Spontaneous CP violation by modulus $\tau$ 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 $\tau$. The source of the CP violation is a non-trivial value of $\mathrm{Re}[\tau ]$ while other parameters of the model are real. The allowed region of $\tau$ is in very narrow one close to the fixed point $\tau =i$ for both normal hierarchy (NH) and inverted ones (IH) of neutrino masses. The CP violating Dirac phase ${\delta }_{CP}$ is predicted clearly in $[98^{\circ },110^{\circ }]$ and $[250^{\circ },262^{\circ }]$ for NH at $3\sigma$ confidence level. On the other hand, ${\delta }_{CP}$ is in $[95^{\circ },100^{\circ }]$ and $[260^{\circ },265^{\circ }]$ for IH at $5\sigma$ confidence level. The predicted $\sum m_i$ is in $[82,102]$ meV for NH and $\sum m_i=[134,180]$ meV for IH. The effective mass $⟨m_{ee}⟩$ for the $0\nu \beta \beta$ decay is predicted in $[12.5,20.5]$ meV and $[54,67]$ meV for NH and IH, respectively.

arXiv comment24 pages, 12 figures, 3 tables. arXiv admin note: text overlap with arXiv:2009.14242

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