Second-order post-Minkowskian scattering in arbitrary 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.