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

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 $\epsilon \equiv \Delta m^2_{21} / \Delta m^2_{31}$, with introduction of the effective $\Delta m^2_{\alpha \alpha}$ ($\alpha = e, \mu, \tau$), in regions of neutrino energy $E$ and baseline $L$ such that $\Delta m^2_{31} L / 2E \sim \pi$. 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 $\epsilon$ and $s_{13} \propto \sqrt{\epsilon}$, we give an affirmative answer to this question and the resultant two-flavor form of the probability is valid to order $\epsilon$. However, we observe a contrived feature of the effective $\Delta m^2_{\alpha \alpha} (a)$ 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 $\Delta m^2_{\mu \mu} (a)$ becomes $L$-dependent, though $\Delta m^2_{ee} (a)$ is not, which casts doubt on adequacy of the concept of effective $\Delta m^2$ in matter. We also find that the appearance probability in vacuum admits, to order $\epsilon$, the similar effective two-flavor form with a slightly different effective $\Delta m^2_{\beta \alpha}$ from the disappearance channel. A general result is derived to describe suppression of the matter effect in the oscillation probability.