# Energy-Density Modeling of Strongly Interacting Matter: Atomic Nuclei and Dense Stars

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## Abstract

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## 1. Introduction

## 2. Description of Cold Static Neutron Stars

## 3. KIDS Framework for the EoS and EDF of Nucleonic Matter and CSS Extension

#### 3.1. General Expression for the KIDS Equation of State (KIDS EoS)

#### 3.2. Standard EoS Parameters in the KIDS Framework

- For symmetric matter: if ${Q}_{0}$ is unspecified, ${\alpha}_{i\ge 3}=0$ and the inversion of the $3\times 3$ system gives$$\left(\right)open="("\; close=")">\begin{array}{c}{\alpha}_{0}{\rho}_{0}\hfill \\ {\alpha}_{1}{\rho}_{0}^{4/3}\hfill \\ {\alpha}_{2}{\rho}_{0}^{5/3}\hfill \end{array}\left(\right)open="("\; close=")">\begin{array}{c}\hfill {\mathcal{E}}_{0}-[\frac{{\hslash}^{2}}{2m}+{C}_{\mathrm{eff}}{\rho}_{0}]{h}_{k}{({\rho}_{0}/2)}^{2/3}\\ \hfill -[2\frac{{\hslash}^{2}}{2m}+5{C}_{\mathrm{eff}}{\rho}_{0}]{h}_{k}{({\rho}_{0}/2)}^{2/3}\\ \hfill {K}_{0}+[2\frac{{\hslash}^{2}}{2m}-10{C}_{\mathrm{eff}}{\rho}_{0}]{h}_{k}{({\rho}_{0}/2)}^{2/3}\end{array}$$If ${Q}_{0}$ is also specified (${\alpha}_{i\ge 4}=0$), inversion of the $4\times 4$ system gives$$\left(\right)open="("\; close=")">\begin{array}{c}{\alpha}_{0}{\rho}_{0}\hfill \\ {\alpha}_{1}{\rho}_{0}^{4/3}\hfill \\ {\alpha}_{2}{\rho}_{0}^{5/3}\hfill \\ {\alpha}_{3}{\rho}_{0}^{2}\hfill \end{array}\left(\right)open="("\; close=")">\begin{array}{c}\hfill {\mathcal{E}}_{0}-(\frac{{\hslash}^{2}}{2m}+{C}_{\mathrm{eff}}{\rho}_{0}){h}_{k}{({\rho}_{0}/2)}^{2/3}\\ \hfill -(2\frac{{\hslash}^{2}}{2m}+5{C}_{\mathrm{eff}}{\rho}_{0}){h}_{k}{({\rho}_{0}/2)}^{2/3}\\ \hfill {K}_{0}+(2\frac{{\hslash}^{2}}{2m}-10{C}_{\mathrm{eff}}{\rho}_{0})]{h}_{k}{({\rho}_{0}/2)}^{2/3}\\ \hfill {Q}_{0}+(-8\frac{{\hslash}^{2}}{2m}+10{C}_{\mathrm{eff}}{\rho}_{0}){h}_{k}{({\rho}_{0}/2)}^{2/3}\end{array}$$
- For the symmetry energy, the inversion of the $4\times 4$ system gives$$\left(\right)open="("\; close=")">\begin{array}{c}{\beta}_{0}{\rho}_{0}\hfill \\ {\beta}_{1}{\rho}_{0}^{4/3}\hfill \\ {\beta}_{2}{\rho}_{0}^{5/3}\hfill \\ {\beta}_{3}{\rho}_{0}^{2}\hfill \end{array}\left(\right)open="("\; close=")">\begin{array}{c}\hfill J-[\sigma (\frac{{\hslash}^{2}}{2m}+{C}_{\mathrm{eff}}{\rho}_{0})+\phi {D}_{\mathrm{eff}}{\rho}_{0}]{h}_{k}{\rho}_{0}^{2/3}\\ \hfill L-[\sigma (2\frac{{\hslash}^{2}}{2m}+5{C}_{\mathrm{eff}}{\rho}_{0})+5\phi {D}_{\mathrm{eff}}{\rho}_{0}]{h}_{k}{\rho}_{0}^{2/3}\\ \hfill {K}_{\mathrm{sym}}+[\sigma (2\frac{{\hslash}^{2}}{2m}-10{C}_{\mathrm{eff}}{\rho}_{0})-10\phi {D}_{\mathrm{eff}}{\rho}_{0}]{h}_{k}{\rho}_{0}^{2/3}\\ \hfill {Q}_{\mathrm{sym}}+[\sigma (-8\frac{{\hslash}^{2}}{2m}+10{C}_{\mathrm{eff}}{\rho}_{0})+10\phi {D}_{\mathrm{eff}}{\rho}_{0}]{h}_{k}{\rho}_{0}^{2/3}\end{array}$$

#### 3.3. KIDS EoS and Skyrme-Like Functional

#### 3.4. From EoS to EDF in Practice

- The most rudimentary option is to split the EoS term ${c}_{2}{\rho}^{5/3}$ into a term $k{c}_{2}{\rho}^{5/3}$, which will provide the parameters ${t}_{1},{t}_{2}$ of the Skyrme-type functional (for ${x}_{1}={x}_{2}=0$), and the rest, $(1-k){c}_{2}{\rho}^{5/3}$, which will provide a genuine density-dependent term. The optimal values of the constant k and at the same time ${W}_{0}$ are determined by a fit to a minimal amount of data (masses and radii of three nuclei). This simple procedure, first explored in [63], typically leads to a ${\mu}_{s}$ close to one. It is good enough for inspecting bulk nuclear and neutron-star properties [23,58,64], but is quite restrictive when looking at, e.g., single-particle spectra and collective excitations.
- A second option is to select, besides the EoS parameters, the desired values for the effective masses; and then, determine ${C}_{12},{D}_{12},{W}_{0}$ by a fit to nuclei. This method has been used, e.g., in the proof-of-concept study of Ref. [58] and in additional applications in Ref. [62]. Once spectroscopic or dynamic properties are considered, the in-medium effective mass should also come into play [16,58,65,66,67,68] and it can be fitted as well. It may be also relevant for a precision fit to nuclear masses, which is underway.

#### 3.5. Summary of Applications

## 4. Constant Speed of Sound Model and Pseudo-Conformal Dense Matter

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CSS | Constant Speed of Sound |

EDF | Energy Density Functional |

EoS | Equation of State |

NSMR | Neutron Star Mass–Radius |

PNM | Pure Neutron Matter |

SNM | Symmetric Nuclear Matter |

TOV | Tolman–Oppenheimer–Volkoff |

## References

- 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. Properties of the Binary Neutron Star Merger GW170817. Phys. Rev. X
**2019**, 9, 011001. [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.
**2019**, 887, L24. [Google Scholar] [CrossRef][Green Version] - Oertel, M.; Hempel, M.; Klähn, T.; Typel, S. Equations of state for supernovae and compact stars. Rev. Mod. Phys.
**2017**, 89, 015007. [Google Scholar] [CrossRef][Green Version] - Pang, P.T.; Dietrich, T.; Coughlin, M.W.; Bulla, M.; Tews, I.; Almualla, M.; Barna, T.; Kiendrebeogo, W.; Kunert, N.; Mansingh, G.; et al. NMMA: A nuclear-physics and multi-messenger astrophysics framework to analyze binary neutron star mergers. arXiv
**2022**, arXiv:2205.08513. [Google Scholar] - Aumann, T.; Bertulani, C.A.; Schindler, F.; Typel, S. Peeling Off Neutron Skins from Neutron-Rich Nuclei: Constraints on the Symmetry Energy from Neutron-Removal Cross Sections. Phys. Rev. Lett.
**2017**, 119, 262501. [Google Scholar] [CrossRef][Green Version] - Roca-Maza, X.; Paar, N. Nuclear equation of state from ground and collective excited state properties of nuclei. Prog. Part. Nucl. Phys.
**2018**, 101, 96–176. [Google Scholar] [CrossRef][Green Version] - Garg, U.; Colò, G. The compression-mode giant resonances and nuclear incompressibility. Prog. Part. Nucl. Phys.
**2018**, 101, 55–95. [Google Scholar] [CrossRef][Green Version] - Adhikari, D.; Albataineh, H.; Androic, D.; Aniol, K.; Armstrong, D.S.; Averett, T.; Barcus, S.; Bellini, V.; Beminiwattha, R.S.; Benesch, J.F.; et al. Accurate Determination of the Neutron Skin Thickness of
^{208}Pb through Parity-Violation in Electron Scattering. Phys. Rev. Lett.**2021**, 126, 172502. [Google Scholar] [CrossRef] - Adhikari, D.; Albataineh, H.; Androic, D.; Aniol, K.A.; Armstrong, D.S.; Averett, T.; Barcus, S.; Bellini, V.; Beminiwattha, R.S.; Benesch, J.F.; et al. Precision Determination of the Neutral Weak Form Factor of
^{48}Ca. Phys. Rev. Lett.**2022**, 129, 042501. [Google Scholar] [CrossRef] - Li, B.A.; Chen, L.W.; Ko, C.M. Recent progress and new challenges in isospin physics with heavy-ion reactions. Phys. Rep.
**2008**, 464, 113–281. [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
**2014**, 41, 093001. [Google Scholar] [CrossRef][Green Version] - Lynch, W.; Tsang, M. Decoding the density dependence of the nuclear symmetry energy. Phys. Lett. B
**2022**, 830, 137098. [Google Scholar] [CrossRef] - Drischler, C.; Bogner, S.K. A Brief Account of Steven Weinberg’s Legacy in ab initio Many-Body Theory. Few-Body Syst.
**2021**, 62, 109. [Google Scholar] [CrossRef] - Gezerlis, A.; Tews, I.; Epelbaum, E.; Gandolfi, S.; Hebeler, K.; Nogga, A.; Schwenk, A. Quantum Monte Carlo Calculations with Chiral Effective Field Theory Interactions. Phys. Rev. Lett.
**2013**, 111, 032501. [Google Scholar] [CrossRef] [PubMed][Green Version] - Newton, W.G.; Crocombe, G. Nuclear symmetry energy from neutron skins and pure neutron matter in a Bayesian framework. Phys. Rev. C
**2021**, 103, 064323. [Google Scholar] [CrossRef] - Xu, J.; Papakonstantinou, P. Bayesian inference of finite-nuclei observables based on the KIDS model. Phys. Rev. C
**2022**, 105, 044305. [Google Scholar] [CrossRef] - Jeong, S.; Papakonstantinou, P.; Ishiyama, H.; Kim, Y. A Brief Overview of RAON Physics. J. Korean Phys. Soc.
**2018**, 73, 516–523. [Google Scholar] [CrossRef] - Colò, G. Nuclear density functional theory. Adv. Phys. X
**2020**, 5, 1740061. [Google Scholar] [CrossRef][Green Version] - Dutra, M.; Lourenço, O.; Sá Martins, J.S.; Delfino, A.; Stone, J.R.; Stevenson, P.D. Skyrme interaction and nuclear matter constraints. Phys. Rev. C
**2012**, 85, 035201. [Google Scholar] [CrossRef][Green Version] - Dutra, M.; Lourenço, O.; Avancini, S.S.; Carlson, B.V.; Delfino, A.; Menezes, D.P.; Providência, C.; Typel, S.; Stone, J.R. Relativistic mean-field hadronic models under nuclear matter constraints. Phys. Rev. C
**2014**, 90, 055203. [Google Scholar] [CrossRef] - Stevenson, P.D.; Goddard, P.M.; Stone, J.R.; Dutra, M. Do Skyrme forces that fit nuclear matter work well in finite nuclei? AIP Conf. Proc.
**2013**, 1529, 262–268. [Google Scholar] - Papakonstantinou, P.; Park, T.S.; Lim, Y.; Hyun, C.H. Density dependence of the nuclear energy-density functional. Phys. Rev. C
**2018**, 97, 014312. [Google Scholar] [CrossRef][Green Version] - Gil, H.; Kim, Y.M.; Hyun, C.H.; Papakonstantinou, P.; Oh, Y. Analysis of nuclear structure in a converging power expansion scheme. Phys. Rev. C
**2019**, 100, 014312. [Google Scholar] [CrossRef][Green Version] - Motta, T.F.; Thomas, A.W. The role of baryon structure in neutron stars. Mod. Phys. Lett. A
**2022**, 37, 2230001. [Google Scholar] [CrossRef] - Kochankovski, H.; Ramos, À.; Vidaña, I. An analytic parametrization of the hypernuclear matter equation of state. Eur. Phys. J. A
**2022**, 58, 31. [Google Scholar] [CrossRef] - Choi, S.; Hiyama, E.; Hyun, C.H.; Cheoun, M.K. Effects of many-body interactions in hypernuclei with Korea-IBS-Daegu-SKKU functionals. Eur. Phys. J. A
**2022**, 58, 161. [Google Scholar] [CrossRef] - Alford, M.G.; Han, S.; Prakash, M. Generic conditions for stable hybrid stars. Phys. Rev. D
**2013**, 88, 083013. [Google Scholar] [CrossRef][Green Version] - Li, A.; Yong, G.C.; Zhang, Y.X. Testing the phase transition parameters inside neutron stars with the production of protons and lambdas in relativistic heavy-ion collisions. arXiv
**2022**, arXiv:2211.04978. [Google Scholar] [CrossRef] - Kojo, T. Phenomenological neutron star equations of state. Eur. Phys. J. A
**2016**, 52, 51. [Google Scholar] [CrossRef][Green Version] - Kojo, T. QCD equations of state and speed of sound in neutron stars. AAPPS Bull.
**2021**, 31, 11. [Google Scholar] [CrossRef] - Kutschera, M.; Kotlorz, A. Maximum quark core in a neutron star for realistic equations of state. Astrophys. J.
**1993**, 419, 752–757. [Google Scholar] [CrossRef] - Aziz, A.; Ray, S.; Rahaman, F.; Khlopov, M.; Guha, B. Constraining values of bag constant for strange star candidates. Int. J. Mod. Phys. D
**2019**, 28. [Google Scholar] [CrossRef][Green Version] - Joshi, S.; Sau, S.; Sanyal, S. Quark cores in extensions of the MIT bag model. J. High Energy Astrophys.
**2021**, 30, 16–23. [Google Scholar] [CrossRef] - Buballa, M. NJL-model analysis of dense quark matter. Phys. Rep.
**2005**, 407, 205–376. [Google Scholar] [CrossRef][Green Version] - Fukushima, K. Critical surface in hot and dense QCD with the vector interaction. Phys. Rev. D
**2008**, 78, 114019. [Google Scholar] [CrossRef][Green Version] - Hell, T.; Weise, W. Dense baryonic matter: Constraints from recent neutron star observations. Phys. Rev. C
**2014**, 90, 045801. [Google Scholar] [CrossRef][Green Version] - Ivanytskyi, O.; Ángeles Pérez-García, M.; Sagun, V.; Albertus, C. Second look to the Polyakov loop Nambu–Jona-Lasinio model at finite baryonic density. Phys. Rev. D
**2019**, 100, 103020. [Google Scholar] [CrossRef][Green Version] - Pinto, M.B. EoS for strange quark matter: Linking the NJL model to pQCD. arXiv
**2022**, arXiv:2211.11071. [Google Scholar] - Peng, G.X.; Li, A.; Lombardo, U. Deconfinement phase transition in hybrid neutron stars from the Brueckner theory with three-body forces and a quark model with chiral mass scaling. Phys. Rev. C
**2008**, 77, 065807. [Google Scholar] [CrossRef][Green Version] - Adam, C.; Martín-Caro, A.G.; Huidobro, M.; Vázquez, R.; Wereszczynski, A. Quantum Skyrmion crystals and the symmetry energy of dense matter. arXiv
**2022**, arXiv:2202.00953. [Google Scholar] [CrossRef] - Ma, Y.L.; Rho, M. Towards the hadron–quark continuity via a topology change in compact stars. Prog. Part. Nucl. Phys.
**2020**, 113, 103791. [Google Scholar] [CrossRef] - Chu, P.C.; Zhou, Y.; Qi, X.; Li, X.H.; Zhang, Z.; Zhou, Y. Isospin properties in quark matter and quark stars within isospin-dependent quark mass models. Phys. Rev. C
**2019**, 99, 035802. [Google Scholar] [CrossRef] - Iida, K.; Itou, E. Velocity of Sound beyond the High-Density Relativistic Limit from Lattice Simulation of Dense Two-Color QCD. 2022. arXiv
**2022**, arXiv:2207.01253. [Google Scholar] - Huang, Y.J.; Baiotti, L.; Kojo, T.; Takami, K.; Sotani, H.; Togashi, H.; Hatsuda, T.; Nagataki, S.; Fan, Y.Z. Merger and Postmerger of Binary Neutron Stars with a Quark-Hadron Crossover Equation of State. Phys. Rev. Lett.
**2022**, 129, 181101. [Google Scholar] [CrossRef] [PubMed] - Rho, M. Pseudo-Conformal Sound Speed in the Core of Compact Stars. Symmetry
**2022**, 14, 2154. [Google Scholar] [CrossRef] - Paeng, W.G.; Kuo, T.T.S.; Lee, H.K.; Ma, Y.L.; Rho, M. Scale-invariant hidden local symmetry, topology change, and dense baryonic matter. II. Phys. Rev. D
**2017**, 96, 014031. [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] - Altiparmak, S.; Ecker, C.; Rezzolla, L. On the Sound Speed in Neutron Stars. Astrophys. J. Lett.
**2022**, 939, L34. [Google Scholar] [CrossRef] - McLerran, L.; Reddy, S. Quarkyonic Matter and Neutron Stars. Phys. Rev. Lett.
**2019**, 122, 122701. [Google Scholar] [CrossRef][Green Version] - Lee, H.K.; Ma, Y.L.; Paeng, W.G.; Rho, M. Cusp in the symmetry energy, speed of sound in neutron stars and emergent pseudo-conformal symmetry. Mod. Phys. Lett. A
**2022**, 37, 2230003. [Google Scholar] [CrossRef] - Braun, J.; Geißel, A.; Schallmo, B. Speed of sound in dense strong-interaction matter. arXiv
**2022**, arXiv:2206.06328. [Google Scholar] - Tolman, R.C. Static Solutions of Einstein’s Field Equations for Spheres of Fluid. Phys. Rev.
**1939**, 55, 364. [Google Scholar] [CrossRef][Green Version] - Oppenheimer, J.R.; Volkoff, G.M. On Massive Neutron Cores. Phys. Rev.
**1939**, 55, 374. [Google Scholar] [CrossRef] - Lim, Y.; Kim, T.; Oh, Y. Nuclear Equation of State and the Structure of Neutron Stars. New Phys. (Sae Mulli)
**2016**, 66, 1571–1577. [Google Scholar] [CrossRef] - Douchin, F.; Haensel, P. A unified equation of state of dense matter and neutron star structure. Astron. Astrophys.
**2001**, 380, 151–167. [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] - Viñas, X.; Gonzalez-Boquera, C.; Centelles, M.; Mondal, C.; Robledo, L.M. Unified Equation of State for Neutron Stars Based on the Gogny Interaction. Symmetry
**2021**, 13, 1613. [Google Scholar] [CrossRef] - Gil, H.; Papakonstantinou, P.; Hyun, C.H.; Oh, Y. From homogeneous matter to finite nuclei: Role of the effective mass. Phys. Rev. C
**2019**, 99, 064319. [Google Scholar] [CrossRef][Green Version] - Fetter, A.L.; Walecka, J.D. Quantum Theory of Many-Particle Systems; Dover Publications: New York, NY, USA, 1971. [Google Scholar]
- Kaiser, N.; Fritsch, S.; Weise, W. Chiral dynamics and nuclear matter. Nucl. Phys. A
**2002**, 697, 255. [Google Scholar] [CrossRef][Green Version] - Hammer, H.-W.; Furnstahl, R.J. Effective field theory for dilute Fermi systems. Nucl. Phys. A
**2000**, 678, 277. [Google Scholar] [CrossRef][Green Version] - Gil, H.; Kim, Y.M.; Papakonstantinou, P.; Hyun, C.H. Constraining the density dependence of the symmetry energy with nuclear data and astronomical observations in the Korea-IBS-Daegu-SKKU framework. Phys. Rev. C
**2021**, 103, 034330. [Google Scholar] [CrossRef] - Gil, H.; Papakonstantinou, P.; Hyun, C.H.; Park, T.S. Nuclear energy density functional for KIDS. Acta Phys. Pol. B
**2017**, 48, 537. [Google Scholar] [CrossRef][Green Version] - Gil, H.; Hyun, C.H. Compression modulus and symmetry energy of nuclear matter with KIDS density functional. New Phys. (Sae Mulli)
**2021**, 71, 242–248. [Google Scholar] [CrossRef] - Gil, H.; Hyun, C.H.; Kim, K. Quasielastic electron scattering with the KIDS nuclear energy density functional. Phys. Rev. C
**2021**, 104, 044613. [Google Scholar] [CrossRef] - Gil, H.; Hyun, C.H.; Kim, K. Inclusive electron scattering in the quasielastic region with the Korea-IBS-Daegu-SKKU density functional. Phys. Rev. C
**2022**, 105, 024607. [Google Scholar] [CrossRef] - Kim, K.; Gil, H.; Hyun, C.H. Quasielastic charged-current neutrino-nucleus scattering with nonrelativistic nuclear energy density functionals. Phys. Lett. B
**2022**, 833, 137273. [Google Scholar] [CrossRef] - Hutauruk, P.T.P.; Gil, H.; Nam, S.i.; Hyun, C.H. Effect of nucleon effective mass and symmetry energy on the neutrino mean free path in a neutron star. Phys. Rev. C
**2022**, 106, 035802. [Google Scholar] [CrossRef] - Papakonstantinou, P. Density dependence of the nuclear symmetry energy and neutron skin thickness in the KIDS framework. HNPS Adv. Nucl. Phys.
**2022**, 28, 36–41. [Google Scholar] [CrossRef] - Gil, H.; Oh, Y.; Hyun, C.; Papakonstantinou, P. Skyrme-Type Nuclear Force for the KIDS Energy Density Functional. New Phys. (Sae Mulli)
**2017**, 67, 456–461. [Google Scholar] [CrossRef] - Gil, H.; Papakonstantinou, P.; Hyun, C.H. Constraints on the curvature of nuclear symmetry energy from recent astronomical data within the KIDS framework. Int. J. Mod. Phys. E
**2022**, 31, 2250013. [Google Scholar] [CrossRef] - Papakonstantinou, P. Nuclear symmetry energy and the PREX-CREX neutron skin puzzle within the KIDS framework. Nucl. Theory
**2022**, 39, 36. [Google Scholar] - Hyun, C.H. Neutron Skin Thickness of
^{48}Ca,^{120}Sn, and^{208}Pb with KIDS Density Functional. New Phys. (Sae Mulli)**2022**, 72, 371–375. [Google Scholar] [CrossRef] - Akmal, A.; Pandharipande, V.R.; Ravenhall, D.G. Equation of state of nucleon matter and neutron star structure. Phys. Rev. C
**1998**, 58, 1804–1828. [Google Scholar] [CrossRef][Green Version] - Brandes, L.; Weise, W.; Kaiser, N. Inference of the sound speed and related properties of neutron stars. Phys. Rev. D
**2023**, 107, 014011. [Google Scholar] [CrossRef] - Glendenning, N.K.; Kettner, C. Possible third family of compact stars more dense than neutron stars. Astron. Astrophys.
**2000**, 353, L9–L12. [Google Scholar] - Alvarez-Castillo, D.E.; Blaschke, D.B.; Grunfeld, A.G.; Pagura, V.P. Third family of compact stars within a nonlocal chiral quark model equation of state. Phys. Rev. D
**2019**, 99, 063010. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**The three nucleonic EoSs used in this work and corresponding results for selected nuclei. The EoS of symmetric nuclear matter (SNM) is the same for all three. For KIDS-P4, the EoS of pure nuclear matter (PNM) was obtained by a fit to the APR EoS [22,58]. The KIDS-46 ($L=46$ MeV) and KIDS-65 ($L=65$ MeV) sets were obtained by adjusting to gross nuclear properties as well as the NSMR relation [62]. Left: energy per particle for SNM, PNM, and $\beta -$stable matter and the symmetry energy as a function of nucleon density. Inset: speed of sound for $\beta -$stable matter. Right: nuclear masses (top) and charge radii (bottom) for the nuclei ${}^{16,28}$O, ${}^{40,48,60}$Ca, ${}^{56,68,78}$Ni, ${}^{90}$Zr, ${}^{100,120,132}$Sn, ${}^{208}$Pb, ${}^{218}$U. Lines are drawn to guide the eye. For the purposes of this illustration, the effective mass and gradient terms of the EDFs are kept fixed to the values shown, while the spin-orbit coupling strength ${W}_{0}$ is fitted to the shown data, resulting in the following values in units of MeV fm${}^{5}$: 114 for KIDS-46, 111 for KIDS-65, and 128 for KIDS-P4.

**Figure 2.**Pressure as a function of the baryonic density and of the energy density and speed of sound as a function of the baryonic density (left), and neutron star mass–radius relations (right) for the EoSs considered in this work, specifically: the KIDS-46, KIDS-65, and KIDS-P4 nucleonic EoSs (see Figure 1) are adopted up to the density where the speed of sound ${c}_{s}^{2}\equiv {u}^{2}/{c}^{2}$ reaches the indicated maximum value ${c}_{s,max}^{2}$ (thick continuous lines); thereafter, the constant speed of sound, pseudo-conformal EoS for ${c}_{s}^{2}=1/3$ is adopted (thin lines), assuming an abrupt crossover (see text). The EoSs are matched at low densities to the crustal EoSs SLy, BPS, or D1M* models, leading to the lowest, middle, and highest radius for a canonical star, respectively.

**Figure 3.**Pressure as a function of the baryonic density and of the energy density and speed of sound as a function of the baryonic density (left), and neutron star mass–radius relations (right) for the EoSs considered in this work, specifically: the KIDS-46, KIDS-65, and KIDS-P4 nucleonic EoSs (see Figure 1) are adopted up to the density where the speed of sound ${c}_{s}^{2}\equiv {u}^{2}/{c}^{2}$ reaches the indicated maximum value (thick continuous lines); thereafter, the constant speed of sound EoS for ${c}_{s}^{2}={c}_{s,max}^{2}$ is adopted (thin lines), assuming a smooth crossover (see text). Also indicated are characteristic values of a heavy neutron star’s central baryonic density. The EoSs are matched at low densities to the crustal EoSs SLy, BPS, or D1M* models, leading to the lowest, middle, and highest radius for a canonical star, respectively.

**Table 1.**Analytical EoS parameters used in this work. The numbers tabulated under “Symm. Energy” correspond to $(J,L,{K}_{\mathrm{sym}},{Q}_{\mathrm{sym}})$ in units of MeV. The numbers tabulated under “Extension($x,y$)” correspond to the values of $({\rho}_{x},B,D)$ in units of (fm${}^{-3}$, MeV fm${}^{3y}$, MeV/fm${}^{3}$), respectively, where $x={c}_{s,max}^{2}$ is the assumed onset condition for the CSS EoS, at which point the baryonic density is ${\rho}_{x}$, while $y=1/3$ or ${c}_{s,max}^{2}$ is the speed of sound at higher densities. The density, energy per particle, and compression modulus of symmetric nuclear matter are set to the respective values, $0.16$ fm${}^{-3}$, $-16$ MeV, and 240 MeV, at saturation.

EoS | Symm. Energy | Extension(0.6,1/3) | Extension(1.0,1/3) | Extension(0.6,0.6) | Extension(1.0,1.0) |
---|---|---|---|---|---|

KIDS-P4 | $(33,49,-156,580)$ | $(0.52,1086,75.9)$ | $(1.19,1343,-153.6)$ | $(0.52,1078,151.3)$ | $(1.19,799,408)$ |

KIDS-46 | $(32,65,-110,650)$ | $(0.60,1100,68.5)$ | $(1.09,1282,-73.8)$ | $(0.60,1051,161.0)$ | $(1.09,808,403)$ |

KIDS-65 | $(30,46,-145,650)$ | $(0.59,1084,79.0)$ | $(1.22,1371,-185.5)$ | $(0.59,1040,168.3)$ | $(1.22,800,411)$ |

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Papakonstantinou, P.; Hyun, C.H.
Energy-Density Modeling of Strongly Interacting Matter: Atomic Nuclei and Dense Stars. *Symmetry* **2023**, *15*, 683.
https://doi.org/10.3390/sym15030683

**AMA Style**

Papakonstantinou P, Hyun CH.
Energy-Density Modeling of Strongly Interacting Matter: Atomic Nuclei and Dense Stars. *Symmetry*. 2023; 15(3):683.
https://doi.org/10.3390/sym15030683

**Chicago/Turabian Style**

Papakonstantinou, Panagiota, and Chang Ho Hyun.
2023. "Energy-Density Modeling of Strongly Interacting Matter: Atomic Nuclei and Dense Stars" *Symmetry* 15, no. 3: 683.
https://doi.org/10.3390/sym15030683