Complete High Temperature Expansions for One-Loop Finite Temperature Effects
Peter N. Meisinger and Michael C. Ogilvie
We develop exact, simple closed form expressions for partition functions associated with relativistic bosons and fermions in odd spatial dimensions. These expressions, valid at high temperature, include the effects of a non-trivial Polyakov loop and generalize well-known high temperature expansions. The key \ technical point is the proof of a set of Bessel function identities which resum low temperature expansions into high temperature expansions. The complete expressions for these partition functions can be used to obtain one-loop finite temperature contributions to effective potentials, and thus free energies and pressures.
hep-ph0108026.tex
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\begin{document}

\title{Complete High Temperature Expansions
for One-Loop Finite Temperature Effects}

\author{Peter N. Meisinger and Michael C. Ogilvie}
\address{ Dept. of Physics
Washington University
St. Louis, MO 63130}

\date{\today}
\maketitle

\begin{abstract}

We develop exact, simple closed form expressions for
partition functions associated with relativistic bosons and fermions in odd
spatial dimensions. These expressions, valid at high temperature, include
the effects of a non-trivial Polyakov loop and generalize well-known high
temperature expansions. The key \ technical point is the proof of a set of
Bessel function identities which resum low temperature expansions into high
temperature expansions. The complete expressions for these partition
functions can be used to obtain one-loop finite temperature contributions to
effective potentials, and thus free energies and pressures.

\end{abstract}

\widetext
\pacs{11.10.Wx,11.15.Bt}

\section{\protect\vspace{1pt}Introduction}

The applications of finite temperature field theory are numerous and
diverse \cite{Kapusta:1989tk,Bellac:1996,Das:1997gg}  
For many applications, a high-temperature expansion of one-loop
contributions to thermodynamic functions is necessary. A typical one-loop
contribution to a $(d+1)$-dimensional effective potential from a bosonic
degree of freedom and its antiparticle has the form 
\begin{equation}
V=\frac{2}{\beta }\int \frac{d^{d}k}{\left( 2\pi \right) ^{d}}\ln \left[
1-e^{-\beta \omega _{k}}\right] 
\end{equation}
where the relativistic energy $\omega _{k}$ is given by $\sqrt{k^{2}+M^{2}}$%
. In many cases, the mass $M$ is a function of other quantities, most
notably vacuum expectation values of fields. When the total effective
potential attains its minimum, $V\,$\ may be identified as a contribution to
the total free energy, and $-V$ as a contribution to the pressure. A more
general case is obtained when there is a non-trivial, but spatially uniform,
Polyakov loop as well as a non-zero chemical potential $\mu $.In this case
we have 
\begin{equation}
V_{B}\left( \theta -i\beta \mu \right) =\frac{1}{\beta }\int \frac{d^{d}k}{%
\left( 2\pi \right) ^{d}}\ln \left[ 1-e^{-\beta \omega _{k}+i\theta +\beta
\mu }\right] +\frac{1}{\beta }\int \frac{d^{d}k}{\left( 2\pi \right) ^{d}}%
\ln \left[ 1-e^{-\beta \omega _{k}-i\theta -\beta \mu }\right] \text{.} 
\end{equation}
Note that the effect of a non-trivial Polyakov loop is to add a phase factor $%
\exp \left( \pm i\theta \right) $ to $\exp \left( -\beta \omega \right) $.\
We will evaluate $V_{B}$ for arbitrary $\theta $. In principle, the effect
of $\mu $ can be included by a careful analytic continuation $\theta
\rightarrow \theta -i\beta \mu $, but we do not consider it here. 

We
will develop a high temperature expansion for $V_{B}\left( \theta \right) $,
valid for $d\,$odd, as well as a similar expression for the corresponding
fermionic quantity $V_{F}(\theta )$. This derivation is simple and exact,
and generalizes the results of Dolan and Jackiw \cite{Dolan:1974qd},
who gave approximate high-temperature
expressions for $V_{B}(\theta=0 )$ and $V_{F}(\theta=0 )$
valid up to order $M^4 ln(\beta M)$ in four dimensions.
The work of Dolan and Jackiw was extended by Haber and Weldon
\cite{Haber1:1982} \cite{Haber2:1982}
who gave a complete expression for the Bosonic case $V_B$ as an infinite sum
over hypergeometric functions. Their work included a non-zero chemical
potential. Later work by Actor showed that similar high-temperature expansions
could be obtained using zeta-function techniques
\cite{Actor:1986zf,Actor:1987cf}.
In both cases, higher-order correction terms are given by infinite series
in $\beta M$. 
Our expressions effectively resum these corrections into
a simpler form. 
Analytical results for 
the case of a non-trivial Polyakov loop, $\theta \neq 0$, were first given
in the case $M=0$ by Gross, Pisarski and Yaffe
and by Weiss \cite{Gross:1981br,Weiss:1981rj,Weiss:1982ev}.
Our work generalizes their results to the case $M \neq 0$.
The higher-order terms in
our expressions are manifestly periodic
in $\theta$.
This periodicity
is important in the application
of these results to our recent work with Miller on
models of the deconfinement transition
\cite{Meisinger:2001cq}.
In this work, the eigenvalues of the Polyakov loop
serve as the order parameters for deconfinement,
a point of view also emphasized recently by
Pisarski\cite{Pisarski:2000eq}.




Before beginning the
derivation, we give some examples of its application.
As a first example, consider a scalar boson in the fundamental
representation of an $SU(N)$ gauge group. The Polyakov loop is an $N\times N$
unitary matrix given in general by 
\begin{equation}
P\left( \overrightarrow{x}\right) ={\cal T} \exp 
\left[ i\int_{0}^{\beta
}d\tau \,A_{0}\left( \overrightarrow{x},\tau \right) \right] 
\end{equation}
where ${\cal T}$ on the right-hand 
side indicates Euclidean time ordering. Here we
assume that the Polyakov loop can be made spatially uniform by an
appropriate choice of gauge. A global unitary transformation then puts $P$
into the diagonal form 
\begin{equation}
P_{jk}=\delta _{jk}\exp \left( i\theta _{j}\right) 
\end{equation}
and the partition function is 
\begin{equation}
\sum_{j}V_{B}\left( \theta _{j}\right) \text{.} 
\end{equation}

As a second example,
consider the case of the gauge bosons themselves, which lie in the
adjoint representation of the gauge group. The Polyakov loop in the adjoint
representation is an $\left( N^{2}-1\right) \times \left( N^{2}-1\right) $
matrix. The partition function for the $N^{2}-1$ particles is 
\begin{equation}
s\frac{1}{2}\sum_{j,k=1}^{N}(1-\frac{1}{N}\delta _{jk})V_{B}\left( \theta
_{j}-\theta _{k}\right) 
\end{equation}
where the $\delta _{jk}$ removes a singlet contribution, and the factor of $%
1/2$ corrects for overcounting since $V_{B}$ has both a particle and
antiparticle contribution. The factor $s$ accounts for spin degeneracy; in $%
3+1$ dimensions $s=2$, a consequence of the two possible polarization states
of gauge bosons .

For our third and final example, consider
the evaluation of fermionic partition functions, which can be reduced to the
general bosonic problem. A typical fermionic contribution of particle and
antiparticle has the form 
\begin{equation}
V_{F}\left( \theta -i\beta \mu \right) =-\frac{1}{\beta }\int \frac{d^{d}k}{%
\left( 2\pi \right) ^{d}}\ln \left[ 1+e^{-\beta \omega _{k}+i\theta +\beta
\mu }\right] -\frac{1}{\beta }\int \frac{d^{d}k}{\left( 2\pi \right) ^{d}}%
\ln \left[ 1+e^{-\beta \omega _{k}-i\theta -\beta \mu }\right] 
\end{equation}
which is easily written as 
\begin{equation}
V_{F}\left( \theta \right) =-V_{B}\left( \pi +\theta \right) \text{.} 
\end{equation}
For fermions in the fundamental representation of $SU(N)$, the partition
function is 
\begin{equation}
s\sum_{j}V_{F}\left( \theta _{j}\right) =-s\sum_{j}V_{B}\left( \pi +\theta
_{j}\right) 
\end{equation}
where the factor $s$ again accounts for spin degeneracy.

In section 2, we review the derivation of low temperature expansions for $%
V_{B}\left( \theta \right) $ and $V_{F}\left( \theta \right) $. Section 3
derives the Bessel function identities which convert these low temperature
expansions to high temperature expansions. Section 4 applies these
identities to the case of three spatial dimensions. A final section gives
brief conclusions. There are two appendices.

\vspace{1pt}

\section{Low Temperature Expansion in $d$ Dimensions}

\vspace{1pt}A low-temperature expansion for $V_{B}\left( \theta \right) $
can be generated for arbitrary spatial dimension $d$ by expanding the
logarithm and integrating term by term, first over the surface of a $d$%
-dimensional sphere, and then over a radial degree of freedom $k$ 
\cite{Actor:1986zf,Actor:1987cf}.
The result, given in terms of modified Bessel functions
\cite{GandR}, is
\begin{eqnarray}
V_{B}\left( \theta \right) &=&\frac{1}{\beta }\int \frac{d^{d}k}{\left( 2\pi
\right) ^{d}}\ln \left[ 1-e^{-\beta \omega _{k}+i\theta }\right] +\frac{1}{%
\beta }\int \frac{d^{d}k}{\left( 2\pi \right) ^{d}}\ln \left[ 1-e^{-\beta
\omega _{k}-i\theta }\right] \\
&=&-\frac{M^{d/2+1/2}}{2^{d/2-3/2}\pi ^{d/2+1/2}\beta ^{d/2+1/2}}%
\sum_{n=1}^{\infty }\frac{1}{n^{d/2+1/2}}K_{\left( d+1\right) /2}\left(
n\beta M\right) \cos \left( n\theta \right)
\end{eqnarray}
which is derived in detail in Appendix A. Each term in the series represents
the contribution of $n$ particles or antiparticles, with a corresponding
phase\ factor of $\exp \left( \pm in\theta \right) $. If the one-loop finite
temperature functional determinant is represented as a functional integral
over a space-time variable $x_{\mu }$, the phase factors are associated with
paths which wind non-trivially in the Euclidean time direction.

For fermions, we have 
\begin{equation}
V_{F}\left( \theta \right) =\frac{M^{d/2+1/2}}{2^{d/2-3/2}\pi
^{d/2+1/2}\beta ^{d/2+1/2}}\sum_{n=1}^{\infty }\frac{\left( -1\right) ^{n}}{%
n^{d/2+1/2}}K_{\left( d+1\right) /2}\left( n\beta M\right) \cos \left(
n\theta \right) \text{.} 
\end{equation}
In a path integral representation, the factors of $\left( -1\right) ^{n}$
are a consequence of fermionic antiperiodic boundary conditions. We next
derive a set of identities which resum these series for $d$ odd.

\section{\protect\vspace{1pt}Bessel Function Identities}

We will derive a set of identities for sums of the form 
\begin{equation}
\sum_{n=1}^{\infty }\frac{1}{n^{p}}K_{p}\left( nr\right) \cos \left( n\phi
\right) 
\end{equation}
\vspace{1pt}for $p$ an even integer. Our starting point is the identity 
\begin{equation}
\sum_{p=1}^{\infty }K_{0}(pr)\cos (p\phi )=\frac{1}{2}\left[ \gamma +\ln
\left( \frac{r}{4\pi }\right) \right] +\frac{\pi }{2}\sum_{l}\,^{^{^{\prime
}}}\left[ \frac{1}{\sqrt{r^{2}+\left( \phi -2\pi l\right) ^{2}}}-\frac{1}{%
2\pi \left| l\right| }\right] 
\end{equation}
which may be found in \cite{GandR};
we provide a derivation in Appendix B which
provides some physical insight into its origin. The notation $%
\,\sum_{l}\hspace{0.00in}^{^{^{\prime }}}\,$is 
used to indicate that singular terms, here
the $1/\left| l\right| $ term, are omitted when $l=0$.

Using the recursion formula 
\begin{equation}
\frac{d}{dz}K_{\nu }(z)=-K_{\nu -1}(z)-\frac{\nu }{z}K_{\nu }(z)\text{,} 
\end{equation}
it follows immediately that 
\begin{equation}
\frac{d}{dz}\sum_{p=1}^{\infty }\frac{z}{p}K_{1}(pz)\cos (p\phi
)=-z\sum_{p=1}^{\infty }K_{0}(pz)\cos (p\phi )\text{.} 
\end{equation}
This in turn implies that 
\begin{equation}
\sum_{p=1}^{\infty }\frac{1}{p}K_{1}(pz)\cos (p\phi )=-\frac{1}{z}\int
dz\,\,\,z\left[ \sum_{p=1}^{\infty }K_{0}(pz)\cos (p\phi )\right] +\frac{%
C(\phi )}{z} 
\end{equation}
where $C(\phi )\,$is an unknown function to be determined. Integration
yields immediately 
\begin{equation}
\sum_{p=1}^{\infty }\frac{1}{p}K_{1}(pz)\cos (p\phi )=-\frac{1}{4}z\left[
\ln \left( \frac{z}{4\pi }\right) +\gamma -\frac{1}{2}\right] -\frac{\pi }{2z%
}\sum_{l}\,^{^{^{\prime }}}\left[ \sqrt{z^{2}+\left( \phi -2\pi l\right) ^{2}%
}-\frac{z^{2}}{4\pi \left| l\right| }\right] +\frac{C(\phi )}{z}\text{.} 
\end{equation}
The function $C(\phi )\,$is determined from the behavior of $K_{\nu}(z)$ for $%
z\rightarrow 0$%
\begin{equation}
K_{\nu }(z) \rightarrow  \frac{1}{2}\Gamma (\nu )\left( \frac{2}{z}\right) ^{\nu
} 
\end{equation}
in combination with the standard result \cite{GandR}
\begin{equation}
\label{eq:ber2}
\sum_{p=1}^{\infty }\frac{\cos (p\phi )}{p^{2}}=\frac{1}{4}\phi ^{2}-\frac{%
\pi }{2}\phi +\frac{\pi ^{2}}{6}\text{,} 
\end{equation}
which is valid for $0\leq \phi <2\pi $. The right hand side of 
Eq.\ (\ref{eq:ber2}) is a
rescaling of the second Bernoulli polynomial; it can be extended to all real
values if $\phi \,$is replaced by $\phi\, mod\, 2\pi $ on the right hand
side of the equation. This implies the leading behavior of the sum as $%
z\rightarrow 0\,$is given by 
\begin{equation}
\lim_{z\rightarrow 0}\,z\sum_{p=1}^{\infty }\frac{1}{p}K_{1}(pz)\cos (p\phi
)=\sum_{p=1}^{\infty }\frac{\cos (p\phi )}{p^{2}}=\frac{1}{z}\left[ \frac{1}{%
4}\phi ^{2}-\frac{\pi }{2}\phi +\frac{\pi ^{2}}{6}\right] \text{,} 
\end{equation}
giving us finally 
\begin{eqnarray}
\sum_{p=1}^{\infty }\frac{1}{p}K_{1}(pz)\cos (p\phi ) &=&-\frac{1}{4}z\left[
\ln \left( \frac{z}{4\pi }\right) +\gamma -\frac{1}{2}\right] +\frac{1}{z}%
\left[ \frac{1}{4}\phi _{+}^{2}-\frac{\pi }{2}\phi _{+}+\frac{\pi ^{2}}{6}%
\right] \nonumber\\
&&-\frac{\pi }{2z}\sum_{l}\,^{^{^{\prime }}}\left[ \sqrt{z^{2}+\left( \phi
-2\pi l\right) ^{2}}-\left| \phi -2\pi l\right| -\frac{z^{2}}{4\pi \left|
l\right| }\right]
\end{eqnarray}
where we have introduced the notation $\phi _{+}$ to represent $\phi \,
mod\, 2\pi $. When discussing fermions, we will also use $\phi _{-}$ to
similarly represent an angle chosen to lie between $-\pi $ and $\pi $. Note
that the last part of this expression is automatically periodic due to the
sum over $l$.

Application of this technique a second time gives 
\begin{eqnarray}
\sum_{p=1}^{\infty }\frac{1}{p^{2}}K_{2}(pz)\cos (p\phi ) &=&\frac{1}{16}%
z^{2}\left[ \ln \left( \frac{z}{4\pi }\right) +\gamma -\frac{3}{4}\right] -%
\frac{1}{2}\left[ \frac{1}{4}\phi _{+}^{2}-\frac{\pi }{2}\phi _{+}+\frac{\pi
^{2}}{6}\right]  \nonumber\\
&&+\frac{2}{z^{2}}\left[ \frac{-1}{48}\phi _{+}^{4}+\frac{\pi }{12}\phi
_{+}^{3}-\frac{\pi ^{2}}{12}\phi _{+}^{2}+\frac{\pi ^{4}}{90}\right]\nonumber\\
&&+\frac{\pi }{2z^{2}}\sum_{l}\,^{^{^{\prime }}}\left\{ \frac{1}{3}\left[
z^{2}+\left( \phi -2\pi l\right) ^{2}\right] ^{3/2}-\frac{1}{3}\left| \phi
-2\pi l\right| ^{3}-\frac{1}{2}\left| \phi -2l\pi \right| z^{2}-\frac{z^{4}}{%
16\pi \left| l\right| }\right\} 
\end{eqnarray}
which is needed for the case $d=3$. We have used the standard result 
\cite{GandR}
\begin{equation}
\sum_{p=1}^{\infty }\frac{\cos (p\phi )}{p^{4}}=\frac{-1}{48}\phi _{+}^{4}+%
\frac{\pi }{12}\phi _{+}^{3}-\frac{\pi ^{2}}{12}\phi _{+}^{2}+\frac{\pi ^{4}%
}{90}\text{.}
\end{equation}
Formulas appropriate for $d=5,7,..$ can also be derived in the same manner.

\section{\protect\vspace{1pt}High-Temperature Expansions for $d=3$}

\vspace{1pt}We now can write complete expressions for $V_{B}\left( \theta
\right) $ and $V_{F}\left( \theta \right) $ in three spatial dimensions:
\begin{eqnarray}
V_{B}\left( \theta \right)  &=&-\frac{M^{2}}{\pi ^{2}\beta ^{2}}%
\sum_{n=1}^{\infty }\frac{1}{n^{2}}K_{2}\left( n\beta M\right) \cos \left(
n\theta \right) \nonumber \\
&=&-\frac{2}{\pi ^{2}\beta ^{4}}\left[ \frac{\pi ^{4}}{90}-\frac{1}{48}%
\theta _{+}^{4}+\frac{\pi }{12}\theta _{+}^{3}-\frac{\pi ^{2}}{12}\theta
_{+}^{2}\right] +\frac{M^{2}}{2\pi ^{2}\beta ^{2}}\left[ \frac{1}{4}\theta
_{+}^{2}-\frac{\pi }{2}\theta _{+}+\frac{\pi ^{2}}{6}\right] \nonumber \\
&&-\frac{1}{2\pi \beta ^{4}}\sum_{l}\,^{^{^{\prime }}}\left\{ \frac{1}{3}%
\left[ \left( \beta M\right) ^{2}+\left( \theta -2\pi l\right) ^{2}\right]
^{3/2}-\frac{1}{3}\left| \theta -2\pi l\right| ^{3}-\frac{1}{2}\left| \theta
-2l\pi \right| \beta ^{2}M^{2}-\frac{\left( \beta M\right) ^{4}}{16\pi
\left| l\right| }\right\} \nonumber \\
&&-\frac{M^{4}}{16\pi ^{2}}\left[ \ln \left( \frac{\beta M}{4\pi }\right)
+\gamma -\frac{3}{4}\right] \text{.}
\end{eqnarray}
Parts of this complete expression have been known for some time. For $\theta
=0$, the leading behavior is 
\begin{equation}
V_{B}\left( \theta =0\right) \approx -2\frac{\pi ^{2}}{90\beta ^{4}}+\frac{%
M^{2}}{12\beta ^{2}}-\frac{M^{3}}{6\pi \beta }-\frac{M^{4}}{16\pi ^{2}}%
\left[ \ln \left( \frac{\beta M}{4\pi }\right) +\gamma -\frac{3}{4}\right] 
\text{,}
\end{equation}
an expression first derived by Dolan and Jackiw \cite{Dolan:1974qd}.
The leading $T^{4}$
behavior for $\theta \neq 0$ was first derived by Gross, Pisarski and Yaffe
and Weiss for $SU(N)$ gauge bosons and for massless fermions
\cite{Gross:1981br,Weiss:1981rj,Weiss:1982ev}. 

Each of the four terms deserves comment. The first is the blackbody free
energy for two degrees of freedom, and depends only on the temperature and
the angle $\theta $. The second term, which is the leading correction due to
the mass $M$, often appears in discussions of symmetry restoration at high
temperatures with $\theta =0$. For example, suppose we are calculating the
effective potential for a complex scalar field $\Phi $. The mass $M^{2}$ is
given by the second derivative of the classical potential, $\partial
^{2}U/\partial \Phi ^{\ast }\partial \Phi $, and depends on the expectation
value of the field $\Phi $. If $U$ has the form $U=-m^{2}\Phi ^{\ast }\Phi
+\lambda \left( \Phi ^{\ast }\Phi \right) ^{2}$, then $M^{2}=-m^{2}+4\lambda
\Phi ^{\ast }\Phi $. For $m^{2}>0$, the $U(1)$ symmetry is spontaneously
broken at low temperature. At high temperature, the $M^{2}/12\beta ^{2}$
term generates a positive mass for the $\Phi $ field of order $T$, restoring
the symmetry. The third term in $V_{B}\left( \theta \right) $ is linear in $T
$, and non-analytic in $M^{2}$ for $\theta =0$. It is closely associated
with the $n=0$ Matsubara mode, which is the most infrared singular
contribution to a finite temperature functional determinant. This term is
responsible for non-analytic behavior in finite temperature perturbation
theory via the summation of ring diagrams. For example, in a scalar theory
it gives rise to the $\lambda ^{3/2}$ contribution to the free energy; in
QED, the contribution is $e^{3}$ \cite{Kapusta:1989tk}
Note how subtractions occur
in the $l\neq 0$ parts of this term to keep these parts subleading.\ The
last term is logarithmic in the dimensionless combination $\beta M$ and
independent of $\theta $. In calculations of effective potentials, it
typically combines with zero-temperature logarithms in such a way that the
temperature $T$ sets the scale of running coupling constants at high $T$.

From the basic result for $V_{B}$, we can build other results. Consider a
complex scalar field in the fundamental representation of $SU(N)$. The
partition function in a constant background Polyakov loop is given by 
\begin{eqnarray}
V_{FT} &=&\sum_{j}V_{B}\left( \theta _{j}\right) \nonumber\\
&=&-\frac{\pi ^{2}N}{45\beta ^{4}}+\frac{2}{\pi ^{2}\beta ^{4}}%
\sum_{j}\left[ \frac{1}{48}\theta _{j}^{4}-\frac{\pi }{12}\theta _{j}^{3}+%
\frac{\pi ^{2}}{12}\theta _{j}^{2}\right] \nonumber \\
&&+\frac{NM^{2}}{12\beta ^{2}}+\frac{M^{2}}{2\pi ^{2}\beta ^{2}}%
\sum_{j}\left[ \frac{1}{4}\theta _{j}^{2}-\frac{\pi }{2}\theta _{j}\right] 
\nonumber\\
&&-\frac{1}{2\pi \beta ^{4}}\sum_{j,l}\,^{^{^{\prime }}}\left\{ \frac{1%
}{3}\left[ \left( \beta M\right) ^{2}+\left( \theta _{j}-2\pi l\right)
^{2}\right] ^{3/2}-\frac{1}{3}\left| \theta _{j}-2\pi l\right| ^{3}-\frac{1}{%
2}\left| \theta _{j}-2l\pi \right| \beta ^{2}M^{2}-\frac{\left( \beta
M\right) ^{4}}{16\pi \left| l\right| }\right\} \nonumber \\
&&-\frac{NM^{4}}{16\pi ^{2}}\left[ \ln \left( \frac{\beta M}{4\pi }\right)
+\gamma -\frac{3}{4}\right] 
\end{eqnarray}
where we assume all $\theta $'s are chosen to lie between $0$ and $2\pi $ in
accordance with the convention for $\theta _{+}$. A similar expression holds
for bosons in the adjoint representation.

\vspace{1pt}For fermions, we have similarly 
\begin{eqnarray}
V_{F}\left( \theta \right) &=&\frac{2}{\pi ^{2}\beta ^{4}}\left[ -\frac{7}{%
720}\pi ^{4}+\frac{1}{24}\pi ^{2}\theta _{-}^{2}-\frac{1}{48}\theta
_{-}^{4}\right] +\frac{M^{2}}{2\pi ^{2}\beta ^{2}}\left[ \frac{1}{12}\pi
^{2}-\frac{1}{4}\theta _{-}^{2}\right] \nonumber\\
&&+\frac{1}{2\pi \beta ^{4}}\sum_{l}\,^{^{^{\prime }}}\left\{ \frac{1}{3}%
\left[ \left( \beta M\right) ^{2}+\left( \theta -\left( 2l-1\right) \pi
\right) ^{2}\right] ^{3/2}-\frac{1}{3}\left| \theta -\left( 2l-1\right) \pi
\right| ^{3}-\frac{1}{2}\left| \theta -\left( 2l-1\right) \pi \right| \beta
^{2}M^{2}-\frac{\left( \beta M\right) ^{4}}{16\pi \left| l\right| }\right\}
\nonumber\\
&&+\frac{M^{4}}{16\pi ^{2}}\left[ \ln \left( \frac{\beta M}{4\pi }\right)
+\gamma -\frac{3}{4}\right]
\end{eqnarray}
with $\theta _{-}$ now used. For fermions in the fundamental representation
of $SU(N)$, we may write 
\begin{eqnarray}
V_{FT} &=&-\frac{7\pi ^{2}N}{180\beta ^{4}}+\sum_{j}\frac{1}{12\pi ^{2}\beta
^{4}}\left[ 2\pi ^{2}\theta _{j}^{2}-\theta _{j}^{4}\right] +\frac{NM^{2}}{%
12\beta ^{2}}-\sum_{j}\frac{M^{2}}{4\pi ^{2}\beta ^{2}}
\theta _{j}^{2}\nonumber \\
&&+\frac{1}{\pi \beta ^{4}}\sum_{j,l}\,^{^{^{\prime }}}\left\{ \frac{1%
}{3}\left[ \left( \beta M\right) ^{2}+\left( \theta _{j}-\left( 2l-1\right)
\pi \right) ^{2}\right] ^{3/2}-\frac{1}{3}\left| \theta _{j}-\left(
2l-1\right) \pi \right| ^{3}-\frac{1}{2}\left| \theta _{j}-\left(
2l-1\right) \pi \right| \beta ^{2}M^{2}-\frac{\left( \beta M\right) ^{4}}{%
16\pi \left| l\right| }\right\} \nonumber\\
&&+\frac{NM^{4}}{8\pi ^{2}}\left[ \ln \left( \frac{\beta M}{4\pi }\right)
+\gamma -\frac{3}{4}\right] \text{.}
\end{eqnarray}
where the angles $\theta _{j}$ must now be chosen to lie between $-\pi $ and 
$\pi $.

\section{Conclusions}

We have found complete, simple expressions for $V_{B}\left(
\theta \right) $ and $V_{F}\left( \theta \right) $ 
in the high-temperature limit
which generalize
previously known expressions. Not only are the expressions simple, their
derivation is direct and relatively elementary. 
Our formulae reflect in a direct way periodicity in $ \theta$, a
property which is lost when analytically continuing power series
in $\beta \mu$ to $\beta \mu + i \theta$.

As a practical matter, it is natural to ask how accurate both the low- and
high-temperature expansions are. The low temperature expansion for $%
V_{B}\left( \theta \right) $ is an infinite series in $n$; using the first $%
10$ terms in the series gives an accuracy better than $1$ part in $10^{3}$
over the entire range $0$ to $2\pi $ for temperatures $T\leq 0.25M$. The
high temperature expansion also involves an infinite series, in the the
parameter $l$. In comparison, the high temperature expansion is within $5\%$
of the exact answer for all values of $\theta $ at $T=0.5M$ when terms up to 
$\left| l\right| =10$ are included. The accuracy improves substantially as $%
T\,\ $increases. Both expansions are more accurate when restricted to $%
\theta =0$.

Our primary interest in these results lies in their application
to the study of systems where a non-trivial Polyakov loop is
expected to occur.
The foremost physical example is QCD at finite temperature.
The high-temperature form of $V_B ( \theta )$ suggests that the
Bernoulli polynomials appear naturally in the free energy
of $SU(N)$ gauge theories with a non-trivial Polyakov loop,
essentially as
polynomials in the Polyakov loop eigenvalues.
In our recent work with Miller\cite{Meisinger:2001cq},
we have used this observation
to construct
a phenomenological free energy for the quark-gluon plasma
which reproduces much of the thermodynamic observed in
lattice simulations.
One can also apply the results obtained here 
to the Savvidy model 
at finite temperature
\cite{Starinets:1994vi,Meisinger:1997jt}.
Savvidy originally proposed a model of
the QCD vacuum in which gluons moved 
in a constant chromomagnetic field
\cite{Savvidy:1977as}.
Using low-temperature expansions, we have shown that a
confining state, where the Polyakov loop expectation value is zero,
minimizes the free energy. 
Using the Bessel function identities proven here,
we have recently
\cite{usrealsoonnow}  developed
a high-temperature expansion for this model which shows
that a non-zero Polyakov line is favored at high temperature.

\appendix
\section{Derivation of Low Temperature Expansions}

We begin by expanding the logarithms and performing the angular
integrations: 
\begin{eqnarray}
V_{B}\left( \theta \right)  &=&\frac{1}{\beta }\int \frac{d^{d}k}{\left(
2\pi \right) ^{d}}\ln \left[ 1-e^{-\beta \omega _{k}+i\theta }\right] +\frac{%
1}{\beta }\int \frac{d^{d}k}{\left( 2\pi \right) ^{d}}\ln \left[ 1-e^{-\beta
\omega _{k}-i\theta }\right]  \nonumber\\
&=&-\frac{4\pi ^{d/2}}{\Gamma (d/2)\left( 2\pi \right) ^{d}\beta }\int
dk\,k^{d-1}\sum_{n=1}^{\infty }\frac{1}{n}e^{-n\beta \omega _{k}}\cos \left(
n\theta \right) \text{.}
\end{eqnarray}
The standard substitution $k=M\sinh (t)$ gives 
\begin{eqnarray}
V_{B}\left( \theta \right)  &=&-\frac{4\pi ^{d/2}}{\Gamma (d/2)\left( 2\pi
\right) ^{d}\beta }\sum_{n=1}^{\infty }\frac{\cos \left( n\theta \right) }{n}%
M^{d}\int_{0}^{\infty }dt\,\cosh t\,\sinh t^{d-1}e^{-n\beta M\cosh t}
\nonumber \\
&=&\frac{4\pi ^{d/2}}{\Gamma (d/2)\left( 2\pi \right) ^{d}}%
\sum_{n=1}^{\infty }\frac{\cos \left( n\theta \right) }{n^{2}\beta ^{2}}M^{d}%
\frac{d}{dM}\int_{0}^{\infty }dt\,\,\sinh t^{d-1}e^{-n\beta M\cosh t} 
\nonumber\\
&=&\frac{4\pi ^{d/2}}{\Gamma (d/2)\left( 2\pi \right) ^{d}}%
\sum_{n=1}^{\infty }\frac{\cos \left( n\theta \right) }{n^{2}\beta ^{2}}M^{d}%
\frac{d}{dM}\left[ \frac{\Gamma \left( d/2\right) }{\sqrt{\pi }}\left( \frac{%
2}{n\beta M}\right) ^{\left( d-1\right) /2}K_{\left( d-1\right) /2}\left(
n\beta M\right) \right] 
\end{eqnarray}
This can in turn be reduced using standard recursion relations for modified
Bessel functions: 
\begin{eqnarray}
V_{B}\left( \theta \right)  &=&\frac{4\pi ^{\left( d-1\right) /2}}{\left(
2\pi \right) ^{d}\beta }M^{d}\sum_{n=1}^{\infty }\frac{\cos \left( n\theta
\right) }{n}\frac{d}{dz}\left[ \left( \frac{2}{z}\right) ^{\nu }K_{\nu
}\left( z\right) \right] _{z=n\beta M,\nu =\left( d-1\right) /2} \nonumber\\
&=&\frac{4\pi ^{\left( d-1\right) /2}}{\left( 2\pi \right) ^{d}\beta }%
M^{d}\sum_{n=1}^{\infty }\frac{\cos \left( n\theta \right) }{n}\left[ \left( 
\frac{2}{z}\right) ^{\nu }\left( \frac{dK_{\nu }\left( z\right) }{dz}-\frac{%
\nu }{z}K_{\nu }\left( z\right) \right) \right] _{z=n\beta M,\nu =\left(
d-1\right) /2} \nonumber\\
&=&\frac{4\pi ^{\left( d-1\right) /2}}{\left( 2\pi \right) ^{d}\beta }%
M^{d}\sum_{n=1}^{\infty }\frac{\cos \left( n\theta \right) }{n}\left[
-\left( \frac{2}{z}\right) ^{\nu }K_{\nu +1}\left( z\right) \right]
_{z=n\beta M,\nu =\left( d-1\right) /2} \nonumber\\
&=&-\frac{4\pi ^{\left( d-1\right) /2}}{\left( 2\pi \right) ^{d}\beta }%
M^{d}\sum_{n=1}^{\infty }\frac{\cos \left( n\theta \right) }{n}\left( \frac{2%
}{n\beta M}\right) ^{\left( d-1\right) /2}K_{\left( d+1\right) /2}\left(
n\beta M\right)  \nonumber\\
&=&-\frac{M^{d/2+1/2}}{2^{d/2-3/2}\pi ^{d/2+1/2}\beta ^{d/2+1/2}}%
\sum_{n=1}^{\infty }\frac{\cos \left( n\theta \right) }{n^{d/2+1/2}}%
K_{\left( d+1\right) /2}\left( n\beta M\right) 
\end{eqnarray}

\section{Proof of a Bessel Function Identity}

\vspace{1pt}In this appendix, we prove the Bessel function identity
\begin{equation}
\sum_{m=1}^{\infty }K_{0}(mr)\cos (m\phi )=\frac{1}{2}\left[ \ln \left( 
\frac{r}{4\pi }\right) +\gamma \right] +\frac{\pi }{2}\sum_{l}\,^{^{^{\prime
}}}\left[ \frac{1}{\sqrt{r^{2}+\left( \phi -2\pi l\right) ^{2}}}-\frac{1}{%
2\pi \left| l\right| }\right] \text{.}
\end{equation}
Using a standard integral representation \cite{GandR}
\begin{equation}
K_{0}(pz)=\int_{0}^{\infty }dt\,\frac{\cos (pt)}{\sqrt{t^{2}+z^{2}}}\text{,}
\end{equation}
we have 
\begin{eqnarray}
&&\sum_{m=1}^{\infty }\cos (m\phi )K_{0}(mr) \nonumber\\
&=&\sum_{m=1}^{\infty }\cos (m\phi )\int_{0}^{\infty }dk_{x}\frac{\cos
(k_{x}r)}{\sqrt{k_{x}^{2}+m^{2}}} \nonumber\\
&=&\frac{1}{2}\sum_{m=1}^{\infty }\cos (m\phi )\int dk_{x}\frac{1}{\sqrt{%
k_{x}^{2}+m^{2}}}e^{ik_{x}r} \nonumber\\
&=&\frac{1}{4\pi }\sum_{m\neq 0}\int dk_{x}dk_{y}\frac{1}{%
k_{x}^{2}+k_{y}^{2}+m^{2}}e^{ik_{x}r+im\phi }
\end{eqnarray}
We introduce a regulating mass $\mu $, which will be taken to zero at the
end of the calculation, obtaining 
\begin{equation}
\frac{1}{4\pi }\sum_{m\neq 0}\int dk_{x}dk_{y}\frac{1}{%
k_{x}^{2}+k_{y}^{2}+m^{2}+\mu ^{2}}e^{ik_{x}r+im\phi }
\end{equation}
We add and subtract the divergent $m=0$ term 
\begin{eqnarray}
&&\frac{1}{4\pi }\sum_{m}\int dk_{x}dk_{y}\frac{1}{%
k_{x}^{2}+k_{y}^{2}+m^{2}+\mu ^{2}}e^{ik_{x}r+im\phi }-\frac{1}{4\pi }\int
dk_{x}dk_{y}\frac{1}{k_{x}^{2}+k_{y}^{2}+\mu ^{2}}e^{ik_{x}r} \nonumber\\
&=&\frac{1}{4\pi }\sum_{m}\int dk_{x}dk_{y}dk_{z}\frac{1}{%
k_{x}^{2}+k_{y}^{2}+k_{z}^{2}+\mu ^{2}}\delta \left( k_{z}-m\right)
e^{ik_{x}r+ik_{z}\phi }-\frac{1}{4\pi }\int dk_{x}dk_{y}\frac{1}{%
k_{x}^{2}+k_{y}^{2}+\mu ^{2}}e^{ik_{x}r}
\end{eqnarray}
Using the Poisson summation technique in the form $\sum_{m}\delta \left(
k_{z}-m\right) =\sum_{n}\exp \left( -2\pi ink_{z}\right) $, we obtain 
\begin{equation}
\frac{1}{4\pi }\sum_{n}\int dk_{x}dk_{y}dk_{z}\frac{1}{%
k_{x}^{2}+k_{y}^{2}+k_{z}^{2}+\mu ^{2}}e^{ik_{x}r+ik_{z}\left( \phi -2\pi
n\right) }-\frac{1}{4\pi }\int dk_{x}dk_{y}\frac{1}{k_{x}^{2}+k_{y}^{2}+\mu
^{2}}e^{ik_{x}r}
\end{equation}
which reads in a compact notation 
\begin{equation}
\frac{\pi }{2}\sum_{n}\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\frac{4\pi 
}{k^{2}+\mu ^{2}}e^{ik_{x}r+ik_{z}\left( \phi -2\pi n\right) }-\frac{1}{4\pi 
}\int d^{2}k\frac{1}{k^{2}+\mu ^{2}}e^{ik_{x}r}
\end{equation}
The first integral gives a sum of screened Coulomb, or Yukawa, potentials 
\begin{equation}
\frac{\pi }{2}\sum_{n}\left[ \frac{e^{-\mu \sqrt{r^{2}+\left( \phi -2\pi
n\right) ^{2}}}}{\sqrt{r^{2}+\left( \phi -2\pi n\right) ^{2}}}\right] -\frac{%
1}{4\pi }\int d^{2}k\frac{1}{k^{2}+\mu ^{2}}e^{ik_{x}r}
\end{equation}
and both terms appear to be problematic as $\mu \rightarrow 0$. The first
term can be made finite in this limit by subtracting the contribution at $%
r=\phi =0$ for $n\neq 0$. Using the notation $\sum_{n}{}^{^{^{\prime }}}$ to
denote a summation over all $n$ with the omission of the singular term when $%
n=0$, we have 
\begin{equation}
\frac{\pi }{2}\sum_{n}\,^{^{^{\prime }}}\left[ \frac{e^{-\mu \sqrt{%
r^{2}+\left( \phi -2\pi n\right) ^{2}}}}{\sqrt{r^{2}+\left( \phi -2\pi
n\right) ^{2}}}-\frac{e^{-\mu 2\pi \left| n\right| }}{2\pi \left| n\right| }%
\right] +\left[ \frac{\pi }{2}\sum_{n}\,^{^{^{\prime }}}\frac{e^{-\mu 2\pi
\left| n\right| }}{2\pi \left| n\right| }-\frac{1}{4\pi }\int d^{2}k\frac{1}{%
k^{2}+\mu ^{2}}e^{ik_{x}r}\right] \text{.}
\end{equation}
The second term in brackets can be evaluated by summing the series 
\begin{equation}
\frac{\pi }{2}\sum_{n}\,^{^{^{\prime }}}\frac{e^{-\mu 2\pi \left| n\right| }%
}{2\pi \left| n\right| }=-\frac{1}{2}\ln \left[ 1-e^{-2\pi \mu }\right] 
\end{equation}
and using the identity 
\begin{equation}
-\frac{1}{4\pi }\int d^{2}k\frac{1}{k^{2}+\mu ^{2}}e^{ik_{x}r}=-\frac{1}{2}%
K_{0}\left( \mu r\right) 
\end{equation}
so we have 
\begin{equation}
\frac{\pi }{2}\sum_{n}\,^{^{^{\prime }}}\left[ \frac{e^{-\mu \sqrt{%
r^{2}+\left( \phi -2\pi n\right) ^{2}}}}{\sqrt{r^{2}+\left( \phi -2\pi
n\right) ^{2}}}-\frac{e^{-\mu 2\pi \left| n\right| }}{2\pi \left| n\right| }%
\right] +\left[ -\frac{1}{2}\ln \left[ 1-e^{-2\pi \mu }\right] -\frac{1}{2}%
K_{0}\left( \mu r\right) \right] \text{.}
\end{equation}
In the limit $\mu \rightarrow 0$, the second term in brackets gives 
\begin{eqnarray}
&&-\frac{1}{2}\ln \left[ 1-e^{-2\pi \mu }\right] -\frac{1}{2}K_{0}\left( \mu
r\right)  \nonumber\\
&\rightarrow &-\frac{1}{2}\ln \left[ 2\pi \mu \right] +\frac{1}{2}\ln \left( 
\frac{\mu r}{2}\right) -\frac{1}{2}\psi (1)=\frac{1}{2}\ln \left( \frac{r}{%
4\pi }\right) +\frac{1}{2}\gamma 
\end{eqnarray}
so we finally obtain 
\begin{equation}
\sum_{m=1}^{\infty }K_{0}(mr)\cos (m\phi )=\frac{\pi }{2}\sum_{n}\,^{^{^{%
\prime }}}\left[ \frac{1}{\sqrt{r^{2}+\left( \phi -2\pi n\right) ^{2}}}-%
\frac{1}{2\pi \left| n\right| }\right] +\frac{1}{2}\ln \left( \frac{r}{4\pi }%
\right) +\frac{1}{2}\gamma \text{.}
\end{equation}

\begin{references} 

%\cite{Kapusta:1989tk}
\bibitem{Kapusta:1989tk}
J.~I.~Kapusta,
%``Finite Temperature Field Theory,''.
{\it Finite Temperature Field Theory}
(Cambridge , 1989).


\bibitem{Bellac:1996}
M.~Le~Bellac,
{\it Thermal Field Theory}
(Cambridge , 1996).

%\cite{Das:1997gg}
\bibitem{Das:1997gg}
A.~Das,
%``Finite Temperature Field Theory,''
{\it Finite Temperature Field Theory}
(World Scientific, Singapore, 1997)

%\cite{Dolan:1974qd}
\bibitem{Dolan:1974qd}
L.~Dolan and R.~Jackiw,
%``Symmetry Behavior At Finite Temperature,''
Phys.\ Rev.\ D {\bf 9}, 3320 (1974).
%%CITATION = PHRVA,D9,3320;%%

\bibitem{Haber1:1982}
H.~E.~Haber and H.~A.~Weldon,
%``Finite Temperature Symmetry Breaking As Bose-Einstein Condensation,''
Phys.\ Rev.\ D {\bf 25}, 502 (1982).
%%CITATION = PHRVA,D25,502;%%

\bibitem{Haber2:1982}
H.~E.~Haber and H.~A.~Weldon,
%``On The Relativistic Bose-Einstein Integrals,''
J.\ Math.\ Phys.\  {\bf 23}, 1852 (1982).
%%CITATION = JMAPA,23,1852;%%

%\cite{Actor:1986zf}
\bibitem{Actor:1986zf}
A.~Actor,
%``Zeta Function Regularization Of High Temperature Expansions In Field Theory,''
Nucl.\ Phys.\ B {\bf 265}, 689 (1986).
%%CITATION = NUPHA,B265,689;%%
 
%\cite{Actor:1987cf}
\bibitem{Actor:1987cf}
A.~Actor,
%``More On Zeta Function Regularization Of High Temperature Expansions,''
Fortsch.\ Phys.\  {\bf 35}, 793 (1987).
%%CITATION = FPYKA,35,793;%%

%\cite{Gross:1981br}
\bibitem{Gross:1981br}
D.~J.~Gross, R.~D.~Pisarski and L.~G.~Yaffe,
%``QCD And Instantons At Finite Temperature,''
Rev.\ Mod.\ Phys.\  {\bf 53}, 43 (1981).
%%CITATION = RMPHA,53,43;%%
 
%\cite{Weiss:1981rj}
\bibitem{Weiss:1981rj}
N.~Weiss,
%``The Effective Potential For The Order Parameter Of Gauge Theories At Finite Temperature,''
Phys.\ Rev.\ D {\bf 24}, 475 (1981).
%%CITATION = PHRVA,D24,475;%%
 
%\cite{Weiss:1982ev}
\bibitem{Weiss:1982ev}
N.~Weiss,
%``The Wilson Line In Finite Temperature Gauge Theories,''
Phys.\ Rev.\ D {\bf 25}, 2667 (1982).
%%CITATION = PHRVA,D25,2667;%% 

\bibitem{Meisinger:2001cq}
P.~N.~Meisinger, T.~R.~Miller and M.~C.~Ogilvie,
%``Phenomenological equations of state for the quark-gluon plasma,''
arXiv:hep-ph/0108009.
%%CITATION = HEP-PH 0108009;%%

\bibitem{Pisarski:2000eq}
R.~D.~Pisarski,
%``Quark-gluon plasma as a condensate of SU(3) Wilson lines,''
Phys.\ Rev.\ D {\bf 62}, 111501 (2000)
[hep-ph/0006205].
%%CITATION = HEP-PH 0006205;%%

\bibitem{GandR}
I.~Gradshteyn and I.~Ryzhik,
{\it Table of Integrals, Series, and Products, 5th ed.}
(Academic, San Diego, 1994).  

%\cite{Starinets:1994vi}
\bibitem{Starinets:1994vi}
A.~O.~Starinets, A.~S.~Vshivtsev and V.~C.~Zhukovsky,
%``Color ferromagnetic state in SU(2) gauge theory at finite temperature,''
Phys.\ Lett.\ B {\bf 322}, 403 (1994).
%%CITATION = PHLTA,B322,403;%%

%\cite{Meisinger:1997jt}
\bibitem{Meisinger:1997jt}
P.~N.~Meisinger and M.~C.~Ogilvie,
%``Finite temperature quark confinement,''
Phys.\ Lett.\ B {\bf 407}, 297 (1997)
[arXiv:hep-lat/9703009].
%%CITATION = HEP-LAT 9703009;%%

%\cite{Savvidy:1977as}
\bibitem{Savvidy:1977as}
G.~K.~Savvidy,
%``Infrared Instability Of The Vacuum State Of Gauge Theories And Asymptotic Freedom,''
Phys.\ Lett.\ B {\bf 71}, 133 (1977).
%%CITATION = PHLTA,B71,133;%%

\bibitem{usrealsoonnow}
P.~N.~Meisinger and M.~C.~Ogilvie,
in preparation.

\end{references} 

\end{document}

hep-ph0108026.extracted_bib
\bibitem{Kapusta:1989tk}
J.~I.~Kapusta,
%``Finite Temperature Field Theory,''.
{\it Finite Temperature Field Theory}
(Cambridge , 1989).



\bibitem{Bellac:1996}
M.~Le~Bellac,
{\it Thermal Field Theory}
(Cambridge , 1996).

%\cite{Das:1997gg}

\bibitem{Das:1997gg}
A.~Das,
%``Finite Temperature Field Theory,''
{\it Finite Temperature Field Theory}
(World Scientific, Singapore, 1997)

%\cite{Dolan:1974qd}

\bibitem{Dolan:1974qd}
L.~Dolan and R.~Jackiw,
%``Symmetry Behavior At Finite Temperature,''
Phys.\ Rev.\ D {\bf 9}, 3320 (1974).
%%CITATION = PHRVA,D9,3320;%%


\bibitem{Haber1:1982}
H.~E.~Haber and H.~A.~Weldon,
%``Finite Temperature Symmetry Breaking As Bose-Einstein Condensation,''
Phys.\ Rev.\ D {\bf 25}, 502 (1982).
%%CITATION = PHRVA,D25,502;%%


\bibitem{Haber2:1982}
H.~E.~Haber and H.~A.~Weldon,
%``On The Relativistic Bose-Einstein Integrals,''
J.\ Math.\ Phys.\  {\bf 23}, 1852 (1982).
%%CITATION = JMAPA,23,1852;%%

%\cite{Actor:1986zf}

\bibitem{Actor:1986zf}
A.~Actor,
%``Zeta Function Regularization Of High Temperature Expansions In Field Theory,''
Nucl.\ Phys.\ B {\bf 265}, 689 (1986).
%%CITATION = NUPHA,B265,689;%%
 
%\cite{Actor:1987cf}

\bibitem{Actor:1987cf}
A.~Actor,
%``More On Zeta Function Regularization Of High Temperature Expansions,''
Fortsch.\ Phys.\  {\bf 35}, 793 (1987).
%%CITATION = FPYKA,35,793;%%

%\cite{Gross:1981br}

\bibitem{Gross:1981br}
D.~J.~Gross, R.~D.~Pisarski and L.~G.~Yaffe,
%``QCD And Instantons At Finite Temperature,''
Rev.\ Mod.\ Phys.\  {\bf 53}, 43 (1981).
%%CITATION = RMPHA,53,43;%%
 
%\cite{Weiss:1981rj}

\bibitem{Weiss:1981rj}
N.~Weiss,
%``The Effective Potential For The Order Parameter Of Gauge Theories At Finite Temperature,''
Phys.\ Rev.\ D {\bf 24}, 475 (1981).
%%CITATION = PHRVA,D24,475;%%
 
%\cite{Weiss:1982ev}

\bibitem{Weiss:1982ev}
N.~Weiss,
%``The Wilson Line In Finite Temperature Gauge Theories,''
Phys.\ Rev.\ D {\bf 25}, 2667 (1982).
%%CITATION = PHRVA,D25,2667;%% 


\bibitem{Meisinger:2001cq}
P.~N.~Meisinger, T.~R.~Miller and M.~C.~Ogilvie,
%``Phenomenological equations of state for the quark-gluon plasma,''
arXiv:hep-ph/0108009.
%%CITATION = HEP-PH 0108009;%%


\bibitem{Pisarski:2000eq}
R.~D.~Pisarski,
%``Quark-gluon plasma as a condensate of SU(3) Wilson lines,''
Phys.\ Rev.\ D {\bf 62}, 111501 (2000)
[hep-ph/0006205].
%%CITATION = HEP-PH 0006205;%%


\bibitem{GandR}
I.~Gradshteyn and I.~Ryzhik,
{\it Table of Integrals, Series, and Products, 5th ed.}
(Academic, San Diego, 1994).  

%\cite{Starinets:1994vi}

\bibitem{Starinets:1994vi}
A.~O.~Starinets, A.~S.~Vshivtsev and V.~C.~Zhukovsky,
%``Color ferromagnetic state in SU(2) gauge theory at finite temperature,''
Phys.\ Lett.\ B {\bf 322}, 403 (1994).
%%CITATION = PHLTA,B322,403;%%

%\cite{Meisinger:1997jt}

\bibitem{Meisinger:1997jt}
P.~N.~Meisinger and M.~C.~Ogilvie,
%``Finite temperature quark confinement,''
Phys.\ Lett.\ B {\bf 407}, 297 (1997)
[arXiv:hep-lat/9703009].
%%CITATION = HEP-LAT 9703009;%%

%\cite{Savvidy:1977as}

\bibitem{Savvidy:1977as}
G.~K.~Savvidy,
%``Infrared Instability Of The Vacuum State Of Gauge Theories And Asymptotic Freedom,''
Phys.\ Lett.\ B {\bf 71}, 133 (1977).
%%CITATION = PHLTA,B71,133;%%


\bibitem{usrealsoonnow}
P.~N.~Meisinger and M.~C.~Ogilvie,
in preparation.

