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REVIEW 4 major objections 5 minor 110 references

Interacting quark matter and $f(Q)$ gravity: A new paradigm in exploring the properties of quark stars

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper derives exact interior solutions for quark stars in f(Q) gravity with a unified interacting quark matter equation of state, and shows the resulting mass-radius curves are stable and match observed radii.

desk verdict A real exact solution is buried under claims the paper's own equations contradict. read the letter →

arxiv 2412.00693 v2 pith:PUOUCEUN submitted 2024-12-01 gr-qc nucl-th

Debadri Bhattacharjee , Koushik Ballav Goswami , Pradip Kumar Chattopadhyay This is my paper · ORCID
classification gr-qcnucl-th MSC 83C1583C5585A15 PACS 04.40.Dg97.60.Jd04.50.Kd
keywords quarkstarsf(Q)gravityinteractingmattercoloursuperconductivityBuchdahl-IansatzTolman-Oppenheimer-Volkoffequationmass-radiusrelationcompactstarradii
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that quark stars—ultra-dense compact objects made of deconfined up, down, and strange quarks—can be described by exact solutions of the field equations in $f(Q)$ gravity, a modified theory in which gravity is carried by non-metricity rather than curvature. Using a unified interacting quark matter equation of state that covers the 2SC, 2SC+s, and color-flavor locked (CFL) phases, together with the Buchdahl-I metric ansatz and a linear $f(Q)=\alpha_0+\alpha_1 Q$, the authors derive closed-form interior solutions and integrate the Tolman-Oppenheimer-Volkoff equations to obtain mass-radius curves. The maximum masses are $1.89\,M_\odot$ for the 2SC phase, $1.84$–$1.89\,M_\odot$ for 2SC+s, and $1.99$–$2.07\,M_\odot$ for CFL, with radii between about 10 and 11.3 km; the predicted radii of five known compact stars fall within the observed bands. If correct, the work shows that strong-interaction effects in quark matter can be combined with a modified theory of gravity to give stable, observationally consistent stellar models.

What carries the argument

Three linked pieces carry the argument. First, the unified interacting quark matter equation of state, $p = (\rho - 4B_{\rm eff})/3$, with $B_{\rm eff}$ built from the bag constant, the color-superconductivity gap $\Delta$, the strange quark mass $m_s$, and the perturbative QCD parameter $a_4$; the dimensionless coefficients in the free energy pick out the 2SC, 2SC+s, and CFL phases. Second, the Buchdahl-I metric ansatz, $e^{2\lambda} = 2(1+\chi r^2)/(2-\chi r^2)$, which converts the field equations into algebraic relations and yields the closed-form density. Third, the linear $f(Q)=\alpha_0+\alpha_1 Q$ action, forced by the constraint equation $Q' f_{QQ}=0$, which makes the exterior vacuum solution the Schwarzschild--(anti-)de Sitter metric and provides the junction conditions at the stellar surface.

What would settle it

Recompute the exact solution (33)--(36) with $B_{\rm eff}$ kept as a function of the chemical potential instead of the constant $B_g$, and re-integrate the TOV equations; if the maximum mass changes by more than the few-hundredths of a solar mass separating the phases, the reported values are not stable. Observationally, a radius measurement of a $1.4\,M_\odot$ compact object to better than about 0.5 km would discriminate, since the model predicts 10.5--11.2 km depending on phase.

Watch

Extended reading notes

Core claim

The central discovery is an exact, singularity-free interior solution of the $f(Q)$ field equations for an anisotropic quark star whose matter obeys the linearized interacting quark matter equation of state $p = \frac{1}{3}(\rho - 4B_{\rm eff})$. With $f(Q)=\alpha_0+\alpha_1 Q$ and the Buchdahl-I ansatz $e^{2\lambda}=2(1+\chi r^2)/(2-\chi r^2)$, the field equations reduce to explicit expressions for $\rho$, $p_r$, $p_t$, and the metric potential $\nu$ (eqs. 33--36). Matching this interior to the vacuum exterior solution fixes $\chi = -4M/[R^2(4M-3R)]$, and numerical integration of the TOV equations gives maximum masses of $1.89\,M_\odot$ (2SC), $1.84$--$1.89\,M_\odot$ (2SC+s), and $1.99$--$2.07\,M_\odot$ (CFL), with corresponding radii $10.04$--$11.27$ km. The same solutions satisfy causality ($v_r^2=1/3$, $0<v_t^2<1$), the strong, weak, null, and dominant energy conditions, the Abreu cracking criterion, and the generalized TOV force balance, and they yield radii for EXO 1745-248, 4U 1820-30, LMC X-4, HER X-1, and PSR J1903+0327 that agree with observations.

Load-bearing premise

The load-bearing premise is that the effective bag parameter $B_{\rm eff}$ in the linearized equation of state can be replaced by the constant bag constant $B_g$ in the exact solution; if the chemical-potential dependence of $B_{\rm eff}$ is retained, all subsequent densities, pressures, masses, and radii change.

Editorial extensions

If this is right

  • If the model is correct, CFL quark matter can support a star up to $2.07\,M_\odot$, so any confirmed compact object above that mass cannot be a quark star within this parameter set.
  • Because predicted radii for a $1.4\,M_\odot$ star range from about 10.5 km (2SC) to 11.2 km (CFL), a radius measurement with sub-kilometer precision would select among the phases.
  • Raising the strange quark mass softens the equation of state and lowers both maximum mass and radius, so the mass-radius curve carries direct information about $m_s$ once the phase is known.
  • The exterior vacuum geometry is Schwarzschild--(anti-)de Sitter because $f(Q)$ is linear, which means the modified-gravity signature appears through the interior solution and stability conditions rather than through a distinct vacuum metric.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: retaining the full chemical-potential dependence of $B_{\rm eff}$ in the exact solution is the natural next computation, and the reported maximum masses would shift if that dependence matters.
  • Beyond the paper: the same construction could be applied to non-linear $f(Q)$ forms or other interior ansätze; the linear action and the Buchdahl-I ansatz are what currently make the problem exactly solvable.
  • Beyond the paper: a measured tidal deformability from a future binary merger would test the CFL equation of state more directly than mass-radius curves alone.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript constructs a quark star model in f(Q) gravity by combining the Buchdahl-I metric ansatz with a linear f(Q) action and a unified interacting quark matter (IQM) equation of state. The authors analytically solve the field equations, then numerically integrate the Tolman-Oppenheimer-Volkoff equations to obtain maximum masses between 1.84 and 2.07 solar masses for the 2SC, 2SC+s, and CFL phases, and they tabulate predicted radii for several known compact stars. They also examine causality, energy conditions, hydrostatic equilibrium, and stability criteria for a representative object (4U 1608-52).

Significance. If the results were internally consistent, the paper would offer a useful exact stellar solution in symmetric teleparallel gravity with a microphysically motivated quark matter EoS, together with explicit analytic expressions (Eqs. 33-36) and numerical tables. The authors also perform standard viability checks. However, the derivation contains a load-bearing inconsistency: the IQM EoS of Eq. (8) is not the EoS used in the exact solution, because B_eff is silently replaced by the constant B_g in Eq. (34). In addition, the linear f(Q) model is acknowledged by the authors themselves to reduce to GR with a cosmological constant, and the radius 'predictions' in Table 2 are obtained by tuning the strange quark mass per object. These issues undermine the central claims of the paper and cannot be treated as mere presentation problems.

major comments (4)
  1. [Section 5, Eq. (34)] The exact solution replaces the effective bag constant B_eff from the IQM EoS (8) with the constant B_g in Eq. (34). Eq. (8) defines B_eff as a function of the quark chemical potential μ through terms proportional to μ^2, and these terms do not vanish for the adopted parameters Δ = 100 MeV and m_s = 0, 50, 100 MeV. Because μ is related to ρ via Eq. (6), retaining the μ-dependence changes the EoS to the non-linear form of Eq. (7). The manuscript provides no justification for dropping these terms, so the solution (33)-(36) corresponds to the non-interacting MIT bag model, not to the unified IQM EoS advertised in the abstract and introduction. All subsequent physical quantities and stability checks built on this solution inherit the inconsistency.
  2. [Section 6, Eqs. (47)-(49)] The paper states that the linear f(Q) action with Λ = α0/α1 effectively reduces to GR, and the exterior solution becomes Schwarzschild-(anti) de Sitter. Since the analysis uses exactly this linear f(Q), the claim of a 'new paradigm in f(Q) gravity' is overstated: the stellar models are solutions of GR with a cosmological constant rather than of a genuinely modified gravitational theory. The authors should either frame the results as GR-with-cosmological-constant models or use a nonlinear f(Q) form that does not trivially reduce to GR.
  3. [Section 7, Table 2] The radius predictions for known compact stars are obtained by choosing a different strange quark mass for each object (e.g., m_s = 250 MeV for 4U 1820-30 to reproduce 9.10 km, m_s = 200 MeV for HER X-1 and PSR J1903+0327, and m_s = 50 MeV for EXO 1745-248). This is parameter fitting rather than prediction, since m_s is a constant of nature and cannot be varied independently from star to star. Moreover, several of the quoted predictions fall outside the observational uncertainties even after this tuning, for example LMC X-4 (predicted 9.72, 8.64, 9.05 km versus 8.301 ± 0.2 km) and PSR J1903+0327 (predicted 10.75, 9.57, 10.36 km versus 9.438 ± 0.03 km). The claimed agreement with observations is therefore not established.
  4. [Section 9.3, Eq. (61)] The adiabatic index is defined as Γ = (ρ + p_r)/p_r · dp_r/dr and then set equal to (ρ + p_r)/p_r · v_r^2. This equality is dimensionally inconsistent because dp_r/dr has dimensions of pressure per unit length, whereas v_r^2 = dp_r/dρ is dimensionless. The correct relativistic adiabatic index is Γ = (ρ + p_r)/p_r · dp_r/dρ. Since the plotted Γ and the stability bound Γ > 4/3 are based on the erroneous expression, the stability conclusion drawn from the adiabatic index is not supported.
minor comments (5)
  1. [Title and throughout] There are numerous formatting and typographical errors, including the missing space in 'andf (Q)' in the title, 'Bef f' in Section 2, 'T angential' and 'Hererra' in Section 9, and 'mode' instead of 'model' in Section 9.1.
  2. [Eq. (38)] The mass function is written as m(r) = 4π ∫_0^R ρ r^2 dr, but the upper limit should be r, not R, for it to define the enclosed mass as a function of radial coordinate.
  3. [Reference [26]] The reference for Zhang and Mann is listed as Phys. Rev. D 103 (2011) 063018, but the actual publication year is 2021; the volume number 103 appeared in 2021.
  4. [Section 8, parameter choices] The choice α0 = 10^-46 km^-2 and α1 = -0.6 is adopted from a previous reference without any dedicated justification; because α0/α1 plays the role of a cosmological constant, a brief consistency check with observational bounds would improve the manuscript.
  5. [Figures 4, 13, 14] Several figure captions do not identify which curve corresponds to which value of m_s, and some figures (e.g., Figs. 13 and 14) lack axis labels, making the plots difficult to interpret.

Circularity Check

1 steps flagged · score 6.0 of 10

The paper's 'predicted radii' for known compact stars are obtained by tuning m_s per object, so the claimed observational agreement is by construction; a separate consistency flaw silently drops the μ-dependent interaction terms in Eq. (34).

  1. fitted input called prediction [Section 7, Table 2 (Radius prediction of some known compact objects)]
    "Radius prediction of some known compact objects from TOV mass-radius relation in 2SC, 2SC+s and CFL phases respectively. ... 4U 1820-30 [96] 1.58+0.06−0.06 9.1+0.4−0.4 10.68 250 9.10 9.77"

    Table 2 assigns a different strange-quark mass m_s to each star: 50 MeV for EXO 1745-248, 250 MeV for 4U 1820-30 and LMC X-4, and 200 MeV for HER X-1 and PSR J1903+0327. No independent constraint fixes these per-object values; Section 3 gives only upper bounds (262 MeV and 308 MeV for 2SC+s and CFL). Since Table 1 and Fig. 4 show that increasing m_s monotonically decreases the predicted radius, m_s acts as a free dial. For 4U 1820-30 the choice m_s=250 yields 9.10 km, exactly the observed central radius 9.1 km. Thus the claimed observational alignment is not a model prediction; it is a parameter choice made to match the data, i.e., a fitted input relabeled as a prediction.

full rationale

Most of the formal derivation chain is self-contained: the f(Q) field equations, the linear f(Q) restriction, the Buchdahl-I ansatz, the junction conditions, and the numerical TOV integrations are standard algebraic consequences of the stated assumptions, and I find no load-bearing self-citation chain. Reference [94] is used only as a supporting comparison for the CFL phase and [82] is a prior metric-ansatz work; neither forces the central results by citation. The one clear circular element is Table 2: the predicted radii of known compact objects are generated by choosing a different m_s for each object, and for 4U 1820-30 the chosen m_s=250 reproduces the observed central radius exactly. Because m_s shifts the mass-radius relation, this agreement is constructed rather than predicted. That warrants a partial-circularity score of 6 rather than a lower score, because the Table 1 maximum-mass results are not themselves fitted to observed radii and the paper's algebraic solution is otherwise internally developed. Separately, there is a serious consistency flaw that I do not count as circularity: Eq. (8) defines B_eff with explicit μ dependence through Δ and m_s, but Eq. (34) substitutes the constant B_g without justification, dropping all interaction terms. This means the exact solution and the physical-viability checks are not demonstrably built from the claimed interacting EoS; however, that is a correctness risk in the derivation chain rather than an equivalence-by-construction, so it does not raise the circularity score further but should be addressed by the authors.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central construction rests on the Zhang-Mann IQM free energy, the Buchdahl-I ansatz, the linear f(Q) ansatz, and an unstated identification of B_eff with B_g. No new particles or fields are introduced.

free parameters (5)
  • Bag constant B_g = 70 MeV/fm^3
    Chosen in Section 3 from the stability window E_B < 930.4 MeV; used in all exact and TOV solutions.
  • Color superconductivity gap Delta = 100 MeV
    Chosen in Section 3 as a representative value; affects all three phases through the IQM free energy.
  • Strange quark mass m_s = 0, 50, 100 MeV in main analysis; 200 and 250 MeV in Table 2
    Main analysis fixes m_s at three values, but Table 2 varies m_s per compact object to bring the predicted radius close to the observed value.
  • f(Q) parameter alpha_0 = 10^-46 km^-2
    Chosen by hand in Section 8 following Maurya et al. [101]; the paper notes only that it is an arbitrary choice.
  • f(Q) parameter alpha_1 = -0.6
    Chosen by hand in Section 8 following Maurya et al. [101]; affects the field equations and the effective cosmological constant.
assumptions (5)
  • domain assumption The IQM free energy of eq. (1) with coefficients from refs. [83-85] describes quark matter phases.
    All three phases (2SC, 2SC+s, CFL) are obtained from this parametrized free energy, which is taken from the cited literature without independent derivation.
  • ad hoc to paper The Buchdahl-I metric ansatz e^{2lambda} = 2(1+chi r^2)/(2-chi r^2) is imposed as the interior solution.
    This ansatz is chosen in Section 5 to obtain closed-form solutions; it is not derived from the field equations or from microphysics.
  • ad hoc to paper A linear f(Q) action f(Q) = alpha_0 + alpha_1 Q is sufficient.
    Chosen in Section 5 as one branch of eq. (27), but Section 6 shows this reduces the theory to general relativity with a cosmological constant, so the modified gravity content is trivial.
  • ad hoc to paper The effective bag constant B_eff in eq. (8) can be replaced by the constant B_g in eq. (34).
    B_eff depends on chemical potential, but the exact solution uses constant B_g without stating or justifying the substitution; all later results depend on this step.
  • domain assumption The exterior junction conditions for f(Q) gravity from Wang et al. [87] apply, giving chi = -4M/(R^2(4M-3R)).
    The boundary matching in Section 6 is adopted from the cited literature and not re-derived here.

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Cite this review

Pith. "Pith review of Interacting quark matter and $f(Q)$ gravity: A new paradigm in exploring the properties of quark stars." pith.science (2026). https://pith.science/paper/PUOUCEUN

@misc{pith2026241200693,
  author       = {Pith},
  title        = {Pith review of: Interacting quark matter and $f(Q)$ gravity: A new paradigm in exploring the properties of quark stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PUOUCEUN}},
  note         = {Machine review of arXiv:2412.00693}
}
abstract

Perturbative Quantum Chromodynamics corrections and the colour superconductivity indicate that strongly interacting matter can manifest unique physical behaviours under extreme conditions. Motivated by this notion, we investigate the interior structure and properties of quark stars composed of interacting quark matter, which provides a comprehensive avenue to explore the strong interaction effects, within the framework of $f(Q)$ gravity. A unified equation of state is formulated to describe various phases of quark matter, including up-down quark matter $(2SC)$, strange quark matter $(2SC+s)$, and the Colour-Flavor Locked $(CFL)$ phase. By employing a systematic reparametrisation and rescaling, the number of degrees of freedom in the equation of state is significantly reduced. Utilising the Buchdahl-I metric ansatz and a linear $f(Q)$ functional form, $f(Q)=\alpha_{0}+\alpha_{1}Q$, we derive the exact solutions of the Einstein field equations in presence of the unified interacting quark matter equation of state. For the $2SC$ phase, we examine the properties of quark stars composed of up-down quark matter. For the $(2SC+s)$and $CFL$ phases, we incorporate the effects of a finite strange quark mass $(m_{s}\neq0)$. The Tolman-Oppenheimer-Volkoff equations are numerically solved to determine the maximum mass-radius relations for each phase. Our results indicate that the model satisfies key physical criteria, including causality, energy conditions, and stability requirements, ensuring the viability of the configurations. Furthermore, the predicted radii for certain compact star candidates align well with observational data. The study highlights that quark stars composed of interacting quark matter within the $f(Q)$ gravity framework provide a robust and physically consistent stellar model across all considered phases.

Figures

Figures reproduced from arXiv: 2412.00693 by the authors.

Figure 1
Figure 1. Energy per baryon, (EB) vs. colour superconductivity, (∆) for 2SC phase colour superconductivity (∆) in 2SC phase. It is observed that the udQM actually possesses a lower energy per baryon number in comparison to SQM, which aids to the higher stability of udQM. Further, from figure 2, we note the influence of strange quark mass (ms) on EB for 2SC + s and CF L phases respectively [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 2
Figure 2. Energy per baryon, (EB) vs. colour superconductivity, (∆) for (a) 2SC + s phase and (b) CF L phase. Here, the solid, dashed and dotdashed lines represent ms = 0, 50 and 100 MeV respectively. 4 Fundamentals of f(Q) gravity In the framework of symmetric teleparallel f(Q) gravity, the action integral is represented as follows: S = Z √ −gd4x " 1 2 f(Q) + λ kij l R l kij + τ ij k T k ij + Lm # , (10) Here, g represents t… view at source ↗
Figure 3
Figure 3. Mass-radius diagram for 2 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: Mass-radius diagram for (a) 2SC + s phase and (b) CF L phase. Here, the solid, dashed and dotdashed lines represent ms = 0, 50 and 100 MeV respectively. mass-radius plot for udQM matter in 2SC phase. Whereas, from figure 4, we note that with increasing strange quark ma…
Figure 5
Figure 5. Figure 5: Radial variation of energy density (ρ) for 2SC phase. 0 2 4 6 8 0.0004 0.0005 0.0006 0.0007 r HKmL Ρ HKm -2 L (a) 0 2 4 6 8 10 0.00030 0.00035 0.00040 0.00045 0.00050 0.00055 0.00060 r HKmL Ρ HKm -2 L (b) [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Radial variation of energy density (ρ) for (a) 2SC + s phase and (b) CF L phase. Here, the solid, dashed and dotdashed lines represent ms = 0, 50 and 100 MeV respectively. • Radial pressure: The radial pressure counterbalances the inward gravitational force which preve…
Figure 7
Figure 7. Figure 7: Radial variation of radial pressure (pr) for 2SC phase. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Radial variation of radial pressure (pr) for (a) 2SC + s phase and (b) CF L phase. Here, the solid, dashed and dotdashed lines represent ms = 0, 50 and 100 MeV respectively. • Tangential pressure: Tangential pressure (pt) actually maintains the spherical shape of the c…
Figure 9
Figure 9. Figure 9: Radial variation of tangential pressure ( [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: Radial variation of tangential pressure ( [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Radial variation of pressure anisotropy (∆ [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: Radial variation of pressure anisotropy (∆ [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: Radial variation of tangential sound velocity ( [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]
Figure 14
Figure 14. Figure 14: Radial variation of tangential sound velocity ( [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]
Figure 15
Figure 15. Figure 15: Radial variation of energy conditions for 2 [PITH_FULL_IMAGE:figures/full_fig_p016_15.png]
Figure 16
Figure 16. Figure 16: Radial variation of energy conditions for (a) 2 [PITH_FULL_IMAGE:figures/full_fig_p016_16.png]
Figure 17
Figure 17. Figure 17: Radial variation of different forces for 2 [PITH_FULL_IMAGE:figures/full_fig_p017_17.png]
Figure 18
Figure 18. Figure 18: Radial variation of different forces for (a) 2 [PITH_FULL_IMAGE:figures/full_fig_p017_18.png]
Figure 19
Figure 19. Figure 19: Radial variation of |v 2 t − v 2 r | for 2SC phase. 0 2 4 6 8 0.07 0.08 0.09 0.10 0.11 0.12 0.13 r HKmL Èvt 2-vr 2 È (a) 0 2 4 6 8 10 0.06 0.07 0.08 0.09 0.10 0.11 r HKmL Èvt 2-vr 2 È (b) [PITH_FULL_IMAGE:figures/full_fig_p018_19.png]
Figure 20
Figure 20. Figure 20: Radial variation of |v 2 t − v 2 r | for (a) 2SC + s phase and (b) CF L phase. Here, the solid, dashed and dotdashed lines represent ms = 0, 50 and 100 MeV respectively. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_20.png]
Figure 21
Figure 21. Figure 21: Radial variation of adiabatic index for 2 [PITH_FULL_IMAGE:figures/full_fig_p019_21.png]
Figure 22
Figure 22. Figure 22: Radial variation of adiabatic index for (a) 2 [PITH_FULL_IMAGE:figures/full_fig_p019_22.png]

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