Stars as Laboratories for Fundamental Physics - MPP Theory Group

172 Chapter 5
cosine of the angle between k a and k T . The angular integration then
averages z to zero and leaves us with
Γ γT →a = g2 aγT
16π . (5.15)
Collective longitudinal oscillations only exist for k L ∼ < k D (Debye screening
scale). Because a momentum transfer |k T −k a | = O(ω T ) is required,
this result applies only for ω T ∼ < k D . This conversion rate agrees with
the low-ω expansion of the Primakoff result Eq. (5.8).
It must be stressed that Eq. (5.15) is not an additional contribution
to the conversion rate, it is the same result derived in a different fashion.
Here, longitudinal plasmons in the static limit were used as the external
electric field in which the Primakoff effect takes place. Before, the static
limit was taken from the start; the collective behavior of the electron
motion was reflected in the screening of the Coulomb potential. These
two paths of performing the calculation in the end extracted the same
information from the electromagnetic polarization tensor which defines
both screening effects and the dispersion behavior of electromagnetic
excitations.
5.2.3 Axion Emission from Electromagnetic Plasma
Fluctuations
These simple calculations of the axion emission rate apply only in the
classical (nondegenerate, nonrelativistic) limit. Even though this is the
most relevant case from a practical perspective it is worth mentioning
how one proceeds for a more general evaluation. To this end note that
the 2γ interaction Eq. (5.1) corresponds to a source term for the axion
wave equation,
( + m 2 a) a = g aγ E · B, (5.16)
where = ∂ µ ∂ µ = ∂t 2 − ∇ 2 . Axions are then emitted by the E · B fluctuations
caused by the presence of thermal electromagnetic radiation
as well as the collective and random motion of charged particles.
The Primakoff calculation in Sect. 5.2.1 used the (screened) electric
field of charged particles and the magnetic field of (transverse) electromagnetic
radiation (photons) as a source. Actually, one can include
the magnetic field of moving charges for this purpose. Then axions are
emitted in the collision of two particles (Fig. 5.4), a process sometimes
referred to as the “electro Primakoff effect.” Unsurprisingly, the emission
rate is much smaller because the magnetic field associated with