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arxiv: 2501.08217 · v2 · submitted 2025-01-14 · ✦ hep-lat · hep-ph

Topological susceptibility and excess kurtosis in SU(3) Yang-Mills theory

Stephan Durr , Gianluca Fuwa This is my paper

Pith reviewed 2026-05-23 05:43 UTC · model grok-4.3

classification ✦ hep-lat hep-ph
keywords topological susceptibilitySU(3) Yang-Millsgradient flowcontinuum extrapolationlattice gauge theorytopological chargekurtosis
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0 comments X

The pith

The topological susceptibility in SU(3) Yang-Mills theory reaches χ_top^{1/4} = 198.1(0.7)(2.7) MeV after continuum extrapolation.

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

This paper computes the topological susceptibility of pure SU(3) gauge theory on the lattice with high precision across seven lattice spacings and seven physical volumes. Two complementary gradient-flow smoothing strategies, one holding flow time fixed in lattice units and one in physical units, are shown to produce the same extrapolated continuum value. The result is reported both in units of the Sommer scale r0 and in physical MeV units. An appendix tracks the excess kurtosis of the topological charge distribution and finds it falls proportionally to L^{-2} at large volumes.

Core claim

Using a gluonic topological charge measured after gradient flow smoothing, the authors establish that the two smoothing strategies yield a universal continuum limit. The extrapolated value is χ_top^{1/4} r_0 = 0.4775(14)(11), or 198.1(0.7)(2.7) MeV. The appendix data indicate that the excess kurtosis ⟨q^4⟩ / ⟨q^2⟩^2 − 3 decreases proportionally to L^{-2} for large box sizes L.

What carries the argument

Gradient flow smoothing of the gluonic topological charge operator, applied with two complementary choices for the flow-time scale.

Load-bearing premise

The two complementary smoothing strategies produce a universal continuum limit.

What would settle it

A simulation at a substantially smaller lattice spacing in which the two smoothing strategies produce statistically incompatible extrapolated values would falsify the claim of a universal continuum limit.

Figures

Figures reproduced from arXiv: 2501.08217 by Gianluca Fuwa, Stephan Durr.

Figure 1
Figure 1. Figure 1: ⟨1−Re Tr(Uµν)/3⟩ of the ensembles used in the continuum extrapolation, unsmeared (left) and with one of the three smoothing strategies (right). L/a β 7 stout flow 0.21 fm flow 0.30 fm 12 5.9421 1.2757(18) 1.3782(57) 1.2337(20) 14 6.0314 1.22881(57) 1.2522(12) 1.15821(41) 16 6.1142 1.1974(10) 1.17807(80) 1.11444(49) 18 6.1912 1.17499(62) 1.13280(43) 1.08579(26) 20 6.2629 1.15682(53) 1.10237(47) 1.06672(17) … view at source ↗
Figure 2
Figure 2. Figure 2: The Zq factors involved, with quadratic fits in (a/r0) 2 (left) and rational fits in g 2 0 (right). L/a β 7 stout flow 0.21 fm flow 0.30 fm 7 stout flow 0.21 fm flow 0.30 fm 12 5.9421 1.4653(78) 1.2158(64) 1.5970(85) 2.453(12) 2.387(12) 2.486(13) 14 6.0314 1.5362(74) 1.4734(73) 1.7419(88) 2.369(11) 2.364(12) 2.365(12) 16 6.1142 1.554(12) 1.633(11) 1.806(11) 2.268(17) 2.294(14) 2.254(14) 18 6.1912 1.592(12)… view at source ↗
Figure 3
Figure 3. Figure 3: Continuum extrapolation of the topological susceptibility with an ansatz linear in a 2 based on the six finest spacings (left) and with an ansatz quadratic in a 2 based on all seven spacings (right). [χtopr 4 0 ]lin. [χtopr 4 0 ]quad. [χ 1/4 topr0]lin. [χ 1/4 topr0]quad. 7 stout 0.05230(57) 0.0506(11) 0.4789(12) 0.4749(22) flow 0.21 fm 0.05187(54) 0.04845(96) 0.4780(12) 0.4703(21) flow 0.30 fm 0.05173(54) … view at source ↗
Figure 4
Figure 4. Figure 4: Same as [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The standard continuum extrapolation as presented in the right panel of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Continuum extrapolation of three varieties of the excess kurtosis (top, middle, bottom) for our three smearing strategies (“7 stout” left, “flow 0.21 fm” middle, “flow 0.30 fm” right). We select our intermediate coupling (β = 6.1912, 184 lattice) of the continuum series, and aug￾ment it with smaller/larger boxes as indicated in Tab. 8. We do fewer updates between adjacent measurements [to establish τint(q … view at source ↗
Figure 7
Figure 7. Figure 7: Volume scaling of the topological susceptibility plotted versus 1/L (left) and 1/V (right). The standard inverse size/volume used in Sec. 4 is marked with a dashed vertical line. The runs of Ref. [6] were performed with a box size ∼ 10% smaller, which coincides roughly with the fifth data point. The seventh data point is out of scale (both horizontally and vertically). more detail (see App. A). As we shall… view at source ↗
Figure 8
Figure 8. Figure 8: Large volume behavior of the three kurtosis varieties with the “7 stout” smoothing strategy. In the left panel one variety is multiplied with (L/r0) 4 , one unchanged, one divided by (L/r0) 4 (top, middle, bottom). In the right panel the respective factors are (L/r0) 2 , (r0/L) 2 and (r0/L) 6 . use a large variety of fit functions and cuts to determine the systematic uncertainty. Our standard volume V = L … view at source ↗
Figure 9
Figure 9. Figure 9: Large volume scaling of the topological susceptibility (top left) and of the three kurtosis varieties (remaining panels) with the “7 stout” strategy. The new data (β = 5.9421, blue triangles) show mild cut-off effects relative to the old ones (β = 6.1912, orange circles), but they extend to larger box sizes. The power-law fits based on the five largest volumes suggest the scaling laws (24). References [1] … view at source ↗
read the original abstract

We present a high-precision study of the topological susceptibility in $SU(3)$ pure gauge theory in four space-time dimensions. The result is based on ensembles at seven lattice spacings and in seven physical volumes to facilitate a controlled continuum and infinite-volume extrapolation. We use a gluonic topological charge measurement, with gradient flow smoothing in the operator. Two complementary smoothing strategies are used (one keeps the flow time fixed in lattice units, one in physical units). Our data support the idea that both strategies yield a universal continuum limit; we find $\chi_\mathrm{top}^{1/4}r_0=0.4775(14)(11)$ or $\chi_\mathrm{top}^{1/4}=198.1(0.7)(2.7)\,\mathrm{MeV}$. Our appendix data suggest that the excess kurtosis $\langle q^4 \rangle / \langle q^2 \rangle^2-3$ decreases $\propto L^{-2}$ for large box sizes $L$.

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

1 major / 1 minor

Summary. The paper presents a high-precision lattice study of the topological susceptibility χ_top in four-dimensional SU(3) Yang-Mills theory. Using gluonic topological charge with gradient-flow smoothing on ensembles at seven lattice spacings and seven physical volumes, the authors perform controlled continuum and infinite-volume extrapolations. Two smoothing strategies (flow time fixed in lattice units versus physical units) are employed; the data are stated to support a universal continuum limit, yielding the result χ_top^{1/4} r_0 = 0.4775(14)(11) or χ_top^{1/4} = 198.1(0.7)(2.7) MeV. An appendix observation that the excess kurtosis decreases proportionally to L^{-2} for large volumes is also reported.

Significance. If the central result holds, the work supplies a benchmark value for χ_top with quantified statistical and systematic errors from multiple spacings and volumes. The dual smoothing cross-check and the volume dependence of kurtosis are useful additions to the literature on topological observables in pure gauge theory.

major comments (1)
  1. [Results and continuum extrapolation discussion] The headline value χ_top^{1/4} r_0 = 0.4775(14)(11) is obtained only after asserting that the two gradient-flow smoothing protocols produce identical a→0 limits. The manuscript states that the data support universality but does not describe an explicit test (separate continuum extrapolations for each strategy versus a joint fit enforcing a shared continuum value). Without this check, residual O(a²) or higher artifacts that differ between the fixed-t/a² and fixed-physical-t strategies could be absorbed into the shared ansatz, making the quoted precision dependent on an unverified assumption.
minor comments (1)
  1. [Appendix] The appendix statement that excess kurtosis decreases ∝ L^{-2} would benefit from an explicit fit or plot showing the coefficient and its uncertainty to allow readers to assess the large-volume regime.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and constructive comments on our manuscript. We address the major comment point by point below and will revise the paper to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [Results and continuum extrapolation discussion] The headline value χ_top^{1/4} r_0 = 0.4775(14)(11) is obtained only after asserting that the two gradient-flow smoothing protocols produce identical a→0 limits. The manuscript states that the data support universality but does not describe an explicit test (separate continuum extrapolations for each strategy versus a joint fit enforcing a shared continuum value). Without this check, residual O(a²) or higher artifacts that differ between the fixed-t/a² and fixed-physical-t strategies could be absorbed into the shared ansatz, making the quoted precision dependent on an unverified assumption.

    Authors: We agree with the referee that an explicit description of the test for universality would strengthen the manuscript. Although our internal analysis included separate extrapolations for each smoothing strategy (which agree within statistical uncertainties) and a joint fit, this was not detailed in the original submission. In the revised manuscript, we will add a dedicated paragraph in the results section describing these checks, including the separate continuum limits for fixed-t/a² and fixed-physical-t strategies, and confirm that they are consistent. This addresses the concern about potential unverified assumptions in the quoted precision. revision: yes

Circularity Check

0 steps flagged

No circularity: result from direct Monte Carlo measurement and extrapolation

full rationale

The paper computes χ_top via direct measurement of the topological charge on Monte Carlo ensembles generated at seven lattice spacings and seven volumes. The quoted value χ_top^{1/4} r_0 = 0.4775(14)(11) follows from a controlled a→0 and L→∞ extrapolation of these measured data points. Two gradient-flow smoothing protocols are compared and stated to share a common continuum limit on the basis of the observed numerical agreement; this is an empirical check, not a reduction of the final number to a fitted parameter by construction. No equations, self-citations, or ansätze are invoked that would make the reported susceptibility equivalent to its own inputs. The kurtosis scaling is likewise an observed finite-volume trend in the appendix data. The derivation chain is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the standard assumptions of lattice regularization of SU(3) Yang-Mills theory, the existence of a well-defined continuum limit, and the validity of gradient flow for defining a topological charge operator; no additional free parameters or invented entities are introduced in the abstract.

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Forward citations

Cited by 3 Pith papers

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

  1. Scaling flow-based approaches for topology sampling in $\mathrm{SU}(3)$ gauge theory

    hep-lat 2025-10 unverdicted novelty 6.0

    Out-of-equilibrium simulations with open-to-periodic boundary switching plus a tailored stochastic normalizing flow enable efficient topology sampling in the continuum limit of four-dimensional SU(3) Yang-Mills theory.

  2. Enhanced Sampling Techniques for Lattice Gauge Theory

    hep-lat 2026-04 unverdicted novelty 5.0

    Metadynamics bias potentials and volume-extrapolation strategies reduce integrated autocorrelation times of topological charge in lattice gauge theories.

  3. Topological Susceptibility and QCD at Finite Theta Angle

    hep-lat 2026-04 unverdicted novelty 1.0

    A pedagogical review summarizing analytic predictions and recent lattice results for theta-dependence and topological susceptibility in QCD.

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