B-L Model with ${\mathbf{S}}_3$ Symmetry: Nearest Neighbor Interaction Textures and Broken $\mu \leftrightarrow \tau$ Symmetry

We make a scalar extension of the B-L gauge model where the ${\mathbf{S}}_3$ non-abelian discrete group drives mainly the Yukawa sector. Motived by the large and small hierarchies among the quark and active neutrino masses respectively, the quark and lepton families are not treated on the same footing under the assignment of the discrete group. As a consequence, the Nearest Neighbor Interactions (NNI) textures appear in the quark sector, leading to the CKM mixing matrix, whereas in the lepton sector, a soft breaking of the $\mu \leftrightarrow \tau$ symmetry in the effective neutrino mass that comes from type I see-saw mechanism, provides a non-maximal atmospheric angle and a non-zero reactor angle.