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.The formulation of the Standard Model (SM) was one of the biggest successes of particle physics, accounting for most interactions of matter known to date. Notwithstanding its success, it is still a theory that leaves unsanswered some questions. Why are there three copies of fermions? Why are their masses hierarchical? What gives rise to the specific mixing patterns observed? These questions (among others) are part of what is known as the flavour problem [1].The inclusion of a flavour symmetry to the SM became a prolific way to attempt to meaningfully answer some of the questions posed by the flavour problem. These flavour symmetries affect not the gauge structure of the model, rather they restrict the way the different particles are able to interact with each other in the Yukawa sector. As such, this strategy has been widely used to predict the leptonic mixing structure, for which the flavour symmetries favoured are typically non-Abelian discrete symmetries. In these flavour models, we can take the SM to be a low-energy version of a more complete model which includes a flavour symmetry that has been broken by either scalar fields that are singlets with respect to the SM gauge structure (these are called flavons and it is this paradigm we will be discussing), or by enlarging the Higgs sector. Then, the vacuum expectation values (VEV) of the scalars dictate the Yukawa structures. In leptonic flavour models, the type-I seesaw mechanism is employed and Sequential Dominance (SD) [2, 3, 4, 5] is a useful paradigm for model building, with Constrained Sequential Dominance (CSD) [6] remaining a viable strategy to produce predictive models [7,8,9,10,11,12].Since in the indirect approach models the mixing structure is defined by the flavon VEVs, the problem trickles down to choosing appropriate VEVs to match the observed mixing pattern. Nevertheless, this is still a question that must be in accordance to the scalar potential of the chosen flavour model (both the flavour symmetry chosen and the number of flavons will influence the outcome). Moreover, cross-terms in the scalar potential that arise from the interaction of the different flavons will, in gene...

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We present the first complete model of the Littlest Modular Seesaw, based on two right-handed neutrinos, within the framework of multiple modular symmetries, justifying the use of multiple moduli fields which take their values at 3 specific stabilizers of Γ4 ≃ S4, including a new phenomenological possibility. Using a semi-analytical approach, we perform a χ2 analysis of each case and show that good agreement with neutrino oscillation data is obtained, including predictive relations between the leptonic mixing angles and the ratio of light neutrino masses, which non-trivially agree with the experimental values. It is noteworthy that in this very predictive setup, the models fit the global fits of the experimental data remarkably well, both with and without the Super-Kamiokande atmospheric data, for both models presented here. By extending the model to include a weighton and the double cover group $$ {\Gamma}_4^{\prime}\simeq {S}_4^{\prime } $$ Γ 4 ′ ≃ S 4 ′ , we are able to also account for the hierarchy of the charged leptons using modular symmetries, without altering the neutrino predictions.

The idea of modular invariance provides a novel explanation of flavour mixing. Within the context of finite modular symmetries ΓN and for a given element γ ∈ ΓN, we present an algorithm for finding stabilisers (specific values for moduli fields τγ which remain unchanged under the action associated to γ). We then employ this algorithm to find all stabilisers for each element of finite modular groups for N = 2 to 5, namely, Γ2 ≃ S3, Γ3 ≃ A4, Γ4 ≃ S4 and Γ5 ≃ A5. These stabilisers then leave preserved a specific cyclic subgroup of ΓN. This is of interest to build models of fermionic mixing where each fermionic sector preserves a separate residual symmetry.