  The self-force expansion allows the study of deviations from geodesic motion
due to the emission of radiation and its consequent back-reaction. We
investigate this scheme within the on-shell framework of semiclassical
scattering amplitudes for particles emitting photons or gravitons on a static,
spherically symmetric background. We first present the exact scalar 2-point
amplitudes for Coulomb and Schwarzschild, from which one can extract classical
observables such as the change in momentum due to geodesic motion. We then
present, for the first time, the 3-point semiclassical amplitudes for a scalar
emitting a photon in Coulomb and a graviton on linearised Schwarzschild,
outlining how the latter calculation can be generalized to the fully non-linear
Schwarzschild metric. Our results are proper resummations of perturbative
amplitudes in vacuum but, notably, are expressed in terms of Hamilton's
principal function for the backgrounds, rather than the radial action.
