Analytical Approximation of the Neutrino Oscillation Matter Effects at large $\theta_{13}$

We argue that the neutrino oscillation probabilities in matter are best understood by allowing the mixing angles and mass-squared differences in the standard parametrization to `run' with the matter effect parameter $a=2\sqrt{2}G_F N_e E$, where $N_e$ is the electron density in matter and $E$ is the neutrino energy. We present simple analytical approximations to these `running' parameters. We show that for the moderately large value of $\theta_{13}$, as discovered by the reactor experiments, the running of the mixing angle $\theta_{23}$ and the CP violating phase $\delta$ can be neglected. It simplifies the analysis of the resulting expressions for the oscillation probabilities considerably. Approaches which attempt to directly provide approximate analytical expressions for the oscillation probabilities in matter suffer in accuracy due to their reliance on expansion in $\theta_{13}$, or in simplicity when higher order terms in $\theta_{13}$ are included. We demonstrate the accuracy of our method by comparing it to the exact numerical result, as well as the direct approximations of Cervera et al., Akhmedov et al., Asano and Minakata, and Freund. We also discuss the utility of our approach in figuring out the required baseline lengths and neutrino energies for the oscillation probabilities to exhibit certain desirable features.