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I think that, by now, it's understood that the gluon propagator in QCD has a dynamically generated mass. Ok, so my question is the following: where does the extra polarization degree of freedom come from? Or, asking in another way: suppose you try to define an S matrix for QCD, apart from the usual problems for doing so, would it be unitary? How?

In the case of a Higgs mechanism, it is clear that the extra degree of freedom comes from "eating" the Higgs, as they say. But and where does is come from in theories where the mass is dynamically generated?

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Is that the gluon propagator with respect to the $\xi$-gauge? Or some other gauge? Because the gluon propagator is gauge-dependent. –  QGR Jan 31 '11 at 15:44

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The asymptotic states of QCD are gauge invariant. They can include mesons which are quark-anti quark bound states and glueballs (which are roughly speaking bound states of gluons) but not gluons themselves. It doesn't really make any sense to say that the gluon propagator has a dynamically generated mass as this is a very gauge dependent statement and gluons are not asymptotic states with a well defined mass. Glueballs are massive and can have various spins, but there no puzzle regarding counting degrees of freedom because one is not starting with a perturbative state with fewer degrees of freedom and adding an additional degree of freedom as in the Higgs mechanism. This is not to say that the mass generation for glueballs is obvious. As a matter of fact, proving that pure glue QCD has a mass gap will win you a million dollars from the Clay Institute.

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I understand, this is exactly what I meant by "the usual problems in trying to define the QCD S matrix". But is this true for all theories with dynamically generated mass? I mean, if I write down a theory which is not confining but has a mass dynamically generated, how is this problem solved? Or is it the case that such theory does not exist? In this case, why not? –  Rafael Jan 30 '11 at 23:15

I'm not sure this is useful, but I suppose that the problem with an S-matrix for gluons is that the gluons are not free. That is, an S-matrix deals with free particles that interact, perhaps exchange bodily fluids (i.e. charge or whatever), and then escape to infinity. But the real questions with gluons have to do with bound states. So the problem is the absence of free particles at the beginning and end of the interaction.

On the other hand, I want to believe that an S-matrix "formalism" exists even with bound particles. Instead of letting the participants escape to infinity, you assume that they eventually return to their initial state. Then perhaps you can derive some restrictions based on the requirement that the initial state repeating. This is, sort of like a dual to the usual S-matrix.

Here's the wikipedia article on S-matrix theory: http://en.wikipedia.org/wiki/S-matrix

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The Wikipedia entry on LSZ reduction is probably more relevant for dealing with a theory like QCD which has only bound states of the fundamental constituents as asymptotic states: en.wikipedia.org/wiki/LSZ_reduction_formula . Unfortunately it isn't really stated there in sufficient generality. You can define the S-matrix for asymptotic states provided only that you can find an operator which has a nonzero amplitude to create the state from the vacuum. Of course a definition is not the same as a computation. –  pho Jan 31 '11 at 0:17

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