Lee Yang edge singularities of QCD in association with Roberge-Weiss phase transition and chiral phase transition

Fei Gao; Yi Lu; Yu-Xin Liu; Zi-Yan Wan

REVIEW 3 major objections 6 minor 67 references

Lee Yang edge singularities of QCD in association with Roberge-Weiss phase transition and chiral phase transition

T0 review · 3 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The Lee-Yang edge singularity trajectory of QCD terminates at the Roberge-Weiss line, $\mu_i = \frac{1}{3}\pi T$, at $T_e = 0.235$ GeV, because the Roberge-Weiss symmetry cuts it off.

desk verdict A serious DSE calculation with a new and plausible observation—LYES terminating at the Roberge-Weiss line—but the reported cutoff temperature is only as trustworthy as the one-loop, vacuum-fitted truncation behind it. read the letter →

arxiv 2504.12964 v1 pith:HYDT5WHB submitted 2025-04-17 hep-ph hep-thnucl-th

Zi-yan Wan , Yi Lu , Fei Gao , Yu-xin Liu This is my paper · ORCID

classification hep-phhep-thnucl-th

keywords Lee-YangedgesingularitiesRoberge-WeisssymmetrychiralphasetransitiondeconfinementgluonicbackgroundfieldPolyakovloopDyson-SchwingerequationsQCDdiagram

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 claims that QCD's Lee-Yang edge singularity, the complex-chemical-potential singularity used to locate the chiral critical endpoint, is cut off by Roberge-Weiss symmetry before it can serve as an extrapolation tool at high temperature. Working with Dyson-Schwinger equations coupled to a constant gluonic background field $\bar A_4$ that represents confinement, the authors solve the quark gap equation and the background-field equation self-consistently and find that the Lee-Yang edge singularity trajectory terminates exactly when its imaginary part reaches $\mu_i = \frac{1}{3}\pi T$, at $T_e = 0.235$ GeV. This cut-off, they argue, is caused by the Roberge-Weiss symmetry, not by the chiral dynamics alone. If correct, this means that above $T_e$ there is no Lee-Yang edge singularity left to extrapolate toward the QCD critical endpoint, while below it the critical scaling exponent $\beta\delta = 1.38$ is unchanged by the presence of the confining background field.

What carries the argument

The central object is the constant gluonic background field $\bar A_4$, diagonalized as $\bar A_4 = \frac{1}{g} 2\pi T(\phi_3 t_3 + \phi_8 t_8)$, whose physical value is fixed by the minimum of the Polyakov-loop potential. The machinery is the coupled set of Dyson-Schwinger equations: the quark gap equation for the three color components, with $\bar A_4$ appearing as a color-dependent shift of the fermionic Matsubara frequencies $\tilde\omega_n = \omega_n + i\mu + g\bar A_4$, together with the one-loop background-field equation $V'(\bar A_4) = 0$ whose propagators and vertex are the same 'functional-lattice' vacuum-fit ans\"atze used in the authors' earlier work. The Roberge-Weiss symmetry, realized as invariance under $\mu_i = \frac{2\pi T}{3} n$ combined with center transformations, selects the physical minimum and is what confines the Lee-Yang edge singularity trajectory to $\mu_i \le \frac{1}{3}\pi T$.

What would settle it

Compute the two-loop contribution to $V'(\bar A_4)$ with the same vacuum-fitted inputs, or refit the gluon propagator and vertex at finite temperature and complex chemical potential, and track the minimum of $\bar A_4$ near $T_e$: if a chiral Lee-Yang edge singularity with $\operatorname{Im}\mu > \pi T/3$ appears for $T > 0.235$ GeV, the cut-off claim fails. On the lattice, an analytic continuation of the chiral susceptibility that locates a singularity above the Roberge-Weiss line in that temperature range would likewise falsify the termination.

Watch

Extended reading notes

Core claim

The central discovery is that a self-consistently determined gluonic background field changes the fate of the Lee-Yang edge singularity trajectory at high temperature while leaving its critical behavior intact. In the coupled scheme, the background field shifts the Matsubara frequencies of the three color components differently, and the one-loop background-field potential, minimized together with the chiral order parameter, produces a first-order Roberge-Weiss transition at imaginary chemical potential $\mu_i = (1+2n)\pi T/3$ above $T \simeq 0.155$ GeV. As temperature rises, the imaginary part of the chiral Lee-Yang edge singularity grows toward this line; at $T_e = 0.235$ GeV it reaches $\mu_i = \frac{1}{3}\pi T$, and no singularity exists beyond it, because the Roberge-Weiss symmetry forces the physical $\bar A_4$ configuration to be mirrored across that axis and the singularity is absorbed into the branch cut of the first-order Roberge-Weiss transition. Below the cut, the trajectory follows the scaling law $\operatorname{Im}\mu_{\mathrm{LYE}} = i c_2 (T - T_{\mathrm{CEP}})^{\beta\delta}$ with $\beta\delta = 1.38$, and the implied endpoint location is consistent with the previous calculation without the background field.

Load-bearing premise

The calculation's load-bearing premise is that a single-loop approximation for the background-field potential, with the gluon propagator and quark-gluon vertex fixed by vacuum fits, correctly decides where the $\bar A_4$ condensate sits at finite temperature and complex chemical potential.

Editorial extensions

If this is right

For $T > T_e = 0.235$ GeV, no chiral Lee-Yang edge singularity exists, so Lee-Yang-based extrapolation to the QCD critical endpoint is invalid in that temperature range.
For $T \lesssim 0.160$ GeV the trajectory still follows the chiral scaling law, so low-temperature extrapolation to the critical endpoint remains plausible; the extracted endpoint is $(T, \mu_B) = (118, 608)$ MeV.
The critical exponent $\beta\delta = 1.38$ is insensitive to the confining background field, indicating that chiral dynamics dominates the Lee-Yang edge singularity critical behavior in the region $T - T_{\mathrm{CEP}} \lesssim 0.04$ GeV.
The Roberge-Weiss transition becomes first order above $T \simeq 0.155$ GeV at $\mu_i = (1+2n)\pi T/3$, and its branch cut is what absorbs the Lee-Yang edge singularity at higher temperature.
Including the background field raises the chiral pseudo-critical temperature from $0.155$ GeV to $0.162$ GeV, showing that confinement back-coupling strengthens chiral symmetry breaking.

Reading between the lines

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

Inference: If this termination is generic, any QCD approach that respects Roberge-Weiss periodicity, including lattice simulations and functional renormalization group studies, should also find no chiral Lee-Yang edge singularity beyond $\mu_i = \pi T/3$; the Roberge-Weiss branch cut would act as a universal wall for singularity-based extrapolations.
Inference: The same wall may apply to singularities in other conserved-charge directions, meaning estimates of critical points based on singularity locations in the baryon plane are confined to temperatures below $T_e$.
Inference: Because the quantitative value $T_e = 0.235$ GeV inherits the one-loop truncation, the symmetry protects only the existence and location of the cut-off line, not the precise temperature; recomputing with two-loop terms or a finite-temperature refit of the gluon is the natural check.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The one thing you should know: this paper reports a genuinely new numerical observation—the Lee-Yang edge singularity trajectory of the chiral transition cuts off at the Roberge-Weiss line mu_i = pi T/3, at T_e = 0.235 GeV, and the authors argue this is due to the RW symmetry, not the chiral dynamics alone. That is absent from their earlier no-background-field work and from the cited lattice/DSE literature. The paper also demonstrates RW symmetry emerging in a DSE framework with a gluonic background field solved self-consistently with the quark gap equation.

What it does well: the calculation is laid out clearly. The equations, parameter tables, and symmetry analysis are sufficiently detailed to follow. The comparison between the with-A4 and without-A4 cases is explicit, and the scaling fit to Im mu_LYE = i c2 (T - T_CEP)^{beta delta} is transparent. The qualitative mechanism—that the RW transition line provides a branch cut which stops the LYES trajectory—is physically plausible and consistent with what we know about first-order transition lines.

The soft spots are real but not fatal. The biggest one is the truncation: the background-field potential V' is evaluated at one loop only, with the gluon propagator, ghost propagator, and quark-gluon vertex frozen at vacuum fits. The paper says this explicitly. That means T_e = 0.235 GeV and the RW transition temperature T_c = 0.155 GeV are all predictions of that fixed-input, one-loop potential. There is no stability check, no parameter variation, no estimate of how higher-loop terms or finite-T/mu refitting would move the numbers. So the exact cutoff temperature is a truncation-dependent model prediction, not a robust QCD statement. The qualitative cutoff, though, is likely robust because RW symmetry itself is exact, and the branch cut reasoning does not depend on the details of the potential—only the temperature at which the LYES hits it does.

Also, the phrase 'fully coupled, self-consistent' overstates what is actually solved: the gluon and vertex are inputs, not solved for. And the explanation of why the LYES cannot cross the RW line is argued in words rather than shown analytically—the statement about singularities on one side mapping to the other side deserves a more explicit treatment. Minor: beta*delta = 1.38 is imported from the authors' own previous paper and the slope c2 is fit in the same framework, so there is some circularity, but it is mild and the RW symmetry check is independent.

Who is this for: specialists in functional QCD (DSE/fRG) and anyone working on locating the QCD critical endpoint via LYES extrapolations. It deserves a serious referee; I would send it to review. A good referee will ask for a truncation-dependence check, but the paper is coherent, honest, and the new observation is worth reporting.

Referee Report

3 major / 6 minor

Summary. The paper extends a previous Dyson-Schwinger equation (DSE) study of QCD phase transitions to include a constant gluonic background field A4. The authors solve the quark gap equation and the background-field equation self-consistently, obtaining the Polyakov loop potential and the chiral condensate. They then continue the calculation to complex chemical potential, identify the Roberge-Weiss (RW) transition, and compute the trajectory of Lee-Yang edge singularities (LYES) of the chiral transition. The central new result is that the LYES trajectory terminates at the RW line mu_i = pi*T/3 at a temperature T_e = 0.235 GeV, which the authors attribute to RW symmetry. They also report that the LYES scaling exponent beta*delta is unchanged from their previous work (beta*delta = 1.38) and extract a slope parameter c2 = 5.3 GeV^(1-beta*delta).

Significance. If the computed cutoff is robust, the paper makes a nontrivial prediction: above T_e approximately 0.235 GeV, the chiral LYES are absent below the RW line, which would restrict the use of LYES extrapolations for locating the QCD critical endpoint. The paper also demonstrates a self-consistent coupling between the A4 background field and the chiral condensate within the DSE framework and explicitly exhibits RW symmetry in the Polyakov loop potential. These are useful contributions. However, the main quantitative result rests on a one-loop truncation of the background-field potential with gluon, ghost, and vertex inputs frozen at vacuum fits, so the significance is conditional on the stability of that truncation.

major comments (3)

[II.B, Eq. (12)] The central quantitative prediction T_e = 0.235 GeV is obtained from the one-loop background-field potential V' in Eq. (12), with the gluon propagator (Eq. (10)), the quark-gluon vertex (Eq. (11)), and the ghost dressing (Eq. (16)) fixed at vacuum fits, as stated in Section II.B: 'we analyze exclusively the one-loop term in Fig. 3.' Because the location of the A4 minima controls both the RW transition temperature and the LYES termination temperature, this truncation is load-bearing: higher-loop contributions, or a finite-T/mu refitting of the propagators and vertex, could shift the minima and hence T_e. The manuscript provides no stability check, such as a parameter variation, a two-loop estimate, or a lattice benchmark of the RW potential. I ask the authors to assess the sensitivity of T_e and of the termination behavior to the truncation and to state the resulting uncertainty on the central claim.
[IV.B] Section IV.B reports a directly computed CEP at (T, mu_B) = (0.095, 0.735) GeV but later states that the scaling analysis yields an extracted CEP location at (118, 608) MeV. These values differ by roughly 23 MeV in T and 127 MeV in mu_B, and the text does not reconcile them. Since the final paragraph recommends the use of LYES scaling for T <= 160 MeV as a plausible way to determine the CEP, this discrepancy is directly relevant to the paper's conclusions. Please clarify which quantity (118, 608) MeV refers to and, if it is an extrapolated value distinct from the direct CEP, discuss the origin of the difference and its implications for the extrapolation method.
[IV.B (termination mechanism)] The claim that the LYES trajectory terminates because of RW symmetry is asserted rather than demonstrated. The text states that RW symmetry entails a reflection of the singularity to mu_i < 1/3 pi T, but no explicit Riemann-surface argument or continuity proof is given to show that the LYES branch point must meet the RW branch cut at T_e = 0.235 GeV rather than continuing onto another sheet. Please provide a more formal description of the analytic structure, or at least a numerical demonstration that the singularity cannot be continued past the RW line, to substantiate the causal interpretation that the cutoff is caused by RW symmetry.

minor comments (6)

[Throughout] There are several typographical and grammatical issues, e.g., 'singulairties' in the Introduction, 'the imaginary of phi8' in Section IV.A, and 'has represented a cut-off temperature' in Section IV.B; the text should be carefully proofread.
[II.B, Eq. (12)] In Eq. (12), the symbol P is used for the momentum sum/integral without definition; please define it explicitly (e.g., T sum_n integral d^3q/(2 pi)^3).
[II.A, Eq. (10)] The name 'functional-lattice' for the gluon propagator in Eq. (10) is not explained; a brief description or a reference to the origin of this parameterization would help the reader.
[Abstract and III.B] The phrase 'demonstrate the emergence of RW symmetry' is imprecise: RW symmetry is an exact property of the theory, and the paper demonstrates its realization or spontaneous breaking in the computed potential. Please rephrase.
[IV.B, Eq. (34)] In the scaling analysis, beta*delta = 1.38 is fixed from previous work rather than extracted from the data. The authors do state 'taking beta*delta = 1.38', but they should explicitly note that the critical exponent is an input and comment on how the conclusions would change if beta*delta were left free in the fit.
[IV.B] The RW transition below T_c is called a 'crossover'; for imaginary chemical potential this term is nonstandard, since there is no order parameter and the transition is analytic. Please rephrase to avoid confusion.

Circularity Check

1 steps flagged · score 2.0 of 10

Imported βδ = 1.38 from the authors' prior paper is used to 'verify' unchanged scaling; the RW-termination result itself is not circular.

other [Section IV.B, CEP-scaling discussion around Eq. (34) and Fig. 10(b)]
"Within the scaling region T-T_{CEP} <= 0.04 GeV, taking βδ = 1.38, both datasets exhibit linear behavior, demonstrating that the critical behaviour of the LYES is not sensitive to the presence of a confining gluon background field."

The value βδ = 1.38 is not derived in this work; it is imported from the authors' earlier paper [1]. The claim that the critical behavior is 'not sensitive' to A4 is therefore not an independent result: the data are plotted against (T-T_CEP)^{1/βδ} with βδ fixed to the previous value, so the plot can only test linearity and determine the slope c2, not the exponent itself. This is a consistency check on the old exponent rather than a prediction of a new one. The central RW cut-off claim at T_e = 0.235 GeV does not depend on this imported exponent, so the circularity is minor and non-load-bearing.

full rationale

The main derivation is self-contained: the one-loop background-field potential (Eq. (12)) is minimized to determine A4, the quark DSE is solved with that A4, and the LYES are located as singularities in the complex μ-plane. RW symmetry is an exact property of the underlying theory and is checked via the potential shape (Sec. III.B), not imposed as the desired termination. The cut-off T_e = 0.235 GeV is a computed crossing of the LYE trajectory with μ_i = πT/3, so it is not equivalent by construction to an input. The only mild circularity is the CEP-scaling discussion: βδ = 1.38 is taken from Ref. [1] and then used to demonstrate that the exponent is unchanged. Truncation and vacuum-fitted parameter concerns are robustness issues, not circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim depends on a chain of truncation and input assumptions: a two-point DSE truncation, a one-loop background-field potential, fixed vacuum-fitted gluon, vertex, and ghost parameterizations, an assumed scaling form for the singularity, and neglect of off-diagonal color components. No new physical entity is introduced; the model borrows all parameters from earlier work, so the ledger is dominated by imported fit parameters rather than novel degrees of freedom.

free parameters (5)

Gluon propagator parameters {a,b,c,d,e,f,g} = a=1 GeV, b=0.735 GeV, c=0.12, d=0.0257 GeV^-1, e=0.081 GeV^-1, f=0.65 GeV, g=0.87 GeV
Fixed in vacuum in Ref. [1] and used in Eq. (10) and in the Polyakov loop potential; not refit at finite T or complex mu.
Quark-gluon vertex parameters {alpha(zeta), Lambda, d1, d2} = alpha(40 GeV)=0.3, Lambda=1.4 GeV, d1=6.8 GeV^2, d2=0.5 GeV^2
Vertex ansatz in Eq. (11) fixed from previous work; controls the chiral dynamics and the resulting LYES trajectories.
Ghost dressing parameters {a1,b1,c1,d1,e1,f1,g1} = a1=0.152 GeV^2, b1=0.697 GeV, c1=0.0055 GeV^2, d1=0.016 GeV, e1=0.045, f1=0.025 GeV^-2, g1=0.0237 GeV^-2
Ghost dressing function in Eq. (16) from Refs. [61,62]; enters the A4 potential in Eq. (12).
LYES scaling exponent beta*delta = 1.38
Taken from the authors' previous work [1] and imposed in Eq. (34); the current data are checked against it rather than used to determine it.
LYES scaling slope c2 = 5.3 GeV^(1-beta*delta)
Obtained by a linear fit to the current LYES trajectory near the CEP in Fig. 10(b); it is a fitted value, not a parameter-free prediction.

assumptions (6)

domain assumption The DSE truncation to two-point functions, together with the background field method in Landau-deWitt gauge, captures the relevant dynamics.
Section II.A; the central claim depends on this truncation.
domain assumption The one-loop term of the background field equation, Eq. (12), is sufficient to determine the physical A4 condensate.
Section II.B: 'we analyze exclusively the one-loop term in Fig. 3, whose physical viability is demonstrated in [19]'.
domain assumption The fixed functional-lattice gluon propagator and vertex ansatz remain valid at finite temperature, finite chemical potential, and imaginary chemical potential.
Eqs. (10)-(11); parameters are not refit for the complex-mu region where the LYES are extracted.
domain assumption The analytic continuation to imaginary chemical potential and the scaling form Im mu_LYE = i c2 (T - T_CEP)^(beta*delta) describe the Lee-Yang edge singularity.
Eq. (34); the scaling form is assumed from universality and prior work, not derived here.
domain assumption Neglect of off-diagonal color components of the dressed quark propagator is justified.
Section II.A: 'we neglect the non-diagonal term across all momentum ranges'.
standard math Roberge-Weiss symmetry is an exact consequence of center symmetry and the periodicity of the imaginary chemical potential.
Section III.B; this is a symmetry argument rather than an empirical input.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Lee Yang edge singularities of QCD in association with Roberge-Weiss phase transition and chiral phase transition." pith.science (2026). https://pith.science/paper/HYDT5WHB

@misc{pith2026250412964,
  author       = {Pith},
  title        = {Pith review of: Lee Yang edge singularities of QCD in association with Roberge-Weiss phase transition and chiral phase transition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HYDT5WHB}},
  note         = {Machine review of arXiv:2504.12964}
}

abstract

We study the Quantum Chromodynamics (QCD) phase transitions in the complex chemical potential plane in the framework of Dyson-Schwinger equation approach, in the presence of a constant gluonic background field that represents confining dynamics. We solve the quark gap equation and the background field equation self consistently, which allows us to directly explore the confinement phase transition and furthermore, evaluate the impact of the back-coupling of confinement on chiral symmetry breaking. Moreover, within such a coupled framework towards the complex chemical potential region, we demonstrate the emergence of Roberge-Weiss (RW) symmetry and investigate the trajectory of Lee-Yang edge singularities (LYES). Our analysis reveals that the LYES scaling behavior is similar to our previous findings without the background field condensate. However, a significant difference from our earlier work is that the trajectory of LYES terminates when the imaginary part of the singularity becomes $1/3 \, \pi T$. We elaborate that this cut-off behavior is caused by the RW symmetry that is symmetric to the imaginary chemical potential $\mu_i=1/3 \, \pi T$.

Figures

Figures reproduced from arXiv: 2504.12964 by the authors.

**Figure 3.** FIG. 3. One loop truncated DSE for the background gluon [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** FIG. 4. The minima of Polyakov loop potential at specific imaginary chemical potential [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: FIG. 5. The two panels display contour plots of the Polyakov [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 9.** Figure 9: FIG. 9. Temperature dependence of the imaginary part of [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

**Figure 10.** Figure 10: FIG. 10. Upper panel: the LYES trajectory with (black [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

67 extracted references · 9 canonical work pages

[1]

Z.-Y. Wan, Y. Lu, F. Gao, and Y.-X. Liu, Eur. Phys. J. C 84, 899 (2024), arXiv:2401.04957 [hep-ph]

arXiv 2024
[2]

Aarts et al

G. Aarts et al. , Prog. Part. Nucl. Phys. 133, 104070 (2023), arXiv:2301.04382 [hep-lat]

arXiv 2023
[3]

Fu, Commun

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

arXiv 2022
[4]

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

arXiv 2019
[5]

On the QCD phase structure from effective models

B.-J. Schaefer and M. Wagner, Prog. Part. Nucl. Phys. 62, 381 (2009), arXiv:0812.2855 [hep-ph]

work page Pith review arXiv 2009
[6]

Bors´ anyi, Z

S. Bors´ anyi, Z. Fodor, J. N. Guenther, R. Kara, P. Parotto, A. P´ asztor, L. Pirelli, and C. H. Wong, Phys. Rev. D 111, 014506 (2025)

2025
[7]

Philipsen, Symmetry 13, 2079 (2021), arXiv:2111.03590 [hep-lat]

O. Philipsen, Symmetry 13, 2079 (2021), arXiv:2111.03590 [hep-lat]

arXiv 2021
[8]

H. T. Ding et al. (HotQCD), Phys. Rev. Lett. 123, 062002 (2019), arXiv:1903.04801 [hep-lat]

arXiv 2019

Show all 67 references

[9]

Gao and J

F. Gao and J. M. Pawlowski, Phys. Rev. D 102, 034027 (2020), arXiv:2002.07500 [hep-ph]

2020 arXiv
[10]

Gao and J

F. Gao and J. M. Pawlowski, Phys. Lett. B 820, 136584 (2021), arXiv:2010.13705 [hep-ph]

2021 arXiv
[11]

W.-j. Fu, J. M. Pawlowski, and F. Rennecke, Phys. Rev. D 101, 054032 (2020), arXiv:1909.02991 [hep-ph]

2020 arXiv
[12]

P. J. Gunkel and C. S. Fischer, Phys. Rev. D104, 054022 (2021), arXiv:2106.08356 [hep-ph]

2021 arXiv
[13]

Braun et al., (2023), arXiv:2310.19853 [hep-ph]

J. Braun et al., (2023), arXiv:2310.19853 [hep-ph]

2023 arXiv
[14]

C. S. Fischer, J. Phys. G 32, R253 (2006), arXiv:hep- ph/0605173

2006
[15]

Greensite, An introduction to the confinement problem, Vol

J. Greensite, An introduction to the confinement problem, Vol. 821 (2011)

2011
[16]

Reinosa, Eur

U. Reinosa, Eur. Phys. J. C 85, 199 (2025), arXiv:2404.06118 [hep-ph]

2025 arXiv
[17]

Dupuis, L

N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, Phys. Rept. 910, 1 (2021), arXiv:2006.04853 [cond-mat.stat-mech]

2021 arXiv
[18]

Braun, H

J. Braun, H. Gies, and J. M. Pawlowski, Phys. Lett. B 684, 262 (2010), arXiv:0708.2413 [hep-th]

2010 arXiv
[19]

Fister and J

L. Fister and J. M. Pawlowski, Phys. Rev. D 88, 045010 (2013), arXiv:1301.4163 [hep-ph]

2013 arXiv
[20]

Reinosa, J

U. Reinosa, J. Serreau, M. Tissier, and N. Wschebor, Phys. Lett. B 742, 61 (2015), arXiv:1407.6469 [hep-ph]

2015 arXiv
[21]

Connelly, G

A. Connelly, G. Johnson, F. Rennecke, and V. Skokov, Phys. Rev. Lett. 125, 191602 (2020), arXiv:2006.12541 [cond-mat.stat-mech]

2020 arXiv
[22]

Rennecke and V

F. Rennecke and V. V. Skokov, Annals Phys.444, 169010 (2022), arXiv:2203.16651 [hep-ph]

2022 arXiv
[23]

Johnson, F

G. Johnson, F. Rennecke, and V. V. Skokov, Phys. Rev. D 107, 116013 (2023), arXiv:2211.00710 [hep-ph]

2023 arXiv
[24]

Ihssen and J

F. Ihssen and J. M. Pawlowski, SciPost Phys. 15, 074 (2023), arXiv:2207.10057 [hep-th]

2023 arXiv
[25]

Zambello, D

K. Zambello, D. A. Clarke, P. Dimopoulos, F. Di Renzo, J. Goswami, G. Nicotra, C. Schmidt, and S. Singh, PoS LATTICE2022, 164 (2023), arXiv:2301.03952 [hep-lat]

2023 arXiv
[26]

D. A. Clarke, P. Dimopoulos, F. Di Renzo, J. Goswami, C. Schmidt, S. Singh, and K. Zambello, PoS LAT- TICE2023, 168 (2024), arXiv:2401.08820 [hep-lat]

2024 arXiv
[27]

A. Adam, S. Bors´ anyi, Z. Fodor, J. N. Guenther, P. Parotto, A. P´ asztor, D. Peszny´ ak, L. Pirelli, and C. H. Wong, PoS LATTICE2024, 178 (2025), arXiv:2502.03211 [hep-lat]

2025 arXiv
[28]

Roberge and N

A. Roberge and N. Weiss, Nucl. Phys. B 275, 734 (1986)

1986
[29]

Sakai, K

Y. Sakai, K. Kashiwa, H. Kouno, and M. Yahiro, Phys. Rev. D 77, 051901 (2008), arXiv:0801.0034 [hep-ph]

2008 arXiv
[30]

Kashiwa, T

K. Kashiwa, T. Hell, and W. Weise, Phys. Rev. D 84, 056010 (2011), arXiv:1106.5025 [hep-ph]

2011 arXiv
[31]

C. S. Fischer, J. Luecker, and J. M. Pawlowski, Phys. Rev. D 91, 014024 (2015), arXiv:1409.8462 [hep-ph]

2015 arXiv
[32]

Cuteri, J

F. Cuteri, J. Goswami, F. Karsch, A. Lahiri, M. Neu- mann, O. Philipsen, C. Schmidt, and A. Sciarra, Phys. Rev. D 106, 014510 (2022), arXiv:2205.12707 [hep-lat]

2022 arXiv
[33]

Dimopoulos, L

P. Dimopoulos, L. Dini, F. Di Renzo, J. Goswami, G. Nicotra, C. Schmidt, S. Singh, K. Zambello, and F. Ziesch´ e, Phys. Rev. D 105, 034513 (2022), arXiv:2110.15933 [hep-lat]

2022 arXiv
[34]

C. S. Fischer, L. Fister, J. Luecker, and J. M. Pawlowski, Phys. Lett. B 732, 273 (2014), arXiv:1306.6022 [hep-ph]

2014 arXiv
[35]

C. S. Fischer, J. Luecker, and C. A. Welzbacher, Phys. Rev. D 90, 034022 (2014), arXiv:1405.4762 [hep-ph]. 11

2014 arXiv
[36]

Y. Lu, F. Gao, Y.-x. Liu, and J. M. Pawlowski, (2025), arXiv:2504.05099 [hep-ph]

2025 arXiv
[37]

Braun, L

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

2011 arXiv
[38]

Braun, A

J. Braun, A. Eichhorn, H. Gies, and J. M. Pawlowski, Eur. Phys. J. C70, 689 (2010), arXiv:1007.2619 [hep-ph]

2010 arXiv
[39]

Marhauser and J

F. Marhauser and J. M. Pawlowski, (2008), arXiv:0812.1144 [hep-ph]

2008 arXiv
[40]

T. K. Herbst, J. Luecker, and J. M. Pawlowski, (2015), arXiv:1510.03830 [hep-ph]

2015 arXiv
[41]

Reinosa, J

U. Reinosa, J. Serreau, and M. Tissier, Phys. Rev. D 92, 025021 (2015), arXiv:1504.02916 [hep-th]

2015 arXiv
[42]

Maelger, U

J. Maelger, U. Reinosa, and J. Serreau, Phys. Rev. D 97, 074027 (2018), arXiv:1710.01930 [hep-ph]

2018 arXiv
[43]

D. M. van Egmond, U. Reinosa, J. Serreau, and M. Tissier, SciPost Phys. 12, 087 (2022)

2022
[44]

D. M. van Egmond, O. Oliveira, U. Reinosa, J. Serreau, P. J. Silva, and M. Tissier, (2024), arXiv:2412.07930 [hep-lat]

2024 arXiv
[45]

V. T. Mari Surkau and U. Reinosa, (2025), arXiv:2503.22373 [hep-ph]

2025 arXiv
[46]

C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33, 477 (1994), arXiv:hep-ph/9403224

1994 arXiv
[47]

Alkofer and L

R. Alkofer and L. von Smekal, Phys. Rept. 353, 281 (2001), arXiv:hep-ph/0007355

2001 arXiv
[48]

C. D. Roberts and S. M. Schmidt, Prog. Part. Nucl. Phys. 45, S1 (2000), arXiv:nucl-th/0005064

2000 arXiv
[49]

Rennecke, W.-j

F. Rennecke, W.-j. Fu, and J. M. Pawlowski, PoS CORFU2018, 182 (2019), arXiv:1907.08179 [hep-ph]

2019 arXiv
[50]

A. M. Polyakov, Phys. Lett. B 72, 477 (1978)

1978
[51]

Svetitsky and L

B. Svetitsky and L. G. Yaffe, Nucl. Phys. B 210, 423 (1982)

1982
[52]

Bazavov et al

A. Bazavov et al. , Phys. Rev. D 85, 054503 (2012), arXiv:1111.1710 [hep-lat]

2012 arXiv
[53]

P. N. Meisinger and M. C. Ogilvie, Phys. Lett. B 379, 163 (1996), arXiv:hep-lat/9512011

1996 arXiv
[54]

Schaefer, J

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

2007 arXiv
[55]

L. F. Abbott, Acta Phys. Polon. B 13, 33 (1982)

1982
[56]

M. E. Peskin and D. V. Schroeder, An Introduction to quantum field theory (Addison-Wesley, Reading, USA,
[57]

Ellis, Comput

J. Ellis, Comput. Phys. Commun. 210, 103 (2017), arXiv:1601.05437 [hep-ph]

2017 arXiv
[58]

Dohse, (2018), arXiv:1802.00689 [cs.OH]

M. Dohse, (2018), arXiv:1802.00689 [cs.OH]

2018 arXiv
[59]

F. Gao, J. Papavassiliou, and J. M. Pawlowski, Phys. Rev. D 103, 094013 (2021), arXiv:2102.13053 [hep-ph]

2021 arXiv
[60]

Weiss, Phys

N. Weiss, Phys. Rev. D 24, 475 (1981)

1981
[61]

A. K. Cyrol, M. Mitter, J. M. Pawlowski, and N. Strodthoff, Phys. Rev. D 97, 054006 (2018), arXiv:1706.06326 [hep-ph]

2018 arXiv
[62]

Zheng, Y

H.-w. Zheng, Y. Lu, F. Gao, S.-x. Qin, and Y.-x. Liu, Phys. Rev. D109, 114013 (2024), arXiv:2312.00382 [hep- ph]

2024 arXiv
[63]

Kouno, Y

H. Kouno, Y. Sakai, K. Kashiwa, and M. Yahiro, J. Phys. G 36, 115010 (2009), arXiv:0904.0925 [hep-ph]

2009 arXiv
[64]

Scheffler, M

D. Scheffler, M. Buballa, and J. Wambach, Acta Phys. Polon. Supp. 5, 971 (2012), arXiv:1111.3839 [hep-ph]

2012 arXiv
[65]

Fukushima, Phys

K. Fukushima, Phys. Rev. D 79, 074015 (2009), arXiv:0901.0783 [hep-ph]

2009 arXiv
[66]

Bollweg, J

D. Bollweg, J. Goswami, O. Kaczmarek, F. Karsch, S. Mukherjee, P. Petreczky, C. Schmidt, and P. Scior (HotQCD), Phys. Rev. D 105, 074511 (2022), arXiv:2202.09184 [hep-lat]

2022 arXiv
[67]

github.io/

fQCD collaboration, https://fqcd-collaboration. github.io/

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Z.-Y. Wan, Y. Lu, F. Gao, and Y.-X. Liu, Eur. Phys. J. C 84, 899 (2024), arXiv:2401.04957 [hep-ph]

arXiv 2024

[2] [2]

Aarts et al

G. Aarts et al. , Prog. Part. Nucl. Phys. 133, 104070 (2023), arXiv:2301.04382 [hep-lat]

arXiv 2023

[3] [3]

Fu, Commun

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

arXiv 2022

[4] [4]

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

arXiv 2019

[5] [5]

On the QCD phase structure from effective models

B.-J. Schaefer and M. Wagner, Prog. Part. Nucl. Phys. 62, 381 (2009), arXiv:0812.2855 [hep-ph]

work page Pith review arXiv 2009

[6] [6]

Bors´ anyi, Z

S. Bors´ anyi, Z. Fodor, J. N. Guenther, R. Kara, P. Parotto, A. P´ asztor, L. Pirelli, and C. H. Wong, Phys. Rev. D 111, 014506 (2025)

2025

[7] [7]

Philipsen, Symmetry 13, 2079 (2021), arXiv:2111.03590 [hep-lat]

O. Philipsen, Symmetry 13, 2079 (2021), arXiv:2111.03590 [hep-lat]

arXiv 2021

[8] [8]

H. T. Ding et al. (HotQCD), Phys. Rev. Lett. 123, 062002 (2019), arXiv:1903.04801 [hep-lat]

arXiv 2019