Stars as Laboratories for Fundamental Physics - MPP Theory Group

Two-Photon Coupling of Low-Mass Bosons 169
This cross section exhibits the usual forward divergence from the
long-range Coulomb interaction. For m a ≠ 0 it is cut off in vacuum by
the minimum necessary momentum transfer q min = m 2 a/2ω (m a ≪ ω);
the total cross section is then σ γ→a = Z 2 g 2 aγ [ 1 2 ln(2ω/m a) − 1 4 ].
In a plasma, the long-range Coulomb potential is cut off by screening
effects; according to Sect. 6.4 the differential cross section is modified
with a factor q 2 /(k 2 S + q 2 ). In a nondegenerate medium the screening
scale is given by the Debye-Hückel formula
k 2 S = 4πα
T n B
(
Ye + ∑ j
Z 2 j Y j
)
, (5.7)
where n B = ρ/m u (atomic mass unit m u ) is the baryon density while
Y e and Y j are the number fractions per baryon of the electrons and
various nuclear species j. With this modification the total scattering
cross section is easily calculated (Raffelt 1986a). Summing over all
targets one may derive an expression for the transition rate (“decay
rate”) of a photon of frequency ω into an axion of the same energy,
Γ γ→a = g2 aγT k 2 S
32π
[( ) ( ) ]
1 + k2 S
ln 1 + 4ω2 − 1 , (5.8)
4ω 2 kS
2
where the plasma “mass” of the initial-state photon and the axion mass
were neglected relative to the energy ω. In the limit ω ≪ k S this expression
expands as Γ γ→a = g 2 aγω 2 T/16π which is entirely independent
of the density and chemical composition.
For a stellar plasma, however, this approximation is usually not
justified. Ignoring the plasma frequency for the initial-state photons,
the energy-loss rate per unit volume is
∫
Q =
2 d 3 k γ
(2π) 3
Γ γ→a ω
e ω/T − 1 = g2 aγT 7
4π F (κ2 ), (5.9)
where κ ≡ k S /2T and
F (κ 2 ) = κ2 ∫ [
(
∞
dx (x 2 + κ 2 ) ln
2π 2 0
1 + x2
κ 2 )
− x 2 ]
x
e x − 1 , (5.10)
with x = ω/T . This function is shown in Fig. 5.3. In a standard solar
model κ 2 ≈ 12 throughout the Sun with a variation of less than 15%. In
the core of an HB star with ρ = 10 4 g/cm 3 and T = 10 8 K it is κ 2 ≈ 2.5.
One finds F = 0.98 and 1.84 for κ 2 = 2.5 and 12, respectively.