arxiv: 2604.20196 · v1 · submitted 2026-04-22 · ✦ hep-ph · hep-th· nucl-th

Recognition: unknown

Chiral first order phase transition at finite baryon density and zero temperature from self-consistent pole masses in the linear sigma model with quarks

Alejandro Ayala , Bruno El-Bennich , Ricardo L. S. Farias , Luis A. Hern\'andez , Bruno S. Lopes , Luis C. Parra L. , Renato Zamora

Authors on Pith no claims yet

Pith reviewed 2026-05-10 00:35 UTC · model grok-4.3

classification ✦ hep-ph hep-thnucl-th

keywords chiral phase transitionlinear sigma model with quarksfinite baryon densityzero temperaturefirst-order transitionpole masseseffective potentialspeed of sound

0 comments

The pith

In the two-flavor linear sigma model with quarks the chiral phase transition at finite baryon density and zero temperature is first order and occurs when the chemical potential reaches the vacuum quark mass.

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

This paper treats the linear sigma model with quarks as an effective description of QCD and solves the one-loop self-consistent equations for the pole masses of the sigma, pions, and quarks. The chemical-potential dependence of the chiral order parameter is found by minimizing the one-loop effective potential. The calculation shows that the transition is first order for the chosen parameters, occurring precisely when the quark chemical potential equals the vacuum quark mass. This produces discontinuous jumps in the chiral condensate, the particle masses, and the couplings. The square of the speed of sound also jumps at the transition point before rising smoothly toward the conformal limit.

Core claim

We find that the phase transition is of first order, and occurs when the quark chemical potential reaches the value of the vacuum quark mass for the chosen set of parameters. The first order nature of the transition is signaled by the discontinuous behavior of the chiral condensate, the masses and the couplings. The square of the speed of sound exhibits a discontinuity at the phase transition and then smoothly approaches the conformal limit from below.

What carries the argument

The system of self-consistent one-loop pole-mass equations for sigma, pions and quarks, whose solutions are inserted into the one-loop effective potential that is then minimized with respect to the chiral condensate at each value of the chemical potential.

If this is right

The chiral condensate jumps discontinuously at the transition.
Particle masses and couplings exhibit discontinuous changes.
The square of the speed of sound jumps at the transition and then increases toward the conformal limit from below.
Thermodynamic quantities such as pressure and energy density can be computed directly from the minimized potential and the pole masses.

Where Pith is reading between the lines

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

The abrupt jump in the order parameter implies a sudden softening or stiffening of the equation of state for cold dense matter, which could affect the maximum mass or radius of neutron stars.
The same self-consistent pole-mass method can be extended to finite temperature to map the full phase diagram and locate the critical endpoint.
Because the approach avoids the ring-diagram approximation it remains applicable at arbitrarily large chemical potentials where conventional expansions break down.

Load-bearing premise

The linear sigma model with quarks remains a valid effective description of QCD at finite baryon density and zero temperature, and the one-loop self-consistent pole-mass equations plus minimization of the effective potential capture the correct non-perturbative physics without higher-order corrections or additional degrees of freedom.

What would settle it

A direct computation of the order parameter or effective potential in a different effective model, or a future lattice QCD simulation at finite density, that shows either a continuous transition or a transition at a chemical potential different from the vacuum quark mass.

Figures

Figures reproduced from arXiv: 2604.20196 by Alejandro Ayala, Bruno El-Bennich, Bruno S. Lopes, Luis A. Hern\'andez, Luis C. Parra L., Renato Zamora, Ricardo L. S. Farias.

**Figure 2.** Figure 2: FIG. 2. (top) Chiral condensate [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Parameters [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Speed of sound squared [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

read the original abstract

We use the two-flavor Linear Sigma Model with quarks as an effective description of QCD to investigate the nature of the chiral phase transition at finite baryon chemical potential and zero temperature. We work at one-loop order to set up and solve the system of self-consistent coupled equations for the particle pole masses. The chemical potential-dependent value of the chiral order parameter is obtained by minimizing the one-loop effective potential. This treatment goes beyond the conventional ring-diagram approximation and provides a description valid for arbitrary values of the chemical potential. We find that the phase transition is of first order, and occurs when the quark chemical potential reaches the value of the vacuum quark mass for the chosen set of parameters. The first order nature of the transition is signaled by the discontinuous behavior of the chiral condensate, the masses and the couplings. The thermodynamics of the system is readily implemented and in particular, we find that the square of the speed of sound exhibits a discontinuity at the phase transition and then smoothly approaches the conformal limit from below.

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.

Desk Editor's Note private letter to a colleague

This paper finds a first-order chiral transition at T=0 in the linear sigma model when mu hits the vacuum quark mass, via self-consistent one-loop pole masses, but the outcome is tied to parameters and the approximation.

read the letter

The main result is a first-order chiral transition at zero temperature that sets in exactly when the quark chemical potential reaches the vacuum constituent mass for the parameters they picked. They signal it with discontinuities in the condensate, the meson and quark masses, and even the couplings, plus a jump in the square of the speed of sound that then rises toward the conformal limit. The setup solves the coupled gap equations for the pole masses self-consistently at one loop, folds the chemical potential into the Fermi-sea integrals, and minimizes the effective potential to locate the minimum. This is a step past the usual ring-diagram truncation and lets them treat any value of mu without extra approximations. The numerics look reproducible from the description, and the thermodynamics section is straightforward to implement once the masses are known. Within the linear sigma model literature this is a clean technical extension. The limitation is that the first-order character and the precise location both rest on the chosen sigma mass, couplings, and vacuum scales, with no shown variation or stability checks. One-loop effective potentials in these models often get softened or shifted by two-loop terms or meson loops, so the order could change under a more complete treatment. The coincidence with the vacuum mass also raises the question of how much is input versus output. This is the sort of calculation that matters for people building model equations of state for neutron stars or scanning effective-model phase diagrams at high density. It is worth sending to referees because the equations are explicit, the numerics are standard, and the community can judge the robustness for themselves.

Referee Report

2 major / 2 minor

Summary. The manuscript uses the two-flavor linear sigma model with quarks at one-loop order to study the chiral phase transition at T=0 and finite baryon chemical potential. Self-consistent equations for pole masses are solved and the effective potential is minimized to obtain the chiral order parameter as a function of mu. The central claim is that a first-order transition occurs precisely when mu equals the vacuum quark mass for the chosen parameters, signaled by discontinuities in the condensate, masses, and couplings; the square of the speed of sound is also reported to jump at the transition before approaching the conformal limit from below.

Significance. If the result is robust, the work supplies a concrete prediction within this effective model for both the location and first-order character of the high-density chiral transition, together with a thermodynamic observable (speed of sound) that could be compared to other approaches. The self-consistent treatment of pole masses is a technical step beyond the conventional ring-diagram approximation and permits an in-principle description at arbitrary mu. The significance is nevertheless reduced by the absence of robustness checks against parameter variation and by the reliance on a one-loop truncation whose ability to fix the order of the transition is not independently verified.

major comments (2)

[Abstract and results discussion] The statement that the transition occurs exactly when the chemical potential reaches the vacuum quark mass is presented for a specific parameter set without any variation of those parameters or explicit demonstration that the coincidence is independent of the input scales (e.g., sigma mass or couplings). This leaves open the possibility that the reported location is largely fixed by the choice of vacuum mass rather than emerging as a dynamical prediction.
[Methodology (one-loop effective potential and gap equations)] The claim that the one-loop treatment with self-consistent pole masses is valid for arbitrary chemical potentials rests on an untested assumption that higher-order corrections (two-loop diagrams, vertex corrections, or meson fluctuations) do not alter the order of the transition. At T=0 the quark-loop integrals contain sharp Fermi-sea step functions; in comparable effective models such corrections frequently convert a mean-field first-order jump into a crossover or shift the critical mu by O(10-20 %). No test or estimate of these effects is provided.

minor comments (2)

The numerical values of all model parameters (sigma mass, couplings, etc.) and the precise fitting procedure used to fix the vacuum quark mass should be collected in a dedicated table or subsection to permit reproducibility and parameter-variation studies.
Notation for the self-consistent pole-mass equations and the effective potential should be clarified, including explicit display of the coupled gap equations and the minimization condition, so that the numerical procedure can be followed without ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We respond to each major comment below and indicate the revisions planned for the next version.

read point-by-point responses

Referee: [Abstract and results discussion] The statement that the transition occurs exactly when the chemical potential reaches the vacuum quark mass is presented for a specific parameter set without any variation of those parameters or explicit demonstration that the coincidence is independent of the input scales (e.g., sigma mass or couplings). This leaves open the possibility that the reported location is largely fixed by the choice of vacuum mass rather than emerging as a dynamical prediction.

Authors: We thank the referee for this observation. The manuscript already qualifies the result as holding 'for the chosen set of parameters,' which are fixed by reproducing vacuum meson masses, the pion decay constant, and the sigma mass. Within the model at T=0, the coincidence arises dynamically from the structure of the self-consistent gap equations and the one-loop effective potential: the quark contributions involve step-function Fermi-sea integrals that only become active once mu exceeds the pole mass, which itself is determined by the chiral condensate. Thus the jump in the order parameter occurs precisely when mu equals the vacuum quark mass. To address the concern we will add a short explanatory paragraph in the results section deriving this feature from the T=0 limit and the self-consistency condition, while reiterating that the parameters are constrained by vacuum phenomenology. An exhaustive parameter scan lies beyond the scope of the present methodological focus, but the reported location is not an arbitrary input. revision: partial
Referee: [Methodology (one-loop effective potential and gap equations)] The claim that the one-loop treatment with self-consistent pole masses is valid for arbitrary chemical potentials rests on an untested assumption that higher-order corrections (two-loop diagrams, vertex corrections, or meson fluctuations) do not alter the order of the transition. At T=0 the quark-loop integrals contain sharp Fermi-sea step functions; in comparable effective models such corrections frequently convert a mean-field first-order jump into a crossover or shift the critical mu by O(10-20 %). No test or estimate of these effects is provided.

Authors: We agree that the one-loop truncation is an approximation whose quantitative predictions can be affected by higher-order terms, and that no explicit test of two-loop or meson-fluctuation corrections is performed. Our technical advance is the self-consistent solution for the pole masses rather than the conventional ring-diagram resummation; this already incorporates momentum-dependent propagators and goes beyond simple mean-field. Nevertheless, a full two-loop calculation at finite density is computationally demanding and outside the present scope. We will add a paragraph in the methodology and conclusions sections acknowledging this limitation, citing literature on higher-order effects in related models, and noting that while O(10-20 %) shifts in the critical mu are possible, the first-order character is expected to persist for the parameter set used. This frames the result as a concrete benchmark within the improved one-loop framework. revision: partial

Circularity Check

1 steps flagged

Transition location reported at input vacuum quark mass scale

specific steps

fitted input called prediction [Abstract]
"We find that the phase transition is of first order, and occurs when the quark chemical potential reaches the value of the vacuum quark mass for the chosen set of parameters."

The location of the transition is identified with the vacuum quark mass, an input fixed by the model parameters chosen to reproduce vacuum phenomenology. The one-loop effective potential at T=0 naturally produces a feature at μ equal to the constituent mass, so the reported critical value is forced by the input scale rather than derived as a genuine prediction.

full rationale

The paper computes the first-order nature of the transition via minimization of the one-loop effective potential after solving self-consistent pole-mass gap equations. This procedure is independent of the specific location. However, the reported critical chemical potential is stated to equal the vacuum quark mass for the chosen parameters. In the T=0 limit of these models the quark-loop contribution to the effective potential turns on precisely when μ exceeds the σ-dependent constituent mass, so the coincidence is a direct consequence of the model structure and parameter choice rather than an independent prediction. No other load-bearing steps reduce to self-definition, self-citation, or renaming of known results.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model relies on standard linear-sigma-model parameters (sigma mass, pion decay constant, quark-meson couplings) that are fitted to vacuum phenomenology; the transition location is reported to match the vacuum quark mass for those parameters.

free parameters (1)

model parameters (sigma mass, couplings, etc.)
Chosen to reproduce vacuum properties; the transition point is stated to occur at mu equal to the resulting vacuum quark mass.

axioms (2)

domain assumption Linear sigma model with quarks is a valid effective theory for two-flavor QCD at finite density and T=0
Invoked in the opening sentence of the abstract as the framework for the entire calculation.
domain assumption One-loop effective potential plus self-consistent pole masses suffices to determine the order of the transition
Stated as the computational method that goes beyond ring diagrams.

pith-pipeline@v0.9.0 · 5516 in / 1549 out tokens · 20638 ms · 2026-05-10T00:35:36.411390+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

76 extracted references · 67 canonical work pages · 1 internal anchor

[1]

For subsequent𝜇 𝑖 values, the previous solution serves as the initial guess

An initial guess for𝑣is provided for𝜇=𝜇 max. For subsequent𝜇 𝑖 values, the previous solution serves as the initial guess
[2]

The self-consistent masses are computed for the current 𝑣, and𝑉 eff(𝑣)is evaluated
[3]

A step𝜖 𝑣 =0.5 MeV is taken to the left (𝑣→𝑣−𝜖 𝑣 ), and the masses and potential are recomputed
[4]

If the new potential is lower than the previous one, the step is repeated in the same direction
[5]

9 300 400 500 600 [MeV] 10 5 0 5 10 15 20 Couplings g a2/a2 0 FIG

If the new potential is larger, the step size is halved (𝜖𝑣 →𝜖 𝑣 /2), and a step to the right is attempted. 9 300 400 500 600 [MeV] 10 5 0 5 10 15 20 Couplings g a2/a2 0 FIG. 4. Parameters𝜆,𝑔, and renormalized𝑎 2/𝑎2 0 with𝑎 0 = 415.45 MeV as functions of the quark chemical potential𝜇at𝑇=0. The vacuum values are𝜆 0 =22.71,𝑔 0 =3.37. These parameters, as we...
[6]

This process of alternating directions and reducing step size is repeated𝑁 recursive =20 times, yielding a final precision in𝑣of𝜖 𝑣 /2𝑁recursive. This algorithm efficiently locates the global minimum of 𝑉eff(𝑣)without requiring explicit computation of the deriva- tive, which is advantageous given the implicit dependence of the masses on𝑣. C. Parameters an...

2025
[7]

The phase diagram of dense QCD

K. Fukushima and T. Hatsuda, Rept. Prog. Phys.74, 014001 (2011), arXiv:1005.4814 [hep-ph]

work page Pith review arXiv 2011
[8]

R. D. Pisarski and F. Rennecke, Phys. Rev. Lett.127, 152302 (2021), arXiv:2103.06890 [hep-ph]

work page arXiv 2021
[9]

Rennecke, R

F. Rennecke, R. D. Pisarski, and D. H. Rischke, Phys. Rev. D 107, 116011 (2023), arXiv:2301.11484 [hep-ph]

work page arXiv 2023
[10]

Arslandoket al., Hot QCD White Paper, (2023), arXiv:2303.17254 [nucl-ex]

M. Arslandoket al., (2023), arXiv:2303.17254 [nucl-ex]

work page arXiv 2023
[11]

X. Luo, Q. Wang, N. Xu, and P. Zhuang, eds.,Properties of QCD Matter at High Baryon Density(Springer, 2022)

2022
[12]

M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Phys. Rev. D60, 114028 (1999), arXiv:hep-ph/9903292. 11

work page arXiv 1999
[13]

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

work page arXiv 2009
[14]

Luo and N

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

work page arXiv 2017
[15]

Mroczek, A

D. Mroczek, A. R. Nava Acuna, J. Noronha-Hostler, P. Parotto, C. Ratti, and M. A. Stephanov, Phys. Rev. C103, 034901 (2021), arXiv:2008.04022 [nucl-th]

work page arXiv 2021
[16]

W.-j. Fu, C. Huang, J. M. Pawlowski, F. Rennecke, R. Wen, and S. Yin,Strangeness neutrality and the QCD phase diagram, Tech. Rep. (arXiv, 2026) arXiv:2603.13455 [hep-ph]

work page arXiv 2026
[17]

W.-j. Fu, J. M. Pawlowski, and F. Rennecke, Phys. Rev. D101, 054032 (2020), arXiv:1909.02991 [hep-ph]

work page arXiv 2020
[18]

Gutierrez, A

E. Gutierrez, A. Ahmad, A. Ayala, A. Bashir, and A. Raya, J. Phys. G41, 075002 (2014), arXiv:1304.8065 [hep-ph]

work page arXiv 2014
[19]

B. B. Brandt, G. Endr ˝odi, and G. Mark ´o, PoSLATTICE2024, 176 (2025), arXiv:2411.12918 [hep-ph]

work page arXiv 2025
[20]

Ayala, A

A. Ayala, A. Bashir, A. Raya, and E. Rojas, AIP Conf. Proc. 857, 311 (2006)

2006
[21]

Braun, L

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

work page arXiv 2011
[22]

Fukushima and C

K. Fukushima and C. Sasaki, Prog. Part. Nucl. Phys.72, 99 (2013), arXiv:1301.6377 [hep-ph]

work page arXiv 2013
[23]

J. O. Andersen, W. R. Naylor, and A. Tranberg, Rev. Mod. Phys. 88, 025001 (2016), arXiv:1411.7176 [hep-ph]

work page arXiv 2016
[24]

R. L. S. Farias, K. P. Gomes, G. I. Krein, and M. B. Pinto, Phys. Rev. C90, 025203 (2014), arXiv:1404.3931 [hep-ph]

work page arXiv 2014
[25]

V. A. Miransky and I. A. Shovkovy, Phys. Rept.576, 1 (2015), arXiv:1503.00732 [hep-ph]

work page arXiv 2015
[26]

Adhikari, J

P. Adhikari, J. O. Andersen, and P. Kneschke, Eur. Phys. J. C 79, 874 (2019), arXiv:1904.03887 [hep-ph]

work page arXiv 2019
[27]

Endrodi, QCD with background electromagnetic fields on the lattice: A review, Prog

G. Endrodi, Prog. Part. Nucl. Phys.141, 104153 (2025), arXiv:2406.19780 [hep-lat]

work page arXiv 2025
[28]

Adhikari, M

P. Adhikariet al., Prog. Part. Nucl. Phys.146, 104199 (2026), arXiv:2412.18632 [nucl-th]

work page arXiv 2026
[29]

Ding, J.-B

H.-T. Ding, J.-B. Gu, A. Kumar, and S.-T. Li, Phys. Rev. D112, 094508 (2025), arXiv:2508.07532 [hep-lat]

work page arXiv 2025
[30]

Hamada, M

Y. Hamada, M. Nitta, and Z. Qiu,QCD phase diagram in a magnetic field with baryon and isospin chemical potentials, Tech. Rep. (DESY, 2026) arXiv:2602.11762 [hep-ph]

work page arXiv 2026
[31]

Schmidt,Selected topics on the QCD phase diagram at finite temperature and density, PoSLATTICE2024(2025) 005 [2504.00629]

C. Schmidt, PoSLATTICE2024, 005 (2025), arXiv:2504.00629 [hep-lat]

work page arXiv 2025
[32]

Ayala, B

A. Ayala, B. S. Lopes, R. L. S. Farias, and L. C. Parra, Phys. Lett. B864, 139396 (2025), arXiv:2409.19406 [hep-ph]

work page arXiv 2025
[33]

B. S. Lopes, S. S. Avancini, A. Bandyopadhyay, D. C. Duarte, and R. L. S. Farias, Phys. Rev. D103, 076023 (2021), arXiv:2102.02844 [hep-ph]

work page arXiv 2021
[34]

B. S. Lopes, D. C. Duarte, R. L. S. Farias, and R. O. Ramos, Phys. Rev. D112, L091903 (2025), arXiv:2507.14343 [hep-ph]

work page arXiv 2025
[35]

Ayala, B

A. Ayala, B. S. Lopes, R. L. S. Farias, and L. C. Parra, Eur. Phys. J. A60, 250 (2024), arXiv:2310.13130 [hep-ph]

work page arXiv 2024
[36]

Ayala, A

A. Ayala, A. Bandyopadhyay, R. L. S. Farias, L. A. Hern´andez, and J. L. Hern ´andez, Phys. Rev. D107, 074027 (2023), arXiv:2301.13633 [hep-ph]

work page arXiv 2023
[37]

A. E. B. Pasqualotto, R. L. S. Farias, W. R. Tavares, S. S. Avancini, and G. Krein, Phys. Rev. D107, 096017 (2023), arXiv:2301.10721 [hep-ph]

work page arXiv 2023
[38]

L. A. Hern ´andez, J. D. Mart´ınez-S´anchez, and R. Zamora, Eur. Phys. J. C86, 56 (2026), arXiv:2508.03133 [hep-th]

work page arXiv 2026
[39]

Abdallahet al.(STAR), Phys

M. Abdallahet al.(STAR), Phys. Rev. Lett.127, 262301 (2021), [Erratum: Phys.Rev.Lett. 134, 139903 (2025)], arXiv:2105.14698 [nucl-ex]

work page arXiv 2021
[40]

Aboonaet al.(STAR), Phys

B. Aboonaet al.(STAR), Phys. Rev. Lett.130, 082301 (2023), [Erratum: Phys.Rev.Lett. 134, 139901 (2025)], arXiv:2207.09837 [nucl-ex]

work page arXiv 2023
[41]

M. S. Abdallahet al.(STAR), Phys. Rev. Lett.128, 202303 (2022), arXiv:2112.00240 [nucl-ex]

work page arXiv 2022
[42]

Abdallahet al.(STAR), Phys

M. Abdallahet al.(STAR), Phys. Rev. C107, 024908 (2023), arXiv:2209.11940 [nucl-ex]

work page arXiv 2023
[43]

Search for the QCD Critical Point in High Energy Nuclear Collisions: A Status Report

Y. Zhang, Z. Wang, X. Luo, and N. Xu (2026) arXiv:2602.08356 [nucl-ex]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[44]

Pandav, EPJ Web Conf.296, 01016 (2024)

A. Pandav, EPJ Web Conf.296, 01016 (2024)

2024
[45]

Abgaryanet al.(MPD), Eur

V. Abgaryanet al.(MPD), Eur. Phys. J. A58, 140 (2022), arXiv:2202.08970 [physics.ins-det]

work page arXiv 2022
[46]

Abdulinet al.(MPD), Rev

R. Abdulinet al.(MPD), Rev. Mex. Fis.71, 041201 (2025), arXiv:2503.21117 [nucl-ex]

work page arXiv 2025
[47]

Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations

M. Troyer and U.-J. Wiese, Phys. Rev. Lett.94, 170201 (2005), arXiv:cond-mat/0408370

work page Pith review arXiv 2005
[48]

Nagata, Finite-density lattice QCD and sign prob- lem: current status and open problems, Prog

K. Nagata, Prog. Part. Nucl. Phys.127, 103991 (2022), arXiv:2108.12423 [hep-lat]

work page arXiv 2022
[49]

Schmidt, J

C. Schmidt, J. Subatomic Part. Cosmol.3, 100057 (2025), arXiv:2501.19336 [hep-lat]

work page arXiv 2025
[50]

Basar,QCD critical point, Lee-Yang edge singularities, and Padé resummations,Phys

G. Basar, Phys. Rev. C110, 015203 (2024), arXiv:2312.06952 [hep-th]

work page arXiv 2024
[51]

Z.-Y. Wan, Y. Lu, F. Gao, and Y.-X. Liu, Eur. Phys. J. C84, 899 (2024), arXiv:2401.04957 [hep-ph]

work page arXiv 2024
[52]

C. S. Fischer, Prog. Part. Nucl. Phys.105, 1 (2019), arXiv:1810.12938 [hep-ph]

work page Pith review arXiv 2019
[53]

Bernhardt, C

J. Bernhardt, C. S. Fischer, P. Isserstedt, and B.-J. Schaefer, Phys. Rev. D104, 074035 (2021), arXiv:2107.05504 [hep-ph]

work page arXiv 2021
[54]

Tripolt, N

R.-A. Tripolt, N. Strodthoff, L. von Smekal, and J. Wambach, Phys. Rev. D89, 034010 (2014), arXiv:1311.0630 [hep-ph]

work page arXiv 2014
[55]

Fu, QCD at finite temperature and density within the fRG approach: an overview, Commun

W.-j. Fu, Commun. Theor. Phys.74, 097304 (2022), arXiv:2205.00468 [hep-ph]

work page arXiv 2022
[56]

Parotto, M

P. Parotto, M. Bluhm, D. Mroczek, M. Nahrgang, J. Noronha- Hostler, K. Rajagopal, C. Ratti, T. Sch¨afer, and M. Stephanov, Phys. Rev. C101, 034901 (2020), arXiv:1805.05249 [hep-ph]

work page arXiv 2020
[57]

Kahangirwe, S

M. Kahangirwe, S. A. Bass, E. Bratkovskaya, J. Jahan, P. Moreau, P. Parotto, D. Price, C. Ratti, O. Soloveva, and M. Stephanov, Phys. Rev. D109, 094046 (2024), arXiv:2402.08636 [nucl-th]

work page arXiv 2024
[58]

Critelli, J

R. Critelli, J. Noronha, J. Noronha-Hostler, I. Portillo, C. Ratti, and R. Rougemont, Phys. Rev. D96, 096026 (2017), arXiv:1706.00455 [nucl-th]

work page arXiv 2017
[59]

Grefa, J

J. Grefa, J. Noronha, J. Noronha-Hostler, I. Portillo, C. Ratti, and R. Rougemont, Phys. Rev. D104, 034002 (2021), arXiv:2102.12042 [nucl-th]

work page arXiv 2021
[60]

R. L. S. Farias, G. Dallabona, G. Krein, and O. A. Battistel, Phys. Rev. C73, 018201 (2006), arXiv:hep-ph/0510145

work page arXiv 2006
[61]

W.-j. Fu, X. Luo, J. M. Pawlowski, F. Rennecke, and S. Yin, Phys. Rev. D111, L031502 (2025), arXiv:2308.15508 [hep-ph]

work page arXiv 2025
[62]

Ayala, L

A. Ayala, L. A. Hern ´andez, M. Loewe, J. C. Rojas, and R. Zamora, Eur. Phys. J. A56, 71 (2020), arXiv:1904.11905 [hep-ph]

work page arXiv 2020
[63]

Ayala, M

A. Ayala, M. Hentschinski, L. A. Hernandez, M. Loewe, and R. Zamora, Phys. Rev. D98, 114002 (2018), arXiv:1809.04728 [hep-ph]

work page arXiv 2018
[64]

D. C. Duarte, R. L. S. Farias, and R. O. Ramos, Phys. Rev. D 99, 016005 (2019), arXiv:1811.10598 [hep-ph]

work page arXiv 2019
[65]

B. S. Lopes, R. L. S. Farias, V. Dexheimer, A. Bandyopad- hyay, and R. O. Ramos, Phys. Rev. D106, L121301 (2022), arXiv:2206.01631 [hep-ph]

work page arXiv 2022
[66]

R. M. Nunes, R. L. S. Farias, W. R. Tavares, and V. S. Tim´oteo, Phys. Rev. D111, 056026 (2025), arXiv:2412.14541 [hep-ph]

work page arXiv 2025
[67]

R. L. S. Farias and W. R. Tavares, Phys. Rev. D112, L051902 (2025), arXiv:2507.08130 [hep-ph]

work page arXiv 2025
[68]

Scavenius, A

O. Scavenius, A. Mocsy, I. N. Mishustin, and D. H. Rischke, Phys. Rev. C64, 045202 (2001), arXiv:nucl-th/0007030

work page arXiv 2001
[69]

The Phase Diagram of the Quark-Meson Model

B.-J. Schaefer and J. Wambach, Nucl. Phys. A757, 479 (2005), arXiv:nucl-th/0403039. 12

work page Pith review arXiv 2005
[70]

Schaefer, J

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

work page arXiv 2007
[71]

T. K. Herbst, J. M. Pawlowski, and B.-J. Schaefer, Phys. Lett. B696, 58 (2011), arXiv:1008.0081 [hep-ph]

work page arXiv 2011
[72]

The low-temperature behavior of the quark-meson model

R.-A. Tripolt, B.-J. Schaefer, L. von Smekal, and J. Wambach, Phys. Rev. D97, 034022 (2018), arXiv:1709.05991 [hep-ph]

work page Pith review arXiv 2018
[73]

Ayala, B

A. Ayala, B. A. Zamora, J. J. Cobos-Mart ´ınez, S. Hern´andez- Ortiz, L. A. Hern ´andez, A. Raya, and M. E. Tejeda-Yeomans, Eur. Phys. J. A58, 87 (2022), arXiv:2108.02362 [hep-ph]

work page arXiv 2022
[74]

Ayala, L

A. Ayala, L. A. Hern ´andez, M. Loewe, and C. Villavicencio, Eur. Phys. J. A57, 234 (2021), arXiv:2104.05854 [hep-ph]

work page arXiv 2021
[75]

Ayala, J

A. Ayala, J. D. Casta ˜no-Yepes, J. J. Cobos-Mart ´ınez, S. Hern´andez-Ortiz, A. Julia Mizher, and A. Raya, Int. J. Mod. Phys. A31, 1650199 (2016), arXiv:1510.08548 [hep-ph]

work page arXiv 2016
[76]

Ayala, S

A. Ayala, S. Hernandez-Ortiz, and L. A. Hernandez, Rev. Mex. Fis.64, 302 (2018), arXiv:1710.09007 [hep-ph]

work page arXiv 2018