Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 304 / 4 / a / Solution

Substitution of the gauge transformations and $U{U}^{†}=I$ givesA direct commutator calculation givesCovariance of the left side then implies $F_{\mu \nu }^{\prime }=U{F}_{\mu \nu }{U}^{†}$. The cyclic property of the trace makes the traces of $F_{\mu \nu }{F}^{\mu \nu }$, $D_{\mu }\varphi D^{\mu }\varphi$, and ${\varphi }^3$ invariant, so the Lagrangian is gauge invariant.