Strong coupling constant from moments of quarkonium correlators revisited

We revisit previous determination of the strong coupling constant from moments of quarkonium correlators in (2+1)-flavor QCD. We use previously calculated moments obtained with Highly Improved Staggered Quark (HISQ) action for five different quark masses and several lattice spacings. We perform a careful continuum extrapolations of the moments and from the comparison of these to the perturbative result we determine the QCD Lambda parameter, $\Lambda_{\overline{MS}}^{n_f=3}=332 \pm 17 \pm 2(scale)$ MeV. This corresponds to $\alpha_s^{n_f=5}(\mu=M_Z)=0.1177(12)$.

moments can be calculated in perturbation theory in MS scheme G n = g n (α s (µ), µ/m h ) am n−4 h (µ m ) .
Here µ is the MS renormalization scale. The scale, µ m at which the MS heavy-quark mass is defined could be different from µ in general [12]. The coefficient g n (α s (µ), µ/m h ) is calculated up to 4-loop, i.e. including the term of order α 3 s [13,14,15].
In lattice calculations it is more practical to consider the reduced moments [3] R n = where G (0) n is the moment calculated from the free correlation function. The lattice artifacts largely cancel out in these reduced moments.
It is straightforward to write down the perturbative expansion for R n : From the above equations it is clear that R 4 is suitable for the extraction of the strong coupling constant α s (µ) at scale proportional to the heavy-quark mass, m h , while the ratios R n /m h0 with n ≥ 6 are suitable for extracting the heavy-quark mass once α s (µ) is determined. One can also use the ratios of the reduced moments, namely R 6 /R 8 and R 8 /R 10 to determine α s . We will discuss these ratios in the Appendix.

Continuum extrapolations of the reduced moments of quarkonium correlators
In our analysis we used previously published lattice QCD results for the reduced moments in (2+1)-flavor QCD obtained for heavy-quark masses m h = m c , 1.5m c , 2m c , 3m c and 4m c , Table 1 The continuum results for the reduced moments of quarkonium correlators at different heavy-quark masses. The last column shows the α s values extracted from R 4 with µ = m h . The first, second, and third errors in α s correspond to the lattice error, the perturbative error, and the error due to the gluon condensate, respectively, see text.
1.0m c 1.2778 (20) 1.0200 (16) 0.9166 (17) 0.8719 (21) 0.3798(28)(31) (22) 1.5m c 1.2303(30) 1.0792 (20) 0.9860 (20) 0.9462 (23) 0.3151(43) (  with m c being the charm-quark mass [8]. The lattice calculations have been performed using HISQ action at several values of the lattice spacing [8]. The lattice spacing has been fixed through the r 1 parameter from the static quark-antiquark potential [9,10,11], and the value r 1 = 0.3106(18) fm (7) obtained from the pion decay constant was used [16]. Furthermore, the calculations have been performed at two values of the light quark masses corresponding to the pion mass of 161 MeV and 320 MeV in the continuum limit, and no dependence on the light quark mass of the reduced moments was found within errors [8]. The lattice results on R n , n = 4, 6, 8, 10 are found in Tables VII-XI of Ref. [8] for different lattice spacings. The errors in the tables include statistical errors, errors related to mistuning of the charm-quark mass and finite volume errors. All the errors have been added in quadrature. The bare charm-quark masses are found in Table I of [8].
Because the tree-level lattice artifacts cancel out for the reduced moments the lattice spacing dependence can be parameterized as where α b s = g 2 0 /(4πu 4 0 ), g 2 0 = 10/β is the boosted gauge coupling. We performed joint fits of the lattice results on R 4 and R n /m h0 obtained at different quark masses to Eq. (8) and Eq. (9) setting d i jk = e i jk = 0. The reason for setting the coefficients of the log terms to zero was to avoid having too many poorly constrained parameters since the logarithmic dependence on am h0 is much weaker than the power-law dependence. For the continuum extrapolations of R 4 , where the lattice spacing dependence is the most prominent, we also performed fits allowing for a few terms proportional to log(am h0 ). These fits are discussed 7 in the Appendix. Furthermore, the maximal number of terms N and M i in Eqs. (8) and (9) should be sufficiently large so that higher order terms have negligible impact on the continuum extrapolation. The continuum extrapolations of R 4 based on the joint fits of m h = m c − 4m c lattice data turned out to be much more stable with respect to fit range variations than the extrapolations performed in Ref. [8] separately for each value of m h . In particular, there was no problem incorporating 4m c data in the analysis unlike in Ref. [8], where this was not possible. A sample fit of the data on R 4 is shown in Fig. 1 As a cross-check we also use Akaike information criterion (AIC) [18,19] to obtain continuum result for R 4 from the performed fits. First, we calculate the AIC weights for each fit and then calculate the weighted average of the fit results with the corresponding weights.
Interestingly, this resulted in central values of R 4 very similar to those shown in Table 1.
Furthermore, as mentioned above we also performed the continuum extrapolations, which allow for a few terms proportional to log(am h0 ). The corresponding continuum results for R 4 are not significantly different from the ones in Table 1, see Appendix.
To obtain continuum results for R n /m h0 , n ≥ 6 it is sufficient to consider fits with N = 1 and M 1 = 2. This is because the errors on R n /m h0 are much larger than for R 4 . These errors are dominated by the uncertainties in m h0 , which are essentially the uncertainties in m c0 multiplied by the corresponding constant (3/2, 2, 3 and 4). The uncertainties in m c0 come from the errors in tuning the charm-quark mass in the lattice calculations due to the errors of the ground state charmonium mass and the error in the lattice spacing [8]. The errors on m c0 are given in the sixth column of Table 1 in Ref. [8]. We used several values of (am h0 ) max in our fits. The differences in the central values of (R n /m h0 ) cont corresponding to the fits with various (am h0 ) max turned out to be much smaller than the statistical errors. So one can choose any of these fit results. For the final continuum estimate we choose the fits with the smallest χ 2 /d f , which turned out to be the fits with (am h0 ) 2 max = 1.0 for R 6 /m h0 and R 8 /m h0 , and the fit with (am h0 ) 2 max = 0.8 for R 10 /m h0 . The corresponding results are given in Table 1.
The new continuum results agree very well with the previous ones but have smaller errors.
For some cases the error reduction is significant.
Having determined the continuum limit of the reduced moments of quarkonium correlators we are in the position to obtain the strong coupling constant. As discussed in section II the value of α s (µ) can be obtained by comparing R 4 calculated in perturbation theory to the continuum extrapolated lattice result. The renormalization scale µ has to be of the order of the heavy-quark mass. We also need to fix the renormalization scale µ m at which the heavy-quark mass is defined. The most natural choice is µ m = µ and will be used throughout in this paper. We consider the following choices of the renormalization scale,  Table 1. There is a perturbative error due to missing higher order corrections. We estimated this error in the same way as in Ref. [8], namely we added a term proportional to r 43 (α s /π) 4 with the coefficient which was varied between −5 and +5, to be conservative. Finally, there is an error due to the gluon condensate contribution which was estimated in Ref. [8] by varying the poorly know gluon condensate by factor two.
For larger heavy-quark mass the perturbative error is smaller because the corresponding α s (µ) is smaller. The lattice error on the other hand is larger for larger m h since R 4 is closer to one and the relative error on R 4 increases. The error due to the gluon condensate rapidly decreases with increasing m h and is negligible for m h > 2m c . As an example we show the values of α s for µ = m h in Table 1 are several uncertainties in this determination. One is due to the error in the continuum extrapolated value of R n /m h0 . The second source of the uncertainty is due to the missing higher order perturbative corrections in R n . There is also an uncertainty due to the gluon condensate contribution. These have been estimated in the same manner as in the case of R 4 . Finally there is an uncertainty due to the error in α s . The latter in turn is also affected by the error due to the gluon condensate contribution, which is correlated with the gluon condensate error in R 6 . This correlation should be taken into account. The perturbative error in R 4 and R n , n ≥ 6 can be assumed to be uncorrelated. The statistical errors in R 4 and The charm-quark masses obtained from R 10 /m c0 are not shown since they appear to be very close to the ones obtained from R 8 /m c0 . The values of m c determined for different µ/m h but same value of µ in GeV also agree with each other except for µ/m h = 3, which are about two sigma lower than the ones for µ/m h = 1, for µ < 5 GeV. This fact may indicate that the above procedure of estimating the perturbative error due to missing higher order terms was not sufficiently conservative for µ < 5 GeV. Since the dependence of α s on the charm-quark mass is logarithmic the above small inconsistency in m c determination is insignificant compared to other sources of errors. We also note that because of the uncertainty in the absolute scale we could not improve significantly the charm-quark mass determination compared to the result of Ref. [8] despite the reduced errors in the continuum values for R n /m c0 , n ≥ 6.
Having determined the charm-quark mass we are in a position to obtain the running coupling constant and the Λ-parameter for three-flavor QCD. Using the values of α s shown in Fig. 2 Table 2. In this Table we show Our error estimate for the Λ n f =3 MS parameter is quite conservative and is significantly larger than the one by HPQCD collaboration [4]. This is due to the absence of Bayesian priors and different choices of the renormalization scale µ and not just µ = 3m h . We compare our results for the Λ-parameter with other three-flavor lattice determinations, namely from the moments of quarkonium correlators [4], from the static quark anti-quark potential [22,23,24], from step-scaling analysis by ALPHA collaboration [25], from ghost-gluon vertex in Landau gauge [26], and from the light quark vector current correlator [27]. This comparison is shown Fig. 5. We see that our result is consistent with other lattice determinations. to the present determination. The result of Ref. [22] are not shown as these are superseded by Ref. [23].
bottom quark mass. The running charm-quark mass in MS scheme is obtained from R 8 as This value of α n f =5 s agrees with other determinations from the moments of quarkonium correlators within errors [3,4,5,7,28]. It also agrees with the averaged α n f =5 s from lattice determinations [1,2].
Before closing this section let us discuss the determination of the strong coupling constant from the ratios R 6 /R 8 and R 8 /R 10 . As mentioned in section II the heavy-quark mass drops out in these ratios and therefore they are well suited for the determination of α s .
Naively, one would expect that the continuum extrapolation of these ratios is simpler than for R 4 as the higher order moments are less sensitive to the short distance physics. Using such reasonings in Ref. [7] the strong coupling constant was determined using only these ratios. The ratios R 6 /R 8 and R 8 /R 10 have been also used in an attempt to determine α s with additional cross-checks [8]. It turns out, however, that the cutoff dependence of R 6 /R 8 and R 8 /R 10 is far from simple, and it is challenging to describe it quantitatively. Furthermore, the finite volume effects are also significant for these ratios. The continuum extrapolations for R 6 /R 8 and R 8 /R 10 from simultaneous fits of the lattice data at different quark masses are discussed in the Appendix. We explain there why these continuum extrapolations are difficult.
It turns out that in order to obtain consistent α s determination from these ratios additional priors have to be imposed. The continuum results for R 6 /R 8 at m h = m c and m h = 1.5m c are consistent with the previous results [8]. However, it is not possible to obtain reliable continuum results for R 6 /R 8 for m h > 2m c . In the case of R 8 /R 10 the finite volume effects are quite severe for m h = m c and therefore, reliable continuum results can be obtained only for m h ≥ 1.5m c . The continuum results for R 8 /R 10 turned out to be systematically larger than in Ref. [8]. As the result the corresponding α s values are larger than the α s values obtained from R 8 /R 10 in Ref. [8] and agree well with the corresponding ones obtained from R 4 . On the other side the strong coupling constant extracted from R 8 /R 10 has larger error and therefore, does not improve the precision of our α s determination. Nevertheless, it does provide a useful cross-check of our analysis.

Conclusion
In this paper we revisited the determination of the strong coupling constant from the moments of quarkonium correlators. Using previously published lattice results on the reduced moments in (2+1)-flavor QCD with heavy-quark masses m h = m c , 1.5m c , 2m c , 3m c and 4m c at several lattice spacings we estimated the continuum results on the fourth moment.
These estimates were based on simultaneous fits of the lattice spacing dependence of the reduced moments at several quark masses, similar to the analysis of HPQCD collaboration [4,5]. The new continuum estimates turned out to be much more robust compared to the ones obtained from fits of the cutoff dependence of R 4 performed separately for each quark mass [8]. While both studies use the same form to parameterize the cutoff dependence of R 4 , there is an essential difference. The present analysis strongly relies on the specific form of the cutoff dependence given by Eqs. (8) and (12), while in Ref. [8] it is just an effective way to parameterize the lattice spacing dependence of these quantities, and is not essential for the final continuum result. In this study we constrain the lattice spacing dependence at each heavy-quark mass with the lattice spacing dependence of all other heavy-quark masses, while the previous analysis in Ref. [8] permitted independent variation of the coefficients at different heavy-quark masses. The continuum results at m c and 1.5m c are in good agreement in these approaches. This is reassuring for controlling the continuum extrapolation of these quantities at least for the two lower values of the quark masses. We also revisited the continuum extrapolations of R 6 /R 8 and R 8 /R 10 using simultaneous fits of the lattice results at different quark masses. We have shown that the apparent weaker cutoff dependence of these ratios is misleading, and reliable continuum extrapolations are challenging. We were able to obtain reliable continuum extrapolations for R 6 /R 8 only for m h ≤ 2m c . For

A Continuum extrapolation of the ratios and α s determination
In this appendix we discuss continuum extrapolations for R 4 which allow for terms proportional to log(am h0 ).
Furthermore, we discuss the continuum extrapolations of the ratios R 6 /R 8 and R 8 /R 10 and the determination of α s from these ratios.
As discussed in the main text including terms proportional to log(am h0 ) is challenging as the logarithmic dependence on am h0 is much weaker than the power-law dependence. Therefore, only a few logarithmic terms can be included in the fits to avoid over-fitting and the number of terms in Eq. (8) should be also reduced. We   As discussed in the main text it is also possible to determine the strong coupling constant from the ratios R 6 /R 8 and R 8 /R 10 as the heavy-quark mass drops out in these ratios. The apparent cutoff dependence of the ratios R 6 /R 8 and R 8 /R 10 calculated on the lattice is indeed smaller than for R 4 [8]. As we have seen above, to describe the cutoff dependence of R 4 many powers of am h0 are needed and the coefficients often have opposite signs from one order in (am h0 ) 2 to the next one. Therefore, the apparent cutoff dependence of R 4 turns out to be smaller as we increase the heavy-quark mass contrary to the naive expectations. The situation could be similar for R 6 /R 8 and R 8 /R 10 . Furthermore, the cutoff dependence of the numerator and denominator, while being significant, could cancel out in the ratios, thus fooling one into thinking that cutoff effects are small and can be modeled with a low order polynomial in (am h0 ) 2 . We should keep these issues in mind when performing continuum extrapolations of the ratios.
To obtain the continuum result for R 6 /R 8 we perform simultaneous fits of the lattice data at different quark masses to As in Ref. [8] we omit data on fine lattices to avoid finite volume effects when performing fits. The χ 2 /d f of the fit is large unless we use high order polynomials in (am h0 ) 2 . However, using high order polynomials in the fit results in many poorly constrained parameters. Furthermore, a closer look at the lattice data reveals that the slope of the (am h0 ) 2 dependence is quite different for various m h , explaining why χ 2 /d f is large. The apparent Table 4 The continuum values of the ratios R 6 /R 8 and R 8 /R 10 and the corresponding coupling constants α s (µ = m h ) for different values of the heavy-quark masses m h , see text. deal with these problems we omit lattice results for m h ≥ 3m c . Including these data will require adding many more parameters in Eq. (12)  The fit for (am h0 ) 2 max = 0.6 as well as the main features of the lattice data for R 6 /R 8 are demonstrated in Fig.   6. No significant dependence of the continuum result on (am h0 ) 2 max have been found. We choose results of fits with (am h0 ) 2 max = 0.6 for the final continuum estimate, which are shown in Table 4. We also used AIC to obtain the continuum values and these were very close to the central value from the above fits. The new continuum estimate for R 6 /R 8 agree with the results of Ref. [8] within errors.
From the continuum results on R 6 /R 8 at m c and 1.5m c we determine the corresponding α s (m h ) by comparing to the 4-loop perturbative results, which are also given in Next we perform the continuum extrapolation for R 8 /R 10 . As for R 6 /R 8 we fit the cutoff dependence of the lattice results at different quark masses with Eq. (12). The finite volume effects are the largest for R 10 and thus for R 8 /R 10 , and it is possible that the finite volume errors in Ref. [8] were not adequate for many of the β values, especially in the case of m h = m c . This may explain why the α s values obtained from R 8 /R 10 were systematically lower [8]. in β for 6.74 ≤ β ≤ 7.28. We interpret this as indication that the finite volume errors are not under control for β > 6.88 and m h = m c . We note that the low central value of R 8 /R 10 is not unique to Ref. [8] but has been seen in other works [3,6,7] as well with the exception of Ref. [4]. The non-monotonic dependence of R 8 /R 10 with increasing β is also observed for m h = 1.5m c and 2m c but the maximum is shifted to significantly larger values of β. Finally, for m h = 3m c and 4m c this non-monotonic behavior cannot be clearly observed because of the large errors on the finest lattices. The above differences in the cutoff dependence of R 8 /R 10 at different quark masses make a simultaneous fit of the cutoff dependence very difficult. This difficulty is likely related to the finite volume effects. To solve this problem we discard data on R 8 /R 10 with small spatial extent. For m h = m c the finite volume effects are under control for β up to β = 6.88, which corresponds to the bare charm-quark mass am c0 = 0.48 and spatial extent N s = 48 (c.f. Table I Table 4. We also applied the AIC to different fit results and the resulting continuum estimates turned out to be close to the central value of the fit with (am h0 ) 2 max = 0.8. From the continuum values for R 8 /R 10 in Table 4 we determine α s (m h ) by comparing