January 2025 - Phases in YM-TD

I'm reading this one again mostly in order to get the terminology straight for myself. Most of the aspects of this paper, I suspect I've already seen in various levels of detail, but considering my lack of background for cosmology, I should at least be able to not have to think actively about the thermodynamics of it all.

SU(2) YM-TD happens in three phases: Deconfining, preconfining and confining in order of descending temperature. The temperature separating deconfining and preconfining phases is considered the "critical temperature". The thermal ground state for the deconfining phase can be derived through coarse-graining on a trivial-holonomy (anti-)caloron field, derived through one-loop quantum fluctuations. The solutions to the fundamental YM equation on the Euclidean cylinder have neither pressure, nor energy-density, they are nonpropagating. Spatial coarse-graining leads to an integral over the 3D spatial ball (center) at any value of effective time τ. The singular gauge that the (anti)caloron is constructed in, the center locates the topological charge in the sense of a 3D surface integral over the maximum of the action density of the Chern-Simons current. It represents the topological charge of the (anti)caloron independently of the minimum radius. The averaged radius is the normalization, cubically rising with the upper cutoff, and producing a harmonic τ-dependence. When these centers are densely packed, they gain an inert, temporally winding adjoint scalar field, which breaks the fundamental gauge symmetry from SU(2) to U(1). The YM scale Λ for the deconfining phase determines the field modulus. Under dense packing, the (anti)caloron peripheries overlap with one another, and with centers, so coarse graining delivers the position of (anti)caloron centers through a pure-gauge solution to the effective YM equations. The peripheral overlap at delivers the ground-state energy density and pressure, which characterize the thermal ground state. On microscopic level, the overlap introduces light nontrivial holonomy to the caloron, which leads to negative ground-state pressure. The adjoint Higgs mechanism invoked by the scalar field has two of three propagating gauge field components dependent on the quasiparticle mass T, hence being massive. Under de-winding of the gauge field, the Polyakov loop on it exchanges its center element, dynamically breaking the electric center symmetry. The associated modes of the effective excitation are either purely quantum thermal for all frequencies (massive) and classical, off-shell, or quantum thermal, depending on frequency (massless). The one-loop approximation is accurate under these considerations for TD bulk quantities.

If the one-loop fluctuations and thermal ground-state estimate are consistent under Legendre transform, gain ordinary differential equations. Assume a << 1

This violates the assumption of a << 1, invalidating the approach at small temperatures. The exact solution grows with decreasing λ. The singularity behaves logarithmically, which implies masslessness for isolated screened monopoles, liberated by the dissociation of large-holonomy (anti)calorons and collectively described by effective radiative corrections. The holonomy of typical (anti)caloron becomes large as λ decreases to its critical value.

In the preconfining phase extends from a λ value of 13.87 to 11.57. At 12.15, the preconfining phase energy density matches supercooled massless photonic exciation phase tunneling. Microscopically, the thermal ground state associates with massless magnetic (anti-)monopoles, assumed to be electric. The dense packing of their cores is described by a complex scalar field, which breaks the remaining gauge symmetry U(1) dynamically. The overlap then is described by an effective pure-gauge configuration. In the preconfining phase, the Polyakov loop on the ground-state gauge field is unity in winding and unitary gauge, so this phase already confines infinitely heavy, fundamental test charges. The pressure is around 0, which is a superconductor characteristic, reigned in by the phase mixing of the field. Still, electric conductance is expected to be very high compared to either of the other phases.

In confining phase, the unstable defects of the massive loops at selfintersections become massless and metastable. They are solitons, which break the magnetic center symmetry dynamically upon condensation after shrinking to massless points. This requires a new degenerate ground state, which confines the fundamental test charges, not supporting thermal gauge-mode excitation. Energy density and pressure at this point is zero. Above the ground state, the center-vortex loop enters Hagedorn transition, a phase change by ±π of a complex scalar field, determined by symmetry, ground-state continuity, and pressure. Collisions of loops lead to twistings and formation of stable regions of selfintersections, characterized by the blobs of maximal radius. The density of states for these soliton solutions are subject to an arbitrary number of selfintersections, rising more than exponentially with energy. There is no partition function for λ around the point where the massive loops become massless, so thermodynamics will not work as a description.