# A non-perturbative exploration of the high energy regime in \(N_{\mathrm{f}}=3\) QCD

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## Abstract

Using continuum extrapolated lattice data we trace a family of running couplings in three-flavour QCD over a large range of scales from about 4 to 128 GeV. The scale is set by the finite space time volume so that recursive finite size techniques can be applied, and Schrödinger functional (SF) boundary conditions enable direct simulations in the chiral limit. Compared to earlier studies we have improved on both statistical and systematic errors. Using the SF coupling to implicitly define a reference scale \(1/L_0\approx 4\) GeV through \(\bar{g}^2(L_0) =2.012\), we quote \(L_0 \Lambda ^{N_\mathrm{f}=3}_{{\overline{\mathrm{MS}}}} =0.0791(21)\). This error is dominated by statistics; in particular, the remnant perturbative uncertainty is negligible and very well controlled, by connecting to infinite renormalization scale from different scales \(2^n/L_0\) for \(n=0,1,\ldots ,5\). An intermediate step in this connection may involve any member of a one-parameter family of SF couplings. This provides an excellent opportunity for tests of perturbation theory some of which have been published in a letter (ALPHA collaboration, M. Dalla Brida et al. in Phys Rev Lett 117(18):182001, 2016). The results indicate that for our target precision of 3 per cent in \(L_0 \Lambda ^{N_\mathrm{f}=3}_{{\overline{\mathrm{MS}}}}\), a reliable estimate of the truncation error requires non-perturbative data for a sufficiently large range of values of \(\alpha _s=\bar{g}^2/(4\pi )\). In the present work we reach this precision by studying scales that vary by a factor \(2^5= 32\), reaching down to \(\alpha _s\approx 0.1\). We here provide the details of our analysis and an extended discussion.

## 1 Introduction

The Standard Model seems to describe all high energy physics experiments carried out to date, in some cases with extraordinary accuracy (cf. [2] for the most recent PDG review). For processes involving the strong interactions the precision is usually less impressive, due to our limited ability to extract quantitative information from QCD. One of the main tools is perturbation theory (PT) in the strong coupling, \(\alpha _s\), and there has been significant progress in high order perturbative QCD calculations, with renormalization group functions now available up to 5-loop order in the \({\overline{\mathrm{MS}}}\)-scheme [3, 4, 5, 6, 7]. However, before a perturbative result can be confronted with experimental observables, the transition from quarks and gluons to hadronic degrees of freedom needs to be modelled in some way. Such models come in various shapes and forms, from “hadronisation Monte Carlo” in jet physics to “quark hadron duality” in QCD sum rules. A common problem then consists in assigning systematic errors to the model assumptions. A further issue is the reliability of PT itself, given that the series is only asymptotic. To some extent, the reliability can be assessed within PT itself, by comparing different orders of the expansion, or by increasing the energy scale, \(\mu \), such that \(\alpha _s(\mu )\) becomes small, due to asymptotic freedom. Unfortunately, the rapidly increasing complexity of higher order calculations means that typically only a few terms in the perturbative series are available. In addition, the energy scale is often defined by the kinematics of the physical process under consideration. The variation of the scale is then rather limited and to assign an error to the perturbative result is difficult.^{1}

In this work we carry out a systematic investigation into the reliability of PT. We do this by directly comparing non-perturbative QCD observables to their perturbative expansions, over a wide range of scales. Lattice QCD, together with a careful treatment of the continuum limit, is currently the only way to obtain such non-perturbative results, subject only to standard assumptions such as locality and universality. The main reason why this is rarely done is the usual limitation of any numerical approach: on a finite system it is very expensive to simultaneously resolve very different length scales. Most lattice QCD projects aim at hadronic low energy physics, and the space-time volume, \(L^4\), must then measure several femto metres across in order not to distort the hadronic states of interest. At least for massive single particle states, the finite volume effects are exponentially suppressed [9] and one may then pretend to be in infinite space time volume, up to a systematic error which is often below the percent level. On the other hand, with current lattice resolutions of, say, \(L/a < 100\) this means that the cutoff scale set by the inverse lattice spacing, 1 / *a*, reaches a few GeV at most, and the deep perturbative high energy regime seems out of reach. It is important to realize that this limitation is only due to the requirement that the lattice covers a physically large space-time volume. If this constraint is lifted, there is nothing that prevents simulations at very high energies, albeit in physically tiny space-time volumes. The observables^{2} we consider in this situation are all normalized as effective couplings, which run with *L*, the scale set by the finite space-time volume. In order to achieve this we set all quark masses to zero and scale all other dimensionful parameters proportionally to *L*, thereby obtaining a mass-independent scheme. In the high-energy regime, PT can be used to relate to more commonly used schemes such as the \({\overline{\mathrm{MS}}}\) scheme of dimensional regularization. Moreover, by combining the idea of a finite volume scheme for the coupling with recursive step-scaling techniques [10, 11], one may both determine the scale *L* in units of some hadronic scale, and reach the perturbative high energy regime without ever requiring enormous lattice resolutions, *L* / *a*. Obviously, the finite space-time volume constitutes an integral part in the definition of these observables. PT must then be adapted to this situation. While the Euclidean space-time signature used in lattice QCD is advantageous in PT, all the sophisticated tools of standard PT in (infinitely extended) momentum space are of limited use.

The technique, used earlier for between \(N_\mathrm{f}=0\) and \(N_\mathrm{f}=4\) quark flavours [18, 19, 20, 21], allows one to non-perturbatively verify the close-to perturbative running of the coupling and observe the small effects of dynamical quarks, as illustrated in Fig. 1. A preview of our final result is included in the figure, demonstrating our advanced precision.

The paper is organized as follows: Sect. 2 uses a continuum language to explain how our QCD observables are defined and collects the relevant perturbative results from the literature. We also comment on “non-perturbative effects” which are associated with secondary minima of the action. Section 3 then presents the lattice set-up, the numerical simulations and statistics produced, and discusses the perturbative improvement of the data. The impatient reader might skip this section and directly pass to Sect. 4. There, after the discussion of the continuum extrapolated results and associated systematic errors, the comparison to renormalized perturbation theory is performed before we conclude in Sect. 5. Finally, a technical appendix presents the models we used for the sensitivity of the data to a variation of the two O(*a*) boundary counterterm coefficients \(c_\mathrm{t}\) and \(\tilde{c}_\mathrm{t}\).

## 2 SF couplings

In order to apply the recursive step-scaling techniques to lattice QCD, it is desirable to define renormalized QCD couplings in a finite space-time volume, \(L^4\), and in the chiral limit. Such finite volume renormalization schemes are quark mass independent by construction [22], and the renormalization scale is set by \(\mu =1/L\). It is then possible to apply recursive finite size scaling methods and trace the scale evolution over a wide range without the need for very large lattice sizes, *L* / *a* [10]. Still, these requirements leave many options, such as the boundary conditions for the fields and the exact choice of observable. We here choose Schrödinger functional boundary conditions [23, 24]: these introduce a gap in the spectrum of the Dirac operator, so that numerical simulations can be performed directly at zero quark masses, without the need for any chiral extrapolation. Moreover, perturbation theory remains tractable in this framework, as the absolute minimum of the action is unique up to gauge equivalence. For the observable we choose the traditional SF coupling [25, 26] and a 1-parameter family of close relatives [27]. The most important reason for this choice is the existence of a 2-loop calculation in this case [14, 15], which, in combination with [16, 17] allows to infer the 3-loop \(\beta \)-function for these schemes. Furthermore, the values of the 3-loop \(\beta \)-function coefficients are reasonable and enable us to make contact with the asymptotic perturbative regime at energy scales in the range O(10–100) GeV.

In the future one might also consider the more recent coupling definitions based on the gradient flow [28, 29]. The QCD 3-loop \(\beta \)-function is currently known in the case of infinite space-time volume [30], and there is progress for the case of a finite volume with SF boundary conditions [29] using numerical stochastic perturbation theory [31, 32, 33]. These results seem to point to a 3-loop \(\beta \)-function coefficient which is significantly larger than in the \({\overline{\mathrm{MS}}}\)- and SF-schemes. This indicates that gradient flow couplings may not be ideal for matching with the asymptotic perturbative regime. Furthermore, cutoff effects are typically larger with the GF couplings than with the traditional SF coupling [13], so that larger lattice sizes are required. This partially offsets other computational advantages. Obviously, further studies are required and one should re-assess the situation once more perturbative information becomes available.

### 2.1 \(\hbox {SF}_\nu \) schemes

*L*-periodic boundary conditions are imposed on all fields. At the time boundaries the fermionic fields satisfy [24]

*n*,

*n*is then subject to the usual instanton bound [23]

*n*, with \(\Lambda \) in Eq. (2.6) replaced by fixed representatives \(\Lambda _n\) for each topological sector. In particular one often sets \(\Lambda _0 = 1\).

### 2.2 \(\beta \)-functions and perturbative relations to the \({\overline{\mathrm{MS}}}\)-coupling

Coefficients in the asymptotic expansion of the \(\beta \)-function in different schemes. Note that the universal coefficients for \(N_\mathrm{f}=3\) are \((4\pi )b_0 \approx 0.716197\), \((4\pi )^2b_1\approx 0.405285\)

Scheme | \((4\pi )^3b_2\) | \((4\pi )^4b_3\) | \((4\pi )^5b_4\) |
---|---|---|---|

SF (\(\nu =-0.5\)) | \(\phantom {-}0.5655\) | – | – |

SF (\(\nu =0\)) | \(-0.064\) | – | – |

SF (\(\nu =0.3\)) | \(-0.4417\) | – | – |

\(\overline{\mathrm{MS}}\) | 0.324447 | 0.484842 | 0.416059 |

Comparing the \({\overline{\mathrm{MS}}}\) to the \(\hbox {SF}_\nu \) scheme we note that the respective 3-loop \(\beta \)-functions coincide for \(\nu \approx -0.3\). In general, \(\nu \)-values of O(1) are reasonable from a perturbative point of view.

### 2.3 Perturbation theory and the \(\Lambda \)-parameter

*exact*relationship

### 2.4 On exponentially suppressed corrections to perturbation theory

*t*, corresponding to \(c=\sqrt{8t}/L =10\); the gradient flow is a smoothing operation and drives the gauge field towards a local minimum of the action. At large flow times we selected the gauge field configurations in the \(Q=0\) sector.

^{3}Apart from the background field, Eq. (2.17), we have indeed found a single further local minimum. In order to check for its stability and to obtain its continuum limit, we have performed similar simulations on finer lattices with \(L/a=12,16,24\), and bare couplings such as to keep \(\bar{g}^2(L/2) =2.77\) approximately fixed. After extrapolation to the continuum and in the temporal gauge we find that this secondary minimum corresponds to the spatially constant Abelian field,

## 3 Lattice set-up and simulations

In this section we briefly describe the main elements of the lattice set-up chosen for this study and discuss some details pertaining to the error treatment.

### 3.1 Lattice action

*a*) improved Wilson fermions. The lattice action is then given by \(S=S_g+S_f\), with

*p*on the lattice, with the weights

*w*(

*p*), and the parallel transporters

*U*(

*p*) around

*p*. With the gauge field boundary conditions given in terms of the Abelian fields, Eq. (2.15),

### 3.2 Lattice observables

^{4}(3.1, 3.2). The lattice version of the Abelian background field takes the form,

*k*in order to ensure \(\bar{g}^2=g_0^2\)

*exactly*at lowest order. Note that this also holds for \(\bar{g}_\nu ^2\), since \(\bar{v}\) vanishes identically at tree level.

*u*of the coupling \(\bar{g}^2(L)\) at vanishing quark mass defines our approach to the continuum limit, and other lattice observables are then well-defined functions of

*u*. In particular \(\bar{v}\) gives rise to 2 lattice observables

*L*/

*a*and 2

*L*/

*a*, at the same bare parameters. In particular, the simulations on the 2

*L*/

*a*-lattices are performed at the bare mass parameters for which the PCAC mass vanishes on the

*L*/

*a*lattice. Finally, we also consider the lattice step-scaling functions for \(\bar{g}^2_\nu \),

### 3.3 Perturbatively improved lattice observables

^{5}

*a*/

*L*is due to the incomplete cancellation of the O(

*a*) boundary effects and could be eliminated by a non-perturbative determination of \(c_\mathrm{t}\) and \(\tilde{c}_\mathrm{t}\). We will come back to the question of remnant O(

*a*) effects in Sect. 3.7.

Values of the coefficients for \(N_\mathrm{f}=3\) and the relevant lattice sizes, as required for perturbative cancellation of lattice artefacts up to 2-loop order in \(\Sigma \), and to one-loop order in \(\Sigma _\nu \), \(\Omega \) and \(\tilde{\Omega }\), cf. text

L/a | \(\delta _1 \times 10^2\) | \(\delta _{2}\times 10^2\) | \((\delta ^\nu _{1}-\delta _1)/\nu \times 10^2\) | \(\epsilon _1\times 10^2\) | \(\tilde{\epsilon }_1(a/2L)\times 10^2\) |
---|---|---|---|---|---|

4 | \(-\)1.02700 | 0.28560 | \(-\)3.94211 | 33.26842 | 5.71494 |

6 | \(-\)0.43600 | 0.02510 | \(-\)1.44433 | 12.20048 | 2.10529 |

8 | \(-\)0.22700 | \(-\)0.01380 | \(-\)0.61453 | 5.42725 | 1.13201 |

10 | \(-\)0.13800 | \(-\)0.01260 | \(-\)0.32597 | 2.99507 | 0.71670 |

12 | \(-\)0.09400 | \(-\)0.00960 | \(-\)0.20334 | 1.91825 | 0.49698 |

### 3.4 Simulation parameters and statistics

Using the openQCD code [42, 43] we have simulated lattice sizes \(L/a=4,6,8,10,12\) around 9 values of the coupling \(\bar{g}^2(L)=u\) in the range 1.1–2.0, cf. Table 3. At the same bare coupling \(g_0^2=6/\beta \) and bare quark mass \(am_0 = 1/(2\kappa )-4\) we then doubled the lattice sizes and simulated for \(2L/a=8,12,16\) and, in 3 cases also for \(2L/a=24\), cf. Table 4. Starting from the \(L/a=12\) lattices we have tried to approximately match the values of the coupling for \(\nu =0\) at \(L/a=4,6,8\), so as to be able to do continuum extrapolations of the step-scaling function at individual values of the coupling, without the necessity for large interpolations of the data.

Simulation parameters and results on the *L*-lattices. The hopping parameter \(\kappa \) was tuned such that the PCAC mass *m*(*L*), Eq. (3.15), vanishes

| \(\beta \) | \(\kappa \) | \(\bar{g}^2\) | \(\Delta \bar{g}^2\) | \(\bar{g}^2_{\nu =0.3}\) | \(\Delta \bar{g}_{\nu =0.3}^2\) | \(\bar{v}\) | \(\Delta \bar{v}\) |
---|---|---|---|---|---|---|---|---|

6 | 6.2650 | 0.13558688 | 2.0194 | 0.0032 | 2.1991 | 0.0042 | 0.1349 | 0.0016 |

6 | 6.5964 | 0.13499767 | 1.7983 | 0.0025 | 1.9448 | 0.0033 | 0.1396 | 0.0017 |

6 | 6.9283 | 0.13444591 | 1.6247 | 0.0021 | 1.7462 | 0.0029 | 0.1427 | 0.0018 |

6 | 7.2604 | 0.13393574 | 1.4799 | 0.0015 | 1.5831 | 0.0020 | 0.1467 | 0.0016 |

6 | 7.5769 | 0.13348828 | 1.3680 | 0.0011 | 1.4568 | 0.0015 | 0.1485 | 0.0014 |

6 | 7.8935 | 0.13307660 | 1.2703 | 0.0009 | 1.3487 | 0.0013 | 0.1526 | 0.0014 |

6 | 8.2103 | 0.13269801 | 1.1864 | 0.0009 | 1.2552 | 0.0012 | 0.1540 | 0.0015 |

6 | 8.5271 | 0.13234995 | 1.1125 | 0.0008 | 1.1723 | 0.0011 | 0.1528 | 0.0016 |

8 | 6.4575 | 0.13525498 | 2.0201 | 0.0034 | 2.1875 | 0.0046 | 0.1262 | 0.0017 |

8 | 6.7900 | 0.13467912 | 1.7943 | 0.0031 | 1.9341 | 0.0043 | 0.1343 | 0.0022 |

8 | 7.1225 | 0.13414622 | 1.6173 | 0.0025 | 1.7291 | 0.0034 | 0.1332 | 0.0021 |

8 | 7.4550 | 0.13365676 | 1.4783 | 0.0019 | 1.5728 | 0.0026 | 0.1354 | 0.0020 |

8 | 7.7721 | 0.13322862 | 1.3629 | 0.0016 | 1.4440 | 0.0021 | 0.1374 | 0.0020 |

8 | 8.0891 | 0.13283568 | 1.2657 | 0.0009 | 1.3374 | 0.0013 | 0.1412 | 0.0014 |

8 | 8.4062 | 0.13247474 | 1.1845 | 0.0008 | 1.2473 | 0.0011 | 0.1417 | 0.0014 |

8 | 8.7232 | 0.13214306 | 1.1122 | 0.0013 | 1.1674 | 0.0017 | 0.1419 | 0.0024 |

10 | 6.6046 | 0.13498493 | 2.0259 | 0.0124 | 2.1879 | 0.0167 | 0.1218 | 0.0059 |

10 | 6.6073 | 0.13498022 | 2.0129 | 0.0035 | 2.1803 | 0.0049 | 0.1271 | 0.0017 |

10 | 7.6010 | 0.13344250 | 1.4898 | 0.0069 | 1.5799 | 0.0090 | 0.1276 | 0.0067 |

10 | 7.6063 | 0.13343533 | 1.4794 | 0.0019 | 1.5719 | 0.0025 | 0.1326 | 0.0018 |

10 | 8.8675 | 0.13198989 | 1.1118 | 0.0030 | 1.1638 | 0.0040 | 0.1341 | 0.0056 |

10 | 8.8755 | 0.13198218 | 1.1093 | 0.0010 | 1.1633 | 0.0013 | 0.1395 | 0.0019 |

12 | 6.7300 | 0.13475901 | 2.0123 | 0.0037 | 2.1725 | 0.0048 | 0.1221 | 0.0018 |

12 | 7.7300 | 0.13326291 | 1.4805 | 0.0020 | 1.5752 | 0.0026 | 0.1355 | 0.0019 |

12 | 9.0000 | 0.13185703 | 1.1089 | 0.0014 | 1.1614 | 0.0018 | 0.1358 | 0.0024 |

Simulation parameters and results on the doubled lattices. The hopping parameter \(\kappa \) was tuned such that the PCAC mass *m*(*L* / 2) vanishes, cf. Eq. (3.15)

| \(\beta \) | \(\kappa \) | \(\bar{g}^2\) | \(\Delta \bar{g}^2\) | \(\bar{g}^2_{\nu =0.3}\) | \(\Delta \bar{g}_{\nu =0.3}^2\) | \(\bar{v}\) | \(\Delta \bar{v}\) |
---|---|---|---|---|---|---|---|---|

8 | 6.0522 | 0.13546638 | 2.4124 | 0.0044 | 2.6281 | 0.0057 | 0.1134 | 0.0014 |

8 | 6.3757 | 0.13492039 | 2.0955 | 0.0039 | 2.2648 | 0.0052 | 0.1189 | 0.0018 |

8 | 6.7145 | 0.13437600 | 1.8586 | 0.0035 | 1.9989 | 0.0047 | 0.1259 | 0.0021 |

8 | 7.0275 | 0.13390509 | 1.6756 | 0.0020 | 1.7919 | 0.0026 | 0.1291 | 0.0015 |

8 | 7.3496 | 0.13345482 | 1.5286 | 0.0018 | 1.6277 | 0.0024 | 0.1328 | 0.0017 |

8 | 7.6782 | 0.13303090 | 1.3997 | 0.0013 | 1.4855 | 0.0018 | 0.1376 | 0.0015 |

8 | 7.9822 | 0.13266902 | 1.3014 | 0.0011 | 1.3743 | 0.0015 | 0.1359 | 0.0015 |

8 | 8.3130 | 0.13230601 | 1.2128 | 0.0010 | 1.2784 | 0.0013 | 0.1410 | 0.0016 |

12 | 6.2650 | 0.13558688 | 2.4568 | 0.0060 | 2.6788 | 0.0081 | 0.1124 | 0.0018 |

12 | 6.5964 | 0.13499767 | 2.1287 | 0.0042 | 2.2995 | 0.0054 | 0.1163 | 0.0018 |

12 | 6.9283 | 0.13444591 | 1.8780 | 0.0029 | 2.0175 | 0.0039 | 0.1227 | 0.0017 |

12 | 7.2604 | 0.13393574 | 1.6839 | 0.0024 | 1.8045 | 0.0033 | 0.1323 | 0.0019 |

12 | 7.5769 | 0.13348828 | 1.5378 | 0.0019 | 1.6370 | 0.0026 | 0.1314 | 0.0018 |

12 | 7.8935 | 0.13307660 | 1.4148 | 0.0016 | 1.5010 | 0.0021 | 0.1353 | 0.0018 |

12 | 8.2103 | 0.13269801 | 1.3114 | 0.0017 | 1.3860 | 0.0022 | 0.1369 | 0.0022 |

12 | 8.5271 | 0.13234995 | 1.2210 | 0.0014 | 1.2880 | 0.0019 | 0.1420 | 0.0022 |

16 | 6.4575 | 0.13525498 | 2.4540 | 0.0056 | 2.6708 | 0.0072 | 0.1103 | 0.0016 |

16 | 6.7900 | 0.13467912 | 2.1251 | 0.0043 | 2.2970 | 0.0057 | 0.1174 | 0.0018 |

16 | 7.1225 | 0.13414622 | 1.8810 | 0.0039 | 2.0230 | 0.0051 | 0.1244 | 0.0021 |

16 | 7.4550 | 0.13365676 | 1.6863 | 0.0029 | 1.8017 | 0.0039 | 0.1265 | 0.0021 |

16 | 7.7721 | 0.13322862 | 1.5375 | 0.0022 | 1.6370 | 0.0029 | 0.1317 | 0.0019 |

16 | 8.0891 | 0.13283568 | 1.4164 | 0.0018 | 1.5011 | 0.0024 | 0.1328 | 0.0020 |

16 | 8.4062 | 0.13247474 | 1.3090 | 0.0017 | 1.3825 | 0.0022 | 0.1353 | 0.0021 |

16 | 8.7232 | 0.13214306 | 1.2204 | 0.0014 | 1.2842 | 0.0019 | 0.1358 | 0.0021 |

24 | 6.7300 | 0.13475901 | 2.4517 | 0.0067 | 2.6732 | 0.0087 | 0.1126 | 0.0019 |

24 | 7.7300 | 0.13326291 | 1.6847 | 0.0033 | 1.7980 | 0.0042 | 0.1246 | 0.0023 |

24 | 9.0000 | 0.13185703 | 1.2232 | 0.0022 | 1.2892 | 0.0029 | 0.1394 | 0.0032 |

All statistical errors were determined using the \(\Gamma \)-method [44]. For our observables, one even has to be careful that one sums up the autocorrelation function sufficiently far. Still the final autocorrelation times range from values somewhat below 2 MDU for weak coupling and small *L* / *a*, to about 8 MDU at larger coupling and \(L/a=24\). Further details on the performance of our algorithms will be reported in [45].

### 3.5 Treatment of statistical errors

*u*the central value of the estimate of \(\bar{g}^2(L)\), via

*u*we first perform a rough continuum extrapolation neglecting the errors on

*u*. The resulting polynomial fit function

*L*/

*a*-dependence of the derivative.

### 3.6 Quality of tuning to the chiral limit

### 3.7 Lattice artefacts linear in *a* / *L*

*a*) improved bulk action the very presence of the time boundaries in the Schrödinger functional creates lattice artefacts linear in

*a*. In principle these could be cancelled by an appropriate non-perturbative tuning of the improvement coefficients \(c_\mathrm{t}\) and \(\tilde{c}_\mathrm{t}\), Eqs. (3.1, 3.7). In practice, however, we are currently limited to the use of perturbative estimates, Eqs. (3.8, 3.9). Hence some remnant linear

*a*-effects in our data cannot be excluded. Instead of including a corresponding term in the fit ansatz for the continuum extrapolations we try to estimate the size of these uncertainties and include them as an additional systematic error. Using a combination of simulations and perturbation theory we have produced a model for the sensitivity of our data to a variation of \(c_\mathrm{t}\) and \(\tilde{c}_\mathrm{t}\). The details are deferred to Appendix A, where we obtain linearized shifts of the data, for instance,

There are several options for the inclusion of this systematic error. We chose to proceed as follows: we first perform continuum extrapolations ignoring potential O(*a*) errors in both the original and the shifted data. We then take the spread of a given observable as an additional systematic error and add it in quadrature. Obviously this assumes that this systematic error is subdominant. We have therefore dismissed all continuum extrapolations where this turned out not to be the case. We will discuss the impact of these variations on the continuum extrapolations in the next section.

## 4 Continuum results

### 4.1 Continuum extrapolation of the step-scaling function

*u*-values between different

*L*/

*a*, and this is seen in the vertical line-up of the data. The fact that the data are so close together at given

*u*-value illustrates that cutoff effects in the SF scheme with the chosen lattice regularization are generally small, even without perturbative improvement.

*u*-value by

*u*-value, we have done this only as a cross-check. Our preferred strategy is to simultaneously fit all data to a global ansatz of the form

*u*-values is not required. More importantly, however, the expected smooth

*u*-dependence of the step-scaling function both on the lattice and in the continuum limit, is automatically built into this ansatz. As anticipated in the last section, we assume leading cutoff effects to start at O(\(a^2\)), with the linear

*a*-effects being treated as systematic errors. Our fit ansätze for the cutoff effects thus are of the form,

*u*. The fit has an excellent \(\chi ^2/\mathrm{d.o.f} = 14.5/15\) with the continuum parameters and their covariance given by

*F*(with \(n_\rho =1\)) indicates that this may be an accident. In fact the \(\chi ^2\)-values are not a sharp criterion in our case, as these strictly refer only to the statistical errors of the data and the given fit functions used, and thus do not account for the systematic uncertainties from cutoff effects linear in

*a*.

Overview of the continuum fit functions and results. The naming convention is the same as in Ref. [1]. The two errors in the fit parameters are the statistical and the total error respectively, where the total error includes the systematic uncertainty from a variation of \(c_\mathrm{t}\) and \(\tilde{c}_\mathrm{t}\), added in quadrature

\(\nu \) | Fit | i | \(\big [\tfrac{L}{a}\big ]_{\mathrm {min}}\) | \(n_\rho \) | \(n_c\) | \(c_1\times 10^4\) | \(b_{3,\nu }^{\mathrm {fit}}\) \(\times (4\pi )^{4}\) | \(\chi ^2\) | dof |
---|---|---|---|---|---|---|---|---|---|

0 | A | 0 | 6 | 2 | 1 | 6(2)(3) | 14.7 | 16 | |

0 | B | 1 | 6 | 2 | 1 | 5(3)(3) | 14.2 | 16 | |

0 | \(\hbox {B}'\) | 1 | 6 | 1 | 1 | 8(2)(2) | 18.4 | 17 | |

0 | C | 2 | 6 | 2 | 1 | 6(3)(3) | 14.5 | 16 | |

0 | D | 2 | 6 | 2 | 2 | cf. Eq. (4.6) | 14.5 | 15 | |

0 | E | 2 | 6 | 2 | 1 | 4(2)(2) | 14.6 | 16 | |

0 | F | 2 | 8 | 1 | 1 | 4(3)(3) | 12.7 | 9 | |

0 | G | 2 | 8 | 0 | 2 | cf. Eq. (4.7) | 13.0 | 9 | |

0 | H | 1 | 6 | 2 | 1 | 3(2)(3) | 14.1 | 16 | |

0.3 | A | 0 | 6 | 2 | 1 | 3(2)(3) | 21.2 | 16 | |

0.3 | B | 1 | 6 | 2 | 1 | 1(2)(3) | 20.0 | 16 | |

0.3 | \(\hbox {B}'\) | 1 | 6 | 1 | 1 | 3(2)(2) | 20.8 | 17 | |

0.3 | H | 1 | 6 | 2 | 1 | 0(2)(2) | 20.0 | 16 | |

\(-0.5\) | A | 0 | 6 | 2 | 1 | 12(5)(5) | 11.6 | 16 | |

\(-0.5\) | B | 1 | 6 | 2 | 1 | 15(5)(5) | 10.4 | 16 | |

\(-0.5\) | \(\hbox {B}'\) | 1 | 6 | 1 | 1 | 24(4)(4) | 18.4 | 17 | |

\(-0.5\) | H | 1 | 6 | 2 | 1 | 11(5)(5) | 10.4 | 16 |

While the total errors for most fits are dominated by the statistical error, this is not the case of fit G, where the total errors are predominantly systematic, cf. Table 6. This indicates that fits with \(n_\rho =0\) are too rigid to account for the O(*a*) variation of the data. While \(n_\rho =1\) fits B\('\) and F are acceptable, we settled for fit ansätze with \(n_\rho =2\) and \(n_c=1\) to data with \(L/a\ge 6\) as our preferred choice (fits A, B, C, E, H). Then, using the 2-loop improved data leaves us with fits C and E, which are essentially equivalent, and Fig. 2 shows \(\sigma (u)\) from fit C with its error band.

### 4.2 The SF coupling for \(\nu =0\) at scales \(L_n=L_0/2^n\)

*u*for which the step-scaling function is known. In physical units the scale \(L_0\) has been determined to be around \(1/(4\,\mathrm{GeV})\) [12]. We note that \(\sigma (2.012)\) defines the coupling \(\bar{g}^2(2L_0)\), so that the lowest energy scale reached with the SF coupling is around 2 GeV.

^{6}

Results for the couplings \(u_n=\bar{g}^2_\nu (L_n)\), the \(\Lambda \)-parameter evaluated at \(u_n\), cf. Eq. (4.15), in units of the reference scale, \(L_0\) (4.9), and the effective \(\beta \)-function coefficient, \(b_3^\mathrm{eff}\) (4.14), for most fits of Table 5. Results for \(L_0\Lambda \) obtained with fits E, F and H are given in Table 7

| \(\nu = 0\) | |||||
---|---|---|---|---|---|---|

fit A | fit B | fit \(\hbox {B}'\) | fit C | fit D | fit G | |

\(u_n\): | ||||||

0 | 2.012 | 2.012 | 2.012 | 2.012 | 2.012 | 2.012 |

1 | 1.712(3) | 1.714(3) | 1.710(3) | 1.712(3) | 1.712(3)(3) | 1.711(1)(5) |

2 | 1.493(4) | 1.495(4) | 1.490(3) | 1.493(4) | 1.493(4)(5) | 1.492(2)(7) |

3 | 1.326(4) | 1.327(4) | 1.322(3) | 1.325(4) | 1.325(5)(6) | 1.324(2)(8) |

4 | 1.193(4) | 1.194(4) | 1.190(3) | 1.193(4) | 1.192(5)(6) | 1.191(2)(8) |

5 | 1.085(3) | 1.086(4) | 1.082(3) | 1.085(4) | 1.084(5)(6) | 1.084(3)(8) |

\(-1\) | 2.450(10) | 2.447(10) | 2.458(8) | 2.451(10) | 2.451(10)(11) | 2.457(5)(12) |

\(L_0 \Lambda \times 10^2\): | ||||||

0 | 3.14 | 3.14 | 3.14 | 3.14 | 3.14 | 3.14 |

1 | 3.10(3) | 3.11(3) | 3.08(2) | 3.10(3) | 3.10(2)(3) | 3.09(1)(4) |

2 | 3.07(4) | 3.09(5) | 3.04(4) | 3.07(5) | 3.07(5)(6) | 3.05(2)(8) |

3 | 3.05(6) | 3.08(6) | 3.01(5) | 3.05(6) | 3.05(7)(8) | 3.03(3)(11) |

4 | 3.04(7) | 3.06(7) | 2.98(5) | 3.03(7) | 3.03(9)(11) | 3.02(4)(14) |

5 | 3.03(7) | 3.06(8) | 2.97(6) | 3.02(8) | 3.01(12)(14) | 3.00(5)(17) |

\(b_3^\mathrm{eff}\times (4\pi )^4\): | ||||||

0 | 3(2) | 2(2) | 4(1) | 3(2) | 2(5)(5) | 5(2)(2) |

1 | 3(2) | 2(2) | 5(2) | 3(2) | 3(5)(5) | 5(2)(2) |

2 | 4(3) | 3(3) | 6(2) | 4(3) | 4(4)(4) | 6(2)(3) |

3 | 4(3) | 3(3) | 7(2) | 4(3) | 4(3)(4) | 6(2)(3) |

4 | 5(3) | 3(3) | 7(3) | 5(3) | 5(3)(3) | 7(1)(4) |

5 | 5(3) | 4(3) | 8(3) | 5(3) | 5(3)(4) | 7(1)(5) |

| \(\nu = 0.3\) | \(\nu =-0.5\) | ||||
---|---|---|---|---|---|---|

fit A | fit B | fit \(\hbox {B}'\) | fit A | fit B | fit \(\hbox {B}'\) | |

\(u_n\): | \(u_n\): | |||||

0 | 2.169 | 2.169 | 2.169 | 1.795 | 1.795 | 1.795 |

1 | 1.828(4) | 1.832(4) | 1.829(3) | 1.550(3) | 1.548(4) | 1.542(3) |

2 | 1.582(5) | 1.587(5) | 1.584(4) | 1.366(5) | 1.363(5) | 1.356(4) |

3 | 1.396(5) | 1.401(5) | 1.398(4) | 1.223(5) | 1.220(5) | 1.212(4) |

4 | 1.250(4) | 1.255(4) | 1.252(3) | 1.108(5) | 1.106(5) | 1.098(4) |

5 | 1.133(4) | 1.136(4) | 1.134(3) | 1.014(4) | 1.012(4) | 1.004(3) |

\(-1\) | 2.677(12) | 2.665(13) | 2.672(10) | 2.145(10) | 2.151(10) | 2.168(8) |

\(L_0 \Lambda \times 10^2\): | \(L_0 \Lambda \times 10^2\): | |||||

0 | 3.05 | 3.05 | 3.05 | 3.34 | 3.34 | 3.34 |

1 | 3.02(3) | 3.05(3) | 3.03(2) | 3.28(4) | 3.25(4) | 3.19(3) |

2 | 3.00(4) | 3.05(5) | 3.02(4) | 3.23(7) | 3.19(7) | 3.09(5) |

3 | 2.99(6) | 3.04(6) | 3.01(5) | 3.20(8) | 3.15(8) | 3.01(6) |

4 | 2.98(7) | 3.04(7) | 3.00(5) | 3.17(10) | 3.12(10) | 2.96(7) |

5 | 2.97(7) | 3.04(8) | 3.00(6) | 3.15(11) | 3.09(11) | 2.91(8) |

\(b_3^\mathrm{eff}\times (4\pi )^4\): | \(b_3^\mathrm{eff}\times (4\pi )^4\): | |||||

0 | 2(2) | 0(2) | 1(1) | 5(4) | 7(4) | 13(3) |

1 | 2(2) | 0(2) | 1(2) | 7(4) | 9(4) | 17(3) |

2 | 2(2) | 0(2) | 1(2) | 8(5) | 11(5) | 19(4) |

3 | 3(2) | 0(3) | 2(2) | 9(5) | 12(6) | 21(4) |

4 | 3(3) | 0(3) | 2(2) | 9(6) | 13(6) | 23(5) |

5 | 3(3) | 0(3) | 2(2) | 10(6) | 13(6) | 24(5) |

### 4.3 Effective and fitted \(\beta \)-function

*u*yields

*u*where it is measured. Extracting this coefficient at different values of

*u*should yield consistent results in the perturbative regime, and this is indeed the case for the \(\nu =0\) data, cf. Table 6.

This motivates a different parameterization of our fits with a single continuum parameter, namely via a 4-loop coefficient \(b^\mathrm{fit}_3\) in the \(\beta \)-function as a fit parameter.^{7} This is the purpose of fits E, F and H, cf. Table 5, where we have taken \(\sigma (u)\) to be defined by Eq. (4.12) with \(b(u) = b_\mathrm{3loop}(u) + b_3^\mathrm{fit} u^5\) and inserted \(\sigma (u)\) into Eq. (4.1). The resulting values for the fit parameter \(b_3^\mathrm{fit}\) are given in Table 5. This representation of our continuum results is very practical. While the fit function in Eq. (4.4) allows us to find the couplings at scales which are separated by a factor 2, the \(\beta \)-function readily yields the scale ratio separating two given couplings.

### 4.4 Determination of the \(\Lambda \)-parameter

*n*and the way the integral is evaluated. For completeness we note that our default evaluation consists in the direct numerical integration, using the truncated 3-loop \(\beta \)-function without expansion of the integrand or the exponential function. The results for \(\Lambda \) in units of \(L_0\) are given in Table 6, where Eq. (4.15) is evaluated for the coupling at scales \(L_n\), for \(n=0,\ldots ,5\) and for the various fit functions.

### 4.5 Continuum extrapolation of \(\Omega \) and \(\tilde{\Omega }\)

*L*/

*a*cover the range 6–24, i.e. a factor of 4 in scale and thus allow for an excellent control of the continuum limit.

Results of the combined fits A and B for \(\Omega ^{(i)}(u,a/L)\) and \(\tilde{\Omega }^{(i)}(u,a/L)\) with (\(i=1\)) and without (\(i=0\)) improvement. The 2 errors given are the statistical and the total error, respectively, where the latter includes an estimate of the remnant uncertainty due to linear *a*-effects

fit | \(\{\Omega ^{(i)}\), \(\tilde{\Omega }^{(i)}\}\) | \(\chi ^2/\hbox {dof}\) | \(\omega (1.11)\) | \(\omega (1.5)\) | \(\omega (2.012)\) | \(\omega (2.45)\) |
---|---|---|---|---|---|---|

A | \(i=1, L/a\ge 6\) | 47.8/45 | 0.1368(8)(9) | 0.1307(7)(8) | 0.1201(8)(9) | 0.1123(13)(13) |

A | \(i=1, L/a\ge 8\) | 33.5/37 | 0.1385(10)(10) | 0.1319(8)(9) | 0.1199(9)(10) | 0.1117(13)(13) |

A | \(i=0, L/a\ge 6\) |
| 0.1350(9)(9) | 0.1290(7)(8) | 0.1193(9)(10) | 0.1118(12)(12) |

A | \(i=0, L/a\ge 8\) | 33.5/37 | 0.1379(11)(11) | 0.1311(8)(9) | 0.1193(10)(10) | 0.1115(12)(13) |

B | \(i=1, L/a\ge 6\) | 47.8/44 | 0.1368(10)(11) | 0.1307(7)(8) | 0.1201(9)(9) | 0.1123(13)(13) |

B | \(i=1, L/a\ge 8\) | 33.5/36 | 0.1385(12)(13) | 0.1319(9)(9) | 0.1199(10)(10) | 0.1117(13)(13) |

B | \(i=0, L/a\ge 6\) |
| 0.1344(12)(12) | 0.1291(7)(8) | 0.1192(9)(10) | 0.1120(12)(13) |

B | \(i=0, L/a\ge 8\) | 33.5/36 | 0.1381(15)(15) | 0.1311(8)(9) | 0.1194(10)(10) | 0.1115(13)(13) |

*a*/

*L*for either data set, with coefficients \(\rho _{1,2}\) and \(\tilde{\rho }_{1,2}\), e.g.

*u*are chosen according to the expectation from perturbation theory. As in the case of the step-scaling function, linear terms in

*a*/

*L*will be treated as systematic errors.

*u*,

*a*) counterterm coefficients \(c_\mathrm{t}\) and \(\tilde{c}_\mathrm{t}\) (cf. Sect. 3.7), we here proceed in complete analogy with the analysis of the step-scaling function. In Table 8 we quote 2 errors, the first statistical, the second including the effect of a \(c_\mathrm{t}\) and \(\tilde{c}_\mathrm{t}\)-variation. This only marginally increases the errors, as is evident from Table 8.

The fits to the unimproved data have higher \(\chi ^2/\mathrm{d.o.f.}\) values, emphasized in bold face in Table 8, unless the \(L/a=6\) data are dropped. As mentioned above, \(\chi ^2\) is not the full story, given that our fits assume the absence of *a* / *L* effects and this effect is taken into account afterwards by our \(c_\mathrm{t},\tilde{c}_\mathrm{t}\)-variation. However, we do see that (1) these variations have a tiny effect on the continuum values and (2) still, for example, \(\omega (1.11)\) of the large \(\chi ^2\) fits is off significantly. These fits have to be discarded. The other ones, which cover a remarkable range of lattice spacings, are entirely consistent.

These observations allow us to conclude that perturbative improvement works very well in our coupling range, our treatment of \(c_\mathrm{t},\tilde{c}_\mathrm{t}\)-variations is safe (maybe overly conservative), and most importantly, resolutions \(a/L \le 1/6\) are sufficient to apply our continuum extrapolations which assume that \(\mathrm{O}((a/L)^3)\) effects have a negligible effect. All this makes us very confident also in the continuum extrapolations of \(\Sigma \), where the very small lattice spacings are not available, but where we have 2-loop perturbative improvement at our disposal.

### 4.6 The step-scaling function for \(\nu \ne 0\) and tests of perturbation theory

*E*,

*F*and

*H*, cf. Table 5. Not surprisingly, the resulting fit coefficients roughly agree with the effective 4-loop coefficients, Eq. (4.14), given in Table 6. We also note that schemes at different \(\nu \)-values behave quite differently; the 2 chosen non-zero values of \(\nu \) illustrate this: while \(\nu =0.3\) data shows no significant remnant \(\alpha \)-dependence even up to \(\alpha \approx 0.2\), the slope in \(\alpha ^2\) is very pronounced for \(\nu =-0.5\). Therefore, it is a strong consistency check for our analysis that all values for \(L_0\Lambda \) are compatible around \(\alpha =0.1\), despite considerable deviations at larger couplings. This means we can confidently extract \(L_0\Lambda \) in this regime. Our final value is obtained from fit C, taking the \(n=4\) estimate at \(\nu =0\), viz

*a*) boundary effects, cf. Sect. 3.7. For an even more conservative error estimate one could take fit D, again at \(n=4\) and \(\nu =0\), which yields \(L_0\Lambda =0.0303(11)\).

Using the fits E, F and H, in terms of the fitted \(\beta \)-function, the values in Table 7 are obtained. The fact that these are all compatible, with very similar central values further boosts the confidence that our final result is very robust. Finally, coming back to the question raised in Sect. 2 about exponentially suppressed contributions, we emphasize that the consistency of our analysis with fits taking the same functional form as higher order perturbative terms provides indirect evidence for the absence of such non-standard terms within our numerical precision.

### 4.7 Alternative tests

So far, our strategy has been to first determine \(\Lambda \) in the SF scheme and then convert it to \(\Lambda _{{\overline{\mathrm{MS}}}}\). However, one might also match the SF to the \({\overline{\mathrm{MS}}}\)-coupling at 2-loop order using Eq. (2.41) and then extract the \(\Lambda \)-parameter within the \({\overline{\mathrm{MS}}}\)-scheme. While the perturbative precision is parametrically the same as before, we present this alternative view here, as it is closer to the strategy often used in phenomenological applications.

*s*is an additional scale parameter and \(p_i^{\nu }(s) = c_i^\nu (s)/(4\pi )^i\), cf. Eq. (2.41). The unknown 3-loop and higher order terms in the argument of \(\varphi _{{\overline{\mathrm{MS}}}}\) will be neglected in the following. The function \(\varphi _{{\overline{\mathrm{MS}}}}\), Eq. (2.47) can be evaluated using up to 5-loop order for the \(\beta \)-function. For our range of \(\alpha \)-values, the numerical difference between 4- or 5-loop order evaluation is found to be negligible. The dominant uncertainty is due to the 2-loop truncation of the perturbative conversion to the \({\overline{\mathrm{MS}}}\) coupling,

We now use the non-perturbative results for the \(\hbox {SF}_\nu \)-couplings from Table 6 as input in Eq. (4.28) and study the dependence of the \(\Lambda \)-parameter estimates on the choice of scale \(L_n\), the scale factor *s* and the parameter \(\nu \). Figure 5 shows some typical results; at fixed \(\nu \) and *s* we observe again an approximate linearity in \(\alpha ^2\), with the asymptotic values being compatible with our best estimate, Eq. (4.27). However, we note that the slope varies significantly as a function of *s* and \(\nu \).

We find that the choice of \(s=s^\star \), Eq. (2.44), which eliminates the one-loop term in the matching, Eq. (2.45), is often (but not always) a good one. For the cases \(\nu =0\) and \(\nu =-0.5\), Fig. 6 shows the 1- and 2-loop matching coefficients to the \({\overline{\mathrm{MS}}}\)-coupling, Eq. (2.41), as functions of the scale factor *s*. The values for \(s^\star \) are roughly around 3, 5 and 2 for \(\nu =0\), \(-0.5\) and 0.3, respectively. While for \(\nu =0\) (similarly for \(\nu =0.3\)) the two-loop coefficient is near minimal around \(s^\star \) and stays positive (Fig. 6, left panel), a more complicated pattern is seen for \(\nu =-0.5\) (Fig. 6, right panel).

However, in all cases we note that for \(\alpha \sim 0.1\), the systematic uncertainty of the matching with perturbation theory, obtained by varying *s* is well below the statistical uncertainties. Moreover the latter are in line with the errors obtained with our previous strategy. This further reinforces our previous conclusions: thanks to the high energies reached with the step scaling method, our uncertainties are fully dominated by statistical errors, systematic uncertainties being negligible. The spread of results obtained by the variation of the perturbative matching scale provides a way to assess the systematic uncertainties which works well with the SF schemes at \(\nu =0,0.3\), even at \(\alpha \approx 0.2\) (although the systematic uncertainty there is large). But the failure of this method for the SF scheme with \(\nu =-0.5\) indicates that this method may not always be reliable, particularly if the coupling is not small and cannot be varied. This illustrates that perturbative truncation errors are very difficult to estimate within perturbation theory, and that reaching high energies is crucial for a robust determination of the strong coupling. Indeed we see that for values \(\alpha \approx 0.1\) there is nice agreement between all schemes and reasonable choices of the scale factor *s* within errors, which clearly allow us to meet the target accuracy of 3 per cent for the \(\Lambda \)-parameter.

## 5 Conclusions and outlook

Using numerical simulations and finite volume step-scaling techniques, we have studied a family of SF couplings, parameterized by \(\nu \), over a range of scales corresponding to energies of 4–128 GeV, thus differing by a scale factor 32. This, together with an unprecedented control of statistical and systematic errors represents a luxury which we have exploited to test the accuracy of perturbation theory. Choosing the \(\Lambda \)-parameter for \(\nu =0\) in units of \(L_0 \approx 1/(4\,\text {GeV})\) as a reference, its evaluation requires the knowledge of a coupling and its \(\beta \)-function between a finite and the infinite energy scale, where the coupling vanishes by asymptotic freedom. Perturbation theory to 3-loop order is available for the asymptotic scale dependence beyond an energy scale 1 / *L*, which can be chosen anywhere in the range covered by our non-perturbative data, provided the ratio \(L/L_0\) is known. By looking at the spread of values for \(L_0\Lambda \) one therefore tests the accuracy of perturbation theory at the scale 1 / *L*. Moreover, the exact relation between \(\Lambda \)-parameters of different schemes requires a one-loop matching of couplings which is known in all cases considered. It is therefore also possible to test the robustness of the \(\Lambda \)-parameter determination by using SF-schemes at various values of \(\nu \) as an intermediate step. The result is neatly illustrated in Fig. 4, where all data points should coincide up to a parametric uncertainty of order \(\alpha ^2\). We conclude that a target precision of better than \(3\%\) for \(L_0\Lambda \) (which approximately corresponds to a \(0.5\%\) precision for \(\alpha _s(m_Z)\)) requires non-perturbative data for a large enough range of couplings so that the perturbative truncation errors can be safely estimated. Our range of scales 4–128 GeV reaching down to \(\alpha \approx 0.1\) allows us to reach such a precision. While some schemes may give compatible results even at \(\alpha \approx 0.2\), it seems all but impossible to anticipate the quality of a given scheme if the coupling cannot be varied significantly.

With the hindsight of our \(2.3\%\) precision result for \(L_0\Lambda \), Eq. (4.27), we have also looked at an alternative test, which is close to procedures widely used in phenomenology. Namely, we have converted our non-perturbative observable, an \(\hbox {SF}_\nu \)-coupling with some choice for \(\nu \) and *L*, to the \({\overline{\mathrm{MS}}}\)-coupling where we allowed for a relative scale factor *s* in this perturbative conversion. Given the coupling in the \({\overline{\mathrm{MS}}}\)-scheme the full machinery with up to 5-loops for the \(\beta \)-function [3, 4, 5, 6, 7] is available to extract the \(\Lambda \)-parameter. However, as in phenomenological applications, the limiting factor is the perturbative order in the conversion to the \({\overline{\mathrm{MS}}}\)-scheme. We can perform this step at 2-loop order; for comparison we note that the 5-loop, O(\(\alpha ^4\)) result for the *R*-ratio [46] translates to 3-loop order when formulated as a conversion between couplings. Looking at the dependence of the scale factor, a common method consists in identifying an “optimal scale factor”, \(s^\star \), and then vary this factor between \(s^\star /2\) and \(2s^\star \) to obtain a systematic error estimate (c.f. the review of QCD in [2]). It is a bit of an art to determine the “optimal scale factor”, and some appeal to the kinematics or the physics of a given observable is often made in this context [2]. We here applied such a procedure, choosing \(s^\star \) close to the ratio of \(\Lambda \)-parameters, which means the one-loop coefficient in the conversion to the \({\overline{\mathrm{MS}}}\) scheme is made very small. As illustrated in Fig. 7, this procedure gives an error that shrinks proportionally to \(\alpha ^2\) and often brackets the correct result. However, we have also found cases (e.g. \(\nu =-0.5\)) where this procedure does not work and underestimates the systematic effect substantially, even at couplings around \(\alpha =0.15\). We interpret this result as a warning: estimating errors within perturbation theory is notoriously difficult, and one may chance one’s luck by being too aggressive in this step.

The work presented in this paper constitutes a major step in the \(\alpha _s\)-determination by the ALPHA-collaboration [12]. Despite considerable improvements in the precision, this step currently still contributes the largest single error in this project. One may therefore hope for further progress, perhaps by combining the \(\hbox {SF}_\nu \) schemes with alternative schemes. Gradient flow couplings are obvious candidates, provided the problems with large cutoff effects can be solved [13, 47], and the perturbative information is pushed at least to the same level as for the SF coupling. The latter step is possible based on numerical stochastic perturbation theory [31, 32, 33]. Finally we note that, given the coupling results, similar non-perturbative tests of perturbation theory might also be performed using the quark mass parameters [48].

## Footnotes

- 1.
For a recent discussion in the context of \(\alpha _s\)-determinations cf. Ref. [8].

- 2.
Here by observable we mean some finite quantity defined by the Euclidean path integral, that can be estimated in a Monte Carlo simulation of lattice QCD.

- 3.
In practice we defined this to mean gauge configurations for which \(|Q|<0.5\), with

*Q*defined as in Ref. [28]. - 4.
- 5.
Here and in the following the \(i=0\) label refers to unimproved data, for instance \(\Sigma ^{(0)}(u,a/L) = \Sigma (u,a/L)\), etc.

- 6.
The recursion towards larger

*n*requires a numerical inversion of the step-scaling function. This is not a problem given that the step-scaling function is, in practice, found to be a monotonously increasing function for the range of couplings considered here. - 7.
In Ref. [1] this fit parameter was denoted \(b_3^\mathrm{eff}\).

## Notes

### Acknowledgements

We thank our colleagues of the ALPHA collaboration, in particular C. Pena, S. Schaefer, H. Simma and U. Wolff for many useful discussions. We thank the computer centres at HLRN (bep00040) and NIC at DESY, Zeuthen for providing computing resources and support. S.S. is grateful to the CERN Theory group for the hospitality extended to him. P.F. acknowledges financial support from the Spanish MINECO’s “Centro de Excelencia Severo Ochoa” Programme under Grant SEV-2012-0249, as well as from the Grant FPA2015-68541-P (MINECO/FEDER). This work is based on previous work [49] supported strongly by the Deutsche Forschungsgemeinschaft in the SFB/TR 09.

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