Einstein Gauss-Bonnet Theories as Ordinary, Wess-Zumino Conformal Anomaly Actions

TL;DR

This paper explores the connection between Einstein Gauss-Bonnet theories and conformal anomaly actions, showing how certain regularizations lead to Wess-Zumino actions and proposing a purely gravitational, nonlocal effective theory.

Contribution

It demonstrates that the regularized Einstein Gauss-Bonnet action corresponds to a Wess-Zumino anomaly action and introduces a scale-free, quadratic dilaton effective action that can be purely gravitational and nonlocal.

Findings

Regularization yields Wess-Zumino anomaly action.

Finite renormalization produces a quadratic, scale-free effective action.

Pure gravitational nonlocal theory can be derived without the dilaton.

Abstract

Recently, the possibility of evading Lovelock's theorem at $d=4$, via a singular redefinition of the dimensionless coupling of the Gauss-Bonnet term, has been very extensively discussed in the cosmological context. The term is added as a quadratic contribution of the curvature tensor to the Einstein-Hilbert action, originating theories of "Einstein Gauss-Bonnet" (EGB) type. We point out that the action obtained by the dimensional regularization procedure, implemented with the extraction of a single conformal factor, correspond just to an ordinary Wess-Zumino anomaly action, even though it is deprived of the contribution from the Weyl tensor. We also show that a purely gravitational version of the EGB theory can be generated by allowing a finite renormalization of the Gauss-Bonnet topological contribution at $d= 4 + \epsilon$, as pointed out by Mazur and Mottola. The result is an effective action which is quadratic, rather then quartic, in the dilaton field, and scale free, compared to the previous derivations. The dilaton, in this case can be removed from the spectrum, leaving a pure gravitational theory, which is nonlocal. We comment on the physical meaning of the two types of actions, which may be used to describe such topological terms both below and above the conformal breaking scale.