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

===== 1. Compute the one-loop box kernel Kμνρσ(ki)\mathcal{K}^{\mu\nu\rho\sigma}(k_i)Kμνρσ(ki). Extract the low-momentum expansion coefficients (keep leading k0k^0k0 and k2k^2k2 terms). =====
# Derive the induced quartic photon term from the determinant and write it as λ (AA−14ηA2)2+⋯\lambda\, (AA - \tfrac14\eta A^2)^2 + \cdotsλ(AA−41ηA2)2+⋯. Provide explicit λ\lambdaλ as function of e,me,me,m.
# Perform the Hubbard–Stratonovich transform to introduce hμνh_{\mu\nu}hμν; integrate out photons at one loop to get Dμνρσ(k)\mathcal{D}^{\mu\nu\rho\sigma}(k)Dμνρσ(k) (inverse propagator for hhh).
# Compute Gμν,ρσ(k)\mathcal{G}_{\mu\nu,\rho\sigma}(k)Gμν,ρσ(k) and project on spin-2 sector: Πμν,ρσ(2)(k)=12(PμρPνσ+PμσPνρ)−13PμνPρσ,Pμν=ημν−kμkνk2.\Pi^{(2)}_{\mu\nu,\rho\sigma}(k)=\tfrac12(P_{\mu\rho}P_{\nu\sigma}+P_{\mu\sigma}P_{\nu\rho})-\tfrac13P_{\mu\nu}P_{\rho\sigma},\quad P_{\mu\nu}=\eta_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}.Πμν,ρσ(2)(k)=21(PμρPνσ+PμσPνρ)−31PμνPρσ,Pμν=ημν−k2kμkν. Check for a 1/k21/k^21/k2 pole and extract coefficient κ~\tilde\kappaκ~.
# Numerics: evaluate κ~\tilde\kappaκ~ for electron mass mem_eme; compute required amplification NαN^\alphaNα (coherent cell count scaling) so that Geff(N)G_{\rm eff}^{(N)}Geff(N) matches measured GGG. Provide sensitivity analysis.