Impact of approximate oscillation probabilities in the analysis of three neutrino experiments

As neutrino oscillation data becomes ever more precise, the use of approximate formulae for the oscillation probabilities ${\mathcal P}_{\alpha\beta}$ must be examined to ensure that the approximation is adequate. Here, the oscillation probability ${\mathcal P}_{ee}$ is investigated in the context of the Daya Bay experiment; the oscillation probability ${\mathcal P}_{\mu\mu}$ is investigated in terms of the T2K disappearance experiment; and the probability ${\mathcal P}_{\mu e}$ is investigated in terms of the T2K appearance experiment. Daya Bay requires ${\mathcal P}_{ee}$ in vacuum and thus the simple analytic formula negates the need for an approximate formula. However, improved data from T2K will soon become sensitive to the hierarchy, and thus require a more careful treatment of that aspect. For the other cases, we choose an expansion by Akhmedov et al. which systematically includes all terms through second order in $\sin\theta_{13}$ and in $\alpha =: \Delta_{21}/\Delta_{31}$ ($\Delta_{jk} =: m^2_j - m^2_k$). For the T2K disappearance experiment the approximation is quite accurate. However, for the T2K appearance experiment the approximate formula is not precise enough for addressing such questions as hierarchy or the existence of CP violation in the lepton sector. We suggest the use of numerical calculations of the oscillation probabilities, which are stable, accurate, and efficient and eliminate the possibility that differences in analyses are emanating from different approximations.