General theory of weak processes involving neutrinos. II. Pure leptonic decays

Abstract

We present a general theory of leptonic decays which consistently incorporates possibility of nonzero neutrino masses and associated lepton mixing. We calculate differential decay distribution $\frac{{d}^{2}\ensuremath{\Gamma}}{d{E}_{b}d}$ $cos\ensuremath{\theta}$ and ${l}_{b}$ polarization for the decay ${l}_{a}\ensuremath{\rightarrow}{\ensuremath{\nu}}_{\mathrm{la}}{l}_{b}{\overline{\ensuremath{\nu}}}_{\mathrm{lb}}$, i.e., in general incoherent sum of decays ${l}_{a}\ensuremath{\rightarrow}{\ensuremath{\nu}}_{i}{l}_{b}{\overline{\ensuremath{\nu}}}_{j}$ into all allowed mass eigenstates ${\ensuremath{\nu}}_{i}$ and ${\overline{\ensuremath{\nu}}}_{j}$. Expressions are also given for average quantities $〈cos\ensuremath{\theta}〉({E}_{b})$, $〈cos\ensuremath{\theta}〉$ (integrated over ${E}_{b}$), $〈{E}_{b}〉(\ensuremath{\theta})$, and $〈{E}_{b}〉$ for individual ($i$,$j$) decay modes and observed sum. The total rate for massive-neutrino modes is calculated for relevant experimental cases. These results are applied to analyze what observable characteristics of massive neutrinos and lepton mixing would be in leptonic ${l}_{a}$ decay. We show that $\frac{d\ensuremath{\Gamma}}{d{E}_{b}}$ would in general contain kinks at intermediate energies and carry out a search for these in existing $\ensuremath{\mu}$ and $\ensuremath{\tau}$ decay data. We further show that conventional determination of Lorentz structure of weak leptonic couplings via measurement of spectral parameters $\ensuremath{\rho}$, $\ensuremath{\eta}$, $\ensuremath{\xi}$, and $\ensuremath{\delta}$ is not applicable in presence of massive neutrinos and lepton mixing; a deviation of observed parameters (with radiative corrections extracted) from their conventional $V\ensuremath{-}A$ values could be caused either by non-($V\ensuremath{-}A$) Lorentz structure or by massive-neutrino decay modes and lepton mixing. Thus, past measurements of spectral parameters yield information only on combined effects of underlying Lorentz structure of couplings and on possible neutrino masses and mixing, but not on either of these in isolation. The appropriate generalized formalism for analysis of Lorentz structure in leptonic decays is given, and a quantitative study is performed of effects of neutrino masses and mixing on spectral parameters. We propose methods to distinguish between these effects and those due to possible non-($V\ensuremath{-}A$) Lorentz structure; these methods can be applied in a reanalysis of old $\ensuremath{\mu}$ and $\ensuremath{\tau}$ decay data, and can serve as part of a generalized framework for analysis of forthcoming data. Within context of standard electroweak theory we apply our results to existing data to obtain new upper bounds on possible contributions of massive neutrino modes. Finally, we determine optimal ways in which, and corresponding sensitivity with which, forthcoming experiments on $\ensuremath{\mu}$ and $\ensuremath{\tau}$ decay can search for massive neutrinos and lepton mixing.