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Cobimaximal mixing predicts $\pi/4$ and $3\pi/2$ for the atmospheric angle and the Dirac CP-violating phase, respectively. These values are in tension with the neutrino globals fits. If this pattern was behind the lepton mixings, then it would have to be broken. In that case, in this paper, we explore the $\mathbf{S}_{3}$ flavor symmetry within the $B-L$ gauge model where the aforementioned scheme comes from the neutrino sector but the charged lepton contribution breaks the well known predictions so the mixing observables as well as the $m_{ee}$ mass can be accommodated quite well according to the available data. Notably, the predicted regions for the Dirac CP-violating phase would allow us to test the model in future experiments., Comment: 16 pages, 7 figures. Minor corrections, references added and the conclusions are unchanged. Version to be published in Revista Mexicana de F\'isica

We propose extended 3HDM and 4HDM models where the SM gauge symmetry is enlarged by the spontaneously broken $S_4$ group, the preserved $Z_2$ and broken $Z_4$ cyclic symmetries. Each model has three active $SU(2)$ scalar doublets, in addition, the first one has an extra inert scalar singlet, whereas the second model has an inert scalar doublet. Furthermore, each model has several extra gauge singlet scalars, which are triplets under $S_4$. Both models yield the same structure of the mass matrices for the fermion sector, where a radiative seesaw generates the tiny light active neutrinos masses at one-loop level, through the $S_4$ triplets. The presence of flavor changing neutral currents mediated by heavy scalars allowed us to study the $(K^0-\overline{K}^0)$ and $(B_{d,s}^0-\overline{B}_{d,s}^0)$ meson mixings, in the parameter space that currently satisfies the experimental constraints. On the other hand, due to the preserved $Z_2$ symmetry, our proposed models have stable scalar and fermionic dark matter candidates. Furthermore, these models are consistent with the current pattern of SM fermion masses and mixings, with the measured dark matter relic abundance and successfully accommodate the constraints arising from meson oscillations and oblique parameters. The extra scalars in our models provide radiative corrections to the oblique parameters, where due to the presence of the scalar inert doublet, renders the 4HDM less restrictive than the 3HDM one., Comment: 34 pages, 19 figures

The $\mu \leftrightarrow \tau$ symmetry has been ruled out by its predictions on the reactor and atmospheric angles, nevertheless, a breaking of this symmetry might provide correct values. For that reason, we build a non-renormalizable lepton model where the mixings arise from the spontaneous breaking of the $\mathbf{S}_{4}\otimes \mathbf{Z}_{2}$ discrete group, subsequently the $\mu \leftrightarrow \tau$ symmetry is broken in the effective neutrino mass matrix, that comes from the type II see-saw mechanism. As main result, the reactor and atmospheric angles are corrected and their values are in good agreement with the experimental data for the inverted hierarchy. Furthermore, we point out a link between the atmospheric angle and reactor one. In the quark sector, under certain assumptions, the generalized Fritzsch textures shape to the quark mass matrices so that the CKM matrix values are guaranteed., Comment: 19 pages, 6 figures. Minor changes, accepted in Revista Mexicana de F\'isica

The lepton sector is studied within a flavored non-renormalizable model where the $\mathbf{S}_{3}\otimes \mathbf{Z}_{2}$ flavor symmetry drives the Yukawa couplings. In this framework, the effective neutrino mass, that comes from the type II see-saw mechanism, as well as the charged lepton mass matrices are hierarchical and possessed (under a benchmark in the charged sector) a kind of Fritzsch textures that achieve to accommodate the mixing angles in good agreement with the last experimental data. The model predicts a CP-violating phase consistent with the experimental data and the branching ratio for the lepton flavor violation process, $\mu\rightarrow e\gamma$, is well below the current bound., Comment: Extended discussion and we add one additional figure. arXiv admin note: substantial text overlap with arXiv:2010.15370

We propose a predictive $Q_4$ flavored 2HDM model, where the scalar sector is enlarged by the inclusion of several gauge singlet scalars and the fermion sector by the inclusion of right handed Majorana neutrinos. In our model, the $Q_4$ family symmetry is supplemented by several auxiliary cyclic symmetries, whose spontaneous breaking produces the observed pattern of SM charged fermion masses and quark mixing angles. The light active neutrino masses are generated from an inverse seesaw mechanism at one loop level thanks to a remnant preserved $Z_2$ symmetry. Our model succesfully reproduces the measured dark matter relic abundance and is consistent with direct detection constraints for masses of the DM candidate around $\sim$ 6.3 TeV. Furthermore, our model is also consistent with the lepton and baryon asymmetries of the Universe as well as with the muon anomalous magnetic moment, Comment: Substantial improvement to the overall content of the paper specially in sections regarding dark matter, analysis of fermion and scalar sectors. Detailed explanation of the possible UV completion of the model as well as about the muon electric dipole moment. Discussions enlarged, new figures, references and appendices added. 35 pages. Matches published version in EPJP

We construct a multiscalar and non-renormalizable model where the $\mathbf{S}_{3}$ flavor symmetry drives mainly the Yukawa couplings. In the quark sector, the Nearest Neighbor Interaction (NNI) textures are behind the CKM mixing matrix so that this is fitted in good agreement with the last available results. In the lepton sector, an almost diagonal charged lepton mass matrix and extended Fritzsch mass textures in the effective neutrino mass matrix, that comes from the type II see-saw mechanism, provide consistent values for the mixing angles where the normal hierarchy is favored. The model predicts a CP-violating phase consistent with the experimental data and the BR for the lepton flavor violation process, $\mu\rightarrow e\gamma$, is well below the current bound., Comment: Some references added

From mass textures point of view, we present a comparative study of the $\mathbf{S}_{3}$ flavor symmetry in the left-right symmetry model (LRSM) and the baryon minus lepton model (BLM) taking into account their predictions on the CKM mixing matrix. To do this, we recover the already studied quark mass matrix, that comes from some published papers, and under certain strong assumption, one can show that there are predictive scenarios in the LRSM and BLM where the modified Fritzsch and nearest neighbor interaction (NNI) textures drive respectively the quark mixings. As main result, the CKM mixing matrix is in good agreement with the last experimental data in the flavored BLM model., Comment: 15 pages, 8 figures. Minor chances, references and some comments added

We propose a multiscalar singlet extension of the singlet-triplet Higgs model capable of explaining the SM fermion mass spectrum and mixing parameters. Our model is based on the $\Delta \left( 27\right) $ family symmetry, supplemented by cyclic symmetries, which are spontaneously broken thus yielding the observed hierarchy of the SM charged fermion masses and quark mixing angles. The masses of the light active neutrinos are produced by type-II seesaw mechanism mediated by the neutral component of the $SU(2)_{L}$ scalar triplet. The model symmetries lead to the extended Gatto-Sartori-Tonin relations between the quark masses and mixing angles., Comment: Revised version. Discussion enlarged, text improved. References added. 19 pages, 2 figures

A complete study on the fermion masses and flavor mixing is presented in a non-minimal left-right symmetric model (NMLRMS) where the ${\bf S}_{3}\otimes {\bf Z}_{2}\otimes {\bf Z}^{e}_{2}$ flavor symmetry drives the Yukawa couplings. In the quark sector, the mass matrices possess a kind of the generalized Fritzsch textures that allow us to fit the CKM mixing matrix in good agreement to the last experimental data. In the lepton sector, on the other hand, a soft breaking of the $\mu\leftrightarrow \tau$ symmetry provides a non zero and non maximal reactor and atmospheric angles, respectively. The inverted and degenerate hierarchy are favored in the model where a set of free parameters is found to be consistent with the current neutrino data., Comment: 23 pages, 9 figures. references added, typos corrected, JCGI added his current institution;conclusions rewrote and unchanged results. Version published in European Physical Journal C

We make a scalar extension of the B-L gauge model where the ${\bf S}_{3}$ 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., Comment: 20 pages, 6 figures. Improved version and scalar discussion extended and references added. Version published in EPJC

In the simplest version of a $\mathbf{Q}_{6}$ flavored supersymmetric model, we analyze the leptonic masses and mixings in the framework of a soft breaking of the $\mu \leftrightarrow \tau$ symmetry. This breaking is controlled by the inequality $m_{e\tau}\neq m_{e\mu}$ in the effective neutrino mass. As a consequence of this breaking, the reactor and atmospheric angle are deviate from $0^{\circ}$ and $45^{\circ}$, respectively. Such deviations can be enhanced or suppressed by the CP parities in the Majorana phases, so that an analytic study is carried out to remark their importance to constrain the free parameters that accommodate the mixing angles. The normal hierarchy is completely discarded in this model, the inverted hierarchy is less favored than the degenerate one where the reactor and atmospheric angles are in good agreement with the experimental data. Additionally, the model predicts defined regions for the effective neutrino mass decay, the neutrino mass scale and the sum of the neutrino mass in the inverted and degenerate mass spectrum. Thus, this model may be testable by future experiments that focus in neutrinoless double beta decay., Comment: 21 pages, 20 figures. Improved version, new figures and references added. New section included. Conclusions unchanged. Version accepted for publication in International Journal of Modern Physics A

We propose a non-minimal left-right symmetric model (LRSM) with Parity Symmetry where the fermion mixings arise as result of imposing an ${\bf S}_{3}\otimes {\bf Z}_{2}$ flavor symmetry, and an extra ${\bf Z}^{e}_{2}$ symmetry is considered to suppress some Yukawa couplings in the lepton sector. As a consequence, the effective neutrino mass matrix possesses approximately the $\mu-\tau$ symmetry. The breaking of the $\mu-\tau$ symmetry induces sizable non zero $\theta_{13}$, and the deviation of $\theta_{23}$ from $45^{\circ}$ is strongly controlled by an $\epsilon$ free parameter and the complex neutrino masses. Then, an analytic study on the extreme Majorana phases is done since these turn out to be relevant to enhance or suppress the reactor and atmospheric angle. So that we have constrained the parameter space for the $\epsilon$ parameter and the lightest neutrino mass that accommodate the mixing angles. The highlighted results are: a) the normal hierarchy is ruled out since the reactor angle comes out being tiny, for any values of the Majorana phases; b) for the inverted hierarchy there is one combination in the extreme phases where the values of the reactor and atmospheric angles are compatible up to $2, 3~\sigma$ of C. L., but the parameter space is tight; c) the model favors the degenerate ordering for one combination in the extreme Majorana phases. In this case, the reactor and atmospheric angle are compatible with the experimental data for a large set of values of the free parameters. Therefore, this model may be testable by the future result that the Nova and KamLAND-Zen collaborations will provide., Comment: Version three. I added references and corrected typos. 19 pages. All figures replaced, section II changed. Analytic study improved. Conclusions unchanged

We analyze the leptonic sector in the left-right symmetric model dressed with a $(Z_{2})^{3}$ discrete symmetry which realizes, after weak spontaneous breaking, a small broken $\mu\lra\tau$ symmetry that is suggested to explain observable neutrino oscillation data. $\mu\lra\tau$ symmetry is broken at tree level in the effective neutrino mass matrix due to the mass difference $\widetilde{m}_{\tau}\neq \widetilde{m}_{\mu}$ in the diagonal Dirac mass terms, whereas all lepton mixings arise from a Majorana mass matrix. In the limit of a small breaking we determined $\theta_{13}$, and the deviation from the maximal value of $\theta_{ATM}$, in terms of the light neutrino hierarchy scale, $m_{3}$, and a single free parameter $h_{s}$ of the model., Comment: 9 pages, 4 figures. Title changed and we have added one new section: CP phase contribution. Minor corrections and references updated. Published version

Neutrino oscillations data indicates that neutrino mixings are consistent with an apparent $\nu_\mu - \nu_\tau$ exchange symmetry in neutrino mass matrix. We observe that in the mininimally extended standard model with the see-saw mechanism, one can impose $\mu\leftrightarrow\tau$ symmetry at the tree level on all Lagrangian terms, but for the mass difference among $\mu$ and $\tau$ leptons. In the absence of any new extra physics, this mass difference becomes the only source for the breaking of such a symmetry, which induces, via radiative corrections, small but predictable values for $\theta_{13}$, and for the deviation of $\theta_{ATM}$ from maximallity. In the CP conserving case, the predictions only depend on neutrino mass hierarchy and may provide a unique way to test for new physics with neutrino experiments., Comment: Typos corrected. References and short discussions added. Final version, 6 pages, 2 figures