Eigenvalues: the Rosetta Stone for Neutrino Oscillations in Matter

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.