Alpha_s(M_Z) from hadronic tau decays

)FromHadronicτDecaysαs(MZ

K.Maltman∗

DepartmentofMathematicsandStatistics,YorkUniversity,

4700KeeleSt.,Toronto,ONCANADAM3J1P3†

T.Yavin‡

arXiv:0807.0650v1 [hep-ph] 3 Jul 2008DepartmentofPhysicsandAstronomy,YorkUniversity,4700KeeleSt.,Toronto,ONCANADAM3J1P3

(Dated:July3,2008)

Weperformanextractionofαsbasedonsumrulesinvolvingisovectorhadronicτdecaydata.Theparticularsumrulesemployedareconstructedspecificallytosuppresscontributionsassociatedwithpoorlyknownhigherdimensioncondensates,andhencereducetheoreticalsystematicuncertaintiesassociatedwiththetreatmentofsuchcontributionswhichareshowntobepresentinearlierrelatedanalyses.

2)=0.1187±Runningourresultsfromthenf=3tonf=5regimewefindαs(MZ

0.0016,inexcellentagreementwiththerecentlyupdatedglobalfittoelectroweakdataattheZscale,0.1191±0.0027,andotherhigh-scaledirectdeterminations.

PACSnumbers:12.38.-t,13.35.Dx,11.55.Hx

I.INTRODUCTION

Thevalueoftherunningstrongcoupling,αs(µ2),atsomeconventionallychosenref-erencescale,µref,isoneofthefundamentalparametersoftheStandardModel(SM).Inwhatfollows,weadheretostandardconventionandquoteresultsatthescaleµref=MZ,fornf=5,inthe

∗†

kmaltman@yorku.ca

CSSM,Univ.ofAdelaide,Adelaide,SA5005AUSTRALIA‡

t˙yavin@yorku.ca2

recentdeterminationsyielding

2

αs(MZ)

2

αs(MZ)

=0.1170±0.0012(lattice)

=0.1212±0.0011(τdecay)(1)(2)

forthelattice[3]andτdecay[6]determinations,respectively.

Inthispaperwerevisitthehadronicτdecayextraction,focussingonalternateFESRchoicesdesignedspecificallytoreducetheoreticalsystematicuncertaintiesnotincludedintheerrorassessmentofEq.(2)andassociatedwithpossiblesmallhigherdimension(D>8)OPEcontributionsassumednegligibleintheanalysesreportedinRefs.[4,6].

2

Wefindashiftintheresultsforαs(MZ)inexcessofthepreviouslyquotederror,andobtainalsoanimprovementintheagreement(i)betweentheτdecayanddirecthigh-scaledeterminationsand(ii)amongsttheseparateτdecayextractionsobtainedfromthevector(V),axialvector(A),andvector-plus-axial-vector(V+A)channelanalyses.Therestofthepaperisorganizedasfollows.InSectionIIwe(i)outlinethegeneralFESRapproachtoextractingαsfromhadronicτdecaydata,(ii)discusstherelevantfeaturesofexistinganalyses,(iii)pointoutpotentialadditionaltheoreticaluncertaintiesinthoseanalyses,associatedwiththeneglectofD>8OPEcontributions,(iv)establishexplicitlythepresenceofsuchcontributionsatalevelnotnegligibleonthescaleofthepreviouslyquotederrors,and(v)discussalternatesumrulechoiceswhichsignificantlyreducetheseuncertainties.InSectionIIIweusethesealternatesumrulestoperformseparateV,AandV+Aanalyses,employingeithertheALEPH[4,6,7,8]orOPAL[9]

2

isovectorhadronicτdecaydatasets.OurfinalresultsforαS(MZ),togetherwithadiscussionoftheseresults,aregiveninSectionIV.

II.

HADRONICτDECAYEXTRACTIONSOFαs

ThekinematicsofτdecayintheSMallowstheinclusiverateforhadronicτdecaysmediatedbytheflavorij=ud,us,VorAcurrentstobewrittenasasumofkinematically

(J)

weightedintegralsoverthespectralfunctionsρV/A;ij(s),associatedwiththespinJ=0,1componentsoftherelevantcurrent-currenttwo-pointfunctions[10].DefiningRV/A;ij≡Γ[τ−→ντhadronsV/A;ij(γ)]/Γ[τ−→ντe−ν¯e(γ)]andyτ≡s/m2τ,onehas

󰀆1󰀈󰀊

(0+1)(0)222

RV/A;ij=12π|Vij|SEWdyτ(1−yτ)(1+2yτ)ρV/A;ij(s)−2yτρV/A;ij(s)(3)

0

withVijtheflavorijCKMmatrixelement,SEWashort-distanceelectroweakcorrec-(0+1)(1)(0)

tion[11,12,13],andρV/A;ij(s)≡ρV/A;ij(s)+ρV/A;ij(s).Weconcentratehereontheisovector(ij=ud)case.

(0)(0)

Forij=ud,apartfromtheπpolecontributiontoρA;ud,allcontributionstoρV;ud(s),(0)

ρA;ud(s),areofO([md∓mu]2),andhencenumericallynegligible,allowingthesumoftheflavorudVandAspectralfunctionsρV+A;ud(s)tobedetermineddirectlyfromexperimentalresultsfordRV+A;ud/ds.FurtherseparationintoVandAcomponentsis

(0+1)

3

¯unambiguousfornπstates,butrequiresadditionalinputforKKnπ(n>0)states.

ErrorsontheexperimentaldistributionarethusreducedbyworkingwiththeV+Asum.DataandcovariancematricesforthespectraldistributionsdRV;ud/ds,dRA;ud/dsanddRV+A;ud/dshavebeenprovidedbyboththeALEPH[4,7,8]andOPAL[9]collaborations.TheALEPHcovariancesleadtoweightedspectralintegralswithnon-normalization-inducederrorsafactorof∼2smallerthanthoseobtainedusingtheOPALresults.Inaddition,ALEPHhasrecentlyprovidedpreviouslyunavailableinformation

¯distribution[6],amodeforwhichseparateinformationisnotavail-ontheV+AKKπ

ablefromOPAL.ThisisofrelevancetoperformingtheseparateVandAanalysessince

¯electroproductioncross-sections[14],recentBaBardeterminationsoftheisovectorKKπ

combinedwithCVC,allowforasignificantimprovementinthetreatmentoftheV/A

¯channel[6],whichchanneldominatestheuncertaintyintheseparationintheKKπ

V/Aseparationfornon-strangehadronicτdecays.Inviewoftheseadvantages,wewillfocusourdiscussionontheALEPHdata[15],thoughwewillalsoperformalternateindependentanalysesusingtheOPALdataasinput,asafurtherconsistencycheck.

(0+1)

Thespectralfunctions,ρV/A;ij(s),correspondtoscalarcorrelatorcombinations,ΠV/A;ij(s)≡ΠV/A;ij(s)+ΠV/A;ij(s),havingnokinematicsingularities.Foranysuchcorrelator,Π(s),withspectralfunctionρ(s),andanyw(s)analyticin|s|s0,analyticityimpliesthefiniteenergysumrule(FESR)relation

󰀆s0

1

w(s)ρ(s)ds=−

0

(0+1)

(1)

(0)

s

2

whereQ=−sandv(s)=dsw(s),withv(s0)=0.Inthisform,potentiallylargelogarithmscanbesummeduppoint-by-pointalongthecontourthroughthescalechoiceµ2=Q2.Theresulting“contour-improved”(CIPT)evaluationimprovestheconver-genceoftheintegratedD=0series[19].Analternateevaluation,referredtoas“fixedorderperturbationtheory”(FOPT),involveschoosingacommonfixedscale(suchasµ2=s0)forallpointsonthecontour.Largelogarithmsarethenunavoidableoveratleastsomeportionofthecontour.DetailedargumentsinfavoroftheCIPTprescription

󰀉

󰀃

DT(Q)

2

󰀅

D=0

,(5)

4

havebeenpresentedinRef.[6],andwewilltaketheCIPTevaluationasourcentralone.However,thedifferencebetweentheCIPTandFOPTevaluations,bothtruncatedatthesamegivenorder,liesentirelyincontributionsofyethigherorder.TheCIPT-FOPTdifferencethusservesasonepossiblemeasureoftheD=0seriestruncationuncertainty.Itturnsoutthatthisdifferenceis,inmostcases,significantlylargerthanotherpossibleestimatesofthesameuncertainty.WewillthusadoptaconservativeviewandincludethefullCIPT-FOPTdifferenceasonecomponentofourtruncationuncertaintyestimate.

4

TheD=0contributiontoDV/A;ijisknowntoO(αs),andgivenby

󰀃

DV/A;ij(Q)

2

󰀅

D=0

=

1

MSscheme,and,fornf=3,d0=d1=1,d2=1.63982,d3=6.37101and(0)(0)

d4=49.07570[5,20].Thenextcoefficient,d5,hasbeenestimatedtobe∼275[5]usingmethodsknowntohave(i)workedwellsemi-quantitativelyforthecoefficientsoftheD=0series[21]and(ii)produced,inadvanceoftheactualcalculation,anaccuratepredictionfortherecentlycomputedO(a3)D=2coefficientofthe(J)=(0+1)V+Acorrelatorsum[22].

ItisthestrongnumericaldominanceoftypicalOPEintegralsbyD=0contributionsatscalesaboves0∼2GeV2thatallowsthecorrespondingweightedspectralintegralstobeusedinmakingaprecisiondeterminationofαs.TheimpactofuncertaintiesinthesmallresidualhigherDOPEtermscanbeunderstoodbynotingthat,forallw(s),theD=0contributiontothew(s)-weightedOPEintegral,expandedasaseriesina0≡a(s0),hastheformCw[1+a0+O(a20)],wherebothCwandthecoefficients

2

occurringintheO(a20)contributiondependonw(s).Sincea(mτ)∼0.1,weseethatahigherDcontributionwithafractionaluncertaintyrrelativetothedominantD=0termwillproduceacorrespondingfractionaluncertainty∼10ronαs(m2τ).(Thefactorof10isreducedsomewhat(to∼5−6)whenoneincludestheeffectofhigherorderterms.)

2

Thus,e.g.,toachieveadeterminationofαs(MZ)accurateto∼1%(whichcorrespondstoadeterminationofαs(m2τ)accurateto∼3%)oneneedstoreducetheuncertaintiesinthedeterminationofthehigherDcontributions,relativetotheOPEtotal,tothesub-0.5%level.Howeasyitistosatisfythisrequirementdependsstronglyonthechoiceofweightw(s).Wewillreturntothispointbelow.

AmongtheremainingD>0OPEcontributions,theD=2contributionsareeither

22

O(m2u,d)orO(αsms)[23]andnumericallynegligibleatthescalesweconsider.TheD=4OPEtermsare,uptonumericallytinyO(m4s)corrections,determinedbytheRG

¯󰀁RGI,󰀂mssinvariantlightquark,strangequarkandgluoncondensates,󰀂mℓℓℓ¯s󰀁RGIand

󰀃󰀅OPE

󰀂aG2󰀁RGI.ExplicitexpressionsforΠV/A(Q2)D=4maybefoundinRefs.[23,24].

D≥6OPEcontributionsarepotentiallymoreproblematicsincetherelevantconden-satesareeitherpoorlyknownorphenomenologicallyundetermined.ALEPH[4,6,7,8]andOPAL[9]dealtwiththisproblemby(i)assumingD>8contributionscouldbeneglectedforallw(s)employedintheiranalyses,and(ii)fittingeffectiveD=6,8con-(8)

densatecombinations,δ(6)=−24π2C6/m6=−16π2C8/m8τandδτtodatausinga

(0)(0)(0)(0)

5

rangeofw(s).HeretheeffectivecondensatecombinationsC6,C8,···aredefinedsuchthat󰀄󰀃󰀅2OPE

Π(Q)D>4≡CD/QD(7)

D=6,8,···

uptologarithmicproportionaltoαslog(Q2/µ2).󰀇corrections,

Forw(s)=m=0bmym,theD≥6contributionstotheRHSofEq.(4)aregivenby

b2

C6

s30

+b4

C10

s50

+···,

(8)

againuptologarithmiccorrections,proportionaltoαs[26].IntegratedOPEcontribu-k+1

tionsofD=2k+2thusscaleas1/sk0(uptologarithms[27]),andhenceas1/s0relativetotheleadingD=0contribution.Forpinchedweights,theintegralsofthelogarithmiccorrectionstoEq.(7)aresuppressed,notjustbyadditionalfactorsofαs,butalso󰀋bysmallnumericalfactorswhichresultfromthestructureofthelogarithmicin-tegrals,|s|=s0dsykℓn(Q2/µ2)/QD,andcancellationsinherentinthepinchingcondition󰀇

mbm=0.

InRefs.[4,6,7,9],αs,󰀂aG2󰀁RGI,δ(6)andδ(8)(equivalentlyC6andC8),weredeterminedaspartofacombinedfittothes0=m2τversionsofthe(km)=(00),(10),(11),(12),(13)“spectralweightsumrules”,FESRsbasedontheweights,w(km)(y)=(1−y)kymw(00)(y),wherew(00)(y)=(1−y)2(1+2y)isthekinematicweightoccuringontheRHSofEq.(3).ALEPH[4,6,7,8]performedindependentversionsofthisfitforeachoftheV,AandV+Achannels,whileOPAL[9]performedindependent

(km)(km)

fitsfortheV+AandcombinedV,Achannels.Since(b2,···,b7)=(−3,2,0,0,0,0),(−3,5,−2,0,0,0),(−1,−3,5,−2,0,0),(1,−1,−3,5,−2,0)and(0,1,−1,−3,5,−2)for(km)=(00),(10),(11),(12),(13),respectively,weseethatallsixofthequantities,C6,···,C16,wouldinprinciplecontributetothesetofsumrulesemployed,makingacombinedfitimpossiblewithoutadditionalassumptions.

TheneglectofC10throughC16intheALEPHandOPALanalysescreatesatheoreticalsystematicuncertaintynotincludedintheerrorassessmentsofRefs.[4,6,7,8,9].Sincethefitsareperformedwithasingles0(s0=m2τ),thedifferings0-dependencesofintegratedcontributionsofdifferentDplaynorole,andhenceneglectofnon-negligibleD>8contributionscanbecompensatedforbyshiftsinthevaluesoffittedparametersrelevanttolowerDcontributions[28].Indicationsthatsuchacompensationmay,indeed,beatworkareprovidedby(i)thelackofagreementbetweenthevaluesfor󰀂aG2󰀁RGIobtainedfromtheseparateALEPHVandAanalyses[4,6],(ii)thefactthatthecentralfittedvaluesof󰀂aG2󰀁RGIobtainedintheV,AandV+ACIPTanalysesofbothgroupsareuniformlylowerthanoftheupdatedcharmoniumsumruleanalysisofRef.[25],and(iii)thepoorqualityofthe2005ALEPHAandV+Afits(χ2/dof=4.97/1and3.66/1,respectively)and2008ALEPHAfit(χ2/dof=3.57/1).

AfurtherindicationthattheneglectofD>8contributions(whichareinprinciplepresentinthe(km)=(10),(11),(12)and(13)spectralweightFESRs)ispotentiallydangerousisprovidedbyaconsiderationoftherelativesizesoftheD=6,8andD=0termscorrespondingtotheresultsoftheearlierALEPHandOPALfits.Oneshouldbear

6

inmindthattheadditionalfactorsofyintheweightsw(1m)(y),m≥1,stronglysuppressthecorrespondinglyweightedD=0integrals,butproducenosuchsuppressionsoftheintegratedhigherDcontributions,causingtheD>4contributionstoplayamuchlargerrelativerolefortheseweightsthantheydoforthe(00)and(10)weightcases.Takingthe2005ALEPHVfitasanexample,wefindthat

•forthe(11)spectralweightFESR,theD=6andD=8contributions(whichinclude,asperEq.(8),thepolynomialcoefficientfactors−1and−3,respectively)represent,respectively,5.2%and7.4%oftheleadingD=0contribution,whileD=10and12contributions(whichwouldbeweightedbythecoefficients5and−2fromw(11))areassumednegligible;•forthe(12)spectralweightFESR,theD=6andD=8contributions(weightedbypolynomialcoefficients1and−1,respectively)represent,respectively,−13.7%and6.5%oftheD=0contribution,whileD=10,12and14contributions(whichwouldbeaccompaniedbythew(12)polynomialcoefficients−3,5and−2)areagainassumednegligible;and•forthe(13)spectralweightFESR,theD=8contribution(weightedbypolynomialcoefficient1)represents−14.3%oftheD=0contribution,whileD=10,12,14and16contributions(whichwouldbeaccompaniedbythew(13)polynomialcoefficients−1,−3,5and−2,respectively)areoncemoreassumednegligible.Giventhe<0.5%toleranceinthesumofD>4relativetoD=0contributionsrequired

2

fora∼1%determinationofαs(MZ),theneglectofD>8contributionsappearstoustorepresentaratherstrongassumption.

Aquantitativetestofwhetherornotsuchcontributionscan,infact,besafelyne-glectedforalloftheweightsemployedintheALEPHandOPALanalysescanbeobtainedbystudyingthequalityofthefittedOPErepresentationsofthew(km)(y)-weightedspec-tralintegralsasafunctionofs0.Theutilityofthistestfollowsfromthefact,alreadynotedabove,thatintegratedcontributionsofdifferentDscaledifferentlywiths0.Thus,ifthefittedvaluesofαs,󰀂aG2󰀁RGI,C6andC8areunphysicalasaresultofshiftsinducedbytheneedtocompensateformissingD>8contributionsinoneormoreoftheFESRsemployed,thefactthatthiscompensationoccursinlowerdimensioncontributions,whichscalemoreslowlywiths0thandothecontributionstheyarereplacing,willshowupasadeteriorationofthefitqualityass0isdecreasedbelowthesinglevalues0=m2τusedintheALEPHandOPALanalyses.Incontrast,werethefitqualitytobemaintainedatlowers0,thiswouldprovidesignificantevidenceinsupportoftheprescriptionofne-glectingD>8contributionsinthesetofFESRsemployedinthoseanalyses.Wethusdefinethes0-dependentfit-qualities,

wFT(s0)

≡

ww

Ispec−IOPE(s0)

7

andscales0underconsideration.TheassumptionthatD>8OPEcontributionscanbe

w

safelyneglectedcorrespondstotheexpectationthat|FT(s0)|shouldremainlessthan∼1

2

forarangeofs0belowmτ,andforallofthew(s)employedintheanalysisinquestion.ItturnsoutthatneithertheALEPHnortheOPALfitssatisfythisexpectation.

w

Toillustratethispoint,weshow,inFig.1,thefitqualities,FV(s0),correspondingtothe2005ALEPHdataandfit[4],foraselectionofthe(km)spectralweights.Inthefigure,thedotted,medium-dashedandlong-dashedlinescorrespond,respectively,tothe(00),(12)and(13)spectralweights,whilethesolidlinesindicatetheboundariesFV(s0)=±1withinwhichwewouldexpectcurvescorrespondingtoaphysicallymean-ingfulfittolie.Weremindthereaderthat,althoughtheoriginal2005ALEPHs0=m2τ

22

AandV+Afitshadχ/dofsignificantly>1,theχ/doffortheVchannelfitwas0.52/1.Thetestisthusbeingappliedtothemostsuccessfulofthepreviousfits.

w

AlsoshowninthefigurearetheVchannelfitqualities,FV(s0),forthreeadditionalweights,w2(y)=(1−y)2(shortdot-dashedline),w3(y)=1−3(longdot-dashed2

line)andw(y)=y(1−y)2(double-dot-dashedline).Theweightsw2andw3arethefirsttwomembersofaseries,

wN(y)=1−

N

N−1

yN

(10)

towhichwewillreturninourownanalysisbelow.FromEq.(8),weseethattheonly

).TheD>4contributiontothew2(respectively,w3)FESRisC6

2s30

w2(respectivelyw3)FESRthusprovidesausefulindependenttestofthevalueofC6(respectivelyC8)obtainedintheearlierfits.Thew(y)=y(1−y)2FESR,withD>4OPEcontribution−2C6,providesanothersuchtestsincethislinearcombinations30

isindependentofthatappearinginthe(0,0)spectralweightFESR.ThestrengthofthetestisenhancedinthiscasebecausethefactoryintheweightleadstoasignificantsuppressionoftheD=0integral,makingthey(1−y)2FESRrelativelymoresensitivetoD>4contributions.IftheneglectofD>8contributionsintheearlieranalyseswasactuallyjustified,thes08contaminationinatleastsomeoftheoriginalfittedFESRs.Thedeteriorationinthefitqualityass0isdecreasedbelowm2τseenforallcasesshowninthefigureisinfactageneralfeature,andisfoundforalloftheweightsdiscussedandallthreeofthechannelsinvestigatedinthispaper.

Onecould,ofcourse,attempttousethes0dependenceofthew(km)-weightedspectralintegralstoaidinachievinganimprovedfitfortheD>4CD.Itisimportanttobearinmind,however,thattherangeofs0thatcanbeemployedinsuchafitislimited:to

2

s08

FIG.1:Fitqualitiesforthe2005ALEPHVfit,fortheweightsw(00),w(12),w(13),w2,w3andw(y)=y(1−y)2.Allnotationisasdescribedinthetext.

105FV(s0)0w-5-102.22.42.82.62s0 (GeV)3window,thenumberofindependentparametersthatcanbesuccessfullyfittedislimited.The(km)spectralweightFESRsthusrepresentnon-optimalchoicesforananalysisofthistypesincetheirOPEsidestypicallyinvolve,inadditiontotheparameterαs(m2τ)weareprimarilyinterestedindetermining,acombinationofseveraloftheunknownD>4CD.Itisalsoworthstressingthatthe(11),(12)and(13)spectralweightFESRsusedinthepreviousanalyseshaveanotherfeaturewhichmakesthemnon-optimalforananalysiswhosemaingoalisthedeterminationofαs.Optimizationofsuchadeterminationisachievedbyusingsumruleswhichenhance,asmuchaspossible,therelativecontributionoftheintegratedD=0series,sinceitisinthiscontributionthatthedominantdependenceonαslies.The(1m),m≥1,spectralweights,however,doexactlytheopposite,theadditionalfactorsofyproducingratherstrongsuppressionsoftheleadingD=0OPEintegrals(byfactorsof∼6.5,17,and37relativetothecorresponding(00)integralforthe(11),(12)and(13)cases,respectively)withoutanyaccompanyingsuppressionofhigherDcontributions(beyondthatwhichmay(ormaynot)bepresentinthecorrelatoritself).

Inviewoftheproblemsdisplayedbythe(km)spectralweightFESRanalyses,we

9

turntoFESRsbasedontheweights,wN(y)introducedalreadyinEq.(10)above.ThewNareconstructedtosharewiththe(0,0)spectralweightthepresenceofadoublezeroats=s0andtheresultingsuppressionofOPE-violatingcontributionsnearthetimelikepointontheOPEcontour.Forourproblemtheyhave,inaddition,thefollowingpositivefeatures,notsharedbythesetof(k,m)spectralweightsemployedintheALEPHandOPALanalyses:

•theD=0integralsgrowmoderatelywithNratherthandecreasingstronglyaswasthecasewhenonewentfromthelowertothehigherspectralweights;•atthesametime,thecoefficientgoverningtheonlyunsuppressedD>4contribu-tion(thatwithD=2N+2)decreaseswithN,furtherenhancingD=0relativetoD>4contributions;•becauseeachwNFESRinvolvesonlyasingleunsuppressedD>4contribution,thecollectionofwNFESRsiswell-adaptedtomostefficientlyimplementingthecon-straintsassociatedwiththes0dependenceofthecorrespondinglyweightedspectralintegralsinthefittingoftheunknownD>4OPEparameters;and

+1

scalingofthesingleunsuppressedD=2N+2•asNisincreased,the1/sN0

contributionrelativetotheleadingD=0contributionvariesmoreandmorestronglywiths0,increasingtheleverageforfittingC2N+2(thoughtheeffectisofcourseoffsettosomeextentbythedecreasewithNofthepolynomialcoefficient,1/(N−1),presentintheintegratedformoftheD=2N+2contribution).

ToquantifytheextenttowhichthelevelofD=0dominanceofthewNFESRsrepresentsanimprovementoverthatofthe(km)spectralweightFESRs,weintroducethedoubleratio,RD[wN,w(km),s0],definedby

R[wN,w

D

(km)

,s0]=

Drw(s0)N

w

[IOPE(s0)]D=0

.(12)

RD[wN,wkm,s0]representsthesuppressionofthefractionalcontributionofdimensionD

inthewNFESRrelativetothatinthew(km)FESRand,byconstruction,isindependentofCD.Takings0=m2τtobespecific,wefindthat•R6[w2,w(km),m2τ]=−1/2.1,−1/2.9,−1/4.4,and−1/12for(km)=(00),(10),

(11)and(12),respectively;•R8[w3,w(km),m2τ]=1/3.1,1/11,−1/25,−1/26and−1/58for(km)=(00),(10),(11),(12)and(13),respectively;•R10[w4,w(km),m2τ]=−1/6.8,1/79,−1/126,and−1/91for(km)=(10),(11),(12)and(13),respectively;

10

•R12[w5,w(km),m2τ]=−1/44,1/288and−1/379for(km)=(11),(12)and(13),respectively;and•R14[w6,w(km),m2τ]=−1/149and1/814for(km)=(12)and(13),respectively.NeglectofD>8contributionswouldthusbebetween∼1and3ordersofmagnitudesaferforthew4,w5andw6FESRsthanitwouldforthe(10),(11),(12)and(13)spectralweightsumrules.Haditbeensafeforthelatter,thenitwouldcertainlyalsobesafefortheformer.Fromourfitsbelow,however,wefindsmall,butnotentirelynegligible,D=10,12,14contributionstothew4,w5andw6FESRs,respectively.Theanalogouscontributions,whichplayamuchlargerrelativeroleinthehigherspectralweightFESRs,accountfortheproblemsoftheALEPHandOPALspectralweightFESRfitsseeninthefitqualityplotabove.

III.

THEwNFESRANALYSES

AsNgetslarge,thedifferentwN(y)becomelessandlessindependent,approaching1−yinthelimitthatN→∞.Theapproachto1−yalsoweakensthelevelofthedesiredsuppressionofcontributionsfromthevicinityofthetimelikepointontheOPEcontour.Inaddition,thereductionoftheunsuppressedintegratedD=2N+2contributionbythefactor1/(N−1)meansthatthesecontributionswilleventuallybedrivendowntotheleveloftheother,numericallyandαs-suppressed,contributionsofD>4havingD=2N+2[32].Forthesereasonswefocus,inwhatfollows,onthoseFESRscorrespondingtothelimitedsetofweightsw2,···,w6.AcleardemonstrationoftheindependenceoftheresultsassociatedwiththedifferentwNinthissetwillbegiveninSectionIV.

Thevaluesofanyinputparameters,togetherwithdetailsofourtreatmentofthespectralandOPEintegralsidesofthewNFESRs,aregiveninSubsectionsIIIAandIIIB,respectively.ResultsfortheALEPH-basedV,AandV+AandOPAL-basedV+Afits,aswellasabreakdownofthecontributionstothetheoreticalerrorsonthefittedparameters,αs(m2τ)andCD,D=6,8···14,aregiveninsubsectionIIIC.AfinalassessmentanddiscussionoftheresultsisdeferredtoSectionIV.

A.

ThewN-weightedspectralintegrals

OnthespectralintegralsideofthewNFESRs,weemployforourmainanalysisthepubliclyavailable2005ALEPHV,AandV+Aspectraldataandcovariancematrices[4,7].OurcentralresultswillalsofollowRef.[6]inincorporating,intheVandAchannels,

¯modemadetheimproveds-dependentV/AseparationofthecontributionfromtheKKπ

possiblebytherecentBaBarisovectorelectroproductioncross-sectionmeasurements[14]

¯distributionpresentedinRef.[6].IndependentandthedetailsontheV+AKKπ

analysesusingthe1999OPALV,AandV+Adataandcovariancematriceshavealsobeen

¯distributionperformed,thoughinthiscasewedonothavetheinformationontheKKπ

11

neededtomaketheimprovedV/AseparationforthatmodeandsowillreportresultsbelowonlyfortheV+Aanalysis.

WeemployasinputtothedeterminationoftheisovectorspectralfunctionfromtheALEPHorOPALdistributionsthevalues

SEW=1.0201(3)Be=0.17818(32)|Vud|=0.97408(26)

(13)(14)(15)

whereSEWistakenfromRef.[12],thelepton-universality-constrainedresultforBefromRef.[34],andtheresultfor|Vud|fromthemostrecentupdateofthe0+→0+superallowednuclearβdecayanalysis[35].TheπpolecontributiontotheAandV+Aspectralinte-gralsisevaluatedusingtheveryaccuratedeterminationoffπ|Vud|fromtheπµ2width[2].AsmallglobalrenormalizationmustalsobeappliedtotheALEPHandOPALdataasaresultofsmallchangestoBe,SEW,|Vud|andthetotalτstrangebranchingfraction,Bs,(whichentersthemostprecisedeterminationoftheoverallV+Anormalization,Rud;V+A)sincetheoriginalpublications.WiththefullsetofrecentBaBarandBelleupdatestothebranchingfractionsofvariousstrangemodes[36],weobtainRud;V+A=3.478(11).ItisassumedthattheV,AandV+Adistributionsarealltoberescaledbythesamecommonfactor.TheuncertaintyinRud;V+Astronglydominatestheoverallnormalizationuncertaintyonthespectralintegrals.

B.

ThewN-weightedOPEintegrals

FortheD=0contributionweemploytheCIPTevaluationasourcentraldetermina-tion.WetruncatetheD=0AdlerfunctionseriesatO(¯a5),usingtheknowncoefficients

(0)

fortermsuptoO(¯a4)andtheestimated5=275±275ofRef.[5]forthecoefficientofthelastterm.AnindependentevaluationusingthealternateFOPTevaluationisalso

(0)

performedandthevariationinducedbytheuncertaintyind5andtheCIPT-FOPTdifferenceaddedinquadraturetoproducethefulltruncationuncertaintyestimate.Ananalogousprocedure,usinghowevertheaverageoftheCIPTandFOPTdeterminationsascentralvalue,andhalfthedifferenceasthecorrespondingcomponentofthetruncation

(0)

uncertaintyestimate(addedlinearlytotheuncertaintygeneratedbythatond5),wasemployedinRef.[5].OurestimateyieldsaD=0truncationuncertaintyassessmentsimilartothatofRef.[5],butsignificantlymoreconservativethanthealternateestimates

(0)

basedonacombinationofthed5uncertaintyandresidualscaledependencewhichhasalsobeenemployedintheliterature.

InevaluatingtherunningcouplingovertheOPEcontourweemploytheexactanalyticsolutionassociatedwiththe4-loop-truncatedβfunction[37].Thereferencescaleinputneededtospecifythissolution,takenheretobeαs(m2τ),istobedeterminedaspartofthefittingprocedure.

22

TheD=2contributions,asalreadynoted,areeitherO([md±mu]2)orO(αsms),andhenceexpectedtobenumericallynegligible.Ourcentralvaluescorrespondtoneglecting

12

thementirely.TheO([md±mu]2)contributionsshould,infact,beneglectedinanycase,asamatterofconsistency.Thereasonisthat,evenatthehighestscale,s0=m2τ,allowedbykinematics,theOPErepresentationofthe“longitudinal”(J=0)contributiontotheexperimentalspectraldistribution(inthe(J)=(0+1)/(0)decompositionofEq.(3))iscompletelyoutofcontrol.NotonlydothevariouslyweightedintegratedD=2OPEseriesdisplayextremelybadconvergence,butalltruncationschemesforthesebadlybehavedseriesemployedintheliteraturebadlyviolateconstraintsassociatedwithspectralpositivity[38].ItisthusimpossibletousethelongitudinalOPEtoestimatetheO([md±mu]2)longitudinalcontributionstothespectraldistribution,whichmeansthat

(0+1)

thespectralfunctionsρud;V/A(s)canbedeterminedonlyuptouncertaintiesofO([md∓mu]2),respectively.ItwouldthusbeinconsistenttoexplicitlyincludecontributionsofthissameorderontheOPEsideofthe0+1FESRs.Wehave,inanycase,verified,bydirectcomputation,thatincludingtheintegratedD=2J=0+1OPEcontributionswouldhaveanegligibleimpactonouranalysis,inagreementwiththeresultsforthesecontributionsquotedintheearlieranalyses.TheJ=0+1,D=2computationemployedtheexactsolutionfortherunningmassescorrespondingtothe4-looptruncatedβ[37]andγ[39]functions,withPDG06valuesforthe

¯󰀁RGI󰀂ℓℓ

=1.1±0.6,(18)

obtainedbyupdatingtheanalysisofRef.[43],usingtherangeofrecentnf=2+1

latticeresultsforfBs/fBasinput[44].AlthoughthisvalueofrcisnearlytwicethatemployedintheearlierALEPHandOPALanalyses(whosevalues,however,arebasedonsomewhatout-of-dateinput),thedifferencebetweenthetwohasnegligibleimpactonthefinalanalysissincetheintegratedD=4contributionsarebothsmallatthescalesemployedand,inanycase,dominatedbythegluoncondensatecontribution.Thesizableuncertaintywequoteonrc,forthesamereason,playsanegligibleroleinourfinaltheoreticalerrorestimate.

13

D>4contributionsarehandledbytreatingthevariousC2N+2asfitparameters.C2N+2isfitted,togetherwithαs(m2τ),tothesetofIwN(s0)correspondingtoarangeofs0.Therequirementthatthevaluesofαs(m2τ)obtainedinthismannerfromthedifferentwNFESRsshouldbeconsistentprovidesanon-trivialcheckonthereliabilityoftheanalysis.WediscussthisissuefurtherinSectionIV.

FortheALEPH-basedfits,weworkwithanequallyspacedsetofs0values,s0=(2.15+0.2k)GeV2,k=1,···,6,adaptedtotheALEPHexperimentalbins.Wealsostudythestabilityofourfitsbyeitherremovingthe2.15GeV2pointoradding,inaddition,s0=1.95GeV2.FortheOPAL-basedfits,theanalogouss0setiss0=(2.176+0.192k)GeV2,k=1,···,6,withstabilitystudiedbyeitherremovingthelowestpoint,oraddinganadditionalpointwiths0=1.984GeV2.

C.

Results

ResultsfortheV,AandV+AfitsbasedontheALEPHdataarepresentedintheupperportionofTableI.Inthetable,wedisplay,foreachofthewN,N=2,···,6,FESRs,thefittedvaluesofαs(m2τ)andtherelevantD>4coefficient,C2N+2,thelatter

2N+2

quotedinthedimensionlessform,C2N+2/mτ.Weremindthereaderthat,inarriving

¯mode,atthesevalues,wehaveimplementedtheimprovedV/AseparationfortheKKπ

discussedalreadyabove.Thisimprovementproducesanupward(downward)shiftof0.0013inthecentralvalueoftheA(V)determinationsofαs(m2τ),improvingfurthertheconsistencybetweentheresultsoftheseparateV,AandV+Aanalyses.Thelevelofconsistency,evenbeforethisimprovement,issignificantlybetterthanthatdisplayedbythe(km)spectralweightanalysisresultsreportedinRef.[6].

ThelowerportionofTableIcontainsthecorrespondingresultsfortheOPAL-basedV+Afits.TheresultsfortheseparateVandAfitsarenotdisplayedinthiscase,since

¯contributiontotheinclusivedistributionrequiredwelacktheinformationontheKKπ

toperformtheimprovedV/Aseparation.Forcompleteness,however,wementionthatthecentralvaluesofαs(m2τ)obtainedwithoutthiscorrectionlie0.003lower(higher)fortheV(A)fits.TheimprovedV/Aseparation,ofcourse,playsnoroleintheV+Afit.TheALEPH-andOPAL-basedresultsareseentobeinverygoodagreementwithinerrors.

Theexperimentalerrorsquotedinthetablecontainacomponentassociatedwiththe0.32%normalizationuncertainty,whichis100%correlatedforalloftheseparateanalyses.Thetheoryerrorisobtainedbyaddinginquadratureuncertaintiesassociatedwith(i)thetruncationoftheD=0series(itselfthequadraturesumofthedifferenceof

(0)

theCIPTandFOPTfitresultsandtheuncertaintyproducedbytakingd5=275±275),(ii)theuncertaintiesontheD=4inputcondensatesand(iii)the“stability”uncertainty,generatedbyvaryingtheloweredgeofthefitwindowemployed,asdescribedabove.Individualcontributionstothetheoreticalerrorsonthefittedparameters,αs(m2τ)and

2N+2

C2N+2/mτ,obtainedfromthewN-weighted,ALEPH-basedV+AFESRs,areshown,intheupperandlowerhalvesofTableII,respectively.ResultsfortheOPAL-basedV+AandALEPH-basedVandAfitsarenotquotedseparately,thedecompositionsbeing

14

2N+2obtainedusingeitherTABLEI:ResultsofthewNFESRfitsforαs(m2τ)andC2N+2/mτ

theALEPHorOPALdataandcovariances.Inallentries,thefirsterrorisexperimentalandthesecondtheoretical.

Dataset

V

Weight

0.321(7)(8)0.321(7)(10)0.321(7)(11)0.321(7)(12)0.321(7)(12)

w2w3w4w5w6

V+A

0.320(5)(8)0.320(5)(9)0.320(5)(10)0.320(5)(11)0.320(5)(12)0.322(7)(8)0.322(7)(10)0.322(7)(11)0.322(7)(12)0.322(8)(12)

2N+2C2N+2/mτ

−0.000072(24)(60)

0.000182(28)(71)−0.000216(27)(70)0.000201(23)(66)−0.000166(19)(59)

V+A

15

2N+2ob-TABLEII:Contributionstothetheoreticaluncertaintiesonαs(m2τ)andC2N+2/mτtainedinthefitstowNV+AFESRsbasedontheALEPHdataandcovariances.

Observable

w2w3w4w5w6

2N+2C2N+2/mτ

FOPT

δd5

(0)

δ󰀂aG2󰀁

stability

0.000069

0.0000900.0000780.0000630.0000510.0000190.0000160.0000130.0000120.0000080.0000840.0000720.0000580.0000450.0000350.0000270.0000440.0000530.0000580.000062

forthekinematicweight,w(00),theFOPTexpansion,truncatedatagivenorder,wasshowntooscillateaboutthecorrespondinglytruncatedCIPTexpansionwithaperiodofabout6perturbativeorders[45].StudyingtheFOPT-CIPTdifferenceasafunctionoftruncationorderforthevariouswNwefindevidenceforasimilaroscillatorypattern,butwiththetruncationorderatwhichthecross-overbetweenthetwotruncatedsumsoccursdependentonN.WethusconsiderthesmallFOPT-CIPTdifferenceforw2anartifactoftheparticulartruncationorderofourcentralresults,andexpectthedifferencetogrowforthenextfewtruncationorders.Forthisreason,tobeconservative,wetakethelargestoftheFOPT-CIPTdifferences(thatforw6)asourestimateoftheFOPTvs.CIPTcomponentofthetruncationuncertaintyforαs(m2τ)forallofthewNFESRsstudied.Thisprescriptionleadstoacommontheoreticalerrorof±0.012forallofourdeterminationsofαs(m2τ).

Theresultsquotedsofartakeintoaccountshort-distanceelectroweakcorrectionsbutdonotincludelong-distanceelectromagnetic(LDEM)effects.SuchLDEMcorrec-tions,thoughbelievedtobesmall,havebeeninvestigatedindetailonlyfortheππfinalhadronicstate[46,47].WestudytheimpactoftheππLDEMcorrectionsontheVandV+AchannelanalysesusingtheformofthesecorrectionsgiveninRef.[46](whichim-plementationincorporatesaresonancecontributionnotincludedintheearlierstudiesofRefs.[47]).Wefindthatthecorrectionraisesαs(m2τ)by0.0002−0.0003(0.0001−0.0002)forthevariousV(V+A)channelwNFESRanalyses.Inarrivingatourfinalassessment,reportedinthenextsection,wehaveincludedtheππLDEMcorrection,assigningitanuncertaintyof100%,inviewoftheas-yet-undeterminedcorrectionsassociatedwithhighermultiplicitymodes.Evenwereonetoexpandthisuncertaintyseveral-fold,theimpactonourfinalerrorwouldremainentirelynegligible.

16

IV.

FINALRESULTSANDDISCUSSION

A.

Results

InordertoavoidtheadditionaluncertaintiesassociatedwiththeseparationoftheobservedV+AspectraldistributionintoitsVandAcomponents,webaseourfinalresultsforαsontheV+AwNFESRanalyses.Asseenabove,theagreementoftheALEPH-andOPAL-basedV+Aresultsisexcellent.TheindividualALEPHVandAfitsare,inaddition,inextremelygoodagreementwiththecorrespondingV+Aresults,though,ofcourse,withlargerexperimentalerrors.TheagreementoftheALEPHV,AandV+AcentralvaluesisconsiderablycloserthanthatobtainedfromthespectralweightanalysisofRef.[6].Itshouldbestressedthattheagreementinthepresentcaseisobtainedusingthevalueof󰀂aG2󰀁RGIdeterminedindependentlyinRef.[25],insharpcontrasttotheAandV+AfitsofRef.[6],whichrequireincompatible,andunambiguouslynegative,values.

AveragingtheV+Aresults,usingthenon-normalizationcomponentoftheexperi-mentalerrors,weobtain

αs(m2(19)τ)=0.3209(46)(118)wherethefirsterrorisexperimental(nowincludingthenormalizationuncertainty)and

thesecondtheoretical.TheexperimentalerrorisidenticaltothatobtainedinthespectralweightanalysisofRef.[6],whileourtheoreticalerrorislargerasaresultofthemoreconservativetreatmentoftheD=0truncationuncertainty.Thetheoreticalerroroftheearlieranalyses,ofcourse,doesnotincludetheadditionalcontributionidentifiedabove,associatedwiththeneglectofD>8OPEcontributions.

2

Thenf=5result,αs(MZ),isobtainedfromthenf=3resultgiveninEq.(19)usingthestandardself-consistentcombinationof4-looprunningwith3-loopmatchingattheflavorthresholds[48].AsshowninRef.[5],takingmc(mc)=1.286(13)GeVandmb(mb)=4.1(25)GeV[49],thematchingthresholdstobermc,b(mc,b)withrvaryingbetween0.7and3,andincorporatinguncertaintiesassociatedwiththetruncatedrunning

2

andmatching,producesacombinedevolutionuncertaintyof0.0003onαs(MZ).Ourfinalresultisthen

2

αs(MZ)=0.1187(3)(6)(15)(20)wherethefirstuncertaintyisduetoevolution,thesecondisexperimentalandthethird

theoretical.Thedifferencebetweenthisvalueandthatobtainedintheearlierspectralweightanalysis,0.1212(11),servestoquantifytheimpactoftheD>8contributionsneglectedinthepreviousanalysis.

Theresult,Eq.(20),isingoodagreementwithanumberofrecentindependentexperimentaldeterminations,specifically,

•the2008updateoftheglobalfittoelectroweakobservablesattheZscale,quoted

2

inRef.[6],whichyieldsαs(MZ)=0.1191(27)exp(1)th;•thecombinedNLOfittotheinclusivejetcross-sectionsmeasuredbyH1and

2

ZEUS[50],whichyieldsαs(MZ)=0.1198(19)exp(26)th;

17

•theNLOfittohigh-Q21-,2-and3-jetcross-sectionsmeasuredbyH1(presented

2

atDIS2008󰀁andthe2008HERA-LHCworkshop[51])whichyieldsαs(MZ)=󰀂+41

0.1182(8)exp−31scales(18)pdf;•theNNLOfittoeventshapeobservablesine+e−→hadronsatLEP[52],which

2

yieldsαs(MZ)=0.1240(33);

•theSCETanalyis,includingresummationofnext-to-next-to-next-toleadingloga-rithms,ofALEPHandOPALthrustdistributionsine+e−→hadrons[53],which

2

yieldsαs(MZ)=0.1172(13)exp(17)th;and•thefittoe+e−→hadronscross-sectionsbetween2GeVand10.6GeVCMen-󰀁󰀂2ergy[54],whichyieldsαs(MZ)=0.119+9−11.

TheagreementwiththerecentupdatedanalysisofΓ[Υ(1s)→γX]/Γ[Υ(1s)→X][55],whichreplacestheolder󰀁+6󰀂analysisusuallycitedinthePDGQCDreviewsection,andyields2

αs(MZ)=0.119−5,isalsogood.Notethattheτdecayextractionisconsiderablymoreprecisethananyoftheotherexperimentaldeterminations.Inaddition,theτdecayandlatticeresults,whosediscrepancywasnotedattheoutset,arenowseentobecompatiblewithinerrors,acompatibilitywhichis,infact,furtherimprovedbytheresultsofarecentstudy[56]whichrevisitsthelatticedetermination,incorporatinglatticedataatawiderrangeofscalesthanthatemployedinRef.[3].

B.

Discussion

Inthissubsectionwediscussfurtherthereliabilityandconsistencyofourextractionofαs,compareourresultsfortheCDwiththoseofotheranalyses,andcommentonanumberofotherrelevantpoints.

WebeginbydiscussingwhatimpacttherecentlyreleasedBelleτ→ππντdata[57]mighthaveonourconclusions.Notethattheππbranchingfraction,Bππ,measuredbyBelleisingoodagreementwiththepreviousτmeasurementsreportedbyALEPH[7],OPAL[9],CLEO[58],L3[59]andDELPHI[60].Theunit-normalizednumberdistribu-tion,however,differsslightlyinshapefromthatobtainedbyALEPH,beingsomewhathigher(lower)thanALEPHbelow(above)theρpeak.Suchadifferencewillleadtonormalizationands0-dependenceshiftsintheweightedVandV+Aspectralintegrals,causing,ingeneral,shiftsinthefittedvaluesofbothαs(m2τ)andtheC2N+2.Toinvesti-gatethesizeoftheseeffects,weusethenewworldaverageforBππ(includingtheBelleresult)tofixtheoverallnormalizationoftheBelleππdistributionand,afteraddingthedifferenceoftheweightedBELLEandALEPHππspectralintegralcomponentstotheALEPHspectralintegrals,performaseriesof“Belle-ππ-modified”wNFESRfits.SincewelackthecovarianceinformationneededtofullyreplacetheALEPHππwithBelleππdata,weemploytheALEPHcovariancematrix,withoutchange,inthefit.Theresultsthusrepresentonlyanexplorationofthemagnitudeoftheshiftinαslikelytobeassociatedwithsuchashiftintheshapeoftheππdistribution.Wefindthatthe

18

2

Belle-ππ-modifiedVchannel(respectively,V+Achannel)fitsyieldαs(MZ)valueslowerthanthoseobtainedusingtheALEPHdataaloneby∼0.00007(respectively,0.00013),showingthattheimpactonourcentralresult(obtainedfromtheV+Achannelfits)isnegligibleonthescaleofourotheruncertainties.Itwouldnonethelessbeextremelyinter-estingtohavemeasuredversionsofthefullnon-strangespectraldistribution,includingtheimprovedV/Aseparationmadepossiblebythemuchhigherstatistics,fromtheBfactoryexperiments.

Withregardtothereliabilityandconsistencyofourresults,wenotefirstthat,foreachoftheV,AandV+Aanalyses,thesamequantity,αs(m2τ),isobtainedfromfiveindependentFESRfits.IneachoftheV,AandV+Achannels,wefindthatthere-sultsfromthedifferentwNanalysesareinexceedinglygoodagreement,thevariationacrossthedifferentweightchoicesbeingatthe±0.0001level,andhenceinvisibleattheprecisiondisplayedinTableI.ThefittingoftheD>4OPEcoefficients,CD,andcon-commitantidentificationofthesmallD>4OPEcontributionsiscrucialtoachievingthislevelofagreement,ascanbeseenfromTableIII,whichshowstheALEPHV+Afitvaluesforαs(m2τ)alreadyquotedabove,togetherwiththecorrespondingresultsob-tainedbyignoringtherelevantD>4contribution,andworkingatthehighestavailablescale,s0=m2τ.InassessingtheimprovementinconsistencyproducedbyincludingtheCDinthefits,oneshouldbearinmindthatthenon-normalizationcomponentoftheexperimentaluncertainty(whichisstillcorrelatedbut,unlikethenormalizationandthe-oreticaluncertainties,not100%correlatedamongstthedifferentweightcases)is0.003.TheimpactofincludingtheD>4contributionsis,notsurprisingly,greatestforthew2FESR,wherethesuppressionoftheD=6contributionbythepolynomialcoefficientfactor1/(N−1)(=1inthiscase)istheleaststrongofallthecasesstudied.TheresultsofthetablealsoshowthatuseofthewNFESRshas(asintended)beensuccessfulinsuppressingD>4relativetoD=0OPEcontributions,aneffectdesirableforopti-mizingtheaccuracyofourαsdetermination.ThetableinfactshowsthattheimpactofthefullD>4contribution,inallbutthew2case,isatalevellessthan∼50%ofthedominanttheoreticalcomponentoftheoveralluncertainty,makingtheimpactofhigherordercorrectionstothetreatmentoftheintegratedD>4contributionssafelynegligible[27].

WhilethelackofconsistencyoftheresultsforαsinthelimitthatalltheCDaresettozeroestablishestheindependenceofthedifferentwN-weightedFESRs,andhencethenon-trivialnatureoftheconsistencyobservedoncetheCDareincludedinthefits,anevenmorecompellingcaseforthedegreeofindependenceofthedifferentFESRsisprovidedbytheresultsobtainedbyfittingthewN-weightedOPEintegralstothesetofwM-weightedspectralintegrals,withN=M.Theresultsforαs(m2τ)obtainedfromthisexercise,usingtheALEPHdataintheV+Achannel,areshowninTableIV,whoserow(respectively,column)headingsgivetheweightemployedforthespectral(respectively,OPE)integrals.Blankentriesinthetabledenotecaseswherenominimumcouldbefoundfortheχ2functionhavingpositiveαs(m2τ).ItisevidentfromthetablethattheconstraintsonαsassociatedwiththesetofwNemployedinouranalysisenjoyahighdegreeofindependence.

FurtherevidenceforthereliabilityofourfitsforαsandtheCDisprovidedbythe

19

TABLEIII:ImpactoftheinclusionofD>4OPEcontributionsonthefittedvaluesforαs(m2τ)fortheALEPH-basedanalyses.ThecolumnheadedfullfitrepeatsthevaluesquotedaboveforthevariouswN-weightedV+AFESRs,whilethatheadednoD>4containsthecorrespondingvaluesobtainedbyworkingatthemaximumscales0=m2τandneglectingthecontributionofdimensionD=2N+2ontheOPEside.

Channel

w2w3w4w5w6

A

0.3190.3190.3190.3190.319

w2w3w4w5w6

0.3140.3120.3140.3160.318

fullfit

noD>4

w2w3w40.3200.175—w3

w5—w6—

0.4990.3840.3200.2770.243w5

—

0.4500.3880.3490.320

factthat,unlikethefitqualitiesassociatedwiththeALEPHfitparametersets,thoseassociatedwithourfitsremainbetween−1and1forallthreechannels,allfivewN,andalls0inourfitwindow.ThisisillustratedfortheVchannelinFig.2,whichshowsthewFV(s0)correspondingtoourfitsforthefourweightsdiscussedabove(w(00),w2,w3andw(y)=y(1−y)2)whoseOPEintegralsdonotdependonanyoftheCD>8.Alsoshown,forcomparison,arethecorrespondingALEPHfitresultsforthissamesetofweightsand

20

TABLEV:Thefittedvaluesforαs(m2τ)obtainedfromtheALEPH-basedwN-weightedV+AanalysesasafunctionoftheD=0truncationorder,M,whereMherespecifiesthatthelasttermkeptintheD=0seriesfortheAdlerfunctionisthatproportionaltodM[αs(Q2)]M.OurcentralanalysesabovecorrespondtoM=5.

Mαs(m2τ)

21

FIG.2:Comparisonofthefitqualitiescorrespondingto(i)ourfitsand(ii)the2005ALEPHfit,fortheVchannelandtheweightsw(00),w2,w3andw(y)=y(1−y)2.Allnotationasdescribedinthetext.

105FV(s0)0w-5-102.22.42.6s0 (GeV)Asaresult,theprecisioninourdeterminationsofmostoftheCDisnothigh.InTableVIwecompareourresults(withtheexperimentalandtheoreticalerrorsnowcombinedinquadrature)withthoseofALEPH,OPALandtwootherrecentcondensatestudies[33,61],focussingonthequantitiesC6,8obtainedinthoseearlierstudies.IntheALEPHandOPALcases,theerrorsshownarethenominalonesquotedintheoriginalpublications,anddonotincludethesizeableadditionaluncertaintyassociatedwiththeneglectofD>8contributionsdiscussedalreadyabove.InthecaseofRef.[33],whichemploysfitsusingtheweightsw(y)=1−yN(whichhaveazerooforder1aty=1),wequoteonlythevaluesconsideredreliablebytheauthorsthemselves,andofthese,onlytheonescorrespondingtoΛ=350MeV,sinceitisthisvaluewhichliesclosesttothat(346MeV)associatedwithourcentralfitresultabove.InthecaseofRef.[61]wequoteonlytheAchannelC6result,sincethiswastheonlyonetodisplaydemonstrablestability,withinerrors,ingoingfromthe2-parameterfit(includingcontributionsupto

22.8322

wN

FIG.3:ThefitqualitiesFV+A(s0)correspondingtoourfittedOPEparametersfor,fromtopthroughbottomontheright,thew2throughw6FESRs.

105(s0)V+A0FwN-5-102.5s0 (GeV)23D=6)tothe3-parameterfit(includingcontributionsuptoD=8)[62].

Wenotethat,fortheVchannel,wheretheALEPHfitqualitywasbetter,ourC8valuesactuallyagreewellwiththoseofALEPHandOPAL,whileourC6centralvaluesaresomewhatlarger,butofthesamegeneralsize.FortheAchannel,wheretheALEPHfitqualitywaspoorer,wehave,instead,significantdisagreementforC6,notjustinmagnitude,butalsointhesignofthecentralvalue.ThesignificantdifferencesfortheAchannelarealsoseenintheV+Achannel,asonewouldexpect.SinceourvaluesleadtoextremelygoodOPErepresentationsforthew(00),w2,w3andw(y)=y(1−y)2spectralintegralsinallthreechannels,whiletheALEPHandOPALfitsdonot,itisnosurprisethatsignificantdifferencesbetweenourfitsandtheirsshouldbefound.We

A

notethatthedisagreementinsignforC6confirmstheresultfoundinRefs.[33,61].Aspointedoutinthosereferences,thefitresultsimplyasignificantbreakdownofthevacuumsaturationapproximation(VSA)forthefour-quarkD=6condensates,sinceVSAvaluesfortheVandAchannelareintheratio−7:11.Whileitistruethat,

A

giventhesizeoftheerrors,thesignofC6isnotfirmlyestablishedbyeitherourfitsor

23

TABLEVI:ComparisonofourresultsforC6andC8withthoseofRefs.[6](ALEPH),[9](OPAL),[33](DS)and[61](AAS).C6isgiveninunitsof10−3GeV6andC8inunitsof10−3GeV8.Theerrorsquotedareasdescribedinthetext.

ReferenceALEPH

−3.4(5)

DS

—−5.9(2.0)

—6.0(7.0)5.0(8)

−4.3(3.0)

—

—−8.4(3.8)

—25.1(13.2)

AC.6(3)

AC8−6.0(3)

−0.3(1.5)1.3(4.2)

thoseofRefs.[33,61],nonethelesstherelativemagnitudesoftheVandAresultsarefarfromsatisfyingtheVSArelation.ToimproveontheaccuracyofthedeterminationsoftheCD,andinvestigatesuchissuesfurther,wouldrequireworkingwithadifferentsetofweightfunctions,choseninsuchawayastosuppressD=0andemphasizehigherDcontributions.

C.

Finalsummaryandcomments

Tosummarize,wehaveperformedanumberofrelatedFESRanalysesdesignedspecif-icallytoreducetheimpactofpoorlyknownD>4OPEcontributionsontheextractionofαsusinghadronicτdecaydata.Ourresultsshowahighdegreeofconsistencyandsatisfyconstraintsnotsatisfiedbyotherτdecaydeterminations.Ourfinalresultis

2

αs(MZ)=0.1187±0.0016

(21)

wheretheevolution,experimentalandtheoreticalerrorshavenowbeencombinedin

quadrature.Theresultisinexcellentagreementwith(andmoreprecisethan)alternateindependenthigh-scaleexperimentaldeterminations.Itis,however,significantlylowerthanthevaluesobtainedintheearlierALEPHandOPALhadronicτdecayanalyses.Wehaveprovidedclearevidencethatthesourceofthisdiscrepancyliesinthecontaminationoftheseearliercombinedspectralweightanalysesbyneglected,butnon-negligible,D>8OPEcontributions.

Atechnicalpointworthemphasizingfromthediscussionaboveistheimportanceofworkingwitharangeofs0ratherthanjustthesinglevalues0=m2τ,andtheutility,inthiscontext,ofusingweightsdefinedintermsofthedimensionlessvariabley=s/s0.Forsuchweights,thes0-dependenceoftheresultingweightedspectralintegralsallowsonetostraightforwardlytestanyassumptionsmadeaboutthevaluesofD>4OPEcoefficients,or,betteryet,toattemptactualfitstoobtainthesevaluesusingdata.Suchs0-dependencestudiesseemtousunavoidableifonewishestodemonstratethatD>4OPEcontributionshaveindeedbeenbroughtundercontrolatthelevel(∼0.5%ofthefull

2

spectralintegrals)requiredfora∼1%precisiondeterminationofαs(MZ).Fortunately,

24

aswehaveshown,suchcontrolisnotdifficulttoachieve,andwehavedisplayedanumberofweightswhichareusefulforthispurpose.Theweights,wN(y),whichisolateindividualintegratedD=2N+2contributions,arerelatedtothekinematicweight,w(00)(y),byslowlyvaryingmultiplicativefactors[63],andhenceproduceerrorsonthespectralintegralsthatarecomparableto,orbetterthan,thoseforw(00).

Westressthattheoreticalerrorsnowdominatetheuncertaintyinthehadronicτ

2

decaydeterminationofαs(MZ),theD=0OPEtruncationerrorbeingthelargestamongthese.Furtherreductioninexperimentalerrors,andinparticular,improvementsintheV/Aseparation,arelikelytobepossibleusingdatafromtheBfactories,andsuchimprovementswouldbeusefulforfurthertestingtheconsistencyoftheV,AandV+Adeterminations.Giventhecurrentsituation,however,reducedexperimentalerrors

2

wouldhavelittleimpactonthetotalerroronαs(MZ).NOTEADDED:Afterthecompletionoftheworkdescribedinthispaper,anewex-plorationoftheextractionofαsfromhadronicτdecaydatawasposted[].Thisstudyemploysa5-parametermodelfortheBoreltransformoftheD=0componentoftheAdlerfunction,onewhosestructureincorporatestheformoftheknownleadingUVrenormalonandtwoleadingIRrenormalonsingularities.Theparametersofthemodelarefixedusing

(0)

theknowncoefficients,d(0),···,d4,oftheD=0Adlerfunctionseriesexpansion,to-(0)

getherwiththeestimatedvalued5=283.Thestudymakestheworkingassumptionthatthetrueall-ordersresultwillbewellapproximatedbytheBorelsumofthecorrespondingmodelAdlerfunctionseries.TheresultsgeneratedusingthemodelarethenarguedtofavortheuseofFOPToverCIPTfortheD=0OPEcontribution.ItisnotcleartouswhetherextendedansatzefortheBoreltransform,involvingadditionalparameters,wouldleadtothesameordifferentconclusions.Wedocomment,however,thattheresultsforαs(m2τ)obtainedfromourFOPTfits,thoughyieldingrepresentationsofthespectralinte-graldatawhichareofnearlyasgoodqualityasthoseproducedbythecorrespondingCIPTfits,aresignificantlylessconsistentthanthoseobtainedusingtheCIPTprescription,theresultsfortheV+Achannelrangingfrom0.320forw2to0.312forw6.WhetheroneviewsthisasanempiricalargumentinfavorofsofteningtheconclusionsofRef.[]ornot,theargumentsofthatreferenceclearlysupporttakingaconservativeapproachtoassessingtheD=0truncationuncertainty.

ForreadersinclinedtoadopttheFOPTdeterminationasthecentralone(inspiteofthereducedconsistencyofitsoutput),wecommentthattheαs(m2τ)obtainedfromthew2

2

throughw6V+Afitscorrespondtovaluesofαs(MZ)lyingbetween0.1186and0.1176.TheCIPTresult,asitturnsout,notonlydisplaysbetterconsistency,butisalsoinbetteragreementwiththesoon-to-be-reportedresultsoftheupdatedlatticeanalysisofRef.[56].

2

Regardingthevaluesforαs(m2τ)andαs(MZ)quotedinRef.[],thereadershouldbearinmindthattheseresultfromaw(00)-weightedV+AFESRanalysisrestrictedto

V+A

andthesinglevalues0=m2τ.Withonlyasingles0,itisnotpossibletofitC6

V+A

C8,andcentralvalues(anderrors)mustthereforebeassumedforthesequantities.

V+A

tobegivenbytwicetheVSATheauthorsofRef.[]takethecentralvalueforC6

V+A

resultandthatforC8tobe0.OurfifthorderFOPTfitsinfactreturnsignificantly

V+A

differentvalues.ItispossibletotesttheconsistencyoftheassumedvaluesforC6

25

wFIG.4:ThefitqualitiesFV+A(s0)correspondingtotheALEPHdataandtheOPEparameters

ofRef.[]forthew(00),w2,w3andw(y)=y(1−y)2.Allnotationasinthetext.

105FV+A(s0)w0-5-102.5s0 (GeV)V+A

andC8withtheresultingextractedvalueofαs(m2τ),asabove,bystudyingthes0-dependenceofthematchbetweentheOPEandspectralintegralsidesofthew2,w3,w(00)andw(y)=y(1−y)2FESRs,whoseOPEsidesdonotdependonanyoftheCD>8.

(0)

Thereader,here,shouldbearinmindthat,inRef.[],slightlydifferentvaluesofd5

(0)V+A

and󰀂aG2󰀁RGIwereemployedthanthoseusedabove.Usingthed5,󰀂aG2󰀁RGI,C6

V+A

andC8valuesofRef.[],togetherwiththeresultingO(¯a5)-truncatedFOPTfitvalue

w

forαs(m2τ),wefindthefitqualities,FV+A(s0),displayedinFig.4,wherethedotted,dashed,dot-dashedanddouble-dot-dashedlinescorrespondtow(00),w2,w3andy(1−y)2,

w(00)22

respectively.FV+A(s0)is,ofcourse,smallnears0=mτsincethevalueofαs(mτ)

(00)

employedinthecalculationswasfixedusingthes0=m2FESR.τversionofthew

Thedeteriorationinthefitqualityforw(00)ass0isdecreased,aswellastheverypoorfitqualitiesfortheotherthreeweights,clearlydemonstratesthatthevaluesassumedforV+AV+AC6andC8areproblematic.Thevalueobtainedforαs(m2τ)usingthesevaluesasinputshouldthusalsobetreatedwithcaution.Wehavealreadynotedtheresultsofour

2326

ownFOPTfitsabove.Sincetheαs(m2τ)valuesobtainedfromthew2andw3FESRsdonotshowthesamedegreeofconsistencyaswasobservedintheCIPT-basedfit,itwouldbenecessarytoperformacombinedfit,usinganumberofthedegree≤3weights,toimprovefurtherontheFOPTdetermination.

Acknowledgments

KMwouldliketoacknowledgethehospitalityoftheCSSM,UniversityofAdelaide,andtheongoingsupportoftheNaturalSciencesandEngineeringCouncilofCanada.TYwishestoacknowledgehelpfuldiscussionswithR.KoniukandC.Wolfewhilethisworkwasinprogress.

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(1)

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󰀃0󰀅2

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29

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