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

The first three terms are the scalar kinetic, mass, and $n$-point interaction terms. The remaining terms are the field-strength, mass, and coupling counterterms. The momentum-space rules arefor a propagator, an $n$-leg interaction vertex, a two-leg counterterm insertion, and an $n$-leg counterterm vertex, respectively, together with momentum conservation at every vertex.

Put $d=6-ϵ$. The one-loop two-point bubble has symmetry factor $1/2$ and, after a Feynman parameter and momentum shift, its pole is proportional toCancelling the pole in the minimal subtraction scheme gives, with $ϵ=6-d$,If one defines dimensional regularization by $d=6-2ϵ$, these same poles are written with $2ϵ$ in place of $ϵ$.

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 ${\delta }_Z$.