Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 135
However, what exactly does one mean by a vanishing energy transfer
The nucleons in the ambient medium constantly scatter with each
other so that their individual energies are uncertain to within about
1/τ coll with a typical time between collision τ coll . Therefore, one would
expect that the structure of S σ (ω), which was calculated on the basis
of free nucleons which interact only once, is smeared out over scales
of order ∆ω ≈ τcoll −1 . In particular, this smearing-out effect naturally
regulates the low-ω behavior of the structure function (Raffelt and
Seckel 1991).
Because Γ σ ≈ τcoll −1 is a typical nucleon-nucleon collision rate, a simple
ansatz for a modified S σ (ω) is a Lorentzian (Raffelt and Seckel 1991)
S σ (|ω|) →
Γ σ
ω 2 + Γ 2 σ/4 s(ω/T ) = 1 T
γ σ
s(x), (4.36)
x 2 + γσ/4 2
with x = ω/T . For γ σ ≪ 1 one has ∫ +∞
−∞ dω S σ(ω) = 2π + O(γ σ ) for the
modified S σ (ω). Thus, for γ σ ≪ 1 essentially S σ (ω) = 2πδ(ω) so that
the total “inelastic” scattering rate Eq. (4.35) reproduces the elastic
scattering one. At the same time S σ (ω) has wings which, for ω ≫ Γ σ ,
give the correct inelastic scattering rate which is of order γ σ .
This simple ansatz can be expected to give a reasonable approximation
to the true S σ (ω) only in the limit γ σ ≪ 1. For practical
applications in cooling calculations of young SN cores, however, one
has to confront the opposite limit γ σ ≫ 1 causing the smearing-out effect
by multiple nucleon collisions to be a dominating feature of S σ (ω)
even for ω ≫ T . This implies that for typical thermal energies of
neutrinos and nucleons a reasonably clean separation between elastic
and inelastic scattering processes is not logically possible—there is only
one structure function S σ (ω), broadly smeared out, which governs all
axial-vector scattering, emission, and absorption processes.
This observation has important ramifications not only for neutrino
scattering, but also for the bremsstrahlung emission of axions and neutrino
pairs from a nuclear medium. Earlier it seemed that one did not
have to worry about details of the behavior of S σ (ω) near ω = 0 because
the low-ω part was suppressed by axion or neutrino phase-space
factors. However, since the notion of “low energy” presently means
ω < ∼ Γ σ , and because T ≪ Γ σ , all relevant energy transfers are low in
this sense, and even the emission processes are dominated by multiplescattering
effects. Consequences of this behavior are explored in more
detail below after a formal introduction of the structure functions and
their general properties.