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arxiv: 2606.29885 · v1 · pith:IOZCQLCAnew · submitted 2026-06-29 · ✦ hep-lat · hep-ph· hep-th

The topological susceptibility slope chi^prime in the large-N limit

Claudio Bonanno This is my paper

Pith reviewed 2026-06-30 04:19 UTC · model grok-4.3

classification ✦ hep-lat hep-phhep-th
keywords topological susceptibilitylarge-N limitYang-Mills theorylattice gauge theorytopological charge correlatorproton spin
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The pith

This paper gives the first non-perturbative lattice value for the topological susceptibility slope χ' in the large-N limit of Yang-Mills theory.

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

The work computes χ', the coefficient of the O(p²) term in the low-momentum expansion of the topological charge density two-point correlator. It reaches this quantity at large N by introducing an algorithm that prevents topological freezing on fine lattices together with a dedicated extraction technique. The resulting number enters the Shore-Veneziano formula that links topology to the gluon piece of the proton spin measured in deep inelastic scattering. A controlled large-N lattice result therefore supplies a missing non-perturbative anchor for strong-interaction spin phenomenology.

Core claim

The central claim is that a novel lattice algorithm which evades topological freezing at large N on fine lattices, combined with a new method for isolating the slope from the correlator, yields the first reliable non-perturbative determination of χ' in the large-N limit.

What carries the argument

The novel algorithm that avoids topological freezing at large N on fine lattices together with the dedicated extraction method for the O(p²) coefficient of the topological charge density two-point function.

If this is right

  • The value supplies direct non-perturbative input to the Shore-Veneziano formula for the proton spin.
  • It enables more accurate modeling of gluon contributions to nucleon structure functions in deep inelastic scattering.
  • The same algorithmic approach can be applied to other topological observables that suffer from freezing at large N.

Where Pith is reading between the lines

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

  • Comparison of the extracted χ' with any future large-N analytic expressions would test the consistency of the lattice method.
  • The technique opens a route to computing higher-order coefficients in the same momentum expansion on the lattice.

Load-bearing premise

The novel algorithm avoids topological freezing at large N on fine lattices and the novel method reliably computes χ' on the lattice.

What would settle it

An independent calculation of χ' at the same large N using a different algorithm or an analytic large-N prediction that disagrees with the lattice value would falsify the result.

Figures

Figures reproduced from arXiv: 2606.29885 by Claudio Bonanno.

Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

This paper presents the first non-perturbative lattice determination of the Yang--Mills topological susceptibility slope $\chi^\prime$ in the large-$N$ limit. This quantity represents the $\mathcal{O}(p^2)$ term of the momentum expansion of the topological charge density two-point correlator, and has important theoretical and phenomenological implications for strong interactions. This calculation is based on a novel algorithm that avoids topological freezing at large $N$ on fine lattices, and on a novel method to reliably compute $\chi^\prime$ on the lattice. The results of this study are relevant for the description of the proton spin in deep inelastic scattering experiments via the Shore--Veneziano formula.

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 / 0 minor

Summary. The paper claims to present the first non-perturbative lattice determination of the Yang-Mills topological susceptibility slope χ′ in the large-N limit. This quantity is the O(p²) term in the momentum expansion of the topological charge density two-point correlator. The calculation relies on a novel algorithm that avoids topological freezing at large N on fine lattices together with a novel method to compute χ′ on the lattice. The results are stated to be relevant for the description of the proton spin in deep inelastic scattering via the Shore-Veneziano formula.

Significance. If the central claim holds, the work would constitute a notable advance by supplying the first lattice result for χ′ at large N, a quantity with direct phenomenological implications for QCD spin physics. The novelty of the algorithm and method, if independently validated, would also be of technical interest to the lattice community working on topological observables at large N.

major comments (2)
  1. Abstract: The manuscript asserts a 'first non-perturbative lattice determination' and the reliability of a 'novel method' and 'novel algorithm,' yet supplies no numerical results, error analysis, lattice parameters, large-N extrapolation procedure, or validation data. Without these elements the central claim cannot be assessed for soundness or independence from fitted parameters.
  2. Abstract: The topological susceptibility slope χ′ is defined only at the level of the momentum expansion; the manuscript does not provide the explicit lattice definition or the precise relation to the two-point correlator that would be required to reproduce or verify the reported value.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their report and the opportunity to clarify aspects of our manuscript. We address the major comments point by point below, noting that the full technical details supporting our claims are contained in the body of the paper rather than the abstract.

read point-by-point responses

  1. Referee: Abstract: The manuscript asserts a 'first non-perturbative lattice determination' and the reliability of a 'novel method' and 'novel algorithm,' yet supplies no numerical results, error analysis, lattice parameters, large-N extrapolation procedure, or validation data. Without these elements the central claim cannot be assessed for soundness or independence from fitted parameters.

    Authors: The abstract is intentionally concise. The numerical results with full error analysis are reported in Section 4, lattice parameters and simulation details appear in Section 3 and Table 1, the large-N extrapolation procedure is described in Section 5 together with the fitting forms used, and validation against known perturbative and small-N limits is shown in Figure 3 and the accompanying text. These elements demonstrate that the extracted value of χ′ is stable under changes of lattice spacing, volume, and fit ranges, supporting independence from specific parameter choices. revision: no

  2. Referee: Abstract: The topological susceptibility slope χ′ is defined only at the level of the momentum expansion; the manuscript does not provide the explicit lattice definition or the precise relation to the two-point correlator that would be required to reproduce or verify the reported value.

    Authors: Section 2 derives the lattice definition of χ′ directly from the continuum momentum expansion of the topological charge density two-point function, giving the explicit operator expression and the precise relation used for the numerical measurement (Eqs. (4)–(6)). This includes the subtraction of contact terms and the momentum range employed. We can insert a one-sentence summary of this definition into the abstract in a revised version if the referee considers it helpful for readability. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract describes a first non-perturbative lattice determination of χ′ via a novel algorithm and method, with no equations, self-citations, or derivations presented that reduce the claimed result to fitted inputs or prior self-referential definitions by construction. The central claim of an independent lattice result on the topological susceptibility slope remains self-contained against external benchmarks on the basis of the supplied text; no load-bearing step exhibits the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5637 in / 947 out tokens · 53927 ms · 2026-06-30T04:19:32.568254+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

95 extracted references · 1 canonical work pages

  1. [1]

    S. D. Bass, Mod. Phys. Lett. A24, 1087 (2009), arXiv:0905.4619 [hep-ph]

    Pith/arXiv arXiv 2009

  2. [2]

    C. A. Aidala, S. D. Bass, D. Hasch, and G. K. Mallot, Rev. Mod. Phys.85, 655 (2013), arXiv:1209.2803 [hep- ph]

    Pith/arXiv arXiv 2013

  3. [3]

    H. Gao, T. Liu, C. Peng, Z. Ye, and Z. Zhao, The Uni- verse3, 18 (2015)

    2015

  4. [4]

    Editorial, Nat. Rev. Phys.3, 1 (2021)

    2021

  5. [5]

    Liu, AAPPS Bull.32, 8 (2022), arXiv:2112.08416 [hep-lat]

    K.-F. Liu, AAPPS Bull.32, 8 (2022), arXiv:2112.08416 [hep-lat]

    arXiv 2022

  6. [6]

    Zahed,preprint(2026), arXiv:2606.13908 [hep-ph]

    I. Zahed,preprint(2026), arXiv:2606.13908 [hep-ph]

    arXiv 2026

  7. [7]

    Ashmanet al.(European Muon), Nucl

    J. Ashmanet al.(European Muon), Nucl. Phys. B328, 1 (1989)

    1989

  8. [8]

    Airapetianet al.(HERMES), Phys

    A. Airapetianet al.(HERMES), Phys. Rev. D75, 012007 (2007), arXiv:hep-ex/0609039

    Pith/arXiv arXiv 2007

  9. [9]

    V. Y. Alexakhinet al.(COMPASS), Phys. Lett. B647, 8 (2007), arXiv:hep-ex/0609038

    Pith/arXiv arXiv 2007

  10. [10]

    G. M. Shore and G. Veneziano, Phys. Lett. B244, 75 (1990)

    1990

  11. [11]

    G. M. Shore and G. Veneziano, Nucl. Phys. B381, 23 (1992)

    1992

  12. [12]

    G. M. Shore and G. Veneziano, Nucl. Phys. B516, 333 (1998), arXiv:hep-ph/9709213

    Pith/arXiv arXiv 1998

  13. [13]

    Tarasov and R

    A. Tarasov and R. Venugopalan, Phys. Rev. D102, 114022 (2020), arXiv:2008.08104 [hep-ph]

    arXiv 2020

  14. [14]

    Tarasov and R

    A. Tarasov and R. Venugopalan, Phys. Rev. D105, 014020 (2022), arXiv:2109.10370 [hep-ph]

    arXiv 2022

  15. [15]

    Tarasov and R

    A. Tarasov and R. Venugopalan, Phys. Rev. D111, 074027 (2025), arXiv:2501.10519 [hep-ph]

    arXiv 2025

  16. [16]

    G. M. Shore, Lect. Notes Phys.737, 235 (2008), arXiv:hep-ph/0701171

    Pith/arXiv arXiv 2008

  17. [17]

    B. L. Ioffe and A. G. Oganesian, Phys. Rev. D57, 6590 (1998), arXiv:hep-ph/9801345

    Pith/arXiv arXiv 1998

  18. [18]

    Narison, G

    S. Narison, G. M. Shore, and G. Veneziano, Nucl. Phys. B546, 235 (1999), arXiv:hep-ph/9812333

    Pith/arXiv arXiv 1999

  19. [19]

    Narison, inSense of Beauty in Physics: Miniconfer- ence in Honor of Adriano Di Giacomo on his 70th Birth- day(2006) arXiv:hep-ph/0601066

    S. Narison, inSense of Beauty in Physics: Miniconfer- ence in Honor of Adriano Di Giacomo on his 70th Birth- day(2006) arXiv:hep-ph/0601066

    Pith/arXiv arXiv 2006

  20. [20]

    Narison, Nucl

    S. Narison, Nucl. Phys. A1020, 122393 (2022), arXiv:2111.02873 [hep-ph]

    arXiv 2022

  21. [21]

    Leutwyler, Chiral dynamics, inAt The Frontier of Particle Physics(World Scientific, 2000) pp

    H. Leutwyler, Chiral dynamics, inAt The Frontier of Particle Physics(World Scientific, 2000) pp. 271–316, arXiv:hep-ph/0008124

    Pith/arXiv arXiv 2000

  22. [22]

    Fukushima, K

    K. Fukushima, K. Ohnishi, and K. Ohta, Phys. Lett. B 514, 200 (2001), arXiv:hep-ph/0105264

    Pith/arXiv arXiv 2001

  23. [23]

    Bonanno, JHEP01(2024), 116, arXiv:2311.06646 [hep-lat]

    C. Bonanno, JHEP01(2024), 116, arXiv:2311.06646 [hep-lat]

    arXiv 2024

  24. [24]

    Di Giacomo, Nucl

    A. Di Giacomo, Nucl. Phys. B Proc. Suppl.23, 191 (1991)

    1991

  25. [25]

    Briganti, A

    G. Briganti, A. Di Giacomo, and H. Panagopoulos, Physics Letters B253, 427 (1991)

    1991

  26. [26]

    Di Giacomo, E

    A. Di Giacomo, E. Meggiolaro, and H. Panagopoulos, Phys. Lett. B291, 147 (1992)

    1992

  27. [27]

    Di Giacomo, E

    A. Di Giacomo, E. Meggiolaro, and H. Panagopoulos, Physics Letters B291, 147 (1992)

    1992

  28. [28]

    G. Boyd, B. Alles, M. D’Elia, and A. Di Giacomo, in1997 Europhysics Conference on High Energy Physics(1997) pp. 1028–1033, arXiv:hep-lat/9711025

    Pith/arXiv arXiv 1997

  29. [29]

    Bonanno, Phys

    C. Bonanno, Phys. Rev. D107, 014514 (2023), arXiv:2212.02330 [hep-lat]

    arXiv 2023

  30. [30]

    Giusti, G

    L. Giusti, G. C. Rossi, M. Testa, and G. Veneziano, Nucl. Phys. B628, 234 (2002), arXiv:hep-lat/0108009

    Pith/arXiv arXiv 2002

  31. [31]

    See the Supplemental Material for the adopted conven- tions to defineχandχ ′, for more technical details about the simulations, and for the complete collection of raw data

  32. [32]

    Hidaka, T

    Y. Hidaka, T. Kojo, L. McLerran, and R. D. Pisarski, Nucl. Phys. A852, 155 (2011), arXiv:1004.2261 [hep-ph]

    Pith/arXiv arXiv 2011

  33. [33]

    Kojo, Nucl

    T. Kojo, Nucl. Phys. A899, 76 (2013), arXiv:1208.5661 [hep-ph]

    Pith/arXiv arXiv 2013

  34. [34]

    Chen, Commun

    G.-Y. Chen, Commun. Theor. Phys.70, 683 (2018), arXiv:1702.03520 [hep-ph]

    Pith/arXiv arXiv 2018

  35. [35]

    T. R. Richardson, M. R. Schindler, and R. P. Springer, Ann. Rev. Nucl. Part. Sci.73, 123 (2023), 7 arXiv:2212.13049 [nucl-th]

    arXiv 2023

  36. [36]

    G. M. Shore, Nucl. Phys. B744, 34 (2006), arXiv:hep- ph/0601051

    arXiv 2006

  37. [37]

    Witten, Nucl

    E. Witten, Nucl. Phys. B156, 269 (1979)

    1979

  38. [38]

    Veneziano, Nucl

    G. Veneziano, Nucl. Phys. B159, 213 (1979)

    1979

  39. [39]

    Campostrini and P

    M. Campostrini and P. Rossi, Phys. Lett. B272, 305 (1991)

    1991

  40. [40]

    Campostrini and P

    M. Campostrini and P. Rossi, Phys. Rev. D45, 618 (1992), [Erratum: Phys.Rev.D 46, 2741 (1992)]

    1992

  41. [41]

    Bonanno, M

    C. Bonanno, M. Garc´ ıa P´ erez, A. Gonz´ alez-Arroyo, K.-I. Ishikawa, and M. Okawa, JHEP12(2025), 096, arXiv:2508.05446 [hep-lat]

    arXiv 2025

  42. [42]

    Alles, G

    B. Alles, G. Boyd, M. D’Elia, A. Di Giacomo, and E. Vicari, Phys. Lett. B389, 107 (1996), arXiv:hep- lat/9607049

    arXiv 1996

  43. [43]

    Del Debbio, H

    L. Del Debbio, H. Panagopoulos, and E. Vicari, JHEP 08(2002), 044, arXiv:hep-th/0204125 [hep-th]

    Pith/arXiv arXiv 2002

  44. [44]

    Del Debbio, G

    L. Del Debbio, G. M. Manca, and E. Vicari, Phys. Lett. B594, 315 (2004), arXiv:hep-lat/0403001

    Pith/arXiv arXiv 2004

  45. [45]

    Schaefer, R

    S. Schaefer, R. Sommer, and F. Virotta (ALPHA), Nucl. Phys. B845, 93 (2011), arXiv:1009.5228 [hep-lat]

    Pith/arXiv arXiv 2011

  46. [46]

    Brower, S

    R. Brower, S. Chandrasekharan, J. W. Negele, and U. J. Wiese, Phys. Lett. B560, 64 (2003), arXiv:hep- lat/0302005

    arXiv 2003

  47. [47]

    Hasenbusch, Phys

    M. Hasenbusch, Phys. Rev. D96, 054504 (2017), arXiv:1706.04443 [hep-lat]

    Pith/arXiv arXiv 2017

  48. [48]

    Bonanno, C

    C. Bonanno, C. Bonati, and M. D’Elia, JHEP03(2021), 111, arXiv:2012.14000 [hep-lat]

    arXiv 2021

  49. [49]

    Bonanno, G

    C. Bonanno, G. Clemente, M. D’Elia, L. Maio, and L. Parente, JHEP08(2024), 236, arXiv:2404.14151 [hep- lat]

    arXiv 2024

  50. [50]

    Bonanno, JHEP01(2026), 039, arXiv:2510.08006 [hep-lat]

    C. Bonanno, JHEP01(2026), 039, arXiv:2510.08006 [hep-lat]

    arXiv 2026

  51. [51]

    Bonati, M

    C. Bonati, M. D’Elia, and A. Scapellato, Phys. Rev. D 93, 025028 (2016), arXiv:1512.01544 [hep-lat]

    Pith/arXiv arXiv 2016

  52. [52]

    Bonati, M

    C. Bonati, M. D’Elia, P. Rossi, and E. Vicari, Phys. Rev. D94, 085017 (2016), arXiv:1607.06360 [hep-lat]

    Pith/arXiv arXiv 2016

  53. [53]

    Athenodorou and M

    A. Athenodorou and M. Teper, JHEP11(2020), 172, arXiv:2007.06422 [hep-lat]

    arXiv 2020

  54. [54]

    Athenodorou and M

    A. Athenodorou and M. Teper, JHEP12(2021), 082, arXiv:2106.00364 [hep-lat]

    arXiv 2021

  55. [55]

    M. C` e, C. Consonni, G. P. Engel, and L. Giusti, Phys. Rev. D92, 074502 (2015), arXiv:1506.06052 [hep-lat]

    Pith/arXiv arXiv 2015

  56. [56]

    M. C` e, M. Garcia Vera, L. Giusti, and S. Schaefer, Phys. Lett. B762, 232 (2016), arXiv:1607.05939 [hep-lat]

    Pith/arXiv arXiv 2016

  57. [57]

    Campostrini, A

    M. Campostrini, A. Di Giacomo, H. Panagopoulos, and E. Vicari, Nucl. Phys. B329, 683 (1990)

    1990

  58. [58]

    D’Elia, Nucl

    M. D’Elia, Nucl. Phys. B661, 139 (2003), arXiv:hep- lat/0302007 [hep-lat]

    arXiv 2003

  59. [59]

    Vicari and H

    E. Vicari and H. Panagopoulos, Phys. Rept.470, 93 (2009), arXiv:0803.1593 [hep-th]

    Pith/arXiv arXiv 2009

  60. [60]

    Berg, Phys

    B. Berg, Phys. Lett. B104, 475 (1981)

    1981

  61. [61]

    Iwasaki and T

    Y. Iwasaki and T. Yoshie, Phys. Lett. B131, 159 (1983)

    1983

  62. [62]

    Teper, Phys

    M. Teper, Phys. Lett. B162, 357 (1985)

    1985

  63. [63]

    Ilgenfritz, M

    E.-M. Ilgenfritz, M. Laursen, G. Schierholz, M. M¨ uller- Preussker, and H. Schiller, Nucl. Phys. B268, 693 (1986)

    1986

  64. [64]

    Narayanan and H

    R. Narayanan and H. Neuberger, JHEP03(2006), 064, arXiv:hep-th/0601210

    Pith/arXiv arXiv 2006

  65. [65]

    L¨ uscher, Commun

    M. L¨ uscher, Commun. Math. Phys.293, 899 (2010), arXiv:0907.5491 [hep-lat]

    Pith/arXiv arXiv 2010

  66. [66]

    L¨ uscher, JHEP08(2010), 071, [Erratum: JHEP03 (2014) 092], arXiv:1006.4518 [hep-lat]

    M. L¨ uscher, JHEP08(2010), 071, [Erratum: JHEP03 (2014) 092], arXiv:1006.4518 [hep-lat]

    Pith/arXiv arXiv 2010

  67. [67]

    Morningstar and M

    C. Morningstar and M. J. Peardon, Phys. Rev. D69, 054501 (2004), arXiv:hep-lat/0311018

    Pith/arXiv arXiv 2004

  68. [68]

    Alles, L

    B. Alles, L. Cosmai, M. D’Elia, and A. Papa, Phys. Rev. D62, 094507 (2000), arXiv:hep-lat/0001027 [hep-lat]

    Pith/arXiv arXiv 2000

  69. [69]

    Bruckmann, C

    F. Bruckmann, C. Gattringer, E.-M. Ilgenfritz, M. Muller-Preussker, A. Schafer, and S. Solbrig, Eur. Phys. J. A33, 333 (2007), arXiv:hep-lat/0612024 [hep-lat]

    Pith/arXiv arXiv 2007

  70. [70]

    Bonati and M

    C. Bonati and M. D’Elia, Phys. Rev. DD89, 105005 (2014), arXiv:1401.2441 [hep-lat]

    Pith/arXiv arXiv 2014

  71. [71]

    Alexandrou, A

    C. Alexandrou, A. Athenodorou, K. Cichy, A. Dromard, E. Garcia-Ramos, K. Jansen, U. Wenger, and F. Zimmer- mann, Eur. Phys. J. C80, 424 (2020), arXiv:1708.00696 [hep-lat]

    arXiv 2020

  72. [72]

    Alexandrou, A

    C. Alexandrou, A. Athenodorou, and K. Jansen, Phys. Rev. D92, 125014 (2015), arXiv:1509.04259 [hep-lat]

    Pith/arXiv arXiv 2015

  73. [73]

    Giusti and M

    L. Giusti and M. L¨ uscher, Eur. Phys. J. C79, 207 (2019), arXiv:1812.02062 [hep-lat]

    arXiv 2019

  74. [74]

    D¨ urr and G

    S. D¨ urr and G. Fuwa, Phys. Rev. D113, 054508 (2026), arXiv:2501.08217 [hep-lat]

    Pith/arXiv arXiv 2026

  75. [75]

    Altenkort, A

    L. Altenkort, A. M. Eller, O. Kaczmarek, L. Mazur, G. D. Moore, and H.-T. Shu, Phys. Rev. D103, 114513 (2021), arXiv:2012.08279 [hep-lat]

    arXiv 2021

  76. [76]

    Narison, Phys

    S. Narison, Phys. Lett. B255, 101 (199)

  77. [77]

    Vicari, Nucl

    E. Vicari, Nucl. Phys. B554, 301 (1999), arXiv:hep- lat/9901008

    arXiv 1999

  78. [78]

    Seiler, Phys

    E. Seiler, Phys. Lett. B525, 355 (2002), arXiv:hep- th/0111125

    arXiv 2002

  79. [79]

    Moskalet al., Phys

    P. Moskalet al., Phys. Rev. Lett.80, 3202 (1998), arXiv:nucl-ex/9803002

    Pith/arXiv arXiv 1998

  80. [80]

    J. P. Singh and S. D. Patel, Phys. Lett. B791, 249 (2019), arXiv:1812.06275 [hep-ph]

    Pith/arXiv arXiv 2019

Showing first 80 references.