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REVIEW 3 major objections 6 minor 65 references

The nucleon properties in finite temperature and density with Gaussian fluctuations

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

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Pith's one-line read Adding Gaussian fluctuations to the quark meson model makes the nucleon mass drop faster with temperature, rebound in an intermediate range, and makes the nucleon radius grow faster than mean-field theory predicts.

desk verdict A genuinely new combination—Gaussian-fluctuation thermodynamics plus chiral-soliton nucleon properties—with credible numerics, but the homogeneous-background treatment of fluctuations is a load-bearing approximation that a referee should probe. read the letter →

arxiv 2412.19982 v2 pith:BBBGNHJN submitted 2024-12-28 hep-th hep-phnucl-th

Peixin Weng , Bingtao Li , Yiming Lyu , Song Shu , Hui Zhang This is my paper · ORCID
classification hep-thhep-phnucl-th
keywords chiralsolitonmodelnucleonmassGaussianfluctuationsquarkmesonfinitetemperaturedensityradiusphasetransition
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 what happens to a nucleon's mass and size when the hot, dense medium around it is described beyond the usual mean-field approximation. In the two-flavor quark meson model, the nucleon is a chiral soliton, a bound state of three valence quarks in a sigma–pion field. The authors add Gaussian fluctuations of the meson fields on top of the mean field and solve the soliton equations at finite temperature and baryon chemical potential. They report that fluctuations make the nucleon mass drop more steeply with temperature, with an upward bump in an intermediate range, and make the RMS radius expand faster than mean-field theory gives. This matters because heavy-ion collision simulations usually keep hadron masses and sizes fixed during hadronization, and the results suggest those properties are changing substantially near the chiral phase transition.

What carries the argument

The key machinery is the averaged thermodynamic potential $\langle\Omega(\sigma,\pi,T,\mu)\rangle$ built from the mean-field potential plus Gaussian meson fluctuations. The meson fields are split into classical values plus fluctuations $\Delta$ and $\vec{\delta}$; odd fluctuation moments vanish by symmetry, and the potential is replaced by its Gaussian average, computed through the explicit Gaussian-integration formula in the appendix. Meson masses $m_\sigma$, $m_\pi$ and the variances $\langle\Delta^2\rangle$, $\langle\delta^2\rangle$ are fixed by coupled equations together with the gap equation for the chiral condensate. On top of this background sits a hedgehog chiral soliton for the nucleon: three valence quarks in radial wave functions $u(r)$, $v(r)$ plus $\sigma$ and pion profiles $\sigma(r)$, $\pi(r)$, solved from radial field equations in which the potential derivatives are $\partial\langle\Omega\rangle/\partial\sigma$ and $\partial\langle\Omega\rangle/\partial\pi$. The nucleon mass is the energy functional $M_B = N\epsilon + 4\pi\int dr\,r^2[\cdots]$ with a subtraction $B(T,\mu)$ that zeroes the thermal vacuum pressure, and the radius is $R = \sqrt{4\pi\int dr\,r^4(u^2+v^2)}$.

What would settle it

Re-solve the radial soliton equations with $m_\sigma$, $m_\pi$, $\langle\Delta^2\rangle$, and $\langle\delta^2\rangle$ evaluated at the local $\sigma(r)$, $\pi(r)$ values instead of the uniform-background values, and check whether the intermediate-temperature rise in $M_B$ and the faster radius growth survive; if they vanish, the central claims are artifacts of the uniform-background approximation.

Watch

Extended reading notes

Core claim

The paper's central claim is that mesonic Gaussian fluctuations beyond the mean field markedly change the static properties of the chiral-soliton nucleon in a hot, dense medium. With fluctuations included, the nucleon mass $M_B$ decreases more rapidly with temperature than the mean-field result, and in an intermediate temperature range $M_B$ rises instead of continuing to fall; the authors attribute this non-monotonic bump to a fast rise in the quark eigenenergy term in the soliton energy. The RMS radius $R$ likewise grows faster with temperature and chemical potential than mean-field theory predicts, which the paper interprets as an effective repulsive force acting like the repulsive Casimir force seen in the gold-bromobenzene-silica system. The same pattern appears when the chemical potential is varied at fixed temperature. The paper also reports that Gaussian fluctuations alter the phase diagram itself, producing both crossover and first-order transitions rather than a purely first-order line, with the critical end point near $T_c = 137$ MeV and $\mu_c = 255$ MeV.

Load-bearing premise

The load-bearing assumption is that the meson masses and fluctuation variances entering the soliton equations are computed in the homogeneous thermal background and are taken unchanged inside the soliton, where the sigma and pion fields vary with radius; if local fluctuations inside the soliton differ enough from background values, the predicted non-monotonic mass and faster-growing radius could be artifacts of that embedding.

Editorial extensions

If this is right

  • Nucleon mass and radius become strongly temperature- and density-dependent near the chiral phase boundary, so heavy-ion hadronization simulations that keep hadron masses fixed would need to fold in this variation.
  • The non-monotonic rise of $M_B$ in an intermediate temperature range would modify the energy balance in a hot medium, potentially affecting yields and momentum distributions if it persists in more complete models.
  • The faster radius growth and the narrowing gap between $M_B$ and $3M_q$ indicate that the baryon softens and becomes less bound as the medium heats or densifies.
  • Gaussian fluctuations change the phase structure of the model from a single first-order line to a first-order segment plus a crossover segment, with a critical end point at $T_c=137$ MeV, $\mu_c=255$ MeV.
  • Lowering the sigma mass weakens the non-monotonic mass bump, tying the predicted effect to the strength of sigma fluctuations.

Reading between the lines

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

  • A direct numerical check would recompute the soliton equations with $m_\sigma$, $m_\pi$, $\langle\Delta^2\rangle$, and $\langle\delta^2\rangle$ evaluated at the local $\sigma(r)$, $\pi(r)$ values; if the intermediate-temperature bump vanishes, the reported effect is an artifact of the uniform-background embedding.
  • The same Gaussian averaging could be applied to a Polyakov-loop-extended or three-flavor quark meson model, where the shift in the phase transition order would change the predicted location of the critical endpoint.
  • Lattice QCD calculations of nucleon spectral functions at finite temperature could test whether a bound baryon survives with the mass and width behavior the soliton model predicts, rather than only the static soliton energy.
  • If the mass bump translates into hadrochemistry, observables like hadron yield ratios and flow harmonics could show a non-monotonic dependence on collision energy that constant-mass simulations would miss.
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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

3 major / 6 minor

Summary. The paper studies the two-flavor quark-meson model at finite temperature T and quark chemical potential μ, going beyond the mean-field approximation by including Gaussian fluctuations of the meson fields. The authors compute the thermodynamic potential with these fluctuations, solve the coupled equations for the chiral condensate, meson masses, and fluctuation variances in a homogeneous medium, and then embed a chiral soliton into this medium. From the soliton solution they extract the nucleon mass M_B and RMS radius R. The central claims are that Gaussian fluctuations make M_B decrease more rapidly with T than in mean-field theory, produce a non-monotonic increase of M_B in an intermediate temperature range, and cause R to grow more rapidly with T and μ; the non-monotonic behavior is traced to the quark eigenenergy contribution.

Significance. If the central claims hold, the results indicate that hadron masses and radii can change substantially near the chiral phase boundary, which is relevant for modeling hadronization in heavy-ion collisions and for the in-medium modification of hadron properties. The paper is clearly written, the comparison between mean-field and Gaussian-fluctuation results uses the same fitted parameters, and the identification of the quark eigenenergy as the driver of the non-monotonic behavior in Fig. 4 is a useful diagnostic. However, the significance is conditional: the fluctuation feedback into the soliton equations is treated through r-independent homogeneous background values, and this approximation is not tested in the manuscript. Because the central non-monotonic mass feature is attributed to exactly this fluctuation feedback, the robustness of the qualitative conclusions is not yet established.

major comments (3)
  1. [Sec. 2.3, Eqs. (33)-(34) and (18)-(19)] The soliton equations use the averaged potential ⟨Ω(σ(r),π(r),T,μ)⟩ with the meson masses m_σ^2, m_π^2 and fluctuation variances ⟨Δ^2⟩, ⟨δ^2⟩ held at their homogeneous background values. These quantities are solved once from Eqs. (A.7)-(A.9) and (18) in a uniform medium and then treated as r-independent constants. Inside the soliton, σ(r) runs from O(f_π) at the core to σ_v(T,μ) at infinity, so the local curvature of the effective potential and the local fluctuation widths are not the background values; the only r-dependent fluctuation effect is a Gaussian smearing with fixed widths. Since Fig. 4 identifies the quark eigenenergy as the source of the non-monotonic M_B, and the eigenenergy depends on the σ(r),π(r) profile, this approximation is directly load-bearing. Please test the robustness of the central claim, for example by recomputing m_σ^2, m_π^2, ⟨Δ^2⟩, and ⟨δ^2⟩ in a local-density approximation at each r, or by showing quantitatively how much these quantities vary across the soliton at the T and μ values where the non-monotonic feature appears.
  2. [Sec. 2.2, Eqs. (16)-(18)] The treatment of the zero-point energy is inconsistent as written. Eqs. (16)-(17) include the term E_σ/2 and E_π/2 in Ω_σ and Ω_π, but Eq. (18) defines the variances from the thermal part only, omitting the derivative of the zero-point term. The zero-point part is divergent and is neither renormalized nor explicitly subtracted; if it is meant to be dropped, the Ω_m appearing in Eq. (19) should be the thermal-only expression, while if it is kept, Eq. (18) should contain the corresponding vacuum contribution. This choice affects the meson masses, the phase boundary in Fig. 1, and hence the soliton solutions, so the paper should state the regularization/subtraction scheme and apply it consistently.
  3. [Appendix A, Eqs. (A.6)-(A.9)] Eq. (A.6) states that derivatives of averaged quantities contain terms proportional to ∂⟨Δ^2⟩/∂α and ∂⟨δ^2⟩/∂α, but the gap equation (A.9) and the mass equations (A.7)-(A.8) contain no such terms. The authors appear to differentiate at fixed variances, which is a legitimate Hartree-type approximation, but the convention is not stated and it is not clear that the same convention is used when taking the field derivatives in Eqs. (33)-(34). Please clarify this convention; if the variances are not held fixed, the homogeneous background used for the soliton embedding is not a stationary point of ⟨Ω⟩ and the phase boundary in Fig. 1 would shift.
minor comments (6)
  1. [Fig. 6, left panel] The vertical axis is labeled "MB [fm]" but M_B is a mass; the unit should be MeV.
  2. [Sec. 2.1, after Eq. (9)] The sentence "We find that the system is primarily determined by constituent quarks, while the contribution from scalar fields is always zero" is unclear, since σ has a nonzero expectation value; the authors presumably mean the pion condensate.
  3. [Sec. 2.3, Eqs. (28)-(29)] The boundary condition σ(∞)=σ_v is first defined with σ_v as the zero-temperature vacuum value, while the text later replaces it by the thermal expectation value; define σ_v(T,μ) consistently.
  4. [Abstract and Sec. 4] The analogy between the radius growth and the repulsive Casimir force in the gold-bromobenzene-silica system is not derived or quantified; it should be removed or properly qualified.
  5. [Sec. 3, numerical details] The numerical section does not provide the grid size, convergence criterion, or momentum cutoff used in the integrals; these details are needed for reproducibility.
  6. [Abstract] The abstract states non-monotonic behavior "as a function of temperature and density," but the non-monotonic increase is demonstrated in Fig. 3 as a function of T; the μ-dependence in Fig. 6 is described as a rapid monotonic decrease, so the wording should be narrowed.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the thermal behavior of the nucleon mass and radius is obtained by solving coupled gap, meson-mass, fluctuation, and soliton equations with parameters fixed at vacuum; the only self-citation is a non-load-bearing technical subtraction.

full rationale

The paper's central quantities, M_B(T,mu) and R(T,mu), are not fitted inputs. The model parameters (f_pi = 93 MeV, m_pi = 138 MeV, m_q = 500 MeV, m_sigma = 1200 MeV) are fixed at vacuum and are used identically in the mean-field and Gaussian-fluctuation calculations. The Gaussian-fluctuation scheme is a self-consistent system: m_sigma^2 and m_pi^2 are defined as second derivatives of the averaged potential (Eq. 12), the variances <Delta^2> and <delta^2> follow from the meson partition function (Eq. 18), and the gap equation (A.9) closes the system. These coupled equations are solved simultaneously before the soliton profile is obtained from Eqs. (31)-(34). The reported non-monotonic nucleon mass and faster-growing radius emerge from the numerical solution of the soliton equations, not from a parameter adjusted to reproduce those curves. The one self-citation, Ref. [38] by H. Zhang and S. Shu, is used only for the subtraction factor B(T,mu) in Eq. (37), a standard vacuum subtraction that shifts the soliton energy and does not encode the fluctuation-induced T and mu dependence. The Gaussian-fluctuation method itself is imported from the independent Ref. [36]. The homogeneous-background treatment of the fluctuation variances is a modeling approximation and a possible correctness risk, but it is not a circular reduction: the central predictions are not equal to the inputs by construction. Therefore no load-bearing circular step is present; the score reflects only the minor, non-load-bearing self-citation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The calculation rests on standard finite-temperature field theory plus the hedgehog soliton ansatz, and on two model-specific approximations: the Gaussian truncation of meson fluctuations and the use of a homogeneous background for meson masses and variances. The two fitted meson parameters, m_q and m_sigma, are carried over from previous fits to vacuum nucleon properties, so the absolute scale of the nucleon mass is partly constrained by the target observable.

free parameters (2)
  • constituent quark mass m_q in vacuum = 500 MeV
    Chosen following Refs. [50,51] to reproduce the vacuum nucleon mass of 1086 MeV and the magnetic moments; the temperature and density dependence inherits this choice.
  • sigma meson mass m_sigma in vacuum = 1200 MeV
    Chosen following Refs. [50,51] to fit nucleon properties; controls the strength of Gaussian fluctuations, and the authors show the non-monotonic mass behavior weakens as m_sigma decreases.
assumptions (4)
  • domain assumption Finite-temperature Matsubara formalism with neglect of the quark zero-point energy term
    Eq. (9) says 'We neglect this term in our calculations and focus on the contribution from the Gaussian fluctuations'; the absolute scale of the nucleon mass depends on this renormalization choice.
  • domain assumption Hedgehog ansatz for the baryon wave function and fields
    Eqs. (24)-(26) restrict the soliton to spherical symmetry with one spin-isospin locked channel; this is standard for chiral soliton models but limits the configuration space.
  • domain assumption Gaussian truncation of meson fluctuations
    Eq. (13) linearizes the potential and Appendix A averages only the second moments; higher-order correlations are discarded, which is an uncontrolled approximation near the phase boundary.
  • ad hoc to paper Homogeneous background for meson masses and fluctuation variances
    Sec. 2.3 embeds a soliton in a homogeneous thermal background; m_sigma, m_pi, <Delta^2>, and <delta^2> are background quantities not recomputed from the local soliton profile in Eqs. (33)-(34).

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Cite this review

Pith. "Pith review of The nucleon properties in finite temperature and density with Gaussian fluctuations." pith.science (2026). https://pith.science/paper/BBBGNHJN

@misc{pith2026241219982,
  author       = {Pith},
  title        = {Pith review of: The nucleon properties in finite temperature and density with Gaussian fluctuations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BBBGNHJN}},
  note         = {Machine review of arXiv:2412.19982}
}
read the original abstract

We investigate the properties of nucleons at finite temperature and density using a two-flavor quark meson model with Gaussian fluctuations that extend beyond the mean-field approximation. Our findings suggest that Gaussian fluctuations lead to a non-monotonic behavior of the nucleon mass as a function of temperature and density, which may play an important role in the study of the hadronization process of relativistic heavy-ion collisions. Moreover, we observe an increase in the nucleon radius due to Gaussian fluctuations, suggesting an effective repulsive force akin to the Casimir effect, as observed in the gold-bromobenzene-silica system. This study offers new insights into how temperature, density, and quantum fluctuations affect the structure and properties of nucleons under extreme conditions.

Figures

Figures reproduced from arXiv: 2412.19982 by the authors.

Figure 1
Figure 1. illustrates the order parameter as a function of temperature and density, highlighting this feature. In the right panel of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. The RMS radius R as a function of T with (solid lines) and without (dashed lines) Gaussian fluctuations, for µ = 0, 100, 200, 300 MeV. In [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The nucleon mass MB as a function of T with (solid lines) and without (dashed lines) Gaussian fluctuations, for µ = 0, 100, 200, 300 MeV. A crucial consideration is the stability of the triple quark system. In our calculations, a baryon is considered as a stable bound state only if its mass is less than the combined mass of its three constituent quarks. In [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Eigenengergy (left) and fluctuations (right) as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: The nucleon mass MB (dashed lines) and three free constituent quark mass 3Mq (solid lines) as a function of T, for µ = 0, 100, 200, 300 MeV. In [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Nucleon mass MB (left) and RMS radius R (right) as a function of µ with (solid lines) and without (dashed lines) Gaussian fluctuations, for T = 10, 50, 100, 120, 150 MeV. more rapid decrease in the nucleon mass relative to mean-field results. The non-monotonic trend in…

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Works this paper leans on

65 extracted references · 22 canonical work pages

  1. [1]

    Braun-Munzinger and J

    P. Braun-Munzinger and J. Wambach, Rev. Mod. Phys. 81, 1031-1050 (2009) doi:10.1103/RevModPhys.81.1031 [arXiv:0801.4256 [hep-ph]]

    arXiv 2009

  2. [2]

    Luo, Nucl

    X. Luo, Nucl. Phys. A 956, 75-82 (2016) doi:10.1016/j.nuclphysa.2016.03.025 [arXiv:1512.09215 [nucl-ex]]

    arXiv 2016

  3. [3]

    Collins, Camb

    J. Collins, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 32, 1-624 (2011) Cambridge University Press, 2011, ISBN 978-1-009-40184-5, 978-1-009-40183-8, 978-1-009-40182-1 doi:10.1017/9781009401845

    doi:10.1017/9781009401845 2011

  4. [4]

    Ghiglieri, A

    J. Ghiglieri, A. Kurkela, M. Strickland and A. Vuorinen, Phys. Rept. 880, 1-73 (2020) doi:10.1016/j.physrep.2020.07.004 [arXiv:2002.10188 [hep-ph]]

    arXiv 2020

  5. [5]

    Brock et al

    R. Brock et al. [CTEQ], Rev. Mod. Phys. 67, 157-248 (1995) doi:10.1103/RevModPhys.67.157

    doi:10.1103/revmodphys.67.157 1995

  6. [6]

    A. Deur, S. J. Brodsky and G. F. de Teramond, Nucl. Phys. 90, 1 (2016) doi:10.1016/j.ppnp.2016.04.003 [arXiv:1604.08082 [hep-ph]]

    arXiv 2016

  7. [7]

    Experimental Tests of Asymptotic Freedom

    S. Bethke, Prog. Part. Nucl. Phys. 58, 351-386 (2007) doi:10.1016/j.ppnp.2006.06.001 [arXiv:hep-ex/0606035 [hep-ex]]

    work page Pith review arXiv 2007

  8. [8]

    Hagler et al

    P. Hagler et al. [LHPC], Phys. Rev. D 77, 094502 (2008) doi:10.1103/PhysRevD.77.094502 [arXiv:0705.4295 [hep-lat]]

    arXiv 2008

Show all 65 references
  1. [9]

    Hagler, Phys

    P. Hagler, Phys. Rept. 490, 49-175 (2010) doi:10.1016/j.physrep.2009.12.008 [arXiv:0912.5483 [hep-lat]]

    2010 arXiv

  2. [10]

    Liang et al

    J. Liang et al. [XQCD], Phys. Rev. D 101, no.11, 114503 (2020) doi:10.1103/PhysRevD.101.114503 [arXiv:1906.05312 [hep-ph]]

    2020 arXiv

  3. [11]

    S. P. Klevansky, Rev. Mod. Phys. 64, 649-708 (1992) doi:10.1103/RevModPhys.64.649

    1992 doi

  4. [12]

    Hatsuda and T

    T. Hatsuda and T. Kunihiro, Phys. Rept. 247, 221-367 (1994) doi:10.1016/0370-1573(94)90022-1 [arXiv:hep-ph/9401310 [hep-ph]]

    1994 arXiv

  5. [13]

    Fukushima, Phys

    K. Fukushima, Phys. Lett. B 591, 277-284 (2004) doi:10.1016/j.physletb.2004.04.027 [arXiv:hep-ph/0310121 [hep-ph]]

    2004 arXiv

  6. [14]

    Fukushima, Phys

    K. Fukushima, Phys. Rev. D 77, 114028 (2008) [erratum: Phys. Rev. D 78, 039902 (2008)] doi:10.1103/PhysRevD.77.114028 [arXiv:0803.3318 [hep-ph]]

    2008 arXiv

  7. [15]

    Chodos, R

    A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn and V. F. Weisskopf, Phys. Rev. D 9, 3471-3495 (1974) doi:10.1103/PhysRevD.9.3471

    1974 doi

  8. [16]

    M. G. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B 422, 247-256 (1998) doi:10.1016/S0370-2693(98)00051-3 [arXiv:hep-ph/9711395 [hep-ph]]

    1998 arXiv

  9. [17]

    Johnson, Acta Phys

    K. Johnson, Acta Phys. Polon. B 6, 865 (1975) MIT-CTP-494

    1975

  10. [18]

    Hagedorn, Nuovo Cim

    R. Hagedorn, Nuovo Cim. Suppl. 3, 147-186 (1965) CERN-TH-520

    1965

  11. [19]

    Becattini, J

    F. Becattini, J. Cleymans, A. Keranen, E. Suhonen and K. Redlich, Phys. Rev. C 64, 024901 (2001) doi:10.1103/PhysRevC.64.024901 [arXiv:hep-ph/0002267 [hep-ph]]

    2001 arXiv

  12. [20]

    Andronic, P

    A. Andronic, P. Braun-Munzinger and J. Stachel, Nucl. Phys. A 772, 167-199 (2006) doi:10.1016/j.nuclphysa.2006.03.012 [arXiv:nucl-th/0511071 [nucl-th]]

    2006 arXiv

  13. [21]

    Huovinen and P

    P. Huovinen and P. Petreczky, Nucl. Phys. A 837, 26-53 (2010) doi:10.1016/j.nuclphysa.2010.02.015 [arXiv:0912.2541 [hep-ph]]

    2010 arXiv

  14. [22]

    Weinberg, Physica A 96, no.1-2, 327-340 (1979) doi:10.1016/0378-4371(79)90223-1

    S. Weinberg, Physica A 96, no.1-2, 327-340 (1979) doi:10.1016/0378-4371(79)90223-1

    1979 doi

  15. [23]

    Gasser and H

    J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984) doi:10.1016/0003-4916(84)90242-2

    1984 doi

  16. [24]

    Ecker, Prog

    G. Ecker, Prog. Part. Nucl. Phys. 35, 1-80 (1995) doi:10.1016/0146-6410(95)00041-G [arXiv:hep-ph/9501357 [hep-ph]]

    1995 arXiv

  17. [25]

    Scherer, Adv

    S. Scherer, Adv. Nucl. Phys. 27, 277 (2003) [arXiv:hep-ph/0210398 [hep-ph]]

    2003 arXiv

  18. [26]

    R. S. Hayano and T. Hatsuda, Rev. Mod. Phys. 82, 2949 (2010) doi:10.1103/RevModPhys.82.2949 [arXiv:0812.1702 [nucl-ex]]

    2010 arXiv

  19. [27]

    Collaboration et al

    A. Collaboration et al. [ALICE], Nature 588, 232-238 (2020) [erratum: Nature 590, E13 (2021)] doi:10.1038/s41586-020- 3001-6 [arXiv:2005.11495 [nucl-ex]]

    2020 arXiv

  20. [28]

    J. Chen, J. H. Chen, X. Dong, X. He, X. H. He, H. Huang, H. Z. Huang, F. Liu, X. Luo and X. F. Luo, et al. Nucl. Sci. Tech. 35, no.12, 214 (2024) doi:10.1007/s41365-024-01591-2 [arXiv:2407.02935 [nucl-ex]]

    2024 arXiv

  21. [29]

    Q. Y. Shou, Y. G. Ma, S. Zhang, J. H. Zhu, Y. X. Mao, H. Pei, Z. B. Yin, X. M. Zhang, D. C. Zhou and X. Y. Peng, et al. [arXiv:2409.17964 [nucl-ex]]

    arXiv

  22. [30]

    G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Song and T. Takatsuka, Rept. Prog. Phys. 81, no.5, 056902 (2018) doi:10.1088/1361-6633/aaae14 [arXiv:1707.04966 [astro-ph.HE]]

    2018 arXiv

  23. [31]

    M. I. Abdulhamid et al. [STAR], JHEP 10, 139 (2024) doi:10.1007/JHEP10(2024)139 [arXiv:2407.10110 [nucl-ex]]

    2024 arXiv

  24. [32]

    Petropoulos, J

    N. Petropoulos, J. Phys. G 25, 2225-2241 (1999) doi:10.1088/0954-3899/25/11/305 [arXiv:hep-ph/9807331 [hep-ph]]

    1999 arXiv

  25. [33]

    Nemoto, K

    Y. Nemoto, K. Naito and M. Oka, Eur. Phys. J. A 9, 245-259 (2000) doi:10.1007/s100500070042 [arXiv:hep-ph/9911431 [hep-ph]]

    2000 arXiv

  26. [34]

    Scavenius, A

    O. Scavenius, A. Mocsy, I. N. Mishustin and D. H. Rischke, Phys. Rev. C 64, 045202 (2001) doi:10.1103/PhysRevC.64.045202 [arXiv:nucl-th/0007030 [nucl-th]]

    2001 arXiv

  27. [35]

    Baacke and S

    J. Baacke and S. Michalski, Phys. Rev. D 67, 085006 (2003) doi:10.1103/PhysRevD.67.085006 [arXiv:hep-ph/0210060 [hep-ph]]

    2003 arXiv

  28. [36]

    Mocsy, I

    A. Mocsy, I. N. Mishustin and P. J. Ellis, Phys. Rev. C 70, 015204 (2004) doi:10.1103/PhysRevC.70.015204 [arXiv:nucl- th/0402070 [nucl-th]]

    2004

  29. [37]

    E. S. Bowman and J. I. Kapusta, Phys. Rev. C 79, 015202 (2009) doi:10.1103/PhysRevC.79.015202 [arXiv:0810.0042 [nucl-th]]

    2009 arXiv

  30. [38]

    Zhang, R

    H. Zhang, R. Dong and S. Shu, Int. J. Mod. Phys. E 24, no.04, 1550025 (2015) doi:10.1142/S0218301315500251 [arXiv:1504.02248 [hep-ph]]

    2015 arXiv

  31. [39]

    Zhang, D

    H. Zhang, D. Hou, T. Kojo and B. Qin, Phys. Rev. D 96, no.11, 114029 (2017) doi:10.1103/PhysRevD.96.114029 [arXiv:1709.05654 [hep-ph]]. 11

    2017 arXiv

  32. [40]

    Osman, D

    M. Osman, D. Hou, W. Wang and H. Zhang, [arXiv:2402.15474 [hep-ph]]

    arXiv

  33. [41]

    B. J. Schaefer, J. M. Pawlowski and J. Wambach, Phys. Rev. D 76, 074023 (2007) doi:10.1103/PhysRevD.76.074023 [arXiv:0704.3234 [hep-ph]]

    2007 arXiv

  34. [42]

    Skokov, B

    V. Skokov, B. Stokic, B. Friman and K. Redlich, Phys. Rev. C 82, 015206 (2010) doi:10.1103/PhysRevC.82.015206 [arXiv:1004.2665 [hep-ph]]

    2010 arXiv

  35. [43]

    Skokov, B

    V. Skokov, B. Friman and K. Redlich, Phys. Rev. C 83, 054904 (2011) doi:10.1103/PhysRevC.83.054904 [arXiv:1008.4570 [hep-ph]]

    2011 arXiv

  36. [44]

    T. K. Herbst, J. M. Pawlowski and B. J. Schaefer, Phys. Lett. B 696, 58-67 (2011) doi:10.1016/j.physletb.2010.12.003 [arXiv:1008.0081 [hep-ph]]

    2011 arXiv

  37. [45]

    T. K. Herbst, J. M. Pawlowski and B. J. Schaefer, Phys. Rev. D 88, no.1, 014007 (2013) doi:10.1103/PhysRevD.88.014007 [arXiv:1302.1426 [hep-ph]]

    2013 arXiv

  38. [46]

    Fukushima and V

    K. Fukushima and V. Skokov, Prog. Part. Nucl. Phys.96, 154-199 (2017) doi:10.1016/j.ppnp.2017.05.002 [arXiv:1705.00718 [hep-ph]]

    2017 arXiv

  39. [47]

    Tanihata, H

    I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N. Yoshikawa, K. Sugimoto, O. Yamakawa, T. Kobayashi and N. Takahashi, Phys. Rev. Lett. 55, 2676-2679 (1985) doi:10.1103/PhysRevLett.55.2676

    1985 doi

  40. [48]

    Abrahamyan, Z

    S. Abrahamyan, Z. Ahmed, H. Albataineh, K. Aniol, D. S. Armstrong, W. Armstrong, T. Averett, B. Babineau, A. Barbieri and V. Bellini, et al.Phys. Rev. Lett. 108, 112502 (2012) doi:10.1103/PhysRevLett.108.112502 [arXiv:1201.2568 [nucl-ex]]

    2012 arXiv

  41. [49]

    Gallimore and J

    D. Gallimore and J. Liao, Nucl. Phys. A 1055, 123012 (2025) doi:10.1016/j.nuclphysa.2024.123012 [arXiv:2405.07077 [hep-ph]]

    2025

  42. [50]

    M. C. Birse and M. K. Banerjee, Phys. Lett. B 136, no.4, 284-288 (1984) doi:10.1016/0370-2693(84)91162-6

    1984 doi

  43. [51]

    M. C. Birse and M. K. Banerjee, Phys. Rev. D 31, 118 (1985) doi:10.1103/PhysRevD.31.118

    1985 doi

  44. [52]

    Alberto, E

    P. Alberto, E. Ruiz Arriola, M. Fiolhais, F. Grummer, J. N. Urbano and K. Goeke, Phys. Lett. B 208, 75-78 (1988) doi:10.1016/0370-2693(88)91206-3

    1988 doi

  45. [53]

    Bernard and U

    V. Bernard and U. G. Meissner, Nucl. Phys. A 489, 647-670 (1988) doi:10.1016/0375-9474(88)90114-5

    1988 doi

  46. [54]

    T. S. T. Aly, J. A. McNeil and S. Pruess, Phys. Rev. D 60, 114022 (1999) doi:10.1103/PhysRevD.60.114022 [arXiv:hep- ph/9809473 [hep-ph]]

    1999

  47. [55]

    Naar and M

    E. Naar and M. C. Birse, J. Phys. G 19, 555-568 (1993) doi:10.1088/0954-3899/19/4/011

    1993 doi

  48. [56]

    Abu-Shady and H

    M. Abu-Shady and H. M. Mansour, Phys. Rev. C 85, 055204 (2012) doi:10.1103/PhysRevC.85.055204

    2012 doi

  49. [57]

    H. Mao, T. Wei and J. Jin, Phys. Rev. C 88, 035201 (2013) doi:10.1103/PhysRevC.88.035201 [arXiv:1301.6227 [hep-ph]]

    2013 arXiv

  50. [58]

    Zhang and S

    H. Zhang and S. Shu, Chin. Phys. C 39, no.9, 094104 (2015) doi:10.1088/1674-1137/39/9/094104 [arXiv:1401.1021 [hep- ph]]

    2015 arXiv

  51. [59]

    Abu-Shady and H

    M. Abu-Shady and H. M. Mansour, J. Phys. G 43, no.2, 025001 (2016) doi:10.1088/0954-3899/43/2/025001 [arXiv:1507.02214 [hep-ph]]

    2016 arXiv

  52. [60]

    Jin and H

    J. Jin and H. Mao, Phys. Rev. C 93, no.1, 015202 (2016) doi:10.1103/PhysRevC.93.015202 [arXiv:1508.03920 [hep-ph]]

    2016 arXiv

  53. [61]

    Y. Li, J. Hu and H. Mao, Phys. Rev. C 97, no.5, 054313 (2018) doi:10.1103/PhysRevC.97.054313 [arXiv:1801.09202 [hep-ph]]

    2018 arXiv

  54. [62]

    J. Wang, Z. Yu and H. Mao, Chin. Phys. C 48, no.5, 053105 (2024) doi:10.1088/1674-1137/ad2a4b [arXiv:2309.13529 [hep-ph]]

    2024 arXiv

  55. [63]

    Skokov, B

    V. Skokov, B. Friman, E. Nakano, K. Redlich and B. J. Schaefer, Phys. Rev. D 82, 034029 (2010) doi:10.1103/PhysRevD.82.034029 [arXiv:1005.3166 [hep-ph]]

    2010 arXiv

  56. [64]

    J. N. Munday, F. Capasso and V. A. Parsegian, Nature 457, no.7226, 170-173 (2009) doi:10.1038/nature07610

    2009 doi

  57. [65]

    R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022) doi:10.1093/ptep/ptac097 12

    2022 doi

Pith tools

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