kaon condensation in a nambu–jona-lasinio model at high density

PHYSICAL REVIEW D 72, 094032 (2005)

Kaon condensation in a Nambu–Jona-Lasinio model at high density

Michael McNeil ForbesCenter for Theoretical Physics, Department of Physics, MIT, Cambridge, Massachusetts 02139, USA

(Received 26 July 2005; published 30 November 2005)

1550-7998=20

We demonstrate a fully self-consistent microscopic realization of a kaon-condensed color-flavor lockedstate (CFLK0) within the context of a mean-field Nambu–Jona-Lasinio (NJL) model at high density. Theproperties of this state are shown to be consistent with the QCD low-energy effective theory once theproper gauge neutrality conditions are satisfied, and a simple matching procedure is used to compute thepion decay constant, which agrees with the perturbative QCD result. The NJL model is used to comparethe energies of the CFLK0 state to the parity even CFL state, and to determine locations of the metal/insulator transition to a phase with gapless fermionic excitations in the presence of a nonzero hyperchargechemical potential and a nonzero strange quark mass. The transition points are compared with resultsderived previously via effective theories and with partially self-consistent NJL calculations. We find thatthe qualitative physics does not change, but that the transitions are slightly lower.

DOI: 10.1103/PhysRevD.72.094032 PACS numbers: 12.39.Fe, 05.70.Fh, 12.38.Lg, 14.40.Aq

I. INTRODUCTION

Recently there has been interest in the structure ofmatter at extremely high densities, such as might be foundin the cores of neutron stars. At large enough densities, thenucleons are crushed together and the quarks become therelevant degrees of freedom. The asymptotic freedom ofQCD ensures that the theory is weakly-coupled at highenough densities. This allows one to perform weak-coupling calculations at asymptotically high densities.Such calculations have established that the structure ofthe ground state of quark matter is a color superconductor(see for example [1–15]). In particular, at densities highenough that the three lightest quarks can be treated asmassless, the ground state is the color-flavor–locked(CFL) state in which all three colors and all three flavorsparticipate in maximally (anti)-symmetric pairing[7,10,16,17].

Determination of the QCD phase structure at moderatedensities and in the presence of nonzero quark masseshas proceeded in several ways. One approach has beento formulate a chain of effective theories, and then tomatch coefficients across several energy scales throughthese effective theories to perturbative calculations.Coefficients in the low-energy chiral effective theory [18]are matched to calculations performed in high-densityeffective theories (HDET) [9,19,20] which in turn arematched to weakly-coupled QCD. This allows one todetermine the properties of the Goldstone bosons anddetermine the effects of small quark masses [21–30].Within this framework, it has been noted that, in thepresence of a finite strange quark mass, neutral ‘‘kaons’’(the lightest pseudo-Goldstone modes at high density withthe same quantum numbers as their vacuum counterpart)can Bose-condense in the CFL state to form a kaon-condensed CFLK0 phase with lower condensation energy[31–33].

05=72(9)=094032(19)$23.00 094032

Unfortunately, the low-energy effective theory is onlyreliable for small perturbations and at moderate densitiesthe strange quark mass is not a small perturbation. A recentattempt has been made to extrapolate to large strange quarkmass (ms) [34], but this approach has not dealt with addi-tional complications in the condensate structure that allowdifferent gap parameters for each pair of quarks.

To deal with moderate quark masses, another approachhas been to study Nambu–Jona-Lasinio (NJL) models[35,36] of free quarks with contact interactions that modelinstanton interactions or single gluon exchange. Thesemodels are amenable to a mean-field treatment and exhibita similar symmetry breaking pattern to QCD which resultsin CFL ground states [5,6].

Within these models, one can study the effects of mod-erate quark masses through self-consistent solutions of themean-field gap equations. This has led to a plethora ofphases. In particular, several analyses show a transition to acolor-flavor locked phase with gapless fermionic excita-tions (the gCFL phase). These include both NJL-basedcalculations [37–40] and effective-theory–based calcula-tions [34,41]. Until recently, however, the NJL calculationshave excluded the possibility of kaon condensation (seehowever [42] which considers kaon condensation in theNJL model at low density), while the effective theories donot consider the complicated patterns in which the con-densate parameters evolve at finite quark masses.

The goal of this paper is to show that one can combinethe analysis of the low-energy effective theories, whichexhibit kaon condensation, with the self-consistent mean-field analysis of the NJL model, which accounts for the fullcondensate structure. In particular, we use an NJL modelbased on single gluon exchange to find self-consistentsolutions that correspond to the CFLK0 phase; we showthat these phases agree with the predictions of the low-energy effective theory; and we determine how and where

-1 © 2005 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.72.094032

MICHAEL MCNEIL FORBES PHYSICAL REVIEW D 72, 094032 (2005)

the zero-temperature phase transition to a gapless CFLphase occurs as one increases the strange quark mass. Inaddition, unlike previous work on the NJL model, ournumerical solutions are fully self-consistent: we includeall condensates and self-energy corrections required toclose the gap equations.

We first describe the pattern of symmetry breaking thatleads to the CFL and CFLK0 states (Sec. II). Then wepresent our numerical results, demonstrating some proper-ties of these states and determining the locations of thezero-temperature phase transitions (Sec. III). After a care-ful description of our model (Sec. IV) we derive the low-energy effective theory, paying particular attention to thedifferences between QCD and the NJL model (Sec. V).Here we demonstrate that, for small perturbations, ournumerical solutions are well described by the effectivetheory, and we use our numerical results to compute thepion decay constant f which agrees with the perturbativeQCD results. Specific numerical details about our calcu-lations and a full description of our self-consistent parame-trization are given in the Appendix A.

We leave for future work the consideration of finitetemperature effects, the analysis of the gapless CFLK0

(gCFLK0), the inclusion of instanton effects, the inclusionof up and down quark mass effects, and the possibility ofother forms of meson condensation.

II. COLOR FLAVOR LOCKING (CFL)

QCD has a continuous symmetry group of U1B SU3L SU3R SU3C. In addition, there is an ap-proximate U1A axial flavor symmetry that is explicitlybroken by anomalies. At sufficiently high densities, how-ever, the instanton density is suppressed and this symmetryis approximately restored.

The CFL ground state spontaneously breaks these con-tinuous symmetries through the formation of a diquarkcondensate [7]

h Ca5 bi / 3kabk 6ab

b

a : (1)

The symmetry breaking pattern (including the restoredaxial U1A symmetry) is thus1

U3L U3R SU3CZ3

! SU3LRC Z2 Z2;

(2)

where the Z2 symmetries correspond to L ! L and R ! R. It has been noted that the symmetry breakingpattern at high density (2) is the same as that for hyper-

1The Z3 factor mods out the common centers. See (12) for theexplicit representation.

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nuclear matter at low density [43]. This leads one toidentify the low-energy pseudoscalar degrees of freedomin both theories. We shall refer to the pseudoscalarGoldstone bosons in the high-density phase as ‘‘pions’’and kaons etc. when they have the same flavor quantumnumbers as the corresponding low-density particles.

The CFL state (1) preserves parity, and is preferred wheninstanton effects are considered. Excluding instanton ef-fects, there is an uncountable degeneracy of physicallyequivalent CFL ground states that violate parity. Theseare generated from the parity-even CFL by the brokensymmetry generators.

The symmetry breaking pattern (2) breaks 18 generators.The quarks, however, are coupled to the eight gluonsassociated with the SU3C color symmetry and to thephoton of the U1EM electromagnetism (which is a sub-group of the vector flavor symmetry). Eight of these gaugebosons acquire a mass through the Higgs mechanism andthe colored excitations are lifted from the low-energyspectrum. There remain 10 massless Nambu-Goldstoneexcitations: a pseudoscalar axial flavor octet of mesons, ascalar superfluid boson associated with the broken U1Bbaryon number generator, and a pseudoscalar 0 bosonassociated with broken axial U1A generator. There re-mains one massless gauge boson that is a mixture of theoriginal photon and one of the gluons [7,44]. With respectto this ‘‘rotated electromagnetism’’ U1 ~Q the CFL stateremains neutral [45].

The degeneracy of the vacuum manifold is lifted bythe inclusion of a nonzero strange quark mass ms. In theabsence of instanton effects and other quark masses, theground state is not near to the parity-even CFL state (1), butrather, is a kaon rotated state CFLK0. As ms ! 0 this stateapproaches a state on the vacuum manifold that is a purekaon rotation of the parity-even CFL (1).

Even in the absence of quark masses, the vacuum mani-fold degeneracy is partially lifted by the anomalous break-ing of the U1A axial symmetry which we have neglected:Instanton effects tend to disfavor kaon condensation byfavoring parity-even states, and thus delay the onset of theCFLK0 until ms reaches a critical value (possibly exclud-ing it). The effects of anomaly and instanton contributionshave been well studied [6,16,26,46–48] and play an im-portant quantitative role in the phase structure of QCD.Nonzero up and down quark masses also tend to disfavorkaon condensation.

For the purposes of this paper, we shall neglect both theeffects of instantons, and the effects of finite up and downquark masses. This will ensure that kaon condensationoccurs for arbitrarily small ms. Both of these effects openthe possibility of a much richer phase structure, includingcondensation of other mesons (see for example [33,49]).Future analyses should take these numerically importanteffects into account, both in the effective theory and in theNJL model.

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KAON CONDENSATION IN A NAMBU-JONA-LASINIO . . . PHYSICAL REVIEW D 72, 094032 (2005)

The primary source of for kaon condensation is the finitestrange quark mass. To lowest-order, this behaves as achemical potential [31–33] (see (3) and (4)). In this paper,we also consider the addition of a hypercharge chemicalpotential as this removes many complications associatedwith masses and leads to a very clean demonstration ofkaon condensation.

III. SELF-CONSISTENT SOLUTIONS

We consider four qualitatively different phases: Two areself-consistent mean-field solutions to the NJL model witha finite hypercharge chemical potential parameter Y ; theother two are self-consistent mean-field solutions to theNJL model with a finite strange quark mass parameter ms.In each of these cases, one solution corresponds to a parity-even CFL phase and the other corresponds to a kaon-condensed CFLK0 phase. Our normalizations and a com-plete description of the model are presented in Sec. IV. Afull description of all the parameters required to describethese phases along with some typical values is presented inthe Appendix A.

A. Finite hypercharge chemical potential

The CFL phase in the presence of a hypercharge chemi-cal potential corresponds to the fully gapped CFL phasediscussed in [37]. Here one models the effects of thestrange quark through its shift on the Fermi surface pF q of the strange quarks. This can be seen by expandingthe free-quark dispersion

p2 M2

s

q jpj

M2s

2q (3)

or, more carefully, by integrating out the antiparticles toformulate the High Density Effective Theory. (See forexample [9,19,20,30].) These leading order effects areequivalent to adding a hypercharge chemical potential ofmagnitude

Y M2s

2q: (4)

and a baryon chemical potential shift of

B M2s

q: (5)

We consider only the effect of the hypercharge chemicalpotential here, holding B fixed. Note that the relevantparameters are Ms and q rather than ms and s B=3.Ms is the constituent quark mass that appears in the dis-persion relation whereas ms is the bare quark mass pa-rameter; likewise, q is the corrected quark chemicalpotential that determines the Fermi surface whereas s B=3 is the bare baryon chemical potential. (These dis-

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tinctions are important because our model takes into ac-count self-energy corrections.)

The CFL phase responds in a trivial manner to a hyper-charge chemical potential: the quasiparticle dispersionsshift such that the physical gap in the spectrum becomessmaller; none of the other physical properties change. Inparticular, as the hypercharge chemical potential increases,the colored chemical potential 8 Y decreases tomaintain neutrality. The values of all of the gap parameters,the self-energy corrections, the densities and the thermo-dynamic potential remain unchanged until the physical gapin the spectrum vanishes. (The apparent change in themagnitude of the gap parameters in the first figure of [37]is due to the shift in the baryon chemical (5) which occursif one uses the strange quark chemical potential shift srather than a hypercharge shift Y .) This is a consequenceof the ~Q neutrality of the CFL state [50]. In particular, theelectric chemical potential remains zero e 0 and thestate remains an insulator until the onset of the gaplessmodes. The same phenomena has also been noticed in thetwo-flavor case [51–53].

As such, we can analytically identify the phase transitionto the gCFL phase which occurs for the critical chemicalpotential

cY 0 (6)

where 0 3 6 is the physical gap in the spectrum inthe absence of any perturbations. Throughout this paper weuse parameters arbitrarily chosen so that c

Y 0 25 MeV to correspond with the parameter values in[37,38,40]. We show typical quasiparticle dispersion rela-tions for this state in Fig. 1.

The splitting of the dispersions can also be easily under-stood from the charge neutrality condition (50) and theleading order effects are summarized in Table I. Aftersetting 8 Y , the chemical potentials for the rs andgs quarks shift byY whereas for the bu and bd quarks itshifts by Y . Thus, the (gs,bd) and (rs,bu) pairs are thefirst to become gapless.

The kaon-condensed hypercharge state is more compli-cated. One can again use the appropriate charge neutralityconditions (50) to estimate how the quarks will be affectedby Y , but the naıve results hold only to lowest order. Inparticular, the condensates of the CFLK0 state also vary asY increases (see Table III). These higher order effectsbreak all the degeneracy between the quark species andFig. 2 has nine independent dispersions.

We shall compare the thermodynamic potentials of thesetwo states later (see Figs. 7 and 8), but we point out herethat the transition to a gapless color-flavor–locked statewith kaon condensation (gCFLK0) occurs at a larger hy-percharge chemical potential than the CFL/gCFL transi-tion. This can be most easily seen in Fig. 3. This is inqualitative agreement with [34,41], but in quantitativedisagreement.

-3

TABLE I. Leading order shifts in the chemical potentials ofthe various quarks in the CFL and CFLK0 states in the presenceof a hypercharge chemical potential shift Y . This followsdirectly from (50).


In the CFL/gCFL transition, two modes become gaplesssimultaneously: the lower branches of the (rs,bu) and(gs,bd) pairs. One of these modes is electrically neutral(gs,bd) and it crosses the zero-energy axis giving rise to a

ru gd bs rd gu rs bu gs bd

CFL 0 0 0 0 0 1 1 1 1CFLK0 0 1

2 12 0 1

2 1 12 1

2 12

0

20

40

60

80

450 500 550(MeV) Y = 0

p(M

eV)

0

20

40

60

80

450 500 550(MeV) Y = c

Y 2

p(M

eV)

0

20

40

60

80

450 500 550(MeV) Y = c

Y

p(M

eV)

FIG. 1. Lowest lying quasiparticle dispersions about theFermi momentum pF q 500 MeV for the CFL phasewith different values of the hypercharge chemical. All disper-sions have left-right degeneracy: we now consider the color-flavor degeneracy. In the top plot Y 0, and the lowestdispersion has an eight-fold degeneracy and a gap of 0 25 MeV. The upper band contains a single quasiparticle pairing(ru,gd,bs) with a gap of 46 23 54 MeV. In the middleplot, Y c

Y=2 12:5 MeV, and (rs,bu) and (gs,bd) pairs areshifting as indicated in Table I. In the last plot, two pairs havebecome gapless marking the CFL/gCFL transition.

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‘‘breach’’ in the spectrum. The other mode is electricallycharged: as soon as in crosses, the electric chemical po-tential must rise to enforce neutrality. The state now con-tains gapless charged excitations and becomes a conductor.The result is that the the neutral gapless mode has twolinear dispersions while the charged gapless mode has a

0

20

40

60

80

450 500 550(MeV) Y = c

Y 2

p(M

eV)

0

20

40

60

80

450 500 550(MeV) Y = 1 20 c

Y

p(M

eV)

FIG. 2. Lowest lying quasiparticle dispersions about the Fermimomentum pF q 500 MeV for the CFLK0 phase withdifferent values of the hypercharge chemical. (The Y 0dispersions are the same as in the top of Fig. 1.) Again, alldispersion have a left-right degeneracy. In the top plot at Y cY=2 12:5 MeV, the eight-fold degenerate lowest band has

split into eight independent dispersions. To leading order in theperturbation, the splitting is described by Table I, but the lack ofdegeneracy indicates that there are also higher order effects. Thelower plot at Y 1:20c

Y 30 MeV is close to theCFLK0/gCFLK0 transition. The gapless band now containsonly a single mode and is charged.

-4

min

pp

∆0

YcY

00

1

0 2 0 4

0 5

0 6 0 8 1 0 1 2

FIG. 3. Physical gap of the lowest lying excitation as afunction of the hypercharge chemical potential. The dottedline corresponds to the CFL phase: the phase transition to thegCFL occurs at Y c

Y where the gap vanishes. The solid linecorresponds to the CFLK0 state. The transition to a gapless phaseis delayed by a factor of 1:22.

0

20

40

60

80

450 500 550(MeV) 2

s (2 q) = 0 50 cY

p(M

eV)

0

20

40

60

80

450 500 550(MeV) 2

s (2 q) = 0 83 cY

p(M

eV)

FIG. 4. Lowest lying quasiparticle dispersion relationshipsabout the Fermi momentum pF q 500 MeV for theCFL phase with two different values of the strange quarkmass. (The Ms 0 dispersions are the same as in the top ofFig. 1.) Qualitatively this has the same structure as Fig. 1 exceptthat middle dispersion is now split by higher order mass effects.

min

pp

∆0

2s (2 q

cY )

00

1

0 2 0 4

0 5

0 6 0 8 1 0 1 2

FIG. 5. Physical gap of the lowest lying excitation as a func-tion of the strange quark mass. The dotted line corresponds to theCFL phase and the solid line corresponds to the CFLK0 phase.We have normalized the axes in terms of c

Y 0 for compari-son with the hypercharge chemical potential case.The CFL/gCFL transition occurs at a slightly smaller value ofM2s=q 45:4 MeV than the value of 46.8 MeV in [37,38,40].

This is due to the effects of the other parameters on thequasiparticle dispersion relations. We note that, as with Y ,the transition from the CFLK0 to a gapless phase is delayedrelative to the CFL/gCFL transition, but by a slightly reducedfactor of 1.2. This is in qualitative agreement but quantitativedisagreement with the factor of 4=3 found in [34]. The is most-likely the result of our fully self-consistent treatment of thecondensate parameters.

0

20

40

60

80

500 550(MeV) 2

s (2 q) = 0 50 cY

p(M

eV)

0

20

40

60

80

450 500 550(MeV) 2

s (2 q) = 0 84 cY

p(M

eV)

FIG. 6. Lowest lying quasiparticle dispersion relationshipsabout the Fermi momentum pF q 500 MeV for theCFLK0 phase with two different values of the strange quarkmass. (The Ms 0 dispersions are the same as in the top ofFig. 1.) Qualitatively this has the same structure as Fig. 2.


094032-5


virtually quadratic dispersion when electric neutrality isenforced. (This was discovered in [37] and is explained indetail in [38].)

In the CFLK0/gCFLK0 transition, a single charged modebecomes gapless.2 Thus, immediately beyond the transi-tion, the corresponding gCFLK0 state will also be a con-ductor but there will be a single charged gapless mode withalmost quadratic dispersion. Additional modes will con-tinue to lower until either more modes become gapless, or afirst order phase transition to a competing phase occurs.

B. Finite strange quark mass

The second pair of CFL/CFLK0 states that we considerare self-consistent solutions to the gap equation in thepresence of a finite strange quark mass. Qualitatively weexpect to see similar features to the states at finite hyper-charge chemical potential and indeed we do as shown inFigs. 4–6.

Quantitatively, we notice a few differences with pre-vious analyses concerning the locations of the phase tran-sitions to gapless states. Our parameters have been chosento match the parameters in [37,38,40]. They find that thegCFL/CFL transition occurs at M2

s= 46:8 MeV, butthe CFL/gCFL transition happens noticeably earlier withour model at M2

s= 43:9 MeV. This is due to a corre-sponding six-percent reduction in the condensate parame-ters and represents the effects of performing a fully self-consistent calculation.

Another difference concerns the appearance of gaplessmodes in the CFLK0 state (see Fig. 6). This transitionoccurs at M2

s= 52:5 MeV in our model—a factor of1:2 larger than the CFL/gCFL transition. This is some 10%smaller than the factor of 4=3 derived in [34]. This is likelydue to the more complicated condensate structure we con-sider and the inclusion of self-energy corrections.

3Here the matrices A are the eight 3 3 Gell-Mann matricesand the are the Dirac matrices which we take in the chiral

IV. NJL MODEL

We base our analysis on the following Hamiltonian forthe NJL model

H Z d3 ~p23

y~p ~ ~p 0M ~p Hint: (7)

Here we consider 9 species of quarks 3 colors 3 flavors: Including the relativistic structure, there are 36quark operators in the vector . The matrices and M arethe quark chemical potentials and masses, respectively.

2This mode pairs rs, gu, and bu quarks in quite a nontrivialmanner. In the CFL, the quasiparticles form a nice block-diagonal structure in which the quarks exhibit definite pairingbetween two species. In the CFLK0, the block structure is morecomplicated and the pairing cannot be simply described: thelowest lying quasiparticle is a linear combination of the three rs,gu, and bu quark.

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We take the interaction to be a four-fermion contactinteraction with the quantum numbers of single gluonexchange:3

Hint gZ A

A : (8)

The Gell-Mann matrices act on the color space and theflavor structure is diagonal. We point out that this form ofNJL interaction has the desirable feature of explicitlybreaking the independent color SU3CL left and SU3CR

right symmetries that some NJL models preserve. This isimportant because the condensation pattern (1) does notexplicitly link left and right particles: Our model thus hasthe same continuous symmetries as QCD, and the onlycomplication to deal with is the gauging of the single colorSU3C symmetry.

Our goal here is to provide a nonperturbative model todiscuss the qualitative features of QCD at finite densities.We model the finite density by working in the grandthermodynamic ensemble and introducing a baryon chemi-cal potential for all of the quarks:

B

31: (9)

With only this chemical potential and no quark masses, ourmodel has an U3L U3R SU3C=Z3 continuousglobal symmetry in which the left-handed quarks trans-form as 3; 1; 3 and the right handed quarks transform as1; 3; 3. In the chiral basis we have explicitly4

L R

!

eiLFL C 00 eiRFR C

L R

; (10)

where F and C are SU(3) matrices. For an attractiveinteraction, this NJL model exhibits the same symmetrybreaking pattern as QCD (2) with a restored axial symme-try. The difference between this NJL model and QCD isthat the NJL model contains no gauge bosons. Thus, thereare 18 broken generators which correspond to masslessGoldstone bosons, and none of these is eaten. To effec-tively model QCD, we must remove the extra coloredGoldstone bosons. At the mean-field level, this is doneby imposing gauge neutrality conditions [45,54,55]. Oncethe appropriate chemical potentials are introduced, thedependence on the vacuum expectation values of the col-

basis. Our normalizations and conventions are

Tr AB 12

AB; 5 i0123 1 00 1

;

C i20:

We also use natural units where c @ kB 1.4From this explicit representation we can see how the centers

of the color and flavors overlap giving rise to the Z3 factor.

-6

5To derive the full equations, one introduces the augmentedstructure while imposing constraints on the matrices throughoutthe variation. For example, one must ensure that H CHTCwhere C C. To derive the proper momentum structure,one simply attaches momentum indices: for homogeneous statesall correlations have the form hy~p ~qi /

3 ~p ~q and themomentum structure follows trivially.


ored Goldstone modes is cancelled and the low-energyphysics of the NJL model matches that of QCD.

The usual NJL model has a local interaction, but this isnot renormalizable and needs regulation. For the purposesof this paper, we introduce a hard cutoff on each of themomenta ~p k ~pk to mimic the effects ofasymptotic freedom at large momenta:

Hint g29

Zd3 ~pd3 ~p0d3 ~qd3 ~q0pp0qq03

~p ~p0 ~q ~q0 ~pA ~p0 ~q

A ~q0 :

To study this model we perform a variational calculationby introducing the quadratic Hamiltonian

H0 Z d3 ~p23

y~pE ~p ~p T~pB ~p y~pBy ~p; (11)

where

E ~p ~ ~p 0MA; (12)

and then computing the following upper bound [56] on thethermodynamic potential of the full system:

0 hH H0i0: (13)

0 is the thermodynamic potential of the quadratic modeland the expectation value hi0 is the thermal average withrespect to the quadratic ensemble defined by H0. In prin-ciple, the quadratic model is exactly solvable, thus theupper bound can be computed. One then varies the parame-ters A and B to minimize this upper bound, obtaining avariational approximation for the true ensemble. In thezero-temperature limit, this is equivalent to simply mini-mizing the expectation value of the Hamiltonian over theset of all Gaussian states.

In practice, it is difficult to vary with respect to allpossible quadratic models since the space is of uncountabledimensionality. In this paper we restrict ourselves to min-imizing over homogeneous and isotropic systems. This isequivalent to performing a fully self-consistent mean-fieldanalysis. The condition for the right hand side of (13) to bestationary with respect to the variational parameters isequivalent to the self-consistent gap equation.

The microscopic analysis presented in this paper con-sists of choosing reasonable parametrizations of A (whichincludes the chemical potentials, masses and related cor-rections) and B (which includes the gap parameters ) thatare closed under the self-consistency condition, and nu-merically finding stationary points of this system of equa-tions. (As A and B are arbitrary 36 36 matrices subjectonly to A Ay and B BT , a full parametrizationconsists of 2556 parameters and was too costly for thepresent author to consider. However, the parametrizationchosen is quite natural and fully closed.) Once the parame-ters A and B are found, the properties of the ensemble canbe computed by diagonalizing the quadratic Hamiltonian.

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As discussed in Sec. V E and [45], we must impose theappropriate gauge charge neutrality conditions. This isdone by introducing bare gauge chemical potentials intothe model and choosing these to ensure the final solution isneutral.

To impose a charge neutrality condition, we instead varyR (along with with the other parameters) to obtain aneutral solution (again we note that the total charge andother correlations of the state depend only on the correctedparameters R). Once this solution is found, is com-puted and the required bare chemical potential R determined. Despite the fact that the self-energy cor-rections depend only on the corrected parameters (R etc.),the thermodynamic potential depends on both the correctedand the bare parameters and so this last step is important.

One must also be careful about which thermodynamicpotential is used to compare states when neutrality con-ditions are enforced as we are no longer in the grandensemble. The differences between the potentials of therelevant ensembles are proportional to terms of the formQ, however, so for neutrality conditions, Q 0, and thethermodynamic potential may still be used to comparestates.

A. Numerical techniques

We sketch here the method used to calculate the ther-modynamic potential and perform the variational minimi-zation. First, we express the Hamiltonian H in thefollowing simplified form

H yH gyyy (14)

where H is a Hermitian matrix. In order to do this andinclude the ‘‘anomalous’’ correlations h i, we must usean augmented ‘‘Nambu-Gorkov’’ spinor

C

; (15)

where C is the charge conjugated spinor. This doubling ofthe degrees of freedom requires careful attention to avoiddouble counting.

To simplify the presentation of our method in this sec-tion, we shall ignore this complication and assume that contains a single set of operators with no duplicate degreesof freedom. We also consider only a single interactionterm, and subsume the momentum structure into the matrixstructure. Explicitly dealing with these complications isstraightforward and the details are presented in [57].5

-7


We now express the variational Hamiltonian as

H0 yH yH0: (16)

The matrix of variational parameters may be thought ofas the self-energy corrections. All of the two-point corre-lations are determined from the corrected Hamiltonianmatrix H0, with the anomalous correlations being foundoff the diagonal:

hyi F; hTi FT;

F 1

1 eH0:

At finite temperatures, there is a one-to-one relationshipbetween E and the matrix F 1 F. Armed with thisresult, the variational bound takes the explicit form6

1

Tr lnF TrF gTryFTrF

TryFF: (17)

From this, one may find the stationary points by differ-entiating with respect to F. At finite temperature this isformally equivalent to finding the stationary points byvarying with respect to . Differentiating with respect to is complicated by the functional dependence and theresult is not expressible as a simple matrix equation. Theconditions @=@Fij 0 yield the fully self-consistentSchwinger-Dyson equations which may be expressed as:

gyTrF TryF yF Fy:

(18)

In principle, one may derive analytic expressions for theseequations, but, in practise, the matrices are 72 72 andcomputing F analytically—even when many approxima-tions are made—is quite tedious. Instead, we simply usethese expressions numerically. The required diagonaliza-tion is then efficiently performed using standard numericallinear algebra tools. The traces involved include momen-tum integrals, but for homogeneous states these are one-dimensional and thus also quite efficient. The biggestchallenge is to solve simultaneously the equations presentin (18). This is done by first projecting out the limitedsubspaces describe in the Appendix A and then employinga multidimensional root-finder.

Since the search space is large ( 45 parameters for theCFLK0 states), traditional root-finders are prohibitive be-cause they recompute the Jacobian at each step. Here weuse a modified Broyden algorithm [58,59] to provide a

6For a fully self-consistent analysis, we must include the termghyyihi. These correlations vanish in this simplifiedanalysis, but are included when the full augmented structure (15)is considered as discussed in [57]. Note also that momentumintegration is implicit in the matrix multiplication and traces.

094032

secantlike update to the Jacobian requiring far fewer func-tion evaluations per step of the algorithm.

In many cases, the Schwinger-Dyson Eq. (18) convergesthrough simple iteration. With charge neutrality con-straints, this is often no longer the case, but the Broydenupdate is sufficient to restore convergence.

B. CFL at ms 0.

As an example, consider the parity-even CFL state. Theself-consistency conditions are fully closed when one in-cludes four variational parameters. There are two gapparameters 3 and 6 corresponding to the diquark con-densate (1), one chemical potential correction B to thebaryon chemical potential and an induced off-diagonalchemical potential oct. The quadratic Hamiltonian (11)can thus be expressed

H0 y ~ ~p 13B

12

TC5 h:c:

where

ab 3kabk 6

a

b

b

a ; (19)

13B oct; (20)

and

octab oct

X8

A1

AaAb 80a

0b

!:

Most of this structure is well-known and discussed manytimes in the literature, however, there has been no mentionof the parameter oct because most analyses neglect theself-energy corrections.

Neglecting the correction to the baryon chemical poten-tial is reasonable since it has little physical significance: itsimply enters as a Lagrange multiplier to establish a finitedensity. As such, the effective common quark chemicalpotential

q 13

effB

13B B (21)

is the relevant physical parameter defining the Fermi sur-face. To compare states in the grand ensemble, however,one must fix the bare rather than the effective chemicalpotentials. This is what we have done in our calculations.Numerically, we find that the corrections B cause q tovary by only a few percent as we vary the perturbationparameters Y and ms.

There is no bare parameter corresponding to oct. Thus,it is spontaneously induced and should be treated on thesame footing as . To see that such a parameter must exist,consider changing to the ‘‘octet’’ basis using the aug-mented Gell-Mann matrices

~ A 2Aa a (22)

where 0 1=6p

. In this basis, the off-diagonal conden-

-8


sate becomes diagonal with one singlet parameter 46 23 and eight octet parameters 6 3:

~

46 23

6 3

. ..

6 3

0BBBB@

1CCCCA: (23)

It is clear that in the CFL, the singlet channel decouplesfrom the octet channel: there is no symmetry relating theseand the two gap parameters are related by the numericalvalue of the coupling g. This decoupling is also present inthe chemical potential corrections. One linear combinationcorresponds to the identity: this corrects the baryon chemi-cal potential B. The other is the induced oct.

Numerically, we calibrate our model with this CFLsolution. In particular, we chose our parameters to repro-duce the results of [40]. We use a hard cutoff at 800 MeV, and a coupling constant chosen so that, withan effective quark chemical potential of q 500 MeVone has a physical gap in the spectrum of 0 3 6 25 MeV. This fixes the following parametervalues which we hold fixed for all of our calculations:

800 MeV; (24a)

g2 1:385; (24b)

B=3 549:93 MeV: (24c)

With these parameters fixed, the fully self-consistentmean-field CFL solution has the following variationalparameters:

3 25:6571 MeV; 6 0:6571 MeV;

B=3 49:93 MeV; oct 0:03133 MeV:

(25)

As first noted in [7], and discussed in [60], the parameter6 is required to close the gap equation, but is smallbecause the sextet channel is repulsive. In weakly-coupledQCD, 6 is suppressed by an extra factor of the coupling.This effect is numerically captured in the NJL model. Theparameter oct is also required to close the gap equationwhen the Hartree-Fock terms are included. It is also nu-merically suppressed. Recent calculations often omit 6

and oct: we see that this is numerically justified.The physical gap in the spectrum also defines the critical

hypercharge chemical potential for the CFL/gCFL transi-tion (6):

cY 0 25:00 MeV: (26)

C. CFL at Y;ms 0

Once one introduces a strange quark mass, one mustintroduce additional parameters. A simple way to deter-mine which parameters are required is to add the mass,then compute the gap equation and see which entries in the

094032

self-energy matrix are nonzero. By doing this for a varietyof random values of the parameters, one can determine thedimension of the subspace required to close the gap equa-tion and introduce the required parameters.

In the case of the CFL state with nonzero hyperchargechemical potential, one only needs to introduce the pa-rameters Y and 8 to ensure gauge neutrality: As dis-cussed in Sec. III A none of the other parameters change.To go beyond the transition into the gCFL phase, however,or to extend the results to nonzero temperature, one mustintroduce additional parameters. These include the pertur-bation Y , the gauge chemical potentials 3, 8, and erequired to enforce neutrality, as well as nine gap parame-ters 12, 23, 13, 45, 67, 89, 11, 22, and 33 thatfully parametrize the triplet and sextet diquark conden-sates. (These latter nine parameters correspond to theparameters i, ’i and i defined in Ref. [39].) The addi-tional parameters are chemical potentials similar to oct

which are induced by the gap equations. The full set ofparameters in discussed in the Appendix A.

Adding a strange quark mass is more complicated. Firstof all, we need to introduce additional Lorentz structure.For homogeneous and isotropic systems, there are eightpossible relativistic structures:

A 1 5 5 0 m 05 m5;

B C5 C 5 0C5 0C 5:

Introducing quark masses requires one to introduce theadditional Lorentz structure [60] to close the gap equa-tions, but these are found to be small. In total, one requiresabout 20 parameters to fully parametrize the CFL in thepresence of a strange quark mass (see Table IV).

With the inclusion of a bare quark massms one induces achiral condensate h i which in turn generates a correc-tion to the quark mass. The resulting parameter in H0 (16)is the constituent quark mass Ms which appears in thedispersion relationships for the quarks. It is this constituentquark mass that must be used when calculating the effec-tive chemical potential shift (4). Generally the constituentquark mass is quite a bit larger than the bare quark massparameter ms. For example, close to the phase transition,we havems 83 MeV while the constituent quark mass isMs 150 MeV (see Table IV). We have checked that ourcalculations are quantitatively consistent with the calcula-tions presented in [61] in this regard.

D. CFLK0

Applying a kaon rotation to the CFL state breaks theparity of the state, and mixes the parity-even parameters ,m, and with their parity odd counterparts 5, m5, 5

and 5. The full set of parameters and typical numericalvalues is presented in the Appendix A.

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V. LOW-ENERGY EFFECTIVE THEORY

To describe the low-energy physics of these models, wefollow a well established procedure: identify the low-energy degrees of freedom and their transformation prop-erties, identify the expansion parameters (power countingscheme), write down the most general action consistentwith the symmetries and power counting, and determinethe arbitrary coefficients by matching to experiment oranother theory. In our case, we will match onto themean-field approximation of the NJL model. The resultinglow-energy effective theory has been well studied [18,62]:we use this presentation to establish our conventions, andto contrast the effective theory of QCD with that of themicroscopic NJL model.

A. Degrees of freedom

The coset space in the NJL model is isomorphic toU3 U3. This can be fully parametrized with twoSU(3) matrices X and Y and two physical phases A andV which one can physically identify with the condensates:

VyAp

Xc / abch aL bL i; (27a)

VyAyp

Yc / abch aR bR i: (27b)

These thus transform as follows:

X! FLXCy; (28a)

Y ! FRYCy; (28b)

A! e2iRLA; (28c)

V ! e2iRLV: (28d)

Note that the condensation pattern X Y 1, A V 1 is unbroken by the residual symmetry where FL FR C and also by the Z2 symmetries where L; R . Thisis the reason for the extra factor of 2 in the phases. In QCDthe degrees of freedom are similar, but one must consideronly color singlet objects. Thus, the low-energy theory forQCD should include only the color singlet combination

XYy ! FLFyR (29)

and the color singlet phases A and V. Note also that thesehave the following transformation properties under parity

X $ Y; A$ Ay; $ y: (30)

The field content of the effective theories is thus:
H, 0 T wo singlet fields corresponding to the U(1) phases
of A and V. The field associated with V is a scalarboson associated with the superfluid baryon numbercondensation. We shall denote this field H.The field associated with A is a pseudoscalar bosonassociated with the axial baryon number symmetryand shall be identified with the 0 particle. Asdiscussed in Sec. II, the axial symmetry symmetry

094032-10

is anomalously broken in QCD and the 0 is notstrictly massless due to instanton effects, but theseare suppressed at high density. We ignore theseeffects. Our NJL model thus contains no instantonvertex and our low-energy theory will contain noWess-Zumino-Witten terms [63,64]. It would beinteresting to include both of these terms and repeatthis calculation as these effects are likely not small[48]

a E
ight pseudoscalar mesons a corresponding to thebroken axial flavor generators. As color singletsthese remain as propagating degrees of freedom inboth QCD and NJL models. These have the quan-tum numbers of pions, kaons and the eta and trans-form as an octet under the unbroken symmetry.
a E
ight scalar bosonsa corresponding to the brokencolored generators. These are eaten by the gaugebosons in QCD and are removed from the low-energy theory. This gives masses to eight of thegauge bosons and decouples them from low-energyphysics. In the NJL model these bosons still remainas low-energy degrees of freedom, but decouplefrom the color singlet physics when one properlyenforces color neutrality.
There are additional fields and effects that should beconsidered as part of a complete low-energy theory, butthat we neglect:

(1) T
he appropriately ‘‘rotated electromagnetic field’’associated with the unbroken U1 ~Q symmetry re-mains massless. Both the CFL and CFLK0 statesremain neutral with respect to this field, however,and we do not explicitly include it in ourformulation.
(2) T
he leptons are not strictly massless, but the elec-tron and muon are light enough to consider in thelow-energy physics. In particular, they contribute tothe charge density in the presence of an electricchemical potential and at finite temperature. In thispaper, leptonic excitations play no role since weconsider only T 0 and both CFL and CFLK0
quark matter is electrically neutral for e 0.The leptons play an implicit role in fixing ~Q suchthat e 0 in both insulating phases.

To be explicit, we relate all of the dimensional physicalfields H, 0, a and a to the phase angles through theirdecay constants: H fH ~H, 0 f0 ~

0, a f ~a, anda f ~a. The two U(1) phases angles have a slightlydifferent normalization because of the normalization of thegenerators. This normalization is chosen to match thekinetic terms in the original theory and matches [22,23]:~0

6pR L, and ~H0

6pR L. The realiza-

tion of these transformations in the microscopic theory is

! expi ~H1 ~05

26p ~ara ~afaA

; (31)


where

faR;L 1 5 a 1=2; (32a)

ca 1 1 a; (32b)

faA faR faL 5 a 1; (32c)

faV faR faL 1 a 1; (32d)

ra faV ca=2 (32e)

and the corresponding realization in the effective theory is

X expfi~aag expfi ~aag; (33a)

Y expfi~aag expfi ~aag; (33b)

A expf2i~0=6pg; (33c)

V expf2i ~H=6pg; (33d)

expf2i~aag: (33e)

B. Power counting

In addition to QCD which separates the three lightquarks from the heavy quarks, there are two primary scalesin high-density QCD: the quark chemical potential q andthe gap . In the NJL model there is also a cutoff and thecoupling constant: these are related by the gap equationwhen one holds and fixed and the qualitative physics isnot extremely sensitive to the remaining renormalizationparameter.

Our low-energy theory is an expansion in the energy/momentum of the Goldstone fields. Thus, the expansion isin powers of the derivatives with respect to the scales and. In this paper, we shall only consider leading orderterms: Systematic expansions have been discussed else-where (see for example [65]).

C. Kinetic terms

To construct the low-energy theory we follow [18] andintroduce colored currents

J X Xy@

vX! CJXCy;

JY Yy@vY ! CJY Cy;

J JX JY ! CJCy:

In the presence of a finite density, we no longer havemanifest Lorentz invariance and must allow for additionalconstants into our spatial derivatives

@v @

0; v@i

to account for the differing speeds of sound. This paperwill be concerned with static properties, so we can neglectthese. In principle, one must also match these coefficientsv. In QCD this matching, along with other coefficients, hasbeen made with perturbative calculations at asymptotic

094032

densities [22,23]. Our theory and states still maintain rota-tional invariance. Thus, to lowest order we have [18]

L eff L0 L LH L ;

3f2

0

4@v0

Ay@v0 Af2

4TrJJ

3f2

H

4@vH

Vy@vHV f2

4TrJJ ;

12@v0

0@v0 0 1

2@v

a@va

12@vH

H@vHH 12@v

a@va :

The neglected terms are of higher order in the derivativeexpansion. Note that our normalizations have been chosenso that this expression is canonically normalized to qua-dratic order in terms of the dimensionful fields.

The division of Leff is natural [18] because it separatesout the color singlets. L depends only on for example:

L f2

4TrJJ

f2

4Tr@

vy@v:

Thus, with the exceptions noted above, the lowest-orderlow-energy effective theory of massless Nf 3 QCD is

L QCD L LH L0 ; (34)

whereas the NJL model proper must also include L.

D. Perturbations

We shall now consider two types of perturbations:chemical potentials and quark masses. To deal with theseperturbations, we note that they enter the microscopicLagrangian as

L SB yLL L

yR

R R yRM L

yLMy R:

These terms break the original symmetries of the theory,but one can restore these symmetries by imparting thefollowing spurion transformations to the masses andchemical potentials

M! FR CMFL CyeiRL; (35a)

L ! FL CLFL Cy; (35b)

R ! FR CRFR Cy: (35c)

The transformation M! M preserves the residual Z2

symmetries. This prevents odd powers of the mass termsfrom appearing in the chiral effective theory. In particular,the linear term dominant in the vacuum is forbidden,resulting in an inverse mass-ordering of the mesons[22,23] with the kaon being the lightest particle at highdensity.

All these symmetries must be restored in the effectivetheory: we are only allowed to couple these parameters tothe fields in ways that preserve the global symmetries. To

-11


lowest order, this greatly limits the possible terms in theeffective theory.

In the case of the chemical potentials, we can go one stepfurther by noting that the perturbations always appears incombination with the time derivative

L yi@0 : (36)

One can thus promote the chemical potentials to a temporalcomponent of a spurion gauge field and render the sym-metries local in time:

! F C i@0F Cy: (37)

The effective theory must also maintain these local sym-metries. One concludes that the chemical potential pertur-bations can only appear through the introduction ofcovariant derivatives in the effective theory. In particular,consider adding independent color and flavor chemicalpotential terms:

L;R L;R1 1 L;RF 1 1 C; (38)

where F and C are traceless 3 3 matrices. From thesewe may construct the following quantities that transformcovariantly:

r0X @0X iLFX iXC; (39a)

r0Y @0Y iRFY iYC; (39b)

r0 @0 iLF iR

FT; (39c)

r0V @0 2iVV; (39d)

r0A @0 2iAA; (39e)

where V R L is a small adjustment of the baryonchemical potential B=3 and A R L is the ‘‘axialbaryon’’ chemical potential. For the rest of this paper, weshall only consider vector chemical potentials that are realand symmetric: L;RF F

F

yF etc. With these

restrictions, the static potential in the effective theory is

V f2

2TryFF 2

F 3f2HV

2 3f20 A

2

f2

4TrXyFX YyFY 2C

2 (40)

to lowest order. The terms omitted include terms of higherorder in the perturbation and small corrections due to theexplicit violation of the ‘‘local’’ spurion symmetries by thecutoff.

E. Charge neutrality

As discussed in [45,54,55], the gauge invariance of QCDimplies that homogeneous states must be color neutral.Nonzero static color sources A0

C ~p 0 cancel the tadpolediagrams ensuring neutrality. These sources enter the NJLcalculation as Lagrange multipliers to enforce neutrality.

One can see explicitly how these arise in the context ofthe effective theory. The gauge fields effect the localsymmetry and thus couple through the derivatives in ex-

094032

actly the same way as the spurion colored chemical poten-tials: C / gsA0

C. Enforcing gauge-invariance induces aneffective colored chemical potential that makes (40) sta-tionary with respect to variations of the gauge field, andthus equivalently,with respect to traceless variations ofC.Thus, we see that, to lowest order [18,54,55]

C 12X

yFX YyFY: (41)

Inserting this into (40), and considering only tracelessperturbations, we see that the color dependence drops outof the effective theory and we are left with the staticeffective potential involving only the color singlet fields:

V f2

2TryFF 2

F . . . : (42)

In order to reproduce the physics of this in the NJL model,however, we must remove the colored degrees of freedom.This is done by introducing color chemical potentials to theNJL model as Lagrange multipliers and using them toimpose color neutrality [45,54,55]. This removes the colordependence in the NJL model to all orders in the same wayas it removes the color dependence in (41) to lowest order.(In general, it is not sufficient to impose color neutrality:one must also project onto color singlet states (as well asstates of definite baryon number). This projection is im-portant for small systems, but likely has negligible cost forthermodynamically large systems such as neutron stars.See [66] for an explicit demonstration of this in the two-flavor case.)

The quarks also couple to the photon, and so we alsomust enforce electric neutrality. Enforcing electromagneticgauge invariance will likewise induce an electric chemicalpotentiale that ensures electric neutrality. It turns out thatboth the CFL and the CFLK0 quark matter are neutralunder a residual charge ~Q (both are ~Q insulators). Thismeans that one has some freedom in choosing the chemicalpotentials used to enforce neutrality. In particular, prior tothe onset of gapless modes, one may choose these combi-nations so that e 0. This is naturally enforced byincluding charged leptons in the calculation.

Once a charged excitation becomes gapless, the materialbecomes a conductor and a nonzero e is required toenforce neutrality. The phase transition to the gCFL andgCFLK0 is defined by exactly such a charged excitation. Inthis paper, we shall only consider the insulating phases, andthus simply set e 0. For further discussions of themetal/insulator properties of the CFL and gCFL we referthe reader to [37,39,40].

F. Kaon condensation

We are now in a position to argue for the existence of akaon-condensed state. Consider performing an axial K0

rotation on the parity-even CFL state. This is effected using(31) in the microscopic theory and using (33) in the effec-tive theory with the parameter ~6 . Such a state is now

-12


described by

e2i6

1cos i sini sin cos

0@

1A: (43)

In the presence of a hypercharge chemical potential, theeffective potential becomes [31–33]

V f22

Y

2cos2 1 : (44)

We see that this has a minimum for =2: this is thestate with maximal K0 condensation. We can also directlycompute the difference in the thermodynamic potentialdensities between the CFL state and the CFLK0 state:

CFLK0 CFL f22

Y

2: (45)

Armed with this relationship, we can now turn to themicroscopic calculation and determine the coefficient f.In Fig. 7 we plot our numerical results so that the linearrelationship (45) is evident. From the slope of the relation-ship we find that

f 0:19q: (46)

(ΩK

0Ω

CF

L)

(q

c Y)2

( YcY )2

0

0

0 01

0 02

0 03

0 040 5 1 0 1 5

FIG. 7. Numerical difference in energy densities between thekaon-condensed CFLK0 state and the CFL state at finite hyper-charge potential Y obtained from our microscopic NJL calcu-lation. The units are scaled in terms of the quark chemicalpotential q 500 MeV and the critical hypercharge chemicalpotential c

Y 25 MeV. The quantities plotted were chosen sothat the relationship will be linear if our calculation agrees withthe effective theory result (45). The slope of the line is m f2

=22q 0:018 from which we can determine the effective

theory parameter f 0:19q. This is in good numericalagreement with the perturbative QCD result f 0:209q

[22,23]. The dashed extension shows the comparison betweenthe CFLK0 potential and the CFL potential, but beyond 1.0, theCFL becomes the gCFL and the energy dependence changes. Wehave not calculated the gCFL potential in this paper, but plot thisextension to emphasis that the CFLK0 persists beyond the CFL/gCFL transition point at 1.0.

094032

We note that this is in good numerical agreement with theperturbative QCD result [22,23] of f 0:209q. Thisstriking agreement between two very different modelsarises from the fact that this coefficient is not very sensitiveto the effects of the cutoff (which is different in the twotheories) and gives encouraging support to the use of theNJL model to study QCD.

The equivalent relationship in the case of a strange quarkmass requires one to include mass terms in the effectivetheory (see for example [30]), but the leading order effectcan be determined by using the ‘‘effective’’ strange quarkchemical potential Y M2

s=2q that follows from (3):

CFLK0 CFL f2

2

M2s

2q

2: (47)

It is important here to note, however, that the strange quarkmass affects the solution in such a way that the gapparameters change and self-energy corrections modifythe quark chemical potential and the constituent quarkmass. It is the renormalized parameters that appear inthis relation and in the perturbative QCD result. Thus, asa function of the bare parameter ms, we have Ms / ms andq s / m2

s . Thus, we should see a linear relationshipbetween CFLK0 CFL and M4

s . We plot this relationshipin Fig. 8 and extract the slope which gives the relationshipf 0:21q. This is in qualitative agreement with our

(ΩK

0Ω

CF

L)

(q

c Y)2

[ 2s (2 q

cY )]2

0

0

0 01

0 02

0 03

0 040 5 1 0 1 5

FIG. 8. Numerical difference in energy densities between thekaon-condensed CFLK0 state and the CFL state at finite strangequark mass ms obtained from our microscopic NJL calculation.The units are scaled in terms of the renormalized quark chemicalpotential q 500 MeV and the critical hypercharge chemicalpotential c

Y 25 MeV to facilitate comparison with Fig. 7 andto emphasize the linear relationship implicit in (47). The slope ofthe line is m f2

=22q 0:028 which gives an effective

f 0:21q which is consistent with our previous results. Incomparison with Fig. 7, the CFL! gCFL transition occurssomewhat earlier because the gap parameters are reduced withincreasing strange quark mass. The curve cannot be extended asin Fig. 7 because the free-energy of the CFL is no longer aconstant as it was with a hypercharge perturbation.

-13

23

Y=

48

Y

YcY

0 0 50 8

1 0

1 0

1 2

1 5

FIG. 9. Chemical potentials required by the NJL model toenforce color neutrality in the CFLK0 phase with finite hyper-charge chemical potentials. Note that the effective-theory rela-tionship (50) is satisfied from small chemical potentials. Thelinear deviation seen here reflects the missing terms in theeffective theory that are of higher order in the perturbation Ywith a linear deviation here corresponds to 2

Y terms missingin (50).

Y(

c Y2 q)

YcY

00

0 05

0 5 1 0 1 5

FIG. 10. Hypercharge density of the CFLK0 state in thepresence of a hypercharge chemical potential Y as obtainedfrom our microscopic NJL model calculation. The units arescaled as in Fig. 7 so that the relationship will be linear if theNJL model calculation agrees with the effective-theory predic-tion (51). By determining the slope of this relationship we haveanother way of determining the coefficient f in the effectivetheory. The slope is f2

=2q 0:037 which agrees with our

previous determination of f 0:19q.


previous result. The slight numerical disagreement is dueto effects of the strange quark mass that are not captured bythe chemical potential shift (3).

We pause here to point out a discrepancy between ourresults and similar work by Buballa [67]. Our resultsshown in Fig. 8 suggests that kaon condensation occursfor all values of ms in this simple model with mu md 0 whereas Buballa finds that kaon condensation is onlyfavored for ms sufficiently large. If the chemical potentialshift were the only effect of a strange quark mass, then thiswould be inconsistent with (40). This is not, however, thecorrect expansion. Instead, one has, for maximal kaoncondensation and mu md 0:

CFLK0 CFL 4a6 a8m2s cm4

s Om6s:

(48)

A proper discussion of this is beyond the scope of thispaper, but will be discussed elsewhere [68]. The term withcoefficient c should be identified with the leading ordercontribution from (3); the term with coefficient a6 arisesfrom the sextet gap contribution 6 and is discussed in[49]; the term with coefficient a8 is of higher order inperturbative QCD and so is usually neglected [22,23,30].

Thus, Buballa’s results are consistent with the effectivetheory. The discrepancy is due to a different choice ofparameters 100 MeV and 600 MeV comparedwith our parameters 25 MeV and 800 MeV.With the large gap, one is further from the perturbativeQCD regime and the quadratic term appears to play asignificant role. In our analysis, the term 4a6 a8m2

sis small compared with the cm4

s term. Using Buballa’sparameters, however, we qualitatively reproduce his re-sults. A further discussion of these effects will be presentedshortly [68].

There are a couple of other consequences that followdirectly from the effective theory. One is the value of thecolored chemical potentials required to enforce neutrality.In our microscopic model, we have fixed the gauge (unitarygauge) by setting X Yy

p

for the axial rotations.The CFL state has X Y 1 while the CFLK0 state has

X Yy 12p

2p

1 ii 1

0B@1CA: (49)

From (41) we have the following relationships required toenforce neutrality [54]

8 Y; 3 0; CFL; (50a)

8 1

4Y; 3

1

2Y; CFLK0: (50b)

We plot these relationships in Fig. 9. Note that they onlyhold for small perturbations where the effective theory isvalid: this plot also demonstrates a departure from the

094032

lowest-order effective theory as the perturbation isincreased.

As a final demonstration of the effective theory, wecalculate the hypercharge density. This is obtained byvarying the thermodynamic potential with respect to thehypercharge chemical potential:

nY @

@Y f2

Ycos2 1: (51)

-14


There should be no hypercharge density in the CFL stateand a density of nY f2

Y in the CFLK0 state. Indeed,the CFL supports no hypercharge density with nu nd ns. The hypercharge density of the CFLK0 phase is shownin Fig. 10 and provides another method of extracting f 0:19q.

G. A note on the meaning of V

We make a few remarks here about the meaning of theeffective potential V. In particular, one might betempted to try and compute the functional form of Vin the microscopic theory to facilitate matching with theeffective theory. Such an approach will generally fail be-cause one is allowed to pick an arbitrary parametrization ofthe Goldstone fields as long as they leave the kinetic termsunaltered [69,70]. Physical quantities must be invariantunder this change of parametrization: thus the spectrumabout the minimum, densities, and energy differences arereasonable quantities to compare in each theory. The gen-eral form of the effective potential away from the sta-tionary points, however, is rather arbitrary.

As an example: consider starting with the parity-evenCFL state in the presence of a finite Y . This state corre-sponds to a stationary point of the effective potential and isa self-consistent solution to the gap equations. One canthen form a continuum of ‘‘kaon rotated’’ states ji byapplying the broken symmetry generators to this state. Onemight expect to find V by computing the energy of thesestates, but instead one finds an expression that is only validlocally about the stationary point. The reason is twofold:First, there is not a unique ‘‘kaon rotated’’ state ji. Thisstate has many other parameters corresponding to other‘‘directions’’ (such as the gap parameters , the chemicalpotential corrections etc.) The only way to uniquely deter-mine these is to solve the gap equations, and these onlyhave well-defined solutions at stationary points. Second,the generators of the pseudo-Goldstone bosons in the pres-ence of perturbations are not the same as the generators ofthe true Goldstone bosons in the unbroken model: thepseudo-Goldstone bosons have some admixture of theseother directions.

This becomes even more evident when you analyze theCFLK0 state with a large perturbation: one can try to‘‘undo’’ the kaon rotation by applying the appropriatesymmetry generators to minimize the parity violating con-densates, but one finds that there is no way to do this. Onemust also transform the other parameters in order convert aCFLK0 state back to a parity-even CFL state.

VI. CONCLUSION

We have explicitly found self-consistent solutionswithin a microscopic NJL model exhibiting the feature ofkaon condensation in a color-flavor–locked state. Usingthese solutions, we have demonstrated that by properly

094032

enforcing gauge neutrality, one can remove the extraneouscolored degrees of freedom from the NJL model andeffectively model kaon condensation in high-densityQCD. In particular, the microscopic calculations can bematched onto the low-energy effective theory of QCD. Wedetermined f 0:19q which is in good numericalagreement with the perturbative QCD result.

Furthermore, our solutions are fully self-consistent: noapproximations have been made beyond the mean-fieldapproximation and restricting our attention to isotropicand homogeneous states. We find that our results agreequalitatively with both the expected properties of theCFLK0 phase based on effective theory calculations, andwith the previous numerical calculations of the CFL/gCFLtransition.

Quantitatively we find that the phase transitions occur atslightly smaller parameter values than previously found inthe literature. Concerning the CFL/gCFL transition, wefind that the gap parameters are reduced by a few percentcompared with those presented in [40], and subsequently,the critical Ms is also a few percent lower. Concerning theCFLK0/gCFLK0 transition, we find that the transition oc-curs about a factor of 1:2 higher than the CFL/gCFLtransition. This is in qualitative agreement but quantitativedisagreement with the factor of 4=3 calculated in [34].

The next step is to use this microscopic model to deter-mine the phase structure of high-density QCD in the regionwhere the gapless modes appear. We suspect that thegCFLK0 state will survive somewhat longer than thegCFL state on account of its lower condensation energy,but a quantitative comparison is required. Extrapolationto finite temperature is also a trivial extension in ourformalism.

A somewhat more challenging direction is to considerthe effects of instantons and finite up and down quarkmasses and investigate other forms of meson condensation.Preliminary investigations indicate, however, that the num-ber of parameters required to close the gap equations inthe presence of arbitrary meson rotations may be prohibi-tively large to continue with fully self-consistent calcula-tions. This should still be tractable with carefully madeapproximations.

ACKNOWLEDGMENTS

I would like to thank K. Rajagopal and F. Wilczek forsuggesting this problem, and P. Bedaque, K. Fukushima,D. K. Hong, D. Kaplan, C. Kouvaris, J. Kundu, M. J.Savage, T. Schafer, I. Shovkovy, I. Stewart, and A. R.Zhitnitsky for useful discussions. Related work has beendone independently by M. Buballa [67], and I am gratefulto him for postponing his preprint submission while thefirst version of this work was completed. This work issupported in part by funds provided by the U.S.Department of Energy (D.O.E.) under cooperative researchagreement #DF-FC02-94ER40818.

-15


APPENDIX: FULL PARAMETRIZATION

In this appendix, we give the full parametrization used toanalyze the K0 condensed states. First, we must introduce afull set of diagonal chemical potentials. One approachwould be to introduce the 9 individual quark chemicalpotentials, but certain linear combinations couple to rele-vant physics. We fix the overall density by fixing thebaryon chemical potential B. Then we must enforcegauge neutrality, so we introduce e which couples tothe electromagnetic field, and the diagonal color chemicalpotentials 3 and 8. The rest of the chemical potentialsare chosen to be orthogonal to these. Here then are thediagonal elements of the diagonal chemical potentials ex-pressed as tensor products of the flavor and color structure :

B 1; 1; 1 1; 1; 1=3; (A1a)

e 2;1;1 1; 1; 1=3; (A1b)

3 1; 1; 1 1;1; 0=2; (A1c)

8 1; 1; 1 1; 1;2=3; (A1d)

f 0; 1;1 1; 1; 1; (A1e)

e3 2;1;1 1;1; 0; (A1f)

e8 2;1;1 1; 1;2; (A1g)

f3 0; 1;1 1;1; 0; (A1h)

f8 0; 1;1 1; 1;2: (A1i)

An alternative set of chemical potentials includes the hy-percharge chemical potential Y instead of f. These areno longer orthogonal, but are still linearly independent.

B 1; 1; 1 1; 1; 1=3; (A2a)

e 2;1;1 1; 1; 1=3; (A2b)

3 1; 1; 1 1;1; 0=2; (A2c)

8 1; 1; 1 1; 1;2=3; (A2d)

Y 1; 1;2 1; 1; 1=3; (A2e)

e3 2;1;1 1;1; 0; (A2f)

e8 2;1;1 1; 1;2; (A2g)

f3 0; 1;1 1;1; 0; (A2h)

f8 0; 1;1 1; 1;2: (A2i)

The diagonal mass corrections (chiral condensates) do notcouple to any external physics, so we simply use the ninequark mass corrections (mur, mug, mub, mdr, mdg,mdb, msr, msg, msb).

The rest of the parameters are described in the followingmatrices. These appear more condensed when expressed inthe basis described in [40] where the quarks are ordered(ru, gd, bs, rd, gu, rs, bu, gs, bd). In this basis, the matricescorresponding to the variational parameters

A 1 5 5 0 m 05 m5;

B C5 C 5 0C5 0C 5:

(A3)

094032

In order to allow for a computer to enumerate the parame-ters, we introduce a systematic method for labelling theparameters. First, we use one of the names , 5, m, m5,, 5, , or 5 corresponding to the structure given above.We then use a two-digit index to specify which elementsare nonzero and an i indicates that the specified element is irather than simply 1. The symmetric entry must also be setso that the resulting matrix is either Hermitian orantisymmetric depending on whether or not it parametrizesA or B respectively. In total, there are 666 independentmatrices. For example

12 512 m12 12 5

12 12

0 1 0

1 0 0

0 0 0

..

. . ..

0BBBBB@

1CCCCCA;

m512

512

0 1 0

1 0 0

0 0 0

..

. . ..

0BBBBB@

1CCCCCA;

12i 512i m12i

512i

0 i 0

i 0 0

0 0 0

..

. . ..

0BBBBB@

1CCCCCA;

m512i 12i 5

12i 12i

0 i 0

i 0 0

0 0 0

..

. . ..

0BBBBB@

1CCCCCA:

(A4)

The reason that m5 and 5 behave differently than theothers is that, while 1, 5, and 0 are Hermitian, 05 isanti-Hermitian. Likewise, while C5, C, and 0C5 areantisymmetric, 0C is symmetric. Again, recall that theseare all specified in the ‘‘unlocking’’ basis which is orderedas

ru ; gd; bs; rd; gu; rs; bu; gs; bd: (A5)

The parity-even CFL state with no mass or hypercharge isexpressed in terms of this parametrization as

12 13 23 3 6=2; (A6a)

45 67 89 6 3=2; (A6b)

11 22 33 6; (A6c)

12 13 23 3oct; (A6d)

e3 3e8 f3 f8 3oct=4: (A6e)

-16

TABLE II. Parameters required for a self-consistent parity-even CFL solution in the presence of a hypercharge chemicalpotential. These values correspond to the dispersions shown inFig. 1. All values are in MeV. The first column labeled ‘‘Bare’’gives the fixed bare parameters that enter the Hamiltonian (7).The column labeled ‘‘Correction’’ is the contribution from theself-energy. The sum of the columns is the value that enters thequadratic Hamiltonian (11).

Y 0:50cY Y c

YParameter Bare Correction Bare Correction

B=3 549:93 49:93 549:93 49:938 12:5 0 25 0Y 12:5 0 25 0oct 0 0:031 332 0 0:031 332

3 0 25:657 0 25:6576 0 0:65 709 0 0:65 709


Here are some comparisons with other conventions in theliterature. Alford, Kouvaris, and Rajagopal [37] introduce1, 2 and 3 which are all related to the attractiveantisymmetric 3 channel:

TABLE III. Parameters required for a self-consistent CFLK0 soluvalues correspond to the dispersions shown in Fig. 2. All values aparameters that enter the Hamiltonian (7). The column labeled Corcolumns is the value that enters the quadratic Hamiltonian (13).

Y 0:50cY

Parameter Bare Cor

B=3 549:93 4Y 12:5 1e 0 0:3 6:4772 0:8 3:2386 0:e3 0 0:e8 0 0:0f3 0 0:0f8 0 0:012 5

18i 0 0:013 5

19i 0 0:23 89 0 0:05

28i 0 0:05

29i 538i 0 0:0

539i 0 0:0

546i 0 0:

11 0 0:22 88 5

28i 0 0:33 99 5

39i 0 0:12 5

18i 0 913 5

19i 0 823 89 0 145 5

56i 0 967 5

47i 0 85

29i 538i 0 0:

094032

23 89 1; (A7a)

13 67 2; (A7b)

12 45 3: (A7c)

Ruster, Shovkovy, and Rischke [39] introduces the pa-rameters , ’ and which include the repulsive symmet-ric 6 channel parameters:

23 ’1; 13 ’2; 12 ’3; (A8a)

89 1; 67 2; 45 3; (A8b)

11 1; 22 2; 33 3: (A8c)

Finally, Buballa [67] uses only the following parameters toparametrize the meson condensed phases:

12 45 s22=4; (A9a)

13 67 s55=4; (A9b)

23 89 s77=4; (A9c)

519i 5

47i p25=4; (A9d)

518i 5

56i p52=4: (A9e)

tion in the presence of a hypercharge chemical potential. Thesere in MeV. The first column labeled Bare gives the fixed barerection is the contribution from the self-energy. The sum of the

Y 1:20cY

rection Bare Correction

9:932 549:93 49:947:0687 30 2:5931534 36 0 1:2965000 00 16:346 3:331 107

000 00 8:1729 1:665 107

024 21 0 0:027 85608 069 9 0 0:009 285 335 998 0 0:084 80111 999 0 0:028 26716 852 0 0:047 357119 02 0 0:196 9446 967 0 0:046 61788 616 0 0:145 2646 967 0 0:046 617

03 023 4 0 0:065 811114 79 0 0:275 14

644 68 0 0:5851322 65 0 0:315 66335 23 0 0:345 23:6383 0 10:138:9893 0 8:57462:789 0 12:605:1811 0 9:7012:5229 0 8:1128332 42 0 0:350 25

-17

TABLE IV. Parameters required for a self-consistent parity-even CFL solution in the presence of a strange quark mass.These values correspond to the dispersions shown in Fig. 4. Allvalues are in MeV. The first column labeled Bare gives the fixedbare parameters that enter the Hamiltonian (7). The columnlabeled Correction is the contribution from the self-energy.The sum of the columns is the value that enters the quadraticHamiltonian (13). For example, the right set of data (just slightlybefore the CFL/gCFL transition) has a bare (current) strangequark mass of 80 MeV. This corresponds to a constituent quarkmass of 80 64 144 MeV. (Note that there is a slight differ-ence for the blue constituent quark masses because of thepresence of the colored chemical potential 8 required toenforce neutrality.)

M2s=2 0:50c

Y M2s =2 0:83c


B=3 549:93 48:952 549:93 48:3168 12:649 0:000 00 20:95 1:469 107

e3 0 0:022 617 0 0:022 026e8 0 0:007 454 2 0 0:007 208 7f3 0 0:022 617 0 0:022 026f8 0 0:022 362 0 0:021 62612 0 0:090 467 0 0:088 10313 23 0 0:088 963 0 0:085 669mur mdg 0 0:157 78 0 0:193 66mug mdr 0 0:172 55 0 0:2117mub mdb 0 0:156 04 0 0:191 55msr msg 61:843 50:029 80 64:267msb 61:843 50:079 80 64:329m12 0 0:014 765 0 0:018 037m13 m23 0 0:026 496 0 0:032 723

11 22 0 0:620 77 0 0:5974533 0 0:640 43 0 0:629 2612 0 12:914 0 12:75313 23 0 12:639 0 12:30245 0 12:293 0 12:15567 89 0 12:011 0 11:693 11 0 3:8762 106 0 5:1011 106

22 0 3:8762 106 0 5:0102 106

33 0 0:078 913 0 0:098 773 12 0 0:001 723 4 0 0:002 092 3 13 23 0 0:527 51 0 0:660 61 45 0 0:001 719 5 0 0:002 087 2 67 89 0 0:488 35 0 0:611 84

TABLE V. Parameters required for a self-consistent CFLK0

solution in the presence of a strange quark mass. These valuescorrespond to the dispersions shown in Fig. 5. All values are inMeV.

M2s=2 0:50c

Y M2s=2 0:84c


B=3 549:93 48:951 549:93 48:093 6:6057 0:000 00 13:002 8:7 108

8 3:3029 0:000 00 6:5008 4:35 108

f 0 0:539 78 0 1:0238e3 0 0:023 555 0 0:025 582e8 0 0:007 851 6 0 0:008 527 4f3 0 0:035 852 0 0:066 184f8 0 0:011 951 0 0:022 06112 5

18i 0 0:014 623 0 0:026 92413 5

19i 0 0:116 15 0 0:162 6523 89 0 0:044 694 0 0:042 9085

28i 0 0:086 431 0 0:119 035

29i 538i 0 0:044 644 0 0:042 927

539i 0 0:001 146 0 0:047 235

546i 0 0:115 28 0 0:216 22

mug mub 0 0:007 747 6 0 0:009 312 5mdr 0 0:170 93 0 0:215 03mdg mdb 0 0:155 29 0 0:195 36msr 61:637 50:279 85 69:188msg msb 61:637 50:335 85 69:301m12 m

518i 0 0:010 709 0 0:013 162

m13 m519i 0 0:029 548 0 0:053 931

m23 m89 0 0:0134 0 0:017 695m5

29i m538i 0 0:012 525 0 0:016 59

m546i 0 0:007 770 7 0 0:009 337

11 0 0:607 01 0 0:538 9622 88 0 0:304 09 0 0:282 6333 99 0 0:326 68 0 0:324 9412 5

18i 0 9:4718 0 9:639213 5

19i 0 8:614 0 8:010423 89 0 12:282 0 11:72645 5

56i 0 9:0411 0 9:244567 5

47i 0 8:1684 0 7:58475

28i 0 0:303 82 0 0:282 085

29i 538i 0 0:317 99 0 0:316 04

539i 0 0:320 38 0 0:313 21

22 88 0 1:7029 105 0 1:3167 105

33 99 0 0:040 639 0 0:055 178 12 5

18i 0 0:000 660 95 0 0:000 901 86 13 5

19i 0 0:360 95 0 0:463 24 23 89 0 0:506 26 0 0:663 74 45

556i 0 0:000 750 37 0 0:001 152

67 547i 0 0:333 13 0 0:4268

529i

538i 0 0:019 786 0 0:026 852


In Tables II, III, IV, and V, we give the numerical values ofthe parameters for each of the states displayed in Figs. 1, 2,4, and 5, respectively. We only list the nonzero parameters:the other parameters are zero.

094032-18


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kaon condensation in a nambu–jona-lasinio model at high density

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