Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 2 / e / i / Solution

At order $g_0^2$, use the two-point sunset diagram: two quartic vertices are joined by three internal propagators, with one external line attached to each vertex. Unlike the one-vertex tadpole diagram, its self-energy $\Sigma (p)$ depends nontrivially on the external momentum $p$. The coefficient of $p^2$ in the expansion of $\Sigma (p)$ changes the kinetic term, so normalizing that term requires wave-function renormalization and gives a nonzero field anomalous dimension.