One-particle-irreducible vertex (${\Gamma }^{(n)}$)

A functional derivative of the quantum effective action gives an amputated one-particle-irreducible correlation function. Its perturbative graphs stay connected after cutting any single internal propagator. A free quadratic effective action has no vertices of order greater than two; interactions generate higher vertices.

Use the Minkowski generating functional $Z=e^{iW}$, source $+J\varphi$, mean field $\phi =\delta W/\delta J$, and quantum effective action $\Gamma =W-\int J\phi$. Then $W^{(2)}=i{G}_2^c$, $W^{(3)}=-G_3^c$, and ${\Gamma }^{(2)}=-(W^{(2)}{)}^{-1}$. Differentiating this inverse gives ${\Gamma }^{(3)}=(W^{(2)}{)}^{-1\otimes 3}{W}^{(3)}=-i(G_2^c{)}^{-1\otimes 3}{G}_3^c$, with integration over the repeated spacetime arguments. Thus amputation of external propagators gives $i{\Gamma }^{(3)}$ in this convention. The full connected two-point functions, rather than bare propagators, perform the exact amputation.