Resumming Scattering Amplitudes for Waveforms

TL;DR

This work addresses the challenge of obtaining non-perturbative gravitational-wave observables from two-body scattering across arbitrary trajectories by formulating a projector-based framework that maps perturbative on-shell amplitudes to two effective potentials, $V_{ m PM}$ and $R_{ m PM}$, which govern conservative dynamics and graviton emission. By combining Feshbach projection, Born subtractions, and the WKB/classical limit, the authors construct non-perturbative wavefunctions and derive a resummed 5-point amplitude, then compute gravitational waveforms via the KMOC formalism, confirming that the waveform can be obtained by integrating the radiation potential along the classical trajectory. The paper provides explicit leading-order expressions for the radiation potential and a compact integral representation for the non-perturbative amplitude and waveform, applicable to generic bent worldline dynamics. This framework extends EFT-style matching of the gravitational two-body potential to radiative phenomena, enabling extraction of GW source terms directly from perturbative on-shell amplitudes and paving the way for all-orders resummation in the gravitational coupling and mass ratio, with potential connections to EOB and black-hole perturbation theory.

Abstract

We develop a formalism to compute non-perturbative 5-point scattering amplitudes and apply it to gravitational waveforms in the two-body problem for arbitrary trajectories. Drawing inspiration from Feshbach's projector formalism in nuclear physics, we introduce effective potentials governing graviton emission and relate them to perturbative scattering amplitudes at arbitrary order in the gravitational coupling and mass ratio. Once these potentials are determined, the corresponding non-perturbative amplitudes in the classical limit are obtained by iterative insertions and subsequently translated into gravitational waveforms using the KMOC formalism. As an application, we compute the gravitational waveform emitted by a two-body system moving along a generic, potentially highly bent, trajectory. Importantly, our formalism extends effective field theory matching of the gravitational two-body potential to radiative phenomena, enabling the extraction of gravitational-wave source terms directly from perturbative on-shell scattering amplitudes.

Figures (6)

• Figure 1: The problem of GWs can be split into three stages. In our framework, each is described by the post-Minkowskian two-body potential $V_{\rm PM}$, the radiation potential $R_{\rm PM}$, and the wavefunction of the outgoing graviton $\Psi^-_{\bm{k}}$. This paper focuses on the generation, with only brief discussions of propagation.
• Figure 2: Ladders
• Figure 3: DWBA
• Figure 5: Typical diagrams in effective potentials.
• Figure 6: Iterated topologies are resummed by using wavefunctions.