 Solution

Choose the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$. Write $\rho =f+h$, $r=1+h/f$ and $B_{\mu }={\partial }_{\mu }\pi /f+e{A}_{\mu }$. The polar parametrization of the Abelian Higgs field givesHence the full Lagrangian density isAssume $f>0$ and $\lambda >0$, so the minimum has $|\varphi |=f/\sqrt{2}$. In this convention the local gauge symmetry acts asThe combination ${\partial }_{\mu }\pi +ef{A}_{\mu }$, the radial field $h$, and the gauge field strength are each invariant, proving invariance of the displayed Lagrangian density.

This parametrization follows the circle of classical minima: $h$ measures distance away from the vacuum manifold, whereas $\pi$ moves along a gauge orbit. It separates the massive radial Higgs mode from the would-be Goldstone boson and makes its removal by the Higgs mechanism transparent. At quadratic order,The polar parametrization of the Abelian Higgs field is a local description near the chosen nonzero vacuum expectation value; it is singular at $\varphi =0$ and is not a global coordinate system for configurations with zeros or winding.