{'fraction_non_alphanumeric': 0.048106448311156604, 'fraction_numerical': 0.02770620830833304, 'mean_word_length': 4.0416370106761565, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 8, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The Realistic Lattice Determination of alpha_s(M_Z) Revisited

We revisit the earlier determination of alpha_s(M_Z) via perturbative analyses of short-distance-sensitive lattice observables, incorporating new lattice data and performing a modified version of the original analysis. We focus on two high-intrinsic-scale observables, log(W_11) and log(W_12), and one lower-intrinsic scale observable, log(W_{12}/u_0^6), finding improved consistency among the values extracted using the different observables and a final result, alpha_s(M_Z)=0.1192(11), 2 sigma higher than the earlier result, in excellent agreement with recent non-lattice determinations and, in addition, in good agreement with the results of a similar, but not identical, re-analysis by the HPQCD Collaboration. A discussion of the relation between the two re-analyses is given, focussing on the complementary aspects of the two approaches.


I. INTRODUCTION
The strong coupling α s is usually characterized by giving the value, α s (M Z ), in the MS scheme at the conventionally chosen n f = 5 reference scale µ = M Z . A high precision determination of α s (M Z ) based on the perturbative analysis of short-distance-sensitive lattice observables computed using the a ∼ 0.09, 0.12 and 0.18 fm n f = 2 + 1 MILC data was presented in Ref. [1]. The result, α s (M Z ) = 0.1170 (12), plays a dominant role in fixing the central value of the current PDG assessment [2], α s (M Z ) = 0.1176 (20).
Over the last year, a number of improved non-lattice determinations of α s (M Z ) have appeared, in a variety of independent processes, over a wide range of scales [4,5,6,7,8,9,10,11,12,13]. The results, given in Table I (with all errors combined in quadrature), yield a weighted average, α s (M Z ) = 0.1190 (10), ∼ 2σ higher than the lattice determination.  [4,5] 0.1191 ± 0.0027 H1+ZEUS NLO inclusive jets [6] 0.1198 ± 0.0032 H1 high-Q 2 NLO jets [7] 0.1182 ± 0.0045 NNLO LEP event shapes [8] 0.1240 ± 0.0033 NNNLL ALEPH+OPAL thrust distributions [9] 0.1172 ± 0.0022 σ[e + e − → hadrons] (2-10.6 GeV) [10] 0.1190 +0.0090 −0.0110 Γ[Υ(1s)→γX] Γ[Υ(1s)→X] [11] 0.1190 +0.0060 −0.0050 hadronic τ decay [12,13,14] 0.1187 ± 0.0016 This difference, though not large, motivates revisiting the lattice analysis, especially in light of the existence of new high-scale (a ∼ 0.06 fm) lattice data not available at the time of the earlier study. We perform such an extended re-analysis in this paper. The rest of the paper is organized as follows. In Section II, we outline the original analysis, specify our own strategy for implementing the underlying approach, and clarify the difference between our implementation and that of the earlier study and recent HPQCD re-analysis. In Section III, we discuss the details of, and input to, our version of the analysis. Finally, in Section IV, we present and discuss our results.

II. THE LATTICE DETERMINATION OF α s (M Z )
A. The original HPQCD/UKQCD analysis In Ref. [1], α s (M Z ) was extracted by studying perturbative expansions for a number of UV-sensitive lattice observables, O k . The generic form of this expansion is where 2 ) have been computed in 3-loop lattice perturbation theory [16], and, with the corresponding d k , tabulated for a number of O k in Refs. [1,16,17]. In Eq. (1), α V (µ) is a coupling with the same expansion to O(α 3 s ) (with α s the MS coupling) as the heavy quark potential coupling, α p V , but differing from it, beginning at O(α 4 s ), in a way that will be specified below. The expansion coefficients are known to O(α 4 s ), and hence the β function of α V , defined in our conventions by µ 2 da V (µ)/dµ 2 = − n=0 β V n a n+2 V (µ), with a V ≡ α V /π, is determined to 4 loops by the known coefficients, β 0 , · · · , β 3 , of the 4-loop MS β function [18]. The coefficientsc 3 tabulated in Refs. [1,16,17] are valid for expansions of the O k in terms of any variable, α T , sharing the same expansion as α V out to O(α 3 s ).
With only the known, third order terms in the expansions of the O k , no value for the reference scale coupling, α V (7.5 GeV) ≡ α 0 V , was found to produce a simultaneous fit to the data at all three lattice spacings employed [1]. In consequence, terms out to tenth order in the expansion of Eq. (1) were incorporated, the unknown coefficientsc (k) 4,···,10 being fitted using input Bayesian prior constraints. The 4-loop version of β V was used to run α 0 V to the scales Q k relevant to each of the given observables at each of the three lattice spacings. Linear extrapolation in the quark masses was employed, and possible residual mass-independent non-perturbative (NP) contributions estimated, and subtracted, using the known leading-order gluon condensate contributions to the relevant Wilson loops [19].
The scales r 1 /a and r 1 , which determine the lattice spacing, a, in physical units, as well as the gluon condensate, α s G 2 /π , required for the mass-independent NP subtraction, were determined as part of the independent fit performed for each of the O k . This was accomplished using an augmented χ 2 function in which the squared deviations of the relevant parameters from their input central values were scaled by the squares of the input prior widths. For r 1 /a and r 1 the central values and widths were provided by the measured values and their uncertainties. For α s G 2 /π , a central value 0 and uncertainty ±0.010 GeV 4 (∼ the conventional SVZ value 0.012 GeV 4 [20]) were employed [21]. While this procedure allows r 1 /a and r 1 (which should be characteristic of the lattice under consideration) to take on values which vary slightly with the O k being analyzed, one should bear in mind that the measured uncertainties, which set the range of these variations, are small compared to the variation of scales across the a ∼ 0.09, 0.12 and 0.18 fm lattices employed in the analysis. The impact of any potential unphysical observable-dependence of the physical scales on the fitted α 0 V andc (k) n should thus be safely negligible. The situation with regard to the independent fitting of α s G 2 /π for each O k is potentially more complicated, and will be discussed further below.
The resulting best fit value for α 0 V , averaged over the various observables, was then matched to the n f = 3 MS coupling, and the corresponding n f = 5 result, α s (M Z ), obtained via standard running and matching at the flavor thresholds [22,23], yielding the result, α s (M Z ) = 0.1170(12), already quoted above.
Regarding the conversion from α V to α s , one should bear in mind that, while the expansion for α V in terms of α s is, in principle, defined to all orders (see below for more on this point), the coefficients beyond O(α 4 s ) involve the currently unknown MS β function coefficients β 4 , β 5 , · · ·. The n f = 3 conversion step is thus subject to a (hopefully small) higher order perturbative uncertainty. As will be explained in Section II C, with the definition of α V employed in Ref. [1], the higher order perturbative uncertainties are, in fact, entirely isolated in the V → MS conversion step of the analysis.
B. An alternate implementation of the HPQCD/UKQCD approach The higher order perturbative uncertainty encountered in matching α V to α s can be removed entirely by working with any expansion parameter, α T , whose expansion in α s is fully specified. We take α T to be defined by the third-order-truncated form of the relation between α p V (µ 2 ) and α s (µ 2 ) [24] which, for n f = 3, yields α T (µ 2 ) = α s (µ 2 ) 1 + 0.5570α s (µ 2 ) + 1.702α 2 s (µ 2 ) .
The β function for α T , β T , is then determined to 4-loops by the known values of β 0 , · · · , β 3 . With all coefficients on the RHS positive, α T runs much faster than α s , a fact reflected in the significantly larger values of the non-universal β function coefficients, β T 2 = 33.969 and β T 3 = −324.393. This makes running α T using the 4-loop-truncated β T function typically unreliable at the BLM scales corresponding to the coarsest (a ∼ 0.18 fm) lattices considered here. Since, however, the 4-loop-truncated MS running of α s remains reliable down to these scales, and the relation, Eq. (2) is, by definition, exact, the running of α T may be performed by converting from α T to α s at the initial scale, running α s to the final scale, and then converting back to α T . This proceedure will be especially reliable for ) k like log(W 11 ) and log(W 12 ) with lowest BLM scales > 3 GeV.
Though the conversion from the fitted reference scale α T value to the equivalent MS coupling α s can be accomplished without perturbative uncertainties, higher order perturbative uncertainties do remain in the analysis. To see where, define α 0 ≡ α T (Q 0 ), with Q 0 = Q max k = d k /a min the maximum of the BLM scales (corresponding to the finest of the lattice spacings, a min ) for the observable in question. Expanding the couplings at those BLM scales corresponding to coarser lattices, but the same observable, in the standard manner as a power series in where the known numerical values of β T 0 , · · · , β T 3 have been employed, and we display only terms involving one or more of the unknown quantities β T , · · ·. Running the MS coupling numerically using the 4-loop-truncated β function is equivalent to keeping terms involving β 0 , · · · , β 3 to all orders, and setting β 4 = β 5 = · · · = 0. The neglect of β 4 , β 5 · · · means that β T 4 , β T 5 , · · · also do not take on their correct physical values, leading to an alteration of the true t k -dependence, beginning at O(α 6 0 ). Since it is the scale-dependence of O k which is used to fit the unknown coefficients c  as well, and, in consequence, a further compensating shift in α 0 . From Eq. (3), the size of such effects, associated with the truncation of the running, and unavoidable at some level, can be minimized by taking Q 0 as large as possible (achieved by working with the observable with the highest intrinsic BLM scale) and keeping t k from becoming too large (achieved by restricting one's attention, if possible, to a subset of finer lattices) [25].

C. More on the relation between the two implementations
For n f = 3, in our notation, the relation between α p V and α s , (3), and κ 1 (x) = [7 + 4β 0 log(x)] /4π. Our expansion parameter, α T (q 2 ) is defined to be equal to the RHS of Eq. (4) with µ 2 = q 2 , leading to the numerical result given in Eq. (2). The conversion from α T to α s can be performed exactly but the absence in β T 4,5,··· of terms ∝ β 4,5,··· induces a perturbative uncertainty in the values of our fitted parameters, one which can, however, be reduced by working with high scale observables and fine lattices. It is also possible to test for its presence by expanding the fits to include coarser lattices, where the effects of the omitted contributions will be larger.
The construction of the expansion parameter α V is somewhat more complicated, but turns out to be equivalent to the following [26]. One first takes the RHS of Eq. (4), with . The corresponding β function, β ′ , is then determined to 4-loops by β 0 , · · · , β 3 . The higher order coefficients, β ′ 4,5,··· , however, depend on the presently unknown β 4,5,··· , are hence are themselves unknown. The final HPQCD coupling, α V , is obtained from α ′ V by adding terms of O(α 5 s ) and higher with coefficients chosen to make β V 4 = β V 5 = · · · = 0. Since β 4,5,··· are not known, the values of the coefficients needed to implement these constraints are also not known. The coupling is nonetheless, in principle, well-defined, with higher order coefficients computable as soon as the corresponding higher order β k become available. Since the 4-loop-truncated β V function is, by defnition, exact, the distortions of the fit parameters induced, in general, by the 4-loop truncation of the running are absent for the α V coupling. The price to be paid for this advantage is the unknown perturbative uncertainty in the relation between α V and α s , which affects the conversion and running to α s (M Z ). With this definition, α V differs from α T beginning at O(α 4 s ). The other difference between the two re-analyses lies in the treatment of r 1 /a, r 1 , and α s G 2 /π . In Ref. [1], these are allowed to vary independently, though within the range of the input prior constraints, for each O k , whereas in our analysis, they are treated as fixed external input, and have the same central values for all O k . As noted above, the difference in the treatment of r 1 /a and r 1 is expected to have a negligible impact. The impact of the differing treatments of α s G 2 /π should be similarly negligible for observables with intrinsic scales high enough that the associated correction is small.
The two different implementations of the original HPQCD/UKQCD approach will thus, when restricted to high-scale observables, correspond to isolating residual higher order perturbative uncertainties in different sectors of the analysis. If these uncertainties are, as desired, small in both cases, the two analyses should be in good agreement. Such agreement (which is, in fact, observed, provided comparison is made to the very recent HPQCD update) serves to increase confidence in the results of both analyses.

III. DETAILS OF OUR RE-ANALYSIS
In our analysis, we have calculated the desired Wilson loops using the publicly available a ∼ 0.09, 0.12, 0.15 and 0.18 fm MILC n f = 2+1 ensembles and incorporated information on W 11 and W 12 for the three a ∼ 0.06 fm USQCD ensembles provided to us by Doug Toussaint of the collaboration.
We follow the basic strategy of the earlier analysis, using the same 3-loop perturbative input, but with the following differences in implementation. First, we employ the expansion parameter α T throughout. All running of α T is carried out using exact 4loop-truncated running of the intermediate variable, α s , whose relation to α T is given by Eq. (2). Second, to minimize the effect of our incomplete knowledge of the running of α T beyond 4-loop order, the impact of which will be larger for coarser lattices, we perform "central" 3-fold versions of our fits using the three finest lattices, with a ∼ 0.12, 0.09 and 0.06 fm. Expanded 5-fold fits then serve as a way of studying the impact of the truncated running, as well as of the truncation of the perturbative expansion for the O k . Since we do not currently have access to the actual a ∼ 0.06 fm configurations, we are restricted to analyzing the three observables indicated above. One of these, log(W 12 /u 6 0 ), has a significantly lower BLM scale, and hence is particularly useful for studying the impact of these truncations. As in Ref. [1], we extrapolate linearly in the quark masses [27], and estimate (and subtract) residual mass-independent NP effects using the known form of the leading order gluon condensate contributions to the relevant Wilson loops.
Regarding the mass extrapolation, the sets of configurations for different mass combinations am ℓ /am s corresponding to approximately the same lattice spacing a in fact have slightly different measured r 1 /a. Since the O k we study are themselves scale-dependent, full consistency requires converting the results corresponding to the different am ℓ /am s to a common scale before extrapolation. This could be done with high accuracy if the parameters appearing in the perturbative expansion of the O k were already known. Since, however, some of these parameters are to be determined as part of the fit, the extrapolation and fitting procedure must be iterated. With sensible starting points, convergence is achieved in a few iterations. The dominant uncertainty in the converged iterated extrapolated values is that associated with the uncertainties in r 1 /a. There is also a 100%correlated global scale uncertainty associated with that on r 1 . We employ r 1 = 0.318 (7) fm, as given in the MILC Lattice 2007 pseudoscalar project update [28].
The mass-independent NP subtractions are estimated using the leading order (LO) D = 4 gluon condensate contribution, δ g W mn , to the m × n Wilson loop, W mn [19] δ g W mn = −π 2 36 m 2 n 2 a 4 α s G 2 /π (5) and the central value, α s G 2 /π = (0.009 ± 0.007) GeV 4 , of the updated charmonium sum rule analysis [29]. Since the error here is already close to 100%, we take the difference between results obtained with and without the related subtraction as a measure of the associated uncertainty. This should be sufficiently conservative if the correction is small. If not, the measured O k values may contain additional non-negligible mass-independent contributions, of dimension D > 4, which we do not know how to estimate and subtract. O k for which this occurs will thus provide a less reliable determination of α s . Fortunately, for the observables we consider, the gluon condensate correction is, as desired, small. For O k = log(W 11 ), the corrections required for the 3-fold (5-fold) fit do not exceed ∼ 0.1% (∼ 0.5%). The corrections remain small (less than ∼ 0.4% (∼ 1.8%)) for O k = log(W 12 ). The effect is somewhat larger for log(W 12 /u 6 0 ), as a consequence of cancellations encountered in combining the uncorrected log(W 11 ) and log(W 12 ) values, but still reaches only ∼ 1.3% (∼ 5.6%) for the 3-fold (5-fold) fit [30].
In line with what was seen in Ref. [1], we find that the known terms in the perturbative expansions of the O k are insufficient to provide a description of the observed scale-dependence, even when only the three finest lattices are considered. When c (k) 3 is added to the fit, however, we find very good fits, with χ 2 /dof < 1 (very significantly so for the 3-fold fits). With current errors, it is thus not possible to sensibly fit additional coefficients c (k) m>3 . This raises concerns about possible truncation uncertainties. Comparison of the results of the 3-fold and 5-fold fits provides one handle on such an uncertainty since the relative weight of higher order to lower order terms grows with decreasing scale. If neglected higher order terms are in fact not negligible, the growth with decreasing scale of the resulting fractional error should show up as an instability in the values of the parameters extracted using the different fits. We see no signs for such an instability within the errors of our fits, but nonetheless include the difference of central values obtained from the 3-fold and 5-fold fits as a component of our error estimate.

IV. RESULTS
Central inputs for our fits are the measured lattice observables (whose errors are tiny on the scale of the other uncertainties), the computed D k , c  [1,16], r 1 /a, r 1 and α s G 2 /π , and the choice of the 3-fold fitting procedure. In addition to the uncertainties generated by the errors on r 1 /a, r 1 and α s G 2 /π , are those due to uncertainties in numerical evaluations of the D k , c We construct an "overall scale uncertainty error" by adding linearly the fit uncertainties generated by those on r 1 and the r 1 /a. This combined error is added in quadrature to (1) uncertainties produced by varying the c  [31], and r allowed to vary between 1 and 3. These uncertainties in the matching thresholds, together with standard estimates for the impact of the truncated running and matching, produce an evolution contribution to the uncertainty on α s (M Z ) of ±0.0003 [4].   Table II. For comparison, the results for α s (M Z ) obtained in Ref. [1] were 0.1171(12), 0.1170 (12) and 0.1162 (12), for log(W 11 ), log(W 12 ) and log(W 12 /u 6 0 ), respectively. Our α s (M Z ) are significantly larger, and in closer mutual agreement. The recent HPQCD update [17] also finds significantly larger values. (We will return to a more detailed comparison of the two updates below.) The very good agreement between the α s (M Z ) values obtained in our fits using both lowand high-scale observables suggests that the effects of the truncated running, present at some level in all such fits, are small in the cases we have studied.
One-sided versions of the various components of the total errors on α s (M Z ) are displayed in Figure 1. The difference of the 3-fold and 5-fold determinations is ∼ 0.0004, significantly smaller than the ∼ 0.0009 overall scale uncertainty. The results thus show no evidence for any instability associated with opening up the fit to lower scales.
While the total error on α s (M Z ) is the same for all three O k considered, the general arguments above lead us to believe that the most reliable determination is that obtained using the highest-scale observable, log(W 11 ), and highest-scale (3-fold fit) analysis window. Our final assessment, is in excellent agreement with the non-lattice average and the result, 0.1184 ± 0.0009, of the independent HPQCD analysis. The various results are shown for comparison in Figure 2. A more detailed discussion of the relation between our re-analysis and that of HPQCD may be found in the Appendix.  [17] with the average of recent non-lattice determinations.

APPENDIX: MORE ON THE RELATION TO THE HPQCD RE-ANALYSIS
After the completion of the work reported in this paper, the HPQCD Collaboration posted an update of their earlier 2005 analysis [17]. This update works with a subset of 11 of the available MILC ensembles, spanning the a ∼ 0.18, 0.15, 0.12, 0.09 and 0.06 fm lattices and a range of am ℓ /am s . The fits follow the strategy of the earlier analysis [1], employing the expansion parameter α V , and fitting the unknownc (k) n using priors. Linear mass extrapolation has been employed, and mass-independent NP D = 4 contributions estimated and subtracted using the LO formula for δ g W mn . The fitting of r 1 /a, r 1 and α s G 2 /π , observable by observable, using central input and prior widths, is also as in the earlier analysis, with the exception that the central value and width for α s G 2 /π are now 0 and ±0.012 GeV 4 , respectively.
The HPQCD implementation differs from ours in the choice of expansion parameter, and in the implementation of the input information on r 1 /a, r 1 and αs π G 2 . For the reasons discussed above, we expect the impact on α s (M Z ) of the observable-by-observable fitting of r 1 /a, r 1 and α s G 2 /π in the HPQCD approach to be small for O k having small gluon condensate corrections. Since the different choices of expansion parameter correspond to different ways of isolating residual higher-order perturbative uncertainties, one expects the results of the two analyses to be in good agreement so long as (i) one is working with O k having small mass-independent NP corrections, (ii) the same input values are used for both, and (iii) residual NP and higher-order perturbative uncertainties are indeed small. The situation is likely to be more complicated for O k with sizeable estimated D = 4 gluon condensate corrections.
The results of the HPQCD fit for the three O k we consider are α s (M Z ) = 0.1186(9), 0.1186(9) and 0.1183 (8) for log(W 11 ), log(W 12 ) and log(W 12 /u 6 0 ), respectively [17]. All are in good agreement within errors with the corresponding results from our analysis. This agreement is further improved if one takes into account the small difference in input r 1 values. Were we to switch from r 1 = 0.318 fm to the central value of the HPQCD determination, 0.321(5) fm, all three of our α s (M Z ) results would decrease by 0.0002. Note also that use of the central charmonium sum rule input for α s G 2 /π in our calculation raises the output α s (M Z ) obtained from log(W 11 ), log(W 12 ) and log(W 12 /u 6 0 ) by 0.0001, 0.0004 and 0.0005, respectively. Our fitted values would thus be in even closer agreement with those of the HPQCD update were we to impose the HPQCD central zero value of α s G 2 /π in our fits. Since the fitted α s G 2 /π values obtained by analyzing the various O k are not quoted in Ref. [17], it is not possible to quantify further the role of this effect. The agreement for the three observables under discussion is in any case good, within expectations, independent of this question.
We now turn to a more detailed discussion of the issue of the subtraction of the mass-independent NP contributions. If the estimated LO, D = 4 gluon condensate subtraction represents only a small fraction of the measured O k at the scales under consideration, analogous mass-independent NP contributions with D > 4 should be even smaller, and hence safely negligible. If, however, the estimated D = 4 correction is sizeable, analogous D > 4 corrections can no longer be expected to be small. These necessarily scale differently with lattice spacing than do the D = 0 perturbative and D = 4 NP contributions and hence, if not included when fitting the data, are likely to force shifts in both α s and α s G 2 /π if present at a non-negligible level.
We deal with this potential problem by focussing on O k for which the impact of the estimated D = 4 gluon condensate subtraction is small compared to the variation of the O k in question over the lattice scales employed in the fit. In the initial version of the HPQCD re-analysis, mass-independent NP subtractions were estimated using only the D = 4 gluon condensate form, even for observables where the estimated correction is sizeable. In the more recent update, additional terms, scaling as would mass-independent contributions of D > 4, are added to the fit function for each observable, and the accompanying coefficients extracted as part of the augmented Bayesian fit. The impact of including the D > 4 terms is, as expected, small for those observables having small values of the estimated D = 4 subtraction. For observables with larger D = 4 subtractions, the fit errors are increased (by factors of ∼ 2 for those observables having the largest D = 4 corrections) and some shifts in α s (M Z ) of order 1/2 to 1 times the smaller preliminary errors are observed. The shifts serve to reduce the spread of α s (M Z ) values compared to that seen in the original version of the re-analysis. The values of α s G 2 /π obtained from the independent fits to the different observables are not quoted in Ref. [17], but a useful test of the self-consistency of the approach would be to verify that the inclusion of the D > 4 contributions has brought these values into good agreement with one another.
It is worth noting that the observables, log(W 23 /u 10 0 ), log(W 14 /W 23 ), and log(W 11 W 23 /W 12 W 13 ), which produce the three smallest results for α s (M Z ), have estimated D = 4 corrections significantly larger than those for any of the other observables. The magnitudes of the corrections in these cases represent ∼ 50 − 100% of the variation with scale of the uncorrected O k between the lightest mass a ∼ 0.06 and a ∼ 0.12 fm ensembles. (This variation-with-scale provides a suitable measure for use in assessing the importance of NP corrections since it is the variation with scale which provides the input needed to fix the fit parameters, and, as explained in Ref. [17], the a ∼ 0.06, 0.09 and 0.12 fm ensembles which dominate the HPQCD re-analysis.) These observables are thus, for the purposes of the analysis, rather non-perturbative. Were one to exclude observables with larger NP contributions from the HPQCD average, on the grounds that the related subtractions introduce additional theoretical systematic uncertainties, the HPQCD result would be brought into even closer agreement with ours, though the resulting shift would in fact be small (at the ∼ quarter σ level).
We stress that, independent of these questions, our results agree well within errors with those of the HPQCD update. This agreement is further improved by a shift to common input. We argue that the non-zero central value for α s G 2 /π obtained from the updated charmonium sum rule analysis represents our best present knowledge of this quantity, and hence also the best choice as input for evaluating the small mass-independent NP subtractions needed for extracting α s (M Z ). In addition, for the reasons just discussed, we believe that the most reliable determinations of α s (M Z ) are those based on those observables for which the D = 4 correction is as small as possible. Such an assessment produces the results already noted above, which are in extremely good agreement with what it known from other sources.