Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 3 / c / Solution

With the convention that the fields have charges $q_{\Phi }=2$ and $q_{\Psi }=1$, the Noether current may be writtenup to an overall sign convention for the generator. The field equations, including the invariant Yukawa interaction, give ${\partial }_{\mu }{J}^{\mu }=0$. Subject to vanishing flux at spatial infinity, the Noether chargeis conserved and generates the global $U(1)$ transformations.