Stars as Laboratories for Fundamental Physics - MPP Theory Group

322 Chapter 9
neutrinos are important only in young SN cores where one may ignore
antineutrinos. With the total neutrino matrix of densities
∫
ρ ≡ dp ρ p , (9.20)
and with G S = 1 in the standard model, the neutrino contribution
to the refractive energy shift is Ω S p = √ 2 G F ρ. The trace term was
dropped because it does not contribute to the commutator in the equation
of motion. The diagonal entries of ρ are the neutrino densities.
However, in the presence of mixing and oscillations ρ also has offdiagonal
elements, i.e. there are “off-diagonal refractive indices” as first
realized by Pantaleone (1992b).
In a SN core the complete first-order equation of motion for ρ p is
then
i ˙ρ p = [ Ω 0 p, ρ p
]
+
√
2 GF
[
(Nl + ρ), ρ p
]
, (9.21)
which is intrinsically nonlinear. Interestingly, if one integrates both
sides over dp one obtains an equation for ρ which is linear as the neutrino
term drops out from the commutator. Therefore, even though
individual modes of the neutrino field oscillate differently in the presence
of other neutrinos, the instantaneous rate of change of the overall
flavor polarization is as if they were absent.
In a SN core the refractive effects are dominated by nonneutrino
particles, notably by electrons. However, above the neutrino sphere
the flow of neutrinos itself represents a particle density exceeding that
of the background medium (Sect. 11.4). Also, the medium of the early
universe is dominated by neutrinos so that self-interactions and the
corresponding nonlinearities of the neutrino flavor oscillations must be
carefully included. For recent studies of primordial neutrino oscillations
see Samuel (1993), Kostolecký, Pantaleone, and Samuel (1993), and
references to the earlier literature given there.
9.3.3 Kinetic Terms
The refractive term (first order) of Eq. (9.13) is just a sum over different
medium components, whereas the collision term (second order) in general
contains interference terms between different target species. However,
if they are uncorrelated, corresponding to ⟨B a µ Bb ν ⟩ = ⟨B a µ ⟩⟨Bb ν ⟩
for a ≠ b, these interference terms only contribute to second-order
forward-scattering effects which are neglected. The collision term is
then an incoherent sum over all target species so that in the following
one may suppress the subscript a for simplicity.