# Strong coupling constant and heavy quark masses in 2+1 flavor QCD

###### Abstract

We present a determination of the strong coupling constant and heavy quark masses in 2+1-flavor QCD using lattice calculations of the moments of the pseudo-scalar quarkonium correlators at several values of the heavy valence quark mass with Highly Improved Staggered Quark (HISQ) action. We determine the strong coupling constant in scheme at four low energy scales corresponding to , , and , with being the charm quark mass and from these we obtain MeV, which is equivalent to . For the charm and bottom quark masses in scheme we obtain: GeV and GeV.

###### pacs:

12.38. Gc, 12.38.-t, 12.38.Bx## I Introduction

In recent years there was an extensive effort toward accurate determination of QCD parameters since the precise knowledge of these parameters is important for testing the predictions of the Standard Model. Two important examples are the sensitivity of Higgs branching ratios to the heavy quark masses and the strong coupling constant Dawson et al. (2013); Lepage et al. (2014) and the stability of the Standard Model vacuum Buttazzo et al. (2013); Espinosa (2014). Lattice calculations play an increasingly important role in the determination of the QCD parameters as these calculations become more and more precise with advances in computational approaches.

The strong coupling constant has been known for a long time. The current Particle Data Group (PDG) values Tanabashi et al. (2018) has small errors suggesting that the uncertainties are well under control. However, there are several phenomenological determinations of that give much smaller central values and have small errors Moch et al. (2014). Therefore, the errors of determinations may not be fully understood. The lattice average of provided by Flavor Lattice Averaging Group (FLAG) agrees well with the PDG average Aoki et al. (2017). However, there are several lattice determinations reporting smaller values Bazavov et al. (2012, 2014a); Maezawa and Petreczky (2016). The running of the strong coupling constant at lower energy scales is also interesting. For example, for testing the weak coupling approach to QCD thermodynamics through comparison to lattice QCD results Bazavov et al. (2018a, b, 2016); Berwein et al. (2016); Ding et al. (2015); Haque et al. (2014); Bazavov et al. (2013) one needs to know the coupling constant at a relatively low energy scale , with being the temperature. The analysis of the decay offers the possibility of extraction at a low energy scale but there are large systematic uncertainties due to different ways of organizing the perturbative expansion in this method (see Refs. Pich and RodrÃez-SÃ¡hez (2016); Boito et al. (2017a, b) for a recent work on this topic and references therein). Lattice QCD calculations on the other hand are well suited to map out the running of at low energies.

Lattice determination of the charm quark mass significantly improved over the years. Though, the ETMC collaborations reported much larger value for the charm quark mass Carrasco et al. (2014). Some lattice QCD calculations use 2 or 3 flavors of dynamical quarks Durr and Koutsou (2012); Davies et al. (2010a); Yang et al. (2015), while others use 4 dynamical flavors Chakraborty et al. (2015); Carrasco et al. (2014); Bazavov et al. (2014b). Therefore, additional calculations of the charm quark mass could be useful. These calculations can also help to better understand the dependence of the charm quark mass on the number of active quark flavors.

Determination of the bottom quark mass is difficult in the lattice simulations due to the large discretization errors caused by powers of , where is the bare mass of the heavy quarks. One needs a small lattice spacing to control the corresponding discretization errors. One possibility to deal with this problem is to perform calculations with heavy quark masses smaller than the bottom quark mass and extrapolate to the bottom quark mass guided by heavy quark effective theory McNeile et al. (2010); Bazavov et al. (2018c). Several determinations of the bottom to charm quark mass ratio have been reported and slight inconsistency has been found. The ratio recently obtained by the ETMC collaboration Bussone et al. (2016) shows smaller value than that previously determined by the HPQCD collaboration McNeile et al. (2010); Chakraborty et al. (2015) as well as the recent determination of the Fermilab, MILC and TUMQCD collaborations Bazavov et al. (2018c).

In this paper we report on the calculations of and the heavy quark masses in 2+1 flavor QCD using the Highly Improved Staggered Quark (HISQ) action and moments of pseudo-scalar quarkonium correlators. We extend the previous work reported in Ref. Maezawa and Petreczky (2016). This paper is organized as follows. In section II we introduce the details of the lattice setup. In section III we discuss the moments of quarkonium correlators and our main numerical results. The extracted values of the strong coupling constant and heavy quark masses are discussed in section IV and compared to other lattice and non-lattice determinations. The paper is concluded in Sec. V. Some technical details of the calculations are given in the Appendices.

## Ii Lattice setup and details of analysis

To determine the heavy quark masses and the strong coupling constant we calculate the pseudo-scalar quarkonium correlators in -flavor lattice QCD. As in our previous study we take advantage of the gauge configurations generated using the tree level improved gauge action Luscher and Weisz (1985) and the Highly Improved Staggered Quark (HISQ) action Follana et al. (2007) by HotQCD collaboration Bazavov et al. (2014c). The strange quark mass, , was fixed to its physical value, while for the light (u and d) quark masses the value was used. The latter corresponds to the pion mass MeV in the continuum limit, i.e. the sea quark masses are very close to the the physical value. This is the same set of gauge configurations as used in Ref. Maezawa and Petreczky (2016). We use additional HISQ gauge configurations with light sea quark masses , corresponding to the pion mass of MeV at five lattice spacings corresponding to lattice gauge coupling and , generated for the study of the QCD equation of state at high temperatures Bazavov et al. (2018b). This allows to perform calculations at three smaller lattice spacings, namely , and fm, and, also check for sensitivity of the results to the light sea quark masses. As we will see later, the larger than the physical sea quark mass has no effect on the moments of quarkonium correlators.

For the valence charm and bottom quarks we use the HISQ action with the so-called -term Follana et al. (2007), which removes the tree-level discretization effects due to the large quark mass up to . The HISQ action with -term turned out to be very effective for treating the charm quark on the lattice Follana et al. (2007); Bazavov et al. (2014b, 2015); Mohler et al. (2015). The lattice spacing in our calculations has been fixed using scale defined in terms of the energy of static quark anti-quark pair as

(1) |

We use the value of determined in Ref. Bazavov et al. (2010) using the pion decay constant as an input:

(2) |

In the above equation all the sources of errors in Ref. Bazavov et al. (2010) have been added in quadrature. The above value of corresponds to the value of the scale parameter determined from the Wilson flow fm Bazavov et al. (2014c). This agrees very well with determination of Wilson flow parameter by BMW collaboration fm Borsanyi et al. (2012). It is also consistent with the value fm reported by HPQCD collaboration within errors Davies et al. (2010b). It turns out that the value of does not change within errors when increasing the sea quark mass from to Bazavov et al. (2018b). Therefore, the same value can be used to set the scale for calculations.

We calculate pseudo-scalar meson correlators for different heavy quark masses using random color wall sources Chakraborty et al. (2015). This reduces the statistical errors in our analysis by an order of magnitude compared the previous study Maezawa and Petreczky (2016). Here we consider several quark masses in the region between the charm and bottom quark, namely and . This helps us to study the running of at low energies and provides additional cross-checks on the error analysis. The bare quark mass that corresponds to the physical charm quark mass has been determined in Ref. Maezawa and Petreczky (2016). We use the same values of in this work. For the three largest values of we determined the bare charm quark mass by requiring that the mass of meson obtained on the lattice in physical units agrees with the corresponding PDG value. We determine the quark mass at each value of gauge coupling by performing calculations at several values of the heavy quark mass near the quark mass and linearly interpolating to find the quark mass at which the pseudo-scalar mass is equal to the physical mass of from PDG. In these calculations we use both random color wall sources and corner wall sources. In Appendix A we summarize the gauge ensembles used in our study, the number of measurements for each as well as the values of the bare charm and bottom quark masses.

In Fig. 1 we show our results for the ratio as function of . For the smallest we have , and thus, we may expect large discretization errors, which one can indeed see in Fig. 1. For the largest three values the ratio seems to be constant within errors. Fitting the corresponding data with a constant we obtain

(3) |

The above result agrees with the previous 2+1 flavor HISQ analysis that was based on extrapolations to the bottom quark mass region and resulted in Maezawa and Petreczky (2016). It also agrees with the HPQCD result Chakraborty et al. (2015) but significantly lower than the recent Fermilab-MILC-TUMQCD result Bazavov et al. (2018c).

## Iii Moments of quarkonium correlators and QCD parameters

We consider the moments of the pseudo-scalar quarkonium correlator, which are defined as

(4) |

Here is the pseudo-scalar current and is the lattice heavy quark mass. To take into account the periodicity of the lattice of temporal size the above definition of the moments can be generalized as follows:

(5) |

The moments are finite for ( even) since the correlation function diverges as for small . Furthermore, the moments do not need renormalization because the explicit factors of the quark mass are included in their definition Allison et al. (2008). They can be calculated in perturbation theory in scheme

(6) |

Here is the renormalization scale. The scale at which the heavy quark mass is defined can be different from Dehnadi et al. (2015), though most studies assume . The coefficient is calculated up to 4-loop, i.e. up to order Sturm (2008); Kiyo et al. (2009); Maier et al. (2010). Given the lattice data on one can extract and from the above equation. However, as it was pointed out in Ref. Allison et al. (2008) it is more practical to consider the reduced moments

(7) |

where is the moment calculated from the free correlation function. The lattice artifacts largely cancel out in these reduced moments. Our numerical results on and some of the relevant ratios, e.g. , are given in Appendix A. From the tables one can clearly see that the statistical errors are tiny and can be neglected. The dominant errors in our calculations are the errors due to finite size of the lattices and the errors induced by mistuning of the heavy quark mass. We estimated the finite size errors in the free theory and included them in the analysis. The free theory estimate provides an upper bound on the finite size errors. It is known that the finite size effects in the interacting theory are much smaller McNeile et al. (2010); Chakraborty et al. (2015). To estimate the errors due to the mistuning of the heavy quark mass we performed interpolation of in the heavy quark mass and examined the changes in when the value of the heavy quark mass is varied by one sigma. These systematic errors are summarized in Appendix A.

It is straightforward to write down the perturbative expansion for :

(8) | |||||

(9) |

From the above equations it is clear that as well as the ratios and are suitable for the extraction of the strong coupling constant , while the ratios with are suitable for extracting the heavy quark mass . In our analysis we choose the renormalization scale . With this choice the expansion coefficients, are just simple numbers that are given in Table 1. This choice of the renormalization scale has the advantage that the expansion coefficients are never large. If the renormalization scale is different from the scale dependence of needs to be taken into account, which increases the uncertainty of the perturbative result Dehnadi et al. (2015).

n | |||
---|---|---|---|

4 | 2.3333 | -0.5690 | 1.8325 |

6 | 1.9352 | 4.7048 | -1.6350 |

8 | 0.9940 | 3.4012 | 1.9655 |

10 | 0.5847 | 2.6607 | 3.8387 |

There are also non-perturbative contribution to the moments proportional to gluon condensate Broadhurst et al. (1994). We included this contribution at tree level using the value

(10) |

from the analysis of decay Geshkenbein et al. (2001).

In order to extract and the heavy quark masses from continuum extrapolation needs to be performed. Since tree-level lattice artifacts cancel out in the reduced moments we expect that discretization errors are proportional to . Therefore, we fitted the lattice spacing dependence of , , and of the ratios and with the form

(11) |

Here for we use the boosted lattice coupling defined as

(12) |

where is a bare lattice gauge coupling and is an averaged link valuable defined by the plaquette . We performed fits using up to and in Eq. (11). We use data corresponding to to avoid uncontrolled cutoff effects. We note that the radius of convergence of the series expansion in for the free theory is Bazavov et al. (2018d), and thus, our upper limit on is well within the radius of convergence of the expansion. The coefficients of higher order terms in are not well constrained by the data and therefore in many cases we assumed that they are equal to a certain fraction of the lowest order coefficient. The fit range in was also varied. We find that for larger values of and larger range of lattice spacings more terms in Eq. (11) had to be included, as expected. The lattice spacing (cutoff) dependence of turned out to be the most complicated. This is not completely surprising as being the lowest moment is most sensitive to short distance physics. Here we had to use up to five order polynomial in and at least two powers of to describe the lattice data on . Simpler fit forms only worked for the lowest mass and very small value of the lattice spacings. For the ratios and we also had to use high order polynomials in though the leading order in turned out to be sufficient. On the other hand the lattice spacing dependence of the ratios are described by the leading order () or leading order plus next-to-leading order () forms even for large values of . To demonstrate these features we show sample continuum extrapolations for the moments in Fig. 2 and Fig. 3. In Fig. 2 we show the results for at and for . As one can see from the left panel of the figure the slope of dependence of increases with decreasing . Therefore, if there are no data points at small the continuum limit may be underestimated. The leading order fit only works for the four smallest lattice spacings but agrees with the fit that uses fifth order polynomial in with , and extends to the whole range of the lattice spacings. Thus, the additional three lattice spacings included in this study are important for cross-checking the validity of continuum extrapolation, although the correct continuum result for can be obtained without these additional data points. For -dependence of the ratio we see the opposite trend, the slope decreases at small . Not having lattice results at small lattice spacing may lead to an overestimated continuum result. This difference in the two quantities provides an important cross-check for the determination of . In the right panel of Fig. 2 we show the fits using fourth (solid line) and third order (dashed line) polynomials in and leading order in . We see that the two fits give very similar result and we find that higher terms in do no have a big impact here. Because high order polynomials are needed for extrapolations, continuum results for , and could only be obtained for . For larger values of the quark masses we simply do not have enough data satisfying to perform the continuum extrapolations.

The lattice spacing dependence of is well described by the next-to-leading order form for all quark masses as can be seen for example in the right panel of Fig. 3. For even leading dependence is sufficient to describe the data, cf. left panel of Fig. 3. Including higher order terms in has no effect in this case.

1.279(4) | 1.1092(6) | 1.0485(8) | |

1.228(2) | 1.0895(11) | 1.0403(10) | |

1.194(2) | 1.0791(7) | 1.0353(5) | |

1.158(6) | 1.0693(10) | 1.0302(5) |

For each quantity of interest we performed many continuum extrapolations using different ranges in the lattice spacing and different fit forms. We only consider fits that have of around one or smaller and take a weighted average of the corresponding results to obtain the final continuum value. We use the scattering of different fits around this averaged value to estimate the error of our continuum result. When the scattering in the central value of different fits around the average is considerably smaller than the errors of the individual fits we take the typical errors of the fits as our final error estimate. In Appendix B we give the details of this procedure. Our continuum results for , and are shown in Table 2. In Table 3 we give our continuum results for for . These two tables present the main result of this study.

As will be discussed in the following section using the continuum results on the reduced moments and their ratios presented in Tables 2 and 3 one can obtain the strong coupling constant as well as the values of the heavy quark masses, and may perform many important cross-checks. However, before discussing the determination of and the quark masses let us compare our continuum results for the moments and their ratios with other lattice determinations.

1.0195(20) | 0.9174(20) | 0.8787(50) | |

0.7203(35) | 0.6586(16) | 0.6324(13) | |

0.5584(35) | 0.5156(17) | 0.4972(17) | |

0.3916(23) | 0.3647(19) | 0.3527(20) | |

0.3055(23) | 0.2859(12) | 0.2771(23) | |

0.2733(17) | 0.2567(17) | 0.2499(16) |

In Fig. 4 we show the comparison of our continuum results on , and , which can be used for determination, with other lattice calculations for . Our result on agrees with HPQCD results, published in 2008 Allison et al. (2008) and 2010 McNeile et al. (2010) and labeled as HPQCD 08 and HPQCD 10, but is higher than the continuum result from Ref. Maezawa and Petreczky (2016), denoted as MP 16. This is due to the fact that in Maezawa and Petreczky (2016) simple and continuum extrapolations have been used, which cannot capture the correct dependence on the lattice spacing as we now understand. The statistical errors on in those calculations were much larger and the inadequacy of simple and extrapolations was not apparent. Our result for agrees with the JLQCD determination Nakayama et al. (2016) (JLQCD 16) but is smaller than the HPQCD results published in 2008 and 2010 (labeled as HPQCD 08 and HPQCD 10). This could be due to the fact that in the analysis of HPQCD only few data points were available for small enough . As discussed above not having data for small may lead to an overestimated continuum limit for . For the same reason the MP 16 result for is larger. In Ref. Maezawa and Petreczky (2016) the continuum extrapolations was performed using the simplest form with lattice results limited to . Because of much larger statistical errors this fit was acceptable. However, with the new extended and more precise data a simple continuum extrapolation is no longer appropriate. Finally for all lattice results agree within errors maybe with the exception of HPQCD 10 result.

In Figure 5 we compare our continuum results on , and for with other lattice studies, including JLQCD Nakayama et al. (2016), Maezawa and Petrezky Maezawa and Petreczky (2016) and HPQCD Allison et al. (2008); McNeile et al. (2010). As one can see from the figure our results agree with other lattice works within errors. Perhaps, this is not too surprising as the -dependence of these reduced moments is well described by a simple form.

## Iv Strong coupling constant and heavy quark masses

From the continuum results on and the ratio of the reduced moments, and we can extract the value of the strong coupling constant. As discussed in the previous section we choose the renormalization scale to be and solve the nonlinear equations to obtain the value of . To estimate the error due to the truncation of the perturbative series in we assume that the coefficient of the unknown term varies between and , i.e. . This error estimate should be sufficiently conservative. We also take into account the non-perturbative contribution to the reduced moments at tree level according to Ref. Broadhurst et al. (1994) and the value of the gluon condensate given in Eq. (10). In Table 4 we give our results for for different quark masses. We see that both the perturbative uncertainty as well as the uncertainty due to the gluon condensate drastically decrease with increasing . The values determined from and have much larger perturbative uncertainties than the ones from . As one can see from the table the strong coupling constants obtained from , and do not always agree with each other. Only for we find that all three determination agree within errors. We usually find that determined from is systematically low. To obtain our final estimate of for and we performed weighted average of the results obtained from , and . These are given in fifth column of Table 4. The uncertainty of this averaged values was determined such that it agrees with all individual extraction within the estimated errors.

av. | MeV | ||||
---|---|---|---|---|---|

0.3815(55)(30)(22) | 0.3837(25)(180)(40) | 0.3550(63)(140)(88) | 0.3788(65) | 315(9) | |

0.3119(28)(4)(4) | 0.3073(42)(63)(7) | 0.2954(75)(60)(17) | 0.3099(48) | 311(10) | |

0.2651(28)(7)(1) | 0.2689(26)(35)(2) | 0.2587(37)(34)(6) | 0.2649(29) | 285(8) | |

0.2155(83)(3)(1) | 0.2338(35)(19)(1) | 0.2215(367)(17)(1) | 0.2303(150) | 284(48) |

Using the continuum results for , given in Table 3 together with the corresponding perturbative expression for , and the averaged value of given in the fifth column of Table 4 we obtain the values of in scheme at . These are presented in Table 5. The heavy quark masses obtained from , and agree with each other very well. Therefore, we calculated the corresponding weighted average to obtain our final estimates and the errors for the heavy quark masses. These are given in the last column of Table 5. The errors on the heavy quark masses in the table do not contain the overall scale error.

av. | ||||
---|---|---|---|---|

1.2740(25)(17)(11)(61) | 1.2783(28)(23)(00)(43) | 1.2700(72)(46)(13)(33) | 1.2754(39) | |

1.7147(83)(11)(03)(60) | 1.7204(42)(14)(00)(40) | 1.7192(35)(29)(04)(30) | 1.7191(38) | |

2.1412(134)(07)(01)(44) | 2.1512(71)(10)(00)(29) | 2.1531(74)(19)(02)(21) | 2.1507(52) | |

2.9788(175)(06)(00)(319) | 2.9940(156)(08)(00)(201) | 3.0016(170)(16)(00)(143) | 2.9949(153) | |

3.7770(284)(06)(00)(109) | 3.7934(159)(08)(00)(68) | 3.8025(152)(15)(00)(47) | 3.7956(110) | |

4.1888(260)(05)(00)(111) | 4.2045(280)(07)(00)(69) | 4.2023(270)(14)(00)(47) | 4.1985(163) |

Combining the information from the above table with the value of in the fifth column of Table 4 we can obtain the values of which are given in the last column of Table 4. To obtain the from the value of the coupling at we use the implicit scheme Chetyrkin et al. (2000). We also calculated the in explicit scheme and the small differences between the two schemes have been treated as systematic errors. Finally, we included the error in scale determination in the values of and the error in . All these errors have been added in quadrature. We see that the value of determined from the data is significantly lower that the ones obtained from the and data. If one fits the values of from Table 4 to a constant the is larger than two. One way to obtain a of about one for this fit is to assume that the error of has been underestimated by a factor two. However, it is not obvious how to justify such an assumption. Therefore, to obtain our final estimate for we take a weighted average of the data in the last column of Table 4 and use the spread around this central value as our (systematic) error:

(13) |

The too low value of obtained for is of some concern. One could imagine that the continuum extrapolation of at this quark mass is not reliable and the corresponding should not be considered in the analysis. If we determine using only the results for and we obtain a value for , which is only one tenth sigma smaller than the value in Table 4. Finally if we take only from we obtain MeV resulting in a weighted average of MeV, which lies well within the uncertainty of the above result.

Using Eq. (13) for we can calculate and and also determine the corresponding quark masses. These are presented in the last two rows of Table 5. Again we see that the heavy quark masses obtained from , and are very consistent with each other suggesting that the quark mass determination from the reduced moments is under control even for the largest values of the heavy quark masses.

With all the above information we can now test the running of the strong coupling constant and the heavy quark masses. To study the running of the heavy quark mass we consider the ratio , where . We can think about this quantity as the charm quark mass at different scales. The running of and the heavy quark mass is shown in Fig. 6. In the figure we show the coupling constant determined in other lattice studies, including the determination from the static quark anti-quark energy Bazavov et al. (2014a) and moments of quarkonium correlators Allison et al. (2008); McNeile et al. (2010); Chakraborty et al. (2015). The results of Ref. Allison et al. (2008); Chakraborty et al. (2015) correspond to the four flavor theory, however, the difference between four flavors and three flavors is small at the low energy scales considered in the figure. Our running coupling constant is consistent with the value obtained from energy of static quark anti-quark pair but is significantly lower than the HPQCD results Allison et al. (2008); McNeile et al. (2010); Chakraborty et al. (2015). This is somewhat unexpected given the fact that the value of for agrees within errors. The reason of this discrepancy could be the different strategy of extracting from the reduced moments. We can also see from Fig. 6 that the running of the heavy quark mass follows the expectation very well.

We can convert our result on to for at scale by including the contribution of the charm and bottom quarks to the running of the coupling constant. We do this by using the RunDeC package Chetyrkin et al. (2000) and first match at the charm threshold which we choose to be GeV and then at the bottom quark threshold, which we choose to be GeV and finally evolve to . We get

(14) |

This result agrees with obtained in Ref. Maezawa and Petreczky (2016) that used the same lattice setup within the error, but the error increased despite much smaller statistical errors and more lattice data points. The use of many fit forms, several heavy quark masses and of the ratios of reduced moments resulted in a more conservative error estimate. Let us now compare our results for given in Eq. (14) with other determinations. Our result is significantly smaller than the PDG average Tanabashi et al. (2018) as well as the FLAG average Aoki et al. (2017). It is also smaller that the determination of by ALPHA collaboration using Schrödinger function method Bruno et al. (2017). Two recent lattice determinations, one from the combined analysis of , and Nakayama et al. (2016) and another one from hadronic vacuum polarization Hudspith et al. (2018) give values and , respectively. These agree with our results though the central values are higher. The very recent analysis of the static quark anti-quark energy resulted in and , depending on the analysis strategy Takaura et al. (2018a, b). These again agree with our result within errors. Finally, a recent phenomenological estimate based on bottomonium spectrum gave Mateu and Ortega (2018), which is again compatible with our result.

Now let us compare the determination of the charm quark mass, . Using the result from Table 5 and adding the scale uncertainty from we have GeV. Combining the errors in quadrature and performing the matching to four flavor theory with RunDeC we obtain

(15) |

Here we also included the uncertainty in from Eq. (13). This result agrees with the value obtained by HPQCD Collaboration Chakraborty et al. (2015). It is customary to quote the result for the charm quark mass at scale GeV. Evolving our result for to GeV and with the RunDeC package we obtain

(16) |

As before the matching to the four flavor theory was carried out at GeV. The uncertainty in given Eq. (13) has a significant effect on the evolution and thus leads to larger error on at this scale. Our result agrees well with the HPQCD determinations that relies on the moments of quarkonium correlators: Chakraborty et al. (2015), the HQPCD result based on RI-SMOM scheme, Lytle et al. (2018), as well as with the Fermilab-MILC-TUMQCD result based on MRS scheme, Bazavov et al. (2018c). Furthermore, we also agree with the value reported by JLQCD collaboration, GeV.

Finally, we discuss the determination of the bottom quark mass, . Using the result from Table 5 and taking into account the scale error we get GeV. We estimate the bottom quark mass for five flavors by evolving the result with the RunDec package as before, which results in

(17) |

The above error also includes the uncertainty in the value of . This value for is in good agreement with other lattice determinations by HQPCD Collaboration, Chakraborty et al. (2015), by ETMC Collaboration, Bussone et al. (2016), as well as with the new result from Fermilab-MILC-TUMQCD, . Moreover, our result also agrees with the value GeV obtained from bottomonium phenomenology Mateu and Ortega (2018). We could have also determined the bottom quark mass from the value of and the ratio obtained in section II. As one can see from Fig. 6 this would have resulted in a value which is compatible with the above result but has significantly larger error at scale .

## V Conclusion

In this paper we calculated the moments of quarkonium correlators for several heavy quark masses in 2+1 flavor QCD using HISQ action. From the moments of quarkonium correlators we extracted the strong coupling constant and the heavy quark masses. Our main results are given in Tables 2 and 3 and by Eqs. (13), (14), (15) and (17). We improved and extended the previous 2+1 flavor HISQ analysis published in Ref. Maezawa and Petreczky (2016). We drastically reduced the statistical errors on the moments by using random color wall sources, extended the calculations to smaller lattice spacings and considered several values of the heavy quark masses in the region between the charm and bottom quark mass. The calculations of the reduced moments at several heavy quark masses enabled us to map out the running of the coupling constant at low energy scales and better control the systematic errors due to the truncation of the perturbative series. Having many lattice spacings enabled us to study the cutoff dependence of the moments in detail and control the errors due continuum extrapolation. Furthermore, to have a handle on the uncertainties due to continuum extrapolations the strong coupling constant and the quark masses have been extracted using different moments and/or their ratios. The heavy quark masses in scheme obtained from various moments and different valence quark masses are remarkably consistent with each other. On the other hand there is some tension in the determination of the strong coupling constant performed using different quantities and different valence heavy quark masses. To deal with this tension the errors on the -parameter have been chosen accordingly.

Evolving the low energy determination of to we obtain the value , which agrees with the previous result Maezawa and Petreczky (2016) but has larger error. Our result for is lower than many lattice QCD determinations as well as the PDG average. On the other hand it agrees with the determination of static energy Bazavov et al. (2014a).

From the sixth, eighth and tenth moments we determined the charm and bottom quark masses. Our results on the heavy quark masses agree well with the previous 2+1 flavor HISQ determination Maezawa and Petreczky (2016) but have smaller errors. We also found that our results agree well with other lattice determinations that are based on various approaches. Thus, our analysis suggests that lattice determination of the heavy quark masses is under control.

## Acknowledgments

This work was supported by U.S. Department of Energy under Contract No. DE-SC0012704. This research was supported by the DFG cluster of excellence “Origin and Structure of the Universe” (www.universe-cluster.de). The simulations have been carried out on the computing facilities of the Computational Center for Particle and Astrophysics (C2PAP) as well as on the SuperMUC at the Leibniz-Rechenzentrum (LRZ). The lattice QCD calculations have been performed using the publicly available MILC code. PP would like to thank V. Mateu for useful discussions on the the gluon condensate contributions to the moments of quarkonium correlators, K. Nakayama for correspondence on the lattice results with domain wall fermions and R. Sommer for eliminating the smallness of the coefficient of the reduced moments due to the choice of the renormalization scale .

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## Appendix A Ensembles and numerical results on the reduced moments

In this Appendix we present the details of the gauge ensembles used in our calculations as well as the numerical results on the reduced moments . In Table 6 we present the lattice parameters used in our study, including the bare gauge couplings, lattice spacings, lattices volumes as well as the values of bare charm and bottom quark masses. In Table 7 we show the numerical results for the reduced moments, for . In Tables 8-11 we present the numerical results for the moments for the larger values of the quark masses, and . In these tables we show three errors for the moments: the statistical, the finite size errors and the errors due to mistuning of the heavy quark mass. The last one was estimated by fitting the quark mass dependence of the reduced moments by a polynomial and estimating the changes in the moments from this fits when the heavy quark mass is changed by one sigma. The finite size errors have been estimated using the free theory calculations. In Table 12 we give the results of our calculations of moments at the bottom quark mass. Finally in Tables 13-18 we give our numerical results for the ratios and for and . As one can see from Tables 7-18 the results for the same but different light sea quark masses agree within the statistical errors. Thus, the use of heavier than the physical light sea quark masses has no effect on our results.

vol | GeV | ||||
---|---|---|---|---|---|

6.740 | 0.05 | 1.81 | 0.5633(10) | ||

6.880 | 0.05 | 2.07 | 0.4800(10) | ||

7.030 | 0.05 | 2.39 | 0.4047(9) | ||

7.150 | 0.05 | 2.67 | 0.3547(9) | ||

7.280 | 0.05 | 3.01 | 0.3086(13) | ||

7.373 | 0.05 | 3.28 | 0.2793(5) | ||

7.596 | 0.05 | 4.00 | 0.2220(2) | 1.019(8) | |

7.825 | 0.05 | 4.89 | 0.1775(3) | 0.7985(5) | |

7.030 | 0.20 | 2.39 | 0.4047(9) | ||

7.825 | 0.20 | 4.89 | 0.1775(3) | 0.7985(5) | |

8.000 | 0.20 | 5.58 | 0.1495(6) | 0.6710(6) | |

8.200 | 0.20 | 6.62 | 0.1227(3) | 0.5519(6) | |

8.400 | 0.20 | 7.85 | 0.1019(27) | 0.4578(6) |

# corr. | ||||||
---|---|---|---|---|---|---|

6.740 | 0.05 | 1601 | 1.19152(15)(11)(30) | 1.02463(8)(27)(11) | 0.94348(5)(8)(20) | 0.89969(4)(16)(23) |

6.880 | 0.05 | 1619 | 1.20299(7)(12)(30) | 1.00224(4)(5)(11) | 0.91580(3)(2)(20) | 0.87198(2)(3)(23) |

7.030 | 0.05 | 1967 | 1.21414(11)(12)(31) | 0.97833(7)(4)(14) | 0.88936(4)(5)(21) | 0.84685(3)(38)(22) |

7.150 | 0.05 | 1317 | 1.22165(14)(13)(37) | 0.96015(7)(5)(18) | 0.87067(4)(3)(25) | 0.82887(4)(0)(26) |

7.280 | 0.05 | 1343 | 1.22960(12)(14)(38) | 0.94222(8)(8)(19) | 0.85290(5)(5)(24) | 0.81220(4)(13)(26) |

7.373 | 0.05 | 1541 | 1.23459(16)(18)(26) | 0.92688(9)(16)(13) | 0.84089(5)(8)(17) | 0.80120(5)(5)(18) |

7.596 | 0.05 | 1585 | 1.24527(18)(9)(10) | 0.90377(9)(43)(5) | 0.81797(6)(213)(6) | 0.78352(4)(678)(6) |

7.825 | 0.05 | 1589 | 1.25410(22)(20)(20) | 0.88364(14)(345)(10) | 0.8052(1)(110)(1) | 0.7812(1)(252)(1) |

7.030 | 0.20 | 597 | 1.21440(19)(12)(31) | 0.97833(12)(4)(14) | 0.88924(8)(5)(21) | 0.84666(6)(38)(22) |

7.825 | 0.20 | 298 | 1.25322(39)(20)(20) | 0.88313(21)(345)(10) | 0.8049(1)(110)(1) | 0.7811(1)(252)(1) |

8.000 | 0.20 | 462 | 1.26020(69)(112)(50) | 0.8740(3)(109)(2) | 0.8054(2)(271)(2) | 0.7919(2)(509)(1) |

8.200 | 0.20 | 487 | 1.27035(65)(398)(28) | 0.8744(4)(297)(1) | 0.8191(3)(581)(0) | 0.8160(2)(916)(0) |

8.400 | 0.20 | 495 | 1.27594(115)(945)(350) | 0.8850(6)(596)(1) | 0.8416(4)(980)(5) | 0.845(0)(138)(1) |

# corr. | ||||||
---|---|---|---|---|---|---|

6.880 | 0.05 | 1619 | 1.13022(4)(12)(35) | 1.03704(2)(5)(17) | 0.98313(1)(3)(38) | 0.95055(1)(2)(48) |

7.030 | 0.05 | 1967 | 1.14320(5)(12)(36) | 1.02203(3)(5)(24) | 0.95878(2)(2)(42) | 0.92274(1)(2)(49) |

7.150 | 0.05 | 1317 | 1.15241(7)(12)(43) | 1.00802(4)(5)(34) | 0.93945(2)(2)(53) | 0.90251(2)(2)(58) |

7.280 | 0.05 | 1343 | 1.16151(9)(12)(43) | 0.99220(6)(5)(37) | 0.91994(4)(2)(54) | 0.88308(3)(2)(57) |

7.373 | 0.05 | 723 | 1.16774(12)(12)(30) | 0.98062(8)(5)(27) | 0.90671(5)(2)(37) | 0.87024(4)(1)(39) |

7.596 | 0.05 | 642 | 1.18017(16)(9)(11) | 0.95412(9)(47)(18) | 0.87889(5)(12)(14) | 0.84368(4)(25)(14) |

7.825 | 0.05 | 627 | 1.19047(15)(14)(24) | 0.93016(10)(1)(22) | 0.85552(6)(42)(25) | 0.82239(6)(177)(21) |

7.030 | 0.20 | 597 | 1.14348(7)(12)(36) | 1.02214(5)(5)(24) | 0.95879(3)(2)(43) | 0.92269(3)(2)(49) |

7.825 | 0.20 | 298 | 1.19049(28)(14)(23) | 0.93003(18)(1)(22) | 0.85539(12)(42)(25) | 0.82228(10)(177)(21) |

8.000 | 0.20 | 462 | 1.19752(25)(9)(58) | 0.91320(15)(38)(48) | 0.84039(10)(199)(46) | 0.81108(9)(650)(32) |

8.200 | 0.20 | 487 | 1.20592(29)(10)(32) | 0.89663(17)(263)(17) | 0.82892(12)(902)(12) | 0.8079(1)(219)(0) |

8.400 | 0.20 | 495 | 1.21132(63)(90)(118) | 0.88617(38)(968)(79) | 0.8279(2)(252)(2) | 0.8171(2)(488)(12) |

# corr. | ||||||
---|---|---|---|---|---|---|

6.880 | 0.05 | 1619 | 1.108989(3)(11)(24) | 1.04248(2)(5)(2) | 1.01130(1)(3)(16) | 0.99245(1)(2)(26) |

7.030 | 0.05 | 1967 | 1.10192(3)(11)(26) | 1.03622(2)(5)(7) | 0.99575(1)(3)(24) | 0.97037(1)(2)(33) |

7.150 | 0.05 | 1317 | 1.11140(4)(11)(33) | 1.02855(2)(5)(15) | 0.98066(2)(3)(34) | 0.95131(1)(2)(43) |

7.280 | 0.05 | 1343 | 1.12118(6)(12)(34) | 1.01794(4)(5)(20) | 0.96304(3)(2)(38) | 0.93114(2)(2)(45) |

7.373 | 0.05 | 723 | 1.12784(9)(12)(24) | 1.00898(6)(5)(16) | 0.95002(4)(2)(28) | 0.91725(3)(2)(31) |

7.596 | 0.05 | 625 | 1.14168(8)(11)(9) | 0.98582(5)(5)(8) | 0.92098(3)(2)(11) | 0.88817(2)(1)(12) |

7.825 | 0.05 | 627 | 1.15324(13)(11)(19) | 0.96256(9)(5)(17) | 0.89567(6)(2)(22) | 0.86378(4)(3)(23) |

7.030 | 0.20 | 597 | 1.10207(5)(11)(26) | 1.03630(3)(5)(7) | 0.99579(2)(3)(24) | 0.97038(2)(2)(33) |

7.825 | 0.20 | 298 | 1.15332(19)(11)(19) | 0.96265(12)(5)(18) | 0.89573(7)(2)(22) | 0.86383(6)(3)(23) |

8.000 | 0.20 | 462 | 1.16101(15)(11)(46) | 0.94509(10)(4)(44) | 0.87797(7)(6)(53) | 0.84709(6)(44)(52) |

8.200 | 0.20 | 487 | 1.16967(23)(11)(24) | 0.92603(13)(11)(22) | 0.85960(8)(85)(24) | 0.83121(7)(328)(20) |

8.400 | 0.20 | 495 | 1.17615(40)(4)(297) | 0.91003(26)(106)(242) | 0.84664(17)(445)(212) | 0.8239(1)(125)(12) |

# corr. | ||||||
---|---|---|---|---|---|---|

6.880 | 0.05 | 1619 | 1.10541(1)(11)(30) | 1.03418(1)(5)(7) | 1.01888(0)(3)(6) | 1.01193(0)(2)(15) |

7.030 | 0.05 | 1967 | 1.10597(1)(11)(33) | 1.03505(1)(5)(1) | 1.01754(5)(3)(19) | 1.00841(0)(2)(33) |

7.150 | 0.05 | 1317 | 1.06590(3)(11)(27) | 1.03405(2)(5)(1) | 1.01248(1)(3)(18) | 0.99975(1)(2)(29) |

7.280 | 0.05 | 1343 | 1.07382(3)(41)(29) | 1.03159(2)(9)(7) | 1.00425(1)(5)(26) | 0.98690(1)(25)(38) |

7.373 | 0.05 | 723 | 1.08016(5)(42)(21) | 1.02872(4)(7)(8) | 0.99657(2)(6)(22) | 0.97587(2)(25)(30) |

7.596 | 0.05 | 629 | 1.09489(5)(11)(8) | 1.01736(3)(5)(6) | 0.97396(2)(2)(11) | 0.94755(1)(2)(13) |

7.825 | 0.05 | 630 | 1.10813(6)(11)(17) | 1.00076(5)(5)(16) | 0.94910(3)(2)(24) | 0.92077(2)(1)(27) |

7.030 | 0.20 | 597 | 1.05968(2)(11)(33) | 1.03507(2)(5)(1) | 1.01755(1)(3)(19) | 1.00841(1)(2)(33) |

7.825 | 0.20 | 298 | 1.10808(10)(11)(17) | 1.00072(7)(5)(16) | 0.94907(4)(2)(24) | 0.92073(4)(1)(27) |

8.000 | 0.20 | 462 | 1.11732(10)(84)(41) | 0.98579(7)(51)(45) | 0.93034(5)(2)(61) | 0.90200(4)(37)(65) |

8.200 | 0.20 | 487 | 1.12727(14)(90)(22) | 0.96768(10)(63)(26) | 0.90997(6)(14)(32) | 0.88213(5)(48)(33) |

8.400 | 0.20 | 495 | 1.13531(24)(50)(114) | 0.95106(16)(27)(324) | 0.89267(10)(3)(375) | 0.86562(9)(18)(351) |

# corr. | ||||||
---|---|---|---|---|---|---|

7.150 | 0.05 | 1317 | 1.04559(2)(11)(27) | 1.02832(1)(5)(8) | 1.01534(1)(3)(4) | 1.00922(1)(2)(12) |

7.280 | 0.05 | 1343 | 1.05008(2)(41)(22) | 1.02913(1)(9)(4) | 1.01432(1)(5)(6) | 1.00649(1)(25)(14) |

7.373 | 0.05 | 818 | 1.05437(3)(42)(17) | 1.02883(2)(7)(1) | 1.01142(1)(6)(8) | 1.00129(1)(25)(15) |

7.596 | 0.05 | 629 | 1.06662(3)(10)(7) | 1.02559(2)(5)(2) | 0.99950(1)(2)(6) | 0.98250(1)(2)(9) |

7.825 | 0.05 | 630 | 1.07976(5)(10)(15) | 1.01734(3)(5)(8) | 0.98113(2)(2)(19) | 0.95833(2)(2)(23) |

7.825 | 0.20 | 298 | 1.07965(7)(10)(15) | 1.01727(4)(5)(8) | 0.98109(3)(2)(19) | 0.95830(2)(2)(23) |

8.000 | 0.20 | 462 | 1.08972(6)(8)(38) | 1.00735(4)(42)(30) | 0.96424(2)(6)(52) | 0.93912(2)(25)(59) |

8.200 | 0.20 | 487 | 1.10053(8)(59)(20) | 0.99275(6)(25)(20) | 0.94397(4)(4)(29) | 0.91810(3)(17)(31) |

8.400 | 0.20 | 495 | 1.10942(13)(94)(84) | 0.97756(9)(66)(276) | 0.92590(5)(9)(360) | 0.90027(5)(50)(377) |

# corr. | ||||||
---|---|---|---|---|---|---|

7.596 | 0.05 | 638 | 1.05521(2)(66)(6) | 1.02587(1)(39)(0) | 1.00650(1)(2)(3) | 0.99434(1)(3)(6) |

7.825 | 0.05 | 641 | 1.06916(3)(26)(6) | 1.02083(2)(1)(2) | 0.99128(1)(7)(5) | 0.97197(1)(14)(7) |

7.825 | 0.20 | 298 | 1.06919(5)(26)(5) | 1.02084(4)(0)(2) | 0.99128(3)(7)(5) | 0.97197(2)(14)(7) |

8.000 | 0.20 | 462 | 1.07927(5)(21)(8) | 1.01323(3)(48)(5) | 0.97609(2)(1)(10) | 0.95318(2)(27)(12) |

8.200 | 0.20 | 487 | 1.09013(10)(17)(9) | 1.00109(7)(19)(8) | 0.95704(5)(8)(13) | 0.93234(4)(11)(14) |

8.400 | 0.20 | 495 | 1.10994(10)(78)(13) | 0.98692(8)(46)(13) | 0.93871(5)(0)(18) | 0.91382(5)(33)(19) |

# corr. | ||||

6.740 | 0.05 | 1601 | 1.08601(4)(36)(17) | 1.04867(2)(10)(7) |

6.880 | 0.05 | 1619 | 1.09438(2)(3)(17) | 1.05026(1)(5)(7) |

7.030 | 0.05 | 1967 | 1.10004(3)(11)(15) | 1.05020(1)(42)(5) |

7.150 | 0.05 | 1317 | 1.10277(4)(3)(16) | 1.05043(1)(4)(5) |

7.280 | 0.05 | 1343 | 1.10472(4)(3)(14) | 1.05011(2)(23)(5) |

7.373 | 0.05 | 1541 | 1.10560(5)(9)(9) | 1.04955(2)(73)(3) |

7.596 | 0.05 | 1585 | 1.10489(5)(235)(3) | 1.04397(2)(637)(0) |

7.825 | 0.05 | 1589 | 1.0975(1)(108)(0) | 1.0307(0)(197)(0) |

7.030 | 0.20 | 597 | 1.10018(5)(11)(15) | 1.05029(2)(42)(5) |

7.825 | 0.20 | 298 | 1.0972(1)(108)(0) | 1.0305(0)(197)(0) |

8.000 | 0.20 | 462 | 1.0851(2)(236)(0) | 1.0170(0)(326)(1) |

8.200 | 0.20 | 487 | 1.0676(2)(414)(1) | 1.0038(1)(436)(1) |

8.400 | 0.20 | 495 | 1.0515(3)(539)(10) | 0.9964(1)(466)(9) |

# corr. | ||||
---|---|---|---|---|

6.880 | 0.05 | 1619 | 1.05484(1)(3)(20) | 1.03428(0)(1)(10) |

7.030 | 0.05 | 1967 | 1.06596(1)(3)(19) | 1.03906(1)(1)(8) |

7.150 | 0.05 | 1317 | 1.07299(2)(3)(21) | 1.04094(1)(1)(8) |

7.280 | 0.05 | 1343 | 1.07855(2)(3)(19) | 1.04173(1)(1)(6) |

7.373 | 0.05 | 723 | 1.08151(3)(3)(12) | 1.04192(2)(1)(4) |

7.596 | 0.05 | 642 | 1.08560(5)(75)(4) | 1.04173(2)(16)(1) |

7.825 | 0.05 | 627 | 1.08724(4)(57)(5) | 1.04028(2)(176)(2) |

7.030 | 0.20 | 597 | 1.06607(2)(3)(19) | 1.03913(1)(1)(8) |

7.825 | 0.20 | 298 | 1.08726(7)(57)(5) | 1.04026(3)(176)(2) |

8.000 | 0.20 | 462 | 1.08663(7)(212)(5) | 1.03614(3)(590)(12) |

8.200 | 0.20 | 487 | 1.08169(8)(868)(3) | 1.0261(0)(170)(1) |

8.400 | 0.20 | 495 | 1.0703(2)(213)(10) | 1.0132(1)(308)(8) |

# corr. | ||||
---|---|---|---|---|

6.880 | 0.05 | 1619 | 1.03082(1)(3)(13) | 1.01900(0)(1)(9) |

7.030 | 0.05 | 1967 | 1.04064(1)(3)(16) | 1.02616(0)(1)(9) |

7.150 | 0.05 | 1317 | 1.04884(1)(3)(20) | 1.03085(0)(1)(10) |

7.280 | 0.05 | 1343 | 1.05700(2)(3)(19) | 1.03426(1)(1)(9) |

7.373 | 0.05 | 723 | 1.06207(2)(3)(13) | 1.03573(1)(1)(5) |

7.596 | 0.05 | 625 | 1.07040(2)(3)(4) | 1.03695(1)(1)(1) |

7.825 | 0.05 | 627 | 1.07469(3)(3)(7) | 1.03692(2)(6)(2) |

7.030 | 0.20 | 597 | 1.04068(1)(3)(16) | 1.02618(1)(1)(9) |

7.825 | 0.20 | 298 | 1.07471(6)(3)(7) | 1.03692(3)(6)(2) |

8.000 | 0.20 | 462 | 1.07644(4)(12)(15) | 1.03646(2)(46)(2) |

8.200 | 0.20 | 487 | 1.07729(7)(94)(5) | 1.03415(3)(307)(3) |

8.400 | 0.20 | 495 | 1.07488(11)(441)(1) | 1.0276(0)(104)(9) |

# corr. | ||||
---|---|---|---|---|

6.880 | 0.05 | 1619 | 1.01502(0)(3)(11) | 1.00687(0)(1)(8) |

7.030 | 0.05 | 1967 | 1.01721(0)(3)(16) | 1.00905(0)(1)(11) |

7.150 | 0.05 | 1317 | 1.02131(1)(3)(14) | 1.01273(0)(1)(10) |

7.280 | 0.05 | 1343 | 1.02722(1)(6)(17) | 1.01759(0)(23)(11) |

7.373 | 0.05 | 723 | 1.03226(1)(13)(12) | 1.02122(0)(21)(7) |

7.596 | 0.05 | 629 | 1.04456(1)(2)(5) | 1.02787(0)(1)(2) |

7.825 | 0.05 | 630 | 1.05444(2)(3)(9) | 1.03077(1)(1)(3) |

7.030 | 0.20 | 597 | 1.01722(1)(3)(16) | 1.00906(0)(1)(11) |

7.825 | 0.20 | 298 | 1.05443(3)(3)(9) | 1.0309(0)(7) |

8.000 | 0.20 | 462 | 1.05960(3)(70)(13) | 1.03141(1)(42)(6) |

8.200 | 0.20 | 487 | 1.06343(4)(97)(8) | 1.03156(2)(41)(2) |

8.400 | 0.20 | 495 | 1.06541(8)(37)(13) | 1.03125(3)(26)(4) |

# corr. | ||||
---|---|---|---|---|

7.150 | 0.05 | 1317 | 1.01279(0)(3)(10) | 1.00606(0)(1)(7) |

7.280 | 0.05 | 1343 | 1.01460(0)(6)(10) | 1.00778(0)(23)(7) |

7.373 | 0.05 | 818 | 1.001721(1)(7)(9) | 1.01012(0)(6)(6) |

7.596 | 0.05 | 629 | 1.02610(1)(2)(4) | 1.01731(0)(1)(3) |

7.825 | 0.05 | 630 | 1.03691(1)(3)(10) | 1.02379(0)(1)(5) |

7.825 | 0.20 | 298 | 1.03688(2)(3)(10) | 1.02378(1)(1)(5) |

8.000 | 0.20 | 462 | 1.04472(2)(53)(22) | 1.02675(1)(23)(8) |

8.200 | 0.20 | 487 | 1.05167(2)(31)(10) | 1.02818(1)(26)(3) |

8.400 | 0.20 | 495 | 1.05580(3)(95)(99) | 1.02847(2)(49)(28) |

# corr. | ||||
---|---|---|---|---|

7.596 | 0.05 | 638 | 1.01924(1)(47)(3) | 1.01223(0)(2)(3) |

7.825 | 0.05 | 641 | 1.02981(1)(5)(4) | 1.01986(3)(7)(2) |

7.825 | 0.20 | 298 | 1.02982(1)(5)(4) | 1.01987(0)(7)(2) |

8.000 | 0.20 | 462 | 1.03805(1)(55)(5) | 1.02404(1)(33)(2) |

8.200 | 0.20 | 487 | 1.04602(3)(26)(5) | 1.02649(1)(3)(2) |

8.400 | 0.20 | 495 | 1.05136(3)(62)(6) | 1.02724(1)(39)(2) |

## Appendix B Details of continuum extrapolations

In this Appendix we discuss further details of the continuum extrapolations. As mentioned in the main text we performed a variety of continuum extrapolations using Eq. (11) and keeping different number of terms in the sum. In doing so we varied the fit interval such the is close to or below one. Fewer terms in Eq. (11) usually means more restricted interval in .

We first discuss the continuum extrapolation of the fourth reduced moment, , which is the most challenging. Here we find that including terms up to in Eq. (11) is important if we want to obtain good fits in extended region in . For , the coefficients in Eq.( 11) could be treated as free fit parameter as we have many data points for relatively small to have stable fits. We also consider fits where was fixed to some value between and but the continuum extrapolated value did not change much. For the value of the coefficient of term was varied between and , but setting the coefficient to zero did not change the result much. In Fig. 7 we show continuum estimates for obtained from different fits. The central values of these estimates show some scattering, although most of them agree within errors. We take the weighted average of these estimates to obtain our final continuum result for , which is shown as the solid horizontal line in Fig. 7. We then assign an error to this continuum result. The size of the error band is defined in such way that all individual fits agree with the final continuum estimate or it represents the typical error of the fits, whichever is the largest. The errors on the final continuum estimate are represented by dashed vertical bands in Fig. 7. We estimated the finite volume errors on the reduced moments using the free theory result, see above. As one can see from Fig. 2 the finite size effects estimated this way are fairly large for the three finest lattices. To exclude the possibility that finite volume errors are underestimated we also performed fits omitting the data points corresponding to the three smallest lattice spacings for , where finite volume effects are the largest. We find that doing so does not affect the final continuum estimate within errors.

A similar analysis has been performed also for the ratios of the reduced moments, and . Here including terms up to in Eq. (11) turned out to be less important and good fits could be obtained already with in the entire interval. The corresponding results are shown in Fig. 8 and Fig. 9.

In Fig. 10 we show different continuum extrapolations for at