Gradient flow step-scaling function for SU(3) with twelve flavors
GGradient flow step-scaling function for SU(3) with twelve flavors
A. Hasenfratz, C. Rebbi, and O. Witzel ∗ Department of Physics, University of Colorado, Boulder, Colorado 80309, USA Department of Physics and Center for Computational Science, Boston University, Boston, Massachusetts 02215, USA (Dated: December 19, 2019)We calculate the step scaling function, the lattice analog of the renormalization group β -function,for an SU(3) gauge theory with twelve flavors. The gauge coupling of this system runs very slowly,which is reflected in a small step scaling function, making numerical simulations particularly chal-lenging. We present a detailed analysis including the study of systematic effects of our extensivedata set generated with twelve dynamical flavors using the Symanzik gauge action and three timesstout smeared M¨obius domain wall fermions. Using up to 32 volumes, we calculate renormalizedcouplings for different gradient flow schemes and determine the step-scaling β function for a scalechange s = 2 on up to five different lattice volume pairs. Our preferred analysis is fully O ( a )Symanzik improved and uses Zeuthen flow combined with the Symanzik operator. We find an in-frared fixed point within the range 5 . ≤ g c ≤ . c = 0 .
250 finite volume gradient flowscheme. We account for systematic effects by calculating the step-scaling function based on alter-native flows (Wilson or Symanzik) as well as operators (Wilson plaquette, clover) and also explorethe effects of the perturbative tree-level improvement.
I. INTRODUCTION
The renormalization group β -function characterizesthe nature of gauge-fermion systems with gauge group G and N f fermion flavors in representation R . Choosinge.g. fermions in the fundamental representation and theSU(3) gauge group, the system is chirally broken witha fast running coupling for a small number of flavors.A particularly well studied case is Quantum Chromody-namics (QCD). This is a chirally broken theory whichexhibits only the Gaussian fixed point (GFP) at barecoupling g = 0. When increasing the number of fla-vors, predictions based on perturbation theory suggestthat a second, infrared fixed point (IRFP) develops atsome definite g [1]. For an even larger number offlavors, the theory becomes IR free. Theories exhibitingan IRFP are conformal. Of special interest is the lowestnumber of flavors, N cf , for which a gauge-fermion sys-tem ( G , R ) is conformal because this denotes the onsetof the conformal window. Perturbative methods to de-termine the nature of a system with N f flavors may notbe reliable because the IRFP can occur at large g and may lie outside the trustworthy region of perturba-tion theory. Inside the conformal window, the IRFP isexpected to move to stronger couplings as the numberflavors approaches N cf . Thus a perturbative estimate of N cf is particularly troublesome and warrants us to usenonperturbative methods [5–8].In this work we focus on the SU(3) gauge systemwith twelve flavors in the fundamental representation and ∗ [email protected] Theories just below the conformal window are promising candi-dates to describe physics beyond the Standard Model e.g. com-posite Higgs scenarios as discussed in Refs. [2–4] and referenceswithin. present details of our nonperturbative, large-scale investi-gation using lattice field theory techniques. We evaluatethe gradient flow (GF) step-scaling function, the latticeanalogue of the renormalization group (RG) β -function inthe infinite volume continuum limit. Perturbation theoryup to the 4-loop level predicts that the system is confor-mal. The 5-loop β -function, however, does not predict afixed point [5]. Since the 5-loop correction is very large,Refs. [7] suggested to improve the convergence of the per-turbative series by using Pad´e approximation. Variousforms of the Pad´e series predict that the system is con-formal. The scheme-independent form [8] also finds anIRFP. There are several nonperturbative lattice studiesof this system based on staggered fermions [9–21]. Theresults of lattice studies are controversial as Ref. [18] findsa FP in the gradient flow c = 0 .
25 scheme at g ≈ . O ( a ) Symanzikimproved [29, 30] and discretization effects are further a r X i v : . [ h e p - l a t ] D ec suppressed by the perturbative tree-level normalizationintroduced in Ref. [31]. We demonstrate that this com-bination has indeed small discretization errors which re-sult in mild continuum limit extrapolations. Using GFrenormalization schemes c = 0 . c = 0 .
250 scheme we observe the IRFP in the range of5 . ≤ g c ≤ . β -function using different renor-malization schemes c for our preferred combination ofZeuthen flow with Symanzik operator with and withouttree-level normalization (short: nZS and ZS). Alternativedeterminations using different gradient flows, operatorswith and without tree-level normalization are presentedin Sec. IV B. Additional details are collected in the ap-pendices: Appendix A lists the tree-level normalizationfactors for Symanzik gauge action and our three gradientflows with the three different operators, in Appendix Bwe present the renormalized couplings for our preferred(n)ZS analyses and in Appendix C we show comparisonplots including additional continuum extrapolations. Fi-nally we summarize our results and conclude in Sec. V. II. THE CONTINUUM LIMIT OF THESTEP-SCALING FUNCTION
The finite volume step scaling function, or discrete β function of scale change s is defined by β c,s ( g c ; L ) = g c ( sL ; a ) − g c ( L ; a )log( s ) , (1)where g c ( L ; a ) is a renormalized coupling at the energyscale set by the volume µ = ( cL ) − which is related tothe gradient flow time t by µ = 1 / √ t . The parameter c denotes the renormalization scheme corresponding to arenormalized gradient flow coupling given by g c ( L ; a ) = 128 π N −
1) 1 C ( c, L/a ) (cid:10) t E ( t ) (cid:11) , (2)with N = 3 for SU(3), E ( t ) the energy density at gra-dient flow time t , and C ( c, L/a ) a perturbatively com-puted tree-level improvement term [31]. Without tree-level improvement C ( c, L/a ) is replaced by the term Numerical values for C ( c, L/a ) are listed in Table III. g c ( L ) N f =12, nZS, c=0.25 L=8L=10L=12L=14L=16L=20L=24L=28L=32
FIG. 1. Renormalized couplings g c in the c = 0 .
250 schemefor bare couplings 4 . ≤ β ≤ .
00 on hypercubic (
L/a ) volumes with L/a ranging from 8 to 32. The values shownare obtained using Zeuthen flow with the Symanzik operatorand the tree-level normalization (nZS). Statistical errors areshown but barely visible. / (1 + δ ( t/L )) that compensates for the zero modes ofthe gauge fields in periodic volumes [36].The dependence on the lattice spacing a reflects cutoffeffects that arise because simulations are not performedwith a “perfect action,” i.e. along the renormalized tra-jectory (RT) emerging from the Gaussian FP. As the gra-dient flow time t/a increases, irrelevant operators die outand cutoff effects are reduced. Thus the continuum limitis approached by increasing L/a while keeping c and therenormalized coupling g c fixed. In asymptotically freetheories this forces the bare coupling toward zero (theGaussian fixed point) or β ≡ /g → ∞ . If all but oneirrelevant operators are negligible the remaining cutoffeffects are proportional to ( √ t/a ) α or ( L/a ) α where α is the scaling exponent of the least irrelevant operator.In the vicinity of the Gaussian FP α = −
2, thus an ex-trapolation in a /L at fixed g c predicts the continuumlimit.In practice simulations are performed at many barecoupling values, “daisy-chaining” them to cover the in-vestigated renormalized coupling range. In slowly run-ning (“walking”) systems a very large scale change is re-quired to cover even a moderate change in the renormal-ized coupling. Figure 1 shows g c evaluated with Zeuthenflow, Symanzik operator and tree-level normalization inthe c = 0 .
250 scheme as the function of the bare couplingon lattice volumes ranging from 8 to 32 . The 16 barecoupling values cover the range g c ∈ (1 . , .
5) in roughlyuniform ∆ g c ≈ . N f = 12 cou-pling runs very slowly, prohibitively large volumes wouldbe needed to reach the strong coupling regime as the barecoupling is tuned toward the GFP. When approaching anIRFP, β c,s → L/a → ∞ limit pre-dicts the infinite-cutoff step scaling function correctly, aslong as the GF flow time is large enough to approach thevicinity of the RT. This reflects the fact that the renor-malized trajectory describes continuum physics.Cutoff effects scale with the scaling exponent of theleast irrelevant operator. In the vicinity of the GaussianFP α = −
2, thus an extrapolation in a /L removes thecutoff effects at weak coupling. Along the RT α can how-ever change and the a /L continuum extrapolation canbecome incorrect at stronger gauge couplings. Since itis very difficult to determine the unknown exponent α numerically, an elegant way out is to chose an improvedsetup where cutoff effects are suppressed and the infinitevolume extrapolation is mild enough not to depend on theexact extrapolation form. We will show that our favoredsetup of Symanzik action, Zeuthen flow, and Symanzikoperator has this property in the range of couplings cov-ered by our simulations.After extrapolating the discrete β -function β c,s ( g c ; L )to the continuum limit (i.e. L/a → ∞ ) at fixed g c ,the continuum β -function β c,s ( g c ) depends only on therenormalized coupling g c . Therefore it is expected that β c,s ( g c ) is free of effects from irrelevant operators intro-duced by the lattice regularization and depends only onthe renormalization scheme c and scale change s . III. NUMERICAL SIMULATION DETAILS
Our nonperturbative determination of the gradientflow β function requires the generation of dynamicalgauge field configurations on ( L/a ) hypercubic volumes.Choosing tree-level improved Symanzik (L¨uscher-Weisz)gauge action [27, 28] and M¨obius domain wall fermions(MDWF) [25] with three levels of stout-smearing [26]( (cid:37) = 0 .
1) for the fermion action, we generate ensem-bles of gauge field configurations using the hybrid MonteCarlo (HMC) update algorithm [39] with six masslesstwo-flavor fermion fields and trajectories of length τ =2 in molecular dynamics time units (MDTU). MDWFprovide a prescription to simulate chiral fermions withcontinuum-like flavor symmetries by adding a fifth di-mension to separate the chiral, physical modes of four di-mensional space-time. In practice, the extent of the fifthdimension, L s , is finite which results in a residual chiralsymmetry breaking, conventionally parametrized by anadditive mass term am res . To study the β function fortwelve flavors in the chiral limit, we set the input quarkmass to zero and choose L s such that am res < · − .As can be seen in Fig. 2, the residual mass increases whenapproaching strong coupling but the numerically deter-mined values of am res are largely independent of the four This combination of actions has already demonstrated its goodproperties for simulations in QCD [37, 38]. -8 -6 -4 -2 a m r e s N f =12, L=24N f =12, L=32 L s =12L s =16L s =24L s =324.1 4.2 4.310 -5 -4 FIG. 2. Residual chiral symmetry breaking, measured interms of the residual mass am res , as function of the bare gaugecoupling β using ( L/a ) volumes with L/a = 24 and 32. Onlystatistical errors are shown.TABLE I. Values of the extent of the fifth dimension L s for theensembles entering our main analysis as function of the baregauge coupling β and the length L/a of the four dimensional(
L/a ) volume. L/aβ dimensional volume. In our step-scaling calculation wetherefore increase L s from 12 at weak coupling up to 32for simulations at our strongest couplings to ensure thatany effect from nonzero am res is negligible. The specificvalues of L s for each value of β and L/a are listed inTable I. At the strongest gauge couplings we performedadditional simulations with alternative choices of L s toverify that the L s values listed in Table I are sufficient forthe flow times ( c values) used in this analysis. Values ofthe renormalized couplings on these additional ensemblesare listed in Appendix B in Table V. In all cases we findthat renormalized couplings for the same choice of L/a and β but different choices of L s agree at the 1 σ level.We further found consistent results for β c,s ( g c , L ) whensubstituting ensembles with different L s .We set the domain wall height M = 1 and have Pauli-Villars terms of mass one. Simulations are performedwith the same boundary conditions (BC) in all four di-rections: periodic BC for the gauge field and antiperiodicBC for the fermion fields. The latter triggers a gap in theeigenvalues of the Dirac operator and thus allows simu-lations with zero input quark mass. In order to explore a v g . p l aque tt e FIG. 3. Average plaquette (normalized to 1) as the functionof the bare gauge coupling β on 32 volumes. Statistical errorsare shown but barely visible. renormalized couplings up to g c ∼ .
5, we create en-sembles for a set of bare couplings starting in the weakcoupling limit with β = 7 .
00 and going down to β = 4 . We choose hypercubic (
L/a ) volumes with L/a = 8, 10, 12, 14, 16, 20, 24, 28, and 32and typically generate 6-10k (2-4k) thermalized MDTUfor most ensembles with
L/a ≤
24 (
L/a = 28 , GRID [40, 41] and agauge field configuration is saved every five trajectories(10 MDTU). In Fig. 3 we demonstrate that our combi-nation of actions leads to sufficiently smooth gauge fieldconfigurations with the average plaquette, the smallest1 × QLUA [42, 43]. For all threeflows, we estimate the energy density using three differ-ent operators: the Wilson-plaquette (W), clover (C), andSymanzik (S) operator. As described in Sec. II, the en-ergy densities are proportional to the renormalized cou-pling which in turn lead to the step-scaling β function.In Appendix B we show the full details of our determi-nations of the energy densities for our preferred analysesbased on Zeuthen flow with Symanzik operator (with andwithout tree-level normalization). Besides presenting g c values in the renormalization schemes c = 0 . For
L/a = 8 and 16 we also simulated at β = 4 . and 0.300, we also list the total number of measurementsfor each ensemble as well as the integrated autocorrela-tion time determined using the Γ-method [44].In addition we use the gradient flow measurements todetermine the topological charge and confirm it vanishesas expected for massless simulations. IV. GRADIENT FLOW β FUNCTION FORTWELVE FUNDAMENTAL FLAVORS
As discussed above, we have calculated energy densi-ties using different gradient flows and operators for all ofour ensembles of gauge field configurations. This allowsus to determine the gradient flow β function in multipleways and check for possible systematic effects. A. Preferred (n)ZS analysis
Our preferred analysis is based on Zeuthen flow and theSymanzik operator. Since our ensembles are generatedwith Symanzik gauge action, this combination is fully O ( a ) improved and we indeed find small discretizationeffects. In the weak coupling limit, discretization effectscan be further suppressed by applying the perturbativelycalculated tree-level normalization factors (see Eq. (2)).A priori the range of validity in g c for perturbatively cal-culated coefficients is not known. Therefore, we presentthe results for our preferred analysis with and withouttree-level normalization. We refer to the two analysisas nZS and ZS, respectively. Our results are presentedfor the renormalization schemes c = 0 . β c,s functions forfive different volume pairs with scale change s = 2. Theresulting values are denoted by the colored data symbolsin the plots in the top panels of Figs. 4–6. By simulat-ing a set of bare couplings β for all volumes, we directlyobtain these statistically independent data points.We interpolate values for each lattice volume pair us-ing a polynomial Ansatz motivated by the perturbativeexpansion β c,s ( g c ; L ) = n (cid:88) i =0 b i g ic . (3)In practice we find that n = 3 is sufficient to describe ourdata and obtain fits with good p -value. In the case ofnZS data, discretization effects are sufficiently small thatwe omit the constant term with coefficient b . Lookingat the data, it does not seem justified to force a zerointercept for ZS although numerically our fit does notresolve b . We list the results of our interpolation inTable II where we also quote the χ /dof as well as the p -value of the fits. The p -values reflect that the fitted linesincluding statistical uncertainties pass through almost all -0.15-0.10-0.050.000.050.100.150.200.25 c , s ( g c ) N f =12, nZS, c=0.25, s=2 g c2 p - v a l ue linearquadratic -0.10-0.050.000.050.10 c , s ( g c = . ) nZS quadratic: c,scont =0.034(5) [23%]nZS linear: c,scont =0.037(5) [93%]ZS quadratic: c,scont =0.030(6) [33%]ZS linear: c,scont =0.053(7) [44%] N f =12 c=0.250s=2 (a/L) -0.100.000.10 c , s ( g c = . ) nZS quadratic: c,scont =0.056(8) [19%]nZS linear: c,scont =0.054(9) [93%]ZS quadratic: c,scont =0.044(8) [11%]ZS linear: c,scont =0.082(9) [48%] N f =12 c=0.250s=2 -0.20-0.15-0.10-0.050.000.050.100.150.20 c , s ( g c ) N f =12, ZS, c=0.25, s=2 g c2 p - v a l ue linearquadratic -0.200.000.20 c , s ( g c = . ) nZS quadratic: c,scont =0.05(1) [7%]nZS linear: c,scont =0.03(1) [29%]ZS quadratic: c,scont =0.02(1) [3%]ZS linear: c,scont =0.07(1) [44%] N f =12 c=0.250s=2 (a/L) -0.200.000.20 c , s ( g c = . ) nZS quadratic: c,scont =-0.01(3) [15%]nZS linear: c,scont =-0.06(3) [12%]ZS quadratic: c,scont =-0.06(3) [8%]ZS linear: c,scont =-0.02(3) [32%] N f =12 c=0.250s=2 FIG. 4. Discrete step-scaling β -function in the c = 0 .
250 gradient flow scheme for our preferred nZS (left) and ZS (right) datasets. The symbols in the top row show our results for the finite volume discrete β function with scale change s = 2. The dashedlines with shaded error bands in the same color of the data points show the interpolating fits. We perform two continuumextrapolations: a linear fit in a /L to the three largest volume pairs (black line with gray error band) and a quadratic fit in a /L to all five volume pairs (gray dash-dotted line). The p -values of the continuum extrapolation fits are shown in the plotsin the second row. Further details of the continuum extrapolation at selected g c values are presented in the small panels at thebottom where the legend lists the extrapolated values in the continuum limit with p -values in brackets. Only statistical errorsare shown. data points. Hence the fit Ansatz Eq. (3) leads to a gooddescription of our data. Only the 8 →
16 volume paircorresponding to the smallest volumes used exhibits alow p -value around 5%. This data set suffers most fromdiscretization effects and also exhibits a few “outliers” inthe strong coupling limit which cause a larger χ (lower p -value).The interpolating fits predict finite volume discretestep-scaling functions β c,s ( g c ; L ). These discrete step-scaling functions are shown in the top row panels ofFigs. 4–6 by dashed lines with shaded error band in thesame color as the corresponding data points. The inter- polation for nZS starts at g c = 0 due to the constraint b = 0 of the interpolating polynomial, whereas for ZS itbegins with our data at the weakest coupling. The upperend of the interpolation depends on the range where wehave data and varies slightly depending on the chosenscheme c .Subsequently, we perform infinite volume continuumlimit extrapolations using the interpolated finite-volume β -functions β c,s ( g c ; L ), which are continuous in g c . Weconsider two Ans¨atze for the extrapolation • a linear fit in a /L to the three largest volumepairs, -0.15-0.10-0.050.000.050.100.150.200.25 c , s ( g c ) N f =12, nZS, c=0.275, s=2 g c2 p - v a l ue linearquadratic -0.10-0.050.000.050.10 c , s ( g c = . ) nZS quadratic: c,scont =0.041(7) [32%]nZS linear: c,scont =0.044(7) [92%]ZS quadratic: c,scont =0.037(8) [40%]ZS linear: c,scont =0.055(9) [50%] N f =12 c=0.275s=2 (a/L) -0.100.000.10 c , s ( g c = . ) nZS quadratic: c,scont =0.07(1) [10%]nZS linear: c,scont =0.07(1) [65%]ZS quadratic: c,scont =0.06(1) [6%]ZS linear: c,scont =0.09(1) [42%] N f =12 c=0.275s=2 -0.20-0.15-0.10-0.050.000.050.100.150.20 c , s ( g c ) N f =12, ZS, c=0.275, s=2 g c2 p - v a l ue linearquadratic -0.200.000.20 c , s ( g c = . ) nZS quadratic: c,scont =0.07(1) [11%]nZS linear: c,scont =0.06(1) [46%]ZS quadratic: c,scont =0.06(2) [7%]ZS linear: c,scont =0.09(2) [44%] N f =12 c=0.275s=2 (a/L) -0.200.000.20 c , s ( g c = . ) nZS quadratic: c,scont =0.03(4) [26%]nZS linear: c,scont =-0.01(4) [10%]ZS quadratic: c,scont =0.00(4) [36%]ZS linear: c,scont =0.01(4) [30%] N f =12 c=0.275s=2 FIG. 5. Discrete step-scaling β -function in the c = 0 .
275 gradient flow scheme for our preferred nZS (left) and ZS (right) datasets. The symbols in the top row show our results for the finite volume discrete β function with scale change s = 2. The dashedlines with shaded error bands in the same color of the data points show the interpolating fits. We perform two continuumextrapolations: a linear fit in a /L to the three largest volume pairs (black line with gray error band) and a quadratic fit in a /L to all five volume pairs (gray dash-dotted line). The p -values of the continuum extrapolation fits are shown in the plotsin the second row. Further details of the continuum extrapolation at selected g c values are presented in the small panels at thebottom where the legend lists the extrapolated values in the continuum limit with p -values in brackets. Only statistical errorsare shown. • a quadratic fit in a /L to all five volume pairs,which are motivated by the expected O ( a ) discretiza-tion effects in the weak coupling limit. Similar to theuse of tree-level normalization, the validity of an extrap-olation proportional to a /L is limited to sufficientlyweak couplings. Moreover correction terms of higher or-der may only be resolved if the statistical uncertaintiesare small enough. We therefore monitor the p -value ofthe extrapolation fits as a criteria to judge the validity ofthe continuum limit extrapolations. p -values are shownas a function of g c in the plots in the second row panels ofFigs. 4–6. In addition we show details of the continuum limit extrapolations at g c = 2 .
0, 3.4, 4.8, and 6.2 in thefour smaller plots at the bottom of the three figures. Theresulting continuum step-scaling functions are shown inthe top row plots by the solid black line with gray errorband (linear fit) and the gray dash-dotted line (quadraticfit).Overall most extrapolation exhibit excellent p -valuesover the full range in g c covered by our simulations andcorrections to an a /L extrapolation seem to be small.Performing a linear extrapolation in a /L to our threelargest volume pairs is well justified for all data sets al-though a noticeable drop of the p -value can be observed -0.15-0.10-0.050.000.050.100.150.200.25 c , s ( g c ) N f =12, nZS, c=0.3, s=2 g c2 p - v a l ue linearquadratic -0.10-0.050.000.050.10 c , s ( g c = . ) nZS quadratic: c,scont =0.049(8) [46%]nZS linear: c,scont =0.051(9) [92%]ZS quadratic: c,scont =0.05(1) [54%]ZS linear: c,scont =0.06(1) [48%] N f =12 c=0.300s=2 (a/L) -0.100.000.10 c , s ( g c = . ) nZS quadratic: c,scont =0.09(1) [8%]nZS linear: c,scont =0.09(1) [44%]ZS quadratic: c,scont =0.08(1) [7%]ZS linear: c,scont =0.10(1) [30%] N f =12 c=0.300s=2 -0.20-0.15-0.10-0.050.000.050.100.150.20 c , s ( g c ) N f =12, ZS, c=0.3, s=2 g c2 p - v a l ue linearquadratic -0.200.000.20 c , s ( g c = . ) nZS quadratic: c,scont =0.10(2) [21%]nZS linear: c,scont =0.09(2) [67%]ZS quadratic: c,scont =0.09(2) [17%]ZS linear: c,scont =0.11(2) [48%] N f =12 c=0.300s=2 (a/L) -0.200.000.20 c , s ( g c = . ) nZS quadratic: c,scont =0.06(5) [31%]nZS linear: c,scont =0.02(5) [9%]ZS quadratic: c,scont =0.05(5) [62%]ZS linear: c,scont =0.03(5) [31%] N f =12 c=0.300s=2 FIG. 6. Discrete step-scaling β -function in the c = 0 .
300 gradient flow scheme for our preferred nZS (left) and ZS (right) datasets. The symbols in the top row show our results for the finite volume discrete β function with scale change s = 2. The dashedlines with shaded error bands in the same color of the data points show the interpolating fits. We perform two continuumextrapolations: a linear fit in a /L to the three largest volume pairs (black line with gray error band) and a quadratic fit in a /L to all five volume pairs (gray dash-dotted line). The p -values of the continuum extrapolation fits are shown in the plotsin the second row. Further details of the continuum extrapolation at selected g c values are presented in the small panels at thebottom where the legend lists the extrapolated values in the continuum limit with p -values in brackets. Only statistical errorsare shown. for nZS data around g c ∼ . c values con-sidered. However, even for our strongest couplings thelinear fit still gives a good p -value greater than 10%. Incase of the ZS data, the p -values for linear extrapolationsare mostly well above the 20% level. They do howevershow a similar behavior like in case of the nZS fits withlower values around g c ∼ . →
16 volumepair. Discretization effects for the 8 →
16 data set arequite significant for c = 0 .
25 but decrease for c = 0 . c values require larger lattice volumes [45].Using tree-level normalization, this effect is attenuated.For nZS, the p -value of the quadratic extrapolation evenat c = 0 .
250 remains at the acceptable 4–5% level for thefull range in g c covered, whereas the quadratic extrap-olation for ZS at c = 0 .
250 exhibits vanishing p -valuesaround g c ∼ . a /L dependence for all c valuesand in fact can also be described by fitting only a con- TABLE II. Results of the interpolation fits for the five lattice volume pairs for our preferred (n)ZS analysis using renormalizationschemes c = 0 .
250 (top panel), 0.275 (middle panel), and 0.300 (bottom panel). Since discretization effects are sufficiently smallfor nZS, we constrain the constant term b = 0 in Eq. (3) and perform fits with 13 degrees of freedom (dof) or in case of the8 →
16 volume pair with 14 dof. For ZS the intercept b is fitted and we have 12 and 13 dof, respectively. In addition we listthe χ / dof as well as the p -value. nZS ZS χ /dof p -val. b b b χ /dof p -val. b b b b →
16 1.738 0.042 -0.00220(20) 0.0188(14) -0.0024(24) 1.687 0.056 -0.00136(42) 0.0124(47) -0.061(16) -0.011(17)10 →
20 1.385 0.157 -0.00269(22) 0.0162(17) -0.0013(29) 1.407 0.154 -0.00281(55) 0.0198(60) -0.045(20) 0.016(19)12 →
24 1.507 0.106 -0.00269(23) 0.0147(18) -0.0021(32) 1.591 0.086 -0.00290(67) 0.0181(72) -0.028(24) 0.013(23)14 →
28 0.905 0.547 -0.00267(34) 0.0140(26) -0.0002(45) 0.971 0.474 -0.0023(10) 0.011(11) 0.002(34) -0.009(31)16 →
32 0.875 0.579 -0.00209(43) 0.0106(32) 0.0045(53) 0.933 0.512 -0.0025(12) 0.015(13) -0.016(43) 0.016(41)8 →
16 1.620 0.066 -0.00283(23) 0.0201(17) -0.0040(29) 1.660 0.062 -0.00221(53) 0.0165(59) -0.048(20) -0.003(20)10 →
20 1.344 0.179 -0.00278(27) 0.0159(21) -0.0013(35) 1.410 0.153 -0.00291(70) 0.0190(75) -0.033(25) 0.013(24)12 →
24 1.439 0.132 -0.00242(29) 0.0126(23) 0.0008(40) 1.547 0.100 -0.00248(86) 0.0140(93) -0.014(30) 0.005(29)14 →
28 0.815 0.644 -0.00269(45) 0.0143(34) 0.0001(59) 0.857 0.592 -0.0019(13) 0.007(14) 0.019(44) -0.023(41)16 →
32 0.863 0.593 -0.00188(56) 0.0094(41) 0.0073(68) 0.920 0.525 -0.0024(16) 0.016(17) -0.017(54) 0.021(52)8 →
16 1.709 0.047 -0.00319(27) 0.0208(20) -0.0046(34) 1.787 0.039 -0.00273(64) 0.0186(71) -0.036(24) -0.001(23)10 →
20 1.274 0.220 -0.00282(33) 0.0158(25) -0.0011(43) 1.365 0.174 -0.00279(87) 0.0166(94) -0.019(31) 0.005(30)12 →
24 1.423 0.140 -0.00212(37) 0.0106(28) 0.0040(50) 1.535 0.103 -0.0020(11) 0.010(12) -0.002(38) -0.000(37)14 →
28 0.732 0.732 -0.00278(56) 0.0153(43) -0.0002(75) 0.754 0.699 -0.0017(17) 0.004(18) 0.034(56) -0.035(52)16 →
32 0.884 0.570 -0.00168(70) 0.0085(52) 0.0103(84) 0.930 0.515 -0.0027(20) 0.020(21) -0.030(67) 0.036(64) stant [33]. For the smaller volumes at stronger couplings,corrections to a /L arise and the data show an upwardcurvature. In contrast to that the large volume ZS dataclearly exhibit a slope in a /L for all c -values. An ex-planation could be that corrections at weak coupling arecaused by higher order terms which can essentially be re-moved by the tree-level normalization, whereas the cor-rections at strong coupling indicate that perturbative im-provement and extrapolations in a /L are subject tononperturbative corrections. B. Alternative flow/operators
In order to estimate systematic effects, we considerthe step-scaling β function obtained with different flows(Wilson and Symanzik flow) as well as different operators(Wilson plaquette and clover). We repeat the same stepsof the analysis as for our preferred (n)ZS data i.e. we firstobtain data points for the discrete β functions for our fivelattice volume pairs and next perform an interpolatingfit. Again we constrain b to be zero when using tree-levelimprovement but fit b otherwise. Finally we again carryout an infinite volume continuum limit extrapolation us-ing a linear Ansatz for the three largest volume pairsand a quadratic Ansatz using all five volume pairs. Asexamples for our alternative determinations, we presentin Fig. 7 the determination for the step-scaling β func-tion using Symanzik flow and the clover operator and inFig. 8 Wilson flow with the Wilson plaquette operator.The plots on the left show the analysis with tree-levelnormalization factors, the plots on the right the analysiswithout tree-level improvement. Again we show results(top to bottom) for c = 0 . β c,s ( g c ) obtained from different flow/operator combina-tions are expected to agree, the finite volume predictionsof β c,s ( g c , L ) are subject to discretization effects andhence depend on the specific flow/operator combination.At finite volume, different flow/operator combinationspredict different renormalized couplings even on the sameset of gauge field ensembles evaluated using the samescheme c . Only after taking the continuum limit, the ob-tained β c,s ( g c ) are expected to be free of discretizationeffects and can be meaningfully compared. The accessi-ble range of g c and β c,s ( g c ) varies for each flow/operatorcombination at each bare coupling β ≡ /g and ( L/a ) volume. How different flow/operator combinations ap-proach the same continuum limit can be seen by com-paring Figs. 4–8. Another way to understand this effect is to consider the gra-dient flow as a continuous renormalization group (RG) trans-formation where the continuum limit is reached as the flow ap-proaches the renormalized trajectory (RT) [46, 47]. Since the RTin bare parameter space depends on the RG transformation, dif-ferent flow/operator combinations approach their correspondingRT differently, predicting different g c on the same ensembles. g c2 -0.15-0.10-0.050.000.050.100.150.200.25 c , s ( g c ) N f =12, nSC, c=0.25, s=2 g c2 -0.200.000.200.400.600.801.001.20 c , s ( g c ) N f =12, SC, c=0.25, s=2 g c2 -0.15-0.10-0.050.000.050.100.150.200.25 c , s ( g c ) N f =12, nSC, c=0.275, s=2 g c2 -0.200.000.200.400.600.801.001.20 c , s ( g c ) N f =12, SC, c=0.275, s=2 g c2 -0.15-0.10-0.050.000.050.100.150.200.25 c , s ( g c ) N f =12, nSC, c=0.3, s=2 g c2 -0.200.000.200.400.600.801.001.20 c , s ( g c ) N f =12, SC, c=0.3, s=2 FIG. 7. Alternative determination of the discrete β function using Symanzik flow with clover operator. Plots on the left showthe analysis including the tree-level improvement (nSC), plots on the right without (SC). From top to bottom we present resultsfor the renormalization scheme c = 0 . In total we obtain 36 different continuum limit predic-tions for the step-scaling function based on nine differentflow/operator combinations, analyzing the data with andwithout tree-level normalization, and performing two dif-ferent extrapolations to the 18 data sets. Choosing fourrepresentative couplings for the range of g c simulated, wepresent in Fig. 9 an overview of our results. The plots inone column correspond to g c = 2 .
0, 3.4, 4.8 and 6.2, whilethe plots in a row have the same c value, from top to bot-tom c = 0 . β c,s on the abscissa. The upperhalf of each plot shows the results without tree-level im-provement, the lower half with improvement. The differ-ent colors denote the three different flows: blue Zeuthen,red Symanzik, and green Wilson, whereas a circle marksthe Symanzik operator, a square the Wilson plaquette op-erator, and the triangle the clover operator. Open sym-bols indicate continuum extrapolations with a p -value be-low 5%. The two strongest couplings ( g c = 4 . , g c2 -0.15-0.10-0.050.000.050.100.150.200.25 c , s ( g c ) N f =12, nWW, c=0.25, s=2 g c2 -0.80-0.60-0.40-0.200.000.20 c , s ( g c ) N f =12, WW, c=0.25, s=2 g c2 -0.15-0.10-0.050.000.050.100.150.200.25 c , s ( g c ) N f =12, nWW, c=0.275, s=2 g c2 -0.80-0.60-0.40-0.200.000.20 c , s ( g c ) N f =12, WW, c=0.275, s=2 g c2 -0.15-0.10-0.050.000.050.100.150.200.25 c , s ( g c ) N f =12, nWW, c=0.3, s=2 g c2 -0.80-0.60-0.40-0.200.000.20 c , s ( g c ) N f =12, WW, c=0.3, s=2 FIG. 8. Alternative determination of the discrete β function using Wilson flow with Wilson plaquette operator. Plots on theleft show the analysis including the tree-level improvement (nWW), plots on the right without (WW). From top to bottom wepresent results for the renormalization scheme c = 0 . fore, e.g. no red Symanzik flow symbols are shown in therightmost plots. The different continuum extrapolationsare indicated by a solid (linear fit) or dashed (quadraticfit) line denoting the statistical uncertainty. Highlightedby the vertical, blue-shaded bands are the results of ourpreferred (n)ZS analysis.At the weakest coupling shown ( g c = 2 . →
16 volume pair. Looking at stronger cou-pling ( g c = 3 . c = 0 .
250 dif-1 - c,s (g c2 =2.0) nZS linnZS quadnZW linnZW quadnZC linnZC quadnSS linnSS quadnSW linnSW quadnSC linnSC quadnWS linnWS quadnWW linnWW quadnWC linnWC quadZS linZS quadZW linZW quadZC linZC quadSS linSS quadSW linSW quadSC linSC quadWS linWS quadWW linWW quadWC linWC quad N f =12c=0.250s=2 - c,s (g c2 =3.4)N f =12c=0.250s=2 - c,s (g c2 =4.8)N f =12c=0.250s=2 -0.15 -0.1 -0.05 0 - c,s (g c2 =6.2) nZS linnZS quadnZW linnZW quadnZC linnZC quadnSS linnSS quadnSW linnSW quadnSC linnSC quadnWS linnWS quadnWW linnWW quadnWC linnWC quadZS linZS quadZW linZW quadZC linZC quadSS linSS quadSW linSW quadSC linSC quadWS linWS quadWW linWW quadWC linWC quad N f =12c=0.250s=2 - c,s (g c2 =2.0) nZS linnZS quadnZW linnZW quadnZC linnZC quadnSS linnSS quadnSW linnSW quadnSC linnSC quadnWS linnWS quadnWW linnWW quadnWC linnWC quadZS linZS quadZW linZW quadZC linZC quadSS linSS quadSW linSW quadSC linSC quadWS linWS quadWW linWW quadWC linWC quad N f =12c=0.275s=2 - c,s (g c2 =3.4)N f =12c=0.275s=2 - c,s (g c2 =4.8)N f =12c=0.275s=2 -0.1 -0.05 0 0.05 0.1 - c,s (g c2 =6.2) nZS linnZS quadnZW linnZW quadnZC linnZC quadnSS linnSS quadnSW linnSW quadnSC linnSC quadnWS linnWS quadnWW linnWW quadnWC linnWC quadZS linZS quadZW linZW quadZC linZC quadSS linSS quadSW linSW quadSC linSC quadWS linWS quadWW linWW quadWC linWC quad N f =12c=0.275s=2 - c,s (g c2 =2.0) nZS linnZS quadnZW linnZW quadnZC linnZC quadnSS linnSS quadnSW linnSW quadnSC linnSC quadnWS linnWS quadnWW linnWW quadnWC linnWC quadZS linZS quadZW linZW quadZC linZC quadSS linSS quadSW linSW quadSC linSC quadWS linWS quadWW linWW quadWC linWC quad N f =12c=0.300s=2 - c,s (g c2 =3.4)N f =12c=0.300s=2 - c,s (g c2 =4.8)N f =12c=0.300s=2 -0.05 0 0.05 0.1 0.15 - c,s (g c2 =6.2) nZS linnZS quadnZW linnZW quadnZC linnZC quadnSS linnSS quadnSW linnSW quadnSC linnSC quadnWS linnWS quadnWW linnWW quadnWC linnWC quadZS linZS quadZW linZW quadZC linZC quadSS linSS quadSW linSW quadSC linSC quadWS linWS quadWW linWW quadWC linWC quad N f =12c=0.300s=2 FIG. 9. Systematic effects due to tree-level improvement, different flows and operators as well as linear extrapolations of thethree largest volume pairs vs. quadratic extrapolation of all volume pairs. The columns show our continuum limit results atselective g c values of 2.0, 3.4, 4.8, and 6.2; the rows correspond to renormalization schemes c = 0 . p -value below 5% and the vertical shaded bands highlight our preferred (n)ZS analysis. σ from either of our preferred (n)ZSdeterminations. In both cases, the continuum extrapola-tion is less reliable because it covers a very large range(more than an order of magnitude) as can be seen e.g. inthe upper right plot of Fig. 7. The larger scatter for theintermediate couplings could be related to the observa-tion that most of our extrapolations around g c ∼ p -value, hence continuum predictions may be lessreliable. At the strongest coupling, g c = 6 .
2, statisticalerrors are significantly larger and within these all (avail-able) results agree at the 1 σ level.Comparing the different step-scaling results for differ-ent c values, the overall consistency improves significantlywhen c increases. While at c = 0 .
250 our preferred deter-minations based on nZS and ZS exhibit a small tensionat weak and intermediate couplings, both determinationsare consistent at the 1 σ level for c = 0 .
275 and 0.300.Being aware that c = 0 .
250 might be affected by dis-cretization effects on our smaller lattice volume pairs andperturbative improvement may not be fully justified atthe stronger couplings, we therefore choose to quote theenvelope covering both, nZS and ZS, determinations asour final result which also accounts for systematic effects.Systematic effects due to different flow/operator combi-nations are not resolved within the combined nZS+ZSuncertainty. In fact only a few extrapolations for alter-native flow/operators at c = 0 .
250 lead to results differ-ing by more than 1 σ . Such extrapolations cover a largerange and are less reliable because extrapolating all fivepoints using the quadratic Ansatz tends to differ by sev-eral sigmas from extrapolating the three largest volumepairs using a linear Ansatz. While the analyzed choiceof flow/operator combinations is by no means complete,in fact an arbitrary number of operators and flows couldbe considered, we do expect to have studied a represen-tative set. Finally, it is noteworthy that the agreementat weakest and strongest couplings is better than at theintermediate range.Our final continuum limit results for the gradientflow step-scaling function obtained in renormalizationschemes c = 0 . g c > . c = 0 .
250 scheme indicates a fixedpoint near the 4-loop MS value, whereas for c = 0 . c . Our pre-dicted range of couplings for an IRFP of an SU(3) gaugetheory with twelve flavors is consistent with predictionsby Ryttov and Shrock who improve perturbative conver-gence using Pad´e approximants [7, 48]. V. CONCLUSION
We have presented details of our gradient flow step-scaling calculation for SU(3) with twelve dynamical fla-vors. Our calculations are based on gauge field ensem-bles generated with Symanzik gauge action and threetimes stout-smeared M¨obius domain wall fermions. Us-ing Zeuthen flow combined with the Symanzik opera-tor, we determine renormalized couplings and predict thestep-scaling β -function. We assign a systematic uncer-tainty to our numerical predictions that covers the dif-ference between the continuum limit extrapolated resultsof various flows and operators, different extrapolationforms, and also the effect of the tree-level normalization.For the c = 0 .
250 scheme we can identify a sign changeof the β -function (infrared fixed point) in the range of5 . ≤ g c ≤ .
4, while for schemes c = 0 .
275 ( c = 0 . . ≤ g c (5 . ≤ g c ). It appears that the step scalingfunction and the value of the fixed point exhibit a milddependence on the renormalization scheme c , however,further data at even stronger couplings with greater pre-cision are required to resolve such a dependence. Theset of available gauge field ensembles restricts the max-imum g c value a given flow/operator combination canreach. Less than half of the 36 combinations we considerreach g c = 6 .
2, which unfortunately limits the control ofour systematic error estimate around g c ∼ . c = 0 . g c depends on the flow/operator combination, the converseargument implies that on a given set of gauge field ensem-bles, an IRFP can be identified for some flow/operatorcombinations but missed for others.Our presented result are consistent with perturbativepredictions obtained at 3- or 4-loop in the MS scheme.Finally, we compare our continuum limit results to other,nonperturbative determinations published in the litera-ture. So far only calculations based on staggered fermionshave been performed [17–21]. The work by Hasenfratzand Schaich [18] extends from the weak couplings at g c ∼ . g c ∼ . g c ∼ . c = 0 .
25 and c = 0 . c val-ues; for g c (cid:38) . to the prediction of the step-scaling function by Hasenfratz/Schaich [18] and LatHC[19–21]. Reference [18] presents results for the renormal-ization schemes c = 0 .
250 and 0 .
300 and accounts forsystematic effects due to the extra- and interpolations.References [19–21] only use c = 0 .
250 without quantify-ing systematic effects. Extending what has so far been ASCII files containing our final results (blue shaded bands) areuploaded as Supplemental Material. β -functionpredicted by Hasenfratz and Schaich in Ref. [18]. Whilefor c = 0 . ∼ σ for in-termediate and strong couplings, the differences almostvanish for c = 0 . c = 0 .
250 are underestimated. Incontrast, the results of Refs. [19–21] predict a qualita-tively different step scaling function that is nearly con-stant in a wide g c coupling range, without a sign of anIRFP. Further investigations are needed to track downthe origin of this disagreement. ACKNOWLEDGMENTS
We are very grateful to Peter Boyle, Guido Cossu,Anontin Portelli, and Azusa Yamaguchi who develop the
GRID software library providing the basis of this workand who assisted us in installing and running
Grid ondifferent architectures and computing centers. A.H. andO.W. acknowledge support by DOE Grant No. DE-SC0010005 and C.R. by DOE Grant No. DE-SC0015845.A.H. would like to acknowledge the Mainz Institute forTheoretical Physics (MITP) of the Cluster of ExcellencePRISMA+ (Project ID 39083149). O.W. acknowledgespartial support by the Munich Institute for Astro- andParticle Physics (MIAPP) of the DFG cluster of excel-lence “Origin and Structure of the Universe”.Computations for this work were carried out in part onfacilities of the USQCD Collaboration, which are fundedby the Office of Science of the U.S. Department of Energyand the RMACC Summit supercomputer [49], which issupported by the National Science Foundation (awardsACI-1532235 and ACI-1532236), the University of Col-orado Boulder, and Colorado State University. Thiswork used the Extreme Science and Engineering Dis-covery Environment (XSEDE), which is supported byNational Science Foundation grant number ACI-1548562[50] through allocation TG-PHY180005 on the XSEDEresource stampede2 . This research also used resourcesof the National Energy Research Scientific ComputingCenter (NERSC), a U.S. Department of Energy Office ofScience User Facility operated under Contract No. DE-AC02-05CH11231. We thank Fermilab, Jefferson Lab,NERSC, the University of Colorado Boulder, Texas Ad-vanced Computing Center, the NSF, and the U.S. DOEfor providing the facilities essential for the completion ofthis work. c2 -0.15-0.1-0.0500.050.10.150.20.25 c , s ( g c ) N f =12, c=0.25, s=2 nZSZSnZS + ZS2-loop3-loop4-loop5-loopdata1 c2 -0.15-0.1-0.0500.050.10.150.20.25 c , s ( g c ) N f =12, c=0.275, s=2 nZSZSnZS + ZS2-loop3-loop4-loop5-loopdata1 c2 -0.15-0.1-0.0500.050.10.150.20.25 c , s ( g c ) N f =12, c=0.3, s=2 nZSZSnZS + ZS2-loop3-loop4-loop5-loopdata1 FIG. 10. Continuum extrapolations of our preferred (n)ZSdata set for c = 0 . c2 -0.15-0.1-0.0500.050.10.150.20.25 c , s ( g c ) N f =12, c=0.25, s=2 domain wall fermionstaggered [18]staggered [19-21]2-loop3-loop4-loop5-loop c2 -0.15-0.1-0.0500.050.10.150.20.25 c , s ( g c ) N f =12, c=0.3, s=2 domain wall fermionstaggered [18]2-loop3-loop4-loop5-loop FIG. 11. Comparison of our continuum extrapolations in-cluding an estimate for systematic effects for renormalizationschemes c = 0 .
250 and 0.300 to nonperturbative determina-tions based on staggered fermions. Reference [18] identifiesan IRFP at somewhat stronger coupling than our determina-tion and presents results for the c = 0 .
250 and 0.300 schemes.Using only c = 0 .
250 to determine β c,s , Refs. [19–21] reporta basically flat β function for the range 6 . (cid:46) g c (cid:46) .
2. Allpredictions use s = 2. Appendix A: Tree-level normalization factors
TABLE III: Tree-level normalization coefficients C ( c, L/a ) for renormal-ization schemes c = 0 . L/a C (0 . , L/a ) C (0 . , L/a ) C (0 . , L/a )ZS 8 1.098653 1.054568 1.023078ZS 10 1.028937 1.008921 0.992696ZS 12 1.006523 0.994355 0.982724ZS 14 0.997574 0.988333 0.978506ZS 16 0.993296 0.985414 0.976448ZS 20 0.989702 0.982950 0.974707ZS 24 0.988394 0.982052 0.974073ZS 28 0.987826 0.981662 0.973798ZS 32 0.987549 0.981472 0.973664ZW 8 0.991436 0.967277 0.949748ZW 10 0.963060 0.954041 0.945790ZW 12 0.960664 0.955833 0.949679ZW 14 0.963470 0.959650 0.953905ZW 16 0.966875 0.963204 0.957414ZW 20 0.972488 0.968506 0.962350ZW 24 0.976300 0.971920 0.965417ZW 28 0.978873 0.974170 0.967403ZW 32 0.980657 0.975710 0.968749ZC 8 0.692642 0.720334 0.740038ZC 10 0.773982 0.795212 0.809212ZC 12 0.827175 0.843095 0.852580ZC 14 0.863400 0.875159 0.881220ZC 16 0.888948 0.897503 0.900980ZC 20 0.921408 0.925563 0.925559ZC 24 0.940295 0.941715 0.939585ZC 28 0.952163 0.951796 0.948291ZC 32 0.960073 0.958485 0.954048SS 8 1.004010 0.981303 0.964323SS 10 0.975429 0.965750 0.956887SS 12 0.970635 0.964851 0.957992SS 14 0.971361 0.966657 0.960279SS 16 0.973194 0.968759 0.962429SS 20 0.976744 0.972206 0.965663SS 24 0.979337 0.974545 0.967756SS 28 0.981140 0.976124 0.969139SS 32 0.982411 0.977218 0.970087SW 8 0.913774 0.905341 0.898897SW 10 0.916176 0.915408 0.913243SW 12 0.928013 0.928589 0.926585SW 14 0.939067 0.939241 0.936594SW 16 0.947869 0.947315 0.943949SW 20 0.960001 0.958090 0.953542SW 24 0.967478 0.964576 0.959217SW 28 0.972316 0.968720 0.962808SW 32 0.975597 0.971509 0.965210SC 8 0.657535 0.687715 0.710135SC 10 0.744876 0.769007 0.785720SC 12 0.803525 0.822215 0.834139SC 14 0.844127 0.858375 0.866552SC 16 0.873090 0.883830 0.889124 flow/operator L/a C (0 . , L/a ) C (0 . , L/a ) C (0 . , L/a )SC 20 0.910297 0.916108 0.917444SC 24 0.932164 0.934852 0.933730SC 28 0.945990 0.946612 0.943886SC 32 0.955241 0.954442 0.950622WS 8 1.417786 1.309105 1.228189WS 10 1.212382 1.153760 1.111117WS 12 1.124191 1.089496 1.061970WS 14 1.081241 1.056847 1.035972WS 16 1.056513 1.037441 1.020202WS 20 1.029792 1.016056 1.002600WS 24 1.016164 1.005008 0.993423WS 28 1.008214 0.998522 0.988011WS 32 1.003155 0.994380 0.984546WW 8 1.255986 1.183080 1.127496WW 10 1.123465 1.083497 1.053546WW 12 1.067677 1.043763 1.023836WW 14 1.041450 1.024261 1.008595WW 16 1.026736 1.012918 0.999521WW 20 1.011175 1.000644 0.989555WW 24 1.003383 0.994402 0.984431WW 28 0.998885 0.990771 0.981433WW 32 0.996043 0.988465 0.979523WC 8 0.808940 0.829206 0.841227WC 10 0.869590 0.880908 0.886360WC 12 0.903618 0.910174 0.911931WC 14 0.924867 0.928370 0.927770WC 16 0.938990 0.940417 0.938230WC 20 0.955954 0.954837 0.950722WC 24 0.965340 0.962790 0.957597WC 28 0.971059 0.967628 0.961775WC 32 0.974797 0.970786 0.964499 Appendix B: Renormalized couplings g c TABLE IV: Details of our preferred analysis based on Zeuthen flow and Symanzik operator. For each ensemble specified by thespatial extent
L/a and bare gauge coupling β we list N , the number of measurements, as well as the renormalized couplings g c for the analysis with (nZS) and without tree-level improvement (ZS) for the three renormalization schemes c = 0 . c = 0 . c = 0 . c = 0 . L/a β N g c (nZS) g c (ZS) τ int g c (nZS) g c (ZS) τ int g c (nZS) g c (ZS) τ int c = 0 . c = 0 . c = 0 . L/a β N g c (nZS) g c (ZS) τ int g c (nZS) g c (ZS) τ int g c (nZS) g c (ZS) τ int
14 4.40 1002 4.5157(54) 4.5634(55) 0.66(9) 4.5201(67) 4.5529(68) 0.7(1) 4.5198(83) 4.5431(84) 0.8(1)14 4.50 1002 4.1451(63) 4.1889(63) 1.0(2) 4.1559(79) 4.1862(79) 1.1(2) 4.1633(96) 4.1848(97) 1.1(2)14 4.60 1001 3.8283(50) 3.8688(50) 0.8(1) 3.8374(61) 3.8653(62) 0.9(1) 3.8426(76) 3.8624(76) 0.9(2)14 4.70 1004 3.5659(54) 3.6036(55) 1.0(2) 3.5746(69) 3.6006(70) 1.1(2) 3.5796(86) 3.5982(86) 1.2(2)14 4.80 1003 3.3349(39) 3.3701(40) 0.69(10) 3.3423(50) 3.3666(50) 0.7(1) 3.3467(63) 3.3640(63) 0.8(1)14 5.00 1002 2.9738(38) 3.0052(38) 0.70(10) 2.9825(48) 3.0041(48) 0.8(1) 2.9885(58) 3.0039(58) 0.8(1)14 5.20 1001 2.6755(30) 2.7038(30) 0.59(8) 2.6810(37) 2.7005(37) 0.63(9) 2.6838(45) 2.6977(45) 0.69(10)14 5.50 1002 2.3444(26) 2.3691(26) 0.60(8) 2.3509(32) 2.3680(33) 0.64(8) 2.3554(39) 2.3675(39) 0.67(9)14 6.00 1001 1.9407(23) 1.9612(23) 0.7(1) 1.9449(29) 1.9591(29) 0.8(1) 1.9476(37) 1.9577(38) 0.9(1)14 6.50 1002 1.6569(17) 1.6744(18) 0.58(7) 1.6586(22) 1.6707(22) 0.61(8) 1.6586(26) 1.6672(26) 0.64(8)14 7.00 1003 1.4516(16) 1.4669(16) 0.63(8) 1.4536(21) 1.4642(21) 0.71(10) 1.4547(26) 1.4622(26) 0.8(1)16 4.13 608 6.491(13) 6.532(13) 0.9(2) 6.454(16) 6.482(16) 1.0(2) 6.427(19) 6.446(19) 1.1(3)16 4.15 657 6.219(12) 6.257(12) 1.0(2) 6.200(15) 6.227(15) 1.2(2) 6.187(19) 6.206(19) 1.3(3)16 4.17 961 5.9924(83) 6.0297(84) 0.8(1) 5.976(10) 6.002(10) 0.9(1) 5.965(13) 5.983(13) 1.0(2)16 4.20 922 5.679(10) 5.715(10) 1.3(2) 5.667(12) 5.691(12) 1.3(2) 5.654(15) 5.671(15) 1.4(3)16 4.25 886 5.3213(77) 5.3545(77) 0.8(1) 5.3192(97) 5.3421(98) 0.9(1) 5.317(12) 5.333(12) 0.9(2)16 4.30 889 5.0244(77) 5.0557(77) 0.9(1) 5.0273(100) 5.049(10) 1.0(2) 5.030(12) 5.045(12) 1.1(2)16 4.40 896 4.5468(63) 4.5751(64) 0.9(2) 4.5554(80) 4.5749(80) 1.0(2) 4.5634(98) 4.5773(99) 1.0(2)16 4.50 877 4.1536(68) 4.1795(68) 1.1(2) 4.1592(84) 4.1771(84) 1.2(2) 4.163(10) 4.176(10) 1.2(2)16 4.60 783 3.8457(53) 3.8697(54) 0.66(9) 3.8529(66) 3.8694(66) 0.7(1) 3.8578(82) 3.8696(83) 0.8(1)16 4.70 675 3.5801(73) 3.6023(73) 1.4(3) 3.589(10) 3.604(10) 1.9(5) 3.597(14) 3.608(14) 2.3(6)16 4.80 602 3.3545(74) 3.3754(75) 1.4(3) 3.3619(97) 3.3764(98) 1.6(4) 3.368(12) 3.378(12) 1.9(5)16 5.00 621 2.9900(49) 3.0086(49) 0.9(2) 2.9983(62) 3.0112(62) 0.9(2) 3.0052(76) 3.0144(76) 1.0(2)16 5.20 514 2.6931(46) 2.7099(47) 0.7(1) 2.6998(60) 2.7114(61) 0.8(2) 2.7052(78) 2.7135(79) 1.0(2)16 5.50 561 2.3487(34) 2.3633(35) 0.6(1) 2.3529(43) 2.3630(43) 0.7(1) 2.3556(50) 2.3628(51) 0.7(1)16 6.00 537 1.9470(37) 1.9591(37) 0.9(2) 1.9503(46) 1.9587(47) 1.1(2) 1.9522(57) 1.9582(57) 1.1(2)16 6.50 543 1.6695(31) 1.6799(32) 1.0(2) 1.6737(39) 1.6809(39) 1.0(2) 1.6771(47) 1.6822(48) 1.1(2)16 7.00 471 1.4536(20) 1.4626(20) 0.43(6) 1.4537(24) 1.4599(25) 0.47(6) 1.4525(29) 1.4569(29) 0.50(8)20 4.15 701 6.134(12) 6.150(12) 1.2(2) 6.118(15) 6.129(15) 1.3(3) 6.108(18) 6.116(18) 1.4(3)20 4.17 642 5.941(14) 5.956(14) 1.6(4) 5.931(19) 5.942(19) 2.0(5) 5.928(24) 5.935(24) 2.3(6)20 4.20 499 5.678(14) 5.692(14) 1.3(3) 5.669(17) 5.679(17) 1.5(4) 5.662(22) 5.669(22) 1.7(4)20 4.25 902 5.3314(93) 5.3452(94) 1.3(2) 5.334(12) 5.344(12) 1.4(3) 5.341(15) 5.348(15) 1.6(3)20 4.30 923 5.0325(89) 5.0455(89) 1.3(3) 5.036(11) 5.045(11) 1.4(3) 5.042(13) 5.048(13) 1.4(3)20 4.40 851 4.562(12) 4.573(12) 2.6(7) 4.570(15) 4.578(15) 2.9(7) 4.580(18) 4.586(18) 3.1(8)20 4.50 985 4.1592(70) 4.1699(70) 1.2(2) 4.1640(90) 4.1714(90) 1.4(3) 4.169(11) 4.174(11) 1.5(3)20 4.60 863 3.8684(97) 3.8784(97) 2.2(5) 3.877(13) 3.884(13) 2.5(6) 3.885(16) 3.890(16) 2.9(7)20 4.70 759 3.6016(65) 3.6109(65) 1.2(2) 3.6112(83) 3.6177(83) 1.3(3) 3.621(10) 3.626(10) 1.4(3)20 4.80 751 3.3777(75) 3.3864(75) 1.6(4) 3.3865(94) 3.3926(94) 1.8(4) 3.395(11) 3.399(11) 1.8(4)20 5.00 801 2.9961(44) 3.0038(44) 0.9(1) 3.0017(57) 3.0070(57) 1.0(2) 3.0066(75) 3.0104(75) 1.3(3)20 5.20 691 2.7139(40) 2.7209(40) 0.9(2) 2.7219(53) 2.7268(53) 1.0(2) 2.7295(70) 2.7329(70) 1.3(3)20 5.50 621 2.3736(51) 2.3797(51) 1.4(3) 2.3796(65) 2.3839(65) 1.6(4) 2.3849(81) 2.3879(81) 1.7(4)20 6.00 606 1.9601(40) 1.9652(41) 1.2(3) 1.9632(51) 1.9667(51) 1.3(3) 1.9653(65) 1.9678(65) 1.5(4)20 6.50 610 1.6738(30) 1.6782(30) 1.1(2) 1.6743(37) 1.6773(37) 1.1(2) 1.6735(44) 1.6757(44) 1.1(2)20 7.00 505 1.4654(27) 1.4692(27) 0.9(2) 1.4669(35) 1.4695(35) 1.0(2) 1.4673(44) 1.4692(44) 1.1(3)24 4.15 435 6.094(18) 6.102(18) 1.8(5) 6.091(22) 6.096(22) 2.0(6) 6.094(28) 6.098(28) 2.2(7)24 4.17 615 5.893(14) 5.900(14) 1.6(4) 5.893(18) 5.899(18) 1.8(5) 5.901(23) 5.905(23) 2.0(5)24 4.20 552 5.675(17) 5.683(17) 2.3(6) 5.679(22) 5.684(22) 2.5(7) 5.687(27) 5.691(27) 2.7(8)24 4.25 663 5.324(13) 5.330(13) 1.8(5) 5.334(17) 5.339(17) 2.0(5) 5.350(21) 5.353(21) 2.2(6)24 4.30 840 5.041(12) 5.047(12) 2.0(5) 5.055(15) 5.059(15) 2.3(5) 5.072(19) 5.075(19) 2.6(7)24 4.40 1056 4.5645(91) 4.5702(91) 2.0(4) 4.574(12) 4.578(12) 2.2(5) 4.584(15) 4.587(15) 2.5(6)24 4.50 932 4.1981(86) 4.2034(86) 1.8(4) 4.213(11) 4.217(11) 2.0(4) 4.230(14) 4.233(14) 2.2(5)24 4.60 910 3.8613(78) 3.8662(79) 1.7(4) 3.8653(98) 3.8687(98) 1.8(4) 3.868(12) 3.871(12) 2.0(5)24 4.70 786 3.6073(56) 3.6118(56) 1.0(2) 3.6139(71) 3.6171(71) 1.2(2) 3.6200(91) 3.6222(91) 1.4(3)24 4.80 642 3.3858(74) 3.3900(74) 1.5(4) 3.3888(93) 3.3917(93) 1.6(4) 3.389(11) 3.391(11) 1.7(4)24 5.00 556 3.0222(58) 3.0260(58) 1.1(2) 3.0307(75) 3.0334(75) 1.2(3) 3.039(10) 3.040(10) 1.6(4)24 5.20 610 2.7263(66) 2.7297(66) 1.8(5) 2.7335(88) 2.7359(88) 2.2(6) 2.740(12) 2.742(12) 2.6(8)24 5.50 502 2.3910(53) 2.3940(53) 1.3(3) 2.3988(68) 2.4008(68) 1.4(4) 2.4058(87) 2.4073(87) 1.6(4)24 6.00 483 1.9684(68) 1.9708(68) 2.9(9) 1.9723(87) 1.9740(87) 3(1) 1.976(11) 1.977(11) 3(1)24 6.50 502 1.6848(34) 1.6869(34) 1.1(3) 1.6892(43) 1.6906(43) 1.2(3) 1.6932(55) 1.6942(55) 1.5(4)24 7.00 502 1.4721(33) 1.4740(33) 1.3(3) 1.4741(41) 1.4754(41) 1.4(3) 1.4752(51) 1.4761(51) 1.5(4)28 4.15 254 6.075(40) 6.079(40) 4(2) 6.077(49) 6.080(49) 5(2) 6.088(58) 6.090(58) 4(2)28 4.17 372 5.889(23) 5.893(23) 2.3(8) 5.894(30) 5.897(30) 2.6(9) 5.907(38) 5.909(38) 3(1) c = 0 . c = 0 . c = 0 . L/a β N g c (nZS) g c (ZS) τ int g c (nZS) g c (ZS) τ int g c (nZS) g c (ZS) τ int
28 4.20 433 5.673(18) 5.677(18) 2.2(7) 5.685(24) 5.687(24) 2.5(8) 5.702(31) 5.703(31) 2.8(10)28 4.25 472 5.327(21) 5.331(21) 3(1) 5.340(27) 5.342(27) 3(1) 5.356(33) 5.357(33) 4(1)28 4.30 341 5.042(21) 5.045(21) 3(1) 5.060(26) 5.063(26) 3(1) 5.082(33) 5.083(33) 3(1)28 4.40 322 4.579(14) 4.582(14) 1.3(4) 4.590(17) 4.592(17) 1.4(5) 4.600(21) 4.602(21) 1.6(5)28 4.50 362 4.195(14) 4.198(14) 2.1(7) 4.205(18) 4.207(18) 2.3(8) 4.215(22) 4.217(22) 2.6(9)28 4.60 361 3.890(12) 3.893(12) 1.6(5) 3.903(16) 3.905(16) 2.1(7) 3.917(23) 3.918(23) 2.8(10)28 4.70 334 3.637(11) 3.639(11) 1.7(6) 3.650(14) 3.651(14) 2.0(7) 3.662(19) 3.663(19) 2.6(9)28 4.80 361 3.419(10) 3.421(10) 1.7(5) 3.434(13) 3.436(13) 1.9(6) 3.450(17) 3.451(17) 2.3(8)28 5.00 340 3.0554(84) 3.0575(84) 1.6(5) 3.072(11) 3.074(11) 1.8(6) 3.090(14) 3.091(14) 2.0(7)28 5.20 323 2.7563(77) 2.7582(77) 1.6(5) 2.7685(98) 2.7698(98) 1.8(6) 2.780(13) 2.781(13) 2.2(7)28 5.50 329 2.3881(99) 2.3898(99) 2.7(10) 2.395(13) 2.396(13) 3(1) 2.402(18) 2.403(18) 4(2)28 6.00 360 1.9800(80) 1.9813(80) 3(1) 1.984(11) 1.985(11) 3(1) 1.986(14) 1.987(14) 4(2)28 6.50 361 1.6940(70) 1.6951(70) 3(1) 1.6967(91) 1.6975(91) 3(1) 1.698(11) 1.699(11) 4(1)28 7.00 361 1.4809(42) 1.4819(42) 1.7(5) 1.4843(57) 1.4850(57) 2.1(7) 1.4872(72) 1.4877(72) 2.3(8)32 4.15 263 6.069(44) 6.071(44) 6(3) 6.076(54) 6.077(54) 6(3) 6.086(64) 6.087(64) 7(3)32 4.17 302 5.906(27) 5.908(27) 3(1) 5.920(36) 5.921(36) 3(1) 5.937(46) 5.939(46) 4(2)32 4.20 446 5.686(20) 5.689(20) 2.4(8) 5.708(25) 5.710(25) 2.6(9) 5.735(32) 5.736(32) 3(1)32 4.25 371 5.351(23) 5.353(23) 3(1) 5.372(31) 5.373(31) 4(2) 5.397(38) 5.398(38) 4(2)32 4.30 347 5.063(18) 5.065(18) 2.3(8) 5.081(25) 5.083(25) 3(1) 5.101(32) 5.102(32) 3(1)32 4.40 322 4.606(15) 4.608(15) 2.1(7) 4.631(19) 4.633(19) 2.3(8) 4.659(23) 4.660(23) 2.4(9)32 4.50 321 4.236(17) 4.237(17) 2.2(7) 4.253(22) 4.254(22) 2.4(9) 4.272(28) 4.273(28) 3(1)32 4.60 478 3.898(16) 3.899(16) 4(1) 3.910(21) 3.912(21) 4(1) 3.923(26) 3.924(26) 5(2)32 4.70 318 3.656(20) 3.657(20) 3(1) 3.672(26) 3.673(26) 4(2) 3.687(34) 3.688(34) 5(2)32 4.80 319 3.423(14) 3.424(14) 3(1) 3.433(17) 3.434(17) 3(1) 3.440(21) 3.441(21) 3(1)32 5.00 339 3.062(17) 3.064(17) 5(2) 3.080(22) 3.081(22) 5(2) 3.099(28) 3.099(29) 6(3)32 5.20 350 2.753(19) 2.754(19) 7(3) 2.763(24) 2.764(24) 7(3) 2.773(30) 2.773(30) 8(4)32 5.50 319 2.407(12) 2.408(12) 4(2) 2.416(15) 2.417(15) 4(2) 2.424(18) 2.424(18) 5(2)32 6.00 351 2.0029(58) 2.0038(58) 1.8(6) 2.0134(76) 2.0139(76) 2.2(7) 2.024(10) 2.024(10) 3(1)32 6.50 351 1.6998(52) 1.7005(52) 1.6(5) 1.7041(68) 1.7046(68) 1.9(6) 1.7078(87) 1.7081(87) 2.2(8)32 7.00 351 1.4875(61) 1.4881(61) 3(1) 1.4924(75) 1.4928(75) 3(1) 1.4972(89) 1.4975(89) 3(1) TABLE V. Renormalized couplings determined on additional ensembles generated with an alternative choice of L s to testeffects of residual chiral symmetry breaking. For easier comparison we also list and highlight in bold face the ensembles usedin our main analysis. Ensembles are specified by the spatial extent L/a , the bare gauge coupling β , and the value of the fifthdimension L s . As above we list N , the number of measurements, as well as the renormalized couplings g c for the analysis with(nZS) and without tree-level improvement (ZS) for the three renormalization schemes c = 0 . c = 0 . c = 0 . c = 0 . L/a β L s N g c (nZS) g c (ZS) τ int g c (nZS) g c (ZS) τ int g c (nZS) g c (ZS) τ int
10 4.15 12 501 6.151(14) 6.411(15) 0.7(1) 6.180(16) 6.355(16) 0.7(1) 6.185(17) 6.307(18) 0.7(1)
10 4.15 16 901 6.190(10) 6.452(11) 0.60(8) 6.228(12) 6.404(12) 0.61(8) 6.237(13) 6.360(14) 0.64(8)
10 4.20 12 941 5.6509(81) 5.8901(84) 0.58(7) 5.6845(93) 5.8451(96) 0.58(8) 5.700(11) 5.812(11) 0.59(8)
10 4.20 16 902 5.6467(84) 5.8858(88) 0.61(8) 5.6778(98) 5.838(10) 0.61(8) 5.690(11) 5.802(12) 0.63(8)
12 4.15 12 543 6.215(14) 6.337(14) 0.8(2) 6.207(17) 6.290(17) 0.9(2) 6.193(21) 6.252(21) 1.1(2)
12 4.15 16 1012 6.2402(96) 6.3627(98) 0.72(10) 6.230(11) 6.314(11) 0.71(10) 6.212(12) 6.271(12) 0.7(1)
12 4.20 12 909 5.6942(72) 5.8060(74) 0.58(8) 5.6993(85) 5.7757(86) 0.59(8) 5.697(10) 5.751(10) 0.61(8)
12 4.20 16 1385 5.7112(75) 5.8233(77) 0.9(1) 5.7173(92) 5.7940(93) 1.0(1) 5.716(11) 5.770(11) 1.0(2)
16 4.15 12 521 6.166(12) 6.205(12) 0.9(2) 6.152(15) 6.178(15) 1.0(2) 6.142(19) 6.161(19) 1.1(3)
16 4.15 24 657 6.219(12) 6.257(12) 1.0(2) 6.200(15) 6.227(15) 1.2(3) 6.187(19) 6.206(19) 1.3(3)
16 4.15 32 806 6.230(13) 6.269(13) 1.5(3) 6.208(16) 6.235(16) 1.7(4) 6.194(20) 6.213(20) 1.9(4)16 4.20 12 535 5.694(10) 5.730(10) 0.8(2) 5.689(13) 5.714(13) 0.9(2) 5.686(17) 5.704(17) 1.2(3)
16 4.20 16 922 5.679(10) 5.715(10) 1.3(2) 5.667(12) 5.691(12) 1.3(2) 5.654(15) 5.671(15) 1.4(3)
16 4.20 24 19 5.688(65) 5.724(66) 2(1) 5.683(81) 5.707(82) 2(1) 5.686(96) 5.703(96) 2(1)20 4.20 12 702 5.658(13) 5.673(13) 1.6(4) 5.649(17) 5.659(17) 1.9(5) 5.644(22) 5.651(22) 2.1(5)
20 4.20 16 499 5.678(14) 5.692(14) 1.3(3) 5.669(17) 5.679(17) 1.5(4) 5.662(22) 5.669(22) 1.7(4)
24 4.15 16 97 6.147(55) 6.155(55) 3(2) 6.154(66) 6.159(66) 3(2) 6.164(80) 6.168(80) 3(2)
24 4.15 24 435 6.094(18) 6.102(18) 1.8(5) 6.091(22) 6.096(22) 2.0(6) 6.094(28) 6.098(28) 2.2(7)
24 4.15 32 417 6.134(26) 6.142(27) 3(1) 6.135(32) 6.140(32) 3(1) 6.142(39) 6.145(39) 3(1)32 4.30 12 132 5.092(31) 5.094(31) 3(1) 5.125(40) 5.126(40) 3(1) 5.164(51) 5.165(51) 3(2)
32 4.30 16 347 5.063(18) 5.065(18) 2.3(8) 5.081(25) 5.083(25) 3(1) 5.101(32) 5.102(32) 3(1) Appendix C: Continuum limit extrapolations -0.4-0.20.00.20.40.6 c , s ( g c = . ) SC linear: =0.053(8) [45%] quadratic: =0.056(8) [58%]nSC linear: =0.044(6) [83%] quadratic: =0.041(5) [29%]nZS linear: =0.037(5) [93%] quadratic: =0.034(5) [23%]nWW linear: =0.033(5) [99%] quadratic: =0.036(5) [39%]ZS linear: =0.053(7) [44%] quadratic: =0.030(6) [33%]WW linear: =0.048(6) [50%] quadratic: =0.040(6) [87%]N f =12 c=0.250, s=2 -0.6-0.30.00.30.60.91.2 c , s ( g c = . ) SC linear: =0.09(1) [96%] quadratic: =0.12(1) [88%]nSC linear: =0.073(9) [72%] quadratic: =0.076(9) [26%]nZS linear: =0.054(9) [93%] quadratic: =0.056(8) [19%]nWW linear: =0.043(8) [97%] quadratic: =0.061(8) [42%]ZS linear: =0.082(9) [48%] quadratic: =0.044(8) [11%]WW linear: =0.069(9) [42%] quadratic: =0.063(8) [78%]N f =12 c=0.250, s=2 -0.4-0.20.00.20.40.6 c , s ( g c = . ) nSC linear: =0.06(1) [25%] quadratic: =0.08(1) [14%]nZS linear: =0.03(1) [29%] quadratic: =0.05(1) [7%]nWW linear: =0.01(1) [24%] quadratic: =0.06(1) [37%]ZS linear: =0.07(1) [44%] quadratic: =0.02(1) [3%]WW linear: =0.04(1) [55%] quadratic: =0.04(1) [66%]N f =12 c=0.250, s=2 (a/L) -0.6-0.4-0.20.00.2 c , s ( g c = . ) nZS linear: =-0.06(3) [12%] quadratic: =-0.01(3) [15%]nWW linear: =-0.09(3) [7%] quadratic: =0.00(3) [28%]ZS linear: =-0.02(3) [32%] quadratic: =-0.06(3) [8%]WW linear: =-0.06(2) [15%] quadratic: =-0.07(2) [13%]N f =12 c=0.250, s=2 FIG. 12. Details of the continuum extrapolation for our pre-ferred (n)ZS data in comparison to alternative determinationsbased on (n)SC and (n)WW for c = 0 . β c,s and corresponding p -values of the extrapolationare quoted in the legend. -0.4-0.20.00.20.40.6 c , s ( g c = . ) SC linear: =0.06(1) [38%] quadratic: =0.058(10) [51%]nSC linear: =0.049(7) [84%] quadratic: =0.047(7) [39%]nZS linear: =0.044(7) [92%] quadratic: =0.041(7) [32%]nWW linear: =0.041(7) [98%] quadratic: =0.041(7) [38%]ZS linear: =0.055(9) [50%] quadratic: =0.037(8) [40%]WW linear: =0.049(8) [60%] quadratic: =0.036(8) [54%]N f =12 c=0.275, s=2 -0.6-0.30.00.30.60.91.2 c , s ( g c = . ) SC linear: =0.10(1) [82%] quadratic: =0.11(1) [56%]nSC linear: =0.09(1) [51%] quadratic: =0.09(1) [17%]nZS linear: =0.07(1) [65%] quadratic: =0.07(1) [10%]nWW linear: =0.06(1) [72%] quadratic: =0.07(1) [14%]ZS linear: =0.09(1) [42%] quadratic: =0.06(1) [6%]WW linear: =0.08(1) [40%] quadratic: =0.06(1) [17%]N f =12 c=0.275, s=2 -0.4-0.20.00.20.40.6 c , s ( g c = . ) SC linear: =0.12(2) [17%]nSC linear: =0.09(2) [36%] quadratic: =0.10(1) [25%]nZS linear: =0.06(1) [46%] quadratic: =0.07(1) [11%]nWW linear: =0.05(1) [43%] quadratic: =0.07(1) [18%]ZS linear: =0.09(2) [44%] quadratic: =0.06(2) [7%]WW linear: =0.06(2) [51%] quadratic: =0.04(2) [12%]N f =12 c=0.275, s=2 (a/L) -0.6-0.4-0.20.00.2 c , s ( g c = . ) nZS linear: =-0.01(4) [10%] quadratic: =0.03(4) [26%]nWW linear: =-0.04(3) [8%] quadratic: =0.02(3) [27%]ZS linear: =0.01(4) [30%] quadratic: =0.00(4) [36%]WW linear: =-0.02(3) [15%] quadratic: =-0.04(3) [11%]N f =12 c=0.275, s=2 FIG. 13. Details of the continuum extrapolation for our pre-ferred (n)ZS data in comparison to alternative determinationsbased on (n)SC and (n)WW for c = 0 . β c,s and corresponding p -values of the extrapolationare quoted in the legend. -0.4-0.20.00.20.40.6 c , s ( g c = . ) SC linear: =0.06(1) [33%] quadratic: =0.06(1) [49%]nSC linear: =0.055(9) [87%] quadratic: =0.054(8) [52%]nZS linear: =0.051(9) [92%] quadratic: =0.049(8) [46%]nWW linear: =0.048(9) [97%] quadratic: =0.048(8) [46%]ZS linear: =0.06(1) [48%] quadratic: =0.05(1) [54%]WW linear: =0.05(1) [60%] quadratic: =0.041(10) [54%]N f =12 c=0.300, s=2 -0.6-0.30.00.30.60.91.2 c , s ( g c = . ) SC linear: =0.11(2) [61%] quadratic: =0.12(2) [39%]nSC linear: =0.10(1) [36%] quadratic: =0.10(1) [13%]nZS linear: =0.09(1) [44%] quadratic: =0.09(1) [8%]nWW linear: =0.08(1) [50%] quadratic: =0.08(1) [7%]ZS linear: =0.10(1) [30%] quadratic: =0.08(1) [7%]WW linear: =0.09(1) [30%] quadratic: =0.07(1) [7%]N f =12 c=0.300, s=2 -0.4-0.20.00.20.40.6 c , s ( g c = . ) SC linear: =0.13(3) [15%]nSC linear: =0.11(2) [54%] quadratic: =0.12(2) [44%]nZS linear: =0.09(2) [67%] quadratic: =0.10(2) [21%]nWW linear: =0.08(2) [67%] quadratic: =0.09(2) [18%]ZS linear: =0.11(2) [48%] quadratic: =0.09(2) [17%]WW linear: =0.09(2) [56%] quadratic: =0.07(2) [11%]N f =12 c=0.300, s=2 (a/L) -0.6-0.4-0.20.00.2 c , s ( g c = . ) nZS linear: =0.02(5) [9%] quadratic: =0.06(5) [31%]nWW linear: =-0.01(4) [8%] quadratic: =0.05(4) [29%]ZS linear: =0.03(5) [31%] quadratic: =0.05(5) [62%]WW linear: =0.01(4) [18%] quadratic: =0.00(3) [30%]N f =12 c=0.300, s=2 FIG. 14. 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