Emergent Structure in QCD
EEmergent Structure in QCD
James
Biddle , ∗ , Waseem
Kamleh , ∗∗ , and Derek
Leinweber , ∗∗∗ Centre for the Subatomic Structure of Matter, Department of Physics,The University of Adelaide, SA 5005, Australia
Abstract.
The structure of the
S U (3) gauge-field vacuum is explored throughvisualisations of centre vortices and topological charge density. Stereoscopicvisualisations highlight interesting features of the vortex vacuum, especiallythe frequency with which singular points appear and the important connectionbetween branching points and topological charge. This work demonstrates howvisualisations of the QCD ground-state fields can reveal new perspectives ofcentre-vortex structure.
Quantum Chromodynamics (QCD) is the fundamental relativistic quantum field theory un-derpinning the strong interactions of nature. The gluons of QCD carry colour charge andself-couple. This self-coupling makes the empty vacuum unstable to the formation of non-trivial quark and gluon condensates. These non-trivial ground-state “QCD-vacuum” fieldconfigurations form the foundation of matter.There are eight chromo-electric and eight chromo-magnetic fields composing the QCDvacuum. An stereoscopic illustration of one of these chromo-magnetic fields is provided inFig. 1. Animations of the fields are also available [1–3].The essential, fundamentally-important, nonperturbative features of the QCD vacuumfields are the dynamical generation of mass through chiral symmetry breaking, and the con-finement of quarks. But what is the fundamental mechanism of QCD that underpins thesephenomena? What aspect of the QCD vacuum causes quarks to be confined? Which aspect isresponsible for dynamical mass generation? Do the underlying mechanisms share a commonorigin?One of the most promising candidates is the centre vortex perspective [4, 5]. This per-spective describes the nature of the QCD vacuum in terms of the most fundamental centre ofthe
S U (3) gauge group, characterised by the three values of √
1. Centre vortices on the latticehave been demonstrated to give rise to indicators of confinement such as the linear static quarkpotential [6–10] and enhancement of the infrared gluon propagator [11–14]. They also repro-duce indicators of dynamical chiral symmetry breaking through enhancement of the infraredquark mass function [15, 16] and mass splitting in the low-lying hadron spectrum [16–18]. ∗ e-mail: [email protected] ∗∗ e-mail: [email protected] ∗∗∗ e-mail: [email protected] a r X i v : . [ h e p - l a t ] S e p igure 1. Stereoscopic image of one of the eight chromo-magnetic fields composing the nontrivialvacuum of QCD. Hints for stereoscopic viewing are provided in the text.
As such, it is interesting to ask, what do these structures look like? To this end, we presentvisualisations of centre vortices and topological charge density as identified on lattice gauge-field configurations. These visualisations are presented as stereoscopic images. To see the3D image, try the following:1. If you are viewing the image on a monitor, ensure the image width is 12 to 13 cm.2. Bring your eyes very close to one of the image pairs.3. Close your eyes and relax.4. Open your eyes and allow the (blurry) images to line up. Tilting your head from sideto side will move the images vertically.5. Move back slowly until your eyes are able to focus. There’s no need to cross your eyes!
Figure 2.
QR code to download the aug-mented reality app.
This work also features Josh Charvetto’s aug-mented reality app for lattice QCD. Get the appusing the QR code in Fig. 2 and see a whole newview of these visualisations through the camera’seye.
Centre vortices are identified through a gauge fix-ing procedure designed to bring the lattice linkvariables as close as possible to the identity ma-trix multiplied by a phase equal to one of the threecube-roots of one [7, 19, 20].he links U µ ( x ) are then projected to the centre elements of S U (3), U µ ( x ) → Z µ ( x ) , where Z µ ( x ) = e π i m µ ( x ) / I , and m µ ( x ) = − , , , (1)such that the gluon field is characterised by the most fundamental aspect of the S U (3) linkvariable, the centre. This “vortex-only” field, Z µ ( x ), can be examined to learn the extent towhich centre vortices alone capture the essence of nonperturbative QCD.Vortices are identified by the centre charge, z , given by the product of the vortex-onlyfield around an elementary square or plaquette on the lattice z =
13 Tr (cid:89) (cid:3) Z µ ( x ) = e π i m / . (2)If z (cid:44)
1, a vortex with charge z characterised by m pierces the plaquette. Vortex Lines:
The plaquettes with nontrivial centre charge are plotted as red or blue jets
Figure 4.
An example of a familiar vor-tex. The position of the vortex is charac-terised by the line running along the cen-tre of the rotation. piercing the centre of the plaquette in Figure 3.Here we are considering a 3D slice of the 4Dspace-time lattice at fixed time. The orientationand colour of the jets reflect the value of the non-trivial centre charge. Using a right-hand rule forthe direction, plaquettes with m = + m = − z = e π i / throughspatial plaquettes. They are analogous to the linerunning down the centre of a vortex as illustratedin Fig. 4.Figure 3 exhibits rich emergent structure in thenonperturbative QCD ground-state fields. Branching Points or Monopoles: In S U (3) gauge theory, three vortex lines can mergeinto or emerge from a single point. Their prevalence is surprising, as is their correlation withtopological charge density.
Vortex Sheet Indicator Links:
As the vortex line moves in time, it creates a vortex sheetin 4D spacetime. This movement is illustrated by arrows along the links of the lattice (shownas cyan and orange arrows in Fig. 3 indicating centre charge flowing through the suppressedtime direction.
Singular Points:
When the vortex sheet spans all four space-time dimensions, it cangenerate topological charge. Lattice sites with this property are called singular points and areillustrated by spheres. The sphere colour indicates the number of times the sheet adjacent toa point can generate a topological charge contribution [21].
Topological Charge Density:
Non-trivial topological charge is often associated withinstanton-like field configurations which dynamically generate the mass the proton and other igure 3.
Stereoscopic image of centre vortices as identified on the lattice. Vortex features includ-ing vortex lines (jets), branching points (3-jet combinations), crossing points (4 jets), indicator links(arrows) and singular points (spheres) are described in the text. hadrons. Topological charge density is plotted as red through yellow volumes for positivecharge density and blue through green volumes for negative charge density in Fig. 5.
Vortices & Topological Charge:
Vortices are somewhat correlated with the positions ofsignificant topological charge density, but not in a strong manner. However, the percolation ofvortex structure is significant and the removal of these vortices destroys most instanton-likeobjects.
Visualisations of centre-vortex phenomena provide a fascinating glimpse into the structureof the QCD ground-state fields. Patterns and correlations that remain hidden within giga-bytes of data become clear as one examines the data drawing on our highly-evolved sense ofvision. It provides an extremely powerful form of data analysis. Animations of centre-vortexstructure are also available [1, 22, 23].A more detailed presentation of this work providing access to the full 3D models is avail-able in Ref. [21]. It is interesting to reflect on the discoveries made in Ref. [21] throughvisualisation. These include: • The high density of centre vortices and the complexity of their structure. • The long-distance structure of the centre vortices as they percolate through the vacuumfields. • The proliferation of branching points in
S U (3) gauge theory. • The correlation of branching points with topological-charge-density peaks, revealing thevital role of vortex branching in generating topological charge density. igure 5.
Stereoscopic image illustrating the correlation of topological charge density in the gluonfield with the positions of centre vortices. Positive charge density is rendered red through yellow andnegative charge density is rendered blue through blue. • Understanding the sheet-like structure of centre vortices in four dimensions as they pierceone 3D time-slice of the lattice and traverse to the next. • The speed with which centre vortices move through the lattice. • The rapid loss of centre vortices under gauge-field smoothing routines.The new understanding of centre-vortex branching and topological charge is important. Thismechanism governs the size of instanton-like field configurations which generate a density ofnear zero-modes in the Dirac operator, dynamically breaking chiral symmetry in the ground-state fields and generating the mass of visible matter.
Acknowledgements
We thank Daniel Trewartha for his contributions to the gauge ensembles underlying thisinvestigation. This research is supported with supercomputing resources provided by thePhoenix HPC service at the University of Adelaide and the National Computational In-frastructure (NCI) supported by the Australian Government. This research is supportedby the Australian Research Council through Grants No. DP190102215, DP150103164,DP190100297 and LE190100021.
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