On the K μ3 and K e3 decay modes

Abstract

On the basis of the usual local current-current coupling between the lepton fields and the boson fields some general expressions are obtained for the muon (electron) and pion spectra and the muon polarization in an arbitrary direction taken as a function of the muon energy. The derivation of the muon spectrum and polarization involves a phase space integration over the pion momenta, on which, however, the two unknown form factors f ± ( p π p K) describing the effect of strong interactions are dependent. By means of a covariant integration technique originally due to Berman † the phase space integral is recast into a suns of products of factors depending on the muon and kaon momenta alone and very simple integrals over the form factors. These integrals depend on the muon and kaon momenta only through their upper and lower limits. In the CMS of the decaying kaon the factors depend solely on E 1, the muon or electron energy. The general expressions are applied to two special examples, viz. constant form factors and form factors derived by MacDowell by means of a dispersion technique. The results for these cases are very much alike.