BRS symmetry for Yang-Mills theory with exact renormalization group

TL;DR

This paper demonstrates how to fix boundary conditions in the exact renormalization group flow of Yang-Mills theory to satisfy Slavnov-Taylor identities, ensuring gauge invariance in the infrared regime.

Contribution

It shows perturbatively that physical couplings can be chosen to satisfy all Slavnov-Taylor identities within the RG framework for Yang-Mills theory.

Findings

Boundary conditions can be fixed to satisfy ST identities perturbatively.

Three physical couplings are determined by wave function normalization and vertex at subtraction point.

Remaining couplings are either vanishing or expressed by irrelevant vertices.

Abstract

In the exact renormalization group (RG) flow in the infrared cutoff $\Lambda$ one needs boundary conditions. In a previous paper on $SU(2)$ Yang-Mills theory we proposed to use the nine physical relevant couplings of the effective action as boundary conditions at the physical point $\Lambda=0$ (these couplings are defined at some non-vanishing subtraction point $\mu \ne 0$). In this paper we show perturbatively that it is possible to appropriately fix these couplings in such a way that the full set of Slavnov-Taylor (ST) identities are satisfied. Three couplings are given by the vector and ghost wave function normalization and the three vector coupling at the subtraction point; three of the remaining six are vanishing (\eg the vector mass) and the others are expressed by irrelevant vertices evaluated at the subtraction point. We follow the method used by Becchi to prove ST identities in the RG framework. There the boundary conditions are given at a non-physical point $\Lambda=\Lambda' \ne 0$, so that one avoids the need of a non-vanishing subtraction point.