Lepton Mixing Predictions from Infinite (1) Group Series D with Generalized CP 9n,3n Cai-Chang Li∗, Chang-Yuan Yao†, Gui-Jun Ding‡ 6 1 0 2 Department of Modern Physics, University of Science and Technology of China, n Hefei, Anhui 230026, China a J 4 2 ] h Abstract p - (1) We have performed a comprehensive analysis of the type D group D as flavor p 9n,3n e symmetry and the generalized CP symmetry. All possible residual symmetries and h their consequences for the prediction of the mixing parameters are studied. We find [ that only one type of mixing pattern is able to accommodate the measured values of 1 the mixing angles in both “direct” and “variant of semidirect” approaches, and four v 3 types of mixing patterns are phenomenologically viable in the “semidirect” approach. 9 The admissible values of the mixing angles as well as CP violating phases are studied 3 (1) 6 in detail for each case. It is remarkable that the first two smallest D9n,3n groups with 0 n = 1,2 can fits the experimental data very well. The phenomenological predictions . 1 for neutrinoless double beta decay are discussed. 0 6 1 : v i X r a ∗E-mail: [email protected] †E-mail: [email protected] ‡E-mail: [email protected] 1 Introduction The precise measurement of the reactor mixing angle θ [1–5] encourages the pursuit 13 of the still missing results on leptonic CP violation and neutrino mass ordering as well as the characteristic neutrino nature. Some low-significance hints for a maximally CP-violating value of the Dirac phase δ (cid:39) 3π/2 have been observed [6]. The global fits to lepton mixing CP parameters [7–9] also provide weak evidence for the existence of Dirac type CP violation in neutrino oscillation. In the case that neutrinos are Majorana particles, two more Majorana CP phases α and α would be present, and they are crucial to the neutrinoless double 21 31 beta decay process. However, the present experimental data don’t impose any constraint on the values of the Majorana phases. Finite discrete non-abelian flavor symmetries have been widely used to make predictions for lepton flavor mixing. Assuming the original flavor symmetry group is spontaneously broken to distinct abelian residual symmetries in the neutrino and charged lepton sectors at a low energy scale, one can then determine mixing patterns from the residual symmetries and the structure of discrete flavor symmetry groups. Please see Refs. [10–12] for review on discrete flavor symmetries and the application in model building. For Majorana neutrinos, if the residual symmetries of the charged lepton and neutrino mass matrices originate from a finite flavor group, the lepton mixing matrix would be fully determined by residual symme- tries up to independent row and column permutations. It turns out that the possible forms of the PMNS matrix are strongly constrained in this scenario such that the mixing patterns compatible with the data are of trimaximal form, and the Dirac CP phase is predicted to be 0 or π [13]. The same conclusion is reached for neutrinos being Dirac particles [14]. We note that the neutrino masses are not constrained in this approach and consequently the both Majorana phases α and α are undetermined. Their values can be fixed by consid- 21 31 ering a specific model. If the residual flavor symmetries of the neutrino and charged lepton mass matrix are partially contained in the underlying flavor group, the PMNS matrix would contains at least two free continuous parameters. As a result, the predictivity of the model would be lessened to a certain extent. Besides the extensively discussed residual flavor symmetries, the neutrino and charged lepton mass matrices also admit residual CP transformations, and the residual CP symme- tries can be generated by performing two residual CP transformations [15–17]. Analogous to residual flavor symmetries, the residual CP transformations can also constraint the lep- ton flavor mixing in particular the CP violating phases [15]. The simplest nontrivial CP transformation is known as µ−τ reflection which gives rise to maximal atmospheric mixing and maximal Dirac phase [18–20]. The deviation from maximal atmospheric mixing and non-maximal Dirac CP violation can be naturally obtained from the so-called generalized µ−τ reflection [21]. Recently the flavor symmetry has been extended to combine with the generalized CP symmetry [22,23]. This can lead to rather predictive scenario where both mixing angles and CP phases determined by a small number of (frequently only one) input parameters [22]. In this case, the CP transformation matrix is generally non-diagonal and it is also called generalized CP. The generalized CP symmetry and the corresponding constraints on quark mass matrices have been exploited about thirty year ago [24,25]. In this case the interplay between CP and flavor symmetries has to be carefully treated in order to make the theory consistent [22,23,26]. There have been some models and model independent analysis of CP and flavor symmetries, such as A [27], S [22,28–32], ∆(27) [33], ∆(48) [34], A [35–37], 4 4 5 ∆(96) [38], and the group series ∆(3n2) [39,40] and ∆(6n2) [39,41,42] for general integer n. It is notable that smaller group for instance A [27], S [22,28–32] and A [35–37] can 4 4 5 already describe the experimentally measured values of the mixing angles, and the Dirac CP 1 phase is predicted to be conserved or maximal while the Majorana phases are trivial. On the other hand, all the three CP violating phases generally depend on the free real parameter θ for ∆(3n2) [39,40] and ∆(6n2) [39,41,42] flavor symmetries. In the present work, we shall thoroughly analyze the lepton mixing patterns which can be obtained from the breaking of D(1) flavor symmetry and generalized CP. All possible 9n,3n residual symmetries in the “direct”, “semidirect” and “variant of semidirect” approaches and their consequences for the prediction of the mixing parameters are studied. We shall perform a detailed numerical analysis for all the possible mixing patterns. The admissible values of the mixing parameters for each n and the possible values of the effective mass |m | ee will be explored. The outline of this paper is as follows. In section 2 we find the class-inverting auto- morphism of the D(1) group and the corresponding physically well-defined generalized CP 9n,3n transformations are determined by solving the consistency condition. In section 3 we review the approach to determining the lepton flavor mixing from residual flavor and CP symme- tries of the neutrino and the charged lepton sectors. All possible residual symmetries and the consequences for the prediction of the flavor mixing are studied in the method of the direct approach in section 4. The PMNS matrix is determined to be of the trimaximal form, both Dirac phase δ and the Majorana phase α are conserved, and the values of CP 31 α are integer multiple of 2π/(3n). We investigate the possible mixing patterns which can 21 be derived from the semidirect approach and variant of semidirect approach in section 5 and section 6. The analytical expressions of the PMNS matrices, mixing angles and CP invariants are presented, the admissible values of the mixing angles and CP violation phases are analyzed numerically in detail, and phenomenological predictions for neutrinoless double beta decay are studied. For the lowest order D(1) group with n = 1,2, we find all the 9n,3n mixing patterns that can describe the experimentally measured values of the mixing an- gles, and a χ2 analysis is performed. Finally we summarize and present our conclusions in section 7. The group theory of D(1) is presented in Appendix A including the conjugacy 9n,3n classes, the irreducible representations, the character table, the Kronecker products and the Clebsch-Gordan coefficients. (1) 2 Generalized CP consistent with D family symme- 9n,3n try The finite subgroups of SU(3) have been systematically classified by mathematicians [43] (see Refs. [44–46] for recent work). It is well-established that all discrete subgroups of SU(3) can be divided into five categories: type A, type B, type C, type D, and type E [45,46]. The type D group turns out to be particularly significant in flavor symmetry theory [13,47]. Type D group is isomorphic to (Z ×Z )(cid:111)S , and it can be generated by four generators m n 3 a, b, c and d subject to the following rules [45]: a3 = b2 = (ab)2 = cm = dn = 1, cd = dc, aca−1 = ckd, ada−1 = c−m/nd−(k+1), bcb−1 = cd, bdb−1 = d−1. (2.1) It is found that the type D group exists only for [45] k = 0,m = n or k = 1,m = 3n. (2.2) In the case of k = 0, m = n, the corresponding group denoted as D(0) is exactly the n,n well-known ∆(6n2) group [48]. For another case of k = 1, m = 3n, the corresponding 2 n G GAP-Id Inn(G ) Out(G ) f f f 1 D(1) [162,14] ((Z ×Z )(cid:111)Z )(cid:111)Z Z 9,3 3 3 3 2 6 2 D(1) [648,259] ((Z ×Z )(cid:111)Z )(cid:111)Z Z 18,6 6 6 3 2 6 3 D(1) [1458,659] ((Z ×Z )(cid:111)Z )(cid:111)Z Z 27,9 9 9 3 2 18 Table 1: The automorphism groups of the D(1) group with n = 1,2,3, where Inn(G ) and Out(G ) 9n,3n f f denote inner automorphism group and outer automorphism group of G respectively. Note that each of f these three groups has a unique class-inverting outer automorphism. type D group denoted as D1) is isomorphic to Z ×∆(6n2) if n is not divisible by 3 [45]. 3n,n 3 Therefore the representation of D(1) can be obtained by multiplying the representation 3n,n matrices of ∆(6n2) with 1, e2πi/3 and e4πi/4 for 3 (cid:45) n. As a consequence, the D(1) group 3n,n for 3 (cid:45) n would give rise to the same set of lepton flavor mixing as ∆(6n2) group no matter whether the generalized CP symmetry is considered or not. The ∆(6n2) as flavor symmetry group has been comprehensively explored in the literature [39,41,47], we shall focus on the second independent type D infinite series of groups D(1) where n is any positive integer. 9n,3n It is remarkable that D(1) can generate experimentally viable lepton and quark mixing 9n,3n simultaneously [14]. In the present work, we shall include the generalized CP symmetry compatible with D(1) and investigate its predictions for lepton mixing angles and CP 9n,3n violating phases. The group theory of D(1) , its irreducible representations and the Clebsch- 9n,3n Gordan coefficients are presented in Appendix A. It is highly nontrivial to introduce the generalized CP symmetry in the presence of a discrete flavor symmetry G . In order to consistently combine the generalized CP symmetry f with flavor symmetry, the following consistency condition has to be fulfilled [22,23,26], X ρ∗(g)X† = ρ (g(cid:48)), g,g(cid:48) ∈ G , (2.3) r r r r f where ρ (g) is the representation matrix of the element g in the irreducible representation r r of G , and X is the generalized CP transformation. Obviously the CP transformation X f r r maps g into another group element g(cid:48). Therefore the generalized CP symmetry corresponds the automorphism group of G . Moreover, it was shown that the physically well-defined f CP transformations should be given by class-inverting automorphism of G [26]. We have f exploited the computer algebra system GAP [49] to calculate the automorphism group of the first three D(1) groups with n = 1,2,3, the results are listed in table 1. Notice that larger 9n,3n D(1) group for n ≥ 4 is not stored in GAP at present. We see that the automorphism group 9n,3n of D(1) is quite complex but each one of D(1), D(1) and D(1) has a unique class-inverting 9n,3n 9,3 18,6 27,9 outer automorphism. Furthermore, we find a generic class-inverting automorphism u of the D(1) group, and its actions on the generators a, b, c, d are as follows 9n,3n a (cid:55)−u→ a, b (cid:55)−u→ b, c (cid:55)−u→ c−1, d (cid:55)−u→ d−1. (2.4) Itiseasytocheckthatuindeedmapseachelementintotheclassofitsinverseelementforany value of the parameter n. We denote the physical CP transformation corresponding to the automorphism u as X (u), and its explicit form is determined by the following consistency r equations: X (u)ρ∗(a)X†(u) = ρ (u(a)) = ρ (a), r r r r r 3 X (u)ρ∗(b)X†(u) = ρ (u(b)) = ρ (b) , r r r r r (cid:0) (cid:1) X (u)ρ∗(c)X†(u) = ρ (u(c)) = ρ c−1 , r r r r r (cid:0) (cid:1) X (u)ρ∗(d)X†(u) = ρ (u(d)) = ρ d−1 . (2.5) r r r r r In our working basis shown in Appendix A, the representation matrices of a and b are real while the representation matrices of c and d are complex and diagonal for any irreducible representations of D(1) . Therefore the CP transformation X (u) is a unit matrix, i.e. 9n,3n r X (u) = 1 . (2.6) r r Given this CP transformation X (u), the matrix ρ (g)X (u) = ρ (g) is also an admissible r r r r CP transformation for any g ∈ D(1) . It corresponds to performing a conventional CP 9n,3n transformation followed by a group transformation ρ (g). As a consequence, we conclude r that the generalized CP transformation compatible with the D(1) family symmetry is of 9n,3n the same form as the flavor symmetry transformation in our basis, i.e. X = ρ (g), g ∈ D(1) . (2.7) r r 9n,3n Note that other possible CP transformations can also be defined if a model contains only a subset of irreducible representations. Lepton mixing can be derived from the remnant sym- metries in the charged lepton and neutrino mass matrices, while the mechanism of symmetry breaking is irrelevant. The basic procedure and the resulting master formulae are given in Refs. [15,16,27,28,41]. In the following, we shall consider all possible remnant symmetries of the neutrino and charged lepton sectors and discuss the predictions for the PMNS matrix and the lepton mixing parameters. 3 Framework In the present work, the family symmetry is taken to be D(1) , and the generalized CP 9n,3n symmetry is considered in order to predict the lepton mixing parameters including the CP violating phases. Without loss of generality, we assume that the three left-handed leptons transform as a triplet 3 under D(1) . For brevity we shall denote the faithful irreducible 1,0 9n,3n representation 3 as 3. The representation matrices of the generators a, b, c and d in 3 1,0 1,0 are given in Eq. (A.35). The light neutrinos are assumed to be Majorana particles. From the bottom-upperspective, themostgeneralsymmetryofagenericchargedleptonmassmatrices is U(1)×U(1)×U(1), which has finite subgroups isomorphic to a cyclic group Z for any m integer m or a direct product of several cyclic groups [14–16]. On the other hand, the largest possible symmetry of the neutrino mass matrix is Z ×Z [14–16,50]. Moreover the neutrino 2 2 and charged lepton mass matrices are invariant under a set of CP transformations, and both the U(1)×U(1)×U(1) symmetry group of the charged-lepton mass term and the Z ×Z 2 2 symmetry of the neutrino mass terms can be generated by performing two CP symmetry transformations [15,16]. Conversely, the lepton mass matrices are strongly constrained by the postulated remnant symmetry such that the lepton mixing matrix can be derived from the remnant symmetries in the charged lepton and neutrino sectors, while the mechanism of dynamically realizing the assumed remnant symmetries is irrelevant [15,16]. From the view of the top-down method, the remnant flavor and CP symmetries of the neutrino and charged lepton mass matrices may originate from certain symmetry group implemented at high energy scales. In the present work, both flavor symmetry D(1) and the generalized CP 9n,3n are imposed, i.e., the parent symmetry is D(1) (cid:111)H , where H denotes the generalized 9n,3n CP CP 4 CP transformations consistent with D(1) and it is given by Eq. (2.7). D(1) (cid:111) H is 9n,3n 9n,3n CP assumed to be broken down into G (cid:111)Hl and G ×Hν in the charged lepton and neutrino l CP ν CP sectors respectively. The allowed forms of the neutrino and charged lepton mass matrices are constrained by the remnant symmetries, and subsequently we can diagonalize them to get the PMNS matrix. The requirement that a subgroup G (cid:111)Hl is preserved at low energies entails that the l CP combination m†m has to fulfill l l ρ†(g )m†m ρ (g ) = m†m , g ∈ G , 3 l l l 3 l l l l l (cid:16) (cid:17)∗ X† m†m X = m†m , X ∈ Hl , (3.1) l3 l l l3 l l l3 CP where the charged lepton mass matrix m is given in the convention lcm l. The hermi- l l tian combination m†m is diagonalized by the unitary transformation U with U†m†m U = l l l l l l l diag(m2,m2,m2). The three charged leptons have distinct masses m (cid:54)= m (cid:54)= m . From e µ τ e µ τ Eq. (3.1), it is straightforward to derive that the remnant symmetry G (cid:111)Hl leads to the l CP following constraints on U l U†ρ (g )U = ρdiag(g ), g ∈ G , l 3 l l 3 l l l U†X U∗ = Xdiag, X ∈ Hl , (3.2) l l3 l l3 l3 CP where both ρdiag(g ) and Xdiag are diagonal phase matrices. As a consequence, we see that U 3 l l3 l also diagonalizes the residual flavor symmetry transformation matrix ρ (g ), the residual CP 3 l transformation X is a symmetric matrix, and the following restricted consistency condition l3 should be satisfied [32], X ρ∗(g )X−1 = ρ (g−1), g ∈ G , X ∈ Hl . (3.3) lr r l lr r l l l lr CP In the same fashion, the neutrino mass matrix is invariant under the action of the elements of the residual subgroup G ×Hν : ν CP ρT(g )m ρ (g ) = m , g ∈ G , 3 ν ν 3 ν ν ν ν XT m X = m∗, X ∈ Hν . (3.4) ν3 ν ν3 ν ν3 CP Wedenotetheunitarydiagonalizationmatrixofm asU fulfillingUTm U = diag(m ,m ,m ). ν ν ν ν ν 1 2 3 Then U would be subject to the following constraints from the postulated residual symme- ν try [15–17]: U†ρ (g )U = diag(±1,±1,±1), ν 3 ν ν U†X U∗ = diag(±1,±1,±1), (3.5) ν ν3 ν where the “±” signs can be chosen independently. Therefore the residual CP transformation X is a symmetric unitary matrix as well, and the restricted consistency condition on the ν3 neutrino sector takes the form [15–17,22]: X ρ∗(g )X−1 = ρ (g ), g ∈ G , X ∈ Hν . (3.6) νr r ν νr r ν ν ν νr CP Obviously X maps any element g of the neutrino residual flavor symmetry G into itself. νr ν ν Hence the mathematical structure of the remnant subgroup comprising G and Hν is ν CP generally a direct product instead of a semidirect product. Given a pair of well-defined remnantsymmetriesG (cid:111)Hl andG ×Hν forwhichtheconsistencyequationsinEqs.(3.3, l CP ν CP 3.6) are fulfilled, the allowed forms of the mass matrices m†m and m can be determined l l ν 5 from Eqs. (3.1, 3.4), and subsequently the prediction for the PMNS matrix U = U†U PMNS l ν can be obtained by diagonalizing m†m and m . l l ν For two pair of remnant symmetry subgroups (cid:8)G (cid:111)Hl ,G ×Hν (cid:9) and (cid:8)G(cid:48) (cid:111)Hl(cid:48) , l CP ν CP l CP G(cid:48) ×Hν(cid:48) (cid:9), if G , G and G(cid:48), G(cid:48) are related by a similarity transformation, for example if ν CP l ν l ν they are conjugate, G(cid:48) = hG h−1, G(cid:48) = hG h−1, h ∈ D(1) . (3.7) l l ν ν 9n,3n The remnant CP would also be related by Hl(cid:48) = ρ (h)Hl ρT(h), Hν(cid:48) = ρ (h)Hν ρT(h) (3.8) CP r CP r CP r CP r in order to fulfill the consistency conditions in Eqs. (3.3, 3.6). That is to say the elements of Hl(cid:48) and Hν(cid:48) are given by ρ (h)X ρT(h) and ρ (h)X ρT(h) respectively, where X ∈ Hν CP CP r lr r r νr r νr CP and X ∈ Hl . Notice that all the possible remnant CP transformations compatible with lr CP the remnant flavor symmetry have been considered in this work. Hence if G (cid:111) Hl and l CP G × Hν fix the charged lepton and neutrino mass matrices to be m†m and m , then ν CP l l ν m(cid:48)†m(cid:48) ≡ ρ (h)m†m ρ†(h) and m(cid:48) ≡ ρ∗(h)m ρ†(h) would be invariant under the remnant l l 3 l l 3 ν 3 ν 3 symmetriesG(cid:48)(cid:111)Hl(cid:48) andG(cid:48) ×Hν(cid:48) respectively. Asaresult, twopairofremnantsymmetries l CP ν CP (cid:8)G (cid:111)Hl ,G ×Hν (cid:9) and (cid:8)G(cid:48) (cid:111) Hl(cid:48) , G(cid:48) × Hν(cid:48) (cid:9) would yield the same results for the l CP ν CP l CP ν CP PMNS matrix U . In this work, we shall perform a comprehensive analysis of the mixing PMNS patterns which can be derived from the group D(1) (cid:111)H . It is sufficient to only analyze 9n,3n CP a few representative remnant symmetries which give rise to different results for U and PMNS lepton mixing parameters, as other possible choices for the remnant symmetry groups are related to the representative ones by similarity transformation and consequently no new results are obtained. 4 Lepton mixing from direct approach In the direct approach, the residual flavor symmetry G is a Klein four subgroup, and ν the residual flavor symmetry G is a cyclic group Z with index m ≥ 3 or a product l m of cyclic groups. We assume that the residual flavor symmetry group G can distinguish l the three generations of charged lepton. In other words, the restricted representation of the triplet representation 3 on G should decompose into three inequivalent 1-dimensional l representations of G . From Eq. (3.1) and Eq. (3.2), we see that U not only diagonalizes l l the mass matrix m†m but also the residual flavor symmetry transformation matrix ρ (g ) l l 3 l with g ∈ G . As a result, the requirement that U†ρ (g )U = ρdiag(g ) is diagonal allows l l l 3 l l 3 l us to determine U and no knowledge of m†m is necessary. Notice that the remnant CP l l l invariant condition in Eq. (3.1) is automatically satisfied, the reason is that the residual CP transformation X has to be compatible with residual flavor symmetry and its allowed form l3 is strongly constrained by the restricted consistency condition of Eq. (3.3). As shown in the Appendix A, the group structure of the D(1) has been studied in detail. 9n,3n The residual subgroup G is an abelian subgroup, and it can be generated by the generator l csdt, bcsdt, acsdt, a2csdt, abcsdt or a2bcsdt with s = 0,1,...,9n−1,t = 0,1,...,3n−1. The diagonalization of ρ (g ) determines the unitary transformation U up to permutations and 3 l l phases of the column vectors if ρ (g ) has non-degenerate eigenvalues, where g can be taken 3 l l to be the generator of G . The explicit form of U for different G and the corresponding l l l remnant CP transformations compatible with G are summarized in table 2. If the eigen- l values of ρ (g ) are degenerate so that its diagonalization matrix U can not be determined 3 l l uniquely, we would extend G from a single cyclic subgroup to a product of cyclic groups, l 6 G U Constraints Hl l l CP 1 0 0 t (cid:54)= 0 (cid:104)csdt(cid:105) U(1) = 0 1 0 s−t (cid:54)= 0 mod(3n) {cγdδ} l 0 0 1 s−2t (cid:54)= 0 mod(3n) √   2 0 0 {c2t−s+2δ+3nτdδ, (cid:104)bcsdt(cid:105) Ul(2) = √12  0 −eiπ(23tn−s) eiπ(23tn−s) s (cid:54)= 0,3n,6n bc2δ+3nτdδ} 0 1 1  e−29iπns ω2e−29iπns ωe−29iπns  {bc−2t+3nτd−t, (cid:104)acsdt(cid:105) Ul(3) = √13 e2iπ(39tn−2s) ωe2iπ(39tn−2s) ω2e2iπ(39tn−2s) — abcs−2t+3nτds−2t, 1 1 1 a2bct−s+3nτ} e2iπ(39tn−2s) ωe2iπ(39tn−2s) ω2e2iπ(39tn−2s)  {bc2(t−s)+3nτdt−s, (cid:104)a2csdt(cid:105) Ul(3(cid:48)) = √13  e2iπ(93nt−s) ω2e2iπ(93nt−s) ωe2iπ(93nt−s)  — abc−t+3nτd−t, 1 1 1 a2bc2t−s+3nτ}   iπ(t−s) iπ(t−s) e 3n √0 −e 3n (cid:104)abcsdt(cid:105) U(4) = √1  0 2 0  s−3t (cid:54)= 0,3n,6n {cγdγ+s−t,abcγdγ} l 2   1 0 1 −e−i3πnt e−i3πnt 0  (cid:104)a2bcsdt(cid:105) U(5) = √1  1 1 0  2s−3t (cid:54)= 0,3n,6n {cγd−t,a2bcγ} l 2 √ 0 0 2 Table 2: The form of U for different residual subgroup G generated by a single element g, and here we l l denote G = (cid:104)g(cid:105). Hl is the residual CP transformations consistent with G . The allowed values of the l CP l parameters s, t, γ, δ and τ are t,δ = 0,1,...3n−1, s,γ = 0,1,...,9n−1 and τ = 0,1,2. The parameter ω is the cube root of unit with ω = e2πi/3. Note that because (cid:0)ac2s−3tds−t(cid:1)2 = a2csdt holds, the U for l G =(cid:104)a2csdt(cid:105) can be obtained from the that corresponding to G =(cid:104)acsdt(cid:105) by the replacement s→2s−3t l l andt→s−t. Theconstraintsontheparameterssandtistoremovethedegeneracyamongtheeigenvalues. for example G = G ×G where the generators of G and G should be commutable with l 1 2 1 2 each other. If G (or G ) is sufficient to distinguish among the generations such that its 1 2 eigenvalues are not degenerate, then another subgroup G (or G ) would not impose any 2 1 new constraint on the lepton mixing. On the other hand, if the three eigenvalues of the generator of either G or G are completely degenerate, e.g. G ( or G ) = (cid:104)c3n(cid:105), its three- 1 2 1 2 dimensional representation matrix would be proportional to a unit matrix. As a result, we shall concentrate on the case that the representation matrices of both G and G have two 1 2 degenerate eigenvalues, therefore either G or G alone fixes only a column of U and the 1 2 l third column can be determined by unitary condition. The possible extension of remnant flavor symmetry group G , the corresponding remnant CP transformations and the unitary l transformations U are collected in table 3. We see that the diagonalization matrix U can l l only take five distinct forms U(1), U(2), U(3), U(4) or U(5) such that the constraints on s and l l l l l t shown in table 2 are relaxed. In the direct approach, the flavor symmetry group D(1) group is broken down to a Klein 9n,3n four subgroup in the neutrino sector. From Appendix A, we see that D(1) for even n has 9n,3n only four Klein four subgroups: K(c9n/2,d3n/2) ≡ (cid:8)1,c9n/2,d3n/2,c9n/2d3n/2(cid:9), K(d3n/2,bdx) ≡ (cid:8)1,d3n/2,bdx,bdx+3n/2(cid:9), 4 4 K(c9n/2d3n/2,abc3ydy) ≡ (cid:8)1,c9n/2d3n/2,abc3ydy,abc3y+9n/2dy+3n/2(cid:9), 4 7 G G Constraints on group parameters Form of U Hl 1 2 l CP (cid:40) s−2t = 0 (mod 3n) s(cid:48) −t(cid:48) = 0 (mod 3n) (cid:40) s−2t = 0 (mod 3n) or (cid:104)csdt(cid:105) (cid:104)cs(cid:48)dt(cid:48)(cid:105) t(cid:48) = 0 (mod 3n) U(1) {cγdδ} l (cid:40) s−t = 0 (mod 3n) or t(cid:48) = 0 (mod 3n) (s ↔ s(cid:48), t ↔ t(cid:48)) s(cid:48) −2t(cid:48) = 0 (mod 3n) (cid:104)cs(cid:48)dt(cid:48)(cid:105) s = 0 (mod 3n) {c2t+2δ+3nτdδ, (cid:104)bcsdt(cid:105) U(2) (s−s(cid:48))−2(t−t(cid:48)) = 3l n (mod 6n) l bc2δ+3nτdδ} (cid:104)bcs(cid:48)dt(cid:48)(cid:105) 1 s = 3l n (mod 6n), s(cid:48) = 3l n (mod 6n) 2 3 s(cid:48) −t(cid:48) = 0 (mod 3n) (cid:104)cs(cid:48)dt(cid:48)(cid:105) 3t−s = 0 (mod 3n) {cγdγ+2t, (cid:104)abcsdt(cid:105) (s−s(cid:48))−(t−t(cid:48)) = 3l1n (mod 6n) U(4) abcγdγ} l (cid:104)abcs(cid:48)dt(cid:48)(cid:105) 3t−s = 3l n (mod 6n) 2 3t(cid:48) −s(cid:48) = 3l n (mod 6n) 3 t(cid:48) = 0 (mod 3n) (cid:104)cs(cid:48)dt(cid:48)(cid:105) 2s−3t = 0 (mod 3n) {cγd−t, (cid:104)a2bcsdt(cid:105) t−t(cid:48) = 3l1n (mod 6n) U(5) a2bcγ} l (cid:104)a2bcs(cid:48)dt(cid:48)(cid:105) 2s−3t = 3l n (mod 6n) 2 2s(cid:48) −3t(cid:48) = 3l n (mod 6n) 3 Table3: TheproductextensionoftheremnantflavorsymmetryG =G ×G ,theremnantCPtransforma- l 1 2 tion compatible with G , and the corresponding unitary transformation U . We require the column vectors l l fixed by G and G be different. Consequently we have the parameters l = 0,1 and l +l +l = 1,3. 1 2 1,2,3 1 2 3 The values of parameters s, t, s(cid:48), t(cid:48), γ, δ and τ are s,s(cid:48),γ = 0,1,··· ,9n−1, t,t(cid:48),δ = 0,1,··· ,3n−1 and τ =0,1,2. G X ν ν K(c9n/2,d3n/2) ρ (cγdδ) 4 r K(d3n/2,bdx) ρ (c2δ+2x+3nτdδ),ρ (bc2δ+3nτdδ) 4 r r K(c9n/2d3n/2,abc3ydy) ρ (cδ−2y−3nτdδ),ρ (abcδ−3nτdδ) 4 r r K(c9n/2,a2bc3zd2z) ρ (cγd−2z),ρ (a2bcγ) 4 r r Table 4: The K subgroups of the D(1) group and eligible remnant CP transformations, where the 4 9n,3n superscript of the K subgroup denotes its generators. The allowed values of the parameters are γ = 4 0,1,...,9n−1, x,y,z,δ =0,1,...,3n−1, and τ =0,1,2. 8 K(c9n/2,a2bc3zd2z) ≡ (cid:8)1,c9n/2,a2bc3zd2z,a2bc3z+9n/2d2z(cid:9) , (4.1) 4 where x,y,z = 0,1,...,3n−1. We note that K(c9n/2,d3n/2) is a normal subgroup of D(1) , 4 9n,3n and the remaining three K subgroups are conjugate: 4 (a2cy−x+2δdδ)K(d3n/2,bdx)(a2cy−x+2δdδ)−1 = K(c9n/2d3n/2,abc3ydy), 4 4 (ac−z−x+2δdδ)K(d3n/2,bdx)(ac−z−x+2δdδ)−1 = K(c9n/2,a2bc3zd2z), (4.2) 4 4 with δ = 0,1,...,3n − 1. Furthermore, the residual CP symmetry Hν in the neutrino CP sector has to be compatible with the remnant K symmetry, and the following restricted 4 consistency condition must be fulfilled, X ρ∗(g)X−1 = ρ (g), g ∈ K . (4.3) νr r νr r 4 Solving this equation, we can straightforwardly find the eligible remnant CP transformations fordifferentK subgroups. Theresultsarecollectedintable4. Thenweproceedtodetermine 4 the neutrino mass matrix m invariant under the action of both remnant CP and remnant ν flavor symmetry for each case, i.e., m is subject to the constraints in Eq. (3.4). ν • G = K(c9n/2,d3n/2), X = ρ (cγdδ) ν 4 νr r In our working basis, the representation matrices for both a and c are diagonal with     −1 0 0 1 0 0 ρ3(c9n/2) =  0 −1 0, ρ3(d3n/2) = 0 −1 0  . (4.4) 0 0 1 0 0 −1 Consequently the residual flavor symmetry enforces the neutrino mass matrix to be diag- onal as well. Taking into account the remnant CP symmetry further, we find   m e−2iπ γ 0 0 11 9n m =  0 m e−2iπγ−3δ 0  , (4.5) ν  22 9n  0 0 m e2iπ2γ−3δ 33 9n wherem ,m andm arerealparameters. Wecanreadouttheneutrinodiagonalization 11 22 33 matrix U as ν (cid:16) (cid:17) U = diag eiπ γ ,eiπγ−3δ,e−iπ2γ−3δ Q , (4.6) ν 9n 9n 9n ν where Q is a diagonal phase matrix with entry being ±1 or ±i, and it encodes the CP ν parity of the neutrino states. The light neutrino mass eigenvalues are m = |m |, m = |m |, m = |m | . (4.7) 1 11 2 22 3 33 Obviouslythelightneutrinomassesdependononlythreerealparameters,andtheorderof the light neutrino masses can not be fixed by remnant symmetries. Therefore the unitary transformation U is determined up to independent row and column permutations in the ν present framework, and the neutrino mass spectrum can be wither normal ordering (NO) or inverted ordering (IO). • G = K(d3n/2,bdx), X = (cid:8)ρ (c2δ+2x+3nτdδ),ρ (bc2δ+3nτdδ)(cid:9) ν 4 νr r r 9