Stars as Laboratories for Fundamental Physics - MPP Theory Group

142 Chapter 4
small. In this “long-wavelength limit” one is only interested in the
small-k structure functions
S ρ,σ (ω) ≡ lim
k→0
S ρ,σ (ω, k). (4.57)
It should be stressed that this quantity is not identical with S ρ,σ (ω, 0).
For example, in Eq. (4.47) for k = 0 the interference term does not
average to zero. The structure function becomes N 2 B/V and thus coherently
enhanced because the momentum transfer is so small that a
target consisting of many particles in a volume V cannot be resolved.
The limit k → 0 is understood such that |k| −1 remains much smaller
than the geometrical dimension V 1/3 of the system.
In the long-wavelength limit the normalization Eq. (4.50) and the
f-sum rule Eq. (4.56) yield for the spin-density structure function
∫ +∞
−∞
∫ +∞
−∞
∫
dω
∞
2π S dω
σ(ω) =
0 2π (1 + e−ω/T )S σ (ω) = 1 + 4
∫
dω
∞
2π ω S dω
σ(ω) =
0 2π ω (1 − e−ω/T )S σ (ω) = 4
⟨
∑ NB
3n B
i,j=1
i≠j
⟨
∑ NB
3n B
i,j=1
i≠j
⟩
σ i · σ j ,
3
V T
2 ij
⟩
.
(4.58)
These relations will be of great use to develop a general understanding
of the behavior of S σ (ω) at high densities. The second column of expressions
follows from the first by detailed balance. Because S σ (ω) ≥ 0 it is
evident that all of these expressions are always positive, independently
of details of the medium interactions.
In a noninteracting medium the operators ρ(t, k) and s(t, k) are
constant so that S ρ,σ (ω) = 2πδ(ω), allowing for scattering (zero energy
transfer), but not for the emission of radiation. This behavior is familiar
from a gas of free particles which can serve as targets for collisons, but
which cannot emit radiation because of energy-momentum constraints.
In an interacting medium the density correlator retains this property
because in the long-wavelength limit it depends on ρ(t, k→0) =
V −1 ∫ d 3 r ρ(t, r) which remains constant. Therefore, even in an interacting
medium one expects S ρ (ω) = 2πδ(ω), in agreement with the
finding that the neutrino vector current does not contribute to bremsstrahlung
in the nonrelativistic limit relative to the axial-vector current
(Friman and Maxwell 1979).
The relevant quantity for the latter is V −1 ∫ d 3 r s(t, r) = 1 ∑ NB
2 i=1 σ i
with σ i the individual nucleon spins. If the evolution of different spins