CP violation in cold dense quark matter and axion effects on the non-radial oscillations of neutron stars

Deepak Kumar; Hiranmaya Mishra

REVIEW 2 major objections 5 minor 1 cited by

CP violation in cold dense quark matter and axion effects on the non-radial oscillations of neutron stars

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Axions could make neutron star cores ring measurably louder, this paper argues, by changing the quark matter equation of state.

desk verdict A technically sound NJL+RMF hybrid-star calculation whose headline axion effect rests entirely on θ=π, a point the paper's own effective potential identifies as a maximum. read the letter →

arxiv 2411.17828 v3 pith:6PDSP22X submitted 2024-11-26 hep-ph astro-ph.HEnucl-th

Deepak Kumar , Hiranmaya Mishra This is my paper · ORCID

classification hep-phastro-ph.HEnucl-th

keywords axionstrongCPviolationNambu-Jona-Lasiniomodelhybridneutronstarshadron-quarkphasetransitionGibbsconstructionf-modeoscillationshyperonicmatter

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 tries to establish that axion-induced CP violation in cold dense quark matter can change the hadron-quark phase transition enough to matter for neutron star structure and gravitational-wave asteroseismology. Within a three-flavor Nambu-Jona-Lasinio (NJL) model with the axion angle $\theta=\langle a\rangle/f_a$ coupled through the flavor-mixing determinant term, a nonzero $\theta$ lowers the critical chemical potential for chiral restoration, and at $\theta=\pi$ it shifts the onset of the hadron-quark mixed phase to lower baryon densities. Combining this quark-matter equation of state with a hyperonic relativistic mean-field hadronic equation of state through a Gibbs construction, the paper finds stable hybrid neutron stars with a pure quark matter core and a hyperonic outer core that satisfy modern NICER and GW170817 constraints, provided quark vector repulsion is included. The paper further claims that the quadrupolar fundamental oscillation mode ($f$-mode) of such hybrid stars is enhanced by roughly 110 Hz relative to a nucleonic star of the same mass, making the composition potentially audible in gravitational waves.

What carries the argument

The load-bearing object is the zero-temperature effective potential $\Omega(I_s^i,I_p^i,\theta,\mu)$ of the three-flavor Nambu-Jona-Lasinio model, whose flavor-mixing determinant interaction carries the axion phase $e^{\pm i\theta}$. It generates coexisting scalar and pseudoscalar quark condensates, whose jumps mark the first-order chiral transition, and its dependence on $\theta$ gives the axion potential: degenerate minima at $\theta=2n\pi$ and maxima at $\theta=(2n+1)\pi$. This potential supplies the quark-matter equation of state that is joined to a hyperonic relativistic mean-field hadronic equation of state by a Gibbs construction enforcing $\beta$ equilibrium and global charge neutrality; the resulting speed-of-sound profile feeds the Cowling-approximation equations for the non-radial $f$-mode oscillations.

What would settle it

Compute the minimum of the zero-temperature finite-density axion effective potential without fixing $\theta$; if the global minimum remains at $\theta=2n\pi$ for all neutron-star densities, the predicted equation-of-state softening and f-mode enhancements disappear. Observationally, a precise measurement of the f-mode frequency of a $\sim2\,M_\odot$ neutron star would test the predicted $\sim110$ Hz offset from a nucleonic star of the same mass, since the offset is the paper's concrete, mass-matched signature.

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Extended reading notes

Core claim

The central claim is that at $\theta=\pi$, where CP violation is maximal, the quark matter equation of state in the three-flavor NJL model with vector coupling $G_v=0.1\,G_s$ supports hybrid neutron stars with a maximum mass of $2.05\,M_\odot$, in agreement with the NICER measurement of PSR J0740+6620 and the GW170817 tidal constraints. The star has a pure quark matter core of radius $\sim 1.6$ km, quark matter in a mixed phase out to $\sim 6.5$ km, and a hyperonic outer core; without the vector term the same $\theta=\pi$ equation of state gives only $1.84\,M_\odot$ and fails the two-solar-mass constraint. The $\theta=\pi$ case also lowers the onset of the mixed phase from about $0.44$ fm$^{-3}$ (at $\theta=0$) to $0.35$ fm$^{-3}$ for $G_v=0.1\,G_s$. For the maximum-mass hybrid star, the paper computes an $f$-mode frequency near $2.1$ kHz, about $110$ Hz above a nucleonic star of the same mass, with hyperons and the enlarged quark core contributing separately to the shift.

Load-bearing premise

The load-bearing premise is that the axion angle can be held at $\theta=\pi$, where the paper's own effective potential has a maximum rather than a minimum; a dynamical axion would normally settle at $\theta=2n\pi$ and erase the reported equation-of-state and f-mode effects.

Editorial extensions

If this is right

If $\theta=\pi$ with $G_v=0.1\,G_s$, hybrid neutron stars with a pure quark matter core satisfy the $2\,M_\odot$ mass constraint; if $G_v=0$, they do not.
Axion presence lowers the onset density of the hadron-quark mixed phase, so quark matter can appear well below the hyperon threshold or coexist with hyperons in the same star.
The $f$-mode frequency shift (about 110 Hz for the $\theta=\pi$, $G_v=0.1\,G_s$ maximum-mass star) turns the composition of the core into a potentially observable gravitational-wave signature.
A nonmonotonic speed of sound with sharp drops at phase boundaries is a generic prediction of these Gibbs-constructed hybrid equations of state, linking the model to the broad behavior inferred from neutron star observations.

Reading between the lines

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

The paper's representative case $\theta=\pi$ is a maximum, not a minimum, of its own axion effective potential; a dynamical axion would relax to $\theta=2n\pi$, where the CP-violating effects vanish, so the reported signals require an unspecified mechanism to pin the angle.
Even if the 110 Hz enhancement were observed, it would not identify axions uniquely: the paper itself shows hyperons already raise $f$-mode frequencies by about 260 Hz for a $2.35\,M_\odot$ star, so mass-radius-tidal joint measurements would be needed to separate axionic quark cores from merely hyperonic stars.
One testable extension is to give the axion a finite relaxation time: if the dense-matter vacuum is metastable at $\theta=\pi$ on neutron-star timescales, the $f$-mode shift should depend on stellar age or formation history, which could be searched for in populations of young versus old pulsars.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is a clean, internally consistent model calculation: three-flavor NJL with axion coupling, hyperonic RMF hadronic EOS, Gibbs mixed phase, and Cowling-approximation f-modes. The new bit is combining the axionic 3-flavor NJL with hyperonic RMF in a Gibbs construction and pushing it through to oscillation frequencies. That combination is genuinely new, the numerics look careful, and the parameters are fitted to external low-energy data rather than to the target results, so there is no circular fitting. Credit where due: the gap equations, the Gibbs treatment, and the f-mode machinery are standard but competently executed, and the paper is upfront about the Cowling approximation's ~15-20% frequency overestimate. The soft spot is load-bearing. The central physical claim—that axions stabilize hybrid stars and boost f-modes—is computed at θ=π, yet the paper's own Fig. 7 shows the thermodynamic potential has degenerate minima at θ=2nπ and maxima at θ=(2n+1)π, for every chemical potential studied. The text explicitly says the potential is still a maximum at θ=π. For a dynamical QCD axion, the field relaxes to the minimum, θ=0 mod 2π, where the reported effects vanish. The paper cites the experimental bound θ<0.7×10⁻¹¹ in the introduction, which makes θ=π an even harder sell. No mechanism—pinning, domain walls, boundary conditions, or a density-induced shift of the minimum—is proposed. So the abstract's claim about axions producing stable hybrid stars with pure quark cores is conditional on a hand-imposed, non-equilibrium angle. The calculation is fine as a parametric study of θ-vacuum effects; it is not fine as a statement about physical axions in neutron stars. Minor issues: no code or data shipped, the error assessment is a literature estimate, and the 15-20% Cowling systematic could affect the magnitude of the 110 Hz enhancement, though less so the model-to-model difference. Who gets value from this: people working on θ-vacuum effects in dense QCD and on hybrid-star asteroseismology as a model exercise. It deserves a serious referee, but the referee should demand either a dynamical story for θ=π or a reframing of the results as θ-vacuum rather than axion effects. My own verdict is skeptical: the physical claim is not supported as stated, but the model machinery is usable and the paper is honest about its limitations.

Referee Report

2 major / 5 minor

Summary. This paper studies the effects of a CP-violating axion angle theta on cold dense quark matter within a three-flavor Nambu-Jona-Lasinio model, and uses the resulting equation of state, combined with a relativistic mean-field hadronic EOS via a Gibbs construction, to compute hybrid neutron star mass-radius relations and quadrupolar f-mode frequencies in the Cowling approximation. The main claims are that theta=pi lowers the chiral transition density, allows a pure quark matter core in stable hybrid stars when the vector coupling Gv=0.1 Gs, increases the maximum mass to 2.05 solar masses, and enhances f-mode frequencies by about 110 Hz relative to nucleonic stars.

Significance. If the theta=pi input were physically justified, the paper would provide a plausible model exploration of how axion-induced CP violation could alter hybrid star structure and oscillation signatures. The gap equations and thermodynamic potential treatment are standard and carefully reduced to zero temperature, and the Gibbs construction and Cowling f-mode formalism are applied consistently. The paper also makes a clear comparison with NICER and GW170817 mass-radius constraints. However, the central physical claim depends entirely on the axion field sitting at a maximum of its own effective potential, which no mechanism in the paper stabilizes. Since a dynamical QCD axion would relax to theta=0, where the reported effects largely disappear, the significance for axion physics is currently not established.

major comments (2)

[Sec. VI A, Fig. 7; Sec. VI C] The paper's own effective potential, shown in Fig. 7 and discussed in Sec. VI A, has degenerate minima at theta=2n pi and maxima at theta=(2n+1) pi; the text explicitly states that at theta=(2n+1) pi 'the thermodynamic potential is still a maximum.' All axion-dependent results—the EOS in Fig. 8, the mass-radius curves and core profiles in Figs. 9 and 10, and the f-mode frequencies in Fig. 12—are computed at theta=pi. For the QCD axion, which the paper identifies with theta=<a>/f_a and which is designed to relax to CP-conserving values, the field would settle at a minimum, theta=0 mod 2 pi, not at the maximum theta=pi. No pinning mechanism, domain-wall stabilization, or density-induced shift of the minimum is provided. Given that the introduction itself cites the experimental bound theta<0.7 x 10^-11, the representative choice theta=pi is both dynamically and phenomenologically unjustified, and the central claim of axion-induced stabilization of hybrid stars with pure quark cores collapses unless a mechanism to maintain theta=pi is supplied.
[Abstract; Sec. VI C, Figs. 3(b), 12] The claimed axion-specific enhancements are not robust under a dynamically selected theta. At theta=0 with Gv=0.1 Gs, the model does not produce a pure quark matter core (Fig. 3(b)), and the f-mode enhancement relative to a nucleonic star is 90 Hz rather than 110 Hz (Fig. 12). The difference between theta=0 and theta=pi is therefore not only quantitatively modest but also rests on the unphysical theta=pi choice. The abstract's statement that 'with the presence of axions, it is possible to have stable hybrid neutron stars having an inner core of quark matter both in pure quark matter phase as well as in a mixed phase with hyperonic matter' is thus conditional on an input that the paper's own dynamics rule out; as it stands, it overstates what the calculation establishes.

minor comments (5)

[Sec. II, Eq. (14)] The text contains 'we we have introduced'; please correct to 'we have introduced'.
[Sec. VI A, figures captions] The in-text references to 'Fig 5 b' and 'Fig 5 c' in the discussion of the strange quark condensates are mislabeled; they should refer to Figs. 6(c) and 6(d).
[Sec. VI C] The sentence 'the maximum mass becomes 2.05 M_sun ans satisfies the maximum mass constraint' contains a typo: 'ans' should be 'and'.
[Fig. 9 caption] The caption describes 'The dark gray and light gray region here correspond to 50% and 90% confidence interval', while the figure legend states '90% (solid) and 50% (dashed)'; please align the caption with the legend.
[Sec. VI C, inner crust] The claim that the inner-crust polytrope treatment yields radius differences of less than 0.5 km compared to a unified description is stated without a quantitative reference; please cite the specific figure or table from Ref. [69].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: model parameters are fitted to vacuum meson data and hyperon potentials, while the f-mode frequencies and maximum masses are computed outputs rather than fitted inputs.

full rationale

The derivation chain is self-contained and does not reduce to its inputs by construction. The NJL parameters (Λ = 602.3 MeV, GsΛ² = 1.835, KΛ⁵ = 12.36, mu = md = 5.5 MeV, ms = 140.7 MeV) are fixed in Section II by fitting the pion decay constant and the masses of pion, kaon, and eta-prime; the hadronic NL3ωρ parameters are taken from Ref. [59], and hyperon couplings are set by hyperon potential depths. None of these targets are the f-mode frequencies or the 2.05 solar-mass maximum that the paper reports. The f-mode frequencies are obtained by numerically solving the coupled differential equations (60)-(61) with the boundary condition (63) for each EOS, so the quoted 110 Hz enhancement relative to a nucleonic star of the same mass is a computed output. Self-citations (Refs. [29,30,36,41]) provide the NJL thermodynamic potential and the Cowling-approximation oscillation formalism, but the relevant equations are displayed in full in this paper and the cited works are not invoked to exclude alternatives. The one substantive concern is that Section VI A states the thermodynamic potential has maxima at θ = (2n+1)π, while the axion-dependent results are computed at θ = π; however, this is a physical-consistency (and possibly dynamical-relaxation) concern about whether a QCD axion would sit at a maximum, not a circularity. The paper does not fit θ to the reported frequencies or masses, and it explicitly compares with θ = 0 results to show the dependence of the outputs on the chosen input.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The central results depend on the NJL model parameters fitted to vacuum meson masses, on the hand-picked vector coupling Gv=0.1 Gs, and on the imposed axion angle theta=pi. The most consequential input is theta=pi, which is not the equilibrium value of the axion effective potential computed in the same paper. No new particles or forces are introduced; the axion is the standard QCD axion and theta is a parameter.

free parameters (8)

NJL cutoff Lambda = 602.3 MeV
Chosen to reproduce vacuum meson masses in the Rehberg parameter set.
Scalar coupling Gs = Gs Lambda^2 = 1.835
Fitted to pion decay constant and meson masses.
KMT coupling K = K Lambda^5 = 12.36
Fitted to eta-eta' mass splitting.
Current quark masses mu=md, ms = 5.5 MeV, 140.7 MeV
Inputs from the parameter set fitted to the meson spectrum.
Vector coupling Gv = 0.1 Gs, also 0
Chosen by hand; the value 0.1 Gs is needed to make the theta=pi hybrid star reach 2.05 solar masses and satisfy the maximum mass constraint.
Axion angle theta = pi, with 0 and pi/2 for comparison
Set to maximize CP-violating effects. Theta=pi is a maximum of the axion effective potential, not the equilibrium value.
Hyperon potential depths U_Lambda, U_Sigma, U_Xi = -28 MeV, +30 MeV, -18 MeV
Inputs used to fix scalar hyperon couplings in the RMF model, following Refs. [57,58].
Inner crust polytrope constants a,b = determined by matching BPS and core EOS
Used to connect the outer crust and outer core, following the standard procedure of Ref. [68].

assumptions (6)

domain assumption The 3-flavor NJL model with KMT determinant is a valid effective description of low-energy QCD at finite density.
Used throughout Section II to compute the axion potential and quark equation of state; not derived from QCD.
domain assumption The axion field can be replaced by its vacuum expectation value theta = <a>/fa with fa around 10^15 GeV.
Stated in Section II after Eq. (2); ignores axion dynamics and fluctuations.
ad hoc to paper The CP-violating angle theta=pi is a physically relevant configuration for neutron star matter.
All axion effects are evaluated at theta=pi, yet Fig. 7 shows theta=pi is a maximum of the effective potential, not a minimum.
domain assumption The hadron-quark transition is described by Gibbs construction with globally conserved charge.
Section IV, Eqs. (49)-(53); requires a small surface tension, which is plausible but not established.
domain assumption Non-radial f-modes can be computed in the Cowling approximation with errors no more than 15-20%.
Section V and Section VI C; metric perturbations are ignored, and the error estimate is taken from earlier literature.
domain assumption Beta equilibrium and charge neutrality hold in neutron star matter.
Imposed in Sections II and III to construct the equation of state, appropriate for cold isolated neutron stars.

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Pith. "Pith review of CP violation in cold dense quark matter and axion effects on the non-radial oscillations of neutron stars." pith.science (2026). https://pith.science/paper/6PDSP22X

@misc{pith2026241117828,
  author       = {Pith},
  title        = {Pith review of: CP violation in cold dense quark matter and axion effects on the non-radial oscillations of neutron stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6PDSP22X}},
  note         = {Machine review of arXiv:2411.17828}
}

abstract

Charge-conjugation and parity violation in strong interaction for cold dense quark matter is studied with axions of quantum chromodynamic within the three flavor Nambu--Jona-Lasinio model that includes the coupling of axions to quarks. We first calculate the effective potential for axions at finite baryon density and zero temperature including the effects of a first order chiral phase transition. Using the equation of state for quark matter with axions and a hadronic matter equation of state in the ambit of a relativistic mean field theory in quantum hadrodynamics, we discuss the hadron-quark phase transition. Inclusion of axions reduces the critical density for chiral transition. We use a Gibbs construct for the hadron-quark phase transition satisfying the constraints of beta equilibrium and charge neutrality as appropriate for the neutron star matter. The equation of state so obtained is used to investigate the structure of hybrid neutron stars. It is found that with the presence of axions, it is possible to have stable hybrid neutron stars having an inner core of quark matter both in pure quark matter phase as well as in a mixed phase with hyperonic matter along with a outer core of hyperonic matter and is in agreement with modern astrophysical constraints. We also discuss the properties of non-radial oscillations of such hybrid neutron stars. It is observed that the quadrupolar fundamental modes ($f$-modes) for such hybrid neutron stars get substantial enhancements both due to a larger quark core in the presence of axions and from the hyperons as compared to a canonical nucleonic neutron stars.

Figures

Figures reproduced from arXiv: 2411.17828 by the authors.

**Figure 2.** FIG. 2: Population densities of different species for charge neutral hypronic matter as a function of baryon density. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** FIG. 3: Upper panel: Quark matter fraction as a function of baryon density in Gibbs construction for the mixed [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4: Upper panel: The variations of normalized scalar condensate, [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5: : Different contribution to the constituent quark masses as a function of quark chemical potential [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: FIG. 6: Upper panel: The variations of scalar condensate, [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: FIG. 7: Axion potential at finite density. For various quark chemical potentials, the dashed line corresponds to [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: FIG. 8: Fig. 8a :The variation of pressure as a function of energy density with and without phase transitions i.e. [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: FIG. 9: The mass radius curves for different EOS depicted in the FIG. [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 10.** Figure 10: FIG. 10: The profiles of the energy density [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

**Figure 11.** Figure 11: FIG. 11: The variation of speed of sound inside a compact star as a function of radial distance from the center of a [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

**Figure 12.** Figure 12: FIG. 12: The frequencies of quadruple fundamental modes ( [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]

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Reviewed August 12, 2026 · model on record in the stance chip above.

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