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

TL;DR

This paper demonstrates that the approximate neutrino oscillation formula expanded in the ratio of mass squared differences remains accurate near the solar resonance in matter, due to cancellations of singularities, impacting current experimental analyses.

Contribution

It analytically explains why the expansion remains valid near the solar resonance and calculates the actual error, removing previous energy constraints for experiments.

Findings

The formula is accurate near the solar resonance due to singularity cancellations.

The paper provides the first analytical computation of the formula's error.

The validity of the approximation extends to energies close to the solar resonance.

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.