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It is known in vacuum that the three-flavor neutrino survival probability canbe 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 neutrinoenergy $E$ and baseline $L$ such that $\Delta m^2_{31} L / 2E \sim \pi$. Here,we investigate the question of whether the similar effective two-flavorapproximation can be formulated for the survival probability in matter. Using aperturbative framework with the expansion parameters $\epsilon$ and $s_{13}\propto \sqrt{\epsilon}$, we give an affirmative answer to this question andthe 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 fundamentalparameters and has energy dependence, which may be legitimate because it comesfrom 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 doubton adequacy of the concept of effective $\Delta m^2$ in matter. We also findthat the appearance probability in vacuum admits, to order $\epsilon$, thesimilar effective two-flavor form with a slightly different effective $\Deltam^2_{\beta \alpha}$ from the disappearance channel. A general result is derivedto describe suppression of the matter effect in the oscillation probability.

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