Fermion masses and mixings from dihedral flavor symmetries with preserved subgroups

Abstract

We perform a systematic study of dihedral groups used as flavor symmetry. The key feature here is the fact that we do not allow the dihedral groups to be broken in an arbitrary way, but in all cases some (nontrivial) subgroup has to be preserved. In this way we arrive at only five possible (Dirac) mass matrix structures which can arise, if we require that the matrix has to have a nonvanishing determinant and that at least two of the three generations of left-handed (conjugate) fermions are placed into an irreducible two-dimensional representation of the flavor group. We show that there is no difference between the mass matrix structures for single- and double-valued dihedral groups. Furthermore, we comment on possible forms of Majorana mass matrices. As a first application we find a way to express the Cabibbo angle, i.e. the Cabibbo-Kobayashi-Maskawa matrix element $|{V}_{us}|$, in terms of group theory quantities only, the group index $n$, the representation index j and the index ${m}_{u,d}$ of the different preserved subgroups in the up and down quark sector: $|{V}_{us}|=|\mathrm{cos}(\frac{\ensuremath{\pi}({m}_{u}\ensuremath{-}{m}_{d})\mathrm{j}}{n})|$ which is $|\mathrm{cos}(\frac{3\ensuremath{\pi}}{7})|\ensuremath{\approx}0.2225$ for $n=7$, $\mathrm{j}=1$, ${m}_{u}=3$ and ${m}_{d}=0$. We prove that two successful models which lead to maximal atmospheric mixing and vanishing ${\ensuremath{\theta}}_{13}$ in the lepton sector are based on the fact that the flavor symmetry is broken in the charged lepton, Dirac neutrino and Majorana neutrino sector down to different preserved subgroups whose mismatch results in the prediction of these mixing angles. This also demonstrates the power of preserved subgroups in connection with the prediction of mixing angles in the quark as well as in the lepton sector.