  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", $b=b_c(v)$, marking the threshold between scattering and plunge.
Near the separatrix, the scattering angle diverges as $\sim\log(b-b_c)$. The
self-force correction to the scattering angle (at fixed $v,b$) diverges even
faster, like $\sim A_1(v)b_c/(b-b_c)$. Here we numerically calculate the
divergence coefficient $A_1(v)$ in a scalar-charge toy model. We then use our
knowledge of $A_1(v)$ 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-Minkowskian calculations for black-hole
scattering.
