Sterile Neutrinos: Propagation in Matter and Sensitivity to Sterile Mass Ordering

We analytically calculate the neutrino conversion probability \(P_{\mu e}\) in the presence of sterile neutrinos, with exact dependence on \(\Delta m^2_{41}\) and with matter effects explicitly included. Using perturbative expansion in small parameters, the terms involving the small mixing angles \(\theta_{24}\) and \(\theta_{34}\) can be separated out, with \(\theta_{34}\) dependence only arising due to matter effects. We express \(P_{\mu e}\) in terms of the quantities of the form \(\sin(x)/x\), which helps in elucidating its dependence on matter effects and a wide range of \(\Delta m^2_{41}\) values. Our analytic expressions allow us to predict the effects of the sign of \(\Delta m^2_{41}\) at a long baseline experiment like DUNE. We numerically calculate the sensitivity of DUNE to the sterile mass ordering and find that this sensitivity can be significant in the range \(|\Delta m^2_{41}| \sim (10^{-4} - 10^{-2})\) eV\(^2\), for either mass ordering of active neutrinos. The dependence of this sensitivity on the value of \(\Delta m^2_{41}\) for all mass ordering combinations can be explained by investigating the resonance-like terms appearing due to the interplay between the sterile sector and matter effects.