Supersymmetric $SO(10)$ grand unified models with renormalizable Yukawa couplings involving only $\mathbf{10}$ and $\overline{\mathbf{126}}$ Higgs fields have been shown to realize the fermion masses and mixings economically. In previous works, the sum rule of the fermion mass matrices is given by inputting the quark matrices, and the neutrino mixings are predicted in this framework. Now the three neutrino mixings have been measured, and in this paper, we give the sum rule by inputting the lepton mass matrices, which makes clear certain features of the solution, especially if the vacuum expectation values of $\mathbf{126}+\overline{\mathbf{126}}$ ($v_R$) are large and the right-handed neutrinos are heavy. We perform the ${\chi }^2$ analyses to fit the fermion masses and mixings using the sum rule. In previous works, the best fit appears at $v_R\sim 10^{13}\mathrm{GeV}$, and the fit at the large $v_R$ scale ($\sim 10^{16}\mathrm{GeV}$) has been less investigated. Our expression of the sum rule has a benefit to understand the flavor structure in the large $v_R$ solution. Using the fit results, we perform the calculation of the $\mu \to e\gamma$ process and the electric dipole moment of electron, and the importance of $v_R$ dependence emerges in low energy phenomena. We also show the prediction of the $CP$ phase in the neutrino oscillations, which can be tested in the near future.

• Received 3 September 2015

DOI:https://doi.org/10.1103/PhysRevD.94.075018