Improved bilinears in lattice QCD with non-degenerate quarks

Jackson M. S. Wu; Rajan Gupta; Stephen R. Sharpe; Tanmoy Bhattacharya; Weonjong Lee

REVIEW

Improved bilinears in lattice QCD with non-degenerate quarks

Tanmoy Bhattacharya, Rajan Gupta, Weonjong Lee, Stephen R. Sharpe, Jackson M. S. Wu

High Energy Physics - Lattice · Submitted 4 Nov 2005 · 28 references

Pith's verdict: A careful, honest bookkeeping paper that maps which Ward identities fix which O(a) improvement coefficients for non-degenerate Wilson fermions — the algebra holds up, and the self-correction in Appendix C is the most valuable part.

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arXiv IDhep-lat/0511014 (version 3)

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CitationJackson M. S. Wu, Rajan Gupta, Stephen R. Sharpe, Tanmoy Bhattacharya, Weonjong Lee. "Improved bilinears in lattice QCD with non-degenerate quarks." arXiv:hep-lat/0511014 (2005). https://pith.science/paper/FIHMYLQH

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This is a bookkeeping paper, and it's a good one. If you do 2+1 or 2+1+1 flavor Wilson fermion work and care about O(a m_s) or O(a m_c) in bilinear matrix elements, this is the reference that tells you which coefficient each Ward identity actually pins down. The enumeration for non-degenerate masses, the singlet sector with the r_O normalization ratios, and the identity-to-coefficient map in Tables I and II are all new relative to the degenerate-quark work of Refs. [1, 8, 17]. The tables cross-check each other, and the paper is explicit about which identities give new information and which are just consistency checks. That kind of honesty about its own redundancy is rarer than it should be. The two coefficients that stay undetermined (d-bar_A and b-bar_T + d-bar_T) are named in the conclusion, so 'nearly all' in the abstract [abstract:s2] is not overselling. The Nf=2 appendix is a genuinely useful negative result: eight undetermined combinations, and you can't even fix the normalization of the non-singlet axial current away from the chiral limit. Appendix C is the best part. They show their own earlier proceedings claim about vector Ward identities was wrong, trace the error to a missed contact term proportional to tr(O), and explain why the equations-of-motion mnemonic of Refs. [7, 8] doesn't capture it. Disclosing that in the abstract and introduction is the right behavior. Soft spots, in proportion: the sufficiency step is done by hand. The paper lists which combinations each identity yields and then asserts you can solve for the individual coefficients, but never writes down the coefficient matrix or its rank. Given that this same hand-counting already missed a term once, an explicit rank check would have cost a paragraph and closed the gap. I'd call that a legitimate minor ask, not a flaw in the result — the algebra in eq. (75) and the cross-checks make a hidden degeneracy unlikely. Second, the practical question they raise themselves — whether anyone will actually carry out these determinations, given the disconnected contractions — is left open, and that's fine; they flag it rather than pretend otherwise. Who should read it: lattice people doing Wilson fermion systematics, and anyone writing a review of O(a) improvement. Worth a serious referee.

Referee report

Significance

This is the reference analysis that defines what O(a) improvement of bilinears costs when the sea quarks are not degenerate, which is the realistic case for 2+1 and 2+1+1 simulations. It is a methods paper with no new data, but its value is that it converts a vague statement ('more coefficients are needed') into an explicit, identity-by-identity target list that any future non-perturbative determination can work from, and it identifies precisely which quantities Ward identities cannot reach and which therefore require non-perturbative renormalization or short-distance matching. Two features deserve explicit credit. First, the paper is scrupulous about distinguishing new information from cross-checks; for example, in §IV.D it states that several of the S and P combinations obtained from three-point identities are related to the two-point results by Table I and therefore 'obtain a check of these relations, but no new information on the constants themselves.' Second, Appendix C is a genuine self-correction: the authors identify the flaw in their own earlier argument, name the missing contact operator, and explain why the equations-of-motion mnemonic fails rather than patching over it.

Major objections (2)

[§IV.C–§IV.D and Table II] The central claim of the abstract — that for three or four flavors 'nearly all can be determined' — is a rank statement about the linear system obtained after imposing the Table I relations, but the sufficiency step is argued identity by identity in prose rather than demonstrated. For instance, §IV.D asserts 'we can now separately determine b_P, f_P, b_m and d_m as well as d_P' without displaying the coefficient matrix or counting unknowns against independent combinations at that stage. This matters because the same hand-counting has previously failed: Appendix C documents that Ref. [12] concluded f_O was determinable from a vector Ward identity and that this was wrong. I want to be clear that I checked several of the algebraic steps and they are correct — Eq. (75)'s coefficients of trM and (m_i - m_j) follow from Eq. (74) once the ~m_ij = 0 constraint is imposed, Eqs. (30) and (31) follow from Eqs. (18) and (23), and Eqs. (77)–(78) are consistent — so this is not a suspicion of error. It is a request that the sufficiency argument be made checkable, for example by a short table giving, at each step, the number of unknowns carried in and the number of independent combinations extracted, with a note on which combinations are redundant with earlier steps.
[§IV.D, Eq. (93)] The determination of f_T rests entirely on the assertion that exactly two contact operators are allowed in the T^(jj) + T^(kk) identity, namely a ~m_jk (δT^(jj) + δT^(kk)) and a ~m_jk tr(δT). The paper states this enumeration rather than deriving it, and it is precisely the kind of enumeration that Appendix C shows to be error-prone: the authors themselves write that 'there is no such factorization of contact terms' and that the mnemonic 'needs to be generalized.' If a third operator with the same quantum numbers and dimension existed, f_T would not be determinable and the final row of Table II would overstate what the identity delivers. Since f_T is one of the quantities the paper claims as newly determined, I would ask the authors to spell out the enumeration: list the quantum numbers of the operator product P^(jk) T^(kj), list the candidate dimension-four operators, and state why each excluded candidate is forbidden (by CPS, by flavor, or by being related to the two retained operators through the equations of motion).

Minor objections (8)

[§III, second paragraph] The text states that the singlet coefficients 'we denote them c-bar_A, c-bar_V and d-bar_T, respectively,' but Eq. (12) and every later occurrence (§IV.D, Table II, Table III) use c-bar_T. The d-bar_T in this sentence appears to be a typographical slip and should be corrected to c-bar_T.
[Table II (p. 16)] Table II is placed well before the notation it uses is introduced: the symbol δ^(ij)_A for the axial variation of the action is only defined in §IV.D (p. 26), and the reader encountering the table on p. 16 has no way to parse rows such as 'δ^(ij)_A T^(jk) = T^(ik)'. A one-line definition in the caption, or a forward reference to Eq. (69), would make the table usable on first reading.
[§IV.C] The sentence 'one can only determine seven combinations of the ten improvement constants which enter (we exclude b_A and f_A ...)' requires the reader to reconstruct the set of ten by inspecting Eqs. (60)–(68). A parenthetical list of the ten constants would let the count be verified without that reconstruction.
[Footnote 9] The footnote asserts that quark-disconnected diagrams give vanishing contributions if one uses the charge built from the conserved vector current, but gives no argument or reference. Since this is what makes the practical determination of b_V and b-bar_V tractable, a sentence of justification would be worthwhile.
[§IV.D, discussion of b_g] The paper claims twice (§II and §IV.D) that the scheme is self-contained with respect to b_g because g_S can be determined in the chiral limit. The logical ordering is not spelled out: Table I was derived at fixed effective coupling g-tilde_0, which presupposes b_g, so the reader must infer that the chiral-limit determination of g_S (and hence b_g = 2 g_0^2 g_S) is to be carried out first and the mass-dependent identities only afterwards. An explicit statement of the order of operations would remove the appearance of circularity.
[§III, first paragraph] The sentence ends 'described in the appendices..' with a doubled period.
[Eq. (100)] The equation is written as a definition ('≡') of the right-hand side via the left-hand side, but the qualifier '~m_jk = 0' is appended after the O(a^2) term, which reads awkwardly. Restating it as a two-line definition with the constraint stated first would be clearer.
[§V, first paragraph] The statement that 'the only scale independent quantities which are left undetermined are d-bar_A and b-bar_T + d-bar_T' should be explicitly qualified as applying to Nf = 3 and 4, since the abstract's 'nearly all' is flavor-dependent and the Nf = 2 count of eight undetermined combinations appears only in Appendix B.

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The paper

The paper claims

The paper extends Symanzik O(a) improvement of Wilson-fermion bilinears from degenerate to non-degenerate dynamical quarks, and argues that for three or four flavors the resulting proliferation of improvement coefficients is almost entirely

The reading

Lattice QCD with Wilson fermions carries discretization errors linear in the lattice spacing $a$, and removing them requires adding every allowed higher-dimension operator to the action and to each quark bilinear, with coefficients fixed by symmetry conditions rather than by fitting. That programme was worked out for degenerate quarks; this paper asks what happens in the realistic case where the up, down, strange and charm quarks have different masses, as in $N_f = 2+1$ and $N_f = 2+1+1$ simulations. Treating the mass matrix as a symmetry-breaking spurion, the authors enumerate the extra improvement and normalization coefficients that appear for flavor non-singlet and flavor singlet bilinears and for the quark masses themselves, and find far more of them than in the degenerate case. They then show which of these can be pinned down by enforcing vector charge conservation and axial Ward identities, concluding that for three flavors (and, with more cross-checks, four) nearly all can, while for two flavors many combinations remain undetermined. The practical stake is stated plainly: hadronic matrix elements used to extract CKM elements carry errors proportional to $am_s$ and especially $am_c$, and the paper's own estimate is that a tree-level $b_A = 1$ may suffice for $f_K$-type quantities at $a^{-1} = 2$ GeV but leaves roughly a 10% error for $f_D$ in a four-flavor theory.

What carries the argument

The engine is a spurion enumeration combined with Ward identity bookkeeping. The mass matrix $M$ is treated as transforming in the adjoint of the flavor group, which restricts the allowed $O(a)$ terms to the anticommutator structure $tr({{lambda, M} O})$ and forbids the commutator by CP; this yields the master formulas eqs. (15), (23), (24) and (25) for non-singlet bilinears, singlet bilinears, non-singlet masses and the singlet mass. The second ingredient is the relation $partial m_H / partial hat{m}_j = V langle H | hat{S}^{(jj)} | H rangle / langle H | H rangle$, enforced not only in the continuum limit but at finite lattice spacing; because the fermion action depends on the bare masses only through $a m_j S^{(jj)}$, the derivative of the hadron mass with respect to a bare mass is a known scalar matrix element, and the gluon term supplies $b_g$. Chaining Jacobians from improved to bare masses and matching coefficients produces Table I. The third ingredient is the classification of contact terms: when the axial variation of the action meets another operator at the same point, extra operators of the appropriate dimension and quantum numbers appear, and the paper's rule is to enumerate them by symmetry rather than by the equations-of-motion mnemonic used earlier, which the authors state cannot be trusted because contact terms do not factorize.

Extended reading notes

Core claim

The central claim is a counting and determination result. Away from the chiral limit, each non-singlet bilinear needs three improvement coefficients beyond its chiral-limit normalization, written $b_O$, $b_O$-bar and $f_O$ in the general ansatz of eq. (15); each singlet bilinear needs two, $d_O$ and $d_O$-bar, plus a ratio $r_O$ of singlet to non-singlet normalization; and the quark masses need their own set, $r_m$, $b_m$, $b_m$-bar, $d_m$, $d_m$-bar. The authors show that for $N_f = 3$ the vector charge normalization fixes all vector coefficients except $c_V$ and $c_V$-bar, that matching the mass derivative of hadron masses to the scalar matrix element fixes the relations collected in Table I (including $b_S = -2b_m$, $b_S$-bar $= -b_m$-bar, and $b_g = 2g_0^2 g_S$), and that two-point and three-point axial Ward identities then determine essentially everything else. Only two scale-independent combinations survive: $d_A$-bar and $b_T$-bar $+ d_T$-bar; the scale-dependent $Z_T$, $Z_S Z_P$ and $r_A$ are also out of reach, as they must be, since Ward identities carry no renormalization scale. For $N_f = 2$ the same machinery leaves eight combinations undetermined, including $b_A$-bar, so that even the normalization of the non-singlet axial current away from the chiral limit is not fixed. The paper is explicitly a correction of an earlier version of the same analysis: enforcing vector Ward identities was previously claimed to determine more than it does, and Appendix C traces the error to a contact term proportional to $tr(O)$ that had been missed, which is exactly what allows the $f_O$ term to exist without being forced to vanish.

Load-bearing premise

The whole determination rests on being able to list every $O(a)$ contact term that can appear when the axial variation of the action meets another operator, and the authors show their own earlier list was incomplete, so the corrected list, obtained by counting symmetries rather than by deriving the terms, is the least controlled step.

Editorial extensions

If this is right

For a 2+1 flavor simulation, the coefficients needed for the flavor off-diagonal axial current ($Z_A$, $b_A$, $b_A$-bar) and for the electromagnetic current ($Z_V$, $b_V$, $b_V$-bar, $f_V$) are all obtainable from Ward identities, with the vector ones requiring quark-disconnected contractions.
Because $b_g = 2g_0^2 g_S$, determining $g_S$ from the axial transformation of the singlet scalar gives an independent, self-contained determination of $b_g$, so the improvement programme does not have to assume $b_g$ is known in advance.
The improved quark condensate can be obtained by evaluating the axial variation of the singlet pseudoscalar at $tilde{m}_{jk} = 0$, which removes the power-divergent identity-operator contribution by explicit subtraction.
Two scale-independent combinations, $d_A$-bar and $b_T$-bar $+ d_T$-bar, plus the scale-dependent $Z_T$, $Z_S Z_P$ and $r_A$, must come from non-perturbative renormalization or from matching short-distance correlators to perturbation theory.
For two flavors, eight combinations of scale-independent coefficients remain unknown, so the normalization of the non-singlet axial, pseudoscalar and tensor bilinears away from the chiral limit is not determined by Ward identities alone.

Reading between the lines

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

The determination scheme's real vulnerability is not the algebra but the completeness of the contact-term list: a numerical 2+1 flavor check that the tabulated coefficient set makes three-point axial Ward identity ratios independent of source and separation to $O(a^2)$ would validate the whole chain, and a residual $O(a)$ drift would locate exactly where the enumeration is short.
The contrast between three and two flavors suggests the operative resource is the number of independently variable mass handles rather than the number of flavors, which makes a graded or partially quenched extension the natural next test, though the paper itself cautions that a non-unitary sea weakens the Symanzik basis.
If $d_A$-bar and $b_T$-bar $+ d_T$-bar are genuinely invisible to Ward identities, a two-loop perturbative estimate of both would settle whether the gap matters numerically at $a^{-1}$ between 2 and 4 GeV, in the same spirit as the paper's own tree-level estimate for $b_A$.
The paper's remark that Wilson averaging and maximally twisted mass make these coefficients unnecessary at the cost of requiring an even number of flavors frames the result as a benchmark: the improvement coefficients computed here are precisely the size of the effect those approaches are designed to cancel.

Original abstract

We describe the extension of the improvement program for bilinear operators composed of Wilson fermions to non-degenerate dynamical quarks. We consider two, three and four flavors, and both flavor non-singlet and singlet operators. We find that there are many more improvement coefficients than with degenerate quarks, but that, for three or four flavors, nearly all can be determined by enforcing vector and axial Ward identities. The situation is worse for two flavors, where many more coefficients remain undetermined.

The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,547 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.91 · human participants 0.01 · controlled comparison 0.16 · randomized assignment 0.01 · states quantitative result 0.24 · compares with prior results 0.75 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We describe the extension of the improvement program for bilinear operators composed of Wilson fermions to non-degenerate dynamical quarks. We consider two, three and four flavors, and both flavor non-singlet and singlet operators. We find that there are many more improvement coefficients than with degenerate quarks, but that, for three or four flavors, nearly all can be determined by enforcing vector and axial Ward identities. The situation is worse for two flavors, where many more coefficients remain undetermined.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We describe the extension of the improvement program for bilinear operators composed of Wilson fermions to non-degenerate dynamical quarks.	contribution 1.00	0.89	0.09	asserted 0.94
s1	We consider two, three and four flavors, and both flavor non-singlet and singlet operators.	contribution 0.67	0.49	0.04	asserted 0.87
s2	We find that there are many more improvement coefficients than with degenerate quarks, but that, for three or four flavors, nearly all can be determined by enforcing vector and axial Ward identities.	result 1.00	0.95	0.67	asserted 1.00
s3	The situation is worse for two flavors, where many more coefficients remain undetermined.	result 0.63	0.79	0.10	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.80 · evidence from proof or derivation 0.73.

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sources read

arXiv listing abstract v3

not examined

full text
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P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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hep-lat 2017-03 conditional novelty 6.0 of 10

A lattice QCD calculation covering all three light quark flavors, with disconnected loops and nonperturbative singlet renormalization, gives g_A^s = -0.0240(21)(11) and shows that disconnected diagrams dominate the in...
Lattice simulations with $N_f=2+1$ improved Wilson fermions at a fixed strange quark mass
hep-lat 2016-06 conditional novelty 6.0 of 10

An order-a improved recipe keeps the axial-Ward-identity strange quark mass constant to sub-percent precision across two lattice spacings in Nf=2+1 Wilson-fermion simulations.
Direct determinations of the nucleon and pion $\sigma$ terms at nearly physical quark masses
hep-lat 2016-03 conditional novelty 6.0 of 10

Two-flavour lattice QCD at pion masses down to 150 MeV yields a pion-nucleon sigma term of 35(6) MeV and a strange sigma term of 35(12) MeV, using a new boundary-condition trick that removes a dominant excited-state c...
Dimension-5 CP-odd operators: QCD mixing and renormalization
hep-ph 2015-02 accept novelty 6.0 of 10

The one-loop mixing matrix and a lattice-compatible RI-SMOM renormalization scheme, with MS-bar matching coefficients in both NDR and 't Hooft-Veltman, are given for the flavor-diagonal quark chromo-electric dipole op...
QCD thermodynamics with continuum extrapolated Wilson fermions I
hep-lat 2012-05 conditional novelty 6.0 of 10

Continuum-extrapolated Wilson-fermion lattice QCD reproduces the temperature dependence of three thermodynamic observables obtained from staggered fermions, at a pion mass of 545 MeV.
Lattice QCD at the physical point: Simulation and analysis details
hep-lat 2010-11 accept novelty 6.0 of 10

Using 2+1 flavor lattice QCD with pion masses down to 120 MeV at five lattice spacings, the authors obtain m_s and m_ud to better than 2% and their ratio to better than 1%, with a quenched cross-check against an indep...
Window observable for the hadronic vacuum polarization contribution to the muon $g-2$ from lattice QCD
hep-lat 2022-06 accept novelty 5.0 of 10

Lattice QCD gives the intermediate-window HVP contribution as 237.30(1.46) x 10^-10, which is 3.9 sigma above the data-driven value of 229.4(1.4) x 10^-10.
Lattice calculation of the pion transition form factor with $N_f=2+1$ Wilson quarks
hep-lat 2019-03 accept novelty 5.0 of 10

Lattice QCD with 2+1 dynamical quarks yields the pion transition form factor and a 6%-precision estimate of the pion-pole contribution to muon g-2 hadronic light-by-light scattering, (59.7 ± 3.6) × 10^-11.
Non-perturbative renormalization and O$(a)$-improvement of the non-singlet vector current with $N_{\mathrm{f}}=2+1$ Wilson fermions and tree-level Symanzik improved gauge action
hep-lat 2018-11 conditional novelty 5.0 of 10

The authors determine non-perturbatively the renormalization factor and O(a) improvement coefficients of the local and conserved non-singlet vector currents for Nf=2+1 Wilson fermions, leaving only fV undetermined.
Simulation of QCD with N_f=2+1 flavors of non-perturbatively improved Wilson fermions
hep-lat 2014-11 accept novelty 5.0 of 10

A production-scale set of N_f=2+1 Wilson-fermion gauge ensembles is generated and validated using open time boundaries and twisted-mass reweighting, with parameters, autocorrelations, reweighting factors and t_0, m_pi...
The strange quark mass and Lambda parameter of two flavor QCD
hep-lat 2012-05 conditional novelty 5.0 of 10

Two-flavor lattice QCD gives Lambda_MS = 310(20) MeV and m_s = 138(3)(1) MeV, with r_0 = 0.503(10) fm, using the kaon decay constant to set the scale.
Strangeness Contribution to the Proton Spin from Lattice QCD
hep-lat 2011-12 conditional novelty 5.0 of 10

Two-flavour lattice QCD gives the strangeness contribution to proton spin as Delta s = -0.020(10)(4) at 7.4 GeV, small and negative.
The strange and light quark contributions to the nucleon mass from Lattice QCD
hep-lat 2011-11 conditional novelty 5.0 of 10

The strange quark contributes at most a few percent of the nucleon mass, and the pion-nucleon sigma term is 38(12) MeV at the physical point, from a direct lattice calculation of connected and disconnected scalar matr...
Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks
hep-lat 2011-02 conditional novelty 5.0 of 10

Holding the average of the up, down and strange quark masses fixed along the path to the physical point, combined with a group-theoretic classification of allowed mass polynomials, gives tightly constrained fits that ...
Tuning the strange quark mass in lattice simulations
hep-lat 2010-03 conditional novelty 5.0 of 10

Holding the flavour-singlet quark mass fixed while varying the up-down/strange splitting gives a smoother lattice tuning path and baryon masses within a few percent of experiment.
Setting the scale for the CLS $2 + 1$ flavor ensembles
hep-lat 2016-08 accept novelty 4.0 of 10

Using the pion-kaon decay-constant combination f_piK as experimental input, the paper fixes sqrt(8 t0) = 0.415(4)(2) fm and the lattice spacings of the CLS 2+1 flavor ensembles.
Lattice QCD and the anomalous magnetic moment of the muon
hep-lat 2018-07 conditional novelty 2.0 of 10

Lattice QCD now pins the hadronic vacuum polarisation contribution to the muon g-2 at the 2-3% level and the light-by-light contribution at roughly 10-25%, still short of the sub-percent precision needed to arbitrate ...

How later papers used it

60 later papers cite it

Evidence: substantial. 52 of the citing papers share no author with it, and 16 of those carry a confident label from Pith's citation classifier.

Pith read 25 of the 126 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 later papers by other authors are listed below: 1 built on it and 2 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Builds on it 1 paper

Combined the argument for (a) improvement for TM with an extension of this paper to Wilson-like theories, extending work originally formulated only for Wilson fermions.

Heavy-quark physics with a tmQCD valence action (2018) · 0.98 · paraphrased: the licence does not allow quoting

Uses its method or tool 3 of 15 papers with this label · 1 by the same authors

(23) of [ref] , we obtain [equation] Again, we replaced the lattice decay constants within the $𝒪(a)$ terms by partially improved ones.

Masses and decay constants of the $\eta$ and $\eta^\prime$ mesons from lattice QCD (2021) · Renormalization and $𝒪(a)$ improvement · 0.99
The renormalised $O(a)$ improved current reads [ref] [equation] where $m_qc$ and $M$ denote the bare vector Ward identity quark mass combinations [equation] The hopping parameter for quark flavour $q$ is denoted by $κ_q$ and $κ_crit$ labels its critical value.

$\mathrm{D}$ and $\mathrm{D_s}$ decay constants in $N_{\rm f}=2+1$ QCD with Wilson fermions (2024) · Observables · 0.84

By the same authors

We will do this using the notation and results given in Ref. [ref] for an $N_f$ flavor theory with two light degenerate flavors, $m_u = m_d ≡m_l$, and $N_f-2$ heavier flavors denoted generically by $m_s$.

The pion-nucleon sigma term from lattice QCD (2021) (same authors) · Renormalization · 0.90

How the literature uses it

58 arXiv papers cite this in 118 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 89 sentences without a confident label

Contests it 3 papers

We would therefore like to point out a subtlety that was overlooked in [ref] .

Heavy Wilson Quarks and O($a$) Improvement: Nonperturbative Results for $b_{\rm g}$ · Chiral Ward identities and $$ · contest 0.80
For a detailed derivation, which also corrects an oversight in [ref] , we refer to our paper [ref] .

A non-perturbative determination of $b_{\rm g}$ · Improvement condition for $$ · contest 0.79
This introduces an additional term $∝(g_0^2)$ in the derivation which was possibly overlooked in the original publication [ref] .

$\mathrm{O}(a)$ improvement of the flavour singlet scalar density in a setup with Wilson fermions · Conclusion and Outlook · contest 0.76

Uses its method or tool 13 papers

For the currents relevant for this study, we perform the substitutions [ref] ($a=1,…,8$): [equation] where we re-expressed the last equation using the singlet AWI in the massless case.[Note that $c_P^s=-16π^2 g_P/√(6)$ to leading order in $g^2$.

Masses and decay constants of the $\eta$ and $\eta^\prime$ mesons from lattice QCD · Gauge ensembles · 0.93
This ratio can be determined non-perturbatively from the slope of the axial Ward identity quark mass ($m̃_q$) as a function of the vector Ward identity mass ($m_q$), see, e.g, Ref. [ref] : [equation] where the quark masses are defined as [equation] $A_4^I$ and $P$ denote the $𝒪(a)$ improved axial-vector current and the pseudoscalar operator, respectively, with corresponding renormalization factors $Z_A^ns$ and $Z_P^ns$. $κ_c,sea$ denotes the critical mass parameter along the isosymmetric line for which the quark mass is zero.

Direct determinations of the nucleon and pion $\sigma$ terms at nearly physical quark masses · Renormalization · 0.93
The renormalised $O(a)$ improved current reads [ref] [equation] where $m_qc$ and $M$ denote the bare vector Ward identity quark mass combinations [equation] The hopping parameter for quark flavour $q$ is denoted by $κ_q$ and $κ_crit$ labels its critical value.

$\mathrm{D}$ and $\mathrm{D_s}$ decay constants in $N_{\rm f}=2+1$ QCD with Wilson fermions · Observables · 0.84

Compares with it 1 paper

Finally we compare these results with those obtained in [ref] , to see whether we can match the $4$ improvement terms found in eq.

Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks · $𝐎(𝐚)$ improvement of quark masses · 1.00

Cites it as background 6 papers

We follow the standard notation in the literature [ref] where $c_*$ terms cancel $(a)$ cutoff effects and $b_*$, $b̅_*$ the leading mass-dependent effects of order $am$, induced from the valence or sea quark sector, respectively.

Mass-improvement of the vector current in three-flavor QCD · Prelude · 0.85
The $𝒪(a)$ improvement of quark bilinears has been worked out for Wilson fermions in [ref] .

Masses and decay constants of the $\eta$ and $\eta^\prime$ mesons from lattice QCD · Renormalization and $𝒪(a)$ improvement · 0.80
For those cases, the mass-dimension 4 operator basis relevant for $(a)$ corrections has been discussed earlier [ref] with explicit improvement in mind and not its impact on scaling due to the powers $Γ̂_i^$.

Lattice artifacts of local fermion bilinears up to $\mathrm{O}(a^2)$ · Introduction · 0.83

Discussion

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