An analysis of systematic effects in finite size scaling studies using the gradient flow
DDESY 20-127IFIC/20-38
An analysis of systematic effects in finite size scaling studies using thegradient flow
Alessandro Nada a and Alberto Ramos ba John von Neumann Institute for Computing (NIC), DESY, Platanenallee 6, 15738 Zeuthen,Germany b Instituto de Física Corpuscular (IFIC), CSIC-Universitat de Valencia, 46071, Valencia, Spain
Abstract
We propose a new strategy for the determination of the step scaling function σ ( u ) in finite size scalingstudies using the Gradient Flow. In this approach the determination of σ ( u ) is broken in two pieces:a change of the flow time at fixed physical size, and a change of the size of the system at fixed flowtime. Using both perturbative arguments and a set of simulations in the pure gauge theory we showthat this approach leads to a better control over the continuum extrapolations. Following this newproposal we determine the running coupling at high energies in the pure gauge theory and re-examinethe determination of the Λ -parameter, with special care on the perturbative truncation uncertainties. Contents σ ( u ) . . . . . . . . . . . . . 42.2.1 Leading order perturbationtheory . . . . . . . . . . . . 62.2.2 Non-perturbative study . . 7 J and J J . . . . . . . . . 14 4.1.2 The case J . . . . . . . . . 144.2 The quantity √ t × Λ MS . . . . . 154.2.1 The scale µ ref . . . . . . . . 154.2.2 The extraction of Λ MS . . . 164.2.3 Results and discussion . . . 194.2.4 Scale uncertainties . . . . . 20 O ( a ) effects 23B Matching with the SF scheme 23References 24 a r X i v : . [ h e p - l a t ] J u l Introduction
In recent years the gradient flow [1, 2] (GF) has found many successful applications. In particular, whencombined with the ideas of finite-size scaling [3], it provides a powerful tool for the determination ofthe running coupling in asymptotically-free strongly-coupled gauge theories [4, 5, 6]. The applicationsinclude the determination of the strong coupling in QCD, and the study of near conformal systems(see [7] for a recent review).Coupling definitions based on the gradient flow have some properties that make them very attrac-tive. First, the relevant observables show a small variance. A modest numerical effort allows theirdetermination with a sub-percent precision. Second, the gradient flow coupling is given directly asan expectation value. Its determination only involves the numerical integration of the flow equations,something that in practice can be done with arbitrary precision without having to perform any fits,or taking any limit. This means that finite-size scaling studies using the GF only have one source ofsystematic effect: the continuum extrapolation.Nevertheless, it has been shown that these systematic effects are difficult to keep under control(see [8] for a review). It was soon noticed [9] that seemingly innocent extrapolations could cause largesystematic effects. Although the same work suggested to simply use larger flow times as a way to havea better control of the continuum extrapolation, this comes with an increase in the variance of theobservable. One of the strong points of GF studies, the high statistical precision, had to be sacrificed.Some efforts were made in order to understand the anatomy of these cutoff effects, first in a perturbativecontext to tree level [10], and later more systematically from an effective field theory point of view [11].It became clear that the integration of the flow equations and the evaluation of the relevant observablesat positive flow time could be performed in a way that no O ( a ) effects are produced: it is enough touse classically improved discretizations [11]. Still, the use of these improved observables did not reducesubstantially the observed scaling violations (see for example [12]). Unavoidably large lattices have tobe simulated in GF studies, and the continuum extrapolation will remain the main source of concern.In this work we study the main sources of systematic effects in finite-size scaling studies using theGF. We will argue that changes in the flow time t are responsible for the scaling violations. On thecontrary, when different finite volume couplings are determined at the same value of the flow time t , the scaling violations are very small. This observation will be supported by a non-perturbativestudy. Moreover it will allow us to propose a new strategy for the determination of the step scalingfunction by breaking it up into two pieces: first a change in the flow time, without any change in thevolume, second a change in the volume without any change in the flow time. Since the first step can beperformed without having to double the lattices , the continuum extrapolation can be performed muchmore accurately. We will discuss in detail the advantages of this approach. Finally we will apply thisalternative strategy to the case of pure gauge SU(3) . Using the same datasets of [13], we will revisitthe most crucial part of this work: the running at high energies and the matching with the asymptoticperturbative regime.We also include an analysis of the variance of flow observables, allowing us to predict the dependenceof the statistical uncertainties with the flow time t . The gradient flow provides a set of renormalized observables with small variance that are easy tocompute via numerical simulations. The key idea consists in adding an extra time coordinate to thegauge field, the flow time t . The dependence of the gauge field B µ ( t, x ) on the flow time is given by2 / 25 √ t Figure 1: The GF coupling in finite volume ¯ g ( µ ) is measured by computing the action density of the flowfield B µ ( t, x ) smeared over a distance √ t = µ − (see eq. (2.3)). The renormalization scale µ and the sizeof the system L are linked by the relation √ t = cL (with c a constant). the first order diffusion equation ∂ t B µ ( t, x ) = D ν G νµ ( t, x ) ; B µ (0 , x ) = A µ ( x ) . (2.1) G µν ( t, x ) = D µ B ν ( t, x ) − D ν B µ ( t, x ) ; D µ = ∂ µ + [ B µ , · ] . (2.2)The flow time has dimensions of length squared, and therefore introduces a scale into the problem(see figure 1). Gauge-invariant composite operators defined at positive flow time are automaticallyrenormalized [14]. In particular, a renormalized coupling at scale µ = 1 / √ t can be defined by usingthe action density [2] ¯ g ( µ ) (cid:12)(cid:12)(cid:12) µ =1 / √ t ∝ t (cid:104) E ( t, x ) (cid:105) . (cid:0) E ( t, x ) = G aµν ( t, x ) G aµν ( t, x ) (cid:1) (2.3)In finite volume schemes where the invariance under Euclidean time translations is broken, like theSchrödinger Functional (SF) or open-SF boundary conditions, the coupling is only measured at thetime-slice x = T / and usually only certain components of the field strength are used E m ( t, x ) = G aij ( t, x ) G aij ( t, x ) (cid:12)(cid:12)(cid:12) x = T/ , (2.4a) E e ( t, x ) = G a j ( t, x ) G a j ( t, x ) (cid:12)(cid:12)(cid:12) x = T/ . (2.4b)The coupling defined using E = E m is usually referred as magnetic and the one using E = E e as electric .Most applications of the gradient flow until today derive from these coupling definitions. In infinitevolume the scale µ = 1 / √ t at which the coupling ¯ g ( µ ) takes some pre-defined value is used todefine the reference scale t . In the context of finite-size scaling, the renormalization scale µ = 1 / √ t is linked with the linear size of the system µ − = √ t = cL , (2.5)3 / 25nd therefore the coupling “runs” with the size L (see fig 1). The constant c defines a scheme by fixingthe ratio of the length of the system and the flow time scale √ t . The full definition of the GF couplingin finite volume reads ¯ g c ( µ ) = N ( c ) t (cid:104) E ( t, x ) (cid:105) (cid:12)(cid:12)(cid:12) µ − = √ t = cL . (2.6)The constant N ( c ) has been determined for different choices of boundary conditions [4, 5, 6, 15].When computing the gradient flow coupling on the lattice there are several choices to make, beyondthe action chosen for the simulation. First, the flow equation (2.1) has to be translated to the lattice.Second, there are many valid discretizations of the energy density E ( t, x ) . It is well understood thatusing the Zeuthen flow to integrate the flow equations and using any classically improved discretizationof the action density E ( t, x ) guarantees that no further O ( a ) effects are generated. The only remaining O ( a ) effects are those of the choice of action for the simulation and an additional counterterm, onlyaffecting flow quantities [11].On the lattice, where we can work only with dimensionless quantities, the finite volume gradientflow coupling (equation (2.6)) can be measured by determining ¯ g ( µ ) at the flow time (in lattice units) √ t/a = cL/a . It is also common to use a lattice version of the normalization factor N ( c, a/ √ t ) instead of the N ( c ) of eq. (2.6), in order to ensure that the leading order perturbative relation ¯ g c ( µ ) µ →∞ ∼ ¯ g ( µ ) + O (¯ g ) , (2.7)is exact for any lattice size.The key character in finite-size scaling studies is the step scaling function σ ( u ) = ¯ g c ( µ/ (cid:12)(cid:12)(cid:12) ¯ g c ( µ )= u . (2.8)It measures how much the coupling changes under a variation in the renormalization scale of a factortwo . Its determination in lattice simulations is performed via a matching of the bare parameters. At afixed value of the bare coupling g (and therefore fixed value of the lattice spacing a ) , one determinesthe GF coupling on a lattice of size L/a (resulting in ¯ g ( µ ) ), and on a lattice L/a (resulting in ¯ g ( µ/ ). This allows the determination of a lattice approximation of the step scaling function: Σ( u, a/ √ t ) = ¯ g c ( µ/ (cid:12)(cid:12)(cid:12) ¯ g c ( µ )= u a/ √ t → ∼ σ ( u ) . (2.9)In the rest of this section we will be concerned with the continuum extrapolation of Σ( u, a/ √ t ) ; beforethat however, let us insist on two points: • The direct determination of Σ , as suggested above, requires the determination of the GF couplings at two different renormalization scales µ and µ/ , on lattices of two different sizes L/a and L/a . • It is clear that once c (eq. (2.5)) is fixed, taking a/L → is equivalent to taking a/ √ t → , sothe usual notation that uses a/L as the variable to parametrize cutoff effects is fully justified.Nevertheless, it is a/ √ t = aµ the natural variable to measures the size of cutoff effects. Notethat for the typical choices of c ∈ [0 . − . we have scale √ t < L .With this choice of the renormalization scale, it is clear that, at fixed L/a , larger values of c would lead to smaller cutoff effects. This notation will also be convenient for the discussion thatfollows. σ ( u ) As already pointed out, the determination of the lattice step scaling function involves two steps: achange in the renormalization scale by a factor two, and change in the lattice size by the same factor. Other scale factors, like 3/2, are less common, but obviously possible. The discussion in this paper also applies toany other choice. One also needs to specify the value of the quark masses. Simulations on a finite volume make it possible to directlysimulate at m = 0 , and this is typically the additional condition that completes the line of constant physics. J J Figure 2: The determination of the step scaling function σ involves a change in the renormalization scaleand in the size of the system of a factor two. These two steps do not need to be performed at the simetime. The figure shows that σ can be determined as the composition of the function J that changes therenormalization scale µ → µ/ at fixed size L , with the function J that changes the size L → L at fixedrenormalization scale µ . In previous works these two changes have been performed at the same time in a single step, butconceptually there is no need to do so.Figure 2 shows that the value of the step scaling function σ ( u ) can be determined by the compositionof two functions. First we have J ( u ) = ¯ g c ( µ/ (cid:12)(cid:12)(cid:12) ¯ g c ( µ )= u . (2.10)This function changes the renormalization scale by a factor two µ → µ/ at fixed physical size L .Second, we need to determine J ( u ) = ¯ g c ( µ/ (cid:12)(cid:12)(cid:12) ¯ g c ( µ/ u . (2.11)This function changes the lattice size L → L keeping constant the renormalization scale µ − = √ t .The relation σ = J ◦ J , (2.12)is now exact, and provides an alternative method to determine σ .Our main assumption is that large scaling violations come with changes in the renormalizationscale . We will later provide evidence that this is the case, but for the moment let us discuss whythis opens up the possibility to improve the quality of the continuum extrapolations. Let us start byexplaining how these functions are computed in practice: • The determination of J involves measuring how much the coupling changes when the renor-malization scale is varied as µ → µ/ at constant physical size L . This is simply achieved bymeasuring on a lattice simulation the value of the GF coupling at two different flow times (i.e. t → t , see eq. (2.5)). Crucially, this determination does not require to double the lattice size,allowing precise results without the need of very large values of L/a . • The determination of J requires to change the physical size L without varying the renormaliza-tion scale. In practice one fixes the bare coupling g at a given value, and then measures the GFcoupling on a L/a lattice at flow time √ t/a = cL/a and on a L/a lattice at the same t . Thisstep requires to change the lattice size, but since the renormalization scale remains the same, oneexpects reduced cutoff effects. In some sense the determination of J corresponds to measuringthe finite volume effects in ¯ g c ( µ ) (see figure 2).In the rest of this section we will provide evidence of these statements, but before that let us furthercomment on two points: 5 / 25 The determination of the functions J and J can be done on the lattice just by applyingthe definitions equation (2.10) and (2.11). Note however that in this case the functions willcarry a dependence on the cutoff. We will label these functions ˆ J ( u, a/ √ t ) and ˆ J ( u, a/ √ t ) respectively. • In this work all numerical results make use of the same discretizations used in [13]. We encouragethe reader to consult this reference for more details. Here it is enough to say that we define theGF coupling with SF boundary conditions and that our preferred setup, based on theoreticalexpectations, uses the Zeuthen flow and an improved definition of E ( t, x ) . This preferred setupwill also be compared with the more common combination Wilson flow/Clover observable.Moreover, we will focus in the rest of the text on the magnetic definition of the GF coupling (seeequations (2.4)) . Of course, our discussion is general, and does not depend on these particularchoices. As a first look at the proposal we examine the leading order perturbative relation. We use the contin-uum norm N ( c ) in the evaluation of the finite volume couplings ¯ g c and examine, to leading order inperturbation theory, the quantities ˆ C ( a/ √ t ) = Σ( u, a/ √ t ) u , ˆ C ( a/ √ t ) = ˆ J ( u, a/ √ t ) u , ˆ C ( a/ √ t ) = ˆ J ( u, a/ √ t ) u . (2.13)Note that since we are working at leading order in ¯ g , and thanks to the normalization by the constantfactor u , all these quantities are one in the continuum.In this example we will examine a typical case where we consider data for L/a = 8 , , , , , , , , , , . (2.14)We will use c = 0 . (see eq. (2.5)). Let us note a few basic points: • The determination of ˆ C ( a/ √ t ) and ˆ C ( a/ √ t ) requires to double the lattice size. This meansthat with our data, lattice estimates for these functions will be available only for a factor 3change from the finest to the coarsest lattice spacings: → , → , → , → , → , → , → . • On the other hand, the determination of ˆ C ( a/ √ t ) only requires the measurement of the GFcoupling at different values of the flow time t . This can be done on all lattices, and our datasetprovides a factor change from the coarsest ( L/a = 8 ) to the finest (
L/a = 48 ) lattice spacing.Figure 3 shows the perturbative results. As the reader can see, the cutoff effects in ˆ C ( a/ √ t ) arevery similar to those in ˆ C ( a/ √ t ) . This can be understood in a simple way, since both these functionsinvolve a change in the renormalization scale by a factor two. The main difference between both casesis that ˆ C ( a/ √ t ) can be determined using lattice spacings that are a factor two smaller, since itsdetermination does not require any change in the lattice size. The determination of ˆ C ( a/ √ t ) doesnot involve any change in the renormalization scale, and it shows cutoff effects that are one order ofmagnitude smaller than either ˆ C ( a/ √ t ) or ˆ C ( a/ √ t ) .In the next section we will show that indeed these properties hold non-perturbatively, and thatthey are not a coincidence of leading order perturbation theory. Reference [13] showed a perfect agreement between the electric and magnetic schemes. Focusing on one choice keepsthe discussion and the notation more simple. . . . . . .
951 0 0 .
05 0 . .
15 0 . .
25 0 . .
35 0 . a/ √ t ) ˆ C ( a/L ) ˆ C ( a/L ) ˆ C ( a/L ) Figure 3: Cutoff effects in the usual step scaling function ( ˆ C ( a/ √ t ) ), compared to those in J (see ˆ C ( a/ √ t ) ) and J (see ˆ C ( a/ √ t ) ). See text for more details. Here we show the case of the Zeuthenflow/improved observable with plaquette gauge action (i.e. the same setup that will be used in our non-perturbative study). In this section we will describe the non-perturbative results used in this study, first to support theclaim that the numerical determination of ˆ J has very small scaling violations, due to the fact thatthe renormalization scale is not changed (i.e. the determination of ˆ J amounts to measuring finitevolume effects in the coupling). Then, we want to show that the determination of J can be performedaccurately even at values of c that are too small to allow for a conservative estimate of the step scalingfunction.All the analysis have been performed using two different analysis codes: one [16] based on the Γ -method [16, 17, 18, 19], and the other using a jackknife resampling technique. Description of the data set
For our non-perturbative study we are going to use exactly the same dataset of [13]. This setupincludes simulations of the pure gauge theory with the Wilson plaquette action on a lattice of size L and lattice spacing a . We have several resolutions, L/a = 8 , , , , , , , , at a large range oflattice spacings a and with Schrödinger Functional (SF) boundary conditions. The setup is the sameas the one used for the perturbative study in section 2.2.1.We have measurements of the GF coupling at values of c = 0 . , . , . with two different discretiza-tions: the usual Wilson flow/Clover observable and the Zeuthen flow/improved observable (more detailscan be found in [13]). Our target will be to determine the running non-perturbatively in the scheme de-fined by c = 0 . by computing the associated step scaling function σ ( u ) . Note that in reference [13] thevalue c = 0 . was used because the large scaling violations at c = 0 . did not allow for a determinationof σ ( u ) .We will revisit this attempt at a direct determination of σ ( u ) here. Moreover, together with thedata at c = 0 . , we will be able to determine both J , J , and compare their composition with the7 / 25 . . . . . .
03 1 1 . . J ( u ) − u u Continuum
L/a = 8
L/a = 10
L/a = 12
L/a = 16
L/a = 24 (a) Results for the combination of eq. (2.15) and contin-uum extrapolation from global fit. . .
97 0 0 .
05 0 . .
15 0 . .
25 0 . .
35 0 . ˆ J ( u = . , a / √ t ) / u ( a/ √ t ) Zeuthen FlowWilson Flow0 . .
97 0 0 .
05 0 . .
15 0 . .
25 0 . .
35 0 . (b) Results for the ˆ J ( u, a/ √ t ) /u ratio for u = 1 . fordifferent discretizations. The continuum extrapolation isalso included. Figure 4: Results for the ˆ J ( u, a/ √ t ) function and continuum extrapolation. direct determination of σ .Finally, the data with c = 0 . will be used in section 4 to compare the results of [13] (where the Λ parameter is obtained by using a direct determination of the step scaling function with c = 0 . ) withour new strategy.We will focus our investigations on the high energy regime, where ¯ g ∼ − . Note that this regionshowed significant scaling violations, and in fact turns out to be the most delicate part of the analysisin the extraction of Λ (see [13] for more details). Scaling violations in J Let us start by investigating the scaling violations of J . It is convenient to study the combination J ( u, a/ √ t ) − u = f ( u, a/ √ t ) . (2.15)The continuum limit of the right hand side, f ( u, , has an asymptotic expansion (in perturbationtheory) as a polynomial in u , starting with a constant term. Note that our data set allows to determine ˆ J ( u, a/ √ t ) for a factor three change in a/ √ t .The numerical raw data for ˆ J ( u, a/ √ t ) is shown in figure 4a. We also include in the plot thecontinuum extrapolation. At this point we defer the discussion on how this continuum curve is deter-mined to section 4, and focus on the key element: the non-perturbative data for ˆ J show very smallscaling violations for all values of the coupling under study. The continuum curve is at most twostandard deviations away from the coarser lattice data ( √ t/a ≈ . ), and the two finest lattices with √ t/a ≈ . , . show no significant deviation from the continuum value.One can look in more detail at the previous statement by interpolating the data with different L/a to a common value of u , and then look at the continuum extrapolation of ˆ J . We choose the value u = 1 . , where we have several points at each L/a , and therefore the necessary interpolations can beperformed in the safest conditions available. Figure 4b shows that the Zeuthen flow data shows nosignificant scaling violations in the whole range of lattice spacing. The Wilson flow data show somescaling violations, but they are rather mild, with the finest lattice being almost compatible with thecontinuum value. Extrapolations of the data with both discretizations are in full agreement with eachother.
Scaling violations in J . − . − . − . − . − . − . . . J ( u ) − u u Continuum
L/a = 8
L/a = 10
L/a = 12
L/a = 16
L/a = 20
L/a = 24
L/a = 32
L/a = 48 (a) Results for the combination of eq. (2.16) and contin-uum extrapolation from global fit. . . . . .
24 0 0 .
05 0 . .
15 0 . .
25 0 . .
35 0 . ˆ J ( u = . , a / √ t ) / u ( a/ √ t ) Zeuthen FlowWilson Flow1 . . . . .
24 0 0 .
05 0 . .
15 0 . .
25 0 . .
35 0 . (b) Results for the ˆ J ( u, a/ √ t ) /u ratio for u = 1 . fordifferent discretizations. The continuum extrapolation isalso included. Figure 5: Results for the ˆ J ( u, a/ √ t ) function and continuum extrapolation. Once more, it is convenient to study the quantity J ( u, a/ √ t ) − u = f ( u, a/ √ t ) . (2.16)The crucial difference with the previous case is that the determination of J involves a change inrenormalization scale µ → µ/ , so we expect significant scaling violations. On the other hand, itsdetermination does not require to double the lattice sizes. In practice we have a range in latticespacing that spans a factor 2 further.Figure 5a shows a comparison of the raw data with the continuum curve (see section 4 for adiscussion on its determination). In contrast with the case of ˆ J , we observe significant scaling vi-olations, confirming our hypothesis that such violations are mainly a result of changes in the renor-malization scale. Moreover, they show a complicated functional form: the three coarser lattices with √ t/a = 1 . , . , . do not show a monotonous pattern. The data is several standard deviations awayfrom the continuum curve. Figure 5b shows again the continuum extrapolation of J ( u ) at the fixedvalue u = 1 . . The plot confirms that scaling violations are significant, with the different discretizationsbased on the Wilson/Zeuthen flow showing differences of several standard deviations for √ t/a < . .Still, one can obtain an accurate extrapolation of J . In order to do so, the large range of latticespacings available to us, from L/a = 8 to L/a = 48 , is crucial. This is of course possible only becausethe determination of J does not require to double the lattice size . Figure 5a shows that the two finestlattices are in agreement with the continuum curve. Note however that these are very fine lattices with √ t/a ≈ . , . . A comparison with a direct determination of σ . ( u ) Finally, let us compare our new strategy with the direct determination of the step scaling function σ c =0 . . As an illustration we choose the value u = 1 . as in the previous cases. Figure 6 shows thatthe data for the direct determination of σ c =0 . ( u ) is more precise, but it also displays large scalingviolations, similar to the case of J . However, since the lattice size has to be doubled, we lack the dataat √ t/a = 9 . , . and the finest lattice has √ t/a = 4 . , a factor two smaller. The final uncertaintyin the continuum value of σ c =0 . ( u ) is very difficult to assess due to the missing data at finer latticespacing. This findings are consistent with earlier works: reference [12] already showed that controllingthe continuum extrapolation is difficult if one relies on data with √ t/a ≤ . . Here we see that9 / 25 . . . . . .
19 0 0 .
05 0 . .
15 0 . .
25 0 . .
35 0 . Σ c = . ( u = . , a / √ t ) / u ( a/ √ t ) Zeuthen FlowWilson Flow1 . . . . . .
19 0 0 .
05 0 . .
15 0 . .
25 0 . .
35 0 . Figure 6: Results for the Σ c =0 . ( u, a/ √ t ) /u ratio for u = 1 . . The continuum extrapolation for the twodifferent discretizations of the gradient flow is included. In the case of the Wilson flow, only the latticeswith √ t/a ≥ . are used for the extrapolation, while for the Zeuthen flow both those with √ t/a ≥ . and with √ t/a ≥ . .
10 / 25 . . . . . . . . . . . . . V a r [ ¯ g ] F ¯ g ¯ g L/a = 12
L/a = 16
L/a = 20
L/a = 24
L/a = 32
L/a = 48
Figure 7: Results for the quantity of eq. (3.4) for three different datasets and six different lattice spacings.Orange symbols are for the “magnetic” definition of the GF coupling; black symbols are for the average of“magnetic” and “electric” definitions of the GF coupling; the blue symbols are from the dataset of ref. [30]. there is poor agreement between the continuum value obtained by the two different discretizationsof the gradient flow with respect to the excellent agreement found for ˆ J and ˆ J in figures 5 and 4respectively. Despite the higher statistical accuracy, the large systematic effects overshadow the betterstatistical precision. It has been argued that a simple way to improve the scaling properties of GF couplings consists in usinglarge values of c (see for example the discussion in [20]). Unfortunately, it is well known that this comesat the cost of increased statistical uncertainties. In this section we want to make this last statementmore precise. We will present a simple model for the understanding of the scaling of the statisticaluncertainties of the GF coupling and then we will show how the results of numerical simulations agreewith this naive approach. L √ t L Figure 8: A simple model to explain thescaling of the variance of the GF coupling.
Let us first focus our discussion on schemes that fullypreserve the translational invariance, like the case of peri-odic [4] or twisted [21] boundary conditions. The gradientflow smears the original gauge field A µ ( x ) over a distance d ∼ √ t . Due to the invariance under translations, eachfour dimensional ball of radius √ t provides an estimate ofthe quantity (cid:104) E ( x, t ) (cid:105) (see figure 8). Under this assumptionthe volume average on a lattice L × L × L × L will makethe variance of the observable (cid:104) E ( x, t ) (cid:105) proportional to F = (cid:89) µ =0 √ tL µ . (3.1)Note that in the common situation of an L lattice withthe same length in all directions one has F = c (see equa-tion (2.5)). This gives a quantitative explanation to the factthat the statistical uncertainties are large at large values of c . 11 / 25n schemes where the invariance under translations isbroken in the time direction, like Schrödinger Functional(SF) [5] or open-SF [15] boundary conditions on a box ofsizes L × L × L × L , a similar argument applies, except that in these cases the coupling is onlymeasured at a single time-slice x = L / . Therefore in this case we expect a factor F = (cid:89) µ =1 √ tL µ , (3.2)e.g. on a symmetric lattice F = c . When do we expect this model to break down? For the volumeaverage argument to make sense, the region that is smeared by the flow must be much smaller thanthe size of the lattice, so we require √ tL µ (cid:28) / . (3.3)Note that for the case of an L lattice this condition just means c (cid:28) . . The typical values used inthe literature are c = 0 . − . , so we can only expect the scaling of the variance to be approximate.In order to see how good this approximation is, it is useful to have a look at the quantity Var[¯ g ] F ¯ g ≈ K (¯ g ) . (3.4)If our hypothesis is correct, we expect this combination to be independent on the lattice size and onthe values of √ t/L µ . Figure 7 shows this quantity in three data sets.First, in orange we plot the usual magnetic coupling definition that we have been using for ournon-perturbative study (section 2.2.2). This data includes values of c = 0 . , . , . , . . . , . for lattices of sizes L/a = 12 , , , , , at several values of the bare coupling β = 6 /g . Notethat, despite the fact that the lattice size changes by a factor of four, and that the values of c changeby a factor two, the combination in equation (3.4) shows a very mild variation in all the range ofcouplings ¯ g = 1 − . The plot shows some variation, but to a reasonably good approximation we cansay that Q (¯ g ) ≈ . . An even better description of the data can be obtained by using a simple linearapproximation.Second, in black, we have another definition of the coupling in the same datasets (in particularthe lattice sizes and values of c are the same as in the previous case). The data corresponds to thecoupling definition based on the space-time components of the Energy density (i.e. the average betweenthe “magnetic” and the “electric” components). Despite the high correlation between the electric andmagnetic energy densities, the average shows a significant smaller variance. In this the combination ofequation (3.4) can also be reasonably well described by a linear function.Finally, in blue, we have data with twisted boundary condition on an asymmetric lattice [30]. Inthis case the simulations are done on volumes L × ( L/ (see [22] for the theoretical motivationbehind this particular geometry). We use data with √ t/L = 0 . , . , . . . , . and lattice sizes L/a = 12 , , . Note that in this particular case the condition equation (3.3) is flagrantly violated inthe two short directions, since √ t/L = 0 . − . . This may explain why this dataset shows a muchlarger dispersion. For this case the combination in equation (3.4) shows a larger dependence on detailslike the particular choices of L/a and √ t and not only on ¯ g . Still, the variation is not large takinginto account the disparity of scales (varying by several factors) involved in the data (note that naivelythe variance changes by more than two orders of magnitude).It is also worth mentioning that the variance at weak coupling is very similar for the three datasets,differing by at most a factor three. Together with the observation that being a flow observable, thequantity in equation (3.4) has a well defined continuum limit, we can conclude that the function inequation (3.4) is universal . Details like the choice of boundary conditions or the choices of discretiza-tions induce relatively small scaling violations, especially at the weakest couplings. 12 / 25 The high energy regime of Yang-Mills revisited
As a further test on our proposal, we will re-examine the high energy regime of Yang-Mills. Let usfirst recall the relevant points of the work [13]. • The determination of the Λ MS parameter in units of √ t is divided in two fundamental pieces.First, a high energy part where contact with perturbation theory is made. This results in adetermination of Λ MS /µ ref , with µ ref being defined by ¯ g c =0 . ( µ ref ) = 0 . π . Second, a low energypart where the dimensionless ratio µ ref × √ t is determined. • Most of the error in the dimensionless ratio Λ MS × √ t = Λ MS µ ref × ( µ ref √ t ) , (4.1)comes from the first piece (i.e. the high energy part). The total uncertainty in Λ MS × √ t is1.57%, while the uncertainty in Λ MS /µ ref is already 1.37%. • The result for Λ MS ×√ t = 0 . shows a significant discrepancy with other determinations:in particular the very precise determination of FlowQCD [23] Λ MS ×√ t = 0 . lies about3 sigma away from the value of [13].Given that the pure gauge determination of Λ MS has to face the very same challenges as thedetermination of the strong coupling α s ( M Z ) in QCD, we think that revising the crucial part of thework [13] with the new method proposed in this work is fully justified. We recall that our dataset isexactly the same as the one used in [13] (see section 2.2.2). J and J In order to obtain the functions J and J in the continuum, the best strategy consists in combiningthe continuum extrapolation with a parametrization of the function in the continuum. In particularwe are going to use the parametrization J i ( u, a/ √ t ) − u = n c (cid:88) n =0 c ( i ) n u n + (cid:18) a √ t (cid:19) n ρ (cid:88) n =0 ρ ( i ) n u n . (4.2)Note that the coefficients c ( i ) n parametrize the continuum function J i ( u ) , while the coefficients ρ ( i ) n parametrize the O ( a ) cutoff effects in the function ˆ J i . There are several assumptions hidden in thisparametrization. First we assume that the continuum function / ˆ J i ( u, − /u can be well describedby a polynomial. This is certainly the case in perturbation theory to all orders , and we expect thatthe non-perturbative functions can be well described by a polynomial ansatz.A more delicate assumption is that in our functional form of equation (4.2) all scaling violationsare quadratic (i.e. a / (8 t ) ). First, due to the breaking of translational invariance in the SchrödingerFunctional, we expect cutoff effects linear in the lattice spacing. They are expected to be small, dueto the localization of the GF coupling at the timeslice x = T / . Moreover the extrapolations insection 2.2.2 have completely ignored these effects, and our data in fact seem to scale like O ( a ) afterdropping the coarser lattices. But due to the high precision of our data, these O ( a ) effects cannotbe completely ignored, especially if we take into account the fact that our strategy uses data at large In fact the first two coefficients c ( i )0 , can be computed with the help of the k i coefficients calculated in [24]: c (1)0 = 0 . , c (1)1 = − . .c (2)0 = − . , c (2)1 = 0 . .
13 / 25
Parameters 1.125 1.250 1.500 1.750 2.000
L/a ≥ n c = 3 , n ρ = 2 n c = 4 , n ρ = 2 L/a ≥ n c = 3 , n ρ = 2 n c = 4 , n ρ = 2 L/a ≥ n c = , n ρ = n c = 3 , n ρ = 3 n c = 3 , n ρ = 4 n c = 4 , n ρ = 2 Table 1: Values of the continuum function J ( u ) for different fit parametrizations and cuts (see equa-tion (4.2)). In bold we show our preferred fit. See text for more details. values of √ t/T = 0 . , where these effects are expected to be larger than in [13], where √ t/T = 0 . was used. For this reason we include a generous estimate of these linear effects in the error. The detailsare explained in Appendix A.Our data set has also higher order cutoff effects, of the form a n for n > , and logarithmic correctionsas well [25]. The effect of these terms in our extrapolations will be estimated by changing the cutsused to fit the coefficients c ( i ) n , ρ ( i ) n . J In the exploratory study in section 2.2.2 the Zeuthen flow/improved observable discretization alreadydisplayed better scaling properties, but still we had to discard all lattices with
L/a < . Since theWilson flow/Clover combination would require even more stringent cuts, we will only use the improvedsetup to quote final results.This is confirmed by looking at table 1, where the values in the continuum of J ( u, are shownat a few representative values of u . As the reader can see, the effect of varying the number of fitparameters ( n c and n ρ in equation (4.2)) is negligible. On the other hand, the cut in L/a has asmall effect on the extrapolations. If lattices with
L/a = 12 are included, the continuum value of J seems to be systematically higher, but still compatible within errors. Also the statistical errors aresmaller for these analysis. A conservative approach consists in just taking a fit with L/a ≥ (i.e. a/ √ t < / ), so that the continuum value has a larger uncertainty. Note that since the computationof J does not require to double the lattice sizes, even with this stringent cut our dataset still offersmore than a factor two in lattice spacing. Among these fits there is very little difference betweendifferent parametrizations. Moreover the fit quality is very similar in all cases. All in all we just chooseone of these fits ( n c = 3 , n ρ = 2 , bold in table 1) as our final result. J The computation of J requires to double the lattice sizes, and then our datasets offers only halfthe lever arm in lattice spacing for the continuum extrapolations. Our hypothesis is that the scalingviolations are small for J because its determination does not involve a change in renormalizationscale. Our preliminary investigation of section 2.2.2 has also confirmed this hypothesis. Table 2shows that this is indeed the case. Even including the coarser lattices with L/a = 8 (corresponding to a/ √ t = 1 / . ), the results are in agreement within errors. It is clear that the choice of parametrizationhas very little effect. We just settle for one particular fit with L/a ≥ (represented in bold in table 2)that we will use for any further analysis. 14 / 25 Parameters 1.125 1.250 1.500 1.750 2.250All
L/a n c = 3 , n ρ = 2 n c = 4 , n ρ = 2 L/a ≥ n c = 3 , n ρ = 2 n c = 4 , n ρ = 2 L/a ≥ n c = , n ρ = n c = 3 , n ρ = 3 n c = 3 , n ρ = 4 n c = 4 , n ρ = 2 Table 2: Values of the continuum function J ( u ) for different fit parametrizations and cuts (see equa-tion (4.2)). In bold we show our preferred fit. See text for more details. √ t × Λ MS µ ref As we have already mentioned, the original work [13] determined the dimensionless combination √ t × Λ MS as the product of two factors. First the low energy factor √ t µ ref = 7 . . , (4.3)that has a very small uncertainty . The other factor Λ MS /µ ref was much more delicate to determine. Itis precisely this last quantity that we want to determine once more with our new strategy. A first stepconsists in dealing with the factor µ ref . This was defined in the scheme with c = 0 . by the condition ¯ g c =0 . ( µ ref ) = 4 π ≈ . . . . . (4.4)Since our new strategy provides the step scaling function σ ( u ) for c = 0 . , we must first determine thevalue of our coupling ¯ g c =0 . ( µ ref ) . The procedure is completely analogous to the determination of J .We first define ˆ J ( u, a/ √ t ) = ¯ g c =0 . (2 µ/ (cid:12)(cid:12)(cid:12) ¯ g c =0 . ( µ )= u . (4.5)We choose to fit our data to the model J ( u, a/ √ t ) − u = n c (cid:88) n =0 c (3) n u n + (cid:18) a √ t (cid:19) n ρ (cid:88) n =0 ρ ( i ) n u u . (4.6)The same considerations discussed in section 4.1.1 apply to the determination, in the continuum, ofthe relation between ¯ g c =0 . ( µ ) and ¯ g c =0 . ( µ ) . In this case, however, we expect the scaling violations tobe smaller, since the change in renormalization scale is not a factor two, but only a factor 3/2.We performed several fits, changing the number of fit parameters n c and n ρ , and using differentcuts for our data, and the overall analysis results in a consistent value for ¯ g c =0 . ( µ ref ) as long as datawith L/a ≥ is used.We choose to quote the result with n c = 3 and n ρ = 2 and L/a ≥ g c =0 . ( µ ref ) = 2 . . (4.7)Despite the high precision, the result should be actually considered conservative, as this particular fithas one of the largest uncertainties of all the combinations that we tried (see figure 9). The quantity √ t µ ref was determined in two different schemes, and for each scheme, using two different strategies.All analysis resulted in completely negligible differences.
15 / 25 .
174 2 .
175 2 .
176 2 .
177 2 . L/a ≥ L/a ≥ L/a ≥ n c = , n ρ = ¯ g c =0 . ( µ ref ) n c = 3 , n ρ = 2 n c = 4 , n ρ = 2 n c = 3 , n ρ = 2 n c = 4 , n ρ = 2 n c = 3 , n ρ = 3 n c = 3 , n ρ = 4 n c = 4 , n ρ = 2 Figure 9: Determination of ¯ g c =0 . ( µ ref ) using different parametrizations and cuts. In black and bold facethe result of equation (4.7)). Λ MS The Λ s -parameter in the scheme defined by the coupling ¯ g s is given by the expression Λ s µ = (cid:2) b ¯ g s ( µ ) (cid:3) − b b e − b g s ( µ ) exp {− I s ( ¯ g s ( µ )) } , I s ( g ) = (cid:90) g d x (cid:20) β s ( x ) + 1 b x − b b x (cid:21) . (4.8)Note that this expression is exact, and valid beyond perturbation theory, as long as the non-perturbative β -function, defined by µ dd µ ¯ g s ( µ ) = β s (¯ g ) . (4.9)is known. If two renormalized couplings are related to one-loop by the expression ¯ g s (cid:48) ( µ ) = ¯ g s ( µ ) + c ss (cid:48) ¯ g s ( µ ) + . . . (4.10)the corresponding Λ -parameters are related by Λ s (cid:48) Λ s = exp (cid:18) − c ss (cid:48) b (cid:19) . (4.11)This last formula allows for a non-perturbative definition of Λ MS , even if the MS scheme is intrinsicallyperturbative.All in all, the determination of Λ MS requires the determination of the integral in equation (4.8) ina scheme that is non-perturbatively defined. The lower limit of the integral is zero, which requires todetermine the β -function up to infinite energy. In practice this can only be achieved by a limit process.One first defines K s (¯ g s ( µ ) , g PT ) = (cid:90) ¯ g s ( µ ) g PT d x (cid:20) β s ( x ) + 1 b x − b b x (cid:21) + (cid:90) g PT d x (cid:34) β ( l ) s ( x ) + 1 b x − b b x (cid:35) , (4.12)very similar to the previous function I s (¯ g s ( µ )) . The only difference is that the integral in equation (4.8)for values of the coupling smaller than g PT is determined by substituting the β s -function by its l -loopperturbative approximation β ( l ) s ( x ) β s ( x ) x → ∼ β ( l ) s ( x ) = − x l (cid:88) n =0 b n x n + O ( x l +1) ) . (4.13)16 / 25 F SF
MS ( s = 1) MS ( s = 2) n ¯ g ( µ n ) √ t Λ MS ¯ g (0 . µ n +1 ) √ t Λ MS ¯ g (0 . µ n +1 ) √ t Λ MS ¯ g (0 . µ n +2 ) √ t Λ MS ∞ Linear 0.631(15) 0.621(16) 0.618(14) 0.621(16) ∞ Constant - 0.6260(74) - 0.6234(73)
Table 3: Sequence of couplings in different schemes and at different scales ( µ n = 2 n µ ref ) and the corre-sponding values of √ t Λ MS (see text for more details). The values of ¯ g ( µ n ) are obtained via a recursiveapplication of the step scaling function in the GF scheme (equation (4.17)). The conversion from the GFscheme to the SF scheme is performed non-perturbatively and detailed in appendix B. The conversion tothe MS scheme is done by using the perturbative relation with the SF scheme (equation (4.19)). The lasttwo rows show possible extrapolations of √ t Λ MS using the last four values ( n = 2 , , , ). We showboth an extrapolation linear in the ¯ g and a extrapolation to a constant for the cases that this behavior iscompatible with the data. The first two coefficients b = 11 / (4 π ) and b = 102 / (4 π ) are scheme-independent, while the values b n for n > depend on the chosen scheme. It is now clear that K s (¯ g s ( µ ) , g PT ) g PT → ∼ I s (¯ g s ( µ )) + O ( g l − ) , (4.14)In practice for most finite volume schemes, the β s -function is known up to three loops, and thereforethe corrections are O ( g ) . The value of the coupling g PT delimits the energy region (from µ PT to ∞ )where perturbation theory is used via the relation ¯ g s ( µ PT ) = g . (4.15)Ideally one would like to estimate the Λ -parameter by taking the following limit Λ s µ = lim g PT → (cid:26)(cid:2) b ¯ g s ( µ ) (cid:3) − b b e − b g s ( µ ) exp {− K s ( ¯ g s ( µ ) , g PT ) } (cid:27) . (4.16)Since the value of the coupling ¯ g s ( µ ) runs logarithmically with µ , it is technically a challenge to probea large range of energy scales so that the corrections O ( g l − ) vary substantially and the limit canbe taken accurately.Of course finite-size scaling was designed to explore such large ranges of energy scales. Startingfrom the scale µ ref (see section 4.2.1), and with the knowledge of the step scaling function σ = J ◦ J (see section 4.1), one can define the sequence of couplings u = ¯ g ( µ ref ) , u n = σ − ( u n − ) = J − (cid:0) J − ( u n − ) (cid:1) = ¯ g (2 n µ ref ) . (4.17)The energy scales reached by this procedure increase geometrically. Contact with perturbation theorycan be made at each step by choosing g = u n in equation (4.14), and one can indeed check that thecorrections O ( u n ) are small and decrease as they should. For a long time, the challenge was mainlyto maintain a high precision, but the most recent works [26, 13] have shown that when one reaches ahigh precision, the corrections can be significant in some schemes even at very high energy scales.For this reason reaching high energies alone is not enough . The limit in equation (4.16) has to betaken seriously and the systematics well estimated. Fortunately our dataset allows us to study thematching with perturbation theory at energy scales µ n = 2 n µ ref , ( n = 0 , . . . , . (4.18)17 / 25oreover we will explore several options to match with perturbation theory: GF:
This is just the direct application of equation (4.8) using g = u n to determine I GF (¯ g ( µ n )) (cf.equation (4.14)). Schematically: u n = ¯ g ( µ n ) β (3)GF −−−−−−→ ( eq. (4 . Λ GF µ ref ΛMSΛGF −−−→ Λ MS µ ref . In this case the matching with perturbation theory is performed in the GF scheme at a scale µ PT = µ n = 2 n µ ref . SF:
Reference [13] showed that schemes based on the GF show a very poor perturbative convergence.The same reference suggested to match non-perturbatively to the traditional SF coupling [27]with background field. The details of this matching are explained in appendix B. Schematically: u n = ¯ g ( µ n ) GF → SF −−−−−→ ( ap . B ) ¯ g (0 . µ n +1 ) β (3)SF −−−−−−→ ( eq. (4 . Λ SF µ ref ΛMSΛSF −−−→ Λ MS µ ref . In this case matching with perturbation theory is performed in the SF scheme at a scale µ PT =0 . × µ n +1 = 0 . × n +1 µ ref . MS:
One can convert the values of the SF coupling to the MS scheme using the perturbative rela-tion [28] ¯ g ( sµ ) = ¯ g ( µ ) + c ( s )4 π ¯ g ( µ ) + c ( s )(4 π ) ¯ g ( µ ) + . . . . (4.19)where c ( s ) = − πb log s + 1 . , (4.20a) c ( s ) = c ( s ) − π b log( s ) + 1 . . (4.20b)Once the value of the coupling in the MS scheme is known, one can use the known 5-loop β -function [29] to determine directly Λ MS . Even if the running is known much more accurately inthe MS scheme, this procedure carries the same parametric uncertainty O (¯ g ) as the others,since the limiting factor is represented by the known orders in the perturbative relation betweencouplings, equation (4.19) (see [26]). Schematically we have u n = ¯ g ( µ n ) GF → SF −−−−−→ ( ap . B ) ¯ g (0 . µ n +1 ) SF → MS −−−−−−−→ ( eq. (4.19) ) ¯ g ( s . µ n +1 ) β (5)MS −−−−−−→ ( eq. (4 . Λ MS µ ref . In this case the scale of matching with perturbation theory is performed in the SF scheme at ascale µ PT = s . × n +1 µ ref , but the RG evolution is done in the MS scheme. The value of s is inprinciple arbitrary, but if taken too large the perturbative coefficients of equation (4.20) becomelarge, and one expects a bad asymptotic convergence of the perturbative series. We will exploretwo choices: first the simple s = 1 , and then the value s = 2 , that is very close to the value of fastest apparent convergence .The values of Λ MS /µ ref can be multiplied by the factor √ t µ ref (cf. equation (4.3)) to producethe results for √ t Λ MS reported in table 3 according to the different procedures. In the next sectionwe will comment on the results. The scale of fastest apparent convergence is defined by a vanishing 1-loop coefficient in the relation between thecoupling and the MS scheme (i.e. c ( s ) = 0 in equation (4.20)).
18 / 25 . . . . . . . . . .
005 0 .
01 0 .
015 0 .
02 0 .
025 0 .
03 0 .
035 0 .
04 0 . √ t × Λ M S α ≡ g / (4 π ) Dalla Brida, Ramos ’19FlowQCD GFSF
MS( s = 1)MS( s = 2) Figure 10: The dimensionless product √ t × Λ MS as a function of g PT (see equation (4.16)). The emptysymbols represent the data of table 3 for n = 0 , . . . , , while the filled symbols are extrapolations g PT → (shifted for better visibility) of the different approaches to the perturbative matching (see text for moredetails). The gray band is the result of reference [13], while the data point labeled FlowQCD is the resultof reference [23]. We refer the reader once more to table 3. The values for √ t Λ MS differ for the different treatmentsof perturbation theory. There are two important points worth mentioning:1. Even at scales where ¯ g ≈ (corresponding to α ≈ . ), different treatments of perturbationtheory produce values of √ t Λ MS that vary as much as 3%.2. There are two particular treatments of perturbation theory (labeled SF and MS( s = 2) ), wherethe value of √ t Λ MS is constant within errors when extracted over a range of energy scales thatvary by a factor 32.These results are also plotted in figure 10. Qualitatively we see that the variations between differenttreatments of perturbation theory roughly scale as expected (i.e. decrease proportionally to α ).A more quantitative picture is obtained by looking at the two last rows of table 3. They showpossible extrapolations of the quantity √ t Λ MS (see equation (4.16)): the deviation from the finalresult of reference [13] √ t Λ MS = 0 . of any of the possible extrapolations is below thestatistical uncertainties (about . ). This is half of the differences present at scales where ¯ g ≈ .All extrapolations g PT → agree well (last two rows of table 3). In particular even the extrapo-lations that assume that the higher order terms proportional to g are negligible show quite a smalluncertainty. Still, the error band covers the central values of all other extrapolations. Note howeverthat the size of the uncertainties depends strongly on how much data one decides to include. A veryconservative approach (such as the one used in reference [13]) would consist in just quoting as final re-sult the value at the most perturbative point. This is justified since the data labeled SF and MS ( s = 2) shows basically no dependence on the value of g PT .Note however that the methodology in these two works is very different. In particular they deal withthe systematic associated with the continuum extrapolations in a very different way. Reference [13] usesthe GF coupling with c = 0 . in order to perform the non-perturbative running. On the other hand weuse the step scaling function with c = 0 . , determined as the composition of the functions J and J asdescribed in section 2.2 order to do the non-perturbative running. Even with the highly conservativeapproach that we used in section 4.1 to perform the continuum extrapolations of ˆ J and ˆ J , we find a19 / 25nal uncertainty on √ t Λ MS of the same size of the one obtained in reference [13]. Moreover, the factthat the central values are in perfect agreement in the two calculations provides very strong evidencethat the systematic effects associated with the continuum extrapolations are completely under control,and well below our statistical uncertainties.Finally, the extrapolations that assume a linear dependence in α show larger uncertainties. Still,it is very important to see that the g PT → extrapolation substantially improves the agreementbetween all treatments of the perturbative matching.It is also worth mentioning that the propagation of the linear O ( a ) effects (see appendix A) representabout a 15% of the final error squared in √ t Λ MS . This is about a 50% larger than in the extractionof reference [13] and can be understood noting that making use of values of the coupling at c = 0 . ,we increase the boundary effects.All in all our approach shows a remarkable agreement between very different treatments of thematching with perturbation theory, and thanks to our new proposal, we are able to also show avery good agreement with previous works that have rather different systematics associated with thecontinuum extrapolation. .
005 0 .
01 0 .
015 0 .
02 0 .
025 0 . δ Λ M S [ % ] α δδ ? Figure 11: Scale uncertainties for Λ associated to dif-ferent renormalization scales for the coupling ¯ g ( sµ ) .The solid curve (labeled δ (cid:63) ) shows the perturbativeuncertainty estimated by varying the scale around thescale of fastest apparent convergence. The dashedcurve (labeled δ ) shows the perturbative uncertaintyestimated by varying the scale around the physicalscale. See text for more details. The approach labeled MS in section 4.2.2 is veryclose to many phenomenological extractions ofthe strong coupling. The value of ¯ g is extractedfrom a measurement, in this case from the value ofthe SF coupling obtained in a simulation, thanksto its perturbative expansion ¯ g ( sµ ) = ¯ g ( µ )+ c ( s )4 π ¯ g ( µ )+ c ( s )(4 π ) ¯ g ( µ )+ . . . . Different renormalization scales sµ can be usedfor each value of the physical scale µ . The differ-ences between different renormalization scales arean estimate of the truncation errors (i.e. an esti-mate of the O ( g l − ) effects in equation (4.14)).In particular, in phenomenology, it is very com-mon to vary the renormalization scale a factortwo above/below some chosen value.Figure 11 shows such estimate of the uncer-tainties propagated to the Λ -parameter. δ (cid:63) is ob-tained by varying the renormalization scale of afactor two above/below the scale of fastest appar-ent convergence (i.e. the average difference be-tween the values of Λ obtained after using s = 1 and s = 2 and then s = 2 and s = 4 ).From figure 11 it is clear that that scales uncertainties are rather large in the pure gauge theory .Even at α ≈ . , corresponding to the highest scales reached in our study, they are around 2%. Onemight question the results of our works, that claim a significantly smaller uncertainty. The key toclaim smaller errors than the scale uncertainties lies in the limit definition of Λ , equation (4.16). Oncethe limit g PT → is properly taken, and its systematic estimated, one does not need to talk aboutthe uncertainties at non-zero g PT . Of course taking such a limit is hard: data at different values of In fact they are even larger if one varies the scale around the physical scale (by a factor three), instead of using thevalue of fastest convergence. See figure 11.
20 / 25 PT is required. Due to the logarithmic running of the coupling with the physical energy scale, theapparently innocent limit of equation (4.16) requires to solve a hard multi-scale problem. Even withour datasets, that spans a factor 32 in energy scales (a change in the coupling g by more than afactor two), we have seen that some assumptions on the scaling as g PT → are needed in order toreach the 1.4% precision on Λ .We consider our approach to treat perturbative uncertainties very conservative. Still, future worksin the pure gauge theory might want to explore even larger energy scales. In this work we have examined the main sources of uncertainties present in finite-size scaling studiesusing the Gradient Flow: the continuum extrapolation and the statistical uncertainties. We haveargued that scaling violations are a result of exploring changes in the flow time. This observation hasbeen supported both by a perturbative study and by non-perturbative numerical results.The determination of the step scaling function σ ( u ) , the crucial observable in step scaling studies,involves both a change in the flow time and a change in the size of the system. We propose to dividethe determination of σ ( u ) in two pieces: first, a change in the renormalization scale at a constant size ofthe system (the function J ), followed by a change of the size of the system at constant renormalizationscale (the function J ). The advantage is that, according to our hypothesis, only the first step showssignificant cutoff effects. By breaking up the determination in two pieces, the scaling violations can bestudied much more accurately. Modest datasets allow to explore a change of the renormalization scaleat constant physical volume with lattice spacings varying by factors 4-6. In section 2.2 we have seenthat this strategy usually comes at the cost of larger statistical uncertainties, especially in schemes likethe Schrödinger Functional that break translation invariance and have to deal with O ( a ) systematiceffects. Thus, the proposal trades the large systematic associated with the continuum extrapolationspresent in many GF studies with a statistical uncertainty. Since the latter are much easier to control,we think that the proposed strategy shows a clear advantage. In section 3 we have shown that statisticaluncertainties can be well understood and predicted with a simple model.We think that this strategy can shed some light in many problems that are currently being stud-ied where systematic effects of the continuum extrapolation are relevant (see for example the recentdiscussion in [7]). A detailed study of the scaling violations of J , that according to our hypothesisare very similar to those of the step scaling function σ , should become a standard way to assess thequality of the continuum determination of the step scaling function.We have also re-examined the determination of the Λ -parameter in the pure gauge theory. Themost crucial step is the high energy region and the matching with the asymptotic perturbative regime.We have used the step scaling function with c = 0 . , determined using our new proposal. The matchingwith perturbation theory is performed in different schemes and using different procedures. Our datasetsallows to match with perturbation theory at energy scales µ PT where α ( µ PT ) (cid:46) . . Even at this largeenergy scales the perturbative truncation effects are large, corresponding to about a uncertainty in Λ . The size of this uncertainty is also confirmed by a scale variation analysis. All in all, perturbativeerrors are large in the pure gauge theory, making the determination of Λ rather challenging in thisaspect, in particular when compared with the corresponding determination in QCD . Fortunately, ourdataset explores a large range of energy scales, and gives us the possibility to explore the limit α ( µ PT ) → (corresponding to µ PT → ∞ ). Once this limit is properly taken, the perturbative uncertainties at α ( µ PT ) estimated using scale variation or any other procedure are irrelevant. Of course taking sucha limit is very challenging. The corrections, O ( α ( µ PT )) , decrease very slowly due to the logarithmicrunning of the coupling at high energies. Note however that the coupling runs faster in the pure gauge theory. For a fixed range of scales, the pure gaugetheory allows to study larger variations in the coupling.
21 / 25epending on the assumptions made in the extrapolation α ( µ PT ) → , the uncertainty in Λ variesin the range − . All in all, our results show a perfect agreement with the final result of [13]( √ t × Λ MS = 0 . ), obtained with c = 0 . , that quotes an uncertainty ≈ . . We stressonce more that the method that we used in this work provides a careful control on the continuumextrapolations, leading to a final uncertainty on √ t × Λ MS of the same size of [13] even when usinga very conservative approach. Due to the different treatments of perturbation theory and the useof different schemes, it seems clear that, despite some discrepancies with other works (see discussionin [13]), the systematic effects in [13] are well under control. Acknowledgments
We want to thank specially Rainer Sommer for many useful discussions, sharing his ideas with us anda careful reading of the manuscript.We thank M. Dalla Brida for his contribution in producing the datasets of reference [13] that wasessential for the results presented here, and E. Bribian and M. Garcia-Perez for sharing the data ofreference [30] used in section 3. 22 / 25 = 0 . c = 0 . c = 0 . a ( c ) a ( c ) -0.11(2) -0.26(3) -0.43(3) Table 4: Values of the fit coefficients a ( c ) and a ( c ) that parameterize the sensitivity of ¯ g c to the boundaryimprovement coefficient c t . A Boundary O ( a ) effects The procedure to estimate the O ( a ) cutoff effects is completely analogous as in [13]. In fact we usethe same datasets. We determine numerically the dependence of the GF coupling at different valuesof c = 0 . , . , . with the boundary parameter c t . This is done using simulations at different valuesof the improvement parameter c t close to its 2-loop value c (cid:63)t ( g ) = 1 − . g − . g + O ( g ) , ( g = 6 /β ) , (A.1)for lattice sizes L/a = 8 , , . This allows to obtain ∂ ¯ g c ∂c t (cid:12)(cid:12)(cid:12) c t = c (cid:63)t = aL (cid:2) a ( c )¯ g c + a ( c )¯ g c (cid:3) . (A.2)Table 4 shows the result of the parameters a ( c ) , a ( c ) .Now we need an estimate of how much the true, non-perturbative, value of c t differs from its 2-loop value. There is no information available, but the extrapolations at constant u of section 2.2.2completely ignored this linear effects and the data showed no significant deviation from a O ( a ) scaling.This suggests that these effects are smaller than our statistical uncertainties. With this insight, aconservative approach consists in using the full 2-loop contribution as an estimate of the differencebetween the 2-loop value and the correct non-perturbative one.In summary we add to all our data, in quadratures, the error δ ¯ g c = aL (cid:2) a ( c )¯ g c + a ( c )¯ g c (cid:3) × . g . (A.3)in order to account for a possible mis-tuning of the boundary O ( a ) -improvement parameter c t . B Matching with the SF scheme
Since the strategy is the same as the one used in [13], we refer the interested reader to this referencefor more details.Here it is enough to say that we performed a set of SF simulations with background field andlattice sizes
L/a = 6 , , , , and at the same values of the bare coupling g = 6 /β as the availablemeasurements of the GF coupling in twice the lattice size L/a = 12 , , , , . We collect between × and × measurements of the SF coupling ¯ g . We further remove all cutoff effects up to2 loops from ¯ g (i.e. the leading cutoff effects are O ( g ) ). The non-perturbative data is fitted to afunctional form of the type g ( µ ) − g c =0 . ( µ/ (2 c )) = f ( u ) + (cid:16) aL (cid:17) ˜ ρ ( u ) , ( u = ¯ g c =0 . ( µ/ (2 c ))) , (B.1)where both f ( u ) and ˜ ρ ( u ) are simple polynomials in u . For the GF coupling we use the measurementsat c = 0 . . The matching with other values of c is performed thanks to the knowledge of the function J of equation (4.5).To quote all our results we use the same fit used in reference [13], where the coarser lattice isdropped from the fit and the functions f ( u ) , ˜ ρ ( u ) are degree two polynomials. 23 / 25 eferences [1] R. Narayanan and H. Neuberger, “Infinite N phase transitions in continuum Wilson loopoperators,” JHEP (2006) 064, arXiv:hep-th/0601210 [hep-th] .[2] M. Lüscher, “Properties and uses of the Wilson flow in lattice QCD,”
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