Abstract
We discuss the modular A4 invariant model of leptons combining with the generalized CP symmetry. In our model, both CP and modular symmetries are broken spontaneously by the vacuum expectation value of the modulus τ. The source of the CP violation is a non-trivial value of Re[τ] while other parameters of the model are real. The allowed region of τ is in very narrow one close to the fixed point τ = i for both normal hierarchy (NH) and inverted ones (IH) of neutrino masses. The CP violating Dirac phase δCP is predicted clearly in [98°, 110°] and [250°, 262°] for NH at 3 σ confidence level. On the other hand, δCP is in [95°, 100°] and [260°, 265°] for IH at 5 σ confidence level. The predicted ∑mi is in [82, 102] meV for NH and ∑mi = [134, 180] meV for IH. The effective mass 〈mee〉 for the 0νββ decay is predicted in [12.5, 20.5] meV and [54, 67] meV for NH and IH, respectively.
Highlights
The source of the CP violation is a non-trivial value of Re[τ ] while other parameters of the model are real
We investigate two possible cases of neutrino masses mi, which are the normal hierarchy (NH), m3 > m2 > m1, and the inverted hierarchy (IH), m2 > m1 > m3
We show the numerical result of two samples for NH and IH, respectively
Summary
Let us start with discussing the generalised CP symmetry [92, 99]. The CP transformation is non-trivial if the non-Abelian discrete flavor symmetry G is set in the Yukawa sector of a Lagrangian. Where xP = (t, −x) and Xr is a unitary transformations of ψ(x) in the irreducible representation r of the discrete flavor symmetry G. If Xr is the unit matrix, the CP transformation is the trivial one This is the case for the continuous flavor symmetry [99]. 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 [101, 102]. The general Kähler potential consistent with the modular symmetry possibly contains additional terms [111].
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