Three-Neutrino Oscillations of Atmospheric Neutrinos, theta13, Neutrino Mass Hierarchy and Iron Magnetized Detectors

We derive predictions for the Nadir angle (theta_n) dependence of the ratio Nmu-/Nmu+ of the rates of the mu- and mu+ multi-GeV events, and for the mu- - mu+ event rate asymmetry, A_{mu-mu+}=[Nmu- - Nmu+]/[Nmu- + Nmu+], in iron-magnetized calorimeter detectors (MINOS, INO) in the case of 3-neutrino oscillations of the atmospheric nu_mu and antinu_mu, driven by one neutrino mass squared difference, |Delta m^2_{31}|>>Delta m^2_{21}. The asymmetry A_{mu- mu+} (the ratio Nmu-/Nmu+) is shown to be particularly sensitive to the Earth matter effects in the atmospheric neutrino oscillations, and thus to the values of sin^2(theta13) and sin^2(theta23), theta13 and theta23 being the neutrino mixing angles limited by the CHOOZ and Palo Verde experiments and that responsible for the dominant atmospheric nu_mu ->nu_tau (antinu_mu ->antinu_tau) oscillations. It is also very sensitive to the type of neutrino mass spectrum which can be with normal (Delta m^2_{31}>0) or with inverted (Delta m^2_{31}<0) hierarchy. We find that for sin^2(theta23)>0.50, sin^2(2 theta13)>0.06 and Delta m^2_{31}=(2-3) 10^{-3} eV^2, the Earth matter effects produce a relative difference between the integrated asymmetries barA_{mu- mu+} and barA^{2nu}_{mu- mu+}$ in the mantle (cos(theta_n)=0.30-0.84) and core (cos(theta_n)=0.84-1.0) bins, which is bigger in absolute value than ~15%, can reach the values of (30-35)%, and thus can be sufficiently large to be observable. The sign of the indicated asymmetry difference is anticorrelated with the sign of Delta m^2_{31}. An observation of the Earth matter effects in the Nadir angle distribution of the asymmetry A_{mu- mu+} (ratio Nmu-/Nmu+) would clearly indicate that sin^2(2 theta13)>0.06 and sin^2(theta23)>0.50, and would lead to the determination of the sign of Delta m^2_{31}.


Introduction
There has been a remarkable progress in the studies of neutrino oscillations in the last several years. The experiments with solar, atmospheric and reactor neutrinos [1,2,3,4,5,6,7] have provided compelling evidences for the existence of neutrino oscillations driven by nonzero neutrino masses and neutrino mixing. Evidences for oscillations of neutrinos were obtained also in the first long baseline accelerator neutrino experiment K2K [8]. It was predicted already in 1967 [9] that the existence of solar neutrino oscillations would cause a deficit of solar neutrinos detected on Earth. The hypothesis of solar neutrino oscillations, which in one variety or another were considered starting from the late 60's as the most natural explanation of the observed [1,2] solar neutrino deficit (see, e.g., refs. [9,10,11,12]), has received a convincing confirmation from the measurement of the solar neutrino flux through the neutral current reaction on deuterium by the SNO experiment [4,5], and by the first results of the KamLAND experiment [7]. The analysis of the solar neutrino data obtained by Homestake, SAGE, GALLEX/GNO, Super-Kamiokande and SNO experiments showed that the data favor the Large Mixing Angle (LMA) MSW solution of the solar neutrino problem (see, e.g., ref. [4]). The first results of the KamLAND reactor experiment [7] have confirmed (under the very plausible assumption of CPT-invariance) the LMA MSW solution, establishing it essentially as a unique solution of the solar neutrino problem.
The latest addition to this magnificent effort is the evidence presented recently by the Super-Kamiokande (SK) collaboration for an "oscillation dip" in the L/E−dependence, of the (essentially multi-GeV) µ−like atmospheric neutrino events 1 [13], L and E being the distance traveled by neutrinos and the neutrino energy. As is well known, the SK atmospheric neutrino data is best described in terms of dominant 2-neutrino ν µ → ν τ (ν µ →ν τ ) vacuum oscillations with maximal mixing, sin 2 2θ 23 ∼ = 1. The observed dip is predicted due to the oscillatory dependence of the ν µ → ν τ andν µ →ν τ oscillation probabilities, P (ν µ → ν τ ) ∼ = P (ν µ →ν τ ), on L/E. The dip in the observed L/E distribution corresponds to the first oscillation minimum of the ν µ (ν µ ) survival probability, P (ν µ → ν µ ) = 1 − P (ν µ → ν τ ), as L/E increases starting from values for which ∆m 2 31 L/(2E) ≪ 1 and P (ν µ → ν µ ) ∼ = 1, ∆m 2 31 being the neutrino mass squared difference responsible for the atmospheric ν µ andν µ oscillations. This beautiful result represents the first ever observation of a direct effect of the oscillatory dependence on L/E of the probability of neutrino oscillations in vacuum.
The interpretation of the solar and atmospheric neutrino, and of KamLAND data in terms of neutrino oscillations requires the existence of 3-neutrino mixing in the weak charged lepton current (see, e.g., ref. [14]): Here ν lL , l = e, µ, τ , are the three left-handed flavor neutrino fields, ν jL is the left-handed field of the neutrino ν j having a mass m j and U is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix [15], (2) where we have used a standard parametrization of U with the usual notations, s ij ≡ sin θ ij , c ij ≡ cos θ ij , and δ is the Dirac CP-violation phase 2 . If one identifies ∆m 2 21 > 0 and ∆m 2 31 with the neutrino mass squared differences which drive the solar and atmospheric neutrino oscillations, the data suggest that |∆m 2 31 | ≫ ∆m 2 21 . In this case θ 12 and θ 23 , represent the neutrino mixing angles responsible for the solar and the dominant atmospheric neutrino oscillations, θ 12 , θ 23 , while θ 13 is the angle limited by the data from the CHOOZ and Palo Verde experiments [19,20].
The 3-neutrino oscillations of the solar ν e depend in the case of interest, |∆m 2 31 | ≫ ∆m 2 21 , not only on ∆m 2 21 and θ 12 , but also on θ 13 . A combined 3-neutrino oscillation analysis of the solar neutrino, CHOOZ and KamLAND data showed [21] that for sin 2 θ 13 < ∼ 0.05 the allowed ranges of the solar neutrino oscillation parameters do not differ substantially from those derived in the 2-neutrino oscillation analyzes (see, e.g., refs. [5,22]). A description of the indicated data in terms of ν e → ν µ,τ andν e →ν µ,τ oscillations is possible (at 99.73% C.L.) for sin 2 θ 13 < ∼ 0.075. The data favor the LMA-I MSW solution (see, e.g., refs. [21,22]) with ∆m 2 21 ∼ = 7.2 × 10 −5 eV 2 and sin 2 θ 12 ∼ = 0.30. The LMA-II solution, corresponding to ∆m 2 21 ∼ = 1.5 × 10 −4 eV 2 and approximately the same value of sin 2 θ 12 , is severely constrained by the fact [23] that the ratio of the rates of the CC and NC reactions on deuterium, measured with a relatively high precision in SNO during the salt phase of the experiment [5], turned out to be definitely smaller than 0.50. This solution is currently allowed by the data only at 99.13% C.L. [21].
Let us note that the atmospheric neutrino and K2K data do not allow one to determine the signs of ∆m 2 31 , and of cos 2θ 23 if sin 2 2θ 23 = 1.0. This implies that in the case of 3neutrino mixing one can have ∆m 2 31 > 0 or ∆m 2 31 < 0. The two possibilities correspond to two different types of neutrino mass spectrum: with normal hierarchy (NH), m 1 < m 2 < m 3 , and with inverted hierarchy (IH), m 3 < m 1 < m 2 . The fact that the sign of cos 2θ 23 is not determined when sin 2 2θ 23 = 1.0 implies that when, e.g., sin 2 2θ 23 = 0.92, two values of sin 2 θ 23 are possible, sin 2 θ 23 ∼ = 0.64 or 0.36.
The precise limit on the angle θ 13 from the CHOOZ and Palo Verde data is ∆m 2 31 − dependent (see, e.g, ref. [27]). Using the 99.73% allowed range of ∆m 2 31 = (1.1 − 3.2) × 10 −3 eV 2 , from ref. [25], one gets from a combined 3-neutrino oscillation analysis of the solar neutrino, CHOOZ and KamLAND data [21]: The global analysis of the solar, atmospheric and reactor neutrino data performed in ref. [28] gives sin 2 θ 13 < 0.054 at 99.73% C.L. It is difficult to overestimate the importance of getting more precise information about the value of the mixing angle θ 13 , of determining the sign of ∆m 2 31 , or the type of the neutrino mass spectrum (with normal or inverted hierarchy), and of measuring the value of sin 2 θ 23 with a higher precision, for the future progress in the studies of neutrino mixing. Although this has been widely recognized, let us repeat the arguments on which the statement is based.
If the neutrinos with definite mass are Majorana particles (see, e.g., ref. [11]), the predicted value of the effective Majorana mass parameter in neutrinoless double β−decay depends strongly in the case of normal hierarchical or partially hierarchical neutrino mass spectrum on the value of sin 2 θ 13 (see, e.g., ref. [36]).
The sign of ∆m 2 31 determines, for instance, which of the transitions (e.g., of atmospheric neutrinos) ν µ → ν e and ν e → ν µ , orν µ →ν e andν e →ν µ , can be enhanced by the Earth matter effects [37,38,39]. The predictions for the neutrino effective Majorana mass in neutrinoless double β−decay depend critically on the type of the neutrino mass spectrum (normal or inverted hierarchical) [36,40]. The knowledge of the value of θ 13 and of the sign of ∆m 2 31 is crucial for the searches for the correct theory of neutrino masses and mixing as well.
Somewhat better limits on sin 2 θ 13 than the existing one can be obtained in the MINOS, OPERA and ICARUS experiments [41,42]. Various options are being currently discussed (experiments with off-axis neutrino beams, more precise reactor antineutrino and long baseline experiments, etc., see, e.g., ref. [43]) of how to improve by at least an order of magnitude, i.e., to values of ∼ 0.005 or smaller, the sensitivity to sin 2 θ 13 . The sign of ∆m 2 31 can be determined in very long baseline neutrino oscillation experiments at neutrino factories (see, e.g., refs. [31,32]), and, e.g, using combined data from long baseline oscillation experiments at the JHF facility and with off-axis neutrino beams [44]. If the neutrinos with definite mass are Majorana particles, it can be determined by measuring the effective neutrino Majorana mass in neutrinoless double β−decay experiments [36,40]. Under certain rather special conditions it might be determined also in experiments with reactorν e [45].
In the present article we study possibilities to obtain information on the value of sin 2 θ 13 and on the sign of ∆m 2 31 using the data on atmospheric neutrinos, which can be obtained in experiments with detectors able to measure the charge of the muon produced in the charged current (CC) reaction by atmospheric ν µ orν µ . It is a natural continuation of our similar study for water-Čerenkov detectors [46]. In the experiments with muon charge identification it will be possible to distinguish between the ν µ andν µ induced events. As is well known, the water-Čerenkov detectors do not have such a capability. Among the operating detectors, MINOS has muon charge identification capabilities for multi-GeV muons [41]. The MINOS experiment is currently collecting atmospheric neutrino data. The detector has relatively small mass, but after 5 years of data-taking it is expected to collect about 440 atmospheric ν µ and about 260 atmosphericν µ multi-GeV events (having the interaction vertex inside the detector). There are also plans to build a 30-50 kton magnetized tracking iron calorimeter detector in India within the India-based Neutrino Observatory (INO) project [47]. The INO detector will be based on MONOLITH design [48]. The primary goal is to study the oscillations of the atmospheric ν µ andν µ . This detector is planned to have efficient muon charge identification, high muon energy resolution (∼ 5%) and muon energy threshold of about 2 GeV. It will accumulate sufficiently high statistics of atmospheric ν µ andν µ induced events in several years, which would permit to search for effects of the subdominant ν µ → ν e (ν e → ν µ ) andν µ →ν e (ν e →ν µ ) transitions.
For ∆m 2 31 > 0, the ν µ → ν e (ν µ →ν e ) and ν e → ν µ (ν e →ν µ ) transitions in the Earth lead to a reduction of the rate of the multi-GeV µ − events observable in MINOS, INO, etc., with respect to the case of absence of these transitions (see, e.g., refs. [29,49,50,51,53]). If ∆m 2 31 < 0, the µ + event rate will be reduced. Correspondingly, as observable which is sensitive to the Earth matter effects, and thus to the value of sin 2 θ 13 and the sign of ∆m 2 31 , as well as to sin 2 θ 23 , we can consider the Nadir-angle distribution of the ratio N(µ − )/N(µ + ) of the multi-GeV µ − and µ + event rates, or, equivalently, of the µ − − µ + event rate asymmetry The systematic uncertainty, in particular, in the Nadir angle dependence of N(µ − )/N(µ + ), and correspondingly in the asymmetry A µ − µ + , can be smaller than those on the measured Nadir angle distributions of the rates of µ − and µ + events, N(µ − ) and N(µ + ).
We have obtained predictions for the Nadir-angle distribution of A µ − µ + in the case of 3-neutrino oscillations of the atmospheric ν µ ,ν µ , ν e andν e , both for neutrino mass spectra with normal (∆m 2 31 > 0) and inverted (∆m 2 , and for sin 2 θ 23 = 0.64; 0.50; 0.36. These are compared with the predicted Nadir-angle distributions of the same asymmetry in the case of 2-neutrino ν µ → ν τ andν µ →ν τ oscillations of the atmospheric ν µ andν µ (i.e., for sin 2 θ 13 = 0), A 2ν µ − µ + . Predictions for the three types of asymmetries indicated above of the suitably integrated Nadir angle distributions of the µ − and µ + multi-GeV event rates are also given. Our results show, in particular, that for sin 2 θ 23 > ∼ 0.50 and sin 2 2θ 13 > ∼ 0.06 the effects of the Earth matter enhanced subdominant transitions of the atmospheric neutrinos, ν µ → ν e and and ν e → ν µ , orν µ →ν e andν e →ν µ , can be sufficiently large to be observable with INO and possibly with MINOS detectors. Conversely, if the indicated effects are observed in the MINOS and/or INO experiments, that would imply that sin 2 2θ 13 > ∼ 0.05, sin 2 θ 23 > ∼ 0.50 and at the same time would permit to determine the sign of ∆m 2 31 and thus to answer the fundamental question about the type of hierarchy -normal or inverted, the neutrino mass spectrum has.
Let us note that the Earth matter effects in atmospheric neutrino oscillations have been widely studied (for recent detailed analyzes see, e.g., refs. [52,46], which contain also a rather complete list of references to earlier work on the subject). A rather detailed analysis for the MONOLITH detector has been performed in ref. [53]. A large number of studies have been done for the Super-Kamiokande detector, or more generally, for water-Čerenkov detectors. In ref. [46], in particular, the magnitude of the Earth matter effects in the Nadir angle distribution of the ratio of the multi-GeV µ−like and e−like events, measured in water-Čerenkov detectors, N µ /N e , has been investigated. This Nadir angle distribution is the observable most sensitive to the matter effects of interest. It was concluded that for sin 2 θ 23 > ∼ 0.50, sin 2 θ 13 > ∼ 0.01 and ∆m 2 31 > 0, the effects of the Earth matter enhanced ν µ → ν e and ν e → ν µ transitions of the atmospheric ν µ and ν e , might be observable with the Super-Kamiokande detector. However, determining the sign of ∆m 2 31 would be quite challenging in this experiment (or its bigger version -Hyper-Kamiokande [54]). In general, the matter effects in the Nadir angle distribution of the ratio of the multi-GeV µ−like and e−like events, N µ /N e , which can be measured in the Super-Kamiokande or other water-Cerenkov detectors, are smaller than the matter effects in the Nadir angle distribution of the ratio of the multi-GeV µ − and µ + events, N(µ − )/N(µ + ), which can be measured in MINOS, INO, or any other atmospheric neutrino experiment with sufficiently good muon charge identification. The reason is that in the case of water-Čerenkov detectors, approximately 2/3 of the rate of the multi-GeV µ − like events is due to ν µ , and ∼ 1/3 is due toν µ ; similar partition is valid for the multi-GeV e−like events. Depending on the sign of ∆m 2 31 , the matter effects enhance either the neutrino transitions, ν µ → ν e and and ν e → ν µ , or the antineutrino transitions,ν µ →ν e andν e →ν µ , but not both types of transitions. Correspondingly, because the ν µ,e − andν µ,e − induced events are indistinguishable in water-Cerenkov detectors, only ∼ 2/3 or ∼ 1/3 of the events in the multi-GeV µ−like and e−like samples collected in these detectors are due to neutrinos whose transitions can be enhanced by matter effects. This effectively reduces the magnitude of the matter effects in the samples of multi-GeV µ−like and e−like events. Obviously, such a "dilution" of the magnitude of the matter effects does not take place in the samples of the multi-GeV µ − and µ + events, which can be collected in MINOS and INO experiments, i.e., in the experiments with muon charge identification.

Subdominant 3-ν Oscillations of Multi-GeV Atmospheric Neutrinos in the Earth
In the present Section we review the physics of the subdominant 3-neutrino oscillations of the multi-GeV atmospheric neutrinos in the Earth (see, e.g., ref. [46]).
The fluxes of atmospheric ν e,µ of energy E, which reach the detector after crossing the Earth along a given trajectory specified by the value of θ n , Φ νe,µ (E, θ n ), are given by the following expressions in the case of the 3-neutrino oscillations under discussion [30]: is the ν e(µ) flux in the absence of neutrino oscillations and The interpretation of the SK atmospheric neutrino data in terms of ν µ → ν τ oscillations requires the parameter s 2 23 to lie approximately in the interval (0.30 -0.70), with 0.5 being the statistically preferred value. For the predicted ratio r(E, θ n ) of the atmospheric ν µ and ν e fluxes for i) the Earth core crossing and ii) only mantle crossing neutrinos, having trajectories for which 0.3 < ∼ cos θ n ≤ 1.0, one has [56,57,58] Obviously, the effects of interest are much larger for the multi-GeV neutrinos than for the sub-GeV neutrinos. They are also predicted to be larger for the flux of (and event rate due to) multi-GeV atmospheric ν e than for the flux of (and event rate due to) multi-GeV atmospheric ν µ .
The same conclusions are valid for the effects of oscillations on the fluxes of, and event rates due to, atmospheric antineutrinosν e andν µ . The formulae for anti-neutrino fluxes and oscillation probabilities are analogous to those for neutrinos: they can be obtained formally from eqs. (6) -(13) by replacing the neutrino related quantities -probabilities, κ, A 2ν (ν τ → ν τ ) and fluxes, with the corresponding quantities for antineutrinos: P 2ν (∆m 2 31 , νe,µ (E, θ n ) and r(E, θ n ) →r(E, θ n ) (see refs. [30,46]). Equations (6) -(9), (11) -(12) and the similar equations for antineutrinos imply that in the case under study the effects of the ν µ → ν e ,ν µ →ν e , and ν e → ν µ(τ ) ,ν e →ν µ(τ ) , oscillations i) increase with the increase of s 2 23 and are maximal for the largest allowed value of s 2 23 , ii) should be substantially larger in the multi-GeV samples of events than in the sub-GeV samples, and iii) in the case of the multi-GeV samples, for ∆m 2 31 > 0 they lead to a decrease of the µ − event rate, while if ∆m 2 31 < 0, the µ + event rate will decrease. The last point follows from the fact that the magnitude of the effects we are interested in depends also on the 2-neutrino oscillation probabilities, P 2ν andP 2ν , and that P 2ν orP 2ν (but not both probabilities) can be strongly enhanced by the Earth matter effects. In the case of oscillations in vacuum we have P 2ν =P 2ν ∼ sin 2 2θ 13 . Given the existing limits on sin 2 2θ 13 , the probabilities P 2ν andP 2ν cannot be large if the oscillations take place in vacuum.
If sin 2 θ 13 = 0, the Earth matter effects can resonantly enhance either the ν µ → ν e and ν e → ν µ , or theν µ →ν e andν e →ν µ transitions, depending on the sign of ∆m 2 31 . The enhancement mechanisms are discussed briefly in the next subsection.

Enhancing Mechanisms
As is well-known, the Earth density distribution in the existing Earth models is assumed to be spherically symmetric 3 and there are two major density structures -the core and the mantle, and a certain number of substructures (shells or layers). The core radius and the depth of the mantle are known with a rather good precision and these data are incorporated in the Earth models. According to the Stacey 1977 and the more recent PREM models [59,60], which are widely used in the calculations of the probabilities of neutrino oscillations in the Earth, the core has a radius R c = 3485.7 km, the Earth mantle depth is approximately R man = 2885.3 km, and the Earth radius is R ⊕ = 6371 km. The mean values of the matter densities and the electron fraction numbers in the mantle and in the core read, respectively:ρ man ∼ = 4.5 g/cm 3 ,ρ c ∼ = 11.5 g/cm 3 , and Numerical calculations show [29,34] that, e.g., the ν e → ν µ oscillation probability of interest, calculated within the two-layer model of the Earth withρ man (orN man e ) andρ c (orN c e ) for a given neutrino trajectory determined using the PREM (or the Stacey) model, reproduces with a remarkably high precision the corresponding probability, calculated by solving numerically the relevant system of evolution equations with the much more sophisticated Earth density profile of the PREM (or Stacey) model.
In the two-layer model, the oscillations of atmospheric neutrinos crossing only the Earth mantle (but not the Earth core), correspond to oscillations in matter with constant density. The relevant expressions for P 2ν , κ and A 2ν (ν τ → ν τ ) are given by (see, e.g., ref. [46]): 3 Let us note that because of the approximate spherical symmetry of the Earth, a given neutrino trajectory through the Earth is completely specified by its Nadir angle.
is the mass difference between the two mass-eigenstate neutrinos in the mantle, θ ′ m is the mixing angle in the mantle, L is the distance the neutrino travels in the mantle,ρ man and ρ res man are the mean density along the neutrino trajectory and the resonance density in the mantle, For a neutrino trajectory which is specified by a given Nadir angle θ n we have: where R ⊕ = 6371 km is the Earth radius (in the PREM [60] and Stacey [59] models) 4 . Consider for definiteness the case of ∆m 2 31 > 0. It follows from eqs. (11) and (12) that the oscillation effects of interest will be maximal if P 2ν ∼ = 1. The latter is possible provided i) the well-known resonance condition [38,39], leading to sin 2 2θ m ∼ = 1, is fulfilled, and ii) cos If the first condition is satisfied, the second determines the length of the path of the neutrinos in the mantle for which one can have P 2ν ∼ = 1: It follows from the above simple analysis [33] that the Earth matter effects can amplify P 2ν significantly when the neutrinos cross only the mantle i) for E ∼ (6 − 11) GeV, i.e., in the multi-GeV range of neutrino energies, and ii) only for sufficiently long neutrino paths in the mantle, i.e., for cos θ n > ∼ 0.3. The magnitude of the matter effects of interest increases with increasing of sin 2 θ 13 .
In the case of atmospheric neutrinos crossing the Earth core, new resonant effects become apparent. For sin 2 θ 13 < 0.05 and ∆m 2 31 > 0, we can have P 2ν ∼ = 1 only due to the effect of maximal constructive interference between the amplitudes of the the ν e → ν ′ τ transitions in the Earth mantle and in the Earth core [29,49,50]. The effect differs from the MSW one [29] and the enhancement happens in the case of interest at a value of the energy between the resonance energies corresponding to the density in the mantle and that of the core. The mantle-core enhancement effect is caused by the existence (for a given neutrino trajectory through the Earth core) of points of resonance-like total neutrino conversion, P 2ν = 1, in the corresponding space of neutrino oscillation parameters [49,50]. The points where P 2ν = 1 are determined by the conditions [49,50]: where the signs are correlated and cos 2θ ′′ m cos(2θ ′′ m −4θ ′ m ) ≤ 0. In eq. (23) 2φ ′ and 2φ ′′ are the oscillation phases (phase differences) accumulated by the (two) neutrino states after crossing respectively the first mantle layer and the core, and θ ′′ m is the neutrino mixing angle in the core. A rather complete set of values of ∆m 2 31 /E and sin 2 2θ 13 for which both conditions in eq. (23) hold and P 2ν = 1 for the Earth core-crossing atmospheric ν µ and ν e having trajectories with Nadir angle θ n = 0; 13 0 ; 23 0 ; 30 0 was found in ref. [50]. The location of these points determines the regions where P 2ν is large, P 2ν > ∼ 0.5. These regions vary slowly with the Nadir angle, they are remarkably wide in the Nadir angle and are rather wide in the neutrino energy [50], so that the transitions of interest produce noticeable effects: we have δE/E ∼ = 0.3 for the values of sin 2 θ 13 of interest [30,50].  Table 2 in ref. [50]). The first solution corresponds to [29] cos 2φ ′ ∼ = −1, cos 2φ ′′ ∼ = −1 and 5 sin 2 (2θ ′′ m − 4θ ′ m ) = 1. For ∆m 2 31 = 2.0 (3.0) × 10 −3 eV 2 , the total neutrino conversion occurs in the case of the first solution at E ∼ = (2.8 − 3.1) GeV (E ∼ = (4.2 − 4.7) GeV). The values of sin 2 2θ 13 at which the second solution takes place are marginally allowed. If, e.g., ∆m 2 31 = 2.5 × 10 −3 eV 2 , one has P 2ν = 1 for this solution for a given θ n in the interval 0 < ∼ θ n < ∼ 23 0 at E lying in the interval E ∼ = (5.3 − 6.7) GeV.
The effects of the mantle-core enhancement of P 2ν (orP 2ν ) increase rapidly with sin 2 2θ 13 as long as sin 2 2θ 13 < ∼ 0.06, and should exhibit a rather weak dependence on sin 2 2θ 13 for 0.06 < ∼ sin 2 2θ 13 < 0.19. If 3-neutrino oscillations of atmospheric neutrinos take place, the magnitude of the matter effects in the multi-GeV µ−like and e−like event samples, produced by neutrinos crossing the Earth core, should be larger than in the event samples due to neutrinos crossing only the Earth mantle (but not the core). This is a consequence of the fact that in the energy range of interest the atmospheric neutrino fluxes decrease rather rapidly with energy -approximately as E −2.7 , while the neutrino interaction cross section rises only linearly with E, and that the maximum of P 2ν (orP 2ν ) due to the resonancelike mantle-core interference effect takes place at approximately two times smaller energies than that due to the MSW effect for neutrinos crossing only the Earth mantle (e.g., at E ∼ = (3.5 − 3.9) GeV and E ∼ = 8.3 GeV, respectively, for ∆m 2 31 = 2.5 × 10 −3 eV 2 ). The same results, eqs. (21) and (22), and conclusions are valid for the antineutrino oscillation probabilityP 2ν in the case of ∆m 2 31 < 0. As a consequence, a preferable detector for distinguishing the type of mass hierarchy would be the one with muon charge discrimination, such that neutrino interactions can be distinguished from those due to antineutrinos.

Results
It follows from the preceding analysis that in the case of detectors with muon charge identification, as observable which is most sensitive to the Earth matter effects, and thus to the value of sin 2 θ 13 and the sign of ∆m 2 31 , as well as to sin 2 θ 23 , we can consider the Nadir-angle (θ n ) distribution of the ratio N(µ − )/N(µ + ) of the multi-GeV µ − and µ + event rates, or equivalently the Nadir-angle distribution of the µ − − µ + event rate asymmetry We have obtained predictions for the cos θ n distribution of the ratio N(µ − )/N(µ + ) and the asymmetry A µ − µ + in the case of 3-neutrino oscillations of the atmospheric ν µ ,ν µ , ν e andν e , both for neutrino mass spectra with normal (∆m 2 31 > 0) and inverted (∆m 2 31 < 0) hierarchy, and for sin 2 θ 23 = 0.64; 0.50; 0.36, and sin 2 2θ 13 = 0.05; 0.10. These are compared with the predicted Nadir-angle distributions of the same ratio and asymmetry in the case of 2neutrino (sin 2 θ 13 = 0) vacuum ν µ → ν τ andν µ →ν τ oscillations of the atmospheric ν µ and ν µ , A 2ν µ − µ + . In the calculations we have used the predictions for the Nadir angle and energy distributions of the atmospheric neutrino fluxes given in ref. [58]. The interactions of the atmospheric neutrinos are described by taking into account only the ν µ andν µ deep inelastic scattering (DIS) cross sections. The latter are calculated using the GRV94 parton distributions given in ref. [62]. We present here results for the asymmetry A µ − µ + 6 . They are shown graphically in Figs. 1 -9. The figures correspond to three different intervals of integration over the energies of the atmospheric ν µ andν µ , and of the µ − and µ + they produce in the detector, E = [2,10], [2,20], [5,20] GeV, and 7 thus to three different possible event samples. Figures 1 -4 show the the asymmetries A µ − µ + and A 2ν µ − µ + as functions of cos θ n for two "reference" values of |∆m 2 31 |, ∆m 2 31 = ±2 × 10 −3 eV 2 and ∆m 2 31 = ±3 × 10 −3 eV 2 , while in 6 It is interesting to note that the ratio N (µ − )/N (µ + ) exhibits essentially the same dependence on cos θ n as the asymmetry A µ − µ + . This is a consequence of the special form of the dependence of A µ − µ + on N (µ − )/N (µ + ) and of the fact that typically one finds N (µ − )/N (µ + ) ∼ (1.5 − 2.4) for the ranges of the values of the parameters of interest. Correspondingly, the following approximate relation holds (within ∼ 20% and typically with much higher precision) for the range of values of the parameters of interest: The iron-magnetized calorimeter detectors allow to reconstruct with a certain precision the initial neutrino energy as well, see refs. [41,47,48].
Figs. 5 -9 we present results for the asymmetries in the rates of the multi-GeV µ − and µ + events, integrated over cos θ n in the intervals [0.30,0.84] (mantle bin) and [0.84,10] (core bin), A µ − µ + andĀ 2ν µ − µ + . The dependence of the latter on sin 2 2θ 13 for ∆m 2 31 = ±2 × 10 −3 eV 2 , and on ∆m 2 31 for sin 2 2θ 13 = 0.10, is shown for three values of sin 2 θ 23 = 0.36; 0.50; 0.64. As Figs. 1 -9 indicate, the Earth matter effects can produce a noticeable deviations of A µ − µ + from the 2-neutrino vacuum oscillation asymmetry A 2ν µ − µ + at cos θ n > ∼ 0.3. As a quantitative measure of the magnitude of the matter effects one can use the deviation of the asymmetry A µ − µ + in the case of 3-neutrino oscillations, sin 2 2θ 13 = 0, sin 2 2θ 13 > ∼ 0.04, from the asymmetry, A 2ν µ − µ + , predicted in the case of 2-neutrino oscillations, i.e., for sin 2 2θ 13 = 0, or the relative difference between the two asymmetries, The magnitude of the matter effects, or the relative difference ∆, depends critically on the value of sin 2 θ 23 : |∆| increases rapidly with the increasing of sin 2 θ 23 . This is clearly seen in Figs. 1 -9. The matter effects in A µ − µ + are hardly observable for sin 2 θ 23 < ∼ 0.30. For cos θ n ≤ 0.84, i.e., in the mantle bin, the asymmetry difference |∆| increases practically linearly with sin 2 2θ 13 . In the case of 0.84 ≤ cos θ n ≤ 1.0, i.e., in the core bin, |∆| increases rapidly with sin 2 2θ 13 until the latter reaches the value of sin 2 2θ 13 ∼ = 0.06. For values of sin 2 2θ 13 ∼ = (0.06 − 0.15), ∆ is essentially independent of sin 2 2θ 13 and is given by its value at sin 2 2θ 13 ∼ = 0.06 (Figs. 5 -9). The magnitude of the asymmetry difference ∆ depends weakly on ∆m 2 31 taking values in the interval ∼ (2 − 3) × 10 −3 eV 2 , as long as the energy integration interval is sufficiently wide to include the energy regions where the Earth matter effects enhance strongly the subdominant transition probabilities. If this is not the case, a noticeable dependence on ∆m 2 31 can be present. This is illustrated e.g, in Fig. 5, which corresponds to E = [2,10] GeV. The asymmetry difference in the mantle bin diminishes monotonically as |∆m 2 31 | increases starting from the value of ∼ 1.3 × 10 −3 eV 2 and becomes rather small at |∆m 2 31 | > ∼ 3 × 10 −3 eV 2 . This behavior can be easily understood: for |∆m 2 31 | ∼ = 2 × 10 −3 eV 2 , the region of enhancement of the subdominant neutrino oscillations lies in the region of energy integration, while for |∆m 2 31 | > 3 × 10 −3 eV 2 the enhancement region is practically outside the region of integration over the neutrino energy.
For the ranges considered of the three oscillation parameters, sin 2 θ 23 , sin 2 2θ 13 and |∆m 2 31 |, the magnitude of the asymmetry difference |∆| depends weakly on the maximal neutrino (and muon) energy, E max , for the chosen event sample as long as E max > ∼ 10 GeV. By increasing the minimal energy of the neutrinos contributing to the event sample, E min , from 2 GeV to, e.g., 5 GeV, one could diminish the asymmetry in the core bin substantially (Fig. 4). In that case, a large fraction of the region of enhancement is not included within the interval of integration.
Our results show that the Earth matter effects in the Nadir-angle distribution of the ratio N(µ − )/N(µ + ) of the rates of multi-GeV µ − and µ + events, or equivalently in the Nadir-angle distribution of the µ − − µ + event rate asymmetry A µ − µ + , eq. (24), can be sufficiently large to be observable in the current and planned experiments with iron magnetized calorimeter detectors which have muon charge identification capabilities (MINOS, INO, etc.).

Conclusions
We have studied the possibilities to obtain information on the values of sin 2 θ 13 and sin 2 θ 23 , and on the sign of ∆m 2 31 using the data on atmospheric neutrinos, which can be obtained in experiments with detectors able to measure the charge of the muon produced in the charged current (CC) reaction by atmospheric ν µ orν µ (MINOS, INO, etc). The indicated oscillation parameters control the magnitude of the Earth matter effects in the subdominant oscillations, ν µ → ν e (ν e → ν µ ) andν µ →ν e (ν e →ν µ ), of the multi-GeV (E ∼ (2 − 10) GeV) atmospheric neutrinos. As observable which is most sensitive to the Earth matter effects, and thus to the value of sin 2 θ 13 and and the sign of ∆m 2 31 , as well as to sin 2 θ 23 , we have considered the Nadir-angle (θ n ) distribution of the ratio N(µ − )/N(µ + ) of the multi-GeV µ − and µ + event rates, and the corresponding µ − − µ + event rate asymmetry A µ − µ + , eq. (24). The systematic uncertainty, in particular, in the Nadir angle dependence of N(µ − )/N(µ + ) and of the asymmetry A µ − µ + , can be smaller than those on the measured Nadir angle distributions of the rates of µ − and µ + events, N(µ − ) and N(µ + ). We have obtained predictions for the cos θ n distribution of the asymmetry A µ − µ + (and of the ratio N(µ − )/N(µ + )) in the case of 3-neutrino oscillations of the atmospheric ν µ ,ν µ , ν e andν e , both for neutrino mass spectra with normal (∆m 2 31 > 0) and inverted (∆m 2 31 < 0) hierarchy, and for sin 2 θ 23 = 0.64; 0.50; 0.36, and sin 2 2θ 13 = 0.05; 0.10. These are compared with the predicted Nadir-angle distribution of the same ratio and asymmetry in the case of 2-neutrino (sin 2 θ 13 = 0) vacuum ν µ → ν τ andν µ →ν τ oscillations of the atmospheric ν µ andν µ , A 2ν µ − µ + .