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

It is shown that a partially conserved $\Delta S=0\mathrm{}$ axial-vector current (${\partial }_{\lambda }{J_{\lambda }}^A=C{\varphi }_{\pi }$) 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 $A^{\pi N(+)\mathrm{}}$, the pionic form factor of the nucleon $K^{\mathrm{NN}\pi }$, and the rationalized, renormalized pion-nucleon coupling constant $g_r$: $\frac{{g_r}^2}{M}=\frac{A^{\pi N(+)\mathrm{}}(\nu =0, {\nu }_B=0, k^2=0)}{K^{\mathrm{NN}\pi }(k^2=0)}.$ [$M$ is the nucleon mass and $-k^2$ the ${\mathrm{}\left(\mathrm{mass}\right)\mathrm{}}^2$ of the initial pion. The final pion is on mass shell; the energy and momentum transfer variables $\nu$ and ${\nu }_B$ 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 $\pi \pi$ and the $\pi \Lambda$ scattering amplitudes are stated.