The LMA MSW Solution of the Solar Neutrino Problem, Inverted Neutrino Mass Hierarchy and Reactor Neutrino Experiments

In the context of three-neutrino oscillations, we study the possibility of using antineutrinos from nuclear reactors to explore the 10^{-4} {\rm eV^2}<\ms \ltap 8\times 10^{-4} {\rm eV^2} region of the LMA MSW solution of the solar neutrino problem and measure $\ms$ with high precision. The KamLAND experiment is not expected to determine $\ms$ 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 \bar{\nu}_e oscillations in vacuum, we show that the optimal baseline of such an experiment is L \sim (20 - 25) km. Furthermore, for 10^{-4} {\rm eV^2}<\ms \ltap 5\times 10^{-4} {\rm eV^2}, 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- \Delta m^2 driven oscillations; for larger values of \ms not exceeding 8.0\times 10^{-4} eV^2, a shorter baseline, L \cong 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 \Delta m^2, a sufficiently large \sin^2\theta and a non-maximal \sin^22\theta_{\odot}, where \theta and \theta_{\odot} 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 \bar{\nu}_e + p \to e^{+} + n.


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, ν e , ν µ and ν τ . 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.
The atmospheric neutrino data can be explained by dominant ν µ → ν τ andν µ →ν τ oscillations, characterized by large, possibly maximal, mixing, and a mass squared difference, ∆m 2 atm , having a value in the range [1] (99% C.L.): 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 sin 2 2θ ⊙ and mass-squared difference ∆m 2 ⊙ lying in the regions (99.73% C.L.): The best fit value of ∆m 2 ⊙ found in the independent analyses [5,6,7,9] is spread in the interval (4.3 − 6.3) × 10 −5 eV 2 . The results obtained in [5,6,7,9] show that values of ∆m 2 ⊙ > 10 −4 eV 2 are allowed already at 90% C.L. Values of cos 2θ ⊙ < 0 (for ∆m 2 ⊙ > 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 anti-neutrinos, were obtained in the CHOOZ and Palo Verde disappearance experiments with reactorν e [11,12]. The CHOOZ and Palo Verde experiments were sensitive to values of neutrino mass squared difference ∆m 2 10 −3 eV 2 , which includes the corresponding atmospheric neutrino region, eq. (1). No disappearance of the reactorν e was observed. Performing a two-neutrino oscillation analysis, the following rather stringent upper bound on the value of the corresponding mixing angle, θ, was obtained by the CHOOZ collaboration 1 [11] at 95% C.L. for ∆m 2 ≥ 1.5 × 10 −3 eV 2 : The precise upper limit in eq. (4) is ∆m 2 -dependent: it is a decreasing function of ∆m 2 as ∆m 2 increases up to ∆m 2 ≃ 6 · 10 −3 eV 2 with a minimum value sin 2 θ ≃ 10 −2 . The upper limit becomes an increasing function of ∆m 2 when the latter increases further up to ∆m 2 ≃ 8 · 10 −3 eV 2 , where sin 2 θ < 2 · 10 −2 . Somewhat weaker constraints on sin 2 θ have been obtained by the Palo Verde collaboration [12]. In the future, sin 2 θ might be further constrained or determined, e.g., in long baseline neutrino oscillation experiments [13]. The long baseline experiment with reactorν e 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 ∆m 2 ⊙ and sin 2 2θ ⊙ . Due to the long baseline of the experiment, L ∼ 180 km, however, ∆m 2 ⊙ can be determined with a relatively good precision only if ∆m 2 ⊙ ∼ < 10 −4 eV 2 .
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: where ν lL , l = e, µ, τ , are the three left-handed flavour neutrino fields, ν jL is the left-handed field of the neutrino ν j having a mass m j > 0 and U is a 3 × 3 unitary mixing matrix -the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix [17,18]. The three neutrino masses m 1,2,3 can obey the so-called normal hierarchy (NH) relation m 1 < m 2 < m 3 , or that of the inverted hierarchy (IH) type, m 3 < m 1 < m 2 . 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 β-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 nonnegligible [21,22,23]. 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 paper, in the context of three-neutrino oscillations, we study the possibility of using antineutrinos from nuclear reactors to explore the ∆m 2 ⊙ > 10 −4 eV 2 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 ∆m 2 ⊙ due to the fact that ∆m 2 ⊙ > 10 −4 eV 2 [25,26,27]. We determine the optimal baseline of the possible experiment with reactorν e , which would provide a precise measurement of ∆m 2 ⊙ in the region 10 −4 eV 2 < ∆m 2 ⊙ ∼ < 8 × 10 −4 eV 2 . 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 ∆m 2 , i.e., by ∆m 2 ⊙ and ∆m 2 atm . For the optimal baseline found earlier, L ∼ = (20 − 25) km, the indicated effect could be relevant for 10 −4 eV 2 < ∆m 2 ⊙ ∼ < 5 × 10 −4 eV 2 . For larger values of ∆m 2 ⊙ 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 reactionν e + p → e + + n (i.e., a high statistics experiment with sufficiently good energy resolution), a measurement of ∆m 2 atm with very high precision, sin 2 2θ ⊙ = 1.0, e.g., sin 2 2θ ⊙ ∼ < 0.9, and a sufficiently large value of the angle θ, which for ∆m 2 ⊙ ≪ ∆m 2 atm controls, e.g., the oscillations of the atmospheric ν e andν e and is constrained by the CHOOZ and Palo Verde data.

Theν e 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 ν j , j = 1, 2, 3, in such a way that their masses obey m 1 < m 2 < m 3 . Then the cases of NH and IH neutrino mass spectrum differ, in particular, by the relation between the mixing matrix elements |U ej |, j = 1, 2, 3, and the mixing angles θ ⊙ and θ (see further). With the indicated choice one has ∆m 2 jk > 0 for j > k. Let us emphasize that we do not assume any of the relations m 1 ≪ m 2 ≪ m 3 , or m 1 ∼ < m 2 ≪ m 3 , or m 1 ≪ m 2 ∼ = m 3 , 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 reactorν e oscillations will not be affected by Earth matter effects when theν e travel between the source (reactor) and the detector. If 3-neutrino mixing takes place, eq. (5), theν e would take part in 3-neutrino oscillations in vacuum on the way to the detector.
We shall obtain next the expressions for the reactorν e 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: and where θ 12 and θ 13 being two of the three mixing angles in the standard parameterization of the PMNS matrix (see, e.g., [16]). Note that |U e3 | 2 is constrained by the CHOOZ and Palo Verde results. It is not difficult to derive the expression for theν e survival probability in the case under discussion: where E ν 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., [28,20,16]): and The mixing matrix element constrained by the CHOOZ and Palo Verde data is now |U e1 | 2 : The expression for theν e survival probability can be written in the form [29]: Several comments concerning the expressions for theν e 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" ∆m 2 31 are accounted for. The CHOOZ and Palo Verde experiments are primarily sensitive to this term and their results limit sin 2 θ. 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 ∆m 2 ⊙ ≪ ∆m 2 31 = ∆m 2 atm , ∆m 2 ⊙ ∼ < 10 −4 eV 2 , only one of the indicated two terms leads to an oscillatory dependence of theν e survival probability for the ranges of L/E ν characterizing the CHOOZ and Palo Verde, and the KamLAND experiments: on the sourcedetector distance L of the CHOOZ and Palo Verde experiments the oscillations due to ∆m 2 ⊙ cannot develop, while on the distance(s) traveled by theν e in the KamLAND experiment ∆m 2 atm 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 sin 2 θ ⊙ , while the corresponding term in eq. (13) is proportional to cos 2 θ ⊙ [29]. This is the only difference between P N H (ν e →ν e ) and P IH (ν e →ν e ), that can be used to distinguish between the two cases of neutrino mass spectrum in an experiment with reactorν e . Obviously, if cos 2θ ⊙ = 0, we have P N H (ν e →ν e ) = P IH (ν e →ν e ) and the two types of spectrum would be indistinguishable in the experiments under discussion. For vanishing sin 2 θ, 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 P N H (ν e →ν e ) and P IH (ν e →ν e ), can take. The allowed region of values of ∆m 2 31 , ∆m 2 ⊙ , sin 2 θ ⊙ and θ should be determined in a global 3-neutrino oscillation analysis of the solar, atmospheric and reactor neutrino oscillation data, in which, in particular, ∆m 2 ⊙ should be allowed to take values in the LMA solution region, including the interval 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 quoted 2 in eqs. (2) and (3) as long as ∆m 2 31 ∼ > 1.5 × 10 −3 eV 2 . The reason is that for ∆m 2 31 ∼ > 1.5 × 10 −3 eV 2 and ∆m 2 ⊙ ∼ < 6.0 × 10 −4 eV 2 , the solar ν e 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) ∆m 2 31 . Thus, ∆m 2 ⊙ and θ ⊙ 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 ∆m 2 ⊙ in the LMA MSW solution region (see, e.g., [6,31]): For ∆m 2 ⊙ ∼ < 1.0 × 10 −4 eV 2 , the CHOOZ and solar neutrino data imply the upper limit on sin 2 θ given in eq. (4). For ∆m 2 ⊙ ∼ (2.0 − 6.0) × 10 −4 eV 2 of interest, the upper limit on sin 2 θ as a function of ∆m 2 31 ∼ > 10 −3 eV 2 for given ∆m 2 ⊙ and sin 2 2θ ⊙ 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 ∆m 2 31 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 ∆m 2 31 in the two cases of neutrino mass spectrum. We expect the regions of allowed values 2 Let us note that the LMA MSW solution values of ∆m 2 ⊙ and θ⊙ we quote in eqs. (2) and (3) were obtained by taking into account the CHOOZ and Palo Verde limits as well.
of the mixing angle θ atm , which controls the dominant atmospheric ν µ → ν τ andν µ →ν τ oscillations, to differ somewhat in the two cases. Note, however, that this mixing angle does not enter the expression for theν e survival probability we are interested in.
For ∆m 2 ⊙ ∼ < 1.0 × 10 −4 eV 2 and sufficiently small values of sin 2 θ, ∆m 2 31 coincides effectively with ∆m 2 atm of the two-neutrino ν µ andν µ oscillation analyses of the SK atmospheric neutrino data. If sin 2 θ > 0.01, a three-neutrino oscillation analysis of the atmospheric neutrino and CHOOZ data, performed under the assumption of ∆m 2 ⊙ ∼ < 1.0 × 10 −4 eV 2 [31], gives regions of allowed values of ∆m 2 atm = ∆m 2 31 , which are correlated with the value of sin 2 θ. 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 ∆m 2 ⊙ allowed to take values up to ∼ (6.0 − 7.0) × 10 −4 eV 2 , i.e., in the region where deviations from the two-neutrino approximation could be nonnegligible, is lacking in the literature. Therefore in what follows we will use representative values of ∆m 2 31 which lie in the region given by eq. (1).
3 The Difference between P N H (ν e →ν e ) and P IH (ν e →ν e ) Let us discuss next in greater detail the difference between theν e surviving probabilities in the two cases of neutrino mass spectrum of interest, P N H (ν e →ν e ) and P IH (ν e →ν e ). While the terms in the first two lines in eqs. (9) and (13) describe oscillations in L/E ν with frequencies ∆m 2 31 /4π and ∆m 2 ⊙ /4π, 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: This is a modulated oscillation with approximately the same frequency of the first term in eqs. (9) and (13) (∆m 2 31 /4π) and amplitude oscillating between 0 and 2 sin 2 θ ⊙ of the amplitude of the first term itself. The beat frequency is equal to the frequency of the dominant oscillation (∆m 2 ⊙ /4π). The modulation is exactly in phase with the ∆m 2 ⊙ −driven dominant oscillation of interest, so that the maximum of the oscillation amplitude of the interference term (third lines in the expressions for P N H (ν e →ν e ) and P IH (ν e →ν e )) is reached in coincidence with the points of maximal decreasing of theν e survival probability, where ∆m 2 ⊙ L/4 E = π/2, and vice versa -this amplitude vanishes at the local maxima of the survival probability. At the minima of theν e survival probability, for instance at ∆m 2 ⊙ L/4 E ν = π/2, P N H(IH) (ν e →ν e ) takes the value: From eqs. (9), (13) and (15) one deduces that: • for maximal mixing, cos 2θ ⊙ = 0, the last term cancels, and P N H = P IH ; • for very small mixing angles, cos 2θ ⊙ ≃ 1, the terms describing the oscillations driven by ∆m 2 31 in the NH and IH cases have opposite signs: the two waves are exactly out of phase.
• for intermediate values of cos 2θ ⊙ from the LMA MSW solution region, cos 2θ ⊙ ∼ = (0.3 − 0.6), the ∆m 2 31 −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 2θ ⊙ sin 2 θ.
The net result of these properties is that in the region of the minima of theν e survival probability due to ∆m 2 ⊙ , where ∆m 2 ⊙ L /(2E) = π(2k + 1), k = 0, 1, . . . , the difference between P N H (ν e →ν e ) and P IH (ν e →ν e )) is maximal. In contrast, at the maxima of P N H (ν e →ν e ) and P IH (ν e →ν e )) determined by ∆m 2 ⊙ L /(2E) = 2πk, we have, for any sin 2 θ ⊙ , P N H (ν e →ν e ) = P IH (ν e →ν e ). The two-neutrino oscillation approximation used in the analysis of the CHOOZ and Palo Verde data is rather accurate as long as ∆m 2 ⊙ is sufficiently small [29]: for ∆m 2 ⊙ ∼ < 10 −4 eV 2 , the L/E ν values characterizing these experiments, chosen to ensure maximal sensitivity to ∆m 2 31 ∼ > 10 −3 eV 2 , are much smaller than the value at which the first minimum of P N H(IH) (ν e →ν e ) due to the ∆m 2 ⊙dependent oscillating term occurs. Correspondingly, the effect of the interference term is strongly suppressed by the beats. For ∆m 2 ⊙ ∼ > 2 × 10 −4 eV 2 this is no longer valid and the interference term under discussion has to be taken into account in the analyses of the CHOOZ and Palo Verde data [29].

Measuring Large ∆m 2 ⊙ at Reactor Facilities
As is well-known, nuclear reactors are intense sources of low energyν e (E ν ∼ < 8 MeV), emitted isotropically in the β-decays of fission products with high neutron density [32]. Anti-neutrinos can then be detected through the positrons produced by inverse β-decay on nucleons. The reactorν e energy spectrum has been accurately measured and is theoretically well understood 3 [33]: it essentially consists of a bell-shaped distribution in energy centered around E ν ∼ 4 MeV, having a width of approximately 3 MeV. CHOOZ, Palo Verde and KamLAND are examples of experiments with reactorν e , 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 ∆m 2 ⊙ 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 * ≡ 2π E ν /∆m 2 ⊙ . This implies that for E ν = 4 MeV, the optimal length to test neutrino oscillations with reactor experiments is: The best sensitivity of KamLAND, for instance, is in the range of 2 ÷ 3 × 10 −5 eV 2 . We will discuss next in greater detail the distances L which could be used to probe the LMA MSW solution region at ∆m 2 ⊙ > 10 −4 eV 2 , in order to extract ∆m 2 ⊙ from these oscillation experiments.

Total Event Rate Analysis
One of the signatures of theν e −oscillations would be a substantial reduction of the measured total event rate due to the reactorν e in comparison with the predicted one in the absence of oscillations. In order to compute the expected total event rate one has to integrate theν e survival probability multiplied by theν e energy spectrum over E ν . In Fig. 1 we show this averaged survival probability for different values of L as a function of ∆m 2 ⊙ , using the "best fit" values [1,5,6,7] for ∆m 2 31 and sin 2 2θ ⊙ . When averaging over theν e 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 δE ν of the energy spectrum is such that the integration runs over more than one period, i.e., approximately for: Since δE ν ∼ 3 MeV, at KamLAND this happens approximately for ∆m 2 ⊙ ∼ > 7 × 10 −5 eV 2 . The corresponding curve in Fig. 1 indicates that the actual sensitivity extends to somewhat larger values of ∆m 2 ⊙ than what is expected on the basis on the above estimate, but the total event rate becomes flat for ∆m 2 ⊙ ∼ > 10 −4 eV 2 . 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 ∆m 2 ⊙ only if ∆m 2 ⊙ ∼ < 10 −4 eV 2 . If ∆m 2 ⊙ ∼ > 2 × 10 −4 eV 2 , it would be possible to obtain only a lower bound on ∆m 2 ⊙ and a new experiment might be required to determine ∆m 2 ⊙ . Fig. 1 shows that as L decreases, the sensitivity region moves to larger ∆m 2 ⊙ . These results imply that a reactorν e experiment with L ∼ = (20 − 25) km can probe the range 0.8 × 10 −4 eV 2 < ∆m 2 ⊙ ∼ < 6 × 10 −4 eV 2 . One finds that for ∆m 2 should use the information about the e + −spectrum distortion due to theν e −oscillations. By measuring the e + −spectrum with a sufficient precision it would be possible to cover the whole interval i.e., to determine ∆m 2 ⊙ if it lies in this interval, by performing an experiment at L ∼ = (20 − 25) km from the reactor(s) 4 (see the next sub-section). 4 The fact that if ∆m 2 ⊙ ∼ = 3.2 × 10 −4 eV 2 , a reactorνe experiment with L ∼ = 20 km would allow to measure ∆m 2 ⊙ with a high precision was also noticed recently in [27]. Applying eq. (17) with ∆m 2 = ∆m 2 31 , one sees that for the ranges of L which allow to probe ∆m 2 ⊙ from the LMA MSW solution region, the total event rate is not sensitive to the oscillations driven by ∆m 2 31 ∼ > 1.5 × 10 −3 eV 2 . Thus, the total event rate analysis would determine ∆m 2 ⊙ which would be the same for both the normal and inverted hierarchy neutrino mass spectrum.

Energy Spectrum Distortions
An unambiguous evidence of neutrino oscillations would be the characteristic distortion of thē ν e 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 theν e energy spectrum is essentially a direct test of theν e oscillations. It is more effective than the total rate analysis since it is not affected, e.g., by the overall normalization of the reactorν e 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 theν e spectrum expected for ∆m 2 ⊙ = 2 × 10 −4 eV 2 and ∆m 2 ⊙ = 6 × 10 −4 eV 2 and the spectrum in the absence ofν e 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 reactorν e spectrum [36].
As Fig.  2 illustrates, theν e spectrum in the case of oscillation is well distinguishable from that in the absence of oscillations. Moreover, for ∆m 2 ⊙ lying in the interval 10 −4 eV 2 < ∆m 2 ⊙ ∼ < 8.0 × 10 −4 eV 2 , the shape of the spectrum exhibits a very strong dependence on the value of ∆m 2 ⊙ . A likelihood analysis of the data would be able to determine the value of ∆m 2 ⊙ 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 ∆m 2 ⊙ thus determined should coincide with value obtained from the analysis of the total event rate and should be independent of ∆m 2 31 .

Normal vs. Inverted Hierarchy
In Fig. 2 we show the deformation of the reactorν e 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 5 , sin 2 θ is not too small and ∆m 2 31 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 ∆m 2 31 in long (very long) baseline neutrino oscillation experiments [21,22,23] 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 θ ⊙ : the different behavior of the two survival probabilities is due to the difference between sin 2 θ ⊙ and cos 2 θ ⊙ ; correspondingly, the effect vanishes for maximal mixing; thus, the more the mixing deviates from the maximal the larger the effect; • the value of sin 2 θ, which controls the magnitude of the sub-leading effects due to ∆m 2 31 on the ∆m 2 ⊙ −driven oscillations: the effect of interest vanishes in the decoupling limit of sin 2 θ → 0; • the value of ∆m 2 ⊙ (see Fig. 1): for given L and ∆m 2 ⊙ 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 ∆m 2 ⊙ L = 2πE ν : The ratio could be rather large: the factor in front of the cos π ∆m 2 31 /∆m 2 ⊙ is about 25% for sin 2 2θ ⊙ = 0.8 and sin 2 θ = 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 ∆E ν must satisfy: 5 It would be impossible to distinguish between the normal and inverted hierarchy neutrino mass spectrum if for given ∆m 2 ⊙ > 10 −4 eV 2 and sin 2 2θ⊙ = 1, the LMA solution region is symmetric with respect to the change θ⊙ → π/2 − θ⊙ (cos 2θ⊙ → − cos 2θ⊙). While the value of sin 2 2θ⊙ is expected to be measured with a relatively high precision by the KamLAND experiment, the sign of cos 2θ⊙ will not be fixed by this experiment. However, the θ⊙ − (π/2 − θ⊙) ambiguity can be resolved by the solar neutrino data. Note also that the current solar neutrino data disfavor values of cos 2θ⊙ < 0 in the LMA solution region (see, e.g., [5,6,10]).
For L ∼ 1 km this condition could be satisfied for δE ν ≃ ∆E ν , but at L ∼ = (15 − 20) Km, for ∆m 2 31 = 2.5 × 10 −3 eV 2 and E ν in the interval (3 − 5) MeV, one should have ∆E ν ∼ < 0.5 MeV. Our discussion so far was performed for simplicity in terms of the reactorν e energy spectrum, while in the experiments of interest one measures the energy of the positron emitted in the inverse βdecay, E e . The relation between E e and E ν 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: E ν ∼ = E e + (E th ν − m e ). The maximal ∆E ν allowed in order to make the effect observable can be then directly compared to the experimental positron energy resolution ∆E e 6 .
For ∆m 2 ⊙ ∼ < 10 −4 eV 2 , 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 smaller. This means that for ∆m 2 ⊙ ≪ ∆m 2 31 , 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 < E ν < 7.2 MeV into 15 bins, with ∆E ν = 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. 6 In the CHOOZ experiment, for instance, the binning in Ee was ∆Ee ≃ 0.40 MeV [11]. KamLAND is expected to have a resolution better than ∆Ee/Ee = 10%/ √ Ee, where Ee is in MeV [37] .
As our results show and Fig. 3  should be done with a smaller baseline, L ∼ = 10 km. If, however, sin 2 θ ∼ < 0.01, and/or sin 2 2θ ⊙ ∼ > 0.9, and/or sin 2 2θ ⊙ ∼ < 0.9 but the LMA solution admits equally positive and negative values of cos 2θ ⊙ , the difference between the spectra in the two cases becomes hardly observable. Further, in obtaining Fig. 3 we have implicitly assumed that ∆m 2 31 is known with negligible uncertainty. Actually, for the difference between the spectra under discussion to be observable, ∆m 2 31 has to be determined, according to our estimates, with a precision of ∼ 10% or better 8 : given the values of ∆m 2 ⊙ , sin 2 2θ ⊙ and sin 2 θ, a spectrum in the NH case corresponding to a given ∆m 2 31 can be rather close in shape to the spectrum in the IH case for a different value of ∆m 2 31 . There is no similar effect when varying ∆m 2 ⊙ .

Conclusions
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 might be required to determine ∆m 2 ⊙ with high precision if the results of the KamLAND experiment show that ∆m 2 ⊙ > 10 −4 eV 2 . Performing a three-neutrino oscillation analysis of both the total event rate suppression and the e + −energy spectrum distortion caused by thē ν e −oscillations in vacuum, we show that a value of ∆m 2 ⊙ from the interval 10 −4 eV 2 < ∆m 2 ⊙ ∼ < 8.0 × 10 −4 eV 2 could be determined with a high precision in experiments with L ∼ = (20 − 25) km if the e + −energy spectrum is measured with a sufficiently good accuracy. Furthermore, if ∆m 2 ⊙ ∼ = (1.0 − 5.0) × 10 −4 eV 2 , 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 ∆m 2 ⊙ not exceeding 8.0 × 10 −4 eV 2 , 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 ∆E e ∼ < 0.40 MeV) and the observed event rate due to the reactorν e 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 θ must be sufficiently large (sin 2 θ ∼ 0.03 − 0.05), and the "solar" mixing angle θ ⊙ should not be maximal (sin 2 2θ ⊙ ∼ < 0.9). In addition, the value of ∆m 2 31 , which is responsible for the dominant ν µ → ν τ andν µ →ν τ 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]. 7 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. 8 The analysis, e.g, of the MINOS potential for a high precision determination of ∆m 2 31 in the case of ∆m 2 ⊙ ∼ < 10 −4 eV 2 yields very encouraging results (see, e.g., [38]). For ∆m 2 ⊙ ∼ = (2.0 − 8.0) × 10 −4 eV 2 , such analysis is lacking in the literature.