Variational Calculation of Relativistic Meson-Nucleon Scattering In Zeroth Order

Abstract

We extend the polaron variational treatment previously developed for the propagator to the case where one nucleon and $n$ external mesons are present. Using the particle representation of the scalar Wick-Cutkosky model this is done in lowest order of an expansion of the exact action around a retarded quadratic trial action. In particular, we evaluate the form factor for scattering of mesons from the scalar nucleon and determine the radius of the dressed particle. After analytic continuation to Minkowski space we study elastic meson-nucleon scattering both analytically and numerically near threshold and show that it is essential to incorporate the correct behaviour of the retardation function at large proper times. Only if this is done the optical theorem is approximately fulfilled over a range of energies and coupling constants.