Wilson loops with oppositely oriented plaquettes as a probe of center vortex structure

Abstract

We study Wilson loops with a nontrivial orientation structure in lattice gauge theory as a probe of the center vortex picture. The observable is a single Wilson loop containing two plaquettes with opposite orientations, realized in two geometries referred to as the vertical and parallel configurations. The vertical case behaves consistently with expectations from the vortex picture. In contrast, the parallel configuration shows a deviation from the naive area-law expectation which cannot be explained solely by the opposite orientations of the plaquettes. We introduce a simple qualitative vortex model which accounts for this behavior and shows that the observed effect can still be understood within the vortex framework.

Figures (15)

• Figure 1: When the phase contributed by a center vortex differs in sign, it can be denoted as the center vortex piercing the plane of the Wilson loop from opposite directions. Specifically, we can denoted that when the phase contributed by a center vortex piercing the Wilson loop is negative, it corresponds to an exit from the plane containing the Wilson loop (the left panel); conversely, a positive phase corresponds to an entry into that plane (the right panel) Mickley:2024vkm.
• Figure 2: If we define the oriented area enclosed by a Wilson loop within a plane as the area on its right side, then the Wilson loop in the left panel, composed of two small Wilson loops with the same orientation (denoted as $\langle W_{s.o.}\rangle$), has an oriented area that can be viewed as twice the area of a small plaquette. The Wilson loop in the right panel, composed of two small Wilson loops with opposite orientations (denoted as $\langle W_{o.o.}\rangle$), has an oriented area that can be regarded as zero. It is worth noting that the Wilson loop in the left panel is the boundary of a Möbius strip, while the one in the right panel corresponds to the boundary of an orientable rectangular strip (a cylindrical strip with a cut). In the following, the Wilson loops in this figure are denoted as 'vertical direction'.
• Figure 3: Same as Fig. \ref{['fig:fold']} but for case that the two plaquettes are in a same plane. In the following, the Wilson loops in this figure are denoted as 'parallel direction'.
• Figure 4: $\chi _P$ and $\chi _c$ as functions of temperature in the quenched approximation (the left panel) and when the dynamical fermions are turned on (the right panel).
• Figure 5: $\langle W_{o.o}\rangle$ and $\langle W_{s.o.}\rangle$ in the case of vertical direction (as shown in Fig. \ref{['fig:fold']}) at high and low temperatures in quenched approximation. The top-left panel corresponds to $l=1$, the top-right panel corresponds to $l=2$, The bottom-left panel corresponds to $l=3$, and the bottom-right panel corresponds to $l=4$, respectively.