Double Froggatt--Nielsen Mechanism

We present a doubly parametric extension of the standard Froggatt--Nielsen (FN) mechanism. As is well known, mass matrices of the up- and down-type quark sectors and the charged lepton sector in the standard model can be parametrized well by a parameter $\lambda$ which is usually taken to be the sine of the Cabibbo angle ($\lambda = \sin\theta_\text{C} \simeq 0.225$). However, in the neutrino sector, there is still room to realize the two neutrino mass squared differences $\Delta m_\text{sol}^2$ and $\Delta m_\text{atm}^2$, two large mixing angles $\theta _{12}$ and $\theta _{23}$, and non-zero $\theta _{13}$. Then we consider an extension with an additional parameter $\rho$ in addition to $\lambda$. Taking the relevant FN charges for a power of $\lambda~(=0.225)$ and additional FN charges for a power of $\rho$, which we assume to be less than one, we can reproduce the ratio of the two neutrino mass squared differences and three mixing angles. In the normal neutrino mass hierarchy, we show several patterns for taking relevant FN charges and the magnitude of $\rho$. We find that if $\sin \theta_{23}$ is measured more precisely, we can distinguish each pattern. This is testable in the near future, for example in neutrino oscillation experiments. In addition, we predict the Dirac CP-violating phase for each pattern.