Table Of ContentMelting Pattern of Diquark Condensates in Quark Matter
K. Iida,1 T. Matsuura,2 M. Tachibana,1 and T. Hatsuda2
1
The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan
2
Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
Thermal color superconducting phase transitions in high density three-flavor quark matter are
investigatedin theGinzburg-Landauapproach. Effectsofnonzerostrange quarkmass, electric and
colorchargeneutrality,anddirectinstantonsareconsidered. Weakcouplingcalculations showthat
aninterplaybetweenthemassandelectricneutralityeffectsnearthecriticaltemperaturegivesrise
to three successive second-order phase transitions as the temperature increases: a modified color-
flavor locked (mCFL) phase (ud, ds, and us pairings) → a “dSC” phase (ud and ds pairings) →
an isoscalar pairing phase (ud pairing) → a normal phase (no pairing). The dSC phase is novel in
thesensethatwhilealleight gluonsareMeissner screenedasin themCFL phase,threeoutof nine
quarkquasiparticles are gapless.
4
0
PACSnumbers: 12.38.-t,12.38.Mh,26.60.+c
0
2
n Unraveling the phase structure at high baryon den- d,s quarks and u quarks differ. Through this chemi-
a
sity is one of the most challenging problems in quantum cal potential difference, the GL potential acquires an-
J
1 chromodynamics(QCD). Among others,colorsupercon- other ms dependence, whichis the essentialoriginof the
ductivity in cold dense quark matter has been discussed dSC phase. Thirdly, the instanton contribution gives an
2
fromvariousviewpoints[1,2]. Inrelationtorealsystems ms dependence totheGLpotentialthroughtheeffective
2 suchasnewlyborncompactstarsinstellarcollapse,itis four-fermion interaction proportional to m . We finally
s
v important to study the color superconductivity not only note that in weak coupling, color neutrality makes neg-
3
as a function of the quark chemical potential µ but also ligible difference in the GL potential near the transition
6
as a function of the temperature T. This is because the temperatures [4].
3
2 possible presence of color superconducting quark matter We assume that diquark pairing takes place in the
1 in a star affects the star’s thermal evolution [3]. color-flavorantisymmetric channel with JP =0+, which
3 ThepurposeofthisLetteristoinvestigatephasetran- is predicted to be the most attractive channel in weak
0
sitionsincolorsuperconductingquarkmatterwith three coupling [9]. In this case, the pairing gap of a quark
/
h flavors (uds) and three colors (RGB) near the transi- of color b and flavor j with that of color c and fla-
p tion temperatures. We consider a realistic situation in vor k has the form φbcjk = ǫabcǫijk(da)i [4]. Here
p- whichnonzerostrangequark massms, electric andcolor the 3× 3 matrix (da)i transforms as a vector under
e chargeneutrality,anddirectinstantonstakeeffect. Aswe G = SU(3)C SU(3)L+R U(1)B and belongs to the
v:h nshoanlzlesreoeminwaenadkecloeuctprliicngne(umtrsa,ΛlitQyCaDre≪imµp),ortthaenetffiencttshaotf (3∗F,o3r∗)arephreosm×enotgaetnioenouosf SsUy×s(t3e)mC ×oSfUm(3a)sLs+leRs.s quarks
s
Xi they induce multiple phase transitions that change the (mu,d,s = 0), the GL potential is invariant under
pattern of diquark pairing as T increases. In particular, SU(3)C SU(3)L+R and valid near the critical temper-
r ×
a wefindanewphase,whichwecall“dSC,”asaninterface ature, Tc, common to all states belonging to the channel
between a modified type of color-flavor locked (mCFL) considered here. This potential reads [4, 10]
phase and an isoscalar two-flavor(2SC) phase. S =α¯ d 2+β ( d 2)2+β d ∗ d 2, (1)
a 1 a 2 a b
Throughout this Letter, we adopt the Ginzburg- Xa | | Xa | | Xab | · |
Landau(GL)approachnearthetransitiontemperatures,
which was previously used to study the massless three- where(da)i =(dua,dda,dsa)andtheinnerproductistaken
flavorcase[4,5,6]andisamoreadvantageousframework for flavorindices. In the weak coupling regime, the coef-
to weak coupling calculations than other mean-field ap- ficients are [4]
proaches [7, 8]. In a realistic situation, the GL potential
7ζ(3)
acquires the following corrections. First of all, nonzero β1 =β2 = 8(πT )2N(µ)≡β, α¯ =4N(µ)t≡α0t, (2)
c
m affects the potential through the s quark propagator
s
[8] in such a way as to lower the temperature at which a whereN(µ)=µ2/2π2isthedensityofstatesattheFermi
diquark condensate with s quarks dissolves. This is be- surface,andt=(T T )/T is the reducedtemperature.
c c
−
cause the pairing interaction due to one-gluon exchange Withtheparameters (2),onefindsasecondorderphase
is effectively diminished by m if the pair contains the s transition at T = T from the CFL phase (di δi) to
s c a ∝ a
quark. Secondly, when quark matter with nonzero m is the normal phase (di =0) in mean-field theory [4].
s a
beta equilibrated and neutralized by electrons near the Let us now consider the effect of a nonzero m in
s
transitiontemperatures,thechemicalpotentialsbetween the quark propagator on the GL potential. We assume
2
m = 0 for simplicity and consider the high density pairinggapaffectonlythequartictermsintheGLpoten-
u,d
regime, m µ. Near T the leading effect of m is tial. By combining this estimate with the weak coupling
s c s
≪
to modify the quadratic term in the GL potential (1). expression for η/µ given in Ref. [4], we obtain
e
The corrections to the quartic terms are subleading and
m2 µ
negligible. Since ms affects only us and ds pairings, the η ≃α08µs2 ln(cid:18)T (cid:19)∼α0σ. (7)
correction to the quadratic term has the form c
Since η >0, ds pairingis more favorablethan udand us
ǫ (|dua|2+|dda|2)=ǫ (|da|2−|dsa|2). (3) pairings. Thisfeaturestemsfromthemodificationbyδµi
Xa Xa to the exponential factor of Tc, exp[ 3π2/(√2g)], which
−
tends to increase (decrease) the critical temperature for
Note that this correction induces an asymmetry in the
ij pairing when δµ +δµ >0(<0).
i j
flavor structure of the CFL phase.
We consider color neutrality of the system as well. In
In weak coupling, ǫ can be calculated by including m
s contrast to the case at T = 0, however, it affects only
in the Nambu-Gor’kov quark propagator when evaluat-
the quartic terms in the GL potential through possi-
ing the thermodynamic potential. Following Ref. [4], we
ble chemical potential differences between colors [4, 11],
expand the thermodynamic potential not only in di but
a and in weak coupling its magnitude is suppressed by
also in ms up to O(m2s), and obtain ((Tc/gµ)2) compared to the leading quartic terms.
O
Thuscolorneutralityhasnoessentialconsequencetothe
m2 µ
ǫ α s ln 2α σ. (4) phase transitions considered in this Letter. A major dif-
≃ 04µ2 (cid:18)T (cid:19)∼ 0
c ference between corrections of the charge neutrality and
the color neutrality is that the former shifts the quark
Here the dimensionless parameter σ is given by
chemical potentials even in the normal phase, while the
3π2 m2 latter works only when the pairing occurs. This is why
σ = s , (5) the former is more important than the latter near T .
(cid:18)8√2(cid:19)gµ2 c
The direct instanton at nonzero m , which induces
s
an effective four-fermion interaction between u and d
wheregistheQCDcouplingconstant. Aslongasσ 1,
≪ quarks [12], leads to a quadratic term in the GL poten-
which is relevant at asymptotically high density, the fol-
tial, ζ ds 2. An explicit weak coupling calculation
lowing GL analysis near Tc is valid. In the latter es- a| a|
timate in Eq. (4) we use the leading-order result in g, shows tPhat ζ α0(ms/µ)(ΛQCD/µ)9(1/g)14. The neg-
∼−
ln(T /µ) 3π2/(√2g) [9]. This behavior of T orig- ative sign indicates that the instanton favors ud pairing
c c
∼ − as does one-gluonexchange [see Eq. (3)], but the magni-
inates from the long-range color magnetic interaction
tudeofζ ishighlysuppressedathighdensities. Hereafter
which prevails in the relativistic regime. As a result of
we will thus ignore instanton effects.
themodificationbym toln(T /µ),ǫhasapositivevalue
s c
Since the two effects of nonzero m , characterized by
such that ud pairing is favored over us and ds pairings. s
Eqs.(3)and(6), favorudpairingandds pairing,respec-
Consequently,theCFLphasebecomesasymmetricinfla-
tively, the finite temperature transition from the CFL
vorspace and its criticaltemperature is lowered,leading
totheappearanceofthe2SCphase(di δis)justbelow to the normal phase at ms = 0 is expected to be signifi-
a ∝ cantlymodified. Infact,successivecolor-flavorunlocking
T [8]. Note that Eq.(3) has no effect on the 2SC phase.
c
transitionstakeplaceinsteadofasimultaneousunlocking
WealsonotethatT itselfismodifiedbym throughthe
c s
of all color-flavor combinations. To describe this hierar-
modification of the normal medium as T (1+ (gσ)).
c
O chical thermal unlocking, it is convenient to introduce a
We turn to discuss effects of charge neutrality which
parameterization,
alsodependonm asmentionedabove. Underbetaequi-
s
librium and charge neutrality, the electron chemical po- ∆ 0 0
1
tential µe and the shift δµi of the chemical potential of dia =0 ∆2 0 , (8)
flwaitvhoreilefcrtormictchheaargveerqag.e(Wµ)easreeerferloamtedRbeyf.δ[µ4i]=th−atqitµhee, 0 0 ∆3
i
where∆ areassumedwithoutlossofgeneralitytobe
thermodynamic-potential correction due to δµ has the 1,2,3
i
real. We also name the phases for later convenience as
form
∆ =0 : mCFL,
η (13 |da|2− |dua|2). (6) ∆11,2=,306, ∆2,3 6=0 : uSC, (9)
Xa Xa ∆2 =0, ∆1,3 =0 : dSC,
6
∆ =0, ∆ =0 : 2SC,
1,2 3
In weak coupling, where one may regard normal quark 6
matter and electrons as noninteracting Fermi gases, µ where dSC (uSC) stands for superconductivity in which
e
is related to m as µ = m2/4µ. This estimate is valid for d (u) quarks all three colors are involved in the pair-
s e s
in the vicinity of T where corrections to µ by a finite ing.
c e
3
normal
mCFL
T T 1 3
c c TIII
c dSC
CFL 2SC 2SC 2
4
2SC
TII 2
c 5 D3
TS
c 3
dSC
4
mCFL TcI 3 D21
mCFL
2
D2
(a) (b) (c)
I II III
FIG. 1: Transition temperatures of the three-flavor color su- Tc Tc Tc T
perconductor in weak coupling: (a) all quarks are massless; FIG. 2: A schematic illustration of the gaps squared as a
(b) nonzero ms in the quark propagator is considered; (c) function of T.
electric charge neutrality is further imposed. The numbers
attached tothe arrows are in unitsof σTc.
allowsthe2SCphasetoappearattemperaturesbetween
Ts andT . Asonegoesfrom(b)to(c),chargeneutrality
c c
Intermsofthe parameterization(8),the GLpotential setsin,whichactstodecreasethechemicalpotentialofu
with corrections of (m2) to the quadratic term, Eqs. quarksby(2/3)µ andincreasethatofdandsquarksby
O s e
(3) and (6), reads (1/3)µ . Since µ >0, the average chemical potential of
e e
quarksinvolvedinds pairingincreases,whilethoseinud
S = α¯′(∆2+∆2+∆2) ǫ∆2 η∆2
1 2 3 − 3− 1 and us pairings decrease equally. Accordingly, the tran-
+ β1(∆21+∆22+∆23)2+β2(∆41+∆42+∆43),(10) sition temperatures further change from Tc to TcIII and
from Ts to TI and TII.
where α¯′ =α¯+ǫ+ η. c c c
3 Now we examine in more detail how the color-flavor
We proceed to analyse the phase structure dictated
unlocking in case (c) proceeds with increasing T from
by Eq. (10) with the weak coupling parameters (2), (4), the region below TI.
c
and (7) up to leading order in g. In Figs. 1 and 2 the (i)JustbelowTI,wehaveaCFL-likephase,butthethree
c
results obtained by solving the coupled algebraic equa-
gaps take different values, with an order ∆ > ∆ >
3 1
tions, ∂S/∂∆ = 0, are summarized. Figure 1(a)
1,2,3 ∆2 =0 (the mCFL phase). The reasonwhy this orderis
shows the second-order phase transition, CFL nor- 6
realized can be understood from the GL potential (10).
→
mal for m = 0. Figures 1(b,c) represent how the phase
s The ǫ-termandη-terminEq.(10) tendto destabilizeus
transitions and their critical temperatures bifurcate as
pairing (∆ ) relative to ud pairing (∆ ) and ds pairing
2 3
we introduce (b) effects of a nonzero m in the quark
s (∆1). Since ǫ > η(> 0), ds pairing is destabilized more
propagator and then (c) effects of charge neutrality. In
effectively than ud pairing. The value of each gap in the
case(b),twosecond-orderphasetransitionsarise,mCFL
mCFL phase reads
(with ∆ = ∆ ) 2SC at T = Ts (1 4σ)T , and
1 2 → c ≡ − c
2suScCce→ssivneorsmecaolnadt-oTrd=erTpch.aIsnectraasnes(icti)o,ntsh,emreCaFriLses thdrSeCe ∆23 = 8αβ0 (cid:18)TcT−T + 38σ(cid:19), (11)
c
→
aatt TT ==TTcIc,IIdI.SCSh→ow2nSCinaFtiTg.=2TiscIIt,haendT-2dSeCpe→ndennocremoafl ∆21 = 8αβ0 (cid:18)TcT−T − 34σ(cid:19), (12)
c
the gaps ∆ for case (c). All the gaps are continuous
1,2,3
α T T 16
functions of T, but their slopes are discontinuous at the ∆2 = 0 c− σ . (13)
2 8β (cid:18) T − 3 (cid:19)
critical points, which reflects the second order nature of c
the transitions in the mean-field treatment of Eq. (10).
The mCFL phase has only a global symmetry
We may understand the bifurcation of the transition
U(1) U(1) in contrastto the globalsym-
C+L+R C+L+R
temperatures in weak coupling as follows. In the mass- ×
metry SU(3) in the CFL phase with m = 0.
C+L+R u,d,s
less case (a), T is degenerate betweenthe CFL and 2SC
c TherearenogaplessquarkexcitationsinbothmCFLand
phases,thechemicalpotentialiscommontoallthreefla-
CFL phases. As T increases, the first unlocking transi-
vors and colors, and the CFL phase is more favorable
tion, the unlocking of ∆ (the pairing between Bu and
2
than the 2SC phase below T . As one goes from (a)
c Rs quarks), takes place at the critical temperature,
to (b), nonzero m sets in, which tends to suppress the
s
pairing interactions including the s quark. The critical 16
TI = 1 σ T . (14)
temperature for the CFL phase is then lowered, which c (cid:18) − 3 (cid:19) c
4
(ii) For TI <T <TII, ∆ =0 and So far, we have studied the phase transitions in the
c c 2
mean-field level. In weak coupling, as shown in Ref. [6]
∆2 = α0 Tc−T + 2σ , (15) in the massless limit, thermally fluctuating gauge fields
3 6β (cid:18) Tc 3 (cid:19) could change the order of the transitions described in
α T T 7 Figs. 1 and 2. A detailed account on this effect will be
∆2 = 0 c− σ . (16)
1 6β (cid:18) T − 3 (cid:19) reported elsewhere [14]; here we recapitulate the impor-
c
tant results. First, the second order transition, mCFL
In this phase, we have only ud and ds pairings (the dSC dSC,remainssecondordereveninthepresenceofthe
→
phase), andthere is a manifest symmetry,U(1) thermalgluonfluctuations. This is because alleightglu-
C+L+R
×
U(1) U(1) U(1) . Bydiagonaliz- ons are Meissner screened at T = TI and thus cannot
C+L+R× C+V+B× C+V+B c
ing the inverse quark propagator in the Nambu-Gor’kov induce a cubic term with respect to the order parameter
representation,wefindthreegaplessquarkexcitationsin in the GL potential. On the other hand, the transitions,
thecolor-flavorcombinations: Bu,Rs,andalinearcom- dSC 2SC and 2SC normal, become weak first or-
→ →
binationofRuandBs. AtT =TII,thesecondunlocking der since some gluons, which are massless in the high
c
transition, the unlocking of ∆ (the pairing between Gs temperaturephase,become Meissnerscreenedinthe low
1
and Bd quarks), takes place at the critical temperature, temperature phase (Table I).
In summary, we have investigated color-flavor un-
7
TII = 1 σ T . (17) lockings at finite temperatures taking into account the
c (cid:18) − 3 (cid:19) c strange quark mass and charge neutrality in the GL ap-
proach. We find three successive unlocking transitions,
(iii) For TII <T <TIII, one finds the 2SC phase, which
c c mCFL dSC 2SC normal, occurring in weak cou-
has only ud pairing with → → →
pling. Mostremarkably,the dSC phase appearsbetween
themCFLand2SCphases. Inthisphasealleightgluons
α T T 1
∆2 = 0 c− σ . (18) areMeissnerscreenedandthethreequarkexcitationsare
3 4β (cid:18) T − 3 (cid:19)
c gapless. The questionofhowthe phasestructureofneu-
tral quark matter obtained near T is connected to that
The 2SC phase has a symmetry SU(2) SU(2) c
C L+R
× × at T = 0, which approaches the CFL phase in the limit
U(1) U(1) [6]. In this phase the s quark
C+B L+R+B
× of high density, is an interesting open problem.
andB quarkexcitationsaregapless. The finalunlocking
We are grateful to G. Baym and H. Abuki for helpful
transition where ∆ (the pairing between Rd and Gu
3
discussions. Thisworkwassupportedinpartby RIKEN
quarks) vanishes occurs at
Special Postdoctoral Researchers Grant No. A11-52040
1 and No. A12-52010, and by the Grants-in-Aid of the
TIII = 1 σ T . (19)
c (cid:18) − 3 (cid:19) c Japanese Ministry of Education, Culture, Sports, Sci-
ence, and Technology (No. 15540254).
Above TIII, the system is in the normal phase.
c
InTableI,wesummarizethesymmetryandthegapless
quark modes in each phase discussed above. The num-
ber of gluons having nonzero Meissner masses, which is
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[3] S. Reddy, M. Sadzikowski, and M. Tachibana, Nucl.
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Symmetry Gapless Numberof ibid. 66, 059903(E) (2002).
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[6] T. Matsuura, K. Iida, T. Hatsuda, and G. Baym, Phys.
mCFL [U(1)]2 none 8
Rev. D (in press); hep-ph/0312042.
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5
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