Light-by-light scattering
$\gamma\gamma \to \gamma\gamma$
Photon-photon scattering, forbidden classically because Maxwell's equations are linear. In QED it proceeds through a virtual electron box at one loop. First directly observed at the LHC in 2017.
No tree-level diagrams exist because there is no direct photon-photon vertex in QED. The leading contribution is the one-loop electron box, making this an intrinsically quantum effect with no classical analog.
Order $\alpha^{2}$ diagrams
1 diagram.