Question of hierarchy: Matter effects with atmospheric neutrinos and antineutrinos

It is by now established that neutrinos mix, have (different) nonzero masses, and therefore oscillate. The oscillation parameters themselves, however, are not all well-known. An open problem is that of the neutrino mass hierarchy. We study the possibility of determining the neutrino mass hierarchy with atmospheric neutrinos using an iron calorimeter detector capable of charge identification such as the proposed monolith and ical/ino detectors. We find that such detectors are sensitive to the sign of the mass-squared difference, ${\delta }_{32}=m_3^2-m_2^2$, provided the as-yet unknown mixing angle between the first and third generations, ${\theta }_{13}$, is greater than $6°$ (${sin}^{2}2{\theta }_{13}>0.04$). A result with a significance greater than $90\%$ CL requires large exposures (more than 500 kton-years) as well as good energy and angular resolution of the detected muons (better than $15\%$), especially for small ${\theta }_{13}$. Hence obtaining definitive results with such a detector is difficult, unless ${\theta }_{13}$ turns out to be large. In contrast, such detectors can establish a clear oscillation pattern in atmospheric neutrinos in about 150 kton-years, therefore determining the absolute value of ${\delta }_{32}$ and ${sin}^{2}2{\theta }_{23}$ to within $10\%$.