Hamiltonian Approach to QCD: The effective potential of the Polyakov loop

0.11 MB·English

by  Hugo Reinhardt

Hamiltonian Approach to QCD: The effective
3 potential of the Polyakov loop
1
0
2

n
a
J

1
HugoReinhardt
1 ∗
UniversitätTübingen,InstitutfürTheoretischePhysik

]
AufderMorgenstelle14,72076Tübingen,Germany
h
t E-mail:[email protected]
-
p
e JanHeffner
h UniversitätTübingen,InstitutfürTheoretischePhysik
[
AufderMorgenstelle14,72076Tübingen,Germany

1
v
0 The effective potential of the order parameter for confinement is calculated within the Hamil-
1
tonianapproachto Yang–Millstheory. Compactifyingonespatial dimensionandusing a back-
5
2 groundgaugefixingthispotentialisobtainedbyminimizingtheenergydensityforagivenback-
.
1 groundfield. UsingGaussiantypetrialwavefunctionalsIestablishananalyticrelationbetween
0
the propagatorsin the backgroundgaugeat finite temperatureand the correspondingzero tem-
3
1 perature propagators in Coulomb gauge. In the simplest truncation, neglecting the ghost and
:
v using the ultravioletform of the gluonenergyone recoversthe Weiss potential. Fromthe fully
i
X non-perturbativepotential(withtheghostincluded)oneextractsacriticaltemperatureofthede-
r confinementphasetransitionof270MeVforthegaugegroupSU(2).
a
XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
HamiltonianApproachtoQCD:TheeffectivepotentialofthePolyakovloop HugoReinhardt
1. Introduction
Understanding the deconfinement phase transition is one of the major challenges of particle
physics. Inquenched QCDreliable resultsareobtained withinthelattice approach. Thisapproach
fails, however,atlargebaryon density duetothenotorious fermionsignproblem. Therefore alter-
native non-perturbative approaches tocontinuum QCDare desirable. In recent years a variational
approachtoYang–MillstheoryinCoulombgaugewasdeveloped[1],whichhasprovidedadecent
description of the infrared sector of the theory [2, 3, 4, 5, 6, 7]. Recently this approach was ex-
tendedtofinitetemperature[8]andalsotofullQCD[9]. InthistalkIwillreportonthecalculation
oftheeffectivepotentialoftheconfinementorderparameterwithintheHamiltonianapproach[10].
Inquantumfieldtheorythetemperature T ismosteasilyintroduced bycompactifying theEu-
clidean timeandinterpreting thelength Lofthecompactified timeintervalasinversetemperature.
InfinitetemperatureYang-Millstheorytheorderparameter ofconfinementistheexpectationvalue
ofthePolyakovloop
1
P[A0]= trPe− 0Ldx0A0(x0,~x). (1.1)
N
R
Thequantity P[A0](~x) exp[ F¥ (~x)L]isrelated tothefreeenergy ofa(infinitely heavy) quark
h i∼ −
at spatial position~x. In the confined phase this quantity vanishes by center symmetry while it is
non-zero in the deconfined phase, where center symmetry is broken. In continuum Yang-Mills
theory the Polyakov loop is most easily calculated in Polyakov gauge ¶ A =0,A = diagonal.
0 0 0
In the fundamental modular region 0<A L/2<p the Polyakov loop P[A ] is a unique function
0 0
of the field A , which, for SU(2), is given by P[A ] = cos(A L/2). As a consequence of this
0 0 0
relation and ofJenssen’s inequality one can useinstead of P[A ] alternatively P[ A ]or A as
0 0 0
h i h i h i
order parameter of confinement, Refs. [11, 12]. The order parameter of confinement can be most
easilyobtainedbycalculatingtheeffectivepotentiale[a ]ofatemporalbackgroundfielda chosen
0 0
in the Polyakov gauge and by calculating the Polyakov line (1.1) from the field configuration a¯
0
which minimizes e[a ], i.e. P[A ] P[a¯ ]. The effective potential e[a ] was first calculated in
0 0 0 0
h i≃
Refs. [13, 14] in 1-loop perturbation theory and is shown in Fig. 1. This potential is minimal for
a vanishing field and the order parameter accordingly yields P[a¯ =0]=1, which indicates the
0
deconfining phase. The aim of the present work is to give a non-perturbative evaluation of e[a ]
0
[10]intheHamiltonapproach toYang-Millstheory[1].
It is obvious that the effective potential of A cannot be straightforwardly evaluated in the
0
h i
Hamiltonian approach sincetheletterassumesWeylgauge A =0. However,wecanexploit O(4)
0
invariance of Euclidean quantum field theory and compactify instead of the time one spatial axis
(for example the x -axis) to a circle and interpret the length L of the compactified dimension as
3
inverse temperature. Therefore we will consider in the following Yang-Mills theory at a finite
compactified length L in a constant color diagonal background field a and calculate the effective
3
potentiale[a ]. IntheHamiltonianapproachtheeffectivepotentiale[~a]ofaspatialbackgroundfield
3
~a is given by the minimum of the energy density H /V calculated under the constraint ~A =~a.
h i h i
Thisminimalpropertyoftheeffectivepotential callsfora variational calculation.
2. Hamiltonapproach inbackground gauge
In the presence of an external constant background field~a the Hamiltonian approach can be
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HamiltonianApproachtoQCD:TheeffectivepotentialofthePolyakovloop HugoReinhardt
mostconveniently formulated inthebackground gauge
d~,~A =0, d~=~¶ +~a (2.1)
h i
whereallfieldsaretakenintheadjointrepresentation. Thisgaugeallowsforanexplicitresolution
ofGauss’law,whichresults inthegaugefixedHamiltonian
1
H = d3x J 1~P (~x)J ~P (~x)+~B2(~x) +H , (2.2)
2 A− A· C
Z
(cid:16) (cid:17)
where~P = id /d ~Aisthemomentumoperatorofthegaugefixedfieldand
−
J =Det ~D d~ , ~D=~¶ +~A (2.3)
A
− ·
(cid:16) (cid:17)
istheFaddeev-Popovdeterminant. Furthermore,
g2
H = d3xd3yJ 1r a(~x)J Fab(~x,~y)r b(~y) (2.4)
C 2 A− A
Z
istheanalogue oftheso-called Coulombtermwhichresultsfrom thekinetic termofthe“longitu-
dinal”partofthemomentumoperator. Here
r a= ~D ~P = ~A ~a ~P (2.5)
− · − − ·
(cid:16) (cid:17)
isthecolorcharge densityofthegluons, whichinteracts throughthekernel
1 1
F = ~D d~ − d~ d~ ~D d~ − . (2.6)
− · − · − ·
(cid:16) (cid:17) (cid:16) (cid:17)(cid:16) (cid:17)
Foravanishingbackgroundfield~a=0thegauge(2.1)reducestotheordinaryCoulombgaugeand
the Hamiltonian H (2.2) becomes the familiar Yang-Mills Hamiltonian in Coulomb gauge, [15].
Furthermore,aswasshowninRef.[8]theCoulombterm H (2.4)isnegligibleinthegluonsector.
C
Therefore wewillignorethisterminthefollowing.
Weareinterested hereintheenergydensity inthestate y [A]minimizing
a
H := y H y (2.7)
h ia h a| | ai
under the constraint ~A =~a. Forthis purpose weperform avariational calculation with the trial
a
h i
wavefunctional
y a[A]=JA−1/2y˜[A−a], y˜[A]=N e−12 Aw A, (2.8)
R
which already fulfills the constraint ~A =~a. For ~a = 0 this ansatz reduces to the trial wave
a
h i
functionalusedinCoulombgauge[1]. ProceedingasinthevariationalapproachinCoulombgauge
[1],from H minonederivesasetofcoupledequations forthegluonandghost propagators
a
h i →
D = AA = 1w 1, G= (~Dˆ +~aˆ)d~ˆ −1 . (2.9)
a=0 −
h i 2 −
(cid:28) (cid:29)a=0
(cid:16) (cid:17)
Using the same approximation as in Ref. [8] in Coulomb gauge, i.e. restricting to two loops in
theenergy, whileneglecting H (2.4)andalsothetadpolearisingfromthenon-Abelian part ofthe
C
magneticenergy, onefindsfromtheminimizationof H thegapequation
a
h i
w 2= d~ d~+c 2, (2.10)
− ·
3
HamiltonianApproachtoQCD:TheeffectivepotentialofthePolyakovloop HugoReinhardt
where1
1 d 2lnJ[A+a] 1
c (1,2)= = Tr[GG (1)GG (2)] (2.11)
−2 d A(1)d A(2) 2 0
(cid:28) (cid:29)a=0
is the ghost loop (referred to as “curvature”) with G and G being the bare and full ghost-gluon
0
vertex. The gap equation (2.10) has to be solved together with the Dyson-Schwinger equation
(DSE)fortheghostpropagator
G 1= d~ d~ G (1)GG (2)D(2,1). (2.12)
− 0
− · −
Due to the presence of the background field these equations have a non-trivial color structure.
Fortunately, due to the choice of the background gauge (2.1), the background field enters these
equations onlyinformofthecovariant derivative d=¶ +a.
Weareinterested intheeffective potential ofabackground fieldinthe Cartanalgebra, which
forSU(2)hastheform~aT . Thisfieldbecomes diagonal inthe Cartanbasis defined bythe eigen-
3
vectors of the generators of the Cartan subgroup T s = is s ,s =0, 1. By analyzing our
3
| i − | i ±
equations ofmotion,i.e. thegapequationandtheghostDyson-Schwingerequation, onefindsthat
theseequations havesolutions, whicharediagonal intheCartanbasis
Dst (~p)=d st Ds (~p), Gst (~p)=d st Gs (~p). (2.13)
Thisisnotsurprising sincethesourceofthenon-trivial colorstructure isthebackground fieldand
if this field ischosen to be diagonal in color space the same should be true forall propagators. In
addition, onecanshowthatthepropagators inthepresence ofthebackground fieldinbackground
gaugearerelatedtothepropagatorsinCoulombgauge(i.e. intheabsenceofthebackgroundfield),
D(~p),G(~p),by
Ds (~p)=D(~ps ), Gs (~p)=G(~ps ), (2.14)
where
~ps =~p s ~a (2.15)
−
is the momentum shifted by the background field. In this way the results of the variational calcu-
lation in Coulomb gauge (in the absence of the background field) are sufficient to determine the
propagators inthepresence ofthebackground fieldinbackground gauge.
Latticecalculation [16]ofthegluonpropagatorinCoulombgaugeshowthatthegluonenergy
canbenicelyfittedbyGribov’sformula[17]
w (~p)= ~p2+M4/~p2. (2.16)
q
A full self-consistent solution of the gap equation (2.10) and the ghost DSE (2.12) reveals that
w (~p) contains in addition sub-leading UV-logs, which on the lattice are found to be small. Using
Gribov’sformula(2.16)for w (~p)andsolving thegapequation (2.10)for c (~p)yields
c (~p)=M2/~p , (2.17)
| |
which is indeed the correct IR-behavior obtained in a full solution [8] of the coupled ghost DSE
andgapequation butwhichmissesthesub-leading UV-logs.
1WeuseherethecompactnotationA(1)≡Aai11(~x1). ForLorentzscalarsliketheghost,theindex“1”standsforthe
colorindexa1andthespatialposition~x1.Repeatedindicesaresummed/integratedover.
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HamiltonianApproachtoQCD:TheeffectivepotentialofthePolyakovloop HugoReinhardt
0.9 0
0.8
0.7 -0.5
4L(e[a]-e[a=0])UVUV 0000....3456 4L (e[a]-e[a=0])IRIR -1-.15
0.2 -2
0.1
0 -2.5
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
x x
Figure1:TheWeisspotentiale multipliedby Figure 2: The infrared potential e multiplied
UV IR
L4 as a function of the dimensionless field x= byL4 fordifferentL 1.
−
aL/(2p ).
3. The effective potential
Compactifying the3-axis toacircle withcircumference Land choosing thebackground field
alongthecompactifieddimension~a=a~e theshiftedmomentum(2.15)becomes
3
~ps =~p +(p s a)~e , p =2p n/L, (3.1)
n 3 n
⊥ −
where ~p is the projection of ~p into the 1-2-plane and p is the Matsubara frequency. In the
n
⊥
Hamiltonianapproachtheeffectivepotentialoftheconstantbackgroundfieldisgivenbytheenergy
density in the state minimizing H under the constraint A =a [18]. Using the gap equation
h ia h ia
2.10onefindsfortheenergydensitypertransversaldegreeoffreedominthepresentapproximation
[10]
e(a,L)=(cid:229) L1 (cid:229)¥ (d22pp)⊥2(w (~ps )−c (~ps )). (3.2)
s n= ¥ Z
−
Byshifting thesummationindexnoneverifiestheperiodicity
e(a+2p /L,L)=e(a,L), (3.3)
which is a necessary property for the effective potential of the confinement order parameter by
center symmetry. Neglecting c (~p) Eq. (3.2) gives the energy of a non-interacting Bose gas with
single-particle energy w (~p). This quasi-particle picture is a consequence of the Gaussian ansatz
(2.8)forthewavefunctional. Thequasi-particle energy w (~p)is,however,highlynon-perturbative,
see forexample Eq.(2.16). Thecurvature c (~p)inEq.(3.2)arises from theFaddeev-Popov deter-
minantinthekineticpartoftheYang–MillsHamiltonian(2.2).
In certain limiting cases and for 0 aL/2p 1 the energy density (3.2) can be calculated
≤ ≤
analytically. Neglecting c (~p) and assuming the perturbative expression for the gluon energy
w (~p)= ~p onefindsfrom(3.2)theWeisspotential originally obtained in[14]
| |
4p 2 aL 2 aL 2
e (a,L)= 1 (3.4)
UV 3L4 2p 2p −
(cid:18) (cid:19) (cid:18) (cid:19)
5
HamiltonianApproachtoQCD:TheeffectivepotentialofthePolyakovloop HugoReinhardt
0 1000
-0.2
100
-0.4
a=0]) -0.6 10
e[
e[a]- -0.8 1
4L(
-1
0.1
-1.2 |c |(p)
w (p)
-1.4 0.01
0 0.2 0.4 0.6 0.8 1 0.001 0.01 0.1 1 10 100 1000 10000
x p/m
Figure 3: The simplified potential (3.7) multi- Figure4: Thenumericalsolutionsofthe gluon
pliedbyL4 fordifferenttemperatures L 1. The energyw andthecurvaturec obtainedin[8].
−
criticaltemperatureisT 485MeV.
c
≃
shown in Fig. 1. Neglecting c (~p) and using the infrared expression for the gluon energy w (~p)=
M2/~p (seeEq.(2.16)),oneobtains[10]
| |
M2 aL 2 aL
e (a,L)=2 (3.5)
IR L2 2p −2p
" #
(cid:18) (cid:19)
showninFig.2. ThisexpressiondrasticallydiffersfromtheWeisspotential(3.4): Whilee (a,L)
UV
is minimal fora=0, the minimum ofe (a,L) occurs at a=p /L corresponding toa centersym-
IR
metricgroundstate. Accordingly e (a,L)yieldsforthePolyakovloop P =P[A =0]=1while
UV 0
h i
e (a,L)yields P =P[A =p /L]=0.
IR 0
h i
The deconfinement phase transition results from the interplay between the confining infrared
potential,Eq.(3.5)andthedeconfiningUV-potential,Eq.(3.4). Toillustratethisletusapproximate
thegluonenergy w (~p)(2.16)by
w (~p) ~p +M2/~p . (3.6)
≈| | | |
This expression agrees with the Gribov formula (2.16) in both, the IR and UV but deviates from
itinthemid-momentum regime, whichinfluences thedeconfinement phase transition. With w (~p)
givenbyEq.(3.6)andwithc (~p)=0theenergydensity(3.2)becomes
4p 2 aL
e(a,L)=e (a,L)+e (a,L)= f , (3.7)
IR UV 3L4 2p
(cid:18) (cid:19)
3M2L2
f(x)=x2(x 1)2+cx(x 1), c= .
− − 2p 2
For small temperatures L 1, e (a,L) dominates and the system is in the confined phase. As L 1
− IR −
increases the center symmetric minimum at x = 1/2 eventually turns into a maximum and the
system undergoes thedeconfinement phasetransition, seeFig.3. Inthedeconfinedphase f(x)has
twodegenerate minimaand,starting inthedeconfined phase, thephasetransition occurswhenthe
threerootsof f (x)degenerate. Thisoccursfor c=1/2,i.e. foracriticaltemperature
′
T =L 1=√3M/p . (3.8)
c −
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HamiltonianApproachtoQCD:TheeffectivepotentialofthePolyakovloop HugoReinhardt
0.02
0 1
-0.02
0.8
0]) -0.04
a]-e[a= --00..0086 P[a]min 0.6
e[ 0.4
4L( -0.1
-0.12 0.2
-0.14
0
-0.16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 0.2 0.4 0.6 0.8 1 T/T
c
x
Figure 5: The numerically evaluated effective Figure6:ThePolyakovloop P[a] evaluatedat
h i
potential. The critical temperature is T 269 theminimuma=a ofthefulleffectivepoten-
c min
≃
MeV. tial(figure(a))asafunctionofT/T.
c
With the lattice result M =880 MeV this corresponds to a critical temperature of T 485 MeV,
c
≃
which is much too high. This value is only slightly reduced to T =432 MeV when the correct
c
Gribov formula (2.16) is used instead of the approximation (3.4). The reason why the transition
temperature comes out too high is the neglect of the ghost loop c (p)=0. This can be seen from
Fig.4,wherethegluonenergy w (p)andthecurvature c (p)obtained fromthevariational calcula-
tion in Coulomb gauge are shown. In the deep infrared the gluon energy w (p) and the curvature
c (p) agree, while inthe ultraviolet c (p)has the opposite sign ofw (p) and isonly suppressed by
a log compared to w (p). Therefore, neglecting the ghost loop enhances the infrared contribution
to the potential, which favors confinement, while at the same time it reduces the ultraviolet con-
tribution, which favors deconfinement. Both effects add coherently and push the deconfinement
transition toahighertemperature.
In a full numerical evaluation of the effective potential (3.2) using for w (~p) and c (~p) the
numerical solution of the variational approach in Coulomb gauge obtained in Ref. [8], one finds
the effective potential shown in Fig. 5. From this potential one extracts a critical temperature for
the deconfinement phase transition of T 269 MeV, which is close to the lattice predictions of
c
≃
T =290 MeV.This value is also close to the range of critical temperatures T =275...290 MeV
c c
obtained in Ref. [8] from the grand canonical ensemble of Yang–Mills theory in Coulomb gauge.
Letusalsomentionthatifoneusesforw (~p)theGribovformula(2.16)andinaccordwiththegap
equation (2.10) for c (~p)itsinfrared expression (2.17) onefindsacritical temperature ofT 267
c
≃
MeV, which is only slightly smaller than the value obtained with the full numerical solution for
w (~p) and c (~p). This shows that it is the infrared part of the curvature (neglected in Eq. (3.7)),
which is crucial for the critical temperature. In view of the ghost dominance in the IR this is not
surprising. Fig.6showsthePolyakovloop P[a]calculated fromtheminimuma ofthepotential
min
(3.2). At the phase-transition P[a ] rapidly changes from P=0 to P=1, which is typical for a
min
second orderphasetransition.
Inthepresentapproach thedeconfinementphasetransition isentirelydeterminedbythezero-
temperaturepropagators,whicharedefinedasvacuumexpectationvalues. Consequently,thefinite-
temperature behavior of the theory and, in particular, the dynamics of the deconfinement phase
7
HamiltonianApproachtoQCD:TheeffectivepotentialofthePolyakovloop HugoReinhardt
transition must be fully encoded in the vacuum wave functional. The results obtained above are
encouraging foranextension ofthepresentapproach tofull QCDatfinitetemperature andbaryon
density.
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