YM-Thermodynamics 2024, 27: Effective Action and a priori Estimate of Thermal Ground State
Fluctuations in the topologically trivial sector are formally possible, but have not yet been considered for the model. Due to the scale of maximal resolution, an effective theory can be established even in their presence. The effective action density is first subject to a maximal resolution |ϕ|.
This action is gauge-invariant, and the gravitation term demanded by perturbative renormalizability is present, so that integrating out k = 0 fluctuations down to a certain resolution retains the form of the YM-action. The nonlocal terms are expandable into powers of D, so their contributions can be excluded, and the nonlocal-invariant terms can be ignored. The coupling between a and ϕ contained in the squared derivative of ϕ may be ignored, since no momentum transfer occurs between them for the coupling. This defines the field as a pure-gauge configuration of a. The effective action as written then is complete.
The fluctuating field a is integrated out loop-expanding the logarithm of the partition function about the free quasiparticles. This is nontrivial. Momentum transfer in the resulting 3- and 4-vertices is constrained further by the existence of the maximal resolution. The effective coupling e is determined ultimately by the invariance of Legendre transformations between thermodynamic quantities under the applied coarse-graining (usually defined up some loop order). The ground-state estimate for SU(2) is given a priori via the configuration and pure-gauge configurations.
These terms include temperature-dependent cosmological constants, and the interactions between calorons, as well as their radiative corrections, associated with momentum transfers larger than |ϕ|² lift the energy density of the BPS estimate to a nonzero-value, dependent on temperature and YM-scale. This completes the thermal ground state estimate.
The ground-state pressure, however, is negative. This is microscopically explained by (anti-)calorons of small holonomy having their BPS magnetic (anti-)monopole constituents, attracting one another under the radiative corrections. The excitation of large caloron holonomy leads to the dissociation of the associated pairs, and further to the monopole-screening. The Polyakov-loop under the periodic gauge transform shows the electric Z2 degeneracy of the thermal ground-state estimate, which characterizes deconfinement