Diagonal reflection symmetries, four-zero texture, and trimaximal mixing with predicted $\theta_{13}$ in an $A_{4}$ symmeric model

In this paper, we impose a magic symmetry on the neutrino mass matrix $m_{\nu}$ 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 has only six free parameters, physical observables of leptons are all determined from the charged leptons masses $m_{ei}$, the neutrino mass differences $\Delta m_{i1}^{2}$, and the mixing angle $\theta_{23}$. This scheme predicts $\sin \theta_{13} = 0.149$, that is almost equal to the latest best fit, as a function of the lepton masses $m_{e, \mu}$ and the mass differences $\Delta m_{i1}^{2}$. Moreover, even if the mass matrix has perturbations that break the magic symmetry, the prediction of $\sin \theta_{13}$ is retained with good accuracy for the four-zero texture with diagonal reflection symmetries.