Very long baseline neutrino oscillation experiments and the MSW effect

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

Very long baseline neutrino oscillation experiments and the MSW e!ect夽 M. Freund , M. Lindner , S.T. Petcov
*, 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 to verify and test the MSW e!ect and to measure the lepton mixing angle h . We suppose that intense neutrino and  antineutrino beams will become available in the so-called neutrino factories. We "nd that the most promising and statistically signi"cant results can be obtained by studying m Pm and m Pm oscillations which lead to matter  l  l enhancements and suppressions of wrong-sign muon rates. We show the h ranges where matter e!ects could be  observed as a function of the baseline. We discuss the scaling laws of rates, signi"cances and sensitivities with the relevant mixing angles and experimental parameters. Our analysis includes #uxes, event rates and statistical aspects so that the conclusions should be useful for the planning of experimental setups. We discuss the subleading *m e!ects in the case of  the LMA MSW solution of the solar problem, showing that they are small for ¸97000 km. For shorter baselines, *m  e!ects can be relevant and their dependence on ¸ o!ers a further handle for the determination of the CP-violation phase d. Finally, we comment on the possibility to measure the speci"c distortion of the energy spectrum due to the MSW e!ect.  2000 Elsevier Science B.V. All rights reserved.

1. Introduction

夽

Work supported in part by `Sonderforschungsbereich 375 fuK r Astro-Teilchenphysika der Deutschen Forschungsgemeinschaft, by the TMR Network under the EEC Contract No. ERBFMRX}CT960090 and by the Italian MURST under the program `Fisica Teorica delle Interazioni Fundamentalia. For the 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), romanino@thphys. ox.ac.uk (A. Romanino).  Also at: INRNE, Bulgarian Academy of Sciences, 1789 So"a, Bulgaria.

The long-term aim to build muon colliders o!ers the very attractive intermediate possibility for `neutrino factoriesa [1}3] with uniquely intense and precisely characterized neutrino and antineutrino beams. This requires only one muon beam at intermediate energies such that neutrino factories are rather realistic medium-term projects which constitute also a useful step in accelerator technology towards a muon collider. The current knowledge of neutrino masses and mixing implies for typical setups very promising very long baseline neutrino 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

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

a three neutrino framework the potential to verify and test the MSW e!ect and to measure or limit h in terrestrial very long baseline experiments  with neutrino factories where the neutrino spectrum and #uxes are rather well known and under control [1]. Calculating the oscillation probabilities and event rates for di!erent channels and comparing with those for oscillations in vacuum we "nd that the asymmetry between the m m and m m  l  l oscillations is a very promising tool to test and verify the MSW e!ect. The reason is, as we will see, that matter e!ects lead to measurably enhanced event rates due to the m Pm transitions, while the  l rates due to the m Pm transitions are equally  l suppressed. The asymmetry between the event rates associated with these two channels would therefore be very sensitive to the MSW e!ect since the matter-induced changes have opposite e!ect on the two rates thus amplifying the `signala, while at the same time common backgrounds would drop out. We analyze event rates and we include statistical aspects such that the results are directly applicable for the planning of optimal experimental setups. We discuss the capabilities of a neutrino factory experiment as a function of the distance between the neutrino source and the detector and of the muon source energy for the optimal observation of the MSW e!ect. Moreover, we determine the sensitivity to the value of h for di!erent experimental  con"gurations. Demonstrating and testing the MSW e!ect directly is of fundamental importance since this e!ect plays a basic role in di!erent neutrino physics scenarios. The MSW mechanism provides, for example, the only clue for understanding the solar neutrino de"cit with a neutrino mass squared di!erence within a few orders of magnitude from that inferred from the atmospheric neutrino data. Atmospheric neutrinos can undergo matter-enhanced transitions in the earth. The matter e!ects in neutrino oscillations will play an important role in the interpretation of the results of a neutrino factory experiment using an ¸91000 km baseline. They are essential in

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

19

the searches for CP-violation in such experiments [2,4], since matter e!ects generate an asymmetry between the two relevant CP-conjugated appearance channels [4]. Knowing the asymmetry caused by matter e!ects is therefore essential for obtaining information on the CP-violation originating from the lepton mixing matrix. Matter-enhanced neutrino transitions can play important role in astrophysics as well. Neutrino factories have been extensively discussed in the literature [1}3,5}8]. Either muons or anti-muons are accelerated to an energy E and l decay then in straight sections of a storage ring like l\Pe\#m #m or l>Pe>#m #m so that  l l  a very pure neutrino beam containing m and m or  l m and m , respectively, is produced. The muon  l energy E could be in a wide range from 10 to l 50 GeV or more and a neutrino #ux corresponding to 2;10 muon decays per year in the straight section of the ring pointing to a remote detector could be achieved. Higher #uxes are also currently under discussion [11]. The neutrino #uxes are therefore very intense and can be easily calculated from the decay spectrum at rest. For unpolarized muons and negligible beam divergence one "nds for a baseline of ¸"730 km a neutrino #ux of K4.3;10 yr\ m\ and for a baseline ¸" 10 000 km a neutrino #ux K2.3;10 yr\ m\. Note also that the m and m #uxes depend sizably   on the beam polarization, that will be assumed to be vanishing in this paper. Altogether a neutrino factory would provide pure and high-intensity neutrino beams with a well-known energy spectrum that in turn would allow a wide physical program including precise measurement of mixing parameters [2,3], matter e!ects [5,8] and, in case of LMA solution of the solar problem, leptonic CP-violation [6,7]. The produced neutrino beam will be directed towards 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 ¸910 km, as we shall assume, matter e!ects become important in neutrino oscillations. For ¸410 km the beam traverses the earth along a trajectory in the earth mantle without crossing the earth core where the density is substantially higher. According to the earth models [12,13], the

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

average matter density along the neutrino trajectories with ¸"(10}10) km lies in the interval &(2.9}4.8) g cm\. The matter density changes along each trajectory, but the variation is relatively small } by about 1.5}2.0 g cm\, and even the largest takes place over relatively big distances of several thousand kilometers. As a consequence, one can approximate the earth mantle density pro"le by a constant average density distribution. The constant density can be chosen to be equal to the average density along every trajectory. For the calculation of the neutrino oscillation probabilities in the case of interest, the indicated constant density model provides a very good approximation to the somewhat more complicated density structure of the earth mantle. Let us note also that the earth mantle is with a good precision isotopically symmetric: > "0.494 [12,13], where > is the electron   fraction number in the mantle. Since the beam consists always either of m  together with m or m in combination with m there l  l are, in principle, eight di!erent appearance experiments and four di!erent disappearance experiments which could be performed. From an experimental point of view, however, at present the four channels with muon neutrinos or antineutrinos in the "nal state, m Pm , m Pm , m Pm , l l l l  l m Pm seem to be most promising. A very impor l tant issue is in this context the ability of the detector to discriminate between neutrinos and antineutrinos, namely the ability to measure the charge of the leptons produced by the neutrino charged current interactions. Note that very good discrimination capability is required for a measurement of the appearance probabilities since they produce `wrong signa muon events in the detector and have to be discriminated from the much larger number of events associated with the m and m surl l vival probabilities. The channels with electron neutrinos or antineutrinos in the "nal state are problematic from this point of view due to the di$culty of telling e> from e\ in a large highdensity detector. On the contrary, a very good l>/l\ discrimination could be obtained in a large properly oriented magnetized detector [14]. The paper is organized as follows. In Section 2 we give the analytic formulae for three neutrino oscillations in matter which contain the essential

physics relevant for our study. In Section 3 we discuss event rates, their parameter dependence, their scaling behavior and we show results from our numerical calculations. In Section 4 we de"ne the sensitivity to matter e!ects in a statistical sense and discuss the results of our numerical calculations including parameter uncertainties. This is followed by a discussion in Section 5 of the e!ects of a nonvanishing *m . In Section 6 the possibility of  detecting matter e!ects by looking for the enhancement, the broadening and the shift of the MSW resonance energy is discussed and we conclude in Section 7.

2. Three neutrino oscillation probabilities in matter We will assume in this paper the existence of three #avor neutrino mixing:  "m 2" ; "m 2, l"e,l,s J JI I I

(1)

where "m 2 is the state vector of the (left-handed) J #avor neutrino m , "m 2 is the state vector of a neuJ I trino m possessing a de"nite mass m , m Om , I I I H kOj"1,2,3, m (m (m , and ; is a 3;3 uni   tary matrix } the lepton mixing matrix. It is natural to suppose in this case that one of the two independent neutrino mass-squared di!erences, say *m ,  is relevant for the vacuum oscillation (VO), small or large mixing angle MSW solutions (SMA MSW and LMA MSW) of the solar neutrino problem with values in the intervals [15,16] VO: 5.0;10\ eV:*m :5.0;10\ eV  (2a) SMA MSW: 4.0;10\ eV:*m :9.0;10\ eV  (2b) LMA MSW: 2.0;10\ eV:*m :2.0;10\ eV  (2c) while *m is responsible for the dominant atmos pheric m m oscillations and lies in the interval l s ATM: 10\ eV:*m :8.0;10\ eV. 

(3)

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

For Em 51 GeV, ¸410 km, the *m-hierarchy *m ;*m (4)   and *m ;10\ eV, the probabilities of three neutrino oscillations in vacuum of interest reduce e!ectively to two-neutrino vacuum oscillation probabilities [17,18]: Pm (m Pm )"P (m Pm )  J JY J JY




2"; ""; " 1!cos J JY lOl"e,l,s m



*m ¸  , 2E (5)

m

P (m Pm )"P (m Pm )  J J  J J

1!2"; "(1!"; ") J J *m ¸  , l"e,l,s. (6) ; 1!cos 2E












21

playing the role of the corresponding two-neutrino oscillation parameters [20,21] (for recent reviews see, e.g., Refs. [22,23]). For "; " satisfying limit (7),  however, this dependence is rather weak and cannot be used to further constrain or determine "; ". In  general, under condition (4) and for *m ;  10\ eV the relevant solar neutrino transition probability depends only on the absolute values of the elements of the "rst row of the lepton mixing matrix, i.e., on "; ", i"1,2,3, while the vacuum G oscillations of the (atmospheric) m , m , m and m on l l   earth distances are controlled by the elements of the third column of ;, "; ", l"e,l,s. The other eleJ ments of ; are not accessible to direct experimental determination. Moreover, the CP- and T-violation e!ects in the oscillations of neutrinos are negligible. For our analysis we use a standard parametrization of the lepton mixing matrix ;:



; ; ; c c s c s e\ B         ; ; ; " !s c !c s s e B c c !s s s e B s c l l l             ; ; ; s s !c c s e B !c s !s c s e B c c s s s            

Under conditions (1) and (4) the element "; " of  the lepton mixing matrix, which controls the m Pm , m Pm , m Pm and the m Pm oscil ls  ls l  l  lations, is tightly constrained by the CHOOZ experiment [19] and the oscillation interpretation of the solar and atmospheric neutrino data: for 3.0;10\ eV4*m 48.0;10\ eV one has  "; ":0.025. (7)  The CHOOZ upper limit is less stringent for 1.0;10\ eV4*m (3.0;10\ eV where  values of "; " 0.05 are allowed.  Note that under condition (4), the VO or MSW transitions of solar neutrinos depend on "; " and  on the two-neutrino VO or MSW transition probability with *m and h , where   "; ""; "   sin 2h "4 ,  ("; "#"; ")   "; "!"; "  cos 2h "  (8)  "; "#"; "  

(9)

where c ,cos h , s ,sin h and d is the Dirac GH GH GH GH CP-violation phase. The angles h and h in   Eq. (9) are constrained to lie within rather narrow intervals by the solar and atmospheric neutrino data for each of the di!erent solutions, Eqs. (2a)}(2c), of the solar neutrino problem. With the accumulation of data the uncertainties in the knowledge of h and h will diminish, while only upper   limits on s like Eq. (7) have been obtained so far.  The mixing angle h is one of the 4 (or 6 } depend ing on whether the massive neutrinos are of Dirac or Majorana type [24,25,18]), fundamental parameters in the lepton mixing matrix. It controls the probabilities of the m (m )Pm (m ), m (m )Pm (m ), l   l l   l m Pm and m Pm oscillations and the CP- and  s  s T-violation e!ects in neutrino oscillations depend

 We have not written explicitly the possible Majorana CPviolation phases which do not enter into the expressions for the oscillation probabilities [24,25].

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on it. Obviously, one of the main goals of the future neutrino oscillation experiments should be to determine the value of h or to obtain a more  stringent experimental upper limit than the existing one (9). Under condition (4), with *m ;10\ eV and  in the constant density approximation, the oscillation probabilities of interest take the following simple form (see, e.g., Refs. [22,23,26]): Pm(m Pm )"Pm(m Pm ) # l  #  l

s Pm(*m , sin 2h )  #  

(10a)

Pm(m Pm ) # l l

c #s [1!Pm(*m , sin 2h )]   #   # 2c s Re[e\ GAm(*m , sin 2h )] (10b)   #   Pm(m Pm ) 2c s [1!Pm(*m , sin 2h ) # l s    #   ! Re (e\ GAm(*m , sin 2h ))] #   (10c) where Pm(*m , sin 2h )"[1!cos *E ¸] sin 2hK #    K  (11) and Am(*m , sin 2h )"1#(e\ #K *!1) cos hK  #   (12) are respectively the two-neutrino transition probability and probability amplitude of neutrino survival in matter with constant density and



¸ *m  #
K 2 2E



(13)

is a phase. In Eqs. (11)}(13) *E and hK are the K  neutrino energy di!erence and mixing angle in matter,






*m 1 2E< *E "  C , cos 2hK " cos 2h ! K  C  *m 2E > >  (14)

where






2E<  2E< C " 1G sin h $4 !  *m *m   and

(15)

<"(2G NM  (16) $  is the matter term, NM  being the average electron  number density along the neutrino trajectory in the earth mantle. The probability Pm(m Pm ) can #  s be obtained from Eq. (10a) by replacing the factor s with c . The corresponding anti  neutrino transition and survival probabilities have the same form and can formally be obtained from Eqs. (10a)}(10c) by changing the sign of the matter term, i.e. NM P!NM  in the expressions for i,   *E cos 2hK , Eqs. (13) and (14), and by replacing K  C by C . > \ Several comments are in order. First, the analytic expressions (10a)}(10c) represent excellent approximations in the case of the VO and SMA MSW solutions of the solar neutrino problem and for values of *m :5;10\ eV from the LMA MSW solu tion region. For 5;10\ eV(*m :2; 10\ eV,  the corrections due to *m can be non-negligible  and we are going to discuss them in Section 5. Second, as it is clear from Eqs. (10a) and (11), the probabilities Pm(m Pm ) and Pm(m Pm ) cannot #  l #  l exceed s &0.5. The maximal value s can be   reached only if the MSW resonance condition sin 2hK "1 and the condition cos *E ¸"!1  K are simultaneously ful"lled. At the MSW resonance, however, one has *E 1.23p;10\ km\ K tan 2h NM (N cm\), where NM  is in units of     N cm\, N being the Avogadro number, and for   s 40.025 the second condition cos *E¸"!1  K can only be satis"ed for ¸510 km. If s "0.05  this condition requires ¸58;10 km. Given the fact that the neutrino #uxes decrease with the distance as ¸\, the above discussion suggests that the maximum of the event distribution in E and ¸ in the transition m Pm should take place  l approximately at the MSW resonance energy, but at values of ¸ smaller than those at which max(Pm(m Pm )) s . The MSW resonance en#  l  ergy is given by E 6.56*m (10\ eV)   cos 2h (NM (cm\ N ))\ GeV, where *m and     NM  are in units of 10\ eV and cm\ N ,  

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

respectively. For *m (10\ eV)"3.5, 6.0,  8.0 and, e.g., NM (cm\N )"2, we have   E 11.5;19.7;26.2 cos 2h GeV. The m Pm and    l m Pm oscillations will be a!ected substantially by  l the MSW e!ect if the energy of the parent muon beam E 'E . This condition can be satis"ed l  for any value of *m from the interval (3)  for E 930 GeV. For the value of *m l  (10\ eV)"3.5 which is currently `preferreda by the Super-Kamiokande data we have E 9 l 15 GeV. Finally, we note also that the hierarchy *m ;"*m " does not necessarily correspond to   a situation where m 'm 9m , namely to a situ   ation where the two closest mass eigenstates (m ,  m ) are lighter than the third one (m ). As long as   matter and CP-violation e!ects are not taken into account, a phenomenologically equivalent situation is given when the two closest mass eigenstates are heavier than the third one, i.e., m 9m 'm .    In this case we still have *m ;"*m " but   *m (0. Such a change in the sign of *m re  verses the sign of N which opens the possibility to  determine this sign via matter e!ects as recently emphasized by Barger et al. [8]. 3. Event rates We discuss now the e!ects of matter on the total rate of m Pm , m Pm , m Pm , m Pm events. If  l  l l l l l for example l> are accumulated in the storage ring then the neutrino beam contains m and m . Since  l the neutrino}antineutrino transitions have a negligible rate, the beam-induced l> events in the detector must be attributed in this case to the charged current interactions of unoscillated m . The total l number n > (l>) of l> events measures therefore an l averaged m survival probability. Wrong-sign l l\ events must be attributed to m generated by l oscillations of the initial m , so that their total number  n > (l\) measures the averaged m Pm oscillation l  l probability. Analogously, for l\ in the storage ring, the total numbers of l\ and l> events n \ (l\), l n \ (l>) measure the averaged m survival probabill l ity and m Pm oscillation probability, respectively.  l  This scheme implies an approximate degeneracy of either m and m or all the mass eigenstates.  

23

The total number of events in each channel are given by 10N E  l n > (l\)"N > N l l 2 mp ¸ l #l ; fm ml (E)Pm(m Pm )(E)(dE/E ) #  l l #  (17a)



10N E  l n \ (l>)"N \ N l l 2 mp ¸ l #l ; fm  m l (E)Pm(m Pm )(E)(dE/E ) #  l l #  (17b)



10N E  l n \ (l\)"N > N l l 2 mp ¸ l #l ; fml ml (E)Pm(m Pm )(E)(dE/E ) # l l l # 



(17c)

10N E  l n > (l>)"N \ N l 2 mp ¸ l l #l ; fm l m l (E)Pm(m Pm )(E)(dE/E ) # l l l #  (17d)



where Pm denotes the oscillation probability de# scribed in Section 2, N > (N \ ) is the number of l l `usefula l> (l\) decays, namely the number of decays occurring in the straight section of the storage ring pointing to the detector, N is the size of 2 the detector in kilotons, 10N is the number of  nucleons in a kiloton, E is the energy of the muons l in the ring and E "3 GeV is a lower cut on the

 neutrino energies that helps a good detection e$ciency. Since low-energy events are suppressed by the low initial #ux and the low cross section (see below), the results do not depend signi"cantly on the precise value of E for E (5 GeV. The



 functions f averaging the probabilities are given by fm ml (E)"gm (E/E ) (pml (E)/E )e \ (E) l l l fm  m l (E)"gm  (E/E ) (pm l (E)/E )e > (E) l l l fml ml (E)"gml (E/E ) (pml (E)/E )e \ (E) l l l fm l m l (E)"gm l (E/E ) (pm l (E)/E )e > (E) l l l

(18a) (18b) (18c) (18d)

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

and take into account the appropriately normalized initial spectrum of m-neutrinos produced in the decay of unpolarized muons, gm (E/E ), the l charged current cross section per nucleon, pml m l (E),   and the e$ciency for the detection of l\ (l>), e \ > (E) (we neglect here the "nite resolution of the l l  detector). For the numerical calculations we use gm (x)"gm  (x)"12x(1!x), gml (x)"gm l (x)"2x(3!2x)

(19a)

pml (E)"0.67;10\E cm GeV\, pm l (E)"0.34;10\E cm GeV\

(19b)

and e \ (E)"e > (E)"e for E'E so that l l

 fm ml (E)/fm  m l (E)"fml ml (E)/fm l m l (E)"2 independent of the energy. The contribution of the background to the number of muons observed in the detector has been neglected in Eqs. (17). That background includes muons from the decay of charm quarks produced by charged and neutral current neutrino interactions in the detector and from the decay of s produced by m interactions. Both these sources can be s kept under control in di!erent ways [8,27]. Moreover, Eqs. (17a)}(17d) neglect the divergence of the muon beam in the straight section of the storage ring. We can discuss now the dependence of n > (l\), l n \ (l>), n \ (l\), n > (l>) on the mixing parameters l l l and some of the experimental parameters. In the *m ;10\ eV approximation discussed above,  the only physical mixing parameters are h and  h . In the leading approximation the dependence  of the total event rates in matter (and in vacuum) on those parameters is n \ (l\), n > (l>)Jsin 2h (20a) l  l n > (l\), n \ (l>)Jsin h sin 2h . (20b) l l   Higher-order corrections in h are constrained by  the CHOOZ limit. Despite the resonant matter enhancement of the mixing due to h , such correc tions become only sizable for h close to its upper  limit and very long baselines where they can reach about 20% in the resonant channels. Since Eq. (20b) will turn out to be useful when discussing the sensitivity to matter e!ects, we discuss

it in greater detail. Let us write Eq. (10a) in the form Pm(m Pm ,m Pm ) #  l  l






sin(* C )   ! (21) C ! where C is given by Eq. (15) and corresponds to > neutrinos, C to antineutrinos (the opposite for \ *m (0) and * "*m ¸/(4E). In the limit in    which sin h can be neglected on the right-hand  side of Eq. (15), Eq. (21) shows that n > (l\) and l n \ (l>) are indeed proportional to sin 2h . Del  spite the CHOOZ limit, sin h :0.025, the second  term in Eq. (15) can be relevant when the "rst term vanishes around the resonance. For 2E<"*m  we have in fact C "2sin h and >  Pm(m Pm ) #  l ¸< sin(sin h ¸<)   "sin h sin 2h (22)   4 sin h ¸<  whereas by neglecting the sin h term in Eq. (15)  we would get C "0 and > ¸< Pm(m Pm )"sin h sin 2h . (23) #  l   4 "sin h sin 2h  






A comparison of Eqs. (22) and (23) shows however that the approximation works even at the resonance provided ¸:p/(4< sin h )&7000 km  (0.15/sin h ). The e!ect of the sin h term is   therefore maximal in Eq. (15) for very long baselines and h close to the experimental limit. In this  case it can a!ect the averaged probability sizably, while it is negligible for smaller h or smaller  baseline. For a better approximation one can use n > (l\), n \ (l>) l l






sin(sin h ¸<)   (24) sin h ¸<  which deviates less than 5% from the exact result in the whole parameter space (¸:10 000 km, 20 GeV :E :50 GeV). In most cases ¸:p/(4< sin h ) l  holds and the oscillating term in Eq. (24) can be expanded, giving Eq. (20b) or higher-order approximations. We stress that the baseline ¸ appears as part of the correction to the h , h   scaling only. The dependence on ¸ of the rates is more involved, especially for very large baselines, Jsin h sin 2h  

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

25

Fig. 1. Appearance event rates n > (l\), n \ (l>) in matter against baseline ¸ (solid lines) compared with the corresponding event rates in l l vacuum (dashed lines) for E "20 GeV (left) and E "50 GeV (right). Both plots assume N "2;10, e"50%, sin 2h "1 and l l l  sin 2h "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 the dependence on the beam intensity, detector size and e$ciency is trivial: n > (l\), n \ (l\)JN > N e > l l l 2 l

(25a)

n \ (l>), n > (l>)JN \ N e \ . l l l 2 l

(25b)

We present now quantitative results for the total rates in matter for *m ;10\ eV and we com pare them with the results one would obtain in vacuum. The statistical signi"cance of the matter e!ects will be discussed in the following section and e!ects of larger *m will be covered in Section 5.  The total event rates depend as already discussed in a transparent way on the experimental parameters N ! , N , e ! and on the mixing parameters l 2 l h , h . We focus our discussion therefore on the   less transparent dependence on the baseline and muon energy. For that we use the central value of *m "3.5;10\ eV, and we assume *m '0.   In the *m ;10\ eV approximation, in which 

CP-violation e!ects are negligible, the results for *m (0 can be obtained by simply interchanging  neutrinos and antineutrinos. The total number of events in the two appearance channels m Pm , m Pm is shown for  l  l E "20 GeV and E "50 GeV in Fig. 1 as a funcl l tion of the baseline (solid lines) in comparison with the event rates one would get if the neutrinos did not interact with matter (dashed lines). Fig. 2 shows the two disappearance channels m Pm , m Pm . l l l l Both "gures correspond to a `defaulta experimental setup providing N > "N \ "N " l l l 2;10 useful muon decays (e.g. in one year of running) and to a detector with N "10 kt and an 2 e$ciency e > "e \ "e"50% in both channels. l l The rates depend only on the combination N N e l 2 which is in our case N N e"10. Moreover, l 2 we assume for these "gures sin 2h "1 and  sin 2h "0.01, one order of magnitude below the  experimental limit. The rates for di!erent values of N , N , e, h , h can be obtained by using Eqs. l 2   (25), (20a) and (20b) or (24).

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Fig. 3. Di!erential appearance rates for the channels m Pm (left) and m Pm (right) over ¸ and Em showing for large ¸ the  l  l enhancement and suppression due to the MSW mechanism. The parameters are identical to those of Fig. 1.

The vacuum event rates in the neutrino channels are twice as big as the rates in the antineutrino channels. This is because the oscillation probabilities in CP-conjugated channels in the *m ;10\ eV approximation are the same  while (due to the larger cross section) the functions averaging the probabilities are larger by a factor 2 in the neutrino channels. The disappearance channels shown in Fig. 2 are essentially independent of matter e!ects since these e!ects come only with the h corrections to the oscillation probabil ities. In contrast, Fig. 1 shows the drastic enhancement (depletion) of the event rates in the m Pm  l (m Pm ) channel for very long baselines. The  l growth of the total rates with the muon energy which is obvious from Eq. (17) can also be seen in Fig. 1. Fig. 3 shows in more detail the di!erential event rates of the appearance channels m Pm and  l m Pm as three-dimensional plots over ¸ and Em .  l The "gures show nicely the enhancement (suppression) due to the MSW mechanism in the m Pm (m Pm ) channel for large baseline ¸ at  l  l Em K10 GeV. The baseline and muon energy dependence of matter e!ects will be further discussed in the next section in connection with a quantitative analysis of the signi"cance of the e!ects shown in Figs. 1 and 2. 4. Statistical signi5cance of matter e4ects We have seen in the previous section that matter e!ects change the total event rates in the appearance

 As discussed above e!ects as large as 20% can occur for h close to the CHOOZ limit and very large baselines. 

channels m Pm and m Pm in very long baseline  l  l experiments in a drastic way. Such experiments would therefore o!er unique possibilities to observe matter e!ects and to test the predictions of the MSW theory. In order to study the capabilities of a neutrino factory experiment quantitatively, we must "rst de"ne the meaning of `observing matter e!ectsa. One of the most interesting possibilities would be a detailed observation of the shape of the neutrino energy spectrum which is modi"ed by the MSW e!ect in a very characteristic way. This would allow to test non-trivial predictions of the MSW theory and would allow to unambiguously attribute the enhancement/depletion of the total number of neutrino interactions to matter e!ects. High di!erential event rates and a good calibration of the detector would however be necessary for this option. We will discuss this possibility in Section 6. In this section we con"ne ourselves to a discussion of the signi"cance of MSW e!ects in total event rates. Matter e!ects produce deviations of the total number of wrong-sign muon events n > (l\), l n \ (l>) from what is expected in the absence of l interaction with matter. The discussion above clearly shows that such deviations occur in opposite directions in the appearance channels. Suppose that a certain number of wrong-sign muon events in the neutrino (antineutrino) appearance channel n > (l\) (n \ (l>)) is measured, while 1n>(l\)2 l l l (1n\(l>)2) would be expected in the absence of l matter e!ects. We want to determine the con"dence level at which n > (l\) and n \ (l>) could represent l l statistical #uctuations around the expected values in vacuum 1n>(l\)2, 1n\(l>)2. We follow the l l procedure proposed by the Particle Data Book

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[28] and calculate con"dence levels by using s"2[1n>(l\)2!n > (l\)] l l n > (l\) #2n > (l\)log l l 1n>(l\)2 l # 2[1n\(l>)2!n \ (l>)] l l n \ (l>) #2n \ (l>)log l . (26) l 1n\(l>)2 l The corresponding `number of standard deviationsa is given by n ,(s. This prescription incorporates N the available information in both the measured numbers n > (l\), n \ (l>) in the most complete way. l l Before we show the numerical results, let us discuss the qualitative dependence of n on the relN evant parameters. The dependence of n on the N intensity of the muon source and the detector size and e$ciency, as well as the dependence on the mixing parameters follows simply from the previous section: n J(N N e ) and n Jsin h sin 2h . N l 2 l N   (27) The dependence of n on the baseline ¸ and the N muon energy E is less trivial. A very long baseline l is essential for the observation of matter e!ects and the oscillating factor in Eq. (21) can be expanded for ¸:3000 km and not too small muon energy in a series of its argument. Matter e!ects cancel [2] in the leading order of this expansion. Matter e!ects are therefore in this regime small such that baselines shorter than about 3000 km are better suited for CP-violation measurements [2,7]. The importance of matter e!ects grows however quadratically with the baseline [7] until the argument * C in  ! Eq. (21) approaches and exceeds p/2, making matter e!ects large.  This procedure can be understood intuitively by de"ning the asymmetry A " : */&, where * " : n > (l\)!2n \ (l>) and l l &" : n > (l\)#2n \ (l>), which is obviously very sensitive to l l e!ects which go in opposite direction, while at the same time a number of common systematic e!ects drop out. For small &  (i.e. small beam energies) this method is related to comparing the absolute asymmetry expected in matter with the expected #uctuations of this quantity in the vacuum case s"*/d* . For large &  this is equivalent to doing the same with the asymmetry: s"A/dA .

Fig. 4. Contour lines of n in the ¸}E plane. We use N l *m "3.5;10 eV and N N e"10 and the solid con l 2 tour lines correspond to n "100 sin 2h ) +1,2,4,8,16,. N 

The dependence of n on ¸ and E is shown N l in Fig. 4 for *m "3.5;10\ eV and  N N e"10, where the contour lines of n are l 2 N plotted in the ¸}E plane. The solid contour lines l correspond to n "100 sin 2h ) +1,2,4,8,16,. The N  main muon energy requirement is for this central value of *m and for "xed very long baselines  E 920 GeV. Below this energy the neutrino enl ergy spectrum does not fully cover the MSW resonance so that n decreases rapidly. The number of N neutrinos which fall into the resonance region decreases for E beyond approximately 20 GeV, rel ducing thus n even though the total rates are N growing. Beyond E 950 GeV the m Pm (and for l  l very large E and ¸ also m Pm ) suppression starts l  l to dominate thus increasing n again. N The signi"cance of the e!ect, i.e. the number of standard deviations n , depends most crucially on N h , ¸. We illustrate this dependence in Fig. 5 again  for *m "3.5;10 eV, where the contour lines  corresponding to n "1}5 are plotted in the N sin 2h }¸ plane for two values of the product  N N e: 10 (left plot) and 10 (right plot). The l 2 muon energy in both cases of Fig. 5 is E "20 GeV, while Fig. 6 shows the same plots l with identical parameters for E "50 GeV. The l vertical dashed lines represent in all these "gures the upper limit on sin 2h . Figs. 5 and 6 show that  matter e!ects could be observed in the total event

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Fig. 5. Contour lines of n corresponding to n "1}5 in the sin 2h }¸ plane for E "20 GeV and the two di!erent values N N  l N N e"10 (left plot) and N N e"10 (right plot). l I2 l I2

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

rates for given baseline ¸ in a rather large sin 2h  interval, while non-observation implies very strong upper bounds on sin 2h . Figs. 5 and 6 show in  other words the sin 2h range where the enhance ment/depletion of the total appearance rates in vacuum due to matter e!ects is statistically signi"cant for a given con"dence level. In those ranges one can not only observe the deviations from the results expected in vacuum, but also the deviation

from the results which one would get in matter if *m were reverted. A measurement of the sign of  *m would therefore be possible in the sin 2h   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 sensitivity of the statistical signi"cance to the chosen value of *m . Fig. 7 is exactly the same plot as Fig. 4  with minimal and maximal *m . In the left plot we 

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Fig. 7. Sensitivity of the statistical signi"cance of matter e!ects to the value of *m analogous to Fig. 4. The left plot uses  *m "1.0;10 eV while for the right plot *m "8.0;10 eV with otherwise unchanged parameters. As in Fig. 4 the solid contour   lines correspond to n "100 sin 2h ) +1,2,4,8,16,. N 

wrong-sign muon events would add important information to the simple counting of events. A detailed discussion of muon energy optimization depends therefore on the way that information will be exploited. 5. Subleading *m -e4ects 

Fig. 8. Sensitivity of the statistical signi"cance of matter e!ects to the value of *m for "xed ¸"8000 km as a function of E .  l The lines show 100 sin 2h n for di!erent *m values.  N 

have *m "1.0;10 eV and in the right plot  *m "8.0;10 eV. Fig. 8 shows the statistical  signi"cance (i.e. 100 sin 2h n ) for di!erent *m  N  values for ¸"8000 km as a function of E . For l very long baselines, one can see from Fig. 8 that increasing *m mainly shifts the resonance energy  to higher values thus demanding higher beam energies to reach optimal statistical signi"cance. The local maximum in Fig. 7 shifts to 10 GeV for minimal *m and to 40 GeV for maximal *m . Im  proved knowledge of *m would thus in principle  allow to discuss optimization issues, but such a study should also include systematics and backgrounds. Moreover, the energy distribution of

The results shown so far were obtained in the limit *m "0 which is, as already explained,  a perfect approximation for *m ;10\ eV or  sin 2h ;1, i.e. for the VO and SMA MSW solu tions of the solar neutrino problem. The LMA MSW solution, that we will consider in this section, allows however *m values up to 2;10\ eV  and prefers sin 2h K0.8 [29] so that e!ects asso ciated to *m can become important, especially  for the largest *m values in the LMA range. Two  more mixing parameters, namely h and d, be come relevant for *m O0 [30,31] (see also, e.g.,  Ref. [32]). While h is rather constrained, so that  we will use sin 2h "0.8 in the following numer ical results, any value of d in its range 04d(2p is allowed at present. In order to calculate the e!ects associated with a non-vanishing *m , the value of  d must be speci"ed. In Fig. 9, the total rates in the appearance channels m Pm (left) and m Pm  l  l (right) for *m "0 (solid line) are compared with  the total rates for *m "10\ eV and four pos sible values of d in its range, d"0, p/2, p, 3p/2

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Fig. 9. Appearance event rates n > (k\), n \ (k>) in matter with subleading *m "10\ eV and four possible values of the CP-phase I I  d"0, p/2, p, 3p/2 against baseline ¸ (dashed lines) compared with the corresponding event rates with negligible *m (solid lines) for  the channels m Pm (left) and l Pl (right). Both plots assume N "2;10, e"50%, E "20 GeV, sin 2h "1, sin 2h "0.8  l  l l l   and sin 2h "0.1. 

Fig. 10. Same as Fig. 9 but for sin 2h "0.01. 

(dashed lines). The size of the e!ects depends crucially on the value of h . Fig. 9 assumes a value of  h at its upper limit, i.e. sin 2h "0.1, whereas   Fig. 10 shows the e!ects for a sin 2h one order of  magnitude smaller, sin 2h "0.01. Both "gures  assume N N e"10. l 2 Figs. 9 and 10 illustrate several interesting features of the *m e!ects. First of all, a comparison  of Figs. 9 and 10 con"rms that the relative size of the e!ects grows when h gets smaller [4]. This is  because the zeroth-order approximations (in *m )  for the appearance probabilities have a sin 2h  suppression, whereas the linear *m corrections  (CP-conserving and violating) are suppressed by only one power of sin 2h and the corrections  quadratic in *m (CP-conserving) are not sup pressed by h at all. Unlike the appearance chan nels, the disappearance channels are dominated by the transitions to m (m ) and are therefore not s s

suppressed by h or *m , so that the h -sup   pressed *m corrections are much less signi"cant  in this case. Figs. 9 and 10 show also that the range of the *m corrections is essentially determined by  h , but the precise size and sign of these correc tions within this range is unknown if d is unconstraint. This can be seen also explicitly in Figs. 9 and 10 where the *m corrections show up as  oscillations around the leading *m contribution  to the rates (dashed line), whose initial phase depends on d. *m e!ects represent consequently in  the present LMA scenario in a high statistics longbut-not-too-long baseline measurement of h an  important source of systematic error, unless d is measured [4,7]. On the other hand, sin d could be measured or constrained in this scenario by comparing the rates in the two CP-conjugated channels m Pm and m Pm if very high intensity sources  l  l and large detectors will become available [7]. By

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comparing e.g. the left and right plots of Fig. 10 one can see that the *m correction has the same sign  in the two channels for the two CP-conserving values d"0, p, when sin d"0, but has opposite sign 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 on a comparison of CP-conjugated rates at a single value of ¸ would not be enough in order to keep the *m e!ects under control. Suppose, for example,  that d"0 or p. Then the *m e!ects would be the  same in the two channels and an ideal experiment looking for CP-violation would measure sin d"0. The value of d would, however, still be undetermined for the simple reason that both d"0 and p give sin d"0. As a consequence, it would be still unknown whether the *m corrections in both  channels add (d"0) or subtract (d"p) to the leading *m contribution to the rates and this  ambiguity would translate, e.g., in an uncertainty on a measurement of h . Such an ambiguity could  be resolved by comparing rates measured at di!erent distances ¸. Fig. 10 for sin 2h "0.01 shows  in the m Pm channel that the change in rates  l between ¸"3000 and 700 km allows to discriminate between d"0 and p. Fig. 9 shows that the di!erent ¸ dependence is also signi"cant for sin 2h "0.1, allowing also to distinguish be tween the two possibilities d"0 and p. If CPviolation were maximal, "sin d""1, the comparison of the rates at di!erent baselines would be less signi"cant but still helpful. Figs. 9 and 10 show "nally also that the *m e!ects become smaller  when ¸ is increased. At very long baselines the e!ects are smaller than they would be in vacuum. The results obtained in the previous sections hold therefore within a good approximation also in the case of the LMA scenario and large *m if  ¸97000 km.

6. Matter e4ects in the energy spectrum Motivated by the small di!erential event rates, we discussed so far only the in#uence of matter e!ects on the total rates of wrong-sign muon events. We demonstrated in the previous sections that statistically signi"cant deviations from the

31

total event rates in vacuum represent already a good test of the MSW theory. A signi"cant test would, however, be also given by a detailed measurement of MSW e!ects in the neutrino energy spectrum, which is modi"ed in a very characteristic way. The m Pm and m Pm disappearance l l l l channels are again dominated by transitions to m and m while m and m transitions are only small s s   corrections. These channels are therefore mostly insensitive to matter e!ects in the di!erential event rate spectrum and will not be discussed further. To understand the e!ects in the appearance spectrum of m Pm and m Pm we use again the approx l  l imation *m "0. Matter e!ects have no in#uence  on the angle h in this case, whereas they modify  the mixing due to h signi"cantly. One obtains  thus for the m Pm and m Pm appearance  l  l channels the usual two #avor picture where sin 2h Psin 2hK "sin 2h /C , and where C    ! > corresponds to neutrinos, C to antineutrinos. \ The enhancement of hK is maximal in the neutrino  channel when the neutrino energy E coincides with the MSW resonance energy E de"ned via  2E <  "cos 2h . (28)  *m  Note, however, that the maximum of the event spectrum in the m Pm channel, in general, does  l not coincide with E since the MSW oscillation  probabilities are folded with the #uxes and cross sections. The maximum of the event spectrum is thus determined by the maximization of






sin(* C )   ! fm ml (E/E ) (29) l C ! but the resulting maximum is still around E for  the muon energies under consideration. The o!set depends in a rough approximation on the di!erence between E and the maximum of the #ux  which lies roughly at an energy of the order E . l This has to be compared with the vacuum case where the oscillation probabilities are also folded with #uxes and cross sections and where the

 Note that C and C as de"ned in Eq. (15) must be > \ interchanged for *m (0. 

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Fig. 11. Modi"cations in the di!erential event rate spectrum (events per GeV) due to matter e!ects for E "20 GeV. The solid lines are l in matter while the dashed lines are without matter showing the `broadeninga, `shifta and `enhancementa due to MSW e!ects. The assumed parameters are N N e"10, ¸"6596 km (CERN-MINOS) and sin 2h "0.01. l 2 

maxima of the event rates are also not precisely at the maxima of the oscillation probabilities. The event rate spectrum is thus due to this folding in both cases with and without matter a rather complicated function of Em which depends in a nontrivial way on ¸, E and *m . Nevertheless for l  given ¸, E and for given *m , h , h measured l    with a suitable long baseline experiment, one can predict the shape of the di!erential event rate spectrum in all channels and compare it with the spectrum of oscillations una!ected by matter. This results in a very good opportunity to detect speci"c details of the MSW e!ects which arise when the oscillation parameters are chosen such that the "rst maximum of vacuum oscillation coincides roughly with E . The point is that the MSW e!ect changes  the probabilities compared to vacuum in three genuine ways: The "rst maximum of the oscillation

probability as the energy decreases is enhanced, its width is broadened and its center is shifted to lower energies. Similarly one has an `anti-MSW e!ecta in the antineutrino appearance channel which implies for the "rst oscillation maximum a reduction in height, again a broadening and a shift to lower energies. These `genuine MSW e!ectsa are demonstrated in Fig. 11 (where E "20 GeV) and Fig. 12 (with l E "50 GeV) showing the modi"cations in the l energy spectrum (events per GeV) due to matter e!ects. The solid lines are in matter while the dashed lines are without matter and the assumed parameters are N N e"10 as before, l 2 ¸"6596 km and sin 2h "0.01. Fig. 11 shows  already all e!ects due to the MSW mechanism, namely the broadening (the last oscillation in matter covers two oscillations in vacuum), the shift (the

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Fig. 12. Same as in Fig. 11 but for E "50 GeV where the MSW e!ects are harder to extract since E is already sizably above the l l resonance energy E . For details see text. 

maximum in matter lies almost in the minimum in vacuum) and the enhancement or suppression compared to vacuum. It is interesting to look at the modi"cations when the muon energy becomes higher, e.g., for E "50 GeV as shown in Fig. 12. The point is that l the beam energy is already rather far away from the MSW resonance energy and the importance of the weight function fm ml (and of the Em scaling of unoscillated events) in the determination of the shape of the spectrum becomes apparent since the genuine broadening, shift and enhancement/suppression effects become harder to distinguish. For the m Pm  l channel the e!ect could be hard to distinguish from uncertainties (with low statistics) in the spectrum. This brings up the general issue that one has to have enough statistics for such an analysis. In order to have a chance to see such e!ects one has to be lucky and h should be at the upper experimental 

limit (see scaling laws). Otherwise N N e must be l 2 increased correspondingly which implies a more intense muon source, a larger detector or both. Although a more quantitative analysis of the significance of e!ects in the di!erential neutrino event rate spectrum is beyond the scope of this paper, an analysis of the di!erential event rate spectrum would clearly provide extremely valuable additional information which would allow to test some of the characteristic features of the MSW mechanism.

7. Conclusions Assuming three-neutrino mixing we studied in this paper the possibility to test the MSW e!ect in terrestrial very long baseline neutrino oscillation experiments which become possible with neutrino

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factories. Such direct tests are important since the MSW mechanism is widely used in di!erent scenarios of neutrino physics and astrophysics. The correct analysis and interpretation of the data from terrestrial very long baseline neutrino oscillation experiments, ¸91000 km, is in fact impossible without a proper treatment of matter e!ects. The latter is also crucial for the searches of CP-violation in neutrino oscillations generated by the lepton mixing matrix, since matter e!ects create an asymmetry between the two CP-conjugated appearance channels. Studies of the m Pm and m Pm oscillations  l  l are by far most promising for the detection of the matter e!ects since the corresponding total event rates are a!ected in a drastic way by these e!ects. We considered for the present study neutrino trajectories through the earth which cross the mantle, but do not pass through the earth core, which corresponds to neutrino path lengths ¸:10 000 km. Using analytic expressions for the three neutrino oscillation probabilities in matter in the constant average density and small *m ap proximations and including #uxes, cross sections and detection e$ciencies allows to describe the relevant event rates analytically. This permitted a full analytic understanding of our numerical results. By considering the asymmetry between the m Pm and m Pm induced wrong-sign muon  l  l event rates, we studied the statistical signi"cance of the observation of matter e!ects as a function of the neutrino oscillation parameters h and *m as   well as its dependence on the experimental conditions via the parent muon beam energy E , the path l length ¸ and the product of useful muons, detector size and e$ciency. The scaling of rates, statistical signi"cances and sensitivities with the relevant mixing angles, in particular, with h , the intensity of  the muon source and with the detector size and e$ciency have been given, so that the results for any value of those parameters can easily be obtained. The sign of the asymmetry depends on whether the two closest neutrino mass eigenstates are lighter (*m '0) or heavier (*m (0) than   the third one, thus providing a way of determining which of these two possibilities is realized. Figs. 5 and 6 show the conservative ranges of sin 2h 

where such a determination would be signi"cant at a given con"dence level from the statistical point of view as a function of the baseline. We analyzed, in particular, the statistical signi"cance of matter e!ects as a function of E and ¸. l The most important requirement regarding the muon energy is that for given *m it has to be  greater than the MSW resonance energy. For, e.g., *m 46.0(8.0);10\ eV, this implies E 920  l (30) GeV. The value of E &30 GeV (and l ¸97000 km) is practically optimal for *m K  (4.0}8.0);10\ eV, while if *m K(2.0}3.5);  10\ eV, E K20 GeV would be preferable (see l Figs. 4, 7 and 8). If, however, *m K10\ eV and  sin 2h :0.01, then establishing matter e!ects (or  obtaining a stringent upper limit on sin 2h ) re quires E K(40}50) GeV. It is clear from the above l results that the optimal value of E depends signi"l cantly on the precise value of *m .  Our analysis shows that a higher sensitivity to the MSW e!ect in the case of relatively small values of sin 2h is achieved at ¸97000 km. We showed  that this conclusion holds also when subleading *m e!ects are included. These e!ects can be sig ni"cant in the case of the LMA MSW solution with *m K(0.5}2.0);10\ eV. At ¸97000 km, the  indicated *m -induced e!ects are considerably  smaller than in vacuum and essentially negligible. Thus, the results obtained in the limit of *m "0  are su$ciently accurate for ¸97000 km. This is valid even in the case of the LMA MSW solution with *m K(0.5!2.0);10\ eV. For shorter  baselines, *m e!ects are non-negligible and a de termination of the CP-violating phase d would be necessary in order to know their precise magnitude. We have found that the ¸ dependence of the *m  e!ects o!ers the possibility to determine the CPphase d, especially when sin d is small. For sin 2h "0.01, for instance, the event rates due  to the m Pm and m Pm transitions change  l  l considerably when ¸ changes from &700 km to &3000 km. Thus, a measurement of these rates, e.g., at the indicated two distances would allow to determine the value of d with a certain precision. Finally, we discussed the matter e!ects in the di!erential event rate spectrum as a function of the neutrino energy, and showed that they lead to very

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characteristic distortions. The observation of these distortions would allow very detailed tests of the MSW theory. To conclude, our study shows that the predictions of the MSW theory can be tested in a statistically reliable way in a large region of the corresponding parameter space by a simple analysis of the total event rates in a very long baseline, ¸97000 km, neutrino oscillation experiment. This can allow to determine the sign of *m as well, as  was recently noticed also in Ref. [8]. Not seeing the matter e!ects would lead to impressive upper limits on the mixing angle h down to sin 2h K10\   or even better. Acknowledgements

[12] [13] [14]

[15] [16]

[17] [18] [19]

A.R. and S.P. wish to thank the Institute T30d at the Physics Department of the Technical University of Munich for warm hospitality.

[20] [21] [22]

References [1] S. Geer, Phys. Rev. D 57 (1998) 6989; erratum } Phys. Rev. D 59 (1999) 039903. [2] A. De Rujula, M.B. Gavela, P. Hernandez, Nucl. Phys. B 547 (1999) 21. [3] V. Barger, S. Geer, K. Whisnant, e-Print Archive: hepph/9906487. [4] K. Dick, M. Freund, M. Lindner, A. Romanino, e-Print Archive: hep-ph/9903308, Nucl. Phys. B (2000), to appear. [5] M. Campanelli, A. Bueno, A. Rubbia, e-Print Archive: hepph/9905240. [6] A. Donini, M.B. Gavela, P. Hernandez, S. Rigolin, e-Print Archive: hep-ph/9909254. [7] A. Romanino, e-Print Archive: hep-ph/9909425, Nucl. Phys. B (2000), to be published. [8] V. Barger, S. Geer, R. Raja, K. Whisnant, e-Print Archive: hep-ph/9911524. [9] P. Lipari, e-Print Archive: hep-ph/9903481. [10] M. Narayan, S.U. Sankar, Phys. Rev. D 61 (2000) 013003. [11] Nufact '99 Workshop, July 5}9, Lyon. See e.g. the Summary of Detector/Neutrino Beam Parameters by D. Harris, http://lyopsr.in2p3.fr/nufact99/talks/harrisconc.

[23] [24] [25] [26] [27]

[28]

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THEORY