We revisit the problem of plane monochromatic waves impinging upon a Schwarzschild black hole from complex angular momentum techniques. We focus more particularly on the differential scattering cross sections associated with scalar and electromagnetic waves. We derive an exact representation of the corresponding scattering amplitudes by replacing the discrete sum over integer values of the angular momentum which defines their partial wave expansions by a background integral in the complex angular momentum plane plus a sum over the Regge poles of the $S$-matrix involving the associated residues. We show that, surprisingly, the background integral is numerically negligible for intermediate and high reduced frequencies (i.e., in the short-wavelength regime) and, as a consequence, that the cross sections can be reconstructed, for arbitrary scattering angles, in terms of Regge poles with very good agreement. We show in particular that, for large values of the scattering angle, a small number of Regge poles permits us to describe the black hole glory and that, by increasing the number of Regge poles, we can reconstruct very efficiently the differential scattering cross sections for small and intermediate scattering angles and therefore describe the orbiting oscillations. In fact, in this wavelength regime, the sum over Regge poles allows us to extract by resummation the physical information encoded in the partial wave expansion defining a scattering amplitude and, moreover, to overcome the difficulties linked to its lack of convergence due to the long-range nature of the fields propagating on the black hole. As a consequence, from asymptotic expressions for the lowest Regge poles and the associated residues based on the correspondence Regge poles, “surface waves” propagating close to the photon sphere, we can provide an analytical approximation describing with very good agreement both the black hole glory and a large part of the orbiting oscillations. We finally discuss the role of the background integral for low reduced frequencies (i.e., in the long-wavelength regime).

• Received 13 January 2019

DOI:https://doi.org/10.1103/PhysRevD.99.104079