Pion-Nucleon Vertex Functions

Abstract

A quantitative study of the pion-nucleon vertex function with one nucleon off the mass shell is presented using dispersion-theoretical techniques. The discussion is based on the unitarity conditions for the vertex function and for the one-nucleon irreducible parts of the pion-nucleon scattering amplitudes with $I=J=\frac{1}{2}$. It is shown in terms of the $\frac{N}{D}$ method that the vertex function has a sharp maximum in the low-energy region. Its existence is found to be crucial for a certain inequality, which is due to the requirement of no ghosts in the theory, to be satisfied. The nucleon propagator is strongly suppressed at the energy of this maximum. Reasonable results obtained seem to suggest that ghosts are not present in pion physics, although the present calculation is still incomplete. The pion-nucleon vertex function with pion off the mass shell is also discussed and some early attempts are re-examined. The importance of a pole or a resonance-like behavior in this vertex is suggested.