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

TL;DR

This work computes one-loop four-graviton amplitudes with a massive Proca field in the loop using the Landau-Lifshitz tensor density formulation and shows that the resulting contact-term-free $n$-gon numerators admit a manifest double-copy structure when external gravitational trees satisfy the tree-level double-copy property. By exploiting a tree-level double copy for the external currents, the authors derive a factorized loop numerator form and obtain explicit scalar- and vector-loop amplitudes, including all-plus, mostly-plus, and MHV configurations, with large-mass expansions that agree with known results. They analyze single- and double-copy constructions, clarifying how ghost contributions and partial CK-dual behavior are required to reproduce correct Yang-Mills and gravity amplitudes from abelian preliminaries. The results provide a transparent, scalable framework for one-loop double-copy relations in gravity with massive matter, with potential extensions to higher-point and higher-loop calculations and applications to dark-sector scenarios.

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.