Lepton and quark mixing patterns from finite flavor symmetries

We perform a systematical and analytical study of lepton mixing which can be derived from the subgroups of $SU(3)$ under the assumption that neutrinos are Dirac particles. We find that type D groups can predict lepton mixing patterns compatible with the experimental data at the $3\sigma$ level. The lepton mixing matrix turns out to be of the trimaximal form, and the Dirac $CP$ violating phase is trivial. Moreover, we extend the flavor symmetry to the quark sector. The Cabibbo mixing between the first two generations of quarks can be generated by type D groups. Since all the finite subgroups of $U(3)$ that are not the subgroups of $SU(3)$ have not been classified, an exhaustive scan over all finite discrete groups up to order 2000 is performed with the help of the computer algebra system gap. We find that only 90 (10) groups for Dirac (Majorana) neutrinos can generate the lepton mixing angles in the experimentally preferred ranges. The lepton mixing matrix is still the trimaximal pattern and the Dirac $CP$ phase remains trivial. The smallest groups that lead to viable mixing angles are [162, 10], [162, 12], and [162, 14]. For quark flavor mixing, the correct order of magnitude of the Cabibbo-Kobayashi-Maskawa matrix elements cannot be generated. Only the Cabibbo mixing is allowed even if we impose very loose constraints $0.1\le |V_{us}|\le 0.3$ and $|V_{ub}|\le |V_{cb}|<|V_{us}|$. The group $\mathrm{\Delta }(6\times 7^2)$ can predict a Cabibbo angle ${\theta }_q=\pi /14$ in good agreement with the best fit value. The observed Cabibbo mixing angle can easily be accommodated if the first two left-handed quark fields are assigned to a doublet. The groups that can give rise to both phenomenologically viable lepton mixing angles and acceptable Cabibbo angles are discussed, and the groups $\mathrm{\Delta }(6\times 9^2)$, [648, 259], [648, 260], [648, 266], and $\mathrm{\Delta }(6\times {14}^2)$ are especially promising in the case of the triplet assignment for both quark and lepton sectors. The three groups [496, 19], [496, 21], and [496, 23] are interesting candidates for flavor symmetry if the left-handed lepton and quark fields are assigned to triplet and doublet plus singlet, respectively.

Figure 1

The correlation between $\sin {\theta }_{13}$ and ${\sin }^2{\theta }_{12}$ for the mixing pattern in Eq. (3.45). The pink region represents the possible values of the mixing angles when $\theta$ and $\delta$ freely vary in the region of $[-\pi ,\pi ]$. The $1\sigma$ and $3\sigma$ bounds are adopted from Ref. [4].

Figure 2

Correlations between the mixing angles for the viable mixing pattern in Eq. (3.47), where the flavor symmetry is a group of type D.

Figure 3

The viable values of $\sin {\theta }_{13}$, ${\sin }^2{\theta }_{12}$, and ${\sin }^2{\theta }_{23}$ for $D_{n,n}^{(0)}$ flavor symmetry with respect to the group index $n$, where the predicted mixing angles that fall outside of the $3\sigma$ limits are not shown. To distinguish the predictions for the Dirac and Majorana neutrinos, we present the data in different colors, all the red and black points refer to the predictions for Dirac neutrinos while the red points are for Majorana neutrinos. The experimentally preferred $1\sigma$ and $3\sigma$ regions are adapted from Ref. [4].

Figure 4

The viable values of $\sin {\theta }_{13}$, ${\sin }^2{\theta }_{12}$, and ${\sin }^2{\theta }_{23}$ for $D_{9n^{\prime },3n^{\prime }}^{(1)}\cong (Z_{9n^{\prime }}\times Z_{3n^{\prime }})⋊S_3$ flavor symmetry with respect to the group index $n$, where the predicted mixing angles that fall outside of the $3\sigma$ limits are not shown. To distinguish the predictions for the Dirac and Majorana neutrinos, we present the data in different colors; all the red and black points refer to the predictions for Dirac neutrinos while the red points are for Majorana neutrinos. The experimentally preferred $1\sigma$ and $3\sigma$ regions are adapted from Ref. [4].

Figure 5

The possible values of $|V_{us}|=|\sin \theta |$ for $D_{n,n}^{(0)}\cong \mathrm{\Delta }(6n^2)$ flavor symmetry group. The green band represents the regions allowed by the experimental data [1], and the blue band denotes the region $0.1\le |V_{us}|\le 0.3$. If the present experimental data of lepton mixing angles cannot be accommodated by a $\mathrm{\Delta }(6n^2)$ group, the corresponding predictions for $|V_{us}|$ are plotted with red points.

Figure 6

The possible values of $|V_{us}|=|\sin \theta |$ for $D_{9n^{\prime },3n^{\prime }}^{(1)}$ flavor symmetry group. The green band represents the regions allowed by the experimental data [1], and the blue band denotes the region $0.1\le |V_{us}|\le 0.3$.

Figure 7

The groups generated by the residual symmetries ${\mathcal{G}}_{\nu }$ and ${\mathcal{G}}_l$ in the lepton sector. If a subgroup is generated, it is linked to the parent group by an arrowed line and the arrow points to the parent group. The blue boxes denote that all the possible residual subgroups $\{{\mathcal{G}}_{\nu },{\mathcal{G}}_l\}$ do not generate the entire group. The yellow boxes imply that the group can be generated by at least one set of the remnant symmetries. The similar figure including all the 90 groups in Table 6 looks quite complex, and it is available at our Web site [28].

Figure 8

The values of the Cabibbo mixing angle predicted by discrete flavor symmetry groups with order less than 2000. The blue dots represent the results shown in Eq. (5.10), and the red star refers to the experimental best fit value.

Figure 9

The groups generated by the residual symmetries $\{{\mathcal{G}}_{\nu },{\mathcal{G}}_l,{\mathcal{G}}_U,{\mathcal{G}}_D\}$ for the discrete flavor symmetry groups listed in Table 10. If a subgroup is generated, it is linked to the parent group by an arrowed line and the arrow points to the parent group.