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

In four dimensions the action is dimensionless and $[{\partial }_{\mu }]=1$. The kinetic terms giveso $[\Phi ]=1$ and $[\Psi ]=3/2$.

Up to a constant and total derivatives, the most general renormalizable Lagrangian invariant underisHere $\lambda >0$ is required for stability at large field. The last line is the allowed charge-conserving scalar-fermion Yukawa interaction; its precise two-component expression depends on the chosen spinor notation.