very long baseline neutrino oscillation experiments and the msw effect

qWork supported in part by `Sonderforschungsbereich 375fuK r Astro-Teilchenphysika der Deutschen Forschungsgemein-schaft, by the TMR Network under the EEC Contract No.ERBFMRX}CT960090 and by the Italian MURST under theprogram `Fisica Teorica delle Interazioni Fundamentalia. Forthe complete and "nal version of this study see hep-ph/9912457.

*Corresponding author.E-mail addresses: [email protected]

(M. Freund), [email protected] (M.Lindner), [email protected] (S.T. Petcov), [email protected] (A. Romanino).

1Also at: INRNE, Bulgarian Academy of Sciences, 1789 So"a,Bulgaria.

Nuclear Instruments and Methods in Physics Research A 451 (2000) 18}35

Very long baseline neutrino oscillation experiments andthe MSW e!ectq

M. Freund!, M. Lindner!, S.T. Petcov",*,1, A. Romanino#

!Theoretische Physik, Physik Department, Technische Universita( t Mu( nchen, James-Franck-Strasse, D-85748 Garching, Germany"Scuola Internazionale Superiore di Studi Avanzati, and INFN } Sezione de Trieste, I-34014 Trieste, Italy

#Department of Physics, Theoretical Physics, University of Oxford, Oxford OX13NP, UK

Abstract

Assuming three-neutrino mixing, we study the capabilities of very long baseline neutrino oscillation experiments toverify and test the MSW e!ect and to measure the lepton mixing angle h

13. We suppose that intense neutrino and

antineutrino beams will become available in the so-called neutrino factories. We "nd that the most promising andstatistically signi"cant results can be obtained by studying m

%Pm

land m6

%Pm

loscillations which lead to matter

enhancements and suppressions of wrong-sign muon rates. We show the h13

ranges where matter e!ects could beobserved as a function of the baseline. We discuss the scaling laws of rates, signi"cances and sensitivities with the relevantmixing angles and experimental parameters. Our analysis includes #uxes, event rates and statistical aspects so that theconclusions should be useful for the planning of experimental setups. We discuss the subleading *m2

21e!ects in the case of

the LMA MSW solution of the solar problem, showing that they are small for ¸Z7000km. For shorter baselines, *m221

e!ects can be relevant and their dependence on ¸ o!ers a further handle for the determination of the CP-violation phased. Finally, we comment on the possibility to measure the speci"c distortion of the energy spectrum due to the MSWe!ect. ( 2000 Elsevier Science B.V. All rights reserved.

1. Introduction

The long-term aim to build muon colliders o!ersthe very attractive intermediate possibility for`neutrino factoriesa [1}3] with uniquely intenseand precisely characterized neutrino and antineut-rino beams. This requires only one muon beam atintermediate energies such that neutrino factoriesare rather realistic medium-term projects whichconstitute also a useful step in accelerator technol-ogy towards a muon collider. The current knowl-edge of neutrino masses and mixing implies fortypical setups very promising very long baselineneutrino experiments. We study in this paper in

0168-9002/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 3 6 9 - 7

2The possibility to detect matter e!ects in long baselineneutrino oscillation experiments with ¸K730 km (MINOS,CERN } GS) was discussed e.g. in Refs. [9,10].

a three neutrino framework the potential to verifyand test the MSW e!ect and to measure or limith13

in terrestrial very long baseline experiments2with neutrino factories where the neutrino spec-trum and #uxes are rather well known and undercontrol [1]. Calculating the oscillation probabilit-ies and event rates for di!erent channels and com-paring with those for oscillations in vacuum we "ndthat the asymmetry between the m

%%m

land m6

%%m6

loscillations is a very promising tool to test andverify the MSW e!ect. The reason is, as we will see,that matter e!ects lead to measurably enhancedevent rates due to the m

%Pm

ltransitions, while the

rates due to the m6%Pm6

ltransitions are equally

suppressed. The asymmetry between the event ratesassociated with these two channels would thereforebe very sensitive to the MSW e!ect since the mat-ter-induced changes have opposite e!ect on the tworates thus amplifying the `signala, while at the sametime common backgrounds would drop out. Weanalyze event rates and we include statistical as-pects such that the results are directly applicable forthe planning of optimal experimental setups. Wediscuss the capabilities of a neutrino factory experi-ment as a function of the distance between theneutrino source and the detector and of the muonsource energy for the optimal observation of theMSW e!ect. Moreover, we determine the sensitiv-ity to the value of h

13for di!erent experimental

con"gurations.Demonstrating and testing the MSW e!ect dir-

ectly is of fundamental importance since this e!ectplays a basic role in di!erent neutrino physics scen-arios. The MSW mechanism provides, for example,the only clue for understanding the solar neutrinode"cit with a neutrino mass squared di!erencewithin a few orders of magnitude from that inferredfrom the atmospheric neutrino data. Atmosphericneutrinos can undergo matter-enhanced transitionsin the earth. The matter e!ects in neutrino oscilla-tions will play an important role in the interpreta-tion of the results of a neutrino factory experimentusing an ¸Z1000km baseline. They are essential in

the searches for CP-violation in such experiments[2,4], since matter e!ects generate an asymmetrybetween the two relevant CP-conjugated appear-ance channels [4]. Knowing the asymmetry causedby matter e!ects is therefore essential for obtaininginformation on the CP-violation originating fromthe lepton mixing matrix. Matter-enhanced neu-trino transitions can play important role in astro-physics as well.

Neutrino factories have been extensively dis-cussed in the literature [1}3,5}8]. Either muons oranti-muons are accelerated to an energy E

land

decay then in straight sections of a storage ring likel~Pe~#m6

%#m

lor l`Pe`#m6

l#m

%so that

a very pure neutrino beam containing m6%

and ml

orm%

and m6l, respectively, is produced. The muon

energy El

could be in a wide range from 10 to50 GeV or more and a neutrino #ux correspondingto 2]1020 muon decays per year in the straightsection of the ring pointing to a remote detectorcould be achieved. Higher #uxes are also currentlyunder discussion [11]. The neutrino #uxes aretherefore very intense and can be easily calculatedfrom the decay spectrum at rest. For unpolarizedmuons and negligible beam divergence one "ndsfor a baseline of ¸"730 km a neutrino #ux ofK4.3]1012 yr~1 m~2 and for a baseline ¸"

10 000 km a neutrino #ux K2.3]1010 yr~1 m~2.Note also that the m6

%and m

%#uxes depend sizably

on the beam polarization, that will be assumed tobe vanishing in this paper. Altogether a neutrinofactory would provide pure and high-intensity neu-trino beams with a well-known energy spectrumthat in turn would allow a wide physical programincluding precise measurement of mixing param-eters [2,3], matter e!ects [5,8] and, in case of LMAsolution of the solar problem, leptonic CP-viola-tion [6,7].

The produced neutrino beam will be directedtowards a remote detector at a given Nadir angle h,which corresponds to an oscillation baseline¸"2R

^cos(h), where R

^"6371 km is the earth

radius. If ¸Z103 km, as we shall assume, mattere!ects become important in neutrino oscillations.For ¸4104 km the beam traverses the earth alonga trajectory in the earth mantle without crossingthe earth core where the density is substantiallyhigher. According to the earth models [12,13], the

M. Freund et al. / Nuclear Instruments and Methods in Physics Research A 451 (2000) 18}35 19

THEORY

average matter density along the neutrino trajecto-ries with ¸"(103}104) km lies in the interval&(2.9}4.8) g cm~3. The matter density changesalong each trajectory, but the variation is relativelysmall } by about 1.5}2.0 g cm~3, and even thelargest takes place over relatively big distances ofseveral thousand kilometers. As a consequence, onecan approximate the earth mantle density pro"leby a constant average density distribution. Theconstant density can be chosen to be equal to theaverage density along every trajectory. For the cal-culation of the neutrino oscillation probabilities inthe case of interest, the indicated constant densitymodel provides a very good approximation to thesomewhat more complicated density structure ofthe earth mantle. Let us note also that the earthmantle is with a good precision isotopically sym-metric:>

%"0.494 [12,13], where>

%is the electron

fraction number in the mantle.Since the beam consists always either of m

%together with m6

lor m6

%in combination with m

lthere

are, in principle, eight di!erent appearance experi-ments and four di!erent disappearance experi-ments which could be performed. From anexperimental point of view, however, at present thefour channels with muon neutrinos or antineu-trinos in the "nal state, m

lPm

l, m6

lPm6

l, m6

%Pm

l,

m6%Pm6

lseem to be most promising. A very impor-

tant issue is in this context the ability of the de-tector to discriminate between neutrinos andantineutrinos, namely the ability to measure thecharge of the leptons produced by the neutrinocharged current interactions. Note that very gooddiscrimination capability is required for a measure-ment of the appearance probabilities since theyproduce `wrong signa muon events in the detectorand have to be discriminated from the much largernumber of events associated with the m

land m6

lsur-

vival probabilities. The channels with electron neu-trinos or antineutrinos in the "nal state areproblematic from this point of view due to thedi$culty of telling e` from e~ in a large high-density detector. On the contrary, a very goodl`/l~ discrimination could be obtained in a largeproperly oriented magnetized detector [14].

The paper is organized as follows. In Section2 we give the analytic formulae for three neutrinooscillations in matter which contain the essential

physics relevant for our study. In Section 3 wediscuss event rates, their parameter dependence,their scaling behavior and we show results from ournumerical calculations. In Section 4 we de"ne thesensitivity to matter e!ects in a statistical sense anddiscuss the results of our numerical calculationsincluding parameter uncertainties. This is followedby a discussion in Section 5 of the e!ects of a non-vanishing *m2

21. In Section 6 the possibility of

detecting matter e!ects by looking for the enhance-ment, the broadening and the shift of the MSWresonance energy is discussed and we conclude inSection 7.

2. Three neutrino oscillation probabilities in matter

We will assume in this paper the existence ofthree #avor neutrino mixing:

DmlT"

3+k/1

;lkDm

kT, l"e,l,s (1)

where DmlT is the state vector of the (left-handed)

#avor neutrino ml, Dm

kT is the state vector of a neu-

trino mk

possessing a de"nite mass mk, m

kOm

j,

kOj"1,2,3, m1(m

2(m

3, and ; is a 3]3 uni-

tary matrix } the lepton mixing matrix. It is naturalto suppose in this case that one of the two indepen-dent neutrino mass-squared di!erences, say *m2

21,

is relevant for the vacuum oscillation (VO), small orlarge mixing angle MSW solutions (SMA MSWand LMA MSW) of the solar neutrino problemwith values in the intervals [15,16]

VO: 5.0]10~11 eV2[*m221

[5.0]10~10 eV2

(2a)

SMA MSW: 4.0]10~6 eV2[*m221

[9.0]10~6 eV2

(2b)

LMA MSW: 2.0]10~5 eV2[*m221

[2.0]10~4 eV2

(2c)

while *m231

is responsible for the dominant atmos-pheric m

l%m

soscillations and lies in the interval

ATM: 10~3 eV2[*m231

[8.0]10~2 eV2. (3)

20 M. Freund et al. / Nuclear Instruments and Methods in Physics Research A 451 (2000) 18}35

A;

%1;

%2;

%3;

l1;

l2;

l3;

s1;

s2;

s3B"A

c12

c13

s12

c13

s13

e~*d!s

12c23

!c12

s23

s13

e*d c12

c23

!s12

s23

s13

e*d s23

c13

s12

s23

!c12

c23

s13

e*d !c12

s23

!s12

c23

s13

e*d c23

c13B (9)

3We have not written explicitly the possible Majorana CP-violation phases which do not enter into the expressions for theoscillation probabilities [24,25].

For Em51 GeV, ¸4104 km, the *m2-hierarchy

*m221

;*m231

(4)

and *m221

;10~4 eV2, the probabilities of three-neutrino oscillations in vacuum of interest reducee!ectively to two-neutrino vacuum oscillationprobabilities [17,18]:

P3m

7!#(m

lPm

l{)"P7!#(m6

lPm6

l{)

+2D;l3

D2D;l{3

D2A1!cos*m2

31¸

2E B,lOl@"e,l,s (5)

P3m

7!#(m

lPm

l)"P3m

7!#(m6

lPm6

l)

+1!2D;l3

D2(1!D;l3

D2)

]A1!cos*m2

31¸

2E B, l"e,l,s. (6)

Under conditions (1) and (4) the element D;%3

D ofthe lepton mixing matrix, which controls them%Pm

l(s), m6

%Pm6

l(s), m

lPm

%and the m6

lPm6

%oscil-

lations, is tightly constrained by the CHOOZ ex-periment [19] and the oscillation interpretation ofthe solar and atmospheric neutrino data: for3.0]10~3 eV24*m2

3148.0]10~3 eV2 one has

D;%3

D2[0.025. (7)

The CHOOZ upper limit is less stringent for1.0]10~3 eV24*m2

31(3.0]10~3 eV2 where

values of D;%3

D2+0.05 are allowed.Note that under condition (4), the VO or MSW

transitions of solar neutrinos depend on D;%3

D2 andon the two-neutrino VO or MSW transition prob-ability with *m2

21and h

12, where

sin2 2h12

"4D;

%1D2D;

%2D2

(D;%1

D2#D;%2

D2)2,

cos 2h12

"

D;%1

D2!D;%2

D2D;

%1D2#D;

%2D2

(8)

playing the role of the corresponding two-neutrinooscillation parameters [20,21] (for recent reviewssee, e.g., Refs. [22,23]). For D;

%3D2 satisfying limit (7),

however, this dependence is rather weak and cannotbe used to further constrain or determine D;

%3D2. In

general, under condition (4) and for *m221;

10~4 eV2 the relevant solar neutrino transitionprobability depends only on the absolute values ofthe elements of the "rst row of the lepton mixingmatrix, i.e., on D;

%iD2, i"1,2,3, while the vacuum

oscillations of the (atmospheric) ml, m6

l, m

%and m6

%on

earth distances are controlled by the elements of thethird column of ;, D;

l3D2, l"e,l,s. The other ele-

ments of ; are not accessible to direct experimentaldetermination. Moreover, the CP- and T-violatione!ects in the oscillations of neutrinos are negligible.

For our analysis we use a standard parametriz-ation of the lepton mixing matrix ;:

where cij,cos h

ij, s

ij,sin h

ijand d is the Dirac

CP-violation phase.3 The angles h12

and h23

inEq. (9) are constrained to lie within rather narrowintervals by the solar and atmospheric neutrinodata for each of the di!erent solutions, Eqs.(2a)}(2c), of the solar neutrino problem. With theaccumulation of data the uncertainties in the knowl-edge of h

12and h

23will diminish, while only upper

limits on s213

like Eq. (7) have been obtained so far.The mixing angle h

13is one of the 4 (or 6 } depend-

ing on whether the massive neutrinos are of Diracor Majorana type [24,25,18]), fundamental param-eters in the lepton mixing matrix. It controls theprobabilities of the m

l(m

%)Pm

%(m

l), m6

l(m6

%)Pm6

%(m6

l),

m%Pm

sand m6

%Pm6

soscillations and the CP- and

T-violation e!ects in neutrino oscillations depend


THEORY

on it. Obviously, one of the main goals of the futureneutrino oscillation experiments should be todetermine the value of h

13or to obtain a more

stringent experimental upper limit than the existingone (9).

Under condition (4), with *m221

;10~4 eV2 andin the constant density approximation, the oscilla-tion probabilities of interest take the followingsimple form (see, e.g., Refs. [22,23,26]):

P3m

E(m

lPm

%)"P3m

E(m

%Pm

l)

+s223

P2m

E(*m2

31, sin2 2h

13) (10a)

P3m

E(m

lPm

l)

+c423

#s423

[1!P2m

E(*m2

31, sin2 2h

13)]

#2c223

s223

Re[e~*iA2m

E(*m2

31, sin2 2h

13)] (10b)

P3m

E(m

lPm

s)+2c2

23s223

[1!12P2m

E(*m2

31, sin2 2h

13)

!Re (e~*iA2m

E(*m2

31, sin2 2h

13))]

(10c)

where

P2m

E(*m2

31, sin2 2h

13)"1

2[1!cos*E

m¸] sin2 2hm

13(11)

and

A2m

E(*m2

31, sin2 2h

13)"1#(e~**EmL!1) cos2 hm

13(12)

are respectively the two-neutrino transition prob-ability and probability amplitude of neutrino sur-vival in matter with constant density and

i+¸

2C*m2

312E

#<!*EmD (13)

is a phase. In Eqs. (11)}(13) *Em

and hm13

are theneutrino energy di!erence and mixing angle inmatter,

*Em"

*m231

2EC

`, cos2hm

13"

1

C`Acos 2h

13!

2E<

*m231B

(14)

where

C2B"A1G

2E<

*m231B

2$4

2E<

*m231

sin2h13

(15)

and

<"J2GFNM .!/

%(16)

is the matter term, NM .!/%

being the average electronnumber density along the neutrino trajectory inthe earth mantle. The probability P3m

E(m

%Pm

s) can

be obtained from Eq. (10a) by replacing thefactor s2

23with c2

23. The corresponding anti-

neutrino transition and survival probabilities havethe same form and can formally be obtained fromEqs. (10a)}(10c) by changing the sign of the matterterm, i.e. NM .!/

%P!NM .!/

%in the expressions for i,

*Em

cos 2hm13

, Eqs. (13) and (14), and by replacingC

`by C

~.

Several comments are in order. First, the analyticexpressions (10a)}(10c) represent excellent approxi-mations in the case of the VO and SMA MSWsolutions of the solar neutrino problem and for valuesof *m2

21[5]10~5 eV2 from the LMA MSW solu-

tion region. For 5]10~5 eV2(*m221

[2] 10~4 eV2,the corrections due to *m2

21can be non-negligible

and we are going to discuss them in Section 5.Second, as it is clear from Eqs. (10a) and (11), theprobabilities P3m

E(m

%Pm

l) and P3m

E(m6

%Pm6

l) cannot

exceed s223

&0.5. The maximal value s223

can bereached only if the MSW resonance conditionsin2 2hm

13"1 and the condition cos*E

m¸"!1

are simultaneously ful"lled. At the MSW reson-ance, however, one has *E3%4

m+1.23p]10~4 km~1

tan 2h13

NM .!/%

(NA

cm~3), where NM .!/%

is in units ofN

Acm~3, N

Abeing the Avogadro number, and for

s213

40.025 the second condition cos*E3%4m¸"!1

can only be satis"ed for ¸5104 km. If s213

"0.05this condition requires ¸58]103 km. Given thefact that the neutrino #uxes decrease with the dis-tance as ¸~2, the above discussion suggests that themaximum of the event distribution in E and¸ in the transition m

%Pm

lshould take place

approximately at the MSW resonance energy, butat values of ¸ smaller than those at whichmax(P3m

E(m

%Pm

l))+s2

23. The MSW resonance en-

ergy is given by E3%4

+6.56*m231

(10~3 eV2)cos 2h

13(NM .!/

%(cm~3 N

A))~1 GeV, where *m2

31and

NM .!/%

are in units of 10~3 eV2 and cm~3 NA,


4This scheme implies an approximate degeneracy of eitherm

1and m

2or all the mass eigenstates.

respectively. For *m231

(10~3 eV2)"3.5, 6.0,8.0 and, e.g., NM .!/

%(cm~3N

A)"2, we have

E3%4

+11.5;19.7;26.2 cos 2h13

GeV. The m%Pm

land

m6%Pm6

loscillations will be a!ected substantially by

the MSW e!ect if the energy of the parent muonbeam E

l'E

3%4. This condition can be satis"ed

for any value of *m231

from the interval (3)for E

lZ30 GeV. For the value of *m2

31(10~3 eV2)"3.5 which is currently `preferredaby the Super-Kamiokande data we have E

lZ

15 GeV. Finally, we note also that the hierarchy*m2

21;D*m2

31D does not necessarily correspond to

a situation where m3'm

2Zm

1, namely to a situ-

ation where the two closest mass eigenstates (m1,

m2) are lighter than the third one (m

3). As long as

matter and CP-violation e!ects are not taken intoaccount, a phenomenologically equivalent situ-ation is given when the two closest mass eigenstatesare heavier than the third one,4 i.e., m

2Zm

1'm

3.

In this case we still have *m221

;D*m231

D but*m2

31(0. Such a change in the sign of *m2

31re-

verses the sign of N3%4%

which opens the possibility todetermine this sign via matter e!ects as recentlyemphasized by Barger et al. [8].

3. Event rates

We discuss now the e!ects of matter on the totalrate of m6

%Pm

l, m6

%Pm6

l, m

lPm

l, m6

lPm6

levents. If

for example l` are accumulated in the storage ringthen the neutrino beam contains m

%and m6

l. Since

the neutrino}antineutrino transitions have a negli-gible rate, the beam-induced l` events in the de-tector must be attributed in this case to the chargedcurrent interactions of unoscillated m6

l. The total

number nl`(l`) of l` events measures therefore an

averaged m6l

survival probability. Wrong-signl~ events must be attributed to m

lgenerated by

oscillations of the initial m%, so that their total number

nl`(l~) measures the averaged m6

%Pm

loscillation

probability. Analogously, for l~ in the storage ring,the total numbers of l~ and l` events n

l~(l~),

nl~(l`) measure the averaged m

lsurvival probabil-

ity and m6%Pm6

loscillation probability, respectively.

The total number of events in each channel aregiven by

nl`(l~)"N

l`N

,T

109NA

m2lp

E3l

¸2

]PEl

E.*/

fm%m

l(E)P3m

E(m

%Pm

l)(E)(dE/E

l)

(17a)

nl~(l`)"N

l~N

,T

109NA

m2lp

E3l

¸2

]PEl

E.*/

fm6%m6

l(E)P3m

E(m6

%Pm6

l)(E)(dE/E

l)

(17b)

nl~(l~)"N

l`N

,T

109NA

m2lp

E3l

¸2

]PEl

E.*/

fmlm

l(E)P3m

E(m

lPm

l)(E)(dE/E

l)

(17c)

nl`(l`)"N

l~N

,T

109NA

m2lp

E3l

¸2

]PEl

E.*/

fm6lm6

l(E)P3m

E(m6

lPm6

l)(E)(dE/E

l)

(17d)

where P3m

Edenotes the oscillation probability de-

scribed in Section 2, Nl` (N

l~) is the number of

`usefula l` (l~) decays, namely the number ofdecays occurring in the straight section of the stor-age ring pointing to the detector, N

,Tis the size of

the detector in kilotons, 109NA

is the number ofnucleons in a kiloton, E

lis the energy of the muons

in the ring and E.*/

"3 GeV is a lower cut on theneutrino energies that helps a good detection e$-ciency. Since low-energy events are suppressed bythe low initial #ux and the low cross section (seebelow), the results do not depend signi"cantly onthe precise value of E

.*/for E

.*/(5 GeV. The

functions f averaging the probabilities are given by

fm%m

l(E)"gm

%(E/E

l) (pm

l(E)/E

l)e

l~(E) (18a)

fm6%m6

l(E)"gm6

%(E/E

l) (pm6

l(E)/E

l)e

l`(E) (18b)

fmlm

l(E)"gm

l(E/E

l) (pm

l(E)/E

l)e

l~(E) (18c)

fm6lm6

l(E)"gm6

l(E/E

l) (pm6

l(E)/E

l)e

l`(E) (18d)


THEORY

and take into account the appropriately nor-malized initial spectrum of m-neutrinos produced inthe decay of unpolarized muons, gm(E/E

l), the

charged current cross section per nucleon, pml (m6 l )

(E),and the e$ciency for the detection of l~ (l`),el~(l`)

(E) (we neglect here the "nite resolution of thedetector). For the numerical calculations we use

gm%(x)"gm6

%(x)"12x2(1!x),

gml(x)"gm6

l(x)"2x2(3!2x) (19a)

pml(E)"0.67]10~38E cm2 GeV~1,

pm6l(E)"0.34]10~38E cm2 GeV~1 (19b)

and el~(E)"e

l`(E)"e for E'E

.*/so that

fm%m

l(E)/fm6

%m6

l(E)"fm

lm

l(E)/fm6

lm6

l(E)"2 independent of

the energy.The contribution of the background to the num-

ber of muons observed in the detector has beenneglected in Eqs. (17). That background includesmuons from the decay of charm quarks producedby charged and neutral current neutrino interac-tions in the detector and from the decay of s pro-duced by m

sinteractions. Both these sources can be

kept under control in di!erent ways [8,27]. More-over, Eqs. (17a)}(17d) neglect the divergence of themuon beam in the straight section of the storagering.

We can discuss now the dependence of nl`(l~),

nl~(l`), n

l~(l~), n

l`(l`) on the mixing parameters

and some of the experimental parameters. In the*m2

12;10~4 eV2 approximation discussed above,

the only physical mixing parameters are h23

andh13

. In the leading approximation the dependenceof the total event rates in matter (and in vacuum) onthose parameters is

nl~(l~), n

l`(l`)Jsin2 2h

23(20a)

nl`(l~), n

l~(l`)Jsin2 h

23sin2 2h

13. (20b)

Higher-order corrections in h13

are constrained bythe CHOOZ limit. Despite the resonant matterenhancement of the mixing due to h

13, such correc-

tions become only sizable for h13

close to its upperlimit and very long baselines where they can reachabout 20% in the resonant channels.

Since Eq. (20b) will turn out to be useful whendiscussing the sensitivity to matter e!ects, we discuss

it in greater detail. Let us write Eq. (10a) in the form

P3m

E(m

%Pm

l,m6

%Pm6

l)

"sin2 h23

sin2 2h13A

sin(*31

CB

)

CB

B2

(21)

where C`

is given by Eq. (15) and corresponds toneutrinos, C

~to antineutrinos (the opposite for

*m231

(0) and *31

"*m231

¸/(4E). In the limit inwhich sin2 h

13can be neglected on the right-hand

side of Eq. (15), Eq. (21) shows that nl` (l~) and

nl~(l`) are indeed proportional to sin2 2h

13. De-

spite the CHOOZ limit, sin2 h13

[0.025, the secondterm in Eq. (15) can be relevant when the "rst termvanishes around the resonance. For 2E<"*m2

31we have in fact C

`"2sin h

13and

P3m

E(m

%Pm

l)

"sin2 h23

sin2 2h13

¸2<2

4 Asin(sin h

13¸<)

sin h13

¸< B2

(22)

whereas by neglecting the sin2 h13

term in Eq. (15)we would get C

`"0 and

P3m

E(m

%Pm

l)"sin2 h

23sin2 2h

13

¸2<2

4. (23)

A comparison of Eqs. (22) and (23) shows how-ever that the approximation works even at theresonance provided ¸[p/(4< sin h

13)&7000 km

(0.15/sinh13

). The e!ect of the sin2 h13

term istherefore maximal in Eq. (15) for very long base-lines and h

13close to the experimental limit. In this

case it can a!ect the averaged probability sizably,while it is negligible for smaller h

13or smaller

baseline. For a better approximation one can use

nl`(l~), n

l~(l`)

Jsin2 h23

sin2 2h13A

sin(sinh13

¸<)

sin h13

¸< B2

(24)

which deviates less than 5% from the exact result inthe whole parameter space (¸[10 000km, 20GeV[E

l[50 GeV). In most cases ¸[p/(4< sin h

13)

holds and the oscillating term in Eq. (24) can beexpanded, giving Eq. (20b) or higher-order approx-imations. We stress that the baseline ¸ appearsas part of the correction to the h

13, h

23scaling only. The dependence on ¸ of the rates ismore involved, especially for very large baselines,


Fig. 1. Appearance event rates nl`(l~), n

l~(l`) in matter against baseline ¸ (solid lines) compared with the corresponding event rates in

vacuum (dashed lines) for El"20 GeV (left) and E

l"50 GeV (right). Both plots assume N

l"2]1020, e"50%, sin2 2h

23"1 and

sin2 2h13

"0.01. The scaling of the rates with these parameters is described in the text.

Fig. 2. Same as in Fig. 1 but for the disappearance channels. The dashed lines coincide almost perfectly with the solid lines.

and will be described in Figs. 1 and 2. Note that thedependence on the beam intensity, detector sizeand e$ciency is trivial:

nl`(l~), n

l~(l~)JN

l`N

,Tel` (25a)

nl~(l`), n

l`(l`)JN

l~N

,Tel~ . (25b)

We present now quantitative results for the totalrates in matter for *m2

21;10~4 eV2 and we com-

pare them with the results one would obtain invacuum. The statistical signi"cance of the mattere!ects will be discussed in the following section ande!ects of larger *m2

21will be covered in Section 5.

The total event rates depend as already discussedin a transparent way on the experimental param-eters N

lB , N

,T, e

lB and on the mixing parameters

h23

, h13

. We focus our discussion therefore on theless transparent dependence on the baseline andmuon energy. For that we use the central value of*m2

31"3.5]10~3 eV2, and we assume *m2

31'0.

In the *m221

;10~4 eV2 approximation, in which

CP-violation e!ects are negligible, the results for*m2

31(0 can be obtained by simply interchanging

neutrinos and antineutrinos.The total number of events in the two appear-

ance channels m%Pm

l, m6

%Pm6

lis shown for

El"20GeV and E

l"50GeV in Fig. 1 as a func-

tion of the baseline (solid lines) in comparison withthe event rates one would get if the neutrinos didnot interact with matter (dashed lines). Fig. 2 showsthe two disappearance channels m

lPm

l, m6

lPm6

l.

Both "gures correspond to a `defaulta experi-mental setup providing N

l`"N

l~"N

l"

2]1020 useful muon decays (e.g. in one year ofrunning) and to a detector with N

,T"10 kt and an

e$ciency el`"e

l~"e"50% in both channels.

The rates depend only on the combination NlN

,Te

which is in our case NlN

,Te"1021. Moreover,

we assume for these "gures sin2 2h23

"1 andsin2 2h

13"0.01, one order of magnitude below the

experimental limit. The rates for di!erent values ofN

l, N

,T, e, h

23, h

13can be obtained by using Eqs.

(25), (20a) and (20b) or (24).


THEORY

Fig. 3. Di!erential appearance rates for the channels m%Pm

l(left) and m6

%Pm6

l(right) over ¸ and Em showing for large ¸ the

enhancement and suppression due to the MSW mechanism. The parameters are identical to those of Fig. 1.

5As discussed above e!ects as large as 20% can occur forh13

close to the CHOOZ limit and very large baselines.

The vacuum event rates in the neutrino channelsare twice as big as the rates in the antineutrinochannels. This is because the oscillation probabilit-ies in CP-conjugated channels in the*m2

21;10~4 eV2 approximation are the same

while (due to the larger cross section) the functionsaveraging the probabilities are larger by a factor2 in the neutrino channels. The disappearancechannels shown in Fig. 2 are essentially indepen-dent of matter e!ects since these e!ects come onlywith the h

13corrections to the oscillation probabil-

ities.5 In contrast, Fig. 1 shows the drastic enhance-ment (depletion) of the event rates in the m

%Pm

l(m6

%Pm6

l) channel for very long baselines. The

growth of the total rates with the muon energywhich is obvious from Eq. (17) can also be seen inFig. 1. Fig. 3 shows in more detail the di!erentialevent rates of the appearance channels m

%Pm

land

m6%Pm6

las three-dimensional plots over ¸ and Em .

The "gures show nicely the enhancement (sup-pression) due to the MSW mechanism in them%Pm

l(m6

%Pm6

l) channel for large baseline ¸ at

EmK10GeV. The baseline and muon energy de-pendence of matter e!ects will be further discussedin the next section in connection with a quantitat-ive analysis of the signi"cance of the e!ects shownin Figs. 1 and 2.

4. Statistical signi5cance of matter e4ects

We have seen in the previous section that mattere!ects change the total event rates in the appearance

channels m%Pm

land m6

%Pm6

lin very long baseline

experiments in a drastic way. Such experimentswould therefore o!er unique possibilities to ob-serve matter e!ects and to test the predictions of theMSW theory. In order to study the capabilities ofa neutrino factory experiment quantitatively, wemust "rst de"ne the meaning of `observing mattere!ectsa. One of the most interesting possibilitieswould be a detailed observation of the shape of theneutrino energy spectrum which is modi"ed by theMSW e!ect in a very characteristic way. Thiswould allow to test non-trivial predictions of theMSW theory and would allow to unambiguouslyattribute the enhancement/depletion of the totalnumber of neutrino interactions to matter e!ects.High di!erential event rates and a good calibration ofthe detector would however be necessary for thisoption. We will discuss this possibility in Section 6.In this section we con"ne ourselves to a discussionof the signi"cance of MSW e!ects in total eventrates.

Matter e!ects produce deviations of the totalnumber of wrong-sign muon events n

l`(l~),

nl~(l`) from what is expected in the absence of

interaction with matter. The discussion aboveclearly shows that such deviations occur in oppo-site directions in the appearance channels. Supposethat a certain number of wrong-sign muon eventsin the neutrino (antineutrino) appearance channeln

l`(l~) (n

l~(l`)) is measured, while Sn7!#

l` (l~)T

(Sn7!#l~ (l`)T) would be expected in the absence of

matter e!ects. We want to determine the con"dencelevel at which n

l`(l~) and n

l~(l`) could represent

statistical #uctuations around the expected valuesin vacuum Sn7!#

l` (l~)T, Sn7!#

l~ (l`)T. We follow the

procedure proposed by the Particle Data Book


6This procedure can be understood intuitively by de"ningthe asymmetry A :"*/&, where * :"n

l` (l~)!2n

l~ (l`) and

& :"nl`(l~)#2n

l~(l`), which is obviously very sensitive to

e!ects which go in opposite direction, while at the same timea number of common systematic e!ects drop out. For small&7!# (i.e. small beam energies) this method is related to compar-ing the absolute asymmetry expected in matter with the expected#uctuations of this quantity in the vacuum case s"*/d*7!#. Forlarge &7!# this is equivalent to doing the same with the asym-metry: s"A/dA7!#.

Fig. 4. Contour lines of np in the ¸}El

plane. We use*m2

31"3.5]103 eV2 and N

lN

,Te"1021 and the solid con-

tour lines correspond to np"100 sin 2h13

) M1,2,4,8,16N.

[28] and calculate con"dence levels by using

s2"2[Sn7!#l` (l~)T!n

l`(l~)]

#2nl`(l~)log

nl`(l~)

Sn7!#l` (l~)T

# 2[Sn7!#l~ (l`)T!n

l~ (l`)]

#2nl~(l`)log

nl~(l`)

Sn7!#l~ (l`)T

. (26)

The corresponding `number of standard deviationsais given by np,Js2. This prescription incorporatesthe available information in both the measured num-bers n

l`(l~), n

l~(l`) in the most complete way.6

Before we show the numerical results, let us dis-cuss the qualitative dependence of np on the rel-evant parameters. The dependence of np on theintensity of the muon source and the detector sizeand e$ciency, as well as the dependence on themixing parameters follows simply from the pre-vious section:

npJ(NlN

,Tel)1@2 and npJsin h

23sin 2h

13.

(27)

The dependence of np on the baseline ¸ and themuon energy E

lis less trivial. A very long baseline

is essential for the observation of matter e!ects andthe oscillating factor in Eq. (21) can be expanded for¸[3000km and not too small muon energy ina series of its argument. Matter e!ects cancel [2] inthe leading order of this expansion. Matter e!ectsare therefore in this regime small such that base-lines shorter than about 3000km are better suitedfor CP-violation measurements [2,7]. The import-ance of matter e!ects grows however quadraticallywith the baseline [7] until the argument *

31C

Bin

Eq. (21) approaches and exceeds p/2, making mat-ter e!ects large.

The dependence of np on ¸ and El

is shownin Fig. 4 for *m2

31"3.5]10~3 eV2 and

NlN

,Te"1021, where the contour lines of np are

plotted in the ¸}El

plane. The solid contour linescorrespond to np"100 sin2 2h

13) M1,2,4,8,16N. The

main muon energy requirement is for this centralvalue of *m2

31and for "xed very long baselines

ElZ20 GeV. Below this energy the neutrino en-

ergy spectrum does not fully cover the MSW reson-ance so that np decreases rapidly. The number ofneutrinos which fall into the resonance region de-creases for E

lbeyond approximately 20 GeV, re-

ducing thus np even though the total rates aregrowing. Beyond E

lZ50 GeV the m6

%Pm6

l(and for

very large Eland ¸ also m

%Pm

l) suppression starts

to dominate thus increasing np again.The signi"cance of the e!ect, i.e. the number of

standard deviations np , depends most crucially onh13

, ¸. We illustrate this dependence in Fig. 5 againfor *m2

31"3.5]103 eV2, where the contour lines

corresponding to np"1}5 are plotted in thesin2 2h

13}¸ plane for two values of the product

NlN

,Te: 1021 (left plot) and 1022 (right plot). The

muon energy in both cases of Fig. 5 isE

l"20 GeV, while Fig. 6 shows the same plots

with identical parameters for El"50 GeV. The

vertical dashed lines represent in all these "guresthe upper limit on sin2 2h

13. Figs. 5 and 6 show that

matter e!ects could be observed in the total event


THEORY

Fig. 5. Contour lines of np corresponding to np"1}5 in the sin2 2h13}¸ plane for E

l"20 GeV and the two di!erent values

NlN

kTe"1021 (left plot) and N

lN

kTe"1022 (right plot).

Fig. 6. Same as in Fig. 5 but for El"50 GeV.

rates for given baseline ¸ in a rather large sin2 2h13

interval, while non-observation implies very strongupper bounds on sin2 2h

13. Figs. 5 and 6 show in

other words the sin2 2h13

range where the enhance-ment/depletion of the total appearance rates invacuum due to matter e!ects is statistically signi"-cant for a given con"dence level. In those rangesone can not only observe the deviations from theresults expected in vacuum, but also the deviation

from the results which one would get in matter if*m2

31were reverted. A measurement of the sign of

*m231

would therefore be possible in the sin2 2h13

range where matter e!ects are statistically signi"-cant at a given con"dence level.

Finally we show in Figs. 7 and 8 and the sensitiv-ity of the statistical signi"cance to the chosen valueof *m2

31. Fig. 7 is exactly the same plot as Fig. 4

with minimal and maximal *m231

. In the left plot we


Fig. 7. Sensitivity of the statistical signi"cance of matter e!ects to the value of *m231

analogous to Fig. 4. The left plot uses*m2

31"1.0]103 eV2 while for the right plot *m2

31"8.0]103 eV2 with otherwise unchanged parameters. As in Fig. 4 the solid contour

lines correspond to np"100 sin 2h13

) M1,2,4,8,16N.

Fig. 8. Sensitivity of the statistical signi"cance of matter e!ectsto the value of *m2

31for "xed ¸"8000 km as a function of E

l.

The lines show 100 sin 2h13

np for di!erent *m231

values.

have *m231

"1.0]103 eV2 and in the right plot*m2

31"8.0]103 eV2. Fig. 8 shows the statistical

signi"cance (i.e. 100 sin 2h13

np) for di!erent *m231

values for ¸"8000 km as a function of El. For

very long baselines, one can see from Fig. 8 thatincreasing *m2

31mainly shifts the resonance energy

to higher values thus demanding higher beam ener-gies to reach optimal statistical signi"cance. Thelocal maximum in Fig. 7 shifts to 10 GeV for min-imal *m2

31and to 40 GeV for maximal *m2

31. Im-

proved knowledge of *m231

would thus in principleallow to discuss optimization issues, but sucha study should also include systematics and back-grounds. Moreover, the energy distribution of

wrong-sign muon events would add important in-formation to the simple counting of events. A de-tailed discussion of muon energy optimizationdepends therefore on the way that information willbe exploited.

5. Subleading *m221

-e4ects

The results shown so far were obtained in thelimit *m2

21"0 which is, as already explained,

a perfect approximation for *m221

;10~4 eV2 orsin2 2h

12;1, i.e. for the VO and SMA MSW solu-

tions of the solar neutrino problem. The LMAMSW solution, that we will consider in this section,allows however *m2

21values up to 2]10~4 eV2

and prefers sin2 2h12

K0.8 [29] so that e!ects asso-ciated to *m2

21can become important, especially

for the largest *m221

values in the LMA range. Twomore mixing parameters, namely h

12and d, be-

come relevant for *m221

O0 [30,31] (see also, e.g.,Ref. [32]). While h

12is rather constrained, so that

we will use sin2 2h12

"0.8 in the following numer-ical results, any value of d in its range 04d(2p isallowed at present. In order to calculate the e!ectsassociated with a non-vanishing *m2

21, the value of

d must be speci"ed. In Fig. 9, the total rates in theappearance channels m

%Pm

l(left) and m6

%Pm6

l(right) for *m2

21"0 (solid line) are compared with

the total rates for *m221

"10~4 eV2 and four pos-sible values of d in its range, d"0, p/2, p, 3p/2


THEORY

Fig. 9. Appearance event rates nk`(k~), nk~(k`) in matter with subleading *m221

"10~4 eV2 and four possible values of the CP-phased"0, p/2, p, 3p/2 against baseline ¸ (dashed lines) compared with the corresponding event rates with negligible *m2

21(solid lines) for

the channels m%Pm

l(left) and l6

%Pl6

l(right). Both plots assume N

l"2]1020, e"50%, E

l"20 GeV, sin2 2h

23"1, sin2 2h

12"0.8

and sin2 2h13

"0.1.

Fig. 10. Same as Fig. 9 but for sin2 2h13

"0.01.

(dashed lines). The size of the e!ects depends cru-cially on the value of h

13. Fig. 9 assumes a value of

h13

at its upper limit, i.e. sin2 2h13

"0.1, whereasFig. 10 shows the e!ects for a sin2 2h

13one order of

magnitude smaller, sin2 2h13

"0.01. Both "guresassume N

lN

,Te"1021.

Figs. 9 and 10 illustrate several interesting fea-tures of the *m2

21e!ects. First of all, a comparison

of Figs. 9 and 10 con"rms that the relative size ofthe e!ects grows when h

13gets smaller [4]. This is

because the zeroth-order approximations (in *m221

)for the appearance probabilities have a sin2 2h

13suppression, whereas the linear *m2

21corrections

(CP-conserving and violating) are suppressed byonly one power of sin 2h

13and the corrections

quadratic in *m221

(CP-conserving) are not sup-pressed by h

13at all. Unlike the appearance chan-

nels, the disappearance channels are dominated bythe transitions to m

s(m6

s) and are therefore not

suppressed by h13

or *m221

, so that the h13

-sup-pressed *m2

21corrections are much less signi"cant

in this case. Figs. 9 and 10 show also that the rangeof the *m2

21corrections is essentially determined by

h13

, but the precise size and sign of these correc-tions within this range is unknown if d is uncon-straint. This can be seen also explicitly in Figs.9 and 10 where the *m2

21corrections show up as

oscillations around the leading *m231

contributionto the rates (dashed line), whose initial phase de-pends on d. *m2

21e!ects represent consequently in

the present LMA scenario in a high statistics long-but-not-too-long baseline measurement of h

13an

important source of systematic error, unless d ismeasured [4,7]. On the other hand, sin d could bemeasured or constrained in this scenario by com-paring the rates in the two CP-conjugated channelsm%Pm

land m6

%Pm6

lif very high intensity sources

and large detectors will become available [7]. By


7Note that C`

and C~

as de"ned in Eq. (15) must beinterchanged for *m2

31(0.

comparing e.g. the left and right plots of Fig. 10 onecan see that the *m2

21correction has the same sign

in the two channels for the two CP-conservingvalues d"0, p, when sin d"0, but has oppositesign for the two CP-violating values d"p/2, 3p/2,i.e. when sin d"$1.

Note, however, that a measurement of d based ona comparison of CP-conjugated rates at a singlevalue of ¸ would not be enough in order to keep the*m2

21e!ects under control. Suppose, for example,

that d"0 or p. Then the *m221

e!ects would be thesame in the two channels and an ideal experimentlooking for CP-violation would measure sin d"0.The value of d would, however, still be undeter-mined for the simple reason that both d"0 andp give sin d"0. As a consequence, it would be stillunknown whether the *m2

21corrections in both

channels add (d"0) or subtract (d"p) to theleading *m2

31contribution to the rates and this

ambiguity would translate, e.g., in an uncertaintyon a measurement of h

13. Such an ambiguity could

be resolved by comparing rates measured at di!er-ent distances ¸. Fig. 10 for sin2 2h

13"0.01 shows

in the m%Pm

lchannel that the change in rates

between ¸"3000 and 700 km allows to discrimi-nate between d"0 and p. Fig. 9 shows that thedi!erent ¸ dependence is also signi"cant forsin2 2h

13"0.1, allowing also to distinguish be-

tween the two possibilities d"0 and p. If CP-violation were maximal, Dsin dD"1, the comparisonof the rates at di!erent baselines would be lesssigni"cant but still helpful. Figs. 9 and 10 show"nally also that the *m2

21e!ects become smaller

when ¸ is increased. At very long baselines thee!ects are smaller than they would be in vacuum.The results obtained in the previous sections holdtherefore within a good approximation also in thecase of the LMA scenario and large *m2

21if

¸Z7000km.

6. Matter e4ects in the energy spectrum

Motivated by the small di!erential event rates,we discussed so far only the in#uence of mattere!ects on the total rates of wrong-sign muonevents. We demonstrated in the previous sectionsthat statistically signi"cant deviations from the

total event rates in vacuum represent alreadya good test of the MSW theory. A signi"cant testwould, however, be also given by a detailedmeasurement of MSW e!ects in the neutrino en-ergy spectrum, which is modi"ed in a very charac-teristic way. The m

lPm

land m6

lPm6

ldisappearance

channels are again dominated by transitions tom

sand m6

swhile m

%and m6

%transitions are only small

corrections. These channels are therefore mostlyinsensitive to matter e!ects in the di!erential eventrate spectrum and will not be discussed further.

To understand the e!ects in the appearance spec-trum of m

%Pm

land m6

%Pm6

lwe use again the approx-

imation *m221

"0. Matter e!ects have no in#uenceon the angle h

23in this case, whereas they modify

the mixing due to h13

signi"cantly. One obtainsthus for the m

%Pm

land m6

%Pm6

lappearance

channels the usual two #avor picture wheresin 2h

13Psin 2hm

13"sin 2h

13/C

B, and where C

`corresponds to neutrinos, C

~to antineutrinos.7

The enhancement of hm13

is maximal in the neutrinochannel when the neutrino energy E coincides withthe MSW resonance energy E

3%4de"ned via

2E3%4<

*m231

"cos 2h13

. (28)

Note, however, that the maximum of the eventspectrum in the m

%Pm

lchannel, in general, does

not coincide with E3%4

since the MSW oscillationprobabilities are folded with the #uxes and crosssections. The maximum of the event spectrum isthus determined by the maximization of

Asin(*

31C

B)

CB

B2fm

%m

l(E/E

l) (29)

but the resulting maximum is still around E3%4

forthe muon energies under consideration. The o!setdepends in a rough approximation on the di!er-ence between E

3%4and the maximum of the #ux

which lies roughly at an energy of the order El.

This has to be compared with the vacuum casewhere the oscillation probabilities are also foldedwith #uxes and cross sections and where the


THEORY

Fig. 11. Modi"cations in the di!erential event rate spectrum (events per GeV) due to matter e!ects for El"20 GeV. The solid lines are

in matter while the dashed lines are without matter showing the `broadeninga, `shifta and `enhancementa due to MSW e!ects. Theassumed parameters are N

lN

,Te"1021, ¸"6596 km (CERN-MINOS) and sin2 2h

13"0.01.

maxima of the event rates are also not precisely atthe maxima of the oscillation probabilities. Theevent rate spectrum is thus due to this folding inboth cases with and without matter a rather com-plicated function of Em which depends in a non-trivial way on ¸, E

land *m2

31. Nevertheless for

given ¸, El

and for given *m231

, h23

, h13

measuredwith a suitable long baseline experiment, one canpredict the shape of the di!erential event rate spec-trum in all channels and compare it with the spec-trum of oscillations una!ected by matter. Thisresults in a very good opportunity to detect speci"cdetails of the MSW e!ects which arise when theoscillation parameters are chosen such that the "rstmaximum of vacuum oscillation coincides roughlywith E

3%4. The point is that the MSW e!ect changes

the probabilities compared to vacuum in threegenuine ways: The "rst maximum of the oscillation

probability as the energy decreases is enhanced, itswidth is broadened and its center is shifted to lowerenergies. Similarly one has an `anti-MSW e!ecta inthe antineutrino appearance channel which impliesfor the "rst oscillation maximum a reduction inheight, again a broadening and a shift to lowerenergies.

These `genuine MSW e!ectsa are demonstratedin Fig. 11 (where E

l"20 GeV) and Fig. 12 (with

El"50 GeV) showing the modi"cations in the

energy spectrum (events per GeV) due to mattere!ects. The solid lines are in matter while thedashed lines are without matter and the assumedparameters are N

lN

,Te"1021 as before,

¸"6596km and sin2 2h13

"0.01. Fig. 11 showsalready all e!ects due to the MSW mechanism,namely the broadening (the last oscillation in mat-ter covers two oscillations in vacuum), the shift (the


Fig. 12. Same as in Fig. 11 but for El"50 GeV where the MSW e!ects are harder to extract since E

lis already sizably above the

resonance energy E3%4

. For details see text.

maximum in matter lies almost in the minimum invacuum) and the enhancement or suppression com-pared to vacuum.

It is interesting to look at the modi"cationswhen the muon energy becomes higher, e.g., forE

l"50 GeV as shown in Fig. 12. The point is that

the beam energy is already rather far away from theMSW resonance energy and the importance of theweight function fm

%m

l(and of the E3m scaling of unos-

cillated events) in the determination of the shape ofthe spectrum becomes apparent since the genuinebroadening, shift and enhancement/suppression ef-fects become harder to distinguish. For the m

%Pm

lchannel the e!ect could be hard to distinguish fromuncertainties (with low statistics) in the spectrum.This brings up the general issue that one has tohave enough statistics for such an analysis. In orderto have a chance to see such e!ects one has to belucky and h

13should be at the upper experimental

limit (see scaling laws). Otherwise NlN

,Te must be

increased correspondingly which implies a moreintense muon source, a larger detector or both.Although a more quantitative analysis of the signif-icance of e!ects in the di!erential neutrino eventrate spectrum is beyond the scope of this paper,an analysis of the di!erential event rate spec-trum would clearly provide extremely valuableadditional information which would allow to testsome of the characteristic features of the MSWmechanism.

7. Conclusions

Assuming three-neutrino mixing we studied inthis paper the possibility to test the MSW e!ect interrestrial very long baseline neutrino oscillationexperiments which become possible with neutrino


THEORY

factories. Such direct tests are important since theMSW mechanism is widely used in di!erent scen-arios of neutrino physics and astrophysics. Thecorrect analysis and interpretation of the data fromterrestrial very long baseline neutrino oscillationexperiments, ¸Z1000 km, is in fact impossiblewithout a proper treatment of matter e!ects. Thelatter is also crucial for the searches of CP-violationin neutrino oscillations generated by the leptonmixing matrix, since matter e!ects create an asym-metry between the two CP-conjugated appearancechannels.

Studies of the m%Pm

land m6

%Pm6

loscillations

are by far most promising for the detection of thematter e!ects since the corresponding total eventrates are a!ected in a drastic way by these e!ects.We considered for the present study neutrinotrajectories through the earth which cross themantle, but do not pass through the earth core,which corresponds to neutrino path lengths¸[10000 km. Using analytic expressions for thethree neutrino oscillation probabilities in matter inthe constant average density and small *m2

21ap-

proximations and including #uxes, cross sectionsand detection e$ciencies allows to describe therelevant event rates analytically. This permitteda full analytic understanding of our numericalresults.

By considering the asymmetry between them%Pm

land m6

%Pm6

linduced wrong-sign muon

event rates, we studied the statistical signi"cance ofthe observation of matter e!ects as a function of theneutrino oscillation parameters h

13and *m2

31as

well as its dependence on the experimental condi-tions via the parent muon beam energy E

l, the path

length ¸ and the product of useful muons, detectorsize and e$ciency. The scaling of rates, statisticalsigni"cances and sensitivities with the relevant mix-ing angles, in particular, with h

13, the intensity of

the muon source and with the detector size ande$ciency have been given, so that the results forany value of those parameters can easily be ob-tained. The sign of the asymmetry depends onwhether the two closest neutrino mass eigenstatesare lighter (*m2

31'0) or heavier (*m2

31(0) than

the third one, thus providing a way of determiningwhich of these two possibilities is realized. Figs.5 and 6 show the conservative ranges of sin2 2h

13

where such a determination would be signi"cant ata given con"dence level from the statistical point ofview as a function of the baseline.

We analyzed, in particular, the statistical signi"-cance of matter e!ects as a function of E

land ¸.

The most important requirement regarding themuon energy is that for given *m2

31it has to be

greater than the MSW resonance energy. For,e.g., *m2

3146.0(8.0)]10~3 eV2, this implies E

lZ20

(30) GeV. The value of El&30 GeV (and

¸Z7000 km) is practically optimal for *m231

K

(4.0}8.0)]10~3 eV2, while if *m231K(2.0}3.5)]

10~3 eV2, ElK20 GeV would be preferable (see

Figs. 4, 7 and 8). If, however, *m231

K10~3 eV2 andsin2 2h

13[0.01, then establishing matter e!ects (or

obtaining a stringent upper limit on sin2 2h13

) re-quires E

lK(40}50) GeV. It is clear from the above

results that the optimal value of El

depends signi"-cantly on the precise value of *m2

31.

Our analysis shows that a higher sensitivity tothe MSW e!ect in the case of relatively small valuesof sin2 2h

13is achieved at ¸Z7000 km. We showed

that this conclusion holds also when subleading*m2

21e!ects are included. These e!ects can be sig-

ni"cant in the case of the LMA MSW solution with*m2

21K(0.5}2.0)]10~4 eV2. At ¸Z7000 km, the

indicated *m221

-induced e!ects are considerablysmaller than in vacuum and essentially negligible.Thus, the results obtained in the limit of *m2

21"0

are su$ciently accurate for ¸Z7000 km. This isvalid even in the case of the LMA MSW solutionwith *m2

21K(0.5!2.0)]10~4 eV2. For shorter

baselines, *m221

e!ects are non-negligible and a de-termination of the CP-violating phase d would benecessary in order to know their precise magnitude.We have found that the ¸ dependence of the *m2

21e!ects o!ers the possibility to determine the CP-phase d, especially when sin d is small. Forsin2 2h

13"0.01, for instance, the event rates due

to the m%Pm

land m6

%Pm6

ltransitions change

considerably when ¸ changes from &700 kmto &3000 km. Thus, a measurement of theserates, e.g., at the indicated two distances wouldallow to determine the value of d with a certainprecision.

Finally, we discussed the matter e!ects in thedi!erential event rate spectrum as a function of theneutrino energy, and showed that they lead to very


characteristic distortions. The observation of thesedistortions would allow very detailed tests of theMSW theory.

To conclude, our study shows that the predic-tions of the MSW theory can be tested in a statist-ically reliable way in a large region of thecorresponding parameter space by a simple analy-sis of the total event rates in a very long baseline,¸Z7000 km, neutrino oscillation experiment. Thiscan allow to determine the sign of *m2

31as well, as

was recently noticed also in Ref. [8]. Not seeing thematter e!ects would lead to impressive upper limitson the mixing angle h

13down to sin2 2h

13K10~4

or even better.

Acknowledgements

A.R. and S.P. wish to thank the Institute T30d atthe Physics Department of the Technical Univer-sity of Munich for warm hospitality.

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THEORY

very long baseline neutrino oscillation experiments and the msw effect

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