 Solution

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.