One-Loop Renormalization and Asymptotic Behaviour of a Higher-Derivative Scalar Theory in Curved Spacetime

TL;DR

This paper investigates a higher-derivative scalar field theory in curved spacetime, deriving one-loop counterterms, renormalization group equations, and analyzing conditions for asymptotic freedom and conformal invariance.

Contribution

It provides the first derivation of one-loop counterterms and renormalization group equations for a general sigma-model type scalar theory in curved spacetime, exploring asymptotic behaviors.

Findings

Possible construction of asymptotically free theories depending on coupling sign

Identification of conditions for asymptotic conformal invariance at high or low curvature

Connection to effective theories of the conformal factor in quantum gravity

Abstract

A higher-derivative, interacting, scalar field theory in curved spacetime with the most general action of sigma-model type is studied. The one-loop counterterms of the general theory are found. The renormalization group equations corresponding to two different, multiplicatively renormalizable variants of the same are derived. The analysis of their asymptotic solutions shows that, depending on the sign of one of the coupling constants, we can construct an asymptotically free theory which is also asymptotically conformal invariant at strong (or small) curvature. The connection that can be established between one of the multiplicatively renormalizable variants of the theory and the effective theory of the conformal factor, aiming at the description of quantum gravity at large distances, is investigated.