Soft Pion Emission Induced by Electromagnetic and Weak Interactions

We consider some extensions of the relation which has been obtained by Nambu and Lurié between an elastic process and the accompanying soft pion emission process under the assumption of approximate ${\gamma }_5$ invariance. In particular, a generalized formula is found which enables one to describe the electropion production $e+N\to N+e+\pi$ and the neutrino-pion production $\nu +N\to N+e+\pi$ at the threshold of $N\pi$ system in terms of the vector (Hofstadter) and the axial-vector form factors. Explicit forms of the cross sections are given.