Earth Matter Effects at Very Long Baselines and the Neutrino Mass Hierarchy

We study matter effects which arise in the muon neutrino oscillation and survival probabilities relevant to atmospheric neutrino and very long baseline beam experiments. The inter-relations between the three probabilities P_{\mu e}, P_{\mu \tau} and P_{\mu \mu} are examined. It is shown that large and observable sensitivity to the neutrino mass hierarchy can be present in P_{\mu \mu} and P_{\mu \tau}. We emphasize that at baselines of > 7000 Km, matter effects in P_{\mu \tau} can be large under certain conditions. The muon survival rates in experiments with very long baselines thus depend on matter effects in both P_{\mu \tau} and P_{\mu e}. We indicate where these effects are sensitive to \theta_{13}, and identify ranges of E and L where the event rates increase with decreasing \theta_{13}, providing a handle to probe small \theta_{13}. The effect of parameter degeneracies in the three probabilities at these baselines and energies is studied in detail. Realistic event rate calculations are performed for a charge discriminating 100 kT iron calorimeter which demonstrate the possibility of realising the goal of determining the neutrino mass hierarchy using atmospheric neutrinos. It is shown that a careful selection of energy and baseline ranges is necessary in order to obtain a statistically significant signal, and that the effects are largest in bins where matter effects in both P_{\mu e} and P_{\mu \tau} combine constructively. Under these conditions, upto a 4\sigma signal for matter effects is possible (for \Delta_{31}>0) within a timescale appreciably shorter than the one anticipated for neutrino factories.