# Doubts about quark confinement, the pole mass and the quark gluon plasma

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I have seen written a few times that the notion of a pole mass for a quark contradicts the quark confinement picture and that non-perturbatively it is expected that the quark pole mass be infinite. Nevertheless, I have also heard about the quark gluon plasma where quarks and gluons are deconfined. This means that at least in some situations speaking about deconfined quark masses does make sense and that the argument pointing out that the pole mass should be infinite for quarks because of confinement trembles. How does one reconcile these ideas? Is the quark pole mass expected to be infinite or not?

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The overall picture is the following:

Reinterpreting nonrelativistic scattering theory in terms of quantum field theory one finds that the finite poles of the renormalized propagator are the masses of the asymptotic particle states (corresponding to the bound states in the nonrelativistic theory) of the theory with the appropriate quantum numbers. Since quarks are confined, there are no such asymptotic states for the quark field, hence the renormalized quark propagator should have no finite poles. But the concept of asymptotic states is appropriate only at zero (or sufficiently low) temperatures.

At higher temperatures the role of the asymptotic states is taken instead by quasiparticle states. In quantum field theory, these correspond to representations of the observable algebra different from the vacuum representation. Here the spectrum of the Hamiltonian is not bounded below, and one has instead to satisfy the so-called KMS condition. This changes everything.

The representation becomes temperature dependent, and at sufficiently high temperature a phase transition is observed numerically that changes the quasiparticle spectrum and results in quasiparticles with the quantum numbers of quarks. The corresponding 2-point functions have complex poles describing the mass and lifetime of the quasiparticles in a similar way as, in the nonrelativistic case, complex poles of the analytically continued resolvent of a Hamiltonian describes energy and lifetime of unstable particles and resonances.

In the literature, all these things look somewhat nebulous since there is no clear common language mediating between the relativistic and the nonrelativistic case, and everything is argued in terms of analogies.

answered Jan 18, 2016 by (15,747 points)

Thanks for the answer. I want to say something though, is it possible to@ArnoldNeumaier  have a picture of this without having to resort to temperatures? I mean,  temperature has never stricken me as a fundamental concept in field theory. I figure that in the study of quark gluon plasmas the concept of temperature will appear quite naturally but it's a little dissatisfying (for my taste) having an answer about the pole mass of a quark using temperatures.

@DmitryhandmetheKalashnikov: in a loose sense, you can interpret the temperature as an energy scale (convert it using Boltzmann's constant). But temperature is absolutely essential for bulk matter except if the latter is a crystal. in all other cases you cannot calculate anything about bulk matter based on the vacuum expectation values that governs few-particle physics when treated by QFT.

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