24 December 1998

Ž .Physics Letters B 444 1998 379–386

Implications of CHOOZ results for the decoupling of solar andatmospheric neutrino oscillations

S.M. Bilenky a,b, C. Giunti b

a Joint Institute for Nuclear Research, Dubna, Russiab INFN, Sezione di Torino, and Dipartimento di Fisica Teorica, UniÕersita di Torino, Via P. Giuria 1, I–10125 Torino, Italy`

Received 27 February 1998; revised 4 August 1998Editor: R. Gatto

Abstract

We have considered the results of solar and atmospheric neutrino oscillation experiments in the scheme of mixing ofŽ 2 2 y3 2 .three neutrinos with a mass hierarchy. It is shown that if m ym )10 eV the recent results of the CHOOZ3 1

< < 2 Ž .experiment imply that U <1 U is the neutrino mixing matrix , that the oscillations of solar neutrinos are described bye3

the two-generation formalism and that the oscillations of solar and atmospheric neutrinos decouple. It is also shown that if< < 2 < <not only U <1 but also U <1, then the oscillations of atmospheric neutrinos do not depend on matter effects ande3 e3

are described by the two-generation formalism. In this case, with an appropriate identification of the mixing parameters, thetwo-generation analyses of solar and atmospheric neutrino data provide direct information on the mixing parameters of three

< < 2neutrinos. We discuss the possibility to get information on U in long-baseline neutrino oscillation experiments. q 1998e3

Published by Elsevier Science B.V. All rights reserved.

PACS: 14.60.Pq; 26.65.q t; 95.85.Ry

1. Introduction

w x w xThe existence of the solar 1 and atmospheric 2neutrino problems have received in the last monthsimpressive confirmations from the preliminary re-

w xsults of the SuperKamiokande experiment 3,4 . Thew xtraditional analyses of solar 5–8,3 and atmospheric

w x9–11,4 neutrino data in terms of neutrino oscilla-Ž w x.tions see 12–14 have been performed under the

assumption of two-neutrino mixing. In particular,this assumption has been adopted in the most recent

w x w xanalyses of the solar 15,16 and atmospheric 17neutrino data, which include preliminary data from

w xSuperKamiokande 3,4 . However, we know that

there are three light flavor neutrinos, n , n , n , thate m t

can participate to the oscillations of solar and atmo-spheric neutrinos and it is reasonable to ask whatinformation on the mixing of three neutrinos can beobtained from the two-generation analyses of solarand atmospheric data.

Here we consider a scheme of mixing of threemassive neutrinos with the mass hierarchy

m <m <m , 1Ž .1 2 3

w xwhich is motivated by the see-saw mechanism 18and by the analogy with the mass spectra of chargedleptons and up and down quarks. The masses mk

0370-2693r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01418-X

( )S.M. Bilenky, C. GiuntirPhysics Letters B 444 1998 379–386380

Ž .ks1,2,3 are associated to the massive neutrinofields n , whose left-handed components n arek k L

Žconnected with the left-handed components n aaL.se,m,t of the flavor neutrino fields by the relation

3

n s U n , 2Ž .ÝaL a k k Lks1

where U is a 3=3 unitary mixing matrix. In thisscheme there are two independent mass-squared dif-ferences, Dm2 'm2 ym2 and Dm2 'm2 ym2,21 2 1 31 3 1

that generate neutrino oscillations for two differentscales of the ratio ErL, where E is the neutrinoenergy and L is the distance of propagation. Thesolar and atmospheric neutrino anomalies can beexplained in terms of neutrino oscillations only ifDm2 and Dm2 are relevant, respectively, for the21 31

woscillations of solar and atmospheric neutrinos 19–x24 .In Sections 2 and 3 we will show that in the

scheme under consideration, if Dm2 )10y3 eV 2,31w xthe recent results of the CHOOZ experiment 25

< < 2imply that U <1, that the oscillations of solare3

neutrinos are described by the two-generation for-malism and that the oscillations of solar and atmo-spheric neutrinos depend on different and indepen-dent elements of the neutrino mixing matrix, i.e.they decouple. In Section 3 it is shown that if not

< < 2 < <only U <1 but also U <1, then the oscilla-e3 e3

tions of atmospheric neutrinos do not depend onmatter effects and are described by the two-genera-tion formalism. In this case, with an appropriateidentification of the mixing parameters, the formal-ism used in the two-generation analyses of solar andatmospheric neutrino data is appropriate also in thecase of three-neutrino mixing and the results of theseanalyses provide direct information on the mixingparameters of three neutrinos. In Section 4 we dis-cuss the implications of the present solar and atmo-spheric neutrino data for the values of the elementsof the neutrino mixing matrix. As shown in Section

< <5, the hypothesis U <1 could be tested by futuree3

accelerator long-baseline neutrino oscillation experi-ments.

In this paper we do not consider the indication infavor of neutrino oscillations obtained by the LSND

w xexperiment 26 , which needs the enlargement of theneutrino mixing scheme with the introduction of a

Ž w x.sterile neutrino see 27,28 . A possible decoupling

of solar and atmospheric neutrino oscillations in thisenlarged scheme will be discussed elsewhere.

2. CHOOZ and solar neutrinos

The first reactor long-baseline neutrino oscillationexperiment CHOOZ found no evidence for neutrino

w xoscillations 25 . The CHOOZ collaboration has pub-lished an exclusion curve in the plane of the two-generation mixing parameters which shows that

sin2 2q F0.18 for Dm2 R10y3 eV 2 .CHOOZ CHOOZ

3Ž .

In the three-neutrino scheme under considerationŽ w x.these parameters are given by see 23,24

2 < < 2 < < 2sin 2q s4 U 1y UŽ .CHOOZ e3 e3

and Dm2 sDm2 . 4Ž .CHOOZ 31

If Dm2 )10y3 eV 2, as indicated by the solution of31w xthe atmospheric neutrino anomaly 9 , the upper

Ž . 2bound 4 on sin 2q implies thatCHOOZ

< < 2 y2 < < 2U F5=10 or U G0.95 . 5Ž .e3 e3

< < 2A large value of U does not allow to explaine3

the solar neutrino problem with neutrino oscillationsŽ w x.see 29 . Indeed, in the scheme under considerationthe averaged survival probability of solar electron

w xneutrinos is given by 1922 4sun Ž1 ,2.< < < <P E s 1y U P E q U ,Ž . Ž .Ž .n ™ n e3 n ™ n e3e e e e

6Ž .Ž1,2. Ž .where E is the neutrino energy and P E is then ™ ne e

two-generation survival probability of solar n ’se

which depends on

< <Ue22 2Dm sDm and sinq s . 7Ž .sun 21 sun 2< <(1y Ue3

Ž . sun Ž . < < 4The expression 6 implies that P E G Un ™ n e3e e

< < 2 sun Ž .and for U G0.95 we have P E G0.90.e3 n ™ ne e

With such a high and practically constant value ofsun Ž .P E it is not possible to explain the suppres-n ™ ne e

sion of the solar n flux measured by all experimentseŽ w x w x w xHomestake 5 , Kamiokande 6 , GALLEX 7 ,

w x w x.SAGE 8 and SuperKamiokande 3 with respect tow xthat predicted by the Standard Solar Model 30–32 .

( )S.M. Bilenky, C. GiuntirPhysics Letters B 444 1998 379–386 381

Therefore, we are led to the conclusion that the< < 2results of the CHOOZ experiment imply that Ue3

is small:

< < 2 y2U F5=10 . 8Ž .e3

Ž . Ž .Furthermore, from Eqs. 6 and 7 one can see< < 2that for such small values of U we havee3

sun Ž1 ,2. < <P E ,P E and sinq , U .Ž . Ž .n ™ n n ™ n sun e2e e e e

9Ž .

Hence the two-generation analyses of the solar neu-trino data are appropriate in the three-neutrino schemewith a mass hierarchy and they give information onthe values of

2 2 < <Dm sDm and U ,sinq . 10Ž .21 sun e2 sun

According to the most recent analysis of the solarw xneutrino data 16 , which include preliminary data

w xfrom SuperKamiokande 3 , the ranges of the mixingparameters allowed at 90% CL for the small and

w xlarge mixing angle MSW 33 solutions and for thevacuum oscillation solution are, respectively,

4=10y6 eV 2 QDm2 Q1.2=10y5 eV 2 ,sun

3=10y3 Qsin2 2q Q1.1=10y2 , 11Ž .sun

8=10y6 eV 2 QDm2 Q3.0=10y5 eV 2 ,sun

0.42Qsin2 2q Q0.74 , 12Ž .sun

6=10y11 eV 2 QDm2 Q1.1=10y10 eV 2 ,sun

0.70Qsin2 2q F1 . 13Ž .sun

< < 2Therefore, taking into account that U is smalle3< < < < Žand assuming that U F U this choice is neces-e2 e1

.sary only for the MSW solutions , we have

< <U ,1 ,e1

< <U ,0.03y0.05 small mixing MSW , 14Ž . Ž .e2

< <U ,0.87y0.94 ,e1

< <U ,0.35y0.49 large mixing MSW , 15Ž . Ž .e2

< <U ,0.71y0.88 ,e1

< <U ,0.48y0.71 vacuum oscillations . 16Ž . Ž .e2

Notice that these ranges are statistically rather stable.For example, the range of sin2 2q allowed at 99%sun

CL in the case of the large mixing angle MSW

2 w xsolution is 0.36Qsin 2q Q0.85 16 , which implysun< < < < ŽU , 0.83 y 0.95, U , 0.32 y 0.55 confronte1 e2

Ž ..with Eq. 15 .

3. Atmospheric neutrinos

The evolution equation for the flavor amplitudesŽ .c ase,m,t of atmospheric neutrinos propagat-a

Žing in the interior of the Earth can be written as seew x.34,24

d 12 †i Cs U M U qA C , 17Ž . Ž .

d t 2 E

with

ce

2 2 2cC' , M 'diag 0,Dm ,Dm ,Ž .m 21 31� 0ct

A'diag A ,0,0 , 18Ž . Ž .CC

'and A '2 EV , where V s 2 G N is theCC CC CC F e

charged-current effective potential which depends onŽthe electron number density N of the medium Ge F

is the Fermi constant and for anti-neutrinos ACC.must be replaced by A syA . If the squared-CC CC

mass difference Dm2 is relevant for the explanation21

of the solar neutrino problem, we have

Dm2 R21 [<1 , 19Ž .

2 E

where R s6371 Km is the radius of the Earth.[Notice, however, that caution is needed for low-en-ergy atmospheric neutrinos if Dm2 R10y5 eV 2, as21

in the case of the large mixing angle MSW solutionof the solar neutrino problem and marginally in the

Žcase of the small mixing angle MSW solution seeŽ . Ž .. 2 y5 2Eqs. 12 and 11 . Indeed, if Dm R10 eV we21

have Dm2 R r2 E<1 only for E4150 MeV. In21 [this case, in order to get information on the three-neutrino mixing matrix with a two-generation analy-sis it is necessary to analyze the atmospheric neu-trino data with a cut in energy such thatDm2 R r2 E<1. In order to be on the safe side,21 [when we will consider the case of the MSW solu-tions of the solar neutrino problem we will take intoaccount the information obtained from the two-gen-eration fit of the preliminary SuperKamiokande

Ž w x.multi-GeV data alone Fig. 11 of 17 .

( )S.M. Bilenky, C. GiuntirPhysics Letters B 444 1998 379–386382

Ž .The inequalities 19 imply that the phase gener-ated by Dm2 can be neglected for atmospheric21

neutrinos and M 2 can be approximated with

M 2 ,diag 0,0,Dm2 . 20Ž .Ž .31

ŽIn this case taking into account that the phases ofthe matrix elements U can be included in thea 3

.charged lepton fields we have

2 †X

2 < < < <XU M U ,Dm U U . 21Ž . Ž .a a 31 a 3 a 3

Ž . Ž .Comparing this expression with Eqs. 9 and 10 ,one can see that the oscillations of solar and atmo-spheric neutrinos depend on different and indepen-dent Dm2 ’s and on different and independent ele-ments of the mixing matrix, i.e. they are decoupled.

< < Ž . Ž .Strictly speaking U in Eqs. 9 and 10 is note2< <independent from U because of the unitarity con-e3

< < 2 < < 2 < < 2 Ž .straint U q U q U s1, but the limit 8e1 e2 e3< < 2on U implies that its contribution to the unitaritye3

constraint is negligible.Hence, we have shown that the smallness of

< < 2U inferred from the results of the CHOOZ exper-e3

iment imply that the oscillations of solar and atmo-spheric neutrinos are decoupled.

Ž . Ž .From Eqs. 17 and 21 one can see that unless< <U <1, the evolution equations of the atmospherice3

electron neutrino amplitude c and those of thee

muon and tau neutrino amplitudes c and c arem t

coupled. In this case matter effects can contribute toŽ w x.the dominant n ™n oscillations see 35 and them t

atmospheric neutrino data must be analyzed with theŽ .three generation evolution Eq. 17 .

From the results of the CHOOZ experiment it< < 2follows that the quantity U is small and satisfye3

Ž . Ž 2 y3 2 .the inequality 8 for Dm R10 eV . However,31< < Ž .the upper bound for U implied by Eq. 8 is note3

< <very strong: U -0.22. In the following we wille3< < 2assume that not only U <1, but also the ele-e3

< <ment U that connects the first and third genera-e3< < Žtions is small: U <1 let us remind that in thee3

y2 < < y2 .quark sector 2=10 F V F5=10 . We willub

consider the other elements of the mixing matrix asfree parameters and we will see that these parameterscan be determined by the two-neutrino analyses ofthe solar and atmospheric neutrino data. In Section 5

< <it will be shown that the hypothesis U <1 can bee3

tested in future long-baseline neutrino oscillationexperiments.

< < Ž .If U <1, for the evolution operator in Eq. 17e3

we have the approximate expression

U M 2 U † qA

ACC0 02Dm31

2,Dm 22Ž .231 < < < < < <0 U U Um3 m3 t 3� 02< < < < < <0 U U Ut 3 m3 t 3

which shows that the evolution of n is decouplede

from the evolution of n and n . Thus, the survivalm t

probability of atmospheric n ’s is equal to one ande

n ™n transitions are independent from matter ef-m t

fects and are described by a two-generation formal-ism. In this case, the two-generation analyses of theatmospheric neutrino data in terms of n ™n arem t

appropriate in the three-neutrino scheme under con-sideration and yield information on the values of theparameters

2 2 < <Dm sDm and U ssinq . 23Ž .31 atm m3 atm

w xAccording to a recent analysis 17 of the atmo-spheric neutrino data, the ranges of Dm2 andatm

sin2 2q for n ™n oscillations allowed at 90%atm m t

CL by the SuperKamiokande multi-GeV data and byall data are, respectively,

4=10y4 eV 2 QDm2 Q8=10y3 eV 2 ,atm

0.72Qsin2 2q F1 , 24Ž .atm

4=10y4 eV 2 QDm2 Q6=10y3 eV 2 ,atm

0.76Qsin2 2q F1 . 25Ž .atm

< < < <Thus, assuming that U F U and taking intom3 t 3Ž .account the comments after Eq. 19 , we have

< <U ,0.49y0.71 ,m3

< <U ,0.71y0.87 MSW , 26Ž . Ž .t 3

< <U ,0.51y0.71 ,m3

< <U ,0.71y0.86 vacuum oscillations . 27Ž . Ž .t 3

Ž . Ž .As in the case of the ranges 14 – 16 , also theŽ . Ž .ranges 26 – 27 are statistically rather stable. For

example, the range of sin2 2q allowed at 99% CLsun

by all the atmospheric neutrino data is 0.66Q2 w x < <sin 2q F1 17 , which imply U ,0.46y0.71,sun m3

< < Ž Ž ..U ,0.71y0.89 confront with Eq. 27 .t

( )S.M. Bilenky, C. GiuntirPhysics Letters B 444 1998 379–386 383

4. The mixing matrix

Taking into account the unitarity of the mixingŽ . Ž . Ž .matrix, the information in Eqs. 14 – 16 and 26 –

Ž . < <27 , together with the assumption U <1, allowe3< <to infer the allowed ranges for the values of U ,m1

< < < < < <U , U and U . The simplest way to do it is tom2 t 1 t 2

start from the Maiani parameterization of a 3=3w xmixing matrix 36 :

c c s c s12 13 12 13 13

ys c y c s s c c y s s s s cU s ,12 23 12 23 13 12 23 12 23 13 23 13ž /s s y c c s yc s y s c s c c12 23 12 23 13 12 23 12 23 13 23 13

28Ž .where c 'cosq and s 'sinq and we havei j i j i j i j

Žomitted possible CP-violating phases one for Dirac.neutrinos and three for Majorana neutrinos on which

there is no information.< < < <A very small U implies that s <1. In thise3 13

case we have

c s <112 12

U, . 29Ž .ys c c c s12 23 12 23 23� 0s s yc s c12 23 12 23 23

< < < < < < < <Using the information on s , U and s , U12 e2 23 m3Ž . Ž . Ž . Ž .given by Eqs. 14 – 16 and 26 – 27 , for the mod-

uli of the elements of the mixing matrix we obtain:

,1 0.03 y 0.05 < 1Small mixing MSW : ,0.02 y 0.05 0.71 y 0.87 0.49 y 0.71ž /0.01y 0.04 0.48 y 0.71 0.71 y 0.87

30Ž .0.87y 0.94 0.35 y 0.49 < 1

Large mixing MSW : ,0.25 y 0.43 0.61 y 0.82 0.49 y 0.71ž /0.17y 0.35 0.42 y 0.66 0.71 y 0.87

31Ž .0.71y 0.88 0.48 y 0.71 < 1

Vacuum oscillations: .0.34 y 0.61 0.50 y 0.76 0.51 y 0.71ž /0.24y 0.50 0.36 y 0.62 0.71 y 0.86

32Ž .Let us remark that in the case of the small mixingangle MSW solution of the solar neutrino problem< <U <1 could be of the same order of magnitude ase3< <U .e2

It is interesting to notice that, because of the largemixing of n and n with n , the transitions of solarm t 2

n ’s in n ’s and n ’s are of comparable magnitude.e m t

However, this phenomenon and the values of theŽ . Ž .entries in the n ,n – n ,n sector of the mixingm t 1 2

matrix cannot be checked with solar neutrino experi-ments because the low-energy n ’s and n ’s comingm t

from the sun can be detected only with neutral-cur-rent interactions, which are flavor-blind. Moreover,

< <it will be very difficult to check the values of U ,m1< < < < < <U , U and U in laboratory experiments be-m2 t 1 t 2

cause of the smallness of m .2Ž . Ž .In the derivation of Eqs. 30 – 32 we have as-

< < < < < < < <sumed that U F U and U F U . The othere2 e1 m3 t 3< < < < < < < <possibilities, U G U and U G U , are equiv-e2 e1 m3 t 3

alent, respectively, to an exchange of the first andsecond columns and to an exchange of the second

Ž . Ž .and third rows in the matrices 30 – 32 . Unfortu-nately, these alternatives are hard to distinguish ex-perimentally because of the above mentioned diffi-

< < < <culty to measure directly the values of U , U ,m1 m2< < < < < < < <U and U . Only the choice U F U , which ist 1 t 2 e2 e1

necessary for the MSW solutions of the solar neu-trino problem, could be confirmed by the results ofthe new generation of solar neutrino experimentsŽSuperKamiokande, SNO, ICARUS, Borexino, GNO

w x.and others 37 if they will allow to exclude thevacuum oscillation solution.

5. Accelerator long-baseline experiments

Future results from reactor long-baseline neutrinoŽ w xoscillation experiments CHOOZ 25 , Palo Verde

w x w x.38 , Kam-Land 39 could allow to improve theŽ . < < 2upper bound 8 on U . In this section we discusse3

how an improvement of this upper bound could beobtained by future accelerator long-baseline neutrinooscillation experiments that are sensitive to n ™nm e

Ž w x w x w xtransitions K2K 40 , MINOS 41 , ICARUS 42w x.and others 43 .

If matter effects are not important, in the schemeunder consideration the parameter sin2 2q mea-me

sured in n ™n long-baseline experiments is givenm eŽ w x.by see 23,242 < < 2 < < 2sin 2q s4 U U . 33Ž .me e3 m3

If accelerator long-baseline neutrino oscillation ex-periments will not observe n ™n transitions andm e

will place an upper bound sin2 2q Fsin2 2q Žmax ., itme m e

will be possible to obtain the limit

sin2 2q Žmax .m e2< <U F , 34Ž .e3 2< <4 U Žmin .m3

( )S.M. Bilenky, C. GiuntirPhysics Letters B 444 1998 379–386384

< < 2 < < 2where U is the minimum value of UŽmin .m3 m3

allowed by the solution of the atmospheric neutrinoanomaly and by the observation of n ™n long-m t

< < 2baseline transitions. For example, taking U sŽmin .m3Ž Ž .. < < 2 2 Žmax .0.25 see Eq. 27 we have U Fsin 2q . Ife3 m e

a value of sin2 2q Žmax .,10y3, that corresponds tome

the sensitivity of the ICARUS experiment for onew xyear of running 42 , will be reached, it will be

< < y2possible to put the upper bound U Q3=10 .e3

The observation of n ™n transitions in long-m t

baseline experiments will allow to establish a lower< < 2 2bound for U because the parameter sin 2q ism3 mt

Žgiven in the scheme under consideration by seew x.23,24

2 < < 2 < < 2sin 2q s4 U U . 35Ž .mt m3 t 3

< < 2 < < 2 < < 2From the unitarity relation U q U q U s1e3 m3 t 3

it follows that an experimental lower bound sin2 2qmt2 Žmin . < < 2Gsin 2q allows to constraint U in themt m3

range

1 12Ž .2 min < <1y 1ysin 2q F U F( mt m3ž /2 2

= Ž .2 min1q 1ysin 2q . 36Ž .( mtž /If sin2 2q Žmin . is found to be close to one, as sug-mt

gested by the solution of the atmospheric neutrinoŽ Ž . Ž ..problem see Eqs. 24 and 25 , the lower bound

12 Ž .2 min< <U s 1y 1ysin 2q(Žmin .m3 mtž /2

is close to 1r2.If matter effects are important, the extraction of

< < 2an upper bound for U from the data of n ™ne3 m e

accelerator long-baseline experiments is more com-plicated. In this case the probability of n ™n oscil-m e

Ž w x.lations is given by see 24

< < 2 < < 24 U Ue3 m3P sn ™ n 2m e A ACC CC2< <1y q4 Ue32 2ž /Dm Dm31 31

22Dm L A A31 CC CC22 < <sin 1y q4 U ,) e32 2ž /4E Dm Dm� 031 31

37Ž .

where E is the neutrino energy and L is the distanceof propagation. This probability depends on the neu-trino energy not only through the explicit E in thedenominator of the phase, but also through the en-ergy dependence of A '2 EV . For long-base-CC CC

line neutrino beams propagating in the mantle of theEarth the charged-current effective potential VCC

y3's 2 G N is practically constant: N ,2 N cmF e e AŽ .N is the Avogadro number and V ,1.5=A CC

10y13 eV.If long-baseline experiments will not observe nm

Ž™n transitions or will find that they have ane.extremely small probability for neutrino energies

2 < < 2such that A QDm , it will mean that U isCC 31 e3

small and a fit of the experimental data with theŽ .formula 37 will yield a stringent upper limit for

< < 2 Ž < < 2U taking into account the lower limit U Ge3 m3< < 2U obtained from the solution of the atmo-Žmin .m3

spheric neutrino anomaly and from the observation.of n ™n long-baseline transitions . On the otherm t

hand, the non-observation of n ™n transitions form e

neutrino energies such that A 4Dm2 does notCC 31< < 2provide any information on U because in thise3Ž .case the transition probability 37 is suppressed by

the matter effect. Hence, in order to check the hy-< <pothesis U <1, as well as to have some possibil-e3

ity to observe n ™n transitions if this hypothesism e

is wrong, it is necessary that a substantial part of theenergy spectrum of the neutrino beam lies below

Dm2 Dm231 31

,30 GeV . 38Ž .y2 2ž /2V 10 eVCC

This requirement will be satisfied in the acceleratorŽlong-baseline experiments under preparation K2K

w x w x w x w x.40 , MINOS 41 , ICARUS 42 and others 43 ifDm2 is not much smaller than 10y2 eV 2.31

6. Conclusions

We have considered the scheme of mixing ofŽ .three neutrinos with the mass hierarchy 1 and with

Dm2 and Dm2 relevant, respectively, for the oscil-21 31

lations of solar and atmospheric neutrinos.We have shown that in the framework of this

scheme, if Dm2 )10y3 eV 2, the recent results of31w x < < 2the CHOOZ experiment 25 imply that U ise3

small, the oscillations of solar neutrinos are de-

( )S.M. Bilenky, C. GiuntirPhysics Letters B 444 1998 379–386 385

scribed by the two-generation formalism and theoscillations of solar and atmospheric neutrinos de-pend on different and independent elements of theneutrino mixing matrix, i.e. they decouple. We have

< < 2also shown that if not only U <1 but alsoe3< <U <1, then the oscillations of atmospheric neutri-e3

nos do not depend on matter effects and are alsodescribed by the two-generation formalism. In this

Ž . Ž .case, with the identifications 10 and 23 the two-generation analyses of solar and atmospheric neu-trino data provide direct information on the mixing

Ž Ž . Ž ..parameters of three neutrinos see Eqs. 30 – 32 .Let us notice that the independence of the oscilla-

tions of atmospheric neutrinos from matter effectscan be checked by comparison of the transitionprobabilities of neutrinos and antineutrinos in futureaccelerator long-baseline experiments.

Ž w xIf future results from reactor CHOOZ 25 , Palow x w x. ŽVerde 38 , Kam-Land 39 and accelerator K2K

w x w x w x w x.40 , MINOS 41 , ICARUS 42 and others 43long-baseline neutrino oscillation experiments will

Ž . < < 2confirm and improve the upper bound 8 for Ue3

obtained from the first results of the CHOOZ experi-w xment 25 , the indications in favor of a decoupling of

solar and atmospheric neutrino oscillations and oftheir accurate description by the two-generation for-malism will be strengthened. In this case the distinc-tion of the three allowed solutions of the solar

Žneutrino problem small and large mixing angle MSW.and vacuum oscillations , which is a goal of the new

Žgeneration of solar neutrino experiments Super-Kamiokande, SNO, ICARUS, Borexino, GNO and

w x.others 37 could provide an indication of the actualvalues of the elements of the neutrino mixing matrix

Ž . Ž .selecting one of the three possibilities 30 – 32 .

Acknowledgements

C.G. would like to thank V. Berezinsky for astimulating discussion on neutrino physics and astro-physics.

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