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 $(-1,1,1)$. These generalized $CP$ 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 $U(1)_{\rm PQ}$ symmetry and a flavored axion. The axion scale is suggested to be $\langle \theta_{u,d} \rangle \sim \Lambda_{\rm GUT} \, \sqrt{m_{u,d} \, m_{c,s}} / v \sim 10^{12} \,$[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].