Stars as Laboratories for Fundamental Physics - MPP Theory Group

Two-Photon Coupling of Low-Mass Bosons 173
Fig. 5.4. Example for the electro Primakoff effect.
nonrelativistic moving particles is small. (For an explicit calculation
see Raffelt 1986a.) Put another way, the dominant contribution to
magnetic field fluctuations in a nonrelativistic plasma is from photons,
not from moving charges.
In order to illustrate the relationship between field fluctuations and
axion emission consider the amplitude for the conversion of a classical
transverse electromagnetic wave into a classical axion wave in the presence
of an electric field configuration E(x). One finds from Eq. (5.16)
f(Ω) = g aγ
4π (ϵ × k) · ∫
d 3 x e −iq·x E(x) = g aγ
4π (ϵ × k) · E(q), (5.17)
where k and ϵ are the wave and polarization vector of the incident
wave, respectively, q is the “momentum” transfer to the axion, and
E(q) is a Fourier component of E(x). The differential transition rate
is dΓ/dΩ = |f(Ω)| 2 so that
dΓ γ→a
dΩ = g2 aγ
(4π) 2 (ϵ × k) i(ϵ × k) j E i (−q)E j (q), (5.18)
where it was used that E(x) is real so that E ∗ (q) = E(−q). If E(x) is
a random field configuration one needs to take an ensemble average so
that the transition rate is proportional to ⟨E i E j ⟩ q ≡ ⟨E i (−q)E j (q)⟩.
The electric and magnetic field fluctuations of a plasma are intimately
related to the medium response functions to electric and magnetic
fields, i.e. to the polarization tensor. For a plasma at temperature
T one can show on general grounds (Sitenko 1967)
⟨E i E j ⟩ q =
×
∫ +∞
−∞
[
qi q j
q 2
dω
2π
2
e ω/T − 1
(
Im ϵ L
|ϵ L | + δ 2 ij − q )
iq j
q 2
]
Im ϵ T
, (5.19)
|ϵ T − q 2 /ω 2 | 2
where ϵ L,T (ω, q) are the longitudinal and transverse dielectric permittivities
of the medium (Sect. 6.3.3). A quantity such as Im ϵ L /|ϵ L | 2 is
known as a spectral density—here of the longitudinal fluctuations.