Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 304 / 2 / b / iii

The only two-point diagram through second loop order is the double tadpole from one sextic vertex. It has no route by which the external momentum can pass through an internal propagator, so its value is independent of $p$ and renormalizes only the mass. A field-strength counterterm is determined by the coefficient of $p^2$ in the two-point 1PI function, hence ${\delta }_Z=0$ at one and two loops. The first momentum-dependent two-point topology uses two sextic vertices joined by five internal lines and has four loops.