Physics Letters B 533 (2002) 94-106

www. elsevier. com/locate/npe

The LMA MSW solution of the solar neutrino problem, inverted neutrino mass hierarchy and reactor neutrino experiments

S.T. Petcov1,M. Piai

SISSA/INFN, Via Beirut 2-4, I-34014 Trieste, Italy Received 11 December 2001; received in revised form 14 March 2002; accepted 18 March 2002

Editor: G.F. Giudice

Abstract

In the context of three-neutrino oscillations, we study the possibility of using antineutrinos from nuclear reactors to explore the 10-4 eV2 < AmQ < 8 x 10-4 eV2 region of the LMA MSW solution of the solar neutrino problem and measure AmQ with high precision. The KamLAND experiment is not expected to determine AmQ if the latter happens to lie in the indicated region. By analysing both the total event rate suppression and the energy spectrum distortion caused by ve oscillations in vacuum, we show that the optimal baseline of such an experiment is L ~ (20-25) km. Furthermore, for 10-4 eV2 < AmQ < 5 x 10-4 eV2, the same experiment might be used to try to distinguish between the two possible types of neutrino mass spectrum—with normal or with inverted hierarchy, by exploring the effect of interference between the atmospheric- and solar-Am2 driven oscillations; for larger values of AmQ not exceeding 8.0 x 10-4 eV2, a shorter baseline, L = 10 km, would be needed for the purpose. The indicated interference effect modifies in a characteristic way the energy spectrum of detected events. Distinguishing between the two types of neutrino mass spectrum requires, however, a high precision determination of the atmospheric Am2, a sufficiently large sin2 0 and a non-maximal sin2 20 q , where 0 and 0q are the mixing angles, respectively, limited by the CHOOZ and Palo Verde data and characterizing the solar neutrino oscillations. It also requires a relatively high precision measurement of the positron spectrum in the reaction ve + p ^ e+ + n. © 2002 Elsevier Science B.V. All rights reserved.

PACS: 14.60.Pq; 13.15.+g

1. Introduction

In recent years the experiments with solar and atmospheric neutrinos collected strong evidences in favor of the existence of oscillations between the flavour neutrinos, v^, vM and vT. Further progress in our understanding of the neutrino mixing and oscillations requires, in particular, precise measurements of the parameters entering into the oscillation probabilities—the neutrino mass-squared differences and mixing angles, and the reconstruction of the neutrino mass spectrum.

E-mail address: petcov@he.sissa.it (S.T. Petcov).

1 Also at: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria.

0370-2693/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S0370-2693(02)01591-5

The atmospheric neutrino data can be explained by dominant vM ^ vT and vM ^ vT oscillations, characterized

by large, possibly maximal, mixing, and a mass squared difference, Am^, having a value in the range [1] (99% C.L.):

1.3 x 10-3 eV2 < |Amytal < 5 x 10-3 eV2. (1)

The first results from the Sudbury Neutrino Observatory (SNO) [2], combined with the mean event rate data from the Super-Kamiokande (SK) experiment [3], provide a very strong evidence for oscillations of the solar neutrinos [4-10]. Global analyses of the solar neutrino data, including the SNO results and the SK data on the e--spectrum and day-night asymmetry, show that the data favor the large mixing angle (LMA) MSW solution of the solar neutrino problem, with the corresponding neutrino mixing parameter sin2 20© and mass-squared difference Am© lying in the regions (99.73% C.L.):

2 x 10-5 eV2 < Am© < 8 x 10-4 eV2, (2)

0.6 < sin2 20© < 1. (3)

Thebestfitvalue of Am© found in the independent analyses [5-7,9] is spread in the interval (4.3-6.3) x 10-5 eV2. The results obtained in [5-7,9] show that values of Am© > 10-4 eV2 are allowed already at 90% C.L. Values of cos 20© < 0 (for Am© > 0) are disfavored by the data.

Important constraints on the oscillations of electron (anti)neutrinos, which play a significant role in our current understanding of the possible patterns of oscillations of the three flavour neutrinos and antineutrinos, were obtained in the CHOOZ and Palo Verde disappearance experiments with reactor ve [11,12]. The CHOOZ and Palo Verde experiments were sensitive to values of neutrino mass squared difference Am2 > 10-3 eV2, which includes the corresponding atmospheric neutrino region, Eq. (1). No disappearance of the reactor ve was observed. Performing a two-neutrino oscillation analysis, the following rather stringent upper bound on the value of the corresponding mixing angle, 0, was obtained by the CHOOZ Collaboration2 [11] at 95% C.L. for Am2 > 1.5 x 10-3 eV2:

sin2 0< 0.09. (4)

The precise upper limit in Eq. (4) is Am2-dependent: it is a decreasing function of Am2 as Am2 increases up to Am2 ~ 6 x 10-3 eV2 with a minimum value sin2 0 ~ 10-2. The upper limit becomes an increasing function of Am2 when the latter increases further up to Am2 ~ 8 x 10-3 eV2, where sin2 0 < 2 x 10-2. Somewhat weaker constraints on sin2 0 have been obtained by the Palo Verde Collaboration [12]. In the future, sin2 0 might be further constrained or determined, e.g., in long baseline neutrino oscillation experiments [13].

The long baseline experiment with reactor ve KamLAND [14] has been designed to test the LMA MSW solution of the solar neutrino problem. This experiment is planned to provide a rather precise measurement of Am© and sin2 20©. Due to the long baseline of the experiment, L ~ 180 km, however, Am© can be determined with a relatively good precision only if Am© < 10-4 eV2.

The explanation of both the atmospheric and solar neutrino data in terms of neutrino oscillations requires, as is well-known, the existence of 3-neutrino mixing in the weak charged lepton current:

Vll = UljVjl, (5)

where vlL, l = e,^,t, are the three left-handed flavour neutrino fields, Vj l is the left-handed field of the neutrino Vj having a mass mj > 0 and U is a 3 x 3 unitary mixing matrix—the Pontecorvo-Maki-Nakagawa-Sakata (PMNS)

2 The possibility of large sin2 0 > 0.9 which is admitted by the CHOOZ data alone is incompatible with the neutrino oscillation interpretation of the solar neutrino deficit (see, e.g., [15,16]).

neutrino mixing matrix [17,18]. The three neutrino masses m12 3 can obey the so-called normal hierarchy (NH) relation m1 <m2 <m3, or that of the inverted hierarchy (IH) type, m3 < m1 < m2. Thus, in order to reconstruct the neutrino mass spectrum in the case of 3-neutrino mixing, it is necessary to establish, in particular, which of the two possible types of neutrino mass spectrum is actually realized. This information is particularly important for the studies of a number of fundamental issues related to lepton mixing, as like the possible Majorana nature of massive neutrinos, which can manifest itself in the existence of neutrino-less double j-decay (see, e.g., [19,20]). It would also constitute a critical test for theoretical models of fermionic mass matrices and flavor physics in general.

It would be possible to determine whether the neutrino mass spectrum is with normal or inverted hierarchy in terrestrial neutrino oscillation experiments with a sufficiently long baseline, so that the neutrino oscillations take place in the Earth and the Earth matter effects in the oscillations are non-negligible [21-23]. The ambiguity regarding the type of the neutrino mass spectrum might be resolved by the MINOS experiment [13], although on the baseline of this experiment the matter effects are relatively small [21]. This might be done in an experiment with atmospheric neutrinos, utilizing a detector with a sufficiently good muon charge discrimination [24]. The experiments at neutrino factories would be particularly suitable for the indicated purpose [22,23].

In this Letter, in the context of three-neutrino oscillations, we study the possibility of using antineutrinos from nuclear reactors to explore the Am© > 10 - 4 eV2 region of the LMA MSW solution. Such an experiment might be of considerable interest if, in particular, the results of the KamLAND experiment will confirm the validity of the LMA MSW solution of the solar neutrino problem, but will allow to obtain only a lower bound on Am© due to the fact that Am© > 10-4 eV2 [25-27]. We determine the optimal baseline of the possible experiment with reactor ve, which would provide a precise measurement of Am© in the region 10-4 eV2 < Am© < 8 x 10-4 eV2. Furthermore, the same experiment might be used to try to distinguish between the two types of neutrino mass spectrum—with normal or with inverted hierarchy. This might be done by exploring the effect of interference between the amplitudes of neutrino oscillations, driven by the solar and atmospheric Am2, i.e., by Am© and Ami;tm. For the optimal baseline found earlier, L = (20-25) km, the indicated effect could be relevant for 10-4 eV2 < Am© < 5 x 10 4 eV2. For larger values of Am© within the interval (2), the effect could be relevant at L = 10 km. Distinguishing between the two possible types of neutrino mass spectrum requires a relatively high precision measurement of the positron spectrum in the reaction ve + p ^ e+ + n (i.e., a high statistics experiment with sufficiently good energy resolution), a measurement of Am^m with very high precision, sin2 20© = 1.0, e.g., sin2 20© < 0.9, and a sufficiently large value of the angle 0, which for Am© ^ Am^m controls, e.g., the oscillations of the atmospheric ve and ve and is constrained by the CHOOZ and Palo Verde data.

2. The ve survival probability

We shall assume in what follows that the 3-neutrino mixing described by Eq. (5) takes place. We shall number (without loss of generality) the neutrinos with definite mass in vacuum Vj, j = 1,2, 3, in such a way that their masses obey m1 < m2 < m3. Then the cases of NH and IH neutrino mass spectrum differ, in particular, by the relation between the mixing matrix elements | Ue j |, j = 1, 2, 3, and the mixing angles 0© and 0 (see further). With the indicated choice one has Amjk> 0 for j > k. Let us emphasize that we do not assume any of the relations m1 ^ m2 ^ m3, or m1 < m2 ^ m3, or m1 ^ m2 = m3, to be valid in what follows.

Under the conditions of the experiment we are going to discuss, which must have a baseline L considerably shorter than the baseline ~ 180 km of the KamLAND experiment, the reactor ve oscillations will not be affected by Earth matter effects when the ve travel between the source (reactor) and the detector. If 3-neutrino mixing takes place, Eq. (5), the ve would take part in 3-neutrino oscillations in vacuum on the way to the detector.

We shall obtain next the expressions for the reactor ve survival probability of interest in terms of measurable quantities for the two types of neutrino mass spectrum. In the case of normal hierarchy between the neutrino masses

we have

AmQ = Am2i,

| Uel | = cos 0Q^/l-\Ue3\2, \Ue2\ = sin0QiJ 1 — \Ue312,

Oq = 012,

|^e3|2 = sin2 O = sin2 013,

0i2 and 0i3 being two of the three mixing angles in the standard parameterization of the PMNS matrix (see, e.g., [16]). Note that |^e3|2 is constrained by the CHOOZ and Palo Verde results. It is not difficult to derive the expression for the Ve survival probability in the case under discussion:

Pnh(Ve ^ Ve)

1 - 2sin2O cos20( 1 - cos

Am\xL 2 Ev

4 AmiL - - cos4 0 sin 20© 1 - cos-22 °V 2 Ev

+ 2sin2 O cos2 O sin2 0Q ( cos

Am\xL 2 Ev

AmQL 2 Ev

Am231 L 2 Ev

where Ev is the neutrino energy and we have made use of Eqs. (6), (7) and (8).

If the neutrino mass spectrum is with inverted hierarchy one has (see, e.g., [16,20,28]):

AmQ = Am^2,

\Ue2\ = cosOq-^ 1 |Uq\ p, |C7e3| = sin00yi-|C7ei|2. The mixing matrix element constrained by the CHOOZ and Palo Verde data is now | ^e112 : |^e!|2 = sin2 0.

The expression for the Ve survival probability can be written in the form [29]:

PIH( Ve ^ Ve )

t T / AmiL

= 1 — 2 sin 0 cos 0(1— cos - 31

--cos 0 sin 200 1 - cos

2 °V 2 Ev

+ 2sin2 O cos2 O cos2 0Q ( cos

Am21L AmQL

Amj1L^ 2 Ev

Several comments concerning the expressions for the ve survival probability, Eqs. (9) and (13), follow. In the first lines in the right-hand side of Eqs. (9) and (13), the oscillations of the electron (anti-)neutrino driven by the "atmospheric" Am^1 are accounted for. The CHOOZ and Palo Verde experiments are primarily sensitive to this term and their results limit sin2 O. The second lines in the expressions in Eqs. (9) and (13) contain the solar neutrino

oscillation parameters. This is the term KamLAND should be most sensitive to. For Am© ^ Am21 = Am2tm, Am© < 10-4 eV2, only one of the indicated two terms leads to an oscillatory dependence of the ve survival probability for the ranges of L/Ev characterizing the CHOOZ and Palo Verde, and the KamLAND experiments: on the source-detector distance L of the CHOOZ and Palo Verde experiments the oscillations due to Am© cannot develop, while on the distance(s) traveled by the ve in the KamLAND experiment Am^m causes fast oscillations which average out and are not predicted to lead, e.g., to specific spectrum distortions of the KamLAND event rate.

The terms in the third lines in Eqs. (9) and (13) are not present in any two-neutrino oscillation analysis. They represent interference terms between the amplitudes of neutrino oscillations, driven by the solar and atmospheric neutrino mass squared differences. The term in Eq. (9) is proportional to sin2 0©, while the corresponding term in Eq. (13) is proportional to cos2 0© [29]. This is the only difference between Pnh( ve ^ ve) and fIH( ve ^ ve), that can be used to distinguish between the two cases of neutrino mass spectrum in an experiment with reactor ve. Obviously, if cos20© = 0, we have Pnh( ve ^ ve) = fIH( ve ^ ve) and the two types of spectrum would be indistinguishable in the experiments under discussion. For vanishing sin2 0, only the terms in the second line of Eqs. (9) and (13) survive, and the two-neutrino mixing formula for solar neutrino oscillations in vacuum is exactly reproduced.

Let us discuss next the ranges of values the different oscillation parameters, which enter into the expressions for the probabilities of interest Pnh( ve ^ ve) and fIH( ve ^ve), can take. The allowed region of values of Am21, Am©, sin2 0© and 0 should be determined in a global 3-neutrino oscillation analysis of the solar, atmospheric and reactor neutrino oscillation data, in which, in particular, Am© should be allowed to take values in the LMA solution region, including the interval Am© ~ (1.0-6.0) x 10-4 eV2. Such an analysis is lacking in the literature. However, as was shown in [30], a global analysis of the indicated type would not change essentially the results for the LMA MSW solution we have quoted3 in Eqs. (2) and (3) as long as Am21 > 1.5 x 10-3 eV2. The reason is that for Am^1 > 1.5 x 10-3 eV2 and Am© < 6.0 x 10-4 eV2, the solar ve survival probability, which determines the level of suppression of the solar neutrino flux and plays a major role in the analyses of the solar neutrino data, depends very weakly on (i.e., is practically independent of) Am^1. Thus, Am© and 0© are uniquely determined by the solar neutrino and CHOOZ and Palo Verde data, independently of the atmospheric neutrino data and of the type of the neutrino mass spectrum. The CHOOZ and Palo Verde data lead to an upper limit on Am© in the LMA MSW solution region (see, e.g., [6,31]): Am© < 7.5 x 10-4 eV2. For Am© < 1.0 x 10-4 eV2, the CHOOZ and solar neutrino data imply the upper limit on sin2 0 given in Eq. (4). For Am© ~ (2.0-6.0) x 10-4 eV2 of interest, the upper limit on sin2 0 as a function of Am21 > 10-3 eV2 for given Am© and sin2 20© is somewhat more stringent [29].

Would a global 3-neutrino oscillation analysis of the solar, atmospheric and reactor neutrino oscillation data lead to drastically different results for Am21 in the two cases of normal and inverted neutrino mass hierarchy? Our preliminary analysis shows that given the existing atmospheric neutrino data from the Super-Kamiokande experiment, such an analysis (i) would not be able to discriminate between the two cases of neutrino mass spectrum, and (ii) would give essentially the same allowed region for Am^1 in the two cases of neutrino mass spectrum. We expect the regions of allowed values of the mixing angle 0atm, which controls the dominant atmospheric vM ^ vT and ^ vT oscillations, to differ somewhat in the two cases. Note, however, that this mixing angle does not enter the expression for the ve survival probability we are interested in.

For Am© < 1.0 x 10-4 eV2 and sufficiently small values of sin2 0, Am21 coincides effectively with Am^lm of the two-neutrino vM and oscillation analyses of the SK atmospheric neutrino data. If sin2 0 > 0.01, a three-neutrino oscillation analysis of the atmospheric neutrino and CHOOZ data, performed under the assumption of

3 Let us note that the LMA MSW solution values of Am© and 0© we quote in Eqs. (2) and (3) were obtained by taking into account the CHOOZ and Palo Verde limits as well.

AmQ < 1.0 x 10 4 eV2 [31], gives regions of allowed values of Am2tm = Am^, which are correlated with the value of sin2 0. The latter must satisfy the CHOOZ and Palo Verde constraints.

At present, as we have already indicated, a complete three-neutrino oscillation analysis of the atmospheric neutrino and CHOOZ data with Am22 allowed to take values up to ~ (6.0-7.0) x 10-4 eV2, i.e., in the region where deviations from the two-neutrino approximation could be non-negligible, is lacking in the literature. Therefore, in what follows we will use representative values of Am31 which lie in the region given by Eq. (1).

3. The difference between PNH(ve ^ ve) and PIH(ve ^ ve)

Let us discuss next in greater detail the difference between the ve surviving probabilities in the two cases of neutrino mass spectrum of interest, PNH( ve ^ ve) and PIH( ve ^ ve). While the terms in the first two lines in Eqs. (9) and (13) describe oscillations in L/Ev with frequencies Am21/4n and Am2Q/4n, respectively, the third term has the shape of beats, being produced by the interference of two waves, with the same amplitude but slightly different frequencies:

( Am?, L

cosl -^

V 2 Ev

AmQL 2 Ev

- cos -

Am\xL 2 Ev

= 2 sin -

Am^jL Am^L

— 2 sin -

AmQL ( Am^i L

This is a modulated oscillation with approximately the same frequency of the first term in Eqs. (9) and (13) (Am31/4n) and amplitude oscillating between 0 and 2 sin2 02 of the amplitude of the first term itself. The beat frequency is equal to the frequency of the dominant oscillation (AmQ/4n). The modulation is exactly in phase with the AmQ -driven dominant oscillation of interest, so that the maximum of the oscillation amplitude of the interference term (third lines in the expressions for fNH( ve ^ ve) and PIH( ve ^ ve)) is reached in coincidence with the points of maximal decreasing of the ve survival probability, where Am2QL/4E = n/2, and vice versa— this amplitude vanishes at the local maxima of the survival probability. At the minima of the ve survival probability, for instance at Am2QL/4Ev = n/2, PNH(IH)(ve ^ ve) takes the value:

pnh(ih)(-e

lit Ev

T7.. —1

= 1 - 2sin2 e cos2 e - cos4 e sin2 29q

(+) 2 2 Am?,

- cos20©2sin 0cos 0COS7T

From Eqs. (9), (13) and (15) one deduces that:

for maximal mixing, cos2e© = 0, the last term cancels, and P = P

forvery small mixing angles, cos2e0 ~ 1, the terms describing the oscillations driven by Am?1 in the NH and

IH cases have opposite signs: the two waves are exactly out of phase; for intermediate values of cos202 from the LMAMSW solution region, cos202 = (0.3-0.6), the Am21-driven contributions in the cases of normal and inverted hierarchy have still opposite signs and the magnitude of the effect is proportional to 2 cos 202 sin2 0.

The net result of these properties is that in the region of the minima of the ve survival probability due to AmQ, where Am22L/(2E) = n(2k + 1), k = 0,1,..., the difference between PNH( ve ^ ve) and PIH( ve ^ ve) is maximal. In contrast, at the maxima of PNH( ve ^ ve) and PIH( ve ^ ve) determined by Am22L/(2E) = 2nk, we have, for any sin2 0G, Pnh(ve ^ ve) = Pih(ve ^ ve).

The two-neutrino oscillation approximation used in the analysis of the CHOOZ and Palo Verde data is rather accurate as long as Am© is sufficiently small [29]: for Am© < 10-4 eV2, the L/Ev values characterizing these experiments, chosen to ensure maximal sensitivity to Am21 > 10-3 eV2, are much smaller than the value at which the first minimum of fNH(iH)(ve ^ ve) due to the Am©-dependent oscillating term occurs. Correspondingly, the effect of the interference term is strongly suppressed by the beats. For Am© > 2 x 10-4 eV2 this is no longervalid and the interference term under discussion has to be taken into account in the analyses of the CHOOZ and Palo Verde data [29].

4. Measuring large at reactor facilities

As is well-known, nuclear reactors are intense sources of low energy ve (E v < 8 MeV), emitted isotropically in the j-decays of fission products with high neutron density [32]. Anti-neutrinos can then be detected through the positrons produced by inverse j -decay on nucleons. The reactor ve energy spectrum has been accurately measured and is theoretically well understood4 [33]: it essentially consists of a bell-shaped distribution in energy centered around Ev ~ 4 MeV, having a width of approximately 3 MeV CHOOZ, Palo Verde and KamLAND are examples of experiments with reactor ve, the main difference being the distance between the source and the detector explored (L ~ 1 km for CHOOZ and Palo Verde, and L ~ 180 km for KamLAND).

The best sensitivity to a given value of Am© of the experiment of interest is at L at which the maximum reduction of the survival probability is realized. As can be seen from Eqs. (9)-(13), this happens for L around L* = 2nEv/Am©. This implies that for Ev = 4 MeV, the optimal length to test neutrino oscillations with reactor experiments is:

5 x 10-3

L =-~-km. (16)

(Am©/eV)

-5 0w2

The best sensitivity of KamLAND, for instance, is in the range of (2-3) x 10-5 eV2. We will discuss next in greater detail the distances L which could be used to probe the LMA MSW solution region at Am© > 10-4 eV2, in order to extract Am© from these oscillation experiments.

4.1. Total event rate analysis

One of the signatures of the ve -oscillations would be a substantial reduction of the measured total event rate due to the reactor ve in comparison with the predicted one in the absence of oscillations. In order to compute the expected total event rate one has to integrate the ve survival probability multiplied by the ve energy spectrum

over E . In Fig. 1 we show this averaged survival probability for different values of L as a function of Am2

using the "best fit" values [1,5-7] for Am3>1 and sin2 20©. When averaging over the ve energy spectrum, oscillatory effects with too short a period are washed out, and the experiment is sensitive only to the average amplitude. This happens when the width SEv of the energy spectrum is such that the integration runs over more than one period, i.e., approximately for:

^ 4nE2 4 x 104eV3

&EV>----. (17)

v ~ Am2L Am2(L/km)

4 By reactor ve energy spectrum we mean here and in what follows the product of the ve production spectrum and the inverse j-decay cross-section, which gives the "detected" neutrino spectrum in the no oscillation case. The ve production spectrum is known with larger uncertainties at ve energies Ev < 2 MeV, but this range is not of interest due to the threshold energy Ev = 1.8 MeV of the inverse j-decay reaction [34]. Certain known time dependence at the level of a few percent is also present up to 3.5 MeV [35] and should possibly be taken into account in the analysis of the experimental data.

0.1 0.2 0.5 1 2 5 10 Am2 (10~4 eV2)

Fig. 1. The reactor ve survival probability, averaged over the ve energy spectrum, for Am^^ = 2.5 x 10-3 eV2, sin2 20© = 0.8, sin2 0 = 0.05, as a function of Am©. The curves correspond to L = 180 km (long dashed), L = 50 km (dashed), L = 20 km (thick) and L = 10 km (dotted), respectively.

Since SEv ~ 3 MeV, at KamLAND this happens approximately for Am© > 7 x 10-5 eV2. The corresponding curve in Fig. 1 indicates that the actual sensitivity extends to somewhat larger values of Am© than what is expected on the basis on the above estimate, but the total event rate becomes flat for Am© > 10-4 eV2. This means that KamLAND will be able, through the measurement of the total even rate, to test all the region of the LMA MSW solution and determine whether the latter is the correct solution of the solar neutrino problem, but will provide a precise measurement of Am© only if Am© < 10-4 eV2. If Am© > 2 x 10-4 eV2, it would be possible to obtain only a lower bound on Am© and a new experiment might be required to determine Am©.

Fig. 1 shows that as L decreases, the sensitivity region moves to larger Am©. These results imply that a reactor ve experiment with L = (20-25) km can probe the range 0.8 x 10-4eV2 < Am© < 6 x 10-4eV2. One finds that for Am© = 2 x 10-4 eV2 and Am^ = 2.5 x 10-3 eV2, the best sensitivity is at L = 20 km. Moreover, with L = (20-25) km, the predicted total event rate deviates from being flat (in Am©) actually for Am© as large as ~ (5-6) x 10-4 eV2. In order to have a precise determination of Am© with L = (20-25) km for the largest values given in Eq. (2), Am© = (7-8) x 10-4 eV2, one should use the information about the e+-spectrum distortion due to the ve-oscillations. By measuring the e+-spectrum with a sufficient precision it would be possible to cover the whole interval

1.0 x 10-4 eV2 < Am© < 8.0 x 10-4eV2, (18)

i.e., to determine Am© if it lies in this interval, by performing an experiment at L = (20-25) km from the reactor(s)5 (see the next subsection).

Applying Eq. (17) with Am2 = Am^, one sees that for the ranges of L which allow to probe Am© from the LMA MSW solution region, the total event rate is not sensitive to the oscillations driven by Am^x > 1.5 x 10-3 eV2. Thus, the total event rate analysis would determine Am© which would be the same for both the normal and inverted hierarchy neutrino mass spectrum.

5 The fact that if Am© = 3.2 x 10 4 eV2, a reactor ve experiment with L = 20 km would allow to measure Am© with a high precision was also noticed recently in [27].

70 60 50 40 30 20 10

; / ? \\ \\

. f t/

2 3 4 5 6 7 8 Ey (MeV)

Fig. 2. The reactor ve energy spectrum at distance L = 20 km from the source, in the absence of ve oscillations (double-thick solid line) and in the case of ve oscillations characterized by Am2i = 2.5 x 10-3 eV2, sin2 26© = 0.8 and sin2 6 = 0.05. The thick lines are obtained for Am'q = 2 x 10-4 eV2 and correspond to NH (light grey) and IH (dark grey) neutrino mass spectrum. Shown is also the spectrum for Am| = 6 x 10-4 eV2 in the NH (dotted) and IH (dashed) cases.

4.2. Energy spectrum distortions

An unambiguous evidence of neutrino oscillations would be the characteristic distortion of the ve energy spectrum. This is caused by the fact that, at fixed L, neutrinos with different energies reach the detector in a different oscillation phase, so that some parts of the spectrum would be suppressed more strongly by the oscillations than other parts. The search for distortions of the ve energy spectrum is essentially a direct test of the ve oscillations. It is more effective than the total rate analysis since it is not affected, e.g., by the overall normalization of the reactor ve flux. However, such a test requires a sufficiently high statistics and sufficiently good energy resolution of the detector used.

Energy spectrum distortions can be studied, in principle, in an experiment with L = (20-25) km. In Fig. 2 we show the comparison between the ve spectrum expected for Am22 = 2 x 10-4 eV2 and Am22 = 6 x 10-4 eV2 and the spectrum in the absence of ve oscillations. No averaging has been performed and the possible detector resolution is not taken into account. The curves show the product of the probabilities given by Eqs. (9) and (13) and the predicted reactor ve spectrum [36]. As Fig. 2 illustrates, the ve spectrum in the case of oscillation is well distinguishable from that in the absence of oscillations. Moreover, for Am22 lying in the interval 10-4 eV2 < Am22 < 8.0 x 10-4 eV2, the shape of the spectrum exhibits a very strong dependence on the value of Am22. A likelihood analysis of the data would be able to determine the value of Am22 from the indicated interval with a rather good precision. This would require a precision in the measurement of the e+-spectrum, which should be just not worse than the precision achieved in the CHOOZ experiment and that planned to be reached in the KamLAND experiment. If the energy bins used in the measurement of the spectrum are sufficiently large, the value of Am22 thus determined should coincide with value obtained from the analysis of the total event rate and should be independent of Am31.

5. Normal vs. inverted hierarchy

In Fig. 2 we show the deformation of the reactor ve spectrum both for the normal and inverted hierarchy neutrino mass spectrum: as long as no integration over the energy is performed, the deformations in the two cases of neutrino mass spectrum can be considerable, and the sub-leading oscillatory effects driven by the atmospheric mass squared difference (see the first and the third line of Eqs. (9)-(13)) can, in principle, be observed. They could be used to

distinguish between the two hierarchical patterns, provided the solar mixing is not maximal,6 sin2 0 is not too small and Am21 is known with high precision. It should be clear that the possibility we will be discussing next poses remarkable challenges.

The experiment under discussion could be in principle an alternative to the measurement of the sign of Am21 in long (very long) baseline neutrino oscillation experiments [21-23] or in the experiments with atmospheric neutrinos (see, e.g., [24]).

The magnitude of the effect of interest depends, in particular, on three factors, as we have already pointed out:

• the value of the solar mixing angle 0©: the different behavior of the two survival probabilities is due to the difference between sin2 0© and cos2 0©; correspondingly, the effect vanishes for maximal mixing; thus, the more the mixing deviates from the maximal the larger the effect;

• the value of sin2 0, which controls the magnitude of the subleading effects due to Am21 on the Am©-driven oscillations: the effect of interest vanishes in the decoupling limit of sin2 0 ^ 0;

• the value of Am© (see Fig. 1): forgiven L and Am© the difference between the spectrum in the cases of normal and inverted hierarchy is maximal at the minima of the survival probability, and vanishes at the maxima.

A rough estimate of the possible difference between the predictions of the event rate spectrum for the two hierarchical patterns, is provided by the ratio between the difference and the sum of the two corresponding probabilities at Am©L = 2nEv:

Pnh - Pih 2cos20© sin2 0 cos2 0 Am2,

-=-9-o-a-9-C0S7r-T"- (19)

Pnh + Pih 1 - 2sin2 0 cos2 0 - cos4 0 sin2 20© Am©

The ratio could be rather large: the factor in front of the cos nAm^1/Am© is about 25% for sin2 20© = 0.8 and sin2 0 = 0.05.

The actual feasibility of the study under discussion depends crucially on the integration over (i.e., the binning in) the energy: for the effect not to be strongly suppressed, the energy resolution of the detector AE v must satisfy

^ 4nE2 (2-6) x 104eV3

A Ev<--^-• (20)

Am21L Am21(L/km)

For L ~ 1 km this condition could be satisfied for SE v ~ AE v, but at L = (15-20) km, for Am:. = 2.5 x 10-3 eV2 and Ev in the interval (3-5) MeV, one should have AEv < 0.5 MeV

Our discussion so far was performed for simplicity in terms of the reactor ve energy spectrum, while in the experiments of interest one measures the energy of the positron emitted in the inverse j -decay, Ee. The relation between Ee and Ev is well-known (see, for instance, [36]), and, up to corrections of at most few per cent, consists just in a shift due to the threshold energy of the process: Ev = Ee + (Eth - me). The maximal AEv allowed in order to make the effect observable can be then directly compared to the experimental positron energy resolution AEe 7

For Am© < 10-4 eV2, the first (most significant) minimum of the survival probability can be explored if L ~ 180 km. In this case, due to the bigger distance L, the energy resolution required would be by a factor of ten

6 It would be impossible to distinguish between the normal and inverted hierarchy neutrino mass spectrum if for given Am© > 10-4 eV2 and sin2 20© = 1, the LMA solution region is symmetric with respect to the change 0© ^ n/2 - 0© (cos 20© ^ - cos 20©). While the value of sin2 20© is expected to be measured with a relatively high precision by the KamLAND experiment, the sign of cos 20© will not be fixed by this experiment. However, the 0© - (n/2 - 0©) ambiguity can be resolved by the solar neutrino data. Note also that the current solar neutrino data disfavor values of cos 20© < 0 in the LMA solution region (see, e.g., [5,6,10]).

7 In the CHOOZ experiment, for instance, the binning in Ee was AEe ~ 0.40 MeV [11]. KamLAND is expected to have a resolution better than AEe/Ee = 10%/jE~e, where Ee is in MeV [37].

Ey (MeV) Ey (MeV)

Fig. 3. Comparison between the predicted event rate spectrum at L = 20 km, measured in energy bins having a width of AEV = 0.3 MeV in the cases of normal (light grey) and inverted (dark grey) neutrino mass hierarchy. The two upper and the lower left figures are for Am© = 2 x 10-4 eV2, sin2 20© = 0.8, sin2 0 = 0.05, and Am2i = 1.3; 2.5; 3.5 x 10-3 eV2, respectively. The lower right figure was obtained for Am© = 6 x 10-4 eV2 and Am\l = 2.5 x 10-3 eV2.

smaller. This means that for Am© ^ Am^, it is practically impossible to realize the condition of maximization of the difference between the survival probabilities in the two cases of neutrino mass spectrum without strongly suppressing the magnitude of the difference by the binning of the energy spectrum.

In order to illustrate what are the concrete possibilities in the case of the experiment under discussion, we have divided the energy interval 2.7 MeV < EV < 7.2 MeV into 15 bins, with AEV = 0.3 MeV, and calculated the value of the product of the survival probability and the energy spectrum in each of the bins. The results are shown in Fig. 3.

As our results show and Fig. 3 indicates, for Am^1 = (1.5—3.0) x 10-3 eV2,

Am© = (2.0-5.0) x 10-4 eV2, (21)

sin2 20© = 0.8 and sin2 0 = (0.02-0.05), it might be possible to distinguish the two cases of neutrino mass spectrum by a high precision measurement of the positron energy spectrum in an experiment with reactor Ve with a baseline of L = (20-25) km. This should be a high statistics experiment (not less than about 2000 ve-induced events per year) with a sufficiently good energy resolution.8 For larger values of Am© not exceeding 8.0 x 10-4 eV2, and Am21 = (1.5-3.0) x 10 3 eV2, the experiment should be done with a smaller baseline, L = 10 km. If, however, sin2 0 < 0.01, and/or sin2 20© > 0.9, and/or sin2 20© < 0.9 but the LMA solution admits equally positive and negative values of cos 20©, the difference between the spectra in the two cases becomes hardly observable. Further, in obtaining Fig. 3 we have implicitly assumed that Am3>1 is known with negligible uncertainty. Actually, for the difference between the spectra under discussion to be observable, Am\1 has to be determined, according to our estimates, with a precision of ~ 10% or better:9 given the values of Am©, sin2 20© and sin2 0, a spectrum in the

8 Preliminary estimates show that a detector of the type of KamLAND and a system of nuclear reactors with a total power of approximately 5-6 GW might produce the required statistics and precision in the measurement of the positron spectrum.

9 The analysis, e.g, of the MINOS potential for a high precision determination of Am21 in the case of Am© < 10-4 eV2 yields very encouraging results (see, e.g., [38]). For Am2 = (2.0-8.0) x 10-4 eV2, such analysis is lacking in the literature.

NH case corresponding to a given Am\x can be rather close in shape to the spectrum in the IH case for a different value of Am\x. There is no similar effect when varying Am2.

6. Conclusions

might be required to determine AmQ with high precision if the results of the KamLAND experiment show that AmQ > 10-4 eV2. Performing a three-neutrino oscillation analysis of both the total event rate suppression and the e+-energy spectrum distortion caused by the ve -oscillations in vacuum, we show that a value of AmQ from the interval 10-4 eV2 < Am2 < 8.0 x 10-4 eV2 could be determined with a high precision in experiments

Reactor experiments have the possibility to test the LMA MSW solution of the solar neutrino problem.

While the KamLAND experiment should be able to test this solution, a new experiment with a shorter baseline

- 1 (\ — 4a\T2 Darf^minr, o tl

lanai'mr rMQ/i+Mim /4ir+/M*fic\fi ^mira/4 Vvrr n Ar/iilloti/Mir -in TroAinun tttq rlimir +V»o+ o ttoIiiq a-P A -ti,^ ^

^ 2 ^ C H x 1 r^ 4 f2 /■'Auld Ko /lotoifnino/1 iTdtii o In rrli nm/^i ciah m ovr*i

with L = (20-25) km if the e+-energy spectrum is measured with a sufficiently good accuracy. Furthermore, if Am0 = (1.0-5.0) x 10-4 eV2, such an experiment with L = (20-25) km might also be able to distinguish between the cases of neutrino mass spectrum with normal and inverted hierarchy; for larger values of AmQ not exceeding 8.0 x 10-4 eV2, a shorter baseline, L = 10 km, should be used for the purpose. The indicated possibility poses remarkable challenges and might be realized for a limited range of values of the relevant parameters. The corresponding detector must have a good energy resolution (allowing a binning in the positron energy with AEe < 0.40 MeV) and the observed event rate due to the reactor ve must be sufficiently high to permit a high precision measurement of the e+-spectrum. Further, the mixing angle constrained by the CHOOZ and Palo Verde data 0 must be sufficiently large (sin2 0 ~ 0.03-0.05), and the "solar" mixing angle 00 should not be maximal (sin2 200 < 0.9). In addition, the value of Am2j, which is responsible for the dominant vM ^ vT and ^ oscillations of the atmospheric neutrinos, should be known with a high precision. However, as it is well known, "only those who wager can win" [39].

Acknowledgements

S.T.P. would like to thank S.M. Bilenkly and D. Nicolo for very useful discussions, and S. Schoenert for discussions and for bringing to his attention the reference [26]. S.T.P. acknowledges with gratefulness the hospitality and support of the SLAC Theoretical Physics Group, where part of the work on the present study was done. This work was supported in part by the EEC grant ERBFMRXCT960090 and by the RTN European Program HPRN-CT-2000-00148.

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