One-loop Vilkovisky-DeWitt Counterterms for 2D Gravity plus Scalar Field Theory

TL;DR

This paper computes the one-loop divergences in a 2D gravity plus scalar field theory using Vilkovisky-DeWitt formalism, revealing conditions for finiteness and renormalizability both on-shell and off-shell.

Contribution

It provides gauge-independent one-loop counterterms for 2D gravity coupled to scalar fields with arbitrary potentials, including Liouville theory, using Vilkovisky-DeWitt approach.

Findings

Liouville theory is finite on shell.

A class of renormalizable potentials includes Liouville potential.

Liouville theory is finite off shell when $R=0$.

Abstract

The divergent part of the one-loop off-shell effective action is computed for a single scalar field coupled to the Ricci curvature of 2D gravity ($c \phi R$), and self interacting by an arbitrary potential term $V(\phi)$. The Vilkovisky-DeWitt effective action is used to compute gauge-fixing independent results. In our background field/covariant gauge we find that the Liouville theory is finite on shell. Off-shell, we find a large class of renormalizable potentials which include the Liouville potential. We also find that for backgrounds satisfying $R=0$, the Liouville theory is finite off shell, as well.