Scattering of massive scalars by Schwarzschild black holes

The Klein-Gordon equation for the wave function of a single massive scalar is written in spherical and parabolic coordinates in the presence of a Schwarzschild background, and some semi-classical techniques for deriving asymptotic results are considered. In addition to the well-known logarithmic phase shift due to the tortoise radial coordinate, it is found that there is a mass-dependent logarithmic phase shift at infinity. This is found to be necessary to match the Newtonian cross-section in the non-relativistic limit. Finally, by imposing suitable boundary conditions near the horizon, the phase shift is calculated.