Ward-Takahashi identities for the radiative axial-vector vertex and low-energy theorem for the pion electroproduction.

Abstract

Using the assumption of a partially conserved axial-vector current and the conservation law for the electromagnetic current, we rederive a low-energy theorem for the pion electroproduction. In contrast to the traditional approach, we include all effects of off-shell nucleons and pions. We parametrize the axial-vector current in its most general form and apply the minimal substitution prescription to obtain the radiative axial-vector vertex that is required for gauge invariance. We split the full radiative axial-vector vertex into the isolated-pole contribution, the minimal-coupling interaction, and the remainder. The Ward-Takahashi identities are translated into the constraints on the pion electroproduction amplitude. The rigorous low-energy theorem is obtained for the zero four-momentum pion electroproduction. It is found that the off-shell matrix elements of the axial-vector current affect the isolated-pole term and the gauge term so that the low-energy theorem is at variance with the standard one given in the literature.