Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 303 / 2 / f

One cubic vertex cannot leave four external legs. Two cubic vertices can be joined by one contracted fast-field line, leaving four slow-field legs, so the first correction to ${\lambda }_0$ involving ${\alpha }_0$ isIt arises from the second term of the cumulant expansion.

A correction with three external legs can be made from one cubic and one quartic vertex by contracting four of their seven fields into internal lines. Consequently the first correction to ${\alpha }_0$ involving ${\lambda }_0$ isA purely quartic interaction cannot generate an odd interaction because its ${\mathbb{Z}}_2$ symmetry forbids odd powers of $\varphi$.