Analytic approximations for three neutrino oscillation parameters and probabilities in matter

The corrections to neutrino mixing parameters in the presence of matter of constant density are calculated systematically as series expansions in terms of the mass hierarchy $\Delta m_{21}^2/\Delta m_{31}^2.$ The parameter mapping obtained is then used to find simple, but nevertheless accurate formulas for oscillation probabilities in matter including $\mathrm{CP}$ effects. Expressions with one to one correspondence to the vacuum case are derived, which are valid for neutrino energies above the solar resonance energy. Two applications are given to show that these results are a useful and powerful tool for analytical studies of neutrino beams passing through the Earth mantle or core: First, the “disentanglement problem” of matter and $\mathrm{CP}$ effects in $\mathrm{CP}$ asymmetry is discussed and second, estimations of the statistical sensitivity to the $\mathrm{CP}$ terms of the oscillation probabilities in neutrino factory experiments are presented.