# Nonextensive Quasiparticle Description of QCD Matter

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Short Reminder of the $\mathit{z}$-QPM

## 3. Formulation of the $\mathit{qz}$-QPM

## 4. Results

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Limitations of the Allowed Phase Space in the Nonextensive Approach

## Appendix B. Approximate Calculation of ${z}_{q}$

## Appendix C. Some Selected First Order Expansions in ($q-1$)

## Appendix D. Expansion of Pressure in Chemical Potential $\mu $

## References

- Walecka, J.D. A Theory of Highly Condensed Matter. Ann. Phys.
**1974**, 83, 491–529. [Google Scholar] [CrossRef] - Chin, S.A.; Walecka, J.D. An Equation of State for Nuclear and Hihger-Density Matter Based on a Relativistic Mean-Field Theory. Phys. Lett. B
**1974**, 52, 24–28. [Google Scholar] [CrossRef] - Serot, B.D.; Walecka, J.D. The Relativistic Nuclear Many Body Problem. Adv. Nucl. Phys.
**1986**, 16, 1–327. [Google Scholar] - Nambu, Y.; Jona-Lasinio, G. Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I. Phys. Rev.
**1961**, 122, 345–358. [Google Scholar] [CrossRef] - Nambu, Y.; Jona-Lasinio, G. Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. II. Phys. Rev.
**1961**, 124, 246–254. [Google Scholar] [CrossRef] - Klevansky, S.P. The Nambu-Jona-Lasinio model of quantum chromodynamics. Rev. Mod. Phys.
**1992**, 64, 649–708. [Google Scholar] [CrossRef] - Rehberg, P.; Klevansky, S.P.; Hüfner, J. Hadronization in the SU(3) Nambu–Jona-Lasinio model. Phys. Rev. C
**1966**, 53, 410–429. [Google Scholar] [CrossRef] - Randrup, J. Phase transition dynamics for baryon-dense matter. Phys. Rev. C
**2009**, 79, 054911. [Google Scholar] [CrossRef] - Palhares, L.F.; Fraga, E.S.; Kodama, T. Chiral transition in a finite system and possible use of finite-size. scaling in relativistic heavy ion collisions. J. Phys. G
**2011**, 38, 085101. [Google Scholar] [CrossRef] - Skokov, V.V.; Voskresensky, D.N. Hydrodynamical description of first-order phase transitions: Analytical treatment and numerical modeling. Nucl. Phys. A
**2009**, 828, 401–438. [Google Scholar] [CrossRef] - Wilk, G.; Włodarczyk, Z. Consequences of temperature fluctuations in observables measured in high-energy collisions. Eur. Phys. J. A
**2012**, 48, 161. [Google Scholar] [CrossRef] - Wilk, G.; Włodarczyk, Z. Quasi-power laws in multiparticle production processes. Chaos Solitons Fractals
**2015**, 81, 487–496. [Google Scholar] [CrossRef] [Green Version] - Tsallis, C. Introduction to Nonextensive Statistical Mechanics; Springer: New York, NY, USA, 2009. [Google Scholar]
- Tsallis, C. Thermodynamics and statistical mechanics for complex systems—Foundationsa and applications. Acta Phys. Pol. B
**2015**, 46, 1089–1101. [Google Scholar] [CrossRef] - Santos, A.P.; Pereira, F.I.M.; Silva, R.; Alcaniz, J.S. Consistent nonadditive approach and nuclear equation of state. J. Phys. G
**2014**, 41, 055105. [Google Scholar] [CrossRef] [Green Version] - Rożynek, J.; Wilk, G. Nonextensive Nambu—Jona-Lasinio Model of QCD matter. Eur. Phys. J. A
**2016**, 52, 13. [Google Scholar] [CrossRef] - Megias, E.; Menezes, D.P.; Deppman, A. Non extensive thermodynamics for hadronic matter with finite chemical potentials. Physica A
**2015**, 421, 15–24. [Google Scholar] [CrossRef] - Lavagno, A.; Pigato, D. Nonextensive nuclear liquid-gas phase transition. Physica A
**2013**, 392, 5164–5171. [Google Scholar] [CrossRef] - Maroney, O.J.E. Thermodynamic constraints on fluctuation phenomena. Phys. Rev. E
**2009**, 80, 061141. [Google Scholar] [CrossRef] - Biró, T.S.; Ürmösy, K.; Schram, Z. Thermodynamics of composition rules. J. Phys. G
**2010**, 37, 094027. [Google Scholar] [CrossRef] - Biró, T.S.; Vań, P. Zeroth law compatibility of nonadditive thermodynamics. Phys. Rev. E
**2011**, 83, 061147. [Google Scholar] [CrossRef] - Biró, T.S. Is There a Temperature? Conceptual Challenges at High Energy, Acceleration and Complexity; Springer: New York, NY, USA; Dordrecht, The Netherlands; Heidelberg, Germany; London, UK, 2011. [Google Scholar]
- Teweldeberhan, A.M.; Plastino, A.R.; Miller, H.G. On the cut-off prescriptions associated with power-law generalized thermostatistics. Phys. Lett. A
**2005**, 343, 71–78. [Google Scholar] [CrossRef] [Green Version] - Teweldeberhan, A.M.; Miller, H.G.; Tegen, R. Generalized statistics and the formation of a quark-gluon plasma. Int. J. Mod. Phys. E
**2003**, 12, 395–405. [Google Scholar] [CrossRef] - Conroy, J.M.; Miller, H.G.; Plastino, A.R. Thermodynamic consistency of the q-deformed Fermi–Dirac distribution in nonextensive thermostatics. Phys. Lett. A
**2010**, 374, 4581–4584. [Google Scholar] [CrossRef] - Cleymans, J.; Worku, D. Relativistic thermodynamics: Transverse momentum distributions in high-energy physics. Eur. Phys. J. A
**2012**, 48, 160. [Google Scholar] [CrossRef] - Biyajima, M.; Mizoguchi, T.; Nakajima, N.; Suzuki, N.; Wilk, G. Modified Hagedorn formula including temperature fluctuation: Estimation of temperatures at RHIC experiments. Eur. Phys. J. C
**2006**, 48, 597–603. [Google Scholar] [CrossRef] [Green Version] - Peshier, A.; Kampfer, B.; Soff, G. From QCD lattice calculations to the equation of state of quark matter. Phys. Rev. D
**2002**, 66, 094003. [Google Scholar] [CrossRef] - Peshier, A.; Kampfer, B.; Pavlenko, O.P.; Soff, G. Massive quasiparticle model of the SU(3) gluon plasma. Phys. Rev. D
**1996**, 54, 2399–2402. [Google Scholar] [CrossRef] - Fukushima, K. Chiral effective model with the Polyakov loop. Phys. Lett. B
**2004**, 591, 277–284. [Google Scholar] [CrossRef] [Green Version] - Tsai, H.M.; Müller, B. Phenomenology of the three-flavor PNJL model and thermal strange quark production. J. Phys. G
**2009**, 36, 075101. [Google Scholar] [CrossRef] [Green Version] - Chandra, V.; Ravishankar, V. Quasiparticle description of (2+1)- flavor lattice QCD equation of state. Phys. Rev. D
**2011**, 84, 074013. [Google Scholar] [CrossRef] - Chandra, V. Bulk viscosity of anisotropically expanding hot QCD plasma. Phys. Rev. D
**2011**, 84, 094025. [Google Scholar] [CrossRef] - Chandra, V. Transport properties of anisotropically expanding quark-gluon plasma within a quasiparticle model. Phys. Rev. D
**2012**, 86, 114008. [Google Scholar] [CrossRef] - Jamal, M.Y.; Mitra, S.; Chandra, V. Collective excitations of hot QCD medium in a quasiparticle description. Phys. Rev. D
**2017**, 95, 094022. [Google Scholar] [CrossRef] [Green Version] - Mitra, S.; Chandra, V. Transport coefficients of a hot QCD medium and their relative significance in heavy-ion collisions. Phys. Rev. D
**2017**, 96, 094003. [Google Scholar] [CrossRef] [Green Version] - Mitra, S.; Chandra, V. Covariant kinetic theory for effective fugacity quasiparticle model and first order transport coefficients for hot QCD matter. Phys. Rev. D
**2018**, 97, 034032. [Google Scholar] [CrossRef] [Green Version] - Gorenstein, M.I.; Yang, S.N. Gluon plasma with a medium-dependent dispersion relation. Phys. Rev. D
**1995**, 52, 5206. [Google Scholar] [CrossRef] - Schneider, R.A.; Weise, W. Quasiparticle description of lattice QCD thermodynamics. Phys. Rev. C
**2001**, 64, 055201. [Google Scholar] [CrossRef] - Ivanov, Y.B.; Skokov, V.V.; Toneev, V.D. Equation of state of deconfined matter within a dynamical quasiparticle description. Phys. Rev. D
**2005**, 71, 014005. [Google Scholar] [CrossRef] - Luo, L.J.; Cao, J.; Yan, Y.; Sun, W.M.; Zong, H.S. A thermodynamically consistent quasi-particle model without density-dependent infinity of the vacuum zero-point energy. Eur. Phys. J. C
**2013**, 73, 2626. [Google Scholar] [CrossRef] [Green Version] - Bannur, V.M. Landau’s statistical mechanics for quasi-particle models. Int. J. Mod. Phys. A
**2014**, 29, 1450056. [Google Scholar] [CrossRef] - Cheng, M.; Ejiri, S.; Hegde, P.; Karsch, F.; Kaczmarek, O.; Laermann, E.; Mawhinney, R.D.; Miao, C.; Mukherjee, S.; Petreczky, P.; et al. Equation of state for physical quark masses. Phys. Rev. D
**2010**, 81, 054504. [Google Scholar] [CrossRef] - Bazavov, A.; Bhattacharya, T.; Cheng, M.; Christ, N.H.; DeTar, C.; Ejiri, S.; Gottlieb, S.; Gupta, R.; Heller, U.M.; Huebner, K.; et al. Equation of state and QCD transition at finite temperature. Phys. Rev. D
**2009**, 80, 014504. [Google Scholar] [CrossRef] [Green Version] - Borsanyi, S.; Endrodi, G.; Fodor, Z.; Jakovac, A.; Katz, S.D.; Krieg, S.; Ratti, C.; Szabo, K.K. The QCD equation of state with dynamical quarks. J. High Energy Phys.
**2010**, 11, 077. [Google Scholar] [CrossRef] - Aarts, G. Introductory lectures on lattice QCD at nonzero baryon number. J. Phys. Conf. Ser.
**2016**, 706, 022004. [Google Scholar] [CrossRef] [Green Version] - Ratti, C. Lattice QCD: Bulk and transport properties of QCD matter. Nucl. Phys. A
**2016**, 956, 51–58. [Google Scholar] [CrossRef] - Rożynek, J.; Wilk, G. An example of the interplay of nonextensivity and dynamics in the description of QCD matter. Eur. Phys. J. A
**2016**, 52, 294. [Google Scholar] [CrossRef] - Rożynek, J. Non-extensive distributions for a relativistic Fermi gas. Physica A
**2015**, 440, 27–32. [Google Scholar] [CrossRef] [Green Version] - Biró, T.S.; Schram, Z. Lattice gauge theory with fluctuating temperature. Eur. Phys. J. Web Conf.
**2011**, 13, 05004. [Google Scholar] [CrossRef] [Green Version] - Frigori, R.B. Nonextensive lattice gauge theories: Algorithms and methods. Comput. Phys. Commun.
**2014**, 185, 2232–2239. [Google Scholar] [CrossRef] [Green Version] - Osada, T.; Wilk, G. Nonextensive hydrodynamics for relativistic heavy-ion collisions. Phys. Rev. C
**2008**, 78, 069903. [Google Scholar] [CrossRef]

**Figure 1.**(Color online) Upper panels: Results for ${z}_{q}^{\left(q\right)}\left(\tau \right)$ and ${z}_{q}^{\left(g\right)}\left(\tau \right)$ as a function of the scaled temperature $\tau =T/{T}_{c}$ (calculated for $\mu =0$). Lower panels: As above, but shown in more detail and with an enlarged range of the nonextensivity parameter q.

**Figure 2.**(Color online) Illustration of the changes introduced by the chemical potential $\mu $ for $q=1.01$ and $q=0.99$.

**Figure 3.**(Color online) Left panel: Relative density, ${\rho}_{q}/{\rho}_{q=1}$, of quarks and gluons as a function of the nonextensivity q for $\mu =0$. Right panel: Dependence of the relative density, ${\rho}_{q}/{\rho}_{q=1}$, on the chemical potential $\mu $.

**Figure 4.**(Color online) Left panel: Results for the ratio ${M}_{D}/{M}_{D}^{I}$ of the Debye masses (as defined by Equations (21) and (22)), respectively) in the nonextensive environment as functions of the scaled temperature, $\tau =T/{T}_{c}$ for $q=0.96,\phantom{\rule{3.33333pt}{0ex}}1,\phantom{\rule{3.33333pt}{0ex}}1.04$, calculated for $\mu =0$. Right panel: The same as above, but shown in more detail and with an enlarged range of the nonextensivity parameter q.

**Figure 5.**(Color online) Dependencies of relative pressure ${P}_{q}/{P}_{q=1}$ and density ${\rho}_{q}/{\rho}_{q=1}$ on the nonextensivity parameter q at fixed temperature (left panel) and on the temperature T for fixed q (right panel).

**Figure 6.**(Color online) The behavior of the change in the trace anomaly in the $qz$-quasi-particle model (QPM) with ${z}_{q}={z}_{q=1}$ as a function of $\tau $ for some selected values of q (left panel) and the same, but for some selected values of the chemical potential $\mu $ (right panel).

**Figure 7.**(Color online) The behavior of the ratio ${M}_{D}/{M}_{D}^{I}$ of the Debye masses (as defined by Equations (21) and (22) as a function of $\tau $ for some selected values of q (left panel) and the same, but for some selected values of the chemical potential $\mu $ (right panel). The dynamics is the same as in the z-QPM, i.e., ${z}_{q}={z}_{q=1}=z$.

q | $\left(\mathit{i}\right)$ | ${\mathit{a}}_{\left(\mathit{i}\right)}$ | ${\mathit{b}}_{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{\left(\mathit{i}\right)}^{\prime}$ | ${\mathit{b}}_{\left(\mathit{i}\right)}^{\prime}$ |
---|---|---|---|---|---|

$q=1$ | $i=g$ | $0.803$ | $1.837$ | $0.978$ | $0.942$ |

$q=1$ | $i=q$ | $0.810$ | $1.721$ | $0.960$ | $0.846$ |

q | $\left(\mathit{i}\right)$ | ${\mathit{a}}_{\left(\mathit{i}\right)}$ | ${\mathit{b}}_{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{\left(\mathit{i}\right)}^{\prime}$ | ${\mathit{b}}_{\left(\mathit{i}\right)}^{\prime}$ |
---|---|---|---|---|---|

$q=0.96$ | $i=g$ | $0.985$ | $1.581$ | $1.168$ | $0.860$ |

$q=0.96$ | $i=q$ | $1.030$ | $1.510$ | $1.200$ | $0.747$ |

$q=0.98$ | $i=g$ | $0.897$ | $1.702$ | $1.078$ | $0.870$ |

$q=0.98$ | $i=q$ | $0.924$ | $1.603$ | $1.073$ | $0.770$ |

$q=0.99$ | $i=g$ | $0.850$ | $1.760$ | $1.028$ | $0.904$ |

$q=0.99$ | $i=q$ | $0.867$ | $1.662$ | $1.018$ | $0.799$ |

$q=1.01$ | $i=g$ | $0.753$ | $1.916$ | $0.927$ | $0.990$ |

$q=1.01$ | $i=q$ | $0.751$ | $1.791$ | $0.896$ | $0.879$ |

$q=1.02$ | $i=g$ | $0.704$ | $2.006$ | $0.876$ | $1.059$ |

$q=1.02$ | $i=q$ | $0.694$ | $1.862$ | $0.835$ | $0.925$ |

$q=1.04$ | $i=g$ | $0.600$ | $2.221$ | $0.766$ | $1.180$ |

$q=1.04$ | $i=q$ | $0.580$ | $2.061$ | $0.712$ | $1.050$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rożynek, J.; Wilk, G.
Nonextensive Quasiparticle Description of QCD Matter. *Symmetry* **2019**, *11*, 401.
https://doi.org/10.3390/sym11030401

**AMA Style**

Rożynek J, Wilk G.
Nonextensive Quasiparticle Description of QCD Matter. *Symmetry*. 2019; 11(3):401.
https://doi.org/10.3390/sym11030401

**Chicago/Turabian Style**

Rożynek, Jacek, and Grzegorz Wilk.
2019. "Nonextensive Quasiparticle Description of QCD Matter" *Symmetry* 11, no. 3: 401.
https://doi.org/10.3390/sym11030401