# Quark Cluster Expansion Model for Interpreting Finite-T Lattice QCD Thermodynamics

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## Abstract

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## 1. Introduction

## 2. Cluster Virial Expansion to Quark-Hadron Matter

#### 2.1. Beth–Uhlenbeck Model for HRG with Mott Dissociation

#### 2.2. Polyakov-Loop Improved Nambu–Jona-Lasinio (PNJL) Model

#### 2.3. Perturbative Contribution

## 3. Stationarity Condition for the Polyakov Loop

## 4. Results

#### 4.1. Polyakov Loop

#### 4.2. Pressure

#### 4.3. Quark Number Susceptibilities

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Pressure as a function of temperature for the HRG model with stable hadrons (red line) and for the MHRG model with Mott dissociation of hadrons according to the simple phase shift model (4) employed in the present work. These results are compared to the LQCD data from the HotQCD Collaboration [4] (green band) and the Wuppertal–Budapest Collaboration [3] (blue band).

**Figure 3.**Two-loop diagram for the contribution of the one-gluon exchange interaction to the thermodynamic potential of quark matter.

**Figure 4.**The traced Polyakov loop $\varphi $ from the solution of the stationarity condition (23) on the thermodynamical potential as a function of temperature (magenta solid line) compared with the lattice results for the renormalized Polyakov loop the TU Munich QCD (TUMQCD) Collaboration [32] (green band) and the Wuppertal–Budapest Collaboration [1] (blue symbols).

**Figure 5.**The temperature derivatives of the chiral condensate (chiral susceptibility $d{\Delta}_{l,s}/dT$, red solid line) and of the Polyakov loop (Polyakov-loop susceptibility $d\varphi /dT$, ) as functions of temperature. The vertical lines indicate their almost coincident peak positions at ${T}_{\chi}=156.5$ MeV and ${T}_{\varphi}=159.0$ MeV, respectively.

**Figure 6.**The temperature dependence of the total scaled pressure (red solid line) and it’s constituents: MHRG (coral dotted line), quark (dashed magenta line), Polyakov-loop potential $\mathcal{U}(T;\varphi )$ (dash–dotted green line) and perturbative QCD contribution (dash-dotted blue line) compared to the lattice QCD data: HotQCD Collaboration [4] (green band) and Wuppertal–Budapest Collaboration [3] (blue band).

**Figure 7.**The temperature dependence of the total scaled pressure (red solid line) and its constituents: MHRG (coral dotted line), quark (dashed magenta line), Polyakov-loop potential $U(\varphi ,T)$ (dash-dotted green line) and perturbative QCD contribution (dash-dotted blue line) compared to the lattice QCD data: HotQCD Collaboration [4] (green band) and Wuppertal–Budapest Collaboration [3] (blue band), and the high-temperature result [33] (magenta band).

**Figure 8.**The dimensionless ratio of quark number density to quark number susceptibility ${R}_{12}\left(T\right)={n}_{q}\left(T\right)/\left({\mu}_{q}{\chi}_{q}\left(T\right)\right){|}_{{\mu}_{q}=0}$ as a function of temperature for ${\mu}_{q}/T=0.4$ (red solid line) and ${\mu}_{q}/T=0.8$ (blue dash-dotted line) compared to the lattice QCD data [34] ${\mu}_{q}/T=0.4$ (red band), ${\mu}_{q}/T=0.8$ (blue band). For details, see text.

${\mathit{a}}_{0}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{b}}_{3}$ | ${\mathit{b}}_{4}$ |
---|---|---|---|---|---|

6.75 | −1.95 | 2.625 | −7.44 | 0.75 | 7.5 |

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**MDPI and ACS Style**

Blaschke, D.; Devyatyarov, K.A.; Kaczmarek, O.
Quark Cluster Expansion Model for Interpreting Finite-T Lattice QCD Thermodynamics. *Symmetry* **2021**, *13*, 514.
https://doi.org/10.3390/sym13030514

**AMA Style**

Blaschke D, Devyatyarov KA, Kaczmarek O.
Quark Cluster Expansion Model for Interpreting Finite-T Lattice QCD Thermodynamics. *Symmetry*. 2021; 13(3):514.
https://doi.org/10.3390/sym13030514

**Chicago/Turabian Style**

Blaschke, David, Kirill A. Devyatyarov, and Olaf Kaczmarek.
2021. "Quark Cluster Expansion Model for Interpreting Finite-T Lattice QCD Thermodynamics" *Symmetry* 13, no. 3: 514.
https://doi.org/10.3390/sym13030514