Gluon propagator on a centre-vortex background

The impact of $SU(3)$ centre vortices on the gluon propagator in Landau gauge is investigated on original, vortex-removed and vortex-only lattice gauge field configurations. Vortex identification is found to partition the gluon propagator into short range strength on the vortex removed configurations and long range strength on the vortex only configurations. The effect of smoothing vortex-only configurations is also studied, and a regime for recovering the form of the smoothed original propagator from vortex-only configurations is introduced. The results reinforce the significance of centre vortices in a fundamental understanding of QCD vacuum structure.


I. INTRODUCTION
There is now significant evidence supporting the role of centre vortices in confinement and dynamical chiral symmetry breaking in QCD. It is well established that for SU (2) gauge theories, centre vortices account for both of these properties [1][2][3] and thus it would seem intuitive that the same would hold for SU(3) theories as well. Previous work has successfully shown that vortex removal in SU (3) corresponds to a loss of dynamical mass generation and string tension [4,5]. However recovering these properties on vortex only backgrounds has proven difficult due to the roughness of the gauge fields after centre projection [6].
In this paper we explore the behaviour of the Landau gauge gluon propagator [7][8][9] on original, vortexremoved and vortex-only configurations. Suppression of the gluon propagator after vortex removal has been previously demonstrated [10,11]. However, the scalar propagator has not yet been examined on a vortex-only background.
Analysis of the overlap fermion quark propagator has shown that smoothing is an essential step when analysing vortex-only configurations in the quark sector [5,12,13]. Through the process of smoothing, the thin P-vortices that are obtained from the maximal centre gauge projection evolve into thick vortices, the fully fledged topological objects that are present in the physical vacuum. These studies found that by starting from the vortexonly fields, which consist only of the centre elements of SU (3), through the application of cooling we are able to reproduce all the salient features of QCD, including confinement, dynamical mass generation, and the lowlying hadron spectrum. In this work, we extend this line of investigation by examining the effect of cooling and smearing on the gluon propagator obtained from vortexmodified gauge field configurations.

II. LANDAU GAUGE GLUON PROPAGATOR
The momentum space gluon propagator on a finite lattice with four-dimensional volume V is given by where A a µ are the Hermitian gluon fields (see the Appendix for more details). In the continuum, the Landaugauge momentum-space gluon propagator has the following form [14,15] where D(q 2 ) is the scalar gluon propagator. Contracting Gell-Mann index b with a and Lorentz index ν with µ one has such that the scalar function can be obtained from the gluon propagator via where n c = 3 is the number of colours. As the lattice gauge links U µ (x) naturally reside in the fundamental representation of SU (3), it is convenient to work with the corresponding 3 × 3 matrix representation of the gauge potential A µ = A a µ (λ a /2), where λ a are the eight Gell-Mann matrices. Using the orthogonality relation Tr(λ a λ b ) = δ ab for the Gell-Mann matrices, it is straightforward to see that which can be substituted into equation 4 to obtain the final expression for the lattice scalar gluon propagator, Following the formalism of Ref. [14], we calculate the lattice gluon propagator using the mid-point definition of the gauge potential in terms of the lattice link variables [16], arXiv:1806.04305v1 [hep-lat] 12 Jun 2018 The gluon fields U µ (x) are first gauge-fixed by maximizing an O(a 2 )-improved functional using a Fourieraccelerated algorithm [17][18][19]. The gauge potential in momentum space is then obtained by taking the discrete Fourier transform, The gauge fields used in this analysis are created using the O(a 2 )-improved Lüscher and Weisz action [20]. It is known that the continuum continuum propagator has the form as p 2 → ∞. To preserve this behaviour on the lattice, it is necessary to make use of the momentum variable q µ defined by the tree-level form of the O(a 2 )-improved gluon propagator [15,21].
where p µ are the usual lattice momentum variables This choice of a O(a 2 )-improved action, gauge-fixing functional [18] and momentum ensures that we reduce the sensitivity of the gluon propagator to finite latticespacing effects [15]. We follow the tradition of examining q 2 D(q 2 ) such that at large q 2 we observe q 2 D(q 2 ) trending towards a constant. We then renormalise such that q 2 D(q 2 ) = 1 for qa = 2.81 on the original configurations, and apply this same renormalisation factor to all subsequent vortexmodified propagators.

III. CENTRE VORTEX PROJECTION
In the centre vortex model of confinement [22,23], centre vortices are associated with regions of an SU (N ) gauge field that have a non-trivial topology. In a four-dimensional Euclidean space-time, the physical centre vortices present in the QCD vacuum form threedimensional volumes. These physical or thick vortices are distinguished from the concept of thin vortices. A thin vortex corresponds to the two-dimensional boundary surface of the three-dimensional volume formed by a thick vortex, such that the thin and thick vortex are gauge equivalent, i.e. related by a gauge transformation [24,25].
A key property of the thin vortices is that Wilson loops which enclose a vortex line acquire a non-trivial centre phase e n2πi/N , n = 0, 1, . . . , N −1. An illustration of a Wilson loop that is pierced by a thin centre vortex is shown in Figure 1. Three of the four space-time dimensions are shown. The thin vortex is represented by the dashed line. In this instance, the extent of the two-dimensional vortex surface in the fourth dimension is suppressed, and only the intersection with the threedimensional space is shown. The oriented solid circle represents a Wilson loop which lies within the vertical plane, and is pierced by the vortex line at a single point. The vortex intersection causes the Wilson loop to acquire a non-trivial centre phase.
On the lattice, the trace of the elemental plaquettes represent the smallest non-trivial Wilson loops. We decompose the SU (3) lattice gauge links in such a way that all vortex information is captured in the field of centre-projected elements Z µ (x), with the remaining short-range fluctuations described by the vortex-removed field R µ (x). The centre-projected plaquettes in the Z µ (x) configurations with a nontrivial flux around the boundary form the thin vortices (or Pvortices) that are embedded within the thick vortices of the original Monte Carlo configurations.
To identify centre vortices in the Monte Carlo generated configurations we follow the maximal centre gauge (MCG) centre projection procedure as described in Refs. [26,27]. We seek a gauge transformation Ω(x) that minimises where Z µ (x) are the centre elements of SU (3). This transformation is performed by maximising the so-called "mesonic" functional [6] Once the configurations are fixed to maximal centre gauge, each link can be projected onto the nearest centre element. We define these projected configurations Z µ (x) as the vortex-only configurations. This projection also allows us to define the vortex removed configurations Hence, we refer to the three different gauge field ensembles created as follows:

A. Survey of configurations
We calculate the gluon propagator on 100 configurations of a 20 3 × 40 SU (3) lattice with spacing a = 0.125 fm, as used in Refs. [4,5]. Following the procedure of Ref. [14,15] all results are plotted after a momentum half-cut and a cylinder cut of radius pa = 2 lattice units have been performed. Additionally, we can take advantage of the rotational symmetry of the scalar propagator to perform Z(3) averaging over the Cartesian coordinates. This means that we average over all points with the same Cartesian radius; for example, we would average across the points (n x , n y , n z ) = (2, 1, 1), (1, 2, 1) and (1, 1, 2). Calculating the scalar propagator on untouched and vortex-removed configurations gives the results illustrated in Fig. 2. The vortex removed configurations display the expected behaviour, with vortex removal corresponding to significant infrared suppression of the propagator when compared to the untouched propagator, in agreement with the results of Ref. [11]. The increased roughness of the gauge fields after vortex removal is evidenced by the enhancement of the propagator at large q. This reflects the increase in short-distance fluctuations that have been introduced to the gauge fields by the vortex removal procedure.
The results for all three field ensembles are illustrated in Fig. 3. It is clear that the projected vortex-only configurations are too rough to accurately recreate the gluon propagator and have lost all resemblance to the untouched propagator. Motivated by our previous results using smoothed vortex-only fields [5,12,13], in order to proceed we again apply smoothing to the projected centre vortex configurations before calculating the gluon propagator.

B. Smoothing
As smoothing will play an important role for the vortex-only configurations, we investigate the effect of both O(a 4 )-improved cooling [28] and over-improved stoutlink smearing [29]. Following the results of Ref. [29], the over-improved smearing parameters are ρ = 0.06 and = 0.25 to best preserve the size of instantons on the lattice. To accomplish a similar preservation of topological objects under cooling, we used the three-loop improved algorithm as described in Ref. [28].
We The gluon propagator after cooling or improved smearing. We see that the shape of the plot changes minimally between the smoothing routines. However cooling requires fewer sweeps to produce the same effect when compared to smearing. and 8 sweeps of cooling in Fig. 4. We observe the expected removal of short distance fluctuations that is typical of smoothing, resulting in a suppressed propagator at large q. This is complemented by an amplification in the infra-red region which can be attributed to the increase in low momentum modes arising from the smoothing of the gauge fields.
To compare the effects of cooling and over-improved smearing, the untouched gluon propagator is plotted in Fig. 5 after either over-improved smearing or cooling. By comparing the smeared and cooled propagator we can see that cooling has a more rapid effect, related to the wellknown fast removal of action from the lattice. The qualitative shape of the propagator remains the same however, and it can be seen that, for example, 4 smearing sweeps produces a propagator remarkably similar to 1 cooling sweep. More generally, we observe that in regards to the shape of the propagator, n sm ≈ 4 n cool . Following the observation made in Ref. [30] that the number of overimproved stoutlink smearing sweeps is related to the gradient flow time by we deduce that the relationship between gradient flow time and cooling is We choose to use cooling as the smoothing algorithm for the results presented in this paper, however it is worth noting that similar results can be obtained with the use of over-improved smearing.

C. Role of centre vortices
After performing 10 sweeps of cooling on the untouched, vortex-removed and vortex-only ensembles, we obtain the results shown in Fig. 6. After cooling we see that our gauge fields are sufficiently smooth to observe the familiar structure on the vortex-only configurations. As is typical of cooling, the removal of short range structures means that all three ensembles tend to zero as q → ∞. There is now a noticeable improvement in the agreement between the untouched and vortex only configurations; however there is still a difference present, especially in the qa ≈ 0.5 and qa ≈ 1.5 regions.  The amplification of the untouched propagator in the infra-red region and the suppression in the qa ≈ 1.5 region is reminiscent of the difference we see in Fig. 4 between the propagator after different amounts of smoothing. As the peak in the vortex-only propagator lies below the untouched propagator, further cooling on the vortexonly configurations should align the two propagators, following the trend seen in Fig. 4. Given that we know the vortex-only configurations are initially much rougher [5], we anticipate that additional sweeps of cooling should be needed to bring them in line with the untouched configurations.
We take the average O(a 4 ) three-loop improved action of the lattice divided by the single instanton action S 0 = 8π 2 β 6 , denotedS/S 0 , to be a measure of roughness. We observe that the vortex-only configurations have a significantly higher action than their untouched counterparts after the same number of sweeps of cooling, as illustrated in Fig. 7. We therefore seek to find the number of sweeps required to best match the action between the vortex-only and untouched configurations.
The results of this procedure are shown in Table I. If we now plot these matched configurations, we obtain the results shown in Fig. 8. Here we have truncated the plot at large qa to better show the agreement in the mid-qa region. By matching the actions as closely as possible with an integer number of cooling sweeps, we see that there is a high level of agreement between the untouched and vortex-only gluon propagators.

V. CONCLUSIONS
By making use of smoothing on untouched, vortexremoved and vortex-only gauge field configurations, we have demonstrated the significant role centre vortices have in regards to the structure of the gluon propagator. We investigated the effect of smoothing on the gluon propagator, and determined that both cooling and overimproved smearing produce similar suppression of high frequency modes and amplification of infrared behaviour. Removal of vortices leads to suppression of the gluon propagator when compared to the untouched configurations.
After applying smoothing, the untouched and vortexonly configurations have similar shape, and by using the average action as a measure of roughness we see that the it is possible to recover the strength of the propagator on the vortex-only configurations when compared to the untouched configurations. The accuracy to which the smoothed vortex-only configurations are able to recreate the gluon propagator on similarly smooth original configurations is remarkable, as illustrated in Fig. 8. This work motivates further exploration of the impact of centre vortices in full QCD, where we would anticipate infrared screening of the propagator [21]. Furthermore, we wish to investigate the impact of centre vortices and smoothing in the continuum limit. The results of this work contribute further numerical evidence that centre vortices are the fundamental mechanism underpinning QCD vacuum structure.

ACKNOWLEDGMENTS
This work was supported with supercomputing resources provided by the Phoenix HPC service at the University of Adelaide. Additional computing resources used to assist this investigation were provided by the National Computational Infrastructure (NCI), which is supported by the Australian Government. This research was supported by the Australian Research Council through Grants No. DP140103067, DP150103164, and LE160100051.

Appendix: Notes on the gluon propagator
Here we briefly recall the derivation of the expression used to calculate the momentum space gluon propagator on the lattice [7,8,10]. It is instructive to reference the coordinate space form, The propagator in momentum space is simply related by the discrete Fourier transform, Noting that the coordinate space propagator D ab µν (x − y) only depends on the difference x − y, we can make use of translational invariance to average over the fourdimensional volume to obtain the form for the momentum space propagator given in Eq. 1, D ab µν (p) = 1 V x,y e −ip·x A a µ (x + y) A b ν (y) = 1 V x,y e −ip·(x+y) A a µ (x + y) e +ip·y A b ν (y) On a discrete lattice we prefer to make use of the midpoint definition of the gauge potential, as this yields a symmetric local Landau gauge condition [16], This local gauge condition in momentum space is then µ 2i sin p µ 2 A µ (p) = 0, (A.5) which is free from O(a) errors. The Landau gauge fixing functional that we use to transform the links U µ (x) is further improved by taking a combination of one-and two-link terms to eliminate O(a 2 ) errors [18].
The scalar propagator is obtained from the Lorentz diagonal components of the gluon propagator. In the case µ = ν, it is straightforward to replace y → y +μ/2 in equation (A.3) to derive D aa µµ (p) in terms of the mid-point definition of the A µ fields. The propagator itself is calculated directly from the potential in momentum space to avoid the problem of statistical noise in the coordinate space propagator at large separations |x − y| [7,8,10].