Diagonal reflection symmetries and universal four-zero texture

In this paper, we consider a set of new symmetries in the SM, {\it diagonal reflection} symmetries $R , m_{u,\nu}^{} , R = m_{u,\nu}, ~ m_{d,e}^{} = m_{d,e}$with$R =$diag$(-1,1,1)$\mathrm{.}Thesegeneralized$CP$ symmetries predict the Majorana phases to be $\alpha_{2,3} /2 \sim 0$or$\pi /2$. A realization of reflection symmetries suggests a broken chiral $U(1){\rm PQ}$symmetryandaflavoredaxion\mathrm{.}Theaxionscaleissuggestedtobe$\langle \theta{u,d} \rangle \sim \Lambda_{\rm GUT} , \sqrt{m_{u,d} , m_{c,s}} / v \sim 10^{12} , $[GeV]. By combining the symmetries with the four-zero texture, the mass eigenvalues and mixing matrices of quarks and leptons are reproduced well. This scheme predicts the normal hierarchy, the Dirac phase $\delta_{CP} \simeq 203^{\circ},$and$|m_{1}| \simeq 2.5$or$6.2 , $[meV]. In this scheme, the type-I seesaw mechanism and a given neutrino Yukawa matrix $Y_{\nu }$ completely determine the structure of right-handed neutrino mass $M_R$. An $u-\nu$ unification predicts mass eigenvalues to be $ (M_{R1} , , M_{R2} , , M_{R3}) = (O (10^{5}) , , O (10^{9}) , , O (10^{14})) , $[GeV].