Neutrino masses in the economical 3-3-1 model

We show that, in frameworks of the economical 3-3-1 model, the suitable pattern of neutrino masses arises from the three quite different sources - the lepton-number conserving, the spontaneous lepton-number breaking and the explicit lepton-number violating, widely ranging over the mass scales including the GUT one: $u\sim O(1) \mathrm{GeV}$, $v\approx 246 \mathrm{GeV}$, $\om\sim O(1) \mathrm{TeV}$ and $\mathcal{M}\sim \mathcal{O}(10^{16}) \mathrm{GeV}$. At the tree-level, the model contains three Dirac neutrinos: one massless, two large with degenerate masses in the order of the electron mass. At the one-loop level, the left-handed and right-handed neutrinos obtain Majorana masses $M_{L,R}$ in orders of $10^{-2}-10^{-3} \mathrm{eV}$ and degenerate in $M_R=-M_L$, while the Dirac masses get a large reduction down to $\mathrm{eV}$ scale through a finite mass renormalization. In this model, the contributions of new physics are strongly signified, the degenerations in the masses and the last hierarchy between the Majorana and Dirac masses can be completely removed by heavy particles. All the neutrinos get mass and can fit the data.