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A bstract We study scalar Non-Standard Neutrino Interactions (NSI) induced by moduli or flavon exchange between electrons and neutrinos. In a region with non-vanishing electron number density, they are known to determine a shift of the neutrino mass matrix. We review and extend the relevant formalism, and we update the existing limits on electron and neutrino scalar couplings. We explore the observability of scalar NSI in models of lepton masses based on flavour symmetries. We analyze models where the scalar couplings are constrained either by abelian symmetries or by modular invariance. We highlight regions of the parameter space where observable effects can occur.

A bstract We perform a systematical analysis of the A 4 modular models with generalized CP for the masses and flavor mixing of quarks and leptons, and the most general form of the quark and lepton mass matrices is given. The CP invariance requires all couplings real in the chosen basis and thus the vacuum expectation value of the modulus τ uniquely breaks both the modular symmetry and CP symmetry. The phenomenologically viable models with minimal number of free parameters and the results of fit are presented. We find 20 models with 7 real free parameters that can accommodate the experimental data of lepton sector. We then apply A 4 modular symmetry to the quark sector to explain quark masses and CKM mixing matrix, the minimal viable quark model is found to contain 10 free real parameters. Finally, we give two predictive quark-lepton unification models which use only 16 real free parameters to explain the flavor patterns of both quarks and leptons.

A bstract We extend the framework of modular invariant supersymmetric theories to encompass invariance under more general discrete groups Γ, that allow the presence of several moduli and make connection with the theory of automorphic forms. Moduli span a coset space G / K , where G is a Lie group and K is a compact subgroup of G , modded out by Γ. For a general choice of G , K , Γ and a generic matter content, we explicitly construct a minimal Kähler potential and a general superpotential, for both rigid and local N = 1 supersymmetric theories. We also specialize our construction to the case G = Sp(2 g, ℝ), K = U( g ) and Γ = Sp(2 g, ℤ), whose automorphic forms are Siegel modular forms. We show how our general theory can be consistently restricted to multi-dimensional regions of the moduli space enjoying residual symmetries. After choosing g = 2, we present several examples of models for lepton and quark masses where Yukawa couplings are Siegel modular forms of level 2.

A bstract We extend the even weight modular forms of modular invariant approach to general integral weight modular forms. We find that the modular forms of integral weights and level N can be arranged into irreducible representations of the homogeneous finite modular group Γ N ′ which is the double covering of Γ N . The lowest weight 1 modular forms of level 3 are constructed in terms of Dedekind eta-function, and they transform as a doublet of Γ 3 ′ ≅ T′ . The modular forms of weights 2, 3, 4, 5 and 6 are presented. We build a model of lepton masses and mixing based on T′ modular symmetry.

A bstract The formalism of non-holomorphic modular flavor symmetry is developed, and the Yukawa couplings are level N polyharmonic Maaß forms satisfying the Laplacian condition. We find that the integer (even) weight polyharmonic Maaß forms of level N can be decomposed into multiplets of the finite modular group Γ N ′ (Γ N ). The original modular invariance approach is extended by the presence of negative weight polyharmonic Maaß forms. The non-holomorphic modular flavor symmetry can be consistently combined with the generalized CP symmetry. We present three example models for lepton sector based on the Γ 3 ≅ A 4 modular symmetry, the charged lepton masses and the neutrino oscillation data can be accommodated very well, and the predictions for the leptonic CP violation phases and the effective Majorana neutrino mass are studied.

A bstract We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as Γ( N′ )/Γ( N″ ), and the modular group SL(2 , ℤ) is ex- tended to a principal congruence subgroup Γ( N′ ). The original modular invariant theory is reproduced when N′ = 1. We perform a comprehensive study of Γ 6 ′ modular symmetry corresponding to N′ = 1 and N″ = 6, five types of models for lepton masses and mixing with Γ 6 ′ modular symmetry are discussed and some example models are studied numerically. The case of N′ = 2 and N″ = 6 is considered, the finite modular group is Γ(2)/Γ(6) ≅ T′ , and a benchmark model is constructed.

A bstract We revisit the modular flavor symmetry from a more general perspective. The scalar modular forms of principal congruence subgroups are extended to the vector-valued modular forms, then we have more possible finite modular groups including Γ N and Γ N ′ as the flavor symmetry. The theory of vector-valued modular forms provides a method of differential equation to construct the modular multiplets, and it also reveals the simple structure of the modular invariant mass models. We review the theory of vector-valued modular forms and give general results for the lower dimensional vector-valued modular forms. The general finite modular groups are listed up to order 72. We apply the formalism to construct two new lepton mass models based on the finite modular groups A 4 × Z 2 and GL(2, 3).

A bstract We combine SO(10) Grand Unified Theories (GUTs) with A 4 modular symmetry and present a comprehensive analysis of the resulting quark and lepton mass matrices for all the simplest cases. We focus on the case where the three fermion families in the 16 dimensional spinor representation form a triplet of Γ 3 ≃ A 4 , with a Higgs sector comprising a single Higgs multiplet H in the 10 fundamental representation and one Higgs field ∆ ¯ in the 126 ¯ for the minimal models, plus one Higgs field Σ in the 120 for the non-minimal models, all with specified modular weights. The neutrino masses are generated by the type-I and/or type II seesaw mechanisms and results are presented for each model following an intensive numerical analysis where we have optimized the free parameters of the models in order to match the experimental data. For the phenomenologically successful models, we present the best fit results in numerical tabular form as well as showing the most interesting graphical correlations between parameters, including leptonic CP phases and neutrinoless double beta decay, which have yet to be measured, leading to definite predictions for each of the models.

A bstract Texture zeros in fermion mass matrices have been widely considered in tackling the Standard Model flavour puzzle. In this work, we perform a systematic analysis of texture zeros in lepton mass matrices in the framework of Γ 3 ′ ≅ T ′ modular symmetry. Assuming that the lepton fields transform as irreducible representations of T ′, we obtain all possible texture-zero patterns for both charged-lepton and neutrino mass matrices which can be achieved from T ′ modular symmetry. We provide representative models for the phenomenologically-viable textures which can accommodate the experimental data. The predictions for lepton mixing angles, CP-violating phases, light neutrino masses and effective neutrino mass relevant for neutrinoless double beta decay, are discussed. We find that the minimal viable lepton model depends on only 7 real free parameters including the modulus τ (the corresponding charged-lepton mass matrix contains 4 vanishing entries, and the neutrino mass matrix has 1 texture zero). Finally, we study in detail three benchmark models, one for each neutrino mass generation mechanism considered (Dirac, Majorana via Weinberg operator and Majorana via minimal type-I seesaw mechanism).

A bstract We present a neutrino mass model based on modular symmetry with the fewest input parameters to date, which successfully accounts for the 12 lepton masses and mixing parameters through 6 real free parameters including the modulus. The neutrino masses are predicted to be normal ordering, the atmospheric angle θ 23 is quite close to maximal value and the Dirac CP phase δ CP is about 1 . 34 π . We also study the soft supersymmetry breaking terms due to the modulus F -term in this minimal model, which are constrained to be the non-holomorphic modular forms. The radiative lepton flavor violation process μ → eγ is discussed.