New Relations for Gauge-Theory and Gravity Amplitudes at Loop Level
Song He, Oliver Schlotterer
TL;DR
The paper addresses extending tree-level KLT/BCJ double-copy relations to loop level by developing a manifestly gauge- and diffeomorphism-invariant framework. It introduces partial integrands as gauge-invariant building blocks for one-loop gauge-theory amplitudes and formulates one-loop BCJ relations among them, along with a KLT-type double-copy formula that expresses gravity integrands as bilinears of partial integrands via a momentum-kernel S[π|ρ]ℓ. The construction is universal across particle content, spacetime dimensions, and supersymmetry, and it naturally extends to Einstein–Yang–Mills (EYM) with explicit relations and examples. The work provides a practical, invariant path to compute one-loop gravity amplitudes from gauge theories, with clear avenues for higher-loop generalizations and connections to CHY methods and other double-copy theories.
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.