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\begin{document}


\vspace{1cm}

\title{
\begin{flushright} 
{\small SUNY-NTG-02/32}
\end{flushright} 
 Fluctuation Induced Critical Behavior at Non-Zero Temperature
\\  and Chemical Potential}  


\author{K.\ Splittorff$^{\, (a)}$, J.\ T.\ Lenaghan$^{(b)}$, and 
J.\ Wirstam$^{(c)}$\footnote{Former address: Department of 
Physics, Brookhaven National Laboratory, Upton, {\sl New York 11973}}}


\address{
$^{(a)}$ Department of Physics and Astronomy, SUNY, Stony
Brook, {\sl New York 11794}\\ $^{(b)}$ Department
of Physics, University of Virginia, 382 McCormick Rd.,
Charlottesville, {\sl VA 22904-4714} \\ $^{(c)}$ Swedish Defense
Research Agency (FOI), S-172 90 Stockholm, Sweden}
\maketitle

\begin{abstract}
The presence of a chemical potential explicitly breaks Lorentz
invariance and may additionally break other internal symmetries.  This
introduces new subtleties in the determination of the critical
properties of field theories at nonzero temperature.  In this article,
we discuss the occurrence of phase transitions 
in relativistic systems as a function of both
chemical potential and temperature.  We find that the $\epsilon$-expansion at
nonzero chemical potential is not affected by the explicit breaking of
Lorentz invariance but is sensitive to the breaking of additional
symmetries by the chemical potential.  We derive effective three
dimensional theories at criticality and study the effect of the
chemical potential on the fixed point structure of the
$\beta$-functions within several different contexts:
first for a scalar field theory with a global $U(1)$ symmetry, then
for the chiral phase transition in QCD with two colors and two
flavors, and finally, we consider QCD with
three colors at nonzero baryon and isospin chemical potential. 
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}
\label{sec:intro}

The concept of universality is a powerful tool in the description of
second--order phase transitions.  In many cases, a mean field
treatment of the
Landau theory for the order parameter provides a sufficient
description of the universal properties of the phase transition.
However, in some cases, a mean--field analysis is not reliable since
higher-order corrections to the Landau theory may result in
$\beta$-functions which do not have stable fixed points.  
The form of
the higher-order corrections depends directly upon the detailed
symmetries of the problem.
One can use the $\epsilon$-expansion to determine the
stability of the fixed points of the $\beta$-functions in the
Landau theory \cite{BKM,BD,amit}.  If there are no stable fixed 
points, the transition is first order induced by fluctuations
\cite{BKM,BD,amit,cw,yamagishi}.  This approach 
has lead to a successful description of some temperature induced phase
transitions in both solid state systems \cite{BKM} and relativistic
systems \cite{PS,robfrank,arnoldyaffe}. This success is perhaps somewhat
surprising given the fact that at high temperature the systems are
effectively three dimensional and the renormalization group analysis
is consequently based upon the $\epsilon$-expansion for $\epsilon \to
1$.

In this article, we discuss the influence of chemical potentials on
the fixed point structure of the $\beta$-functions of systems
which are relativistically invariant at zero temperature and zero
chemical potential. Both the temperature and the chemical potential 
explicitly break the O(4) relativistic invariance to
O(3).\footnote{Throughout this article, we will work in Euclidean space.}
The presence of a chemical potential may also break additional
symmetries both explicitly and spontaneously through, for example,
Bose condensation or Cooper pairing. Hence, there are three
characteristic effects of the 
chemical potential: {\sl 1)} explicit breaking of Lorentz
invariance, {\sl 2)} explicit breaking of symmetries in addition
to Lorentz invariance, and {\sl 3)} spontaneous breaking of symmetries.

The relativistic system we will focus on here is strongly interacting
systems as described by quantum chromodynamics (QCD) and the
spontaneous breaking of symmetries induced by the chemical potential
which we will consider are through Bose condensation. The possibility
of superconducting phases has been studied using renormalization group
techniques in \cite{BCSreno}. In that case gauge fields can play an
important role and are incorporated into a Landau-Ginzburg theory. 

We discuss the effects of chemical potentials by studying three
examples, provided by  
a complex scalar field theory with a global $U(1)$ symmetry and the
Landau theories for the chiral symmetry restoring
phase transition in QCD -- with two and three
colors at nonzero chemical potentials and zero quark masses.
These examples are chosen to illustrate the three distinct scenarios for the
role played by the chemical potential.  

In the $U(1)$ model and in two
color QCD the order parameter field is charged and the chemical
potential, $\mu$, couples directly to the Landau degrees of freedom.  In the
$U(1)$ model no additional symmetries are explicitly broken, whereas
in two color QCD the baryonic chemical potential explicitly breaks the
$U(2N_{\! f})$ flavor invariance to $U(N_{\! f})\times U(N_{\!
f})$. This symmetry is 
in turn broken spontaneously to $Sp(N_{\! f})\times Sp(N_{\!
f})\times U(1)$ by
the formation of a diquark condensate.  After taking into account
this symmetry breaking we find that, under certain restrictions to
be discussed, the theories dimensionally reduce
in the standard way to effective three dimensional theories even if
$\mu>T$. 

The case of ordinary QCD with three colors is somewhat
special in that the baryonic chemical potential couples only indirectly
to the effective theory since the order parameter is not charged under
baryon number. 
This implies that the $\beta$-functions are not affected by the presence of
a baryonic chemical potential. This point was previously raised in
\cite{HS}, and here we discuss the region in the $(\mu,T)$-plane
where such a simplification applies. A nonzero isospin chemical
potential however couples directly to the order parameter and in this
case we determine the influence of the chemical potential on the order
of the phase transition. This example also allows us to compare
with existing lattice simulations.

Before we start our analysis let us emphasize that the presence of 
fixed points within the $\epsilon$-expansion only implies that the phase
transition can be of second order, since higher dimensional operators
may render the phase transition first order. Additionally, the
possible existence of a nonperturbative fixed point may drive a phase
transition to second order. 


The paper is organized as follows. In the next section, we discuss the
case where the introduction of a chemical potential only leads to a
breaking of the Lorentz invariance (or, in Euclidean space, of the
O(4) rotational symmetry). In sec.\ III we extend the analysis to 
$N_{\! c}=2$ QCD, where, as mentioned above, additional 
global flavor symmetries are broken. Three-color QCD is discussed
in sec.\ IV and sec.\ V, and we end with our conclusions in sec.\ VI.
 
\section{Dimensional reduction at nonzero $\mu$}
\label{sec:dimreduct}

We study temperature induced phase transitions in relativistic
systems with a nonzero chemical potential from the
perspective of the $\epsilon$-expansion. The approach begins with the
four dimensional Landau theory for the relevant order parameter
$\Phi$. This effective theory is the most general renormalizable
Lagrangian consistent with the relevant symmetries.  Because of the
explicit breaking of Lorentz invariance the kinetic term may take a
non-standard form ($B$ is the charge matrix defined below): 
\be
\Tr\left[\d_0\Phi^\dagger\d_0\Phi\right] 
+v^2(\mu,T)\Tr\left[\d_i\Phi^\dagger\d_i\Phi\right]+\mu
q_1(\mu,T)\Tr\left[B\Phi^\dagger\d_0\Phi\right]+ \mu q_2(\mu,T)\Tr\left[B\Phi\d_0\Phi^\dagger\right] 
\ .  
\ee 
The functions $q_1$ and $q_2$ are constrained by the 
symmetries of the underlying microscopic Lagrangian and determine the
conserved current for the charge to which the chemical potential is
associated.  This is analogous to how the chemical potential
enters in chiral perturbation theory
\cite{KST,KSTVZ,SS,SSS,KT,LSS,STV1,STV2}, the difference being mainly
that the Goldstone field has fewer components than the generalized
order parameter field of the Landau theory. 

To discuss how the chemical potential affects the renormalization group
equations we consider the simplest possible example, a complex scalar
field theory with a $U(1)$ symmetry to which we couple a chemical
potential.  The Lagrangian is 
\be\label{LagrangianU(1)} 
L_{4d}  & = &
\del_0\Phi^*\del_0\Phi+v^2\del_i\Phi^*\del_i\Phi+\mu \left[\Phi\del_0
\Phi^*-(\del_0
\Phi)\Phi^*\right]+(m^2-\mu^2)\Phi^*\Phi+\lambda(\Phi^*\Phi)^2 \ . 
\ee 
The Lagrangian reveals the standard coupling of $\mu$ to the
zeroth component of the conserved current. It is the lowest order
coupling of the chemical potential to the order parameter field
consistent with the $U(1)$ invariance.

The analysis of the fluctuation induced critical behavior proceeds in
four steps: 
{\sl Step 1:} Fourier decompose the fields and integrate over
$x_0\in [0,1/T]$.  
{\sl Step 2:} Determine the propagators in the resulting three
dimensional theory.
{\sl Step 3:} Integrate out the massive Matsubara modes to get the
effective three dimensional theory.
{\sl Step 4:} Study the stability of the fixed points of the
$\beta$-functions in the effective three dimensional theory. 

Writing out the Lagrangian in Eq.\ (\ref{LagrangianU(1)}) in terms of
the real components of the order parameter $\Phi\equiv a+ib$, we find 
\be
L_{4d} & = & \del_0 a\del_0 a+v^2\del_i a\del_i a  +\del_0 b\del_0
b+v^2\del_i b\del_i b - 2i\mu \,
\left(a\del_0b-b\del_0 a\right)
+\left(m^2-\mu^2\right)\left(a^2+b^2\right) +\lambda \,
\left(a^2+b^2\right)^2 \ .  
\ee 
Since the temporal direction is
compact, we may Fourier decompose the fields as
\be 
a(x_0,\vec{x}) = T \sum_{n=-\infty}^\infty e^{i\omega_n x_0} a_n(\vec{x}) \ \ \ \ {\rm and}
 \ \ \ \ b(x_0,\vec{x}) = T \sum_{n=-\infty}^\infty e^{i\omega_n x_0} b_n(\vec{x}) \,\,.
\ee 
The dimensional reduction then follows by inserting these
expressions into the corresponding action and integrating $x_0$ from 0
to $1/T$. Using that $a_{-n}=a_n^*$ and $b_{-n}=b_n^*$, we get 
\be
L_{3d} & = & T \sum_{n=-\infty}^\infty \left[v^2\del_i a_n\del_i a_n^*
+v^2\del_i b_n\del_i 
b_n^*+\left\{\omega_n^2+m^2-\mu^2\right\}(a_na_n^*+b_nb_n^*)
-2\mu\omega_n(a_nb_n^*-b_na_n^*)\right]+\lambda-{\rm terms} \ .   
\ee 

Next we write the three dimensional Lagrangian
in terms of the real fields $a_n\equiv c_n+id_n$ and $b_n\equiv
e_n+if_n$ 
\be 
L_{3d} & = & T \sum_{n=-\infty}^\infty \big[v^2((\del_i c_n)^2+(\del_i d_n)^2+(\del_i e_n)^2
+(\del_if_n)^2)+\left\{\omega_n^2+m^2-\mu^2\right\}(c_n^2+d_n^2+e_n^2+f_n^2) \\
&& - 4 i \mu \omega_n (d_ne_n-c_nf_n)\big]+\lambda-{\rm terms} \
. \nonumber 
\ee 
The propagator matrix in the $(c_n,f_n)$-sector and
equivalently in the $(e_n,d_n)$-sector is 
\be\label{propU(1)}
\left( \matrix{ 
v^2 p^2+\omega_n^2+m^2-\mu^2 & -2i\mu\omega_n \cr
-2i\mu\omega_n & v^2 p^2+\omega_n^2+m^2-\mu^2} \right) \ .  
\ee

Given the propagator, we can derive the effective three dimensional 
theory by integrating out the massive Matsubara modes which in the
present case are all the nonzero modes. In doing so we
extend Ginsparg's analysis \cite{ginsparg} to $\mu\neq0$. We first
choose the vacuum expectation value of the order
parameter to be in the direction of $a_0$, i.e. $b$ is the Goldstone
boson. The leading order contribution in $\lambda$ from the nonzero
Matsubara modes in the effective three dimensional theory of the
massless zeroth Matsubara mode is then the one-loop correction to
the mass of the $a_0$. Schematically in going from four dimensions to
three dimensions we have
\be
 (m^2-\mu^2)a_0^2 \to (m^2-\mu^2)a_0^2+ \lambda T
\left(3\sum_{n\neq0}a_na_{-n}+\sum_{n\neq0}b_nb_{-n}\right)a_0^2 \ .
\ee
Using the propagator in Eq.\ (\ref{propU(1)}), each of the summations
over $n\neq0$ leads to a correction term 
\be M^2(T,m^2,\mu^2)
& = & \frac{1}{2}\lambda T\sum_{n\neq 0}\int \frac{{\rm d}^{d-1}
p}{(2\pi)^{d-1}}\left(\frac{1}{v^2 p^2+m^2-(\mu+i\om_n)^2}+
\frac{1}{v^2  p^2+m^2-(\mu-i\om_n)^2} \right) \ .
\ee 
Inserting 
\be 
\frac{1}{v^2 p^2+m^2-(\mu\pm i\om_n)^2} = \int_0^\infty {\rm d}t \
e^{-(v^2 p^2+m^2-(\mu\pm i\om_n)^2)t} 
\ee 
and integrating first over $p$ then over $t$ we get
\be 
M^2(T,m^2,\mu^2) & = & \frac{\lambda
T}{2}\frac{\pi^{(d-1)/2}}{(2\pi v)^{d-1}}\Gamma\left(\frac{3}{2}-\frac{d}{2}
\right)
\sum_{n\neq0}\left[(m^2-(\mu+i\om_n)^2)^{(d-3)/2}+(m^2-(\mu-i\om_n)^2)^{(d-3)/2}\right]
\ .  
\ee 
Assuming that $T\gg |m|$, we can expand the argument of the sum in
powers of $m^2/\om_n^2$. In the limit $d\to4$ we find (ignoring the
regularization of the next to leading order terms)
\be 
M^2(T,m^2,\mu^2) & = & -\frac{\lambda T^2}{4\pi v^3} \sum_{n=1}^\infty
2\pi n \left[2+\frac{1}{1+(\mu/\om_n)^2}\frac{m^2}{\om_n^2}
-\frac{1-3(\mu/\om_n)^2}{4(1+(\mu/\om_n)^2)^3}\frac{m^4}{\om_n^4}
+{\cal O}\left(\frac{m^6}{\om_n^6}\right)\right] \ .  
\ee 
The leading order term, $M^2(T,m^2,\mu^2) =  \lambda T^2/(12v^3)(1+{\cal
O}(m^2/(2\pi T)^2))$, is independent of both $m$ and $\mu$. Moreover,
this holds independently of the ratio\footnote{This is not the case if we
consider an imaginary chemical potential. In that case the nonzero
Matsubara modes become massless when $\mu=\om_n$. Consequently, they
must be included in the effective theory for the phase transition.} of
$\mu$ and $T$.
In other words, at leading order in $m^2/(2\pi T)^2$, the correction takes
the same form as at $\mu=0$ and we get 
\be\label{LU(1)zero-mode} 
L_{3d-{\rm eff}} =
\del_i a_0\del_i a_0 +\del_i b_0\del_i b_0+\left(m^2-\mu^2+\frac{\lambda
T^2}{3 v^3}\right)a_0^2+\lambda \left(a_0^2+b_0^2\right)^2 \ .  
\ee 
Hence, if the $U(1)$ symmetry is broken at zero temperature, we
reproduce the standard result for the temperature at the 
symmetry restoration (see eg. \cite{HW,kapustaPRD} and \cite{BBD} where the
relation to the effective potential is also explained in detail). 
Of course, for the theory to be self--consistent at the phase
transition temperature $T_c$, the condition $T_c\gg |m|$ must be
fulfilled. From Eq.\ (\ref{LU(1)zero-mode}) we see that $T_c^2 = 
3v^3(\mu^2-m^2)/\lambda$. If $m^2<0$ and $\lambda \ll 1$ the consistency
requirement is always fulfilled. If the coupling constant increases
to $\lambda\simeq 1$ while $m^2<0$, we still have $T_c\gg |m|$ if
$\mu^2\gg -m^2$: however, the one--loop approximation is no longer
trustworthy for such large values of $\lambda$, and the results not reliable. 
On the other hand, when $m^2>0$ the situation becomes
different. Since $T^2_c>0$, a necessary condition is $\mu \geq m$, and
unless $\lambda$ is {\em extremely} small, we must have
$\mu\gg m$ for consistency (i.e. to satisfy $T_c\gg |m|$).

The general lesson to be learned from this is that the presence of the
chemical potential in the effective three dimensional theory is felt
only through the direct term, $-\mu^2 \Phi^*\Phi$, in the
potential. At criticality the quadratic mass-like terms vanish, and the
effective theory in three dimensions relevant at the phase transition 
contains only the spatial derivative terms and the quartic couplings.
Specifically, in the $U(1)$--model considered above, the effective
Lagrangian is reduced to, 
\be
L_{3d-{\rm eff}}(T_c) = \del_i a_0\del_i a_0 +\del_i b_0\del_i
b_0+\lambda(a_0^2+b_0^2)^2 \ .  
\ee 
This Lagrangian has stable fixed
points for the $\beta$-function and consequently the phase transition
is second order \cite{GZ-J}. Again we want to emphasize that
this holds true independently of the value of $\mu$ as
long as $T_c\gg |m|$.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{When $\mu$ breaks internal symmetries}
\label{sec:Nf=Nc=2}

We now consider the case in which the chemical potential explicitly
breaks symmetries in addition to Lorentz invariance. This is
exemplified by QCD with two colors 
and two flavors at nonzero baryonic chemical potential. For simplicity
we ignore the effect of the $U_A(1)$ axial anomaly. The general case 
including the anomaly and any even number of flavors is treated in Ref.\
\cite{WLS}. 
QCD with two colors and two massless flavors enjoys a $U(2N_{\! f})$
classical invariance. This invariance is explicitly broken to 
$U(N_{\! f})\times U(N_{\! f})$ when $\mu\neq0$. This remaining symmetry is
spontaneously broken down to $Sp(N_{\! f})\times Sp(N_{\! f})\times U(1)$ for
any nonzero value of $\mu$ by the formation of a diquark condensate
(see eg. \cite{KST,KSTVZ}).
   
We assume that the order parameter field can be represented by a
complex anti-symmetric matrix,
\be 
\Phi\equiv \left( \matrix{
X_1 & X_2 \cr -X_2^T & X_3} \right) \label{definematrix}
\ee 
where $X_1^T=-X_1$ and $X_3^T=-X_3$. In this basis, the baryonic
charge matrix is 
\be 
B\equiv \left( \matrix{1 & 0 \cr 0 & -1} \right) \ .
\ee 

The four dimensional Landau theory for QCD with two
colors which is relevant for the renormalization group analysis of the
temperature induced phase transition must be invariant under the same
$U(2N_{\! f})$ symmetry as the microscopic Lagrangian, and so is given
by (cf. \cite{KST,KSTVZ})
\be 
L_0 & = & 
\frac{1}{2}\Tr[(\del_\nu\Phi^\dagger ) (\del_\nu\Phi
)-2\mu\{\Phi,B\}\d_0\Phi^\dagger-2\mu^2(\Phi\Phi^\dagger+\Phi
B\Phi^\dagger B)] + V \ ,  
\ee 
where the potential is,
\be \label{pot} V= \frac{m^2}{2} {\rm
Tr} \left [\Phi^{\dagger} \Phi \right ] +\lambda_1 \bigl ( \Tr \left [
\Phi^{\dagger} \Phi \right ] \bigr )^2 + \lambda_2 {\rm Tr} \bigl [
\left (\Phi^{\dagger} \Phi \right )^2 \bigr ]\ .  
\ee  
In the four dimensional theory, the quadratic part of the Lagrangian
becomes, by using the representation in Eq.\ (\ref{definematrix}),  
\be && \frac{1}{2}\Tr[(\del_{\nu}\Phi^{\dagger} )
(\del_{\nu}\Phi )]+\frac{m^2}{2} {\rm Tr} \left [\Phi^{\dagger} \Phi
\right ]-\mu^2{\rm Tr} \left [\Phi\Phi^\dagger+\Phi B\Phi^\dagger
B\right ]-\mu{\rm Tr} \left [\{\Phi,B\}\d_0\Phi^\dagger\right ] \\ 
& = & \frac{1}{2}\Tr[\del_\nu
X_1^\dagger\del_\nu X_1]+\Tr[\del_\nu X_2^\dagger\del_\nu
X_2]+\frac{1}{2}\Tr[\del_\nu X_3^\dagger\del_\nu X_3] \nn \\ 
&& +m^2 \Tr[X_2^\dagger X_2]+\frac{1}{2}(m^2-4\mu^2)\Tr[X_1^\dagger
X_1+X_3^\dagger X_3]-2\mu\Tr[X_1 \d_0 X_1^\dagger-X_3 \d_0
X_3^\dagger] \ .  \nn
\ee 
Note, the explicit breaking of $U(2N_f)$ at $\mu\neq0$. We now 
proceed with the steps {\sl 1} to {\sl 4} introduced in the previous
section. For simplicity we focus on two flavors. From the equation
above it is clear that the $X_2$ modes do not couple directly to the
chemical potential. The minimum of the potential for the diquark order
parameter 
\be
\Phi_0=\phi_0\left(\begin{array}{cccc}0&-i&0&0 \\ i&0&0&0 \\ 0&0&0&-i
\\ 0&0&i&0 \end{array}\right)
\ee
however changes with $\mu$. This imply that the $X_2$ modes 
remain massive at the phase transition. Hence all Matsubara modes
from $X_2$ must be integrated out. The $X_1$ and $X_3$ modes do couple
to the chemical potential and we now derive their dispersion relations
in the three dimensional theory. Setting 
\be 
X_1 \equiv
\frac{1}{\sqrt{2}}\left( 
\matrix{0 & a_1+i b_1 \cr -a_1- i
 b_1 & 0} \right) 
\ee 
and likewise for $X_3$, then Fourier decomposing the real fields and
integrating over $x_0\in[1,1/T]$ we get the three dimensional theory.
We find that the dispersion relations in the $X_1$ (or $X_3$) sector
are given by   
\be 
\left( \matrix{ 
p^2+\omega_n^2+m^2-4\mu^2 & -4i\mu\omega_n \cr
-4i\mu\omega_n & p^2+\omega_n^2+m^2-4\mu^2} \right) \ .  
\ee 
This is precisely the same dispersion relations as we found in the $U(1)$ 
model, eq. (\ref{propU(1)}), except that the charge of the modes is two 
and not one. When $m^2= (2\mu)^2$, there are four massless modes in
the three dimensional theory, namely the four zeroth Matsubara modes of $X_1$
and $X_3$. In the $U(1)$ model all modes are charged and all zeroth
Matsubara modes become massless at the phase transition. In the
present case only the $X_1$ and $X_3$ modes have nonzero baryonic charge
and these are exactly the fields who's zeroth Matsubara modes becomes
massless at the phase transition. 
The effective theory at the phase transition is given
in terms of these zeroth Matsubara modes and reads   
\be 
L_{3d-{\rm eff}}(T_c) & = & \frac{1}{2}\left[\d_j
a^{(1)}_0\d_j a^{(1)}_0 +\d_j a^{(3)}_0\d_j a^{(3)}_0 + \d_j
b^{(1)}_0\d_j b^{(1)}_0+ \d_j b^{(3)}_0\d_j b^{(3)}_0\right] \\ &&
+\lambda_1
\left[\left(a_0^{(1)}\right)^2+\left(b_0^{(1)}\right)^2+\left(a_0^{(3)}\right)^2+\left(b_0^{(3)}\right)^2\right]^2 \nn \\ && 
+\frac{\lambda_2}{2}\left[\left(\left(a_0^{(1)}\right)^2+\left(b_0^{(1)}\right)^2\right)^2+\left(\left(a_0^{(3)}\right)^2+\left(b_0^{(3)}\right)^2\right)^2\right]
\nn \ .  
\ee 
This Lagrangian has an $O(2)\times O(2)$ symmetry and
from the results of \cite{PS} we know that the $\beta$-functions have
a marginally stable fixed point.  Hence, the order of the phase
transition is not determined at one-loop level.
If one takes the axial anomaly into account, the symmetry is reduced
to O(2) and the $\beta$-function develops a fixed point
\cite{WLS}. Consequently, the phase transition is of second order in
the presence of the axial anomaly. At zero chemical potential but
$c\neq0$, the phase transition remains second order 
, however, now with O(6) critical exponents \cite{wirstam}. 

The effect of the chemical potential on the fixed point structure of
the $\beta$-functions is only through the explicit breaking of the
flavor symmetries. This result holds as long as $T_c\gg |m|$, and the
consistency requirement is fulfilled for all values of $\mu$ if
$m^2<0$. On the other hand, when $m^2>0$ self-consistency requires that
$\mu\gg m$.  The case
$m^2<0$ mimics the phase structure for massless quarks while the case
with $m^2>0$ resembles two color QCD when the quarks have a common,
nonzero mass. Some caution must be exercised when considering
infinitesimally small values of $\mu$, since in that case the masses of
$X_2$-modes in $\Phi$ is not well separated from those of $X_1$ and
$X_3$. The results given above only apply when the  mass scales are
well separated. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Three color QCD at nonzero baryonic chemical potential}
\label{sec:Nc=3}

We now consider the case in which a chemical potential is
introduced for a charge which is not carried by the order parameter
field. This is exemplified by the chiral phase transition in QCD with
three colors and massless quarks at nonzero baryonic chemical
potential. 
For not too large values ($\mu_B< M_{\rm Nucleon}/3$) the baryonic
chemical potential does not induce a charged condensate and hence the
order parameter does not change.
Following \cite{robfrank} we assume that this chiral symmetry breaking 
order parameter field can be parametrized by a complex
$N_f\times N_f$ $\Phi$ for the values of $\mu_B$ under
consideration. Because the baryonic chemical potential 
explicitly break Lorentz invariance the allowed form of the kinetic
term is  
\be
\Tr[\d_0\Phi^\dagger\d_0\Phi]+v^2(\mu,T)\Tr[\d_i\Phi^\dagger\d_i\Phi]
+i\mu q(\mu,T)\left(\Tr[\Phi^\dagger\d_0\Phi]-\Tr[\Phi\d_0\Phi^\dagger]\right)
\ . 
\ee
However, since all components of the order parameter field have zero
baryonic charge the current carries no charge. In chiral perturbation
theory this is analogous: The pions have baryonic charge zero and
consequently the baryonic chemical potential does not appear in that
context.  
At the chiral phase transition it is possible that the charge function
$q(\mu,T)$ is nonzero for $\mu\neq0$ since the current term is not
excluded by the  global symmetries. However, whether or not $q(\mu,T)$
is numerically small, as we have shown above, its influence on the
fixed point structure is negligible as long as $T_c\gg |m|$.  In
\cite{HS}, Hsu and Schwetz considered the $\beta$-functions for massless
QCD at nonzero baryonic chemical potential. They suggested that the linear
derivative terms can be neglected along the entire phase transition in
the $(\mu,T)$-phase diagram. Consequently they found that the
order of the chiral phase transition does not change in the
$(\mu,T)$-plane.  Based on our analysis here, we agree with 
their arguments as long as $T_c\gg |m|$.  For $T_c\sim m$, the
consistency of the approach breaks down and an alternative expansion
scheme must be employed. Such a theory must interpolate between the
critical behavior in the three dimensional theory relevant for
$T_c\gg|m|$ and the full four dimensional Landau theory relevant at
$T=0$. We are not aware of the existence of such a scheme. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Three color QCD at nonzero isospin chemical potential}
\label{sec:Nc=3muI}

Finally, we discuss the influence on the fixed point structure 
of the $\beta$-functions if we allow for different chemical potentials 
for different flavors. A physically relevant case is given by $N_c=3$
with two massless quarks at nonzero baryonic chemical potential,
$\mu_B=\mu_u+\mu_d$, and nonzero isospin chemical potential,
$\mu_I=\mu_u-\mu_d$. The isospin chemical potential breaks the flavor
invariance explicitly and we are therefore in a similar situation as
in Sec. \ref{sec:Nf=Nc=2}. 

At $\mu_I=0$ the flavor symmetries is $SU(2)\times SU(2)\times
U_A(1)\times U_V(1)$ and again following \cite{robfrank} we assume
that the chiral symmetry breaking order parameter field can be
parametrized by a complex $N_f\times N_f$ matrix $\Phi$ transforming as 
\be
\Phi \to U \Phi V \ ,
\ee
where $U,V\in U(N_f)$.
At nonzero isospin chemical potential the four dimensional Landau theory
reads (compare with \cite{SS})
\be 
L_0 & = & 
\frac{1}{2}\Tr[(\del_\nu\Phi^\dagger ) (\del_\nu\Phi)]
-\frac{\mu_I}{4}\Tr[[\tau_3,\Phi] \d_0\Phi^\dagger-h.c.]
+\frac{\mu_I^2}{4}\Tr[\Phi\tau_3\Phi^\dagger\tau_3-\Phi\Phi^\dagger] 
+ V \ ,\ee 
where the potential is given by 
\be
V = \frac{m^2}{2}\Tr[\Phi^{\dagger} \Phi]+\lambda_1(\Tr[\Phi^{\dagger}
\Phi])^2 +\lambda_2\Tr[(\Phi^{\dagger} \Phi)^2]
+c\det(\Phi+\Phi^*) \ .
\ee
The chiral condensate breaks $SU(2)\times SU(2)$ to $SU(2)$. The
remaining $SU(2)$ is broken explicitly when the isospin chemical
potiential is nonzero.  Furthermore if $\mu_I>|m|$ then the $U_V(1)$
is spontaneously broken as a pion condensate forms and picks a
direction in isospin space, say 
\be
\Phi_0\equiv i\phi_0\tau_2 \ .
\ee
Chiral perturbation theory at nonzero $\mu_I$ has been discussed in
\cite{SS,SSS,KT,STV2}.    

As for two color QCD at nonzero baryonic chemical potential, the isospin
chemical potential in 3 color QCD splits the masses of the charged and
the noncharged modes. At criticality only zeroth Matsubara modes of the
charged modes are massless. The effective three dimensional theory at the
phase transition in terms of these zeroth Matsubara modes reads 
\be \label{L3dmuI}
L_{\rm 3d-eff}(T_c) & = & 
\frac{1}{2}\Tr[(\del_i\Phi^\dagger ) (\del_i\Phi)]
+\lambda_1(\Tr[\Phi^\dagger \Phi])^2 
+\lambda_1\Tr[(\Phi^\dagger \Phi)^2] +c\det(\Phi+\Phi^*) 
\ee 
where $\Phi$ now has four real components $a$, $b$, $d$, and $f$
\be
\Phi=\left(\begin{array}{cc} 0 & a+i b \\ d+if & 0 \end{array}\right) \ .
\ee
In order to determine the one loop $\beta$-functions in three dimensions
we expand the Lagrangian about the vacuum $\Phi\to\Phi_0+\Phi$ and
make use of the background field method at one-loop level. The
$\beta$-functions when the axial anomaly is not present at the phase
transition are ($\kappa$ is the arbitrary mass scale) 
\be 
\beta_1 = &&\kappa\frac{\del\lambda_1}{\del\kappa} =
-\epsilon\lambda_1 +\frac{1}{\pi^2}\biggl [ 
6\lambda_1^2+4\lambda_1\lambda_2\biggr ]\label{betafunctions1} \ , \\
\beta_2 = &&\kappa\frac{\del\lambda_2}{\del\kappa} 
= -\epsilon\lambda_2 +\frac{1}{\pi^2}\biggl [
5\lambda_2^2+6\lambda_1\lambda_2 \biggr ] \ . \label{betafunctions2}
\ee
The fixed points ($\lambda_1^*,\lambda_2^*$) to order $\epsilon$ and
the eigenvalues of the stability matrix $S$ at the fixed point are 

\begin{itemize}

\item $\lambda_1^*=\lambda_2^*=0$. \ \ \ \ \ \ \ \ \ The eigenvalues of
$S$ at $(\lambda_1^*,\lambda_2^*)$ are $-\epsilon$ and $-\epsilon$ and
hence this fixed point is not stable.    

\item $\lambda_1^*=0$, $\lambda_2^*=\epsilon\pi^2/5$. The eigenvalues of
$S$  at $(\lambda_1^*,\lambda_2^*)$ are $-\epsilon/5$ and $-\epsilon$
and hence this fixed point is not stable.   

\item $\lambda_1^*=\epsilon\pi^2/6$, $\lambda_2^*=0$. The eigenvalues of
$S$  at $(\lambda_1^*,\lambda_2^*)$ are $0$ and $\epsilon$ and hence
this fixed point is marginally stable.    

\end{itemize}

As indicated there is one marginally stable fixed point and this implies
that the order of the phase transition is not determined at one-loop when
the axial anomaly is not present at $T_c$. 

The anomaly term in (\ref{L3dmuI}) is a mass-like operator since we
consider $N_f=2$. The explicit breaking of the $U_A(1)$ splits the
masses of the modes with the effect that only $2$ Matsubara modes are
massless at criticality. The effective theory of these massless modes
at $T_c$ is the familiar $O(2)$ symmetric $\phi^4$ theory. 
Hence the phase transition is second order, see \cite{GZ-J} and references
therein. 

Let us compare this result to the one at $\mu_I=0$ derived in
\cite{robfrank}. For $c=0$ and $\mu_I=0$ the phase transition is first
order induced by fluctuation. Turning on $\mu_I$ a marginally stable 
fixed point appears. For $c\neq0$ the phase transition may be second
order. If so it is characterized by $O(4)$ critical exponents at $\mu_I=0$
and by $O(2)$ at $\mu_I\neq0$.

Euclidean QCD at nonzero isospin chemical potential does not suffer
from the notorious sign problem. Hence it is possible to conduct first 
principle numerical computations at nonzero isospin chemical potential
using the standard methods \cite{AKW}. Such simulations have exposed 
\cite{Kogut:2002tm,KS2} a rich phase diagram in the
$(\mu_I,T)$-plane. For $T=0$ the pion condensate sets in when $\mu_I$
exceeds the pion mass. This transition 
into the pion phase is second order. For temperatures 
on the order of the pion mass and higher the phase transition changes
from second order to first order. This scenario has been explained
within the context of chiral perturbation theory \cite{STV2}. 
Making a direct comparison to the predictions in the present paper is 
delicate since the lattice simulations necessarily work at a nonzero
quark mass. However, we expect that if $\mu_I\gg m_\pi$ the results
should be independent of the small common quark mass. If the phase
transition remains first order for $\mu_I\gg m_\pi$ then for consistency
with our results the axial anomaly must be restored below $T_c$.
We stress however that in our approach we can not exclude
first order phase transitions which are driven by higher dimensional
operators, that is we can not conclude from this approach that a phase
transition is definitely second order.  
 
Introducing the baryonic chemical potential in addition to the
isospin chemical potential does not affect the order of the
chiral phase transition. The baryonic chemical potential does not 
explicitly break any additional symmetries and in the range we
consider it does not lead to additional condensates.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusions}
\label{sec:conc}

We have described the effect of a chemical potential on the order of 
the temperature induced phase transitions in relativistic systems. We
have focused on the 
stability of the fixed points of the $\beta$-functions and have
studied three examples.  Our examination of the $U(1)$ model shows
that the effect of Lorentz breaking in the Landau theory for
the order parameter does not affect the renormalization group
equations as long as $T\gg |m|$. The effect of the chemical potential
on the $\beta$-functions is only through the breaking of internal
symmetries in addition to Lorentz invariance. We have illustrated this
by examining QCD with two colors and two flavors. The existence of
fluctuation induced phase transitions in two color QCD is studied in
further detail in \cite{WLS}. As another example of how the chemical
potential affects the stability of the $\beta$-functions, we
considered the chiral phase transition in ordinary three color QCD at
nonzero baryonic chemical 
potential. In this case the order parameter field is neutral with
respect to the charge and the Lorentz breaking in the effective theory
is suppressed.  In the range of temperatures and chemical potentials
under consideration no additional symmetries are broken by the
baryonic chemical potential. In agreement with \cite{HS} we conclude
that the order of the phase transition in QCD does not
change. However, we stress that this result is only self consistent
for $T_c\gg |m|$. The situation is quite different when we consider 
a nonzero isospin chemical potential in 3 color QCD. In that case 
part of order parameter field does have nonzero third component of
isospin. It is only thise components which lead to massless zeroth Matsubara
modes in the effective three dimensional theory at the phase
transition. The modes with zero third component of isospin remain
massive at the phase transition when the isospin chemical potential
is nonzero. Since the number of degree of freedom at the phase
transition changes the $\beta$-functions change. This leads to a new 
stability pattern of the fixed point and finally to a different prediction for 
the order of the phase transition.     

The examples studied illustrates three main effects of the chemical
potential: {\sl 1)} the explicit breaking of Lorentz invariance, {\sl
2)}  the explicit breaking of global symmetries and {\sl 3)}
spontaneous breaking of symmetries through Bose condensation. The
discussion is in this way relevant for all Landau theories which are
relativistically invariant at zero temperature and chemical potential. 

Finally let us stress that the
analysis as performed here does not address all caveats associated
with fluctuation induced phase transitions. For example fixed points
outside the reach of perturbation theory can change the conclusions.     


\acknowledgements It is our pleasure to thank A.D. Jackson and
R.D. Pisarski for useful conversations and comments on the
manuscript. K.S.\ wishes to thank T. Sch\"afer and J.J.M. Verbaarschot
for pointing out an error at a crucial point.  
The work of K.S. was supported by the Rosenfeld foundation. 
J.T.L.\ is supported by the U.S.\ DOE grant DE-FG02-97ER41027.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{document}


