Search for a Lee-Yang edge singularity in high-statistics Wuppertal-Budapest data

Alexander Adam; Attila P\'asztor; Chik Him Wong; D\'avid Peszny\'ak; Jana N. Guenther; Ludovica Pirelli; Paolo Parotto; Szabolcs Bors\'anyi; Zolt\'an Fodor

REVIEW 4 major objections 6 minor 2 cited by

Search for a Lee-Yang edge singularity in high-statistics Wuppertal-Budapest data

T0 review · 4 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read High-statistics lattice data place the Lee-Yang edge critical temperature at a broad median of 34 MeV.

desk verdict A transparent but fragile Padé-based Lee-Yang edge search on high-statistics WB data; the honest result is a wide CDF, not a value of T_c. read the letter →

arxiv 2502.03211 v1 pith:EVEMLID4 submitted 2025-02-05 hep-lat

Alexander Adam , Szabolcs Borsányi , Zoltán Fodor , Jana N. Guenther , Paolo Parotto , Attila Pásztor , Dávid Pesznyák , Ludovica Pirelli

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Chik Him Wong

This is my paper · ORCID

classification hep-lat PACS 12.38.Gc05.70.Jk25.75.Nq

keywords Lee-YangedgesingularityQCDcriticalendpointPadéapproximantsbaryonsusceptibilities3DIsinguniversalitylatticeanalyticcontinuation

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

The paper attempts to locate the QCD critical endpoint by extrapolating the Lee-Yang edge singularity—the complex chemical potential where the partition function's zeros cluster closest to the real axis—from lattice QCD data. Using about two million configurations per temperature on a $16^3\times 8$ lattice, it builds [1,2] Padé approximants from the baryon susceptibilities $\chi_2,\chi_4,\chi_6,\chi_8$ to extract the singularity's position at each temperature, then extrapolates the imaginary part to zero assuming 3D Ising scaling. Combining all observables, four scaling ansatzes, and all fit ranges that include the 140–150 MeV window, the resulting cumulative distribution has a median of $T_c^{50}=34$ MeV, with the 84th percentile at 67 MeV and the 16th percentile at $-127$ MeV. The paper's main message is that even with this high-quality dataset, the systematic uncertainties from the ansatz and fit-range choices dominate, so the method does not yet give a reliable critical-endpoint prediction.

What carries the argument

The core object is the Lee-Yang edge singularity, the endpoint of the locus of complex baryon chemical potentials where the QCD partition function vanishes; its distance from the real axis is governed by universal scaling near a critical point. To estimate it, the paper uses the baryon susceptibilities $\chi_{2n}(T)$ as Taylor coefficients of the pressure in $\hat{\mu}=\mu_B/T$, and algebraically converts the truncated Taylor series into a [1,2] Padé approximant (a rational function with numerator $a_1 x + a_2 x^2$ and denominator $1+b_1 x + b_2 x^2$ in $x=\hat{\mu}^2$) for the pressure, density, and susceptibility. The roots of the denominator serve as the Lee-Yang zeros. The approach of these zeros to the real axis is then fitted with the 3D Ising scaling law $\operatorname{Im}[f(\mu_B^{LY},T)]^{1/(\beta\delta)} = \kappa (T-T_c)$, using four different scaling variables $f=\mu_B$, $\mu_B/T$, $\mu_B^2$, $\mu_B^2/T^2$; the temperature axis intercept of this linear relation is the claimed $T_c$.

What would settle it

Recomputing the pole locations on the same susceptibilities with increasingly higher-order Padé approximants, for instance a [2,3] or [3,4] in $\hat{\mu}$, would settle whether the [1,2] poles are genuine: if the pole nearest the real axis moves by more than the jackknife uncertainty as the order increases, the identification with a Lee-Yang zero is falsified. A second decisive check is to exclude the 140–150 MeV window from all fits; the paper shows this shifts the extrapolated $T_c$ by dozens of MeV, and if a particular ansatz then yields a result inconsistent with the quoted 34 MeV median, the central estimate is not stable.

Watch

Extended reading notes

Core claim

The central claim is that, under the assumption of a 3D Ising critical point, the Lee-Yang edge singularity extrapolated from high-statistics lattice QCD data lands at a critical temperature whose combined statistical and systematic distribution is centered near 34 MeV but is extremely broad. This is not a sharp determination: the distribution extends to negative temperatures at its 16th percentile and to 67 MeV at its 84th percentile, so the data are consistent with a very wide range of critical temperatures, including no critical point at all in the scanned range. The paper also shows that the extrapolation is highly sensitive to the choice of scaling variable and to which temperatures are included in the fit—using the same observable but different temperature windows shifts the result from about 41 MeV to about 110 MeV. The intended contribution is therefore a cautionary, high-statistics test of the Lee-Yang edge method: with the current lattice data and a [1,2] Padé ansatz, the approach does not produce a precise or stable critical-endpoint location.

Load-bearing premise

The result depends on the assumptions that the poles of a simple [1,2] Padé built from only four cumulants are genuine Lee-Yang zeros rather than truncation artifacts, that the 140–150 MeV temperature window lies within the 3D Ising asymptotic scaling region, and that the QCD critical point is indeed of 3D Ising type.

Editorial extensions

If this is right

If the result is taken at face value, the QCD critical endpoint, assuming it exists, is not detectable by the Lee-Yang edge method with this dataset: the 16th percentile at negative $T_c$ means the data are consistent with no critical point in the simulated range.
The strong dependence of $T_c$ on the fit window—shifting from about 41 MeV to 110 MeV when the high-temperature points are included or excluded—implies that any future Lee-Yang edge search must first establish the range of validity of the scaling ansatz, which the current analysis does not do.
Since the paper uses only a single Padé order, the systematic error estimate does not include the uncertainty from the choice of the rational approximant; a conservative interpretation is that the true uncertainty is even larger than the quoted 84th-versus-16th percentile spread.
The comparison of the three observables (pressure, density, susceptibility) giving different pole locations shows that the choice of observable is a significant systematic, so a reliable extraction would require the pole positions from all observables to agree.

Reading between the lines

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

One testable extension would be to repeat the analysis with higher-order Padé approximants and with a finer lattice to see whether the pole positions converge; if the median $T_c$ moves systematically toward the crossover temperature with improving precision, the 'edge singularity' being seen is likely the crossover's analytic structure rather than a true critical endpoint.
A more conservative reading than the paper's is that the data place only an upper bound on the Lee-Yang edge temperature, since 16% of the accepted fits already give negative $T_c$; a careful upper limit, rather than a central value, might be the more robust output of this method.
The slope change near 160 MeV reported by the authors and also visible in Dyson-Schwinger calculations [8] suggests that the scaling variable's linear form breaks down at the crossover, so restricting the fit to temperatures below 140 MeV may be necessary; a dedicated test would be to see whether the extrapolated $T_c$ stabilizes under that restriction.

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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 — quick take: this is a legitimate methods exercise, not a measurement of T_c. What's new: the Wuppertal-Budapest team applies the known [1,2] Padé extraction to their high-statistics 16^3x8 dataset, compares four scaling ansatze, and scans fit ranges systematically. The paper is unusually candid about its own weak spots: it states that only one Padé order was used and that systematic effects from the ansatz order are not accounted for. The conclusion section explicitly says the analysis does not account for all systematics. Credit where due: the jackknife error treatment and the visualization of pole locations are careful, and the CDF presentation makes the uncertainty visible rather than hiding it.

That said, the central estimate is not robust. The [1,2] Padé is built from exactly four cumulants, so the denominator poles are algebraically forced; there is no evidence that the pole tracks a branch-point singularity. The paper itself notes this. The extrapolation assumes 3D Ising scaling, and the median T_c = 34 MeV lies far below the lowest fitted temperature (110 MeV) — the fit is being extended well outside its support. The 5% p-value cut and the arbitrary -500 MeV cutoff remove about 10% of fits; these choices are disclosed but post hoc. The 16th percentile at -127 MeV should be read as 'the method has no constraining power in that direction,' not as a physical temperature. I also note no data or repository is provided, which blocks independent reproduction of the poles and fits.

The stress-test worry about Padé-order dependence is on target; the paper would be stronger if it had checked, say, a [2,2] or [3,3] approximant with the same cumulants or with higher-order chi's. Without that check, the CDF is really a distribution over scaling ansatz and fit-range choices, not over true systematic uncertainty. That said, I don't think this is circular — the Lee-Yang zero positions are derived from Taylor coefficients, and the target T_c is not inserted as input. The model dependence is real but the logic is linear.

Who is this for? People working on Padé methods for QCD analytic continuation, and anyone building on the WB dataset. It's a useful cautionary reference, not a result to quote as a T_c constraint.

Recommendation: send to peer review — it deserves a serious referee, and referees can push for the missing Padé-order systematics and data release. But I would not cite it as a measurement; I'd cite it, if at all, as a demonstration of how sensitive this method is to choices.

Referee Report

4 major / 6 minor

Summary. The paper applies [1,2] Padé approximants to the pressure, net-baryon density, and baryon susceptibility, with coefficients built from the Taylor cumulants chi_2, chi_4, chi_6, and chi_8 from Wuppertal-Budapest simulations on a 16^3 x 8 lattice. The poles of these approximants are identified with Lee-Yang zeros, and their imaginary parts are extrapolated to zero using an assumed 3D Ising scaling form with one of four scaling variables. Fit ranges are varied while always including 140–150 MeV, accepted fits pass a 5% p-value cut and a Tc > -500 MeV cut, and all results are combined with flat weighting into a cumulative distribution. The quoted result is a median Tc = 34 MeV with an 84th percentile of 67 MeV and a 16th percentile of -127 MeV. The paper explicitly acknowledges that only a single Padé order was used and that the corresponding systematic uncertainty is not accounted for.

Significance. If the result were robust, it would provide an interesting data-driven constraint on the QCD critical endpoint from a high-statistics lattice dataset. The paper is transparent about several limitations: it uses jackknife errors, compares three observables and four scaling variables, and explicitly states the omission of Padé-order systematics. However, the central numerical estimate rests on the unvalidated identification of a low-order Padé pole with a Lee-Yang branch point, and the quoted CDF is sensitive to arbitrary selection cuts. I therefore view the paper as a useful methods demonstration rather than a definitive CEP determination; the requested revision should convert the acknowledged omissions into quantitative systematics.

major comments (4)

[Sec. 3, paragraph beginning "It should be noted..."] The paper explicitly states that only a [1,2]-Padé was used and that systematic effects from the order of the functional ansatz are not accounted for. Since all Lee-Yang zero locations in Figs. 2 and 3 come from this single rational form, the CDF in Sec. 4 is conditional on this ansatz. Without a comparison to other rational approximants (for example [2,1] or [2,2] Padé forms in the same variable) or a validation on a synthetic free energy with a known branch-point singularity, there is no evidence that the denominator roots track the true Lee-Yang edge rather than truncation artifacts. This is the central load-bearing issue and must be addressed before the numerical result can be considered reliable.
[Sec. 3, CDF construction paragraph] The final distribution depends on several post-hoc choices: the 5% p-value cut, the requirement that every fit range include 140–150 MeV, and the arbitrary Tc > -500 MeV cut that removes about 10% of fits. The sensitivity of the reported median (34 MeV) and 84th percentile (67 MeV) to these choices is not quantified. Figure 5 shows that changing only the upper boundary of the fit range changes the extrapolated Tc from 41.3 MeV to 110.1 MeV, so the range selection is not innocuous; a stability scan over the p-value cut and the Tc outlier cut, or an explicit accounting of these choices as systematic weights, is needed.
[Sec. 4 and Fig. 6] The quoted result includes a 16th percentile of -127 MeV and the CDF reaches 34% at Tc = 0. Since a negative physical critical temperature is not meaningful in this context, this large probability mass below zero indicates that a substantial fraction of the accepted fits are inconsistent with the assumed scaling behavior, rather than being a genuine physics outcome. The paper should report the fraction of fits with Tc > 0, discuss what the negative-intercept fits imply about the validity of the scaling ansatz over the fitted ranges, and test whether excluding the scaling variables or ranges that produce negative intercepts materially changes the median.
[Sec. 2, Eq. (8) and Eq. (10)] The identification of Padé poles with the Lee-Yang edge singularity is assumed rather than demonstrated. The Lee-Yang edge is a branch-point singularity, and a single pole pair from a four-parameter rational function need not reproduce its location. Additionally, Eq. (10) assumes that the 3D Ising linear scaling form is already valid in the fitted temperature range (roughly 110–175 MeV), even though the extrapolated median Tc is approximately 34 MeV, far below the lowest fitted temperature. The paper should at least test the linear-in-T form against a known equation of state and quantify how much of the final CDF comes from extrapolations many tens of MeV beyond the fitted domain.

minor comments (6)

[Fig. 1 caption] The caption says the susceptibilities are plotted over the temperature range 110–175 MeV, but the axis labels end at 160 MeV; please reconcile the figure and the text.
[Fig. 4 caption and axis labels] The y-axis label of the first panel, "Im( LY/[MeV])(1/ ) 2", appears to have a missing superscript, and the meaning of the quoted "red:0.69707" values is not defined in the caption; please clarify.
[Eq. (11) and Fig. 4] The list of scaling observables in Eq. (11) includes mu^2/T^2, while the fourth panel of Fig. 4 appears to plot Im((mu_LY/T)^2); please make the notation uniform.
[Fig. 3 caption] In the caption, "nominator" should be "numerator".
[Acknowledgments] The word "Schorlarship" should be "Scholarship".
[Sec. 2, after Eq. (8)] Please clarify which four coefficients in Eq. (8) are matched by the four free parameters a1, a2, b1, b2; the text says n ranges from zero to three, but it would help to spell out the resulting four equations explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Tc estimate is an extrapolation from lattice Taylor coefficients through a Padé proxy and an assumed 3D Ising scaling form, neither of which is equivalent to its own output.

full rationale

The derivation chain is not self-referential in a way that forces the reported Tc. The input is a fixed set of lattice Taylor coefficients chi_2 through chi_8 (Sec. 2, Eq. 1). The [1,2]-Padé denominator roots are algebraically determined from those coefficients (Eqs. 5-8), so the Lee-Yang zero locations are derived quantities, not fitted to Tc. The subsequent scaling extrapolation (Eqs. 9-10) assumes the 3D Ising universal scaling form and uses Im[f(mu_LYZ,T)] as the y-variable; the target Tc appears only as the x-axis intersection of a linear fit and is not inserted anywhere as an input or fit parameter. The paper explicitly acknowledges that the 3D Ising scaling is assumed and that only a single Padé order is used, which is an admitted model-dependence and limitation rather than circular reasoning. The one self-citation with author overlap, Ref. [9], is used only as background for radius-of-convergence estimates and is not load-bearing for the central extrapolation. The broad CDF and negative 16th percentile are consequences of unstable extrapolations, not of a circular construction. Therefore the analysis is self-contained against the data and does not reduce, by definition or by construction, to its own inputs.

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

The central claim depends on four stated or implicit assumptions: the 3D Ising scaling hypothesis, the fidelity of low-order Padé poles to true Lee-Yang zeros, the adequacy of one lattice volume and spacing, and the sufficiency of the truncated Taylor input. No new entities are postulated. The only fitted numerical parameter in the scaling law is kappa, and several hand-selected cuts (p-value, fit range, Tc floor) enter the final distribution.

free parameters (4)

Scaling constant kappa = not reported (linear fit slope in Eq. 10)
Fit parameter per scaling ansatz and temperature range; not fixed by theory.
Fit range boundaries in temperature = varied, always including 140-150 MeV
Hand-selected ranges; Fig. 5 shows shifting the upper boundary changes Tc from 41.3 to 110.1 MeV.
p-value cut = 5%
Selected before combining fits; removes low-quality fits and affects the CDF.
Tc outlier cutoff = -500 MeV
Arbitrary cutoff that removes roughly 10% of fits; the authors state it is necessary because the Tc distribution is non-Gaussian.

assumptions (4)

domain assumption 3D Ising universality class and scaling exponents beta, delta at the QCD critical endpoint.
Invoked in Sec. 2 after Eq. 8 and in the abstract; the entire Tc extrapolation is conditional on it.
ad hoc to paper [1,2] Padé approximant with only leading nonzero coefficients reproduces the true analytic structure of the QCD free energy near the Lee-Yang edge.
Sec. 2, Eqs. 5-8; the authors note that only one Padé order is used, so order-dependent systematics are not covered.
domain assumption 16^3 x 8 lattice data are close enough to thermodynamic and continuum limits for this analysis.
Sec. 3 mentions finite volume effects hinder low-temperature LYZs from approaching the real axis; no continuum extrapolation is given.
domain assumption Taylor coefficients up to chi_8 are sufficient to construct the Padé approximant.
Sec. 2 uses chi_2, chi_4, chi_6, chi_8; truncation error is not quantified.

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Cite this review

Pith. "Pith review of Search for a Lee-Yang edge singularity in high-statistics Wuppertal-Budapest data." pith.science (2026). https://pith.science/paper/EVEMLID4

@misc{pith2026250203211,
  author       = {Pith},
  title        = {Pith review of: Search for a Lee-Yang edge singularity in high-statistics Wuppertal-Budapest data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EVEMLID4}},
  note         = {Machine review of arXiv:2502.03211}
}

abstract

Near a critical endpoint the Lee-Yang edge singularity approaches the real axis in the complex chemical potential plane. In the vicinity of the critical point the functional form of this approach depends on the universality class. Assuming a three dimensional Ising critical point in the QCD phase diagram the location of the critical endpoint can be extrapolated provided that the position of the Lee-Yang edge singularity is known at multiple temperatures. A popular method to estimate the position of a singularity is to model the free energy as a rational function of the baryon chemical potential $\mu_\text{B}$. The parameters of this model can be constrained by the cumulants of the net baryon density taken at $\mu_\text{B}^2\leq0$ . Using high-statistics simulations on a lattice $16^3\times8$ by the Wuppertal-Budapest Collaboration we estimate the location of the closest singularity in the QCD phase diagram. We also compare various models for the functional form of the free energy and discuss the predictive power of this approach.

Figures

Figures reproduced from arXiv: 2502.03211 by the authors.

**Figure 1.** The baryon susceptibilities used in the analysis of the Lee-Yang edge singularity singularity plotted against the temperature range of 110 MeV to 175 MeV (Equation 1) can provide insight into the proximity of singularities by determining the radius of convergence (Refs. [9, 10]). In this work we will search for the Lee-Yang edge singularity by determining the LYZs. The LYZs themselves are estimated by using a ration… view at source ↗

**Figure 2.** The location of the LYZs extracted using a Padé ansatz for the susceptibility 𝜒2 (𝑇), see Equation 7. The left plot shows the entire temperature range of 110 MeV to 175 MeV while the right plot depicts a reduced range of 110 MeV to 155 MeV. The ellipses represent the 68% confidence region based on the covariance matrix computed from the jackknife samples. the underlying observables with their respective Padé formula… view at source ↗

**Figure 3.** Location of the LYZs in a reduced range of 110 MeV to 155 MeV, each extracted with a [1,2] Padé ansatz with the trivial lower order coefficients in the nominator removed. The left plot uses the density 𝜒1 (𝑇) as the underlying observable, while the right plot uses the pressure difference Δ𝑝(𝑇) as the underlying observable. See Equation 5 and Equation 6 for the specific Padé used. 5 [PITH_FULL_IMAGE:figures/full_fig… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Selected scaling fits for the imaginary part of the Lee-Yang zeros are shown, using different scaling ansatzes across various data ranges. Here, the susceptibility 𝜒2 (𝑇) is employed for the Padé ansatz. The red points indicate the LYZs used for the fit while the gray …

**Figure 5.** Figure 5: Scaling fits for the imaginary part of the LYZs using the scaling ansatz Im(𝜇 2 𝐿𝑌 ) 1/𝛽𝛾 showing systematic variation with different temperature selections for the extrapolation: red (110 MeV to 150 MeV), blue (155 MeV to 175 MeV). employed ansatz. Additionally, finit…

**Figure 6.** Figure 6: The probability distribution function (PDF) and the corresponding cumulative distribution function (CDF) derived from the results of the systematic analysis. The data selection is based on a p-value cut of 5%, a fit range that included at least four temperatures, alway…

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Forward citations

Cited by 2 Pith papers

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An improved lattice QCD equation of state, combined with entropy contours continued from imaginary chemical potential, excludes a QCD critical point below μB = 450 MeV at 2σ confidence.
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hep-ph 2026-03 conditional novelty 4.0 of 10

Adding a dynamical charm quark moves the predicted QCD critical endpoint from (102.9 MeV, 618.8 MeV) to (104.3 MeV, 600.1 MeV).

Reference graph

Works this paper leans on

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Pith tools

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

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