Residual Symmetries and Scalar Multiplet Vacuum Alignment in Non-Abelian Flavour Models

TL;DR

The study tackles how vacuum alignments in non-Abelian flavour models relate to residual discrete flavour symmetries in the broken phase. It introduces a one-to-one correspondence principle between broken residual flavour symmetries $\mathcal{G}_{\text{RFS}}$ and vacuum alignment corrections, and validates this using $S_4$ toy models before applying the insights to realistic $A_4$ AF and $\Delta(27)$ UTZ constructions. Through renormalizable single-flavon potentials and perturbative corrections (à la Altarelli–Feruglio), the work shows that RFS-preserving operators leave leading alignment directions intact up to universal rescalings, while RFS-violating operators induce realignments that alter IR Yukawa textures. These results highlight a potential source of fine-tuning in flavour models and provide a practical framework to assess the stability of vacuum alignments against higher-dimensional corrections in EFTs.

Abstract

We demonstrate that, upon minimizing a renormalizable, single-scalar potential invariant under a non-Abelian symmetry, special orientations in the associated vacuum alignment of the scalar multiplet correspond to the preservation of a discrete residual flavour symmetry in the broken phase of the theory. Conversely, we show that these special scalar alignments are perturbed when additional Lagrangian operators (e.g. renormalizable, multi-flavon operators and/or effective, higher-dimensional operators) are present that break said residual symmetry, leading to a vacuum reorientation and phenomenological consequences. We therefore construct a one-to-one correspondence principle between broken residual symmetries and vacuum alignment corrections, providing a mechanism to identify (and correct) a subtle but persistent form of phenomenologically relevant fine-tuning embedded in -- but often ignored by -- most successful non-Abelian flavour models. We first establish this correspondence in a set of toy models based on the S4 permutation symmetry, and then apply the lessons learned to the more realistic A4 Altarelli-Feruglio and $Δ(27)$ Universal Texture Zero models.