Stars as Laboratories for Fundamental Physics - MPP Theory Group

94 Chapter 3
3.2.3 Pseudoscalars
Next, turn to the photoproduction of low-mass pseudoscalars ϕ which
couple to electrons by the interaction
L int = (1/2f) ψ e γ µ γ 5 ψ e ∂ µ ϕ or L int = −ig ψ e γ 5 ψ e ϕ, (3.6)
where f is an energy scale and g a dimensionless coupling constant.
Both interaction laws yield the same Compton cross section with the
identification g = m e /f (see the discussion in Sect. 14.2.3).
If a pseudoscalar (frequency ω, wavevector k) is emitted in a transition
between the nonrelativistic electron states |i⟩ and |f⟩, the matrix
element is
M pseudoscalar =
g 1 ⟨ ∣ √ ∣∣e f ik·r ∣
σ · k∣i ⟩ . (3.7)
2m e 2ω
This is to be compared with the corresponding matrix element for photon
transitions,
M photon = −e
2m e
1
√
2ω
⟨
f
∣ ∣∣e ik·r [2ϵ · p + σ · (k × ϵ)] ∣ ∣ ∣i
⟩
, (3.8)
with the photon polarization vector ϵ and the electron momentum operator
p. Therefore, transitions involving pseudoscalars closely compare
with photonic M1 transitions, a fact that was used to scale nuclear or
atomic photon transition rates to those involving axions (Donelly et al.
1978; Dimopoulos, Starkman, and Lynn 1986a,b).
The relativistic Compton cross section for massive pseudoscalars
was first worked out by Mikaelian (1978). The most general discussion
of the matrix element was provided by Brodsky et al. (1986) and by
Chanda, Nieves, and Pal (1988) who also included an effective photon
mass relevant for a stellar plasma. For the present purpose it is enough
to consider massless photons and pseudoscalars. With σ 0 as defined in
Eq. (3.2) one finds
( ln(ŝ)
σ = σ 0
ŝ − 1 − 3ŝ − 1 )
, (3.9)
2ŝ 2
shown in Fig. 3.2 (dotted line).
For ω ≫ m e one finds σ = (παα ′ /2ω 2 ) [ln(2ω) − 3 ], similar to the
4
scalar case. For ω ≪ m e , however,
σ = 4 3 σ 0 (ω/m e ) 2 , (3.10)
so that the cross section is suppressed at low energies. This reduction
is related to the M1 nature of the transition.