Broken Democracy with Intermediate $mathbb S_2 times mathbb S_2$ Residual Symmetry and Random Perturbations

The democratic mass matrix is an intriguing possibility for explaining the observed fermion mixings due to its inherent hierarchical mass eigenvalues and large mixing angles. Nevertheless, two of the three mass eigenvalues are zero if the flavor democracy is exact, in obvious contradiction with the experimental observations. One possibility is breaking the flavor democracy with anarchical perturbations as we proposed in an earlier work. However, even within the first two generations, the charged fermion masses are also hierarchical which may not be a coincidence. The democratic $mathbb S^L_3 times mathbb S^R_3$ symmetry of the three generations may first be broken down to an intermediate $mathbb S^L_2 times mathbb S^R_2$ symmetry among the first two generations to regulate the sequential hierarchies, followed by random perturbations, that generate the correct size of all measured observables. Unique predictions for neutrinoless double beta decay are also found.