Natural Vacuum Alignment from Group Theory: The Minimal Case

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, cross-couplings 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 A4, T7, S4, T' or \Delta(27). We have found a number of candidate groups and present a model based on one of the smallest extension of A4, namely Q8 \rtimes A4. We show that the most general non-supersymmetric 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.