Order Parameters for the Higgs Phase 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).
 A: This is a real, and often encountered problem in general in condensed matter. As the comments in Lubos' answer discusses, it's not easy to invent an "order parameter" which unambiguously yields the correct phase since symmetries can't really be broken, and 2nd order correlators tend to display quantum/thermal fluctuations anyway. In condensed matter theory, a frequently used construct is the Off-Diagonal Long Ranged Order, $\lim_{r \rightarrow \infty} \langle \psi(0)\psi(r) \rangle$, which gives a measure of correlation. In practice however, real systems are finite and it is hard to take the limit physically. Anthony Leggett has though long and hard about this problem in the context of cold atoms and quantum condensation, and suggests something based around the single particle density matrix; it is unclear whether this generalises into things like QCD.
A: Dear Moshe, your question seems to demand the answer to be very complicated and unexpected and the obvious answer seems to be forbidden in between the lines except that I think that the obvious answer is right. 
The Higgs field's vacuum condensate is the order parameter of the vacuum in the electroweak theory. And if you define it in a gauge-invariant way,
$$\sqrt{H^\dagger H},$$
then it is gauge-invariant, too. Obviously, this quantity is still the same $v$ we all know and love. This fact, that the Higgs field is the order parameter, lies behind the fact that the Higgs mechanism uses the same maths as Landau's theory of phase transitions.
The vev can't be measured "directly" as a mass of something but it may still be measured indirectly - e.g. from the masses it gives to other particles (because the couplings may be measured, too).
