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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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Pµν=δµν

p2+n2(1 + ξp2)

(np)2

pµpν

p2

−1

p2

(nµpν+ nνpµ)

np

. (2.14)

It displays the usual IR poles proportional to 1/p2. We observe additional divergences for

momenta orthogonal to nµ. These poles appear explicitly up to second order in 1/(np) and

can even be of higher order for certain (np)-dependent choices of ξ. For the planar gauge

ξp2= −1, the spurious divergences appear only up to first order.

This artifact makes the application of perturbative techniques very cumbersome as an

additional regularisation for these spurious singularities has to be introduced. We argued

in [7] that these spurious singularities are missing in the flow equation. Here, we further

the discussion, also providing some information about the intricate limit where the cut-off

is removed. First of all we derive a bound on the flow (2.10). Then, we argue that this

bound results in weak constraints on the decay behaviour of the regulator function r for

large momenta. This is sufficient for providing a well-defined RG flow.

We start with an analysis of the momentum dependence of the propagator in the presence

of the regulator. To that end we set the background field to zero,¯A = 0, and specify the

regulator as

Rab

k,µν(p) = δab?

r(p2)p2δµν− ˜ r(p2)pµpν

?

.(2.15)

The IR/UV limits of r, ˜ r can be read-off from (2.12). In (2.15) we did not introduce terms

with tensor structure (nµpν+ nνpµ) and nµnν. For the present purpose, the discussion of

spurious singularities, the choice (2.15) suffices. Indeed, even ˜ r plays no rˆ ole for the absence

of spurious singularities in the flow equation approach. The only important term for the

discussion of spurious singularities is that proportional to the term p2δµνδab. It is this term,

proportional to the identity operator, that guarantees the suppression of all momentum

modes for large cut-off. The other tensor structures are proportional to projection opera-

tors and cannot lead to a suppression of all modes. With a regulator obeying (2.15) the

propagator takes the form

Pk,µν= a1δµν

p2+ a2pµpν

p4

+ a3nµpν+ nνpµ

p2(np)

+ a4nµnν

n2p2, (2.16)

with the dimensionless coefficients

a1= 1/(1 + r),

a2= (1 + ˜ r)(1 + ξp2(1 + r))/z ,

a3= −(1 + ˜ r)s2/z ,

a4= −(r − ˜ r)/z,

(2.17a)

(2.17b)

(2.17c)

(2.17d)

and

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s2= (np)2/(n2p2),

z = (1 + r)[(1 + ˜ r)s2+ (r − ˜ r)(1 + p2ξ(1 + r))].

(2.17e)

(2.17f)

Now we evaluate the different limits in p2and k important for the approach. To keep things

simple we restrict ourselves to the case ˜ r = 0 and a regulator r leading to a mass-like IR

behaviour: limp2/k2→0r(p2) = k2/p2. For this choice we deduce from (2.16) and (2.12) that

Pk,µνhas the limits

lim

p2/k2→∞Pk,µν= Pµν, lim

p2/k2→0Pk,µν=

1

k2

?

δµν+nµnν

n2

1

1 + ξk2

?

,(2.18)

with Pµνdefined in (2.14). By construction, the propagator (2.16) is IR finite for any k > 0.

Now, the important observation is the following: in contrast to the perturbative propagator

Pµν, the limit of Pk,µνfor np → 0 is finite. This holds true even for an arbitrary choice of

ξ(p,n) and leads to

Pk,µν=

1

1 + r

δµν

p2+

1

(1 + r)r

pµpν

p4

−

1

(1 + r)(1 + p2ξ(1 + r))

nµnν

n2p2. (2.19)

Thus (2.19) is well-behaved and finite for all momenta p. The plain spurious divergences are

already absent as soon as the infra-red behaviour of the propagator is under control. This

holds for R with the most general tensor structure as long as it obeys the limits (2.12). For

example, it is easily extended to non-zero ˜ r as long as the regulators r and ˜ r have not been

chosen to be identical. Already in the infrared region ˜ r has to be smaller than r in order to

have a suppression of longitudinal modes at all. So we discard the option of identical r and

˜ r.

Still, for np = 0 and large momenta squared y = p2the regulator tends to zero and the

second term in (2.19) diverges in the limit y → ∞ proportional to y−1(r − ˜ r)−1> yd/2−1,

following from (2.12). Hence, even though the term only diverges for y → ∞, a more careful

analysis is needed for proving the finiteness of the flow equation. We emphasise that the

remaining problem is the integration over large momenta in the flow equation and not an

IR problem at vanishing momentum. Thus, by showing that this problem is absent in the

flow equation for all k it cannot reappear at k = 0. Indeed, we shall see that finiteness of

the flow for all k implies a stronger decay of the regulator for large momenta as in (2.12).

In turn, one may expect problems for regulators with weaker decay.

C. Finiteness of the RG flow

Here, finiteness of the flow equation is proven by deriving an upper bound for the flow

following a bootstrap approach. The derivation of the flow equation is based on the existence

of a finite renormalised Schwinger functional for the full theory [21]. In the present context

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this only implies the existence of a renormalisation procedure for axial gauges, the form

of which is then determined by integrating the flow. An explicit systematic constructive

renormalisation procedure is not required. The latter is a problem in perturbative field

theory: no renormalisation procedure is known, which can be proven to be valid to all

orders of perturbation theory.

In the present approach, spurious singularities could spoil the finiteness of (2.10) due to

infinities arising from the integration of the large momentum domain. For the derivation

of a bound we can safely assume, that for all k and large momenta p2the full propagator

Γ(2)

k

is dominated by its classical part (possibly with some multiplicative renormalisation

constants). Hence for large momenta we can estimate Γ(2)

C[A,¯A] > 0. Consequently the field independent part of the flow provides a bound on the

full flow. The only terms that could produce divergences are related to the terms in (2.16)

proportional to a2and a3, the source for divergences being z−1. The coefficient a4of the

last term in (2.16) also contains z−1but also an additional factor r. Hence the limit np → 0

can be safely done in the term a4.

k(S(2)+ S(2]

gf)−1< C[A,¯A] with

We do not go into the details of the computation. A more detailed derivation and

discussion is given elsewhere. We quote the result for ˜ r = 0. Upon integrating the angular

s-part of the momentum integration we get an estimate from the part of TrPk∂tRkwith

the slowest decay for y → ∞

??????

where the square root terms stem from an integration

potential problem only occurs from an integration over large momenta squared y = p2, we

have restricted the y-integral to y ≥ a where a is at our disposal. It can be chosen the

same for all k. This ensures that the limit k → 0 can be taken smoothly. The bound (2.20)

stems from the second term in (2.16) proportional to a2. Eq. (2.20) is finite for regulators

r that decay faster than y−5. Without spurious singularities, r has to decay stronger than

y−2, see (2.12). Hence we have a mild additional constraint due to the fact that the full

propagator Gk does not introduce an additional suppression. Typically, the regulator is

chosen to decay exponentially for large momenta. Similar finite integrals as in (2.20) also

occur in field dependent terms in the flow, as we shall see later in Sect. V.

bound ∝

?∞

a

dy y2

√1 + yξ

1 + aξ

r′(y)

?

r(y)

??????, (2.20)

?1

−1ds/[s2+ (1 + ξy)r(y)]. Since the

This analysis shows the finiteness of the flow (2.10) and supports the claim that the flow

equation provides a consistent quantisation procedure for gauge theories in axial gauges. The

bound also marks the use of Callan-Symanzik (CS) type flows (Rk∝ k2and r(y) ∝ y−1)

as questionable in axial gauges. It has been already mentioned in [3] that such a choice

requires an additional renormalisation. The presence of contributions from all momenta at

every flow step makes the limit k → 0 an extremely subtle one. This limit is very sensitive

to a proper fine-tuning. In axial gauges, this problem for CS flows gets even worse due

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to the spurious singularities. We know that a consistent renormalisation procedure in the

axial gauge is certainly non-trivial. For CS-type flows, one is back to the original problem

of spurious singularities in perturbation theory, but with a more difficult propagator and

additional renormalisation problems. A recent calculation of perturbative corrections to the

Wilson loop has indeed shown that formulations in axial gauge with a mass term for the

gauge field meet problems [25,26]. The massless limit of this observable did not coincide

with the well-known result. In turn, for regulators which decay faster than r(y) ∼ y−5, the

problem is cured.

III. SYMMETRIES

In this section, the issue of gauge invariance of physical Greens functions, controlled by

modified Ward-Takahashi identities, is studied. We discuss the role of background fields,

which, in contrast to the usual background field method [22,8], will only be introduced for

the Wilsonian regulator term. The Ward-Takahashi identities for the quantum and the

background field are derived. We define a gauge-invariant effective action as it follows from

the present formalism, and discuss its background field dependence. Finally, we discuss the

background field dependent regularisation.

A. Modified Ward-Takahashi Identities

We now address the issue of gauge invariance for physical Greens functions. The problem

to face is that the presence of a regulator term quadratic in the gauge fields is, a priori, in

conflict with the requirements of a (non-linear) gauge symmetry. This question has been

addressed earlier for Wilsonian flows within covariant gauges [4,5,6,7,9]. The resolution

to the problem is that modified Ward-Takahashi identities (as opposed to the usual ones)

control the flow such that physical Greens functions, obtained from Γkat k = 0, satisfy the

usual Ward-Takahashi identities.

The same line of reasoning applies in the present case even though in the presence of the

background field¯A some refinement is required [8]. In this particular point it is quite similar

to the symmetry properties of the full background field formalism as discussed in [10]. The

background field makes it necessary to deal with two kinds of modified Ward-Takahashi

Identities. The first one is related to the requirement of gauge invariance for physical Green

functions, and is known as modified Ward Identity (mWI). The second one has to do with

the presence of a background field¯A in the regulator term Rk, and will be denoted as the

background field Ward-Takahashi Identity (bWI).

To simplify the following expressions let us introduce the abbreviation δωand¯δωfor the

generator of gauge transformations on the fields A and¯A respectively:

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δωA = D(A)ω

¯δωA = 0

δω¯A = 0

¯δω¯A = D(¯A)ω.

(3.1a)

(3.1b)

The action of the gauge transformations δωand¯δωon the effective action Γkcan be com-

puted straightforwardly. It is convenient to define

Wk[A,¯A;ω] ≡ δωΓk[A,¯A] − Tr(nµ∂µω)

¯ Wk[A,¯A;ω] ≡¯δωΓk[A,¯A] −1

In terms of (3.2), the behaviour of Γk[A,¯A] under the transformations δωand¯δω, respec-

tively, is given by

1

n2ξnνAν+1

2Trω

?

Gk[A,¯A],Rk[¯A]

?

(3.2a)

2Trω

?

Gk[A,¯A],Rk[¯A]

?

. (3.2b)

Wk[A,¯A;ω] = 0

¯ Wk[A,¯A;ω] = 0

(3.3a)

(3.3b)

Eq. (3.3b) is valid for regulators Rkthat transform as tensors under δω,

¯δωRk[¯A] =

?

Rk[¯A],ω

?

.(3.4)

Eq. (3.3a) is referred to as the modified Ward-Takahashi identity, and (3.3b) as the back-

ground field Ward-Takahashi identity.

Let us show that (3.3) is consistent with the basic flow equation (2.10). With consistency,

we mean the following. Assume, that a functional Γkis given at some scale k which is a

solution to both the mWI and the bWI. We then perform a small integration step from k

to k′= k−∆k, using the flow equation, and ask whether the functional Γk′ again fulfils the

required Ward identities (3.3). That this is indeed the case is encoded in the following flow

equations for (3.3), namely

∂tWk[A,¯A;ω] = −1

∂t¯ Wk[A,¯A;ω] =1

2Tr

?

Gk∂Rk

Gk∂Rk

∂tGk

δ

δA⊗

δ

δA⊗

δ

δA

δ

δA

?

Wk[A,¯A;ω](3.5a)

2Tr

?

∂tGk

?

¯ Wk[A,¯A;ω],(3.5b)

where

mWI is satisfied for the initial scale. The required consistency follows from the fact that the

flow is proportional to the mWI itself (3.5a), which guarantees that (3.3a) is a fixed point

of (3.5a). The same follows for the bWI by using (3.5b). There is no fine-tuning involved

in lifting a solution to (3.3a) to a solution to (3.3b). It also straightforwardly follows from

(3.5a) and (3.5b).

?

δ

δA⊗

δ

δA

?ab

µν(x,y) =

δ

a(x) δAµ

δ

b(y). Eq. (3.5) states that the flow of mWI is zero if the

δAν

We close with a brief comment on the use of mass term regulators. Such a regulator

corresponds simply to Rk = k2and leads to a Callan-Symanzik flow. The regulator is

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momentum-independent, which implies that the loop term in (3.2a) vanishes identically.

Hence one concludes that the modified Ward identity reduces to the usual one for all scales

k. This happens only for an axial gauge fixing [7].

B. Gauge invariant effective action

Returning to our main line of reasoning and taking advantage of the results obtained

in the previous section, we define a gauge invariant effective action only dependent on A

by identifying¯A = A. It is obtained for a particular choice of the background field, and

provides the starting point for our formalism.

It is a straightforward consequence of the mWI (3.3a) and the bWI (3.3b) that the effec-

tive action Γk[A,¯A] is gauge invariant – up to the gauge fixing term – under the combined

transformation

(δω+¯δω)Γk[A,¯A] = Trnµ(∂µω)1

n2ξnνAν. (3.6)

We define the effective actionˆΓk[A] as

ˆΓk[A] = Γk[A,¯A = A].(3.7)

The actionˆΓk[A] is gauge invariant up to the gauge fixing term, to wit

δωˆΓk[A] = Tr

?

nµ(∂µω)1

n2ξnνAν

?

. (3.8)

This follows from (3.6). Because of (2.13b), the effective actionˆΓk=0[A] is the full effective

action. The flow equation forˆΓk[A] can be read off from the basic flow equation (2.10),

∂tˆΓk[A] =1

2Tr{Gk[A,A]∂tRk[A]}, (3.9)

Notice that the right-hand side of (3.9) is not a functional ofˆΓk[A]. The flow depends on

the full propagator Gk[A,A], which is the propagator of A in the background of¯A taken at

¯A = A. Thus for the flow ofˆΓk[A] one needs to know the flow (of a subset) of vertices of

δ2Γk[A,¯A]/(δA)2at¯A = A. Still, approximations, where this difference is neglected are of

some interest [27].

We argue that (3.8) has far reaching consequences for the renormalisation procedure of

ˆΓk[A] as is well-known for axial gauges and the background field formalism. Γk[A] is gauge

invariant up to the breaking due to the gauge fixing term. We define its gauge invariant

part as

Γk,inv[A] = Γk[A] − Sgf[A]

δωΓk,inv[A] = 0 .

(3.10a)

(3.10b)

10

Page 12

Eq. (3.10) implies that the combination gA is invariant under renormalisation, ∂t(gA) = 0.

If one considers wave function renormalisation and coupling constant renormalisation for A

and g respectively

A → Z1/2

g → Zgg

FA (3.11a)

(3.11b)

we conclude that

Zg= Z−1/2

F

.(3.12)

C. Background field dependence

By construction, the effective action Γk[A,¯A] at some finite scale k ?= 0 will depend on

the background field¯A. This dependence disappears for k = 0. The effective actionˆΓk[A] is

the simpler object to deal with as it is gauge invariant and only depends on one field. As we

have already mentioned below (3.9), its flow depends on the the propagator δ2

A =¯A. Eventually we are interested in approximations where we substitute this propagator

by δ2

background field dependence of Γk[A,¯A]. The flow of the background field dependence of

Γk[A,¯A] can be derived in two ways. δ ¯ A∂tΓkcan be derived from the flow equation (2.10),

AΓk[A,¯A] at

AˆΓk. The validity of such an approximation has to be controlled by an equation for the

δ

δ¯A∂tΓk[A,¯A] =1

2

δ

δ¯ATr

?

Gk[A,¯A]∂tRk[¯A]

?

. (3.13)

The flow ∂tδ ¯ AΓkfollows the observation that the only background field dependence of Γk

originates in the regulator. Thus, δ ¯ AΓkis derived along the same lines as the flow itself and

we get

∂t

δ

δ¯AΓk[A,¯A] =1

2Tr∂t

?

Gk[A,¯A]δ

δ¯ARk[¯A]

?

,(3.14)

which turns out to be important also for the derivation of the universal one loop β-function

in Sect. VB. The difference of (3.13) and (3.14) has to vanish

[δ

δ¯A, ∂t]Γk[A,¯A] = 0 .(3.15)

Eq. (3.15) combines the flow of the intrinsic¯A-dependence of Γk[A,¯A] (3.14) with the¯A-

dependence of the flow equation itself (3.13). It provides a check for the validity of a given

approximation. Using the right hand sides of (3.13) and (3.14) the consistency condition

(3.15) can be turned into

11

Page 13

Tr

GkδΓ(2)

k

δ¯AGk∂tRk

= Tr

?

GkδRk

δ¯AGk∂tΓ(2)

k

?

, (3.16)

where

Γ(2)

k[A,¯A]

ab

µν(x,x′) =

δ2Γk[A,¯A]

δAµ

a(x)δAν

b(x′). (3.17)

With (3.16), we control the approximation

δ2Γk[A,¯A]

δA δA

?????¯ A=A

=δ2ˆΓk[A]

δA δA+ sub-leading terms(3.18)

For this approximation the flow (3.9) is closed and can be calculated without the knowledge

of Γ(2)

proper-time flows, where the use of heat-kernel methods is even more natural [28]. This is

discussed in [29] (see also [27]). Let us finally comment on the domain of validity for the

approximation (3.18). In the infrared k → 0, the dependence of the effective action Γk[A,¯A]

on the background field¯A becomes irrelevant, because the regulator Rk[¯A] tends to zero.

Therefore we can expect that (3.18) is reliable in the infrared, which is the region of interest.

k, but withˆΓ(2)

k. Amongst others, the approximation (3.18) is implicitly made within

D. Regulators

We have seen that the symmetries of the effective action Γkand the flow crucially depend

on the properties of Rk[¯A], in particular the construction of a gauge invariant effective action.

The regulator has to transform as a tensor under gauge transformations of¯A, (3.4). Here we

specify a general class of regulators which has this property and is well-suited for practical

applications. As already argued in section IIB, the infrared regularisation is provided by r,

whereas ˜ r ?= 0 only gives different weights to the longitudinal degrees of freedom, see (2.15).

In the following we set ˜ r ≡ 0. We choose

Rk[¯A] =¯DTr(¯DT)(3.19)

with the yet unspecified function r. We introduced DT, the Laplace operator for spin 1,

Dab

T,µν(A) := −(DρDρ)ab(A)δµν− 2gFab

µν(A)(3.20)

and¯DT = DT(¯A). For vanishing background field the Laplacean DT reduces to the free

Laplacean DT(0) = p2. In this case we have Rk= p2r(p2). Written in terms of some general

Laplace operator P2(¯A), a typical example for the regulator functions Rk(P2) and r(P2) is

Rk(P2) =

P2

expP2/k2− 1,r(P2) =

1

expP2/k2− 1

(3.21)

12

Page 14

which meets the general properties as described in (2.12). Eq. (3.21) is an example for a

regulator with a mass-like IR behaviour, γ = 1. More generally the IR/UV conditions for

Rkin (2.12) translates into

lim

k2/p2→0

?p2

k2

?2

r = 0, lim

P2→0r ∼

?k2

p2

?γ

(3.22)

for the function r.

IV. ANALYTIC METHODS

In this section we develop analytical methods to study flow equations for gauge theories

in general axial gauges. The flow equation is a one-loop equation which makes it possible

to use heat kernel techniques for its solution. The main obstacles, technically speaking,

are the constraint imposed by the modified Ward identity and the necessity to come up

with a closed form for the full propagator. We first derive such an expression for the case

of covariantly constant fields within general axial gauges. In addition a generic expansion

procedure in powers of the fields is discussed. Finally, we give the basic heat kernels to be

employed in the next section.

A. Propagator for covariantly constant fields

We derive an explicit expression for the full propagator for specific field configura-

tions. This is a prerequisite for the evaluation of the flow equation (2.10). To that end

we restrict ourselves to field configurations with covariantly constant field strength (see

e.g. [30]), namely DµFνρ = 0.This is a common procedure within the algebraic heat

kernel approach. We also use the existence of the additional Lorentz vector to demand

nµAµ= nµFµν= 0. That this can be achieved is proven by the explicit example of nµ= δµ0

and (Aµ) = (A0= 0,Ai(? x)). These constraints lead to

[Dµ,Fνρ] = 0, (4.1a)

nµAµ= 0 (4.1b)

nµFµν= 0. (4.1c)

To keep finiteness of the action of such configurations we have to go to a theory on a finite

volume. However, the volume dependence will drop out in the final expressions and we

smoothly can take the limit of infinite volume. For the configurations satisfying (4.1) we

derive the following properties

[D2,Dµ] = −2gFµρDρ,

DT,µρDρ= −DµD2,

[nρDρ,Dµ] = 0.

(4.2a)

(4.2b)

(4.2c)

13

Page 15

Defining the projectors Pnand PDwith

Pn,µν=nµnν

n2, (4.3a)

PD,µν= Dµ

1

D2Dν

(4.3b)

we establish that

PDDT= −PDD2PD,PnDT= −PnD2

(4.4)

holds true. After these preliminary considerations we consider the gauge-fixed classical

action given in (2.1).We need the propagator on tree level to obtain the traces on one-loop

level. The initial action reads

ΓΛ[A] = SA+ Sgf. (4.5)

From (4.5) we derive the full inverse propagator as

Γ(2)ab

k,µν[A,A] =

?

Dab

T,µν+ (DµDν)ab+

1

ξn2nµnνδab

?

+ O(g2;DT,DµDν) .(4.6)

The inverse propagator (4.6) is an operator in the adjoint representation of the gauge group.

We now turn to the computation of the propagator (2.11) for covariantly constant fields.

Using (4.6), (4.1) and (4.2), we find

Gk[A,A]ab

µν= −

??a1

DT

?

µν

+ Dµa2

D4Dν+ nµ

a3

D2(nD)Dν+ Dµ

a3

D2(nD)nν+nµa4nν

n2D2

?

, (4.7)

with the dimensionless coefficient functions

a1=

1

1 + rT

,(4.8a)

a2=1 − ξD2(1 + rD)

(1 + rD)

s2

(1 + rD)

rD

(1 + rD)

?

s2+ rD[1 − D2ξ(1 + rD)]

?−1

,(4.8b)

a3= −

?

?

s2+ rD[1 − D2ξ(1 + rD)]

?−1

?−1

, (4.8c)

a4= −

s2+ rD[1 − D2ξ(1 + rD)]

. (4.8d)

Notice that a1is a function of DT while a2, a3and a4are functions of both D2and (nD)2.

We also introduced the convenient short-hands

rT≡ rk(DT),rD≡ rk(−D2),s2≡(nD)2

(n2D2).(4.9)

The regulator function, as introduced in (3.19), depends on DT. The dependence on D2, as

apparent in the terms a2, a3and a4, comes into game due to the conditions (4.1) and (4.2).

They imply

14

Page 16

rk(DT)Dµ= Dµrk(−D2),rk(DT)nµ= nµrk(−D2) ,(4.10)

which can be shown term by term for a Taylor expansion of rkabout vanishing argument.

For vanishing field A = 0 the propagator (4.7) reduces to the one already discussed in [7].

There, it has been shown that the regularised propagator (4.7) (for r ?= 0) is not plagued by

the spurious propagator singularities as encountered within standard perturbation theory,

and in the absence of a regulator term (r = 0). For the axial gauge limit ξ = 0 the expression

(4.7) simplifies considerably. With (4.6) and (4.9) we get

Gk,µν[A] =

?

1

DT(1 + rT)

?

µν

nD

− Dµ

1

D4(1 + rD)(s2+ rD)Dν+nµ

nν

n2+

n2

nD

D4(1 + rD)(s2+ rD)Dν

+Dµ

D4(1 + rD)(s2+ rD)

rD

D2(1 + rD)(s2+ rD)Pn,µν. (4.11)

The propagators (4.7) and (4.11) are at the basis for the following computations. Notice that

this analysis straightforwardly extends to approximations for Γk[A,¯A] beyond the one-loop

level. Indeed, it applies for any Γk[A,¯A] such that Γ(2)

k,µν[A,A] is of the form

Γ(2)

k,µν[A,A] = fDT

k

DTµν+ DµfDD

k

Dν+ nµfnD

k

nDDν+ DµfnD

k

nDnν+ nµfnn

k nν. (4.12)

Here, the scale-dependent functions fDT

turn, the functions fnD

expression for the full propagator, similar to (4.7), follows from (4.12). Such approximations

take the full (covariant) momentum dependence of the propagator into account. The inverse

propagator (4.6) corresponds to the particular case fDT= fDD= 1, fnD= 0, and fnn=

1/ξ.

k

and fDD

k

can depend on DT, D2and nD. In

can depend only on D2and nD. An explicit analytical

k

and fnn

k

B. Expansion in the fields

Even for analytic calculations one wishes to include more than covariantly constant

gauge fields, and to expand in powers of the fields, or to make a derivative expansion.

Eventually one has to employ numerical methods where it is inevitable to make some sort

of approximation. Therefore it is of importance to have a formulation of the flow equation

which allows for simple and systematic expansions.

In this section we are arguing in favour for a different splitting of the propagator which

makes it simple to employ any sort of approximation one may think of. For this purpose

we employ the regulator Rk[D2(¯A)]. This is an appropriate choice since it has no negative

eigenvalues. We split the inverse propagator into

Γ(2)ab

k,µν[A] = ∆ab

µν−

?

2gFab

µν− (DµDν)ab?

(4.13)

15

Page 17

with

∆ab

µν=

?

−D2(1 + rD)

?abδµν+

1

ξn2nµnνδab. (4.14)

The operator ∆ can be explicitly inverted for any field configuration (and A =¯A). We have

∆−1= −

1

D2(1 + rD)1l +

1

D2(1 + rD)

1

1 + ξD2(1 + rD)Pn. (4.15)

With (4.13) and (4.15) we can expand the propagator as

Gk[A,A] = ∆−1

∞

?

n=0

?

(2gF − D ⊗ D)∆−1?n. (4.16)

where (D ⊗ D)ab

written as

µν(x,y) = Dac

µDcb

νδ(x − y). For ξ = 0 (the axial gauge), ∆−1can be neatly

∆−1(ξ = 0) = −

1

D2(1 + rD)(1l − Pn),(4.17)

which simplifies the expansion (4.16). The most important points in (4.16) concern the

fact that it is valid for arbitrary gauge field configurations and each term is convergent for

arbitrary gauge fixing parameter ξ. Moreover such an expansion is not possible in the case

of covariant gauges. Both facts mentioned above are spoiled in this case.

C. Heat kernels

We present closed formulae for the heat-kernel of the closely related operators DT and

−D2= DT+ 2gF. These are needed in order to evaluate the traces in (5.15). We define

the heat-kernels as KO(τ) = exp{τO}(x,x)

d4p

(2π)4eτXµXµ,

K−DT(τ) = e2τFKD2(τ),

KD2(τ) =

?

(4.18a)

(4.18b)

where Xµ= ipµ+Dµin the corresponding representation. Here we used that 2gF commutes

with Xµfor covariantly constant fields. All kernels are tensors in the Lie algebra ( K−DTis

also a Lorentz tensor because of the prefactor). For the calculation of the momentum integral

we just refer the reader to the literature (e.g. [30]) and quote the result for covariantly

constant field strength

KD2(τ) =

1

16π2τ2det

?

τgF

sinhτgF

?1/2

, (4.19a)

K−DT(τ) = exp(2τgF) KD2(τ) . (4.19b)

16

Page 18

Here, the determinant is performed only with respect to the Lorentz indices. For the com-

putation of the one-loop beta function we need to know K(τ) in (4.19) up to order F2

(equivalently to order τ0). Expanding KD2 in τgF we get

KD2(τ) =

1

16π2

?1

τ2−1

12g2(F2)ρρ

?

+ O[τ,(gF)3]. (4.20)

With (4.20) and the expansion (exp2τgF)µν= 1 + 2τgFµν+ 2τ2g2(F2)µν+ O[τ,(gF)3] we

read off the coefficient of the K(τ) proportional to F2,

TrKD2|F2 = −

1

16π2

1

16π2

4

3Ng2SA[A] ,

20

3Ng2SA[A] ,

(4.21a)

TrK−DT|F2 =

(4.21b)

where the trace Tr denotes a sum over momenta and indices. We have also used that

SA[A] =1

2

representation the trace Tr includes tradwith 2Ntrftatb= tradtatb.

?trfF2with trftatb= −1

2δab. Since the operators DT and D2carry the adjoint

V. APPLICATIONS

In order to put the methods to work we consider in this section the full one-loop effective

action for SU(N) Yang-Mills theory which entails the universal one-loop beta function for

arbitrary regulator function.

A. Effective action

For the right hand side of the flow we need

Γk[A,¯A] =1

2

?

ZF(t)trfF2(A) + Sgf[A] + O[(gA)5,g2∂A], trftatb= −1

2δab

(5.1)

where trRdenotes the trace in the representation R, R = f stands for the fundamental rep-

resentation, R = ad for the adjoint representation. Only the classical action can contribute

to the flow, as n-loop terms in (5.1) lead to n + 1-loop terms in the flow, when inserted on

the right hand side of (3.9). This Ansatz leads to the propagator (4.11) which together with

our choice for the regulator (3.19) is the input in the flow equation (3.9). We also use the

following in the evaluation of the different terms in (3.9):

trD2= 4trD ⊗ D(5.2)

With this we finally arrive at

∂tˆΓk=1

2Tr

?

∂tr(DT)

1 + r(DT)−1

2

∂tr(−D2)

1 + r(−D2)+14

∂tr(−D2)

s2+ r(−D2)

?

,(5.3)

17

Page 19

where the trace Tr contains also the Lorentz trace and the adjoint trace trad in the Lie

algebra. The first term on the right-hand side in (5.3) has a non-trivial Lorentz structure,

while the two last terms are proportional to δµν. We notice that the flow equation (5.3)

is well-defined in both the IR and the UV region. We apply the heat-kernel results of

section IVC to the calculation of (5.3). To that end we take advantage of the following

fact: Given the existence (convergence, no poles) of the Taylor expansion of a function f(x)

about x = 0 we can use the representation

f(−O) = f(−∂τ)exp{τO}|τ=0

(5.4)

Due to the infrared regulator the terms in the flow equation (5.3) have this property, where

O = DT,D2. Hence we can rewrite the arguments DT and −D2in (5.3) as derivatives

w.r.t. τ of the corresponding heat kernels K−DT(τ) and KD2(τ). Applying this to the flow

equation (5.3) we arrive at

∂tˆΓk=1

2

?

∂tr(−∂τ)

1 + r(−∂τ)TrK−DT(τ) −1

+1

4p2

2

∂tr(−∂τ)

1 + r(−∂τ)TrKD2(τ)

n− ∂τ)

n− ∂τ)r(p2

?

dpn

(p2

n+ (p2

n− ∂τ)∂tr(p2

n− ∂τ)

τ1/2

√πTrKD2(τ)

?

τ=0

(5.5)

The two terms in the first line follow from (5.3). The last term is more involved because

it depends on both D2and nD due to s2≡ (nD)2/n2D2. We note that nD = (n∂) holds

for configurations satisfying (4.1a) and only depends on the momentum parallel to nµ.

Furthermore it is independent of the gauge field. Now we use the splitting of (pµ) = (pn,? p)

where pn= Pnp and ? p = (1 − Pn)p. The heat kernel related to?D2follows from the one for

D2via the relation K?D2(τ) =τ1/2

the pn-direction.

√πKD2(τ) as can be verified by a simple Gaußian integral in

With these prerequisites at hand, we turn to the full effective action at the scale k, which

is given by

ˆΓk=ˆΓΛ+

?k

Λdk′∂ˆΓk′

∂k′,(5.6)

where Λ is some large initial UV scale. We start with the classical action ΓΛ= SA+ Sgf.

Performing the k-integral in (5.6) we finally arrive at

ˆΓk[A] =

?

+Sgf[A] +

1 +Ng2

16π2

?

22

3− 7(1 − γ)

∞

?

?

lnk/Λ

?

SA[A]

m=1

Cm(k2/Λ2) ∆Γ(m)[gF/k2] + const.(5.7)

The combination SA+ Sgf on the right-hand side of (5.7) is the initial effective action.

All further terms stem from the expansion of the heat kernels (4.19) in powers of τ. The

terms ∼ τ−2give field-independent contributions, while those ∼ τ−1are proportional to

18

Page 20

trF and vanish. The third term on the right-hand side of (5.7) stems from the τ0coef-

ficient of the heat kernel. This term also depends on the regulator function through the

coefficient γ (3.22). All higher order terms ∼ τm,m > 0 are proportional to the terms

Cm(k2/Λ2)∆Γ(m)[gF/k2]. These terms have the following structure: They consists of a

prefactor

Cm(x) = −1

4m

(−)m

m!

(1 − xm) (5.8a)

and scheme-dependent functions of the field strength, ∆Γ(m)[gF], each of which is of the

order 2 + m in the field strength gF. They are given explicitly as

∆Γ(m)[gF] = BDT

m TrK(m)

−DT(0) +

?

BD2

m + BnD

m

?

TrK(m)

D2 (0) . (5.8b)

Here, K(m)

the following identity

D2 (0) and K(m)

−DT(0) denote the expansion coefficients of the heat kernels. We use

f(m)(0) = f(∂τ)τm|τ=0, (5.9)

and f(m)(x) = (∂x)mf(x). In addition, the terms in (5.8b) contain the scheme-dependent

coefficients

BDT

m =

?

˙ r1

1 + r1

?(m)

(0) , (5.10a)

BD2

m= −1

m =(−1)m+1

2BDT

m , (5.10b)

BnD

4

?∞

0

dx

?

∂x−1

xα∂α

?m+1

˙ r1(x)

??

r1(x)r1(x) + α

??????

α=1

.(5.10c)

The coefficients BDT, BD2and BnDfollow from the first, second and third term in (5.3).

We introduced dimensionless variables by defining r1(x) = r(xk2) and ˙ r1(x) ≡ ∂tr1(x) =

−2xk2r′(xk2) = −2xr′

k-dependence into (5.8a). The explicit derivation of BnDis tedious but straightforward and

is given – together with some identities useful for the evaluation of the integral and the

derivatives – in appendix A. All coefficients BDT, BD2and BnDare finite. The appearance

of roots in the coefficient BnDis not surprising after the discussion of the absence of spurious

singularities in section IIB.

1(x), in order to simplify the expressions and to explicitly extract the

In particular, we can read off the coefficients for m = 0 which add up to the prefactor

of the classical action in (5.7):

BDT

0

= 2γ,BD2

0

= −γ,BnD

0

= −1

2(1 − γ),(5.11)

where we have used (A.5) in the appendix. Together with the heat kernel terms proportional

to τ0given in (4.21) this leads to (5.7).

19

Page 21

This application can be extended to include non-perturbative truncations. The flow of

the coefficients (5.8b) becomes non-trivial, and regulator-dependent due to the regulator-

dependence of the coefficients (5.10). Then, optimisation conditions for the flow can be

employed to improve the truncation at hand [31].

Finally, we discuss the result (5.7) in the light of the derivative expansion. Typically,

the operators generated along the flow have the structure F fk[(D2+k2)/Λ2]F, and similar

to higher order in the field strength. For dimensional reasons, the coefficient function fk(x)

of the operator quadratic in F develops a logarithm ∼ lnx in the infrared region. An

additional expansion of this term in powers of momenta leads to the spurious logarithmic

infrared singularity as seen in (5.7). To higher order in the field strength, the coefficient

function behave as powers of 1/(D2+ k2), which also, at vanishing momenta, develop a

spurious singularity in the IR, and for the very same reason. All these problems are absent

for any finite external gluon momenta, and are an artifact of the derivative expansion.

A second comment concerns the close similarity of (5.7) with one-loop expressions found

within the heat-kernel regularisation. In the latter cases, results are given as functions of the

proper-time parameter τ and a remaining integration over dlnτ. Expanding the integrand

in powers of the field strength and performing the final integration leads to a structure as

in (5.7), after identifying τ ∼ k−2. In particular, these results have the same IR structure

as found in the present analysis.

B. Running coupling

We now turn to the computation of the beta function at one loop. We prove that

the result is independent of the choice of the regulator and agrees with the standard one.

However, it turns out that the actual computation depends strongly on the precise small-

momentum behaviour of the regulator, which makes a detailed discussion necessary.

Naively we would read-off the β-function from the t-running of the term proportional to

the classical action SAin (5.7). Using (3.12) leads to ∂tlnZg= −1

(5.7)

2∂tlnZF. We get from

ZF=

?

22

3− 7(1 − γ)

?Ng2

16π2t→∂tlnZg= −

?

11

3−7

2(1 − γ)

?Ng2

16π2+ O(g4). (5.12)

We would like to identify β = ∂tlnZg. This relation, however, is based on the assumption

that at one loop one can trade the IR scaling encoded in the t-dependence of this term

directly to a renormalisation group scaling. This assumption is based on the observation that

the coefficient of SA[A] is dimensionless and at one loop there is no implicit scale dependence.

It is the latter assumption which in general is not valid. A more detailed analysis of this fact

is given in [21]. Here, we observe that the background field dependence of the cut-off term

inflicts contributions to ∂tZFScl. These terms would be regulator-dependent constants for

20

Page 22

a standard regulator without¯A. As mentioned below (2.5), one should see the background

field as an index for a family of different regulators. We write the effective action as

Γk[A,¯A] = Γk,1[A] + Γk,2[¯A] + Γk,3[A,¯A] . (5.13)

The second term only depends on¯A and is solely related to the¯A-dependence of the regula-

tor. The last term accounts for gauge invariance of Γkunder the combined transformation

δω+¯δω. This term vanishes in the present approximation, because of the observation that

our Ansatz is invariant – up to the gauge fixing term – under both δωand¯δωseparately.

The physical running of the coupling is contained in the flow of Γk,1[A]. This leads to

β = −1

2∂tZF+1

2∂tZF,2, (5.14)

where ZF,2is the scale dependence of Γk,2∝ ZF,2SA[A]. We rush to add that this procedure

is only necessary because we are interested in extracting the universal one-loop β-function

from the flow equation. For integrating the flow itself this is not necessary since for k = 0

the background field dependence disappears anyway. For calculating ∂tlnZF,2we use (4.11)

and (5.2) and get

∂t

δ

δ¯Aa

µ

Γk[A,¯A = A] =1

2Tr∂t

?

R′

k[DT]

DT+ Rk[DT]

R′

(−nD)2+ Rk[−D2]

δDT

δ¯Aa

µ

δD2

δ¯Aa

+1

2

R′

k(−D2)

−D2+ Rk[−D2]

δD2

δ¯Aa

µ

−1

4

k[−D2]

µ

?

,(5.15)

where we have introduced the abbreviation

R′

k(x) = ∂xRk(x). (5.16)

For the derivation of (5.15) one uses the cyclycity of the trace and the relations (4.2). We

notice that (5.15) is well-defined in both the IR and the UV region. The explicit calculation

is done in appendix B. Collecting the results (B.2),(B.3),(B.4) we get

∂tδ ¯ AΓk[A,¯A = A]|F2 = −Ng2

16π27(1 − γ)δASA[A] → ∂tZF,2= −Ng2

16π27(1 − γ)(5.17)

We insert the results (5.12) for ∂tZF and (5.17) for ∂tZF,2in (5.14) and conclude

β = −11

3

Ng2

16π2+ O(g4).(5.18)

which is the well-known one-loop result. For regulators with a mass-like infrared limit, γ = 1,

there is no implicit scale dependence at one loop. It is also worth emphasising an important

difference to Lorentz-type gauges within the background field approach. In the present case

only the physical degrees of freedom scale implicitly with t = lnk for γ ?= 0. This can be

deduced from the prefactor 7(1 − γ) in (5.17). Within the Lorentz-type background gauge,

21

Page 23

this coefficient is

gauge one has no auxiliary fields but only the physical degrees of freedom. In a general

gauge, this picture only holds true after integrating-out the ghosts. This integration leads

to non-local terms. They are mirrored here in the non-local third term on the right hand

side of the flow (5.5) and in the third term on the right hand side of (5.15) [see also (B.4)].

22

3(1 − γ) [21]. The difference has to do with the fact that in the axial

VI. CONCLUSIONS

We have shown how the exact renormalisation group can be used for gauge theories

in general axial gauges. We have addressed various conceptual points, in particular the

absence of spurious singularities and gauge invariance, which are at the basis for a reliable

application of this approach. We have shown that spurious singularities are absent provided

that the regulator Rkdecays stronger than (p2)−4for large momenta. In turn, regulators

with milder decay are highly questionable. At least they are subject to a renormalisation

of the flow itself, which implicitly brings back the problem of spurious singularities. This

concerns in particular the mass regulator Rk= k2, see also [3].

Our main goal was to develop methods which allow controlled and systematic analytical

considerations. The formalism has the advantage that ghost fields are not required. Also,

no additional regularisation – in spite of the axial gauge fixing – is needed. This is a

positive side effect of the Wilsonian regulator term. In addition, we worked in a background

field formulation, which is helpful in order to construct a gauge invariant effective action.

Also, it allows to expand the flow equation around relevant field configurations. Instead of

relying on the standard background field gauge, we have introduced the background field

only in the regulator term. The axial gauge fixing is independent on the background field.

This way, it is guaranteed that the background field dependence vanishes in the IR limit.

It is important to discuss how this differs from the usual background field approach to

Wilsonian flows. In both cases, applications of the flow require an approximation, where

derivatives w.r.t. the background field are neglected, cf. (3.18). In the present approach, this

approximation improves in the infrared, finally becoming exact for k = 0 as the background

field dependence disappears. For the background field gauge this does not happen, because

the full effective action still depends non-trivially on the background field.

As an application, the full one-loop effective action and the universal beta-function have

been computed. This enabled us to address some of the more subtle issues of the formalism

like the implicit scale dependence introduced by the cutoff, which has properly to be taken

into account for the computation of universal quantities, and the scheme independence of

the beta-function. The equation which controls the additional background field dependence

introduced by the cutoff contains the related information.

These results are an important step towards more sophisticated applications, both nu-

merically and analytically. A natural extension concerns dynamical fermions. The present

22

Page 24

formalism is also well-adapted for QCD at finite temperature T, where the heat-bath singles-

out a particular Lorentz vector. Here, an interesting application concerns the thermal pres-

sure of QCD.

ACKNOWLEDGEMENTS

We thank P. Watts for helpful discussions.

financial support. DFL has been supported by the European Community through the

Marie-Curie fellowship HPMF-CT-1999-00404.

JMP thanks CERN for hospitality and

A. EVALUATION OF THE ONE LOOP EFFECTIVE ACTION

The calculation of the last term in (5.7) is a bit more involved. Note that the following

argument is valid for m ≥ −1, m > −1 is of importance for the evaluation of (5.7), m = −1

will be used in Appendix B. We first convert the factor τm+1/2appearing in the expansion

of the heat kernel using τ1/2+m= (−1)m+1 τ

√π

?dz∂m+1

n− ∂τ)

z2

e−τz2. We further conclude that

BnD

m =

1

4π

?

dpndz

(p2

n+ (p2

n− ∂τ)∂tr(p2

n− ∂τ)r(p2

dpndz ∂m+1

z2

p2

n− ∂τ)τm+1e−τz2|τ=0

∂tr(p2

n+ (p2

∂tr(z2+ p2

p2

n

z2+p2

=(−1)m+1

4π

?

?

n− ∂τ)

p2

n− ∂τ)r(p2

n)

n− ∂τ)(p2

n− ∂τ)e−τz2

?????τ=0

=(−1)m+1

4π

dpndz∂m+1

z2

n+ r(z2+ p2

n),(A.1)

The expression in (A.1) can be conveniently rewritten as

BnD

m =(−1)m+1

8π

?∞

0

dx

?2π

∂x−1

0

dφ

?

∂x−1

xα∂α

?m+1

∂tr(x)

αsin2φ + r(x)

?????α=1

=(−1)m+1

4

?∞

0

dx

?

xα∂α

?m+1

∂tr(x)

??

r(x)r(x) + α

??????

α=1

.(A.2)

where x = z2+ p2

representation of ∂z2 on sin2φ = p2

x only. The expression in (A.2) is finite for all m ≥ 0. Evidently it falls of for x → ∞. For

the behaviour at x = 0 the following identity is helpful:

nand sin2φ = p2

n/(z2+ p2

n/(z2+p2

n). It is simple to see that −(1/x)α∂α is a

n) and ∂xa representation of ∂z2 on functions of

?

∂x−1

xα∂α

?m+1

=

m+1

?

i=0

(−1)m+1−i

?m + 1

i

?

∂i

x

?α

x

?m+1−i

∂m+1−i

α

,(A.3)

Eq. (A.3) guarantees that the integrand in (A.2) only contains terms of the form

23

Page 25

∂i

x

?

˙ r

√r√1 + r(x + xr)i−m−1

?

(A.4)

with i = 0,...,m + 1. For x → 0 one has to use that ∂tr → 2nr and r →k2n

integrand in (A.2) as displayed in (A.4) are finite for x = 0.

We are particularly interested in BnD

effective action (5.7). With (A.2) it follows

xn. The terms of

0

relevant for the coefficient of SAin the one loop

BnD

0

= −1

4

?∞

0

dx

?

∂x−1

xα∂α

?

∂tr(x)

?

r(x)

?

r(x)r(x) + α

??????

α=1

= −1

4

∂tr(x)

??

r(x)1 + r(x)

− 2

?

1 + r(x)

?

x=∞

x=0

= −1

2(1 − γ),(A.5)

where we have used ∂tr(z) = −2z∂zr(z) and the limits for ∂tr(z → 0) = 2γz−γ,r(z → 0) =

z−γ,r(z → ∞) = 0.

B.¯A-DERIVATIVES

For the calculation of (5.15) the following identity is useful:

Tr

?

δ

δAa

µ

O

?

eτO=1

τTr

δ

δAa

µ

eτO, (B.1)

where we need (B.1) for O = D2and O = −DT. Now we proceed in calculating the first

term in (5.15) by using a similar line of arguments as in the calculation of (5.7) and in

Appendix A. We make use of the representation of τ−1=

?∞

0dz exp−τz and arrive at

1

τδAa

µ

?Ng2

16π2

3 δAa

µ

1

2Tr∂t

?

R′

k[DT]

DT+ Rk[DT]

δDT

δAa

µ

?

=1

2Tr∂t

?

R′

k(−∂τ)

−∂τ+ Rk[−∂τ]

dx

x∂t

1 + r[x]

δ

K−DT(τ)

?

τ=0

=1

2

?∞

0

?

R′

k[x]20

δ

(SA[A] + O[g])

= −Ng2

16π2

20

3(1 − γ)

δ

δAa

µ

(SA[A] + O[g]).(B.2)

Note that ∂tacts as −2x∂xon functions which solely depend on x/k2. The term R′/(1+r)

is such a function. The second term can be calculated in the same way leading to

1

4Tr ∂t

?

−R′

k[D2]

−D2+ Rk[−D2]

δ

δAa

µ

D2

?

=1

4

?∞

0

dx

x

∂t

?

R′

k[x]

1 + r[x]

?Ng2

16π2

4

3

δ

δAa

µ

(SA[A] + O[g])

= −Ng2

16π2

2

3(1 − γ)

δ

δAa

µ

(SA[A] + O[g]).(B.3)

24

Page 26

The calculation of the last term in (5.15) is a bit more involved, but boils down to the same

structure as for the other terms. Along the lines of Appendix A it follows that this term

can be written as

1

8Tr∂t

?

−R′

k[−D2]

(−nD)2+ Rk[−D2]

δD2

δAa

µ

?

=1

8Tr ∂t

??

dx

x∂t

dpn

R′

k[p2

n− ∂τ]

n− ∂τ]

Ng2

16π2

p2

n+ Rk[p2

R′

k

√r√1 + r

δ

δAa

µ

τ−1/2

√π

δ

δAa

µ

KD2(τ)

?

τ=0

= −1

=Ng2

16π2

8

?∞

0

4

3

δ

δAa

µ

(SA[A] + O[g]),

1

3(1 − γ)

(SA[A] + O[g])(B.4)

Note that when rewriting the left hand side of (B.4) as a total derivative w.r.t. A this also

includes a term which stems from

δ

δA(nD)2. This, however, vanishes because it is odd in pn.

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