Probing the neutrino mass hierarchy and the 13-mixing with supernovae

We consider in detail the effects of the 13-mixing (sin² θ₁₃) and of the type of mass hierarchy/ordering (sign[Δm₁₃²]) on neutrino signals from the gravitational collapses of stars. The observables (characteristics of the energy spectra of v_e and v̄_e events) sensitive to sin ² θ₁₃ and sign[Δ₁₃²] have been calculated. They include the ratio of average energies of the spectra, r _E ≡ 〈E〉 /〈Ē〉, the ratio of widths of the energy distributions, r_Γ ≡ Γ/Γ̄, the ratios of total numbers of v_e and v̄_e events at low energies, S, and in the high energy tails, R_tail. We construct and analyse scatter plots which show the predictions for the observables for different intervals of sin² θ₁₃ and signs of Δm₁₃², taking into account uncertainties in the original neutrino spectra, the star density profile, etc. Regions in the space of observables r_E, r_Γ, S, R_tail exist in which certain mass hierarchy and intervals of sin² θ₁₃ can be identified or discriminated. We elaborate on the method of the high energy tails in the spectra of events. The conditions are formulated for which sin² θ₁₃ can be (i) measured, (ii) restricted from below, (iii) restricted from above. We comment on the possibility to determine sin² θ₁₃ using the time dependence of the signals due to the propagation of the shock wave through the resonance layers of the star. We show that the appearance of the delayed Earth matter effect in one of the channels (v_e or v̄_e) in combination with the undelayed effect in the other channel will allow us to identify the shock wave appeareance and determine the mass hierarchy.