Stars as Laboratories for Fundamental Physics - MPP Theory Group

312 Chapter 9
A certain “damping” of flavor oscillations occurs even without collisions
when the neutrinos are not monochromatic because then different
modes oscillate with different frequencies. This “dephasing” effect was
shown in Fig. 8.4 and is repeated in Fig. 9.1 (dashed line). While the
dephasing washes out the oscillation pattern, it does not lead to flavor
equilibrium: the probability for ν e ends at a constant of 1 − 1 2 sin2 2θ.
For the rest of this chapter “damping of oscillations” never refers to
this relatively trivial dephasing effect.
The interplay of collisions and oscillations leads to flavor equilibrium
between mixed neutrinos. In a SN core the concentration of electron
lepton number is initially large so that the ν e form a degenerate Fermi
sea. The other flavors ν µ and ν τ are characterized by a thermal distribution
at zero chemical potential. However, if they mix with ν e they
will achieve the same large chemical potential. In a SN core heat and
lepton number are transported mostly by neutrinos; the efficiency of
these processes depends crucially on the degree of neutrino degeneracy
for each flavor. Therefore, it is of great interest to determine the time
it takes a non-ν e flavor to equilibrate with ν e under the assumption
of mixing (Maalampi and Peltoniemi 1991; Turner 1992; Pantaleone
1992a; Mukhopadhyaya and Gandhi 1992; Raffelt and Sigl 1993).
Moreover, if ν e mixes with a sterile neutrino species, conversion into
this inert state leads to the loss of energy and lepton number from the
inner core of a SN. The observed SN 1987A neutrino signal may thus
be used to constrain the allowed range of masses and mixing angles
(Kainulainen, Maalampi, and Peltoniemi 1991; Raffelt and Sigl 1993;
see also Shi and Sigl 1994).
These applications are discussed in Sects. 9.5 and 9.6 below. A
simple estimate of the rate of flavor conversion and the emission rate
of sterile neutrinos from a SN core requires not much beyond the approximate
rate 1 2 sin2 2θ Γ. However, a proper kinetic treatment of the
evolution of a neutrino ensemble under the simultaneous action of oscillations
and collisions is an interesting theoretical problem in its own
right. Notably, it is far from obvious how to treat degenerate neutrinos
in a SN core because the different flavors will suffer different Pauli
blocking factors. Does this effect break the coherence between mixed
flavors in neutral-current collisions
The bulk of this chapter is devoted to the derivation and discussion
of a general kinetic equation for mixed neutrinos (Dolgov 1981;
Rudzsky 1990; Raffelt, Sigl, and Stodolsky 1993; Sigl and Raffelt 1993).
This equation provides a sound conceptual and quantitative framework
for dealing with various aspects of coherent and incoherent neutrino