Logarithmic corrections to O($a^2$) effects in lattice QCD with unrooted Staggered quarks

Nikolai Husung

REVIEW 2 major objections 6 minor 59 references

Logarithmic corrections to O($a^2$) effects in lattice QCD with unrooted Staggered quarks

T0 review · 2 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For unrooted staggered quarks, the leading lattice-spacing artifact in spectral quantities is not simply $a^2$ but $a^2$ times a power of the running coupling, with exponents as negative as $-2.614$ for $N_\mathrm{f}=12$.

desk verdict A serious one-loop SymEFT calculation that likely gets the staggered exponents right, but the unproven completeness of the 28-operator basis is a load-bearing assumption the referee needs to check. read the letter →

arxiv 2501.17036 v2 pith:Y466BL3D submitted 2025-01-28 hep-lat

Nikolai Husung This is my paper · ORCID

classification hep-lat

keywords latticeQCDstaggeredquarksunrootedartifactsanomalousdimensionmatrixcontinuumextrapolationtastesymmetrybreakinglogarithmiccorrections

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 pins down how fast lattice-spacing errors vanish in lattice QCD simulations that use unrooted staggered quarks. The leading artifact in hadron masses and other spectral quantities is shown to take the form $a^2[2b_0\bar{g}^2(1/a)]^{\gamma_i}$: a plain $a^2$ suppression multiplied by a power of the running coupling, with the smallest power $\gamma_i$ equal to $-0.273,-0.301,-0.913,-2.614$ for $N_\mathrm{f}=0,4,8,12$. The more flavours, the more negative the power, so continuum extrapolations become logarithmically harder as $N_\mathrm{f}$ grows. The derivation builds a minimal on-shell operator basis for the effective theory of lattice artifacts, computes its one-loop anomalous dimension matrix, and clarifies that mass-dimension-five operators vanish only on shell, which is what guarantees automatic $O(a)$ improvement.

What carries the argument

The central object is the one-loop anomalous dimension matrix of the minimal on-shell operator basis for the lattice-artifact effective action, computed in a strictly local taste representation in which the four tastes of a staggered hypercube play the role of continuum quark flavours. The quoted exponents are the eigenvalues of the ordinary part of that matrix divided by the one-loop $\beta$-function coefficient, read off from its Jordan normal form; where the matrix is not diagonalisable, explicit logarithms multiply the leading power. The basis is obtained by imposing the full set of exact symmetries of the interacting staggered action, and operators that vanish by the continuum equations of motion are separated out as EOM-vanishing terms, which is what secures automatic $O(a)$ improvement while leaving $O(a^2)$ matching and composite fields affected.

What would settle it

An independent one-loop computation of the same operator mixing in a different renormalisation scheme should reproduce the same Jordan-normal-form exponents; a discrepancy in the eigenvalues would refute the prediction. On the numerical side, compute a hadron mass with unrooted staggered quarks at three or more lattice spacings and test whether the $a^2$-scaled deviation from the continuum tracks $[2b_0\bar{g}^2(1/a)]^{\hat{\gamma}_i}$ with the quoted $\hat{\gamma}_i$; a statistically clear deviation rules out the one-loop exponents.

Watch

Extended reading notes

Core claim

Within the lattice-artifact effective theory, the paper claims that the asymptotic $a\to 0$ behaviour of spectral quantities with unrooted staggered quarks is $a^2[2b_0\bar{g}^2(1/a)]^{\hat{\gamma}_i}$, with the leading exponents determined by the Jordan normal form of the one-loop anomalous dimension matrix. The values $\min_i\hat{\gamma}_i\approx -0.273,-0.301,-0.913,-2.614$ for $N_\mathrm{f}=0,4,8,12$ are obtained before any suppression from matching coefficients, and the quenched case $N_\mathrm{f}=0$ also carries an explicit logarithm multiplying one leading term because the matrix is not fully diagonalisable. The paper further claims that a strictly local taste representation yields a minimal on-shell basis with no dimension-five operators, while dimension-five operators that vanish by the equations of motion do exist; the discrete flavour symmetry of the continuum theory then guarantees automatic $O(a)$ improvement, and the EOM-vanishing terms matter for $O(a^2)$ matching and for composite local fields.

Load-bearing premise

The calculation assumes that the listed symmetries, including those realised only with a field redefinition, exhaust the exact symmetries of the interacting staggered action and can be restored to their ordinary continuum form step by step; if any symmetry is missing, the operator basis and the exponents change.

Editorial extensions

If this is right

Continuum extrapolations of staggered spectral quantities should be fit with $a^2[2b_0\bar{g}^2(1/a)]^{\hat{\gamma}_i}$ (or its log-modified form for $N_\mathrm{f}=0$) rather than a pure $a^2$ once the lattice spacing is small enough for one-loop running to apply.
For $N_\mathrm{f}=8$ and $12$, the predicted exponents $-0.913$ and $-2.614$ mean the $a^2$ suppression is significantly weaker, so reaching a given precision requires smaller lattice spacings than a plain $a^2$ analysis would suggest.
Because the chirality-breaking four-quark operators behind the most negative exponents are expected to be suppressed at tree level, the practically relevant exponents may be shifted by roughly $+1$ through matching.
Automatic $O(a)$ improvement holds for spectral quantities, but the dimension-five EOM-vanishing operators affect $O(a^2)$ artifacts and must be included in any matching or in analyses of local composite fields such as currents.
Any complete artifact analysis for staggered quarks needs the 28 taste-breaking four-quark operators of the enlarged basis; the smaller basis used in an earlier work is not closed under one-loop renormalisation.

Reading between the lines

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

If the one-loop exponents survive higher-order corrections, the effective power observed at current couplings ($\bar{g}^2(1/a)/(4\pi)\sim0.2$) could deviate noticeably from the asymptotic values, so fitting the leading-log form over a limited set of lattice spacings may require next-to-leading-log matching to be reliable.
The same symmetry analysis applied to rooted staggered quarks is not directly available because rooting is nonlocal at finite lattice spacing; reconciling the rooting limit with these continuum-limit exponents is a natural next step.
A direct numerical test could measure a spectral quantity at three or more lattice spacings and check whether the ratio of $a^2$-scaled artifacts follows $[2b_0\bar{g}^2(1/a)]^{\hat{\gamma}_i}$ with the predicted exponents; a clear mismatch would prompt a revision of the operator basis or of the one-loop computation.

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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 paper extends Husung's SymEFT program from Wilson and Ginsparg-Wilson quarks to unrooted staggered quarks, delivering the first leading-log scaling exponents for O(a^2) artifacts in that formulation. The main result is that the asymptotic lattice-spacing dependence goes as a^2 [2b0 gbar^2(1/a)]^gamma_i, with min gamma_i about -0.27, -0.30, -0.91, -2.61 for Nf=0,4,8,12. If right, this directly affects how continuum extrapolations are fit in staggered simulations, especially at Nf=8 and 12 where the suppression is much softer than naive a^2.

What is genuinely new: a local taste representation for staggered quarks, a proposed 28-operator taste-breaking basis at mass dimension six, and new one-loop anomalous dimension matrices. The paper also clarifies the status of mass-dimension five operators and the role of shift-symmetry field redefinitions. On the positive side, the work is careful and honest: the author states the limitations, supplies the Mathematica notebook and references public FORM scripts, and cross-checks the Ginsparg-Wilson blocks against prior work. The RG solution that turns the matrix into the exponents is standard and the quoted eigenvalues are consistent with the printed matrices.

The soft spot is exactly the one the stress test flags. Closure under one-loop renormalization -- that all listed operators appear as counterterms -- proves the set has no linear dependencies, but it does not prove the set spans every operator allowed by the lattice symmetries. An allowed operator that does not mix with the listed ones at one loop would be missed, and it would change the anomalous dimension matrix and hence all the exponents. The author acknowledges this in footnote 7 but downplays it. The disagreement with Lee-Sharpe, which found four fewer operators, shows the counting is nontrivial, and the symmetry transformations in Section 3 are partly schematic with terms deferred to the notebook. The argument that field redefinitions affect the action only through EOM-vanishing operators is a sketch, not a proof. None of this makes me think the exponents are wrong; it makes me think they are not yet established with certainty. The Nf=12 row also carries the usual near-conformal caveat.

This is a paper for lattice practitioners who use staggered quarks and care about continuum extrapolations. It deserves a serious referee, and the referee should push for an independent enumeration of the operator basis or a proof of completeness. The calculation itself is solid, reproducible work; the weakness is a missing argument, not a fudge.

Recommendation: send to peer review. Ask the author to either prove basis completeness or soften the claim to a conjecture supported by the closed-mixing check.

Referee Report

2 major / 6 minor

Summary. This paper derives the leading logarithmic corrections to O(a^2) lattice-spacing effects for unrooted staggered quarks in the framework of Symanzik effective theory. Using a strictly local taste representation, the author constructs a minimal on-shell operator basis of mass dimension up to six, computes the one-loop anomalous-dimension matrix of the taste-breaking four-quark operators, and obtains the asymptotic form a^2 [2b0 gbar^2(1/a)]^{gamma_hat_i} for spectral quantities. The reported minimal exponents are gamma_hat_min ≈ -0.273, -0.301, -0.913, -2.614 for Nf = 0, 4, 8, 12, implying that the a^2 suppression is softened by logarithms in all cases and substantially so at Nf = 8 and 12. The paper also discusses EOM-vanishing dimension-five operators, automatic O(a) improvement via a discrete flavour symmetry, and the extension to rooted staggered quarks.

Significance. If the operator basis and the anomalous-dimension calculation are correct, this is a useful and nontrivial result: it replaces the naive a^2 scaling by a log-modified power for staggered-spectrum continuum extrapolations, and the Nf = 8 and 12 exponents are strong quantitative predictions that could be tested in lattice perturbation theory or against non-perturbative data. The method is reproducible in principle: the supplemental Mathematica files, FORM scripts, and public git repository are described, no data fitting is involved (the exponents are eigenvalues of a computed matrix), and the counting of taste-breaking operators is cross-checked against the HPQCD/UKQCD basis. The main caveats are the unproven completeness of the operator basis and the sketchiness of the field-redefinition argument, both of which are load-bearing for the exponents.

major comments (2)

[§3.2.2, footnote 7; Eqs. (3.7), (4.4)–(4.5), Table 1] The central result depends on the completeness of the 28-operator basis in Eq. (3.7). The only evidence offered for completeness is that one-loop renormalization requires all listed operators as counterterms; as the author states in footnote 7, this guarantees minimality and closure but not completeness. If an allowed O(a^2) operator is missing, the anomalous-dimension matrix in Eqs. (4.4)–(4.5) is a submatrix of the true mixing matrix, and the exponents in Table 1 would change. The disagreement with Ref. [42], which found four fewer operators, shows that the counting is nontrivial. Please provide an independent enumeration of the basis (for example, by automated symmetry projection or an explicit case-by-case construction) and a detailed reconciliation with Ref. [42].
[§3.1 and Appendix D, Eqs. (3.1d)–(3.1f), (D.3)] The argument that field redefinitions (especially Shift symmetry) affect the SymEFT action only through EOM-vanishing operators, and that the canonical form of the symmetry can be restored order by order, is a sketch. Equation (D.3) shows that the leading incompatible term is EOM-vanishing, but the induction step assumes that the redefinition which removes it at O(a^d) does not generate new non-EOM operators at O(a^{d+1}) beyond those already in the basis. In addition, the field redefinitions in Eqs. (3.1d)–(3.1f) are written only schematically ('1+a^2...') and deferred to a Mathematica notebook, so the symmetry constraints that define the basis cannot be checked by the reader. Please give the explicit O(a^2) field redefinitions and complete the induction at least through O(a^2), or show directly that the on-shell O(a^2) basis is stable under these redefinitions.

minor comments (6)

[§4, Eq. (4.4)] The full anomalous-dimension matrix uses the Ginsparg-Wilson blocks γ_L|R, γ_L|R,m, and γ_m from Ref. [8], which are not reproduced. Since the eigenvalues in Table 1 are the main quantitative result, please either reproduce the full matrix in an appendix or clearly state that it is contained in the supplementary file staggered.wl and give the file path; as written, the reader cannot reproduce Table 1 from the paper alone.
[Table 1] The typographical annotations (underdotted, underlined, bold) are not legible in the manuscript text; they appear as strings of dots. Please typeset the table properly and include a clear legend, so that the mass-dependent, chirally symmetric, and mixed contributions can be identified.
[§5, Eq. (3.3)] The claim of automatic O(a) improvement via the discrete symmetry in Eq. (3.3) is asserted in analogy to twisted-mass QCD but not derived for the staggered action. A more explicit argument would be useful, especially because the lattice symmetry involves field redefinitions and the analogy may not be immediate.
[Footnote 8] The discussion of whether the renormalization scale should be μ = 1/a or μ = 1/(2a) affects the argument of the running coupling and hence the prefactor of the asymptotic form, although not the exponents. Please comment on the numerical impact on the predictions in Table 1.
[Abstract and §1] The paper should state prominently that Nf is the number of tastes (with Nf/4 physical flavours in the rooted case) and that all results are for unrooted staggered quarks; the present notation is easy to misread.
[§3.2.2, Eq. (3.7)] The shorthand (\barΨ Γ ⊗ τ Ψ)^2 with summation over the free index in the second pair is compact but can be ambiguous; spelling out the contraction for one representative operator would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the exponents are computed eigenvalues of a one-loop anomalous-dimension matrix, not fitted outputs; the self-citation to [8] is code-reproduced and not load-bearing.

full rationale

The paper's derivation chain is: interacting staggered lattice symmetries -> local taste representation -> minimal on-shell SymEFT operator basis -> one-loop off-shell renormalization in MS -> anomalous-dimension matrix -> Jordan normal form -> powers gamma_i. At no point is a parameter fitted to the target exponents. The gamma_i in Table 1 are eigenvalues of the 1-loop anomalous-dimension matrix in Eqs. (4.4)-(4.5), computed from Feynman-diagram counterterms; matching coefficients omega_i are left free and are not needed for the exponents. The main self-citation, [8] (N. Husung), supplies the method and the common blocks gamma_m, gamma_L|R, and gamma_L|R,m. The paper states the blocks were re-checked after adapting the FORM scripts to taste-space matrices and that the scripts/notebooks are publicly available, which qualifies as code-reproduced independent support under the review rules; it therefore does not make the argument circular. The residual weakness is basis completeness: Section 5 and footnote 7 admit that one-loop closure 'only guarantees that our basis is minimal in the sense of absence of linear dependencies but it may still be incomplete.' That is an acknowledged correctness risk, not a circular reduction: the exponents do not define the basis, nor is the basis defined as the set that reproduces the exponents. The disagreement with Lee-Sharpe over operator counting is an independent technical discrepancy. No step reduces by construction to its inputs, so the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The calculation introduces no new particles or interactions and fits no numerical data. The load-bearing assumptions are the validity of the Symanzik expansion, the completeness and correctness of the local taste symmetry mapping, and the completeness of the operator basis. The last of these is explicitly flagged in the paper as not fully proven.

assumptions (6)

domain assumption Symanzik EFT expansion: lattice artifacts of local lattice actions can be described by a local continuum effective Lagrangian order by order in the lattice spacing.
Standard SymEFT framework used throughout the paper; foundational for the operator expansion and the scaling analysis.
domain assumption The local taste representation in eq. (2.9) correctly maps the one-component staggered theory to four tastes and preserves locality.
Section 2; all symmetry constraints and operator classification depend on this mapping and on the locality of the resulting taste action.
domain assumption The symmetry transformations in eqs. (3.1) are the complete set of exact symmetries of the interacting staggered action, and field redefinitions can be restored to canonical form order by order.
Section 3.1 and Appendix D; underpins the minimal operator basis. The proof in Appendix D is sketched, not formal.
standard math One-loop renormalization in MS with background field gauge gives the correct anomalous dimension matrix.
Section 4; standard technique from earlier literature, used here for the staggered operator basis.
ad hoc to paper The on-shell operator basis is complete.
Footnote 7: 'it may still be incomplete. The latter seems unlikely given that no counterterms are missing.' The calculation assumes completeness even though it is only supported by a negative check.
standard math Perturbative running of the coupling at leading logarithm determines the asymptotic lattice-spacing dependence.
Section 4, Eq. (4.6); standard RG solution adopted from the author's prior work.

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Cite this review

Pith. "Pith review of Logarithmic corrections to O($a^2$) effects in lattice QCD with unrooted Staggered quarks." pith.science (2026). https://pith.science/paper/Y466BL3D

@misc{pith2026250117036,
  author       = {Pith},
  title        = {Pith review of: Logarithmic corrections to O($a^2$) effects in lattice QCD with unrooted Staggered quarks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y466BL3D}},
  note         = {Machine review of arXiv:2501.17036}
}

abstract

We derive the asymptotic lattice-spacing dependence $a^2[2b_0\bar{g}^2(1/a)]^{\hat{\gamma}_i}$ relevant for spectral quantities of lattice QCD, when using unrooted Staggered quarks. Without taking any effects from matching into account we find $\min_i\hat{\gamma}_i\approx -0.273, -0.301, -0.913, -2.614$ for $N_\mathrm{f}=0,4,8,12$ respectively. Common statements in the literature on the absence of mass-dimension~5 operators from the on-shell basis of the Symanzik Effective Field Theory action are being clarified for a description using strictly local tastes, here playing the role of continuum quark flavours. Potential impact of $\mathrm{O}(a)$ EOM-vanishing terms beyond spectral quantities is being discussed.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

59 extracted references · 41 canonical work pages

[42]

Lee and S

W.-J. Lee and S. R. Sharpe,Partial flavor symmetry restoration for chiral staggered fermions, Phys. Rev. D60 (1999) 114503 [hep-lat/9905023]

arXiv 1999
[1]

Finkenrath,Review on Algorithms for dynamical fermions, PoS LATTICE2022 (2023) 227 [2402.11704]

J. Finkenrath,Review on Algorithms for dynamical fermions, PoS LATTICE2022 (2023) 227 [2402.11704]

arXiv 2023
[2]

Symanzik,Cutoff dependence in latticeϕ4 4 theory, NATO Sci

K. Symanzik,Cutoff dependence in latticeϕ4 4 theory, NATO Sci. Ser. B59 (1980) 313

work page 1980
[3]

Symanzik,Some Topics in Quantum Field Theory, inMathematical Problems in Theoretical Physics

K. Symanzik,Some Topics in Quantum Field Theory, inMathematical Problems in Theoretical Physics. Proceedings, 6th International Conference on Mathematical Physics, West Berlin, Germany, August 11-20, 1981, pp. 47–58, 1981

work page 1981
[4]

Symanzik,Continuum Limit and Improved Action in Lattice Theories

K. Symanzik,Continuum Limit and Improved Action in Lattice Theories. 1. Principles and ϕ4 Theory, Nucl. Phys.B226 (1983) 187

work page 1983
[5]

Symanzik,Continuum Limit and Improved Action in Lattice Theories

K. Symanzik,Continuum Limit and Improved Action in Lattice Theories. 2. O(N) Nonlinear Sigma Model in Perturbation Theory, Nucl. Phys.B226 (1983) 205

work page 1983
[6]

Renormalization and lattice artifacts

P. Weisz,Renormalization and lattice artifacts, inModern perspectives in lattice QCD: Quantum field theory and high performance computing. Proceedings, International School, 93rd Session, Les Houches, France, August 3-28, 2009, pp. 93–160, 2010,1004.3462

work page Pith review arXiv 2009
[7]

J. B. Kogut and L. Susskind,Hamiltonian Formulation of Wilson’s Lattice Gauge Theories, Phys. Rev. D11 (1975) 395

work page 1975

Show all 59 references

[8]

Husung,Logarithmic corrections to O(a) and O(a2) effects in lattice QCD with Wilson or Ginsparg–Wilson quarks, Eur

N. Husung,Logarithmic corrections to O(a) and O(a2) effects in lattice QCD with Wilson or Ginsparg–Wilson quarks, Eur. Phys. J. C83 (2023) 142 [2206.03536]

2023 arXiv
[9]

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

1974
[10]

H. B. Nielsen and M. Ninomiya,Absence of Neutrinos on a Lattice. 1. Proof by Homotopy Theory,
[11]

H. B. Nielsen and M. Ninomiya,Absence of Neutrinos on a Lattice. 2. Intuitive Topological Proof, Nucl. Phys. B193 (1981) 173

1981
[12]

H. B. Nielsen and M. Ninomiya,No Go Theorem for Regularizing Chiral Fermions, Phys. Lett. B105 (1981) 219

1981
[13]

K. G. Wilson,Quarks and Strings on a Lattice, inNew Phenomena in Subnuclear Physics: Proceedings, International School of Subnuclear Physics, Erice, Sicily, Jul 11-Aug 1 1975. Part A, p. 99, 1975

1975
[14]

P. H. Ginsparg and K. G. Wilson,A Remnant of Chiral Symmetry on the Lattice, Phys. Rev.D25 (1982) 2649. 18

1982
[15]

D. B. Kaplan,A Method for simulating chiral fermions on the lattice, Phys. Lett. B 288 (1992) 342 [hep-lat/9206013]

1992 arXiv
[16]

Furman and Y

V. Furman and Y. Shamir,Axial symmetries in lattice QCD with Kaplan fermions, Nucl. Phys. B439 (1995) 54 [hep-lat/9405004]

1995 arXiv
[17]

Neuberger,Exactly massless quarks on the lattice, Phys

H. Neuberger,Exactly massless quarks on the lattice, Phys. Lett. B417 (1998) 141 [hep-lat/9707022]

1998 arXiv
[18]

Neuberger,More about exactly massless quarks on the lattice, Phys

H. Neuberger,More about exactly massless quarks on the lattice, Phys. Lett. B427 (1998) 353 [hep-lat/9801031]

1998 arXiv
[19]

H. S. Sharatchandra, H. J. Thun and P. Weisz,Susskind Fermions on a Euclidean Lattice, Nucl. Phys. B192 (1981) 205

1981
[20]

M. F. L. Golterman and J. Smit,Selfenergy and Flavor Interpretation of Staggered Fermions, Nucl. Phys. B245 (1984) 61

1984
[21]

Kluberg-Stern, A

H. Kluberg-Stern, A. Morel, O. Napoly and B. Petersson,Flavors of Lagrangian Susskind Fermions, Nucl. Phys. B220 (1983) 447

1983
[22]

Daniel and T

D. Daniel and T. D. Kieu,On the Flavor Interpretations of Staggered Fermions, Phys. Lett. B175 (1986) 73

1986
[23]

Jolicoeur, A

T. Jolicoeur, A. Morel and B. Petersson,Continuum Symmetries of Lattice Models With Staggered Fermions, Nucl. Phys. B274 (1986) 225

1986
[24]

Verstegen,Symmetry Properties of Fermionic Bilinears, Nucl

D. Verstegen,Symmetry Properties of Fermionic Bilinears, Nucl. Phys. B249 (1985) 685

1985
[25]

H. J. Rothe,Lattice Gauge Theories : An Introduction (Fourth Edition), vol. 43. World Scientific Publishing Company, 2012, 10.1142/8229

2012 doi
[26]

Kähler,Der innere Differentialkalkül, pp

E. Kähler,Der innere Differentialkalkül, pp. 160–258. Springer Berlin Heidelberg, Berlin, Heidelberg, 2011. 10.1007/978-3-642-10952-2_6

2011 doi
[27]

Graf,Differential Forms as Spinors, Ann

W. Graf,Differential Forms as Spinors, Ann. Inst. H. Poincare Phys. Theor.29 (1978) 85

1978
[28]

J. M. Rabin,Homology Theory of Lattice Fermion Doubling, Nucl. Phys. B201 (1982) 315

1982
[29]

Mitra and P

P. Mitra and P. Weisz,On Bare and Induced Masses of Susskind Fermions, Phys. Lett. B126 (1983) 355

1983
[30]

Husung,Lattice artifacts of local fermion bilinears up toO(a2), 2409.00776

N. Husung,Lattice artifacts of local fermion bilinears up toO(a2), 2409.00776

arXiv
[31]

Frezzotti, P

R. Frezzotti, P. A. Grassi, S. Sint and P. Weisz,A Local formulation of lattice QCD without unphysical fermion zero modes, Nucl. Phys. Proc. Suppl.83 (2000) 941 [hep-lat/9909003]

2000 arXiv
[32]

Alpha collaboration, Lattice QCD with a chirally twisted mass term, JHEP 08 (2001) 058 [hep-lat/0101001]. 19

2001 arXiv
[33]

Frezzotti, G

R. Frezzotti, G. Martinelli, M. Papinutto and G. C. Rossi,Reducing cutoff effects in maximally twisted lattice QCD close to the chiral limit, JHEP 04 (2006) 038 [hep-lat/0503034]

2006 arXiv
[34]

Aoki and O

S. Aoki and O. Bär,Automatic O(a) improvement for twisted-mass QCD, PoS LAT2006(2006) 165 [hep-lat/0610098]

2006 arXiv
[35]

Sint,Lattice QCD with a chiral twist, inWorkshop on Perspectives in Lattice QCD Nara, Japan, October 31-November 11, 2005, 2007, hep-lat/0702008, DOI

S. Sint,Lattice QCD with a chiral twist, inWorkshop on Perspectives in Lattice QCD Nara, Japan, October 31-November 11, 2005, 2007, hep-lat/0702008, DOI

2005 arXiv
[36]

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]

1996 arXiv
[37]

Capitani, M

S. Capitani, M. Göckeler, R. Horsley, P. E. L. Rakow and G. Schierholz,On-shell and off-shell improvement for Ginsparg-Wilson fermions, Nucl. Phys. B Proc. Suppl. 83 (2000) 893 [hep-lat/9909167]

2000 arXiv
[38]

Capitani, M

S. Capitani, M. Göckeler, R. Horsley, H. Perlt, P. E. L. Rakow, G. Schierholz et al., Renormalization and off-shell improvement in lattice perturbation theory, Nucl. Phys. B 593 (2001) 183 [hep-lat/0007004]

2001 arXiv
[39]

Luo,Improvement of the staggered fermion operators, Phys

Y.-b. Luo,Improvement of the staggered fermion operators, Phys. Rev. D55 (1997) 353 [hep-lat/9604025]

1997 arXiv
[40]

Sheikholeslami and R

B. Sheikholeslami and R. Wohlert,Improved Continuum Limit Lattice Action for QCD with Wilson Fermions, Nucl. Phys.B259 (1985) 572

1985
[41]

HPQCD, UKQCD collaboration, Highly improved staggered quarks on the lattice, with applications to charm physics, Phys. Rev. D75 (2007) 054502 [hep-lat/0610092]

2007 arXiv
[43]

B. S. DeWitt,Quantum Theory of Gravity. 2. The Manifestly Covariant Theory, Phys. Rev.162 (1967) 1195

1967
[44]

Kluberg-Stern and J

H. Kluberg-Stern and J. B. Zuber,Renormalization of Nonabelian Gauge Theories in a Background Field Gauge. 1. Green Functions, Phys. Rev.D12 (1975) 482

1975
[45]

L. F. Abbott,The Background Field Method Beyond One Loop, Nucl. Phys.B185 (1981) 189

1981
[46]

Lüscher and P

M. Lüscher and P. Weisz,Background field technique and renormalization in lattice gauge theory, Nucl. Phys.B452 (1995) 213 [hep-lat/9504006]

1995 arXiv
[47]

S. D. Joglekar and B. W. Lee,General Theory of Renormalization of Gauge Invariant Operators, Annals Phys.97 (1976) 160

1976
[48]

J. C. Collins and R. J. Scalise,The Renormalization of composite operators in Yang-Mills theories using general covariant gauge, Phys. Rev.D50 (1994) 4117 [hep-ph/9403231]. 20

1994 arXiv
[49]

’t Hooft and M

G. ’t Hooft and M. J. G. Veltman,Regularization and Renormalization of Gauge Fields, Nucl. Phys.B44 (1972) 189

1972
[50]

’t Hooft,Dimensional regularization and the renormalization group, Nucl

G. ’t Hooft,Dimensional regularization and the renormalization group, Nucl. Phys. B61 (1973) 455

1973
[51]

W. A. Bardeen, A. J. Buras, D. W. Duke and T. Muta,Deep Inelastic Scattering Beyond the Leading Order in Asymptotically Free Gauge Theories, Phys. Rev.D18 (1978) 3998

1978
[52]

Balog, F

J. Balog, F. Niedermayer and P. Weisz,Logarithmic corrections to O(a2) lattice artifacts, Phys. Lett.B676 (2009) 188 [0901.4033]

2009 arXiv
[53]

Balog, F

J. Balog, F. Niedermayer and P. Weisz,The Puzzle of apparent linear lattice artifacts in the 2d non-linear sigma-model and Symanzik’s solution, Nucl. Phys.B824 (2010) 563 [0905.1730]

2010 arXiv
[54]

S. A. Gottlieb, W. Liu, D. Toussaint, R. L. Renken and R. L. Sugar,Hybrid Molecular Dynamics Algorithms for the Numerical Simulation of Quantum Chromodynamics, Phys. Rev. D35 (1987) 2531

1987
[55]

MILC collaboration, Chiral logs in the presence of staggered flavor symmetry breaking, Phys. Rev. D65 (2002) 054031 [hep-lat/0111051]

2002 arXiv
[56]

Aubin and C

C. Aubin and C. Bernard,Staggered chiral perturbation theory, Nucl. Phys. B Proc. Suppl. 129 (2004) 182 [hep-lat/0308036]

2004 arXiv
[57]

Bernard,Staggered chiral perturbation theory and the fourth-root trick, Phys

C. Bernard,Staggered chiral perturbation theory and the fourth-root trick, Phys. Rev. D 73 (2006) 114503 [hep-lat/0603011]

2006 arXiv
[58]

Bernard,Order of the chiral and continuum limits in staggered chiral perturbation theory, Phys

C. Bernard,Order of the chiral and continuum limits in staggered chiral perturbation theory, Phys. Rev. D71 (2005) 094020 [hep-lat/0412030]

2005 arXiv
[59]

J. A. M. Vermaseren,New features of FORM, math-ph/0010025. 21

arXiv

Pith tools

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

[1] [42]

Lee and S

W.-J. Lee and S. R. Sharpe,Partial flavor symmetry restoration for chiral staggered fermions, Phys. Rev. D60 (1999) 114503 [hep-lat/9905023]

arXiv 1999

[2] [1]

Finkenrath,Review on Algorithms for dynamical fermions, PoS LATTICE2022 (2023) 227 [2402.11704]

J. Finkenrath,Review on Algorithms for dynamical fermions, PoS LATTICE2022 (2023) 227 [2402.11704]

arXiv 2023

[3] [2]

Symanzik,Cutoff dependence in latticeϕ4 4 theory, NATO Sci

K. Symanzik,Cutoff dependence in latticeϕ4 4 theory, NATO Sci. Ser. B59 (1980) 313

work page 1980

[4] [3]

Symanzik,Some Topics in Quantum Field Theory, inMathematical Problems in Theoretical Physics

K. Symanzik,Some Topics in Quantum Field Theory, inMathematical Problems in Theoretical Physics. Proceedings, 6th International Conference on Mathematical Physics, West Berlin, Germany, August 11-20, 1981, pp. 47–58, 1981

work page 1981

[5] [4]

Symanzik,Continuum Limit and Improved Action in Lattice Theories

K. Symanzik,Continuum Limit and Improved Action in Lattice Theories. 1. Principles and ϕ4 Theory, Nucl. Phys.B226 (1983) 187

work page 1983

[6] [5]

Symanzik,Continuum Limit and Improved Action in Lattice Theories

K. Symanzik,Continuum Limit and Improved Action in Lattice Theories. 2. O(N) Nonlinear Sigma Model in Perturbation Theory, Nucl. Phys.B226 (1983) 205

work page 1983

[7] [6]

Renormalization and lattice artifacts

P. Weisz,Renormalization and lattice artifacts, inModern perspectives in lattice QCD: Quantum field theory and high performance computing. Proceedings, International School, 93rd Session, Les Houches, France, August 3-28, 2009, pp. 93–160, 2010,1004.3462

work page Pith review arXiv 2009

[8] [7]

J. B. Kogut and L. Susskind,Hamiltonian Formulation of Wilson’s Lattice Gauge Theories, Phys. Rev. D11 (1975) 395

work page 1975