Introducing vortices in the continuum using direct and indirect methods

Inspired by direct and indirect maximal center gauge methods which confirm the existence of vortices in lattice calculations and by using the connection formalism, we show that under some appropriate gauge transformations vortices and chains appear in the QCD vacuum of the continuum limit. In the direct method, by applying center gauge transformation and “center projection,” QCD is reduced to a gauge theory including vortices, which corresponds to the nontrivial first homotopy group ${\mathrm{\Pi }}_1(\mathrm{SO}(3))=Z_2$. On the other hand, using the indirect method, in addition to the center gauge transformation and center projection, an initial step called Abelian gauge transformation and then Abelian projection are applied. Therefore, instead of single vortices, chains that contain monopoles and vortices appear in the theory.

Figure 1

Linking between the Wilson loop $C$ and a vortex with hypersurface $\mathrm{\Sigma }$. The boundary of the hypersurface is shown by $\partial \mathrm{\Sigma }$, indicating a thin vortex.

Figure 2

Appearance of a vortex and an antivortex under center gauge transformation.

Figure 3

Appearance of a monopole accompanying a Dirac-string, an anti-Dirac string, and an antimonopole as a result of an Abelian gauge transformation.

Figure 4

Appearance of a monopole attached to a Dirac string and a vortex after Abelian gauge transformation, Abelian projection, and center gauge transformation.

Figure 5

Appearance of an anti-Dirac string and an antivortex after Abelian gauge transformation, followed by an Abelian projection and then a center gauge transformation.

Figure 6

Appearance of a chain after Abelian gauge transformation, Abelian projection, center gauge transformation, and center projection.

Figure 7

Vortex field strength: (a) before gauge fixing, (b) after maximal Abelian gauge fixing in the horizontal $\pm {\sigma }^3$ direction, and (c) after Abelian projection [24]. As shown in this figure, two vortex enter a monopole or an antimonopole, in agreement with what we have introduced in this paper as a chain configuration in Fig. 6.

Figure 8

Illustration of the connection between Dirac string shown in plot (a) and the center vortex shown in plot (b) [41]. The interpretation of a chain represented in Fig. 6 is the same as this figure where two vortex enter a monopole.