Generalized Potential for the Pion-Nucleon System

Abstract

The problem of obtaining the generalized potential in the physical region for a given partial wave from the experimentally known phase shifts for that channel is examined. We consider the $\pi N$ system, where the phase shifts are now well known up to reasonably high energies in various partial waves, and compute the generalized potentials for the $P_{33}$, $P_{11}$, and $D_{13}$ partial waves. These generalized potentials obtained from the experimental phase shifts turn out to be rather insensitive to the asymptotic behavior of the phase shifts. We compare our results with generalized potentials obtained by the elementary-particle exchanges and also with those obtained by imposing the correct Regge asymptotic behavior. The single-particle exhange potential is in good agreement with the "experimental" value for the $P_{33}$ channel, while for the $P_{11}$ and $D_{13}$ channels the single-particle exchange potentials do not reproduce the potentials computed from the phase shifts.