Two Vacuum Poles and Pion-Nucleon Scattering

Abstract

A general expression is given for the pion-nucleon non-charge-exchange scattering amplitude for arbitrary energy and small momentum transfer on the assumption that only the vacuum pole $P$ and the second vacuum pole $P^{\prime }$ exist in the upper half $J$ plane. We derive sum rules for non-spin-flip and spin-flip amplitudes and use them, combined with the analysis of the high-energy $\pi -N$ cross sections in terms of Regge poles, to investigate the behavior of $P$ and $P^{\prime }$ trajectories near $t\approx 0$. For this purpose the importance of a precise measurement of the low-energy partial-wave phase shifts is emphasized. A sum rule for the $S$-wave pion-nucleon non-charge-exchange scattering length can be satisfied with ${\alpha }_{P^{\prime }}\approx 0.5$.