How does photon-photon interaction manifest in QED?

In classical electrodynamics the electromagnetic field only interacts with charged particles, but in quantum electrodynamics (QED), there is a very weak interaction between photons. The lowest order contribution comes from this Feynman diagram:

Having four vertices, we see that its contribution is extremely low compared to for example electron-electron interaction, it is non-zero though. Computing the contribution to the $$S$$-matrix element would give a first approximation of the scattering cross-section.

However, could we compute how this interaction would manifest itself more precisely? Will it act as an attractive or a repulsive force between photons, or even something else altogether?

• Commented Aug 23, 2021 at 17:06
• @AlmostClueless thank you! Commented Aug 23, 2021 at 21:34

I am fully disclosing off the bat here that I do not know of a plausible way/gimmick to attribute an "attractive-versus-repulsive-force" feature to these amplitudes, a subject discussed amply on this site, as the relevant 2-to-2 scattering cross sections cannot be used for these purposes (C-conjugation). You might have to strain to define this feature here. Photons scatter off each other in precise ways impervious to any attractive-vs-repulsive cartoon interpretations.

A state-of-the art friendly discussion of the (1936) Euler–Heisenberg effective quartic Lagrangian describing the process and the Delbrück scattering and quantum birefringence phenomena involved, is to be found in Thiescheffer's (2017) thesis, and especially the discussion around his (3.14).

The full differential cross section for $$γγ\to γγ$$ was calculated by Karplus & Neuman (1951), who dealt with all polarization states, angular distributions, etc... The cross-section goes as $$α^4/s$$, in contrast to $$γγ\to e^+e^-$$, which goes as $$α^2/s$$, so it's four orders of magnitude weaker! (α=1/137.)

You could pretend you read off the 4γ effective contact interaction from the quartic "potential" term of the E-H lagrangian, $$\mathcal{L} = \frac{1}{2}\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right)+\frac{2\alpha^{2}}{45 m_e^{4}}\left[\left(\mathbf{E}^2 - \mathbf{B}^2\right)^{2} + 7 \left(\mathbf{E}\cdot\mathbf{B}\right)^{2}\right],$$ the Maxwell quadratic one being the "kinetic" term.

• Why can you only pretend to read off the effective 4$\gamma$ effective contact interaction? It should be the dominant contribution at low energies below the electron mass, no? Commented Aug 24, 2021 at 18:11
• Of course. Pretend because it’s computable directly from QED, and this effective theory follows it, instead. Commented Aug 24, 2021 at 18:24
• Got it. Thanks! Commented Aug 24, 2021 at 18:27