A closed electron loop with four photon legs. The simplest topology for photon-photon scattering. UV finite by gauge invariance (the tensor structure forces enough powers of external momentum in the numerator).
The low-energy limit gives the Euler-Heisenberg effective Lagrangian (1936). First directly observed at the LHC by ATLAS in 2017 in ultra-peripheral Pb-Pb collisions.
The computational chain
From diagram to number—every step of the amplitude calculation.
The diagram
A closed electron loop with four photon legs — one at each vertex. No tree-level diagrams exist for $\gamma\gamma \to \gamma\gamma$ because QED has no direct photon self-interaction. This is an intrinsically quantum process.
Feynman rules expression
$$i\mathcal{M} = (-1)(-ie)^4\int\frac{d^4k}{(2\pi)^4}\,\mathrm{Tr}\left[\gamma^{\mu_1}\frac{i(\not{k}+m)}{k^2-m^2}\gamma^{\mu_2}\frac{i(\not{k}+\not{q}_2+m)}{(k+q_2)^2-m^2}\gamma^{\mu_3}\cdots\right]\prod_i\epsilon_{\mu_i}$$
The $(-1)$ is from the closed fermion loop. The trace runs over four gamma matrices and four fermion propagators — an 8th-rank tensor in the loop momentum. There are $3! = 6$ inequivalent orderings of the four photon legs around the loop.
UV finiteness by gauge invariance
Naively the box integral diverges as $\sim \int d^4k/k^4$ (logarithmic). But the Ward identity $q_{i\mu_i}\mathcal{M}^{\mu_1\mu_2\mu_3\mu_4} = 0$ forces the amplitude to vanish when any polarization vector is replaced by its momentum. This requires the tensor structure to carry extra powers of external momenta in the numerator, which improves the UV behavior enough to make the integral finite. No renormalization needed.
Low-energy limit: Euler-Heisenberg
$$\mathcal{L}_{\text{EH}} \supset \frac{\alpha^2}{90m^4}\left[(\mathbf{E}^2-\mathbf{B}^2)^2 + 7(\mathbf{E}\cdot\mathbf{B})^2\right]$$
When all photon energies are much less than $m$, the box integral can be expanded in powers of external momenta. The leading term gives the Euler-Heisenberg effective Lagrangian (1936) — nonlinear corrections to Maxwell's equations from virtual electron-positron pairs. This is the origin of vacuum birefringence and photon-photon scattering.
Experimental observation
Light-by-light scattering was first directly observed by the ATLAS collaboration at the LHC in 2017, in ultra-peripheral lead-lead collisions. The strong Coulomb fields of the lead nuclei act as sources of quasi-real photons (the Weizsäcker-Williams approximation), which scatter via the electron box. The measured cross section agrees with the QED prediction.