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UWThPh-2004-19 OCHA-PP-244 January 19, 2005 Non-vanishing U and cos2θ from a broken Z symmetry e3 23 2 Walter Grimus,a Anjan S. Joshipura,b Satoru Kaneko,c ∗ † ‡ 5 0 Lu´ıs Lavoura,d Hideyuki Sawanaka e and Morimitsu Tanimotof § ¶ k 0 2 n aInstitut fu¨r Theoretische Physik, Universita¨t Wien, Boltzmanngasse 5, A–1090 Wien, Austria a J bPhysical Research Laboratory, Ahmedabad 380009, India 9 1 cDepartment of Physics, Ochanomizu University, Tokyo 112-8610, Japan 4 v dInstituto Superior Técnico, Universidade Técnica de Lisboa, P–1049-001 Lisboa, Portugal 3 2 eGraduate School of Science and Technology, Niigata University, 950-2181 Niigata, Japan 1 8 f Department of Physics, Niigata University, 950-2181 Niigata, Japan 0 4 0 / ABSTRACT h p - It is shown that the neutrino mass matrices in the flavour basis yielding a vanishing U are p e3 e characterized by invariance under a class of Z symmetries. A specific Z in this class also leads h 2 2 : v to a maximal atmospheric mixing angle θ . The breaking of that Z can be parameterized by two 23 2 i X dimensionlessquantities, ǫ and ǫ; theeffects of ǫ,ǫ = 0are studiedperturbatively and numerically. r ′ ′ 6 a The induced value of U strongly depends on the neutrino mass hierarchy. We find that U is e3 e3 | | | | less than 0.07 for a normal mass hierarchy, even when ǫ,ǫ 30%. For an inverted mass hierarchy ′ ∼ U tends to be around 0.1 but can be as large as 0.17. In the case of quasi-degenerate neutrinos, e3 | | U could be close to its experimental upper bound 0.2. In contrast, cos2θ can always reach e3 23 | | | | its experimental upper bound 0.28. We propose a specific model, based on electroweak radiative corrections in the MSSM, for ǫ and ǫ. In that model, both U and cos2θ , could be close to ′ e3 23 | | | | their respective experimental upper bounds if neutrinos are quasi-degenerate. ∗e-mail: [email protected] †e-mail: [email protected] ‡e-mail: [email protected] §e-mail: [email protected] ¶e-mail: [email protected] ke-mail: [email protected] 1 Introduction In recent years, the observation of solar [1, 2] and atmospheric [3] neutrino oscillations has dramat- ically improved our knowledge of neutrino masses and lepton mixing. The neutrino mass-squared differences ∆ and ∆ , and themixing angles tan2θ andsin22θ , are now quite well deter- sun atm sun atm mined. The third mixing angle, represented by the matrix element U of the lepton mixing matrix e3 U (MNS matrix [4]), is constrained to be small by the non-observation of neutrino oscillations at the CHOOZ experiment [5]. In spite of all this progress, the available information on neutrino masses and lepton mixing is not sufficient to uncover the mechanism of neutrino mass generation. In particular, we do not yet know whether theobserved features of lepton mixing aredueto some underlyingflavour symmetry, or they are mere mathematical coincidences [6] of the seesaw mechanism. Two features of lepton mixing which would suggest a definite symmetry are the small magnitudes of U and cos2θ , e3 23 where θ is one of the angles in the standard parameterization of the MNS matrix and coincides 23 withtheatmosphericmixingangleθ whenU = 0.Thebest-fitvalueforθ inatwo-generation atm e3 23 analysis [3] of the atmospheric data is θ = π/4, corresponding to cos2θ = 0. Likewise, U is 23 23 e3 | | required to be small: U 0.26 at 3σ from a combined analysis of the atmospheric and CHOOZ e3 | |≤ data [7]. This smallness strongly hints at some flavour symmetry. There are many examples of symmetries which can force U and/or cos2θ to vanish. Both e3 23 quantities vanish in the extensively studied bi-maximal mixing Ansatz [8, 9, 10, 11], which can be realized through a symmetry [12]. One can also make both U and cos2θ zero while leaving the e3 23 solar mixing angle arbitrary [13, 14]. Alternatively, it is possible to force only U to be zero, by e3 imposingadiscrete Abelian [15]or non-Abelian [16]symmetry; conversely, one can obtain maximal atmospheric mixing but a free U by means of a non-Abelian symmetry or a non-standard CP e3 symmetry [17]. The symmetries mentioned above need not be exact. It is important to consider perturbations of those symmetries from the phenomenological point of view and to study quantitatively [18] the magnitudes of U and cos2θ possibly generated by such perturbations. e3 23 This paper is a study of a special class of symmetries and of the consequences of their perturbative violation. We show in section 2 that U vanishes if the neutrino mass matrix in the flavour e3 2 basis is invariant under a class of Z symmetries. The solar and atmospheric mixing angles, as well 2 as the neutrino masses, remain unconstrained by these Z symmetries. Those Z symmetries thus 2 2 constitute a general class of symmetries leading only to a vanishing U . We point out that there e3 is a special Z in this class which leads, furthermore, to maximal atmospheric mixing. We consider 2 more closely that specific Z in section 3, wherein we study departures from the symmetric limit. 2 We parameterize perturbations of the Z -invariant mass matrix in terms of two complex param- 2 eters, and derive general expressions for U and cos2θ in terms of those parameters; we also e3 23 present detailed numerical estimates of U and cos2θ . Section 4 is devoted to the study of the e3 23 specific perturbation which is induced by the electroweak radiative corrections to a Z -invariant 2 neutrino mass matrix defined at a high scale. We discuss a specific model for this scenario. In the concludingsection 5 we make a comparison of thepredictions for U and cos2θ obtained within e3 23 | | various frameworks. 2 Vanishing U from a class of Z symmetries e3 2 The neutrino masses and lepton mixing are completely determined by the neutrino mass matrix in the flavour basis—the basis where the charged-lepton mass matrix is diagonal—which we denote as . In this section we look for effective symmetries of which may lead to a vanishing νf νf M M U . e3 One knows [19] that the lepton-number symmetry L L L implies (i) a vanishing solar e µ τ − − mass-squared difference ∆ , (ii) a maximal solar mixing angle θ , and (iii) a vanishing U , sun 23 e3 while it keeps the atmospheric mixing angle unconstrained; one must introduce [20] a significant breaking of L L L in order to correct the predictions (i) and (ii). A better symmetry seems e µ τ − − to be the µ–τ interchange symmetry [13], which implies vanishing U and maximal θ , but leaves e3 23 both the neutrino masses and the solar mixing angle unconstrained; this is consistent with the present experimental results. The µ–τ interchange symmetry can be physically realized in a model based on the discrete non-Abelian group D [14]; a variation of this model [16] keeps the prediction 4 U = 0 but leaves the atmospheric mixing angle arbitrary. Recently, Low [15] has considered e3 models wherein has, due to a discrete Abelian symmetry, a structure leading to U = 0. νf e3 M We now show that there exists a class Z (γ,α) of discrete symmetries of the Z type which 2 2 3 encompasses all the models discussed above and enforces a form of leading to U = 0. This νf e3 M class is parametrized by an angle γ (0 < γ < 2π) and a phase α (0 α < 2π). The symmetry ≤ Z (γ,α) is defined by the 3 3 matrix 2 × 1 0 0 S(γ,α) = 0 cosγ e iαsinγ . (1) −   0 eiαsinγ cosγ −   This matrix is unitary; indeed, it satisfies [S(γ,α)]2 = 1 , (2) 3 3 × T [S(γ,α)] = [S(γ,α)]∗. (3) Equation (2) meansthat S(γ,α) is arealization of thegroup Z . We definetheZ (γ,α) invariance 2 2 of by νf M T [S(γ,α)] S(γ,α) = . (4) νf νf M M If one writes X˜ A˜ B˜ = A˜ C˜ D˜ , (5) νf M   B˜ D˜ E˜   where all the matrix elements are complex in general, then equation (4) is equivalent to B˜ γ e iαtan = 0, A˜ − − 2 (6) eiαE˜ e iαC˜ sinγ+2D˜ cosγ = 0. − − (cid:16) (cid:17) Let us first prove that the Z (γ,α) invariance of implies U = 0. The matrix S(γ,α) has 2 νf e3 M a unique eigenvalue 1 corresponding to the eigenvector − 0 v =  exp( iα/2)sin(γ/2) . (7) −  exp(iα/2)cos(γ/2)   −    T Equation (4), together with S(γ,α)v = v, imply that [S(γ,α)] ( v) = ( v). Then, νf νf − M − M equation (3), together with the fact that the eigenvalue 1 of S(γ,α) is unique, implies that − v v . Now, determines the lepton mixing matrix—MNS matrix—U according to νf ∗ νf M ∝ M = U diag(m ,m ,m )U , (8) νf ∗ 1 2 3 † M where m , m , and m are the (real and non-negative) neutrino masses. Thus, if we write U = 1 2 3 (u ,u ,u ), then the column vectors u satisfy u = m u for j = 1,2,3. The fact that 1 2 3 j Mνf j j ∗j 4 v v therefore means that, apart from a phase factor, v is one of the columns of the MNS νf ∗ M ∝ matrix, hence U =0, q.e.d. e3 Let us next prove the converse of the above, i.e. that U = 0 implies that there is some angle e3 γ and phase α such that is Z (γ,α)-invariant. If U = 0 then U may be parametrized by νf 2 e3 M two angles ϑ and five phases χ as 1,2 1,2,3,4,5 eiχ1cosϑ eiχ2sinϑ 0 1 1 U = eiχ3sinϑ cosϑ ei(χ2+χ3 χ1)cosϑ cosϑ eiχ4sinϑ . (9)  1 2 − 1 2 2  − eiχ5sinϑ sinϑ ei(χ2+χ5 χ1)cosϑ sinϑ ei(χ4+χ5 χ3)cosϑ 1 2 − 1 2 − 2  −    When one computes through equation (8) one then finds that it satisfies equations (6) with νf M γ/2 = ϑ and α = χ χ +π, q.e.d. 2 5 3 − One has thus proved the equivalence of U = 0 with the existence of some angle γ and phase e3 α such that is Z (γ,α)-invariant. νf 2 M It should be stressed that Z (γ,α) will not usually be a symmetry of the full model, nor is 2 it necessarily the remaining symmetry of some larger symmetry operating at a high scale. Some examples may help making this clear: The µ–τ interchange symmetry [13], which corresponds to cosγ = 0, eiαsinγ = 1, cannot be • a symmetry of the full theory, since the masses of the µ and τ charged leptons are certainly different; thus, that symmetry must be broken in the charged-lepton mass matrix, but that breaking must occur in such a way that it remains unseen—at least at tree level—in the form of . Moreover, the µ–τ interchange symmetry predicts cos2θ = 0 together with νf 23 M U = 0. e3 Many models based on L = L L L lead to [19] e µ τ • − − x y ry = y z rz . (10) νf   M ry rz r2z     In this case cosγ = 1 r 2 / 1+ r 2 and eiαsinγ = 2r / 1+ r 2 . The symmetry ∗ −| | | | | | Z (γ,α) is not a subg(cid:0)roup of(cid:1)th(cid:0)e origina(cid:1)l L symmetry, rather (cid:0)it occurs(cid:1) accidentally as a 2 consequence of the specific particle content of the models and of the particular way in which L is softly broken. The mass matrix in equation (10) predicts m = 0 together with U =0. 3 e3 5 The softly-broken D model [16] has 4 • x y t 1 = y z 0 , (11) M−νf   t 0 z     together with the condition argy2 = argt2. In this case cosγ = y2 t2 / y2+t2 and − eiαsinγ = 2yt/ y2+t2 . The fact that the (µ,τ) matrix element o(cid:0)f 1(cid:1)is (cid:0)zero, an(cid:1)d the M−νf fact that its (µ,µ(cid:0)) and ((cid:1)τ,τ) matrix elements remain equal, are just reflections of the limited particle content used to break the original D symmetry softly. 4 Thus, the symmetry Z (γ,α) may be fundamental, effective, or accidental, depending on the 2 specific model at hand. Consideringequation (9) more carefully one notices that the phaseα = χ χ +π is physically 5 3 − meaningless, since it can be removed through a rephasing of the charged-lepton fields. Let us then set α = 0. In that case, the satisfying equations (6) can be written in the form νf M X √2Acos(γ/2) √2Asin(γ/2) = √2Acos(γ/2) B +Ccosγ Csinγ . (12) νf M   √2Asin(γ/2) Csinγ B Ccosγ −   The eigenvalue corresponding to the eigenvector in equation (7) is B C. − Specific choices of the parameters in equation (12) give different models. The model with B = C = X = 0 corresponds to L L L symmetry [19]. The model with γ = π/2 corresponds e µ τ − − to µ–τ interchange symmetry [13]. The D model in [16] has X˜ = A˜2/D˜. Likewise, various models 4 in [15] can be shown to have a which is formally identical to the matrix in equation (12). νf M In this paper we modify the standard parametrization for U by multiplying its third row by 1, i.e. we use − c c c s s e iδ 13 12 13 12 13 − U = c s s s c eiδ c c s s s eiδ s c diag eiρ, eiσ, 1 . (13) 23 12 23 13 12 23 12 23 13 12 23 13 − − − × s23s12+c23s13c12eiδ s23c12+c23s13s12eiδ c23c13 (cid:16) (cid:17) − −   Then, if we let U = s e iδ = 0, equation (8) reduces to equation (12) with γ/2 = θ and e3 13 − 23 X = c2 m e 2iρ+s2 m e 2iσ, 12 1 − 12 2 − c s A = 12 12 m e 2iρ m e 2iσ , 1 − 2 − − √2 − 1 (cid:0) (cid:1) (14) B = s2 m e 2iρ+c2 m e 2iσ +m , 2 12 1 − 12 2 − 3 1(cid:0) (cid:1) C = s2 m e 2iρ+c2 m e 2iσ m . 2 12 1 − 12 2 − − 3 (cid:0) (cid:1) 6 3 Non-zero U , cos2θ from Z breaking e3 23 2 Models with U = 0 can be divided in two different categories: e3 Those in which the solar scale also vanishes, along with U . These are obtained by setting e3 • m = m in equations (14). In these models, the perturbation which generates the solar scale 1 2 can be expected to also generate U , and one may find [18, 21] correlations between them. e3 Models in which the solar scale is present already at the zeroth order. These are represented • by equation (12) without additional restrictions on its parameters, except possibly γ = π/4. We consider here the more general second category, but fix γ = π/4, i.e. we consider models with vanishing U and cos2θ . can be explicitly written in this case as e3 23 νf M Mνf = U0∗diag(m1,m2,m3)U0†, (15) where U is obtained from equation (13) by setting s = 0 and θ = π/4. One then has 0 13 23 X A A = A B C . (16) νf M   A C B   Considerageneral perturbationδ toequation (16). Thematrixδ isageneralcomplex νf νf M M symmetric matrix, but part of it can be absorbed through a redefinition of the parameters in equation (16). The remaining part can be written, without loss of generality, as 0 ǫ ǫ 1 1 − δ = ǫ ǫ 0 . (17) νf 1 2 M   ǫ 0 ǫ 1 2 − −   The perturbation is controlled by two parameters, ǫ and ǫ , which are complex and model- 1 2 dependent. We want to study their effects perturbatively, i.e. we want to assume ǫ and ǫ to 1 2 be small. This smallness can be quantified by saying either that they are smaller than the largest element in , or that the perturbation to a given matrix element of is smaller than the νf νf M M element itself. We adopt the latter alternative and define two dimensionless parameters: ǫ ǫA, ǫ ǫB. (18) 1 2 ′ ≡ ≡ Thus, we have the neutrino mass matrix with Z breaking as follows: 2 X A(1+ǫ) A(1 ǫ) − = A(1+ǫ) B(1+ǫ) C , (19) νf ′ M   A(1 ǫ) C B(1 ǫ) ′ − −   7 where we shall assume ǫ and ǫ to be small, ǫ , ǫ 1. ′ ′ | | | |≪ One finds that, to first order in ǫ and ǫ, the only effect of the δ in equation (17) is to ′ νf M generate non-zero U and cos2θ . The neutrino masses, as well as the solar angle, do not receive e3 23 any corrections. U and cos2θ are of the same order as ǫ and ǫ. Define e3 23 ′ mˆ m e 2iρ, (20) 1 1 − ≡ mˆ m e 2iσ, (21) 2 2 − ≡ and ǫ (mˆ mˆ )ǫ, (22) 1 2 ≡ − mˆ s2 +mˆ c2 +m ǫ 1 12 2 12 3 ǫ. (23) ′ ′ ≡ 2 Then, we get s c Ue3 = m212 1m22 ǫs212mˆ∗2+ǫ∗s212m3−ǫ′mˆ∗2−ǫ′∗m3 3s− c2 (cid:16) (cid:17) +m212 1m22 ǫc212mˆ∗1+ǫ∗c212m3+ǫ′mˆ∗1+ǫ′∗m3 , (24) 3− 1 (cid:16) (cid:17) 2c2 2s2 cos2θ23 = Re(m23−12m22 (cid:16)ǫs212−ǫ′(cid:17)(mˆ2+m3)∗− m23−12m21 (cid:16)ǫc212+ǫ′(cid:17)(mˆ1+m3)∗). (25) The meaningful phases in are the ones of rephasing-invariant quartets. Since is νf νf M M symmetric, there are three such phases which are linearly independent. (Correspondingly, there are three physical phases in the MNS matrix: δ, 2ρ, and 2σ.) One easily sees that, in the first- orderapproximation in ǫ and ǫ, theimaginary parts of those two small parameters aremeaningless ′ when taken separately; only Im(2ǫ ǫ) is physically meaningful to this order. Indeed, one can ′ − manipulate equations (24) and (25) to obtain c2 cos2θ = 12 m2+c2 m2+s2 Re(mˆ mˆ +mˆ m )+ 1+c2 Re(mˆ m ) 23 (m22−m23 h 3 12 2 12 1 ∗2 1 3 (cid:16) 12(cid:17) 2 3 i s2 + 12 m2+s2 m2+c2 Re(mˆ mˆ +mˆ m )+ 1+s2 Re(mˆ m ) Reǫ m21−m23 h 3 12 1 12 2 ∗1 2 3 (cid:16) 12(cid:17) 1 3 i) ′ m2 Re(mˆ mˆ +mˆ m mˆ m ) +2c2 s2 2− 1 ∗2 1 3− 2 3 12 12" m22−m23 m2 Re(mˆ mˆ mˆ m +mˆ m ) + 1− 1 ∗2− 1 3 2 3 Reǫ m21−m23 # c2 s2 m2 m2 Im(mˆ mˆ +mˆ m +mˆ m ) + 12 12 1−m22 m2 1m2∗2 m21 3 ∗2 3 Im 2ǫ−ǫ′ , (26) (cid:0) 3(cid:1)− 1 3− 2 (cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1) 8 U 1 1 e3 = s2 mˆ mˆ +s2 mˆ m + 1+c2 mˆ m +m2+c2 m2 c s 2 m2 m2 12 1 ∗2 12 ∗1 3 12 ∗2 3 3 12 2 12 12 (cid:26) 2− 3 h (cid:16) (cid:17) i 1 + c2 mˆ mˆ +c2 mˆ m + 1+s2 mˆ m +m2+s2 m2 Reǫ m2 m2 12 ∗1 2 12 ∗2 3 12 ∗1 3 3 12 1 ′ 3− 1 h (cid:16) (cid:17) i(cid:27) s2 + 12 m2 mˆ mˆ mˆ m +mˆ m "m22−m23 (cid:16) 2− 1 ∗2− ∗1 3 ∗2 3(cid:17) c2 + 12 m2 mˆ mˆ +mˆ m mˆ m Reǫ m23−m21 (cid:16) 1− ∗1 2 ∗1 3− ∗2 3(cid:17)# i s2 + 12 m2 mˆ mˆ +mˆ m mˆ m 2 "m22−m23 (cid:16) 2− 1 ∗2 ∗1 3− ∗2 3(cid:17) c2 + 12 m2 mˆ mˆ mˆ m +mˆ m Im 2ǫ ǫ . (27) m23−m21 (cid:16) 1− ∗1 2− ∗1 3 ∗2 3(cid:17)# (cid:0) − ′(cid:1) The induced values of U and cos2θ are strongly correlated to neutrino mass hierarchies. e3 23 | | | | This makes it possible to draw some general conclusions even if we do not know the magnitudes of ǫ,ǫ. In Table 1 we give expressions and values for U and cos2θ in case of the hierarchical ′ e3 23 | | | | (m < m < m ), inverted (m m √∆ m ) and quasi-degenerate neutrino spectrum. 1 2 3 1 2 atm 3 ≈ ∼ ≫ CP conservation is assumed but we distinguish two different cases (a) the Dirac solar pair corresponding to σ = ρ = 0 and the Pseudo-Dirac solar pair with1 ρ = π/2,σ = 0. We have also given approximate values in some cases assuming the common degenerate mass m 0.3 eV. ∼ It follows from the Table 1 and equations (26,27) that: The first order contribution to U given in equation (27) vanish identically if mˆ = mˆ . As e3 1 2 • a consequence of this, U gets suppressed by a factor (∆sun) for the inverted or quasi- e3 O ∆atm degenerate spectrum with ρ = σ = 0. Similar suppression also occurs in case of the normal neutrino mass hierarchy even when ρ= σ. U need not be suppressed in other cases and can e3 6 be large. In contrast to U , cos2θ is almost as large as ǫ,ǫ if neutrino mass spectrum is normal or e3 23 ′ • inverted. It gets enhanced compared to these parameters if the spectrum is quasi-degenerate. In case of the quasi-degenerate spectrum, both cos2θ and U can become quite large 23 e3 • | | | | and reach the present experimental limits. Especially, the enhancement factors are large in case of thepseudo-Dirac solar pair (ρ = π/2, σ = 0). U and cos2θ are in fact proportional e3 23 1The physically different case with ρ=0,σ=π/2 has similar results. 9 Normal Hierarchy U c s ∆sun(ǫ+ ǫ′) 0.09(ǫ+ ǫ′) | e3|≈ 12 12 ∆atm 2 ≈ 2 q m m ;m2 ∆ ;m2 ∆ cos2θ ǫ 1 ≪ 2 2 ≈ sun 3 ≈ atm | 23| ≈ ′ Inverted Hierarchy U ∆sun s c (ǫ ǫ′) 0.009(ǫ ǫ′) | e3|≈ 2∆atm 12 12 − 2 ≈ − 2 σ = 0;ρ = 0 cos2θ ǫ 23 ′ | | ≈ σ = 0;ρ = π/2 U 1 sin4θ (ǫ ǫ′) 0.4(ǫ ǫ′) | e3|≈ 2 12 − 2 ≈ − 2 cos2θ 2(ǫsin22θ + ǫ′ cos22θ ) | 23| ≈ 12 2 12 Quasi-Degenerate U 2ǫc s m2 ∆sun 1.6ǫ | e3|≈ ′ 12 12∆atm∆atm ≈ σ = 0;ρ = 0 cos2θ 4 m2 ǫ 180ǫ | 23| ≈ ∆atm ′ ≈ ′ σ = 0;ρ = π/2 U 4 m2 c s (ǫs2 + ǫ′c2 ) 81(ǫs2 + ǫ′c2 ) | e3|≈ ∆atm 12 12 12 2 12 ≈ 12 2 12 cos2θ 8 m2 c2 (ǫs2 + ǫ′c2 ) 259(ǫs2 + ǫ′c2 ) | 23| ≈ ∆atm 12 12 2 12 ≈ 12 2 12 Table 1: Leading order predictions for U , cos2θ in case of different neutrino mass hierar- e3 23 | | | | chies with CP conservation. The numerical estimates are based on the best fit values of neutrino parameters and the quasi-degenerate mass m = 0.3eV. to each other in this particular case. The parameters ǫ,ǫ are constrained to be lower than ′ 10 2 for the quasi-degenerate spectrum. − The perturbative expressions given above may not be reliable for some values of ǫ,ǫ due to large ′ enhancement factor of ( m2 ) and one should do a numerical analysis. We now discuss results O ∆atm of such analysis in various circumstances. Scattered plots of the predicted values for cos2θ 23 | | and U are given in Figure 1 in the case of normal neutrino mass hierarchy. CP conservation e3 | | (ρ= σ = 0, real ǫ,ǫ) is assumed. Neutrinomasses and θ do notreceive any corrections at O(ǫ,ǫ) ′ 12 ′ and hence do not appreciablly change by perturbations. We therefore randomly varied these input parameters in the experimentally allowed regions. m was varied up to m . On the other hand, 1 2 ǫ,ǫ are unknown unless the symmetry breaking is specified, so these are varied randomly in the ′ range 0.3 0.3 with the condition that the output parameters should lie in the 90% CL limit − ∼ [2, 7]: 0.33 tan2θ 0.49 , 7.7 10 5 ∆ 8.8 10 5 eV2, 90%C.L. , sun − sun − ≤ ≤ × ≤ ≤ × 0.92 sin22θ , 1.5 10 3 ∆ 3.4 10 3 eV2, 90%C.L. . (28) atm − atm − ≤ × ≤ ≤ × 10