Table Of Content3 Hamiltonian Dyson–Schwinger Equations of QCD
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J
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t InstituteforTheoreticalPhysics,UniversityofTübingen
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p AufderMorgenstelle14,72076Tübingen,Germany
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E-mail:[email protected]
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Thegeneralmethodfortreatingnon-Gaussianwavefunctionalsin theHamiltonianformulation
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6 ofaquantumfieldtheory,whichwaspreviouslydevelopedandappliedtoYang–Millstheoryin
5 Coulomb gauge, is generalized to full QCD. The Hamiltonian Dyson–Schwinger equations as
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1 wellasthequarkandgluongapequationsarederivedandanalysedintheIRandUVmomentum
1. regime.Theback-reactionofthequarksonthegluonsectorisinvestigated.
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XthQuarkConfinementandtheHadronSpectrum,
October8–12,2012
TUMCampusGarching,Munich,Germany
Speaker.
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(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
HamiltonianDyson–SchwingerEquationsofQCD DavideCampagnari
1. Introduction
Coulomb gauge Yang–Mills theory has attracted a considerable amount of attention over the
last few years. In the continuum, both the Hamiltonian [1, 2, 3, 4] and the Lagrangian [5, 6, 7, 8]
approach havebeeninvestigated. IntheHamiltonian approach, inparticular, theuseofvariational
methodswithGaussianwavefunctionalshasledtotheso-calledgapequationfortheinverseequal-
time gluon propagator. The analytical and numerical solutions show an inverse gluon propagator
which in the UV behaves like the photon energy but diverges in the IR, signalling confinement.
The obtained propagator also compares favourably with the available lattice data [9]; deviations
in the mid-momentum regime (and minor ones in the UV), which can be attributed to the gluon
loopescaping theGaussianwavefunctionals, canbetakenintoconsideration bygoingbeyondthe
purely Gaussian form. Totreat these non-Gaussian functionals the authors developed in Ref. [10]
a method relying on Dyson–Schwinger Equations (DSEs) by exploiting the formal similarity be-
tween vacuum expectation values in the Hamiltonian formalism and correlation functions in Eu-
clideanquantumfieldtheory. Inthistalkwereportabouttheimplementationofthesetechniquesin
thequarksectorofthetheory, inordertoinvestigate issueslikethespontaneous breaking ofchiral
symmetry.
The topic of chiral symmetry breaking in Coulomb gauge has been studied for example in
Ref. [11]. While it has been shown that an infrared enhanced potential can account for chiral
symmetrybreaking, thecalculated physical quantities, suchasthedynamicalmassandchiralcon-
densate, turn out tobe fartoo small. It has been recently shown [12] that using a wave functional
which includes the coupling of the quarks to the transverse gluon field improves the results to-
wards the phenomenological findings. Weapply here thetechniques developed inRef. [10]tothe
wavefunctional proposed inRef.[12]. Whilewedonotexpect considerably differentresultsfrom
Ref. [12]for physical quantities, the approach presented here has abroader range of applications,
andcouldeasilybeappliedtomorecomplicated wavefunctionals.
2. HamiltonianDyson–Schwinger Equations
The derivation of the Hamiltonian DSEs in pure Yang–Mills theory has been presented in
Ref. [10]. We summarize here briefly the essential steps leading from the choice of the wave
functional tothepertinent DSEs.
IntheSchrödingerpictureoftheYang–Millssectorwerepresentthegaugefieldandthecanon-
icalmomentumoperators inthebasis A oftheeigenstates ofthefieldoperatorby
| i
d
A Aˆ f =AF [A], A Pˆ f = F [A], (2.1)
h | | i h | | i id A
where F [A] A F isaphysical state. Thevacuum expectation value(VEV)ofanoperator K is
≡h | i
therefore givenby
K[A,P ] = DAJ F [A]K[A, d ]F [A]. (2.2)
h i Z A ∗ id A
InEq.(2.2) thefunctional integration runs overtransverse fieldconfigurations satisfying theCou-
lomb gauge condition, ¶ Aa=0, and is restricted to the first Gribov region; J =Det(G 1) is the
i i A −A
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HamiltonianDyson–SchwingerEquationsofQCD DavideCampagnari
Faddeev–Popov determinant whicharisesfromthegaugefixing,and
GaAb(~x,~y) −1= d ab¶ 2 gfacbAci(~x)¶ i d (~x ~y) (2.3)
− − −
(cid:2) (cid:3) (cid:0) (cid:1)
is the inverse Faddeev–Popov operator, with g being the coupling constant and fabc being the
structure constants ofthe su(N )algebra.
c
In a similar manner, a state F in the fermionic Fock space possesses a “coordinate” repre-
| i
sentation
x F F (x †,x ), (2.4)
h | i≡
where x ,x † arecomplexGrassmannfields. TheVEVofafermionic operator inthisstateisgiven
by
O[yˆ,yˆ†] = Dx †Dx e−x †(L +−L −)x F ∗(x ,x †)O yˆ,yˆ† F (x †,x ), (2.5)
Z
(cid:10) (cid:11) (cid:2) (cid:3)
wheretheL aretheprojectorsontostatesofpositiveandnegativeenergyofthefreeDiractheory.
±
TheDiracfieldoperators actonfunctionals according to
d
yFˆ (x †,x )= L x +L F (x †,x ),
(cid:18) − +dx †(cid:19)
(2.6)
d
yˆ†F (x †,x )= L x †+L F (x †,x ).
(cid:18) + −dx (cid:19)
Wehaveputexplicitlyahatoverthefermionoperators yˆ†inEq.(2.6)todistinguishthemfromthe
Grassmann (classical) fields x ,x † used inthe“coordinate” representation. Theexponential factor
occurring inthefermionicfunctional integration inEq.(2.5)arisesfromthecompleteness relation
forfermioniccoherent states, seee.g.Ref.[13].
ThevacuumstateofQCDcanbeassumedtobeoftheform
1
Y [A,x ,x †]=:exp S [A] S [x ,x †,A] , (2.7)
A f
(cid:26)−2 − (cid:27)
where S is afunctional ofthe gauge fieldonly, while S contains the fermion and fermion-gluon
A f
interaction parts. Dyson–Schwingerequations arederived bystarting fromtheidentity [10]
d
0= DADx †Dx df JAe−x †(L +−L −)x Y ∗[A,x ,x †]K[A,x ,x †]Y [A,x ,x †] (2.8)
Z
n o
withf A,x ,x † .
∈{ }
Toproceedfurther,weneedanexplicitansatzforthevacuum wavefunctionalEq.(2.7). Since
weareinterested heremainlyinthequarksector,wechoosethesimpleGaussianform
S = d3xd3yAa(~x)w (~x,~y)Aa(~y) (2.9)
A i i
Z
forthe Yang–Mills part. Itshould bestressed, however, that this isin no way necessary, and non-
Gaussian generalization can betaken aswell[10];werestrict ourselves totheform Eq.(2.9) only
forpedagogical reasons.
Forthequarkwavefunctional wechoosetheansatz[12]
S = d3xd3yx m†(~x)Kmn(~x,~y)x n(~y), x =L x . (2.10)
f + A
Z − ± ±
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HamiltonianDyson–SchwingerEquationsofQCD DavideCampagnari
−1 −1
=2 + =2 + +
−
Figure1: HamiltonianDSEsforthequark(left)andgluonpropagator(right). Filleddotsstandfordressed
propagatorsandemptycirclesforproperfunctions.Thesmallemptyboxesrepresentthevariationalkernels.
Curlylinesrepresentgluons,dashedlinesghosts,andstraightlinesfermions.
The kernel K contains both the purely fermionic part and the coupling ofthe quarks tothe trans-
A
versegluons,
Kmn(~x,~y)=d mnb s(~x,~y)+g d3zV(~x,~y;~z)~a ~Aa(~z)tmn. (2.11)
A a
Z ·
Here, b and~a are the usual Dirac matrices, t are the generators of the group in the fundamental
a
representation (or, in general, in the representation to which the fermion fields belong), and the
functions sandV arethevariational kernels.
With these choices for the wave functional the resulting DSEs are represented diagrammati-
callyinFig.1.
3. VariationalApproach to QCD
The Hamilton operator of QCD in Coulomb gauge [14], resulting from the resolution of
Gauss’slawinthecanonically quantized theory, canbewrittenas
1 d d 1
H = d3xJ 1 J + d3xFa(~x)Fa(~x)
QCD −2Z A− d Aa(~x) Ad Aa(~x) 4Z ij ij
i i
+ d3xy m†(~x) ia ¶ +b m y m(~x) g d3xy m†(~x)a Aa(~x)tmny n(~x)
i i i i a
Z − − Z
(cid:2) (cid:3)
+ d3xd3yJ 1r a(~x)J Fab(~x,~y)r b(~y). (3.1)
Z A− A A
InEq.(3.1)
Fa(~x)=¶ Aa(~x) ¶ Aa(~x)+gfabcAb(~x)Ac(~x) (3.2)
ij i j j i i j
−
isthespatialpartofthefieldstrength tensor,
Fab(~x,~y)= d3zGac(~x,~z)( ¶ 2)Gcb(~z,~y) (3.3)
A A z A
Z −
istheso-calledCoulombkernel,whicharisesfromthelongitudinalcomponentoftheelectricfield,
and
d
r a(~x)=y m†(~x)tmny n(~x)+ fabcAb(~x) (3.4)
a i id Ac(~x)
i
isthetotalcolourchargedensity.
The energy density can be evaluated as expectation value of the Hamiltonian Eq. (3.1). The
Hamiltonian DSEs in the general form Eq. (2.8) enter this calculation twice: first to express the
VEVsoftheoperators yˆ,yˆ† intermsoftheGrassmannfieldsx ,x †,andsecondtoturntheresult-
ing expressions, which involve propagators and vertex functions, in afunctional of the variational
kernels. Inthis calculation wehave tointroduce approximations, and weevaluate theenergy den-
sityuptotwoloops(seeRef.[10]formoredetails).
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HamiltonianDyson–SchwingerEquationsofQCD DavideCampagnari
4. GapEquations
Thevariation oftheenergydensity withrespecttoV fixesthevectorkernel
1+s(~p)s(~q)
V(~p,~q)= , (4.1)
−W (~p+~q)+|~p|1−s2(1~p+)+s22(s~p()~p)s(~q)+|~q|1−s2(1~q+)+s22(s~q()~p)s(~q)
whereW = 1 AA 1 istheinversegluonpropagator. Replacing thekernelsoccurring ontheright-
2h i−
handsidebytheirtree-levelforms,i.e. W (~p)= ~p ands(~p)=0,weobtain
| |
1
V (~p,~q)= , (4.2)
0
− ~p+~q + ~p + ~q
| | | | | |
which isexactly theleading-order perturbative expression forthe quark-gluon vertexformassless
fermions[15].
Equation (4.1) can be inserted back into the expression for the energy density, and taking
functional derivatives withrespect to w (~p)ands(~p)weobtainthegluonandquarkgapequations.
Asafirstapproximation,wewillignorethes-dependenceofthedenominatorofEq.(4.1). Keeping
thewholedenominatorstructurewouldgiverisetomorecomplicatedexpressionswhich,however,
havethesameIRandUVasymptoticbehaviour.
Theapproximated gapequation forthescalarquarkkernel sreads
g2C d3q s(~q) 1 s2(~p) pˆ qˆs(~p) 1 s2(~q)
F
~p s(~p)= F(~p ~q) − − · −
| | 2 Z (2p )3 − (cid:2) 1(cid:3)+s2(~q) (cid:2) (cid:3)
(4.3)
g2C d3q X(~p,~q) 1+s(~p)s(~q) s(~q) s(~p)
F
+ − ,
2 Z (2p )3 W (~p+~q) 1+(cid:2) s2(~q) W (~p(cid:3)+(cid:2)~q)+ ~q +(cid:3)~p
| | | |
(cid:2) (cid:3)(cid:2) (cid:3)
where F = F is the Coulomb propagator and X is a tensor structure arising from the traces in
A
h i
Dirac space. The first term on the right-hand side of Eq. (4.3) was obtained by Adler and Davis
[11]; the second term is due to the coupling of the quarks to the transverse gluons. Asmentioned
intheIntroduction, inRef.[12]thecouplingtotransverse gluonswasalsotakenintoaccount. The
authors of Ref. [12] used the same wave functional Eq. (2.10) as in this work but did a quenched
calculation. As a consequence the effect of the Yang–Mills kinetic energy operator on the quarks
wsneglected.
Trading the scalar kernel s in favour of the mass function M, Eq. (4.3) takes the following
simpleform
C d3q ~p M(~q) pˆ qˆ ~q M(~p)
~p M(~p)=g2 F F(~p ~q)| | − · | |
| | 2 Z (2p )3 − E(~q)
(4.4)
C d3q X(~p,~q) ~p M(~q) ~qM(~p)
+g2 F | | −| | ,
2 Z (2p )3 W (~p+~q) W (~p+~q)+ ~p + ~q E(~q)
| | | |
(cid:2) (cid:3)
with
E(~p)= ~p2+M2(~p). (4.5)
p
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HamiltonianDyson–SchwingerEquationsofQCD DavideCampagnari
0.5
Gribov
0.45
withquarks
0.4
0.35
0.3
D(p) 0.25
0.2
0.15
0.1
0.05
0
0 1 2 3 4 5 6 7 8
p/√σ
C
Figure2: GluonpropagatorD=1/(2W )inthepresenceofquarksforthegapequation(4.6)assumingthe
GribovformfortheYang–Millspart. PhysicaldimensionsaresetbytheCoulombstringtensions .
C
Thevariationoftheenergydensitywithrespecttothegluonkernelw combinedwiththegluon
propagator DSEyieldsthegapequation forinversegluonpropagator W
W 2(~p)=~p2+Yang–Millsterms
d3q 1+s(~q)s(~p+~q) 2 2W (~p)+ ~q + ~p+~q
g2 | | | |
− Z (2p )3 1+(cid:2)s2(~q) 1+s2(~p+(cid:3) ~q) W (~p)+ ~q + ~p+~q 2
| | | |
(cid:2) (cid:3)(cid:2) (cid:3)(cid:2) ~p ~q(cid:3)+(pˆ ~q)2
1+ · · , (4.6)
×(cid:20) ~q ~p+~q (cid:21)
| || |
wheretheusualYang–Millsterms(ghostloop,Coulombinteraction, andpossiblyadditionalterms
stemmingfromanon-Gaussian ansatz)canbefoundinRef.[10].
Equations(4.3)and(4.6)formacoupled system,toghetherwiththeDSEfortheghostpropa-
gator and the Coulomb form factor. The analysis of the full coupled set of equations is subject of
ongoing work. Asa first estimate, we investigate the effect of the quark loop on the gluon propa-
gator: wetaketheGribovformulaforthepureYang–Millsgluonpropagatoranduseforthescalar
kernel s a form fitted from the data in Ref. [12]. The result is plotted in Fig. 2. As we see, the
gluon propagatorlosessomestregthinthemid-momentum regime. Thisbehaviour isknownfrom
Landaugaugestudies, andCoulombgaugeinvestigations onthelatticeconfirmthis[16].
5. Conclusions
Thegeneralmethodtotreatnon-Gaussianwavefunctionals intheHamiltonianformulationof
a quantum field theory presented in Ref. [10] has been applied here to QCD in Coulomb gauge.
Wehaveusedthequarkwavefunctional suggestedinRef.[12],whichincludesthecouplingofthe
quarks tothetransverse gluons. Theresulting quark gapequations aresimilarinbothapproaches;
the numerical solution of the equations and the comparison to the findings of Ref. [12] are the
subject of ongoing work. As a first application we have presented here the effect of the back-
coupling of the quarks on the gluon propagator. In future work we intend to expand the present
approach toQCDatfinitetemperature [17]andbaryondensity.
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HamiltonianDyson–SchwingerEquationsofQCD DavideCampagnari
Acknowledgments
The authors are grateful to M. Pak and P. Watson for useful discussions. This work was
supported bytheDFGundercontracts No.Re856/9-1andbyBMBF06TU7199.
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