YM-Thermodynamics 2024, 39: Evolving Center-Vortex Loops, n = 0

With the divergence of mass of the dual gauge field containing collapsing, closed and unresolved magnetic flux lines of finite core-size in center-vortex loops, it's implied that the core-size vanishes. The center-vortex loops with nonvanishing selfintersection number then become stable solitons. Through the limit of negative pressure for the vanishing vortex core, isolated center-vortex loops become particle-like excitations, classified through their knot-topology and distribution of magnetic charge direction over selfintersection. The case of n = 0 has no topological reason for stability, so it shrinks to nothingness within finite time. For low-energy SU(2) YM-theory, closed curves describe the late-time behaviour of center-vortex loops with n = 0. The n = 0 however does not come with the energy required to form the intersection point, hence the shrinking out of existence. A planar center-vortex loop of finite length L and selfintersection number n = 0 maintains its mass by frequent interactions with a noisy environment. Its finite resolving power shrinks with the center-vortex loop, and to a point at finite resolution. Since the mass for n > 0 is also finite, a gap exists in the mass spectrum of the theory when probing the system with average resolution.

For an n = 0 center-vortex loop of finite core-size of an isolated SU(2) YM, embedded onto a 2D surface, an outside observer in the plane will measure a directed curvature along some sector of the vector line. This indicates pressure on either side of the line. The speed of the resulting motion is a monotonic function of the curvature, shrinking the center-vortex loop.

This effective action is defined geometrically, with the parameter σ as the environment, the presence of which leads to an asymptotic mass gap. The critical value τ = T marks the end of the flow that begins at τ = 0.

The Wilsonian renormalization group flow for the curve-shrinking is a partition function, as a statistical average over n = 0 center-vortex loop, that remains invariant under a change of the resolution expressed by τ. A geometric ansatz for the effective action derives from integrals over local densities. The action can be factorized into Euclidean point-symmetrical parts that are invariant and not invariant under the shift of scaling symmetry x → λx. Since the evolution generates circles in the limit of T, higher derivatives of k rapidly vanish. The conformally invariant expressions are derived through factors or the inverse of integrals involving lower derivatives.

The partition function numerically describing the effective action is the average over the ensemble of curves E.