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Neutrino mass hierarchy and matter effects

Matter effects modify the mixing and the effective masses of neutrinos in a way which depends on the neutrino mass hierarchy. Consequently, for normal and inverted hierarchies the oscillations and flavor conversion results are different. Sensitivity to the mass hierarchy appears whenever the matter effects on the 1-3 mixing and mass splitting become substantial. This happens in supernovae in wide energy range and in the matter of the Earth. The Earth density profile is a multi-layer medium where the resonance and parametric enhancements of oscillations occur. The enhancement is realized in neutrino (antineutrino) channels for the normal (inverted) mass hierarchy. Multi-megaton scale under ice (water) atmospheric neutrino detectors with low energy threshold can establish mass hierarchy with $(3 - 10) σ$ confidence level in few years. The main challenges of these experiments are discussed and various ways to improve sensitivity are outlined. In particular, inelasticity measurements will allow to increase significance of the hierarchy identification by $20 - 50 \%$ .