## Fermion Masses and Mixings from Dihedral Flavor Symmetries with Preserved Subgroups

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Blum, A.

Hagedorn, C.

Lindner, M.

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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
(non-trivial) 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 non-vanishing 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 CKM 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}|=|cos(pi(m_{u}-m_{d})j/n)| which is |cos(3 pi/7)| = 0.2225 for n=7,
j=1, m_{u}=3 and m_{d}=0. We prove that two successful models which lead to
maximal atmospheric mixing and vanishing 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.

Comment: 25 pages

Comment: 25 pages

##### Keywords

High Energy Physics - Phenomenology