The renormalization group for non-renormalizable theories: Einstein gravity with a scalar field

TL;DR

This paper develops a renormalization-group framework for non-renormalizable theories, specifically applied to Einstein gravity coupled with a scalar field, revealing asymptotic behaviors and a Weyl-invariant phase at high energies.

Contribution

It introduces a novel RG formalism for non-renormalizable theories and derives equations for scalar-dependent functions in Einstein gravity, highlighting asymptotic freedom and Weyl invariance.

Findings

Derived one-loop counterterms for the theory.

Established RG equations for functions U, G, V.

Identified a Weyl-invariant asymptotic phase at high energies.

Abstract

We develop a renormalization-group formalism for non-renormalizable theories and apply it to Einstein gravity theory coupled to a scalar field with the Lagrangian $L=\sqrt{g} [R U(\phi)-{1/2} G(\phi) g^{\mu\nu} \partial_{\mu}\phi \partial_{\nu}\phi- V(\phi)]$, where $U(\phi), G(\phi)$ and $V(\phi)$ are arbitrary functions of the scalar field. We calculate the one-loop counterterms of this theory and obtain a system of renormalization-group equations in partial derivatives for the functions $U, G$ and $V$ playing the role of generalized charges which substitute for the usual charges in multicharge theories. In the limit of a large but slowly varying scalar field and small spacetime curvature this system gives the asymptotic behaviour of the generalized charges compatible with the conventional choice of these functions in quantum cosmological applications. It also demonstrates in the over-Planckian domain the existence of the Weyl-invariant phase of gravity theory asymptotically free in gravitational and cosmological constants.