Mass hierarchy determination via future atmospheric neutrino detectors

We study the problem of determination of the sign of $\Delta m_{31}^2$, or the neutrino mass hierarchy, through observations of atmospheric neutrinos in future detectors. We consider two proposed detector types: (a) Megaton sized water C̆erenkov detectors, which can measure the event rates of ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ and ${\nu }_e+{\overline{\nu }}_e$ and (b) 100 kton sized magnetized iron detectors, which can measure the event rates of ${\nu }_{\mu }$ and ${\overline{\nu }}_{\mu }$. For energies and path lengths relevant to atmospheric neutrinos, these rates obtain significant matter contributions from $P_{\mu e}$, $P_{\mu \mu }$ and $P_{ee}$, leading to an appreciable sensitivity to the hierarchy. We do a binned ${\chi }^2$ analysis of simulated data in these two types of detectors which includes the effect of smearing in neutrino energy and direction and incorporates detector efficiencies and relevant statistical, theoretical and systematic errors. We also marginalize the ${\chi }^2$ over the allowed ranges of neutrino parameters in order to accurately account for their uncertainties. Finally, we compare the performance of both types of detectors vis a vis the hierarchy determination.