Spinning Black Hole Binary Dynamics, Scattering Amplitudes and Effective Field Theory

TL;DR

This work develops an amplitudes-based framework to derive conservative spin-dependent two-body dynamics for spinning compact binaries, using an arbitrary-spin field theory and an EFT that matches to full gravity amplitudes at ${ m O}(G^2)$ with all velocity dependence. Spin bilinears are organized via a compact six-operator basis, and the two-body Hamiltonian is obtained by explicit matching, with observables computed from the eikonal phase. The paper also uncovers double-copy structures for three-point vertices and a KLT-like factorization in the gravitational Compton amplitude, and validates results against Kerr stress tensors and known PN/PM results. This approach yields a gauge-invariant, all-velocity description of spin-spin interactions suitable for inclusion in waveform models and EOB formalisms, and sets the stage for higher-order spin and loop studies. It also highlights a surprisingly direct link between scattering amplitudes and classical observables through the eikonal phase, suggesting broader applicability of amplitude methods in classical GR.

Abstract

We describe a systematic framework for finding the conservative potential of compact binary systems with spin based on scattering amplitudes of particles of arbitrary spin and effective field theory. An arbitrary-spin formalism is generally required in the classical limit. By matching the tree and one-loop amplitudes of four spinning particles with those of a suitably-chosen effective field theory, we obtain the spin1-spin2 terms of a two-body effective Hamiltonian through O(G^2) and valid to all orders in velocity. Solving Hamilton's equations yields the impulse and spin changes of the individual bodies. We write them in a surprisingly compact form as appropriate derivatives of the eikonal phase obtained from the amplitude. It seems likely this structure persists to higher orders. We also point out various double-copy relations for general spin.

Figures (11)

• Figure 1: The previously known results in PN and PM expansions of the (bilinear in spin) spin$_1$-spin$_2$ interactions in the two-body potential, are outlined in horizontal (green) and vertical (blue) direction respectively. The new results in this paper at ${\mathcal{O}}(G^2)$ and all orders in velocity are correspond to the shaded (red) region. Each horizontal row corresponds to the same order in $G$, or the PM expansion. The velocity expansion is indicated by $v^n$. Each vertical column corresponds to the same PN order for the spin$_1$-spin$_2$ interaction, where the leading order (LO), next-to-leading order (NLO), next-to-next leading order (NNLO), and the static part at $G^4$ are known up to quadratic in spins.
• Figure 2: The three vertex labels. All momenta are outgoing.
• Figure 3: The tree-level Feynman diagram containing the ${\mathcal{O}}(G)$ spin interactions. Because we are interested only in long range interactions contact terms where the graviton propagator cancel can be ignored.
• Figure 4: The two-particle cut needed for extracting classical dynamics. The blobs represent on-shell tree amplitudes and the exposed lines indicate that the propagators are replaced with on-shell conditions.
• Figure 5: The tree-level Feynman diagram for gravitational Compton scattering. For integer-spin electrodynamics diagram (d) is absent. Here the internal lines represent Feynman propagators.