# Eigenvalues: the Rosetta Stone for Neutrino Oscillations in Matter

## TL;DR

This paper introduces a new exact method for calculating neutrino oscillation probabilities in matter using eigenvalues, simplifying the determination of mixing angles and CP phase, and improves the accuracy of perturbative eigenvalues.

## Contribution

The paper presents a novel exact approach leveraging eigenvalues for neutrino oscillations in matter and demonstrates enhanced accuracy of perturbative eigenvalues.

## Key findings

- Eigenvalues enable simple calculation of mixing angles in matter.
- Perturbative eigenvalues converge five orders of magnitude faster than expected.
- Updated speed versus accuracy plot for oscillation probabilities.

## Abstract

We present a new method of exactly calculating neutrino oscillation probabilities in matter. We leverage the "eigenvector-eigenvalue identity" to show that, given the eigenvalues, all mixing angles in matter follow surprisingly simply. The CP violating phase in matter can then be determined from the Toshev identity. Then, to avoid the cumbersome expressions for the exact eigenvalues, we have applied previously derived perturbative, approximate eigenvalues to this scheme and discovered them to be even more precise than previously realized. We also find that these eigenvalues converge at a rate of five orders of magnitude per perturbative order which is the square of the previously realized expectation. Finally, we provide an updated speed versus accuracy plot for oscillation probabilities in matter, to include the methods of this paper.