# Color superconductivity in dense quark matter

###### Abstract

Matter at high density and low temperature is expected to be a color superconductor, which is a degenerate Fermi gas of quarks with a condensate of Cooper pairs near the Fermi surface that induces color Meissner effects. At the highest densities, where the QCD coupling is weak, rigorous calculations are possible, and the ground state is a particularly symmetric state, the color-flavor locked (CFL) phase. The CFL phase is a superfluid, an electromagnetic insulator, and breaks chiral symmetry. The effective theory of the low-energy excitations in the CFL phase is known and can be used, even at more moderate densities, to describe its physical properties. At lower densities the CFL phase may be disfavored by stresses that seek to separate the Fermi surfaces of the different flavors, and comparison with the competing alternative phases, which may break translation and/or rotation invariance, is done using phenomenological models. We review the calculations that underlie these results, and then discuss transport properties of several color-superconducting phases and their consequences for signatures of color superconductivity in neutron stars.

^{†}

^{†}preprint: MIT-CTP-3861

###### Contents

- I Introduction
- II Matter at the highest densities
- III Below CFL densities
- IV Weak-coupling QCD calculations
- V Effective theories of the CFL phase
- VI NJL model comparisons among candidate phases below CFL densities
- VII Transport properties and neutrino processes
- VIII Color superconductivity in neutron stars

## I Introduction

### i.1 General outline

The study of matter at ultra-high density is the “condensed matter physics of quantum chromodynamics”. It builds on our understanding of the strong interaction, derived from experimental observation of few-body processes, to predict the behavior of macroscopic quantities in many-body systems where the fundamental particles of the standard model—quarks and leptons—become the relevant degrees of freedom. As in conventional condensed-matter physics, we seek to map the phase diagram and calculate the properties of the phases. However, we are in the unusual position of having a sector of the phase diagram where we can calculate many properties of quark matter rigorously from first principles. This sector is the region of “asymptotically high” densities, where quantum chromodynamics is weakly coupled. We will review those rigorous results and describe the progress that has been made in building on this solid foundation to extend our understanding to lower and more phenomenologically relevant densities. Quark matter occurs in various forms, depending on the temperature and quark chemical potential (see Fig. 1). At high temperatures () entropy precludes any pattern of order and there is only quark-gluon plasma (QGP), the phase of strongly interacting matter that has no spontaneous symmetry breaking, and which filled the universe for the first microseconds after the big bang. Quark-gluon plasma is also being created in small, very short-lived, droplets in ultrarelativistic heavy ion collisions at the Relativistic Heavy Ion Collider.

In this review we concentrate on the regime of relatively low temperatures, , where we find a rich variety of spontaneous symmetry breaking phases. To create such material in nature requires a piston that can compress matter to super-nuclear densities and hold it while it cools. The only known context where this might happen is in the interior of neutron stars, where gravity squeezes the star to an ultra-high density state where it remains for millions of years. This gives time for weak interactions to equilibrate, and for the temperature of the star to drop far below the quark chemical potential. We do not currently know whether quark matter exists in the cores of neutron stars. One of the reasons for studying color superconductivity is to improve our understanding of how a quark matter core would affect the observable behavior of a neutron star, and thereby resolve this uncertainty.

When we speak of matter at the highest densities, we shall always take the high density limit with up, down and strange quarks only. We do so because neutron star cores are not dense enough (by more than an order of magnitude) to contain any charm or heavier quarks, and our ultimate goal is to gain insight into quark matter at densities that may be found in nature. For the same reason we focus on temperatures below about ten MeV, which are appropriate for neutron stars that are more than a few seconds old.

As we will explain in some detail, at low temperatures and the highest
densities we expect to find a degenerate liquid
of quarks, with Cooper pairing near the Fermi surface that spontaneously
breaks the color gauge symmetry (“color superconductivity”). Speculations
about the existence of a quark matter phase at high density go back to the
earliest days of the quark model of hadrons
Carruthers (1973); Ivanenko and Kurdgelaidze (1965); Pacini (1966); Boccaletti
*et al.* (1966); Itoh (1970),
and the possibility of quark Cooper pairing was noted even before
there was a comprehensive theory of the strong interaction
Ivanenko and Kurdgelaidze (1969, 1970). After the development of
quantum chromodynamics (QCD), with its property of asymptotic freedom
Gross and Wilczek (1973); Politzer (1973), it became clear that a quark matter
phase would exist at sufficiently high density
Collins and Perry (1975); Kislinger and Morley (1976); Freedman and McLerran (1977, 1978); Baym and Chin (1976); Chapline and Nauenberg (1976, 1977)
and the study of quark Cooper
pairing was pioneered by Barrois and Frautschi
Barrois (1977, 1979); Frautschi (1978), who first used
the term “color superconductivity”, and by Bailin and Love
Bailin and Love (1979, 1984), who classified many of the possible
pairing patterns. Iwasaki and Iwado Iwasaki and Iwado (1995); Iwasaki (1995)
performed mean-field calculations of single-flavor pairing in a
Nambu-Jona-Lasinio (NJL) model, but it was not until the prediction
of large pairing gaps Alford *et al.* (1998); Rapp *et al.* (1998) and the
color-flavor locked (CFL) phase Alford
*et al.* (1999b) that the phenomenology
of color-superconducting quark matter became widely studied. At present
there are many reviews of the subject from various stages in its development
Shovkovy (2005); Nardulli (2002); Huang (2005); Buballa (2005a); Ren (2004); Hsu (2000); Bailin and Love (1984); Rajagopal and Wilczek (2000); Alford (2001); Hong (2001); Rischke (2004); Schäfer (2003b); Reddy (2002); Alford and Rajagopal (2006),
and the reader may wish to consult them for alternative presentations
with different emphases.
As these reviews make clear, the last decade has seen dramatic
progress in our understanding of dense matter. We are now able to
obtain, directly from QCD,
rigorous and quantitative answers to the basic
question: “What happens to matter if you squeeze it to arbitrarily
high density?”. In Sec. IV we will show how QCD becomes
analytically tractable at arbitrarily high density: the coupling
is weak and the physics of confinement never arises, since
long-wavelength magnetic interactions are cut off, both by Landau
damping and by the Meissner effect.
As a result, matter at the highest densities is known to be in the
CFL phase, whose properties (see Sec. II)
are understood rigorously from first principles. There is
a well-developed effective field theory describing the low energy
excitations of CFL matter (see Sec. V), so
that at any density at which the
CFL phase occurs, even if this density is not high enough for a
weak-coupling QCD calculation to be valid, many phenomena can
nevertheless be described quantitatively in terms of a few parameters,
via the effective field theory.

It should be emphasized that QCD at arbitrarily high density is more fully understood than in any other context. High energy scattering, for example, can be treated by perturbative QCD, but making contact with observables brings in poorly understood nonperturbative physics via structure functions and/or fragmentation functions. Or, in quark-gluon plasma in the high temperature limit much of the physics is weakly-coupled but the lowest energy modes remain strongly coupled with nonperturbative physics arising in the nonabelian color-magnetic sector. We shall see that there are no analogous difficulties in the analysis of CFL matter at asymptotic densities.

If the CFL phase persists all the way down to the transition to
nuclear matter then we have an exceptionally good theoretical
understanding of the properties
of quark matter in nature. However, less
symmetrically paired phases of quark matter may well intervene
in the intermediate density region between nuclear and CFL matter
(Sec. I.5).
We enumerate some of the possibilities in Sec. III.
In principle this region could also be understood
from first principles, using brute-force numerical methods (lattice QCD) to
evaluate the QCD path integral, but unfortunately current lattice QCD
algorithms are defeated by the fermion sign problem in the
high-density low-temperature regime Schmidt (2006).^{1}^{1}1
Condensation of Cooper pairs of quarks has been studied
on the lattice in 2-color QCD
Nishida *et al.* (2004); Kogut *et al.* (1999, 2000); Fukushima and Iida (2007); Hands *et al.* (1999); Kogut *et al.* (2001); Hands *et al.* (2006); Kogut *et al.* (2002); Alles *et al.* (2006),
for high isospin
density rather than baryon density Kogut and Sinclair (2002); Splittorff
*et al.* (2001); Son and Stephanov (2001) and
in NJL-type models Hands and Walters (2004).
This means
we have to use models, or try to derive information from astrophysical
observations.
In Sec. VI
we sketch an example of a (Nambu–Jona-Lasinio) model analysis
within which one can compare some of the possible intermediate-density
phases suggested in Sec. I.5.
We finally discuss the observational approach, which involves
elucidating the properties of the suggested phases of quark matter
(Secs. VI.3 and VII), and then finding
astrophysical signatures by
which their presence inside neutron stars might be established or ruled
out using astronomical observations (Sec. VIII).

### i.2 Inevitability of color superconductivity

At sufficiently high density and low temperature it is a good starting point to imagine that quarks form a degenerate Fermi liquid. Because QCD is asymptotically free — the interaction becomes weaker as the momentum transferred grows — the quarks near the Fermi surface are almost free, with weak QCD interactions between them. (Small-angle quark-quark scattering via a low-momentum gluon is no problem because it is cut off by Landau damping, which, together with Debye screening, keeps perturbation theory at high density much better controlled than at high temperature Pisarski and Rischke (1999a); Son (1999).) The quark-quark interaction is certainly attractive in some channels, since we know that quarks bind together to form baryons. As we will now argue, these conditions are sufficient to guarantee color superconductivity at sufficiently high density.

At zero temperature, the thermodynamic potential (which we will loosely refer
to as the “free energy”) is , where is
the total energy of the system, is the chemical potential, and
is the number of fermions.
If there were no interactions then the energy required to add a particle
to the system would be the Fermi energy , so adding or subtracting
particles or holes near the Fermi surface would cost zero free energy.
With a weak attractive interaction in any channel, if we
add a pair of particles (or holes)
with the quantum numbers of the attractive channel, the free energy
is lowered by the potential energy of their attraction.
Many such pairs will therefore be created in the modes near the Fermi
surface, and these pairs, being bosonic, will form a condensate. The
ground state will be a superposition of states with different numbers
of pairs, breaking the fermion number symmetry. This argument,
originally developed by Bardeen, Cooper,
and Schrieffer (BCS) Bardeen *et al.* (1957) is
completely general, and can be applied to electrons in a metal, nucleons
in nuclear matter, He atoms, cold fermionic atoms in a trap, or
quarks in quark matter.

The application of the BCS mechanism to pairing in dense quark matter is in a sense more direct than in its original setting. The dominant interaction between electrons in a metal is the repulsive Coulomb interaction, and it is only because this interaction is screened that the attraction mediated by phonons comes into play. This means that the effective interactions that govern superconductivity in a metal depend on band structure and other complications and are very difficult to determine accurately from first principles. In contrast, in QCD the “color Coulomb” interaction is attractive between quarks whose color wave function is antisymmetric, meaning that superconductivity arises as a direct consequence of the primary interaction in the theory. This has two important consequences. First, at asymptotic densities where the QCD interaction is weak we can derive the gap parameter and other properties of color superconducting quark matter rigorously from the underlying microscopic theory. Second, at accessible densities where the QCD interaction is stronger the ratio of the gap parameter to the Fermi energy will be much larger than in conventional BCS superconducting metals. Thus, superconductivity in QCD is more robust, both in the theoretical sense and in the phenomenological sense, than superconductivity in metals.

It has long been known
that, in the absence of pairing,
an unscreened static magnetic interaction results in a
“non-Fermi-liquid” Holstein *et al.* (1973); Ipp *et al.* (2006); Chakravarty
*et al.* (1995); Baym *et al.* (1990); Vanderheyden and
Ollitrault (1997); Manuel (2000a, b); Brown
*et al.* (2000a); Boyanovsky and
de Vega (2001a, b); Ipp *et al.* (2004); Gerhold *et al.* (2004); Polchinski (1994); Nayak and Wilczek (1994). However, in QCD the
magnetic interaction is screened at nonzero frequency (Landau damping)
and this produces a particularly mild form of non-Fermi-liquid behavior,
as we describe in Sec. V.1.2. In the absence of pairing but in the
presence of interactions, there are still quark quasiparticles and
there is still a “Fermi surface”, and the BCS argument goes through.
This argument is rigorous at high densities, where the QCD coupling
is small. The energy scale below which non-Fermi liquid effects would
become strong enough to modify the quasiparticle picture qualitatively
is parametrically of order whereas the BCS gap
that results from pairing is parametrically larger, of order
as we shall see in Sec. IV.
This means that pairing occurs
in a regime where the basic logic of the BCS argument remains
valid.

Since pairs of quarks cannot be color singlets, the Cooper pair condensate in quark matter will break the local color symmetry , hence the term “color superconductivity”. The quark pairs play the same role here as the Higgs particle does in the standard model: the color-superconducting phases can be thought of as Higgs phases of QCD. Here, the gauge bosons that acquire a mass through the process of spontaneous symmetry breaking are the gluons, giving rise to color Meissner effects. It is important to note that quarks, unlike electrons, have color and flavor as well as spin degrees of freedom, so many different patterns of pairing are possible. This leads us to expect a panoply of different possible color superconducting phases.

As we shall discuss in Sec. II, at the highest densities we can achieve an ab initio understanding of the properties of dense matter, and we find that its preferred state is the CFL phase of three-flavor quark matter, which is unique in that all the quarks pair (all flavors, all colors, all spins, all momenta on the Fermi surfaces) and all the nonabelian gauge bosons are massive. The suppression of all of the infrared degrees of freedom of the types that typically indicate either instability toward further condensation or strongly coupled phenomena ensures that, at sufficiently high density, the CFL ground state, whose only infrared degrees of freedom are Goldstone bosons and an abelian photon, is stable. In this regime, quantitative calculations of observable properties of CFL matter can be done from first principles; there are no remaining nonperturbative gaps in our understanding.

As the density decreases, the effect of the strange quark mass becomes more noticeable, imposing stresses that may modify the Cooper pairing and the CFL phase may be replaced by other forms of color superconducting quark matter. Furthermore, as the attractive interaction between quarks becomes stronger at lower densities, correlations beyond the two-body correlation that yields Cooper pairing may become important, and at some point the ground state will no longer be a Cooper-paired state of quark matter, but something quite different. Indeed, by the time we decrease the density down to that of nuclear matter, the average separation between quarks has increased to the point that the interactions are strong enough to bind quarks into nucleons. It is worth noting that quark matter is in this regard different from Cooper-paired ultracold fermionic atoms (to be discussed in Sec. III.9). For fermionic atoms, as the interaction strength increases there is a crossover from BCS-paired fermions to a Bose-Einstein condensate (BEC) of tightly-bound, well-separated, weakly-interacting di-atoms (molecules). In QCD, however, the color charge of a diquark is the same as that of an antiquark, so diquarks will interact with each other as strongly as quarks, and there will not be a literal analogue of the BCS/BEC crossover seen in fermionic atoms. In QCD, the neutral bound states at low density that are (by QCD standards) weakly interacting are nucleons, containing three quarks not two.

We shall work with colors throughout. In the limit with fixed (i.e fixed ), Cooper
pairing is not necessarily energetically preferred. A strong
competitor for the large- ground state is the chiral density wave
(CDW), a condensate of quark-hole pairs, each with total momentum
Deryagin *et al.* (1992).
Quark-hole scattering is enhanced by a factor of over quark-quark
scattering, but, unlike Cooper pairing, it only uses a small
fraction of the Fermi surface, and in the case of short range forces
the CDW phase is energetically favored in one-dimensional
systems, but not in two or more spatial dimensions
Shankar (1994). However, in QCD in the large limit
the equations governing the CDW state become effectively one-dimensional
because the gluon propagator is not modified by the medium,
so the quark-hole
interaction is dominated by almost collinear scattering.
Since pairing gaps are exponentially small in the coupling but medium
effects only vanish as a power of , the CDW state requires an
exponentially large number of colors. It is estimated that
for GeV, quark-hole pairing becomes favored over
Cooper pairing when Shuster and Son (2000).
Recent work McLerran and Pisarski (2007) discusses aspects
of physics at large at lower densities that may
also be quite different from
physics at .

Before turning to a description of CFL pairing in Sec. II and less-symmetrically paired forms of color superconducting quark matter in Sec. III, we discuss some generic topics that arise in the analysis of color-superconducting phases: the gap equations, neutrality constraints, the resultant stresses on Cooper pairing, and the expected overall form of the phase diagram.

### i.3 Quark Cooper pairing

The quark pair condensate can be characterized in a gauge-variant way by the expectation value of the one-particle-irreducible quark-quark two-point function, also known as the “anomalous self-energy”,

(1) |

Here is the quark field operator, color indices range over red, green, and blue (), flavor indices range over up, down and strange (), and are the spinor Dirac indices. The angle brackets denote the one-particle-irreducible part of the quantum-mechanical ground-state expectation value. In general, both sides of this equation are functions of momentum. The color-flavor-spin matrix characterizes a particular pairing channel, and is the gap parameter which gives the strength of the pairing in this channel. A standard BCS condensate is position-independent (so that in momentum space the pairing is between quarks with equal and opposite momentum) and a spin singlet (so that the gap is isotropic in momentum space). However, as we will see later, there is good reason to expect non-BCS condensates as well as BCS condensates in high-density quark matter.

Although (1) defines a gauge-variant quantity, it is still of physical relevance. Just as electroweak symmetry breaking is most straightforwardly understood in the unitary gauge where the Higgs vacuum expectation value is uniform in space, so color superconductivity is typically analyzed in the unitary gauge where the quark pair operator has a uniform color orientation in space. We then relate the gap parameter to the spectrum of the quark-like excitations above the ground state (“quasiquarks”), which is gauge-invariant.

In principle, a full analysis of the phase structure of quark matter in the - plane would be performed by writing down the free energy , which is a function of the temperature, the chemical potentials for all conserved quantities, and the gap parameters for all possible condensates, including the quark pair condensates but also others such as chiral condensates of the form . We impose neutrality with respect to gauge charges (see Sect. I.4 below) and then within the neutral subspace we minimize the free energy with respect to the strength of the condensate:

(2) |

We have written this gap equation and stability condition somewhat schematically since for many patterns of pairing there will be gap parameters with different magnitudes in different channels. The free energy must then be minimized with respect to each of the gap parameters, yielding a coupled set of gap equations. The solution to (2) with the lowest free energy that respects the neutrality constraints discussed below yields the favored phase.

### i.4 Chemical potentials and neutrality constraints

Why do we describe “matter at high density” by introducing a large
chemical potential for quark number but no chemical potentials
for other quantities? The answer is that this reflects the
physics of neutron stars, which are the main physical arena that we consider.
Firstly, on the long timescales relevant to neutron stars,
the only global charges that are conserved
in the standard model are quark number and lepton number, so only these
can be coupled to chemical potentials (we shall discuss gauged charges
below). Secondly, a neutron star is permeable to
lepton number because neutrinos are so light and weakly-interacting
that they can quickly escape from the star, so the chemical potential
for lepton number is zero.
Electrons are present because they carry
electric charge, for which there is a nonzero potential.
In the first few seconds of
the life of a neutron star the neutrino mean free path may be short
enough to sustain a nonzero lepton number chemical potential, see for
instance Ruester *et al.* (2006); Kaplan and Reddy (2002); Laporta and Ruggieri (2006); Berdermann
*et al.* (2004), but we will
not discuss that scenario.

Stable bulk matter must be neutral under all gauged charges, whether
they are spontaneously broken or not. Otherwise, the net charge
density would create large electric fields, making the energy
non-extensive. In the case of the electromagnetic gauge symmetry,
this simply requires zero charge density, . The correct formal
requirement concerning the color charge of a large lump of matter is
that it should be a color singlet, i.e., its wavefunction should
be invariant under a general color gauge transformation. However,
it is sufficient for us to impose color neutrality,
meaning equality in the numbers of red, green, and
blue quarks. This is a less stringent constraint
(singletneutral but neutralsinglet)
but the projection of a color neutral
state onto a color singlet costs no extra free energy in the
thermodynamic limit Amore *et al.* (2002).
(See also Elze *et al.* (1983, 1984).)
In general there are 8 possible
color charges, but because the
Cartan subalgebra of is two-dimensional
it is always possible to transform to a gauge
where all are zero except and , the charges associated
with the diagonal generators
and
in space
Buballa and Shovkovy (2005); Rajagopal and Schmitt (2006). In this review, we only
discuss such gauges. So to impose color neutrality we just require
.

In nature, electric and color neutrality are enforced by the dynamics of the electromagnetic and QCD gauge fields, whose zeroth components serve as chemical potentials coupled to the charges , and which are naturally driven to values that set these charges to zero Iida and Baym (2001); Alford and Rajagopal (2002); Gerhold and Rebhan (2003); Kryjevski (2003); Dietrich and Rischke (2004). In an NJL model with fermions but no gauge fields (see Sec. VI) one has to introduce the chemical potentials , and by hand in order to enforce color and electric neutrality. The neutrality conditions are then

(3) |

(Note that we define an electrostatic potential that is coupled to the negative electric charge , so that in typical neutron star conditions, where there is a finite density of electrons rather than positrons, is positive.)

Finally we should note that enforcing local neutrality is appropriate for uniform phases, but there are also non-uniform charge-separated phases (“mixed phases”), consisting of positively and negatively charged domains which are neutral on average. These are discussed further in Sec. III.8.

### i.5 Stresses on BCS pairing

The free energy argument that we gave in Sec. I.2 for the inevitability of BCS pairing in the presence of an attractive interaction relies on the assumption that the quarks that pair with equal and opposite momenta can each be arbitrarily close to their common Fermi surface. However, as we will see in Sec. II, the neutrality constraint, combined with the mass of the strange quark and the requirement that matter be in beta equilibrium, tends to pull apart the Fermi momenta of the different flavors of quarks, imposing an extra energy cost (“stress”) on the formation of Cooper pairs involving quarks of different flavors. This raises the possibility of non-BCS pairing in some regions of the phase diagram.

To set the stage here, let us discuss a simplified example: consider two massless species of fermions, labeled and , with different chemical potentials and , and an attractive interaction between them that favors cross-species BCS pairing with a gap parameter . It will turn out that to a good approximation the color-flavor locked pairing pattern contains three such sectors, so this example captures the essential physics we will encounter in later sections. We define the average chemical potential and the stress parameter

(4) |

As long as the stress is small enough relative to ,
BCS pairing between species 1 and 2 can occur,
locking their Fermi surfaces together and ensuring that
they occur in equal numbers.
At the Chandrasekhar-Clogston point Clogston (1962); Chandrasekhar (1962),
where ,
the two-species model undergoes a first-order transition to
the unpaired phase.
At this point BCS pairing still exists as a locally stable
state, with a completely gapped spectrum of quasiparticles.
When reaches the spectrum becomes gapless at momentum
, indicating that
cross-species BCS pairing is no longer favored at all momenta
Alford
*et al.* (2004b).
If the two species are part of a larger pairing pattern, the
Chandrasekhar-Clogston transition can be shifted, and we shall
see that in the two-species subsectors of the CFL pattern it
is shifted to .
The onset of gaplessness is therefore the relevant threshold
for our purposes, and it always occurs at ,
independent of the larger context in which the two flavors pair.
This follows from the fact that BCS pairing
only occurs if the energy gained from turning a quark into a
quark with the same momentum (namely ) is less
than the cost of breaking the Cooper pair formed by these quarks,
which is Rajagopal and Wilczek (2001). Thus the - Cooper
pairs are energetically stable (or metastable) as long as . A more detailed treatment of this illustrative example can
be found in Alford and Wang (2005).

This example uses massless quarks, but it can easily be modified to include the leading effect of a quark mass . A difference in the masses of the pairing quarks also stresses the pairing, because it gives them different Fermi momenta at the same chemical potential, so the quarks in a - Cooper pair, which have equal and opposite momenta, will not both be close to their Fermi energies. The leading-order effect is easily calculated, since for a quark near its Fermi surface it acts like a shift in the quark chemical potential by (given that Fermi momentum to this order).

Returning from our toy model to realistic quark matter, the quark flavors that are potentially relevant at neutron-star densities are the light up () and down () quarks, with current masses and that are , and a medium-weight flavor, the strange () quark, with current mass . Their effective “constituent” masses in the vacuum are hundreds of MeV larger, but are expected to decrease with increasing quark density. We shall refer to the density-dependent constituent masses as and shall typically neglect and . As our toy model has illustrated, however, the strange quark mass will contribute to stresses on cross-flavor pairing, and those stresses will become more severe as the density (and hence ) decreases. This will be a major theme of later sections.

### i.6 Overview of the quark matter phase diagram

Fig. 1 shows a schematic phase diagram for QCD that is consistent with what is currently known. Along the horizontal axis the temperature is zero, and the density is zero up to the onset transition where it jumps to nuclear density, and then rises with increasing . Neutron stars are in this region of the phase diagram, although it is not known whether their cores are dense enough to reach the quark matter phase. Along the vertical axis the temperature rises, taking us through the crossover from a hadronic gas to the quark-gluon plasma. This is the regime explored by high-energy heavy-ion colliders.

At the highest densities we find the color-flavor locked
color-superconducting phase,^{2}^{2}2
As explained in Sec. I.1, we fix
at all densities, to maintain relevance to neutron star interiors.
Pairing with arbitrary has been studied
Schäfer (2000a). For a multiple
of three one finds multiple copies of the CFL pattern; for
the pattern is more complicated.
in which the strange quark participates
symmetrically with the up and down quarks in Cooper
pairing. This is
described in more detail in Secs. II, IV, and V.
It is not yet clear what
happens at intermediate density, and in Secs. III and VI we will
discuss the factors that disfavor the CFL phase at intermediate
densities, and survey the color superconducting phases that
have been hypothesized to occur there.

Various aspects of color superconductivity at high temperatures
have been studied, including the phase structure
(see Sec. VI.1), spectral functions, pair-forming and -breaking fluctuations,
possible precursors to condensation such as pseudogaps,
and various collective phenomena
Kitazawa *et al.* (2007); Kitazawa
*et al.* (2005a); Kitazawa *et al.* (2002); Kitazawa
*et al.* (2005b); Fukushima and Iida (2005); Abuki *et al.* (2002); Kitazawa *et al.* (2004); Voskresensky (2004); Yamamoto *et al.* (2007); Hatsuda *et al.* (2006).
However, this review centers on quark matter at neutron star
temperatures, and
throughout Secs. II and III we restrict
ourselves to the
phases of quark matter at zero temperature. This is because most of
the phases that we discuss are expected to persist up to critical
temperatures that are well above the core temperature of a typical
neutron star, which drops below 1 MeV within seconds of its birth
before cooling down through the keV range over millions of years.

## Ii Matter at the highest densities

### ii.1 Color-flavor locked (CFL) quark matter

Given that quarks form Cooper pairs, the next question is who pairs with
whom? In quark matter at sufficiently high densities, where the up, down
and strange quarks can be treated on an equal footing and the disruptive
effects of the strange quark mass can be neglected, the most symmetric and
most attractive option is the color-flavor locked phase, where
quarks of all three colors and all
three flavors form conventional zero-momentum spinless Cooper pairs.
This pattern,
anticipated in early studies of
alternative condensates for zero-density chiral symmetry breaking
Srednicki and Susskind (1981), is encoded in
the quark-quark self-energy Alford
*et al.* (1999b)

(5) |

The symmetry breaking pattern is

(6) |

Color indices and flavor indices run from 1 to 3,
Dirac indices are suppressed, and is the Dirac charge-conjugation
matrix. Gauge symmetries are in square brackets.
is the CFL gap parameter. The Dirac structure is a Lorentz
singlet, and corresponds to parity-even spin-singlet pairing, so it is
antisymmetric in the Dirac indices. The two quarks in the Cooper pair
are identical fermions, so the remaining color+flavor structure must
be symmetric. The dominant color-flavor component in (5)
transforms as , antisymmetric in both. The
subdominant term, multiplied by , transforms as .
It is almost certainly not energetically favored on its own
(all the arguments in Sec. II.1.5 for the color triplet imply repulsion
for the sextet),
but in the presence of the dominant pairing it
breaks no additional symmetries, so is in general small but
not zero
Alford
*et al.* (1999b); Schäfer (2000a); Shovkovy and Wijewardhana (1999); Pisarski and
Rischke (1999c).

#### ii.1.1 Color-flavor locking and chiral symmetry breaking

A particularly striking feature of the CFL pairing pattern is that it breaks
chiral symmetry. Because of color-flavor locking, chiral symmetry
remains broken up to arbitrarily high densities in three-flavor quark
matter. The mechanism is quite different from the formation of the
condensate that breaks chiral symmetry in the
vacuum by pairing left-handed (L) quarks with right-handed (R) antiquarks.
The CFL condensate pairs L quarks with each other and R quarks with each
other (quarks in a Cooper pair have opposite momentum, and zero net spin,
hence the same chirality) and so it might naively appear chirally symmetric.
However, the Kronecker deltas in (5) connect color indices
with flavor indices, so that the condensate is not invariant under
color rotations, nor under flavor rotations, but only under
simultaneous, equal and opposite, color and flavor
rotations. Color is a vector symmetry, so the compensating flavor rotation
must be the same for L and R quarks, so
the axial part of the flavor group, which is
the chiral symmetry, is broken by the
locking of color and flavor
rotations to each other Alford
*et al.* (1999b). Such locking is familiar from other
contexts, including the QCD vacuum, where a condensate of
quark-antiquark pairs locks to breaking chiral
symmetry “directly”, and the B phase of superfluid He, where
the condensate transforms nontrivially under rotations of spin and
orbital angular momentum, but is invariant under simultaneous
rotations of both.

The breaking of the chiral symmetry is associated with an expectation value for a gauge-invariant order parameter with the structure (see Sec. V). There is also a subdominant “conventional” chiral condensate Schäfer (2000a). These gauge-invariant observables distinguish the CFL phase from the QGP, and if a lattice QCD algorithm applicable at high density ever becomes available, they could be used to map the presence of color-flavor locking in the phase diagram.

We also expect massless Goldstone modes associated with chiral symmetry breaking (see Secs. II.1.4 and V). In the real world there is small explicit breaking of chiral symmetry from the current quark masses, so the order parameters will not go to zero in the QGP, and the Goldstone bosons will be light but not massless.

#### ii.1.2 Superfluidity

The CFL pairing pattern spontaneously breaks
the exact global baryon number symmetry ,
leaving only a discrete symmetry under
which all quark fields are multiplied by . There is
an associated gauge-invariant 6-quark order parameter with the
flavor and color structure of two Lambda baryons,
where . This order parameter distinguishes the CFL phase
from the QGP, and there is an associated massless Goldstone boson that makes
the CFL phase a superfluid, see Sec. V.3.2.
The vortices that result
when CFL quark matter is rotated have been studied in Balachandran
*et al.* (2006); Nakano *et al.* (2007); Iida and Baym (2002); Forbes and Zhitnitsky (2002).

#### ii.1.3 Gauge symmetry breaking and electromagnetism

As explained above, the CFL condensate breaks the symmetry down to the diagonal group of simultaneous color and flavor rotations. Color is a gauge symmetry, and one of the generators of is the electric charge, which generates the gauge symmetry. This means that the unbroken contains one gauged generator, corresponding to an unbroken which consists of a simultaneous electromagnetic and color rotation. The rest of the color group is broken, so by the Higgs mechanism seven gluons and one gluon-photon linear combination become massive via the Meissner effect. The orthogonal gluon-photon generator remains unbroken, because every diquark in the condensate has . The mixing angle is where and are the QED and QCD couplings. Because the angle is close to zero, meaning that the photon is mostly the original photon with a small admixture of gluon.

The photon is massless. Given small but nonzero quark masses, there are no gapless -charged excitations; the lightest ones are the pseudoscalar pseudo-Goldstone bosons and (see Secs. II.1.4 and V), so for temperatures well below their masses (and well below the electron mass Shovkovy and Ellis (2003)) the CFL phase is a transparent insulator, in which -electric and magnetic fields satisfy Maxwell’s equations with a dielectric constant and index of refraction that can be calculated directly from QCD Litim and Manuel (2001),

(7) |

(This result is valid as long as .) Apart from the fact that , the emergence of the photon is an exact QCD-scale analogue of the TeV-scale spontaneous symmetry breaking that gave rise to the photon as a linear combination of the and hypercharge gauge bosons, with the diquark condensate at the QCD scale playing the role of the Higgs condensate at the TeV scale.

If one could shine a
beam of ordinary light on a lump of CFL matter in vacuum, some would be
reflected and some would enter, refracted, as a beam of -light.
The reflection and refraction coefficients are known
Manuel and Rajagopal (2002) (see also Alford and Good (2004)).
The static limit of this academic result is relevant: if a volume of CFL
matter finds itself in a static magnetic field as within a neutron star,
surface currents are induced such that a fraction of this field is expelled
via the Meissner effect for the non- component of , while a
fraction is admitted as -magnetic field Alford
*et al.* (2000b). The
magnetic field within the CFL volume is not confined to flux tubes, and is
not frozen as in a conducting plasma: CFL quark matter is a color
superconductor but it is an electromagnetic insulator.

All Cooper pairs have zero net -charge, but some
have neutral constituents (both quarks -neutral)
and some have charged constituents (the two quarks have opposite
-charge). The -component of an external magnetic field
will not affect the first type, but it will affect the pairing
of the second type, so external magnetic fields
can modify the CFL phase to the so-called magnetic CFL (“MCFL”)
phase. The MCFL phase has a different gap structure Ferrer *et al.* (2005, 2006)
and a different effective theory Ferrer and de la
Incera (2007b). The original analyses of the MCFL phase
were done for rotated magnetic fields large enough that all quarks are in the lowest Landau
level; solving the gap equations at lower shows that
the gap parameters in the MCFL phase exhibit de Haas-van Alphen
oscillations, periodic in Noronha and Shovkovy (2007); Fukushima and Warringa (2007).

#### ii.1.4 Low-energy excitations

The low-energy excitations in the CFL phase are: the 8 light pseudoscalars arising from broken chiral symmetry, the massless Goldstone boson associated with superfluidity, and the -photon. The pseudoscalars form an octet under the unbroken color+flavor symmetry, and can naturally be labeled according to their -charges as pions, kaons, and an . The effective Lagrangian that describes their interactions, and the QCD calculation of their masses and decay constants will be discussed in Sec. V. We shall find, in particular, that even though the quark-antiquark condensate is small, the pion decay constant is large, .

The symmetry breaking pattern (6) does not include the
spontaneous breaking of the
“symmetry” because it is explicitly broken by instanton
effects. However, at large densities these effects become
arbitrarily small, and the spontaneous breaking of
will have an associated order parameter and a
ninth pseudo-Goldstone boson with the quantum numbers of the .
This introduces the possibility of a second type
of vortices Forbes and Zhitnitsky (2002); Son *et al.* (2001).

Among the gapped excitations, we find the quark-quasiparticles which
fall into an of the unbroken
global , so there are two gap parameters and .
The singlet has the larger gap .
We also find an octet of massive vector mesons, which are the
gluons that have acquired mass via the Higgs mechanism.
The symmetries of the 3-flavor CFL phase are the
same as those one would expect for 3-flavor hypernuclear matter,
and even the pattern of gapped excitations is remarkably similar,
differing only in
the absence of a ninth massive vector meson. It is therefore
possible that there is no phase transition between hypernuclear
matter and CFL quark matter Schäfer and
Wilczek (1999c). This hadron-quark
continuity can arise in nature only if the strange quark is so light
that there is a hypernuclear phase, and this phase is
characterized by proton-, neutron- and -
pairing, which can then continuously evolve into CFL
quark matter upon further increasing the density Alford
*et al.* (1999a).

#### ii.1.5 Why CFL is favored

The dominant component of the CFL pairing pattern is
the color , flavor , and Dirac
(Lorentz scalar).
There are many reasons to expect the color to be
favored. First, this is the most attractive channel
for quarks interacting via
single-gluon exchange which is the dominant interaction at high densities
where the QCD coupling is weak; second, it is also the most attractive channel for
quarks interacting via the instanton-induced ’t Hooft interaction, which
is important at lower densities; third, qualitatively, combining two quarks
that are each separately in the color- representation to obtain a
diquark that is a color- lowers the color-flux at large
distances; and, fourth, phenomenologically, the idea that baryons can
be modeled as bound states of a quark and a color-antisymmetric
diquark, taking advantage of the attraction in this diquark channel,
has a long history and has had a recent renaissance
Jaffe and Wilczek (2003); Selem and Wilczek (2006); Close and Tornqvist (2002); Jaffe (1977); Anselmino *et al.* (1993).

It is also easy to understand why pairing in the Lorentz-scalar channel
is favorable:
it leaves rotational invariance unbroken, allowing for quarks at all angles
on the entire Fermi-sphere to participate coherently in the pairing.
Many calculations have shown that pairing is weaker in channels
that break rotational symmetry
Iwasaki and Iwado (1995); Alford *et al.* (1998); Schäfer (2000b); Buballa *et al.* (2003); Alford *et al.* (2003); Schmitt *et al.* (2002). There is also a rotationally invariant
pairing channel with negative parity described by the order parameter
. Perturbative gluon exchange interactions
do not distinguish between positive and negative parity diquarks, but
non-perturbative instanton induced interactions do, favoring the
positive parity channel Alford *et al.* (1998); Rapp *et al.* (1998, 2000).

Once we have antisymmetry in color and in Dirac indices, we are forced to antisymmetrize in flavor indices, and the most general color-flavor structure that the arguments above imply should be energetically favored is

(8) |

CFL pairing corresponds to , and this is the only
pattern that pairs all the quarks
and leaves an entire global symmetry unbroken.
The 2SC pattern is ,
in which only and quarks of two
colors pair Barrois (1979); Bailin and Love (1984); Alford *et al.* (1998); Rapp *et al.* (1998),
see Sec. III.1.
As long as the
strange quark mass can be neglected (the parametric criterion turns out to
be , see Sec. III.2)
calculations comparing patterns of the structure
(8) always find the CFL phase to have the highest
condensation energy, making it the favored pattern.
This has been confirmed in weak-coupling QCD calculations valid at
high density Schäfer (2000a); Evans *et al.* (2000); Shovkovy and Wijewardhana (1999),
in the Ginzburg-Landau
approximation Iida and Baym (2001), and in many calculations using
Nambu–Jona-Lasinio models
Alford
*et al.* (1999b); Rapp *et al.* (2000); Schäfer and
Wilczek (1999c); Alford
*et al.* (1999a); Malekzadeh (2006).
In the high-density limit where and
we can expand in powers of and
explicitly compare CFL to 2SC pairing.
The CFL condensation energy is
which is when
(see Sec. II.1.4)
whereas the condensation energy in
the 2SC phase is only .
We shall see later that
the 2SC gap parameter turns out to be larger than the CFL gap parameter
by a factor of , so up to corrections of order
the CFL condensation
energy is larger than that in the 2SC phase by a factor of . At lower densities the condensation energies become
smaller, and we cannot neglect negative terms which
are energy penalties induced by the neutrality requirement.
Their coefficient is larger for CFL than for 2SC, partly (but
usually not completely) cancelling the
extra condensation energy—see Fig. 3
and Sec. III.1.

### ii.2 Intermediate density: stresses on the CFL phase

As we noted in section I.5, BCS pairing between two species is suppressed if their chemical potentials are sufficiently different. In real-world quark matter such stresses arise from the strange quark mass, which gives the strange quark a lower Fermi momentum than the down quark at the same chemical potentials and , and from the neutrality requirement, which gives the up quark a different chemical potential from the down and strange quarks at the same and . Once flavor equilibrium under the weak interactions is reached, we find that all three flavors prefer to have different Fermi momenta at the same chemical potentials. This is illustrated in Fig. 2, which shows the Fermi momenta of the different species of quarks.

In the unpaired phase (Fig. 2, left panel), the strange quarks have a lower Fermi momentum because they are heavier, and to maintain electrical neutrality the number of down quarks is correspondingly increased. To lowest order in the strange quark mass, the separation between the Fermi momenta is , so the splitting becomes larger as the density is reduced, and smaller as the density is increased. The phase space at the Fermi surface is proportional to , so the resultant difference in quark number densities is , so their charge density is parametrically of order , meaning that they are unimportant in maintaining neutrality. . Electrons are also present in weak equilibrium, with

In the CFL phase all the colors and flavors pair with each other,
locking all their Fermi momenta together at a common value
(Fig. 2, right panel). This is possible as long
as the energy cost of forcing all species to have the same Fermi
momentum is compensated by the pairing energy that is released by
the formation of the Cooper pairs. Still working to lowest
order in , we can say that parametrically the cost is
, and the pairing energy is
, so we expect CFL pairing to become
disfavored when .
In fact, the CFL phase remains favored over the unpaired phase as long as
Alford and Rajagopal (2002), but already becomes
unstable against unpairing when
(see Sec. III.2).
NJL model calculations
Alford and Rajagopal (2002); Alford
*et al.* (2005c); Fukushima *et al.* (2005); Abuki *et al.* (2005); Blaschke *et al.* (2005); Rüster
*et al.* (2005)
find that if the attractive interaction were strong enough to induce
a 100 MeV CFL gap when
then the CFL phase would survive all the way down to the transition
to nuclear matter. Otherwise, there must be a transition to some other
quark matter phase: this is the “non-CFL” region shown schematically
in Fig. 1. When the stress is small, the CFL pairing can
bend rather than break, developing a condensate of mesons,
described in Sec. II.3 below. When the stress is
larger, however, CFL pairing becomes disfavored. A comprehensive
survey of possible BCS pairing patterns shows that all of them suffer
from the stress of Fermi surface splitting Rajagopal and Schmitt (2006),
so in the intermediate-density “non-CFL” region we expect more
exotic non-BCS pairing patterns. In Sec. III we
give a survey of possibilities that have been explored.

### ii.3 Kaon condensation: the CFL- phase

Bedaque and Schäfer Bedaque and Schäfer (2002) showed that when the stress is not too large (high density), it may simply modify the CFL pairing pattern by inducing a flavor rotation of the condensate. This modification can be interpreted as a condensate of “” mesons. The meson carries negative strangeness (it has the same strangeness as a quark), so forming a condensate relieves the stress on the CFL phase by reducing its strangeness content. At large density kaon condensation occurs for , where is mass of the light ( and ) quarks. At moderate density the critical strange quark mass is increased by instanton contribution to the kaon mass Schäfer (2002a). Kaon condensation was initially demonstrated using an effective theory of the Goldstone bosons, but with some effort can also be seen in an NJL calculation Buballa (2005b); Forbes (2005). The CFL- phase is a superfluid; it is a neutral insulator; all its quark modes are gapped (as long as ); it breaks chiral symmetry. In all these respects it is similar to the CFL phase. Once we turn on small quark masses, different for all flavors, the symmetry of the CFL phase is reduced by explicit symmetry breaking to just , with a linear combination of a diagonal color generator and hypercharge. In the CFL- phase, the kaon condensate breaks spontaneously. This modifies the spectrum of both quarks and Goldstone modes, and thus can affect transport properties.

## Iii Below CFL densities

As we discussed in the introduction (end of Sec. I.1) and above (Sec. II.2), at intermediate densities the CFL phase suffers from stresses induced by the strange quark mass, combined with beta-equilibration and neutrality requirements. It can only survive down to the transition to nuclear matter (occurring at quark chemical potential ) if the pairing is strong enough: roughly , ignoring strong interaction corrections, which are presumably important in this regime. It is therefore quite possible that other pairing patterns occur at intermediate densities, and in this section we survey some of the possibilities that have been suggested.

Fig. 3 shows a comparison of the free energies of some of these phases. We have chosen , so there is a window of non-CFL pairing between nuclear density and the region where the CFL phase becomes stable. (For stronger pairing, , there would be no such window.) The curves for the CFL, 2SC, gCFL, g2SC, and crystalline phases (2PW, CubeX and 2Cube45z) are obtained from an NJL model as described in Sec. VI. The curves for the CFL- and meson supercurrent (curCFL-) phases are calculated using the CFL effective theory with parameters chosen by matching to weak-coupling QCD, as described in Sec. V, except that the gap was chosen to match . The phases displayed in Fig. 3 are discussed in the following sections.

### iii.1 Two-flavor pairing: the 2SC phase

After CFL, 2SC is the most straightforward less-symmetrically paired
form of quark matter, and was one of the first patterns to be analyzed
Barrois (1979); Bailin and Love (1979, 1984); Alford *et al.* (1998); Rapp *et al.* (1998).
In the 2SC phase, quarks with two out of three colors (red and
green, say) and two out of three flavors, pair in the standard BCS
fashion. The flavors with the most phase space near their Fermi
surfaces, namely and , are the ones that pair, leaving the
strange and blue quarks unpaired (middle panel of Fig. 2).
According to NJL models, if the coupling is weak then there is no 2SC region
in the phase diagram Steiner *et al.* (2002).
This can be understood by an
expansion in powers of , which finds that the CFL2SC transition
occurs at the same point as the 2SCunpaired transition, leaving no
2SC window Alford and Rajagopal (2002) (this is the
situation in Fig. 3).
However, NJL models with stronger coupling
leave open the possibility of a 2SC window in the “non-CFL” region of the
phase diagram Rüster
*et al.* (2005); Abuki and Kunihiro (2006). (These calculations have
to date not included the possibility of meson current or crystalline color
superconducting phases, discussed below, that may prove more favorable.)

The 2SC pairing pattern, corresponding to in (8), is , where the symmetry breaking pattern, assuming massless up and down quarks, is

(9) |

using the same notation as in Eq. (6). The unpaired massive
strange quarks introduce a symmetry.
The color gauge symmetry is broken down
to an red-green gauge symmetry, whose confinement distance
rises exponentially with density,
as
Rischke *et al.* (2001) (see also Casalbuoni
*et al.* (2002b); Ouyed and Sannino (2001)).
An interesting
feature of 2SC pairing is that no global symmetries are broken. The
condensate is a singlet of the flavor symmetry,
and baryon number survives as , a linear combination of the
original baryon number and the broken diagonal color generator.
Electromagnetism, originally a linear combination of , ,
and (isospin), survives as an unbroken linear combination
of , , and .
2SC quark matter is therefore a color superconductor but
is neither a superfluid nor an electromagnetic superconductor, and there
is no order parameter that distinguishes it from the unpaired phase or
the QGP Alford *et al.* (1998). With respect to the unbroken
gauge symmetry, the 2SC phase is a conductor not an insulator because
some of the ungapped blue and strange quarks are -charged.

### iii.2 The unstable gapless phases

As was noted in Sec. II.2, and can be seen in
Fig. 3, the CFL phase becomes unstable
when . At this point the pairing
in the - sector suffers the instability
discussed in Sec. I.5, and it becomes
energetically favorable to convert quarks into quarks
(both near their common Fermi momentum).^{3}^{3}3The
onset of gaplessness occurs at the at which
, as explained
in Sec. I.5. Note that in the CFL phase
, twice its value in unpaired
quark matter because of the nonzero color chemical potential
required by color
neutrality in the presence of CFL pairing Steiner *et al.* (2002); Alford and Rajagopal (2002).
If we restrict ourselves to diquark condensates that are
spatially homogeneous, the result is a modification of the pairing
in which there is still pairing in all the color-flavor channels
that characterize CFL, but - Cooper pairing
ceases to occur in a range of momenta near the Fermi surface
Alford
*et al.* (2004b, 2005c); Fukushima *et al.* (2005).
In this range of momenta there are quarks but no quarks,
and quark modes at the edges of this range are ungapped,
hence this is called a gapless phase (“gCFL”).
Such a phenomenon was first proposed for
two flavor quark matter (“g2SC”) Shovkovy and Huang (2003), see also
Gubankova *et al.* (2003). It has been confirmed in
NJL analyses such as those in
Alford
*et al.* (2004b, 2005c, a); Rüster
*et al.* (2004); Fukushima *et al.* (2005); Alford
*et al.* (2005b); Rüster
*et al.* (2005); Abuki and Kunihiro (2006); Abuki *et al.* (2005),
which predict that at densities too low for CFL pairing there will be gapless
phases.

In Fig. 3, where , we see the transition from CFL to gCFL at . (It is interesting to note that, whereas the CFL phase is a -insulator, the gCFL phase is a -conductor, because it has a small electron density, balanced by unpaired quarks from a very thin momentum shell of broken - pairing; the CFLgCFL transition is the analogue of an insulator to metal transition at which a “band” that was unfilled in the insulating phase drops below the Fermi energy, making the material a metal.) The gCFL phase then remains favored beyond the value at which the CFL phase would become unfavored relative to completely unpaired quark matter Alford and Rajagopal (2002).

However, it turns out that in QCD the gapless phases,
both g2SC Huang and Shovkovy (2004b); Giannakis and
Ren (2005a) and gCFL
Casalbuoni
*et al.* (2005b); Fukushima (2005), are unstable
at zero temperature. (Increasing the temperature above a critical
value removes the instability; the critical value varies dramatically
between phases, from a fraction of an MeV to of order 10 MeV
Fukushima (2005).) The instability manifests itself in an
imaginary Meissner mass for some of the
gluons. is the low-momentum current-current two-point
function, and (where the strong interaction
coupling is ) is the coefficient of the gradient term in the
effective theory of small fluctuations around the ground-state
condensate, so a negative value indicates an instability towards
spontaneous breaking of translational invariance
Reddy and Rupak (2005); Huang (2006); Iida and Fukushima (2006); Hashimoto (2006); Fukushima (2006).
Calculations in a simple two-species model Alford and Wang (2005) show
that gapless charged fermionic modes generically lead to imaginary
.

The instability of the gapless phases indicates that there must be other phases of even lower free energy, that occur in their place in the phase diagram. The nature of those phases is not reliably determined at present; likely candidates are discussed below.

### iii.3 Crystalline color superconductivity

The Meissner instability of the gCFL phase points to a breaking
of translational invariance, and crystalline color superconductivity
represents a possible resolution of that instability. The basic idea,
first proposed in condensed matter physics
Larkin and Ovchinnikov (1965); Fulde and Ferrell (1964) and more recently analyzed
in the context of color superconductivity
Alford
*et al.* (2001a); Bowers and Rajagopal (2002); Casalbuoni and Nardulli (2004), is to allow
the different quark flavors to have different Fermi momenta, thus accommodating
the stress of the strange quark mass, and to form
Cooper pairs with nonzero momentum, each quark lying close to its
respective Fermi surface. The price one must pay for this arrangement
is that only fermions in certain regions on the Fermi surface
can pair. Pairs with nonzero momenta chosen from some set of wave vectors
yield condensates that vary in position space like , forming a crystalline pattern whose
Bravais lattice is the set of .

Analyses to date have focused on - and - pairing,
neglecting pairing of and because the separation of their Fermi momenta
is twice as large (Fig. 2).
If the
condensate includes only pairs with a single
nonzero momentum , this means that in position space the
condensate is a single plane-wave and means that in momentum space
pairing is allowed on a single ring on the Fermi surface and a
single ring on the opposite side of the Fermi surface. The simplest
“crystalline” phase of three-flavor quark matter that has been analyzed
Casalbuoni
*et al.* (2005a); Mannarelli
*et al.* (2006b) includes two such single-plane
wave condensates (“2PW”), one and one . The favored orientation of the two ’s is parallel,
keeping the two “pairing rings” on the Fermi surface (from the
and condensates) as far
apart as possible Mannarelli
*et al.* (2006b). This simple pattern of
pairing leaves much of the Fermi surfaces unpaired, and it is much
more favorable to choose a pattern in which the
and condensates each include pairs with more
than one -vector, thus more than one ring and more than one
plane wave. Among such more realistic pairing patterns, the two that
appear most favorable have either
four ’s per condensate
that together point at the eight corners of a cube in momentum space
(“CubeX”) or
eight ’s per condensate
that each point at the
corners of separate cubes, rotated relative to each other by 45
degrees (“2Cube45z”) Rajagopal and
Sharma (2006b).
It has been shown that the chromomagnetic instability is no longer
present in these phases Ciminale *et al.* (2006).
The free energies
of the 2PW, CubeX and 2Cube45z phases as calculated within an NJL model
(see Sec. VI) are shown in Fig. 3.
The calculation is an expansion in powers of
which in the CubeX and 2Cube45z phases turns out to be of order
a tenth to a quarter. According to results obtained in a calculation
done to third order in this expansion parameter,
the CubeX and 2Cube45z condensation energies are large enough that
one or other of them is favored over a wide range of
as illustrated in Fig. 3. The uncertainty in each is of
the same order as the difference between them, so one cannot yet
say which is favored, but the overall scale is plausible (one would
expect condensation energies an order of magnitude
bigger than that of the 2PW state).
We discuss crystalline color superconductivity in greater detail
in Sec. VI.

### iii.4 Meson supercurrent (“curCFL-”)

Kaon condensation alone does not remove the gapless modes that
occur in the CFL phase when becomes large enough, but it
does affect the number of gapless modes and the onset value of
. In the CFL- phase, the electrically charged ()
mode becomes gapless at (compared
to in the CFL phase), and the electrically neutral
() mode becomes gapless for
Kryjevski and Schäfer (2005); Kryjevski and Yamada (2005). (In an NJL model
analysis Forbes (2005), the charged mode
in the CFL- phase becomes gapless at for MeV as in Fig. 3).
The gapless CFL- phase has an instability which is similar
to the instability of the gCFL phase. This instability can be
viewed as a tendency towards spontaneous generation of Goldstone
boson (kaon) currents Schäfer (2006); Kryjevski (2005). The
currents correspond to a spatial modulation of the kaon condensate.
There is no net transfer of any charge because the Goldstone
boson current is counterbalanced by a backflow of
ungapped fermions. The meson supercurrent ground state is lower in
energy than the CFL- state and the magnetic screening
masses are real Gerhold *et al.* (2007). Because the ungapped
fermion mode is electrically charged, both the magnitude of the
Goldstone boson current needed to stabilize the phase
and the magnitude of the resulting energy gain relative to
the phase without a current are very small.
Goldstone boson currents can also be generated in the gCFL phase
without condensation. In this case gauge invariance implies that
the supercurrent state is equivalent to a single plane-wave LOFF
state, but the analyses can be carried out in the limit that the gap
is large compared to the magnitude of the
current Gerhold and Schäfer (2006). This analysis is valid near the onset
of the gCFL phase, but not for larger mismatches, where states with
multiple currents are favored.

### iii.5 Single-flavor pairing

If the stress due to the strange quark mass is large enough then there may be a range of quark matter densities where no pairing between different flavors is possible, whether spatially uniform or inhomogeneous. From Fig. 3 we can estimate that this will occur when , so it requires a large effective strange quark mass and/or small CFL pairing gap. The best available option in this case is Cooper pairing of each flavor with itself. Single-flavor pairing may also arise among the strange quarks in a 2SC phase, since they are not involved in two-flavor pairing. We will discuss these cases separately below.

To maintain fermionic antisymmetry of the Cooper pair wavefunction,
single-flavor pairing phases have to either be symmetric in color,
which greatly weakens or eliminates the attractive interaction,
or symmetric in Dirac indices, which compromises the uniform
participation of the whole Fermi sphere.
As a result, they
have much lower critical temperatures than multi-flavor phases
such as the CFL or 2SC phases, perhaps as large as a few MeV, more
typically in the eV to many keV range
Alford *et al.* (1998); Schäfer (2000b); Buballa *et al.* (2003); Alford *et al.* (2003); Schmitt *et al.* (2002); Schmitt (2005).

Matter in which each flavor only pairs with itself
has been studied using NJL models and weakly-coupled QCD.
These calculations agree that the energetically favored state
is color-spin-locked (CSL) pairing for each flavor
Bailin and Love (1979); Schäfer (2000b); Schmitt (2005). CSL pairing
involves all 3 colors, with
the color direction of each Cooper pair correlated with its spin
direction, breaking .
The phase is isotropic, with
rotational symmetry surviving as a group of simultaneous spatial and color
rotations. Other possible phases exhibiting spin-one, single-flavor,
pairing include the polar, planar, and A phases described in
Schmitt (2005); Schäfer (2000b) (for an NJL model treatment see
Alford *et al.* (2003)). Some of these phases exhibit point
or line nodes in the energy gap at the Fermi surface, and
hence do break rotational symmetry.

If 2SC pairing occurs with strange quarks present, one might expect the strange quarks of all three colors to undergo CSL self-pairing, yielding an isotropic “2SC+CSL” pattern. However, the 2SC pattern breaks the color symmetry, and in order to maintain color neutrality, a color chemical potential is generated, which splits the Fermi momentum of the blue strange quarks away from that of the red and green strange quarks. This is a small effect, but so is the CSL pairing gap, and NJL model calculations indicate that the color chemical potential typically destroys CSL pairing of the strange quarks Alford and Cowan (2006). The system falls back on the next best alternative, which is spin-one pairing of the red and green strange quarks.

Because their gaps and critical temperatures can range as low as
the eV scale, single-flavor pairing phases in compact stars would
appear relatively late in the life of the star, and might cause
dramatic changes in its behavior. For example, unlike the CFL
and 2SC phases, many single-flavor-paired phases are electrical
superconductors Schmitt *et al.* (2003), so their appearance
could significantly affect the magnetic field dynamics of the star.

### iii.6 Gluon condensation

In the 2SC phase (unlike in the CFL phase) the magnetic instability
arises at a lower value of the stress on the BCS pairing than that
at which the onset of gapless pairing occurs. In this 2SC regime,
analyses done using a Ginzburg-Landau approach indicate that the
instability can be cured by the appearance of a chromoelectric
condensate Gorbar
*et al.* (2006a, b); Hashimoto and Miransky (2007); Gorbar *et al.* (2007). The 2SC condensate
breaks the color group down to the red-green
subgroup, and five of the gluons become massive vector bosons via the
Higgs mechanism. The new condensate involves some of these massive
vector bosons, and because they transform non-trivially under
it now breaks that gauge symmetry. Because they are
electrically charged vector particles, rotational symmetry is also broken,
and the phase is an electrical superconductor.
Alternatively, it has been suggested Ferrer and de la
Incera (2007a)
that the gluon condensate may be inhomogeneous with a large
spontaneously-induced magnetic field.

### iii.7 Secondary pairing

Since the Meissner instability is generically associated with the presence of gapless fermionic modes, and the BCS mechanism implies that any gapless fermionic mode is unstable to Cooper pairing in the most attractive channel, one may ask whether the instability could be resolved without introducing spatial inhomogeneity simply by “secondary pairing” of the gapless quasiparticles, which would then acquire their own gap Hong (2005); Huang and Shovkovy (2003). Furthermore, there is a mode in the gCFL phase whose dispersion relation is well approximated as quadratic, , yielding a greatly increased density of states at low energy (diverging as ), so its secondary pairing is much stronger than would be predicted by BCS theory: for an effective four-fermion coupling strength , as compared with the standard BCS result Hong (2005). This result is confirmed by an NJL study in a two-species model Alford and Wang (2006), but the secondary gap was found to be still much smaller than the primary gap , so it does not generically resolve the magnetic instability (in the temperature range , for example).

### iii.8 Mixed phases

Another way for a system to deal with a stress on its pairing pattern
is to form a mixed phase, which is a charge-separated state
consisting of positively and negatively
charged domains which are neutral on average. The coexisting phases
have a common
pressure and a common value of the charge chemical potential
which is not equal to the neutrality value for either phase
Ravenhall *et al.* (1983); Glendenning (1992).
The size of the domains is determined by a balance between
surface tension (which favors large domains) and electric field
energy (which favors small domains).
Separation of color charge is expected
to be suppressed by the very high energy cost of color electric fields,
but electric charge separation is quite possible, and may occur at the
interface between color-superconducting quark matter and nuclear matter
Alford
*et al.* (2001b) and an interface between quark matter and the
vacuum Jaikumar
*et al.* (2006a); Alford *et al.* (2006), just as it occurs at
interfaces between nuclear matter and a nucleon gas
Ravenhall *et al.* (1983). Mixed phases are a
generic phenomenon, since, in the approximation where
Coulomb energy costs are neglected,
any phase can always lower its free
energy density by becoming charged (this follows from the fact
that free energies are concave functions of chemical potentials).
In this approximation, if two phases A and B can coexist at the same
pressure with opposite charge densities then
such a mixture will always be favored over a
uniform neutral phase of either A or B. For a pedagogical discussion,
see Alford
*et al.* (2005c). Surface and Coulomb energy costs can cancel this
energy advantage, however, and have to be calculated on a case-by-case
basis.

In quark matter it has been found that as long
as we require local color neutrality such mixed phases are not the
favored response to the stress imposed by the strange quark mass
Alford
*et al.* (2004b, a). Phases involving color charge
separation have been studied Neumann *et al.* (2003) but it seems likely
that the energy cost of the color-electric fields will disfavor them.

### iii.9 Relation to cold atomic gases

An interesting class of systems in which stressed superconductivity
can be studied experimentally is trapped atomic gases in which two
different hyperfine states (“species”) of the atom
pair with each other Giorgini *et al.* (2007).
This is a useful experimental model
because the stress and interaction strength are both under
experimental control, unlike quark matter where one physical variable
() controls both the coupling strength and the stress.
The atomic pairing stress can be adjusted by changing the relative
number of atoms of the two species (“polarization”). The scattering
length of the atoms can be controlled using Feshbach resonances,
making it possible to vary the strength of the inter-atomic attraction
from weak (where BCS pairing occurs) through the unitarity limit
(where a bound state forms) to strong (Bose-Einstein condensation of
diatomic molecules).

The theoretical expectation is that, in the weak coupling limit, there will be BCS pairing as long as , the chemical potential difference between the species, is small enough. The BCS phase is unpolarized because the Fermi surfaces are locked together. A first-order transition from BCS to crystalline (LOFF) pairing is expected at , where is the BCS gap at ; then at a continuous transition to the unpaired phase Clogston (1962); Chandrasekhar (1962); Larkin and Ovchinnikov (1965); Fulde and Ferrell (1964). For the single plane wave LOFF state , but for multiple plane wave states may be larger.

Experiments with cold trapped atoms near the unitary limit
(strong coupling) have seen phase separation between an unpolarized
superfluid and a polarized normal state Chin *et al.* (2006); Zwierlein *et al.* (2006); Partridge *et al.* (2006).
If one ignores the crystalline phase (perhaps only favored
at weak coupling Sheehy and Radzihovsky (2007, 2006); Mannarelli
*et al.* (2006a))
this is consistent with the theoretical expectation for the
BCS regime: the net polarization forces the system to
phase separate, yielding a mixture of BCS and unpaired phases
with fixed at the first order transition
between them Bedaque *et al.* (2003); Carlson and Reddy (2005).
It remains an exciting possibility that crystalline superconducting
(LOFF) phases of cold atoms may be observed: this may require experiments closer
to the BCS regime.

In the strong coupling limit the superfluid consists of tightly bound molecules. Adding an extra atom requires energy . For the atomic gas is a homogeneous mixture of an unpolarized superfluid and a fully polarized Fermi gas, so the system is a stable gapless superfluid. This means that in strong coupling polarization can be carried by a gapless superfluid, whereas in weak coupling even a small amount of polarization leads to the appearance of a mixed BCS/LOFF phase. It is not known what happens at intermediate coupling, but one possibility is that the gapless superfluid and the LOFF phase are connected by a phase transition Son and Stephanov (2006). This transition would correspond to a magnetic instability of the gapless superfluid.

## Iv Weak-coupling QCD calculations

We have asserted in Secs. I and II that at sufficiently high densities it is possible to do controlled calculations of properties of CFL quark matter directly from the QCD Lagrangian. We describe how to do such calculations in this section. We shall focus on the calculation of the gap parameter, but shall also discuss the critical temperature for the transition from the CFL phase to the quark-gluon plasma and the Meissner and Debye masses that control color-magnetic and color-electric effects in the CFL phase. Phenomena that are governed by the massless Goldstone bosons and/or the light pseudo-Goldstone bosons are most naturally described by first constructing the appropriate effective theory and then, if at sufficiently high densities, calculating its parameters directly from the QCD Lagrangian. We defer these analyses to Sec. V.

Although the weak-coupling calculations that we describe in this section are only directly applicable in the CFL phase, we shall present them in a sufficiently general formalism that they can be applied to other spatially homogeneous phases also, including for example the 2SC and CSL phases. These phases can be analyzed at weak-coupling either just by ansatz, or by introducing such a large strange quark mass that CFL pairing is disfavored even at enormous densities. Such calculations provide insights into the properties of these phases, even though they do not occur in the QCD phase diagram at high enough densities for a weak-coupling approach to be applicable. To keep our notation general, we shall refer to the gap parameter as ; in the CFL phase, .

We shall see that at weak coupling the expansion parameter that controls the calculation of is at best , certainly not . (The leading term is of order ; the and terms have also been calculated. The and terms are nonzero, and have not yet been calculated. Beyond , it is possible that fractional powers of may arise in the series.) We therefore expect the weak-coupling calculations to be quantitatively reliable only at densities for which , which corresponds to densities many orders of magnitude greater than that at the centers of neutron stars. Indeed, it has been shown Rajagopal and Shuster (2000) that some of the contributions start to decline in magnitude relative to the term only for which corresponds, via the two-loop QCD beta function, to MeV meaning densities 15-16 orders of magnitude greater than those at the centers of compact stars. The reader may therefore be tempted to see this section as academic. From a theoretical point of view, it is exceptional to have an instance where the properties of a superconducting phase can be calculated rigorously from a fundamental short-distance theory, making this exploration a worthy pursuit even if academic. From a practical point of view, the quantitative understanding that we derive from calculations reviewed in this section provides a completely solid foundation from which we can extrapolate downwards in . The effective field theory described in Sec. V gives us a well-defined way of doing so as long as we stay within the CFL phase, meaning that we can come down from all the way down to . Finally, we shall gain qualitative insights into the CFL phase and other color superconducting phases, insights that guide our thinking at lower densities.

The QCD Lagrangian is given by

(10) |

Here, is the quark spinor in Dirac, color, and flavor space, i.e., a -component spinor, and . The covariant derivative acting on the fermion field is , where is the strong coupling constant, are the gauge fields, () are the generators of the gauge group , and are the Gell-Mann matrices. The field strength tensor is with the structure constants . The chemical potential and the quark mass are diagonal matrices in flavor space. If weak interactions are taken into account flavor is no longer conserved and there are only two chemical potentials, one for quark (baryon) number, , and one for electric charge, . At the very high densities of interest in this section, the constituent quark masses are essentially the same as the current quark masses , and meaning that we need not distinguish between them. Furthermore, at asymptotic densities we can neglect even the strange quark mass, so throughout most of this section we shall set .

If the coupling is small then the natural starting point is a free Fermi gas of quarks. In a degenerate quark gas all states with momenta are occupied, and all states with are empty. Because of Pauli-blocking, interactions mainly modify states in the vicinity of the Fermi surface. Since the Fermi momentum is large, typical interactions between quarks near the Fermi surface involve large momentum transfer and are governed by the weak coupling . Interactions in which quarks scatter by only a small angle involve only a small momentum transfer and are therefore potentially dangerous. However, small momenta correspond to large distances, and medium modifications of the exchanged gluons are therefore important. In a dense medium, electric gluons are Debye screened at momenta . The dominant interaction for momenta below the screening scale is due to unscreened, almost static, magnetic gluons. In a hot quark-gluon gas, interactions between magnetic gluons become nonperturbative for momenta less than . This phenomenon does not take place in a very dense quark liquid, and gluon exchanges with arbitrarily small momenta remain perturbative. On a qualitative level this can be attributed to the absence of Bose enhancement factors in soft gluon propagators. A more detailed explanation will be given in Sec. V.1.2. The unscreened magnetic interactions nevertheless make the fluid a “non-Fermi liquid” at temperatures above the critical temperature for color superconductivity. We shall discuss this also in Sec. V.1.2, where we shall see that these non-Fermi liquid effects do not spoil the basic logic of the BCS argument that diquark condensation must occur in the presence of an attractive interaction, but are crucial in the calculation of the gap that results.

### iv.1 The gap equation

As discussed in Sec. I.2, any attractive interaction in a
many-fermion system leads to Cooper pairing. QCD at high density provides an
attractive interaction via one-gluon exchange. In terms of quark
representations of , the attractive channel is the antisymmetric
anti-triplet , appearing by “pairing” two color triplets:
. Consequently,
only quarks of different colors form Cooper pairs. There is an induced
pairing in the symmetric sextet channel . However, this pairing
is much weaker Alford
*et al.* (1999b); Schäfer (2000a); Shovkovy and Wijewardhana (1999); Pisarski and
Rischke (1999c); Alford
*et al.* (1999a),
and we shall largely neglect it in the following. As in an electronic
superconductor, Cooper pairing results in an energy gap in the
quasiparticle excitation
spectrum. Its magnitude at zero temperature is crucial for the
phenomenology of a superconductor. In addition, it also sets the scale for
the critical temperature of the phase transition which can be expected
to be of the same order as (in BCS theory, ). Over
the course of the next five subsections, we shall discuss the QCD gap equation,
which is used to determine both and .

Our starting point is the partition function

(11) |

with the action and the Lagrangian (10).
In the following we shall only sketch the derivation of the gap equation. Details
following the same lines can be found in
Manuel (2000b); Pisarski and
Rischke (1999b, 2000a); Schmitt *et al.* (2002); Rischke (2004); Schmitt (2004, 2005).

We begin by introducing Nambu-Gorkov spinors. This additional two-dimensional
structure proves convenient in the theoretical description of a superconductor
or a superfluid, see for instance Abrikosov *et al.* (1963); Fetter and Walecka (1971). It allows for the
introduction of a source that couples to quark bilinears (as opposed to
quark-anti-quark bilinears). Spontaneous symmetry breaking is realized by taking
the limit of a vanishing source. The Nambu-Gorkov basis is given by

(12) |

where is the charge-conjugate spinor, obtained by multiplication with the charge conjugation matrix . In a free fermion system, the new basis is a pure doubling of degrees of freedom with the inverse fermion propagator consisting of the original free propagators,

(13) |

where . Here and in the following capital letters denote four-vectors,
e.g., . The effect of a nonzero diquark condensate can now be
taken into account through adding a suitable source term to the action and computing
the effective action as a functional of the gluon and fermion propagators
and
Abuki (2003); Rüster and Rischke (2004); Rischke (2004); Schmitt (2004); Miransky *et al.* (2001); Takagi (2003):

(14) | |||||

This functional is called the “2PI effective action” since the contribution
consists of all two-particle irreducible diagrams
Luttinger and Ward (1960); Baym (1962); Cornwall *et al.* (1974). This formalism is particularly
suitable for studying spontaneous symmetry breaking in a self-consistent way. The
ground state of the system is obtained by finding the stationary point of the
effective action. The stationarity conditions yield Dyson-Schwinger equations for
the gauge boson and fermion propagators,

(15a) | |||||

(15b) |

where the gluon and fermion self-energies are the functional derivatives of at the stationary point,

(16) |

Writing the second of these equations as , we can then use the Dyson-Schwinger equation (15b) to evaluate the fermionic part of the effective action at the stationary point, obtaining the pressure

(17) |

We shall return to this expression for the pressure in Sec. IV.3.

Here, we proceed to analyze the Dyson-Schwinger equation (15b) for the fermion propagator. We denote the entries of the 22 matrix in Nambu-Gorkov space as

(18) |

where the off-diagonal elements are related via . One can invert the Dyson-Schwinger equation formally to obtain the full fermion propagator in the form

(19) |

where the fermion propagators for quasiparticles and charge-conjugate quasiparticles are

(20) |

and the so-called anomalous propagators, typical for a superconducting system, are given by

(21) |

They can be thought of as describing the propagation of a charge-conjugate particle (i.e., a hole) with propagator that is converted into a particle with propagator , via the condensate . (Or, a particle that is converted into a hole via the condensate.) The essence of superconductivity or superfluidity is the existence of a difermion condensate that makes the quasiparticle excitations superpositions of elementary states with fermion-number ; we see the formalism accommodating this phenomenon here.

We shall approximate by only taking into account two-loop diagrams. Upon taking the functional derivative with respect to , this corresponds to a one-loop self-energy . We show diagrammatically in Fig. 4. We shall argue later that this approximation is sufficient to calculate up to terms of order . Upon making this approximation, the gap equation takes the form shown in the lower panel of Fig. 4, namely

(22) |

in momentum space, where is the gluon propagator.

Note that in the derivation of the gap equation we have assumed the system
to be translationally invariant. This assumption fails for crystalline color
superconductors, see Sec. VI. There has been some work on analyzing
a particularly simple crystalline phase in QCD at
asymptotically high densities and weak coupling Leibovich *et al.* (2001), but the formalism we are
employing does not allow us to incorporate it into our presentation
and, anyway, this subject remains to date largely unexplored.

### iv.2 Quasiparticle excitations

Before we proceed with solving the gap equation, it is worthwhile to derive the dispersion relations for the fermionic quasiparticle excitations in a color superconductor. That is, we suppose that the gap parameter(s) have been obtained in the manner that we shall describe below and ask what are the consequences for the quasiparticle dispersion relations. Based on experience with ordinary superconductors or superfluids, we expect (and shall find) gaps in the dispersion relations for the fermionic quasiparticles. We may also expect that in some color superconducting phases, quasiparticles with different colors and flavors, or different linear combinations of color and flavor, differ in their gaps and dispersion relations. Indeed, some gaps may vanish or may be nonzero only in certain directions in momentum space.

The quasiparticle dispersion relations are encoded within the anomalous self-energy , defined in (18), which satisfies the gap equation (22). We shall assume that can be written in the form

(23) |

where is a matrix in color, flavor and Dirac space, and are projectors onto states of positive () or negative () energy. The corresponding gap functions are denoted as and will be determined by the gap equation. Here and in the following the energy superscript is denoted in parentheses to distinguish it from the superscript that denotes components in Nambu-Gorkov space. In our presentation we shall assume that is momentum-independent, corresponding to a condensate of Cooper pairs with angular momentum , but the formalism can easily be extended to allow a momentum-dependent as required for example in the analysis of the CSL phase and we shall quote results for this case also. Note that in Eq. (23) we are assuming that every nonzero entry in is associated with the same gap functions ; the formalism would have to be generalized to analyze phases in which there is more than one independent gap function, as for example in the gCFL phase.

We shall analyze color superconducting phases whose color, flavor and Dirac structure takes the form

(24) |

where the Dirac structure selects a positive parity condensate, where, as described in Secs. I and II, the antisymmetric color matrix is favored since QCD is attractive in this channel and the antisymmetric flavor matrix is then required, and where is a matrix. We note that because the full flavor symmetry is the chiral symmetry, the matrix is actually a pair . In this section we shall assume . The case , which corresponds to a meson condensate in the CFL phase, is discussed in Sec. V.3.

The excitation spectrum is given by the poles of the propagator in (19). (We shall see that the diagonal and the off-diagonal entries in have the same poles.) It will turn out that the Hermitian matrix determines which quasiparticles are gapped and determines the ratios among the magnitudes of (possibly) different gaps. It is convenient to write this matrix via its spectral representation

(25) |

where are the eigenvalues and the projectors onto the corresponding eigenstates.

The final preparation that we must discuss prior to computing the
propagator is that we approximate the diagonal elements of the
quark self-energy as Brown
*et al.* (2000c); Manuel (2000b); Gerhold and Rebhan (2005)

(26) |

where is the square of the effective gluon mass at finite density, and is the Euler constant. The expression (26) is the low energy approximation to the one-loop self-energy, valid for , for the positive energy () states. Taking the low energy approximation to and neglecting the self-energy correction to the negative energy states will prove sufficient to determine up to order .

With all the groundwork in place, we now insert , from (26), and from (23) into (20) and (21), and hence (19), and use (25) to simplify the result. We find that the diagonal entries in the fermion propagator are given by

(27) |

while the anomalous propagators are

(28a) | |||||

(28b) |

In writing these expressions, we have defined the wave function renormalization factor

(29) |

for the positive energy components, originating from the self-energy (26). (By neglecting the negative energy contribution to in (26), we are setting the negative energy wave function renormalization .) We have furthermore defined

(30) |

The ’th quasiparticle and antiquasiparticle energies are then given by solving for . To leading order in , wave function renormalization can be neglected and the quasiparticle and antiquasiparticle energies are given by the themselves. We see from (30) that the antiparticles have — in fact, for near they have . They therefore never play an important role at high density. This justifies our neglect of the negative energy