The Cervera et al. formula, the best known approximate formula of neutrino oscillation probability for long-baseline experiments, can be regarded as a second-order perturbative formula with small expansion parameter ǫ ≡ ∆m 2 21 /∆m 2 31 ≃ 0.03 under the assumption s 13 ≃ ǫ. If θ 13 is large, as suggested by a candidate ν e event at T2K as well as the recent global analyses, higher order corrections of s 13 to the formula would be needed for better accuracy. We compute the corrections systematically by formulating a perturbative framework by taking θ 13 as s 13 ∼ √ ǫ ≃ 0.18, which guarantees its validity in a wide range of θ 13 below the Chooz limit. We show on general ground that the correction terms must be of order ǫ 2 . Yet, they nicely fill the mismatch between the approximate and the exact formulas at low energies and relatively long baselines. General theorems are derived which serve for better understanding of δ-dependence of the oscillation probability. Some interesting implications of the large θ 13 hypothesis are discussed.Typeset by REVT E X One of the most important progresses in particle physics in the last decades is the discovery of neutrino masses [1] and the lepton flavor mixing [2]. It was done through observing neutrino oscillation phenomena and it constitutes, up until this moment, only available experimental method for measuring lepton mixing parameters. ∆m 2 32 and θ 23 are determined by atmospheric neutrino observation by Super-Kamiokande [3][4][5], and then by accelerator neutrino experiments [6,7]. ∆m 2 21 and θ 12 are measured independently by two types of experiments, the KamLAND reactor experiment [8-10] and the solar neutrino observation using various experimental techniques. For the latest results and for a review of the solar neutrino experiments see e.g., [11,12] and [13], respectively. The remaining mixing angle θ 13 is being explored by the ongoing and the upcoming accelerator [14,15] and reactor neutrino experiments [16][17][18]. If it turned out that θ 13 is not too small, we may proceed to measure CP violation by the lepton Kobayashi-Maskawa (KM) [19] phase δ ℓ , to which we refer just δ in this paper.It is expected that precision measurement is required to determine δ because CP violation effect is tiny due to suppression by the two small factors, ∆m 2 21 /∆m 2 31 and the Jarlskog coefficient J ≡ c 12 s 12 c 23 s 23 c 2 13 s 13 [20]. Therefore, understanding of full complexity of neutrino oscillation phenomena would be of some help e.g., to design future experiments. An example of such is the parameter degeneracy [21-23], the problem of multiple copy of the solutions of mixing parameters allowed by given sufficient but limited numbers of experimental data. See [24] for a comprehensive overview of this phenomenon. To facilitate understanding of qualitative features of the neutrino oscillation, it is crucially important to have analytic formula, albeit approximate, for the oscillation probability. For relatively short baseline experiments, such as low-energy superbeam [25][26][...
