An Effective Two-Flavor Approximation for Neutrino Survival Probabilities in Matter

Abstract

It is known in vacuum that the three-flavor neutrino survival probability can be approximated by the effective two-flavor form to first orders in $ε\equiv Δm^2_{21} / Δm^2_{31}$, with introduction of the effective $Δm^2_{αα}$ ($α= e, μ, τ$), in regions of neutrino energy $E$ and baseline $L$ such that $Δm^2_{31} L / 2E \sim π$. Here, we investigate the question of whether the similar effective two-flavor approximation can be formulated for the survival probability in matter. Using a perturbative framework with the expansion parameters $ε$ and $s_{13} \propto \sqrtε$, we give an affirmative answer to this question and the resultant two-flavor form of the probability is valid to order $ε$. However, we observe a contrived feature of the effective $Δm^2_{αα} (a)$ in matter. It ceases to be a combination of the fundamental parameters and has energy dependence, which may be legitimate because it comes from the matter potential. But, it turned out that $Δm^2_{μμ} (a)$ becomes $L$-dependent, though $Δm^2_{ee} (a)$ is not, which casts doubt on adequacy of the concept of effective $Δm^2$ in matter. We also find that the appearance probability in vacuum admits, to order $ε$, the similar effective two-flavor form with a slightly different effective $Δm^2_{βα}$ from the disappearance channel. A general result is derived to describe suppression of the matter effect in the oscillation probability.