Loop-level double copy relations from forward limits

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 $\phi^3$ 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).