Interplay between $\mu$-$\tau$ reflection symmetry, four-zero texture and universal texture

Abstract

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 $\nu_{L}$. By a bi-maximal transformation, all the mass matrices lead to $\mu-\tau$ reflection symmetric forms, which seperately satisfy $T_{u} \, m_{u,\nu}^{*} \, T_{u} = m_{u,\nu}$ and $T_{d} \, m_{d,e}^{*} \, T_{d} = m_{d,e}$. Reconciliation between the $\mu-\tau$ reflection symmetries and observed $\sin \theta_{13}$ predicts $\delta_{CP} \simeq 203^{\circ}$. Moreover, imposition of universal texture $(m_{f})_{11} = 0$ for $f=u,d,\nu,e$ predicts the normal hierarchy with the lightest neutrino mass $|m_{1}| = 6.26$ or $2.54$ meV.