Update on two-level sampling for glueball observables in quenched QCD

Francesco Knechtli; Jacob Finkenrath; Juan Andr\'es Urrea-Ni\~no; Lorenzo Barca; Michael Peardon; Stefan Schaefer

REVIEW 3 major objections 6 minor 1 cited by

Update on two-level sampling for glueball observables in quenched QCD

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Two-level sampling, combined with distillation, cuts the statistical error of disconnected quark-loop correlation functions in quenched QCD, giving a variance that falls as $1/N_1^2$ when the loops lie in different dynamical regions.

desk verdict A clean, honest methods progress report with a real statistical gap: the corrected estimator's error does not achieve the advertised 1/N1^2 scaling because the bias correction is one-level. read the letter →

arxiv 2501.17988 v1 pith:WL5ROTZH submitted 2025-01-29 hep-lat

Lorenzo Barca , Jacob Finkenrath , Francesco Knechtli , Michael Peardon , Stefan Schaefer , Juan Andrés Urrea-Niño This is my paper · ORCID

classification hep-lat PACS 11.15.Ha12.38.Gc02.70.Uu

keywords two-levelsamplingmultilevelalgorithmdistillationdisconnecteddiagramsquarkloopsquenchedQCDglueballobservablessignal-to-noiseratio

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper combines distillation with a two-level sampling algorithm to reduce the statistical error of disconnected quark-loop correlation functions — the pieces of glueball and singlet-meson observables that are the most demanding to compute — in quenched QCD. By factorizing the quark propagator into overlapping domains with Dirichlet boundary conditions, the authors sub-average quark loops living in different temporal regions independently, then correct for the factorization bias with a one-level term. On a single $16^3 \times 64$ ensemble at $\beta=6.0$ ($m_\pi \approx 760$ MeV) they show the variance of the corrected two-level estimator falls as $1/N_1^2$ when the loops lie in different dynamical regions, compared with $1/N_1$ for standard sampling. The result extends the pure-gauge glueball speedup to fermionic disconnected diagrams and pushes the effective-mass signal of the would-be $f_0$ state to larger time separations.

What carries the argument

The load-bearing mechanism is the factorization of the quark propagator by domain decomposition: the temporal extent is split into four regions $\Lambda=\Lambda_0\oplus\Lambda_1\oplus\Lambda_2\oplus\Lambda_3$, with $\Lambda_0$ and $\Lambda_2$ updated (dynamical) and $\Lambda_1$ and $\Lambda_3$ frozen, and the propagator is approximated on the overlapping domains $\Omega_0$ and $\Omega_1$ with Dirichlet boundary conditions. This approximation is justified by exponential locality, $\|D^{-1}(y,x)\|\sim e^{-\frac12 m_\pi |y-x|}$, which makes distant gauge-field fluctuations irrelevant for a given quark loop. Distillation enters through the elementals $\phi$ and perambulators $\tau$ that compute the quark-loop traces, and the two-level estimator together with the bias correction $\delta$ carries the variance reduction: the $1/N_1^2$ scaling appears because sub-averages over independent level-1 updates in different dynamical regions decorrelate, as in the pure-gauge analysis.

What would settle it

Increase $N_1$ with $N_0$ fixed on the same ensemble (for example to $N_1=1000$) and plot the error of the corrected estimator $C^{2\text{-lvl}}$; if the error stops following the $1/N_1^2$ line and instead flattens toward the $1/\sqrt{N_1}$ behavior of the correction term, then the bias correction is the bottleneck and the exponential error reduction claimed for fermionic disconnected diagrams fails in that regime.

Watch

Extended reading notes

Core claim

The paper establishes that a two-level estimator built from an approximated, domain-decomposed quark propagator reduces the statistical error of disconnected quark-loop two-point functions in quenched QCD. The central object is the corrected estimator $C^{2\text{-lvl}}(t_1,t_0) = \tilde{C}^{2\text{-lvl}}(t_1,t_0) + \delta(t_1,t_0)$, where $\tilde{C}^{2\text{-lvl}}$ sub-averages the quark loops in separate dynamical regions using the approximated propagator, and $\delta = C^{1\text{-lvl}} - \tilde{C}^{1\text{-lvl}}$ removes the bias of the approximation using full-propagator one-level statistics. The paper reports that when the two loops are placed in different dynamical regions the variance of the two-level part scales as $1/N_1^2$, with corrections that fall exponentially with distance from the frozen regions, and that adding $\delta$ does not visibly bias this particular observable. The net effect is a reduction in the error of the disconnected scalar correlator and a longer usable window in the effective mass.

Load-bearing premise

The load-bearing premise is that the bias correction $\delta$, computed with one-level full-propagator statistics, stays small enough and accurate enough that it does not re-introduce the one-level $1/\sqrt{N_1}$ noise; the paper checks this only by observing no significant bias for this particular observable.

Editorial extensions

If this is right

With $N_0=101$ and $N_1=200$ sub-updates per level-0 configuration, two-level sampling extends the disconnected scalar correlator signal by a few more time slices than standard sampling; the paper notes the statistics are not yet sufficient for a reliable effective-mass plateau.
Because the variance scales as $1/N_1^2$ when the quark loops are in different dynamical regions, increasing the sub-measurements by a factor of 5 would reduce the error by a similar factor for $t>11a$, a route the authors identify toward a visible plateau.
The same corrected two-level estimator applies to the other singlet channels $\Gamma=1,\gamma_5,\gamma_\mu,\gamma_4\gamma_5,\gamma_i\gamma_j$, with the connected pieces still computed at one level.
For full QCD the method requires a local factorization of the fermion determinant, which the paper frames as the next step toward glueball studies in full QCD.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct consequence of the reported $1/N_1^2$ scaling is that any disconnected diagram built from quark loops should inherit the same variance reduction, not just the scalar channel, provided the loops sit in distinct dynamical regions and $\delta$ stays subdominant.
A testable extension would be to estimate $\delta$ from two independent one-level samples; if $\delta$'s variance dominates at larger $N_1$, the method would need a multi-level estimator for the correction itself, which the paper does not provide.
Because the factorization rests on exponential locality, the speedup should strengthen at heavier quark masses and weaken near the chiral limit; the present $m_\pi\approx760$ MeV setup is a favorable case, so the low-mass regime remains open.
The Dirichlet-boundary effects visible near the interfaces are repaired only through the one-level $\delta$; a fully local two-level scheme would need to control these boundary effects directly, for instance by enlarging the overlap regions.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing you should know: this is a genuinely new combination—distillation plus two-level sampling for disconnected quark-loop correlators—and it is written clearly and honestly. But the headline error-reduction claim is only rigorously demonstrated for the biased estimator in Eq. (18). Once the bias correction δ (Eq. 19) is added, the variance floor from that one-level correction prevents the 1/N1^2 scaling advertised, and Fig. 3 shows it: the filled circles sit above the empty circles for much of the time range.

What is actually new: the paper shows, on one quenched ensemble (β=6.0, 16^3×64, mπ~760 MeV), that approximated quark propagators from domain decomposition can be combined with distillation and two-level sampling to reduce the error of disconnected scalar two-point functions when the loops are in different dynamical regions. The unbiased estimator (Eq. 20) does still gain over standard sampling, but the gain is downgraded relative to the biased one. The authors are admirably candid that no effective-mass plateau is reached; they do not oversell this as a physics result.

The soft spots, in proportion: (1) The bias correction is a one-level estimator. Its variance has an N1-independent piece from the level-0 fluctuations, so the final estimator cannot scale as 1/N1^2 unless δ is negligible or itself factorized. The paper only tests bias, not variance, and only for one observable. This is a load-bearing gap in the central claim. (2) The demonstration is a single ensemble, single channel, no release of data or code mentioned. That is fine for a proceedings, but it limits the strength of the conclusion. (3) The references and context are fine; self-citation is to prior pure-gauge work on the same method and is appropriate.

Who this is for: lattice practitioners working on multilevel algorithms and glueball/fermionic disconnected diagrams. It deserves a serious referee, but the referee should push on the bias-correction variance. I would recommend the authors either show that δ is numerically negligible where the two-level gain matters (and use Eq. 18 alone), or redesign the correction so it is also two-level. Conditionally acceptable.

Referee Report

3 major / 6 minor

Summary. This proceedings contribution reports an exploratory combination of two-level sampling with distillation for disconnected quark-loop correlators in quenched QCD. On a single 16^3x64 ensemble at beta=6.0 with m_pi ~ 760 MeV, the authors compare one-level, approximate two-level, and bias-corrected two-level estimators (Eqs. 16-20) for scalar disconnected two-point functions. They find that the approximate two-level estimator reduces statistical errors when the quark loops lie in different dynamical regions, with variance reportedly scaling as 1/N1^2, and they show a modest extension of the disconnected-signal effective mass to larger separations. The central caveat is that the bias-corrected estimator (Eq. 20) uses a correction delta (Eq. 19) whose variance has an N1-independent one-level component, so the advertised variance reduction is not established for the estimator actually used in the physics results.

Significance. If the variance reduction for disconnected diagrams is confirmed, the method would be a useful step toward affordable glueball and singlet-meson spectroscopy with fermions. The paper is a concrete first demonstration that distillation and domain-decomposed propagators can be combined with multilevel integration, and the error analysis uses the Gamma-method with explicit configuration counts. However, the central quantitative claim currently rests on the uncorrected two-level estimator, while the corrected estimator used for the effective-mass comparison has not been shown to retain the 1/N1^2 scaling. The issue is load-bearing rather than cosmetic, and it needs to be addressed before the stated conclusions can be considered supported.

major comments (3)

[Section 4.2, Eqs. (19)-(20), Fig. 3] The bias correction delta in Eq. (19) is the difference of two one-level estimators, C^{1-lvl} and \tilde C^{1-lvl}, both averaged over the N0 x N1 configurations. For a fixed level-0 configuration i, the full quark loop in Eq. (14) depends on the frozen-region gauge fields, so the average over j does not remove the level-0 fluctuation of delta. Consequently Var(delta) has an N1-independent component of order 1/N0, and adding delta to \tilde C^{2-lvl} introduces a noise floor into C^{2-lvl} that prevents the advertised 1/N1^2 scaling in general. The empirical consequence is visible in Fig. 3, where the filled circles (corrected estimator) lie above the empty circles (uncorrected estimator) over much of the time range. The sentence in Section 4.2 stating that 'the variance scales with 1/N1^2' is therefore only supported for \tilde C^{2-lvl} in Eq. (18), not for the corrected estimator used in Fig. 4. The statement 'we do not observe a significant bias' addresses the expectation value of delta, not its variance, so it does not resolve this concern.
[Section 5, Conclusion] The conclusion that 'the two-level integration reduces substantially the statistical error of the disconnected contributions' is demonstrated for the uncorrected estimator in Eq. (18), but the corrected estimator in Eq. (20) is the one used for the physical results in Fig. 4. Until either delta is shown to be negligible in the relevant Delta t range, or delta is recomputed with a factorized estimator whose variance is also reduced by the two-level integration, the final estimator's gain is not established. A concrete test would be to plot Var(C^{2-lvl}) versus N1 alongside Var(\tilde C^{2-lvl}) and compare the slopes, including a fit to a floor term.
[Section 4.2, Fig. 3] The claim that the variance scales as 1/N1^2 is made without a quantitative fit. With only N1 values of 1, 50, and 200 and N0=101, the data are sufficient to fit Var(N1) = a/N1^2 + b, which would directly reveal whether the correction floor b is significant. Adding such a fit, or at least showing the ratio of the filled-to-empty circle errors, would make the scaling statement precise and would quantify how much of the two-level gain survives in the corrected estimator.

minor comments (6)

[Section 4.2] The text refers to 'C_I(t)' for the scalar correlator, while Eqs. (16)-(20) use the subscript Gamma; please define the relation between these notations.
[Fig. 2] The orange data points are said to be shifted along the x-axis for visibility, but the shift amount is not stated; this makes the comparison of the full and approximated loops at a given time slice ambiguous.
[Eq. (10) and Section 4.1] The symbol m_pi denotes the mass of the lightest pseudoscalar state in the quenched approximation; this is a valence-quark mass parameter, not the physical pion mass, and the text should state this explicitly.
[Section 4.1, footnote 1] The footnote states that autocorrelation effects are suppressed, but no integrated autocorrelation times are reported; providing these estimates for the loop observables would strengthen the error analysis.
[Section 2, Eq. (7)-(8)] The truncation to 10 Laplacian eigenvectors is introduced without a convergence study; a short check showing the stability of the disconnected correlator as a function of the number of eigenvectors would help assess the systematic error from distillation.
[Fig. 3] The figure legend is difficult to parse: the labels include both \tilde sigma_C and sigma_C for several N1 values, and the correspondence between symbols and estimators is not fully specified in the caption. Please make the caption self-contained.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the variance-scaling prediction is re-tested against new numerical data, and the bias correction is not fitted to the target result.

full rationale

The claimed derivation chain is not circular. The two-level estimator in Eq. (18) is a genuine factorization over local integrations of the approximated quark propagator, and the scaling expectation is imported from the authors' earlier pure-gauge study Ref. [5] as a conjecture to be checked on the new fermionic ensemble, not as a fitted input. The correction delta in Eq. (19) is estimated from the same N0 x N1 configurations and added in Eq. (20), but it is an unbiased additive correction estimated with the usual one-level variance; it is not a parameter fitted to the final effective mass, so the central physics result is not forced by construction. Self-citations (Refs. [4,5,9]) are present, but they are not load-bearing: Ref. [5] is an independent numerical result that is re-examined with new data, and Ref. [9] provides the domain-decomposition factorization technique rather than the error-reduction claim. The skeptic's concern that delta's one-level variance prevents the corrected estimator Eq. (20) from scaling as 1/N1^2 is a statistical correctness question about whether the claimed variance reduction survives the correction; it is not circularity because the paper's definitions do not make the 1/N1^2 scaling true by construction. The paper's own Fig. 3 even displays the corrected error (filled circles) above the uncorrected two-level error, which is an empirical check, not a tautology. The limitation acknowledged in Sec. 4.2 ('we do not observe a significant bias ... at least for this particular observable') concerns the expectation of delta, not its variance; this is a legitimate caveat about the estimator's performance, but it does not make the derivation circular. Overall, the derivation is self-contained against the new numerical data, with at most minor non-load-bearing self-citation, so the circularity score is 2.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim depends on standard lattice QCD assumptions plus the locality of the quark propagator, which justifies the domain decomposition. The paper introduces no new particles or forces; the main free parameters are the quark mass, the distillation truncation, the region geometry, and the statistics. The most fragile assumption is the exponential locality bound, whose numerical validity is only indirectly checked through the agreement of the approximated and full quark loops in Fig. 2.

free parameters (4)

kappa (hopping parameter) = 0.13393
Quark mass tuned so the lightest pseudoscalar has m_pi ≈ 760 MeV and the lowest non-interacting pi-pi energy is close to the pure-gauge scalar glueball mass (Section 4.1).
N_eig (number of Laplacian eigenvectors) = 10
Distillation truncation; a fixed low-rank approximation to the quark field used in all operators (Section 2).
Frozen-region boundary slices = Λ3: t/a in {0,1,61,62,63}; Λ1: t/a in {29,30,31,32,33}
Chosen by hand to define the domain decomposition for two-level integration (Section 4.1).
N0 and N1 = N0=101, N1=200
Numbers of level-0 and level-1 gauge configurations; chosen to control statistics and autocorrelations (Section 4.1).

assumptions (5)

domain assumption Quenched approximation: the fermion determinant is set to unity, so the gauge ensemble is pure SU(3) and fermionic Wick contractions are evaluated on quenched gauge fields.
Used throughout; the two-point function in Eq. (2) integrates only over the pure gauge action S_g.
domain assumption Exponential locality of the quark propagator: ||D^-1(y,x)|| ~ exp(-1/2 m_pi |y-x|), 'supported by empirical arguments'.
Section 3, Eq. (10); the entire domain decomposition and the approximation D^{-1}_{Ωr} rely on this locality.
domain assumption Domain-decomposed propagator with Dirichlet boundary conditions approximates the full propagator away from boundaries.
Section 4.1; the paper uses the full propagator near boundaries to compensate for Dirichlet artifacts (Section 4.2, Fig. 2).
standard math Standard lattice QCD framework: Wilson gauge action at β=6.0, Wilson clover fermions, periodic boundary conditions, Gamma-method error analysis.
Background setup stated in Section 4.1; error analysis follows Ref. [17].
ad hoc to paper Distillation projection onto the lowest 10 eigenvectors of the 3D Laplacian captures the relevant physics of the quark loops.
Section 2; a truncation specific to this work, with no systematic check of N_eig dependence.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Update on two-level sampling for glueball observables in quenched QCD." pith.science (2026). https://pith.science/paper/WL5ROTZH

@misc{pith2026250117988,
  author       = {Pith},
  title        = {Pith review of: Update on two-level sampling for glueball observables in quenched QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WL5ROTZH}},
  note         = {Machine review of arXiv:2501.17988}
}

abstract

We report our progress in combining a two-level sampling algorithm with distillation techniques for calculations of disconnected diagrams in quenched QCD. The simulations are performed on a single ensemble at $\beta=6.0$ and volume $V=16^3\times 64$, and at a pion mass of $m_\pi\approx 760~\mathrm{MeV}$.

Figures

Figures reproduced from arXiv: 2501.17988 by the authors.

**Figure 1.** Domain decomposition used in this analysis. The blue and white regions are the frozen and dynamical regions, respectively. 3. Two-level sampling for fermionic observables In the quenched approximation, the two-point functions in eq. (2) depend on the pure gauge action 𝑆𝑔 [𝑈] and the Wick contractions ⟨O(𝑡1)O¯ (𝑡0)⟩𝐹. The pure gauge action 𝑆𝑔 [𝑈] is local in the gauge fields as the action is constructed in terms of W… view at source ↗

**Figure 2.** Comparison between the 1-point functions of scalar quark loops using the approximated (blue) and full (orange) quark propagator. The scalar channel has non-vanishing expectation value because it has vacuum quantum numbers. The blue vertical bands highlight the location of the frozen regions Λ1 and Λ3. These estimates are computed using eqs. (14)-(15) with 𝑖 = 1, ..., 𝑁0 and 𝑗 = 1, ..., 𝑁1, and the errors are estimat… view at source ↗

**Figure 3.** Statistical errors of the two-level estimators for the disconnected scalar two-point functions before (empty circles) and after (filled circles) adding the correction in eq. (19). The empty (filled) circles correspond to the error of the estimator in eq. (18) ((20)) in the scalar channel. to obtain a corrected two-level estimator for the two-point functions. For the error analysis, we use the Γ-method [17] to comput… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: (left) Comparison between a 1-level (16) and 2-level (20) estimator for the weighted average disconnected two-point functions of scalar quark loops; (right) Comparison between the two sampling techniques for the effective masses of scalar disconnected two-point functi…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Lattice evidence that scalar glueballs are small
hep-lat 2025-08 conditional novelty 8.0 of 10

First lattice extraction of scalar glueball gravitational form factors gives a mass radius of 0.263(31) fm, smaller than typical hadrons.

Reference graph

Works this paper leans on

17 extracted references · 6 canonical work pages · cited by 1 Pith paper

[1]

Parisi,The strategy for computing the hadronic mass spectrum,Phys

G. Parisi,The strategy for computing the hadronic mass spectrum,Phys. Rept.103 (1984) 203

work page 1984
[2]

Lepage,The Analysis of Algorithms for Lattice Field Theory, inTheoretical Advanced Study Institute in Elementary Particle Physics, p

G.P. Lepage,The Analysis of Algorithms for Lattice Field Theory, inTheoretical Advanced Study Institute in Elementary Particle Physics, p. 96, 6, 1989

work page 1989
[3]

Luscher and P

M. Luscher and P. Weisz,Locality and exponential error reduction in numerical lattice gauge theory,JHEP 09(2001) 010 [hep-lat/0108014]

arXiv 2001
[4]

Performance of two-level sampling for the glueball spectrum in pure gauge theory

L. Barca, F. Knechtli, M.J. Peardon, S. Schaefer and J.A. Urrea-Niño,Performance of two-level sampling for the glueball spectrum in pure gauge theory, PoS LATTICE2023 (2024) 030 [2312.11372]

work page Pith review arXiv 2024
[5]

Barca, S

L. Barca, S. Schaefer, F. Knechtli, J.A. Urrea-Niño, S. Martins and M. Peardon,Exponential error reduction for glueball calculations using a two-level algorithm in pure gauge theory, Phys. Rev. D110(2024) 054515 [2406.12656]

arXiv 2024
[6]

Bruno, M

M. Bruno, M. Hansen and A. Rago,Towards glueball scattering in lattice Yang-Mills theory, PoS LATTICE2024(2025) 127

work page 2025
[7]

R.Abbott,D.C.Hackett,D.A.Pefkou,F.Romero-LópezandP.Shanahan, Gravitationalform factors of glueballs in Yang-Mills theory, PoS LATTICE2024(2025) 459 [2410.02706]

arXiv 2025
[8]

Particle Data Groupcollaboration, Review of particle physics,Phys. Rev. D110 (2024) 030001

2024

Show all 17 references

[9]

M. Cè, L. Giusti and S. Schaefer,Domain decomposition, multi-level integration and exponential noise reduction in lattice QCD, Phys. Rev. D93 (2016) 094507 [1601.04587]

2016 arXiv
[10]

M.Luscher, ANewapproachtotheproblemofdynamicalquarksinnumericalsimulationsof lattice QCD, Nucl. Phys. B418 (1994) 637 [hep-lat/9311007]

1994 arXiv
[11]

M. Cè, L. Giusti and S. Schaefer,A local factorization of the fermion determinant in lattice QCD, Phys. Rev. D95(2017) 034503 [1609.02419]

2017 arXiv
[12]

M.DallaBrida,L.Giusti,T.HarrisandM.Pepe, Multi-levelMonteCarlocomputationofthe hadronic vacuum polarization contribution to(𝑔𝜇− 2), Phys. Lett. B816 (2021) 136191 [2007.02973]

2021 arXiv
[13]

Hadron Spectrum Collaborationcollaboration, Novel quark-field creation operator construction for hadronic physics in lattice QCD, Phys. Rev. D80 (2009) 054506

2009
[14]

Falcioni, M.L

M. Falcioni, M.L. Paciello, G. Parisi and B. Taglienti,Again on SU(3) glueball mass,Nucl. Phys. B251 (1985) 624

1985
[15]

Athenodorou and M

A. Athenodorou and M. Teper,The glueball spectrum of SU(3) gauge theory in 3 + 1 dimensions,JHEP 11 (2020) 172 [2007.06422]. 9 Update on two-level sampling for glueball observables in quenched QCD Lorenzo Barca

2020 arXiv
[16]

Luscher,Solution of the Dirac equation in lattice QCD using a domain decomposition method,Comput

M. Luscher,Solution of the Dirac equation in lattice QCD using a domain decomposition method,Comput. Phys. Commun.156 (2004) 209 [hep-lat/0310048]

2004 arXiv
[17]

ALPHAcollaboration, Monte Carlo errors with less errors, Comput. Phys. Commun.156 (2004) 143 [hep-lat/0306017]. 10

2004 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Parisi,The strategy for computing the hadronic mass spectrum,Phys

G. Parisi,The strategy for computing the hadronic mass spectrum,Phys. Rept.103 (1984) 203

work page 1984

[2] [2]

Lepage,The Analysis of Algorithms for Lattice Field Theory, inTheoretical Advanced Study Institute in Elementary Particle Physics, p

G.P. Lepage,The Analysis of Algorithms for Lattice Field Theory, inTheoretical Advanced Study Institute in Elementary Particle Physics, p. 96, 6, 1989

work page 1989

[3] [3]

Luscher and P

M. Luscher and P. Weisz,Locality and exponential error reduction in numerical lattice gauge theory,JHEP 09(2001) 010 [hep-lat/0108014]

arXiv 2001

[4] [4]

Performance of two-level sampling for the glueball spectrum in pure gauge theory

L. Barca, F. Knechtli, M.J. Peardon, S. Schaefer and J.A. Urrea-Niño,Performance of two-level sampling for the glueball spectrum in pure gauge theory, PoS LATTICE2023 (2024) 030 [2312.11372]

work page Pith review arXiv 2024

[5] [5]

Barca, S

L. Barca, S. Schaefer, F. Knechtli, J.A. Urrea-Niño, S. Martins and M. Peardon,Exponential error reduction for glueball calculations using a two-level algorithm in pure gauge theory, Phys. Rev. D110(2024) 054515 [2406.12656]

arXiv 2024

[6] [6]

Bruno, M

M. Bruno, M. Hansen and A. Rago,Towards glueball scattering in lattice Yang-Mills theory, PoS LATTICE2024(2025) 127

work page 2025

[7] [7]

R.Abbott,D.C.Hackett,D.A.Pefkou,F.Romero-LópezandP.Shanahan, Gravitationalform factors of glueballs in Yang-Mills theory, PoS LATTICE2024(2025) 459 [2410.02706]

arXiv 2025

[8] [8]

Particle Data Groupcollaboration, Review of particle physics,Phys. Rev. D110 (2024) 030001

2024