Black hole scattering near the transition to plunge: Self-force and resummation of post-Minkowskian theory

Geodesic scattering of a test particle off a Schwarzschild black hole can be parameterized by the speed-at-infinity v and the impact parameter b, with a "separatrix", b=bc(v), marking the threshold between scattering and plunge. Near the separatrix, the scattering angle diverges as ~log(b-bc). The self-force correction to the scattering angle (at fixed $v,b$) diverges even faster, like ~ A₁(v)bc/(b-bc). Here we numerically calculate the divergence coefficient A₁(v) in a scalar-charge toy model. We then use our knowledge of A₁(v) to inform a resummation of the post-Minkowskian expansion for the scattering angle, and demonstrate that the resummed series agrees remarkably well with numerical self-force results even in the strong-field regime. We propose that a similar resummation technique, applied to a mass particle subject to a gravitational self-force, can significantly enhance the utility and regime of validity of post-Minkowsian calculations for black-hole scattering.