Second-order Post-Minkowskian scattering in arbitrary dimensions

TL;DR

This work extends the gravitational scattering program to arbitrary spacetime dimensions by deriving the long-range two-body potential at 2PM from elastic amplitudes using both the Lippmann–Schwinger framework and EFT matching. A key novel feature in $D>4$ is a nonzero classical contribution from box diagrams and a quadratic-in-tree-amplitude term in the $p^2(r,G_N)$ relation, which nevertheless preserves a linear dependence of the deflection angle on the classical amplitude at 2PM, in agreement with eikonal results. The analysis emphasizes unitarity as the underlying reason for eikonal exponentiation and the reality of the potential, and provides explicit closed-form deflection-angle expressions for dimensions $4$ through $10$. Together with cross-checks against the eikonal approach and multiple derivations (LS, EFT, and Damour-type Hamiltonians), the paper furnishes a coherent, dimensionally general picture of classical GR observables from quantum amplitudes, with practical formulas for the scattering angle in several dimensions.

Abstract

We extract the long-range gravitational potential between two scalar particles with arbitrary masses from the two-to-two elastic scattering amplitude at 2nd Post-Minkowskian order in arbitrary dimensions. In contrast to the four-dimensional case, in higher dimensions the classical potential receives contributions from box topologies. Moreover, the kinematical relation between momentum and position on the classical trajectory contains a new term which is quadratic in the tree-level amplitude. A precise interplay between this new relation and the formula for the scattering angle ensures that the latter is still linear in the classical part of the scattering amplitude, to this order, matching an earlier calculation in the eikonal approach. We point out that both the eikonal exponentiation and the reality of the potential to 2nd post-Minkowskian order can be seen as a consequence of unitarity. We finally present closed-form expressions for the scattering angle given by leading-order gravitational potentials for dimensions ranging from four to ten.