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For quantum electrodynamics, $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$ is an operator built from a dynamical Dirac field. Integrating over $\psi ,\bar{\psi }$ generates arbitrarily high powers of $e$, so the Gaussian-source truncation fails.

For the connected photon two-point function, the required diagrams through $e^4$ are: the bare photon line at $e^0$; one fermion-loop photon vacuum polarization insertion at $e^2$; and at $e^4$, a photon line with two successive one-loop polarization insertions together with the two-loop one-particle-irreducible fermion loop whose loop contains one internal photon chord. The latter includes the cyclic placements conventionally interpreted as self-energy and vertex corrections. Counterterm insertions must be added in a renormalized calculation. Disconnected vacuum bubbles cancel against the vacuum normalization.

Solved by gpt-5.6-sol high.