Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== Consider two distant localized currents J(1)J^{(1)}J(1) and J(2)J^{(2)}J(2) (e.g., two atoms or charges). The amplitude for exchanging two photons (lowest order that can produce a symmetric rank-2 kernel) is dominated by box/loop diagrams where the electron loop ties two photon lines to two other photon lines. Symbolically: ===

Two-photon exchange amplitude between currents:

M∼∫ ⁣d4k(2π)4  Jμ(1)(−k) Jρ(2)(k)  Kμνρσ(k)  Jν(1)(−k) Jσ(2)(k),\mathcal{M} \sim \int\! \frac{d^4k}{(2\pi)^4}\;
J^{(1)}_\mu(-k)\,J^{(2)}_\rho(k)\; \mathcal{K}^{\mu\nu\rho\sigma}(k)\; J^{(1)}_\nu(-k)\,J^{(2)}_\sigma(k),M∼∫(2π)4d4kJμ(1)(−k)Jρ(2)(k)Kμνρσ(k)Jν(1)(−k)Jσ(2)(k),
where Kμνρσ(k)\mathcal{K}^{\mu\nu\rho\sigma}(k)Kμνρσ(k) is built from combinations of photon propagators and the fermion box kernel (4-point fermion loop). In a condensed/coherent limit we can reinterpret the pair of currents and the box kernel as sourcing and propagating an effective hμνh_{\mu\nu}hμν.

Low-momentum expansion (∣k∣≪m|k|\ll m∣k∣≪m) of the box gives a local effective operator; in particular you will find terms schematically like

Leff⊃e4m4 (FμαFνα)(FμβFνβ)+⋯ ,\mathcal{L}_{\rm eff} \supset \frac{e^4}{m^4}\,(F_{\mu\alpha}F_{\nu}{}^{\alpha})(F^{\mu\beta}F^{\nu}{}_{\beta}) + \cdots,Leff⊃m4e4(FμαFνα)(FμβFνβ)+⋯,
and—after appropriate averaging/coarse graining—these quartic terms produce an effective quadratic action for hμνh_{\mu\nu}hμν.