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arxiv: 2606.09633 · v1 · pith:X5FOK3LInew · submitted 2026-06-08 · ✦ hep-ph · nucl-th

Effective QCD model with consistent quasi-gluon treatment : formulation and application

Chowdhury Aminul Islam , Munshi G. Mustafa , Rajarshi Ray , Pracheta Singha This is my paper

Pith reviewed 2026-06-27 16:01 UTC · model grok-4.3

classification ✦ hep-ph nucl-th
keywords quasiparticle modelPNJL modelQCD thermodynamicsPolyakov loopgluon quasi-particlestransport coefficients
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The pith

The PNJL model is reformulated with gluon quasi-particles beyond the saddle-point approximation to produce a consistent quasiparticle description of QCD thermodynamics.

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

The paper takes the Polyakov loop enhanced Nambu-Jona-Lasinio model and adds gluon quasi-particles to the existing quark ones. The treatment for the gluon sector is extended past the saddle-point approximation. This produces a quasiparticle model for QCD thermodynamics that avoids earlier physical inconsistencies. The method is then used to compute transport coefficients in the light quark sector. A reader would care because standard quasiparticle approaches to hot QCD have long suffered from thermodynamic inconsistencies that this change aims to remove.

Core claim

Reformulating the PNJL model in terms of both quark and gluon quasi-particles, with the gluon sector treated beyond the saddle-point approximation, yields a physically consistent quasiparticle model for QCD thermodynamics whose advantages appear in calculations of transport coefficients.

What carries the argument

The reformulation of the PNJL Lagrangian and thermodynamic potential that incorporates gluon quasi-particles without restricting to the saddle-point approximation, thereby enforcing consistency between the quasi-particle dispersion relations and the thermodynamic relations.

If this is right

  • Thermodynamic quantities and transport coefficients become calculable within a single consistent quasiparticle framework for both quarks and gluons.
  • The light-quark transport coefficients obtained from the model satisfy the same consistency requirements as the underlying thermodynamics.
  • The approach removes the need for ad-hoc fixes that previous quasiparticle models required to restore thermodynamic consistency.
  • The same framework can be applied to other observables that depend on quasi-particle properties at finite temperature.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may improve agreement with lattice data on the QCD equation of state once the quasi-gluon masses are fixed.
  • Similar extensions could be tested in models that include strange quarks or finite baryon density.
  • If the consistency holds, the model could serve as input for hydrodynamic simulations of heavy-ion collisions without additional patching.

Load-bearing premise

Extending the PNJL model with gluon quasi-particles beyond the saddle-point approximation removes physical inconsistencies without introducing new uncontrolled approximations or parameter dependencies.

What would settle it

A calculation of a thermodynamic quantity such as pressure or energy density, or a transport coefficient such as shear viscosity, within this model that still violates thermodynamic consistency relations or deviates systematically from lattice QCD results in a way that cannot be fixed by parameter choice would falsify the consistency claim.

Figures

Figures reproduced from arXiv: 2606.09633 by Chowdhury Aminul Islam, Munshi G. Mustafa, Pracheta Singha, Rajarshi Ray.

Figure 1
Figure 1. Figure 1: FIG. 1. Effective gluon mass as function of temperature fit [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Scaled pressure as a function of temperature. Lattice [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Thermal evolution of the temperature derivatives [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Temperature variations of scaled conformal measure [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Thermal evolution of (a) [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Temperature variation of (a) scaled specific heat and [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The order parameters (a) and their thermal deriva [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Comparative study of (a) thermal derivative of ad [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Momentum distributions of quark and gluonic quasi [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Temperature distribution of quark and gluon quasi [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Temperature variation of shear viscosity. (a) Quark [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Temperature variation of bulk viscosity. (a) Quark [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Temperature variation of shear viscosity with tem [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Temperature variation of bulk viscosity with tem [PITH_FULL_IMAGE:figures/full_fig_p014_16.png] view at source ↗
read the original abstract

The Polyakov loop enhanced Nambu-Jona-Lasinio model is reformulated in terms of the gluon quasi-particles in addition to the already existing quark quasi-particles. The formulation goes beyond the saddle point approximation for the gluon sector. The framework provides a physically consistent quasiparticle model for QCD thermodynamics. The ensuing advantages of this formulation is discussed using transport coefficients in the light quark sector.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper reformulates the Polyakov-loop enhanced Nambu–Jona-Lasinio (PNJL) model by incorporating gluon quasi-particles in addition to the existing quark quasi-particles, extending the gluon-sector treatment beyond the saddle-point approximation. It claims that this yields a physically consistent quasiparticle framework for QCD thermodynamics and demonstrates advantages through calculations of transport coefficients in the light-quark sector.

Significance. If the beyond-saddle-point gluon treatment indeed preserves exact thermodynamic relations (pressure obtained from the effective potential and entropy from its temperature derivative) without introducing new uncontrolled approximations or parameter dependencies, the resulting model would strengthen the internal consistency of effective QCD descriptions used for the equation of state and transport properties in the quark-gluon plasma regime.

major comments (1)
  1. [Abstract] Abstract: the central claim that the reformulation 'provides a physically consistent quasiparticle model' requires that the gluon-sector extension beyond the saddle-point approximation maintains thermodynamic consistency and introduces no new inconsistencies or parameter dependencies. The abstract states the reformulation occurs and advantages are shown via transport coefficients but supplies no explicit construction, derivation, or verification that the beyond-saddle-point step preserves the required relations (pressure from effective potential, entropy from derivative) without additional assumptions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address the concern about the abstract's claim of thermodynamic consistency below, providing clarification on where the derivations appear in the main text while agreeing to strengthen the abstract.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the reformulation 'provides a physically consistent quasiparticle model' requires that the gluon-sector extension beyond the saddle-point approximation maintains thermodynamic consistency and introduces no new inconsistencies or parameter dependencies. The abstract states the reformulation occurs and advantages are shown via transport coefficients but supplies no explicit construction, derivation, or verification that the beyond-saddle-point step preserves the required relations (pressure from effective potential, entropy from derivative) without additional assumptions.

    Authors: The explicit construction of the gluon quasi-particle treatment beyond the saddle-point approximation, together with the verification that thermodynamic relations are preserved (pressure obtained directly from the effective potential and entropy from its temperature derivative), is derived in Sections II and III without introducing new parameters or uncontrolled approximations. The same set of PNJL parameters is retained. We agree that the abstract would benefit from a brief reference to this verification and will revise it accordingly in the next version. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected; derivation presented as independent model reformulation

full rationale

The paper describes a reformulation of the PNJL model by adding gluon quasi-particles treated beyond the saddle-point approximation, claiming this yields a physically consistent quasiparticle description of QCD thermodynamics with advantages illustrated through transport coefficients. No equations or steps are quoted that reduce a claimed prediction or result to a fitted input by construction, nor is there evidence of load-bearing self-citation chains, uniqueness theorems imported from the same authors, or ansatzes smuggled via prior work that would make the central consistency claim equivalent to its inputs. The extension is framed as a new formulation whose consistency follows from the reformulation itself, with transport coefficients serving as an application rather than a re-labeled fit. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities can be extracted or audited.

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discussion (0)

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Reference graph

Works this paper leans on

126 extracted references · 91 linked inside Pith

  1. [1]

    In our numerical analysis to reproduce results for usual PNJL model, we have used ΩPoly with the parameters given in table.I, taken from Ref

    Polynomial potential approach In this approach, aZ(3) symmetric polynomial po- tential is constructed in terms of Φ, ¯Φ and the first or- der phase transition is captured through the temperature dependent parameters,α i, following the usual Ginsburg- Landau approach [20].U(Φ, ¯Φ) can be written as [21–28], U(Φ, ¯Φ) T 4 =− α1(T) 2 Φ¯Φ− α2 6 (Φ3+ ¯Φ3)+ α3 4...

  2. [2]

    The thermodynamic potential, Ωgqp can be written as, Ωgqp = = 2T Z d3p (2π)3 ln det 1− ˆLAe− |⃗ p| T = 2T Z d3p (2π)3 ln 1 + 8X n=1 ane− n|⃗ p| T ! ,(13) where ˆLA is given in Eq

    Quasi-particle approach In the quasi-particle model [76–79, 81], the gluon quasi- particles are considered to be swarming in the back- ground of Polyakov field. The thermodynamic potential, Ωgqp can be written as, Ωgqp = = 2T Z d3p (2π)3 ln det 1− ˆLAe− |⃗ p| T = 2T Z d3p (2π)3 ln 1 + 8X n=1 ane− n|⃗ p| T ! ,(13) where ˆLA is given in Eq. (5). The coeffic...

  3. [3]

    The standard approach Nambu—Jona-Lasinio model [65–70], is a chiral effec- tive model where the gauge field is integrated out and re- placed by the local four-point interaction of quark degrees of freedom. To incorporate the confinement properties in this chiral model, the quarks are considered to be coupled to the static background gauge field and gluon ...

  4. [4]

    gluon mass data

    Quasiparticle approach Our proposal in this work is to couple the gluon quasi- particle model [76–79, 81] to the 2 flavour NJL model. In the usual saddle point approximation the thermodynamic potential is given as, Ωqp PNJL = Ωq + Ωqp ,(22) where, Ω q is given in Eq.(20), and Ω qp is given in Eq. (10), with Ωgqp given in Eq.(13). This model can pro- vide ...

    2065

  5. [5]

    Adamczyket al.(STAR), Nature548, 62 (2017), arXiv:1701.06657 [nucl-ex]

    L. Adamczyket al.(STAR), Nature548, 62 (2017), arXiv:1701.06657 [nucl-ex]

    Pith/arXiv arXiv 2017

  6. [6]

    G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Leg- eland, M. Lutgemeier, and B. Petersson, Nucl. Phys. B469, 419 (1996), arXiv:hep-lat/9602007 [hep-lat]

    Pith/arXiv arXiv 1996

  7. [7]

    Engels, O

    J. Engels, O. Kaczmarek, F. Karsch, and E. Laermann, Nucl. Phys. B558, 307 (1999), [Erratum: Nucl.Phys.B 576, 657–657 (2000)], arXiv:hep-lat/9903030

    Pith/arXiv arXiv 1999

  8. [8]

    Fodor and S

    Z. Fodor and S. D. Katz, Phys. Lett. B534, 87 (2002), arXiv:hep-lat/0104001

    Pith/arXiv arXiv 2002

  9. [9]

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

    Pith/arXiv arXiv 2002

  10. [10]

    Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, Nature443, 675 (2006), arXiv:hep-lat/0611014

    Pith/arXiv arXiv 2006

  11. [11]

    Bazavovet al., Phys

    A. Bazavovet al., Phys. Rev. D85, 054503 (2012), arXiv:1111.1710 [hep-lat]

    Pith/arXiv arXiv 2012

  12. [12]

    P. N. Meisinger and M. C. Ogilvie, Phys. Lett. B379, 163 (1996), arXiv:hep-lat/9512011

    Pith/arXiv arXiv 1996

  13. [13]

    Levai and U

    P. Levai and U. W. Heinz, Phys. Rev. C57, 1879 (1998), arXiv:hep-ph/9710463

    Pith/arXiv arXiv 1998

  14. [14]

    Karsch, Adv

    F. Karsch, Adv. Ser. Direct. High Energy Phys.6, 61 (1990), [,61(1989)]

    1990

  15. [15]

    Ali Khanet al.(CP-PACS), Phys

    A. Ali Khanet al.(CP-PACS), Phys. Rev. D64, 074510 (2001), arXiv:hep-lat/0103028

    Pith/arXiv arXiv 2001

  16. [16]

    Borsanyi, G

    S. Borsanyi, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, JHEP07, 056 (2012), arXiv:1204.6184 [hep-lat]

    Pith/arXiv arXiv 2012

  17. [17]

    Petreczky, Nucl

    P. Petreczky, Nucl. Phys. B Proc. Suppl.140, 78 (2005), arXiv:hep-lat/0409139

    Pith/arXiv arXiv 2005

  18. [18]

    V. V. Braguta, A. Kotov, A. Roenko, and D. Sychev, PoSLATTICE2022, 190 (2023), arXiv:2212.03224 [hep-lat]

    arXiv 2023

  19. [19]

    V. V. Braguta, M. N. Chernodub, I. E. Kudrov, A. A. Roenko, and D. A. Sychev, Phys. Atom. Nucl.86, 1249 (2023)

    2023

  20. [20]

    V. V. Braguta, M. N. Chernodub, and A. A. Roenko, Phys. Rev. D111, 114508 (2025), arXiv:2503.18636 [hep-lat]

    arXiv 2025

  21. [21]

    R. D. Pisarski, Phys. Rev.D62, 111501 (2000), arXiv:hep-ph/0006205 [hep-ph]

    Pith/arXiv arXiv 2000

  22. [22]

    Dumitru and R

    A. Dumitru and R. D. Pisarski, Phys. Lett.B504, 282 (2001), arXiv:hep-ph/0010083 [hep-ph]

    Pith/arXiv arXiv 2001

  23. [23]

    R. D. Pisarski (2002) pp. 353–384, arXiv:hep- ph/0203271 [hep-ph]

    arXiv 2002

  24. [24]

    Landau, CERN Document Server53(08)(1965)

    L. Landau, CERN Document Server53(08)(1965)

    1965

  25. [25]

    Fukushima, Physics Letters B591, 277 (2004)

    K. Fukushima, Physics Letters B591, 277 (2004)

    2004

  26. [26]

    Megias, E

    E. Megias, E. Ruiz Arriola, and L. L. Salcedo, Phys. Rev. D74, 065005 (2006), arXiv:hep-ph/0412308

    Pith/arXiv arXiv 2006

  27. [27]

    Ratti, M

    C. Ratti, M. A. Thaler, and W. Weise, Phys. Rev. D 73, 014019 (2006)

    2006

  28. [28]

    S. K. Ghosh, T. K. Mukherjee, M. G. Mustafa, and R. Ray, Phys. Rev. D73, 114007 (2006), arXiv:hep- ph/0603050

    arXiv 2006

  29. [29]

    Megias, E

    E. Megias, E. Ruiz Arriola, and L. L. Salcedo, Phys. Rev. D74, 114014 (2006), arXiv:hep-ph/0607338

    Pith/arXiv arXiv 2006

  30. [30]

    Mukherjee, M

    S. Mukherjee, M. G. Mustafa, and R. Ray, Phys. Rev. D75, 094015 (2007), arXiv:hep-ph/0609249

    Pith/arXiv arXiv 2007

  31. [31]

    S. K. Ghosh, T. K. Mukherjee, M. G. Mustafa, and R. Ray, Phys. Rev. D77, 094024 (2008), arXiv:0710.2790 [hep-ph]

    Pith/arXiv arXiv 2008

  32. [32]

    Schaefer, J

    B.-J. Schaefer, J. M. Pawlowski, and J. Wambach, Phys. Rev. D76, 074023 (2007), arXiv:0704.3234 [hep- ph]

    Pith/arXiv arXiv 2007

  33. [33]

    Kashiwa, H

    K. Kashiwa, H. Kouno, M. Matsuzaki, and M. Yahiro, Phys. Lett. B662, 26 (2008), arXiv:0710.2180 [hep-ph]

    Pith/arXiv arXiv 2008

  34. [34]

    Ciminale, R

    M. Ciminale, R. Gatto, N. D. Ippolito, G. Nardulli, and M. Ruggieri, Phys. Rev.D77, 054023 (2008), arXiv:0711.3397 [hep-ph]

    Pith/arXiv arXiv 2008

  35. [35]

    Fukushima, Phys

    K. Fukushima, Phys. Rev.D77, 114028 (2008), [Er- ratum: Phys. Rev.D78,039902(2008)], arXiv:0803.3318 [hep-ph]

    Pith/arXiv arXiv 2008

  36. [36]

    P. Deb, A. Bhattacharyya, S. Datta, and S. K. Ghosh, Phys. Rev.C79, 055208 (2009), arXiv:0901.1992 [nucl- th]

    Pith/arXiv arXiv 2009

  37. [37]

    Schaefer, M

    B.-J. Schaefer, M. Wagner, and J. Wambach, Phys. Rev. D81, 074013 (2010), arXiv:0910.5628 [hep-ph]

    Pith/arXiv arXiv 2010

  38. [38]

    Bhattacharyya, P

    A. Bhattacharyya, P. Deb, S. K. Ghosh, and R. Ray, Phys. Rev. D82, 014021 (2010), arXiv:1003.3337 [hep- ph]

    Pith/arXiv arXiv 2010

  39. [39]

    Bhattacharyya, P

    A. Bhattacharyya, P. Deb, A. Lahiri, and R. Ray, Phys. Rev. D82, 114028 (2010), arXiv:1008.0768 [hep-ph]

    Pith/arXiv arXiv 2010

  40. [40]

    Bhattacharyya, P

    A. Bhattacharyya, P. Deb, A. Lahiri, and R. Ray, Phys. Rev. D83, 014011 (2011), arXiv:1010.2394 [hep-ph]

    Pith/arXiv arXiv 2011

  41. [41]

    Bhattacharyya, S

    A. Bhattacharyya, S. K. Ghosh, S. Majumder, and R. Ray, Phys. Rev. D86, 096006 (2012), arXiv:1107.5941 [hep-ph]

    Pith/arXiv arXiv 2012

  42. [42]

    Bhattacharyya, P

    A. Bhattacharyya, P. Deb, S. K. Ghosh, R. Ray, and S. Sur, Phys. Rev. D87, 054009 (2013), arXiv:1212.5893 [hep-ph]

    Pith/arXiv arXiv 2013

  43. [43]

    Bhattacharyya, S

    A. Bhattacharyya, S. K. Ghosh, A. Lahiri, S. Ma- jumder, S. Raha, and R. Ray, Phys. Rev. C89, 064905 (2014), arXiv:1212.6134 [hep-ph]

    Pith/arXiv arXiv 2014

  44. [44]

    S. K. Ghosh, A. Lahiri, S. Majumder, M. G. Mustafa, S. Raha, and R. Ray, Phys. Rev. D90, 054030 (2014), arXiv:1407.7203 [hep-ph]

    Pith/arXiv arXiv 2014

  45. [45]

    S. K. Ghosh, S. Raha, R. Ray, K. Saha, and S. Upad- haya, Phys. Rev. D91, 054005 (2015), arXiv:1411.2765 [hep-ph]

    Pith/arXiv arXiv 2015

  46. [46]

    Bhattacharyya, R

    A. Bhattacharyya, R. Ray, and S. Sur, Phys. Rev. D 91, 051501 (2015), arXiv:1412.8316 [hep-ph]

    Pith/arXiv arXiv 2015

  47. [47]

    Bhattacharyya, S

    A. Bhattacharyya, S. K. Ghosh, R. Ray, K. Saha, and S. Upadhaya, EPL116, 52001 (2016), arXiv:1507.08795 [hep-ph]

    Pith/arXiv arXiv 2016

  48. [48]

    Bhattacharyya, S

    A. Bhattacharyya, S. K. Ghosh, S. Maity, S. Raha, R. Ray, K. Saha, and S. Upadhaya, Phys. Rev. D95, 054005 (2017), arXiv:1609.07882 [hep-ph]

    Pith/arXiv arXiv 2017

  49. [49]

    Singha, A

    P. Singha, A. Abhishek, G. Kadam, S. Ghosh, and H. Mishra, DAE Symp. Nucl. Phys.62, 862 (2017)

    2017

  50. [50]

    Bhattacharyya, S

    A. Bhattacharyya, S. K. Ghosh, S. Maity, S. Raha, R. Ray, K. Saha, S. Samanta, and S. Upadhaya, Phys. Rev. C99, 045207 (2019), arXiv:1708.04549 [hep-ph]

    Pith/arXiv arXiv 2019

  51. [51]

    Singha, A

    P. Singha, A. Abhishek, G. Kadam, S. Ghosh, and H. Mishra, J. Phys. G46, 015201 (2019), arXiv:1705.03084 [nucl-th]

    Pith/arXiv arXiv 2019

  52. [52]

    Bhattacharyya, P

    A. Bhattacharyya, P. Deb, S. K. Ghosh, S. Maity, S. Raha, R. Ray, K. Saha, and S. Upadhaya, Phys. Rev. D102, 074006 (2020), arXiv:1912.07019 [hep-ph]

    arXiv 2020

  53. [53]

    W.-j. Fu, Z. Zhang, and Y.-x. Liu, Phys. Rev. D77, 014006 (2008), arXiv:0711.0154 [hep-ph]

    Pith/arXiv arXiv 2008

  54. [54]

    Costa, M

    P. Costa, M. C. Ruivo, and C. A. de Sousa, Phys. Rev. D77, 096001 (2008), arXiv:0801.3417 [hep-ph]. 16

    Pith/arXiv arXiv 2008

  55. [55]

    Buballa, A

    M. Buballa, A. G. Grunfeld, A. E. Radzhabov, and D. Scheffler, Prog. Part. Nucl. Phys.62, 365 (2009), arXiv:0811.4543 [hep-ph]

    Pith/arXiv arXiv 2009

  56. [56]

    Costa, H

    P. Costa, H. Hansen, M. C. Ruivo, and C. A. de Sousa, Phys. Rev. D81, 016007 (2010), arXiv:0909.5124 [hep- ph]

    Pith/arXiv arXiv 2010

  57. [57]

    Lourenco, M

    O. Lourenco, M. Dutra, A. Delfino, and M. Malheiro, Phys. Rev. D84, 125034 (2011), arXiv:1201.1239 [nucl- th]

    Pith/arXiv arXiv 2011

  58. [58]

    Inagaki, D

    T. Inagaki, D. Kimura, H. Kohyama, and A. Kvinikhidze, Phys. Rev. D86, 116013 (2012), arXiv:1202.5220 [hep-ph]

    Pith/arXiv arXiv 2012

  59. [59]

    A. V. Friesen, Y. L. Kalinovsky, and V. D. Toneev, Int. J. Mod. Phys. A27, 1250013 (2012), arXiv:1111.3042 [hep-ph]

    Pith/arXiv arXiv 2012

  60. [60]

    Davoudiasl, T

    H. Davoudiasl, T. G. Rizzo, and A. Soni, Phys. Rev. D 77, 036001 (2008), arXiv:0710.2078 [hep-ph]

    Pith/arXiv arXiv 2008

  61. [61]

    Sakai, H

    Y. Sakai, H. Kouno, and M. Yahiro, J. Phys. G37, 105007 (2010), arXiv:0908.3088 [hep-ph]

    Pith/arXiv arXiv 2010

  62. [62]

    Morita, V

    K. Morita, V. Skokov, B. Friman, and K. Redlich, Phys. Rev. D84, 076009 (2011), arXiv:1107.2273 [hep-ph]

    Pith/arXiv arXiv 2011

  63. [63]

    Tsai and B

    H.-M. Tsai and B. Muller, J. Phys. G36, 075101 (2009), arXiv:0811.2216 [hep-ph]

    Pith/arXiv arXiv 2009

  64. [64]

    Braun, L

    J. Braun, L. M. Haas, F. Marhauser, and J. M. Pawlowski, Phys. Rev. Lett.106, 022002 (2011), arXiv:0908.0008 [hep-ph]

    Pith/arXiv arXiv 2011

  65. [65]

    Megias, E

    E. Megias, E. Ruiz Arriola, and L. L. Salcedo, Phys. Rev. D89, 076006 (2014), arXiv:1311.2814 [hep-ph]

    Pith/arXiv arXiv 2014

  66. [66]

    Singha, V

    P. Singha, V. E. Ambrus, S. Busuioc, A. Bandyopad- hyay, and M. N. Chernodub, (2025), arXiv:2508.20237 [nucl-th]

    arXiv 2025

  67. [67]

    Singha, S

    P. Singha, S. Busuioc, V. E. Ambrus, and M. N. Chernodub, Phys. Rev. D112, 094031 (2025), arXiv:2503.17291 [nucl-th]

    Pith/arXiv arXiv 2025

  68. [68]

    Singha, V

    P. Singha, V. E. Ambrus, and M. N. Chernodub, Phys. Rev. D110, 094053 (2024), arXiv:2407.07828 [hep-ph]

    arXiv 2024

  69. [69]

    Nambu and G

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

    1961

  70. [70]

    Nambu and G

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

    1961

  71. [71]

    Hatsuda and T

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

    Pith/arXiv arXiv 1994

  72. [72]

    Vogl and W

    U. Vogl and W. Weise, Prog. Part. Nucl. Phys.27, 195 (1991)

    1991

  73. [73]

    Klevansky, Rev

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

    1992

  74. [74]

    Buballa, Phys

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

    arXiv 2005

  75. [75]

    D. U. Jungnickel and C. Wetterich, Phys. Rev. D53, 5142 (1996), arXiv:hep-ph/9505267

    Pith/arXiv arXiv 1996

  76. [76]

    Berges, D

    J. Berges, D. U. Jungnickel, and C. Wetterich, Int. J. Mod. Phys. A18, 3189 (2003), arXiv:hep-ph/9811387

    Pith/arXiv arXiv 2003

  77. [77]

    Tetradis, Nucl

    N. Tetradis, Nucl. Phys. A726, 93 (2003), arXiv:hep- th/0303244

    arXiv 2003

  78. [78]

    Schaefer and J

    B.-J. Schaefer and J. Wambach, Phys. Rev. D75, 085015 (2007), arXiv:hep-ph/0603256

    Pith/arXiv arXiv 2007

  79. [79]

    Schaefer and J

    B.-J. Schaefer and J. Wambach, Phys. Part. Nucl.39, 1025 (2008), arXiv:hep-ph/0611191

    Pith/arXiv arXiv 2008

  80. [80]

    Sasaki and K

    C. Sasaki and K. Redlich, Phys. Rev.D86, 014007 (2012), arXiv:1204.4330 [hep-ph]

    Pith/arXiv arXiv 2012

Showing first 80 references.