Relativistic generalizations of simple pion-nucleon models

Abstract

A relativistic, partial wave $\frac{N}{D}$ dispersion theory is developed for low energy pion-nucleon elastic scattering. The theory is simplified by treating crossing symmetry only to lowest order in the inverse nucleon mass. The coupling of elastic scattering to inelastic channels is included by taking the necessary inelasticity from experimental data. Three models are examined: pseudoscalar coupling of pions and nucleons, pseudovector coupling, and a model in which all intermediate antinucleons are projected out of the amplitude. The phase shifts in the dominant ${P}_{33}$ channel are quantitatively reproduced for ${P}_{\mathrm{lab}}\ensuremath{\le}1.2$ GeV/c with a pion-nucleon vertex of range 1110 MeV/c. We find that there are large (not of the order of the inverse nucleon mass) kinematic corrections to Chew-Low models, and that the Chew-Low model is successful because a reduction in the pion-nucleon cutoff provides a remarkable compensation for the large kinematic corrections. The intermediate antinucleon states are found to provide a significant fraction of the interaction in both $S$ and $P$ waves, and the model which explicitly removes them is incompatible with the ${P}_{33}$ phase shifts. Thus a model of the pion-nucleon interaction which does not include antinucleon degrees of freedom is found to be unphysical.NUCLEAR REACTIONS Pion-nucleon scattering, kinematic corrections, role of intermediate antinucleon states.