Properties of a kaon-condensed phase in hyperon-mixed matter with three-baryon forces

Takumi Muto

REVIEW 3 major objections 5 minor 163 references

Properties of a kaon-condensed phase in hyperon-mixed matter with three-baryon forces

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A hadronic phase containing both a kaon condensate and hyperons can still support neutron stars as massive as 2.2 solar masses, provided a universal three-baryon repulsion stiffens the high-density equation of state.

desk verdict Careful extension of the group's earlier kaon-hyperon EOS work, but the abstract's 2.1–2.2 Msun consistency claim partly rests on acausal Mmax branches and should be reined in. read the letter →

arxiv 2411.09967 v2 pith:IWOK4V54 submitted 2024-11-15 nucl-th astro-ph.HE

Takumi Muto This is my paper · ORCID

classification nucl-thastro-ph.HE PACS 26.60.-c21.65.-f97.60.Jd

keywords kaoncondensationhyperon-mixedmatterneutronstarequationofstateuniversalthree-baryonrepulsionchiraleffectiveLagrangianrelativisticmean-fieldtheorysymmetryenergyslopemassivestars

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 tries to establish that the coexistence of a kaon condensate and hyperons, the ($Y$+$K$) phase, in neutron-star matter does not rule out massive neutron stars if the high-density equation of state is stiffened by a universal three-baryon repulsion. The author builds a hadronic model from an effective chiral Lagrangian for kaon-baryon and kaon-kaon interactions, a minimal relativistic mean-field baryon interaction, and density-dependent effective two-body potentials for three-baryon repulsion and three-nucleon attraction. With this model, the ($Y$+$K$) phase supports maximum gravitational masses of 2.12 and 2.20 solar masses for $L=65$ and $70$ MeV (for $\Sigma_{Kn}=300$ MeV), and 2.08 and 2.16 solar masses for $\Sigma_{Kn}=400$ MeV, with mass-radius curves that pass through the observed constraints on PSR J0740+6620 and PSR J1810+1744. The paper thereby argues that the hyperon puzzle and kaon softening can both be overcome within a purely hadronic picture.

What carries the argument

The central machinery is the coupled field theory for the ($Y$+$K$) phase: an effective chiral $SU(3)_L \times SU(3)_R$ Lagrangian with a classical kaon field $K^\pm = (f/\sqrt{2})\theta e^{\pm i\mu_K t}$, where $\theta$ is the chiral angle and $\mu_K$ the kaon chemical potential, joined to a minimal relativistic mean-field baryon sector with $\sigma$, $\sigma^*$, $\omega$, $\rho$, and $\phi$ meson exchanges. Three-baryon forces enter as density-dependent effective two-body potentials: the universal three-baryon repulsion in the SJM2 string-junction form $U_{\rm SJM2}(r;\rho_B)=V_r\rho_B(1+c_r\rho_B/\rho_0)\exp[-(r/\lambda_r)^2]$ and a phenomenological three-nucleon attraction. Kaon condensation sets in where the lowest $K^-$ energy $\omega_K(\rho_B)$ equals the charge chemical potential $\mu$, and the competition between kaons and hyperons is carried by the $V$-spin charge combination $X_0 = (2f^2)^{-1}(\rho_p + \rho_n/2 - \rho_{\Sigma^-}/2 - \rho_{\Xi^-})$ that controls the $s$-wave vector attraction and by the kaon-baryon $\sigma$ terms $\Sigma_{Kb}$ that control the scalar attraction.

What would settle it

Re-solve the TOV equations with the same model but with the three-baryon repulsion computed from lattice QCD or chiral effective field theory rather than assumed universal; if the maximum gravitational mass drops below 2.08 solar masses, the paper's central claim fails.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the long-expected softening from kaon condensation and hyperon mixing can be compensated by two repulsive mechanisms acting together: the slope $L$ of the symmetry energy controls the two-baryon repulsion from vector-meson exchange beyond saturation, and the universal three-baryon repulsion adds a strong, flavor-blind repulsion at high baryon densities. In the resulting ($Y$+$K$) phase, $\Lambda$ hyperons always appear before kaon condensation, and their presence weakens the kaon-nucleon vector attraction; as a consequence the onset density of kaon condensation rises with $L$, and the $s$-wave kaon-baryon scalar attraction is self-suppressed as baryon effective masses drop. Once these effects are included, the equation of state stays stiff enough that for $L=65$ and $70$ MeV the maximum neutron-star mass reaches 2.12 to 2.20 solar masses, and the mass-radius branches are compatible with the massive-pulsar observations. The author also finds that a 2.0-solar-mass star can have a substantial ($Y$+$K$) core, while a 1.4-solar-mass star's core contains only nucleons and leptons.

Load-bearing premise

The load-bearing premise is that the three-baryon repulsion acts with equal strength among all baryon species, with the specific SJM2 form and parameters ($V_r=95$ MeV fm$^3$, $c_r=0.024$, $\lambda_r=0.86$ fm), so that if this repulsion is weaker or flavor-dependent, the high-density softening from hyperons and kaons may not be compensated and the two-solar-mass conclusion fails.

Editorial extensions

If this is right

For $L=65$ and $70$ MeV, the ($Y$+$K$)-phase equation of state gives maximum neutron-star masses of 2.12 and 2.20 solar masses for $\Sigma_{Kn}=300$ MeV, and 2.08 and 2.16 solar masses for $\Sigma_{Kn}=400$ MeV.
The mass-radius curves for these cases pass through the observed constraints on PSR J0740+6620 and PSR J1810+1744, so the phase is compatible with the most massive known pulsars.
In all cases $\Lambda$ hyperons appear before kaon condensation, and a larger symmetry-energy slope $L$ raises the kaon-condensation onset density because abundant hyperons suppress the kaon-nucleon vector attraction.
A 2.0-solar-mass neutron star can contain a sizable ($Y$+$K$) core with radius up to about 6.8 km for $L=65$ MeV and $\Sigma_{Kn}=400$ MeV, while a 1.4-solar-mass star's core contains only nucleons and leptons.
The model's equation of state violates causality above $\rho_B \approx 1.03$ fm$^{-3}$ for $L=60$ and $65$ MeV before the maximum mass is reached, indicating the need for an improved Lorentz-scalar three-baryon repulsion or a transition to quark matter at high density.

Reading between the lines

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

If the universality of the three-baryon repulsion is not confirmed by microscopic calculations, the same framework with flavor-dependent three-body forces would likely bring the maximum mass back below 2 solar masses, making the coexistence claim testable against lattice QCD and chiral effective field theory.
The predicted nearly lepton-free ($Y$+$K$) core and the kaon-induced Urca process imply distinctive rapid cooling; comparing predicted cooling curves with surface temperatures of massive neutron stars could indirectly confirm or exclude the phase.
The self-suppression of the kaon-baryon scalar attraction suggests that kaon condensation develops gradually rather than as a strong first-order transition, so future gravitational-wave tidal-deformability measurements could distinguish this scenario from models with abrupt phase transitions.
Because the causal limit coincides with the density at which baryon repulsive cores touch, a natural completion of the model is a smooth crossover to quark matter near $\rho_B \sim 5\rho_0$, which would preserve the two-solar-mass result while restoring causality.

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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 solid, fully specified model study of the (Y+K) phase in neutron-star matter. What is genuinely new here relative to Ref. [57] is the inclusion of muons, the systematic scan over L = 60–70 MeV and Sigma_Kn = 300/400 MeV, and the more detailed discussion of kaon-condensation onset in hyperon-mixed matter. The derivations are careful, the ground-state equations are written out in full, and the observational comparison is not circular: couplings are fixed by saturation properties, scattering lengths, and hypernuclear potentials, not by NICER masses. The paper also deserves credit for openly flagging the UTBR universality assumption and the causality limit as caveats.

The main soft spot is exactly where the abstract overreaches. The paper itself states that for L = 60 and 65 MeV the sound speed exceeds c beyond rho_B ~ 1.03 fm^-3, before the maximum mass is reached. Yet Table IV lists Mmax = 2.124 Msun for L = 65, Sigma_Kn = 300 MeV and 2.076 Msun for L = 65, Sigma_Kn = 400 MeV, both with central densities above that causal limit. Those entries are not admissible evidence for two-solar-mass consistency. The physically meaningful numbers are the TOV masses truncated at the causal boundary, and the paper does not report them. For L = 65, Sigma_Kn = 400 the truncated mass may be near or below 2.0 Msun, which would weaken the abstract's claim. This is a real, load-bearing flaw in the presentation, though not in the underlying formalism.

Other concerns are softer. The UTBR strength and the Sigma_Kn choices are phenomenological inputs drawn from broad ranges, and no uncertainty propagation is attempted. That is a limitation but not a fatal one, and the paper says as much. The lack of EOS tables or code also makes independent verification harder, but the equations are sufficiently explicit that reproduction should be possible.

Bottom line: this is a useful, honest paper for the dense-matter and neutron-star community, but it should be revised to restrict the consistency claim to the causal part of the EOS branch and to report the truncated masses. A serious referee should engage with it; I would not desk-reject. I would bring it to a reading group only if someone is actively working on kaon condensation or hyperon EOS.

Referee Report

3 major / 5 minor

Summary. This paper studies the coexistence of kaon condensates and hyperons (the (Y+K) phase) in beta-equilibrated, charge-neutral neutron-star matter, using an effective chiral Lagrangian for kaon-baryon and kaon-kaon interactions combined with a minimal relativistic mean-field baryon interaction, a universal three-baryon repulsion (UTBR), and a phenomenological three-nucleon attraction (TNA). The author derives the mean-field ground-state equations, fixes the meson couplings to nuclear saturation properties and hypernuclear potential depths, and scans the symmetry-energy slope L=60, 65, 70 MeV and the kaon-nucleon sigma term Sigma_Kn=300, 400 MeV. The resulting equations of state are used to compute particle compositions, onset densities, sound speeds, mass-radius relations, and density profiles. The central claim is that the (Y+K) EOS becomes stiff enough for L=65 and 70 MeV to be consistent with massive pulsar and NICER constraints, with maximum masses of about 2.08-2.20 solar masses.

Significance. The paper is a serious, falsifiable phenomenological construction: it states the full mean-field and ground-state equations, includes both electrons and muons, compares the UTBR+TNA saturation mechanism with the NLSI alternative, and gives concrete M-R and density-profile predictions. The explicit self-criticism in Sec. IX concerning the universality of the UTBR and the causality limit is a strength. However, the headline claim is currently oversold: for L=60 and L=65 MeV the quoted maximum masses are reached after the model's own sound speed has exceeded the speed of light, so those Table IV entries cannot be used as evidence for the two-solar-mass statement. Only the L=70 cases are causal at the maximum-mass point, and they do support the claim; the L=65 cases need to be re-evaluated by truncating the TOV integration at the causal limit.

major comments (3)

[Sec. VII C; Tables III and IV] Section VII C states that the sound speed exceeds c beyond rho_B = 1.03 fm^-3 and that for L=60 and L=65 MeV this happens before the maximum mass is reached. Nevertheless Table III reports rho_B,center(Mmax)=1.07 fm^-3 (L=65, Sigma_Kn=300) and 1.16 fm^-3 (L=65, Sigma_Kn=400), and Table IV lists Mmax=2.124 and 2.076 Msun for those configurations. An EOS with v_s>c cannot be integrated through the acausal region, so these Mmax entries are not admissible evidence for compatibility with PSR J0740+6620 or PSR J1810+1744. Please compute the TOV mass evaluated at the highest central density where the causality condition is satisfied, report that truncated mass for every L and Sigma_Kn, mark the acausal entries in Table IV, and use the truncated values in the abstract and in the Sec. VIII A compatibility statements. The L=65 truncated masses may still clear 2.08 Msun, but that must be shown explicitly.
[Sec. III B, Eq. (23); Sec. IX] The high-density stiffening that drives the conclusion that the (Y+K) phase supports roughly 2.1 Msun stars is dominated by the UTBR potential USJM2 of Eq. (23), with V_r=95 MeV fm^3, c_r=0.024, and lambda_r=0.86 fm, and by the assumption that this repulsion acts universally among NNN, YNN, and YYY triplets. Since Sec. IX itself states that the universality should be tested against chiral effective field theory, quark models, and lattice QCD, the manuscript needs a sensitivity study that varies the UTBR strength (or suppresses the hyperonic three-baryon components) and reports the resulting changes in Mmax and in the NICER compatibility. Without such a test, the quantitative two-solar-mass claim is contingent on an unquantified and explicitly acknowledged model assumption.
[Abstract and Sec. VIII A] The abstract's unconditional statement that the EOS with the (Y+K) phase 'becomes stiff enough to be consistent with recent observations of massive neutron stars' is too strong given the paper's own causality analysis. The causal part of the L=70 branch is sufficient to support the claim, but the L=65 branch is not presented in a way that separates the causal and acausal parts of the M-R and M-rho_B,center curves. Please restrict all mass consistency statements to the causal portion of each branch or add an explicit statement that the quoted Mmax for L=65 and L=60 are formal maxima of an acausal extension and are not observational evidence.

minor comments (5)

[Table III] In the row for L=70 MeV and Sigma_Kn=400 MeV, the value of rho_c_B(Xi^- in Lambda) is given as "(0.516)" in parentheses; this convention is not explained and should be defined in the caption or removed.
[Sec. VII C] The word "chracteristic" in the discussion of oscillation modes should read "characteristic".
[Eq. (43)] The chemical equilibrium relations are written with mu_e throughout; because muons are included, the paper should state explicitly that mu_e = mu_mu in beta equilibrium, or write the muon relations separately.
[Reference [54]] Reference [54] gives Phys. Rev. D 100, 02315 (2019); the standard article number appears to be 023015 and should be corrected.
[Fig. 14] The asterisk marking the direct Urca threshold is described only in the text; adding a legend entry would make the figure self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: model parameters are fixed by external nuclear/hypernuclear inputs, and the neutron-star predictions are checked against independent NICER and pulsar observations.

full rationale

The derivation is self-contained and non-circular. The baryon-interaction parameters are fixed by saturation properties (B0 = 16.3 MeV, K0 = 240 MeV, S0 = 31.5 MeV) and chosen values of L in its empirical range; hyperon couplings follow from hypernuclear potential depths (V_Y^N); kaon parameters are fixed by on-shell K-N scattering lengths (Appendix X B). The UTBR and TNA are adopted from independent phenomenological models (SJM2 and NTH/LP), not fitted to neutron-star masses. The resulting EOS is then integrated with the TOV equation and compared with external NICER and pulsar constraints, which were not used to determine the model constants. The paper's admitted causal-limit violation for L = 60 and 65 MeV is a physical consistency caveat, not a circular step, and the abstract's strongest claim should be read as restricted to the causal part of the branch. Self-citations to previous works by the author are present, but they are historical or provide details of coupling estimation; the load-bearing equations and comparisons are given in this paper and rest on external data and observations.

Assumptions & free parameters 7 free parameters · 8 assumptions · 1 invented entities

The central calculation depends on a moderately large set of model inputs: TNA and meson-nucleon parameters are fitted to saturation properties, Sigma_Kn is chosen from a broad empirical range, hyperon couplings come from hypernuclear phenomenology, and the UTBR strength is taken from a specific prior model. None of these are fitted to the neutron star mass-radius observations used for validation, so the comparison is not circular, but the parameter choices create model dependence. The most consequential assumptions are the universality of UTBR and the large Sigma_Kn values.

free parameters (7)

TNA parameters gamma_a and eta_a = gamma_a = -1662.63, -1597.67, -1585.48 MeV fm^6 and eta_a = 17.18, 18.25, 19.82 fm^3 for L = 60, 65, 70 MeV
Chosen to reproduce the empirical saturation density, binding energy 16.3 MeV, incompressibility K0 = 240 MeV, and chosen L. They control the attraction near saturation density.
Meson-nucleon coupling constants g_sigmaN, g_omegaN, g_rhoN = For L = 60, 65, 70 MeV: g_sigmaN = 5.27, 5.71, 6.07; g_omegaN = 8.16, 9.07, 9.77; g_rhoN = 3.29, 3.35, 3.41
Determined from the same saturation conditions plus symmetry energy S0 = 31.5 MeV and L. They set the two-body repulsion that dominates the high-density stiffness.
K-neutron sigma term Sigma_Kn = 300 MeV and 400 MeV
Selected as two typical cases from Fig. 1 because the empirical range is broad. This parameter drives the strength of s-wave kaon-baryon attraction and strongly affects the kaon onset density.
UTBR parameters V_r, c_r, lambda_r = V_r = 95 MeV fm^3, c_r = 0.024, lambda_r = 0.86 fm
Taken from the SJM2 string-junction model of Tamagaki and previous applications. They are not fitted to neutron star observables in this paper, but they are model inputs that determine the high-density repulsion.
Hyperon scalar couplings via potential depths = V_N(Lambda) = -27 MeV, V_N(Sigma-) = 23.5 MeV, V_N(Xi-) = -14 MeV
Used to fix g_sigmaY from hypernuclear phenomenology. These values carry experimental uncertainty that is not propagated into the neutron star results.
g_sigma*Lambda, g_sigma*Xi-, g_sigma*Sigma- = 7.2, 4.0, 0
Fixed to reproduce hypernuclear separation energies such as B_LambdaLambda for 11_LambdaLambda Be and the Kiso event for Xi hypernuclei. The values are empirical but model dependent.
K-N range and Lambda(1405) parameters = g_Lambda* = 0.583, gamma_Lambda* = 12.4 MeV, d_p = 0.351 - Sigma_Kp/m_K, d_n = 0.130 - Sigma_Kn/m_K
Determined from empirical s-wave K-N scattering lengths. In the final self-energy formula (59) the range and Lambda* terms are omitted as subdominant, but they were used to set the input framework.

assumptions (8)

domain assumption The kaon-baryon and kaon-kaon interactions are described by the effective chiral SU(3)_L x SU(3)_R Lagrangian in the mean-field approximation.
Introduced in Sec. II, Eq. (1). This is the standard chiral symmetry framework for s-wave kaon condensation, but it is a model assumption about which terms dominate at high density.
domain assumption The kaon condensate is spatially uniform with spatial momentum k = 0.
Stated in Sec. II around Eq. (3). This restricts the calculation to s-wave condensates and excludes p-wave kaon condensation.
domain assumption Baryon-baryon interactions are described by minimal relativistic mean-field theory with only sigma, sigma*, omega, rho, and phi exchange, without nonlinear self-interacting meson terms.
Defined in Sec. III A, Eq. (17). This is a deliberate modeling choice that shifts all many-body repulsion into the phenomenological three-body potentials.
ad hoc to paper The universal three-baryon repulsion acts identically among nucleons and hyperons with the SJM2 density-dependent potential.
Introduced in Sec. III B, Eq. (23). The paper acknowledges in Sec. IX that the universality of UTBR still needs to be checked against chiral EFT, quark models, and lattice QCD.
domain assumption The three-nucleon attraction has the Nishizaki-Takatsuka-Hiura density-dependent form with isospin dependence.
Introduced in Sec. III C, Eq. (24). This phenomenological form is used to reproduce saturation properties of symmetric nuclear matter.
domain assumption The ground state is in beta equilibrium with neutrinos free, with charge neutrality and baryon number conservation imposed.
Stated in Sec. IV C, Eq. (40) through Eq. (43). This is the standard cold neutron star matter assumption.
domain assumption Meson mean fields and the classical kaon field are uniform and depend only on total baryon density.
Stated in Sec. IV B. This is standard in relativistic mean-field treatments of neutron star matter.
standard math Neutron star structure follows the Tolman-Oppenheimer-Volkoff equations with a low-density crust from Baym-Pethick-Sutherland.
Used in Sec. VIII to compute mass-radius relations. This is the standard general-relativistic hydrostatic equilibrium framework.

invented entities (1)

Universal three-baryon repulsion (UTBR), SJM2 form
purpose: Provides a density-dependent repulsive effective two-body potential that stiffens the equation of state at high densities and counteracts the softening from hyperons and kaon condensation.
The paper assumes the repulsion is spin-flavor independent and universal across baryon species, motivated by string-junction quark confinement. It is compared against neutron star observations, but the observations do not isolate the UTBR from other high-density repulsion mechanisms, so the entity itself has no independent falsifiable handle in this paper.

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Cite this review

Pith. "Pith review of Properties of a kaon-condensed phase in hyperon-mixed matter with three-baryon forces." pith.science (2026). https://pith.science/paper/IWOK4V54

@misc{pith2026241109967,
  author       = {Pith},
  title        = {Pith review of: Properties of a kaon-condensed phase in hyperon-mixed matter with three-baryon forces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IWOK4V54}},
  note         = {Machine review of arXiv:2411.09967}
}

abstract

Coexistent phase of kaon condensates and hyperons [($Y$+$K$) phase] in beta equilibrium with electrons and muons is investigated as a possible form of dense hadronic phase with multi-strangeness. The effective chiral Lagrangian for kaon-baryon and kaon-kaon interactions is utilized within chiral symmetry approach in combination with the interaction model between baryons. For the baryon-baryon interactions, we adopt the minimal relativistic mean-field theory with exchange of scalar mesons and vector mesons between baryons without including the nonlinear self-interacting meson field terms. In addition, the universal three-baryon repulsion and the phenomenological three-nucleon attraction are introduced as density-dependent effective two-body potentials. The repulsive effects leading to stiff equation of state at high densities consist of both the two-baryon repulsion via the vector-meson exchange and the universal three-baryon repulsion. Interplay of kaon condensates with hyperons through chiral dynamics in dense matter is clarified, and resulting onset mechanisms of kaon condensation in hyperon-mixed matter and the equation of state with the ($Y$+$K$) phase and characteristic features of the system are presented. It is shown that the slope $L$ of the symmetry energy controls the two-baryon repulsion beyond the saturation density and resulting stiffness of the equation of state. The stiffness of the equation of state in turn controls admixture of hyperons and the onset and development of kaon condensates as a result of competing effect between kaon condensates and hyperons. The equation of state with the ($Y$+$K$) phase becomes stiff enough to be consistent with recent observations of massive neutron stars. Static properties of neutron stars with the ($Y$+$K$) phase are discussed.

Figures

Figures reproduced from arXiv: 2411.09967 by the authors.

**Figure 2.** FIG. 2. The total energy per nucleon, [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3. The total energy per baryon, [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (14 more)

**Figure 2.** Figure 2: The deficit attraction in the E (two-body) is compensated with more attractive TNA than the case of LP (1981). For ρB & ρ0, the E (TNA) in both cases decrease in magnitude and become negligible for ρB & 0.4 fm−3 . C. Dependence of the EOS for SNM and PNM on the slope L…

**Figure 4.** Figure 4: FIG. 4. The total energy per nucleon, [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5. The pressure in SNM (black curves) and in PNM [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: FIG. 6. (a) The Λ potential [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: FIG. 7. (a) The lowest [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: FIG. 8. (a) The particle fractions in the ( [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: FIG. 9. (a) The scalar densities for baryons [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 10.** Figure 10: FIG. 10. (a) The total energy per unit of baryon [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

**Figure 11.** Figure 11: FIG. 11. (a) The pressure, [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]

**Figure 12.** Figure 12: FIG. 12. (a) The sound velocity [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]

**Figure 13.** Figure 13: FIG. 13. The gravitational mass [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]

**Figure 15.** Figure 15: FIG. 15. (a) The density profiles of a compact star with [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]

**Figure 16.** Figure 16: FIG. 16. (a) The same as Fig. 15 (a) but for [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]

**Figure 17.** Figure 17: FIG. 17. The result for the energy contributions with the [PITH_FULL_IMAGE:figures/full_fig_p030_17.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

163 extracted references · 65 canonical work pages

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Incompressibility K0 K0 = 9 ρ0 ( ∂2E/∂ρ2 B ) ρ B=ρ 0 = K (kin) 0 +K (σ ) 0 +K (ω ) 0 +K (UTBR) 0 +K (TNA) 0 ,(67) where K (kin) 0 = 9 ρ0 π2 2 1 pF~E(pF ) , (68a) K (σ ) 0 = − (gσNM ∗ N/~E(pF ))2 m2 σ +g2 σNI (pF ) , (68b) K (ω ) 0 = (gωN mω )2 , (68c) K (UTBR) 0 = (2688 .12)π3/ 2(~λr)3ρ2 0 , (68d) K (TNA) 0 = 27 γaρ2 0e−η aρ 0 (6 − 6ηaρ0 +η2 aρ2
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MeV in the case of Σ Kn = 300 MeV, and (1.45, 1.74, 1.92)M⊙ forL = (60, 65, 70) MeV in the case of Σ Kn = 400 MeV, as is also seen from Figs. 13 and 14. Constraints on mass and radius from recent observa- tions of some neutron stars are indicated in Fig. 13. 22 400 600 800 1000 1200 1400 (MeV fm−3 ) P (MeV fm−3 ) L = 60 MeV (a) Σ Kn = 300 MeV L = 65 MeV L...
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K-baryon sigma term

MeV. Thus the eﬀects of muons on the onset of KC and structure of compact stars are small. B. Density distribution in the core for typical masses 2.0 M⊙ and 1.4 M⊙ The density proﬁles of a compact star with M = 2.0 M⊙ and M = 1.4 M⊙, as a function of the radial distancer from the center, are shown in Figs. 15 and 16. The former is for L = 65 MeV with (a) ...
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Symmetry energy S0 S0 = 1 8 (∂2E ∂x2 p ) ρ B =ρ 0 = S(kin) 0 +S(ρ) 0 +S(TNA) 0 , (70) where S(kin) 0 = p2 F 6~E(pF ) , (71a) S(ρ) 0 = 1 2 (gρN mρ )2 ρ0 , (71b) S(TNA) 0 = −2γaρ2 0e−η aρ 0 . (71c)
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[4] [3]

K-baryon sigma term

MeV. Thus the eﬀects of muons on the onset of KC and structure of compact stars are small. B. Density distribution in the core for typical masses 2.0 M⊙ and 1.4 M⊙ The density proﬁles of a compact star with M = 2.0 M⊙ and M = 1.4 M⊙, as a function of the radial distancer from the center, are shown in Figs. 15 and 16. The former is for L = 65 MeV with (a) ...

[5] [4]

Baym and D

G. Baym and D. K. Campbell, Mesons and Nuclei , ed. M. Rho and D. H. Wilkinson (North Holland, Am- sterdam, 1979),Vol. III, p.1031

1979

[6] [5]

The integral I (pF ) in Eq

(68e) with ~E(pF ) ≡ √ p2 F +M ∗2 N ,M ∗ N ( =MN −gσN ⟨σ⟩0) be- ing the eﬀective nucleon mass at ρ0. The integral I (pF ) in Eq. (68b) is given by I (pF ) ≡ 2 π2 ∫ pF 0 dp p4 (p2 +M ∗2 N )3/ 2 = 3 ( ρS 0 M ∗ N − ρ0 ~E(pF ) ) (69) with ρS 0 being the nuclear scalar density at ρ0 in SNM (ρS 0 ≡ρS p +ρS n with ρS p = ρS n for SNM)

[7] [6]

Symmetry energy S0 S0 = 1 8 (∂2E ∂x2 p ) ρ B =ρ 0 = S(kin) 0 +S(ρ) 0 +S(TNA) 0 , (70) where S(kin) 0 = p2 F 6~E(pF ) , (71a) S(ρ) 0 = 1 2 (gρN mρ )2 ρ0 , (71b) S(TNA) 0 = −2γaρ2 0e−η aρ 0 . (71c)

[8] [7]

Slope L of the symmetry energy L ≡ 3ρ0 (dS(ρB) dρB ) ρ B =ρ 0 = 3 ρ0 PPNM(ρ0) = L(kin) +L(ρ) +L(TNA) , (72) where L(kin) = p2 F 6~E(pF ) [ 2 − ( pF ~E(pF ) )2 (73a) + 2 π2 (gσNM ∗ N )2(pF/~E(pF ))3 m2 σ +g2 σNI(pF ) ] , (73b) L(ρ) = 3 2 (gρN mρ )2 ρ0 , (73c) L(TNA) = 6 γaρ2 0(ηaρ0 − 2)e−η aρ 0 . (73d) D. The relativistic mean-ﬁeld model with the nonlinear...