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We analytically calculate the neutrino conversion probability Pμe in the presence of sterile neutrinos, with exact dependence on $$ \Delta {m}_{41}^2 $$ ∆ m 41 2 and with matter effects explicitly included. Using perturbative expansion in small parameters, the terms involving the small mixing angles θ24 and θ34 can be separated out, with θ34 dependence only arising due to matter effects. We express Pμe 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 $$ \Delta {m}_{41}^2 $$ ∆ m 41 2 values. Our analytic expressions allow us to predict the effects of the sign of $$ \Delta {m}_{41}^2 $$ ∆ m 41 2 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 $$ \mid \Delta {m}_{41}^2\mid \sim \left({10}^{-4}-{10}^{-2}\right) $$ ∣ Δ m 41 2 ∣ ∼ 10 − 4 − 10 − 2 eV2, for either mass ordering of active neutrinos. The dependence of this sensitivity on the value of $$ \Delta {m}_{41}^2 $$ ∆ m 41 2 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.

We present compact analytic expressions for 3-flavor neutrino oscillation probabilities with invisible neutrino decay, where matter effects have been explicitly included. We take into account the possibility that the oscillation and decay components of the effective Hamiltonian do not commute. This is achieved by employing the techniques of inverse Baker-Campbell-Hausdorff (BCH) expansion and the Cayley-Hamilton theorem applied in the 3-flavor framework. If only the vacuum mass eigenstate ν3 decays, we show that the treatment of neutrino propagation may be reduced to an effective 2-flavor analysis in the One Mass Scale Dominance (OMSD) approximation. The oscillation probabilities for Pμμ, Pee, Peμ and Pμe — relevant for reactor, long baseline and atmospheric neutrino experiments — are obtained as perturbative expansions for the case of only ν3 decay, as well as for the more general scenario where all components of the decay matrix are non-zero. The analytic results thus obtained match the exact numerical results for constant density matter to a high precision and provide physical insights into possible effects of the decay of neutrinos as they propagate through Earth matter. We find that the effects of neutrino decay are most likely to be observable in Pμμ. We also point out that at any long baseline, the oscillation dips in Pμμ can show higher survival probabilities in the case with decay than without decay, and explain this feature using our analytic approximations.

We analyze invisible decay of neutrinos in the presence of oscillation and matter effects. The inclusion of decay can be accommodated by a non-Hermitian effective Hamiltonian, with the Hermitian component giving rise to oscillations, and the anti-Hermitian component leading to the invisible decay of neutrinos. We consider the possibility that the oscillation and decay matrix may not commute; in fact, in matter, they will invariably become non-commuting. This would lead to a mismatch between the effective mass eigenstates and the decay eigenstates. Employing a resummation of the Zassenhaus expansion, we develop a formalism for calculating the neutrino oscillation probabilities in the two-flavor scenario. This technique can easily be extended to three flavors.

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