  We perform an old school, one-loop renormalization of the Abelian-Higgs model
in the Unitary and $R_\xi$ gauges, focused on the scalar potential and the
gauge boson mass. Our goal is to demonstrate in this simple context the
validity of the Unitary gauge at the quantum level, which could open the way
for an until now (mostly) avoided framework for loop computations. We indeed
find that the Unitary gauge is consistent and equivalent to the $R_\xi$ gauge
at the level of $\beta$-functions. Then we compare the renormalized, finite,
one-loop Higgs potential in the two gauges and we again find equivalence. This
equivalence needs not only a complete cancellation of the gauge fixing
parameter $\xi$ from the $R_\xi$ gauge potential but also requires its
$\xi$-independent part to be equal to the Unitary gauge result. We follow the
quantum behaviour of the system by plotting Renormalization Group trajectories
and Lines of Constant Physics, with the former the well known curves and with
the latter, determined by the finite parts of the counter-terms, particularly
well suited for a comparison with non-perturbative studies.
