Conservative Binary Dynamics with a Spinning Black Hole at $\mathcal{O}(G^3)$ from Scattering Amplitudes

TL;DR

This paper derives the conservative Hamiltonian for a spinning black hole binary system at third order in gravitational constant G, using advanced scattering amplitude techniques to include all velocity orders and spin effects.

Contribution

It presents the first complete calculation of the two-body Hamiltonian with a spinning black hole at rd order in G, incorporating all velocity and spin contributions through novel amplitude methods.

Findings

Derived the rd order G Hamiltonian including spin effects.

Applied modern scattering amplitude techniques to gravitational two-body problem.

Explicitly evaluated divergent integrals in the amplitude and phase-space calculations.

Abstract

We compute the conservative two-body Hamiltonian of a compact binary system with a spinning black hole through $\mathcal{O}(G^3)$ to all orders in velocity, including linear and quadratic spin terms. To obtain our results we calculate the classical limit of the two-loop amplitude for the scattering of a massive scalar particle with a massive spin-1 particle minimally coupled to gravity. We employ modern scattering amplitude and loop integration techniques, in particular numerical unitarity, integration-by-parts identities, and the method of regions. The conservative potential in terms of rest-frame spin vectors is extracted by matching to a non-relativistic effective field theory. We also apply the Kosower-Maybee-O'Connell (KMOC) formalism to calculate the impulse in the covariant spin formalism directly from the amplitude. We work systematically in conventional dimensional regularization and explicitly evaluate all divergent integrals that appear in full- and effective-theory amplitudes, as well as in the phase-space integrals that arise in the KMOC formalism.