Flow equations for Yang-Mills theories in general axial gauges

We present a formulation of non-Abelian gauge theories in general axial gauges using a Wilsonian (or ’Exact’) Renormalisation Group. No ’spurious’ propagator divergencies are encountered in contrast to standard perturbation theory. Modified Ward identities, compatible with the flow equation, ensure gauge invariance of physical Green functions. The axial gauge nA = 0 is shown to be a fixed point under the flow equation. Possible non-perturbative approximation schemes and further applications are outlined. E-Mail: Litim@tholos.ecm.ub.es E-Mail: jmp@stp.dias.ie 1. General axial gauges [1]-[7] have enjoyed considerable attention amongst the noncovariant gauges, especially for computations in QCD at vanishing [2],[3] or non-vanishing temperature [2],[4]. The main reason for their popularity stems from the fact that the ghost sector decouples. The number of Feynman diagrams in a perturbative loop expansion is reduced, leading to an important simplification from a technical point of view. Furthermore the problem of possible Gribov copies [8], generically present in covariant gauges, is absent [2]. The price to pay is that the (perturbative) propagator receives ’spurious’ poles, which have to be dealt with separately. The question about how to regularise the propagator as to allow for a consistent loop expansion stimulated extensive investigations [7]. The intricacies concerning these regularisations partly spoil the advantage of having fewer diagrams to calculate. Nevertheless it has been an appealing gauge to e.g. calculate expectation values of Wilson loops which serve as order parameters for confinement. In the strong coupling limit they are expected to fulfil Wilson’s area law which is correlated to a linear quark potential. The proper calculation of these expectation values may also necessitate the inclusion of topologically non-trivial configurations like instantons [9]. Wilson loop calculations have also been used as a testing ground for the consistency of calculations in general axial gauges. Thus it would be interesting to see whether an alternative approach (other than standard perturbation theory) to gauge theories in general axial gauges is available which preserves the above mentioned benefits without encountering ’spurious’ divergencies. In this Letter, we shall argue that these ’spurious’ poles are indeed an artifact of a perturbative loop expansion. The remedy we propose is known as the ’Exact’ (or Wilsonian) Renormalisation Group [10]-[12]. This approach has already been applied to scalar [13], Abelian Higgs theories [14]-[16] and non-Abelian gauge theories [17]-[21] in covariant gauges, and is particularly useful in cases where perturbative expansion parameters tend to be large. The key difference to a perturbative loop expansion consists in the fact that flow equations integrate-out quantum fluctuations mode by mode while integrating over infinitesimal momentum shells. This connects continuously the classical with the full quantum effective action and can be interpreted as a ’coarse-graining’ of the microscopic field theory. The perturbative loop expansion lacks the notion of ’coarse-graining’ as all modes will be integrated-out in one step within a given loop order. For the issues addressed in this Letter fermions will act only as spectators. Thus for the sake of simplicity we concentrate on the pure non-Abelian gauge theories. 2. Let us shortly review the appearance of ’spurious’ propagator poles related to an axial gauge fixing in standard perturbation theory. We will start with the Euclidean action for a pure non-Abelian gauge theory, given in d dimensions by