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.
Highlights
The high energy limit of scattering amplitudes in gravitational theories has been thoroughly studied as a gedanken-experiment that provides a nontrivial test of the consistency of the gravitational theory
The D-dimensional case is slightly more intricate than the 4D one as we find that the contribution from the scalar box integral contributes to the exponentiation of the first post-Minkowskian (1PM) result, and yields nontrivial subleading terms that have to be combined with the triangle contributions to obtain the full 2PM eikonal
II and III; in Appendix A we evaluate the box and the triangle integrals in the limit s, m2i ≫ jtj, while in Appendix B we derive the deflection angle through a classical geodesic calculation in the background of a D-dimensional Schwarzschild black hole
Summary
The high energy limit of scattering amplitudes in gravitational theories has been thoroughly studied as a gedanken-experiment that provides a nontrivial test of the consistency of the gravitational theory. A tractable regime is the Regge limit, where both the energies and the impact parameter are large and unitarity is preserved due to a resummation of Feynman diagrams which reproduces the effect of a classical geometry [1,2,3,4] These early studies focused on the case of external massless states whose high energy Regge scattering matches the gravitational interaction of two well-separated shock waves. In the limit mentioned above the perturbative amplitude at a fixed order in GN is divergent creating tension with unitarity These divergent terms should exponentiate when resumming the leading contributions at a large energy at different orders in GN. II and III; in Appendix A we evaluate the box and the triangle integrals in the limit s, m2i ≫ jtj, while in Appendix B we derive the deflection angle through a classical geodesic calculation in the background of a D-dimensional Schwarzschild black hole
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