Probing the Energy Evolution and Hadronic Cascade Time Dependence of Forward-Backward Multiplicity Correlations in pp Collisions at $\sqrt{s} = 0.9$--$13.6$ TeV with AMPT

Dibakar Dhar; Prabir Kumar Haldar; Tumpa Biswas

REVIEW

Probing the Energy Evolution and Hadronic Cascade Time Dependence of Forward-Backward Multiplicity Correlations in pp Collisions at $\sqrt{s} = 0.9$--$13.6$ TeV with AMPT

Dibakar Dhar, Tumpa Biswas, Prabir Kumar Haldar

High Energy Physics - Phenomenology · Submitted 18 Oct 2026

Pith's verdict: A solid, honest AMPT study whose new tau_HC scan is internally consistent, but whose "primarily partonic imprint" conclusion overreaches the cascade-time range actually scanned.

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arXiv ID2610.25154 (version 2)

Pith ID5BMOG4L3, Pith's permanent record of this paper and its review

CitationDibakar Dhar, Prabir Kumar Haldar, Tumpa Biswas. "Probing the Energy Evolution and Hadronic Cascade Time Dependence of Forward-Backward Multiplicity Correlations in pp Collisions at $\sqrt{s} = 0.9$--$13.6$ TeV with AMPT." arXiv:2610.25154 (2026). https://pith.science/paper/5BMOG4L3

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Editor's note

read the letter

Read this if you work with AMPT or with forward-backward correlations in small systems; otherwise it is a solid model study whose central claim is slightly over-claimed. The new piece is the hadronic cascade time scan, and it is worth having.

The paper computes b_corr with AMPT default (DF) and string melting (SM) modes at six LHC energies, compares with ALICE, and — the genuinely new part — scans tau_HC in SM mode at 0.6, 5, 15, 30 fm/c across six energies and four window widths. The scan is internally consistent: monotonic decrease of b_corr with tau_HC everywhere, no crossing of the curves, and the effect is statistically solid (errors ~0.002 on 2–7M events). The two extraction procedures agree. The DF/SM comparison is reported honestly, including the energy-dependent switch in which configuration describes ALICE better and the low-pT deviations. Validation against dN/deta and pion pT spectra is a plus. Citation pattern looks fine.

Soft spots, in proportion: First, the load-bearing interpretation is the weak link. The scan starts at tau_HC = 0.6 fm/c; there is no pre-hadronic baseline. If most hadronic dilution happens within the first 0.6 fm/c — early resonance decays, rescattering, redistribution across the |eta| < 0.8 and 0.3 < pT < 1.5 GeV/c acceptance — the flat-looking 0.6-to-30 fm/c behaviour would be expected even if the hadronic stage removed a substantial part of the correlation. The "only mildly diluted" conclusion is an extrapolation, not a measurement. The paper does limit its own claim to qualitative and says a quantitative statement needs an SRC/LRC decomposition, so this is a cap on the conclusion rather than a fatal flaw. Second, b_corr is a ratio. A change in the mean or variance of the window multiplicities with tau_HC would move b_corr without any change in the underlying source correlation, and those quantities are not reported as functions of tau_HC. A cheap check, and it would strengthen Section 6 considerably. Third, "data available on request" with no parameter files makes exact reproduction depend on the authors. AMPT itself is public, so this is minor.

None of these sink the paper. The scan is new, the statistics are large, and the internal consistency is good. What is not established is the quantitative strength of the partonic-imprint interpretation.

Verdict: send it to peer review. I would want the interpretation softened or a pre-cascade baseline added, plus the mean/variance check. I would bring it to a reading group only if someone there is actively doing AMPT or small-system correlations, and I would not cite it in my own work next year.

Referee report

Significance

If the central interpretive claim is supported, the paper would provide a useful AMPT-based phenomenological statement about the relative roles of partonic and hadronic stages in shaping long-range FB multiplicity correlations in pp collisions. The statistical analysis is sound in its basics: b_corr is computed from the standard expression in Eq. (1.2), the two extraction procedures are cross-checked, the event samples are large (2.1e6 to 6.7e6 events), and no parameter is fitted to reproduce the experimental b_corr values. The comparison with ALICE data at 0.9, 2.76 and 7 TeV is a genuine model-versus-data test, and the energy-dependent crossover between the DF and SM descriptions is a noteworthy observation. These strengths make the paper a credible phenomenological survey; however, the main physical conclusion depends on an interpretation of the tau_HC scan that is not fully established by the presented quantities.

Major objections (3)

[Section 6.3] The central interpretive claim, that the FB correlation is 'primarily imprinted at the partonic stage and only mildly diluted' by hadronic evolution, is not supported by the tau_HC scan alone because the scan starts at tau_HC = 0.6 fm/c. The paper never measures the pre-cascade baseline: the correlation before the ART hadronic cascade begins, or equivalently at tau_HC = 0. If a substantial fraction of the hadronic dilution happens in the first 0.6 fm/c through early resonance decays, rescattering, and redistribution of particles across the |eta| < 0.8 and 0.3 < pT < 1.5 GeV/c acceptance, then the flat behaviour between 0.6 and 30 fm/c in Figures 15 and 16 would be fully consistent with a large hadronic-stage effect. A direct computation of b_corr with the cascade switched off, or with tau_HC set to a value much smaller than 0.6 fm/c, is required before the 'mild diluter' conclusion can be drawn.
[Eq. (1.2), Section 6] Because b_corr is a ratio of covariance to variance of the window multiplicities, a change in the mean multiplicity, in the variance of n_F, or in the acceptance occupancy with tau_HC can shift b_corr even if the underlying source correlation is unchanged. The paper reports only b_corr as a function of tau_HC and does not present <n_F>, <n_B>, Var(n_F), or the covariance as functions of tau_HC. Without these moments, the monotonic reduction shown in Figures 15 and 16 cannot be uniquely attributed to dilution of the correlation, rather than to a tau_HC-dependent normalization or to particle migration across the pseudorapidity or pT boundaries. Reporting these moments, or the covariance-to-variance ratio directly, would make the interpretation testable.
[Section 6, Section 6.1] The paper treats tau_HC as an isolated control parameter that dilutes correlations without altering the rest of the collision dynamics, but it does not address how the early termination of the ART cascade affects particle redistribution near the acceptance boundaries. A longer cascade allows more rescattering, but it also changes which particles end up inside |eta| < 0.8 and 0.3 < pT < 1.5 GeV/c, so the observed ordering of the tau_HC curves could in principle be a consequence of how the cascade is stopped rather than of rescattering physics alone. The authors correctly note in Section 2.1.2 that varying tau_HC in DF mode would conflate the cascade effect with initial-state differences, but the analogous acceptance-redistribution check for the SM scan is not discussed. A simple diagnostic would be to plot the mean charged multiplicity and its variance inside the counting windows versus tau_HC.

Minor objections (4)

[Section 6.1] The quoted statistical uncertainties (about 0.002) and the numerical values for b_corr at tau_HC = 0.6 and 30 fm/c are given in the text, but no error bars are visible in Figures 15 and 16. Adding error bars or shaded uncertainty bands would make the claimed separation between the curves directly visible in the figures.
[Section 5] The caption of Figure 12 says the pT intervals have the same <n_c>, but the text does not state how this condition is implemented or how the intervals were chosen. A sentence explaining the equal-<n_c> construction would help the reader assess the comparison with ALICE.
[Acknowledgements] There is a typo in the acknowledgements: 'greatful' should be 'grateful'.
[References] The reference list is generally complete, but the ALICE 13.6 TeV pseudorapidity-density reference [54] is an arXiv preprint with a 2026 date; the authors should check whether a journal version has appeared and cite the published version if so.

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The paper

The paper claims

Using the AMPT model, the paper claims that forward-backward multiplicity correlations in proton-proton collisions are set mainly at the partonic stage: lengthening the hadronic cascade from 0.6 to 30 fm/c lowers the correlation…

The reading

The paper tries to establish where in the space-time evolution of a proton-proton collision the strength of forward-backward multiplicity correlations is decided. Using the AMPT transport model, it computes the correlation coefficient $b_{\rm corr}$ for windows separated in pseudorapidity and azimuth across six LHC energies, then scans the hadronic cascade time $\tau_{\rm HC}$ from 0.6 to 30 fm/c while keeping the partonic stage fixed. The central result is that $b_{\rm corr}$ falls monotonically as the cascade is extended, but only by about 0.01 over the full range, with the cascade-time curves never crossing at any energy or window width. The authors read this as evidence that the correlation is imprinted at the partonic stage and only mildly diluted by hadronic rescattering. If true, it sharpens the long-standing idea that long-range multiplicity correlations encode the earliest stage of the collision.

What carries the argument

The carrier of the analysis is the forward-backward correlation coefficient $b_{\rm corr}$, the slope of the linear regression of mean backward multiplicity on forward multiplicity, computed event-by-event from $b_{\rm corr} = (\langle n_F n_B\rangle - \langle n_F\rangle\langle n_B\rangle)/(\langle n_F^2\rangle - \langle n_F\rangle^2)$. The probe is the hadronic cascade time $\tau_{\rm HC}$, a parameter of the hadronic transport stage that controls how long the post-hadronisation hadron gas rescatters in AMPT. The procedure holding the argument together is to fix the string melting initial state and partonic cascade while varying $\tau_{\rm HC}$ over $0.6$, $5$, $15$, and $30$ fm/$c$, which is meant to isolate the dilution of pre-existing correlations by rescattering from everything else in the collision. The ordering-preserving, approximately parallel curves of $b_{\rm corr}$ versus $\sqrt{s}$ for the four cascade times are the observable signature that carries the interpretation of the cascade as a uniform, energy-independent dilution stage.

Extended reading notes

Core claim

Forward-backward correlation strength $b_{\rm corr}$ — the slope in $\langle n_B\rangle = a + b_{\rm corr}\, n_F$ — rises substantially with collision energy from $\sqrt{s}=0.9$ to $13.6$ TeV, falls gradually with the pseudorapidity gap while keeping a pedestal, and rises non-linearly with window width. The paper's new claim comes from the cascade-time scan: in the string melting configuration, where a partonic cascade precedes hadronisation, increasing $\tau_{\rm HC}$ from $0.6$ through $5$, $15$, and $30$ fm/$c$ reduces $b_{\rm corr}$ monotonically at all six energies and all four window widths, with no crossing between the curves and approximately constant spacing when plotted against $\sqrt{s}$. The reduction is modest — at most about $0.01$ — but well above the $\sim 0.002$ statistical precision of the simulation. Because the dilution is small and roughly energy-independent, the authors conclude that the long-range FB correlation is imprinted before hadronisation and that the hadronic cascade acts as a mild, uniform diluter. Two supporting results: the default mode, which omits a partonic scattering stage, gives systematically larger $b_{\rm corr}$ than string melting, and the configuration that matches the ALICE data better switches with energy — SM at $0.9$ and $2.76$ TeV, DF at $7$ TeV — which the paper reads as a shift in the balance of short- and long-range contributions.

Load-bearing premise

The argument assumes that lengthening the hadronic cascade dilutes correlations only through genuine rescattering, and that the way AMPT stops the cascade does not itself move particles across the window or acceptance boundaries — so the monotonic drop in $b_{\rm corr}$ measures correlation erasure rather than a bookkeeping artifact.

Editorial extensions

If this is right

At all six energies and all four window widths, extending $\tau_{\rm HC}$ from 0.6 to 30 fm/c lowers $b_{\rm corr}$ monotonically, so within AMPT the hadronic cascade always acts as a diluter of FB correlations, never as a source of them.
Because the four cascade-time curves stay roughly parallel against $\sqrt{s}$, the fractional dilution is nearly independent of collision energy, making $\tau_{\rm HC}$ a usable control parameter for separating pre-hadronic from hadronic contributions.
The energy dependence of $b_{\rm corr}$ (a rise of roughly 0.26 from 0.9 to 13.6 TeV) is far stronger than the cascade-time spread (at most about 0.01), so hadronic dilution is a small correction on top of the energy baseline rather than a competing effect.
Which AMPT configuration describes the ALICE data better switches with energy — string melting at 0.9 and 2.76 TeV, default at 7 TeV — indicating that the relative weight of partonic and hadronic stages shifts with $\sqrt{s}$.
The largest model-data deviations sit in the lowest transverse momentum interval (0.3–0.4 GeV/c), which the paper attributes to a less complete description of the short-range correlation component in AMPT.

Reading between the lines

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

A direct test the paper leaves open (it states that a quantitative decomposition is beyond its scope): split $b_{\rm corr}$ into short- and long-range parts as a function of $\tau_{\rm HC}$. The partonic-imprint claim predicts the long-range piece stays nearly flat while the short-range piece absorbs the drop.
The monotonic ordering could in part be an acceptance effect: if stopping the cascade early or late moves particles across the $|\eta|<0.8$ boundary rather than erasing correlations, then widening the counting windows to cover the full acceptance should shrink the $\tau_{\rm HC}$ spread; this can be checked with the same generated events.
The same $\tau_{\rm HC}$ scan applied to Pb–Pb collisions would test whether the mild dilution survives in a denser hadron gas where rescattering is stronger; if the drop steepens there, the partonic-imprint claim would need qualification as a property of dilute pp systems.

Original abstract

Forward-backward (FB) multiplicity correlations are studied in proton--proton collisions at centre-of-mass energies of $\sqrt{s} = 0.9$, $2.76$, $5.02$, $7$, $13$, and $13.6$~TeV using the AMPT model in its default (DF) and string melting (SM) configurations. The coefficient $b_{\rm corr}$ is obtained for windows separated in pseudorapidity and azimuth, and its variation with the pseudorapidity gap, window width, collision energy, transverse momentum interval, and hadronic cascade time $\tau_{\rm HC}$ is investigated. The simulated results are compared with the available ALICE measurements, and the model is validated against experimental pseudorapidity densities and charged-pion transverse momentum spectra. The correlation strength is found to decrease gradually with increasing pseudorapidity gap while retaining a substantial pedestal, to rise non-linearly with the window width, and to increase substantially with collision energy. The DF configuration yields systematically larger values of $b_{\rm corr}$ than the SM configuration, while the SM configuration provides the better description of the ALICE data at $0.9$ and $2.76$~TeV and the DF configuration at $7$~TeV. In the lowest transverse momentum intervals the deviations from the data are largest for both configurations. The hadronic cascade time is varied within the SM configuration over $\tau_{\rm HC} = 0.6$, $5$, $15$, and $30$~fm/$c$. A monotonic reduction of $b_{\rm corr}$ with increasing $\tau_{\rm HC}$ is observed at all energies and for all window widths, with the ordering of the curves preserved throughout. The effect is modest but statistically significant. These results indicate that the FB correlation strength is primarily imprinted at the partonic stage and only mildly diluted by the subsequent hadronic evolution.

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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-20: 1 request, 7,962 input tokens.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.70 · human participants 0.01 · controlled comparison 0.27 · randomized assignment 0.01 · states quantitative result 0.64 · compares with prior results 0.96 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Forward-backward (FB) multiplicity correlations are studied in proton--proton collisions at centre-of-mass energies of $\sqrt{s} = 0.9$, $2.76$, $5.02$, $7$, $13$, and $13.6$~TeV using the AMPT model in its default (DF) and string melting (SM) configurations. The coefficient $b_{\rm corr}$ is obtained for windows separated in pseudorapidity and azimuth, and its variation with the pseudorapidity gap, window width, collision energy, transverse momentum interval, and hadronic cascade time $\tau_{\rm HC}$ is investigated. The simulated results are compared with the available ALICE measurements, and the model is validated against experimental pseudorapidity densities and charged-pion transverse momentum spectra. The correlation strength is found to decrease gradually with increasing pseudorapidity gap while retaining a substantial pedestal, to rise non-linearly with the window width, and to increase substantially with collision energy. The DF configuration yields systematically larger values of $b_{\rm corr}$ than the SM configuration, while the SM configuration provides the better description of the ALICE data at $0.9$ and $2.76$~TeV and the DF configuration at $7$~TeV. In the lowest transverse momentum intervals the deviations from the data are largest for both configurations. The hadronic cascade time is varied within the SM configuration over $\tau_{\rm HC} = 0.6$, $5$, $15$, and $30$~fm/$c$. A monotonic reduction of $b_{\rm corr}$ with increasing $\tau_{\rm HC}$ is observed at all energies and for all window widths, with the ordering of the curves preserved throughout. The effect is modest but statistically significant. These results indicate that the FB correlation strength is primarily imprinted at the partonic stage and only mildly diluted by the subsequent hadronic evolution.

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	Forward-backward (FB) multiplicity correlations are studied in proton--proton collisions at centre-of-mass energies of $\sqrt{s} = 0.9$, $2.76$, $5.02$, $7$, $13$, and $13.6$~TeV using the AMPT model in its default (DF) and string melting (SM) configurations.	contribution 0.71	0.92	0.03	asserted 0.97
s1	The coefficient $b_{\rm corr}$ is obtained for windows separated in pseudorapidity and azimuth, and its variation with the pseudorapidity gap, window width, collision energy, transverse momentum interval, and hadronic cascade time $\tau_{\rm HC}$ is investigated.	method 0.78	0.94	0.03	asserted 0.89
s2	The simulated results are compared with the available ALICE measurements, and the model is validated against experimental pseudorapidity densities and charged-pion transverse momentum spectra.	method 0.89	0.87	0.03	asserted 0.99
s3	The correlation strength is found to decrease gradually with increasing pseudorapidity gap while retaining a substantial pedestal, to rise non-linearly with the window width, and to increase substantially with collision energy.	result 1.00	0.95	0.17	asserted 1.00
s4	The DF configuration yields systematically larger values of $b_{\rm corr}$ than the SM configuration, while the SM configuration provides the better description of the ALICE data at $0.9$ and $2.76$~TeV and the DF configuration at $7$~TeV.	result 1.00	0.92	0.11	asserted 1.00
s5	In the lowest transverse momentum intervals the deviations from the data are largest for both configurations.	result 0.89	0.86	0.02	asserted 1.00
s6	The hadronic cascade time is varied within the SM configuration over $\tau_{\rm HC} = 0.6$, $5$, $15$, and $30$~fm/$c$.	method 0.97	0.93	0.02	asserted 0.97
s7	A monotonic reduction of $b_{\rm corr}$ with increasing $\tau_{\rm HC}$ is observed at all energies and for all window widths, with the ordering of the curves preserved throughout.	result 1.00	0.95	0.13	asserted 1.00
s8	The effect is modest but statistically significant.	result 0.95	0.90	0.06	asserted 1.00
s9	These results indicate that the FB correlation strength is primarily imprinted at the partonic stage and only mildly diluted by the subsequent hadronic evolution.	interpretation 1.00	0.83	0.82	hedged possibility 0.97

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

What was examined, and what was not

sources read

arXiv listing abstract v1

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.** Pseudorapidity density of charged particles as a function of η in pp collisions at √ s = 0.9–13.6 TeV. Results from the AMPT default (DF) and string melting (SM) modes are compared with the corresponding experimental measurements; the references are listed in

**Figure 2.** Transverse momentum distributions of charged pions, π ±, produced in pp collisions at (a) √ s = 7 TeV and (b) 13 TeV, within the mid-rapidity interval |y| < 0.5. Theoretical predictions obtained with the AMPT default (DF) and string melting (SM) modes are presented alongside the corresponding ALICE measurements, whose references are listed in

**Figure 3.** Schematic representation of the pseudorapidity observables adopted in this study. (a) Definition of the variables δη, ηgap, and ηsep. (b) Arrangement of η-windows with varying widths δη and separation gaps ηgap

**Figure 4.** Layout of the eight azimuthal sector configurations considered in this analysis. Each forward and backward pseudorapidity window, of width δη = 0.2, is further divided into eight azimuthal sectors of width δφ = π/4. The grey sectors belong to the first window of the FB pair and the yellow sectors to the second. The quantity φsep denotes the azimuthal separation between the centres of the two sectors. – 7 –

**Figure 5.** Representative forward and backward raw multiplicity distributions for windows of width δη = 0.8 and ηgap = 0.0 at √ s = 13.6 TeV, obtained with the AMPT default mode in (a) and the string melting mode in (c). The corresponding correlation functions are shown in (b) for the default mode and (d) for the string melting mode. 2.2 Definition of counting windows Two intervals of variable width δη, ranging from 0.2 to 0.8 and placed symmetrically about η = 0, are labelled as forward (F, η > 0) and backward (B, η < 0). Correlations between the multiplicities of charged particles, n, are…

**Figure 6.** Forward-backward correlation strength bcorr as a function of the pseudorapidity gap ηgap for window widths δη = 0.2, 0.4, 0.6, and 0.8 in pp collisions at √ s = 0.9, 2.76, and 7 TeV. Results from the AMPT default (DF) and string melting (SM) modes are compared with the ALICE measurements. Filled symbols denote the ALICE data and open symbols the AMPT results, with δη = 0.2 (red), 0.4 (blue), 0.6 (green), and 0.8 (magenta). The lower panels display the corresponding ratio plots. non-linearly with δη in both the AMPT default (DF) and string melting (SM) configurations. The results…

**Figure 7.** Forward-backward correlation strength bcorr as a function of the pseudorapidity gap ηgap for window widths δη = 0.2, 0.4, 0.6, and 0.8 in pp collisions at √ s = 5.02, 13, and 13.6 TeV. Results from the AMPT default (DF) and string melting (SM) modes are shown, with δη = 0.2 (red), 0.4 (blue), 0.6 (green), and 0.8 (magenta). 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 corr b pp, s = 0.9 TeV = 0 gap η 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.6 0.8 1 1.2 1.4 AMPT / ALICE pp, s = 2.76 TeV ALICE AMPT (DF) AMPT (SM) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 δη pp, s = 7 TeV 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

**Figure 8.** Correlation strength bcorr as a function of the window width δη for a pseudorapidity gap ηgap = 0 in pp collisions at √ s = 0.9, 2.76, and 7 TeV. Results from the AMPT string melting (open triangles) and default (open circles) modes are compared with the ALICE measurements. The lower panels display the ratio of bcorr between ALICE and AMPT. correlation strengths extracted under different experimental conditions. In particular, it accounts for the small values of bcorr measured by the STAR collaboration at RHIC in pp – 11 –

**Figure 9.** Correlation strength bcorr for separated η–φ window pairs in pp collisions at √ s = 0.9 TeV as a function of the pseudorapidity separation, for fixed window widths δη = 0.2 and δφ = π/4. The panels correspond to different azimuthal separation distances between the two sectors: φsep = 0, π/4, π/2, 3π/4, and π. Results from the AMPT default (open squares) and string melting (open triangles) modes are compared with the ALICE measurements (filled circles). The lower panels display the ratio of bcorr between the ALICE and AMPT results. 0.05 0.1 0.15 0.2 corr b = 0 sep ϕ pp, s = 7 TeV δη…

**Figure 10.** Correlation strength bcorr for separated η–φ window pairs in pp collisions at √ s = 7 TeV as a function of the pseudorapidity separation, for fixed window widths δη = 0.2 and δφ = π/4. Results from the AMPT default (open squares) and string melting (open triangles) modes are compared with the ALICE measurements (filled circles). The lower panels display the ratio of bcorr between the ALICE and AMPT results. 4 Multiplicity correlations in separated η and φ windows Multiplicity correlations are also examined for different configurations of forward and backward azimuthal sectors.…

**Figure 11.** Correlation strength bcorr for separated η–φ window pairs in pp collisions at √ s = 13.6 TeV as a function of the pseudorapidity separation, for fixed window widths δη = 0.2 and δφ = π/4. Results from the AMPT default (open squares) are presented. In all cases considered, the AMPT default (DF) configuration yields higher values of bcorr than the string melting (SM) configuration. In Figures 9 and 10, the results are compared with the ALICE measurements. At √ s = 0.9 TeV, the SM configuration matches the ALICE data more closely, whereas at √ s = 7 TeV the DF configuration provides…

**Figure 12.** Correlation strength bcorr in pp collisions at √ s = 7 TeV for separated pseudorapidity window pairs, measured in pT intervals with the same ⟨nc⟩, as a function of the pseudorapidity gap ηgap. The window width is fixed at δη = 0.2. The left and right panels contain the same data points, with the lines corresponding to the AMPT default (left) and string melting (right) modes. pT intervals, around 0.3–0.4 GeV/c, where the model predictions are furthest from the measurements. This behaviour can be understood in terms of the transverse-momentum dependence of the short-range (SR) and…

**Figure 13.** Correlation strength bcorr in pp collisions at √ s = 7 TeV for separated η–ϕ windows in different pT intervals with the same ⟨nch⟩. Five values of ϕsep are shown as a function of the pseudorapidity separation ηsep, with window widths δη = 0.2 and δϕ = π/4. The upper panels contain the same ALICE experimental data as symbols, while the lines correspond to the AMPT default (DF) results. The lower panels display the corresponding ratio plots. 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 corr b = 0 sep ϕ corr b 4 π = sep ϕ δ η = 0.2 corr b 2 π = sep ϕ pp, s = 7 TeV corr b ALICE…

**Figure 14.** Correlation strength bcorr in pp collisions at √ s = 7 TeV for separated η–ϕ windows in different pT intervals with the same ⟨nch⟩. Five values of ϕsep are shown as a function of the pseudorapidity separation ηsep, with window widths δη = 0.2 and δϕ = π/4. The upper panels contain the same ALICE experimental data as symbols, while the lines correspond to the AMPT string melting(SM) results. The lower panels display the corresponding ratio plots. model’s treatment of partonic scattering and string dynamics, the agreement with the data improves. – 15 –

**Figure 15.** Correlation strength bcorr as a function of the window width δη for a pseudorapidity gap ηgap = 0 in pp collisions at √ s = 0.9 TeV (a) and 13.6 TeV (b), for different hadronic cascade times τHC = 0.6 fm/c (black), 5 fm/c (red), 15 fm/c (blue), and 30 fm/c (green). The lines joining the points are drawn to guide the eyes. 0.8 1 2 3 4 5 6 7 8 910 s (TeV) 0.45 0.5 0.55 0.6 0.65 0.7 corr b pp, AMPT (SM) = 0, δη = 0.8 gap η = 0.6 fm/c HC τ = 5 fm/c HC τ = 15 fm/c HC τ = 30 fm/c HC τ

**Figure 16.** Correlation strength bcorr as a function of the centre-of-mass energy √ s for a window width δη = 0.8 and a pseudorapidity gap ηgap = 0 in pp collisions, for different hadronic cascade times τHC = 0.6 fm/c (black), 5 fm/c (red), 15 fm/c (blue), and 30 fm/c (green).The lines joining the points are drawn to guide the eyes. – 16 –

Figures reproduced from arXiv:2610.25154 by the authors.

Discussion

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