Quadratic-in-Spin Hamiltonian at $\mathcal{O}(G^2)$ from Scattering Amplitudes

TL;DR

This paper derives the quadratic-in-spin conservative Hamiltonian for binary systems in General Relativity at second order in the gravitational constant, extending scattering amplitude methods to include non-minimal coupling effects.

Contribution

It introduces a framework that incorporates non-minimal coupling in scattering amplitude calculations for spinning bodies, providing a more complete Hamiltonian at $ ext{O}(G^2)$.

Findings

Validated the formula relating impulse, spin kick, and eikonal phase.

Extended the scattering amplitude approach to non-minimal coupling.

Derived the quadratic-in-spin Hamiltonian at second order in G.

Abstract

We obtain the quadratic-in-spin terms of the conservative Hamiltonian describing the interactions of a binary of spinning bodies in General Relativity through $\mathcal{O}(G^2)$ and to all orders in velocity. Our calculation extends a recently-introduced framework based on scattering amplitudes and effective field theory to consider non-minimal coupling of the spinning objects to gravity. At the order that we consider, we establish the validity of the formula proposed in \cite{Bern:2020buy} that relates the impulse and spin kick in a scattering event to the eikonal phase.