YM-Thermodynamics 2024, 34: Condensation of Magnetic Monopole-Antimonopole Pairs

The magnetic charge is obtained through the t'Hooft tensor

In unitary gauge, where the orientation of the adjoint Higgs field in su(2) all have a shared fixed direction, the flux over the B-field is compensated by a Dirac string in position space, and monopoles centrally trapped by a sufficiently large S2, through which the magnetic flux vanishes. This is a very classical string-theory construction, wherein the S2 experiences the Hedgehog's lemma from topology. Static (anti-)monopoles outside S2 of infinite radius that lie in a distance of b away from its surface with Dirac strings not piercing the surface, then

For SU(2), the unitary gauge imposes an isolated system of an (anti-)monopoles pairs at rest and not interacting outside of an S2 with infinite radius. Their Dirac strings are characterized by unit vectors pointing away from the cores of the (anti-)monopole pair, and some plane perpendicular to S2 with an intersection line L = P ∩ S2 coincides with that of S2 and the plane spanned by the Dirac string vectors. Whether or not this system contributes to the magnetic flux through S2 depends on the angle between the planes, and the angle of projection of the unit vectors onto P-forms with L. In the absence of a heat bath, give the flux as by the probability of measuring it

The interaction of the monopoles can be modeled using Bose-Condensates by coupling the isolated system to a surrounding heat bath, then projecting onto zero-spatial-momentum states of the system, so that each constituent carries no spatial momentum. The massless limit below the critical temperature gives the phase describing interaction. Doing this gives

The transition between deconfining and preconfining SU(2) YM-thermodynamics from above in a specified way, becomes complicated through the effective value of coupling. Starting from the a priori estimate of the deconfining thermal ground state, the computation of thermodynamic quantities is organized into a rapidly converging loop expansion in terms of effective gauge-field fluctuations of trivial topology, so at high temperatures, the unresolved stable, screened (anti)monopoles of a number density scales with the third power of temperature. Slightly above the critical temperature, the collective dynamics of the defects induces either screening or antiscreening on the tree-level massless, effective gauge mode. Numerically,

Based on a model of noninteracting Bose-particles with mass m in a condensate,

When the critical temperature is rapidly approached from above, an adiabatic approximation takes into account the static screening effects imposed at high temperature, and neglects the propagating dual gauge modes close to the critical temperature. The pole in the coupling is logarithmic, hence interacting, but statistically equilibrated. The nonrelativistic nature of (anti)monopole pairs gives an expression for μ

So either pairs of stable (anti)monopoles condense when becoming massless (at pole position) for effective coupling (scenario for infinitely slow approach to critical temperature), or the effective coupling has a lower bound (scenario for infinitely fast approach to critical temperature).

The coarse-grained condensate of (anti)monopoles come with a previously fixed differential operator D, which annihilates the phase of the complex scalar field.

The field is BPS-saturated, which allows solving the differential equation by introducing a mass scale. A scale of maximal resolution in the effective theory for the preconfining phase follows from it. The monopole in the interior of the S2 construction can then be separated from the antimonopole in the exterior, then experience the same magnetic flux as an S2 for infinite radius. The (anti)monopoles can't probe the finite curvature, and the infinite-surface limit is trivially saturated in the spatial coarse-graining process.