Simple model of pion-nucleon elastic scattering

A nonstatic model for elastic pion-nucleon scattering is developed using knowledge of the analytic structure of the scattering amplitude and partial wave $\frac{N}{D}$ dispersion techniques. In $S$, $D$, and $F$ partial wave channels the model is equivalent to a separable potential model which includes the coupling to inelastic channels. In $P$-wave channels the nucleon pole which arises from absorption and emission of the pion is included. A more general model which incorporates both a nucleon pole term and a separable potential term is developed for the $P_{11}$ channel. This model is capable of reproducing the change of sign which is present in the $P_{11}$ phase shifts. The necessary information concerning the coupling of the inelastic scattering to the elastic scattering is taken from the experimentally determined phase shifts as input. If simple kinematic factors are included explicitly, quite simple analytic functions for the form factors in the model are able to reproduce the experimental phase shifts for pion laboratory kinetic energies up to 1.2 GeV. The model produces off-shell behavior which is different from previous models. In particular, the inclusion of the nucleon pole alters substantially the range of the interaction in all $P$-wave channels.