How the Polyakov loop and the regularization affect strangeness and restoration of symmetries at finite T

0.15 MB·English

by  M. C. Ruivo

How the Polyakov loop and the regularization affect
strangeness and restoration of symmetries at finite T
M. C. Ruivo∗, P. Costa∗, C. A. de Sousa∗, H. Hansen† and W. M. Alberico∗∗
∗CentrodeFísicaComputacional
0 DepartamentodeFísica,UniversidadedeCoimbra,Portugal
1 †Univ.Lyon/UCBL,CNRS/IN2P3,IPNL,69622VilleurbanneCedex,France
0
∗∗DipartimentodiFisicaTeorica,UniversityofTorinoandINFN,SezionediTorino,viaP.Giuria1,I-10125
2
Torino,Italy

n
a
Abstract. TheeffectsofthePolyakovloopandof aregularizationprocedure thatallowsthepresenceof highmomentum
J
quarkstatesatfinitetemperatureisinvestigatedwithinthePolyakov-loop extendedNambu-Jona-Lasiniomodel. Thechar-
8
acteristictemperatures,aswellasthebehavior ofobservablesthatsignaldeconfinement andrestorationofchiralandaxial
1
symmetries, areanalyzed, paying special attentiontothebehavior of strangeness degrees of freedom. Weobserve that the

cumulativeeffectsofthePolyakovloopandoftheregularizationprocedurecontributetoabetterdescriptionofthethermody-
]
namics,ascomparedwithlatticeestimations.Wefindafasterpartialrestorationofchiralsymmetryandtherestorationofthe
h
axialsymmetryappearsasanaturalconsequenceofthefullrecoveringofthechiralsymmetrythatwasdynamicallybroken.
p
TheseresultsshowtherelevanceoftheeffectsoftheinterplayamongthePolyakovloopdynamics,thehighmomentumquark
-
p satesandtherestorationofthechiralandaxialsymmetriesatfinitetemperature.
e
h Keywords: PNJLmodel,Restorationofchiralsymmetry,Deconfinement,UA(1)anomaly
PACS: 11.10.Wx,11.30.Rd,12.38.Aw,12.38.Mh,14.65.Bt,25.75.Nq
[

1
v INTRODUCTION
2
7 ThestudyoftheQCDphasediagraminthe(T−m )-planeandthesearchforsignaturesofthequark-gluonplasmahave
0
attractedanintensiveinvestigationoverthelastdecades.Theoutputofthisresearchisexpectedtoplayanimportant
3
. roleinourunderstandingoftheevolutionoftheearlyuniverseandofthephysicsofheavy-ioncollisionsattheBNL
1
andatLHC(CERN).
0
PhasetransitionsassociatedtodeconfinementandrestorationofchiralandaxialU (1)symmetriesareexpectedto
0 A
1 occurat highdensityand/ortemperatureand the investigationof observablesthatsignalsuch criticalbehaviorsis a
: challenging problem. As an approach complementaryto first-principle lattice simulations, we consider an effective
v
model that can treat both the chiral and the deconfinementphase transitions, the Polyakov loop extended Nambu–
i
X Jona-Lasinio(PNJL)model[1,2,3,4,5].ThismodelhastheadvantageofallowingtointerpretfairlythelatticeQCD
r resultsandtoextrapolateintoregionsnotyetaccessibletolatticesimulations.
a
AnontrivialquestioninNJLtypemodelsisthechoiceoftheregularizationprocedure.Infact,forsomeintegrals
thethreedimensionalcutoffisonlynecessaryatzerotemperature,thedroppingofthiscutoffatfiniteTallowsforthe
presenceofhighmomentumquarkstates, leadingto interestingphysicalconsequences,as ithasbeenshownin[6],
wheretheadvantagesanddrawbacksofthisregularizationhavebeendiscussed,intheframeworkoftheNJLmodel.
We will enlarge the use of this procedure to the PNJL model and discuss its influence on the behavior of several
relevantobservables.Let us notice that the choice of a regularizationprocedureis part of the effective modelingof
QCDthermodynamic.Indeedthepresenceofhighmomentumquarksisrequiredtoensurethattheentropyscalesas
T3athightemperature.
Moreover,weshallinvestigatetheroleoftheU (1)anomalywhich,asiswellknown,isresponsiblefortheflavor
A
mixing effect that removes the degeneracy among several mesons and for the non zero value of the topological
susceptibility, hence the restoration of axial symmetry should have relevant consequences on the behavior of these
observables.TheimplementationofthenewcutoffprocedureintheNJLmodel[6]lowersthecriticaltemperaturefor
thephasetransition,andshowsthatrestorationofchiralandaxialsymmetriescanalsobeaphenomenonrelevantin
thestrangesector. Infact,the dynamicallybrokenchiralsymmetryiscompletelyrecovered,inboththestrangeand
nonstrangesectors,leadingtotherestorationoftheaxialsymmetryataboutthesametemperature:thequarkmasses
gotothecurrentvalues,thequarkcondensatesandtopologicalsusceptibilityvanish,themixinganglesgototheideal
valuesandtheU (1)chiralpartnersconverge.
A
On theotherhand,we found[5]thatthe presenceofthe Polyakovloopbringsthecriticaltemperatureto a better
agreement with the most recent results of lattice calculations, and deconfinement and partial restoration of chiral
symmetryoccuratveryclosetemperatures.TherestorationofchiralsymmetryisfasterinthePNJLmodelthaninthe
NJL one, leading to several meaningful effects: the meson-quark coupling constants show a remarkable difference
in both models, there is a faster tendency to recover the Okubo-Zweig-Iizuka rule, and, finally, the topological
susceptibility nicely reproducesthe lattice results around T/T ≈1.0. Moreover,due to the strange components of
c
some mesons, the behavior of the strange quark mass (that decreases faster due to the Polyakov loop) is important
for their properties, as well as for other observables related to the axial anomaly (as noticed in [7] the topological
susceptibilityisstronglyinfluencedbythestrangesector).
TheaimofthispaperistoinvestigatethecumulativeeffectsofthePolyakovloopandoftheinfinitecutoffatfinite
temperature,havinginmindtoanalyzetheinterplayamongtheU (1)anomaly,thePolyakovloopdynamics,thehigh
A
momentumquarkstatesandtherestorationofchiralsymmetry,payingspecialattentiontothebehaviorofthestrange
quark degrees of freedom. In this concern, the characteristic temperatures, the spectrum of scalar and pseudoscalar
mesonchiralpartnersandthetopologicalsusceptibilitywillbeanalyzed.
MODELAND FORMALISM
WeperformourcalculationsintheframeworkofanextendedSU (3)PNJLLagrangian,whichincludesthe’tHooft
f
instanton induced interaction term that breaks the U (1) symmetry; moreover quarks are coupled to a (spatially
A
constant)temporalbackgroundgaugefieldrepresentingthePolyakovloop[8,9,10]:
LPNJL = q¯(ig m Dm −mˆ)q+1gS (cid:229) 8 [(q¯l aq)2 + (q¯ig5l aq)2]
2
a=0
+ g {det[q¯(1+g )q]+det[q¯(1−g )q]}−U F [A],F¯[A];T . (1)
D 5 5
(cid:0) (cid:1)
Here q=(u,d,s) is the quark field with three flavors (N =3) and three colors (N =3), mˆ =diag(m ,m ,m ) is
f c u d s
the currentquark mass matrix and l a are the flavor SU (3) Gell–Mann matrices (a=0,1,...,8), with l 0= 2I.
f 3
The covariant derivative is defined as Dm =¶ m −iAm , with Am =d m A0 (Polyakov gauge); in Euclidean noqtation
0
A0=−iA .ThestrongcouplingconstantG isabsorbedinthedefinitionofAm (x)=G Am (x)la,whereAm
4 Strong Strong a 2 a
isthe(SU (3))gaugefieldandl arethe(color)Gell–Mannmatrices.
c a
The Polyakov loop field F appearing in the potential term of (1) is related to the gauge field through the gauge
covariantaverageofthePolyakovline:
1 b
F (~x)=<<l(~x)>>= Tr <<L(~x)>>, with L(~x)=Pexp i dt A (~x,t ) . (2)
c 4
Nc (cid:20) Z0 (cid:21)
ThePolyakovloopisanorderparameterfortherestorationoftheZ (thecenterofSU (3))symmetryofQCD and
3 c
is related to the deconfinementphase transition: Z is broken in the deconfined phase (F →1) and restored in the
3
confined one (F →0). Several effective potentials for the field F are available in the literature. Here we use the
followingpotential[11],whichisknowntogivesensibleresults:
U F ,F¯;T a(T)
=− FF¯ +b(T)ln[1−6FF¯ +4(F¯3+F 3)−3(FF¯ )2] (3)
(cid:0) T4 (cid:1) 2
with:
T T 2 T 3
0 0 0
a(T)=a +a +a , b(T)=b , (4)
0 1 2 3
(cid:18)T (cid:19) (cid:18)T (cid:19) (cid:18)T (cid:19)
where:a =3.51,a =−2.47,a =15.2,b =−1.75and T =270MeV. The parameterset used for the pure NJL
0 1 2 3 0
sectoristhesameasin[7].
ConcerningthevalueofT ,somecommentsareinorder.Sinceoneofthepurposesofthepresentpaperistomake
0
a comparativestudy of the effects of two types of regularization,we choose to plot the quantities under study on a
relative temperature scale T/T, where T is a characteristic temperature. As noticed by [4] the dependence of the
c c
results on T is mild and in this contextthe physicaloutcomesare notdramaticallymodifiedwhen one changesT .
0 0
ThechoiceT =270MeVappearstobethebetteroneforthisworkbecauseitensuresanalmostexactcoincidence
0
betweenchiralcrossoveranddeconfinementatzerochemicalpotential,asobservedinlatticecalculations.Themean
field equations are obtained by minimizing the thermodynamicalpotential W , which now depends on Fermi-Dirac
distributionfunctionsmodifiedbythePolyakovloop,withrespecttothequarkcondensateshq¯qiandtothefieldsF
i i
andF¯.Themesonmassspectrumiscalculatedusingthesameproceduredescribedin[5]andreferencestherein.
RESULTS AND DISCUSSION
Westartouranalysisbyidentifyingthecharacteristictemperatureswhichseparatethedifferentthermodynamicphases
inthePNJLmodel[4],withtworegularizationproceduresatfinitetemperature:
(i) RegularizationI,wherethecutoffisusedonlyintheintegralsthataredivergent(L →¥ intheconvergentones).
(ii) RegularizationII,wheretheregularizationconsistsintheuseofthecutoffL inallintegrals.
T
600 1.0 ¶/ 7 fi ¥ 7
500 M s 0.8 ¶T, 6 ¶ /¶ T 6
M (MeV)q 340000 M i Teff / Tc 00..46 ¶¶T, <ss>/ 345 ¶¶¶ <<us/s¶u> >T//¶¶ TT 345 ¶¶ / T
120000 fi ¥ 0.2 ¶ uu>/ 12 12
= const <
¶
0 0.0 0 0
0.0 0.5 1.0 1.5 2.0 0.00 0.25 0.50 0.75 1.00 1.25 1.50

T/T T/T
c c
FIGURE1. Leftpanel:comparisonofthequarkmassesinPNJLmodelwithregularizationI(solidlines)andregularizationII
(dashedlines)asfunctionsofthereducedtemperatureT/Tc;thePolyakovloopcrossoverisalsoshown.AtTeff =1.6Tc,Mi=mi.
Rightpanel:derivativesofthequarkcondensatesandofthePolyakovloopfieldF ,withregularizationI.
TABLE1. Characteristictemperatures.
c F
Tc Tc Tc
[MeV] [MeV] [MeV]
I:L →¥ 222 210 216
II:L =const. 258 234 246
Let us analyze the comparative effects of the two regularizationson the quark masses and the field F . The critical
temperature related to the “deconfinement”1 phase transition is TF and the chiral phase transition characteristic
c c
temperature, T , signals partial restoration of chiral symmetry. These temperatures are given, respectively, by the
c
inflexionpointsofF andofthechiralcondensateshq¯qi,andT isdefinedastheaveragevaluebetweenTF andTc .
i i c c c
ThemaineffecttobenoticedisthatregularizationIlowersthecharacteristictemperatures(seeTable1)anddecreases
F c
the gap T −T (24MeV→12MeV), leading therefore to better agreement with lattice results. It is interesting to
c c
notice (Fig.1, right panel) that the inflexion points of F and of the strange quark condensate are almost coincident
1 Theterminology“deconfinement”inourmodelisusedtodesignatethetransitionbetweenF ≃0andF ≃1.
andslightlyseparatedfromtheinflexionpointofhq¯q i.Asalreadyshownin[5],thePolyakovloopleadstoafaster
u u
decreaseofthequarkmassesaroundT .HereweseethatregularizationIenhancesthiseffectandatT =1.6T,it
c eff c
leads to the completerecoveringof the chiralsymmetry that was dynamicallybroken:the quarkmasses go to their
currentvaluesandthequarkcondensatesvanish.Weremarkthattheeffectofallowinghighmomentumquarkstates
isstrongerforthestrangequarkmass.2
0 0 fi ¥
1200 f = const f
V) 1000 0’ 0’ Teff / Tc
e 800 a a
M
( 600
M

M
400
200
0
0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0

T/T T/T
c c
FIGURE2. Comparison of the pseudoscalar and scalar mesons masses inPNJLwithregularization I andregularization II as
functionsofthereducedtemperatureT/Tc.
Since in our model chiral symmetry is explicitly broken by non zero current quark masses, chiral symmetry is
realizedthroughparitydoublingratherthanbymasslessquarks.Theeffectiverestorationofchiralsymmetryshould
besignaledbythedegeneracyofmesonchiralpartners.InFig.2weplotthemassesofscalarandpseudoscalarmesons
withthetworegularizations.OnenoticesthatwithregularizationIIthechiralpartners(s ,p )and(a ,h )degenerate,
0
butnotthechiralpartners(f ,h ′).Theanalysisofthescalarandpseudoscalarmixinganglesshowsthattheyexhibita
0
tendencytogototheiridealvaluesandthats andh mesonsbecamelessstrange,whiletheh ′becomesmorestrange.
TheU (1)partners(p ,h )and(a ,s )becomeclosebutdonotconverge.Inconclusion,chiralsymmetryiseffectively
A 0
restoredinthenonstrangesectorandaxialsymmetryisnotrestored.ThesituationchangeswhenregularizationIis
used:(s ,p )and(a ,h )degeneratefasterands andh becamepurelynonstrange,while(f ,h ′)almostconvergeand
0 0
h ′ becomespurelystrange.TheU (1)chiralpartnersconvergeatT =1.6T andthemixinganglesattaintheideal
A eff c
values.So,chiralandaxialsymmetriesareeffectivelyrestoredbothinthenonstrangeandinthestrangesectors.
TABLE2. Motttemperatures.
T s p h
Mott/Tc
PNJL:L →¥ 1.004 1.050 1.004
PNJL:L =const. 0.963 1.085 1.000
NJL:L =const. 0.820 1.086 0.920
Asitiswellknown,attheMotttemperaturethemesonsdissociateintoqq¯pairsandceasetobeboundstates.The
Polyakovloop has the effect of increasing the Mott temperature (see [5]) of the mesons whose strangeness content
decreases,like the s and theh . We foundthatregularizationI enhancesthis effect(see Table 2),whichmeansthat
thesemesonssurviveasboundstatesfortemperaturesslightlyhigherthanthecriticaltemperature.Finallyweanalyze
the topological susceptibility (Fig.3). We notice that with both regularizations the topological susceptibility has a
sharp decrease around T and there is a nice adjustment to the first lattice points. This effect is mainly due to the
c
2 WeremarkthataboveTeff =1.6Tcthequarkmassesbecomelowerthattheircurrentvaluesandthequarkcondensateschangesign.Toavoidthis
unphysicaleffectweusetheapproximationofimposingbyhandtheconditionthat,aboveTeff,Mi=miandhq¯iqii=0.
200
PNJL NJL
)150
V
e
M
(100

4
1/
50 fi ¥ fi ¥

= const = const
0
0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0
T / T T / T
C C
FIGURE3. Thetopologicalsusceptibility,ascomparedwithlatticeresults,inPNJLandNJLmodels,withtworegularizations.
Polyakovloop,whichisdominantaroundT ;asthetemperatureincreasesregularizationIleadstoasharperdecrease
c
ofthetopologicalsusceptibility,thatvanishesatT =1.6T,thetemperatureatwhichthedynamicallybrokenchiral
eff c
symmetryiscompletelyrestored.Acloseconnectionbetweentherestorationofchiralandaxialsymmetriesisfound.
Indeed,theinflexionpointofthetopologicalsusceptibility,c ,coincideswiththeoneofhq¯q iandisveryclosetothe
u u
oneofhq¯q i(seeFig.1,rightpanel)andthevanishingofthesequantitiesoccursimultaneouslyatT =1.6T (when
s s eff c
M =m).ItcanbenoticedthatthecomparativeeffectofthetworegularizationsisqualitativelysimilarinPNJLand
i i
NJLmodel(Fig.3,leftpanel).ThePolyakovloopfastentheeffectsobservedinNJLmodelandleadstoanicefitto
thefirstfourlatticepoints.
Inconclusion,weobservethat,whiletheeffectsofthePolyakovlooparedominantaroundthecriticaltemperature,
therelevanceofthepresenceofhighmomentquarkstatesisseenalsoathighertemperaturesandinfluencesmainly
thestrangesector.By combiningbotheffects,we geta betterdescriptionofthethermodynamics,ascomparedwith
lattice estimations; we find a faster partial restorationof chiral symmetryand the restorationof the axial symmetry
appearsas a naturalconsequenceof the fullrecoveringof the chiralsymmetry thatwas dynamicallybroken.These
resultsshowtherelevanceoftheinterplayamongU (1)anomaly,thePolyakovloopdynamics,thehighmomentum
A
quarkstatesandtherestorationofthechiralsymmetryatfinitetemperature.
ACKNOWLEDGMENTS
WorksupportedbyprojectCERN/FP/83644/2008,FCT.
REFERENCES
1. P.N.MeisingerandM.C.Ogilvie,Phys.Lett.B379163(1996)
2. R.D.PisarskiPhys.Rev.D62,111501(R)(2000).
3. C.Ratti,M.A.Thaler,andW.Weise,Phys.Rev.D73014019(2006).
4. H.Hansen,W.M.Alberico,A.Beraudo,A.Molinari,M.Nardi,andC.Ratti,Phys.Rev.D75065004(2007).
5. P.Costa,M.C.Ruivo,C.A.deSousa,H.Hansen,andW.M.Alberico,Phys.Rev.D79,116003(2009).
6. P.Costa,M.C.Ruivo,andC.A.deSousa,Phys.Rev.D77,096009(2008).
7. P.Costa,M.C.Ruivo,C.A.deSousa,andYu.Kalinovsky,Phys.Rev.D70,116013(2004);71(2005).
8. Wei-jieFu,Z.Zhang,andYu-xinLiu,Phys.Rev.D77,014006(2008).
9. M.Ciminale,R.Gatto,N.D.Ippolito,G.Nardulli,andM.Ruggieri,Phys.Rev.D77,054023(2008).
10. K.Fukushima,Phys.Rev.D77,114028(2008).
11. S.Roessner,C.Ratti,andW.Weise,Phys.Rev.D75,034007(2007).