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 --

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by  H. Abuki

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
↔