Stars as Laboratories for Fundamental Physics - MPP Theory Group

324 Chapter 9
Equation (9.22) and the corresponding result for ρ p were derived
by Sigl and Raffelt (1993) whose exposition I have closely followed. In
the nondegenerate limit where (1 − ρ p ) → 1 it agrees with a kinetic
equation of Dolgov (1981) and Barbieri and Dolgov (1991). Moreover,
a similar equation was derived by Rudzsky (1990) which can be shown
to be equivalent to Eq. (9.22) in the appropriate limits.
The relatively complicated collision term that follows from the neutrino-neutrino
Hamiltonian Eq. (9.16) has been worked out by Sigl and
Raffelt (1993). However, in a SN core the collisions of neutrinos with
each other are negligible relative to interactions with nucleons and electrons.
In the limit of a single neutrino flavor, or several unmixed flavors,
the role of ρ p is played by the usual occupation numbers f p while the
matrix G is unity, or the unit matrix. Then Eq. (9.22) is
∫
f˙
p,coll =
dp ′ [ W P ′ ,P f p ′(1 − f p ) − W P,P ′ f p (1 − f p ′)
+ W −P ′ ,P (1 − f p )(1 − f p ′) − W P,−P ′ f p f p ′]
(9.27)
which is the usual Boltzmann collision integral. The main difference
to Eq. (9.22) is the appearance there of “nonabelian Pauli blocking
factors” which involve noncommuting matrices of neutrino occupation
numbers and coupling constants.
9.3.4 Recovering Stodolsky’s Formula
The damping of neutrino oscillations becomes particularly obvious in
the limit where a typical energy transfer ∆ 0 in a neutrino-medium interaction
is small relative to the neutrino energies themselves. This
would be the case for “heavy” and thus nonrelativistic background
fermions. Then pair processes may be ignored and neutrinos change
their direction of motion in a collision, but not the magnitude of their
momentum, which also implies W (P, P ′ ) ≈ W (P ′ , P ). If the neutrino
ensemble is isotropic one then has ρ p = ρ p ′ under the integral
in Eq. (9.22). For the matrix structure of the collision term this leaves
2Gρ p G − GGρ p − ρ p GG = −[G, [G, ρ p ]] which puts the nature of the
collision term as a double commutator in evidence—see also Eq. (9.13).
One may define a total scattering rate for nondegenerate neutrinos of
momentum p by virtue of
∫
Γ p = dp ′ W (P ′ , P ) . (9.28)