Inelastic exponentiation and classical gravitational scattering at one loop

We calculate the inelastic 2 → 3 one-loop amplitude for the scattering of two point-like, spinless objects with generic masses involving the additional emission of a single graviton. We focus on the near-forward, or classical, limit. Our results include the leading and subleading orders in the soft-region expansion, which captures all non-analytic contributions in the transferred momentum and in the graviton’s frequency. This allows us to check the first constraint arising from the inelastic exponentiation put forward in refs. [ 1 – 3 ], and to calculate the 2 → 3 one-loop matrix element of the N -operator, linked to the S -matrix by S = e iN , showing that it is real, classical and free of infrared divergences. We discuss how our results feature in the calculation of the 𝒪 ( G 3 ) corrections to the asymptotic waveform.