In 4D Yang-Mills theory, perturbation theory generates an infrared cut-off for magnetic screening, that is too weak for convergence of thermodynamical quantities. The nontrivial thermal ground state on the propagation and interaction is affected by this effect even at higher temperatures. It's constructed of gauge field configurations, approximated by the perturbative expansion and will add mass to the effective theory. The apparent gauge-symmetry breaking is independent of a small coupling assumption. It leads to a fast convergence of loop expansions, evading the perturbative infrared problem. Motivation for the approach is taken from replacing the renormalization scale by the temperature at asymptotic freedom.
Matsubara frequencies of course dictates the modes in terms of mass. The vanishing ones express massless 3D gauge modes, while the rest are associated with the massive ones, which behave linearly with temperature. Massless 3D modes are given by the effective action, and can be expanded to four dimensions via perturbative matching. It contains some free parameters, in form of coefficients
The consideration for three dimensions is less computationally intensive, but requires truncation at low values of M, and a convergence behavior at nonzero M, to match the 4D behavior. In 4 dimensional perturbation theory, the mass-scale is generated by the gauge coupling constant g, The momenta (i.e. powers of g) of order T are categorized into hard, soft and ultrasoft hierarchy for the modes. Integrating out the momentum scales starting from hard and working down the orders, will experience screening of softer modes by the harder ones. The Interaction monomial I of the YM-Lagrangian is hierarchically smaller than the expectation value K. The magnetic modes experience weak screening. K~g⁶T⁴, I = g⁶T⁴, so K ~ I. Starting at sixth order of g, the loop expansion of the pressure (sans free energy) contains contributions otherwise not determinable in perturbation theory. This describes an effective Lagrangian density. On the other hand, below this order, YM-theory is considered somewhat incomplete. Matching of the 3D coefficients to the full theory happens perturbatively.
To address the inherent global symmetries, take the Polyakov loop as a central field variable in the 3D effective theory. It's a Wilson line of the gauge field along a compact time interval, and transforms under 4D gauge transformations induced by the native group elements. As an element of the center group, its elements are distinct, and commute with any SU(N) group element.
For SU(2), the perturbative definition of the gauge field without absorption of the fundamental coupling g is defined through its Wilson line on a Euclidean background
with the path-ordering operator P. The gauge transformations are associated with a center jump along the time-axis. Through the invariance under singular, periodic gauge transformations induced by the group, the YM action density partition function is also invariant.
This remains true for all SU(N) under the same generalizations of the center jump. This is a global symmetry. The Polyakov loop in its totality can be considered an order parameter for deconfinement, associated with the free energy of an infinitely heavy test "quark", transforming under the fundamental representation of SU(N). This is somewhat problematic, as it can be constructed to have infinite free energy, and so far no mechanism is included to stop the particle to gain infinite mass as well. This should be rejected as a test particle in a thermodynamical system. Outright rejection should categorically apply to isolated heavy quarks - it's only allowed to exist in a compound state with an antiquark, in which the gauge charge is completely screened at long distances. For Dirac fermions and its standard action, the gauge-field component is gauge-rotated into the one-direction of the Lie algebra, so that the heavy test-charge is located at spatial origin. The density is not in itself gauge-invariant. Under the effects of the full partition function, the thermal YM and static test-charge must be related to an expression invariant under smooth gauge transformations.
As Z should be finite, the vanishing of the average indicates that the electric and global center symmetry of the effective 3D theory remains unbroken, and the test-quarks are confined. If the average does not vanish, the N-fold center degenerate entails a finite force. This is the deconfinement scenario.
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