Realistic fermion mass and mixing in $$\mathbf {U(1)_L}$$ model with $$\mathbf {A_4}$$ flavor symmetry for majorana neutrino

Abstract

We propose a \(U(1)_L\) model based on \(A_4\) symmetry aiming to explain the smallness of neutrino masses as well as the quark and lepton mixing patterns. The smallness of Majorana neutrino mass is reproduced through the combination of type-I and -II seesaw mechanisms. The model can accommodate the current observed patterns of lepton and quark mixing in which the solar neutrino mixing angle and the Dirac CP violating phase are in \(3\sigma \) range for both normal hierarchy and inverted hierarchy, the Majorana violating phases are predicted to be \(\eta _{1} \in (0.00, 9.76)^\circ \) and \(\eta _{2} \in (68.70, 270.00)^\circ \) for normal hierarchy while \(\eta _1\in (0.00, 10.83)^\circ \) and \(\eta _{2} \in (78.60, 90.00)^\circ \) for inverted hierarchy. The obtained sum of neutrino mass and the effective Majorana neutrino mass are in good consistent with the recent limits. For quark sector, all the quark masses can get the best-fit values and all the elements of the quark mixing matrix are in agreement with the experimental constraints except the element \( (V_{{{\text{CKM}}}} )_{{21}} \) with a deviation about \(0.25\,\%\).