Spontaneous CP violation by modulus τ in A4 model of lepton flavors

A bstract We discuss the modular A 4 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 δ CP is predicted clearly in [98°, 110°] and [250°, 262°] for NH at 3 σ confidence level. On the other hand, δ CP is in [95°, 100°] and [260°, 265°] for IH at 5 σ confidence level. The predicted ∑ m i is in [82, 102] meV for NH and ∑ m i = [134, 180] meV for IH. The effective mass 〈 m ee 〉 for the 0 νββ decay is predicted in [12 . 5, 20 . 5] meV and [54 , 67] meV for NH and IH, respectively.