TU-TP/98-01 January, 1998 Analysis of neutrino oscillation in three-flavor neutrinos T. Teshima∗ , T. Sakai and O. Inagaki 8 9 9 Department of Applied Physics, Chubu University, Kasugai 487, Japan 1 n a J 3 1 Abstract 1 v 6 We analyzed the solar, terrestrial and atmospheric neutrinos experiments us- 7 2 1 ingthethree-flavorneutrinoframeworkandgottheallowedregionsforparam- 0 8 eters(∆m2 , sin22θ , ∆m2 , θ , θ ). Insolarneutrinoexperiments,wegot 9 12 12 23 13 23 / h thelargeanglesolution(∆m2 , sin22θ ) =(4 10−6 7 10−5eV2, 0.6 0.9) 12 12 p × − × − - p and small angle solution (3 10−6 1.2 10−5eV2, 0.003 0.01) for × − × − e h θ = 0◦ 20◦. From the terrestrial and atmospheric neutrino experiments, : 13 − v i we got the allowed regions (θ < 4◦, 24◦ < θ < 26◦) for ∆m2 = 2eV2, X 13 23 23 r a (θ < 4◦, 24◦ < θ < 45◦)for∆m2 = 0.2eV2 and(θ < 14◦, θ 40◦)for 13 23 23 13 23 ∼ ∆m2 = 0.02eV2. It seems that the large angle solution for (∆m2 , sin22θ ) 23 12 12 is favored than the small angle slution from the analysis of zenith angle de- pendence in atmospheric neutrino sub-GeV experiment. I. INTRODUCTION The problem of neutrino masses and oscillations is one of the most interesting issues to study physics beyond the standard model (SM) [1]. In many experiments which are under way, indications in favor of neutrino masses and oscillations have been obtained. In these experiments, the solar neutrino experiments [2–5] measure the event rates significantly 1 lower than the ones predicted by the standard solar model, and the atmospheric neutrino experiments [6–8] observe an anomaly referred to as the unexpected difference between the measured and predicted µ-like and e-like neutrinos. Another indication in favor of non-zero neutrino masses is in the cosmological analysis by dark matter [9]. On the other hand, terrestrial neutrino experiments searching for the neutrino masses and oscillations are under way. The E531 [10], CHORUS and NOMAD [11] experiments using the beam from accelerator search for ν appearance in a ν , and E776 [12], KARMEN τ µ [13] and LSND [14] experiments using the accelerator beams are searching for ν ν µ e → and ν¯ ν¯ oscillations. The experiments using nuclear power reactor [15] search for µ e → the disappearance of ν¯ , in which ν¯ ν¯ (X = µ, τ) transitions are expected. These e e X → experiments do not observe significantly large neutrino transitions. In this paper, we analyze the data of current solar, terrestrial and atmospheric exper- iments in a framework where the three neutrinos have masses and mix each other, and search the allowed regions of parameters (∆m2 , sin22θ , ∆m2 , θ , θ ) characteriz- 12 12 23 13 23 ing the masses and mixing of three-flavor neutrinos. Although there are many analyses which study the solar, atmospheric and terrestrial neutrino problems in three-flavor neutri- nos framework [16–18], we study more thoroughly these problems including the analysis of zenith angle dependence in recent SuperKamiokande atmospheric experiment [19,20]. After the analyses of experiments for neutrino oscillation, we will present a matrix of neutrino mixing ascribed from the allowed regions of parameters obtained. II. NEUTRINO OSCILLATION Weak currents for the interactions producing and absorbing neutrinos are described as 3 ¯ J = 2 l γ U ν , (1) µ Lα µ lαβ Lβ α,β=1 X where l (l = e, l = µ, l = τ) represents the leptonflavor, ν the neutrino mass eigenstate α 1 2 3 β andU istheleptonmixingmatrix. U istheunitarymatrixcorrespondingtotheCKMmatrix V† for quarks defined by CKM U = U U†, (2) l ν where the unitary matrices U and U transform mass matrices M for charged leptons and l ν l M for neutrinos to diagonal mass matrices as ν U MlU−1 = diag[m ,m ,m ], l l e µ τ (3) U MνU−1 = diag[m ,m ,m ]. ν ν 1 2 3 We present the unitary matrix neglecting the CP violation phases as 2 U = eiθ23λ7eiθ13λ5eiθ12λ2 cν cν sν cν sν 12 13 12 13 13   = sν cν cν sν sν cν cν sν sν sν sν cν , (4) − 12 23 − 12 23 13 12 23 − 12 23 13 23 13       sν sν cν cν sν cν sν sν cν sν cν cν   12 23 − 12 23 13 − 12 23 − 12 23 13 23 13     cν = cosθν, sν = sinθν,  ij ij ij ij where λ ’s are Gell-Mann matrices. i The probabilities for transitions ν ν are written as lα → lβ P(ν ν )= < ν (t) ν (0) > 2 = δ +p12 S +p23 S +p31 S , lα → lβ | lβ | lα | lαlβ νlα→νlβ 12 νlα→νlβ 23 νlα→νlβ 31 p12 = 2δ (1 2U2 )+2(U2 U2 +U2 U2 U2 U2 ), νlα→νlβ − lαlβ − lα3 lα1 lβ1 lα2 lβ2 − lα3 lβ3 p23 = 2δ (1 2U2 )+2( U2 U2 +U2 U2 +U2 U2 ), νlα→νlβ − lαlβ − lα1 − lα1 lβ1 lα2 lβ2 lα3 lβ3 p31 = 2δ (1 2U2 )+2(U2 U2 U2 U2 +U2 U2 ), νlα→νlβ − lαlβ − lα2 lα1 lβ1 − lα2 lβ2 lα3 lβ3 (5) where S is the term representing the neutrino oscillation; ij ∆m2 S = sin21.27 ijL, (6) ij E in which ∆m2 = m2 m2 , E and L are measured in units eV2, MeV and m, respectively. ij | i − j| From the unitarity of U, we get relations pij +pij +pij = 0, i, j = 1,2,3, l = e,µ,τ. (7) νlα→νe νlα→νµ νlα→ντ α The values of neutrino masses are not known precisely, but we know that if one identifies thedark matter of universe (or atleast itshot dark matter component) with neutrino matter one has [9] m +m +m several eV. (8) 1 2 3 ∼ Wedonotusethisvaluestrictlyinpresent analysis, butwe consider thatthesumofneutrino masses is not so small. In two-flavor neutrinos analyses in which one mass parameter ∆m2 appears for solar neutrino experiments, the result that ∆m2 is 10−4 10−5eV2 or 10−10eV2 − ∼ isobtained[21]. Foratmosphericexperiments, ∆m2 isobtainedas10−1 10−2eV2 [21]. Then − it seems that two neutrinos masses inthree neutrinos arevery close andanother oneisrather far away from them. Then we assume that three neutrino masses have such a mass hierarchy as m m m . (9) 1 2 3 ≈ ≪ In the the mass hierarchy Eq. (9), ∆m2 ∆m2 ∆m2 , the expression Eq. (5) for the 12 ≪ 23 ≃ 13 transition probabilities P(ν ν ) are rewritten as lα → lβ 3 P(ν ν ) = 1 2(1 2U2 U4 U4 +U4 )S 4U2 (1 U2 )S , e → e − − e3 − e1 − e2 e3 12 − e3 − e3 23 P(ν ν ) = 1 2(1 2U2 U4 U4 +U4 )S 4U2 (1 U2 )S , µ → µ − − µ3 − µ1 − µ2 µ3 12 − µ3 − µ3 23 P(ν ν ) = 1 2(1 2U2 U4 U4 +U4 )S 4U2 (1 U2 )S , τ → τ − − τ3 − τ1 − τ2 τ3 12 − τ3 − τ3 23 P(ν ν ) = P(ν ν ) = 2(U2 U2 +U2 U2 U2 U2 )S +4U2 U2 S , µ → e e → µ µ1 e1 µ2 e2 − µ3 e3 12 e3 µ3 23 P(ν ν ) = P(ν ν ) = 2(U2 U2 +U2 U2 U2 U2 )S +4U2 U2 S , τ → e e → τ τ1 e1 τ2 e2 − τ3 e3 12 e3 τ3 23 P(ν ν ) = P(ν ν ) = 2(U2 U2 +U2 U2 U2 U2 )S +4U2 U2 S . τ → µ µ → τ τ1 µ1 τ2 µ2 − τ3 µ3 12 µ3 τ3 23 (10) III. NUMERICAL ANALYSES OF NEUTRINO OSCILLATION IN THREE-FLAVOR NEUTRINOS A. Solar neutrinos We first analyze the solar neutrino experiments. Considering the matter effects (MSW effect [22]) in three-flavor neutrinos, the transition probability for ν ν is expressed as e e → [17] PMSW(∆m2 , θ , θ , E) = cos4θ PMSW(∆m2 , θ , θ , E)+sin4θ C (11a) 3ν 12 12 13 13 2ν 12 12 13 13 1 1 PMSW(∆m2 , θ , θ , E) = + Θ(Acos2θ ∆m2 cos2θ )P (θ ,E) 2ν 12 12 13 2 2 − 13 − 12 12 c 12 (cid:18) (cid:19) cos2θ cos2θm, (11b) × 12 12 π P (θ , E) = exp( γ(θ ,E)), (11c) c 12 12 −2 ∆m2 sin22θ γ(θ , E) = 12 12 , (11d) 12 2Ecos2θ dN /N dx 12 e e 0 | | whereΘisthethetafunction, A = 2√2G N E (G isFermiconstant, N numberofelectron F e F e per cm3 at the production point of neutrinos in the sun and E the energy of neutrino), P c the Landau-Zener-Stueckerberg crossing probability, γ is the adiabaticity parameter ( 0 |···| represents the value at the production point of neutrinos) and θm the mixing angle at 12 the production point. PMSW is the transition probability with a replacement of N 2ν e → N cos2θ in two-flavor neutrinos transition probability. This expression is obtained from e 13 an approximation; π (m2 Λ/2) sin22θ Θ[A (m2 Λ/2)cos2θ ]exp 3 − 13 1, (12) − 3 − 13 −4 dNe/Nedx 0E cos2θ13 ! ≪ | | where Λ = ((m2+m2) (m2 m2)cos2θ ). This approximation isreasonable for present as- 1 2 − 2− 1 12 sumptionofmasshierarchyEq.(9)becauseofA m2 Λ/2andm2 Λ/2 dN /N dx E. ≪ 3− 3− ≫ | e e |0 4 The ratios of the detected e neutrino fluxes to the expected e neutrino fluxes deduced from the standard solar model (SSM) [23] are expressed by using the transition probability PMSW as 3ν EmaxPMSW(E)f(E)dE R = Emin 3ν , (13) R Emaxf(E)dE Emin R where f(E) is the product of the spectral function of neutrino flux and detector sensitivity. The neutrino flux is the sum of many neutrino fluxes produced by the various nuclear fusion reaction at the center of the sun. The detector sensitivity also depends to the neutrino energy. Numerical results for neutrino flux and detector sensitivity are analyzed precisely in Ref. [23]. For f(E) used in this paper, see Appendix A. In our analysis, we use the following three experimental data for R: Ga experiment [2,3]: R = 0.534 0.087, (14a) ± Cl experiment [4]: R = 0.274 0.046, (14b) ± water Cherenkov experiment [5]: R = 0.437 0.092. (14c) ± In Ga experiment, we combined the SAGE [2] and GALLEX [3] data. First we estimate the numerical values R for θ = 0 which corresponds to two-flavor neutrinos case using the 13 Eq. (13), and show the contours of R on sin22θ ∆m2 plane in Fig. 1. Each contour 12 12 − denoted as Ga, Cl and Kam corresponds to the upper and lower values of R in Eq. (14) for Ga, Cl and Kamiokande’s water Cherenkov experiment. There are two solutions which are denoted as common areas enclosed by each two contours of Ga, Cl and Kam. These are called as large solution and small solution. This result is similar to the one obtained in various analyses [24]. Next we estimate the numerical values R for the case of θ = 0. In Fig. 2, we show the 13 6 allowed regions of the combined Ga, Cl and Kam experiments using the χ-square, where the solid thin, solid thick and dotted lines define the regions allowed at 99%(χ2 = 9.2), 95%(χ2 = 6.0) and 90%(χ2 = 4.5) C.L., respectively. Fig. 2(a) - Fig. 2(h) show the allowed regions for θ = 0◦ 50◦. Fig. 2(a) shows the θ = 0 case, thus this shows the two-flavor 13 13 − neutrinos’ solution. These results are similar to the ones obtained by Ref. [18]. Numerically, we show the allowed regions in 95% C.L.; for θ = 0◦ 20◦ 13 − ∆m2 = 4 10−6 7 10−5eV2 12 × − × large angle solution (15a) sin22θ12 = 0.6 0.9  −  5 ∆m2 = 3 10−6 1.2 10−5eV2 12 × − × small angle solution (15b) sin22θ12 = 0.003 0.01  − for θ13 = 25◦ −40◦  ∆m2 = 2 10−6 3 10−5eV2 12 × − × small angle solution (15c) sin22θ12 = 0.001 0.01  − for θ13 = 45◦ −50◦  ∆m2 = 2 10−6 3 10−5eV2 12 × − ×  (15d) sin22θ12 = 0.001 0.7  −  The characteristic feature of three-flavor neutrinos’ solution is as follows; increasing θ from 13 0◦ to 25◦, the small mixing solution and the large mixing solution merge in a single solution, further increasing θ to 50◦, the allowed region becomes broader and next shrinks and lastly 13 disappears. B. Terrestrial neutrinos In terrestrial experiments, there are two types of experiment: short baseline and long baseline experiment. In present study, we analyze the short baseline experiment. In the short baseline experiments, there are E531 [10], CHORUS and NOMAD [11] accelerator experiments searching for ν appearance in ν . We use the data of E531, CHORUS and τ µ NOMAD experiments; P(ν ν ) < 2 10−3 (90% C.L.), (16) µ τ → × L/E 0.02. ∼ For the experiments searching for ν ν and ν¯ ν¯ oscillations, there are E776 [12], µ e µ e → → KARMEN [13] and LSND [14] accelerator experiments; P(ν ν ) < 3 10−3 (90% C.L.), E776 (17a) µ e → × L = 1km, E 1GeV, ∼ P(ν¯ ν¯ ) < 3.1 10−3 (90% C.L.), KARMEN (17b) µ e → × L = 17.5m, E < 50MeV, P(ν¯ ν¯ ) = 3.4+2.0 0.7 10−3, LSND (17c) µ → e −1.8 ± × L = 30m, E 36 60MeV. ∼ − Furthermore, we analyse the experiments using nuclear power reactor [15] searching for the disappearance of ν¯ , e 6 1 P(ν¯ ν¯ ) < 10−2 (90% C.L.), (18) e e − → L = 15, 40, 95m, E 1 6MeV. ∼ − < In the short baseline experiments, the neutrino propagation length L is about L 1km, ∆m2 ∼ then S = sin21.27 12L is very small because ∆m2 10−5 10−4eV2 suggested from 12 E 12 ∼ − last solar neutrino analyses. Then, the S term is dominant in the transition probability 23 Eq.(10). SeeingthatthemixingpartsproportionaltoS are4U2 (1 U2 ),4U2 (1 U2 )and 23 e3 − e3 µ3 − µ3 4U2 (1 U2 ), the transition probabilities in present short baseline terrestrial experiments τ3 − τ3 seem to be insensitive to the parameters; ∆m2 and θ . 12 12 We show the contour plots of the allowed regions on (tan2θ , tan2θ ) plane determined 13 23 by the probability P expressed in Eq. (10) and above experimental data Eqs. (16), (17) and (18) in Fig. 3. Allowed regions are corners, left and right hand sides surrounded by curves. Curves represent the boundary of 90 % C.L. of P. We fixed the parameters ∆m2 and 12 θ as ∆m2 = 10−5eV2 and sin22θ = 0.8, and the parameter ∆m2 to be various values 12 12 12 13 from 0.02eV2 to 20eV2. Although we fix the parameters ∆m2 and θ as ∆m2 = 10−5eV2 12 12 12 and sin22θ = 0.01, the results are not changed. Dotted lines show the allowed regions 12 restricted by the LSND data. From these results, we see that the numerical allowed regions on (θ , θ ) without the LSND data are as follows; 13 23 for ∆m2 = 20eV2 23 (< 4◦, < 2◦), (< 2◦, > 88◦), (> 86◦ 88◦, any), (19a) − for ∆m2 = 2eV2 23 (< 4◦, < 26◦), (< 2◦, > 65◦), (> 86◦ 88◦, any), (19b) − for ∆m2 = 0.2eV2 23 (< 4◦, any), (> 86◦, any), (19c) for ∆m2 = 0.02eV2 23 (< 12◦, any), (> 78◦, any) (19d) for ∆m2 < 0.005eV2 23 all regions are allowed. (19e) If we combine the LSND data with the above analyses, the allowed region disappears lower than 0.2eV2 of the ∆m2 value. Furthermore, combining the solar neutrino solutions 23 with this terrestrial ones, the allowed regions larger than 50◦ of θ on terrestrial neutrino 13 are excluded. C. Atmospheric neutrinos The evidence for an anomaly in atmospheric neutrino experiments was pointed out by the Kamiokande Collaboration [6,7] and IMB Collaboration [8] using the water-Cherencov 7 experiments. More recently, SuperKamiokande Collaboration [19,20] reports the more pre- cise results on anomaly in atmospheric neutrino. We tabulate these results in Table II. That the double ratio, R(µ/e) R (µ/e)/R (µ/e), is less than 1 is the atmospheric expt MC ≡ neutrino’s anomaly. In Table II, sub-GeV experiments detect the visible-energy less than 1.33GeV, and in the second column (total exposure), left number represents the detector exposure in which fully contained events are detected and right numbers partially contained events. The ratios R (µ/e) and R (µ/e) are defined as expt MC ǫ (E )σ (E ,E )F (E ,θ )P(ν ν )dE dE dθ R (µ/e) = α µ µ µ α µ α α α α → µ µ α α, (20a) expt ǫ (E )σ (E ,E )F (E ,θ )P(ν ν )dE dE dθ P αR e e e α e α α α α e e α α → ǫ (E )σ (E ,E )F (E ,θ )dE dE dθ R (µ/e) = Pα R µ µ µ α µ α α α µ α α, (20b) MC ǫ (E )σ (E ,E )F (E ,θ )dE dE dθ PαR e e e α e α α α e α α where the summationP Rα are taken in µ, e neutrino and µ, e untineutrino. ǫβ(Eβ) is the detection efficiency of the detector for β-type charged lepton with energy E , σ is the dif- P β β ferential cross section of ν and F (E ,θ ) is the incident ν flux with energy E and zenith β α α α α α angle θ . P(ν ν ) is the transition probability Eq. (10) and it depends on the energy E α α β α → and the distance Lwhich depends onzenith angleθ as L = (r +h)2 r2sin2θ rcosθ , α α α − − where r is the radius of the Earth and h is the altitude of pqroduction point of atmospheric neutrino. Although informations of F (E ,θ ) etc. are given in Refs. [25–27], we use the MC α α α predictions for f (E ,θ ) ǫ (E )σ (E ,E )F (E ,θ )dE in Ref. [6] for sub-GeV α α α α µ µ µ α µ α α α µ ≡ experiment and Ref. [7] for multi-GeV experiment. Explicit E dependence of f (E ,θ ) P R α α α α are shown in Appendix B. Since P(ν ν ) and P(ν ν ) are the functions of α µ α e → → (∆m2 ,∆m2 ,θ ,θ ,θ ,L,E), the double ratio R(µ/e) which is integrated in neutrino 12 23 12 13 23 energy E and zenith angle θ (related to L) is the function of (∆m2 ,∆m2 ,θ ,θ ,θ ). We 12 23 12 13 23 estimate the R(µ/e) fixing the parameters (δm2 ,sin22θ) on the allowed values Eq. (15) 12 predicted by the solar neutrino experiments; δm2 = 3 10−5eV2, sin22θ = 0.7 which cor- 12 12 × responds to large angle solution and δm2 = 10−5eV2, sin22θ = 0.005 which corresponds 12 12 to small angle solution. In Fig. 4, we showed the contour plots of double ratio R(µ/e) on tan2θ tan2θ plane 13 23 − for various ∆m2 . Contoure lines correspond the upper and lower values of R(µ/e) in Table 23 II. We showed the plots of sub-GeV experiment in Fig. 4(a)-(d) and plots of multi-GeV one in Fig. 4(e)-(h), and in these figures solid lines denote the large angle solution plots and dotted lines the small angle solution plots. In Fig. (a)-(c), the allowed regions are surrounded by two solid lines (or dotted lines) and in Fig. (d), by two outer lines (or dotted lines) and inner solid line (or dotted line). In Fig. (e)-(h), dotted lines are close in solid lines. In Fig. (e)-(g), allowed regions are surrounded by two outer solid lines (or dotted lines) and inner solid line (or dotted line) and in Fig. (h), by two solid lines (or dotted lines). Observing the allowed regions obtained terrestrial neutrino experiments Fig. 3 and the present allowed regions obtained atmospheric neutrino experiments, we obtain the allowed 8 regions satisfying all experiments. First, we show the allowed regions of (θ , θ ) for the 13 23 large angle solution and small angle solution in sub-GeV experiment: large angle solution small angle solution for ∆m2 = 20eV2 no allowed region no allowed region (21a) 23 (θ < 4◦, 24◦ < θ < 26◦) 13 23 for ∆m223 = 2eV2 (θ13 < 4◦, 18◦ < θ23 < 26◦)  (21b)  (θ13 < 2◦, 64◦ < θ23 < 66◦) for ∆m223 = 0.2eV2 (θ13 < 4◦, 18◦ < θ23 < 62◦) (θ13 < 4◦, 24◦ < θ23 < 66◦) (21c) (θ < 12◦, 17◦ < θ < 38◦) (θ < 14◦, 23◦ < θ < 41◦) 13 23 13 23 for ∆m2 = 0.02eV2   (21d) 23 (θ13 < 12◦, 47◦ < θ23 < 63◦)  (θ13 < 14◦, 49◦ < θ23 < 67◦)  (θ < 29◦, 27◦ < θ < 52◦)  (θ < 34◦, 34◦ < θ < 56◦) 13 23 13 23 for ∆m2 = 0.002eV2  (21e) 23 (39◦ < θ13 < 59◦, 76◦ < θ23)  (34◦ < θ13 < 56◦, 56◦ < θ23) second, in multi-GeVexperiment:  large angle solution small angle solution for ∆m2 = 20eV2 no allowed region no allowed region (22a) 23 (θ < 4◦, 24◦ < θ < 26◦) (θ < 4◦, 24◦ < θ < 26◦) 13 23 13 23 for ∆m2 = 2eV2   (22b) 23  (θ13 < 2◦, 64◦ < θ23 < 66◦) (θ13 < 2◦, 64◦ < θ23 < 66◦) for ∆m223 = 0.2eV2 (θ13 < 4◦, 24◦ < θ23 < 66◦) (θ13 < 4◦, 24◦ < θ23 < 66◦) (22c) for ∆m2 = 0.02eV2 (θ < 12◦, 41◦ < θ < 49◦) (θ < 12◦, 41◦ < θ < 49◦) (22d) 23 13 23 13 23 for ∆m2 = 0.002eV2 (22◦ < θ < 68◦, 45◦ < θ ) (22◦ < θ < 68◦, 45◦ < θ ) (22e) 23 13 23 13 23 If we combine the LSND experiment with above terrestrial data, allowed regions are restricted as follows, sub-GeV case: large angle solution small angle solution (2◦ < θ < 4◦, 13  for ∆m223 = 2eV2 (2◦ < θ13 < 4◦,18◦ < θ23 < 26◦)(0.8◦2<4◦θ<13θ<232<◦,26◦) (23a) for ∆m223 = 0.2eV2 no allowed region no allow64e◦d<reθg2io3n< 66◦) (23b) multi-GeV case: 9 large angle solution small angle solution (2◦ < θ < 4◦, (2◦ < θ < 4◦, 13 13   for ∆m223 = 2eV2  (0.8◦ 2<4◦θ1<3 <θ232◦<, 26◦)  (0.8◦2<4◦θ<13θ<232<◦,26◦) (24a) for ∆m223 = 0.2eV2 no allo6w4e◦d<reθg23io<n 66◦) no allow64e◦d<reθg2io3n< 66◦) (24b) Although there are allowed regions satisfying both terrestrial and atmospheric exper- imental restrictions, all of these solutions do not satisfy the zenith angle dependence of R(µ/e) for atmospheric neutrino experiments. From resent SuperKamiokande experiments [19,20], zenith angle dependence seems to be more definite than that obtained previously [6,7]: the double ratio R(µ/e) for sub-GeV experiment seems to decrease monotonically as zenith angle θ increases from θ = 0◦ to θ = 180◦ and R(µ/e) for multi-GeV experi- ments seems to decrease as zenith angle θ increases from θ = 0◦ to θ = 180◦. Among the allowed solutions obtained above, Eqs. (21) and (22) (or (23) and (24)), the large angle (sin22θ 0.7) solutions with θ < 45◦ satisfy the monotonic decreasing of the R(µ/e) 12 23 ∼ in sub-GeV neutrino experiment. This solution also decreases in multi-GeV experiment as zenith angle increases. In Fig. 5(a), we showed the zenith angle (cosθ) dependence of R(µ/e) in sub-GeV exper- iment on the typical parameters for large angle solution (∆m2 = 3 10−5eV2, sin22θ = 12 × 12 0.7, ∆m = 0.2eV2, θ = 4◦, θ = 25◦) by solid curve and small angle solution 23 13 23 (∆m2 = 10−5eV2, sin22θ = 0.007, ∆m = 0.2eV2, θ = 4◦, θ = 25◦) by dotted curve. 12 12 23 13 23 Experimental data is referred to SuperKamiokande [20]. Fig. 5(b) represents the zenith an- gle (cosθ) dependence of R(µ/e) in multi-GeV experiment on the parameters for large angle solution (∆m2 = 3 10−5eV2, sin22θ = 0.7, ∆m = 0.2eV2, θ = 4◦, θ = 30◦) (solid 12 × 12 23 13 23 curve) and small angle solution (∆m2 = 10−5eV2, sin22θ = 0.007, ∆m = 0.2eV2, θ = 12 12 23 13 4◦, θ = 30◦) (dotted curve). 23 We summarize the solution satisfying the solar, terrestrial and atmospheric experiments as follows; for ∆m2 = 2eV2 23 ∆m2 = 4 10−6 7 10−5eV2, sin22θ = 0.6 0.9,  12 × − × 12 − (25a) θ13 < 4◦, 24◦ < θ23 < 26◦, for∆m223 = 0.2eV2 ∆m2 = 4 10−6 7 10−5eV2, sin22θ = 0.6 0.9,  12 × − × 12 − (25b) θ13 < 4◦, 24◦ < θ23 < 45◦. for∆m223 = 0.02eV2 10