Universal bilinear form of quark and lepton mass matrices

In the so-called Yukawaon model, the (effective) Yukawa coupling constants $Y_f^{\text{eff}}$ are given by vacuum expectation values (VEVs) of scalars $Y_f$ (Yukawaons) with $3\times 3$ components. In the present model, all of the VEV matrices $⟨Y_f⟩$ are given by a bilinear form of VEVs of flavons ${\mathrm{\Phi }}_f$, ${{⟨Y_f⟩}_i}^j=k_f⟨{\mathrm{\Phi }}_f{⟩}_{ik}⟨{\overline{\mathrm{\Phi }}}_f{⟩}^{kj}$, where ${\mathrm{\Phi }}_f$ is assigned to $\mathbf{6}$ of U(3) family symmetry. As input parameters with family number–dependent values, we use only charged lepton mass values. Under this formulation, we can give reasonable values of quark and lepton masses and their mixings. A $CP$-violating phase ${\delta }_{CP}^{\ell }=\pm 26°$ in the lepton sector is predicted. The effective Majorana neutrino mass is also predicted.

Figure 1

(a) ${\alpha }_{\nu }$ dependence of the lepton mixing parameters ${\sin }^{2}2{\theta }_{12}$, ${\sin }^{2}2{\theta }_{23}$, ${\sin }^{2}2{\theta }_{13}$, and the neutrino mass-squared difference ratio $R_{\nu }$. We draw curves of these as functions of ${\alpha }_{\nu }$ for the case of ${\xi }_{\nu }=-0.20$ taking $a_{\nu }=-3.5$ (dotted), $-3.53$ (solid), and $-3.56$ (dashed). (b) $a_{\nu }$ dependence of the lepton mixing parameters ${\sin }^{2}2{\theta }_{12}$, ${\sin }^{2}2{\theta }_{23}$, ${\sin }^{2}2{\theta }_{13}$, and the neutrino mass-squared difference ratio $R_{\nu }$. We draw curves of these as functions of $a_{\nu }$ for the case of ${\xi }_{\nu }=-0.20$ taking ${\alpha }_{\nu }=7.0°$ (dotted), 8.7° (solid), and 10° (dashed).

Figure 2

Allowed region in the (${\varphi }_1$, ${\varphi }_2$) parameter plane obtained by the observed values of the CKM-mixing matrix elements $|V_{us}|$, $|V_{cb}|$, $|V_{ub}|$, and $|V_{td}|$. We draw allowed regions obtained from the observed constraints of the CKM-mixing matrix elements shown in Eq. (3.15), taking ${\xi }_u=-0.001467$, $a_u=-1.467$, $a_d=-1.477$, and ${\xi }_d^{\prime }=0.0237$. Here we take $2\sigma$ errors for all the observed values of the CKM-mixing matrix elements. We find that the parameter set around $({\varphi }_1,{\varphi }_2)=(21.8°,-4.9°)$ indicated by a star is consistent with all the observed values.