Pith. sign in

REVIEW 5 major objections 6 minor 1 cited by

The order of the chiral phase transition in massless many-flavour lattice QCD

T0 review · 5 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read The paper claims that the first-order chiral transition observed on coarse lattices is a cutoff effect for up to seven quark flavours, leaving a second-order transition in the continuum chiral limit.

desk verdict A useful step in mapping the many-flavour chiral transition, but the all-Nf<=7 conclusion reaches beyond the fits shown. read the letter →

arxiv 2501.19251 v2 pith:5VF2Q5L3 submitted 2025-01-31 hep-lat

Jan Philipp Klinger , Reinhold Kaiser , Owe Philipsen This is my paper · ORCID
classification hep-lat MSC 81T2581V05 PACS 11.15.Ha12.38.Gc11.30.Rd
keywords chiralphasetransitionmany-flavourQCDtricriticalpointlimitlatticestaggeredfermionsZ2boundaryconformalwindow
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 asks whether the chiral phase transition in massless QCD turns first order as the number of quark flavours grows, and answers no for every flavour number up to seven. On coarse lattices a first-order region does appear, but the authors show it is a cutoff effect: for each Nf <= 7, the boundary line between the first-order and crossover regions terminates at a nonzero tricritical lattice spacing, so the continuum extrapolation reaches a second-order transition. The critical temperature at the tricritical point falls as Nf increases, and the data are consistent with a physical tricritical point at vanishing temperature, which would coincide with the onset of the conformal window. If correct, the result rules out a first-order chiral transition in the continuum and ties the order of the transition to conformal dynamics.

What carries the argument

The load-bearing object is the Z2 boundary: the line of second-order transitions that separates the first-order region from the crossover in the extended parameter space of gauge coupling, quark mass, flavour number, and lattice spacing. The boundary is located by finite-size scaling of the Binder cumulant of the chiral condensate, using the 3D Ising critical value B4 = 1.6044(10) and the Ising exponents y_t and y_h. Its extrapolation to the chiral limit is governed by tricritical scaling, aT_c(am,Nf) = aT^tric(Nf) + A(Nf)(am)^{2/5} + B(Nf)(am)^{4/5} + ..., whose intercept aT^tric is the tricritical lattice spacing below which the transition becomes second order. That intercept is what decides whether the first-order region reaches the continuum.

What would settle it

Run the same Z2-boundary determination on finer lattices (N_tau = 12 or 16) for Nf = 6, where three lattice spacings already support leading-order scaling. If the critical masses on those finer lattices fall on the fitted curve and keep the intercept at the same nonzero aT^tric, the paper's conclusion holds; if the intercept moves toward zero as N_tau grows, the first-order region survives in the continuum.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that with standard staggered fermions and the Wilson gauge action, the first-order chiral transition seen for many flavours on coarse lattices does not survive the continuum limit. Fitting the Z2-boundary critical masses with the tricritical scaling form aT_c(am,Nf) = aT^tric(Nf) + A(Nf)(am)^{2/5} + B(Nf)(am)^{4/5} + ... gives nonzero intercepts aT^tric(Nf) for each Nf from 3 to 7, meaning the first-order region pinches off at a finite lattice spacing; the continuum chiral limit is therefore second order for all Nf <= 7. For Nf = 8 the same analysis is blocked by a bulk transition, but the observed first-order region is again disconnected from the continuum. Temperatures at the tricritical points decrease with rising Nf, and the authors find the data compatible with T = 0 at the physical tricritical point, located between seven and eight flavours.

Load-bearing premise

The result assumes that the tricritical scaling formula aT_c = aT^tric + A(am)^{2/5} + B(am)^{4/5} accurately describes the Z2 boundary over the simulated mass range; if it does not, the nonzero intercepts do not prove the first-order region vanishes in the continuum.

Editorial extensions

If this is right

  • For every Nf <= 7, the first-order region is a lattice artifact, and the continuum chiral limit transition is second order.
  • The physical tricritical point, where the tricritical line meets the continuum, has T = 0 and lies between Nf = 7 and 8, possibly coinciding with the onset of the conformal window.
  • No first-order chiral transition occurs for any number of flavours, unless an unknown first-order region appears at very small lattice spacings.
  • The tricritical temperature decreases with flavour number, confirming the expected trend and connecting the transition to conformal scaling.
  • For Nf = 8, the thermal Z2 line ends at the bulk transition, so no tricritical scaling applies, but the first-order region is still not connected to the continuum.

Reading between the lines

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

  • If the T = 0 tricritical point is real, the chiral transition order becomes a clean observable for locating the conformal-window onset in many-flavour gauge theories.
  • The two-point fits for Nf = 3 and 4 leave the extrapolation unconstrained; finer lattices or additional masses would provide a direct test of the assumed scaling.
  • The same technique of extending the Columbia plot into the lattice-spacing direction could be used to separate other thermal transitions from bulk artifacts.
  • A confrontation with improved lattice actions at Nf = 8 would help distinguish the bulk transition from any physical signal, sharpening the boundary of the claim.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The paper studies the order of the chiral phase transition in massless many-flavour QCD using unimproved staggered fermions. For each N_f in [2,8] and lattice spacings N_tau in {4,6,8,10}, the authors map the Z2 boundary separating the first-order region from the crossover in the lattice parameter space, extract critical masses via finite-size scaling of the kurtosis with 3D Ising critical exponents, and then fit the tricritical scaling form Eq. (4) to obtain a tricritical lattice spacing aT^tric(N_f). The fitted nonzero intercepts for N_f <= 7 lead the authors to conclude that the first-order region seen on coarse lattices is a cutoff effect and that the continuum chiral-limit transition is second order for all N_f <= 7. For N_f = 8 the apparent critical points at N_tau = 8 and 10 are attributed to a lattice bulk transition. Scale setting via the Sommer parameter is used to convert the results to physical temperatures, yielding tricritical temperatures T^tric(N_f) that decrease toward zero, consistent with a physical tricritical point at T=0 between N_f=7 and 8.

Significance. If the central claim is correct, it is an important result: the first-order chiral transition observed on coarse lattices at several flavour numbers would not survive the continuum limit, and the chiral phase transition in massless QCD would be second order for all N_f below the conformal window. The paper also gives a falsifiable prediction of a physical tricritical point between 7 and 8 flavours at T=0. The approach is sound in conception: the finite-size scaling analysis uses standard 3D Ising critical values, the tricritical scaling form Eq. (1) is externally motivated by tricritical theory, the N_f=6 data show leading-order scaling over three lattice spacings in Fig. 3b, and the codebase CL2QCD is openly available. The main weakness is that the central extrapolation is not documented quantitatively: no fit parameters, chi^2 values, fit ranges, or stability checks are reported, and for N_f=3 and 4 no extrapolation is possible with only two data points.

major comments (5)
  1. [Section 3.1] The conclusion that the first-order region is absent in the continuum for 'all flavours N_f <= 7' is not supported for N_f=3 and 4. The same section states that for these flavours 'we only have two data points making an extrapolation impossible.' No aT^tric values are obtained by fitting for N_f=3 and 4, yet these flavours are included in the summary claim. Either additional data or an explicitly restricted conclusion (e.g., N_f=5,6,7) is needed.
  2. [Section 3.1, Eq. (4)] The paper does not report the fit parameters, chi^2/dof, fit ranges, or fit-range stability for the aT^tric(N_f) extrapolations. The NLO form in Eq. (4) has three free parameters (aT^tric, A, B); with only two or three data points per flavour, the fit is underdetermined or has zero degrees of freedom, so the plotted lines in Fig. 3b cannot be quantitatively assessed. The authors should provide a table of fit results and demonstrate that the nonzero intercepts are stable under variations of the fit range.
  3. [Section 3.2, Eq. (5)] The same fit-documentation problem applies to the T^tric(N_f) values used to support the statement that T^tric(N_f) decreases to zero. In addition, the scale-setting procedure relies on the improved Sommer parameter, but no lattice spacing values, statistical or systematic uncertainties, or checks of the N_f-independence of r_1 are presented. The claim T^tric(N_f) -> 0 is therefore not quantitatively supported by the reported data.
  4. [Section 4] The final caveat that a first-order region could still emerge 'at very small lattice spacings' is logically in tension with the earlier claim that a first-order transition in the chiral limit 'can be excluded for all numbers of flavours.' If the caveat stands, the data constrain only the simulated lattice spacings, and the conclusion should be rephrased as a statement about the absence of evidence in the investigated range, not an exclusion.
  5. [Section 3.3] For N_f=8, the conclusion that the observed first-order region is not connected to the continuum rests on identifying the N_tau=8 and 10 points as a bulk transition and on the general expectation that bulk transitions are lattice artifacts. This is plausible, but it does not demonstrate that no thermal Z2 line exists at finer lattice spacings. The statement that the first-order region is 'not connected to the continuum for any N_f' therefore exceeds what the N_f=8 data alone can show.
minor comments (6)
  1. [Figure 2 caption] The word 'Szenario' should be 'Scenario'.
  2. [Section 3.1, Figure 3b] The x-axis label aT = N_\tau^{-1} is introduced only in the text; the figure caption should state that aT is in lattice units and that the continuum limit is at the origin.
  3. [Section 3.1, Figure 3b caption] The caption mentions 'dotted lines are LO-fits, while dashed lines are NLO-fits,' but the fit ranges and the number of points included in each fit are not specified; this information should be given in the caption or in the text.
  4. [General] All numerical results are presented only through figures; a table of critical masses, critical couplings, lattice spacings, and fit results would substantially improve reproducibility and allow independent verification.
  5. [Section 3.3] The sentence stating that critical beta-values for N_tau=8 and 10 are identical but 'not shown' should either be supported by a table or figure or be removed.
  6. [Figure 5] The axis label 'm^{2/5} [r1]^{2/5}' should be explained; the mass variable and its units are not defined in the figure caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: critical masses are measured and extrapolated with externally derived scaling forms; the nonzero tricritical intercepts are fitted outputs, not inputs, and the self-citations are contextual rather than load-bearing.

full rationale

The derivation chain is: (i) locate Z2-boundary masses by finite-size scaling with the 3D Ising kurtosis value and exponents taken from external references [34,35]; (ii) for each Nf, fit the measured boundary in the (aT, (am)^{2/5}) plane with Eq. (4), whose constant term aT^tric(Nf) is a fitted parameter; (iii) infer from the fitted nonzero intercept that the first-order region does not connect to the continuum origin. This is an extrapolation, not a definitional identity: nothing in the input forces the intercept to be nonzero, and the alternative physical scenario (boundary reaching the origin) is not excluded by construction. The temperatures T^tric(Nf) in Sec. 3.2 come from separately scale-set lattice spacings (Sommer r1), not from the same fit that defines aT^tric(Nf). The self-citations [6,18] describe prior work by the same group and supply context, but the new all-Nf<=7 statement rests on the new Ntau=4,6,8,10 data and fits in Fig. 3, so no load-bearing reduction to those citations can be exhibited. The paper's own admission that for Nf=3,4 'we only have two data points making an extrapolation impossible' is a data-support weakness rather than circularity, and the unvalidated tricritical scaling ansatz Eq. (4) is a modeling/correctness concern, not a circular reduction. No step was found in which a fitted parameter is renamed a prediction or in which an equation reduces to its own input.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the tricritical scaling ansatz and on 3D Ising critical values taken from the literature, plus per-flavor fit coefficients that are not tabulated. There are no invented physical entities; the tricritical line is an extrapolated phase boundary, not a new object.

free parameters (4)
  • A(Nf), tricritical scaling amplitude = not tabulated
    Fitted per Nf in Eqs. (4) and (5) to the critical masses; controls the intercept aT^tric and Ttric, and hence the main conclusion.
  • B(Nf), next-to-leading amplitude = not tabulated
    Fitted per Nf in the same scaling fits; its value affects the extrapolated intercept.
  • aT^tric(Nf) = nonzero for Nf <= 7, plots only
    Intercept from Eq. (4); the central quantity separating first-order from second-order lattice behavior.
  • Ttric(Nf) = decreasing trend, plots only
    Intercept from Eq. (5) converted to MeV; used for the T=0 compatibility statement.
assumptions (4)
  • domain assumption The Z2-boundary near the chiral limit follows tricritical scaling with exponents 2/5 and 4/5 (Eqs. (1) and (4), ref [32]).
    This functional form is assumed, not derived in the paper; all fitted intercepts aT^tric and Ttric depend on it.
  • domain assumption On the Z2-boundary the transition is in the 3D Ising universality class, with B4^Z2 = 1.6044(10), y_t = 1.5870(10), y_h = 2.4818(3), from refs [34,35].
    Used in Eq. (3) to identify critical masses from Binder cumulant finite-size scaling; wrong universality would shift the critical masses.
  • domain assumption The bulk transition seen for Nf=8 at Ntau=8,10 is a discretization artifact and not a physical transition (refs [40,41,42,43]).
    Used to exclude Nf=8 from tricritical scaling and to extend the cutoff-artifact conclusion to Nf=8.
  • domain assumption The Sommer scale r1 is robust against changes in quark mass and flavor (refs [38,39]).
    Used to convert lattice spacings and temperatures to physical units for different Nf, even though different Nf are different theories.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The order of the chiral phase transition in massless many-flavour lattice QCD." pith.science (2026). https://pith.science/paper/5VF2Q5L3

@misc{pith2026250119251,
  author       = {Pith},
  title        = {Pith review of: The order of the chiral phase transition in massless many-flavour lattice QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5VF2Q5L3}},
  note         = {Machine review of arXiv:2501.19251}
}
abstract

The nature of the QCD phase transition in the chiral limit presents a challenging problem for lattice QCD. However, its study provides constraints on the phase diagram at the physical point. In this work, we investigate how the order of the chiral phase transition depends on the number of light quark flavours. To approach the lattice chiral limit, we map out and extrapolate the chiral critical surface that separates the first-order region from the crossover region in an extended parameter space, which includes the gauge coupling, the number of quark flavours, their masses, and the lattice spacing. Lattice simulations with standard staggered quarks reveal that for each $N_f < 8$, there exists a tricritical lattice spacing $a^\text{tric}(N_f)$, at which the chiral transition changes from first order ($a>a^\text{tric}$) to second order ($a<a^\text{tric}$). Thus, the first-order region is merely a lattice artifact and not connected to the continuum. By determining the associated temperatures $T(N_f^\text{tric},a ^\text{tric})$ at these tricritical points, we confirm the expected decrease in the critical temperature as the number of flavours increases. The obtained temperatures define a tricritical line which is connected to the continuum and terminates at a physical $ N_f^\text{tric}(a=0) $. Our data is compatible with a vanishing temperature at that point, $T(N_f^\text{tric}(a=0))=0 $.

Figures

Figures reproduced from arXiv: 2501.19251 by the authors.

Figure 1
Figure 1. Columbia plots. Every point represents a phase boundary with an implicitly associated (pseudo-) critical temperature 𝑇𝑐. Figures are taken from [6]. 1. Introduction The chiral limit refers to QCD in the presence of massless quarks. As a controllable deformation of QCD, it offers valuable insights into fundamental principles of the strong interaction and provides relevant constraints for physical QCD. Particularly wi… view at source ↗
Figure 2
Figure 2. Comparison of phase diagrams for possible scenarios for the chiral limit depending on whether a first-order transition emerges for higher 𝑁𝑓 or not. Figures are taken from [6]. distinct states (a vanishing, positive and negative chiral condensate at the critical temperature). The onset of the triple line is marked by a tricritical point. For a detailed description see [6]. All flavours below 𝑁 tric 𝑓 exhibit a secon… view at source ↗
Figure 3
Figure 3. The Z2-boundary {𝛽 Z2 𝑐 , 𝑎𝑚 Z2 𝑐 , 𝑁𝑓 , 𝑁𝜏 } projected onto different planes. Every point represents a phase boundary with an implicitly tuned 𝛽 Z2 𝑐 (𝑎𝑚, 𝑁𝑓 , 𝑁𝜏). The lines correspond to fits according to Eq.(1) for the left panel and Eq.(4) for the right panel. On the right, dotted lines are LO-fits, while dashed lines are NLO-fits. The points for 𝑁𝑓 = 8 at 𝑁𝜏 = 8 and 10 do not correspond to the Z2-boundary, but… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The lattice spacings of the Z2-boundary in physical units. 0 50 100 150 200 T [MeV] 0.0 0.2 0.4 0.6 0.8 m 2/5 [r 1]2/5 broken sym. Nf = 3 Nf = 4 Nf = 5 Nf = 6 Nf = 7 Nf = 8 T tric(Nf = 5) T tric(Nf = 6) T tric(Nf = 7) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 6
Figure 6. Figure 6: Lattice phase diagram in the chiral limit. Tricritical points {𝑁 tric 𝑓 , 𝑇tric, 𝑎tric} in red separate the first order from the second-order region. Only the latter is connected to the continuum 𝑎 = 0. 4. Conclusion For unimproved staggered fermions, a first-order tra…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. RG-Invariant Symmetry Ratio for QCD: A Study of $U(1)_A$ and Chiral Symmetry Restoration

    hep-lat 2026-02 conditional novelty 6.0 of 10

    In the continuum limit, κ_PS, κ_VA and κ_TX are all consistent with zero and with each other down to 164 MeV, implying concurrent chiral and U(1)_A restoration in the quark-connected nonsinglet sector.

Reference graph

Works this paper leans on

43 extracted references · 13 canonical work pages · cited by 1 Pith paper

  1. [1]

    F.R.Brown,F.P.Butler,H.Chen,N.H.Christ,Z.-h.Dong,W.Schafferetal., Ontheexistence of a phase transition for QCD with three light quarks, Phys. Rev. Lett.65(1990) 2491

    work page 1990

  2. [2]

    Y. Aoki, G. Endrodi, 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 (2006) 675 [hep-lat/0611014]

    arXiv 2006

  3. [3]

    G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lütgemeier et al., Thermodynamics of𝑆𝑈(3) lattice gauge theory, Nucl. Phys. B469 (1996) 419 [hep-lat/9602007]

    arXiv 1996

  4. [4]

    Cuteri, O

    F. Cuteri, O. Philipsen, A. Schön and A. Sciarra,Deconfinement critical point of lattice qcd with𝑁f = 2 wilson fermions,Phys. Rev. D103 (2021) 014513 [2009.14033]

    arXiv 2021

  5. [5]

    WHOT-QCD Collaborationcollaboration, End point of the first-order phase transition of qcd in the heavy quark region by reweighting from quenched qcd, Phys. Rev. D101 (2020) 054505 [1912.10500]

    work page Pith review arXiv 2020

  6. [6]

    Cuteri, O

    F. Cuteri, O. Philipsen and A. Sciarra,On the order of the QCD chiral phase transition for different numbers of quark flavours, JHEP 11(2021) 141 [2107.12739]

    arXiv 2021

  7. [7]

    Bazavov, H.-T

    A. Bazavov, H.-T. Ding, P. Hegde, F. Karsch, E. Laermann, S. Mukherjee et al.,Chiral phase structure of three flavor qcd at vanishing baryon number density,Phys. Rev. D95(2017) 074505 [1701.03548]

    arXiv 2017

  8. [8]

    L.Dini, P.Hegde, F.Karsch, A.Lahiri, C.SchmidtandS.Sharma, Chiralphasetransitionin three-flavor QCD from lattice QCD, Phys. Rev. D105 (2022) 034510 [2111.12599]

    arXiv 2022

Show all 43 references
  1. [9]

    Kuramashi, Y

    Y. Kuramashi, Y. Nakamura, H. Ohno and S. Takeda,Nature of the phase transition for finite temperature𝑁f = 3 qcd with nonperturbativelyO(𝑎) improved wilson fermions at𝑁t= 12, Phys. Rev. D101(2020) 054509 [2001.04398]

    2020 arXiv

  2. [10]

    Zhang, Y

    Y. Zhang, Y. Aoki, S. Hashimoto, I. Kanamori, T. Kaneko and Y. Nakamura,Finite temperature QCD phase transition with 3 flavors of Mobius domain wall fermions,PoS LATTICE2022(2023) 197 [2212.10021]

    2023 arXiv

  3. [11]

    Zhang, Y

    Y. Zhang, Y. Aoki, S. Hashimoto, I. Kanamori, T. Kaneko and Y. Nakamura,Exploring the QCD phase diagram with three flavors of Möbius domain wall fermions, PoS LATTICE2023(2024) 203 [2401.05066]

    2024 arXiv

  4. [12]

    Zhang, Y

    Y. Zhang, Y. Aoki, S. Hashimoto, I. Kanamori, T. Kaneko and Y. Nakamura,Three flavor QCD phase transition with Möbius domain wall fermions,PoS LATTICE2024(2025) 193 [2501.15494]

    2025 arXiv

  5. [13]

    Fejős,Second-order chiral phase transition in three-flavor quantum chromodynamics?, Phys

    G. Fejős,Second-order chiral phase transition in three-flavor quantum chromodynamics?, Phys. Rev. D105(2022) L071506 [2201.07909]. 9 The order of the chiral phase transition in massless many-flavour lattice QCD Jan Philipp Klinger

    2022 arXiv

  6. [14]

    Fejős and T

    G. Fejős and T. Hatsuda,Order of the𝑆𝑈(𝑁𝑓)× 𝑆𝑈(𝑁𝑓) chiral transition via the functional renormalization group,Phys. Rev. D110 (2024) 016021 [2404.00554]

    2024 arXiv

  7. [15]

    Braun et al.,Soft modes in hot QCD matter, 2310.19853

    J. Braun et al.,Soft modes in hot QCD matter, 2310.19853

    arXiv

  8. [16]

    Resch, F

    S. Resch, F. Rennecke and B.-J. Schaefer,Mass sensitivity of the three-flavor chiral phase transition,Phys. Rev. D99(2019) 076005 [1712.07961]

    2019 arXiv

  9. [17]

    D 108 (2023) 114018 [2309.06737]

    J.BernhardtandC.S.Fischer, Qcdphasetransitionsinthelightquarkchirallimit ,Phys.Rev. D 108 (2023) 114018 [2309.06737]

    2023 arXiv

  10. [18]

    Cuteri, O

    F. Cuteri, O. Philipsen and A. Sciarra,Qcd chiral phase transition from noninteger numbers of flavors,Phys. Rev. D97(2018) 114511 [1711.05658]

    2018 arXiv

  11. [19]

    Braun and H

    J. Braun and H. Gies,Scaling laws near the conformal window of many-flavor QCD,JHEP 05 (2010) 060 [0912.4168]

    2010 arXiv

  12. [20]

    Miransky and K

    V.A. Miransky and K. Yamawaki,Conformal phase transition in gauge theories,Phys. Rev. D 55(1997) 5051 [hep-th/9611142]

    1997 arXiv

  13. [21]

    Banks and A

    T. Banks and A. Zaks,On the phase structure of vector-like gauge theories with massless fermions, Nucl. Phys. B196 (1982) 189

    1982

  14. [22]

    Hasenfratz, E.T

    A. Hasenfratz, E.T. Neil, Y. Shamir, B. Svetitsky and O. Witzel,Infrared fixed point of the SU(3) gauge theory with Nf=10 flavors,Phys. Rev. D108 (2023) L071503 [2306.07236]

    2023 arXiv

  15. [23]

    Hasenfratz and C.T

    A. Hasenfratz and C.T. Peterson,Infrared fixed point in the massless twelve-flavor SU(3) gauge-fermion system,Phys. Rev. D109 (2024) 114507 [2402.18038]

    2024 arXiv

  16. [24]

    Deuzeman, M.P

    A. Deuzeman, M.P. Lombardo and E. Pallante,Evidence for a conformal phase inSU(𝑁) gauge theories,Phys. Rev. D82(2010) 074503 [0904.4662]

    2010 arXiv

  17. [25]

    Deuzeman, M.P

    A. Deuzeman, M.P. Lombardo and E. Pallante,The Physics of eight flavours,Phys. Lett. B 670 (2008) 41 [0804.2905]

    2008 arXiv

  18. [26]

    LombardoandA

    A.Y.Kotov, M.P. LombardoandA. Trunin,GlidingDown theQCD TransitionLine, from𝑁𝑓 = 2 till the Onset of Conformality, Symmetry 13 (2021) 1833 [2111.00569]

    2021 arXiv

  19. [27]

    Miura, M.P

    K. Miura, M.P. Lombardo and E. Pallante,Chiral phase transition at finite temperature and conformal dynamics in large Nf QCD,Phys. Lett. B710 (2012) 676 [1110.3152]

    2012 arXiv

  20. [28]

    Miura and M.P

    K. Miura and M.P. Lombardo,Lattice monte carlo study of pre-conformal dynamics in strongly flavoured qcd in the light of the chiral phase transition at finite temperature, Nucl. Phys. B871 (2013) 52 [1212.0955]

    2013 arXiv

  21. [29]

    Braun and H

    J. Braun and H. Gies,Chiral phase boundary of QCD at finite temperature, JHEP 06(2006) 024 [hep-ph/0602226]

    2006 arXiv

  22. [30]

    Hasenfratz, C

    A. Hasenfratz, C. Rebbi and O. Witzel,Gradient flow step-scaling function for SU(3) with Nf=8 fundamental flavors, Phys. Rev. D107 (2023) 114508 [2210.16760]. 10 The order of the chiral phase transition in massless many-flavour lattice QCD Jan Philipp Klinger

    2023 arXiv

  23. [31]

    Witzel and A

    O. Witzel and A. Hasenfratz,Investigating SU(3) with𝑁𝑓 = 8 fundamental fermions at strong renormalized coupling,PoS LATTICE2024(2025) 146 [2412.10322]

    2025 arXiv

  24. [32]

    Lawrie and S

    I. Lawrie and S. Sarbach,Theory of tricritical points, inPhase transitions and critical phenomena, C. Domb and J. Lebowitz, eds., vol. 9, p. 1 (1984)

    1984

  25. [33]

    Sciarra, C

    A. Sciarra, C. Pinke, M. Bach, F. Cuteri, L. Zeidlewicz, C. Schäfer et al., CL2QCD,Zenodo (2021)

    2021

  26. [34]

    Blote, E

    H.W.J. Blote, E. Luijten and J.R. Heringa,Ising universality in three dimensions: a monte carlo study, J. Phys. A28 (1995) 6289 [cond-mat/9509016]

    1995 arXiv

  27. [35]

    Pelissetto and E

    A. Pelissetto and E. Vicari,Critical phenomena and renormalization-group theory, Phys. Rept. 368 (2002) 549 [cond-mat/0012164]

    2002 arXiv

  28. [36]

    Sommer,A new way to set the energy scale in lattice gauge theories and its application to the static force and𝛼𝑠 in𝑆𝑈(2) Yang-Mills theory, Nucl

    R. Sommer,A new way to set the energy scale in lattice gauge theories and its application to the static force and𝛼𝑠 in𝑆𝑈(2) Yang-Mills theory, Nucl. Phys. B411 (1994) 839 [hep-lat/9310022]

    1994 arXiv

  29. [37]

    Sommer,Scale setting in lattice QCD,PoS LATTICE2013(2014) 015 [1401.3270]

    R. Sommer,Scale setting in lattice QCD,PoS LATTICE2013(2014) 015 [1401.3270]

    2014 arXiv

  30. [38]

    ALPHAcollaboration, On the𝑁𝑓-dependence of gluonic observables, PoS LATTICE2013 (2014) 321 [1311.5585]

    2014 arXiv

  31. [39]

    ALPHAcollaboration, The shape of the static potential with dynamical fermions, PoS LATTICE2011(2011) 315 [1112.1246]

    2011 arXiv

  32. [40]

    A.Deuzeman,E.PallanteandM.P.Lombardo, TheBulktransitionofmany-flavourQCDand the search for a UVFP at strong coupling,PoS LATTICE2010(2010) 067 [1012.5971]

    2010 arXiv

  33. [41]

    Deuzeman, M.P

    A. Deuzeman, M.P. Lombardo, T. Nunes da Silva and E. Pallante,Bulk transitions of twelve flavor QCD and𝑈𝐴(1) symmetry, PoS LATTICE2011(2011) 321 [1111.2590]

    2011 arXiv

  34. [42]

    Deuzeman, M.P

    A. Deuzeman, M.P. Lombardo, T. Nunes Da Silva and E. Pallante,The bulk transition of QCD with twelve flavors and the role of improvement, Phys. Lett. B720 (2013) 358 [1209.5720]

    2013 arXiv

  35. [43]

    Cheng, A

    A. Cheng, A. Hasenfratz and D. Schaich,Novel phase in𝑆𝑈(3) lattice gauge theory with 12 light fermions,Phys. Rev. D85 (2012) 094509 [1111.2317]. 11

    2012 arXiv

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.