Pion-Nucleon Scattering in the Mandelstam Representation

Abstract

Using the analytic properties of partial wave scattering amplitudes, as derived from Mandelstam's representation, we have studied the $J=\frac{3}{2}$, $P$ state of the pion-nucleon system. The method used is covariant, it incorporates unitarity, and the effect of a possible pion-pion resonance has been investigated. Using the "single nucleon term" and the low-energy scattering properties of the "crossed states," we obtain a resonance in the $J=\frac{3}{2}$, $T=\frac{3}{2}$ pion-nucleon state without the aid of a cutoff. We have also investigated the scattering in the $T=\frac{1}{2}$ state. The pion-pion resonance appears to have only a very small effect in the $T=\frac{3}{2}$ state whereas in the $T=\frac{1}{2}$ state it increases the phase shift by a factor of 2.The resonance obtained in the $T=\frac{3}{2}$ state occurs at too low an energy. There are several factors which may account for this: We have not been able to include fully the contributions from crossed states, and we have not systematically included inelastic scattering.