On a model with two zeros in the neutrino mass matrix

We consider a Majorana neutrino mass matrix $\mathcal{M}_\nu$ with $(\mathcal{M}_\nu)_{\mu\mu} = (\mathcal{M}_\nu)_{\tau\tau} = 0$, in the basis where the charged-lepton mass matrix is diagonal. We show that this pattern for the lepton mass matrices can be enforced by extending the Standard Model with three scalar SU(2) triplets and by using a horizontal symmetry group $\mathbbm{Z}_4$. The Ma--Sarkar (type-II seesaw) mechanism leads to very small vacuum expectation values for the triplets, thus explaining the smallness of the neutrino masses; at the same time, that mechanism renders the physical scalars originating in the triplets very heavy. We show that the conditions $(\mathcal{M}_\nu)_{\mu\mu} = (\mathcal{M}_\nu)_{\tau\tau} = 0$ allow both for a normal neutrino mass spectrum and for an inverted one. In the first case, the neutrino masses must be larger than $0.1 {eV}$ and the atmospheric mixing angle $\theta_{23}$ must be practically equal to $45^\circ$. In the second case, the product $\sin{\theta_{13}} | \tan{2 \theta_{23}} |$ must be of order one or larger, thus correlating the large or maximal atmospheric neutrino mixing with the smallness of the mixing angle $\theta_{13}$.