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

It is shown that a partially conserved ΔS=0 axial-vector current (∂_λJ_λA=Cφ_π) 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^πN(+), the pionic form factor of the nucleon K^NNπ, and the rationalized, renormalized pion-nucleon coupling constant gᵣ: gᵣ²M=A^πN(+)(ν=0, νB=0, k²=0)K^NNπ(k²=0). [M is the nucleon mass and -k² the (mass)² of the initial pion. The final pion is on mass shell; the energy and momentum transfer variables ν and νB 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 ππ and the πΛ scattering amplitudes are stated.