Extensive study of phase diagram for charge neutral homogeneous quark matter affected by dynamical chiral condensation – unified picture for thermal unpairing transitions from weak to strong coupling – Hiroaki Abuki1,∗ and Teiji Kunihiro1,† 1Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: February 2, 2008) Westudythephasestructuresofchargeneutralquarkmatterundertheβ-equilibriumforawide range of the quark-quark coupling strength within a four-Fermion model. A comprehensive and unifiedpictureforthephasetransitionsfromweaktostrongcouplingispresented. Wefirstdevelop atechniquetodealwith thegap equation andneutrality constraintswithout recourse tonumerical derivatives, and show that the off-diagonal color densities automatically vanish with the standard assumption for the diquark condensates. The following are shown by the numerical analyses: (i) 6 The thermally-robustest pairing phase is the two-flavor pairing (2SC) in any coupling case, while 0 the second one for relatively low density is the up-quark pairing (uSC) phase or the color-flavor 0 locked(CFL)phasedependingonthecouplingstrengthandthevalueofstrangequarkmass. (ii)If 2 thediquarkcouplingstrengthislargeenough,thephasediagramismuchsimplifiedandisfreefrom theinstabilityproblemsassociated withimaginary Meissner massesin thegapless phases. (iii) The n interplaybetweenthechiralanddiquarkdynamicsmaybringanon-trivialfirstordertransitioneven a in the pairing phases at high density. We confirm (i) also by using the Ginzburg-Landau analysis J expanding the pair susceptibilities up to quartic order in the strange quark mass. We obtain the 8 analytic expression for the doubly critical density where the two lines for the second order phase transitions merge, and below which the down-quark pairing (dSC) phase is taken over by the uSC 3 phase. Also we study how the phase transitions from fully gapped states to partially ungapped v states are smeared at finite temperatureby introducing theorder parameters for these transitions. 2 7 PACSnumbers: 12.38.-t,25.75.Nq 1 9 0 I. INTRODUCTION matter in compact stars, the effect of (ii) gives so strong 5 0 constraintbecause of a long range nature of gauge inter- / actions that such an exotic phase may exist stably. The h Surprisingly rich phase structure of quark matter is gCFLphasehasbeenextensivelystudiedandclaimedto p being revealedby extensive theoretical studies [1–3]. On be the most promising candidate of next phase down in - the basis of the asymptotic-free nature of QCD and the p density at vanishing temperature [11] and is also known attraction between quarks due to a gluon exchange, it e to lead an interesting astrophysical consequence on the h is now believed that the ground state of the sufficiently cooling history of an aged compact star [12]. It should : cold and extremely dense matter is in the color-flavor v be, however,noted here that the gapless phases are usu- locked (CFL) phase where all the quark species equally i allyaccompaniedbysometransversegluonicmodeswith X participate in pairing [4]. In contrast, it remains still imaginaryMeissnermassatlowtemperature[13,14],in- r controversial what phase sets in next to the CFL as the a dicatingtheexistenceofmorestableexoticstates[15–21]. density is decreased and/or the temperature is raised. Although the resolution of the instability or finding the At zero temperature (T = 0), the following two in- possible new stable state is now one of the central issues gredients play crucial roles for determining the second in this field, we won’t deal with this problem; neverthe- densest phase in QCD; (i) the strange quark mass [5– less we will give some suggestions to possible resolution 7], and (ii) the charge neutrality constraints as well as on the basis of the results obtained in this work. the β-equilibrium condition [8–10]. The former tends to bring a Fermi-momentum mismatch between the light The neutrality constraints are also known to bring about an interesting phase even at finite temperature and strange quarks, while the latter in the paired phase tendstomatchtheFermi-momentabytuning thecharge (T =0). Infact,aGinzburg-Landauanalysisshowsthat 6 chemical potentials. Some theoretical studies suggest the d quark pairing phase with the u-d and d-s pairings, that, when the baryon density is decreased, the effect denotedby“dSC”,appearsasthesecondphasewhenthe (ii) in the CFL phase cancelsthe effect (i) downto some 2SC (CFL) phase is cooled (heated) [22]. The existence critical density, at which the CFL phase turns into the of such a kind of intermediate phase has also been con- gapless CFL (gCFL) phase [11]. In the realistic quark firmedusingtheNJLmodel[23]. Alsothegaplessversion ofthedSCwithanadditionalgaplessmodeisalsoexam- ined for finite temperature [22] andfor zero temperature [24]; the gaplessdSC is shownto be alwaysa metastable ∗E-mail:[email protected] atzerotemperature,whileitmayexiststablyfornonzero †E-mail:[email protected] temperature. It should be noted, however,that the gap- 2 less phase for nonzero temperature cannot be thermo- replaced by simple algebraic equations. In Sec. III, we dynamically distinguished from the fully gapped one; in discuss the points listed above based on the numerical fact, it has no distinct phase boundary with the gapped results. Alsothe Ginzburg-Landauanalysisis performed one,andthusisoflessphysicalinterestthanthatatzero andsomeinterestingaspectsofthethermalphasetransi- temperature. tions are given at the end of this section. The summary In the most of the literature concerning the pairing and outlook are provided in Sec. IV. In Appendix A, phases of the quark matter [11, 23, 25], the strange we derive the Ginzburg-Landau potential expanded up quark mass is treated as a parameter just like an ex- to quarticorderin the strangequarkmass. InAppendix ternal magnetic field applied to a metallic superconduc- B,weshowthattheoff-diagonalcolordensitiesautomat- tor [26]. In QCD, however, the strong attraction exists ically vanish under the standard ansatz for the diquark in the scaler quark-antiquark channel which leads to a condensates. non-perturbative phenomenon called the dynamical chi- ral condensation in the low density regime. It is inter- esting to investigate how the incorporation of the dy- namical formation of the chiral condensates affects color superconducting phases [27, 28]. The competition be- II. FORMULATION tween the chiral and diquark condensations under the charge neutrality constraints was first investigated in a In this section, we introduce our model and formu- four-Fermion model [24]; it was shown that the second late the meanfield approximationunder the kinetic con- densest phase next to the CFL strongly depends on the straints. In Sec. IIB, we present some useful algebraic quark-quark (diquark) coupling strength, and in partic- techniques to solve the gap equation and constraints ular,thegaplessCFLphasemaybewashedoutfromthe without recourse to numerical derivatives. phase diagram except for an extremely weak coupling case. Their investigation is extended to finite tempera- tures[29,30]andrecentlyalsoappliedtothesystemwith A. Model finite neutrino density [31]. Some of the results obtained in [29, 30], however, seemingly contradict with the pre- We start with the following Lagrangianas in [24] vious claim by the authors of [22, 23] on the appearance of the dSC phase. This is one of puzzles indicating that 3 a further investigation is needed for a systematic under- = q¯(i∂/ m +µγ )q+ Gd (q¯Ptq¯)(tqP¯ q) standing of the phases of quark matter and transitions L − 0 0 16 η η η=1 among them. X(cid:2) (cid:3) G A main aim of this paper is to explore the phase di- + s (q¯λ q)2+(q¯iγ λ q)2 . (1) F 5 F 8N agram for a wide range of diquark coupling systemat- c (cid:2) (cid:3) ically and thereby give a unified picture for the phase Here, λ = 2/31,~λ are the unit matrix and the transitions from weak to strong coupling. We clarify the F { F} Gell-Mann matrices in the flavor space. P is defined as detailed mechanism and features of thermal phase tran- p η [11] sitions, and make clear the relations among the previous studies [22, 23, 29, 30]. In particular, we shall study (P )ab =iγ Cǫηabǫ no sum over index η (2) the the following. (i) How the (µ,T)-domain accommo- η ij 5 ηij dating the pairing grows and that for the gapless phases and P¯ = γ (P )†γ . a,b, and i,j, represent the vanishes as the coupling in the scaler diquark channel is η 0 η 0 ··· ··· color and flavor indices, respectively. The second term increased. (ii) Howwe candrawa unifiedpicture for the in Eq. (1) simulates the attractive interaction in the thermalunpairing phase transitionsfromweakto strong coloranti-triplet,flavoranti-tripletandJP =0+ channel coupling; for this purpose, we extend the previous study in QCD. m = diag. m ,m ,m is the current-quark [22] by making re-analysis of Ginzburg-Landau expan- 0 { u d s} mass matrix. In order to impose the color and electric sion focusing on the effects of the terms quartic in the neutralities, we have introduced in Eq. (1) the chemical strange quark mass. (iii) How and why non-trivial first potential matrix µ in the color-flavorspace as order phase transitions could be caused by the compe- tition between the pairing and chiral dynamics in the µab = µ µ Q +µ Tab+µ Tab strange quark sector. (iv) How the transition from the ij − e ij 3 3 8 8 fully gapped phase to the partially ungapped (gapless) +µ1T1ab+µ2T2ab+µ4T4ab (3) one is smeared at finite temperature by introducing a +µ Tab+µ Tab+µ Tab. 5 5 6 6 7 7 definite order parameter. The paper is organized as follows. In Sec. II, we in- Q = diag.(2/3, 1/3, 1/3) counts electric charge of − − troduce the model and derive the gap equation under each quark species. T = diag.(1/2, 1/2,0) and T = 3 8 − the neutrality constraints. Some technical developments diag.(1/3,1/3, 2/3)are the diagonalchargesfor quarks − arepresented,by whichthe numericalderivativescanbe inthefundamentalrepresentationofthecolorSU(3). We 3 haveincludedthesecondandthirdlinesinEq.(3)which defined by stand for the chemical potentials for off-diagonal color cphoatergnetsia:lIstshhaosulbdeebnerineccelundtleydcfloarimtheedctohmatptlheteesecoclhoermniecua-l S−1(iωn,p)=(cid:18)/p+µηγP¯0η−∆ηM t/pP−µηPγ0η∆+ηM (cid:19), (8) tralities [32]. We can prove,however,that if the diquark with p/ = iω γ p Pγ. Finally, Ω is the contribution condensates have the color-flavor structure as given by n 0 e − · from massless electrons Eq. (2), then the off-diagonal color densities must auto- matically vanish for µ1 = µ2 = µ4 = µ5 = µ6 = µ7 = 0; µ4 µ2T2 7π2T4 see Appendix B for the detail of the proof. Thus, we Ωe = −12πe2 − e6 − 180 . (9) can safely adopt the usual diagonalansatz for the chem- ical potential matrix. The explicit forms of the diagonal The optimal values of the variationalparameters∆ , M η elements (µ =µaa) are as follows. and M must satisfy the stationary condition (the gap ai ii s equations); µ = µ 2µ + 1µ + 1µ , ru − 3 e 2 3 3 8 µgd = µ+ 31µe− 21µ3+ 13µ8, ∂∂∆Ω =0, ∂M∂Ω =0 and ∂∂MΩ =0. (10) µ = µ+ 1µ 2µ , η u,d s bs 3 e− 3 8 Our task is to search the minimum of the effective po- µ = µ+ 1µ + 1µ + 1µ , tential by solving these gap equations under the local rd 3 e 2 3 3 8 electric and color charge neutrality conditions; µ = µ 2µ 1µ + 1µ , gu − 3 e− 2 3 3 8 (4) ∂Ω ∂Ω ∂Ω µrs = µ+ 31µe+ 21µ3+ 13µ8, ∂µe =0, ∂µ3 =0 and ∂µ8 =0. (11) µ = µ 2µ 2µ , bu − 3 e− 3 8 For a later convenience, we define here the electron den- sity by µ = µ+ 1µ 1µ + 1µ , gs 3 e− 2 3 3 8 µbd = µ+ 31µe− 32µ8 . ne =−∂∂Ωµe = 3µπ3e2 + µe3T2. (12) e We treat the diquark coupling constant G as a sim- d The formulation made above is a straightforwardexten- ple parameter, although the perturbative one-gluon ex- sionofourpreviouswork[24]tothe(T =0)case[29,30]. change vertex int = (g2/2)q¯γµ(λa/2)qq¯γµ(λa/2)q, 6 L − which is valid at extremely high density, tells us that G /G =1/2 with N =3 [33–35]. As a measure of cou- d s c B. Representation of gap equation and kinetic pling constant G , we shall mainly use the gap energy d constraints in terms of quasi-quark wave functions (∆ ) in the pure CFL phase at µ=500MeV and T =0 0 in the chiral SU(3) limit, as in [11, 23, 24]. Inthis section,we presentsomeanalyticalwayto deal Weevaluatethethermodynamicpotentialinthemean- with the gap equation and neutrality constraints. We field approximation; shall show that the gap equation and neutrality con- Ω = Ω +Ω , straints derived above can be further simplified with the q e aidofthequasi-quarkwavefunctions(spinors). Bydoing 3 3 4 N this, not only the physical meanings of these equations Ω = ∆2 + c (M m )2 q G η G i− i (5) become transparent, but also the numerical derivatives d s TXη=1 dp Xi=1 canbecircumvented. Inparticular,thelatterhasaprac- trLog S−1(iω ,p) , tical advantage because the computations of matrix ele- −2 (2π)3 n n Z mentsaremorefavorablethanthenumericalderivatives. X (cid:2) (cid:3) First, we introduce the Nambu-Gor’kov mean field where Hamiltonian density (p;µ,∆,M) following [11, 23], G H ∆ = d tqP q , (6) η 8 h η i Γ S−1(iω ,p)Γ¯ =iω 1 (p;µ,∆,M). (13) 0 n 0 n 72 −H M 0 0 u Here we have defined M = 0 M 0 d   0 0 Ms γ 1 0 1 0  u¯u 0 0 Γ0 = 0⊗0 9 136 , Γ¯0 = 036 γ0 19 . (14) = m Gs h 0 i d¯d 0 , (7) (cid:18) (cid:19) (cid:18) ⊗ (cid:19) 0− 2N  h i  1 denotes the 72-dimensional unit matrix and is c 0 0 s¯s 72 H h i Hamiltonian density which is also 72 72 matrix in   × are the gap parameter and constituent quark mass ma- the color, flavor and spinor space. In the same man- trices. S denotesthe72 72Nambu-Gor’kovpropagator ner as shown in [29], we can lift the spin degeneracy × 4 away from as = (p) P ( p) P give the explicit form of these matrices below, 36 + 36 − H H H ⊗ ⊕ H − ⊗ with P 1±pˆ·σ being the helicity projectors. (p) hHa8(sbda,sgb)±lo⊕c≡kH-d8(sira2,gubn)a⊕lizHed8(ufgo,rdmr).liFkuerHth3e6r(mp)or=e,Hth1(eu2rN,dHga,m3sb6b)u⊕- H4αβ = Mαp0−µα −Mi∆αpα−βµα M−βi∆0+αµββ −i∆0pαβ , Gor’kovdegeneracyismanifestinthelatterthreeblocks; −  i∆αβ 0 p M +µ  (α,β) = (α,β) (α,β) . Thusweneedtoknowonly  − − β β  H8 H4 ⊕ −H4  (15) two matrix structures of (α,β) αβ and (ur,dg,sb) where∆αβ =∆ ,∆ and∆ for(α,β)=(bd,sg),(sr,ub) (cid:2) H4(cid:3) ≡H4 H12 ≡ 1 2 3 uds for the evaluation of the effective potential. We and (ug,dr), respectively, and H12 M µ p 0 0 0 0 0 i∆ 0 0 0 i∆ u ur 3 2 − − − p M µ 0 0 0 0 i∆ 0 0 0 i∆ 0 u ur 3 2  0 − 0− M +µ p 0 i∆ −0 0 0 i∆ −0 0  u ur 3 2 −  0 0 p Mu+µur i∆3 0 0 0 i∆2 0 0 0   0 0 −0 − i∆3 Md µdg p 0 0 0 0 0 i∆1   0 0 i∆ −0 −p M µ 0 0 0 0 i∆ −0  Hu12ds = i∆0 i∆03 −00 3 00 00 − d00− dMg d+pµdg M−+pµ i∆0 i∆01 −00 1 00 . (16)  03 0 0 i∆ 0 0 −0 − di∆ dMg µ1 p 0 0   − 2 − 1 s− sb   0 0 i∆ 0 0 0 i∆ 0 p M µ 0 0   − 2 − 1 − s− sb   0 i∆ 0 0 0 i∆ 0 0 0 0 M +µ p   2 1 s sb −   i∆ 0 0 0 i∆ 0 0 0 0 0 p M +µ   2 1 − − s sb   Because the Nambu-Gor’kovdegeneracy is not removed, and equating the result to zero to obtain, theeigenvaluesofthismatrixhavesixsetsofthedoublet 8 dp 1 ∂ (ǫ, ǫ) corresponding to the energies for a quasi-quark ∆ = T tr H36 . and−its Nambu-Gor’kov partner (anti-quasi-quark). Gd η − n Z (2π)3 (cid:20)iωn136−H36 ∂∆η (cid:21) X Let us write 18(= 72 2 2) independent eigen- (17) vthaeluelasbeolf fHor tahse{qǫuαa}si-wqiut÷harktsh÷e(αin=dex1,2α, red,e1fi8n)edfrotmo dHeenrseitHy3d6efiinsetdhbeyrreedmucoevdinNgathmebsup-iGnodre’kgoevneHraacmyiflrtoomnian that for the direct products of color and··fl·avor (α = as was introduced above. Also we note that the reduceHd ur,dg,sb,db,sg,sr,ub,ug,dr). Thenthe thermodynamic Hamiltonian density takes the form potential can be simplified to = +M φ +M φ +M φ H36 H0 u Mu d Md s Ms +∆ φ +∆ φ +∆ φ (18) 4 N 1 1 2 2 3 3 Ωq = Gd ∆2η+ Gsc (Mi−mi)2 −3µB−µeQe−µ3T3−µ8T8, 2 X18 dp X|ǫα| +T log 1+e−|ǫα|/T with H0 = H36|µ=0,∆=0,M=0 being the Hamiltonian − (2π)3 2 density for the system with nine free massless quarks. αX1=81Z dp p(cid:20) (cid:16) (cid:17)(cid:21) cφoMnis,taφn∆tηmanatdriBx,eQleem,Ten3t,s8.arTehthesee3m6×atr3i6cemsactarnicebsewoitbh- +2 . (2π)32 tainedbydifferentiatingthereducedHamiltonianmatrix αX=1Z with respect to Mu,d, Ms, ∆1,2,3, µ, µe,3,8 as follows. We have subtracted the vacuum contribution in the sys- φ = ∂H36 , φ = ∂H36, Mu,d ∂Mu,d Ms ∂Ms tem with the nine massless quarks. It is difficult to nu- φ = ∂H36, φ = ∂H36, φ = ∂H36, mericallysearchthe minimawithrespectingtheneutral- 1 ∂∆1 2 ∂∆2 3 ∂∆3 (19) ityconstraintsfromthiseffectivepotentialalone. There- B =−31∂∂Hµ36, Qe =−∂∂Hµ3e6, fore, we search them with the aid of the gap equation T = ∂H36, T = ∂H36. which is the ∆-derivative of the effective potential. If 3 − ∂µ3 8 − ∂µ8 possible,numericalderivativesshouldbeavoidedbecause Because is an Hermitian matrix for any momen- 36 the numerical errors associated with them are not well H tum,wecandefinethecompletesetofspinors p,α, controllable. To avoid them, we express the gap equa- {| ±i} by the eigenvalue equation tions in terms of the eigenvectors (eigen-spinors) of . H First, we simply differentiate Eq. (5) with respect to ∆ p,α, = ǫ (p)p,α, , (20) η 36 α H | ±i ± | ±i 5 α=1, ,18and denotetheNambu-Gor’kovcharges. basis of the broken charges (X,Y) in addition to unbro- ··· ± Then we have ken Q˜ as [10], ∂ǫ hp,α,±|φη|p,α,±i=±∂∆α , (η =1,2,3). (21) X = Q+T3−4T8, η Y = Q T , Thus we find that Eq. (17) is reduced to − 3 8 18 dp ǫ the chemical potentials for these charges become ∆ = tanh α p,α,+φ p,α,+ , Gd η αX=1Z(2π)3 (cid:16)2T(cid:17)h | η| i µQ˜ = −94 µe+µ3+ 12µ8 , wheretheMatsubarasummationhasbeenperformed. In (cid:0) (cid:1) µ = 1 ( µ +µ 4µ ), much the same way, we obtain X 18 − Q 3− 8 2Nc(Mi mi) µY = 12(−µQ−µ3). G − s = 18 dp tanh ǫα p,α,+φ p,α,+ . We find that the matrix representation of Q˜ operator (2π)3 2T h | Mi| i inthe32-dimensionalcolor-flavormixedNambu-Gor’kov αX=1Z (cid:16) (cid:17) base is given by These formulae enable us to evaluate the gap equation Q˜ = ∂H36 = Q T 1T . (23) solelybycomputingthe18-dimensionaleigen-spinorsde- − ∂µQ˜ − e− 3− 2 8 finedbyEq.(20)andsomematrixelementsinthesebases Becausethis commutes with the Nambu-Gor’kovHamil- [36] without recourse to numerical derivatives done in tonian density as it should be, the quasi-particles [11, 23, 29]. 36 (eigen-spinors) Hhave definite Q˜-charges which can be Thechargeneutralityconstraintscanbealsoexpressed shown to be of integers (+1,0, 1) [3]. It is also to be in terms of the matrix elements in the basis composed − noted here, that thermodynamic potential in the quark of these eigen-spinors. A straightforward application of th∂eΩab/o∂vµe =methnodantod ρρ3,8== 1−∂∂ΩΩ/q/∂∂µµl3e,a8ds= 0, ρqe = sCeFctLorphΩaqsedosoesthnaott idteipsenadQ˜o-innsµuQl˜ationrt[h1e0].fuBllyecgaaupsepeodf − q e − e B −3 q this, the value of µ in the CFL phase (T = 0) can- Q˜ 18 dp not be determined by the Q˜ neutrality condition in the 0 = (2f (ǫ ) 1) p,α,+Q p,α,+ , (2π)3 F α − h | 3,8| i quark sector; it should be determined completely by the α=1Z X vanishing-pointofverygentleslopeofpotentialcurvature 18 dp coming from the electron sector (µ3/3π2 =0). n = (2f (ǫ ) 1) p,α,+Q p,α,+ , e − e (2π)3 F α − h | e| i α=1Z X 18 dp ρ = (2f (ǫ ) 1) p,α,+B p,α,+ . B (2π)3 F α − h | | i α=1Z X III. NUMERICAL RESULTS AND Here, the Fermi-Dirac distribution function f (ǫ) = F DISCUSSIONS 1/(eǫ/T +1) is introduced. At zero temperature, f (ǫ) = θ( ǫ) so that the net F − Inthis section,we presentournumericalresults ofthe charge will be accumulated in the blocking region where solution of the gap equations, and provide the phase di- ǫ<0. Fortheneutralityconstraintstobesatisfied,there agrams in the (µ,T)-plane for several values of diquark must be the opposite chargedensity coming from a non- coupling G . quarksectororfromthequarksectorwiththefiniteback- d Before that, we fix our model parameters. We take groundchargedensity dp p,α,+Qp,α,+ =0, (2π)3 αh | | i6 the chiral SU(2) limit for the u, d current quark masses which aresupplied by tuning the chargechemicalpoten- (m = m = 0) and m = 80MeV [24]. These R P u d s tials µe, µ3 and µ8. values might slightly underestimate the effect of the For a later convenience, we introduce here a charge current masses because m (2GeV) = 3-4MeV and u,d generator m (2GeV) = 80-100MeV according to the full lattice s QCD simulation [37]. Also we restrict the variational 1 Q˜ = Q T T , (22) space by putting M = M = M for simplicity. This − − 3− 2 8 u d simplificationdoesnotmatterinthechiralsymmetryre- the operation of which keeps the CFL state invariant stored phase [29, 30]. (neutral) and thus represents an unbroken U(1) symme- For comparison with the previous work [29], we write try in the CFL phase [3]. If we choose the orthogonal down the dimensionless parameters adopted in this 6 Gap and mass Conditions Gapless quark and Q˜ charge parameters for (ur-dg-sb) (db-sg) (ub-sr) (ug-dr) Phase ∆ (ds) ∆ (us) ∆ (ud) M M chemical potentials 0, 0, 0 0, 0 +1, 1 +1, 1 1 2 3 s − − CFL (9) ∆ ∆ ∆ M [µ =0] all quark modes are fully gapped 1 2 3 s e gCFL (8) ∆ ∆ ∆ M δµ + Ms2 >∆ db 8 1 2 3 s dbsg 4µ 1 gCFL (7) ∆ ∆ ∆ M δµ + Ms2∼>∆ db ub 1 2 3 s dbsg(ubsr) 4µ 1(2) ∼ uSC(6) ∆ ∆ M [µ =0] dg-sb(1) (db,sg) 2 3 s e guSC (5) ∆ ∆ M δµ + Ms2 >∆ dg-sb(1) (db,sg) ub 2 3 s ubsr 4µ 2 ∼ 2SC (4) ∆ M [µ =0] sb (db,sg) (ub, sr) 3 s 3 g2SC (2) ∆ M [µ =0], δµ =δµ >∆ dg,sb (db,sg) (ub, sr) dr 3 s 3 dgur drug 3 dSC(6) ∆ ∆ M ur-sb (1) (ub, sr) 1 3 s gdSC (5) ∆ ∆ M δµ + Ms2 >∆ ur-sb (1) db (ub, sr) 1 3 s dbsg 4µ 1 ∼ 2SCus (4) ∆ M dg (db,sg) (ub, sr) 2 s UQM(0) M [µ =µ =0] all quarksare ungapped. s 3 8 χSB (0) M M [µ =µ =0] all quarksare massive. s 3 8 TABLEI:Thenonzerogapparameters,someconditionsbetweengapsandchemicalpotentials,andthegaplessquarksineach phase. The figure in the parenthesis in the first column represents the number of gapped quasi-quark mode. We have defined the (ai-bj) relative chemical potential by µ (µ µ )/2. “dg-sb (1)” means that one of the linear combinations, the aibj ai bj ≡ − dg quark and the sb hole remains gapless. The equation for chemical potentials in a bracket must necessarily hold for some symmetry or kinetic reason. study; the following notice is in order here. (i) Our strange quark mass m /Λ = 0.1 is about one half of the value s mu,d/Λ=0, ms/Λ=0.1, ms/Λ = 0.23 adopted in [29]. (ii) We did not included the six-quark interaction which effectively increases the G Λ2 GSΛ2 s =2.17. scalarcoupling GS. These two differences make our case ≡ 8Nc favorable to the pairing phases rather than the unpaired quark matter (UQM) phase or the chiral-symmetry bro- The value of G is chosen so that the dynamical quark s ken (χSB) phase. In fact, we will see that the phase massatµ=0is400MeVforthecutoffΛ=800MeVjust diagrams for our weak coupling and intermediate cou- for comparison with our previous work [24]. Note, how- pling cases seem more or less to correspond to those for ever, that this coupling is a little larger than the value G /G =3/4 and G /G =1 in [29], respectively. extracted in the NJL model analysis of the meson spec- D S D S WeconsiderthecandidatesofphaselistedinTABLEI troscopy with instanton induced six-quark coupling [38], as in [24] in the numerical analyses below. i.e.,G Λ2 =1.835whichisadoptedin[29]. Weshallper- S form the calculation with following five different values of the diquark coupling (G G /16): D d ≡ A. Phases for extremely weak coupling 1. extremely weak coupling: ∆ =25MeV G Λ2 =0.91 (G /G =0.42) 0 D D S ↔ Let us first discuss the extremely weak coupling case 2. weak coupling: (∆0 = 25MeV). The phase structure in this case is dis- ∆ =80MeV G Λ2 =1.37 (G /G =0.63) played in Fig. 1. At a first glance, we notice that the 0 D D S ↔ UQM phase without any symmetry breaking dominates 3. intermediate coupling: the phase diagram pushing the superconducting phases ∆0 =125MeV GDΛ2 =1.69 (GD/GS =0.78) to the high density regime. This is because the energy ↔ gain due to the s¯s condensation in the UQM phase is 4. strong coupling: h i larger than the paring energy under the stress as is clar- ∆ =160MeV G Λ2 =1.92 (G /G =0.88) 0 ↔ D D S ified in[24]. There arealsoseveralthermalphase transi- 5. extremely strong coupling: tions. The thermally-robustest pairing phase is the 2SC ∆ =200MeV G Λ2 =2.17 (G /G =1.00) and the second phase in this case is the uSC as is found 0 D D S ↔ in [29, 30]. In the following, we shall discuss the fea- ThevaluesofG /G fortheintermediateandextremely tures of these phase transitions in detail. We first make D S strong coupling cases are similar to those employed in a close examination on the zero temperature case, and [29], i.e., G /G = 3/4 and G /G = 1. However, then investigate the finite temperature case. D S D S 7 FIG. 1: The phase diagram for the extremely weak coupling (∆ =25MeV). Sold(dashed)linesmeanfirst(second)order 0 transitions. Large dot placed on the µ-axis (µ=581.1MeV) representsthetransitionpointatwhichtheCFLphaseonthe high density side continuously turns into the gCFL phase on thelow density side. a. CFL/gCFL and gCFL/UQM transitions at T =0: The phases realized at T = 0 are the CFL, gCFL and UQM states. In Fig. 2(a), we show the gaps, masses, andchemicalpotentialsalongthe (g)CFLsolutionofthe gap equation. For µ > 558.4MeV we confirm that the condensation energy in the (g)CFL solution is largest among those for all the candidates. The excitation gaps of the nine quasi-quarks in the CFL phase are shown in FIG. 2(b). We can see from the figure that one quasi- quark has the largest excitation gap, and other quarks haverelativelysmallgaps. Thelattereightmodesarethe remnantsofthecolor-flavoroctetmodesinthepureCFL phase at M = 0. In the CFL phase, the non-vanishing s µ and M are dynamically realized so that the original 8 s symmetry of color-flavordiagonalSU(3) is explicitly C+V broken down to color SU(2) (color-flavor isospin) in C+V the Lagrangian level; because of this, the octet modes split into the isospin-singlet mode (like eta) and two set ofdoublet(kaonic)modes andthe triplet(pionic)modes as8 3(+1,0, 1)+2(+1,0)+2(0, 1)+1(0)where → − − the associated Q˜-charge is indicated in the parenthesis. FIG. 2: (a) The (g)CFL solution of the gap equation under One of the most striking features of the CFL phase is theneutralityconstraints. The(g)CFL isenergetically taken the absence of electrons; the electric neutrality is real- over by the UQM phase on the left side of µ = 558.4MeV ized solely by the quark sector n = n = n as can be u d s denotedbytheverticaldashedline. (b) Theexcitationgaps seeninFig.2(c). ThisCFLphasebehavesasQ˜-insulator in the nine quasi-quark spectra as a function of µ. The Q˜- because of the absence of gapless Q˜-carriers [11]. charge associated with each mode is also indicated. (c) The As the density is decreased, the stress energy quark and electron densities are displayed. The lines on the µ (µ)/2 + M2(µ)/4µ [11] becomes large in the CFL lefthandsidearedensitiesintheUQMphase,whilethoseon − 8 s theright hand side are densities along the(g)CFL solution. state. When it reaches ∆ = ∆ , the first qualitative 1 2 8 change takes place; this happens when the chemical po- tential is decreased down to µ = 581.1MeV. At this point, the CFL state continuously turns into the gCFL phase. Just at the gCFL onset, the excitation gaps for the one doublet modes reaches zero as can be seen in Fig. 2(b). The situation is similar to the K0 condensa- tion (with a small fraction of K+ condensation) [39, 40] because the ub-sr and db-sg modes belong to the color- isospin SU(2) doublet, and have Q˜ = +1 and 0, re- C+V spectively. When the chemical potential is decreased further, the gap parameters split into three different values al- though they are all still finite. Accordingly, the isospin SU(2) symmetry gets broken by µ and µ so that C+V e 3 the gaps in quasi-quarkdispersions alltake different val- ues (see FIG. 2(b)). We remark that the electron chem- ical potential or its density actually serves as an order parameteroftheCFL/gCFL(insulator/metal)transition as claimed in [11]. The gCFL phase continues to be the ground state down to µ = 558.4MeV below which the UQM phase is more favorable in terms of the thermodynamic poten- tial. When the transition gCFL UQM takes place, → there should be the large re-configuration of the flavor contents as can be seen in FIG. 2(c). Thus, this transi- tionrequiresalotofelectro-weakprocesseswhichinclude the d production like ug(ur) dg(dr)+e+ +ν in ad- e → dition to the decay to the gapless modes accompanied by the electron production db ub+e−+ν¯ and the s e → quark decay sb+u db+u. We should note that there → still remains an open interesting question how the UQM droplets are dynamically formed in the gCFL phase and grow against the surface tension. Wehaveseenthat,asthedensityisdecreased,theCFL phase turns into the gCFL phase, and then the gCFL phase gets taken over by the UQM phase. Accordingly, the number of gapped quasi-quark modes decrease as 9 (CFL) 7 (gCFL) 0 (UQM) at T = 0 as shown in → → the previous work [24]. Next, we will discuss how the situation is changed in the T =0 case. 6 b. Phase transitions and crossovers for finite temper- ature: We show here that the quark matter undergoes a sequence of the transitions, CFL(9) gCFL (8) 8 → → uSC(6) guSC(5) 2SC(4) g2SC(2) UQM(0) → → → → as T becomes large; the number of the gapped modes in FIG.3: (a) Thegapandmassparameters,andthechemical each phase is indicated with a parenthesis. potentials as a function of T at µ = 590MeV so that the In FIG. 3(a), we show the T-dependence of the gaps, systematT =0staysintheCFLphase(seeFIG.2). (b) The masses, and chemical potentials. The figure shows that nine quasi-quark gaps versus T. Nonzero Q˜-charge is also the ∆1 first melts and then ∆2 disappears as T is in- indicated by 1. (c) Just an enlargement of the figure(b). creased. The thermally-robustest pairing is the 2SC, ± while second one is the uSC, which is in agreement with the result in [29]. This conclusion will be found to be wehavealsoplottedtheT-dependenceofthegapsinthe also consistentwith the Ginzburg-Landauanalysisgiven nine quasi-quark spectra in FIG. 3(b). FIG. 3(c) is just in Sec. IIIE, where we extend the previous study [22] an enlargement of (b). The three large points on the and see that the large value of the strange quark mass horizontal line indicate T = T (η = 1,2,3) at which cη actually disfavors the dSC phase. ∆ vanishes (the same as the large points in FIG. 3(a)). η Inordertostudythethermaltransitioninmoredetail, When the temperature is increasedfrom T =0, the first 9 FIG.4: Thephasediagram fortheweakcoupling(∆ =80MeV)(a),andfortheintermediatecoupling(∆ =125MeV)(b). 0 0 qualitative change occurs at T 15.8MeV where one mode. When ∆ 0, these two modes get unpaired 2 ∼ → of the db-sg dispersion becomes gapless and the gCFL to become the bare ub and sr quarks. This guSC(5) 8 → phase (gCFL in [31]) sets in. Unlike the gCFL phase 2SC(4) phase transition is of second order. at T = 0, the ub-sr dispersion does not become gapless As T is increased further, the crossover 2SC(4) → at this point. From the viewpoint of the insulator-to- g2SC(2) takes place at T 26.6MeV, and finally the ∼ metal transition, however, the sharp phase transition is g2SC(2) is taken over by the UQM(0) phase through a smoothen not only due to the absence of the gapless ub- second order phase transition at T 27.4MeV. sr mode with Q˜ = 1, but also to the thermally-excited We have found that the excitati∼on gaps behave in a on-shell quasi-quarks at finite temperature as the latter more complicated manner than the gap parameters as ingredient is already noticed in [23]. In fact, from the functions of T. However, as we explained, there are no FIG. 3(b), we can see that the ub-sr (Q˜ = +1) mode sharp boundary with thermodynamical singularity be- is lighter than the db-sg (Q˜ = 1) mode for T < 5MeV tween CFL(9) and gCFL8(8), uSC(6) and guSC(5), and andaccordinglythereisalittle−excessofthequas∼i-quarks 2SC(4) and 2SC(2) transitions. For this reason, we did with Q˜ = +1 in the system. In order to achieve the notindicate these crossoverboundariesin the phase dia- neutrality, there must be the equal amount of electrons gram of FIG. 1. sothatµ takespositivefinitevalueaslongasT isfinite. e As a result, the CFL-gCFL (insulator-metal) transition 8 becomes a smooth crossover. B. Phases for weak and intermediate coupling When T is increased further beyond the gCFL onset, 8 ∆ disappears at T 18.4MeV. This is a second order Let us next examine the weak and intermediate cou- 1 ∼ phasetransitionofgCFL uSC.Asaconsequence,two plingcases(∆ =80and∆ =125MeV). InFIG.4,the 8 0 0 → quasi-quarkmodesbecomegaplessasisseeninFIG.3(b); phase diagrams for the both cases are shown. one is the sg-db mode i.e., the partner of the gapless bd- sg mode, while the other is the isospin singlet dg-sb-ur mode. 1. Phases for the intermediate coupling Next qualitative change occurs when T is increased to T 21.2MeV. At this point, the ub-sr mode with We first discuss the case of the intermediate coupling Q˜ =+∼1 becomes gapless and the guSC(5) phase sets in. ∆ =125MeV, in advance of the weak coupling case. 0 Notice that because the thermally-excited quasi-quarks FIG. 4(b) shows that the UQM and gCFL phases dis- arealreadypresentinthesystemovertherange(Eub-sr < appear at T =0 and the g2SC (CFL) phase exist in the T), there is no sharp boundary between the uSC(6) an∼d intermediate (high) density regime [24]. We notice that guSC(5) phases. thefullygapped2SCphaseexistsinasmallµ-regionbe- At a little higher temperature T 21.8MeV, ∆2 van- tweentheCFLandg2SCphases. Alsowenoticethatthe ∼ ishes and above which the 2SC(4) phase is realized. At transition between the 2SC and g2SC phases is of a first this transition point, the sr-ub mode having Q˜ =+1 be- order accompanied by a jump in the dynamical strange comes gapless. Through the gap parameter ∆ , this sr- quarkmassandotherphysicalquantities. Thisisincon- 2 ubmode is pairedwithits partner,i.e., the gaplessub-sr trast to the usual 2SC/g2SC transition without strange 10 FIG. 6: The occupation number for u and d quarks in the pairing(r,g)sectorasafunctionofmomentumpintheg2SC phase at ∆ =125MeV, µ=450MeV and T =0. 0 FIG. 7: The strange quark density as a function of µ in the 2SC sector. The vertical dashed line is the same as that in FIG. 5. At µ=498.4MeV, the strange quark density jumps accompanying thefirst order 2SC/g2SC transition. quarks. We will later discuss this point in detail. The phases for T = 0 also differ from the extremely 6 FIG.5: (a) Thegapandmassparameters,andthechemical weak coupling case: (i) The window for the uSC phase potentialsversusµalongthe(g)2SCsolution. Thedashedline is pushed away to higher density side and is confined placed on µ = 496.4MeV represents the point above which in a small region. The cross placed on the dashed line the 2SC state is metastable against the CFL. (b) The four represents the point at which the window for the uSC quasi-quark gaps as a function of µ. (c) The fraction of phase opens. (ii) The large dot putted on the CFL/2SC the isospin density (na na) to the iso-scaler density a=r,g d− u transitionlineindicatesthe criticalendpoint; the dashed (na+na);bythesummation,thisquantityismadeof a=r,g u d P linebetweenthelargedotandthecrossshowsthatthere neutral with respect to theremaining SU(2) charges. P color existsthe continuousphasetransitionof2SC CFL.In ↔