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

Corrections quadratic in the cubic-anisotropy coupling use two ${\lambda }_0{\sum }_a{\varphi }_a^4$ vertices joined by two fast propagators. Because all four legs at each anisotropic vertex carry the same component index, the internal propagators force the two vertices to have that same index. The resulting one-loop bubble renormalizes the anisotropic tensor ${\sum }_a{\varphi }_a^4$, hence $\lambda$, but does not generate the distinct-index structure in $({\mathbf{\varphi }}^2{)}^2$ needed for a correction to $g$. There is no freely summed closed component index, so the diagram is not proportional to $N$.

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 ${\lambda }_0{g}_0$.