Naturalness and UV sensitivity in Kaluza-Klein theories

More than twenty years ago a paradigm emerged according to which a UV-insensitive Higgs mass mH and (more generally) a UV-insensitive Higgs effective potential V1l(0) are obtained from higher -dimensional theories with compact extra dimensions and Scherk-Schwarz supersymmetry breaking. Since then, these ideas have been applied to different models of phenomenological interest, including recent applications to the dark energy problem. A thorough analysis of the framework on which such a paradigm is based allows us to show that a source of strong UV sensitivity for mH and V1l(0), intimately connected to the nontrivial topology of these models' spacetime, was missed. The usual picture of the Scherk-Schwarz mechanism and its physical consequences need to be seriously reconsidered.