Stars as Laboratories for Fundamental Physics - MPP Theory Group

150 Chapter 4
Fig. 4.9. Schematic variation of the neutrino scattering rate per nucleon
(axial-vector interaction only) with Γ σ /T , taking Eq. (4.72) for the spindensity
structure function. The dashed line is the naive rate without the
inclusion of multiple-scattering effects, i.e. it is based on S σ (ω) = 2πδ(ω).
that fluctuate on faster time scales cannot be resolved. Therefore, if
a nucleon spin flips many times within the time scale ω −1 , the probe
“sees” a vanishing average spin and the scattering rate is reduced accordingly.
For the vector current, the relevant zeroth component does not fluctuate
so that this contribution to neutrino scattering is not suppressed
(Sect. 4.6.4). This observation is intimately tied to the absence of
bremsstrahlung emission of neutrino pairs by the vector current. Both
of these effects are summarized in the statement that the vector-current
structure function S ρ (ω) is always given as 23 2πδ(ω), allowing for scattering
(energy transfer ω = 0), but not for bremsstrahlung.
23 This statement is based on the assumption that there are no spatial correlations
among the nucleons—possibly a poor approximation in a dense medium. Moreover,
collective oscillations may occur so that it can be too simplistic to treat the medium
as consisting of essentially free, individual nucleons (Iwamoto and Pethick 1982).
Calculations of the long-wavelength structure factor by Sawyer (1988, 1989) in a
specific nucleon interaction model showed a substantial suppression of S ρ (0) and
thus, of the neutrino mfp. Other related works are those of Haensel and Jerzak
(1987) and of Horowitz and Wehrberger (1991a,b) who calculated the dynamic
structure functions within certain interaction models. Unfortunately, these works
do not shed much light on the high-density behavior of the quantities which are of
prime interest to this book such as the axion emission rate from a hot SN core.