We discuss the 3.55 keV X-ray line anomaly reported by XMN-Newton X-ray observatory using data of various galaxy clusters and Andromeda galaxy in a radiative neutrino model, in which the mixing between the active neutrino and the dark matter is gener
ated at two-loop level after the spontaneous breaking of $Z_2$ symmetry. It might provide us a natural explanation of its tiny mixing ${cal O}(10^{-10})$, which is observed by their experiments. Such an Abelian discrete symmetry plays a crucial role in differentiating the TeV scale Majorana field from our dark matter, whose mass is expect to be around 7.1 keV.
Following the way proposed recently by Hernandez and Smirnov, we seek possible residual symmetries in the quark sector with a focus on the von Dyck groups. We begin with two extreme cases in which both $theta_{13}$ and $theta_{23}$ or only $theta_{13
}$ are set to zero. Then, cases where all the Cabibbo-Kobayashi-Maskawa parameters are allowed to take nonzero values are explored. The $Z_7$ symmetry is favorable to realize only the Cabibbo angle. On the other hand, larger groups are necessary in order to be consistent with all the mixing parameters. Possibilities of embedding the obtained residual symmetries into the $Delta(6N^2)$ series are also briefly discussed.
Inspired by the small mass-squared difference measured in the solar neutrino oscillation experiments and by the testability, we suggest that a limit of the partial mass degeneracy, in which masses of the first two generation fermions are degenerate,
may be a good starting point for understanding the observed fermion mass spectra and mixing patterns. The limit indicates the existence of a two-dimensional rotation symmetry, such as $O(2)$, $D_N$ and so on, in flavor space of the first two generations. We propose simple models for the lepton sector based on $D_N$ and show that the models can successfully reproduce the experimental data without imposing unnatural hierarchies among dimensionless couplings, although at least $10%$ tuning is necessary in order to explain a large atmospheric mixing. It is especially found that the $Z_2$ subgroup of the $D_N$ symmetry plays an important role in understanding the smallness of the electron mass and $theta_{13}^{rm PMNS}$. We also discuss testability of the models by the future neutrinoless-double-beta-decay experiments and cosmological observations.
