= Solution
The following are the leading <quantum electrodynamics> diagrams. Solid arrows indicate fermion-number flow, and wavy lines indicate <photons>. The heavy source line represents a charged <nucleus>; its recoil allows <four-momentum conservation>.
\Image[/past-exam-of-the-nuclear-and-particle-physics-course-of-the-university-of-oxford/2022-qed-diagrams.png]
{title=Leading Compton, electron-line bremsstrahlung and nuclear-field pair-production diagrams}
For <Compton scattering>, $e^-\gamma\to e^-\gamma$, there are two diagrams: absorption of the incident <photon> followed along the <electron> line by emission of the outgoing <photon>, and the opposite ordering. These are the $s$- and $u$-channel <electron> <propagators>. Both <scattering amplitudes> are order $e^2$.
For <bremsstrahlung>, $e^-Z\to e^-Z\gamma$, a virtual <photon> transfers <momentum> between the <electron> and the source. The real <photon> is emitted before or after this interaction along the <electron> line. These two dominant external-field diagrams have <scattering amplitudes> of order $Ze^3$. A full finite-mass charged-target calculation also includes radiation from the target; it is suppressed in the heavy stationary-source approximation used here.
For nuclear-field <pair production>, $\gamma Z\to e^-e^+Z$, crossing the preceding diagrams gives two order-$Ze^3$ contributions, with the incoming real <photon> and the exchanged virtual <photon> in either order along the pair's fermion line. An isolated real <photon> cannot produce a massive pair in vacuum: its invariant squared <mass> is zero, whereas the pair has invariant squared <mass> at least $4m_e^2$. The source absorbs recoil <momentum>.
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