Abstract
The observed neutrino mixing, having a near maximal atmospheric neutrino mixing angle and a large solar mixing angle, is close to tri-bi-maximal. This structure may be related to the existence of a discrete non-Abelian family symmetry. In this paper the family symmetry is the non-Abelian discrete group Δ(27), a subgroup of SU(3) with triplet and anti-triplet representations. Different frameworks are constructed in which the mixing follows from combining fermion mass terms with the vacuum structure enforced by the discrete symmetry. Mass terms for the fermions originate from familon triplets, anti-triplets or both. Vacuum alignment for the family symmetry breaking familons follows from simple invariants.
Highlights
In terms of use in particle physics ∆(27), has an interesting history
Different frameworks are constructed in which the mixing follows from combining fermion mass terms with the vacuum structure enforced by the discrete symmetry
It was used over 50 years ago in hadron physics in [4], and over 30 years ago to obtain spontaneous geometrical CP violation in [5], where the vacuum expectation values (VEVs) display a CP violating phase that is related to the group structure and independent of arbitrary phases in the Lagrangian
Summary
An important advantage of building family symmetry models in SUSY frameworks is the holomorphic superpotential. One solution forces the 3 entries of the VEV to be equal φ123 = (c, c, c) , σ01 = − c00 It isn’t used further in this paper, it is relevant to note that using singlet familons 1ij with i, j = 0 in this type of term enables the alignment of directions such as (ω, 1, 1) in a SUSY framework. This class of VEV is relevant in that it leads to spontaneous geometrical CP violation [5, 18]. In this sense, having both VEV directions is often not necessary in ∆(27) frameworks, as demonstrated by the specific example in section 2.3, featuring a minimal alignment superpotential SV
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