# Revisiting the 2PM eikonal and the dynamics of binary black holes

## TL;DR

This paper analyzes the classical scattering of two black holes using the eikonal approximation in various spacetime dimensions, deriving the 2PM order scattering angle and discussing implications for higher-order calculations.

## Contribution

It provides a detailed derivation of the 2PM scattering angle from the eikonal approach, including subleading contributions and their implications for unitarity and future 3PM calculations.

## Key findings

- Derived the leading and subleading eikonal in general spacetime dimensions.
- Calculated the 2PM classical scattering angle between black holes.
- Identified a log-divergent contribution at subsubleading order affecting unitarity.

## Abstract

In this paper we study the two-body gravitational scattering of massive scalars with different masses in general spacetime dimensions. We focus on the Regge limit (eikonal regime) of the resulting scattering amplitudes and discuss how to extract the classical information representing the scattering of two black holes. We derive the leading eikonal and explicitly show the resummation of the first leading energy contribution up to second order in Newton's gravitational constant. We also calculate the subleading eikonal showing that in general spacetime dimensions it receives a non-trivial contribution from the box integral. From the eikonal we extract the two-body classical scattering angle between the two black holes up to the second post-Minkowskian order (2PM). Taking various probe-limits of the two-body scattering angles we are able to show agreement between our results and various results in the literature. We highlight that the box integral also has a log-divergent (in energy) contribution at subsubleading order which violates perturbative unitarity in the ultra-relativistic limit. We expect this term to play a role in the calculation of the eikonal at the 3PM order.