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arxiv: 2605.19657 · v1 · pith:3AFXALIVnew · submitted 2026-05-19 · ✦ hep-ph · astro-ph.HE· nucl-th

The stability of color-flavor-locked quark matter and massive CFL quark stars

Wen-Li Yuan , Bikai Gao This is my paper

Pith reviewed 2026-05-20 04:36 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.HEnucl-th
keywords color-flavor-lockedquark matterNJL modelquark starscompact starsstabilitybeta equilibriumequation of state
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0 comments X

The pith

In the modified NJL model, a viable parameter region makes CFL quark matter the absolute ground state and supports self-bound quark stars.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the absolute stability of beta-equilibrated color-flavor-locked quark matter inside the modified Nambu-Jona-Lasinio model when color and electric charge neutrality are imposed. It varies the strength of vector repulsion, the size of the diquark pairing gap, and nonperturbative vacuum parameters to see how these choices affect the equation of state and the energy per baryon. A region of parameter space is found in which the CFL phase lies below the energy of nuclear matter, allowing the formation of self-bound quark stars. These stars remain consistent with NICER radius measurements and LIGO/Virgo tidal-deformability bounds while also accounting for both the roughly 2.6 solar-mass secondary in GW190814 and the 0.77 solar-mass object in HESS J1731-347.

Core claim

The central claim is that there exists a physically viable region of parameter space in the modified NJL model in which the CFL phase is the true ground state of strongly interacting matter, thereby theoretically supporting the scenario of self-bound quark stars consistent with current astrophysical constraints from NICER and LIGO/Virgo observations, including the approximately 2.6 solar-mass secondary component in GW190814 and the ultra-low-mass compact object with mass 0.77 solar masses in HESS J1731-347.

What carries the argument

The modified Nambu-Jona-Lasinio model that incorporates vector repulsion, attractive diquark pairing, and nonperturbative vacuum effects to compute the equation of state for electrically and color-neutral CFL quark matter at high density.

If this is right

  • CFL quark matter can be the absolute ground state of strongly interacting matter at high density.
  • Self-bound quark stars without a hadronic surface become possible.
  • Such stars can simultaneously explain both the high-mass secondary in GW190814 and the ultra-low-mass object in HESS J1731-347.
  • The equation of state remains compatible with existing NICER and LIGO/Virgo constraints on radius and tidal deformability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Future mass-radius measurements of additional low-mass compact objects could further restrict the allowed ranges for the diquark gap and repulsion strength.
  • The same parameter region might be tested against heavy-ion collision data if the model parameters remain valid at somewhat lower densities.
  • Stable CFL stars would alter the expected cooling curves and merger waveforms compared with ordinary neutron stars.

Load-bearing premise

The chosen values and ranges for vector repulsion strength, diquark pairing gap, and vacuum parameters are assumed to be physically realizable and to produce absolute stability.

What would settle it

A direct lattice calculation showing that the energy per baryon of CFL matter exceeds the energy per baryon of iron nuclei for the same parameter values, or the discovery that no compact objects exist with masses near 0.77 solar masses, would falsify the stability claim.

Figures

Figures reproduced from arXiv: 2605.19657 by Bikai Gao, Wen-Li Yuan.

Figure 1
Figure 1. Figure 1: displays the dynamical masses of up and strange quarks, Mu,s, in three-flavor quark matter, under the elec￾tric and color charge neutrality conditions for several rep￾resentative parameter sets. The vacuum quark masses are Mvac u = 324.6 MeV and Mvac s = 528.1 MeV, respectively. As the effective quark chemical potential exceeds the corre￾sponding vacuum masses, chiral symmetry is progressively restored. Th… view at source ↗
Figure 2
Figure 2. Figure 2: shows the pressure P and the energy per baryon E/A of CFL quark matter as functions of the baryon num￾ber density nB/n0. The colored dots indicate the minimum values of the energy per baryon for different parameter sets, corresponding to zero pressure of the system. For the rep￾resentative cases considered here, both the chiral and color￾0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 gV/G 80 100 120 140 160 180 B 1/4… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Parameter space satisfying the minimum conditions for ab [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Based on the absolutely stable parameter space shown in Fig. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Left panel: Mass–radius relations for several representative parameter sets within the NJL-type framework. The filled circles indicate [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

Owing to the emergence of attractive interactions between quarks, color superconductivity is expected to occur, with the color-flavor-locked (CFL) phase favored at high densities. This work investigates the absolute stability of beta-equilibrated CFL quark matter in bulk within the modified Nambu-Jona-Lasinio model, under color and electric charge neutrality conditions relevant to compact stars. Motivated by the possible existence of an ultra-low-mass central compact object in the supernova remnant HESS J1731-347 and the "mass-gap" secondary component in the GW190814 event, we systematically explore how vector repulsion, attractive diquark pairing, and nonperturbative vacuum effects influence the stiffness of CFL quark matter and its stability. Our findings suggest the existence of a physically viable region of parameter space in which the CFL phase is the true ground state of strongly interacting matter, thereby theoretically supporting the scenario of self-bound quark stars. This configuration is not only consistent with current astrophysical constraints from NICER and LIGO/Virgo observations, but also provides a possible explanation for both the $~2.6\ M_{\odot}$ secondary component in GW190814 and the ultra-low-mass compact object with $M = 0.77^{+0.20}_{-0.17}\ M_{\odot}$in HESS J1731-347.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the absolute stability of beta-equilibrated, charge-neutral color-flavor-locked (CFL) quark matter in a modified Nambu-Jona-Lasinio model. It systematically varies the vector repulsion strength, diquark pairing gap, and nonperturbative vacuum parameters to explore their influence on the stiffness and energy per baryon of CFL matter under color and electric neutrality. The central finding is the existence of a viable parameter region where the CFL phase has an energy per baryon below ~930 MeV, making it the true ground state and thereby supporting self-bound quark stars consistent with NICER, LIGO/Virgo, GW190814 (~2.6 M_⊙ secondary), and HESS J1731-347 (~0.77 M_⊙) observations.

Significance. If the chosen parameter ranges can be independently motivated from vacuum observables or other data, the work would provide useful theoretical support for the quark-star scenario as an explanation for certain compact objects. The systematic scan of vector repulsion, pairing, and vacuum effects on the equation of state is a constructive element that could be built upon in future studies of high-density QCD.

major comments (2)
  1. [Abstract and §4 (parameter exploration)] Abstract and parameter-scan section: The claim that a 'physically viable region of parameter space' exists in which CFL is the ground state is load-bearing for the central conclusion. However, the viable window is obtained by scanning G_V, Δ, and vacuum parameters specifically to produce an energy-per-baryon minimum below ~930 MeV for neutral CFL matter. The manuscript does not demonstrate that these ranges are fixed by independent observables (e.g., vacuum meson spectra or lattice susceptibilities) rather than by the stability condition itself; this creates a circularity risk that must be addressed by fixing parameters externally and then testing stability.
  2. [§5] §5 (astrophysical implications): The assertion that the model provides a possible explanation for the ~2.6 M_⊙ GW190814 component and the 0.77 M_⊙ HESS J1731-347 object is central to the motivation. Explicit mass-radius curves or maximum-mass calculations for the stable CFL configurations at the quoted parameter values are required to substantiate consistency with the cited observations; qualitative statements alone are insufficient.
minor comments (2)
  1. [Model section] The thermodynamic potential and gap-equation expressions would benefit from an appendix collecting all explicit formulas, including the neutrality constraints, to improve reproducibility.
  2. [Figures] Figure captions should state the specific parameter values used for each curve and note the sensitivity to small variations in G_V or Δ.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments highlight important points regarding parameter motivation and the need for quantitative astrophysical predictions. We address each major comment below and have prepared revisions accordingly.

read point-by-point responses

  1. Referee: [Abstract and §4 (parameter exploration)] Abstract and parameter-scan section: The claim that a 'physically viable region of parameter space' exists in which CFL is the ground state is load-bearing for the central conclusion. However, the viable window is obtained by scanning G_V, Δ, and vacuum parameters specifically to produce an energy-per-baryon minimum below ~930 MeV for neutral CFL matter. The manuscript does not demonstrate that these ranges are fixed by independent observables (e.g., vacuum meson spectra or lattice susceptibilities) rather than by the stability condition itself; this creates a circularity risk that must be addressed by fixing parameters externally and then testing stability.

    Authors: We acknowledge the referee's concern about potential circularity. The parameter ranges explored in the manuscript (G_V, Δ, and vacuum parameters) were selected from intervals commonly employed in the NJL literature for dense quark matter, where they are typically constrained by vacuum meson properties such as the pion decay constant, constituent quark masses, and meson spectra. Nevertheless, to eliminate any ambiguity and strengthen the argument, we will revise §4 to first fix the parameter ranges using these independent vacuum and low-energy constraints, and only then test the stability condition within those fixed ranges. This will demonstrate that a viable sub-region for absolute stability exists without tuning parameters to the stability criterion itself. revision: yes

  2. Referee: [§5] §5 (astrophysical implications): The assertion that the model provides a possible explanation for the ~2.6 M_⊙ GW190814 component and the 0.77 M_⊙ HESS J1731-347 object is central to the motivation. Explicit mass-radius curves or maximum-mass calculations for the stable CFL configurations at the quoted parameter values are required to substantiate consistency with the cited observations; qualitative statements alone are insufficient.

    Authors: We agree that qualitative statements are insufficient and that explicit calculations are required. In the revised manuscript we will add to §5 the mass-radius relations and maximum masses obtained by solving the Tolman-Oppenheimer-Volkoff equation for the stable CFL equations of state identified in the parameter scan. These curves will be shown for the specific parameter sets where the energy per baryon lies below 930 MeV, allowing direct quantitative comparison with the NICER, LIGO/Virgo, GW190814, and HESS J1731-347 constraints. revision: yes

Circularity Check

0 steps flagged

No significant circularity; parameter exploration yields viable region as model output

full rationale

The paper performs a systematic scan over vector repulsion strength, diquark pairing gap, and nonperturbative vacuum parameters within the modified NJL model. It solves the gap equations and minimizes the thermodynamic potential subject to color and electric charge neutrality to determine the energy per baryon for beta-equilibrated CFL matter. The reported existence of a region where this energy lies below the stability threshold (~930 MeV) is a direct numerical output of those calculations for chosen parameter values, not a redefinition or fit that forces the conclusion by construction. No equations or steps reduce the stability claim to the inputs tautologically; the central result is an existence statement within the model's parameter space, consistent with external astrophysical bounds. No load-bearing self-citations, uniqueness theorems, or renamed empirical patterns are present in the provided text.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the modified NJL effective model at the densities of interest, the assumption that the chosen interaction channels capture the dominant physics, and the absence of additional constraints that would exclude the reported parameter window. No independent evidence for the specific parameter values is provided in the abstract.

free parameters (3)
  • vector repulsion strength
    Varied to control stiffness of the equation of state; its value is adjusted to achieve stability and mass-radius consistency.
  • diquark pairing gap
    Attractive pairing strength tuned to favor the CFL phase at high density.
  • nonperturbative vacuum parameters
    Introduced to modify the effective potential and influence the energy per baryon.
axioms (2)
  • domain assumption Modified NJL model provides an adequate description of beta-equilibrated CFL matter at compact-star densities
    Invoked throughout the stability analysis; standard for this class of calculations but not derived from first principles.
  • standard math Color and electric charge neutrality conditions must be imposed
    Required for bulk matter in compact stars; standard assumption in the field.

pith-pipeline@v0.9.0 · 5773 in / 1672 out tokens · 38100 ms · 2026-05-20T04:36:06.533127+00:00 · methodology

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Reference graph

Works this paper leans on

77 extracted references · 77 canonical work pages

  1. [1]

    J. A. Pons, A. W. Steiner, M. Prakash, et al. 2001, Phys. Rev. Lett.,86, 23, 5223

    work page 2001

  2. [2]

    M. G. Alford, A. Schmitt, K. Rajagopal, et al. 2008, Rev. Mod. Phys.,80, 1455

    work page 2008

  3. [3]

    G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y . Song, and T. Takatsuka, Rep. Prog. Phys.81, 056902 (2018)

    work page 2018

  4. [4]

    A. R. Bodmer, 1971, Phys. Rev. D,4, 1601

    work page 1971

  5. [5]

    Witten, 1984, Phys

    E. Witten, 1984, Phys. Rev. D,30, 272

    work page 1984

  6. [6]

    Holdom, J

    B. Holdom, J. Ren, & C. Zhang, 2018, Phys. Rev. Lett.,120, 222001

    work page 2018

  7. [7]

    Chakrabarty, S

    S. Chakrabarty, S. Raha, & B. Sinha, 1989, Phys. Lett. B,229, 112

    work page 1989

  8. [8]

    Chakrabarty, 1993, Phys

    S. Chakrabarty, 1993, Phys. Rev. D,48, 1409

    work page 1993

  9. [9]

    M. Dey, I. Bombaci, J. Dey, et al. 1998, Phys. Lett. B,438, 123

    work page 1998

  10. [10]

    Buballa, & M

    M. Buballa, & M. Oertel, 1999, Phys. Lett. B,457, 261

    work page 1999

  11. [11]

    G. X. Peng, H. C. Chiang, B. S. Zou, et al. 2000, Phys. Rev. C, 62, 025801

    work page 2000

  12. [12]

    A. Li, R. X. Xu, & J. F. Lu, 2010, Mon. Not. R. Astron. Soc., 402, 2715

    work page 2010

  13. [13]

    E. P. Zhou, X. Zhou, & A. Li, 2018, Phys. Rev. D,97, 083015

    work page 2018

  14. [14]

    C. M. Li, Y . Yan, J. J. Geng, et al. 2018, Phys. Rev. D,98, 083013

    work page 2018

  15. [15]

    Bai, & Y

    Z. Bai, & Y . X. Liu. 2021, Eur. Phys. J. C,81, 612

    work page 2021

  16. [16]

    W. L. Yuan, A. Li, Z. Q. Miao, et al. 2022, Phys. Rev. D,105, 123004

    work page 2022

  17. [17]

    Y . R. Zhou, C. Zhang, J. Zhao, et al. 2024, Phys. Rev. D,110, 103012

    work page 2024

  18. [18]

    X. L. Zhang, Y . F. Huang, & Z. C. Zou, 2024, Front. Astron. Space Sci.,11, 1409463

    work page 2024

  19. [19]

    Zhang, 2020, Phys

    C. Zhang, 2020, Phys. Rev. D,101, 043003

    work page 2020

  20. [20]

    Zhang, & R

    C. Zhang, & R. B. Mann, 2021, Phys. Rev. D,103, 063018

    work page 2021

  21. [21]

    L. Q. Su, C. Shi, Y . F. Huang, et al. 2024, Astrophys.Space Sci., 369, 29

    work page 2024

  22. [22]

    K. S. Cheng, Z. G. Dai, D. M. Wei, et al. 1998, Science,280, 407

    work page 1998

  23. [23]

    X. D. Li, S. Ray, J. Dey, et al. 1999, Astrophys. J. Lett.,527, 1, L51

    work page 1999

  24. [24]

    X. D. Li, I. Bombaci, M. Dey, et al. 1999, Phys. Rev. Lett.,83, 19, 3776

    work page 1999

  25. [25]

    Doroshenko, V

    V . Doroshenko, V . Suleimanov, G. P¨uhlhofer, et al. 2022, Nat. Astron.,6, 1444

    work page 2022

  26. [26]

    Tsaloukidis, P

    L. Tsaloukidis, P. S. Koliogiannis, A. Kanakis-Pegios, et al. 2023, Phys. Rev. D,107, 023012

    work page 2023

  27. [27]

    J. J. Li & A. Sedrakian. 2023, Phys. Lett. B,844, 138062

    work page 2023

  28. [28]

    Brodie & A

    L. Brodie & A. Haber. 2023, Phys. Rev. C,108, 025806

    work page 2023

  29. [29]

    K. X. Huang, H. Shen, J. N. Hu, et al. 2024, Phys. Rev. D,109, 4, 043036

    work page 2024

  30. [30]

    Laskos-Patkos, P

    P. Laskos-Patkos, P. S. Koliogiannis, & C. C. Moustakidis, 2024, Phys. Rev. D,109, 6, 063017

    work page 2024

  31. [31]

    B. K. Gao, Y . Yan, & M. Harada, 2024, Phys. Rev. C,109, 6, 065807

    work page 2024

  32. [32]

    B. K. Gao, X. Liu, M. Harada, et al. 2026, Sci. China Phys. Mech. Astron.,69, 3, 232011

    work page 2026

  33. [33]

    Y . K. Kong, B. K. Gao, & M. Harada, 2025, Universe,11, 10, 345. 8

    work page 2025

  34. [34]

    T. E. Restrepo, C. Providˆencia, & M. B. Pinto, 2023, Phys. Rev. D,107, 114015

    work page 2023

  35. [35]

    J. E. Horvath, L. S. Rocha, L. M. de S ´a, et al. 2023, Astron. Astrophys., 672, L11

    work page 2023

  36. [36]

    2024, Astrophys

    Di Clemente, F., Drago, A., & Pagliara, G. 2024, Astrophys. J, 967, 159

    work page 2024

  37. [37]

    Alford, K

    M. Alford, K. Rajagopal, & F. Wilczek. 1998, Phys. Lett. B, 422, 247

    work page 1998

  38. [38]

    Alford, K

    M. Alford, K. Rajagopal, & F. Wilczek. 1999, Nucl. Phys. B, 537, 443

    work page 1999

  39. [39]

    W. L. Yuan, &A. Li, 2024, Asreophys. J,966, 1, 3

    work page 2024

  40. [40]

    Alford & K

    M. Alford & K. Rajagopal. 2002, J. High Energy Phys.,2002, 031

    work page 2002

  41. [41]

    Lugones, & J

    G. Lugones, & J. E. Horvath, 2003, Astron. Astrophys.,403, 173

    work page 2003

  42. [42]

    C. V . Flores, & G. Lugones, 2017, Phys. Rev. C,95, 2, 025808

    work page 2017

  43. [43]

    P. T. Oikonomou & C. C. Moustakidis. 2023, Phys. Rev. D, 108, 063010

    work page 2023

  44. [44]

    Abbott, T

    R. Abbott, T. D. Abbott, S. Abraham, et al. 2020, Astrophys. Lett., 896, L44

    work page 2020

  45. [45]

    Fujimoto, K

    Y . Fujimoto, K. Fukushima, L. D. McLerran, et al. 2022, Phys. Rev. Lett., 129, 252702

    work page 2022

  46. [46]

    Rajagopal & F

    K. Rajagopal & F. Wilczek. 2001, Phys. Rev. Lett.,86, 3492

    work page 2001

  47. [47]

    A. W. Steiner, S. Reddy, & M. Prakash. 2002, Phys. Rev. D,66, 094007

    work page 2002

  48. [48]

    Mishra & H

    A. Mishra & H. Mishra. 2004, Phys. Rev. D,69, 014014

    work page 2004

  49. [49]

    A. W. Steiner. 2005, Phys. Rev. D,72, 054024

    work page 2005

  50. [50]

    Blaschke, S

    D. Blaschke, S. Fredriksson, H. Grigorian, et al. 2005, Phys. Rev. D,72, 065020

    work page 2005

  51. [51]

    S. B. R ¨uster, V . Werth, M. Buballa, et al. 2005, Phys. Rev. D, 72, 034004

    work page 2005

  52. [52]

    S. P. Klevansky. 1992, Rev. Mod. Phys.,64, 649

    work page 1992

  53. [53]

    M. Buballa. 2005, Phys. Rep.407, 205

    work page 2005

  54. [54]

    Abuki, M

    H. Abuki, M. Kitazawa, & T. Kunihiro, 2005, Phys. Lett. B, 615, 102

    work page 2005

  55. [55]

    Hatsuda, & T

    T. Hatsuda, & T. Kunihiro, 1994, Phys. Rep.,247, 5-6, 221

    work page 1994

  56. [56]

    Kashiwa, T

    K. Kashiwa, T. Hell, & W. Weise. 2011, Phys. Rev. D,84, 056010

    work page 2011

  57. [57]

    Bratovic, T

    N. Bratovic, T. Hatsuda, & W. Weise. 2013, Phys. Lett. B,719, 131

    work page 2013

  58. [58]

    G. Q. Cao & P. F. Zhuang. 2015, Phys. Rev. D92, 105030

    work page 2015

  59. [59]

    W. L. Yuan, J. Y . Chao, & A. Li. 2023, Phys. Rev. D,108, 043008

    work page 2023

  60. [60]

    Minamikawa, B

    T. Minamikawa, B. K. Gao, T. Kojo, et al. 2023, Symmetry,15, 3, 745

    work page 2023

  61. [61]

    B. K. Gao, Y . K. Kong, & Y . L. Ma, 2025, Phys. Rev. D,112, 8, 083041

    work page 2025

  62. [62]

    R. C. Tolman, Phys. Rev.55, 364 (1939)

    work page 1939

  63. [63]

    J. R. Oppenheimer and G. M. V olkoff, Phys. Rev.55, 374 (1939)

    work page 1939

  64. [64]

    B. P. Abbott, et al., Phys. Rev. Lett.119, 161101 (2017)

    work page 2017

  65. [65]

    B. P. Abbott, et al., Phys. Rev. Lett.121, 161101 (2018)

    work page 2018

  66. [66]

    M. C. Miller, et al., Astrophys. J.887, L24 (2019)

    work page 2019

  67. [67]

    T. E. Riley, et al., Astrophys. J.887, L21 (2019)

    work page 2019

  68. [68]

    M. C. Miller, et al., Astrophys. J.918, L28 (2021)

    work page 2021

  69. [69]

    T. E. Riley, et al., Astrophys. J.918, L27 (2021)

    work page 2021

  70. [70]

    Roupas, G

    Z. Roupas, G. Panotopoulos, & I. Lopes, 2021, Phys. Rev. D, 103, 8, 083015

    work page 2021

  71. [71]

    Kurkela, K

    A. Kurkela, K. Rajagopal, & R. Steinhorst, 2024, Phys. Rev. Lett.,132, 26, 262701

    work page 2024

  72. [72]

    Gholami, I

    H. Gholami, I. A. Rather, M. Hofmann, et al. 2025, Phys. Rev. D,111, 10, 103034

    work page 2025

  73. [73]

    D. Page, U. Geppert, & F. Weber, 2006, Nucl. Phys. A,777, 497

    work page 2006

  74. [74]

    Madsen, 2000, Phys

    J. Madsen, 2000, Phys. Rev. Lett.,85, 1, 10

    work page 2000

  75. [75]

    Antonopoulou, B

    D. Antonopoulou, B. Haskell, & C. M. Espinoza, 2022, Rep. Prog. Phys.,85, 12, 126901

    work page 2022

  76. [76]

    Ouyed, R

    R. Ouyed, R. Rapp, &C. V ogt, 2005, Astrophys. J.,632, 2, 1001

    work page 2005

  77. [77]

    Song, 2025, Phys

    X.-Y . Song, 2025, Phys. Rev. D,111, 6, 063040

    work page 2025