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arXiv:hep-th/0203005v2 25 Sep 2002

CERN-TH-2002-049

FAU-TP3-02-06

Renormalisation group flows for gauge theories in axial gauges

Daniel F. Litim∗and Jan M. Pawlowski†

∗Theory Division, CERN

CH-1211 Geneva 23.

†Institut f¨ ur Theoretische Physik III

Universit¨ at Erlangen, D-91054 Erlangen.

Abstract

Gauge theories in axial gauges are studied using Exact Renormalisation Group flows.

We introduce a background field in the infrared regulator, but not in the gauge fixing,

in contrast to the usual background field gauge. We discuss the absence of spurious

singularities and the finiteness of the flow. It is shown how heat-kernel methods can

be used to obtain approximate solutions to the flow and the corresponding Ward

identities. New expansion schemes are discussed, which are not applicable in covari-

ant gauges. As an application, we derive the one-loop effective action for covariantly

constant field strength, and the one-loop β-function for arbitrary regulator.

∗Daniel.Litim@cern.ch

†jmp@theorie3.physik.uni-erlangen.de

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I. INTRODUCTION

The perturbative sector of QCD is very well understood due to the weak coupling of glu-

ons in the ultraviolet (UV) limit, known as asymptotic freedom. In the infrared (IR) region,

however, the quarks and gluons are confined to hadronic states and the gauge coupling is

expected to grow large. Thus the IR physics of QCD is only accessible with non-perturbative

methods. The exact renormalisation group (ERG) provides such a tool [1,2]. It is based on a

regularised version of the path integral for QCD, which is solved by successively integrating-

out momentum modes.

ERG flows for gauge theories have been formulated in different ways (for a review, see

[3]). Within covariant gauges, ERG flows have been studied in [4,5,6], while general ax-

ial gauges have been employed in [7,8]. In these approaches, gauge invariance of physical

Greens functions is controlled with the help of modified Ward or Slavnov-Taylor identities

[5,6,7,8,9,10]. A different line has been followed in [11] based on gauge invariant variables,

e.g. Wilson loops. Applications of these methods to gauge theories include the physics of su-

perconductors [12], the computation of instanton-induced effects [13], the heavy quark effec-

tive potential [14,15], effective gluon condensation [16], Chern-Simons theory [17], monopole

condensation [18], chiral gauge theories [19], supersymmetric Yang-Mills theories [20], and

the derivation of the universal two-loop beta function [21].

In the present paper, we use flow equations to study Yang-Mills theories within a back-

ground field method. In contrast to the usual background field formalism [22], we use a

general axial gauge, and not the covariant background field gauge. The background field

enters only through the regularisation, and not via the gauge fixing. Furthermore, in axial

gauges no ghost degrees of freedom are present and Gribov copies are absent. Perturba-

tion theory in axial gauges is plagued by spurious singularities of the propagator due to an

incomplete gauge fixing, which have to be regularised separately. Within an exact renormal-

isation group approach, and as a direct consequence of the Wilsonian cutoff, these spurious

singularities are absent [7]. The resulting flow equation can be used for applications even

beyond the perturbative level. This formalism has been used for a study of the propagator

[23], for a formulation of Callan-Symanzik flows in axial gauges [24], and for a study of

Wilson loops [25,26].

Here, we continue the analysis of [7,8] and provide tools for the study of Yang-Mills

theories within axial gauges.First we detail the discussion of the absence of spurious

singularities.Then a framework for the evaluation of the path integral for covariantly

constant fields is discussed. We use an auxiliary background field which allows us to define

a gauge invariant effective action. The background field is introduced only in the regulator,

in contrast to the usual background field formalism. This way it is guaranteed that all

background field dependence vanishes in the infrared limit, where the cutoff is removed.

We employ heat kernel techniques for the evaluation of the ERG flow. The heat kernel is

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used solely as a technical device, and not as a regularisation. The flow equation itself is by

construction infra-red and ultra-violet finite and no further regularisation is required. As

an explicit application, we compute the full one-loop effective action for non-Abelian gauge

theories. This includes the universal β-function at one loop for arbitrary regulator. We also

discuss new expansions of the flow, which are not applicable for covariant gauges.

The work is organised as follows. We begin with a brief review of the Wilsonian approach

for gauge theories. This includes a derivation of the flow equation. We discuss the absence

of spurious singularities and the finiteness of the flow. This leads to a mild restriction on

the fall-off behaviour of regulators at large momenta. (Section II). Next, we consider the

implications of gauge symmetry. This includes a discussion of the Ward-Takahashi identities,

the construction of a gauge-invariant effective action, and the study of the background field

dependence. Explicit examples for background field dependent regulators are also given

(Section III). We derive the propagator for covariantly constant fields, and explain how

expansions in the fields and heat kernel techniques can be applied in the present framework

(Section IV). We compute the full one loop effective action using heat kernel techniques.

We also show in some detail how the universal beta function follows for arbitrary regulator

functions (Section V). We close with a discussion of the main results (Section VI) and leave

some more technical details to the Appendices.

II. WILSONIAN APPROACH FOR GAUGE THEORIES

In this section we review the basic ingredients and assumptions necessary for the con-

struction of an exact renormalisation group equation for non-Abelian gauge theories in

general axial gauges. This part is based on earlier work [7,8]. New material is contained in

the remaining subsections, where we discuss the absence of spurious singularities and the

finiteness of the flow.

A. Derivation of the flow

The starting point for the derivation of an exact renormalisation group equation are

the classical action SAfor a Yang-Mills theory, an appropriate gauge fixing term Sgfand a

regulator term ∆Sk, which introduces an infra-red cut-off scale k (momentum cut-off). This

leads to a k-dependent effective action Γk. Its infinitesimal variation w.r.t. k is described

by the flow equation, which interpolates between the gauge-fixed classical action and the

quantum effective action, if ∆Skand Γksatisfy certain boundary conditions at the initial

scale Λ. The classical action of a non-Abelian gauge theory is given by

SA[A] =1

4

?

d4xFa

µν(A)Fa

µν(A)(2.1)

with the field strength tensor

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Fa

µν(A) = ∂µAa

ν− ∂νAa

µ+ gfa

bcAb

µAc

ν

(2.2)

and the covariant derivative

Dab

µ(A) = δab∂µ+ gfacbAc

µ,[tb,tc] = fabcta.(2.3)

A general axial gauge fixing is given by

Sgf[A] =1

2

?

d4xnµAa

µ

1

ξn2nνAa

ν.(2.4)

The gauge fixing parameter ξ has the mass dimension −2 and may as well be operator-

valued [7]. The particular examples ξ = 0 and ξp2= −1 are known as the axial and the

planar gauge, respectively. The axial gauge is a fixed point of the flow [7].

The scale-dependent regulator term is

∆Sk[A,¯A] =1

2

?

d4xAa

µRkab

µν[¯A]Ab

ν.(2.5)

It is quadratic in the gauge field and leads to a modification of the propagator. We have

introduced a background field¯A in the regulator function. Both the classical action and

the gauge fixing depend only on A. The background field serves as an auxiliary field which

can be interpreted as an index for a family of different regulators Rk,¯ A. Its use will become

clear below.

The scale dependent Schwinger functional Wk[J,¯A], given by

expWk[J,¯A] =

?

DAexp

?

−Sk[A,¯A] +

?

d4xAa

µJa

µ

?

,(2.6)

where

Sk[A,¯A] = SA[A] + Sgf[A] + ∆Sk[A,¯A]. (2.7)

We introduce the scale dependent effective action Γk[A,¯A] as the Legendre transform of

(2.6)

Γk[A,¯A] =

?

d4xJa

µAa

µ− Wk[J,¯A] − ∆Sk[A,¯A],Aa

µ=δWk[J,¯A]

δJa

µ

. (2.8)

For later convenience, we have subtracted ∆Skfrom the Legendre transform of Wk. Thus

Γk[A,¯A] is given by the integro-differential equation

exp−Γk[A,¯A] =

?

Da exp

?

−SA[a] − Sgf[a] − ∆Sk[a − A,¯A] +

δ

δAΓk[A,¯A](a − A)

?

. (2.9)

The corresponding flow equation for the effective action

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∂tΓk[A,¯A] =1

2Tr

?

Gk[A,¯A]∂tRk[¯A]

?

(2.10)

follows from (2.9) by using ?a − A? = 0. The trace sums over all momenta and indices,

t = lnk. Gkis the full propagator of the field A, whereas¯A is not propagating. Its inverse

is given by

?

Gk[A,¯A]

?−1ab

µν(x,x′) =

δ2Γk[A,¯A]

δAµ

a(x)δAν

b(x′)+ Rk[¯A]

ab

µν(x,x′).(2.11)

There are no ghost terms present in (2.10) due to the axial gauge fixing. For the regulator

Rkwe require the following properties at¯A = 0.

lim

p2/k2→∞p2Rk= 0,lim

p2/k2→0Rk∼ p2

?k2

p2

?γ

,(2.12)

where p2is plain momentum squared. Regulators with γ = 1 have a mass-like infra-red

behaviour with Rk(0) ∼ k2. The example in (3.21) has γ = 1. In turn, regulator with γ > 1

diverge for small momenta. The latter condition in (2.12) implies that Rkintroduces an IR

regularisation into the theory. The first condition in (2.12) ensures the UV finiteness of the

flow in case that Gk∝ p−2for large p2. For covariant gauges this is guaranteed. Within

axial gauges, additional care is necessary because of the presence of spurious singularities.

It is seen by inspection of (2.9) and (2.12) that the saddle-point approximation about A

becomes exact for k → ∞. Here, Γkapproaches the classical action. For k → 0, in turn, the

cut-off term disappears and we end up with the full quantum action. Hence, we confirmed

that the functional Γk indeed interpolates between the gauge-fixed classical and the full

quantum effective action:

lim

k→∞Γk[A,¯A] ≡ S[A] + Sgf[A],

lim

(2.13a)

k→0Γk[A,¯A] ≡ Γ[A].(2.13b)

Notice that both limits are independent of¯A supporting the interpretation of¯A as an index

for a class of flows. It is worth emphasising that both the infrared and ultraviolet finiteness

of (2.10) are ensured by the conditions (2.12) on Rk.

B. Absence of spurious singularities

The flow equation (2.10) with a choice for the initial effective action ΓΛat the initial

scale Λ serves upon integration as a definition of the full effective action Γ = Γk=0. It

remains to be shown that (2.10) is finite for all k thus leading to a finite Γ. In particular

this concerns the spurious singularities present in perturbation theory: the propagator Pµν

related to S = SA+ Sgfis

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