Stars as Laboratories for Fundamental Physics - MPP Theory Group

238 Chapter 6
Explicit expressions in terms of the electron phase-space integrals
were derived, e.g. by D’Olivo, Nieves, and Pal (1989) and by Altherr
and Salati (1994),
Λ αβ
V
Λ αβ
A
= 4e C V
∫
d 3 p [
fe −(p) + f
2E(2π) 3 e +(p) ]
× (P · K)2 g αβ + K 2 P α P β − P · K (K α P β + K α P β )
(P · K) 2 − 1 4 (K2 ) 2 ,
= 2ie C A ϵ αβµν ∫
d 3 p [
fe −(p) − f
2E(2π) 3 e +(p) ]
×
K 2 P µ K ν
(P · K) 2 − 1 4 (K2 ) 2 , (6.99)
where f e ±(p) is the electron and positron phase-space distribution with
P = (E, p) the electron or positron four-momentum.
Instead of a neutrino pair, another photon can be thought of as
being coupled to the electron line in Fig. 6.11 or 6.12, a process which
represents photon forward scattering. Therefore, apart from overall
coupling constants Λ αβ
V is identical with the electronic contribution to
the photon polarization tensor Π αβ studied earlier in this chapter
Λ αβ
V = (C V /e) Π αβ . (6.100)
In an isotropic plasma, Π αβ is characterized by the two medium characteristics
π T (ω, k) and π L (ω, k) which are functions of the photon fourmomentum
K = (ω, k) with k = |k|. For the antisymmetric piece, in
an isotropic medium the phase-space integration averages the spatial
part of P α to zero so that
Λ αβ
A = 2ieC A ϵ αβµ0 K µ a(ω, k) (6.101)
in terms of a single medium characteristic a.
The single most important application of the effective neutrino electromagnetic
coupling is the photon decay process γ → νν that was
studied in the previous section. It turns out that Λ αβ
A contributes very
little to the decay process so that Λ αβ
V , or rather the polarization tensor
Π αβ determines all aspects of the plasma process.
Another possible process is the Cherenkov emission of photons by
neutrinos. Of course, because in a plasma transverse photons acquire an
“effective mass,” i.e. their dispersion relation is time-like (ω 2 − k 2 > 0),