Pion-nucleon form factor in the Chew-Low theory

Abstract

We find a solution to the static Chew-Low theory of pion-nucleon scattering, avoiding the "one-meson approximation." Our basic equation is crossing symmetric and may be solved for phase shifts $\delta \mathrm{}\left(p\right)\mathrm{}$ by standard numerical techniques, upon specifying a form factor $\nu \mathrm{}\left(p\right)\mathrm{}$ and a set of inelasticities. With $\nu \mathrm{}\left(p\right)=\exp \left(\frac{-p^2}{30}\right)$ we reproduce experimental $\delta \mathrm{}\left(p\right)\mathrm{}$ for $p_L\le 1.2 \frac{\mathrm{GeV}}{c}$ in the (3,3) state; in the (1,3) states and (3,1) states $\delta \mathrm{}\left(p\right)\mathrm{}$ compare well on the average but in the (1,1) state $\delta \mathrm{}\left(p\right)\mathrm{}$ have opposite signs. We show the importance of crossing symmetry and the coupling to inelastic channels, and we discuss the possibility of determining $\nu \mathrm{}\left(p\right)\mathrm{}$ directly from elastic scattering by an inverse scattering formula.