Low-Energy Theorem for the Weak Axial-Vector Vertex

A low-energy theorem is derived for the weak axial-vector vertex. The theorem enables one to calculate from strong or electromagnetic processes the two leading terms in the expansion of the axial-vector vertex in powers of the leptonic four-momentum transfer. Applications to weak pion production, $K_{e4\mathrm{}}$ decay, and radiative $\mu$ capture are discussed. In particular, we express the radiative $\mu$-capture matrix element, up to and including contributions linear in the leptonic four-momentum transfer and the photon four-momentum, in terms of the elastic weak form factors and pion photoproduction amplitudes.