Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 303 / 3 / e / ii / Solution

At order ${\lambda }_0{g}_0$, use one anisotropic quartic vertex and one $O(N)$-invariant quartic vertex joined by two fast propagators. With four equal external component labels, the bubble renormalizes $\lambda$; with external labels in two equal pairs, it also renormalizes $g$. The anisotropic vertex fixes the component index carried by the internal lines, so none of these mixed diagrams contains an independent component loop. Consequently no correction proportional to $N$ occurs at order ${\lambda }_0^2$ or ${\lambda }_0{g}_0$.