Hamiltonian Approach to QCD in Coulomb Gauge: Perturbative Treatment of the Quark Sector

# Hamiltonian Approach to QCD in Coulomb Gauge: Perturbative Treatment of the Quark Sector

Davide R. Campagnari\thanksrefe1 Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany    Hugo Reinhardt\thanksrefe2 Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
19th August 2019
###### Abstract

We study the static gluon and quark propagator of the Hamiltonian approach to Quantum Chromodynamics in Coulomb gauge in one-loop Rayleigh–Schrödinger perturbation theory. We show that the results agree with the equal-time limit of the four-dimensional propagators evaluated in the functional integral (Lagrangian) approach.

###### pacs:
11.10.Ef 12.38.Aw
journal: Eur. Phys. J. C\thankstext

e1e-mail: d.campagnari@uni-tuebingen.de \thankstexte2e-mail: hugo.reinhardt@uni-tuebingen.de

## 1 Introduction

Over the years there has been increased activity in Quantum Chromodynamics (QCD) in Coulomb gauge, starting with the seminal works of Gribov Gribov:1977wm and Zwanziger Zwanziger:1998ez . The use of Coulomb gauge is motivated by the fact that it is a so-called “physical gauge”. In fact, in QED Coulomb gauge fixing yields immediately the gauge invariant (transverse) part of the gauge field, i.e. physical degrees of freedom. Although this is not the case in QCD, one still expects that the transverse components of the gauge field contain the dominant part of the physical degrees of freedom.111This is indeed confirmed by variational calculations within the Hamiltonian approach to Yang–Mills theory in Coulomb gauge, where the longitudinal part of the momentum operator, i.e. of the kinetic energy, is completely irrelevant compared to the transverse part Heffner:2012sx .

Coulomb gauge has been mainly used in two approaches to QCD: i) in the Dyson–Schwinger equations (DSEs) based on the functional integral formulation of QCD Zwanziger:1998ez ; Baulieu:1998kx ; Watson:2006yq ; Watson:2007mz ; Watson:2008fb ; Popovici:2008ty ; Watson:2011kv ; Popovici:2010mb and ii) in a variational approach based on the Hamiltonian formulation Schutte:1985sd ; Szczepaniak:2001rg ; Feuchter:2004mk ; Reinhardt:2004mm ; Epple:2006hv ; Epple:2007ut ; Campagnari:2010wc ; Pak:2011wu ; Reinhardt:2011hq ; Reinhardt:2012qe . The general formulation of Yang–Mills theory within the Dyson–Schwinger approach in Coulomb gauge was set up in Ref. Watson:2006yq and treated in one-loop perturbation theory in Refs. Watson:2007mz . Thereby the results of covariant gauges were reproduced.

Since QCD is an asymptotically free theory, one expects its high-energy behaviour to be dominated by the perturbative results. The understanding of perturbation theory is therefore necessary for the regularization and renormalization of non-perturbative approaches. The perturbative treatment of the Yang–Mills sector of QCD within the Hamiltonian formulation was given in Refs. Campagnari:2009km ; Campagnari:2009wj . In the present paper we extend the perturbative analysis to the quark sector of QCD. Some of the results presented below have been already obtained in Ref. PhDCampagnari . Using the familiar Rayleigh–Schrödinger perturbation theory we calculate the static quark and gluon propagators to one-loop order within the Hamiltonian approach and show that they agree with the equal-time limit of the four-dimensional propagators evaluated in the more traditional functional integral approach Popovici:2008ty . The perturbative results obtained in the present paper for the static propagators are essential ingredients for the renormalization of the (non-perturbative) variational approach to the Hamiltonian formulation of QCD in Coulomb gauge to be presented elsewhere Campagnari:tbp .

## 2 Perturbative Expansion of the QCD Hamilton Operator

The Hamilton operator of QCD in Coulomb gauge Christ:1980ku reads in space dimensions

 (1)

where and are the Dirac matrices, (with ) are the generators of the algebra in the fundamental representation, and

 JA=DetG−1A,[GabA(→x,→y)]−1=(−δab∂2−gfacbAci(→x)∂i)δ(→x−→y) (2)

is the Faddeev–Popov determinant, with being the bare coupling and the structure constants of . Furthermore, is the non-Abelian magnetic field, and

 FabA(→x,→y)=∫ddzGacA(→x,→z)(−∂2→z)GcbA(→z,→y) (3)

is the so-called Coulomb kernel, which arises from the resolution of Gauss’s law in Coulomb gauge: It describes the Coulomb-like interaction between colour charges, whose density is given by

 ρa(→x)=ρaQ(→x)+ρaA(→x)=ψm†(→x)tamnψn(→x)+fabcAbi(→x)δiδAci(→x)

to which both the quarks and the gluons contribute.

The fermion field operator can be expanded in terms of the eigenspinors , of the free Dirac Hamiltonian

 h0(→p)=→α⋅→p+βm

in the standard way

 ψm(→x)=∫\djpei→p⋅→xψm(→p),ψm(→p)=1√2E→p[u(→p,s)bm(→p,s)+v(−→p,s)dm†(−→p,s)], (4)

where the index accounts for the two spin degrees of freedom. Furthermore, we have introduced the abbreviation

 ∫\djp≡∫ddp(2π)d.

The spinors , satisfy the eigenvalue equation

 h0(→p)u(→p,s)=E→pu(→p,s),h0(→p)v(−→p,s)=−E→pv(−→p,s), (5)

with , and are normalized to

 u†(→p,s)u(→p,s′)=2E→pδss′=v†(→p,s)v(→p,s′),u†(→p,s)βu(→p,s′)=2mδss′=−v†(→p,s)βv(→p,s′),u†(→p,s)v(−→p,s′)=0. (6)

The expansion coefficients , are annihilation and creation operators satisfying the usual anti-commutation relations

 {bm(→p,s),bn†(→q,t)}=δmnδst(2π)3δ(→p−→q), {dm(→p,s),dn†(→q,t)}=δmnδst(2π)3δ(→p−→q),

which, with the normalization Eq. (6), ensure that the Fermi field in coordinate space has the required anticommutation relation

 {ψm(→x),ψn†(→y)}=δmnδ(→x−→y).

For later convenience it is useful to introduce the following orthogonal projectors

 Λ±(→p):=12±h0(→p)2E→p, (7)

which are (colour diagonal) Dirac matrices satisfying

 Λ+(→p)+Λ−(→p)=1,Λ+(→p)Λ−(→p)=0, 2=Λ±(→p).

Furthermore, from Eqs. (5) and (7) follows

 Λ+(→p)u(→p,s) =u(→p,s), Λ+(→p)v(−→p,s) =0, Λ−(→p)v(−→p,s) =v(−→p,s), Λ−(→p)u(→p,s) =0.

The projectors are related to the Dirac spinors by the following completeness relations

 ∑su(→p,s)⊗u†(→p,s)2E→p=Λ+(→p),∑sv(→p,s)⊗v†(→p,s)2E→p=Λ−(−→p). (8)

The Hamiltonian Eq. (1) can be perturbatively expanded in powers of the coupling constant ,

 HQCD=H0+gH1+g2H2+…

Since the perturbative treatment of the gluon sector within the Hamiltonian approach in Coulomb gauge was already given in Ref. Campagnari:2009km we will focus here on the perturbative treatment of the quark sector. The “unperturbed” Hamiltonian for the quarks is the free Dirac Hamiltonian, i.e. the third term on the r.h.s. of Eq. (1). Using the decomposition Eq. (4) of the quark field and the orthogonality relations (6), it acquires the standard form

 HQ0=EQ0+∫\djpE→p[bm†(→p,s)bm(→p,s)+dm†(→p,s)dm(→p,s)],

where is the (negative divergent) zero-point energy. The vacuum state of the free Dirac theory is annihilated by the operators and ,

 bm(→p,s)|0⟩Q=0,dm(→p,s)|0⟩Q=0, (9)

and their Hermitian conjugate operators and generate the eigenstates of , e.g.

 HQ0bm†(→p,s)|0⟩Q =(EQ0+E→p)bm†(→p,s)|0⟩Q, HQ0dm†(→p,s)|0⟩Q =(EQ0+E→p)dm†(→p,s)|0⟩Q.

The gauge field operator can be expanded as

 Aci(→x)=∫\djpei→p⋅→xAci(→p),Aci(→p)=1√2|→p|[aci(→p)+ac†i(−→p)] (10)

where and are bosonic ladder operators satisfying

 [aci(→p),ab†j(→q)]=δcbtij(→p)(2π)dδ(→p−→q),

with

 tij(→p)=δij−pipj→p2

being the transverse projector in momentum space. The unperturbed gluon Hamiltonian becomes

 HYM0=EYM0+∫\djp|→p|ac†i(→p)aci(→p)

where is the irrelevant diverging zero-point energy of the gluons. The vacuum state of the free Yang–Mills sector is annihilated by the operators

 aci(→p)|0⟩YM=0 (11)

and the eigenstates of are generated by .

The unperturbed QCD vacuum state is given by the tensor product

 |0⟩=|0⟩YM⊗|0⟩Q,

with the quark and gluonic vacuum defined, respectively, by Eq. (9) and Eq. (11). Expectation values of products of field operators obviously factorize in products of fermionic and gluonic expectation values, e.g.

 ⟨0|Aψ†ψψAψ†|0⟩=YM⟨0|AA|0⟩YM×Q⟨0|ψ†ψψψ†|0⟩Q,

for which Wick’s theorem holds: therefore, in perturbation theory all matrix elements can be expressed by the free static gluon

 ⟨0|Aai(→p)Abj(→q)|0⟩=δabtij(→p)2|→p|(2π)dδ(→p+→q), (12)

and quark Green functions

 ⟨0|ψmα(→p)ψn†β(→q)|0⟩=δmn(2π)dδ(→p−→q)[Λ+(→p)]αβ,⟨0|ψm†α(→p)ψnβ(→q)|0⟩=δmn(2π)dδ(→p−→q)[Λ−(→p)]βα, (13)

where the spinor indices have been written out explicitly.

The first-order perturbation is given by the minimal coupling term , the fourth term on the r.h.s. of Eq. (1), and reads in momentum space

 H1=∫\djp1\djp2ψk†(→p1)[−taklαiAai(→p1−→p2)]ψl(→p2). (14)

The second-order perturbation arises from the non-Abelian Coulomb interaction, last term in Eq. (1). Since this operator comes with a factor of , we can replace the Coulomb kernel by its lowest-order expression, which is given by the negative inverse Laplacian [see Eqs. (2) and (3) with ], yielding

 H2=tamntakl12∫\djp1\djp2\djp3ψm†(→p1)ψn(→p2)×1(→p1−→p2)2ψk†(→p3)ψl(→p1−→p2+→p3). (15)

We have included here only the Coulomb interaction between fermionic charges: the coupling between the fermionic and gluonic colour charge through the Coulomb kernel does not contribute to the propagators to one-loop order and will henceforth be discarded.

## 3 Perturbative Corrections to the Vacuum State

In Rayleigh–Schrödinger perturbation theory the vacuum state is expanded in a power series

 |0⟩QCD∼|0⟩+g|0⟩(1)+g2|0⟩(2)+O(g3). (16)

and the perturbative corrections to the wave function are chosen to be orthogonal to the unperturbed state

 ⟨0|0⟩(n)=0,n≥1.

The first- and second-order corrections to the vacuum state are

 |0⟩(1)=−∑⟨N|H1|0⟩EN−E0|N⟩|0⟩(2)=−∑⟨N|H2|0⟩+⟨N|H1|0⟩(1)EN−E0|N⟩ (17)

where stands for a generic -particle state222For bosonic -particle states a factor has to be included to avoid multiple counting. with energy . Furthermore, is the energy of the perturbative QCD vacuum, which cancels, however, in the energy denominators. The series given in Eq. (16) is not normalized. The normalized state reads to the desired order

 |0⟩QCD=(1−g22(1)⟨0|0⟩(1))|0⟩+g|0⟩(1)+g2|0⟩(2)+O(g3). (18)

It will not be necessary to evaluate explicitly: this term merely cancels disconnected diagrams occurring in the evaluation of the propagators.

With the help of the projectors [Eq. (7)] and of the sum rules (8) we obtain

 bm†(→p,s)|0⟩⟨0|bm(→p,s)=[Λ+(→p)]αβψm†α(→p)|0⟩⟨0|ψmβ(→p),dm†(→p,s)|0⟩⟨0|dm(→p,s)=[Λ−(→p)]αβψmβ(→p)|0⟩⟨0|ψm†α(→p). (19)

Analogously, in view of Eq. (10) we have

 ac†i(→p)|0⟩⟨0|aci(→p)=2|→p|Aci(−→p)|0⟩⟨0|Aci(→p), (20)

where we have used . These relations allow us to express the matrix elements occurring in Eq. (17) in terms of field operators only, for which we can then use Wick’s theorem together with Eqs. (12) and (13).

The evaluation of the matrix elements in Eq. (17) with the operators and given by Eqs. (14) and (15) is now straightforward. As an example, we sketch here the evaluation of the first-order correction. From the form of the first-order perturbation [Eq. (14)] follows immediately that we need to consider in the sum in Eq. (17) the states with one gluon, one quark, and one antiquark. With Eqs. (19) and (20) we have therefore

 |0⟩(1)=−∫\djℓ1\djℓ2\djℓ3⟨0|Acj(→ℓ3)ψn†γ(→ℓ2)ψmβ(→ℓ1)H1|0⟩E→ℓ1+E→ℓ2+|→ℓ3|×[Λ+(→ℓ1)]αβ[Λ−(→ℓ2)]γδ2|→ℓ3|ψm†α(→ℓ1)ψnδ(→ℓ2)Acj(−→ℓ3)|0⟩.

If we now insert the explicit form Eq. (14) of the first-order perturbation into the above expression, we are led to an expression containing the matrix element

 ⟨0|Acj(→ℓ3)ψn†γ(→ℓ2)ψmβ(→ℓ1)ψk†(→p1)Aai(→p1−→p2)ψl(→p2)|0⟩.

The fermionic and gluonic expectation values factorize and can be evaluated with the help of Eqs. (13) and (12). For the first-order correction to the QCD vacuum wave functional we finally obtain

 |0⟩(1)=∫\djp\djqtamnAai(→p−→q)×ψm†(→p)Λ+(→p)αiΛ−(→q)ψn(→q)E→p+E→q+|→p−→q||0⟩. (21)

Analogously one calculates the higher-order contributions. The second-order perturbative correction to the vacuum wave functional contains a fermionic part

 |0⟩(2)qq=CF∫\djp\djq1E→pψm†(→p)Λ+(→p)[−S0(→q)2(→p−→q)2+tij(→p−→q)2|→p−→q|αiS0(→q)αjE→p+E→q+|→p−→q|]Λ−(→p)ψm(→p)|0⟩, (22)

where is the quadratic Casimir of the fundamental representation, as well as a term involving gauge field operators

 |0⟩(2)AA=14∫\djp\djqtr[Λ+(→p)αiΛ−(→q)αj]E→p+E→q+|→p−→q|×1|→p−→q|Aai(→q−→p)Aaj(→p−→q)|0⟩. (23)

In Eq. (22), is the tree-level static quark propagator

 S0(→p)=12[Λ+(→p)−Λ−(→p)]=→α⋅→p+βm2E→p.

Other second-order corrections to the wave functional do not contribute to the propagators at one-loop order and will be henceforth ignored. Equation (22) contributes only to the quark propagator, while Eq. (23) yields a quark-loop term to the gluon propagator. The first-order term Eq. (21) gives a one-loop correction to both the gluon and the quark propagator.

With the perturbative corrections to the QCD vacuum state given above it is straightforward to carry out the perturbative expansion of the quark and gluon propagators analogously to the perturbative treatment of the Yang-Mills sector given in Ref. Campagnari:2009km .

## 4 One-Loop Perturbative Propagators

### 4.1 Gluon Propagator

The perturbative correction to the gluon propagator

 QCD⟨0|Aai(→p)Abj(→q)|0⟩QCD=:δab(2π)dδ(→p+→q)tij(→p)D(→p)

can be evaluated by inserting Eq. (18) with the corrections given by Eqs. (21)–(23) into the above expression. The evaluation of the resulting matrix elements is straightforward. To one-loop level the gluon propagator is obtained in the form

 D(→p)=12|→p|[1+g2D(1)A(→p)+g2D(1)Q(→p)+O(g4)],

where

 D(1)A(→p)=g2Nc(d−1)→p2∫\djq1−(^→p⋅^→q)2|→q||→p+→q||→p|+|→q|(|→p|+|→q|+|→p+→q|)2×[(d−1)(→p2+→q2)+(d−2)→p2→q2+(→p⋅→q)2(→p+→q)2]−g2Nc4(d−1)→p2∫\djqd−2+(^→p⋅^→q)2|→q|→q2−→p2(→p+→q)2

represents the one-loop correction arising from the Yang–Mills sector which was calculated in Refs. Campagnari:2009km ; Campagnari:2009wj , while

 D(1)Q(→p)=tij(→p)2(d−1)→p2∫\djq2|→p|+E→q+E→p+→q(|→p|+E→q+E→p+→q)2×tr[Λ+(→p+→q)αiΛ−(→q)αj]

is the contribution of the dynamical quarks. The Dirac trace can be explicitly taken and results in

 D(1)Q(→p)=1(d−1)→p2∫\djq|→p|+E→qE→qE→p+→q×(d−1)[E→qE→p+→q+→q⋅(→p+→q)+m2]−2qitij(→p)qj(|→p|+E→q+E→p+→q)2. (24)

As shown in Refs. Campagnari:2009km ; Campagnari:2009wj , the result of the Rayleigh–Schrödinger perturbation theory can be compared with the result Popovici:2008ty ; Watson:2007mz of the more conventional perturbation theory in the Lagrangian (functional integral) approach in Coulomb gauge. The quark-loop contribution to the gluon form factor evaluated in Ref. Popovici:2008ty reads

 W(1)Q(p)=2(d−1)p2∫dd+1q(2π)d+1×(d−1)[q24+q4p4+→q⋅(→q+→p)+m2]−2qitij(→p)qj(q2+m2)[(p+q)2+m2], (25)

where is the squared Euclidean four-momentum. The dressing function of the equal-time propagator is related to the dressing function Eq. (25) of the four-dimensional (energy-dependent) propagator by

 D(1)Q(→p)=2|→p|∫dp42πW(1)Q(p)p2. (26)

Inserting Eq. (25) into Eq. (26) and performing the integrals over and , Eq. (24) is indeed recovered. Furthermore, from the non-renormalization of the ghost-gluon vertex the well-known first coefficient of the QCD function

 β(g)=μ2∂∂μ2g2(μ)=β0(4π)2g4+…, β0=−11Nc−2Nf3

in presence of quarks is recovered in the present Hamiltonian approach.

### 4.2 Quark Propagator

The quark propagator is defined in the Hamiltonian approach by

 12QCD⟨0|[ψmα(→p),ψn†β(→q)]|0⟩QCD=:Smnαβ(→p)(2π)dδ(→p−→q). (27)

The commutator in Eq. (27) arises from the equal-time limit of the time-ordering present in the definition of the time-dependent quark Green function. The perturbative expansion of the quark propagator can be obtained by inserting Eq. (16) into Eq. (27), yielding

 (28)

The evaluation of the matrix elements arising from the insertion of Eqs. (21) and (22) into Eq. (28) is somewhat lengthy but straightforward, and results in

 (29)

If the perturbed propagator is parameterized by

 S(→p)=A(→p)→α⋅→p+βB(→p)2E→p (30)

from Eq. (29) we can extract the one-loop expressions for the form factors and by taking the appropriate traces. This results in

 A(→p)=1−g2m2CF2→p2E2→p∫ddq(2π)d→p⋅(→p+→q)E→q(→p+→q)2−g2CF2→p2E2→p∫ddq(2π)d1E→q|→p+→q|(E→p+E→q+|→p+→q|)2×{[(d−3)→p⋅→q+2[→p⋅(→p+→q)][→q⋅(→p+→q)](→p+→q)2]×[m2(E→p+E→q+|→p+→q|)−→p2E→p]+(d−1)→p2[E2→pE→q+m2(2E→p+E→q+|→p+→q|)]} (31a) for the form factor of the kinetic term, and B(→p)=m+mg2CF2E2→p∫\djq→p⋅(→p+→q)E→q(→p+→q)2+mg2CF2E2→p∫\djq1E→q|→p+→q|(E→p+E→q+|→p+→q|)2×{(2E→p+E→q+|→p+→q|)[(d−3)→p⋅→q+2[→p⋅(→p+→q)][→q⋅(→p+→q)](→p+→q)2]+(d−1)[(→p2−m2)E→p+→p2|→p+→q|−m2E→q]} (31b)

for the mass term.

As for the gluon propagator, these form factors can be compared with the results of the Lagrangian approach. In Ref. Popovici:2008ty , the quark propagator (in Euclidean space) was parameterized in the form333In the Hamiltonian approach we work with rather than . The formulae presented here differ from the ones in Ref. Popovici:2008ty by an overall matrix .

 S(→p,p4)=p4Ft(p)+→α⋅→pFs(p)+βM(p)p24+→p2+m2 (32)

where the dressing functions , , and are functions of and . In this parameterization the component has been discarded, since it does not arise at one-loop level.444Lattice calculations Burgio:2012ph indicate that this component vanishes. The static propagator is obtained from the energy-dependent one [Eq. (32)] by integrating out the temporal component of the four-momentum

 S(→p)=∫dp42πS(→p,p4). (33)

Since the dressing function is an even function of , this component does not contribute to the equal-time propagator. Inserting Eq. (32) into Eq. (33) we find from Eq. (30) the identification

 ∫dp42πFs(p)p24+→p2+m2=A(→p), ∫dp42πM(p)p24+→p2+m2=B(→p).

In fact, inserting here for and the results found in Ref. Popovici:2008ty and performing the integration over we recover for and