Pion-NucleonP-Wave Scattering Amplitudes and TheirS-Matrix Calculations

Abstract

The effective-range formulas for the $P$-wave scattering amplitudes, $I=\frac{1}{2}$ and $I=\frac{3}{2}$, of pions on nucleons are derived for the low-energy region using the requirements of analyticity, unitarity, and crossing symmetry and a sum rule. The $S$-matrix calculations are carried out by the generalized one-channel Chew-Mandelstam method in which the denominator function $D$ may, generally, contain a Castillejo-Dalitz-Dyson (CDD) pole, while the function $N$ is considered in the three-pole effective-range approximation. The effective-range parameters are then determined by the usual procedure in which the input values are the $\ensuremath{\rho}$-meson and (3,3)-isobar ${N}^{*}$ discontinuities and the high-energy contributions in the crossed channels, and the nucleon pole in the direct channel. The amplitudes ${P}_{13}$ and ${P}_{33}$, derived in this way as functions of the CDD-pole term, are then compared with experiment by means of the calculated and experimental values of the scattering lengths ${a}_{13}$ and ${a}_{33}$. It is shown, as one of the main results, that the low-energy ${P}_{13}$ amplitude does not contain any CDD pole and produces a scattering length which is in excellent agreement with the experimental value, while the ${P}_{33}$ amplitude necessarily contains a nonzero pole term. These results are in agreement with the conclusions made on the basis of a quite different approach-the "crankshaft analysis" - in the papers by Atkinson et al.