Neutrino masses generation in a $Z_4$ model

We present a renormalizable flavor model with $Z_4$ as flavor symmetry in both the quark and lepton sectors. The model is constructed with a minimal approach and no right-handed neutrinos are introduced. In this approach a minimum number of two SU(2) Higgs doublets and one scalar singlet are required in order to obtain the Nearest Neighbor Interaction form for charged fermions and to generate neutrino masses radiatively. For the quark sector we follow the charge assignations made by Branco et al. in reference [G. C. Branco, D. Emmanuel-Costa, and C. Simoes, Phys. Lett. B 690, 62 (2010).]. All fermion masses and mixing angles in the model are in agreement with current experimental data and only the inverted hierarchy for the neutrino mass spectrum is allowed. Since neutrinos are Majorana the contribution to neutrinoless double beta decay is also analyzed.

Figure 1

Diagram for radiative corrections to neutrino masses ($i$, $j=1$, 2, $i\ne j$, and $m_{dc}$ are the matrix elements of $M_l$).

Figure 2

Angles that repeat the experimental mass difference ratio for the neutrino sector (Inverted hierarchy).

Figure 3

Points in the $m_{\beta \beta }-m_3$ plane corresponding to the consistent range of mixing angles in the model.

Figure 4

Allowed range for $|m_{\beta \beta }|$ from oscillation data (region between gray lines), with the small spot being the points in Fig. 3. The dashed line represents the smallest (most optimistic) upper limit from the absence of a direct observation of $0\nu \beta \beta$ decay in ${}^{76}\mathrm{Ge}$ ($|m_{\beta \beta }|<(0.20–0.32)\mathrm{eV}$) [26].