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REVIEW 4 major objections 5 minor 54 references

Neutron star in Logarithmic model of Cartan $F(R)$ gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read In logarithmic Cartan $F(R)$ gravity, the scalaron field raises neutron-star masses, lifting the minimum mass toward one solar mass for a potential scale near 80 MeV, which matches the observed absence of lighter neutron stars.

desk verdict A competent but flawed application: the claimed effect comes from parameters that make the exterior an asymptotically de Sitter space, not Schwarzschild, so the central result is not about isolated neutron stars. read the letter →

arxiv 2412.03219 v3 pith:467WYY2O submitted 2024-12-04 gr-qc

Hiroki Sakamoto , Masahiko Taniguchi This is my paper · ORCID
classification gr-qc PACS 04.50.Kd97.60.Jd
keywords CartanF(R)gravityscalar-tensortheoryneutronstarmass-radiusrelationTolman-Oppenheimer-Volkoffequationfour-fermioninteractionchiralsymmetrybreakinglogarithmicmodelminimummass
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

This paper asks whether a scalar field that appears in a specific extension of general relativity, Cartan $F(R)$ gravity, can explain why no neutron stars lighter than about one solar mass have been observed, even though standard equations of state allow stars down to roughly $0.2\,M_\odot$. The authors treat the scalaron field as a dynamical gravitational degree of freedom, couple it to neutrons through a four-fermion interaction, and integrate out the fermions to obtain an effective potential with a chiral-symmetry-breaking minimum. Solving the modified Tolman-Oppenheimer-Volkoff equations with the SLy and APR equations of state, they find that the scalar field increases neutron-star masses: the minimum mass rises to roughly $0.7\,M_\odot$ at a potential scale $U_1^{1/4}=78$ MeV and above $1\,M_\odot$ at 92 MeV, which the authors read as consistent with the observed floor near one solar mass. The paper thus proposes a modified-gravity explanation for the neutron-star mass gap, provided a new chiral condensate at the 100$-$1000 TeV scale exists. If the effect is real, standard mass-radius predictions for low-mass neutron stars need revision.

What carries the argument

The load-bearing object is the logarithmic Cartan $F(R)$ model, $F(R)=R-\alpha R\ln(1+R/R_0)$, whose modified Cartan equation replaces the torsion degree of freedom with a scalar field $\phi$ (the scalaron) and a four-fermion interaction. The paper converts the four-fermion term into an NJL-type interaction, introduces auxiliary fields, integrates out the fermions with a cutoff regulator, and obtains the effective potential $U(\phi) = -U_1\phi/M_{\rm Pl} + U_2(2e^{\sqrt{2/3}\,\phi/M_{\rm Pl}} - e^{2\sqrt{2/3}\,\phi/M_{\rm Pl}})$, which has a minimum at negative $\phi$ when $U_2 \gtrsim U_1$. This potential supplies the scalar field's energy density and pressure, plus a scalar-field equation, which are added to the Tolman-Oppenheimer-Volkoff equations and solved numerically with the SLy and APR equations of state; the boundary condition places $\phi$ near the potential minimum at the stellar surface.

What would settle it

A confirmed neutron star with mass below $\sim0.7\,M_\odot$ would contradict the paper's favored parameter set, where the minimum mass is $0.71\,M_\odot$ for both SLy and APR equations of state; standard equations of state without the scalar field predict the minimum near $0.2\,M_\odot$. Alternatively, a definitive demonstration that no chiral condensation occurs between 100 and 1000 TeV would remove the input that gives the potential its minimum.

Watch

Extended reading notes

Core claim

The paper's central claim is that in logarithmic Cartan $F(R)$ gravity, the scalar degree of freedom couples to spinor matter through a four-fermion interaction, and after chiral symmetry breaking the resulting effective potential acts as an extra source term that increases the mass of a neutron star. Solving the modified TOV equations with the SLy and APR equations of state, the authors find that the mass-radius curves shift upward compared with unmodified general relativity. With the linear-potential parameter $\tilde U_1 = 5\times10^{-2}$, corresponding to $U_1^{1/4}\simeq78$ MeV, the minimum neutron-star mass increases from about $0.2\,M_\odot$ to $0.71\,M_\odot$ for both equations of state; larger values push the minimum above one solar mass and become disfavored by observation. The paper concludes that a potential scale near 80 MeV, together with a chiral condensate at 100$-$1000 TeV, can explain why observed neutron stars are all near or above one solar mass.

Load-bearing premise

The whole effect depends on an assumed fermion condensate at a scale of 100$-$1000 TeV and a potential strength $U_1^{1/4}$ of a few tens of MeV; neither is derived from the theory or observed, and the paper states that without the condensate the scalar-field effect disappears.

Editorial extensions

If this is right

  • At the favored parameter range ($U_1^{1/4}\simeq78$ MeV), the predicted minimum neutron-star mass is $\sim0.7\,M_\odot$, up from $\sim0.2\,M_\odot$ in unmodified general relativity with the same equations of state.
  • Values above $\sim90$ MeV push the minimum mass above the observed range, so the neutron-star mass gap becomes a constraint on the model parameter $U_1$.
  • The maximum mass also grows moderately with $U_1$ (for SLy, from 2.05 to 2.29 $M_\odot$ over the scanned range), so the mechanism can simultaneously affect the upper end of the mass distribution.
  • The mass-increasing effect requires the chiral-condensate term in the potential; with only the linear potential from the logarithmic model, the scalar-field contribution is negligible and would fade with time.
  • The scalar field's mass at the potential minimum is of order MeV, making it a possible dark-matter candidate, a direction the paper explicitly leaves for future work.

Reading between the lines

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

  • A direct observational test is implied: if a neutron star with mass below about $0.7\,M_\odot$ and radius larger than about 12 km is found, it would contradict the paper's favored parameter set, whereas standard equations of state allow such objects.
  • The 100$-$1000 TeV chiral condensate is put in by hand and is far above any observed scale; a negative search for new TeV-scale strong dynamics would remove the input that makes the potential minimum exist, even if the gravity side of the calculation is correct.
  • The paper scans $U_1$ and selects the range that matches the observed mass gap; if the logarithmic-model parameter $\alpha$ is pinned down independently from inflation observations, $U_1$ becomes a prediction and the neutron-star mass floor becomes a test rather than a fit.
  • The same modified-TOV machinery could be applied to other equations of state or to tidal-deformability measurements from gravitational-wave events, which would sharpen or shift the allowed window of $U_1$.
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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 paper studies neutron stars in a Cartan F(R) gravity model, working in the equivalent scalar-tensor description with a scalar field (scalaron) whose effective potential is derived from a four-fermion interaction using NJL-type auxiliary-field methods. The authors integrate the modified Tolman-Oppenheimer-Volkoff equations with SLy and APR equations of state and obtain mass-radius relations. They report that the scalar field increases the neutron star mass, and for a potential scale U1^(1/4) near 80 MeV the minimum neutron star mass rises to about one solar mass, which they claim is consistent with the absence of observed neutron stars below ~1 Msun.

Significance. If the central claim were sound, the paper would offer a concrete astrophysical target for Cartan F(R) gravity and a possible explanation for the neutron star mass gap. The numerical work is performed carefully with realistic equations of state, and the internal computation appears consistent for the equations stated. The paper, however, does not deliver an independent prediction: the two potential parameters are scanned by hand, and the observed minimum mass is used to select the preferred range. More importantly, the exterior matching to Schwarzschild is inconsistent with the nonzero potential minimum at the parameters that produce the claimed effect. The manuscript also states explicitly that the effect disappears without a tuned TeV-scale fermion condensate, a new physics ingredient that is neither derived nor observed. These issues undermine the central claim rather than being mere presentation problems.

major comments (4)
  1. [Sec. 5 / Eq. (4.16)] For the central parameter set \tilde U1 = 5e-2, \tilde U2 = 5e-1.4, the potential minimum at ϕ_min ≈ -2.035 gives U(ϕ_min) ≈ 0.17 ρ0 c², i.e., about 0.17 times nuclear saturation energy density. Since the potential does not vanish at its minimum, the exterior vacuum contains a positive effective cosmological constant of this order. The unique asymptotically flat vacuum solution does not exist; the exterior form of Eqs. (4.12)–(4.15) does not admit h→1 and M→constant as s→∞, and matching to Schwarzschild at the surface imposes a boundary condition that the equations do not satisfy. The same issue affects all Table 1 entries, whose potential minima are many orders of magnitude above the observed dark-energy density. Consequently, the mass-radius curves in Figs. 3 and 5, including the rise of M_min to ~1 Msun, are not curves for an isolated neutron star in asymptotically flat spacetime, and the central claim is not supported.
  2. [Sec. 3 / Eq. (3.1)] The Fierz transformation is stated without derivation, and the sign of the (ψ̄ψ)² term is crucial because it determines whether the effective coupling λ = λ0 + Ω can exceed the critical value needed for chiral symmetry breaking. The last term in Eq. (3.1) is dropped by assuming that ⟨ψ̄γ^μψ⟩ vanishes, but the paper gives no justification for this assumption in a dense neutron star medium, where Lorentz invariance is broken by the background. Since the entire effective potential (3.8) is built on this identity and this truncation, the scalar potential used in the TOV calculation is not established on a firm basis.
  3. [Sec. 5 / Table 1] The parameters \tilde U1 and \tilde U2 are scanned by hand (U1^(1/4) = 52, 78, 92, 102 MeV), and the observed minimum neutron star mass is then used to select U1^(1/4) ≈ 80 MeV as consistent with observations. This is parameter fitting rather than an independent prediction. The text itself states that a chiral condensate at the 100–1000 TeV scale is required for the effect to appear and that without this tuned condensate the effect disappears; that new ingredient is neither derived from the model nor independently observed. The claimed agreement with the observed mass gap is therefore a restatement of the chosen parameters, not a success of the theory.
  4. [Sec. 4 / Sec. 5] Even setting aside the exterior matching problem, the parameter values that produce the claimed effect imply a vacuum energy density at the potential minimum of order 0.1 ρ0 c², about 10^36 times the observed dark-energy density. The paper does not explain how such a large vacuum energy avoids catastrophic consequences for cosmology or for the spacetime outside the star. No screening mechanism is discussed, and the model is not shown to be compatible with standard cosmological observations for any of the parameter sets in Table 1.
minor comments (5)
  1. [Throughout] There are several typographical slips, including 'neuron star' in Sec. 4, 'Fiertz' for 'Fierz' in Sec. 3, and 'T olman' in the Introduction; these should be corrected.
  2. [Eq. (4.11)] The coefficients a1 to a18 for the SLy and APR equations of state are not given; without these coefficients or a clear reference to the tabulated values, the numerical results are not reproducible.
  3. [Eq. (4.16) vs Eq. (3.8)] Eq. (4.16) omits the (1+Π)^(1/2) factors that appear in Eq. (3.8). If Π is negligible, as the text suggests, this should be stated explicitly at the point where the dimensionless potential is introduced.
  4. [Eq. (4.17)] The conversion from \tilde U1 to U1^(1/4) in MeV is not shown explicitly; the expression in Eq. (4.17) mixes powers of eV and is difficult to follow. A direct formula connecting \tilde U1 and the reported MeV values in Table 1 would improve clarity.
  5. [References] Reference [48] is cited for the APR equation of state, but the given source is a paper on magnetic neutron star cooling; the original APR EoS reference (Akmal, Pandharipande, Ravenhall 1998) appears to be missing.

Circularity Check

2 steps flagged · score 5.0 of 10

The claimed consistency with the observed ~1 solar-mass minimum is a parameter fit: U1 is scanned by hand and the observation selects the allowed range, so it is not an independent prediction.

  1. fitted input called prediction [Sec. 5, around Figure 5/Table 1 and the paragraph following Table 1]
    "The main contribution in potential is parameter ˜U1. Therefore, we numerically analyze the neutron star for several values of ˜U1. ... Conversely, for scales above 90 MeV the minimum mass exceeds the solar mass and is inconsistent with observations. Therefore, for scales below 80 MeV, the Cartan F(R) gravity effect on neutron stars is consistent with observations. In particular, for scales near 80 MeV, the theoretical limits are close to observational constraints."

    U1 is a free parameter of the model: the theory does not fix it to the 80 MeV scale, and the dark-energy scale value gives negligible effects. The paper scans U1 and then uses the observed absence of neutron stars below about 1 M⊙ to select the range U1^(1/4) ≈ 78–92 MeV as 'consistent'. This is using the same observation to fit the parameter and then citing agreement with that observation as support. The quantitative conclusion is therefore a selection effect rather than a prediction.

  2. ansatz smuggled in via citation [Sec. 2.1, Eqs. (2.15)–(2.18)]
    "we focus on the logarithmic model [31] defined by f(R) = −αR ln(1 + R/R0) ... In Ref. [31], the accelerated expansion in the inflation and dark energy era can be explained simultaneously in this model. To obtain the suitable CMB observables we relate the parameter R0 to the inflation ... Thus, we adopt the potential in Eq.(2.18)."

    The logarithmic potential and the inflation-parameter relation on which the whole neutron-star calculation relies are imported from the authors' own prior paper [31] (Inagaki and Taniguchi), not derived or independently verified here. The citation provides no external evidence because that prior work itself postulates the same logarithmic Cartan F(R) ansatz. The central model choice is therefore carried by a self-citation chain, although it is an explicitly stated model assumption rather than a disguised one.

full rationale

Most of the derivation chain—Cartan F(R) to scalar-tensor form, the NJL-type effective potential, and the TOV equations—is algebraically self-contained and is not circular: the mass-radius curves are obtained by numerical solution of the stated ODEs, not read off from the target observable. However, the paper's central quantitative claim of consistency with the observed lower mass bound is a fitted-input-called-prediction. U1 is scanned by hand, U2 is explicitly tuned so that ϕ_min ≈ −2, and then the observed 1 M⊙ minimum is used to select the 'allowed' U1 range. The abstract's statement that the scalar field 'could give theoretical plausibility to the current observations' therefore rests on parameter selection, not on an independent prediction. I also note the self-citation issue: the logarithmic potential is adopted from the same authors' prior work [31], so the model choice is not externally anchored. The paper itself concedes that an additional chiral-condensation mechanism at 100–1000 TeV is required, further showing that the scenario is assembled from free inputs. I do not count the exterior-matching/vacuum-energy inconsistency as a circularity; it is a serious physical-consistency defect but not a definitional loop. Overall the derivation is not equivalent to its inputs, but the headline observational agreement is substantially a fit, giving a moderate circularity score.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central claim rests on two hand-chosen energy scales: the linear potential scale U1 (52 to 102 MeV) and the fermion condensate scale U2 (around 300 TeV). These are not derived from the logarithmic model's parameters nor observed independently. The model also assumes a vanishing vector condensate and a Schwarzschild exterior, and it does not address the huge vacuum energy implied by the chosen parameters.

free parameters (4)
  • U1 (linear potential scale) = U1^(1/4) from 52 to 102 MeV in scans (dimensionless U1_tilde = 10^-2 to 1.5e-1)
    The energy scale of the linear part of the scalar potential is scanned by hand to reproduce the observed neutron star mass gap. Not derived from the logarithmic model parameters alpha or R0.
  • U2 (fermion condensate coupling) = U2_tilde = 10^-1.4 to 1.5e-0.4, corresponding to <psi-psi>^(1/3) around 300 TeV
    The strength of the four-fermion induced potential is tuned so that the potential has a minimum and the effect on mass is significant. Requires a chiral condensate at hundreds of TeV, which is not predicted by the Standard Model.
  • surface density = 10^10 g/cm^3
    The density at the neutron star surface is assumed, based on the outer crust range. The paper argues the effect on radius is small.
  • boundary fluctuation Delta = 0 to 1.0
    The scalar field at the surface is set to (1+Delta) times phi_min instead of phi_min. The paper shows the effect on mass is small, so this is not a critical parameter.
assumptions (5)
  • domain assumption The torsion is on-shell (algebraic equation), allowing torsion to be eliminated in favor of spin density and derivatives of F'(R).
    Invoked in Sec. 2.1 to derive Eq. (2.10). This is standard in ECSK-type theories.
  • ad hoc to paper The vector condensate <psi-bar gamma^mu psi> vanishes, ignoring the last term in the Fierz identity (3.1).
    Stated in Sec. 3: 'For simplicity, we ignore the last term by assuming its VEV vanishes.' This is needed to obtain the scalar-pseudoscalar form.
  • standard math The one-loop effective potential (3.4) from Ref. [45] is valid and is truncated at one loop.
    The potential is taken from the cited work without re-derivation in this paper.
  • domain assumption The exterior of the neutron star is Schwarzschild, with a constant scalar field at the potential minimum.
    Used in Sec. 5 for the shooting method. This ignores the energy density of the scalar field outside the star, which is nonzero and large for the interesting parameter values.
  • ad hoc to paper The parameters U1 and U2 can take values far above the dark energy and QCD scales without upsetting cosmology.
    The large U1 (MeV scale) and U2 (TeV scale) are chosen to match the mass gap, but the vacuum energy at the minimum is a huge cosmological constant that is not discussed.
invented entities (1)
  • TeV-scale chiral condensate (fermion condensate at 100 to 1000 TeV)
    purpose: Provides a large U2 so that the scalar potential has a minimum at phi about -2 and the mass of neutron stars is raised.
    The paper explicitly requires 'a mechanism of chiral condensation at the 100-1000 TeV scale' (Sec. 5, after Table 1). No Standard Model or experimental evidence supports such a condensate.

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Cite this review

Pith. "Pith review of Neutron star in Logarithmic model of Cartan $F(R)$ gravity." pith.science (2026). https://pith.science/paper/467WYY2O

@misc{pith2026241203219,
  author       = {Pith},
  title        = {Pith review of: Neutron star in Logarithmic model of Cartan $F(R)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/467WYY2O}},
  note         = {Machine review of arXiv:2412.03219}
}
abstract

Cartan $F(R)$ gravity introduces the equivalent scalar-tensor theory by extending the gravity sector. From the solution of the modified Cartan equation leads to the interaction with the scalar field and the fermion. We derived the effective potential of the scalar field using the auxiliary field method, which is commonly used in studies of spontaneous chiral symmetry breaking. Using this effective potential we investigated the mass-radius relation of neutron star to solve the Tolman-Oppenheimer-Volkoff equation. By the numerical computation, we found that the contribution of the scalar field increases the mass of a neutron star. This could give theoretical plausibility to the current observations.

Discussion (0). Continue with ORCID to comment.

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