Abstract
We study the phenomenological implications of the modular symmetry Γ(3) ≃ A4 of lepton flavors facing recent experimental data of neutrino oscillations. The mass matrices of neutrinos and charged leptons are essentially given by fixing the expectation value of modulus τ, which is the only source of modular invariance breaking. We introduce no flavons in contrast with the conventional flavor models with A4 symmetry. We classify our neutrino models along with the type I seesaw model, the Weinberg operator model and the Dirac neutrino model. In the normal hierarchy of neutrino masses, the seesaw model is available by taking account of recent experimental data of neutrino oscillations and the cosmological bound of sum of neutrino masses. The predicted sin2θ23 is restricted to be larger than 0.54 and δCP = ±(50°-180°). Since the correlation of sin2θ23 and δCP is sharp, the prediction is testable in the future. It is remarkable that the effective mass mee of the neutrinoless double beta decay is around 22 meV while the sum of neutrino masses is predicted to be 145 meV. On the other hand, for the inverted hierarchy of neutrino masses, only the Dirac neutrino model is consistent with the experimental data.
Highlights
Those vacuum expectation values (VEVs) determine the flavor structure of quarks and leptons
We study the phenomenological implications of the modular symmetry Γ(3) A4 of lepton flavors facing recent experimental data of neutrino oscillations
We study the phenomenological implications of the modular symmetry Γ(3) A4 facing recent experimental data of neutrino oscillations
Summary
We give a brief review on the modular symmetry on the torus and its low-energy effective field theory. The modulus parameter transforms as aτ + b τ −→ τ = They satisfy the following algebraic relations, S2 = I , (ST )3 = I. Under the modular transformation eq (2.2) are called modular forms of weight k. The superpotential should be invariant under the modular symmetry. Should have vanishing modular weight in global supersymmetric models, while the superpotential in supergravity should be invariant under the modular symmetry up to the Kahler transformation. By use of η(τ ) and its derivative, A4 triplet modular forms (Y1, Y2, Y3) of modular weight 2 are written by [29], i η (τ /3) η ((τ + 1)/3) η ((τ + 2)/3) 27η (3τ ). The overall coefficient in eq (2.10) is one choice and cannot be determined essentially
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