Natural vacuum alignment from group theory: the minimal case

Abstract

Discrete flavour symmetries have been proven successful in explaining the leptonic flavour structure. To account for the observed mixing pattern, the flavour symmetry has to be broken to different subgroups in the charged and neutral lepton sector. However, crosscouplings via non-trivial contractions in the scalar potential force the group to break to the same subgroup. We present a solution to this problem by extending the flavour group in such a way that it preserves the flavour structure, but leads to an 'accidental' symmetry in the flavon potential.We have searched for symmetry groups up to order 1000, which forbid all dangerous cross-couplings and extend one of the interesting groups A 4 , T 7 , S 4 , T or ∆(27). We have found a number of candidate groups and present a model based on one of the smallest extensions of A 4 , namely Q 8 A 4 . We show that the most general nonsupersymmetric potential allows for the correct vacuum alignment. We investigate the effects of higher dimensional operators on the vacuum configuration and mixing angles, and give a see-saw-like UV completion. Finally, we discuss the supersymmetrization of the model. Additionally, we release the Mathematica package Discrete providing various useful tools for model building such as easily calculating invariants of discrete groups and flavon potentials.

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.…”

Higgs →μτ in Abelian and non-Abelian flavor symmetry models

“…[36]. Obviously these fields break the discrete symmetry group A 4 down to the subgroup T |T 3 = E ∼ = Z 3 , while simultaneously breaking the electroweak gauge group SU(2) L × U(1) Y down to the electromagnetic U(1) em .…”

“…Its form has been observed before in alignment models with driving fields [22] and non-trivial group extensions [36]. Contrary to the philosophy employed in those references, we do not assume ε 1, and therefore rather use the parametrization 5…”

“…Here we aim to employ non-Abelian discrete symmetries with a slightly different point of view, namely we want to emphasize the possibility of additional phenomenology of non-Abelian flavor symmetries at the electroweak scale [3,5,15,18,[27][28][29][30][31][32][33][34][35][36]. Thus, instead of only concentrating on predicting mixing angles, we have additional tests of models at our disposal, e.g.…”

“…(2.2) for a certain choice of parameters (see for example Ref. [36] and references therein). In the following, we will always present results in the limit λ χA = 0, which simplifies the mixing in the scalar sector.…”