Proof of a three-loop relation between the Regge limits of four-point amplitudes in $ \mathcal{N} $ = 4 SYM and $ \mathcal{N} $ = 8 supergravity

A previously proposed all-loop-orders relation between the Regge limits of four-point amplitudes of $$ \mathcal{N} $$N = 4 supersymmetric Yang-Mills theory and $$ \mathcal{N} $$N = 8 supergravity is established at the three-loop level. We show that the Regge limit of known expressions for the amplitudes obtained using generalized unitarity simplifies in both cases to a (modified) sum over three-loop ladder and crossed-ladder scalar diagrams. This in turn is consistent with the result obtained using the eikonal representation of the four-point gravity amplitude. A possible exact three-loop relation between four-point amplitudes is also considered.