Completeness and consistency of renormalisation group flows

TL;DR

This paper compares various renormalisation group flows, analyzing their structures, approximations, and relations, especially focusing on their dependence on propagators and their accuracy in reproducing perturbation theory.

Contribution

It provides a detailed comparison of exact, proper-time, and Callan-Symanzik renormalisation group flows, including a new exact proper-time flow within a background field formalism.

Findings

Proper-time flows are approximations to Callan-Symanzik flows.

Standard perturbation theory is not reproduced by certain flows.

A generalized exact proper-time flow is introduced within a background field formalism.

Abstract

We study different renormalisation group flows for scale dependent effective actions, including exact and proper-time renormalisation group flows. These flows have a simple one loop structure. They differ in their dependence on the full field-dependent propagator, which is linear for exact flows. We investigate the inherent approximations of flows with a non-linear dependence on the propagator. We check explicitly that standard perturbation theory is not reproduced. We explain the origin of the discrepancy by providing links to exact flows both in closed expressions and in given approximations. We show that proper-time flows are approximations to Callan-Symanzik flows. Within a background field formalism, we provide a generalised proper-time flow, which is exact. Implications of these findings are discussed.