Dispersion Relations for Pion-Nucleon Scattering: No-Spin-Flip Amplitude

Abstract

It is shown that the principle of microscopic causality is sufficient and in general also necessary for the existence of dispersion relations for the derivatives of the no-spin-flip amplitude with respect to $cos\ensuremath{\vartheta}$, taken at zero angle. Physical dispersion relations are derived for these quantities, neglecting the nucleon recoil.The dispersion formulas for the first derivative reduce to the no-spin-flip parts of Low's $P$-wave equations, if one neglects in addition to the recoil all inelastic processes and contributions from partial waves with $l>1$. In combination with corresponding approximate relations, obtained from the exact dispersion formulas for the forward scattering amplitude, one can derive integral equations for $S$-waves only. The inhomogeneous terms of these equations contain the zero-energy scattering lengths.Finally, it is shown that in general one cannot derive normal dispersion relations for the amplitudes corresponding to individual angular momenta.