arXiv:hep-th/0203005v2 25 Sep 2002
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.
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.
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
). 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  based on gauge invariant variables,
e.g. Wilson loops. Applications of these methods to gauge theories include the physics of su-
perconductors , the computation of instanton-induced effects , the heavy quark effec-
tive potential [14,15], effective gluon condensation , Chern-Simons theory , monopole
condensation , chiral gauge theories , supersymmetric Yang-Mills theories , and
the derivation of the universal two-loop beta function .
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 , 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 . 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
, for a formulation of Callan-Symanzik flows in axial gauges , 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
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
with the field strength tensor
µν(A) = ∂µAa
and the covariant derivative
µ(A) = δab∂µ+ gfacbAc
µ,[tb,tc] = fabcta.(2.3)
A general axial gauge fixing is given by
The gauge fixing parameter ξ has the mass dimension −2 and may as well be operator-
valued . 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 .
The scale-dependent regulator term is
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
The scale dependent Schwinger functional Wk[J,¯A], given by
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
µ− Wk[J,¯A] − ∆Sk[A,¯A],Aa
For later convenience, we have subtracted ∆Skfrom the Legendre transform of Wk. Thus
Γk[A,¯A] is given by the integro-differential equation
−SA[a] − Sgf[a] − ∆Sk[a − A,¯A] +
δAΓk[A,¯A](a − A)
The corresponding flow equation for the effective action
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
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.
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:
k→∞Γk[A,¯A] ≡ S[A] + Sgf[A],
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
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.
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
e−τz2. We further conclude that
n+ r(z2+ p2
The expression in (A.1) can be conveniently rewritten as
αsin2φ + r(x)
r(x)r(x) + α
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). It is simple to see that −(1/x)α∂α is a
n) and ∂xa representation of ∂z2 on functions of
?m + 1
Eq. (A.3) guarantees that the integrand in (A.2) only contains terms of the form
√r√1 + r(x + xr)i−m−1
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
relevant for the coefficient of SAin the one loop
r(x)r(x) + α
r(x)1 + r(x)
1 + r(x)
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.
For the calculation of (5.15) the following identity is useful:
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 + r[x]
(SA[A] + O[g])
3(1 − γ)
(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 + r[x]
(SA[A] + O[g])
3(1 − γ)
(SA[A] + O[g]).(B.3)
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
√r√1 + r
(SA[A] + O[g]),
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
δA(nD)2. This, however, vanishes because it is odd in pn.
 K. G. Wilson, I. G. Kogut, Phys. Rep. 12 (1974) 75;
F. Wegner, A. Houghton, Phys. Rev. A 8 (1973) 401.
 J. Polchinski, Nucl. Phys. B 231 (1984) 269;
C.Wetterich, Phys. Lett. B301 (1993) 90;
U. Ellwanger, Z. Phys. C 62 (1994) 503;
T. R. Morris, Int. J. Mod. Phys. A9 (1994) 2411.
 D. F. Litim and J. M. Pawlowski, hep-th/9901063.
 M. Reuter and C. Wetterich, Nucl. Phys. B417 (1994) 181; Nucl. Phys. B 427 (1994) 291.
 U. Ellwanger, Phys. Lett. B335 (1994) 364 [hep-th/9402077].
 M. Bonini, M. D’Attanasio and G. Marchesini, Nucl. Phys. B 421 (1994) 429 [hep-th/9312114].
 D. F. Litim and J. M. Pawlowski, Phys. Lett. B435 (1998) 181 [hep-th/9802064]; Nucl. Phys.
Proc. Suppl. 74 (1999) 329 [hep-th/9809023].
 D. F. Litim and J. M. Pawlowski, Nucl. Phys. Proc. Suppl. 74 (1999) 325 [hep-th/9809020].
 M. D’Attanasio and T. R. Morris, Phys. Lett. B 378 (1996) 213 [hep-th/9602156].
 F. Freire, D. F. Litim and J. M. Pawlowski, Phys. Lett. B 495 (2000) 256 [hep-th/0009110];
Int. J. Mod. Phys. A 16 (2001) 2035 [hep-th/0101108].
 T. R. Morris, Nucl. Phys. B 573 (2000) 97 [hep-th/9910058]; JHEP 0012 (2000) 012 [hep-
 B. Bergerhoff, F. Freire, D. F. Litim, S. Lola and C. Wetterich, Phys. Rev. B53 (1996) 5734;
B. Bergerhoff, D. F. Litim, S. Lola and C. Wetterich, Int. J. Mod. Phys. A11 (1996) 4273;
D. F. Litim, C. Wetterich and N. Tetradis, Mod. Phys. Lett. A 12 (1997) 2287.
 J. M. Pawlowski, Phys. Rev. D 58 (1998) 045011 [hep-th/9605037]; Nucl. Phys. Proc. Suppl.
49 (1996) 151;
E. Meggiolaro and C. Wetterich, Nucl. Phys. B 606 (2001) 337 [hep-ph/0012081].
 U. Ellwanger, M. Hirsch and A. Weber, Z. Phys. C69 (1996) 687 [hep-th/9506019]; Eur. Phys.
J. C1 (1998) 563 [hep-ph/9606468].
 B. Bergerhoff and C. Wetterich, Phys. Rev. D 57 (1998) 1591 [hep-ph/9708425].
 M. Reuter and C. Wetterich, Phys. Lett. B334 (1994) 412 [hep-ph/9405300]; Phys. Rev. D56
(1997) 7893 [hep-th/9708051].
 M. Reuter, Phys. Rev. D 53 (1996) 4430 [hep-th/9511128].
 U. Ellwanger, Nucl. Phys. B 531 (1998) 593 [hep-ph/9710326]; Nucl. Phys. B 560 (1999) 587
F. Freire, Phys. Lett. B 526 (2002) 405 [hep-th/0110241].
 M. Bonini and F. Vian, Nucl. Phys. B 511 (1998) 479 [hep-th/9707094].
 S. Falkenberg and B. Geyer, Phys. Rev. D 58 (1998) 085004 [hep-th/9802113].
 J. M. Pawlowski, Int. J. Mod. Phys. A 16 (2001) 2105; in preparation.
 L. F. Abbott, Nucl. Phys. B185 (1981) 189.
 K. Geiger, Phys. Rev. D60 (1999) 034012 [hep-ph/9902289].
 M. Simionato, Int. J. Mod. Phys. A15 (2000) 2153 [hep-th/9810117]; hep-th/0005083.
 A. Panza and R. Soldati, Phys. Lett. B 493 (2000) 197 [hep-th/0006170].
 A. Panza and R. Soldati, Int. J. Mod. Phys. A 16 (2001) 2101.
 D. F. Litim and J. M. Pawlowski, hep-th/0208216 (PLB, in press).
 S. B. Liao, Phys. Rev. D 56 (1997) 5008 [hep-th/9511046].
 D. F. Litim and J. M. Pawlowski, Phys. Rev. D 66 (2002) 025030 [hep-th/0202188]; Phys.
Rev. D 65 (2002) 081701 [hep-th/0111191]; Phys. Lett. B 516 (2001) 197 [hep-th/0107020].
 I. N. McArthur and T. D. Gargett, Nucl. Phys. B 497 (1997) 525 [hep-th/9705200].
 D. F. Litim, Phys. Lett. B486 (2000) 92 [hep-th/0005245]; Phys. Rev. D 64 (2001) 105007