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REVIEW 3 major objections 5 minor 70 references

Polyakov linear-sigma model in mean-field approximation and optimized perturbation theory

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

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Pith's one-line read In the Polyakov linear-sigma model, optimized perturbation theory follows lattice QCD more closely than the mean-field approximation does, and the gap grows with the order of the conserved-charge moments.

desk verdict A legitimate but under-supported comparison of OPT vs MFA in PLSM for high-order cumulants, undermined by unspecified Polyakov potential parameters and purely visual agreement claims. read the letter →

arxiv 1908.05939 v1 pith:KUXTE4S6 submitted 2019-08-16 hep-ph hep-th

Abdel Nasser Tawfik (Nile U. , ECTP , Johann Wolfgang Goethe-Universitat) , Carsten Greiner (Johann Wolfgang Goethe-Universitat) , Abdel Magied Diab (Egyptian Ctr. Theor. Phys. , Cairo , WLCAPP , Cairo)
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M.T. Ghoneim H. Anwer (Cairo U.)
This is my paper · ORCID
classification hep-phhep-th PACS 11.30.Rd11.10.Wx12.39.Fe02.60.-x
keywords chiralsymmetriestransitionLagrangiannumericalapproximationandanalysisoptimizedperturbationtheorymean-fieldPolyakovlinear-sigmamodelconserved-chargecumulants
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether optimized perturbation theory (OPT), a variationally tuned resummation scheme, improves on the standard mean-field approximation (MFA) when both are applied to the Polyakov linear-$\sigma$ model, a low-energy effective description of the quark-hadron transition. It establishes that the two schemes give nearly identical chiral condensates, subtracted condensates, and pseudo-critical temperatures at small chemical potential, but that OPT is closer to lattice QCD for the thermodynamic pressure and for fluctuations and correlations of conserved charges. The gap widens with the order of the cumulant: the ratios $\chi_6^B/\chi_2^B$ and $\chi_8^B$ computed in OPT agree with lattice QCD above about $1.2\,T_\chi$, where the mean-field curves drift away. Because these high-order moments are the proposed experimental signatures of the deconfinement crossover, the result matters for interpreting heavy-ion collision data with an affordable effective model.

What carries the argument

The engine is the $\delta$-expansion of the PLSM Lagrangian. The full Lagrangian is rewritten as $L = L_0(\eta) + \delta \,[L - L_0(\eta)]$, where $L_0(\eta)$ is a solvable free Lagrangian containing an arbitrary mass parameter $\eta$; results are evaluated at $\delta = 1$ after truncating at order $\delta^2$. The variational parameter $\eta$ is not chosen by hand but fixed by the principle of minimal sensitivity, $\partial F_{\mathrm{OPT}}/\partial \eta = 0$ at $\delta = 1$, which makes the final answer approximately independent of the artificial mass. This $\eta$ enters the quark dispersion relation $E_f = [|\mathbf{P}|^2 + (m_f + \eta)^2]^{1/2}$ and produces an extra term in the quark-antiquark thermodynamic potential that is absent in the mean-field limit, recovered by taking $\eta = 0$ at $\delta = 1$. The Polyakov-loop potential $U(\varphi,\varphi^*,T)$ is left outside the optimization and imported unchanged from a prior fit.

What would settle it

Tighter lattice QCD data on $\chi_8^B$ in the range $T/T_\chi \approx 1.2$ to $2.0$—the current comparison is made within large error bars—would settle whether OPT really sits closer to the central values than MFA does. A second check would be to recompute the high-order cumulants with the gluon-sector potential also included in the optimization; if the OPT advantage disappears or reverses, the paper's conclusion would not survive.

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Extended reading notes

Core claim

The central claim is that, within the Polyakov linear-$\sigma$ model, the optimized perturbation theory—a $\delta$-expansion of the Lagrangian with an arbitrary mass parameter $\eta$ fixed by the principle of minimal sensitivity—provides a systematically better account of the QCD equation of state and of conserved-charge fluctuations than the mean-field limit of the same model. The chiral order parameters and the crossover temperature are largely unaffected by the choice of approximation, but the story changes for derivatives of the free energy with respect to chemical potentials. For the quadratic susceptibilities of baryon number, electric charge, and strangeness, and for their mutual correlations, OPT sits closer to the lattice data than MFA does. The most distinctive result is at high order: $\chi_6^B/\chi_2^B$ and $\chi_8^B$, computed in OPT, match the lattice QCD bands at $T/T_\chi \gtrsim 1.2$, a region where the MFA curves are visibly less accurate. The paper's conclusion is that moving from lower- to higher-order moments makes the OPT approach progressively more reliable than MFA.

Load-bearing premise

The paper's conclusion rests on trusting the truncated optimization procedure, with its adjustable mass fixed by a minimal-sensitivity condition, to be a faithful approximation of the model's free energy, while the gluon-sector potential is taken from an earlier fit and left unoptimized; if either piece misrepresents the theory, the claimed improvement of OPT over MFA could be an artifact.

Editorial extensions

If this is right

  • Within the Polyakov linear-sigma model at finite temperature and chemical potential, OPT is the approximation to use for the pressure and the cumulants of conserved charges, while MFA remains adequate for the order parameters and for the crossover line at small $\mu_B$.
  • The ratios $\chi_6^B/\chi_2^B$ and $\chi_8^B$ become practical discriminators between approximation schemes: OPT places them inside the lattice-QCD bands for $T/T_\chi \gtrsim 1.2$, where MFA drifts away.
  • At large baryon chemical potential the two schemes disagree on the pseudo-critical temperature, with OPT giving a slightly higher $T_\chi$, so estimates of the phase boundary in that regime should carry the difference as a systematic uncertainty.
  • Because OPT reproduces the off-diagonal correlations $\chi_{11}^{QS}$ and $-\chi_{11}^{BS}$ better than MFA, these correlations become reliable tests of whether the model is in the right regime.

Reading between the lines

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

  • Not pursued in the paper: applying the same $\delta$-expansion to the gluon-sector Polyakov-loop potential rather than importing it from a prior fit; if the remaining gap between OPT and lattice QCD shrinks or grows, it would reveal how much of the agreement is carried by the unoptimized gluon part.
  • If the paper's ordering is correct—OPT improves with cumulant order—the method should be tested on even higher moments such as $\chi_{10}^B$ and on strangeness-related high-order cumulants, where lattice data with small errors could separate the schemes more decisively.
  • An interpretation the paper does not make explicit is that the variational mass $\eta$ resums quark thermal self-energy contributions that the mean-field approximation drops, which would explain why the improvements show up in chemical-potential derivatives of the free energy rather than in the order parameters themselves.
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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 / 5 minor

Summary. The manuscript compares two approximation schemes for the Polyakov linear-sigma model (PLSM): the mean-field approximation (MFA) and optimized perturbation theory (OPT), a δ-expansion with the variational mass parameter fixed by the principle of minimal sensitivity. The authors compute chiral condensates, Polyakov-loop expectation values, pseudo-critical temperatures, the thermodynamic pressure, and second- and higher-order cumulants of conserved charges, and compare these with lattice QCD results. The central claim is that, as one moves from lower- to higher-order moments, OPT agrees with lattice QCD better than MFA, with the improvement highlighted for χ6_B/χ2_B and χ8_B. The analysis is performed at finite temperature and finite quark chemical potential, and the paper also maps out the QCD phase diagram in both approximations.

Significance. If the central claim is supported, the paper would provide a useful demonstration that a truncated δ-expansion with PMS can outperform the mean-field treatment for higher-order fluctuations in a chiral effective model with Polyakov-loop dynamics. The comparison uses external lattice data that are not used to fit the OPT parameters, so the agreement is not circular in the strict sense. The paper also contains explicit expressions for the OPT free energy and the gap equations, which can be adapted by other practitioners. However, the significance is currently limited by the lack of a quantitative closeness metric, by the absence of specified values for the Polyakov-loop potential parameters a and b in Eq. (5), and by an internal tension between the pressure results (where OPT is farther from the Stefan-Boltzmann limit than MFA) and the claimed higher-order improvement. These issues are addressable, but they must be resolved before the headline conclusion can be accepted.

major comments (3)
  1. [Section II, Eq. (5)] The parameters a and b of the Polyakov-loop potential U(φ,φ*,T) are nowhere given numerical values, and they are not listed among the parameters fixed at m_σ = 800 MeV. Since U enters the free energy in Eq. (9) and therefore all cumulant ratios, the absolute scale of χ6_B/χ2_B and χ8_B, and hence their visual agreement with lattice data, depends on these unstated inputs. The manuscript cannot be reproduced from the information provided, and no sensitivity analysis is given for the dependence of the higher-order moments on a and b. This is load-bearing for the central claim that OPT becomes closer to QCD at high order.
  2. [Section III D 2, Fig. 8] The headline conclusion that OPT agrees 'excellently' with lattice QCD for χ6_B/χ2_B and χ8_B is based entirely on visual inspection of Fig. 8. There is no quantitative measure of closeness, no error estimate or uncertainty band on the model curves, and no statement of the temperature range over which the agreement is claimed. Given that the lattice points carry sizable error bars and that the two model curves are close to each other in several regions, a numerical comparison (for example, a weighted squared deviation or a chi-square per degree of freedom over a stated T/Tχ interval) is needed to support the claim that OPT is actually better than MFA.
  3. [Section III C and Section IV] There is an internal consistency issue between the pressure results and the overall conclusion. In Section III C the authors state that at T ≤ 2.5Tχ the gap to the Stefan-Boltzmann limit is 31.8% for MFA and 34.9% for OPT, meaning OPT is farther from the ideal-gas limit for the pressure. Yet Section IV concludes that 'when moving to lower- to higher-order moments, the OPT approach becomes more and more reliable than the MFA.' The paper does not explain why a worse low-order quantity (pressure) is consistent with better high-order cumulants, or whether the pressure discrepancy is a signature of the truncation or of the fixed Polyakov-loop potential. This needs to be addressed explicitly, since the central narrative depends on the higher-order improvement being meaningful rather than accidental.
minor comments (5)
  1. [Abstract and Section III D 2] The phrase 'OPT becomes more closer to QCD' is ungrammatical; it should read 'OPT becomes closer to QCD' or 'OPT moves closer to QCD'.
  2. [Figure 2, panel (d)] The label 'µf=0.150 MeV' appears to be a typo; it should presumably read 'µf=150.0 MeV' to match the other panels.
  3. [Figure 3, caption] The caption repeats the label '(a)' and the lattice reference twice, and the embedded text in the figure image duplicates the MFA and OPT labels; the figure should be cleaned up.
  4. [Section II, Eq. (11)] The sentence stating that the second term in Eq. (11) 'is approximately equal the derivative of the first term' is missing a word ('to') and would benefit from a more precise mathematical statement of the approximate relation.
  5. [References] Reference [21], 'A. N. Tawfik, A. M. Diab, M. T. Ghoneim, and H. Anwer, (2019)', lacks a journal or preprint identifier and cannot be located by the reader; a complete citation should be provided.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the OPT-versus-MFA comparison is anchored by external lattice data, and no load-bearing parameter is fitted to the compared quantities.

full rationale

The paper's central claim is that PLSM in optimized perturbation theory becomes closer to lattice QCD than the mean-field approximation for higher-order conserved-charge cumulants. The comparison is made against external lattice data (e.g. refs. [53] and [61] in Fig. 8), and the OPT parameters are not fitted to those data. The OPT free energy is derived from a δ-expansion with the variational mass η fixed by the principle of minimal sensitivity, Eq. (6), while the mean-field result is the δ→1, η=0 limit of the same free energy, Eqs. (9)-(11). Both calculations share the same Polyakov-loop potential U, Eq. (5); as the paper states, 'the Polyakov Lagrangian, Eq. (5), isn't directly impacted by the OPT approach.' Thus the difference between OPT and MFA is generated by the quark-meson sector rather than by a re-fit of lattice results. The manuscript does not specify numerical values for a and b in Eq. (5), and it cites earlier PLSM work by the same authors for the model setup, but these are transparency and provenance concerns rather than circular reductions: the lattice data are not used as inputs to determine the reported OPT curves, and no derived quantity is equal by construction to a fitted input. The self-citations establish the model framework but are not the load-bearing evidence for the claimed agreement with QCD, which rests on direct comparison with independent lattice simulations. Therefore no circular step meeting the required standard of explicit reduction is present.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a model with a dozen or so unspecified parameters inherited from prior fits; the paper does not state their values, so the reader cannot tell whether the reported OPT advantage is robust or an artifact of a particular parameter choice.

free parameters (3)
  • LSM tree-level parameters (m^2, lambda1, lambda2, c, hl, hs) = Not stated; fixed by m_sigma=800 MeV
    These are set by vacuum hadron properties such as the sigma mass and pion decay constant, not by finite-T lattice data. Their values determine the chiral transition and all subsequent observables.
  • Polyakov-loop potential coefficients a and b = Not stated
    Appear in Eq. (5). The paper does not define their values; in the PLSM literature they are typically fixed by lattice pure-gauge thermodynamics, which would make part of the agreement with lattice QCD inherited rather than predictive.
  • Yukawa coupling g = Not stated
    Determines the constituent quark mass m_q = g f_pi; not listed in the paper, though it enters the fermion sector Eq. (3) and affects all chiral condensates and fluctuations.
assumptions (4)
  • standard math The path-integral representation and Matsubara summation used to derive the Fermi-Dirac distribution functions in Eqs. (12)-(13) are valid.
    Standard finite-temperature field theory; not proven in the paper but accepted background.
  • domain assumption The Polyakov-loop potential form in Eq. (5) is the correct effective potential for gluonic degrees of freedom at finite T and mu.
    The paper uses this specific logarithmic form without derivation, and its parameters a and b are not given.
  • domain assumption The O(4) linear-sigma model with Nf=2+1 quark flavors captures the relevant chiral dynamics of QCD in the temperature and chemical potential range studied.
    Effective model assumption common to PLSM; not derived from QCD.
  • domain assumption The principle of minimal sensitivity, Eq. (6), uniquely fixes the variational mass eta and that truncation at O(delta^2) is adequate.
    OPT/PMS is an approximation scheme with no controlled error estimate; the paper assumes the delta-expansion converges at second order.

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Pith. "Pith review of Polyakov linear-sigma model in mean-field approximation and optimized perturbation theory." pith.science (2026). https://pith.science/paper/KUXTE4S6

@misc{pith2026190805939,
  author       = {Pith},
  title        = {Pith review of: Polyakov linear-sigma model in mean-field approximation and optimized perturbation theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KUXTE4S6}},
  note         = {Machine review of arXiv:1908.05939}
}
read the original abstract

We compare results from the Polyakov linear-sigma model (PLSM) in optimized perturbation theory (OPT) with the mean-field approximation (MFA). At finite temperatures and chemical potentials, the chiral condensates and the decofinement order parameters, the thermodynamic pressure, the pseudo-critical temperatures, the subtracted condensates, the second- and high-order moments of various conserved charges (cumulants) obtained in MFA are compared with OPT and also confronted to available lattice QCD simulations. We conclude that when moving from lower- to higher-order moments of various quantum charges, OPT becomes more closer to QCD.

Figures

Figures reproduced from arXiv: 1908.05939 by the authors.

Figure 7
Figure 7. At high temperatures, the temperature dependence of [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗

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Works this paper leans on

70 extracted references · 64 canonical work pages

  1. [1]

    Second-Order Moments 11

    work page

  2. [2]

    Conclusions 14 Acknowledgements 14 References 14 ∗ Electronic address: tawfik@itp.uni-frankfurt.de 2 I

    Higher-Order Moments 12 IV. Conclusions 14 Acknowledgements 14 References 14 ∗ Electronic address: tawfik@itp.uni-frankfurt.de 2 I. INTRODUCTION In theory of quantized fields [1–6], the linear-sigma model [7] with a sp inless scalar field σa [8] and triplet pseudoscalar fields πa was introduced to describe the pion-nucleon interactions and the c hiral degrees...

    work page 2010

  3. [3]

    In the calculations presented in Fig

    Second-Order Moments With this regard, the PLSM results for the second-order fluctuat ions are deduced at i +j +k = 2. In the calculations presented in Fig. 6, we consider the MF A and OPT approa ches at µf = 0. The PLSM results are compared with recent LQCD calculations. 0 0.1 0.2 0.3 0.4 0.5 (a) χ2 B HOTQCD: PRD 86, 034509 (2012) 0 0.1 0.2 0.3 0.4 0.5 (a...

    work page 2012

  4. [4]

    Schwinger, Phys

    J. Schwinger, Phys. Rev. 92, 1283 (1953)

    work page 1953

  5. [5]

    Higher-Order Moments 0 0.2 0.4 0.6 0.8 1 1.2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (a) Net-Baryonic κσ 2 T/Tχ HOQCD: PRD 95, 054504 (2017) 0 0.2 0.4 0.6 0.8 1 1.2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (a) Net-Baryonic κσ 2 T/Tχ HOQCD: PRD 95, 054504 (2017) PLSM in MFA PLSM in OPT 0 0.5 1 1.5 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (b) Net-Charge κσ 2 T/Tχ LQCD: PRD 79, 074505 (2009) 0...

    work page 2017

  6. [6]

    Schwinger, Phys

    J. Schwinger, Phys. Rev. 94, 1362 (1954)

    work page 1954

  7. [7]

    J. S. Schwinger, Phys. Rev. 82, 914 (1951)

    work page 1951

  8. [8]

    J. S. Schwinger, Phys. Rev. 91, 713 (1953)

    work page 1953

Show all 70 references
  1. [9]

    Schwinger, Phys

    J. Schwinger, Phys. Rev. 91, 728 (1953)

    1953

  2. [10]

    Roder, J

    D. Roder, J. Ruppert, and D. H. Rischke, Phys. Rev. D68, 016003 (2003)

    2003

  3. [11]

    Schwinger, Phys

    J. Schwinger, Phys. Rev. 93, 615 (1954)

    1954

  4. [12]

    A. N. Tawfik and A. M. Diab, Phys. Rev. C91, 015204 (2015)

    2015

  5. [13]

    Gell-Mann and M

    M. Gell-Mann and M. Levy, Nuovo Cim. 16, 705 (1960)

    1960

  6. [14]

    J. S. Schwinger, Annals Phys. 2, 407 (1957)

    1957

  7. [15]

    M. C. Birse, J. Phys. G20, 1537 (1994)

    1994

  8. [16]

    A. N. Tawfik, A. M. Diab, and M. T. Hussein, Chin. Phys. C43, 034103 (2019)

    2019

  9. [17]

    Gallas, F

    S. Gallas, F. Giacosa, and D. H. Rischke, Phys. Rev. D82, 014004 (2010)

    2010

  10. [18]

    A. N. Tawfik, A. M. Diab, and T. M. Hussein, Int. J. Adv. Res . Phys. Sci. 3, 4 (2016)

    2016

  11. [19]

    C. Wesp, H. van Hees, A. Meistrenko, and C. Greiner, Eur. Phys. J. A54, 24 (2018)

    2018

  12. [20]

    A. N. Tawfik, A. M. Diab, and M. T. Hussein, J. Phys. G45, 055008 (2018)

    2018

  13. [21]

    A. M. Abdel Aal Diab and A. N. Tawfik, EPJ Web Conf. 177, 09005 (2018)

    2018

  14. [22]

    Randrup, Nucl

    J. Randrup, Nucl. Phys. A616, 531 (1997)

    1997

  15. [23]

    A. N. Tawfik, A. M. Diab, N. Ezzelarab, and A. G. Shalaby, A dv. High Energy Phys. 2016, 1381479 (2016)

    2016

  16. [24]

    M. H. Thoma and H. J. Mang, Z. Phys. C44, 349 (1989)

    1989

  17. [25]

    A. N. Tawfik, A. M. Diab, and M. T. Hussein, J. Exp. Theor. P hys. 126, 620 (2018)

    2018

  18. [26]

    A. N. Tawfik, A. M. Diab, and M. T. Hussein, Int. J. Mod. Phy s. A31, 1650175 (2016)

    2016

  19. [27]

    A. N. Tawfik, A. M. Diab, M. T. Ghoneim, and H. Anwer, (2019 )

    2019

  20. [28]

    P. M. Stevenson, Phys. Rev. D23, 2916 (1981)

    1981

  21. [29]

    Kunihiro and T

    T. Kunihiro and T. Hatsuda, Prog. Theor. Phys. 71, 1332 (1984)

    1984

  22. [30]

    Kneur, M

    J.-L. Kneur, M. B. Pinto, and R. O. Ramos, Phys. Rev. C81, 065205 (2010)

    2010

  23. [31]

    M. H. Thoma, Z. Phys. C44, 343 (1989)

    1989

  24. [32]

    Weinberg, The Quantum theory of fields

    S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations , Cambridge University Press, 2005

    2005

  25. [33]

    Schaefer and M

    B.-J. Schaefer and M. Wagner, Phys. Rev. D79, 014018 (2009)

    2009

  26. [34]

    Dmitrasinovic, H

    V. Dmitrasinovic, H. J. Schulze, R. Tegen, and R. H. Lemm er, Annals Phys. 238, 332 (1995)

    1995

  27. [35]

    P. M. Stevenson, Phys. Rev. D24, 1622 (1981)

    1981

  28. [36]

    T. E. Restrepo, J. C. Macias, M. B. Pinto, and G. N. Ferrar i, Phys. Rev. D91, 065017 (2015)

    2015

  29. [37]

    Schaefer, M

    B.-J. Schaefer, M. Wagner, and J. Wambach, Phys. Rev. D81, 074013 (2010). 15

    2010

  30. [38]

    H. Mao, J. Jin, and M. Huang, J. Phys. G37, 035001 (2010)

    2010

  31. [39]

    S. P. Klevansky, Rev. Mod. Phys. 64, 649 (1992)

    1992

  32. [40]

    L. M. Haas, R. Stiele, J. Braun, J. M. Pawlowski, and J. Sc haffner-Bielich, Phys. Rev. D87, 076004 (2013)

    2013

  33. [41]

    Oertel, M

    M. Oertel, M. Buballa, and J. Wambach, Phys. Lett. B477, 77 (2000)

    2000

  34. [42]

    R. F. Dashen, Phys. Rev. 183, 1245 (1969)

    1969

  35. [43]

    Hansen et al., Phys

    H. Hansen et al., Phys. Rev. D75, 065004 (2007)

    2007

  36. [44]

    Costa, M

    P. Costa, M. C. Ruivo, C. A. de Sousa, H. Hansen, and W. M. A lberico, Phys. Rev. D79, 116003 (2009)

    2009

  37. [45]

    P. Cea, L. Cosmai, and M. D’Elia, Nucl. Phys. Proc. Suppl . 129, 751 (2004), [,751(2003)]

    2004

  38. [46]

    Das, (2014), [EPJ Web Conf.90,08007(2015)]

    S. Das, (2014), [EPJ Web Conf.90,08007(2015)]

    2014

  39. [47]

    Gell-Mann, R

    M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175, 2195 (1968)

    1968

  40. [48]

    Borsanyi et al., Nature 539, 69 (2016)

    S. Borsanyi et al., Nature 539, 69 (2016)

    2016

  41. [49]

    Borsanyi et al., JHEP 09, 073 (2010)

    S. Borsanyi et al., JHEP 09, 073 (2010)

    2010

  42. [50]

    Bellwied et al., Phys

    R. Bellwied et al., Phys. Lett. B751, 559 (2015)

    2015

  43. [51]

    Bazavov et al., Phys

    A. Bazavov et al., Phys. Lett. B795, 15 (2019)

    2019

  44. [52]

    The left panels of Fig

    (solid-filled bands) and [53] (solid curve). The left panels of Fig. 6 show the susceptibilities for the net- baryon numbers, the net-electric charge, and the net-strange ness in panels (a), (b), and (c), respectively. The phase transition (crossover) is defined where the mesoni...

  45. [53]

    Andronic, P

    A. Andronic, P. Braun-Munzinger, K. Redlich, and J. Sta chel, Nature 561, 321 (2018)

    2018

  46. [54]

    Borsanyi et al., J

    S. Borsanyi et al., J. Phys. G38, 124060 (2011)

    2011

  47. [55]

    We find that at low temperatures χB 4/χB 2 starts from unity, where the mesonic contributions are dominant within the hadr on phase

    (closed symbols) and HotQCD collaboration [53]. We find that at low temperatures χB 4/χB 2 starts from unity, where the mesonic contributions are dominant within the hadr on phase. With the increase in T , a rapid 13 drop in χB 4/χB 2 =κσ 2 takes place, due to the rapid increas...

    2017

  48. [56]

    J. I. Kapusta and C. Gale, Finite-temperature field theory: Principles and applicati ons, Cambridge University Press, 2006, UK

    2006

  49. [57]

    Luo and N

    X. Luo and N. Xu, Nucl. Sci. Tech. 28, 112 (2017)

    2017

  50. [58]

    Borsanyi et al., JHEP 01, 138 (2012)

    S. Borsanyi et al., JHEP 01, 138 (2012)

    2012

  51. [59]

    Bazavov et al., Phys

    A. Bazavov et al., Phys. Rev. D86, 034509 (2012)

    2012

  52. [60]

    Bazavov et al., Phys

    A. Bazavov et al., Phys. Rev. D95, 054504 (2017)

    2017

  53. [61]

    Cheng et al., Phys

    M. Cheng et al., Phys. Rev. D79, 074505 (2009)

    2009

  54. [62]

    V. Koch, A. Majumder, and J. Randrup, Phys. Rev. Lett. 95, 182301 (2005)

    2005

  55. [63]

    Jeon and V

    S. Jeon and V. Koch, Phys. Rev. Lett. 85, 2076 (2000)

    2000

  56. [64]

    Friman, F

    B. Friman, F. Karsch, K. Redlich, and V. Skokov, Eur. Phy s. J. C71, 1694 (2011)

    2011

  57. [65]

    Adamczyk et al., Phys

    L. Adamczyk et al., Phys. Lett. B785, 551 (2018)

    2018

  58. [66]

    M. A. Stephanov, Phys. Rev. Lett. 102, 032301 (2009)

    2009

  59. [67]

    Borsanyi et al., JHEP 10, 205 (2018)

    S. Borsanyi et al., JHEP 10, 205 (2018)

    2018

  60. [68]

    Esha, Nucl

    R. Esha, Nucl. Phys. A967, 457 (2017)

    2017

  61. [69]

    Esha, PoS CPOD2017, 003 (2018)

    R. Esha, PoS CPOD2017, 003 (2018)

    2018

  62. [70]

    L. Chen, Z. Li, F. Cui, and Y. Wu, Nucl. Phys. A957, 60 (2017)

    2017

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