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

TL;DR

This work tackles the problem of obtaining the classical gravitational two-body dynamics at 3PM order by deriving the full 3PM eikonal from the quantum amplitude in ${\cal N}=8$ supergravity and extending it to General Relativity. The authors systematically evaluate the classical limit of the two-loop elastic amplitude in the full soft region using IBP and differential equations with near-static boundary conditions, and they supplement this with a direct-unityarity analysis of three-particle cuts to fix the imaginary part and radiation effects. They provide a real-analytic, crossing-symmetric formulation of the amplitudes, revealing precise relationships between radiation-reaction contributions to the real part of the eikonal and the IR structure of the imaginary part, ensuring a universal ultrarelativistic limit consistent with analyticity and crossing. The results yield the complete conservative and dissipative 3PM eikonal for massive scattering in ${\cal N}=8$ and in GR, including waveforms and spectra for emitted massless fields, and establish a robust amplitude-based framework that can be extended to higher PM orders and to waveform reconstruction. Overall, the paper demonstrates that incorporating the full soft region is essential to recover correct high-energy behavior and radiation effects, providing a bridge between quantum scattering amplitudes and classical gravitational dynamics with practical implications for gravitational-wave phenomenology.

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.

Figures (7)

• Figure 1: Schematic representation of a $2\to2$ scattering process involving two objects with masses $m_1$ (blue) and $m_2$ (green).
• Figure 2: The box diagram.
• Figure 3: The double box diagram.
• Figure 4: The nonplanar box diagram.
• Figure 5: The H diagram.