Past exam of the quantum field theory course of the University of Oxford / 2016 / 3 / a / Solution

Let brackets denote mass dimension. In natural units, $[x]=-1$ and $[{\partial }_{\mu }]=1$, so a dimensionless action requires $[\mathcal{L}]=D$. The kinetic term fixesThe cubic scalar field theory interaction then requires $[\lambda ]+3[\varphi ]=D$, givingThe coupling constant is dimensionless at this critical dimension of a monomial scalar interaction. For $D<6$ it has positive mass dimension, while for $D>6$ it has negative mass dimension, the usual power counting distinction between super-renormalizable and non-renormalizable cubic interactions.