We have studied the lepton and quark mixing patterns which can be derived from the dihedral group in combination with CP symmetry. The left-handed lepton and quark doublets are assigned to the direct sum of a singlet and a doublet of . A unified description of the observed structure of the quark and lepton mixing can be achieved if the flavor group and CP are broken to in neutrino, charged lepton, up quark and down quark sectors, and the minimal group is . We also consider another scenario in which the residual symmetry of the charged lepton and up quark sector is while remains preserved by the neutrino and down quark mass matrices. Then can give the experimentally favored values of CKM and PMNS mixing matrices.

CP violation Discrete Symmetries Neutrino Physics

Article funded by SCOAP3

Introduction

Lepton and quark mixing from residual symmetry

Dihedral group and CP symmetry

1$. A regular polygon with$n$sides has$2n$different symmetries:$n$rotational symmetries and$n$reflection symmetries, therefore the group order of$D_{n}$is$2n$. All$D_{n}$are non-ablelian permutation groups for$n>2$,$D_{1}$is isomorphic to$Z_{2}$and$D_{2}$is isomorphic to$Z_{2}\times Z_{2}$. The group$D_{n}$is the semidirect product$Z_{n}\rtimes Z_{2}$of the cyclic groups$Z_{n}$and$Z_{2}$. The dihedral group can be conveniently defined by two generators$R$and$S$which obey the relations~\cite{Blum-ml-2007jz,lomont1959applications,Georgi-ml-1999wka}, \begin{equation} \label{eq:Dn_multi_rules} R^{n}=S^{2}=(RS)^2=1\,, \end{equation} where$R$refers to rotation and$S$is the reflection. As a consequence, all the group elements of$D_{n}$can be expressed as \begin{equation} g=S^{\alpha}R^{\beta}\, \end{equation} \looseness=-1 where$\alpha=0, 1$and$\beta=0, 1, \dots, n-1$. Then it is straightforward to determine the conjugacy classes of the dihedral group. Depending on whether the group index$n$is even or odd, the$2n$group elements of$D_n$can be classified into three or five types of conjugacy classes. \begin{itemize} \item{$nis odd} \begin{equation} \begin{aligned} 1C_{1}&=\{1\}\,,\\ 2C_{m}^{(\rho)}&=\{R^{\rho},R^{-\rho}\}\,,~~\text{with}~~\rho=1,\dots,\frac{n-1}{2}\,,\\ nC_{2}&=\{S,SR,SR^{2},\dots,SR^{n-1}\}\,, \end{aligned} \end{equation} wherem$is minimal integer such that the identity$m\rho=0 \,(\mathrm{mod}~n)$is satisfied, and$kC_l$denotes a conjugacy class of$k$elements whose order are$l$. \item{$nis even} \begin{equation} \begin{aligned} 1C_{1}&=\{1\}\,,\\ 1C_{2}&=\{R^{n/2}\}\,,\\ 2C_{m}^{(\rho)}&=\{R^{\rho},R^{-\rho}\}\,,~~\text{with}~~\rho=1,\dots,\frac{n-2}{2}\,,\\ \frac{n}{2}C_{2}&=\{S,SR^{2},SR^{4},\dots,SR^{n-4},SR^{n-2}\}\,,\\ \frac{n}{2}C_{2}&=\{SR,SR^{3},\dots,SR^{n-3},SR^{n-1}\}\,, \end{aligned} \end{equation} \end{itemize} The group structure ofD_n$is simple, and the subgroups of$D_{n}group turn out to be either dihedral or cyclic group. The explicit expressions of all the subgroups are \begin{align} \nonumber Z_{j}&= \langle R^{\frac{n}{j}}\rangle \,& \text{with}~~j|n&\,,\\ \nonumber Z_{2}^{(m)}&=\langle SR^{m}\rangle \,& \text{with}~~m&=0,1,\dots,n-1\,,\\ D_{j}^{(m)}&=\langle R^{\frac{n}{j}},SR^{m}\rangle \,&\text{with}~~j|n,~ m&=0,1,\dots,\frac{n}{j}-1\,. \end{align} Hence the total number of cyclic subgroups generated by certain power ofR$is equal to the number of positive divisors of$n$, and the total number of dihedral subgroups is the sum of positive divisors of$n$. The group$D_{n}$only has real one-dimensional and two-dimensional irreducible representations. The number of irreducible representations is dependent on the parity of the group index$n$. \begin{itemize} \item{$n$is odd} If the index$n$is an odd integer, the group$D_{n}$has two singlet representations$\mathbf{1}_{i}$and$\frac{n-1}{2}$doublet representations$\mathbf{2}_{j}$, where the indices$i$and$j$are$i=1, 2$and$j=1,\dots,\frac{n-1}{2}$. We observe that the sum of the squares of the dimensions of the irreducible representations is \begin{equation} 1^2+1^2+2^{2}\times \frac{n-1}{2}=2n\,, \end{equation} which is exactly the number of elements in$D_n$group. In the one-dimensional representations, we have \begin{equation} \mathbf{1}_{1}:~R=S=1\,, \quad \mathbf{1}_{2}:~R=1,~S=-1\,. \end{equation} For the two-dimensional representations, the generators$R$and$S$are represented by \begin{equation} \mathbf{2}_{j}:~R=\left(\begin{array}{cc} e^{2\pi i\frac{j}{n}} & 0 \\ 0 & e^{-2\pi i\frac{j}{n}} \end{array} \right)\,, \quad S=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)\,, \end{equation} with$j=1,\dots,\frac{n-1}{2}$. \item{$n$is even} For the case that the index$n$is an even integer, the group$D_{n}$has four singlet representations$\mathbf{1}_{i}$with$i=1,2,3,4$and$\frac{n}{2}-1$doublet representations$\mathbf{2}_{j}$with$j=1,\dots,\frac{n}{2}-1$. It can be checked that the squared dimensions of the inequivalent irreducible representations add up to the group order as well, \begin{equation} 1^2+1^2+1^2+1^2+2^{2} \times \left(\frac{n}{2}-1\right)=2n\,. \end{equation} The generators$R$and$Sfor the one-dimensional representations are given by \begin{equation} \begin{aligned} \mathbf{1}_{1}:~R&=S=1\,, &\mathbf{1}_{2}:~R&=1,~S=-1\,,\\ \mathbf{1}_{3}:~R&=-1,~S=1\,, & \mathbf{1}_{4}:~R&=S=-1\,. \end{aligned} \end{equation} The explicit forms of these generators in the irreducible two-dimensional representations are \begin{equation} \mathbf{2}_{j}:~R=\left(\begin{array}{cc} e^{2\pi i\frac{j}{n}} & 0 \\ 0 & e^{-2\pi i\frac{j}{n}} \end{array} \right)\,, \quad S=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)\,, \end{equation} withj=1,\dots,\frac{n}{2}-1$. Notice that the doublet representation$\mathbf{2}_{j}$and the complex conjugate$\bar{\mathbf{2}}_{j}$are unitarily equivalent, and they are related through change of basis, i.e.,$R^{*}=URU^{-1}$and$S^{*}=USU^{-1}$where$U=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. Hence all the two-dimensional representations of$D_n$are real representations, although the representation matrix of$R$is complex in the chosen basis. Moreover, if$a=\left(a_1, a_2\right)^{T}$is a doublet transforming as$\mathbf{2}_j$, the complex conjugate$\bar{a}=\left(a^{*}_1, a^{*}_2\right)^{T}$doesn't transform as$\mathbf{2}_j$, but rather$\left(a^{*}_2, a^{*}_1\right)^{T}$transform as$\mathbf{2}_j$under$D_n$. \end{itemize} In order to consistently combine a flavor symmetry$G_f$with the CP symmetry, the subsequent action of the CP transformation, an element of the flavor group and the CP transformation should be equivalent to the action of another element of the flavor group. In other word, the so-called consistency condition has to be fulfilled~\cite{Grimus-ml-1995zi,Feruglio-ml-2012cw,Holthausen-ml-2012dk,Chen-ml-2014tpa} \begin{equation} \label{eq:consis_cond}X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(g)X^{\dagger}_{\mathbf{r}}=\rho_{\mathbf{r}}(g^{\prime}),\quad g,g^{\prime}\in G_{f}\,, \end{equation} where$\rho_{\mathbf{r}}(g)$is the representation matrix of the element$g$in the representation$\mathbf{r}$, and$X_{\mathbf{r}}$is the CP transformation. Here$g$and$g'$are generally different group elements, consequently the CP transformation$X_{\bf r}$is related to an automorphism which maps$g$into$g^{\prime}$. In addition, ref.~\cite{Chen-ml-2014tpa} showed that physical CP transformations should be a class-inverting automorphism of$G_f$, i.e.\$g^{-1}$and$g'$which is the image of$g$under the automorphism should be in the same conjugacy class. We find that the$D_n$groups really have a class-inverting outer automorphism$\mathfrak{u}$, and its action on the generators is \begin{equation} \label{eq:class_inverting_aut_Dn} R\stackrel{\mathfrak{u}}{\longmapsto}R^{-1}\,,\quad S\stackrel{\mathfrak{u}}{\longmapsto}S\,. \end{equation} The CP transformation corresponding to$\mathfrak{u}$is denoted by$X^0_{\mathbf{r}}$, its concrete form is determined by the following consistency conditions, \begin{eqnarray} \nonumber X^0_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(R)X^{0\dagger}_{\mathbf{r}}&=&\rho_{\mathbf{r}}\left(\mathfrak{u}\left(R\right)\right)=\rho_{\mathbf{r}}\left(R^{-1}\right)\,,\\ \label{eq:cons_eq} X^0_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(S)X^{0\dagger}_{\mathbf{r}}&=&\rho_{\mathbf{r}}\left(\mathfrak{u}\left(S\right)\right)=\rho_{\mathbf{r}}\left(S\right)\,. \end{eqnarray} In our working basis shown above,$X^0_{\mathbf{r}}$is fixed to be a unit matrix up to an overall irrelevant phase, \begin{equation} \label{eq:gcp_trans}X^0_{\mathbf{r}}=\mathbb{1}\,. \end{equation} Furthermore, including the inner automorphisms, the full set of CP transformations compatible with$D_{n}$flavor symmetry are \begin{equation} \label{eq:GCP_all}X_{\mathbf{r}}=\rho_{\mathbf{r}}(g)X^0_{\mathbf{r}}=\rho_{\mathbf{r}}(g),\quad g\in D_{n}\,. \end{equation} Hence the CP transformations compatible with$D_n$are of the same form as the flavor symmetry transformations in the chosen basis. ]]> Mixing patterns from \texorpdfstring{\boldmath$D_n$}{D(n)} and CP symmetry breaking to two distinct \texorpdfstring{\boldmath$Z_2\times CP$}{Z(2) x CP} subgroups Mixing patterns from \texorpdfstring{\boldmath$D_n$}{D(n)} and CP symmetry breaking to \texorpdfstring{\boldmath$Z_2$}{Z(2)} and \texorpdfstring{\boldmath$Z_2\times CP$}{Z(2) x CP} subgroups Summary and conclusions Acknowledgments Flavor mixing from the residual symmetry \texorpdfstring{\boldmath$Z_{2}\times CP\$}{Z(2) x CP}

https://doi.org/10.1103/PhysRevD.98.030001 https://doi.org/10.1007/JHEP01%282019%29106 https://doi.org/10.1103/PhysRevD.96.092006 https://doi.org/10.1103/PhysRevD.98.032012 https://doi.org/10.1016/j.physletb.2018.06.019 https://doi.org/10.1016/j.ppnp.2018.05.005 https://doi.org/10.1016/j.physletb.2007.09.032 https://doi.org/10.1103/PhysRevD.77.076004 https://doi.org/10.1103/PhysRevD.88.033018 https://doi.org/10.1103/PhysRevD.88.096002 https://doi.org/10.1103/PhysRevD.92.096010 https://doi.org/10.1142/S0217751X17500476 https://doi.org/10.1007/JHEP07%282013%29027 https://doi.org/10.1007/JHEP04%282013%29122 https://doi.org/10.1016/j.nuclphysb.2014.03.023 https://doi.org/10.1103/PhysRevD.91.033003 https://doi.org/10.1103/PhysRevD.92.073002 https://doi.org/10.1007/JHEP04%282015%29069 https://doi.org/10.1103/PhysRevD.96.035030 https://doi.org/10.1016/j.physletb.2015.12.069 https://doi.org/10.1103/PhysRevD.94.033002 https://doi.org/10.1007/JHEP07%282018%29077 https://doi.org/10.1103/PhysRevD.98.055019 https://doi.org/10.1007/JHEP03%282019%29036 https://doi.org/10.1007/JHEP12%282013%29006 https://doi.org/10.1016/j.nuclphysb.2016.09.005 https://doi.org/10.1007/JHEP05%282013%29084 https://doi.org/10.1140/epjc/s10052-014-2753-2 https://doi.org/10.1016/j.nuclphysb.2014.02.002 https://doi.org/10.1007/JHEP08%282015%29017 https://doi.org/10.1103/PhysRevD.95.015012 https://doi.org/10.1007/JHEP12%282017%29022 https://doi.org/10.1103/PhysRevD.92.036007 https://doi.org/10.1016/j.nuclphysb.2015.07.024 https://doi.org/10.1088/1674-1137/39/2/021001 https://doi.org/10.1007/JHEP06%282014%29023 https://doi.org/10.1007/JHEP05%282015%29100 https://doi.org/10.1007/JHEP08%282015%29037 https://doi.org/10.1103/PhysRevD.92.093008 https://doi.org/10.1103/PhysRevD.92.116007 https://arxiv.org/abs/1811.09662 https://doi.org/10.1103/PhysRevD.89.093020 https://doi.org/10.1016/j.nuclphysb.2014.12.013 https://doi.org/10.1103/PhysRevD.93.025013 https://doi.org/10.1007/JHEP12%282014%29007 https://doi.org/10.1007/JHEP05%282016%29007 https://doi.org/10.1007/JHEP03%282016%29206 https://doi.org/10.1016/j.nuclphysb.2017.03.015 https://doi.org/10.1103/PhysRevD.96.075005 https://doi.org/10.1007/JHEP02%282018%29038 https://doi.org/10.1103/PhysRevD.98.055011 https://arxiv.org/abs/1811.07750 https://arxiv.org/abs/1811.09262 https://doi.org/10.1016/j.physletb.2017.12.049 https://doi.org/10.1103/PhysRevD.94.073006 J. Lomont, Applications of finite groups, Academic Press, New York, NY, U.S.A. and London, U.K. (1959). H. Georgi, Lie algebras in particle physics, Front. Phys. 54 (1999) 1 [https://inspirehep.net/search?p=find+recid+1236686]. https://doi.org/10.1016/S0370-1573%2896%2900030-0 https://doi.org/10.1103/PhysRevLett.55.1039 https://doi.org/10.1103/RevModPhys.84.515 https://doi.org/10.1016/0370-2693%2886%2990307-2 https://doi.org/10.1103/PhysRevD.36.315 https://doi.org/10.1103/PhysRevD.64.076005 https://doi.org/10.1088/0954-3899/43/3/030401 https://doi.org/10.1016/j.nuclphysBPS.2015.06.024 https://arxiv.org/abs/1601.05471 https://arxiv.org/abs/1512.06148 https://arxiv.org/abs/1601.05823 https://arxiv.org/abs/1601.02984 https://arxiv.org/abs/1309.0184 https://arxiv.org/abs/1412.4673 https://doi.org/10.1093/ptep/pty044 https://doi.org/10.1103/PhysRevD.57.6989 https://doi.org/10.1016/S0550-3213%2899%2900070-X https://doi.org/10.1088/0034-4885/72/10/106201 https://doi.org/10.1103/PhysRevLett.109.032505 https://doi.org/10.1038/nature13432 https://doi.org/10.1103/PhysRevLett.110.062502 https://arxiv.org/abs/1807.06209 http://www.utfit.org/UTfit/ResultsSummer2018SM