Loop Amplitudes in an Extended Gravity Theory

TL;DR

This work investigates how extending gauge and gravitational theories by minimal three-point interactions alters their one-loop ultraviolet behavior, using on-shell unitarity as a diagnostic. It shows that extended Yang–Mills renormalizes the cubic $F^3$ vertex at leading order, while extended gravity requires $R^4$ counterterms to cancel one-loop divergences associated with an $R^3$ extension. Importantly, the necessary $R^4$ counterterm can be chosen orthogonal to the supersymmetric combination, illustrating a precise structure of UV completions in gravity. The study also highlights that three-point data alone does not fully fix the theory at higher orders, as polynomial ambiguities and the need for additional four-point information emerge, underscoring the utility and limits of constructibility and unitarity in organizing higher-dimension operators in gravity.

Abstract

We extend the $S$-matrix of gravity by the addition of the minimal three-point amplitude or equivalently adding $R^3$ terms to the Lagrangian. We demonstrate how Unitarity can be used to simply examine the renormalisability of this theory and determine the $R^4$ counter-terms that arise at one-loop. We find that the combination of $R^4$ terms that arise in the extended theory is complementary to the $R^4$ counter-term associated with supersymmetric Lagrangians.

Figures (3)

• Figure 1: The two-particle cut of the all-plus amplitude.
• Figure 2: The bubbles for single minus
• Figure 3: Factorisations of the four-point MHV amplitude at $\alpha^2$.