Spontaneous CP violation by modulus $\tau$ in $A_4$ model of lepton flavors

TL;DR

This paper presents a model where spontaneous CP violation arises from the modulus $ au$ in an $A_4$ flavor symmetry framework, predicting specific ranges for CP phases and neutrino masses consistent with experimental constraints.

Contribution

It introduces a novel $A_4$ modular invariant model with spontaneous CP violation driven by the modulus $ au$, providing precise predictions for neutrino CP phases and masses.

Findings

Predicted $ au$ near fixed point $ au=i$ for both hierarchies.

CP phase $ heta_{CP}$ in specific ranges for NH and IH.

Neutrino mass sums and $0 uetaeta$ effective masses within experimental bounds.

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 ${\rm Re} [\tau]$ 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 $\langle m_{ee}\rangle$ for the $0\nu\beta\beta$ decay is predicted in $[12.5,\,20.5]$ meV and $[54,\,67]$ meV for NH and IH, respectively.