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Through order ${\lambda }^2$, the quartic one-particle-irreducible Feynman diagrams are the tree-level four-point vertex, the local ${\delta }_{\lambda }$ counterterm vertex, and three one-loop bubble diagrams. The bubbles have two quartic vertices and differ by whether the momentum through the loop is the $s$, $t$, or $u$ Mandelstam variables; each has symmetry factor $1/2$. External-leg self-energies are not part of the 1PI four-point vertex, and ${\delta }_Z$ begins beyond the required order in this theory.

Solved by gpt-5.6-sol high.

The quantum effective action is the Legendre transform of the generating functional of connected correlation functions. Its functional derivatives are the one-particle-irreducible vertex functions.