We construct a new perturbative framework to describe neutrino oscillation in matter with the unique expansion parameter epsilon, which is defined as Delta m^2_{21} / Delta m^2_{ren} with the renormalized atmospheric Delta m^2_{ren} equiv Delta m^2_{31} - s^2_{12} Delta m^2_{21}. It allows us to derive the maximally compact expressions of the oscillation probabilities in matter to order epsilon in the form akin to those in vacuum. This feature allows immediate physical interpretation of the formulas, and facilitates understanding of physics of neutrino oscillations in matter. Moreover, quite recently, we have shown that our three-flavor oscillation probabilities P( u_alpha rightarrow u_beta) in all channels can be expressed in the form of universal functions of L/E. The u_e disappearance oscillation probability P( u_e rightarrow u_e) has a special property that it can be written as the two-flavor form which depends on the single frequency. This talk is based on the collaborating work with Stephen Parke [1].
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