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

Promoting $\alpha$ to a spacetime-dependent parameter makes the scalar kinetic term vary by both terms linear and quadratic in the gauge field. Merely adding $e{J}^{\mu }{A}_{\mu }$ accounts for the linear term but is not invariant because the scalar current itself changes under a local transformation.

Withminimal coupling givesThe missing term is the seagull vertex $4e^2{A}^2|\Phi {|}^2$. This agrees exactly with replacing partial derivatives by gauge covariant derivatives.

The Dirac kinetic term is first order in derivatives:Its expansion is only linear in $A_{\mu }$, so the current coupling already completes the gauge-invariant fermion kinetic term and no $A^2\overline{\Psi }\Psi$ term arises.