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Hybrid stars with large quark cores within the parity doublet model and modified NJL model

Wen-Li Yuan, Bikai Gao, Yan Yan, Renxin Xu

TL;DR

This work addresses whether neutron stars can host large quark cores by linking a chiral-symmetric hadronic EOS from the parity doublet model to a modified NJL quark EOS via a sharp first-order transition calculated with Maxwell construction. By varying the chiral-invariant mass $m_0$, the NJL exchange-weight parameter $\alpha$, and the QCD vacuum pressure $B$, the authors map the stability regions for two-flavor and (2+1)-flavor quark cores and compute the resulting mass–radius and tidal-deformability predictions. They find substantial parameter space allowing large $\sim 1\ M_\odot$ two-flavor quark cores in $\sim 2\ M_\odot$ hybrid stars, while three-flavor cores are more constrained and tend to require $m_0\approx600$ MeV to satisfy GW170817 tidal constraints; the maximum hybrid-star mass remains around $2.2\ M_\odot$ for both cases. Overall, the study demonstrates the viability of hybrid stars with large quark cores within chiral-symmetric effective theories and highlights how GW170817 and NICER data constrain the allowed parameter space, informing the physical interpretation of dense QCD matter.

Abstract

Using the parity doublet model (PDM) for hadronic matter and a modified Nambu-Jona-Lasinio (NJL) model for quark matter, we investigate the potential existence of two- and three-flavor quark matter in neutron star cores. Both models respect chiral symmetry, and a sharp first-order phase transition is implemented via Maxwell construction. We find stable neutron stars with quark cores within a specific parameter space that satisfies current astronomical observations. Typical neutron stars with masses around $1.4 \ M_\odot$ may possess deconfined quark matter in their centers. The hybrid star scenario with a two-flavor quark core offers enough parameter space to allow the neutron stars with large quark cores exceeding $\sim 1\ M_\odot$, and allow the early deconfinement position before $2\ ρ_0$, where $ρ_0$ is the nuclear saturation density. The observations of gravitational wave event GW170817 suggest a relatively large chiral invariant mass $m_0=600\ \rm MeV$ in the PDM for scenarios involving three-flavor quark matter cores. The maximum mass of the hybrid star with a quark core is found to be approximately $2.2\ M_\odot$ for both two- or three-flavor quark matter in their centers.

Hybrid stars with large quark cores within the parity doublet model and modified NJL model

TL;DR

This work addresses whether neutron stars can host large quark cores by linking a chiral-symmetric hadronic EOS from the parity doublet model to a modified NJL quark EOS via a sharp first-order transition calculated with Maxwell construction. By varying the chiral-invariant mass , the NJL exchange-weight parameter , and the QCD vacuum pressure , the authors map the stability regions for two-flavor and (2+1)-flavor quark cores and compute the resulting mass–radius and tidal-deformability predictions. They find substantial parameter space allowing large two-flavor quark cores in hybrid stars, while three-flavor cores are more constrained and tend to require MeV to satisfy GW170817 tidal constraints; the maximum hybrid-star mass remains around for both cases. Overall, the study demonstrates the viability of hybrid stars with large quark cores within chiral-symmetric effective theories and highlights how GW170817 and NICER data constrain the allowed parameter space, informing the physical interpretation of dense QCD matter.

Abstract

Using the parity doublet model (PDM) for hadronic matter and a modified Nambu-Jona-Lasinio (NJL) model for quark matter, we investigate the potential existence of two- and three-flavor quark matter in neutron star cores. Both models respect chiral symmetry, and a sharp first-order phase transition is implemented via Maxwell construction. We find stable neutron stars with quark cores within a specific parameter space that satisfies current astronomical observations. Typical neutron stars with masses around may possess deconfined quark matter in their centers. The hybrid star scenario with a two-flavor quark core offers enough parameter space to allow the neutron stars with large quark cores exceeding , and allow the early deconfinement position before , where is the nuclear saturation density. The observations of gravitational wave event GW170817 suggest a relatively large chiral invariant mass in the PDM for scenarios involving three-flavor quark matter cores. The maximum mass of the hybrid star with a quark core is found to be approximately for both two- or three-flavor quark matter in their centers.

Paper Structure

This paper contains 17 sections, 30 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Formulism
  3. Hadronic matter within PDM model
  4. Quark matter within modified NJL model
  5. two-flavor modified NJL model
  6. (2+1)-flavor modified NJL model
  7. quark condensate and quark number density at zero temperature
  8. QCD vacuum pressure as free parameter
  9. EOS and Hybrid star structure with large quark core
  10. Pressure versus baryon chemical potential
  11. Hybrid star with large two-flavor quark matter core
  12. The influence of $B^{1/4}$
  13. The impact of $\alpha$
  14. Hybrid star with (2+1)-flavor quark core
  15. Tidal deformability
  16. ...and 2 more sections

Figures (7)

  • Figure 1: The dynamical quark mass $M$ of $u$, $d$ and $s$ quarks versus the quark chemical potential $\mu_q$, as well as quark number density $\rho_f$ ($f=u, d, s$) versus $\mu_q$ for two-flavor and (2+1)-flavor NJL modified NJL models with $\alpha=0.0$ and $\alpha=0.8$. The red curves represent the results for $\alpha=0.0$, while the dot-dashed blue curves exhibit the results for $\alpha=0.8$.
  • Figure 2: Pressure $P$ as a function of baryon chemical potential $\mu_B$ for hadronic matter and quark matter. In each panel, the PDM calculations for hadronic matter with chiral invariant masses of $m_0 = 500~\mathrm{MeV}$ and $m_0 = 600~\mathrm{MeV}$ are shown as gray and black curves, respectively. For quark matter, curves with the same style represent results for the same value of $\alpha$, while different colors correspond to different vacuum pressures. For instance, in the left panel, the blue curves denote results from the two-flavor modified NJL model with $B^{1/4} = 120~\mathrm{MeV}$ at various values of $\alpha=0.8, \ 0.9$.
  • Figure 3: The pressure $P$ as functions of energy density $\epsilon$, derived from PDM and predicted from $\chi$EFT, are shown. The gray, black, and sandy brown lines represent hadronic EOSs derived from the PDM with $m_0 = 500~\mathrm{MeV}$, $m_0 = 600~\mathrm{MeV}$, and $m_0 = 700~\mathrm{MeV}$, respectively. The red band denotes the $1\sigma$ uncertainty in $\chi$EFT 2021PhRvC.103d5808D2023PhRvL.130g2701K. The other data-driven results incorporating constraints from $\chi$EFT within the $1\sigma$ uncertainty are also shown together: the blue and violet-red bands represent the results of Bayesian analyses 2021ApJ...918L..29R and EOSs inferred from data-driven deep learning methods 2021JHEP...03..273F, respectively. The green band shows the Bayesian analyses results mainly constrained by neutron star observations Ozel:2015fiaBogdanov:2016nle.
  • Figure 4: Upper panel: The influence of the vacuum pressure $B^{1/4}$ on the results of the hybrid EOS and on the properties of hybrid stars with a two-flavor quark core. Pressure versus density (in units of nuclear saturation density $\rho_0$) for different choices of NJL parameter sets and PDM with different $m_0$ are shown in the upper left panel. The central densities $\rho_{\rm center}$ of the corresponding maximum mass hybrid stars are given. The mass-radius relations for the corresponding parameter sets are presented in the right panel. The arrows indicate the position where the quark matter begins to appear. The available mass-radius constraints from the NICER mission (PSR J0030 + 0451 2019ApJ...887L..24M2019ApJ...887L..21R and PSR J0740 + 6620 2021ApJ...918L..28M2021ApJ...918L..27R) at the 90$\%$ confidence level and the binary tidal deformability constraint from LIGO/Virgo (GW170817 2017PhRvL.119p1101A2018PhRvL.121p1101A) at the 90$\%$ confidence level are also shown together. Lower panel: The results of the effects of $\alpha$ are displayed with several specific parameter sets. The size of quark cores obtained are: (Upper panel) $1.51\,M_{\odot}$ ($\alpha=0.8, B^{1/4}=120\;\text{MeV}, \rm PDM500$), $0.53\,M_{\odot}$ ($\alpha=0.8, B^{1/4}=130\;\text{MeV}, \rm PDM500$), $0.08\,M_{\odot}$ ($\alpha=0.8, B^{1/4}=140\;\text{MeV}, \rm PDM600$); (Lower panel) $1.07\,M_{\odot}$ ($\alpha=0.7, B^{1/4}=120\;\text{MeV}, \rm PDM500$), $1.51\,M_{\odot}$ ($\alpha=0.8, B^{1/4}=120\;\text{MeV}, \rm PDM500$), $1.07\,M_{\odot}$ ($\alpha=0.9, B^{1/4}=120\;\text{MeV}, \rm PDM500$). The corresponding properties of the star are displayed in Table \ref{['table:PDM_2f_NJL']} in detail.
  • Figure 5: Pressure versus density (left panel) in units of nuclear saturation density $\rho_0$ and mass-radius relations (right panel) with strange quark matter core for selected parameter sets. The arrows in the right panel indicate the position where the quark matter begins to appear. The size of quark cores obtained are: $0.35\,M_{\odot}$ ($\alpha=0.7, B^{1/4}=120\;\text{MeV}, \rm PDM600$), $0.01\,M_{\odot}$ ($\alpha=0.8, B^{1/4}=115\;\text{MeV}, \rm PDM600$), $0.28\,M_{\odot}$ ($\alpha=0.8, B^{1/4}=120\;\text{MeV}, \rm PDM500$). The center densities corresponding to the maximum mass are: $5.8\rho_0$ ($\alpha=0.7, B^{1/4}=120\;\text{MeV}, \rm PDM600$), $5.5\rho_0$ ($\alpha=0.8, B^{1/4}=115\;\text{MeV}, \rm PDM600$), $4.8\rho_0$ ($\alpha=0.8, B^{1/4}=120\;\text{MeV}, \rm PDM500$). The available mass-radius constraints from the NICER mission (PSR J0030 + 0451 2019ApJ...887L..24M2019ApJ...887L..21R and PSR J0740 + 6620 2021ApJ...918L..28M2021ApJ...918L..27R) at the 90$\%$ confidence level and the binary tidal deformability constraint from LIGO/Virgo (GW170817 2017PhRvL.119p1101A2018PhRvL.121p1101A) at the 90$\%$ confidence level are also shown together.
  • ...and 2 more figures