Topological Corrections and Conformal Backreaction in the Einstein Gauss-Bonnet/Weyl Theories of Gravity at D=4

TL;DR

This paper explores the effects of topological corrections and conformal backreaction in four-dimensional Einstein-Gauss-Bonnet and Weyl gravity theories, focusing on regularization issues, nonlocal formulations, and their implications for conformal anomalies.

Contribution

It introduces nonlocal gravitational realizations of Einstein-Gauss-Bonnet theories, showing how they can be derived through finite renormalization of topological densities and expanded around flat space.

Findings

Nonlocal, scale-deprived EGB theories are possible via renormalization.

Such theories can be expanded in terms of $R \, \Box^{-1}$ and anomaly variations.

The approach clarifies the structure of conformal anomalies in modified gravity.

Abstract

We investigate the gravitational backreaction, generated by coupling a general conformal sector to external, classical gravity, as described by a conformal anomaly effective action. We address the issues raised by the regularization of the topological Gauss-Bonnet and Weyl terms in these actions and the use of dimensional regularization (DR). We discuss both their local and nonlocal expressions, as possible IR and UV descriptions of conformal theories, below and above the conformal breaking scale. Our discussion overlaps with several recent studies of dilaton gravities - obtained via a certain singular limit of the Einstein-Gauss-Bonnet (EGB) theory - originally introduced as a way to bypass Lovelock's theorem. We show that nonlocal, purely gravitational realizations of such EGB theories, quadratic in the dilaton field, beside their local quartic forms, are possible by a finite renormalization of the Euler density. Such nonlocal versions, which are deprived of any scale, can be expanded, at least around flat space, in terms of the combination $R \Box^{-1}$ times multiple variations of the anomaly functional, as pointed out in recent studies at $d=4$. Similar conclusions can be drawn for the proposed nonlocal EGB theory. The expansion emerges from previous investigations of the anomalous conformal Ward identities that constrain such theories around the flat spacetime limit in momentum space.