Comparison of post-Minkowskian and self-force expansions: Scattering in a scalar charge toy model

We compare numerical self-force results and analytical fourth-order post-Minkowskian (PM) calculations for hyperbolic-type scattering of a pointlike particle carrying a scalar charge Q off a Schwarzschild black hole, showing a remarkably good agreement. Specifically, we numerically compute the scattering angle including the full O(Q2) scalar-field self-force term (but ignoring the gravitational self-force), and we compare with analytical expressions obtained in a PM framework using scattering-amplitude methods. This example provides a nontrivial, high-precision test of both calculation methods and illustrates the complementarity of the two approaches in the context of the program to provide high-precision models of gravitational two-body dynamics. Our PM calculation is carried out through 4PM order, i.e., including all terms through O(Q2G3). At the fourth post-Minkowskian order the point-particle description involves two a priori undetermined coefficients, due to contributions from tidal effects in the model under consideration. These coefficients are chosen to align the post-Minkowskian results with the self-force ones.