arxiv: 2604.01097 · v2 · submitted 2026-04-01 · ✦ hep-lat

Recognition: 1 theorem link

· Lean Theorem

Strong coupling constant from the 1-loop improved static energy

Viljami Leino , Alexei Bazavov , Nora Brambilla , Georg von Hippel , Andreas S. Kronfeld , Julian Mayer-Steudte , Peter Petreczky , Sipaz Sharma

show 3 more authors

Sebastian Steinbei{\ss}er Antonio Vairo Johannes H. Weber

Authors on Pith no claims yet

Pith reviewed 2026-05-13 21:08 UTC · model grok-4.3

classification ✦ hep-lat

keywords strong coupling constantlattice QCDstatic energyWilson loopperturbation theory improvementalpha_s extractiondiscretization artifacts

0 comments

The pith

1-loop lattice perturbation theory improvement of the Wilson loop sharpens the extraction of the strong coupling constant α_s from the static energy.

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

The paper demonstrates that applying a one-loop correction derived from lattice perturbation theory to the Wilson loop reduces discretization errors in the static energy observable. This correction enables a more precise comparison between lattice data and continuum perturbative predictions of the static energy up to three loops with ultrasoft resummation. A preliminary reanalysis of existing (2+1)-flavor QCD simulation data shows the resulting improvement in the determined value of α_s.

Core claim

The static energy extracted from Wilson line correlators on the lattice can be improved at short distances by incorporating one-loop lattice perturbation theory corrections to the Wilson loop; this reduces discretization artifacts and yields a more accurate matching to the three-loop perturbative expression, thereby improving the determination of α_s when applied to (2+1)-flavor lattice data.

What carries the argument

The 1-loop lattice perturbation theory improved Wilson loop, which supplies additive corrections that suppress leading lattice artifacts in the static energy at short distances.

If this is right

The improved static energy allows reliable use of shorter lattice distances in the α_s fit, increasing the lever arm against perturbative uncertainties.
Systematic errors from lattice artifacts are reduced, leading to smaller total uncertainty in the extracted α_s.
The method can be directly applied to future higher-statistics (2+1+1)-flavor ensembles without changing the continuum matching procedure.
Consistency between lattice and perturbative static energy improves at the distances used for the fit.

Where Pith is reading between the lines

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

Extending the improvement to two-loop order could further suppress residual cutoff effects and tighten the α_s error budget.
The same correction technique may be portable to other short-distance observables involving Wilson lines, such as heavy-quark potentials or quarkonia correlators.
If the improvement proves robust, it could become a standard preprocessing step for any lattice determination of α_s that relies on static-energy matching.

Load-bearing premise

The one-loop lattice perturbation theory correction is assumed to capture and remove the dominant discretization artifacts in the Wilson loop without introducing new biases that would distort the short-distance matching to continuum perturbation theory.

What would settle it

If the α_s value extracted after applying the one-loop improvement deviates by more than the quoted uncertainty from independent determinations such as those from the static potential at larger distances or from τ-decay data, the improvement claim would be falsified.

Figures

Figures reproduced from arXiv: 2604.01097 by Alexei Bazavov, Andreas S. Kronfeld, Antonio Vairo, Georg von Hippel, Johannes H. Weber, Julian Mayer-Steudte, Nora Brambilla, Peter Petreczky, Sebastian Steinbei{\ss}er, Sipaz Sharma, Viljami Leino.

**Figure 1.** Figure 1: Left: The leading factor normalized static energy 𝑅 = −𝑟𝐸0/𝐶F for the two different approaches to reduce the renormalon contribution. Right: The difference between the two methods. ultrasoft logarithms ln[ 1 2 𝐶A𝛼s(𝜇us)]. We set the soft scale to be 𝜇 = 1/𝑟 and the ultrasoft scale 𝜇us = 𝐶A𝛼s(𝜇)/2𝑟. The coefficients 𝑥ˆ𝑘 describe the rotation symmetry breaking discretization effects arising from lattice pert… view at source ↗

**Figure 2.** Figure 2: Left: the interpolation of residual finite mass effects for 1-loop lattice perturbation theory. Right: The final interpolation reconstruction of the fermionic contribution (x’s) compared to data from HPsrc (solid symbols). The solid line shows the perturbative curve that we are expected to approach asymptotically. 1 2 3 4 5 6 7 8 r/a −0.20 −0.18 −0.16 −0.14 rE0 x1 ΛV/Λlat =42.44(28) 1 2 3 4 5 6 7 8 r/a 0.9… view at source ↗

**Figure 3.** Figure 3: Left: Example of the final 1-loop lattice perturbation theory contribution for a single ensemble and a fit to Λ-ratio. Right: The final 1-loop improvement compared to the tree-level improvement. an arbitrary mass as 𝐴 = log(𝐴2𝑚 2 𝑞 + 𝐴1𝑚𝑞 + 𝐴0) with 𝐴𝑖 being fit parameters. Next, we note that most of mass dependence is already captured by the perturbative curve and we define a reduced quantity and then int… view at source ↗

**Figure 4.** Figure 4: Example of the Λ extraction from a joint fit over 4 finest ensembles with 1-loop improved distances. in the proceedings, we do not include any further terms to describe lattice artifacts beyond the tree-level and one-loop improvement. To keep the scale dependence down and to stay in the perturbative range, we limit the fits to maximum separation of 𝑟 < 0.15 fm and fit over all possible data ranges. The fit… view at source ↗

read the original abstract

The static energy is an excellent observable for extracting the strong coupling $\alpha_s$ on the lattice. For short distances, the static energy can be calculated both on the lattice using Wilson line correlators, and with perturbation theory up to three loop accuracy with leading ultrasoft log resummation. Comparing the perturbative expression and lattice data allows for precise determination of $\alpha_s$. We present early results for 1-loop lattice perturbation theory improvement of the Wilson loop and show how it improves the $\alpha_s$ extraction. We present a preliminary reanalysis of the TUMQCD (2+1)-flavor QCD data.

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

Preliminary 1-loop lattice PT improvement on static energy data gives a cleaner short-distance match to 3-loop PT, but the manuscript is still too early to size the actual gain.

read the letter

This paper reports early results on correcting Wilson loop data for the static energy with 1-loop lattice perturbation theory, then reanalyzing the existing TUMQCD (2+1)-flavor ensembles to extract alpha_s. The central move is straightforward: compute the 1-loop lattice coefficients for the Wilson loop and apply them so the short-distance lattice points sit closer to the continuum three-loop expression that includes ultrasoft log resummation. They show that this correction visibly changes the fit, which is the concrete new piece. The setup itself is clean; they reuse high-quality ensembles and keep the improvement calculation independent of the alpha_s fit, so circularity stays low. The approach is a natural extension of existing lattice PT tools to an observable where continuum PT is already at three loops, and that combination is worth documenting. The soft spot is the preliminary character. No numerical shift in alpha_s is given, no error budget appears, and there is no direct test (finer lattices or 2-loop comparison) for residual O(a^2) artifacts after the 1-loop term. The stress-test worry about those residuals biasing the matching is therefore still open. Until those checks are shown, it is hard to know whether the improvement is large enough to matter at the target precision. This is for lattice groups already working on static potentials or precision alpha_s. A reader in that niche will see how the correction behaves on real data and can judge whether to adopt it. The paper deserves peer review because the method is grounded and the community needs these incremental steps recorded, but the referees will have to insist on the missing quantitative results and artifact tests before acceptance.

Referee Report

2 major / 2 minor

Summary. The paper presents early results for applying 1-loop lattice perturbation theory to improve the Wilson loop observable used in the static energy, and demonstrates via a preliminary reanalysis of TUMQCD (2+1)-flavor ensembles how this correction enhances the extraction of the strong coupling α_s by matching short-distance lattice data to three-loop continuum perturbation theory with ultrasoft log resummation.

Significance. If the 1-loop improvement demonstrably reduces discretization errors without introducing new biases, the work would strengthen lattice determinations of α_s at short distances by providing a more controlled matching to high-order perturbative expressions, potentially reducing systematic uncertainties in a key input for precision QCD phenomenology.

major comments (2)

[Abstract and Results] Abstract and Results section: The central claim that the 1-loop lattice PT improvement 'improves the α_s extraction' is asserted without any quantitative results, such as the numerical shift in the fitted α_s value, changes in fit quality (e.g., χ²/dof), or explicit error budgets before versus after the correction. This leaves the magnitude of the improvement unassessed.
[Results] Results section: The preliminary reanalysis does not quantify residual O(a²) lattice artifacts after the 1-loop correction (e.g., via direct comparison to 2-loop lattice PT coefficients or to finer lattice spacings), nor propagate them into the final uncertainty on α_s. If these residuals are comparable to the ultrasoft resummation or 3-loop terms, they could shift the extracted α_s by an amount comparable to the target precision.

minor comments (2)

[Methods] The notation for the improved Wilson loop (e.g., distinction between bare and tadpole-improved versions) should be defined explicitly in the Methods section to avoid ambiguity when comparing to continuum PT.
[Figures] Figure captions for the static energy plots should include the specific r/a ranges used in the α_s fit and the corresponding perturbative orders shown.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the major comments point by point below, noting that this work presents early and preliminary results. We will revise the manuscript to strengthen the quantitative presentation where feasible.

read point-by-point responses

Referee: [Abstract and Results] Abstract and Results section: The central claim that the 1-loop lattice PT improvement 'improves the α_s extraction' is asserted without any quantitative results, such as the numerical shift in the fitted α_s value, changes in fit quality (e.g., χ²/dof), or explicit error budgets before versus after the correction. This leaves the magnitude of the improvement unassessed.

Authors: We agree that the preliminary nature of the reanalysis means the current draft lacks explicit numerical comparisons. In the revised version we will add direct before/after values for the fitted α_s, the change in χ²/dof, and a concise error-budget table that isolates the effect of the 1-loop Wilson-loop improvement. revision: yes
Referee: [Results] Results section: The preliminary reanalysis does not quantify residual O(a²) lattice artifacts after the 1-loop correction (e.g., via direct comparison to 2-loop lattice PT coefficients or to finer lattice spacings), nor propagate them into the final uncertainty on α_s. If these residuals are comparable to the ultrasoft resummation or 3-loop terms, they could shift the extracted α_s by an amount comparable to the target precision.

Authors: The present results are preliminary and we do not yet have a complete estimate of the remaining O(a²) artifacts. We will add a comparison of the 1-loop improved data against the two available finer TUMQCD spacings and will include a conservative estimate of the residual discretization uncertainty in the final error budget on α_s. revision: partial

Circularity Check

0 steps flagged

Minor self-citation of prior ensembles; central α_s fit remains independent of inputs

full rationale

The paper computes the 1-loop lattice PT improvement of the Wilson loop as a separate perturbative calculation and then matches the corrected lattice static energy to an independent three-loop continuum PT expression (with ultrasoft log resummation) to extract α_s. The reanalysis uses existing TUMQCD ensembles as input data rather than deriving the coupling or the improvement factor from the same fitted quantities by construction. No load-bearing step reduces to a self-citation chain or to a fitted parameter renamed as a prediction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The extraction rests on the validity of three-loop perturbative QCD with ultrasoft resummation at short distances and on the assumption that 1-loop lattice PT accurately corrects the dominant lattice artifacts; no new entities are introduced.

free parameters (1)

α_s
The strong coupling is the single parameter fitted by matching the improved lattice static energy to the perturbative expression.

axioms (1)

domain assumption Perturbative expansion of the static energy up to three loops plus leading ultrasoft log resummation is reliable at the short distances considered.
Invoked when comparing lattice data to the perturbative formula for α_s extraction.

pith-pipeline@v0.9.0 · 5442 in / 1291 out tokens · 41060 ms · 2026-05-13T21:08:39.733397+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlphaDerivationExplicit.lean alphaProvenanceCert unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We present early results for 1-loop lattice perturbation theory improvement of the Wilson loop and show how it improves the α_s extraction. We present a preliminary reanalysis of the TUMQCD (2+1)-flavor QCD data.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 6 internal anchors

[1]

G. S. Bali,QCD forces and heavy quark bound states,Phys. Rept.343(2001) 1 [hep-ph/0001312]

work page Pith review arXiv 2001
[2]

Brambilla, A

N. Brambilla, A. Pineda, J. Soto and A. Vairo,The infrared behavior of the static potential in perturbative QCD,Phys. Rev. D60(1999) 091502 [hep-ph/9903355]

work page arXiv 1999
[3]

The Renormalization Group Improvement of the QCD Static Potentials

A. Pineda and J. Soto,The renormalization group improvement of the QCD static potentials, Phys. Lett. B495(2000) 323 [hep-ph/0007197]

work page internal anchor Pith review Pith/arXiv arXiv 2000
[4]

Brambilla, X

N. Brambilla, X. Garcia i Tormo, J. Soto and A. Vairo,The logarithmic contribution to the QCD static energy at N4LO,Phys. Lett. B647(2007) 185 [hep-ph/0610143]

work page arXiv 2007
[5]

The QCD static energy at NNNLL

N. Brambilla, A. Vairo, X. Garcia i Tormo and J. Soto,The QCD static energy at N3LL, Phys. Rev. D80(2009) 034016 [0906.1390]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[6]

Anzai, Y

C. Anzai, Y. Kiyo and Y. Sumino,Static QCD potential at three-loop order,Phys. Rev. Lett. 104(2010) 112003 [0911.4335]

work page arXiv 2010
[7]

A. V. Smirnov, V. A. Smirnov and M. Steinhauser,Three-loop static potential,Phys. Rev. Lett.104(2010) 112002 [0911.4742]. [8]Flavour Lattice A veraging Group (FLAG)collaboration, Y. Aoki et al.,FLAG Review 2024,2411.04268

work page arXiv 2010
[8]

d’Enterria et al.,The strong coupling constant: State of the art and the decade ahead,J

D. d’Enterria et al.,The strong coupling constant: State of the art and the decade ahead,J. Phys. G51(2024) 090501 [2203.08271]

work page arXiv 2024
[9]

Brambilla, X

N. Brambilla, X. Garcia i Tormo, J. Soto and A. Vairo,Precision determination of𝑟0ΛMS from the QCD static energy,Phys. Rev. Lett.105(2010) 212001 [1006.2066]. 8 Strong coupling from the one-loop improved static energyViljami Leino

work page arXiv 2010
[10]

Husung, M

N. Husung, M. Koren, P. Krah and R. Sommer,SU(3) Yang Mills theory at small distances and fine lattices,EPJ Web Conf.175(2018) 14024 [1711.01860]

work page arXiv 2018
[11]

Brambilla, V

N. Brambilla, V. Leino, J. Mayer-Steudte and A. Vairo,Static force from generalized Wilson loops on the lattice using the gradient flow,Phys. Rev. D109(2024) 114517 [2312.17231]. [13]ETMcollaboration, K. Jansen, F. Karbstein, A. Nagy and M. Wagner,ΛMS from the static potential for QCD with𝑛𝑓 =2dynamical quark flavors,JHEP01(2012) 025 [1110.6859]

work page arXiv 2024
[12]

Karbstein, A

F. Karbstein, A. Peters and M. Wagner,Λ (𝑛 𝑓 =2) MS from a momentum space analysis of the quark-antiquark static potential,JHEP09(2014) 114 [1407.7503]

work page arXiv 2014
[13]

Karbstein, M

F. Karbstein, M. Wagner and M. Weber,Determination ofΛ (𝑛 𝑓 =2) MS and analytic parametrization of the static quark-antiquark potential,Phys. Rev. D98(2018) 114506 [1804.10909]

work page arXiv 2018
[14]

Bazavov, N

A. Bazavov, N. Brambilla, X. Garcia i Tormo, P. Petreczky, J. Soto and A. Vairo, Determination of𝛼𝑠 from the QCD static energy,Phys. Rev. D86(2012) 114031 [1205.6155]

work page arXiv 2012
[15]

Bazavov, N

A. Bazavov, N. Brambilla, X. Garcia i Tormo, P. Petreczky, J. Soto and A. Vairo, Determination of𝛼s from the QCD static energy: An update,Phys. Rev. D90(2014) 074038 [1407.8437]

work page arXiv 2014
[16]

Takaura, T

H. Takaura, T. Kaneko, Y. Kiyo and Y. Sumino,Determination of𝛼s from static QCD potential: OPE with renormalon subtraction and lattice QCD,JHEP04(2019) 155 [1808.01643]. [19]TUMQCDcollaboration, A. Bazavov, N. Brambilla, X. Garcia i Tormo, P. Petreczky, J. Soto, A. Vairo et al.,Determination of the QCD coupling from the static energy and the free energy,P...

work page arXiv 2019
[17]

Ayala, X

C. Ayala, X. Lobregat and A. Pineda,Determination of𝛼(𝑀𝑍 )from an hyperasymptotic approximation to the energy of a static quark-antiquark pair,JHEP09(2020) 016 [2005.12301]

work page arXiv 2020
[18]

J. M. Mena-Valle, V. Mateu and P. G. Ortega,A Precise𝛼𝑠 Determination from the R-improved QCD Static Energy,2510.24846. [22]TUMQCDcollaboration, V. Leino, A. Bazavov, N. Brambilla, A. S. Kronfeld, J. Mayer-Steudte, P. Petreczky et al.,Strong coupling in (2+1+1)-flavor QCD,PoS LATTICE2024(2025) 298 [2502.01453]

work page arXiv 2025
[19]

Brambilla, V

N. Brambilla, V. Leino, O. Philipsen, C. Reisinger, A. Vairo and M. Wagner,Lattice gauge theory computation of the static force,Phys. Rev. D105(2022) 054514 [2106.01794]. [24]TUMQCDcollaboration, N. Brambilla, J. Komijani, A. S. Kronfeld and A. Vairo,Relations between heavy-light meson and quark masses,Phys. Rev. D97(2018) 034503 [1712.04983]. 9 Strong co...

work page arXiv 2022
[20]

Komijani,A discussion on leading renormalon in the pole mass,JHEP08(2017) 062 [1701.00347]

J. Komijani,A discussion on leading renormalon in the pole mass,JHEP08(2017) 062 [1701.00347]

work page arXiv 2017
[21]

A. S. Kronfeld,Factorial growth at low orders in perturbative QCD: control over truncation uncertainties,JHEP12(2023) 108 [2310.15137]

work page arXiv 2023
[22]

Curci, P

G. Curci, P. Menotti and G. Paffuti,Symanzik’s Improved Lagrangian for Lattice Gauge Theory,Phys. Lett. B130(1983) 205

work page 1983
[23]

U. M. Heller and F. Karsch,One Loop Perturbative Calculation of Wilson Loops on Finite Lattices,Nucl. Phys. B251(1985) 254

work page 1985
[24]

Weisz and R

P. Weisz and R. Wohlert,Continuum Limit Improved Lattice Action for Pure Yang-Mills Theory. 2.,Nucl. Phys. B236(1984) 397

work page 1984
[25]

J. R. Snippe,Computation of the one loop Symanzik coefficients for the square action,Nucl. Phys. B498(1997) 347 [hep-lat/9701002]

work page internal anchor Pith review Pith/arXiv arXiv 1997
[26]

Computation of the b-quark Mass with Perturbative Matching at the Next-to-Next-to-Leading Order

G. Martinelli and C. T. Sachrajda,Computation of the b quark mass with perturbative matching at the next-to-next-to-leading order,Nucl. Phys. B559(1999) 429 [hep-lat/9812001]

work page internal anchor Pith review Pith/arXiv arXiv 1999
[27]

Athenodorou and H

A. Athenodorou and H. Panagopoulos,Large Wilson loops with overlap and clover fermions: Two-loop evaluation of the b-quark mass shift and the quark-antiquark potential,Nucl. Phys. B799(2008) 1 [hep-lat/0509039]

work page arXiv 2008
[28]

G. S. Bali and P. Boyle,Perturbative Wilson loops with massive sea quarks on the lattice, hep-lat/0210033. [34]HPQCDcollaboration, E. Follana, Q. Mason, C. Davies, K. Hornbostel, G. P. Lepage, J. Shigemitsu et al.,Highly improved staggered quarks on the lattice, with applications to charm physics,Phys. Rev. D75(2007) 054502 [hep-lat/0610092]

work page internal anchor Pith review Pith/arXiv arXiv 2007
[29]

A. Hart, G. M. von Hippel, R. R. Horgan and E. H. Muller,Automated generation of lattice QCD feynman rules,Comput. Phys. Commun.180(2009) 2698 [0904.0375]

work page arXiv 2009
[30]

Effective Field Theory Approach to Pionium

D. Eiras and J. Soto,Effective field theory approach to pionium,Phys. Rev. D61(2000) 114027 [hep-ph/9905543]

work page internal anchor Pith review Pith/arXiv arXiv 2000
[31]

G. M. von Hippel, V. Leino and S. Steinbeißer,One loop improvement of the static potential with HISQ quarks,In preparation: TUM-EFT 171/22(2026)

work page 2026
[32]

Larsen, S

R. Larsen, S. Mukherjee, P. Petreczky, H.-T. Shu and J. H. Weber,Scale Setting and Strong Coupling Determination in the Gradient Flow Scheme for 2+1 Flavor Lattice QCD, 2502.08061

work page arXiv
[33]

W. I. Jay and E. T. Neil,Bayesian model averaging for analysis of lattice field theory results, Phys. Rev. D103(2021) 114502 [2008.01069]. 10

work page arXiv 2021