Lattice determination of the critical point of QCD at finite T and \mu

S.D. Katz; Z. Fodor

REVIEW

Lattice determination of the critical point of QCD at finite T and \mu

Z. Fodor, S.D. Katz

High Energy Physics - Lattice · Submitted 8 Jun 2001 · 22 references

Pith's verdict: First ab initio lattice estimate of the QCD critical endpoint, with a clean determinant construction; the endpoint fit and its error budget are the parts to press on.

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arXiv IDhep-lat/0106002 (version 2)

Pith IDWCN2FBR5, Pith's permanent record of this paper and its review

CitationS.D. Katz, Z. Fodor. "Lattice determination of the critical point of QCD at finite T and \mu." arXiv:hep-lat/0106002 (2001). https://pith.science/paper/WCN2FBR5

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Editor's note

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Worth your time, with a caveat about the error bar rather than the central value. This is the first lattice QCD estimate of the critical endpoint coordinates rather than a model guess, and the machinery is the interesting part: reweighting from mu=0 transition ensembles combined with the characteristic-polynomial determinant, det M = e^{-3 L^3 L_t mu} prod (e^{L_t mu} - lambda_i), which fixes the Riemann sheet of the fractional power unambiguously. That construction is clean, and it is what makes n_f=2+1 with a physical strange mass tractable here. The Lee-Yang zero analysis is a legitimate way to separate crossover from first order, and the electroweak precedent they lean on is real. Soft spots, in order of how much they matter. The endpoint: mu_end = 0.375(20) comes from a linear fit to Im(beta_inf) versus mu. At mu=0.4 the fit gives -0.25(10), a negative imaginary part of a Lee-Yang zero, which is unphysical. The authors say so (slightly overshoots) and cite the electroweak case, but they never show the crossing is stable under a curved fit. Near a critical point the zero should vanish as (mu_E - mu)^beta with beta not unity, so the linear crossing is a proxy. The quoted +/-0.20 is dominated by a systematic read off that same overshoot: the residual sets the error while the central value still comes from the fit that produced the residual. That is the one place I would push a referee to. The V to infinity extrapolation is the second: three volumes with a leading 1/V ansatz. At mu=0.1 the L_s=6 point sits well above the line through 4 and 8; it is cleaner near the endpoint where the errors are larger. Fine for exploratory work, not fine as a final number. Everything else is acknowledged: L_t=4, m_u,d four times heavy, no continuum or R-algorithm step-size extrapolation. The conclusions state all of this plainly, so the classifier note that no limitations are stated is wrong on the text. Who it is for: anyone on finite-density lattice QCD or the beam energy scan. The method is the durable contribution; the numbers have since moved. Peer review: yes, send it out. The method deserves scrutiny and the endpoint fit is a specific, answerable question for a referee.

Referee report

Major objections (3)

[§3, Figure 1 and Table 1] The endpoint μ_end = 0.375(20) is read off a linear fit of Im(β_0^∞) against μ. The four fitted points (1.02, 1.12, 0.74, −0.25 at μ = 0.1, 0.2, 0.3, 0.4) are visibly non-monotonic: they rise from μ = 0.1 to 0.2 and then fall. A linear fit to non-monotonic data has a crossing that depends on the fit range and functional form; dropping the μ = 0.1 point or adding a quadratic term will move the crossing. The paper acknowledges the overshoot ('slightly overshoots') and cites the electroweak precedent, but does not demonstrate that the central value is stable. Please report μ_end for at least (i) the linear fit to all four points, (ii) a linear fit to μ = 0.2, 0.3, 0.4, and (iii) a quadratic fit, and state the expected critical scaling of Im(β_0^∞) near μ_E (the Lee–Yang zero vanishes with an exponent, not necessarily linearly) so the reader can judge whether the linear fit is a leading-order approximation. If the spread across these fits is within the quoted ±0.20, the result stands; if not, the error should be enlarged.
[§3, Table 1] The V → ∞ extrapolation β_0(V) = β_0^∞ + ζ/V rests on three volumes (L_s = 4, 6, 8). At μ = 0.1 the L_s = 6 value (2.66 ± 0.06) lies above the line through L_s = 4 (5.56 ± 0.08) and L_s = 8 (1.38 ± 0.06), so leading 1/V scaling is not validated at small μ; it is better near the endpoint only because the errors there are larger. Since the endpoint fit consumes these β_0^∞ values, the extrapolation systematic propagates directly into μ_end. Please show the effect of (i) fitting only L_s = 6 and 8, (ii) including a 1/V² term, and (iii) quoting χ²/dof for the three-point fits. In a paper whose central claim is a finite-size-scaling statement, the scaling ansatz should be tested rather than assumed.
[§3 and §4, error budget] The abstract quotes T_E = 160 ± 3.5 MeV, μ_E = 725 ± 35 MeV and T_c = 172 ± 3 MeV, but these errors appear to be statistical plus finite-volume only. The paper itself lists, in §4, the R-algorithm step-size extrapolation, the chiral extrapolation (m_π/m_ρ ≈ 0.73 versus the physical value, with m_{u,d} four times too heavy) and the continuum extrapolation (L_t = 4, O(a) staggered) as outstanding. In addition, the scale setting in §3 ('weighted average of R_0, m_ρ, √σ') is underspecified: as quoted in Table 1, the three quantities give lattice spacings that differ by up to ~17% at β = 5.208 (R_0 = 1.87, m_ρ = 1.22, √σ = 0.58 against 1/403, 770 and 440 MeV), and the reader cannot tell whether this spread enters the quoted errors. Please spell out the scale-setting procedure and state explicitly in the abstract that the quoted errors are statistical plus finite-volume and do not include the systematics listed in §4.

Minor objections (9)

[Figure 1] State the fit range and which μ points enter the dashed line, and label the fit form in the caption; as drawn, the reader cannot tell whether the line is a four-point or a two-point fit.
[Figure 2] Give the Re(β_0) → T mapping used to draw the crossover and first-order lines; the reader cannot reproduce the phase diagram from the text as written.
[Table 1 caption] The β_0^∞ row is an extrapolation, not a measurement; label it as such and give the fit form and the volumes entering it.
[Eqs. (2.4)–(2.7)] The derivation is compressed. Add one line for the sign factor (−1)^{6L_s^3} in passing from det(P − e^{L_t μ}) to the product form of Eq. (2.7), and state explicitly that the eigenvalues of P are gauge invariant so the result does not depend on the partial gauge fixing.
[§1] The sentence 'no ab initio, lattice analysis based on QCD was done to locate the endpoint' should be softened to 'to our knowledge' and should cite the searches the authors are aware of, so the novelty claim is checkable.
[§3] The phrase 'the former remains the same' has no clear antecedent; specify which cost is unchanged when going to physical masses.
[§3] Clarify the notation 'at least upto V = 4·12^3' — whether this means four spatial volumes of 12^3 or something else.
[§1 and §3] The abstract's 'semi-realistic masses' would be more informative with the actual ratio, e.g. m_π/m_ρ ≈ 0.73 versus ≈ 0.18 in the real world, so the reader can judge how far the result is from the chiral limit.
[Throughout] Typographical: 'performes' (§1), 'three times larger then' (§3), 'upto' (§3), 'analyzes' (§2), and the mangled author affiliation 'E¨ otv¨ os' should be corrected.

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The paper

The paper claims

A lattice calculation places the QCD critical endpoint at $T_E = 160$ MeV and baryonic chemical potential $725$ MeV, by watching Lee–Yang zeros of the partition function close on the real axis.

The reading

QCD is expected to have a critical point in the temperature–chemical-potential plane, where a line of first-order phase transitions ends and the smooth crossover seen at zero density begins; heavy-ion collisions passing near it would show characteristic non-monotonic fluctuations, but no first-principles lattice calculation had located it. This paper reports one. The authors build an ensemble of gauge configurations at zero chemical potential near the transition temperature, reweight them to non-zero $T$ and $T$ using the exact dependence of the Dirac determinant on $T$, and diagnose the transition from the volume scaling of Lee–Yang zeros of the partition function on the complex gauge-coupling plane. The extrapolated imaginary part of the zero shrinks as $T$ grows and is consistent with zero at $T_{end}=0.375(20)$ in lattice units, which the authors convert into $T_E = 160$ MeV and $T_E = 725$ MeV, together with $T_c = 172$ MeV at $T=0$. The calculation uses $n_f = 2+1$ staggered quarks with the strange mass near its physical value and the light quarks roughly four times too heavy, on $L_t = 4$ lattices.

What carries the argument

Three pieces carry the argument. First, a reweighting identity: the partition function at $(T, T)$ is written as an expectation value over an ensemble generated at $(T_w, T_w)$ with $T_w=0$, so that the ratio of bosonic actions and the ratio of determinant powers is measured configuration by configuration, with Ferrenberg–Swendsen reweighting in $T$. Second, an exact reduction of the fermion determinant: after gauge fixing to $A_0=0$ on all but the last timeslice, the determinant is manipulated into $T M = e^{-3L_s^3 L_t T} T (e^{L_t T} - T_i)$, the characteristic polynomial of a $6L_s^3$-dimensional matrix $T$ whose eigenvalues fix both the value of the determinant and the Riemann sheet of the fractional power in the $n_f/4$ exponent — the point that makes the complex-$T$ weight unambiguous. Third, the Lee–Yang zero analysis: zeros of the partition function in the complex $T$ plane are located on each volume, extrapolated as $T_0(V) = T_0^T + T/V$, and the vanishing or non-vanishing of the infinite-volume imaginary part distinguishes a first-order transition from a crossover.

Extended reading notes

Core claim

The paper claims the first ab initio lattice determination of the QCD critical endpoint. Working with $n_f=2+1$ dynamical staggered quarks at $m_s$ near its physical value and $m_{u,d}$ about four times heavier than in nature, on $L_t=4$ lattices with spatial sizes $L_s = 4, 6, 8$, the authors measure the complex Lee–Yang zeros of the partition function on the $T$ plane at several real chemical potentials. Extrapolating each zero to infinite volume via $T_0(V) = T_0^T + T/V$, they find that the imaginary part of the limiting zero decreases with $T$ and crosses zero at $T_{end} = 0.375(20)$ in lattice units, which they read as the disappearance of the crossover and the onset of a genuine first-order transition. Converting to physical units with a scale set by a weighted average of $R_0$, $m_$ and $T$, they report $T_E = 160$ MeV, $T_E = 725$ MeV for the baryonic chemical potential, and $T_c = 172$ MeV at vanishing $T$. The authors are explicit that this is an exploratory result: larger volumes, finer lattices and smaller quark masses are needed for a final answer, and they expect the endpoint to move closer to the $T=0$ axis once the light masses are reduced.

Load-bearing premise

The endpoint is read off the zero crossing of a straight-line fit to the extrapolated imaginary part of the Lee–Yang zero as a function of chemical potential, using three volumes for the $1/V$ extrapolation and a fit whose own residual at the largest $T$ is negative and whose size is then used to set the systematic error.

Editorial extensions

If this is right

The QCD phase diagram acquires a concrete, non-perturbative anchor: a first-order line at large baryonic chemical potential terminating at $T_E \approx 160$ MeV, $T_E \approx 725$ MeV, with a crossover below it.
Heavy-ion programs can compare their collision trajectories against a calculated endpoint rather than a model estimate, since the endpoint's location follows from the QCD Lagrangian rather than from a chiral-model ansatz.
The reweighting-plus-eigenvalue technique extends to any number of staggered quarks and to finer lattices, where the eigenvalue cost scales as $L_s^9$ and stays subdominant to configuration generation.
The zero-density transition temperature $T_c = 172 \pm 3$ MeV is fixed on the same lattices and with the same scale setting, giving an internally consistent baseline for the endpoint.

Reading between the lines

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

Because the light quark masses here are about four times physical, the reported $T_E$ is plausibly an upper bound: the authors themselves argue the endpoint moves toward the $T=0$ axis as $m_{u,d}$ decrease, so a physical-mass calculation would likely land below $725$ MeV.
The same machinery could be run at $L_t = 6$ or $8$ to test whether the endpoint survives continuum extrapolation; if $T_E$ drifts by more than the quoted few-percent error, the $L_t=4$ value is a lattice artifact rather than a QCD number.
The method's viability beyond $m_\pi/2$ — the point where Glasgow-style reweighting fails — suggests the overlap problem is controlled by reweighting between transition configurations rather than between vacuum ones, a diagnostic other finite-density approaches could adopt.
A sharper test than the central values would be to check the predicted critical fluctuation signatures against a beam-energy scan whose freeze-out curve passes near $(T_E, T_E)$.

Original abstract

Based on universal arguments it is believed that there is a critical point (E) in QCD on the temperature (T) versus chemical potential (\mu) plane, which is of extreme importance for heavy-ion experiments. Using finite size scaling and a recently proposed lattice method to study QCD at finite \mu we determine the location of E in QCD with n_f=2+1 dynamical staggered quarks with semi-realistic masses on $L_t=4$ lattices. Our result is T_E=160 \pm 3.5 MeV and \mu_E= 725 \pm 35 MeV. For the critical temperature at \mu=0 we obtained T_c=172 \pm 3 MeV.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.20 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Based on universal arguments it is believed that there is a critical point (E) in QCD on the temperature (T) versus chemical potential (\mu) plane, which is of extreme importance for heavy-ion experiments. Using finite size scaling and a recently proposed lattice method to study QCD at finite \mu we determine the location of E in QCD with n_f=2+1 dynamical staggered quarks with semi-realistic masses on $L_t=4$ lattices. Our result is T_E=160 \pm 3.5 MeV and \mu_E= 725 \pm 35 MeV. For the critical temperature at \mu=0 we obtained T_c=172 \pm 3 MeV.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Based on universal arguments it is believed that there is a critical point (E) in QCD on the temperature (T) versus chemical potential (\mu) plane, which is of extreme importance for heavy-ion experiments.	background 1.00	0.04	0.04	hedged possibility 0.77
s1	Using finite size scaling and a recently proposed lattice method to study QCD at finite \mu we determine the location of E in QCD with n_f=2+1 dynamical staggered quarks with semi-realistic masses on $L_t=4$ lattices.	method 0.60	0.96	0.11	asserted 1.00
s2	Our result is T_E=160 \pm 3.5 MeV and \mu_E= 725 \pm 35 MeV.	result 1.00	0.97	0.88	asserted 1.00
s3	For the critical temperature at \mu=0 we obtained T_c=172 \pm 3 MeV.	result 1.00	0.97	0.09	asserted 1.00

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

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sources read

arXiv listing abstract v2

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full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 2.** The phase diagram in physical units. Direct results are with errorbars. Dotted line illustrates the crossover, solid line the first order phase transition. The small box shows the uncertainties of the endpoint. the method approaching the chiral and continuum limits. In the present analysis the evaluation of the eigenvalues was somewhat less costly than the production of the configurations. For physical mu,d the latter would need an additional factor of O(50) (the former remains the same). Thus, for physical masses at least upto V=4·123 the cost of the eigenvalue determination is…

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References

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In a solvable one-site Hubbard model, reproducing the exact step-function density requires interference among many Lefschetz thimbles, not just the dominant one, and the number needed grows linearly with inverse temperature.
Complex Langevin dynamics for dynamical QCD at nonzero chemical potential: a comparison with multi-parameter reweighting
hep-lat 2015-08 conditional novelty 5.0 of 10

Complex Langevin simulations of four-flavor lattice QCD at nonzero chemical potential agree with multi-parameter reweighting for lattice spacings below about 0.15 fm, and break down on coarser lattices.
Curvature of the chiral pseudo-critical line in QCD: continuum extrapolated results
hep-lat 2015-07 conditional novelty 5.0 of 10

Continuum-extrapolated lattice QCD simulations at imaginary chemical potential give a chiral pseudo-critical line curvature kappa = 0.0135(20), about twice earlier Taylor-expansion values.
Phase structure of three and four flavor QCD
hep-ph 2014-05 conditional novelty 5.0 of 10

Adding a charm quark to a Dyson-Schwinger description of QCD changes the finite-temperature, finite-chemical-potential phase diagram by only a few MeV, leaving the critical end point near (115, 168) MeV.
Dynamic critical phenomena at a holographic critical point
hep-th 2011-08 conditional novelty 5.0 of 10

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hep-ph 2008-03 conditional novelty 5.0 of 10

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hep-lat 2026-06 unverdicted novelty 4.0 of 10

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Information theoretical view of QCD effective model with heavy quarks
hep-ph 2020-09 conditional novelty 4.0 of 10

The transfer mutual information in a 3D 3-state Potts model peaks at the confinement-deconfinement transition and distinguishes behavior below and above the Roberge-Weiss endpoint.
Thermodynamics of QCD-inspired theories
hep-ph 2006-04 conditional novelty 4.0 of 10

A PhD thesis calculating the 1/N-corrected effective potential and pressure of QCD-inspired sigma and NJL models, finding that temperature-dependent divergences vanish at the minimum and that pseudoscalar condensation...
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hep-ph 2010-05 accept novelty 3.0 of 10

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hep-lat 2018-04 accept novelty 2.0 of 10

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nucl-th 2015-10 accept novelty 2.0 of 10

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hep-lat 2015-12 accept novelty 1.0 of 10

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hep-lat 2012-03 accept novelty 1.0 of 10

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How later papers used it

102 later papers cite it

Evidence: substantial. 87 of the citing papers share no author with it, and 13 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 125 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 1 used its method or tool and 1 compared its own results with it.

Uses its method or tool 1 of 7 papers with this label

Derived a formula needed to use Wilson fermions for multiparameter reweighting by Fodor and Katz without Taylor expansion, as in this paper, and for the canonical expression.

Wilson fermions with imaginary chemical potential (2009) · 0.88 · paraphrased: the licence does not allow quoting

Compares with it 1 of 6 papers with this label

Obtained T of about 190 for N_f = 3 and f_pi = 93, calling it a reasonably good approximation to this paper's lattice result of T of about 164, the two differing by only about 15%.

Six Flavor Quark Matter (2018) · 0.89 · paraphrased: the licence does not allow quoting

How the literature uses it

95 arXiv papers cite this in 118 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

We locate the critical endpoint (E) of QCD on the Re(\mu)-T plane.

Lattice QCD at finite T and \mu and the critical point of QCD · 0.88

Papers citing it · 71 sentences without a confident label

Builds on it 1 paper

Second, we present a reduction formula for the quark determinant of the Wilson action, which is an extension of the reduction formula given in Ref [ref] .

Wilson fermions with imaginary chemical potential · Introduction · 0.99

Uses its method or tool 8 papers

Using the multi-parameter reweighting technique [ref] we studied dynamical QCD with $n_f$=2+1 staggered quarks of physical masses on $N_t$=4 lattices [ref] .

Recent Result in QCD Thermodynamics from the Lattice · Results at non-vanishing baryonic chemical potential · 0.97
In this paper, as in Refs. [ref] , we adopt the following prescription.

Lattice simulations of the QCD chiral transition at real baryon density · B.1. Staggered rooting at finite chemical potential · 0.99
Similarly to our previous work in Refs. [ref] we used the reduced matrix formalism [ref] to obtain the configuration-specific expansion coefficients to arbitrary order.

High-precision baryon number cumulants from lattice QCD in a finite box: cumulant ratios, Lee-Yang zeros and critical endpoint predictions · Lattice observables and simulation details · 1.00

Uses its data or values 4 papers

The Standard Model predicts that the quark-hadron transition at $μ_= 0$ is a continuous crossover, and lattice simulations infer the approximate temperature to be $T_≃164 ±2$ [ref] .

Six Flavor Quark Matter · First order quark-hadron phase transition · 0.98
The critical temperature of the QCD phase transition of the SM (considering two flavours only), determined using QCD lattice simulations, is $T_c,0 ≃( 172 ±3 ) MeV$ [ref] .

Baryogenesis and Dark Matter in Multiple Hidden Sectors · Exotic Sectors · 1.00
Lattice QCD calculations [ref] (see Fig. [ref]) indicate a rapid increase of the energy density around $T≈160 - 170$ MeV.

Jet Propagation and Mach-Cone Formation in (3+1)-dimensional Ideal Hydrodynamics · The Phase Diagram of QCD · 0.80

Compares with it 6 papers

For $N_f = 3$ and $f_π= 93$ we obtain $T_≃190$, which is a reasonably good approximation to the lattice result, $T_≃164$ [ref] , and the two values differ by only $∼15%$.

Six Flavor Quark Matter · First order quark-hadron phase transition · 0.89
The plot of transition temperature versus chemical potential seems to agree with lattice results [ref] in both the models.

Thermodynamics of soft wall AdS/QCD at finite chemical potential · Conclusion · 0.99
In Fig. [ref], we plot Eq. ([ref]), together with the lattice result [ref] , where $R≡T_c(ρ_q)/T_0$ and $ρ̅_q =ρ_q z_IR^3$.

Deconfinement phase transition in holographic QCD with matter · A Hawking-Page type transitions at finite density · 0.84

Cites it as background 26 papers

Though there have been attempts in extracting information about the location of the supposed CEP from lattice simulations [ref] , these attempts face great difficulties, as extrapolation-type methods have the property that they give reliable results mostly in the immediate vicinity of $μ_B=0$.

The QCD crossover at finite chemical potential from lattice simulations · Introduction · 0.91
These include Taylor expansion at $μ_B=0$ [ref] , analytic continuation from imaginary chemical potential [ref] , and reweighting [ref] .

Towards a reliable lower bound on the location of the critical endpoint · Introduction · 0.85
The possibility of the existence of the critical end point (CEP) in the QCD phase diagram [ref] is considered as one of the most interesting development in the field of relativistic heavy ion research.

Correlation of density fluctuation in a magnetized QCD matter near the critical end point · Hydrodynamic equations in linearised form to estimate the dynamic structure factor · 0.81

Discussion

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