On an asymptotic estimate of the $n$-loop correcti

arXiv:hep-ph/9210237v115Oct1992InstituteofPhysics,Czechosl.Acad.ofSci.PRA-HEP-92/17andSeptember1992NuclearCentre,CharlesUniversityPragueOnanasymptoticestimateofthen-loopcorrectioninperturbativeQCDJ.Ch´yla,J.FischerandP.Kol´aˇrInstituteofPhysics,CzechoslovakAcademyofSciencesPrague,Czechoslovakia∗AbstractArecentlyproposedmethodofestimatingtheasymptoticbehaviourofQCDperturbationtheorycoefficientsiscriticallyreviewedandshowntocontainnu-merousinvalidmathematicaloperationsandunsubstantiatedassumptions.Wediscussindetailwhythisprocedure,basedsolelyonrenormalizationgroup(RG)considerationsandanalyticityconstraints,cannotleadtosuchestimates.Westresstheimportanceofcorrectrenormalizationscheme(RS)dependenceofanymeaningfulasymptoticestimateandarguethattheunambiguoussummationofQCDperturbationexpansionsforphysicalquantitiesrequiresinformationfromoutsideofperturbationtheoryitself.∗Postaladdress:NaSlovance2,18040Prague8,Czechoslovakia1.IntroductionBecauseofcomputationalcomplicationsonlythelowesttwoorthreetermsinQCDperturbationexpansionsareavailableforphenomenologicallyinterestingquantities.Takingintoaccountthattherelevantexpansionparameter(strongcouplingconstantαs/π)isoftheorderof0.1,andforcertainquantitiesevenbigger,wefacethefollowingtwoimportantquestions:1.howtoreliablyestimatehigherordertermsinperturbationexpansionsofphysicalquantitiesand,havingobtainedthem,2.howtoformulateaphysicallywell-motivatedsummationmethodinthesituationwhentheseasymptoticestimatespreventtheuseofthefamiliarBorelsummationtechnique.AnumberoftheoristshavestruggledoverthelasttwodecadestoevaluatemultiloopFeynmandiagramsinQCDandothertheoriesandweseemcurrentlytobeattheborderofwhatcanbedonewiththeavailableanalyticalaswellasnumericaltools.ThestateoftheartinthisfieldhasrecentlybeenreachedbythefirstthreeloopQCDcalculationofaphysicalquantity,thefamiliarR-ratioine+e−annihilationintohadrons[1]R(q2)=σ(e+e−→hadrons)σ(e+e−→μ+μ−)=3nfXi=1Q2i!∞Xn=0rn(q2/μ2)αs(μ2)π!n,(1)whereq2denotesthesquareofthecenterofmassenergyine+e−collisionsandQiistheelectricchargeoftheproducedquarkwithflavouri.These,aswellastheanalogousresultsoncertainstructurefunctionssumrules[2]havebeenusedinphenomenologicalanalysesofexperimentaldataandtheinclusionofNNLOcor-rectionhasbeenshowntoreducetheoreticaluncertainties[3,4]and,inthecaseoftheGross-Llewelyn-Smithsumrule,alsotoimproveagreementwithexperiment[4].Thereseems,however,tobelittlechanceofgoinginaforeseeablefuturetostillhigherordersforanyofthephysicallymeasurablequantities.Thequestforatleastareliableestimateofthesehigherordercontributionsisthereforehighlycommendable,inparticularinviewofthefactthattherelevantperturbationseriesareexpectedtobefactoriallydivergentwithasymptoticallysign-definiteterms.InfactthequestionofthepossibledivergenceofQCD(aswellasotherfieldtheories)perturbationexpansions,whichgoesbacktotheoriginalargumentofDyson[5],stilldefiesdefiniteandclearanswer.TheoriginalconjecturethatthedivergenceofperturbationexpansionsisdirectlyrelatedtothediscontinuitiesofGreen’sfunctionsinunphysicaldomainsofthecouplingconstant[6]hasbeenseriouslyquestionedbyseveralauthors[7,8,9].Theknowledgeofthedisconti-nuityitselfissimplyinsufficientforthedeterminationoflargeorderbehaviourof2perturbationtheorycoefficientsandoneneedsmuchmoredetailedinformationonthebehaviourofGreen’sfunctionsinthevicinityofthecorrespondingcutinordertoestablishthisrelation.WesharetheviewpointofStevenson[8]thatthereislittleevidencethatQCDperturbationexpansionsforphysicalquantitiesdoindeeddiverge.Theexamplesdiscussedin[8]indicatethatinfactmostoftheinformationontheanalyticpropertiesofrelevantGreen’sfunctionsnearg2=0[10,11]mayactuallybeinvisibleinperturbationexpansionsthemselves.Moreover,thefact,containedalreadyin[7]andlaterrediscoveredandadvocatedbyStevenson[8],thatthesummabilityofperturbationexpansionsisnotthesameasthepossibilityofrecoveringfromsuchaseriesthecorresponding“full”physicalquantity,hasnowgainedwideracceptance.AsthesummabilityofQCDperturbationexpansionsisdeterminedbythelargeorderbehaviouroftheircoefficientsweprimarilyneedtoknowwhichkindofinformationisnecessaryandsufficienttoderiveit.Quiterecentlytwonewattemptshavebeenundertakeninordertoderivetheasymptoticbehaviouroftheperturbation-theorycoefficientsforcertainphysicalquantities[9,12].Theyusecompletelydifferentideasbutbotharriveatthesameconclusion:QCDperturbationexpansionsareindeedfactoriallydivergent.Thenovelargumentof[9]istailoredforthespecialcaseofparticlefractionsine+e−annihilationandcannotbeusedforsuchsimplequantityas(1).Contrarytotheoriginalapproachof[6]it,however,doesn’temployanyinformationonanalyticpropertiesofcorrespondingGreen’sfunctions.Althoughitisbasedonseveralstrongandquestionableassumptions,whichrenderitsconclusionsopentodoubt,wecanatleastimaginethatmoresophisticatedprocedurecanbeformulatedalongsimilarlines.Thisisnotthecasewiththeproceduresuggestedin[12],whichisclaimedtoanswerbothoftheabovequestions.Inasubsequentpaper[13],Westuseshisestimatestoformulateasummationprocedurewithtrulyremarkableproperties.Inparticulartheymaketheuseofafewlowes