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

The invariant interaction isbecause ${\Phi }^{\ast }$ has charge $-2$ while the fermion pair has charge $+2$. Once $\Phi$ has a vacuum expectation value, it generates a Majorana-type fermion mass and can split the two Majorana components of the original Dirac field.

The operator has dimensionso its coupling is classically marginal in four dimensions. By comparison, $\overline{\Psi }\Psi$ has dimension three and its coefficient is a relevant mass parameter.