Constraints on Skyrme Equations of State from Doubly Magic Nuclei, Ab-Initio Calculations of Low-Density Neutron Matter, and Neutron Stars

B. A. Brown; C. Y. Tsang; F.J. Fattoyev; M. B. Tsang; W. G. Lynch

REVIEW 3 major objections 5 minor 61 references

Constraints on Skyrme Equations of State from Doubly Magic Nuclei, Ab-Initio Calculations of Low-Density Neutron Matter, and Neutron Stars

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper shows that requiring Skyrme equations of state to produce a 2.1-solar-mass neutron star forces the neutron effective mass at saturation density down to 0.60–0.65, and that this constrained family then predicts a 1.4-solar-mass…

desk verdict A solid, honest model-dependent constraint: within the Skyrme family, m*_n(ρ0)=0.60–0.65 is needed to reach 2.1 M⊙, but the wording oversells it as a universal requirement. read the letter →

arxiv 1908.11842 v1 pith:MMNJXBHA submitted 2019-08-30 nucl-th

C. Y. Tsang , B. A. Brown , F.J. Fattoyev , W. G. Lynch , M. B. Tsang This is my paper · ORCID

classification nucl-th PACS 21.10.Dr21.30.Fe21.60.Jz21.65.-f

keywords Skyrmeenergydensityfunctionalneutroneffectivemassstarequationofstatetidaldeformabilitymaximumdoublymagicnucleilow-densitymatterskin

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 argues that the neutron effective mass, a poorly pinned-down parameter of nuclear theory, can be fixed by combining three independent inputs: the measured binding energies and radii of doubly magic nuclei, ab initio calculations of low-density neutron matter, and the existence of neutron stars with masses near 2.1 solar masses. Within the Skyrme energy-density functional, meeting the 2.1-solar-mass maximum requires the neutron effective mass at ordinary nuclear density to be 0.60–0.65 times the bare neutron mass, noticeably below the commonly used 0.85. With that value, the same functionals predict a radius of $12.4(1)$ km and a dimensionless tidal deformability $\Lambda = 423(+35/-40)$ for a 1.4-solar-mass star, together with a symmetry-energy slope $L = 65(7)$ MeV and neutron skins near 0.19 fm in $^{208}$Pb and 0.18 fm in $^{48}$Ca. If correct, this converts existing nuclear data plus one astrophysical mass scale into sharp predictions for radii and tidal deformabilities that current gravitational-wave and X-ray observations can test.

What carries the argument

The central object is the Skyrme energy-density functional, specifically its analytic neutron-matter equation of state $E(\rho) = a_n\rho^2 + b_n\rho^{2+\sigma} + c_n\rho^{5/3} + d_n\rho^{8/3}$, where the last term comes from the p-wave interaction. The neutron effective mass is related to that last term by $m^*_n(\rho)/m = c_n/(c_n + d_n\rho)$, so the high-density pressure of neutron matter and the effective mass at ordinary density are tied together in one parameter combination $d_n$. The paper's argument works by re-fitting the Skyrme parameters to all the nuclear-data and low-density-neutron-matter constraints while adjusting the effective mass until the Tolman–Oppenheimer–Volkoff equations yield a maximum neutron star mass near 2.1 solar masses.

What would settle it

A clean falsifier would be a precise measurement of a 1.4-solar-mass neutron star radius outside $12.4(1)$ km, or an ab initio neutron-matter pressure at two to three times saturation density that lies outside the fitted Skyrme band; either would show that the low effective mass is an artifact of the assumed functional form.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a two-way calibration: the Skyrme neutron equation of state is re-fitted to nuclear data and low-density ab initio results, and the requirement that the same equation of state produce a maximum neutron star mass of about 2.1 solar masses fixes the neutron effective mass at saturation density at $[m^*_n/m](\rho_0) = 0.60$–$0.65$. This low effective mass carries over to neutron-star observables: the predicted radius of a 1.4-solar-mass star is $12.4(1)$ km and the tidal deformability is $\Lambda = 423(+35/-40)$, both consistent with the GW170817 tidal constraint. The same fits give $L = 65(7)$ MeV and neutron skins $R_{\rm skin}(^{208}{\rm Pb}) = 0.194(7)$ fm and $R_{\rm skin}(^{48}{\rm Ca}) = 0.178(3)$ fm.

Load-bearing premise

The load-bearing premise is that the analytic Skyrme formula for the neutron-matter equation of state, together with the effective-mass formula tied to its last term, keeps the same form up to several times normal nuclear density; if that form is wrong at high density, the extracted effective mass and the radius and tidal predictions lose their force.

Editorial extensions

If this is right

If the constrained Skyrme functionals are correct, a 1.4-solar-mass neutron star has radius $12.4(1)$ km and tidal deformability $\Lambda = 423(+35/-40)$, values directly checkable with future merger and X-ray observations.
The symmetry-energy slope and neutron skins become sharp predictions—$L = 65(7)$ MeV, $R_{\rm skin}(^{208}{\rm Pb}) = 0.194(7)$ fm, $R_{\rm skin}(^{48}{\rm Ca}) = 0.178(3)$ fm—testable by neutron-skin and parity-violating electron-scattering experiments.
Since the maximum mass is so sensitive to the neutron effective mass, the observed pulsars near 2 solar masses already disfavor Skyrme functionals with $[m^*_n/m](\rho_0) = 0.85$ unless the functional form changes at high density.
If the low effective mass is physical, neutron-star thermal properties—heat capacity and neutrino luminosity—would be affected, which would show up in cooling curves.

Reading between the lines

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

Beyond the paper: the same logic suggests that in any equation-of-state family that connects the low-density neutron matter constraint to a 2-solar-mass maximum, the effective-mass-like parameter at saturation will be driven low; the extracted value is not Skyrme-specific.
Beyond the paper: the identification of $m^*_n/m = 0.60$–$0.65$ depends on the four-term analytic form; a future precise measurement of a neutron-star radius or neutron skin that disagrees would indicate that the effective mass is standing in for missing high-density physics, as the authors themselves note.
Beyond the paper: because the argument pins $L$ and the skins through the same fits, a parity-violating measurement of the $^{208}$Pb neutron skin near 0.19 fm would either corroborate or undermine the whole chain.

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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 does what it says: it takes 12 Skyrme functionals that already fit nuclear data and low-density neutron matter, refits them under the condition that the maximum neutron star mass is about 2.1 M⊙, and finds the neutron effective mass at saturation drops from 0.85 to 0.60–0.65. That is a real, quantitative result within the Skyrme framework. The refit is tested against binding energies, charge radii, and single-particle energies, and the rms deviations stay small, so the constraint is not vacuous: the same Skyrme family can satisfy both the nuclear data and the astrophysical limit only at this lower effective mass. I also give credit for the explicit caveat near Fig. 1—that the effective mass parameter may be “mocking up” dense-neutron-matter physics. That is the right thing to say, and it is the key weakness.

The soft spot is precisely that caveat. The maximum mass in these functionals is tuned through the d_n ρ^{8/3} term, which is the same term that sets the effective mass via Eq. (6). So the statement that m* = 0.60–0.65 is “required” is true only given the Skyrme power-law extrapolation from saturation to several times saturation. The low-density anchors do not pin the functional form at 2–5 ρ0. A different high-density mechanism—a phase transition, a speed-of-sound plateau, a different momentum dependence—could produce a 2.1 M⊙ star with a higher m* at ρ0. The reader’s circularity concern is partly fair but not damning: the authors impose Mmax, not m* directly, and the nuclear-data checks are independent. Still, the abstract’s “required” should be read as “required within this family of functionals,” and the quoted radius and Lambda uncertainties are spreads over 12 functionals, not systematic errors from the form choice.

Minor points: the crust treatment is a simple spline rather than a full pasta calculation, but the paper cites work showing tidal deformability is insensitive to the inner crust, so this is acceptable. No code or table of all refitted parameters is released beyond the d_n values, which limits reproducibility but is not disqualifying.

Who gets value: nuclear structure people and astrophysicists who use Skyrme EOSs. It is a useful data point showing what this particular EOS family can and cannot tell us, and it sharpens predictions for neutron skins, radii, and tidal deformability that can be checked against future measurements.

Yes, send it to peer review. It is a serious paper with a clear logical chain, honest caveats, and a result worth publishing, provided the wording is softened to make the model-dependence explicit in the abstract.

Referee Report

3 major / 5 minor

Summary. The paper refits a family of 12 Skyrme energy-density functionals to a common dataset that includes binding energies, charge radii, and single-particle energies of doubly magic nuclei, plus ab initio low-density neutron matter calculations, and then uses the resulting equations of state to compute neutron star mass-radius relations and tidal deformabilities. The central finding is that, in order to reproduce a maximum neutron star mass of about 2.1 solar masses, the neutron effective mass at saturation density must be [m*_n/m](rho0) = 0.60-0.65, rather than the previously used 0.85. With this effective mass, the constrained functionals predict R_1.4 = 12.4(1) km and Lambda_1.4 = 423(+35/-40), consistent with the GW170817 tidal deformability constraint. The quoted uncertainties are spreads across the 12 selected functionals.

Significance. If the central result held as a robust constraint, it would be significant: it would tightly link nuclear structure data, low-density ab initio neutron matter, and neutron star observables, and would sharpen predictions for the symmetry energy, neutron skins, and tidal deformability that can be tested with ongoing experiments (PREX-II/CREX) and gravitational-wave detections. Strengths of the paper include the use of a 12-member functional family with a common refit to a diverse dataset, the explicit comparison with GW170817, the check of causality in the resulting EOSs, and the authors' transparent caveat that the effective mass may be 'mocking up' dense-matter physics. The limitation is that the headline effective-mass constraint is not an independent determination: it follows from imposing Mmax ~ 2.1 solar masses as an input within a single analytic EOS family, so its significance as a constraint on dense matter is conditional on the assumed Skyrme form.

major comments (3)

[Abstract and final paragraph (after Eq. (6))] The abstract and the concluding paragraph state that [m*_n/m](rho0) = 0.60-0.65 is 'required' to obtain a maximum neutron star mass of 2.1 solar masses. This is better described as a consequence of imposing Mmax ~ 2.1 solar masses as an input in the refit: through Eqs. (5)-(6), a smaller m*_n(rho0) corresponds to a larger d_n, which stiffens the EOS at high density and raises Mmax. The nuclear and low-density neutron matter data anchor the fit at or below saturation density, so the extracted effective-mass range is not an independent constraint but a property of the assumed Skyrme family. The authors partly acknowledge this in their 'mocking up' caveat, but the abstract and the final paragraph should be qualified explicitly, for example, 'within the Skyrme EDF family used here.'
[Figs. 2-3 and text near 'narrowed down to 423...'] The quoted errors for R_1.4 = 12.4(1) km and Lambda = 423(+35/-40) are the spreads across the 12 selected Skyrme functionals, not full systematic uncertainties. Because all 12 functionals share the same analytic form (Eq. (5)) and are fit to the same dataset, the spread does not account for uncertainty in the functional form, the crust-core matching, or the high-density extrapolation. Please label these as 'spread across the chosen Skyrme family' and avoid presenting them as total theoretical errors, especially in the abstract's claim of a narrowed constraint.
[Eq. (5) and the paragraph beginning 'It is possible that...'] The central effective-mass constraint relies on a single high-density term d_n rho^{8/3} in Eq. (5) remaining valid up to the central densities of maximum-mass stars (several times rho0). The paper's own caveat that the effective mass parameter may be 'mocking up some aspect of dense neutron matter' identifies this as the key assumption. To make the headline claim robust, I ask for a concrete sensitivity test: for example, a comparison with a different EOS family (a relativistic mean-field model or a piecewise polytrope) fit to the same low-density constraints, or an explicit bound on the density range over which Eq. (5) is trusted. Without such a test, the statement that m*_n(rho0) = 0.60-0.65 is 'required' is only a statement about the Skyrme family.

minor comments (5)

[Paragraph before Eq. (5)] The phrase 'up to the E/N of 0.04 neutron/fm^3' should be reworded: the ab initio calculations constrain the energy per particle up to a density of 0.04 fm^-3, not 'up to the E/N of 0.04 neutron/fm^3.'
[Table I and Eq. (5)] The header for the d_n column lists units 'MeV fm5'; given the rho^{8/3} term in Eq. (5), this is dimensionally consistent only if rho is measured in fm^-3, but the header for b_n ('MeV fm3γ') is ambiguous. Please define all units explicitly and consistently.
[Fig. 3] The GW170817 constraint is shown as a single rectangular box; please specify the confidence level and the prior used (e.g., low-spin prior, 90% credible interval), since the posterior from Ref. [11] is not rectangular.
[Second section, discussion of maximum masses] The text states that the maximum mass obtained with m* = 0.85 is smaller than the '2.01(4) solar mass neutron star observed in [48,49].' Reference [48] reports 1.97(4) M_sun, while Ref. [49] reports 2.01(4) M_sun; please state the values separately or provide the combined value with a clear provenance.
[End of Section II] The rms deviations quoted for binding energies and charge radii appear only in the text; a small table or a statement of the number of degrees of freedom in the fit would help the reader judge the quality of the refit against the previous analysis in Ref. [2].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the m* constraint is an explicit inversion of the imposed Mmax=2.1 condition within a stated analytic family, and the radius/tidal outputs are genuine predictions.

full rationale

The paper's central statement, '[m*_n/m](rho0)=0.60-0.65 is required to obtain a maximum neutron star mass of 2.1 solar masses,' is not presented as an independent prediction of Mmax; the preceding text states explicitly that the authors 'refit the Skyrme parameters to all of the nuclear data and low-density neutron EOS constraints considered in [2], with the additional constraint that the maximum neutron star mass comes out to be about 2.1 solar masses. The outcome is that the neutron effective mass at rho0 is reduced from 0.85 to 0.60-0.65.' Thus m* is the inferred value of a fitted coefficient (Eq. (6) defines m* in terms of the d_n rho^{8/3} term), obtained by inverting the imposed mass condition within the stated Skyrme family. That is a constraint/deduction, not circularity. The radius R_{1.4}=12.4(1) km and Lambda=423(+35/-40) are genuine outputs: they are not fit to GW170817 or to the radius, but computed from the resulting EOS and then compared with observations. The paper's caveat that 'the effective mass parameter in Skyrme is mocking up some aspect of dense neutron matter that cannot be extrapolated from normal nuclear density EOSs' is an honest limitation about the extrapolation of the assumed functional form; it is a correctness/robustness risk, not a circular step. Self-citations (Refs. [2], [35]) supply the starting parameter set and the analytic neutron EOS form, both of which were constrained in earlier work by external nuclear data and ab-initio neutron matter; they are not invoked as circular loading. No step reduces by construction to its own input.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the Skyrme functional form, the ab initio low-density input, the nucleonic composition assumption, and the imposed 2.1 Msun target. No new particles, forces, or dimensions are introduced. The effective mass is the key fitted output.

free parameters (3)

Target maximum neutron star mass Mmax = ~2.1 Msun
Chosen by the authors as the constraint to refit the EDFs; motivated by the 2.01 Msun pulsar and GW170817-based upper limits, but not an independently measured quantity for this fit.
Skyrme parameters for 12 EDFs (a_n, b_n, c_n, d_n, sigma) = Values given in Table I
Each of the 12 functionals is refit to nuclear data, low-density neutron matter, and the Mmax constraint; the effective mass is derived from these parameters via Eq. 6.
Neutron effective mass m*_n/m at rho0 = 0.60-0.65
The headline output, derived from the refit parameters; it is the quantity adjusted (through the parameters) to satisfy the imposed Mmax constraint, so it is a fitted value.

assumptions (5)

domain assumption The Skyrme energy density functional, specifically the neutron EOS form in Eq. 5 and the effective mass relation in Eq. 6, remains valid when extrapolated to densities several times nuclear saturation.
The whole analysis connects low-density nuclear constraints to high-density neutron star properties through this functional form. The authors flag the possibility that the effective mass 'mocks up' unknown dense-matter physics.
domain assumption The ab initio low-density neutron matter results of Refs [44-47] are correct and applicable as constraints.
These calculations are treated as fixed inputs without independent verification in this paper.
domain assumption Neutron star core matter is composed of nucleons, electrons, and muons in beta equilibrium.
The EOS for the core uses only these degrees of freedom; no exotic matter is included.
standard math General relativity (TOV equations) correctly determines neutron star structure.
Standard physics used as the framework for deriving mass, radius, and tidal deformability.
domain assumption The BPS crust EOS plus a cubic spline interpolation between crust and core is adequate for computing tidal deformability.
The paper cites Ref [13] showing tidal deformability is insensitive to inner crust details.

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Cite this review

Pith. "Pith review of Constraints on Skyrme Equations of State from Doubly Magic Nuclei, Ab-Initio Calculations of Low-Density Neutron Matter, and Neutron Stars." pith.science (2026). https://pith.science/paper/MMNJXBHA

@misc{pith2026190811842,
  author       = {Pith},
  title        = {Pith review of: Constraints on Skyrme Equations of State from Doubly Magic Nuclei, Ab-Initio Calculations of Low-Density Neutron Matter, and Neutron Stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MMNJXBHA}},
  note         = {Machine review of arXiv:1908.11842}
}

abstract

We use properties of doubly-magic nuclei, ab-initio calculations of low-density neutron matter, and of neutron stars to constrain the parameters of the Skyrme energy-density functional. We find all of these properties can be reproduced within a constrained family of Skyrme parameters. The maximum mass of a neutron star is found to be sensitive to the neutron effective mass. A value of [$ m^{*}_{\rm n}/m](\rho_0) = 0.60-0.65 $ is required to obtain a maximum neutron star mass of 2.1 solar masses. Using the constrained Skyrme functional with the aforementioned effective mass, the predicted radius for a neutron star of 1.4 solar masses is 12.4(1) km and $\Lambda$ = 423(40).

Figures

Figures reproduced from arXiv: 1908.11842 by the authors.

**Figure 1.** FIG. 1. The EOS in the form of Pressure versus Density used [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** FIG. 2. Mass-vs-Radius relation predicted by the two groups [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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