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 $J={\Gamma }^{\prime }(\varphi )$ gives $G^{\prime \prime }=1/{\Gamma }^{\prime \prime }$ and $G^{\prime \prime \prime \prime }=-{\Gamma }^{\prime \prime \prime \prime }/({\Gamma }^{\prime \prime }{)}^4+3({\Gamma }^{\prime \prime \prime }{)}^2/({\Gamma }^{\prime \prime }{)}^5$. Dividing by the four full external propagators proves the formula.