CP Symmetries as Guiding Posts: revamping tri-bi-maximal Mixing. Part II
Peng Chen, Salvador Centelles Chuliá, Gui-Jun Ding, Rahul Srivastava, José W. F. Valle
UUSTC-ICTS-19-09
CP Symmetries as Guiding Posts: revamping tri-bi-maximal Mixing. Part II
Peng Chen, ∗ Salvador Centelles Chuli´a, † Gui-Jun Ding, ‡ Rahul Srivastava, § and Jos´e W. F. Valle ¶ College of Information Science and Engineering,Ocean University of China, Qingdao 266100, China AHEP Group, Institut de F´ısica Corpuscular – C.S.I.C./Universitat de Val`encia, Parc Cient´ıfic de Paterna.C/ Catedr´atico Jos´e Beltr´an, 2 E-46980 Paterna (Valencia) - SPAIN Interdisciplinary Center for Theoretical Study and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China
In this follow up of arXiv:1812.04663 we analyze the generalized CP symmetries of the chargedlepton mass matrix compatible with the complex version of the Tri-Bi-Maximal (TBM) lepton mixingpattern. These symmetries are used to “revamp” the simplest TBM
Ansatz in a systematic way.Our generalized patterns share some of the attractive features of the original TBM matrix and areconsistent with current oscillation experiments. We also discuss their phenomenological implicationsboth for upcoming neutrino oscillation and neutrinoless double beta decay experiments.
I. INTRODUCTION
The structure of the lepton sector and the properties of neutrinos stand out as a key missing link in particlephysics, whose understanding is required for the next leap forward. Neutrino oscillation studies have alreadygiven us a first hint for CP violation in the lepton sector [1], though further experiments are needed to improveour measurement of leptonic CP violation [2–4]. Moreover we need information on the elusive Majorana phasesthat would show up in the description of lepton number violating processes such as neutrinoless double-betadecay [5, 6], whose discovery would establish the self-conjugate nature of neutrinos. Improving the currentexperimental sensitivities [7–12] is, again, necessary for the next step.It is a timely moment to make theory predictions for mixing parameters and CP violating phases charaterizingthe lepton sector. The most promising tool is to appeal to symmetry considerations [13]. As benchmarks we haveideas such as mu-tau symmetry and the Tri-Bi-Maximal (TBM) neutrino mixing [14]. Thanks to the reactormeasurements of non-zero θ , these benchmarks can not be the final answer [15–17]. However, they capture animportant part of the truth, providing a valid starting point for building viable patterns of lepton mixing [18–20].Flavor symmetries can be implemented within different approaches. For example, one can build specific theoriesfrom scratch [21–23]. Alternatively, one can adopt a model-independent framework based on the imposition ofresidual symmetries, irrespective of how the mass matrices actually arise from first principles [24–30]. As a ∗ [email protected] † salcen@ific.uv.es ‡ [email protected] § rahulsri@ific.uv.es ¶ valle@ific.uv.es a r X i v : . [ h e p - ph ] J un step in this direction one may consider complexified versions of the standard Tri-Bi-Maximal (TBM) neutrinomixing pattern. By partially imposing generalized CP symmetries one can construct non-trivial variants of thestandard TBM Ansatz (or any other) in a systematic manner. Depending on the type and number of preservedCP symmetries, one obtains several different mixing matrices. Such “revamped” variants of the TBM
Ansatz have in general non-zero θ as well as CP violation, as currently indicated by the oscillation data. Examples ofthis procedure have already been given in [30, 31]. The prospects for probing various neutrino mixing scenariosin present and future oscillation experiments have been discussed extensively in the literature [4, 19, 32–38].This paper is a follow-up of Ref. [30]. Throughout this work we will adopt the basis in which the neutrino massmatrix is diagonal and therefore the leptonic mixing matrix U lep = U † cl , where U cl is the matrix which diagonalizesthe charged lepton mass matrix. Here we take the famous Ansatz of TBM mixing [14] in the charged leptonsector as a starting point, i.e. U cl = U † T BM → U lep = U † cl = U T BM and seek for solutions which satisfy only apartial symmetry with respect to the full TBM symmetries. The paper is structured as follows. We start bybriefly discussing remnant generalized CP symmetries in Section II and the CP and flavour symmetries of the U † T BM mixing matrix in Section III. We will then focus in the cases in which the mass matrix satisfies two CPsymmetries in Section IV and one CP symmetry in Section V. We also discuss the phenomenological predictionsfrom these matrices and give a brief sum-up discussion at the end.
II. PRELIMINARIES
In this section, we shall briefly review the remnant generalized CP symmetry and flavor symmetry of thecharged lepton sector. We assume that the neutrinos are Majorana particles. The neutrino and charged leptonmass term can be written as L mass = − l R m l l L + 12 ν TL C T m ν ν L + h.c., (1)where C is the charge-conjugation matrix, l L and l R are the three generations of the left and right-handed chargedlepton fields respectively. In this paper we work in the neutrino diagonal basis without loss of generality. As aconsequence, m ν is the diagonal neutrino mass matrix and ν L stands for the three generation left-handed neutrinomass eigenstates.A generalized CP symmetry combines the canonical CP transformation with a flavor symmetry. Under theaction of a generalized CP transformation, a generic fermion multiplet field ψ transforms as ψ → iXγ C ψ T , (2)where X is a unitary symmetric matrix in the flavour space which characterizes the generalized CP transformation.Notice that for the conventional CP transformation X is just the identity matrix and does not mix the flavours.Studying the remnant generalized symmetries, i.e., the surviving generalized CP symmetries after spontaneoussymmetry breaking, provides a powerful method to study the mixing pattern of leptons. Given a mixing patternone can extract all the remnant CP symmetries of the corresponding lepton mass matrices [25, 39]. On the otherhand, one can also invert the procedure and extract the possible mixing matrices (up to some freedom) thatrespect one or more generalized CP symmetries. In this work we will study some aspects of such generalized CPtransformations acting on the charged lepton sector.We denote the charged lepton mass matrix as m cl , and the hermitian mass matrix M ≡ m † cl m cl can bediagonalized by a unitary transformation U cl , U † cl M U cl = diag( m e , m µ , m τ ) , (3)where m e , m τ and m τ are the charged lepton masses. If the charged lepton mass term is invariant under thegeneralized CP transformation of Eq. (2), X and M should satisfy the following relation X † M X = M ∗ . (4)If this is the case, we say that X is a remnant CP symmetry of the mass matrix M . As shown in several previousworks [24–26] [30], X can be written in terms of U cl as X = U cl diag( e iδ , e iδ , e iδ ) U Tcl , (5)where the δ i are arbitrary real parameters which label the CP transformation. In other words, given a massmatrix one can extract an infinite set of X matrices satisfying Eq. (4) labeled by the three real parameters δ , δ and δ . Eq. (5) allows us to build all the possible generalized CP symmetries from a given mass matrix ormixing pattern.As stated before, one can also invert the logic and construct the mixing matrix given a remnant CP symmetrymatrix X . If the squared mass matrix M satisfies Eq. (4) it can be shown [24–26] [30] that, after Takagidecomposing the symmetric CP transformation as X = Σ Σ T , the resulting relation between Σ and U cl is U cl = Σ O T × Q − / , (6)where O × is a generic real orthogonal matrix and Q is a general unitary diagonal matrix. If U ν is the unitarymatrix that diagonalizes the neutrino mass matrix, then the lepton mixing matrix is given as U lep = U † cl U ν . (7)As mentioned above, we shall work in the neutrino mass diagonal basis throughout this paper. Hence U ν is aunit matrix and the lepton mixing matrix can be written as U lep = U † cl .We can also follow a similar procedure to obtain the flavour symmetries of the mass matrix M . We define aremnant flavour transformation as ψ → Gψ , (8)where again G is a 3 × G is a residual flavour symmetry of the mass matrix and it satisfies G † M G = M . (9)We can see from Eq. (9) that the flavour symmetries are all the matrices G which commute with M and thereforethey share a common basis of eigenvectors. Since the eigenvalues of M are the squared charged lepton masses,which are non-degenerate, in the basis in which M is diagonal G will also be diagonal. Therefore both G and M are diagonalized by the same unitary matrix U cl , and we can build its explicit form: U † cl G U cl = diag( e iα , e iβ , e iγ ) → G = U cl diag( e iα , e iβ , e iγ ) U † cl , (10)where we used the fact that the eigenvalues of a unitary matrix must have norm 1, so we write them as e iα , e iβ and e iγ , with α , β and γ real. As a side remark that will become important in Section IV, note that the previousargument implies that if we impose a flavour symmetry with non-degenerate eigenvalues then the mixing matrixwould be uniquely determined up to permutation of its column vectors. This holds since, if the eigenvalues of G are ( e iα , e iβ , e iγ ) with α (cid:54) = β (cid:54) = γ , then both M and G are diagonalized by the same unitary matrix.However, the above argument does not hold if the eigenvalues of G are at least partially degenerate, sincein this case in a basis in which G is diagonal, the matrix M may not be diagonal. There will be a subspacegenerated by the eigenvectors associated to the degenerate eigenvalues which won’t be diagonal in general. Wewill exploit this feature in Section IV.As a final comment, let us mention that there is a relation between flavour symmetries and CP symmetries. Iftwo CP symmetries X = U cl diag( e iδ , e iδ , e iδ ) U Tcl and X = U cl diag( e iδ , e iδ , e iδ ) U Tcl are preserved by thecharged lepton sector, then a flavor symmetry can be induced by successively performing two CP transformations G = X X ∗ = U cl diag( e iα , e iβ , e iγ ) U † cl , (11)with α = δ − δ , β = δ − δ and γ = δ − δ . In other words, applying two CP symmetries automatically impliesthe existence of a flavour symmetry. III. CP AND FLAVOUR SYMMETRIES OF TRI-BIMAXIMAL MIXING
The goal of this paper is to modify the Tri-Bi-Maximal mixing (TBM) pattern based on the charged leptonCP symmetries. In this section we will use the results obtained in section II to extract the CP and flavoursymmetries of the celebrated TBM
Ansatz [14]. This will be useful in the following sections, in which we willstart from a charged lepton mass matrix satisfying the full TBM symmetry, and add a perturbation term whichwill satisfy only a partial symmetry.The standard TBM mixing pattern [14] is the
Ansatz in which the three mixing angles take the following valuessin θ = 1 √ , θ = 0 , θ = π/ . (12)The vanishing of one of the mixing angles, in our case θ , implies that the Dirac CP phase δ CP of the leptonmixing matrix is unphysical [40]. Assuming zero Majorana phases, we can write the “real TBM mixing” as, U rT BM = U ( π/ , U (arcsin(1 / √ ,
0) = (cid:113)
23 1 √ − √ √ √ √ − √ √ (13)where U ij ( θ, φ ) is a complex rotation in the ( ij ) − plane of angle θ and phase φ . For example, U ( θ, φ ) = cos θ sin θe − iφ − sin θe iφ cos θ
00 0 1 . (14)We can generalize this Ansatz so as to include non-zero Majorana phases, thus defining the “complex TBMmixing” pattern as U cT BM = U ( π/ , σ ) U (arcsin(1 / √ , ρ ) = (cid:113) e − iρ √ − e iρ √ √ e − iσ √ e i ( ρ + σ ) √ − e iσ √ √ . (15)Note that, apart from the choice ( ρ, σ ) = (0 , ρ and σ which also lead toreal mixing patterns: ( ρ, σ ) = ( π, , (0 , π ) and ( π, π ).If the lepton mixing matrix is the complex TBM matrix U lep = U cT BM , then the charged lepton mixing matrixwill be U cl = U † cT BM in the neutrino diagonal basis. We can explicitly build the charged lepton mass matrixdiagonalized by U † cT BM as M cT BM = m e √ e − iρ √ e iρ + m µ −√ e − iρ −√ e − i ( ρ + σ ) −√ e iρ √ e − iσ −√ e i ( ρ + σ ) √ e iσ (16)+ m τ −√ e − iρ √ e − i ( ρ + σ ) −√ e iρ −√ e − iσ √ e i ( ρ + σ ) −√ e iσ Using Eq. (5) we can easily extract the CP symmetries of the matrix M , i.e. all the matrices X that satisfyEq. (4), X = U † cT BM diag( e iδ , e iδ , e iδ ) U ∗ cT BM = e iδ √ e iρ √ e iρ e iρ
00 0 0 + e iδ e − iρ −√ e − iρ −√ e − i ( ρ − σ ) −√ e − iρ √ e iσ −√ e − i ( ρ − σ ) √ e iσ e iσ + e iδ e − i ( ρ + σ ) −√ e − i ( ρ +2 σ ) √ e − i ( ρ + σ ) −√ e − i ( ρ +2 σ ) e − iσ −√ e − iσ √ e − i ( ρ + σ ) −√ e − iσ . (17)The CP symmetries which correspond to the real TBM limit in Eq. (13) can be constructed from Eq. (17) justby going to the limit ρ → σ →
0. In this case the CP matrix takes the simpler form X = U † rT BM diag( e iδ , e iδ , e iδ ) U ∗ rT BM = e iδ √ √ + e iδ −√ −√ −√ √ −√ √ + e iδ −√ √ −√ −√ √ −√ . (18)We now turn to the residual flavour symmetries of the mixing matrix U † cT BM . Again we will extract all thematrices G , labeled by the three real parameters α , β and γ , that satisfy Eq. (9). Using Eq. (10), we find thatthese matrices take the form G = e iα √ e − iρ √ e iρ + e iβ −√ e − iρ −√ e − i ( ρ + σ ) −√ e iρ √ e − iσ −√ e i ( ρ + σ ) √ e iσ (19)+ e iγ −√ e − iρ √ e − i ( ρ + σ ) −√ e iρ −√ e − iσ √ e i ( ρ + σ ) −√ e iσ . As before, we can recover the real TBM limit just by going to the limit ρ → σ → G = e iα √ √ + e iβ −√ −√ −√ √ −√ √ + e iγ −√ √ −√ −√ √ −√ . (20)Note that if the mass matrix is real then M = M ∗ and there is no difference between Eq. (4) and Eq. (9). Thisis why in the real TBM case the flavour symmetry matrices are identical to the CP symmetry matrices. Thisstatement not only applies to the case ( ρ, σ ) = (0 , ρ, σ ) = (0 , π ),( ρ, σ ) = ( π,
0) and ( ρ, σ ) = ( π, π ).Although the TBM
Ansatz provides an interesting starting point, note that neither the real nor the complexvariants of the TBM mixing are viable lepton mixing patterns. Indeed, recent reactor measurements [15–17] haveestablished that θ is non-zero with high significance. In the same spirit as [30] here we show that, starting fromthe cTBM matrix in the charged lepton sector, and using the generalized CP symmetries, one can systematicallyconstruct and analyze realistic neutrino mixing matrices with non-zero reactor angle. An appealing feature ofthis method is that the resulting mixing patterns will share many properties with the original TBM Ansatz , whileavoiding the unwanted θ = 0 prediction.As a starting point we will assume neutrinos to be Majorana-type and will work in a basis in which neutrinosare diagonal. We will then start with the complex TBM matrix of Eq. (15). The real TBM matrix can alwaysbe obtained from it by simply taking the limit ρ, σ →
0. In what follows we will take this limit at various stagesof our discussion.
IV. CHARGED LEPTON MASS MATRIX CONSERVING TWO CP SYMMETRIES
We will start our analysis by taking as a starting point a charged lepton mass matrix diagonalized by U † cT BM ,which will be of the form shown in Eq. (16). We will then add a small perturbation which only preserves tworemnant CP symmetries, and study the resulting mixing pattern, namely, M = M cT BM + δM , (21)where δM is the perturbation matrix and is therefore expected to be small. As explained in the previous section,a CP symmetry in the charged lepton sector compatible with U cT BM will be of the form shown in Eq. (17) andwill satisfy Eq. (4). We impose, in the perturbation term, two such symmetries given by X = U † cT BM diag( e iδ , e iδ , e iδ ) U ∗ cT BM , X = U † cT BM diag( e iδ , e iδ , e iδ ) U ∗ cT BM , (22)where in general the δ i can be different. As explained in section II, two CP symmetries generate automaticallya flavour symmetry G l = X X ∗ satisfying G † l M G l = M and given by G l = U † cT BM diag( e iα , e iβ , e iγ ) U cT BM , (23)with α = δ − δ , β = δ − δ , and γ = δ − δ . It is clear that U cT BM G l U † cT BM = diag( e iα , e iβ , e iγ ) . (24)One sees that G l is diagonalized by U † cT BM and its eigenvalues are e iα , e iβ , e iγ . As shown in section II, when α (cid:54) = β (cid:54) = γ the eigenvalues of G l are non-degenerate and U † cT BM will diagonalize both G l and M . In this casethe lepton mixing matrix would be U lep = U † cl U ν = U cT BM with θ = 0, hence inconsistent with experiment.Here we study the particular scenarios in which G l is partially degenerate, i.e. G l = U † cT BM P l diag( e iα , e iα , e iβ ) P Tl U cT BM , (25)where α (cid:54) = β and P l is a permutation matrix which parametrizes the three possible orderings, ( e iα , e iα , e iβ ),( e iα , e iβ , e iα ) or ( e iβ , e iα , e iα ). These permutation matrices can be given as: P = , P = , P = , (26)and correspond to three possible different cases respectively, as follows:case ααβ : P l = P → G l = U † cT BM diag( e iα , e iα , e iβ ) U cT BM , (27)case αβα : P l = P → G l = U † cT BM diag( e iα , e iβ , e iα ) U cT BM , (28)case βαα : P l = P → G l = U † cT BM diag( e iβ , e iα , e iα ) U cT BM . (29)In the following, we will consider the cases ααβ and αβα . We discard the third scenario, the βαα case, as itleads to the prediction θ = 0 and is therefore not viable.
1. Case ααβ
We start our analysis by imposing the flavour symmetry G l = U † cT BM diag( e iα , e iα , e iβ ) U cT BM into the per-turbation term δM . Note that this implies δ − δ = δ − δ = α , while β = δ − δ (cid:54) = α . Instead of six CPparameters, going to the ααβ case restricts the situation to five CP parameters. It is clear that the eigenvectorassociated to the eigenvalue e iβ of G l will also be an eigenvector of M . However, two independent eigenvectorsof G l with the degenerate eigenvalue e iα will span a subspace of eigenvectors of G l of which only a particular com-bination is also eigenvector of M . Since G l is diagonalized by U † cT BM , then, after getting rid of the unphysicalphases via redefinition of the phases of the charged lepton fields we get U cT BM ( M cT BM + δM ) U † cT BM = M δ M e iφ δ M e − iφ M
00 0 m τ , (30)where M , M , δ M and φ are real parameters. Notice that the form of the mass matrix is not dependenton the particular values of α and β , only on the choice of the degenerate (12) sector in this case. Then theimposition of X implies φ = δ − δ with δ being completely unphysical. Regarding the CP labels, note thatonly the combination δ − δ is physical, but not the two phases δ and δ independently. As a consequence,the generalized CP symmetries enforce the charged lepton mass matrix with perturbation to be of the followingform, U cT BM ( M cT BM + δM ) U † cT BM = M δ M e i δ − δ δ M e − i δ − δ M
00 0 m τ . (31)As a consistency check, we can see that imposing X does not add any new information, since we have alreadyimposed the flavour symmetry parametrized by α = δ − δ = δ − δ and β = δ − δ . We can now see that themass matrix in Eq. (30) can be diagonalized by diag( e i δ , e i δ , e i δ ) U T ( θ,
0) withtan 2 θ = − δ M M − M , δ M = −
12 ( m µ − m e ) sin 2 θ , M = 12 [ m e (1 + cos 2 θ ) + m µ (1 − cos 2 θ )] , M = 12 [ m e (1 − cos 2 θ ) + m µ (1 + cos 2 θ )] . (32)Note that θ will always be in the first or fourth quadrant and is expected to be small, since we are perturbingthe mass matrix. Therefore the charged lepton diagonalization matrix is given by U cl = U † cT BM diag( e i δ , e i δ , e i δ ) U T ( θ, . (33)Consequently the lepton mixing matrix U lep = U † cl U ν = U † cl will be given by U lep = U ( θ,
0) diag( e − i δ , e − i δ , e − i δ ) U cT BM . (34)Moreover, we can exploit the relation U ( θ,
0) diag( e − i δ , e − i δ , e − i δ ) = diag( e − i δ , e − i δ , e − i δ ) U ( θ, δ ) , (35)with δ = ( δ − δ ) /
2. Since the phases on the left side of a mixing matrix can be absorbed by redefinitions of thecharged lepton fields, only the combination of CP labels given by δ ≡ ( δ − δ ) / U lep = U ( θ, δ ) U cT BM . (36)where we remind that θ is a free real parameter and δ is the meaningful CP label. Now we can proceed to extractthe mixing parameters from the lepton mixing matrix in Eq. (36). In the simplifying limit ρ → σ →
0, i.e.when the CP symmetries of the real TBM matrix U † rT BM and not U † cT BM are imposed, the mixing parametersare given by the following expressionssin θ = sin θ , sin θ = 2 + 2 sin 2 θ cos δ θ + 3 , sin θ = cos θ cos θ + 1 , sin δ CP = sign(sin 2 θ ) (cid:0) θ + 2 (cid:1) sin δ (cid:113)(cid:0) θ − θ cos δ (cid:1)(cid:0) θ cos δ (cid:1) , cos δ CP = sign(sin 2 θ ) (cid:0) sin 2 θ + (cid:0) θ − (cid:1) cos δ (cid:1)(cid:113)(cid:0) θ − θ cos δ (cid:1)(cid:0) θ cos δ (cid:1) , tan δ CP = (2 cos θ + 2) sin δ sin 2 θ + (6 cos θ −
2) cos δ , sin 2 φ = 3 sin θ (5 cos θ + 3 cos 3 θ ) sin δ + 6 sin θ cos θ sin 2 δ (5 + 3 cos 2 θ − θ cos δ ) (1 + sin 2 θ cos δ ) , cos 2 φ = 1 − θ sin δ (5 + 3 cos 2 θ − θ cos δ )(1 + sin 2 θ cos δ ) , sin 2 φ = 8 sin 2 δ cos θ − δ sin 2 θ θ − θ cos δ , cos 2 φ = 1 −
16 cos θ sin δ θ − θ cos δ . (37)The expressions for the general scenario in which ρ and σ are nonzero and the initial symmetry is the complexTBM mixing matrix U † cT BM are particularly lenghty, specially concerning the CP parameters. However, we canrecover the general scenario just by doing the following substitutions in the previous equations δ → δ − ρ , φ → φ + ρ , φ → φ + ρ + σ . (38)As a result, the modified variant of the complex TBM matrix predicts the same relations between mixing anglesand the Dirac CP phase as the modified variant of the real TBM case. These predictions can be neatly summarizedas cos θ cos θ = 1 / , cos δ CP = 3 cos 2 θ (2 − θ ) + cos θ θ sin θ √ θ − . (39) FIG. 1. Correlations between θ and θ when two CP symmetries are preserved by the charged lepton sector. For the ααβ case (blue dashed line), they come from Eq. 39 (left) and the αβα case (red dashed line), they come from Eq. 46 (left).The boxes represent the 3 σ and 1 σ allowed ranges for normal ordered neutrino masses [1]. The star is the best-fit valueof sin θ and sin θ . Note that fixing θ in its 3 σ allowed range greatly restricts the allowed values of the atmosphericangle θ , nearly maximal in both cases. The ααβ and αβα cases correspond to first and second octant, respectively. This is because, if we only look at the oscillation observables, all the CP parameters ( ρ , σ , δ , δ and δ )are either unphysical for oscillations ( δ and σ ), or can be rewritten in terms of just one meaningul CP label, δ (cid:48) ≡ ( δ − δ ) / − ρ , while the expressions of the oscillation parameters ( θ ij and δ CP ) will take the same formas in Eq. (37) just replacing δ by δ (cid:48) .From Eq. (37) we see that both θ and θ only depend on the free parameter θ . Hence the possible valuesof the free parameter θ are strongly constrained due to a very good measurement of the reactor angle θ . Thisimplies a very sharp prediction for the atmospheric angle θ independent of the particular values of the CPlabels δ and δ . The good determination of θ [1, 41–43] and the fact that the atmospheric mixing angle θ is a slowly varying function of θ in the allowed 3 σ range, leads to a tight prediction, θ ∈ [44 . ◦ , . ◦ ], asshown in Fig. 1.Turning to the solar angle θ Eq. (37) indicates that θ and the Dirac CP phase δ CP depend on both thefree parameter θ and the CP parameter δ . The correlation between sin θ and δ CP is displayed in Fig. 2,where the values of the CP label δ CP are indicated by the color shadings. Requiring the solar angle θ inthe experimentally preferred 3 σ range [1], we can read out the allowed region of δ CP is 1 . < δ CP /π < . θ and the CP label δ in the ranges[ − π/ , π/
2] and [0 , π ] respectively, keeping only the points for which the lepton mixing angles θ ij and δ CP areconsistent with experimental data at 3 σ level. The resulting predictions for the lepton mixing angles and CPviolation phases are displayed in Fig. 3. We observe strong correlations among the solar angle sin θ , the DiracCP phase δ CP and the Majorana phase φ and φ in this case.The situation changes for the Majorana phases, since non-zero ρ and σ will shift the Majorana phases, as canbe seen in Eq. (38). Therefore, the difference between the symmetries of the real and the complex versions ofTBM can be seen only in neutrinoless double beta decay experiments, as discussed in Sec. VI B.0 FIG. 2. Correlations between the mixing angle θ and the CP phase δ CP when two CP symmetries are preserved by thecharged lepton sector. For the ααβ case (left panel), they come from Eq. 39 (right), and in the αβα case (right panel),they come from Eq. 46 (right). The boxes are the 3 σ and 1 σ allowed ranges respectively for normal ordered neutrinomasses [1]. The star is the best-fit value of sin θ and δ CP . The reactor angle θ is assumed to lie in the 3 σ region ofthe global fit [1].The shaded colour indicates the value of the CP label δ .
2. Case αβα
Following an analogous argument as in the ααβ case, we can now study the case in which the flavour symmetryis G l = U † cT BM diag( e iα , e iβ , e iα ) U cT BM . In this case the perturbation δM satisfies U cT BM ( M cT BM + δM ) U † cT BM = M δ M e i ( δ − δ )0 m µ δ M e − i ( δ − δ ) 0 M , (40)where M , M , δ M and M are real parameters and m µ is the muon mass. Eq. (40) can be diagonalizedby diag( e iδ / , e iδ / , e iδ / ) U ( θ, T , withtan 2 θ = − δ M M − M , δ M = −
12 ( m τ − m e ) sin 2 θ M = 12 [ m e (1 + cos 2 θ ) + m τ (1 − cos 2 θ )] , M = 12 [ m e (1 − cos 2 θ ) + m τ (1 + cos 2 θ )] , (41)where θ is expected to be small. Using similar arguments to those of the previous case, the lepton mixing matrix U lep = U † cl U ν is given by U lep = U ( θ, δ ) U cT BM , (42)where we have defined δ ≡ ( δ − δ ) /
2. Extracting the mixing parameters in the real TBM case where ρ → σ →
0, we find them to besin θ = sin θ , sin θ = 2 − θ cos δ θ + 3 , sin θ = 1cos θ + 1 , (43)1 FIG. 3. Correlations between mixing angles and CP phases when two CP symmetries are preserved by the charged leptonsector in the case ααβ . We treat both parameters θ and δ as random numbers, the three lepton mixing angles and DiracCP phase δ CP are required to lie within their 3 σ ranges [1]. The blue, green and magenta regions correspond to the CPviolation phases δ CP , φ and φ , respectively. while the CP phases are given bysin δ CP = sign(sin 2 θ ) (cid:0) θ + 2 (cid:1) sin δ (cid:113)(cid:0) θ + 4 sin 2 θ cos δ (cid:1)(cid:0) − θ cos δ (cid:1) , cos δ CP = − sign(sin 2 θ ) (cid:0) sin 2 θ − (cid:0) θ − (cid:1) cos δ (cid:1)(cid:113)(cid:0) θ + 4 sin 2 θ cos δ (cid:1)(cid:0) − θ cos δ (cid:1) , tan δ CP = − θ + 1) sin δ sin 2 θ − θ −
1) cos δ , sin 2 φ = − θ (5 cos θ + 3 cos 3 θ ) sin δ + 6 sin θ cos θ sin 2 δ (5 + 3 cos 2 θ + 4 sin 2 θ cos δ )(1 − sin 2 θ cos δ ) , cos 2 φ = 1 − θ sin δ (5 + 3 cos 2 θ + 4 sin 2 θ cos δ )(1 − sin 2 θ cos δ ) , sin 2 φ = 8 sin δ cos θ + 4 sin δ sin 2 θ θ + 4 sin 2 θ cos δ , FIG. 4. Correlations between mixing angles and CP phases when two CP symmetries are preserved by the charged leptonsector for the αβα case. The blue, green and magenta regions correspond to δ CP , φ and φ , respectively. cos 2 φ = 1 −
16 cos θ sin δ θ + 4 sin 2 θ cos δ . (44)Similar to the previous case, the general scenario in which ρ and σ are nonzero can be recovered by making thefollowing substitutions δ → δ − ρ − σ , φ → φ + ρ , φ → φ + ρ + σ . (45)Again, as in the previous case, the good measurement of θ severely restrict the allowed values of the freeparameter θ . This gives a very sharp prediction for the atmospheric angle θ ∈ [45 . ◦ , . ◦ ], as can be seenin Fig. 1. Moreover, we find the following analytical correlations between the physical parameterssin θ cos θ = 1 / , cos δ CP = 3 cos 2 θ (3 cos θ − − cos θ θ sin θ √ θ − , (46)which also hold in the case where ρ and σ are non-zero. Notice that, in contrast to the ααβ case, this time θ falls in the second octant, although again very close to the maximal mixing value. It is worth noting that the twocases show a very similar behaviour, as can be seen from figures 1 and 2. In both cases the angle θ is very closeto the maximal mixing value. As seen in Fig. 1 the first octant corresponds to the ααβ case, while the αβα case3is associated to the second octant. The correlation between θ and δ CP is similar to the previous case but ofopposite shape. From here we can read off the allowed range for the Dirac CP phase, i.e. 1 . π ≤ δ CP ≤ . π .The numerical results for the lepton mixing and CP phase parameters for the real TBM case are shown in Fig. 4,where the three mixing angles and the Dirac CP phase are required to lie inside the current 3 σ range [1]. Similarto the ααβ case, the modified variants of the complex and real TBM obey the same relations given in Eq. (46)and hence their predictions for the three mixing angles and the Dirac CP phase are the same. However theydiffer in their predictions for neutrinoless double beta decay, as discussed in Sec. VI B. V. CHARGED LEPTON MASS MATRIX CONSERVING ONE CP SYMMETRY
We will now study the case in which only one CP symmetry is preserved in the perturbation term. The CPsymmetries compatible with U cT BM in the charged sector are X = U † cT BM diag( e iδ , e iδ , e iδ ) U ∗ cT BM (47)where δ i are CP parameters labelling the CP transformation. The charged lepton squared mass matrix satisfiesthe relation X † M X = M ∗ , (48)where M = M cT BM + δM . Then, the matrix form of U cT BM M U † cT BM can be written as U cT BM ( M cT BM + δM ) U † cT BM = M δ M e i ( δ − δ δ M e i ( δ − δ δ M e − i ( δ − δ M δ M e i ( δ − δ δ M e − i ( δ − δ δ M e − i ( δ − δ M , (49)where M , M , M and δ M ij are real parameters, unconstrained by the residual symmetry. The matrix U cT BM M U † cT BM in Eq. (49) can be diagonalized by diag( e i δ , e i δ , e i δ ) O T × . Here O × is a real orthogonalmatrix, its matrix elements are determined by the M , M , M and δ M ij [30], and it can be parametrizedas O × = U ( θ , U ( θ , U ( θ , . (50)Then, one can write U cl = U † cT BM diag( e i δ , e i δ , e i δ ) O T , (51)and therefore the lepton mixing matrix can be written as U lep = U † cl U ν = O × diag( e − i δ , e − i δ , e − i δ ) U cT BM . (52)Extracting the mixing angles in the simple limit of real TBM, i.e. ρ = 0 = σ yieldssin θ = 12 (cid:16) θ sin θ cos δ − δ − cos θ cos θ (cid:17) , sin θ = 2 (cid:0) − sin 2 θ (cid:0) cos θ cos δ − δ + sin θ cos δ − δ (cid:1) + sin 2 θ cos θ cos δ − δ (cid:1) (cid:0) − sin 2 θ sin θ cos δ − δ + cos θ cos θ (cid:1) , sin θ = sin θ − − δ − δ sin 2 θ cos θ cos θ − θ cos θ + sin 2 θ sin θ sin 2 θ − sin 2 θ sin θ cos δ − δ + cos θ cos θ ) . (53)One sees that in this case there is no predictivity for the mixing angles, since all parameters are completely free.Since the expressions for the CP parameters are too lengthy to be enlightening, we will not show them here.4 VI. PHENOMENOLOGICAL IMPLICATIONS
In this section we shall study the phenomenological implications of the above CP symmetries for neutrinooscillation as well as neutrinoless double beta decay experiments. We will focus on the case where two remnantCP symmetries are preserved in the charged lepton sector, as this is the most predictive situation.
A. Neutrino oscillations
The observation of neutrino oscillations indicates that neutrinos are massive and that neutrino flavor eigenstatesmix with each other. The three lepton mixing angles and the neutrino mass-squared differences have beenprecisely measured. However, we have not yet established with high significance whether CP is violated in thelepton sector. Moreover, we still don’t know whether the neutrino mass spectrum has normal ordering (NO)or inverted ordering (IO), nor whether the atmospheric angle lies in the first or second octant. The upcomingreactor and long-baseline experiments such as JUNO, DUNE, T2HK should be able to shed light on these issuesand they expect to bring us increased precision on the oscillation parameters θ , θ and δ CP . As shown inprevious sections, lepton mixing parameters are predicted to lie in narrow regions and correlations among themixing parameters are obtained when two residual CP symmetries are preserved. This would translate intophenomenological implications for the expected neutrino and anti-neutrino appearance probabilities in neutrinooscillation experiments.The ν µ → ν e neutrino oscillation probability in matter can be expanded to second order in the mass hierarchyparameter α ≡ ∆ m / ∆ m and the reactor angle sin θ , as follows from [44]: P µe = α sin θ c sin A ∆ A + 4 s s sin ( A − A − + 2 α s sin 2 θ sin 2 θ cos(∆ + δ CP ) sin A ∆ A sin( A − A − , (54)with ∆ = ∆ m L E and A = EV ∆ m , where L is the baseline length and E is the energy of the neutrino beam. Thematter-induced effective potential is V (cid:39) . × − ρ g / cm Y e eV where Y e = 0 . ρ = 3g / cm is assumed. The oscillation probability for antineutrinos is related to thatfor neutrinos by P ¯ µ ¯ e = P µe ( δ CP → − δ CP , V → − V ). The oscillation probability asymmetry between neutrinosand anti-neutrinos is defined as: A µe = P µe − P ¯ µ ¯ e P µe + P ¯ µ ¯ e . (55)In order to illustrate our points we focus, for definiteness, on the revamped ααβ scenario. We display the resultsfor the neutrino appearance oscillation probability P µe and the CP asymmetry A µe in Figs. 5, 6 and 7. Theseare given in terms of the neutrino energy at a fixed distance of L = 295 km, L = 810 km and L = 1300 km,corresponding to the baselines of the T2K, NOvA and DUNE experiments, respectively.Note that our generalized CP symmetries lead to correlations involving the mixing and CP violation parameters.In particular, the three mixing angles and the CP phase can be given in terms of only two parameters. Thistranslates into restrictions on the attainable ranges for the neutrino oscillation probabilities. In Figs. 5, 6 and 7the cyan bands are the generically expected regions obtained when the oscillation parameters are varied withintheir current 3 σ ranges, while the yellow bands denote the predictions for the ααβ neutrino mixing pattern.The solid black lines correspond to the current best fit point predictions, where we have chosen the Dirac phase5 FIG. 5. The appearance probability P µe (left panel) and the CP asymmetry A µe (right panel) as functions of the beamenergy when the baseline is fixed to 295km (T2K experiment). In both panels the predicted band in cyan is the genericallyexpected region obtained by varying the oscillation parameters within their 3 σ ranges [1], while the yellow band is theprediction for the ααβ neutrino mixing pattern, see text for explanation.FIG. 6. Same as Fig. 5 when the baseline is fixed to 810km (NOvA experiment), see text for explanation. δ CP = 1 . π . As a final comment let us mention that, by looking at Eqs. 37 and 38 one sees that the oscillationresults obtained by taking real or complex TBM as the starting point before revamping, i.e. taking ρ and σ inEq.15 to be non-zero, are essentially the same.6 FIG. 7. Same as Fig. 5 when the baseline is fixed to 1300km (DUNE experiment), see text for explanation.
B. Neutrinoless double decay
The neutrinoless double beta (0 νββ ) decay (
A, Z ) → ( A, Z + 2) + 2 e − is the unique probe of the Majorananature of neutrinos [45]. There are many experiments currently searching for 0 νββ decay, or in various stages ofplanning and construction. The sensitivity to this rare process should improve significantly, with good prospectsfor probing the whole region of parameter space associated with the inverted ordering spectrum. The 0 νββ decay provides another test of the CP symmetries of TBM. For our ααβ Ansatz in the general scenario wherethe phases ρ and σ are non-zero, we find the analytical expression of the effective Majorana mass m ee is given by m cT BMee,ααβ = 16 (cid:12)(cid:12)(cid:12)(cid:12) m e iρ (cid:16) sin θ − e i ( δ − ρ ) cos θ (cid:17) + 2 m (cid:16) e i ( δ − ρ ) cos θ + sin θ (cid:17) + 3 m e − iσ sin θ (cid:12)(cid:12)(cid:12)(cid:12) . (56)Notice that ρ , σ and δ are CP parameters, i.e, they label the CP symmetries respected by the mass matrix, while θ is a completely free parameter, which represents the degree up to which the mass matrix is not determined bythe remnant CP symmetry. If all the three labels ρ , σ and δ are treated as free parameters, i.e. scanning over thefull class of mass matrices which allow some preserved CP symmetry of the complex TBM Ansatz , there is noprediction for the 0 νββ decay, since the correlations between Majorana phases and mixing angles disappear. Itis easy to understand this behaviour from Eq. (38). It is worth noting that in the real TBM case, i.e. ρ = 0 = σ ,the analytical expression of the effective Majorana mass m ee simplifies to, m rT BMee,ααβ = 16 (cid:12)(cid:12)(cid:12) m (cid:0) sin θ − e iδ cos θ (cid:1) + 2 m (cid:0) e iδ cos θ + sin θ (cid:1) + 3 m sin θ (cid:12)(cid:12)(cid:12) . (57)Limiting ourselves to the real TBM case we can not only generate CP violation as shown in the previoussections, but also obtain very stringent predictions for neutrinoless double beta decay. The allowed values of m ee are shown in Fig. 8, where the oscillation parameters are required to lie in the currently preferred 3 σ ranges [1].One sees that the real TBM prediction for m ee in the IO case corresponds to the upper boundary of the genericIO region, very close to the sensitivities of the upcoming 0 νββ decay experiments. Notice the existence of alower bound for m ee in the real TBM scenario also for the NO case, where generically it is absent due to possible7cancellations.We can also study the predictions for neutrinoless double beta decay within the αβα scenario. The effectivemass m ee parameters in this case is given by m cT BMee,αβα = 16 (cid:12)(cid:12)(cid:12)(cid:12) m (cid:16) θe iδ + sin θe i ( ρ + σ ) (cid:17) + 2 m (cid:16) e i ( δ − ρ ) cos θ − e iσ sin θ (cid:17) + 3 m sin θ (cid:12)(cid:12)(cid:12)(cid:12) , (58)which simplifies to the following expression in the limit of ρ = 0 = σ , m cT BMee,αβα = 16 (cid:12)(cid:12)(cid:12) m (cid:0) θe iδ + sin θ (cid:1) + 2 m (cid:0) e iδ cos θ − sin θ (cid:1) + 3 m sin θ (cid:12)(cid:12)(cid:12) . (59)Imposing the current restrictions from neutrino oscillations [1] one finds nearly identical results for the 0 νββ decay amplitude parameter m ee also for this case. Indeed, m ee in the αβα scenario only differs from the ααβ case by a slightly different value for θ (see Fig. 1) and a slightly different correlation for θ vs δ CP (see Fig. 2).The atmospheric angle is nearly maximal in both cases, but in different octant. FIG. 8. The effective Majorana mass m ee versus the lightest neutrino mass for the ααβ (left) and the αβα cases (right).The thick colored regions correspond to the predictions of the complex TBM scenario, while the thin magenta and purplebands are for the real TBM case, with ρ = σ = 0. The red and blue dashed lines delimit the most general allowed regionsfor IO and NO neutrino mass spectra, and are obtained by varying the mixing parameters over their 3 σ ranges [1]. Themost stringent current upper limit m ee < .
061 eV from KamLAND-ZEN [7] and EXO-200 [46] is shown by horizontalgrey band. The vertical grey band refers to the current sensitivity of cosmological data from the Planck collaboration [47].
VII. SUMMARY
Starting from the complex version of the Tri-Bi-Maximal lepton mixing pattern we have examined the general-ized CP symmetries of the charged lepton mass matrix. These symmetries are employed in order to ’revamp’ thesimplest TBM
Ansatz for the lepton mixing matrix in a systematic manner. The resulting generalized patternsshare some of the attractive features of the original TBM matrix, while being consistent with current oscillation8experiments indicating non-vanishing θ . We have explicitly examined the case where two CP symmetries arepreserved in the charged lepton sector, the resulting predictions given in Eqs. 39 and 46 and Figs. 1-4. We havealso briefly discussed some of the phenomenological implications of our new mixing patterns, both for neutrinooscillation as well as neutrinoless double beta decay search experiments, illustrated in Figs.5, 6, 7 and 8. Finally,we mention that the same systematic procedure may be employed in order to ’revamp’ other a priori unrealisticpatterns of neutrino mixing. VIII. ACKNOWLEDGMENTS
This work is supported by National Natural Science Foundation of China under Grant Nos 11835013, 11522546and 11847240 and China Postdoctoral Science Foundation under Grant Nos 2018M642700 and the Spanish grantsFPA2017-85216-P (AEI/FEDER, UE), SEV-2014-0398 and PROMETEO/2018/165 (Generalitat Valenciana).We thank the support of the Spanish Red Consolider MultiDark FPA2017-90566-REDC. S.C.C is also supportedby the FPI grant BES-2016-076643. [1] P. F. de Salas et al. , “Status of neutrino oscillations 2018: 3 σ hint for normal mass ordering and improved CPsensitivity,” Phys. Lett.
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