# Phase Diagram, Scalar-Pseudoscalar Meson Behavior and Restoration of Symmetries in (2 + 1) Polyakov-Nambu-Jona-Lasinio Model

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Model and Formalism

#### 2.1. The PNJL Model

#### 2.2. Gap Equations

#### 2.3. Pseudoscalar and Scalar Meson Nonets

#### 2.4. Thermodynamics

#### 2.5. Model Parameters and Regularization Procedure

## 3. The Phase Diagram in the PNJL Model

#### 3.1. Characteristic Temperatures at Zero Density

#### 3.2. Finite Temperature and Chemical Potential

#### 3.3. Nernst Principle and Isentropic Trajectories

## 4. Scalar and Pseudoscalar Mesons in the PNJL Model

#### 4.1. Mesons Properties at Finite Temperature

#### 4.1.1. Mesonic Masses and Mixing Angles

#### 4.1.2. Pion and Kaon Coupling Constants

#### 4.2. Mesons at Zero Temperature

#### 4.3. Mesons Properties in Different Regions of the Phase Diagram

#### 4.3.1. Meson Masses in the Crossover Region

#### 4.3.2. Mesons through the CEP

#### 4.3.3. Mesons through the First-Order Transition

#### 4.3.4. Mesons along the Isentropic Trajectory That Passes over the CEP

#### 4.4. Effective Restoration of Chiral Symmetry and Mott Dissociation of $\pi $ and $\sigma $ along the Phase Diagram

## 5. Conclusions

- (i)
- the survivability of some meson modes, especially the pion, as a bound state after the transition to the QGP (this tendency to a slightly longer survival as bound state is also shown by the behavior of meson-quark coupling constants for $\pi $, $\sigma $ and $\eta $ mesons);
- (ii)
- the change of identity between $\eta $ and ${\eta}^{\prime}$ at finite density for scenarios at lower temperatures;
- (iii)
- the meson masses change abruptly when choosing a path that passes through the CEP (this can be very important for the signatures of the CEP);
- (iv)
- in relation to kaons, with the exception of the limiting cases for $T=0$ and ${\mu}_{B}=0$, a kaon charge splitting before critical temperature/baryonic chemical potential occurs. At CEP and first-order cases, kaons first degenerate with the respective chiral partners and only then with charge multiplet, contrary to the crossover scenario where charge multiplets degenerate first. At the CEP, there is a accentuated splitting for kaons, with ${K}^{+}$ sharply increasing, a splitting that is still pronounced just after the CEP;
- (v)
- above certain critical values of temperature and chemical potentials (${T}_{eff}^{\chi}$, ${{\mu}_{B}}_{eff}^{\chi}$) the masses of the chiral partners [$\pi $,$\sigma $] will degenerate, meaning that chiral symmetry is effectively restored. All quantities that violate chiral symmetry are guaranteed to be already sufficiently small.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Quark condensates and respective derivatives (

**left panel**) and quark masses (

**right panel**) in the PNJL model as functions of the temperature; the Polyakov loop field, $\Phi $, is also shown, together with its derivative given by the thin blue line in the left panel.

**Figure 2.**(

**Left panel**) Deconfinement transition (red dots) and chiral transition with first-order (full blue line), CEP (black dot) and crossover regions (dashed black line) and spinodal lines (dashed blue line). (

**Right panel**) Deconfinment and chiral transitions alongside isentropic lines (full green lines).

**Figure 3.**(

**Left panel**) Masses of the pseudoscalar (full lines) and scalar mesons (dashed lines) as function of the reduced temperature. (

**Right panel**) Pseudoscalar and scalar mixing angles as a function of the reduced temperature.

**Figure 4.**Coupling constants (

**left panel**) and decay constants (

**right panel**) as a function of the reduced temperature.

**Figure 5.**Quark masses at $T=0$ as a function of the baryonic chemical potential (

**left panel**) and reduced baryonic density (

**right panel**). The dashed lines represent the quark masses in the first-order region. ${\mu}_{B}^{crit}$ is the baryonic critical chemical potential of the first-order phase transition, and ${\rho}_{B}^{crit}/{\rho}_{0}$ the respective critical reduced density (${\rho}_{0}=0.16$ fm${}^{-3}$).

**Figure 6.**Masses of the neutral pseudoscalar and scalar mesons as a function of ${\mu}_{B}$ (panel (

**a**)) and as a function of ${\rho}_{B}/{\rho}_{0}$ (panel (

**b**)); the pseudoscalar and scalar mixing angles as a function of ${\mu}_{B}$ (panel (

**c**)) and as a function of ${\rho}_{B}/{\rho}_{0}$ (panel (

**d**)); masses of the charged pseudoscalar and scalar mesons as a function of ${\mu}_{B}$ (panel (

**e**)) and as a function of ${\rho}_{B}/{\rho}_{0}$ (panel (

**f**)). All results are at $T=0$ (see text for details).

**Figure 7.**Masses of the pseudoscalar and scalar mesons as function of the temperature (panels (

**a**,

**c**)) and as function of ${\mu}_{B}$ (panels (

**b**,

**d**)) along the path $T=0.5{\mu}_{B}$, the crossover region. ${T}^{\mathrm{cross}}=213$ MeV and ${\mu}_{B}^{\mathrm{cross}}=426$ MeV are, respectively, the temperature and the baryonic chemical potential for the crossover.

**Figure 8.**Masses of the pseudoscalar and scalar mesons as function of the temperature (panels (

**a**,

**c**)) and as function of ${\mu}_{B}$ (panels (

**b**,

**d**)) along the path that crosses the CEP.

**Figure 9.**Masses of the pseudoscalar and scalar mesons as function of the temperature (panels (

**a**,

**c**)) and as function of ${\mu}_{B}$ (panels (

**b**,

**d**)) along the the path $T=0.05{\mu}_{B}$, crossing the first-order region.

**Figure 10.**Masses of the pseudoscalar and scalar mesons as function of the temperature (

**left panel**) and as function of ${\mu}_{B}$ (

**right panel**) along the the path $T=0.05{\mu}_{B}$, crossing the first-order region.

**Figure 11.**Line of effective restoration of chiral symmetry (brown curve) and the Mott dissociation line for $\pi $ (magenta) and $\sigma $ (green) along the ($T-{\mu}_{B}$)-plane.

**Table 1.**Physical quantities in the vacuum state and the parameter set used in this work. The asterisk signalize the results of the model for such physical quantities.

Physical Quantities | Parameter Set and Constituent Quark Masses |
---|---|

${f}_{\pi}=92.4$ MeV | ${m}_{u}={m}_{d}=5.5$ MeV |

${M}_{\pi}=135.0$ MeV | ${m}_{s}=140.7$ MeV |

${M}_{K}=497.7$ MeV | $\Lambda =602.3$ MeV |

${M}_{{\eta}^{\prime}}=957.8$ MeV | ${g}_{S}{\Lambda}^{2}=3.67$ |

${M}_{\eta}=$$514.8$ MeV ${}^{*}$ | ${g}_{D}{\Lambda}^{5}=-12.36$ |

${f}_{K}=93.1$ MeV ${\phantom{\rule{3.33333pt}{0ex}}}^{*}$ | ${M}_{u}{=M}_{d}=367.7$ MeV ${}^{*}$ |

${M}_{\sigma}=728.9$ MeV ${}^{*}$ | ${M}_{s}=549.5$ MeV ${}^{*}$ |

${M}_{{a}_{0}}=880.2$ MeV ${}^{*}$ | |

${M}_{\kappa}=1050.5$ MeV ${}^{*}$ | |

${M}_{{f}_{0}}=1198.3$ MeV ${}^{*}$ | |

${\theta}_{P}=-5.8$${}^{\circ}\phantom{\rule{3.33333pt}{0ex}}{}^{*}$ ; ${\theta}_{S}=16$${}^{\circ}\phantom{\rule{3.33333pt}{0ex}}{}^{*}$ |

**Table 2.**Characteristic and Mott temperatures in the PNJL model at ${\mu}_{B}$ with ${T}_{0}=195$ MeV.

${\mathit{T}}_{\mathit{c}}^{\mathit{\chi}}$ [MeV] | ${\mathit{T}}_{\mathit{c}}^{\mathbf{\Phi}}$ [MeV] | ${\mathit{T}}_{\mathit{eff}}^{\mathit{\chi}}$ [MeV] | ${\mathit{T}}_{\mathit{\pi}}^{\mathit{Mott}}$ [MeV] | ${\mathit{T}}_{\mathit{\eta}}^{\mathit{Mott}}$ [MeV] | ${\mathit{T}}_{\mathit{K}}^{\mathit{Mott}}$ [MeV] | ${\mathit{T}}_{\mathit{\sigma}}^{\mathit{Mott}}$ [MeV] |
---|---|---|---|---|---|---|

231 | 170 | 280 | 239 | 211 | 243 | 197 |

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**MDPI and ACS Style**

Costa, P.; Pereira, R.
Phase Diagram, Scalar-Pseudoscalar Meson Behavior and Restoration of Symmetries in (2 + 1) Polyakov-Nambu-Jona-Lasinio Model. *Symmetry* **2019**, *11*, 507.
https://doi.org/10.3390/sym11040507

**AMA Style**

Costa P, Pereira R.
Phase Diagram, Scalar-Pseudoscalar Meson Behavior and Restoration of Symmetries in (2 + 1) Polyakov-Nambu-Jona-Lasinio Model. *Symmetry*. 2019; 11(4):507.
https://doi.org/10.3390/sym11040507

**Chicago/Turabian Style**

Costa, Pedro, and Renan Pereira.
2019. "Phase Diagram, Scalar-Pseudoscalar Meson Behavior and Restoration of Symmetries in (2 + 1) Polyakov-Nambu-Jona-Lasinio Model" *Symmetry* 11, no. 4: 507.
https://doi.org/10.3390/sym11040507