Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 3 / c / ii / Solution

Because normal ordering removes contractions within either exponential, Wick theorem sums all cross-contractions intoUnder $z\mapsto \lambda z$, this scales as ${\lambda }^{-k^2}$. Each operator has scaling dimension $\Delta =h+\tilde{h}=k^2/2$, so a two-point function of two such primaries must scale as ${\lambda }^{-2\Delta }={\lambda }^{-k^2}$. The two calculations agree.