The local potential approximation in the background field formalism

TL;DR

This paper examines the local potential approximation in the background field formalism for scalar fields, identifies issues with the common identification of total and background fields, and proposes a modified approach to restore universality and fix pathologies.

Contribution

It introduces a consistent treatment of the background field using the modified shift Ward identity within the local potential approximation.

Findings

Identification of pathologies when total and background fields are equated

Modified shift Ward identity restores universality of physical quantities

Similarities found with f(R) approximation in asymptotic safety for gravity

Abstract

Working within the familiar local potential approximation, and concentrating on the example of a single scalar field in three dimensions, we show that the commonly used approximation method of identifying the total and background fields, leads to pathologies in the resulting fixed point structure and the associated spaces of eigenoperators. We then show how a consistent treatment of the background field through the corresponding modified shift Ward identity, can cure these pathologies, restoring universality of physical quantities with respect to the choice of dependence on the background field, even within the local potential approximation. Along the way we point out similarities to what has been previously found in the f(R) approximation in asymptotic safety for gravity.