Modulus $\tau$ linking leptonic CP violation to baryon asymmetry in $A_4$ modular invariant flavor model

TL;DR

This paper presents an $A_4$ modular invariant lepton flavor model linking CP violation to baryon asymmetry, predicting specific ranges for neutrino parameters and the Dirac CP phase, with implications for leptogenesis and neutrino mass hierarchy.

Contribution

It introduces a novel $A_4$ modular invariant model that connects leptonic CP violation with baryon asymmetry through the modulus $ au$, providing specific predictions for neutrino parameters and leptogenesis.

Findings

Predicted Dirac CP phase ranges for different $ au$ regions.

Baryon asymmetry $Y_B$ consistent with observed values in certain $ au$ ranges.

No allowed $ au$ region for inverted hierarchy at 3$\sigma$ confidence level.

Abstract

We propose an $A_4$ modular invariant flavor model of leptons, in which both CP and modular symmetries are broken spontaneously by the vacuum expectation value of the modulus $\tau$. The value of the modulus $\tau$ is restricted by the observed lepton mixing angles and lepton masses for the normal hierarchy of neutrino masses at $3\sigma$ confidence level. The predictive Dirac CP phase $\delta_{CP}$ is in the ranges $[0^\circ,50^\circ]$, $[170^\circ,175^\circ]$ and $[280^\circ,360^\circ]$ for ${\rm Re}\, [\tau]<0$, and $[0^\circ,80^\circ]$, $[185^\circ,190^\circ]$ and $[310^\circ,360^\circ]$ for ${\rm Re}\,[\tau]>0$ at $3\sigma$ confidence level. The sum of three neutrino masses is predicted in $[60, 84]$ meV, and the effective mass for the $0\nu\beta\beta$ decay is in [0.003, 3] meV. On the other hand, there is no allowed region of the modulus $\tau$ for the inverted hierarchy of neutrino masses at $3\sigma$ confidence level. The modulus $\tau$ links the Dirac CP phase to the cosmological baryon asymmetry (BAU) via the leptogenesis. Due to the strong wash-out effect, the predictive baryon asymmetry $Y_B$ can be at most the same order of the observed value. Then, the lightest right-handed neutrino mass is restricted in the range of $M_1 =[1.5,\,6.5] \times 10^{13}$ GeV. We find the correlation between the predictive $Y_B$ and the Dirac CP phase $\delta_{CP}$. Only two predictive $\delta_{CP}$ ranges, $[5^\circ,40^\circ]$ (${\rm Re}\,[\tau]>0$) and $[320^\circ,355^\circ]$ (${\rm Re}\,[\tau]<0$) are consistent with the BAU.