Locality of the weak interaction of a lepton pair and heavy lepton decays

Pionie decays of a polarized $\tau$ lepton $\tau\to\nu_\tau+n\pi$ in which one or two pions are detected are investigated in general form. Particular attention is devoted to the consequences of locality of the weak interaction of leptons. The specific form of the hadronic weak current is unimportant, but use is made of the circumstance that the production of an even number of pions is governed by the conserved vector weak current and the production of an odd number of pions by only the axial part of the weak current. It is shown that the asymmetry of decays with the formation of an even number of pions, integrated over all the pions (for fixed value of their total $4$-momentum), depends only on the mass of the neutrino $\nu_\tau$ and not on the structure of the hadronic current. It follows from the requirement of locality that the probability of decays with the detection of one or two pions must be a quadratic function of their energies.