Diffusion of Wilson loops

A phenomenological analysis of the distribution of Wilson loops in SU(2) Yang-Mills theory is presented in which Wilson loop distributions are described as the result of a diffusion process on the group manifold. It is shown that, in the absence of forces, diffusion implies Casimir scaling and, conversely, exact Casimir scaling implies free diffusion. Screening processes occur if diffusion takes place in a potential. The crucial distinction between screening of fundamental and adjoint loops is formulated as a symmetry property related to the center symmetry of the underlying gauge theory. The results are expressed in terms of an effective Wilson loop action and compared with various limits of SU(2) Yang-Mills theory.

Figure 1

Wilson loop distribution for rectangular loops with aspect ratio two of different areas in SU(2) lattice gauge theory (left) and for an ensemble of field configurations containing 500 merons (right). These results have been fitted with distributions resulting from diffusion, Eq. (11), denoted by dashed lines.

Figure 2

Values of Wilson loops in different planes centered at the origin as a function of the area of the loops (in ${\mathrm{f}\mathrm{m}}^2$) for a single configuration in SU(2) lattice gauge theory (left) and for a configuration of 500 merons (right).

Figure 3

Approximate Casimir scaling for Wilson loops on a ${32}^4$ lattice with lattice constant $a=0.13$ fm. Left: Lattice Wilson loop distributions for loops of different aspect ratios with sizes given in lattice units. The solid lines are obtained from Eq. (10) with time $t$ determined by the expectation values. Right: Logarithm of Wilson loop as function of the area (in lattice units) for $j=1/2,1,3/2,2$ representations. The straight lines are scaled by the corresponding ratio of the Casimir operators.