Table Of ContentThermal unpairing transitions affected by
neutrality constraints and chiral dynamics
Hiroaki Abuki∗ and Teiji Kunihiro∗
∗YukawaInstituteforTheoreticalPhysics,KyotoUniversity,Kyoto606-8502,Japan
6
0
0 Abstract. Wediscussthephasestructureofhomogeneousquarkmatterunderthechargeneutrality
2 constraints,andpresentaunifiedpictureofthethermalunpairingphasetransitionsforawiderange
n ofthequarkdensity.WesupplementourdiscussionsbydevelopingtheGinzburg-Landauanalysis.
a
J
7 Surprisingly rich phases of quark matter under high pressure are being revealed
by extensive efforts [1, 2, 3, 4, 5, 6]. The key ingredients which are important at
1
v moderatedensitycanbedividedintotwocategories;(i)thedynamicaleffectsand(ii)the
5
kinematical effects. The dynamical effects include strong coupling effects [7, 8, 9, 10],
5
0 an interplay between the pairing and chiral dynamics [3, 4, 5, 6] and so on. In this
1 talk, we focus on the competition of the pairing with the chiral dynamics and the
0
kinematical effects such as (1) the strange quark mass and (2) the charge neutrality
6
0 constraints under the b -equilibrium. The kinematical effects will make a stress on the
/ pairingandbringabouttheexoticphasecalled“gapless”superconductivityatmoderate
h
p density [1, 2]; the gapless CFL (gCFL) [2] is one of such examples. The neutrality
- constraints are also known to lead to an interesting complication in the phase diagram
p
e even at finite temperature: For instance, the Ginzburg-Landau analysis [11] shows that
h
the down-quark pairing phase (dSC phase) consisting of only the u-d and d-s pairings
:
v may become the second coldest phase at high density. However, this conclusion seems
i
X tobemodel-dependent.Infact,theNJLanalyseswhichincorporatethechiraldynamics
r [5,6]showthattheseconddensestphaseistheup-quarkpairing(uSC)phaseconsisting
a
of only the u-d and u-s pairing. In this talk, we report our recent work [4] which gives
a systematic and unified picture of the thermal unpairing transitions under the charge
neutralityconstraintsforawiderangeofthequark density.
We start with theLagrangian densityL =q¯(i¶/ −mˆ +mgˆ )q+L withL being
0 0 int int
thefollowing4-fermioncoupling[3]
Lint =GD(cid:229) 3h =1(cid:2)(q¯Phtq¯)(tqP¯h q)(cid:3)+GS(cid:229) a8=0(cid:2)(q¯l Fa q)2+(q¯ig5l Fa q)2(cid:3). (1)
The first term simulates the attractive interaction in the color (flavor) anti-triplet and
JP = 0+ channel in QCD, i.e., (Ph )aijb = ig5Ce h abe h ij. See [4], for the other details of
notation. We take the chiral SU(2) limit mˆ =diag.{0,0,m } with m =80MeV fixed.
0 s s
mˆ in the Lagrangian contains the charge chemical potentials (m ,m ,m ) which couple
e 3 8
totheelectricandtwo diagonalcolorcharge densitiesas
mˆab =m −m Q +m Tab+m Tab+(off-diagonalpart), (2)
ij e ij 3 3 8 8
(a) (b) (c)
100 100
D0 = 25 [MeV] (cid:2)(cid:3)(cid:4)(cid:5)(cid:4)M(cid:6)e(cid:3)V(cid:8) Stress energy Ms2/m [MeV]
(cid:4)(cid:7) 10 20 30 40
80 80 Tc01.0 1 UQM
T [MeV] 642000 cSB UQM 2SCuSC T [MeV] 642000 (cid:9)SB g2SC 2SC uCSCFL Temperature in unit of 0.9 CFLdSC TTcc12DCP 6 TTcc122SCuSCTc3
(g)CFL 0.8
0350 400 450 500 550 600 650 0350 400 450 500 550 600 650 High density (cid:230)Ms2(cid:246) Low density
m [MeV] (cid:1) [MeV] Ł(cid:231)m ł(cid:247)DCP
FIGURE1. Phase diagramcalculated with the extremelyweak (qq) coupling(a) and that with weak
coupling (b). In (a), there is a small region for the realization of gCFL phase at T =0 (bold red line).
(c) shows Tch evaluated by the Ginzburg-Landau analysis with the parameter set GD/GS = 0.42 and
m =500MeV.Whenthecouplingstrengthisincreased,(M2/m ) ,theintersectionofT andT moves
s DCP c2 c1
tohighervalue,whiletheratioofT −T :T −T =1:7wouldnotbeaffected.
c0 c3 c0 DCP
where we have defined Q, T and T as usual [4]. It can be shown that the chemical
3 8
potentials for off-diagonal color densities are unnecessary for the standard diagonal
ansatz for the diquark condensate, i.e., hqaqbi ∼ (Ph )ab with h = 1,2,3 [4]. We de-
i j ij
termine the phases in (m ,T)-plane by calculating the effective potential through the
mean field approximation with the condensate fields, D h = G8dhtqPh qi, and Mˆ −mˆ0 =
− Gs diag.(hu¯ui,hd¯di,hs¯si). We perform the calculation with several values of G /G
2Nc D S
with G set so as to reproduce the dynamical quark mass M = 400MeV in the chiral
S
limit at m =T =0 for a cutoff L =800MeV; we have G L 2 =2.17. Fig. 1(a) and (b)
S
showthephasediagram forG /G ∼=0.42(theextremely weak couplingcase)and that
D S
forG /G ∼=0.63(theweakcouplingcase).Thec SBdenotesthechiral-symmetrybro-
D S
ken phase and the UQM is an abbreviation of the “unpaired quark matter”. For other
phases,see[4].Eachofthesuperconductingphases,CFL, 2SCanduSC, hasitsgapless
version,gCFL, g2SC, and guSC, where somequasi-quarks become gapless in thepres-
ence of the finite background charge chemical potentials. As the value of the diquark
couplingis increased, these“premature” gaplessphases tend to disappear, and thefully
gapped phases dominate the phase diagram. In fact, we can see that the gCFL phase at
T =0 in Fig. 1(a) is taken overby theUQM phase in (b), which can beinterpreted as a
consequenceofthecompetitionbetween thedynamicaland kinematicaleffects [4].
The reason why the uSC shows up in the phase diagram instead of the dSC can be
nicely understood by the Ginzburg-Landau (GL) analysis. We can expand the effective
potentialin terms of the gap parameters near the critical temperature T which denotes
c0
theCFL →UQM transitiontemperatureinthesymmetricmatterwithm =0:
s
L =4N[m ] − f (M )D 2+1g (M )D 2D 2+··· , (3)
GL (cid:8) i s i 2 ij s i j (cid:9)
where N[m ] = m 2/2p 2 is the density of state. It is important that under the neutrality
constraints,theGLcoefficients becomefunctionsofM , whichmay beexpandedas
s
f (M ,T)=a +a M2+a M4+···, g (M ,T)=b +b M2+···. (4)
i s 0i 2i s 4i s ij s 0ij 2ij s
Wecan calculateallthecoefficients usingtheFeynmandiagrams.Theultravioletdiver-
gence appears only in a as a consequence of the singularity in the diquark propagator
0i
at T =T for M =0. We can regularize it by subtracting the equation expressing the
c0 s
Thouless condition which serves as the mass counter term that guarantees the second
order transition at T =T . We can show a =−t (i=1,2,3) witht being the reduced
c0 0i
temperature T−Tc0,and b =d 7z (3) . a was derivedin[11], anda , (b )was ob-
Tc0 0ij ij16p 2Tc20 2i 4i 2ij
tained1 in [4]. As we show below, a (a ,b ) causes a split of the order of M2 (M4)
2i 4i 2ij s s
in the melting temperature; Tc0 → (Tc1,Tc2,Tc3) where D h vanishes at Tch . When T is
increased, the first CFL-to-non-CFL transition with D h → 0 takes place at T = Tch .
1 1
WecandetermineTch bysolvingTch =min.{T1,T2,T3}withTh definedbytherootof
1 1
D h2(Th )=0where D h2(T)=gh−1j fj(Ms,T)is thesolutionof ¶ L¶ ~DGL(cid:12) =~0. Wehave
(cid:12)T
(cid:12)
TTch0 = 1+a2h Ms2+(cid:16)a4h +176zp (23T)c20(cid:229) jb 2−h1j(a2h −a2j)(cid:17)Ms4+···, (5)
by which Tch as a function of Ms can be determined. To find the next melting order
1
parameter D h , we put D h = 0 into Eq. (3) and repeat the same procedure in two
2 1
order parameter space. Finally, we obtain the order of hierarchical melting transitions
(Tch <Tch <Tch ).WeconfirmedthatD 3survivesathighesttemperaturesothath 3=3
1 2 3
irrespective of the value of M . In contrast, which of D and D first vanishes with
s 1 2
increasing T depends on M ; in fact, we have found the doubly critical strange quark
s
massMDCP above(below)whichtheuSC(dSC)isrealized asthesecondcoldestphase.
s
In Fig. 1(c), we have given the phase diagram in the (M2/m ,T)-plane, obtained by the
s
GLanalysisforG /G ∼=0.42and m =500MeV.
D S
In conclusion,wehavemadean extensiveanalysisof thephasediagram ofthe quark
matter and given a unified view on the thermal unpairing transitions. By extending the
earlier work [11] with the higher order effects of the strange quark mass on the pairing
taken into account, we have shown how the window for the dSC-realizaion in the high
density regime tends to close towards lower density. It should also be stressed that an
analyticexpressionforthedoublycriticalpoint[12]can bederivedinourframework.
REFERENCES
1. I.ShovkovyandM.Huang,Phys.Lett.B564,205(2003).
2. M.Alford,C.KouvarisandK.Rajagopal,Phys.Rev.Lett.92,222001(2004).
3. H.Abuki,M.KitazawaandT.Kunihiro,Phys.Lett.B615,102(2005)[arXiv:hep-ph/0412382].
4. H.AbukiandT.Kunihiro,arXiv:hep-ph/0509172.
5. S. B. Ruster, V. Werth, M. Buballa, I. A. Shovkovyand D. H. Rischke, Phys. Rev. D 72, 034004
(2005)[arXiv:hep-ph/0503184].
6. D. Blaschke, S. Fredriksson, H. Grigorian, A. M. Oztas and F. Sandin, Phys. Rev. D 72, 065020
(2005)[arXiv:hep-ph/0503194].
7. H.Abuki,T.HatsudaandK.Itakura,Phys.Rev.D65,074014(2002)[arXiv:hep-ph/0109013].
8. Y.NishidaandH.Abuki,Phys.Rev.D72,096004(2005)[arXiv:hep-ph/0504083].
9. M.Kitazawa,T.Kunihiro,T.KoideandY.Nemoto,Prog.Theor.Phys.114,117(2005).
10. L.He,M.JinandP.Zhuang,arXiv:hep-ph/0511300.
11. K.Iida,T.Matsuura,M.TachibanaandT.Hatsuda,Phys.Rev.Lett.93,132001(2004).
12. K.Fukushima,C.KouvarisandK.Rajagopal,Phys.Rev.D71,034002(2005).
1 Onehastotakecarethatthereisafeedbacktob fromtheFermigaspartofthermodynamicpotential.
2ij
