**3. Results**

Here, we present the calculations of the center vortex string tension for different values of *Tr*max at *β* = 2.3. Similar results were obtained for *β* = 2.4 and *β* = 2.5. In the following, only the Creutz ratios of the center-projected configurations *<sup>χ</sup>*(*R*)*Z*2 are of relevance. The Creutz ratios of the full SU(2) theory *<sup>χ</sup>*(*R*)*SU*(2) and the calculations of the string tension based on the vortex density are calculated for comparison. They are only shown for the sake of completeness. All data was calculated with SU(2) Wilson action.

The Creutz ratios tend towards the literature value of the string tension with increasing number of simulated annealing steps with a *Tr*max = −0.985, whereas they clearly underestimate the string tension when center regions are ignored, see Figure 9.

**Figure 9.** By preserving center regions, the Creutz ratios tend towards the literature value of string tension during the simulated annealing procedure. The data was calculated at *β* = 2.3 in a 12<sup>4</sup> lattice with 100 configurations taken into account per datapoint. Displayed is the mean of *<sup>χ</sup>*(2), *<sup>χ</sup>*(3), and *<sup>χ</sup>*(4). The increased error bars when center regions are preserved might be because the algorithm does not reach the exact local maxima, but fluctuates around it.

The full string tension can be easily recovered, although the value of the gauge functional is reduced, see Figure 10.

The upper three graphs show the calculations done for optimizing the value of *Tr*max. The final results, shown in the left graph in the lower row, are calculated with a value of *Trnax* = −0.985, that is, a value between the respective values of the left and middle graph in the upper line. The final results are compared with raw simulated annealing, that is, without preserving center regions shown in the right graph of the lower row. The large errors using center regions might result from fluctuations of the gauge functional around the maxima, which can not be reached due to the constraint of the preservation of center regions: further approaches to the local maxima of the gauge functional are therefore prevented.

**Figure 10.** Optimization of *Tr*max in the upper line and final results for the guided simulated annealing in the lower row at *β* = 2.3. The Creutz ratios were calculated with 300 Wilson configurations at *β* = 2.3 in lattices of sizes 12<sup>4</sup> in the upper-left graph and 14<sup>4</sup> for the other graphs. The error bars are calculated with the one-deletion-Jackknife method. The optimal value of *Tr*max was identified by taking into account the behavior of the Creutz ratios and found to be around *Tr*max ≈ −0.985, reducing the value of gauge functional from *R* = 0.871 to 0.862.
