Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 303 / 2 / b / Solution

Requiring $\int d^{d}x(\nabla \varphi {)}^2$ to be dimensionless gives the engineering dimensionsandThus ${\mu }_0$ is relevant. The cubic coupling is relevant, marginal, or irrelevant for $d<6$, $d=6$, or $d>6$; the quartic coupling has the same classification around $d=4$; and the sextic coupling has it around $d=3$.