The Amplitude for Classical Gravitational Scattering at Third Post-Minkowskian Order

TL;DR

This work computes the classical gravitational scattering amplitude for two massive scalars to third order in the Post-Minkowskian expansion using amplitude techniques and eikonal exponentiation. By organizing the two-loop integrand with velocity cuts and a shared master-integral basis, the authors obtain all terms necessary to derive the classical scattering angle, including radiation-reaction contributions, and confirm a key real-imaginary relation for the radiation piece. The results reproduce Damour's radiation-reaction findings and reveal a tight link between the amplitude method and world-line formalisms, with explicit $b$-space factorization showing exponentiation. The study also highlights a minimal integral basis that persists from maximal supergravity to Einstein gravity, suggesting a scalable path to higher PM orders and deeper connections to semiclassical gravitational dynamics.

Abstract

We compute the scattering amplitude for classical black-hole scattering to third order in the Post-Minkowskian expansion, keeping all terms needed to derive the scattering angle to that order from the eikonal formalism. Our results confirm a conjectured relation between the real and imaginary parts of the amplitude by Di Vecchia, Heissenberg, Russo, and Veneziano, and are in agreement with a recent computation by Damour based on radiation reaction in general relativity.

Figures (9)

• Figure 1: Double-box and crossed double-box diagrams.
• Figure 2: Box-triangle graphs.
• Figure 3: Double-triangle graphs.
• Figure 4: The $H$ diagram.
• Figure 5: The box-bubble diagram.