Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 2 / e / iv / Solution

In the displayed connected Feynman diagram, each sextic vertex carries two slow external legs and the four remaining legs at each vertex are paired across the vertices. The second cumulant expansion has a factor $-1/2$, the slow legs can be selected in ${\left(\genfrac{}{}{0ex}{}{6}{2}\right)}^2$ ways, and the four cross-contractions can be paired in $4!$ ways. The coefficient is thereforeWriting ${\mathcal{S}}_{\zeta }=\{q:\Lambda /\zeta <|q|<\Lambda \}$, the local zero-external-momentum contribution iswith the integral restricted further so that $q_1+q_2+q_3\in {\mathcal{S}}_{\zeta }$. The three independent internal momenta agree with the diagram's loop order $L=4-2+1=3$.