Nucleon as a Composite Particle State

Abstract

A general scheme is presented for approximately determining the mass $M$ and coupling with pions $\frac{g^2}{4\pi }$ of the nucleon, and the position and width of the $\pi N 33$ resonance. No cutoffs are required. Contributions from distant left-hand singularities to the $I=J=\frac{1}{2},\frac{3}{2}$ $p$-wave amplitudes are approximated by Balázs-type pole terms, and those from nearby left-hand singularities, by crossing symmetry. Balázs-pole residues are determined by requiring crossing symmetry in a form relating the $\pi N$ partial-wave amplitude on the nearby left-hand cut to physical $\pi N$ scattering. As a first step in the investigation of our general scheme, we use experimental data on $\pi N$ scattering in the $I=J=\frac{3}{2}$ $p$-wave state to determine the nucleon mass and pionnucleon coupling constant. We find $M=1.03\mathrm{}$ in units of the physical-nucleon mass and $\frac{g^2}{4\pi }=18.6\mathrm{}$. The latter appears to be weakly dependent on the choice of Balázs-pole positions. The calculated nucleon mass, on the other hand, appears to be practically independent of the choice of pole positions.