# Simple and Precise Factorization of the Jarlskog Invariant for Neutrino Oscillations in Matter

## TL;DR

This paper presents a precise factorization of the Jarlskog invariant in neutrino oscillations in matter, revealing a simple product of vacuum and matter resonance factors with minimal corrections, enhancing understanding of CP violation effects.

## Contribution

The paper introduces a novel, highly accurate factorization of the Jarlskog invariant in matter, simplifying the analysis of CP violation in neutrino oscillations.

## Key findings

- Factorization of the Jarlskog invariant into three parts with 0.04% accuracy
- The inverse Jarlskog squared is a fourth order polynomial in matter potential
- The factorized form simplifies the understanding of matter effects in neutrino oscillations

## Abstract

For neutrino propagation in matter, we show that the Jarlskog invariant, which controls the size of true CP violation in neutrino oscillation appearance experiments, factorizes into three pieces: the vacuum Jarlskog invariant times two simple two-flavor matter resonance factors that control the matter effects for the solar and atmospheric resonances independently. If the solar effective matter potential and the atmospheric effective $\Delta m^2$ are chosen carefully for these two resonance factors, then the fractional corrections to this factorization are an impressive 0.04\% or smaller. We also show that the inverse of the square of the Jarlskog in matter ($1/\hat{J}^2$) is a fourth order polynomial in the matter potential which guarantees that it can be factored into two quadratics which immediately implies the functional form of our approximate, factorized expression.