Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 217
as a resonance with a finite width Γ L . The approximate uncertainty
±Γ L (k) of the energy ω(k) is shown in Fig. 6.4 as a shaded area.
The damping rate Eq. (6.50) does not show a threshold effect at
k = ω because it was calculated nonrelativistically so that the highenergy
tail of the electron distribution contains particles with velocities
exceeding the speed of light. Tsytovich (1961) calculated a relativistic
result with the correct threshold behavior. However, because the
damping rate is exceedingly small for k ≪ k D , the correction is very
small if ω P ≪ k D , equivalent to T ≪ m e . For the degenerate case,
explicit expressions for the imaginary parts of π T,L (ω, k) were derived
by Altherr, Petitgirard, and del Río Gaztelurrutia (1993).
The general expressions Eq. (6.38) for the real parts of π L and
π T were derived without the need to assume that K was time-like.
Therefore, they should also apply “below the light cone,” even though
Braaten and Segel (1993) confined their discussion to the region k < ω.
As expected, these expressions break down for k > ω/v ∗ where Landau
damping becomes strong.
6.3.6 Renormalization Constants Z T,L
Armed with the dispersion relation one may determine the vertex renormalization
constants Z T,L relevant for the coupling of external photons
or plasmons to an electron in the medium (Sect. 6.2.3),
ZT,L −1 = 1 − ∂π ∣
T,L(ω, k) ∣∣∣∣ω
. (6.51)
∂ω 2 2 −k 2 =π T,L (ω,k)
With the same approximations as before, Braaten and Segel (1993)
found an analytic representation accurate to O(α),
Z T =
Z L =
2ω 2 (ω 2 − v 2 ∗k 2 )
ω 2 [3ω 2 P − 2 (ω 2 − k 2 )] + (ω 2 + k 2 )(ω 2 − v 2 ∗k 2 ) ,
2 (ω 2 − v 2 ∗k 2 )
3ω 2 P − (ω 2 − v 2 ∗k 2 )
ω 2
ω 2 − k 2 . (6.52)
In each case ω and k are “on shell,” i.e. they are related by the dispersion
relation relevant for the T and L case, respectively.
Inspection of Eq. (6.52) reveals that Z T is always very close to unity,
as expected for excitations with only a small deviation from a massiveparticle
dispersion relation. The contours in Fig. 6.7 confirm that Z T
never deviates from unity by more than a few percent.