Lepton mixing predictions from infinite group series D9n, 3n(1) with generalized CP

We have performed a comprehensive analysis of the type D group D9n, 3n(1) as flavor symmetry and the generalized CP symmetry. All possible residual symmetries and their consequences for the prediction of the mixing parameters are studied. We find that only one type of mixing pattern is able to accommodate the measured values of the mixing angles in both “direct” and “variant of semidirect” approaches, and four types of mixing patterns are phenomenologically viable in the “semidirect” approach. The admissible values of the mixing angles as well as CP violating phases are studied in detail for each case. It is remarkable that the first two smallest D9n, 3n(1) groups with n = 1, 2 can fit the experimental data very well. The phenomenological predictions for neutrinoless double beta decay are discussed.


Introduction
The precise measurement of the reactor mixing angle θ 13 [1][2][3][4][5][6][7] encourages the pursuit of the still missing results on leptonic CP violation and neutrino mass ordering as well as the characteristic neutrino nature. Some low-significance hints for a maximally CP-violating value of the Dirac phase δ CP 3π/2 have been observed [8]. The global fits to lepton mixing parameters [9][10][11] also provide weak evidence for the existence of Dirac type CP violation in neutrino oscillation. In the case that neutrinos are Majorana particles, two more Majorana CP phases α 21 and α 31 would be present, and they are crucial to the neutrinoless double beta decay process. However, the present experimental data don't impose any constraint on the values of the Majorana phases.
Finite discrete non-abelian flavor symmetries have been widely used to make predictions for lepton flavor mixing. Assuming the original flavor symmetry group is spontaneously broken to distinct abelian residual symmetries in the neutrino and charged lepton sectors at a low energy scale, one can then determine mixing patterns from the residual symmetries and the structure of discrete flavor symmetry groups. Please see refs. [12][13][14][15][16] for review on discrete flavor symmetries and the application in model building. For Majorana neutrinos, if the residual symmetries of the charged lepton and neutrino mass matrices originate from a finite flavor group, the lepton mixing matrix would be fully determined by residual symmetries up to independent row and column permutations. It turns out that the possible forms of the PMNS matrix are strongly constrained in this scenario such that the JHEP05(2016)007 mixing patterns compatible with the data are of trimaximal form, and the Dirac CP phase is predicted to be 0 or π [17]. The same conclusion is reached for neutrinos being Dirac particles [18]. We note that the neutrino masses are not constrained in this approach and consequently the both Majorana phases α 21 and α 31 are undetermined. Their values can be fixed by considering a specific model. If the residual flavor symmetries of the neutrino and charged lepton mass matrices are partially contained in the underlying flavor group, the PMNS matrix would contains at least two free continuous parameters. As a result, the predictivity of the model would be lessened to a certain extent.
Besides the extensively discussed residual flavor symmetries, the neutrino and charged lepton mass matrices also admit residual CP transformations, and the residual CP symmetries can be generated by performing two residual CP transformations [19][20][21]. Analogous to residual flavor symmetries, the residual CP transformations can also constraint the lepton flavor mixing in particular the CP violating phases [19]. The simplest nontrivial CP transformation is known as µ−τ reflection which gives rise to maximal atmospheric mixing and maximal Dirac phase [22][23][24][25][26][27]. The deviation from maximal atmospheric mixing and non-maximal Dirac CP violation can be naturally obtained from the so-called generalized µ − τ reflection [28].
In the present work, we shall thoroughly analyze the lepton mixing patterns which can be obtained from the breaking of D (1) 9n,3n flavor symmetry and generalized CP . All possible residual symmetries in the "direct", "semidirect" and "variant of semidirect" approaches and their consequences for the prediction of the mixing parameters are studied. We shall perform a detailed numerical analysis for all the possible mixing patterns. The admissible values of the mixing parameters for each n and the possible values of the effective mass |m ee | will be explored.
The outline of this paper is as follows. In section 2 we find the class-inverting automorphism of the D (1) 9n,3n group and the corresponding physically well-defined generalized CP transformations are determined by solving the consistency condition. In section 3 we JHEP05(2016)007 review the approach to determining the lepton flavor mixing from residual flavor and CP symmetries of the neutrino and the charged lepton sectors. All possible residual symmetries and the consequences for the prediction of the flavor mixing are studied in the method of the direct approach in section 4. The PMNS matrix is determined to be of the trimaximal form, both Dirac phase δ CP and the Majorana phase α 31 are conserved, and the values of α 21 are integer multiple of 2π/(3n). We investigate the possible mixing patterns which can be derived from the semidirect approach and variant of semidirect approach in section 5 and section 6. The analytical expressions of the PMNS matrices, mixing angles and CP invariants are presented, the admissible values of the mixing angles and CP violation phases are analyzed numerically in detail, and phenomenological predictions for neutrinoless double beta decay are studied. For the lowest order D (1) 9n,3n group with n = 1, 2, we find all the mixing patterns that can describe the experimentally measured values of the mixing angles, and a χ 2 analysis is performed. Finally we summarize and present our conclusions in section 7. The group theory of D (1) 9n,3n is presented in appendix A including the conjugacy classes, the irreducible representations, the character table, the Kronecker products and the Clebsch-Gordan coefficients.
2 Generalized CP consistent with D (1) 9n,3n family symmetry The finite subgroups of SU (3) have been systematically classified by mathematicians [56] (see refs. [57][58][59] for recent work). It is well-established that all discrete subgroups of SU (3) can be divided into five categories: type A, type B, type C, type D, and type E [58,59]. The type D group turns out to be particularly significant in flavor symmetry theory [17,60]. Type D group is isomorphic to (Z m × Z n ) S 3 , and it can be generated by four generators a, b, c and d subject to the following rules [58]: It is found that the type D group exists only for [58] k = 0, m = n or k = 1, m = 3n .
In the case of k = 0, m = n, the corresponding group denoted as D n,n is exactly the well-known ∆(6n 2 ) group [61]. For another case of k = 1, m = 3n, the corresponding type D group denoted as D (1) 3n,n is isomorphic to Z 3 × ∆(6n 2 ) if n is not divisible by 3 [58]. Therefore the representation of D (1) 3n,n can be obtained by multiplying the representation matrices of ∆(6n 2 ) with 1, e 2πi/3 and e 4πi/4 for 3 n. As a consequence, the D (1) 3n,n group for 3 n would give rise to the same set of lepton flavor mixing as ∆(6n 2 ) group no matter whether the generalized CP symmetry is considered or not. The ∆(6n 2 ) as flavor symmetry group has been comprehensively explored in the literature [52,54,60], we shall focus on the second independent type D infinite series of groups D (1) 9n,3n where n is any positive JHEP05(2016)007 [1458,659] ((Z 9 × Z 9 ) Z 3 ) Z 2 Z 18 Table 1. The automorphism groups of the D (1) 9n,3n group with n = 1, 2, 3, where Inn(G f ) and Out(G f ) denote inner automorphism group and outer automorphism group of G f respectively. Note that each of these three groups has a unique class-inverting outer automorphism.
integer. It is remarkable that D (1) 9n,3n can generate experimentally viable lepton and quark mixing simultaneously [18]. In the present work, we shall include the generalized CP symmetry compatible with D (1) 9n,3n and investigate its predictions for lepton mixing angles and CP violating phases. The group theory of D (1) 9n,3n , its irreducible representations and the Clebsch-Gordan coefficients are presented in appendix A.
It is highly nontrivial to introduce the generalized CP symmetry in the presence of a discrete flavor symmetry G f . In order to consistently combine the generalized CP symmetry with flavor symmetry, the following consistency condition has to be fulfilled [29,30,37], where ρ r (g) is the representation matrix of the element g in the irreducible representation r of G f , and X r is the generalized CP transformation. Obviously the CP transformation X r maps g into another group element g . Therefore the generalized CP symmetry corresponds the automorphism group of G f . Moreover, it was shown that the physically well-defined CP transformations should be given by class-inverting automorphism of G f [37]. We have exploited the computer algebra system GAP [62] to calculate the automorphism group of the first three D (1) 9n,3n groups with n = 1, 2, 3, the results are listed in table 1. Notice that larger D (1) 9n,3n group for n ≥ 4 is not stored in GAP at present. We see that the automorphism group of D (1) 9n,3n is quite complex but each one of D (1) 9,3 , D 18, 6 and D (1) 27,9 has a unique class-inverting outer automorphism. Furthermore, we find a generic class-inverting automorphism u of the D (1) 9n,3n group, and its actions on the generators a, b, c, d are as follows (

2.4)
It is easy to check that u indeed maps each element into the class of its inverse element for any value of the parameter n. We denote the physical CP transformation corresponding to the automorphism u as X r (u), and its explicit form is determined by the following consistency equations: X r (u) ρ * r (a)X † r (u) = ρ r (u (a)) = ρ r (a) , X r (u) ρ * r (b)X † r (u) = ρ r (u (b)) = ρ r (b) , X r (u) ρ * r (c)X † r (u) = ρ r (u (c)) = ρ r c −1 , X r (u) ρ * r (d)X † r (u) = ρ r (u (d)) = ρ r d −1 . (2.5)

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In our working basis shown in appendix A, the representation matrices of a and b are real while the representation matrices of c and d are complex and diagonal for any irreducible representations of D (1) 9n,3n . Therefore the CP transformation X r (u) is a unit matrix, i.e.
X r (u) = 1 r . (2.6) Given this CP transformation X r (u), the matrix ρ r (g)X r (u) = ρ r (g) is also an admissible CP transformation for any g ∈ D (1) 9n,3n . It corresponds to performing a conventional CP transformation followed by a group transformation ρ r (g). As a consequence, we conclude that the generalized CP transformation compatible with the D (1) 9n,3n family symmetry is of the same form as the flavor symmetry transformation in our basis, i.e. (2.7) Note that other possible CP transformations can also be defined if a model contains only a subset of irreducible representations. Lepton mixing can be derived from the remnant symmetries in the charged lepton and neutrino mass matrices, while the mechanism of symmetry breaking is irrelevant. The basic procedure and the resulting master formulae are given in refs. [19,20,38,39,54]. In the following, we shall consider all possible remnant symmetries of the neutrino and charged lepton sectors and discuss the predictions for the PMNS matrix and the lepton mixing parameters.

Framework
In the present work, the family symmetry is taken to be D (1) 9n,3n , and the generalized CP symmetry is considered in order to predict the lepton mixing parameters including the CP violating phases. Without loss of generality, we assume that the three left-handed leptons transform as a triplet 3 1,0 under D (1) 9n,3n . For brevity we shall denote the faithful irreducible representation 3 1,0 as 3. The representation matrices of the generators a, b, c and d in 3 1,0 are given in eq. (A.37). The light neutrinos are assumed to be Majorana particles. From the bottom-up perspective, the most general symmetry of a generic charged lepton mass matrices is U(1) × U(1) × U(1), which has finite subgroups isomorphic to a cyclic group Z m for any integer m or a direct product of several cyclic groups [18][19][20]. On the other hand, the largest possible symmetry of the neutrino mass matrix is Z 2 × Z 2 [18][19][20]63]. Moreover the neutrino and charged lepton mass matrices are invariant under a set of CP transformations, and both the U(1) × U(1) × U(1) symmetry group of the charged-lepton mass term and the Z 2 × Z 2 symmetry of the neutrino mass term can be generated by performing two CP symmetry transformations [19,20]. Conversely, the lepton mass matrices are strongly constrained by the postulated remnant symmetry such that the lepton mixing matrix can be derived from the remnant symmetries in the charged lepton and neutrino sectors, while the mechanism of dynamically realizing the assumed remnant symmetries is irrelevant [19,20]. From the view of the top-down method, the remnant flavor and CP symmetries of the neutrino and charged lepton mass matrices may originate from certain symmetry group implemented at high energy scales. In the present work, both H CP is assumed to be broken down into G l H l CP and G ν × H ν CP in the charged lepton and neutrino sectors respectively. The allowed forms of the neutrino and charged lepton mass matrices are constrained by the remnant symmetries, and subsequently we can diagonalize them to get the PMNS matrix.
The requirement that a subgroup G l H l CP is preserved at low energies entails that the combination m † l m l has to fulfill where the charged lepton mass matrix m l is given in the convention l c m l l. The hermitian combination m † l m l is diagonalized by the unitary transformation U l with U † l m † l m l U l = diag(m 2 e , m 2 µ , m 2 τ ). The three charged leptons have distinct masses m e = m µ = m τ . From eq. (3.1), it is straightforward to derive that the remnant symmetry G l H l CP leads to the following constraints on U l where both ρ diag 3 (g l ) and X diag l3 are diagonal phase matrices. As a consequence, we see that U l also diagonalizes the residual flavor symmetry transformation matrix ρ 3 (g l ), the residual CP transformation X l3 is a symmetric matrix, and the following restricted consistency condition should be satisfied [43], In the same fashion, the neutrino mass matrix is invariant under the action of the elements of the residual subgroup G ν × H ν CP : We denote the unitary diagonalization matrix of m ν as U ν fulfilling U T ν m ν U ν = diag (m 1 , m 2 , m 3 ). Then U ν would be subject to the following constraints from the postulated residual symmetry [19][20][21]: where the "±" signs can be chosen independently. Therefore the residual CP transformation X ν3 is a symmetric unitary matrix as well, and the restricted consistency condition on the neutrino sector takes the form [19][20][21]29]:

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Obviously X νr maps any element g ν of the neutrino residual flavor symmetry G ν into itself. Hence the mathematical structure of the remnant subgroup comprising G ν and H ν CP is generally a direct product instead of a semidirect product. Given a pair of welldefined remnant symmetries G l H l CP and G ν × H ν CP for which the consistency equations in eqs. (3.3), (3.6) are fulfilled, the allowed forms of the mass matrices m † l m l and m ν can be determined from eqs. (3.1), (3.4), and subsequently the prediction for the PMNS matrix U PMNS = U † l U ν can be obtained by diagonalizing m † l m l and m ν . For two pairs of remnant symmetry subgroups G l H l CP , G ν × H ν CP and G l H l CP , G ν × H ν CP , if G l , G ν and G l , G ν are related by a similarity transformation, for example if they are conjugate, The remnant CP would also be related by in order to fulfill the consistency conditions in eqs. (3.3), (3.6). That is to say the elements of H l CP and H ν CP are given by ρ r (h)X lr ρ T r (h) and ρ r (h)X νr ρ T r (h) respectively, where X νr ∈ H ν CP and X lr ∈ H l CP . Notice that all the possible remnant CP transformations compatible with the remnant flavor symmetry have been considered in this work. Hence if G l H l CP and G ν × H ν CP fix the charged lepton and neutrino mass matrices to be m † l m l and m ν , H CP . It is sufficient to only analyze a few representative remnant symmetries which give rise to different results for U PMNS and lepton mixing parameters, as other possible choices for the remnant symmetry groups are related to the representative ones by similarity transformation and consequently no new results are obtained.

Lepton mixing from direct approach
In the direct approach, the residual flavor symmetry G ν is a Klein four subgroup, and the residual flavor symmetry G l is a cyclic group Z m with index m ≥ 3 or a product of cyclic groups. We assume that the residual flavor symmetry group G l can distinguish the three generations of charged lepton. In other words, the restricted representation of the triplet representation 3 on G l should decompose into three inequivalent one-dimensional representations of G l . From eq. (3.1) and eq. (3.2), we see that U l not only diagonalizes the mass matrix m † l m l but also the residual flavor symmetry transformation matrix ρ 3 (g l ) with g l ∈ G l . As a result, the requirement that U † l ρ 3 (g l )U l = ρ diag 3 (g l ) is diagonal allows us to determine U l without knowledge of m † l m l . Notice that the remnant CP invariant condition in eq. (3.1) is automatically satisfied, the reason is that the residual CP transformation JHEP05(2016)007 X l3 has to be compatible with residual flavor symmetry and its allowed form is strongly constrained by the restricted consistency condition of eq. (3.3).
As shown in the appendix A, the group structure of the D (1) 9n,3n has been studied in detail. The residual subgroup G l is an abelian subgroup, and it can be generated by the generators c s d t , bc s d t , ac s d t , a 2 c s d t , abc s d t or a 2 bc s d t with s = 0, 1, . . . , 9n − 1, t = 0, 1, . . . , 3n − 1. The diagonalization of ρ 3 (g l ) determines the unitary transformation U l up to permutations and phases of the column vectors if ρ 3 (g l ) has non-degenerate eigenvalues, where g l can be taken to be the generator of G l . The explicit form of U l for different G l and the corresponding remnant CP transformations compatible with G l are summarized in table 2. If the eigenvalues of ρ 3 (g l ) are degenerate so that its diagonalization matrix U l can not be determined uniquely, we would extend G l from a single cyclic subgroup to a product of cyclic groups, for example G l = G 1 × G 2 where the generators of G 1 and G 2 should be commutable with each other. If G 1 (or G 2 ) is sufficient to distinguish among the generations such that its eigenvalues are not degenerate, then another subgroup G 2 (or G 1 ) would not impose any new constraint on the lepton mixing. On the other hand, if the three eigenvalues of the generator of either G 1 or G 2 are completely degenerate, e.g. G 1 ( or G 2 ) = c 3n , its three-dimensional representation matrix would be proportional to a unit matrix. As a result, we shall concentrate on the case that the representation matrices of both G 1 and G 2 have two degenerate eigenvalues, therefore either G 1 or G 2 alone fixes only a column of U l and the third column can be determined by unitary condition. The possible extension of remnant flavor symmetry group G l , the corresponding remnant CP transformations and the unitary transformations U l are collected in table 3. We see that the diagonalization matrix U l can only take five distinct forms U such that the constraints on s and t shown in table 2 are relaxed. In the direct approach, the flavor symmetry group D (1) 9n,3n is broken down to a Klein four subgroup in the neutrino sector. From appendix A, we see that D (1) 9n,3n for even n has only four Klein four subgroups: where x, y, z = 0, 1, . . . , 3n − 1. We note that K 9n,3n , and the remaining three K 4 subgroups are conjugate: with δ = 0, 1, . . . , 3n − 1. Furthermore, the residual CP symmetry H ν CP in the neutrino sector has to be compatible with the remnant K 4 symmetry, and the following restricted Table 2. The form of U l for different residual subgroup G l generated by a single element g, and here we denote G l = g . H l CP is the residual CP transformations consistent with G l . The allowed values of the parameters s, t, γ, δ and τ are t, δ = 0, 1, · · · , 3n − 1, s, γ = 0, 1, · · · , 9n − 1 and τ = 0, 1, 2. The parameter ω is the cube root of unit with ω = e 2πi/3 . Note that because ac 2s−3t d s−t 2 = a 2 c s d t holds, the U l for G l = a 2 c s d t can be obtained from that corresponding to G l = ac s d t by the replacement s → 2s − 3t and t → s − t. The constraints on the parameters s and t is to remove the degeneracy among the eigenvalues. consistency condition must be fulfilled,  , X νr = ρ r (c γ d δ ): in our working basis, the representation matrices for both a and c are diagonal with JHEP05(2016)007 2s − 3t = 3l 2 n (mod 6n) 2s − 3t = 3l 3 n (mod 6n) Table 3. The product extension of the remnant flavor symmetry G l = G 1 × G 2 , the remnant CP transformation compatible with G l , and the corresponding unitary transformation U l . We require the column vectors fixed by G 1 and G 2 be different. Consequently we have the parameters l 1,2,3 = 0, 1 and l 1 + l 2 + l 3 = 1, 3. The values of parameters s, t, s , t , γ, δ and τ are s, s , γ = 0, 1, · · · , 9n − 1, t, t , δ = 0, 1, · · · , 3n − 1 and τ = 0, 1, 2.

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Consequently the residual flavor symmetry enforces the neutrino mass matrix to be diagonal as well. Taking into account the remnant CP symmetry further, we find where m 11 , m 22 and m 33 are real parameters. We can read out the neutrino diagonalization matrix U ν as where Q ν is a diagonal phase matrix with entry being ±1 or ±i, and it encodes the CP parity of the neutrino states. The light neutrino mass eigenvalues are Obviously the light neutrino masses depend on only three real parameters, and the order of the light neutrino masses can not be fixed by remnant symmetries. Therefore the unitary transformation U ν is determined up to independent row and column permutations in the present framework, and the neutrino mass spectrum can be either normal ordering (NO) or inverted ordering (IO).
: in the same fashion as previous case, we find that the light neutrino mass matrix takes the following form: where the matrix Q ν is omitted for simplicity and we will also not explicitly write out this factor hereafter. The light neutrino masses are , X νr = ρ r (c δ−2y−3nτ d δ ), ρ r (abc δ−3nτ d δ ) : in this case, we find that the light neutrino mass matrix takes the form The light neutrino masses are , X νr = ρ r (c γ d −2z ), ρ r (a 2 bc γ ) : the light neutrino mass matrix m ν is constrained by the remnant symmetry to be of the form where m 11 , m 12 and m 33 are real. The unitary matrix U ν diagonalizing the above neutrino mass matrix is determined to be The neutrino masses are given by  Then we proceed to discuss the possible mixing patterns achievable in direct approach by combining the different remnant symmetries of the charged lepton sector with those of the neutrino sector. As shown in section 3, two pairs of subgroups {G l , G ν } and {G l , G ν } would yield the same results for the PMNS matrix after considering all the eligible residual CP transformations, if these two pairs of groups are conjugate. Notice the conjugate relations between distinct K 4 subgroups in eq. (4.1) and the identities for any integer , we find it is sufficient to only consider eight kinds of remnant symmetries with G l = c s d t , bc s d t , ac s d t , . In this scenarios, all mixing parameters including Majorana phases are completely fixed by remnant symmetries.
, X νr = {ρ r (c γ d δ )}: in this case, the unitary transformation U l is a unit matrix, as shown in table 2. U ν is a diagonal phase matrix and it is given by eq. (4.6). As a result, the PMNS matrix is also a diagonal matrix up to row and column permutations, and obviously it doesn't agree with the present neutrino oscillation data [9][10][11].

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, X νr = {ρ r (c γ d δ )}: in this case, the postulated residual subgroups lead to the mixing pattern with The lepton mixing angles are θ 13 = θ 12 = 0, θ 23 = 45 • , and therefore large corrections to both θ 12 and θ 13 are necessary in order to be compatible with the experimental data.
, X νr = {ρ r (c γ d δ )}: this residual symmetry allows us to pin down the lepton mixing matrix as: where This pattern leads to sin 2 θ 12 = sin 2 θ 23 = 1/2, sin 2 θ 13 = 1/3 and a maximal Dirac CP phase |δ CP | = π/2. The solar as well as the reactor mixing angles have to acquire appropriate corrections in order to be in accordance with the experimental data.

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All possible predictions of sin θ 13 for each D (1) 9n,3n of even n are displayed in figure 1. It is remarkable that viable reactor mixing angle θ 13 can always be achieved for each n. Moreover, the three mixing angles are closely related as follows 3 sin 2 θ 12 cos 2 θ 13 = 1, Inputting the experimentally preferred 3σ range 0.0176 ≤ sin 2 θ 13 ≤ 0.0295 [9], we obtain predictions for solar as well as atmospheric mixing angles: From the PMNS matrix of eq. (4.24), we can also extract the CP violating phases where the contribution of the CP parity matrix Q ν is considered. We see that both Dirac phase δ CP and the Majorana phase α 31 are trivial, and another Majorana phase α 21 is The admissible values of α 21 are which are plotted in figure 1. Note that here the predictions for the CP phases are consistent with the general results of ref. [20].
, ρ r (bc 2δ+3nτ d δ )}: in this case, we find the lepton mixing matrix is the well-known bimaximal pattern The bimaximal mixing can be a valid first approximation in a model where corrections of order of the Cabibbo angle can naturally arise [43,64].

Lepton mixing from semidirect approach
In the semidirect approach, the original symmetry D H CP is broken at low energies into G l H l CP in the charged lepton sector and to Z 2 × H ν CP in the neutrino sector. The PMNS matrix turns out to depend on only a single real parameter in this scenario. It is generally assumed that the residual flavor symmetry G l is able to distinguish the three generations of charged leptons such that the unitary matrix U l can be determined from the requirement that all the generators of G l should be simultaneously diagonalized by U l . The possible candidates for the subgroup G l , the remnant CP transformations compatible with G l and the corresponding unitary transformation U l are summarized in table 2 and table 3.
9n,3n group with even n ≤ 50 when the remnant symmetries are G l = ac s d t , The light blue region denotes the 3σ bound of sin θ 13 , which is taken from ref. [9].
Then we turn to the neutrino sector. From the multiplication rules given in eq. (A.1), we see that the order 2 elements of the D (1) 9n,3n group are bd x , abc 3y d y , a 2 bc 3z d 2z , x, y, z = 0, 1, . . . , 3n − 1 , (5.1) and additionally c 9n for even n. The residual CP transformation X νr is a symmetric unitary matrix, and it should map the element of the neutrino residual flavor symmetry to itself, The eligible residual CP transformations for different Z 2 subgroups are collected in table 5. Furthermore, we notice that all the Z 2 elements in eq. (5.1) are conjugate: Similarly the three elements in eq. (5.2) are also conjugate to each other: Since only a Z 2 subgroup instead of a full Klein subgroup is preserved by the neutrino mass matrix, the postulated remnant flavor symmetries can only fix one column of the PMNS matrix. We list the explicit forms of the determined columns for different remnant flavor symmetries in table 6. Global analysis of the neutrino oscillation data gives the 3σ ranges on the absolute values of the elements of the PMNS matrix [9]: It is obvious that none entry of the PMNS matrix is vanishing [9][10][11]. Therefore if one element of the fixed column is predicted to be zero, it would be excluded by the experimental data. From table 6 we see that only three independent cases are viable with the residual flavor symmetries (G ν , In the following, the contribution of all admissible remnant CP transformations will be included further. We shall find the neutrino mass matrix invariant under the residual flavor and CP symmetries, and then the unitary transformation U ν as well as the PMNS matrix U PMNS will be presented for each case. : the residual symmetry transformation G ν × H ν CP of the neutrino fields leaves the neutrino JHEP05(2016)007 Table 6. The column vector of the PMNS matrix determined by the residual flavor symmetries G ν and G l . If one (or two) element of the fixed column is vanishing, we would use the notation "" to indicate that it is disfavored by the present experimental data, otherwise the notation "" is labelled to indicate that agreement with the experimental data could be achieved. Notice that two pair of subgroups (G ν , G l ) = (Z bd x 2 , abc s d t ) and (Z bd x 2 , a 2 bc s d s−t ) are conjugate under the element bc 2x d 2x . mass term invariant. Therefore the neutrino mass matrix m ν must satisfy In our working basis, it is straightforward to find that the neutrino mass matrix is constrained to take the form where m 11 , m 12 , m 22 and m 23 are real. It follows that the neutrino mass matrix m ν can be diagonalized by where the angle θ is The factor Q ν is a diagonal phase matrix with elements equal to ±1 and ±i, and it is necessary to make the light neutrino masses positive definite. The neutrino mass eigenvalues are given by We see that the neutrino masses depend on four parameters m 11 , m 12 , m 22 and m 23 , the experimentally measured mass squared differences could be easily accommodated. The order of the three neutrino masses m 1 , m 2 and m 3 can not be pinned down in the present framework, hence the unitary matrix U ν is determined up to permutations of the columns (the same holds true in the following cases), and the neutrino mass spectrum can be either normal ordering or inverted ordering. Taking into account the corresponding charged lepton diagonalization matrix U l listed in table 2 and table 3, we find the PMNS matrix U PMNS ≡ U † l U ν up to row and column permutations is Both ϕ 1 and ϕ 2 are determined by the postulated remnant symmetries, they are independent of each other, and their values can be multiple of π 9n and π 3n respectively We see that one column of the PMNS matrix is determined to be

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in this case. As the neutrino mass ordering isn't constrained in the present framework, this column vector can be any of the three column of the PMNS matrix. As a consequence, the PMNS matrix can take the following three possible forms: The effect of row permutation is equivalent to redefinitions of the parameters θ, ϕ 1 and ϕ 2 , and no new possible values of ϕ 1 and ϕ 2 beyond those in eq. (5.14) are obtained. These mixing patterns U I,1 PMNS , U I,2 PMNS and U I,3 PMNS can also be derived from ∆(6n 2 ) H CP with different expressions for ϕ 1 and ϕ 2 [54], Hence all the mixing patterns predicted by D 9n,3n H CP can be obtained from ∆(6(9n) 2 ) H CP , and further the patterns in ∆(6(3n) 2 ) H CP are achievable from D (1) 9n,3n H CP . This fact can be easily understood from the group relation ∆(6(3n) 2 ) < D (1) 9n,3n < ∆(6(9n) 2 ), as shown in appendix A.

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Obviously ϕ 1 should be around π/2 or 3π/2. Furthermore, the three CP rephasing invariants J CP , I 1 and I 2 are predicted to be sin 2 ϕ 1 sin ϕ 2 √ 2 cos 2 θ cos ϕ 2 − sin 2θ cos ϕ 1 , where J CP is well-known Jarlskog invariant, and I 1 and I 2 are defined for the Majorana phases with sin 2θ 12 sin 2θ 13 sin 2θ 23 cos θ 13 sin δ CP , where α 31 ≡ α 31 − 2δ CP , δ CP is the Dirac CP violating phase, α 21 and α 31 are the Majorana CP phases in the standard parameterization of the PMNS matrix [65]. We show the absolute values of J CP , I 1 and I 2 in eq. (5.20), the reason is because the sign of the J CP depends on the ordering of rows and columns and the sign of I 1 and I 2 could be changed by the CP parity matrix Q ν . Moreover, if the lepton doublet fields are assigned to the triplet 3 9n−1,0 instead of 3 1,0 , the prediction for U PMNS would be complex conjugated such that the signs of J CP , I 1 and I 2 are all inversed. We show the possible predictions for the mixing parameters sin 2 θ 12 , sin θ 13 , sin 2 θ 23 as well as |sin δ CP |, |sin α 21 | and |sin α 31 | for each D 9n,3n group in figure 2, where all the admissible values of ϕ 1 and ϕ 2 shown in eq. (5.14) are considered and all the three mixing angles are required to lie in the 3σ allowed regions adapted from [9]. It is notable that the solar mixing angle is predicted to be within the narrow interval of 0.313 ≤ sin 2 θ 12 ≤ 0.344. The near future mediumbaseline reactor neutrino oscillation experiments, such as JUNO [66] and RENO-50 [67] are expected to make very precise, sub-percent measurements of the solar mixing angle θ 12 . They provide one of the most significant test of this mixing pattern. The allowed values of the CP violation phases increase with group index n and they are strongly constrained for smaller n. The possible values of sin 2 θ 12 , sin θ 13 , sin 2 θ 23 , |sin δ CP |, |sin α 21 | and |sin α 31 | with respect to n for the mixing pattern U I,1 PMNS in the case I, where the three lepton mixing angles are required to be within the experimentally preferred 3σ ranges. The 1σ and 3σ regions of the three neutrino mixing angles are adapted from global fit [9].
for the mixing angles are We see that the solar and reactor mixing angles are correlated as 3 sin 2 θ 12 cos 2 θ 13 = 2 sin 2 ϕ 1 .  PMNS in case I, where the first four smallest D (1) 9n,3n group with n = 1, 2, 3, 4 are considered. The 1σ and 3σ regions of the three neutrino mixing angles are adapted from global fit [9].
In order to accommodate the experimental results on θ 12 and θ 13 , ϕ 1 should vary in the interval: (5.24) Consequently we have We see that both (22) and (32) entries of U I,2 PMNS are not in agreement with the experimental data given by eq. (5.6). Hence this mixing pattern is phenomenologically disfavored.
For the third possible arrangement of the rows and columns, the PMNS matrix is U I,3 PMNS . In this case, the third column of the PMNS matrix doesn't depend on the continuous parameter θ and it is completely fixed by the remnant flavor symmetry. It is JHEP05(2016)007 straightforward to extract the mixing angles: sin 2 θ 13 = 2 3 sin 2 ϕ 1 , sin 2 θ 23 = 1 + sin (π/6 + 2ϕ 1 ) 2 + cos 2ϕ 1 , The experimental data 0.0176 ≤ sin 2 θ 13 ≤ 0.0295 at 3σ level [9] can be accommodated for the following values of the parameter ϕ 1 : As both θ 13 and θ 23 depend on a single parameter ϕ 1 , we can derive a sum rule between them, Given the experimental best fitting value of the reactor mixing angle sin 2 θ 13 = 0.0234 [9], we have sin 2 θ 23 0.391, or sin 2 θ 23 0.609 , (5.29) which is within the 3σ range although it is non-maximal. For a given D 9n,3n group, the atmospheric and reactor mixing angles can only take a set of discrete values. The possible values of sin 2 θ 23 and sin θ 13 for the first four smallest n = 1, 2, 3, 4 are displayed in figure 3. We see that the values ϕ 1 = ±π/18, ±17π/18 in the case of n = 2, 4 lead to (θ 13 , θ 23 ) = (8.151 • , 50.813 • ) or (8.151 • , 39.187 • ) which are compatible with the present experimental data [9]. The next generation of superbeam neutrino oscillation experiments would provide a high-precision determination of θ 23 . If no significant deviations from maximal mixing of θ 23 will be detected, our present scheme will be excluded. Furthermore, we find that the CP invariants are cos ϕ 1 sin ϕ 2 4 cos 2θ cos ϕ 1 cos ϕ 2 − √ 2 sin 2θ cos 2ϕ 1 , Furthermore, we study the admissible values of mixing angles and CP phases for each D 9n,3n group. The numerical results are displayed in figure 4. We easily see that the atmospheric mixing angle θ 23 is not maximal and it is around the 3σ upper or lower bound. Similar to the ∆(6n 2 ) group [54], maximal value of the Majorana phase α 31 can not be achieved in this case and it is found to be in the range of |sin α 31 | ≤ 0.910 while almost any values of δ CP and α 21 can be possible for large n.
As a concrete example, we shall study the first two smallest D  Figure 4. The possible values of sin 2 θ 12 , sin θ 13 , sin 2 θ 23 , |sin δ CP |, |sin α 21 | and |sin α 31 | with respect to n for the mixing pattern U I,3 PMNS in the case I, where the three lepton mixing angles are required to be within the experimentally preferred 3σ ranges. The 1σ and 3σ regions of the three neutrino mixing angles are adapted from global fit [9]. following symmetry properties: where the diagonal matrix can be absorbed into the matrix Q ν . Similar relations are satisfied for the PMNS matrix U I,3 PMNS . Note that the PMNS matrix would become its complex conjugation if the three generations of leptons are assigned to the triplet 3 9n−1,0 ∼ = 3 * 1,0 . As a result, without loss of generality, we shall focus on the case of 0 ≤ ϕ 1 ≤ π and JHEP05(2016)007 0 ≤ ϕ 2 ≤ π/2. A conventional χ 2 analysis is performed. Notice that we don't include the information of the Dirac CP phase δ CP into the χ 2 function, since the evidence for a preferred value of δ CP coming from both present experiments and the global fitting is rather weak. The numerical results are reported in table 7, where we exclude all patterns that can not accommodate the experimental data at the best fitting point θ = θ bf for which the χ 2 function is minimized. Since the global fit results of the mixing angles are slightly distinct for NO and IO neutrino mass spectrums [9], the χ 2 function has been defined for NO and IO respectively. The values in the parentheses are the results for the IO case.
Applying the symmetry transformations in eq. (5.31), we can obtain other values of ϕ 1 and ϕ 2 which yield the same best fit values for the mixing angles such that the same χ 2 min is obtained. For both mixing patterns U I,1 PMNS and U I,3 PMNS , we can check that the formulae in eqs. (5.17), (5.26) for the mixing angles sin 2 θ 12 and sin 2 θ 13 are invariant while sin 2 θ 23 turns into cos 2 θ 23 under the transformation ϕ 1 → π − ϕ 1 , θ → π − θ. As a result, the sum of the best fitting value θ bf for ϕ 1 and π − ϕ 1 is approximately equal to π. It is remarkable that even the smallest D (1) 9n,3n group with n = 1 allows a reasonable fit to the experimental data, for instance, the mixing patterns with (ϕ 1 , ϕ 2 ) = (4π/9, 0), (4π/9, π/3), (5π/9, 0) and (5π/9, π/3) can describe the experimentally measured values of the mixing angles, as can be seen from table 7. In particular, the CP violating phases are neither conserved nor maximal in the case of (ϕ 1 , ϕ 2 ) = (4π/9, π/3) and (5π/9, π/3). The PMNS matrix U I,1 PMNS for n = 2 as well as (ϕ 1 , ϕ 2 ) = (π/2, π/2) give rise to maximal atmospheric mixing and maximal Dirac phase. On the other hand, the group index n should be equal or greater than 2 in order to obtain phenomenologically viable mixing pattern of the form U I, 3 PMNS . Scrutinizing all the admissible cases listed in table 7, we find that the predictions for θ 13 are almost the same, nevertheless θ 12 , θ 23 and δ CP are predicted to be considerably different. The JUNO experiment will be capable of reducing the error of sin 2 θ 12 to about 0.1 • or around 0.3% [66]. Future long baseline experiments such as DUNE [68,69], LBNO [70][71][72][73], T2HK [74] and possibly ESSνSB [75,76] at the European Spallation Source can make very precise measurements of the oscillation parameters θ 12 , θ 23 and δ CP . Therefore future neutrino facilities have the potential to discriminate between the above possible cases, or to rule them out entirely. Furthermore, we expect that a more ambitious facility such as the neutrino factory [77][78][79] could provide a more stringent test of our approach.
Since the Majorana CP violating phases can be predicted in the present framework, we now discuss its phenomenological implications in the neutrinoless double beta (0νββ) decay. It is well-known that the 0νββ decay process is the most sensitive probe for Majorana neutrinos. Its observation would establish the Majorana nature of neutrinos irrespective of the underlying mass generation mechanism. The 0νββ decay rate is proportional to the square of the effective Majorana mass |m ee | which is given by [65] |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 iα 31 .   Table 7. Results of the χ 2 analysis for n = 1, 2 in the case I. The χ 2 function has a global minimum χ 2 min at the best fit value θ bf for θ. We give the values of the mixing angles and CP violation phases for θ = θ bf . The values given in parentheses denote the results for the IO neutrino mass spectrum.

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mixing pattern U I, 1 PMNS , |m ee | is of the form where q 1 , q 2 = ±1 appear due to the undetermined CP parity of the neutrino states encoded in the matrix Q ν . For another admissible mixing pattern U I,3 PMNS , |m ee | is given by The achievable values of the effective mass |m ee | for both n → ∞ and n = 2 are plotted in figure 5. Here we require the three mixing angle be within their 3σ allowed values while the neutrino mass-squared splittings are fixed at their best-fit values from ref. [9]. We see that the majority of the experimentally allowed 3σ region of |m ee | can be reproduced in the limit n → ∞. In the case of n = 2, it is remarkable that the effective mass |m ee | obtained from U I,1 PMNS is found to be around 0.0155eV, 0.0175eV, 0.0279eV, 0.0423eV, or 0.0484eV for IO neutrino mass spectrum. These predictions are beyond the reach of the present 0νββ experiments such as GERDA [80], EXO-200 [81,82] and KamLAND-ZEN [83]. However, the proposed facilities nEXO and KamLAND2-Zen [84] etc aim to increase the sensitivity to cover the full IO region, such that all of our patterns with this mass spectrum could be tested. For NO the effective mass |m ee | is much smaller than the IO case and it can even vanish for certain values of the lightest neutrino mass because of a cancellation between different terms in eq. (5.32). Obviously exploring the NH region experimentally is beyond the reach of any planned 0νββ experiment. Even if the signals of 0νββ decays are not observed and the neutrino masses spectrum are measured to be NO by upcoming neutrino oscillation experiments [66,67], one can still extract useful information on the Majorana phases α 21 and α 31 by combining the cosmological data on the absolute neutrino mass scale and the improved measurement of θ 12 , θ 23 and δ CP from a number of complementary neutrino oscillation experiments.
: this case differs from the previous one in the residual flavor symmetry G l . From table 2  and table 3, we know that the charged lepton diagonalization matrix is exactly U (4) l . Since the neutrino mass matrix preserves the same remnant symmetry as case I, the neutrino mass matrix should take the form of eq. (5.8), and it is diagonalized by the unitary transformation U ν in eq. (5.10). Using the freedom in exchanging rows and columns, we find the phenomenologically viable lepton mixing matrix is 9n,3n group with n = 2. Notice that the purple (green) region overlaps the orange (cyan) one. The present most stringent upper limits |m ee | < 0.120 eV from EXO-200 [81,82] and KamLAND-ZEN [83] is shown by horizontal grey band. The vertical grey exclusion band represents the current bound coming from the cosmological data of m i < 0.230 eV at 95% confidence level obtained by the Planck collaboration [85]. where

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We see that the second column of the PMNS matrix is (1, 1, − √ 2)/2 or (1, − √ 2, 1)/2 in this case. For the mixing pattern U II,1 PMNS , the three lepton mixing angles are found to be sin 2 θ 13 = 1 8 3 − cos 2θ − 2 √ 2 sin 2θ cos ϕ 3 , sin 2 θ 12 = 2 5 + cos 2θ + 2 √ 2 sin 2θ cos ϕ 3 , sin 2 θ 23 = 3 − cos 2θ + 2 √ 2 sin 2θ cos ϕ 3 5 + cos 2θ + 2 √ 2 sin 2θ cos ϕ 3 , (5.41) which fulfill the following sum rules 4 sin 2 θ 12 cos 2 θ 13 = 1, cos 2 θ 13 cos 2 θ 23 = cos 2θ 13 + 2 cos 2 ϕ 3 ± 2 cos ϕ 3 6 sin 2 θ 13 − 8 sin 4 θ 13 − sin 2 ϕ 3 1 + 8 cos 2 ϕ 3 . (5.42) Given the 3σ range of θ 13 , the solar mixing angle θ 12 is determined to lie in the region of 0.254 ≤ sin 2 θ 12 ≤ 0.258 which is rather close to its 3σ lower limit 0.259 [9]. However, this mixing pattern is a good leading order approximation because accordance with the experimental data could be easily achieved in a concrete model after higher order corrections contributions are included. We plot the 1σ, 2σ and 3σ contour regions for sin 2 θ ij with ij = 12, 13, 23 in the ϕ 3 −θ plane in figure 6. Obviously the most stringent constraint comes from the precisely measured reactor mixing angle θ 13 . Moreover, the three CP rephasing invariants are given by The three CP violation phases extracted from these invariants depend on θ and ϕ 3 . The predictions for | sin δ CP |, | sin α 21 | and | sin α 31 | are plotted in figure 7, where the black areas represent the regions in which all three lepton mixing angles are in the experimentally preferred 3σ ranges. To accommodate the experimental data of mixing angles [9], both δ CP and α 21 can not be maximal. The values of | sin δ CP | and | sin α 21 | are bounded from above with | sin δ CP | ≤ 0.895 and | sin α 21 | ≤ 0.545. The second PMNS matrix U II,2 PMNS can be obtained from U II,1 PMNS by exchanging the second and third rows. Therefore U II,2 PMNS and U II,1 PMNS give rise to the same reactor and solar mixing angles and the Majorana phases, while the atmospheric mixing angle changes from θ 23 to π/2 − θ 23 and the Dirac phase changes from δ CP to π + δ CP . The achievable values of the mixing parameters for each D (1) 9n,3n group are displayed in figure 8. For the first two smallest D (1) 9n,3n group with n = 1, 2. The possible values of ϕ 3 are 0, π 3 , . . . , 5π 3 for n = 1 and 0, π 6 , . . . , 11π 6 for n = 2. We find that agreement with experimental JHEP05(2016)007 Figure 6. The contour regions of the three mixing angles in the case II. The red, blue and green areas denote the predictions for sin 2 θ 13 , sin 2 θ 12 and sin 2 θ 23 respectively. The allowed 1σ, 2σ and 3σ regions of each mixing angle are represented by different shadings. Here we take the 3σ lower limit of sin 2 θ 12 to be 0.254 instead of 0.259 given by ref. [9]. The best fit values of the mixing angles are indicated by dashed lines.  Table 8. Results of the χ 2 analysis for n = 1, 2 in the case II. The χ 2 function has a global minimum χ 2 min at the best fit value θ bf for θ. We give the values of the mixing angles and CP violation phases for θ = θ bf . The values given in parentheses denote the results for the IO neutrino mass spectrum. data can be achieved for ϕ 3 = 0 or π. Due to symmetry relation in eq. (5.40), ϕ 3 = 0 and ϕ 3 = π should give rise to the same predictions for the mixing parameters. Therefore it is sufficient to focus on ϕ 3 = 0, and the best fitting results are listed in table 8. Notice that all the three CP phases are predicted to take CP conserving values {δ CP , α 21 , α 31 } ⊆ {0, π}. The same conclusion can be drawn from figure 8.
As regards the neutrinoless double beta decay, both U II,1 PMNS and U II,2 PMNS yield the same effective Majorana mass: with q 1 , q 2 = ±1. We show the predicted values of |m ee | in figure 9. Notice that for IO spectrum |m ee | can be either 0.0233eV or 0.0483eV which are accessible to the next JHEP05(2016)007 generation 0νββ experiments. In the case of NO spectrum, |m ee | strongly depends on the lightest neutrino mass and CP parity, and it can be vanishing for certain values of the lightest neutrino mass.
(iii) G l = ac s d t , G ν = Z c 9n/2 2 , X νr = ρ r (c γ d δ ) : in this case, n should be even in order to have a Z 2 subgroup generated by c 9n/2 . The neutrino mass matrix invariant under the assumed residual symmetry is found to take the form  Figure 8. The possible values of sin 2 θ 12 , sin θ 13 , sin 2 θ 23 , |sin δ CP |, |sin α 21 | and |sin α 31 | with respect to n for the mixing pattern U II,1 PMNS and U II,2 PMNS in the case II, where the three lepton mixing angles are required to be within the experimentally preferred 3σ ranges. The 1σ and 3σ regions of the three neutrino mixing angles are adapted from global fit [9]. Here we take the 3σ lower limit of sin 2 θ 12 to be 0.254 instead of 0.259 given by ref. [9].
9n,3n group with n = 2. Notice that the purple (green) region overlaps the orange (cyan) one. The present most stringent upper limits |m ee | < 0.120 eV from EXO-200 [81,82] and KamLAND-ZEN [83] is shown by horizontal grey band. The vertical grey exclusion band represents the current bound coming from the cosmological data of m i < 0.230 eV at 95% confidence level obtained by the Planck collaboration [85].
9n,3n group until n = 50 are plotted in figure 12. Then we proceed to study the phenomenologically viable mixing patterns which can be derived from the D (1) 9n,3n group with n = 2. Note that the index n has to be even in this case. We can check that the PMNS matrix given by eq. (5.48) has the following symmetry  Figure 12. The possible values of sin 2 θ 12 , sin θ 13 , sin 2 θ 23 , |sin δ CP |, |sin α 21 | and |sin α 31 | with respect to n for the mixing pattern U III PMNS in the case III, where the three lepton mixing angles are required to be within the experimentally preferred 3σ ranges. The 1σ and 3σ regions of the three neutrino mixing angles are adapted from global fit [9]. properties where the diagonal matrix on the right-handed side can be absorbed into Q ν . That is to say, both U III PMNS (θ, π + ϕ 4 , ϕ 5 ) and U III PMNS (θ, ϕ 4 , π/2 + ϕ 5 ) give rise to the same predictions for the lepton mixing parameters as U III PMNS (θ, ϕ 4 , ϕ 5 ) up to redefinition of the free parameter θ. Hence we can take the fundamental intervals of ϕ 4 and ϕ 5 to be [0, π) and [0, π/2) respectively. The allowed values of ϕ 4 are 0, π/18, π/9, . . ., 17π/9, 35π/18. However, only ϕ 4 (mod π) = 0, π/18, π/9, 8π/9 and 17π/18 are within the range of eq. (5.54) such JHEP05(2016)007 Figure 13. The possible values of the effective Majorana mass |m ee | as a function of the lightest neutrino mass in the case III. The red (blue) dashed lines indicate the most general allowed regions for IO (NO) neutrino mass spectrum obtained by varying the mixing parameters over the 3σ ranges [9]. The orange (cyan) areas denote the achievable values of |m ee | in the limit of n → ∞ assuming IO (NO) spectrum. The purple and green regions are the theoretical predictions for the D (1) 9n,3n group with n = 2. Notice that the purple (green) region overlaps the orange (cyan) one. The present most stringent upper limits |m ee | < 0.120 eV from EXO-200 [81,82] and KamLAND-ZEN [83] is shown by horizontal grey band. The vertical grey exclusion band represents the current bound coming from the cosmological data of m i < 0.230 eV at 95% confidence level obtained by the Planck collaboration [85]. that they can give a good fit to the experimental data. The results of the χ 2 analysis are summarized in table 9. Notice that the best fitting values of the mixing angles and | sin δ CP |, | sin α 31 | are dependent on ϕ 4 while the best fitting value of | sin α 21 | depends on ϕ 4 as well as ϕ 5 . The mixing patterns with the same ϕ 4 but different ϕ 5 are expected to be distinguished by some rare processes which are sensitive to the Majorana phases such as the neutrinoless double decay and the radiative emission of neutrino pair in atoms [86]. In this case, the effective Majorana mass |m ee | is predicted to be where q 1 , q 2 = ±1. The numerical results are shown in figure 13.
(iv) G l = ac s d t , G ν = Z c 9n/2 2 , X νr = ρ r (a 2 bc γ ) : this case differs from the case III in the residual CP transformation of the neutrino sector. The group index n has to be an even integer as well. In the same way, the neutrino mass matrix invariant under the assumed remnant symmetry is determined to be   Table 9. Results of the χ 2 analysis for n = 2 in the case III. The χ 2 function has a global minimum χ 2 min at the best fit value θ bf for θ. We give the values of the mixing angles and CP violation phases for θ = θ bf . The values given in parentheses denote the results for the IO neutrino mass spectrum. Notice that θ = π/2 − θ bf gives rise to the same results for the mixing parameters except | sin α 21 |, because the PMNS matrix U III where m 11 , m 12 , m 33 and θ are real parameters. The unitary transformation U ν which diagonalizes m ν is of the form

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The light neutrino masses are given by Then we find that the PMNS matrix takes the form Notice that the mixing matrix depends on the combination ϕ 6 + θ/2 so that the value of ϕ 6 is irrelevant and it can be absorbed into the free parameter θ by redefinition θ → θ − 2ϕ 6 . We remind that this kind of mixing pattern can also be obtained from ∆(6n 2 ) H CP with [54] ∆(6n 2 ) H CP : ϕ 7 (mod 2π) = 0, 1 n π, 2 n π, . . . , 2n − 1 n π .
9n,3n group with n = 2. Notice that the purple (green) region overlaps the orange (cyan) one. The present most stringent upper limits |m ee | < 0.120 eV from EXO-200 [81,82] and KamLAND-ZEN [83] is shown by horizontal grey band. The vertical grey exclusion band represents the current bound coming from the cosmological data of m i < 0.230 eV at 95% confidence level obtained by the Planck collaboration [85].
The predictions on |m ee | are plotted in figure 14. For the IO spectrum and n = 2, we find |m ee | can take a few discrete values and these results can be tested in forthcoming 0νββ experiments.

Lepton mixing from a variant of semidirect approach
In contrast with semidirect approach discussed in section 5, we shall assume that the original symmetry D H CP is broken down to Z 2 × CP in the charged lepton sector, and the residual symmetry of the neutrino mass matrix is K 4 × H ν CP , where K 4 is a Klein subgroup of D . In this variant of the semidirect approach, the JHEP05(2016)007 Table 10. The column vector of the PMNS matrix determined by the residual flavor symmetries G ν and G l , where x, x , y, z = 0, 1, . . . , 3n − 1. If one (or two) element of the fixed column is vanishing, we would use the notation "" to indicate that it is disfavored by the present experimental data, otherwise the notation "" is labelled to indicate that agreement with the experimental data could be achieved. Because two pair of subgroups (G ν , G l ) = (K PMNS matrix turns out to depend on only one real continuous parameter besides the discrete parameters specifying the remnant symmetries, and one row of the PMNS matrix would be completely fixed by the assumed remnant symmetries. The fixed row vectors for different representative residual flavor symmetries are listed in table 10. We find that essentially only one type of residual symmetry with (G ν , G l ) = (K (c 9n/2 ,a 2 bc 3z d 2z ) 4 , Z bd x 2 ) is phenomenologically viable in this scenario.
and X νr = ρ r (c γ d −2z ), ρ r (a 2 bc γ ) : here we would like to recall that the residual CP transformations are determined by the restricted consistency conditions in eqs. as well as the residual CP transformation X νr = ρ r (c γ d −2z ), ρ r (a 2 bc γ ) have been studied in section 4. The light neutrino mass matrix m ν and its diagonalization matrix U ν are found to be given by eq. (4.14) and eq. (4.15) respectively. Then we proceed

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to the charged lepton sector. The invariance of the charged lepton mass matrix under the residual symmetry G l = {1, bd x } and X lr = ρ r (c 2x+2δ+3nτ d δ ), ρ r (bc 2x+2δ+3nτ d x+δ ) implies that the hermitian matrix m † l m l has to fulfill the invariant condition of eq. (3.1), i.e.  Notice that the order of the masses m 2 l 1 , m 2 l 2 and m 2 l 3 can not be pinned down by remnant symmetry, therefore the matrix U l in eq. (6.3) is determined up to permutations and phases of its column vectors. The lepton flavor mixing originates from the mismatch between the unitary transformations U l in eq. (6.3) and U ν in eq. (4.15), and the PMNS matrix can take the form where
mixing angles are within the experimentally preferred 3σ ranges. Furthermore, we find the following expressions for the CP invariants, |I 2 | = sin 2 θ 8 √ 2 sin 2θ sin(2ϕ 9 − ϕ 8 ) + 2 cos 2 θ sin 2(ϕ 9 − ϕ 8 ) + sin 2 θ sin 2ϕ 9 , (6.16) from which we know that both δ CP and α 21 are only dependent on θ and ϕ 8 , while the value of α 31 depends on three parameters θ, ϕ 8 and ϕ 9 . We display the predictions for | sin δ CP | and | sin α 21 | in the ϕ 8 − θ plane in figure 16. One can see that both δ CP and α 21 can not be maximal if the three mixing angles are required to be consistent with the experimental data. In analogy to previous cases, we numerically study the possible values of the mixing parameters for each D 9n,3n group. We can read from figure 17 that a bit larger θ 12 (still in the 3σ range) is favored with 0.328 ≤ sin 2 θ 12 ≤ 0.359, and the atmospheric mixing angle sin 2 θ 23 is predicted to be around 0.487 and 0.513. These results can be testable at forthcoming neutrino oscillation facilities. The same conclusions on CP phases are reached as those from figure 16. We find the upper bounds of | sin δ CP | and | sin α 21 | are |sin δ CP | ≤ 0.594 and |sin α 21 | ≤ 0.399 respectively. On the other hand, any value of the Majorana phase α 31 is possible for large value of n. group with n = 2. Note that the smallest D (1) 9n,3n group for n = 1 doesn't comprise the required Klein subgroup. The PMNS matrices U V,1 PMNS and U V,2 PMNS fulfill the following relations

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where U V PMNS refers to U V,1 PMNS and U V,2 PMNS . The diagonal matrices on the left and righthand sides can be absorbed by the charged lepton fields and Q ν respectively. Therefore the shifts of ϕ 8 into ϕ 8 + π and ϕ 9 into ϕ 9 + π/2 don't lead to physically new results. For n = 2 the values of ϕ 8 and ϕ 9 can be 0, π/6, π/3, . . ., 11π/6. Considering the constraint on the parameter ϕ 8 given by eq. (6.14), we find only ϕ 8 (mod π) = 0, π/6 and 5π/6 can describe the data on lepton mixing. The results of our χ 2 analysis are displayed in table 11. Since the mixing angles sin 2 θ ij and the CP invariants J CP and I 1 are expressed in terms of θ and ϕ 8 , and the parameter ϕ 9 only enters into the expression of I 2 , the relation in eq. (6.17c) implies that ϕ 8 and π − ϕ 8 give rise to the same best fitting values of mixing parameters except | sin α 31 |. This is exactly the reason why the numerical results for ϕ 8 = π/6 and ϕ 8 = 5π/6 are only different in the values of | sin α 31 |. Finally we plot the predictions for the effective mass |m ee | with respect to the lightest neutrino mass in figure 18. One can see that the values of |m ee | are rather close to the lower or upper boundary of the 3σ region for IO.

Conclusions
The type D finite subgroup of SU(3) has two independent series: D  Figure 17. The possible values of sin 2 θ 12 , sin θ 13 , sin 2 θ 23 , |sin δ CP |, |sin α 21 | and |sin α 31 | with respect to n for the mixing pattern U V,1 PMNS and U V,2 PMNS in the case V, where the three lepton mixing angles are required to be within the experimentally preferred 3σ ranges. The 1σ and 3σ regions of the three neutrino mixing angles are adapted from global fit [9]. Note that the group index n should be even in this case. and its predictions for the lepton flavor mixing have been discussed in the literature. In the present work, we have performed a comprehensive analysis of the mixing patterns which can be derived from another type D group series D (1) 9n,3n and the generalized CP . The phenomenological consequence of the "direct" approach, "semidirect" approach and "variant of semidirect" approach are studied in a model independent way. The three approaches differ in the residual symmetries preserved by the neutrino and charged lepton sectors.
The mathematical structure of D (1) 9n,3n has been investigated. Using the method of induced representations, we find all the irreducible representations of D   Table 11. Results of the χ 2 analysis for n = 2 in the case V. The χ 2 function has a global minimum χ 2 min at the best fit value θ bf for θ. We give the values of the mixing angles and CP violation phases for θ = θ bf . The values given in parentheses denote the results for the IO neutrino mass spectrum. Because of the symmetry relations in eq. (6.17), only the results for 0 ≤ ϕ 8 < π and 0 ≤ ϕ 9 < π/2 are shown here.
the Clebsch-Gordan coefficients. These details would be necessary and particularly useful for model builders aiming at construction of flavor models based on the group D (1) 9n,3n . Furthermore, we have identified the class-inverting automorphisms of the D (1) 9n,3n group, and show that the corresponding CP transformations are of the same form as the flavor symmetry transformations in our working basis.
In the "direct" approach, the original symmetry D H CP is broken down to K 4 × H ν CP in the neutrino sector and to G l H l CP in the charged lepton sector, where G l is an abelian subgroup which allows to distinguish the three generations of leptons. In this scenario, all the lepton mixing parameters including the Majorana CP phases are completely fixed by the residual symmetries. We have considered all the possible residual subgroups K 4 , G l and the residual CP transformations that can be consistently combined. We find that the lepton mixing matrices compatible with the data are of the trimaximal form. Both Dirac phase δ CP and the Majorana phase α 31 are predicted to be conserved, and the values of the Majorana phase α 21 are 0, 2 3n π, 4 3n π, . . ., 6n−2 3n π. In contrast with the "direct" approach, the residual symmetry preserved by the neutrino mass matrix is Z 2 × H ν CP in the "semidirect" approach. Since the remnant flavor JHEP05(2016)007 9n,3n group with n = 2. Notice that the purple (green) region overlaps the orange (cyan) one. The present most stringent upper limits |m ee | < 0.120 eV from EXO-200 [81,82] and KamLAND-ZEN [83] is shown by horizontal grey band. The vertical grey exclusion band represents the current bound coming from the cosmological data of m i < 0.230 eV at 95% confidence level obtained by the Planck collaboration [85]. symmetry of the neutrino sector is Z 2 instead of K 4 , it would fix only one column of the PMNS matrix. Taking into account the remnant CP transformations further, all the lepton mixing angles as well as the CP violating phases would be predicted in terms of a continuous free parameter θ besides the parameters characterizing the residual symmetries. We find that only four types of mixing patterns named as cases I, II, III and IV can accommodate the experimental data on lepton mixing angles for certain values of the continuous parameter θ and the discrete parameter ϕ i determined by the postulated residual symmetries. For cases III and IV, the residual Z 2 subgroup is chosen to be generated by the element c 9n/2 such that the group index n has to be even. We have performed a detailed analytical and numerical analysis. It is remarkable that either the solar mixing angle θ 12 or the atmospheric mixing angle θ 23 is bounded within certain intervals for arbitrary n. As a consequence, these predictions can be testable by the next generation of reactor neutrino experiments and long baseline experiments. The admissible values of the mixing angles and CP phases for each D (1) 9n,3n group until n = 50 have been studied. Interestingly enough, the first two smallest D (1) 9n,3n groups with n = 1, 2 already allow a good fit to the data on lepton mixing angles, and the CP violating phases can be conserved, maximal or some other irregular values. Moreover, the phenomenological predictions for the neutrinoless double beta decay are exploited.

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In the so-called "variant of semidirect" approach, the remnant symmetries of the neutrino and the charged lepton mass matrices are assumed to be K 4 × H ν CP and Z 2 × H l CP respectively. We find only one type of mixing pattern named as case V is phenomenologically viable in this scenario. One row of the PMNS matrix is determined to be (1/2, 1/2, −e iϕ 9 / √ 2). The solar mixing angle is predicted to lie in the interval 0.328 ≤ sin 2 θ 12 ≤ 0.359, and the atmospheric mixing angle is in the range of 0.510 ≤ sin 2 θ 23 ≤ 0.515 or 0.485 ≤ sin 2 θ 23 ≤ 0.490. Moreover, both Dirac phase and the Majorana phase α 21 are bounded from above |sin δ CP | ≤ 0.594 and |sin α 21 | ≤ 0.399 respectively. D (1) 9n,3n is a subgroup of ∆(6(9n) 2 ) group, and it has a subgroup isomorphic to ∆(6(3n) 2 ). However, both ∆(6(9n) 2 ) and ∆(6(3n) 2 ) don't admit a class-inverting automorphism [52,54]. The generalized CP symmetry can be consistently defined in the context of ∆(6(9n) 2 ) or ∆(6(3n) 2 ) flavor symmetry only if the fields transforming as the two dimensional representations 2 2 , 2 3 or 2 4 are absent in a model [54]. The absence of these doublet representations has been assumed in [54]. As a consequence, all the mixing patterns found in this work could be obtained from the ∆(6(9n) 2 ) group with CP. Similarly the mixing matrices predicted in ∆(6(3n) 2 ) H CP can be achieved from D (1) 9n,3n H CP . In the limit of large n, the predictions for mixing patterns and CP phases of the three groups ∆(6(3n) 2 ), D (1) 9n,3n , ∆(6(9n) 2 ) with CP would be quite close to each other. In our framework, the obtained results for lepton flavor mixing only depend on the structure of flavor symmetry group and the postulated residual symmetries, and they are independent of the breaking mechanism that how the required vacuum alignment needed to achieve the remnant symmetries is dynamically realized. It would be interesting to construct concrete models in which the breaking of the symmetry group to the residual symmetries are spontaneous due to the non-vanishing vacuum expectation values of some flavon fields.

Acknowledgments
The group D (1) 9n,3n for a generic integer n is a non-Abelian finite subgroup of SU(3) of type D [58]. Its order is 162n 2 . It is isomorphic to the semidirect product of the S 3 , the smallest non-Abelian finite group, with (Z 9n × Z 3n ), i.e. D (1) 9n,3n 9n,3n group can be defined in terms of four generators a, b, c and d fulfilling the following relations [18,58]: One can see that a and b generate S 3 , and c and d generate the Z 9n and Z 3n subgroups respectively. Any group element g ∈ D 9n,3n group belong to the following conjugacy classes: where the quantity on the left of the colon denotes the number of classes and the quantity on the right of the colon refers to the number of elements contained in the classes. The parameters ρ and σ in the conjugacy class 6C (ρ,σ) 1 can take the values ρ = 0, 1, . . . , 9n − 1, σ = 0, 1, . . . , 3n − 1, and the following possibilities are excluded, As a result, the D 9n,3n group totally has 1 + 2 + (9n − 3) + 27n(n−1)+6 6 + 3 + 9n = (3n + 1)(3n + 8)/2 different conjugacy classes. Furthermore, we can check that the center of the D (1) 9n,3n group is Z(D (1) 9n,3n ) = 1, c 3n , c 6n . In the end, we recall that the well studied ∆(6n 2 ) can be generated by four generators a , b , c and d obeying the relations [54]:

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One can straightforwardly check that the multiplication rules of D (1) 9n,3n in eq. (A.1) are fulfilled by a = a , b = b , c = c 2β d β , d = c −3β d −3β for n = 9n where β is an arbitrary nonzero integer. This implies that D (1) 9n,3n is a subgroup of the ∆(6(9n) 2 ) group, i.e. D (1) 9n,3n < ∆(6(9n) 2 ). In the same fashion, we can show the relations in eq. (A.6) are fulfilled by a = a, b = b, c = c 3γ d γ , d = c −3γ d −2γ for n = 3n and γ = 0. Hence ∆(6(3n) 2 ) is a subgroup of D (1) 9n,3n . Consequently we summarize ∆(6(3n) 2 ) < D (1) 9n,3n < ∆(6(6n) 2 ). Hence D 9n,3n group has six singlet representations given by As far as we know, the representations of the D 9n,3n group has not been worked out in the literature. It is a nontrivial task. In the following, we shall use the method of induced representations to build the remaining irreducible representations. The induced representation can be commonly found in the literature. In the following, we first briefly review the basic idea of the induced representation. Let G be a finite group and H any subgroup of G with index n. The index of H in G is the number of cosets of H in G, i.e. n = |G|/|H| where |G| and |H| denote the order of G and H respectively. We denote x 1 , x 2 , . . ., x n as a full set of representatives in G of the cosets in G/H, i.e.

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The induced representation can be thought of as acting on the following space: where each x i V is an isomorphic copy of the vector space V . The basis vector of the space W can be taken to be x k e i ≡ e k,i , with k = 1, 2, . . . , n, i = 1, 2, . . . , d .
According to the definition of coset, any g ∈ G will then send each x k to a unique x m h with h ∈ H such that gx k = x m h where k, m = 1, 2, . . . , n. In the induced representation, an element g ∈ G acts on the vector space W as follows Thus we see that G acts linearly on W , and its action is thus represented by a (dn × dn) matrix. Notice that the induced representation is not necessarily irreducible. We now apply this method to the group D 9n,3n = G, and take the subgroup to be H = Z 9n ×Z 3n . The index of H in G is n = 6. Since H is an abelian subgroup, its irreducible representations can only be one-dimensional. e 1 is the basis for the representation space of H, the generators c and d act on e 1 as follows where η = e ae 1 = e 3 , ae 2 = e 1 , ae 3 = e 2 , ae 4 = e 5 , ae 5 = e 6 , ae 6 = e 4 , be 1 = e 4 , be 2 = e 5 , be 3 = e 6 , be 4 = e 1 , be 5 = e 2 , be 6 = e 3 , ce 1 = η l e 1 , ce 2 = η l−3k e 2 , ce 3 = η −2l+3k e 3 , ce 4 = η l−3k e 4 , ce 5 = η −2l+3k e 5 , ce 6 = η l e 6 , de 1 = η −3k e 1 , de 2 = η −3l+6k e 2 ,

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Then we can read out the representation matrices as follows where 1 3 refers to a 3 × 3 unit matrix, and the different submatrices are given by The above different representations labelled by (l, k) may be equivalent. If we perform the similarity transformations generated by the representations matrices for a and b are kept intact while the diagonal elements of both c and d are interchanged. As a result, the same representation is labeled in six different ways The six pairs above can be compactly written into the form Now we proceed to study whether the six-dimensional representations constructed by the induced representation method are irreducible or not by the famous Mackey theorem in math [87][88][89][90][91]. If any one of them is reducible, we further decompose it into the direct sum of the irreducible representations of D 9n,3n group.
Corollary. Suppose H is a normal subgroup of G, then we have H s = H and Res Hs H ( ) = . In order that Ind G H ( ) be irreducible, it is necessary and sufficient that is irreducible and not isomorphic to any of its conjugate ρ s for s / ∈ H.
This implies that the representation Ind G H ( ) would be reducible if there is a s ∈ G \ H leading to s (H) ∼ = (H) for normal subgroup H. The corollary can be exploited to determine whether the six-dimensional representations 6 (l,k) of the D (1) 9n,3n group in eq. (A.19) are reducible or not. The subgroup H = Z 9n × Z 3n is a normal subgroup of D (1) 9n,3n , and it is abelian such that its irreducible representation is one-dimensional and specified by eq. (A.15). From the above corollary of the Mackey theorem, we know that the six-dimensional representation 6 (l,k) is reducible if and only if s (H) and (H) are equivalent representations for an element s ∈ D (1) 9n,3n \ H. In order to obtain the conditions in which the six-dimensional representations 6 (l,k) is reducible, we only need to consider the value of s is b, ab, a 2 b, a and a 2 respectively. The results are collected in table 12.
(A. 28) In the case of 3k = 0 and l = 0, 3n, 6n, the three vectors e 1 , e 2 and e 3 can be distinguished from each other by the actions of c and d, and they must be closed under the action of a and b. Considering the effect of a, we find ae 1 = e 3 , ae 2 = e 1 , ae 3 = e 2 , (A.29) which yield where the similarity transformation Ω reads In the new basis, the representation matrices of the generators are of the following form This means that the six-dimensional representation 6 (l,0) breaks up into two threedimensional irreducible representations 3 l,0 and 3 l,1 which differ in the overall sign of b. Notice that the values of l = 0, 3n, 6n should be excluded, since both triplet representations 3 l,0 and 3 l,1 then could be decomposed into one-dimensional and two-dimensional representation.