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High Energy Physics - Phenomenology · arXiv:hep-ph/0502212 v2 · submitted 22 Feb 2005 · 42 references

A Determination of the Lambda Parameter from Full Lattice QCD

M. Göckeler, R. Horsley, A.C. Irving, D. Pleiter, P.E.L. Rakow, G. Schierholz, H. Stüben

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The paper says

“We present a determination of the QCD parameter Lambda in the quenched approximation (n_f=0) and for two flavours (n_f=2) of light dynamical quarks.”

more of the abstract

The calculations are performed on the lattice using O(a) improved Wilson fermions and include taking the continuum limit. We find Lambda_n_f=0 = 259(1)(20) MeV and Lambda_n_f=2 = 261(17)(26) MeV}, using r_0 = 0.467 fm to set the scale. Extrapolating our results to five flavours, we obtain for the running coupling constant at the mass of the Z boson alpha_s(m_Z) = 0.112(1)(2). All numbers refer to the MSbar scheme.

from the abstract

Pith says

Machine review · deepseek-v4.1-flash · 13 Nov 2026 · what was read was not recorded · not checked by a person

Reviewed in 2026 with later literature in view.

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Holds up, with conditions. A careful, honest update of the collaboration's own Lambda determination: the nf=2 number is solid and quotable, but the alpha_s(mZ) headline carries an error budget that quietly omits the flavour-matching systematic. This line is wrong

4 major objections · 6 minor

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The paper says, at §4.2, Eq. (44), Table 4

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Pith says: Objection 1 of 4 · major

The systematic error 0.021 quoted in Eq. (44) is described as 'the spread in the results between method III and IIIP', and indeed the continuum row of Table 4 gives III = 0.6376(412) and IIIP = 0.6170(395), a difference of 0.0206.

more of objection 1

But methods II and III differ by the trajectory used to run c_sw from g=0 to g=g_□, and their continuum values are II = 0.6085(352) and III = 0.6376(412), a spread of 0.0291. That trajectory spread is comparable to the quoted systematic and is not folded into it. Since the choice of IIIP over IIP shifts the central value by roughly one sigma (0.617 versus 0.582), the reader needs either an enlarged systematic that includes the II–III spread, or an explicit argument for why the Padé spread is the appropriate estimator and the trajectory spread is not. As it stands the error budget of the headline n_f=2 number, and hence of the two-sigma tension with the collider average, is understated.

The paper says, at §5.3, Eq. (53)

The review did not quote this passage.

Pith says: Objection 2 of 4 · major

Eq. (53) quotes r0Λ_3 = 0.616(29)(19), obtained from Eq. (44) by multiplication by the matching ratio Λ_3/Λ_2 = 0.999. Multiplication by a factor of 0.999 cannot reduce the statistical error from 0.040 in Eq. (44) to 0.029 in Eq. (53); simple propagation gives 0.040. Either the errors are correlated in a way that is not explained, or there is an arithmetic slip. Because Λ_3 is the input to the α_s(m_Z) running, this needs to be corrected or explained explicitly.

The paper says, at §4.2, Fig. 6

The review did not quote this passage.

Pith says: Objection 3 of 4 · major

The n_f=2 continuum extrapolation rests on four points spanning only a^{-1} ≈ 2–3 GeV, and the linearity in (a/r0)^2 is therefore much less tested than in the quenched case, where the fit spans a^{-1} ≈ 2–6.5 GeV. The paper selects IIIP partly because 'the results from method IIIP appear to be insensitive to the particular form of the continuum extrapolation', which risks a selection bias: the method is being preferred on the flatness of its own extrapolation.

more of objection 3

Please show the extrapolation with the coarsest point dropped, and with a combined II+III fit, so that the reader can see how much of the quoted central value and error depends on the fit window rather than on the data.

The paper says, at §5.3

The review did not quote this passage.

Pith says: Objection 4 of 4 · major

The paper states that it has 'not attempted to estimate the systematic error induced by the matching procedure', and §5.2 acknowledges that r0Λ ≈ 0.6 is not small, so the perturbative matching at scale r0 is being used outside its comfortable domain.

more of objection 4

The effect is probably small, since Λ_3/Λ_2 = 0.999, but the α_s(m_Z) number runs through n_f=3 and the matching uncertainty is currently absent from the error budget. A bound obtained by varying the matching scale (for example r0/2) or by taking the 1-loop versus 3-loop spread visible in Fig. 7 would close this gap at modest cost.

6 minor objections

The paper says, at Abstract

Pith says: minor objection 1 of 6

The abstract contains a stray closing brace in '261(17)(26) MeV}'. Please remove it.

The paper says, at §4.1, Fig. 3

Pith says: minor objection 2 of 6

The text says the smallest a value (β = 6.92) is excluded 'as it appears to deviate a little' but that including it 'would not have changed the extrapolated value much'. Please quantify the shift, since excluding the finest lattice point from a continuum extrapolation is exactly the kind of choice a reader will want to see justified numerically.

show 4 more

The paper says, at §4.2, Eq. (42)

Pith says: minor objection 3 of 6

The global fit ansatz for ln(r0/a) has A1 linear in β and A2, A3 quadratic in β, i.e. five free parameters fitted to fifteen points across four β values. Please report the χ²/dof and the parameter count so that the reader can judge whether the ansatz is over-parameterised, and state whether the fit error on r0/a at κ_c is propagated into the final error on r0Λ_MS.

The paper says, at §6, Eq. (57)

Pith says: minor objection 4 of 6

The systematic error (2) in α_s(m_Z) = 0.112(1)(2) should be decomposed: how much comes from the scale r0 = 0.467 fm, how much from the continuum extrapolation, and how much from the threshold matching? The reader cannot currently tell which of these dominates.

The paper says, at References

Pith says: minor objection 5 of 6

Reference [39] is listed as 'talk at Lattice 2004, and paper to appear'. Since it appears only in the comparison figure (Fig. 9) and is not load-bearing, this is not a blocker, but it should be updated or marked as preliminary.

The paper says, at Fig. 8

Pith says: minor objection 6 of 6

The caption notes that the n_f=3 and n_f=4 entries 'have been displaced horizontally'. Please state the size of the displacement in the caption so that the figure is not misread as showing a physical n_f dependence.

What would settle it

Compute the 3-loop conversion coefficient t3^box (or the 3-loop lattice beta-function coefficient b3^box) and redo the method II/III comparison. If the method dependence shrinks when t3 is included, the truncation concern is resolved; if it persists, the systematic error should be enlarged. A cheaper complementary check: redo the continuum extrapolation for IIIP with an added a^4 term and with the coarsest beta dropped, and see whether the intercept shifts by more than the quoted systematic (0.021).

from the skeptic's reading

How the review went

4 readings by deepseek-v4.1-flash · 13 Nov 2026

  1. Reader holds up, with conditions confidence high
  2. Skeptic partly agrees 4 min 21 s · 18,197 tokens read, 12,778 written
  3. Editor wrote the verdict above 33 s · 19,302 tokens read, 1,820 written
  4. Referee minor revision: 4 major objections, 6 minor 1 min 59 s · 18,566 tokens read, 6,438 written

Not run on this paper: the simulated authors' defence.

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The review in full

The review in brief, machine-written

This is the QCDSF-UKQCD update of their own Lambda determination: more statistics at beta = 5.20, 5.25, 5.29, a new beta = 5.40 ensemble reaching a ~ 0.07 fm, and a systematic comparison of four schemes for converting the boosted lattice coupling to MS-bar. The nf=2 result, r0*Lambda = 0.617(40)(21) (eq. 44), is a real number and the paper earns it. Everything is tabulated (Tables 1-4), the chiral fits are shown (Figs. 4, 5), and the continuum extrapolations against (a/r0)2 are linear over a wide range of lattice spacings. The quenched update to beta = 6.92 is a genuine improvement over ref. [2]. The method comparison is the honest part: they run I, II, IIP, III, IIIP and show the spread rather than quietly picking one, and they say outright that the trajectory dependence should vanish at higher order but does not yet (section 4.2). That is the right way to write this up. On the stress-test note: it is half right. The quoted 0.021 systematic is exactly the III-IIIP spread, and it does not include the II-III trajectory difference of 0.029. But II and III agree within errors (about 0.5 sigma), so folding the trajectory spread in widens the total to roughly 0.055 without moving the central value. That is an error-budget quibble, not a problem with the physics, and the authors already flag the underlying issue. The softer spot is the one they admit and then leave: the nf=3 matching systematic is not estimated (section 5.3), and the headline alpha_s(mZ) = 0.112(1)(2) does not visibly include it. Since that is the number most readers will cite, it deserves more than a sentence. The 7% uncertainty on r0 = 0.467 fm dominates the MeV errors, but that is a community-wide problem, not theirs. The two-sigma tension with the collider average is real, the paper flags it, and the Wilson-below-staggered pattern is worth tracking. Who should read it: anyone who needs a citable Lambda_nf=2 or a lattice cross-check on alpha_s(mZ), and anyone working on flavour-threshold matching, where the unestimated systematic is the open question. I would cite the nf=2 Lambda value; I would not cite alpha_s(mZ) without checking the matching error myself. Worth a serious referee.

The strongest objection

The central claim is the nf=2 continuum value r0*Lambda_MS = 0.617(40)(21) (eq. 44). The systematic error 0.021 is estimated 'by the spread in the results between method III and IIIP' (section 4.2). But methods II and III differ by the trajectory used to run c_sw from g=0 to g=g_box: method II holds c_sw fixed, method III runs it along the improvement path. The continuum values are II: 0.6085(352), III: 0.6376(412), IIP: 0.5819(329), IIIP: 0.6170(395). The II-III spread (0.029) is comparable to the quoted systematic (0.021) and is not included in it. The paper justifies choosing IIIP on theoretical grounds (method III gives constant beta-function coefficients, section 3.3) and on the flatness of its continuum extrapolation, but the choice shifts the central value by ~1 sigma relative to IIP. If the trajectory spread were folded into the systematic, the total error would be ~0.055 rather than 0.045, and the significance of the two-sigma tension with the collider average would weaken. Counter-argument: the continuum values of II and III agree within statistical errors (difference 0.029, combined error 0.054, ~0.5 sigma), so the trajectory spread is not statistically significant at the continuum level. The concern is about the error budget, not the central value.

The paper's central claim is the nf=2 continuum value r0*Lambda_MS = 0.617(40)(21) (eq. 44), which feeds into Lambda_MS = 261(17)(26) MeV and alpha_s(m_Z) = 0.112(1)(2). The reader's weakest_assumption correctly identifies the perturbative conversion (eq. 27) as the least secure step: t1 and t2 are known but t3 is not, and the result depends on the trajectory used to run c_sw. The paper acknowledges this ('Ideally, the result should not depend on the choice of trajectory... However, at the order to which we have the series this is not yet so,' section 4.2) and justifies choosing IIIP on theoretical grounds (method III gives constant beta-function coefficients, section 3.3) and on the flatness of its continuum extrapolation. The numerical spread between IIP (0.582) and IIIP (0.617) is 0.035, which is smaller than the quoted statistical error (0.040) but larger than the quoted systematic (0.021). The systematic is estimated only from the III-IIIP (Pade) spread, not the II-III (trajectory) spread. This is a real but modest gap: folding in the trajectory spread would raise the total error from ~0.045 to ~0.055, weakening but not overturning the two-sigma tension with the collider average. The nf=3 matching (section 5) is a separate acknowledged limitation that affects the alpha_s claim more than the nf=2 Lambda claim. Overall, the paper's methodology is sound and its limitations are largely acknowledged; the residual concern is about the completeness of the error budget, not the central value. I therefore agree partially with the reader: the right step is identified, but the numerical significance is overstated, since the continuum values of II and III agree within statistical errors (0.6085(352) vs 0.6376(412), ~0.5 sigma). The verdict should remain CONDITIONAL.

The referee's report

If the result holds, this is a first-principles determination of QCD's only dimensionful parameter with two dynamical flavours, and an independent lattice cross-check of the collider α_s average. The paper has real methodological strengths: the raw simulation parameters (β, κ, c_sw, volumes, r0/a, plaquette) are fully tabulated in Tables 1–3, the chiral extrapolations are displayed (Figs. 4 and 5), and five different conversion strategies are carried through to the continuum limit and compared (Table 4, Fig. 6), which is more than most lattice papers of this era provide. The one-loop matching relation Eq. (48), Λ_3/Λ_2 = (Λ_2/Λ_0)^{11/18}, follows correctly from Eq. (47) and is a useful check on the flavour extrapolation. The paper is also candid about the two-sigma tension with the phenomenological α_s(m_Z) and about the modelling that underlies the comparison values. The main weaknesses are in the error budget rather than in the central derivation.

Pith's reading of the paper (a machine's summary, not the authors')

The paper's target is Λ, the single dimensionful parameter left in QCD once quark masses are fixed, the number that sets how fast the strong coupling runs. The authors compute it directly from lattice simulations rather than from experiment, working in the quenched approximation and with two flavours of light dynamical quarks, using O(a) improved Wilson fermions and, for the first time in this collaboration's programme, a genuine continuum extrapolation supported by data down to lattice spacing a ≈ 0.07 fm. The chain is: measure the force scale r₀ and the average plaquette, form a 'boosted' lattice coupling from them, convert that coupling to the MS-bar coupling with known perturbative coefficients, and then run to physical units. They obtain r₀Λ = 0.614(2)(5) for zero flavours and 0.617(40)(21) for two flavours, which with r₀ = 0.467 fm becomes Λ = 259(1)(20) MeV and 261(17)(26) MeV. Matching the static force across flavour thresholds to reach five flavours gives α_s(m_Z) = 0.112(1)(2), about two standard deviations below the phenomenological average of 0.1182(27) — a discrepancy the authors report plainly rather than explain away.

What carries the argument. The load-bearing device is the 'boosted' coupling g²_✷ ≡ g²₀/u⁴₀, where u⁴₀ = ⟨Tr U_✷⟩/3 is the average plaquette. Lattice perturbation theory converges poorly in the bare coupling, so the authors reorganise the series in g²_✷ and in the tadpole-improved clover coefficient c^✷_sw = c_sw u³₀, which shrinks the n_f-dependent part of the two-loop coefficient t^✷₂ by roughly two orders of magnitude. The conversion formula 1/g²_MS(μ) = 1/g²_✷(a) + 2b₀ ln(aμ) − t^✷₁ + (2b₁ ln(aμ) − t^✷₂) g²_✷(a) + … carries the scheme change, with t^✷₁ fixing the ratio Λ_MS/Λ_✷ and t^✷₂ fixing the three-loop lattice β-function coefficient. Two refinements matter: choosing the scale μ* = a⁻¹ exp(t^✷₁/2b₀) to kill the O(1) term, and Padé-improving the β function as β^✷_[1/1], which estimates the unknown four-loop coefficient b^✷₃ ≈ (b^✷₂)²/b₁. Separately, the extrapolation from two to three flavours is done by holding the static force fixed at r₀ and matching Λ ratios at one, two and three loops.

Core claim. The central claim is that a first-principles lattice determination of Λ in full QCD with two dynamical quark flavours is now under control: statistical errors on r₀Λ are a few percent, and the continuum extrapolation in (a/r₀)² is linear over a wide range of lattice spacings, a⁻¹ ≈ 2–6.5 GeV in the quenched case and 2–3 GeV unquenched. The authors emphasise that the conversion from lattice to MS-bar coupling must be done after extrapolating to the chiral limit, and they compare several conversion strategies — fixed versus running tadpole-improved clover coefficient, and with or without Padé improvement of the β function — finding that the two main strategies agree in the continuum limit even though their slopes in a² differ substantially. Their preferred value, from the running-cₛw trajectory with Padé improvement, is r₀Λ^MS = 0.617(40)(21) at n_f = 2, and the resulting Λ values for n_f = 0, 2, 3 are nearly flavour-independent, 259, 261 and 260 MeV. The headline phenomenological output is α_s^MS(m_Z) = 0.112(1)(2).

The premise it rests on. The conversion from the boosted lattice coupling to the MS-bar coupling is truncated at two loops, and the third coefficient is unknown, so the answer depends on which path is taken from zero coupling to the simulated coupling — a dependence the paper itself shows to be about 6% in the continuum limit, larger than the quoted statistical error.

If the paper is right (Pith's inferences, not the authors')

Λ in the MS-bar scheme is fixed to roughly 260 MeV for zero, two and three light flavours, so the strong coupling at any scale can be predicted once the flavour number is specified.

Running the three-flavour result through the charm and bottom thresholds with four-loop running and three-loop matching gives α_s^MS(m_Z) = 0.112(1)(2), an independent lattice cross-check of the collider average.

Because the quenched and unquenched values of r₀Λ agree within errors, the dominant systematic is not the number of sea flavours but the scale r₀ and the continuum extrapolation.

The result supplies a benchmark that any future lattice calculation with finer spacings or non-perturbative coupling definitions must reproduce in the continuum limit.

The two-standard-deviation tension with the phenomenological α_s(m_Z) = 0.1182(27) is worth watching: if the lattice value survives improved systematics, either the perturbative conversion or some part of the experimental extraction carries an unaccounted shift.

The paper's observation that the fixed-cₛw and running-cₛw trajectories give continuum values differing by about 6% (0.582 versus 0.617) suggests that a non-perturbative determination of the lattice-to-MS-bar conversion, rather than a higher-order perturbative one, is the cleanest way to settle the remaining spread.

Since Λ comes out nearly flavour-independent here, while threshold-crossing in the MS-bar scheme produces large jumps when quark masses are far above Λ, a useful cross-check would be to repeat the matching at a smaller distance where r²f(r) is smaller and the perturbative treatment is better justified.

The same machinery applied to finer lattices would let the a² extrapolation be tested against an a⁴ term, which is currently indistinguishable from a straight line over the available range.

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The paper

Abstract

We present a determination of the QCD parameter Lambda in the quenched approximation (n_f=0) and for two flavours (n_f=2) of light dynamical quarks. The calculations are performed on the lattice using O(a) improved Wilson fermions and include taking the continuum limit. We find Lambda_n_f=0 = 259(1)(20) MeV and Lambda_n_f=2 = 261(17)(26) MeV}, using r_0 = 0.467 fm to set the scale. Extrapolating our results to five flavours, we obtain for the running coupling constant at the mass of the Z boson alpha_s(m_Z) = 0.112(1)(2). All numbers refer to the MSbar scheme.

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. No sentence in this record was written by a model.

abstract compiled · 2026-09-23

The authors' main claim, as the classifier picked it out 0.78

“We find Lambda_{n_f=0} = 259(1)(20) MeV and Lambda_{n_f=2} = 261(17)(26) MeV}, using r_0 = 0.467 fm to set the scale.”

How the main claim is stated: asserted 1.00

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.51 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.07 · mentions code or data availability 0.02

The abstract, sentence by sentence

This table labels the abstract only (arXiv listing abstract v2). The number is the probability the classifier gave the label.

SentenceRoleThe authors' own workStates the main conclusionStrength
We present a determination of the QCD parameter Lambda in the quenched approximation (n_f=0) and for two flavours (n_f=2) of light dynamical quarks. contribution 1.00 0.94 0.09 asserted 0.99
The calculations are performed on the lattice using O(a) improved Wilson fermions and include taking the continuum limit. method 1.00 0.93 0.03 asserted 1.00
We find Lambda_{n_f=0} = 259(1)(20) MeV and Lambda_{n_f=2} = 261(17)(26) MeV}, using r_0 = 0.467 fm to set the scale. result 1.00 0.97 0.65 asserted 1.00
Extrapolating our results to five flavours, we obtain for the running coupling constant at the mass of the Z boson alpha_s(m_Z) = 0.112(1)(2). result 0.99 0.94 0.29 asserted 1.00
All numbers refer to the MSbar scheme. method 0.43 0.35 0.03 asserted 0.99

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00

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Sources read

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Figures (9)
Figure 1 of the paper
Figure 1. α MS s (µ) versus µ/Λ MS for nf = 0 (left picture) and nf = 2 (right picture), using successively more and more coefficients of the β function. and p S A = − b1 4b 2 0 − b 2 1 − 2b0b S 2 4b 2 0 p b 2 1 − 4b0b S 2 , p S B = − b1 4b 2 0 + b 2 1 − 2b0b S 2 4b 2 0 p b 2 1 − 4b0b S 2 . (8) Results are usually given in the MS scheme, with the scale M being replaced by µ, and thus Λ MS µ = F MS(gMS(µ)). (9) In this scheme the next two β function coefficients are known [9, 10, 11]: b MS 2 = 1 (4π) 6  2857 2 − 5033 18 nf + 325 54 n 2 f  , b MS 3 = 1 (4π) 8  149753 6 + 3564 ζ3 −  1078361…
Figure 2 of the paper
Figure 2. F MS(gMS) for g 2 MS = 2 versus β function coefficient number n. The nf = 0 values are filled circles, while the nf = 2 values are filled squares. The [1/1] Pad´e approximations are given as open symbols. To evaluate F LAT , we need to know the b LAT i s. They can be found by expanding gMS as a power series in g0 as 1 g 2 MS(µ) = 1 g 2 0 (a) + 2b0 ln aµ − t LAT 1 + (2b1 ln aµ − t LAT 2 )g 2 0 (a) + [−2b0b1 ln2 aµ + 2(b MS 2 + b1t LAT 1 ) ln aµ − t LAT 3 ] g 4 0 (a) + . . . . (15) To have consistency between eqs. (9) and (14) we need t LAT 1 = 2b
Figure 3 of the paper
Figure 3. The quenched r0Λ MS points versus (a/r0) 2 , together with a linear ex￾trapolation to the continuum limit for method IIP. The filled circles are used for the extrapolation. The star represents the extrapolated value. 11
Figure 4 of the paper
Figure 4. The plaquette P (filled symbols) plotted against the bare quark mass amq for β = 5.20 (lower curve) until β = 5.40 (upper curve). The fits use eq. (41), giving the extrapolated values in the chiral limit (open symbols). 0 0.02 0.04 0.06 0.08 amq 4 5 6 7 8 r 0/a
Figure 5 of the paper
Figure 5. The force parameter r0/a plotted against amq. The same notation as in
Figure 6 of the paper
Figure 6. The unquenched r0Λ MS points (filled circles) versus (a/r0) 2 , together with a linear extrapolation to the continuum limit for methods IIP and IIIP. Stars represent the extrapolated values. not depend on the choice of trajectory. The way this should work, as mentioned before, is that although the coefficients t ✷ i will be different depending on the path one might choose, the sum 1 g 2 ✷(a) − t ✷ 1 − t ✷ 2 g 2 ✷(a) + . . . (43) should not. However, at the order to which we have the series this is not yet so. The difference between methods II and III is that we have replaced c ✷ sw…
Figure 7 of the paper
Figure 7. The ratio ΛMS 3 /Λ MS 2 against the ratio ΛMS 2 /Λ MS 0 from 1-loop (dotted line), 2-loop (dashed line) and 3-loop (solid line) matching. then gqq(r) is independent of nf too. We can compare the qq¯ scheme Λs by using rΛ qq 0 = F qq (gqq(r), nf = 0), rΛ qq 2 = F qq (gqq(r), nf = 2), rΛ qq 3 = F qq (gqq(r), nf = 3). (52) We can take ratios of these equations to cancel r and find equations for Λ ratios. These qq¯ scheme Λ ratios can then be converted into MS by using t qq¯ 1 from eq. (50). This gives us a way of making a parametric plot of Λ ratios by varying gqq and calculating all…
Figure 8 of the paper
Figure 8. Values of ΛMS versus number of quark flavours nf . The filled circles are our nf = 0, 2 results, and the open circle is our extrapolated value. The inner error bars give the statistical errors, while the outer error bars give the total errors. The square is from a 3-loop analysis of the non-singlet structure functions [32]. The triangles are taken from [33]. The open triangles are evaluated using the 4-loop expansion of αs and 3-loop matching at the quark thresholds. The entries at nf = 3 and 4 have been displaced horizontally. the charm and bottom quark masses within reasonable…
Figure 9 of the paper
Figure 9. Comparison of α MS s (mZ) from this work (solid circle) with other lat￾tice results [35, 36, 37, 2, 38, 4, 39, 21] (from top to bottom). The circles are from Wilson fermions and the squares from staggered fermions. The dashed line indicates the mean experimental value [33]. 7 Conclusions Due to substantial improvements of the performance of our hybrid Monte Carlo algorithm [41], we were able to extend our dynamical simulations to smaller quark masses and to larger values of β. Our smallest lattice spacing now is a ≈ 0.07 fm. This enabled us to perform a chiral and continuum extrapol

Figures are the authors', reproduced from the paper.

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  20. The infrared behavior of lattice QCD Green's functions hep-lat · Sep 2006
  21. Non--Singlet QCD Analysis of Deep Inelastic World Data at \boldmath $O(\alpha_s^3)$ hep-ph · Jul 2006
  22. Moments of pseudoscalar meson distribution amplitudes from the lattice hep-lat · Jun 2006
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How the literature uses it

32 arXiv papers cite this in 41 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 · 11 sentences without a confident label

Uses its data or values 13 papers
  • Up to $𝒪(ϵ)$ the renormalization matrix reads in the mRI scheme: [equation] Here we have used the renormalized strong coupling $α_s(μ)=g_R(μ)^2/4π$ and adopted the following conventions for the dimensional regularization: [equation] In the numerical evaluation of $α_s$ we use $Λ^MS= 261 MeV$ [ref] .

    Non-perturbative renormalization of three-quark operators · Scheme Matching to $MS$ · 0.98

  • Rescaling lattice results in terms of $m_0 = 3.405(21) √(σ)$ [ref] , we find Λ_MS= m_0/6.43 [ref] , T_c = m_0/5.30 [ref] , m_1 = 1.78 m_0 [ref] ,

    Dark Matter from Dark Glueball Dominance · Glueball Data from the Lattice · 1.00

  • The MC data points have been taken from [ref] .

    Wilson loops in very high order lattice perturbation theory · Non-perturbative gluon condensate · 0.99

Compares with it 7 papers
Cites it as background 5 papers

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