Lepton Mixing in $A_5$ Family Symmetry and Generalized CP

We study lepton mixing patterns which can be derived from the $A_5$ family symmetry and generalized CP. We find five phenomenologically interesting mixing patterns for which one column of the PMNS matrix is $(\sqrt{\frac{5+\sqrt{5}}{10}},\frac{1}{\sqrt{5+\sqrt{5}}},\frac{1}{\sqrt{5+\sqrt{5}}})^{T}$ (the first column of the golden ratio mixing), $(\sqrt{\frac{5-\sqrt{5}}{10}},\frac{1}{\sqrt{5-\sqrt{5}}},\frac{1}{\sqrt{5-\sqrt{5}}})^{T}$ (the second column of the golden ratio mixing), $(1,1,1)^{T}/\sqrt{3}$ or $(\sqrt{5}+1,-2,\sqrt{5}-1)^{T}/4$. The three lepton mixing angles are determined in terms of a single real parameter $\theta$, and agreement with experimental data can be achieved for certain values of $\theta$. The Dirac CP violating phase is predicted to be trivial or maximal while Majorana phases are trivial. We construct a supersymmetric model based on $A_5$ family symmetry and generalized CP. The lepton mixing is exactly the golden ratio pattern at leading order, and the mixing patterns of case III and case IV are reproduced after higher order corrections are considered.

) T (the second column of the golden ratio mixing), (1, 1, 1) T / √ 3 or ( √ 5+1, −2, √ 5−1) T /4. The three lepton mixing angles are determined in terms of a single real parameter θ, and agreement with experimental data can be achieved for certain values of θ. The Dirac CP violating phase is predicted to be trivial or maximal while Majorana phases are trivial. We construct a supersymmetric model based on A 5 family symmetry and generalized CP. The lepton mixing is exactly the golden ratio pattern at leading order, and the mixing patterns of case III and case IV are reproduced after higher order corrections are considered.
It is known that the flavor symmetry group should be of the von Dyck type [34]. The finite von Dyck groups include S 3 , A 4 , S 4 , A 5 and dihedral groups [35]. Since S 3 and dihedral groups don't have irreducible three dimensional representations, they are not suitable as flavor symmetry otherwise two mixing angles would vanish. The phenomenological consequences of A 4 and S 4 flavor symmetries combined with generalized CP have been studied [20,[23][24][25][26][27][28]. In the present work, we shall investigate the A 5 flavor symmetry and CP symmetry. We shall perform a model independent analysis of possible lepton flavor mixing obtained from breaking of the original symmetry A 5 H CP . We find five phenomenologically interesting mixing patterns summarized in Table 1. The three mixing angles turn out to depend on only one free parameter θ and good agreement with their measured values can be achieved for certain values of θ, the Dirac CP phase is conserved or maximal and the Majorana CP phases are trivial. Furthermore, we construct a model based on A 5 H CP . The lepton mixing is exactly the golden ratio (GR) texture at leading order (LO). A non-zero θ 13 is generated by the next-to-leading-order (NLO) corrections, and the mixing patterns of cases III and IV discussed in the model independent analysis are generated.
The layout of the rest of this paper is as follows. In section 2, the physical CP transformations compatible with the A 5 family symmetry are found. In section 3, we perform a model independent analysis of possible lepton mixing patterns achievable from the underlying symmetry group A 5 H CP . In section 4, we present our A 5 H CP model, the LO structure, vacuum alignment and the NLO corrections are discussed. Section 5 concludes the paper. In Appendix A, we review the group theory of A 5 and the Clebsch-Gordan coefficients in our working basis are reported. In Appendix B, we present the possible mixing patterns arising from the A 5 flavor symmetry without CP symmetry, where the residual flavor symmetry in the neutrino sector is either Klein or Z 2 subgroup of A 5 . Compared with section 3, we see that generalized CP is really a powerful method of predicting CP phases as well as lepton mixing angles.

Approach
Both family symmetry and CP symmetry acts on the flavor space in a non-trivial way, and the interplay between them should be carefully treated. In order to consistently combine a family symmetry G f with a CP symmetry which is represented by unitary CP transformation matrix X, X must be related to an automorphism u : G f → G f . To be precise, the CP transformation X should be a solution to the consistency equation [20,21] Xρ * (g)X −1 = ρ (u(g)) , ∀g ∈ G f , where ρ is a representation of G f with ρ : G → GL (N, C), and it is generally reducible. We can easily check that the automorphism associated with ρ(h)X for any h ∈ G f is an composition of u and an inner automorphism µ h : g → hgh −1 with h, g ∈ G f [28,31]. Therefore the effects of inner automorphism can be easily included, and it is equivalent to a family symmetry transformation. As a consequence, we could firstly focus on the the outer automorphism of G f . Furthermore, it has been shown that u has to a class-inverting automorphism for X to be a physical CP transformation [36]. In other words, u should map each irreducible representation r of G f into its own complex conjugate. Hence the consistency condition in Eq. (2.1) takes a more restricted form: where the subscript "r" refers to the representation space acted on. The CP transformation X in Eq. (2.1) is given by the direct sum of the X r corresponding to the particle content of the model. Notice that the consistency conditions of Eq. (2.2) can also be derived from the requirement that the Lagrangian is invariant under both CP symmetry and flavor symmetry [37].
In the present work, we are interested in the family symmetry G f = A 5 . The group theory of A 5 , its representation and all the Clebsch-Gordan coefficients are reported in Appendix A. The structure of the automorphism group of A 5 is quite simple and is very clear in mathematica.
where Z(A 5 ), Aut(A 5 ), Inn(A 5 ) and Out(A 5 ) denote the center, automorphism group, inner automorphism group and outer automorphism group of A 5 respectively. We see that the outer automorphism group of A 5 is isomorphic to Z 2 . Consequently there is only one nontrivial outer automorphism u with The order of u is really 2, i.e., u 2 = id, where id represents the trivial automorphism id(g) = g, ∀g ∈ A 5 . One can straightforwardly check that u acts on the A 5 conjugacy classes as follows It interchanges the classes 12C 5 and 12C 5 . Since the inverse of each A 5 conjugacy class is equal to itself, u is not a class-inverting automorphism, and the corresponding CP transformation is unphysical. In terms of representations, the two different three-dimensional irreducible representations 3 and 3 are exchanged not mapped into their conjugate under the action of u. The generalized CP symmetry related with u can only be consistently defined if fields transforming as 3 and 3 are absent in a model. As a result, we conclude that only the CP transformation associated with the trivial outer automorphism (i.e., the inner automorphism) can be compatibly imposed on the theory with A 5 family symmetry. Now we consider the representative inner automorphism µ T 3 ST 2 ST 3 S : (S, T ) → (S, T 4 ). The corresponding generalized CP transformation X 0 r is fixed by the consistency equations: X 0 r ρ * r (S)(X 0 r ) −1 = ρ r (S), X 0 r ρ * r (T )(X 0 r ) −1 = ρ r (T 4 ) . (2.6) From the representation matrices given in Appendix A, we see that for any representation ρ * r (S) = ρ r (S), ρ * r (T ) = ρ r (T 4 ) .
(2.7) Therefore X 0 r is an identity matrix up to an overall phase, i.e., X 0 r = 1 . (2.8) Including the contribution of the remaining inner automorphisms in the manner stated below Eq. (2.1), the most general CP transformation consistent with A 5 family symmetry is of the form X r = ρ r (g)X 0 r = ρ r (g), g ∈ A 5 . (2.9) This means that the generalized CP transformation consistent with A 5 is of the same form as the family group transformation in our working basis while they act on the a field multiplet in different ways: ϕ(x) g −→ ρ r (g)ϕ(x), g ∈ A 5 versus ϕ(x) CP −→ X r ϕ * (x P ) = ρ r (g)ϕ * (x P ), where x P = (t, − x).
In this work, the phenomenological implications of A 5 family symmetry combined with the generalized CP symmetry would be investigated in a systematical and comprehensive way. The parent symmetry is A 5 H CP at high energy scale, where the element of H CP is the CP transformation compatible with A 5 and its explicit form is given by Eq. (2.9). In this setup, lepton mixing can be predicted from A 5 H CP breaking into different remnant symmetries G l H l CP and G ν H ν CP in the charged lepton and neutrino masses respectively, where G l , G ν and H l CP , H ν CP denote residual family symmetries and residual CP symmetries respectively. It is notable that the predictions for the lepton flavor mixing only depend on the assumed symmetry breaking patterns and are independent of the details of a specific implementation scheme, such as the possible additional symmetries of the model and the involved flavon fields and their assignments etc. In practice, the three generations of lefthanded leptons doublets are embedded into the faithful three-dimensional representation 3 of A 5 . Since 3 is related to 3 by the outer automorphism u, the results would be the same and no additional results would be found, if we assign the three left-handed leptons to the representation 3 instead. The requirement that G l H l CP is preserved by the charged lepton mass term implies that the hermitian combination m † l m l must be invariant under the remnant symmetry G l H l CP , i.e., where the mass matrix m l is defined in the convention l R m l l L . Once G l and H l CP are specified, the most general form of m † l m l can be straightforwardly constructed from Eqs. (2.10a, 2.10b). In the present work, we shall assume neutrinos are Majorana particles. In the same fashion, requiring that G ν H ν CP is a symmetry of the neutrino mass matrix m ν implies that m ν should be invariant under the action of The prediction for the PMNS matrix can be obtained by further diagonalizing the reconstructed mass matrices m † l m l and m ν . Please see Ref. [15] for an alternative way of directly extracting the PMNS matrix from the representation matrices of the remnant symmetries without resorting to the mass matrices. As the order of neutrino and charged lepton masses is indeterminate in our framework, it is only possible to determine the PMNS matrix up to independent row and column permutations. From the remnant symmetry invariant conditions of Eqs. (2.10a, 2.10b), we can see that X lr and ρ r (g l )X lr with g l ∈ G l lead to the same constraint on m † l m l . Furthermore, the residual CP transformation X lr should be a symmetric matrix otherwise the charged lepton masses would be restricted to be partially degenerate [15,28]. The same comments apply to X νr and ρ r (g ν )X νr with g ν ∈ G ν . Notice that the same result for PMNS matrix would be obtained [23,28,31], if a pair of subgroups {G l , G ν } is conjugated to the pair of subgroups {G l , G ν } under an element of A 5 , i.e., The reason is that remnant CP symmetries determined by restricted consistency condition of Eqs. (2.12a, 2.12b) are strongly correlated in the two cases such that lepton mass matrices {m † l m l , m ν } for the new primed residual symmetry are related to {m † l m l , m ν } by a similarity transformation ρ 3 (g) [23,28,31]. In this way, it is sufficient to only discuss the independent pairs of {G l , G ν } which are not related by group conjugation and subsequently all possible residual CP compatible with the residual family symmetry should be included .

Lepton mixing from remnant symmetries of A 5 H CP
Neutrino are assumed to be Majorana particles here, therefore the remnant flavor symmetry G ν must be a Klein four K 4 ∼ = Z 2 × Z 2 subgroup or a single Z 2 subgroup of A 5 . G l can be any abelian subgroups of A 5 with order equal or greater than 3. A complete or partial degeneracy of the charged lepton mass spectrum would be produced if G l had a non-abelian character. In the case of G ν = K 4 , the lepton mixing matrix U P M N S is fully determined by the mismatch between the remnant family symmetry G l and G ν . As shown in Appendix B, U P M N S can take four possible forms such as the golden ratio mixing, democratic mixing and so on. However, none of them is compatible with experimental data. Then we turn to the scenario of G ν = Z 2 . With this setting, U P M N S is partially constrained, and only one column of the lepton mixing matrix is fixed up to reordering and rephasing of the elements. The explicit forms of the fixed column vectors for all the independent residual flavor symmetries are summarized in Table 4. We find that four cases are viable: is the golden ratio. The phenomenological implications of each case are explored in Appendix B, and the lepton mixing matrix U P M N S turns out to depend on two free parameters up to indeterminant Majorana phases. We see that the measured values of the three mixing angles can be accommodated very well, but the allowed values of Dirac CP phase δ CP scatter in a quite large range. Furthermore, the breaking patterns with (G l , G ν ) = (Z 2 , K 4 ) are studied as well. Accordingly a row of the lepton mixing matrix U P M N S is determined to be 1 2 (κ, 1, κ − 1) or (1, 0, 0) which are not in the experimentally preferred regions.
In order to be able to predict the values of CP phases, we extend the A 5 family symmetry to include the generalized CP. In the following, we shall perform a thorough analysis of lepton mixing patterns for the possible residual symmetries G l H l CP and G ν H ν CP in the charged lepton and neutrino sectors, where the remnant family symmetries G l and G ν would be restricted to the four viable cases listed in Table 4, and the remnant CP symmetries H l CP and H ν CP are determined by consistency condition of Eqs. (2.12a,2.12b). In this setup, U P M N S as well as all mixing angles and all CP phases generically depend on a free parameters θ whose value can be fixed by the measured value of θ 13 . As a consequence, all observables are strongly correlated. For the concerned A 5 family symmetry, the Dirac phase would be predicted to be trivial or maximal while both Majorana phases are trivial after generalized CP symmetry is imposed. In order to evaluate how well the predicted mixing patterns agree with the experimental data on mixing angles, we shall perform a usual χ 2 analysis which uses the global fit results of Ref. [5]. We begin to discuss all possible cases one by one.
In this case, the parent symmetry A 5 H CP is broken down to Z T 5 H l CP and Z S 2 × H ν CP subgroups in the charged lepton and neutrino sectors, respectively. The residual CP symmetry H l CP must be consistent with the residual flavor symmetry Z T 5 in the charged lepton sector. That is to say the element X lr of H l CP should fulfill the consistency equation of Eq. (2.12b), Then we find only 10 choices out of the 60 CP transformations of H CP listed in Eq. (2.9) are acceptable As shown in Eq. (2.10a), the residual family symmetry Z T 5 impose the following constraint on the charged lepton mass matrix: In our working basis, the representation matrix of the generator T is diagonal with ρ 3 (T ) = diag(1, ω 5 , ω 4 5 ). Consequently the hermitian combination m † l m l of charged lepton mass matrix is also diagonal, i.e., m † l m l = diag m 2 e , m 2 µ , m 2 τ , (3.4) where m e , m µ and m τ represent the electron, muon and tau masses respectively. Furthermore, we can check that the remnant CP invariant condition of Eq. (2.10b) is automatically satisfied for X lr = ρ r (1), ρ r (T ), ρ r (T 2 ), ρ r (T 3 ), ρ r (T 4 ). However, the mass degeneracy m µ = m τ arises for the remaining values The reason is that all remnant CP transformations except ρ r (T 3 ST 2 ST 3 S) are not symmetric. Generally speaking, any remnant CP transformation must be a symmetric matrix to avoid degenerate masses [15,28]. This case is obviously not viable, and will be disregarded hereafter. Now we turn to the neutrino sector. The residual CP transformations X νr of H ν CP is specified by the consistency condition: which can be easily obtained by applying the general consistency condition of Eq. (2.12a). We see that the CP transformation X νr commutes with flavor symmetry transformation ρ r (S), and therefore remnant symmetry is Z S 2 × H ν CP in the neutrino sector in this case. Notice that the semi-direct product structure between residual flavor and CP symmetries generally reduces to a direct product if the residual flavor symmetry is a Z 2 subgroup [23,24]. It is easy to check that X νr can only take 4 possible values, The neutrino mass matrix m ν respects the residual symmetry Z S 2 × H ν CP , satisfying We find that the most general neutrino mass matrix invariant under the residual family symmetry Z S 2 , takes the following form (3.8) where α, β, γ and δ are generally complex parameters, and they are further constrained to be real or pure imaginary by residual CP. This neutrino mass matrix m ν can be simplified into a quite simple form by performing a golden ratio transformation, is the golden ratio mixing pattern [38] which can be naturally derived in A 5 models [39]. The neutrino mass matrix m ν is further diagonalized by a unitary rotation U ν in the (2,3)-plane, The next step is to explore the constraint of remnant CP on m ν . Two different phenomenological predictions arise for the four possibe X νr shown in Eq. (3.6), as ρ r (S)X νr and X νr lead to the same predictions.
(I) X νr = ρ r (1), ρ r (S) Obviously we have m ν = m * ν such that all the four parameters α, β, γ and δ are real. As a consequence, the neutrino mass matrix m ν is a real symmetric matrix. The unitary transformation U ν is of the form: where K ν is a diagonal phase matrix with elements equal to ±1 or ±i which makes the neutrino masses m 1,2,3 positive. The effect of K ν is a possible change of the Majorana phases by π, and it would be omitted hereinafter for the other cases. The parameter θ is given by The light neutrino mass eigenvalues are (3.14) Given the diagonal charged lepton mass matrix, the lepton mixing matrix takes the form One can straightforwardly extract the lepton mixing angles and CP phases as follows, sin 2 θ 13 = 3 − κ 5 sin 2 θ , sin 2 θ 12 = 1 + cos 2θ 3 + 2κ + cos 2θ , where δ CP is the Dirac CP phase, α 21 and α 31 are the Majorana CP phases in the standard parameterization [1]. There is no CP violation in this case as the neutrino mass matrix is real. Expressing θ in terms of θ 13 , correlations among the three mixing angles follow immediately, For the measured reactor mixing angles sin 2 θ 13 0.0234 [5], we have sin 2 θ 23 0.258 or 0.742 which is outside of the experimentally favored 3σ region [5] although sin 2 θ 12 0.259 is acceptable. As a consequence, this mixing pattern isn't viable. This point remains even after permutation of rows and columns is considered.
Solving the residual CP invariant condition in Eq. (3.7), we find α, β and γ are real while δ is pure imaginary. The unitary diagonalization matrix U ν is where the diagonal matrix K ν multiplied from the right-hand side has been omitted, and the rotation angle θ fulfills sin 2 θ 13 = 3 − κ 5 sin 2 θ , sin 2 θ 12 = 1 + cos 2θ 3 + 2κ + cos 2θ , Here we present the absolute value of sin δ CP , since the sign of sin δ CP depends on the ordering of rows and columns. We see that both atmospheric angle θ 23 and Dirac CP phase δ CP are maximal while Majorana phases are conserved. Given the weak evidence of δ CP ∼ 3π/2 from T2K [8], this pattern is slightly preferred. The prediction of maximal Dirac CP can be tested by next generation long-baseline neutrino oscillation experiments such as the proposed LBNE [9], LBNO [10] and HyperKamiokande [11], which aim to search for leptonic CP violation. Moreover, the correlation between θ 13 and θ 12 is of the same form as that of case I, and it is plotted in Fig. 1. The results of the χ 2 analysis are reported in Table 1. We see that the experimental data [5] on lepton mixing angles can be accommodated very well. Notice that the solar mixing angle θ 12 is predicted to be around the present 3σ lower bound. As far as we known, the JUNO experiment can measure θ 12 with high accuracy [44]. If significant deviations sin 2 θ 12 from 0.259 was detected, this mixing pattern would be excluded. It is well-known that leptonic CP phases can play a crucial role in the rare process neutrinoless double beta ((ββ) 0ν −) decay. The dependence of the (ββ) 0ν −decay amplitude on the neutrino mixing parameters is characterized by the effective Majorana mass |m ee | [1] with the definition: |m ee | = m 1 cos 2 θ 12 cos 2 θ 13 + m 2 sin 2 θ 12 cos 2 θ 13 e iα 21 + m 3 sin 2 θ 13 e i(α 31 −2δ CP ) . For the predicted mixing parameters in Eq. (3.22), we have On the left panel, the best fitting value is labelled with a red pentagram, and the points for θ = 0, π/6, π/3 and π/2 are marked with a cross to guide the eye. The 1σ and 3σ ranges of the mixing angles are taken from Ref. [5]. On the right panel, the orange and green bands denote the 3σ regions for normal ordering and inverted ordering mass spectrum respectively. The red and purple areas are the predictions for the lepton mixing matrix in Eq. (3.21). The present most strict bound |m ee | < (0.120−0.250) eV from EXO-200 [40,41] combined with KamLAND-ZEN [42] is represented by the horizontal dashed line, and the upper limit on m min from the latest Planck result m 1 + m 2 + m 3 < 0.230 eV at 95% confidence level [43] is shown by vertical dashed line.
where k 2 , k 3 = ±1 originates from the ambiguity of the matrix K ν . The prediction for the effective mass |m ee | with respect to the lightest neutrino mass is shown Fig. 1. We find that |m ee | is close to 0.022eV or the upper bound 0.045eV in case of IO neutrino mass spectrum, which are within the future sensitivity of planned (ββ) 0ν −decay experiments. However, in case of NO spectrum, |m ee | strongly depends on lightest neutrino mass m min , and it can even be approximately vanishing for particular value of m min .

G
The charged lepton sector preserves the same remnant symmetry Z T

5
H l CP as that discussed in section 3.1. Therefore the charged lepton mass is subject to the same constraint, and m † l m l should be diagonal as well. In neutrino sector, the residual CP symmetry H ν CP has to be compatible with the residual family symmetry It is easy to check that only 4 generalized CP transformations are acceptable, is of the form

Analytic expression
Best fitting sin 2 θ 13 sin 2 θ 12 sin 2 θ 23 θ bf χ 2 min sin 2 θ 13 sin 2 θ 12 sin 2 θ 23  Table 1: Summary of the predictions for the lepton mixing angles and their best fitting values for all viable cases in the framework of A 5 H CP . In case VII, the mixing patterns for θ 23 in the first and second octant are related through the exchange of the second and third rows of the PMNS matrix. Notice that all the three CP phases are independent of θ in all cases: Dirac phase is trivial or maximal, and both Majorana phases are trivial.

II
where the parameters α, β, γ and δ are generically complexes, and they are further constrained by the remnant CP. After performing a GR transformation, m ν becomes In the following, we proceed to investigate the constraints imposed by the remnant CP transformations shown in Eq. (3.26). The four possible X νr can be divided into two classes.
In this case, the residual flavor and residual CP transformations are of the same form. As a result, the four parameters α, β, γ and δ are all real. The neutrino mass matrix m ν can be diagonalized by a unitary transformation The three neutrino masses are The absolute neutrino mass scale can not be predicted. Then the PMNS matrix reads Note that the second column vector is T which coincides with the second column of the GR mixing. The lepton mixing parameters are predicted to be , We see that θ 23 deviates from maximal mixing and all the three CP violating phases are trivial due to a common CP transformation ρ r (1) of the charged lepton and neutrino sectors. The mixing angles θ 12 , θ 13 and θ 23 only depend on the parameter θ, and they fulfill the following relations, which are plotted in Fig. 2. Obviously the mixing angles can be very close to the their measured values for certain values of the parameter θ. The global minimum of the χ 2 function is rather small, as shown in Table 1. The predictions for the effective mass |m ee | are also displayed in Fig. 2.
Invariance of the neutrino mass matrix m ν under the action of these residual CP transformations implies that α, β, γ are real while δ is pure imaginary. The diagonalization matrix of m ν is .
The neutrino masses are given by  Figure 2: The correlation among sin 2 θ 12 , sin 2 θ 23 and sin θ 13 (the former three panels) and the allowed values of the effective mass |m ee | (the last panel) in case III. The global minimum of the χ 2 function is labelled with a red pentagram, and the points for θ = 0, π/6, π/3, π/2, 2π/3 and 5π/6 are marked with a cross to guide the eye. The 1σ and 3σ ranges of the mixing angles are taken from Ref. [5]. In the last panel, the orange and green bands denote the 3σ regions for normal ordering and inverted ordering mass spectrum respectively. The red and purple areas are the predictions for the lepton mixing matrix in Eq. (3.32). The present most strict bound |m ee | < (0.120 − 0.250) eV from EXO-200 [40,41] combined with KamLAND-ZEN [42] is represented by the horizontal dashed line, and the upper limit on m min from the latest Planck result m 1 + m 2 + m 3 < 0.230 eV at 95% confidence level [43] is shown by vertical dashed line.
The PMNS matrix is of the form The second column has the same form as for the GR mixing. The lepton mixing angles and CP phases are determined to be The orange and green bands denote the 3σ regions for normal ordering and inverted ordering mass spectrum respectively. The red and purple areas are the predictions for the lepton mixing matrix in Eq. (3.38). The present most strict bound |m ee | < (0.120 − 0.250) eV from EXO-200 [40,41] combined with KamLAND-ZEN [42] is represented by the horizontal dashed line, and the upper limit on m min from the latest Planck result m 1 + m 2 + m 3 < 0.230 eV at 95% confidence level [43] is shown by vertical dashed line. Note that the correlation between sin 2 θ 12 and sin θ 13 is the same as that of case III and can be found in Fig. 2.
We see that both θ 23 and δ CP are maximal and the two Majorana CP phases α 21 and α 31 are trivial. Similar to case III, the relation sin 2 θ 12 cos 2 θ 13 = (3 − κ)/5 is satisfied as well.
The best fitting results for the three mixing angles are listed in Table 1. The predictions for the (ββ) 0ν −decay effective mass |m ee | are shown in Fig. 3.
In the charged lepton sector, the remnant CP transformation H l CP is determined by the consistency condition We find that there are 6 possible solutions for X lr , i.e., The charged lepton mass matrix should respect both the remnant family symmetry Z T 3 ST 2 S 3 and the remnant CP symmetry H l CP : For the remaining ones X lr =ρ r (ST 3 S), ρ r (T 3 ), ρ r (T 3 ST 2 ST 3 ), the hermitian combination m † l m l is constrained to take the following form where a, b and c are real parameters. It can be diagonalized by the unitary matrix where the charged lepton masses are The neutrino mass matrix preserving the remnant family symmetry 47) where parameters α, β, γ and δ are generally complex, and they are further constrained to be either real or imaginary by CP symmetry. It is convenient to firstly perform a constant unitary transformation U GRP and yield Next we discuss the constraints of the residual CP symmetry on the neutrino mass matrix m ν .
In this case, α, β, γ and δ are determined to be real. Then neutrino mass matrix m ν is a real symmetric matrix, and it can be diagonalized by a rotation matrix U ν in the (2,3) sector, The three light neutrino masses are given by , The lepton mixing matrix is of the form We see that the second column of the PMNS matrix is (1, 1, 1) T / √ 3, which frequently appears in discrete flavor symmetry models. The leptonic mixing parameters read as 2 |sin δ CP | = 1, sin α 21 = sin α 31 = 0 . The measured 3σ range 0.0176 ≤ sin 2 θ 13 ≤ 0.0295 [5] gives rise to 0.339 ≤ sin 2 θ 12 ≤ 0.343 which can be directly tested by JUNO in near future [44]. The correlation between θ 12 and θ 13 and the predictions for the (ββ) 0ν −decay are displayed in Fig. 4. All the three mixing angles can agree within 3σ with the experimental data for certain values of θ. The best fitting results are listed in Table 1, and the minimum values of the χ 2 functions are 3.987 and 7.480 for IO and NO, respectively.
The requirement of real α, β, γ and pure imaginary δ follows immediately from the remnant CP invariant condition. In the same way as previous cases, the PMNS mixing matrix is found to be On the left panel, the best fitting value is labelled with a red pentagram, and the points for θ = 0, π/6 and 2π/3 are marked with a cross to guide the eye. The 1σ and 3σ ranges of the mixing angles are taken from Ref. [5]. On the right panel, the orange and green bands denote the 3σ regions for normal ordering and inverted ordering mass spectrum respectively. The red and purple areas are the predictions for the lepton mixing matrix in Eq. (3.53). The present most strict bound |m ee | < (0.120 − 0.250) eV from EXO-200 [40,41] combined with KamLAND-ZEN [42] is represented by the horizontal dashed line, and the upper limit on m min from the latest Planck result m 1 + m 2 + m 3 < 0.230 eV at 95% confidence level [43] is shown by vertical dashed line.
The expressions for the lepton mixing parameters are as follows, where α 31 = α 31 − 2δ CP . It is remarkable that all the three CP violating phases nontrivially depend on the parameter θ. However, we see that in case of θ = π/4 the minimum value of θ 13 is obtained with sin 2 θ 13 | θ=π/4 = (2 − √ 3)/6 0.0447 which is outside the 3σ range [5]. Furthermore, we note that the atmospheric angle θ 23 is the complementary angle of θ 12 or is equal to θ 12 if the second and the third rows of the PMNS matrix is interchanged. As a result, this mixing pattern is not compatible with experimental data and consequently we don't included it in Table 1.
In the last case, the residual symmetries are assumed to be K to hold, the mass matrix m † l m l has to fulfill Then m † l m l is constrained to take the form where a, b and c are real. It is diagonalized by the unitary matrix The mass matrix m † l m l is also subject to the constraint of the residual CP symmetry H l CP . It is straightforward to determine that H l CP can take the value The twelve CP transformations can be classified into two categories. For l m l ) * is automatically satisfied, and therefore no additional constraint is produced. Nevertheless, the remaining eight CP transformations X lr = ρ r (ST 2 ST ), ρ r (ST 3 ), ρ r ((T 2 S) 2 T 3 ), ρ r (T 2 ST 4 ), ρ r (T 3 S), ρ r (T 3 (ST 2 ) 2 ), ρ r (T 4 ST 2 ) and ρ r (T ST 2 S) are not viable, as they require b = c = 0 so that the charged lepton mass spectrum is completely degenerate with m 2 e = m 2 µ = m 2 τ = a. In neutrino sector, the remnant symmetry Z S 2 × H ν CP and its phenomenological implications have been studied in section 3.1. The neutrino mass matrix m ν is found to be given by Eq. (3.8), where the parameters α, β and γ are real while δ is real or pure imaginary depending on the residual CP transformation X νr .
In this case, the neutrino mass matrix is diagonalized by the unitary matrix in Eq. (3.21). Combining the unitary transformation U l in Eq. (3.60) from the charged lepton sector, we obtain the lepton flavor mixing matrix: where the parameter θ is specified by Eq. (3.19). The lepton mixing parameters are predicted to be sin 2 θ 13 = (cos θ − κ sin θ) 2 4κ 2 , sin 2 θ 12 = (κ cos θ + sin θ) 2 4κ 2 − (cos θ − κ sin θ) 2 , We find all the three CP violating phases δ CP , α 21 and α 31 are conserved, this is because that a common CP transformation ρ r (T 3 ST 2 ST 3 S) is shared by the neutrino and charged lepton sectors. In addition, θ 23 deviates from maximal value. After some tedious calculations, we find the following relations between the mixing angles 4 cos 2 θ 12 cos 2 θ 13 = 1 + κ , 5 sin 2 θ 23 = 3 − κ + (1 + 2κ) tan 2 θ 13 ± 2κ tan θ 13 2 + κ − (2 + 3κ) tan 2 θ 13 , (3.65) which is plotted in Fig. 5. For the 3σ interval 0.0176 ≤ sin 2 θ 13 ≤ 0.0295 [5], we have 0.326 ≤ sin 2 θ 12 ≤ 0.334 and 0.454 ≤ sin 2 θ 23 ≤ 0.511, which are in the experimentally favored ranges [5]. The global minimum of the χ 2 function is rather small 3.503 (1.626) for NO (IO) neutrino mass spectrum, therefore this mixing pattern can describe the experimental data very well. Moreover, we note that the best fitting value of θ 23 is in the first octant with sin 2 θ 23 (θ bf ) = 0.480 (0.486) for NO (IO) spectrum. Agreement with experimental data can also be achieved if the second and third rows of the PMNS matrix in Eq. (3.63) are exchanged. Then the atmospheric mixing angle θ 23 changes to  Table 1, we see that this case gives rise to the smallest χ 2 min for both NO and IO. The above predictions for solar and atmospheric mixing angles could be tested directly in near future, since the next generation neutrino oscillation experiments are expected to reduce the experimental error on θ 12 and θ 23 to few degrees. The theoretical results for the (ββ) 0ν −decay effective mass |m ee | are displayed in Fig. 5. Note that interchanging the second and third rows does't matter since |m ee | is independent of θ 23 . Again, the predictions for IO neutrino spectrum are within the sensitivity of forthcoming experiments.
(VIII) X νr = ρ r (1), ρ r (S) The neutrino mass matrix is diagonalized by the unitary transformation in Eq. (3.15). The PMNS matrix is found to take the following form We see that the solar mixing angle θ 12 has a lower bound given by sin 2 θ 12 ≥ (5 − √ 5)/10 0.276, and the experimental data on θ 12 can be accommodated for particular values of θ. Both θ 13 and θ 23 are independent of θ, and they are outside the 3σ ranges [5]. Furthermore, 6 × 6 =36 possible permutations of rows and columns of this mixing pattern are considered. However, none of them can give rise to three mixing angles in the experimentally preferred 3σ range [5].  Figure 5: The results for sin 2 θ 12 , sin 2 θ 23 and sin θ 13 (the former three panels) and the allowed values of the effective mass |m ee | (the last panel) in case VII. The global minimum of the χ 2 function is labelled with a red pentagram, and the points for θ = 0, π/6, π/3, π/2, 2π/3 and 5π/6 are marked with a cross to guide the eye. The black solid lines and blue dashed lines in the upper-right and lower-left panels represent the two solutions for θ 23 shown in Eq. (3.64) and Eq. (3.66) respectively. The corresponding PMNS matrices are related through a exchange of the second and third rows. The 1σ and 3σ ranges of the mixing angles are taken from Ref. [5]. In the last panel, the orange and green bands denote the 3σ regions for normal ordering and inverted ordering mass spectrum respectively. The red and purple areas are the predictions for the lepton mixing matrix in Eq. (3.63). The present most strict bound |m ee | < (0.120 − 0.250) eV from EXO-200 [40,41] combined with KamLAND-ZEN [42] is are represented by the horizontal dashed line, while the upper limit on m min from the latest Planck result m 1 + m 2 + m 3 < 0.230 eV at 95% confidence level [43] is shown by dashed line.

Model building
In previous section, we have performed a model-independent analysis of the lepton mixing patterns which can be derived from A 5 H CP . As summarized in Table 1, we find five new mixing patterns which are compatible with current experimental data. In this section, we shall construct a concrete model with both A 5 family symmetry and generalized CP symmetry, the symmetry breaking patterns studied in section 3.2 are implemented, and therefore the lepton flavor mixings given by Eqs. (3.32, 3.38) in case III and case IV are realized. Note that it would be also interesting to implement other cases such as case VII in Field l ν c e c µ c τ c h u,d ϕ φ ψ ξ ζ χ ρ ∆ σ 0 φ 0 ψ 0 ξ 0 χ 0 ρ 0 ∆ 0 A 5 3 3 1 1 1 1 3 3 5 1 1 3 3 5 1 4 5 1 3 3 5 1 ω 6 ω 2 6 ω 2 6 1 1 1 1 1 ω 4 6 ω 3 6 ω 4 6 1 1 1 1 U (1) R 1 1 1 1 1 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 and ω 6 = e  Table 2. In the following, we first discuss the vacuum alignment of the model, then specify the structure of the model at leading order and next-to-leading order. As we shall show, the lepton mixing is exactly the GR at LO, and a non-vanishing value of the reactor mixing angle θ 13 is generated by higher order corrections. Consequently θ 13 is naturally of the correct order in our model.

Vacuum alignment
We utilize the standard supersymmetric driving field mechanism [45] to solve the vacuum alignment problem. A global U (1) R continuous symmetry is assumed in this approach, and the usual R−parity is a discrete group of this U (1) R . The matter fields have R−charge equal to one, both flavon fields and Higgs are chargeless and the driving fields carry two units of R−charge. At LO the most general driving superpotential w d invariant under A 5 × Z 3 × Z 4 × Z 6 with R = 2 can be written as with where (. . .) R denotes a contraction into the A 5 irreducible representation R according to the Clebsch-Gordan coefficients listed in Appendix A. Notice that all the couplings f i (i = 1, . . . , 4), g i (i = 1, . . . , 8) and the mass parameters M ψ , M ξ , M ∆ are real, since the theory is invariant under the generalized CP defined in Eq. (2.9). In the SUSY limit, the vacuum alignment is achieved via the requirement of vanishing F −terms of the driving fields. In the charged lepton sector, the equations for the vanishing of the derivatives of w l d with respect to each component of the driving fields are: We find one solution to those equations, with v ϕ undetermined. A common order of magnitude for the VEVs (scaled by the cutoff Λ) is expected. In order to generate the mass hierarchies among the charged lepton, we assume where λ c 0.23 is the Cabibbo angle [1]. In the neutrino sector, the minimization equations for the vacuum are A solution to those equations with each flavon acquiring non-zero VEV is given by These VEVs are related through v ξ = 10(κ − 3)g 5 g 7 where v χ is undetermined. It is easy to check that the VEVs of ξ, ζ and ∆ break the A 5 family symmetry down to K is preserved by vacuum of χ and ρ. Furthermore, Eq. (4.10) implies that v 2 ζ , v 2 χ , v 2 ρ , v ξ and v 1 have the same phase up to π, since all couplings are real. In our model, the GR mixing is reproduced exactly and a non-zero reactor mixing angle θ 13 is generated after subleading order contributions are included. In order to obtain the correct size of θ 13 , we could choose

Leading order results
The charged lepton mass terms, which are invariant under the imposed family symmetry A 5 × Z 3 × Z 4 × Z 6 , can be written as where dots stand for higher dimensional operators which will be discussed later. Note that all couplings here are real due to the generalized CP symmetry. After the electroweak and flavor symmetries breaking by the VEVs shown in Eq. (4.5), we obtain a diagonal charged lepton mass matrix, and the three charged lepton masses are We see that the realistic mass hierarchies m e : m µ : m τ λ 4 c : λ 2 c : 1 is generated for the order of magnitude of the flavon VEVs in Eq. (4.7). Furthermore, as both m l and ρ 3 (T ) are diagonal, obviously we have ρ † 3 (T )m † l m l ρ 3 (T ) = m † l m l , i.e., the residual flavor symmetry of m † l m l is Z T 5 . Next let's discuss the neutrino sector. Neutrino masses are generated by type I see-saw mechanism in this work. The LO superpotential for neutrino masses is where the coupling constants y 1 , y 2 and the mass M are enforced to be real by the generalized CP symmetry. The Dirac mass matrix is obtained from the first two terms in Eq. (4.14) and it is given by where v u = h u , and the parameters a, b are The common phase of a and b can be absorbed by field redefinition, consequently both a and b can considered as real. The last term of Eq. (4.14) leads to the Majorana mass matrix: Therefore the three right-handed neutrinos are completely degenerate with mass equal to M . The light neutrino mass matrix is then given by the see-saw relation: We find that the neutrino mass matrix m ν in Eq. (4.18) is of the same form as the general mass matrix in Eq. (3.27) with δ = 0. Therefore m ν is exactly diagonalized by the GR mixing pattern, i.e., where the phase matrix K ν which encodes the CP parity of the neutrino state, has been omitted. The mass eigenvalues m 1,2,3 are Since the charged lepton mass matrix is diagonal in LO, the lepton mixing is exactly the GR mixing pattern. Here the reason why the GR mixing is produced is because that the flavor symmetry A 5 is broken to K

Next-to-leading-order corrections
At LO our model gives rise to the GR mixing pattern U GR which predicts a vanishing reactor mixing angle (θ 13 = 0 • ). Hence substantial next-to-leading-order corrections are needed to bring the model to agree with the experimental data on θ 13 . We will demonstrate in the following that a non-zero θ 13 can be obtained after the NLO contributions are included. Moreover, the LO remnant symmetry K of neutrino sector is further broken down to Z T 3 ST 2 ST 3 2 such that the mixing patterns of case III and case IV discussed in section 3.2 are realized. Firstly we consider the corrections to the flavon superpotential w l d in Eq. (4.2) which determines the vacuum alignment of the charged lepton sector. The symmetry allowed NLO operators are of the following form where all possible A 5 contractions should be considered, and all dimensionless coupling constants are omitted with Ψ 0 l ≡ {σ 0 , ψ 0 }, Ψ l ≡ {φ, ψ}, Ψ ν ≡ {ξ, ∆} and Ψ ν ≡ {ζ, χ}. Note that δw l d is suppressed by λ 2 c with respect to the LO superpotential w l d in Eq. (4.2). The NLO vacuum configuration is determined by searching for the zeros of the F −terms of w l d + δw l d with respect to the driving fields σ 0 , φ 0 and ψ 0 . We find that the NLO vacuum of ϕ, φ and ψ are given by where i (i = 1, . . . , 10) are general complex numbers with absolute values of order one. The higher dimensional operators contributing to the charged lepton masses are: The charged lepton mass matrix can be obtained by inserting the NLO VEVs of Eq. (4.23) into the LO mass terms plus the contribution of δw l evaluated with the LO VEVs of Eq. (4.5). We find that the NLO charged lepton mass matrix is of the following form: Therefore the contributions of charged lepton sector to the lepton mixing angles is of order λ 2 c and can be neglected. We proceed to discuss the subleading corrections in the neutrino sector. The higher order corrections to the flavon superpotential of ξ, ζ, χ, ρ and ∆ read where all couplings g i (i = 9, . . . , 17) are real because of the generalized CP symmetry. The resulting contributions to the F −terms of the driving fields σ 0 , ρ 0 , χ 0 and ∆ 0 are suppressed by Ψ /Λ ∼ λ c (Ψ ≡ {ζ, χ, ρ}) compared to the contribution from the LO terms in Eq. (4.3). Hence they induce shifts in the VEVs of ξ, ζ, χ, ρ and ∆ at relative order λ c with respect to the LO results. After some straightforward algebra, the new vacuum configuration can be written as Obviously the vacuum of χ is kept intact, ρ acquires O(λ c ) corrections in the same direction, while the alignment of ∆ is tilted. Moreover, from the relations in Eq. (4.10), we see that the shifts δv ξ , δv ζ , δv ρ and δv ∆ carry the same phase as v ξ , v ζ , v ρ and v 1 up to π, respectively. The light neutrino mass matrix receives corrections from both the modified vacuum and the higher dimensional operators in the superpotential w ν . It is easy to check that the NLO corrections to the Majorana mass terms are suppressed by 1/Λ 4 which can be safely neglected. The subleading operators contributing to the neutrino Dirac masses are as follows As a consequence, the corrected Dirac mass matrix becomes where the four parameters a, b, c and d are Notice that the three parameters a, b and c have the same phase with v 2 χ up to π, while the phase difference between d and v 2 χ is 0, π or ± π 2 depending on the product g 1 M ∆ (g 2 g 8 + g 3 g 7 )g 2 4 g 8 M ∆ − 5g 4 g 5 g 7 g 2 8 M ξ being positive or negative. Since the phase of v χ can be factorized out as an overall phase of the neutrino mass matrix m ν , the VEV v χ can be taken to be real without loss of generality. As a result, a, b and c are all real and the parameter d is real for g 1 M ∆ (g 2 g 8 + g 3 g 7 )g 2 4 g 8 M ∆ − 5g 4 g 5 g 7 g 2 8 M ξ < 0 or pure imaginary for g 1 M ∆ (g 2 g 8 + g 3 g 7 )g 2 4 g 8 M ∆ − 5g 4 g 5 g 7 g 2 8 M ξ > 0. In addition, we see that d are suppressed by λ c with respect to a, b and c, i.e., Utilizing the see-saw formula, we find the light neutrino mass matrix m ν is of the same form as Eq. (3.27) with Note that the term proportional to δ spoils the LO GR mixing, and it is of relative order λ c compared with α, β and γ since it is induced by the NLO corrections. Therefore the correct size of the reactor mixing angle θ 13 can be naturally achieved in our model. After extracting the overall phase of v χ , the parameters α, β and γ are real while δ is real or pure imaginary. In the case of g 1 M ∆ (g 2 g 8 + g 3 g 7 )g 2 4 g 8 M ∆ − 5g 4 g 5 g 7 g 2 8 M ξ < 0, δ is real such that the neutrino mass matrix m ν has the most general form compatible with the preservation of the remnant symmetry }. This is the case III investigated in the model independent analysis of section 3.2. The lepton mixing matrix U P M N S and the corresponding preditions for the lepton mixing parameters are given by Eq. (3.32) and Eq. (3.33) respectively. There is no CP violation in this case.
In the case of g 1 M ∆ (g 2 g 8 + g 3 g 7 )g 2 4 g 8 M ∆ − 5g 4 g 5 g 7 g 2 8 M ξ > 0, the parameter δ becomes imaginary. The origin symmetry in the neutrino sector. The neutrino mass matrix m ν has the same form as that of case IV discussed in section 3.2. Both atmospheric mixing angle and Dirac CP phase are predicted to be maximal while Majorana CP phases are conserved, as shown in Eq. (3.39). In short, our model reproduces the GR mixing at LO, and realistic value of θ 13 is obtained after higher order contributions are taken into account. Depending on the overall sign of the product g 1 M ∆ (g 2 g 8 + g 3 g 7 )g 2 4 g 8 M ∆ − 5g 4 g 5 g 7 g 2 8 M ξ , either case III or case IV can be realized.

Conclusions
Combining a discrete flavor symmetry with a CP symmetry is a very promising approach of predicting both lepton mixing angles and CP phases. In this work we have performed a comprehensive analysis of the A 5 family symmetry and CP symmetry. Since the inverse of each conjugacy class of A 5 is equal to itself, all the inner automorphisms of A 5 are classinverting while the unique nontrivial outer automorphism of A 5 is not. As a result, the physical CP transformations are defined by the inner automorphisms of A 5 . In our working basis, the CP transformations are found to be of the same form as the flavor symmetry transformations.
Assuming neutrinos are Majorana particles, we have analyzed the possible symmetry breaking patterns of A 5 H CP and the corresponding predictions for the PMNS matrix as well as the lepton mixing parameters in a model independent way. We find five phenomenologically interesting mixing patterns summarized in Table 1, and one column of the PMNS matrix is fixed to be (− κ )/2 is the golden ratio. All the three mixing angles are determined in terms of a single real parameter θ, and their measured values can be accommodated for certain values of θ. In particular, the Dirac CP violating phase δ CP is predicted to be trivial or maximal while the Majorana phases are trivial. In contrast, δ CP is quite weakly constrained and Majorana phases can not be predicted if CP symmetry is not considered, as shown in Appendix B. Our theoretical predictions can be tested by forthcoming long-baseline neutrino oscillation experiments such as LBNE, LBNO and HyperKamiokande. The predicted mixing patterns would be ruled out, if significant deviations of δ CP from trivial and maximal values were detected. Furthermore, the phenomenological predictions for the (ββ) 0ν −decay are investigated. The present experimental bounds are saturated, and the effective mass |m ee | is found to be within the sensitivity of future (ββ) 0ν −decay experiments for inverted ordering neutrino mass spectrum. Guided by above model independent analysis, we construct a flavor model with both A 5 flavor symmetry and generalized CP symmetry. The lepton mixing is exactly the GR pattern at LO, the observed mass hierarchies among charged lepton are generated, and the three light neutrino masses effectively depend on two real parameters which can be fixed by the measured values of the mass-squared splittings. Therefore the neutrino mass spectrum can only be normal ordering and the absolute neutrino masses are predicted. The model is built in such a way that the GR mixing is modified by NLO contributions and only the second column of GR mixing matrix is kept. A non-vanishing value of θ 13 is generated at NLO and it is naturally of the correct order λ c in our model. In case of g 1 M ∆ (g 2 g 8 + g 3 g 7 )g 2 4 g 8 M ∆ − 5g 4 g 5 g 7 g 2 8 M ξ < 0, Dirac CP phase δ CP is 0 or π, consequently the mixing pattern of case III of general analysis in section 3.2 is reproduced exactly. In case of g 1 M ∆ (g 2 g 8 + g 3 g 7 )g 2 4 g 8 M ∆ − 5g 4 g 5 g 7 g 2 8 M ξ > 0, Dirac CP phase δ CP is maximal with δ CP = ±π/2, the mixing pattern of case IV is generated. In other words, our model provides an explicit dynamical realization of the assumed symmetry breaking pattern in section 3.2.
It is interesting to implement any of the remaining cases II, V and VII in Table 1 in a concrete model. Moreover, the group I , which is the double cover of A 5 , may deserve to be studied in a similar fashion. Since I has doublet representations [46], quark masses and mixing should be easily reproduced.

Appendix A Group Theory of A 5
A 5 is the group of even permutations of five objects, and it has 5!/2 = 60 elements. Geometrically it is the symmetry group of a regular icosahedron. A 5 group can be generated by two generators S and T which satisfy the multiplication rules [47]: The 60 element of A 5 group are divided into 5 conjugacy classes: where nC k denotes a class with n elements which have order k. The group structure of A 5 has been elaborately analyzed in Ref. [47]. Following the convention of Ref. [47], we find that A 5 group has thirty-six abelian subgroups in total: fifteen Z 2 subgroups, ten Z 3 subgroups, five K 4 subgroups and six Z 5 subgroups. In terms of the generators S and T , the concrete forms of these abelian subgroups are as follows: All the above fifteen Z 2 subgroups are conjugate to each other.
• Z 3 subgroups The ten Z 3 subgroups are related with each other by group conjugation.
All the five K 4 subgroups are conjugate as well.
• Z 5 subgroups All the six Z 5 subgroups are related to each other under group conjugation.
Here the superscript of a subgroup denotes its generator (generators). The A 5 group has five irreducible representations: one singlet representation 1, two three-dimensional representations 3 and 3 , one four-dimensional representation 4 and one five-dimensional representation 5. In the present work, we choose the same basis as that of Ref. [47]. The explicit forms of the generators S and T in the five irreducible representations are as follows where ω 5 = e 2πi 5 . The character table of A 5 group is reported in Table 3. We can straightforwardly obtain the Kronecker products between various representations: where R represents any irreducible representation of A 5 , and 4 1 , 4 2 , 5 1 and 5 2 stand for the two 4 and two 5 representations that appear in the Kronecker products. We now list the Clebsch-Gordan coefficients for our basis. We use the notation α i (β i ) to denote the elements of the first (second) representation. The subscript "S" ("A") refers to symmetric (antisymmetric) combinations.
B Lepton flavor mixing from A 5 family symmetry without CP In this section, we investigate the possible lepton mixing patterns which can be derived from only A 5 family symmetry without CP symmetry imposed. As usual, the three generations of left-handed leptons are assigned to the triplet representation 3, and A 5 is broken into two different abelian subgroups G l and G ν in the charged lepton and neutrino sector respectively. The residual flavor symmetry G ν can only be a Z 2 or K 4 subgroup of A 5 since we assume neutrinos are Majorana particles here. In this approach, the PMNS matrix can be obtained by simply diagonalizing the representation matrices of the generators of G l and G ν without resorting to the mass matrix [12,48,49]. For G ν = K 4 and G l is capable of distinguishing the three generations of charged lepton, i.e., the eigenvalues of the generators of G l aren't degenerate, the PMNS matrix would be completely fixed up to row and column permutations. However, only one column would be fixed by the remnant flavor symmetries G l and G ν in case of G ν = Z 2 . In the following, the scenario of G l = Z 2 and G ν = K 4 shall be discussed as well, and one row would be fixed instead. It is noteworthy that two pairs of subgroups (G l , G ν ) and (G l , G ν ) lead to the same result for the PMNS matrix, if they are conjugate under an element of the A 5 group.
From Appendix A, we know that G l can be a Z 3 , Z 5 or K 4 subgroup of A 5 . In case of G l = Z 5 , all 6 × 5 = 30 possible combinations of G l and G ν give rise to the same mixing matrix which is the well-known golden ratio mixing pattern. The mixing angles are determined to be sin 2 θ 12 = (3 − κ) /5 0.276, sin 2 θ 23 = 1/2 and sin 2 θ 13 = 0. Obviously θ 13 should acquire moderate corrections to accommodate the measured non-vanishing value of the reactor angle although θ 12 and θ 23 are in the experimentally favored 3σ ranges [5].
In case of G l = Z 3 , we find two mixing patterns can be obtained. For the representative , the elements of G l and G ν generate an A 4 subgroup instead of the full flavor symmetry group A 5 . The resulting mixing matrix is given by the familiar democratic mixing in which all elements have the same absolute value [50], i.e., The mixing angles are sin 2 θ 12 = sin 2 θ 23 = 1/2 and sin 2 θ 13 = 1/3. Large corrections to θ 12 and θ 13 are needed to be compatible with experimental data. For another representative , the parent group A 5 can be generated by G l and G ν . The mixing matrix is found to be of the form: which leads to the following mixing angles: sin 2 θ 12 = (2 − κ) /3 0.127, sin 2 θ 23 = 1/2 and sin 2 θ 13 = 0. Notice that both θ 12 and θ 13 are outside of the 3σ ranges [5]. The same results have been obtained in Refs. [49,51]. For the last case of G l = K 4 , where G ν and G l are not the same Klein group, only one mixing pattern can be derived, We can extract the mixing angles: sin 2 θ 12 = (3 − κ) /5 0.276, sin 2 θ 23 = (2 + κ) /5 0.724 and sin 2 θ 13 = (2 − κ) /4 0.0955. Both θ 13 and θ 23 are too large to be acceptable. This mixing pattern has also been found in Ref. [49]. In summary, no mixing matrix in agreement with experimental data can be obtained if the full Klein symmetry is preserved by the neutrino mass matrix. In the following, we consider the situation with a single residual Z 2 flavor symmetry in the neutrino sector or in the charged lepton sector.
In this case, only one column or one row of the PMNS matrix would be determined up to permutations and phases of its elements by the remnant flavor symmetries G l and G l G ν Fixed column or row (1, 0, 0) Table 4: The possible form of one column (row) of the PMNS matrix determined by the residual flavor symmetry G ν = Z 2 (G l = Z 2 ) within the framework of A 5 flavor symmetry. The notation "" denotes that the relevant lepton mixing is compatible with the experimental data at 3σ level [5]. The notation "" implies the resulting mixing is not viable.
G ν [34,52]. This method generally allows us to obtain relations between mixing parameters and a non-zero θ 13 . We have scanned all independent combinations of G l and G ν , and the corresponding explicit forms of the fixed column or row vector are presented in Table 4.
Comparing with the present 3σ confidence level ranges of the moduli of the elements of the leptonic mixing matrix [5] |U we find that neither of the two possible row vectors can be accommodated by the data, and only four cases are viable. The remnant symmetries can be chosen to be (G l , , Z S 2 ) without loss of generality, and the fixed column are (− κ (1, 1, 1) T and 1 2 (κ, −1, κ − 1) T respectively. These column vectors can fit the first or the second column of the PMNS matrix. The resulting lepton mixing matrix can be obtained from U GR , U DM and U RC by multiplying a unitary matrix U 23 or U 13 from the right-hand side with where θ and δ are real, and a arbitrary phase matrix in the right-hand side of U 13 and U 23 is omitted, since they can be absorbed into the Majorana phases which are not constrained by flavor symmetry. The multiplication of U 13 (U 23 ) corresponds to performing a unitary linear transformation of the 1st (2nd) and 3rd columns. In the following, we shall discuss the predictions for the PMNS matrix and lepton mixing parameters in each case.
The lepton mixing matrix U P M N S is predicted to have one column (− κ which coincides with the first column of the GR mixing. The other two columns should be orthogonal to it, and they can be obtained by making a unitary rotation of the 2nd and 3rd columns of U GR . . The corresponding PMNS matrix is given by Eq. (B.12). The pink regions denote the possible values of the parameters when both θ and δ freely vary in the whole region of [0, 2π]. The dark green areas represent the regions allowed by the current experimental data for three neutrino mixing angles at 3σ level [5]. The red and yellow pentagrams denote the best fitting values of case III and case IV discussed in section 3.2, where the generalized CP symmetry is considered. Notice that the red pentagrams almost coincides with the yellow one in the first panel, since the best fitting values of sin 2 θ 12 and sin θ 13 are nearly the same in case III and case IV. sin 2 θ 23 = 1 2 − √ 2 + κ sin 2θ cos δ 3 + κ + (1 + κ) cos 2θ , J CP = √ 2 + κ 20 sin 2θ sin δ, sin δ CP = 2(2 + κ) (3κ − 2 + κ cos 2θ) sign(sin 2θ) sin δ (13 + 4κ)(3 + cos 4θ) + 4(7 + 6κ) cos 2θ − 20 sin 2 2θ cos 2δ . (B.13) We have a relation between θ 12 and θ 13 , 5 sin 2 θ 12 cos 2 θ 13 = 3 − κ . (B.14) The solar mixing angle θ 12 is restricted by the observed value of θ 13 such as 0.281 ≤ sin 2 θ 12 ≤ 0.285 which is in the 3σ range [5]. We display the allowed regions of the mixing angles, J CP and δ CP in Fig. 7. No dependence of δ CP on sin θ 13 is observed, and δ CP can take any value in the whole range of [0, 2π]. However, δ CP can only be conserved or maximally broken if generalized CP is considered, as shown in section 3.2. Note that the mixing pattern in Eq. (B.12) has been discussed in Ref. [53,55].
The chosen remnant symmetry leads to a trimaximal column 1 (B.15) Such a mixing pattern as a minimal modification to the tri-bimaximal has been widely discussed in the literature [53,55,57], and it can also be naturally reproduced from simple flavor symmetries A 4 [23,58] and S 4 [20,24,58]. The predictions for the lepton mixing parameters are given by which generically holds true for trimaximal mixing. Inserting the experimental bound of θ 13 [5], we obtain 0.339 ≤ sin 2 θ 12 ≤ 0.343. A numerical analysis similar to previous cases is performed, as shown in Fig. 8. We see that no prediction for δ CP can be made. Recalling that δ CP would be constrained to be maximal by generalized CP symmetry discussed in section 3.3.

B.2.4 G
One column is fixed to be 1 2 (κ, −1, κ − 1) T in this case, and it can only be the first column of the PMNS matrix in order to be consistent with the experimental data. As a result, U P M N S is of the form  . The corresponding PMNS matrix is given by Eq. (B.15). The pink regions denote the possible values of the parameters when both θ and δ freely vary in the whole region of [0, 2π]. The dark green areas represent the regions allowed by the current experimental data for three neutrino mixing angles at 3σ level [5]. The red pentagrams refer to the best fitting values of case V discussed in section 3.3, after the generalized CP is imposed.
A relation between θ 12 and θ 13 follows immediately 4 cos 2 θ 12 cos 2 θ 13 = 1 + κ . (B.20) The solar mixing angle is predicted as 0.326 ≤ sin 2 θ 12 ≤ 0.334 which is in the experimental 3σ bound [5]. The Jarlskog invariant J CP is given by and G ν = Z S 2 . The corresponding PMNS matrix is given by Eq. (B.18). The pink regions denote the possible values of the parameters when both θ and δ freely vary in the whole region of [0, 2π]. The dark green areas represent the regions allowed by the current experimental data for three neutrino mixing angles at 3σ level [5]. The red pentagrams refer to the best fitting values of case VII with θ 23 (θ bf ) < 45 • discussed in section 3.4, after the generalized CP is imposed.
The numerical results are displayed in Fig. 9. We see that δ CP is predicted to be in the range of [0, 1.043] ∪ [5.240, 2π], and the atmospheric mixing angle θ 23 mostly is less than 45 • (i.e., in the first octant) in order to be compatible with experimental data of θ 13 . The scenario of θ 23 in the second octant can be achieved, if the second and third rows of the PMNS matrix in Eq. (B.18) are exchanged. Then the predictions for the solar and reactor mixing angles in Eq. (B.19) remain, δ CP becomes π + δ CP , and θ 23 becomes π/2 − θ 23 . Consequently both J CP and sin δ CP change into their opposite, and the expression of sin 2 θ 23 in Eq. (B.19) is replaced by sin 2 θ 23 = κ( √ 5 − cos 2θ + 2 sin 2θ cos δ) 5 + κ + (κ − 1)(cos 2θ + 2 sin 2θ cos δ) . (B.23) The predictions for sin 2 θ 23 and δ CP versus sin θ 13 are shown in Fig. 10. As expected, θ 23 is really larger than 45 • to accommodate the measured values of θ 13 , and the CP phase δ CP is in the range of [2.099, 4.185]. Notice that generalized CP would constrain δ CP to be trivial, The dark green areas represent the regions allowed by the current experimental data for three neutrino mixing angles at 3σ level [5]. The red pentagrams refer to the best fitting values of case VII with θ 23 (θ bf ) > 45 • discussed in section 3.4, after the generalized CP is imposed.
as studied in section 3.4. In summary, if a single Z 2 subgroup of the A 5 flavor symmetry is preserved by the neutrino mass matrix, only one column of the PMNS matrix can be determined and agreement with experimental data can be achieved. However, the Majorana phases cannot be predicted by flavor symmetry, and the Dirac phase δ CP is constrained very weakly. On the other hand, if we extend the A 5 family symmetry to include the generalized CP, δ CP is predicted to be trivial or maximal and Majorana phases are trivial.