Partially Conserved Axial-Vector Current Restrictions on Pion Photoproduction and Electroproduction Amplitudes

We discuss numerically the restrictions imposed by the partially conserved axial-vector current (PCAC) on the pion photoproduction amplitude ${V_1}^{\mathrm{}(+)\mathrm{}}\mathrm{}\left(0\right)\mathrm{}$ and on the pion electroproduction amplitude ${V_6}^{\mathrm{}(-)\mathrm{}}\mathrm{}\left(0\right)\mathrm{}$. We find that the magnetic-dipole dominance and the narrow-resonance approximations are unreliable. The nonresonant $s$ waves make an important contribution to ${V_1}^{\mathrm{}(+)\mathrm{}}\mathrm{}\left(0\right)\mathrm{}$, and we find that the PCAC prediction for this amplitude is reasonably well satisfied. The electric and longitudinal multipoles appear to make a much bigger contribution to ${V_6}^{\mathrm{}(-)\mathrm{}}\mathrm{}\left(0\right)\mathrm{}$ than does the magnetic dipole $M_{1+}$, which is strongly suppressed by the kinematics.