arxiv: 2604.17418 · v1 · submitted 2026-04-19 · ✦ hep-ph

Recognition: unknown

Entropy and mean multiplicity from dipole models in the high energy limit

Krzysztof Kutak , S\'andor L\"ok\"os

Authors on Pith no claims yet

Pith reviewed 2026-05-10 05:56 UTC · model grok-4.3

classification ✦ hep-ph

keywords dipole modelentropymultiplicityhigh energy limitproton-proton collisionsMueller modelQCD

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The pith

The generalized dipole model describes entropy and mean multiplicity in high-energy proton-proton collisions more accurately than the 1D Mueller model.

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

This paper studies the 1D Mueller dipole model, its high-energy limit, and a generalization of it as tools for describing particle production in QCD. To sidestep ambiguities that arise when experiments choose different pseudorapidity windows, the authors treat the entropy plotted against the logarithm of average multiplicity as a universal observable that can be extracted directly from measured multiplicity distributions. They solve the models for both entropy and average charged-particle multiplicity, fit the resulting curves to proton-proton data, and find that the generalized version reproduces the measurements substantially better than the basic 1D Mueller model. A reader should care because the approach supplies a practical, cut-independent way to test and refine models of high-energy particle production. The work thereby strengthens the link between dipole-based pictures of QCD evolution and real collision data.

Core claim

By solving the generalized dipole model in the high-energy limit, the authors obtain the entropy S(ln ⟨n⟩) and the mean multiplicity ⟨n⟩ that, when compared to charged-particle multiplicity distributions measured in proton-proton collisions, describe the data significantly better than the corresponding quantities extracted from the 1D Mueller dipole model.

What carries the argument

The generalized dipole model (an extension of the 1D Mueller dipole model) whose solutions are used to compute the entropy S(ln ⟨n⟩) as the central observable for data comparison.

If this is right

Entropy extracted as S(ln ⟨n⟩) can be read directly from any measured multiplicity distribution and used as a model benchmark.
Model parameters are fixed by fitting the computed S(ln ⟨n⟩) and ⟨n⟩ to existing proton-proton data.
The high-energy limit of the dipole equations supplies concrete, calculable predictions for both entropy and multiplicity.
The improved agreement supports the idea that the generalized model incorporates more of the relevant high-energy QCD dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same S(ln ⟨n⟩) construction could be applied to other collision systems or energies without redefining rapidity intervals.
The method offers a route to constrain dipole-model parameters more tightly before they are used in event generators.
Extending the comparison to nuclear collisions would test whether the same generalized dynamics hold in denser environments.

Load-bearing premise

The entropy S(ln ⟨n⟩) remains the same regardless of the particular pseudorapidity range chosen in the experimental measurements.

What would settle it

New multiplicity distributions measured in the same collision system but with markedly different pseudorapidity cuts would produce S(ln ⟨n⟩) curves that deviate from those obtained in the original data sets, or the generalized model's fitted curves would fail to describe fresh high-energy data.

Figures

Figures reproduced from arXiv: 2604.17418 by Krzysztof Kutak, S\'andor L\"ok\"os.

**Figure 2.** Figure 2: The new observable compared to data 4. Summary and outlook We described the 1D Mueller dipole model and its generalized version. We introduced a new observable, S(ln⟨n⟩), that has been compared to pp data spanning 2 orders of magnitude in energy and covers a large pseudorapidity interval. We found that the generalized model can describe the data while the Mueller model deviates from the data points at low… view at source ↗

read the original abstract

The 1D Mueller dipole model, its high energy limit, and its generalization were investigated. To address the ambiguity stemming from different definitions of the pseudorapidity ranges in experimental measurements, we propose the entropy as the function of the logarithm of the average multiplicity, $S(\ln\langle n\rangle$, as a universal observable. From the solutions of the models, we calculate both the entropy and the average charged particle multiplicity and compare to data measured in proton-proton collisions. We obtained these quantities directly from the measured charged particle multiplicity distributions and determine the model parameters via fits. We find that the generalized dipole model provides a significantly better description of the data than the 1D Mueller model.

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 suggests S(ln ⟨n⟩) as a way around pseudorapidity ambiguities and claims the generalized dipole model fits multiplicity data better, but the fits are to the same data and the universality claim lacks a cross-check on different rapidity windows.

read the letter

The main point is that they use entropy plotted against the log of average multiplicity to compare models in a way that should be less sensitive to how experiments slice their pseudorapidity. They compute S and ⟨n⟩ both from the measured charged particle multiplicity distributions in pp collisions and from the solutions of the 1D Mueller dipole model and its generalization. After fitting parameters, the generalized version matches the data noticeably better.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the 1D Mueller dipole model, its high-energy limit, and a proposed generalization. To circumvent ambiguities arising from different experimental definitions of pseudorapidity ranges, the authors introduce the entropy S as a function of the logarithm of the mean multiplicity, S(ln ⟨n⟩), as a universal observable. They extract S and ⟨n⟩ both from measured charged-particle multiplicity distributions P(n) in proton-proton collisions and from solutions of the dipole models, fit the model parameters to these quantities, and conclude that the generalized dipole model yields a significantly better description of the data than the original 1D Mueller model.

Significance. If the universality of S(ln ⟨n⟩) can be established and the fitting procedure shown to be non-circular, the work would supply a practical, range-independent metric for discriminating among dipole-based models of high-energy multiplicity distributions. The reported superiority of the generalized model would then constitute a concrete, testable improvement in the phenomenological description of soft QCD processes.

major comments (2)

[the section presenting the comparison to data and the definition of the universal observable] The central claim that the generalized dipole model provides a significantly better description rests on treating S(ln ⟨n⟩) as universal and independent of the specific pseudorapidity window. No explicit test is shown that the functional form or normalization of S(ln ⟨n⟩) remains invariant when the same underlying data or model output is restricted to different η intervals (e.g., |η|<2.5 versus |η|<1.0). This validation is load-bearing for the model comparison and must be supplied.
[the section describing the fitting procedure and results] Model parameters are determined by fits to the same multiplicity distributions P(n) that are subsequently used to assess the quality of the description. This procedure introduces circularity: the reported improvement may simply reflect the additional flexibility of the generalized model rather than genuine predictive power. Cross-validation on independent data sets or explicit reporting of fit uncertainties and goodness-of-fit metrics (including error bars on the extracted S and ⟨n⟩) is required to substantiate the claim.

minor comments (2)

The abstract states that quantities are obtained 'directly from the measured charged particle multiplicity distributions' but provides no details on binning, acceptance corrections, or statistical uncertainties; these should be clarified in the main text.
Notation for the entropy function is introduced as S(ln ⟨n⟩) in the abstract; consistency of this notation throughout the manuscript (including any figures) should be checked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

Referee: [the section presenting the comparison to data and the definition of the universal observable] The central claim that the generalized dipole model provides a significantly better description rests on treating S(ln ⟨n⟩) as universal and independent of the specific pseudorapidity window. No explicit test is shown that the functional form or normalization of S(ln ⟨n⟩) remains invariant when the same underlying data or model output is restricted to different η intervals (e.g., |η|<2.5 versus |η|<1.0). This validation is load-bearing for the model comparison and must be supplied.

Authors: We agree that an explicit test of the invariance of S(ln ⟨n⟩) under different pseudorapidity cuts is necessary to support the claim of universality. In the revised manuscript we will add a new figure and accompanying text that extracts S(ln ⟨n⟩) from both the experimental multiplicity distributions and the model solutions for several distinct η intervals (including |η|<2.5 and |η|<1.0) and demonstrates that the functional form and normalization remain consistent within the reported uncertainties. This addition directly addresses the referee's concern. revision: yes
Referee: [the section describing the fitting procedure and results] Model parameters are determined by fits to the same multiplicity distributions P(n) that are subsequently used to assess the quality of the description. This procedure introduces circularity: the reported improvement may simply reflect the additional flexibility of the generalized model rather than genuine predictive power. Cross-validation on independent data sets or explicit reporting of fit uncertainties and goodness-of-fit metrics (including error bars on the extracted S and ⟨n⟩) is required to substantiate the claim.

Authors: We acknowledge that fitting to quantities derived from the same P(n) distributions used for the final comparison raises a legitimate concern about circularity. In the revision we will report the full covariance matrices, uncertainties on the extracted S and ⟨n⟩ values, and standard goodness-of-fit measures (χ²/dof) for both models. We will also perform a limited cross-validation by holding out data from one collision energy and refitting on the remainder. At the same time we note that the models are not purely phenomenological; their functional forms are fixed by the dipole evolution equations, so the improvement is not due to unrestricted flexibility but to the specific generalization introduced. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper proposes S(ln ⟨n⟩) as a universal observable to address pseudorapidity ambiguities, extracts S and ⟨n⟩ directly from measured P(n) distributions, fits model parameters to these quantities, and compares the resulting description quality between the generalized dipole model and the 1D Mueller model. This is a standard fitting-based model comparison with direct calculation of observables from model solutions; no step reduces by construction to its inputs (e.g., no fitted quantity renamed as independent prediction, no self-definitional loop, and no load-bearing self-citation chain). The universality claim is presented as a proposal without circular self-justification, and the central result remains an empirical fit-quality statement independent of the inputs by tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit list of assumptions or parameters; typical dipole models rely on standard QCD evolution equations and fitted initial conditions whose details are not visible here.

free parameters (1)

model parameters
Fitted to multiplicity distributions to achieve the reported better description.

axioms (1)

domain assumption Dipole models correctly capture high-energy QCD dynamics in the given limit
Invoked to justify solving the models for entropy and multiplicity predictions.

pith-pipeline@v0.9.0 · 5411 in / 1213 out tokens · 49518 ms · 2026-05-10T05:56:00.499154+00:00 · methodology

discussion (0)

Reference graph

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