Dispersion Relation for the Axial-Vector Vertex and a Sum Rule for the Axial-Vector Coupling-Constant Renormalization

Abstract

A dispersion relation for the nucleon axial-vector vertex, in which the subtraction term at infinite nucleon mass is given by the unrenormalized vertex, is derived from the axial current-field commutation relations. Using the hypothesis of partially conserved axial-vector current, a sum rule for the axial-vector coupling-constant renormalization $g_A$ is obtained in terms of the $\pi -N$ form factor. In the elastic-unitarity approximation, the sum rule involves the integral over the difference of the $P_{11}$ and $S_{11}$ phase shifts. Certain experimental phase shifts up to 1-GeV pion kinetic energy give $g_A\approx 1.1$. However, because of the lack of consistent experimental data beyond 500 MeV, and also because of the lack of reliable theory to estimate the important contribution from inelastic channels, all we can conclude at the moment is that the sum rule is not inconsistent with the experimental data.