We compute $\Lambda_{\bar{MS}}$ for two flavors of light dynamical quarks
using non-perturbatively $O(a)$ improved Wilson fermions. We improve on a
recent calculation by employing Pad\'e-improved two-loop and three-loop
perturbation theory to convert the lattice numbers to the $\bar{MS}$ scheme.Comment: Contribution to Lattice 2001 (matrix elements), typo correcte

Booth, S.

Göckeler, M.

Horsley, R.

Irving, A. C.

Joo, B.

Pickles, S.

Pleiter, D.

Rakow, P. E. L.

Schierholz, G.

Sroczynski, Z.

Stüben, H.

English

arXiv

We compute $\Lambda_{\bar{MS}}$ for two flavors of light dynamical quarks
using non-perturbatively $O(a)$ improved Wilson fermions. We improve on a
recent calculation by employing Pad\'e-improved two-loop and three-loop
perturbation theory to convert the lattice numbers to the $\bar{MS}$ scheme

Booth, S

Göckeler, M

Horsley, R

Irving, AC

Joo, B

Pickles, S

Pleiter, D

Rakow, PEL

Schierholz, G

Sroczynski, Z

Stüben, H

University of Liverpool Repository

arXiv:hep-lat/0111006v2  16 Nov 20011DESY 01–170October 2001The strong coupling constant from lattice QCD with Nf = 2 dynamicalquarks∗S. Bootha, M. Go¨ckelerb, R. Horsleyc, A.C. Irvingd, B. Jooe, S. Picklesf , D. Pleiterc, P.E.L. Rakowb,G. Schierholzc,g, Z. Sroczynskih, and H. Stu¨beniaEdinburgh Parallel Computing Center EPCC, University of Edinburgh, Edinburgh EH9 3JZ, UKbInstitut fu¨r Theoretische Physik, Universita¨t Regensburg, D-93040 Regensburg, GermanycNIC/DESY, D-15735 Zeuthen, GermanydTheoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, LiverpoolL69 3BX, UKeDepartment of Physics, Columbia University, New York, NY 10027, USAfComputer Services for Academic Research CSAR, University of Manchester, Manchester M13 9PL, UKgDeutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, GermanyhFachbereich Physik, Universita¨t Wuppertal, D-42097 Wuppertal, GermanyiKonrad-Zuse-Zentrum fu¨r Informationstechnik Berlin, D-14195 Berlin, GermanyWe compute ΛMSfor two flavors of light dynamical quarks using non-perturbatively O(a) improved Wilsonfermions. We improve on a recent calculation by employing Pade´-improved two-loop and three-loop perturbationtheory to convert the lattice numbers to the MS scheme.1. THE METHODThe running of the strong coupling constant αsis parameterized by the Λ parameter. In a recentletter [1] we have computed ΛMS from the av-erage plaquette and the scale parameter r0. Anessential step in our calculation was the conver-sion of the boosted coupling g to the renormal-ized coupling gMS . This was done in two-looptadpole improved perturbation theory, which re-sulted in an (estimated) systematic uncertaintyof the order of the statistical error.We can do better now. In this talk we shall ap-ply Pade´ techniques to approximate an unknownthree-loop coefficient in the conversion formula.This reduces the systematic uncertainty to an in-significant amount, as we shall see.The calculation of ΛMS proceeds in three steps:∗Talk presented by G. Schierholz at Lattice 2001, Berlin,GermanyFirst we compute the average plaquette P =(1/3) 〈TrU〉 ≡ u40 from which we obtain g,1g2(a)=Pg2(a), (1)where g(a) is the bare lattice coupling. Note thatthe non-perturbative contribution to the plaque-tte, i.e. the difference between the lattice numberand the perturbative series, is less than 10−3 [2].In the second step we compute gMS from g.This is the main task. To three loops we obtain,following the notation of [1],1g2MS(µ)=1g2(a)+ 2b0 ln aµ− t1+ (2b1 ln aµ− t2 ) g2(a)+(− 2b0b1 ln2aµ+ 2(bMS2 + b1t1 ) ln aµ− t3 + p1t2)g4(a), (2)2Figure 1. The four-loop β function for the MSscheme compared with various approximations.where ti = ti−pi. We know all coefficients exceptfor t3 . (See [1] for references.) The latter canbe computed from the difference of the four-loopcoefficients, bMS3 − b3 , of the β functionβS(gS)=−g3S(b0 + b1g2S + bS2 g4S + bS3 g4S + · · ·),(3)where S =MS, denotes the scheme. We findt3 =(bMS3 − b3 + 2(bMS2 t1 − b2 p1)+ b1(t21 − p21))/2b0. (4)We know bMS3 , but we do not know b3 . In theMS scheme the four-loop β function turns outto be very well reproduced by the Pade´-improvedthree-loop β functionβS[1,1](gS) = −g3Sb0b1 + (b21 − b0bS2 )g2Sb1 − bS2 g2S. (5)In Fig. 1 we compare (5) with the four-loop ex-pression. We see that in the region g2MS. 2,which is the region relevant to our calculation,the Pade´-improved three-loop result differs by lessthan 0.02% from the four-loop result. We expectthat the same is true for the boosted β functionβ, which converges at a similar rate as βMS .Thus we may approximate β by β[1,1]. Expand-ing (5) gives us thenb3 =b 22b1. (6)We can improve the convergence of the series (2)further by re-expressing it in terms of the tadpoleimproved coefficientsc˜SW = cSWu30, am˜ = am/u0. (7)This amounts, for example, to replacing everycSW byc˜SW(1 +34p1 g2+34(p˜2 −16p21)g4). (8)We have already seen in [1] that tadpole im-provement reduces the coefficients of the two-loopfermionic contribution. We find that tadpole im-provement is equally successful in reducing thecoefficients of the three-loop contributions.In the final step we compute ΛMS from gMS(µ)using the renormalization group equationµΛMS=(b0g2MS) b12b20 exp( 12b0g2MS+∫ gMS0dξ( 1βMS(ξ)+1b0ξ3−b1b20ξ)). (9)For βMS we use the Pade´-improved four-loop βfunctionβMS[1,2] (gMS) = −g3MS(b0 (b21 − b0bMS2 ) + (b31− 2b0b1bMS2 + b20 bMS3 ) g2MS)/(b21− b0bMS2 + (b0bMS3 − b1b2) g2MS+ (bMS 22 − b1bMS3 ) g4MS), (10)which is a better approximation than the four-loop β function. The result should be indepen-dent of µ if our approximations of βMS in (2)and (9) are consistent.2. RESULTSWe have varied µ between 4 and 10 GeV andfound indeed that ΛMS changes by a fraction ofa percent only. Finally we have chosen aµ = 2.5,which corresponds to µ ≈ 5 GeV. We may esti-mate the error arising from our approximation (6)by replacing bMS3 by bMS 22 /b1 as well. We foundthat this changes our results by at most 0.05%.To convert our numbers to a physical scale wehave used the force parameter r0. The results are3β κsea V cSW P r0/a am ΛMS r05.20 0.1355 16332 2.0171 0.536294(9) 5.041(40) 0.02364(16) 0.4860(39)5.20 0.1350 16332 q 0.533676(9) 4.754(40) 0.04586(19) 0.4704(40)5.25 0.1352 16332 1.9603 0.541135(24) 5.137(49) 0.04268(17) 0.4757(46)5.26 0.1345 16332 1.9497 0.539732(9) 4.708(52) 0.07196(20) 0.4430(49)5.29 0.1355 24348 1.9192 0.547081(26) 5.62(9) 0.03495(12) 0.4917(79)5.29 0.1350 16332 q 0.545520(29) 5.26(7) 0.05348(19) 0.4679(62)5.29 0.1340 16332 q 0.542410(9) 4.813(45) 0.09272(29) 0.4428(42)Table 1The results for ΛMS r0, together with r0/a, P and the quark masses am, and the parameters of thesimulation.presented in Table 1. The numbers have increasedby ≈ 2% relative to our previous values [1], whichis well inside the systematic error that was esti-mated in that paper. We fit our data byFigure 2. ΛMS r0(m = 0) (top) and ΛMS r0(a =0) (bottom) together with the fit.ΛMSr0 =A(1+Bam)(1+Cmr0)+D(a/r0)2.(11)(For details of the fit see [1].) In Fig. 2 we showΛMS r0(m = 0) = ΛMS r0 − A(Bam + Cmr0 +BCammr0) and ΛMS r0(a = 0) = (ΛMS r0 −D(a/r0)2)/(1 +Bam) which, if the fit is success-ful, should collapse the data points onto a singleline each. We see that this is indeed the case. Wealso see that the largest uncertainty arises fromthe continuum extrapolation, while the data de-pend only weakly on the quark mass. In the chiraland continuum limit our fit givesΛMSr0 = 0.553(34) (12)(= A). Using r0 = 0.5 fm, we finally obtainΛNf=2MS= 218(13)MeV. (13)This is to be compared with our previous resultΛNf=2MS= 217(16)(11)MeV.3. CONCLUSIONSOur calculation faces two problems. One is theconversion from the boosted to the MS scheme.The other is the extrapolation to the chiral andcontinuum limit. We have solved the first prob-lem, and with time and more data we hope toimprove on the extrapolation.REFERENCES1. S. Booth et al., Phys. Lett. B 519 (2001) 229(hep-lat/0103023).2. R. Horsley, P.E.L. Rakow and G. Schierholz,hep-lat/0110210.

The strong coupling constant from lattice QCD with N_f=2 dynamical quarks

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