We present an update on our previous determination of the Lambda parameter in
QCD. The main emphasis is on results for two flavours of light dynamical
quarks, where we can now almost double the amount of data used, including
values at smaller lattice spacings. The calculations are performed using $O(a)$
improved Wilson fermions. Little change is found to previous numerical values.Comment: Lattice2004(spectrum), Fermilab, June 21-26, 2004, 3 page

A.C. Irving

Allton

Bali

Bethke

Blümlein

Booth

D. Pleiter

G. Schierholz

H. Stüben

Lüscher

M. Göckeler

Necco

P.E.L. Rakow

R. Horsley

van Ritbergen

English

arXiv

We present an update on our previous determination of the Lambda parameter in
QCD. The main emphasis is on results for two flavours of light dynamical
quarks, where we can now almost double the amount of data used, including
values at smaller lattice spacings. The calculations are performed using $O(a)$
improved Wilson fermions. Little change is found to previous numerical values

Göckeler, M

Horsley, R

Irving, AC

Pleiter, D

Rakow, PEL

Schierholz, G

Stüben, H

QCDSF-UKQCD Collaboration

University of Liverpool Repository

arXiv:hep-lat/0409166v1  30 Sep 2004DESY 04-167Edinburgh 2004/17LTH 631LU-ITP 2004/030September 2004Determination of Lambda in quenched and full QCD – an update∗M. Go¨ckelera,b, R. Horsleyc, A. C. Irvingd, D. Pleitere, P. E. L. Rakowd, G. Schierholze,f and H. Stu¨beng– QCDSF–UKQCD CollaborationaInstitut fu¨r Theoretische Physik, Universita¨t Leipzig, D-04109 Leipzig, GermanybInstitut fu¨r Theoretische Physik, Universita¨t Regensburg, D-93040 Regensburg, GermanycSchool of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UKdDepartment of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UKeJohn von Neumann Institute NIC / DESY Zeuthen, D-15738 Zeuthen, GermanyfDeutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, GermanygKonrad-Zuse-Zentrum fu¨r Informationstechnik Berlin, D-14195 Berlin, GermanyWe present an update on our previous determination of the Lambda parameter in QCD. The main emphasisis on results for two flavours of light dynamical quarks, where we can now almost double the amount of dataused, including values at smaller lattice spacings. The calculations are performed using O(a) improved Wilsonfermions. Little change is found to previous numerical values.The Λ parameter is one of the fundamantal pa-rameters of QCD, setting the scale for the run-ning coupling constant αs. In this contributionwe shall update our previous work, [1], both forquenched (nf = 0) and unquenched (nf = 2)O(a) improvedWilson (‘clover’) fermions. Specif-ically we are now able to use for• quenched fermions, the force scale r0/a upto β = 6.92, [2] (previously β ≤ 6.4),• unquenched fermions, improved statisticsand additional quark masses at the previousβ values of 5.20, 5.25, 5.29 for r0/a togetherwith new results at β = 5.40 (at three quarkmasses).The ‘running’ of the QCD coupling constant asthe scale changes is controlled by the β-function∂gS(M)∂ logM= βS (gS(M)) ,where, perturbativelyβS (gS) = −b0g3S − b1g5S − bS2 g7S − bS3 g9S − . . . ,∗Talk given by R. Horsley at Lat04, Fermilab, USA.renormalisation having introduced a scale M to-gether with a scheme S. Integrating this equationgivesΛSM= exp[−12b0gS(M)2] [b0gS(M)2]− b12b20 ×exp{−∫ gS(M)0dξ[1βS(ξ)+1b0ξ3−b1b20ξ]},≡ FS(gS(M)) ,where ΛS , the integration constant, is the funda-mental scheme dependent QCD parameter. Re-sults are usually given in the MS scheme, withthe scale M being denoted by µ. In this schemethe first four β-function coefficients are known,bMS3 being found in [3]. The running couplingαMSs (µ) ≡ gMS(µ)2/4pi is plotted in Fig. 1 fornf = 2, by solving the previous equation (nu-merically) using successively more and more co-efficients of the β-function. The figure showsan apparently fast convergent series (cf 3- to 4-loop), certainly in the range we are interested in,µ/ΛMS ∼ 8. A very similar result holds for nf = 0but with slightly lower curves.121.0 10.0 100.0 1000.0µ / ΛMS−0.00.10.20.30.40.5αMS−(µ)one looptwo loopthree loopfour loopnf=2Figure 1. αMSs (µ) versus µ/ΛMS for nf = 2.On the lattice we also have a Λ parameter,aΛ2 = F2(g2(a)) ,where to help convergence of lattice perturbativeexpansions we use g22≡ g2(a)/u40 with u40 the av-erage plaquette value. To calculate ΛMS, we shallcompute gMS at some appropriate scale µ∗ fromg2(a) and then using the r0 scale, extrapolater0ΛMS ≡(r0a)FMS(gMS(µ∗))aµ∗ ,to the continuum limit.Equating lattice and continuum expressions[FMS(gMS(µ))]−1 = aµΛ2ΛMS[F2(g2(a))]−1 ,and expanding as1g2MS(µ)=1g22(a)+[2b0 ln aµ− t21 ] + [2b1 ln aµ− t22 ]g22(a) . . . ,gives t21 = 2b0 ln ΛMS/Λ2 and b22 = bMS2 + b1t21 −b0t22 . For (hopefully) good convergence of thisseries we choose the scale so that the O(1) termvanishes, aµ∗ = exp(t21 /2b0).For t21 the general expression is known for nf ,csw and linear terms in nfamq, while for t22 thenfamq dependence is not known, [1] and refer-ences therein. We can estimate the scales asµ∗ = 2.63/a, nf = 0 and µ∗ ∼ 1.4/a for nf = 2.t23 (the gMS(µ)4, ln aµ independent term) is notknown. So equivalently b23 is not known. Howevera Pade´ estimate gives bS3 ≈ (bS2 )2/b1, and is smalland in reasonable agreement with the known co-efficient in the MS scheme, [1]. Assuming thisalso holds for b23 gives little change to the resultspresented here. For complete O(a) cancellation,[4], we need g˜2 = g2(1 + bgamq) where pertur-batively bg = 0.01200nfg2 + O(g4), which withcsw = 1 + O(g2) then gives no mass dependencein t21 . This indicates little quark mass depen-dence in the fit formulae (indeed there is more inthe numerical data). Finally to further improvethe convergence of the series, we tadpole improvethe t2i coefficients cTIsw = cswu30 (for t21 + t22 g22)further reducing the size of the nf term in t22 .In Fig. 2 we show the quenched (nf = 0) re-sults. The data lies on a straight line (as a func-0.00 0.01 0.02 0.03 0.04 0.05(a/r0)20.500.550.600.65r 0ΛMS−Figure 2. r0ΛMS versus (a/r0)2 for nf = 0, togetherwith a linear extrapolation to a = 0. The last point hasnot been included in the fit.tion of (a/r0)2) at least over a−1 ∼ 2 – 6.5GeVor µ ∼ 5 – 17GeV, using the value for r0 ofr0 = 0.5 fm. This gives a result of r0ΛMS =0.613(2)(25) or ΛMS(0) = 242(1)(10)MeV wherethe first error is statistical and to estimate thesystematic uncertainty, the second error takes ag4 coeff. = 25% × g2 coeff. (which is very muchgreater than when using the Pade´ b23 estimate).For unquenched (nf = 2) fermions, due tothe sea quark, the fit ansatz is not so sim-ple as we must consider both chiral and con-tinuum extrapolations. We take for finite a,aΛMS|mq 6=0,a6=0 = aΛMS|mq=0,a6=0+Damq+ . . . or3r0ΛMS|mq 6=0,a6=0 = r0ΛMS|mq=0,a6=0+Dr0mq+. . ..After chiral extrapolation we would thus expectr0ΛMS|mq=0,a6=0 = r0ΛMS|mq=0,a=0 + B(a/ρ)2 +. . . with ρ ≡ r0|mq=0. Together with (a/r0)2 =(a/ρ)2 + Eamq + . . . this gives our fit ansatz asr0ΛMS = A+B(a/r0)2 + Camq +Dr0mq .So by subtracting out the B and C terms fromr0ΛMS we can consider the chiral extrapolationand similarly by subtracting out the C and Dterms we may consider the continuum extrapola-tion1. In Fig. 3 we show the results. a−1 ranges0.00 0.01 0.02 0.03 0.04 0.05(a/r0)20.450.500.550.600.0 0.1 0.2 0.3 0.4 0.50.450.500.550.60r0mqr0ΛMS− − B(a/r0)2 − Camqr0ΛMS− − Camq − Dr0mqFigure 3. r0ΛMS versus r0mq (upper picture) and ver-sus (a/r0)2 (lower picture) for nf = 2, together with ap-propriate extrapolations (amq from [5]).at least over a−1 ∼ 2 – 3GeV or µ ∼ 3 – 4GeVThis gives a result of r0ΛMS = 0.563(10)(70) orΛMS(2) = 222(4)(28)MeV where again the firsterror is statistical and the second error is obtainedby taking a g4 coeff. = 25%×g2 coeff. which againis much larger than the error found when using aPade´ b23 estimate, setting csw = 1 + O(g2) or in-cluding an additional (amq)2 fit term. Note thatthis result is consistent with that obtained in [6].Finally in Fig. 4 we present results for differentnf . Our result lies somewhat low in comparisonwith phenomenological results. Alternatively us-ing the matching procedure as in [1] we find fornf = 5, αMSs (mZ) = 0.1084(6)(38).1An alternative procedure is first to extrapolate both r0/aand u40to the chiral limit, evalute r0ΛMS and then extrap-olate to the continuum limit; this gives similar results, [5].0 1 2 3 4 5 6nf050100150200250300350400450500ΛMS−Figure 4. ΛMS(nf ) versus nf The open diamonds arefrom [7], using αMSs (MZ ) = 0.1183(27) (nf = 5) to matchto nf = 4 and nf = 3, while the open square is from [8].The filled circles are the results reported here.ACKNOWLEDGEMENTSThe numerical calculations have been per-formed on the Hitachi SR8000 at LRZ (Munich),on the Cray T3E at EPCC (Edinburgh) underPPARC grant PPA/G/S/1998/00777, [9], on theCray T3E at NIC (Ju¨lich) and ZIB (Berlin), aswell as on the APE1000 and Quadrics at DESY(Zeuthen). This work is supported in part by theEU Integrated Infrastructure Initiative HadronPhysics (I3HP) and by the DFG (ForschergruppeGitter-Hadronen-Pha¨nomenologie).REFERENCES1. S. Booth et al., Phys. Lett. B519 (2001) 229,hep-lat/0103023; Nucl. Phys. Proc. Suppl.106(2002) 308, hep-lat/0111006.2. S. Necco et al., Nucl. Phys. B622 (2002) 328,hep-lat/0108008.3. T. van Ritbergen et al., Phys. Lett. B400(1997) 379, hep-ph/9701390.4. M. Lu¨scher et al., Nucl. Phys. B478 (1996)365, hep-lat/9605038.5. M. Go¨ckeler et al., in preparation.6. G. Bali et al., hep-lat/0210033.7. S. Bethke, hep-ex/0211012.8. J. Blu¨mlein et al., hep-ph/0407089.9. C. R. Allton et al., Phys. Rev. D65 (2002)054502, hep-lat/0107021.

Determination of Lambda in quenched and full QCD - an update

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Determination of Lambda in quenched and full QCD – an update

We present an update on our previous determination of the Lambda parameter in QCD. The main emphasis is on results for two flavours of light dynamical quarks, where we can now almost double the amount of data used, including values at smaller lattice spacings. The calculations are performed using $O(a)$ improved Wilson fermions. Little change is found to previous numerical values

Determination of Lambda in quenched and full QCD - an update

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