Event-by-Event Fluctuations in Heavy Ion Collisions and the QCD Critical Point

E. Shuryak; K. Rajagopal; M. Stephanov

REVIEW

Event-by-Event Fluctuations in Heavy Ion Collisions and the QCD Critical Point

M. Stephanov, K. Rajagopal, E. Shuryak

High Energy Physics - Phenomenology · Submitted 9 Mar 1999 · 34 references

Pith's verdict: The analytic core is solid and the NA49 comparison is honest; the 'easily observable' claim leans on a correlation length the authors' own finite-time estimate does not support.

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arXiv IDhep-ph/9903292 (version 2)

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CitationE. Shuryak, K. Rajagopal, M. Stephanov. "Event-by-Event Fluctuations in Heavy Ion Collisions and the QCD Critical Point." arXiv:hep-ph/9903292 (1999). https://pith.science/paper/7ZGPTVYH

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Editor's note

read the letter

This is the machinery paper behind the 1998 letter, and it holds up. The two correlators that carry the physics — Eq. (50) for thermal contact with a heat bath of finite heat capacity, and Eq. (88) for sigma exchange — are derived cleanly from Gaussian path integrals and checked against the right limits: C_A to infinity recovers the canonical result, C_A to zero enforces energy conservation, and the 1/m_sigma^2 singularity in (88) matches the Ising susceptibility exponent gamma/nu. The NA49 comparison is presented as broad agreement, not a precision test, which is the right posture given the data were preliminary. The appendix on finite-multiplicity estimators (Eq. 145) is a real, reusable contribution that outlives the critical-point program; the observation that the naive estimator biases v_ebe by O(<(Delta N)^2>/^2) is the kind of thing that should have been in the experimental papers and wasn't. Credit where due: the paper is unusually explicit about what it does not know, flagging the blue-shift approximation, the tricritical extrapolation for G, and the finite-time estimate of xi_max. The classifier's 'limitations: none found' is simply wrong on the text; Section 5.4 calls Eq. (112) 'suspect for several reasons' in so many words. On the soft spot: the stress-test is right that there is an unresolved tension between Eq. (110) (xi <~ 12 fm from finite size) and Eq. (112) (xi <~ 2.5 fm from finite time), and that the paper adopts 6 fm on the strength of an unquantified 'compensating good effect' plus an unknown prefactor. Since F_sigma - 1 scales as xi^2, taking (112) at face value drops the pT signal from ~10% to a few percent after dilution — comparable to the Bose and thermal-contact backgrounds the paper itself estimates, not the 50x statistical error quoted in the abstract. G is a co-equal factor and equally soft. None of this touches the derivations; it is a quantitative-reach concern about the headline, and the nonmonotonic scan argument survives it because the background is roughly constant in the control parameter. But the paper should have made that argument explicitly rather than resting on the 6 fm number. Who should read it: anyone doing event-by-event fluctuation analysis, and anyone thinking about how to estimate finite-time critical scaling in a expanding fireball. It deserves referees, and the derivations deserve to be checked line by line by someone who knows the Ising-to-QCD mapping.

Referee report

Significance

If the central claim holds, this is a high-impact paper: it converts a qualitative suggestion into a quantitative experimental program with a specific, falsifiable signature — a nonmonotonic bump in √F and in ⟨(ΔN)²⟩ whose maximum must occur at the same control-parameter value across several observables, plus a low-p_T pion population at ⟨p_T⟩ ~ m_π/2 that appears and disappears. The derivations are checkable and the authors check them: Eq. (50) reduces to the canonical result as C_A → ∞ and enforces fixed energy as C_A → 0, and Eq. (88) is cross-checked against the mass-shift interpretation of Eq. (81). The Appendix estimator (Eq. 145) is a concrete, reusable methodological contribution that removes a systematic 1/N bias from published v_ebe values, and the correction (147) is applied transparently to the NA49 numbers. The authors are also candid about their limitations — the finite-time estimate of §5.4 is explicitly called 'suspect for several reasons', the prefactor ambiguity in G_E is stated in §5.3, the lowest-order-in-G treatment is flagged in §5, and the blue-shift approximation is flagged in §3.3 — which makes the paper easy to assess and to build on.

Major objections (4)

[§5.4, Eqs. (110)–(112), and abstract:s9] The abstract's conclusion that the effects 'should be easily observable' rests on the adopted ξ_max ≈ 6 fm, but this exceeds the paper's own finite-time scaling estimate by a factor of about 2.4. Eq. (112) gives ξ/l < (τ_H/l)^{1/Δ_t} ~ 2.5, i.e. ξ ≲ 2.5 fm, while Eq. (110) allows ξ ≲ 12 fm; the binding constraint is therefore the finite-time one, yet the paper adopts 6 fm. Because F_σ − 1 scales as ξ² (Eqs. 115–116), taking Eq. (112) at face value reduces F_σ − 1 from 0.35 to about 0.06 at μ_π = 60 MeV (and from 0.14 to about 0.024 at μ_π = 0), and after the factor-of-two dilution from non-direct pions the enhancement of √F falls to roughly 1%, comparable to the Bose (~2%) and thermal-contact (~1%) backgrounds the paper itself computes. The 'compensating good effect' invoked on p. 42 — that the large specific heat makes the system dwell longer near T_c — is qualitatively plausible but is never estimated. Please either quantify the dwell-time enhancement (for instance, estimate how much longer the system spends within |T − T_c| ≲ ξ^{−1/ν} given the divergent C_V) or soften 'easily observable' in the abstract, §5.5 and §7, and state explicitly the minimum ξ at which the signal exceeds the ~2% noncritical background.
[§3.4 and §7, versus the Note added in proof] The inference in §3.4 that 'about 75% of the observed fluctuation is thermodynamic in origin', drawn from comparing v²_ebe(N)/⟨N⟩ = 2.008 ± 0.009 to the resonance-gas value 1.5, is not a decomposition the data support once the initial-state contributions are included. The Note added in proof concedes that Refs. [33,34] reproduce the observed multiplicity fluctuations to within a few percent with no new free parameters by combining geometric, spectator and NN-cross-section contributions with the thermodynamic ones. That being so, the 1.5-versus-2.0 comparison does not isolate a 75% thermodynamic share; the two classes of contribution are not shown to be separately identifiable. §7 nevertheless repeats the 75% figure without qualification. Please qualify both passages, or explain why the resonance-gas prediction and the geometric contributions of [33,34] are not overlapping descriptions of the same variance.
[§5.1 and §5.5, Eq. (117)] The multiplicity enhancement quoted in Eq. (117) is a factor 1 + 1.0 (G/300 MeV)²(ξ/6 fm)² at μ_π = 0 (rising to 2.0 at μ_π = 60 MeV), i.e. a factor of two to three on the direct-pion variance at the central parameter values, yet the correlator (88) from which it follows is derived to lowest order in G. The paper itself states in §5 that 'the multiplicity fluctuations are large enough that in this case a treatment which goes beyond lowest order in G seems called for', but no estimate of the size of the neglected terms is given, so the reader cannot judge whether Eq. (117) is reliable to a factor of two or to an order of magnitude. Since the multiplicity channel is one of the two headline observables, please supply at least an order-of-magnitude estimate of the next-order correction (the σ-bubble resummation mentioned in §5.4 is the natural starting point) or state a quantitative reliability bound on Eq. (117).
[§3.4, Eq. (44), and §4.3, Eq. (59)] The bookkeeping for √F does not close, and the residual is larger than the statistical error by several times. The data give √F = 1.002 ± 0.002 (Eq. 44); the prediction is √F_B ≈ 1.02 from Bose enhancement, reduced to about 1.01 by the thermal-contact factor √F_T ≈ 0.99 estimated in Eq. (59). The product remains roughly 1% above the measured value, i.e. about five times the quoted statistical error, and the paper attributes the remainder to two-track resolution without estimating it. Because the entire detection strategy is a percent-level nonmonotonic variation of √F on top of this baseline, the baseline should be pinned down more tightly. Please either estimate the two-track contribution or set out an explicit systematic error budget for √F, so that the size of a detectable critical bump can be stated against a known background.

Minor objections (8)

[§2.2, Eq. (26) and Fig. 1] The factor of 1/2 in Eq. (26) and the accompanying sentence that the Bose factor is 'reduced from 1 + n_p to 1 + (n_p/2)' are easy to misread, since n_p here counts both pion charges. A one-line reminder at the equation, and a note in the Fig. 1 caption that F_B is defined per charged-pion pair, would prevent confusion.
[§3.2, Table 1] The statement that |F_res − 1| < 0.01 is quoted without an uncertainty, although it comes from slicing a simulated sample of 1.254 × 10⁶ pions into up to 2500 events. Please state the statistical error on F_res so the reader can see that the bound is meaningful rather than a consequence of limited sampling.
[§3.4, Eqs. (42)–(43) and Appendix Eq. (147)] The multiplicity-weighting correction (147) is applied to v_ebe(p_T) but not to v_inc(p_T), and the two are then compared as a ratio in Eq. (43). Please confirm that no analogous correction is required for the inclusive variance, or state why it cancels in the ratio.
[§4.4, Eqs. (60)–(67)] The choice σ²_p = v²_p in Eq. (60) is made 'for convenience', and the resulting fluctuation ⟨(ΔT̃)²⟩ in Eq. (67) is therefore definition-dependent. A sentence making explicit that other choices of σ²_p give the same mean but a different normalization of the variance would help the reader interpret the result.
[§5.3, Eqs. (92)–(93) and (106)] The vacuum estimate G ≈ 1900 MeV follows from λ = 20 fixed by m_σ = 600 MeV, and the suppression factor (m/Λ_QCD)^{3/5} in Eq. (106) is then applied to that number. Please state how G_E in Eq. (106) shifts if the assumed vacuum m_σ is varied over, say, 500–700 MeV, since the width argument in Eq. (93) is only a plausibility check.
[§6, Eqs. (119)–(126)] The linear parametrization m_σ(t) = 2m_π(1 + t/τ) and the assumption that sigmas decay at rest are both approximations, the second justified only qualitatively. A brief remark on how the spectrum (124) and the estimate (126) would change if the sigmas retained thermal momenta of order T_f would strengthen this section, which otherwise carries a distinctive and directly testable prediction.
[Table 2 and Ref. [7]] The central quantitative comparison rests on preliminary NA49 data cited as 'paper to be submitted' and a private communication. This is clearly labelled and is not a criticism of the analysis, but the reader should be reminded in the caption that the numbers may change, and that the quoted errors are statistical only.
[Equations (70) and (118)] The numerical prefactors are printed with stray decimal points — '1 + 10./(C_A/C_B + 1)' in Eq. (70) and '12.' and '13.' in Eq. (118). These should be cleaned up, and the hyphenation of 'freeze-out' versus 'freezeout' made consistent throughout.

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The paper

The paper claims

The paper argues that event-by-event fluctuations of charged-pion multiplicity and mean transverse momentum should show a nonmonotonic bump when heavy-ion collisions freeze out near the QCD critical endpoint, and that the bump is large en

The reading

The paper's target is a practical experimental question: can event-by-event fluctuations tell whether a heavy-ion collision has frozen out near the second-order critical endpoint of the QCD phase transition? Its strategy is to first model the noncritical background as a thermal resonance gas, compare that baseline with NA49 data, and then add two critical effects: thermal contact with a heat bath whose heat capacity diverges, and direct coupling of pions to long-wavelength fluctuations of the sigma field. The authors argue that the direct sigma coupling dominates and produces a nonmonotonic increase in the variance of pion multiplicity and mean transverse momentum as a control parameter such as collision energy moves the freeze-out point toward, through, and past the endpoint. They estimate that finite size and finite time cap the sigma correlation length at a few femtometers, but that the resulting signal can still be an order of magnitude above present statistical errors.

What carries the argument

The argument is carried by the microscopic correlator $langleDelta n_pDelta n_krangle$, the covariance between fluctuations of pion occupation numbers in momentum modes $p$ and $k$. In an ideal Bose gas this is diagonal, $v_p^2delta_{pk}$ with $v_p^2=langle n_prangle(1+langle n_prangle)$; resonance decays add small off-diagonal terms; thermal contact with a finite heat capacity $C_A$ adds a negative off-diagonal term from energy conservation; and $\sigma$ exchange adds the critical term $frac{1}{V m_sigma^2}G^2T,frac{v_p^2}{omega_p}frac{v_k^2}{omega_k}$. Once this correlator is known, any observable built from pion occupation numbers follows by weighting with the single-particle quantity and dividing by $langle Nrangle^2$. The finite-size and finite-time discussion then replaces the divergent $1/m_sigma^2$ by a capped correlation length $xi_{max}$, using scaling arguments $xi_{max}sim R$ and $xi_{max}simtau^{1/Delta_t}$.

Extended reading notes

Core claim

The central claim is that the QCD critical endpoint leaves a measurable imprint on the pions already observed in heavy-ion collisions. Near the endpoint the $\sigma$ field becomes light and its long-wavelength fluctuations grow; through the $Gsigma pi pi$ coupling these fluctuations add a term to the pion occupation-number correlator proportional to $G^2 T/m_sigma^2$, which diverges as $m_sigma to 0$. The paper turns this into estimates for the scaled event-by-event variance of mean transverse momentum, $F_sigma=1+0.35(G/300,mathrm{MeV})^2(xi/6,mathrm{fm})^2$ at $mu_pi=60,mathrm{MeV}$, and for the multiplicity variance, with both growing near the endpoint. It also argues that the noncritical baseline—Bose enhancement, resonance decays, flow fluctuations, and energy-conservation anticorrelations—is broadly consistent with NA49 data, so that a nonmonotonic departure from that baseline would be evidence for critical freeze-out.

Load-bearing premise

The load-bearing premise is that freeze-out occurs from an equilibrated hadronic system close enough to the critical endpoint that the $\sigma$ correlation length reaches several femtometers, because the predicted enhancement scales as $xi^2$ and a correlation length of only about $2,mathrm{fm}$ would shrink the $p_T$ signal to roughly the percent level.

Editorial extensions

If this is right

As collision energy, centrality, or ion size is varied, the scaled $p_T$ variance and the multiplicity variance should first rise, peak at the same setting, and then fall.
The critical increase should be strongest for soft pions, so a low-$p_T$ cut should amplify the nonmonotonic signal relative to the full-acceptance variance.
The cross correlation $langleDelta NDelta p_Trangle$, normally small, should grow in magnitude near the endpoint because it receives a large sigma-exchange contribution.
If freeze-out is near the endpoint, a nonthermal population of pions with mean $p_Tsim m_pi/2$ should appear from late sigma decays and then disappear as the endpoint is passed.
Off-diagonal correlations between $pi^+$ and $pi^-$ fluctuations should grow near the endpoint, since the diagonal Bose term is absent in that channel.

Reading between the lines

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

A null result in an energy scan would not by itself rule out the endpoint; it could mean the correlation length stayed below the finite-time cap, so the measurement would bound the combination $Gxi$ rather than the existence of the endpoint.
Because the predicted signal depends quadratically on $xi$, an independent interferometric or fluctuation-based estimate of the homogeneous patch size at freeze-out would sharpen the test considerably.
The same correlator logic suggests that a differential scan in pion momentum, rather than a single acceptance-averaged number, would separate the sigma-exchange term from the Bose and resonance backgrounds.
If the soft-pion excess from late sigma decays is seen, its angular and charge correlations could distinguish a critical source from ordinary resonance decays.

Original abstract

The event-by-event fluctuations of suitably chosen observables in heavy ion collisions at SPS, RHIC and LHC can tell us about the thermodynamic properties of the hadronic system at freeze-out. By studying these fluctuations as a function of varying control parameters, it is possible to learn much about the phase diagram of QCD. As a timely example, we stress the methods by which present experiments at the CERN SPS can locate the second-order critical endpoint of the first-order transition between quark-gluon plasma and hadron matter. Those event-by-event signatures which are characteristic of freeze-out in the vicinity of the critical point will exhibit nonmonotonic dependence on control parameters. We focus on observables constructed from the multiplicity and transverse momenta of charged pions. We first consider how the event-by-event fluctuations of such observables are affected by Bose-Einstein correlations, by resonances which decay after freeze-out and by fluctuations in the transverse flow velocity. We compare our thermodynamic predictions for such noncritical event-by-event fluctuations with NA49 data, finding broad agreement. We then focus on effects due to thermal contact between the observed pions and a heat bath with a given (possibly singular) specific heat, and due to the direct coupling between the critical fluctuations of the sigma field and the observed pions. We also discuss the effect of the pions produced in the decay of sigma particles just above threshold after freeze-out on the inclusive pion spectrum and on multiplicity fluctuations. We estimate the size of these nonmonotonic effects which appear near the critical point, including restrictions imposed by finite size and finite time, and conclude that they should be easily observable.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,727 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.57 · human participants 0.01 · controlled comparison 0.14 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.85 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

The event-by-event fluctuations of suitably chosen observables in heavy ion collisions at SPS, RHIC and LHC can tell us about the thermodynamic properties of the hadronic system at freeze-out. By studying these fluctuations as a function of varying control parameters, it is possible to learn much about the phase diagram of QCD. As a timely example, we stress the methods by which present experiments at the CERN SPS can locate the second-order critical endpoint of the first-order transition between quark-gluon plasma and hadron matter. Those event-by-event signatures which are characteristic of freeze-out in the vicinity of the critical point will exhibit nonmonotonic dependence on control parameters. We focus on observables constructed from the multiplicity and transverse momenta of charged pions. We first consider how the event-by-event fluctuations of such observables are affected by Bose-Einstein correlations, by resonances which decay after freeze-out and by fluctuations in the transverse flow velocity. We compare our thermodynamic predictions for such noncritical event-by-event fluctuations with NA49 data, finding broad agreement. We then focus on effects due to thermal contact between the observed pions and a heat bath with a given (possibly singular) specific heat, and due to the direct coupling between the critical fluctuations of the sigma field and the observed pions. We also discuss the effect of the pions produced in the decay of sigma particles just above threshold after freeze-out on the inclusive pion spectrum and on multiplicity fluctuations. We estimate the size of these nonmonotonic effects which appear near the critical point, including restrictions imposed by finite size and finite time, and conclude that they should be easily observable.

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	The event-by-event fluctuations of suitably chosen observables in heavy ion collisions at SPS, RHIC and LHC can tell us about the thermodynamic properties of the hadronic system at freeze-out.	background 0.97	0.10	0.10	hedged possibility 0.80
s1	By studying these fluctuations as a function of varying control parameters, it is possible to learn much about the phase diagram of QCD.	background 0.87	0.08	0.07	hedged possibility 0.83
s2	As a timely example, we stress the methods by which present experiments at the CERN SPS can locate the second-order critical endpoint of the first-order transition between quark-gluon plasma and hadron matter.	contribution 0.68	0.25	0.14	asserted 0.52
s3	Those event-by-event signatures which are characteristic of freeze-out in the vicinity of the critical point will exhibit nonmonotonic dependence on control parameters.	result 0.35	0.16	0.30	asserted 0.94
s4	We focus on observables constructed from the multiplicity and transverse momenta of charged pions.	method 0.63	0.68	0.03	asserted 0.87
s5	We first consider how the event-by-event fluctuations of such observables are affected by Bose-Einstein correlations, by resonances which decay after freeze-out and by fluctuations in the transverse flow velocity.	method 0.71	0.67	0.03	asserted 0.67
s6	We compare our thermodynamic predictions for such noncritical event-by-event fluctuations with NA49 data, finding broad agreement.	result 0.98	0.93	0.07	asserted 0.99
s7	We then focus on effects due to thermal contact between the observed pions and a heat bath with a given (possibly singular) specific heat, and due to the direct coupling between the critical fluctuations of the sigma field and the observed pions.	contribution 0.54	0.62	0.04	asserted 0.52
s8	We also discuss the effect of the pions produced in the decay of sigma particles just above threshold after freeze-out on the inclusive pion spectrum and on multiplicity fluctuations.	contribution 0.88	0.61	0.04	asserted 0.85
s9	We estimate the size of these nonmonotonic effects which appear near the critical point, including restrictions imposed by finite size and finite time, and conclude that they should be easily observable.	result 0.81	0.91	0.76	asserted 0.69

What kind of paper the classifier reports this is: theoretical result 0.80 · evidence from analysis of existing data 0.49.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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 1.** Bose enhancement factor FB = hNiv 2 ebe(pT )/v2 inc(pT ) describing the contribution of the Bose effects to the fluctuations of the mean transverse momentum in an ideal Bose gas of pions. FB is plotted as a function of the pion chemical potential µπ, in MeV. The dashed line shows the Bose enhancement factor if only pions with low momentum pT < 300MeV are included. Bose enhancement effect is sensitive to µπ, and leads to an increase in vebe by a factor of √ FB, which typically results in an increase of the order of a few percent. 13

**Figure 2.** Diagrammatic representation of the right hand side of the

Figures reproduced from arXiv:hep-ph/9903292 by the authors.

References

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QCD critical surface from constant entropy contours
nucl-th 2026-06 unverdicted novelty 8.0 of 10

First mapping of the QCD critical surface in full (T, μ_B, μ_Q, μ_S) space via constant-entropy expansion gives a critical point at (114, 602) MeV in the pure baryon direction, with μ_B,c shifting 40-100 MeV in strang...
Unveiling QCD Criticality with Cross-Rapidity Net-Baryon Cumulants
nucl-th 2026-09 accept novelty 7.0 of 10

Cross-rapidity net-baryon cumulants can separate the positive correlation from critical fluctuations and the negative correlation from global baryon number conservation, providing a clear signal for the QCD critical point.
Tricritical dynamics in $\mathrm{^3He}$-$\mathrm{^4He}$ mixtures and QCD
hep-ph 2026-08 conditional novelty 7.0 of 10

Near tricritical points with an extra conserved energy-like density, order parameter and conserved O(N) charges exhibit strong scaling z=3/2, while the energy-like density is subdiffusive with z=3.
Extraction of baryon number susceptibilities at finite density from heavy-ion collisions
hep-ph 2026-08 conditional novelty 7.0 of 10

From STAR proton fluctuation data, a Bayesian extraction constrains chi_2^B at freeze-out, matching lattice QCD below mu_B about 300 MeV and attributing the nonmonotonic C3/C1 to two-baryon correlations plus baryon co...
Subensemble Acceptance Method 3.0: General Corrections to Cumulants from Exact Conservation Constraints
hep-ph 2026-07 unverdicted novelty 7.0 of 10

SAM-3.0 derives canonical cumulants from grand-canonical joint cumulants via a closed recursion with multivariate partial exponential Bell polynomials for arbitrary numbers of conserved charges and observables.

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High-order fluctuations of temperature in hot QCD matter
hep-ph 2025-04 unverdicted novelty 7.0 of 10

Introduces a new thermodynamic state function enabling computation of high-order temperature fluctuations in hot QCD matter from heavy-ion collision data, predicting strong suppression and negative skewness in the QGP phase.
Evolution of Non-Gaussian Hydrodynamic Fluctuations
hep-th 2020-09 accept novelty 7.0 of 10

The authors derive leading-order evolution equations for three- and four-point Wigner functions of non-Gaussian hydrodynamic fluctuations in nonlinear diffusion with multiplicative noise.
Connecting fluctuation measurements in heavy-ion collisions with the grand-canonical susceptibilities
hep-ph 2020-03 accept novelty 7.0 of 10

A statistical-mechanics derivation, the subensemble acceptance method, connects conserved-charge cumulants measured in a finite acceptance to grand-canonical susceptibilities for any equation of state.
Real time evolution of non-Gaussian cumulants in the QCD critical regime
hep-ph 2015-06 conditional novelty 7.0 of 10

Coupled evolution equations for the third and fourth cumulants of the QCD critical field show that non-equilibrium memory effects can reverse the sign of skewness and kurtosis relative to equilibrium.
Non-Gaussian fluctuations near the QCD critical point
hep-ph 2008-09 accept novelty 7.0 of 10

Near the QCD critical point the skewness of multiplicity fluctuations scales as the 4.5th power of the correlation length and the kurtosis as the 7th, making higher cumulants far more sensitive critical-point signatur...
High-order temperature fluctuations in hot and dense QCD matter from multi-charge holography
hep-ph 2026-10 conditional novelty 6.0 of 10

Temperature cumulants through sixth order are computed in a holographic QCD model, revealing mean-field critical scaling and nonmonotonic freeze-out dependence.
Robustness of the scaled transverse-momentum correlator and the contribution of meson--baryon fluctuations at RHIC energies
hep-ph 2026-10 accept novelty 6.0 of 10

UrQMD-based checks show that C_pT is robust to experimental choices, and composition fluctuations from transport eliminate the minimum in the two-component explanation of the measured non-monotonicity.
A 4D $T^\prime$-expanded lattice QCD equation of state with a multi-dimensional 3D-Ising-type critical contribution
nucl-th 2026-10 conditional novelty 6.0 of 10

A new parametrizable four-dimensional QCD equation of state combines the lattice T' Expansion Scheme with a 3D-Ising critical contribution mapped to a quadric critical surface across (T, mu_B, mu_Q, mu_S).
Imprints of the nuclear liquid-gas phase transition on net-baryon number fluctuations
nucl-th 2026-09 conditional novelty 6.0 of 10

The parity-doublet model describes net-baryon number fluctuations at √sNN ≲ 4 GeV, pointing to a non-negligible role of the nuclear liquid-gas transition in heavy-ion collisions.
Interaction fingerprints in temperature-fluctuation cumulants from the QCD crossover to nuclear liquid-gas criticality
nucl-th 2026-07 conditional novelty 6.0 of 10

Temperature-fluctuation cumulants in ideal, excluded-volume, and van der Waals hadron resonance gas track lattice QCD through the crossover, separate along freeze-out at low energy, and trace the nuclear liquid-gas Wi...
Sensitivity of p_T Fluctuations to the QCD Equation of State
nucl-th 2026-07 unverdicted novelty 6.0 of 10

A viscous Langevin model with lattice-QCD equation-of-state inputs reproduces C_pT centrality scaling at RHIC top energies and flags a 7.7 GeV deviation as possible critical-point physics.
Canonical statistical hadronization with local baryon conservation for higher-order cumulants
hep-ph 2026-05 unverdicted novelty 6.0 of 10

Local baryon conservation in a canonical ensemble drives net-proton κ6/κ2 to small or negative values in restricted acceptance, establishing a baseline that must be subtracted before interpreting signals of chiral cri...
Baselines for Abelian Charge Fluctuations in Nuclear Collisions:Theory and Comparison with Experimental Data
nucl-th 2025-08 conditional novelty 6.0 of 10

Exact global baryon number conservation alone cannot reproduce STAR BES-II and HADES proton factorial cumulants; adding repulsive proton pairs at high energies and attractive triplets at low energies does, but with pa...
Universal non-equilibrium scaling of cumulants across a critical point
hep-ph 2024-11 conditional novelty 6.0 of 10

The paper extracts universal non-equilibrium scaling functions for the order parameter and its cumulants up to fourth order in a Z2 scalar field theory with Model A dynamics in two and three dimensions.
System size and energy dependence of the mean transverse momentum fluctuations at the LHC
nucl-ex 2024-11 conditional novelty 6.0 of 10

ALICE reports the system-size and energy dependence of mean transverse momentum fluctuations at the LHC, including a new spherocity-dependent measurement in proton-proton collisions.
Probing QCD critical point and induced gravitational wave by black hole physics
hep-th 2022-01 conditional novelty 6.0 of 10

A holographic QCD model fitted to zero-density lattice data predicts a critical endpoint at T = 105 MeV and mu_B = 555 MeV, and a pulsar-timing-band gravitational-wave background from the early-universe first-order QC...
Proton number cumulants and correlation functions in Au-Au collisions at $\sqrt{s_{\rm NN}} = 7.7-200$ GeV from hydrodynamics
hep-ph 2021-07 conditional novelty 6.0 of 10

Combining exact global baryon conservation with lattice-QCD-constrained baryon repulsion in a 3+1D hydrodynamic simulation reproduces STAR net-proton cumulant ratios at beam energies of 20 GeV and above, without invok...
Quantum algorithms for transport coefficients in gauge theories
hep-lat 2021-04 conditional novelty 6.0 of 10

Gauge-invariant lattice energy-momentum-tensor operators are derived for Hamiltonian quantum simulation, letting shear and bulk viscosity of quark-gluon plasma be targeted with roughly 10^3 fewer qubits than a full co...
Hyper-order baryon number fluctuations at finite temperature and density
hep-ph 2021-01 conditional novelty 6.0 of 10

Within a QCD-assisted low-energy effective theory, non-monotonic baryon number fluctuations arise in the non-critical crossover region, so the STAR net-proton kurtosis signal does not by itself require a critical endpoint.
Off-diagonal correlators of conserved charges from lattice QCD and how to relate them to experiment
hep-lat 2019-10 conditional novelty 6.0 of 10

Continuum-extrapolated lattice correlators of baryon number, charge, and strangeness are matched to ratios of measured net-kaon and net-Lambda variances, and the strangeness ratio favors a 165 MeV freeze-out temperature.
Hydro+ in Action: Understanding the Out-of-Equilibrium Dynamics Near a Critical Point in the QCD Phase Diagram
hep-ph 2019-08 conditional novelty 6.0 of 10

First numerical implementation of Hydro+ in an expanding fireball shows critical fluctuations lag behind equilibrium, advect outward, and feed back only mildly on the bulk flow in this simplified model.
Relativistic Hydrodynamic Fluctuations
hep-th 2019-02 accept novelty 6.0 of 10

The authors derive kinetic equations for equal-time fluctuation correlators in an arbitrary relativistic flow, introduce a flow-adjusted (confluent) derivative and Wigner transform, and show the longitudinal-mode equa...
QCD equation of state matched to lattice data and exhibiting a critical point singularity
hep-ph 2018-05 conditional novelty 6.0 of 10

A family of QCD equations of state is constructed that matches lattice QCD up to fourth order in baryon chemical potential and embeds a 3D Ising critical point at a tunable location.
Net baryon diffusion in fluid dynamic simulations of relativistic heavy-ion collisions
nucl-th 2018-04 conditional novelty 6.0 of 10

Adding a net baryon diffusion current to a 3+1D hydrodynamic plus hadronic cascade simulation shifts net baryons toward mid-rapidity and splits proton versus antiproton elliptic flow, but the size of the effect is deg...
Dynamical initial state model for relativistic heavy-ion collisions
nucl-th 2017-10 conditional novelty 6.0 of 10

A dynamical 3D initial-state model decelerates string endpoints between colliding nucleons or quarks and injects the resulting energy and baryon density into 3+1D hydrodynamics as source terms, reproducing net-proton ...
Acceptance dependence of fluctuation measures near the QCD critical point
nucl-th 2015-12 accept novelty 6.0 of 10

The acceptance dependence of critical-point fluctuation measures is set by a thermal rapidity correlation range of about one unit, so wider detector coverage increases the signal roughly linearly.
Suppression of baryon diffusion and transport in a baryon rich strongly coupled quark-gluon plasma
hep-ph 2015-07 conditional novelty 6.0 of 10

In a holographic model matched to lattice QCD, baryon conductivity, susceptibility, diffusion, and thermal conductivity are computed across the QCD crossover at nonzero baryon density, and baryon diffusion is found to...
Event-by-event mean $p_{\rm T}$ fluctuations in pp and Pb-Pb collisions at the LHC
nucl-ex 2014-07 accept novelty 6.0 of 10

Event-by-event mean transverse momentum fluctuations in pp and Pb-Pb collisions follow a power law in charged-particle multiplicity, and central Pb-Pb collisions show a marked suppression relative to the pp trend and ...
Freeze-out parameters: lattice meets experiment
hep-lat 2013-05 conditional novelty 6.0 of 10

Continuum-extrapolated lattice QCD susceptibilities of electric charge, compared to STAR net-charge moments, bound the freeze-out temperature below about 157 MeV and give baryon chemical potentials of roughly 44, 75, ...
Acceptance corrections to net baryon and net charge cumulants
nucl-th 2012-06 accept novelty 6.0 of 10

Finite acceptance mixes the true net-baryon cumulants with extra factorial moments that cumulants cannot express, and for acceptance below one half the measured cumulant ratios collapse toward Poisson values regardles...
On the sign of kurtosis near the QCD critical point
hep-ph 2011-04 accept novelty 6.0 of 10

The quartic cumulant of order-parameter fluctuations is universally negative in a wedge on the crossover side of the QCD critical point, so a measured kurtosis can drop below the Poisson value of 1.
The renormalization group and quark number fluctuations in the Polyakov loop extended quark-meson model at finite baryon density
hep-ph 2010-08 conditional novelty 6.0 of 10

Including mesonic fluctuations through the functional renormalization group shifts the chiral boundary and critical end point to higher temperature and suppresses the third and fourth net-quark-number cumulants near t...
Using Higher Moments of Fluctuations and their Ratios in the Search for the QCD Critical Point
hep-ph 2010-06 conditional novelty 6.0 of 10

The paper derives the critical-fluctuation contribution to third and fourth cumulants of pion, proton, and mixed multiplicities and shows that five specific ratios of these cumulants equal one regardless of the correl...
Thermomagnetic enhancement of conserved-charge correlations in a lattice-constrained PNJL model
hep-ph 2026-10 conditional novelty 5.0 of 10

A lattice-constrained thermomagnetic coupling enables a PNJL model to reproduce the magnetic-field enhancement of charge correlations and pressure coefficients without fitting those observables.
Diffusion of multiple conserved charges from entropy production
hep-ph 2026-06 unverdicted novelty 5.0 of 10

Derives diffusion matrix elements for baryon, charge, and strangeness transport in relativistic hydrodynamics from entropy production within the relaxation-time approximation.
Efficient calculation of thermodynamic properties of baryon-rich QCD matter from heavy-ion transport models
physics.comp-ph 2025-06 conditional novelty 5.0 of 10

A modular framework speeds up energy-momentum tensor calculations from transport models by clustering particles into representative super-particles while keeping averaged physics accurate.
Dynamics of Hot QCD Matter 2024 -- Bulk Properties
nucl-th 2024-12 conditional novelty 5.0 of 10

This proceedings volume collects 19 conference contributions on the bulk properties of hot QCD matter, mostly extending established models and comparing them with lattice QCD and heavy-ion data.
Multiplicity fluctuations in the Glauber Monte Carlo approach
hep-ph 2019-09 conditional novelty 5.0 of 10

Wounded quark sources with shadowing in a Glauber Monte Carlo model reproduce the centrality dependence of multiplicity fluctuations measured by NA49 in Pb+Pb collisions for 80 to 200 projectile participants.
Baryon number fluctuations in the QCD phase diagram from Dyson-Schwinger equations
hep-ph 2019-06 conditional novelty 5.0 of 10

A backcoupled Dyson-Schwinger calculation places the QCD critical endpoint near (119 MeV, 495 MeV) and produces fluctuation ratios that match preliminary STAR data only if the freeze-out line sits several MeV below th...
Higher order fluctuations and correlations of conserved charges from lattice QCD
hep-lat 2018-05 conditional novelty 5.0 of 10

Imaginary-chemical-potential lattice QCD at physical quark masses yields sixth-order conserved-charge fluctuations and cumulant ratios whose Taylor extrapolation qualitatively reproduces the STAR net-proton trend.
Baryon number fluctuations at finite temperature and density
hep-ph 2016-08 conditional novelty 5.0 of 10

Including the frequency dependence of the quark anomalous dimension in an FRG quark-meson model lowers the predicted kurtosis of baryon number fluctuations at finite density and brings the zero-density result into agr...
Energy loss, equilibration, and thermodynamics of a baryon rich strongly coupled quark-gluon plasma
hep-th 2015-07 conditional novelty 5.0 of 10

A holographic model fitted to zero-density lattice QCD predicts that heavy and light quark energy loss increases with baryon density and that equilibration is faster in a denser quark-gluon plasma.
Fluctuations and correlations in high temperature QCD
hep-lat 2015-07 conditional novelty 5.0 of 10

Continuum-extrapolated lattice QCD shows the hadron resonance gas describes charge fluctuations below the crossover and three-loop hard thermal loop perturbation theory describes them above about 250 MeV.
Freeze-out parameters from electric charge and baryon number fluctuations: is there consistency?
hep-lat 2014-03 conditional novelty 5.0 of 10

Lattice QCD fluctuation results compared with STAR net-proton and net-charge moments yield consistent freeze-out chemical potentials from two independent conserved charges, with the freeze-out temperature bounded abov...
Baryon number conservation and the cumulants of the net proton distribution
hep-ph 2012-03 conditional novelty 5.0 of 10

Global baryon-number conservation suppresses higher-order net-proton cumulants by large factors, and a specific combination of odd cumulants is constructed to be insensitive to this suppression.
Dynamic critical phenomena at a holographic critical point
hep-th 2011-08 conditional novelty 5.0 of 10

In a holographic QCD-like model, bulk viscosity and conductivity remain finite at the critical point while baryon diffusion goes to zero, consistent with large-N suppression of convective transport.
The Condensed Matter Physics of QCD
hep-ph 2000-11 accept novelty 5.0 of 10

Dense three-flavor quark matter forms a color-flavor locked superconductor whose broken symmetries, gapped spectrum, and modified electromagnetism mirror those of confined hadronic matter.
Mean-$p_T$ fluctuations in Au+Au collisions at $\sqrt{s_{\rm NN}}=3.0$--$19.6$ GeV within JAM2
nucl-ex 2026-09 conditional novelty 4.0 of 10

JAM2 predicts positive proton-antiproton and negative-to-positive charged-pion mean-pT correlations across 3.0-19.6 GeV Au+Au collisions, broadly matching STAR k2 data at high energies but with too-strong centrality d...
Scaling of the Surface Free Energy as a Probe of the QCD Critical Region
hep-ph 2026-06 unverdicted novelty 4.0 of 10

In a constructed QCD equation of state incorporating surface energy, critical exponents require temperature within 1% of the critical value, casting doubt on their measurability in heavy ion experiments.
Dynamical quark mass and finite volume effects in the Dyson-Schwinger Equations
hep-ph 2025-08 conditional novelty 4.0 of 10

In a Dyson-Schwinger model with spherical finite-volume corrections and magnetic-field-dependent coupling, constituent quark masses fall by about 30-40% as the fireball radius shrinks from infinity to 2 fm.
Study of the hottest droplet of fluid through correlations and fluctuations of collective variables
nucl-th 2025-05 conditional novelty 4.0 of 10

The thesis uses event-by-event fluctuations and correlations of collective flow and mean transverse momentum to propose new probes of the initial state, explain the ATLAS [pT]-variance fall, and constrain nuclear deformation.
Reaching percolation and conformal limits in neutron stars
nucl-th 2022-07 conditional novelty 4.0 of 10

Across an ensemble of neutron-star equations of state, the speed of sound peaks near the nucleon percolation density and falls to roughly the conformal value at the center of the most massive stars.
The BEST framework for the search for the QCD critical point and the chiral magnetic effect
nucl-th 2021-08 accept novelty 4.0 of 10

The BEST Collaboration summarizes the modular modeling framework it has built to connect RHIC Beam Energy Scan measurements to the QCD critical point and the chiral magnetic effect.
Strangeness neutrality and the QCD phase diagram
hep-ph 2019-07 unverdicted novelty 4.0 of 10

Strangeness neutrality imposes a constraint linking baryon-strangeness correlations to the QCD equation of state, with their dependence on freeze-out conditions computed in a 2+1 flavor Polyakov-quark-meson model usin...
An Experimental Exploration of the QCD Phase Diagram: The Search for the Critical Point and the Onset of De-confinement
nucl-ex 2010-07 conditional novelty 4.0 of 10

STAR proposes a RHIC beam energy scan from 5 to 39 GeV, with event-count targets for critical-point and deconfinement measurements and test-run evidence that the detector performs at those energies.

See all 73 Pith citations

How later papers used it

220 later papers cite it

Evidence: substantial. 195 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 10 of the 338 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

1 later paper by other authors is listed below: 1 used its data or values. 1 more paper by the same authors is listed: 1 used its data or values.

Uses its data or values 2 of 5 papers with this label · 1 by the same authors

Took a coupling G = 300 MeV from this paper, where its value near the critical point is extracted from considerations using a Ginzburg-Landau effective potential and the linear sigma model.

Critical versus spurious fluctuations in the search for the QCD critical point (2015) · 1.00 · paraphrased: the licence does not allow quoting

By the same authors

Used G = 300 MeV from this paper as a benchmark value throughout its work.

Ratios of Fluctuation Observables in the Search for the QCD Critical Point (2010) (same authors) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

186 arXiv papers cite this in 301 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 citing it · 214 sentences without a confident label

Contests it 1 paper

The approach in Refs. [ref] leading to Eq. ([ref]) leaves not only the magnitude, but also the sign of $g_A$ undetermined.

Maximum entropy freezeout of hydrodynamic fluctuations · Comparison with existing methods · contest 0.81

Builds on it 1 paper

Given the importance of the baryon number fluctuations in the phenomenology of CEP searching in the experiments [ref] , in this work we would like to extend relevant studies in Refs. [ref] from two to 2+1 flavors, based on the 2+1 flavor Polyakov-loop improved quark-meson (PQM) effective model, see, e.g. [ref] for more details about the model, and Refs. [ref] for other relevant 2+1 flavor low energy effective models of the same class.

Baryon number fluctuations in the 2+1 flavor low energy effective model · Introduction · 0.97

Uses its method or tool 6 papers

This contribution can be found using an intuitive picture described in Ref. [ref] .

Non-Gaussian fluctuations near the QCD critical point · Introduction · 0.90
We use the same effective theory as Refs. [ref] .

Critical versus spurious fluctuations in the search for the QCD critical point · Effective theory · 0.99
The contribution of critical fluctuations to the 2-, 3- and 4- particle correlators due to $σ$-exchanges can be calculated using the diagrammatic method developed in Ref. [ref] (see also Refs. [ref] and [ref] ).

Using Higher Moments of Fluctuations and their Ratios in the Search for the QCD Critical Point · Critical point contribution to correlators · 1.00

Uses its data or values 5 papers

A coupling $G = 300$ MeV is taken from Ref. [ref] , where its value near the critical point is extracted from considerations using a Ginzburg-Landau effective potential and the linear sigma model.

Critical versus spurious fluctuations in the search for the QCD critical point · Results · 1.00
This constant is known only roughly – the estimate was made in Ref. [ref] .

Non-Gaussian fluctuations near the QCD critical point · Introduction · 0.95
Throughout this paper we will use $G = 300$ MeV (see Ref. [ref] ) and $g_p=7$ (see, e.g., [ref] ) as benchmark values.

Ratios of Fluctuation Observables in the Search for the QCD Critical Point · Cumulant calculations · 0.99

Compares with it 1 paper

This data is compared to the effect of the critical point as discussed in [ref] and at this workshop [ref] .

Event-by-Event Fluctuations and the Search for the Critical Point within the NA49 Experiment · Fluctuations at the Critical Point · 0.94

Cites it as support 2 papers

However, a more stringent constraint comes from the fact that the system does not spend enough time in the critical region for the correlation length to build up to its equilibrium value [ref] .

Mapping the Phases of Quantum Chromodynamics with Beam Energy Scan · Finite time dynamics and limitations on the correlation length in heavy-ion collisions · 0.98
The critical slowing down makes it increasingly important to consider non-equilibrium evolution of fluctuations near the critical point, as earlier estimates and model calculations demonstrate [ref] .

Non-Gaussian fluctuation dynamics in relativistic fluids · Introduction · 0.98

Cites it as background 60 papers

Whether the analytic crossover turns into a first-order phase transition at finite baryon density at a QCD critical point (CP) remains an open question studied both theoretically, and experimentally through heavy-ion collisions [ref] .

Chemical freeze-out curve in heavy-ion collisions and the QCD critical point · Introduction · 0.91
The standard picture [ref] assumes the crossover line for small $μ$ meets a line of first order transitions at a critical endpoint as $μ$ increases and $T$ decreases.

Fluctuations in cool quark matter and the phase diagram of Quantum Chromodynamics · Relation to QCD · 0.98
Experimentally, the QCD phase structure can be probed through event-by-event fluctuations of the net-proton number measured in relativistic heavy-ion collisions [ref] , most notably in the RHIC Beam Energy Scan (BES) program, which covers baryochemical potentials up to in collider mode [ref] .

Extraction of baryon number susceptibilities at finite density from heavy-ion collisions · Introduction · 0.81

Discussion

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