Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 149
and may even decrease at large densities, although such large values for
Γ σ may never be reached in a nuclear medium as will become clear below.
The high-density downturn of the axion emission rate can be interpreted
in terms of the Landau-Pomeranchuk-Migdal effect (Landau
and Pomeranchuk 1953a,b; Feinberg and Pomeranchuk 1956; Migdal
1956) as pointed out by Raffelt and Seckel (1991). The main idea is
that collisions interrupt the radiation process. The formation of a radiation
quantum of frequency ω takes about a time ω −1 according to the
uncertainty principle and so if collisions are more frequent than this
time, the radiation process is suppressed. Classically, this effect was
demonstrated in the language of current correlators in Sect. 4.6.5.
The impact of the high-density behavior of S σ (ω) on the neutralcurrent
neutrino opacity is crudely estimated by the inverse mean free
path given in Eq. (4.35), averaged over a thermal energy spectrum of
the initial neutrino. Moreover, all expressions become much simpler if
one replaces the Fermi-Dirac occupation numbers with the Maxwell-
Boltzmann expression e −ωi/T ; the resulting error is small for nondegenerate
neutrinos. The relevant quantity is then
⟨ ⟩ ∫ ∞ ∫ ∞
λ
−1
∝ dω 1 dω 2 ω1 2 ω2 2 e −ω1/T S σ (ω 1 − ω 2 ). (4.74)
0
0
One integral can be done explicitly, leaving one with an integral over
the energy transfer alone. With Eq. (4.72) for the structure function
one finds
⟨ ⟩ ∫ ∞
λ
−1
∝ dω Γ σ (T 2 + T ω/2 + ω 2 /12) e −ω/T
. (4.75)
0
ω 2 + Γ 2 /4
This expression is constant for Γ σ ≪ T where Γ = Γ σ and thus the
structure function is essentially 2πδ(ω). Indeed, the average scattering
cross section (or mean free path) is not expected to depend on the
density.
For dense media (Γ σ ≫ T ), however, the broadening of S σ (ω) beyond
a delta function leads to a decreasing average scattering rate. This
means that at a fixed temperature the medium becomes more transparent
to neutrinos with increasing density, even without the impact
of degeneracy effects. This behavior is shown in Fig. 4.9 in analogy to
the axion emission rate Fig. 4.8.
A decreasing cross section is intuitively understood if one recalls
that the nucleon spin is typically flipped in a collision with other nucleons
because the interaction potential couples to the spin. Neutrino scattering
with an energy transfer ω implies that properties of the medium