Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 304 / 4 / c / Solution

Write $\Psi ={\Psi }^a{T}_a$. The infinitesimal form of $\Psi \mapsto U\Psi U^{†}$ isThe adjoint gauge covariant derivative iswhich transforms as $D_{\mu }\Psi \mapsto U(D_{\mu }\Psi )U^{†}$. A gauge-invariant Lagrangian is thereforeThe trace and cyclicity make each term invariant under conjugation.