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Motivated by tremendous progress in neutrino oscillation experiments, wederive a new set of simple and compact formulas for three-flavor neutrinooscillation probabilities in matter of a constant density. A useful definitionof the $\eta$-gauge neutrino mass-squared difference $\Delta^{}_* \equiv \eta\Delta^{}_{31} + (1-\eta) \Delta^{}_{32}$ is introduced, where $\Delta^{}_{ji}\equiv m^2_j - m^2_i$ for $ji = 21, 31, 32$ are the ordinary neutrinomass-squared differences and $0 \leq \eta \leq 1$ is a real and positiveparameter. Expanding neutrino oscillation probabilities in terms of $\alpha\equiv \Delta^{}_{21}/\Delta^{}_*$, we demonstrate that the analytical formulascan be remarkably simplified for $\eta = \cos^2 \theta^{}_{12}$, with$\theta_{12}^{}$ being the solar mixing angle. As a by-product, the mappingfrom neutrino oscillation parameters in vacuum to their counterparts in matteris obtained at the order of ${\cal O}(\alpha^2)$. Finally, we show that ourapproximate formulas are not only valid for an arbitrary neutrino energy andany baseline length, but also still maintaining a high level of accuracy.

Looking into Analytical Approximations for Three-flavor Neutrino Oscillation Probabilities in Matter