Renormalization of the Abelian-Higgs Model in the R-xi and Unitary gauges and the physicality of its scalar potential

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.