The Effective $\Delta m^2_{ee}$ in Matter

In this paper we generalize the concept of an effective $\Delta m^2_{ee}$ for $\nu_e/\bar{\nu}_e$ disappearance experiments, which has been extensively used by the short baseline reactor experiments, to include the effects of propagation through matter for longer baseline $\nu_e/\bar{\nu}_e$ disappearance experiments. This generalization is a trivial, linear combination of the neutrino mass squared eigenvalues in matter and thus is not a simple extension of the usually vacuum expression, although, as it must, it reduces to the correct expression in the vacuum limit. We also demonstrated that the effective $\Delta m^2_{ee}$ in matter is very useful conceptually and numerically for understanding the form of the neutrino mass squared eigenstates in matter and hence for calculating the matter oscillation probabilities. Finally we analytically estimate the precision of this two-flavor approach and numerically verify that it is precise at the sub-percent level.