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arxiv: 2506.16326 · v2 · submitted 2025-06-19 · ✦ hep-lat

Bounding statistical errors in lattice field theory simulations

Mattia Bruno , Gabriele Morandi This is my paper

Pith reviewed 2026-05-19 09:03 UTC · model grok-4.3

classification ✦ hep-lat
keywords lattice field theoryMonte Carloautocorrelation functionstatistical errorserror estimationwindowingmaster field
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The pith

Upper and lower bounds on the autocorrelation function yield an automatic stopping criterion for unbiased statistical error estimates in lattice Monte Carlo simulations.

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

Lattice field theory simulations rely on Markov chain Monte Carlo methods whose statistical errors require integrating the autocorrelation function. Truncating that integral at a finite window can introduce bias, so the paper defines upper and lower bounds on the autocorrelation function to create a reliable stopping rule. These bounds are then used to build an automatic windowing procedure that removes the need for manual tuning. The procedure is examined for both conventional analysis and the master-field approach and is checked on toy models.

Core claim

By constructing upper and lower bounds on the autocorrelation function, a stopping criterion can be imposed that determines the integration window automatically; the resulting error estimates remain unbiased for both standard Monte Carlo and master-field analyses of lattice field theories.

What carries the argument

Automatic windowing procedure that uses simultaneous upper and lower bounds on the autocorrelation function as a stopping criterion for the integrated autocorrelation time.

Load-bearing premise

Upper and lower bounds on the autocorrelation function can be computed or estimated in practice and produce a stopping rule whose error estimates stay unbiased when moving from toy models to full lattice simulations.

What would settle it

Applying the automatic windowing procedure to a toy model with exactly known variance and finding that the reported error bars differ from the true statistical uncertainty by more than the expected fluctuation.

Figures

Figures reproduced from arXiv: 2506.16326 by Gabriele Morandi, Mattia Bruno.

Figure 1
Figure 1. Figure 1: FIG. 1: Numerical example with synthetic data containing three modes. [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Application of the bounding method to synthetic autocorrelated data; the shaded bands represent the statistical [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Bounding method applied to the Monte Carlo autocorrelation function of [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: In fact, its steep rise for t ≲ x0 originates from the behavior of the effective mass below the discontinuity at t = x0. We show it as a warning and in numerical applications we explicitly sum the autocorrelation function up to a minimal t and apply the bounding method for larger values, where Eq. (4) is valid. In the right panel of [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Bounding method applied to the MF autocorrelation functions of the observables [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Autocorrelation function along the MF and MC directions, labeled by [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Example of application of the Bounding method to the MC autocorrelation function of [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Autocorrelation modes obtained from the GEVP and the pGEVP applied to the MC autocorrelation function of the [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
read the original abstract

Simulations of strongly interacting lattice field theories are typically performed using Markov chain Monte Carlo algorithms. Therefore estimators of statistical errors must incorporate the effect of autocorrelations by integrating the corresponding autocorrelation function. Since in practical calculations its integral is truncated to a finite window, in this work we propose a stopping criterion based on upper and lower bounds of the autocorrelation function. We examine its application to both traditional Monte Carlo analysis and the recently introduced master-field approach. By leveraging both bounds, we introduce an automatic windowing procedure which we test on numerical simulations of a few simplified toy models.

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 proposes a stopping criterion for truncating the integrated autocorrelation function in statistical error estimation for lattice field theory Markov chain Monte Carlo simulations. Upper and lower bounds on the autocorrelation function are used to define an automatic windowing procedure, which is examined for both conventional analysis and the master-field approach, with numerical tests performed on simplified toy models.

Significance. If the bounds prove computable and the resulting errors remain unbiased when applied to realistic lattice simulations, the method could offer a systematic improvement over ad-hoc window choices. The manuscript explicitly tests the procedure on toy models, providing a reproducible starting point for assessing its performance.

major comments (1)
  1. [§4] §4 (numerical tests on toy models): the central claim is that the bounds enable an automatic windowing procedure yielding unbiased error estimates in lattice field theory simulations, yet all validation is restricted to a few simplified toy models. This leaves open whether the bounds stay sufficiently tight (and the stopping criterion unbiased) for complex actions, gauge fields, or near-critical dynamics referenced in the introduction.
minor comments (1)
  1. [Methods] The precise construction of the upper and lower bounds (including any assumptions needed for their practical computation) should be stated more explicitly in the methods section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the constructive comment on the scope of our numerical tests. We respond to the major comment below.

read point-by-point responses

  1. Referee: [§4] §4 (numerical tests on toy models): the central claim is that the bounds enable an automatic windowing procedure yielding unbiased error estimates in lattice field theory simulations, yet all validation is restricted to a few simplified toy models. This leaves open whether the bounds stay sufficiently tight (and the stopping criterion unbiased) for complex actions, gauge fields, or near-critical dynamics referenced in the introduction.

    Authors: We agree that the numerical tests are performed exclusively on simplified toy models. These models were deliberately selected because they permit direct comparison against analytically known results, thereby allowing us to verify that the automatic windowing procedure produces unbiased error estimates in a fully controlled setting. The upper and lower bounds on the autocorrelation function are derived from general properties of stationary Markov processes and do not rely on any assumption of simplicity in the underlying action or dynamics. Consequently the stopping criterion itself is expected to remain valid for the more complex cases mentioned in the introduction. To make this generality clearer, we will add a dedicated paragraph in the revised Section 4 that discusses the anticipated behavior for gauge-field theories and near-critical dynamics, together with a brief outline of how the same bounds-based procedure can be applied in those settings. We view this as a partial revision that addresses the referee’s concern without expanding the computational scope of the present work. revision: partial

Circularity Check

0 steps flagged

No circularity: new algorithmic stopping criterion derived from proposed bounds, independent of fitted inputs or self-citation chains

full rationale

The paper proposes an automatic windowing procedure that uses upper and lower bounds on the autocorrelation function as a stopping criterion for error estimation in Monte Carlo and master-field analyses. This is presented as a new algorithmic choice rather than a quantity obtained by fitting parameters to the paper's own data or by reducing to prior self-citations. The derivation chain begins from the standard truncation of the integrated autocorrelation and introduces bounds as an external input that can be estimated or computed separately; the resulting procedure is then tested on toy models without the bounds themselves being defined in terms of the window size or error estimate. No step equates a prediction to its own fitted inputs by construction, and the central claim remains self-contained against external benchmarks for the toy-model regime.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and practical usability of bounds for the autocorrelation function in Markov chains; no free parameters or invented entities are evident from the abstract.

axioms (1)
  • domain assumption Autocorrelation functions in lattice Monte Carlo admit computable upper and lower bounds that can be used for truncation decisions.
    Invoked in the proposal of the stopping criterion.

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Works this paper leans on

68 extracted references · 68 canonical work pages · 21 internal anchors

  1. [1]

    , Nobs) the eigen- values and eigenvectors respectively

    at fixedtP we diagonalize the matrixΓαβ(tP ) and denote withln(tP ) and un(tP ) (n = 1, . . . , Nobs) the eigen- values and eigenvectors respectively

    work page

  2. [2]

    we select the most relevantNP eigenvectors by checking that the corresponding eigenvalues are statistically significant and different from zero

    work page

  3. [3]

    , kNP we construct the Nobs × NP projection matrix P(tP ) = [uk1(tP ), uk2(tP ),

    using the selected eigenvectors labeled by the indicesk1, k2, . . . , kNP we construct the Nobs × NP projection matrix P(tP ) = [uk1(tP ), uk2(tP ), . . . , ukNP (tP )] ; (E3)

    work page

  4. [4]

    operators

    we calculate the projectedNP × NP correlator matrixeΓ(t; tP) ≡ PT (tP)Γ(t) P(tP) and solve the corresponding GEVP. The projection preserves the spectrum and the matrix elements of the original “operators” are obtained by replacing the numerator in Eq. (E2) witheEnt[ev PT Γ][ΓPev ], where theev’s are the eigenvectors of the pGEVP. A numerical comparison be...

    work page

  5. [5]

    Aoyamaet al., Phys

    T. Aoyamaet al., Phys. Rept.887, 1 (2020), arXiv:2006.04822 [hep-ph]

    work page arXiv 2020

  6. [6]

    The anomalous magnetic moment of the muon in the Standard Model: an update

    R. Alibertiet al., (2025), arXiv:2505.21476 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  7. [7]

    Critical slowing down of topological modes

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

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  8. [8]

    Madras and A

    N. Madras and A. D. Sokal, J. Statist. Phys.50, 109 (1988)

    work page 1988

  9. [9]

    Monte Carlo errors with less errors

    U. Wolff (ALPHA), Comput. Phys. Commun. 156, 143 (2004), [Erratum: Comput. Phys. Commun.176,383(2007)], arXiv:hep-lat/0306017 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  10. [10]

    Critical slowing down and error analysis in lattice QCD simulations

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

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  11. [11]

    288,108750(2023),arXiv:2209.14371 [hep-lat]

    F.Joswig, S.Kuberski, J.T.Kuhlmann, andJ.Neuendorf,Comput.Phys.Commun. 288,108750(2023),arXiv:2209.14371 [hep-lat]

    work page arXiv 2023

  12. [12]

    Automatic differentiation for error analysis of Monte Carlo data

    A. Ramos, Comput. Phys. Commun.238, 19 (2019), arXiv:1809.01289 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  13. [13]

    mbruno46/pyobs: v1.5.1,

    M. Bruno, “mbruno46/pyobs: v1.5.1,” (2024)

    work page 2024

  14. [14]

    Bruno and R

    M. Bruno and R. Sommer, Comput. Phys. Commun.285, 108643 (2023), arXiv:2209.14188 [hep-lat]

    work page arXiv 2023

  15. [15]

    Christ, R

    N. Christ, R. Eranki, and C. Kelly (RBC, UKQCD), Phys. Rev. D111, 074514 (2025), arXiv:2409.11379 [hep-lat]

    work page arXiv 2025

  16. [16]

    Kelly and T

    C. Kelly and T. Wang, PoSLA TTICE2019, 129 (2019), arXiv:1911.04582 [hep-lat]

    work page arXiv 2019

  17. [17]

    Efron, The Jackknife, the Bootstrap and Other Resampling Plans(Society for Industrial and Applied Mathematics, Philadelphia, 1982)

    B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans(Society for Industrial and Applied Mathematics, Philadelphia, 1982). 21

    work page 1982

  18. [18]

    M. H. Quenouille, The Annals of Mathematical Statistics20, 355 (1949)

    work page 1949

  19. [19]

    Tukey, Annals of Mathematical Statistics29, 614 (1958)

    J. Tukey, Annals of Mathematical Statistics29, 614 (1958)

    work page 1958

  20. [20]

    Efron, Annals Statist.7, 1 (1979)

    B. Efron, Annals Statist.7, 1 (1979)

    work page 1979

  21. [21]

    M. Lüscher,Proceedings, 35th International Symposium on Lattice Field Theory (Lattice 2017): Granada, Spain, June 18-24, 2017, EPJ Web Conf.175, 01002 (2018), arXiv:1707.09758 [hep-lat]

    work page arXiv 2017

  22. [22]

    Giusti and M

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

    work page arXiv 2019

  23. [23]

    Fritzsch, J

    P. Fritzsch, J. Bulava, M. Cè, A. Francis, M. Lüscher, and A. Rago, PoSLA TTICE2021, 465 (2022), arXiv:2111.11544 [hep-lat]

    work page arXiv 2022

  24. [24]

    Bruno, M

    M. Bruno, M. Cè, A. Francis, P. Fritzsch, J. R. Green, M. T. Hansen, and A. Rago, JHEP11, 167 (2023), arXiv:2307.15674 [hep-lat]

    work page arXiv 2023

  25. [25]

    Blumet al.(RBC, UKQCD), Phys

    T. Blumet al.(RBC, UKQCD), Phys. Rev. D108, 054507 (2023), arXiv:2301.08696 [hep-lat]

    work page arXiv 2023

  26. [26]

    Bruno, M

    M. Bruno, M. Cè, A. Francis, J. R. Green, M. Hansen, and S. Zafeiropoulos, PoS LA TTICE2022, 368 (2023), arXiv:2212.09533 [hep-lat]

    work page arXiv 2023

  27. [27]

    Lehner, inLGT2016 at BNL(2016)

    C. Lehner, inLGT2016 at BNL(2016)

    work page 2016

  28. [28]

    Slope and curvature of the hadron vacuum polarization at vanishing virtuality from lattice QCD

    S. Borsanyi, Z. Fodor, T. Kawanai, S. Krieg, L. Lellouch, R. Malak, K. Miura, K. K. Szabo, C. Torrero, and B. Toth, Phys. Rev. D96, 074507 (2017), arXiv:1612.02364 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  29. [29]

    Yamamoto, P

    S. Yamamoto, P. Boyle, T. Izubuchi, L. Jin, C. Lehner, and N. Matsumoto, PoS LA TTICE2024, 034 (2025), arXiv:2502.05452 [hep-lat]

    work page arXiv 2025

  30. [30]

    Schwarz-preconditioned HMC algorithm for two-flavour lattice QCD

    M. Luscher, Comput. Phys. Commun.165, 199 (2005), arXiv:hep-lat/0409106

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  31. [31]

    Duane, A

    S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, Phys. Lett. B195, 216 (1987)

    work page 1987

  32. [32]

    S. R. Sharpe, inWorkshop on Perspectives in Lattice QCD(2006) arXiv:hep-lat/0607016

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  33. [33]

    Bruno, T

    M. Bruno, T. Izubuchi, C. Lehner, and A. S. Meyer, PoSLA TTICE2019, 239 (2019), arXiv:1910.11745 [hep-lat]

    work page arXiv 2019

  34. [34]

    Blumet al.(RBC, UKQCD), Phys

    T. Blumet al.(RBC, UKQCD), Phys. Rev. Lett.134, 201901 (2025), arXiv:2410.20590 [hep-lat]

    work page arXiv 2025

  35. [35]

    Djukanovic, G

    D. Djukanovic, G. von Hippel, S. Kuberski, H. B. Meyer, N. Miller, K. Ottnad, J. Parrino, A. Risch, and H. Wittig, JHEP 04, 098 (2025), arXiv:2411.07969 [hep-lat]

    work page arXiv 2025

  36. [36]

    Bazavovet al., (2024), arXiv:2412.18491 [hep-lat]

    A. Bazavovet al., (2024), arXiv:2412.18491 [hep-lat]

    work page arXiv 2024

  37. [37]

    Lüscher and U

    M. Lüscher and U. Wolff, Nucl. Phys. B339, 222 (1990)

    work page 1990

  38. [38]

    On the generalized eigenvalue method for energies and matrix elements in lattice field theory

    B. Blossier, M. Della Morte, G. von Hippel, T. Mendes, and R. Sommer, JHEP04, 094 (2009), arXiv:0902.1265 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  39. [39]

    D’Adda, M

    A. D’Adda, M. Lüscher, and P. Di Vecchia, Nuclear Physics B146, 63 (1978)

    work page 1978

  40. [40]

    Campostrini, P

    M. Campostrini, P. Rossi, and E. Vicari, Phys. Rev. D46, 2647 (1992)

    work page 1992

  41. [41]

    Boyle et al., “Grid,”

    P.A. Boyle et al., “Grid,”

    work page

  42. [42]

    P. A. Boyle, G. Cossu, A. Yamaguchi, and A. Portelli, PoSLA TTICE 2015, 023 (2016)

    work page 2015

  43. [43]

    Grid Python Toolkit (GPT), 2024-10,

    C. Lehner, M. Bruno, D. Richtmann, M. Schlemmer, R. Lehner, D. Knüttel, T. Wurm, L. Jin, S. Bürger, A. Hackl, and A. Klein, “Grid Python Toolkit (GPT), 2024-10,” (2024)

    work page 2024

  44. [44]

    Witten, Nucl

    E. Witten, Nucl. Phys. B149, 285 (1979)

    work page 1979

  45. [45]

    D’Adda, M

    A. D’Adda, M. Luscher, and P. Di Vecchia, Nucl. Phys. B146, 63 (1978)

    work page 1978

  46. [46]

    Rabinovici and S

    E. Rabinovici and S. Samuel, Phys. Lett. B101, 323 (1981)

    work page 1981

  47. [47]

    Di Vecchia, A

    P. Di Vecchia, A. Holtkamp, R. Musto, F. Nicodemi, and R. Pettorino, Nucl. Phys. B190, 719 (1981)

    work page 1981

  48. [48]

    Berg and M

    B. Berg and M. Luscher, Nucl. Phys. B190, 412 (1981)

    work page 1981

  49. [49]

    G. P. Engel and S. Schaefer, Comput. Phys. Commun.182, 2107 (2011), arXiv:1102.1852 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  50. [50]

    Monte Carlo simulation of lattice ${\rm CP}^{N-1}$ models at large N

    E. Vicari, Phys. Lett. B309, 139 (1993), arXiv:hep-lat/9209025

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  51. [51]

    Worm Algorithm for CP(N-1) Model

    T. Rindlisbacher and P. de Forcrand, Nucl. Phys. B918, 178 (2017), arXiv:1610.01435 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  52. [52]

    Precision study of critical slowing down in lattice simulations of the CP^{N-1} model

    J. Flynn, A. Juttner, A. Lawson, and F. Sanfilippo, (2015), arXiv:1504.06292 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  53. [53]

    Fighting topological freezing in the two-dimensional CP$^{N-1}$ model

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

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  54. [54]

    Campostrini and P

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

    work page 1991

  55. [55]

    K. G. Wilson, Phys. Rev. D10, 2445 (1974)

    work page 1974

  56. [56]

    Properties and uses of the Wilson flow in lattice QCD

    M. Lüscher, JHEP08, 071 (2010), [Erratum: JHEP 03, 092 (2014)], arXiv:1006.4518 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  57. [57]

    Trivializing maps, the Wilson flow and the HMC algorithm

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

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  58. [58]

    Perturbative analysis of the gradient flow in non-abelian gauge theories

    M. Luscher and P. Weisz, JHEP02, 051 (2011), arXiv:1101.0963 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  59. [59]

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

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  60. [60]

    Topological susceptibility and the sampling of field space in $N_f=2$ lattice QCD simulations

    M. Bruno, S. Schaefer, and R. Sommer (ALPHA), JHEP08, 150 (2014), arXiv:1406.5363 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  61. [61]

    Athenodorou and M

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

    work page arXiv 2020

  62. [62]

    Computational Strategies in Lattice QCD

    M. Luscher, inLes Houches Summer School: Session 93: Modern perspectives in lattice QCD: Quantum field theory and high performance computing(2010) pp. 331–399, arXiv:1002.4232 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  63. [63]

    Seabrook and L

    E. Seabrook and L. Wiskott, Neural Computation35, 1713–1796 (2023)

    work page 2023

  64. [64]

    Non-renormalizability of the HMC algorithm

    M. Luscher and S. Schaefer, JHEP04, 104 (2011), arXiv:1103.1810 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  65. [65]

    H. E. Haber, I. Hinchliffe, and E. Rabinovici, Nucl. Phys. B172, 458 (1980)

    work page 1980

  66. [66]

    Irving and C

    A. Irving and C. Michael, Nuclear Physics B371, 521 (1992)

    work page 1992

  67. [67]

    Bulava, F

    J. Bulava, F. Knechtli, V. Koch, C. Morningstar, and M. Peardon, Phys. Lett. B854, 138754 (2024), arXiv:2403.00754 [hep-lat]

    work page arXiv 2024

  68. [68]

    Blum et al

    T. Blum et al. (RBC, UKQCD), Phys. Rev. D 107, 094512 (2023), [Erratum: Phys.Rev.D 108, 039902 (2023)], arXiv:2301.09286 [hep-lat]

    work page arXiv 2023