Connected shell-contraction expansion ($-\log ⟨e^{-V}{⟩}_{>}=⟨V{⟩}_{>}-\frac{1}{2}⟨V^2{⟩}_{>,c}+\cdots$)

The cumulant expansion of the shell integral keeps precisely connected Feynman diagrams built from shell-restricted scalar propagators. The external low fields are backgrounds, not propagators. Connected graphs with bridges can contribute, so the expansion is not limited to one-particle-irreducible Feynman diagrams.

If a bridge is the only line carrying momentum away from a low external field, tadpole pairs do not change its momentum and the shell-restricted scalar propagator has zero support there. A bridge attached to a product such as ${\varphi }_{<}^3$ can survive because the sum of three low momenta can enter the shell. This matters before a local derivative expansion.