Stars as Laboratories for Fundamental Physics - MPP Theory Group

Oscillations of Trapped Neutrinos 331
of freedom, and one may use the weak-damping limit where neutrino
oscillations are much faster than their rates of collision or absorption.
Then one finds for two flavors (Raffelt and Sigl 1993)
˙ f x p = (1 − f x p) P x P − f x p A x P
+ 1 4 s2 p[
(2 − f
x
p − f e p)(P e P − P x P ) − (f x p + f e p)(A e P − A x P ) ] ,
(9.48)
where f e p and f x p are the occupation numbers for ν e and ν x as in
Sect. 9.3.5. The corresponding equation for ν e is found by exchanging
e ↔ x everywhere.
For ν x = ν µ flavor conversion can build up a nonvanishing muon density
in a SN core because they are light enough to be produced initially
when the electron chemical potential is on the order of 200−300 MeV.
For ν x = ν τ or some sterile flavor, the direct production or absorption
is not possible, A x P = P x P = 0. This simplifies Eq. (9.48) considerably,
˙ f x p = 1 4 s2 p[
(2 − f
x
p − f e p) P e P − (f x p + f e p) A e P
]
. (9.49)
If neither ν e nor ν τ are occupied because, for example, the medium
is transparent to neutrinos so that they escape after production, one
has f ˙ p x = 1 2 s2 pPP
e so that the production rate of ν x is that of ν e times
1
2 sin2 2θ p as one would have expected.
In a SN core where normal neutrinos are trapped Eq. (9.49) is
more complicated as backreaction and Pauli blocking effects must be
included. It becomes simple again if s 2 p ≪ 1, because then the production
of ν x causes only a small perturbation of β equilibrium. Therefore,
one may use the detailed-balance condition (1 − fp)P e P e − fpA e e P = 0.
Inserting this into Eq. (9.49) leads to
˙ f x p = 1 4 s2 p[
(1 − f
x
p ) P e P − f x p A e P
]
, (9.50)
so that now the ν x follow a Boltzmann collision equation with rates
of gain and loss given by those of ν e times 1 4 sin2 2θ p . If the ν x are
sterile they escape without building up so that their production rate is
1
4 sin2 2θ p that of ν e .