Centre Vortex Structure of QCD-Vacuum Fields and Confinement

The non-trivial ground-state vacuum fields of QCD form the foundation of matter. Using modern visualisation techniques, this presentation examines the microscopic structure of these fields. Of particular interest are the centre vortices identified within the ground-state fields of lattice QCD. Our current focus is on understanding the manner in which dynamical fermions in the QCD vacuum alter the centre-vortex structure. The impact of dynamical fermions is significant and provides new insights into the role of centre vortices in underpinning both confinement and dynamical chiral symmetry breaking in QCD.


Centre Vortex Identification
In identifying centre vortices one commences with a gauge fixing procedure which brings the lattice link variables as close as possible to the identity times a phase. Here, the original Monte-Carlo generated configurations are considered. They are gauge transformed directly to Maximal Centre Gauge [15,23,24], without preconditioning [25]. The brings the lattice link variables U µ (x) close to the centre elements of SU (3), One considers gauge transformations Ω such that, and then projects the link variables to the centre and m µ (x) = −1, 0, 1. The product of these centre-projected links around an elementary 1×1 square on the lattice reveals the centre charge associated with that plaquette. The centre-line of an extended vortex Z y (n +x) x y z Z y (n +x) in three dimensions is identified by tracing the presence of nontrivial centre charge, z, through the spatial lattice A right-handed ordering of the dimensions is selected in calculating and illustrating the centre charge. If z = 1, no vortex pierces the plaquette. If z = 1 a vortex with charge z pierces the plaquette. We refer to the centre charge of a vortex via the value of m = ±1.

Centre Vortex Visualisation
The centre lines of extended vortices are illustrated on the dual lattice by rendering a jet piercing the plaquette producing the nontrivial centre charge. The orientation of the jet follows the right-handed coordinate system. Figure 1 provides an illustration of this assignment. For example, with reference to Eq. (4), an m = +1 vortex in the x-y plane is plotted in the +ẑ direction as a blue jet. Similarly, an m = −1 vortex in the x-y plane is plotted in the −ẑ direction. As the centre charge transforms to its complex conjugate under permutation of the two dimensions describing the plaquette, the centre charge can be thought of as the directed flow of charge z = exp(2πi/3). Our current focus is to understand the impact of dynamical-fermion degrees of freedom on the centre-vortex structure of a gluon field. Here we consider the PACS-CS (2 + 1)-flavour full-QCD ensembles [26], made available through the ILDG [27]. These 32 3 × 64 lattice ensembles employ a renormalisation-group improved Iwasaki gauge action with β = 1.90 and non-perturbatively O(a)-improved Wilson quarks, with C SW = 1.715. In this section, their lightest uand d-quark-mass ensemble identified by a pion mass of 156 MeV [26] is considered. The scale is set using the Sommer parameter with r 0 = 0.4921 fm providing a lattice spacing of a = 0.0933 fm [26].
For comparison, a matched 32 3 × 64 pure-gauge ensemble has been generated using the same improved Iwasaki gauge action with β = 2.58 providing a Sommer-scale spacing of a = 0.100 fm. This spacing facilitates comparisons with all the PACS-CS ensembles.
The centre-vortex structure of pure-gauge and dynamical-fermion ground-state vacuum fields is illustrated in Figs. 2 and 3 respectively. The vortex flow displays a rich structure. One observes the continuous flow of centre charge and the presence of monopole or anti-monopole contributions, where three jets emerge from or converge to a point. We refer to the latter as branching points in general. Upon introducing dynamical fermions, the structure becomes more complicated, both in the abundance of nontrivial centre charge and in the increased abundance of branching points.
These figures provide interactive illustrations which can be activated in Adobe Reader 1 by clicking on the image. Once activated, click and drag to rotate, Ctrl-click to translate, Shiftclick or mouse wheel to zoom, and right click to access the "Views" menu. Several views have been created to facilitate and inspection of the centre-vortex structure.
Both Figs. 2 and 3 contain a percolating vortex cluster, a characteristic feature of the confining phase [28]. These illustrations are representative of the ensemble in that the vortex vacuum is typically dominated by a single large percolating cluster. This single large cluster is accompanied by several smaller loops or loop clusters. However, the most important observation is how dynamical fermions significantly increase the number of vortices observed.
For an ensemble of 200 configurations with 32 three-dimensional volume slices each, the average number of vortices composing the primary cluster in these 32 2 × 64 spatial slices is 3, 277 ± 156 vortices in the pure gauge theory, versus 5, 924 ± 239 vortices in full QCD. Since there are 32 2 × 64 × 3 = 196, 608 spatial plaquettes on these lattices, the presence of a vortex is a relatively rare occurrence.
Similarly, Figs. 4 and 5 illustrate the secondary loop structures left behind as one removes the single large percolating structure. Again, the introduction of dynamical fermions increases the complexity of the structure through a proliferation of branching points (or monopoles [29]).

Static Quark Potential
With an understanding the impact of dynamical-fermion degrees of freedom on the centrevortex structure of ground-state vacuum fields, we turn our attention to confinement as realised in the static quark potential. The results presented here are supported by complimentary studies of the nonperturbative gluon propagator.
The static quark potential is accessed via consideration of the expectation value of Wilson loops, 〈W (r, t)〉, with spatial separation r and temporal extent t, The relevant static quark potential is given by the lowest α = 0 state. We use a variational analysis of several spatially-smeared sources to isolate this state. With knowledge of the vortex content of a configuration, contained in Z µ (x) of Eq. (3), we can analyse two vortex-modified ensembles in addition to the original untouched configuration, U µ (x). We refer to these as the vortex-only, Z µ (x), and vortex-removed, The static quark potential for the original untouched configurations is expected to follow a Cornell potential composed of a Coulomb term, dominant at short distances, and a linear term, dominant at large distances. As centre vortices are anticipated to encapsulate the non-perturbative longrange physics, the vortex-only results should give rise to a linear potential [5,12,30]. On the other hand, the vortex-removed results are expected to capture the short-range behaviour. To analyse the linearity of the potential at large distances we plot a sliding local linear fit to the potential with extent r ± 3 2 a.    We commence with preliminary results for the pure-gauge ensemble illustrated in Fig. 6. Qualitatively, centre vortices account for the long-distance physics. The lower plot illustrates how removal of centre vortices completely removes the long-range potential. However, the phenomenology is not perfect. The value of the string tension produced in the vortex-only analysis is once again only 60 % of the original string tension.
Upon introducing dynamical fermions with light quark masses corresponding to a pion mass of m π = 701 MeV, the preliminary results shown in Fig. 7 are observed. Comparing with the pure-gauge sector of Fig. 6, we observe a screening of the original string tension with the introduction of dynamical fermions, in accord with expectations. Again, the effect of vortex removal is to remove confinement. The sliding average lies in excellent agreement with 0 at large distances.
While the centre-vortex phenomenology is similar to the pure gauge sector, this time the vortex-only string tension is in excellent agreement with the original untouched ensemble. This is illustrated in Fig. 7, in the lower plot where the local slopes of the untouched and vortexonly ensembles agree at large distances. This new agreement arises from significant modifications in the centre-vortex structure of ground state fields induced by dynamical fermions, even at relatively large quark masses.

Conclusion
In summary, centre-vortex structure is complex. Each ground-state configuration is dominated by a long-distance percolating centre-vortex structure. In SU(3) gauge field theory, a proliferation of branching points is observed, with further enhancement as light dynamical fermion degrees of freedom are introduced in simulating QCD. There is an approximate doubling in the number of nontrivial centre charges in the percolating vortex structure as one goes from the pure-gauge theory to full QCD. An enhancement in the number of small vortex paths is also observed upon introducing dynamical fermions. Increased complexity in the vortex paths is also observed as the number of monopole-antimonopole pairs is significantly increased with the introduction of dynamical fermions. In short, dynamical-fermion degrees of freedom radically alter the centre-vortex structure of the ground-state vacuum fields.
With regard to the static quark potential and confinement, we find that centre vortices now quantitatively capture the string tension in full QCD, unlike the pure-gauge sector. This represents a significant advance in centre-vortex phenomenology. Moreover, vortex removal continues to eliminate the long distance potential. These encouraging results are also reflected in more recent studies of the gluon propagator in full QCD. In summary, the results presented here show a significant advance in the ability of centre vortices to capture the salient nonperturbative features of QCD.