Neutrino Oscillations in Matter using the Adjugate of the Hamiltonian

TL;DR

This paper introduces a simple, general method using the adjugate matrix to compute eigenvectors of neutrino Hamiltonians in matter, enabling easier analysis of oscillation probabilities across multiple scenarios.

Contribution

It presents a novel approach employing the adjugate matrix for eigenvector calculation, simplifying neutrino oscillation analysis in complex matter potentials.

Findings

Derived matter potential invariant quantities, including a generalized Naumov-Harrison-Scott identity.

The method is applicable to any Hamiltonian, regardless of matter potential complexity.

Identified how non-standard matter effects influence these invariants.

Abstract

We revisit neutrino oscillations in constant matter density for a number of different scenarios: three flavors with the standard Wolfenstein matter potential, four flavors with standard matter potential and three flavors with non-standard matter potentials. To calculate the oscillation probabilities for these scenarios one must determine the eigenvalues and eigenvectors of the Hamiltonians. We use a method for calculating the eigenvalues that is well known, determination of the zeros of determinant of matrix $(\lambda I -H)$, where H is the Hamiltonian, I the identity matrix and $\lambda$ is a scalar. To calculate the associated eigenvectors we use a method that is little known in the particle physics community, the calculation of the adjugate (transpose of the cofactor matrix) of the same matrix, $(\lambda I -H)$. This method can be applied to any Hamiltonian, but provides a very simple way to determine the eigenvectors for neutrino oscillation in matter, independent of the complexity of the matter potential. This method can be trivially automated using the Faddeev-LeVerrier algorithm for numerical calculations. For the above scenarios we derive a number of quantities that are invariant of the matter potential, many are new such as the generalization of the Naumov-Harrison-Scott identity for four or more flavors of neutrinos. We also show how these matter potential independent quantities become matter potential dependent when off-diagonal non-standard matter effects are included.