Generalised CP and $\Delta (6n^2)$ Family Symmetry in Semi-Direct Models of Leptons

We perform a detailed analysis of $\Delta (6n^2)$ family symmetry combined with a generalised CP symmetry in the lepton sector, breaking to different remnant symmetries $G_{\nu}$ in the neutrino and $G_{l}$ in the charged lepton sector, together with different remnant CP symmetries in each sector. We discuss the resulting mass and mixing predictions for $G_{\nu}=Z_2$ with $G_{l}=K_4,Z_p,p>2$ and $G_{\nu}=K_4$ with $G_{l}=Z_2$. All cases correspond to the preserved symmetry smaller than the full Klein symmetry, as in the semi-direct approach, leading to predictions which depend on a single undetermined real parameter, which mainly determines the reactor angle. We focus on five phenomenologically allowed cases for which we present the resulting predictions for the PMNS parameters as a function of $n$, as well as the predictions for neutrinoless double beta decay.


Introduction
Following the pioneering measurements of the reactor mixing angle θ 13 by the Daya Bay [1], RENO [2], and Double Chooz [3] reactor neutrino experiments, the three lepton mixing angles θ 12 , θ 23 , θ 13 and both mass-squared differences ∆m 2 sol and ∆m 2 atm have by now been measured to quite good accuracy, as recently summarised, for example, at the Neutrino Oscillation Workshop 2014 [4].
However the Dirac CP violating oscillation phase has not been measured so far and the neutrino mass squared ordering remains ambiguous. Moreover, if neutrinos are Majorana particles, there are two more unmeasured Majorana CP phases which play a role in determining the neutrinoless double-beta decay mass observable |m ee |, which is the (1, 1) element of the neutrino mass matrix in the flavour basis. Determining the neutrino mass squared ordering and measuring the Dirac CP violating phase are primary goals of the next generation neutrino oscillation experiments. The CP violation has been firmly established in the quark sector and therefore it is natural to expect that CP violation occurs in the lepton sector as well. Indeed hints of a nonzero δ CP ∼ 3π/2 have begun to show up in global analyses of neutrino oscillation data [5][6][7], although we emphasise that such a hint could equally well be due to an upward statistical fluctuation in the T2K electron appearance measurements [4] which are largely responsible for the effect.
In recent years, much effort has been devoted to explaining the structure of the lepton mixing angles through the introduction of a non-abelian discrete family symmetry group. For recent reviews of model building and relevant group theory aspects (see for example [8,9]). Three different model building approaches have been identified, known as "direct", "semidirect" and "indirect" [9]. In the "direct" approach the Klein symmetry Z 2 × Z 2 of the Majorana neutrino mass matrix is identified as a subgroup of the family symmetry, while in the "semi-direct" approach only part of the Klein symmetry, typically Z 2 , is contained in the family symmetry group, allowing the reactor angle to be fitted with one free parameter. In the "indirect approach" the neutrino mass matrix is constructed from "vacuum alignments" which are orthogonal to the symmetry preserving ones, allowing the Klein symmetry to arise as an indirect consequence of the family symmetry (for a recent example of this approach see e.g. [10]).
Inspired by the predictive power of discrete family symmetry, it is conceivable to extend the family symmetry approaches to include a generalised CP symmetry H CP [11,12] in order to allow the prediction of not only the mixing angles but also the CP phases as well. The idea was first explored in the context of CP symmetry combined with µ − τ reflection symmetry, resulting in the atmospheric mixing angle θ 23 and the Dirac CP phase δ CP both being determined to be maximal, as discussed in [13][14][15]. More recently, the phenomenological consequences of imposing both an S 4 family symmetry and a generalised CP symmetry have been investigated in a model-independent way [16][17][18][19][20][21]. Assuming the symmetry breaking of S 4 H CP to Z 2 × CP in the neutrino sector and to some abelian subgroup of S 4 in the charged lepton sector, all the three lepton mixing angles and CP phases were then given in terms of only one free parameter, where this free parameter results from the fact that only part of the Klein symmetry comes from the discrete family symmetry which is the case for the "semi-direct" approach. Concrete "semi-direct" S 4 family models with a generalised CP symmetry have been constructed in Refs. [16][17][18][19][20][21] where the spontaneous breaking of the S 4 H CP down to Z 2 × CP in the neutrino sector was implemented. A similar generalised analysis has also been considered for A 4 family symmetry [22]. The typical prediction for the Dirac CP phase for the "semi-direct" approach to S 4 and A 4 is | sin δ CP | = 0 or 1 in each case, although other model dependent predictions are also possible.
Other models with a family symmetry and a generalised CP symmetry can also be found in Refs. [23][24][25][26]. The interplay between flavor symmetries and CP symmetries has been generally discussed in [27,28]. In addition, there are other theoretical approaches involving both family symmetry and CP violation [29][30][31][32].
A generalised CP analysis of ∆(6n 2 ) has been performed recently [33] based on a "direct" approach with the full Klein symmetry Z 2 × Z 2 preserved in the neutrino sector and a Z 3 preserved in the charged lepton sector. However the result of this analysis is that | sin δ CP | = 0 corresponding to the Dirac CP phase being either zero or ±π. In fact, this result was obtained originally without imposing any CP symmetry [34], the difference being that in the earlier approach the Majorana phases were not determined. Clearly this result is rather disappointing from an experimental point of view, given the large effort going into measuring the Dirac CP phase and the difficulty of proving that | sin δ CP | = 0. Therefore it is of interest to consider the analogous situation for the "semi-direct" approach. Here we shall focus on the infinite series of finite groups ∆(6n 2 ) but relax the requirement of having the full Klein symmetry in the neutrino sector, and in addition consider more general preserved symmetries in the charged lepton sector. In other words we follow the "semi-direct" approach for the infinite series of groups based on ∆(6n 2 ).
In this paper, then, we study generalised CP symmetry for all ∆(6n 2 ) family symmetry groups where the CP symmetry is assumed to exist at a high energy scale and the requirement of having the full Klein symmetry is relaxed, as in the so-called "semi-direct" approach. The work here follows on from a similar analysis of the semi-direct approach performed by two of us based on the group ∆(96) [35]. Here we investigate the lepton mixing parameters which can be obtained from the original symmetry ∆(6n 2 ) H CP breaking to different remnant symmetries in the neutrino and charged lepton sectors, namely G ν and G l subgroups in the neutrino and the charged lepton sector respectively, while the remnant CP symmetries from the breaking of H CP are H ν CP and H l CP , respectively. The generalised CP transformation compatible with an ∆(6n 2 ) family symmetry is defined, and a model-independent analysis of the lepton mixing matrix is performed by scanning all the possible remnant subgroups in the neutrino and charged lepton sectors. Relaxing the requirement of having the full Klein symmetry in the neutrino sector given by a subgroup of ∆(6n 2 ), as in the semi-direct approach, we are led to a large number of possibilities where the results depend on a single parameter, expressed as an angle which determines the reactor angle. We systematically discuss all such possibilities consistent with existing phenomenological data, then analyse in detail the resulting predictions for mixing parameters. Our results divide into two cases, the first in which the residual symmetry in the neutrino sector is Z 2 × CP and the second in which the same residual symmetry Z 2 × CP is preserved by the charged lepton sector. More precisely, we discuss the resulting mass and mixing predictions for all possible cases where the family symmetry ∆(6n 2 ) enhanced with generalised CP is broken to G ν = Z 2 with G l = K 4 , Z p , p > 2 and G ν = K 4 with G l = Z 2 . We are led to six phenomenologically allowed mixing patterns and present the resulting predictions for PMNS parameters as a function of n, as well as the predictions for neutrinoless double beta decay.
While this paper was being prepared, a study of generalised CP within the semi-direct approach appeared based on the infinite series of finite groups ∆(6n 2 ) and ∆(3n 2 ) [36]. Where the results overlap for ∆(6n 2 ) they appear to be broadly in agreement, although the case that the residual symmetry Z 2 × CP is preserved by the charged lepton sector was not considered in [36]. The present paper focuses exclusively on ∆(6n 2 ), and, apart from considering extra cases not previously considered, presents the numerical results in a quite different and complementary way. Many of the numerical results contained in this paper, for example, the predictions for neutrinoless double beta decay, were not previously considered at all.
The remainder of this paper is organised as follows. In Section 2 we consider Generalised CP with ∆(6n 2 ). In Section 3 we consider possible lepton mixing from "semi-direct" approach with residual symmetry Z 2 × CP in the neutrino sector. In Section 4 we consider possible lepton mixing from "semi-direct" approach with residual symmetry Z 2 × CP in the charged lepton sector. The phenomenological predictions of the neutrinoless double beta decay for all the viable cases are presented in Section 5, Finally Section 6 concludes the paper.
2 Generalised CP with ∆(6n 2 ) Let us consider a theory with both family symmetry G f and generalized CP symmetry at high energy scale. A field multiplet ϕ r embedded into the representation r of G f transforms under the action of the family symmetry group G f as where ρ r (g) is the representation matrix of g in the representation r. Furthermore, the most general CP transformations act on the field ϕ r as: where x P = (t, −x), X r is a unitary matrix, and it is the so-called generalized CP transformation. The generalized CP symmetry has to be consistent with the family symmetry. It has been firmly established that the generalized CP symmetry can only be compatible with the family symmetry if the following consistency equation is satisfied [11,12,16,27]: Hence the generalized CP transformation is related to an automorphism which maps g into g . Furthermore, it was recently shown that physical CP transformations have to be given by class-inverting automorphism of G f [28]. In this work, we shall investigate the ∆(6n 2 ) series as the family symmetry group. The group theory of ∆(6n 2 ) is presented in Appendix A.
With the help of the computer algebra program system GAP [37][38][39][40], we have studied the automorphism group of the ∆(6n 2 ) until n = 19 1 . The results are collected in Table 1. We find that the outer automorphism groups of members of the ∆(6n 2 ) series are generally non-trivial except for ∆(6) ∼ = S 3 and ∆(24) ∼ = S 4 . However, there is only one class-inverting outer automorphism for n = 3Z while no class-inverting automorphism exists for n = 3Z. In fact, we find an outer automorphism u acting on the generators as: It can be straightforwardly checked that this automorphism u maps each conjugacy class into the inverse one for n = 3Z. In the case of n = 3Z, we have 2 are not mapped into their inverse classes although the latter is still true for the remaining classes. As a result, we conjecture that the ∆(6n 2 ) group with n = 3Z admits a unique class-inverting automorphism given by Eq. (2.4) although we can not prove it in a strict mathematical manner so far. In the following, we shall concentrate on the n = 3Z case without mention. The generalized CP transformation corresponding to u, which is denoted by X r (u), would be physically well-defined, as suggested in Ref. [28]. Its concrete form is fixed by the consistency equations as follows: In our basis, presented in section A, we can determine that Furthermore, including inner automorphisms, the full set of generalized CP transformations compatible with ∆(6n 2 ) family symmetry is Consequently the generalized CP transformations are of the same form as the family symmetry transformations in the chosen basis. In particular, we see that the conventional CP transformation with ρ r (1)=1 is allowed. As a consequence, all coupling constants would be real in a ∆(6n 2 ) model with imposed CP symmetry since all the CG coefficients are real, as shown in Appendix B. In the case of n = 3Z, the consistency equations of Eq. (2.6) are also satisfied except for r being the doublet representations 2 2 , 2 3 or 2 4 . Hence the generalized CP transformations in Eq. (2.7) can also be imposed on a model with n = 3Z if the fields transforming as 2 2 , 2 3 or 2 4 are absent.  Table 1: The automorphism groups of the ∆(6n 2 ) group series, where Inn(G f ) and Out(G f ) denote inner automorphism group and outer automorphism group of the family symmetry group G f respectively. The last column gives the number of class-inverting outer automorphisms. Note that the inner automorphism group of ∆(6n 2 ) with n = 3Z is isomorphic to ∆(6n 2 )/Z 3 since its center is the Z 3 subgroup generated by c n 3 d 3 Possible lepton mixing from "semi-direct" approach with residual symmetry Z 2 × CP in the neutrino sector In the following, we shall analyze all possible lepton mixing patterns in the "semi-direct" method [9,[17][18][19]22,26,35]. In this context, both family symmetry ∆(6n 2 ) and the consistent generalized CP symmetry are imposed onto the theory at high energy scales. The full symmetry group is ∆(6n 2 ) H CP , which is broken down to G l H l CP and Z 2 × H ν CP residual symmetries in the charged lepton and neutrino sectors respectively. G l is usually taken to be an abelian subgroup of ∆(6n 2 ) of order larger than 2 to avoid degenerate charged lepton masses. The misalignment between the two residual symmetries generates the PMNS matrix. In this approach, only remnant symmetries are considered and we do not discuss how the required symmetry breaking is dynamically achieved as there are generally more than one mechanism and many possible specific model realizations. It is remarkable that one column of the lepton mixing matrix can be fixed and the resulting lepton mixing parameters are generally constrained to depend on only one free parameter in this approach. We shall consider all the possible remnant symmetries G l H l CP and Z 2 ×H ν CP , and predictions for the lepton mixing angles and CP phases would be investigated. As usual the three generation of the left-handed lepton doublet fields are assigned to the faithful representation 3 1,1 which is denoted by 3 for simplicity in the following.

Charged lepton sector
The remnant symmetry G l implies that the charged lepton mass matrix is invariant under the transformation L → ρ 3 (g l ) L , where L stands for the three generations of left-handed lepton doublets, g l is the generator of G l , and ρ 3 (g l ) is the representation matrix of g l in the triplet representation 3. As a consequence, the charged lepton mass matrix satisfies where the charged lepton mass matrix m l is defined in the convention, c m l L . Let us denote the diagonalization matrix of m † l m l by U l , i.e.
where m e , m µ and m τ are the electron, muon and tau masses respectively. Substituting Eq. (3.2) into Eq. (3.1), we obtain One can see that U † l ρ 3 (g l )U l has to be diagonal. Therefore U l not only diagonalizes m † l m l but also the matrix ρ 3 (g l ). As a result, the unitary diagonalization matrix U l is completely fixed by the remnant flavor symmetry G l once the eigenvalues of ρ 3 (g l ) are non-degenerate. In the present work, we shall focus on the case that G l is a cyclic subgroup of ∆(6n 2 ). Hence the generator g l of G l could be of the form 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, t = 0, 1, . . . , n−1. If the eigenvalues of ρ 3 (g l ) are degenerate such that its diagonalization matrix U l can not be fixed uniquely, we could extend G l from a single cyclic subgroup to the product of several cyclic subgroups. This scenario is beyond the scope of this work except that the simplest K 4 extension is included. Given the explicit form of the representation matrices listed in Appendix A, the charged lepton diagonalization matrices U l for different cases of G l can be calculated straightforwardly, and the results are summarized in the following. Since the charged lepton masses can not be constrained at all in the present approach (in other word, the order of the eigenvalues of ρ 3 (g l ) is indeterminate), U l can undergo rephasing and permutations from the left.
Note that the parameters s and t should be subject to the following constraints s + t = 0 mod n, s − 2t = 0 mod n, t − 2s = 0 mod n , (3.5) otherwise the eigenvalues of c s d t would be degenerate and consequently U l can not be determined uniquely. For the value of s = t = n/2, the remnant symmetry could be chose to be K (c n/2 ,d n/2 ) 4 ≡ 1, c n/2 , d n/2 , c n/2 d n/2 instead, and then corresponding unitary transformation U l is still a unit matrix. The constraints of Eq. (3.5) will be assumed for the subgroup G l = c s d t in the following.
To avoid degenerate eigenvalues, we should exclude the values s − t = 0, n/3, 2n/3 mod n . where ω = e 2iπ/3 = − 1 2 + i √ 3 2 is the third root of unity. Notice that the order of the element ac s d t is three regardless of the values of s and t, and its eigenvalues are 1, ω and ω 2 .  Note that because (ac t d t−s ) 2 = a 2 c s d t holds, this U l can be obtained from the one in Eq. (3.8) by the replacement s → t, t → t − s.
Non-degeneracy of the eigenvalues of abc s d t requires t = 0, n/3, 2n/3. In the case of t = 0, the degeneracy can be avoided by expanding G l to the Klein four subgroup K (c n/2 ,abc s ) 4 ≡ 1, c n/2 , abc s , abc s+n/2 , whose diagonalization matrix is of the same form as Eq. (3.10) with t = 0.
Here the parameter s can not be equal to 0, n/3 or 2n/3, otherwise two eigenvalues of a 2 bc s d t would be identical. For the extended residual symmetry G l = K (d n/2 ,a 2 bd t ) 4 ≡ 1, d n/2 , a 2 bd t , a 2 bd t+n/2 , the corresponding unitary transformation is still given by Eq. (3.11) with s = 0.

Neutrino sector
In the present work, we assume the light neutrinos are Majorana particles. As a consequence, the remnant flavor symmetry G ν in the neutrino sector can only be a K 4 or Z 2 subgroup. The phenomenological consequence of G ν = K 4 has been studied in Refs. [33] by two of us. Here we shall concentrate on G ν = Z 2 case with generalized CP symmetry which allows us to predict CP phases. The Z 2 subgroups of ∆(6n 2 ) can be generated by bc x d x , abc y , a 2 bd z , x, y, z = 0, 1 . . . n − 1 (3.12) and additionally c n/2 , d n/2 , c n/2 d n/2 (3.13) for n = 2Z. It is notable that the Z 2 elements in Eq. (3.12) and Eq. (3.13) are conjugate to each other respectively: The remnant generalized CP symmetry should be compatible with the residual Z 2 symmetry in the neutrino sector, and therefore the corresponding consistency equation should be satisfied, i.e., X νr ρ * r (g)X −1 νr = ρ r (g), g ∈ Z 2 , (3. 15) which means that the residual CP and residual flavor transformations are commutable with each other [16,17] in the neutrino sector. For a given solution X νr of Eq. (3.15), one can check that ρ r (g)X νr is also a solution. The remnant CP symmetries consistent with the Z 2 elements in Eq. (3.12) and Eq. (3.13) are summarized as follows. (3.16) • g = abc y , y = 0, 1, 2 . . . n − 1 • g = d n/2 X νr = ρ r (c γ d δ ), ρ r (a 2 bc γ d δ ), γ, δ = 0, 1, 2 . . . n − 1 . (3.20) • g = c n/2 d n/2 X νr = ρ r (c γ d δ ), ρ r (bc γ d δ ), γ, δ = 0, 1, 2 . . . n − 1 . As we shall demonstrate in the following, the remnant CP symmetry should be symmetric to avoid degenerate lepton masses. Then the viable CP transformations would be constrained to be ρ r (abc γ d 2γ ), ρ r (a 2 bc 2δ d δ ) and ρ r (bc γ d −γ ) together with ρ r (c γ d δ ) for g = c n/2 , d n/2 and c n/2 d n/2 respectively. The full symmetry ∆(6n 2 ) H CP is broken down to Z 2 × H ν CP in the neutrino sector. The invariance of the light neutrino mass matrix m ν under the remnant family symmetry G ν = Z 2 and the remnant CP symmetry H ν CP leads to from which we can construct the explicit form of m ν and then diagonalize it.
The light neutrino mass matrix satisfying Eq.
where U ν is where K ν is a diagonal unitary matrix with entries ±1 and ±i which encode the CP parity of the neutrino states and renders the light neutrino masses positive. We shall omit the factor K ν in the following cases for simplicity of notation. The angle θ is given by The light neutrino masses are Here the order of the three eigenvalues m 1 , m 2 and m 3 can not be pinned down, consequently the unitary matrix U ν is determined up to permutations of the columns (the same turns out to be true in the following cases), and the neutrino mass spectrum can be either normal ordering or inverted ordering. Moreover, as four parameters m 11 , m 12 , m 13 and m 22 are involved in the neutrino masses, the measured mass squared splitting can be accounted for easily.
In this case, the light neutrino mass matrix takes the form: where m 11 , m 12 , m 13 and m 33 are real. The unitary matrix U ν which diagonalizes the above neutrino mass matrix is given by The light neutrino mass matrix, which is invariant under both remnant family symmetry and remnant CP symmetry, is of the form: where m 11 , m 12 , m 22 and m 23 are real parameters. The neutrino diagonalization matrix U ν is given by where the angle θ fulfills Finally the light neutrino masses are The light neutrino mass matrix is constrained to be of the following form The light neutrino masses are determined to be For the case of δ = 2γ mod n, the light neutrino masses would be partially degenerate. This is obviously unviable. The reason is that the corresponding generalized CP transformation matrix is not symmetric 2 . Therefore we shall concentrate on the case of δ = 2γ mod n in the following. The neutrino mass matrix is given by where m 11 , m 12 , m 33 and φ are real free parameters. The neutrino diagonalization matrix is The light neutrino mass eigenvalues are The ordering of the neutrino masses can not be determined as well. ( In the basis in which the neutrino mass matrix is diagonal with m ν = diag(m 1 , m 2 , m 3 ), the general CP transformation X which leaves m ν invariant: X T m mu X = m * ν , should be of the form X = diag(±1, ±1, ±1). One can go to an arbitrary basis and define the corresponding CP symmetry transformation X = V † XV * as a symmetry of the general neutrino mass matrix, where V is the basis transformation. As a result, the remnant CP symmetry X in the neutrino sector should be symmetric. The same conclusion has been obtained in Ref. [16].
The light neutrino mass matrix is constrained by residual family and residual CP symmetries to be where m 11 , m 22 , m 23 and m 33 are real. The neutrino diagonalization matrix is The light neutrino masses take the form As has been shown above, X νr has to be symmetric. Then the requirement γ = 2δ mod n follows immediately, otherwise the light neutrino masses would be partially degenerate. In this case, the neutrino mass matrix takes the form: where m 11 , m 22 , m 23 and φ are real. It is diagonalized by the unitary matrix The light neutrino masses are The light neutrino mass matrix invariant under both the residual family and residual CP symmetries is where m 11 , m 13 , m 22 and m 33 are real parameters. The unitary transformation U ν is given by The light neutrino mass eigenvalues are In the case of γ + δ = 0 mod n, the generalized CP transformation ρ r (bc γ d δ ) is not symmetric. As a consequence, the light neutrino masses are partially degenerate. In the following, we shall focus on the case of γ + δ = 0 mod n. The neutrino mass matrix is determined to be of the following form: where m 11 , m 13 , m 22 and φ are real. The neutrino diagonalization matrix is Finally the light neutrino masses are given by

Predictions for lepton flavor mixing
Since the possible forms of the neutrino and charged lepton mass matrices and their diagonalization matrices have been worked out in previous sections, the lepton flavor mixing matrix can be pinned down immediately Because the order of the charged-lepton and neutrino masses is indeterminate in the present framework, the PMNS matrix U P M N S is determined up to independent permutations of rows and columns. From Eqs.(3.14a,3.14b), we know that the remnant Z 2 symmetries generated by bc x d x , abc y , a 2 bd z are conjugate to each other, and the same is true for the Z 2 symmetry generated by c n/2 , d n/2 and c n/2 d n/2 . If a pair of residual flavor symmetries (G ν , G l ) is conjugated to the pair of groups (G ν , G l ) under the group element g ∈ ∆(6n 2 ), it has been established that both pairs lead to the same result for U P M N S even after the generalized CP symmetry is included [22]. As a consequence, we only need to need to consider the representative residual symmetry for normal ordering neutrino mass spectrum, and a very similar result is obtained for inverted ordering spectrum. We see that no entry of the PMNS matrix can be zero. As a result, the mixing patterns with a zero element have been ruled out by experimental data of neutrino mixing. In the following, we shall concentrate on the viable case in which no element of the fixed column is zero, and the predictions for the lepton flavor mixing parameters will be investigated for various remnant CP symmetries compatible with remnant flavor symmetry. (3.56) These two parameters ϕ 1 and ϕ 2 are interdependent of each other, and they can take the discrete values The determined form of one column of the PMNS matrix for different remnant symmetries G ν and G l which are Z 2 and abelian subgroups of ∆(6n 2 ) family symmetry groups respectively. The symbol "" denotes that the resulting lepton mixing is ruled out since there is at least one zero element in the fixed column, and the symbol "" denotes that the resulting mixing is viable. Note that for , the cases of G l = abc s d t and G l = a 2 bc s d t are not independent as we have b (abc Obviously one column of the PMNS matrix is fixed to be For the group ∆(24) ≡ S 4 , we have Note that the first column of the tri-bimaximal mixing matrix is reproduced for ϕ 1 = ±π/2. For the group ∆(96), we have It is interesting that the first and the third columns (up to permutations) of the Toorop-Feruglio-Hagedorn mixing [41][42][43] can be obtained in the case of ϕ 1 = ±π/4, ±3π/4. Now we consider the permutations of the rows and the columns, i.e. The PMNS matrix can be multiplied by a 3 × 3 permutation matrix from both the left-hand side and the right-hand side. There are six permutation matrices corresponding to six possible orderings of rows (or columns): It is well-known that the atmospheric mixing angle θ 23 becomes π/2 − θ 23 , the Dirac CP phases δ CP becomes π + δ CP and the other mixing parameters are kept intact if the second and third row of the PMNS matrix are exchanged. Hence the permutation of the second and third row will not be explored explicitly in the following. First we study the situation that the constant vector 2/3 (sin ϕ 1 , cos (π/6 − ϕ 1 ) , cos (π/6 + ϕ 1 )) T is in the first column. Then the PMNS matrix can be arranged as follows: Note that exchanging its second and third row does not lead to new mixing patterns. Moreover, we see that the above three arrangements are related with each other: where the phase factor diag (±1, ±1, ±1) can be absorbed by the lepton fields. Hence it is sufficient to only discuss the first PMNS matrix U I,1st P M N S in detail, the phenomenological predictions for the other two can be easily obtained by variable substitution. In this case, the lepton mixing parameters are predicted to be sin 2 θ 13 = 1 3 1 + cos 2 θ cos 2ϕ 1 + √ 2 sin 2θ cos ϕ 2 cos ϕ 1 , sin 2 θ 12 = 1 + sin 2 θ cos 2ϕ 1 − √ 2 sin 2θ cos ϕ 2 cos ϕ 1 2 − cos 2 θ cos 2ϕ 1 − √ 2 sin 2θ cos ϕ 2 cos ϕ 1 , sin 2 θ 23 = 1 − cos 2 θ sin (π/6 + 2ϕ 1 ) − √ 2 sin 2θ cos ϕ 2 sin (π/6 − ϕ 1 ) 2 − cos 2 θ cos 2ϕ 1 − √ 2 sin 2θ cos ϕ 2 cos ϕ 1 , |tan δ CP | = 2 √ 2 sin 2θ sin ϕ 2 (1 + 2 cos 2ϕ 1 ) 2 − cos 2 θ cos 2ϕ 1 − √ 2 sin 2θ cos ϕ 2 cos ϕ 1 2 sin 2 2θ cos 2ϕ 2 (cos 3ϕ 1 − 2 cos ϕ 1 ) + cos ϕ 1 9 − 4 cos 2θ + 3 cos 4θ − 16 cos 2 θ cos 2ϕ 1 −2 √ 2 sin 2θ cos ϕ 2 2 − cos 2 θ(5 + cos 2ϕ 1 + cos 4ϕ 1 ) where α 31 = α 31 − 2δ CP , δ CP is the Dirac CP phase, α 21 and α 31 are the Majorana CP phases in the standard parameterization [44]. If we embed the three generations of lefthanded lepton doublets into the triplet 3 1,n−1 which is the complex conjugate representation of 3 1,1 , all three CP phases δ CP , α 21 and α 31 would become their opposite numbers modulo 2π. Furthermore, the overall sign of tan α 21 and tan α 31 depends on the CP parity of the neutrino states which is encoded in the matrix K ν (please see Eq. (3.25)), and the sign of the Jarlskog invariant J CP depends on the ordering of rows and columns. As a result, all these quantities are presented in terms of absolute values here.
It is notable that all three CP phases depend on both the free continuous parameter θ and the discrete parameters ϕ 1 and ϕ 2 associated with flavor and CP symmetries. Obviously both Dirac CP and Majorana CP are conserved for ϕ 2 = 0. Furthermore, the solar mixing angle θ 12 and reactor angle θ 13 are related by 3 cos 2 θ 12 cos 2 θ 13 = 2 sin 2 ϕ 1 , which is independent of the free parameter θ. Taking into account measured values of θ 12 and θ 13 [6], we obtain the constraint on ϕ 1 as which indicates that ϕ 1 is around ±π/2. This mixing pattern is very interesting, and it can accommodate the present neutrino oscillation data very well. After the measured 3σ ranges of the three lepton mixing angles are imposed, the allowed values of the lepton mixing parameters for n = 2, 3, . . . , 100 are displayed in Fig. 1 and Fig. 2. In the case that n is divisible by 3, the doublet representations 2 2 , 2 3 and 2 4 are assumed to be absent such that generalized CP symmetry in Eq. the allowed values of sin 2 θ 12 , sin θ 13 and sin 2 θ 23 for different n, where the three lepton mixing angles are required to lie in their 3σ ranges. The 1σ and 3σ bounds of the mixing parameters are taken from Ref. [6].
any values for sufficiently large n, while θ 12 is always constrained to be in the range of 0.313 ≤ sin 2 θ 12 ≤ 0.344. Hence this mixing pattern can be tested by precisely measuring the solar mixing angle θ 12 . Notice that θ 12 can be measured with rather good accuracy by JUNO experiment [45]. In the end, the correlations between mixing parameters for n → ∞ and n = 8 are displayed in Fig. 3 where we only show the phenomenologically viable cases for which the measured values of θ 12 , θ 13 and θ 23 can be accommodated for certain values of the parameter θ.
The vector 2/3 (sin ϕ 1 , cos (π/6 − ϕ 1 ) , cos (π/6 + ϕ 1 )) T enforced by the remnant Z 2 symmetry can also be the second column of the PMNS matrix. Disregarding the exchange of  The red filled regions denote the allowed values of the mixing parameters if we take the parameters ϕ 1 and ϕ 2 to be continuous (which is equivalent to taking the limit n → ∞) and the three mixing angles are required to lie in their 3σ regions. Note that the three CP phases δ CP , α 21 and α 31 are not constrained in this limit. The black curves represent the phenomenologically viable correlations for n = 8. The 1σ and 3σ bounds of the mixing parameters are taken from Ref. [6].
This case can be realizable if n is divisible by 2, and the PMNS matrix takes the form   e iϕ 4 cos θ − e iϕ 5 sin θ 1 e iϕ 4 sin θ + e iϕ 5 cos θ ωe iϕ 4 cos θ − ω 2 e iϕ 5 sin θ 1 ωe iϕ 4 sin θ + ω 2 e iϕ 5 cos θ ω 2 e iϕ 4 cos θ − ωe iϕ 5 sin θ 1 ω 2 e iϕ 4 sin θ + ωe iϕ 5 cos θ   , where ϕ 4 = γ + δ + 2s n π, ϕ 5 = 2δ − γ + 2t n π , (3.89) which can take the values ϕ 4 , ϕ 5 mod 2π = 0, 1 n π, 2 n π, . . . , 2n − 1 n π .  where the three lepton mixing angles are required to lie in their 3σ ranges (the 3σ lower bound of sin 2 θ 12 is chosen to be 0.254 instead of 0.259 given in Ref. [6]). The 1σ and 3σ bounds of the mixing parameters are taken from Ref. [6]. where the three lepton mixing angles are required to lie in their 3σ ranges (the 3σ lower bound of sin 2 θ 12 is chosen to be 0.254 instead of 0.259 given in Ref. [6]). The 1σ and 3σ bounds of the mixing parameters are taken from Ref. [6].  The red filled regions denote the allowed values of the mixing parameters if we take the parameter ϕ 3 to be continuous (which is equivalent to taking the limit n → ∞) and the three mixing angles are required to lie in their 3σ ranges (the 3σ lower bound of sin 2 θ 12 is chosen to be 0.254 instead of 0.259 given in Ref. [6]). Note that the Majorana phase α 31 is not constrained in this limit. The black curves represent the phenomenologically viable correlations for n = 8. The 1σ and 3σ bounds of the mixing parameters are taken from Ref. [6] Agreement with experimental data can be achieved only if the vector 1/ √ 3, 1/ √ 3, 1/ √ 3 T is placed in the second column. It is the so-called TM 2 mixing [46]. There are three independent arrangements up to the exchange of the second and the third row, Once can check that they are related as follows, It is enough to study the phenomenological predictions of U III,1st P M N S . The lepton mixing parameters are given by ,  where the three lepton mixing angles are required to lie in their 3σ regions. The 1σ and 3σ bounds of the mixing parameters are taken from Ref. [6]. Note that n should be even in this case.
The allowed values of the mixing parameters for different n are shown in Fig 11 and Fig. 12.
The case of ϕ 4 = ϕ 5 is always viable for any n, and the resulting θ 23 and δ CP are maximal while the Majorana phase α 31 is trivial. Correlations among the mixing parameters are plotted in Fig. 13. The three CP phases can take any values for large n.
(IV) G l = ac s d t , G ν = Z c n/2 2 , X νr = ρ r (abc γ d δ )  Figure 12: Numerical results in case III: the allowed values of |sin δ CP |, |sin α 21 | and |sin α 31 | for different n, where the three lepton mixing angles are required to lie in their 3σ ranges. The 1σ and 3σ bounds of the mixing angles are taken from Ref. [6]. Note that n should be even in this case.  Figure 13: The correlations among mixing parameters in case III. The red filled regions denote the allowed values of the mixing parameters if we take the parameters ϕ 4 and ϕ 5 to be continuous (which is equivalent to taking the limit n → ∞) and the three mixing angles are required to lie in their 3σ ranges. Note that the three CP phases δ CP , α 21 and α 31 are not constrained in this limit. The black curves represent the phenomenologically viable correlations for n = 8. The 1σ and 3σ bounds of the mixing parameters are taken from Ref. [6].
In this case, the PMNS matrix is of the form (3.99) The constant vector 1/ √ 3, 1/ √ 3, 1/ √ 3 T must be the second column to account for the measured values of the lepton mixing angles. The PMNS matrices corresponding to other ordering of rows and columns are related to the above one through redefinition of the free parameter φ. This case differs from case III in the remnant CP symmetry, and the resulting PMNS matrix in Eq. (3.98) is still of TM 2 form. The associated lepton mixing parameters read as: , It is remarkable that the contribution of ϕ 6 can be absorbed into the free parameter φ via redefinition φ → φ+2ϕ 6 , the reason is that the PMNS matrix in Eq. (3.98) and the resulting mixing parameters in Eq. (3.100) depend on the combination φ − 2ϕ 6 . Regarding to the CP violating phases, both δ CP and α 31 are always conserved while α 21 can be any value of 0, 1 n π, 2 n π, . . ., 2n−1 n π in this scenario. Furthermore, the three mixing angles are strongly related with each other as follows: 3 cos 2 θ 13 sin 2 θ 12 = 1, For the best fitting value of sin 2 θ 13 = 0.0234 [6], the solar and atmospheric angles are determined to be sin 2 θ 12 0.341, sin 2 θ 23 0.391 or 0.609 , (3.102) which are compatible with the experimentally allowed regions. These correlations between the three mixing angles are shown in Fig. 14. We see that both θ 12 and θ 23 are constrained to be in a narrow range. The deviation of θ 23 from maximal mixing is somewhat large. Hence this mixing pattern can be checked or ruled by precisely measuring θ 12 and θ 23 in next generation neutrino oscillation experiments.  Figure 14: The correlations among mixing parameters in case IV. The red filled regions denote the allowed values of the mixing parameters if we take the parameters ϕ 6 and ϕ 7 to be continuous (which is equivalent to taking the limit n → ∞), where θ 12 and θ 13 are required to lie in their 3σ ranges. The 1σ and 3σ bounds of the mixing parameters are taken from Ref. [6].
In this section, we shall investigate another scenario: the remnant symmetry Z 2 × CP is preserved in the charged lepton sector and the full symmetry ∆(6n 2 ) H CP is broken down to K 4 H ν CP in the neutrino sector. The phenomenological consequences of this scenario have been analyzed for the simple flavor symmetry group ∆(24) = S 4 in Ref. [19]. All the Z 2 subgroups of ∆(6n 2 ) have been listed in Eq. . Other possible choices of G l and G ν are related to these representative residual symmetry by similarity transformations, and therefore the same lepton mixing matrices are generated.
Following the same procedure demonstrated in section 3, the hermitian combination m † l m l of the charged lepton mass matrix and its diagonalization matrix can be straightforwardly calculated from the invariance under the remnant symmetry although the involved algebraic calculations are somewhat lengthy and tedious. To avoid distractions by too many details, we have left the calculations for this to Appendix C. Comparing with the scenario of Z 2 × H ν CP preserved in the neutrino sector which has been studied in section 3.2, we find that the unitary transformation U l is of the same form as U ν listed in section 3.2 if the both residual flavor symmetry and residual CP symmetry in the two occasions are identical. For the sake of presentation in the following, we briefly recapitulate the predicted form of U l for G l = Z bc x d x 2 and Z c n/2 2 here. In case of G l = Z bc x d x 2 , X lr = {ρ r (c γ d −2x−γ ), ρ r (bc x+γ d −x−γ )}, as shown in Appendix C, the charged lepton mass matrix is diagonalized by For the residual symmetry G l = Z c n/2 2 , X lr = ρ r (c γ d δ ), the unitary matrix U l is of the form The remnant symmetry could also be G l = Z c n/2 2 , X lr = ρ r (abc γ d δ ). The non-degeneracy of the charged lepton masses requires δ = 2γ mod n, and then the diagonalization matrix U l is determined to be where φ is a free real parameter.

Neutrino sector
In this section, we shall assume that the ∆(6n 2 ) flavor symmetry is broken down to K 4 in the neutrino sector. Hence the neutrino diagonalization matrix U ν is entirely fixed by the remnant K 4 , and the residual CP symmetry allows us to further determine the three leptonic CP violating phases up to π. The residual CP symmetry H ν CP in the neutrino sector must be compatible with the remnant K 4 symmetry, and the consistency condition should be satisfied, Solving this equation, we can find the consistent remnant CP symmetries for different K 4 subgroups are as follows: • K (c n/2 ,abc y ) 4 , y = 0, 1, . . . n − 1 with γ = 0, 1, . . . n − 1.
In this case, the light neutrino mass matrix must be subject to the constraints from the remnant family symmetry K 4 and the residual CP symmetry H ν CP : Since the representation matrices of both c n/2 and d n/2 are diagonal, the light neutrino mass matrix is constrained to be diagonal as well. Including the remnant CP symmetry, we find where m 11 , m 22 and m 33 are real parameters. The neutrino diagonalization matrix can be easily read out U ν = diag e iπ γ n , e −iπ γ−δ n , e −iπ δ n K ν , (4.12) where K ν is a diagonal matrix with element ±1 or ±i to set the light neutrino masses being positive. The light neutrino masses are We see that the light neutrino masses depend on only three real parameters, and we would like to stress again that the order of the light neutrino masses can not be fixed here, and therefore U ν here and henceforth is determined up to column permutations. For other residual CP symmetries X νr = ρ r (bc γ d δ ), ρ r (ac γ d δ ), ρ r (a 2 c γ d δ ), ρ r (abc γ d δ ) and ρ r (a 2 bc γ d δ ) with γ, δ = 0, 1, . . . n − 1, the light neutrino masses are partially degenerate such that they are not viable.
(ii) G ν = K where m 11 , m 12 and m 33 are real. It is diagonalized by the unitary matrix U ν with The light neutrino masses are given by For the case of X νr = ρ r (c γ d 2y+2γ+n/2 ), ρ r (abc y+γ d 2y+2γ+n/2 ) , the light neutrino masses are degenerate, and therefore are not discussed here.
The light neutrino mass matrix, which is invariant under both residual flavor and residual CP symmetry, is determined to be  For the value of X νr = ρ r (c 2z+2δ+n/2 d δ ), ρ r (a 2 bc 2z+2δ+n/2 d z+δ ) , the neutrino masses are degenerate.
In this case, we find the light neutrino mass matrix is of the form where m 11 , m 13 and m 22 are real. The unitary matrix U ν diagonalizing this neutrino mass matrix is For the remaining value of X νr = ρ r (c γ d −2x−γ+n/2 ), ρ r (bc x+γ d −x−γ+n/2 ) , the light neutrino masses are degenerate. Table 3: The determined form of one row of the PMNS matrix for different remnant symmetries G ν and G l which are K 4 and Z 2 subgroups of ∆(6n 2 ) family symmetry group respectively. The superscript "T " means transpose. The symbol "" denotes that the resulting lepton mixing is ruled out since there is at least one zero element in the fixed row, and the symbol "" denote that the resulting mixing is viable.
, the cases of G ν = K

Predictions for lepton flavor mixing
As the different residual symmetries related by group conjugation lead to the same predictions for the lepton mixing matrix, we only need to consider the cases of G l = Z bc . Compared with section 3, one row instead of one column of the PMNS matrix is fixed by remnant flavor symmetry in this scenario. The explicit form of this row vector for different remnant symmetry is summarized in Table 3. We see that only one independent case is viable. Taking into account the compatible remnant CP symmetry, we can predict both mixing angles and CP phases in terms of one free parameter.
Combining the unitary transformation U l in Eq. (4.2) and U ν in Eq. (4.15), we can pin down the lepton flavor mixing matrix as follows: with ϕ 8 = 3γ + 2x + 2y n π, ϕ 9 = − 3γ + 2x + 2y n π . (4.24) Here ϕ 8 and ϕ 9 are independent, they are determined by the remnant symmetry, and they can take the values, ϕ 8 , ϕ 9 mod 2π = 0, 1 n π, 2 n π, . . . , 2n − 1 n π . (4.25) In order to be in accordance with the present neutrino oscillation data, 1/2, 1/2, −e iϕ 9 / √ 2 can only be the second or the third row. Note that as usual permutation of the second and the third rows of U V P M N S is also viable. We can read out the lepton mixing parameters, |tan α 21 | = 8 √ 2(1 + 3 cos 2θ) sin 2θ sin ϕ 8 7 + 12 cos 2θ + 13 cos 4θ + 8 sin 2 2θ cos 2ϕ 8 , |tan α 31 | = sin 2 θ sin 2ϕ 9 + √ 2 sin 2θ sin(2ϕ 9 − ϕ 8 ) + 2 cos 2 θ sin(2ϕ 9 − 2ϕ 8 ) sin 2 θ cos 2ϕ 9 + √ 2 sin 2θ cos(2ϕ 9 − ϕ 8 ) + 2 cos 2 θ cos(2ϕ 9 − 2ϕ 8 ) . (4.26) We see that all mixing parameters depend on θ and ϕ 8 except |tan α 31 | which involves ϕ 9 additionally. It is interesting that the mixing angles θ 13 and θ 23 are related as follows 2 cos 2 θ 13 sin 2 θ 23 = 1, or 2 cos 2 θ 13 sin 2 θ 23 = 1 − 2 sin 2 θ 13 , (4.27) where the second relation is for the PMNS matrix obtained by exchanging the second and the third rows of U V P M N S . Moreover, θ 12 and θ 13 are related by cos 2 θ 13 cos 2θ 12 = ±2 sin θ 13 cos 2θ 13 cos ϕ 8 , (4.28) which is relevant to the parameter ϕ 8 . The 3σ bound of sin 2 θ 13 gives the limit on θ: The equation for sin 2 θ 12 in Eq. (4.26) leads to The numerical results are displayed in Fig. 15 and Fig. 16. Note that conserved CP corresponding to ϕ 8 = 0, π is always viable. If we require that all three mixing angles are in their 3σ intervals, we find that 0.141 ≤ sin θ 13 ≤ 0.172, 0.328 ≤ sin 2 θ 12 ≤ 0.359, and sin 2 θ 23 is around 0.488 and 0.512 due to the correlation shown in Eq. (4.27). Note that θ 23 is very close to maximal mixing. Therefore precisely measuring the lepton mixing angles at JUNO or long baseline neutrino experiments can test this mixing pattern directly. For the CP phases, δ CP and α 21 are predicted to be in the intervals of |sin δ CP | ≤ 0.586 and |sin α 21 | ≤ 0.396 while α 31 can be any value for sufficient large n. The correlations between different mixing parameters are shown in Fig. 17.  Figure 15: Numerical results in case V: the allowed ranges of sin 2 θ 12 , sin θ 13 and sin 2 θ 23 for different n, where the three lepton mixing angles are required to lie in the 3σ regions. The 1σ and 3σ bounds of the mixing angles are taken from Ref. [6]. Note that n should be divisible by 2 in this case.  Figure 16: Numerical results in case V: the allowed ranges of |sin δ CP |, |sin α 21 | and |sin α 31 | for different n, where the three lepton mixing angles are required to lie in the 3σ regions. The 1σ and 3σ bounds of the mixing angles are taken from Ref. [6]. Note that n should be divisible by 2 in this case. The red filled regions denote the allowed values of the mixing parameters if we take the parameters ϕ 8 and ϕ 9 to be continuous (which is equivalent to taking the limit n → ∞) and the three mixing angles are required to lie in their 3σ ranges. Note that the Majorana phase α 31 is not constrained in this limit. The black curves represent the phenomenologically viable correlations for n = 8. The 1σ and 3σ bounds of the mixing parameters are taken from Ref. [6].

Neutrinoless Double-Beta Decay
The rare process, neutrinoless double-beta decay (0ν2β),is an important probe for the Majorana nature of neutrino and lepton number violation, a sizable number of new experiments are currently running, under construction, or in the planing phase. The effective mass of neutrinoless double-beta decay is [44] |m ee | = (m 1 c 2 12 + m 2 s 2 12 e iα 21 )c 2 13 + m 3 s 2 13 e iα 31 (5.1) For normal hierarchy, the masses are and for inverted hierarchy where m l denotes the lightest neutrino masses, and δm 2 ≡ m 2 2 − m 2 1 and ∆m 2 ≡ m 2 3 − (m 2 1 + m 2 2 )/2 defined in Ref. [6]. The experimental error on the neutrino mass splitting is not taken into account during the analysis, instead the best fit values from [6] are used: for normal (inverted) hierarchy. In the following, the properties of the effective mass are examined for all viable cases of lepton mixing discussed in this paper. In Fig. 18 the allowed ranges of the effective mass are shown for each case in the limit of n → ∞, where the three mixing angles are required to lie in the measured 3σ intervals [6](As previously mentioned, the 3σ lower bound of sin 2 θ 12 is chosen to be 0.254 instead of 0.259 in case II). Furthermore, the predictions for the representative value n = 8 (n = 5 in case I for the 7th-9th ordering) are plotted in Fig. 19 in order to be read easily. The results for any finite value of n must be part of the shown one corresponding to n → ∞. Moreover, the plotting would change only a little bit if the experimental errors on δm 2 and ∆m 2 are taken into account. Note that only one distinct prediction for the effective mass arises except in case I. One reason for this is that, as discussed before, many of the possible permutations of the mixing matrix can be identified with each other. Furthermore, permuting the second and third row has no effect on the effective mass as θ 23 does not appear in Eq. (5.1). As shown in Fig. 18, for inverted hierarchy neutrino mass spectrum, almost all the 3σ range values of the effective masses |m ee | can be reproduced in the limit n → ∞ in case I, case III and case IV. However, the predictions for |m ee | are around the upper bound (about 0.05eV) or lower bound (about 0.013 eV) in case V. The reason is that the solar mixing angle is in a narrow region 0.328 ≤ sin 2 θ 12 ≤ 0.359 and the Majorana phase α 21 is constrained to be |sin α 21 | ≤ 0.586 in this case, as displayed in Fig. 15 and Fig. 16. Similarly |m ee | is near the upper bound and 0.025 eV in case II. Therefore if the effective mass is measured to be far from 0.013 eV, 0.025 eV and 0.05 eV for inverted hierarchy by future experiments, the mixing patterns in cases II and V could be ruled out.
For normal hierarchy neutrino mass spectrum, a sizable part of the experimentally allowed 3σ region of |m ee | can be generated in all cases, and the effective mass could be rather small. In particular, the prediction in case I with 7th to 9th ordering approximately coincides with the present 3σ region. Unfortunately the predictions for normal hierarchy are still out of reach of projected experiments known to the author. As a result, it is should be generally difficult to test the ∆(6n 2 ) family symmetry and generalized CP symmetry through neutrinoless double beta decay experiments in the case of normal ordering spectrum.

Conclusions
We have performed a detailed analysis of ∆(6n 2 ) family symmetry combined with the generalised CP symmetry H CP in the lepton sector. We have investigated the lepton mixing parameters which can be obtained from the original symmetry ∆(6n 2 ) H CP breaking to different remnant symmetries in the neutrino and charged lepton sectors, namely G ν and G l subgroups in the neutrino and the charged lepton sector respectively, while the remnant CP symmetries from the breaking of H CP are H ν CP and H l CP , respectively. We have assumed a preserved symmetry smaller than the full Klein symmetry, as in the semi-direct approach, leading to predictions which depend on a single undetermined real parameter, which mainly controls the reactor angle. We have discussed the resulting mass and mixing predictions for all possible cases where the ∆(6n 2 ) family symmetry with generalised CP is broken to G ν = Z 2 with G l = K 4 , Z p , p > 2 and G ν = K 4 with G l = Z 2 . We have focused on five phenomenologically allowed cases and have presented the resulting predictions for the PMNS parameters as a function of n, as well as the predictions for neutrinoless double beta decay.
It is remarkable that the CP phases are predicted to take irregular values rather than 0, π or ±π/2. In particular, compared to ∆(6n 2 ) in the direct approach where the full Klein symmetry is identified as a subgroup, the result | sin δ CP | = 0 corresponding to the Dirac CP phase being either zero or ±π is relaxed in the indirect approach followed here. Compared to the indirect approach to S 4 (which corresponds in our notation to n = 2) where | sin δ CP | = 1 is the only available possibility for non-zero CP violation, for larger values of n we find many different possibilities for the oscillation phase, where the number of possible predictions grows rapidly as n is increased. However it is interesting that, even for very large n, the predictions for the PMNS parameters is bounded within certain ranges, providing non-trivial tests of the semi-direct approach followed here. It remains to be seen if any of these possibilities will closely correspond to the observed future precise determination of leptonic mixing angles and CP violating parameters in the future.  Figure 18: The allowed ranges of the effective mass for neutrinoless double-beta decay for all viable cases of lepton mixing in semidirect models with a ∆(6n 2 ) flavour group in the limit of n → ∞. The top row corresponds to case I, with 1st-3rd ordering on the left and 7th to 9th ordering on the right, the middle row contains case II and III, and the bottom row case IV and V. Light blue and yellow areas indicate the currently allowed three sigma region for normal and inverted hierarchy, respectively. Purple regions correspond to predictions assuming inverted hierarchy, green regions to normal hierarchy. The upper bound |m ee | < 0.120 eV is given by measurements by the EXO-200 [48,49] and KamLAND-ZEN experiments [50]. Planck data in combination with other CMB and BAO measurements [51] provides a limit on the sum of neutrino masses of m 1 + m 2 + m 3 < 0.230 eV from which the upper limit on the mass of the lightest neutrino can be derived.  Figure 19: The allowed ranges of the effective mass for neutrinoless double-beta decay for all viable cases of lepton mixing in semidirect models with a ∆(6n 2 ) flavour group. The top row corresponds to case I, with 1st-3rd ordering on the left and 7th to 9th ordering on the right, the middle row contains case II and III, and the bottom row case IV and V. Light blue and yellow areas indicate the currently allowed three sigma region for normal and inverted hierarchy, respectively. Purple regions correspond to predictions assuming inverted hierarchy, green regions to normal hierarchy in the limit of n → ∞. Blue and red regions represent predictions for normal and inverted hierarchy for the value n = 8 (in the top-right panel, we choose n = 5 which is the smallest viable value of n in that case). The upper bound |m ee | < 0.120 eV is given by measurements by the EXO-200 [48,49] and KamLAND-ZEN experiments [50]. Planck data in combination with other CMB and BAO measurements [51] provides a limit on the sum of neutrino masses of m 1 + m 2 + m 3 < 0.230 eV from which the upper limit on the mass of the lightest neutrino can be derived. A The group theory of ∆(6n 2 ) ∆(6n 2 ) is non-abelian finite subgroup of SU (3). The ∆(6n 2 ) is isomorphic to (Z n ×Z n ) S 3 , where S 3 is isomorphic to Z 3 Z 2 , and it can be conveniently defined by four generators a, b, c and d obeying the relations [52]: The elements a and b are the generators of S 3 while c and d generate Z n × Z n , and the last line defines the semidirect product " ". Note that the generator d = bc −1 b −1 is not independent. All the group elements can be written into the form where α = 0, 1, 2, β = 0, 1, γ, δ = 0, 1, 2, . . . n − 1. In the following we list the elements of ∆(6n 2 ) by order of the generated cyclic subgroup.
• Elements of order 2, if n even: c n/2 , d n/2 , c n/2 d n/2 (A.3) • Elements of order 2, always: bc d , abc γ , a 2 bd δ (A. 4) with , γ, δ = 0, . . . , n − 1. The ∆(6n 2 ) group have been thoroughly studied in Ref. [52]. In the following, we shall review the basic aspects, which is relevant to our present work. The conjugacy classes of ∆(6n 2 ) group are of the following forms: The irreducible representations and their representation matrices of the ∆(6n 2 ) group are as follows [52]: In our new basis, all the Clebsch-Gordan (CG) coefficients are real, as is shown in the Appendix B. Hence our basis is the so-called the "CP" basis. The conventional CP transformation ϕ → ϕ * can be consistently imposed onto the theory in our basis, and all the coupling constant would be constrained to be real.
• Three-dimensional representations B Clebsch-Gordan coefficients for ∆(6n 2 ) group with n = 3Z In the following, we shall present all the CG coefficients in the form of x ⊗ y in our chosen basis, x i denotes the element of the left base vectors x, and y i is the element of the right base vectors y. We shall see explicitly that all the CG coefficients are real.
x 2 y 2 − x 1 y 1 x 1 y 2 + x 2 y 1 , 1 1 ∼ x 1 y 1 + x 2 y 2 , 1 2 ∼ x 1 y 2 − x 2 y 1 . (B.1) • 2 ⊗ 6 (k,l) = 6 (k,l) ⊕ 6 (k,l) • 3 1,l ⊗ 3 1,l = 3 1,l+l ⊕ 6 (l,−l ) (B.7) • 3 2,l ⊗ 3 2,l = 3 1,l+l ⊕ 6 (l,−l ) . (B.9) For the case of n = 3Z, the CG-coefficients can be straightforwardly calculated although it is somewhat lengthy. Part of the CG coefficients are complex numbers in our chosen basis, the explicit form would not be reported here since generalized CP transformations can not be consistently defined in generic settings based on such groups unless the doublet representations 2 2 , 2 3 and 2 4 are not introduced in a specific model. Comparing with phenomenological predictions of Z 2 × CP in the neutrino sector analyzed in section 3.2, we see that the diagonalization matrix U l is of the same form as U ν provided the remnant flavor and remnant CP symmetries are the same in the two occasions.