Almost exact diagonal reflection symmetries and three-zero texture

Abstract

In this paper, we consider a three-zero texture with diagonal reflection symmetries in the SM. The three-zero texture has two less assumptions ($(M_{u})_{11} , (M_{\nu})_{11} \neq 0$) than the universal four-zero texture of mass matrices $(M_{f})_{11} = (M_{f})_{13,31} = 0$ for $f = u,d,\nu, e$. The texture allows diagonal reflection symmetries to be almost exact and $d$\,-\,$e$ unification. They reproduce the CKM and MNS matrices with accuracies of $O(10^{-4})$ and $O(10^{-3})$. Some perturbative diagonalizations yield two relations with good accuracy for quark mass and mixing. Although this calculation is done in a special basis, it is a general result in a sense, because other textures and generalized $CP$ symmetries exist by some weak basis transformation. By assuming a $d$\,-\,$e$ unified relation ($M_{d} \sim M_{e}$), we obtain the lightest neutrino mass $m_{1} \simeq 2.12\, - \, 5.64\,$[meV] and the effective mass of the double beta decay $|m_{ee}| \simeq 1.23 - 2.15 \,$[meV].