Precise analytical description of the Earth matter effect on oscillations of low energy neutrinos

We present a formalism for the matter effects in the Earth on low energy neutrino fluxes which is both accurate and has all the advantages of a full analytic treatment. The oscillation probabilities are calculated up to the second order term in $ϵ(x)\equiv 2V(x)E/\Delta m^2$, where $V(x)$ is the neutrino potential at position $x$. We show the absence of large undamped phases which makes the expansion in $ϵ$ well behaved. An improved expansion is presented in terms of the variation of $V(x)$ around a suitable mean value which allows one to treat energies up to those relevant for supernova neutrinos. We discuss also the case of three-neutrino mixing.