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^' 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 π-N cross sections in terms of Regge poles, to investigate the behavior of P and P^' trajectories near t≈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 α_P^'≈0.5.
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Keiji Igi (1963) studied this question.
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