# Influence of Fermions on Vortices in SU(2)-QCD

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## Abstract

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## 1. Introduction

- via the potential calculated from the center degrees of freedom only, in pure gluonic ensembles;
- via the potential calculated from the center degrees of freedom only, in the presence of fermionic fields;
- via the potential in the full theory, in pure gluonic ensembles; and
- via the potential in the full theory, in the presence of fermionic fields.

## 2. Materials and Methods

#### 2.1. Simulation Specifications

#### 2.2. Center Vortex Detection

#### 2.3. Smoothing the Vortex Surface

#### 2.4. Potential Fits and Noise Handling

- for small areas, the loop averages are influenced by short range fluctuations; and
- with increasing area, the data suffer from statistical noise and soon the errors get larger than the signal.

## 3. Summarized Results and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The closed vortex surface is visualized by showing the dual P-plaquettes of three-dimensional lattice slices. Stubs of red lines indicate plaquettes that are not fully part of the lattice slice shown. We clearly see that with fermions (

**left**) an overall smaller amount of P-plaquettes is observed compared with the pure gluonic case (

**right**). In both cases, one big vortex cluster dominates.

**Figure 2.**These probability densities specify the relation between the values of gauge functional and Creutz ratio for individual configurations. This determination is based on 40 configurations with 100 gauge copies for the configurations with dynamical fermions (

**left**) and without (

**right**). For Creutz ratios of small Wilson loops, we observe a nearly linear relation. With increasing size of the Wilson loops this correlation weakens. We marked the average values (star) and the most probable values (circle) of the distributions.

**Figure 3.**The average and most probable values of $\chi (R)$ are compared for simulations with and without fermions. This complements the probability densities of Figure 2. The increasing discrepancy between the two quantities with larger R can probably be explained by the low precision of Creutz ratios of single configurations. Until the values start to deviate from one another, the variations of $\chi (R)$ are of the order of 10%. For comparison, we show also the more precise Creutz ratios ${\chi}_{W}(R)$ extracted from averages of Wilson loops.

**Figure 4.**The effect of the smoothing procedures on the vortex surface is depicted, taken from ([19] Figure 5.8). We distinguish warts (

**left**), bottlenecks (

**middle**) and stumbling blocks (

**right**). The unit cubes are not depicted, which are simply deleted.

**Figure 5.**(

**Left**) Example of the optimal 1-exponential fit of Wilson loops for given R. (

**Right**) Dependence of ${\epsilon}_{0}(R,{T}_{i})$ on the fit region $T\ge {T}_{i}$. The line marks the fit for the optimal value for ${T}_{i}$.

**Figure 6.**(

**Left**) Potential $V(R)$ in lattice units between two sources in the fundamental representation. There is a large difference between the string tensions for pure gluonic configurations (“gluonic”) and in the presence of one species of dynamical fermions. (

**Right**) Potentials extracted from Wilson loops after ensemble averaged maximal center projection are depicted for pure gluonic configurations and for configurations with dynamical fermions. Due to the removal of short range fluctuations the potentials are in both cases almost linearly increasing with the lattice distance R. Data are fitted by linear functions. For gluonic (fermionic) configurations, only data with $R\ge 6(2)$ are fitted.

P-plaquette Reduction | smoothing 0 | smoothing 1 | smoothing 2 | smoothing 3 |
---|---|---|---|---|

With fermions | 12.5% | 10.1% | 24% | 10.2% |

Without fermions | 7% | 10.6% | 27.8% | 10.9% |

**Table 2.**Reduction of P-plaquettes for the percolating vortex cluster for different smoothing procedures.

Reduction within Cluster | smoothing 1 | smoothing 2 | smoothing 3 |
---|---|---|---|

With fermions | 8.6% | 24.5% | 8.8% |

Without fermions | 9.6% | 28.1% | 10% |

Theory | SU(2) | Z_{2} | |||
---|---|---|---|---|---|

Parameter | ${\mathit{V}}_{\mathbf{0}}$ | $\mathit{\sigma}$ | $\mathit{\alpha}$ | ${\mathit{v}}_{\mathbf{0}}$ | ${\mathit{\sigma}}_{\mathrm{CP}}$ |

gluonic | 0.5175(38) | 0.0756(12) | 0.2326(26) | −0.0366(8) | 0.07691(13) |

fermionic | 0.5464(27) | 0.0199(9) | 0.2414(19) | 0.01027(13) | 0.02291(5) |

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**MDPI and ACS Style**

Dehghan, Z.; Deldar, S.; Faber, M.; Golubich, R.; Höllwieser, R.
Influence of Fermions on Vortices in SU(2)-QCD. *Universe* **2021**, *7*, 130.
https://doi.org/10.3390/universe7050130

**AMA Style**

Dehghan Z, Deldar S, Faber M, Golubich R, Höllwieser R.
Influence of Fermions on Vortices in SU(2)-QCD. *Universe*. 2021; 7(5):130.
https://doi.org/10.3390/universe7050130

**Chicago/Turabian Style**

Dehghan, Zeinab, Sedigheh Deldar, Manfried Faber, Rudolf Golubich, and Roman Höllwieser.
2021. "Influence of Fermions on Vortices in SU(2)-QCD" *Universe* 7, no. 5: 130.
https://doi.org/10.3390/universe7050130