New Relations for Gauge-Theory and Gravity Amplitudes at Loop Level

TL;DR

This paper extends fundamental amplitude relations from tree level to loop level in gauge theory and gravity, providing a gauge-invariant formulation of their double-copy relations and revealing universal structures across theories.

Contribution

It introduces a gauge-invariant formulation of loop-level KLT and BCJ relations, and relates Einstein--Yang--Mills integrands to gauge theory partial integrands.

Findings

Loop integrands obey universal BCJ relations.

Gravity integrands expressed via gauge-invariant partial integrands.

Relations hold across dimensions and supersymmetry levels.

Abstract

In this letter, we extend the tree-level Kawai--Lewellen--Tye (KLT) and Bern--Carrasco--Johansson (BCJ) amplitude relations to loop integrands of gauge theory and gravity. By rearranging the propagators of gauge and gravity loop integrands, we propose the first manifestly gauge- and diffeomorphism invariant formulation of their double-copy relations. The one-loop KLT formula expresses gravity integrands in terms of more basic gauge invariant building blocks for gauge-theory amplitudes, dubbed partial integrands. The latter obey a one-loop analogue of the BCJ relations, and both KLT and BCJ relations are universal to bosons and fermions in any number of spacetime dimensions and independent on the amount of supersymmetry. Also, one-loop integrands of Einstein--Yang--Mills (EYM) theory are related to partial integrands of pure gauge theories.