_2=\frac{1}{4}(g^2+g'^2), \
\lambda_3 = \frac{1}{4}(g^2-g'^2),\
\lambda_4 = -\frac{1}{4}g^2
\end{equation}
Replacing (\ref{lambdastree}) into (\ref{higgs2}) one obtains
the tree-level potential of the MSSM, as:
\begin{eqnarray}
\label{potmssm}
V_{\rm MSSM}& = & m_1^2 H_1^\dagger H_1+m_2^2 H_2^\dagger H_2
+m_3^2(H_1 H_2+h.c.) \\
&&+\frac{1}{8}g^2\left(H_2^\dagger \vec{\sigma} H_2+
H_1^\dagger\vec{\sigma}H_1\right)^2
+\frac{1}{8}g'^2\left(H_2^\dagger H_2-H_1^\dagger H_1\right)^2
\nonumber
\end{eqnarray}
This potential, along with the gauge and Yukawa couplings in the
superpotential,
\begin{equation}
\label{superp}
W=h_u Q\cdot H_2 U^c+h_d Q\cdot H_1 D^c+ h_\ell L\cdot H_1 E^c
+\mu H_1\cdot H_2
\end{equation}
determine all couplings and masses (at the tree-level) of the
Higgs sector in the MSSM.
After gauge symmetry breaking,
\begin{eqnarray}
v_1 & = & \langle {\rm Re}\; H_1^0 \rangle \nonumber \\
v_2 & = & \langle {\rm Re}\; H_2^0 \rangle
\end{eqnarray}
the Higgs spectrum contains one neutral CP-odd Higgs $A$
(with mass $m_A$, that will be taken as a free parameter)
\begin{equation}
A=\cos\beta\;{\rm Im}H_2^0+\sin\beta\;{\rm Im}H_1^0
\end{equation}
and one neutral Goldstone $\chi^0$
\begin{equation}
\chi^0=-\sin\beta\;{\rm Im}H_2^0+\cos\beta\;{\rm Im}H_1^0
\end{equation}
with $\tan\beta=v_2/v_1$. It also contains one complex charged
Higgs $H^\pm$,
\begin{equation}
H^+=\cos\beta\; H_2^+ +\sin\beta\;(H_1^-)^*
\end{equation}
with a (tree-level) mass
\begin{equation}
\label{masapm}
m_{H^\pm}^2=M_W^2+m_A^2
\end{equation}
and one charged Goldstone $\chi^\pm$,
\begin{equation}
\chi^+=-\sin\beta\; H_2^+ +\cos\beta\;(H_1^-)^*.
\end{equation}
Finally the Higgs spectrum contains two CP-even neutral Higgs
bosons $H,{\cal H}$ (the light and the heavy mass eigenstates)
which are linear combinations of Re~$H_1^0$ and
Re~$H_2^0$, with a mixing angle $\alpha$ given by
\begin{equation}
\label{mixingHiggs}
\cos 2\alpha=-\cos2\beta\;\frac{m_A^2-M_Z^2}{m_{\cal H}^2-m_H^2}
\end{equation}
and masses
\begin{equation}
\label{masahH}
m^2_{H,{\cal H}}=\frac{1}{2}\left[
m_A^2+M_Z^2\mp\sqrt{(m_A^2+M_Z^2)^2-4m_A^2M_Z^2\cos^2 2\beta}
\right]
\end{equation}
\subsubsection{The Higgs couplings}
All couplings in the Higgs sector are functions of the gauge
($G_F,g,g'$) and Yukawa couplings, as in the SM, and of the
previously defined mixing angles $\beta,\alpha$.
Some relevant couplings are contained in Table~2
where all particle momenta, in squared brackets, are incoming.
\begin{center}
\begin{tabular}{||c|c||}\hline
Vertex & Couplings \\ \hline
& \\
$(H,{\cal H})WW $ & $(\phi WW)_{\rm
SM}[\sin(\beta-\alpha),\cos(\beta-\alpha)]$ \\
& \\
$(H,{\cal H})ZZ $ & $(\phi ZZ)_{\rm
SM}[\sin(\beta-\alpha),\cos(\beta-\alpha)]$ \\
& \\
$(H,{\cal H},A)[p]W^\pm H^\mp [k] $ & $\mp i\frac{g}{2}(p+k)^\mu
[\cos(\beta-\alpha), -\sin(\beta-\alpha),\pm i]$ \\
& \\
$(H,{\cal H},A)u\bar{u} $ & $(\phi u\bar{u})_{\rm
SM}[{\displaystyle \frac{\cos\alpha}{\sin\beta},
\frac{\sin\alpha}{\sin\beta} , -i\gamma_5 \cot\beta]} $ \\
& \\
$(H,{\cal H},A)d\bar{d} $ &$(\phi d\bar{d})_{\rm
SM}[{\displaystyle -\frac{\sin\alpha}{\cos\beta},
\frac{\cos\alpha}{\cos\beta}, -i\gamma_5 \tan\beta] } $ \\
& \\
$H^- u\bar{d} $
& $ {\displaystyle \frac{ig}{2\sqrt{2}M_W}
[(m_d\tan\beta+m_u\cot\beta) - (m_d
\tan\beta -m_u \cot\beta)\gamma_5] } $ \\
& \\
$ H^+ \bar{u} d $ & $ {\displaystyle \frac{ig}{2\sqrt{2}M_W}
[(m_d\tan\beta+m_u\cot\beta) + (m_d
\tan\beta -m_u \cot\beta)\gamma_5] } $ \\
& \\
$(\gamma,Z)H^+[p]H^- [k] $ & $ {\displaystyle
-i(p+k)^\mu\left[e,g\frac{\cos
2\theta_W}{2 \cos\theta_W}\right] } $ \\
& \\
$h[p] A [k] Z $ & ${\displaystyle
-\frac{e}{2\cos\theta_W\sin\theta_W} (p+k)^\mu
\cos(\beta-\alpha) } $ \\
& \\ \hline
\end{tabular}
\end{center}
\vspace{1cm}
\begin{center}
Table 2
\end{center}
\subsubsection{Higgs production at LEP2}
The main mechanisms for production of
neutral Higgs particles at $e^+e^-$
colliders, at the LEP2 energies, are~\cite{LEP2}:
\begin{itemize}
\item
HIGGS-STRAHLUNG: $e^+e^-\rightarrow ZH$, where the Higgs boson
is radiated off the virtual $Z$-boson line. This process is
identical to the SM Higgs-strahlung. [See Fig.~\ref{diag1}.]
\item
ASSOCIATED PAIR PRODUCTION: $e^+ e^- \rightarrow HA$,
$e^+ e^-\rightarrow H^\pm H^\mp$.
The production of $HA$ is mediated by a $Z$-boson in the
s-channel (it uses the coupling hAZ in Table~2). The
production of $H^\pm H^\mp$ can be mediated by either $\gamma$
and $Z$, using the $(\gamma,Z)H^\pm H^\mp$ vertex in Table~2.
\end{itemize}
A detailed analysis of these processes for LEP2 can be found in
Ref.~\cite{LEP2}.
\subsubsection{Higgs production at LHC}
The main mechanisms for production of
neutral Higgs bosons at $pp$ colliders,
at the LHC energies, are~\cite{Ferrando}:
\begin{itemize}
\item
GLUON-GLUON FUSION: $gg\rightarrow (H,{\cal H},A)$,
where two gluons in the sea
of the protons collide through a loop of top-quarks, bottom-quarks,
stops and sbottoms which
subsequently emit a Higgs boson. The contribution of a (s)bottom
loop is only relevant for large values of $\tan\beta$.
[Figs.~\ref{diag4} and \ref{diag5}, where curly lines are
gluons, internal fermion lines quarks $t$ and $b$, internal
boson (dashed) lines squarks $\tilde{t}$ and $\tilde{b}$ and the
dashed line is a Higgs boson $H$, ${\cal H}$ or $A$.]
\begin{figure}[hbt]
\centerline{
\psfig{figure=diag5.ps,width=5cm,bbllx=4.75cm,bblly=10.5cm,bburx=14.25cm,bbury=16.5cm}}
\caption{Gluon-gluon fusion process for Higgs production with a squark loop.}
\label{diag5}
\end{figure}
\item
WW (ZZ)-FUSION: $W^\pm W^\mp \rightarrow (H,{\cal H},A)$,
$ZZ \rightarrow (H,{\cal H},A)$, where the Higgs
boson is formed in the fusion of $WW (ZZ)$, the virtual $W (Z)$'s
being radiated off a quark in the proton beam. [See
Fig.~\ref{diag2} where the external dashed line corresponds to a
Higgs boson $H$, ${\cal H}$ or $A$.]
\item
HIGGS STRAHLUNG: $q \bar{q}\rightarrow Z (H,{\cal H},A)$,
$q \bar{q}'\rightarrow W (H,{\cal H},A)$
where the corresponding Higgs boson
is radiated off the virtual $Z(W)$-boson line. [See
Fig.~\ref{diag1}, where the dashed line is a Higgs boson, $H$,
${\cal H}$ or $A$.]
\item
ASSOCIATED PRODUCTION WITH $t\bar{t},b\bar{b}$:
$gg\rightarrow t \bar{t} (H,{\cal H},A)$,
$gg\rightarrow b \bar{b} (H,{\cal H},A)$
where the gluons from the proton sea exchange a top
(bottom)-quark in the
t-channel, the exchanged top (bottom) quark emitting a Higgs boson.
[See Fig.~\ref{diag3} where the curly lines are gluons, the
fermion line a $t$ or $b$ quark and the dahsed line a Higgs
boson $H$, ${\cal H}$ or $A$.]
\end{itemize}
The production of a charged Higgs boson is through the process
$gg\rightarrow t\bar{t}$, where the gluons exchange a top-quark
in the t-channel, and subsequent decay $t\rightarrow b H^+$.
This process is available only when $M_t>m_{H^+}+M_b$. Otherwise
the detection of the charged Higgs is much more difficult.
[Fig.~\ref{diag6} where curly lines are gluons, the fermion
exchanged between the gluons a $t$ quark, the external fermions
$b$ quarks and the external bosons (dashed) are $H^\pm$.]
A complete analysis of the different production channels can be found,
e.g. in Ref.~\cite{FabioLHC}.
\begin{figure}[hbt]
\centerline{
\psfig{figure=diag6.ps,width=5cm,bbllx=5.75cm,bblly=12.cm,bburx=15.25cm,bbury=19.5cm}}
\caption{Charged higgs production process.}
\label{diag6}
\end{figure}
\subsubsection{Higgs decays}
Assuming R-parity conservation, two-body decays should be into
SM particles, or two supersymmetric partners if the
supersymmetric spectrum is kinematically accesible. Assuming the
supersymmetric spectrum to be heavy enough (a useful working
hypothesis), the decays are always into SM particles.
The main decay modes of the Higgs boson are then:
\begin{itemize}
\item
$(H,{\cal H},A)\rightarrow b\bar{b},c\bar{c},\tau^-\tau^+,t\bar{t},gg,
\gamma\gamma,W^* W^*, Z^* Z^*,Z\gamma$, which is very similar
to the corresponding SM modes.
\item
$H\rightarrow AA$.
\item
${\cal H}\rightarrow hh,AA,ZA$.
\item
$A\rightarrow ZH$:
\item
$H^+\rightarrow c\bar{s},\tau^+\nu_\tau,t\bar{b},W^+ H$.
\end{itemize}
A complete analysis of the decay modes in the MSSM can be found in
Ref.~\cite{LEP2}, for LEP2, and~\cite{FabioLHC} for LHC.
\subsection{\bf Radiative corrections}
All previous Higgs production and decay processes depend on the
Higgs masses $m_H,m_{\cal H},m_A,m_{H^\pm}$, and couplings
$g,g',G_F,\tan\beta,\cos\alpha,h_f,\lambda_1,\dots,\lambda_7$.
We have already given their tree-level values. In particular,
the mass spectrum satisfies at tree-level the following
relations:
\begin{eqnarray}
\label{treerel}
m_H & < & M_Z|\cos 2\beta| \nonumber \\
m_H & < & m_A \\
m_{H^\pm} & > & M_W \nonumber
\end{eqnarray}
which could have a number of very important phenomenological
implications, as it is rather obvious. However, it was
discovered that radiative corrections are important and can
spoil the above tree level relations with a great
phenomenological relevance. A detailed knowledge of radiatively
corrected couplings and masses is necessary for experimental
searches in the MSSM.
The {\bf effective potential} methods to compute the (radiatively
corrected) Higgs mass spectrum in the
MSSM are useful since they allow to {\bf resum}
(using Renormalization Group (RG) techniques) LL,
NTLL,..., corrections to {\bf all orders}
in perturbation theory. These methods~\cite{Effpot,EQ}, as well as the
diagrammatic methods~\cite{Diagram} to compute the Higgs mass spectrum
in the MSSM, were first developed in the early nineties.
Effective potential methods are based on the {\bf run-and-match}
procedure by which all dimensionful and dimensionless couplings
are running with the RG scale, for scales greater than the
masses involved in the theory. When the RG scale
equals a particular mass threshold, heavy fields decouple,
eventually leaving threshold effects in order to match the
effective theory below and above the mass threshold. For
instance, assuming a common soft supersymmetry breaking mass
for left-handed and right-handed stops and sbottoms,
$M_S\sim m_Q\sim m_U\sim m_D$, and assuming for the top-quark mass,
$m_t$, and for the CP-odd Higgs mass, $m_A$, the range
$m_t\leq m_A\leq M_S$, we have: for scales $Q\geq M_S$, the MSSM, for
$m_A\leq Q\leq M_S$ the two-Higgs doublet model (2HDM), and for
$m_t\leq Q\leq m_A$ the SM. Of course there are
thresholds effects at $Q=M_S$ to match the MSSM with the 2HDM, and
at $Q=m_A$ to match the 2HDM with the SM.
\begin{figure}[htb]
\centerline{
\psfig{figure=fval3.ps,height=7.5cm,width=7cm,bbllx=5.5cm,bblly=2.5cm,bburx=15.cm,bbury=15.5cm}}
\caption{Plot of $M_H$ as a function of $M_t$ for $\tan\beta\gg 1$
(solid lines), $\tan\beta=1$ (dashed lines), and $X_t^2=6 M_S^2$ (upper set),
$X_t=0$ (lower set). The experimental band from the CDF/D0 detection is also
indicated.}
\label{fval3}
\end{figure}
As we have said, the neutral Higgs sector of the MSSM contains,
on top of the CP-odd Higgs $A$, two CP-even Higgs mass
eigenstates, ${\cal H}$ (the heaviest one) and $H$ (the lightest one).
It turns out that the larger
$m_A$ the heavier the lightest Higgs $H$. Therefore the case
$m_A\sim M_S$ is, not only a great simplification since the effective
theory below $M_S$ is the SM, but also of great interest, since it
provides the upper bound on the mass of the lightest Higgs
(which is interesting for phenomenological purposes, e.g. at
LEP2). In this case the threshold correction at $M_S$ for the SM
quartic coupling $\lambda$ is:
\begin{equation}
\label{threshold}
\Delta_{\rm th}\lambda=\frac{3}{16\pi^2}h_t^4
\frac{X_t^2}{M_S^2}\left(2-\frac{1}{6}\frac{X_t^2}{M_S^2}\right)
\end{equation}
where $h_t$ is the SM top Yukawa coupling and
$X_t=(A_t-\mu/\tan\beta)$ is the mixing in the stop mass
matrix, the parameters $A_t$ and $\mu$ being the trilinear
soft-breaking coupling in the stop sector and the supersymmetric
Higgs mixing mass, respectively. The maximum of
(\ref{threshold}) corresponds to $X_t^2=6 M_S^2$ which provides
the maximum value of the lightest Higgs mass: this case will be
referred to as the case of maximal mixing.
We have plotted in Fig.~\ref{fval3} the lightest Higgs pole mass
$M_H$, where all NTLL corrections
are resummed to all-loop by the RG,
as a function of $M_t$~\cite{CEQR}. From Fig.~\ref{fval3} we
can see that the present experimental band from CDF/D0 for the
top-quark mass requires $M_H\stackrel{<}{{}_\sim} 140$ GeV, while if we fix
$M_t=170$ GeV, the upper bound $M_H\stackrel{<}{{}_\sim} 125$ GeV
follows. It goes without saying
that these figures are extremely relevant for MSSM Higgs searches
at LEP2.
\subsubsection{An analytical approximation}
We have seen~\cite{CEQR} that,
since radiative corrections are minimized for scales $Q\sim m_t$,
when the LL RG improved Higgs mass expressions are
evaluated at the top-quark mass scale, they reproduce the NTLL value
with a high level of accuracy, for any value of $\tan\beta$ and the
stop mixing parameters~\cite{CEQW}
\begin{equation}
\label{relmasas}
m_{H,LL}(Q^2\sim m_t^2)\sim m_{H,NTLL}.
\end{equation}
Based on the above observation, we can work out a very accurate
analytical approximation to $m_{H,NTLL}$ by just keeping two-loop
LL corrections at $Q^2=m_t^2$, i.e. corrections of order $t^2$, where
$t=\log(M_S^2/m_t^2)$.
Again the case $m_A\sim M_S$ is the simplest, and very illustrative,
one. We have found~\cite{CEQW,HHH} that, in the absence of mixing
(the case $X_t=0$) two-loop corrections resum in the one-loop
result shifting the energy scale from $M_S$ (the tree-level scale)
to $\sqrt{M_S\; m_t}$. More explicitly,
\begin{equation}
\label{resum}
m_H^2=M_Z^2 \cos^2 2\beta\left(1-\frac{3}{8\pi^2}h_t^2\; t\right)
+\frac{3}{2\pi^2 v^2}m_t^4(\sqrt{M_S m_t}) t
\end{equation}
where $v=246.22$ GeV.
In the presence of mixing ($X_t\neq 0$), the run-and-match procedure
yields an extra piece in the SM effective potential
$\Delta V_{\rm th}[\phi(M_S)]$ whose second derivative gives an
extra contribution to the Higgs mass, as
\begin{equation}
\label{Deltathm}
\Delta_{\rm th}m_H^2=\frac{\partial^2}{\partial\phi^2(t)}
\Delta V_{\rm th}[\phi(M_S)]=
\frac{1}{\xi^2(t)}
\frac{\partial^2}{\partial\phi^2(t)}
\Delta V_{\rm th}[\phi(M_S)]
\end{equation}
which, in our case, reduces to
\begin{equation}
\label{masthreshold}
\Delta_{\rm th}m_H^2=
\frac{3}{4\pi^2}\frac{m_t^4(M_S)}{v^2(m_t)}
\frac{X_t^2}{M_S^2}\left(2-\frac{1}{6}\frac{X_t^2}{M_S^2}\right)
\end{equation}
We have compared our analytical approximation~\cite{CEQW}
with the numerical NTLL result~\cite{CEQR} and found a difference
$\stackrel{<}{{}_\sim} 2$ GeV for all values of supersymmetric parameters.
\begin{figure}[ht]
\centerline{
\psfig{figure=fval4.ps,height=9.5cm,width=14cm,angle=90} }
\caption[0]
{The neutral ($H,{\cal H}\equiv H_h$ in the figure)
and charged ($H^+$) Higgs mass spectrum
as a function of the CP-odd Higgs mass $m_A$ for
a physical top-quark mass $M_t =$ 175 GeV and $M_S$ = 1 TeV, as
obtained from the one-loop improved RG evolution
(solid lines) and the analytical formulae (dashed lines).
All sets of curves correspond to
$\tan \beta=$ 15 and large squark mixing, $X_t^2 = 6 M_S^2$
($\mu=0$).}
\label{fval4}
\end{figure}
The case $m_A<M_S$ is a bit more complicated since the effective theory
below the supersymmetric scale $M_S$ is the 2HDM. However since radiative
corrections in the 2HDM are equally dominated by the top-quark, we can
compute analytical expressions based upon the LL approximation
at the scale $Q^2\sim m_t^2$.
Our approximation~\cite{CEQW} differs from
the LL all-loop numerical resummation by $\stackrel{<}{{}_\sim} 3$ GeV, which we
consider the uncertainty inherent in the theoretical calculation,
provided the mixing is moderate and, in particular, bounded by the
condition,
\begin{equation}
\label{condicion}
\left|\frac{m^2_{\;\widetilde{t}_1}-m^2_{\;\widetilde{t}_2}}
{m^2_{\;\widetilde{t}_1}+m^2_{\;\widetilde{t}_2}}\right|\stackrel{<}{{}_\sim} 0.5
\end{equation}
where $\widetilde{t}_{1,2}$ are the two stop mass eigenstates.
In Fig.~\ref{fval4} the Higgs mass spectrum is plotted versus $m_A$.
\subsubsection{Threshold effects}
There are two possible caveats in the
analytical approximation we have just
presented: {\bf i)} Our expansion parameter $\log(M_S^2/m_t^2)$
does not behave properly in the supersymmetric limit $M_S\rightarrow 0$,
where we should recover the tree-level result. {\bf ii)} We have expanded
the threshold function $\Delta V_{\rm th}[\phi(M_S)]$ to order $X_t^4$.
In fact keeping the whole threshold function $\Delta V_{\rm th}[\phi(M_S)]$
we would be able to go to larger values of $X_t$ and to evaluate the
accuracy of the approximation (\ref{threshold}) and (\ref{masthreshold}).
Only then we will be able to
check the reliability of the maximum value of the
lightest Higgs mass (which corresponds to the maximal mixing) as provided
in the previous sections.
This procedure has been properly followed~\cite{CEQW,CQW} for
the most general case $m_Q\neq m_U\neq m_D$.
We have proved that keeping the exact threshold function
$\Delta V_{\rm th}[\phi(M_S)]$, and properly running its value from the
high scale to $m_t$ with the corresponding anomalous dimensions as in
(\ref{Deltathm}), produces two effects: {\bf i)} It makes a resummation
from $M_S^2$ to $M_S^2+m_t^2$ and generates as (physical) expansion
parameter $\log[(M_S^2+m_t^2)/m_t^2]$. {\bf ii)} It generates a whole
threshold function $X_t^{\rm eff}$ such that (\ref{masthreshold})
becomes
\begin{equation}
\label{masthreshold2}
\Delta_{\rm th}m_H^2=
\frac{3}{4\pi^2}\frac{m_t^4[M_S^2+m_t^2]}{v^2(m_t)}
X_t^{\rm eff}
\end{equation}
and
\begin{equation}
\label{desarrollo}
X_t^{\rm eff}=\frac{X_t^2}{M_S^2+m_t^2}
\left(2-\frac{1}{6}\frac{X_t^2}{M_S^2+m_t^2}\right)+\cdots
\end{equation}
The numerical calculation shows~\cite{CQW} that $X_t^{\rm eff}$
has the maximum very close to $X_t^2=6(M_S^2+m_t^2)$,
what justifies
the reliability of previous upper bounds on the lightest Higgs mass.
\subsection{\bf The case of obese supersymmetry}
We will conclude this lecture with a very interesting case,
where the Higgs sector of the
MSSM plays a key role in the detection of supersymmetry.
It is the case where all supersymmetric particles are superheavy
\begin{equation}
M_S \sim 1-10\ {\rm TeV}
\end{equation}
and escape detection at LHC.
In the Higgs sector ${\cal H},A,H^\pm$
decouple, while the $H$ couplings go the SM $\phi$ couplings
\begin{equation}
HXY\longrightarrow (\phi XY)_{\rm SM}
\end{equation}
as $\sin(\beta-\alpha)\rightarrow 1$, or are indistinguisable
from the SM ones
\begin{eqnarray}
h_u\sin\beta & \equiv & h_u^{\rm SM} \nonumber \\
h_{d,\ell}\cos\beta & \equiv & h_{d,\ell}^{\rm SM}
\end{eqnarray}
In this way the $\tan\beta$ dependence of the couplings, either
disappears or is absorbed in the SM couplings.
\begin{figure}[htb]
\centerline{
\psfig{figure=fval5.ps,height=7.5cm,width=7cm,bbllx=5.cm,bblly=2.cm,bburx=14.5cm,bbury=15cm}}
\caption{SM lower bounds on $M_H$ (thick lines) as a function of
$M_t$, for $\Lambda=10^{19}$ GeV, from metastability requirements,
and upper bound on the lightest Higgs boson mass in the MSSM
(thin lines) for $M_S=1$ TeV.}
\label{fval5}
\end{figure}
However, from the previous sections it should be clear
that the Higgs and top mass measurements
could serve to discriminate between the SM and its extensions,
and to provide information about the
scale of new physics $\Lambda$.
In Fig.~\ref{fval5}
we give the SM lower bounds on
$M_H$ for $\Lambda\stackrel{>}{{}_\sim} 10^{15}$ GeV (thick lines) and
the upper bound on the mass of the
lightest Higgs boson in the MSSM (thin lines)
for $M_S\sim 1$ TeV. Taking,
for instance, $M_t=180$ GeV, close to the
central value recently reported by CDF+D0~\cite{top},
and $M_H\stackrel{>}{{}_\sim} 130$ GeV, the SM is
allowed and the MSSM is excluded. On the other hand,
if $M_H\stackrel{<}{{}_\sim} 130$ GeV, then the MSSM is
allowed while the SM is excluded. Likewise there
are regions where the SM is excluded, others
where the MSSM is excluded and others where both are permitted or
both are excluded.
\section{Conclusion}
To conclude, we can say that the search of the Higgs boson at present
and future colliders is, not only an experimental challenge, being the
Higgs boson the last missing ingredient of the Standard Model, but
also a theoretically appealing question from the
more fundamental point of view of
physics beyond the Standard Model. In fact, if we are lucky enough and
the Higgs boson is detected soon (preferably at LEP2) and {\it light},
its detection
might give sensible information about the possible existence of
new physics. In that case, the
experimental search of the new physics should be urgent and compelling,
since the existence of new phenomena
might be necessary for our present understanding of the physics
for energies at reach in the planned accelerators.
\section*{Acknowledgments}
Work supported in part by
the European Union (contract CHRX-CT92-0004) and
CICYT of Spain (contract AEN95-0195).
I wish to thank my collaborators in the subjects whose results
are reported in the present lectures: M.~Carena, J.A.~Casas,
J.R.~Espinosa, A.~Riotto, C.~Wagner and F.~Zwirner. I also want
to thank A.~Riotto for his help in drawing some of the diagrams
contained in this paper.
\section*{References}
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\section{Status of 2HDM.}
\subsection{Introduction.}
The mechanism of spontaneous symmetry breaking proposed as
the source of mass for the gauge and fermion fields in the Standard
Model (SM) leads to a neutral scalar particle,
the minimal Higgs boson. According to the LEP I data,
based on the Bjorken process $e^+e^- \rightarrow H Z^*$,
it should be heavier than 66 GeV\cite{hi},
also
the MSSM neutral Higgs particles have been
constrained by LEP1 data to be heavier than
$\sim$ 45 GeV \cite{lep,susy,hi}. The general two Higgs doublet
model (2HDM) may yet accommodate a very light ($ \:\raisebox{-0.5ex}{$\stackrel{\textstyle<}{\sim}$}\: 45 \; \rm GeV$)
neutral scalar $h$ {\underline {or}} a pseudoscalar $A$ as long as
$M_h+M_A \:\raisebox{-0.5ex}{$\stackrel{\textstyle>}{\sim}$}\: M_Z$~\cite{lep}.
The minimal extension of the Standard Model is to include
a second Higgs doublet to the symmetry breaking
mechanism. In two Higgs doublet models
the observed Higgs sector is enlarged to five scalars: two
neutral Higgs scalars (with masses $M_H$ and $M_h$ for heavier and
lighter particle, respectively), one neutral pseudoscalar
($M_A$), and a pair of charged Higgses ($M_{H^+}$ and $M_{H^-}$).
The
neutral Higgs scalar couplings to quarks, charged leptons and gauge
bosons are
modified with respect to analogous couplings in SM by factors that
depend on additional parameters : $\tan\beta$, which is
the ratio of the vacuum expectation values of the Higgs doublets
$v_2/v_1$,
and the mixing angle in the neutral Higgs sector $\alpha$. Further,
new couplings appear, e.g. $Zh (H) A$ and $ZH^+ H^-$.
In this paper we will focus on the appealing version of the models
with two doublets ("Model II") where one Higgs doublet
with vacuum expectation value $v_2$ couples only to the "up"
components
of fermion doublets while the other one couples to the "down"
components \cite{hunter}.
{{In particular, fermions couple to the pseudoscalar $A$
with a strength proportional to $(\tan \beta)^{\pm1}$
whereas the coupling of the fermions to the scalar $h$
goes as $\pm(\sin \alpha/\cos \beta)^{\pm1}$, where the sign
$\pm$ corresponds to isospin $\mp$1/2 components}}.
In such model FCNC processes are absent
and the $\rho $ parameter retains its SM value at the tree level.
Note that in such scenario
the large ratio $v_2/v_1 \sim m_{top}/m_b\gg 1$ is naturally
expected.
The well known supersymmetric model (MSSM) belongs to this class.
In MSSM the relations among the parameters required by the
supersymmetry appear, leaving only two parameters free
(at the tree level) e.g. $M_A$ and $\tan \beta$.
In general case, which we call the general 2 Higgs Doublet Model
(2HDM), masses and parameters $\alpha$ and $\beta$
are not constrained by the model.
Therefore the same experimental data may lead to very distinct
consequences depending on which
version of two Higgs doublet extension of SM,
supersymmetric or nonsupersymmetric, is considered.
\subsection {Present constraints on 2HDM from LEP I.}
Important constraints on the
parameters of two Higgs doublet extensions of SM were obtained
in the precision measurements at LEP I.
The current mass limit on
{\underline {charged}} Higgs boson $M_{H^{\pm}}$=
44 GeV/c was obtained at LEP I \cite{sob}
from process $Z \rightarrow H^+H^-$,
which is { {independent}}
on the parameters $\alpha$ and $\beta$.
(Note that in
the MSSM version one expect
$M_{H^{\pm}} > M_W$).
For {\underline {neutral}} Higgs particles $h$ and
$A$ there are two
main and complementary sources of information at LEP I. One
is the Bjorken processes $Z \rightarrow Z^*h $
which constrains $g_{hZZ}^2 \sim \sin^2(\alpha-\beta)$,
for $M_h$ below 50-60 GeV..
The second process is $Z\rightarrow hA$,
constraining the $g_{ZhA}^2 \sim cos^2(\alpha-\beta)$
for $M_h+M_A{\stackrel{<}{\sim}} M_Z$
{\footnote {
The off shell production could also be included,
$ {\it e.g.}$ as in \cite{susy}.}}.
This Higgs pair production contribution depends
also on the masses $M_h$, $M_A$ and $M_Z$.
Results on $\sin^2(\alpha-\beta)$ and $\cos^2(\alpha-\beta)$
can be translated into
the limits on neutral Higgs bosons masses $M_h$ and $M_A$.
In the MSSM, due to relations among parameters,
the above data allow to draw limits for the masses
of {\underline {individual}}
particles: $M_h\ge 45$ GeV for any $\tan \beta $
and $M_A \ge$ 45 GeV for $\tan\beta \ge$1 \cite{susy,hi}.
In the general 2HDM the implications are quite different, here
the large portion of the ($M_h$,$M_A$) plane,
where {\underline {both}} masses are in the range between
0 and $\sim$50 GeV, is excluded \cite{lep}.
The third basic process in search of a neutral
Higgs particle at LEP I is the Yukawa process, $ {\it i.e.}$
the bremsstrahlung production of the neutral Higgs
boson $h(A)$ from the heavy fermion,
$e^+e^- \rightarrow f {\bar f} h(A)$, where $f$ means here
{\it b} quark or $\tau$
lepton.
This process plays a very important role since
it constrains the production of a very light pseudoscalar
even if the pair production is forbidden kinematically,
$ {\it i.e.}$ for $M_h+M_A>M_Z$ {\footnote{neglecting the off shell production}}.
It allows also to look for a light scalar, being an additional,
and in case of $\alpha=\beta$ the most important, source of information.
The importance of this process was stressed in many papers\cite{pok,gle},
the recent discussion of the potential of the Yukawa process
is presented in Ref.\cite{kk}.
{{ New analysis of the Yukawa process by
ALEPH collaboration \cite{alef} led to
the exclusion plot (95\%) on the $\tan \beta$ versus the
pseudoscalar mass, $M_A$. (Analysis by L3 collaboration
is also in progress { \cite{l3prep}}.).
It happened that obtained limits are rather weak
{\footnote{Note, that the obtained limits
are much weaker than the limits estimated in Ref.
\cite{kk}.}},
allowing for the existence of a light $A$ with mass below 10 GeV
with $\tan \beta$ = 20--30 , for $M_A$=40 GeV $\tan \beta$ till 100 is allowed
!
For mass range above 10 GeV,
similar exclusion limits should in principle hold also for a
scalar $h$ with
the replacement in coupling $\tan \beta\rightarrow \sin\alpha/\cos\beta$.
Larger differences one would expect however
in region of lower mass, where the production
rate at the same value of coupling
for the scalar is considerably larger than
for the pseudoscalar and therefore more
stringent limits should be obtained \cite{kk}.
\subsection {The 2HDM with a light Higgs particle.}
In light of the above results from precision experiments
at LEP I
there is still the possibility of the
existence of one
light neutral Higgs particle with mass below $\sim$ 40--50 GeV.
As far as other experimental data,
especially from low energy measurements, are concerned
they do not contradict
this possibility as they
cover only part of the parameter space of 2HDM, moreover
some of them like the Wilczek process
have large theoretical uncertainties
both due to the QCD and relativistic corrections \cite{wil,hunter}
(see also discussion in \cite{bk,ames}).
In following we will study the 2HDM assuming
that one light Higgs particle may exist.
Moreover we will assume according to LEP I data
the following mass relation between the lightest
neutral Higgs particles: $M_h+M_A \ge M_Z$.
We specify the model further by choosing particular
values for the parameters $\alpha$ and $\beta$
within the present limits from
LEP I. Since $\sin(\alpha-\beta)^2$ was found \cite{lep,hi}
to be
smaller than 0.1 for the $0{\stackrel{<}{\sim}} M_h{\stackrel{<}{\sim}}$ 50 GeV,
and even below 0.01 for a lighter scalar,
we simply take $\alpha=\beta$.
It leads to equal in
strengths of the coupling of fermions to scalars and pseudoscalars.
For the scenario with
large $\tan\beta \sim {\cal O}(m_t/m_b)$ large
enhancement
in the coupling of both $h$ and $A$ bosons to the down-type
quarks and leptons is expected.
As we described above the existing
limits from LEP I for
a light neutral Higgs scalar/pseudoscalar
boson in 2HDM are rather weak.
Therefore it is extremely important to check if more stringent
limits can be obtained from other measurements.
In Sec.2
we present how one can obtained
the limits on the parameters of the 2HDM from current
precision $(g-2)$ for muon data\cite{pres},
also
the potential of the future E821 experiment \cite{fut} with the accuracy
expected to be more than 20 times better is discussed.
(See Ref.\cite{g22} for details.)
Note that in \cite{g22} we took into account
the full contribution from 2HDM, i.e. exchanges of
$h$, $A$ and $H^{\pm}$ bosons incorporating the present
constraints on Higgs bosons masses from LEP I.
In this talk we present limits on $\tan \beta$ which
can be obtained in a simple
approach (Ref.\cite{ames,deb12} and also \cite{gle}), {\it i.e.} ~from the
individual $h$ or $A$ terms. This approach reproduces the full 2HDM
prediction up to say 30 GeV if the mass difference
between $h$ and $A$ is $\sim M_Z$, in wider range mass if
this difference is larger.
The possible exclusion/discovery potential of the gluon-gluon
fusion at $ep$ collider HERA \cite{bk,ames}(Sec.3)
and of the $\gamma \gamma$
collision at the suggested low energy LC (Sec.4)
will also be discussed {\cite{deb12}}.
In Sec.5 the combined exclusion plot (95 \% C.L.) is presented.
The search of a light neutral Higgs
particle in heavy ion collisions at HERA and LHC are discussed
elsewhere\cite{bol}.
\section{Constraints on the parameters of 2HDM from $(g-2)$.}
\subsection{Present limits.}
The present experimental data limits on $(g-2)$ for muon,
averaged over the sign of the muon electric charge, is given by \cite{data}:
$$a_{\mu}^{exp}\equiv{{(g-2)_{\mu}}\over{2}}=1~165 ~923~(8.4)\cdot 10^{-9}.$$
The quantity within parenthesis, $\sigma_{exp}$, refers to the uncertainty
in the last digit. The expected new high-precision E821 Brookhaven
experiment has design sensitivity of $\sigma_{exp}^{new}=
4\cdot 10^{-10}$ (later even 1--2 $\cdot 10^{-10}$, see Ref.\cite{czar})
instead of the above $84\cdot 10^{-10}$.
It is of great importance to reach similar accuracy in the theoretical
analysis.
The theoretical prediction of the Standard Model for this quantity
consists of the QED, hadronic and EW
contribution:
$$a_{\mu}^{SM}=a_{\mu}^{QED}+a_{\mu}^{had}+a_{\mu}^{EW}.$$
The recent SM calculations of $a_{\mu}$
are based on the QED results from \cite{qed}, hadronic contribution
obtained in
\cite{mar,mk,jeg,wort,ll} and \cite{hayakawa} and
the EW results from \cite{czar,kuhto}.
The uncertainties of these
contributions differ among themselves considerably
(see below and in Ref.\cite{nath,czar,jeg,g22}).
The main discrepancy
is observed for the hadronic contribution,
therefore we will mainly consider case A,
based on Refs.\cite{qed,ki,mar,mk,ll,czar},
with relatively small error in the hadronic
part. For comparison the results for case B (Refs.
\cite{ki,jeg,hayakawa,czar}) with the 2 times
larger error in the hadronic part is also displayed.
(We adopt here the notation from \cite{nath}.)
$$
\begin{array}{lrr}
case &~{\rm {A~[in}}~ 10^{-9}] &~{\rm {B~[in}}~ 10^{-9}] \\
\hline
{\rm QED} &~~~~~~~~1 ~165~847.06 ~(0.02)
&~~~~~~~~~~~~~~~~~1 ~165~847.06 ~(0.02) \\
{\rm had} & 69.70 ~(0.76) & 68.82 ~(1.54) \\
{\rm EW} & 1.51 ~(0.04) & 1.51 ~(0.04) \\
\hline
{\rm tot} &1~165~9 1 8.27 ~(0.76) & 1 ~165~9 1 7.39 ~(1.54)
\end{array}
$$
\vspace{0.5cm}
The room for a new physics is given basically
by the difference between the experimental data and theoretical SM
prediction: $a_{\mu}^{exp}-a_{\mu}^{SM}\equiv \delta a_{\mu}$.
{\footnote {However in the calculation of
$a_{\mu}^{EW}$ the (SM) Higgs scalar
contribution is included(see discussion in\cite{g22}).}}
Below the difference $\delta a_{\mu} $
for these two cases, A and B,
is presented together with
the error $\sigma$, obtained by adding the experimental
and theoretical errors in quadrature:
$$
\begin{array}{lcc}
case &~{\rm {A ~[in}} ~10^{-9}]
&~~~~~~~~~~~~~~~{\rm {B ~[in}}~10^{-9}] \\
\hline
\delta a_{\mu}(\sigma) &4.73 (8.43) &~~~~~~~~~~~~~~5.61 (8.54) \\
\hline
{\rm lim(95\%)} &-11.79\le\delta a_{\mu} \le 21.25 &~~~~~~~~~~~~~~-11.13\le\delta a_{\mu} \le 22.35\\
{\rm lim_{\pm}(95\%)} &-13.46\le\delta a_{\mu} \le 19.94&~~~~~~~~~~~~~~ -13.71\le\delta a_{\mu} \le 20.84
\end{array}
$$
\vspace{0.5cm}
One can see that at 1 $\sigma$ level the difference $\delta a_{\mu}$
can be of positive and negative negative sign.
For that beyond SM scenarios
in which both positive and negative
$\delta a_{\mu}$ may appear,
the 95\% C.L. bound can be calculated straightforward
(above denoted by $lim(95\%)$).
For the model where the contribution of
only {\underline {one}} sign
is physically accessible ($ {\it i.e.}$ positive or negative $\delta a_{\mu}$),
the other sign being unphysical, the 95\%C.L. limits
should be calculated in different way \cite{data}.
These limits calculated separately for the positive and
for the negative contributions
($lim_{\pm}$(95\%)),
lead to the shift in the
lower and upper
bounds by -1.3 $\cdot 10^{-9}$ up to -2.6 $\cdot 10^{-9}$
with respect to the standard (95\%) limits.
\subsection{Forthcoming data.}
Since the dominate uncertainty in $\delta a_{\mu}$
is due to the experimental error,
the role of the forthcoming E821 experiment is crucial
in testing the SM or probing a new physics.
The future accuracy of the $(g-2)_{\mu}$ experiment is expected to be
$\sigma^{new}_{exp}\sim0.4 \cdot 10^{-9}$ or better.
One expects also the improvement
in the calculation of the hadronic contribution
{\footnote {The improvement in the ongoing experiments at low energy
in expected as well.}} such
that the total uncertainty will be basically
due to the experimental error.
Below we will assume that
the accessible range for the beyond SM contribution,
in particular 2HDM with a light scalar or pseudoscalar,
would be smaller by factor 20 as compared with the present
$lim_{\pm}$95\% bounds.
So, we consider the following option for future
measurement (in $10^{-9}$):
$$
\delta a_{\mu}^{new} = 0.24, \hspace{0.5cm}
{\rm and}\hspace{0.5cm}
{\rm lim_{\pm}}^{new}(95\%): -0.69\le\delta a_{\mu} \le 1.00. $$
Assuming above bounds,
we discuss below the potential of future $(g-2)$ measurement
for the constraining the 2HDM.
\subsection{2HDM contribution to $(g-2)_{\mu}$.}
As we mentioned above the difference
between experimental and theoretical value
for the anomalous magnetic moment for muon
we ascribe to the 2HDM contribution, so
we take
$\delta a_{\mu}= a_{\mu}^{(2HDM)}$ and
$\delta a_{\mu}^{new} = a_{\mu}^{(2HDM)}$ for present and future
$(g-2)$ data, respectively.
To $ a_{\mu}^{(2HDM)}$
contributes a scalar $h$ ($a_{\mu}^h$), pseudoscalar $A$ ($a_{\mu}^A$)
and the charged
Higgs boson $H^{\pm}$ ($a_{\mu}^{\pm}$).
The relevant formulae can be found in the Appendix in Ref.\cite{g22}
Each term $a_{\mu}^{\Lambda}$ (${\Lambda}=h,~A {\rm ~or} ~H^{\pm}$)
disappears in the limit of large mass,
at small mass the contribution reaches its maximum (or minimum if negative)
value.
The scalar contribution $a_{\mu}^h(M_h)$ is positive whereas the
pseudoscalar boson $a_{\mu}^A(M_A)$ gives
negative contribution, also
the charged Higgs boson contribution is
negative. Note that since the mass of $H^{\pm}$ is above 44 GeV
(LEP I limit),
its small contribution can show up
only if the sum of $h$ and $A$ contributions is small
(see Ref.\cite{g22} for details).
Here we present results based on a simple
calculation of the $ a_{\mu}^{(2HDM)}$
in two scenarios:
\begin{itemize}
\item {\sl a)} pseudoscalar $A$ is light, and
$$ a_{\mu}^{(2HDM)}(M_A)= a_{\mu}^A(M_A)
~~~~~~~(1a)$$
\item {\sl b)} scalar $h$ is light, and
$$ a_{\mu}^{(2HDM)}(M_h)= a_{\mu}^h(M_h)
~~~~~~~~(1b)$$
\end{itemize}
This simple approach is based on the
LEP I mass limits for charged nad neutral Higgs particles
and it means that $h (A)$ and $H^{\pm}$
are heavy enough in order to neglect
their contributions in (1a(b)).
The full 2HDM predictions for these two scenarios
are studied in Ref.\cite{g22}, and differences between
two approaches start to be significant above mass, say 30 GeV.
Note that the contribution
is for the scenario {\sl b)} positive,
whereas for the scenario {\sl a)}~--
negative.
Therefore we have to include this fact
when the 95\% C.L. bounds of $ a_{\mu}^{(2HDM)}$ are calculated
(limits $lim_{\pm}(95\%)$
introduced in Sec.2.1).
Since the case A gives more stringent $lim_{\pm}(95\%)$ constraints,
this case was used in constraining parameters of the
2HDM.
The obtained 95\%C.L. exclusion plots for $\tan \beta$ for
light $h$ or $A$
is presented in Fig.1, together with others limits.
The discussion of these results will be given in Sec.5.
\begin{figure}[ht]
\vskip 4.45in\relax\noindent\hskip -1.05in
\relax{\special{psfile = mojharyx.ps}}
\vspace{-12.5ex}
\caption{ {\em The 95\% exclusion plots for light
scalar(solid lines) or light pseudoscalar (dashed lines)
in 2HDM.
The limits {derivable} from present
$(g - 2)_{\mu}$ measurement and from existing
LEP I results (Yukawa process) for the pseudoscalar
(dotted line) are shown. The possible exclusions from HERA
measurement (the gluon-gluon fusion via a quark loop with the
$\tau^+\tau^-$ final state)
for luminosity 25 pb$^{-1}$ and 500 pb$^{-1}$ as well from
$\gamma \gamma \rightarrow \mu^+ \mu^-$ at low energy NLC
(10 fb$^{-1}$) are also presented.
Parameter space above the curves can be ruled out.
}}
\label{fig:excl}
\end{figure}
\section{ Gluon-gluon fusion at HERA}
The gluon-gluon fusion
via a quark loop,
$gg \rightarrow h(A)$,
can be a significant source of light non-minimal neutral Higgs bosons
at HERA collider due to the hadronic
interaction of quasi-real photons with protons\cite{bk}.
In addition the production of the neutral Higgs boson via
$\gamma g \rightarrow b {\bar b} h (A)$
may also be substantial\cite{grz,bk}. Note that the latter process
also includes
the lowest order contributions due to the resolved photon,
like $\gamma b\rightarrow bh(A)$, $b {\bar b}\rightarrow h(A)$, $bg \rightarrow h(A)b$
etc.
We study the potential of both $gg$ and $\gamma g$ fusions
at HERA collider.
It was found that for mass below $\sim 30$ GeV the $gg$
fusion via a quark loop clearly dominates the cross section.
In order to detect the Higgs particle
it is useful to
study the rapidity distribution ${d \sigma }/{dy}$ of the Higgs bosons
in the $\gamma p$ centre of mass system.
Note that $y=-{{1}\over{2}}log{{E_h-p_h}\over{E_h+p_h}}
=-{{1}\over{2}}log{{x_{\gamma}}\over{x_p}}$, where $x_p(x_{\gamma})$
are the ratio of energy of gluon to the energy of the proton(photon),
respectively.
The (almost)
symmetric shape of the rapidity distribution found for the signal
is extremely useful
to reduce the
background and to separate the $gg\rightarrow h(A)$ contribution.
The main background for the Higgs mass range between
$\tau \tau$ and $b b$
thresholds is due to $\gamma\gamma\rightarrow \tau^+\tau^-$.
In the region of negative rapidity
the cross section ${d \sigma }/{dy}$ is very large,
{\it e.g.} ~for the $\gamma p$ energy equal to 170 GeV $\sim$ 800 pb
at the edge of phase space $y\sim -4$, then it falls down rapidly
approaching $y=0$. At the same time signal reaches at most 10 pb
(for $M_h$=5 GeV).
The region of positive rapidity is {\underline {not}} allowed
kinematically for this process since here one photon interacts directly with
$x_{\gamma}=1$, and therefore $y_{\tau^+ \tau^-}
=-{{1}\over{2}}log{{1}\over{x_p}}\leq 0$.
Moreover, there is a
relation between rapidity and invariant mass:
$M^2_{\tau^+ \tau^-}=e^{2y_{\tau^+ \tau^-}}S_{\gamma p}$.
Significantly different topology found for
$\gamma\gamma\rightarrow \tau^+\tau^-$ events
than for the signal
allows to get rid of this background.
The other sources of background are
$q\bar q\rightarrow\tau^+\tau^-$ processes.
These processes contribute to positive and negative
rapidity $y_{\tau^+ \tau^-}$, with a flat and
relatively low cross sections in the central region (see \cite{bk}).
Assuming that the luminosity ${\cal L}_{ep}$=250 pb$^{-1}$/y
we predict that $gg$ fusion
will produce approximately thousand events
per annum for $M_h=5$ GeV (of the order of 10 events for
$M_h=30$ GeV).
A clear signature for the tagged case with $\tau^+\tau^-$ final state
at positive centre-of-mass
rapidities of the Higgs particle should be seen, even for the mass
of Higgs particle above the $bb$ threshold
(more details can be found in Ref.\cite{bk}).
To show the potential of HERA collider the exclusion plot
based on the $gg$ fusion via a quark loop
can be obtained. In this case, as we mentioned above,
it is easy to find the
part of the phase space where the background is negligible.
To calculate the 95\% C.L. for allowed value of $\tan \beta$
we take into account signal events corresponding only to
the positive rapidity region (in the $\gamma p$ CM system).
Neglecting here the background the number of events
were taken to be equal to 3.
The results
for the $ep$ luminosity ${\cal L}_{ep}$
=25 pb$^{-1}$ and 500 pb$^{-1}$
are presented in Fig. 1 and will be discused in Sec.5.
\section{Photon-photon fusion at NLC}
The possible search for a {\it very} light Higgs particle may
in principle be performed at low energy option of LC suggested
in the literature.
In the papers \cite{deb12} we addressed this problem
and find that the exclusion based on the $\gamma \gamma$ fusion
into Higgs particle decaying into $\mu \mu$ pair,
at energy $\sqrt {s_{ee}}$=10 GeV, may be very efficient
in probing the value of tan $\beta$ down to 5 at $M_h\sim 3.5$ GeV
and below 15 for $2 \:\raisebox{-0.5ex}{$\stackrel{\textstyle<}{\sim}$}\: M_h\:\raisebox{-0.5ex}{$\stackrel{\textstyle<}{\sim}$}\: 8$ GeV
provided that the luminosity is equal to 10 fb$^{-1}$/y (See Fig.1).
\section{Exclusion plots for 2HDM and conclusion}
In Fig.1 the 95\% C.L. exclusion curves for the $\tan \beta$
in the general 2HDM ("Model II")
obtained by us for a light scalar (solid lines)
and for a light pseudoscalar (dashed lines)
are presented in mass range below 40 GeV.
For comparison results from LEP I analysis presented recently
by ALEPH collaboration for pseudoscalar is also shown (dotted line).
The region of ($\tan \beta, M_{h(A)}$) above curves is excluded.
Constraints on $\tan \beta$
were obtained from the
existing $(g-2)_{\mu}$ data including LEP I mass limits.
We applied here a simple approach,
which reproduces the full 2HDM contributions
studied in Ref.\cite{g22}
below mass of 30 GeV.
We see that already the present $(g-2)_{\mu}$ data
improve limits
obtained recently by ALEPH collaboration
on $\tan \beta$ for low mass of the pseudoscalar: $M_A \le$ 2 GeV.
Similar situation should hold for a 2HDM with a light scalar,
although here the Yukawa process may be more restrictive for $M_h\le$
10 GeV\cite{kk}.
The
future improvement
in the accuracy by factor 20 in the forthcoming
$(g-2)_{\mu}$ experiment may lead to more stringent limits
than provided by LEP I up to mass of a neutral Higgs boson $h$ or $A$
equal to 30 GeV,
if the mass difference between scalar and pseudoscalar is $\sim M_Z$,
or to higher mass for a larger mass difference.
Note however that there is some arbitrarilness in the deriving the
expected bounds for the $\delta a_{\mu}^{new}$.
The search at HERA in the gluon-gluon fusion via a quark loop
search at HERA may lead to even more stringent
limits (see Fig.1) for the mass range 5--15 (5--25) GeV, provided
the luminosity will reach 25 (500) pb$^{-1}$ and the
efficiency for $\tau^+ \tau^-$ final state will be high
enough \footnote{In this analysis the 100\% efficiency
has been assumed. If the efficiency will be 10 \% the corresponding
limits will be larger by factor 3.3}.
The other production mechanisms like the $\gamma g$ fusion
and processes with the resolved photon are expected to improve
farther these limits.
In the very low mass range the
additional limits can be obtained from the low energy
NL $\gamma \gamma$ collider. In Fig.1 the
at luminosity 100 pb$^{-1}$ and 10 fb$^{-1}$.
\vspace{0.5cm}
To conclude, in the framework of 2HDM
a light neutral Higgs scalar or pseudoscalar,
in mass range below 40 GeV,
is not ruled out by the present data.
The future experiments may clarify the status of the
general 2HDM with the light neutral Higgs particle.
The role of the forthcoming g-2 measurement seems to be crucial in
clarifying which scenario of 2HDM is allowed: with light scalar or
with light pseudoscalar.
If the $\delta a_{\mu}$ is positive/negative then the light
pseudoscalar/scalar is no more allowed.
Then farther constraints on the coupling of the allowed light Higgs
particle one can obtained from
the HERA collider, which is very well suitable for this.
The simple estimation based on one particular production mechanism
namely gluon-gluon fusion is already promising,
when adding more of them the situation may improve further\cite{bk}.
It suggests that the discovery/exclusion
potential of HERA collider is very large\cite{hera}.
The very low energy region of mass may be studied
in addition in LC machines.
We found that the exclusion based on the $\gamma \gamma$ fusion
into Higgs particle decaying into $\mu \mu$ pair,
at energy $\sqrt {s_{ee}}$=10 GeV, may be very efficient
in probing the Higgs sector of 2HDM even for luminosity 100 pb$^{-1}$.
It is not clear however if these
low energy options will come into operation.
\section{Acknowledgements}
I am grateful very much to Organizing Committee for their
kind invitation to this interesting Workshop.
The results were obtained in the collaboration with D. Choudhury
and J. \.Zochowski. Some of them are updated according to the
reports presented during the conference ICHEP'96, July 1996, Warsaw.
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proofpile-arXiv_065-699
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