Dihedral flavor group as the key to understand quark and lepton flavor mixing

We have studied the lepton and quark mixing patterns which can be derived from the dihedral group Dn 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 Dn. A unified description of the observed structure of the quark and lepton mixing can be achieved if the flavor group Dn and CP are broken to Z2 × CP in neutrino, charged lepton, up quark and down quark sectors, and the minimal group is D14. We also consider another scenario in which the residual symmetry of the charged lepton and up quark sector is Z2 while Z2 × CP remains preserved by the neutrino and down quark mass matrices. Then D7 can give the experimentally favored values of CKM and PMNS mixing matrices.


Introduction
It is well established that the three generations of quarks are mixed with each other to form mass eigenstates in the standard model. The quark flavor mixing matrix appearing in the weak charged-current interactions is referred to as the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1]. The quark mixing angles exhibit a strongly hierarchical structure, and the largest one is the Cabibbo mixing angle θ c 13 • between the first and the second generation. Observation of neutrino oscillation implies that neutrinos have masses and non-zero mixing. Analogously there should be a lepton mixing matrix in the weak chargedcurrent interactions, and it is usually called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [1]. However, the lepton mixing angles are less hierarchical, both solar and atmospheric mixing angles are large and the reactor angle θ 13 8.61 • is of a similar size to the Cabibbo angle [2]. As regards the CP violation phase, the CP violation in the quark sector has been precisely measured in a variety of meson decay processes. It is confirmed that the single complex phase in the CKM matrix is the dominant source of CP violation, and the angle α in the unitary triangle is determined to be α = (84.5 +5. 9 −5.2 ) • [1]. If neutrinos are Majorana particles, there are additional sources of CP violation in the lepton sector, e.g., the Majorana phases in the PMNS matrix. At present CP conservation in neutrino oscillation is disfavored at 2σ level, and the exact values of the leptonic Dirac CP phase δ CP is unknown although weak evidence for δ CP around 3π/2 is reported by T2K [3] and NOνA [4]. Non-zero δ CP is also preferred by global analysis of neutrino oscillation data [2,5,6].
The origin of the above flavor mixing structures of quarks and leptons is one of the most important problems in standard model. Many proposals have been advanced to explain this JHEP03(2019)056 puzzle in the literature. In particular, the non-abelian discrete flavor symmetries appear to be particularly suitable to reproduce the large flavor mixing angles of the leptons. In this paradigm, the three generations of left-handed lepton doublets are usually assumed to transform as a three-dimensional representation of the discrete flavor group G f which is subsequently broken to different subgroups in the charged lepton and neutrino sectors, the mismatch between the two subgroups allows one to predict the PMNS matrix up to permutations of rows and columns. In a similar way, the mismatch of residual symmetries in up and down quark sectors can be employed to determine the CKM mixing matrix [7][8][9][10][11][12]. However, no finite group has been found that can predict the correct values of the three different quark mixing angles at leading order, and only the Cabibbo angle can be generated [11,12].
A recent progress is to extend the discrete flavor symmetry with CP symmetry [13][14][15]. This approach turns out to be quite powerful and it allows for precise predictions of both lepton mixing angles and CP violating phases. It can lead to very predictive scenarios, where all the mixing angles and CP phases are related to a small number of input parameters [16][17][18][19][20][21][22][23][24]. Many models and analyses of CP and flavor symmetries have been studied so far, e.g. A 4 [25,26], S 4 [27][28][29][30][31][32], ∆ (27) [33,34], ∆ (48) [35,36], A 5 [37][38][39][40][41], ∆(96) [42] and the infinite group series ∆(3n 2 ) [43,44], ∆(6n 2 ) [43,45] and D (1) 9n,3n [46]. Flavor and CP symmetries can also constrain the CP violation in leptogenesis [47][48][49]. Moreover, after including CP symmetry, we can achieve a unified description of quark and lepton flavor mixing from a single flavor symmetry group if the residual symmetries of the charged lepton, neutrino, up quark and down quark sectors are different Z 2 × CP subgroups, and the minimal flavor symmetry is ∆(294) [50]. One could reduce the residual subgroups of the charged lepton and up quark sectors to Z 2 while distinct Z 2 ×CP residual symmetries remain preserved by the neutrino and up quark mass matrices. Then either PMNS or CKM matrices depend on only three real free parameters, and the ∆(294) is still the smallest flavor symmetry group to generate the experimentally preferred quark and lepton mixing patterns [51]. There are other proposals to explain lepton and quark mixing from flavor symmetry and CP symmetry, see [51][52][53] for different perspectives. We observe that the group order of the required flavor symmetry is a bit large.
The dihedral group D n with general n is the group of symmetries of a regular polygon and it doesn't have irreducible three-dimensional representation. It is found that the phenomenologically acceptable Cabibbo angle can be accommodated by the dihedral group D 7 [7,8]. In the present work, we shall extend the dihedral flavor group to involve also CP as symmetry. 1 Both left-handed lepton doublets and quark doublets are assumed to transform as a reducible three-dimensional representation which is the direct sum of a singlet and a doublet representation of D n . We shall analyze the mixing patterns for leptons and quarks arising from the breaking of D n and CP symmetry into Z 2 × CP in all the relevant quark and lepton sectors. It is remarkable that the D 14 group of order 28 can give an acceptable prediction for the PMNS and the CKM matrices at leading order. The possible mixing patterns are also studied for the second scenario in which the residual symmetries of the charged lepton and up quark mass matrices are reduced to Z 2 . Then we find that both PMNS and CKM mixing matrices can be accommodated by the dihedral group D 7 .

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The structure of the paper is as follows: in section 2 we show the general constraints on the mass matrix and how to extract the mixing matrix if a residual symmetry Z g ψ 2 × X ψ or Z g ψ 2 is preserved by the fermion fields ψ ∈ {ν, e, u, d}. In section 3 we present the mathematical properties of the dihedral group D n which is employed as flavor symmetry, and the CP transformations compatible with the D n flavor group are discussed. In section 4 we analyze the predictions for lepton and quark mixing if the flavor symmetry D n and CP symmetry are broken to different Z 2 × CP subgroups in the charged lepton (up quark) and neutrino (down quark) sectors. In section 5 we study the scenario that the residual symmetries of charged lepton and up quark sectors are Z 2 subgroups while the neutrino and down quark mass matrices are invariant under Z 2 × CP . Finally we summarize our main results and conclude in section 6. We show the constraints on the unitary transformation U ψ for a generic Z g ψ 2 × X ψ residual symmetry in appendix A.

Lepton and quark mixing from residual symmetry
In the following, we briefly review how the lepton mixing can be predicted from a flavor symmetry group G f and a CP symmetry which are broken down to two different subgroups of the structure Z 2 × CP in the charged lepton and neutrino sectors [31,50,51]. The quark CKM mixing matrix can be derived in an analogous way using this method. We assume that the three generations of left-handed leptons doublets L ≡ (ν, e) T and quark doublets Q ≡ (u, d) T transform as a three-dimensional representation ρ of G f . Notice that the following results for mixing matrix hold true no matter whether ρ is a reducible or irreducible representation of G f . For each type of fermionic field ψ ∈ {ν, e, u, d}, the corresponding residual symmetry is denoted as Z g ψ 2 × X ψ , where g ψ is the generator of the residual flavor symmetry Z g ψ 2 and it fulfills g 2 ψ = 1. The residual CP transformation X ψ should be a symmetric and unitary matrix otherwise the mass spectrum would be partially or completely degenerate [16,17]. The restricted consistency condition between the residual flavor and CP symmetries reads [13,[16][17][18][19]27] Under the action of residual symmetry, the field ψ transforms as where C is the charge conjugation matrix with Px = (t, − x). For the residual symmetry to hold, the mass matrix should satisfy the following conditions if ψ is Dirac field, where the mass matrix m ψ is defined in the right-left basis ψ c m ψ ψ. If ψ (neutrino in the standard model) is Majorana field, the invariance conditions are

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where the Majorana mass matrix m ψ is defined as 1 2 ψ T m ψ ψ. We can diagonalize the mass matrices m † ψ m ψ and m ψ with a unitary transformation U ψ ,  [31,50,51], and , for Dirac field ψ , diag(±1, ±1, ±1), for Majorana field ψ , where α 1,2,3 are real free parameters, P ψ is a generic three dimensional permutation matrix and it can take six possible forms generated by As shown in [31,50,51,55], one can solve the constraint equations of eqs. (2.6), (2.7) by performing Takagi factorization for the residual CP transformation X ψ with the properties given in appendix A as Then the residual symmetry Z g ψ 2 × X ψ would determine the unitary transformation U ψ to be of the form [31,50,51,55] where the free parameter θ ψ can be limited in the range 0 ≤ θ ψ < π without loss of generality. Since both quark and charged lepton carry non-zero electric charges, only neutrinos in standard model can be Majorana particle. From eq. (2.10) we can see that the difference between Dirac and Majorana neutrinos is in the diagonal matrix Q ψ . In the case of Dirac neutrinos, Q ψ is a general phase matrix and it can be rotated away by appropriate field redefination. For Majorana neutrinos, the diagonal entries of Q ψ are ±1 and ±i and the Majorana phases can be predicted up to π. If only a residual flavor symmetry Z g ψ 2 is preserved by the mass matrix of ψ, the unitary transformation U ψ would be only subject to the constraint in eq. (2.6). Since g ψ is of order JHEP03(2019)056 two and its representation matrix ρ(g ψ ) has two degenerate eigenvalues, the residual flavor symmetry Z g ψ 2 can only distinguish one generation from the other two ones. As a result, only one column of U ψ is fixed and it takes the following form [51] where Σ ψ diagonalizes ρ(g ψ ) as in eq. (2.9), Q ψ is an arbitrary diagonal phase matrix, and U 23 (θ ψ , δ ψ ) is a block diagonal unitary rotation in the (23)-plane with where the angles θ ψ and δ ψ need only be defined over the intervals [0, π/2] and [0, π) respectively. In comparison with the residual symmetry Z g ψ 2 × X ψ , the Majorana phases cannot be predicted for Majorana neutrinos in this case.
As a result, if the flavor and CP symmetries are broken to Z gν 2 × X ν and Z ge 2 × X e in the neutrino and charged lepton sectors respectively, the lepton mixing matrix would be given by (2.14) In this scenario, all the mixing angles and CP phases only depend on two real rotation angles θ e and θ ν in the interval between 0 and π. If neutrinos are Majorana particles, Q ν is a diagonal matrix with elements ±1 and ±i, and without loss of generality it can be parameterized as with k 1,2 = 0, 1, 2, 3. Analogously the residual symmetries Z gu 2 ×X u and Z g d 2 ×X d in the up type quark and down type quark sectors allow us to pin down the CKM mixing matrix as In the following, we shall consider a second scenario in which the residual symmetry of the charged lepton sector is degraded to Z ge 2 such that the neutrino and charged lepton mass matrices exhibit the residual symmetries Z ge 2 and Z gν 2 × X ν respectively. Then the lepton mixing matrix is given by which depends on three free parameters θ e , δ e and θ ν . In the same fashion, if the residual symmetries Z gu 2 and Z g d 2 × X d are preserved in the up and down quark sectors respectively, the quark mixing matrix would be determined by One can also straightforwardly extract the expression of the CKM mixing matrix for the residual symmetry Z gu 2 × X u and Z g d 2 .
Before closing this section, we note that the above schemes can be extended to the grand unification theory if both left-handed quarks and leptons could be assigned to the same representation. In the following sections, we assume neutrinos are Majorana particles.

JHEP03(2019)056 3 Dihedral group and CP symmetry
The dihedral group D n is the symmetry group of an n-sided regular polygon for n > 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 × Z 2 . The group D n is the semidirect product Z n 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 [8,56,57], where R refers to rotation and S is the reflection. As a consequence, all the group elements of D n can be expressed as where α = 0, 1 and β = 0, 1, . . . , 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.
• n is odd where m is minimal integer such that the identity mρ = 0 (mod n) is satisfied, and kC l denotes a conjugacy class of k elements whose order are l.
• n is even 1C 1 = {1} , The group structure of D 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  Hence the total number of cyclic subgroups generated by certain power of R 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.

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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.
• n is odd If the index n is an odd integer, the group D n has two singlet representations 1 i and n−1 2 doublet representations 2 j , where the indices i and j are i = 1, 2 and j = 1, . . . , n−1 2 . We observe that the sum of the squares of the dimensions of the irreducible representations is which is exactly the number of elements in D n group. In the one-dimensional representations, we have For the two-dimensional representations, the generators R and S are represented by • n is even For the case that the index n is an even integer, the group D n has four singlet representations 1 i with i = 1, 2, 3, 4 and n 2 − 1 doublet representations 2 j with j = 1, . . . , n 2 − 1. It can be checked that the squared dimensions of the inequivalent irreducible representations add up to the group order as well, The generators R and S for the one-dimensional representations are given by The explicit forms of these generators in the irreducible two-dimensional representations are with j = 1, . . . , n 2 −1. Notice that the doublet representation 2 j and the complex con-jugate2 j are unitarily equivalent, and they are related through change of basis, i.e., representations of D n are real representations, although the representation matrix of R is complex in the chosen basis. Moreover, if a = (a 1 , a 2 ) T is a doublet transforming as 2 j , the complex conjugateā = (a * 1 , a * 2 ) T doesn't transform as 2 j , but rather (a * 2 , a * 1 ) T transform as 2 j under D n .
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 [13][14][15]58] where ρ r (g) is the representation matrix of the element g in the representation r, and X r is the CP transformation. Here g and g are generally different group elements, consequently the CP transformation X r is related to an automorphism which maps g into g . In addition, ref. [15] 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 u, and its action on the generators is The CP transformation corresponding to u is denoted by X 0 r , its concrete form is determined by the following consistency conditions, (3.14) In our working basis shown above, X 0 r is fixed to be a unit matrix up to an overall irrelevant phase, Furthermore, including the inner automorphisms, the full set of CP transformations compatible with D n flavor symmetry are 4 Mixing patterns from D n and CP symmetry breaking to two distinct Z 2 × CP subgroups In this section, we shall consider the dihedral group D n as flavor symmetry G f which is combined with CP symmetry. The three generations of left-handed lepton and quark JHEP03(2019)056 doublets are assumed to transform as a direct sum of one-dimensional representation 1 i and two-dimensional representation 2 j of D n , Notice that usually the three lepton doublets are assigned to an irreducible threedimensional representation of G f in order to obtain at least two non-vanishing lepton mixing angles. In the following, we shall show that the singlet plus doublet assignment can also accommodate the experimental data on mixing angles after the CP symmetry is considered. Moreover, one can also assign either the second or the third generation lepton (quark) doublet to a one-dimensional representation of D n while the remaining two generations transform as a two-dimensional representation of D n . In the same fashion as section 2, we can see these alternative assignments don't give rise to new predictions for the quark and lepton mixing matrices.
Here we assume that the discrete flavor group D n in combination with CP symmetry is broken down to Z 2 × CP in both charged lepton and neutrino sectors, then one entry of the mixing matrix is completely fixed by the residual symmetry. Considering all possible residual symmetries Z ge 2 × X e and Z gν 2 × X ν , we find the fixed element can only be 0, 1 or cos ϕ 1 , where the value of ϕ 1 is determined by the choice of the residual symmetry. Obviously only the mixing pattern with the fixed element cos ϕ 1 could be in agreement with experimental data for certain values of ϕ 1 characterizing the residual symmetry. As a consequence, we find the viable residual symmetries in the lepton sector are Z ge where z e , z ν = 0, 1, . . . , n−1, x = y = 0 for odd n and x, y = 0, n/2 if the group index n is even. Following the procedures presented in appendix A, we can determine the Takagi factorization of the residual symmetry Z SR z 2 × {R −z+x , SR x } as follow, with σ = 0, 1, z, z + 1 for i = 1, 2, 3, 4 respectively, and the parameter ρ depends on the values of x and i which is the index of the one-dimensional representation 1 i , As a consequence, the lepton mixing matrix reads odd j(zν − ze)π/n + π/2 (zν − ze)π/2 + (2j + n)π/4 + π/2 Table 1. The values of the parameters ϕ 1 and ϕ 2 for the residual symmetries Z ge The parameters ϕ 1 and ϕ 2 are discrete group parameters characterizing the residual symmetry, and all possible values of ϕ 1 and ϕ 2 are summarized in table 1. We can see that ϕ 1 and ϕ 2 can take the following discrete values ϕ 1 (mod 2π) = 0, 1 n π, 2 n π, . . . , 2n − 1 n π , ϕ 2 (mod 2π) = 0, 1 2 π, π, 3 2 π . (4.6) In particular, ϕ 2 would be 0 or π if the group index n is odd or x = y = 0, n/2 for even n. It is easy to check that the mixing matrix in eq. (4.4) has the following symmetry properties, where the above diagonal matrices can be absorbed into the lepton fields. Therefore the parameters ϕ 1 and ϕ 2 can be limited in the ranges 0 ≤ ϕ 1 ≤ π/2 and 0 ≤ ϕ 2 < π JHEP03(2019)056 respectively, and the values of ϕ 2 are 0 and π/2 in the fundamental region. In the case of ϕ 2 = 0, the mixing matrix in eq. (4.4) is real such that all CP phases are conserved. The evidence of CP violation in neutrino oscillation has been reported by T2K [3] and NOνA [4] and nontrivial δ CP is also preferred by global data analysis [2]. Hence we shall focus on ϕ 2 = π/2 in the following.
The absolute value of the Jarlskog invariant J CP [59] is the same for all the nine mixing patterns, i.e. |J CP | = 1 2 | sin 2θ e sin 2θ ν sin 2 ϕ 1 cos ϕ 1 sin ϕ 2 |. The symbols I 1 and I 2 are Majorana CP invariants [60][61][62][63]. The parameter F is defined as F = 1 2 sin 2θ e sin 2θ ν cos ϕ 1 . The allowed region of ϕ 1 in the second column is obtained by requiring the fixed element cos ϕ 1 is in the experimentally preferred 3σ range [2]. The notation n min denotes the minimal value of n which can accommodate the measured values of the lepton mixing angles for ϕ 2 = π/2. Notice that only the value of ϕ 2 = π/2 in its fundamental interval can generate a non-trivial Dirac CP phase in the scenario of section 4.

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The current and upcoming neutrino experiments will be able to significantly reduce the experimental errors on θ 12 and θ 23 , and the next generation long-baseline experiments are expected to considerably improve the sensitivity to the Dirac phase δ CP if running in both the neutrino and the anti-neutrino modes. Thus the above sum rules could be tested and possibly distinguished from each other in future, or to be ruled out entirely.
It is notable that the experimental data on lepton mixing angles can be explained by small dihedral group D 4 which is the symmetry group of a square. For instance, if the residual symmetry is specified by x = 2, y = 0, z ν = 1, z e = 0 and the three left-handed lepton doublets are assigned to 1 1 ⊕ 2 1 , we have ϕ 1 = π/4 and ϕ 2 = π/2, the mixing angles for the mixing matrices U I, 6 and U I,9 can be quite close to their best fit values for certain choices of parameters θ ν and θ e . As a measure for the goodness of fit, we perform a global fit using the χ 2 function which is defined in the usual way. 2 The numerical results are listed in table 3. We can see that the deviation of θ 23 from maximal value can be accommodated and the Dirac phase δ CP could be around 1.5π. Moreover, we display the contour regions for sin 2 θ ij , | sin δ CP |, | sin α 21 | and | sin α 31 | in figure 1. As one can clearly see, the rotation angles θ e and θ ν are strongly constrained to accommodate the three lepton mixing angles θ ij within the experimentally preferred 3σ intervals (black areas in the figure). Therefore the allowed ranges of the mixing angles and CP phases should be rather narrow around the numerical values in table 3 and the present approach is very predictive.
The upcoming reactor neutrino oscillation experiments, such as JUNO [64] and RENO-50 [65], expect to make very precise measurements of the solar neutrino mixing angle θ 12 . They will be capable of reducing the error of θ 12 to about 0.1 • or around 0.3%. Future long baseline experiments DUNE [66][67][68][69], T2HK [70,71], T2HKK [72] can make very precise measurements of the oscillation parameters θ 12 , θ 23 and δ CP . Therefore future neutrino facilities have the potential to test the predictions of our models. Furthermore, we expect that a more ambitious facility such as the neutrino factory [73][74][75] could provide a more stringent test of our approach.
The neutrinoless double beta (0νββ) decay is the unique probe for the Majorana nature of neutrinos, and it explicitly depends on the values of the Majorana CP violation phases. The 0νββ decay experiments can provide valuable information on the neutrino mass spectrum and constrain the Majorana phases. The 0νββ decay rate is proportional to the effective Majorana mass |m ee | which is the (11) element of the neutrino mass matrix in the charged lepton diagonal basis, |m ee | = |m 1 cos 2 θ 12 cos 2 θ 13 + m 2 sin 2 θ 12 cos 2 θ 13 e iα 21 + m 3 sin 2 θ 13 e i(α 31 −2δ CP ) | . (4.13) For the mixing pattern U I,6 with ϕ 1 = π/4 and ϕ 2 = π/2, we show the prediction of the effective Majorana mass |m ee | as a function of the lightest neutrino mass m lightest in figure 2. If the neutrino mass spectrum is IO, the effective mass |m ee | is around 0.046 eV or in the narrow interval [0.020eV, 0.026eV]. Since the next generation 0νββ decay experiments will be able to explore the whole region of the IO parameter space such that these predictions 2 The information of δCP is not included in the χ 2 function because it is measured with large uncertainties at present and the indication of its preferred value from global data analyses is rather weak [2]. could be tested. In the case of NO spectrum, the effective Majorana mass has a lower limit |m ee | ≥ 0.0011 eV.
Regarding the quark mixing, it is described by the CKM matrix which is parameterized by three mixing angles θ q ij and one CP-violating phase δ q CP [1]. with n = 4 and ϕ 1 = π/4, ϕ 2 = π/2. The red (blue) dashed lines indicate the most general allowed regions for IO (NO) neutrino mass spectrum obtained by varying the mixing parameters over their 3σ ranges [2]. The present most stringent upper limits |m ee | < 0.120 eV from EXO-200 [76,77] and KamLAND-ZEN [78] is represented by horizontal grey band. The vertical grey exclusion band denotes the current bound coming from the cosmological data of m i < 0.130 eV at 95% confidence level obtained by the Planck collaboration [79], the dashed and solid black lines are for IO and NO respectively. mixing angles and Jarlskog invariant, given by the UTfit collaboration [80], read as sin θ q 12 = 0.22500 ± 0.00100, sin θ q 13 = 0.003675 ± 0.000095 , sin θ q 23 = 0.04200 ± 0.00059, J q CP = (3.120 ± 0.090) × 10 −5 . (4.14) Similar to the lepton sector, the precisely measured CKM mixing matrix can only be explained by the residual symmetry Z gu and X d = {R −z d +y , SR y } where z u , z d = 0, 1, . . . , n − 1, x = y = 0 for odd n and x, y = 0, n/2 for even n. The CKM matrix is predicted to be of the same form as eq. (4.4) and it reads The values of the parameters ϕ 1 and ϕ 2 are summarized in table 1 where z e and z ν should be replaced by z u and z d respectively. The parameter ϕ 2 can be 0, π/2, π and 3π/2, and it should be π/2 or 3π/2 to explain the observed CP violation in the quark sector. We have numerically analyzed all the D n groups with n ≤ 40 and find that the experimentally measured quark mixing matrix can only be described by the following two permutations  Table 3. Results of the χ 2 analysis for the phenomenologically viable mixing patterns U I, 6 and U I,9 with ϕ 1 = π/4 and ϕ 2 = π/2 which can be achieved from the D 4 flavor group. Since the global fit results of the mixing angles slightly differ for normal ordering (NO) and inverted ordering (IO) neutrino mass spectrums, we consider these two cases separately. The χ 2 function reaches a global minimum χ 2 min at the best fit values (θ e , θ ν ) = (θ bf e , θ bf ν ). We display the values of the mixing angles and CP violation phases at the best fitting points.
Accordingly the smallest value of group index n which can accommodate the experimental data is n = 14. The expressions of the mixing parameters can be read from table 2 by replacing θ e and θ ν with θ u and θ d respectively.
For the mixing pattern V I,1 , the correlations in eq. (4.11a) are also satisfied for the quark mixing angles θ q ij and Jarlskog invariant J q CP . Moreover we can express the quark CP violation phase δ q CP in terms of mixing angles, sin δ q CP sin 2ϕ 1 sin 2θ q 12 cos 2 θ q 13 cos θ q 23 . (4.17) If we assign the quark doublets to 1 1 ⊕2 1 of the D 14 group and choose the residual symmetry x = 7, y = 0, z d = 1, z u = 0, we have ϕ 1 = π/14, ϕ 2 = π/2 and the observed quark flavor mixing parameters can be accommodated for certain choices of the free parameters θ u,d ,  We see that sin θ q 13 , sin θ q 23 and J q CP are compatible with the global fitting results of the UTfit collaboration [80]. The mixing angle sin θ q 12 is about 1% smaller than its measured value and it could be brought into agreement with experimental data in an explicit model since small higher order corrections are generally expected to arise.
(4. 19) We see that sin θ q 23 is in the experimentally preferred region, the relative deviations of sin θ q 12 , sin θ q 13 and J q CP from their best fit values are about 1%. This tiny discrepancy should be easily resolved by higher order corrections or renormalization group evolution effects. Furthermore, the flavor group D 14 in combination with CP symmetry can reproduce the experimentally favored values of lepton mixing angles if it is broken down to Z ge 2 × X e and Z gν 2 × X ν in charged lepton sector and neutrino sector respectively. We could choose the residual symmetry specified by x = 7, y = 0, z ν = 4 and z e = 0, then the discrete parameters are ϕ 1 = 2π/7 and ϕ 2 = π/2. The mixing pattern U I,9 can accommodate the three lepton mixing angles very well, and the best fit values of the mixing parameters are θ bf e = 0.439π, θ bf ν = 0.814π, χ 2 min = 1.841, sin 2 θ 13 = 0.0224, sin 2 θ 12 = 0.310 , (4.20) sin 2 θ 23 = 0.602, δ CP /π = 1.532, α 21 /π = 0.167 (mod 1), α 31 /π = 0.116 (mod 1) .
We would like to remind the readers that the smallest flavor group is ∆(294) which can accommodate quark and lepton flavor mixing simultaneously if both left-handed quarks and leptons are assigned to an irreducible triplet of G f and the residual symmetries are Z 2 × CP [50]. Hence the singlet plus doublet assignment seems better than the triplet assignment after including CP symmetry, the order of the flavor symmetry group can be reduced considerably, i.e. 28 versus 294 in this scheme. In particular, the simple dihedral group D n allows for a unified description of quark and lepton mixing. The dihedral group together with the residual symmetry Z 2 × CP indicated above provides an interesting opportunity for model building.
The dihedral group D n is a subgroup of ∆(6n 2 ) [50], therefore all the results of D n can be obtained from the ∆(6n 2 ) flavor symmetry. We find that the mixing pattern U I in eq. (4.4) coincides with the mixing matrix for the case I of ref. [50] up to the redefinition of the free parameters θ e and θ ν , while the allowed values of ϕ 1 and ϕ 2 are different. For the phenomenologically viable flavor groups D 14 and ∆(294), only the mixing pattern U I, 9 can agree well with the experimental data on lepton mixing angles with non-trivial Dirac CP phase δ CP , and the results of χ 2 analysis are shown in table 4. We see that D 14 and ∆(294) give the same lepton mixing angles as well as the same χ 2 min while the predictions for the CP violation phases depend on the values of ϕ 2 . As a result, at least precise measurement of δ CP is necessary in order to distinguish D 14 from ∆(294).
5 Mixing patterns from D n and CP symmetry breaking to Z 2 and Z 2 × CP subgroups Similar to section 4, the left-haded lepton and quark doublets are assigned to the reducible representation 1 i ⊕ 2 j , as shown in eq. (4.1). In this section, we consider the scenario that the residual symmetries of the neutrino and charged lepton mass matrices are Z gν 2 × X ν and Z ge 2 respectively arising from the flavor group D n and CP. Considering all possible choices for g ν , X ν and g e , we find only the residual symmetry Z ge where δ = ϕ 2 + 2δ e , s e = sin θ e , s ν = sin θ ν , c e = cos θ e , c ν = cos θ ν , the permutation matrices P e,ν and phase matrices Q e,ν are omitted. The parameters ϕ 1 and ϕ 2 are determined by residual symmetry, and their admissible values are summarized in table 5. We can see that ϕ 1 takes the following discrete values ϕ 1 (mod 2π) = 0, 1 n π, 2 n π, . . . , 2n − 1 n π. (5. 2) The second discrete parameter ϕ 2 appears in U II through the combination δ = ϕ 2 + 2δ e , the value of ϕ 2 is irrelevant since it can be absorbed into the continuous free parameter δ e . Comparing eq. (5.1) with eq. (4.4), we see that U II can be obtained from U I by replacing ϕ 2 with δ. Therefore the parameter ϕ 1 can be limited in the interval 0 ≤ ϕ 1 ≤ π/2, and the variation ranges of the free parameters θ e , θ ν and δ can be taken to be 0 ≤ θ e ≤ π/2, 0 ≤ θ ν < π and 0 ≤ δ < π respectively. In this approach, we can not make any prediction for the lepton masses, consequently the lepton mixing matrix is determined up to independent row and column permutations.
In order to show concrete examples and find new interesting mixing, we have numerically scanned over the free parameters θ e , θ ν and δ and all possible values of the discrete parameter ϕ 1 for each integer group index n. We find the smallest dihedral group which can accommodate the data is D 3 ∼ = S 3 . Note that D 3 group with n = 3 is the symmetry group of an equilateral triangle, and then ϕ 1 can be either 0 or π/3 in the fundamental region of ϕ 1 ∈ [0, π/2]. Only the value ϕ 1 = π/3 can generate a viable mixing pattern, and it can  be achieved from the residual symmetry g e = S, g ν = SR, X ν = R 2 , S under the lepton doublets assignment 1 1 ⊕ 2 1 . Accordingly the fixed element is cos ϕ 1 = 1/2 and it can be the (21), (22), (31) or (32) entry of the lepton mixing matrix. Hence only the mixing patterns U II,4 , U II,5 , U II,7 and U II,8 can be compatible with experimental data. Requiring all the three mixing angles θ 12 , θ 13 and θ 23 in the 3σ intervals of global fit [2], we can obtain the allowed regions of the mixing angles and CP violation phases and the numerical results are summarized in table 6. We can find that for different viable mixing patterns, the predictions for θ 13 are almost same while the allowed regions of θ 12 , θ 23 , δ CP , α 21 and α 31 are different. The neutrino oscillation experiments, such as JUNO [64], RENO-50 [65], and the future long baseline experiments DUNE [66][67][68][69], T2HK [70,71], T2HKK [72] allow a measurement of θ 12 , θ 23 and δ CP with significantly improved sensitivities. Thus a discrimination between the above possible cases in table 6 will be possible. Subsequently we extend the above scheme to the quark sector, the flavor symmetry D n and CP symmetry are broken down to Z gu 2 and Z g d 2 × X d in the up quark and down quark sectors respectively. The CKM mixing matrix can be correctly reproduced if the residual symmetry is g u = SR zu , g d = SR z d and X d = {R −z d +x , SR x } with z u , z d = 0, 1, . . . , n − 1, x = 0 for odd n and x = 0, n/2 for even n. The CKM matrix is determined to be up to permutations of rows and columns, where c u = cos θ u , c d = cos θ d , s u = sin θ u , s d = sin θ d and δ = ϕ 2 + 2δ u . The values of the discrete parameters ϕ 1 and ϕ 2 can be read from table 5 by substituting z e and z ν with z u and z d respectively. Furthermore, it is straightforward to check that the same mixing pattern would be obtained if the residual symmetry is instead Z gu Similar to the lepton mixing matrices in eq. (5.3), the row and columns permutations of V II can give rise to nine mixing patterns V II,i (i = 1, . . . , 9). The mixing matrix V II,i can be obtained from U II,i by replacing θ e , θ ν and δ e with θ u , θ d and δ u respectively. We have considered all possible values of the discrete parameters ϕ 1 for each group index n  Table 7. Numerical results of the quark mixing parameters for the permutations of the mixing matrix V II in eq. (5.5), where the residual symmetry is Z gu We have analyzed all the D n groups with n ≤ 40. Here we show the values of sin θ q ij and J q CP which are compatible with the experimental data for certain choices of θ u , δ u , θ d and ϕ 1 .
with n ≤ 40. We scan over the free parameters θ u , δ u and θ d in the range from 0 and π to determine whether a good fit to the experimental data can be achieved. For the D n groups with n ≤ 40, we find six permutations V II,1 , V II,2 , V II,4 , V II,5 , V II,6 , and V II,8 can describe the measured values of the quark mixing parameters shown in eq. (4.14) . The values of n, ϕ 1 and the resulting predictions for sin θ q ij and J q CP at certain benchmark values of θ u , δ u , θ d are summarized in table 7. We see that the smallest group index n which can accommodate the experimental data is n = 7 and accordingly the mixing patterns are V II,2 and V II,4 .
Furthermore, we notice that the D 7 group and CP symmetry can also generate phenomenologically viable lepton mixing patterns if the residual symmetries of the charged lepton and neutrino mass matrices are Z ge 2 = Z SR ze 2 and Z gν 2 = Z SR zν 2 , X ν = {R −zν +x , SR x } respectively. We find that only the mixing patterns U II,4 , U II,5 , U II, 8 and U II,9 can agree well with the experimental data on lepton mixing angles, and the discrete parameter ϕ 1 can be 2π/7 or 3π/7. The continuous parameters θ e , δ and θ ν are freely varied between 0 and π, and the current 3σ bounds of sin 2 θ ij [2] are imposed. The allowed regions of the lepton mixing angles and CP phases are reported in table 8. As an example, we display the correlations among the different mixing parameters for U II,8 ϕ 1 = 2π 7 in figure 3. Comparing with the scenario of Z 2 × CP residual symmetry in both neutrino and charged lepton sectors, we see that the admissible region of the Dirac phase δ CP is generally more larger. It is remarkable that the D 7 flavor symmetry with group order 14 already can give experimentally favored values of PMNS and CKM matrix. For the irreducible triplet assignment of quark and lepton doublets, we would like to mention that ∆(294) is the minimal flavor group to generate realistic quark and lepton flavor mixing patterns in the present scheme [51].

Summary and conclusions
A compelling theory of flavor mixing is still missing. The discrete flavor symmetry and CP symmetry through the mismatch of residual symmetries is a powerful approach to explain the observed flavor mixing structure of quarks and leptons. In previous work, we find that realistic CKM and PMNS matrices can be achieved if the residual symmetry in the neutrino   Figure 3. Correlations between different mixing parameters for the mixing pattern U II,8 with ϕ 1 = 2π/7 which can be achieved from the D 7 flavor group, and the three lepton mixing angles are required to lie in their 3σ ranges [2].  and down quark sectors is Z 2 × CP , and a subgroup Z 2 × CP or Z 2 is preserved by the charged lepton and up quark mass matrices [50,51]. If the three generations of left-handed quark and lepton doublets transform as an irreducible three-dimensional representation of the flavor symmetry group, the minimal group turns out to be ∆(294) [50,51]. The motivation of the present work is to find a smaller flavor group which can give a unified description of quark and lepton flavor mixing.
In this paper, we perform a detailed analysis of the dihedral group D n as flavor symmetry in combination with CP symmetry. We have identified the most general form of the CP transformations compatible with D n . Since the group D n only has one-dimensional and two-dimensional irreducible representations, the left-handed quark and lepton fields are assigned to the direct sum of a singlet and a doublet of D n . If the symmetries D n and CP are broken in such a way that neutrino and charged lepton sectors remain invariant under two different Z 2 × CP subgroups, all the lepton mixing angles and CP phases would depend on only two real free parameters θ e and θ ν . The measured values of the lepton mixing angles can be explained by small group D 4 which is the symmetry group of a square, see table 3 for numerical results. In the same way as presented for leptons, viable quark mixing can be derived under the assumption that the residual symmetries of the up and down quark sectors are Z 2 × CP as well. Moreover, we find that the flavor group D 14 can give the experimentally favored CKM and PMNS mixing matrices.
Furthermore, we consider a second scenario in which the residual symmetries of the charged lepton and up quark sectors are Z 2 instead of Z 2 × CP while the neutrino and down quark mass matrices remain invariant under a Z 2 × CP subgroup. The resulting lepton and quark mixing matrices would depend on three free parameters θ e , θ ν , δ e and θ u , θ d , δ u respectively. The observed patterns of quark and lepton flavor mixing can be accommodated by the D 7 group.

JHEP03(2019)056
As regards the constraint arising from the residual CP transformation X ψ , plugging the Takagi factorization of X ψ in eq. (A.4) into eq. (2.7), we find which implies that the combination Σ T ψ U * ψ Q −1 ψ is a real orthogonal matrix denoted by O 3×3 . Consequently the unitary transformation U ψ takes the following form Subsequently we consider the constraint of the residual flavor symmetry Z Hence the residual symmetry Z g ψ 2 × X ψ would determine the unitary transformation U ψ to be of the form [31,50,51,55] U ψ = Σ ψ S 23 (θ ψ )P ψ Q † ψ . (A.12) Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.