π−NScattering with Inelastic Contributions

The $\frac{N}{D}$ formulation of the unitarity $S$ matrix, valid for reactions involving production processes, is applied to the reactions, $\pi +N\to \pi +N$, $\pi +N\to \pi +\pi +N$, $\pi +\pi +N\to \pi +\pi +N$, in the energy region of the higher resonances. A model is constructed in which the two pions in three-particle states interact in only one state, and the left-hand discontinuities of the elastic and three-particle scattering amplitudes are neglected while the left-hand discontinuities of the production amplitude are approximated by a pole. The elastic amplitude is driven through unitarity by the inelastic amplitude. The general properties of this model are investigated, and the procedure is applied to the $D$- and $F$-wave scattering of the $\pi -N$ system. The comparison between the experimental data and the model presented here is quite favorable.