Effective alignments as building blocks of flavour models

Abstract

Flavour models typically rely on flavons - scalars that break the family symmetry by acquiring vacuum expectation values in specific directions. We develop the idea of effective alignments, i.e.\ cases where the contractions of multiple flavons give rise to directions that are hard or impossible to obtain directly by breaking the family symmetry. Focusing on the example where the symmetry is $S_4$, we list the effective alignments that can be obtained from flavons vacuum expectation values that arise naturally from $S_4$. Using those effective alignments as building blocks, it is possible to construct flavour models, for example by using the effective alignments in constrained sequential dominance models. We illustrate how to obtain several of the mixing schemes in the literature, and explicitly construct renormalizable models for three viable cases, two of which lead to trimaximal mixing scenarios.