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

=== Introduce the (normal-ordered / renormalized) composite operator ===

hμν(x)  =  N(Aμ(x)Aν(x)−14ημνAα(x)Aα(x)).h_{\mu\nu}(x) \;=\; \mathcal{N}\bigg( A_\mu(x) A_\nu(x) - \tfrac14 \eta_{\mu\nu} A_\alpha(x)A^\alpha(x)\bigg).hμν(x)=N(Aμ(x)Aν(x)−41ημνAα(x)Aα(x)).
N\mathcal{N}N is a normalization/renormalisation constant chosen so that hhh has canonical dimensions for a spin-2 field (mass dimension 1 for hhh in 4D if needed).

Goal: compute the connected 2-point function

Gμν,ρσ(x−y)≡⟨hμν(x) hρσ(y)⟩conn.\mathcal{G}_{\mu\nu,\rho\sigma}(x-y) \equiv \langle h_{\mu\nu}(x)\,h_{\rho\sigma}(y)\rangle_{\rm conn}.Gμν,ρσ(x−y)≡⟨hμν(x)hρσ(y)⟩conn.
Expanding hhh in AAA, G\mathcal{G}G is built from connected 4-point photon correlators:

Gμν,ρσ(x−y)∝⟨AμAνAρAσ⟩conn−(traces).\mathcal{G}_{\mu\nu,\rho\sigma}(x-y) \propto
\big\langle A_\mu A_\nu A_\rho A_\sigma \big\rangle_{\rm conn}
* \text{(traces)}.Gμν,ρσ(x−y)∝⟨AμAνAρAσ⟩conn−(traces).
At leading order in a Gaussian photon theory (free photons), Wick’s theorem gives disconnected products of two photon propagators; connected pieces arise once we include fermion loops (box diagrams) or resum coherent photon states.