Abstract
We propose an A4 modular invariant flavor model of leptons, in which both CP and modular symmetries are broken spontaneously by the vacuum expectation value of the modulus τ. The value of the modulus τ is restricted by the observed lepton mixing angles and lepton masses for the normal hierarchy of neutrino masses. The predictive Dirac CP phase δCP is in the ranges [0°, 50°], [170°, 175°] and [280°, 360°] for Re [τ] < 0, and [0°, 80°], [185°, 190°] and [310°, 360°] for Re [τ] > 0. The sum of three neutrino masses is predicted in [60, 84] meV, and the effective mass for the 0νββ decay is in [0.003, 3] meV. The modulus τ links the Dirac CP phase to the cosmological baryon asymmetry (BAU) via the leptogenesis. Due to the strong wash-out effect, the predictive baryon asymmetry YB can be at most the same order of the observed value. Then, the lightest right-handed neutrino mass is restricted in the range of M1 = [1.5, 6.5] × 1013 GeV. We find the correlation between the predictive YB and the Dirac CP phase δCP. Only two predictive δCP ranges, [5°, 40°] (Re [τ] > 0) and [320°, 355°] (Re [τ] < 0) are consistent with the BAU.
Highlights
We propose an A4 modular invariant flavor model of leptons, in which both CP and modular symmetries are broken spontaneously by the vacuum expectation value of the modulus τ
A viable lepton model was proposed in the modular A4 symmetry [80], in which the CP violation is realized by fixing τ, that is, the breaking of the modular symmetry
In the framework of the modular symmetry, the baryon asymmetry of the universe (BAU) has been studied in A4 model of leptons, where the source of CP violation is a complex parameter in the Dirac neutrino mass matrix in addition to the modulus τ [40]
Summary
The CP transformation is non-trivial if the non-Abelian discrete flavor symmetry G is set in the Yukawa sector of a Lagrangian [112, 122]. If Xr is the unit matrix, the CP transformation is the trivial one This is the case for the continuous flavor symmetry [122]. Xrρ∗r(g)X−r 1 = ρr(g ) , g, g ∈ G This equation defines the consistency condition, which has to be respected for consistent implementation of a generalized CP symmetry along with a flavor symmetry [124, 125]. Under the modular transformation of eq (2.4), chiral superfields ψi (i denotes flavors) with weight −k transform as [126], ψi −→ (cτ + d)−kρ(γ)ijψj. The Kähler potential of chiral matter fields ψi with the modular weight −k is given by. The general Kähler potential consistent with the modular symmetry possibly contains additional terms [127]. These modular forms have been explicitly given [22] in the symmetric base of the A4 generators S and T for the triplet representation (see appendix A) in appendix B
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