Stars as Laboratories for Fundamental Physics - MPP Theory Group

Supernova Neutrinos 435
A “normal” mass hierarchy prevents a level crossing among antineutrinos.
Still, if the large-angle or the vacuum solution to the solar
neutrino problem obtain, the mixing angle would be so large that
the spectral swapping could still be uncomfortably large. Smirnov,
Spergel, and Bahcall (1994) have considered in detail the probability
for swapping the ν e with the ν µ or ν τ spectrum. For vacuum oscillations,
relevant for small ∆m 2 ν, the fractional exchange of the spectra
is p = 1 2 sin2 2θ (vacuum mixing angle θ) with a maximum of p = 0.5,
i.e. the observed spectrum could be as much as an equal mixture of the
primary ones. In general, medium refractive effects must be included,
although for a large range of ∆m 2 one may still take p to be independent
of energy. Contours for p in the sin 2 2θ-∆m 2 ν-plane are shown in
Fig. 11.18. In the shaded area (A) the Earth effect is important so that
the amount of conversion is energy dependent and differs between the
Fig. 11.18. Contours for the “swap fraction” p between the ν e and the ν µ
or ν τ fluxes from the protoneutron star cooling phase. The matter effect of
the stellar envelope and of the Earth are included. In (A) the Earth effect is
important; in this area p is an average over neutrino energies while otherwise
it does not depend on the energy. In (B) the exact contours depend on the
detailed matter distribution of the stellar envelope. Black areas indicate
the approximate mixing parameters which would explain the solar neutrino
problem. (Adapted from Smirnov, Spergel, and Bahcall 1994.)