= Polar parametrization of the Abelian Higgs field
{title2=$\phi=(f+h)e^{i\pi/f}/\sqrt2$}
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 $D_\mu=\partial_\mu+ieA_\mu$, 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 $\phi=0$.
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