Looking into Analytical Approximations for Three-flavor Neutrino Oscillation Probabilities in Matter

Understand

Motivated by tremendous progress in neutrino oscillation experiments, we derive a new set of simple and compact formulas for three-flavor neutrino oscillation probabilities in matter of a constant density.

• A useful definition of the $\eta$-gauge neutrino mass-squared difference $\Delta^{}_* \equiv \eta \Delta^{}_{31} + (1-\eta) \Delta^{}_{32}$ is introduced, where $\Delta^{}_{ji} \equiv m^2_j - m^2_i$ for $ji = 21, 31, 32$ are the ordinary neutrino mass-squared differences and $0 \leq \eta \leq 1$ is a real and positive parameter.
• Expanding neutrino oscillation probabilities in terms of $\alpha \equiv \Delta^{}_{21}/\Delta^{}_*$, we demonstrate that the analytical formulas can be remarkably simplified for $\eta = \cos^2 \theta^{}_{12}$, with $\theta_{12}^{}$ being the solar mixing angle.
• As a by-product, the mapping from neutrino oscillation parameters in vacuum to their counterparts in matter is obtained at the order of ${\cal O}(\alpha^2)$.