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.
Highlights
Non-Abelian discrete flavor symmetries have been widely used in models of lepton flavor mixing
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
We start by listing the possible vacuum expectation values (VEVs) alignments that are allowed by the potential of one scalar triplet of the flavor group and which serve as the building blocks of effective alignments (EAs)
Summary
Non-Abelian discrete flavor symmetries have been widely used in models of lepton flavor mixing (see some reviews, e.g., [1,2,3]). We start by listing the possible VEV alignments that are allowed by the potential of one scalar triplet of the flavor group (which coincide with the results of [28,29,30]) and which serve as the building blocks of EAs. We consider multiple flavons and discuss how in higher-order operators, two or more flavons can be combined to form EAs, and how these can be implemented to account for leptonic mixing, using sequential dominance [31,32,33,34] in CSD models (see, e.g., [21,22,23,24,25] for recent examples, and references therein). V we discuss cross-couplings between flavons and how they may affect the previous results
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