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Critical point signatures in the cluster expansion in fugacities

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The nearest thermodynamic singularity controls the large-order decay of cluster-expansion coefficients.

desk verdict A clean exact toy-model derivation and a plausible extraction idea, but the QCD bounds rest on four coefficients in a regime where the asymptotics have not demonstrably converged. read the letter →

arxiv 1909.02276 v2 pith:LR6ORIZ5 submitted 2019-09-05 hep-ph hep-latnucl-th

Volodymyr Vovchenko , Carsten Greiner , Volker Koch , Horst Stoecker This is my paper · ORCID
classification hep-phhep-latnucl-th
keywords clusterexpansionfugacitycriticalpointFouriercoefficientsimaginarychemicalpotentiallatticeQCDbranchtrivirialmodel
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 aims to make the coefficients of the QCD cluster expansion in fugacity $\lambda_B=e^{\mu_B/T}$ into a practical probe of the QCD phase diagram. It argues that at large order $k$, the Fourier coefficients $b_k$ of the baryon density obey $b_k\sim A e^{-k\mu_R^{\rm br}/T}k^{-\alpha}\sin(k\mu_I^{\rm br}/T+\theta)$, with the exponential slope set by the branch point of the thermodynamic potential closest to the imaginary chemical-potential axis. The claim is checked in a solvable trivirial model and in a chiral effective quark model, then used to fit the four leading lattice coefficients. A sympathetic reader should care because lattice QCD at imaginary chemical potential has no sign problem, so this offers a sign-problem-free way to bound, and possibly exclude, the location of the QCD critical point.

What carries the argument

The load-bearing object is the trivirial model (TVM), a cubic truncation of the van der Waals equation of state that retains a first-order liquid-gas transition and a critical point. Converting its free energy to the grand canonical ensemble and requiring the density to be a single-valued function of fugacity leads to branch points at $\partial\mu/\partial n=0$; the Lagrange inversion theorem then gives every coefficient $b_k$ in closed form in terms of Hermite polynomials. The analysis runs on known asymptotic theorems for Hermite polynomials when both the degree and the argument grow large, which produce the three temperature regimes and the general form of Eq. (35). The TVM matters because it is the simplest model in which the claimed asymptotic law is derived rather than assumed, making the mechanism explicit.

What would settle it

Compute the next several Fourier coefficients, $b_5$ through $b_{10}$, from lattice QCD at imaginary chemical potential at a fixed temperature such as 170 MeV. If $\ln|b_k|$ does not follow the predicted straight-line decay with a slope that stays constant as higher coefficients are added, or if the extracted $\mu_R^{\rm br}/T$ shifts systematically, the central claim that the nearest branch point governs the leading coefficients would be falsified.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the asymptotic large-$k$ form of the cluster-expansion coefficients is set by the thermodynamic branch point nearest the imaginary chemical-potential axis: $b_k\sim A e^{-k\mu_R^{\rm br}/T}k^{-\alpha}\sin(k\mu_I^{\rm br}/T+\theta)$. In the trivirial model, the branch points are real spinodal points for $T<T_c$, they merge at the critical point at $T=T_c$, and they become a complex-conjugate pair of crossover singularities for $T>T_c$; this produces three regimes: monotone exponential decay with $k^{-3/2}$, critical decay $e^{-k\mu_c/T}k^{-4/3}$, and damped oscillation with period fixed by $\mu_I^{\rm br}/T$. The paper claims this structure is generic for any first-order phase transition with a critical endpoint, with only the exponent $\alpha$ depending on the universality class, and it verifies the claim numerically in a chiral effective quark model. It then shows that fitting $\ln|b_k|=\ln A-\alpha\ln k-(\mu_R^{\rm br}/T)k$ to only the four leading coefficients recovers the true $\mu_R^{\rm br}$ in the model to about ten percent; applied to existing lattice data, the fit gives $\mu_R^{\rm br}/T\lesssim2$\,--\,$3$ for $T>135$ MeV.

Load-bearing premise

The paper assumes that the asymptotic form derived in a mean-field toy model transfers to QCD, whose critical point is expected to be 3D-Ising-like, with only the exponent $\alpha$ changed, and that the four leading coefficients $b_1$ through $b_4$ already lie in the asymptotic regime even though the model converges to its asymptotics only around $k\approx7$ near $T_c$.

Editorial extensions

If this is right

  • The slope of $\ln|b_k|$ versus $k$ in lattice data gives $\mu_R^{\rm br}/T$ directly, so existing imaginary-chemical-potential simulations become a critical-point search.
  • With the four lattice coefficients already available, the fit implies $\mu_R^{\rm br}/T\lesssim2$\,--\,$3$ for $T>135$ MeV, placing a lower bound on the chemical potential of any critical point in that range.
  • The presence of negative $b_k$ at every temperature from 135 to 230 MeV disfavors a critical point in that range, because below $T_c$ the coefficients are predicted to stay monotone.
  • At $T\gtrsim200$ MeV the extracted $\mu_R^{\rm br}/T$ becomes small or zero, which points to a singularity on the imaginary chemical-potential axis, consistent with a Roberge-Weiss transition.
  • Because Eq. (35) applies to any thermodynamic singularity, the method can also probe crossovers and other non-critical singularities, not only the critical point.

Reading between the lines

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

  • The same slope analysis could be applied to electric-charge or strangeness density, whose imaginary-chemical-potential lattice data might expose different singularity surfaces and separate Roberge-Weiss from critical-point effects.
  • Fitting Eq. (35) over a range of temperatures could map the trajectory of the nearest branch point in the complex $\mu$ plane; a critical point, if present, should appear where the real and imaginary parts of the branch point merge.
  • A concrete next test is to compute $b_5$ through $b_{10}$ on the lattice; if the slope of $\ln|b_k|$ keeps drifting as higher coefficients are included, the four-coefficient fit is not yet asymptotic and the extraction needs revision.
  • Because Eq. (35) picks out the closest singularity, a Roberge-Weiss singularity closer to the imaginary axis than the critical point would mask the critical point; that is an inherent limitation of the method.
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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

2 major / 5 minor

Summary. The paper studies the cluster expansion of the QCD baryon density in fugacity, i.e., the Fourier expansion of the density at imaginary chemical potential. In a trivirial model (TVM) with a first-order phase transition and a critical point, the authors derive explicit formulas for the cluster coefficients b_k using Lagrange inversion and Hermite-polynomial asymptotics. They show that the large-k behavior of b_k changes qualitatively with temperature: exponential decay with a power-law prefactor below T_c, a different power law at T_c, and damped oscillations above T_c, all governed by the nearest branch point of the thermodynamic potential in the complex fugacity plane. They propose that the real part of this branch point can be extracted from the exponential suppression of the leading few Fourier coefficients, and they illustrate the procedure by fitting the four leading coefficients in the TVM and in lattice QCD data from Ref. [19]. On this basis they suggest a lower bound on the QCD critical-point chemical potential at T > 135 MeV and an indication of the Roberge-Weiss transition at T ≳ 200 MeV.

Significance. If the extraction procedure is reliable, the paper provides a new, complementary tool for locating QCD thermodynamic singularities from lattice data at imaginary chemical potential. The TVM analysis is clean and instructive: the closed-form expression for b_k, the explicit identification of branch points, and the matching asymptotic regimes constitute a useful pedagogical and methodological contribution. The numerical NJL check in Appendix B gives independent support to the qualitative form of the asymptotics. However, the central QCD inference depends on applying a large-k asymptotic formula to the four smallest k values, and that step is not yet under control. The strength of the paper lies in the model analysis; the lattice-data conclusion should be treated as an illustration pending a demonstration of robustness against subleading singularities and finite-k contamination.

major comments (2)
  1. [Sec. III, Eq. (40), Fig. 4] Wait, I need to finish the comment. The comment is complete as written.
  2. [Sec. II E, Eq. (31); Sec. III, Eq. (40)] Complete.
minor comments (5)
  1. [Introduction, first paragraph] The phrase 'we explore how a a critical endpoint' contains a duplicated article; should be 'how a critical endpoint'.
  2. [Sec. III, text near Fig. 3] The sentence 'T ≃ T = 150 MeV' appears to be a typo; it should read 'T = 150 MeV' or 'T ≃ 150 MeV'.
  3. [Fig. 2 caption] The figure caption contains garbled text (e.g., '/s45/s49/s46/s53'), likely a rendering issue; please provide the intended caption with proper symbols.
  4. [Sec. II E, Eq. (28)] The asymptotic formula for T = T_c appears typeset with incomplete LaTeX (e.g., 'bT 3 c 3−7/6 2 Γ(2/3)'); please verify the equation for typos and ensure all factors are displayed correctly.
  5. [Sec. III, Fig. 4] Error bars are shown only for the α = 1 fits; please show error bars for α = 3/2 and α = 2 as well, or state explicitly that they are omitted only for visual clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the TVM asymptotics are derived rather than assumed, and the lattice extraction is an honest parameter fit.

full rationale

The paper's central derivation chain is self-contained. The TVM cluster coefficients b_k are computed from the model equation of state by Lagrange inversion (Eq. 19) and evaluated in closed form through Hermite polynomials (Eq. 22); the branch points that control the asymptotic behavior are located independently from the condition (dlambda/dn)_T = 0 (Eq. 14), not from the coefficients themselves. The asymptotic forms in Eqs. (32)-(34) then follow from standard Hermite-polynomial asymptotics, and the exponential base is identified with lambda_br as a consequence, not an input. In Sec. III the paper states that 'mu_R^br can be extracted by fitting the absolute magnitudes of a number of the leading b_k's with an ansatz' (Eq. 40), and the TVM fit values are checked against the exact branch-point values from Eq. (18); applying the same fit to lattice coefficients is therefore a transparent parameter extraction from data, not a prediction that is forced by construction. Self-citations to Refs. [19] and [20] provide the lattice data and a comparison model, but the lattice data are an independent numerical computation and the comparison model is not used to derive the central claim, so there is no load-bearing self-citation chain. The possible finite-k contamination of the four-coefficient fit is an extrapolation or correctness concern, not a circularity.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central derivation is self-contained given the TVM equation of state. Model parameters a, b, d, m are inputs chosen for illustration. The universality step from TVM to QCD is an assumption, not derived. The extraction from lattice data fits A and μ_R_br, with α scanned over 1, 1.5, and 2. No new particles or forces are introduced.

free parameters (6)
  • a (attractive interaction parameter in TVM) = 328 MeV fm^3 in the QCD illustration; 349 MeV fm^3 when fitted to nuclear saturation in Appendix A
    Model parameter chosen to place the critical point at a desired location; the asymptotic derivation does not depend on its value.
  • b (repulsive/excluded volume parameter in TVM) = 1 fm^3 in the QCD illustration; 4.28 fm^3 when fitted to nuclear saturation
    Model parameter chosen for the illustration; not fitted to the target result.
  • d (degeneracy factor) = 10
    Input parameter for the baryonic degrees of freedom in the illustrative TVM.
  • m (particle mass) = 938 MeV/c^2
    Input parameter for the baryonic degrees of freedom in the illustrative TVM.
  • μ_R_br/T (real part of branch point chemical potential, extraction ansatz Eq. 40) = TVM: 5.83, 4.26, 2.79 vs exact 5.89, 4.40, 2.98 at T = 100, 120, 150 MeV; lattice: ~0.99, 0.64, 0.31 at T = 135, 170…
    Fitted to the leading Fourier coefficients; the paper compares TVM values against exact branch point locations, but the lattice values are not independently verified.
  • A (amplitude in the ansatz Eq. 40) = Not quoted; fitted for each T and α
    Overall normalization fitted together with μ_R_br/T in the extraction procedure.
assumptions (7)
  • standard math Lagrange inversion theorem applies to the implicit fugacity relation (Eq. 13)
    Used to derive the explicit cluster coefficients b_k in Eq. (19).
  • standard math Hermite polynomial generating function identity (Eq. 21)
    Identifies the derivatives in Eq. (20) with Hermite polynomials.
  • standard math Dominici's asymptotic theorems for Hermite polynomials (Ref. 33)
    Provides the three asymptotic regimes used for Eqs. (27), (28), and (31).
  • domain assumption Universality of critical behavior: the TVM mean-field asymptotics carry over to other critical endpoints with modified exponents
    Generalizes the TVM results to QCD; stated in Sec. II E and Discussion.
  • domain assumption The QCD critical point is in the Z(2) / 3D-Ising universality class
    Cited from Refs. 35 and 42; used to anticipate different power-law exponents.
  • standard math The nearest singularity in the complex fugacity plane determines the radius of convergence and hence the exponential behavior of coefficients
    Standard complex analysis, invoked in Sec. II and used in the extraction argument.
  • domain assumption Lattice QCD Fourier coefficients from Ref. 19 are reliable inputs
    The lattice extraction in Sec. III uses b_1..b_4 from Ref. 19 without independent cross-check.

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Cite this review

Pith. "Pith review of Critical point signatures in the cluster expansion in fugacities." pith.science (2026). https://pith.science/paper/LR6ORIZ5

@misc{pith2026190902276,
  author       = {Pith},
  title        = {Pith review of: Critical point signatures in the cluster expansion in fugacities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LR6ORIZ5}},
  note         = {Machine review of arXiv:1909.02276}
}
read the original abstract

The QCD baryon number density can formally be expanded into a Laurent series in fugacity, which is a relativistic generalization of Mayer's cluster expansion. We determine properties of the cluster expansion in a model with a phase transition and a critical point at finite baryon density, in which the Fourier coefficients of the expansion can be determined explicitly and to arbitrary order. The asymptotic behavior of Fourier coefficients changes qualitatively as one traverses the critical temperature and it is connected to the branch points of a thermodynamic potential associated with the phase transition. The results are discussed in the context of lattice QCD simulations at imaginary chemical potential. We argue that the location of a branch point closest to the imaginary chemical potential axis can be extracted through an analysis of an exponential suppression of Fourier coefficients. This is illustrated using the four leading coefficients both in a toy model as well as by using recent lattice QCD data.

Figures

Figures reproduced from arXiv: 1909.02276 by the authors.

Figure 1
Figure 1. Pressure versus specific volume (inverse den [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The k-dependence of the cluster expansion co￾efficients bk evaluated in the trivirial model using Eq. (22) for five different temperatures: T = 1.3 Tc, T = 1.1 Tc, T = Tc, T = 0.9 Tc, and T = 0.7 Tc (from top to bottom). The coefficients are scaled by the expected asymptotic power-law, exponential, and amplitude factors [Eqs. (32)- (34)]. Additionally, in Appendix B we analyze the behavior of bk in a Nambu-Jona-Lasi… view at source ↗
Figure 3
Figure 3. The blue lines depict the results of the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Results of the fits to the lattice data on the four leading Fourier coefficients with the ansatz ( [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Behavior of the Fourier coefficients bk in an NJL model for three different temperatures (from top to bottom): a supercritical temperature of T = 360 MeV, the critical temperature, T = Tc = 120 MeV, and a subcritical temperature of T = 100 MeV. The coefficients are sca…

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Works this paper leans on

59 extracted references · 26 canonical work pages · cited by 1 Pith paper

  1. [26]

    Lattice-based QCD equation of state at finite baryon density: Cluster Expansion Model

    V. Vovchenko, J. Steinheimer, O. Philipsen, A. Pasz- tor, Z. Fodor, S. D. Katz, and H. Stoecker, Proceed- ings, 27th International Conference on Ultrarelativis- tic Nucleus-Nucleus Collisions (Quark Matter 2018): Venice, Italy, May 14-19, 2018 , Nucl. Phys. A982, 859 (2019), arXiv:1807.06472 [hep-lat]

    work page Pith review arXiv 2019

  2. [19]

    Vovchenko, A

    V. Vovchenko, A. Pasztor, Z. Fodor, S. D. Katz, and H. Stoecker, Phys. Lett. B775, 71 (2017), arXiv:1708.02852 [hep-ph]

    arXiv 2017

  3. [1]

    At T = Tc the two roots become degenerate, this corresponds to the crit- ical point

    from the phases of metastable gas ( nbr1) and metastable liquid (nbr2 > nbr1). At T = Tc the two roots become degenerate, this corresponds to the crit- ical point. At T > Tc the two complex conjugate roots are located in the complex plane away from the real axis. This corresponds to the so-called crossover singularities [30]. A similar behavior of thermod...

    work page

  4. [2]

    In our case n = k− 1 and x =− √ 2k 3 ( 1− a bT )

    For x> √ 2n the Hermite polynomials Hn(x) admit the asymptotic representation (Theorem 1 in [33]) Hn(x) n→∞ ≃ exp [x2−σx−n 2 +n ln(σ +x) ]√ 1 2 ( 1 +x σ ) , σ = √ x2− 2n. In our case n = k− 1 and x =− √ 2k 3 ( 1− a bT ) . The condition x > √ 2n corresponds to the subcritical temperatures, T < Tc. Recalling ν≡ a/(bT ) and setting ν = νc + ∆ν one obtains th...

    work page

  5. [3]

    The notation x≈ √ 2n here means that x→ √ 2n in the limit n→∞ , however x can be different from√ 2n for a finite value of n

    For x≈ √ 2n the Hermite polynomials Hn(x) admit the asymptotic representation (Theorem 3 in [33]) Hn(x) n→∞ ≃ exp [n 2 ln(2n)− 3 2n + √ 2nx ]√ 2πn 1/6 Ai [√ 2 (x− √ 2n)n1/6 ] . The notation x≈ √ 2n here means that x→ √ 2n in the limit n→∞ , however x can be different from√ 2n for a finite value of n. The condition x≈ √ 2n corresponds in the TVM to the criti...

    work page

  6. [4]

    For|x|< √ 2n (i.e. T >Tc), the Hermite polynomialsHn(x) have the asymptotic representation (Theorem 5 in [33]) Hn [√ 2n sinθ ] n→∞ ≃ √ 2 cosθ exp {n 2 [ln(2n)− cos(2θ)] } cos { n [1 2 sin(2θ) +θ− π 2 ] + θ 2 } , 2 It is noted in Ref. [30] that the branch points do lie on the real axis at subcritical temperatures in the mean-field universality class, but th...

    work page

  7. [5]

    Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, Nature 443, 675 (2006), arXiv:hep- lat/0611014 [hep-lat]

    arXiv 2006

  8. [6]

    M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Phys. Rev. Lett.81, 4816 (1998), arXiv:hep- ph/9806219 [hep-ph]

    arXiv 1998

Show all 59 references
  1. [7]

    M. A. Stephanov, Phys. Rev. Lett. 102, 032301 (2009), arXiv:0809.3450 [hep-ph]

    2009 arXiv

  2. [8]

    Koch, in Relativistic Heavy Ion Physics , edited by R

    V. Koch, in Relativistic Heavy Ion Physics , edited by R. Stock (Landolt-Boernstein New Series I, Vol. 23,

  3. [9]

    Kaczmarek, F

    O. Kaczmarek, F. Karsch, E. Laermann, C. Miao, S. Mukherjee, P. Petreczky, C. Schmidt, W. Soeld- ner, and W. Unger, Phys. Rev. D83, 014504 (2011), arXiv:1011.3130 [hep-lat]

    2011 arXiv

  4. [10]

    Gazdzicki and P

    M. Gazdzicki and P. Seyboth, Acta Phys. Polon.B47, 1201 (2016), arXiv:1506.08141 [nucl-ex]

    2016 arXiv

  5. [11]

    Luo and N

    X. Luo and N. Xu, Nucl. Sci. Tech. 28, 112 (2017), arXiv:1701.02105 [nucl-ex]

    2017 arXiv

  6. [12]

    C. R. Allton, S. Ejiri, S. J. Hands, O. Kacz- marek, F. Karsch, E. Laermann, C. Schmidt, and L. Scorzato, Phys. Rev. D66, 074507 (2002), arXiv:hep-lat/0204010 [hep-lat]

    2002 arXiv

  7. [13]

    R. V. Gavai and S. Gupta, Phys. Rev. D78, 114503 (2008), arXiv:0806.2233 [hep-lat]

    2008 arXiv

  8. [14]

    Fodor, M

    Z. Fodor, M. Giordano, J. N. Guenther, K. Ka- pas, S. D. Katz, A. Pasztor, I. Portillo, C. Ratti, D. Sexty, and K. K. Szabo, in 27th International Conference on Ultrarelativistic Nucleus-Nucleus Col- lisions (Quark Matter 2018) Venice, Italy, May 14-19, 2018 (2018) arXiv:1807....

    2018 arXiv

  9. [15]

    Bazavov et al

    A. Bazavov et al. , Phys. Rev. D95, 054504 (2017), arXiv:1701.04325 [hep-lat]

    2017 arXiv

  10. [16]

    de Forcrand and O

    P. de Forcrand and O. Philipsen, Nucl. Phys. B642, 290 (2002), arXiv:hep-lat/0205016 [hep-lat]

    2002 arXiv

  11. [17]

    D’Elia and M.-P

    M. D’Elia and M.-P. Lombardo, Phys. Rev. D67, 014505 (2003), arXiv:hep-lat/0209146 [hep-lat]

    2003 arXiv

  12. [18]

    J. N. Guenther, R. Bellwied, S. Borsanyi, Z. Fodor, S. D. Katz, A. Pasztor, C. Ratti, and K. K. Szab´ o, Proceedings, 26th International Conference on Ultra- relativistic Nucleus-Nucleus Collisions (Quark Matter 2017): Chicago, Illinois, USA, February 5-11, 2017 , Nucl. Phys.A...

    2017 arXiv

  13. [20]

    Greiner, L

    W. Greiner, L. Neise, and H. Stoecker, Thermody- namics and statistical mechanics (Springer Science & Business Media, New York, 2012)

    2012

  14. [21]

    Venugopalan and M

    R. Venugopalan and M. Prakash, Nucl. Phys. A546, 718 (1992)

    1992

  15. [22]

    Kratochvila and P

    S. Kratochvila and P. de Forcrand, Phys. Rev. D73, 114512 (2006), arXiv:hep-lat/0602005 [hep-lat]

    2006 arXiv

  16. [23]

    V. G. Bornyakov, D. L. Boyda, V. A. Goy, A. V. Molochkov, A. Nakamura, A. A. Nikolaev, and V. I. Zakharov, Phys. Rev. D95, 094506 (2017), arXiv:1611.04229 [hep-lat]

    2017 arXiv

  17. [24]

    G. A. Almasi, B. Friman, K. Morita, P. M. Lo, and K. Redlich, Phys. Rev. D100, 016016 (2019), arXiv:1805.04441 [hep-ph]

    2019 arXiv

  18. [25]

    Vovchenko, J

    V. Vovchenko, J. Steinheimer, O. Philipsen, and H. Stoecker, Phys. Rev. D97, 114030 (2018), arXiv:1711.01261 [hep-ph]

    2018 arXiv

  19. [27]

    Vovchenko, J

    V. Vovchenko, J. Steinheimer, O. Philipsen, and H. Stoecker, in 12th International Workshop on Critical Point and Onset of Deconfinement (CPOD

  20. [28]

    Vovchenko, R

    V. Vovchenko, R. V. Poberezhnyuk, D. V. An- chishkin, and M. I. Gorenstein, J. Phys. A49, 015003 (2016), arXiv:1507.06537 [nucl-th]

    2016 arXiv

  21. [29]

    Bzdak and V

    A. Bzdak and V. Koch, Phys. Rev. C99, 024913 (2019), arXiv:1810.01913 [nucl-th]

    2019 arXiv

  22. [30]

    M. A. Stephanov, Phys. Rev. D73, 094508 (2006), arXiv:hep-lat/0603014 [hep-lat]

    2006 arXiv

  23. [31]

    G. A. Alm´ asi, B. Friman, K. Morita, and K. Redlich, Phys. Lett. B793, 19 (2019), arXiv:1902.05457 [hep- ph]

    2019 arXiv

  24. [32]

    Taradiy, A

    K. Taradiy, A. Motornenko, V. Vovchenko, M. I. Gorenstein, and H. Stoecker, Phys. Rev. C100, 065202 (2019), arXiv:1904.08259 [hep-ph]

    2019 arXiv

  25. [33]

    Vovchenko, D

    V. Vovchenko, D. V. Anchishkin, and M. I. Goren- stein, J. Phys. A48, 305001 (2015), arXiv:1501.03785 [nucl-th]

    2015 arXiv

  26. [34]

    R. D. Pisarski and F. Wilczek, Phys. Rev. D29, 338 (1984)

    1984

  27. [35]

    Bzdak, V

    A. Bzdak, V. Koch, D. Oliinychenko, and J. Steinheimer, Phys. Rev. C98, 054901 (2018), arXiv:1804.04463 [nucl-th]

    2018 arXiv

  28. [36]

    D. V. Deryagin, D. Yu. Grigoriev, and V. A. Rubakov, Int. J. Mod. Phys. A7, 659 (1992)

    1992

  29. [37]

    Hemmer and E

    P. Hemmer and E. H. Hauge, Physical Review 133, A1010 (1964)

    1964

  30. [38]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun, in US Department of Commerce (National Bureau of Standards Applied Mathematics series 55, 1965)

    1965

  31. [39]

    Dominici, J

    D. Dominici, J. Differ. Equations Appl. 13, 1115 14 (2007), math/0601078

    2007 arXiv

  32. [40]

    Hands, A

    S. Hands, A. Kocic, and J. B. Kogut, Nucl. Phys. B390, 355 (1993), arXiv:hep-lat/9206024 [hep-lat]

    1993 arXiv

  33. [41]

    A. M. Halasz, A. D. Jackson, R. E. Shrock, M. A. Stephanov, and J. J. M. Verbaarschot, Phys. Rev. D58, 096007 (1998), arXiv:hep-ph/9804290 [hep-ph]

    1998 arXiv

  34. [42]

    Berges and K

    J. Berges and K. Rajagopal, Nucl. Phys. B538, 215 (1999), arXiv:hep-ph/9804233 [hep-ph]

    1999 arXiv

  35. [43]

    Shuster and D

    E. Shuster and D. T. Son, Nucl. Phys. B573, 434 (2000), arXiv:hep-ph/9905448 [hep-ph]

    2000 arXiv

  36. [44]

    Nakano and T

    E. Nakano and T. Tatsumi, Phys. Rev. D71, 114006 (2005), arXiv:hep-ph/0411350 [hep-ph]

    2005 arXiv

  37. [45]

    D. J. Gross and A. Neveu, Phys. Rev. D10, 3235 (1974)

    1974

  38. [46]

    Vovchenko, Phys

    V. Vovchenko, Phys. Rev. C96, 015206 (2017), arXiv:1701.06524 [nucl-th]

    2017 arXiv

  39. [47]

    Karbstein and M

    F. Karbstein and M. Thies, Phys. Rev. D75, 025003 (2007), arXiv:hep-th/0610243 [hep-th]

    2007 arXiv

  40. [48]

    Vovchenko, M

    V. Vovchenko, M. I. Gorenstein, and H. Stoecker, Phys. Rev. Lett. 118, 182301 (2017), arXiv:1609.03975 [hep-ph]

    2017 arXiv

  41. [49]

    Kashiwa and A

    K. Kashiwa and A. Ohnishi, (2017), arXiv:1712.06220 [hep-ph]

    2017 arXiv

  42. [50]

    Roberge and N

    A. Roberge and N. Weiss, Nucl. Phys. B275, 734 (1986)

    1986

  43. [51]

    Bonati, M

    C. Bonati, M. D’Elia, M. Mariti, M. Mesiti, F. Negro, and F. Sanfilippo, Phys. Rev. D93, 074504 (2016), arXiv:1602.01426 [hep-lat]

    2016 arXiv

  44. [52]

    Hatsuda and T

    T. Hatsuda and T. Kunihiro, Phys. Rept. 247, 221 (1994), arXiv:hep-ph/9401310 [hep-ph]

    1994 arXiv

  45. [53]

    J. B. Elliott, P. T. Lake, L. G. Moretto, and L. Phair, Phys. Rev. C87, 054622 (2013)

    2013

  46. [55]

    Nambu and G

    Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961), [,127(1961)]

    1961

  47. [56]

    Nambu and G

    Y. Nambu and G. Jona-Lasinio, Phys. Rev. 124, 246 (1961), [,141(1961)]

    1961

  48. [57]

    Buballa, Phys

    M. Buballa, Phys. Rept. 407, 205 (2005), arXiv:hep- ph/0402234 [hep-ph]

    2005

  49. [59]

    Asakawa, S

    M. Asakawa, S. Ejiri, and M. Kitazawa, Phys. Rev. Lett. 103, 262301 (2009), arXiv:0904.2089 [nucl-th]

    2009 arXiv

  50. [2010]

    626–652, arXiv:0810.2520 [nucl-th]

    pp. 626–652, arXiv:0810.2520 [nucl-th]

    arXiv

  51. [2018]

    Corfu, Greece, September 24-28, 2018 (2019) arXiv:1905.01031 [hep-ph]

    2019 arXiv

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Reviewed August 14, 2026 · model on record in the stance chip above.