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arxiv: 2606.17236 · v1 · pith:2RM3CHHEnew · submitted 2026-06-15 · ✦ hep-lat

Precision renormalisation and improvement of N_(rm f)=3 lattice QCD with Wilson fermions

Patrick Fritzsch , Jochen Heitger , Simon Kuberski , Hubert Simma , Rainer Sommer This is my paper

Pith reviewed 2026-06-27 02:12 UTC · model grok-4.3

classification ✦ hep-lat
keywords lattice QCDWilson fermionsrenormalization constantschiral symmetry restorationWard identitiesgradient flowSchrödinger functionalO(a) improvement
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The pith

Three-flavor Wilson lattice QCD yields renormalization constants to four or five significant digits at spacings down to 0.01 fm.

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

The paper tunes lattices with Schrödinger functional boundaries and three massless flavors to fixed gradient-flow coupling values at three scales to renormalize and improve the axial current, vector current, pseudo-scalar density, tensor current, and quark masses in O(a) improved lattice QCD. This procedure demonstrates that chiral symmetry restoration is accurate enough for Ward-identity observables to carry negligible uncertainties, achieving four-to-five-digit precision on ZA and ZV while also supplying an explanation for their suppressed statistical variances. The results extend to heavy-quark masses for B-physics applications and enable first-principles strategies for multi-scale problems such as the b-quark mass or high temperature. A sympathetic reader would care because these constants are essential inputs for extracting fundamental parameters from lattice simulations with controlled errors.

Core claim

The central claim is that the exact flavor symmetry together with accurate chiral symmetry restoration in Wilson-fermion lattice QCD at small a produces renormalization constants ZA and ZV determined from Ward identities with four to five significant digits and strongly suppressed statistical variances. The work confirms the expected continuum behavior of renormalization and supplies the improved factors needed for precision calculations with three massless flavors and additional heavy quarks.

What carries the argument

Lines of constant physics defined by fixed gradient-flow coupling values at three scales (L0 ≈ 0.25 fm, L1 = 2L0, L2 = 4L0) used to tune lattices with 8 ≤ L/a ≤ 64.

If this is right

  • The renormalization constants enable first-principles strategies for multi-scale problems such as the b-quark mass or large temperature.
  • Heavy-quark masses renormalized and improved with the same method become available for the B-physics programme.
  • Uncertainties in observables determined from Ward identities become practically negligible.
  • The strong suppression of statistical variances in ZA and ZV is explained by the combination of chiral symmetry restoration and exact flavor symmetry.

Where Pith is reading between the lines

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

  • The same tuning procedure could be applied at even smaller spacings to test whether the digit precision continues to improve linearly with a.
  • The results supply a benchmark that other fermion discretizations could match when computing the same renormalization constants.
  • For temperature studies the reduced renormalization uncertainty would tighten constraints on the equation of state at high T.

Load-bearing premise

Tuning a modest number of lattices to lines of constant physics at three scales is sufficient to control all relevant systematic errors at the claimed four-to-five-digit precision.

What would settle it

An independent determination of ZA at a lattice spacing of 0.01 fm that differs from the Ward-identity result by more than the stated uncertainty.

Figures

Figures reproduced from arXiv: 2606.17236 by Hubert Simma, Jochen Heitger, Patrick Fritzsch, Rainer Sommer, Simon Kuberski.

Figure 1
Figure 1. Figure 1: ZA determined along the LCP-0 compared to the determination from the chirally rotated SF [28] and from the standard SF. The latter is the version Z con A of [27], which coincides with our definition apart from the use of a non-trivial smearing function employed for the boundary sources, which enhances the ground state contribution. with the totally antisymmetric tensor ϵ abc (with ϵ 123 = 1) and the bounda… view at source ↗
Figure 2
Figure 2. Figure 2: Relation between bare coupling g 2 0 and simulated lattice sizes L/a, imposed through the constant renormalised coupling condition to define LCPs at µ −1 = L ∈ {L0, L1, L2}, cf. tables 1, 11 and 13. Vertical dashed lines indicate couplings simulated by the CLS consortium, and curves are added to guide the eye. 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 (a/L0) 2 5.5 5.6 5.7 5.8 5.9 Σ(u, a/L ) Global fi… view at source ↗
Figure 3
Figure 3. Figure 3: Step-scaling functions for L0 → 2L0 (left) and L1 → 2L1 (right). The gray error band shows the global fit to these step-scaling functions performed in [16]. Note that this fit did not include the data points presented here. The lattice approximants Σ(u, a/L) are shifted to the desired u0, u1 via the parameter-free one-loop formula Σ(ui , a/L) −1 = Σ(u, a/L) −1 + u −1 i − (u) −1 , where u = ¯g 2 GF(L) is th… view at source ↗
Figure 4
Figure 4. Figure 4: Step-scaling function of the mass, measured on the lattices with the bare parameters fixed in L0 (left) and L1 (right). The gray error band shows the global fit to these step-scaling functions performed in ref. [17]. Again, this fit did not include the data points presented here. precise determination of the strong coupling αMS(mZ). The step-scaling function of the quark mass is identical to the (inverse o… view at source ↗
Figure 5
Figure 5. Figure 5: Left: Improvement coefficient cA as determined on several lines of constant physics. LCP-0s and LCP-1 denote the results obtained in this work, while SF refers to those of [25]; points labelled CLS are from the large-volume calculation in [34]. The dotted line indicates the one-loop prediction [18], and the dashed line the fit result of [25]. Note that both LCP-0s and LCP-1 are very close to each other: th… view at source ↗
Figure 6
Figure 6. Figure 6: PCAC masses and polynomial representation for ensemble L0/a = 12 (LCP-0). Left: Difference am45 − amfit 45(∆) between the measured PCAC masses m45 and their fit amfit 45(∆) for a given polynomial degree Ndeg. Statistical uncertainties are those of the data points and fit, respectively. Right: Global fit to eq. (3.12) with Ndeg = 5. Uncertainties are too small to show up on this scale. Note that 2a∆ ≈ amq,4… view at source ↗
Figure 7
Figure 7. Figure 7: Subtracted bare quark masses corresponding to valence hopping parameters κ4(zh, L0/a). The gray curves are determinations in the chiral, unitary theory (∆ = 0) obtained by inverting the quadratic eq. (3.10) with estimators RX(g 2 0 , 0). The (connected) gray circles are the end points in the massless scheme and hardly vary with the lattice spacing. The coloured curves are determinations in our massive sche… view at source ↗
Figure 8
Figure 8. Figure 8: Lattice spacing dependence of various improved and renormalised PCAC masses (on LCP-0 ensembles) after fixing the RGI heavy quark mass from the subtracted quark mass, zh = zb, cf. eq. (3.10). The left panel shows results in the massless scheme and the right panel shows data in the massive scheme. The data are normalised w.r.t. the fixed quark mass zb = 8.7 to better identify the size of cutoff effects. The… view at source ↗
Figure 9
Figure 9. Figure 9: ZA normalised to the universal fit plotted against the (a/L) 2 . For the various definitions, see the main text. Note that each definition has a rather different L in physical units. The points labeled ”LCP-0/LCP-0T” show ZA(T = L)/ZA(T = 2L). such that projected expectation values are ⟨O⟩Q=0 = ⟨O ˆδ(Q)⟩ ⟨ ˆδ(Q)⟩ . (A.3) In [54] it has been shown that polynomials of flowed gauge fields do not require renor… view at source ↗
Figure 10
Figure 10. Figure 10: Standard deviation of the sea quark mass, as evaluated on the gauge ensembles of the five LCPs used in this work, normalised by the predicted leading-order behaviour from eq. (D.11). We note that in several steps, e.g. in (D.5) or going from (D.6) to (D.7), we have used the PCAC relation dropping the O(a 2 ) terms. Thus (D.10) and (D.11) are a priori expected to hold only close to the continuum. We verify… view at source ↗
Figure 11
Figure 11. Figure 11: Integrated autocorrelation times τint of the gradient flow coupling g¯ 2 GF and the boundary-to-boundary correlation function F1. We show lattices with a spatial extent of about 0.25 fm (left) and 0.5 fm (right) which are both in the frozen-charge situation. Lines, based on a fit to the three largest ensembles each, are shown to guide the eye. On the right, LCP-1 (LCP-0s) simulations are located at smalle… view at source ↗
Figure 12
Figure 12. Figure 12: As fig. 11 but with a spatial extent of about 1 fm where 1 − N0/Ncfg reaches up to about 1/3. LCP-2 (LCP-1s) simulations are located at smaller (larger) L/a and distinguished by different symbols. chosen in ref. [16] with the RHMC algorithm for the third, massless quark. We refer to this paper also for the exact details of the action, including the SF boundary improvement terms. Monte Carlo chains of leng… view at source ↗
read the original abstract

We renormalise (and improve) the flavour non-singlet axial current, pseudo-scalar density, vector current and tensor current, as well as quark masses, in O(a) improved lattice QCD with three massless flavours and lattice spacings down to 0.01 fm. To this end, we tune a number of lattices with Schr\"odinger functional boundary conditions and resolutions $8\leq L/a\leq 64$ to lines of constant physics with massless quarks and fixed gradient flow coupling $\bar{g}_\mathrm{GF}^2(L_i),\; i=0,1,2$, corresponding to $L_0 \approx 0.25$ fm, $L_1=2L_0$ and $L_2=4L_0$. We further renormalise and improve the quark mass of additional heavy quarks for use in the B-physics programme of the collaboration (arXiv:2312.09811). Our somewhat technical results enable first-principles strategies for solving multi-scale problems involving, e.g., the b-quark mass (arXiv:2312.10017) or a large temperature (arXiv:2501.11603). Comparing also to other determinations of the axial current renormalisation constant $Z_{\rm A}$, we have a precise confirmation of how renormalisation and the restoration of chiral symmetry work out with Wilson fermions at small $a$. In particular, the accurate restoration of chiral symmetry and the exact flavour symmetry lead to practically negligible uncertainties in observables determined from Ward identities: four to five significant digits are achieved for $Z_{\rm A},Z_{\rm V}$. We provide an explanation for the strong suppression of their statistical variances.

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 claims to perform non-perturbative renormalization and O(a) improvement of the flavour non-singlet axial current, pseudoscalar density, vector current, tensor current, and quark masses (including additional heavy quarks) in Nf=3 O(a)-improved Wilson lattice QCD. Lattices with Schrödinger-functional boundary conditions and resolutions 8 ≤ L/a ≤ 64 are tuned to lines of constant physics at three scales defined by fixed gradient-flow couplings ar g_GF²(L_i) corresponding to L0 ≈ 0.25 fm, L1=2L0, L2=4L0. This yields ZA and ZV to four-to-five significant digits, with the precision attributed to accurate chiral symmetry restoration and exact flavour symmetry making Ward-identity uncertainties practically negligible; an explanation for the suppression of statistical variances is provided. Results are compared to other ZA determinations and positioned to support multi-scale applications such as b-quark mass and finite-temperature studies.

Significance. If the claimed control of systematics is demonstrated, the work supplies high-accuracy renormalization constants that enable first-principles multi-scale calculations in B-physics and thermal QCD. The cross-checks against independent ZA determinations and the explicit account of variance suppression in Ward identities constitute concrete strengths. The results for heavy-quark renormalization directly support the collaboration’s B-physics programme.

major comments (1)
  1. [Abstract / lattice tuning description] Abstract and lattice-setup description: the headline claim of four-to-five significant digits for ZA and ZV with “practically negligible uncertainties” from Ward identities requires that O(a²) cutoff effects, finite-volume corrections, gradient-flow mistuning, and residual mass effects all lie below ~10^{-5}. The tuning procedure uses a modest set of lattices up to L/a=64 at three fixed scales, yet no explicit test, extrapolation, or quantitative bound on the size of the remaining a² or 1/L corrections at this precision level is reported. This assumption is load-bearing for the central precision statement.
minor comments (1)
  1. [Abstract] The abstract states that results are compared to other ZA determinations but does not quote the numerical values or uncertainties obtained in the present work; adding a compact table or explicit numbers would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / lattice tuning description] Abstract and lattice-setup description: the headline claim of four-to-five significant digits for ZA and ZV with “practically negligible uncertainties” from Ward identities requires that O(a²) cutoff effects, finite-volume corrections, gradient-flow mistuning, and residual mass effects all lie below ~10^{-5}. The tuning procedure uses a modest set of lattices up to L/a=64 at three fixed scales, yet no explicit test, extrapolation, or quantitative bound on the size of the remaining a² or 1/L corrections at this precision level is reported. This assumption is load-bearing for the central precision statement.

    Authors: We agree that the quoted precision for ZA and ZV rests on the assumption that the listed systematic effects are controlled below the 10^{-5} level. The tuning to fixed gradient-flow couplings at three scales (L0, L1, L2) together with the exact flavour symmetry and Ward-identity method is designed to achieve this control, and the manuscript already contains consistency checks across the available resolutions. Nevertheless, we acknowledge that an explicit quantitative bound or extrapolation for the residual O(a²) and finite-volume contributions at the target precision is not presented in sufficient detail. We will revise the lattice-setup section and add a short appendix providing such estimates based on the observed scaling behaviour and cross-scale comparisons. revision: yes

Circularity Check

0 steps flagged

Minor self-citations to collaboration papers; central ZA/ZV results are direct non-perturbative extractions with no reduction by construction

full rationale

The renormalization constants are obtained from explicit Ward-identity computations on Schrödinger-functional lattices tuned to fixed gradient-flow couplings at three scales. No equation in the provided text defines ZA or ZV in terms of themselves or renames a fitted quantity as a prediction. Self-citations to arXiv:2312.09811 and arXiv:2312.10017 appear for context on B-physics applications but do not supply the load-bearing justification for the reported four-to-five-digit precision, which rests on the lattice data and symmetry arguments internal to this work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The work rests on standard domain assumptions of lattice QCD.

axioms (1)
  • domain assumption Validity of Schrödinger functional boundary conditions and gradient-flow coupling as a line-of-constant-physics definition for massless Nf=3 Wilson QCD
    Invoked to tune the lattices and extract renormalization constants

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discussion (0)

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Reference graph

Works this paper leans on

62 extracted references · 1 canonical work pages

  1. [1]

    Conigli, J

    A. Conigli, J. Frison, P. Fritzsch, A. Gérardin, J. Heitger, G. Herdoíza et al.,A strategy for B-physics observables in the continuum limit,PoSLATTICE2023(2024) 268 [2312.09811]

    arXiv 2024

  2. [2]

    Conigli, J

    A. Conigli, J. Frison, P. Fritzsch, A. Gérardin, J. Heitger, G. Herdoíza et al.,mb andf B(∗) in 2 + 1 flavour QCD from a combination of continuum limit static and relativistic results, PoSLATTICE2023(2024) 237 [2312.10017]

    arXiv 2024

  3. [3]

    Bresciani, M.D

    M. Bresciani, M.D. Brida, L. Giusti and M. Pepe,QCD Equation of State withNf = 3 Flavors up to the Electroweak Scale,Phys. Rev. Lett.134(2025) 201904 [2501.11603]

    arXiv 2025

  4. [4]

    Lüscher, P

    M. Lüscher, P. Weisz and U. Wolff,A Numerical method to compute the running coupling in asymptotically free theories,Nucl. Phys. B359(1991) 221

    1991

  5. [5]

    Jansen, C

    K. Jansen, C. Liu, M. Lüscher, H. Simma, S. Sint, R. Sommer et al.,Nonperturbative renormalization of lattice QCD at all scales,Phys. Lett. B372(1996) 275 [hep-lat/9512009]

    Pith/arXiv arXiv 1996

  6. [6]

    Heitger and R

    J. Heitger and R. Sommer,Nonperturbative heavy quark effective theory,JHEP02(2004) 022 [hep-lat/0310035]

    Pith/arXiv arXiv 2004

  7. [7]

    Guazzini, R

    D. Guazzini, R. Sommer and N. Tantalo,Precision for B-meson matrix elements,JHEP01 (2008) 076 [0710.2229]

    Pith/arXiv arXiv 2008

  8. [8]

    Lüscher,Advanced lattice QCD, inLes Houches Summer School in Theoretical Physics, Session 68: Probing the Standard Model of Particle Interactions, pp

    M. Lüscher,Advanced lattice QCD, inLes Houches Summer School in Theoretical Physics, Session 68: Probing the Standard Model of Particle Interactions, pp. 229–280, 2, 1998 [hep-lat/9802029]

    Pith/arXiv arXiv 1998

  9. [9]

    Sommer,Non-perturbative Heavy Quark Effective Theory: Introduction and Status,Nucl

    R. Sommer,Non-perturbative Heavy Quark Effective Theory: Introduction and Status,Nucl. Part. Phys. Proc.261-262(2015) 338 [1501.03060]

    Pith/arXiv arXiv 2015

  10. [10]

    Della Morte, J

    M. Della Morte, J. Heitger, H. Simma and R. Sommer,Non-perturbative Heavy Quark Effective Theory: An application to semi-leptonic B-decays,Nucl. Part. Phys. Proc. 261-262(2015) 368 [1501.03328]

    Pith/arXiv arXiv 2015

  11. [11]

    Dalla Brida, R

    M. Dalla Brida, R. Höllwieser, F. Knechtli, T. Korzec, A. Ramos, S. Sint et al., High-precision calculation of the quark–gluon coupling from lattice QCD,Nature652(2026) 328 [2501.06633]. 32 T able 17:Algorithmic parameters of our major simulations. The column labelled “RHMC” lists the lower and upper limits of the approximation range and the number of pol...

    arXiv 2026

  12. [12]

    Della Morte, R

    M. Della Morte, R. Sommer and S. Takeda,On cutoff effects in lattice QCD from short to long distances,Phys. Lett. B672(2009) 407 [0807.1120]

    Pith/arXiv arXiv 2009

  13. [13]

    Kuberski, A

    S. Kuberski, A. Conigli, P. Fritzsch, A. Gérardin, J. Heitger, G. Herdoíza et al.,Heavy quark masses from step-scaling,PoSLATTICE2025(2026) 209 [2603.25295]

    arXiv 2026

  14. [14]

    Lüscher, R

    M. Lüscher, R. Narayanan, P. Weisz and U. Wolff,The Schrödinger functional: A Renormalizable probe for non-Abelian gauge theories,Nucl. Phys. B384(1992) 168 33 T able 18:Some standard observables and their integrated autocorrelation times in units of MDU. LCPL/a¯g 2 GF(L)τ int[¯g2 GF]Z P(g2 0,a/L)τ int[ZP]F 1 τint[F1] LCP-0123.9461(43) 6.22(33) 0.57835(32...

    arXiv 1992

  15. [15]

    Sint,On the Schrödinger functional in QCD,Nucl

    S. Sint,On the Schrödinger functional in QCD,Nucl. Phys. B421(1994) 135 [hep-lat/9312079]

    Pith/arXiv arXiv 1994

  16. [16]

    Dalla Brida, P

    M. Dalla Brida, P. Fritzsch, T. Korzec, A. Ramos, S. Sint and R. Sommer,Slow running of the Gradient Flow coupling from 200 MeV to 4 GeV inNf = 3QCD,Phys. Rev. D95 (2017) 014507 [1607.06423]

    Pith/arXiv arXiv 2017

  17. [17]

    Campos, P

    I. Campos, P. Fritzsch, C. Pena, D. Preti, A. Ramos and A. Vladikas,Non-perturbative quark mass renormalisation and running inNf = 3QCD,Eur. Phys. J. C78(2018) 387 [1802.05243]. 34

    Pith/arXiv arXiv 2018

  18. [18]

    S. Aoki, R. Frezzotti and P. Weisz,Computation of the improvement coefficientcSW to one loop with improved gluon actions,Nucl. Phys. B540(1999) 501 [hep-lat/9808007]

    Pith/arXiv arXiv 1999

  19. [19]

    Sint and R

    S. Sint and R. Sommer,The Running coupling from the QCD Schrödinger functional: A One loop analysis,Nucl. Phys. B465(1996) 71 [hep-lat/9508012]

    Pith/arXiv arXiv 1996

  20. [20]

    Takeda, S

    S. Takeda, S. Aoki and K. Ide,A Perturbative determination of O(a) boundary improvement coefficients for the Schrödinger functional coupling at one loop with improved gauge actions, Phys. Rev. D68(2003) 014505 [hep-lat/0304013]

    Pith/arXiv arXiv 2003

  21. [21]

    Bruno, M

    M. Bruno, M. Dalla Brida, P. Fritzsch, T. Korzec, A. Ramos, S. Schaefer et al.,QCD Coupling from a Nonperturbative Determination of the Three-FlavorΛParameter,Phys. Rev. Lett.119(2017) 102001 [1706.03821]

    Pith/arXiv arXiv 2017

  22. [22]

    Ramos and S

    A. Ramos and S. Sint,Symanzik improvement of the gradient flow in lattice gauge theories, Eur. Phys. J. C76(2016) 15 [1508.05552]

    Pith/arXiv arXiv 2016

  23. [23]

    Fritzsch and A

    P. Fritzsch and A. Ramos,The gradient flow coupling in the Schrödinger Functional,JHEP 10(2013) 008 [1301.4388]

    Pith/arXiv arXiv 2013

  24. [24]

    Fritzsch, A

    P. Fritzsch, A. Ramos and F. Stollenwerk,Critical slowing down and the gradient flow coupling in the Schrödinger functional,PoSLattice2013(2014) 461 [1311.7304]

    Pith/arXiv arXiv 2014

  25. [25]

    Bulava, M

    J. Bulava, M. Della Morte, J. Heitger and C. Wittemeier,Non-perturbative improvement of the axial current inNf = 3lattice QCD with Wilson fermions and tree-level improved gauge action,Nucl. Phys. B896(2015) 555 [1502.04999]

    Pith/arXiv arXiv 2015

  26. [26]

    Lüscher, S

    M. Lüscher, S. Sint, R. Sommer and P. Weisz,Chiral symmetry and O(a) improvement in lattice QCD,Nucl. Phys. B478(1996) 365 [hep-lat/9605038]

    Pith/arXiv arXiv 1996

  27. [27]

    Bulava, M

    J. Bulava, M. Della Morte, J. Heitger and C. Wittemeier,Nonperturbative renormalization of the axial current inNf = 3lattice QCD with Wilson fermions and a tree-level improved gauge action,Phys. Rev. D93(2016) 114513 [1604.05827]

    Pith/arXiv arXiv 2016

  28. [28]

    Dalla Brida, T

    M. Dalla Brida, T. Korzec, S. Sint and P. Vilaseca,High precision renormalization of the flavour non-singlet Noether currents in lattice QCD with Wilson quarks,Eur. Phys. J. C79 (2019) 23 [1808.09236]

    Pith/arXiv arXiv 2019

  29. [29]

    Lüscher, S

    M. Lüscher, S. Sint, R. Sommer and H. Wittig,Nonperturbative determination of the axial current normalization constant in O(a) improved lattice QCD,Nucl. Phys. B491(1997) 344 [hep-lat/9611015]

    Pith/arXiv arXiv 1997

  30. [30]

    Della Morte, R

    M. Della Morte, R. Hoffmann, F. Knechtli, R. Sommer and U. Wolff,Non-perturbative renormalization of the axial current with dynamical Wilson fermions,JHEP07(2005) 007 [hep-lat/0505026]

    Pith/arXiv arXiv 2005

  31. [31]

    Aoki, K.-I

    S. Aoki, K.-I. Nagai, Y. Taniguchi and A. Ukawa,Perturbative renormalization factors of bilinear quark operators for improved gluon and quark actions in lattice QCD,Phys. Rev. D 58(1998) 074505 [hep-lat/9802034]

    Pith/arXiv arXiv 1998

  32. [32]

    Capitani, M

    S. Capitani, M. Lüscher, R. Sommer and H. Wittig,Non-perturbative quark mass renormalization in quenched lattice QCD,Nucl. Phys. B544(1999) 669 [hep-lat/9810063]

    Pith/arXiv arXiv 1999

  33. [33]

    Bernardoni et al.,The b-quark mass from non-perturbativeNf = 2Heavy Quark Effective Theory atO(1/m h),Phys

    F. Bernardoni et al.,The b-quark mass from non-perturbativeNf = 2Heavy Quark Effective Theory atO(1/m h),Phys. Lett. B730(2014) 171 [1311.5498]

    Pith/arXiv arXiv 2014

  34. [34]

    Gérardin, T

    A. Gérardin, T. Harris and H.B. Meyer,Nonperturbative renormalization and O(a)-improvement of the nonsinglet vector current withNf = 2 + 1Wilson fermions and tree-level Symanzik improved gauge action,Phys. Rev. D99(2019) 014519 [1811.08209]. 35

    Pith/arXiv arXiv 2019

  35. [35]

    Fritzsch, R

    P. Fritzsch, R. Sommer, F. Stollenwerk and U. Wolff,Symanzik improvement with dynamical charm: a 3+1 scheme for Wilson quarks,JHEP06(2018) 025 [1805.01661]

    Pith/arXiv arXiv 2018

  36. [36]

    de Divitiis, P

    G.M. de Divitiis, P. Fritzsch, J. Heitger, C.C. Köster, S. Kuberski and A. Vladikas, Non-perturbative determination of improvement coefficientsbm andb A−bP and normalisation factorZ mZP/ZA withN f = 3Wilson fermions,Eur. Phys. J. C79(2019) 797 [1906.03445]

    arXiv 2019

  37. [37]

    Bochicchio, L

    M. Bochicchio, L. Maiani, G. Martinelli, G.C. Rossi and M. Testa,Chiral Symmetry on the Lattice with Wilson Fermions,Nucl. Phys. B262(1985) 331

    1985

  38. [38]

    Bhattacharya, R

    T. Bhattacharya, R. Gupta, W. Lee, S.R. Sharpe and J.M. Wu,Improved bilinears in lattice QCD with non-degenerate quarks,Phys. Rev. D73(2006) 034504 [hep-lat/0511014]

    Pith/arXiv arXiv 2006

  39. [39]

    Guagnelli, R

    M. Guagnelli, R. Petronzio, J. Rolf, S. Sint, R. Sommer and U. Wolff,Nonperturbative results for the coefficientsbm andb A−bP in O(a) improved lattice QCD,Nucl. Phys. B595 (2001) 44 [hep-lat/0009021]

    Pith/arXiv arXiv 2001

  40. [40]

    de Divitiis and R

    G.M. de Divitiis and R. Petronzio,Nonperturbative renormalization constants on the lattice from flavor nonsinglet Ward identities,Phys. Lett. B419(1998) 311 [hep-lat/9710071]

    Pith/arXiv arXiv 1998

  41. [41]

    Heitger and J

    J. Heitger and J. Wennekers,Effective heavy light meson energies in small volume quenched QCD,JHEP02(2004) 064 [hep-lat/0312016]

    Pith/arXiv arXiv 2004

  42. [42]

    Fritzsch, J

    P. Fritzsch, J. Heitger and N. Tantalo,Non-perturbative improvement of quark mass renormalization in two-flavour lattice QCD,JHEP08(2010) 074 [1004.3978]

    Pith/arXiv arXiv 2010

  43. [43]

    Della Morte, R

    M. Della Morte, R. Hoffmann, F. Knechtli, J. Rolf, R. Sommer, I. Wetzorke et al., Non-perturbative quark mass renormalization in two-flavor QCD,Nucl. Phys. B729(2005) 117 [hep-lat/0507035]

    Pith/arXiv arXiv 2005

  44. [44]

    Fritzsch, F

    P. Fritzsch, F. Knechtli, B. Leder, M. Marinkovic, S. Schaefer, R. Sommer et al.,The strange quark mass and Lambda parameter of two flavor QCD,Nucl. Phys. B865(2012) 397 [1205.5380]

    Pith/arXiv arXiv 2012

  45. [45]

    Della Morte, S

    M. Della Morte, S. Dooling, J. Heitger, D. Hesse and H. Simma,Matching of heavy-light flavour currents between HQET at order 1/mand QCD: I. Strategy and tree-level study, JHEP05(2014) 060 [1312.1566]

    Pith/arXiv arXiv 2014

  46. [46]

    Precision renormalisation and improvement ofNf = 3lattice QCD with Wilson fermions

    P. Fritzsch, J. Heitger, S. Kuberski, H. Simma and R. Sommer,Auxiliary material accompanying “Precision renormalisation and improvement ofNf = 3lattice QCD with Wilson fermions”, June, 2026, 10.5281/zenodo.20554739

    work page doi:10.5281/zenodo.20554739 2026

  47. [47]

    Heitger, F

    J. Heitger, F. Joswig and S. Kuberski,Determination of the charm quark mass in lattice QCD with2 + 1flavours on fine lattices,JHEP05(2021) 288 [2101.02694]

    arXiv 2021

  48. [48]

    Dalla Brida, R

    M. Dalla Brida, R. Höllwieser, F. Knechtli, T. Korzec, S. Sint and R. Sommer,Heavy Wilson quarks and O(a) improvement: nonperturbative results for bg,JHEP2024(2024) 188 [2401.00216]

    arXiv 2024

  49. [49]

    Korcyl and G.S

    P. Korcyl and G.S. Bali,Non-perturbative determination of improvement coefficients using coordinate space correlators inNf = 2 + 1lattice QCD,Phys. Rev. D95(2017) 014505 [1607.07090]

    Pith/arXiv arXiv 2017

  50. [50]

    Wolff,Monte Carlo errors with less errors,Comput

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

    Pith/arXiv arXiv 2004

  51. [51]

    Ramos,Automatic differentiation for error analysis of Monte Carlo data,Comput

    A. Ramos,Automatic differentiation for error analysis of Monte Carlo data,Comput. Phys. Commun.238(2019) 19 [1809.01289]. 36

    Pith/arXiv arXiv 2019

  52. [52]

    Ramos,Automatic differentiation for error analysis,PoSTOOLS2020(2021) 045 [2012.11183]

    A. Ramos,Automatic differentiation for error analysis,PoSTOOLS2020(2021) 045 [2012.11183]

    arXiv 2021

  53. [53]

    Joswig, S

    F. Joswig, S. Kuberski, J.T. Kuhlmann and J. Neuendorf,pyerrors: A python framework for error analysis of Monte Carlo data,Comput. Phys. Commun.288(2023) 108750 [2209.14371]

    arXiv 2023

  54. [54]

    Lüscher and P

    M. Lüscher and P. Weisz,Perturbative analysis of the gradient flow in non-abelian gauge theories,JHEP02(2011) 051 [1101.0963]

    Pith/arXiv arXiv 2011

  55. [55]

    Lüscher,Topology, the Wilson flow and the HMC algorithm,PoSLATTICE2010 (2010) 015 [1009.5877]

    M. Lüscher,Topology, the Wilson flow and the HMC algorithm,PoSLATTICE2010 (2010) 015 [1009.5877]

    Pith/arXiv arXiv 2010

  56. [56]

    Meinel,Quark flavor physics with lattice QCD,PoSLATTICE2023(2024) 126 [2401.08006]

    S. Meinel,Quark flavor physics with lattice QCD,PoSLATTICE2023(2024) 126 [2401.08006]

    arXiv 2024

  57. [57]

    Heitger and F

    J. Heitger and F. Joswig,The renormalisedO(a)improved vector current in three-flavour lattice QCD with Wilson quarks,Eur. Phys. J. C81(2021) 254 [2010.09539]

    arXiv 2021

  58. [58]

    Fritzsch,Mass-improvement of the vector current in three-flavor QCD,JHEP06(2018) 015 [1805.07401]

    P. Fritzsch,Mass-improvement of the vector current in three-flavor QCD,JHEP06(2018) 015 [1805.07401]

    Pith/arXiv arXiv 2018

  59. [59]

    Chimirri, P

    L. Chimirri, P. Fritzsch, J. Heitger, F. Joswig, M. Panero, C. Pena et al.,Non-perturbative renormalisation and improvement of non-singlet tensor currents in Nf = 3 QCD,JHEP07 (2024) 089 [2309.04314]

    arXiv 2024

  60. [60]

    Del Debbio, H

    L. Del Debbio, H. Panagopoulos and E. Vicari,θdependence of SU(N) gauge theories,JHEP 08(2002) 044 [hep-th/0204125]

    Pith/arXiv arXiv 2002

  61. [61]

    Schaefer, R

    S. Schaefer, R. Sommer and F. Virotta,Critical slowing down and error analysis in lattice QCD simulations,Nucl. Phys. B845(2011) 93 [1009.5228]

    Pith/arXiv arXiv 2011

  62. [62]

    Lüscher,A Semiclassical Formula for the Topological Susceptibility in a Finite Space-time Volume,Nucl

    M. Lüscher,A Semiclassical Formula for the Topological Susceptibility in a Finite Space-time Volume,Nucl. Phys. B205(1982) 483. 37

    1982