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 ${P}^{\ensuremath{'}}$ 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 $\ensuremath{\pi}\ensuremath{-}N$ cross sections in terms of Regge poles, to investigate the behavior of $P$ and ${P}^{\ensuremath{'}}$ trajectories near $t\ensuremath{\approx}0$. 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 ${\ensuremath{\alpha}}_{{P}^{\ensuremath{'}}}\ensuremath{\approx}0.5$.
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