Polar parametrization of the Abelian Higgs field ($\varphi =(f+h)e^{i\pi /f}/\sqrt{2}$)

Near a nonzero vacuum expectation value, the radial variable $h$ describes the Higgs mode and the angular variable $\pi$ describes the would-be Goldstone boson. For $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$, their local gauge transformation is $h\mapsto h$, $\pi \mapsto \pi +f\alpha$, and $A_{\mu }\mapsto A_{\mu }-{\partial }_{\mu }\alpha /e$. The scalar kinetic term is $(\partial h{)}^2/2+(1+h/f{)}^2(\partial \pi +efA{)}^2/2$. These coordinates simplify gauge orbits but obscure ordinary power counting through higher-dimensional derivative vertices and fail at $\varphi =0$.