thetaC from the Dihedral Flavor Symmetries D₇ and D₁4

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In [1] it has been shown that the Cabibbo angle thetaC might arise from a dihedral flavor symmetry which is broken to different (directions of) subgroups in the up and the down quark sector. This leads to a prediction of thetaC in terms of group theoretical quantities only, i.e. the index n of the dihedral group Dₙ, the index j of the fermion representation 2ⱼ and the preserved subgroups indicated by mᵤ and md. Here we construct a low energy model which incorporates this idea. The gauge group is the one of the Standard Model and D₇ x Z₂ (aux) serves as flavor symmetry. The additional Z₂ (aux) is necessary in order to maintain two sets of Higgs fields, one which couples only to up quarks and another one coupling only to down quarks. We assume that D₇ is broken spontaneously at the electroweak scale by vacuum expectation values of SU(2)ₗ doublet Higgs fields. The quark masses and mixing parameters can be accommodated well. Furthermore, the potential of the Higgs fields is studied numerically in order to show that the required configuration of the vacuum expectation values can be achieved. We also comment on more minimalist models which explain the Cabibbo angle in terms of group theoretical quantities, while theta₁₃q and theta₂₃q vanish at leading order. Finally, we perform a detailed numerical study of the lepton mixing matrix VMNS in which one of its elements is entirely determined by the group theory of a dihedral symmetry. Thereby, we show that nearly tri-bi-maximal mixing can also be produced by a dihedral flavor group with preserved subgroups.