The minimum width condition for neutrino conversion in matter

We find that for small vacuum mixing angle θ and low energies (s<<M2Z) the width of matter, d1/2, needed to have conversion probability P≥1/2 should be larger than dmin=π/(2√2 GFtan2θ: d1/2≥dmin. Here GF is the Fermi constant, s is the total energy squared in the center of mass and MZ is the mass of the Z boson. The absolute minimum d1/2=dmin is realized for oscillations in a uniform medium with resonance density. For realistic density distributions (monotonically varying density, castle wall profile, etc.) the required width d1/2 is larger than dmin. The width dmin depends on s, and for Z-resonance channels at s∼M2Z we get that dmin(s) is 20 times smaller than the low energy value. We apply the minimum width condition, d≥dmin, to high energy neutrinos in matter as well as in neutrino background. Using this condition, we conclude that the matter effect is negligible for neutrinos propagating in AGN and GRBs environments. Significant conversion can be expected for neutrinos crossing dark matter halos of clusters of galaxies and for neutrinos produced by cosmologically distant sources and propagating in the universe