Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 304 / 2 / ii / Solution

The kinetic term fixes the mass dimension of the scalar field, and the interaction then fixes that of its coupling:A relevant coupling has positive mass dimension; a marginal coupling has zero mass dimension. ThereforeFor integer dimensions the complete interacting list isIf the noninteracting $n=1,2$ terms are included, their couplings have mass dimensions $(d+2)/2$ and $2$, respectively, and are relevant for every positive $d$. This is a classification by power counting in quantum field theory; it does not assert the stability of odd monomial potentials.