Splitting Ward identity

TL;DR

This paper derives a general splitting Ward identity within the background-field framework, addressing background dependence of the effective action and introducing a modified master equation, with applications to gauge theories.

Contribution

It provides a path integral derivation of the splitting Ward identity that is independent of the splitting method, connecting it to background dependence issues in gauge theories.

Findings

Derived a general splitting Ward identity using path integrals

Connected the identity to background dependence of the effective action

Discussed applications to gauge theories within a geometric framework

Abstract

Within the background-field framework we present a path integral derivation of the splitting Ward identity for the one-particle irreducible effective action in the presence of an infrared regulator, and make connection with earlier works on the subject. The approach is general in the sense that it does not rely on how the splitting is performed. This identity is then used to address the problem of background dependence of the effective action at an arbitrary energy scale. We next introduce the modified master equation and emphasize its role in constraining the effective action. Finally, application to general gauge theories within the geometric approach is discussed.