Loop-Level Double Copy Relations from Forward Limits

TL;DR

The paper shows how to establish one-loop KLT and BCJ double copy relations by reconstructing one-loop YM and gravity integrands from forward limits of tree amplitudes. It demonstrates that shifted cyclic symmetry of forward-limit numerators removes spurious poles and yields quadratic propagators, enabling a gravity YM double copy via an invertible one-loop basis and a universal YM expansion to express gravity in EYMS scalar-loop terms. It provides explicit one-loop KLT formulas and BCJ numerators (including n=3,4,5) and discusses extensions to supergravity and CHY-based perspectives, highlighting a concrete, local framework for loop-level double copy and potential paths to higher-loop generalizations. The results offer a practical route to compute gravity loop integrands from gauge theory data and tie into broader formalisms like CHY and surfaceology.

Abstract

We study double copy relations for loop integrands in gauge theories and gravity based on their constructions from single cuts, which are in turn obtained from forward limits of lower-loop cases. While such a construction from forward limits has been realized for loop integrands in gauge theories, we demonstrate its extension to gravity by reconstructing one-loop gravity integrands from forward limits of trees. Under mild symmetry assumptions on tree-level kinematic numerators (and their forward limits), our method directly leads to double copy relations for one-loop integrands: these include the field-theoretic Kawai-Lewellen-Tye (KLT) relations, whose kernel is the inverse of a matrix with rank $(n{-}1)!$ formed by those in bi-adjoint $φ^3$ theory, and the Bern-Carrasco-Johansson (BCJ) double copy relations with crossing-symmetric kinematic numerators (we provide local and crossing-symmetric Yang-Mills BCJ numerators for $n=3,4,5$ explicitly). By exploiting the "universal expansion" for one-loop integrands in generic gauge theories, we also obtain an analogous expansion for gravity (including supergravity theories).

Figures (5)

• Figure 1: Starting from (kinematic numerators of) YM tree, the single-cut reconstruction based on forward limits produces one-loop integrands and BCJ numerators; similarly from gravity tree, one constructs the one-loop gravity integrands, which in turn can be obtained from one-loop KLT/BCJ double copy.
• Figure 2: Dual variables at one loop: the loop puncture $z$ and dual points $x_i,x_j$ define momenta $\ell_i$ along curves $z\to x_i$, giving $Y_i=\ell_i^2$. Throughout we use the in-going convention, with all external momenta flowing into the loop.
• Figure 3: The one-loop integrand reconstructed from forward limits of trees.
• Figure 4: Two diagrams contributing to $\mathcal{M}_{3}(1,2,3|1,2,3)$.
• Figure 5: Maximal-cut numerator for the $n$-gon. The arrow on the left indicates the orientation of the loop momentum. One can easily prove that the right-hand side is exactly \ref{['eq_maximalnum']}.