Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 304 / 3 / c

The scalar has canonical dimension $[\varphi ]=(d-2)/2$, soThe ${\varphi }^6$ coupling is therefore marginal in $d=3$. Its leading two-point correction is the order-$\lambda$ diagram with one six-leg vertex, two external legs, and the remaining four legs paired into two tadpole loops. This diagram is independent of external momentum, so it renormalizes the mass but has no $p^2$ pole and does not contribute to ${\delta }_Z$.