Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 213
Fig. 6.3. Contours for v ∗ and γ = ω P /T as defined in Eqs. (6.39) and (6.40)
where Y e is the number of electrons per baryon.
Next, with Eq. (6.38) one must solve the transcendental equations
π T,L (ω, k) = ω 2 − k 2 which are explicitly
ω 2 − k 2
[
= ωP
2 1 +
1
2 G(v2 ∗k 2 /ω 2 ) ] Transverse,
ω 2 − v 2 ∗k 2 = ω 2 P
[
1 − G(v
2
∗ k 2 /ω 2 ) ] Longitudinal. (6.44)
In the classical limit this is to lowest order in T/m e
ω 2 = k 2 + ω 2 P
ω 2 = ω 2 P
(
(
1 + 3 k2
ω 2
1 + k2
ω 2
)
T
m e
)
T
m e
Transverse,
Longitudinal. (6.45)
For small temperatures the longitudinal modes oscillate with an almost
fixed frequency, independently of momentum, while the transverse
modes behave almost like massive particles (Fig. 6.4).
The general result Eq. (6.38) and the behavior of the function G(x)
reveal that for transverse excitations ω 2 − k 2 can vary only between ω 2 P
and 3 2 ω2 P. Also, ω 2 − k 2 > 0 so that K 2 is always time-like. For k ≫ ω P
the transverse dispersion relation approaches that of a massive particle
with a fixed mass m T , the “transverse photon mass”. With Eq. (6.38)
and because k/ω → 1 for k ≫ ω P one finds m 2 T = ω 2 P[1 + 1 2 G(v2 ∗)], or