$\Lambda$ and $\Sigma$ potentials in neutron stars, hypernuclei, and heavy-ion collisions

Asanosuke Jinno; Koichi Murase; Yasushi Nara

REVIEW 3 major objections 6 minor 19 references

$\Lambda$ and $\Sigma$ potentials in neutron stars, hypernuclei, and heavy-ion collisions

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A single hyperon three-body parameter choice makes Lambda repulsive at high density and still fits hypernuclear, flow, and Sigma measurements

desk verdict A transparent but low-discrimination consistency check: the paper shows some YNN LEC sets pass all three constraints for one chiral YN potential, with the Sigma-potential result as the genuinely new piece. read the letter →

arxiv 2501.09881 v1 pith:J77JMPPE submitted 2025-01-16 nucl-th

Asanosuke Jinno , Koichi Murase , Yasushi Nara This is my paper · ORCID

classification nucl-th PACS 26.60.Kp21.80.+a25.75.Ld

keywords hyperonpuzzleLambdapotentialSigmathree-bodyforceschiraleffectivefieldtheoryhypernucleidirectedflowneutronstars

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 asks whether the three-baryon force needed to keep $\Lambda$ hyperons out of neutron-star cores is consistent with what experiments tell us about hyperons in ordinary nuclear matter. It finds a range of the two low-energy constants $H_1,H_2$ that make the $\Lambda$ single-particle potential strongly repulsive at high density while still reproducing measured $\Lambda$ separation energies in hypernuclei, the $\Lambda$ directed flow in heavy-ion collisions, and the accepted $\Sigma$ potential at saturation density. If the claim is right, the familiar hyperon puzzle—soft equations of state that fail to support massive neutron stars—has a plausible microscopic resolution without invoking exotic degrees of freedom. The authors stress that the check uses one chiral $YN$ potential, so the consistency is established for that interaction rather than as a universal model-independent statement.

What carries the argument

The load-bearing object is the hyperon–nucleon–nucleon ($YNN$) three-body force, whose contact strength is set by two low-energy constants $H_1$ and $H_2$. In the decuplet-dominance approximation this force is converted into an effective density-dependent $YN$ two-body force, so the same constants control both the $\Lambda$ and $\Sigma$ single-particle potentials. The potentials are computed with the Brueckner–Hartree–Fock method using the chiral $YN$ potential NLO13(500); the $H_1$–$H_2$ values are chosen on the line that fixes $U_\Lambda(\rho_0)=-30$ MeV, and along that line $U_\Sigma(\rho_0)$ varies from roughly 30 to 10 MeV. For the heavy-ion comparison, the momentum dependence is parametrized by a Lorentzian form matched to the chiral EFT result up to either $1.0$ or $2.5\ \mathrm{fm}^{-1}$, and the resulting $v_1$ distinguishes soft from hard extrapolations.

What would settle it

Recompute the $\Lambda$ and $\Sigma$ potentials with NLO19 or N2LO replacement using the same $(H_1,H_2)$ sets; if no set simultaneously gives $U_\Lambda(\rho_0)=-30$ MeV, $U_\Sigma(\rho_0)=30\pm20$ MeV, and strong repulsion at $\rho\gtrsim 2\rho_0$, the central consistency claim fails. A new measurement of $U_\Sigma(\rho_0)$ outside $30\pm 20$ MeV would also break the allowed line.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the low-energy constants of the $YNN$ three-body force are not forced into conflict by the data: certain $(H_1,H_2)$ combinations, taken from the construction of Ref. [2], satisfy all three empirical constraints at once. In the paper's own words, "Some of the 3BF LEC sets of $(H_1,H_2)$ from Ref. [2] turned out to be consistent with the empirical information in all three physics." Concretely, these sets yield a $\Lambda$ potential that is attractive at saturation ($U_\Lambda(\rho_0)\simeq -30$ MeV), becomes strongly repulsive toward $\rho\sim 2$–$3\rho_0$ so that $\Lambda$'s are suppressed in neutron stars, reproduce hypernuclear separation energies as accurately as a conventional attractive potential, match the $\Lambda$ directed-flow data when the momentum dependence is extrapolated softly, and give $U_\Sigma(\rho_0)$ inside the empirical $30\pm 20$ MeV band.

Load-bearing premise

The consistency check rests on one chiral $YN$ potential, NLO13(500); if new chiral potentials (NLO19 or N2LO) change the in-medium $\Lambda$ and $\Sigma$ potentials, the $(H_1,H_2)$ sets that pass all three constraints may fall outside the empirical bands.

Editorial extensions

If this is right

A consistent chiral-EFT description of strangeness in dense matter is available: the same three-body force that suppresses $\Lambda$'s in neutron stars is compatible with hypernuclear experiments and heavy-ion flow.
The empirical $\Sigma$ potential becomes a discriminating observable: it narrows the permitted $(H_1,H_2)$ combinations beyond the $\Lambda$ potential alone.
The directed-flow comparison shows that $\Lambda$ flow is sensitive mainly to the momentum dependence of the $\Lambda$ potential, so flow data help constrain the high-momentum extrapolation rather than the density dependence.
Because the $\Lambda$ and $\Sigma$ potentials diverge at high density, transport models should assign species-dependent potentials to hyperons; the present simulations use a common potential for all hyperons and will be updated.

Reading between the lines

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

The paper does not assert this, but if the same consistency holds with the newer chiral $YN$ potentials NLO19 and N2LO, the hyperon puzzle would be resolved without exotic matter, and the favored $(H_1,H_2)$ line would yield a concrete prediction for $\Sigma$ directed flow in heavy-ion collisions.
The paper does not assert this, but the sharpest unconstrained link is the high-momentum behavior: because Chi3momHard fails while Chi3momSoft passes, a direct measurement of the $\Lambda$ optical potential above $1\ \mathrm{fm}^{-1}$ would decide the extrapolation and tighten the parameter set.
The paper does not assert this, but a more precise $\Sigma$-atom or $(\pi^+,K^+)$ determination of $U_\Sigma(\rho_0)$ could single out one $(H_1,H_2)$ pair, turning the allowed line into a point and sharpening neutron-star predictions.

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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 thing to know about this paper is that it is a short, honest proceedings contribution that checks whether the repulsive Lambda potential from chiral YNN forces (Gerstung et al.) survives three independent constraints: hypernuclear binding energies, Lambda directed flow in heavy-ion collisions, and the Sigma single-particle potential. The genuinely new element is the Sigma potential calculation with the 3BF LEC sets; the hypernuclear and directed-flow parts are updates of the authors' own earlier work (Refs. [8] and [12]).

What the paper does well is stay transparent about its own limitations. The BHF calculation with continuous choice is standard, the parameter sets are clearly listed, and the authors explicitly say in Section 5 that everything rests on a single chiral YN potential, NLO13(500), and that the heavy-ion simulations use the same potential for all hyperon species. That is the right tone for a proceedings paper. The new Sigma result is also meaningful: several LEC combinations that make the Lambda potential strongly repulsive at high density also keep U_Sigma(rho0) within the empirical 30 +/- 20 MeV band.

The soft spots are real but not fatal. The Sigma constraint is wide (10-50 MeV), so passing it is a weak filter. The v1 comparison does not distinguish density dependence, as the authors themselves note; it mostly constrains the momentum dependence at high momentum. One Skyrme parameter is tuned to the 13_Lambda_C binding energy. And the stress-test concern is legitimate: the YNN-induced two-body forces contribute to Lambda-N, Sigma-N, and the Lambda-N to Sigma-N transition, so the predicted U_Sigma is not independent of the chosen YN input. The LECs in Table 1 are fitted to NLO13(500), and the authors admit that N2LO gives more attractive potentials. If a switch to NLO19 or N2LO shifts the Sigma potential outside the empirical band, the central existence claim becomes an artifact of the old YN interaction. The paper does everything it can within that limitation, but the claim is conditional on one potential, not a robust property.

Who gets value from this? Anyone working on the hyperon puzzle, hypernuclear spectroscopy, or strangeness in heavy-ion collisions. It is a useful cross-check but not a decisive one. For a proceedings contribution it is a fair piece of work, and the Sigma result is worth citing. I would send it to a referee with the specific request to probe the single-potential dependence and to ask whether the Sigma constraint actually discriminates among the LEC sets. A serious referee can improve the paper, not reject it on basics.

Recommendation: accept for peer review as a proceedings contribution; require a clearer statement that the claimed consistency is for NLO13(500) only, not a general conclusion.

Referee Report

3 major / 6 minor

Summary. This proceedings contribution examines whether the Lambda single-particle potential obtained in Ref. [2] from chiral YN plus YNN three-body forces, with low-energy constants chosen to make the Lambda potential repulsive at high density and thereby suppress Lambda hyperons in neutron stars, is consistent with Lambda hypernuclear separation energies, the directed flow v1 of Lambda in Au+Au collisions at sqrt(s_NN)=4.5 GeV, and the empirical Sigma single-particle potential at saturation density. Using Skyrme-Hartree-Fock for hypernuclei, the JAM2/RQMDv transport model for v1, and Brueckner-Hartree-Fock for the Sigma potential, the authors find that some (H1,H2) LEC sets from Ref. [2] reproduce all three sets of observations. The paper is explicit that the results are based on a single chiral YN potential, NLO13(500), and that NLO19 and N2LO variants may change the in-medium potentials.

Significance. The paper is useful as a first simultaneous consistency check of a strangeness three-body force across hypernuclear, heavy-ion, and neutron-star settings, and it is commendably transparent about the limitations stated in Section 5. The calculation of the Sigma potential with the same YNN force is a genuinely new element. However, the empirical constraints are wide, the heavy-ion v1 data do not distinguish the repulsive density dependence from a conventional attractive potential, and the entire conclusion rests on one chiral YN interaction. If the authors extend the calculation to NLO19 and N2LO as they promise, the work would become a much stronger constraint; in its present form the central claim is a conditional existence proof rather than a robust property of the chiral three-body force.

major comments (3)

[Section 5 (first paragraph) and Section 4] The load-bearing limitation is stated in Section 5: 'The results presented here are based on a single chiral YN potential, NLO13(500).' Because the (H1,H2) sets in Table 1 are fixed by requiring U_Lambda(rho0) = -30 MeV and strong high-density repulsion for NLO13(500), the predicted U_Sigma(rho0) is not independent of that YN input. Section 5 itself notes that N2LO [19] yields more attractive Lambda and Sigma potentials. As a concrete test, the authors should repeat the Figure 4 calculation with NLO19 [16] and N2LO [19] and show whether any (H1,H2) set still satisfies U_Sigma(rho0) = 30 +/- 20 MeV together with the hyperon-puzzle condition. Without this, the existence claim in the abstract may be an artifact of the chosen YN potential rather than a robust property of the YNN three-body force.
[Section 3 and Figure 2 (right panel)] The directed-flow comparison is presented as one of the three consistency checks, but the text reports that Chi3momSoft and LY-IV reproduce v1 of Lambda with equal accuracy and that v1 is not sensitive to the density dependence of the Lambda potential. Since LY-IV is the conventional attractive potential that does not solve the hyperon puzzle, the v1 data are compatible with both classes of potentials and do not positively single out the strongly repulsive YNN-induced Lambda potential. The abstract's framing should therefore be tempered, or the analysis should quantify which densities and momenta actually contribute to v1 for Chi3momSoft versus LY-IV.
[Section 4, Figure 3, and Table 1] The empirical Sigma constraint U_Sigma(rho0) = 30 +/- 20 MeV is claimed to be reproduced by 'certain sets' of the LECs, but the plotted values in Figure 3 span roughly 10 to 30 MeV and several points lie close to the lower edge of the band. The text does not identify which (H1,H2) sets satisfy the constraint with margin, nor does it estimate the numerical uncertainty of the Brueckner calculation. Given that the abstract claims a simultaneous reproduction of the Sigma potential and the Lambda hyperon-puzzle condition, a quantitative table listing U_Sigma(rho0) and U_Lambda at high density for each LEC set is needed to assess how robust the consistency is.

minor comments (6)

[Figure 1, left panel caption] The caption says Chi3 (solid) and Chi2 (dashed) are fitted to GKW2 and GKW3, respectively, but Section 2 describes Chi3 as the fit including the YNN 3BF and Chi2 as the fit without it; the assignment appears to be reversed and should be corrected.
[Figure 1, right panel caption] The caption says Chi3momSoft reproduces Kohno3 up to 2.5 fm^-1 and Chi3momHard up to 1.0 fm^-1, which is opposite to the description in Section 3; please make the caption consistent with the text.
[Section 2] The sentence 'One parameter that cannot be determined from the uniform-matter results is tuned to reproduce the Lambda binding energy data of 13_Lambda C' should be repeated in the discussion of Figure 2, so that the reader understands that the hypernuclear comparison is partly calibrated to one of the data points shown.
[Section 3] The fitted values of the Lorentzian parameters C and mu in Eq. (2) are not given for Chi3momSoft, Chi3momHard, or LY-IVmomSoft; listing these values would improve reproducibility of the transport results.
[Section 5] The sentence 'The results shown here have been obtained by using the same Lambda potential for all other hyperons, including their resonance states' is an important caveat for the v1 simulation, since Figure 4 shows that Lambda and Sigma potentials differ substantially; this should be discussed as a possible source of bias in the v1 comparison.
[Acknowledgements] The word 'Forshungszentrum' should be 'Forschungszentrum'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Sigma potential and heavy-ion checks are out-of-sample, while Lambda suppression is explicitly an input constraint, not a derived prediction.

full rationale

The derivation chain is not circular. The H1/H2 LEC sets are taken from Ref. [2], where they were selected to satisfy two Lambda-potential conditions: U_Lambda(rho0) ~ -30 MeV and strong high-density repulsion. The paper is transparent about this in Sec. 4, stating that the two LECs 'are fixed by requiring the reproduction of the empirical value of the Lambda potential ... and a strongly repulsive U_Lambda at high density.' The Lambda repulsion is therefore an input constraint, not a predicted outcome, and the paper does not claim to predict it. The new content consists of three out-of-sample checks: hypernuclear separation energies (with one surface parameter tuned to 13C and the remaining systematics compared), Lambda directed flow from JAM2/RQMDv compared with STAR data, and the Sigma single-particle potential evaluated from the same YNN forces and compared with the empirical U_Sigma(rho0) = 30 +/- 20 MeV band. None of these three observables is used to determine H1 or H2, so the agreement is not forced by construction. In fact, the paper notes that the Lambda potential is 'practically constant by construction,' while the Sigma potential varies from about 30 to 10 MeV along the one-parameter family of LECs, and only some sets fall in the empirical band. That is a genuine consistency argument rather than a reduction to the input. The self-citations (Refs. [8] and [12]) point to prior peer-reviewed calculations that are benchmarked against external hypernuclear and heavy-ion data, so they are not unverified, load-bearing premises. The stated limitation that only NLO13(500) is used is a theoretical-uncertainty concern, not a circularity. Accordingly, no circular step can be identified and the score is 0.

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

The paper introduces no new particles, forces, or conserved quantities. The YNN three-body force is a standard interaction from the literature, and the LECs are existing parameters being varied. The free parameters are the LECs (taken from Ref [2] and Gerstung's thesis), the tuned Skyrme parameter, and the two Lorentzian fitting parameters in Eq. (2).

free parameters (5)

H1 (YNN contact LEC) = -2.650 to -0.900 f^-2 (Table 1)
Contact LEC for the YNN three-baryon force; chosen to reproduce U_Lambda(rho0) = -30 MeV and strong high-density repulsion (Ref [2], Table 1).
H2 (YNN contact LEC) = 0.100 to -0.300 f^-2 (Table 1)
Second contact LEC; paired with H1 to satisfy the same constraints on the Lambda potential.
Skyrme parameter tuned to 13-Lambda-C = not quoted
One parameter undetermined by uniform matter is tuned to reproduce the Lambda separation energy of 13-Lambda-C (Section 2).
C (Lorentzian strength) = not quoted
Fitting parameter in Eq. (2) used to represent the momentum-dependent Lambda potential in heavy-ion simulations.
mu (Lorentzian width) = not quoted
Second fitting parameter in Eq. (2), chosen to match the chiral EFT momentum dependence up to 1.0 or 2.5 fm^-1.

assumptions (5)

domain assumption Decuplet dominance approximation
Reduces the YNN force LECs to three by assuming decuplet dominance (Ref [3]); invoked in Section 4 building on Ref [2].
domain assumption Brueckner-Hartree-Fock with continuous choice
Used to compute hyperon single-particle potentials; a many-body approximation whose accuracy is not quantified here (Section 4).
domain assumption NLO13(500) chiral YN potential
The only YN interaction used; the authors note NLO19 and N2LO may change results (Section 4 and Summary).
ad hoc to paper Lorentzian extrapolation of momentum dependence beyond EFT cutoff
Eq. (2) assumes a Lorentzian form with fitted C and mu to extend the potential above about 2.8 fm^-1; the authors compare Chi3momHard and Chi3momSoft to explore the uncertainty.
ad hoc to paper Common Lambda potential for all hyperons and resonances in JAM2
Stated in the Summary as a known limitation of the heavy-ion simulation.

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Cite this review

Pith. "Pith review of $\Lambda$ and $\Sigma$ potentials in neutron stars, hypernuclei, and heavy-ion collisions." pith.science (2026). https://pith.science/paper/J77JMPPE

@misc{pith2026250109881,
  author       = {Pith},
  title        = {Pith review of: $\Lambda$ and $\Sigma$ potentials in neutron stars, hypernuclei, and heavy-ion collisions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J77JMPPE}},
  note         = {Machine review of arXiv:2501.09881}
}

abstract

With an appropriate $YNN$ force, the $\Lambda$ single-particle potential ($\Lambda$ potential) can be made strongly repulsive at high density, and one can solve the hyperon puzzle of neutron stars. We investigate the consistency of such a $\Lambda$ potential, evaluated recently from $YN$ and $YNN$ forces based on chiral effective field theory, with hypernuclear data and heavy-ion collision data. It is found that model calculations with such a $\Lambda$ potential can reproduce the data of the $\Lambda$ hypernuclear spectroscopy and the $\Lambda$ directed flow in heavy-ion collisions. Also, we evaluate the $\Sigma$ potential, which can be calculated by using the same hyperon forces as for the $\Lambda$ potential. Specifically, we show that the low-energy constants characterizing the strength of the $YNN$ force can be chosen to suppress the appearance of the $\Lambda$'s in neutron stars while at the same time the empirical value of the $\Sigma$ potential is reproduced.

Figures

Figures reproduced from arXiv: 2501.09881 by the authors.

**Figure 1.** (left panel) Density dependence of the Λ potential. GKW3 represents the results from 𝜒EFT with 𝑌 𝑁 and 𝑌 𝑁𝑁 forces [2]. GKW2 is also from 𝜒EFT but without the 3BFs [2]. Chi3 (solid line) and Chi2 (dashed line) are fitted to GKW2 and GKW3 up to 𝜌/𝜌0 < 1.5, respectively. LY-IV (dotted line) is a conventional Λ potential [10]. (right panel) Momentum dependence of the Λ potential. Kohno3 represents the result from 𝜒EFT … view at source ↗

**Figure 2.** (left panel) Λ binding energy of the Λ hypernuclei. Experimental data (cross) can be found in Ref. [8]. The figure is adopted from Ref. [8]. (right panel) Directed flow of Λ in mid-central Au+Au collisions at √ 𝑠𝑁 𝑁 = 4.5 GeV. The STAR data are taken from Ref. [11]. The figure is updated from Ref. [12] by using the updated version of JAM2. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Σ (red solid) and Λ (black dashed) potentials in symmetric nuclear matter. The horizontal axis corresponds to the three-body LECs 𝐻1 with 𝐻2 listed in table 1. However, it remains unclear how those 3BFs affect the corresponding Σ potential. The effective two-body forces resulting from the 3BFs considered in Ref. [2] contribute not only to the Λ𝑁 and Σ𝑁 channels but also to the Λ𝑁-Σ𝑁 transition potential. A presentl… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Density dependence of the Λ (left panel) and Σ (right panel) potentials in symmetric nuclear matter. The red solid lines are calculated by using the 3BF LECs in table 1. The only two-body case with the NLO13(500) parameter set is represented by the dashed line. The bla…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [8]

Jinno, K

A. Jinno, K. Murase, Y. Nara and A. Ohnishi,RepulsiveΛpotentials in dense neutron star matter and binding energy ofΛin hypernuclei, Phys. Rev. C108 (2023) 065803 [2306.17452]. 7 Λ andΣ potentials in neutron stars, hypernuclei, and heavy-ion collisions Asanosuke Jinno

arXiv 2023

[2] [12]

Y. Nara, A. Jinno, K. Murase and A. Ohnishi,Directed flow ofΛin high-energy heavy-ion collisions andΛpotential in dense nuclear matter,Phys. Rev. C106 (2022) 044902 [2208.01297]

arXiv 2022

[3] [2]

Gerstung, N

D. Gerstung, N. Kaiser and W. Weise,Hyperon–nucleon three-body forces and strangeness in neutron stars,Eur. Phys. J. A56 (2020) 175 [2001.10563]

arXiv 2020

[4] [19]

Haidenbauer, U.-G

J. Haidenbauer, U.-G. Meißner, A. Nogga and H. Le,Hyperon–nucleon interaction in chiral effective field theory at next-to-next-to-leading order, Eur. Phys. J. A59 (2023) 63 [2301.00722]. 8

arXiv 2023

[5] [16]

Haidenbauer, U.G

J. Haidenbauer, U.G. Meißner and A. Nogga,Hyperon–nucleon interaction within chiral effective field theory revisited, Eur. Phys. J. A56(2020) 91 [1906.11681]

arXiv 2020

[6] [1]

Demorest, T

P. Demorest, T. Pennucci, S. Ransom, M. Roberts and J. Hessels,Shapiro Delay Measurement of A Two Solar Mass Neutron Star,Nature 467 (2010) 1081 [1010.5788]

arXiv 2010

[7] [3]

Density-dependent effective baryon-baryon interaction from chiral three-baryon forces

S. Petschauer, J. Haidenbauer, N. Kaiser, U.-G. Meißner and W. Weise,Density-dependent effective baryon–baryon interaction from chiral three-baryon forces, Nucl. Phys. A957 (2017) 347 [1607.04307]

work page Pith review arXiv 2017

[8] [4]

Haidenbauer, S

J. Haidenbauer, S. Petschauer, N. Kaiser, U.G. Meissner, A. Nogga and W. Weise, Hyperon-nucleon interaction at next-to-leading order in chiral effective field theory, Nucl. Phys. A915(2013) 24 [1304.5339]

arXiv 2013