Dihedral families of quarks, leptons, and Higgs bosons

Abstract

We consider finite groups of small order for family symmetry. It is found that the binary dihedral group Q 6 , along with the assumption that the Higgs sector is of type II, predicts mass matrix of a nearest neighbor interaction type for quarks and leptons. We present a supersymmetric model based on Q 6 with spontaneously induced CP phases. The quark sector contains 8 real parameters with one independent phase to describe the quark masses and their mixing. Predictions in the jV ub j ÿ , jV ub j ÿ sin2 1 , and jV ub j ÿ jV td =V ts j planes are given. The lepton sector contains also 9 parameters. A normal as well as an inverted spectrum of neutrino masses is possible, and we compute V e3 . We find that jV e3 j 2 > 10 ÿ4 in the case of a normal spectrum, and jV e3 j 2 > 8 10 ÿ4 in the case of an inverted spectrum. It is also found that Q 6 symmetry forbids all Baryon number violating terms of d 4, and the contributions to EDMs from the A terms vanish in this model.

Stringy origin of non-Abelian discrete flavor symmetries

“…(θ, 0), develops a VEV. 8 In this case, S 3 of S 3 ⋉ ( 3 × 3 ) is broken as S 3 → S 2 , and ( 3 × 3 ) is broken as 3 × 3 → 3 . Hence, the remaining flavor symmetry is the D 3 = S 2 ⋉ 3 symmetry, which consists of the 6 elements 1 0 0 1 ,…”

“…the origin of the number of generations, the observed mass hierarchies as well as the mixing angles. Many attempts to understand flavor are based on spontaneously broken Abelian [1] and non-Abelian flavor symmetries [2], such as S 3 (≈ D 3 ) [3], S 4 [4], A 4 [5], D 4 [6], D 5 [7], Q 6 [8], ∆-subgroups of SU(3) [9,10,11], governing Yukawa couplings for quarks and leptons. Discrete symmetries are not only useful to understand flavor issues (e.g.…”

Variations on the supersymmetricQ6model of flavor

“…These symmetries, along with Q 6 , reduce significantly the number of parameters in the fermion mass matrices. This reduction of parameters leads to a sum rule involving quark masses and mixings [8]. Moreover, CP violation has a spontaneous origin, which is perhaps more satisfying than the usual assumption of explicit CP violation.…”

“…The group theory of Q 6 is discussed in detail in Ref. [8]. We briefly recall its salient features relevant for model building.…”

“…After Q 6 symmetry breaking, flavor changing operators can be generated, but as shown in Ref. [8,10,11], these are all sufficiently suppressed. Higher dimensional operators are suppressed by the Planck scale and have negligible effects.…”