Solutions of the Static Theory Integral Equations for Pion-Nucleon Scattering in the One-Meson Approximation

Abstract

Numerical solutions of the one-meson approximation of the Low equations for elastic pion-nucleon scattering in the fixed-nucleon, extended-source theory are obtained with a Gaussian cutoff function. The validity of the method requires that the scattering amplitudes have no zeros in the complex plane other than at z = plus or minus 1. The functions obtained for the (3,3) and (1,1) states satisfy the Low equations within the accuracy of the method, but the (1,3) and (3,1) states are only approximately given. This difficulty with the (1,3) and (3,1) states is correlated with the development of a zero in the corresponding scattering amplitude well before physically interesting values of the parameters (coupling constant and cutoff) are reached. A best fit to the (3,3)-state data up to 170-Mev pion laboratory energy requires a coupling constant, f/sup 2/, less than 0.08. The solution is consistent with the (3,1)- state data, but gives a (1,1)-state phase shift of larger magnitude than experiment appears to permit. It is found that the cutoff function does not prevent strong interactions at very high energies. Their occurrence appears to be a property of the static model. The contributions of such interactions to various static-theory calculations is briefiy discussed.more » It is shown that in the p-wave part of the relativistic dispersion relations, where there is no cutoff function, if use is made of the low-energy data, then the very high-energy contributions of the static theory are replaced by recoil terms of order mu /M. (auth)« less