Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 303 / 2 / b

At the Gaussian fixed point, the engineering dimension of the field isThe operator ${\varphi }^n({\nabla }^2\varphi {)}^m$ contains $n+m$ fields and $2m$ derivatives. Since its integral must be dimensionless,It is marginal when this vanishes, namelywhen $n+m\ne 2$. Since $n,m$ are positive, the exceptional case is $n=m=1$: $\varphi {\nabla }^2\varphi$ has a dimensionless coupling in every $d$ and differs from the kinetic term by integration by parts. If the displayed formula gives no positive $d$, there is no positive spatial dimension in which that operator is naively marginal.

At an interacting fixed point, field and composite operators acquire anomalous dimensions, and operators with the same symmetries can mix under renormalization. Their full scaling dimensions can therefore differ from these naive engineering dimensions.