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.