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.

We study the nonstandard decays of the lightest Higgs state within the Exceptional Supersymmetric Standard Model (E 6 SSM). We argued that the SM-like Higgs boson can decay predominantly into dark matter particles while its branching ratios into SM particles varies from 2% to 4%. This scenario also implies the presence of other relatively light Inert chargino and neutralino states in the particle spectrum with masses below 200 GeV. We argue that in this case the decays of the lightest Higgs boson into l + l − + X may play an essential role in the Higgs searches.

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In this talk we show how a natural neutrino mass hierarchy can follow from the type I see-saw mechanism, and a natural neutrino mass degeneracy from the type II see-saw mechanism, where the bi-large mixing angles can arise from either the neutrino or charged lepton sector. We summarize the phenomenological implications of such natural models, and discuss the model building applications of the approach, focussing on the SU (3) × SO(10) model. We also show that in such type II models the leptogenesis asymmetry parameter becomes proportional to the neutrino mass scale, in sharp contrast to the type I case, which leads to an upper bound on the neutrino mass scale, allowing lighter right-handed neutrinos and hence making leptogenesis more consistent with the gravitino constraints in supersymmetric models.