# Neutrino oscillation probabilities through the looking glass

## TL;DR

This paper reviews various methods for calculating neutrino oscillation probabilities in matter, assessing their accuracy and efficiency to guide future experimental and phenomenological analyses.

## Contribution

It provides a comparative analysis of exact and approximate oscillation probability expressions, identifying those suitable for next-generation experiments.

## Key findings

- Many existing expressions lack sufficient precision for upcoming experiments.
- Several approximate formulas balance accuracy and simplicity effectively.
- Guidance is provided for choosing appropriate oscillation expressions in analysis tools.

## Abstract

In this paper we review different expansions for neutrino oscillation probabilities in matter in the context of long-baseline neutrino experiments. We examine the accuracy and computational efficiency of different exact and approximate expressions. We find that many of the expressions used in the literature are not precise enough for the next generation of long-baseline experiments, but several of them are while maintaining comparable simplicity. The results of this paper can be used as guidance to both phenomenologists and experimentalists when implementing the various oscillation expressions into their analysis tools.