Past exam of the quantum field theory course of the University of Oxford / 2018 / 2 / a / Solution

Use mass dimensions with $[x]=-1$, $[{\partial }_{\mu }]=1$ and dimensionless action. The Lagrangian density in Euclidean space has dimension $D$, and the kinetic term fixes $2+2[{\varphi }_0]=D$. ThusA dimensionless coupling constant is thereforeHere $\mu$ is a positive regularization scale with mass dimension one. The interaction is marginal at its upper critical dimension $D_c=2k/(k-1)$, relevant below it and irrelevant above it by this engineering-dimension test. In particular, $d_2=4-D$, while $d_k=4-2k+2(k-1)ϵ$ near $D=4-2ϵ$. At this stage $g_{2k}$ is the dimensionless version of the displayed bare coupling; after introducing counterterms one may use the same scale factor for the renormalized coupling.