From Scattering Amplitudes to Classical Potentials in the Post-Minkowskian Expansion

TL;DR

This paper constructs a systematic bridge between on-shell scattering amplitudes and the classical two-body gravitational potential using an effective field theory for non-relativistic spinless particles. By matching EFT and full-theory amplitudes, it derives a closed-form 2PM potential valid to all orders in velocity and shows exact agreement with established 4PN results, while also providing a gauge-invariant, integrand-level framework. The key advance is an explicit mapping between the classical potential coefficients $c_i$ and unitarity-derived scalar coefficients $d_i$, enabling cross-checks for higher-order calculations and extensions to beyond GR. The approach is general and can be applied to other QFTs, including theories with new light states or higher-dimension operators, with potential benefits for gravitational-wave physics. This work thereby improves both conceptual understanding and practical computation of classical gravitational dynamics from quantum amplitudes.

Abstract

We combine tools from effective field theory and generalized unitarity to construct a map between on-shell scattering amplitudes and the classical potential for interacting spinless particles. For general relativity, we obtain analytic expressions for the classical potential of a binary black hole system at second order in the gravitational constant and all orders in velocity. Our results exactly match all known results up to fourth post-Newtonian order, and offer a simple check of future higher order calculations. By design, these methods should extend to higher orders in perturbation theory.