Interplay between $μ$-$τ$ reflection symmetry, four-zero texture and universal texture

Abstract

In this letter, we consider exact $μ-τ$ 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 $ν_{L}$. By a bi-maximal transformation, all the mass matrices lead to $μ-τ$ reflection symmetric forms, which seperately satisfy $T_{u} \, m_{u,ν}^{*} \, T_{u} = m_{u,ν}$ and $T_{d} \, m_{d,e}^{*} \, T_{d} = m_{d,e}$. Reconciliation between the $μ-τ$ reflection symmetries and observed $\sin θ_{13}$ predicts $δ_{CP} \simeq 203^{\circ}$. Moreover, imposition of universal texture $(m_{f})_{11} = 0$ for $f=u,d,ν,e$ predicts the normal hierarchy with the lightest neutrino mass $|m_{1}| = 6.26$ or $2.54$ meV.