 Nonlinear R-xi gauge in polar Higgs variables

For the polar parametrization of the Abelian Higgs field with $D=\partial +ieA$, adding $-H^2/(2\xi )$ cancels quadratic vector-phase mixing. The cross terms reduce, modulo a total derivative, to $-2e(1+h/f)\pi A^{\mu }{\partial }_{\mu }h$; interactions remain. The Faddeev-Popov operator is proportional to $\square +\xi e^2(f+h{)}^2$, so the Faddeev-Popov ghosts couple to $h$. The vector propagator falls as $k^{-2}$ at fixed finite $\xi$, but the polar kinetic and nonlinear gauge fixing vertices include terms of mass dimension above four. Manifest polynomial power counting instead uses Cartesian scalar variables with conventional R-xi gauge.