Neutrino Oscillations with Two $Δm^2$ Scales

An approximation that is often used in fits to reactor and atmospheric neutrino data and in some studies of future neutrino oscillation experiments is to assume one dominant scale, $\Delta m^2$, of neutrino mass squared differences, in particular, $\Delta m^2_{atm} \sim 3 \times 10^{-3}$ eV$^2$. Here we investigate the corrections to this approximation arising from the quantity $\Delta m^2_{sol}$ relevant for solar neutrino oscillations, assuming the large mixing angle solution. We show that for values of $\sin^2(2\theta_{13}) \sim 10^{-2}$ (in the range of interest for long-baseline neutrino oscillation experiments with either intense conventional neutrino beams such as JHF-SuperK or a possible future neutrino factory) and for $\Delta m^2_{sol} \sim 10^{-4}$ eV$^2$, the contributions to $\nu_\mu \to\nu_e$ oscillations from both CP-conserving and CP-violating terms involving $\sin^2(\Delta m^2_{sol}L/(4E))$ can be comparable to the terms involving $\sin^2(\Delta m^2_{atm}L/(4E))$ retained in the one-$\Delta m^2$ approximation. Accordingly, we emphasize the importance of performing a full three-flavor, two-$\Delta m^2$ analysis of the data on $\nu_\mu \to \nu_e$ oscillations in a conventional-beam experiment and $\nu_e \to \nu_\mu$, $\bar\nu_e \to \bar\nu_\mu$ oscillations at a neutrino factory. We also discuss a generalized analysis method for the KamLAND reactor experiment, and note how the information from this experiment can be used to facilitate the analysis of the subsequent data on $\nu_\mu \to \nu_e$ oscillations. Finally, we consider the analysis of atmospheric neutrino data and present calculations of matter effects in a three-flavor, two-$\Delta m^2$ framework relevant to this data and to neutrino factory measurements.