Composite Particle Model for the Nucleon and the (3,3) Resonance

Partial-wave dispersion relations are used with interaction terms arising from the exchange of a nucleon, the $N^{*}$ and the $\rho$ meson to produce integral equations for the partial-wave amplitudes for pion-nucleon scattering. The solutions of these equations depend on a single arbitrary parameter, the energy at which the dispersion integrals are "cutoff." It is shown that when the value of the cutoff is adjusted to produce the $N^{*}$ at the correct energy, a bound state, the nucleon, appears in the $p$ wave, $I=J=\frac{1}{2}$ amplitude. Thus, the nucleon mass, the pion-nucleon coupling constant, the width of the $N^{*}$, and mass of the $N^{*}$ are all determined by the cutoff parameter. It is shown that when the $N^{*}$ mass has the experimental value the other quantities are in reasonable agreement with experiment. The effect of variations in the coupling constants controlling the interaction terms is also studied. In each case all of the $J\le \frac{3}{2}$ partial-wave amplitudes are calculated and compared with experiment.