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arxiv: 2606.29576 · v1 · pith:UC6FGVD5new · submitted 2026-06-28 · ⚛️ nucl-th · astro-ph.HE· hep-th

Quark and hybrid stars with renormalization group improvement of NNLO perturbative QCD

Lo\"ic Fernandez , Jean-Lo\"ic Kneur , Marcus Benghi Pinto , Constan\c{c}a Provid\^encia , Claudia Ratti , Tulio E. Restrepo This is my paper

Pith reviewed 2026-06-30 01:46 UTC · model grok-4.3

classification ⚛️ nucl-th astro-ph.HEhep-th
keywords quark starshybrid starsperturbative QCDequation of statebeta equilibriumrenormalization group
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0 comments X

The pith

RGOPT-improved NNLO pQCD pressure supports pure quark stars for X=3.08-3.58 and hybrid stars for X=2-2.98.

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

The paper extends an RGOPT-resummed version of NNLO perturbative QCD pressure from symmetric matter to beta equilibrium and charge neutrality, then supplies fitting formulas for the resulting equation of state at different renormalization scale parameters X. Using these equations of state the authors construct models of pure quark stars and of hybrid stars that combine a hadronic outer layer with a quark core. They identify specific intervals of X that produce stars whose masses are compatible with observed pulsars such as PSR J0740+6620 and with gravitational-wave constraints, thereby showing that the improved pressure can be used directly for compact-star phenomenology.

Core claim

The RGOPT framework supplies renormalization-group properties to the NNLO pQCD pressure of cold dense quark matter with arbitrary masses; when this pressure is extended to beta equilibrium and charge neutrality the resulting equation of state permits stable pure quark stars for renormalization-scale parameters X in the range 3.08-3.58 and stable hybrid stars with quark cores for X of order 2 to 2.98, the largest values in that interval producing cores 5-8 km in radius.

What carries the argument

The RGOPT-resummed NNLO pQCD pressure extended to beta equilibrium and charge neutrality, which supplies the equation of state for the stellar models.

If this is right

  • Pure quark stars consistent with astrophysical observations exist for X=3.08-3.58.
  • Stable hybrid stars matching the mass of PSR J0740+6620 exist for X of order 2 to 2.98.
  • The largest compatible X values produce the largest quark cores, with radii 5-8 km.
  • If the low-mass object in GW190814 is a neutron star, pure quark stars require the higher value X=4.10.

Where Pith is reading between the lines

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

  • Radius measurements of pulsars could further narrow the allowed window of X.
  • The same fitting formulas could be inserted into merger simulations to predict gravitational-wave signals from quark-matter phases.
  • If future lattice or functional QCD calculations confirm the RGOPT pressure at intermediate densities, the X intervals found here would become direct constraints on the strong coupling in dense matter.

Load-bearing premise

The RGOPT-improved NNLO pQCD pressure remains a reliable description of the true equation of state once it is extended to beta equilibrium and charge neutrality at the chemical potentials inside compact stars.

What would settle it

An observed compact star whose mass and radius lie outside the ranges produced by the EoS at X=3.08-3.58 for pure quark stars or at X=2-2.98 for hybrid stars would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.29576 by Claudia Ratti, Constan\c{c}a Provid\^encia, Jean-Lo\"ic Kneur, Lo\"ic Fernandez, Marcus Benghi Pinto, Tulio E. Restrepo.

Figure 1
Figure 1. Figure 1: FIG. 1. Feynman graphs contributing to the (infrared safe) NNLO weak coupling expansion. In our case, both matter and [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. NNLO Feynman graphs involving infrared divergences requiring resummations, as indicated in the rightmost graph. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Normalized pressure as a function of the baryon chemical potential [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Pressure versus baryon chemical potential (left) and energy density (right) of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Mass-radius relation of QSs obtained with NNLO pQCD (dash-dotted lines) and RGOPT (continuous lines). Lines [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Normalized pressure, [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Pressure (top) and speed of sound (bottom) as functions of the normalized density [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Hybrid NSs using NNLO RGOPT and NNLO pQCD to describe the quark degrees of freedom for the star core [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Normalized baryon density as a function of the internal radius, [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
read the original abstract

Recently, the NNLO perturbative QCD pressure of cold and dense symmetric matter, with arbitrary quark masses, has been resummed within the renormalization-group-optimized perturbation theory (RGOPT) framework. By being imbued with renormalization group properties, the resulting pressure is less sensitive to renormalization scale ($\Lambda\equiv X \mu_B/3$) variations than the NNLO perturbative QCD pressure. Here, we extend this by considering $\beta$-equilibrium and charge neutrality to evaluate the corresponding equation of state (EoS). We provide a compact ``pocket" fitting formula for the EoS for $N_f=2+1$ massive quarks at different renormalization scale parameter ($X$) values. We describe pure quark stars as well as hybrid stars with quark-cores. Pure quark stars compatible with astrophysical observations were obtained with $X=3.08-3.58$, whereas a larger value (4.10) is needed if the low mass object of the observation GW190814 represents a neutron star. Hybrid stars were built considering three representative hadron models based on a relativistic mean-field description, and chosen to produce soft and stiff EoSs. Stable hybrid stars with masses compatible with the massive pulsar PSR J0740+6620 were obtained considering $X$ of the order of 2 to 2.60-2.98, the largest scale giving rise to hybrid stars with a large quark core with a radius of 5 to 8 km, and the smallest to a small quark core at the center of the star.

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

1 major / 1 minor

Summary. The manuscript extends the RGOPT-resummed NNLO pQCD pressure for cold dense symmetric quark matter (with arbitrary quark masses) to beta equilibrium and charge neutrality. It supplies an explicit compact fitting formula for the resulting EoS of Nf=2+1 massive quarks at several fixed values of the renormalization-scale parameter X ≡ Λ/(μB/3). Using this EoS, the authors construct pure quark stars and hybrid stars matched to three representative relativistic mean-field hadronic models (soft and stiff), and report the X intervals that yield stellar masses and radii compatible with PSR J0740+6620 and the low-mass object in GW190814.

Significance. If the RGOPT improvement genuinely reduces renormalization-scale dependence while remaining reliable at the chemical potentials inside compact stars, the work supplies a practical, scale-improved perturbative EoS together with a ready-to-use fitting formula that can be inserted directly into TOV integrators. The explicit mapping of viable X ranges to observed pulsar masses is a concrete, falsifiable output of the approach.

major comments (1)
  1. [Abstract] Abstract (paragraph on the extension to beta equilibrium and charge neutrality): the central claim that specific X intervals produce observationally compatible stars rests on the unverified step of extending the symmetric-matter RGOPT pressure to asymmetric matter; the manuscript supplies neither the explicit derivation, error estimates from the pQCD truncation, nor cross-checks against lattice data at finite isospin asymmetry.
minor comments (1)
  1. The fitting formula is described as a 'pocket' formula but its explicit functional form, coefficients, and domain of validity in chemical potential are not reproduced even in the abstract; providing the formula itself would increase usability.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful reading of the manuscript and the constructive feedback. We address the single major comment below and will make targeted revisions to improve clarity on the extension procedure while noting limitations inherent to the field.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on the extension to beta equilibrium and charge neutrality): the central claim that specific X intervals produce observationally compatible stars rests on the unverified step of extending the symmetric-matter RGOPT pressure to asymmetric matter; the manuscript supplies neither the explicit derivation, error estimates from the pQCD truncation, nor cross-checks against lattice data at finite isospin asymmetry.

    Authors: The RGOPT pressure functional (derived in the preceding symmetric-matter work) is already expressed in terms of independent quark chemical potentials and masses. The extension to β-equilibrium and charge neutrality consists of imposing the standard relations μ_d = μ_u + μ_e, μ_s = μ_u + μ_e together with the charge-neutrality constraint on the number densities obtained from the same pressure; the resulting EoS is then fitted at fixed X. We agree that an explicit derivation of this step was not sufficiently detailed and will add a dedicated subsection (including the explicit fitting formula and the numerical procedure for solving the neutrality conditions) in the revised manuscript. On truncation errors, the RGOPT resummation was constructed precisely to tame the dominant renormalization-scale uncertainty of NNLO pQCD; we will expand the discussion of residual truncation uncertainty by comparing neighboring perturbative orders where available. Cross-checks against lattice QCD at finite baryon density are not feasible at present. revision: partial

standing simulated objections not resolved
  • Cross-checks against lattice data at finite isospin asymmetry for cold, dense matter cannot be performed because of the fermion sign problem; no such lattice results exist.

Circularity Check

0 steps flagged

No significant circularity; X ranges are compatibility constraints on a free renormalization parameter

full rationale

The paper derives the EoS from RGOPT-resummed NNLO pQCD extended to beta equilibrium and charge neutrality, supplies an explicit fitting formula, and then reports the numerical X intervals for which the resulting TOV solutions match observed masses. X is introduced as the renormalization-scale prefactor λ ≡ X μ_B/3 and is varied parametrically; the reported intervals (e.g., X=3.08-3.58 for pure quark stars) are the direct numerical output of that variation, not a quantity defined by or fitted inside the derivation itself. No self-citation, uniqueness theorem, or ansatz is invoked to force the central result. The procedure is therefore self-contained against external benchmarks (the cited pulsar masses) and does not reduce any claimed prediction to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the prior RGOPT pressure result, the standard assumptions of beta equilibrium and charge neutrality, and the choice of X to achieve observational compatibility; no new particles or forces are postulated.

free parameters (1)
  • X = 2 to 4.1
    Renormalization scale parameter varied across 2-4.1 to obtain star models compatible with observed masses
axioms (2)
  • domain assumption beta equilibrium and charge neutrality
    Standard for modeling neutron-star interiors
  • domain assumption RGOPT-resummed NNLO pQCD remains applicable at the relevant densities
    Inherited from the recent prior work being extended

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discussion (0)

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Reference graph

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