What symmetry is actually broken in the Higgs phase of a gauge-Higgs theory?

Abstract: In SU($N$) gauge-Higgs theories, with a single Higgs field in the fundamental representation, there exists in addition to the local gauge symmetry a global SU(2) symmetry, at $N=2$, and a global U(1) symmetry, for $N \ne 2$. We construct a gauge-invariant order parameter for the breaking of these global symmetries in the Higgs sector, and calculate numerically the transition lines, in coupling-constant space, for SU(2) and SU(3) gauge theories with unimodular Higgs fields. The order parameter is non-local, and therefore its non-analyticity does not violate the theorem proved by Osterwalder and Seiler. We then show that there exists a transition, in gauge-Higgs theories, between two types of confinement: ordinary color neutrality in the Higgs region, and a stronger condition, which we have called "separation-of-charge confinement," in the confinement region. We conjecture that the symmetry-breaking transition coincides with the transition between these two physically different types of confinement.