= Scalar-pair production by fermion annihilation
{title2=$\sigma=\pi\alpha^2\beta^3/(3s)$}
For unpolarized ultrarelativistic charged spin-half particles annihilating through a photon into minimally coupled unit-charge scalars of mass $m$, set $\beta=\sqrt{1-4m^2/s}$. The scalar vertex contains the difference $k_--k_+$ of outgoing four-momenta. Contraction with the spin-averaged initial fermion current gives $\overline{|\mathcal M|^2}=e^4\beta^2\sin^2\theta/2$. The two-body final-state flux factor contributes another $\beta$, giving
$$
\frac{d\sigma}{d\Omega}=\frac{\alpha^2\beta^3}{8s}\sin^2\theta,\qquad
\sigma=\frac{\pi\alpha^2\beta^3}{3s}.
$$
The $\sin^2\theta$ angular dependence and threshold suppression contrast with <electron-positron annihilation into a muon pair>. These predictions assume the specified scalar electrodynamic coupling, rather than an arbitrary spin-zero interaction.
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