Phase transitions are often detected by an order parameter - some quantity which is zero in the "disordered" phase, but becomes non-zero when order is established. For example, if the phase transition is accompanied by a breaking of a global symmetry, an order parameter can be any quantity which transforms non-trivially so that it averages to zero in the disordered phase.
Phases not characterized by their global symmetry are more tricky. For confinement there are well-known order parameters: the area law for the Wilson loop, the Polyakov loop (related to breaking of the center symmetry), and the scaling of the entropy with N in the large N limit.
My question is then about the Higgs phase, which is usually referred to (misleadingly in my mind) as spontaneous breaking of gauge "symmetry". More physical (among other things, gauge invariant) characterization of this phase would be in terms of some order parameter. What are some of the order parameters used in that context?
(One guess would be the magnetic duals to the quantities characterizing confinement, but there may be more).