A perturbation theory for the Coulomb phase infrared-divergence

We construct a perturbation theory which we conjecture to be free of the Coulomb-phase infrared divergence. This perturbation theory is developed for one of the simplest yet prototypical scattering amplitudes which would otherwise exhibit this divergence: the semiclassical scattering of a spinless boson on a background Coulomb field. The perturbation theory is based upon replacing plane waves with Coulomb wavefunctions, and the free-field propagator with the Coulomb propagator, in order to appropriately match the asymptotics of the exact in/out states. We compute the leading-order (LO) and next-to-leading-order (NLO) infrared-finite scattering amplitudes in this framework, which include all-order-in-the-coupling effects, and demonstrate that these amplitudes are in agreement with the known exact amplitude at these orders. We comment on the Runge-Lenz symmetry of the LO amplitude which occurs as a principal series representation of the Euclidean conformal group on the 2-sphere.