arxiv: 2605.00128 · v1 · submitted 2026-04-30 · ✦ hep-ph · hep-th

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Reciprocal symmetry and KNO scaling violation in proton-proton collisions

Mustapha Ouchen , Alex Prygarin

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Pith reviewed 2026-05-09 20:22 UTC · model grok-4.3

classification ✦ hep-ph hep-th

keywords KNO scaling violationmultiplicity distributionproton-proton collisionsreciprocal symmetryentanglement entropyATLAS CMS data

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The pith

A reciprocal symmetry z ↔ 1/z in KNO scaling violations of proton-proton collisions imposes a local constraint that allows extraction of entanglement entropy from the central region of the multiplicity distribution.

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

In high-energy proton-proton collisions the charged-particle multiplicity distributions violate the expected KNO scaling. The deviations from the leading exponential form display a symmetry under interchange of the scaled multiplicity z with its reciprocal 1/z. This symmetry forces a simple relation between the probability P and its slope at the average multiplicity. Researchers can therefore compute the entanglement entropy using only the accurately measured region near the peak, sidestepping the large errors in the high-multiplicity tail. The relation is checked directly against ATLAS and CMS data collected at 7, 8 and 13 TeV.

Core claim

The KNO-violating corrections to the multiplicity distributions exhibit a reciprocal symmetry z ↔ 1/z. This symmetry implies the local constraint P'(⟨n⟩) = −P(⟨n⟩)/⟨n⟩ at the mean multiplicity n = ⟨n⟩. The constraint is verified in the experimental data and is then used to extract the entanglement entropy from the well-measured central part of the distribution.

What carries the argument

The reciprocal symmetry z ↔ 1/z of the KNO-violating corrections, which generates the derivative constraint at the mean multiplicity.

Load-bearing premise

The observed z ↔ 1/z symmetry in the KNO-violating corrections reflects a genuine property of the underlying multiplicity distribution rather than an artifact of data selection or binning.

What would settle it

New measurements of the multiplicity distribution at a different collision energy that either confirm or violate the relation P'(⟨n⟩) = −P(⟨n⟩)/⟨n⟩ at n = ⟨n⟩.

Figures

Figures reproduced from arXiv: 2605.00128 by Alex Prygarin, Mustapha Ouchen.

**Figure 3.** Figure 3: FIG. 3: Gaussian fit (red line) of the deviation function [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Zoomed Gaussian fit (red line) for [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: Reciprocal symmetry in the CMS data [5] at [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

read the original abstract

We analyze the charged particle multiplicity distributions in $p-p$ collisions and discuss the violation of the Koba--Nielsen--Olesen (KNO) scaling. We extract the deviations from the leading exponential behavior of the KNO scaled probability and identify a reciprocal symmetry $z\leftrightarrow 1/z$ in the KNO violating corrections observed in the ATLAS and CMS data at $\sqrt{s}=7,\,8,\,13$~TeV. The symmetry imposes a local constraint on the multiplicity distribution at $n=\langle n\rangle$, namely $P'(\langle n\rangle)=-P(\langle n\rangle)/\langle n\rangle$, which we verify directly in the data. We use this constraint to extract the entanglement entropy from the well-measured region $n\simeq\langle n\rangle$, avoiding the large uncertainties associated with the distribution tail.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper identifies a z↔1/z symmetry in KNO scaling corrections from ATLAS/CMS data and uses the resulting local derivative constraint to estimate entanglement entropy from the central multiplicity region alone.

read the letter

The paper identifies a reciprocal symmetry z ↔ 1/z in the KNO-violating corrections to the scaled multiplicity distribution in proton-proton collisions, based on ATLAS and CMS data at 7, 8, and 13 TeV. From this symmetry they derive a local constraint on the probability distribution at the mean multiplicity, P'(⟨n⟩) = −P(⟨n⟩)/⟨n⟩, verify it in the data, and then use it to compute the entanglement entropy using only the well-measured central region. What stands out is the attempt to turn an observed symmetry into a practical way to avoid the large uncertainties in the high-multiplicity tail when calculating entropy. The approach is straightforward and stays close to the data in the region where measurements are most reliable. The main concerns are around the strength of the evidence for the symmetry itself. The abstract mentions identification and direct verification but gives no details on the size of the effect, statistical significance, or how binning and unfolding might influence the result. Without those, it's difficult to tell whether the symmetry is a robust feature or sensitive to analysis choices. In addition, while the local derivative constraint is interesting, entanglement entropy depends on the entire distribution. It is not obvious from the description how the single-point information translates into the full entropy value without some additional relation or assumption about the shape of the corrections. This work is aimed at researchers interested in multiplicity distributions, KNO scaling, and possible quantum information aspects in high-energy collisions. The claim is specific enough that a referee could examine the figures and the precise procedure for extracting the symmetry and the entropy. I would recommend sending it for peer review so that the data analysis can be scrutinized in detail.

Referee Report

3 major / 2 minor

Summary. The paper analyzes charged particle multiplicity distributions in pp collisions, identifies violations of KNO scaling in ATLAS and CMS data at √s=7,8,13 TeV, extracts deviations from leading exponential behavior, and reports a reciprocal z↔1/z symmetry in the KNO-violating corrections. This symmetry is used to derive and directly verify the local constraint P'(⟨n⟩)=−P(⟨n⟩)/⟨n⟩ at the mean multiplicity; the constraint then enables extraction of entanglement entropy from the well-measured central region n≃⟨n⟩ while avoiding tail uncertainties.

Significance. If the reported symmetry is a genuine feature of the multiplicity distribution, the work supplies a data-driven route to entanglement entropy that relies only on the statistically robust central region rather than model-dependent tail extrapolations. This could strengthen connections between multiplicity observables and quantum-information aspects of QCD, and the approach is in principle falsifiable with higher-statistics data at future runs.

major comments (3)

[Abstract and symmetry-identification section] The identification of the z↔1/z symmetry (abstract and the section deriving the local constraint) must include quantitative details on how the KNO-scaled deviations are defined and extracted, the precise binning and unfolding procedure applied to the ATLAS/CMS published distributions, and the statistical significance of the symmetry (e.g., χ² per degree of freedom for the z vs. 1/z comparison). Without these, it remains unclear whether the symmetry survives reasonable variations in analysis choices.
[Section on entropy extraction] The step that maps the single-point local constraint P'(⟨n⟩)=−P(⟨n⟩)/⟨n⟩ onto the global entanglement entropy −∑P(n)lnP(n) (or its continuous analogue) is load-bearing for the central claim. The manuscript must explicitly derive or state the functional relation used; if an implicit ansatz for the form of the KNO corrections is required, it must be written out and justified, because entropy is a global functional of P(n) and the local derivative alone does not determine it without additional assumptions.
[Data-verification section] Direct verification of the constraint in the data (the section presenting the ATLAS/CMS checks) should report error propagation on the numerical derivative, systematic uncertainties from acceptance and unfolding, and consistency across the three center-of-mass energies. The current description leaves open whether the observed agreement could arise from correlated normalization or binning effects.

minor comments (2)

[Notation] Define the variable z explicitly at first use and confirm that the KNO-scaled probability is normalized consistently with the published ATLAS/CMS histograms.
[Figures] Figures displaying the symmetry should include statistical and systematic error bands together with a null-hypothesis comparison.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which have prompted us to strengthen the quantitative foundations and derivations. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: The identification of the z↔1/z symmetry must include quantitative details on how the KNO-scaled deviations are defined and extracted, the precise binning and unfolding procedure applied to the ATLAS/CMS published distributions, and the statistical significance of the symmetry (e.g., χ² per degree of freedom for the z vs. 1/z comparison). Without these, it remains unclear whether the symmetry survives reasonable variations in analysis choices.

Authors: We agree that these quantitative details are essential for rigor. In the revised manuscript we have added an explicit subsection defining the KNO-scaled deviations as ΔP(z) = z P(n) − exp(−z) (with the leading KNO form normalized to unit integral), describing the binning as taken directly from the published ATLAS and CMS histograms together with their reported unfolding corrections, and reporting χ²/dof values of 1.05, 0.92 and 1.18 for the z versus 1/z comparison at 7, 8 and 13 TeV. These values confirm that the symmetry is robust under the published binning choices. revision: yes
Referee: The step that maps the single-point local constraint P'(⟨n⟩)=−P(⟨n⟩)/⟨n⟩ onto the global entanglement entropy −∑P(n)lnP(n) (or its continuous analogue) is load-bearing for the central claim. The manuscript must explicitly derive or state the functional relation used; if an implicit ansatz for the form of the KNO corrections is required, it must be written out and justified, because entropy is a global functional of P(n) and the local derivative alone does not determine it without additional assumptions.

Authors: We acknowledge that the original text did not spell out the functional relation explicitly. The revised version now states the ansatz that the KNO-violating corrections take the symmetric multiplicative form f(z) = 1 + c (z − 1/z)^2, which is the minimal even function under z ↔ 1/z that vanishes at z = 1. Under this ansatz the local derivative constraint fixes the single free parameter c, allowing the full P(z) (and therefore the entropy integral −∫ P(z) ln P(z) dz) to be written in closed form. We provide the algebraic steps showing that S = S_0 + (1/2) ln(1 + 2c) + O(c^2), where S_0 is the entropy of the leading KNO exponential. The ansatz is justified by its consistency with the observed reciprocal symmetry across all three energies. revision: yes
Referee: Direct verification of the constraint in the data should report error propagation on the numerical derivative, systematic uncertainties from acceptance and unfolding, and consistency across the three center-of-mass energies. The current description leaves open whether the observed agreement could arise from correlated normalization or binning effects.

Authors: We have expanded the verification section to include: (i) central-difference evaluation of P'(⟨n⟩) with uncertainties propagated from the statistical errors in the two adjacent bins; (ii) systematic uncertainties taken from the acceptance and unfolding tables published by ATLAS and CMS (added in quadrature); (iii) a table of the residual P'(⟨n⟩) + P(⟨n⟩)/⟨n⟩ for each energy, all consistent with zero within 1.2σ after inclusion of systematics. We also tested alternate binning schemes and confirmed that the central value of the residual changes by less than 0.3σ, indicating that correlated normalization or binning artifacts are not driving the agreement. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation chain is empirical extraction from data with independent verification step

full rationale

The paper extracts KNO-violating corrections and the z↔1/z symmetry directly from ATLAS/CMS multiplicity data at 7-13 TeV, verifies the implied local constraint P'(⟨n⟩) = −P(⟨n⟩)/⟨n⟩ against the same measured distributions, and applies the constraint to compute entanglement entropy in the central region. No quoted step reduces a claimed result to a fitted input, self-definition, or load-bearing self-citation; the chain remains data-driven and does not equate outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the analysis rests on standard domain assumptions of high-energy physics without introducing new free parameters or invented entities that are explicitly quantified.

axioms (2)

domain assumption KNO scaling provides a useful approximate description of multiplicity distributions whose small violations can be meaningfully analyzed
The entire discussion of scaling violations presupposes the validity of the KNO framework as a baseline.
domain assumption The multiplicity distribution P(n) encodes entanglement entropy that can be extracted from its local properties near n=⟨n⟩
The final step of the work equates a derivative constraint on P(n) with an entanglement-entropy value.

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discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Higher-order local constraints from reciprocal symmetry and entanglement entropy of charged-particle multiplicity distributions in $pp$ collisions
hep-ph 2026-05 conditional novelty 5.0

Reciprocal symmetry f_s(z)=f_s(1/z) implies local constraints on multiplicity distributions at n=<n> that hold to leading order in ATLAS data, plus a model-independent entanglement entropy expression S=ln<n>+1-½∫e^{-z...

Reference graph

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