**1. Introduction**

First proposed by Hooft [1] and Cornwall [2] the *center vortex model* gives an explanation of confinement in non-Abelian gauge theories. It states that the vacuum is a condensate of quantized magnetic flux tubes, the so-called *vortices*. The vortex model is able to explain the following:


but suffers from Gribov copy problems: predictions concerning the string tension depend on the specific implementation of the gauge fixing procedure, see [8,9].

In this work, an explanation of the problem is given before an improvement of the vortex detection is presented.

Center vortices are located by P-vortices, which are identified in direct maximal center gauge, the gauge which maximizes the functional

$$\mathcal{R}^2 = \sum\_{\mathbf{x}} \sum\_{\mu} |\text{Tr}[\mathcal{U}\_{\mu}(\mathbf{x})]|^2. \tag{1}$$

The projection onto the center degrees of freedom

$$Z\_{\mu}(\mathbf{x}) = \text{sign } \text{Tr}[\mathcal{U}\_{\mu}(\mathbf{x})] \tag{2}$$

leads to plaquettes with nontrivial center values, P-plaquettes which form P-vortices, and closed surfaces in dual space. This procedure can be seen as a best fit procedure of a thin vortex configuration to a given field configuration [3,10], see Figure 1.

**Figure 1.** Vortex detection as a best fit procedure of P-Vortices to thick vortices shown in a two-dimensional slice through a four dimensional lattice.

The way P-vortices locate thick vortices is called *vortex finding property*.

Center vortices can be directly related to the string tension: the flux building up the vortex contributes a nontrivial center element to surrounding Wilson loops, see Figure 2.

**Figure 2.** Each P-plaquette contributes a nontrivial center element to surrounding Wilson loops.

The behavior of Wilson loops can be explained and a nonvanishing string tension extracted by using the density *ρ*u of uncorrelated P-plaquettes per unit volume

$$\langle \frac{1}{2} \text{Tr}(\mathcal{W}(\mathbb{R}, T)) \rangle = [-1 \,\rho\_{\text{u}} + 1 \,(1 - \rho\_{\text{u}})]^{\mathbb{R} \times T} = e^{\ln(1 - 2\rho\_{\text{u}}) \cdot \mathbb{R} \times T} \Rightarrow \sigma = -\ln(1 - 2 \,\rho\_{\text{u}}).\tag{3}$$

The string tension can also be calculated by Creutz ratios

$$\chi(R,T) = -\ln \frac{\langle \mathbf{W}(R+1,T+1) \rangle}{\langle \mathbf{W}(R,T+1) \rangle} \frac{\langle \mathbf{W}(R,T) \rangle}{\langle \mathbf{W}(R+1,T) \rangle}. \tag{4}$$

From *W*(*<sup>R</sup>*, *T*) ≈ *e*<sup>−</sup>*<sup>σ</sup> R T*−2 *μ* (*<sup>R</sup>*+*<sup>T</sup>*)+*<sup>C</sup>*, it follows for sufficiently large R and T that *<sup>χ</sup>*(*<sup>R</sup>*, *T*) ≈ *σ*. Creutz ratios for center-projected Wilson loops are expected to give correct values for *σ* if the vortex finding property is given.

The problem with the direct maximal center gauge is that different local maxima of the gauge functional *R* can lead to different predictions concerning the string tension in center-projected configurations [8,9]. An improvement in the value of the gauge functional results in an underestimation of the string tension, as can be seen in Figure 3.

**Figure 3.** The string tension, calculated via Creutz ratios of the full theory *<sup>χ</sup>*(*R*)*SU*(2), the center-projected theory *<sup>χ</sup>*(*R*)*Z*2, and the vortex density. By increasing the number of simulated annealing sweeps, a better value of gauge functional is reached, but the string tension is underestimated by *<sup>χ</sup>*(*R*)*Z*2. The data was calculated in lattices of size 12<sup>4</sup> (**left**),12<sup>4</sup> (**middle**), and 14<sup>4</sup> (**right**) in Wilson action. The vortex density was not corrected for correlated P-plaquettes, hence, it is overestimated.

In fact, preliminary analyses show that the string tension decreases linearly with an improvement in the value of the gauge functional.

We believe that this is caused by a failing gauge-fixing procedure during which the vortex finding property is lost. If the P-vortices fail to locate thick vortices, the string tension will be underestimated by *<sup>χ</sup>*(*R*)*Z*2, see Figure 4.

**Figure 4.** When P-vortices no longer locate thick vortices, we speak of a loss of the *vortex finding property*. The figure shows a two-dimensional slice through a four-dimensional lattice.

A failing vortex detection can result in vortex clusters disintegrating into small vortices consisting only of correlated P-plaquettes. This causes a misleadingly high vortex density.

The loss of the vortex finding property can be avoided by using the information about center regions, that is, regions enclosed by a Wilson loop that evaluate to center elements.

Center regions can be related to a non-Abelian generalization of the Abelian stokes theorem:

$$\begin{aligned} P \exp\left(\mathrm{i} \oint\_{\partial S} A\_{\mu}(\mathbf{x}) \, d\mathbf{x}^{\mu}\right) &= \mathcal{P} \exp\left(\frac{\mathrm{i}}{2} \int\_{S} \mathcal{F}\_{\mu\nu}(\mathbf{x}) \, d\mathbf{x}^{\mu} \, d\mathbf{x}^{\nu}\right), \\ \mathcal{F}\_{\mu\nu}(\mathbf{x}) &= \mathcal{U}^{-1}(\mathbf{x}, \mathcal{O}) \, F\_{\mu\nu}(\mathbf{x}) \, \mathcal{U}(\mathbf{x}, \mathcal{O}), \qquad \mathcal{U}(\mathbf{x}, \mathcal{O}) = P \exp\left(\mathrm{i} \int\_{\mathcal{I}} A\_{\eta}(\mathbf{y}) \, d\mathbf{y}^{\eta}\right), \end{aligned} \tag{5}$$

with *P* denoting path ordering, P "surface ordering", and *l* being a path from the base *O* of *∂S* to *x*, see [11]. The left hand side of (5) can be identified as the evaluation of a Wilson loop spanning the surface *S*. The right-hand side can be expressed using plaquettes: *<sup>U</sup>μν*(*x*) = exp *ia*2*Fμν* + <sup>O</sup>(*a*<sup>3</sup>), with lattice spacing *a*, see [12]. With these ingredients, the non-Abelian stokes theorem reads in the lattice, as shown in Figure 5:

**Figure 5.** Factoring a Wilson into factors of plaquettes using the non-Abelian stokes theorem.

By finding center regions, that is, plaquettes within *S* that combine to bigger regions which evaluate to center elements, the Wilson loop spanning *S* can be factorized into a commuting factor, a center element, and an non-Abelian part, see Figure 6.

**Figure 6.** Center regions explain the coulombic behavior and the linear rise of the quark–antiquark potential as they lead to an area law and a perimeter law for Wilson loops.

The center regions capture the center degrees of freedom and can be directly related to the behavior of Wilson loops. It seems reasonable to demand that their evaluation should not be changed by center gauge or projection on the center degrees of freedom. We show that by preserving nontrivial center regions, the loss of the vortex finding property is prevented and the full string tension can be recovered.

#### **2. Materials and Methods**

The predictions of the center vortex model concerning the string tension in SU(2) gluonic quantum chromodynamic are analyzed by calculating the Creutz ratios after center projection in maximal center gauge. The gauge fixing procedure is based upon simulated annealing, maximizing the functional (1), that is, bringing each link as close to a center element as possible. The simulated annealing algorithms are modified so that the evaluation of center regions is preserved during the procedure: transformations resulting in nontrivial center regions projecting onto the nontrivial center element are enforced, and transformations resulting in nontrivial center regions projecting onto the trivial center element are prevented.

The detection of the nontrivial center regions of one lattice configuration is done by enlarging regions until their evaluation becomes the nearest possible to a nontrivial center element, see Figure 7.

Steps 1–3: Starting with a plaquette that neither belongs to an already identified center region nor has already been taken as origin for growing a region, it is tested, whereby enlargement around a neighboring plaquette brings the region's evaluation nearer to a center element. Enlargement in the best direction is done.

Steps 4–6: If no enlargement leads to further improvement, a new enlargement procedure is started with another plaquette. With this enlargement, it is possible that it would grow into an existing region. The collision-handling described in the following is used to prevent this:

Step 7a: The evaluation of the growing region is nearer to a nontrivial centre element than the evaluation of the old region: delete the old region, only keeping the mark on its starting plaquette, and allow growing.

Step 7b: The growing region evaluates further away from a nontrivial centre element than the existing one: prevent growing in this direction and, if possible, enlarge in second best direction instead. Multiple collisions after growing are possible.

**Figure 7.** The algorithm for detecting center regions repeats these procedures until every plaquette either belongs to an identified region or has been taken once as starting plaquette for growing a region. The arrow marks the direction of enlargement. Plaquettes belonging to a region are colored, plaquettes already used as origin are shaded.

The algorithm starts with sorting the plaquettes of a given configuration by a rising trace of their evaluation. This stack is worked down plaquette by plaquette, enlarging each as far as possible by adding neighboring plaquettes. During this procedure, collisions of growing regions are prevented.

The regions identified this way comprise of many, whose evaluation deviates far from the center of the group. A set of nontrivial center regions has to be selected from the set of identified regions, only regions with traces smaller than *Tr*max are taken into account. This parameter *Tr*max has to be adjusted under consideration of the behavior of Creutz ratios, as shown in Figure 8, which are calculated after gauge-fixing and center projection.

**Figure 8.** *Tr*max can be fine-tuned by looking at the dependency of the Creutz ratios on the loop size *R*.

At low values of *Tr*max, the Creutz ratios are expected to be nearly constant with respect to the loop size. With raising *Tr*max they start to approach their asymptotic value from above and become chaotic with *Tr*max chosen inappropriately high.

As the center degrees of freedom are expected to capture the long-range behavior, the Creutz ratios calculated in center-projected configurations are near to the correct value of the string tension already for small loop sizes. Hence, we chose *Tr*max as high as possible without causing the behavior of the Creutz ratios to approach the string tension from above.

The regions determined by this procedure are then used to guide the gauge-fixing procedure. The influence on the predicted string tension is analyzed by calculating the Creutz ratios in center-projected configurations.
