Lepton Flavor Mixing and CP Symmetry
LLepton Flavor Mixing and CP Symmetry
Peng Chen ∗ , Cai-Chang Li † , Gui-Jun Ding ‡ Department of Modern Physics, University of Science and Technology of China,Hefei, Anhui 230026, China
Abstract
The strategy of constraining the lepton flavor mixing from remnant CP symmetry isinvestigated in a rather general way. The neutrino mass matrix generally admits fourremnant CP transformations which can be derived from the measured lepton mixingmatrix in the charged lepton diagonal basis. Conversely, the lepton mixing matrix canbe reconstructed from the postulated remnant CP transformations. All mixing anglesand CP violating phases can be completely determined by the full set of remnant CPtransformations or three of them. When one or two remnant CP transformations arepreserved, the resulting lepton mixing matrix would depend on three real parameters orone real parameter respectively in addition to the parameters characterizing the remnantCP, and the concrete form of the mixing matrix is presented. The phenomenologicalpredictions for the mixing parameters are discussed. The conditions leading to vanishingor maximal Dirac CP violation are studied. ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] a r X i v : . [ h e p - ph ] D ec Introduction
The origin of flavor mixing is one of longstanding open questions in particle physics. Firstlymotivated by the well-known tri-bimaximal mixing [1], a considerable effort has been devotedto understanding lepton mixing from a discrete flavor symmetry which is spontaneously bro-ken down to two different residual subgroups in the neutrino and the charged lepton sec-tors. Please see Refs. [2–6] for review of discrete flavor symmetries and their application inmodel building aspects. So far a complete classification of all possible lepton mixing whichcould be derived from a finite flavor symmetry group under the hypothesis of Majorana neu-trino has been accomplished [7]. Among the complete list of mixing patterns achievable,only the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix with the second column being(1 , , T / √ θ byT2K [8], MINOS [9], DOUBLE-CHOOZ [10], RENO [11] and DAYA-BAY [12] reactor neutrinoexperiments is one of the most significant discoveries in recent years. The sizable θ ∼ ◦ opensthe gateway to access two remaining unknown parameters in the neutrino sector: the neutrinomass hierarchy and the leptonic Dirac CP phase δ CP . If neutrinos are Majorana particles,there are two additional Majorana CP phases which can play a critical role in the neutrinolessdouble beta decay, and we know nothing about their values so far. The T2K collaborationreported a weak evidence for nonzero δ CP ∼ π/ δ CP are starting to appear in global analysis of neutrino oscillation data [14–16]. Needless to say,probing CP violation in the lepton sector would help deepen our understanding of the universe.Some long-baseline neutrino oscillation experiments such as LBNE [17], LBNO [18] and Hyper-Kamiokande [19] have been proposed to precisely measure the lepton mixing parameters inparticular the Dirac CP phase δ CP .If the signal of CP violation is observed in future neutrino oscillation apparatuses, theparadigm of the flavor symmetry would be disfavored. Moreover, in light of the hints formaximal Dirac CP violation δ CP ∼ π/
2, it is imperative and significative to be able tounderstand the observed lepton mixing angles and meanwhile predict the values of CP phasesfrom certain underlying principles. It is notable that CP symmetry was found to imposestrong constraints on the fermion mass matrices nearly thirty years ago [20, 21]. A typicalsimple CP transformation is the so-called µ − τ reflection under which ν µ and ν τ transforminto the CP-conjugate of each other [22–24]. A neutrino mass matrix fulfilling the µ − τ reflection symmetry immediately gives rise to both maximal atmospheric mixing angle θ and maximal CP violation cos δ CP = 0. In recent years, it is found that the µ − τ reflectioncan naturally appear when CP symmetry is imposed together with the widely studied S flavor symmetry [25–27]. Furthermore, phenomenological models in which the desired breakingpatterns of the flavor and CP symmetries are achieved dynamically have been constructed [27–31]. The interplay between CP symmetry with the flavor symmetries A [32] and T (cid:48) [33,34] hasbeen investigated as well. When CP symmetry is combined with ∆(48) [35] or ∆(96) [36] flavorsymmetry, CP transformations distinct from µ − τ reflection can be produced such that δ CP can be non-maximal. It turns out that both mixing angles and CP phases depend on only onecommon free parameter in that case. Recently the possible lepton mixing patterns derived fromCP symmetry and the ∆(6 n ) or ∆(3 n ) flavor symmetry group series have been analyzed [37–39], the experimentally preferred values of the mixing angles can be accommodated verywell, and the corresponding phenomenological implications in neutrinoless double decay arediscussed [39]. Note that it is highly nontrivial to consistently define the CP symmetry in thecontext of a finite flavor symmetry [40, 41]. There are more than one theoretical approachs1ealing with flavor symmetry and CP violation [42].In this work, we shall only concentrate on CP symmetry, and show that the lepton mixingmatrix can be reconstructed from the remnant CP symmetry. We will derive the explicitform of the PMNS matrix when one or two residual CP transformations are preserved. Phe-nomenological implications for the lepton flavor mixing parameters are discussed in detail.Compared with flavor symmetry paradigm, both mixing angles and CP phases can be pre-dicted by remnant CP, and the observed value of the reactor mixing angles can be easilyaccommodated.The paper is organized as follows. In section 2, remnant flavor symmetry and remnant CPtransformations of the neutrino mass matrix are analyzed in the charged lepton diagonal basis.In section 3, we show that the lepton mixing matrix can be reconstructed from the presumedremnant CP transformations. If two (or one) remnant CP transformations are preserved, theexplicit form of the PMNS matrix is derived, and it depends on one (or three) free real pa-rameters in addition to the parameters of the remnant CP. In section 4, the phenomenologicalpredictions for the lepton mixing parameters are discussed, and we search for conditions ofzero or maximal Dirac CP violation. Finally we summarize our results in section 5. In this section, we shall clarify the remnant flavor symmetry and remnant CP symmetryof the lepton mass matrices. We shall assume throughout this paper that the neutrinos areMajorana particles. The lepton mass terms obtained after symmetry breaking are of thefollowing form: L mass = − l R m l l L + 12 ν TL C − m ν ν L + h.c. , (2.1)where C is the charge-conjugation matrix, l L ≡ ( e L , µ L , τ L ) T and l R ≡ ( e R , µ R , τ R ) T de-note the three generation left and right-handed charged lepton fields respectively, and ν L ≡ ( ν eL , ν µL , ν τL ) T is the three left-handed neutrino fields. The Majorana neutrino mass matrix m ν is symmetric. Since the mixing matrix only relates to left-handed fermions in standardmodel, as usual we construct the hermitian mass matrix M l ≡ m † l m l which connects left-handed charged leptons on both sides. We denote the unitary diagonalization matrix of M l and m ν by U l and U ν respectively, i.e., U † l M l U l = diag (cid:0) m e , m µ , m τ (cid:1) , U Tν m ν U ν = diag ( m , m , m ) ≡ m diag , (2.2)where the light neutrino masses m i ( i = 1 , ,
3) are real and non-negative. The lepton mixingmatrix is the mismatch between neutrino and charged lepton diagonalization matrices, U P MNS = U † l U ν . (2.3)Without loss of generality we shall choose to work in the basis where M l is diagonal. Then U l would reduce to a unit matrix and the lepton mixing completely comes from the neutrinosector with U P MNS = U ν . The general form of m ν is m ν = U ∗ P MNS m diag U † P MNS . (2.4)Firstly let’s determine the remnant flavor symmetries G ν and G l of the neutrino and chargedlepton mass terms. A unitary transformation ν L → G ν ν L of the left-handed Majorana neutrinoleads to the transformation of the neutrino mass matrix m ν → G Tν m ν G ν . G ν is a flavorsymmetry if and only if m ν is invariant, i.e., G Tν m ν G ν = m ν (2.5)2ubstituting the expression of m ν in Eq. (2.4) into this invariant condition and consideringthat the three light neutrino masses m i are non-degenerate , we obtain U † P MNS G ν U P MNS = diag ( ± , ± , ± . (2.6)As an overall − G ν is irrelevant, there are essentially four solutions for G ν , G i = U P MNS d i U † P MNS , i = 1 , , , , (2.7)where d = diag (1 , − , − , d = diag ( − , , − ,d = diag ( − , − , , d = diag (1 , , . (2.8)It is easy to see that G is exactly a trivial identity matrix, and we can further check that G i = 1 , G i G j = G j G i = G k with i (cid:54) = j (cid:54) = k (cid:54) = 4 . (2.9)Hence the residual flavor symmetry of the neutrino mass matrix is Z × Z Klein group. Onthe other hand, given the remnant Klein symmetry in the neutrino sector and the associated3-dimensional unitary representation matrices, one can straightforwardly construct the diag-onalization matrix U ν . Similarly, the remnant flavor symmetry G l of the charged lepton massmatrix satisfies G † l M l G l = M l . (2.10)Since M l is diagonal in the chosen basis and the three charged lepton masses are unequal, G l can only be a unitary diagonal matrix, i.e., G l = diag (cid:0) e iα e , e iα µ , e iα τ (cid:1) , (2.11)where α e,µ,τ are arbitrary real parameters. Hence the charged lepton mass term genericallyadmits a U (1) × U (1) × U (1) remnant flavor symmetry. Conversely, if G l is diagonal with non-degenerate eigenvalues, M l would be forced to be real. The idea of residual symmetries G ν and G l arising from some underlying discrete flavor symmetry group G f has been extensivelyexplored, and many flavor models have been constructed [2–6].In the following, we shall investigate the remnant CP symmetry of the lepton mass terms.They don’t receive enough attention they deserve in the past. The CP transformation of theleft-handed neutrino fields is defined via ν L ( x ) CP (cid:55)−→ iX ν γ C ¯ ν TL ( x P ) , (2.12)where x P = ( t, − (cid:126)x ), and X ν is a 3 × X ν isusually called generalized CP transformation in the literature [20,21,43], since it is an identitymatrix in conventional CP transformation. The Lagrangian of the neutrino mass term inEq. (2.1) would be invariant if the neutrino mass matrix m ν fulfills X Tν m ν X ν = m ∗ ν . (2.13)With the general form of m ν in Eq. (2.4), we obtain (cid:16) U † P MNS X ν U ∗ P MNS (cid:17) T m diag (cid:16) U † P MNS X ν U ∗ P MNS (cid:17) = m diag , (2.14) Here we assume that the three light neutrino masses are non-vanishing. If the lightest neutrino is massless,then one diagonal entry “ ±
1” could be replaced with an arbitrary phase factor in Eq. (2.6). U † P MNS X ν U ∗ P MNS = diag ( ± , ± , ±
1) (2.15)Therefore there are eight possibilities for X ν . However, only four of them are relevant, andthey can chosen to be X i = U P MNS d i U TP MNS , i = 1 , , , . (2.16)The remaining four can be obtained from the above chosen ones by multiplying an over − X ν from − X ν since the minus sign can be absorbedby redefining the neutrino fields. Moreover, we see that the remnant CP transformations X i are symmetric unitary matrices: X i = X Ti , (2.17)otherwise the light neutrino masses would be degenerate. The same constraint that theremnant CP transformations in the neutrino sector should be symmetric is also obtainedin Ref. [26] in another way. It is remarkable that the remnant flavor symmetry can be inducedby the remnant CP symmetry. From Eq. (2.13), it is easy to obtain, X † j X Ti m ν X i X ∗ j = m ν . (2.18)This means that successively performing two CP transformations X i X ∗ j is equivalent to aflavor symmetry transformations. Concretely we have the following relations: X X ∗ = X X ∗ = X X ∗ = X X ∗ = G ,X X ∗ = X X ∗ = X X ∗ = X X ∗ = G ,X X ∗ = X X ∗ = X X ∗ = X X ∗ = G ,X X ∗ = X X ∗ = X X ∗ = X X ∗ = G = 1 . (2.19)As a consequence, once we impose a set of generalized CP transformations onto the theory,there is always an accompanied flavor symmetry generated. Furthermore, Eq. (2.19) impliesthat any residual CP transformation can be expressed in terms of the remaining ones asfollows, X i = X j X ∗ m X n , i (cid:54) = j (cid:54) = m (cid:54) = n . (2.20)In other words, only three of the four remnant CP transformations are independent. In thesame fashion, the CP transformation of the charged lepton fields is l L ( x ) CP (cid:55)−→ iX l γ C ¯ l TL ( x P ) , (2.21)for the symmetry to hold, the mass matrices M l has to satisfy X † l M l X l = M ∗ l (2.22)In the chosen basis where M l is diagonal, X l can only be a diagonal phase matrix, i.e., X l = diag (cid:0) e iβ e , e iβ µ , e iβ τ (cid:1) , (2.23)where β e,µ,τ are real. In short, the remnant CP symmetry can be constructed from the mixingmatrix, and its explicit form can be determined more precisely with the improving measure-ment accuracy of the mixing angles and CP phases. Before closing this section, we presentthe above discussed residual CP symmetry in an arbitrary basis: X i = U ν d i U Tν ( i = 1 , , , , X l = U l diag (cid:0) e iβ e , e iβ µ , e iβ τ (cid:1) U Tl , (2.24)which can be derived in exactly the same way. In the end, we conclude that the remnantsymmetries can be constructed from the mixing matrix which can be measured experimentally.In the following section, we shall demonstrate that the lepton mixing matrix can be constructedfrom the postulated remnant CP transformations.4 Reconstruction of lepton mixing matrix from remnantCP symmetries
As has been shown in section 2, residual CP symmetries can be derived from mixing ma-trix, and vice versa lepton mixing matrix can be constructed from the remnant CP symmetriesin the neutrino and the charged lepton sectors. In concrete models, we can start from a setof CP transformations X CP which the Lagrange respects at high energy scale. Subsequently X CP is spontaneously broken by some scalar fields into different remnant symmetries in theneutrino and the charged lepton sectors. The misalignment between the two remnant sym-metries is responsible for the mismatch of the rotations which diagonalize the neutrino andcharged lepton matrices, and accordingly the PMNS matrix is generated. The remnant CPsymmetries would be assumed hereinafter and we shall not consider how the required vacuumcompatible with the remnant symmetries is dynamically achieved, since the resulting leptonmixing pattern is independent of vacuum alignment mechanism and there are generally morethan one methods realizing the desired symmetry breaking in practical model building.As before we still stick to the charged lepton diagonal basis in the following. If four CPtransformations X Ri ( i = 1 , , ,
4) out of X CP are conserved by the neutrino mass matrix,where the subscript “ R ” denotes remnant. In order to be well-defined, X Ri should be unitarymatrices and satisfy: X Ri = X TRi , X Ri X ∗ Rj = X Rj X ∗ Ri = X Rm X ∗ Rn = X Rn X ∗ Rm , (cid:0) X Ri X ∗ Rj (cid:1) = 1 , (3.1)for i (cid:54) = j (cid:54) = m (cid:54) = n . As shown in Eq. (2.19), a Z × Z remnant flavor symmetry is generatedwith element of the form X Ri X ∗ Rj ( i (cid:54) = j ). It is well-known that the lepton mixing matrix(except the Majorana phases) is fixed by the residual Klein group up to independent permu-tations of rows and columns. If the residual flavor symmetry originates from a finite flavorsymmetry group at high energy, a complete classification of all possible PMNS matrix has beenworked out [7]. An added bonus here is that the Majorana CP phases can also be determinedfrom the postulated remnant CP transformations although they are not constrained by theremnant flavor symmetry at all. If three CP transformations are preserved by the neutrinomass terms, the fourth one can be generated in the way shown in Eq. (2.20). Hence there arestill four residual CP transformations. Given the explicit forms of remnant CP, the leptonmixing matrix can be straightforwardly calculated.In the following, we shall investigate the most interesting case in which the neutrino massmatrix is invariant under the action of two residual CP transformations X R and X R . Inconcrete models, this situation can be realized in two different ways: only X R and X R belong to the beginning CP transformations X CP or X CP contains all the four remnant CPtransformations, but two of them are broken at low energy. To avoid degenerate light neutrinomasses, both X R and X R should be symmetric unitary matrices. Furthermore, a residual Z flavor symmetry is induced with the generator G R ≡ X R X ∗ R = X R X ∗ R . It is easy to checkthat the following consistency equations are fulfilled, X R G ∗ R X − R = G R , X R G ∗ R X − R = G R . (3.2)As we have X R = G R X R , the neutrino mass matrix would be invariant under X R if it isinvariant under both CP transformation X R and flavor transformation G R . As a consequence,from now on we shall focus on the residual symmetry X R and G R for convenience. Firstlywe note that only one column of the diagonalization matrix U ν , which coincides with U P MNS in the working basis, is fixed by the single Z residual flavor symmetry G R . This column isexactly the unique eigenvector of G R with eigenvalue ± G R ) = ± M l still diagonal), each element of this column can always set to be real and non-negative. Hencethe column dictated by G R can be parameterized as v = cos ϕ sin ϕ cos φ sin ϕ sin φ , (3.3)where both ϕ and φ are real parameters in the interval of [0 , π/ G R is givenby G R = 2 v v † − I × = 2 v v T − I × , (3.4)where I × denote a three dimensional unit matrix. Given (postulated) the column vector v ,the remaining two columns of the PMNS matrix can be obtained from any orthonormal pairof basis vectors v (cid:48) and v (cid:48)(cid:48) in the plane orthogonal to v by a unitary rotation. Here we choose v (cid:48) = sin ϕ − cos ϕ cos φ − cos ϕ sin φ , v (cid:48)(cid:48) = φ − cos φ . (3.5)Moreover, since the remnant symmetry can not predict the ordering of the light neutrino masseigenvalues, the three column vectors of U P MNS can be permuted in any way you want beforecomparison with experimental data. Therefore the PMNS matrix is of the form: U P MNS = ( v , v (cid:48) , v (cid:48)(cid:48) ) U P = cos ϕ sin ϕ ϕ cos φ − cos ϕ cos φ sin φ sin ϕ sin φ − cos ϕ sin φ − cos φ U P , (3.6)where P represents any 3 × U is the unitary rotation matrix U = e iβ θe iβ sin θe i ( β + β ) − sin θe i ( β − β ) cos θe iβ , (3.7)where the rotation angle θ and phases β , , , are free real parameters. The values of β , , , would be further constrained by the remnant CP transformation X R .From Eq. (3.4), we see that the residual flavor transformation G R could be chosen to bereal G R = G ∗ R without loss of generality. Then the consistency equation of Eq. (3.2) impliesthat G R X R = X R G R . Hence v is a simultaneous eigenvector of G R and X R . Moreover,given an eigenvector v of a symmetric unitary matrix, it is easy to shown that the complexconjugate v ∗ is also an eigenvector with the same eigenvalue. As a result, one can alwaystake the eigenvectors of a symmetric unitary matrix to be real. Therefore the remaining twoeigenvectors of X R denoted by v and v could be real. They are mutually orthogonal andboth orthogonal to v . Accordingly v , can be written as follows, v = sin ϕ cos ρ − sin φ sin ρ − cos ϕ cos φ cos ρ cos φ sin ρ − cos ϕ sin φ cos ρ = v (cid:48) cos ρ − v (cid:48)(cid:48) sin ρ ,v = sin ϕ sin ρ sin φ cos ρ − cos ϕ cos φ sin ρ − cos φ cos ρ − cos ϕ sin φ sin ρ = v (cid:48) sin ρ + v (cid:48)(cid:48) cos ρ , (3.8)6here ρ is a real parameter. The remnant CP transformation X R can be constructed fromits eigenvectors v , , via X R = e iκ v v T + e iκ v v T + e iκ v v T , (3.9)where e iκ , e iκ and e iκ are the eigenvalues of X R . Note that the eigenvalues of a unitarymatrix must be complex numbers with modulus one. Then another CP transformation X R is X R = G R X R = e iκ v v T − e iκ v v T − e iκ v v T . (3.10)We see that v , v and v are also the eigenvectors of X R while the corresponding eigenvaluesbecome e iκ , − e iκ and − e iκ respectively. Now we turn to explore the constraint of theremnant CP transformation X R . The neutrino mass matrix m ν invariant under the action of X R should fulfill X TR m ν X R = m ∗ ν which leads to X R U ∗ P MNS = U P MNS (cid:98) X R , (3.11)as shown in Eq. (2.15), where (cid:98) X R = diag( ± , ± , ± X R inEq. (3.9) and U P MNS in Eq. (3.6), we obtaindiag (cid:0) e iκ , e iκ , e iκ (cid:1) U (cid:48)∗ P = U (cid:48) P (cid:98) X R , (3.12)where we have defined U (cid:48) = ρ − sin ρ ρ cos ρ U (3.13)Further premultiplying diag (cid:0) e − iκ / , e − iκ / , e − iκ / (cid:1) and post-multiplying (cid:98) X − / R P T on bothsides of Eq. (3.12), we havediag( e i κ , e i κ , e i κ ) U (cid:48)∗ ( P (cid:98) X − R P T ) = diag( e − i κ , e − i κ , e − i κ ) U (cid:48) ( P (cid:98) X R P T ) . (3.14)We can see that the left side of this equation is equal to the complex conjugate of the rightside. Notice that U (cid:48) is a block diagonal matrix and P (cid:98) X / R P T is diagonal matrix. Thisequation is satisfied if and only if the combination diag( e − i κ , e − i κ , e − i κ ) U (cid:48) ( P (cid:98) X R P T ) is ablock diagonal real orthogonal matrix, i.e.,diag (cid:0) e − i κ , e − i κ , e − i κ (cid:1) U (cid:48) ( P (cid:98) X R P T ) = θ sin θ − sin θ cos θ ≡ O , (3.15)where each entry of O is determined up to a possible minus sign which can always be absorbedinto the free parameter θ and (cid:98) X R . Then we have U (cid:48) = diag (cid:0) e i κ , e i κ , e i κ (cid:1) O ( P (cid:98) X − R P T ) . (3.16)Therefore if the neutrino mass term has two remnant CP transformations X R and X R givenby Eqs.(3.9,3.10) in the charged lepton diagonal basis, then lepton mixing matrix would beconstrained to be of the form U P MNS = cos ϕ sin ϕ ϕ cos φ − cos ϕ cos φ sin φ sin ϕ sin φ − cos ϕ sin φ − cos φ ρ sin ρ − sin ρ cos ρ e i κ e i κ
00 0 e i κ × θ sin θ − sin θ cos θ P (cid:98) X − R . (3.17)7ote that the remnant CP transformations X R , depend on six real parameters ϕ , φ , ρ , κ , κ and κ , as show in Eqs.(3.9,3.10). The parameters κ , , are the arguments of theeigenvalues of X R , up to π while the eigenvectors are expressed in terms of ϕ , φ and ρ . Inthis approach, X R , can take any values you want, and accordingly ϕ , φ , ρ and κ , , arefree input parameters. It is remarkable that the resulting PMNS matrix is independent ofhow the remnant CP X R , is dynamically realized. Once the values of X R , (the values of ϕ , φ , ρ and κ , , ) are known (postulated), the PMNS matrix compatible with X R , wouldfollow immediately from Eq. (3.17), and it only depends on one free parameter θ . It isobvious that (cid:98) X − R is a diagonal matrix with entries ± ± i . Its effect is only shifting theMajorana phases by π . The permutation matrix P can take six different values. However,exchanging the second and the third columns is equivalent to the redefinitions θ → θ − π/ (cid:98) X − R → diag(1 , , − (cid:98) X − R . Hence there are only three inequivalent permutations ofthe columns. In other words, the fixed vector (cos ϕ, sin ϕ cos φ, sin ϕ sin φ ) T can be the firstcolumn, second column of the third column of the PMNS matrix. The phenomenologicalpredictions of the three cases will be discussed in section 4.Now we consider the last case in which only one remnant CP transformation X R is preservedby the neutrino mass matrix. Similar to previous cases discussed, X R should be a symmetricunitary matrix. As a consequence, X R can be parameterized as X R = e iκ v v T + e iκ v v T + e iκ v v T , (3.18)where e iκ , e iκ and e iκ are the eigenvalues of X R , v , v and v given by Eqs.(3.3,3.8) representits eigenvectors, and they form a set of most general three-dimensional real orthogonal vectors.The invariance of the neutrino mass matrix under X R leads to the constraint: X R U ∗ ν = U ν (cid:98) X R , (3.19)where (cid:98) X R = diag ( ± , ± , ±
1) and U ν denotes the diagonalization matrix of m ν with U Tν m ν U ν =diag( m , m , m ). This condition can be further simplified intodiag (cid:0) e iκ , e iκ , e iκ (cid:1) (cid:101) U ∗ ν = (cid:101) U ν (cid:98) X R , (3.20)where (cid:101) U ν = ( v , v , v ) † U ν . Premultiplying diag (cid:0) e − iκ / , e − iκ / , e − iκ / (cid:1) and post-multiplying (cid:98) X − / R on both sides of Eq. (3.20), we obtaindiag( e i κ , e i κ , e i κ ) (cid:101) U ∗ ν (cid:98) X − R = diag( e − i κ , e − i κ , e − i κ ) (cid:101) U ν (cid:98) X R , (3.21)Therefore the combination diag( e − iκ / , e − iκ / , e − iκ / ) (cid:101) U ν (cid:98) X R is a generic real orthogonal ma-trix, and it can be expressed as,diag( e − i κ , e − i κ , e − i κ ) (cid:101) U ν (cid:98) X R = O × , (3.22)with O × = θ sin θ − sin θ cos θ cos θ θ − sin θ θ cos θ sin θ − sin θ cos θ
00 0 1 , (3.23)where θ , , are real parameters, and a possible overall minus sign of O × is dropped since itis insignificant. Hence we conclude that the PMNS matrix, which coincides with U ν in ourworking basis, is of the form U P MNS = ( v , v , v ) diag (cid:0) e i κ , e i κ , e i κ (cid:1) O × (cid:98) X − R . (3.24)8omparing with scenario with two residual CP transformations, we still need six input pa-rameters ϕ , φ , ρ and κ , , to specify the explicit form of the assumed CP transformation, andthe resulting lepton mixing matrix U P MNS depends on three free parameters θ , , instead ofone. In this section, we shall present the phenomenological predictions for the lepton mixingparameters when one or two residual CP transformations are conserved by the neutrino massmatrix in the charged lepton diagonal basis. Different independent permutations of the threecolumn vectors of U P MNS would be considered. The possible CP transformations which entailmaximal or vanishing Dirac CP violation in the lepton sector are discussed.
The general expression of U P MNS for two remnant CP is presented in Eq. (3.17). The fistcolumn of U P MNS would be completely fixed the imposed remnant symmetry if the permuta-tion matrix P is chosen to be a unit matrix, i.e. P = I × . Then the PMNS matrix is of theform: U P MNS = cos ϕ sin ϕ ϕ cos φ − cos ϕ cos φ sin φ sin ϕ sin φ − cos ϕ sin φ − cos φ ρ sin ρ − sin ρ cos ρ × e i κ e i κ
00 0 e i κ θ sin θ − sin θ cos θ (cid:98) X − / R . (4.1)For the sake of convenience we introduce κ (cid:48) ≡ κ − κ and κ (cid:48) ≡ κ − κ . The expressions forthe mixing angles and the CP-odd weak basis invariants can be straightforwardly extractedfrom Eq. (4.1) as follows. sin θ = (cid:18) cos ρ sin θ + sin ρ cos θ + 12 sin 2 ρ sin 2 θ cos κ (cid:48) − κ (cid:48) (cid:19) sin ϕ , sin θ = (cid:16) cos ρ cos θ + sin ρ sin θ − sin 2 ρ sin 2 θ cos κ (cid:48) − κ (cid:48) (cid:17) sin ϕ − (cid:16) cos ρ sin θ + sin ρ cos θ + sin 2 ρ sin 2 θ cos κ (cid:48) − κ (cid:48) (cid:17) sin ϕ , sin θ = (cid:110) sin φ sin ρ + cos ϕ cos φ cos ρ −
12 cos ϕ sin 2 φ sin 2 ρ cos 2 θ − cos θ cos 2 ρ (cid:0) cos φ cos ϕ − sin φ (cid:1) −
12 cos κ (cid:48) − κ (cid:48) (cid:2) cos ϕ sin 2 φ cos 2 ρ + (cid:0) sin φ − cos φ cos ϕ (cid:1) sin 2 ρ (cid:3) sin 2 θ (cid:111)(cid:46)(cid:104) − (cid:18) cos ρ sin θ + sin ρ cos θ + 12 sin 2 ρ sin 2 θ cos κ (cid:48) − κ (cid:48) (cid:19) sin ϕ (cid:105) ,J CP = 14 cos ϕ sin ϕ sin 2 φ sin 2 θ sin κ (cid:48) − κ (cid:48) , | I | = 14 sin ϕ (cid:12)(cid:12)(cid:12)(cid:12) cos ρ cos θ sin κ (cid:48) + sin ρ sin θ sin κ (cid:48) −
12 sin 2 ρ sin 2 θ sin κ (cid:48) + κ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) , | I | = 14 sin ϕ (cid:12)(cid:12)(cid:12)(cid:12) cos ρ sin θ sin κ (cid:48) + sin ρ cos θ sin κ (cid:48) + 12 sin 2 ρ sin 2 θ sin κ (cid:48) + κ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) , (4.2) J CP is the Jarlskog invariant [45], J CP = Im [( U P MNS ) ( U P MNS ) ( U ∗ P MNS ) ( U ∗ P MNS ) ]= 18 sin 2 θ sin 2 θ sin 2 θ cos θ sin δ CP , (4.3)where δ CP is the CP-violating Dirac phase in standard parameterization [46]. The invariants I and I associated with the Majorana phases are defined by [43, 47, 48], I = Im (cid:2) ( U P MNS ) ( U P MNS ) ∗ (cid:3) = 14 sin θ cos θ sin α ,I = Im (cid:2) ( U P MNS ) ( U P MNS ) ∗ (cid:3) = 14 sin θ cos θ sin α (cid:48) , (4.4)where α (cid:48) ≡ α − δ CP , α and α are the Majorana CP phases [46]. Note that α (cid:48) enters into the effective mass of the neutrinoless double beta decay. Here both I and I are presented in terms of absolute values since the sign of I and I depends on the entry of (cid:98) X R being +1 or −
1. We see that all the mixing parameters generally depend on both theparameters of CP transformation and the free parameter θ . As the first column of U P MNS is (cos ϕ, sin ϕ cos φ, sin ϕ sin φ ) T which is dictated by the induced flavor symmetry G R ≡ X R X ∗ R = X R X ∗ R , solar mixing angle θ and reactor mixing angle θ are related with eachother by cos θ cos θ = cos ϕ . (4.5)Given the global fit results of 3 σ ranges 0 . ≤ sin θ ≤ .
344 and 0 . ≤ sin θ ≤ . ϕ is constrained to be0 . π ≤ ϕ ≤ . π . (4.6)The allowed regions of the parameters ϕ and φ are displayed in Fig. 1 when both mixing anglesand δ CP vary in their 3 σ (or 1 σ ) intervals [16]. We see that the 1 σ ranges for normal ordering(NO) and inverted ordering (IO) mass spectrums are different although the corresponding 3 σ results can hardly be distinguished. Moreover, the 1 σ regions are drastically shrunk comparedwith the 3 σ ones. Therefore more precisely measurement of the mixing parameters can helpto eventually pin down the values of ϕ and φ . Furthermore, from | ( U P MNS ) / ( U P MNS ) | =cot φ we can find a correlation among δ CP and mixing angles as followscos δ CP = cos 2 θ ( − sin θ + cos θ sin θ ) + cos 2 φ (sin θ + cos θ sin θ )sin 2 θ sin 2 θ sin θ . (4.7)Given an input value of φ , we can predict the Dirac CP phase δ CP from the experimentallymeasured values of the mixing angles. The regions of cos δ CP with respect to φ are plottedin Fig. 2. It is remarkable that cos δ CP determined by Eq (4.7) is larger than 1 or smallerthan − φ , and cos δ CP is restricted to be in a rather narrow strip region ifall mixing parameters are required to vary in the 1 σ interval. The improved accuracy of themixing angles will facilitate the determination of δ CP in this approach.From the expression of Jarlskog invariant J CP in Eq. (4.2), we see that J CP vanishes suchthat the Dirac CP is conserved with δ CP = 0 , π if κ = κ or θ = 0 , π/ , π, π/
2. Noticethat both ϕ and φ can not be equal to 0 or π/ U P MNS wouldbe zero. Inspired by the indication of maximal δ CP from T2K Collaboration [13], we shallexplore the condition of maximal Dirac CP-violation. By straightforward but cumbersomecalculations, we find that cos δ CP = 0 can be achieved for the parameters κ = κ ± π, φ = π , ρ = 0 , π , π or 3 π , (4.8)10 π
10 3 π π π π
10 7 π
20 2 π π π π π π π π φϕ Normal, 1st col. π π π π
10 7 π π π π π π π π φϕ Normal, 2nd col. π π π π
10 7 π π π π π π φϕ Normal, 3rd col. π π
10 3 π π π π
10 7 π
20 2 π π π π π π π π φϕ Inverted, 1st col. π π π π
10 7 π π π π π π π π φϕ Inverted, 2nd col. π π π π
10 7 π π π π π π φϕ Inverted, 3rd col.
Figure 1: The allowed regions of φ and ϕ for mixing parameters in the experimentally preferred 3 σ ranges(pink) and 1 σ ranges (light blue) [16], when two residual CP transformations are conserved by the neutrinomass matrix. The first and second rows are the results for normal ordering and inverted ordering neutrino massspectrum respectively. The left column, middle column and the right column are the corresponding results for(cos ϕ, sin ϕ cos φ, sin ϕ sin φ ) T in the first, second and third column of the U P MNS respectively. The black dotsand the green curves denote the viable values of φ and ϕ given in Eqs. (4.13, 4.21, 4.30) if the residual flavorsymmetry G R = X R X ∗ R = X R X ∗ R arises from a finite flavor symmetry group. Note that the cyan curvesare the results for the cases that the first or the third column of the PMNS matrix are the corresponding onesof the infinite series C in Eq. (4.12). no matter what value θ takes. The corresponding residual CP transformations are of the form X R = e iκ + cos κ (cid:48) − i cos 2 ϕ sin κ (cid:48) − i √ sin 2 ϕ sin κ (cid:48) − i √ sin 2 ϕ sin κ (cid:48) − i √ sin 2 ϕ sin κ (cid:48) − i sin ϕ sin κ (cid:48) cos κ (cid:48) + i cos ϕ sin κ (cid:48) − i √ sin 2 ϕ sin κ (cid:48) cos κ (cid:48) + i cos ϕ sin κ (cid:48) − i sin ϕ sin κ (cid:48) ,X R = e iκ + cos 2 ϕ cos κ (cid:48) − i sin κ (cid:48) √ sin 2 ϕ cos κ (cid:48) √ sin 2 ϕ cos κ (cid:48) √ sin 2 ϕ cos κ (cid:48) sin ϕ cos κ (cid:48) − cos ϕ cos κ (cid:48) − i sin κ (cid:48) √ sin 2 ϕ cos κ (cid:48) − cos ϕ cos κ (cid:48) − i sin κ (cid:48) sin ϕ cos κ (cid:48) , (4.9)in case of ρ = 0 or π , κ = κ ± π and φ = π/
4, where κ + = ( κ + κ ) /
2. The above formulasfor X R and X R are interchanged for the remaining values of ρ = π/ π/ κ = κ ± π and φ = π/
4. In this occasion, the lepton mixing parameters are predicted to besin θ = cos θ sin ϕ − sin θ sin ϕ , sin θ = sin θ sin ϕ, sin θ = , cos δ CP = 0 , tan α = tan α = tan κ (cid:48) . (4.10)We see that both θ and δ CP are maximal, and the Majorana CP phases α and α are equalup to π . Note that the requirement of Eq. (4.8) is a sufficient but not a necessary condition ofmaximal Dirac CP violation. Since generally δ CP depends on both the input parameters ϕ , φ ,11 −8−4048 c o s δ C P π π π π φ Normal, 1st col. c o s δ C P π π π π φ Normal, 2nd col. c o s δ C P π π π π φ Inverted, 1st col. c o s δ C P π π π π φ Inverted, 2nd col.
Figure 2: The range of cos δ CP versus φ predicted by the relations of Eqs.(4.7,4.19) when all three mixing anglesvary in the experimentally preferred 3 σ regions (red) and 1 σ regions (yellow). The constraint of | cos δ CP | ≤ φ values. The panels on the left side and onthe right side correspond to the vector (cos ϕ, sin ϕ cos φ, sin ϕ sin φ ) T in the first column and the second columnof U P MNS respectively. The panels in the first row and the second row are for normal ordering and invertedordering mass spectrum respectively. The pink and light blue bands stand for the 3 σ and 1 σ ranges of φ displayed in Fig. 1. The horizontal dashed lines represents the boundaries of cos δ CP = +1 and cos δ CP = − δ CP for the discrete φ and ϕ given in Eqs. (4.13, 4.21). Thecyan areas are for the case that the first column of U P MNS is the corresponding one of the infinite series C (the last one of Eq. (4.13)). Note that the cyan regions nearly overlap with the corresponding 3 σ yellow part. ρ , κ , , associated with the residual CP transformations and the free parameter θ , cos δ CP = 0can also be achieved for some specific value of θ (not for any value of θ ) even if the conditionin Eq. (4.8) is not fulfilled. This point can be clearly seen from Fig. 2. The key observablefor Majorana phases is the neutrinoless double beta (( ββ ) ν − ) decay. The dependence of the( ββ ) ν − decay amplitude on the neutrino mixing parameters is represented by effective mass | m ee | [46]: | m ee | = (cid:12)(cid:12) m cos θ cos θ + m sin θ cos θ e iα + m sin θ e i ( α − δ CP ) (cid:12)(cid:12) . (4.11)The effective mass | m ee | for the predicted patterns in Eq. (4.10) is illustrated in Fig. 3(a). Wesee that almost all possible values of | m ee | allowed by experimental data at 3 σ level can be12 - - - - - - - - m lightest @ eV D ¨ m ee ¨ @ e V D Disfavoured by 0 nbb D i s f avo u re db y C o s m o l ogy - - - - - - - - m lightest @ eV D ¨ m ee ¨ @ e V D Disfavoured by 0 nbb D i s f avo u re db y C o s m o l ogy (a) (b) - - - - - - - - m lightest @ eV D ¨ m ee ¨ @ e V D Disfavoured by 0 nbb D i s f avo u re db y C o s m o l ogy - - - - - - - - m lightest @ eV D ¨ m ee ¨ @ e V D Disfavoured by 0 nbb D i s f avo u re db y C o s m o l ogy (c) (d) Figure 3: The ( ββ ) ν − decay effective mass | m ee | as a function of the lightest neutrino mass when the conditionsof maximal Dirac phase in Eqs. (4.8, 4.28, 4.32) are fulfilled. The corresponding mixing parameters are givenby Eqs. (4.10, 4.20, 4.29, 4.33). To be specific, panel (a) is for the case of sin( α − α ) = 0, (b) for sin α = 0,(c) for sin α = 0, and (d) for sin α = sin α = 0. The cyan and orange areas represent the currently allowed3 σ regions for normal ordering and inverted ordering mass spectrum respectively [16]. The purple and blueregions are theoretical predictions for different scenarios mentioned above. Measurements of EXO-200 [49, 50]in combination with KamLAND-ZEN [51] give a bound of | m ee | < .
120 eV. The upper limit on the mass ofthe lightest neutrino is derived from the latest Planck result m + m + m < .
230 eV at 95% confidencelevel [52]. reproduced in this case except a quite small portion in case of NO. Therefore it should be verychallenging to testify this texture in ( ββ ) ν − decay.The framework put forward above is very general. ϕ and φ are free input parameters, andthey can take any values. Now we consider another interesting scenario in which the inducedflavor symmetry G R = X R X ∗ R = X R X ∗ R arises from some finite flavor symmetry group. Allpossible PMNS matrix has been derived if the residual flavor symmetry is the full Z × Z Klein group in the neutrino sector [7]. The complete list of the lepton mixing matrices contain17 sporadic | U P MNS | patterns and one infinite series denoted by C with | U P MNS | = 13 σ − Re σ ω σ ) 1 1 − Re ( ω σ )1 + Re ( ωσ ) 1 1 − Re ( ωσ )
13 13 θ ν θ ν (cid:0) θ ν − π (cid:1) (cid:0) θ ν − π (cid:1) (cid:0) θ ν + π (cid:1) (cid:0) θ ν + π (cid:1) , (4.12)where ω = e iπ/ and σ = e iπp/n is any root of unity, and in the second step we havedenoted θ ν ≡ πp/n for simplicity of notation. In the present context the first column vector(cos ϕ, sin ϕ cos φ, sin ϕ sin φ ) T of Eq. (4.1) dictated by the residual flavor symmetry G canbe any column of the PMNS matrices predicted in Ref. [7], once G is assumed to originatefrom some underlying finite flavor symmetry group. By examining all the possible | U P MNS | predicted in Ref. [7] and considering permutations of rows and columns, we find that only thefollowing forms are compatible with the present data at 3 σ level, cos ϕ sin ϕ cos φ sin ϕ sin φ = 18 √ − √ , r r r , , r r r , √ − √ , √ − √ , cos θ ν cos (cid:0) θ ν − π (cid:1) cos (cid:0) θ ν + π (cid:1) , (4.13)where the parameter θ ν should be in the interval of − . π ≤ θ ν ≤ . π to achieveagreement with the experimental data. r , , are the roots of the equation 56 x − x + 14 x − r (cid:39) . r (cid:39) . r (cid:39) .
132 [7].Notice that the third column of C is also viable, nevertheless it is related to the first column of C (the last one in Eq. (4.13)) via θ ν redefinition. Moreover, the former five column vectors areof the same form as the last one with θ ν = arccos √ √ , arctan √ , 0, π − arctan √ and π − arccos √ √ respectively. The values of ϕ and φ for the quantized columns in Eq. (4.13)are plotted in Fig. 1. Utilizing Eqs.(4.5, 4.17) and Eqs.(4.7, 4.19), we can predict the valuesof cos δ CP and sin θ when the mixing angles vary within their 3 σ ranges. These resultsare collected in Table 1. Obviously precise measurement of θ and the Dirac CP phase δ CP can test this scenario. Note that θ can be measured with quite good accuracy by JUNOexperiment [53]. The column permutation P takes the value P = (4.14)The PMNS matrix is given by U P MNS = ϕ sin ϕ sin φ sin ϕ cos φ − cos ϕ cos φ − cos φ sin ϕ sin φ − cos ϕ sin φ cos ρ − sin ρ ρ ρ × e i κ e i κ
00 0 e i κ cos θ − sin θ θ θ (cid:98) X − / R . (4.15)14cos ϕ, sin ϕ cos φ, sin ϕ sin φ ) T cos δ CP sin θ (3 + √ , , − √ T ∈ CD , CD . → . → . r , r , r ) T ∈ C . → . → . (4 , , T ∈ C , C − . → .
723 0 . → . r , r , r ) T ∈ C − → .
134 0 . → . (3 + √ , − √ , T ∈ CD , CD − → − .
454 0 . → . (3 + √ , − √ , T ∈ C − → − .
969 0 . → . (cid:0) cos θ ν , cos (cid:0) θ ν − π (cid:1) , cos (cid:0) θ ν + π (cid:1)(cid:1) T ∈ C , − . π ≤ θ ν ≤ . π − → . → . (1 , , T ∈ C , C − → . → . (10 − √ , √ , √ T ∈ C , C , C C − → . → . Table 1: The predictions for cos δ CP and sin θ by Eqs. (4.7, 4.19) and Eqs.(4.5, 4.17) when the vector(cos ϕ, sin ϕ cos φ, sin ϕ sin φ ) T is placed in the first or the second column of the PMNS matrix and mixingangles vary in the experimentally preferred 3 σ ranges [16]. Here we assume that the induced residual flavorsymmetry G R = X R X ∗ R = X R X ∗ R originates from a finite flavor symmetry group. Consequently the fixedcolumn can be any column of the PMNS matrices listed in Ref. [7]. C i and CD i are notations for mixingpatterns introduced in [7]. We see that the second column is (cos ϕ, sin ϕ cos φ, sin ϕ sin φ ) T which is irrelevant to θ . Thelepton mixing parameters are determined to besin θ = (cid:18) cos ρ cos θ + sin ρ sin θ −
12 sin 2 ρ sin 2 θ cos κ (cid:48) − κ (cid:48) (cid:19) sin ϕ , sin θ = cos ϕ − (cid:16) cos ρ cos θ + sin ρ sin θ − sin 2 ρ sin 2 θ cos κ (cid:48) − κ (cid:48) (cid:17) sin ϕ , sin θ = (cid:110) sin φ sin ρ + cos ϕ cos φ cos ρ + 12 cos ϕ sin 2 φ sin 2 ρ cos 2 θ − sin θ cos 2 ρ (cid:0) cos φ cos ϕ − sin φ (cid:1) + 12 cos κ (cid:48) − κ (cid:48) (cid:2) cos ϕ sin 2 φ cos 2 ρ + (cid:0) sin φ − cos φ cos ϕ (cid:1) sin 2 ρ (cid:3) sin 2 θ (cid:111)(cid:46)(cid:104) − (cid:18) cos ρ cos θ + sin ρ sin θ −
12 sin 2 ρ sin 2 θ cos κ (cid:48) − κ (cid:48) (cid:19) sin ϕ (cid:105) ,J CP = 14 cos ϕ sin ϕ sin 2 φ sin 2 θ sin κ (cid:48) − κ (cid:48) , | I | = 14 sin ϕ (cid:12)(cid:12)(cid:12)(cid:12) cos ρ sin θ sin κ (cid:48) + sin ρ cos θ sin κ (cid:48) + 12 sin 2 ρ sin 2 θ sin κ (cid:48) + κ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) , | I | = 14 sin ϕ (cid:12)(cid:12)(cid:12)(cid:12) sin ρ cos 2 θ sin( κ (cid:48) − κ (cid:48) ) + sin 4 ρ sin 2 θ sin κ (cid:48) − κ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) . (4.16)It is easy to see that θ and θ are correlated assin θ cos θ = cos ϕ , (4.17)which leads to 0 . π ≤ ϕ ≤ . π . (4.18)15he allowed regions of ϕ and φ by present experimental data is shown in Fig. 1. Moreover, arelation among δ CP and three mixing angles is found:cos δ CP = cos 2 θ (cos θ − sin θ sin θ ) − cos 2 φ (cos θ + sin θ sin θ )sin 2 θ sin 2 θ sin θ , (4.19)which allows us to constrain the values of cos δ CP versus φ for three mixing angles in theexperimentally preferred 3 σ ranges, as is plotted in Fig. 2. In the same fashion as section 4.1,we have zero CP violation sin δ CP = 0 for κ = κ or θ = 0, π/ π , 3 π/
2. A sufficientcondition of maximal CP violation cos δ CP = 0 for any value of θ is still given by Eq. (4.8),and correspondingly the lepton mixing parameters are constrained by this symmetry to be ofthe form sin θ = cos ϕ − cos θ sin ϕ , sin θ = cos θ sin ϕ, sin θ = , cos δ CP = sin α = 0 , tan α = − tan κ (cid:48) , (4.20)The predictions for the ( ββ ) ν − decay effective mass | m ee | is displayed in Fig. 3(b). In theend, we consider the scenario where the residual Z flavor symmetry generated by G R = X R X ∗ R = X R X ∗ R is a subgroup of a finite flavor symmetry. Then the possible forms of thecolumn vector (cos ϕ, sin ϕ cos φ, sin ϕ sin φ ) T fixed by G R would be strongly constrained [7].In order to be compatible with experimental data [16], this column can only be cos ϕ sin ϕ cos φ sin ϕ sin φ = 13 or 120 − √
55 + √
55 + √ . (4.21)Both solutions lead to φ = π/
4. The predictions for sin θ and cos δ CP by Eq. (4.17) andEq. (4.19) are listed in Table 1. We see that the solar mixing angle sin θ is determinedto be around 0.34 or 0.28 which can be checked at JUNO [53]. Although cos δ CP is notconstrained at 3 σ level, it is found − . ≤ cos δ CP ≤ .
734 ( − ≤ cos δ CP ≤ − . − . ≤ cos δ CP ≤ .
843 ( − ≤ cos δ CP ≤ − . θ and θ liein the 1 σ regions. In this scenario, we can choose the permutation matrix is P = . (4.22)The PMNS matrix takes the form: U P MNS = sin ϕ ϕ − cos ϕ cos φ sin φ sin ϕ cos φ − cos ϕ sin φ − cos φ sin ϕ sin φ cos ρ sin ρ − sin ρ cos ρ
00 0 1 × e i κ e i κ
00 0 e i κ cos θ sin θ − sin θ cos θ
00 0 1 (cid:98) X − / R . (4.23)16ow the vector (cos ϕ, sin ϕ cos φ, sin ϕ sin φ ) T resides in the third column of U P MNS . Thelepton mixing parameters read assin θ = cos ϕ, sin θ = cos φ , sin θ = 12 (cid:18) − cos 2 ρ cos 2 θ + sin 2 ρ sin 2 θ cos κ (cid:48) − κ (cid:48) (cid:19) , tan δ CP = sin κ (cid:48) − κ (cid:48) sin 2 ρ cot 2 θ + cos 2 ρ cos κ (cid:48) − κ (cid:48) , tan α = − ρ cos 2 θ sin( κ (cid:48) − κ (cid:48) ) + 2 sin 4 ρ sin 2 θ sin κ (cid:48) − κ (cid:48) (3 cos ρ −
1) sin θ + sin ρ (1 + cos θ ) cos( κ (cid:48) − κ (cid:48) ) + sin 4 ρ sin 4 θ cos κ (cid:48) − κ (cid:48) , tan α (cid:48) = − ρ cos θ sin κ (cid:48) + 2 sin ρ sin θ sin κ (cid:48) − sin 2 ρ sin 2 θ sin κ (cid:48) + κ (cid:48) ρ cos θ cos κ (cid:48) + 2 sin ρ sin θ cos κ (cid:48) − sin 2 ρ sin 2 θ cos κ (cid:48) + κ (cid:48) . (4.24)The weak basis invariants are given by J CP = 14 cos ϕ sin ϕ sin 2 φ sin 2 θ sin κ (cid:48) − κ (cid:48) , | I | = 14 sin ϕ (cid:12)(cid:12)(cid:12)(cid:12) sin ρ cos 2 θ sin( κ (cid:48) − κ (cid:48) ) + sin 4 ρ sin 2 θ sin κ (cid:48) − κ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) , | I | = 14 sin ϕ (cid:12)(cid:12)(cid:12)(cid:12) ρ cos θ sin κ (cid:48) + 2 sin ρ sin θ sin κ (cid:48) − sin 2 ρ sin 2 θ sin κ (cid:48) + κ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) . (4.25) In this case, the allowed values of ϕ and φ are strongly constrained by the measured valuesof θ and θ : 0 . π ≤ ϕ ≤ . π, . π ≤ φ ≤ . π , (4.26)at 3 σ confidence level [16]. From the expression of tan δ CP in Eq. (4.24), we see that DiracCP would be maximally violated once the conditioncot 2 θ = − cot 2 ρ cos κ (cid:48) − κ (cid:48) , (4.27)is satisfied. In particular, the parameters κ = κ ± π, ρ = 0 , π , π or 3 π , (4.28)lead to a maximal CP-violating phase with cos δ CP = 0 for any value of θ . The associatedremnant CP transformations take the form X R = e iκ + e i κ (cid:48) − ic ϕ sin κ (cid:48) − i sin 2 ϕ cos φ sin κ (cid:48) − i sin 2 ϕ sin φ sin κ (cid:48) − i sin 2 ϕ cos φ sin κ (cid:48) cos 2 φ e i κ (cid:48) − is ϕ c φ sin κ (cid:48) sin 2 φ (cid:18) e i κ (cid:48) − is ϕ sin κ (cid:48) (cid:19) − i sin 2 ϕ sin φ sin κ (cid:48) sin 2 φ (cid:18) e i κ (cid:48) − is ϕ sin κ (cid:48) (cid:19) − cos 2 φ e i κ (cid:48) − is ϕ s φ sin κ (cid:48) ,X R = e iκ + e − i κ (cid:48) − s ϕ cos κ (cid:48) sin 2 ϕ cos φ cos κ (cid:48) sin 2 ϕ sin φ cos κ (cid:48) sin 2 ϕ cos φ cos κ (cid:48) − cos 2 φ e i κ (cid:48) + 2 s ϕ c φ cos κ (cid:48) − sin 2 φ (cid:18) e − i κ (cid:48) − s ϕ cos κ (cid:48) (cid:19) sin 2 ϕ sin φ cos κ (cid:48) − sin 2 φ (cid:18) e − i κ (cid:48) − s ϕ cos κ (cid:48) (cid:19) cos 2 φ e i κ (cid:48) + 2 s ϕ s φ cos κ (cid:48) , for ρ = 0 , π and κ = κ ± π , where c ϕ , s ϕ , c φ and s φ are the abbreviations of cos ϕ , sin ϕ ,cos φ and sin φ respectively. In case of ρ = π/ , π/ κ = κ ± π , the above expressions17or X R and X R are exchanged. Beside maximal δ CP , the Majorana phase α is enforced tobe zero by the chosen residual CP while α is not constrained, i.e.,cos δ CP = sin α = 0 , tan α = − tan κ (cid:48) . (4.29)The resulting predictions for | m ee | are displayed in Fig. 3(c). Finally if the residual flavorsymmetry G R = X R X ∗ R = X R X ∗ R originates from a finite flavor symmetry group, we findthat only the third column of C is viable, i.e., cos ϕ sin ϕ cos φ sin ϕ sin φ = 23 θ ν (cid:0) θ ν − π (cid:1) (cid:0) θ ν + π (cid:1) , (4.30)where θ ν ∈ ± [0 . π, . π ] to be in accordance with experimental data. In this case, θ and θ are determined to be: 0 . ≤ sin θ ≤ . . ≤ sin θ ≤ .
403 or0 . ≤ sin θ ≤ . θ can be tested by forthcoming long baselineneutrino oscillation experiments. As demonstrated in section 3, if the neutrino mass matrix is invariant under the actionof a generic residual CP transformation X R = e iκ v v T + e iκ v v T + e iκ v v T in the chargedlepton diagonal basis, then the lepton mixing matrix would be of the form U P MNS = ( v , v , v ) diag (cid:0) e i κ , e i κ , e i κ (cid:1) O × (cid:98) X − R . (4.31)where O × given by Eq. (3.23) denotes an arbitrary orthogonal matrix. In this scenario, U P MNS depends on three free parameters θ , , besides the input parameters characterizingthe residual CP transformation. The analytical expressions for the mixing parameters arerather lengthy and hence are omitted here. Since generally θ , , are involved in each entryof U P MNS , the observed values of the three mixing angles can be easily accommodated bya suitable choice of the values of θ , , . Depending on the concrete form of the residual CPtransformation and θ , , , the CP violating phases δ CP , α and α can take any values. Aftersome cumbersome algebraic calculations, we find that δ CP would be maximal for any θ , , ifand only if X R = e iκ a e iκ b e iκ b , (4.32)where a, b = 1 , ,
3. This is a minor generalization of the µ − τ reflection symmetry in which κ a = κ b . Moreover, θ is enforced to be maximal by this residual CP transformation, andboth Majorana phases are trivial, i.e.,sin θ = 12 , cos δ CP = sin α = sin α = 0 , (4.33)while θ and θ are not constrained. Similarly δ CP will be zero (or equal to π ) for any θ , , if X R = e iκ j e iκ m
00 0 e iκ n , (4.34)where j, m, n = 1 , ,
3. The Majorana CP-violating phases are also found to be conserved inthis case, sin δ CP = sin α = sin α = 0 . (4.35)18f the residual CP is distinct from Eq. (4.32) and Eq. (4.34), the Dirac CP would be neitherconserved nor maximally violated except for some special values of θ , , . From the formula of | m ee | in Eq. (4.11), we see that the two patterns in Eq. (4.33)(for conserved δ CP ) and Eq. (4.35)(for maximal δ CP ) leads to the same predictions for the effective mass | m ee | , as is shown inFig. 3(d). Over the past years, much effort has been devoted to understanding lepton flavor mixingangles from some discrete flavor symmetry. In this setup, the mismatch between the residualflavor symmetries in the neutrino and the charged lepton sectors generates the lepton mixingmatrix. To account for the observed sizable θ , the order of the flavor symmetry groupshould be quite large, the Dirac CP violating phase δ CP is predicted to be conserved whilethe Majorana phases can not be determined by flavor symmetry alone [7]. In this work, wepropose to constrain the lepton flavor mixing matrix from residual CP symmetry instead ofresidual flavor symmetry.The remnant CP symmetry can be derived from the experimentally measured mixing ma-trix. In the charged lepton diagonal basis, the neutrino mass matrix is invariant under theaction of CP transformation of the neutrino triplets: ν L ( x ) CP (cid:55)−→ iX ν γ C ¯ ν TL ( x P ) and X ν = U P MNS diag( ± , ± , ± U TP MNS , and the remnant CP transformation of the charged leptonfields is a generic diagonal phase matrix. Performing two remnant CP transformations in suc-cession can generate the well-known residual flavor Klein group U P MNS diag( ± , ± , ± U † P MNS .As a result, the residual CP is more efficient than the residual flavor symmetry in predictingthe lepton flavor mixing.On the other hand, we have showed that the lepton mixing matrix can be constructedfrom the postulated residual CP transformations of the neutrino mass matrix. If the whole setof the residual CP transformations or three of them are preserved, lepton mixing angles andDirac phase are completely fixed by the induced residual flavor symmetry. In addition, theMajorana CP phases are also subject to the constraint of remnant CP. In the case that thereare two remnant CP transformations X R and X R in the neutrino sector, a by-product is aresidual Z flavor symmetry G R ≡ X R X ∗ R = X R X ∗ R . As a result, one column of the PMNSmatrix would be fixed by G R . The PMNS matrix is found to be of the form of Eq. (3.17). Wesee that U P MNS depends on a single real free parameter θ besides the input parameters ϕ , φ , ρ and κ , , specifying X R and X R . Furthermore, if only one remnant CP transformation iskept by the neutrino mass matrix, the PMNS matrix is reconstructed to be given by Eq. (3.24). U P MNS is determined up to an orthogonal matrix and it involves three real free parameters θ , , . All the three CP violating phases δ CP , α and α are in general related to both the freeparameter θ (or θ , , ) and the the parameters characterizing the remnant CP transformations,depending on the values of these parameters, they can take any values from 0 to 2 π .The lepton masses can not be predicted in this approach. Therefore the column vector(cos ϕ, sin ϕ cos φ, sin ϕ sin φ ) T determined by the induced flavor symmetry G R can be in thefirst column, the second column or the third column of U P MNS in case of two remnant CPtransformations. The phenomenological predictions of the mixing parameters are studied foreach arrangement. The allowed regions of ϕ and φ by the experimental data are extracted.Interesting relations of Eqs. (4.7,4.19) among δ CP and mixing angles are found. We see thatrefined measurements of mixing angles can help to narrow or eventually pin down δ CP . Inview of the preliminary result of δ CP ∼ π/ θ (or θ , , ). Note that meanwhile at leastone of the phases α , α and α − α is predicted to be zero or π . The corresponding19redictions for the ( ββ ) ν − decay effctive mass | m ee | are studied. Furthermore, the scenarioof induced flavor symmetry G R arising from a finite flavor symmetry group is discussed. Thepossible form of the column dictated by G R would be strongly constrained in this case, andconsequently the mixing angles are found to lie in quite narrow regions. Comparison withforthcoming experimental data should be able to test this scenario.In this paper the phenomenological implications of residual CP transformations are ana-lyzed in a model-independent way. The remnant CP transformations can be any well-definedones. The physical results only depend on the presumed remnant CP and are independent ofhow the remnant symmetry is dynamically realized. The idea of combining a flavor symmetrywith CP symmetry has recently stimulated some interesting discussions [26–39], the predictionsof the lepton mixing matrix for different symmetry breaking chains can be straightforwardlyextracted via Eq. (3.17) and Eq. (3.24) reported here. Finally it is interesting to explore thephysical consequence of CP symmetry in some extension of the standard model such as thegrand unification theory, where the viable CP transformations at high energy scale as well asremnant CP are strongly constrained by the gauge symmetry. In addition, it is possible torelate the leptonic CP violating phases to the precisely measured quark CP phase. Acknowledgements
This work is supported by the National Natural Science Foundation of China under GrantNos. 11275188 and 11179007.
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