Neutrino decay and spontaneous violation of lepton number

The orders of magnitude of decay rates for relatively light neutrinos are studied in the framework of the SU(2)×U(1) gauge group. The assumption is made that a hierarchy parameter $\epsilon \left(\approx \right(\mathrm{muon}\mathrm{mass})÷[\mathrm{some}\mathrm{new}\mathrm{mass}\mathrm{scale}\left(\mathrm{possibly}\mathrm{much}\mathrm{smaller}\mathrm{than}\mathrm{the}\mathrm{grand}\mathrm{unification}\mathrm{scale}\right)\mathrm{}\left]\right)$ plays a meaningful role in the full theory. For orientation it is first noted that the traditional $\nu \gamma$ decay channel as well as the $3\nu$ decay channel give neutrino lifetimes which for "typical" parameters are substantially longer than the age of the universe. Then we examine in detail some recent proposals which are claimed to result in greatly speeded-up decays into $\nu$+Majoron, where the Majoron is a true Goldstone boson associated with spontaneous breakdown of lepton number. In a theory in which the usual Higgs doublet is augmented by a complex singlet (1-2 model) it is noted that the decay width into $\nu$+Majoron actually vanishes to order ${\epsilon }^5$. In a theory where the doublet is augmented by a complex triplet (2-3 model) this decay is shown to vanish exactly, neglecting radiative corrections. A more general Majoron theory (1-2-3 model) is constructed and shown to also yield a vanishing tree-level decay rate for $\nu$+Majoron decay to order ${\epsilon }^5$. Finally, the tree amplitudes in the 1-2 and 1-2-3 models are shown to give decay widths for $\nu$+Majoron of order ${\epsilon }^9$ which correspond to lifetimes much greater than the age of the universe.