Running couplings and operator mixing in the gravitational corrections to coupling constants

TL;DR

This paper examines the challenges of defining running coupling constants in effective field theories with gravity, highlighting issues with operator mixing and the limited usefulness of gravitational corrections in most theories.

Contribution

It clarifies the conditions under which a meaningful running coupling can be defined in theories with gravity and discusses the importance of operator mixing and renormalization schemes.

Findings

Running couplings are well-defined in high-symmetry theories like lambda phi^4.

In most theories, gravitational effects do not produce a meaningful running of couplings.

Operator mixing and off-shell renormalization are crucial for effective field theories with gravity.

Abstract

The use of a running coupling constant in renormalizable theories is well known, but the implementation of this idea for effective field theories with a dimensional coupling constant is in general less useful. Nevertheless there are multiple attempts to define running couplings including the effects of gravity, with varying conclusions. We sort through many of the issues involved, most particularly the idea of operator mixing and also the kinematics of crossing, using calculations in Yukawa and lambda phi^4 theory as illustrative examples. We remain in the perturbative regime. In some theories with a high permutation symmetry, such as lambda phi^4, a reasonable running coupling can be defined. However in most cases, such as Yukawa and gauge theories, a running coupling fails to correctly account for the energy dependence of the interaction strength. As a byproduct we also contrast on-shell and off-shell renormalizaton schemes and show that operators which are normally discarded, such as those that vanish by the equations of motion, are required for off-shell renormalization of effective field theories. Our results suggest that the inclusion of gravity in the running of couplings is not useful or universal in the description of physical processes.