Stars as Laboratories for Fundamental Physics - MPP Theory Group

174 Chapter 5
One finds explicitly ⟨E i E j ⟩ q = ˆq iˆq j T/(1 + q 2 /kS) 2 in the classical
limit (Sitenko 1967) where kS 2 is the Debye-Hückel wave number of
Eq. (5.7). With this result one easily reproduces the Primakoff transition
rate Γ γ→a (Raffelt 1988a).
The language of spectral densities for the electromagnetic field fluctuations
forms the starting point for a quantum calculation of the axion
emission rate in the framework of thermal field theory. This program
was carried out in a series of papers by Altherr (1990, 1991), Altherr and
Kraemmer (1992), and Altherr, Petitgirard, and del Río Gaztelurrutia
(1994). Naturally, in the classical limit they reproduced the Primakoff
transition rate Γ γT →a of Eq. (5.8).
In the degenerate or relativistic limit their results cannot be represented
in terms of simple analytic formulae. The most important
astrophysical environment to be used for extracting bounds on g aγ are
low-mass stars before and after helium ignition with a core temperature
of about 10 8 K (Sect. 5.2.5). Altherr, Petitgirard, and del Río
Gaztelurrutia (1994) gave numerical results for the energy-loss rate for
this temperature as a function of density shown in Fig. 5.5 (solid line).
The dashed line is the classical limit Eq. (5.9); it agrees well with the
general result in the low-density (nondegenerate) limit. In the degen-
Fig. 5.5. Energy-loss rate of a helium plasma at T = 10 8 K by axion emission
with g aγ = 10 −10 GeV −1 . The solid line is from transverse-longitudinal
fluctuations; the dashed line is the corresponding classical limit. The dotted
line is from transverse-transverse fluctuations, i.e. in the axion source term
g aγ E·B both fields are from transverse fluctuations. (Adapted from Altherr,
Petitgirard, and del Río Gaztelurrutia 1994.)