Fourth Order Corrections to Meson-Nucleon Scattering in Pseudoscalar Meson Theory

The low-energy $S$- and $P$-wave phase shifts for meson-nucleon scattering in the relativistic pseudoscalar coupling theory are calculated in perturbation theory to fourth order in the coupling constant. A comparison is made of the complete relativistic theory and the nonrelativistic approximation to it, obtained by use of the Foldy-Wouthuysen transformation, via a comparison of the low-energy scattering predicted by both theories. It is concluded that the $S$-wave scattering is similar in the two theories but that the $P$-wave scattering is very different in the two theories in fourth order. An analysis of the sources of the differences between the two theories is made by calculating the fourth order scattering in a version of the relativistic theory in which all effects due to the production of pairs are eliminated.