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\title{
\vspace*{-4cm}
\begin{flushright}
\rm MIT-CTP-2715\\
\rm IASSNS-HEP-98/13\\
\vspace*{2.2cm}
\end{flushright}
Color Superconductivity and Signs of its Formation\thanks{
These ideas were first broached
in discussions (in particular with R. Jaffe) at a November workshop 
at the RIKEN-BNL center. The fact that they are now seeing
the light of day owes much to conversations K.R. had at 
QM97  (in particular with R. Seto.)  We are grateful to
the organizers of QM97 and the RIKEN-BNL workshop, as both were
fruitful. The research of M.A. and F.W.
is supported in part by DOE grant DE-FG02-90ER40542;
that of K.R. is supported in part by DOE cooperative
research agreement DE-FC02-94ER40818.}}
\author{Mark Alford,\address{Institute for Advanced Study, 
%        School of Natural Science \\ 
%        Olden Lane, 
Princeton, NJ 08540}
Krishna Rajagopal\address{Massachusetts
%	Center for Theoretical Physics,
%	Laboratory for Nuclear Science and Department of Physics\\
	Institute of Technology, Cambridge, MA 02139}
and Frank Wilczek$^a$}






\begin{document}
% typeset front matter
\maketitle
\thispagestyle{empty}

%\baselineskip14pt

\begin{abstract}
We study finite density QCD
in an approximation in which the interaction
between quarks is modelled on that induced by instantons.
We sketch the mechanism by which chiral
symmetry restoration at finite density
occurs in this model.
%that is by the merging of nucleons which are 
%droplets within which the chiral condensate vanishes.
At all densities high enough that the chirally symmetric phase
fills space, we find that color symmetry is broken
by the formation of a $\langle qq \rangle$ 
condensate of quark Cooper pairs.
The formation of this  color superconductor condensate
lowers the energy of the system most if 
the up and down quark chemical potentials are equal.
This suggests that the formation of such a condensate in
a heavy ion collision may be accompanied
by radiation of negative pions, and its decay may yield
more protons than were present in the incident nuclei.
\end{abstract}

\section{Introduction}


In his talk at Quark Matter '97, K.R. described our recent
work on QCD at finite baryon density 
presented in
Ref. \cite{us}, and this paper should
be consulted by those seeking a description of the talk.
Here, we
focus on a speculation concerning 
signatures of the formation of a color superconducting
state in heavy ion collisions.  


Asymptotic freedom leads us to expect
that at high density quarks behave nearly freely and form large Fermi 
surfaces, with interactions between the 
quasiparticles at the Fermi surfaces which become weak at
asymptotically high density.
Since the quark-quark interaction is attractive in the color
$\bar 3$ channel, BCS pairing of quarks will occur no matter
how weak the interaction.
%No matter how weak the attraction, pairing of the
%BCS type can be expected if there is an attractive channel.
Pairs of quarks cannot be color singlets,
and so a $\langle qq \rangle$ condensate inevitably breaks
color symmetry.   This breaking is analogous to the breaking
of electromagnetic gauge invariance in superconductivity.
Color superconductivity implies that five of the eight gluons
are massive and implies that the $U(1)$ gauge boson
which remains massless is a linear combination of the
photon and a gluon.  
Our goal in \cite{us}
was to explore this phase in a context
that is definite, qualitatively reasonable, and yet sufficiently
tractable that likely patterns of symmetry breaking and rough
magnitudes of their effects can be identified 
at non-asymptotic densities where interactions are not weak.  




%This breaking is analogous to the breaking of
%electromagnetic gauge invariance in superconductivity, and we
%therefore refer to it as
%color superconductivity.  
%In this phase, the Higgs mechanism
%operates and (some) gluons become massive. 
%Symmetry breaking in diquark channels is of course
%quite different from chiral symmetry breaking
%in QCD at zero density, 
%which occurs in color singlet quark-antiquark channels.  


Color superconductivity
requires high densities and low temperatures, so the
most favorable experimental conditions are likely those
at the AGS. 
Conditions at the centers of neutron stars are even
more favorable, but that is not our subject here.
Based on present analyses, it seems unlikely that 
color superconductivity can arise at temperatures
above $100$ MeV.  Therefore, one
must select events from the total data set in an
experiment in a way that focuses on a subset 
of events which happen to
have unusually high density and unusually low
temperature.\cite{seto}  The strategy for doing this would
vary in  different experiments, but one might choose
events with little energy at zero degrees (suggesting
central events in which most of the baryons were stopped
at central rapidity creating high densities) and with
protons at central rapidity with unusually soft transverse
momentum (suggesting that the effective temperature experienced
by the baryons in the dense region was lower than in a typical event.)
Signatures of the formation of a color superconductor phase
can then be sought in the subset of events relative to the entire set.
What, then, should we look for in this sort of an event-by-event
analysis?  We propose one answer to this question in Section 3 below, 
although we expect other answers are also possible.  





\section{A Model and Chiral Symmetry Breaking Therein}


%The regime 
%has been studied\cite{bailin} by approximating the interquark
%interactions by one gluon exchange, which is in fact attractive
%in the color antitriplet channel.
%Perturbative treatments cannot, by their nature,
%do full justice to a problem
%whose main physical interest arises at moderate
%densities.  To get more insight into the phenomena, and in particular
%to make quantitative estimates, it seems appropriate 
%to analyze a tractable, physically motivated model.
We sketch here our variational
treatment of a
two-parameter class of models
having two flavors and three colors of massless quarks.  
The kinetic part of the Hamiltonian is 
that for free quarks, while the
interaction Hamiltonian is a  
four fermion interaction (with coupling $K$) 
which is an idealization
of the  
instanton vertex from QCD. 
%explicitly:
%\beq
%H_I = -\dsp K \int\!d^3x\, 
% \bar\psi_{R1\al}\,{\psi_{Lk}}^\ga \,
% \bar\psi_{R2\be}\,{\psi_{Ll}}^\de \, \ep^{kl} 
% \, (3 {\de^\al}_\ga {\de^\be}_\de - 
% {\de^\al}_\de {\de^\be}_\ga)\ + \ {\rm h.c.}\ ,
%\eeq
%where $1,2,k,l$ are flavor indices,
%$\al,\be,\ga,\de$ are color indices,
%repeated indices are summed, and the spinor indices are
%not shown. 
%Color {\bf 3} indices
%are raised and $\bar{\bf 3}$ indices are lowered.
%$H_I$ is not yet a good representation of the instanton
%interaction in QCD: 
In order to mimic the effects of asymptotic freedom,
we multiply the momentum space interaction by
%
%we must modify it in such a way that the interaction
%decreases with increasing momentum.  We 
%write $H_I$ as a mode expansion in momentum
%space involving creation and
%annihilation operators and spinors, and  
%multiply the result by 
a product of form factors each of the form
$F(p)=[\Lambda^2/(p^2 + \Lambda^2)]^\nu$,
one for each of the momenta of the four fermions.
%This
%factorized form is taken for convenience,
%and is an idealization.   
$\Lambda$, of
course, is some effective QCD cutoff scale, which one might anticipate
should be in the range 300 -- 1000 MeV.  $\nu$ parametrizes the shape
of the form factor; we consider $\nu=1/2$ and $\nu=1$.
%Momentarily we will see how to
%fix one parameter by reference to known quantities.  We choose
%to fix the coupling $K$ for any given $\Lambda$ and $n$;
%we quote results for $n=1/2$ and $1$, and for $\Lambda=$ 300, 500,
%and 700 MeV.
%Since the  
%interaction we have chosen is not necessarily an accurate rendering of QCD, 
%we will have faith only in conclusions that are robust with respect to
%the parameter choices.  

%The color, flavor, and Lorentz structure of our interaction has been
%taken over
%directly from the instanton vertex for two-flavor QCD.  
%There are other four-fermion
%interactions in addition to $H_I$ which respect the unbroken symmetries
%of QCD; 
%using $H_I$ alone is the simplest way of breaking all
%symmetries broken by QCD, and is therefore a good starting point. 



In \cite{us}, we first consider chiral symmetry
breaking at zero density. 
We choose a variational wave
function which pairs particles and
antiparticles with the same flavor and
color but opposite helicity and opposite
three-momentum.  We derive the gap equation
which determines the chiral gap $\Delta_\chi$ 
as a function of quark number density $n$.
We fix the coupling $K$ for each choice of $\Lambda$ and $\nu$
by requiring $\Delta_\chi=400 {\rm ~MeV}$
at $n=0$.  $\Delta_\chi$ decreases with increasing $n$, 
and vanishes at some $n_c$.
With wave function in hand, we then evaluate the 
energy density and pressure
as functions of $n$.
We find that the 
phase with broken chiral symmetry is unstable at any nonzero density,
with this instability being signalled by negative pressure at all
but the lowest densities.
%At all but the lowest densities, this instability is signalled by
%negative pressure which presumably triggers the break-up of the
%uniform state into regions of high density separated by empty space.
%For all values of the parameters that we consider reasonable,
%we find that after a tiny (and irrelevant\cite{us})
%interval of very low densities
%at which the pressure is positive, the pressure  
%becomes negative, and continues to decrease until the
%critical density $n_c$ 
%at which chiral symmetry is restored.  At that point we
%switch over to an essentially free quark phase, 
%and the pressure then begins to increase 
%monotonically as $n$ is increased
%further.  
At $n=n_c$, we switch over to an essentially free quark
phase and the pressure then begins to increase monotonically,
reaching zero at a density $n_0>n_c$ and then becoming positive.
%At some $n_0>n_c$, the pressure is zero and
%at higher densities the pressure becomes positive.
In the presence of
a chiral condensate the negative pressure
associated with increasing vacuum energy
overcompensates the increasing Fermi pressure. 
%There is
%an attractive physical interpretation of this phenomenon.  
The
uniform, chiral symmetry broken,
nonzero density phase is mechanically 
unstable and breaks up into 
stable droplets of high
density $n=n_0$ in which the pressure is zero 
and chiral symmetry is restored, surrounded by  
empty space with chiral symmetry broken.  
%Although our simple calculations do not
%allow us to
%follow the evolution and eventual stabilization of the original quark
%cloud, it is hard  
%to avoid
We identify the droplets of chiral
symmetric phase with physical 
nucleons.  Nothing within the model tells us that
the stable droplets have quark number $3$; nucleons are simply
the only candidates in nature which can be identified with droplets
within which the quark density is nonzero and the chiral condensate
is zero. 
%If correct, this identification is very reminiscent of the MIT
%bag philosophy, 
%here arising in the description of a sharply defined physical
%phenomenon.\footnote{Considerations similar
%to those we describe also lead Buballa\cite{buballa}
%to conclude that 
%in a Nambu Jona-Lasinio model
%with an interaction which differs from the one we use,
%matter with broken chiral symmetry is unstable
%and nucleons 
%can therefore only be viewed as bags within which chiral symmetry
%is restored.}
%%
This physical picture has significant
implications for the phase transition, as a function of density, to
restored chiral symmetry.   
%Since the nucleons are regions where the
%symmetry is already restored, 
The transition should occur by
a mechanism analogous to percolation
as nucleons, seen as pre-formed bags of symmetric phase, merge.  

%This transition should
%be complete once a density characteristic of the center of 
%nucleons is achieved.  
%The fact that some external pressure must
%be imposed in order to induce the nucleons to merge (e.g. the
%fact that in nuclear matter at zero pressure the 
%nucleon droplets remain unmerged)
%must reflect interactions between droplets, which we have not treated here.
%The mechanism of chiral symmetry restoration at finite density but
%zero temperature is quite different from the one we expect at finite
%temperature and zero density: it occurs by percolation 
%among pre-formed bags of symmetric phase.  

\section{Color Superconductivity with Unequal Numbers of u and d Quarks}    

We now turn to physics at densities greater than $n_0$,
at which the model describes a uniform phase with no
chiral condensate.
At high density, pairing of particles near the Fermi surface as in the
original BCS scheme becomes more favorable.  
Our Hamiltonian supports
condensation in quark-quark channels.   
The condensation is now between fermions with the same
helicity, and the Hamiltonian selects
antisymmetry in flavor.  
One can therefore have spin 0 --- antisymmetric in spin and therefore
in color, forming a $\bar {\bf 3}$, or spin 1 --- symmetric in spin and
therefore in color, forming a {\bf 6}.   
Here, we only consider the former, because although the spin-symmetric 
color {\bf 6} can arise, the gap in this channel is very small.\cite{us}



%We first consider the former.  A suitable trial wave function is
%\beq
%|\psi\> = G_L^\ad G_R^\ad |p_F\>
%\eeq
%where
%\beq
%\ba{rcll}
%G_L^\ad &=& \dsp
% \prod_{\alpha,\beta,\bp}
%& \Bigl( \cos(\th^L_{A}(\bp)) + \ep^{\alpha\beta 3}\e^{i\xi^L_A(\bp)}
%\sin(\th^L_{A}(\bp))
% a^\ad_{L\,1\alpha}(\bp)  a^\ad_{L\,2\beta}(-\bp) \Bigr) \\[1ex]
%&&& \Bigl( \cos(\th^R_{B}(\bp)) + \ep^{\alpha\beta 3}\e^{i\xi^R_B(\bp)}
%\sin(\th^R_B(\bp))
% b^\ad_{R\,1\alpha}(\bp)  b^\ad_{R\,2\beta}(-\bp) \Bigr) \\[1ex]
%&&& \Bigl( \cos(\th^R_{C}(\bp)) + \ep^{\alpha\beta 3}\e^{i\xi^R_C(\bp)}
%\sin(\th^R_{C}(\bp))
% a_{R\,1\alpha}(\bp)  a^\ad_{R\,2\beta}(-\bp) \Bigr) \\[1ex]
%%
%G_R^\ad &=& \multicolumn{2}{l}{\hbox{same, with~~}R\leftrightarrow L}.
%\ea
%\label{colwf}
%\eeq
%Here,
%$\alpha$ and $\beta$ are color indices, and we
%have chosen to pair quarks of the first two colors.
%$1$ and $2$ are flavor indices. The first term in (\ref{colwf})
%creates particles above the Fermi surface; the second creates
%antiparticles; the third creates holes below the Fermi surface.

In Ref. \cite{us}, we construct a suitable trial wave function in which
the Lorentz scalar
$\langle q^{i\alpha} \,C\gamma^5 q^{j\beta}\varepsilon_{ij}\,
\varepsilon_{\alpha\beta 3}\rangle$ is nonzero. This 
chooses a preferred
direction in color space and breaks color $SU(3) \rightarrow SU(2)$.
Electromagnetism is spontaneously broken but
there is a     linear combination of electric charge
and color hypercharge under which the condensate is neutral, and
which therefore generates an unbroken
$U(1)$ gauge symmetry.  
No flavor symmetries, not even chiral ones, are broken.
It is not difficult to redo the derivation of the
gap equation
satisfied by the superconducting gap parameter
$\Delta$ for the case when
down and up quarks have different chemical potentials: 
$~~\mu_+ = \bar\mu + \delta\mu~~$ for the down quarks
and $~~\mu_- = \bar\mu - \delta\mu~~$ for
the up quarks. One finds 
\bea
1 = \frac{K}{\pi^2} \Biggl\{
\int_{\mu_+}^\infty p^2 dp \frac{F^4(p)} 
{\sqrt{ F^4(p)\De^2 + (p-\bar\mu)^2}}
&+& \int_0^{\mu_-}p^2 dp 
\frac{F^4(p)}{\sqrt{ F^4(p)\De^2 + (\bar\mu-p)^2}}\nonumber\\
&+& \int_0^\infty p^2 dp 
\frac{ F^4(p)}{\sqrt{ F^4(p)\De^2 + (p+\bar\mu)^2}}\Biggr\}\ .
\label{colorgapeq}
\eea
The three terms arise respectively from 
particles above the 
Fermi surface, holes below the Fermi surface,
and antiparticles.  With equal numbers of up and down quarks, 
$\mu_+ = \mu_- = \bar\mu$ and the particle 
and hole integrals diverge logarithmically
at the Fermi surface 
as $\Delta \rightarrow 0$, which signals 
condensation for arbitrarily weak attraction.
However, for $\delta\mu\neq 0$ there is no logarithmic
divergence, and $\Delta$ may vanish.  
For a given $\bar\mu$, as $\delta\mu$ is increased the
domain of integration moves farther and farther from the
logarithmic singularity, and $\Delta$ must decrease in order
for the gap equation to be satisfied.  
Momenta between $\mu_+$ and $\mu_-$ cannot contribute,
since neither particle-particle nor hole-hole
pairing is possible, given that the condensate necessarily
pairs up quarks with down quarks.


As a concrete example, we take 
$\nu=1$ and $\Lambda = 800$ MeV in the form factor, and we work
at a baryon density eight times that in nuclear matter.
This density, which corresponds to a quark number density $n$
of $4.1$ per fm$^3$,
is greater than $n_0$,
and so space is filled by a chirally symmetric color 
superconductor phase with quark number density $n$ 
and energy density $\varepsilon$ given by:
% and so
%is high enough that the chiral symmetry restored
%phase fills space. 
%In this chirally symmetric, color
%superconducting phase, $n$ and the energy density $\varepsilon$
%are given by:
\vfill
\eject
\bea
n &=& \frac{2}{\pi^2}\Biggl\{ 
\int_{\mu_+}^\infty p^2 dp \Biggl(1- \frac{p-\bar\mu}{\sqrt{
F^4(p)\Delta^2+(p-\bar\mu)^2}}\Biggr)
- \int_{0}^{\mu_-} p^2 dp \Biggl(1- \frac{\bar\mu-p}{\sqrt{
F^4(p)\Delta^2+(\bar\mu-p)^2}}\Biggr)\nonumber\\
&-& \int_{0}^{\infty} p^2 dp \Biggl(1- \frac{\bar\mu+p}{\sqrt{
F^4(p)\Delta^2+(\bar\mu+p)^2}}\Biggr)\Biggr\}+\frac{\mu_+^3 + \mu_-^3}
{\pi^2}\ ,\\ 
\varepsilon &=& \frac{2}{\pi^2}\Biggl\{ 
\int_{\mu_+}^\infty p^3 dp \Biggl(1- \frac{p-\bar\mu}{\sqrt{
F^4(p)\Delta^2+(p-\bar\mu)^2}}\Biggr)
- \int_{0}^{\mu_-} p^3 dp \Biggl(1- \frac{\bar\mu-p}{\sqrt{
F^4(p)\Delta^2+(\bar\mu-p)^2}}\Biggr)\nonumber\\
&+&\int_{0}^{\infty} p^3 dp \Biggl(1- \frac{\bar\mu+p}{\sqrt{
F^4(p)\Delta^2+(\bar\mu+p)^2}}\Biggr)\Biggr\}- \frac{\Delta^2}{K} 
+\frac{3(\mu_+^4 + \mu_-^4)}{4\pi^2}\ .
\label{energyeq}
\eea
Working first with $\delta\mu=0$, that is with $\mu_+=\mu_-=\bar\mu$,
we find that $n=4.1$ fm$^{-3}$ corresponds to $\bar\mu =0.534$
and $\Delta=0.156$, both in GeV.  If we create an excess of
down quarks by
increasing $\delta\mu$ while
keeping $\bar\mu$ fixed (this changes $n$) we find that
$\Delta$ decreases, and vanishes when $\delta\mu = .037$.
What is a reasonable value for $\delta\mu$?  In $^{197}{\rm Au}$ there
are 315 down quarks and 276 up quarks.  Keeping the density fixed
but enforcing this down/up ratio requires $\bar\mu=0.535$ and
$\delta\mu=0.012$.  Under these conditions, $\Delta$ is decreased
to $0.128$.  This phase,
with a down/up ratio appropriate for gold nuclei, has an energy
density which exceeds that of the $\delta\mu=0$ phase
with the same density by $(99.8 {\rm ~MeV})^4\sim 13 {\rm ~MeV}/{\rm fm}^3$.
This, then, is the energy density by which the up--down symmetric
phase is favored for these parameters.
This estimate is model dependent.  By varying 
the form factor $F$, it is easy to get a result which is twice
(or half) as large. 

We have demonstrated that the formation of a
color superconducting state lowers the energy 
most if the matter is up--down symmetric.  
In a heavy ion collision, in which down quarks initially
exceed up quarks, as a color superconductor condensate forms
in the densest region near the center of the collision 
it may therefore expel a few down quarks and up antiquarks
(one each per 7.4 fm$^3$ of condensate for the density considered 
above) into the surrounding less dense regions.
These will eventually become negative pions, although
the (inhomogeneous, nonequilibrium) dynamics involved are 
far from simple.  When the
condensate breaks up late in the collision, it likely
yields equal numbers of protons and neutrons. 
%or at minimum
%a proton/neutron ratio closer to unity than in the incident
%nuclei. 
The total number of protons in the final state will
therefore be more than twice 79. 
We now answer the question posed
in the introduction.
When experimentalists
compare AGS events selected to have unusually high density
and low temperature with events from an entire data set,
they should look for (i) an increase in the number of protons
per event, and (ii) an increase in the $\pi^-/\pi^+$ ratio.
We close by noting that effects of the kind we have 
sketched here, namely equalization of chemical potentials
due to the formation of a color superconductor in high
density regions,
may prove more dramatic when 
the strange quark is included.  This is work
in progress.


\begin{thebibliography}{9}

\bibitem{us}
M. Alford, K. Rajagopal and F. Wilczek,  Phys.
Lett. {\bf B} to appear. 
See this paper for references.  See also
%Color 
%superconductivity has also been treated recently by
R. Rapp {\it et al}, .


%\bibitem{collins}
%J. C. Collins and M. J. Perry, Phys. Rev. Lett. {\bf 34} (1975) 1353.

\bibitem{seto}
R. Seto, private communication.

%\bibitem{BCS}
%J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. {\bf 106}
%(1957) 162; {\bf 108} (1957) 1175.

%\bibitem{bailin}
%D. Bailin and A. Love, Phys. Rept. {\bf 107} (1984) 325,
%and references therein.

%\bibitem{thooft}
%G. 't Hooft, Phys. Rev. {\bf D14} (1976) 3432.

%\bibitem{vaks}
%V. G. Vaks and A. I. Larkin, Sov. Phys. JETP {\bf 13} (1961) 192.


%\bibitem{njl}
%Y. Nambu and G. Jona-Lasinio, Phys. Rev. {\bf 122} (1961) 345; {\bf 124}
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%\bibitem{klevansky}
%For reviews, see S. P. Klevansky, Rev. Mod. Phys. {\bf 64} (1992) 649,
%and T. Hatsuda and T. Kunihiro, Phys. Rept. {\bf 247} (1994) 221.\\
%Studies of finite density physics in Nambu Jona-Lasinio
%models include: V. Bernard, U.-G. Meissner and I. Zahed,
%Phys. Rev. {\bf D36} (1987) 819; T. Hatsuda and T. Kunihiro,
%Phys. Lett {\bf B198} (1987) 126; M. Asakawa and K. Yazaki,
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%\bibitem{instantonliquid}
%For a review, see T. Schaefer and E. V. Shuryak, .
%Extensions to finite density are considered in 
%T. Schaefer,  and in R. Rapp, T. Schaefer,
%E. V. Shuryak and M. Velkovsky, in preparation.

%\bibitem{buballa}
%M. Buballa, Nucl. Phys. {\bf A609} (1996) 519.


%\bibitem{aichelin}
%J. Aichelin, private communication.

%\bibitem{satz}
%Percolation-based approaches to the chiral transition
%as a function of density include: G. Baym, Physica {\bf 96A} (1979)
%131; T. Celik, F. Karsch and H. Satz, Phys. Lett. {\bf 97B} (1980)
%128.

%\bibitem{ogleman}
%See, for example,
%J. M. Lattimer {\it et al}, Astrophys. J. {\bf 425} (1994) 425;
%C. Schaab {\it et al}, Astrophys. J. {\bf 480} (1997) L111.\\
%For a review of the observations, see H. \"Ogleman, 
%in The Lives of the Neutron Stars, M. Alpar {\it et al}, eds.,
%(Kluwer, 1995) 101.


%\bibitem{schaab}
%C. Schaab {\it et al} in \cite{ogleman}.

\end{thebibliography}
\end{document}


%The satisfying picture just discussed is not
%obtained for all parameter values, however.
%For example, for $\nu=1$ and $2.2 < \Lambda < 3.2$  
%the phase at $n=n_0$ has higher energy per baryon than
%that of a  dilute gas of quarks with mass
%$\Delta(0)$.  For $\Lambda > 3.2$ GeV,
%the pressure is positive for all $n$.  In these 
%(fortunately, unreasonable) parameter ranges,
%the model, without further
%modification,  has no reasonable physical interpretation.

%At a quantitative level, a naive implementation of our proposed
%identification of droplets of $n=n_0$ matter with nucleons
%works surprisingly well.   
%One
%might want to identify $n_0$ with the quark density at the
%center of  baryons, or one might require that the energy
%per quark be one third the nucleon mass at $n=n_0$.
%Our toy model treatment cannot meet both criteria simultaneously, which
%is not surprising,
%but we see in \cite{us} that the magnitude of $n_0$
%is very reasonable.
%The 
%vacuum energy, which becomes the bag constant,
%is also of the correct order of magnitude.
%Adding further interactions to $H_I$
%would obviously
%make a quantitative difference, 
%but there is no
%reason to expect the qualitative picture to change.

As in the previous section, 
one can obtain expressions for the energy and the density,
and thus derive the equation of state.  We find that the equation of
state is hardly modified from the free-quark values --- the pressures,
at equal density, are 
equal to within a few per cent.  This makes it very plausible that,
as we assume, 
the color condensation makes only a small change in the 
density $n_0$ at which a stable phase exists at zero pressure. 
To make the argument rigorous,
we must do a calculation in which we consider chiral and
color condensation simultaneously; we should form 
a trial wave function that allows for both possibilities
and allows them to compete.  We have
begun this calculation, but will not report on it here
other than to note that since the two condensates compete
for the same quarks the bigger of the two tends to suppress
the smaller.  This is further evidence that the potential for
the formation of  a color breaking gap does not affect the result
that $n_0>n_c$.
For practical purposes, it appears to be a very good approximation to
treat the condensations separately, as we do here,
because where one is large the other is small, even before they compete.





Note that $n$ is now not given by (\ref{ndef}) because
the operators in (\ref{colwf}) can change particle number.
Varying the expectation value of $H-\mu N$ in this state
with respect to $p_F$ yields $p_F=\mu$, unlike in 
the case of the chiral condensate.  This difference
reflects the fact that 
a gap in a $\<qq\>$ channel does not act as an effective mass
term in the way that a chiral gap                  
does.  Upon adding a quark, the condensate can adjust in such
a way that the energy cost is only $p_F$.   $\Delta$
is, however, a true gap in the sense of
condensed matter physics: the energy cost of 
making a particle-hole excitation is  
$2\Delta$ at minimum.
Varying with respect to 
all the other variational parameters yields
$\xi^R_{A,B,C} + \xi^L_{A,B,C} = \pi$, $\theta^R_{A,B,C}=
\theta^L_{A,B,C}$, and
\beq
\tan(2\theta^L_A(\bp)) = \frac{F^2(p)\Delta}{p-\mu}\ ,
\ \ \
\tan(2\theta^L_B(\bp)) = \frac{F^2(p)\Delta}{p+\mu}\ ,
\ \ \
\tan(2\theta^L_C(\bp)) = \frac{F^2(p)\Delta}{\mu-p}\ .
\label{colorvariation}
\eeq
Here, the gap $\Delta$ satisfies
a self-consistency equation of the form
\beq
\ba{rrl}
1 = \dsp{2K} \biggl\{ &&
 \dsp\int_\mu^\infty \frac{p^2 dp}{2\pi^2} { F^4(p) 
\over\sqrt{ F^4(p)\De^2 + (p-\mu)^2}}\\[3ex]
& + &\dsp\int_0^\infty\frac{p^2 dp}{2\pi^2} 
{ F^4(p) \over \sqrt{ F^4(p)\De^2 + (p+\mu)^2}}\\[3ex]
& + &\dsp\int_0^\mu\frac{p^2 dp}{2\pi^2} 
{F^4(p) \over \sqrt{ F^4(p)\De^2 + (\mu-p)^2}}
\ \ \biggr\}\ .
\ea
\label{colorgapeq}
\eeq
The three terms in this equation arise respectively from 
particles above the 
Fermi surface, antiparticles, 
and particles below the Fermi surface.  
For $\mu >  0 $ the particle and hole integrals diverge logarithmically
at the Fermi surface 
as $\Delta \rightarrow 0$, which signals the possibility of
condensation for arbitrarily weak attraction.



One can form reasonable qualitative expectations for the solution of
the gap equation without detailed calculations.  
Because the numerical  coefficient in
the gap equation is smaller than
the threshold value at which one would
have a nonzero $\Delta$ at $\mu=0$, $\Delta$ would be zero were
it not for the logarithmically divergent contribution
to the integral from the region near $\mu$. 
This means that at small $\mu$, the gap must be small
because the density of
states at the Fermi surface is small.  This has only formal
significance, because the only densities of physical relevance
are $n=0$ and $n\ge n_0$. At intermediate densities, matter
is in an inhomogeneous mixture of the $n=0$ and $n=n_0$
phases. (We are assuming that the 
color breaking condensate does not significantly
affect $n_0$; this will be discussed below.)
As $\mu$ increases, the density of states at
the Fermi surface increases and the gap parameter grows.  Finally, at
large $\mu$ the effect of the form factor $F$ is felt,
the effective coupling decreases, and the
gap parameter goes back down.  For the parameter ranges we have examined
the gap parameter is quite substantial:  $\sim 50-150$ 
MeV at $n_0$, and
peaking at $100-200$ MeV at a density somewhat higher. We plot
$\Delta$ for two sets of parameters in Figure 2.  The density
at which the gap peaks depends on $\Lambda$; the shape
of the curve depends on $\nu$; the height of the curve is
almost independent of both.
\begin{figure}[t]
\centerline{
\epsfysize=3in
\hfill\epsfbox{ColGaps.eps}\hfill
}
\caption{Gap created
by the Lorentz scalar color superconductor 
condensate,
as a function of $\mu=p_F$ for $\nu=1$ and (from left to right) 
$\Lambda=0.4,0.8$ GeV. Each curve begins
where $n$ is given by the appropriate $n_0$.}
\end{figure}


In our model as it stands, color 
is realized as a global symmetry.  Breaking of this symmetry generates
Nambu-Goldstone bosons, formally.  
However,
in reality color
is of course
a gauge symmetry, and the true spectrum does not contain massless
scalars, but rather massive vectors.   Aside from a node along the
equator of the Fermi surface
for one color, there is a gap everywhere on the quark Fermi
surfaces. 
%so no massless excitations are left in the spectrum.  
To this point, we have described the color superconducting
phase as a Higgs phase.  One expects, however, that 
there is a complementary description in which this
is a confining phase, albeit one with two vastly different
confinement lengths, neither of which is related to the
confinement length at zero density.
%
%One can speak of the
%color superconducting phase, in this sense,
%as a Higgs or alternatively as a confined phase, but these usages
%might be misleading: there are no fundmamental Higgs fields, and no
%simple continuation to the zero-density confined phase; furthermore,
%in this connection, the mass scale is both vastly different and 
%density dependent.  However, as a formal matter, 
%t is of some
As a formal matter, it is 
of some interest that the color superconducting phase can be considered a
realization of {\it confinement without chiral symmetry breaking}. 

In looking for signatures of color superconductivity in heavy ion physics
and in neutron stars, it is unfortunate that the equation
of state is almost equal to that for a deconfined phase
with no diquark condensate.
Superconducting condensates do modify the gauge interactions and
this may have implications in heavy ion collisions.
The scalar condensate carries electric as well as color charge.  It
is neutral under a certain combination of electrodynamic and color
hypercharge, so taken by itself it would leave a modified massless
photon.   
If densities above $n_0$ are achieved at
low enough temperatures that the scalar condensate forms,
there will
be a mixture of the ordinary photon and the color hypercharge gauge
boson which is massless.
(This modified photon would acquire a small mass from the 
axial vector condensate if temperatures were low enough
for this condensate to be present.)  
There is also
a residual $SU(2)$ gauge symmetry,
presumably deconfined,
and there are five gluons whose mass is set by the scalar 
condensate.   
Either the modification of the photon
or the loss of massless gluons could 
have consequences, but dramatic effects
do not seem apparent. 


Turning to neutron 
star phenomenology, there is some indication, from the slowness
of observed neutron star cooling rates, that a gap in the excitation
spectrum for quark matter might be welcome\cite{ogleman},  
as this suppresses neutrino emission via weak interaction
processes involving single
thermally excited $u$ and $d$ quarks by $\exp(-\Delta/T)$.
A $400$ keV gap has dramatic consequences\cite{schaab};
the scalar gap is therefore enormous in this context, and the
axial vector gap plays a role too, shutting down
these direct neutrino emission processes completely
once the core cools to temperatures at which
the axial condensate forms. It would also be
worthwhile to explore the effects of
the presence of
macroscopic regions in which an axial vector
condensate is ordered.


\section{Discussion}

Many things were ignored in this analysis.  Most important, perhaps,
is the strange quark.  In the spirit of the analysis, we should
consider the modified instanton vertex including the strange quark as
well.  This adds an incoming left-handed and an outgoing right-handed
leg.  
If the mass of this quark were large, we could connect these legs with
a large coefficient, and reduce to the previous case,
perhaps with an additional four-fermion
vertex involving all
three flavors modelled on one-gluon exchange.  
Whatever the interaction(s), color superconductivity in
a three flavor theory necessarily introduces the new
feature of flavor symmetry breaking. Both the 
condensates considered in this paper are flavor singlets;
this is impossible for a $\langle qq \rangle$ condensate
in a three flavor theory.  One particularly attractive
possibility
is condensation in the 
$\langle q^\alpha_i \, C \gamma^5 q^\beta_j \varepsilon^{ijA}
\varepsilon_{\alpha\beta A}\rangle$
channel, with summation over $A$.
This breaks 
flavor and color in a coordinated fashion,
leaving unbroken the
diagonal subgroup of $SU(3)_{\rm color}\times SU(3)_{\rm flavor}$.
%In this case, and indeed in the two flavor case also, 
%a systematic comparison of all possible condensation
%patterns remains to be done. This will in general require treating
%multiple competing condensates.
%
%
%In reality it
%is not clear this is adequate.  The next simplest possibility is that
%the strange quark remains chirally condensed; and in this case too the
%analysis above is only slightly modified.  However there are numerous
%other possibilities, for example including chiral condensation in
%mixed flavor channels (kaon, or for that matter pion, condensation),
%or mixed color-flavor channels.  It is possible to carry out a
%systematic comparison of these, at least at weak coupling, and we are
%presently implementing such a comparison.  
%Need another sentence here.


Another question concerns the postulated Hamiltonian.  While there are
good reasons to take an effective interaction of the instanton type
as a starting point, there could well be 
significant corrections affecting the more delicate
consequences such as axial vector {\bf 6} condensation.   A specific,
important example is to compare the effective interaction derived from
one-gluon exchange.   It turns out that this interaction has a
similar pattern, for our purposes, to the instanton: it is very attractive in
the $\sigma$ channel, attractive in the color antitriplet scalar, and
neither attractive nor repulsive in the color sextet axial vector.
More generally, it would be desirable to use a renormalization
group treatment to find the interactions which are
most relevant near the Fermi surface.

The qualitative model we have treated suggests a compelling picture
both for the chiral restoration transition and for the color
superconductivity which sets in 
at densities just beyond. It
points toward future
work in many directions:  the percolation transition
must be characterized; consequences in neutron 
star and heavy ion physics remain to be elucidated;
the superconducting ordering patterns
may hold further surprises, particularly
as flavor becomes important. 
The whole subject needs more work; the microscopic
phenomenon is so remarkable, that we suspect our imaginations have
failed adequately to grasp its implications. 






