Towards a realistic model of quarks and leptons, leptonic $CP$ violation, and neutrinoless $\beta \beta$-decay

To explain the fermion masses and mixings naturally, we introduce a specific flavor symmetry and mass suppression pattern that constrain the flavor structure of the fermion Yukawa couplings. Our model describes why the hierarchy of neutrino masses is milder than the hierarchy of charged fermion masses in terms of successive powers of flavon fields. We investigate $CP$ violation and neutrinoless double beta ($0\nu \beta \beta$) decay and show how they can be predicted and constrained in our model by present and upcoming experimental data. Our model predicts that the atmospheric neutrino mixing angle ${\theta }_{23}$ should be within $\sim 1°$ of 45° for the normal neutrino mass ordering and between $\sim 4°$ and $\sim 8°$ degrees away from 45° (in either direction) for the inverted neutrino mass ordering. For both neutrino mass ordering and inverted neutrino mass ordering, our model predicts that a $0\nu \beta \beta$ Majorana mass in the limited range $0.035\mathrm{eV}<|m_{ee}|\lesssim 0.15\mathrm{eV}$, which can be tested in current experiments. Moreover, our model can successfully accommodate flavorless leptogenesis as the mechanism to generate the baryon asymmetry in the Universe, provided the neutrino mass ordering is normal, $|m_{ee}|\simeq 0.072\pm 0.012\mathrm{eV}$, and either ${\theta }_{23}\simeq 44°$ and the Dirac $CP$-violating phase ${\delta }_{CP}\simeq 20°$ or 60° or ${\theta }_{23}\simeq 46°$ and ${\delta }_{CP}\simeq 205°$ or 245°.

Figure 1

Scatter plots showing the location of points in parameter space lying within the $3\sigma$ experimental bounds of Table 2. The upper panel shows the correlation between the input parameters $\tilde{\kappa }(\equiv \lambda |y_R^{\nu }|v_{\chi }/M)$ and $y_1\equiv y_1^{\nu }$. The lower panels plot the dependence of the atmospheric mixing angle ${\theta }_{23}$ on the input parameters $\varphi$ (left plot) and $y_2/y_3$ (right plot). The horizontal dotted lines show the best-fit values (two local minima). The red crosses and the blue dots correspond to NO and IO, respectively.

Figure 2

Plots of $|m_{ee}|$ as a function of $\tilde{\kappa }$ (left) and $\varphi$ (right). The horizontal solid (dotted) lines show the current bounds from (the near-future reach of) Xe-based $0\nu \beta \beta$ experiments.

Figure 3

Plots of $|m_{ee}|$ as a function of $m_{\text{lightest}}$ (left) and $\sum m_i$ (right). The horizontal solid (dotted) lines show the current bounds from (the near-future reach of) Xe-based $0\nu \beta \beta$ experiments, while the vertical solid (dotted) lines indicate the cosmological Planck-I (Planck-II) upper bounds.

Figure 4

Predictions for the leptonic $CP$-violating ${\delta }_{CP}$ as a function of neutrino mixing angle ${\theta }_{23}$. The vertical dotted lines bound the current best-fit value of ${\theta }_{23}$. The blue dots and the red crosses correspond to the IO and the NO, respectively.

Figure 5

Plots of the leptonic $CP$-violating phase ${\delta }_{CP}$ (left) and the $CP$-violating invariant $J_{CP}$ (right) vs $|m_{ee}|$. The vertical solid (dotted) lines indicate the current bounds from (the near-future reach of) Xe-based $0\nu \beta \beta$-decay experiments.

Figure 6

Model predictions for primordial baryogenesis. The plot on the left shows the relation between the baryon asymmetry parameter ${\eta }_B$ and the $0\nu \beta \beta$-decay mass $|m_{ee}|$. The vertical solid line indicates the current bounds from Xe-based $0\nu \beta \beta$-decay experiments, while the dotted line indicates their near-future reach. The plot on the right shows our model predictions for ${\delta }_{CP}$ in terms of the positive values of ${\eta }_B$. The thick dashed line on both plots corresponds to the measured values of the baryon asymmetry in the Universe ${\eta }_B=(6.05\pm 0.07)\times 10^{-10}$ from Planck measurements [1], or ${\eta }_B=(6.2\pm 1.0)\times 10^{-10}$ from D/H measurements [33].

Figure 7

Model prediction of $|m_{ee}|$ in terms of ${\theta }_{23}$. The vertical dotted lines show the best-fit values for ${\theta }_{23}$. The horizontal lines show the current upper bounds from (and near-future reach of) $0\nu \beta \beta$-decay experiments. Blue dots correspond to the inverted mass ordering (IO) and red crosses to the normal mass ordering (NO).