Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 215
Fig. 6.5. Dispersion relation for transverse modes according to Eq. (6.44).
Fig. 6.6. Dispersion relation for longitudinal modes according to Eq. (6.44).
Second, for v ∗ < 1 there is always a wave number k 1 where ω(k)
“crosses the light cone” (ω/k = 1),
k 2 1 = 4α π
= ω 2 P
∫ ∞
3
0
v 2 ∗
p 2 [ ( )
1 1 + v
dp f p
E v log 1 − v
[ ( )
1 1 + v∗
log − 1
2v ∗ 1 − v ∗
]
− 1
]
. (6.48)
The second identity (Braaten and Segel 1993) applies at the same level
of approximation as π T,L in Eq. (6.38). Some analytic limiting cases