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

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 $\mathrm{Re}[\tau ]<0$, and $[0^{\circ },80^{\circ }]$, $[185^{\circ },190^{\circ }]$ and $[310^{\circ },360^{\circ }]$ for $\mathrm{Re}[\tau ]>0$ 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, $[5^{\circ },40^{\circ }]$ ($\mathrm{Re}[\tau ]>0$) and $[320^{\circ },355^{\circ }]$ ($\mathrm{Re}[\tau ]<0$) are consistent with the BAU.

Comments: 30 pages, 14 figures, 7 tables, minor corrections