11institutetext: Centre for the Subatomic Structure of Matter (CSSM),
Department of Physics, University of Adelaide, SA, 5005, Australia

# Dynamical fermions, centre vortices, and emergent
phenomena

Derek Leinweber 11 James Biddle 11 Waseem Kamleh 11 Adam Virgili 11

###### Abstract

The non-trivial ground-state vacuum fields of QCD form the foundation of
matter. Here we examine the centre vortices identified within the ground-state
fields of lattice QCD. We aim to understand the manner in which dynamical
fermions in the QCD vacuum alter the centre-vortex structure. Using modern
visualisation techniques, the centre-vortex structure of pure-gauge and
dynamical-fermion fields is quantified and compared. We then explore the
impact this modified structure has on measures of confinement and dynamical
mass generation. The string tension of the static quark potential, positivity-
violation in the gluon propagator, and dynamical mass generation in the
overlap quark propagator are of particular interest. 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.

##  1 Introduction

The essential, fundamentally-important, nonperturbative features of the QCD
vacuum fields are the dynamical generation of mass through chiral symmetry
breaking, and the confinement of quarks. But what is the fundamental mechanism
of QCD that underpins these phenomena?

One of the most promising candidates is the centre vortex perspective of QCD
vacuum structure. While the ideas of a centre-vortex dominated vacuum were
laid down long ago tHooft:1977nqb ; tHooft:1979rtg ; Nielsen:1979xu , it
wasn’t until 1997 when Jeff Greensite, Manfried Faber, et al. demonstrated
that lattice QCD techniques could be used to explore the importance of these
ideas DelDebbio:1996lih ; Faber:1997rp ; DelDebbio:1998luz ; Bertle:1999tw ;
Faber:1999gu ; Engelhardt:1999xw ; Bertle:2000qv ; Engelhardt:2003wm . Indeed
by the end of the millennium, the field had attracted broad interest, with a
comprehensive review in 2003 Greensite:2003bk .

This perspective describes the nature of the nontrivial vacuum in terms of the
fundamental centre of the gauge group. Herein our focus is on the SU(3) gauge
group where centre vortices are characterised by the three centre phases,
$\sqrt[3]{1}$.

By identifying centre vortices within the ground-state fields and then
removing them, a deep understanding of their contributions can be developed.
Removal of centre vortices from the ground-state fields results in a loss of
dynamical mass generation and restoration of chiral symmetry Trewartha:2015nna
; Greensite:2016pfc ; Trewartha:2017ive ; Virgili:2021jnx , a loss of the
string tension Langfeld:2003ev ; Bowman:2010zr ; Biddle:2022zgw ;
Leinweber:2022dpp , a suppression of the infrared enhancement in the Landau-
gauge gluon propagator Langfeld:2001cz ; Bowman:2010zr ; Biddle:2018dtc ;
Biddle:2022acd , and the possibility that gluons are no longer confined
Biddle:2022acd .

One can also examine the role of the centre vortices alone. Remarkably, centre
vortices produce both a linear static quark potential Langfeld:2003ev ;
OCais:2008kqh ; Trewartha:2015ida ; Biddle:2022zgw ; Leinweber:2022dpp  and
infrared enhancement in the Landau-gauge gluon propagator Biddle:2018dtc ;
Biddle:2022acd . The planar vortex density of centre-vortex degrees of freedom
scales with the lattice spacing providing a well defined continuum limit
Langfeld:2003ev . These results elucidate strong connections between centre
vortices and confinement.

A connection between centre vortices and instantons has been identified
through gauge-field smoothing Trewartha:2015ida . An understanding of the
phenomena linking these degrees of freedom is illustrated in Ref.
Biddle:2019gke . In addition, centre vortices have been shown to give rise to
mass splitting in the low-lying hadron spectrum Trewartha:2017ive ;
Trewartha:2015nna ; OMalley:2011aa .

Still, the picture in pure SU(3) gauge theory is not perfect. The vortex-only
string tension obtained from pure Yang-Mills lattice studies has been
consistently shown to be about $\sim 60\%$ of the full string tension.
Moreover, upon removal of centre vortices the gluon propagator shows a remnant
of infrared enhancement Biddle:2018dtc . In short, within the pure gauge
sector, the removal of long-distance non-perturbative effects via centre-
vortex removal is not perfect.

Understanding the impact of dynamical fermions on the centre-vortex structure
of QCD ground-state fields is a contemporary focus of the centre-vortex field
Biddle:2022zgw ; Biddle:2020isk ; Leinweber:2021keq ; Leinweber:2022dpp ;
Biddle:2022acd ; Virgili:2021jnx . Herein, changes in the microscopic
structure of the vortex fields associated with the inclusion of dynamical
fermions are illustrated. The introduction of dynamical fermions brings the
phenomenology of centre vortices much closer to a perfect encapsulation of the
salient features of QCD, confinement and dynamical mass generation through
chiral symmetry breaking.

As such, it is interesting to ask, what do these centre-vortex structures look
like? To this end, we present visualisations of centre vortices as identified
on lattice gauge-field configurations. Some of these visualisations are
presented as stereoscopic images.

##  2 Centre Vortex Identification

Centre vortices are identified through a gauge fixing procedure designed to
bring the lattice link variables as close as possible to the identity matrix
multiplied by a phase equal to one of the three cube-roots of 1. Here, the
original Monte-Carlo generated configurations are considered. They are gauge
transformed directly to Maximal Centre Gauge DelDebbio:1996mh ;
Langfeld:1997jx ; Langfeld:2003ev . This brings the lattice link variables
$U_{\mu}(x)$ close to the centre elements of SU(3)

$Z=\exp\left(\frac{2\pi i}{3}\,n\right)\,\mathbf{I}\,,$ (1)

with $n=-1$, $0$, or $1$ enumerating the three cube roots of 1 such that the
special property of SU(3) matrices, $\det(Z)=1$, is satisfied. One considers
gauge transformations $\Omega$ such that,

$\sum_{x,\mu}\,\left|\mathrm{tr}\,U_{\mu}^{\Omega}(x)\,\right|^{2}\stackrel{{\scriptstyle\Omega}}{{\to}}\mathrm{max}\,,$
(2)

and then projects the link variables to the centre

$U_{\mu}(x)\to Z_{\mu}(x)\textrm{ where }Z_{\mu}(x)=\exp\left(\frac{2\pi
i}{3}\,n_{\mu}(x)\right)\mathbf{I}\,.$ (3)

Here, $n$ has been promoted to a field, $n_{\mu}(x)$, taking a value of $-1$,
$0$, or $1$ for each link variable on the lattice. In this way, the gluon
field, $U_{\mu}(x)$, is characterised by the most fundamental aspect of the
SU(3) link variable, the centre, $Z_{\mu}(x)$. In the projection step, eight
degrees of freedom are reduced to one of the three centre phases. This
“vortex-only” field, $Z_{\mu}(x)$, can be examined to learn the extent to
which centre vortices alone capture the essence of nonperturbative QCD.

The product of these centre-projected links, $Z_{\mu}(x)$, around an
elementary $1\times 1$ square (plaquette) on the lattice also produces a
centre element of SU(3). The value describes the centre charge associated with
that plaquette

$z=\prod_{\Box}Z_{\mu}(x)=\exp\left(2\pi
i\,\frac{m}{3}\right),\;m=-1,0,\mbox{\,or\,}1\,.$ (4)

The most common value observed has $m=0$ indicating that no centre charge
pierces the plaquette. However, values of $m=\pm 1$ indicate that the centre
line of an extended three-dimensional vortex pierces that plaquette.

Figure 1: Stereoscopic image of centre vortices as identified on the lattice
from Ref. Biddle:2020isk . Vortex features including vortex lines (jets),
branching points (3-jet combinations), crossing points (4 jets), indicator
links (arrows) and singular points (spheres) are described in the text.

The complete centre-line of an extended vortex is identified by tracing the
presence of nontrivial centre charge, $m=\pm 1$, through the spatial lattice.
Figure 1 exhibits rich emergent structure in the nonperturbative QCD ground-
state fields in a stereoscopic image. Here a 3D slice of the 4D space-time
lattice is being considered at fixed time. To experience figure 1 in 3D, try
the following:

  1. 1.

If you are viewing the image on a monitor, ensure the image width is 12 to 13
cm.

  2. 2.

Bring your eyes very close to one of the image pairs.

  3. 3.

Close your eyes and relax.

  4. 4.

Open your eyes and allow the (blurry) images to line up. Tilting your head
from side to side will move the images vertically.

  5. 5.

Move back slowly until your eyes are able to focus. There’s no need to cross
your eyes!

Features of the vortex phenomena include:

### Vortex Lines:

The plaquettes with nontrivial centre charge, characterised by $m=+1$ or $-1$,
are plotted as jets piercing the centre of the plaquette. Both the orientation
and colour of the jets reflect the value of the non-trivial centre charge.
Using a right-hand rule for the direction, plaquettes with $m=+1$ are
illustrated by blue jets in the forward direction, and plaquettes with $m=-1$
are illustrated by red jets in the backward direction. Thus, the jets show the
directed flow of $m=+1$ centre charge, $z=e^{2\pi i/3}$, through spatial
plaquettes. They are analogous to the line running down the centre of a vortex
in a fluid.

Vortices are somewhat correlated with the positions of significant topological
charge density, but not in a strong manner Biddle:2019gke . However, the
percolation of vortex structure is significant and the removal of these
vortices destroys most instanton-like objects.

### Branching Points or Monopoles:

In SU(3) gauge theory, three vortex lines can merge into or emerge from a
single point. Their prevalence is surprising, as is their correlation with
topological charge density Biddle:2019gke .

### Vortex Sheet Indicator Links:

As the vortex line moves through time, it creates a vortex sheet in 4D
spacetime. This movement is illustrated by arrows along the links of the
lattice (shown as cyan and orange arrows in figure 1) indicating centre charge
flowing through space-time plaquettes in the suppressed time direction.

### Singular Points:

When the vortex sheet spans all four space-time dimensions, it can generate
topological charge. Lattice sites with this property are called singular
points Bruckmann:2003yd ; Engelhardt:2010ft ; Engelhardt:2000wc ;
Engelhardt:1999xw  and are illustrated by spheres. The sphere colour indicates
the number of times the sheet adjacent to a point can generate a topological
charge contribution Biddle:2019gke .

##  3 Impact of Dynamical Fermions

First results demonstrating the impact of dynamical fermions on the centre-
vortex structure of QCD ground-state fields were presented in Ref.
Leinweber:2021keq . Here we introduce improved visualisation algorithms and
explore alternative time slices.

Matched lattices are considered, one in pure-gauge and the other a
$2+1$-flavor dynamical-fermion lattice from the PACS-CS Collaboration
Aoki:2008sm . These $32^{3}\times 64$ lattice ensembles employ a
renormalisation-group improved Iwasaki gauge action and non-perturbatively
${\cal O}(a)$-improved Wilson quarks, with $C_{\rm SW}=1.715$.

The lightest $u$\- and $d$-quark-mass ensemble identified by a pion mass of
156 MeV Aoki:2008sm  is presented here. The scale is set using the Sommer
parameter with $r_{0}=0.4921$ fm providing a lattice spacing of $a=0.0933$ fm
Aoki:2008sm . A matched $32^{3}\times 64$ pure-gauge ensemble using the same
improved Iwasaki gauge action with a Sommer-scale spacing of $a=0.100$ fm is
created to enable comparisons with the PACS-CS ensembles.

The centre-vortex structure of pure-gauge and dynamical-fermion ground-state
vacuum fields is illustrated in figures 3 and 3 respectively. These are
interactive plots which can be activated by clicking on the image in Adobe
Reader111To interact with these models, it is necessary to open this document
in Adobe Reader or Adobe Acrobat (requires version 9 or newer). Linux users
may install Adobe Acroread version 9.4.1 (the last edition to have full 3D
support), or use a Windows emulator such as PlayOnLinux. From the Adobe “Edit”
menu, select “Preferences…” and ensure “3D & Multimedia” is enabled and
“Enable double-sided rendering” is selected.. Once activated, click and drag
to rotate, Ctrl-click to translate, Shift-click or mouse wheel to zoom, and
right click to access the “Views” menu. Several views identifying interesting
features have been created to facilitate an inspection of the centre-vortex
structure.

[ noplaybutton, 3Dtoolbar, 3Dmenu, 3Dviews=U3D/Primary-Secondary-PG-24.vws,
3Dcoo = 16 16 32, 3Dc2c=0.245114266872406 0.8673180341720581
0.43321868777275085, 3Droo = 110.0, 3Droll =-98.1, 3Dlights=Default,
width=]U3D/Primary-Secondary-PG-24.u3d

Figure 2:  The centre vortex structure of a ground-state vacuum field
configuration in pure SU(3) gauge theory. The flow of $+1$ centre charge
through the gauge field is illustrated by the jets (see main text for a
description of the plotting conventions). Here, blue jets are used to
illustrate the primary percolating vortex cluster, while other colours
illustrate the secondary clusters. (Click to activate.)

[ noplaybutton, 3Dtoolbar, 3Dmenu, 3Dviews=U3D/Primary-Secondary-DF-05.vws,
3Dcoo = 16 16 32, 3Dc2c=0.245114266872406 0.8673180341720581
0.43321868777275085, 3Droo = 110.0, 3Droll =-98.1, 3Dlights=Default,
width=]U3D/Primary-Secondary-DF-05.u3d

Figure 3:  The centre-vortex structure of a ground-state vacuum field
configuration in dynamical 2+1 flavour QCD with
$m_{\pi}=156\leavevmode\nobreak\ $\mathrm{M}\mathrm{e}\mathrm{V}$$. Symbols
are as described in figure 3. (Click to activate.)

[ noplaybutton, 3Dtoolbar, 3Dmenu, 3Dviews=U3D/Secondary-PG-24.vws, 3Dcoo = 16
16 32, 3Dc2c=0.245114266872406 0.8673180341720581 0.43321868777275085, 3Droo =
110.0, 3Droll =-98.1, 3Dlights=Default, width=]U3D/Secondary-PG-24.u3d

Figure 4:  The centre-vortex structure of the secondary loops identified from
the pure-gauge configuration shown in figure 3. (Click to activate.)

[ noplaybutton, 3Dtoolbar, 3Dmenu, 3Dviews=U3D/Secondary-DF-05.vws, 3Dcoo = 16
16 32, 3Dc2c=0.245114266872406 0.8673180341720581 0.43321868777275085, 3Droo =
110.0, 3Droll =-98.1, 3Dlights=Default, width=]U3D/Secondary-DF-05.u3d

Figure 5:  The centre-vortex structure of the secondary loops identified from
the dynamical-fermion configuration shown in figure 3. (Click to activate.)

In both illustrations, the vortex structure is dominated by a single large
percolating structure, a characteristic feature of the confining phase
Engelhardt:1999fd . Whereas small loops will tend to pierce a Wilson loop
twice with zero effect, it is this extended structure that gives rise to a net
vortex piercing of a Wilson loop and the generation of an area law associated
with confinement. These two illustrations are representative of the ensemble
in that the vortex structure is typically dominated by a single large
percolating cluster.

Closer inspection reveals a continuous flow of centre charge, often emerging
or converging to monopole or anti-monopole vertices where three jets emerge
from or converge to a point. These are also referred to as branching points
Spengler:2018dxt , as a $+1$ centre charge flowing out of a vertex is
equivalent to $+2$ centre charge flowing into the vertex and subsequently
branching to two $+1$ jets flowing out of the vertex.

Figures 5 and 5 illustrate the same time slices as in figures 3 and 3 but with
the primary percolating clusters removed. The vortex structure of the
secondary clusters in the dynamical-fermion case is more complex, typically
featuring branching points in their structure. Again, views have been created
in the interactive figures to identify interesting features and link their
locations between the full and secondary structure illustrations.

With the introduction of dynamical fermions, the structure becomes more
complex, both in the abundance of vortices and branching points. The average
number of vortices composing the primary cluster in these $32^{2}\times 64$
spatial slices roughly doubles from $\sim 3,000$ vortices in the pure gauge
theory to $\sim 6,000$ in full QCD. Still, there are $32^{2}\times 64\times
3=196,608$ spatial plaquettes on these lattices and thus the presence of a
vortex is a relatively rare occurrence.

By counting the number of vortices between branching points one discovers the
distribution is exponential, indicating a constant branching probability. This
probability is higher in full QCD by a ratio of $\sim 3/2$.

##  4 Static Quark Potential

With an understanding of the impact of dynamical-fermion degrees of freedom on
the centre-vortex structure of ground-state vacuum fields, attention has
turned to understanding the impact on confinement. In a variational analysis
of standard Wilson loops composed of several spatially-smeared sources to
isolate the ground state potential, the static quark potential has been
calculated on three ensembles including the original untouched links,
$U_{\mu}(x)$, the vortex-only links, $Z_{\mu}(x)$, and vortex-removed links,
$Z_{\mu}^{\dagger}(x)\,U_{\mu}(x)$ Biddle:2022acd  where the multiplication of
the conjugate of the centre-projected field ensure all plaquettes have $z=0$.

For the original untouched configurations, the static quark potential is
expected to follow a Cornell potential

$V(r)=V_{0}-\frac{\alpha}{r}+\sigma\,r\,.$ (5)

As centre vortices are anticipated to encapsulate the non-perturbative long-
range physics, the vortex-only results should give rise to a linearly rising
potential. On the other hand, the vortex-removed results are expected to
capture the short-range Coulomb behaviour. Figure 6 from Ref. Biddle:2022acd
illustrates the static quark potentials obtained from these three ensembles
for the pure-gauge and dynamical $2+1$-flavor ensemble with a pion mass of 156
MeV Aoki:2008sm .

Figure 6: The static quark potential calculated on the vortex-modified pure-
gauge (left) and $m_{\pi}=156$ MeV dynamical-fermion (right) ensembles. The
lower plot shows the local slope from linear fits of the potentials in the
upper plot over a forward-looking window from $r\mbox{ to }r+4a$.

Qualitatively, centre vortices account for the long-distance physics. The
removal of centre vortices completely removes the confinement potential. And
while the vortex-only string tension is typically $\leavevmode\nobreak\ 60$ %
of the original string tension in the pure gauge sector, the introduction of
dynamical fermions has improved the vortex-only phenomenology significantly.
Vortices alone capture both the screening of the pure-gauge string tension and
the full string tension of the original untouched ensemble. This result is
associated with the significant modification of the centre-vortex structure of
ground-state vacuum fields induced by dynamical fermions.

The vortex-removed potential enables an examination of the Coulomb term in the
ansatz of Eq. (5) and its ability to characterise the short-distance aspects
of QCD in the absence of confinement. We find the long-range aspect of the
Coulomb term to be inconsistent with the vortex-removed potential. This is
illustrated in figure 6 by fading out the lattice QCD results that are
inconsistent with the Coulomb ansatz. Potentials including (anti-)screening
considerations to capture the correct physics are recommended in Ref.
Biddle:2022zgw . Indeed, the standard extraction of the string tension is
spoiled by the persistent nature of the Coulomb term and alternative forms are
discussed in detail Biddle:2022zgw .

##  5 Gluon propagator and positivity violation

The improved separation of perturbative and nonperturbative physics through
the consideration of vortex-removed and vortex-only ensembles in full QCD is
also manifest in the nonperturbative gluon propagator Biddle:2022acd . This
time vortex removal removes the infrared enhancement of the gluon propagator,
leaving a tree-level like structure as illustrated in figure 7 for the gluon-
propagator renormalisation function $Z(q^{2})=q^{2}\,D(q^{2})$. Of course,
finite-volume effects force $Z(q^{2})\to 0$ as $q^{2}\to 0$ as $D(q^{2})$ is
finite. Details of the renormalisation procedure are summarised at the end of
Sec. III A in Ref. Biddle:2022acd .

Figure 7: The pure-gauge gluon propagator (left) is compared with the
$m_{\pi}=156\leavevmode\nobreak\ $\mathrm{M}\mathrm{e}\mathrm{V}$$ dynamical-
fermion gluon propagator (right). The propagators are renormalised as
described in the text. UT refers to the original untouched configurations, VR
to the vortex-removed configurations and VO to the vortex-only configurations.
The “recon” results are a reconstruction of the original propagator obtained
from a linear combination of the vortex-only and vortex-removed propagators.

To explore positivity violation, one considers the Euclidean correlator,
$C(t)$, obtained by taking the Fourier transform of $D(q_{0},\vectorsym{0})$
such that Biddle:2022acd

$C(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dq_{0}\,\int_{0}^{\infty}dm^{2}\,\frac{\rho(m^{2})}{q_{0}^{2}+m^{2}}\,e^{-i\,q_{0}\,t}\,.$
(6)

Extending the $q_{0}$ integral to the complex plane and employing the residue
theorem, one arrives at

$C(t)=\int_{0}^{\infty}dm\,e^{-mt}\,\rho(m^{2})\,.$ (7)

Clearly if $C(t)<0$ for any $t$ then $\rho(m^{2})$ is not positive definite,
and we say that positivity has been violated. This implies that there is no
Källen-Lehmann representation and as such the propagator does not represent a
correlation between physical states. Hence, the states do not appear in the
physical spectrum. In the context of the gluon propagator, this can be taken
as an indication that gluons are confined.

The numerical calculation of Eq. (7) is described in detail in Ref.
Biddle:2022acd , where the subtleties of the finite volume are fully accounted
for. The results are shown in figure 8. As expected Bowman:2007du , the
untouched correlator shows clear signs of positivity violation. Interestingly,
the vortex-only correlators also exhibit robust positivity violation.

The positivity violation present in the pure-gauge vortex-removed result at
large distances is consistent with the observations made in figure 7, where
residual infrared strength in the vortex-removed gluon propagator is apparent.
Thus, the separation of perturbative and non-perturbative physics through
vortex modification is imperfect in the pure-gauge sector.

Figure 8: The pure-gauge Euclidean correlator (left) is compared with the
$m_{\pi}=156\leavevmode\nobreak\ $\mathrm{M}\mathrm{e}\mathrm{V}$$ dynamical-
fermion Euclidean correlator (right). Shown are the results from the untouched
(UT), vortex-removed (VR) and vortex-only (VO) ensembles. A dashed line at
$C(t)=0$ is provided to aid in observing positivity violation.

The results from the dynamical ensemble in the right-hand plot of figure 8
demonstrate an interesting change in behaviour. As with the gluon propagator
results in the previous section, the most striking change is in the vortex-
removed correlator. In this sector we now observe consistency with positivity
in the dynamical-fermion case. This supports the interpretation of the
positivity violation in the vortex-removed pure-gauge results as being related
to the residual non-perturbative infrared strength in the gluon propagator. As
this residual strength is significantly diminished on the dynamical ensembles,
we now see that the residual $q^{2}$ dependence in the VR renormalisation
function may be purely perturbative in origin. In this case, vortex
modification has been successful in separating perturbative and non-
perturbative physics.

In summary, the vortex-only ensembles capture the infrared enhancement of the
gluon propagator and the screening of this enhancement in full QCD. Moreover,
the vortex-only configurations exhibit significant positivity violation, as
would be expected of a confining infrared-dominated theory. Conversely, the
vortex-removed dynamical-fermion configurations show a loss of this positivity
violation, admitting the possibility that they do support a spectral
representation of the propagator constructed from perturbative gluon
interactions. These results provide additional support for the fact that
centre vortices encapsulate the confining aspects of QCD.

##  6 Quark propagator and dynamical mass generation

The overlap Landau-gauge quark propagator $S(p)$ can be written in the form

$S(p)=\frac{Z(p)}{i\not{q}+M(p)}\,,$ (8)

where $Z(p)$ is the renormalisation function and $M(p)$ is the mass function,
and $q$ is a tree-level improved momentum variable minimising lattice
artefacts Virgili:2022wfx . The infrared behaviour of the mass function,
specifically, the presence of dynamical mass generation, provides a clear
signal of dynamical chiral symmetry breaking.

Figure 9 compares the $q^{2}$-dependent overlap-Dirac quark-propagator Landau-
gauge mass function on the light dynamical-fermion ensemble for two different
valence-quark masses. In comparison to the pure-gauge sector dynamical mass
generation is suppressed under vortex removal in full QCD Virgili:2021jnx .

Figure 9: The mass functions, $M(p)$, on untouched and vortex-removed
dynamical-fermion ensembles for valence-quark masses $m_{q}=84$ MeV (left),
and $m_{q}=19$ MeV (right). Explicit chiral symmetry breaking through the
quark mass leaves a remnant of dynamical mass generation.

While explicit chiral symmetry breaking through the quark mass leaves a
remnant of dynamical mass generation, it is anticipated that for sufficiently
light current quark masses, chiral symmetry will be restored Trewartha:2017ive
and dynamical mass generation will be completely eliminated in the vortex-
removed theory.

##  7 Summary

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. Increased complexity in the
vortex paths is also observed as the number of branching points 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. This change in structure acts to
improve the phenomenology of centre vortices better reproducing the string
tension, dynamical mass generation and better removing nonperturbative physics
under vortex removal. This represents a significant advance in the ability of
centre vortices to capture the salient nonperturbative features of QCD.

## Acknowledgements

DBL thanks Jeff Greensite for several interesting conversations on centre
vortices during the conference. We thank the PACS-CS Collaboration for making
their 2 +1 flavour configurations available via the International Lattice Data
Grid (ILDG). This research was undertaken with the assistance of resources
from the National Computational Infrastructure (NCI), provided through the
National Computational Merit Allocation Scheme and supported by the Australian
Government through Grant No. LE190100021 via the University of Adelaide
Partner Share. This research is supported by Australian Research Council
through Grants No. DP190102215 and DP210103706. WK is supported by the Pawsey
Supercomputing Centre through the Pawsey Centre for Extreme Scale Readiness
(PaCER) program.

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