Gauge (in)dependence and background field formalism

TL;DR

This paper clarifies the distinction between gauge invariance and dependence in Yang-Mills theories within the background field formalism and explores constructing a gauge-invariant average effective action.

Contribution

It demonstrates the independence of gauge invariance and dependence issues and proposes a method to build a gauge-invariant average effective action in this formalism.

Findings

Gauge invariance and dependence are independent in background field formalism.

A gauge-invariant average effective action can be constructed.

The gauge-invariant action remains gauge dependent on-shell.

Abstract

It is shown that the gauge invariance and gauge dependence properties of effective action for Yang-Mills theories should be considered as two independent issues in the background field formalism. Application of this formalism to formulate the functional renormalization group approach is discussed. It is proven that there is a possibility to construct the corresponding average effective action invariant under the gauge transformations of background vector field. Nevertheless, being gauge invariant this action remains gauge dependent on-shell.