Kinematic Numerators and a Double-Copy Formula for N = 4 Super-Yang-Mills Residues

Recent work by Cachazo, He, and Yuan shows that connected prescription residues obey the global identities of $\mathcal{N} = 4$ super-Yang-Mills amplitudes. In particular, they obey the Bern-Carrasco-Johansson (BCJ) amplitude identities. Here we offer a new way of interpreting this result via objects that we call residue numerators. These objects behave like the kinematic numerators introduced by BCJ except that they are associated with individual residues. In particular, these new objects satisfy a double-copy formula relating them to the residues appearing in recently-discovered analogs of the connected prescription integrals for $\mathcal{N} = 8$ supergravity. Along the way, we show that the BCJ amplitude identities are equivalent to the consistency condition that allows kinematic numerators to be expressed as amplitudes using a generalized inverse.