Expansions of neutrino oscillation and decay probabilities in matter

Abstract: We consider a simple model for invisible neutrino decay as a sub-leading effect in the standard three-flavor neutrino oscillation framework, and use the Cayley–Hamilton formalism to obtain a full set of neutrino oscillation probabilities in matter. These are given as analytical series expansions in the small parameters α ∼ O(λ^2) and s_13 ∼ O(λ), where λ ≡ 0.2 is a “book-keeping parameter” denoting the order of the expansion. We produce explicit formulas for P_eµ, P_eτ , P_µµ, P_µτ , and P_ττ to order O(λ^3), and for P_ee to order O(λ^2), all having first corrections of order O(λ^4). Moreover, we also present vacuum limits of our expressions, as well as discuss the effect of decay on unitarity. We show that all rows in the unitarity table have corrections of order O(λ^2), with the second and third rows having additional corrections of order O(1). In the limit of no decay, unitarity is restored, and we furthermore recover known results for all probabilities.