Meson-Nucleon Scattering in the Tamm-Dancoff Approximation

An attempt is made to calculate meson-nucleon scattering by using charge-symmetric pseudoscalar meson theory with pseudoscalar coupling and without the use of perturbation theory. The Tamm-Dancoff formalism is used, with all states which are not directly coupled to the one-meson, one-nucleon state omitted. In this approximation an inhomogeneous integral equation in three-dimensional momentum space is derived for $g\left(\mathrm{p}\right)$, the probability amplitude for relative momentum p between meson and nucleon. This equation is reduced to a separate one-dimensional integral equation for each of the six angular momentum and isotopic spin ($T$) states, $S_{\frac{1}{2}}$, $P_{\frac{1}{2}}$, and $P_{\frac{3}{2}}$ for $T=\frac{1}{2}, \frac{3}{2}$. The phase shifts for these states are given by the value of $g\left(p\right)\mathrm{}$ on the energy-shell.

All self-energy and renormalization terms are omitted and the present method is inapplicable to the two states of total angular momentum and $T$ equal to ½. The integral equations are solved by semi-numerical methods for the $S_{\frac{1}{2}}$ and $P_{\frac{3}{2}}$ states with $T=\frac{3}{2}$, obtaining the shape of $g\left(p\right)\mathrm{}$ and the variation of phase shift with energy and coupling constant. It is shown that only for the $P_{\frac{3}{2}}$, $T=\frac{3}{2}$ (corresponding to an attractive potential) is the phase shift much larger than in Born approximation and depends strongly on the coupling constant and on energy. For the other, repulsive, states the phase shift is less than in Born approximation. For energies of about 150 Mev or more and for a large enough coupling constant, the $P_{\frac{3}{2}}$, $T=\frac{3}{2}$ phase shift is larger than any of the others. This is in rough qualitative (but by no means quantitative) agreement with experiment.