Stars as Laboratories for Fundamental Physics - MPP Theory Group

92 Chapter 3
Fig. 3.1. Compton processes for photon scattering as well as for axion and
neutrino pair production (“photoneutrino process”). In each case there is
another amplitude with the vertices interchanged.
Zuber 1980)
[
16
σ = σ 0
(ŝ − 1) + ŝ + 1 + 2 ]
(ŝ2 − 6 ŝ − 3)
ln(ŝ) . (3.1)
2 ŝ 2 (ŝ − 1) 3
Here, ŝ ≡ s/m 2 e with √ s the CM (center of mass) energy. 14 This cross
section is shown in Fig. 3.2 as a function of the CM photon energy ω.
For ω ≪ m e the CM frame is the electron rest frame, ŝ → 1, and one
recovers the Thomson cross section σ = 8 3 σ 0. For ω ≫ m e one has
ŝ ≫ 1 and so σ = (σ 0 /ŝ) [2 ln(ŝ) + 1] = (πα 2 /ω 2 ) [ln(2ω) + 1 4 ] because
in this limit s = (2ω) 2 .
The standard Compton cross section can also be used to study the
photoproduction of novel low-mass vector particles which couple to
electrons in the same way as photons except that the fine-structure
constant must be replaced by the new coupling α ′ . Then
σ 0 ≡ παα ′ /m 2 e with α ′ ≡ g 2 /4π, (3.2)
a definition that pertains to all bosons which couple to electrons with
a dimensionless Yukawa or gauge coupling g.
If novel vector bosons such as “paraphotons” exist (Holdom 1986)
they likely couple to electrons by virtue of an induced magnetic moment
rather than by a tree-level gauge coupling (Hoffmann 1987),
L int = (g/4m e ) ψ e σ µν ψ e F µν , (3.3)
where g is a dimensionless effective coupling constant and F the para-
14 The square of the CM energy is s = (P + K) 2 with P = (E, p) and K = (ω, k)
the four-vectors of the initial-state electron and photon, respectively. The CM frame
is defined by p = −k so that s = (E + ω) 2 = [(m 2 e + ω 2 ) 1/2 + ω] 2 with ω the initialstate
photon energy in the CM frame. In the frame where the target electron is at
rest (p = 0) one finds s = 2ωm e + m 2 e where now ω is the photon energy in the
electron frame. Thus, with ω the photon energy in the respective frames,
ω
m e
=
{
1
2 (ŝ − 1)/√ ŝ in the CM frame,
1
2
(ŝ − 1) in the electron rest frame.