One-loop double copy from gravity coupled to a massive vector field

TL;DR

This paper computes one-loop four-graviton amplitudes with a massive vector field in the loop, revealing a double-copy structure that simplifies the construction of gravity amplitudes and enhances understanding of double-copy relations.

Contribution

It introduces a novel approach using the Landau-Lifshitz formulation to demonstrate a manifest double-copy structure in one-loop amplitudes with a Proca field.

Findings

Contact-terms-free numerators exhibit double-copy structure

Allows direct construction of scalar-in-loop amplitudes

Simplifies evaluation of pure gravity amplitudes in specific sectors

Abstract

In this work, we compute the one-loop four-graviton amplitudes with a massive abelian vector field (Proca) circulating in the loop. Instead of the conventional Einstein-Hilbert formulation, we employ the Landau-Lifshitz metric density approach, coupling gravity to the Proca field. Within this framework, we found that the resulting contact-terms-free $n$-gon numerators exhibit a manifest double-copy structure, provided the external gravitational trees satisfy this property. This structure enables the direct construction of the corresponding amplitudes with a scalar in the loop. We further show that these numerators can be consistently used to evaluate pure gravity amplitudes in the sectors where contact terms vanish, offering a simplified setting to explore double-copy relations at one loop level.