Advances in perturbative thermal field theory - Ultra-relativistic ...

414 U Kraemmer and A Rebhan
k 2 = 0in B and to define
∫
−1
∞
t =− A =
k0 2 − k2 − ˆ A (k 0 , |k|) = dk
0
′
−∞ 2π
ˆρ t (k
0 ′ , |k|)
k
0 ′ − k , (A.1)
0
l =− k2
k 2 B =
∫
−1
∞
k 2 + ˆ B (k 0 , |k|) = −∞
dk ′ 0
2π
ˆρ l (k ′ 0 , |k|)
k ′ 0 − k 0
− 1 k 2 ,
(A.2)
where ˆ A,B are the HTL quantities given in (5.15) and (5.16). The spectral functions are
given by
ˆρ t,l (k 0 , |k|) = Disc t,l (k 0 , |k|) = 2 lim Im t,l (k 0 +iɛ, |k|)
ɛ→0
= 2πε(k 0 )z t,l (|k|)δ(k0 2 − ω2 t,l (|k|)) + β t,l(k 0 , |k|)θ(−k 2 ) (A.3)
with ωt,l 2 (|k|) as shown in figure 9 (for k2 > 0). For small and large values of k 2 , they are
approximated by [217]
ωt 2 ≃ ωpl 2 + 6 5 k2 , ωl 2 ≃ ω2 pl + 3 5 k2 , k 2 ≪ ωpl 2 , (A.4)
( )
ωt 2 ≃ k 2 + m 2 ∞ , ω2 l ≃ k2 +4k 2 exp − k2
− 2 , k 2 ≫ m 2
m 2 ∞ , (A.5)
∞
where ωpl 2 = ˆm 2 D /3 is the plasma frequency common to both modes and m2 ∞ = ˆm2 D /2 is the
asymptotic mass of transverse quasi-particles. The effective thermal mass of mode l (or B)
vanishes exponentially for large k 2 .
The residues z t,l are defined by
[ ] ∣ ∂
∣∣∣
z −1
t,l = ∂k0
2 (− t,l ) −1 (A.6)
−1
t,l =0
and explicitly read
2k 2 ∣
0
z t =
k2 ∣∣∣k0 2k 2 0
ˆm 2 D k2 0 − , z (k2 ) 2 l =
k2
=ω t (|k|) k 2 ( ˆm 2 D − k2 )
∣ (A.7)
k0 =ω l (|k|)
with the following asymptotic limits [218]:
( )
z t ≃ 1 − 4k2
5ωpl
2 , z l ≃ ω2 pl
k 2 1 − 3 k 2
10 ωpl
2 , k 2 ≪ ωpl 2 , (A.8)
(
z t ≃ 1 − m2 ∞
2k 2 ln 4k2 − 2
m 2 ∞
)
, z l ≃ 4k2 exp
m 2 ∞
(
− k2
− 2
m 2 ∞
)
, k 2 ≫ m 2 ∞ . (A.9)
The singular behaviour of z l for k 2 → 0 is in fact only due to the factor k 2 /k 2 in (A.2);
the residue in B approaches 1 in this limit. For k 2 ≫ m 2 ∞ , the residue in l vanishes
exponentially, as mentioned in section 5.1.2.
The Landau-damping functions β t,l are given by
β t (k 0 , |k|) = π ˆm 2 k 0 (−k 2 )
D |
2|k| 3 t (k 0 , |k|)| 2 , β l (k 0 , |k|) = π ˆm 2 k 0
D
|k| | l(k 0 , |k|)| 2 .
(A.10)
These are odd functions in k 0 that vanish at k 0 = 0 and at k0 2 = k2 . For large k 2 and fixed ratio
k 0 /|k|, β t,l decay like 1/k 4 .
The spectral functions ρ t,l satisfy certain sum rules that can be obtained by a Taylor
expansion of (A.1) and (A.2) in k 0 [307, 480, 10]. A special, particularly important case is