Renormalization group flow for SU(2) Yang-Mills theory and gauge invariance

TL;DR

This paper develops a Wilson RG approach for SU(2) Yang-Mills theory that maintains gauge invariance through boundary conditions, avoiding fine-tuning and enabling perturbative calculations consistent with Slavnov-Taylor identities.

Contribution

It introduces a gauge-invariant RG formulation for non-Abelian gauge theories using boundary conditions to enforce Slavnov-Taylor identities, simplifying renormalization.

Findings

Derived perturbative expansion of vertex functions in terms of physical coupling.

Performed one-loop calculations confirming gauge invariance.

Provided a schematic proof of perturbative renormalizability.

Abstract

We study the formulation of the Wilson renormalization group (RG) method for a non-Abelian gauge theory. We analyze the simple case of $SU(2)$ and show that the local gauge symmetry can be implemented by suitable boundary conditions for the RG flow. Namely we require that the relevant couplings present in the physical effective action, \ie the coefficients of the field monomials with dimension not larger than four, are fixed to satisfy the Slavnov-Taylor identities. The full action obtained from the RG equation should then satisfy the same identities. This procedure is similar to the one we used in QED. In this way we avoid the cospicuous fine tuning problem which arises if one gives instead the couplings of the bare Lagrangian. To show the practical character of this formulation we deduce the perturbative expansion for the vertex functions in terms of the physical coupling $g$ at the subtraction point $\mu$ and perform one loop calculations. In particular we analyze to this order some ST identities and compute the nine bare couplings. We give a schematic proof of perturbative renormalizability.