# Effective alignments and the landscape of $S_4$ flavour models

## TL;DR

This paper investigates effective alignments in $S_4$ flavour models, analyzing how flavon contractions create novel directions for model building and identifying viable topologies for lepton flavour models.

## Contribution

It introduces the concept of effective alignments in $S_4$ models and systematically checks their phenomenological viability for the first time.

## Key findings

- Identifies three viable topologies for $S_4$ flavour models.
- Provides examples of models for each viable topology.
- Highlights effective alignments as a useful tool for flavour model building.

## Abstract

We explore the concept of effective alignments: contractions of multiple flavour symmetry breaking flavon fields. These contractions give rise to directions that are hard or impossible to obtain directly by breaking the flavour symmetry. Within this context, and using $S_4$ as the flavour symmetry to exemplify, we perform a phenomenological check of lepton flavour models built from pairing any two effective alignments up to order 2 (in flavon contractions). The check is performed for each pair of effective alignments in a framework with models of constrained sequential dominance type, in a basis where the charged leptons are diagonal. We thus obtain an indication of which effective alignments are interesting for model building, within this so-called $S_4$ landscape. We find three types of viable topologies and provide examples of models realizing this strategy for each topology.