Matter effects in four-neutrino mixing

The evidence in favor of neutrino oscillations implies the existence of at least three independent neutrino mass-squared differences. If the Liquid Scintillation Neutrino Detector (LSND) results are confirmed, transitions into active and sterile neutrinos can take place simultaneously for both solar and atmospheric neutrinos. In this paper we present a formalism for the calculation of the transition probabilities into active and sterile neutrinos for solar ${\nu }_e\mathrm{’}\mathrm{s}$ and atmospheric ${\nu }_{\mu }\mathrm{’}\mathrm{s},$ taking into account the matter effects in the Sun and in the Earth. We find that the solar neutrino transition probabilities depend on just one single additional parameter compared to that of the standard two-generation analysis, while for atmospheric neutrinos two additional mixing angles are necessary to analyze the data in addition to those of the usual two-generation analysis.