Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 219
with v ∗ = 1 the complete factor is Z L = 2ω 2 /[3ωP 2 − (ω 2 − k 2 )] and thus
rises quickly with k because ω ≈ k.
As discussed in the previous section, the dispersion relation crosses
the light cone at k = k 1 , a point at which Z L diverges and changes sign.
The sign change is compensated by the change of the polarization vector
e L (Eq. 6.27) at the light cone where it becomes imaginary. Because
in the squared matrix element a factor e ∗µ
L e ν L appears, and because
this expression changes sign at the light cone, the expression Z L e ∗µ
L e ν L
remains positive.
As for the divergence, it is harmless in reactions of the sort γ L → νν
(plasmon decay), e → eγ L (Cherenkov emission), γ L e → e (Cherenkov
absorption), and γ T γ L → a (plasmon coalescence into axions) which are
of interest in this book. These reactions involving three particles are
constrained by their phase space to either time-like excitations (plasmon
decay), or to space-like ones (Cherenkov and coalescence process).
Therefore, the threshold behavior of the phase space moderates the
divergence in these cases.
6.4 Screening Effects
6.4.1 Debye Screening
Scattering processes in the Coulomb field of charged particles such as
Rutherford scattering, bremsstrahlung, or the Primakoff effect typically
lead to cross sections which diverge in the forward direction because of
the long-range nature of the electrostatic interaction. In a plasma this
divergence is moderated by screening effects which thus are crucial for
a calculation of the cross sections or energy-loss rates.
Screening effects are revealed by turning to the static limit of Maxwell’s
equations in a medium,
[
k 2 + π L (0, k) ] Φ(k) = ρ(k),
[
k 2 + π T (0, k) ] A(k) = J(k), (6.54)
where the current must be transverse in both Coulomb and Lorentz
gauge as ∂ t ρ = 0 in the static limit. Notably, the equation for Φ in
vacuum is the Fourier transform of Poisson’s equation and thus gives
rise to a 1/r Coulomb potential if the source is point-like, ρ(r) = eδ(r).
In a QED plasma π L,T (ω, k) are given by the integrals Eq. (6.37).
In the static limit (ω = 0) one finds π T (0, k) = 0 because all terms
in the integrand involve factors of ω. Therefore, stationary currents