# Hybrid Stars with Color Superconducting Cores in an Extended FCM Model

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## Abstract

**:**

## 1. Introduction

## 2. Quark-Hadron Phase Transition in Neutron Stars

## 3. The Hadronic Phase

## 4. The Quark Phase

#### 4.1. Inclusion of Vector Interactions in the FCM Model

#### 4.2. Effects of Color Superconductivity on the Quark EoS

## 5. Results

#### 5.1. Analysis of the FCM Parameter Space Spanned by ${V}_{1}$, ${G}_{2}$, ${K}_{\mathrm{v}}$, $\Delta $

#### 5.2. Astrophysical Constraints

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

- Lattimer, J.M.; Prakash, M. The Physics of Neutron Stars. Science
**2004**, 304, 536–542. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pons, J.A.; Reddy, S.; Prakash, M.; Lattimer, J.M.; Miralles, J.A. Evolution of Proto-Neutron Stars. Astrophys. J.
**1999**, 513, 780–804. [Google Scholar] [CrossRef] [Green Version] - Potekhin, A.Y. The physics of neutron stars. Phys.-Uspekhi
**2010**, 53, 1235–1256. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. [The LIGO Scientific Collaboration and the Virgo Collaboration]. GW170817: Measurements of Neutron Star Radii and Equation of State. Phys. Rev. Lett.
**2018**, 121, 161101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. [The LIGO Scientific Collaboration and the Virgo Collaboration]. Properties of the binary neutron star merger GW170817. Phys. Rev. X
**2019**, 9, 011001. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abraham, S.; Acernese, F.; Ackley, K.; Adams, C.; Adhikari, R.X.; Adya, V.B.; Affeldt, C.; et al. GW190425: Observation of a Compact Binary Coalescence with Total Mass ∼ 3.4 M
_{⊙}. Astrophys. J. Lett.**2020**, 892, L3. [Google Scholar] [CrossRef] - Riley, T.E.; Watts, A.L.; Ray, P.S.; Bogdanov, S.; Guillot, S.; Morsink, S.M.; Bilous, A.V.; Arzoumanian, Z.; Choudhury, D.; Deneva, J.S.; et al. A NICER View of the Massive Pulsar PSR J0740+6620 Informed by Radio Timing and XMM-Newton Spectroscopy. Astrophys. J. Lett.
**2021**, 918, L27. [Google Scholar] [CrossRef] - Miller, M.C.; Lamb, F.K.; Dittmann, A.J.; Bogdanov, S.; Arzoumanian, Z.; Gendreau, K.C.; Guillot, S.; Ho, W.C.G.; Lattimer, J.M.; Loewenstein, M.; et al. The Radius of PSR J0740+6620 from NICER and XMM-Newton Data. Astrophys. J. Lett.
**2021**, 918, L28. [Google Scholar] [CrossRef] - Baym, G.; Hatsuda, T.; Kojo, T.; Powell, P.D.; Song, Y.; Takatsuka, T. From hadrons to quarks in neutron stars: A review. Rept. Prog. Phys.
**2018**, 81, 056902. [Google Scholar] [CrossRef] [Green Version] - Orsaria, M.G.; Malfatti, G.; Mariani, M.; Ranea-Sandoval, I.F.; García, F.; Spinella, W.M.; Contrera, G.A.; Lugones, G.; Weber, F. Phase transitions in neutron stars and their links to gravitational waves. J. Phys. G
**2019**, 46, 073002. [Google Scholar] [CrossRef] [Green Version] - Weber, F. Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics; Series in High Energy Physics, Cosmology and Gravitation; CRC Press: Boca Raton, FL, USA, 1999. [Google Scholar] [CrossRef] [Green Version]
- Weber, F. Strange quark matter and compact stars. Prog. Part. Nucl. Phys.
**2005**, 54, 193–288. [Google Scholar] [CrossRef] [Green Version] - Demorest, P.; Pennucci, T.; Ransom, S.; Roberts, M.; Hessels, J. Shapiro Delay Measurement of A Two Solar Mass Neutron Star. Nature
**2010**, 467, 1081–1083. [Google Scholar] [CrossRef] - Arzoumanian, Z.; Brazier, A.; Burke-Spolaor, S.; Chamberlin, S.; Chatterjee, S.; Christy, B.; Cordes, J.M.; Cornish, N.J.; Crawford, F.; Cromartie, H.T.; et al. The NANOGrav 11-year Data Set: High-precision Timing of 45 Millisecond Pulsars. Astrophys. J. Suppl. Ser.
**2018**, 235, 37. [Google Scholar] [CrossRef] [Green Version] - Antoniadis, J.; Freire, P.C.; Wex, N.; Tauris, T.M.; Lynch, R.S.; Van Kerkwijk, M.H.; Kramer, M.; Bassa, C.; Dhillon, V.S.; Driebe, T.; et al. A Massive Pulsar in a Compact Relativistic Binary. Science
**2013**, 340, 6131. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cromartie, H.T.; Fonseca, E.; Ransom, S.M.; Demorest, P.B.; Arzoumanian, Z.; Blumer, H.; Brook, P.R.; DeCesar, M.E.; Dolch, T.; Ellis, J.A.; et al. Relativistic Shapiro delay measurements of an extremely massive millisecond pulsar. Nat. Astron.
**2020**, 4, 72–76. [Google Scholar] [CrossRef] [Green Version] - Tews, I.; Margueron, J.; Reddy, S. Critical examination of constraints on the equation of state of dense matter obtained from GW170817. Phys. Rev. C
**2018**, 98, 045804. [Google Scholar] [CrossRef] [Green Version] - Shibata, M.; Zhou, E.; Kiuchi, K.; Fujibayashi, S. Constraint on the maximum mass of neutron stars using GW170817 event. Phys. Rev. D
**2019**, 100, 023015. [Google Scholar] [CrossRef] [Green Version] - Farrow, N.; Zhu, X.J.; Thrane, E. The Mass Distribution of Galactic Double Neutron Stars. Astrophys. J.
**2019**, 876, 18. [Google Scholar] [CrossRef] [Green Version] - Gompertz, B.P.; Cutter, R.; Steeghs, D.; Galloway, D.K.; Lyman, J.; Ulaczyk, K.; Dyer, M.J.; Ackley, K.; Dhillon, V.S.; O’Brien, P.T.; et al. Searching for electromagnetic counterparts to gravitational-wave merger events with the prototype Gravitational-Wave Optical Transient Observer (GOTO-4). Mon. Not. R. Astron. Soc.
**2020**, 497, 726–738. [Google Scholar] [CrossRef] - Riley, T.E.; Watts, A.L.; Bogdanov, S.; Ray, P.S.; Ludlam, R.M.; Guillot, S.; Arzoumanian, Z.; Baker, C.L.; Bilous, A.V.; Chakrabarty, D.; et al. A NICER View of PSR J0030+0451: Millisecond Pulsar Parameter Estimation. Astrophys. J. Lett.
**2019**, 887, L21. [Google Scholar] [CrossRef] [Green Version] - Miller, M.C.; Lamb, F.K.; Dittmann, A.J.; Bogdanov, S.; Arzoumanian, Z.; Gendreau, K.C.; Guillot, S.; Harding, A.K.; Ho, W.C.G.; Lattimer, J.M.; et al. PSR J0030+0451 Mass and Radius from NICER Data and Implications for the Properties of Neutron Star Matter. Astrophys. J. Lett.
**2019**, 887, L24. [Google Scholar] [CrossRef] [Green Version] - Rajagopal, K.; Frank Wilczek, F. The condensed matter physics of QCD. In At the Frontier of Particle Physics; World Scientific: Singapore, 2001; pp. 2061–2151. [Google Scholar]
- Alford, M.G. Color superconducting quark matter. Ann. Rev. Nucl. Part. Sci.
**2001**, 51, 131–160. [Google Scholar] [CrossRef] [Green Version] - Alford, M.G.; Schmitt, A.; Rajagopal, K.; Schäfer, T. Color superconductivity in dense quark matter. Rev. Mod. Phys.
**2008**, 80, 1455–1515. [Google Scholar] [CrossRef] [Green Version] - Bardeen, J.; Cooper, L.N.; Schrieffer, J.R. Microscopic Theory of Superconductivity. Phys. Rev.
**1957**, 106, 162–164. [Google Scholar] [CrossRef] [Green Version] - Ranea-Sandoval, I.F.; Orsaria, M.G.; Han, S.; Weber, F.; Spinella, W.M. Color superconductivity in compact stellar hybrid configurations. Phys. Rev. C
**2017**, 96, 065807. [Google Scholar] [CrossRef] [Green Version] - Lugones, G.; Horvath, J.E. High-density QCD pairing in compact star structure. Astron. Astrophys.
**2003**, 403, 173–178. [Google Scholar] [CrossRef] [Green Version] - Orsaria, M.; Rodrigues, H.; Weber, F.; Contrera, G.A. Quark deconfinement in high-mass neutron stars. Phys. Rev. C
**2014**, 89, 015806. [Google Scholar] [CrossRef] [Green Version] - Klähn, T.; Fischer, T. Vector interaction enhanced bag model for astrophysical applications. Astrophys. J.
**2015**, 810, 134. [Google Scholar] [CrossRef] [Green Version] - Ferreira, M.; Pereira, R.C.; Providência, C. Quark matter in light neutron stars. Phys. Rev. D
**2020**, 102, 083030. [Google Scholar] [CrossRef] - Simonov, Y.A.; Trusov, M.A. Deconfinement transition for nonzero baryon density in the field correlator method. JETP Lett.
**2007**, 85, 598–601. [Google Scholar] [CrossRef] [Green Version] - Nefediev, A.V.; Simonov, Y.A.; Trusov, M.A. Deconfinement and Quark—Gluon plasma. Int. J. Mod. Phys. E
**2009**, 18, 549–599. [Google Scholar] [CrossRef] [Green Version] - Mariani, M.; Orsaria, M.; Vucetich, H. Constant entropy hybrid stars: A first approximation of cooling evolution. Astron. Astrophys.
**2017**, 601, A21. [Google Scholar] [CrossRef] [Green Version] - Malfatti, G.; Orsaria, M.G.; Contrera, G.A.; Weber, F.; Ranea-Sandoval, I.F. Hot quark matter and (proto-) neutron stars. Phys. Rev. C
**2019**, 100, 015803. [Google Scholar] [CrossRef] [Green Version] - Pereira, J.P.; Flores, C.V.; Lugones, G. Phase Transition Effects on the Dynamical Stability of Hybrid Neutron Stars. Astrophys. J.
**2018**, 860, 12. [Google Scholar] [CrossRef] [Green Version] - Voskresensky, D.; Yasuhira, M.; Tatsumi, T. Charge screening at first order phase transitions and hadron quark mixed phase. Nucl. Phys. A
**2003**, 723, 291–339. [Google Scholar] [CrossRef] [Green Version] - Endo, T. Region of hadron-quark mixed phase in hybrid stars. Phys. Rev. C
**2011**, 83, 068801. [Google Scholar] [CrossRef] [Green Version] - Wu, X.; Shen, H. Nuclear symmetry energy and hadron-quark mixed phase in neutron stars. Phys. Rev. C
**2019**, 99, 065802. [Google Scholar] [CrossRef] [Green Version] - Maslov, K.; Yasutake, N.; Blaschke, D.; Ayriyan, A.; Grigorian, H.; Maruyama, T.; Tatsumi, T.; Voskresensky, D.N. Hybrid equation of state with pasta phases, and third family of compact stars. Phys. Rev. C
**2019**, 100, 025802. [Google Scholar] [CrossRef] [Green Version] - Weber, F.; Farrell, D.; Spinella, W.M.; Malfatti, G.; Orsaria, M.G.; Contrera, G.A.; Maloney, I. Phases of Hadron-Quark Matter in (Proto) Neutron Stars. Universe
**2019**, 5, 169. [Google Scholar] [CrossRef] [Green Version] - Annala, E.; Gorda, T.; Kurkela, A.; Nättilä, J.; Vuorinen, A. Evidence for quark-matter cores in massive neutron stars. Nat. Phys.
**2020**, 16, 907–910. [Google Scholar] [CrossRef] - Mariani, M.; Orsaria, M.G.; Ranea-Sandoval, I.F.; Lugones, G. Magnetized hybrid stars: Effects of slow and rapid phase transitions at the quark-hadron interface. Mon. Not. R. Astron. Soc.
**2019**, 489, 4261–4277. [Google Scholar] [CrossRef] - Bombaci, I.; Lugones, G.; Vidana, I. Effects of color superconductivity on the nucleation of quark matter in neutron stars. Astron. Astrophys.
**2007**, 462, 1017–1022. [Google Scholar] [CrossRef] [Green Version] - Haensel, P.; Zdunik, J.L.; Schaeffer, R. Phase transitions in dense matter and radial pulsations of neutron stars. A&A
**1989**, 217, 137–144. [Google Scholar] - Bombaci, I.; Parenti, I.; Vidaña, I. Quark Deconfinement and Implications for the Radius and the Limiting Mass of Compact Stars. Astrophys. J.
**2004**, 614, 314–325. [Google Scholar] [CrossRef] [Green Version] - Bombaci, I.; Logoteta, D.; Panda, P.K.; Providência, C.; Vidaña, I. Quark matter nucleation in hot hadronic matter. Phys. Lett. B
**2009**, 680, 448–452. [Google Scholar] [CrossRef] - Lugones, G.; Grunfeld, A.G. Critical spectrum of fluctuations for deconfinement at protoneutron star cores. Phys. Rev. D
**2011**, 84, 085003. [Google Scholar] [CrossRef] [Green Version] - Bombaci, I.; Logoteta, D.; Vidaña, I.; Providência, C. Quark matter nucleation in neutron stars and astrophysical implications. Eur. Phys. J. A
**2016**, 52, 58. [Google Scholar] [CrossRef] [Green Version] - Glendenning, N.K. Neutron stars are giant hypernuclei? Astrophys. J.
**1985**, 293, 470–493. [Google Scholar] [CrossRef] - Alford, M.; Rajagopal, K. Absence of two-flavor color-superconductivity in compact stars. J. High Energy Phys.
**2002**, 2002, 31. [Google Scholar] [CrossRef] - Typel, S.; Wolter, H.H. Relativistic mean field calculations with density dependent meson nucleon coupling. Nucl. Phys. A
**1999**, 656, 331–364. [Google Scholar] [CrossRef] - Spinella, W.M. A Systematic Investigation of Exotic Matter in Neutron Stars. Ph.D. Thesis, Claremont Graduate University & San Diego State University, Claremont, CA, USA, 2017. [Google Scholar]
- Malfatti, G.; Orsaria, M.G.; Ranea-Sandoval, I.F.; Contrera, G.A.; Weber, F. Delta baryons and diquark formation in the cores of neutron stars. Phys. Rev. D
**2020**, 102, 063008. [Google Scholar] [CrossRef] - Lattimer, J.M.; Lim, Y. Constraining the Symmetry Parameters of the Nuclear Interaction. Astrophys. J.
**2013**, 771, 51. [Google Scholar] [CrossRef] [Green Version] - Lattimer, J.M. Neutron Star Mass and Radius Measurements. Universe
**2019**, 5, 159. [Google Scholar] [CrossRef] [Green Version] - Horowitz, C.J.; Brown, E.F.; Kim, Y.; Lynch, W.G.; Michaels, R.; Ono, A.; Piekarewicz, J.; Tsang, M.B.; Wolter, H.H. A way forward in the study of the symmetry energy: Experiment, theory, and observation. J. Phys. G Nucl. Part. Phys.
**2014**, 41, 093001. [Google Scholar] [CrossRef] - Hofmann, F.; Keil, C.M.; Lenske, H. Application of the density dependent hadron field theory to neutron star matter. Phys. Rev. C
**2001**, 64, 025804. [Google Scholar] [CrossRef] [Green Version] - Simonov, Y.; Trusov, M. Vacuum phase transition at nonzero baryon density. Phys. Lett. B
**2007**, 650, 36–40. [Google Scholar] [CrossRef] [Green Version] - Shovkovy, I.A. Two Lectures on Color Superconductivity*. Found. Phys.
**2005**, 35, 1309–1358. [Google Scholar] [CrossRef] [Green Version] - Baym, G.; Pethick, C.; Sutherland, P. The Ground State of Matter at High Densities: Equation of State and Stellar Models. Astrophys. J.
**1971**, 170, 299. [Google Scholar] [CrossRef] - Baym, G.; Bethe, H.A.; Pethick, C.J. Neutron star matter. Nucl. Phys. A
**1971**, 175, 225–271. [Google Scholar] [CrossRef] - Plumari, S.; Burgio, G.; Greco, V.; Zappala, D. Quark matter in neutron stars within the field correlator method. Phys. Rev. D
**2013**, 88, 083005. [Google Scholar] [CrossRef] [Green Version] - Logoteta, D.; Bombaci, I. Quark deconfinement transition in neutron stars with the field correlator method. Phys. Rev. D
**2013**, 88, 063001. [Google Scholar] [CrossRef] [Green Version] - Burgio, G.; Zappalà, D. Hybrid star structure with the Field Correlator Method. Eur. Phys. J. A
**2016**, 52, 1–14. [Google Scholar] [CrossRef] - Khanmohamadi, S.; Moshfegh, H.; Tehrani, S.A. Structure and tidal deformability of a hybrid star within the framework of the field correlator method. Phys. Rev. D
**2020**, 101, 123001. [Google Scholar] [CrossRef] - Char, P.; Traversi, S.; Pagliara, G. A Bayesian Analysis on Neutron Stars within Relativistic Mean Field Models. Particles
**2020**, 3, 621–629. [Google Scholar] [CrossRef] - Xie, W.J.; Li, B.A. Bayesian Inference of the Symmetry Energy of Superdense Neutron-rich Matter from Future Radius Measurements of Massive Neutron Stars. Astrophys. J.
**2020**, 899, 4. [Google Scholar] [CrossRef] - Tolman, R.C. Static solutions of Einstein’s field equations for spheres of fluid. Phys. Rev.
**1939**, 55, 364–373. [Google Scholar] [CrossRef] [Green Version] - Oppenheimer, J.R.; Volkoff, G.M. On Massive Neutron Cores. Phys. Rev.
**1939**, 55, 374–381. [Google Scholar] [CrossRef] - Ranea-Sandoval, I.F.; Han, S.; Orsaria, M.G.; Contrera, G.A.; Weber, F.; Alford, M.G. Constant-sound-speed parametrization for Nambu–Jona-Lasinio models of quark matter in hybrid stars. Phys. Rev. C
**2016**, 93, 045812. [Google Scholar] [CrossRef] - Alford, M.; Sedrakian, A. Compact Stars with Sequential QCD Phase Transitions. Phys. Rev. Lett.
**2017**, 119, 161104. [Google Scholar] [CrossRef] [Green Version] - Rodríguez, M.; Ranea-Sandoval, I.F.; Mariani, M.; Orsaria, M.G.; Malfatti, G.; Guilera, O. Hybrid stars with sequential phase transitions: The emergence of the g2 mode. J. Cosmol. Astropart. Phys.
**2021**, 2021, 9. [Google Scholar] [CrossRef] - Pagliara, G.; Schaffner-Bielich, J. Stability of color-flavor-locking cores in hybrid stars. Phys. Rev. D
**2008**, 77, 063004. [Google Scholar] [CrossRef] [Green Version] - Bonanno, L.; Sedrakian, A. Composition and stability of hybrid stars with hyperons and quark color-superconductivity. Astron. Astrophys.
**2012**, 539, A16. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(Color online) Hybrid EoS (panel (

**a**)) and mass-radius relationship (panel (

**b**)) for the 2SC+s phase at fixed gap value of $\Delta =35$ MeV, for different values of the ${K}_{\mathrm{v}}$ parameter. In panel (

**a**), the grey region shows the constraints presented in [42]. The solid dots in panel (

**b**) indicate the appearance of the color superconducting phase, just before the maximum mass peak. For rapid conversions, the stellar configurations to the left of each maximum mass are unstable and the existence of HSs is only marginal. For slow conversions, an extended stability branch exists. The stable configurations are shown by continuous lines. The terminal configurations are marked with asterisks.

**Figure 2.**Hybrid EoS (panel (

**a**)) and mass-radius relationship (panel (

**b**)) for the CFL phase at fixed gap value of $\Delta =35$ MeV, for different values of the ${K}_{\mathrm{v}}$ parameter. In panel (

**a**), the grey region shows the constraints presented in [42]. The solid dots in panel (

**b**) indicate the appearance of the color superconducting phase, just before the maximum mass peak. For rapid conversions, the stellar configurations to the left of each maximum mass are unstable and the existence of HSs is only marginal. For slow conversions, an extended stability branch exists. The stable configurations are shown by continuous lines. The terminal configurations are marked with asterisks.

**Figure 3.**Hybrid EoS (panel (

**a**)) and mass-radius relationship (panel (

**b**)) for the 2SC+s phase at fixed gap value of ${K}_{\mathrm{v}}=10$ GeV${}^{-2}$, for different values of the $\Delta $ parameter. In panel (

**a**), the grey region shows the constraints presented in [42]. The solid dots in panel (

**b**) indicate the appearance of the color superconducting phase, just before the maximum mass peak. For rapid conversions, the stellar configurations to the left of each maximum mass are unstable and the existence of HSs is only marginal. For slow conversions, an extended stability branch exists. The stable configurations are shown by continuous lines. The terminal configurations are marked with asterisks.

**Figure 4.**(Hybrid EoS) (panel (

**a**)) and mass-radius relationship (panel (

**b**)) for the CFL phase at fixed gap value of ${K}_{\mathrm{v}}=10$ GeV${}^{-2}$, for different values of the $\Delta $ parameter. In panel (

**a**), the grey region shows the constraints presented in [42]. The solid dots in panel (

**b**) indicate the appearance of the color superconducting phase, just before the maximum mass peak. For rapid conversions, the stellar configurations to the left of each maximum mass are unstable and the existence of HSs is only marginal. For slow conversions, an extended stability branch exists. The stable configurations are shown by continuous lines. The terminal configurations are marked with asterisks.

**Figure 5.**(Text) Maximum mass of stars as a function of $\Delta $ and ${K}_{\mathrm{v}}$, for the 2SC+s quark matter (panel (

**a**)) and CFL quark matter (panel (

**b**)). The white curve marks the maximum mass constraint ${M}_{\mathrm{max}}=2.01{M}_{\odot}$.

**Figure 6.**(Color online) M–R relationship of the selected EoSs (Table 4) of this work. The solid dots indicate the appearance of color superconducting quark matter in HSs, which happens just before the maximum-mass peaks are reached. For a rapid conversion of matter, the stellar configurations to the left of each maximum-mass star are unstable. For slow conversions there exist extended branches of stable stars which end at the locations marked with asterisks. The shaded regions (clouds) correspond to constraints imposed by GW170817, GW190425, and NICER observations of PSR J0030+0451. The horizontal pink stripped bands, indicate constraints imposed by pulsars J0740+6620, J0348+0432, and J1614-2230.

**Figure 7.**(Color online) Dimensionless tidal deformability as a function of gravitational mass, with the constraint obtained from GW170817 [5]. Stable stellar configurations beyond the maximum mass have very small values of $\Lambda $, which are almost independent of mass. The positions of the terminal stars of the twin HSs branch (obtained for slow hadron-quark conversion) are marked with asterisks.

**Figure 8.**Text Dimensionless tidal deformabilities ${\Lambda}_{1}$ and ${\Lambda}_{2}$ for the selected EoSs. The solid black line represents the results obtained for a purely hadronic NS-NS merger with masses consistent with data from GW170817. The dark (light) gray areas represent the 50% (90%) confidence limit of the probability contour of GW170817 and the dotted line corresponds to ${\Lambda}_{1}={\Lambda}_{2}$.

**Table 1.**Parameters of the SW4L parametrization that lead to the properties of symmetric nuclear matter at saturation density shown in Table 2.

Quantity | Numerical Value |
---|---|

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

${m}_{\omega}$ (GeV) | 0.7826 |

${m}_{\rho}$ (GeV) | 0.7753 |

${m}_{{\sigma}^{*}}$ (GeV) | 0.9900 |

${m}_{\varphi}$ (GeV) | 1.0195 |

${g}_{\sigma N}$ | 9.8100 |

${g}_{\omega N}$ | 10.3906 |

${g}_{\rho N}$ | 7.8184 |

${g}_{{\sigma}^{*}N}$ | 1.0000 |

${g}_{\varphi N}$ | 1.0000 |

${\tilde{b}}_{\sigma}$ | 0.0041 |

${\tilde{c}}_{\sigma}$ | −0.0038 |

${a}_{\rho}$ | 0.4703 |

**Table 2.**Energy per nucleon ${E}_{0}$, nuclear compressibility ${K}_{0}$, effective nucleon mass ${m}^{*}$, symmetry energy ${J}_{0}$, and slope of the symmetry energy ${L}_{0}$ of nuclear matter at saturation density, ${n}_{0}$, obtained for the SW4L parametrization.

Saturation Properties | Numerical Values |
---|---|

${n}_{0}$ (fm${}^{-3}$) | 0.15 |

${E}_{0}$ (MeV) | −16.0 |

${K}_{0}$ (MeV) | 250.0 |

${m}_{N}^{*}/{m}_{N}$ | 0.7 |

${J}_{0}$ (MeV) | 30.3 |

${L}_{0}$ (MeV) | 46.5 |

i | r | g | b |

u | 1 | 2 | 5 |

d | 3 | 4 | 6 |

s | 7 | 8 | 9 |

Set | Quark Phase | ${\mathit{K}}_{\mathbf{v}}$ (GeV${}^{-2}$) | $\Delta $ (MeV) |
---|---|---|---|

1 | 2SC+s | 15 | 90 |

2 | 2SC+s | 10 | 30 |

3 | CFL | 10 | 30 |

4 | CFL | 15 | 30 |

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Curin, D.; Ranea-Sandoval, I.F.; Mariani, M.; Orsaria, M.G.; Weber, F.
Hybrid Stars with Color Superconducting Cores in an Extended FCM Model. *Universe* **2021**, *7*, 370.
https://doi.org/10.3390/universe7100370

**AMA Style**

Curin D, Ranea-Sandoval IF, Mariani M, Orsaria MG, Weber F.
Hybrid Stars with Color Superconducting Cores in an Extended FCM Model. *Universe*. 2021; 7(10):370.
https://doi.org/10.3390/universe7100370

**Chicago/Turabian Style**

Curin, Daniela, Ignacio Francisco Ranea-Sandoval, Mauro Mariani, Milva Gabriela Orsaria, and Fridolin Weber.
2021. "Hybrid Stars with Color Superconducting Cores in an Extended FCM Model" *Universe* 7, no. 10: 370.
https://doi.org/10.3390/universe7100370