Fermion Regge-Pole Model for the Structure of Pion-Nucleon Elastic Scattering in the Backward Hemisphere

The $Y=+1\mathrm{}$ fermion resonances that show up strongly in total-cross-section data are classified as Regge recurrences on three straight-line trajectories (namely, ${\Delta }_{\delta }$, $N_{\gamma }$, $N_{\alpha }$) in a Chew-Frautschi plot. From extrapolations of the trajectories, resonance doublets are predicted in the vicinity of 2200 MeV (with $J^P={\frac{7}{2}}^{-} \mathrm{and} {\frac{9}{2}}^{+}$) and 2630 MeV (with $J^P=\frac{11}{2^{-}} \mathrm{and} \frac{13}{2^{+}}$), due to recurrences of the $N_{\gamma }$ and $N_{\alpha }$ trajectories at similar mass values. A model is constructed for ${\pi }^{-}p$ elastic scattering near the backward direction based on interference of the direct-channel resonance amplitude (${\Delta }_{\delta }$, ${\Delta }_{\gamma }$, $N_{\alpha }$) with the amplitude due to fermion Regge exchange (${\Delta }_{\delta }$) in the crossed channel. The predictions of the model compare favorably with existing data on the energy dependence of the ${\pi }^{-}p$ differential cross section at 180° center-of-mass scattering angle and the general shape of the ${\pi }^{-}p$ angular distributions near 180°. The results confirm the consistency of the Regge-recurrence parity assignments with the scattering data. The resonance elasticities used in the calculations are roughly the same as the elasticities determined from total-cross-section data. The model is extended to ${\pi }^{+}p$ elastic scattering at backward angles. In the ${\pi }^{+}p$ process, the direct-channel ${\Delta }_{\delta }$ resonance contribution alone saturates the experimental differential cross section in the backward cone at momenta below 4 $\frac{\mathrm{BeV}}{c}$. Comparison with the ${\pi }^{+}p$ backward-scattering data gives additional confirmation for the proposed ${\Delta }_{\delta }$ Regge-recurrence parity assignments. In addition, the model supports the existence of an $I=\frac{3}{2} s$-wave resonance at 1690 MeV. Finally, the polarization is predicted for ${\pi }^{\pm }p$ elastic scattering in the backward cone.