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

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