Hard-pion techniques are presented for calculating vertex functions involving $\pi$, $\rho$, $A_1$, and $\sigma$ mesons. The development of the method involves the following assumptions: (a) saturation of intermediate sums by single mesons, (b) chiral $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ algebra commutation relations, (c) conservation of vector current, and (d) partial conservation of axial-vector current.

• Received 10 June 1968

DOI:https://doi.org/10.1103/PhysRev.174.1999