Stars as Laboratories for Fundamental Physics - MPP Theory Group

Oscillations of Trapped Neutrinos 325
In the present limit this yields a collision “integral”
˙ρ p,coll = − 1 2 Γ p [G, [G, ρ p ]] , (9.29)
where terms nonlinear in ρ p have disappeared even though the neutrinos
may still be degenerate.
Eq. (9.29) is more transparent in the case of two-flavor mixing where
one may write ρ p = 1 2 f p(1 + P p · σ) and G = 1 2 (g 0 + G · σ). The
total occupation number f p is conserved while the polarization vector
is damped according to
Ṗ p,coll = − 1 2 Γ p G × (G × P p ). (9.30)
The r.h.s. is a vector transverse to G, allowing one to write
Ṗ p,coll = − 1 2 Γ p |G| 2 P p,T . (9.31)
Thus one naturally recovers Stodolsky’s damping term Eq. (9.1) with
D = 1Γ 2 p|G| 2 . For ν e and ν µ and if one writes G = diag(g νe , g νµ ) in
the weak interaction basis D = 1Γ 2 p(g νe − g νµ ) 2 . This representation
reflects that the damping of neutrino oscillations depends on the difference
of the scattering amplitudes: D is the square of the amplitude
difference, not the difference of the squares. If one flavor does not
scatter at all, D is half the scattering rate of the active flavor.
Collisions thus lead to chemical equilibrium as discussed in the introduction
to this chapter and as shown in Fig. 9.1. However, the
simple exponential damping represented by Stodolsky’s formula can be
reproduced only in the limit of vanishing energy transfers in collisions,
an assumption which amounts to separating the neutrino momentum
degrees of freedom from the flavor ones. In a more general case the evolution
is more complicated. In particular, collisions usually lead to a
transient flavor polarization in an originally unpolarized ensemble if the
momentum degrees of freedom were out of equilibrium. Still, the neutrinos
always move toward kinetic and chemical equilibrium under the
action of the collision integral Eq. (9.22) in the sense that the properly
defined free energy never increases (Sigl and Raffelt 1993).
9.3.5 Weak-Damping Limit
Even for two-flavor mixing the general form of the collision integral
Eq. (9.22) remains rather complicated. However, for the conditions
of a SN core one may apply two approximations which significantly
simplify the problem. First, one is mostly concerned with the evolution