Past exam of the quantum field theory course of the University of Oxford / 2017 / 2 / a

In $d$ spacetime dimensions, $[d^{d}x]=-d$ and a dimensionless action requires $[\mathcal{L}]=d$. Since $[{\partial }_{\mu }]=1$, the canonical kinetic term fixes $2+2[\varphi ]=d$. The cubic and quartic terms then giveThese brackets denote mass dimensions. The cubic coupling is dimensionless at $d=6$, whereas the quartic coupling is dimensionless at $d=4$. Hence there is no single dimension in which both nonzero couplings are dimensionless. For example, at $d=4$ the cubic coupling still has mass dimension one.^[ref]