Decay of the Charged Pion

The amplitude for pion decay is related to quantities involved in the meson propagator under the strong $\pi$ coupling alone. For pseudovector $\pi$ coupling the Goldberger and Treiman result follows under certain assumptions, but without an expansion in intermediate states. For pseudoscalar $\pi$ coupling, similar, but somewhat inconsistent, results are obtainable by means of an equivalence relation applied to the weak interaction. An expansion in intermediate states of the absorptive part of the decay amplitude is related to a similar expansion for the meson propagator and in an approximation yields the Goldberger and Treiman result when states containing one nucleon pair and some number of pions are included.