Spontaneous CP violation by modulus τ in A 4 model of lepton flavors

Abstract 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 0νββ decay is predicted in [12.5, 20.5] meV and [54, 67] meV for NH and IH, respectively.