Comparison of post-Minkowskian and self-force expansions: Scattering in a scalar charge toy model
Leor Barack, Zvi Bern, Enrico Herrmann, Oliver Long, Julio Parra-Martinez, Radu Roiban, Michael Ruf, Chia-Hsien Shen, Mikhail P. Solon, Fei Teng, Mao Zeng
Abstract
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 $\mathcal{O}({Q}^{2})$ 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 $\mathcal{O}({Q}^{2}{G}^{3})$. 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.