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

=== Write Fourier transform and focus on long wavelengths k→0k\to 0k→0. The connected 4-photon correlator at one fermion loop + two photon propagators yields, schematically, ===

⟨AμAνAρAσ⟩conn(k)  ∼  Πμνρσ(4)(k)  ∼  e4m2 1k2 Tμνρσ+higher powers of k,\langle A_\mu A_\nu A_\rho A_\sigma\rangle_{\rm conn}(k)
\;\sim\; \Pi^{(4)}_{\mu\nu\rho\sigma}(k) \;\sim\; \frac{e^4}{m^2}\,\frac{1}{k^2}\,T_{\mu\nu\rho\sigma} + \text{higher powers of }k,⟨AμAνAρAσ⟩conn(k)∼Πμνρσ(4)(k)∼m2e4k21Tμνρσ+higher powers of k,
where TμνρσT_{\mu\nu\rho\sigma}Tμνρσ is a rank-4 tensor built from ηαβ\eta_{\alpha\beta}ηαβ and kαk_\alphakα. After the trace subtractions for hhh, you find

Gμν,ρσ(k)≡⟨hμνhρσ⟩conn(k)  ∝  κ~k2  Πμν,ρσ(spin-2)(k)  +  regular,\mathcal{G}_{\mu\nu,\rho\sigma}(k) \equiv \langle h_{\mu\nu} h_{\rho\sigma}\rangle_{\rm conn}(k)
\;\propto\; \frac{\tilde{\kappa}}{k^2}\; \Pi^{(\text{spin-2})}_{\mu\nu,\rho\sigma}(k) \;+\; \text{regular},Gμν,ρσ(k)≡⟨hμνhρσ⟩conn(k)∝k2κ~Πμν,ρσ(spin-2)(k)+regular,
with Π(spin-2)\Pi^{(\text{spin-2})}Π(spin-2) the spin-2 projector (transverse, traceless):

Πμν,ρσ(spin-2)(k)=12(PμρPνσ+PμσPνρ)−13PμνPρσ,Pμν=ημν−kμkνk2.\Pi^{(\text{spin-2})}_{\mu\nu,\rho\sigma}(k)
= \tfrac12\big( P_{\mu\rho}P_{\nu\sigma}+P_{\mu\sigma}P_{\nu\rho}\big) - \tfrac13 P_{\mu\nu}P_{\rho\sigma},
\qquad P_{\mu\nu}=\eta_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}.Πμν,ρσ(spin-2)(k)=21(PμρPνσ+PμσPνρ)−31PμνPρσ,Pμν=ημν−k2kμkν.
Thus, under appropriate conditions (coherent phase averaging, low kkk, dominance of the particular tensor structure), the composite propagator of hhh behaves like a massless spin-2 propagator ∝1/k2\propto 1/k^2∝1/k2 with projector Π(spin-2)\Pi^{(\text{spin-2})}Π(spin-2).