Effective gluon potential and Yang-Mills thermodynamics

0.08 MB·English

by  Chihiro Sasaki

Effective gluon potential and Yang-Mills
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1 thermodynamics
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Chihiro Sasaki∗

] FrankfurtInstituteforAdvancedStudies,D-60438FrankfurtamMain,Germany
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E-mail:[email protected]
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p Krzysztof Redlich
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InstituteofTheoreticalPhysics,UniversityofWroclaw,PL-50204Wrocław,Poland
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[ E-mail:[email protected]

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v WederivethePolyakov-loopthermodynamicpotentialintheperturbativeapproachtopureSU(3)
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4 Yang-Mills theory. The potential expressed in terms of the Polyakov loop in the fundamental
6 representationcorrespondstothatofthestrong-coupling expansion,ofwhichtherelevantcoef-
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ficientsofthegluonenergydistributionarespecifiedbycharactersoftheSU(3)group. Athigh
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temperature,thepotentialexhibitsthecorrectasymptoticbehavior,whereasatlowtemperature,
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3 it disfavors gluons as appropriate dynamical degrees of freedom. To quantify the Yang-Mills
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thermodynamicsinconfinedphase,weintroduceahybridapproachwhichmatchestheeffective
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gluon potential to that of glueballs, constrained by the QCD trace anomaly in terms of dilaton
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fields.
r
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XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
EffectivegluonpotentialandYMthermodynamics ChihiroSasaki
1. Introduction
The structure of the QCD phase diagram and thermodynamics at finite baryon density is of
crucial importance in heavy-ion phenomenology. Due to the sign problem in lattice calculations,
a major approach to a finite density QCD is based on effective Lagrangians possessing the same
global symmetries as the underlying QCD. The SU(N ) Yang-Mills theory has a global Z(N )
c c
symmetrywhichisdynamically brokenathightemperature. Thisischaracterized bythePolyakov
loop that plays a role of an order parameter of the Z(N ) symmetry [1]. Effective models for
c
the Polyakov loop were suggested as a macroscopic approach to the pure gauge theory [2, 3].
Theirthermodynamicsisqualitativelyinagreementwiththatobtainedinlatticegaugetheories[4].
Alternative approaches arebasedonthequasi-particle picture ofthermalgluons [5]. Whengluons
propagating in a constant gluon background are considered, the quasi-particle models naturally
merge with the Polyakov loops, that appear in the partition function, as characters of the color
gaugegroup[6,7,8,9,10].
In this contribution we show, that the SU(3) gluon thermodynamic potential can be derived
directly from the Yang-Mills theory and isexpressed in terms ofthe Polyakov loops in the funda-
mentalrepresentation. Wesummarizeitspropertiesandarguethatathighttemperatures,itexhibits
the correct asymptotic behavior, whereas at low temperatures, it disfavors gluons [11]. We there-
foresuggestahybridapproachtoYang-Millsthermodynamics,whichcombinestheeffectivegluon
potential withglueballs implementedasdilatonfields.
2. Thermodynamics ofhotgluons
Westartfromthepartition function ofthepureYang-Millstheory
Z= DAm DCDC¯ exp i d4xLYM , (2.1)
Z (cid:20) Z (cid:21)
with gluon Am and ghost C fields. Following [3, 12] we employ the background field method to
evaluate the functional integral. The gluon field is decomposed into the background A¯m and the
quantum Aˇm fields,
Am =A¯m +gAˇm . (2.2)
Thepartitionfunction isarranged as
d3p
lnZ=V lndet 1−Lˆ e−|~p|/T +lnM(f ,f ), (2.3)
(2p )3 A 1 2
Z
(cid:16) (cid:17)
where Lˆ is the Polyakov loop matrix in the adjoint representation and the two angular variables,
A
f andf ,represent therankofthe SU(3)group. TheM(f ,f )istheHaarmeasure
1 2 1 2
8 f −f 2f +f f +2f
M(f ,f ) = sin2 1 2 sin2 1 2 sin2 1 2 , (2.4)
1 2 9p 2 2 2 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
forafixedvolumeV,whichisnormalized suchthat
2p 2p
df df M(f ,f )=1. (2.5)
1 2 1 2
0 0
Z Z
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EffectivegluonpotentialandYMthermodynamics ChihiroSasaki
ThefirstterminEq.(2.3)yieldsthegluonthermodynamic potential
d3p
W =2T trln 1−Lˆ e−Eg/T , (2.6)
g (2p )3 A
Z
(cid:16) (cid:17)
where E = |~p|2+M2 is the quasi-gluon energy and the effective gluon mass M is introduced
g g g
fromphenomenological reasons.
q
We define the gauge invariant quantities from the Polyakov loop matrix in the fundamental
representation Lˆ ,as
F
1 1
F = trLˆ , F¯ = trLˆ† . (2.7)
3 F 3 F
Performingthetraceovercolorsandexpressing itintermsofF anditsconjugateF¯,onearrivesat
W =2T d3p ln 1+ (cid:229) 8 C e−nEg/T , (2.8)
g (2p )3 n
Z n=1 !
withthecoefficientsC ,
n
C = 1,
8
C = C =1−9FF¯ ,
1 7
C = C =1−27FF¯ +27 F¯3+F 3 ,
2 6
C3 = C5=−2+27FF¯ −8(cid:0)1 FF¯ 2,(cid:1)
C = 2 −1+9FF¯ −27 F¯3(cid:0)+F 3(cid:1) +81 FF¯ 2 . (2.9)
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h i
Thus,thegluonenergydistributionisidentifiedso(cid:0)lelybyth(cid:1)echara(cid:0)cters(cid:1)ofthefundamentalandthe
conjugate representations ofthe SU(3)gaugegroup.
WeintroduceaneffectivethermodynamicpotentialinthelargevolumelimitfromEq.(2.3)as
follows:
W = W g+W F +c0, (2.10)
whereW isgivenbyEq.(2.8)andtheHaarmeasurepartisfoundas
g
W F = −a0Tln 1−6FF¯ +4 F 3+F¯3 −3 FF¯ 2 . (2.11)
h i
The potential (2.10) has, in general, three free para(cid:0)meters; a(cid:1), c (cid:0)and t(cid:1)he gluon mass M . They
0 0 g
can be chosen e.g. to reproduce the equation of state obtained in lattice gauge theories. It is
straightforward tosee, thattheresultofanon-interacting boson gasisrecovered atasymptotically
hightemperature. Indeed, taking F ,F¯ →1onefinds
d3p
W (F =F¯ =1)=16T ln 1−e−Eg/T . (2.12)
g (2p )3
Z
(cid:16) (cid:17)
On the other hand, for a sufficiently large M /T, as expected near the phase transition, one can
g
approximate thepotential as
W ≃ T2Mg2 (cid:229) 8 CnK (nb M ), (2.13)
g p 2 n 2 g
n=1
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EffectivegluonpotentialandYMthermodynamics ChihiroSasaki
with the Bessel function K (x). In the quasi-particle approach, the above result can also be con-
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sidered as a strong-coupling expansion, regarding the relation M (T)=g(T)T with an effective
g
gaugecoupling g(T).
Theeffective action tothe next-to-leading order ofthe strong coupling expansion is obtained
intermsofgroupcharacters as[10],
S(SC)=l S +l S +l S +l S , (2.14)
eff 10 10 20 20 11 11 21 21
with products of characters S , specified by two integers p and q counting the numbers of fun-
pq
damental and conjugate representations, and couplings l being real functions of temperature.
pq
Making the character expansion of Eq. (2.13), one readily finds the correspondence between S
pq
andC as
n
C =S , C =S , C =S , C =S . (2.15)
1,7 10 2,6 21 3,5 11 4 20
Onthe other hand, taking the leading contribution, exp[−M /T]in the expansion, the “mini-
g
malmodel”isdeduced with
W ≃−F(T,M )FF¯ , (2.16)
g g
wherethenegativesignisrequiredforafirst-ordertransition[10]. ThefunctionF canbeextracted
from Eq. (2.10) and the resulting potential is of the form widely used in the PNJL model [13, 14,
15,16].
3. A hybridapproach
Although the potential (2.10) describes quite well thermodynamics in deconfined phase, it
totally fails in the confined phase. In the confined phase, hF i=0 is dynamically favored by the
ground state,thustheC =1termremainsasthemaincontribution. Consequently
1
d3p
W (F =F¯ =0)≃2T ln 1+e−Eg/T . (3.1)
g (2p )3
Z
(cid:16) (cid:17)
One clearly sees that W does not posses the correct sign in front of exp[−E /T], expected from
g g
the Bose-Einstein statistics. This implies that the entropy and the energy densities are negative.
Ontheotherhand, ifone usesthe approximated form (2.16), thepressure vanishes atanytemper-
ature below T . Obviously, this isan unphysical behavior since there exist color-singlet states, i.e.
c
glueballs, contributing tothermodynamics andtheymustgenerate anon-vanishing pressure.
This aspect is in a striking contrast to the quark sector. The thermodynamic potential for
quarks andanti-quarks with N flavorsisobtained as[13,17]
f
d3p
W =−2N T ln 1+N F +F¯e−E+/T e−E+/T +e−3E+/T +(m →−m ), (3.2)
q+q¯ f (2p )3 c
Z
h (cid:16) (cid:17) i
with E±=E ∓m being the energy of aquark oranti-quark. In the limit, F ,F¯ →0, the one- and
q
two-quark states are suppressed and only the three-quark (“baryonic”) states, ∼exp(−3E(±)/T),
survives. This, on a qualitative level, is similar to confinement properties in QCD thermodynam-
ics [15]. One should, however, keep in mind, that such quark models yield only colored quarks
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EffectivegluonpotentialandYMthermodynamics ChihiroSasaki
beingstatistically suppressed atlowtemperatures. Ontheotherhand,unphysical thermodynamics
below T described by the gluon sector (2.10) apparently indicates, that gluons are physically for-
c
bidden. Interestingly, this property is not spoiled by the presence of quarks. Indeed, in this case
andatT <T thethermodynamic potential isapproximated as
c
T2 M 2N 3M
W +W ≃ M2K g − fM2K q . (3.3)
g q+q¯ p 2 g 2 T 3 q 2 T
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
Assuming that glueballs and nucleons are made from two weakly-interacting massive gluons and
threemassivequarksrespectivelyandputtingempiricalnumbers,M =1.7GeVandM =
glueball nucleon
0.94 GeV, one finds that M =0.85 GeV and M =0.31 GeV. Substituting these mass values in
g q
Eq. (3.3), one still gets the negative entropy density at any temperature and for either N =2 or
f
N =3,asfoundinthepureYang-Millstheory.
f
Theunphysical equationofstate(EoS)inconfinedphasecanbeavoided, whengluondegrees
offreedomarereplacedwithglueballs. Aglueballisintroducedasadilatonfieldc representingthe
mn
gluoncompositehAmn A i,whichisresponsiblefortheQCDtraceanomaly[18]. TheLagrangian
isofthestandard form,
1 B c 4 c 4
Lc = 2¶ m c¶ m c −Vc , Vc = 4 c ln c −1 , (3.4)
0 " 0 #
(cid:18) (cid:19) (cid:18) (cid:19)
withthebagconstantBandadimensionfulquantityc ,tobefixedfromthevacuumenergydensity
0
andtheglueball mass. Onereadilyfindsthethermodynamic potential oftheglueballs as
B d3p
W = W c +Vc + 4, W c =T (2p )3 ln 1−e−Ec /T ,
Z
(cid:16) (cid:17)
¶ 2Vc
Ec = |~p|2+Mc2, Mc2 = ¶c 2 , (3.5)
q
whereaconstant B/4isaddedsothatW =0atzerotemperature.
We propose the following hybrid approach which accounts for gluons and glueballs degrees
offreedom bycombining Eqs.(2.10)and(3.5),
W =Q (T −T)W (c )+Q (T−T )W (F ). (3.6)
c c
For a given M , the model parameters, a and c , are fixed by requiring, that W (F ) yields a first-
g 0 0
order phase transition at T =270 MeV and that W (c ) and W (F ) match at T . The resulting EoS
c c
followsgeneraltrendsseeninlatticedata[11]. Themodelcanbeimprovedfurtherbyintroducing
athermalgluonmass,M (T)∼g(T)T,ascarriedoute.g. in[8].
g
4. Summary
Wehavederivedthethermodynamic potentialinthe SU(3)Yang-Millstheoryinthepresence
of a uniform gluon background field. The potential accounts for quantum statistics and repro-
ducesanidealgaslimitathightemperature. Withinthecharacterexpansion, theone-to-one corre-
spondence to the effective action in the strong-coupling expansion isobtained. Different effective
potentials usedsofarappearaslimitingcasesofourresult.
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EffectivegluonpotentialandYMthermodynamics ChihiroSasaki
The phenomenological consequence is that gluons are disfavored as appropriate degrees of
freedom in confined phase. This property is in remarkable contrast to the description of “con-
finement” within a class of chiral models with Polyakov loops [13, 16], where colored quarks are
activatedatanytemperature. FurtherinvestigationsoftheSU(3)gluodynamicsguidedbyavailable
lattice results with the effective gluon mass and a more realistic description of an effective QCD
thermodynamics withquarksaredesired.
Acknowledgments
C. S. acknowledges partial support by the Hessian LOEWE initiative through the Helmholtz
International Center for FAIR (HIC for FAIR). K.R. acknowledges support by the Polish Science
Foundation (NCN).
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