Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 2 / a / Solution

For massless phi-fourth theory in four dimensions, the momentum-space rules areWritingthe two-point counterterm insertion is $i({\delta }_Z{p}^2-\delta m^2)$ and the four-point counterterm is $-i\delta \lambda$.

The renormalized one-particle-irreducible correlation function ${\Gamma }_4$ through order ${\lambda }^2$ contains the tree quartic vertex, the quartic counterterm, and three one-loop bubble diagrams. The bubbles are the $s$, $t$, and $u$ channels and each has symmetry factor $1/2$.