Matter effects in long-baseline experiments, the flavor content of the heaviest (or lightest) neutrino, and the sign ofΔm2

The neutrinos of long-baseline beams travel inside the Earth’s crust where the density is $\rho \simeq 2.8{\mathrm{g}\mathrm{}\mathrm{cm}}^{-3}.$ If electron neutrinos participate in the oscillations, matter effects will modify the oscillation probabilities with respect to the vacuum case. Depending on the sign of $\Delta m^2,$ a Mikheyev-Smirnov-Wolfenstein resonance will exist for neutrinos or antineutrinos with energy $E_{\nu }^{\mathrm{res}}\simeq 4.7\times |\Delta m^2|{/(10\mathrm{}}^{-3}{\mathrm{eV}}^2)\mathrm{GeV}.$ For $\Delta m^2,$ in the interval indicated by the Super-Kamiokande experiment, this energy range is important for the proposed long-baseline experiments. For positive $\Delta m^2$ the most important effects of matter are a 9% (25%) enhancement of the transition probability $P({\nu }_{\mu }\to {\nu }_e)$ for the KEK to Kamioka (Fermilab to Minos and CERN to Gran Sasso) beam(s) in the energy region where the probability has its first maximum, and an approximately equal suppression of $P({\overline{\nu }}_{\mu }\to {\overline{\nu }}_e).\mathrm{}$ For negative $\Delta m^2$ the effects for neutrinos and antineutrinos are interchanged. The fractional enhancement (or the suppression) of $P({\nu }_{\mu }\to {\nu }_e)$ to a good approximation is independent from the mixing parameters and is detectable for $|U_{e3\mathrm{}}|$ sufficiently large. Producing beams of neutrinos and antineutrinos and measuring the oscillation probabilities for both the ${\nu }_{\mu }\to {\nu }_e$ and ${\overline{\nu }}_{\mu }\to {\overline{\nu }}_e$ transitions can solve the sign ambiguity in the determination of $\Delta m^2.$