Background independence in a background dependent renormalization group

TL;DR

This paper investigates the conditions under which background independence can be achieved in a background-dependent renormalization group framework for conformally reduced gravity, revealing specific constraints and redundancies.

Contribution

It demonstrates the compatibility conditions between Ward identities and flow equations and uncovers a background independent flow equation within a background-dependent setting.

Findings

Ward identities are compatible with flow equations only if the anomalous dimension vanishes or the cutoff is a power law.

A background independent flow equation is found for any cutoff profile.

Beyond six-point interactions, the equations become over-constrained or redundant.

Abstract

Within the derivative expansion of conformally reduced gravity, the modified split Ward identities are shown to be compatible with the flow equations if and only if either the anomalous dimension vanishes or the cutoff profile is chosen to be power law. No solutions exist if the Ward identities are incompatible. In the compatible case, a clear reason is found for why Ward identities can still forbid the existence of fixed points; however, for any cutoff profile, a background independent (and parametrisation independent) flow equation is uncovered. Finally, expanding in vertices, the combined equations are shown generically to become either over-constrained or highly redundant beyond the six-point level.