Consistency Conditions on the Strong Interactions Implied by a Partially Conserved Axial-Vector Current

It is shown that a partially conserved $\ensuremath{\Delta}S=0$ axial-vector current (${\ensuremath{\partial}}_{\ensuremath{\lambda}}{{J}_{\ensuremath{\lambda}}}^{A}=C{\ensuremath{\phi}}_{\ensuremath{\pi}}$) implies consistency conditions involving the strong interactions alone. The most interesting of these is a relation among the symmetric isotopic-spin pion-nucleon scattering amplitude ${A}^{\ensuremath{\pi}N(+)}$, the pionic form factor of the nucleon ${K}^{\mathrm{NN}\ensuremath{\pi}}$, and the rationalized, renormalized pion-nucleon coupling constant ${g}_{r}$: $\frac{{{g}_{r}}^{2}}{M}=\frac{{A}^{\ensuremath{\pi}N(+)}(\ensuremath{\nu}=0, {\ensuremath{\nu}}_{B}=0, {k}^{2}=0)}{{K}^{\mathrm{NN}\ensuremath{\pi}}({k}^{2}=0)}.$ [$M$ is the nucleon mass and $\ensuremath{-}{k}^{2}$ the ${(\mathrm{mass})}^{2}$ of the initial pion. The final pion is on mass shell; the energy and momentum transfer variables $\ensuremath{\nu}$ and ${\ensuremath{\nu}}_{B}$ are defined in the text. By using experimental pion-nucleon scattering data, we find that this relation is satisfied to within 10%. Consistency conditions involving the $\ensuremath{\pi}\ensuremath{\pi}$ and the $\ensuremath{\pi}\ensuremath{\Lambda}$ scattering amplitudes are stated.