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

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We derive predictions for the Nadir angle (thetaₙ) 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, Amu-mu+=[Nmu- - Nmu+]/[Nmu- + Nmu+], in iron-magnetized calorimeter detectors (MINOS, INO) in the case of 3-neutrino oscillations of the atmospheric nuₘu and antinuₘu, driven by one neutrino mass squared difference, |Delta m²₃₁| >> Delta m²₂₁. The asymmetry Amu- 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 sin²(theta13) and sin²(theta23), theta13 and theta23 being the neutrino mixing angles limited by the CHOOZ and Palo Verde experiments and that responsible for the dominant atmospheric nuₘu -> nuₜau (antinuₘu -> antinuₜau) oscillations. It is also very sensitive to the type of neutrino mass spectrum which can be with normal (Delta m²₃₁>0) or with inverted (Delta m²₃₁ < 0) hierarchy. We find that for sin²(theta23)>0.50, sin²(2 theta13)>0.06 and Delta m²₃₁=(2-3) 10⁻³ eV², the Earth matter effects produce a relative difference between the integrated asymmetries barAmu- mu+ and barA2numu- mu+ in the mantle (cos(thetaₙ)=0.30-0.84) and core (cos(thetaₙ)=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 m²₃₁. An observation of the Earth matter effects in the Nadir angle distribution of the asymmetry Amu- mu+ (ratio Nmu-/Nmu+) would clearly indicate that sin²(2 theta13)>0.06 and sin²(theta23)>0.50, and would lead to the determination of the sign of Delta m²₃₁.