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

The leading two-point diagram has one sextic vertex, two external legs, and its remaining four legs paired into two tadpole loops; it determines the mass counterterm. The leading six-point diagram has two sextic vertices joined by three internal lines, leaving three external legs at each vertex; it is a two-loop diagram and determines the coupling counterterm. The corresponding local ${\delta }_m$ and ${\delta }_{\lambda }$ vertices complete the counterterm calculation.

Solved by gpt-5.6-sol high.