Decoherence and oscillations of supernova neutrinos

Supernova neutrinos have several exceptional features which can lead to interesting physical consequences. At the production point their wave packets have an extremely small size $\sigma_x \sim 10^{-11}$ cm; hence the energy uncertainty can be as large as the energy itself, $\sigma_E \sim E$, and the coherence length is short. On the way to the Earth the wave packets of mass eigenstates spread to macroscopic sizes and separate. Inside the Earth the mass eigenstates split into eigenstates in matter and oscillate again. The coherence length in the Earth is comparable with the radius of the Earth. We explore these features and their consequences. (i) We present new estimates of the wave packet size. (ii) We consider the decoherence condition for the case of wave packets with spatial spread and show that it is not modified by the spread. (iii) We study the coherence of neutrinos propagating in a multi-layer medium with density jumps at the borders of layers. In this case coherence can be partially restored due to a "catch-up effect", increasing the coherence length beyond the usual estimate. This catch-up effect can occur for supernova neutrinos as they cross the shock wave fronts in the exploding star or the core of the Earth.