Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 6 / c

At a single interaction insertion, the Dyson series supplies $i{\mathcal{L}}_{\mathrm{int}}=-ig{\phi }^{†}\phi \varphi$. Amputating the three external propagators in b gives the momentum-space ruleIf the momentum-conservation factor is included in the vertex convention, append $(2\pi {)}^4{\delta }^{(4)}(p_{\varphi }+p_{\phi }+p_{{\phi }^{†}})$, with all momenta incoming. There is one charged line entering and one leaving the vertex, and a neutral real-scalar line. The absence of a factorial in the interaction is appropriate because its three fields are distinct. The amputated decay amplitude has magnitude $|g|$ with the usual convention that a graph computes $i\mathcal{A}$.