Simple and Compact Expressions for Neutrino Oscillation Probabilities in Matter

We reformulate perturbation theory for neutrino oscillations in matter with an expansion parameter related to the ratio of the solar to the atmospheric $\Delta m^2$ scales. Unlike previous works, we use a renormalized basis in which certain first-order effects are taken into account in the zeroth-order Hamiltonian. We show that the new framework has an exceptional feature that leads to the neutrino oscillation probability in matter with the same structure as in vacuum to first order in the expansion parameter. It facilitates immediate physical interpretation of the formulas, and makes the expressions for the neutrino oscillation probabilities extremely simple and compact. We find, for example, that the $\nu_e$ disappearance probability at this order is of a simple two-flavor form with an appropriately identified mixing angle and $\Delta m^2$. More generally, all the oscillation probabilities can be written in the universal form with the channel-discrimination coefficient of $0,~\pm1$ or simple functions of $\theta_{23}$. Despite their simple forms they include all order effects of $\theta_{13}$ and all order effects of the matter potential, to first order in our expansion parameter.