# Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order

## TL;DR

This paper computes the classical scattering amplitude for gravitationally interacting massive scalars at third post-Minkowskian order using modern amplitude techniques, deriving the conservative Hamiltonian and scattering angle for binary systems.

## Contribution

It introduces a novel approach combining amplitude methods and effective field theory to derive the third post-Minkowskian Hamiltonian for binary systems.

## Key findings

- Hamiltonian matches fourth post-Newtonian order results
- Derived scattering angle at third post-Minkowskian order
- Confirmed agreement with probe limit at all velocities

## Abstract

We present the amplitude for classical scattering of gravitationally interacting massive scalars at third post-Minkowskian order. Our approach harnesses powerful tools from the modern amplitudes program such as generalized unitarity and the double-copy construction, which relates gravity integrands to simpler gauge-theory expressions. Adapting methods for integration and matching from effective field theory, we extract the conservative Hamiltonian for compact spinless binaries at third post-Minkowskian order. The resulting Hamiltonian is in complete agreement with corresponding terms in state-of-the-art expressions at fourth post-Newtonian order as well as the probe limit at all orders in velocity. We also derive the scattering angle at third post-Minkowskian order.