Conformal renormalization of scalar-tensor theories

TL;DR

This paper develops a conformal renormalization scheme for scalar-tensor theories, fixing counterterms via symmetry requirements, and demonstrates its effectiveness through black hole solutions and their thermodynamics.

Contribution

It introduces a conformal renormalization method for scalar-tensor theories, ensuring finiteness of the on-shell action and extending to theories with conformal gravity and Einstein-AdS actions.

Findings

Counterterms are fixed by symmetry restoration.

The renormalized action is finite and conformally covariant.

Black hole thermodynamics are computed within this framework.

Abstract

We study a conformally coupled scalar-tensor theory with a quartic potential possessing local conformal symmetry up to a boundary term. We show that requiring the restoration of the full local conformal symmetry fixes the counterterms that render the on-shell action finite. The building block of the resulting action is a conformally covariant tensor which is constructed out of the metric and the scalar field and it has the same conformal weight as the Weyl tensor. This allows us to obtain the counterterms for the scalar-tensor sector in a closed form. The finiteness of the conformally complete version of the action is suggestive on the validity of the Conformal Renormalization prescription. We extend this theory by adding the Conformal Gravity action and also the Einstein-AdS action written in McDowell-Mansouri form. Even though the latter breaks the conformal symmetry, we find that the action is still renormalized provided a suitable falloff of the scalar field when considering asymptotically locally anti-de Sitter solutions. Black hole solutions in these theories are studied, for which the Hawking temperature and the partition function to first order in the saddle-point approximation are calculated, providing a concrete example of this renormalization scheme.