# Phase Diagram and Critical Properties within an Effective Model of QCD: The Nambu–Jona-Lasinio Model Coupled to the Polyakov Loop

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## Abstract

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## 1. Introduction

## 2. Symmetries of QCD

#### 2.1. Quantum Chromodynamics

#### 2.2. Chiral Symmetry Breaking

#### 2.3. U${}_{A}$(1) Symmetry Breaking

#### 2.4. The Polyakov Loop and the ${\mathbb{Z}}_{3}$ Symmetry Breaking: Pure Gauge Sector

#### 2.5. QCD Phase Diagram

## 3. The PNJL Model with Three Flavors

#### 3.1. Nambu–Jona-Lasinio Model with Anomaly and Explicit Symmetry Breaking

#### 3.2. Coupling between Quarks and the Gauge Sector: The PNJL Model

#### 3.3. Grand Potential in the Mean Field Approximation

#### 3.4. Equations of State and Response Functions

#### 3.5. Model Parameters and Regularization Procedure

## 4. Equation of State at Finite Temperature

#### 4.1. Characteristic Temperatures

#### 4.2. Thermodynamic Quantities

## 5. Phase Diagram and the Location of the Critical End Point

#### 5.1. Phase Transition at Zero Temperature

#### 5.2. Phase Transition at Finite Temperature and Density/chemical Potential

## 6. Nernst Principle and Isentropic Trajectories

## 7. Effects of Strangeness and Anomaly Strength on the Critical End Point

#### 7.1. Role of the Strangeness in the Location of the CEP/TCP

#### 7.2. Role of the Anomaly Strength in the Location of the CEP

## 8. Susceptibilities and Critical Behavior in the Vicinity of the CEP

## 9. Conclusions

## Acknowledgements

## Appendix

## References

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**Figure 1.**The phase diagram from the experimental point of view: freeze-out points for the different beam energies are showed [92].

**Figure 2.**Upper part: quark masses (left panel) and quark condensates (right panel) in PNJL model as functions of the temperature; the Polyakov loop field $\Phi $ is also shown. At ${T}_{eff}=345$ MeV, ${M}_{i}={m}_{i}$. Lower part: derivatives of the quark condensates and of the Polyakov loop field $\Phi $ (left panel); the topological susceptibility, $\chi $, (right panel) in the PNJL model, compared to corresponding lattice results taken from [101].

**Figure 3.**Scaled pressure $(p)$, energy per particle $(\u03f5)$, and entropy $(s)$ as a function of the temperature at zero chemical potential. The data points are taken from [77]. The pressure reaches 66% of the strength of the Stefan-Boltzmann value at $T=300$ MeV, a value which attains 85% at $T=400$ MeV.

**Figure 4.**Left: grand potential for the stable, metastable and unstable phases, used to determine ${\mu}_{B}^{cr}$ and u quark mass; right: baryonic density as function of ${\mu}_{B}$.

**Figure 5.**Pressure (left) and energy per particle (right) as a function of ${\rho}_{B}/{\rho}_{0}$ at different temperatures (${\rho}_{0}=0.17$ fm${}^{-3}$ is the normal nuclear matter density). The points A and B (left panel) illustrate the Gibbs criteria. Only in the $T=0$ line the zero-pressure point is located at the minimum of the energy per particle.

**Figure 6.**Phase diagram in the SU(3) PNJL model. The left (right) part corresponds to the $T-{\mu}_{B}$ ($T-{\rho}_{B}$) plane. Solid (dashed) line shows the location of the first order (crossover) transition. The dashed lines shows the location of the spinodal boundaries of the two phase transitions (shown by shading in the right plot).

**Figure 7.**Isentropic trajectories in the $(T,{\mu}_{B})$ plane. The following values of the entropy per baryon number have been considered: $s/{\rho}_{B}=1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}3,\phantom{\rule{0.166667em}{0ex}}4,\phantom{\rule{0.166667em}{0ex}}5,\phantom{\rule{0.166667em}{0ex}}6,\phantom{\rule{0.166667em}{0ex}}8,\phantom{\rule{0.166667em}{0ex}}10,\phantom{\rule{0.166667em}{0ex}}15,\phantom{\rule{0.166667em}{0ex}}20$ (anticlockwise direction).

**Figure 8.**Left panel: the phase diagram in the SU(3) PNJL model. The solid lines represent the first order phase transition, the dotted line the second order phase transition, and the dashed line the crossover transition. Right panel: the phase diagram and the “line” of TCPs for different values of ${m}_{s}$ (the dotted lines are just drawn to guide the eye); the TCPs in both figures are obtained in the limit ${m}_{u}={m}_{d}=0$ and ${m}_{s}\ne 0$.

**Figure 9.**Dependence of the location of the CEP on the strength of the ’t Hooft coupling constant ${g}_{D}$.

**Figure 10.**Phase diagram: the size of the critical region is plotted for ${\chi}_{B}/{\chi}_{B}^{free}=1(2)$. The TCP is found for ${m}_{u}={m}_{d}=0$ MeV and.

**Figure 11.**Left panel: Baryon number susceptibility (right panel) as functions of ${\mu}_{B}$ for different temperatures around the CEP: ${T}^{CEP}=155.80$ MeV and $T={T}^{CEP}\pm 10$ MeV. Right panel: Specific heat as a function of T for different values of ${\mu}_{B}$ around the CEP: ${\mu}_{B}^{CEP}=290.67$ MeV and ${\mu}_{B}={\mu}_{B}^{CEP}\pm 10$ MeV.

Symmetry | Transformation | Current | Name | Manifestation in nature |
---|---|---|---|---|

SU${}_{V}$(3) | $q\to exp(-i\frac{{\lambda}_{a}{\alpha}_{a}}{2})q$ | ${V}_{\mu}^{a}=\overline{q}{\gamma}_{\mu}\frac{{\lambda}_{a}}{2}q$ | Isospin | Approximately conserved |

U${}_{V}$(1) | $q\to exp(-i{\alpha}_{V})q$ | ${V}_{\mu}=\overline{q}{\gamma}_{\mu}q$ | Baryonic | Conserved |

SU${}_{A}$(3) | $q\to exp(-i\frac{{\gamma}_{5}{\lambda}_{a}{\theta}_{a}}{2})q$ | ${A}_{\mu}^{a}=\overline{q}{\gamma}_{\mu}{\gamma}_{5}\frac{{\lambda}_{a}}{2}q$ | Quiral | Spontaneously broken |

U${}_{A}$(1) | $q\to exp(-i{\gamma}_{5}{\alpha}_{A})q$ | ${A}_{\mu}=\overline{q}{\gamma}_{\mu}{\gamma}_{5}q$ | Axial | “ U${}_{A}$(1) problem” |

**Table 2.**Parameters for the effective potential in the pure gauge sector (Equation (10)).

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

3.51 | −2.47 | 15.2 | −1.75 |

${\mathit{T}}_{0}$ [MeV] | ${\mathit{T}}_{\mathit{c}}^{\mathit{\chi}}$ [MeV] | ${\mathit{T}}_{\mathit{c}}^{\mathbf{\Phi}}$ [MeV] | ${\mathit{T}}_{\mathit{c}}$ [MeV] |
---|---|---|---|

270 | 222 | 210 | 216 |

210 | 203 | 171 | 187 |

© 2010 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 license (http://creativecommons.org/licenses/by/3.0/)

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**MDPI and ACS Style**

Costa, P.; Ruivo, M.C.; De Sousa, C.A.; Hansen, H.
Phase Diagram and Critical Properties within an Effective Model of QCD: The Nambu–Jona-Lasinio Model Coupled to the Polyakov Loop. *Symmetry* **2010**, *2*, 1338-1374.
https://doi.org/10.3390/sym2031338

**AMA Style**

Costa P, Ruivo MC, De Sousa CA, Hansen H.
Phase Diagram and Critical Properties within an Effective Model of QCD: The Nambu–Jona-Lasinio Model Coupled to the Polyakov Loop. *Symmetry*. 2010; 2(3):1338-1374.
https://doi.org/10.3390/sym2031338

**Chicago/Turabian Style**

Costa, Pedro, Maria C. Ruivo, Célia A. De Sousa, and Hubert Hansen.
2010. "Phase Diagram and Critical Properties within an Effective Model of QCD: The Nambu–Jona-Lasinio Model Coupled to the Polyakov Loop" *Symmetry* 2, no. 3: 1338-1374.
https://doi.org/10.3390/sym2031338