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Register a new userModular symmetry at level 6 and a new route towards finite modular groups
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We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as $Gamma(N)/Gamma(N)$, and the modular group $SL(2,mathbb{Z})$ is extended to a principal congruence subgroup $Gamma(N)$. The original modular invariant theory is reproduced when $N=1$. We perform a comprehensive study of $Gamma_6$ modular symmetry corresponding to $N=1$ and $N=6$, five types of models for lepton masses and mixing with $Gamma_6$ 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 $Gamma(2)/Gamma(6)cong T$, and a benchmark model is constructed.
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We consider for the first time level 7 modular invariant flavour models where the lepton mixing originates from the breaking of modular symmetry and couplings responsible for lepton masses are modular forms. The latter are decomposed into irreducible multiplets of the finite modular group $Gamma_7$, which is isomorphic to $PSL(2,Z_{7})$, the projective special linear group of two dimensional matrices over the finite Galois field of seven elements, containing 168 elements, sometimes written as $PSL_2(7)$ or $Sigma(168)$. At weight 2, there are 26 linearly independent modular forms, organised into a triplet, a septet and two octets of $Gamma_7$. A full list of modular forms up to weight 8 are provided. Assuming the absence of flavons, the simplest modular-invariant models based on $Gamma_7$ are constructed, in which neutrinos gain masses via either the Weinberg operator or the type-I seesaw mechanism, and their predictions compared to experiment.
We study a flavor model with $A_4$ symmetry which originates from $S_4$ modular group. In $S_4$ symmetry, $Z_2$ subgroup can be anomalous, and then $S_4$ can be violated to $A_4$. Starting with a $S_4$ symmetric Lagrangian at the tree level, the Lagrangian at the quantum level has only $A_4$ symmetry when $Z_2$ in $S_4$ is anomalous. We obtain modular forms of two singlets and a triplet representations of $A_4$ by decomposing $S_4$ modular forms into $A_4$ representations. We propose a new $A_4$ flavor model of leptons by using those $A_4$ modular forms. We succeed in constructing a viable neutrino mass matrix through the Weinberg operator for both normal hierarchy (NH) and inverted hierarchy (IH) of neutrino masses. Our predictions of the CP violating Dirac phase $delta_{CP}$ and the mixing $sin^2theta_{23}$ depend on the sum of neutrino masses for NH.
We construct a model to explain the muon anomalous magnetic moment, without considering any lepton flavor violations, in the modular $A_4$ symmetry. We have investigated a predictive radiative seesaw model including dark matter candidate at favorable fixed point of $tau=omega$ obtained by recent analysis of the stabilized moduli values from the possible configurations of the flux compactifications. In the result, we show our predictions on the Dirac CP and Majorana phases, the neutrino masses, the mass range of dark matter as well as the muon anomalous magnetic moment through the $chi^2$ analysis.
We make an investigation of modular $Gamma^{prime}_5 simeq A^{prime}_5$ group in inverse seesaw framework. Modular symmetry is advantageous because it reduces the usage of extra scalar fields significantly. Moreover, the Yukawa couplings are expressed in terms of Dedekind eta functions, which also have a $q$ expansion form, utilized to achieve numerical simplicity. Our proposed model includes six heavy fermion superfields i.e., $mathcal{N}_{Ri}$, $mathcal{S}_{Li}$ and a weighton. The study of neutrino phenomenology becomes simplified and effective by the usage of $A^prime_5$ modular symmetry, which provides us a well defined mass structure for the lepton sector. Here, we observe that all the neutrino oscillation parameters, as well as the effective electron neutrino mass in neutrinoless double beta decay can be accommodated in this model. We also briefly discuss the lepton flavor violating decays $ell_i to ell_j gamma$ and comment on non-unitarity of lepton mixing matrix.
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 $Gamma_3simeq A_4$, with a Higgs sector comprising a single Higgs multiplet $H$ in the ${mathbf{10}}$ fundamental representation and one Higgs field $overline{Delta}$ in the ${mathbf{overline{126}}}$ for the minimal models, plus and one Higgs field $Sigma$ in the ${mathbf{120}}$ 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.
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