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

TL;DR

This paper explores how residual flavor symmetries are preserved or broken in non-Abelian flavor models through scalar vacuum alignments, revealing a mechanism for understanding fine-tuning issues and phenomenological implications.

Contribution

It establishes a one-to-one correspondence between residual symmetry preservation and vacuum alignment corrections in non-Abelian flavor models, with applications to realistic models.

Findings

Residual symmetries correspond to special scalar vacuum alignments.

Additional operators can perturb these alignments, breaking residual symmetries.

The framework applies to models based on S4, A4, and Δ(27) symmetries.

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 $\Delta(27)$ Universal Texture Zero models.