Semidirect product groups, vacuum alignment, and tribimaximal neutrino mixing

Abstract

The neutrino oscillation data are in very good agreement with the tribimaximal mixing pattern: sin 2 θ 23 = 1/2, sin 2 θ 12 = 1/3, and sin 2 θ 13 = 0. Attempts to generate this pattern based on finite family symmetry groups typically assume that the family symmetry is broken to different subgroups in the charged lepton and the neutrino mass matrices. This leads to a technical problem, where the cross-couplings between the Higgs fields responsible for the two symmetry breaking chains force their vacuum expectation values to align, upsetting the desired breaking pattern. Here, we present a class of models based on the semidirect product group (S 3 ) 4 ⋊ A 4 , where the lepton families belong to representations which are not faithful. In effect, the Higgs sector knows about the full symmetry while the lepton sector knows only about the A 4 factor group. This can solve the alignment problem without altering the desired properties of the family symmetry. Inclusion of quarks into the framework is straightforward, and leads to small and arbitrary CKM mixing angles. Supersymmetry is not essential for our proposal, but the model presented is easily supersymmetrized, in which case the same family symmetry solves the SUSY flavor problem.

The role of flavon cross couplings in leptonic flavour mixing

“…These approaches can solve the flavon VEV alignment problem effectively, but the price is that many degrees of freedom have to be introduced into the model. There is another solution by extending the flavour group, H, to a larger group N H [46][47][48]. Here, N H should admit irreducible representations of H such that the Standard Model leptons and one flavon can still transform in H, while the other flavon transforms as a different representation that belongs to N H but not to H.…”

Natural vacuum alignment from group theory: the minimal case

“…This symmetry is explicitly broken to (Q 8 A 4 ) × Z 4 by the couplings to leptons and by higher dimensional operators in the potential. As the accidental symmetry is discrete, there is no pseudo-Goldstone boson, as can easily happen in constructions of this type [21].…”

“…The defining homomorphism is sometimes indicated as index of , i.e. N ϕ H. 13 The same applies for the wreath product S 4 3 A4 introduced by Babu and Gabriel [21].…”

“…This type of model is also not so interesting from a general point of view, as it shares a couple of unpleasant features with the model of Babu and Gabriel [21] when viewed as an effective field theory:…”

“…While these approaches are interesting and worth studying, we will develop further an idea by Babu and Gabriel [21], who have suggested a group-theoretical mechanism to forbid the dangerous cross-couplings and thus a way to realise the VEV alignment without using Rsymmetries in supersymmetry or brane constructions. They proposed an extension of the flavour group A 4 in such a way that the Standard Model leptons only transform under the A 4 subgroup of the full flavour group.…”