BackwardπNScattering: A Regge-Pole-Cut Model without Parity Doubling

We present a very simple model in which the amplitude for backward $\pi N$ scattering is described in terms of the exchange of the $N_{\alpha }$ and ${\Delta }_{\delta }$ Regge trajectories and associated Regge cuts. The trajectory functions $\alpha \mathrm{}\left(u\right)\mathrm{}$ and the positions of the branch points ${\alpha }_c\mathrm{}\left(u\right)\mathrm{}$ are linear functions of $u$, but there are no odd-parity resonances associated with the resonances on the $N_{\alpha }$ and ${\Delta }_{\delta }$ trajectories (no parity doubling). The model provides a good description of the data on backward ${\pi }^{\pm }p$ scattering, including the differential charge exchange cross section and the polarization observed in ${\pi }^{+}p$ scattering.