On the Higgs-Confinement Complementarity

It has been noticed long ago that in Higgs models with `complete symmetry breaking' one can move from the confinement to the Higgs regime without crossing a phase boundary, a fact sometimes called referred to as `complementarity'. In a recent paper some doubt was raised about the correctness of the mathematics underlying this fact and it was claimed that the supposed `flaw' would resolve the `paradox' seen in this complementarity. Here we briefly revisit the facts both from a mathematical and a physical point of view and point out that (a) there is no paradox and (b) there is no flaw in the mathematical reasoning.