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At imaginary chemical potential, the first-order chiral transition seen on coarse lattices disappears before the continuum limit; the paper concludes the continuum chiral transition is second-order and its order is independent of chemical p

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2026-08-03 15:50 UTC pith:FSAMZ5W7

load-bearing objection Solid new data at imaginary chemical potential, but the continuum second-order conclusion is still carried by a tricritical extrapolation the data do not yet force. the 4 major comments →

arxiv 2512.15418 v2 pith:FSAMZ5W7 submitted 2025-12-17 hep-lat

On the nature of the QCD chiral phase transition with imaginary chemical potential

Alfredo D'Ambrosio , Michael Fromm , Reinhold Kaiser , Owe Philipsen This is my paper
classification hep-lat MSC 81T2581V05 PACS 12.38.Gc12.38.Mh
keywords QCD chiral phase transitionimaginary chemical potentialtricritical pointlattice QCDstaggered fermionsfinite temperaturechiral limitcritical line
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

At an imaginary baryon chemical potential, where lattice simulations have no sign problem, the paper asks whether the first-order chiral transition observed on coarse lattices survives the continuum limit. Using non-integer numbers of mass-degenerate quark flavours between 1.8 and 6 on three lattice spacings Nτ=4,6,8, it maps the boundary between first-order and crossover regions and fits that boundary to a tricritical scaling form. The fitted tricritical points move toward larger Nf as the lattice is refined, implying there is a maximal Nτ beyond which the first-order transition is lost. The paper concludes that the continuum chiral limit is a second-order transition for these theories, that the order of the transition is independent of imaginary chemical potential, and therefore (barring non-analyticity) independent of small real chemical potential. If correct, the first-order region of the classic quark-mass–flavour phase diagram is a lattice artefact, and the physical Nf=2+1 transition is a second-order line in the massless limit for any strange-quark mass.

Core claim

On Nτ∈{4,6,8} lattices with unimproved staggered fermions and fixed imaginary chemical potential μ_i=0.81πT/3, the paper determines the chiral critical mass for Nf between 1.8 and 6 and fits the Z2 boundary line to the tricritical scaling form N_f^c(m)=N_f^tric+D1(am)^{2/5}+D2(am)^{4/5}. The fits give N_f^tric=1.54(15), 2.61(17), 3.2(5) for Nτ=4,6,8, showing the first-order region shrinks with decreasing lattice spacing and terminates in a tricritical point before the continuum. The authors conclude that the first-order transitions seen on coarse lattices are not analytically connected to the continuum limit, that the continuum chiral limit is second-order, and that the transition order does

What carries the argument

The central tool is the tricritical scaling ansatz: near a tricritical point the chiral critical line behaves as N_f^c(m)=N_f^tric+D1(am)^{2/5}+D2(am)^{4/5}+..., with mean-field exponents 2/5 and 4/5. The paper locates the Z2 boundary by computing the kurtosis (standardized fourth moment) of the chiral condensate on the phase boundary and fitting it to a finite-size scaling form that uses the critical exponents of the three-dimensional Z2 universality class. Fitting the resulting critical masses to the scaling form yields the tricritical point N_f^tric(Nτ) in each lattice spacing; the trend of those points with Nτ is what shows the first-order region disappearing before the continuum limit.

Load-bearing premise

The conclusion rests on the tricritical scaling ansatz extrapolated from only three lattice spacings, together with the assumption that the staggered action's reduced chiral symmetry does not change the transition order; the paper itself concedes that the leading term is just beginning to be constrained and that it is not yet close to the continuum.

What would settle it

Measure the chiral critical boundary on lattices with Nτ=10 or 12, or with a chirally symmetric lattice action, at the same imaginary chemical potential. Finding a first-order transition at small mass, or a fitted tricritical point N_f^tric(Nτ) that decreases rather than keeps rising, would refute the second-order continuum claim. A cheaper test: at Nτ=8 the leading tricritical coefficient is currently D1=2(9); if additional mass points force it significantly away from zero and N_f^tric below 3, the extrapolation collapses.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • At imaginary chemical potential, the first-order region observed on coarse lattices is a lattice artefact: it vanishes in a tricritical point on finite Nτ and is not connected to the continuum.
  • In the continuum chiral limit, the Nf=2+1 transition is second order for massless up and down quarks at any strange-quark mass, and becomes an analytic crossover as soon as the light quarks are massive.
  • The order of the chiral transition is independent of imaginary chemical potential; if the chiral critical surface is analytic around zero chemical potential, the same holds for small real chemical potentials.
  • The paper's three-dimensional phase diagram leaves room for a QCD critical point only if the chiral critical surface bends away from the chiral limit at large real chemical potential; that non-analytic behaviour is not excluded by these data.

Where Pith is reading between the lines

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

  • If one trusts the tricritical extrapolation, the long-standing sigma-model expectation of a wide first-order region in the quark-mass phase diagram is replaced by a second-order line; the physical crossover at nonzero quark masses would be the smoothed remnant of that line, making the pseudo-critical temperature the reliable quantity rather than critical exponents.
  • A direct test of the paper's central caveat: repeat the same flavour scan with a chirally symmetric lattice action or on finer lattices (Nτ=10 or 12) at the same imaginary chemical potential; a first-order region appearing there would overturn the second-order continuum conclusion.
  • The data at Nτ=8 leave the leading tricritical coefficient only weakly constrained, so the extrapolation could be checked by measuring the critical boundary at smaller masses on Nτ=8; if the (am)^{2/5} term does not stabilise, the claim that the first-order region terminates before the continuum is not yet settled.
  • The paper's scenario pushes a possible QCD critical point to large real chemical potential, which would make experimental searches at lower beam energies the only relevant region for a critical-endpoint signal.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. This paper studies the order of the chiral phase transition in QCD with unimproved staggered fermions at fixed imaginary baryon chemical potential μ_i = 0.81πT/3. Following the non-integer Nf strategy of ref. [12], the authors map the Z2 boundary between first-order and crossover behaviour for Nf ∈ [1.8, 6] on Nτ = 4, 6, 8 lattices using Binder cumulants of the chiral condensate. The critical lines are fitted to tricritical scaling forms in the bare mass (eqs. (8)–(10)), yielding tricritical endpoints N_f^tric(Nτ) = 1.54(15), 2.61(17), 3.2(5) for Nτ = 4, 6, 8. The paper interprets the growth of N_f^tric with decreasing lattice spacing as evidence that the first-order regions seen on coarse lattices are cutoff artefacts, so that the continuum chiral transition is second order at zero and imaginary μ, with consequences for the Nf = 2+1 Columbia plot and the possible QCD critical endpoint.

Significance. The direct numerical material is of high quality: about 150 million trajectories, four independent Markov chains per parameter set, explicit checks that the simulation window brackets the transition (min |B3/σ_B3|_edge > 3), and autocorrelation diagnostics in Appendix A. These are important strengths. If the continuum conclusion is correct, the paper substantially strengthens the case against the classic first-order chiral transition in the Nf = 2+1 Columbia plot and is consistent with improved-staggered, domain-wall and Dyson-Schwinger results. The result is therefore potentially field-changing. The weakness is not in the raw data but in the inferential step from three lattice spacings to the continuum; the paper itself acknowledges this in footnote 3 and Sec. 4. Because the headline claim is conditional on an extrapolation that is not yet uniquely selected by the data, I cannot recommend acceptance in the present form.

major comments (4)
  1. [Sec. 4, Eq. (9), Table 3] The central quantity N_tric_f(8)=3.2(5) is not robustly constrained. At Nτ=8, D1=2(9), statistically indistinguishable from zero, and the fit window am∈[0,0.01] implies x=(am)^{2/5}≤0.158, with the smallest simulated mass at x≈0.063; there are no data near x=0. The intercept is therefore fixed by D2 and the assumed curvature. Footnote 3 concedes that the leading term 'is just beginning to be constrained', and Table 2 shows the same feature for C1. The authors should demonstrate that the data lie in the tricritical scaling window, e.g. by stability under excluding the largest mass, adding O(x^3) terms, or using priors from mean-field scaling. Without this, the tricritical endpoint—and the continuum conclusion built on it—remains a plausible scenario rather than an established result.
  2. [Sec. 4, Eq. (11), Table 4] The statement that polynomial behaviour is 'incompatible with the data' is not supported for all Nf. The NLO+NNLO polynomial fit for Nf=4 has χ²/dof=0.058 (dof=1), and the LO+NLO fit for Nf=6 has χ²/dof=1.87; both are statistically acceptable. The same three-point sets are also compatible with tricritical scaling, so χ² alone cannot select between the two functional forms. A model-selection criterion or additional Nτ values are needed to exclude the continuum-first-order scenario described by Eq. (11). As written, this exclusion claim is stronger than the evidence.
  3. [Sec. 5] The bullet conclusion that the chiral transition order 'does not show any dependence on imaginary chemical potential' goes beyond the data. Only μ_i=0 and μ_i=0.81πT/3 are compared, and Sec. 4 notes a crossing of the relative sizes of the first-order regions between Nτ=4 and Nτ=6/8, indicating cutoff-density mixing. At minimum the wording should be softened to 'no dependence observed at the two values studied'; a stronger claim requires additional μ_i values or a curvature analysis.
  4. [Sec. 4] No continuum extrapolation of N_tric_f(Nτ) itself is presented. The argument that the first-order region is lost before the continuum relies on the monotonic increase 1.54→2.61→3.2, but with only three lattice spacings no extrapolation can be tested. The abstract's caveat 'unless additional first-order transitions are found on finer lattices' is therefore an essential part of the claim; the paper should either provide a quantitative extrapolation or explicitly label the continuum statement as a conjecture.
minor comments (3)
  1. [Sec. 4] 'lattic chiral limit' should read 'lattice chiral limit'.
  2. [Sec. 5] Typo: 'first-oder' should be 'first-order'.
  3. [Fig. 5a] The inset showing LO and NLO interpolations for the error estimate of (aT)_tric is not described in the caption; please add a sentence explaining the procedure.

Circularity Check

0 steps flagged

No significant circularity: the tricritical endpoints are fitted, but the qualitative lattice trend is directly observed, the competing polynomial scenario is explicitly tested, and the continuum inference is expressly conditional.

full rationale

The paper's derivation is an empirical lattice study. Critical bare masses are obtained by Binder-cumulant finite-size scaling (eq. 7) with external 3D Ising exponents [53,54], not from the tricritical ansatz. The tricritical forms (eqs. 8-10) are motivated by mean-field tricritical theory (Lawrie-Sarbach [36]) and are used to locate N_tric^f(Nτ); this is a fit, but the qualitative trend is not an artifact. Independently of the extrapolated endpoints, table 1 shows that the smallest Nf at which a first-order region is found rises from ~1.8-2.3 at Nτ=4, to Nf=3.0 at Nτ=6, to Nf=4.0 at Nτ=8. The paper explicitly considers the competing continuum-first-order (polynomial) scenario, eq. (11), and reports it as incompatible with the data, albeit with caveats visible in table 4; that is a statistical-model comparison, not a circular reduction. The continuum conclusion is expressly conditional: the Abstract and Sec. 5 state 'Unless additional first-order transitions are found on finer lattices or with chiral lattice actions', Sec. 4 admits 'we are not yet close to the continuum', and footnote 3 concedes 'the leading term is just beginning to be constrained'. These are extrapolation limitations and should lower confidence, but they are not instances where the conclusion equals its input. Self-citations to refs. [35,12] supply the μ=0 baseline, the method, and the analogous prior result, but the new imaginary-μ simulations and fits are independent data; the load-bearing comparison in this paper is to the new runs, not to an unverified self-citation. No step in the derivation reduces by construction to its own input.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 0 invented entities

Everything non-standard is listed above: two domain assumptions about universality classes, rooted staggered representativity, and staggered chiral representativity, plus a standard reweighting assumption. The free parameters are the fitted tricritical ansatz quantities (N_tric_f, β_tric, scaling coefficients); the least-constrained of them carry the continuum extrapolation. No new particles, fields, or symmetries are invented; the tricritical line is a fitted property of the known parameter space.

free parameters (6)
  • N_tric_f(Nτ) = 1.54(15), 2.61(17), 3.2(5) at Nτ=4,6,8
    Tricritical endpoint in flavour space, fitted via eq. (9) to the Z2 critical masses; it carries the central conclusion that the first-order region terminates before the continuum.
  • β_tric(Nτ) = 5.347(7), 5.325(24), 5.28(9)
    Tricritical gauge coupling from eq. (8), LO+NLO fits; the Nτ=8 value has a large error (9) in the LO+NLO form.
  • C1, C2 (tricritical β-fit coefficients) = e.g. C1=-10(6), C2=-1.5(1.5) at Nτ=8
    Coefficients of the (am)^{2/5} and (am)^{4/5} terms in eq. (8); the leading term is barely constrained (text, fn. 3).
  • D1, D2 (tricritical Nf-fit coefficients) = e.g. D1=2(9), D2=110(40) at Nτ=8
    Coefficients in eq. (9); D1 is consistent with zero at Nτ=8, so the fit is dominated by the NLO term and the extrapolation is weakly anchored.
  • Fit mass windows in am = [0,0.08], [0,0.03], [0,0.01] for Nτ=4,6,8
    Manual choice of fit ranges in table 2; the ranges shrink with Nτ and affect the extracted tricritical quantities.
  • E1, E2 (aT tricritical interpolation coefficients) = not tabulated
    Eq. (10) interpolations in fig. 5a; the text states that with only three Nτ-values a full NLO fit is not feasible (Sec. 4).
axioms (5)
  • domain assumption The Z2 boundary separating first-order from crossover lies in the 3D Ising universality class, so the kurtosis crossing B4=1.6044(10) [53] with 3D Ising exponents [54] locates am_c (eq. 7).
    Standard for this programme, but the universality class of the chiral critical line in staggered QCD is part of the physics under test.
  • domain assumption Tricritical mean-field scaling with exponents 2/5 and 4/5 (Lawrie-Sarbach [36]) describes the Z2 wing line (eqs. 8-10).
    The fits assume a tricritical point exists; the existence of the tricritical point at finite Nτ is the conclusion the fits are used to draw, so the ansatz partly carries the result.
  • domain assumption Continuous Nf via (det M)^{Nf/4} rooting maps faithfully to integer-flavour physics (eq. 1).
    Rooted staggered fermions with non-integer roots are standard in this programme but carry the well-known rooting caveats about reproducing the correct Nf=2+1 theory.
  • domain assumption The transition order observed with unimproved staggered fermions (which reduce chiral symmetry to a U(1) remnant) is the same as in continuum QCD.
    The paper's QCD-level conclusion is inferred from an action with reduced chiral symmetry; the authors explicitly request chirally symmetric checks (Sec. 5).
  • standard math Multiple-histogram reweighting [52] is valid for interpolating B3 and B4 between simulated β-values.
    Standard reweighting method; validity across the simulated range is assumed without discussion.

pith-pipeline@v1.3.0-alltime-deepseek · 18804 in / 28159 out tokens · 281911 ms · 2026-08-03T15:50:56.940433+00:00 · methodology

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read the original abstract

The order of the thermal chiral phase transition in lattice QCD is known to be strongly cutoff-dependent. A previous study using $N_\mathrm{f}\in[2,6]$ mass-degenerate, unimproved staggered quark flavours on $N_\tau\in\{4,6,8\}$ lattices found that the bare mass regions displaying explicit first-order transitions shrink to zero, with their critical boundary line terminating in a tricritical point before the continuum limit is reached. Here we perform an analogous study for fixed imaginary baryon chemical potential and find the same behaviour: first-order regions observed on coarse lattices disappear in tricritical points with diminishing lattice spacing. These observations are consistent with currently available results from improved staggered discretisations, both at zero and non-zero imaginary chemical potential. Unless additional first-order transitions are found on finer lattices or with chiral lattice actions, this implies a second-order transition in the continuum chiral limit for all these cases, at zero and imaginary chemical potential. Implications for the $N_\mathrm{f}=2+1$ QCD phase diagram at the physical point are discussed.

Figures

Figures reproduced from arXiv: 2512.15418 by Alfredo D'Ambrosio, Michael Fromm, Owe Philipsen, Reinhold Kaiser.

Figure 1
Figure 1. Figure 1: Nature of the Nf = 2 + 1 QCD thermal transition as a function of quark masses. Figures taken from [12]. By contrast, it is much harder to obtain robust lattice results on the order of the chiral transition because of the necessary finite size scaling analysis, in addition to the continuum and chiral extrapolations. For Nf = 2 + 1 QCD, the nature of the thermal transition as a function of quark masses is sc… view at source ↗
Figure 2
Figure 2. Figure 2: Columbia plot for mass-degenerate quarks, scenario analogous to figure [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Columbia plot with chemical potential as observed on coarse lattices. The bottom [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Projections of the critical surface on the ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Critical line for different Nf in the (am, aT)-plane [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

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Reference graph

Works this paper leans on

56 extracted references · 5 canonical work pages · cited by 2 Pith papers

  1. [1]

    Y. Aoki, G. Endr¨ odi, Z. Fodor, S. D. Katz, and K. K. Szabo. The order of the quantum chromodynamics transition predicted by the standard model of particle physics.Nature, 443:675–678, 2006.arXiv:hep-lat/0611014,doi:10.1038/nature05120

    Pith/arXiv arXiv 2006

  2. [2]

    The Chiral phase transition in QCD: Critical phenomena and long wavelength pion oscillations

    Krishna Rajagopal. The Chiral phase transition in QCD: Critical phenomena and long wavelength pion oscillations. pages 484–554, 4 1995.arXiv:hep-ph/9504310,doi:10. 1142/9789812830661_0009

    Pith/arXiv arXiv 1995

  3. [3]

    Adam Miklos Halasz, A. D. Jackson, R. E. Shrock, Misha A. Stephanov, and J. J. M. Verbaarschot. On the phase diagram of QCD.Phys. Rev. D, 58:096007, 1998.arXiv: hep-ph/9804290,doi:10.1103/PhysRevD.58.096007

    Pith/arXiv arXiv 1998

  4. [4]

    Stephen D. H. Hsu and Myckola Schwetz. On the QCD phase transition at finite baryon density.Phys. Lett. B, 432:203–208, 1998.arXiv:hep-ph/9803386,doi:10.1016/ S0370-2693(98)00615-7

    Pith/arXiv arXiv 1998

  5. [5]

    Stephanov, K

    Misha A. Stephanov, K. Rajagopal, and Edward V. Shuryak. Signatures of the tricritical point in QCD.Phys. Rev. Lett., 81:4816–4819, 1998.arXiv:hep-ph/9806219,doi:10. 1103/PhysRevLett.81.4816

    Pith/arXiv arXiv 1998

  6. [6]

    The Condensed matter physics of QCD

    Krishna Rajagopal and Frank Wilczek. The Condensed matter physics of QCD. In M. Shif- man and Boris Ioffe, editors,At the frontier of particle physics. Handbook of QCD. Vol. 1-3, pages 2061–2151. 2000.arXiv:hep-ph/0011333,doi:10.1142/9789812810458_0043

    Pith/arXiv arXiv 2061

  7. [7]

    Universality, the QCD critical / tricritical point and the quark number susceptibility.Phys

    Yoshitaka Hatta and Takashi Ikeda. Universality, the QCD critical / tricritical point and the quark number susceptibility.Phys. Rev. D, 67:014028, 2003.arXiv:hep-ph/0210284, doi:10.1103/PhysRevD.67.014028

    Pith/arXiv arXiv 2003

  8. [8]

    H. T. Ding et al. Chiral Phase Transition Temperature in ( 2+1 )-Flavor QCD.Phys. Rev. Lett., 123(6):062002, 2019.arXiv:1903.04801,doi:10.1103/PhysRevLett.123.062002

    Pith/arXiv arXiv 2019

  9. [9]

    Kotov, Maria Paola Lombardo, and Anton Trunin

    Andrey Yu. Kotov, Maria Paola Lombardo, and Anton Trunin. QCD transition at the physical point, and its scaling window from twisted mass Wilson fermions.Phys. Lett. B, 823:136749, 2021.arXiv:2105.09842,doi:10.1016/j.physletb.2021.136749

    Pith/arXiv arXiv 2021

  10. [10]

    Pisarski and Frank Wilczek

    Robert D. Pisarski and Frank Wilczek. Remarks on the Chiral Phase Transition in Chro- modynamics.Phys. Rev., D29:338–341, 1984.doi:10.1103/PhysRevD.29.338

    doi:10.1103/physrevd.29.338 1984

  11. [11]

    Relevance of the axial anomaly at the finite- temperature chiral transition in QCD.Phys

    Andrea Pelissetto and Ettore Vicari. Relevance of the axial anomaly at the finite- temperature chiral transition in QCD.Phys. Rev. D, 88(10):105018, 2013.arXiv: 1309.5446,doi:10.1103/PhysRevD.88.105018

    Pith/arXiv arXiv 2013

  12. [12]

    On the order of the QCD chiral phase transition for different numbers of quark flavours.JHEP, 11:141, 2021.arXiv: 2107.12739,doi:10.1007/JHEP11(2021)141

    Francesca Cuteri, Owe Philipsen, and Alessandro Sciarra. On the order of the QCD chiral phase transition for different numbers of quark flavours.JHEP, 11:141, 2021.arXiv: 2107.12739,doi:10.1007/JHEP11(2021)141

    Pith/arXiv arXiv 2021

  13. [13]

    Gausterer and S

    H. Gausterer and S. Sanielevici. Can the Chiral Transition in QCD Be Described by a LinearσModel in Three-dimensions?Phys. Lett. B, 209:533–537, 1988.doi:10.1016/ 0370-2693(88)91188-4

    1988

  14. [14]

    On the nature of the finite temperature transition in QCD.JHEP, 08:029, 2003.arXiv:hep-ph/0307036,doi: 10.1088/1126-6708/2003/08/029

    Agostino Butti, Andrea Pelissetto, and Ettore Vicari. On the nature of the finite temperature transition in QCD.JHEP, 08:029, 2003.arXiv:hep-ph/0307036,doi: 10.1088/1126-6708/2003/08/029. 14

    Pith/arXiv arXiv 2003

  15. [15]

    Brown, Frank P

    Frank R. Brown, Frank P. Butler, Hong Chen, Norman H. Christ, Zhi-hua Dong, Wendy Schaffer, Leo I. Unger, and Alessandro Vaccarino. On the existence of a phase transition for QCD with three light quarks.Phys. Rev. Lett., 65:2491–2494, 1990.doi:10.1103/ PhysRevLett.65.2491

    1990

  16. [16]

    Iwasaki, K

    Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai, and T. Yoshie. Finite temperature transitions in lattice QCD with Wilson quarks: Chiral transitions and the influence of the strange quark. Phys. Rev. D, 54:7010–7031, 1996.arXiv:hep-lat/9605030,doi:10.1103/PhysRevD.54. 7010

    Pith/arXiv arXiv 1996

  17. [17]

    Karsch, E

    F. Karsch, E. Laermann, and C. Schmidt. The Chiral critical point in three-flavor QCD. Phys. Lett., B520:41–49, 2001.arXiv:hep-lat/0107020,doi:10.1016/S0370-2693(01) 01114-5

    Pith/arXiv arXiv 2001

  18. [18]

    The QCD phase diagram for three degenerate flavors and small baryon density.Nucl

    Philippe de Forcrand and Owe Philipsen. The QCD phase diagram for three degenerate flavors and small baryon density.Nucl. Phys., B673:170–186, 2003.arXiv:hep-lat/ 0307020,doi:10.1016/j.nuclphysb.2003.09.005

    doi:10.1016/j.nuclphysb.2003.09.005 2003

  19. [19]

    Lattice Constraints on the QCD Chiral Phase Transition at Finite Temperature and Baryon Density.Symmetry, 13(11):2079, 2021.arXiv:2111.03590, doi:10.3390/sym13112079

    Owe Philipsen. Lattice Constraints on the QCD Chiral Phase Transition at Finite Temperature and Baryon Density.Symmetry, 13(11):2079, 2021.arXiv:2111.03590, doi:10.3390/sym13112079

    Pith/arXiv arXiv 2079

  20. [20]

    Aspects of finite temperature QCD towards the chiral limit.PoS, LAT- TICE2021:003, 2022.arXiv:2112.08164,doi:10.22323/1.396.0003

    Anirban Lahiri. Aspects of finite temperature QCD towards the chiral limit.PoS, LAT- TICE2021:003, 2022.arXiv:2112.08164,doi:10.22323/1.396.0003

    Pith/arXiv arXiv 2022

  21. [21]

    Guenther

    Jana N. Guenther. An overview of the QCD phase diagram at finiteTandµ.PoS, LATTICE2021:013, 2022.arXiv:2201.02072,doi:10.22323/1.396.0013

    Pith/arXiv arXiv 2022

  22. [22]

    Nature of the phase transition for finite temperatureN f = 3 QCD with nonperturbatively O(a) improved Wilson fermions atN t = 12.Phys

    Yoshinobu Kuramashi, Yoshifumi Nakamura, Hiroshi Ohno, and Shinji Takeda. Nature of the phase transition for finite temperatureN f = 3 QCD with nonperturbatively O(a) improved Wilson fermions atN t = 12.Phys. Rev. D, 101(5):054509, 2020.arXiv:2001. 04398,doi:10.1103/PhysRevD.101.054509

    work page doi:10.1103/physrevd.101.054509 2020

  23. [23]

    Chiral phase transition in three-flavor QCD from lattice QCD.Phys

    Lorenzo Dini, Prasad Hegde, Frithjof Karsch, Anirban Lahiri, Christian Schmidt, and Sipaz Sharma. Chiral phase transition in three-flavor QCD from lattice QCD.Phys. Rev. D, 105(3):034510, 2022.arXiv:2111.12599,doi:10.1103/PhysRevD.105.034510

    Pith/arXiv arXiv 2022

  24. [24]

    Finite temperature QCD phase transition with 3 flavors of Mobius domain wall fermions.PoS, LATTICE2022:197, 2023.arXiv:2212.10021,doi:10.22323/ 1.430.0197

    Yu Zhang, Yasumichi Aoki, Shoji Hashimoto, Issaku Kanamori, Takashi Kaneko, and Yoshifumi Nakamura. Finite temperature QCD phase transition with 3 flavors of Mobius domain wall fermions.PoS, LATTICE2022:197, 2023.arXiv:2212.10021,doi:10.22323/ 1.430.0197

    Pith/arXiv arXiv 2023

  25. [25]

    Three flavor QCD phase transition with Mobius domain wall fermions.PoS, LATTICE2024:193, 2025.arXiv:2501.15494,doi:10.22323/1.466.0193

    Yu Zhang, Yasumichi Aoki, Shoji Hashimoto, Issaku Kanamori, Takashi Kaneko, and Yoshifumi Nakamura. Three flavor QCD phase transition with Mobius domain wall fermions.PoS, LATTICE2024:193, 2025.arXiv:2501.15494,doi:10.22323/1.466.0193

    Pith/arXiv arXiv 2025

  26. [26]

    Exploring the QCD phase diagram with three flavors of M¨ obius domain wall fermions.PoS, LATTICE2023:203, 2024.arXiv:2401.05066,doi:10.22323/ 1.453.0203

    Yu Zhang, Yasumichi Aoki, Shoji Hashimoto, Issaku Kanamori, Takashi Kaneko, and Yoshifumi Nakamura. Exploring the QCD phase diagram with three flavors of M¨ obius domain wall fermions.PoS, LATTICE2023:203, 2024.arXiv:2401.05066,doi:10.22323/ 1.453.0203

    Pith/arXiv arXiv 2024

  27. [27]

    G. Fejos. Second-order chiral phase transition in three-flavor quantum chromodynamics? Phys. Rev. D, 105(7):L071506, 2022.arXiv:2201.07909,doi:10.1103/PhysRevD.105. L071506. 15

    Pith/arXiv arXiv 2022

  28. [28]

    Pisarski and Fabian Rennecke

    Robert D. Pisarski and Fabian Rennecke. Conjectures about the Chiral Phase Transition in QCD from Anomalous Multi-Instanton Interactions.Phys. Rev. Lett., 132(25):251903, 2024.arXiv:2401.06130,doi:10.1103/PhysRevLett.132.251903

    Pith/arXiv arXiv 2024

  29. [29]

    Pisarski, and Fabian Rennecke

    Francesco Giacosa, Gy˝ oz˝ o Kov´ acs, P´ eter Kov´ acs, Robert D. Pisarski, and Fabian Rennecke. Anomalous U(1)A couplings and the Columbia plot.Phys. Rev. D, 111(1):016014, 2025. arXiv:2410.08185,doi:10.1103/PhysRevD.111.016014

    Pith/arXiv arXiv 2025

  30. [30]

    Fejos and T

    G. Fejos and T. Hatsuda. Order of the SU(Nf)×SU(Nf) chiral transition via the functional renormalization group.Phys. Rev. D, 110(1):016021, 2024.arXiv:2404.00554,doi: 10.1103/PhysRevD.110.016021

    Pith/arXiv arXiv 2024

  31. [31]

    CFTs withU(m)×U(n) global symmetry in 3D and the chiral phase transition of QCD.SciPost Phys., 15(2):075, 2023.arXiv: 2209.02837,doi:10.21468/SciPostPhys.15.2.075

    Stefanos Robert Kousvos and Andreas Stergiou. CFTs withU(m)×U(n) global symmetry in 3D and the chiral phase transition of QCD.SciPost Phys., 15(2):075, 2023.arXiv: 2209.02837,doi:10.21468/SciPostPhys.15.2.075

    Pith/arXiv arXiv 2023

  32. [32]

    Julian Bernhardt and Christian S. Fischer. QCD phase transitions in the light quark chiral limit.Phys. Rev. D, 108(11):114018, 2023.arXiv:2309.06737,doi:10.1103/PhysRevD. 108.114018

    Pith/arXiv arXiv 2023

  33. [33]

    Alfredo D’Ambrosio, Owe Philipsen, and Reinhold Kaiser. The chiral phase transition at non-zero imaginary baryon chemical potential for different numbers of quark flavours.PoS, LATTICE2022:172, 2023.arXiv:2212.03655,doi:10.22323/1.430.0172

    Pith/arXiv arXiv 2023

  34. [34]

    Julian Bernhardt and Christian S. Fischer. Quark mass dependence of a QCD critical point and structure of the Columbia plot. 7 2025.arXiv:2507.21680

    Pith/arXiv arXiv 2025

  35. [35]

    QCD chiral phase transition from noninteger numbers of flavors.Phys

    Francesca Cuteri, Owe Philipsen, and Alessandro Sciarra. QCD chiral phase transition from noninteger numbers of flavors.Phys. Rev. D, 97(11):114511, 2018.arXiv:1711.05658, doi:10.1103/PhysRevD.97.114511

    Pith/arXiv arXiv 2018

  36. [36]

    Lawrie and S

    I.D. Lawrie and S. Sarbach. Theory of tricritical points. In C. Domb and J.L. Lebowitz, editors,Phase transitions and critical phenomena, volume 9, page 1. 1984

    1984

  37. [37]

    Gauge Theories With Imaginary Chemical Potential and the Phases of QCD.Nucl

    Andre Roberge and Nathan Weiss. Gauge Theories With Imaginary Chemical Potential and the Phases of QCD.Nucl. Phys. B, 275:734–745, 1986.doi:10.1016/0550-3213(86) 90582-1

    work page doi:10.1016/0550-3213(86 1986

  38. [38]

    PhD thesis, Goethe U., Frankfurt (Main), 2016

    Alessandro Sciarra.The QCD phase diagram at purely imaginary chemical potential from the lattice. PhD thesis, Goethe U., Frankfurt (Main), 2016

    2016

  39. [39]

    Finite Size and Cut-Off Effects on the Roberge- Weiss Transition inN f = 2 QCD with Staggered Fermions.Phys

    Owe Philipsen and Alessandro Sciarra. Finite Size and Cut-Off Effects on the Roberge- Weiss Transition inN f = 2 QCD with Staggered Fermions.Phys. Rev. D, 101(1):014502, 2020.arXiv:1909.12253,doi:10.1103/PhysRevD.101.014502

    Pith/arXiv arXiv 2020

  40. [40]

    TheN f = 2 QCD chiral phase transition with Wilson fermions at zero and imaginary chemical potential.Phys

    Owe Philipsen and Christopher Pinke. TheN f = 2 QCD chiral phase transition with Wilson fermions at zero and imaginary chemical potential.Phys. Rev., D93(11):114507, 2016.arXiv:1602.06129,doi:10.1103/PhysRevD.93.114507

    Pith/arXiv arXiv 2016

  41. [41]

    Roberge-Weiss transition inN f = 2 QCD with Wilson fermions andN τ = 6.Phys

    Christopher Czaban, Francesca Cuteri, Owe Philipsen, Christopher Pinke, and Alessandro Sciarra. Roberge-Weiss transition inN f = 2 QCD with Wilson fermions andN τ = 6.Phys. Rev., D93(5):054507, 2016.arXiv:1512.07180,doi:10.1103/PhysRevD.93.054507

    Pith/arXiv arXiv 2016

  42. [42]

    Constraining the QCD phase diagram by tricritical lines at imaginary chemical potential.Phys

    Philippe de Forcrand and Owe Philipsen. Constraining the QCD phase diagram by tricritical lines at imaginary chemical potential.Phys. Rev. Lett., 105:152001, 2010. arXiv:1004.3144,doi:10.1103/PhysRevLett.105.152001. 16

    Pith/arXiv arXiv 2010

  43. [43]

    Roberge-weiss endpoint inN f = 2 qcd.Phys

    Claudio Bonati, Guido Cossu, Massimo D’Elia, and Francesco Sanfilippo. Roberge-weiss endpoint inN f = 2 qcd.Phys. Rev. D, 83:054505, Mar 2011. URL:http://link.aps. org/doi/10.1103/PhysRevD.83.054505,doi:10.1103/PhysRevD.83.054505

    work page doi:10.1103/physrevd.83.054505 2011

  44. [44]

    The Chiral critical line ofN f = 2 + 1 QCD at zero and non-zero baryon density.JHEP, 0701:077, 2007.arXiv:hep-lat/0607017, doi:10.1088/1126-6708/2007/01/077

    Philippe de Forcrand and Owe Philipsen. The Chiral critical line ofN f = 2 + 1 QCD at zero and non-zero baryon density.JHEP, 0701:077, 2007.arXiv:hep-lat/0607017, doi:10.1088/1126-6708/2007/01/077

    Pith/arXiv arXiv 2007

  45. [45]

    A QCD chiral critical point at small chemical potential: Is it there or not?PoS, LAT2007:178, 2007.arXiv:0711.0262

    Philippe de Forcrand, Seyong Kim, and Owe Philipsen. A QCD chiral critical point at small chemical potential: Is it there or not?PoS, LAT2007:178, 2007.arXiv:0711.0262

    Pith/arXiv arXiv 2007

  46. [46]

    The Chiral critical point ofN f = 3 QCD at finite density to the order (µ/T) 4.JHEP, 11:012, 2008.arXiv:0808.1096,doi:10.1088/ 1126-6708/2008/11/012

    Philippe de Forcrand and Owe Philipsen. The Chiral critical point ofN f = 3 QCD at finite density to the order (µ/T) 4.JHEP, 11:012, 2008.arXiv:0808.1096,doi:10.1088/ 1126-6708/2008/11/012

    Pith/arXiv arXiv 2008

  47. [47]

    Curvature of the critical line on the plane of quark chemical potential and pseu- doscalar meson mass for three-flavor QCD.Phys

    Xiao-Yong Jin, Yoshinobu Kuramashi, Yoshifumi Nakamura, Shinji Takeda, and Akira Ukawa. Curvature of the critical line on the plane of quark chemical potential and pseu- doscalar meson mass for three-flavor QCD.Phys. Rev., D92(11):114511, 2015.arXiv: 1504.00113,doi:10.1103/PhysRevD.92.114511

    Pith/arXiv arXiv 2015

  48. [48]

    Roberge-Weiss endpoint and chiral symmetry restoration inN f = 2 + 1 QCD.Phys

    Claudio Bonati, Enrico Calore, Massimo D’Elia, Michele Mesiti, Francesco Negro, Francesco Sanfilippo, Sebastiano Fabio Schifano, Giorgio Silvi, and Raffaele Tripiccione. Roberge-Weiss endpoint and chiral symmetry restoration inN f = 2 + 1 QCD.Phys. Rev. D, 99(1):014502, 2019.arXiv:1807.02106,doi:10.1103/PhysRevD.99.014502

    Pith/arXiv arXiv 2019

  49. [49]

    Cuteri, J

    F. Cuteri, J. Goswami, F. Karsch, Anirban Lahiri, M. Neumann, O. Philipsen, Chris- tian Schmidt, and A. Sciarra. Toward the chiral phase transition in the Roberge-Weiss plane.Phys. Rev. D, 106(1):014510, 2022.arXiv:2205.12707,doi:10.1103/PhysRevD. 106.014510

    Pith/arXiv arXiv 2022

  50. [50]

    Cl2qcd, February 2021.doi:10.5281/zenodo.5121917

    Alessandro Sciarra, Christopher Pinke, Matthias Bach, Francesca Cuteri, Lars Zeidlewicz, Christian Sch¨ afer, Tim Breitenfelder, Christopher Czaban, Stefano Lottini, and Paul Fred- erik Depta. Cl2qcd, February 2021.doi:10.5281/zenodo.5121917

    doi:10.5281/zenodo.5121917 2021

  51. [51]

    Bahamas, February 2021.doi:10.5281/zenodo.4577425

    Alessandro Sciarra. Bahamas, February 2021.doi:10.5281/zenodo.4577425

    work page doi:10.5281/zenodo.4577425 2021

  52. [52]

    A. M. Ferrenberg and R. H. Swendsen. New Monte Carlo Technique for Studying Phase Transitions.Phys. Rev. Lett., 61:2635–2638, 1988.doi:10.1103/PhysRevLett.61.2635

    work page doi:10.1103/physrevlett.61.2635 1988

  53. [53]

    H. W. J. Blote, E. Luijten, and J. R. Heringa. Ising universality in three dimensions: a Monte Carlo study.J. Phys. A, 28(22):6289–6313, 1995.arXiv:cond-mat/9509016, doi:10.1088/0305-4470/28/22/007

    Pith/arXiv arXiv 1995

  54. [54]

    Critical phenomena and renormalization group theory.Phys

    Andrea Pelissetto and Ettore Vicari. Critical phenomena and renormalization group theory.Phys. Rept., 368:549–727, 2002.arXiv:cond-mat/0012164,doi:10.1016/ S0370-1573(02)00219-3

    Pith/arXiv arXiv 2002

  55. [55]

    Christian S. Fischer. QCD at finite temperature and chemical potential from Dyson–Schwinger equations.Prog. Part. Nucl. Phys., 105:1–60, 2019.arXiv:1810.12938, doi:10.1016/j.ppnp.2019.01.002

    Pith/arXiv arXiv 2019

  56. [56]

    Pawlowski

    Fei Gao and Jan M. Pawlowski. Chiral phase structure and critical end point in QCD.Phys. Lett. B, 820:136584, 2021.arXiv:2010.13705,doi:10.1016/j.physletb.2021.136584. 17

    Pith/arXiv arXiv 2021