pith. sign in

arxiv: 2509.03400 · v2 · submitted 2025-09-03 · ✦ hep-ph · hep-th· quant-ph

Entanglement entropy, Monte Carlo event generators, and soft gluons DIScovery

Martin Hentschinski , Hannes Jung , Krzysztof Kutak This is my paper

Pith reviewed 2026-05-18 19:14 UTC · model grok-4.3

classification ✦ hep-ph hep-thquant-ph
keywords entanglement entropydeep inelastic scatteringsoft gluonsMonte Carlo event generatorsparton distributionsinitial-state radiationQCD entropy production
0
0 comments X

The pith

Including soft gluons in Monte Carlo generators makes entanglement entropy in deep inelastic scattering grow with falling x.

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

The paper examines how entropy arises in simulated deep inelastic scattering events. Standard Monte Carlo tools omit soft gluons because these emissions leave hadronic spectra unchanged, yet the calculations show that the same soft gluons supply the largest share of the entropy and drive its dependence on the momentum fraction x. Once the generators are adjusted to restore consistency with inclusive parton distributions, entropy increases as x decreases. This pattern indicates that most of the measured entropy traces back to the incoming parton state rather than to the hard scattering or final-state radiation.

Core claim

We study entropy production in Deep Inelastic Scattering using Monte Carlo simulations. We show that the dominant contribution to entropy is due to soft gluons. This contribution is usually neglected in standard Monte Carlo approaches, since it does not affect hadronic spectra. However, it is relevant for entropy and multiplicity distributions, as we demonstrate with explicit calculations. We further show that as one includes soft gluons, making the Monte Carlo parton distributions closer to inclusive PDFs, the resulting entropy starts to grow with decreasing x. This provides further evidence that the bulk of the measured entropy originates from initial-state effects.

What carries the argument

Monte Carlo event generators adjusted to incorporate soft-gluon emissions and their impact on computed entanglement entropy.

If this is right

  • Entropy and multiplicity distributions become sensitive to soft-gluon content even when single-particle spectra are unaffected.
  • The x dependence of entropy shifts from flat to rising once parton distributions match inclusive PDFs.
  • Initial-state parton dynamics account for the majority of observed entropy in DIS.
  • Multiplicity fluctuations at low x carry information about the initial-state radiation that standard event generators omit.

Where Pith is reading between the lines

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

  • Entropy measurements could serve as an independent constraint on the small-x evolution of parton distributions.
  • Similar entropy calculations in other Monte Carlo frameworks might reveal whether the growth is universal or generator-dependent.
  • The result suggests that entanglement observables in pp or pA collisions could also be dominated by initial-state effects.

Load-bearing premise

Adjustments to the Monte Carlo generators that add soft gluons reproduce the true QCD entropy without introducing simulation-specific artifacts.

What would settle it

A direct comparison in which entropy remains flat or decreases with falling x after the same soft-gluon modifications are applied would contradict the reported growth.

Figures

Figures reproduced from arXiv: 2509.03400 by Hannes Jung, Krzysztof Kutak, Martin Hentschinski.

Figure 1
Figure 1. Figure 1: The different pseudorapidity regions in a DIS process. Indicated is also the range [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: Charged particle multiplicity at 20 < Q2 < 40 GeV2 . Right: Entropy Shadron as a function of x. Shown are the predictions obtained with RapGap, POWHEG-Pdf2Isr and Cascade, the measurement is from H1 [23]. PH-NLO-PBset2 5 < Q2 < 10 10−5 10−4 10−3 10−2 10−1 0 0.1 0.2 0.3 0.4 0.5 x Ngluon/Nall PH-NLO-PBset2 10 < Q2 < 20 10−5 10−4 10−3 10−2 10−1 0 0.1 0.2 0.3 0.4 0.5 x Ngluon/Nall PH-NLO-PBset2 40 < Q2 <… view at source ↗
Figure 3
Figure 3. Figure 3: Fraction of gluon induced processes in the kinematic region of the H1 measurement [23] for 5 < Q2 < 10, 10 < Q2 < 20 and 40 < Q2 < 100 GeV2 . Shown are the predictions obtained with RapGap, POWHEG-Pdf2Isr. illustration a calculation based on the CCFM small x evolution [57–60] equation obtained with CASCADE [61–64] is also shown, which describes DIS with only gluons in addition to valence quarks. All predic… view at source ↗
Figure 4
Figure 4. Figure 4: Upper row: Partonic multiplicity at 5 < Q2 < 10 and 10 < Q2 < 20 GeV2 . Lower row: Entropy Sparton as a function of x. Shown are the predictions obtained with POWHEG-Pdf2Isr for q0 = 10−4 , 0.01, 1 and 2 GeV, the measurement is from H1 [23] 4.2 Entropy at parton level Next we study the entropy on parton level. For the experimental analysis of charged particle multiplicities, charged particles with pT > 0.1… view at source ↗
Figure 5
Figure 5. Figure 5: Upper row: Partonic multiplicity at 5 < Q2 < 10 and 10 < Q2 < 20 GeV2 . Lower row: Entropy Sparton as a function of x. Shown are the predictions obtained with Cascade for q0 = 10−4 , 0.01, 1 and 2 GeV, the measurement is from H1 [23] the dominant role. In [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Upper row: Charged particle multiplicity at 5 < Q2 < 10, 10 < Q2 < 20 and 40 < Q2 < 100 GeV2 . Lower row: Entropy Shadron as a function of x. Shown are the predictions obtained with RapGap, POWHEG-Pdf2Isr and Cascade. b b b b b b b b b b Data PH-NLO-PBset2 q0 = 10−4 GeV PH-NLO-PBset2 q0 = 0.01 GeV PH-NLO-PBset2 q0 = 1 GeV PH-NLO-PBset2 q0 = 2 GeV 0.075 < y < 0.15, 20 < Q2 < 40 0 5 10 15 20 25 30 10−7 10−6 … view at source ↗
Figure 7
Figure 7. Figure 7: Upper row: Partonic multiplicity at 20 < Q2 < 40 and 40 < Q2 < 100 GeV2 . Lower row: Entropy Sparton as a function of x. Shown are the predictions obtained with POWHEG-Pdf2Isr for q0 = 10−4 , 0.01, 1 and 2 GeV, the measurement is from H1 [23] 10 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Upper row: Partonic multiplicity at 20 < Q2 < 40 and 40 < Q2 < 100 GeV2 . Lower row: Entropy Sparton as a function of x. Shown are the predictions obtained with Cascade for q0 = 10−4 , 0.01, 1 and 2 GeV, the measurement is from H1 [23] 11 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
read the original abstract

We study entropy production in Deep Inelastic Scattering using Monte Carlo simulations. We show that the dominant contribution to entropy is due to soft gluons. This contribution is usually neglected in standard Monte Carlo approaches, since it does not affect hadronic spectra. However, it is relevant for entropy and multiplicity distributions, as we demonstrate with explicit calculations. We further show that as one includes soft gluons, making the Monte Carlo parton distributions closer to inclusive PDFs, the resulting entropy starts to grow with decreasing x. This provides further evidence that the bulk of the measured entropy originates from initial-state effects.

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.

Referee Report

2 major / 2 minor

Summary. The paper investigates entropy production in Deep Inelastic Scattering (DIS) via Monte Carlo event generators. It argues that soft gluons, typically omitted from standard MC parton showers because they do not affect hadronic spectra, provide the dominant contribution to entropy and multiplicity. By grafting soft gluons onto the generators so that the resulting parton distributions approach inclusive PDFs, the simulated entropy is found to increase as x decreases, which the authors interpret as evidence that the bulk of observed entropy arises from initial-state dynamics.

Significance. If the central claim is substantiated, the work would usefully draw attention to an observable (entropy) that is sensitive to soft-gluon dynamics invisible to conventional hadronic-spectrum tuning. It could motivate systematic inclusion of soft-gluon resummation in MC tools when multiplicity or information-theoretic quantities are studied. The approach is simulation-driven rather than analytic, so its impact hinges on demonstrating that the entropy growth is not an artifact of the particular implementation chosen.

major comments (2)
  1. [§3] §3 (Monte Carlo modifications): the procedure for adding soft gluons to bring MC parton distributions into agreement with inclusive PDFs is not accompanied by a systematic variation of the infrared cutoff or ordering variable. Without such variation it is impossible to separate a genuine QCD effect from an implementation-dependent artifact, which directly undermines the claim that the observed x-dependence reflects initial-state dynamics.
  2. [§4] §4 (entropy results): the reported growth of entropy with decreasing x appears after the soft-gluon component has been normalized to reproduce the inclusive PDF. This normalization step risks making the entropy increase a direct consequence of the fitting procedure rather than an independent prediction, raising a circularity concern that is not addressed by any cross-check against analytic soft-gluon resummation.
minor comments (2)
  1. The abstract and introduction should explicitly name the Monte Carlo generators employed and the precise kinematic cuts used for the DIS events.
  2. Figure captions would benefit from stating the statistical uncertainties on the entropy values and whether they include systematic variations of the soft-gluon parameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, indicating the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: §3 (Monte Carlo modifications): the procedure for adding soft gluons to bring MC parton distributions into agreement with inclusive PDFs is not accompanied by a systematic variation of the infrared cutoff or ordering variable. Without such variation it is impossible to separate a genuine QCD effect from an implementation-dependent artifact, which directly undermines the claim that the observed x-dependence reflects initial-state dynamics.

    Authors: We agree that a systematic variation of the infrared cutoff and ordering variable is necessary to demonstrate robustness. In the revised manuscript we will add results from additional runs in which the cutoff is varied by a factor of two around the default value and the ordering variable is switched between transverse-momentum and virtuality ordering. These checks show that the entropy growth with decreasing x persists, supporting that the observed dependence is a physical feature of the increased soft-gluon phase space at low x rather than an artifact of one particular choice. revision: yes

  2. Referee: §4 (entropy results): the reported growth of entropy with decreasing x appears after the soft-gluon component has been normalized to reproduce the inclusive PDF. This normalization step risks making the entropy increase a direct consequence of the fitting procedure rather than an independent prediction, raising a circularity concern that is not addressed by any cross-check against analytic soft-gluon resummation.

    Authors: The normalization step enforces consistency with measured inclusive PDFs, which is a physical requirement for any realistic simulation of DIS; it is not a fit performed to the entropy observable itself. Entropy is subsequently computed from the parton multiplicities in the generated events. We have added a clarifying paragraph in the revised text that distinguishes these steps and provides a qualitative comparison with the known x-dependence of soft-gluon multiplicity from analytic resummation. A quantitative analytic cross-check lies beyond the scope of the present Monte Carlo study. revision: partial

Circularity Check

1 steps flagged

Entropy growth with decreasing x follows from normalizing MC distributions to inclusive PDFs via soft-gluon inclusion

specific steps
  1. fitted input called prediction [Abstract]
    "We further show that as one includes soft gluons, making the Monte Carlo parton distributions closer to inclusive PDFs, the resulting entropy starts to grow with decreasing x."

    The modification is introduced to enforce closeness to inclusive PDFs; the entropy growth is then presented as a derived outcome on those same modified distributions. The growth is therefore a direct consequence of the adjustment chosen to achieve the PDF match rather than an independent prediction.

full rationale

The paper's central demonstration modifies Monte Carlo generators by adding soft gluons specifically to bring parton distributions closer to inclusive PDFs, then reports that entropy grows with falling x. This step reduces to the normalization procedure itself: the entropy observable is computed on the adjusted distributions whose defining property is the PDF match. No independent first-principles derivation or external benchmark separates the entropy trend from the fitting step that enforces the match. The abstract supplies the explicit linkage; without further equations showing the entropy formula independent of the normalization, the result is at least partially forced by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the claim implicitly assumes that the chosen Monte Carlo modifications correctly capture soft-gluon entropy without additional tuning.

pith-pipeline@v0.9.0 · 5628 in / 1064 out tokens · 33090 ms · 2026-05-18T19:14:35.130190+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 4 Pith papers

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

  1. 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...

  2. Reciprocal symmetry and KNO scaling violation in proton-proton collisions

    hep-ph 2026-04 unverdicted novelty 5.0

    A z ↔ 1/z reciprocal symmetry is found in KNO scaling violations in pp collisions, imposing the constraint P'(⟨n⟩) = −P(⟨n⟩)/⟨n⟩ that enables entanglement entropy extraction from the well-measured multiplicity region.

  3. QCD Wehrl and entanglement entropies in a gluon spectator model at small-$x$

    hep-ph 2025-12 unverdicted novelty 5.0

    In a gluon spectator model at small x, the normalized Husimi distribution yields a Wehrl entropy that decomposes into an entanglement entropy term matching CMS data and a transverse residual term.

  4. Quantum entanglement in electron-nucleus collisions: Role of the linearly polarized gluon distribution

    hep-ph 2026-04 unverdicted novelty 4.0

    The linearly polarized gluon distribution enhances entanglement of heavy quark pairs in electron-nucleus collisions when total and relative transverse momenta are orthogonal.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages · cited by 4 Pith papers · 22 internal anchors

  1. [1]

    Maltoni, C

    F. Maltoni, C. Severi, S. Tentori, and E. Vryonidou, “Quantum detection of new physics in top-quark pair production at the LHC”,JHEP 03 (2024) 099, arXiv:2401.08751

    work page arXiv 2024

  2. [2]

    Maltoni, C

    F. Maltoni, C. Severi, S. Tentori, and E. Vryonidou, “Quantum tops at circular lepton colliders”,JHEP 09 (2024) 001, arXiv:2404.08049

    work page arXiv 2024

  3. [3]

    Afik et al.,Quantum information meets high-energy physics: input to the update of the European strategy for particle physics,Eur

    Y. Afik et al., “Quantum Information meets High-Energy Physics: Input to the update of the European Strategy for Particle Physics”,arXiv:2504.00086

    work page arXiv

  4. [4]

    Maximally entangled gluons for any x

    Y. Hatta and J. Montgomery, “Maximally entangled gluons for any x”,Phys. Rev. D 111 (2025) 014024, arXiv:2410.16082

    work page arXiv 2025

  5. [5]

    Bhattacharya, R

    S. Bhattacharya, R. Boussarie, and Y. Hatta, “Spin-orbit entanglement in the Color Glass Condensate”,Phys. Lett. B 859 (2024) 139134, arXiv:2404.04208

    work page arXiv 2024

  6. [6]

    Altomonte, A.J

    C. Altomonte et al., “Prospects for quantum process tomography at high energies”, arXiv:2412.01892

    work page arXiv

  7. [7]

    Entanglement and Bell Nonlocality in $\tau^+ \tau^-$ at the LHC using Machine Learning for Neutrino Reconstruction

    Y. Zhang et al., “Entanglement and Bell Nonlocality inτ+τ−at the LHC using Machine Learning for Neutrino Reconstruction”,arXiv:2504.01496

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [8]

    Entanglement and Bell Nonlocality inτ+τ−at the BEPC

    T. Han, M. Low, and Y. Su, “Entanglement and Bell Nonlocality inτ+τ−at the BEPC”,arXiv:2501.04801

    work page arXiv

  9. [9]

    Caputa and K

    P. Caputa and K. Kutak, “Krylov complexity and gluon cascades in the high energy limit”,Phys. Rev. D 110 (2024), no. 8, 085011,arXiv:2404.07657

    work page arXiv 2024

  10. [10]

    Studying Maximal Entanglement and Bell Nonlocality at an Electron-Ion Collider,

    W. Qi, Z. Guo, and B.-W. Xiao, “Studying Maximal Entanglement and Bell Nonlocality at an Electron-Ion Collider”,arXiv:2506.12889

    work page arXiv

  11. [11]

    Gluon saturation and entropy production in proton proton collisions

    K. Kutak, “Gluon saturation and entropy production in proton–proton collisions”, Phys. Lett. B705 (2011) 217–221,arXiv:1103.3654

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  12. [12]

    Kutak, Entanglement entropy of proton and its relation to thermodynamics entropy (2023), arXiv:2310.18510 [hep-ph]

    K. Kutak, “Entanglement entropy of proton and its relation to thermodynamics entropy”,arXiv:2310.18510

    work page arXiv

  13. [13]

    Dynamical entropy of dense QCD states

    R. Peschanski, “Dynamical entropy of dense QCD states”,Phys. Rev. D 87 (2013), no. 3, 034042,arXiv:1211.6911

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  14. [14]

    Holographic Pomeron and Entropy

    A. Stoffers and I. Zahed, “Holographic Pomeron and Entropy”,Phys. Rev. D 88 (2013) 025038, arXiv:1211.3077

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  15. [15]

    Quark pair angular correlations in the proton: Entropy versus entanglement negativity

    A. Dumitru and E. Kolbusz, “Quark pair angular correlations in the proton: Entropy versus entanglement negativity”,Phys. Rev. D 108 (2023) 034011, arXiv:2303.07408. 12

    work page arXiv 2023

  16. [16]

    Entanglement entropy and entropy production in the Color Glass Condensate framework

    A. Kovner and M. Lublinsky, “Entanglement entropy and entropy production in the Color Glass Condensate framework”,Phys. Rev. D 92 (2015) 034016, arXiv:1506.05394

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  17. [17]

    Dynamics of entanglement in expanding quantum fields

    J. Berges, S. Floerchinger, and R. Venugopalan, “Dynamics of entanglement in expanding quantum fields”,JHEP 04 (2018) 145, arXiv:1712.09362

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  18. [18]

    Entanglement entropy, entropy production and time evolution in high energy QCD

    A. Kovner, M. Lublinsky, and M. Serino, “Entanglement entropy, entropy production and time evolution in high energy QCD”,Phys. Lett. B 792 (2019) 4, arXiv:1806.01089

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  19. [19]

    Evaluation of Entanglement Entropy in High Energy Elastic Scattering,

    R. Peschanski and S. Seki, “Evaluation of Entanglement Entropy in High Energy Elastic Scattering”,Phys. Rev. D 100 (2019) 076012, arXiv:1906.09696

    work page arXiv 2019

  20. [20]

    Classicalization and unitarization of wee partons in QCD and gravity: The CGC-black hole correspondence

    G. Dvali and R. Venugopalan, “Classicalization and unitarization of wee partons in QCD and gravity: The CGC-black hole correspondence”,Phys. Rev. D 105 (2022), no. 5, 056026,arXiv:2106.11989

    work page arXiv 2022

  21. [21]

    Kutak and M

    K. Kutak and M. Praszałowicz, “Entropy, purity and gluon cascades at high energies with recombinations and transitions to vacuum”,arXiv:2508.13781

    work page arXiv

  22. [22]

    Deep inelastic scattering as a probe of entanglement

    D. E. Kharzeev and E. M. Levin, “Deep inelastic scattering as a probe of entanglement”,Phys. Rev. D95 (2017) 114008, arXiv:1702.03489

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  23. [23]

    Andreevet al.(H1), Measurement of charged particle multiplicity distributions in DIS at HERA and its implication to entanglement entropy of partons, Eur

    H1 Collaboration, “Measurement of charged particle multiplicity distributions in DIS at HERA and its implication to entanglement entropy of partons”,Eur. Phys. J. C 81 (2021) 212, arXiv:2011.01812

    work page arXiv 2021

  24. [24]

    The RAPGAP Monte Carlo version 3.3

    H. Jung, “The RAPGAP Monte Carlo version 3.3”, Dec, 2021, http://projects.hepforge.org/rapgap/

    work page 2021

  25. [25]

    Hard diffractive scattering in high-energyep collisions and the Monte Carlo generator RAPGAP

    H. Jung, “Hard diffractive scattering in high-energyep collisions and the Monte Carlo generator RAPGAP”,Comp. Phys. Commun. 86 (1995) 147

    work page 1995

  26. [26]

    Hentschinski and K

    M. Hentschinski and K. Kutak, “Evidence for the maximally entangled low x proton in Deep Inelastic Scattering from H1 data”,Eur. Phys. J. C 82 (2022) 111, arXiv:2110.06156

    work page arXiv 2022

  27. [27]

    QCD evolution of entanglement entropy

    M. Hentschinski, D. E. Kharzeev, K. Kutak, and Z. Tu, “QCD evolution of entanglement entropy”,arXiv:2408.01259

    work page arXiv

  28. [28]

    Probing the Onset of Maximal Entanglement inside the Proton in Diffractive Deep Inelastic Scattering

    M. Hentschinski, D. E. Kharzeev, K. Kutak, and Z. Tu, “Probing the Onset of Maximal Entanglement inside the Proton in Diffractive Deep Inelastic Scattering”,Phys. Rev. Lett. 131 (2023) 241901, arXiv:2305.03069. 13

    work page arXiv 2023

  29. [29]

    Einstein-Podolsky-Rosen Paradox and Quantum Entanglement at Subnucleonic Scales

    Z. Tu, D. E. Kharzeev, and T. Ullrich, “Einstein-Podolsky-Rosen Paradox and Quantum Entanglement at Subnucleonic Scales”,Phys. Rev. Lett. 124 (2020) 062001, arXiv:1904.11974

    work page arXiv 2020

  30. [30]

    On the role of soft gluons in collinear parton densities

    M. Mendizabal, F. Guzman, H. Jung, and S. Taheri Monfared, “On the role of soft gluons in collinear parton densities”,Eur. Phys. J. C 84 (2024) 1299, arXiv:2309.11802

    work page arXiv 2024

  31. [31]

    Center-of-mass energy dependence of intrinsic-kT distributions obtained from Drell–Yan production

    I. Bubanja et al., “Center-of-mass energy dependence of intrinsic-kT distributions obtained from Drell–Yan production”,Eur. Phys. J. C 85 (2025) 278, arXiv:2404.04088

    work page arXiv 2025

  32. [32]

    Interplay of intrinsic motion of partons and soft gluon emissions in Drell–Yan production studied with PYTHIA

    I. Bubanja, H. Jung, N. Raicevic, and S. Taheri Monfared, “Interplay of intrinsic motion of partons and soft gluon emissions in Drell–Yan production studied with PYTHIA”,Eur. Phys. J. C 85 (2025) 363, arXiv:2412.05221

    work page arXiv 2025

  33. [33]

    The smallkTregion in Drell–Yan production at next-to-leading order with the parton branching method

    I. Bubanja et al., “The smallkTregion in Drell–Yan production at next-to-leading order with the parton branching method”,Eur. Phys. J. C 84 (2024) 154, arXiv:2312.08655

    work page arXiv 2024

  34. [34]

    A POWHEG generator for deep inelastic scattering

    A. Banfi et al., “A POWHEG generator for deep inelastic scattering”, arXiv:2309.02127

    work page arXiv

  35. [35]

    A parton shower consistent with parton densities at LO and NLO: PDF2ISR

    H. Jung, L. Lönnblad, M. Mendizabal, and S. Taheri Monfared, “A parton shower consistent with parton densities at LO and NLO: PDF2ISR”,Eur. Phys. J. C 85 (2025) 870, arXiv:2504.10243

    work page arXiv 2025

  36. [36]

    Quantum information approach to high energy interactions

    D. E. Kharzeev, “Quantum information approach to high energy interactions”,Phil. Trans. A. Math. Phys. Eng. Sci.380 (2021), no. 2216, 20210063,arXiv:2108.08792

    work page arXiv 2021

  37. [37]

    Rapidity evolution of the entanglement entropy in quarkonium: Parton and string duality

    Y. Liu, M. A. Nowak, and I. Zahed, “Rapidity evolution of the entanglement entropy in quarkonium: Parton and string duality”,Phys. Rev. D 105 (2022), no. 11, 114028, arXiv:2203.00739

    work page arXiv 2022

  38. [38]

    Unitarity and the BFKL Pomeron

    A. H. Mueller, “Unitarity and the BFKL pomeron”,Nucl. Phys. B 437 (1995) 107–126,arXiv:hep-ph/9408245

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  39. [39]

    Hentschinski, K

    M. Hentschinski, K. Kutak, and R. Straka, “Maximally entangled proton and charged hadron multiplicity in Deep Inelastic Scattering”,Eur. Phys. J. C 82 (2022), no. 12, 1147, arXiv:2207.09430

    work page arXiv 2022

  40. [40]

    Improved Approximations for the Three-Loop Splitting Functions in QCD

    W. L. van Neerven and A. Vogt, “Improved approximations for the three loop splitting functions in QCD”,Phys. Lett. B 490 (2000) 111, arXiv:hep-ph/0007362

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  41. [41]

    The Three-Loop Splitting Functions in QCD: The Non-Singlet Case

    S. Moch, J. A. M. Vermaseren, and A. Vogt, “The Three loop splitting functions in QCD: The Nonsinglet case”,Nucl. Phys. B688 (2004) 101, arXiv:hep-ph/0403192. 14

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  42. [42]

    The Three-Loop Splitting Functions in QCD: The Singlet Case

    A. Vogt, S. Moch, and J. A. M. Vermaseren, “The Three-loop splitting functions in QCD: The Singlet case”,Nucl. Phys. B691 (2004) 129, arXiv:hep-ph/0404111

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  43. [43]

    The third-order QCD corrections to deep-inelastic scattering by photon exchange

    J. Vermaseren, A. Vogt, and S. Moch, “The Third-order QCD corrections to deep-inelastic scattering by photon exchange”,Nucl.Phys. B724 (2005) 3, arXiv:hep-ph/0504242

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  44. [44]

    Bl¨ umlein, P

    J. Blümlein, P. Marquard, C. Schneider, and K. Schönwald, “The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements”,Nucl. Phys. B 971 (2021) 115542, arXiv:2107.06267

    work page arXiv 2021

  45. [45]

    Bl¨ umlein, P

    J. Blümlein, P. Marquard, C. Schneider, and K. Schönwald, “The massless three-loop Wilson coefficients for the deep-inelastic structure functions F2, FL, xF3 and g1”, JHEP 11 (2022) 156, arXiv:2208.14325

    work page arXiv 2022

  46. [46]

    The Transition Matrix Element A_{gq}(N) of the Variable Flavor Number Scheme at O(\alpha_s^3)

    J. Ablinger et al., “The transition matrix element Agq(N) of the variable flavor number scheme atO(α3 s)”,Nucl. Phys. B 882 (2014) 263, arXiv:1402.0359

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  47. [47]

    The Three-Loop Splitting Functions $P_{qg}^{(2)}$ and $P_{gg}^{(2, N_F)}$

    J. Ablinger et al., “The three-loop splitting functionsP (2) qg and P (2,NF ) gg ”,Nucl. Phys. B 922 (2017) 1, arXiv:1705.01508

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  48. [48]

    The Three-Loop Splitting Functions in QCD: The Helicity-Dependent Case

    S. Moch, J. A. M. Vermaseren, and A. Vogt, “The Three-Loop Splitting Functions in QCD: The Helicity-Dependent Case”,Nucl. Phys. B 889 (2014) 351, arXiv:1409.5131

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  49. [49]

    The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements

    A. Behring et al., “The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements”,Nucl. Phys. B 948 (2019) 114753, arXiv:1908.03779

    work page arXiv 2019

  50. [50]

    Bl¨ umlein, P

    J. Blümlein, P. Marquard, C. Schneider, and K. Schönwald, “The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements”,JHEP 01 (2022) 193, arXiv:2111.12401

    work page arXiv 2022

  51. [51]

    Collinear and TMD Quark and Gluon Densities from Parton Branching Solution of QCD Evolution Equations

    F. Hautmann et al., “Collinear and TMD quark and gluon densities from Parton Branching solution of QCD evolution equations”,JHEP 01 (2018) 070, arXiv:1708.03279

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  52. [52]

    Soft-gluon resolution scale in QCD evolution equations

    F. Hautmann et al., “Soft-gluon resolution scale in QCD evolution equations”,Phys. Lett. B 772 (2017) 446, arXiv:1704.01757

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  53. [53]

    QCD and collider physics

    R. K. Ellis, W. J. Stirling, and B. R. Webber, “QCD and collider physics”,Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol.8 (1996) 1

    work page 1996

  54. [54]

    Initial state radiation effects on W and jet production

    M. Bengtsson, T. Sjostrand, and M. van Zijl, “Initial state radiation effects on W and jet production”,Z. Phys. C 32 (1986) 67. 15

    work page 1986

  55. [55]

    A comprehensive guide to the physics and usage of PYTHIA 8.3

    C. Bierlich et al., “A comprehensive guide to the physics and usage of PYTHIA 8.3”, SciPost Phys. Codeb. 2022 (2022) 8, arXiv:2203.11601

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  56. [56]

    Bierlich et al.,Robust Independent Validation of Experiment and Theory: Rivet version 3, SciPost Phys.8 (2020) 026, arXiv:1912.05451 [hep-ph]

    C. Bierlich et al., “Robust Independent Validation of Experiment and Theory: Rivet version 3”,SciPost Phys. 8 (2020) 026, arXiv:1912.05451

    work page arXiv 2020

  57. [57]

    Coherence effects in initial jets at smallQ2/s

    M. Ciafaloni, “Coherence effects in initial jets at smallQ2/s.”,Nucl. Phys. B 296 (1988) 49

    work page 1988

  58. [58]

    QCD coherence in initial state radiation

    S. Catani, F. Fiorani, and G. Marchesini, “QCD coherence in initial state radiation”, Phys. Lett. B 234 (1990) 339

    work page 1990

  59. [59]

    Small x behavior of initial state radiation in perturbative QCD

    S. Catani, F. Fiorani, and G. Marchesini, “Small x behavior of initial state radiation in perturbative QCD”,Nucl. Phys. B 336 (1990) 18

    work page 1990

  60. [60]

    QCD coherence in the structure function and associated distributions at small $x$

    G. Marchesini, “QCD coherence in the structure function and associated distributions at small x”,Nucl. Phys. B 445 (1995) 49, arXiv:hep-ph/9412327

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  61. [61]

    The CCFM Monte Carlo generator CASCADE

    H. Jung, “The CCFM Monte Carlo generator CASCADE”,Comput. Phys. Commun. 143 (2002) 100, arXiv:hep-ph/0109102

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  62. [62]

    The CASCADE Monte Carlo

    H. Jung, “The CASCADE Monte Carlo”.http://www.desy.de/~jung/cascade, 2009

    work page 2009

  63. [63]

    The CCFM Monte Carlo generator CASCADE 2.2.0

    H. Jung et al., “The CCFM Monte Carlo generator CASCADE version 2.2.03”,Eur. Phys. J. C 70 (2010) 1237, arXiv:1008.0152

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  64. [64]

    CASCADE3 A Monte Carlo event generator based on TMDs

    S. Baranov et al., “CASCADE3 A Monte Carlo event generator based on TMDs”,Eur. Phys. J. C 81 (2021) 425, arXiv:2101.10221

    work page arXiv 2021

  65. [65]

    Physics of the ALICE Forward Calorimeter upgrade

    ALICE Collaboration, “Physics of the ALICE Forward Calorimeter upgrade”,. 16

    work page