Forward-backward correlations: A probe to study dynamical fluctuations

Saikat Biswas; Somen Gope; Supriya Das

REVIEW 4 major objections 6 minor 12 references

Forward-backward correlations: A probe to study dynamical fluctuations

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read In 10 AGeV Au+Au collisions, a transport-plus-hydro hybrid simulation produces forward-backward multiplicity correlations that survive rapidity gaps above one unit, and the Cooper-Frye freeze-out step is needed to generate them.

desk verdict A clean qualitative model study whose central statistical claim about Cooper-Frye freeze-out does not follow from the plotted observable. read the letter →

arxiv 2505.03411 v1 pith:75NPAASA submitted 2025-05-06 hep-ph

Somen Gope , Supriya Das , Saikat Biswas This is my paper · ORCID

classification hep-ph PACS 25.75.-q

keywords forward-backwardcorrelationlong-rangeUrQMD-hydrodynamicalfluctuationsrapiditygapCooper-Fryefreeze-outlengthAu+Aucollisionsat10AGeV

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that hybrid transport-plus-hydro simulations of 10 AGeV Au+Au collisions contain long-range forward-backward multiplicity correlations, not just short-range noise, and that the hydrodynamic stage strengthens them relative to the pure transport calculation. If true, collisions at fixed-target energies around 10 AGeV should show measurable correlations between particles emitted in forward and backward rapidity windows separated by more than one unit of rapidity, with the correlations tracing back to the collective evolution of the fireball. The paper further claims that removing the Cooper-Frye freeze-out step makes the fluctuation measure $\sigma^2$ flat in the rapidity gap, so the freeze-out procedure is what converts hydrodynamic flow into correlated final-state particles. A reader should care because this identifies a specific dynamical mechanism — not just statistical noise — as the source of event-by-event fluctuations at energies planned for new fixed-target experiments.

What carries the argument

The load-bearing quantities are the forward-backward correlation strength, defined as the covariance of forward and backward multiplicities divided by the forward multiplicity variance, and the assumed Gaussian decay $b_{\mathrm{corr}} \propto \exp(-y_{\mathrm{gap}}^2/\delta^2)$ with the conversion $\lambda = 2\delta/\sqrt{\pi}$. A second observable, $\sigma^2$, is built from the forward and backward variances and their covariance and is compared against the Poisson baseline of 1 to classify correlations as short- or long-range. The argument is carried by comparing three simulated data sets — default transport, hybrid with Cooper-Frye freeze-out, and hybrid with freeze-out disabled — where the third run isolates the freeze-out mechanism.

What would settle it

Refit the same $b_{\mathrm{corr}}$ versus $y_{\mathrm{gap}}$ points with an exponential and with a power law using the same number of parameters; if either fits comparably or better, the extracted $\lambda$ is a shape artifact. A second, model-independent check is to measure the rapidity gap at which $b_{\mathrm{corr}}$ crosses a fixed small threshold without assuming any functional form; if that crossing sits below one unit of rapidity, the $\lambda > 1$ long-range conclusion collapses.

Watch

Extended reading notes

Core claim

The central claim is that for minimum-bias Au+Au at 10 AGeV, the hybrid UrQMD-hydro model yields a forward-backward correlation strength $b_{\mathrm{corr}}$ that decreases monotonically with the rapidity gap, and the fitted Gaussian parameter $\delta$ corresponds to correlation lengths $\lambda \approx 1.35$ for the hybrid run and $\lambda \approx 1.50$ for the default transport run. Because both extracted lengths exceed unity in rapidity units, the paper reads them as the signature of long-range correlation: particles separated by more than one unit of rapidity are still correlated. The companion result is that the hybrid run with the Cooper-Frye particle-production step disabled shows $\sigma^2$ flat against $y_{\mathrm{gap}}$, which the paper interprets as an absence of correlation, implying that the Cooper-Frye freeze-out stage is the active ingredient producing the observed long-range correlations in the hybrid calculation.

Load-bearing premise

The entire long-range-correlation conclusion rests on assuming that the correlation strength falls off as a bell-shaped Gaussian in the rapidity gap, an assumption quoted from the authors' own earlier work without derivation; if the true falloff is exponential or some other shape, the reported correlation lengths and the 'greater than one means long-range' reading would not follow.

Editorial extensions

If this is right

At fixed-target energies near 10 AGeV, minimum-bias Au+Au events should show a measurable forward-backward correlation that survives rapidity gaps above one unit, so detector acceptances should be designed to cover forward and backward windows with such gaps.
The hydrodynamic stage raises $b_{\mathrm{corr}}$ relative to the default transport calculation, making the correlation strength a model discriminator sensitive to the equation of state and collective expansion.
Switching off the Cooper-Frye freeze-out flattens $\sigma^2$ versus $y_{\mathrm{gap}}$, so the freeze-out treatment is not merely a detail of particle spectra; it determines whether long-range multiplicity correlations appear in the hybrid calculation at all.
The $\sigma^2 > 1$ values seen in the two runs with active freeze-out indicate deviations from Poisson emission across all measured gaps, so independent single-particle production cannot describe these events by itself.

Reading between the lines

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

Editorial inference: The Gaussian-derived $\lambda$ could be checked against a model-independent length scale, such as the rapidity gap at which $b_{\mathrm{corr}}$ falls to half its peak; a large disagreement would show that the functional-form assumption, not the data, drives the $\lambda > 1$ conclusion.
Editorial inference: Because the paper uses only minimum-bias events, a natural next test is centrality-binned runs; if the correlation strengthens with centrality as it does at higher beam energies, the 10 AGeV result would connect cleanly to the established energy pattern.
Editorial inference: The flat $\sigma^2$ in the run without Cooper-Frye could reflect the absence of collective velocity smearing rather than a true loss of correlation; computing $b_{\mathrm{corr}}$ directly for that run, rather than only $\sigma^2$, would separate the two explanations.
Editorial inference: A direct extension would be to scan beam energies from 10 to 40 AGeV in the same hybrid setup and test whether the extracted correlation length grows with collision energy; this paper reports a single energy.

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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

Quick take: this is a small model study with one clean qualitative result and one unsupported headline claim. Read it for the UrQMD-hydro comparison; don't believe the Cooper-Frye conclusion as written.

What's new: the group compares bcorr and sigma2 for UrQMD-default vs UrQMD-hydro at 10 AGeV Au+Au, and adds a third run with Cooper-Frye disabled. The bcorr(ygap) trend is clean: monotonically decreasing, higher for hydro. That's a legitimate extension of ref. [6]'s UrQMD-only study, and the SIS100 community can use it as a baseline prediction.

Credit where due: they generated 150k/100k/60k events, show the raw 2D correlation, and give fit parameters with errors. That's reproducible enough for a model study.

Soft spots, in order of seriousness:

1. The Cooper-Frye claim. The paper says the no-CF run shows 'lack of correlation' because sigma2 is flat vs ygap. But sigma2, even after fixing the typo in Eq. (5), is the scaled variance of the multiplicity difference, not bcorr. A flat sigma2 doesn't imply zero F-B correlation; bcorr could be nonzero and constant. They never show bcorr(ygap) for the no-CF sample. So the central conclusion – that Cooper-Frye freeze-out is necessary for F-B correlations – is unsupported. That's not a minor caveat; it's the paper's distinctive claim.

2. The Gaussian ansatz. Eqs. (3)-(4) come from their own proceedings ref. [3], with no derivation or external motivation. The reported lambda ~1.35-1.50 are fit parameters of that assumed shape. If the decay is not Gaussian, lambda loses meaning. The 'lambda>1 implies LRC' step is therefore conditional.

3. Missing settings and a typo. The rapidity window width is never stated. Eq. (5) as printed has squared variances/covariances, which is dimensionally wrong and can't reproduce the plotted values. I assume it's a typo for the standard form, but it should be fixed. Figures 3-5 have no statistical uncertainties on the points, which matters when interpreting flatness.

The stress-test note lands. The bcorr result itself is fine; it's the interpretation of sigma2 that doesn't hold up.

Who this is for: people working on FB correlations at SIS100 energies, and anyone using UrQMD-hydro. The paper is short, clear, and could be a useful proceedings-level contribution after major revision.

Recommendation: send to peer review, but the referees should push for a bcorr(ygap) plot for the no-CF run, a statement of the rapidity window, and a derivation or external reference for the Gaussian form. If the authors can't show bcorr for the no-CF sample, they need to soften the Cooper-Frye claim.

Referee Report

4 major / 6 minor

Summary. This manuscript studies forward-backward (F-B) multiplicity correlations in Au+Au collisions at 10 AGeV using three variants of UrQMD-generated events: default UrQMD, UrQMD with an intermediate hydrodynamic stage (UrQMD-hydro), and UrQMD-hydro with the Cooper-Frye (C-F) freeze-out step disabled. The authors define the correlation strength bcorr from the slope of ⟨nB⟩ versus ⟨nF⟩ and from the covariance/variance ratio, fit bcorr as a function of the rapidity gap ygap to a Gaussian ansatz exp(−ygap^2/δ^2), and convert the fitted width δ into a correlation length λ = 2δ/√π. They report λ > 1 for both UrQMD-default and UrQMD-hydro, interpret σ² > 1 versus ygap as evidence for long-range correlations, and observe that σ² is flat for the no-CF sample, which they interpret as a lack of correlation and hence as evidence that the Cooper-Frye mechanism is necessary for generating F-B correlations.

Significance. If established, the central claims would be of interest to the heavy-ion community planning SIS100 measurements, because they provide a concrete model prediction for how F-B correlations and their correlation length behave at lower beam energies and how the hydrodynamic freeze-out stage affects these observables. The paper's strength is that it compares three model variants with fixed collision parameters and presents a quantitative observable that could be checked against future SIS100 data; the analysis is also straightforward to reproduce once the model settings are specified. However, the key quantitative conclusions rest on an unvalidated Gaussian parameterization, a dimensionally inconsistent formula for σ², and an interpretive step that equates a flat σ² with the absence of F-B correlation. These issues currently prevent the results from being regarded as established. The paper does not provide machine-checked proofs or released analysis code; its contribution is an exploratory model comparison.

major comments (4)

[Sec. 2, Eq. (5)] Equation (5) as printed is dimensionally inconsistent: if V_FF, V_BB, and V_FB are variances/covariances of multiplicities (dimension n^2), the numerator contains n^4 terms divided by ⟨n_F + n_B⟩ (dimension n), which cannot yield the order-unity values plotted in Figs. 4 and 5. The standard variance-of-difference form is σ² = (V_FF + V_BB − 2 V_FB)/⟨n_F + n_B⟩, and this is presumably what was computed. Because the σ² > 1 interpretation and the flat-σ² conclusion for the no-CF sample depend on this quantity, the formula must be corrected and the figures re-checked against the corrected expression.
[Sec. 4, Fig. 5] A flat σ² as a function of ygap does not imply the absence of F-B correlation. σ² is a scaled variance of the forward-backward multiplicity difference, not the correlation strength bcorr; a ygap-independent σ² is compatible with a nonzero and even ygap-independent bcorr, and conversely a falling bcorr can coexist with a flat σ². The paper never shows bcorr(ygap) for the UrQMD-hydro without C-F sample, so the conclusion that the Cooper-Frye approach is necessary for generating F-B correlations is unsupported by the presented evidence.
[Sec. 2, Eqs. (3)-(4) and Table 1] The Gaussian parameterization bcorr ∝ exp(−ygap^2/δ^2) and the conversion λ = 2δ/√π are attributed only to the authors' own proceedings reference [3] without derivation or external validation. The reported correlation lengths λ ≈ 1.35–1.50 are therefore fit parameters of an assumed functional form; if the true decay were exponential or otherwise non-Gaussian, the extracted λ values and the statement that λ > 1 indicates long-range correlation would lose their meaning. The authors should either derive the Gaussian form from a model, compare it with alternative fits, or benchmark the extracted λ against a known observable with an established source of correlation.
[Sec. 2 and Figs. 3-5] The manuscript never states the value of the rapidity window width δy used in the analysis, although the definition of ygap and the placement of the forward and backward windows depend on it; without this parameter the results in Table 1 and Figs. 3-5 cannot be reproduced or compared with other analyses. Additionally, the figures show no statistical uncertainties or systematic variations, so the claimed differences between UrQMD-default and UrQMD-hydro (and the flatness of the no-CF σ²) cannot be quantitatively assessed.

minor comments (6)

[Sec. 1, Introduction] The sentence beginning "Let consider the first moment" should read "Let us consider" and the notation ⟨n_B⟩ n_F in the same section is confusing and should be cleaned up.
[Sec. 2, Eq. (1)] The regression form ⟨n_B⟩ = a + bcorr ⟨n_F⟩ should be stated with the random variable n_F (or its conditional average) made explicit, since Eq. (2) then defines bcorr through the covariance/variance ratio; a brief derivation would clarify why the two definitions coincide.
[Fig. 3 caption] Typo: "UrQMD-defaut" should be "UrQMD-default"; also the axis label "corrb" should be "b_corr".
[Throughout] Several typographical errors occur, including "eqaution", "diffrent", and the inconsistent use of "C-F" versus "CF" for the Cooper-Frye approach; these should be corrected.
[Sec. 4, Fig. 2] The color-bar scale in Fig. 2 (event counts up to 24000) makes the scatter plot difficult to read; a density plot with a clear color scale or contours would better support the claim of a forward-backward correlation.
[Sec. 5, Summary] The statement that λ > 1 means "particles are still correlated at the rapidity gap more than one unit" is imprecise, since λ is a parameter of a Gaussian fit and not an operator-defined threshold; the wording should be revised to avoid implying a direct measurement of a physical length.

Circularity Check

1 steps flagged · score 6.0 of 10

The reported correlation length λ is a rescaling of the fitted Gaussian width δ (Eq. 4), so 'λ>1 ⇒ LRC' reduces to the self-cited fit ansatz; the no-CF σ² argument is unsupported rather than circular.

fitted input called prediction [Section 4, Eqs. (3)-(4) and Table 1]
"The plots are fitted with the equation 3. The fitted parameter δ is related to correlation length (λ) by the equation 4. ... The correlation length λ is calculated from the value of δ and the values are greater than unity for both the sets of data. This is the signature of existence of long-range correlation (LRC)."

Equation (4) defines λ as a constant rescaling of the fitted width δ (λ = 2δ/√π). Hence every reported λ value is, by construction, the fit parameter δ from Eq. (3), not an independently measured or predicted quantity. The conclusion 'λ>1 implies LRC' therefore restates the fitted Gaussian width (δ > √π/2) and carries no information beyond the success of the assumed exp(−ygap²/δ²) shape. Equations (3)-(4) are cited only to the authors' own proceedings [3], so the LRC result is not checked against an external benchmark.

full rationale

The paper has two independent strands. First, bcorr(ygap) is measured with the standard formula Eq. (2) and decreases monotonically; this is a genuine, non-circular observation. Second, the correlation length λ and the LRC claim are built from Eqs. (3)-(4): a Gaussian ansatz attributed to the authors' own proceedings [3] is fitted to bcorr, and λ is defined as 2δ/√π. Because λ is a pure rescaling of the fitted δ, the conclusion 'λ>1 implies LRC' is equivalent to the fitted width of that assumed Gaussian; it is not a prediction tested against data or an external benchmark. The Cooper-Frye conclusion based on flat σ² for the no-CF run is an unsupported inference (σ² is not bcorr, and no bcorr(ygap) is shown for that sample), but this is a logical gap rather than circularity. Weighing the fitted-parameter definition at the center of the LRC claim gives partial circularity, score 6.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central quantitative outputs (lambda values) come from fitting the Gaussian ansatz of Eq. (3), and the conversion Eq. (4) is self-cited; the interpretation that sigma2 > 1 implies LRC rests on an unvalidated domain assumption; the analysis window width is unspecified.

free parameters (3)

delta (UrQMD-default) = 1.326 ± 0.005
Width of the Gaussian fit bcorr ∝ exp(-ygap^2/delta^2) in Fig. 3; used to compute lambda = 2*delta/sqrt(pi) reported in Table 1.
delta (UrQMD-hydro) = 1.192 ± 0.004
Same fit for the hybrid model; controls the reported correlation length 1.345 ± 0.005.
Rapidity window width delta_y = not specified
The analysis windows are defined in Fig. 1 with a half-width delta_y, but no numerical value is given; bcorr and sigma2 depend on this choice.

assumptions (4)

domain assumption UrQMD-hydro with the chiral equation of state provides a realistic description of 10 AGeV Au+Au collisions.
The paper treats the hybrid model output as representative of the dynamical fluctuations of interest (Sec. 3) without validating against experimental data or showing that event-by-event fluctuations in the model match the initial energy-density fluctuations it claims to probe.
ad hoc to paper The Gaussian form bcorr ∝ exp(-ygap^2/delta^2) and the relation lambda = 2*delta/sqrt(pi) are valid.
Eqs. (3)-(4) are cited only to the authors' own proceedings ref. [3]; no derivation or external benchmark is given. The quantitative claim lambda > 1 depends on this assumed functional form.
domain assumption sigma2 > 1 indicates long-range correlation dominance without subtracting volume fluctuations.
In Sec. 4, the statement that sigma2 > 1 'means LRC is the dominant' assumes the fluctuation measure isolates dynamical correlations from trivial volume fluctuations; this is not demonstrated for minimum-bias events.
domain assumption Minimum-bias event sample is sufficient without centrality selection.
No centrality binning is applied; volume fluctuations in minimum-bias Au+Au may dominate the fluctuation measure and the bcorr values. The paper does not discuss this.

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Cite this review

Pith. "Pith review of Forward-backward correlations: A probe to study dynamical fluctuations." pith.science (2026). https://pith.science/paper/75NPAASA

@misc{pith2026250503411,
  author       = {Pith},
  title        = {Pith review of: Forward-backward correlations: A probe to study dynamical fluctuations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/75NPAASA}},
  note         = {Machine review of arXiv:2505.03411}
}

read the original abstract

The source of the fluctuations in the final state particles is the initial event-by-event fluctuation in energy density. Forward-Backward (F-B) correlation is one of the important probes to study such fluctuations. The results of F-B correlation are available in a wide range of energy, from STAR of RHIC to ALICE of LHC. It will be more interesting to study the dynamical fluctuation using F-B correlation at SIS100 energy too. In this study, an attempt has been made to investigate the F-B correlation in UrQMD-hydro generated data for 10 AGeV Au+Au collisions.

Figures

Figures reproduced from arXiv: 2505.03411 by the authors.

**Figure 2.** 2D plot of distribution of particles in forward & backward rapidity spaces. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** The plot of variation of correlation coefficient with the rapidity gap for UrQMD [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: σ 2 distribution for UrQMD-default & UrQMD-hydro generated data. However, σ 2 vs. ygap has been plotted to check the type of correlation in the particles produced in heavy ion collision [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: σ 2 distribution for the data of UrQMD-hydro & UrQMD-hydro with out CF. approach. It is observed that the new data set shows no dependency with ygap, indicating lack of correlation. 5. Summary In this work, a systematic study on forward-backward multiplicity correlati…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [6]

J. Thakur et al., Forward-backward multiplicity correlation and fluctua- tion in 197 Au + 197 Au collisions at fixed targetElab = 10A, 20A, 30A and 40A GeV in the UrQMD model, Nuclear Physics A 1035 (2023) 122659

work page 2023

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Gope et al., Investigation of forward-backward correlation in hybrid UrQMD-hydro at 10 AGeV Au+Au collisions, Proceedings of the DAE Symp

S. Gope et al., Investigation of forward-backward correlation in hybrid UrQMD-hydro at 10 AGeV Au+Au collisions, Proceedings of the DAE Symp. on Nucl. Phys. 68 (2024)

work page 2024

[3] [1]

Skoby (for the STAR Collaboration), Forward-Backward Multiplicity Correlations for Identified Particles in Au+Au 200 GeV Collisions,

M. Skoby (for the STAR Collaboration), Forward-Backward Multiplicity Correlations for Identified Particles in Au+Au 200 GeV Collisions, . Phys.: Conf. Ser. 230 (2010) 012030

work page 2010

[4] [2]

Adam et al

J. Adam et al. (ALICE Collaboration), Forward-backward multiplicity correlations in pp collisions at√s= 0.9, 2.76 and 7 TeV, JHEP05(2015)097

work page 2015

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work page 1989

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work page 2009

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Bass et al., Microscopic models for ultrarelativistic heavy ion colli- sions, Prog.Part.Nucl.Phys

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Steinheimer et al., Momentum dependent potentials from a parity dou- bling CMF model in UrQMD:Results on flow and particle production, arXiv:2410.01742v1 [hep-ph] 2 Oct 2024

J. Steinheimer et al., Momentum dependent potentials from a parity dou- bling CMF model in UrQMD:Results on flow and particle production, arXiv:2410.01742v1 [hep-ph] 2 Oct 2024. 8

arXiv 2024