Past exam of the quantum field theory course of the University of Oxford / 2017 / 3 / a / Solution

A free contraction must connect one $\varphi$ and one ${\varphi }^{\ast }$ field. In momentum space,Represent the nonzero propagator by a directed line so the two conjugate ends remain distinguishable; an undirected line alone would incorrectly allow same-charge contractions. Choose one consistent arrow convention, for example pointing from the starred end to the unstarred end.

The interaction contains two identical $\varphi$ fields and two identical ${\varphi }^{\ast }$ fields. Their $2!2!$ assignments cancel the coefficient $1/4$, givingThe vertex has two incoming and two outgoing arrows. Include $(2\pi {)}^d{\delta }^d(\sum p_i)$ at every vertex, integrate $d^{d}k/(2\pi {)}^d$ for each independent loop, and divide by the oriented graph's symmetry factor. There is no fermion-loop sign.