YM-Thermodynamics 2024, 35: The Dual Gauge Field

The field of the deconfining phase doesn't self-interact, so the coarse-grained field becomes affected by the free Abelian action. Local minimal U(1) gauge invariance defines the covariant derivative, by which the magnetic running coupling is defined. There are no higher-mass-dimension operators in the effective action for the fields. The BPS-saturation needs to be retained as a bound, so the fields are actually coupled minimally to keep momentum transfers above its lower bound. The action for the preconfining phase of an SU(2) YM-theory is

Since the partial derivative of the scalar field vanishes, the pure-gauge configuration is a solution to the differential equation. The density and pressure evaluates to

The Polyakov loop of the coarse-grained field evaluates to 1 independently of the chosen gauge, as long as it's valid. As a consequence, the electric Z2 degeneracy no longer persists in the preconfining phase. Analogously, for SU(3)

In the deconfining phase, SU(3)→U(1)² are independent in terms of magnetic monopoles, scalar gauge fields and (anti)monopole condensates, represented by the complex scalar fields. The dynamical breaking of the residual gauge symmetry U(1) and U(1)² is manifested in terms of a quasiparticle mass for the dual gauge field. The scalar field is inert and no "photon" selfinteraction occurs in an Abelian gauge theory, so the excitations in the effective theory for the deconfining phase are free thermal quasiparticles. It follows that the number of degrees of freedom before coarse-graining remains the same after coarse-graining, where one species of massive, dual gauge field and three polarizations remain. Add one degree of freedom before and one degree of freedom after coarse-graining, so for SU(3), there are six degrees of freedom at both stages. The contribution of quantum fluctuations to the thermodynamic pressure is a correction of order -5 to the thermal ground-state result. The ground-state pressure dominates the total pressure in the preconfining phase. At the critical scale, where the effective coupling diverges and the g-coupling vanishes, the total pressure P is continuous for the thermal transition between deconfining and preconfining phase. No higher loop corrections to the one-loop result for the pressure in the deconfining phase take place, since the fluctuations decouple. The SU(3) pressure is about twice the SU(2) pressure.

The invariance of the Legendre transformation between thermodynamic quantities under the applied coarse-graining implies that the derivative of the pressure vanishes for both gauge groups. Note for the energy density:

This defines the temperatures at which a system is more likely to preconfine and those that are more likely to deconfine. A condensate is stable for temperature between two critical temperatures. Whether this is real is dependent on the energy density. If the energy is smaller for a given spatial volume in the preconfining state than in the deconfining state, then a larger Boltzmann weight takes place during monopole-condensation. Topological defects within the (anti)monopole condensate are spatially resolved between the critical temperatures, where the condensate dominates the thermodynamics of the SU(2) YM-system.