Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 147
Equation (4.71) also highlights an important weakness of the quantum
result: It does not include the interference effect from multiple
collisions at |ω| ∼ < Γ σ because the calculation was done assuming individual,
isolated collisions. The normalization condition Eq. (4.58) can
then be satisfied only by accepting a pathological infrared behavior of
S σ (ω) like the one suggested by Sawyer (1995).
The classical and the quantum results thus both violate fundamental
requirements. The quantum calculation applies for |ω| ∼ > Γ σ while the
classical one for |ω| ∼ < T . If Γ σ ≪ T (“dilute medium”) the regimes of
validity overlap for Γ σ ∼ < |ω| ∼ < T and the results agree beautifully. In
this case the two calculations mutually confirm and complement each
other. The compound structure function, where the quantum result is
multiplied with ω 2 /(ω 2 +Γ 2 σ/4), fulfills the detailed-balance requirement
and approximately the normalization condition. Therefore, it probably
is a good first guess for the overall shape of S σ (ω).
4.6.7 High-Density Behavior
In a SN core one is typically in the limit Γ σ ≫ T so that the regime
of overlapping validity Γ < σ ∼ |ω| < ∼ T between a classical and a perturbative
quantum calculation no longer exists. Rather, one is in the
opposite situation where for T < ∼ |ω| < ∼ Γ σ neither approach appears
directly justified. Because the structure function S σ (ω) determines all
axial-vector interaction rates in the long-wavelength limit, it is rather
unclear what their high-density behavior might be.
Still, one has important general information about S σ (ω). The detailed-balance
condition S σ (ω) = S σ (−ω) e ω/T reveals that it is enough
to specify S σ (ω) for positive energy transfers (energy given to the
medium). In the classical limit S σ (|ω|) = Γ σ /ω 2 , while the f-sum rule
informs us that the quantum version must fall off somewhat faster with
large |ω|. Finally, if spin-spin correlations can be neglected, the normalization
condition ∫ ∞
0 dω (1 + e −ω/T ) S σ (ω) = 2π obtains.
In order to illustrate the overall impact of the high-density behavior
on axion or neutrino pair emission rates and on neutrino scattering
rates, it is enough to take the classical limiting case for large |ω|, even
though it does not have an integrable f-sum. Thus a simple ansatz is
Γ σ
S σ (|ω|) =
ω 2 + Γ 2 /4 , (4.72)
where Γ is to be determined by the normalization condition. In the
dilute limit (Γ σ ≪ T ) this implies Γ ≈ Γ σ while in the dense limit
(Γ σ ≫ T ) one finds Γ ≈ Γ σ /2.