Inelastic Effects of theN*andρon Pion-Nucleon Scattering

A formalism is developed for calculating the pion-nucleon scattering amplitude which conveniently includes the inelastic production of a (3,3) isobar, and is extended to include $\rho$-meson production. The exact unitarity relations are found to contain nondiagonal terms which, when simplified, are interpreted as the overlap between resonances. Calculations based on the $\frac{N}{D}$ method are performed in which the unphysical discontinuities are evaluated from single-nucleon or pion-exchange diagrams. The numerical solution of the resulting integral equations is compared to simpler approximations and to experiment. The importance of both the isobar and the $\rho$ is pointed out. For the $D_{\frac{3}{2}}$ channel, a resonance was found for $T=\frac{1}{2}$ but not for $T=\frac{3}{2}$. No resonance was found for either of the $F_{\frac{5}{2}}$ or $D_{\frac{5}{2}}$ channels.