Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 3 / b / ii / Solution

For one complex scalar in the fundamental representation, the most general power-counting-renormalizable gauge-invariant Lagrangian isapart from a constant and the four-dimensional Yang-Mills theta term. Stability requires $\lambda >0$. A cubic candidate ${ϵ}_{ijk}{\Phi }_i{\Phi }_j{\Phi }_k$ vanishes because the scalar components commute.