New class of Majoron-emitting double-$\beta$ decays

Motivated by the excess events that have recently been found near the end points of the double-$\beta$ decay spectra of several elements, we reexamine models in which double-$\beta$ decay can proceed through the neutrinoless emission of massless Nambu-Goldstone bosons (Majorons). Noting that models proposed to date for this process must fine-tune either a scalar mass or a VEV to be less than 10 keV, we introduce a new kind of Majoron which avoids this difficulty by carrying lepton number $L=-2\mathrm{}$. We analyze in detail the requirements that models of both the conventional and our new type must satisfy if they are to account for the observed excess events. We find (1) the electron sum-energy spectrum can be used to distinguish the two classes of models from one another, (2) the decay rate for the new models depends on different nuclear matrix elements than for ordinary Majorons, and (3) all models require a (pseudo) Dirac neutrino, having a mass of a several hundred MeV, which mixes with ${\nu }_e$.