  Motivated by the excess events that have recently been found near the
endpoints of the double beta decay spectra of several elements, we re-examine
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 $L=-2$. 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
$\nu_e$.
