Gravitational amplitudes in the Regge limit: waveforms, shock waves and unitarity cuts

Abstract

Motivated by recent progress in the high-energy description of gravitational scattering, we develop a systematic Regge-theory framework for $2\to 2+n$ amplitudes describing the scattering of two massive particles with $n$ graviton emissions, including spin effects. Working in the ultra-relativistic limit at leading logarithmic accuracy, the massive result smoothly reduces to its massless counterpart. We describe both quantum (Regge trajectory and BFKL $t$-channel evolution) and classical ($s$-channel multi-$H$ evolution) contributions using both an exponential representation of the S-matrix and a shock-wave formalism in light-cone quantisation. In the latter approach, gravitational Wilson lines evolve in rapidity space under a boost-invariant Hamiltonian, providing a space-time realisation of the high-energy dynamics and making contact with recent effective field theory descriptions in the forward limit. As an application, we compute the leading-logarithmic contribution to the massive spinless $2\to 2$ amplitude at 5PM-2SF order, recovering the previously determined massless result, and derive the tree-level $2\to 3$ amplitude and its associated scattering waveform for Kerr black holes in the ultra-relativistic limit.