Interplay between chiral and deconfinement phase transitions

0.28 MB·English

by  Fukun Xu

Interplay between chiral and deconfinement phase transitions
FukunXu1,TamalK.Mukherjee1,2,HuanChen3,andMeiHuang1,2
1 InstituteofHighEnergyPhysics,ChineseAcademyofSciences,Beijing,China
2 TheoreticalPhysicsCenterforScienceFacilities,ChineseAcademyofSciences,Beijing,China
3 IstitutoNazionalediFisicaNucleare(INFN),SezionediCatania,Italy
1
1 Abstract. By using the dressed Polyakov loop or dual chiral condensate as an equivalent order parameter
0
ofthedeconfinement phasetransition,weinvestigatetherelationbetweenthechiralanddeconfinement phase
2
transitionsatfinitetemperatureanddensityintheframeworkofthree-flavorNambu–Jona-Lasinio(NJL)model.

n Itisfoundthatinthechirallimit,thecriticaltemperatureforchiralphasetransitioncoincideswiththatofthe
a dressedPolyakovloopinthewhole(T,µ)plane.Inthecaseofexplicitchiralsymmetrybreaking,itisfoundthat
J the phase transitionsareflavordependent. For eachflavor,the transitiontemperature for chiral restorationTχ
c
5 issmallerthanthatofthedressedPolyakovloopTD inthelowbaryondensityregionwherethetransitionisa
c
1 crossover,and,thetwocriticaltemperaturescoincideinthehighbaryondensityregionwherethephasetransition
isoffirstorder.Therefore,therearetwocriticalendpoints,i.e,Tu,d andTs atfinitedensity.Wealsoexplain
CEP CEP
] thefeatureofTχ =TDinthecaseof1stand2ndorderphasetransitions,andTχ <TDinthecaseofcrossover,
h c c c c
andexpectthisfeatureisgeneralandcanbeextendedtofullQCDtheory.
p
-
p
e 1 Introduction In recent years, three lattice groups, MILC group [21],
h
RBC-Bielefeldgroup[22]andWuppertal-Budapestgroup
[ Theinterplaybetweenchiralsymmetrybreakingandcon-
[23,24,25]havestudiedthechiralanddeconfinementphase
1 finement as well as the chiral and deconfinement phase transitiontemperatureswithalmostphysicalquarkmasses.
v transitionsatfinitetemperatureanddensityareofcontin- TheRBC-Bielefeldgroupclaimedthatthetwocriticaltem-
2 uous interests [1,2,3,4,5,6,7,8,9]. The two transitions are
peraturesforN = 2+1coincideatT = 192(7)(4)MeV.
5 characterizedbythebreakingandrestorationofchiraland f c
The Wuppetal-Budapest group found that for the case of
9 center symmetry, which are well defined in two extreme
N = 2 + 1, there are three transition temperatures, the
2 f
quark mass limits, respectively. In the chiral limit when
. transitiontemperatureforchiralrestorationofu,d quarks
1 thecurrentquarkmassiszerom=0,thechiralcondensate Tχ(ud) =151(3)(3)MeV,thetransitiontemperatureforchi-
0 istheorderparameterforthechiralphasetransition.When c
1 thecurrentquarkmassgoestoinfinitym → ∞,QCDbe- ralrestorationofsquarkTcχ(s) =175(2)(4)MeVandthede-
1 confinement transition temperature Td = 176(3)(4)MeV.
comespuregaugeSU(3)theory,whichiscentersymmet- c
: From this result, we can read that the critical tempera-
v ricinthevacuum,andtheusuallyusedorderparameteris
tures for different transitions are different. According to
i thePolyakovloop[1].
X the Wuppetal-Budapestgroup, this is the consequence of
Therelationbetweenthechiralanddeconfinementphase
r transitionshasattractedmoreinterestrecentlyinstudying thecrossovernature.
a
the phase diagram at high baryon density region [10]. It In the framework of QCD effective models, there is
is conjectured in Ref. [11] that in large N limit, a con- still no dynamical model which can describe the chiral
c
finedbutchiralsymmetricphase,whichis calledquarky- symmetry breakingand confinementsimultaneously. The
onicphasecanexistinthehighbaryondensityregion.Itis maindifficultyofeffectiveQCDmodeltoincludeconfine-
veryinterestingtostudywhetherthisquarkyonicphasecan ment mechanism lies in that it is difficultto calculate the
survivein realQCD phasediagram,andhowit competes Polyakov loop analytically. Currently, the popular mod-
with nuclear matter and the color superconductingphase elsusedtoinvestigatethechiralanddeconfinementphase
[12]. (However,it is worthy of noticing that in Ref. [13], transitions are the Polyakov Nambu-Jona-Lasinio model
it is found that at zero chemical potential, the lattice re- (PNJL)[26,27,28,29,30,31,32,33]andPolyakovlinearsigma
sultsforthethermodynamicalpropertieshaveaverymild model(PLSM)[34,35].However,theshortcomingofthese
dependenceonthenumberofcolors.) modelsisthatthetemperaturedependenceofthePolyakov-
Lattice QCD at the current stage cannot go to very looppotentialis putin byhandfromlattice result, which
high baryon density. For zero chemical potential, previ- cannotbe self-consistentlyextendedto finite baryonden-
ouslattice resultsshow thatthe chiraland deconfinement sity.Recently,effortshavebeenmadeinRef.[36]toderive
phase transitions occur at the same temperature, e.g, in alow-energyeffectivetheoryforconfinement-deconfinement
Ref. [14,15,16,17,18], and also in review papers [19,20]. andchiral-symmetrybreaking/restorationfromfirst-principle.
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Recentinvestigationrevealedthatquarkpropagator,heat where0 ≤ φ < 2πis thephaseangleandβ isthe inverse
kernelscanalsoactasanorderparameterastheytransform temperature.
nontriviallyunderthecentertransformationrelatedtode- DualquarkcondensateΣ isthendefinedbytheFourier
n
confinement transition [37,38,39]. But the exciting result transform(w.r.tthephaseφ)ofthegeneralboundarycon-
isthebehaviorofspectralsumoftheDiracoperatorunder ditiondependentquarkcondensate[42,43,44],
centertransformation[38,40,41,42].Aneworderparame-
ter, called dressed Polyakovloop has been definedwhich 2πdφ
canberepresentedasaspectralsumoftheDiracoperator Σn =−Z 2πe−inφhψ¯ψiφ, (2)
[42]. It has been found the infrared part of the spectrum 0
particularlyplaysaleadingroleinconfinement[38].This
wherenisthewindingnumber.
resultisencouragingsinceitgivesahopetorelatethechi-
Particularcaseofn = 1iscalledthedressedPolyakov
ral phase transition with the confinement-deconfinement
loop which transforms in the same way as the conven-
phasetransition.Theorderparameterforchiralphasetran-
tional thin Polyakovloop under the center symmetry and
sitionisrelatedtothespectraldensityoftheDiracopera-
henceis anorderparameterforthe deconfinementtransi-
torthroughBanks-Casherrelation[4].Therefore,boththe
tion [42,43,44]. It reduces to the thin Polyakov loop and
dressed Polyakov loop and the chiral condensate are re-
tothedualoftheconventionalchiralcondensateininfinite
latedtothespectralsumoftheDiracoperator.
and zero quarkmass limits respectively,i.e., in the chiral
BehaviorofthedressedPolyakovloopismainlystud-
limit m → 0 we get the dual of the conventional chiral
iedintheframeworkofLatticegaugetheory[43,44].Apart
condensateandinthem→∞limitwehavethinPolyakov
from that, studies based on Dyson-Schwinger equations
loop[42,43,44].
[45,46,47]andPNJLmodel[48]havebeencarriedout.In
TheLagrangianofthree-flavorNJLmodel[50]isgiven
thosestudiestheroleofdressedPolyakovloopasanorder
as
parameter is discussed at zero chemical potential. In this
paper,we show ourresults[49] ofinvestigatingthe QCD
phase diagram at finite temperature and density by using L =ψ¯(iγµ∂µ−m)ψ+Gs (ψ¯τaψ)2+(ψ¯iγ5τaψ)2
thedressedPolyakovloopasanequivalentorderparame- Xa (cid:26) (cid:27)
ter inthe Nambu-Jona-Lasinio(NJL) model. Itisknown
− K Det [ψ¯(1+γ )ψ]+Det [ψ¯(1−γ )ψ] . (3)
thatheNJLmodellacksofconfinementandthegluondy- (cid:26) f 5 f 5 (cid:27)
namics is encoded in a static coupling constant for four
pointcontactinteraction.However,assuming thatwe can Where ψ = (u,d,s)T denotes the transpose of the quark
read the information of confinementfrom the dual chiral field,andm=Diag(m ,m ,m )isthecorrespondingmass
u d s
condensate,it wouldbe interestingto see the behaviorof matrix in the flavor space. τ with a = 1,···,N2 −1 are
a f
the dressed Polyakov loop in a scenario without any ex- the eight Gell-Mann matrices, and Det means determi-
f
plicitmechanismforconfinement. nantinflavorspace.Thelasttermisthestandardformof
Inthispaper,weshowthephasetransitionsinthetwo- the’tHooftinteraction,whichisinvariantunderSU(3) ×
L
flavor and three-flavorNJL modelsin (T,µ) plane in chi- SU(3) × U(1) symmetry, but breaks down the U (1)
R B A
rallimit as wellfor smallquarkmasslimit. Thispaperis symmetry.
organizedas follows: We introducethe dressed Polyakov Theφdependentthermodynamicpotentialinthemean
loopas an equivalentorderparameterof confinementde- fieldlevelisgivenasfollowing:
confinement phase transition and the NJL model in Sec.
2. Then in Sec.3, we show the results of two-flavor QCD
Ω = Ω +2G hσi2 −4Khσi hσi hσi (,4)
phase diagram in T − µ plane in the chiral limit and in φ φ,Mf s φ,f φ,u φ,d φ,s
X X
f f
thecaseofexplicitchiralsymmetrybreaking,respectively.
Weofferananalysisontherelationbetweenthechiraland
with
deconfinement phase transitions in Sec. 4. In Sec.5, we
show the phase diagram at finite temperatureand density
d3p 1
faonrdtdhirseceu-sflsaivoonr. case. At the end, we give the conclusion Ωφ,Mf =−2NcZΛ(2π)3(cid:20)Ep,f + βln(1+e−βE−p,f)
1
+ ln(1+e−βE+p,f) . (5)
β (cid:21)
2 Dressed Polyakov loop and the NJL
model Wherethesumisintheflavorspace, Ep,f = p2+Mφ2,f
q
andE± = E ±[µ+i(φ−π)T],withtheconstituentquark
p,f p,f
WefirstlyintroducethedressedPolyakovloop.Todothis mass
wehavetoconsideraU(1)valuedboundaryconditionfor
thefermionicfieldsinthetemporaldirectioninsteadofthe Mφ,i =mi−4Gshσiφ,i+2Khσiφ,jhσiφ,k, (6)
canonicalchoiceofanti-periodicboundarycondition,
where(i, j,k)isthequarkflavorindices(u,d,s),andhσi =
φ,f
ψ(x,β)=e−iφψ(x,0), (1) hψ¯ ψ i .
f f φ
HCBM2010–Hot&ColdBaryonicMatter2010
3 Phase diagram for two-flavorcase
0
-0.005
Wefirstlyshowtheresultsinthetwo-flavorcaseandcon-
-0.01
sider the isospin symmetric limit, i.e, we take N = 2,
f
K = 0 and m = m in Eq.(3). The thermodynamic po- 3] -0.015
u d V
tentialcontainsimaginarypart.Wetakeonlytherealpart e -0.02
G
ofthepotentialandtheimaginaryphasefactorisnotcon- [φ -0.025
sidered in this work. The mean field hσi is obtained by >
minimizing the potential for each valueφof φ ∈ [0,2π) σ< -0.03
for fixed T and µ. The conventional chiral condensate is -0.035
hσiπ = hψ¯ψiπ. The dressed Polyakovloop Σ1 is obtained -0.04
by integrating over the angle. The values of the parame-
-0.045
ters Λ and G are taken as 0.6315GeV and 5.498GeV−2,
s 0 0.5 1 1.5 2
respectively.
φ/π
We investigatephase transitions fortwo cases, i.e., in
thechirallimitandwithexplicitchiralsymmetrybreaking. Fig. 2. Angle variation of hσiφ for different values of tempera-
Fig. 1 and 2 show the behavior of the angle dependence tures and chemical potentials inthe case of m = 5.5MeV. The
of the generalchiralcondensate for variouschemical po- solid line corresponds to T = 150MeV,µ = 100MeV, dashed
tentials and temperatures for m = 0 and m = 5.5MeV, linecorrespondstoT =250MeV,µ=100MeV,dash-dottedline
correspondstoT =20MeV,µ=340MeVanddottedlinecorre-
respectively.Thefourcurvespresentedineachfigurerep-
spondstoT =40MeV,µ=340MeV.
resenttwotemperaturesaboveandbelowthecriticaltem-
perature for two particular values of the chemical poten-
tial.Samequalitativefeatureshavebeenfoundforboththe
is consistent with the expectationof complete restoration
quarkmasses. Thevariationissymmetricalaroundφ = π
ofchiralsymmetryinthechirallimit.
asreportedinotherstudies[45,48].
0.03
0.005
0 0.025 - <σ>π
-0.005
0.02
-0.01 Σ
3] 1
V -0.015 0.015
e
G
-0.02
[φ 0.01
> -0.025
σ
< -0.03 0.005
-0.035
0
-0.04
-0.045 -0.005
0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
φ/π T [GeV]
Fig. 1. Angle variation of hσiφ for different values of tempera- Fig.3.Theconventionalchiralcondensate−hσiπandthedressed
turesandchemicalpotentialsinthecaseofchirallimit.Thesolid PolyakovloopΣ asfunctionsoftemperaturefordifferentvalues
1
linecorrespondstoT =150MeV,µ=100MeV,dashedlinecor- ofthechemicalpotentialsinthechirallimit.Here,−hσi andΣ
π 1
responds to T = 250MeV,µ = 100MeV, dash-dotted line cor- botharemeasuredin[GeV3].Fromrighttoleftthevaluesofthe
respondstoT = 40MeV,µ = 300MeV,anddotcorrespondsto chemicalpotentialare0,200,300MeV,respectively.
T =150MeV,µ=300MeV.
Fig.3and4showthebehavioroftheconventionalchi-
Almostnovariationwithrespecttoangleisfoundfor ral condensate −hσi and the dressed Polyakov loop Σ
π 1
low temperatures.As thetemperatureincreasesthe varia- at different chemical potentials as functions of tempera-
tionovertheanglegrows.Weexpecttheabsolutevalueof ture for m = 0 and m = 5.5MeV, respectively. For both
the chiral condensate decreases with the increase of tem- cases, it is observed there are three temperature regions
perature. However, from the figure, this conventionalbe- for −hσi and Σ . For −hσi , at smaller temperatures it
π 1 π
haviorofthechiralcondensatewithtemperatureonlyper- remainsconstantat a value correspondingto the value of
sists upto acertainangle,beyondwhichtheoppositebe- the conventionalchiral condensate in the vacuum, then it
haviorisobserved.Theplateauaroundφ = πismoreflat rapidly decreases in a small window of temperature and
aboveT in case ofzerocurrentquarkmassandits value eventuallyalmostsaturatestoalowervalue.Therapidde-
c
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0.035 0.2
0.03 - <σ>π
0.16
0.025
0.02
Σ V] 0.12
1 e
0.015 G
0.01 T [
0.08
0.005
from Σ
0 0.04 from <σ>1π
-0.005
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.1 0.2 0.3
T [GeV] µ [GeV]
Fig.4.Theconventionalchiralcondensate−hσi andthedressed Fig.5.Two-flavorphasediagramintheT −µplaneforthecase
π
Polyakov loop Σ as functions of temperature for different val- of chiral limit.Thesolid line isthe critical linefor Σ , and the
1 1
uesofthechemicalpotentialsinthecaseofexplicitchiralsym- dashedlineisthecriticallineforconventionalchiralphasetran-
metry breaking m = 5.5MeV and −hσi , Σ are measured in sition.Thesolidcircleindicatesthecriticalpointforchiralphase
π 1
[GeV3]. From right to left the values of the chemical potential transition.
are0,200,300MeV,respectively.
0.24
creasingoccursatdifferenttemperaturesfordifferentval-
ues of the chemicalpotentials. On the other hand the be- 0.2
haviorforthedressedPolyakovloopisjusttheopposite.It
remainsalmostzeroforsmalltemperaturesandthenrises
0.16
rapidly,finallysaturatestoahighvaluewhichvariesvery V]
slowlywithtemperatures.ThealmostzerovalueofΣ for e
1 G
smalltemperaturesisduetothefactthattheU(1)bound- T [ 0.12
ary condition dependentgeneral quark condensate nearly
doesnotvarywiththeangleφforsmalltemperatures(see 0.08
Eq.2).
from Σ
Forfinitequarkmass,nearthecriticaltemperaturere- 1
gion,both−hσi andΣ changemoreslowlythanthosein 0.04 from <σ>π
π 1
thecaseofchirallimit.
0 0.1 0.2 0.3
Fig. 5 and 6 show the T − µ phase diagram for the
µ [GeV]
case of m = 0 and m = 5.5MeV, respectively. The tran-
sition temperaturesare calculatedfromthe slope analysis Fig.6.Two-flavorphasediagramintheT −µplaneforthecase
oftheconventionalchiralcondensatehσi andthedressed ofm=5.5MeV.ThesolidlineisthecriticallineforΣ ,andthe
π 1
Polyakovloop.Thetransitiontemperaturescalculatedfrom dashedlineisthecriticallineforconventionalchiralphasetran-
theconventionalchiralcondensaterepresentthechiralphase sition.Thesolidcircleindicatesthecriticalendpoint forchiral
transition temperature.On the other handthe behaviorof phasetransition.
the dressedPolyakovloopis supposedto indicatethede-
confinementtransitiontemperature.In ourpresentframe-
work confinement is not accounted for. However, if we differentin the low baryondensityregion.The difference
look at the curves presented in figure 3 and 4, they still however decreases from low to high chemical potential,
show an order parameter like behavior. We would like to and the two critical temperatures start to match around
point out here that for the same reason as stated above the critical end point for chiral phase transition. For zero
we are not concerned here with the order of the phase chemicalpotentialandsmallcurrentquarkmassm=5.5MeV,
transtionfromthedressedPolyakovloopandallthecom- we find about 7MeV difference between Tχ and TD, and
c c
ments about the order of the phase transition below are Tχ <TD.Similartrendhasbeenobservedinanotherstudy
c c
withrespecttothechiralphasetransitiononly. basedonDyson-Schwingerapproach[45],wheretheyfound
Form = 0case,wefindalmostexactmatchingforthe chiral transition to occur about 10 − 20MeV below the
transition temperatures calculated from these two quanti- deconfinement transition. Though these studies are not a
tiesinthewholeT −µplaneasshowninFig.5. complete oneand these differencesmay be due to the ef-
For the case of finite quark mass m = 5.5MeV, it is fectsofcrossovertransition.Aspointedoutin[25]during
observedfromFig.6thatthetwocriticaltemperaturesare crossover,differentobservablesareexpectedtobehavedif-
HCBM2010–Hot&ColdBaryonicMatter2010
ferently and there is no way to define a unique crossover
temperature.
Weextendourstudyfurthertoseewhathappensifwe 2.0
increase the current quark mass further. We find at zero T<Tc
chemical potential, the difference between the transition T=Tc
temperatures calculated from dressed Polyakov loop and 2 V] 1.5 T>Tc
conventional chiral condensate increases as we increase e
G
tihsezecruorrfeonrtzqeuraorkcumrraesnst(qseuearFkigm.a7s)s.Ibnuittiafollrymthe=d2iff0e0rMenecVe dT [ 1.0
/
wefindabout26MeVdifferencebetweenthetwotemper- >
<
atures. It is worthy of mentioning that this result is just d
0.5
forillustrativepurposeastherearelimitationofusingNJL
modelwithsuchahugecurrentquarkmass.
0.0
0.0 0.5 1.0 1.5 2.0
0.36
/
0.34 Fig. 8. Temperature derivative of the general chiral condensate
dhσiφ form = 0andµ = 0atthreetemperaturecases:T < T χ,
0.32 dT c
T =T χandT >T χ,whereT χisthechiraltransitiontempera-
c c c
GeV] 0.3 ture.
mperature [ 0.28 finitecurrentquarkmass)andtheregionwithlargevalues
Transition te 00..2246 gdoefΣtd1s/hdiσtTsiφmd/edacTxriemsahsureimsnkaasst.tTeTmhχepareenrfdaotrutehr,eetihtnewcironetatersgaenrsa.sliItniinoanEllq,ted.m(Σ71p)/eidr.aeT-.
c
0.22 turesTDandTχcoincideinthecaseofsecondorderchiral
c c
phasetransition.
0.2
TrTarnasnitsioitino nte tmempepreartautruer efr ofrmom < σΣ>1
0.18
0 50 100 150 200
Current quark mass [MeV]
Fig.7.ThecriticaltemperaturesTχandTDfordifferentcurrent
c c
masses. 1.2 T<Tc
T=Tc
2 V] 0.9 T>Tc
e
G
4 The relation betweenTχ and TD T [
c c /d 0.6
>
Inthefollowing,weofferapossibleunderstandingonthe <
d
simultaneityofthetransitiontemperaturesfor1stand2nd 0.3
order chiral phase transitions and the apparent difference
betweenthetwoforthecaseofcrossover.
0.0
As mentioned earlier the transition temperatures are
determinedfromtheslopeanalysisoftheconventionalchi- 0.0 0.5 1.0 1.5 2.0
ral condensate and the dressed Polyakov loop. So let us
/
lookatthetemperaturederivativeofthegeneralchiralcon-
Fig. 9. Temperature derivative of the general chiral condensate
densatedhσiφ/dT asfunctionsofφ,theintegralonwhich dhσiφ for m = 0 and µ = 300MeV at three temperature cases:
gives the temperature derivative of the Dressed Polyakov dT
T <T χ,T =T χandT >T χ,whereT χisthechiraltransition
loop c c c c
temperature.
dΣ1 =− 2πdφe−iφdhσiφ. (7)
dT Z 2π dT
0
Inthecaseoffirstorderchiralphasetransition,thesitu-
Fig. 8 shows dhσi /dT at different temperatures in the
φ ationismorecomplicated.Duetothediscontinuityofhσi
casesofsecondorderchiralphasetransitions.AroundTχ, φ
c at the firstorderchiralphase transitionpoint,the temper-
largevaluesofdhσi /dT appeararoundφ = πanddomi-
φ ature derivative of the dressed Polyakov loop can be ex-
nate the integralin Eq.(7). Below Tχ , dhσi /dT around
c φ pressedas
φ = π increases monotonously as temperature increases.
Asaresult,dΣ1/dT increases.AboveTcχ,dhσiφ/dT inthe dΣ1 =− 2πdφe−iφdhσiφ − Cosφc∆hσi dφc, (8)
centerregionbecomeszero(orverysmallinthecasewith dT Z 2π dT π cdT
0
EPJWebofConferences
where,thefirsttermisdeterminedbytheregularbehavior phasetransitioninchirallimit.Thedifferenceisthatd2hσi /dT2
φ
ofdhσi /dT (seeFig.9),thesecondtermisdueto∆hσi , in thecenterregion(seetheminimuminFig. 10),issup-
φ c
thejumpofhσi atthefirstorderphasetransitionpointat pressedbelowT χ,approacheszeroatT χ andchangesits
φ c c
φ = φ . When T < Tχ, the second term vanishes. Now signaboveT χ.ForT ≤ T χ,d2hσi /dT2 aroundthetwo
c c c c φ
weconsidertwolimitingcases.First,inthecaseofaweak maximumsdominatethe integralin Eq.(7) and d2Σ /dT2
1
first order phase transition, ∆hσic is small and dhσiφ/dT doesnotchangeitssign.AboveTcχ,thenegativepartaround
aroundφ=πislarge,asshowedinFig.9.Sothefirstterm theminimumcancelsthecontributionsfromthemaximums,
dominatestheresultofEq.(8)andgivesthesimilarresult anduptoacertaintemperatureTD,thiscancelationleads
c
asthatinthecaseofasecondorderchiralphasetransition. to the zero of the integralin Eq.(9). In all, the zero point
Second,inthecaseofastrongfirstorderchiralphasetran- of d2Σ /dT2 comes from the negativecontributionof the
1
sition,∆hσicislargeanddhσiφ/dT issmall.Sothesecond minimumatT >Tcχ,sothetransitiontemperaturerelated
termdominatestheresultofEq.(8).Then dΣ1 isstrongly with the dressed Polyakov loop must be higher than the
dependentonthedetailedinformationofdφcd/TdT.Ournu- chiraltransitiontemperature,i.e.TcD >Tcχ.
mericalresultsshowthatdφ /dT decreasesastemperature
c
increases, and so the secondterm also givesa decreasing
contribution. In all, it is clear that dΣ /dT gets its maxi-
1
mumatTχ,i.e.TD andTχ coincideinthecaseofaweak 5 Phase diagramfor three-flavorcase
c c c
firstorderchiralphasetransitionduetoremnantsofsecond
order chiral phase transition. The coincidencein the case
We nowconsiderthephasediagramforthree-flavorcase,
ofageneralfirstorderchiralphasetransitionissupported
andtheparametersaretakenfromRef.[51,52]:
byournumericalresultsandcanbegenerallyexpected.
m [MeV] m [MeV] G Λ2 KΛ5
q s s
0.4 χ Chiral-limit 0 0 1.926 12.36
T < T
c finite-mass 5.5 140.7 1.918 12.36
0.3
Table1.Twosetsofparametersin3-flavorNJLmodel:thecur-
V] 0.2 T = T χ rent quark mass mq for up and down quark and ms for strange
Ge c quark,couplingconstantsGandK,withaspatialmomentumcut-
2 [ 0.1 offΛ=602.3MeV.
T
d
/φ 0
>
σ Inthechirallimit,i.e,forthecaseofm = m = m =
< -0.1 u d s
2 0 case, we find almost exact matching for the transition
d χ
T >T temperatures calculated from these two quantities in the
-0.2 c
whole T −µ plane as shown in Fig. 11. The chiral phase
-0.3 transitionisof1storderinthewholeT −µplane.
0 0.5 1 1.5 2 Forthecaseoffinitequarkmassm = m = 5.5MeV
u d
φπ and ms = 140.7MeV, it is observedfromFig. 12thatthe
the chiral and deconfinement phase transitions are flavor
Fig.10.Secondderivativeofthegeneralchiralcondensate d2dhTσ2iφ dependent.Forthe degeneratelightu,d quarks,theresult
for m = 5.5MeV and µ = 0 at three temperature cases: T < ofphasetransitionsaresimilartothatinFig.6.Thecritical
T χ, T = T χ and T > T χ, where T χ is the chiral transition
c c c c temperaturesofchiralanddeconfinementphasetransitions
temperature.
have very small difference in the low baryon density re-
gionwhenthetransitionisofcrossover,andthedifference
vanishesatthecriticalendpoint.HeretheCEPislocated
For the case with finite quark mass and small chemi-
cal potential, we have no phase transitions but crossover. at (Tu,d ,µu,d ) = (91MeV,315MeV), which is different
CEP CEP
Soletusconsiderthesecondtemperaturederivativeofthe fromFig.6.Thisdifferencecomesfrom:1)differentmodel
dressedPolyakovloop parametershave been used, 2) the couplingof s quark to
u,dquarkcontributesoneextraterminthethermodynam-
d2Σ1 =− 2πdφe−iφd2hσiφ, (9) icalpotentialcomparingwiththepuretwo-flavorcase.For
dT2 Z 2π dT2 the relative heavy s quark, it is found that this quark has
0
separatephasetransitionlines.Atlowbaryondensity,the
whosezeropointdeterminesthetransitiontemperatureTD. variation of s quarkcondensatein the crossoverregionis
c
Fig.10showsthesecondderivativesofthegeneralchiral so diffusedthatitisnotpossible toidentifythecrossover
condensate d2hσi /dT2 at three temperature cases: T < temperature. When density increases, the crossover tem-
φ
T χ,T = T χ andT > T χ,whereT χ isthechiraltransi- peratures for chiral and deconfinementcan be effectively
c c c c
tiontemperature.Similartoourpreviousobservation,large extracted and still shows the relation of Tχ < TD. These
c c
values of d2hσi /dT2 appear around φ = π (see the two twocriticaltemperaturesofsquarkcoincideatthecritical
φ
maximums in Fig. 10), as remnants of the second order endpoint(Ts ,µs )=(75MeV,445MeV).
CEP CEP
HCBM2010–Hot&ColdBaryonicMatter2010
gionwherethetransitionisacrossover.Withtheincrease
ofcurrentquarkmassthedifferencebetweenthetwocrit-
ical temperaturesis found to be increasing. However,the

twocriticaltemperaturescoincideinthehighbaryonden-

150
sityregionwherethephasetransitionisoffirstorder.For
three-flavor case, it is observedthat there are two critical
endpointsinthe(T,µ)phasediagram.
V]100 From symmetry analysis, the dressed Polyakov loop
e
M can be regarded as an equivalent order parameter of de-
T [
confinement phase transition for confining theory. In the
NJLmodel,thegluondynamicsisencodedinastaticcou-
50
pling constant for four point contactinteraction. Since in
this work we have included only quark degrees of free-
dom,a quantitativecomparisonwillnotmatchwithother
0 results. But the interesting fact is that the qualitative fea-
0 50 100 150 200 250 300 350
tures(anglevariation,temperaturevariation)ofthedressed
[MeV]
Polyakovloopremainsthesame.Moreover,weexpectthat
Fig. 11. Three-flavor phase diagram in the T −µ plane for the independentoftheinputofgluedynamicstothequarkprop-
caseofchirallimit.ThesolidlineisthecriticallineforΣ1,and agator,itisageneralfeatureforTcχ =TcDinthecaseof1st
the dashed lineis thecritical line for conventional chiral phase and2ndorderphasetransitions,andTχ < TD inthecase
transition. c c
ofcrossover,whichqualitativelyagreeswiththelatticere-
sultinRef.[25].ThismightindicatethatforfullQCD,in
thecrossovercase,thereexistsasmallregionwherechiral
symmetryisrestoredbutthecolordegreesoffreedomare
250
still confined. This result should be checked in other ef-

u,d-quark fectivemodels,e.g.intheframeworkofDyson-Schwinger

200 equations(DSE).
The (T,µ) phase diagram with three flavors and with
U (1)anomalyaswellasdiquarkcondensatewillbestud-
V] 150 s-quark A
Me iedinthenearfuture.
T [
100 =315, T=91 TuC,EdP TsCEP
=445, T=75
Acknowledgments: M.H. thanks P. Levai and local or-
50
ganizers for hospitality, and she also acknowledgesHun-
garian Academy of Sciences for supporting the local ex-
0 penses. The work of M.H. is supportedby CAS program
0 100 200 300 400 500
”Outstandingyoungscientistsabroadbrought-in”,andNSFC
[MeV]
underthenumberof10735040and10875134,andK.C.Wong
Fig.12.Three-flavorphasediagramintheT−µplaneforthecase EducationFoundation,HongKong.
ofm = m = 5MeV andm = 140.7MeV.Thesolidlinesare
u d s
thecriticallineforΣ ,andthedashedlinesarethecriticallinefor
1
conventionalchiralphasetransition.Thesolidcirclesindicatethe
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