YM-Thermodynamics 2024, 28: Free Thermal Quasiparticles

At tree-level in the effective theory, constraints need to be made on the propagation and interaction for k = 0 modes. Excitations, which at tree level have some temperature-independent mass, are associated with quasi-particles. In general, each broken generator of the original gauge symmetry with a mass is associated. Apply the SU(2) case in unitary gauge for mass relations.

The four mass-degenerate directions in su(3) impose unitary gauge from SU(3) to SU(2) via their gauge conditions, and in addition, the Coulomb gauge for the unbroken U(1) leads to a completely fixed, physical gauge. This is a physical solution, because the quasi-particle mass spectrum, the physical number of polarization and the transversality of free, massless, propagating gauge modes in the field are consistent with one another. The massless modes may be imposed for a consistent model. Since coarse-graining does not interfere with the number of degrees of freedoms, one only needs to consider the SU(2) case. In unitary-Coulomb gauge for free quasi-particles, the real-time propagators of the fields associate the polarization indices 1, 2, and separately 3.

u = (1, 0, 0, 0) represents the 4-velocity of the head bath. For the 1, 2 modes, only thermal propagation occurs, and that only in the term proportional to the normal vector contributes.

On the one-loop level in the effective theory, the radiative shift of the potential is negligable. It can be estimated by the contribution of the massless mode, appropriately weighted by the number of polarizations of all 3 fields.

This is a generous upper bound. Ignoring the λ-dependence of the effective coupling for now, the one-loop pressure in SU(2) comes to

For SU(3):

The Legendre-transformation fixes the evolution of the effective coupling at one-loop level, since the partition function in terms of fundamental fields needs to be obeyed after reformulation. An additional non-trivial condition on the temperature evolution counters the temperature-dependent parameters m, P.

This indicates a phase boundary, since the massive modes decouple due to diverging effective gauge coupling leading to the magnetic monopole charges being screened completely, and ending up massless. Screened charges make no contributions, but are described collectively in the effective radiative corrections. An attractor to the evolution a(λ) is given in SU(2) is

The constancy of the effective coupling for small a conserves the monopole during msot of the evolution. The magnetic charge and mass of a screened magnetic monopole vanish. The pressure is negative just shortly above the critical temperatures, where YM-thermodynamics starts to be ground-state dominated.

The trace of the energy-momentum tensor at free quasi-particle levels in the effective theory has a nonzero expectation value. The thermal ground state and quasiparticle masses each have contributions to the underlying nonperturbative effects.

The remainder of the energy-momentum tensor consists of the perturbative effects. In SU(N), at all orders of coupling, it's related to the field-strength tensor.

where the μ (if it's not an index) is the mass scale resolution, applied to the process at which the coupling is extracted. At one-loop level, the Landau pole occurs in the solution to the above equation. A thermal ensemble average leads to the energy.