Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 211
In a higher-order calculation one has to include the proper e ± dispersion
relations which imply that electromagnetic excitations are never
damped by γ → e + e − decay (Braaten 1991), in contrast with statements
found in the previous literature. Dropping the (K 2 ) 2 term in the
denominator of Eq. (6.36) prevents Π from developing an imaginary
part from this decay, even with the vacuum e ± dispersion relations.
Therefore, the “approximate” integral actually provides a better representation
of the O(α) dispersion relations than the “exact” one.
With the approximation K 2 = 0 in the denominator of Eq. (6.36)
the angular integral is trivial. 32 With Eq. (6.35) one finds
π L = 4α ω 2 − k 2 ∫ ∞ p 2 [ ( )
ω ω + kv
dp f
π k 2
p
0 E kv log − ω2 − k 2 ]
ω − kv ω 2 − k 2 v − 1 ,
2
π T = 4α π
ω 2 − k 2 ∫ ∞ p 2 [
dp f
k 2
p
0 E
ω 2
ω 2 − k 2 −
ω ( ω + kv
2kv log ω − kv
)]
, (6.37)
where v = p/E is the e ± velocity and f p represents the sum of their
phase-space distributions.
The remaining integration can be done analytically in the classical,
degenerate, and relativistic limits where one finds
π L = ωP[ 2 1 − G(v
2
∗ k 2 /ω 2 ) ] + v∗k 2 2 − k 2 ,
[
π T = ωP
2 1 +
1
2 G(v2 ∗k 2 /ω 2 ) ] . (6.38)
Here, v ∗ is a “typical” electron velocity defined by
v ∗ ≡ ω 1 /ω P . (6.39)
The plasma frequency ω P and the frequency ω 1 are
ω 2 P ≡ 4α π
∫ ∞
0
∫ ∞
dp f p p (v − 1 3 v3 ),
ω1 2 ≡ 4α dp f p p ( 5 3
π
v3 − v 5 ). (6.40)
0
The function G (Fig. 6.2) is defined by
G(x) = 3 [
1 − 2x x 3 − 1 − x ( √ )] 1 + x
2 √ x log 1 − √ x
= 6
∞∑
n=1
x n
(2n + 1)(2n + 3) . (6.41)
Note that G(0) = 0, G(1) = 1, and G ′ (1) = ∞.
32 In the degenerate limit, Jancovici (1962) has calculated analytically the full
integral without the K 2 = 0 approximation.