# Effects of Hadron-Quark Phase Transitions in Hybrid Stars within the NJL Model

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## Abstract

**:**

## 1. Introduction

## 2. Quark Matter Description within NJL Models

#### 2.1. Local SU(3) PNJL Model

#### Zero Temperature Limit Including Vector Interaction

#### 2.2. Non-Local SU(3) Model

#### Zero Temperature Limit Including Vector Interaction

#### 2.3. QCD Phase Diagram

## 3. Astrophysical Applications of NJL EoS

#### 3.1. Hybrid Stars

#### 3.1.1. Hadronic EoS

#### 3.1.2. Phase Transition Formalism

#### 3.2. Stellar Oscillations

## 4. Results and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Phase diagram of the NJL models considered in this work. As a comparison, in gray we show the results obtained with a non-local NJL parametrization considering ${m}_{s}=95$ MeV. The solid line corresponds to the non-local parametrization with ${m}_{s}=127.77$ MeV. The critical endpoint, CEP, of each phase equilibrium curve, crossover or first order, is denoted by a round dot. The different gray dots indicate some LQCD results and signals of deconfinement from the determination of the chemical freeze-out (see Ref. [17] and references therein).

**Figure 2.**Hybrid EoSs (

**left**) and mass-radius (M-R) relationship (

**right**) for the cases corresponding to the local NJL EoSs. In the EoS curves, the constant pressure regions for the $b=0.0$ cases correspond to the transition phase. The $b=2.0$ cases does not have an abrupt transition so they have a mixed phase region. In the M-R curves, the rounded dot on the solid line indicates where the quark mater core appears and the triangle on the dashed curve indicates the first stellar configuration in which mixed phase is present in its inner core. After the peaks, towards smaller radii, the stars become unstable. The horizontal bars are the measured masses of the $2\phantom{\rule{3.33333pt}{0ex}}{M}_{\odot}$ pulsars with their corresponding errors. The horizontal arrow marks the constraint calculated in Ref. [13] for GW170817, ${R}_{M=1.4{M}_{\odot}}<13.76\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}$.

**Figure 3.**Same as Figure 2 but for the non-local NJL.

**Figure 4.**Frequencies of the f and g-modes (if present) for the hybrid EoSs constructed with the local model are presented in

**left**panel while those obtained using the non-local model are presented in the

**right**one. With solid lines we present models with sharp phase transition and with dashed lines those with a mixed phase. With circle and triangle the frequency of the quark hybrid stars.

**Figure 5.**Frequencies of the ${p}_{1}$-modes as a function of the compact object. In the

**left**panel we show those corresponding to hybrid EoSs constructed with the local model. In the

**right**panel we present those obtained using non-local model. With solid lines we present models with sharp phase transition and with dashed lines those with a mixed phase. With circle and triangle the frequency of the quark hybrid stars.

**Figure 6.**Represented with dots, decimal logarithm of the frequencies of the g-modes obtained in this work as a function of the logarithm of the $\Delta \u03f5/{\u03f5}_{\mathrm{trans}}$. With red dashed line we present the revised universal relationship between these quantities. Using gray scale the results presented in [30].

Saturation Property | GM1L |
---|---|

${n}_{0}$ (fm${}^{-3}$) | 0.153 |

${E}_{0}$ (MeV) | $-16.30$ |

${K}_{0}$ (MeV) | 300.0 |

${m}^{*}$/${m}_{N}$ | 0.70 |

J (MeV) | 32.5 |

${L}_{0}$ (MeV) | 55.0 |

$-{U}_{N}$ (MeV) | 65.5 |

**Table 2.**Parameters of GM1(L) that produce the properties of symmetric nuclear matter at saturation density given in Table 1.

Parameter | GM1L |
---|---|

${m}_{\sigma}$ (GeV) | 0.5500 |

${m}_{\omega}$ (GeV) | 0.7830 |

${m}_{\rho}$ (GeV) | 0.7700 |

${g}_{\sigma N}$ | 9.5722 |

${g}_{\omega N}$ | 10.6180 |

${g}_{\rho N}$ | 8.9830 |

${b}_{\sigma}$ | 0.0029 |

${c}_{\sigma}$ | $-0.0011$ |

${a}_{\rho}$ | 0.3898 |

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## Share and Cite

**MDPI and ACS Style**

Ranea-Sandoval, I.F.; Orsaria, M.G.; Malfatti, G.; Curin, D.; Mariani, M.; Contrera, G.A.; Guilera, O.M.
Effects of Hadron-Quark Phase Transitions in Hybrid Stars within the NJL Model. *Symmetry* **2019**, *11*, 425.
https://doi.org/10.3390/sym11030425

**AMA Style**

Ranea-Sandoval IF, Orsaria MG, Malfatti G, Curin D, Mariani M, Contrera GA, Guilera OM.
Effects of Hadron-Quark Phase Transitions in Hybrid Stars within the NJL Model. *Symmetry*. 2019; 11(3):425.
https://doi.org/10.3390/sym11030425

**Chicago/Turabian Style**

Ranea-Sandoval, Ignacio Francisco, Milva Gabriela Orsaria, Germán Malfatti, Daniela Curin, Mauro Mariani, Gustavo Aníbal Contrera, and Octavio Miguel Guilera.
2019. "Effects of Hadron-Quark Phase Transitions in Hybrid Stars within the NJL Model" *Symmetry* 11, no. 3: 425.
https://doi.org/10.3390/sym11030425