 Scalar electrodynamics

A charged scalar field coupled to an Abelian gauge field through $D_{\mu }={\partial }_{\mu }-ie{a}_{\mu }$. Under $\varphi \mapsto e^{i\alpha }\varphi$ and $a_{\mu }\mapsto a_{\mu }+e^{-1}{\partial }_{\mu }\alpha$, its gauge covariant derivative transforms with the same phase. The potential determines whether the perturbative spectrum has a massless photon or exhibits the Higgs mechanism.