Alternative schemes of predicting lepton mixing parameters from discrete flavor and $CP$ symmetry

We suggest two alternative schemes to predict lepton mixing angles as well as $CP$ violating phases from a discrete flavor symmetry group combined with $CP$ symmetry. In the first scenario, the flavor and $CP$ symmetry is broken to the residual groups of the structure $Z_2\times CP$ in the neutrino and charged lepton sectors. The resulting lepton mixing matrix depends on two free parameters ${\theta }_{\nu }$ and ${\theta }_l$. This type of breaking pattern is extended to the quark sector. In the second scenario, an Abelian subgroup of the flavor group is preserved by the charged lepton mass matrix and the neutrino mass matrix is invariant under a single remnant $CP$ transformation, all lepton mixing parameters are determined in terms of three free parameters ${\theta }_{1,2,3}$. We derive the most general criterion to determine whether two distinct residual symmetries lead to the same mixing pattern if the redefinition of the free parameters ${\theta }_{\nu ,l}$ and ${\theta }_{1,2,3}$ is taken into account. We have studied the lepton mixing patterns arising from the flavor group $S_4$ and $CP$ symmetry which are subsequently broken to all of the possible residual symmetries discussed in this work.

Figure 1

The contour plots of ${\sin }^2{\theta }_{ij}$ in the plane ${\theta }_{\nu }$ vs ${\theta }_l$. The red, blue, and green areas denote the $3\sigma$ regions of ${\sin }^2{\theta }_{13}$, ${\sin }^2{\theta }_{23}$, and ${\sin }^2{\theta }_{12}$ respectively. The dashed (or solid) lines indicate the best fit values of the mixing angles. Notice that the best fit value of ${\sin }^2{\theta }_{23}$ depends on the neutrino mass ordering, and the solid and dashed lines are for NH and IH respectively. The residual flavor symmetry is $(G_l,G_{\nu })=(Z_2^{S{T}^{2}SU},Z_2^{TU})$ in this case. The first row corresponds to $(X_l,X_{\nu },P_l,P_{\nu })=(U,T,P_{12},1)$ on the left panel and $(X_l,X_{\nu },P_l,P_{\nu })=(U,T,P_{12},P_{12})$ on the right panel, and the last row is for $(X_l,X_{\nu },P_l,P_{\nu })=(U,STS,P_{12},1)$, $(U,STS,P_{12},P_{12})$. The foreground and background differ in the values of $P_l$ which are equal to $P_{12}$ and $P_{13}$ respectively.

Figure 2

The contour plots of ${\sin }^2{\theta }_{ij}$ in the plane ${\theta }_{\nu }$ vs ${\theta }_l$. The red, blue, and green areas denote the $3\sigma$ regions of ${\sin }^2{\theta }_{13}$, ${\sin }^2{\theta }_{23}$, and ${\sin }^2{\theta }_{12}$ respectively. The dashed (or solid) lines indicate the best fit values of the mixing angles. Notice that the best fit value of ${\sin }^2{\theta }_{23}$ depends on the neutrino mass ordering, and the solid and dashed lines are for NH and IH respectively. The left and right panels correspond to $(G_l,G_{\nu },X_l,X_{\nu },P_l,P_{\nu })=(Z_2^{S{T}^{2}SU},Z_2^{TU},T^2,T,P_{12},P_{12})$ and $(Z_2^{S{T}^{2}SU},Z_2^S,T^2,TS{T}^{2}U,P_{13},P_{12})$ respectively.

Figure 3

The contour plots of ${\sin }^2{\theta }_{ij}$ in the plane ${\theta }_{\nu }$ vs ${\theta }_l$. The red, blue, and green areas denote the $3\sigma$ regions of ${\sin }^2{\theta }_{13}$, ${\sin }^2{\theta }_{23}$, and ${\sin }^2{\theta }_{12}$ respectively. The dashed (or solid) lines indicate the best fit values of the mixing angles. Notice that the best fit value of ${\sin }^2{\theta }_{23}$ depends on the neutrino mass ordering, and the solid and dashed lines are for NH and IH respectively. The residual flavor symmetry is $(G_l,G_{\nu })=(Z_2^{S{T}^{2}SU},Z_2^S)$ in this case. The first row corresponds to $(X_l,X_{\nu },P_l,P_{\nu })=(T^2,1,P_{12},P_{13})$ on the left panel and $(X_l,X_{\nu },P_l,P_{\nu })=(T^2,SU,P_{12},P_{13})$ on the right panel, and the last row is for $(X_l,X_{\nu },P_l,P_{\nu })=(T^2,TS{T}^{2}U,P_{13},P_{13})$, $(U,1,P_{12},P_{13})$. The foreground and background differ in the values of $P_l$ which are equal to $P_{12}$ and $P_{13}$ respectively.

Figure 4

The contour plots of the $CP$-violation phases ${\delta }_{CP}$, ${\alpha }_{21}$, and ${\alpha }_{31}$ in the plane ${\theta }_{\nu }$ vs ${\theta }_l$. The black areas denote the regions in which the lepton mixing angles are compatible with experimental data at $3\sigma$ level. The residual symmetry is $(G_l,G_{\nu },X_l,X_{\nu })=(Z_2^{S{T}^{2}SU},Z_2^S,T^2,SU)$. The figures on the right-hand and lefthand sides correspond to the row and column permutations $(P_l,P_{\nu })=(P_{12},P_{13})$ and $(P_l,P_{\nu })=(P_{13},P_{13})$ respectively.

Figure 5

The predictions for the possible values of the effective Majorana mass $|m_{ee}|$ as a function of the lightest neutrino mass. The red (blue) dashed lines indicate the most general allowed regions for the IH (NH) neutrino mass spectrum obtained by varying the mixing parameters over the $3\sigma$ ranges [6]. The residual flavor symmetry is $(G_l,G_{\nu })=(Z_2^{S{T}^{2}SU},Z_2^{TU})$ in this case. The first row corresponds to $(X_l,X_{\nu },P_l,P_{\nu })=(U,T,P_{12},1)$ on the left and $(X_l,X_{\nu },P_l,P_{\nu })=(U,T,P_{12},P_{12})$ on the right, the middle row is for $(X_l,X_{\nu },P_l,P_{\nu })=(U,STS,P_{12},1)$, $(U,STS,P_{12},P_{12})$, and the last row is for $(X_l,X_{\nu },P_l,P_{\nu })=(T^2,T,P_{12},P_{12})$. The present most stringent upper limits $|m_{ee}|<0.120\mathrm{eV}$ from EXO-200 [73, 74] and KamLAND-ZEN [75] are shown by a horizontal gray band. The vertical gray exclusion band is the current limit on the lightest neutrino masses from the cosmological data $\sum m_i<0.230\mathrm{eV}$ at 95% confidence level obtained by the Planck Collaboration [76].

Figure 6

The predictions for the effective Majorana mass $|m_{ee}|$, where we use the same conventions as in Fig. 5. The residual flavor symmetry is $(G_l,G_{\nu })=(Z_2^{S{T}^{2}SU},Z_2^S)$ in this case. The top left panel corresponds to $(X_l,X_{\nu },P_l,P_{\nu })=(T^2,TS{T}^{2}U,P_{12},P_{13})$, the top right panel is for $(X_l,X_{\nu },P_l,P_{\nu })=(T^2,TS{T}^{2}U,P_{13},P_{12})$, and the last one is for $(X_l,X_{\nu },P_l,P_{\nu })=(T^2,SU,P_{12},P_{13})$.

Figure 7

The predictions for the effective Majorana mass $|m_{ee}|$, where we use the same conventions as in Fig. 5. The residual flavor symmetry is $(G_l,G_{\nu })=(Z_2^{S{T}^{2}SU},Z_2^S)$ in this case. The panels on the right-hand and left-hand sides correspond to $(X_l,X_{\nu },P_l,P_{\nu })=(T^2,1,P_{12},P_{13})$ and $(X_l,X_{\nu },P_l,P_{\nu })=(U,1,P_{12},P_{13})$ respectively. Notice that $|m_{ee}|$ is invariant under the transformations ${\theta }_l\to \pi -{\theta }_l$, ${\theta }_{\nu }\to {\theta }_{\nu }+\pi /2$ and $k_2\to k_2+1$, and hence the effective mass is independent of $k_2$ in this case.

Figure 8

The contour plot of $\sin {\theta }_{13}^q$, $\sin {\theta }_{12}^q$, and $\sin {\theta }_{23}^q$ in the ${\theta }_d-{\theta }_u$ plane. The blue and red lines denote the central values of $\sin {\theta }_{12}^q$ and $\sin {\theta }_{23}^q$ respectively. The different shading areas from dark green to light green represent three interesting regions of $\sin {\theta }_{13}^q$ such as $0.5(\sin {\theta }_{13}^q{)}_{\mathrm{bf}}\to (\sin {\theta }_{13}^q{)}_{\mathrm{bf}}$, $(\sin {\theta }_{13}^q{)}_{\mathrm{bf}}\to 2(\sin {\theta }_{13}^q{)}_{\mathrm{bf}}$, and $2(\sin {\theta }_{13}^q{)}_{\mathrm{bf}}\to 3(\sin {\theta }_{13}^q{)}_{\mathrm{bf}}$, where we use $(\sin {\theta }_{13}^q{)}_{\mathrm{bf}}=0.00368$.

Figure 9

Correlations between different mixing parameters in the case of $(G_l,X_{\nu })=(Z_3^T,S)$, where the three lepton mixing angles are required to be compatible with the experimental data at $3\sigma$ level [6].

Figure 10

Correlations between different mixing parameters in the case of $(G_l,X_{\nu })=(Z_3^T,SU)$, where the three lepton mixing angles are required to be compatible with the experimental data at $3\sigma$ level [6].

Figure 11

Correlations between different mixing parameters in the case of $(G_l,X_{\nu })=(K_4^{(S,U)},T)$, where the three lepton mixing angles are required to be compatible with the experimental data at $3\sigma$ level [6].

Figure 12

The allowed regions of the effective Majorana mass $|m_{ee}|$ with respect to the lightest neutrino mass. The red (blue) dashed lines indicate the most general allowed regions for the IH (NH) neutrino mass spectrum obtained by varying the mixing parameters over their $3\sigma$ ranges [6]. The top row corresponds to the residual symmetry $(G_l,X_{\nu })=(Z_3^T,1)$ on the left and $(G_l,X_{\nu })=(Z_3^T,S)$ on the right, the middle row is for $(G_l,X_{\nu })=(Z_3^T,U)$ and $(G_l,X_{\nu })=(Z_3^T,SU)$, and the bottom row is for $(G_l,X_{\nu })=(K_4^{(S,U)},T)$. The present most stringent upper limits $|m_{ee}|<0.120\mathrm{eV}$ from EXO-200 [73, 74] and KamLAND-ZEN [75] are shown by the horizontal gray band. The vertical gray exclusion band is the current limit on the lightest neutrino masses from the cosmological data $\sum m_i<0.230\mathrm{eV}$ at 95% confidence level obtained by the Planck Collaboration [76].