Properties of SU(2) Center Vortex Structure in Smooth Configurations

Abstract

New analysis regarding the structure of center vortices is presented: Using data from gluonic SU(2) lattice simulation with Wilson action, a correlation of fluctuations in color space to the curvature of vortex fluxes was found. Finite size effects of the S2-homogeneity hint at color homogeneous regions on the vortex surface.

1. Introduction

The center vortex model of Quantum Chromodynamics [1,2] explains confinement [3] and chiral symmetry breaking [4,5,6] by the assumption that the relevant excitations of the QCD vacuum are Center vortices, closed color magnetic flux lines evolving in time. In four-dimensional space-time, these closed flux lines form closed surfaces in dual space, see Figure 1. In the low-temperature phase, they percolate space-time in all dimensions.

In Maximal Center gauge, the center vortex surface can be detected by the identification of Wilson loops evaluating to non-trivial center elements and by projecting the links to the center degrees of freedom—the piercing of a Wilson loop by a center flux line contributes a non-trivial factor to the numerical value of the loop. The detection of plaquettes pierced by a flux line is schematically shown in Figure 2.

The vortex flux has a finite thickness, but is located by thin P-vortices. As long as these P-vortices successfully locate vortices, we speak of a valid vortex-finding property. A problem concerning this property was discovered by Bornyakov et al. [7]. We were able to resolve these problems using an improved version of the gauge-fixing routines [8,9,10] and could show, for relatively small lattices, that center vortices reproduce the string tension. We found hints towards a color structure existing on the vortex surface [11]. This color structure might be directly related to the topological charge of center vortices and could lead to a more detailed explanation of their chiral properties. Their study is especially complicated by the fact that measurements of the topological charge require sufficiently smooth field configurations, whereas the quality of the center vortex detection suffers with increasing smoothness of the lattice, that is, the vortex-finding property can get lost during smoothing or cooling of the lattice. We want to bring more light to these difficulties by an analysis of the following:

• Correlation between color structure and surface curvature,
• Size of color-homogeneous regions on the surface, and
• Influence of cooling on color-homogeneous regions on the surface, and
• Influence of smoothing the vortex surface on color-homogeneous regions.

As this is the first time these measurements are performed with our algorithms, this work is to be understood as a proof of concept.

2. Materials and Methods

Our lattice simulation is based on gluonic SU(2) Wilson action with inverse coupling $\beta$ covering an interval from $\beta =2.1$ to $\beta =3.6$ in steps of $0.05$. The corresponding lattice spacing a is determined by a cubic interpolation of literature values given in

Table 1. The values of the lattice spacing in fm and the string tension corresponding to the respective value of $\beta$ are taken from the Refs. [12,13,14,15,16], setting the physical string tension to ${\left(440\mathrm{MeV}\right)}^2$.

$\mathit{\beta }$	2.3	2.4	2.5	2.635	2.74	2.85
a [fm]	0.165 (1)	0.1191 (9)	0.0837 (4)	0.05409 (4)	0.04078 (9)	0.0296 (3)
$\sigma$ [lattice]	0.136 (2)	0.071 (1)	0.0350 (4)	0.01459 (2)	0.00830 (4)	0.00438 (8)

and complemented by an extrapolation according to the asymptotic renormalization group equation

$$a\left(\beta \right)={\mathsf{\Lambda }}^{-1}{e}^{-\frac{\beta }{8{\beta }_0}}\mathrm{with}{\beta }_0=\frac{11}{24{\pi }^2}\mathrm{and}\mathsf{\Lambda }=0.015\left(2\right){\mathrm{fm}}^{-1}.$$

(1)

We assume a physical string tension of ${\left(440\mathrm{MeV}\right)}^2$. Our analysis is performed on lattices of size $8^4$, ${10}^4$ and ${12}^4$ and ${16}^4$.

We use an improved version of maximal center gauge to identify the position of the flux lines building up the vortex surface via center projection. The improvements are described in detail in the Refs. [8,9,10] and identify non-trivial center regions by perimeters evaluating to non-trivial center elements. These regions are used as guidance for maximal center gauge. By simulated annealing, we look for gauge matrices $\mathsf{\Omega }\left(x\right)\in$ SU(2) at each lattice point x, so that the functional

$$R_{SA}^2=\sum _x\sum _{\mu }\mid \mathrm{Tr}[{\acute{U}}_{\mu }\left(x\right)]{\mid }^2\mathrm{with}{\acute{U}}_{\mu }\left(x\right)=\mathsf{\Omega }(x+e_{\mu })U_{\mu }\left(x\right){\mathsf{\Omega }}^{†}\left(x\right)\in \mathrm{SU}\left(2\right)$$

(2)

is maximized. After fixing the gauge, we project to center degrees of freedom and identify the non-trivial plaquettes, but save the un-projected lattice for our measurements concerning the color structure.

These measurements are based on the S2-homogeneity which compares the color vector of two plaquettes related to the same lattice point. With the two plaquettes written as $W_j={\sigma }_{0}cos{\alpha }_j+\mathrm{i}{\sum }_{k=1}^3{\left(n_j\right)}_k{\sigma }_{k}sin{\alpha }_j$, the S2-homogeneity is defined as

(3)

It is a gauge-independent quantity proportional to $\cos \frac{\alpha }{2}$, where $\alpha$ is the angle between the two color vectors $\overrightarrow{n_1}$ and $\overrightarrow{n_2}$.

By calculating the S2-homogeneity of two plaquettes that are pierced by a center flux-line, we restrict the measurement to the vortex surface, and can distinguish the two scenarios depicted in Figure 3:

• Parallel plaquettes corresponding to a weakly curved flux-line, that is, plaquettes located in the same plane in dual space.
• Orthogonal plaquettes, plaquettes that are neighbours in dual space but have different orientation, corresponding to strongly curved flux-lines,

As the curvature of the flux-line directly correlates to the curvature of the vortex surface, this allows an analysis of the correlation between S2-homogeneity and curvature of the vortex surface.

Calculating the S2-homogeneity for different lattice spacings and lattice sizes allows to examine the size of homogeneous regions on the vortex surface by exploiting finite-size effects. We assume that for sufficiently big lattices, the S2-homogeneity is independent of the lattice size and, at most, linear-dependent on the lattice spacing. As long as the color structure fits into the lattice, we assume a decoupling of this structure from the lattice parameters. Shrinking the lattice by reducing the lattice spacing allows to identify the physical lattice extent below which finite size effects introduce stronger dependencies on the lattice parameters, see Figure 4.

Varying the physical lattice size by changing the number of lattice sites and repeating the procedure, we can check whether these finite size effects define a physical scale independent of the lattice parameters—at the onset of the finite size effects, the lattice spacing times the lattice size should be a constant defining the size of color-homogeneous regions on the vortex surface. With decreasing lattice sizes, the onset of the finite size effects occurs at lower values of $\beta$; hence, we choose small lattices to perform these measurements. As this procedure requires sufficiently good statistics at high values of $\beta$, we include in the averages all pairs of plaquettes belonging to the same P-vortex.

For measurements of topological properties, sufficiently smooth configurations are required, but smooth configurations complicate the vortex detection. There are many different procedures for generating such smooth configurations or smoothing excitations belonging to specific degrees of freedom. We use Pisa Cooling [17] with a cooling parameter of 0.05. Repeating the aforementioned measurements in cooled configurations, we compare the influence of a direct smoothing of the vortex surface, described in detail in [18,19], with the effects of cooling. The smoothing procedure consists of “cutting out” parts of the vortex surface and “sewing together” the surface again, see Figure 5.

Additional to the three depicted procedures, we have “smoothing 0”, which deletes isolated unit-cubes and is part of all other listed smoothing routines. All smoothing procedures remove short-range fluctuations from the vortex surface and are considered to have no influence on the infra-red degrees of freedom, but smoothing two has an influence on the connectivity of the vortex surface in contrast to the other smoothing methods.

3. Results

We start by presenting our findings concerning the correlation between the color structure and surface curvature of vortices. The measurements were performed on lattices of size ${16}^4$, and the data show a clear distinction between weakly and strongly curved parts of the vortex surface with respect to S2-homogeneity: the more the vortex surface is curved, the smaller the value of the color-homogeneity.

This can be seen in Figure 6, where we show distinct evaluations of the S2-homogeneity $h_{S2}$ for parallel plaquette pairs (larger values) and orthogonal plaquette pairs (smaller values) for various smoothing and cooling steps. The difference between parallel and perpendicular plaquettes indicates a clear correlation of fluctuations in color space to fluctuations in space-time. $h_{S2}$ increases with more intense cooling, especially for parallel plaquettes. Additionally, the mentioned difference and the correlation increases. It is interesting to observe that with cooling, this difference between parallel and perpendicular plaquette pairs increases more for pairs off the vortex (red lines) than for pairs on P-vortices (data points). For 0 cooling steps, the difference between the red lines is roughly 50% of the difference between data points. This difference increases to ≈$100\%$ for five cooling steps. Since the thickness of vortices increases with cooling and P-vortices stay thin, “off the P-vortex” can still be “on the thick vortex” and those pairs “on the thick vortex” contribute to the “off the vortex” averages. This shows that the correlation between curvature and S2-homogeneity is a property of the thick vortices, and not of the surrounding vacuum.

Comparing the different smoothing procedures, it can be seen that smoothing 1 and smoothing 3 lead to quantitatively identical results, with a homogeneity raised above those of smoothing 0. With smoothing 2, the difference between weak and strong curvature is reduced: The homogeneity of weak curvature is decreased, and the homogeneity of strong curvature increased. As smoothing 2 is the only procedure influencing the connectivity of the vortex surface, we conclude that this connectivity might be of importance in further considerations of the color structure, and that smoothing 2 should be used with care. This is strengthened by the fact that smoothing 2 results in the vortex becoming overall more homogeneous than the vacuum, see Figure 7.

A non-trivial color structure is indicated by color inhomogeneities, hence we suspect that smoothing 2 cuts out parts of the vortex surface that might carry potential color structure.

The dependency of the S2-homogeneity on the physical lattice volume allows a more detailed investigation of the color structure on the vortex surface. In Figure 8, we compare the S2-homogeneity along the vortex surface for different lattice sizes and smoothing procedures. We find a loss of the S2-homogeneity for small lattice volumes and give arguments that the loss of the homogeneity is caused by finite size effects, indicating a physical non-vanishing size of color-homogeneous regions. We estimate the size of these homogeneous regions by linear fits to the values of the S2-homogeneity in the large and small volume regions, as shown in Figure 4. This procedure is applied to different lattice sizes and numbers of cooling steps.

Multiplying the lattice spacing at the intersection with the lattice extent, we estimate the size of the homogeneous regions. In Figure 9, it can be seen that the values are compatible within errors for smoothing 0, smoothing 1, and smoothing 3 on lattices of size $8^4$ and ${10}^4$. The measurements in lattices of size ${12}^4$ result in higher values with bigger errors.

The estimate of the size of the homogeneous regions is repeated for different numbers of cooling steps in lattices of size $8^4$ and ${10}^4$. This allows to infer the influence of the smoothness of the lattice on homogeneous regions. In Figure 10, the S2-homogeneities are shown after one cooling step.

Of interest is that with cooling the S2-homogeneities of the different smoothing procedures become more and more similar to those of smoothing 2. In Figure 11 the corresponding sizes of the color homogeneous regions are shown.

After two cooling steps, smoothing 1 results in the S2-homogeneity of the vortex becoming no longer distinguishable from the vacuum, see Figure 12.

A small increase in the size of homogeneous regions is observable, see Figure 13.

After three cooling steps, smoothing 1, smoothing 2, and smoothing 3 result in the S2-homogeneity of the vortex growing above those of the vacuum, while the homogeneity of smoothing 0 is no longer distinguishable from those of the vacuum, see Figure 14. The corresponding size of the color homogeneous regions is shown in Figure 15.

After five cooling steps, all smoothing procedures result in the vortex becoming more homogeneous than the vacuum as is depicted in Figure 16. The corresponding size of the color homogeneous regions is shown in Figure 17.

The calculations with 10 cooling steps were complicated by a loss of the vortex-finding property. In many of the configurations with 10 cooling steps, no vortices could be detected, making the configurations unusable for analyzing color structures on the vortex surface. The cause for this loss of the vortex-finding property and a possible resolution is of special interest, as it hints at possible ways to improve the vortex detection algorithms. This topic is discussed in another article [20]. The S2-homogeneities of the usable part of the data are shown in Figure 18.

The line fits used to identify the onset of the finite size effects became problematic, causing a possible underestimation of the error; hence, data resulting from 10 cooling steps should be taken with care. The corresponding size of the color homogeneous regions is shown in Figure 19.

As the error bars result from intersections of line fits, they are not necessarily symmetric. To combine the data, the error bar was interpreted as two separate datapoints. Using the data from the lattize sizes $8^4$ and ${10}^4$ with smoothing 0, smoothing 1, and smoothing 3, we estimate the size of the homogeneous regions to the values shown in Figure 20.

The estimation starts at $0.85\pm 0.07$ fm without cooling, reaching $1.1\pm 0.2$ fm after 10 cooling steps. This increase seems to have a maximal value, that is, cooling does not let the size of homogeneous regions increase to infinity. The dependency of the S2-homogeneities on the lattice spacing below the onset of the finite size effects showed hints at a stepwise reduction. Distinguishing whether these resulted from random fluctuations of our data or were caused by a quantization of the sizes of the homogeneous regions might be possible with more lattice data.

4. Discussion

Our results show that curvature of flux lines correlates with fluctuations in color space. This correlation becomes stronger when cooling procedures are applied, but weakens when the connectivity of the vortex surface is harmed. The correlation seems to be a special feature along the center vortex surface, as it weakens when restricting the analysis to plaquettes that are not pierced.

As intersections and twistings of vortices can produce topological charge and the twisting of the vortex is correlated to color fluctuations, see [21], our data strengthens the assumption that the color structure of vortices directly relates to topological charge, enabling center vortices to explain chiral symmetry breaking.

Complications concerning measurements of the topological charge and center vortices raise the question of whether the topological properties of center vortices get lost during cooling, or if the properties are kept but the vortex detection fails. Our data indicate the latter because a possible loss of the color structure due to expanding color homogeneous regions on the vortex surface is not probable, as the size of these regions seems to be bound by an upper limit. Cooling increases the S2-homogeneity of the Vortex more than it increases the homogeneity of the vacuum.

The finite size of these color homogeneous regions is of further interest, concerning the question whether the vortex surface covers the ${\mathbb{S}}^2$ defining the color space or not. It indicates that finite areas on this color ${\mathbb{S}}^2$ cannot correspond to arbitrarily big areas on the vortex surface.

To stay sufficiently far away from finite size effects and guarantee a sufficiently high resolution concerning homogeneous regions, our advice is to keep the physical lattice volume well above, and the lattice spacing well below 0.78 fm. As our analysis at a higher number of cooling steps suffered from a loss of the vortex-finding property, further improvements at detecting center vortices in lattice simulations are needed. Further analysis of this loss of the vortex finding property is to be published in [20].