Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 301 / 3 / a / iii

Iterating the Schwinger-Dyson equations once, the leading connected correlator is one scalar propagator and two Dirac propagators meeting at one Yukawa interaction vertex. Amputation givesThus the invariant amplitude, in the convention $S_{fi}=i(2\pi {)}^4{\delta }^{(4)}{\mathcal{M}}_{fi}$, is ${\mathcal{M}}_{s,s^{\prime }}=-\lambda {\bar{u}}^s(p)v^{s^{\prime }}(q)$.