Why Is The Neutrino Oscillation Formula Expanded In $\Delta m_{21}^{2}/\Delta m_{31}^{2}$ Still Accurate Near The Solar Resonance In Matter?

Abstract

The conventional approximate formula for neutrino oscillation in matter which is obtained from the expansion in terms of the ratio of mass square differences $\alpha=\Delta m_{21}^{2}/\Delta m_{31}^{2}\approx0.03$, first proposed by Cervera, et al and Freund, turns out to be an accurate formula for accelerator neutrino experiments. Originally it required the neutrino energy to be well above the solar resonance to validate the expansion but it is found to be still very accurate when the formula is extrapolated to the resonance, which is practically important for the T2K experiment. This paper shows that the accuracy is guaranteed by cancellations of branch cut singularities and also, for the first time, analytically computes the actual error of the formula. The actual error implies that the original requirement can be safely removed in current experiments.Comment: 22 pages,7 figures. Some materials are removed for simplicity. Accepted by JHE