A Precise Determination of the Running Coupling in the SU(3) Yang-Mills Theory

Martin Luescher; Peter Weisz; Rainer Sommer; Ulli Wolff

REVIEW

A Precise Determination of the Running Coupling in the SU(3) Yang-Mills Theory

Martin Luescher, Rainer Sommer, Peter Weisz, Ulli Wolff

High Energy Physics - Lattice · Submitted 13 Sep 1993

Pith's verdict: The SU(3) step-scaling paper: the method works and the continuum extrapolation is clean, but the headline alpha_MS precision is capped by a matching coefficient nobody has computed yet, not by the data.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 11, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 7 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-lat/9309005 (version 1)

Pith ID3GS5UPWL, Pith's permanent record of this paper and its review

CitationMartin Luescher, Peter Weisz, Rainer Sommer, Ulli Wolff. "A Precise Determination of the Running Coupling in the SU(3) Yang-Mills Theory." arXiv:hep-lat/9309005 (1993). https://pith.science/paper/3GS5UPWL

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

This is the SU(3) companion to the group's SU(2) step-scaling work, and it is the paper that made recursive finite-size scaling the standard tool for non-perturbative renormalization. What is actually new: the SU(3) running, a one-loop improved action (eqs. 2.4-2.5) that flattens the a/L dependence to the point where the extrapolation slopes come out consistent with zero, and the switch to r0 as the reference scale. The step scaling function is measured at five couplings, continuum extrapolated linearly, and compared against two-loop RG evolution. That comparison is a real test rather than a fit, because sigma is computed from simulation alone. That part I trust, and the improvement in cutoff behaviour over the Wilson-action SU(2) run is visible in their own figure 2. The paper is also honest about what it does not show: it explicitly refuses to claim perturbation theory works for all quantities up to u = 3.48, only for this coupling. Good instinct, and correct. Where I would push back is on emphasis, and here the stress-test is right and the reader is slightly off. The L_max/r0 = 0.674(50) conversion (eq. 5.7) is the bottleneck for the energy scale, but a 7% shift in q moves alpha only logarithmically, well under the statistical error. The load-bearing uncertainty for the quoted alpha_MS is the unknown two-loop coefficient k2 in the matching (3.17). The (10) in eq. 5.8 is dimensional analysis, and the authors say so plainly: it cannot be reliably assessed until the two-loop computation is done. So the few-percent claim of abstract:s2 is honest but soft in a way only a calculation will fix. That is acknowledged, not hidden; a reader should just know the precision is capped by an uncomputed quantity. Two smaller things. The v-tilde universality check in sect. 5.1 is weak and they admit it, errors too large to be significant. And the disagreement with the alpha_qqbar values of refs. [7-11] at beta = 6.5 is waved off as possible non-perturbative corrections to the matching; plausible, but unresolved. The r0 = 0.5 fm conversion and the resulting sqrt(K) = 481(32) MeV is on the high side; they flag it and explain why. Who should read it: anyone doing non-perturbative renormalization, and anyone who wants the template for an honest continuum extrapolation. Worth a serious referee; the errors are budgeted and the caveats are stated.

Referee report

Significance

This is a benchmark calculation: it demonstrates that a non-perturbative, first-principles lattice computation can bridge the low-energy dynamical scales of a gauge theory and the perturbative scaling regime, and it delivers α_MS in a pure-gauge theory with a controlled error budget. The methodological strengths are real and should be named. The step scaling function is computed directly from simulation and is therefore independent of perturbation theory, so the comparison in Table 4 between the data and the two-loop RG evolution is a genuine external test rather than a fit; the agreement over a factor of ~24 in scale is the central physical result. The use of a one-loop improved action with the boundary coefficient c_t of Eq. (2.5) is shown to reduce cutoff effects to the point where the fitted slopes in Eq. (5.2) are consistent with zero, and the paper is careful to warn (sect. 3.6) that this does not license a general claim of small lattice artifacts, illustrating the point with the quantity v. The independent cross-check in sect. 5.5 against the improved one-loop formula of El-Khadra et al. agrees at the few-percent level, and the SU(2)/SU(3) relation of Eq. (6.1) is a falsifi

Major objections (3)

[Eq. (5.8), Eq. (3.17), abstract] The second error in α_MS(q) = 0.1108(23)(10) is not a computed uncertainty but an estimate of the unknown two-loop coefficient k2 in the matching relation (3.17), taken as α^3 by dimensional analysis and by analogy with the fitted b2_eff of Eq. (5.5). The paper states plainly that this error 'cannot be reliably assessed until the necessary 2-loop computations have been performed', which is honest, but the abstract's unqualified 'estimated total error of a few percent' (abstract:s2) then overstates what is controlled: if k2 is an order of magnitude larger than assumed, the MS-bar value moves by more than the quoted total error. Since this is the paper's headline claim, the abstract and the concluding remarks should carry the qualification explicitly, and it would strengthen the paper to state the range of k2 for which the quoted precision survives (e.g. |k2| ≲ 10), so that a reader can judge the robustness of the 'few percent' statement without reconstructing it from sect. 5.4.
[Eq. (5.7), Table 6, sect. 5.3] The conversion factor L_max/r0 = 0.674(50) is the weakest link in the chain to physical units: it rests on four data points, one of which (β = 6.5) comes from ref. [8] and the private communication [28], and it is obtained from a linear-in-a extrapolation whose quality cannot be judged from the text. The paper says only that 'the data are certainly consistent with this procedure, but the errors are quite large'. Please quote the χ² per degree of freedom of the fit in Fig. 4 and show the effect on Eq. (5.8) of the 7.4% uncertainty separately from the statistical error, since the two enter differently: the scale error shifts the momentum label q logarithmically in α but shifts the quoted q = 37 r0^{-1} linearly. A short breakdown of how the (23) in Eq. (5.8) splits between the step scaling fit and the conversion factor would make the error budget auditable.
[Table 4, sect. 5.2] The continuum step scaling function is quoted at five couplings that are only approximately matched ('up to small mismatches the couplings are arranged so that one can step down the energy scale successively'), and the mismatches are handled by the integration procedure of ref. [3] without further description here. Because the comparison with the two-loop RG in the last column of Table 4 is the paper's central evidence for perturbative evolution, the size of these mismatches and their treatment should be stated in this paper rather than delegated entirely to a cited companion; otherwise the reader cannot tell whether the residual differences at the largest couplings are physics or bookkeeping.

Minor objections (7)

[sect. 5.3] The text refers to '(see fig. 5)' for the extrapolation of L_max/r0 to the continuum limit, but that figure is labelled Fig. 4; Fig. 5 is the α(q) plot. Please correct the cross-reference.
[Table 6, ref. [28]] The r0/a value at β = 6.5 is supported in part by a private communication [28]. Since this point enters the four-point extrapolation of Eq. (5.7), it would be preferable to document the number or to show that dropping it leaves Eq. (5.7) unchanged within errors.
[Eq. (3.20)] The parenthetical '(1)' on v = 0.0694603(1) is unexplained; presumably it denotes truncation of the perturbative series rather than a statistical error. A footnote would avoid confusion with the Monte Carlo errors quoted elsewhere.
[Fig. 3, Eq. (5.6)] The fit shown in Fig. 3 excludes the point at the largest coupling. Please state the χ² per degree of freedom for that fit and quantify how much the fitted effective two-loop coefficient moves if the excluded point is included, so that the reader can see the fit is not being protected by the exclusion.
[Eq. (6.1)] The relation 2α_MS^{SU(2)} = 3α_MS^{SU(3)} is stated to hold 'within errors' at equal q in units of r0. Please specify the momentum at which the comparison is made and whether the SU(2) errors from ref. [3] are included in the 'within errors' statement.
[Abstract] The abstract names the string tension K as the low-energy contact scale, whereas the paper's actual reference scale throughout is r0, with K used only as a cross-check (Eq. (1.2)). Aligning the abstract with the body would avoid implying that the final result is expressed in units of K.
[Table 1, Eq. (3.28)] The sign convention of the relative deviation in Eq. (3.28) and the corresponding sign of the tabulated γ1 should be stated in the caption, since the argument in sect. 3.6 depends on the sign of the one-loop coefficient.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

By recursively doubling and tripling the box size on the lattice, the authors follow the SU(3) Yang–Mills coupling across a factor of 24 in energy and find it settles onto the perturbative prediction, fixing $α_{MS}(q)=0.1108(23)(10)$ at $q

The reading

The paper asks a question that sounds circular but is not: if you fix the parameters of a non-Abelian gauge theory at low energy, can you actually compute what the coupling becomes at high energy, where perturbation theory takes over? Doing so requires bridging a large span of scales without ever leaving the regime where the lattice simulation is controlled. The authors do it by shrinking and growing a finite box: the running coupling is defined at the box size $L$, and the change in the coupling when $L$ is multiplied by 2 (or, at the last step, by 3/2) is measured at fixed bare coupling. Chaining these steps together walks the coupling from $g^2 = 3.48$ down to $1.243$, a factor of about 24 in scale, and the resulting curve agrees with the two-loop renormalization-group prediction to within statistical errors. The payoff is a value of the coupling in the $MS$-bar scheme of dimensional regularization, $α_{MS}(q) = 0.1108(23)(10)$ at $q = 37 r_0^{-1}$ (roughly 14 GeV if $r_0 = 0.5$ fm), with a stated total error of a few percent. The authors are careful that the second error, of order $α^3$, only estimates the size of the unknown two-loop matching correction.

What carries the argument

The step scaling function $Σ(s,u)$ is the object that carries the argument: it is the integrated form of the Callan–Symanzik $β$-function, telling you what happens to the coupling when the box is scaled by $s$. Because it is defined at fixed bare coupling on pairs of lattices of size $L$ and $sL$, it can be computed by simulation and then extrapolated to the continuum with an ansatz $Σ(s,u,a/L) = Σ(s,u)(1 + δ(s,u)a/L)$, without ever needing to know the lattice spacing in physical units. The coupling itself comes from differentiating the Schrödinger functional $Z$ with respect to a one-parameter family of constant Abelian boundary fields, $∂_η ln Z|_{η=0} = k/ḡ^2$, with the normalization $k$ chosen so that $ḡ^2 = g_0^2$ at lowest order; a one-loop improved action with boundary coefficient $c_t = 1 - 0.08900(5)g_0^2$ removes cutoff effects through order $a$. The second ingredient is the scale conversion: the ratio $L_{max}/r_0$, where $r_0$ is the distance at which $r^2F(r) = 1.65$, ties the dimensionless evolution to the low-energy dynamics.

Extended reading notes

Core claim

The central claim is that the running coupling of SU(3) Yang–Mills theory, defined non-perturbatively through the Schrödinger functional on a box of size $L$, evolves according to perturbation theory once the coupling is small enough, and that this evolution can be computed with controlled errors. Concretely, the authors measure the step scaling function $Σ(s,u)$ — the coupling at box size $sL$ given the coupling $u$ at size $L$ — for $s=2$ at four values of $u$ and for $s=3/2$ at the largest coupling, extrapolate each to the continuum limit $a/L → 0$, and then integrate the resulting curve. The lattice data show no statistically significant cutoff dependence, and the integrated evolution matches the two-loop $β$-function essentially point by point; only at the largest couplings is a small deviation visible, which a correlated fit absorbs into an effective three-loop coefficient $b^{eff}_2 = 1.5(8)/(4π)^3$, about twice the known three-loop value in minimal subtraction. Converting to physical units through $L_{max}/r_0 = 0.674(50)$ and the one-loop matching relation $α_{MS} = α + 1.25563(4)α^2 + …$ gives $α_{MS}(q) = 0.1108(23)(10)$ at $q = 37 r_0^{-1}$.

Load-bearing premise

The chain from a dimensionless lattice coupling to a physical number hangs on the conversion factor $L_{max}/r_0 = 0.674(50)$, obtained by linearly extrapolating the static quark force from only four data points of other groups' large-volume simulations, and the authors themselves call the errors large at this step.

Editorial extensions

If this is right

The perturbative scaling regime and the low-energy domain of SU(3) Yang–Mills are smoothly connected, with no intermediate regime where the coupling behaves in a way perturbation theory cannot describe.
A numerical value of $α_{MS}$ at a specified momentum is available from first principles alone, with the dominant uncertainty traceable to the scale-setting ratio $L_{max}/r_0$ rather than to the coupling evolution.
Setting $r_0 = 0.5$ fm gives $L_{max} = 0.337(25)$ fm and $\sqrt{K} = 481(32)$ MeV, a string tension somewhat larger than values extracted from phenomenological heavy-quark potentials fitted to charmonium.
Within errors, $2α_{MS}|_{SU(2)} = 3α_{MS}|_{SU(3)}$ when momenta are measured in units of $r_0$, suggesting that $Nα_{MS}$ depends only weakly on the number of colours.
The comparison with the improved one-loop bare-coupling formula of Parisi–Lepage–Mackenzie agrees at the few-percent level over the range of bare couplings examined, so the two routes to $α_{MS}$ are consistent where they overlap.

Reading between the lines

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

The error budget is lopsided: the evolution itself is under control, while the conversion factor $L_{max}/r_0 = 0.674(50)$ carries a 7 percent uncertainty built from only four data points, one of them an unpublished private communication. A dedicated high-statistics static-force calculation at an intermediate coupling such as $β = 6.8$ would buy more precision than any amount of additional step-sc
Because the same technique with the same scale-setting reproduces the SU(2) result, the natural next check is SU(4) or the large-$N$ limit: if $Nα_{MS}$ is genuinely flat in $N$ at fixed $q r_0$, that is a sharp, cheap test of large-$N$ scaling that does not require new analytic input.
The fitted effective three-loop coefficient $b^{eff}_2 \approx 1.5/(4π)^3$ is a concrete target for perturbation theory: a three-loop computation of the $β$-function for this particular finite-volume coupling would either reproduce that number or expose the fit as absorbing genuine non-perturbative curvature. Either outcome sharpens the matching formula's error estimate, which currently rests on a

Original abstract

A non-perturbative finite-size scaling technique is used to study the evolution of the running coupling (in a certain adapted scheme) in the SU(3) Yang-Mills theory. At low energies contact is made with the fundamental dynamical scales, such as the string tension K, while at larger energies the coupling is shown to evolve according to perturbation theory. In that regime the coupling in the MS-bar scheme of dimensional regularization is obtained with an estimated total error of a few percent.

The compiled record

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. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,977 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.39 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.66 · compares with prior results 0.17 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

A non-perturbative finite-size scaling technique is used to study the evolution of the running coupling (in a certain adapted scheme) in the SU(3) Yang-Mills theory. At low energies contact is made with the fundamental dynamical scales, such as the string tension K, while at larger energies the coupling is shown to evolve according to perturbation theory. In that regime the coupling in the MS-bar scheme of dimensional regularization is obtained with an estimated total error of a few percent.

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	A non-perturbative finite-size scaling technique is used to study the evolution of the running coupling (in a certain adapted scheme) in the SU(3) Yang-Mills theory.	method 0.77	0.91	0.10	asserted 0.99
s1	At low energies contact is made with the fundamental dynamical scales, such as the string tension K, while at larger energies the coupling is shown to evolve according to perturbation theory.	result 0.94	0.87	0.31	asserted 1.00
s2	In that regime the coupling in the MS-bar scheme of dimensional regularization is obtained with an estimated total error of a few percent.	result 1.00	0.91	0.42	asserted 0.99

What kind of paper the classifier reports this is: simulation study 0.95 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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

Figures

**Figure 2.** Extrapolation of the lattice step scaling function to the continuum limit. The points at a=L = 0 represent the extrapolated values. perturbative improvement of the action. Further studies are however needed to completely clarify the issue. We now proceed to discuss the cuto dependence of v, or, more precisely, of the function (g2 ; a=L) [eq.(3.27)]. The data listed in the last two columns of table 2 show that v is hardly changing as a=L is made smaller at xed g2 . Note that in most cases we have results for a=L from 0:2 down to 0:05 (small variations of g2 can here be ignor

**Figure 3.** Plot of v = !(g 2 ). The dashed line is a t, using eq.(5.6) with an adjustable 2-loop coecient, to all data points except the one at the largest coupling. and the running coupling in our scheme may very well turn out to be an exceptional case. 5.3 Physical units We now need to relate Lmax to the dynamical low-energy scales of the theory in in nite volume. As discussed in sect. 1, we decided to take the distance r0 as the basic reference scale. Our rst goal is thus to compute the conversion factor Lmax=r0. The heavy quark force F (r) has previously been calculated through num

**Figure 5.** Comparison of numerically computed values of the running coupling (data points) with perturbation theory. The full line represents the t described in subsect. 5.2, while the dashed (dotted) curve is obtained by integrating eq.(3.21), starting at the right-most point and using the 2- loop (1-loop) {function. the coupling. 21

Showing 3 of the paper's 5 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-lat/9309005 by the authors.

References

Forward citations

Cited by 27 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A lattice construction of the Kormos-Runkel-Watts defect flows in minimal models
hep-th 2026-11 accept novelty 8.0 of 10

The dilute RSOS chain realizes the Kormos-Runkel-Watts defect flow from D_{(2,1)} to a conformal interface; at p=4 the endpoint is the invertible line eta, at p=5 it is a new interface C5 with g=1.503(2) and transmiss...
Supersymmetry Inspired QCD Beta Function
hep-th 2007-11 conditional novelty 7.0 of 10

A conjectured all-orders beta function for nonsupersymmetric gauge theories, modeled on the NSVZ supersymmetric result, yields a conformal window bound of 11/8 times the Casimir over the trace normalization.
Non-perturbative quark mass renormalization in quenched lattice QCD
hep-lat 1998-10 accept novelty 7.0 of 10

A non-perturbative lattice calculation fixes the renormalization factor between bare and renormalization-group-invariant quark masses in quenched QCD, and yields the quenched Lambda parameter to 8 percent.
Isospin-breaking effects in inclusive hadronic $\tau$ data for the muon $(g-2)$ from first principles
hep-lat 2026-07 unverdicted novelty 6.0 of 10

A lattice QCD+QED strategy is outlined for calculating isospin-breaking effects in inclusive tau decays to support high-precision HVP contributions to muon g-2.
Asymptotic behavior of cutoff effects in Yang-Mills theory and in Wilson's lattice QCD
hep-lat 2019-12 conditional novelty 6.0 of 10

The leading logarithmic corrections to lattice cutoff effects in Yang-Mills theory have small positive anomalous dimensions (gamma-hat-1 = 7/11, gamma-hat-2 = 63/55), so discretization errors vanish slightly faster th...

Show 22 more citing papers

The strong coupling from a nonperturbative determination of the $\Lambda$ parameter in three-flavor QCD
hep-lat 2017-06 accept novelty 6.0 of 10

Lattice QCD with nonperturbative step scaling from 0.2 to 70 GeV gives Lambda_MS-bar^(3) = 341(12) MeV and alpha_MS-bar^(5)(m_Z) = 0.11852(84).
Slow running of the Gradient Flow coupling from 200 MeV to 4 GeV in $N_{\rm f}=3$ QCD
hep-lat 2016-07 conditional novelty 6.0 of 10

Non-perturbative lattice step scaling of the gradient-flow coupling in three-flavor QCD gives L_had = 21.86(42) L_0, connecting 4 GeV to roughly 200 MeV with a running that stays near one-loop.
The gradient flow running coupling with twisted boundary conditions
hep-lat 2014-09 conditional novelty 6.0 of 10

A gradient-flow coupling defined on a twisted four-torus gives a precise, automatically O(a)-improved finite-volume renormalization scheme, demonstrated non-perturbatively in SU(2) Yang-Mills over a factor 2^26 in scale.
The gradient flow coupling in the Schr\"odinger Functional
hep-lat 2013-01 accept novelty 6.0 of 10

A gradient-flow definition of the running QCD coupling inside a Schrödinger Functional box is computed to leading order in perturbation theory and shown, on Nf=2 lattice ensembles, to have small cutoff effects and hig...
Lattice determination of the $K \to (\pi\pi)_{I=2}$ Decay Amplitude $A_2$
hep-lat 2012-06 accept novelty 6.0 of 10

A first-principles lattice QCD calculation fixes both the real and imaginary parts of the K to two-pion (I=2) decay amplitude, giving the first Standard Model prediction for the CP-violating imaginary part.
Infrared fixed point in SU(2) gauge theory with adjoint fermions
hep-lat 2011-02 conditional novelty 6.0 of 10

Lattice simulations of SU(2) gauge theory with two adjoint fermions locate an infrared fixed point at 1/g*^2 = 0.20(4)(3) with mass anomalous dimension gamma_m(g*) = 0.31(6), placing the theory inside the conformal window.
Conformal Dynamics for TeV Physics and Cosmology
hep-ph 2009-11 conditional novelty 6.0 of 10

A conjectured all-orders beta function is used to map the conformal window of SU, Sp and SO gauge theories, extended for the first time to chiral Bars-Yankielowicz and Georgi-Glashow models.
Zero of the discrete beta function in SU(3) lattice gauge theory with color sextet fermions
hep-lat 2008-03 conditional novelty 6.0 of 10

The discrete beta function of SU(3) gauge theory with two sextet Wilson fermions crosses zero near g^2 = 2.0, suggesting an infrared fixed point at unexpectedly weak coupling, though the result rests on a single latti...
Schroedinger functional formalism with domain-wall fermion
hep-lat 2006-04 conditional novelty 6.0 of 10

An orbifolding projection imposes Schroedinger functional Dirichlet boundary conditions on domain-wall fermions while keeping a mass gap and reproducing the continuum free propagator at tree level.
On lattice actions for static quarks
hep-lat 2005-06 accept novelty 6.0 of 10

Smeared-link static quark actions (APE, one-link-integral, HYP) reduce the exponential noise growth in static-light lattice correlators, with HYP2 best, and the effect is explained by a smaller static self-energy.
Determining the conformal window: SU(2) gauge theory with N_f = 4, 6 and 10 fermion flavours
hep-lat 2011-11 conditional novelty 5.5 of 10

Improved Wilson-clover lattice simulations show QCD-like running at four fermion flavours and a possible infrared fixed point at ten, while the six-flavour case remains undecided within statistical and cutoff errors.
A note on momentum subtraction schemes for quark bilinears and semileptonic operators
hep-lat 2026-06 unverdicted novelty 5.0 of 10

Extends the RI/SMOM scheme to semi-leptonic operators via Ward-identity-protected vector currents and demonstrates equivalence to the Gorbahn et al. 2023 projectors for Wilson coefficient calculations.
SU(N) gauge theories in 3+1 dimensions: glueball spectrum, string tensions and topology
hep-lat 2021-06 accept novelty 5.0 of 10

The paper delivers continuum and N = infinity glueball masses for all cubic representations of SU(N) Yang-Mills theory at N = 2 to 12, together with string tensions, Lambda_MSbar values and topological susceptibilities.
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.
Running coupling and mass anomalous dimension of SU(3) gauge theory with two flavors of symmetric-representation fermions
hep-lat 2010-06 accept novelty 5.0 of 10

SU(3) gauge theory with two sextet fermions shows slow but non-conformal running up to g^2 about 5, ruling out the previously reported infrared fixed point and bounding the mass anomalous dimension below 0.6.
Precise determination of the strong coupling constant in Nf=2+1 lattice QCD with the Schr\"odinger functional scheme
hep-lat 2009-06 conditional novelty 5.0 of 10

A lattice QCD simulation with two light and one strange quark yields alpha_s(M_Z) = 0.12047(81)(48)(+0/-173) and Lambda_MS^(5) = 239(10)(6)(+0/-22) MeV, using non-perturbative Schroedinger-functional running over five...
Progress in four-dimensional lattice supersymmetry
hep-lat 2009-03 accept novelty 5.0 of 10

Super-QCD discretized with Ginsparg-Wilson gauginos requires between eight and fourteen fine-tuned counterterms, all but one of them bosonic and therefore adjustable offline by reweighting.
Conformal Windows of SP(2N) and SO(N) Gauge Theories
hep-ph 2009-02 conditional novelty 5.0 of 10

Using a conjectured all-orders beta function, a ladder Schwinger-Dyson estimate, and a thermal counting bound, the paper maps the conformal windows of Sp(2N) and SO(N) gauge theories with matter in vector and two-inde...
Light Composite Higgs from Higher Representations versus Electroweak Precision Measurements -- Predictions for LHC
hep-ph 2005-05 conditional novelty 5.0 of 10

A two-color, two-flavor technicolor model in the symmetric representation survives electroweak precision data at 90% CL and yields a composite Higgs lighter than 150 GeV.
The Nf=0 heavy quark potential from short to intermediate distances
hep-lat 2001-08 accept novelty 5.0 of 10

Pure-gauge lattice QCD yields a continuum static quark potential and force over 0.05 to 0.8 fm, with lattice artifacts argued to be quadratic in the lattice spacing and r0/a extended to beta=6.92.
Dynamical Stabilization of the Fermi Scale: Phase Diagram of Strongly Coupled Theories for (Minimal) Walking Technicolor and Unparticles
hep-ph 2008-04 conditional novelty 4.0 of 10

A conjectured all-orders beta function for SU(N) gauge theories is used to map the conformal window across matter representations, and that map is then applied to walking technicolor, unparticle, and unification model...
Non-perturbative thermal QCD at very high temperatures: computational strategy and hadronic screening masses
hep-lat 2024-11 conditional novelty 3.0 of 10

Non-perturbative lattice QCD shows hadronic screening masses deviate from NLO perturbation theory from 1 to 160 GeV, with the baryonic NLO correction computed for the first time.

How later papers used it

69 later papers cite it

Evidence: substantial. 61 of the citing papers share no author with it, and 30 of those carry a confident label from Pith's citation classifier.

Pith read 14 of the 147 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

5 later papers by other authors are listed below: 3 used its method or tool, 1 used its data or values and 1 compared its own results with it.

Uses its method or tool 3 of 24 papers with this label

Defined its SF coupling constant as this paper did, taking the derivative in terms of the parameter η introduced in the background field.

Precise determination of the strong coupling constant in Nf=2+1 lattice QCD with the Schr\"odinger functional scheme (2009) · 0.85 · paraphrased: the licence does not allow quoting
Used this paper's finite volume step scaling function to quantify the running of the coupling.

Infrared fixed point of SU(2) gauge theory with six flavors (2017) · 1.00 · paraphrased: the licence does not allow quoting
Implemented the derivatives by putting the appropriate fixed values in place of the boundary links as in this paper.

Running coupling and mass anomalous dimension of SU(3) gauge theory with two flavors of symmetric-representation fermions (2010) · 0.93 · paraphrased: the licence does not allow quoting

Uses its data or values 1 of 4 papers with this label

Took its blue dots for SU(3) from this paper's lattice data.

Supersymmetry Inspired QCD Beta Function (2007) · 0.85 · paraphrased: the licence does not allow quoting

Compares with it 1 of 10 papers with this label

Used its own calculation and this paper's SU(3) calculation to compare with another work's extrapolation to N → ∞, finding good agreement.

The running of the coupling in SU(N) pure gauge theories (2008) · 0.82 · paraphrased: the licence does not allow quoting

How the literature uses it

68 arXiv papers cite this in 146 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 · 80 sentences without a confident label

Builds on it 2 papers

Thus the aim of the present paper is to go one step further than those of Refs. [ref] and evaluate the strong coupling constant in $N_f=2+1$ QCD.

Precise determination of the strong coupling constant in Nf=2+1 lattice QCD with the Schr\"odinger functional scheme · Introduction · 1.00
Our main concern in this paper is to extend the bottom up algorithm to the Schrödinger functional (SF) [ref] , where translation invariance for the time direction is broken, and there is an abelian color background gauge field.

Automatic generation of Feynman rules in the Schroedinger functional · Introduction · 0.83

Uses its method or tool 24 papers

[ref] , the evolution of the physical coupling as a function of the momentum scale $O(1/L)$ is then defined in an iterative way, in terms of the continuum-extrapolated step-scaling function $σ(s, g^2(L))=g^2(sL)$ that was introduced in ref.

Strong coupling from non-equilibrium Monte Carlo simulations · Numerical implementation · 0.92
The derivation of the RGI scale for the SF scheme proceeds in the following steps [ref] : (i) We start by calculating the step scaling function (SSF) $Σ(u,a/L)$ on the lattice at several box sizes and lattice spacings.

Precise determination of the strong coupling constant in Nf=2+1 lattice QCD with the Schr\"odinger functional scheme · Our strategy · 0.92
To investigate the running of RCs, we use the step scaling method [ref] .

Step scaling in coordinate space: running of the quark mass · X-space renormalization scheme and step scaling · 1.00

Uses its data or values 4 papers

Using the data for the SF from Refs. [ref] and the data for the string tension from Refs. [ref] , we can compute the ratio $Λ_/√(σ)$ for SU(2) and SU(3) gauge theory in analogy with the SU(4) case discussed in the previous section.

The running of the coupling in SU(N) pure gauge theories · Large-$N$ limit · 0.94
These values have been calculated by Lüsher et. al. in [ref] for $2$ and in [ref] for $3$ resulting in $c_t ^(1,0)(2)=-0.0543(5)$ and $c_t ^(1,0)(3)=-0.08900(5)$.

Schr\"odinger functional boundary conditions and improvement of the SU($N$) pure gauge action for $N>3$ · Boundary improvement · 0.95
The green dots (biggest errorbars) correspond to lattice data for $SU(2)$ taken from [ref] , the blue dots to $SU(3)$ [ref] and the red dots (smallest errorbars) to $SU(4)$ [ref] .

Supersymmetry Inspired QCD Beta Function · Pure Yang-Mills and Comparison with Lattice Data · 0.85

Compares with it 10 papers

We focused on $(2)$ and $(3)$ Yang-Mills theories, and showed that the results obtained in our non-equilibrium Monte Carlo simulations are fully compatible with those from standard (equilibrium) lattice simulations [ref] .

Strong coupling from non-equilibrium Monte Carlo simulations · Conclusions · 1.00
We are now in a position to test this using the results herein, together with those of SU(2) and SU(3) in [ref] .

Determination of the running coupling in pure SU(4) Yang-Mills theory · Scaling with $N$ · 0.86
Using our calculation and the calculations for SU(2) and SU(3) given respectively in Refs. [ref] and [ref] , we have performed a comparison with the extrapolation to $N →∞$ of [ref] , finding good agreement.

The running of the coupling in SU(N) pure gauge theories · Conclusions · 0.82

Cites it as background 13 papers

The renormalised coupling in the Schrödinger functional (SF) scheme has been defined in [ref] and its scale evolution has been studied in QCD with zero and two quark flavours [ref] .

The SF running coupling with four flavours of staggered quarks · Introduction · 0.87
The method has since been applied to the running of the coupling in QCD with $=0$ [ref] , $=2$ [ref] , $=3$ [ref] and $=4$ [ref] quark flavours (for a review cf. [ref] ), and in various candidate models of physics beyond the Standard Model (for reviews, cf. [ref] ).

Determination of $\alpha_s(m_Z)$ by the non-perturbative decoupling method · Introduction · 0.86
[ref] for some important examples.]

Past, present, and future of precision determinations of the QCD coupling from lattice QCD · The high-energy regime of $=3$ QCD · 0.94

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.