Stars as Laboratories for Fundamental Physics - MPP Theory Group

318 Chapter 9
Heisenberg’s equation with H = H 0 +H int . Taking an expectation value
with regard to the initial state yields
˙ρ p (t) = −i [ Ω 0 p, ρ p (t) ] + i ⟨[ H int (B(t), Ψ(t)), ˆρ p (t) ]⟩ , (9.10)
and an analogous equation for ρ p (t). These equations are exact, but
they are not a closed set of differential equations for the ρ p and ρ p . To
this end one needs to perform a perturbative expansion.
To first order one may set the interacting fields B(t) and Ψ(t) on the
r.h.s. of Eq. (9.10) equal to the free fields 50 B 0 (t) and Ψ 0 (t). Under the
assumption that the original state contained no correlations between
the neutrinos and the background the expectation value factorizes into
a medium part and a neutrino part. With Wick’s theorem and ignoring
fast-varying terms such as b † b † it can be reduced to an expression which
contains only ρ p ’s and ρ p ’s. The result gives the forward-scattering or
refractive effect of the interaction.
To include nonforward collisions one needs to go to second order
in the perturbation expansion. At a given time t a general operator
ξ(t) = ξ(B(t), Ψ(t)) which is a functional of B and Ψ is to first order
∫ t
ξ(t) = ξ 0 (t) + i dt [ ′ Hint(t 0 − t ′ ), ξ 0 (t) ] , (9.11)
0
where ξ 0 and Hint 0 are functionals of the freely evolving fields B 0 (t)
and Ψ 0 (t). Applying this general iteration formula to the operator
ξ = [H int (B, Ψ), ˆρ p ] which appears on the r.h.s. of Eq. (9.10) one
arrives at
˙ρ p (t) = −i [ Ω 0 p, ρ p (t) ] + i ⟨[ ]⟩
Hint(t), 0 ˆρ 0 p
−
∫ t
0
dt ′ ⟨[ H 0 int(t − t ′ ), [ H 0 int(t), ˆρ 0 p]]⟩
, (9.12)
and similar for ρ p (t). The second term on the r.h.s. is the first-order
refractive part associated with forward scattering. The second-order
term contains both forward- as well as nonforward-scattering effects.
50 These free operators are the solutions of the equations of motion in the absence
of H int . However, internal interactions of the medium such as nucleon-nucleon
scattering are not excluded. Moreover, Ψ(0) = Ψ 0 (0) etc. are taken as initial
conditions for the interacting fields. Also, the mass term is ignored in the definition
of Ψ 0 ; its effect is included only in the first term on the r.h.s. of Eq. (9.10), the
“vacuum oscillation term.” Therefore, the free creation and annihilation operators
vary as a 0 j (p, t) = a j(p, 0)e −ipt etc. for all flavors with p = |p|. This implies that
the operators ˆρ 0 (p) and ˆρ 0 (p), which are constructed from the free a’s and b’s, are
time independent.