Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particles Interacting with Electrons and Baryons 93
photon field tensor. In the nonrelativistic limit this yields the same total
cross section as the interaction with pseudoscalars Eq. (3.6) below.
This is seen if one compares the matrix elements between two electron
states i and f for the two cases. For paraphotons (momentum k, polarization
vector ϵ) it is g ⟨f|e ik·r (k × ϵ) · σ|i⟩ while for pseudoscalars it is
g ⟨f|e ik·r k · σ|i⟩. After an angular average the two expressions are the
same. Of course, one must account for the two paraphoton polarization
states by an extra factor of 2.
Fig. 3.2. Total cross section for the Compton process with a final-state
vector, scalar, or pseudoscalar boson according to Eqs. (3.1), (3.5), and
(3.9), respectively, with σ 0 defined in Eq. (3.2) and ω the CM initial photon
energy.
3.2.2 Scalars
Grifols and Massó (1986) studied the stellar emission of scalars ϕ which
couple according to
L int = g ψ e ψ e ϕ. (3.4)
Integrating their differential cross section I find
[ −16
σ = σ 0
(ŝ − 1) + 1 − 3ŝ
]
(ŝ + 3)2
+ 2 2ŝ 2 (ŝ − 1) ln(ŝ) 3
(3.5)
shown in Fig. 3.2 (dashed line). For small and large photon energies
this is half the cross section for massless vector bosons which have
two polarization degrees of freedom. For intermediate energies the two
results are not related by a simple factor.