Gauge-invariant fields and flow equations for Yang-Mills theories

TL;DR

This paper develops a framework for gauge-invariant fields and flow equations in non-abelian gauge theories, enabling gauge-invariant analysis of effective actions and their renormalization group flow.

Contribution

It introduces a method to construct gauge-invariant fields and extends the flow equations to maintain gauge invariance in Yang-Mills theories.

Findings

Derived a gauge-invariant effective action from physical fluctuations.

Formulated a gauge-invariant flow equation for the effective average action.

Applied the framework to compute the running gauge coupling in SU(N) Yang-Mills.

Abstract

We discuss the concept of gauge-invariant fields for non-abelian gauge theories. Infinitesimal fluctuations around a given gauge field can be split into physical and gauge fluctuations. Starting from some reference field the gauge-invariant fields are constructed by consecutively adding physical fluctuations. An arbitrary gauge field can be mapped to an associated gauge invariant field. An effective action that depends on gauge-invariant fields becomes a gauge-invariant functional of arbitrary gauge fields by associating to every gauge field the corresponding gauge-invariant field. The gauge-invariant effective action can be obtained from an implicit functional integral with a suitable "physical gauge fixing". We generalize this concept to the gauge-invariant effective average action or flowing action, which involves an infrared cutoff. It obeys a gauge-invariant functional flow equation. We demonstrate the use of this flow equation by a simple computation of the running gauge coupling and propagator in pure $SU(N)$-Yang-Mills theory.