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In this paper, we consider the diagonal reflection symmetries and three-zero texture in the SM. The three-zero texture has two less assumptions ($(M_{u})_{11} , (M_{ u})_{11} eq 0$) than the universal four-zero texture for mass matrices $(M_{f})_{11} = (M_{f})_{13,31} = 0$ for $f = u,d, u, e$. The texture and symmetries reproduce the CKM and MNS matrices with accuracies of $O(10^{-4})$ and $O(10^{-3})$. By assuming a $d$-$e$ unified relation ($M_{d} sim M_{e}$), this system predicts the normal hierarchy, the Dirac phase $delta_{CP} simeq 202^{circ},$ the Majorana phases $alpha_{12} = 11.3^{circ}, alpha_{13} = 6.90^{circ}$ up to $pi$, and the lightest neutrino mass $m_{1} simeq 2.97,-,4.72,$[meV]. The effective mass of the double beta decay $|m_{ee}|$ is found to be $1.24 sim 1.77 ,$[meV].
In this paper, we consider a set of new symmetries in the SM, {it diagonal reflection} symmetries $R , m_{u, u}^{*} , R = m_{u, u}, ~ m_{d,e}^{*} = m_{d,e}$ with $R =$ diag $(-1,1,1)$. These generalized $CP$ symmetries predict the Majorana phases to
In this paper, we impose a magic symmetry on the neutrino mass matrix $M_{ u}$ with universal four-zero texture and diagonal reflection symmetries. Due to the magic symmetry, the MNS matrix has trimaximal mixing inevitably. Since the lepton sector ha
Using the seesaw mechanism, we construct a model for the light-neutrino Majorana mass matrix which yields trimaximal lepton mixing together with maximal CP violation and maximal atmospheric-neutrino mixing. We demonstrate that, in our model, the ligh
In this letter, we consider exact $mu-tau$ reflection symmetries for quarks and leptons. Fermion mass matrices are assumed to be four-zero textures for charged fermions $f = u,d,e$ and a symmetric matrix for neutrinos $ u_{L}$. By a bi-maximal transf
CP-odd invariants, independent of basis and valid for any choice of CP transformation are a powerful tool in the study of CP. They are particularly convenient to study the CP properties of models with family symmetries. After interpreting the consequ