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

TL;DR

This work derives the conservative two-body Hamiltonian for spinning compact objects at quadratic order in spin up to ${O}(G^2)$, valid for all velocities, by matching full-theory scattering amplitudes to an EFT with non-minimal gravity couplings. The authors compute tree-level and one-loop amplitudes for a scalar and a spinning particle in the center-of-mass frame, then perform EFT matching to fix long-range spin-dependent potentials and obtain the position-space Hamiltonian, including the Kerr limit ${C_{ES^2}}=1$. They verify consistency with known PN results and test-body limits, reproduce the aligned-spin BH scattering angle, and demonstrate that the Bern:2020buy eikonal-phase relation extends to quadratic-in-spin, strengthening the link between scattering data and classical observables. The results provide a robust framework for incorporating higher-spin and finite-size effects in gravitational binaries and suggest avenues for extending to higher PM orders and tidal corrections.

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.

Figures (5)

• Figure 1: The Feynman vertices used to compute full-theory amplitudes. The three-particle vertex (a) determines the $\mathcal{O}(G)$ dynamics. The Compton amplitude, which requires the contact vertex (b), captures the $\mathcal{O}(G^2)$ dynamics. The straight lines correspond to the spinning particle, while the wiggly lines correspond to gravitons.
• Figure 2: The tree-level amplitude that captures the ${\mathcal{O}}(G)$ spin interactions. The thick (thin) straight line represents the spinning (scalar) particle, while the wiggly line corresponds to the exchanged graviton.
• Figure 3: The one-loop scalar box integrals $I_{\Box}$ (a) and $I_{\bowtie}$ (b) and the corresponding triangle integrals $I_{\bigtriangleup}$ (c) and $I_{\bigtriangledown}$ (d). The bottom (top) solid line corresponds to a massive propagator of mass $m_1$ ($m_2$). The dashed lines denote massless propagators.
• Figure 4: The Compton-amplitude Feynman diagrams. The straight line corresponds to the spinning particle. The wiggly lines correspond to gravitons.
• Figure 5: Appropriate residues of the two-particle cut (a) give the triple cuts (b) and (c), and the quadruple cut (d). The thick straight line corresponds to the spinning particle, the thin straight line to the scalar, and the wiggly lines to the exchanged gravitons. All exposed lines are taken on-shell.