Thermodynamics of the three-flavor Nambu–Jona-Lasinio model: Chiral symmetry breaking and color superconductivity

PHYSICAL REVIEW C, VOLUME 65, 045204

Thermodynamics of the three-flavor Nambu–Jona-Lasinio model: Chiral symmetry breakingand color superconductivity

F. Gastineau,1 R. Nebauer,1,2 and J. Aichelin11SUBATECH, Laboratoire EMN, IN2P3-CNRS et Universite´ de Nantes, F-44072 Nantes Cedex 03, France

2Institute for Theoretical Physics Universita¨t Rostock, Rostock, Germany~Received 26 January 2001; revised manuscript received 28 June 2001; published 21 March 2002!

Employing an extended three flavor version of the Nambu–Jona-Lasinio model, we discuss in detail thephase diagram of quark matter. The presence of quark as well as of diquark condensates gives rise to a richstructure of the phase diagram. We study in detail the chiral phase transition and the color superconductivity aswell as color flavor locking as a function of the temperature and chemical potentials of the system.

DOI: 10.1103/PhysRevC.65.045204 PACS number~s!: 12.38.Mh, 11.30.Rd, 11.10.Wx

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I. INTRODUCTION

At low temperatures and densities all quarks are confiinto hadrons. In this phase the chiral symmetry is spontaously broken by the quark condensates. Raising the tempture, one expects that the chiral symmetry becomes restand that the quarks are free. This state is called a quark gplasma~QGP!. In the QGP all symmetries of the QCD Lagrangian are restored. For QCD at low temperatures anddensities, one expects a phase where the quarks aresuperconducting state@1–4#. All these different phases definthe phase diagram of QCD@5# in the plane of the temperature and density. This phase diagram is not directly accsible. QCD calculations are only possible on a lattice at zbaryon density. In order to explore the finite-temperatand- density region, one has to rely on effective models. Ttypes of such effective models were advanced to studyhigh-density, low-temperature section. The first typemodel includes weak-coupling QCD calculations, includithe gluon propagators@6#. The second type includes instaton @4,7,8# as well as Nambu–Jona-Lasinio~NJL! models@2,9#. These models show a color superconducting phashigh density and low temperature. In this phase the SUC(3)color symmetry of QCD breaks down to an SUC(2) symme-try. Including a third flavor, another phase occurs: the coflavor-locked state of quark matter@9–11#.

The two-flavor results of the instanton approach are repduced by the model of NJL@12,13# if one includes an appropriate interaction as shown by Schwarzet al. @14#. Thismodel was extended by Langfeld and Rho@15#, who in-cluded all possible interaction channels and discoveredeven richer phase structure of the QCD phase diagramcluding a phase where Lorentz symmetry is spontaneobroken.

The choice of the NJL model is motivated by the fact ththis model displays the same symmetries as QCD, and thcorrectly describes the spontaneous breakdown of chsymmetry in the vacuum and its restoration at high tempeture and density. In addition, the NJL model was successfused to describe the meson spectra and thus is able to rduce the low-temperature, low-density phenomena of Q@16–18#. Thus this is a model which starts out in the diretion opposite to the instanton model, which is a high-dens

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approximation of QCD. Therefore, it is interesting to swhether the NJL model is able to describe the other phathe color superconducting phase, and the color flavor-lockphase observed in the instanton approach. The shortcomof the NJL model is the fact that it does not describe cofinement, or more generally any gauge dynamics at all. Hwe will evaluate the thermodynamical properties of tquarks in the NJL model at finite temperature and densand we will discuss the symmetries of the different phasWe present numerical results for the calculation of the dferent condensates. For our study of the phase diagramuse one specific set of parameters. We treat the three-flversion of the model, including an interaction in the quaantiquark channel, a t’Hooft interaction, and an interactionthe diquark channel. We restrict ourself to the scalar/ or psdoscalar sector of these interactions.

The paper is organized as follows: In Sec. I we wbriefly review the NJL model and present the Lagrangianwill use. In chapter Sec. II we study the quark condensand the restoration of chiral symmetry. In Sec. III we addinteraction in the diquark channel, and present the numerresults for the color superconducting sector. We will havcomplete evaluation of the phase diagram of the NJL moincluding a chiral and superconducting phase transitionfinite temperature and~strange and light quark! density. InSec. IV we present our conclusions.

II. MODEL

The model we use is an extended version of the Nmodel, including an interaction in the diquark channel.fact, the NJL model can be shown to be the simplest lenergy approximation of QCD. It describes the interactbetween two quark currents as a pointlike exchange operturbative gluon@19,20#. Applying an appropriate Fierztransformation to this interaction, the Lagrangian separainto two pieces: a color singlet interaction between a quand an antiquark (L(qq)), and a color antitriplet interactionbetween two quarks (L(qq)). The color singlet channel isattractive in the scalar and pseudoscalar sector, and repuin the vector and pseudovector channel. The Lagrangiathe diquark sector has two parts, both attractive: a flaantisymmetric channel and a flavor symmetric channel. Tformer includes Lorentz scalar and pseudoscalar and ve

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F. GASTINEAU, R. NEBAUER, AND J. AICHELIN PHYSICAL REVIEW C65 045204

interactions, the latter a pseudoscalar interaction only.The coupling constants of these different channels are

lated to each other by the Fierz transformation. Due toextreme simplification of the gluon propagator in this aproximation, the resulting model cannot reproduce the cfinement which is described by the infrared behavior ofgluon propagator.

The resulting Lagrangian has a global axial symmeUA(1), and anextra termLA in the form of the t’Hooftdeterminant is added in order to break explicitly this symmtry. The resulting Lagrangian then has the general form

L5L01L(qq)1L(qq)1LA , ~1!

whereL0 is the free kinetic part.The interaction part of the Lagrangian has a global co

flavor, and chiral symmetry. The chiral symmetry is explitly broken by nonzero current quark masses, and the flasymmetry by a mass difference between the flavors.

The different interaction channels of this Lagrangian grise to a very rich structure of the phase diagram, which wcompletely evaluated in the two-flavor case by Langfeld aRho @15#. Here we will concentrate on the three-flavor caThe evaluation of the complete phase structure in the thflavor case is a quite difficult task, and we will concentrahere on the Lorentz scalar and pseudoscalar interactionthe mesonic channel this interaction is responsible forappearance of a quark condensate and for the spontanbreakdown of the chiral symmetry. In the diquark channegives rise to a diquark condensate which can be identiwith a superconducting gap.

Describing the quark fields by the Dirac spinorsq, theLagrangian we will use here has the form

L5q~ i ]”2m0 f !q1GS(a50

8

@~ qlFaq!21~ qig5lF

aq!2#

1GDIQ(k51

3

(g51

3

@~ qi ,ae i jkeabgqj ,bC !

3~ qi 8,a8C e i 8 j 8kea8b8gqj 8,b8!#

1GDIQ(k51

3

(g51

3

@~ qi ,aig5e i jkeabgqj ,bC !

3~ qi 8,aC ig5e i 8 j 8kea8b8gqj 8,b8!#

1GD@detq~12 ig5!q1detq~11 ig5!q#. ~2!

The first term is the free kinetic part, including the flavodependent current quark massesm0 f which explicitly breakthe chiral symmetry of the Lagrangian. The second parthe scalar or pseudoscalar interaction in the mesonic chanit is diagonal in color. The matriceslF act in the flavorspace. The third part describes the interaction in the scalapseudoscalar diquark channel. The charge conjugated qfields are denoted byqC5CqT, and the color (a,b,g) andflavor (i , j ,k) indices are displayed explicitly. We note thdue to the charge conjugation operation the productqig5qC

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is a Lorentz scalar. This interaction is antisymmetric in flavand color, expressed by the completely antisymmetric tene i jk . Finally, we add the six-point interaction in the form othe t’Hooft determinant, which explicitly breaks theUA(1)symmetry of the Lagrangian. The det runs over the fladegrees of freedom, consequently the flavors becomenected.

The NJL model is nonrenormalizable; thus it is not dfined until a regularization procedure has been specifiedwe are interested in the thermodynamical properties ofmodel, calculated with help of the thermodynamical potetial, we will use a three-dimensional cutoff in momentuspace. This cutoff limits the validity of the model to momenta well below the cutoff.

The model contains six parameters: the current masthe light and strange quarks, the coupling constantsGD andGS , and the momentum cutoffL. These are fixed by physical observables: the pion and kaon mass; the pion deconstant; the mass difference betweenh and h8, once themass of the light quarks was fixed; as well as by the vacuvalue of the condensateqq&1/352230 MeV. The last pa-rameter is the coupling constant in the diquark chanGDIQ . For the mesonic sector we will use the parametersRef. @21#: a current light quark massm0q55.5 MeV, a cur-rent strange quark massm0s5140.7 MeV, a three-dimensional ultraviolet cutoffL5620 MeV, a scalar cou-pling constantGS51.835/L2, and a determinant couplingGD512.36/L5. We cannot fix the diquark sector indepedently, because we do not have enough informations abthe baryon masses in the NJL model. Therefore, we userelation between the coupling constants,GDIQ54GS/2,given by the Fierz-transformation~see Appendix A!. Thisparameter set results in effective vacuum quark massemq5367.6 MeV andms5549.5 MeV and the quark condensates are Š^qq&‹5(2242 MeV)23 and Š^ss&‹5(2258 MeV)23.

We perform our calculations in the mean-field approafor an operator product

r1r2'r1Š^r2&‹1Š^r1&‹r22Š^r1&‹Š^r2&‹, ~3!

whereŠ^r&‹ is the thermodynamical average of the operatand the fluctuations around this mean value are supposebe small. We will apply this approximation to the productsquark fields appearing in the interaction part of the Lagraian.

III. CHIRAL PHASE TRANSITION

We start our study with an investigation of the quarantiquark sector and the chiral phase transition. The diqusector is subject of Sec. IV.

The NJL model displays the right features of the chisymmetry breaking. On the one hand, we have an explicbroken chiral symmetry by the inclusion of a small currequark mass. On the other hand, the model correctly descrthe spontaneous breakdown of chiral symmetry: the etence of a quark condensate, responsible for a high effecquark mass and the existence of massless~or very light, if the

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THERMODYNAMICS OF THE THREE-FLAVOR . . . PHYSICAL REVIEW C 65 045204

chiral symmetry is explicitly broken! Nambu-Goldstonebosons. Lattice QCD calculations show that at a temperaof '170 MeV the chiral symmetry is restored~the quarkcondensates melt at increasing temperature!, a result which isreproduced by the NJL model@16,17,22#. As the region offinite density is not accessible to lattice QCD calculatiothe chiral phase transition at high density is a subjectspeculation. The point and the order of the chiral phase tsition in the temperature-density plane define the phasegram. Here we will present such a phase diagram forthree-flavor NJL model and a specified set of parametThis phase diagram can be viewed as an approximatiothe QCD phase diagram, but we have to take into accothat the NJL model does not describe confinement~we al-ways have a gas of quarks and not a gas of hadrons! and thatthe degrees of freedom are not the same as in QCD~themodel contains no gluons!. Here we will focus on the thermodynamical properties of the quarks described in the csinglet channel of the Lagrangian@Eq. ~2!#; this means thethermodynamical properties of the quark condensatesmasses.

For the study of the thermodynamical properties ofquark-antiquark sector we will evaluate the thermodynampotential in the mean-field approximation. We start out frothe Lagrangian in the mean-field approximation,

L MF5q~ i ]”2M !q22GS~a21b21g2!14GDabg,~4!

whereM f is the effective quark mass~defined via the quarkcondensatesŠ^qq&‹)

M f5m0 f24GSŠ^qfqf&‹12GDŠ^qf 1qf 1

&‹Š^qf 2qf 2

&‹

5m0 f1dmf , with f Þ f 1Þ f 2 ~5!

and the quark condensates are written in a shorthand not

a5Š^uu&‹ b5Š^dd&‹ g5Š^ss&‹. ~6!

The mean-field Hamiltonian

HMF5Ed3x (f5$u,d,s%

@qf ig0]0qf12GS~a21b21g2!24GDabg#

~7!

is transformed into an operatorH in second quantizationusing

qf~x!5 (s56

E d3p

~2p!3

3@ apW ,s, fuf ~p,s!e2 ipx1bpW ,s, f† v f ~p,s!eipx#. ~8!

At the moment, the quark condensates are unknown quaties. In order to evaluate them, we calculate the gracanonical potential

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with m being the chemical potential,b the inverse temperature andN the particle number operator:

N5n~pW ,s, f ,c!2 n~pW ,s, f ,c!, ~10!

where n(pW ,s, f ,c), n(pW ,s, f ,c) are the number operatorfor particles and antiparticles with momentumpW , spin s,flavor f, and color c. These operators are definevia the creation and annihilation operators for partic

n(pW ,s, f ,c)5apW ,s, f ,c†

apW ,s, f ,c and antiparticles n(pW ,s, f ,c)

5bpW ,s, f ,c†

bpW ,s, f ,c . We consider the condensates as paramewith respect to which the potential has to be minimized. Tappearance of the quark condensates spontaneously bthe chiral symmetry of the original Lagrangian.

In second quantization the exponent of the chemicaltential reads as

~HMF2mN!/V5 (s, f ,c

E0

Lp2dp

2p2

3@EpW , f2~EpW , f2m f !n~pW ,s, f ,c!

2~EpW , f1m f ! n~pW ,s, f ,c!#

1@2GS~a21b21g2!24GDabg#,

~11!

where V denotes the volume we have integrated out. T

energyEpW , f5AM f21pW 2 depends on the flavor of the quark

and their momentum, but is independent of the color or spThe evaluation of the grand canonical potential in the mefield approximation gives the result

VMF

V52GS~a21b21g2!

24GDabg2Nc

p2 (f 5$u,d,s%

3E0

L

p2dpH EpW , f11

bln@11e2b(EpW , f2m f )#

11

bln@11e2b(EpW , f1m f )#J . ~12!

It has to be minimized with respect to the quacondensates:

]VMF

]Š^qfqf&‹50. ~13!

We obtain three equations, one for each quark condensa

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where we defined the Fermi functionf (x)5„11exp(2bx)…21. The equations for the quark condensatescoupled@see Eq.~5!#. For three flavors we thus have threcoupled gap equations which have to be solved sconsistently. Their solution, displayed in Appendix B, eables us to calculate the quark condensates and quark mat finite temperature and chemical potential~density!.

We have to take care about the limits of the theory: Tregularization cutoff of the theory implies that the chemicpotential always has to be smaller than this cutoff, and tthe temperature must not be too elevated: The Fermi funcwill be smoothly extended to high momenta, and we havetake into account that all states above the cutoff are ignoby the model.

The condensate is responsible for the spontaneous brdown of chiral symmetry at low densities and temperaturAt high temperature and density the quark condensate d~it becomes very small, or zero in the case of zero currquark masses!, and consequently chiral symmetry is restor~up to the current quark masses!. Hence the quark condensais the order parameter of the chiral phase transition. Tphase transitions we are dealing with are—depending onparameters and of the density respective temperaturefirst or second order, or of the so-called crossover type,we can classify the phase transition by means of this oparameter. The first-order phase transition is specified bdiscontinuity in the order parameter. For the second-orphase transition the order parameter is continuous butanalytical at the point of the phase transition. The third tythe crossover, is not a phase transition in the proper seHere the order parameter does not display a nonanalypoint, but shows a smooth behavior.

In a first step we will consider the chiral phase transitias a function of temperature and chemical potential oflight quarks, the strange quark density is supposed tozero. In Fig 1, left-hand side, we plot the mass of the ligand strange quarks as a function of temperature atbaryon density for the parameters presented above.

At zero density we observe a smooth crossover ofchiral phase transition as a function of temperature: attemperature the chiral symmetry is spontaneously brok

FIG. 1. The mass of strange and light quarks as a function oftemperature~left-hand side! and as a function of the light quarchemical potential~right-hand side!.

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with rising temperature the quark condensate melts awaythe quark masses approach the current mass, at least folight quarks. For the strange quarks we observe a msmoother transition, and at the highest temperature wetreat in the framework of the NJL model~approximately230 MeV) their mass is still higher than their current maWe observe this smooth crossover only for the special cof three nonzero current quark masses.

At zero temperature, for our parameter set we observfirst order phase transition. As a function of the chemipotential the light quark mass drops suddenly to a vaclose to the current quark mass. The strange quarks chtheir mass slightly due to the coupling between the flavoFor higher values of the chemical potential the strange qumass is stable. The light quark condensate is too small fchange of the strange quark mass. Only a rise of the chempotential of the strange quarks can drop the strange qumass further, as will be discussed in the last part of tsection, where we present the extension to strange qmatter.

A first-order phase transition is characterized by the extence of metastable phases, the equivalent of, for examoversaturated vapor. These metastable phases are a soof the gap equation, but their thermodynamical potentialarger than for the stable phase. We show this in detail in F2. On the top we display the quark mass~light and strange!,and on the bottom the density of light quarks and the thmodynamical potential. The stable phases which minimthe thermodynamical potential are shown as dark lines,the metastable phases as light lines.

For the mass of the light quarks we observe the transifrom the stable phase at high chemical potential to a swhose mass is larger than its chemical potential; this meto zero density. Increasing the chemical potential yieldfirst-order phase transition, i.e., the mass of the quarks d

e

FIG. 2. Detailed representation of the first-order phase transias a function of the light quark potential at zero temperature. Ontop: the light ~left-hand side! and strange~right-hand side! quarkmasses. Bottom: the density~left-hand side! and the thermodynami-cal potential~right-hand side!. The light lines represent the metastable region, the dark lines the stable region~minimization of thethermodynamical potential!.

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suddenly. This abrupt change in the quark masses givesto a jump in the density—for a constant chemical potensuddenly many more states become accessible. This imat the same time that certain densities do not exist. Incase the normal nuclear matter density is just in this regand there are good explanations for this fact@23,3#. For theinterpretation one has to remember that we are discussiquark gas without confinement. Here, for nuclear mattenormal density, one has to consider a phase which contdense droplets of quarks in which chiral symmetry isstored, surrounded by the vacuum or a very diluted quark~which should be confined in QCD!. The size of these droplets is not given by the theory, but it is not farfetchedidentify these objects with the nucleons.

This for our set of parameters we observe thus a fiorder phase transition as a function of the chemical potenat zero temperature, and a crossover as a function of tperature at zero density. Extrapolating now to the planefinite temperature and chemical potential, there must bpoint where both kinds of phase transitions join; thecalled tricritical point. In Fig. 3 we show this phase diagraat finite temperatures and chemical potentials~on the left-hand side! on the right-hand side they are shown as a fution of the density!. Dark lines display the transition by fromthe stable state~or the transition line for the crossover!, andlight lines the metastable phases. The tricritical point iscated at a temperatureT566 MeV and a chemical potentiaof mq5321 MeV which corresponds to a density ofrq51.88r0.

The location of the tricritical point depends strongly othe choice of the cutoff and the coupling constant@24#.

In Fig. 4 we plot the quark masses@light, ~left-hand side!,and strange~right-hand side!# as a function of the chemicapotential of light and strange quarks at zero temperature.can see the influence of the coupling between the flavorsalready discussed for the light quark chemical potential. Tstrange quark mass drops suddenly at high chemical potials of the strange quarksms and low chemical potentials fothe light quarksmq . Once the chiral phase transition for thlight quarks has taken place~at high values ofmq!, thestrange quark mass shows a crossover transition for highms .For high values ofmq and ms , both quark masses havevalue close to their current quark mass. With increasing te

FIG. 3. Phase diagram for the mass of the light quarks~chiralphase transition! as a function of temperature and the light quachemical potential~left-hand side! and density~right-hand side!.The dark lines represent the transition, the light lines the limitsthe metastable phases in case of a first order phase transition.

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perature, phase transitions will take place at lower valuesthe chemical potentials. This is very pronounced for the ligquarks~see Fig. 1!, and less so for the strange quarks whichange their mass quite slowly with temperature due tohigh current quark mass~compare Fig. 1!.

IV. COLOR SUPERCONDUCTIVITY

In this section we will study the diquark channel. We wsee that quarks which have opposite spins and momentadense in the scalar channel into diquarks. This resemsuperconductivity@25,26#. Here we have, in addition, a complex structure in color and flavor space. In classical supconductivity the condensation occurs close to the Fermi sface. In our case we have to take into account that quawith different flavors may have different Fermi surfaces. Bcause the coupling between the quarks is quite small,condensation will only occur if the Fermi momenta of thtwo quarks are quite close to each other.

In order to calculate the properties of the NJL modelthe superconducting sector, we will apply the generalizthermodynamical approach of the Hartree-Bogolyubtheory to quark matter~see, for example, Ref.@27#! describedby Lagrangian~2!.

Lagrangian~2! in the mean-field approximation, includinthe diquark sector, reads as follows:

L MF5( q~ i ]”2M !q22GS~a21b21g2!

14GDabg1qia

Dkg

2qj b

C

1qiaC Dkg†

2qj b2(

k,g

uDkgu2

4GDIQ. ~15!

Greek indices denote the colors, latin indices the flavorsThe diquark condensate is defined by

Dkg52GDIQig5eabge i jkŠ^qi 8a8ig5e i 8 j 8kea8b8gqj 8b8

C &‹

5 ig5eabge i jkDkg. ~16!

This diquark condensate occurs for all three colors simuneously. We note that as in classical superconductivitybaryon~or particle! number is not conserved. Hence the eletromagneticUem(1) symmetry is spontaneously broken, a

f

FIG. 4. Light ~left-hand side! and strange~right-hand side!quark masses as a function of the chemical potential of light (mq)and strange (ms) quarks at zero temperature.

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Goldstone bosons appear in the form of Cooper pairs.diquark condensate carries a color and a flavor index. Fgiven flavor and color the condensate is completely antismetric in the other two flavors and colors. The condensDsr is created, for example, by green and blue up and doquarks.

The diquark condensate is completely antisymmetricthe color degrees of freedom, a property which is only shaby three of the eight Gell Mann matrices which generateSUC(3). Hence a finite diquark condensate breaks downSUC(3) color symmetry to a SUC(2) symmetry if the massof the strange quark is heavy. The same is true for the flasector if the three flavors are degenerated in mass. Forflavors only the Lagrangian is invariant with respect tochiral transformation. If the diquark condensates coexistall three flavors, the chiral symmetry is spontaneously bken.

Due to the product of two antisymmetric tensors, the symetry is even more reduced if all three quark flavors formdiquark condensate. In order to see this, we first assumeall three colors~for one flavor! are equivalent. Than we caassume without loss of generality thatk5g in Eq. ~16!, andwrite the tensor product as

e i j I eabI5 (i , j ,a,b

~d i ,ad j ,b2d i ,bd j ,a!. ~17!

We see that in this case the rotations in color and flaspace are no longer independent but locked. Hencequarks are in a color-flavor locked phase if all three quflavors participate at the formation of the diquark condesates.d i ,a is the unit matrix of SU(3)C3F in which the ma-trices contain the three flavors as columns and the threeors as rows. The Lagrangian is therefore invariant undeSU(3)C3F transformation, and consequently the SU(color and flavor symmetries are reduced to an SU(3)C3Fsymmetry. For more consequences of the appearance ocondensate for the symmetries, we refer to the literature@11#.Here we will focus on a numerical evaluation of the sizethe condensates and the phase transitions at finite temture and density.

A. Thermodynamics

As before in the case of the chiral phase transition,will evaluate all condensates and the phase diagram byevaluation of the thermodynamical potential. We startwriting the Lagrangian in a more symmetric form, followinNambu, who developed this formalism for the classicalperconductivity@28#. For this purpose we rewrite the Lagrangian as a sum of the original Lagrangian and its chaconjugate:

LNambu5L1L C. ~18!

Then the Lagrangian can be presented as a matrix,

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where we suppressed the indices for convenience and dethe term

Lcond522GS~a21b21g2!14GDabguDkgu2

4GDIQ. ~20!

In order to calculate the thermodynamical potential in tnotation,

V52bTr@ ln exp„2b~HNambu2mN2mNC!…#

we need the particle number operator and its charge cogate,

N5 (pW ,s, f ,c

@ apW ,s†

apW ,s2bpW ,s†

bpW ,s#,

NC52 (pW ,s, f ,c

@ apW ,sapW ,s†

2bpW ,sbpW ,s†

# ~21!

where we suppressed the explicit dependence of the optors on flavor and color degrees of freedom.

When calculating the Hamiltonian in the mean field aproximation, one can see—neglecting a small contributionterms likea†b in the case of different quark masses—thais possible to separateH2mN into two parts, one for thequarks~operatorsa and a†) and another for the antiquark~operatorsb et b†):

H2mN5~H2mN! a1~H2mN! b . ~22!

These two parts yield the explicit expressions

~H2mN! a5 (pW ,s, f ,c

@ apW ,s†

a2pW ,2s#

3F EpW , f2m f 2Dkg†N~p!

DkgN~p! 2EpW , f1m fGF apW ,s

a2pW ,2s† G1Hcond

~23!

and

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olve

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~H2mN! b5 (pW ,s, f ,c

@ bpW ,s†

b2pW ,2s#

3F EpW , f1m f 2Dkg†N~p!

DkgN~p! 2EpW , f2m fGF bpW ,s

b2pW ,2s† G1Hcond.

~24!

We denoted the expression2V* Lcond by Hcond, and usedhere the discrete summation over the momenta. The expsions have a defined structure in flavor and color, the dianal terms are diagonal in flavor and color, the off-diagoterms (D) are antisymmetric in color and flavor and

N~p! f 1 , f 25S 11

p2

~Ef 11mf 1

!~Ef 21mf 2

! D3A~Ef 1

1mf 1!~Ef 2

1mf 2!

4mf 1mf 2

Amf 1mf 2

Ef 1Ef 2

.

~25!

This normalization factor is due to the fact that we deal wproducts of spinors for different species in the off-diagoterms, of courseN(p)51, whenf 15 f 2. The explicit form ofthis matrix including all flavor and color indices is displayein Appendix C.

In order to calculate the thermodynamical potential,have to diagonalize these expressions. This has to be donmeans of a Bogolyubov transformation, which determinthe energies of the quasiparticles and the corresponquasi-particle operators. From the discussion of the symtry of the diquark condensate, we expect two quarks offerent flavor and color to form a diquark condensate wherone quark of the third flavor is not involved in forming thcondensate. This has to be seen in the quasiparticle enand is confirmed if we explicitly evaluate the quasiparticenergies as the eigenvalues of the matrices. The diagonaoperators corresponding to (H2mN) a can be expressed ithe form

~H2mN! aD5(

i 51

5

gi~Ea,i aD,i† aD,i1Ea,i8 aD,i aD,i

† !,

where i runs over the flavors anda over the colors.aD ia

andaD ia† are annihilation and creation operators for the qu

particles:

i Ea ,i gi

1 6ADqq2 1E22 3

2 612 (Z1E22Es

2) 23 6

12 (Z2E21Es

2) 24 AY2X 15 AY1X 1

where

E25E2m, ~26!

04520

s-o-l

l

ebysnge-f-s

gy,

ed

i

Es25Es2ms , ~27!

Z5A4Dqs2 N2~p!1~E21Es

2!2, ~28!

Y5 12 ~Dqq

2 14Dqs2 N2~p!1E221Es

22!, ~29!

X25 14 @Dqq

4 1„8Dqs2 N2~p!1~E21Es

2!2…~E22Es

2!2

12Dqq2„4Dqs

2 N2~p!1~E21Es2!~E22Es

2!…#, ~30!

andgi is the degeneracy. For the calculation of the thermdynamical potential it is not necessary to know the exform of the Bogolyubov transformation which relates tquasiparticle operatorsaD with the original quark operatorsa. The quasiparticles are still fermions, and that is all infomation we need in order to evaluate the sum over the ocpied states. It is just only necessary to assign the right egies to the operators. We evaluate the thermodynampotential for the case of two degenerated light quarks:

V

V5

V0

V2

2

bE0

L dpW

~2p!3

3$6 ln@11exp~2bE12!#14 ln@11exp~2bE22!#

14 ln@11exp~2bE32!#12 ln@11exp~2bE42!#

12 ln@11exp~2bE52!#13bE12

12bE2212bE321bE421bE52%, ~31!

with

V0

V54GS~a21b21g2!1

2uDqsu21uDqqu2

2GDIQ. ~32!

This thermodynamical potential contains the~quark anddiquark! condensates as parameters. In order to evaluthem, we have to minimize

]V

]Š^qfqf&‹50,

]V

]D f 1f 2

50. ~33!

This minimization yields the gap equations for the quacondensates. For the SU~3! case the derivation is given inAppendix D. These equations are coupled, we have to sthem selfconsistently. The resulting^qq& condensates maybe found in Appendix E.

B. Results at finite temperature and density

For this part we decide to take parameters in Ref.@29#:m0,q55.96 MeV, L5592.7 MeV, GS56.92 GeV22,GDIQ /GS53./4. andm0,s5130.7 MeV. We use the relationbetween the coupling constants, (GDIQ53GS/4), given bythe Fierz transformation~see Appendix A!, close to~0.73!@29#.

The condensates~masses! at zero temperature as a funtion of the chemical potentialmq5ms are displayed in Fig. 5.On the left-hand side of this figure we show the light a

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strange quark mass, and on the right-hand side the diqcondensates. First we have to note that quark and diqcondensates compete with each other as they are formethe same quarks. Temperature and density determine wcondensate dominates.

When the chiral phase transition occurs~the quark con-densate disappears!, we observe for the light quarks that thsuperconducting phase transition takes place, and we hadiquark condensate. As the two transitions are related,are of the same order. The same scenario repeats itself fostrange diquark condensate at a higher chemical potentiaFig. 5 we display only the solution which is the global minmum of the thermodynamical potential.

At a quite low chemical potential~the light quarks have avery small mass, the strange quarks are heavy! we have onlythe light diquark condensate; the diquark condensate incing strange quarks is almost zero, as the strange quarks sa strong quark condensate. Only when the strange quarkdensate drops, and the mass of the strange quarks approits current mass, does the strange diquark condensate apHere we have the coexistence of light and strange diqucondensates, this is the regime where the chiral symmetagain broken, and the color and flavor are locked. This hpens at a quite high chemical potential; the decreasingquark condensate for even higher chemical potentials icates that we reach the limit of the model: we are too closthe cutoff. The phase transitions concerning the straquarks are quite close to the limits of the models if we spose the current mass of the strange quark to be aro140 MeV. We note that due to the relatively small differenbetween the quark masses, both diquark condensatesapproximately the same value: for the maximum we obtDqq'Dqs'(120 MeV)3. At zero temperature the chiraphase transition~where the quark condensates disappear! andthe superconducting phase transition~where the diquark condensates appear! are strongly related in of our model. Thchanges at higher temperatures. There the diquark consatesDqq extend to smaller values of the chemical potentwhereas we need higher densities in order to form aDqsdiquark condensate. In addition, the diquark condensatecome smaller with increasing temperature. This is shownFig. 6, where we plot the diquark condensates as a funcof the temperature and chemical potential. For a giv

FIG. 5. The light ~dark line! and strange~light line! quarkmasses and the diquark condensatesDqq ~dark line! andDqs ~lightline! as a function of the chemical potentialmq5ms at zero tem-perature.

04520

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n

chemical potential we observe-as in the classisuperconductivity-a second order phase transition as a ftion of the temperature.

In a next step we consider the diquark condensates inmq2ms plane. As already mentioned, we expect the formtion of a diquark condensate only if there are quarks wsimilar Fermi momenta, independent of their mass. BecaGD is zero the disappearance of the quark condensates^ss&and ^qq& does not depend on the chemical potential ofother species. There is one exception the creation ofstrange diquark condensates lowers the strange quarkdensate and increases the light quark mass. In Fig. 7 wethe strength of the diquark condensates. Because bothquarks have the same chemical potential, a diquark condsateDqq between the two different flavors occurs whenevthe light quark mass is small.

The strange diquark condensate exists only in a bwhere the chemical potentials of the light and strange quaare approximately equal. The slight deformation of this bais due to the different current quark masses. The width ofband is determined by the coupling strength: if the couplin the diquark sector is strong, the quarks can bind and foa condensate even if their chemical potentials are quiteferent. For a small coupling strength, the chemical potentof the two quarks have to be~approximately, in case of dif-

FIG. 6. Strength of the diquark condensates as a function ofchemical potentialmq5ms and the temperature. As color levels wshow the strength of the condensatesDqq ~left-hand side! andDqs

~middle!, and superimpose both as a contour plot~right-hand side!.

FIG. 7. As a function of the chemical potentialsmq andms weshow the quark masses on the upper row~light quarks on the left-hand side, and strange quarks right-hand side! and the diquark con-densates in the lower row (Dqq on the left-hand side andDqs on theright-hand side!.

4-8

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ferent quark masses! equal in order to form a diquark condensate.

Now we should study the feedback of the formation of tdiquark condensates on the quark condensates~or the mass!.In Fig. 8 we display the masses of the light and stranquarks as a function ofmq andms . Dqq appears at the chiraphase transition, when the light quark condensate disappand it is formed by the free light quarks. This behavioralmost independent ofms . Only if Dqs becomes finite doesthe lack of quarks for the quark condensate increaseslight quark mass. The behavior ofDqs is generic: When thediquark condensateDsq is finite it takes quarks from thestrange quark condensate, lowering the mass of the strquark.

V. CONCLUSIONS

In conclusion, we presented the phase diagram ofSU(3) flavor NJL model extended to the diquark sector foset of parameters which reproduces meson masses andpling constants. We found a rich structure of condensatesregions where no condensate exists. The temperaturedensity dependence of quark and diquark condensatescalculated in a mean-field approach by minimizing the thmodynamical potential.

The order of chiral phase transition depends on the vaof T andm where the phase transition occurs. At zero teperature the phase transition is first order, and at zero chcal potential we observe a crossover~due to finite currentquark masses!. Therefore there exists a tricritical point. Nomal nuclear matter density exists only as a mixed phasedense quark phase~where chiral symmetry is partially restored! and a very diluted quark gas or the vacuum~wherechiral symmetry is spontaneously broken!. Finally we ex-tended the chiral phase transition to the plane of finstrange quark density, relevant for the discussion of thequark condensates.

Following the idea that the NJL model can be consideas an approximation of the QCD Lagrangian we extendNJL model by including an interaction in the diquark chanel. We find that this interaction gives rise to a diquark codensation which is responsible for the formation of a supconducting gap. This condensation occurs at low temperaand high density. If this gap is formed by two quarksdifferent flavors, their momenta have to be similar in ab

FIG. 8. As a function of the chemical potentialsmq andms weshow the quark masses~light quarks left-hand side, strange quaron the right-hand side! to point out the effect of the diquark condensate on the quark condensate atT51 MeV.

04520

e

rs,

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ei-

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lute value but opposite in direction and therefore their checal potential may differ. In SUF(3) flavor this condensatebreaks the SUF(3)3SUC(3) flavor down to a SUC3F(3)flavor, a phenomenon already observed in phase diagrbased on instanton Lagrangians, and dubbed ‘‘color-flalocking.’’ We can conclude that two quite differently motvated phenomenological approaches to the QCD Lagranprovide a very similar phase structures.

Diquark condensates do not exist at temperatures wepect to obtain in relativistic heavy-ion collisions. In neutrostars, which have a high density and a very low temperatthey could be of relevance.

Note added in proof.We would also like to thank M.Buballa and M. Oertel for making us aware of a small cotribution of terms likea†b to the quasiparticle energies icase of different quark masses.

ACKNOWLEDGMENTS

This work was supported by the Landesgraduiertforderung Mecklenburg-Vorpommern. One of us~J.A.! ac-knowledges an interesting discussion with K. Rajagopal.thank A. W. Steiner and M. Prakash for having pointed oan error in a formula of a previous version of this paper.

APPENDIX A: FIERZ TRANSFORMATION

Following Ebert@21# we have the following relations incolor and flavor space:

d i j dkl51

3d ikd l j 1

1

2 (a51

8

l ika l l j ~qq channel!, ~A1!

d i j dkl51

2 (a50

8

l i lalk j

a ~qq channel!, ~A2!

(a51

8

l i ja lkl

a 516

9d i l dk j2

1

3 (a51

8

l i lalk j ~qq channel!,

~A3!

(a51

8

l i ja lkl

a 52

3 (a50,1,3,4,6,8

l ika l l j

a 24

3

3 (a52,5,7

l ika l l j

a ~qq channel! ~A4!

L52g(a51

8

~ cgmlCa c!2. ~A5!

This leads to the following relations between the differecoupling constants:

qq channel GSCA58

9g, ~A6!

qq channel GDIQ52

3g. ~A7!

4-9


APPENDIX B: Šqq‹ CONDENSATES

1. Light quark condensate

Š^qq&‹521

2E p2dp

2p2

Mq

EqH 3

E2

E21

1 f 8~E22 !S 1

Z S 4Dqs2 N~p!

]N

]Š^qq&‹U

q

1E21Es2D 11D

1 f 8~E23 !S 1

Z S 4Dqs2 N~p!

]N

]Š^qq&‹U

q

1E21Es2D 21D 1

f 8~E24 !

2E24 F4Dqs

2 N~p!]N

]Š^qq&‹U

q

1E2

21

4X S F ~E22Es2!„8Dqs

2 N21~E21Es2!2

…1~E22Es2!2S 8Dqs

2 N~p!]N

]Š^qq&‹U

q

1~E21Es2!D

1Dqq2 S 8Dqs

2 N~p!]N

]Š^qq&‹U

q

12E2D G D G1f 8~E2

5 !

2E25 F4Dqs

2 N~p!]N

]Š^qq&‹U

q

1E2

11

4X S F ~E22Es2!„8Dqs

2 N21~E21Es2!2

…1~E22Es2!2

3S 8Dqs2 N~p!

]N

]Š^qq&‹U

q

1~E21Es2!D 1Dqq

2 S 8Dqs2 N~p! U ]N

]Š^qq&‹U

q

12E2D G D G1~2 !⇒~1 !J . ~B1!

2. Strange quark condensate

Š^ss&‹52E p2dp

2p2

Ms

EsH E2

E21

1 f 8~E22 !S 1

Z S 4Dqs2 N~p!

]N

]Š^ss&‹U

q

1E21Es2D 21D

1 f 8~E23 !S 1

Z S 4Dqs2 N~p!

]N

]Š^ss&‹U

q

1E21Es2D 11D 1

f 8~E24 !

2E24 F4Dqs

2 N~p!]N

]Š^ ss&‹U

s

1Es22

1

4X S F2~E22Es2!„8Dqs

2 N21~E21Es2!2

…1~E22Es2!2S 8Dqs

2 N~p!]N

]Š^ ss&‹U

s

1~E21Es2!D

1Dqq2 S 8Dqs

2 N~p!]N

]Š^ss&‹U

s

22Es2D G D G ~B2!

1f 8~E2

5 !

2E25 F4Dqs

2 N~p!]N

]Š^ ss&‹U

s

1Es21

1

4X S F2~E22Es2!~8Dqs

2 N21~E21Es2!2!1~E22Es

2!2

3S 8Dqs2 N~p!

]N

]Š^ss&‹U

s

1~E21Es2!D

1Dqq2 S 8Dqs

2 N~p!]N

]Š^ ss&‹U

s

22Es2D G D G1~2 !⇒~1 !]. ~B3!

045204-10

s


APPENDIX C: MATRIX

The total matrix can be separated into four submatrice

a a†

a† A Ca -C† B

These submatrices are given by

uR uG uB dR dG dB sR sG ssB

A5

uR† E2 0 0 0 0 0 0 0 0

uG† 0 E2 0 0 0 0 0 0 0

uB† 0 0 E2 0 0 0 0 0 0

dR† 0 0 0 E2 0 0 0 0 0

dG† 0 0 0 0 E2 0 0 0 0

dB† 0 0 0 0 0 E2 0 0 0

sR† 0 0 0 0 0 0 Es

2 0 0

sG† 0 0 0 0 0 0 0 Es

2 0

sB† 0 0 0 0 0 0 0 0 Es

2

uR† uG

† uB† dR

† dG† dB

† sR† sG

† sB†

B5

uR E1 0 0 0 0 0 0 0 0

uG 0 E1 0 0 0 0 0 0 0

uB 0 0 E1 0 0 0 0 0 0

dR 0 0 0 E1 0 0 0 0 0

dG 0 0 0 0 E1 0 0 0 0

dB 0 0 0 0 0 E1 0 0 0

sR 0 0 0 0 0 0 Es1 0 0

sG 0 0 0 0 0 0 0 Es1 0

sB 0 0 0 0 0 0 0 0 Es1

uR† uG

† uB† dR

† dG† dB

† sR† sG

† sB†

C5

uR† 0 0 0 0 Dqq 0 0 0 Dqs

uG† 0 0 0 2Dqq 0 0 0 0 0

uB† 0 0 0 0 0 0 2Dqs 0 0

dR† 0 2Dqq 0 0 0 0 0 0 0

dG† Dqq 0 0 0 0 0 0 0 Dqs

dB† 0 0 0 0 0 0 0 2Dqs 0

sR† 0 0 2Dqs 0 0 0 0 0 0

sG† 0 0 0 0 0 2Dqs 0 0 0

sB† Dqs 0 0 0 Dqs 0 0 0 0

04520

:

APPENDIX D: THERMODYNAMICAL POTENTIAL

1. Formal derivation of the thermodynamical potential

]V

]a5

]V0

]a2

1

bE d3pW

~2p!3

3F26b]E2

1

]af ~E2

1 !24b]E2

2

]af ~E2

2 !24b

3]E2

3

]af ~E2

3 !22b]E2

4

]af ~E2

4 !22b]E2

5

]af ~E2

5 !

13b]E2

1

]a12b

]E22

]a12b

]E23

]a1b

]E24

]a

1b]E2

5

]a~2 !→~1 !G , ~D1!

]V

]a5

]V0

]a22E d3pW

~2p!3 F3]E2

1

]af 8~E2

1 !12]E2

2

]af 8~E2

2 !

12]E2

3

]af 8~E2

3 !1]E2

4

]af 8~E2

4 !1]E2

5

]af 8~E2

5 !G ,~D2!

where f 8(x)5122 f (x)

2. N„p… derivatives

N~p!5S11p2

~E1m!~Es1ms!DA~E1m!~Es1ms!

4mmsAmms

EEs.

~D3!

Then we write

U5S 11p2

~E1m!~Es1ms!D , ~D4!

V5Amms

EEs, ~D5!

W5A~E1m!~Es1ms!

4mms, ~D6!

]W

]a5

1

2W

~Eq1mq!~Es1ms!

4mq2ms

2 F]E2

]a S ms

mq~mq2Eq! D

~D7!

1]Es

2

]a S mq

ms~ms2Es! D G , ~D8!

]W

]a5

]Eq2

]a

]W

]aUq1

]Es2

]a

]W

]aU

s

, ~D9!

4-11


]V

]a5

1

2V F ]E2

]a S ms

Esmq2

msmq

Eq2Es

D1

]Es2

]a S mq

Eqms2

msmq

Es2Eq

D G]V

]a5

]Eq2

]a

]V

]aUq1

]Es2

]a

]V

]aU

s

, ~D10!

]U

]a5

2p2

~Eq1mq!~Es1ms!

3F]E2

]a

1

mq1

]Es2

]a

1

msG

]U

]a5

]U

]a Uq1]U

]aUs

. ~D11!

So,

]N~p!

]a Uq5]U

]aUq

VW1U]V

]a UqW1UV]W

]a Uq

, ~D12!

]N~p!

]a Us5]U

]aUs

VW1U]V

]aUsW1UV]W

]a Us

. ~D13!

v.

ta

,

04520

APPENDIX E: Šqq‹ CONDENSATES

1. Light diquark condensates

Dqq5Gdiq

p2 E p2dpF3Dqq

E21

f 8~E21 !1

f 8~E24 !

2E24

3S Dqq21

2X„Dqq

3 1Dqq~4Dqs2 ~p!1E222Es

22!…D1

f8~E52!

2E25 SDqq1

1

2X„Dqq

3 1Dqq

3~4Dqs2 ~p!1E222Es

22!…D1~2 !⇒~1 !G . ~E1!

2. Strange diquark condensates

Dqs5Gdiq

2p2E p2dpF4DqsN2~p!

Zf 8~E2

2 !

14DqsN

2~p!

Zf 8~E2

3 !1f 8~E2

4 !

2E24 S 4DqsN

2~p!

24

2X„~E22Es

2!2„DqsN

2~p!…1Dqq2 DqsN

2~p!…D1

f 8~E25 !

2E25 S 4DqsN

2~p!14

2X„~E22Es

2!2„DqsN

2~p!…

1Dqq2 DqsN

2~p!…D1~2 !⇒~1 !G . ~E2!

@1# B. Barrois, Nucl. Phys.B129, 390~1977!; S. Frautschi~unpub-lished!.

@2# D. Bailin and A. Love, Phys. Rep.107, 325 ~1984!.@3# M. Alford et al., Phys. Lett. B422, 247 ~1998!.@4# R. Rapp, T. Scha¨fer, E. Shuryak, and M. Velkovsky, Phys. Re

Lett. 81, 53 ~1998!.@5# K. Rajagopal, Nucl. Phys.A661, 150c ~1999!; M. Alford,

Annu. Rev. Nucl. Part. Sci.~to be published!, hep-ph/0011333;K. Rajagopal and F. Wilczek, hep-ph/0011333;At the Frontierof Particle Physics / Handbook of QCD, edited by M. Shifman~World Scientific, Singapore, in press!; T. Schafer and E.Shuryak, Proceedings of the ECT Workshop on Neutron SInteriors, Trento, Italy, 2000~in press!, nucl-th/0010049.

@6# D. Son, Phys. Rev. D59, 094019~1999!; T. Schafer and F.Wilczek, ibid. 60, 114033~1999!; R. Pisarski, and D. Rischkeibid. 61, 074017~2000!.

@7# J. Berges and K. Rajagopal, Nucl. Phys.B538, 215 ~1999!.@8# G. Carter and D. Diakonov, Phys. Rev. D60, 016004~1999!.@9# M. Alford et al., Nucl. Phys.B537, 443 ~1999!.

@10# K. Rajagopal, Nucl. Phys.A642, 26 ~1998!.@11# T. Schafer and F. Wilczek, Phys. Rev. Lett.82, 3956~1999!.

r

@12# Y. Nambu and G. Jona-Lasinio, Phys. Rev.122, 345 ~1961!.@13# Y. Nambu and G. Jona-Lasinio, Phys. Rev.124, 246 ~1961!.@14# T.M. Schwarzet al., Phys. Rev. C60, 055205~1999!.@15# K. Langfeld and M. Rho, Nucl. Phys.A660, 475 ~1999!.@16# T. Hatsuda and T. Kunihiro, Phys. Rep.247, 221 ~1994!.@17# S.P. Klevansky, Rev. Mod. Phys.64, 649 ~1992!.@18# G. Ripka,Quarks Bound by Chiral Fields~Clarendon, Oxford,

1997!.@19# A. Dhar and S.R. Wadia, Phys. Rev. Lett.52, 959 ~1984!.@20# D. Ebertet al., Prog. Part. Nucl. Phys.33, 1 ~1994!.@21# P. Rehberget al., Phys. Rev. C53, 410 ~1996!.@22# M. Asakawa and K. Yazaki, Nucl. Phys.A504, 668 ~1989!.@23# M. Bulballa, Nucl. Phys.A611, 393 ~1996!.@24# P.-B. Gossiaux~private communication!.@25# J.R. Schrieffer,Theory of Superconductivity~Benjamin, New

York, 1964!.@26# A.F. Fetter and J.D. Walecka,Quantum Theory of Many-

Particle Systems~McGraw-Hill, New York, 1971!.@27# M. Iwasaki and T. Iwado, Prog. Theor. Phys.94, 1073~1995!.@28# Y. Nambu, Phys. Rev.117, 648 ~1960!.@29# W. Bentz and A.W. Thomas, nucl-th/0105022.

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Documents

Thermodynamics of the three-flavor Nambu–Jona-Lasinio model: Chiral symmetry breaking and color superconductivity