Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 304 / 3 / c / Solution

The original cubic vertex gives $g$. At order $g^3$, the one-particle-irreducible correction to the cubic interaction is the triangle diagram: one slow external leg leaves each of three cubic vertices, and three high-mode propagators join the vertices cyclically. Diagrams with a high-momentum bridge carrying only a sum of vanishing external momenta do not contribute to the local zero-momentum coupling; external self-energy diagrams belong to the neglected field rescaling.

The relevant interaction at each vertex is $(g/2)\varphi {\chi }^2$. The third connected cumulant contributesWick's theorem gives $8G_{>}(x-y)G_{>}(y-z)G_{>}(z-x)$. Projecting onto a local interaction therefore givesFor $d=6$, the area of the unit five-sphere is ${\Omega }_5={\pi }^3$, and henceThus