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

The propagator carries a real-species label rather than a charge-flow arrow:Differentiating the interaction with respect to its four external real fields gives the compact tensor ruleThus four ${\varphi }_1$ legs or four ${\varphi }_2$ legs give $-3i{\lambda }_0/2$, while two of each give $-i{\lambda }_0/2$. Vertices with an odd number of either species vanish. For example, $4!({\lambda }_0/16)=3{\lambda }_0/2$, whereas $2!2!(2{\lambda }_0/16)={\lambda }_0/2$. These factors already account for assignments at one vertex; graph symmetry factors must still be included separately. Momentum conservation and loop measures are the same as in part a.