\documentclass[%
prl,aps,amsmath,floatfix,superscriptaddress,showpacs,twocolumn%
]{revtex4}
\usepackage{bm}
\usepackage{amsmath}
\usepackage{epsfig}
\begin{document}
\title{Anisotropic admixture in color-superconducting quark matter}
\author{Michael Buballa}
\affiliation{Institut f{\"u}r Kernphysik, Schlossgartenstr.\ 9,
D-64289 Darmstadt, Germany}
\affiliation{Gesellschaft f{\"u}r Schwerionenforschung, Planckstr.\ 1,
D-64291 Darmstadt, Germany}
\author{Ji\v r\'{\i} Ho\v sek}
\affiliation{Dept. Theoretical Physics, Nuclear Physics Institute,
             25068 \v Re\v z (Prague), Czech Republic }
\author{Micaela Oertel} 
\affiliation{IPN-Lyon, 43 Bd du 11 Novembre 1918,
             69622 Villeurbanne C\'edex, France}

\date{May 3, 2002}
\begin{abstract}
The analysis of color-superconducting two-flavor deconfined quark matter 
at moderate densities is extended to include a particular spin-1 
Cooper pairing of those quarks which do not participate
in the standard spin-0 diquark condensate:
(i) Due to the peculiar form of the gap equation, exhibiting spontaneous
breakdown of rotational symmetry, the corresponding gap parameter $\Delta'$
is extremely sensitive to details of the effective interaction and to
the chemical potential.
(ii) The critical temperature of the anisotropic component is approximately
given by the relation $T_c'\simeq \Delta'(T=0)/3$. 
(iii) In the chiral limit the gas of anisotropic Bogolyubov-Valatin
quasiquarks becomes effectively gapless and two-dimensional. Consequently,
its specific heat depends quadratically on temperature. 
(iv) All collective Nambu-Goldstone excitations of the anisotropic phase
have a linear dispersion law and the whole system remains a superfluid.
\end{abstract}
\pacs{12.39.Ki, 12.38.Aw,11.30.Qc}
\maketitle

Recent investigations suggest that the phase structure of QCD is very rich
\cite{RaWi00,Al01}.  At low temperatures and high densities strongly
interacting matter is expected to be a color superconductor \cite{CoPe75}.  At
asymptotically high densities, where the QCD coupling constant becomes small,
this can be analyzed starting from first principles
\cite{Son99,PiRi}, whereas at more moderate densities, present
(presumably) in the interiors of neutron stars, these methods are no
longer justified.  In this region there are good reasons to expect that the
low-energy dynamics of deconfined quark matter is goverened by an effective
Lagrangian ${\cal L}_{\mathit{eff}}$ which contains local four-fermion interaction
terms, quite analogous to the Landau-Migdal ones of nuclear matter.
The non-confining gluon $SU(3)_c$ gauge fields are then treated as weak
external perturbations, and
neglected in lowest approximation.
Whereas the effective couplings which enter ${\cal L}_{\mathit{eff}}$ must in
principle be determined by experiment and are therefore rather uncertain
at present, the general form of ${\cal L}_{\mathit{eff}}$ is partially
constrained by symmetries.
As the primary QCD Lagrangian at finite chemical potential it should be
$SU(3)_c\times SU(N_f)_L\times SU(N_f)_R\times U(1)_V \times O(3)$ invariant, 
where $N_f$ is the number of (approximately) massless flavors, related
to chiral symmetry. 

In this letter we consider the case of two quark flavors ($N_f = 2$)
which is most likely relevant at chemical potentials just above the
deconfinement phase transition. On physical grounds it is then natural to
assume that ${\cal L}_{\mathit{eff}}$ favors the spontaneous formation of
spin-0 isospin singlet Cooper pair condensates \cite{BaLo84,RaWi00}
%\begin{equation}
$
 \delta \;=\;\langle \psi^T \;C\,\gamma_5\,\tau_2\;\lambda_2\; \psi \rangle~,
$
%\label{qqs}
%\end{equation}
where $\psi$ is a quark field, $C$ the matrix of charge conjugation,
$\tau_2$ a Pauli matrix which acts in flavor space, and $\lambda_2$ a
Gell-Mann matrix which acts in color space. 
Due to the latter $SU(3)_c$ is broken down to $SU(2)_c$, since only
the first two color components of $\psi$ participate in the condensate,
whereas the third one does not.
This has the following consequences
for the physical excitations of the system: 

(i) Corresponding to the mixing of the colors 1 and 2 
%in Eq.~(\ref{qqs}) 
there are two Bogolyubov-Valatin quasiquarks for each flavor
with the dispersion law
\begin{equation}
E_{1}^{\pm}(\vec p) = E_2^{\pm}(\vec p) \equiv E^{\pm}(\vec p) 
= \sqrt{(\epsilon_p\pm\mu)^2 + |\Delta|^2}~.
\label{E1}
\end{equation}
The energy gap $\Delta$ is the solution of a selfconsistent gap equation
and is typically found to be of the order $\sim 100$~MeV in model
calculations \cite{ARW98,RSSV98}.
$\epsilon_p = \sqrt{\vec p^{\,2} + M^2}$, where $M$ is an 
effective Dirac mass, related to the chiral condensate 
$\langle\bar\psi\psi\rangle$ via a selfconsistency equation. It can often be
neglected in the high-density regime.
For each flavor there is an unpaired quark of color 3 
with the dispersion law 
$\epsilon_3^{\pm}(\vec p) \,=\, \epsilon_p \,\pm\,\mu$.

(ii) Because of the spontaneous breaking of $SU(3)_c$ down to $SU(2)_c$
five of the eight gluons receive a mass (Meissner effect), whereas three 
remain massless \cite{RiCD00}.
Since no global symmetry is spontaneously broken 
%($\delta$ 
%is symmetric under $SU(2)_L\times SU(2)_R$ chiral transformations) 
there are no massless Goldstone bosons.
 
According to Cooper's theorem any attractive interaction leads to an 
instability at the Fermi surface which is cured by Cooper pair 
reorganization.
It is therefore rather unlikely, that the Fermi sea of color-3 quarks
stays intact. As only quarks of a single color are involved, 
the pairing must take place in a channel which, unlike $\delta$, 
is symmetric in color. Assuming $s$-wave condensation in an 
isospin-singlet channel, a possible candidate is a spin-1 condensate
%as already suggested in Ref
~\cite{ARW98}.
This letter is devoted to a quantitative analysis of this theoretically 
interesting possibility. To this end we consider the condensate
\begin{equation}
  \delta' \;=\; \langle \psi^T \;C\,\sigma^{03}\;\tau_2\;\hat P_3^{(c)}\;\psi 
  \rangle~,
\label{qqt}
\end{equation}
where $\sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu,\gamma^\nu]$ and
$\hat P_3^{(c)} = \frac{1}{3} - \frac{1}{\sqrt{3}}\lambda_8$ is 
the projector on color 3. 
Theoretical interest in $\delta'$ stems from the fact that it is a 
ground-state expectation value of a complex vector order parameter
$\phi^{0n}\equiv\phi_n$ describing spin 1, and breaking spontaneously the 
rotational invariance of the system. 
In relativistic systems this is  certainly not a very frequent phenomenon.
It is possible only at finite chemical potential, which itself breaks 
the Lorentz invariance explicitly. (Relativistic Cooper pairing into
spin-1 with nonzero angular momentum was considered elsewhere, e.g.,
\cite{BaLo84,Sch00}.)

For the quantitative analysis we have to specify the interaction.
Guided by the structure of instanton-induced interactions 
(see, e.g., \cite{RSSV98}) we consider a quark-antiquark term
\begin{equation}
{\cal L}_{q\bar q} = G \Big\{ (\bar \psi\psi)^2 - (\bar \psi\vec\tau\psi)^2
   - (\bar\psi i\gamma_5\psi)^2 + (\bar \psi i\gamma_5\vec\tau\psi)^2 \Big\}
\label{Lqqbar}
\end{equation}
and a quark-quark term
\begin{alignat}{2}
{\cal L}_{qq}\;=\;&H_s \Big\{& (\bar \psi i\gamma_5 C\tau_2\lambda_A\bar\psi^T)
                              &(\psi^T C i\gamma_5 \tau_2\lambda_A\psi^T)
\nonumber \\
                         && -\;(\bar \psi C\tau_2\lambda_A\bar\psi^T)
                            &(\psi^T C \tau_2\lambda_A\psi^T)\quad\Big\}
\nonumber \\
               - &H_t & (\bar \psi \sigma^{\mu\nu} C\tau_2\lambda_S\bar\psi^T)
                       &(\psi^T C \sigma_{\mu\nu} \tau_2\lambda_S\psi^T)~,
\label{Lqq}
\end{alignat}
where $\lambda_A$ and $\lambda_S$ are the antisymmetric and symmetric
color generators, respectively. For instanton induced interactions
the coupling constants fulfill the relation
$G : H_s : H_t =  1 : \frac{3}{4} : \frac{3}{16}$,
but for the moment we will treat them as arbitrary parameters.
As long as they stay positive, the interaction is attractive in the
channels giving rise to the diquark condensates $\delta$ and 
$\delta'$ as well as to the chiral condensate 
$\langle\bar\psi\psi\rangle$.
It is then straight forward to calculate the mean-field 
thermodynamic potential $\Omega(T,\mu)$ in the presence of these condensates:
\begin{alignat}{1}
    \Omega(T,\mu) \;= &-4 \sum_{i=1}^3 \int\frac{d^3p}{(2\pi)^3}\,
                        \Big[ \,\frac{1}{2}(E_i^-+E_i^+) 
\nonumber\\
    &\hspace{0cm}+ T\ln\Big(1+e^{-E_i^-/T}\Big) + T\ln\Big(1+e^{-E_i^+/T}\Big)\Big]
\nonumber\\
&\hspace{-.5cm}+\frac{1}{4G}(M-m)^2 +\frac{1}{4H_s}|\Delta|^2
+\frac{1}{16H_t}|\Delta'|^2~,
\label{Omega}
\end{alignat}
where $m$ is the bare quark mass, $M = m -2G\langle\bar\psi\psi\rangle$,
$\Delta= -2H_s\delta$, and $\Delta' = 4H_t\delta'$.
$E_{1,2}^\pm$ are given in Eq.~(\ref{E1}), while the dispersion law for quarks of color 3 reads
\begin{equation}
E_3^\mp(\vec p) \;=\; \sqrt{ (\sqrt{M_{\mathit{eff}}^2 + \vec p^{\,2}} \mp \mu_{\mathit{eff}}^2)^2
                            + |\Delta_{\mathit{eff}}'|^2 }~,
\label{E3}
\end{equation}  
where 
$\mu_{\mathit{eff}}^2 = \mu^2 + |\Delta'|^2 \sin^2{\theta}$,
$M_{\mathit{eff}} = M \mu/\mu_{\mathit{eff}}$, and
\begin{equation}
|\Delta_{\mathit{eff}}'|^2 = |\Delta'|^2\,(\cos^2{\theta} + \frac{M^2}{\mu_{\mathit{eff}}^2}\,\sin^2{\theta})~. 
\label{Dpeff}
\end{equation}  
Here $\cos{\theta} = p_3/|\vec p|$. 
Thus, as expected, for $\Delta'\neq 0$, $E_3^\pm(\vec p)$ is an anisotropic function of $\vec p$, 
clearly exhibiting the spontaneous breakdown of rotational invariance. 
For $M = 0$, the gap $\Delta_{\mathit{eff}}'$ vanishes at $\theta = \pi$. 
In general its minimal value is given by
\begin{equation}
    \Delta'_0 = \frac{M |\Delta'|}{\sqrt{\mu^2 + |\Delta'|^2}}~.
\label{M0}
\end{equation}
Expanding $E_3^-$ around its minimum the low-lying quasiparticle spectrum takes 
the form
\begin{equation}
  E_3^-(p_\perp, p_3) \approx \sqrt{ \Delta_0^{\prime 2} + v_\perp^2 (p_\perp - p_0)^2
                                  + v_3^2 p_3^2}~,  
\end{equation}
where $p_\perp^2 = p_1^2 + p_2^2$, and
\begin{equation}
  v_\perp = \sqrt{1-(\frac{\mu M}{\mu^2 + |\Delta'|^2})^2},\;
  v_3 = \frac{\Delta'_0}{M},\; 
  p_0 = \frac{v_\perp}{v_3}|\Delta'|. 
\end{equation}
This leads to a density of states linear in energy: 
\begin{equation}
  N(E) = \frac{1}{2\pi} \frac{\mu^2 + |\Delta'|^2}{|\Delta'|}\;E\;
  \theta(E-\Delta'_0)~.
\label{NE}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% figure 1
\begin{figure}[t]
\epsfig{file=cond.eps, width = 5.8cm}
\caption{$M$ (dotted), $\Delta$ (dashed), and $\Delta'$ (solid) at $T$~=~0
         as functions of the quark chemical potential $\mu$ using parameter
         set 1 (see text). 
         The dashed-dotted line indicates the result for $\Delta'$
         taking parameter set 2.}
\label{cond}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The actual values for $\Delta$, $\Delta'$ and $M$ follow from the
condition that the stable solutions correspond to
the absolute minimum of $\Omega$ with respect to these quantities.
Imposing $\partial\Omega/\partial{\Delta'}^* = 0$ we
obtain the following gap equation for $\Delta'$:
\begin{alignat}{1}
\Delta' = 16H_t\Delta' \int \frac{d^3 p}{(2\pi)^3} \, 
\Big\{\;&(1-\frac{{\vec p}_\perp^{\,2}}{s})\frac{1}{E_3^-}
\tanh{\frac{E_3^-}{2T}} 
\nonumber\\
+ &(1+\frac{{\vec p}_\perp^{\,2}}{s})\frac{1}{E_3^+}
\tanh{\frac{E_3^+}{2T}}\Big\}~, 
\label{Deltapgap}
\end{alignat}  
where $s = \mu_{\mathit{eff}}(\vec p^{\,2} + M_{\mathit{eff}}^2)^{1/2}$.
Similarly one can derive gap equations for $\Delta$ and $M$, 
by the requirements $\partial\Omega/\partial\Delta^* = 0$ and 
$\partial\Omega/\partial M = 0$, respectively. Together with 
Eq.~(\ref{Deltapgap}), they form a set of three coupled equations,
which have to be solved simultaneously.
However, the equations for $\Delta$ and $\Delta'$ are not directly 
coupled, but only through their dependence on $M$, which can 
often be neglected in first approximation.

In our numerical calculations we use a sharp 3-momentum cut-off $\Lambda$
to regularize the divergent integrals.
We then have five model parameters: the bare quark mass $m$,
the cutoff $\Lambda$, and the three coupling constants $G$, $H_s$ and $H_t$.
To get started we choose $m =$~5~MeV, $\Lambda = 600$~MeV, 
and $G\Lambda^2 = 2.4$ -- leading to reasonable vacuum properties,
$M = 393$~MeV and $\langle\bar\psi\psi\rangle = -2(244 \rm{MeV})^3$ --,
and the instanton relation  
to fix $H_s$ and $H_t$ ("parameter set 1").
The resulting values of $M$,  $\Delta$, and $\Delta'$  as functions of $\mu$
at $T$~=~0 are displayed in Fig.~\ref{cond}. The chemical potentials 
correspond to baryon densities of about 4 - 7 times nuclear matter
density. 
In qualitative agreement with earlier expectations \cite{ARW98} $\Delta'$ 
is relatively small, about 2-3 orders of magnitude smaller than $\Delta$
in this regime. 
However, it is strongly
$\mu$-dependent and rises  by more than a factor of 10 between 
$\mu = 400$~MeV and $\mu = 500$~MeV.
Being a solution of a selfconsistency problem, $\Delta'$ is also 
extremely sensitive to the coupling constant $H_t$.
As noted earlier, the effective interaction to be used at moderate
densities is rather uncertain and hence the parameters listed above
are by no means fixed. If we take $H_t$ twice as large as 
before (``parameter set 2''), we arrive at the dashed-dotted line for 
$\Delta'$, which is then much larger and comparable to $\Delta$.
We also find that $\Delta'$ is very sensitive to the cut-off.
This can be traced back to the factor
$(1-{\vec p}_\perp^{\,2}/s)$ in the gap equation
(\ref{Deltapgap}) which can become negative for large momenta. 
It is quite obvious then, that also the form of the regularization,
i.e., sharp cutoff, form factor, etc., will have a strong impact on the
results.

%We now turn to the discussion of temperature effects. 
With increasing
temperatures both condensates, $\delta$ and $\delta'$, are reduced
and eventually vanish in second-order phase transitions at critical
temperatures $T_c$ and $T_c'$, respectively. It has been shown 
\cite{PiRi} that $T_c$ is approximately given by the 
well-known BCS relation $T_c \simeq 0.57\Delta$($T$=0). 
In order to derive a similar relation for the critical temperature $T_c'$ 
we inspect the gap equation (\ref{Deltapgap}) at temperature $T$~=~0 and 
in the limit $T \rightarrow T_c'$. Neglecting quark masses and antiparticle 
contributions one gets 
\begin{alignat}{1}
\int \frac{d^3 p}{(2\pi)^3}
\Big\{&\Big[(1-\frac{{\vec p}_\perp^{\,2}}{s})\frac{1}{E_3^-(\vec p)}
\Big]_{\Delta'(T=0)}
\nonumber\\
   -&(1-\frac{{\vec p}_\perp^{\,2}}{\mu\,|\vec p|})
     \frac{1}{|\mu-|\vec p||} \tanh{\frac{|\mu-|\vec p||}{2T_c'}}\Big\} 
\approx 0~.
\label{Tcest}
\end{alignat}  
Since both, the $T$=0-part and the $T=T_c'$-part of the integrand, 
are strongly peaked near the Fermi surface, 
the $|\vec p|$-integrand must approximately vanish at  $|\vec p| = \mu$,
after the angular integration has been performed. 
From this condition one finds to lowest order in $\Delta'/\mu$:
\begin{equation}
  T_c' \;\approx\; \frac{1}{3}\;\Delta'(T=0)~.
\label{Tcapp}
\end{equation}
For the scalar condensate $\delta$
%(\ref{qqs}) 
the analogous steps
would lead to $T_c/\Delta$($T$=0) $\approx \frac{1}{2}$ instead of the
textbook value of 0.57. This gives a rough idea about the quality of the
approximation.    

Numerical results for the temperature dependence of $\Delta$ and 
$\Delta'$ are shown in Fig.~\ref{condt}. The quantities have been properly
rescaled in order to facilitate a comparison with the above relations
for $T_c$ and $T_c'$. 
We find $T_c/\Delta(0)=$~0.53 and $T_c'/\Delta'(0)=$~0.41, which is in
reasonable agreement with our estimates.
The calculations presented in Fig.~\ref{condt} have been 
performed at $\mu = 450$~MeV using parameter set 1, 
but the results look very similar for other
chemical potentials or parameters.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% figure 2
\begin{figure}[t]
\epsfig{file=condt.eps,width = 5.8cm}
\caption{$\Delta_i/\Delta_i$($T$=0) as function of $T/\Delta_i$($T$=0).
Dashed: $\Delta_i=\Delta$. Solid: $\Delta_i=\Delta'$. The calculations
have been performed at $\mu=450$~MeV for parameter set 1 (see text).
}
\label{condt}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The specific heat is given by 
$c_v = -T \partial^2 \Omega/\partial T^2$\footnote{Strictly, 
$c_v = T/V \partial S/\partial T |_{V,N}$, but
the correction term is small~\cite{FW71}.}.  
For $T \ll T_c$ it is completely dominated by
the quarks of color 3, since the contribution of the first two colors
is suppressed by a factor $exp(-\Delta/T)$.  Neglecting the
$T$-dependence of $M$ and $\Delta'$, and employing the approximate
density of states, Eq.~(\ref{NE}), one finds
\begin{alignat}{1}
  c_v \;\approx\; &\frac{12}{\pi} \frac{\mu^2 +
    |\Delta'|^2}{|\Delta'|}\,T^2
\nonumber\\
            &\hspace{-.5cm}\times\;\Big[1 + \frac{\Delta'_0}{T} + \frac{1}{2} 
                 \left(\frac{\Delta'_0}{T}\right)^2
             + \frac{1}{6} \left(\frac{\Delta'_0}{T}\right)^3\Big]\,e^{-\frac{\Delta'_0}{T}} ~,
\label{cvapp}
\end{alignat}
which should be valid for $T \ll T_c'$. 
In this regime $c_v$ depends quadratically on $T$ for $T \gtrsim \Delta'_0$,  
and is exponentially suppressed at lower temperatures.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% figure 3
\begin{figure}[t!]
\epsfig{file=cvtc.eps,width = 5.8cm}
\caption{Specific heat for parameter set 2 at $\mu$~=~450~MeV as function 
of $T/T_c'$. 
Solid: full calculation,
dashed: result for $M = 0$,
dotted: without spin-1 condensate. The dashed-dotted line indicates the
result of Eq.~(\ref{cvapp}).
}
\label{cv}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To test this relation we evaluate $c_v(T)$ explicitly using
Eq.~(\ref{Omega}). The results for fixed $\mu$~=450~MeV are displayed
in Fig.~\ref{cv}. For numerical convenience we choose parameter set 2
(i.e., $H_t$~= twice the instanton value), leading to a relatively
large $\Delta'$($T$=0)~=~30.8~MeV. The critical temperature is
$T_c'\simeq$~0.40~$\Delta'$($T$=0). For the energy gap we find
$\Delta'_0$~=~0.074~$T_c'$.  It turns out that Eq.~(\ref{cvapp}), evaluated
with constant values of $\Delta'$ and $M$, (dashed-dotted line) is in
almost perfect agreement with the numerical result (solid line) up to
$T \approx T_c'/2$.  The phase transition, causing the discontinouity
of $c_v$ at $T=T_c'$, is of course outside the range of validity of
Eq.~(\ref{cvapp}). For comparison we also display the
specific heat for a system with $\Delta'$~=~0 (dotted line). In this
case $c_v$ exhibits a linear $T$ dependence at low temperatures.
Including the spin-1 condensate, but neglecting the quark mass, we
obtain the dashed line. Since the energy gap vanishes for $M$~=~0,
there is no exponential suppression, and $c_v$ is proportional to
$T^2$ down to arbitrarily low temperatures.  However, even if $M$
is included, the exponential suppression is partially
cancelled by the extra-terms in square brackets of
Eq.~(\ref{cvapp}). 

In Ref.~\cite{RaWi00} it was suggested that the exponential 
suppresion of $c_v$ below $T_c'$ might have observable 
consequences for the neutrino emission of a neutron star. 
This argument has to be somewhat refined because, as seen above, 
$c_v(T)$ first behaves as $T^2$ and the exponential suppression 
sets in only at much smaller temperatures, $T < \Delta'_0$.
Moreover, it has recently been argued~\cite{AR02}, that the constraints
imposed by charge and color neutrality might 
completely prohibit the existence of two-flavor color-superconducting 
matter in neutron stars.
In principle, of course, a hypothetical measurement of $c_v(T)$ similar
to Fig.~\ref{cv} would yield valuable information.

Because of the spontaneously broken 
$U(1)\times O(3)$ symmetry in Eq.~(\ref{qqt}), for $\Delta' \neq 0$ 
there should be collective Nambu-Goldstone excitations in the spectrum. 
However, due to the Lorentz non-invariance of the system there can be
subtleties \cite{NLSS,HoOM98,MiSh01}.
The NG spectrum can be analyzed within an underlying effective Higgs
potential 
\begin{equation}
V(\phi) = -a^2 \phi_n^\dagger\phi_n 
+ \frac{1}{2}\lambda_1(\phi_n^\dagger\phi_n)^2
+ \frac{1}{2}\lambda_2\phi_n^\dagger\phi_n^\dagger\phi_m\phi_m,
\end{equation}
for the complex order parameter $\phi_n$ \cite{HoOM98}, with 
$\lambda_1 + \lambda_2 > 0$ for stability. 
For $\lambda_2 < 0$ the ground state is 
characterized by $\phi_{vac}^{(1)} = (\frac{a^2}{\lambda_1})^{1/2} (0,0,1)$
which corresponds to our ansatz Eq.~(\ref{qqt}) for the BCS-type
diquark condensate $\delta'$. This solution has the property 
$\langle\vec S\rangle^2 
=(\phi_{vac}^{(1)\dagger} \vec S \phi_{vac}^{(1)})^2 = 0$.
The spectrum of small oscillations above $\phi_{vac}^{(1)}$ consists
of 1+2 NG bosons, all with linear dispersion law: one zero-sound
phonon and two spin waves \cite{HoOM98}. Implying a finite
Landau critical velocity, this fact is crucial for a macroscopic
superfluid behavior of the system \cite{MiSh01}.

It is interesting to note, that for $\lambda_2 > 0$ there is a
different solution $\phi_{vac}^{(2)} =
(\frac{a^2}{2(\lambda_1+\lambda_2)})^{1/2} (1,i,0)$ with $\langle\vec
S\rangle^2 = 1$.
In this case the NG spectrum above $\phi_{vac}^{(2)}$ 
consists of one phonon with linear dispersion law
and one spin wave whose energy tends to zero with momentum
squared~\cite{HoOM98}. 
Clearly, there is no way of knowing without an
explicit computation which one of the two anisotropic phases of
deconfined quark matter is energetically favorable for a given
interaction. Work in this direction is in progress.

\begin{acknowledgments} 
We thank D. Blaschke, K. Rajagopal, 
I.A. Shovkovy, and E.V. Shuryak for useful discussions.
J.H. is grateful to Jochen Wambach and IKP TU Darmstadt for generous 
hospitality and support. 
We acknowledge financial support by ECT$^*$ during its 2001 
collaboration meeting on color superconductivity.
This work was supported in part by grant GACR 202/02/0847.
M.O. acknowledges support from the Alexander von
Humboldt-foundation as a Feodor-Lynen fellow. 
\end{acknowledgments} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{*}
%\bibitem{Wi00}   F. Wilczek, .
\bibitem{RaWi00} K. Rajagopal and F. Wilczek,  and references
                 therein.
\bibitem{Al01}   M. Alford, Ann. Rev. Nucl. Part. Sci. {\bf 51}, 131 (2001).
\bibitem{CoPe75} J.C. Collins and M.J. Perry, Phys. Rev. Lett. {\bf 34},
                 1353 (1975).
\bibitem{Son99}  D.T. Son, Phys. Rev. {\bf D59}, 094019 (1999);
                 T. Sch\"afer and F. Wilczek, {\it ibid} {\bf D60}, 114033 (1999);
                 D.K. Hong, V.A. Miransky, I.A. Shovkovy, and L.C.R. Wijewardhana, {\it ibid} {\bf D61},
                 056001 (2000), {\it err.} {\bf D62}, 059903 (2000).
\bibitem{PiRi}   R.D. Pisarski and D.H. Rischke, Phys. Rev. {\bf D60}, 
                 094013 (1999); {\bf D61} 051501 (2000); 
                 {\bf D61} 074017 (2000).
\bibitem{BaLo84} D. Bailin and A. Love, Phys. Rep. {\bf 107}, 325 (1984).
\bibitem{ARW98}  M. Alford, K. Rajagopal, and F. Wilczek,
                 Phys. Lett. B {\bf 422}, 247 (1998).
\bibitem{RSSV98} R. Rapp, T. Sch\"afer, E.V. Shuryak, and M. Velkovsky,
                 Phys. Rev. Lett. {\bf 81}, 53 (1998).
\bibitem{RiCD00} D. Rischke, Phys. Rev. {\bf D62}, 034007 (2000);
                 G.W. Carter and D. Diakonov, Nucl.Phys. B {\bf 582}, 
                 571 (2000).
\bibitem{Sch00}  T. Sch\"afer, Phys. Rev. {\bf D62}, 094007 (2000).
\bibitem{FW71}   A.L. Fetter and J.D. Walecka,
                   Quantum theory of many-particle systems, 
                   Mc Graw-Hill, New York (1971)
\bibitem{AR02}   M. Alford and K. Rajagopal, .
\bibitem{NLSS}   H. Nielsen and S. Chadha, Nucl. Phys. {\bf B105}, 445
                 (1976);
                 H. Leutwyler,  Phys. Rev. {\bf D49}, 3033 (1994);
                 T. Sch\"afer, D.T. Son, M.A. Stephanov, D. Toublan,
                 and J.J. Verbaarschot,
                 Phys. Lett. B {\bf 522}, 67 (2001);
                 F. Sannino and W. Sch\"afer, 
                 Phys. Lett. B {\bf 527}, 142 (2002).
\bibitem{HoOM98} T.-L. Ho, Phys. Rev. Lett. {\bf 81}, 742 (1998);
                 T. Ohmi and K. Machida, J. Phys. Soc. Jpn. {\bf 67},
                 1822 (1998).
\bibitem{MiSh01} V.A. Miransky and I.A. Shovkovy, 
                 Phys. Rev. Lett. {\bf 88}, 111601 (2002).
\end{thebibliography}
\end{document}
















