Leading Nonlinear Tidal Effects and Scattering Amplitudes

TL;DR

This work develops a unified framework to compute leading post-Minkowskian tidal corrections for two-body gravitating systems by encoding finite-size effects as higher-dimension curvature operators in a point-particle EFT. Using scattering amplitudes, generalized unitarity, and position-space Fourier techniques, it derives the leading PM contributions to the conservative Hamiltonian $H(oldsymbol p,oldsymbol r)$ and the eikonal phase $oldsymbol\delta(oldsymbol p,oldsymbol b)$ from infinitely many tidal operators built from the Riemann/Weyl tensors, including linear and nonlinear tides and scenarios with spin. It establishes explicit results for curvature-squared ($E^2$, $B^2$) operators with arbitrary derivatives and extends to general multipoles $E^{(l)}$, $B^{(l)}$ and to nonlinear tides, while showing double-copy relations at leading PM order and applicability to effective-field-theory extensions of GR (e.g., $R^n$ terms). The paper therefore provides closed-form amplitudes, potentials, and eikonal phases for broad operator classes, enabling precise modeling of tidal effects in gravitational-wave physics and tests of fundamental gravity theories. Significantly, the results illuminate the structural simplicity of tidal interactions in the classical limit and pave the way for systematic higher-PM and EFT-extensions analyses in the two-body problem.

Abstract

We present the two-body Hamiltonian and associated eikonal phase, to leading post-Minkowskian order, for infinitely many tidal deformations described by operators with arbitrary powers of the curvature tensor. Scattering amplitudes in momentum and position space provide systematic complementary approaches. For the tidal operators quadratic in curvature, which describe the linear response to an external gravitational field, we work out the leading post-Minkowskian contributions using a basis of operators with arbitrary numbers of derivatives which are in one-to-one correspondence with the worldline multipole operators. Explicit examples are used to show that the same techniques apply to both bodies interacting tidally with a spinning particle, for which we find the leading contributions from quadratic in curvature tidal operators with an arbitrary number of derivatives, and to effective field theory extensions of general relativity. We also note that the leading post-Minkowskian order contributions from higher-dimension operators manifest double-copy relations. Finally, we comment on the structure of higher-order corrections.

Figures (8)

• Figure 1: The generalized cut for leading-order contributions to $E^2$- or $B^2$-type tidal operators. Each blob is an on-shell amplitude, which in this case is local. Each exposed line is taken to be on shell and the blobs represent tree amplitudes. The dark blob contains an insertion of an $E^2$- or $B^2$-type higher-dimension operator with an arbitrary number of additional derivatives. The external momenta are all outgoing and the arrows indicated the direction of graviton momenta.
• Figure 2: The generalized cut for leading order contributions to nonlinear tidal operators. Each blob is simply a (local) on-shell amplitude. The dark blob contains the $X^n$ tidal operator. The direction of graviton momentum flow is indicated by the arrows.
• Figure 3: The $L$-loop fan integral.
• Figure 4: The generalized cuts that need to be evaluated at next to leading order for an $R^n$ type tidal operator.
• Figure 5: Sample diagrams for next-to-leading-order contributions for the $R^3$ tidal operators which are simple to evaluate.