QCD Green's Functions and Phases of Strongly-Interacting Matter

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by  Reinhard Alkofer

QCD Green’s Functions and Phases of Strongly-Interacting Matter
ReinhardAlkofer,a,MarioMitter,b,andBernd-JochenSchaefer,c
Institutfu¨rPhysik,Karl-Franzens–Universita¨tGraz,Universita¨tsplatz5,8010Graz,Austria
Abstract. AfterpresentingabriefsummaryoffunctionalapproachestoQCDatvanishingtemperaturesand
1 densitiestheapplicationofQCDGreen’sfunctionsatnon-vanishing temperatureandvanishingdensityisdis-
1 cussed.Itispointedoutinwhichwaytheinfraredbehaviorofthegluonpropagatorreflectsthe(de-)confinement
0
transition. Numerical results for the quark propagator are given thereby verifying the relation between (de-)-
2
confinement and dynamical chiral symmetry breaking (restoration). Last but not least some resultsof Dyson-

n Schwingerequationsforthecolor-superconductingphaseatlargedensitiesareshown.
a
J

0 1 Introduction:Why Green’s functions? tinuumquantumfieldtheory,inprinciple,aninfinitehier-
1 archyof coupledcomplicatedequationshasto be solved.
The QCD Green’s functions, and thereby especially the AsforQCD, we areofcoursemostinterestedin theirin-
]
h onesintheLandaugauge,havebeeninthefocusofmany fraredbehaviour,i.e.,intheGreen’sfunctionsinthestron-
p recentinvestigations.Inprinciple,theyprovideacomplete gly-interactingdomain.Tothisenditturnsoutthatrestrict-
- descriptionoftheStrongInteraction.Thisimpliesthatthey ingoneselftotheprimitivelydivergentGreen’sfunctions,
p
alsoembodythenon-perturbativephenomenaofQCD,a- which are the most interesting ones, provides an appro-
e
h mongst them most prominently confinement, dynamical priate starting point(see e.g. [4] and referencestherein).
[ chiral symmetry breaking,and the axial anomaly.On the EvenintheLandaugaugewiththeleastnumberofprim-
otherhand,theyserveasinputintohadronphenomenology itively divergentfunctionsa formidable task is left: there
1 based on bound-state equations: mesons and their prop- aresevenprimitivelydivergentfunctionswhileinLandau
v
erties are studied from solutions of Bethe-Salpeter, and gaugeYang-Millstheorytherearestillfiveprimitivelydi-
7
baryonsfromthesolutionsofcovariantFaddeevequations.1 vergentfunctions,namelythegluonandghostpropagators
7
9 Ofcourse,thisraisesimmediatelythequestionwhether as well as the 3-gluon, 4-gluon and the gluon-ghost ver-
1 Green’s functions are suitable to study the properties of tices.Includingquarks,onehastoconsiderinadditionthe
. thedifferentQCDphasesandthephasediagramofQCD. quarkpropagatorandthequark-gluonvertex.
1
Although this question will likeley be answered affirma- Thefunctionalapproachesincludemethodswhichare
0
1 tively,theinvestigations,asdescribedinthefollowing,are basedontheDyson-SchwingerEquations(DSEs),theFunc-
1 definitely at an early stage and require several improve- tional Renormalisation Group Equations (FRG), and the
: mentsbeforetheycanbeconsideredconclusive.Neverthe- n-ParticleIrreducibleActions(nPI).Hereby,theDSEs,be-
v
less,theyarecertainlynotonlyverypromisingbutalsothe ingtheequationsofmotionfortheGreen’sfunctions,are
i
X bestaccesstomapouttheQCDphasediagram. derived straightforwardly from the generating functional.
r Beforecontinuingwith abriefsummaryoffunctional TheFRGisbasedontheideaofemployingenergy-momen-
a approachesto QCD at vanishing temperaturesand densi- tum cutoffsto “integrateout”thehigh-momentummodes
ties, we want to point out that studies of the QCD phase andthenPIareeffectiveactionsallowtoderivesymmetry-
diagram within functional methods have been discussed preservingequationsfortheGreen’sfunctions.
from a broader perspective than here in [3] (see also ref- Theadvantagesoffunctionalmethodsare:
erencestherein).
– Itisstraightforwardtoimplementchiralfermionsand
Goldstone’stheorem.
– Thereexistanalyticalinfraredsolutions.
2 FunctionalApproaches to QCD
– Hadrons are described in terms of their fundamental
substructure.
As stated above,QCD Green’s functionsare a promising
– There is no sign problem at non-vanishing chemical
powerfultool.However,inordertodeterminethemincon-
potential.
a e-mail:[email protected] Compared to the most widely employednon-perturbative
b e-mail:[email protected] first-principle method, the lattice gauge theory, these ad-
c e-mail:[email protected] vantagesareclearbenefits.Butthesedonotcomeforfree,
1 Arecentexampleofsuchcalculationscanbefoundin[1,2] because, on the other hand, functional methods miss the
andreferencestherein. mainadvantagesoflatticegaugetheory:
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[11,12,13].Withtheknowledgethattheghost-gluonvertex
staysregularintheinfraredonecannowanalyzetheghost
propagatorDSE.Power-lawlikeansa¨tzeforthegluonand
ghostrenormalizationfunctionsleadtoarelationbetween
theinfraredexponents[6](thiskindofrelationsarenowa-
days known as scaling relations). Denoting by Z(p2) and
G(p2) the gluon and the ghost renormalization function,
respectively,oneobtainsfor p2 →0
Z(p2)∼(p2)2κ, G(p2)∼(p2)−κ, (1)
intermsoftheghostpropagatorinfraredexponentκ.Asit
Fig. 1. The Landau gauge gluon renormalization function ob-
can beshownthat0.5 < κ < 1 [14] thegluonpropagator
tainedfromdifferentcalculations,seetextfordetails.
isinfraredsuppressedandtheghostpropagatorisinfrared
enhanced.3
– Therearenotruncations. ThistypeofinfraredanalysisofDSEscanbeextended
– Manifestgaugeinvariance. toallYang-Millsvertexfunctions[17,18].4 Employingin
additiontheFRG equations,andrequiringthatthesetwo,
This makes plain that, whenever possible, an intelligent seeminglydifferent,towersofequationshavetogiveiden-
combinationsofmethodswillprovidethemostreliablere-
ticalGreen’sfunctions,allowsonetorestrictthetypeofso-
sults.
lutionsstrongly.Thereisoneuniquescalingsolutionwith
power laws for the Green’s functions [20,21] and a one-
parameterfamilyofsolutions,theso-calleddecouplingso-
3 What do we know for T = 0 and µ = 0? lutions.Atthispointsomenotesareinorder.Quite some
time ago, in the Coulomb gauge, a similar situation has
3.1 InfraredStructureofLandaugaugeYang-Mills been found in variational approaches [22]: A family of
theory infrared trivial solution possesses as an endpoint a criti-
cal solution characterized by some infrared power laws.
In order to demonstrate the progress of functional meth- As it concernsthe Landaugauge:Numericalsolutions of
ods over the last decade it is interesting to compare dif- thedecouplingtype(therecalled“massivesolution”)have
ferentsteps in the calculation of the Landau gaugegluon beenpublishedin[23,24]andreferencestherein.Arecent
propagatorwithcorrespondinglatticeresults.Manylattice detailed description and comparison of these two type of
calculations of the gluon propagator are available, and it solutions has been given in Ref. [9].5 Firstly, one has to
would be beyond the scope of these proceedings to cite notethatthescalingsolutionrespectstheBRSTsymmetry
only the major steps in improving on these calculations. whereasthedecouplingsolutionsbreakthissymmetry[9].
In Fig. 1 we have chosen the lattice results for the gluon Secondly,almostalllatticecalculationsofthegluonprop-
renormalizationfunction(see also Eq.(1))ofRef. [5] for agator favor a decoupling solution. But lattice studies at
comparison,butanyotherrecentcalculationwouldequally strongcoupling[27,28,29]revealtheexistenceofaregime
wellbesuitable.Thelowestlyinglinedisplaystheresults wherethescalingrelationbetweenthegluonandtheghost
of Ref. [6] in which the coupled DSEs for the gluon and propagatorisfulfilled,andthecorrespondinginfraredex-
ghost propagatorshave been solved employing some ap- ponentκisveryclosetothevaluedeterminedintruncated
proximative treatment of the involved angular integrals. continuumstudieswith κ = 0.595.A potentialresolution
ThesecondlowestcurveisfromRef.[7]inwhichthean- of this puzzle has been offered recently in Ref. [30]: The
gularintegralscouldbetreatedwithoutanyapproximation. infraredbehaviouroftheGreen’sfunctiondependsonthe
ThecurveaboveistheresultofRef.[8]wheretheFRGhas non-perturbativecompletionofthegauge.
been used to solve for the propagators.The highest lying
At this pointit has to be emphasized that this discus-
curvewhichbecomesalsoclosesttothelatticedataisare-
sionisoffundamentalimportance(especiallywithrespect
sultobtainedbyJanPawlowskibasedontheworkofRef.
to the role of the BRST symmetry in a non-perturbative
[9]whereDSEandFRGmethodshavebeencombined.2
How has this success of functional methods been a-
chieved?Tothisendoneshouldnotethattheghost-gluon 3 It is an interesting side remark that withthis behaviour the
vertex becomes bare in the limit of vanishing ghost mo- Kugo–Ojima confinement criterion, the Oehme–Zimmermann
mentum due to the transversality of the gluon propaga- superconvergence relation, and the Gribov–Zwanziger horizon
tor. This has been noticed already 40 years ago [10] and condition are fulfilled, see, e.g., the review [15] for a detailed
has been tested in several non-perturbativeinvestigations discussion.
4 NotethatarecentlypublishedMATHEMATICAcodecanbe
2 We thank Jan Pawlowski for discussing with us his work usedtosimplifysignificantlythederivationofthecorresponding
in preparation and Lorenz von Smekal for a compilation of DSEs[19].
the data (see also his talk under http://www.thphys.uni-heidel- 5 Theinfraredanalysisforbothtypeofsolutionsisdescribed
berg.de/∼smp/view/Main/Delta10Programme). inRefs.[25,26].
HotandColdBaryonicMatter–HCBM2010
approach)butthedifferencebetweenthetwotypesofso- chiral symmetry is broken, either explicitely or dynami-
lutionsisphenomenologicallyirrelevant.6 cally.
For completenesswe will give the infrared behaviour Thisself-consistentquarkpropagatorandquark-gluon
ofallone-particleirreducibleGreen’sfunctionsinthescal- vertexsolutionrelatestoquarkconfinementviatheanoma-
ing solution in the simplified case with only one external lousinfraredexponentofthefour-quarkfunction.Thestatic
scale p2 → 0. For a function with n external ghost and quarkpotentialcanbecalculatedfromthisfour-quarkone-
antighostaswellasmgluonlegsonehas: particleirreducibleGreenfunction,whichbehaveslike
(p2)−2 for p2 → 0duetotheinfraredenhancementofthe
Γn,m(p2)∼(p2)(n−m)κ. (2) quark-gluonvertexforvanishinggluonmomentum.Using
thewell-knownrelationforafunctionF ∝(p2)−2onegets
NotethatthissolutionfulfillsallDSEs,allFRGequations
andallSlavnov-Tayloridentities.Inaddition,itverifiesthe d3p
hypothesisof infraredghostdominance[31] and leads to V(r)= F(p0 =0,p)eipr ∼ |r| (3)
Z (2π)3
infrareddiverging3-and4-gluonvertexfunctions.
Asafurthersideremarkwewanttoaddthatthegluon forthestaticquark-antiquarkpotentialV(r).Thereforethe
propagatorviolates positivity [32,33] which is a property infrared divergence of the quark-gluon vertex, as found
related to the confinement of transverse gluons. Further- in the scaling solution of the coupled system of DSEs,
more,inRef.[32]ananalyticstructureforthegluonprop- thevertexovercompensatestheinfaredsuppressionofthe
agatorhasbeenproposedinwhichthegluonpropagatoris gluonpropagatorsuchthatoneobtainsalinearlyrisingpo-
analyticinthecomplexp2-planeexceptthespace-likereal tential.
half-axis. Generally,ifanapproachiscompleteithastodescribe
all phenomena of a given theory. This especially implies
thatalsotheaxialU (1)anomalyshouldbedescribedwith
A
3.2 Quarks:Confinementvs.DχSBandtheUA(1) the Green’s functions, or in other words, that the QCD
anomaly Green’s functionsincorporatethe topologicalsusceptibil-
ityoftheQCDvacuum.Tothisendwewouldliketopoint
Independentofthetypeofthesolutions–scalingordecou- outthattheinfrareddivergenceofthequark-gluonvertex,
pling – the gluon propagator is infrared suppressed, and mentionedabove,indeed generatesan η′ mass and there-
therefore quark confinement cannot be generated by any fore describes the axial anomaly [36]. Following an old
typeofgluonexchangetogetherwithaninfrared-bounded idea[37]whichattributedtheη′ masstotheinfraredslav-
quark-gluonvertex.ToaddquarksoneconsiderstheDSE eryofquarksitwasshownthattheinfrareddivergenceof
forthequarkpropagatortogetherwiththeoneforthequark- the quark four-point function in the soft limit is exactly
gluon vertex in a self-consistent way [34,35]. Hereby, a the one which is needed to find a non-vanishingη′ mass.
significantdifferenceofthequarkscomparedtotheYang- However,the appearenceof the correctinfraredsingular-
Mills fields has to be taken into account:Since they pos- ity in single Feynman diagrams is the case only for the
sess a mass, and since dynamicalchiralsymmetrybreak- scaling solution. To our best knowledge, it is up to now
ing (DχSB) doesoccur,the quarkpropagatorwill always unknownhowtheaxialanomalyisencodedintheelemen-
approachaconstantintheinfrared. tary Green’s functionsof the decouplingsolution as only
In general, the fully dressed quark-gluon vertex can aresummationofinfinitelymanydiagramswillbeableto
beexpandedintwelvelinearlyindependentDiractensors. describetheanomalywhenemployingthesesolutionsfor
Halfofthecoefficientfunctionswouldvanishifchiralsym- theGreen’sfunctions.
metry were realized in the Wigner-Weyl mode. From a
solution of the Dyson-Schwinger equations we infer that
these “scalar” structures are, in the chiral limit, gener-
4 Phasesof strongly-interactingmatter:
atednon-perturbativelytogetherwiththedynamicalquark
How to go to T , 0 and µ , 0 ?
mass functionin a self-consistentfashion. Thus, dynami-
cal chiral symmetry breaking manifests itself not only in
The explorationof the QCD phase diagram, in particular
thepropagatorbutalsointhequark-gluonvertex.
the higher baryon-density regime, within functional ap-
Based on the Yang-Millsscaling solution one obtains
proachesisaveryactivefieldofresearchandofgreatim-
from an infrared analysis an infrared divergent solution
portance for a deeper understanding of the experimental
for the quark-gluonvertex such that the Dirac vector and
data of the running and planned heavy-ion programs. As
“scalar” components of this vertex are infrared divergent
withanexponent−κ− 1 ifeitherallmomenta[34]orthe already argued, in the last years a fruitful interaction be-
2 tweenthedifferentfunctionalmethodshasleadtoalargely
gluon momentum vanish [35]. A numerical solution of a
quantitative understanding of QCD at vanishing temper-
truncatedsetofDSEsconfirmsthisinfraredbehaviour.The
ature and density while an understanding of the confine-
essentialcomponents,toobtainthissolution,arethescalar
ment mechanism and its relation to the DχSB is not yet
Diracamplitudesofthequark-gluonvertexandthescalar
settled. At finite temperature and chemical potential the
part of the quark propagator.Both are only present when
situation is even much less clear. On the one hand, with
6 Maybe with the exception of how to describe the axial thehelpoffunctionaltechniquesresultsforthepureYang-
anomalywithGreen’sfunctions,seebelow. Mills sector of QCD at finite temperature as well as for
EPJWebofConferences
the hadronicsector of QCD could be obtainedbut on the tionoftheQCDGreen’sfunctionwithinfunctionalmeth-
otherhandafullQCDinvestigationishamperedbythefact odswillbedescribed.
that the gaugesector, i.e. the (de-)confinementtransition,
is not fully resolved.Furthermore,for full QCD with dy-
namicalquarksthe(de-)confinementtransitionisexpected 4.1 T,0andµ=0case
to be a smooth crossover as the underlying center sym-
metry of the SU(Nc) gauge group is explicitly broken by AtT ,0theheatbathprovidesapreferedrestframe.Ac-
the quarks. Similarly, the nature of the chiral phase tran-
cordingly, the Green’s functions will not only depend on
sition depends, among other quantities, on the values of
theLorentzinvariantcombinationsofenergyandmomenta.
thecurrentquarkmasseswhichagainbreakchiralsymme-
This,byitself,makesthesolutionoffunctionalequations
tryexplicitly.Itisanimportantobservationthatforsmall
muchmoreinvolved.Inaddition,thenumberoffunctions
chemicalpotentialsbothphasetransitionsliewithrespect
whichspecifyuniquelyagivenGreen’sfunctionincreases.
tothetemperatureremarkablyclosetoeachother.Thisis
Anexceptionistheghostpropagatorwhichstillcanbede-
interesting since the (de-)confinementtransition is driven
scribedbyonlyonefunction.However,forthegluonprop-
by the gluodynamicswhile the chiral transition is govern
agatortheonerenormalizationfunctionsplitsintotwo,
by strongly-interactingquarks. The deeper understanding
ofthisinterrelationisaprimaryfocusoftheapplicationof Z (p2) Z (p2)
functionalapproaches,seealso[3,16]. DGluon = T PT (p)+ L PL (p), (4)
µν p2 µν p2 µν
AfirststeptowardsfullQCDwithfunctionalmethods
at non-vanishingtemperatureand densities is done in the accordingtotransverseandlongitudinalprojectionstothe
framework of effective models which are generally con- heat bath. For further discussions it is important to note
structedfromnon-perturbativeYang-Millseffectivepoten- that the corresponding contributions are essentially chro-
tialsandhadronicpotentials.PureYang-Millstheorycor- momagnetic and chromoelectric in nature. For the quark
responds to the heavy-quarklimit of QCD where the ex- propagatoralsotherenormalizationfunctionsplitsintotwo
pectationvalueofthePolyakovloopservesasanorderpa- suchthatthequarkpropagatorisdescribedbythreefunc-
rameterforthe(de-)confinementtransition.Inthesemod- tions7
els some information about the confining glue sector of
1
QCDisincorporatedinaneffectivePolyakov-looppoten- S(p)= . (5)
tial that is extracted frompure Yang-Millslattice simula- −iγipiA(p)−iγ4ωpC(p)+B(p)
tionsatvanishingchemicalpotential.Thissectoriseffec-
tivelycombinedwiththechiralquark(-meson)mattersec- AtT = 0onehasofcourseZT = ZL = Z andC = A
torleadingtothePolyakov-NJL-typeeffectivechiralmod- whicharefunctionsof p2onlywith p =(ωp,p).AtT ,0
elssuchasthePNJLorPQMmodels,e.g.,[38,39,40].The wedealwithsixpropagatorfunctionsdependingonp2and
mattersectorofthesemodelshasbeenstudiedalsobeyond ωp.AtT →∞oneobtainsathree-dimensionalYang-Mills
mean-field level by taking into account the quark-meson theoryplusascalarfieldoriginatingfromtheA4.
quantum fluctuations within a FRG approach, for recent AtanyT thechromomagneticpartofthe gluonprop-
studiesseeRefs[41,42,43]. agator keeps positivity violating [46,47,48]. This clearly
indicates a kind of partial gluon confinementat any tem-
Themostdifficultproblemwithinthesemodelinvesti-
perature(amoredetaileddiscussionmaybefoundin[49]
gationsistheinclusionofthequantumback-reactionofthe
andreferencestherein).Theresultsoftheseinvestigations
mattersectortothegluonicsector.Thisproblemhasbeen
relate also to the non-perturbative origin of chromomag-
attacked in Ref. [39] which leads to a flavor and chem-
neticscreening,whateverthetemperatureis.Thisisnotat
ical potential dependence of the (de-)confinementtransi-
all surprising because three-dimensional Yang-Mills the-
tion temperature. This first perturbative estimate of Ref.
ory is confining (and thus the infrared behaviour of the
[39] has been confirmed by first-principle QCD calcula-
gluon propagator is genuinely non-perturbative),one has
tionswithintheFRGapproachatrealandimaginarychem-
an arealaw forthe spatialWilsonloop,andthe Coulomb
ical potentials[16] and also by constrainingPNJL model
stringtensionisnon-vanishingatanyT [50].
results with those in the statistical model, see [44] for a
To calculate the quark propagator we solve its DSE
recent review. This already demonstrates that even with
with the finite-temperature gluon propagator and quark-
QCD-basedeffectivemodelscombinedwithfunctionalap-
gluon vertex as input. In our corresponding first calcula-
proaches valuable information about the QCD phase di-
tions we follow Ref. [51] where a fit to the lattice gluon
agram can be provided and certain scenarios can be ex-
propagator and an ansatz for the quark-gluon vertex has
cluded. As an example see [45] concerning the expected
been used. As already stated the chromomagneticpart of
focusingofisentropictrajectoriesinthevicinityofacriti-
the gluonpropagatoris not at all influencedby the phase
calendpointinthephasediagram.Inthissensethesemod-
transition.Ontheotherhand,thechromoelectricpartmay
elscanbeunderstoodascontrolledapproximationstofull
serveasanorderparameter.InFig.2oneseesveryclearly
dynamicalQCD.
Nevertheless, treating the QCD Green’s functions di- 7 Bysymmetriesastructureof thetype γpγ ω D(p) would
i i 4 p
rectlyis,ofcourse,verydesireable.Inthenexttwosubsec- be allowedwhich, however, has turned out sofar inall investi-
tionstwotypesofinvestigationsaimingatadirectcalcula- gatedcasesasnegligiblysmall.
HotandColdBaryonicMatter–HCBM2010
the drastice change in the infrared value of the chromo-
1.4
electricgluonpropgator,D (0),atT =T =277MeV.The
L c
1.2 inverse of the square root of D (0) is the chromoelectric
L
V] 1 screeningmassscale,andfromthefitinFig.2weobtaina
Ge criticalexponentofapproximately0.53.
2 [ 0.8 Theresultingquarkpropagatorfunctionsfromthecor-
-1/0) 0.6 respondingDSEaredisplayedinFigs.3and4.Thescalar
D(L 0.4 function B(p2) is the one reflecting DχSB.8 Correspond-
ingly, in the chiral limit it vanishes identically above T
c
0.2
thereby signaling the chiral phase transition. Of course,
0 thistransitionbecomesacrossoverfornon-vanishingcur-
0 0.5 1 1.5 2 2.5 rentquarkmass,andthefunctionBisslowlyvaryingand
T/Tc mostlydrivenbythecurrentquarkmass.Anotanticipated
behaviouris the factthat B(0)raises with temperatureup
Fig.2.DisplayedistheLandaugaugechromoelectricscreening toTc.ButFig.2makesevidentwhythishappens:Thein-
massscalewithafittothecriticalexponentnearthecriticalpoint. fraredvalueofthechromoelectricpartofthegluonprop-
ThedatapointsaretakenfromRef.[51]. agatorincreaseswithT uptoTc beforeitthensharplyde-
creases.
Putting as usual antiperiodic boundary conditions for
T/T = 0, chiral limit quarks,thechiral-limitquarkcondensate(beingessentially
1.6 c
T/Tc = 0.986, chiral limit thefunctionaltraceoverthequarkpropagator)servesasan
1.4 T/T = 0
c orderparameter.However,playingwiththeboundarycon-
V] 1.2 TT/T/Tc = = 0 .19.8861 ditionsallowstolinkconfinementwithspectralproperties
Ge 1 c of the Dirac operator [52]. Correspondingly,in Ref. [51]
2) [ 0.8 also the dual quark condensate and the dressed Polykaov
p
B( 0.6 loop [52,53,54] have been calculated.9 The trick consists
0.4 in setting U(1)-valuedboundaryconditionsfor the quark
0.2 fieldin“temporal”direction:
0
-3 -2 -1 0 1 2 3 4 q(x,β=1/T)=eiϕq(x,0), (6)
Log(p2) [GeV2] whichamountstoaneffectiveshiftofMatsubarafrequen-
cies,
Fig. 3. The Landau gauge scalar quark function B(p2) (see Eq. ω =2πT(n+ϕ/2π). (7)
n
(5))fordifferenttemperaturesinthechirallimit(solidlines)and
The generalized condensate hq¯qi at fixed ϕ is the corre-
foratypicallight-quarkcurrentquarkmass(dashedlines). ϕ
spondingexpectationvalueoftheDiracoperator,anditis
expandablein a geometricseries containingloopsof link
2.4 A, T/T = 0 variables with increasing winding number. One can then
c
A, T/T = 0.986 definethedual(Gattringer)condensateas
2.2 c
C, T/T = 0.986
c
V] 2 A, T/Tc = 1.81 2π dϕ
Ge 1.8 C, T/Tc = 1.81 Σν =−Z 2πe−iνϕhq¯qiϕ (8)
2) [ 1.6 0
p
C( 1.4 where ν = 1 projects out winding-number-oneloops, the
A, 1.2 so-called dressed Polyakov loops [52]. It is an order pa-
rameterforcentersymmetrybreakingandconfinement.
1
The aim of our future investigations is now to calcu-
0.8
late the dual quark condensate and the dressed Polyakov
-3 -2 -1 0 1 2 3 4
loop with an improvedinput. Substituting the parameter-
Log(p2) [GeV2]
izedgluonpropagatorbyrecentresultsoffunctionalequa-
tions is herebymorea matter of convenienceand numer-
Fig.4.TheLandaugaugequarkrenormalizationfunctionsA(p2) ical precision than of fundamentalprogress.However,an
and C(p2) (see Eq. (5)) for different temperatures in the chiral (atleastpartially)self-consistentdeterminationofthequark
limit. propagatorandthequark-gluonvertextogether,generaliz-
ing the work of Ref. [35] to non-vanishingtemperatures,
8 For conceptional clarity we prefer to discuss the function
B(p2) and not one of themass functions M(p2) = B(p2)/A(p2)
orM(p2)=B(p2)/C(p2),respectively.
9 ForcorrespondingT =0latticeandfunctionalequationssee
Refs.[55,56].
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would be a decisive step towards a first-principle calcu- 250
lationoffinite-temperaturequarkpropertiesincontinuum 2SC - unbroken
uSC - 2SC
QCD. V] gCFL - uSC
Attheendofthissubsectionwewanttopointoutthat e 200 CFL - gCFL
M
one can define a Polyakov-loop related confinement cri- [
teriumfrominfraredexponents[57].TogetherwithanFRG V)
calculationofthePolyakovpotentialthisallowsonetocal- Ge 150
culate the transition temperature.In pureYang-Millsthe- 2
=
oryonefindsherebyforSU(2)asecond-orderphasetran- ν
(
sitionlyingintheIsinguniversalityclassandafirst-order cal
[tr5a8n]s.itionforallSU(Nc ≥3),Sp(2)andfortheE(7)group ms,criti100
50
300 350 400 450 500
4.2 Colorsuperconductingphase µ [MeV]
Ofcourse,itisquiteobvioustoexploittheabsenceofthe Fig.5.Colorsuperconductingphasesinthems−µplane.
signprobleminfunctionalapproachestoextendthestud-
ies to non-vanishing density. Nevertheless, the introduc-
clearlyherethatthemainaspectsoftheQCD phasetran-
tionofachemicalpotentialintofunctionalequationsleads
sitionatfinitedensitywillnotbeuncoveredbyfunctional
to a significant complication of the quark propagator pa-
methods if we will not get more information on the cou-
rameterization,usuallydoneintheNambu-Gorkovformal-
plingofquarkstogluonsalsointhisregionofthephasedi-
ism then,see e.g.Refs.[59,60,61]andreferencestherein.
agram.InthecaseofaDSEstudythisespeciallyincludes
It has to be noted that as long as one uses an Abelian-
moreknowledgeaboutthequark-gluonvertex
type model for the quark-gluon vertex these difficulties
aremerelytechnicalobstacleswhichcanbeovercomeby
combining computer algebra with numerical techniques.
5 Summary and Outlook:What may we
Nevertheless, including medium modifications by quarks
(as done e.g. in Refs. [59,60,61]) and employing the re- expect?
sults of functional equations or a combined phenomeno-
logical and lattice fit to the gluon propagator (as e.g. in The last years have seen a quite dramatic increase in our
Ref.[62])oneobtainsaquiteastonishingbutrobustresult: knowledge on the Landau gauge QCD Green’s functions
Restricting oneself to translationally invariantphases one atvanishingtemperaturesanddensities.Ithasbecomeevi-
concludes that for a realistic renormalized strange quark dent that the propagation of transverse gluons is infrared
massforallchemicalpotentialsabovethephasetransition, suppressed, and that positivity is violated for transverse
quarkmatter isin the colour-flavourlockedcolour-super- gluons.Ithasturnedoutthatghostsarethelong-range“de-
conducting phase. In Fig. 5 a resulting phase diagram in greesoffreedom”,andconcentratingonthemainpointone
the plane of renormalized strange quark mass and quark maysaythatinLandaugaugeQCD,ghostsandtherestric-
chemicalpotentialisgiven.10Theshadedbandinthefigure tiontotheGribovhorizon(i.e.thecorrelationsintroduced
indicatesthe value of the strange quarkmass as givenby bygauge-fixinguniquely)arerelatedtotheoriginofcon-
theParticle DataGroup,the differentlinesarephasesep- finement[31,15].
arationlinesfromcolour-flavour-lockedtogaplesscolour- Itcannotbeemphasizedenoughthatdynamicalchiral
flavour-locked (lowest line), from gapless colour-flavour- symmetry breaking does not only lead to the generation
lockedtotheuSCphase(second-lowestline),andsoon. of constituent quark masses but also to scalar-type cou-
In addition, these calculations demonstrate that there plings between quarks and gluons. There are indications
are huge deviations of the gap functions as compared to thattherelatedscalarconfinementpotentialisevenlarger
thoseextrapolatedfromtheweak-couplingresultuptoche- than the vector component [35]. We also remark that the
micalpotentialsoftheorderofseveralGeV.Evenatsuch axialanomalyisencryptedintheinfraredbehaviourofthe
largechemicalpotentialsperturbationtheoryquantitatively Green’sfunctionsinthescalingsolution[36].
fails.Furthermore,thelightquarksscreeninteractionalso Having set the stage, one is ready to continue to ei-
inthestrangequarksector,aneffect,whichisnotseenin ther non-vanishing temperatures or/and densities. Firstly,
mostmodels. oneshouldnotethatthechromomagneticpartofthegluon
A calculation of the dressed Polyakov loop at finite propagatordoesnotseethephasetransitionatall.Inpar-
chemical potential will certainly shed more light on the ticular,onehaspositivityviolation(andthuspartialgluon
natureofthefinite-densityphasetransition,andthereforea confinement)atanytemperature.Secondly,theinfraredre-
correspondinggeneralizationoftheworkdescribedinRef. gionofthechromoelectricpartofthegluonpropagatordis-
[59,60,61]isinprogress[63].However,ithastobestated playsverynicelythephasetransition[51].Needlesstosay,
that in the chiral limit the quark propagatorchanges also
10 Toobtainthisfigureelectricandcolourneutralityhavebeen accordingly due to the chiral symmetry restoration at the
imposed,butthisis,however,notdecisiveforthemainresult. criticaltemperature.
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