= Amputated connected correlation function
An amputated <connected correlation function> is obtained by removing external propagators. Amputation using full propagators eliminates external <self-energy> decorations; internal exchange diagrams remain. It is distinct from a <one-particle-irreducible correlation function>. In a scalar theory with vanishing one-point function, at zero momentum the four-point quantity satisfies $\mathcal A_4=-\Gamma^{(4)}+3[\Gamma^{(3)}]^2/\Gamma^{(2)}$ with a convention in which connected diagram vertices are minus effective-action derivatives. For a constant scalar source, the Legendre relation $J=\Gamma'(\phi)$ gives $G''=1/\Gamma''$ and $G''''=-\Gamma''''/(\Gamma'')^4+3(\Gamma''')^2/(\Gamma'')^5$. Dividing by the four full external propagators proves the formula.
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