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.

中文翻译：

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精确 μ − τ 反射对称性、四零纹理和通用纹理之间的相互作用

在这封信中，我们考虑夸克和轻子的精确 $\mu-\tau$ 反射对称性。假设费米子质量矩阵是带电费米子 $f = u,d,e$ 的四零纹理和中微子 $\nu_{L}$ 的对称矩阵。通过双极大值变换，所有质量矩阵导致$\mu-\tau$反射对称形式，分别满足$T_{u} \, m_{u,\nu}^{*} \, T_{u } = m_{u,\nu}$ 和 $T_{d} \, m_{d,e}^{*} \, T_{d} = m_{d,e}$。$\mu-\tau$ 反射对称性和观察到的 $\sin \theta_{13}$ 之间的协调预测 $\delta_{CP} \simeq 203^{\circ}$。此外，对 $f=u,d,\nu,e$ 施加通用纹理 $(m_{f})_{11} = 0$ 预测具有最轻中微子质量 $|m_{1}| 的正常层次结构 = 6.26 美元或 2.54 美元 meV。