CP Symmetry and Lepton Mixing from a Scan of Finite Discrete Groups

• Mixing patterns compatible with experimental data

  • I(a)	I(b)	II	III	IV	V	VI	VII	VIII

  • This table displays the resulting mixing patterns which can admit a good agreement with experimental data. These viable mixing patterns can be categorized into several cases. The cases whose names begin with "C" are the mixing patterns with fixed column obtained from the semidirect approach, the case beginning with "R" has a fixed row and it is obtained from the variant of the semidirect approach. The rotation matrix $S_{12}(\theta), S_{13}(\theta), S_{23}(\theta)$ are rotation matrices defined in our paper. We denote the parameter $\omega=e^{2\pi/3}$ in case CII, $\kappa=(\sqrt{5}+1)/2$ in case CIV and case CV, $\varphi=\arctan(2-\sqrt{7})$ in case CVI, and $s_n=\sin(2n\pi/7)$ in case CVII. There are generically many groups leading to the same mixing matrix, we only display a representative group. One can click on the representative group id to see all the associated flavor groups.Clickable link is not provided if the mixing pattern can only be produced by one group up to the order of 2000.

• Group information

  • Group ID

  • Group ID
    Structure
    Multiplication Rules
    3-Dimensional Representation
    Class-inverting Automorphism
    Generalized CP

  • This table shows some properies of the flavor symmetry group $G_{f}$, which can be extracted from GAP software. The group identification in GAP and structure description are given in the first two rows. The third row shows the multiplication rules of the group. The 3-dimensional representation given here is irreducible and faithful, where $\rho(g_i)$ denotes the representation matrix of the generator $g_i$. Notice that we use 1 to denote the identity element of the group, and $e_n$ refers to the nth root of unit with $e_n=e^{i2\pi/n}$. The class-inverting automorphism $u$ maps each group element $g$ to an element $u(g)$ which lies in the same conjugacy class as $g^{-1}$.The explicit form of the physical CP transformation given by class-inverting automorphism $u$ can be determined by solving the consistency condition $X_0\rho^{*}(g_i)X^{\dagger}_0=\rho(u(g_i))$.

• Abelian subgroups and residual CP

  • Abelian subgroups(order>2). # Name Generator(s) Res. $CP$ Degeneracy

    $Z_2$ subgroups. # Name Generator(s) Res. $CP$ Degeneracy

  • The left table shows the Abelian subgroups of $G_f$ with order larger than 2, and the right table shows its $Z_2$ subgroups. We use bold blue lines to divide the subgroups into different categories, and all the subgroups within the same category are related by group conjugation. The residual CP symmetry consistent with each subgroup as a residual flavor symmetry is simply represented by the group elements $g_f$, actually the corresponding CP transformation is given by $\rho(g_{f})X_{0}$. Here we only show one representative residual CP transformation, and all the possible residual CP transformations can be obtained by clicking on the representative one (If there is a clickable link with red color, there would be more than one residual CP transformation. If no residual CP transformation consistent the subgroup can be found,it will be denoted by "---"). In the last column, "Yes" means the 3-dimensional representation matrix of the generator of the subgroup has degenerate eigenvalues so that it can not be regarded as remnant flavor symmetry, while "No" means the subgroup has non-degenerate eigenvalues and consequently it can be used as remnant flavor symmetry.

• Fixed column and row by residual symmetry

  • Fixed column in semidirect approach # Res. Sym. Column Check

    Fixed row in the variant of semidirect approach # Res. Sym. Row Check

  • In the semdirect approach and its variant, one row and one column of the PMNS matrix are fixed by the residual flavor symemtry respectively. Many different residual symmetries could lead to the same colum (row), and we only show a representative one of them in the column named "Rem. Sym.". The notation $(i,j)$ represents that the residual flavor symmetry $G_{l}$ of the charged lepton sector is the $i$th Abelian subgroups(order>2) and the residual flavor symemtry $G_{\nu}$ of the neutrino sector is the $j$th $Z_2$ subgroup in the semidirect approach while $G_{l}$ and $G_{\nu}$ are the $i$th $Z_2$ subgroup and $j$th Abelian subgroups(order>2) respectively in the variant of the semidirect approach. One can obtained all the possible residual symemtries for a column (row) by clicking on the representative residual symmetry. If the column (row) is in the 3-sigma ranges of the global fit [arXiv:1409.5439], we mark with the symbol "$\surd$" otherwise with "$\times$".

• Predictions for lepton mixng matrix

  • PMNS matrix predicted in semidirect approach ($G_l=$Abelian subgroup, $G_{\nu}=Z_2$)

    #	Res Sym.	$\Sigma$	$f_c$	$\theta_{\text{bf}}$	$\sin^2\theta_{13}$	$\sin^2\theta_{12}$	$\sin^2\theta_{23}$	$\delta_{CP}/\pi$	$\alpha_{21}/\pi$ $(\text{mod}~1)$	$\alpha^{\prime}_{31}/\pi$ $(\text{mod}~1)$	$\chi^2_{\text{min}}$	NO\IO	$45^{\circ}$	Check

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  • PMNS matrix predicted in the variant of semidirect approach ($G_l=Z_2$, $G_{\nu}=K_4$)

    #	Res. Sym.	$\Sigma$	$f_r$	$\theta_{\text{bf}}$	$\sin^2\theta_{13}$	$\sin^2\theta_{12}$	$\sin^2\theta_{23}$	$\delta_{CP}/\pi$	$\alpha_{21}/\pi$ $(\text{mod}~1)$	$\alpha_{31}/\pi$ $(\text{mod}~1)$	$\chi^2_{\text{min}}$	NO\IO	$45^{\circ}$	Check

  • These two tables summarize the possible PMNS mixing matrices which can be obtained from the semidirect approach and it variant. The PMNS matrix is $U_{\text{PMNS}}=\Sigma S_{ij}(\theta)$ in semidirect approach and $U_{\text{PMNS}}=S^{T}_{ij}(\theta)\Sigma$ in the variant of the semidirect approach, where $S_{ij}(\theta)$ with $ij=12,13,23$ are block diagonal rotation matrices defined in the paper. Depending on the position of the fixed column (row) shown in the colunmn named $f_c$ ($f_r$), we can determine which one of the three rotation matrices should be used. The PMNS matrices with different #$n$ can not coincide with each other even if the redefinition of $\theta$ and phases matrix redefinition of $Q_l$, $Q_\nu$ and permutations of columns and rows are taken into account. Moreover, the mixing matrices with same #$n$ are related through the permutation of rows annd columns, and they are not the same up to redefinition of $\theta$, $Q_l$ and $Q_\nu$. Plenty of distinct residual symemtries can give rise to the same PMNS matrix, and they can be seen by clicking on the representative residual symemtry shown here. Regarding the notation of the residual symmetry $(i_m,j_n)$, the residual flavor symemtries $G_{l}$ and $G_{\nu}$ are the $i$th Abelian subgroups(order>2) and $j$th $Z_2$ subgroup respectively in the semidirect approach,the subscripts $m$ and $n$ means that the residual CP transformation $X_{l}$ of the charged lepton sector is the $m$th residual CP consistent with $G_l$ and the residual CP transformation $X_{\nu}$ of the neutrino sector is the $n$th one consistent with $G_{\nu}$. In the variant of the semidirect approach, $(i_m,j_n)$ denotes that $G_{l}$ is the $i$th $Z_2$ subgroup, $X_{l}$ is the $m$th residual CP transformation compatioble with $G_{l}$, $G_{\nu}$ is the $j$th Abelian subgroups(order>2) and $X_{\nu}$ is the $n$th one consistent with $G_{\nu}$. Note that the subscript "$a$" refers to any residual CP transformation consistent with given residual flavor symemtry. The $\chi^2$ function has a global minimum $\chi^2_{\text{min}}$ at the best fit value $\theta_{\text{bf}}$ for $\theta$. We give the values of the mixing angles $\sin^2\theta_{\text{ij}}$ and the CP violation phases $\delta_{CP}$, $\alpha_{21}$ and $\alpha_{31}$ for $\theta=\theta_{\text{bf}}$. The neutrino mass hierarchy (NO: normal ordering, IO: inverted ordering) and the possible octant of $\theta_{23}$ ($\theta_{23}>45^\circ,\theta_{23}=45^\circ$ and $\theta_{23}<45^\circ$) are considered in the $\chi^2$ analysis. If the best fit values of mixing angles are in experiemntally preferred $3\sigma$ ranges, we shall mark with "$\surd$" otherwise with "$\times$". In the second table, we use the notation "?" to label these mixing patterns with small $\chi^2_{\text{min}}$ and only $\sin^2\theta_{12}$ slightly above its $3\sigma$ upper limit.