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 = eiN, showing that it is real, classical and free of infrared divergences. We discuss how our results feature in the calculation of the O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O} $$\end{document}(G3) corrections to the asymptotic waveform.