Two-zero Textures of the Majorana Neutrino Mass Matrix and Current Experimental Tests
aa r X i v : . [ h e p - ph ] S e p Two-zero Textures of the Majorana Neutrino Mass Matrix andCurrent Experimental Tests
Harald Fritzsch a , Zhi-zhong Xing b ∗ , Shun Zhou c † a Physik-Department, Ludwig-Maximilians-Universit¨at, 80333 M¨unchen, Germany b Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China c Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut), 80805 M¨unchen, Germany
Abstract
In view of the latest T2K and MINOS neutrino oscillation data which hint at a rela-tively large θ , we perform a systematic study of the Majorana neutrino mass matrix M ν with two independent texture zeros. We show that three neutrino masses ( m , m , m )and three CP-violating phases ( δ, ρ, σ ) can fully be determined from two neutrino mass-squared differences ( δm , ∆ m ) and three flavor mixing angles ( θ , θ , θ ). We findthat seven patterns of M ν (i.e., A , , B , , , and C ) are compatible with current ex-perimental data at the 3 σ level, but the parameter space of each pattern is more strictlyconstrained than before. We demonstrate that the texture zeros of M ν are stable againstthe one-loop quantum corrections, and there exists a permutation symmetry betweenPatterns A and A , B and B or B and B . Phenomenological implications of M ν on the neutrinoless double-beta decay and leptonic CP violation are discussed, and arealization of those texture zeros by means of the Z n flavor symmetries is illustrated. PACS numbers: 14.60.Lm, 14.60.Pq ∗ E-mail: [email protected] † E-mail: [email protected] Introduction
Compelling evidence in favor of neutrino oscillations has been accumulated from a numberof solar, atmospheric, reactor and accelerator neutrino experiments since 1998 [1]. We arenow convinced that the three known neutrinos have nonzero and nondegenerate masses, andtheir flavor states can convert from one kind to another. In particular, two neutrino mass-squared differences ( δm , ∆ m ) and two flavor mixing angles ( θ , θ ) have been determinedto a reasonably good degree of accuracy from currently available experimental data. In spiteof this progress made in neutrino physics, our quantitative knowledge about the properties ofmassive neutrinos remain quite incomplete — for instance, the absolute mass scale of threeneutrinos, the sign of ∆ m and the smallest flavor mixing angle θ are still unknown. Althoughthe nature of massive neutrinos is also an open question, we assume them to be the Majoranaparticles. In this case the 3 × M ν is symmetric, and thecorresponding 3 × V contains three CP-violating phases ( δ, ρ, σ ) whichare entirely unrestricted by current experimental data.The T2K [2] and MINOS [3] accelerator neutrino experiments have recently announcedtheir preliminary data on the appearance ν µ → ν e oscillation, which hints at a relatively largevalue of θ . If this mixing angle is confirmed to be not very small by new data from these twoexperiments and from the upcoming reactor antineutrino experiments (e.g., the Double Choozexperiment in France [4], the Daya Bay experiment in China [5] and the RENO experiment inKorea [6]), it will be a great news to the long-baseline neutrino oscillation experiments whichaim at a determination of the sign of ∆ m and the magnitude of the Dirac CP-violating phase δ . The observed values of relevant neutrino flavor parameters may then allow us to reconstructthe Majorana neutrino mass matrix M ν in the flavor basis where the charged-lepton massmatrix M l is diagonal: M ν = V c M ν V T , where c M ν = Diag { m , m , m } with m i (for i = 1 , , V is just the 3 × M ν may have a few texture zerosas a natural consequence of an underlying flavor symmetry in a given model with or withoutthe seesaw mechanism. Such texture zeros are phenomenologically useful in the sense thatthey guarantee the calculability of M ν from which both the neutrino mass spectrum and theflavor mixing pattern can more or less be predicted. Note that texture zeros of a fermion massmatrix dynamically mean that the corresponding matrix elements are sufficiently suppressedin comparison with their neighboring counterparts [8], and they can help us to establish somesimple and testable relations between flavor mixing angles and fermion mass ratios — if suchrelations turn out to be favored by the experimental data, they might have a fundamentalreason and should originate from the underlying flavor theory. Hence a phenomenologicalstudy of possible texture zeros of M ν does make a lot of sense.As M ν is symmetric, it has six independent complex entries. If n of them are taken to be2anishing (i.e., M ν has n independent texture zeros), then we shall arrive at C n = 6! n ! (6 − n )! (1)different textures. It is easy to show that a texture of M ν with more than two independentzeros (i.e., n ≥
3) is definitely incompatible with current experimental data on neutrino massesand flavor mixing angles [9]. Hence a number of authors have paid particular interest to thetwo-zero textures of M ν [10]—[14] . There are totally fifteen two-zero textures of M ν , whichcan be classified into six categories: A : × × ×× × × , A : × × × × × × ; (2) B : × × × × × × , B : × × × ×× × , B : × × ×× × × , B : × × × × × × ; (3) C : × × ×× ×× × ; (4) D : × × ×× × × , D : × × ×× × × ; (5) E : × ×× ×× × × , E : × ×× × ×× × , E : × ×× × × × ; (6)and F : × × × × × , F : × × × × × , F : × × × ×
00 0 × , (7)in which each “ × ” denotes a nonzero matrix element. Previous analyses of these fifteen patterns(see, e.g., Ref. [13]) led us to the following conclusions: (1) seven patterns (i.e., A , , B , , , It is worth mentioning that the one-zero textures of M ν have much less predictability than the two-zerotextures of M ν , and their phenomenological implications have been discussed in Ref. [15]. C ) were phenomenologically favored; (2) two patterns (i.e., D , ) were only marginallyallowed; and (3) six patterns (i.e., E , , and F , , ) were ruled out. Taking account of the latestT2K and MINOS neutrino oscillation data, one may wonder whether the above conclusionsremain true or not. A fast check tells us that Patterns D , can be ruled out by today’sexperimental data at the 3 σ level, simply because θ > ◦ is no more favored. In addition,Patterns E , , and F , , remain phenomenologically disfavored. The question turns out to bewhether Patterns A , , B , , , and C can survive current experimental tests and coincide witha relatively large value of θ . The main purpose of this paper is just to answer this question.We aim to perform a very systematic study of the aforementioned seven patterns of M ν with two independent texture zeros based on a global fit of current neutrino oscillation datadone by Fogli et al [16] . Because a two-zero texture of M ν is very simple and can reveal thesalient phenomenological features of a given Majorana neutrino mass matrix, we intend to takesuch a systematic analysis as a good example to show how to test possible textures of M ν byconfronting them with more and more accurate experimental data. Hence this work is differentfrom any of the previous ones. In particular, almost all the physical consequences of Patterns A , , B , , , and C are explored in an analytical way in the present work; the stability oftexture zeros and the permutation symmetry between two similar patterns are discussed; thenumerical analysis is most updated and more complete; and a realization of texture zeros bymean of certain flavor symmetries is illustrated. We find that Patterns A , , B , , , and C of M ν can all survive current experimental tests at the 3 σ level, but we believe that some ofthem are likely to be ruled out in the near future.The remaining parts of this paper are organized as follows. In section 2 we show that the fullneutrino mass spectrum and two Majorana CP-violating phases ( ρ, σ ) can all be determinedin terms of three flavor mixing angles ( θ , θ , θ ) and the Dirac CP-violating phase ( δ ) for agiven two-zero texture of M ν . Section 3 is devoted to a complete analytical analysis of Patterns A , , B , , , and C of M ν , and to some discussions about the stability of texture zeros againstthe one-loop quantum corrections and the permutation symmetry between two similar patternsunder consideration. With the help of current neutrino oscillation data we perform a detailednumerical analysis of those typical patterns of M ν in section 4. Section 5 is devoted to somediscussions about how to realize texture zeros of M ν by means of certain flavor symmetries,and section 6 is a summary of this work together with some concluding remarks. In the flavor basis where the charged-lepton mass matrix M l is diagonal, the Majorana neutrinomass matrix M ν can be reconstructed in terms of three neutrino masses ( m , m , m ) and the Note that Schwetz et al have done another global fit of current data [17]. Although their best-fit results ofthree flavor mixing angles are slightly different from those obtained by Fogli et al [16], such differences becomeinsignificant at the 3 σ level. Therefore, we shall mainly use the 3 σ results of [16] in our numerical calculations. V . Namely, M ν = V m m
00 0 m V T . (8)It is convenient to express V as V = U P , where U denotes a 3 × θ , θ , θ ) and one Dirac CP-violating phase ( δ ), and P =Diag { e iρ , e iσ , } is a diagonal phase matrix containing two Majorana CP-violating phases ( ρ, σ ).More explicitly, we adopt the parametrization U = c c s c s − c s s − s c e − iδ − s s s + c c e − iδ s c − c c s + s s e − iδ − s c s − c s e − iδ c c , (9)where s ij ≡ sin θ ij and c ij ≡ cos θ ij (for ij = 12 , ,
13) are defined. The neutrino mass matrix M ν can equivalently be written as M ν = U λ λ
00 0 λ U T , (10)where λ = m e iρ , λ = m e iσ and λ = m . If two independent elements of M ν are vanishing(i.e., ( M ν ) ab = ( M ν ) αβ = 0 with ab = αβ ) as shown in Eqs. (2)—(7), one can obtain [11] λ λ = U a U b U α U β − U a U b U α U β U a U b U α U β − U a U b U α U β ,λ λ = U a U b U α U β − U a U b U α U β U a U b U α U β − U a U b U α U β , (11)from which two neutrino mass ratios are given by ξ ≡ m m = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U a U b U α U β − U a U b U α U β U a U b U α U β − U a U b U α U β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,ζ ≡ m m = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U a U b U α U β − U a U b U α U β U a U b U α U β − U a U b U α U β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (12)and two Majorana CP-violating phases turn out to be ρ = 12 arg " U a U b U α U β − U a U b U α U β U a U b U α U β − U a U b U α U β ,σ = 12 arg " U a U b U α U β − U a U b U α U β U a U b U α U β − U a U b U α U β . (13) Since we have assumed massive neutrinos to be the Majorana particles, there are nine physicalparameters: three neutrino masses ( m , m , m ), three flavor mixing angles ( θ , θ , θ ), and5hree CP-violating phases ( δ, ρ, σ ). By imposing two independent texture zeros on M ν , weobtain four constraint relations as given in Eqs. (12) and (13). With the help of currentexperimental data on three flavor mixing angles and two independent neutrino mass-squareddifferences, defined as [16] δm ≡ m − m , ∆ m = m − (cid:16) m + m (cid:17) , (14)one may determine or constrain both the neutrino mass spectrum and three CP-violatingphases through Eqs. (12) and (13).We take ( θ , θ , θ ) and ( δm , ∆ m ) as observables to see why the neutrino mass spectrumand three CP-violating phases can in principle be determined or constrained. First of all, notethat ξ = m /m and ζ = m /m are functions of the Dirac CP-violating phase δ as shown inEq. (12). Then it is possible to determine or constrain δ from the relation R ν ≡ δm | ∆ m | = 2 ( ζ − ξ ) | − ( ζ + ξ ) | . (15)Once δ is fixed, we can obtain ( ξ, ζ ) and ( ρ, σ ) from Eq. (12) and Eq. (13), respectively. Inaddition we have m = √ δm √ ζ − ξ , m = m ζ , m = m ξ . (16)Thus the neutrino mass spectrum is fully determined.Table 1 shows the global-fit values of ( θ , θ , θ ) and ( δm , ∆ m ) obtained by assumingCP conservation with cos δ = ± σ ranges: 0 . ≤ sin θ ≤ .
359 or 30 . ◦ ≤ θ ≤ . ◦ , . ≤ sin θ ≤ .
640 or 35 . ◦ ≤ θ ≤ . ◦ , . ≤ sin θ ≤ .
044 or 1 . ◦ ≤ θ ≤ . ◦ ; (17)and 6 . × − eV ≤ δm ≤ . × − eV , . × − eV ≤ | ∆ m | ≤ . × − eV . (18)In our numerical calculations we shall treat δ as an unconstrained parameter. Note that thesign of ∆ m remains unknown: ∆ m > m < M ν We first examine the stability of texture zeros of M ν against the one-loop quantum corrections.To be explicit, we consider the unique dimension-5 Weinberg operator of massive Majorana6able 1: The latest global-fit results of three neutrino mixing angles ( θ , θ , θ ) and twoneutrino mass-squared differences δm and ∆ m defined in Eq. (14). Here cos δ = ±
1, andthe old reactor antineutrino fluxes have been assumed [16].Parameter δm (10 − eV ) ∆ m (10 − eV ) θ θ θ Best fit 7 .
58 2 .
35 33 . ◦ . ◦ . ◦ σ range [7 . , .
80] [2 . , .
47] [32 . ◦ , . ◦ ] [38 . ◦ , . ◦ ] [6 . ◦ , . ◦ ]2 σ range [7 . , .
99] [2 . , .
57] [31 . ◦ , . ◦ ] [36 . ◦ , . ◦ ] [5 . ◦ , . ◦ ]3 σ range [6 . , .
18] [2 . , .
67] [30 . ◦ , . ◦ ] [35 . ◦ , . ◦ ] [1 . ◦ , . ◦ ]neutrinos in an effective field theory after the heavy degrees of freedom are integrated out [18]: L d=5 Λ = 12 κ αβ ℓ α L ˜ H ˜ H T ℓ cβ L + h . c . , (19)where Λ is the cutoff scale, ℓ L denotes the left-handed lepton doublet, ˜ H ≡ iσ H ∗ with H being the standard-model Higgs doublet, and κ stands for the effective neutrino couplingmatrix. After spontaneous gauge symmetry breaking, ˜ H gains its vacuum expectation value h ˜ H i = v/ √ v ≈
246 GeV. We are then left with the effective Majorana mass matrix M ν = κv / M ν = κ ( v sin β ) /
2, where tan β denotes the ratio of the vacuum expectation values of twoHiggs doublets. Eq. (19) or its supersymmetric counterpart can provide a way of generatingtiny neutrino masses. There are a number of interesting possibilities of building renormalizablegauge models to realize the effective Weinberg mass operator, such as the well-known seesawmechanisms at a superhigh energy scale Λ [19, 20, 21].The running of M ν from Λ to the electroweak scale µ ≃ M Z (or vice versa) is describedby the renormalization-group equations (RGEs) [22]. In the chosen flavor basis and at theone-loop level, M ν ( M Z ) and M ν (Λ) are related to each other via M ν ( M Z ) = I I e I µ
00 0 I τ M ν (Λ) I e I µ
00 0 I τ , (20)where the RGE evolution function I denotes the overall contribution from gauge and quarkYukawa couplings, and I α (for α = e, µ, τ ) stand for the contributions from charged-leptonYukawa couplings [23]. Because of I e < I µ < I τ as a consequence of m e ≪ m µ ≪ m τ , theycan modify the texture of M ν . In comparison, I = 1 only affects the overall mass scale of M ν . Note, however, that the texture zeros of M ν are stable against such quantum correctionsinduced by the one-loop RGEs. Taking Pattern A of M ν for example, we have M A ν (Λ) = a b ca c d (21)7t Λ, and thus M A ν ( M Z ) = I aI e I τ bI µ cI µ I τ aI e I τ cI µ I τ dI τ (22)at M Z . This interesting feature implies that the important relations obtained in Eqs. (12)and (13) formally hold both at Λ and M Z . In other words, if a seesaw or flavor symmetrymodel predicts a two-zero texture of M ν at Λ, one may simply study its phenomenologicalconsequences at M Z by taking account of the same texture zeros. While the values of neutrinomasses and flavor mixing parameters at M Z turn out to be different from those at Λ, theircorrelations dictated by the texture zeros keep unchanged at any scale between Λ and M Z . As pointed out in Refs. [10]—[14], the seven viable two-zero textures of M ν can be classifiedinto three distinct categories: (1) A and A ; (2) B , B , B and B ; and (3) C . Thephenomenological implications of those patterns in the same category have been found to bealmost indistinguishable. Now we show that there exists a permutation symmetry betweenPatterns A and A , B and B , or B and B . This observation may help understand theirsimilarities in model building and phenomenology.Let us take Patterns A and A in Eq. (2) for example. Note that the location of texturezeros in A can be changed to that in A by a permutation in the 2-3 rows and 2-3 columns.To be explicit, we define the elementary transformation matrix P = . (23)Then the Majorana neutrino mass matrix M A ν can be constructed from M A ν via M A ν = P M A ν P T23 . (24)If M A ν is diagonalized by a unitary matrix V = U P like Eq. (8), then Eq. (24) tells us that M A ν can be diagonalized by the unitary matrix ˜ V = ˜ U P with ˜ U = P U . Parametrizing˜ U in terms of three rotation angles (˜ θ , ˜ θ , ˜ θ ) and one CP-violating phase ˜ δ , just like theparametrization of U in Eq. (9), we immediately obtain the relations˜ θ = θ , ˜ θ = θ , ˜ θ = π − θ , ˜ δ = δ − π . (25)In addition, M A ν and M A ν have the same eigenvalues λ i (for i = 1 , , A can therefore be obtained from those of Pattern A by implementingthe replacements in Eq. (25). It is easy to show that a similar permutation symmetry holdsbetween Patterns B and B or between Patterns B and B .8 .3 Analytical approximations Thanks to the permutation symmetry discussed above, it is only necessary to study Patterns A , B , B and C in detail. The analytical results for A , B and B can be obtained,respectively, from those for A , B and B with the replacements θ → π/ − θ and δ → δ − π . In this subsection we explore the analytical relations among three neutrino masses( m , m , m ), three flavor mixing angles ( θ , θ , θ ) and three CP-violating phases ( δ, ρ, σ ) foreach pattern of M ν by means of Eqs. (11)—(16). The effective mass term of the neutrinolessdouble-beta decay, defined as h m i ee ≡ | ( M ν ) ee | , will also be discussed. • Pattern A with ( M ν ) ee = ( M ν ) eµ = 0. With the help of Eq. (11), we obtain [11] λ λ = + s c s s c c e iδ − s ! ,λ λ = − s c c s s c e iδ + s ! . (26)Since s ≪ ξ = m m ≈ tan θ tan θ sin θ ,ζ = m m ≈ cot θ tan θ sin θ ; (27)and ρ ≈ δ ,σ ≈ δ − π . (28)Without loss of generality, we choose three neutrino mixing angles to lie in the firstquadrat and allow three CP-violating phases to vary in the ranges ρ ∈ [ − π/ , + π/ σ ∈ [ − π/ , + π/
2] and δ ∈ [0 , π ]. Given 0 . ≤ tan θ ≤ .
75, 0 . ≤ tan θ ≤ . . ≤ sin θ ≤ .
21 at the 3 σ level, Eq. (27) yields ξ < ζ <
1. So the normal neutrinomass hierarchy m < m < m follows and ∆ m > δ can completely be fixed if three neutrino mixing angles andtwo neutrino mass-squared differences are known. In the lowest order of s , however, R ν = δm ∆ m ≈ θ sin θ sin 2 θ tan 2 θ , (29)which is actually independent of δ . So we have to go beyond the leading order approxi-mation. To the next-to-leading order we obtain ξ = m m ≈ tan θ tan θ sin θ q − θ cot θ sin θ cos δ ,ζ = m m ≈ cot θ tan θ sin θ q θ cot θ sin θ cos δ ; (30)9nd R ν ≈ ζ − ξ = 4 tan θ sin θ sin 2 θ tan 2 θ (1 + tan 2 θ cot θ sin θ cos δ ) . (31)Finally we arrive at δ ≈ cos − " + tan θ tan 2 θ sin θ sin 2 θ tan 2 θ R ν θ sin θ − ! . (32)Taking the best-fit values of the three neutrino mixing angles (i.e., θ = 33 . ◦ , θ =40 . ◦ and θ = 8 . ◦ ) together with those of two neutrino mass-squared differences (i.e., δm = 7 . × − eV and ∆ m = 2 . × − eV ) from Table 1, one immediatelyobtains the neutrino mass spectrum m ≈ √ ∆ m = 4 . × − eV ,m ≈ m cot θ tan θ sin θ = 8 . × − eV ,m ≈ m tan θ tan θ sin θ = 3 . × − eV ; (33)and the CP-violating phases δ ≈ ◦ , ρ ≈ ◦ and σ ≈ − ◦ . Since ( M ν ) ee = 0 holdsfor Pattern A of M ν , the effective mass h m i ee of the neutrinoless double-beta decay isdefinitely vanishing. • Pattern A with ( M ν ) ee = ( M ν ) eτ = 0. As pointed out in section 3.2, all the analyticalresults of Pattern A can be obtained from those of Pattern A with the replacements θ → π/ − θ and δ → δ − π . So it is straightforward to have ξ = m m ≈ tan θ cot θ sin θ q θ tan θ sin θ cos δ ,ζ = m m ≈ cot θ cot θ sin θ q − θ tan θ sin θ cos δ ; (34)and δ ≈ cos − " − cot θ tan 2 θ sin θ sin 2 θ tan 2 θ R ν θ sin θ − ! . (35)In addition, ρ ≈ δ − π ,σ ≈ δ m ≈ √ ∆ m ,m ≈ m cot θ cot θ sin θ ,m ≈ m tan θ cot θ sin θ . (37) Note again that such best-fit values have been obtained in the assumption of cos δ = ± δ . A , h m i ee = | ( M ν ) ee | = 0 holds in Pattern A . Providedthe same best-fit values of δm , ∆ m , θ , θ and θ are taken, one can easily verifythat cos δ > A with currentexperimental data. When the uncertainties of those neutrino mixing parameters aretaken into account (e.g., at the 3 σ level), however, we actually find no problem withPattern A (see the numerical analysis in section 4). • Pattern B with ( M ν ) µµ = ( M ν ) eτ = 0. With the help of Eq. (11), we obtain λ λ = s c s (2 c s − s c ) − c s ( s s e + iδ + c c e − iδ ) s c s c + ( s − c ) c s e iδ + s c s s (1 + c ) e iδ e iδ ,λ λ = s c s (2 c s − s c ) + c s ( c s e + iδ + s c e − iδ ) s c s c + ( s − c ) c s e iδ + s c s s (1 + c ) e iδ e iδ . (38)In the leading order approximation, ξ = m m ≈ tan θ ,ζ = m m ≈ tan θ ; (39)and ρ ≈ σ ≈ δ − π . (40)In the next-to-leading order approximation, we find m m − m m ≈ + 4 sin θ cos δ sin 2 θ sin 2 θ ,ρ − σ ≈ − θ sin δ sin 2 θ tan 2 θ tan θ . (41)The Dirac CP-violating phase δ can be determined from R ν ≈ θ sin 2 θ | tan 2 θ cos δ | . (42)Since δm > m > m , we have cos δ < δ ≈ ◦ and thus ρ ≈ σ ≈ ◦ . Eq. (41) tells us that thedifference between m /m and m /m is about 0.01, and that between ρ and σ is about4 ◦ . On the other hand, the neutrino mass spectrum is given by m ≈ s ∆ m − tan θ ≈ . × − eV ,m ≈ m ≈ m tan θ ≈ . × − eV ; (43)and the effective mass term of the neutrinoless double-beta decay turns out to be h m i ee ≈ m tan θ ≈ . × − eV. Note that we have input the best-fit value θ = 40 . ◦ [16],11o the normal neutrino mass hierarchy m > m > m is allowed. If the 3 σ range of θ istaken, however, both normal and inverted mass hierarchies are likely. When tan θ > σ or 3 σ level [16], we have the inverted neutrino mass hierarchywith ∆ m <
0. In short, the analytical formulas for m , m and m in Eq. (43) arevalid no matter which mass hierarchy is taken, but the corresponding numerical resultsdepend on the input value of θ . • Pattern B with ( M ν ) ττ = ( M ν ) eµ = 0. By using the permutation symmetry discussedin section 3.2, we obtain ξ = m m ≈ cot θ ,ζ = m m ≈ cot θ ; (44)and ρ ≈ σ ≈ δ − π π for ρ and σ , soEq. (45) is identical to Eq. (40) even if the replacement δ → δ − π is made. In thenext-to-leading order approximation, we have m m − m m ≈ − θ cos δ sin 2 θ sin 2 θ ,ρ − σ ≈ − θ sin δ sin 2 θ tan 2 θ cot θ . (46)The Dirac CP-violating phase is also determined by Eq. (42), because | tan 2 θ | keepsinvariant under the replacement θ → π/ − θ . Now we have cos δ > m > m . So δ ≈ ◦ and ρ ≈ σ ≈ − ◦ . Eq. (46) tells us that the differencebetween m /m and m /m is about 0.01, and that between ρ and σ is about 2 ◦ . Theneutrino mass spectrum turns out to be m ≈ s ∆ m − cot θ ≈ . × − eV ,m ≈ m ≈ m cot θ ≈ . × − eV . (47)In addition, h m i ee ≈ m cot θ ≈ . × − eV. Note that we have input the best-fitvalue θ = 40 . ◦ [16], so the inverted mass hierarchy m > m > m appears. If the3 σ range of θ is taken, however, both normal and inverted mass hierarchies are likely.Hence it is experimentally important to determine the deviation of θ from π/ • Pattern B with ( M ν ) µµ = ( M ν ) eµ = 0. With the help of Eq. (11), we obtain λ λ = − s c · s s − c c s e − iδ s c + c s s e + iδ e iδ ,λ λ = − s c · c s + s c s e − iδ c c − s s s e + iδ e iδ . (48)12n the leading order approximation, ξ = m m ≈ tan θ ,ζ = m m ≈ tan θ ; (49)and ρ ≈ σ ≈ δ − π . (50)In the next-to-leading order approximation, we have m m − m m ≈ − θ sin θ cos δ sin 2 θ sin 2 θ ,ρ − σ ≈ + 2 sin θ sin δ sin 2 θ tan 2 θ . (51)The Dirac CP-violating phase δ can be determined from R ν ≈ θ sin 2 θ tan θ | tan 2 θ cos δ | . (52)The condition m > m leads to cos δ >
0. Taking the best-fit values [16], we obtain δ ≈ ◦ and ρ ≈ σ ≈ − ◦ from Eq. (51). The difference between m /m and m /m is about 0.008, and that between ρ and σ is about 3 ◦ . On the other hand, the neutrinomass spectrum is given by m ≈ s ∆ m − tan θ ≈ . × − eV ,m ≈ m ≈ m tan θ ≈ . × − eV . (53)The effective mass term of the neutrinoless double-beta decay turns out to be h m i ee ≈ m tan θ ≈ . × − eV. Note that the phenomenology of Pattern B is essentiallythe same as that of Pattern B except for the Dirac CP-violating phase δ . Note also thatwe have used the best-fit value θ = 40 . ◦ [16], so the normal neutrino mass hierarchy m > m > m is allowed. If the 3 σ range of θ is taken, however, both normal andinverted neutrino mass hierarchies are likely. • Pattern B with ( M ν ) ττ = ( M ν ) eτ = 0. By using the permutation symmetry discussedin section 3.2, we obtain ξ = m m ≈ cot θ ,ζ = m m ≈ cot θ ; (54)and ρ ≈ σ ≈ δ − π m m − m m ≈ + 4 cot θ sin θ cos δ sin 2 θ sin 2 θ ,ρ − σ ≈ + 2 sin θ sin δ sin 2 θ tan 2 θ . (56)The Dirac CP-violating phase is given by R ν ≈ θ sin 2 θ cot θ | tan 2 θ cos δ | . (57)We find cos δ < m > m . Hence δ ≈ ◦ and ρ ≈ σ ≈ ◦ . Eq. (56)tells us that the difference between m /m and m /m is about 0.015, and that between ρ and σ is about 3 ◦ . The neutrino mass spectrum turns out to be m ≈ s ∆ m − cot θ ≈ . × − eV ,m ≈ m ≈ m cot θ ≈ . × − eV . (58)In addition, h m i ee ≈ m cot θ ≈ . × − eV. Note that we have used the best-fitvalue θ = 40 . ◦ [16], so the inverted neutrino mass hierarchy m > m > m appears.If the 3 σ range of θ is taken, however, both normal and inverted mass hierarchies arelikely. It is obvious that a precise determination of δ and the neutrino mass hierarchy iscrucial to pin down one of the four patterns B , B , B and B . All of them predict anearly degenerate neutrino mass spectrum. • Pattern C with ( M ν ) µµ = ( M ν ) ττ = 0. With the help of Eq. (11), we obtain λ λ = c c s · − c ( s − c ) − s s c s e iδ s c s c − ( s − c )( s − c ) s e iδ + 2 s c s c s e iδ e iδ ,λ λ = s c s · + s ( s − c ) − c s c s e iδ s c s c − ( s − c )( s − c ) s e iδ + 2 s c s c s e iδ e iδ . (59)To the lowest order, ξ = m m ≈ vuut − θ cos δ tan 2 θ sin θ + cot θ tan θ sin θ ,ζ = m m ≈ vuut θ cos δ tan 2 θ sin θ + tan θ tan θ sin θ . (60)Since m > m , Eq. (60) implies tan 2 θ cos δ > m > m . Furthermore, one canverify that tan 2 θ tan 2 θ sin θ cos δ > m > m . Henceonly the inverted mass hierarchy m > m > m is allowed. The Dirac CP-violatingphase δ can be determined from R ν ≈ θ tan θ ) (tan 2 θ tan 2 θ sin θ cos δ − θ tan θ tan 2 θ (tan θ + 2 tan 2 θ sin θ cos δ ) . (61)14aking the best-fit values of three neutrino mixing angles and two neutrino mass-squareddifferences, we obtain δ ≈ ◦ from Eq. (61). The Majorana CP-violating phases ρ and σ turn out to be ρ ≈ δ + 12 tan − " cot θ sin δ tan 2 θ sin θ − cot θ cos δ − π ≈ +13 ◦ ,σ ≈ δ −
12 tan − " tan θ sin δ tan 2 θ sin θ + tan θ cos δ − π ≈ − ◦ . (62)Finally we obtain m ≈ vuut tan θ cot θ sin θ ∆ m θ tan 2 θ sin θ cos δ ≈ . × − eV ,m ≈ m vuut θ cos δ tan 2 θ sin θ + tan θ tan θ sin θ ≈ . × − eV ,m ≈ m vuut − θ cos δ tan 2 θ sin θ + cot θ tan θ sin θ ≈ . × − eV (63)together with h m i ee ≈ m vuut − θ cos δ tan 2 θ sin θ + 4 cot θ tan θ sin θ ≈ . × − eV . (64)Note that both tan 2 θ > δ > θ < δ < σ range of θ is taken into account. Nevertheless, the inverted mass hierarchy m The analytical results obtained above show that the neutrino mass hierarchy is actually relatedto the flavor mixing angle θ in Patterns B , B , B and B of M ν . In particular, it dependson whether θ > ◦ or θ < ◦ . According to Table 1 [16], only θ ≤ ◦ is allowedat the 1 σ level . If a two-zero texture of M ν can accommodate both normal and invertedmass hierarchies, we shall only concentrate on the one dictated by θ < ◦ in our numericalanalysis, because the other possibility is just an opposite and trivial exercise. We have noticedthat the Dirac CP-violating phase δ should be close to π/ π/ B , , , , and thedifferences between two Majorana CP-violating phases ρ and σ in these two cases are distinct.For illustration, we shall only focus on the range of δ around π/ M ν we generate a set of random numbers of ( θ , θ , θ )and ( δm , ∆ m ) lying in their 3 σ ranges given by Eqs. (17) and (18) together with arandom value of δ in the range δ ∈ [0 , π ].2. Given the above random numbers, it is possible to calculate other physical parametersof M ν , including three neutrino mass eigenvalues ( m , m , m ), two Majorana-type CP-violating phases ( ρ, σ ), the effective mass of the neutrinoless double-beta decay h m i ee and the Jarlskog invariant of leptonic CP violation J CP = s c s c s c sin δ . Tojudge whether a pattern of M ν is consistent with current experimental data or not, werequire that the consistency conditions should be satisfied: (a) because of δm > 0, werequire m > m or equivalently ζ − ξ > 0; (b) since only the neutrino mass hierarchies m > m > m and m > m > m are phenomenologically allowed, we further require( ζ − ξ − > ξ > ζ > ξ < ζ < δ is actually fixed by Eq. (15). Instead of solving δ , we require that Eq. (15)should be satisfied up to a reasonable degree of precision (e.g., 10 − ).3. We consider all the points satisfying the consistency conditions, and then have a nine-dimensional parameter space spanned by nine quantities ( θ , θ , θ , δ, ρ, σ, m , m , m ).The low-energy observables such as J CP and h m i ee can accordingly be calculated. Topresent the final results in a simple and clear way, we restrict ourselves to the two-dimensional parameter space and set the x -axis to be the Dirac CP-violating phase δ .Therefore, what we actually show are the allowed ranges of relevant physical parametersof each pattern of M ν .4. Corresponding to the allowed ranges of θ , θ and θ changing with δ , their histogramsare also plotted because they can signify the most probable values of three flavor mixing Note that θ ≥ ◦ seems to be favored at the 1 σ level in the global analysis done by Schwetz et al [17]. M ν is phenomenologically more favored.We stress that the strategy of our numerical analysis can also apply to other textures of theMajorana neutrino mass matrix M ν . The two-zero patterns under discussion will serve as agood example to illustrate this strategy.Our numerical results are presented in Figs. 1—14. Comments and discussions follow. • Pattern A . Fig. 1 tells us that the neutrino mixing angles, in particular θ and θ , are actually insensitive to the Dirac CP-violating phase δ . This point can easily beunderstood with the help of Eq. (31) or Eq. (32), in which δ is only loosely related tothe ratio of two neutrino mass-squared differences R ν due to the smallness of θ . Aninteresting observation from the left panel of Fig. 1 is, that a relatively large value of θ (i.e., θ ≈ ◦ · · · ◦ ), which is quite close to the best-fit value θ = 8 . ◦ [16], is particularlyfavored. On the other hand, the neutrino mass spectrum is weakly hierarchical, as shownin Fig. 2. The dependence of m and m on δ is ascribed to the next-to-leading ordercorrections given in Eq. (30). We have also illustrated the numerical prediction for J CP inFig. 2. One can see the maximal value of J CP is at the percent level and should be able tolead to observable effects of CP violation in a variety of long-baseline neutrino oscillationexperiments. Because δ itself is essentially unconstrained by current experimental dataat the 3 σ level, ρ and σ turn out to be arbitrary as shown in Fig. 2, although theircorrelations with δ are rather sharp. Finally we remark that Pattern A of M ν predicts h m i ee = 0 for the neutrinoless double-beta decay. • Pattern A . As shown in Figs. 3 and 4, the phenomenological implications of Pattern A of M ν are essentially the same as those of Pattern A . For instance, θ ≈ ◦ · · · ◦ is favored and the effective mass term h m i ee is vanishing. Thus it is only necessaryto emphasize their main difference. We have demonstrated a permutation symmetrybetween M A ν and M A ν in section 3.2 and found that the present best-fit values ofneutrino mixing parameters (mainly θ = 40 . ◦ [16]) cannot coincide with Pattern A .If the maximal mixing angle θ = 45 ◦ were finally established, however, it would bealmost impossible to distinguish between Patterns A and A in practice. • Pattern B . Since m > m , we have cos δ < π/ < δ < π/ 2) in this case.In our numerical analysis we only focus on the range δ ∈ [ π/ , π ], because the range δ ∈ [ π, π/ 2] can similarly be analyzed. As shown in Figs. 5 and 6, only a very narrowregion δ ∈ [0 . π, . π ] is phenomenologically allowed. This result obviously originatesfrom Eq. (42), where | cos δ | must be small enough to suppress the magnitude of R ν .Furthermore, we only consider the normal neutrino mass hierarchy corresponding to θ < ◦ . The latter seems to be favored by current data at the 1 σ level [16]. Fig. 5shows that θ ∼ ◦ and θ ∼ ◦ are more likely. A strong correlation between θ and δ θ ≈ ◦ requires themaximal CP-violating phase δ ≈ π/ 2. In addition, a nearly degenerate mass spectrumas shown in Fig. 6 is predicted. There is a lower bound on h m i ee (i.e., h m i ee ≥ . 03 eV),which will be tested in the future experiments of the neutrinoless double-beta decay.Note that h m i ee ≈ . θ ≈ ◦ and δ ≈ π/ 2. The other threepatterns of this category (i.e., B , B and B ) have similar consequences, for which thenumerical results have been given in Figs. 7 and 8, Figs. 9 and 10, and Figs. 11 and 12,respectively. But the details of these patterns, such as the allowed ranges of ( ρ, σ, δ ) and( m , m , m ), are somewhat different. • Pattern C. Fig. 13 shows no significant preference in the allowed ranges of threeneutrino mixing angles. A strong correlation between θ and δ only appears when δ isclose to π/ 2. As shown in Fig. 14 and discussed in section 3.3, the inverted neutrino masshierarchy is allowed in most parts of the parameter space (and a normal mass hierarchyis possible only when θ is very close to 45 ◦ and θ is nonvanishing). Like Patterns B , , , , there is a lower bound on h m i ee in Pattern C (i.e., h m i ee > . 02 eV), and itsmaximal value can saturate the present experimental upper bound h m i ee < . P m i < . 58 eV at the 95% confidence level [25], which has been derived from theseven-year WMAP data on the cosmic background radiation combined with the BaryonAcoustic Oscillations. Therefore, the possibility of δ ∼ π/ B , , , .Finally it is worth pointing out that we have also done a numerical analysis of the two-zerotextures of M ν by using the 1 σ and 2 σ values of δm , ∆ m , θ , θ and θ (see Table 1and Ref. [16]). We find that all the seven patterns discussed above are compatible withcurrent experimental data at the 1 σ or 2 σ level, although the corresponding parameter spaceis somewhat smaller. To be conservative, we take our numerical results obtained at the 3 σ level more seriously. In general, the texture zeros of a Majorana neutrino mass matrix can be realized in variousseesaw models with proper discrete flavor symmetries. It is possible to obtain the zeros inarbitrary entries of a fermion mass matrix by means of the Abelian symmetries (e.g., thecyclic group Z n [26]). To illustrate how to realize the two-zero textures of M ν discussed above,we shall work in the type-II seesaw model, which extends the scalar sector of the standardmodel with one or more SU (2) L scalar triplets [20]. For N scalar triplets, the gauge-invariantLagrangian relevant for neutrino masses reads −L ∆ = 12 X j X α,β (cid:16) Y ∆ j (cid:17) αβ ℓ α L ∆ j iσ ℓ cβ L + h . c . , (66)18here α and β run over e , µ and τ , ∆ j denotes the j -th triplet scalar field (for j = 1 , , · · · , N ),and Y ∆ j is the corresponding Yukawa coupling matrix. After the triplet scalar acquires itsvacuum expectation value h ∆ j i ≡ v ∆ j , the Majorana neutrino mass matrix is given by M ν = X j Y ∆ j v ∆ j , (67)where the smallness of v ∆ j is attributed to the largeness of the triplet scalar mass scale [20].In order to generate the texture zeros in M ν and derive the seven viable patterns A , , B , , , and C , we follow the spirit of Ref. [27] and impose the Z n symmetry on the Lagrangianin Eq. (66). The unique generator of the cyclic group Z n is ̟ = e i π/n , which produces all thegroup elements Z n = { , ̟, ̟ , · · · , ̟ n − } . Now it is straightforward to specify the number ofscalar triplets N (i.e., j = 1 , , · · · , N ), the order of the cyclic group n and the representationsof ℓ α and ∆ j under the symmetry group. • N = 3 and n = 6 for A , and B , . In this case we have to introduce three scalar tripletsand the Z symmetry group, for which the generator is ̟ = e iπ/ . The representationsof lepton doublets ℓ α L (for α = e, µ, τ ) and the scalar triplets ∆ j (for j = 1 , , 3) areassigned as follows.1. For Pattern A : ℓ e L → ̟ ℓ e L , ℓ µ L → ℓ µ L , ℓ τ L → ̟ ℓ τ L , ∆ → ∆ , ∆ → ̟ ∆ , ∆ → ̟ ∆ . (68)Given the above representations of lepton doublets, three scalar triplets are neededto enforce the nonzero elements in the neutrino mass matrix: ∆ for ( M ν ) µµ and( M ν ) ττ , ∆ for ( M ν ) µτ and ∆ for ( M ν ) eτ .2. For Pattern A : ℓ e L → ̟ ℓ e L , ℓ µ L → ℓ µ L , ℓ τ L → ̟ ℓ τ L , ∆ → ∆ , ∆ → ̟ ∆ , ∆ → ̟ ∆ . (69)Note that Eq. (69) differs from Eq. (68) only in the assignment for the triplet ∆ .Such a difference originates from the fact that ( M ν ) eµ = 0 and ( M ν ) eτ = 0 hold forPattern A , while ( M ν ) eτ = 0 and ( M ν ) eµ = 0 hold for Pattern A . It is worthwhileto point out that the assignments in Eq. (69) are by no means unique. As we havediscussed in section 3.2, there exists a permutation symmetry between A and A .Therefore, we may exchange the representations of ℓ µ L and ℓ τ L in Eq. (68) butpreserve those of scalar triplets to obtain A from A .3. For Pattern B : ℓ e L → ℓ e L , ℓ µ L → ̟ ℓ µ L , ℓ τ L → ̟ ℓ τ L , ∆ → ∆ , ∆ → ̟ ∆ , ∆ → ̟ ∆ . (70)19. For Pattern B : ℓ e L → ̟ ℓ e L , ℓ µ L → ℓ µ L , ℓ τ L → ̟ ℓ τ L , ∆ → ∆ , ∆ → ̟ ∆ , ∆ → ̟ ∆ . (71)In addition to the permutation symmetry P discussed in section 3.2, we noticethat the location of texture zeros in Pattern B can be obtained from that inPattern A by a permutation in the 1-2 rows and 1-2 columns. Similarly thereexists a 1-3 permutation symmetry between Pattern B and Pattern A . Althoughthese symmetries cannot lead to simple relations between any two of the neutrinomixing parameters, they are instructive for the assignments of lepton doublets. Forinstance, Eq. (70) and Eq. (71) can be derived from Eq. (68) and Eq. (69) byexchanging the representations of ℓ e L and ℓ µ L and the representations of ℓ e L and ℓ τ L ,respectively. The assignments of scalar triplets should not be changed. • N = 2 and n = 3 for B , . In this case we introduce only two scalar triplets and the Z symmetry group, for which the generator is ̟ = e iπ/ . The assignments of leptondoublets ℓ α L (for α = e, µ, τ ) and the scalar triplets ∆ j (for j = 1 , 2) are as follows.1. For Pattern B : ℓ e L → ℓ e L , ℓ µ L → ̟ℓ µ L , ℓ τ L → ̟ ℓ τ L , ∆ → ∆ , ∆ → ̟ ∆ . (72)2. For Pattern B : ℓ e L → ℓ e L , ℓ µ L → ̟ ℓ µ L , ℓ τ L → ̟ℓ τ L , ∆ → ∆ , ∆ → ̟ ∆ . (73)Note that the assignment in Eq. (73) is slightly different from that in Ref. [27],where ∆ → ̟ ∆ is taken and the representations of lepton doublets are the sameas in Eq. (72). Here we have implemented the 2-3 permutation symmetry betweenPattern B and Pattern B . • N = 3 and n = 4 for Pattern C . In this case we have to introduce three scalar tripletsand the Z symmetry group, for which the generator is ̟ = e iπ/ . The assignments oflepton doublets ℓ α L (for α = e, µ, τ ) and the scalar triplets ∆ j (for j = 1 , , 3) are ℓ e L → ℓ e L , ℓ µ L → ̟ℓ µ L , ℓ τ L → ̟ ℓ τ L , ∆ → ∆ , ∆ → ̟ ∆ , ∆ → ̟ ∆ . (74)Thus all the seven two-zero patterns of M ν can be obtained in this simple symmetry scheme.In all cases we have taken the right-handed charged-lepton singlets E α R to transform in thesame way as the left-handed lepton doublets ℓ α L and taken the standard-model Higgs doublet H to be in the trivial representation. Hence the charged-lepton mass matrix M l is diagonal,as we have chosen from the beginning. The two-zero textures of M ν can also be realized in theseesaw models with three right-handed neutrinos, several Higgs singlets, doublets and triplets,by imposing either Abelian or non-Abelian discrete flavor symmetries [26, 27, 28].20 Summary In view of the latest T2K and MINOS neutrino oscillation data which hint at a relatively largevalue of θ , we have performed a systematic study of the Majorana neutrino mass matrix M ν with two independent texture zeros. It turns out that seven patterns (i.e., A , , B , , , and C ) can survive current experimental tests at the 3 σ level, although they are also compatiblewith the data at the 1 σ or 2 σ level. The following is a brief summary of our main observations: • Given the values of three flavor mixing angles ( θ , θ , θ ) and two neutrino mass-squareddifferences ( δm , ∆ m ), it is in principle possible to fully determine three CP-violatingphases ( δ, ρ, σ ) and three neutrino masses ( m , m , m ). The analytical formulas for thelatter have been derived and listed in Tables 2—4. • By making the analytical approximations and taking the best-fit values of neutrino mixingparameters [16], we find that only Pattern A can be excluded. We have numericallyconfirmed that all the seven patterns of M ν (i.e., A , , B , , , and C ) are compatiblewith current neutrino oscillation data at the 1 σ level, but our numerical results have beenpresented only at the more conservative 3 σ level. • Figs. 1—14 show the main numerical results of our systematic analysis. Some interestingpoints should be emphasized. (1) Both A and A favor a relatively large θ (e.g., θ ∼ ◦ ), B , , , prefer a relatively small θ (e.g., θ ∼ ◦ ), and C shows no significantpreference for the magnitude of θ . (2) The Dirac CP-violating phase δ obtained from B , , , lies in a narrow range around π/ π/ 2, and δ = π/ θ = π/ 4. (3) For δ → π/ θ → π/ 4, the predictions for neutrino masses m i and the effective neutrino mass h m i ee may run into contradiction with their upper boundsset by the cosmological observations and the neutrinoless double-beta decay experiments.(4) The size of J CP may reach the percent level and thus appreciable leptonic CP violationis possible to show up in the future long-baseline neutrino oscillation experiments.In addition we have shown that the texture zeros of the Majorana neutrino mass matrix M ν are stable against the one-loop quantum corrections , and pointed out that there exists apermutation symmetry between A and A , B and B or B and B . In the type-II seesawmodel with two or three scalar triplets we have illustrated how to realize two-zero textures of M ν by using the Z n flavor symmetry.The ongoing and upcoming neutrino oscillation experiments are expected to measure theneutrino mixing parameters, in particular the smallest mixing angle θ , the deviation of θ from π/ δ . The sensitivity of future cosmological observa-tions to the sum of neutrino masses P m i and the sensitivity of the neutrinoless double-betadecay experiments to the effective mass term h m i ee will probably reach ∼ . 05 eV in the near In contrast, the texture zeros of the Dirac neutrino mass matrix are essentially sensitive to quantumcorrections like those of quark mass matrices [29]. M ν might be ex-cluded or only marginally allowed by tomorrow’s data, and those capable of surviving shouldshed light on the underlying flavor structures of massive neutrinos. Acknowledgements This work was supported in part by the National Natural Science Foundation of Chinaunder grant No. 10875131 (Z.Z.X.) and by the Alexander von Humboldt Foundation (S.Z.).22 eferences [1] Particle Data Group, K. Nakamura et al. , J. Phys. G , 075021 (2010).[2] T2K Collaboration, K. Abe et al. , Phys. Rev. Lett. , 041801 (2011).[3] MINOS Collaboration, P. 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Rodejohann, JHEP , 034 (2005).24able 2: Seven viable patterns of the neutrino mass matrix M ν with two texture zeros, andtheir predictions for three CP-violating phases ( δ, ρ, σ ).Pattern Texture of M ν CP-violating phases A × × ×× × × δ ≈ cos − " + tan θ tan 2 θ sin θ sin 2 θ tan 2 θ R ν θ sin θ − ! ρ ≈ δ , σ ≈ δ − π A × × × × × × δ ≈ cos − " − cot θ tan 2 θ sin θ sin 2 θ tan 2 θ R ν θ sin θ − ! ρ ≈ δ − π , σ ≈ δ B × × × × × × δ ≈ cos − " − sin 2 θ R ν θ | tan 2 θ | ρ ≈ σ ≈ δ − π , ρ − σ ≈ − θ sin δ sin 2 θ tan 2 θ tan θ B × × × ×× × δ ≈ cos − " + sin 2 θ R ν θ | tan 2 θ | ρ ≈ σ ≈ δ − π , ρ − σ ≈ − θ sin δ sin 2 θ tan 2 θ cot θ B × × ×× × × δ ≈ cos − " + sin 2 θ cot θ R ν θ | tan 2 θ | ρ ≈ σ ≈ δ − π , ρ − σ ≈ + 2 sin θ sin δ sin 2 θ tan 2 θ B × × × × × × δ ≈ cos − " − sin 2 θ tan θ R ν θ | tan 2 θ | ρ ≈ σ ≈ δ − π , ρ − σ ≈ + 2 sin θ sin δ sin 2 θ tan 2 θ C × × ×× ×× × δ ≈ θ tan θ ) + tan θ tan 2 θ tan θ R ν [1 + (1 − R ν ) tan θ tan θ ] tan 2 θ tan 2 θ sin θ ρ ≈ δ + 12 tan − " cot θ sin δ tan 2 θ sin θ − cot θ cos δ − π σ ≈ δ − 12 tan − " tan θ sin δ tan 2 θ sin θ + tan θ cos δ − π M ν with two texture zeros, andtheir predictions for two neutrino mass ratios ξ ≡ m /m and ζ ≡ m /m .Pattern Texture of M ν Neutrino mass ratios A × × ×× × × ξ ≈ tan θ tan θ sin θ , ζ ≈ cot θ tan θ sin θ A × × × × × × ξ ≈ tan θ cot θ sin θ , ζ ≈ cot θ cot θ sin θ B × × × × × × ξ ≈ ζ ≈ tan θ , ξ − ζ ≈ + 4 sin θ cos δ sin 2 θ sin 2 θ B × × × ×× × ξ ≈ ζ ≈ cot θ , ξ − ζ ≈ − θ cos δ sin 2 θ sin 2 θ B × × ×× × × ξ ≈ ζ ≈ tan θ , ξ − ζ ≈ − θ sin θ cos δ sin 2 θ sin 2 θ B × × × × × × ξ ≈ ζ ≈ cot θ , ξ − ζ ≈ + 4 cot θ sin θ cos δ sin 2 θ sin 2 θ C × × ×× ×× × ξ ≈ − θ cos δ tan 2 θ sin θ + cot θ tan θ sin θ ! / ζ ≈ θ cos δ tan 2 θ sin θ + tan θ tan θ sin θ ! / M ν with two texture zeros, and theirpredictions for the absolute neutrino mass m and the effective mass terms of the neutrinolessdouble-beta decay h m i ee .Pattern Texture of M ν The scales of neutrino masses A × × ×× × × m ≈ √ ∆ m , h m i ee = 0 A × × × × × × m ≈ √ ∆ m , h m i ee = 0 B × × × × × × m ≈ s ∆ m − tan θ , h m i ee ≈ m tan θ B × × × ×× × m ≈ s ∆ m − cot θ , h m i ee ≈ m cot θ B × × ×× × × m ≈ s ∆ m − tan θ , h m i ee ≈ m tan θ B × × × × × × m ≈ s ∆ m − cot θ , h m i ee ≈ m cot θ C × × ×× ×× × m ≈ vuut tan θ cot θ sin θ ∆ m θ tan 2 θ sin θ cos δ h m i ee ≈ m vuut − θ cos δ tan 2 θ sin θ + 4 cot θ tan θ sin θ .0 0.5 1.0 1.5 2.04 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] Figure 1: Pattern A of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 28 .0 0.2 0.4 0.6 0.8 1.04.44.64.85.05.25.4 0.0 0.5 1.0 1.5 2.0-1.0-0.50.00.51.00.0 0.5 1.0 1.5 2.00.80.91.01.11.21.31.4 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.50.0 0.5 1.0 1.5 2.00.20.40.60.81.01.2 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.5 m [ - e V ] m [ - e V ] m [ - e V ] J C P [ - ] [] [] [ ] [ ] Figure 2: Pattern A of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ 12 tan − " tan θ sin δ tan 2 θ sin θ + tan θ cos δ − π M ν with two texture zeros, andtheir predictions for two neutrino mass ratios ξ ≡ m /m and ζ ≡ m /m .Pattern Texture of M ν Neutrino mass ratios A × × ×× × × ξ ≈ tan θ tan θ sin θ , ζ ≈ cot θ tan θ sin θ A × × × × × × ξ ≈ tan θ cot θ sin θ , ζ ≈ cot θ cot θ sin θ B × × × × × × ξ ≈ ζ ≈ tan θ , ξ − ζ ≈ + 4 sin θ cos δ sin 2 θ sin 2 θ B × × × ×× × ξ ≈ ζ ≈ cot θ , ξ − ζ ≈ − θ cos δ sin 2 θ sin 2 θ B × × ×× × × ξ ≈ ζ ≈ tan θ , ξ − ζ ≈ − θ sin θ cos δ sin 2 θ sin 2 θ B × × × × × × ξ ≈ ζ ≈ cot θ , ξ − ζ ≈ + 4 cot θ sin θ cos δ sin 2 θ sin 2 θ C × × ×× ×× × ξ ≈ − θ cos δ tan 2 θ sin θ + cot θ tan θ sin θ ! / ζ ≈ θ cos δ tan 2 θ sin θ + tan θ tan θ sin θ ! / M ν with two texture zeros, and theirpredictions for the absolute neutrino mass m and the effective mass terms of the neutrinolessdouble-beta decay h m i ee .Pattern Texture of M ν The scales of neutrino masses A × × ×× × × m ≈ √ ∆ m , h m i ee = 0 A × × × × × × m ≈ √ ∆ m , h m i ee = 0 B × × × × × × m ≈ s ∆ m − tan θ , h m i ee ≈ m tan θ B × × × ×× × m ≈ s ∆ m − cot θ , h m i ee ≈ m cot θ B × × ×× × × m ≈ s ∆ m − tan θ , h m i ee ≈ m tan θ B × × × × × × m ≈ s ∆ m − cot θ , h m i ee ≈ m cot θ C × × ×× ×× × m ≈ vuut tan θ cot θ sin θ ∆ m θ tan 2 θ sin θ cos δ h m i ee ≈ m vuut − θ cos δ tan 2 θ sin θ + 4 cot θ tan θ sin θ .0 0.5 1.0 1.5 2.04 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] Figure 1: Pattern A of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 28 .0 0.2 0.4 0.6 0.8 1.04.44.64.85.05.25.4 0.0 0.5 1.0 1.5 2.0-1.0-0.50.00.51.00.0 0.5 1.0 1.5 2.00.80.91.01.11.21.31.4 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.50.0 0.5 1.0 1.5 2.00.20.40.60.81.01.2 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.5 m [ - e V ] m [ - e V ] m [ - e V ] J C P [ - ] [] [] [ ] [ ] Figure 2: Pattern A of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ level. 29 .0 0.5 1.0 1.5 2.04 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] Figure 3: Pattern A of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 30 .0 0.2 0.4 0.6 0.8 1.04.44.64.85.05.25.4 0.0 0.5 1.0 1.5 2.0-1.0-0.50.00.51.00.0 0.5 1.0 1.5 2.00.80.91.01.11.21.31.4 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.50.0 0.5 1.0 1.5 2.00.20.40.60.81.01.2 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.5 m [ - e V ] m [ - e V ] m [ - e V ] J C P [ - ] [] [] [ ] [ ] Figure 4: Pattern A of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ 12 tan − " tan θ sin δ tan 2 θ sin θ + tan θ cos δ − π M ν with two texture zeros, andtheir predictions for two neutrino mass ratios ξ ≡ m /m and ζ ≡ m /m .Pattern Texture of M ν Neutrino mass ratios A × × ×× × × ξ ≈ tan θ tan θ sin θ , ζ ≈ cot θ tan θ sin θ A × × × × × × ξ ≈ tan θ cot θ sin θ , ζ ≈ cot θ cot θ sin θ B × × × × × × ξ ≈ ζ ≈ tan θ , ξ − ζ ≈ + 4 sin θ cos δ sin 2 θ sin 2 θ B × × × ×× × ξ ≈ ζ ≈ cot θ , ξ − ζ ≈ − θ cos δ sin 2 θ sin 2 θ B × × ×× × × ξ ≈ ζ ≈ tan θ , ξ − ζ ≈ − θ sin θ cos δ sin 2 θ sin 2 θ B × × × × × × ξ ≈ ζ ≈ cot θ , ξ − ζ ≈ + 4 cot θ sin θ cos δ sin 2 θ sin 2 θ C × × ×× ×× × ξ ≈ − θ cos δ tan 2 θ sin θ + cot θ tan θ sin θ ! / ζ ≈ θ cos δ tan 2 θ sin θ + tan θ tan θ sin θ ! / M ν with two texture zeros, and theirpredictions for the absolute neutrino mass m and the effective mass terms of the neutrinolessdouble-beta decay h m i ee .Pattern Texture of M ν The scales of neutrino masses A × × ×× × × m ≈ √ ∆ m , h m i ee = 0 A × × × × × × m ≈ √ ∆ m , h m i ee = 0 B × × × × × × m ≈ s ∆ m − tan θ , h m i ee ≈ m tan θ B × × × ×× × m ≈ s ∆ m − cot θ , h m i ee ≈ m cot θ B × × ×× × × m ≈ s ∆ m − tan θ , h m i ee ≈ m tan θ B × × × × × × m ≈ s ∆ m − cot θ , h m i ee ≈ m cot θ C × × ×× ×× × m ≈ vuut tan θ cot θ sin θ ∆ m θ tan 2 θ sin θ cos δ h m i ee ≈ m vuut − θ cos δ tan 2 θ sin θ + 4 cot θ tan θ sin θ .0 0.5 1.0 1.5 2.04 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] Figure 1: Pattern A of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 28 .0 0.2 0.4 0.6 0.8 1.04.44.64.85.05.25.4 0.0 0.5 1.0 1.5 2.0-1.0-0.50.00.51.00.0 0.5 1.0 1.5 2.00.80.91.01.11.21.31.4 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.50.0 0.5 1.0 1.5 2.00.20.40.60.81.01.2 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.5 m [ - e V ] m [ - e V ] m [ - e V ] J C P [ - ] [] [] [ ] [ ] Figure 2: Pattern A of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ level. 29 .0 0.5 1.0 1.5 2.04 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] Figure 3: Pattern A of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 30 .0 0.2 0.4 0.6 0.8 1.04.44.64.85.05.25.4 0.0 0.5 1.0 1.5 2.0-1.0-0.50.00.51.00.0 0.5 1.0 1.5 2.00.80.91.01.11.21.31.4 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.50.0 0.5 1.0 1.5 2.00.20.40.60.81.01.2 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.5 m [ - e V ] m [ - e V ] m [ - e V ] J C P [ - ] [] [] [ ] [ ] Figure 4: Pattern A of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ level. 31 .50 0.51 0.52 0.53 0.54 0.55 0.560 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] o o o o o Figure 5: Pattern B of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 32 .50 0.51 0.52 0.53 0.54 0.55 0.560.010.050.090.130.170.21 0.50 0.51 0.52 0.53 0.54 0.55 0.560.00.20.40.60.81.00.50 0.51 0.52 0.53 0.54 0.55 0.560.040.080.120.160.200.24 0.50 0.51 0.52 0.53 0.54 0.55 0.560.000.020.040.060.080.50 0.51 0.52 0.53 0.54 0.55 0.560.010.050.090.130.170.21 0.50 0.51 0.52 0.53 0.54 0.55 0.56-0.04-0.020.000.020.040.06 m o r m [ e V ] m [ e V ] < m > ee [ e V ] J C P [ - ] [] [] [ ] [ ] Figure 6: Pattern B of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σ level. 33 .45 0.46 0.47 0.48 0.49 0.500 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] o o o o o Figure 7: Pattern B of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 34 .45 0.46 0.47 0.48 0.49 0.500.040.080.120.160.200.240.280.32 0.45 0.46 0.47 0.48 0.49 0.500.00.20.40.60.81.00.45 0.46 0.47 0.48 0.49 0.500.010.050.090.130.170.210.250.29 0.45 0.46 0.47 0.48 0.49 0.50-0.06-0.04-0.020.000.020.45 0.46 0.47 0.48 0.49 0.500.040.080.120.160.200.240.280.32 0.45 0.46 0.47 0.48 0.49 0.50-0.06-0.04-0.020.000.02 m o r m [ e V ] m [ e V ] < m > ee [ e V ] J C P [ - ] [] [] [ ] [ ] Figure 8: Pattern B of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σ level. 35 .40 0.42 0.44 0.46 0.48 0.500 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] o o o o o Figure 9: Pattern B of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 36 .40 0.42 0.44 0.46 0.48 0.500.010.050.090.130.170.21 0.40 0.42 0.44 0.46 0.48 0.500.00.20.40.60.81.00.40 0.42 0.44 0.46 0.48 0.500.040.080.120.160.200.24 0.40 0.42 0.44 0.46 0.48 0.50-0.10-0.08-0.06-0.04-0.020.000.40 0.42 0.44 0.46 0.48 0.500.010.050.090.130.170.21 0.40 0.42 0.44 0.46 0.48 0.50-0.10-0.08-0.06-0.04-0.020.00 m o r m [ e V ] m [ e V ] < m > ee [ e V ] J C P [ - ] [] [] [ ] [ ] Figure 10: Pattern B of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ 12 tan − " tan θ sin δ tan 2 θ sin θ + tan θ cos δ − π M ν with two texture zeros, andtheir predictions for two neutrino mass ratios ξ ≡ m /m and ζ ≡ m /m .Pattern Texture of M ν Neutrino mass ratios A × × ×× × × ξ ≈ tan θ tan θ sin θ , ζ ≈ cot θ tan θ sin θ A × × × × × × ξ ≈ tan θ cot θ sin θ , ζ ≈ cot θ cot θ sin θ B × × × × × × ξ ≈ ζ ≈ tan θ , ξ − ζ ≈ + 4 sin θ cos δ sin 2 θ sin 2 θ B × × × ×× × ξ ≈ ζ ≈ cot θ , ξ − ζ ≈ − θ cos δ sin 2 θ sin 2 θ B × × ×× × × ξ ≈ ζ ≈ tan θ , ξ − ζ ≈ − θ sin θ cos δ sin 2 θ sin 2 θ B × × × × × × ξ ≈ ζ ≈ cot θ , ξ − ζ ≈ + 4 cot θ sin θ cos δ sin 2 θ sin 2 θ C × × ×× ×× × ξ ≈ − θ cos δ tan 2 θ sin θ + cot θ tan θ sin θ ! / ζ ≈ θ cos δ tan 2 θ sin θ + tan θ tan θ sin θ ! / M ν with two texture zeros, and theirpredictions for the absolute neutrino mass m and the effective mass terms of the neutrinolessdouble-beta decay h m i ee .Pattern Texture of M ν The scales of neutrino masses A × × ×× × × m ≈ √ ∆ m , h m i ee = 0 A × × × × × × m ≈ √ ∆ m , h m i ee = 0 B × × × × × × m ≈ s ∆ m − tan θ , h m i ee ≈ m tan θ B × × × ×× × m ≈ s ∆ m − cot θ , h m i ee ≈ m cot θ B × × ×× × × m ≈ s ∆ m − tan θ , h m i ee ≈ m tan θ B × × × × × × m ≈ s ∆ m − cot θ , h m i ee ≈ m cot θ C × × ×× ×× × m ≈ vuut tan θ cot θ sin θ ∆ m θ tan 2 θ sin θ cos δ h m i ee ≈ m vuut − θ cos δ tan 2 θ sin θ + 4 cot θ tan θ sin θ .0 0.5 1.0 1.5 2.04 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] Figure 1: Pattern A of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 28 .0 0.2 0.4 0.6 0.8 1.04.44.64.85.05.25.4 0.0 0.5 1.0 1.5 2.0-1.0-0.50.00.51.00.0 0.5 1.0 1.5 2.00.80.91.01.11.21.31.4 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.50.0 0.5 1.0 1.5 2.00.20.40.60.81.01.2 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.5 m [ - e V ] m [ - e V ] m [ - e V ] J C P [ - ] [] [] [ ] [ ] Figure 2: Pattern A of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ level. 29 .0 0.5 1.0 1.5 2.04 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] Figure 3: Pattern A of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 30 .0 0.2 0.4 0.6 0.8 1.04.44.64.85.05.25.4 0.0 0.5 1.0 1.5 2.0-1.0-0.50.00.51.00.0 0.5 1.0 1.5 2.00.80.91.01.11.21.31.4 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.50.0 0.5 1.0 1.5 2.00.20.40.60.81.01.2 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.5 m [ - e V ] m [ - e V ] m [ - e V ] J C P [ - ] [] [] [ ] [ ] Figure 4: Pattern A of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ level. 31 .50 0.51 0.52 0.53 0.54 0.55 0.560 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] o o o o o Figure 5: Pattern B of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 32 .50 0.51 0.52 0.53 0.54 0.55 0.560.010.050.090.130.170.21 0.50 0.51 0.52 0.53 0.54 0.55 0.560.00.20.40.60.81.00.50 0.51 0.52 0.53 0.54 0.55 0.560.040.080.120.160.200.24 0.50 0.51 0.52 0.53 0.54 0.55 0.560.000.020.040.060.080.50 0.51 0.52 0.53 0.54 0.55 0.560.010.050.090.130.170.21 0.50 0.51 0.52 0.53 0.54 0.55 0.56-0.04-0.020.000.020.040.06 m o r m [ e V ] m [ e V ] < m > ee [ e V ] J C P [ - ] [] [] [ ] [ ] Figure 6: Pattern B of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σ level. 33 .45 0.46 0.47 0.48 0.49 0.500 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] o o o o o Figure 7: Pattern B of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 34 .45 0.46 0.47 0.48 0.49 0.500.040.080.120.160.200.240.280.32 0.45 0.46 0.47 0.48 0.49 0.500.00.20.40.60.81.00.45 0.46 0.47 0.48 0.49 0.500.010.050.090.130.170.210.250.29 0.45 0.46 0.47 0.48 0.49 0.50-0.06-0.04-0.020.000.020.45 0.46 0.47 0.48 0.49 0.500.040.080.120.160.200.240.280.32 0.45 0.46 0.47 0.48 0.49 0.50-0.06-0.04-0.020.000.02 m o r m [ e V ] m [ e V ] < m > ee [ e V ] J C P [ - ] [] [] [ ] [ ] Figure 8: Pattern B of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σ level. 35 .40 0.42 0.44 0.46 0.48 0.500 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] o o o o o Figure 9: Pattern B of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 36 .40 0.42 0.44 0.46 0.48 0.500.010.050.090.130.170.21 0.40 0.42 0.44 0.46 0.48 0.500.00.20.40.60.81.00.40 0.42 0.44 0.46 0.48 0.500.040.080.120.160.200.24 0.40 0.42 0.44 0.46 0.48 0.50-0.10-0.08-0.06-0.04-0.020.000.40 0.42 0.44 0.46 0.48 0.500.010.050.090.130.170.21 0.40 0.42 0.44 0.46 0.48 0.50-0.10-0.08-0.06-0.04-0.020.00 m o r m [ e V ] m [ e V ] < m > ee [ e V ] J C P [ - ] [] [] [ ] [ ] Figure 10: Pattern B of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ level. 37 .500 0.505 0.510 0.515 0.520 0.5251 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] o o o o o Figure 11: Pattern B of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 38 .500 0.505 0.510 0.515 0.520 0.5250.040.080.120.160.200.240.280.32 0.500 0.505 0.510 0.515 0.520 0.5250.00.10.20.30.40.50.500 0.505 0.510 0.515 0.520 0.5250.010.050.090.130.170.210.250.29 0.500 0.505 0.510 0.515 0.520 0.525-0.020.000.020.040.500 0.505 0.510 0.515 0.520 0.5250.040.080.120.160.200.240.280.32 0.500 0.505 0.510 0.515 0.520 0.5250.000.010.020.030.04 m o r m [ e V ] m [ e V ] < m > ee [ e V ] J C P [ - ] [] [] [ ] [ ] Figure 12: Pattern B of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ 12 tan − " tan θ sin δ tan 2 θ sin θ + tan θ cos δ − π M ν with two texture zeros, andtheir predictions for two neutrino mass ratios ξ ≡ m /m and ζ ≡ m /m .Pattern Texture of M ν Neutrino mass ratios A × × ×× × × ξ ≈ tan θ tan θ sin θ , ζ ≈ cot θ tan θ sin θ A × × × × × × ξ ≈ tan θ cot θ sin θ , ζ ≈ cot θ cot θ sin θ B × × × × × × ξ ≈ ζ ≈ tan θ , ξ − ζ ≈ + 4 sin θ cos δ sin 2 θ sin 2 θ B × × × ×× × ξ ≈ ζ ≈ cot θ , ξ − ζ ≈ − θ cos δ sin 2 θ sin 2 θ B × × ×× × × ξ ≈ ζ ≈ tan θ , ξ − ζ ≈ − θ sin θ cos δ sin 2 θ sin 2 θ B × × × × × × ξ ≈ ζ ≈ cot θ , ξ − ζ ≈ + 4 cot θ sin θ cos δ sin 2 θ sin 2 θ C × × ×× ×× × ξ ≈ − θ cos δ tan 2 θ sin θ + cot θ tan θ sin θ ! / ζ ≈ θ cos δ tan 2 θ sin θ + tan θ tan θ sin θ ! / M ν with two texture zeros, and theirpredictions for the absolute neutrino mass m and the effective mass terms of the neutrinolessdouble-beta decay h m i ee .Pattern Texture of M ν The scales of neutrino masses A × × ×× × × m ≈ √ ∆ m , h m i ee = 0 A × × × × × × m ≈ √ ∆ m , h m i ee = 0 B × × × × × × m ≈ s ∆ m − tan θ , h m i ee ≈ m tan θ B × × × ×× × m ≈ s ∆ m − cot θ , h m i ee ≈ m cot θ B × × ×× × × m ≈ s ∆ m − tan θ , h m i ee ≈ m tan θ B × × × × × × m ≈ s ∆ m − cot θ , h m i ee ≈ m cot θ C × × ×× ×× × m ≈ vuut tan θ cot θ sin θ ∆ m θ tan 2 θ sin θ cos δ h m i ee ≈ m vuut − θ cos δ tan 2 θ sin θ + 4 cot θ tan θ sin θ .0 0.5 1.0 1.5 2.04 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] Figure 1: Pattern A of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 28 .0 0.2 0.4 0.6 0.8 1.04.44.64.85.05.25.4 0.0 0.5 1.0 1.5 2.0-1.0-0.50.00.51.00.0 0.5 1.0 1.5 2.00.80.91.01.11.21.31.4 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.50.0 0.5 1.0 1.5 2.00.20.40.60.81.01.2 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.5 m [ - e V ] m [ - e V ] m [ - e V ] J C P [ - ] [] [] [ ] [ ] Figure 2: Pattern A of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ level. 29 .0 0.5 1.0 1.5 2.04 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] Figure 3: Pattern A of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 30 .0 0.2 0.4 0.6 0.8 1.04.44.64.85.05.25.4 0.0 0.5 1.0 1.5 2.0-1.0-0.50.00.51.00.0 0.5 1.0 1.5 2.00.80.91.01.11.21.31.4 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.50.0 0.5 1.0 1.5 2.00.20.40.60.81.01.2 0.0 0.5 1.0 1.5 2.0-0.5-0.3-0.10.10.30.5 m [ - e V ] m [ - e V ] m [ - e V ] J C P [ - ] [] [] [ ] [ ] Figure 4: Pattern A of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ level. 31 .50 0.51 0.52 0.53 0.54 0.55 0.560 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] o o o o o Figure 5: Pattern B of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 32 .50 0.51 0.52 0.53 0.54 0.55 0.560.010.050.090.130.170.21 0.50 0.51 0.52 0.53 0.54 0.55 0.560.00.20.40.60.81.00.50 0.51 0.52 0.53 0.54 0.55 0.560.040.080.120.160.200.24 0.50 0.51 0.52 0.53 0.54 0.55 0.560.000.020.040.060.080.50 0.51 0.52 0.53 0.54 0.55 0.560.010.050.090.130.170.21 0.50 0.51 0.52 0.53 0.54 0.55 0.56-0.04-0.020.000.020.040.06 m o r m [ e V ] m [ e V ] < m > ee [ e V ] J C P [ - ] [] [] [ ] [ ] Figure 6: Pattern B of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σ level. 33 .45 0.46 0.47 0.48 0.49 0.500 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] o o o o o Figure 7: Pattern B of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 34 .45 0.46 0.47 0.48 0.49 0.500.040.080.120.160.200.240.280.32 0.45 0.46 0.47 0.48 0.49 0.500.00.20.40.60.81.00.45 0.46 0.47 0.48 0.49 0.500.010.050.090.130.170.210.250.29 0.45 0.46 0.47 0.48 0.49 0.50-0.06-0.04-0.020.000.020.45 0.46 0.47 0.48 0.49 0.500.040.080.120.160.200.240.280.32 0.45 0.46 0.47 0.48 0.49 0.50-0.06-0.04-0.020.000.02 m o r m [ e V ] m [ e V ] < m > ee [ e V ] J C P [ - ] [] [] [ ] [ ] Figure 8: Pattern B of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σ level. 35 .40 0.42 0.44 0.46 0.48 0.500 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] o o o o o Figure 9: Pattern B of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 36 .40 0.42 0.44 0.46 0.48 0.500.010.050.090.130.170.21 0.40 0.42 0.44 0.46 0.48 0.500.00.20.40.60.81.00.40 0.42 0.44 0.46 0.48 0.500.040.080.120.160.200.24 0.40 0.42 0.44 0.46 0.48 0.50-0.10-0.08-0.06-0.04-0.020.000.40 0.42 0.44 0.46 0.48 0.500.010.050.090.130.170.21 0.40 0.42 0.44 0.46 0.48 0.50-0.10-0.08-0.06-0.04-0.020.00 m o r m [ e V ] m [ e V ] < m > ee [ e V ] J C P [ - ] [] [] [ ] [ ] Figure 10: Pattern B of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ level. 37 .500 0.505 0.510 0.515 0.520 0.5251 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] o o o o o Figure 11: Pattern B of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 38 .500 0.505 0.510 0.515 0.520 0.5250.040.080.120.160.200.240.280.32 0.500 0.505 0.510 0.515 0.520 0.5250.00.10.20.30.40.50.500 0.505 0.510 0.515 0.520 0.5250.010.050.090.130.170.210.250.29 0.500 0.505 0.510 0.515 0.520 0.525-0.020.000.020.040.500 0.505 0.510 0.515 0.520 0.5250.040.080.120.160.200.240.280.32 0.500 0.505 0.510 0.515 0.520 0.5250.000.010.020.030.04 m o r m [ e V ] m [ e V ] < m > ee [ e V ] J C P [ - ] [] [] [ ] [ ] Figure 12: Pattern B of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ level. 39 .0 0.1 0.2 0.3 0.4 0.50 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o [ ] [ ] [ ] o o o o o Figure 13: Pattern C of M ν : allowed ranges of flavor mixing angles ( θ , θ , θ ) versus theDirac CP-violating phase δ at the 3 σ level, where the probability distribution of three anglesare shown in the left panel. 40 .0 0.1 0.2 0.3 0.4 0.50.000.040.080.120.160.200.240.280.32 0.0 0.1 0.2 0.3 0.4 0.50.00.20.40.60.81.00.0 0.1 0.2 0.3 0.4 0.50.000.040.080.120.160.200.240.280.32 0.0 0.1 0.2 0.3 0.4 0.5-0.5-0.4-0.3-0.2-0.10.00.0 0.1 0.2 0.3 0.4 0.50.000.040.080.120.160.200.240.280.32 0.0 0.1 0.2 0.3 0.4 0.50.000.020.040.060.080.10 m o r m [ e V ] m [ e V ] < m > ee [ e V ] J C P [ - ] [] [] [ ] [ ] Figure 14: Pattern C of M ν : allowed ranges of the neutrino masses ( m , m , m ), the Jarlskoginvariant J CP and the Majorana CP-violating phases ( ρ, σ ) versus the Dirac CP-violating phase δ at the 3 σσ