Effect of Unstable Particle Production on Scattering Amplitudes

The model for pion-nucleon scattering proposed by Ball, Frazer, and Nauenberg is solved numerically. This model yields unitary scattering amplitudes while including the effects of $\rho$-meson production and the associated anomalous thresholds. The behavior of the solutions as a function of the width of the $\rho$ meson and the pion-nucleon coupling constant is investigated. Cusps as well as broad peaks are generated in the elastic-scattering amplitude by the $\rho$-meson production. The effects of the complex singularities are investigated in detail by comparison with another treatment of pion-nucleon scattering in which anomalous cuts are absent. Also, the existence of poles on unphysical sheets is studied by means of a simplified model.