Renormalization Group for Non-minimal $\phi^2 R$ Couplings and Gravitational Contact Interactions

TL;DR

This paper investigates how non-minimal scalar-gravity couplings affect the renormalization group flow, revealing discrepancies between Jordan and Einstein frames and proposing a method to obtain consistent quantum results.

Contribution

It demonstrates the impact of graviton-induced contact interactions on RG calculations and provides a way to correctly include these effects in quantum analyses.

Findings

Contact interactions modify the RG flow in scalar-gravity theories.

Calculations in Jordan and Einstein frames do not commute without accounting for contact terms.

A method to incorporate contact effects yields consistent quantum results.

Abstract

Theories of scalars and gravity, with an Einstein-Hilbert term and non-minimal interactions, $M^2R/2 -\alpha\phi^2R/12 $, have graviton exchange induced contact interactions. These modify the renormalization group, leading to a discrepancy between the conventional calculations in the Jordan frame that ignore this effect (and are found to be incorrect), and the Einstein frame in which $\alpha$ does not exist. Thus, the calculation of quantum effects in the Jordan and Einstein frames does not generally commute with the transition from the Jordan to the Einstein frame. In the Einstein frame, though $\alpha$ is absent, for small steps in scale $\delta\mu/\mu$ infinitesimal contact terms $\sim \delta\alpha$ are induced, that are then absorbed back into other couplings by the contact terms. This modifies the $\beta$-functions in the Einstein frame. We show how correct results can be obtained in a simple model by including this effect.