pith. sign in

arxiv: 2606.29226 · v1 · pith:ITO5FZAUnew · submitted 2026-06-28 · ✦ hep-ph

An overview of scale invariance in proton structure with holographic insights

Akbari Jahan This is my paper

Pith reviewed 2026-06-30 02:53 UTC · model grok-4.3

classification ✦ hep-ph
keywords scale invarianceproton structureparton distribution functionsholographic QCDself-similaritydeep inelastic scatteringfractal geometrylight-front holography
0
0 comments X

The pith

Self-similar scaling patterns in proton parton distributions correspond qualitatively to geometric features in holographic QCD.

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

The paper reviews self-similarity rooted in scale invariance and fractal geometry as a framework for parton distribution functions at small x in deep inelastic scattering. Phenomenological models built on recursive scaling reproduce main features of measured structure functions. It then compares those patterns to the geometric scaling that appears in light-front holographic QCD. The comparison points to a wider role for scale invariance in proton structure, presented as an interpretive link rather than a derived equivalence.

Core claim

By comparing the scaling behaviour appearing in phenomenological models with the geometric structure underlying holographic QCD, qualitative correspondences emerge that suggest a broader role of scale invariance in proton structure, although the connection remains interpretive rather than derivational.

What carries the argument

The comparison of recursive scaling patterns in self-similar phenomenological models of parton distributions to geometric scaling in light-front holographic QCD.

If this is right

  • Self-similar models reproduce key experimental features of parton distributions in the small-x region.
  • Fractal-inspired descriptions capture recursive patterns that influence partonic dynamics.
  • The interpretive link supplies a complementary perspective connecting fractal scaling seen in DIS to holographic geometry.
  • Scale invariance may operate across both phenomenological and geometric descriptions of QCD.

Where Pith is reading between the lines

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

  • Models that combine self-similar recursion with holographic geometry could be constructed to generate new predictions for structure functions.
  • Future DIS measurements at higher energies could search for scaling signatures that appear in both frameworks.
  • The same interpretive comparison might be applied to other hadrons or processes where scale invariance is observed.

Load-bearing premise

That scaling patterns observed in the self-similar models share a meaningful relation to geometric scaling in holographic QCD that extends past qualitative similarity.

What would settle it

A demonstration that the functional forms or exponents of scaling in the phenomenological models have no corresponding geometric feature derivable within light-front holographic QCD would refute the suggested correspondence.

Figures

Figures reproduced from arXiv: 2606.29226 by Akbari Jahan.

Figure 1
Figure 1. Figure 1: Illustrative behaviour of the self-similar parton density [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Qualitative behaviour associated with the functional form appearing [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic illustration of scale invariance in the AdS framework. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

The concept of self-similarity in the internal structure of the proton, rooted in scale invariance and fractal geometry, provides an intriguing framework for understanding the behaviour of parton distribution functions (PDFs), particularly in the small \textit{x} region probed in deep inelastic scattering (DIS). Phenomenological models based on self-similarity have been shown to reproduce key features of experimental data, suggesting that recursive scaling patterns may play an important role in partonic dynamics. In this work, we present an overview of scale-invariant descriptions of proton structure, focusing on self-similar models developed in earlier studies and their phenomenological implications for structure functions and parton distributions. We then explore possible conceptual connections between these fractal-inspired descriptions and modern holographic approaches to QCD, particularly within the framework of light-front holographic QCD. By comparing the scaling behaviour appearing in phenomenological models with the geometric structure underlying holographic QCD, we highlight qualitative correspondences that suggest a broader role of scale invariance in proton structure. Although the connection remains interpretive rather than derivational, it offers a complementary perspective of how fractal-like scaling observed in DIS may relate to geometric scaling in holographic descriptions of QCD.

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

0 major / 2 minor

Summary. The manuscript is an overview of scale invariance in proton structure. It reviews self-similar phenomenological models for PDFs and structure functions in the small-x regime of DIS, their ability to reproduce experimental features, and then explores conceptual connections to the geometric scaling in light-front holographic QCD. The paper highlights qualitative correspondences in scaling behaviors that suggest a broader role for scale invariance while explicitly characterizing the link as interpretive rather than derivational.

Significance. The synthesis of prior phenomenological models is consistent with the abstract and provides a coherent summary. The cautious interpretive framing is a strength, as it avoids overclaiming derivation or quantitative equivalence. If the qualitative correspondences are accepted at face value, the work offers a complementary perspective that may stimulate discussion on fractal-like features in QCD, though the absence of new derivations, fits, or falsifiable predictions limits its immediate predictive impact.

minor comments (2)
  1. The abstract and introduction would benefit from a brief explicit list or table of the specific scaling exponents or functional forms drawn from the phenomenological models versus those in holographic QCD to make the qualitative comparison more concrete for readers.
  2. A short concluding section summarizing the key correspondences and open questions would improve the manuscript's utility as an overview.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript as an overview of scale invariance in proton structure. The recommendation for minor revision is noted, though no specific major comments requiring changes were provided. We address the observation regarding predictive impact below.

read point-by-point responses

  1. Referee: the absence of new derivations, fits, or falsifiable predictions limits its immediate predictive impact.

    Authors: We agree that the manuscript is an overview synthesizing prior self-similar models and exploring qualitative correspondences with holographic QCD, without introducing new derivations, data fits, or explicit predictions. This scope is intentional, as stated in the abstract and introduction, to provide a coherent summary and interpretive perspective that may stimulate further discussion. The cautious framing (explicitly interpretive rather than derivational) is maintained throughout. revision: no

Circularity Check

0 steps flagged

No significant circularity; interpretive overview only

full rationale

The manuscript is explicitly framed as an overview of existing self-similar phenomenological models and light-front holographic QCD, with the central claim limited to 'qualitative correspondences' that 'suggest a broader role' while stating the link 'remains interpretive rather than derivational.' No derivation chain, fitted-parameter prediction, or uniqueness theorem is asserted. All scaling comparisons are drawn from prior literature without reduction to self-citation or self-definition. The paper therefore contains no load-bearing step that collapses to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper that introduces no new free parameters, axioms, or invented entities; all content summarizes existing models from prior literature.

pith-pipeline@v0.9.1-grok · 5718 in / 1004 out tokens · 55954 ms · 2026-06-30T02:53:05.200535+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

48 extracted references · 1 canonical work pages

  1. [1]

    H. I. Alrebdi and M. Ajaz, Sequential retuning of PYTHIA 8.316 to a global soft- QCD basis in pp collisions at √s= 0.9−13 TeV, arXiv:2603.21364[hep-ph]

    work page arXiv

  2. [2]

    Lastovicka, Self-similar properties of the proton structure at lowx, Eur

    T. Lastovicka, Self-similar properties of the proton structure at lowx, Eur. Phys. J. C 24, 529 (2002)

    2002

  3. [3]

    D. K. Choudhury and R.Gogoi, Self-similarity and a parametrization of proton struc- ture function at smallx, Ind. J. Phys. 80, 823 (2006)

    2006

  4. [4]

    Akbari Jahan and D. K. Choudhury, Fractal inspired models of quark and gluon distributions and longitudinal structure functionF L(x, Q2) at smallx, Ind. J. Phys. 85, 587 (2011)

    2011

  5. [5]

    Akbari Jahan and D. K. Choudhury, Momentum fractions of quarks and gluons in a self-similarity based model of proton, Mod. Phys. Lett. A 27, 1250193 (2012)

    2012

  6. [6]

    Akbari Jahan and D. K. Choudhury, An analysis of momentum fractions of quarks and gluons in a model of proton, Mod. Phys. Lett. A 28, 1350086 (2013)

    2013

  7. [7]

    D. K. Choudhury and Akbari Jahan, A model construction of self-similarity based double parton distribution functions for proton-proton collision at LHC, Int. J. Mod. Phys. A 28, 1350079 (2013)

    2013

  8. [8]

    Akbari Jahan and D. K. Choudhury, Self-similarity and the Froissart bound, Phys. Rev. D 89, 014014 (2014)

    2014

  9. [9]

    Akbari Jahan and D. K. Choudhury, A lower bound on longitudinal structure func- tion at smallxfrom a self-similarity based model of proton, Commun. Theor. Phys. 61, 654 (2014)

    2014

  10. [10]

    Blumlein, The longitudinal structure functionF L(x, Q2) at smallx, J

    J. Blumlein, The longitudinal structure functionF L(x, Q2) at smallx, J. Phys. G: Nucl. Part. Physics 19, 1623 (1993)

    1993

  11. [11]

    Altarelli and G

    G. Altarelli and G. Martinelli, Transverse momentum of jets in electroproduction from quantum chromodynamics, Phys. Lett. B 76, 89 (1978)

    1978

  12. [12]

    Rezaei and G

    B. Rezaei and G. R. Boroun, Longitudinal structure function from the parton pa- rameterization, Eur. Phys. J. A 56, 262 (2020)

    2020

  13. [13]

    D. K. Choudhury and B. Saikia, Parton distribution functions and models of proton structure functions with self-similarity, Int. J. Mod. Phys. A 31, 1650176 (2016)

    2016

  14. [14]

    D. K. Choudhury, B. Saikia and K. Kalita, Momentum fractions carried by quarks and gluons in models of proton structure functions at smallx, Int. J. Mod. Phys. A 32, 1750107 (2017). 11

    2017

  15. [15]

    Saikia and D

    B. Saikia and D. K. Choudhury, An improved singularity free self-similar model of proton structure function, Commun. Theor. Phys. 67, 61 (2017)

    2017

  16. [16]

    S. S. Mohsenabadi, S. A. Tehrani and F. Taghavi-Shahri, Proton structure functions at lowx: the fractal distributions, Int. J. Mod. Phys. A 38, 2350136 (2023)

    2023

  17. [17]

    V. N. Gribov and L. N. Lipatov, Deep inelasticepscattering in perturbation theory, Sov. J. Nucl. Phys. 15, 438 (1972)

    1972

  18. [18]

    L. N. Lipatov, The parton model and perturbation theory, Sov. J. Nucl. Phys. 20, 94 (1975)

    1975

  19. [19]

    Altarelli and G

    G. Altarelli and G. Parisi, Asymptotic freedom in parton language, Nucl. Phys. B 126, 298 (1977)

    1977

  20. [20]

    Yu. L. Dokshitzer, Calculation of the Structure Functions for Deep Inelastic Scatter- ing ande +e− Annihilation by Perturbation Theory in Quantum Chromodynamics, Sov. Phys. JETP 46, 641 (1977)

    1977

  21. [21]

    V. S. Fadin, E. A. Kuraev and L. N. Lipatov, On the pomeranchuk singularity in asymptotically free theories, Phys. Lett. B 60, 50 (1975)

    1975

  22. [22]

    E. A. Kuraev, L. N. Lipatov and V. S. Fadin, Multiregge processes in the Yang-Mills theory, Sov. Phys. JETP 44, 443 (1976)

    1976

  23. [23]

    E. A. Kuraev, L. N. Lipatov and V. S. Fadin, The pomeranchuk singularity in nonabelian gauge theories, Sov. Phys. JETP 45, 199 (1977)

    1977

  24. [24]

    I. I. Balitsky and L. N. Lipatov, The pomeranchuk singularity in quantum chromo- dynamics, Sov. J. Nucl. Phys. 28, 822 (1978)

    1978

  25. [25]

    Abramowicz et al, Combination of measurements of inclusive deep inelastice ±pscattering cross sections and QCD analysis of HERA data, Eur

    H1 and ZEUS Collaborations: H. Abramowicz et al, Combination of measurements of inclusive deep inelastice ±pscattering cross sections and QCD analysis of HERA data, Eur. Phys. J. C 75, 580 (2015)

    2015

  26. [26]

    Ball et al, Parton distributions for the LHC run II, J

    NNPDF Collaboration: R.D. Ball et al, Parton distributions for the LHC run II, J. High Energy Phys. 04, 040 (2015)

    2015

  27. [27]

    NNPDF Collaboration: R. D. Ball et al, The path to N3LO parton distributions, Eur. Phys. J. C 84, 659 (2024)

    2024

  28. [28]

    Harland-Lang, A.D

    L.A. Harland-Lang, A.D. Martin, P. Motylinski and R.S. Thorne, Parton distribu- tions in the LHC era: MMHT 2014 PDFs, Eur. Phys. J. C 75, 204 (2015)

    2014

  29. [29]

    L. A. Harland-Lang and R. S. Thorne, On the consistent use of scale variations in PDF fits and predictions, Eur. Phys. J. C 79, 225 (2019)

    2019

  30. [30]

    S. A. Tehrani, F. Taghavi-Shahri and S. Shoeibi, Self-similar properties of the proton structure at lowxwithin the xFitter framework, J. Holography Appl. Phys. 4, 27 (2024)

    2024

  31. [31]

    J. L. Albacete and C. Marquet, Gluon saturation and initial conditions for relativistic heavy ion collisions, Prog. Part. Nucl. Phys. 76, 1 (2014). 12

    2014

  32. [32]

    Froissart, Asymptotic behaviour and subtractions in the Mandelstam represen- tation, Phys

    M. Froissart, Asymptotic behaviour and subtractions in the Mandelstam represen- tation, Phys. Rev. 123, 1053 (1961)

    1961

  33. [33]

    Polchinski and M

    J. Polchinski and M. J. Strassler, Hard scattering and gauge/string duality, Phys. Rev. Lett. 88, 031601 (2002)

    2002

  34. [34]

    Erlich, E

    J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, QCD and a holographic model of hadrons, Phys. Rev. Lett. 95, 261602 (2005)

    2005

  35. [35]

    S. J. Brodsky, G. F. de T’eramond, H. G. Dosch and J. Erlich, Light-front holographic QCD and emerging confinement, Phys. Rep. 584, 1 (2015)

    2015

  36. [36]

    S. J. Brodsky and G. F. de T´ eramond, Hadronic spectra and light-front wave func- tions in holographic QCD, Phys. Rev. Lett. 96, 201601 (2006)

    2006

  37. [37]

    S. J. Brodsky and G. F. de T´ eramond, QCD and light-front holography, Acta Phys. Polon. B 41, 2605 (2010)

    2010

  38. [38]

    H. G. Dosch, G. F. de T´ eramond and S. J. Brodsky, Holographic light-front QCD, J. Subat. Part. Cosmol. 5, 100339 (2026)

    2026

  39. [39]

    HLFHS Collaboration: G. F. de T´ eramond, H. G. Dosch, T. Liu, R. S. Sufian, S. J. Brodsky and A. Deur, Gluon matter distribution in the proton and pion from extended holographic light-front QCD, Phys. Rev. D 104, 114005 (2021)

    2021

  40. [40]

    Ballon-Bayona, H

    A. Ballon-Bayona, H. Boschi-Filho, L. A. H. Mamani, A. S. Miranda and V. T. Zanchin, Effective holographic models for QCD: Glueball spectrum and trace anomaly, Phys. Rev. D 97, 046001 (2018)

    2018

  41. [41]

    Rabemananjara, Ivan A

    Tanjona R. Rabemananjara, Ivan A. Godino, Eva D.Z. Groenendijk, Parametrisation scale invariance of Parton Distribution Function fits, PoS DIS2025, 512, 025 (2025)

    2025

  42. [42]

    Ahmady, R

    M. Ahmady, R. Sandapen and N. Sharma, Diffractive and production at HERA using a holographic AdS/QCD light-front meson wave function, Phys. Rev. D 94, 074018 (2016)

    2016

  43. [43]

    T. J. Hou et al, New CTEQ global analysis of quantum chromodynamics with high- precision date from the LHC, Phys. Rev. D 103, 014013 (2021)

    2021

  44. [44]

    JAM Collaboration: N. Sato, W. Melnitchouk, S. E. Kuhn, J. J. Ethier and A. Accardi, Iterative Monte Carlo analysis of spin-dependent parton distributions, Phys. Rev. D 93, 074005 (2016)

    2016

  45. [45]

    Cocuzza, N

    JAM Collaboration: C. Cocuzza, N. T. Hunt-Smith, W. Melnitchouk, N. Sato and A. W. Thomas, Global QCD analysis of spin PDFs in the proton with high-xand lattice constraints, Phys. Rev. D 112, 114017 (2025)

    2025

  46. [46]

    Melnitchouk, Spin structure of the proton from global QCD analysis, PoS (QCHSC24), 064 (2024)

    W. Melnitchouk, Spin structure of the proton from global QCD analysis, PoS (QCHSC24), 064 (2024)

    2024

  47. [47]

    J. J. Ethier and E. R. Nocera, Parton distributions in nucleons and nuclei, Ann. Rev. Nucl. Part. Sci. 70, 43 (2020)

    2020

  48. [48]

    Accardi et al, Electron-ion collider: The next QCD frontier, Eur

    A. Accardi et al, Electron-ion collider: The next QCD frontier, Eur. Phys. J. A 52, 268 (2016). 13

    2016