Compton scattering
$e^-\gamma \to e^-\gamma$
Scattering of a photon off an electron. Two tree-level diagrams (s and u channels) give the Klein-Nishina formula. At low energies the s-channel dominates; at high energies both contribute comparably.
Order $\alpha$ diagrams
2 diagrams.
Order $\alpha^{2}$ diagrams
1 diagram.
From diagrams to cross section
How the individual diagram contributions combine into the physical result.
Sum both channels
$$i\mathcal{M} = i\mathcal{M}_s + i\mathcal{M}_u$$
Gauge invariance requires both diagrams. Neither the s-channel nor the u-channel is independently gauge invariant — only the sum satisfies the Ward identity $q_\mu \mathcal{M}^\mu = 0$.
Spin and polarization sums
$$\overline{|\mathcal{M}|^2} = \frac{1}{4}\sum_{\text{spins, pol}}|\mathcal{M}_s + \mathcal{M}_u|^2$$
Average over initial electron spin (1/2) and initial photon polarization (1/2). The photon polarization sum replaces $\sum_\lambda \epsilon_\mu^\lambda \epsilon_\nu^{\lambda*} \to -g_{\mu\nu}$ in Feynman gauge.
The Klein-Nishina formula
$$\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{2m^2}\left(\frac{\omega'}{\omega}\right)^2\left[\frac{\omega}{\omega'} + \frac{\omega'}{\omega} - \sin^2\theta\right]$$
In the lab frame: $\omega$ is the incoming photon energy, $\omega' = \omega/[1+(\omega/m)(1-\cos\theta)]$ the scattered photon energy. At low energy ($\omega \ll m$) this reduces to the Thomson cross section $(\alpha/m)^2(1+\cos^2\theta)/2$. At high energy the cross section falls as $1/\omega$.