The Eikonal Approach to Gravitational Scattering and Radiation at $\mathcal O(G^3)$

TL;DR

This paper derives the 3PM gravitational eikonal in supergravity and Einstein gravity, connecting non-relativistic and ultra-relativistic regimes, including radiation effects and providing a framework for waveforms and spectra of emitted radiation.

Contribution

It introduces a method to compute the 3PM eikonal including radiation-reaction effects using unitarity and classical amplitude limits, unifying supergravity and Einstein gravity approaches.

Findings

Derived the 3PM eikonal including radiation effects.

Connected non-relativistic and ultra-relativistic regimes smoothly.

Provided a framework for computing gravitational waveforms and spectra.

Abstract

Using $\mathcal N=8$ supergravity as a theoretical laboratory, we extract the 3PM gravitational eikonal for two colliding massive scalars from the classical limit of the corresponding elastic two-loop amplitude. We employ the eikonal phase to obtain the physical deflection angle and to show how its non-relativistic (NR) and ultra-relativistic (UR) regimes are smoothly connected. Such a smooth interpolation rests on keeping contributions to the loop integrals originating from the full soft region, rather than restricting it to its potential sub-region. This task is efficiently carried out by using the method of differential equations with complete near-static boundary conditions. In contrast to the potential-region result, the physical deflection angle includes radiation-reaction contributions that are essential for recovering the finite and universal UR limit implied by general analyticity and crossing arguments. We finally discuss the real emission of massless states, which accounts for the imaginary part of the 3PM eikonal and for the dissipation of energy-momentum. Adopting a direct approach based on unitarity and on the classical limit of the inelastic tree-level amplitude, we are able to treat $\mathcal N=8$ and General Relativity on the same footing, and to complete the conservative 3PM eikonal in Einstein's gravity by the addition of the radiation-reaction contribution. We also show how this approach can be used to compute waveforms, as well as the differential and integrated spectra, for the different radiated massless fields.