The Phase diagram of three-flavor quark matter under compact star constraints

arXiv:hep-ph/0503194 v3 23 Mar 2005

D. Blas hke,

∗

S. Fredriksson,

†

H. Grigorian,

‡

A.M. Özta³,

§

and F. Sandin

¶

1

Gesells haft für S hwerionenfors hung mbH (GSI), D-64291 Darmstadt, Germany, and

Bogoliubov Laboratory for Theoreti al Physi s, JINR Dubna, 141980 Dubna, Russia

2

Department of Physi s, Luleå University of Te hnology, SE-97187 Luleå, Sweden

3

Institut für Physik, Universität Rosto k, D-18051 Rosto k, Germany, and

Department of Physi s, Yerevan State University,375025 Yerevan, Armenia

4

Department of Physi s, Ha ettepe University, TR-06532 Ankara, Turkey

Thephase diagram ofthree-avor quarkmatterunder ompa tstar onstraintsis investigated

within a NambuJona-Lasinio model. Lo al olor and ele tri harge neutrality is imposed for

β

-equilibratedsuper ondu tingquarkmatter. The onstituentquarkmassesandthediquark on- densatesaredeterminedself onsistentlyintheplaneoftemperatureandquark hemi alpotential.

Both strong and intermediatediquark oupling strengths are onsidered. We show that inboth

ases,gaplesssuper ondu tingphasesdonoto urattemperaturesrelevantfor ompa tstarevolu-

tion,i.e.,below

T ∼ 50

^{MeV.Thestabilityandstru tureofisothermalquarkstar} ongurationsare evaluated. Forintermediate oupling, quarkstarsare omposedof amixedphaseofnormal(NQ)

andtwo-avorsuper ondu ting(2SC)quarkmatteruptoamaximummassof

1.21 M

⊙^{. Athigher}

entraldensities, aphasetransition tothethree-avor oloravorlo ked(CFL)phaseo ursand

the ongurationsbe omeunstable. Forthestrongdiquark ouplingwendstablestarsinthe2SC

phase,withmassesupto

1.326 M

⊙^{. Ase ondfamilyofmore ompa t} ongurations(twins)with aCFLquarkmatter oreanda2SCshellisalsofoundtobestable. Thetwinshavemassesinthe

range

1.301...1.326 M

⊙^{.We onsideralsohotisothermal} ongurationsattemperature

T = 40

^MeV.

Whenthehotmaximummass onguration oolsdown,duetoemissionofphotonsandneutrinos,

amassdefe tof

0.1 M

⊙^{o ursandtwonalstate} ongurationsarepossible.

PACSnumbers: 12.38.Mh,24.85.+p,26.60.+ ,97.60.-s

I. INTRODUCTION

Theoreti al investigations ofthe QCDphase diagram

athighdensitieshavere entlygainedmomentum dueto

results of non-perturbative low-energy QCD models [1,

2, 3℄ of olor super ondu tivity in quark matter [4, 5℄.

These models predi t that the diquark pairing onden-

satesareoftheorderof100MeVand aremarkablyri h

phasestru turehasbeenidentied[6,7,8,9℄. Themain

motivation for studying the low-temperature domain of

theQCDphasediagram isitspossiblerelevan e forthe

physi sof ompa t stars[10, 11, 12℄. Observableee ts

of olor super ondu ting phasesin ompa tstars is ex-

pe ted,e.g.,inthe oolingbehaviour[13,14,15,16,17℄,

magneti eldevolution[18,19,20,21℄,andinburst-type

phenomena[22,23,24,25℄.

The most prominent olor super ondu ting phases

withlargediquarkpairinggapsarethetwo-avors alar

diquark ondensate (2SC) and the olor-avor lo king

(CFL) ondensate. The latter requires approximate

SU(3) avor symmetry and o urs therefore only at

rather large quark hemi al potentials,

µ

q

> 430 − 500

MeV, of theorderof thedynami ally generatedstrange

∗

Ele troni address: Blas hketheory.gsi.de

†

Ele troni address: Sverker.Fredrikssonltu.se

‡

Ele troni address: Hovik.Grigorianuni-rosto k.de

§

Ele troni address: oztasha ettepe.edu.tr

¶

quarkmass

M

s^{, whereas the 2SCphase anappear al-}

readyatthe hiral restorationtransition for

µ

q

> 330 − 350

^{MeV [26, 27,28℄. Notethat thequark hemi alpo-}

tentialinthe enterofatypi al ompa tstarisexpe ted

tonotex eedavalueof

∼ 500

^{MeVso thevolumefra -}

tionofastrangequarkmatter phasewillbeinsu ient

to entail observable onsequen es. However, when the

strange quark mass is onsidered not dynami ally, but

asafreeparameterindependentofthethermodynami al

onditions,ithasbeenshown thatfor nottoolarge

M

s

the CFL phase dominatesover the 2SCphase [29, 30℄.

Studies of the QCD phase diagram with xed strange

quarkmasshavere entlybeenextendedtothedis ussion

ofgaplessCFL (gCFL)phases[31, 32,33℄. The gapless

phaseso urwhenthe asymmetrybetweenFermilevels

of dierent avors is largeenough to allow for zeroen-

ergy ex itations while anite pairinggap exists. They

havebeenfoundrstforthe2SCphase(g2SC)within a

dynami al hiralquarkmodel[34, 35℄.

Anys enariofor ompa t starevolutionthatis based

onthe o uren e ofquark matterrelies onthe assump-

tionsabout theproperties ofthis phase. It istherefore

ofpriorimportan e toobtainaphasediagram ofthree-

avorquarkmatterunder ompa tstar onstraintswith

self onsistentlydetermineddynami al quarkmasses. In

the present paper we will employ the NambuJona-

Lasinio(NJL)modeltodelineatethedierentquarkmat-

terphasesin theplaneoftemperatureand hemi alpo-

tential. WealsoaddressthequestionwhetherCFLquark

matterandgaplessphasesarelikelytoplayarolein om-

II. MODEL

In this paper, we onsider an NJL model with

quark-antiquark intera tions in the olor singlet

s alar/pseudos alar hannel, and quark-quark inter-

a tions in the s alar olor antitriplet hannel. We

negle t the less attra tive intera tion hannels, e.g.,

the isospin-singlet hannel, whi h ould allow for weak

spin-1 ondensates. Su h ondensatesallow for gapless

ex itations atlowtemperaturesand ouldbeimportant

forthe oolingbehaviourof ompa tstars. However,the

oupling strengths in these hannels are poorly known

and we therefore negle t them here. The Lagrangian

densityisisgivenby

L = ¯q

iα

(i∂/δ

ij

δ

αβ

− M

ij⁰

δ

αβ

+ µ

ij,αβ

γ

⁰

)q

jβ

+ G

S

X

8 a=0

 (¯ qτ

_f^a

q)

²

+ (¯ qiγ

5

τ

_f^a

q)

²

 + G

D

X

k,γ

 (¯ q

iα

ǫ

ijk

ǫ

αβγ

q

_jβ^C

)(¯ q

^C_i′α^′

ǫ

i^′j^′k

ǫ

α^′β^′γ

q

j^′β^′

) + (¯ q

iα

iγ

5

ǫ

ijk

ǫ

αβγ

q

^C_jβ

)(¯ q

_i^C′α^′

iγ

5

ǫ

i^′j^′k

ǫ

α^′β^′γ

q

j^′β^′

) 

,

⁽¹⁾

where

M

_ij⁰

=

^diag

(m

⁰_u

, m

⁰_d

, m

⁰_s

)

^{isthe urrentquarkmass}

matrixinavorspa eand

µ

ij,αβ^{isthe hemi alpotential}

matrixin olorandavorspa e. Duetostrongandweak

intera tions, thevarious hemi al potentialsare notin-

dependent. Inthesuper ondu tingphasesa

U (1)

^gauge

symmetry remains unbroken [36℄, and the asso iated

harge isalinear ombinationof theele tri harge,

Q

^,

and two orthogonal generatorsof the unbroken

SU (2)

c

symmetry. Hen e, there are in total four independent

hemi alpotentials

µ

ij,αβ

= (µδ

ij

+ Qµ

Q

)δ

αβ

+ (T

3

µ

3

+ T

8

µ

8

)δ

ij

,

⁽²⁾

where

Q = diag(2/3, −1/3, −1/3)

^{is the ele tri harge}

in avor spa e, and

T

3

= diag(1, −1, 0)

^and

T

8

= diag(1/ √

3, 1/ √

3, −2/ √

3)

^{are the generators in olor}

spa e. The quarknumber hemi al potential,

µ

^{, is re-}

latedtothebaryon hemi alpotentialby

µ = µ

B

/3

^{. The}

quarkeldsin olor,avorandDira spa esaredenoted

by

q

iα ^and

q ¯

iα

= q

_iα^†

γ

^0.

τ

_f^{a areGell-Mann matri esa t-}

ing in avorspa e. Charge onjugated quarkelds are

denotedby

q

^C

= C ¯ q

^{T and}

q ¯

^C

= q

^T

C

^,where

C = iγ

²

γ

⁰

istheDira harge onjugationmatrix. Theindi es

α

^,

β

and

γ

^{represent olors(}

r = 1

^,

g = 2

^and

b = 3

^),while

i

^,

j

^and

k

^{representavors (}

u = 1

^,

d = 2

^and

s = 3

^).

G

S

and

G

D ^are dimensionful oupling onstants that must bedeterminedbyexperiments.

Typi ally,three-avorNJL modelsuse a't Hooftde-

terminant intera tion that indu es a UA^{(1) symmetry}

breakinginthepseudos alarisos alarmesonse torwhi h

anbeadjustedsu hthatthe

η

^-

η

^{′ massdieren eisde-}

s ribed. Thisrealization oftheUA^{(1) breakingleadsto}

theimportant onsequen ethatthequark ondensatesof

dierentavorse torsget oupled. Thedynami allygen-

eratedstrangequarkmass ontainsa ontribution from

the hiral ondensatesofthelightavors. Thereis,how-

ever, another possible realization of the UA^{(1) symme-}

trybreaking thatdoesnotariseonthemeaneld level,

butonlyforthemesoni u tuationsinthepseudos alar

isos alar hannel. Thisisduetothe ouplingtothenon-

perturbative gluon se torvia the the triangle anomaly,

see e.g. [37, 38, 39℄. This realization of the

η

^-

η

^{′ mass}

dieren egivesno ontributionto thequarkthermody-

nami s at the mean eld level, whi h wewill follow in

this paper. Up to now it is not known, whi h of the

twoUA^{(1) breakingme hanismsis thedominantone in}

nature. In the present exploratory study of the mean

eld thermodynami s of three-avor quark matter, we

willtakethepointofviewthatthe'tHooft termmight

be subdominant and an be disregarded. One possible

wayto disentanglebothme hanismsis dueto theirdif-

ferent response to hiral symmetry restoration at nite

temperaturesanddensities. Whileinheavy-ion ollisions

onlythenitetemperatureaspe t anbesystemati ally

studied[40℄,thestateofmatterinneutronstarinteriors

may besuitable to probe theUA^{(1) symmetry restora-}

tion and its possible impli ations for the quark matter

phase diagram at high densities and low temperatures.

A omparisonof theresultspresentedin thisworkwith

thealternativetreatmentof thephasediagramofthree-

avorquark matter in luding the 't Hooft determinant

term,see [41℄,maythereforebeveryinstru tive.

Themean-eld Lagrangianis

L

^MF

= ¯ q

iα



i∂/δ

ij

δ

αβ

− (M

ij⁰

− 4G

^S

hh¯q

^iα

q

jβ

iiδ

^ij

)δ

αβ

+ µ

ij,αβ

γ

⁰

 q

jβ

− 2G

S

X

i

hh¯q

i

q

i

ii

²

− X

k,γ

|∆

kγ

|

²

4G

D

+ ¯ q

iα

∆ e

kγ

2 q

^C_jβ

+ ¯ q

_iα^C

∆ e

^†_kγ

2 q

jβ

,

⁽³⁾

∆ e

kγ

= 2G

D

iγ

5

ǫ

αβγ

ǫ

ijk

hh¯q

i^′α^′

iγ

5

ǫ

α^′β^′γ

ǫ

i^′j^′k

q

_j^C′β^′

ii = iγ

⁵

ǫ

αβγ

ǫ

ijk

∆

kγ

.

⁽⁴⁾

Wedenethe hiralgaps

φ

i

= −4G

^S

hh ¯ q

i

q

i

ii,

⁽⁵⁾

andthediquarkgaps

∆

kγ

= 2G

D

hh¯q

^iα

iγ

5

ǫ

αβγ

ǫ

ijk

q

^C_jβ

ii.

⁽⁶⁾

The hiral ondensates ontribute to the dynami al

mass of the quarks, the onstituent quarkmass matrix

inavorspa eis

M =

^diag

(m

⁰_u

+ φ

u

, m

⁰_d

+ φ

d

, m

⁰_s

+ φ

s

)

^,

where

m

⁰_i ^{are the urrent quarkmasses. Fornite ur-}

rentquark masses the

U (3)

L

× U(3)

^{R symmetry of the}

Lagrangian is spontaneously broken and only approxi-

matelyrestoredathighdensities.

The diquark gaps,

∆

kγ^{, are} antisymmetri in avor and olor, e.g.,the ondensate orresponding to

∆

ur ^is

reatedbygreenandbluedownandstrangequarks.Due

to this property, the diquark gaps anbedenoted with

theavorindi esoftheintera tingquarks

∆

ur

= ∆

ds

, ∆

dg

= ∆

us

, ∆

sb

= ∆

ud

.

⁽⁷⁾

After reformulating the mean-eld lagrangian in 8-

omponentNambu-Gorkovspinors[42,43℄andperform-

ingthefun tionalintegralsoverGrassmanvariables[44℄

weobtainthethermodynami potential

Ω(T, µ) = φ

²_u

+ φ

²_d

+ φ

²_s

8G

S

+ |∆

^ud

|

²

+ |∆

^us

|

²

+ |∆

^ds

|

²

4G

D

− T X

n

Z d

³

p (2π)

³

1 2 Tr ln

 1

T S

⁻¹

(iω

n

, ~ p)



+ Ω

e

− Ω

⁰

.

⁽⁸⁾

Here

S

⁻¹

(p)

^{istheinversepropagatorofthequarkelds}

atfour momentum

p = (iω

n

, ~ p)

^,

S

⁻¹

(iω

n

, ~p) =

"

p / − M + µγ

⁰

∆ e

kγ

∆ e

^†_kγ

p / − M − µγ

⁰

# ,

⁽⁹⁾

and

ω

n

= (2n + 1)πT

^{aretheMatsubara}frequen iesfor fermions. Thethermodynami potentialofultrarelativis-

ti ele trons,

Ω

e

= − 1

12π

²

µ

⁴_Q

− 1

6 µ

²_Q

T

²

− 7

180 π

²

T

⁴

,

⁽¹⁰⁾

hasbeenaddedtothepotential,andtheva uum ontri-

bution,

Ω

0

= Ω(0, 0) = φ

²_0u

+ φ

²0d

+ φ

²_0s

8G

S

−2N

^c

X

i

Z d

³

p (2π)

³

q

M

_i²

+ p

²

,

⁽¹¹⁾

hasbeensubtra tedinordertogetzeropressurein va -

uum. Using the identity Tr

(

^ln

(D)) =

^ln

(

^det

(D))

^and

evaluatingthedeterminant(seeAppendixA),weobtain

lndet

 1

T S

⁻¹

(iω

n

, ~ p)



= 2 X

18 a=1

ln

 ω

²_n

+ λ

a

(~ p)

²

T

²

 .

(12)

The quasiparti le dispersion relations,

λ

a

(~ p)

^{, are the}

eigenvaluesoftheHermitianmatrix,

M =

"

−γ

⁰

~γ · ~p − γ

⁰

M + µ γ

⁰

∆ e

kγ

C γ

⁰

C e ∆

^†_kγ

−γ

⁰

~γ

^T

· ~p + γ

⁰

M − µ

# ,

(13)

in olor,avor,andNambu-Gorkovspa e. Thisresultis

inagreementwith [30,41℄. Finally,theMatsubarasum

anbeevaluatedon losedform[44℄,

T X

n

ln

 ω

²_n

+ λ

²_a

T

²



= λ

a

+ 2T ln(1 + e

^−λa/T

),

⁽¹⁴⁾

leadingtoanexpressionforthethermodynami potential

ontheform

Ω(T, µ) = φ

²_u

+ φ

²_d

+ φ

²_s

8G

S

+ |∆

^ud

|

²

+ |∆

^us

|

²

+ |∆

^ds

|

²

4G

D

−

Z d

³

p (2π)

³

X

18 a=1

 λ

a

+ 2T ln 

1 + e

^−λa/T



+ Ω

e

− Ω

⁰

.

⁽¹⁵⁾

It should be noted that (14) is an even fun tion of

λ

a^,

sothesignsof the quasiparti ledispersionrelationsare

arbitrary. In this paper, we assume that there are no

trappedneutrinos. Thisapproximationisvalidforquark

matterin neutronstars,after theshort period ofdelep-

tonizationisover.

Equations(10),(11),(13),and (15)form a onsistent

thermodynami modelofsuper ondu tingquarkmatter.

The independent variables are

µ

^and

T

^{. The gaps,}

φ

i^,

and

∆

ij^,arevariationalorderparametersthatshouldbe determinedbyminimizationofthegrand anoni alther-

modynami alpotential,

Ω

^{. Also,quarkmattershouldbe}

lo ally olorandele tri hargeneutral,soatthephysi al

minimaofthethermodynami potentialthe orrespond-

ingnumberdensitiesshouldbezero

n

Q

= − ∂Ω

∂µ

Q

= 0,

⁽¹⁶⁾

n

8

= − ∂Ω

∂µ

3

= 0,

⁽¹⁷⁾

n

3

= − ∂Ω

∂µ

8

= 0.

⁽¹⁸⁾

Thepressure,

P

^{,isrelatedtothe}thermodynami poten- tialby

P = −Ω

^{at the global minimaof}

Ω

^{. The quark}

density, entropy and energy density are then obtained

asderivativesofthethermodynami alpotentialwithre-

spe tto

µ

^,

T

^and

1/T

^,respe tively.

III. RESULTS

Thenumeri alsolutionstobereportedinthisSe tion

areobtainedwiththefollowingsetofmodelparameters,

taken from Table5.2 of Ref. [8℄ for vanishing 't Hooft

intera tion,

m

⁰_u,d

= 5.5 MeV ,

⁽¹⁹⁾

m

⁰_s

= 112.0 MeV ,

⁽²⁰⁾

G

S

Λ

²

= 2.319 ,

⁽²¹⁾

Λ

²

= 602.3 MeV .

⁽²²⁾

With these parameters, the following low-energy QCD

observables anbe reprodu ed:

m

π

= 135

^MeV,

m

K

= 497.7

^MeV,

f

π

= 92.4

^{MeV.Thevalueofthediquark ou-}

pling strength

G

D

= ηG

S ^{is onsideredasafreeparam-}

eter of themodel. Here wepresentresultsfor

η = 0.75

(intermediate oupling)and

η = 1.0

^{(strong oupling).}

300 350 400 450 500 550

µ [MeV]

100 200 300 400 500 600

∆ , M [MeV]

M_s M_u Md

Mu, M

∆_ud d

∆_us, ∆_ds

FIG.1: Gapsanddynami alquarkmassesasafun tionof

µ

atT=0forintermediatediquark oupling,

η = 0.75

^.

300 350 400 450 500 550

µ [MeV]

100 200 300 400 500 600

∆ , M [MeV]

M

_s

M

_u

M

_d

M

_u

, M

_d

∆

_ud

∆

_us

, ∆

_ds

FIG.2: Gapsanddynami alquarkmassesasafun tionof

µ

atT=0forstrongdiquark oupling,

η = 1

^.

A. Quark masses andpairing gaps atzero

temperature

Thedynami ally generated quarkmasses and the di-

quarkpairinggapsaredeterminedself onsistentlyatthe

absoluteminimaofthethermodynami potential,in the

planeoftemperatureandquark hemi alpotential. This

isdoneforboththestrongandtheintermediatediquark

oupling strength. In Figs. 1 and 2 we show the de-

penden e ofmasses andgapsonthequark hemi alpo-

tential at

T = 0

^for

η = 0.75

^and

η = 1.0

^{, resp. A}

300 350 400 450 500 550 600

µ [MeV]

-250 -200 -150 -100 -50 0

µ

Q

, µ

8

[MeV]

µ

_{Q, η=0.75}

µ

_{8, η=0.75}

µ

_{Q, η=1}

µ

_{8, η=1}

FIG.3:Chemi alpotentials

µ

Q^and

µ

8^{atT=0forbothvalues}

of the diquark oupling

η = 0.75

^and

η = 1

^{. All phases}

onsideredinthisworkhavezero

n

3 olor hargefor

µ

3

= 0

, hen e

µ

3 ^{isomittedintheplot.}

hara teristi feature of this dynami al quark model is

that the riti al quark hemi al potentials where light

and strange quark masses jump from their onstituent

massvaluesdowntoalmosttheir urrentmassvaluesdo

not oin ide. Within reasing hemi alpotentialthesys-

temundergoesasequen eoftwotransitions: (1)va uum

→

^{two-avorquarkmatter,(2)two-avor}

→

three-avor quarkmatter. Theintermediatetwo-avorquarkmatter

phase o urs within an interval of hemi al potentials

typi alfor ompa tstarinteriors. Whileatintermediate

ouplingtheasymmetrybetweenof upanddown quark

hemi alpotentialsleadstoamixed NQ-2SCphasebe-

low temperatures of 20-30 MeV, at strong oupling the

pure2SCphase extends down to T=0. Simultaneously,

thelimiting hemi al potentials of the two-avorquark

matterregionareloweredbyabout40MeV.Three-avor

quarkmatterisalwaysintheCFLphasewhereallquarks

are paired. The robustness of the 2SC ondensate un-

der ompa tstar onstraints,withrespe tto hangesof

the oupling strength, as well as to a softening of the

momentum utobyaformfa tor,hasbeenre entlyin-

similartrend: for

η = 0.75

^{andNJL formfa torthe2SC}

ondensatedoesnoto urformoderate hemi alpoten-

tialswhilefor

η = 1.0

^{ito urs}simultaneouslywith hiral symmetry restoration. Fig. 3 shows the orresponding

dependen esofthe hemi alpotentials onjugatetoele -

tri (

µ

Q^{) and olor(}

µ

8^{) harges. All phases onsidered}

in thisworkhavezero

n

3 ^{olor hargefor}

µ

3

= 0

^.

0 200 400

p [MeV]

0 100 200 300 400 500

E [MeV]

ug-dr ub-sr, db-sg ur-dg-sb

0 200 400 600

p [MeV]

ub-sr db-sg

FIG.4: Quark-quarkquasiparti le dispersionrelations. For

η = 0.75

^,

T = 0

^{, and}

µ = 480

^{MeV (left panel) there is a}

forbiddenenergybandabovetheFermisurfa e.Alldispersion

relationsaregappedatthispointinthe

µ−T

^{plane,seeFig.5.}

There is noforbiddenenergyband for the

ub − sr

^,

db − sq

^,

and

ur − dg − sb

quasiparti lesat

η = 1

^,

T = 84

^{MeV, and}

µ = 500

^{MeV (right panel). This point inthe}

µ − T

^plane

onstituteapartofthegaplessCFLphaseofFig.6.

B. Dispersion relations andgaplessphases

In Fig. 4 we show the quasiparti le dispersion rela-

tions of dierent ex itations at twopointsin thephase

diagram: (I) theCFL phase(leftpanel), where there is

a nite energy gap for all dispersionrelations. (II) the

gCFLphase(rightpanel),wheretheenergyspe trumis

shifteddue to theassymetryin the hemi al potentials,

su hthat theCFL gapis zeroand (gapless)ex itations

withzeroenergyarepossible. Inthepresentmodel, this

phenomenon o urs only at rather high temperatures,

wherethe ondensatesarediminishedbythermalu tu-

ations.

C. Phase diagram

Thethermodynami alstateofthesystemis hara ter-

izedbythevaluesoftheorderparametersandtheirde-

penden eonTand

µ

^{. Hereweillustratethisdependen y}

350 400 450 500 550

µ [MeV]

0 10 20 30 40 50 60 70 80

T [MeV]

g2SC

175

NQ-2SC CFL

0.9 0.8

gCFL guSC

2SC

χ_2SC = 1.0

0.7

NQ

M

^s

=200 MeV

FIG.5: Phase diagram ofneutralthree-avor quarkmatter

forintermediatediquark oupling

η = 0.75

^. First-orderphase transitionboundariesare indi atedbybold solidlines,while

thinsolidlines orrespondtose ond-orderphaseboundaries.

Thedashedlinesindi ategaplessphaseboundaries. Thevol-

umefra tion,

χ

²SC^{,ofthe2SC omponentofthemixedNQ-}

2SC phaseis denotedwiththindotted lines,while the on-

stituentstrangequarkmassisdenotedwithbolddottedlines.

inaphasediagram. Weidentifythefollowingphases:

1. NQ:

∆

ud

= ∆

us

= ∆

ds

= 0

^;

2. NQ-2SC:

∆

ud

6= 0

^,

∆

us

= ∆

ds

= 0

^,0<

χ

2SC^<1;

3. 2SC:

∆

ud

6= 0

^,

∆

us

= ∆

ds

= 0

^;

4. uSC:

∆

ud

6= 0

^,

∆

us

6= 0

^,

∆

ds

= 0

^;

5. CFL:

∆

ud

6= 0

^,

∆

ds

6= 0

^,

∆

us

6= 0

^;

andtheirgaplessversions. Theresultingphasediagrams

forintermediateandstrong ouplingaregiveninFigs. 5

and6,resp. and onstitutethemainresultofthis work,

whi hissummarizedin thefollowingstatements:

1. Gapless phases o ur only at high temperatures,

above50 MeV (intermediate oupling)or60 MeV

(strong oupling).

2. CFLphaseso uronlyatratherhigh hemi alpo-

tential,wellabovethe hiralrestorationtransition,

i.e. above464MeV(intermediate oupling)or426

MeV(strong oupling).

3. Two-avorquarkmatterfor intermediate oupling

isatlowtemperatures(T<20-30MeV)inamixed

NQ-2SC phase, at high temperatures in the pure

2SCphase.

4. Two-avorquark matter for strong oupling is in

the 2SC phase with rather high riti al tempera-

turesof

∼ 100

^MeV.