  If the only contribution to neutrinoless double beta decay (NLDBD) are
neutrino masses its discovery may be very difficult, especially if neutrinos
present a normal hierarchy spectrum. However, this is not the only possibility;
NLDBD can in general produce electrons of either chirality, in contrast with
the neutrino induced NLDBD which predicts two left-handed electrons. Using an
effective Lagrangian approach we classify the lepton number violating (LNV)
effective operators with two leptons of either chirality but no quarks, ordered
according to the magnitude of their contribution to NLDBD. We point out that,
for each of the three chirality assignments, $e_{L}e_{L},e_{L}e_{R}$ and
$e_{R}e_{R}$, there is only one LNV operator of the corresponding type to
lowest order, and these have dimensions 5, 7 and 9, respectively. Neutrino
masses are always induced by these extra operators but can be delayed to one
(dimension 7) or two loops (dimension 9). Under the assumption that NLDBD is
dominated by the operators of dimension 7 or 9 we find that the scale of new
physics should be relatively low ($\lesssim 30$ TeV). We also list the SM
additions generating these operators upon integration of the heavy modes, and
discuss simple realistic examples of renormalizable theories for each case. The
phenomenology of a model giving rise to the dimension 9 operator has been
analyzed with some detail: if NLDBD is going to be seen in the next round of
experiments, the doubly charged scalars of the model could be seen at the LHC
and lepton flavour violating (LFV) rates could be at the reach of foreseen
experiments. Moreover neutrino masses, which arise at two loops, display a very
particular structure and are strongly constrained, in fact,
$\sin^{2}\theta_{13}\gtrsim0.008$, when $\mu\rightarrow eee$ is required to lie
below its present experimental limit.
