Relativistic eikonal expansion

Abstract

The generalized ladder series of Feynman diagrams for scattering of two particles by scalar-meson exchange is expanded, using functional methods, to obtain the relativistic eikonal approximation and the next two terms of an expansion about the eikonal limit. The established similarity between nonrelativistic and relativistic eikonal approximations is shown to persist, in part, to the higher-order terms in the relativistic eikonal expansion. The leading-order correction to the eikonal limit differs only kinematically from its nonrelativist $c$ counterpart. In second order, there is again much similarity with nonrelativistic results; however, a part of the second-order eikonal correction explicitly depends on the relative time coordinate of the scattering particles. An approximate relativistic Schr\"odinger equation is found to reproduce the leading corrections to the eikonal limit by means of a simple kinematic generalization of the nonrelativistic potential theory results; however, the relativistic time effect cannot be readily incorporated into a three-dimensional wave equation.