Center Phase Transition from Fundamentally Charged Matter Propagators

0.28 MB·English

by  Mario Mitter

Center Phase Transition from
Fundamentally Charged Matter Propagators
3 Mario Mitter
∗
1 UniversitätHeidelberg,InstitutfürTheoretischePhysik,Philosophenweg16,D-69120
0
Heidelberg,Germany
2
E-mail: [email protected]
n
a
Markus Hopfer
J
0 InstitutfürPhysik,Karl-Franzens-UniversitätGraz,Universitätsplatz5,8010Graz,Austria
3 E-mail: [email protected]

] Bernd-Jochen Schaefer
h
p InstitutfürPhysik,Karl-Franzens-UniversitätGraz,Universitätsplatz5,8010Graz,Austria
- E-mail: [email protected]
p
e
Reinhard Alkofer
h
[ InstitutfürPhysik,Karl-Franzens-UniversitätGraz,Universitätsplatz5,8010Graz,Austria

E-mail: [email protected]
1
v
9
Thecenterphasetransitionatnon-vanishingtemperaturesisinvestigatedinLandaugaugeQuan-
0
3 tum Chromodynamics (QCD) and scalar QCD. For each theory novel order parameters for the
7
transition are introduced. The matter-gluon vertex which occurs in the Dyson-Schwinger equa-
.
1
tionsofthepropagatorshastobemodeledincontemporarystudies. Itisfoundthatthenatureof
0
3 the phase transition depends strongly on the detailed structure of this vertex. Our investigation
1
motivatesaprecisedeterminationofthematter-gluonvertexatnon-vanishingtemperatures.
:
v
i
X
r
a
XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
CenterPhaseTransitionfromMatterPropagators MarioMitter
1. Introduction
Investigations of the phase structure of strongly-interacting matter have received a consider-
ableamountofattentioninthelastyears,both,theoreticallyaswellasexperimentally. Amongthe
most prominent features of the QCD phase diagram is the crossover from a confined phase with
spontaneously broken chiral symmetry to a chirally symmetric and deconfined phase. A confined
phasecanbelinkedtoagroundstatethatrespectscentersymmetry,1 andtheassociatedorderpa-
rameteristhePolyakovloop. Itvanishesinthecentersymmetricphaseandbecomesfiniteassoon
asthecentersymmetryisbroken[1]. Atvanishingquarkchemicalpotentialbothtransitionsoccur
roughlyatthesametemperaturewhichledtotheideaandintroductionofnewdualobservablesin
lattice QCD [3–6]. In particular, the dual chiral condensate pioneered in [3] is constructed from
the chiral condensate, the order parameter of chiral symmetry breaking. More recently, dual ob-
servablehavealsobecomeaccessiblewithinfunctionalmethods[7–9]andhavebeensuccessfully
appliedtoinvestigatethecentertransition. Inthepresentworknovelorderparametersforthecenter
symmetryanditsbreakingareintroducedandanalyzedinQCDaswellasfundamentallycharged
scalarQCD.Theorderparametersaredeterminedbythecorrespondingmatterpropagatorswithout
anyadditionalrenormalization.
2. (Scalar)QuantumChromodynamics
WeaddressthedeconfinementtransitioninordinaryQCDaswellasinscalarQCD,wherethe
quarksarereplacedbyfundamentallychargedscalars(seee.g.[10])bothinLandaugauge.
ThematterpropagatorsarecalculatedbymeansofthecorrespondingDyson-Schwingerequa-
tion (DSE) [11]. As an example, the DSE for the quark propagator is shown diagrammatically in
Fig. 1, where thin lines and dots represent bare propagators and one-particle irreducible vertices
while thick lines and dots denote the corresponding dressed quantities. Accordingly, the DSE for
thescalarpropagator,whichisshowninFig.2,ismoreinvolvedandcontainsmorediagramsdue
tothepresenceofadditionalbareverticessuchasthescalarself-interactionandthequarticscalar-
gluonvertices. Inone-loopapproximationonlythemomentum-independenttadpolediagramsare
leftinadditiontothegluonexchangediagram. Thetadpoles,however,canbetreatedbyadjusting
therenormalizationconstantsappropriately. Hence,inaone-loopapproximation,theDSEforthe
scalarpropagatorisofthesamestructureastheoneforthequarkpropagator,cf.Fig.1.
Explicitly,atfinitetemperatureT theDSEforthequarkpropagatorS(p)reads
(cid:90) d3k
S 1(p)=Z S 1(p) Z C g2T ∑ γµS(k)Γν(k,p;q)Dµν(q) (2.1)
− 2 0− − 1F F (2π)3
ωk(θ)
andcorrespondinglyforthescalarpropagatorD (p)
S
(cid:90) d3k
D 1(p)=Zˆ (p2+Zˆ m2) Zˆ C g2T ∑ (p+k)µD (k)Γν(k,p;q)Dµν(q). (2.2)
S− 2 m 0 − 1F F (2π)3 S S
ωk(θ)
1In a strict sense center symmetry is realized only in the limit of infinitely heavy quarks while in real QCD the
symmetryisalwaysexplicitlybroken,seee.g.[1,2].
2
CenterPhaseTransitionfromMatterPropagators MarioMitter
1 = 1
− −
−
Figure1: DSEforthequarkpropagator.
−1 = −1 − − 12 −
1 1 1
− 2 − 2 − − 2
−
Figure2: DSEforthefundamentallychargedscalarpropagator.
Forthefour-momentaweusek=((cid:126)k,ω (θ))andthegluonmomentumisconstrainedbymomentum
k
conservation to q = p k. The wave function renormalization of the quark (scalar) fields are
−
denotedbyZ (Zˆ ), therenormalizationconstantsofthequark-gluon(scalar-gluon)vertexbyZ
2 2 1F
(Zˆ ) and Z labels the scalar mass renormalization constant. The quadratic Casimir invariant in
1F m
thefundamentalrepresentationofthegaugegroupSU(3)isC =4/3andthecouplingconstantat
F
therenormalizationscaleisgivenbyg.
Ingeneral, exp(iθ)-valuedboundaryconditionsinthefourthspacetimedirectionarerealized
by introducing generalized Matsubara frequencies ω (θ) = (2πn +θ)T, where the sums over
k k
ω (θ) in the DSEs run over the corresponding discrete values n Z. The usual (anti-)periodic
k k
∈
boundaryconditionsfor(fermions)bosonsareobtainedbysetting(θ =π)θ =0,respectively.
BothDSEsdependonthegluonpropagatorDµν aswellasonthecorrespondingmatter-gluon
verticesΓν andΓν. SolutionsoftheDSEfortheLandaugaugegluonpropagatoratnon-vanishing
S
temperatures have been obtained in [12]. In addition, data are available from (quenched) lattice
simulations at finite temperatures and have already been successfully implemented in functional
equations for the quark propagator [8,13]2. In this work we will apply the fit functions for the
gluonpropagatorproposedin[13]andhenceweomitexplicitexpressions.
For the matter-gluon vertices the situation is less satisfactory since the temperature behavior
oftheseverticesisnotknowngenerally. Somemodelinghastobeemployedsuchasin[8]where
thefollowingexpression
(cid:18) (cid:19)
C(k)+C(p) A(k)+A(p)
Γν(k,p;q)=Z˜ δ4νγ4 +δjνγj
3
2 2
(cid:40) d q2 (cid:18)β α(µ)ln(cid:2)q2/Λ2+1(cid:3)(cid:19)2δ(cid:41) (2.3)
1 0
+
× d +q2 q2+Λ2 4π
2
forthequark-gluonvertexcanbefoundandwillbeusedalsointhiswork. TheAnsatzismotivated
by Slavnov-Taylor identities of the Abelian gauge theory (see e.g. [15]) and by the running of the
non-perturbative coupling of the Yang-Mills theory. The purely phenomenological parameters d
1
2Recently, alsounquenchedlatticedatafortheLandaugaugegluonpropagatorareavailable[14], whichwillbe
usedinfuturestudiesofthesystem.
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CenterPhaseTransitionfromMatterPropagators MarioMitter
0.4 0.4
T=0 MeV T=0 MeV
0.35 273 MeV 0.35 273 MeV
-2V] 0.3 277 MeV -2V] 0.3 277 MeV
e 283 MeV e 283 MeV
G 0.25 G 0.25
0)) [ 0.2 π)) [ 0.2
(0 (0
ω 0.15 ω 0.15
(x, S 0.1 (x, S 0.1
D D
0.05 0.05
0 0
-4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5
Log(x/GeV2) Log(x/GeV2)
Figure 3: Solution of the truncated DSE Eq. (2.2) for the scalar propagator as function of(cid:126)p2 =x at the
lowest Matsubara frequency for periodic θ =0 (left panel) and antiperiodic θ =π (right panel) boundary
conditionsatdifferenttemperatureswithrenormalizationscaleµ =4GeVandmassm=1.5GeV.
and d are specified in [13], whereas the anomalous dimension 2δ = 18/44 and β =11N /3
2 0 c
−
ensure a correct perturbative running coupling in the UV regime for SU(N ) gauge theory. The
c
renormalization scale of the Yang-Mills sector has been fixed by α(µ)=0.3 and Λ=1.4 GeV.
ThefactorZ˜ allowstoapplytheSlavnov-TayloridentityZ =Z /Z˜ inLandaugauge[16]which
3 1F 2 3
yields finally only a dependence on the quark wave function renormalization Z . The gluon mo-
2
mentumisdenotedbyqandkand parethein-andoutgoingquarkmomenta,respectively.
Inasimilarcontextweemploy
D 1(p2) D 1(k2)
Γν(k,p;q)=Z˜ −S − −S (p+k)ν
S 3 p2 k2
(cid:40) Λ2 − q2 (cid:18)β α(µ)ln(cid:2)q2/Λ2+1(cid:3)(cid:19)2δ(cid:41) (2.4)
0
d +
× 1 Λ2+q2 q2+Λ2 4π
forthescalar-gluonvertex,whereoneadditionalparameterd =0.53hasbeenintroducedandall
1
remaining parameters are the same as in the quark-gluon vertex. In contrast to the quark-gluon
vertex, the scalar-gluon vertex is dressed only with the vacuum propagators D 1(p2). For the nu-
−S
mericalsolutionofthecorrespondingDSEswerewritethepropagatorsS 1(p)=iγ ω (θ)C(p)+
− 4 p
i(cid:126)/pA(p)+B(p)andD (p)=Z ((cid:126)p2,ω (θ))/((cid:126)p2+ω (θ)2)intermsofdressingfunctionsinastan-
S S p p
dardway. Detailsonthenumericalimplementationaswellasontherenormalizationschemewill
bepublishedelsewhere[17],cf. also[18,19].
Numerical results for the scalar propagator around temperatures of the center transition in
the quenched theory with T 277 MeV are shown in Fig. 3 for periodic (left panel) as well as
c
≈
antiperiodic boundary conditions (right panel). The mass of the scalars has been fixed to m=1.5
GeVwhichresultsinaquiteinertbehavioraroundT . Theseresultsdemonstratethatthereareno
c
direct modifications of the scalar propagator in the vicinity of the transition. This motivates the
constructionofmoresensitiveorderparametersforthecenterphasetransition.
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CenterPhaseTransitionfromMatterPropagators MarioMitter
3. CenterPhaseTransitionandDualOrderParameters
Order parameters for the center phase transition can be composed with functional methods
via dual observables like e.g. the dual chiral condensate [7,8] or the dual density [9]. As has
been discussed in the latter reference, dual quantities can be evaluated in two different ways. In
general,dualorderparametersareconstructedfromsomeboundary-conditiondependentoperator
Oˆ ,whereθ denotesthephaseoftheU(1)-valuedboundaryconditions. Originally,suchoperators
θ
were introduced in lattice calculations [3–6] and evaluated in QCD with the original boundary
conditions,i.e.,with(anti-)periodicboundaryconditionsfor(fermions)bosons.
On the other hand, these operators can also be evaluated with functional methods in various
theorieswithgeneralboundaryconditions,referredtoasQCD in[9]. However,dualobservables
θ
evaluatedinthiswaycanserveasorderparametersonlyifthedeconfinementtransitiontemperature
at physical boundary conditions is a lower bound for the transition temperatures in the different
theoriesQCD [9].
θ
Inpreviousstudiesofthecenterphasetransitionthedualchiralcondensatehasbeencalculated
with functional methods based on the quark propagator [7,8,13]. It is given by an expansion in
complexFouriermodes
(cid:90) 2π dθ
Σ = e inθ ψ¯ψ (3.1)
n − θ
0 2π (cid:104) (cid:105)
withaθ-dependentquarkcondensate
(cid:90) d3p
ψ¯ψ =Z N T ∑ tr S((cid:126)p,ω (θ)). (3.2)
(cid:104) (cid:105)θ 2 c (2π)3 D p
ωp(θ)
For arbitrary n that is not a multiple of N , Σ can then serve as an order parameter for center
c n
symmetry. Usually, the dual chiral condensate Σ is used, also called dressed Polyakov loop.
1
In a lattice formulation it contains contributions from all time-like loops around the torus with
winding number n = 1 [3,4,6] and transforms similar to the ordinary Polyakov loop [1] under
centertransformations.
Hence,thecalculationofthedualchiralcondensaterequiresthechiralcondensatewithgeneral
boundary conditions, which has to be regularized for non-vanishing quark masses. As an already
finitealternativewepropose
(cid:90) 2π dθ (cid:20)1 (cid:21)2
Σ = e iθ Σ , Σ =T ∑ tr S((cid:126)0,ω (θ)) (3.3)
Q − Q,θ Q,θ D p
2π 4
0
ωp(θ)
as an order parameter for the center phase transition. Σ is finite, because the sum scales like
Q,θ
1/ω4 forlargeMatsubaramodes. Similarly,forscalarQCDweproposetheorderparameter
p
(cid:90) 2π dθ
Σ = e iθ Σ , Σ =T ∑ D2((cid:126)0,ω (θ)). (3.4)
S 2π − S,θ S,θ S p
0
ωp(θ)
Moredetailsontheseorderparameterswillbepresentedinanupcomingpublication[17].
5
CenterPhaseTransitionfromMatterPropagators MarioMitter
7
18.4
6
18.2
-3eV)] 18 -3V)] 45
10 G 17.8 0 Ge 3
Σ [(θS, 1177..46 Σ [(1S 12
T=0.273 GeV
17.2 T=0.277 GeV 0
T=0.283 GeV
17 -1
0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6
θ/2π T [GeV]
Figure 4: Left: Σ of Eq. (3.4) as a function of the boundary conditions for temperatures around the
S,θ
transition. Right: ThedualcondensateΣ asafunctionofthetemperature(µ =4GeVandm=1.5GeV).
S
0.16 1.6
0.14 d1d=1=0.05.36 1.4 mm==12..50 GGeeVV
0.12 d1=0.45 1.2 m=1.0 GeV
-3V)] 0.1 -3V)] 0 .18
Ge 0.08 Ge 0.6
0 0.06 0 0.4
1 1
[(S 0.04 [(S 0.2
Σ Σ 0
0.02
-0.2
0 -0.4
-0.02 -0.6
0.1 0.15 0.2 0.25 0.3 0.1 0.15 0.2 0.25 0.3
T [GeV] T [GeV]
Figure5: ParameterdependencyofthedualcondensateΣ asafunctionoftemperatureforthreedifferent
S
valuesofthevertexparameterd (left)andforthreedifferentvaluesofthemassofthescalarfield(right);
1
(µ =4GeVandm=1.5GeV).
4. Results
In order to confirm that both quantities, Σ in Eq. (3.3) and Σ in Eq. (3.4), are well-defined
Q S
orderparametersweinvestigateboththeories,QCDandscalarQCD,atfinitetemperatures.
IntheleftpanelofFig.4theθ-dependenceofΣ forthequenchedscalartheoryisshownfor
S,θ
temperaturesaroundthetransition. FromthedefinitionEq.(3.4)itisclearthatΣ vanishesaslong
S
as the Σ is constant while a θ-dependency is necessary for a non-vanishing order parameter.
S,θ
However, these findings for Σ are qualitatively similar to QCD with finite quark masses [3].
S,θ
In the right panel of Fig. 4 the dual condensate Σ is shown as a function of the temperature
S
which nicely demonstrates its property as an order parameter for the center symmetry. Below the
transition temperature of the quenched theory around T 277 MeV it vanishes and is finite at
c
≈
higher temperatures. In the vicinity of the critical temperature the temperature behavior of the
orderparameterdependscruciallyonthemodelparametersusedforthescalar-gluonvertex. This
isdemonstratedinFig.5,wherethetemperaturebehaviorofΣ isshownfordifferentvaluesofthe
S
d parameterinthescalar-gluonvertexEq.(2.4)(leftpanel)andforvariousvaluesofthemassof
1
thescalar field(rightpanel). Strongerdeviationsfrom avanishingorder parameterslightlybelow
6
CenterPhaseTransitionfromMatterPropagators MarioMitter
〈—ψψ〉
0.2 θ 0.1
Σ
Q,θ
0.08
3] 0.15 3]
V V
Ge Ge 0.06
-1v], [ 0.1 -1v], [ 0.04
e e
G G
[ 0.05 [ 0.02
Σ
0 Σ1
0 Q
0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6
θ/2π T [GeV]
Figure 6: Left panel: The quark condensates ψ¯ψ and Σ , as defined in Eq. (3.3), as a function of
θ Q,θ
(cid:104) (cid:105)
the boundary angle for different temperatures in the chiral limit (dashed lines: T =273 MeV, solid lines:
T =283MeV).Rightpanel: TheorderparametersΣ andΣ asdefinedinEq.(3.1)and(3.3)asafunction
Q 1
ofthetemperature.
the transition temperature are observed for smaller mass. This indicates that the vertex sensitivity
increasestowardssmallermasses.
Finally,theproposedneworderparameterforordinaryQCDshowsasimilarbehaviorwhich
ispresentedintheleftpanelofFig.6. Inthefiguretheθ-dependencyofΣ incomparisontothe
Q,θ
chiralcondensateisplottedasafunctionofthegeneralizedboundaryconditionsfortwodifferent
temperatures. The results have been obtained in the chiral limit of the quenched theory. Above
thetransitiontemperature, Σ showsthesamecharacteristicplateauasthechiralcondensatefor
Q,θ
θ’s close to the physical antiperiodic boundary conditions θ =π. Due to the restoration of chiral
symmetry the condensate vanishes which also affects Σ through the scalar dressing function
Q,θ
B(p) in the quark propagator. The slight variations of Σ in the center symmetric phase below T
Q c
results from a non-constant Σ and can be attributed to lattice artifacts as well as the choice of
Q,θ
the quark-gluon vertex model. However, this effect is more pronounced in Σ than in the chiral
Q,θ
condensate. TherightpanelofFig.6showsthecorrespondingorderparameterΣ incomparison
Q
tothedualchiralcondensate. BothvanishbelowT andjumpimmediatelytonon-vanishingvalues
c
aboveT . Theirdeviationatlargertemperaturescanbeassignedtodifferentdimensionalities.
c
5. Conclusions
We investigated the center phase transition of QCD as well as fundamentally charged scalar
QCDinaquenchedformulation. Novelorderparametersareproposedalongthelinesofpreviously
constructed dual observables accessible by functional methods. Solving the Dyson-Schwinger
equations for the corresponding matter propagators numerical results for these order parameters
are presented. A parameter dependency of the employed matter-gluon vertices on the presented
results is found and motivates a more detailed investigation of its temperature dependence (see
also[20]forcorrespondinginvestigationsatvanishingtemperature).
7
CenterPhaseTransitionfromMatterPropagators MarioMitter
Acknowledgements
We are grateful to the organizers of the Xth Quark Confinement and the Hadron Spectrum
conferenceforalltheireffortswhichmadethisextraordinaryeventpossible.
We thank F. Bruckmann, C.S. Fischer, L. Fister, J. Luecker, A. Maas, P. Maris, J.M. Pawlowski
and L. von Smekal for valuable discussions. This work is supported by the FWF through DK
W1203-N16andgrantP24780-N27.
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