Atmospheric Neutrino Oscillations, θ13 and Neutrino Mass Hierarchy

We derive predictions for the Nadir angle (θn) dependence of the ratio Nμ/Ne of the rates of the μ−like and e−like multi-GeV events measured in water-Čerenkov detectors in the case of 3-neutrino oscillations of the atmospheric νe (ν̄e) and νμ (ν̄μ), driven by one neutrino mass squared difference, |∆m31| ∼ (2.5− 3.0)× 10−3 eV2 ≫ ∆m21. This ratio is particularly sensitive to the Earth matter effects in the atmospheric neutrino oscillations, and thus to the values of sin θ13 and sin 2 θ23, θ13 and θ23 being the neutrino mixing angle limited by the CHOOZ and Palo Verde experiments and that responsible for the dominant atmospheric νμ → ντ (ν̄μ → ν̄τ ) oscillations. It is also sensitive to the type of neutrino mass spectrum which can be with normal (∆m31 > 0) or with inverted (∆m31 < 0) hierarchy. We show that for sin 2 θ13 >∼ 0.01, sin θ23 >∼ 0.5 and at cos θn >∼ 0.4, the Earth matter effects modify substantially the θn− dependence of the ratio Nμ/Ne and in a way which cannot be reproduced with sin 2 θ13 = 0 and a different value of sin θ23. For normal hierarchy the effects can be as large as ∼ 25% for cos θn ∼ (0.5 − 0.8), can reach ∼ 35% in the Earth core bin cos θn ∼ (0.84 − 1.0), and might be observable. They are typically by ∼ 10% smaller in the inverted hierarchy case. An observation of the Earth matter effects in the Nadir angle distribution of the ratio Nμ/Ne would clearly indicate that sin 2 θ13 >∼ 0.01 and sin θ23 >∼ 0.50. Also at: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria.