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

Arnau Koemans Collado, Paolo Di Vecchia, Rodolfo Russo

TL;DR

This work analyzes 2→2 gravitational scattering of massive scalars in general spacetime dimensions within the Regge/eikonal framework to extract classical two-body black hole dynamics at 2PM. It derives the leading (1PM) and first subleading (2PM) eikonals by resumming energy-growing contributions from one- and two-graviton exchange diagrams, including both box and triangle topologies, and verifies results against known geodesic limits in multiple probe regimes. The study reveals a rich D-dimensional structure, with the box integral contributing both to exponentiation and to genuine subleading eikonal corrections, and it identifies a log-divergent subsubleading term suggesting quantum eikonal effects that may influence 3PM, while demonstrating a smooth massless limit. The results bridge amplitude-based approaches with classical GR predictions and raise questions about 3PM consistency, pointing to potential extensions via bootstrap methods and higher-loop analyses.

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.

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

TL;DR

This work analyzes 2→2 gravitational scattering of massive scalars in general spacetime dimensions within the Regge/eikonal framework to extract classical two-body black hole dynamics at 2PM. It derives the leading (1PM) and first subleading (2PM) eikonals by resumming energy-growing contributions from one- and two-graviton exchange diagrams, including both box and triangle topologies, and verifies results against known geodesic limits in multiple probe regimes. The study reveals a rich D-dimensional structure, with the box integral contributing both to exponentiation and to genuine subleading eikonal corrections, and it identifies a log-divergent subsubleading term suggesting quantum eikonal effects that may influence 3PM, while demonstrating a smooth massless limit. The results bridge amplitude-based approaches with classical GR predictions and raise questions about 3PM consistency, pointing to potential extensions via bootstrap methods and higher-loop analyses.

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.

Paper Structure

This paper contains 15 sections, 122 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Massive Scalar Scattering
  3. One Graviton Exchange
  4. Two Graviton Exchanges
  5. Box contribution
  6. Triangle contribution
  7. The Eikonal and Two-Body Deflection Angles
  8. Various Probe Limits in Arbitrary $D$
  9. Various Probe Limits in $D=4$
  10. Discussion
  11. High-Energy Expansion of Various Integrals
  12. Scalar Box Integral
  13. Triangle Integrals
  14. Auxiliary Integrals
  15. Geodesics in a $D$-dimensional Schwarzschild Background

Figures (3)

  • Figure 1: A figure illustrating the procedure outlined at the beginning of section \ref{['sec:amplitude']} and described by equation \ref{['eq:amptree']} for the tree-level amplitude. The solid lines represent massive scalars and the wavy lines represent gravitons. The shaded blob is described by equation \ref{['eq:amp3pt']}.
  • Figure 2: A figure illustrating the procedure outlined at the beginning of section \ref{['sec:amplitude']} and described by equation \ref{['eq:ampmaster']} for the one-loop amplitude. The solid lines represent massive scalars and the wavy lines represent gravitons. The shaded blob is described by equation \ref{['eq:amp4pt']}.
  • Figure 3: The two topologies of integrals that contribute to the two graviton exchange amplitude in the classical limit. In \ref{['fig:amp1']} we have the box topology and in \ref{['fig:amp2']} we have the triangle topology. The integral structure in \ref{['fig:amp2']} receives contributions from various Feynman diagrams, including those with a three-point vertex in the bulk. We can ignore other integral structures, such as bubble and tadpoles, since they do not contribute in the classical limit.