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

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