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arXiv:hep-ph/0304168v116Apr2003PreprinttypesetinJHEPstyle-HYPERVERSIONhep-ph/yymmdddSLAC–PUB–9675UCLA/03/TEP/10DESY03–034April,2003Two-LoopHelicityAmplitudesforQuark-GluonScatteringinQCDandGluino-GluonScatteringinSupersymmetricYang-MillsTheoryZviBern∗DepartmentofPhysicsandAstronomyUCLA,LosAngeles,CA90095-1547,USAE-mail:bern@physics.ucla.eduAbilioDeFreitasDeutschesElektronenSynchrotronDESY,D-15738Zeuthen,GermanyE-mail:dfreitas@ifh.deLanceDixon†StanfordLinearAcceleratorCenter,StanfordUniversityStanford,CA94309,USAE-mail:lance@slac.stanford.eduAbstract:Wepresentthetwo-loopQCDhelicityamplitudesforquark-gluonscattering,andforquark-antiquarkannihilationintotwogluons.Theseamplitudesarerelevantfornext-to-next-to-leadingordercorrectionsto(polarized)jetproductionathadroncolliders.Wegivetheresultsinthe‘tHooft-Veltmanandfour-dimensionalhelicity(FDH)variantsofdimensionalregularization.Thetransitionrulesforconvertingtheamplitudesbetweenthedifferentvariantsaremuchmoreintricatethanforthepreviouslydiscussedcaseofgluon-gluonscattering.Summingourtwo-loopexpressionsoverhelicitiesandcolors,andconvertingtoconventionaldimensionalregularization,givesresultsincompleteagreementwiththoseofAnastasiou,Glover,OleariandTejeda-Yeomans.Wedescribetheamplitudesfor2→2scatteringinpureN=1supersymmetricYang-Millstheory,obtainedfromtheQCDamplitudesbymodifyingthecolorrepresentationandmultiplicities,andverifysupersymmetryWardidentitiesintheFDHscheme.Keywords:QCD,NNLOComputations,Jets,HadronColliders.SubmittedtoJHEP1.IntroductionRecentyearshaveseenrapidprogressinourabilitytocomputetwo-loopmatrixelementswithmorethanasinglekinematicvariable[1,2,3,4,5,6,7,8,9,10,11].Theprogresshasreliedinpartonnewdevelopmentsinloopintegration[12,13,14,15,16,17,18]andinunderstandingtheinfrareddivergencesofthetheory[19].Thenewtwo-loopamplitudeswillbeessentialforreducingtheoreticaluncertaintiesinanumberofphysicalquantities.(Forarecentsummarydescribingthevariousexpectedimprovementsseee.g.ref.[20].)Inparticular,injetphysicsanimportantsourceoftheoreticaluncertaintyisfrommissinghigherordercorrectionstothewidelyusedNLOcalculations[21,22,23],whichhavebeencrucialformatchingtheorytoexperiment.Forthecaseofhadroncolliders,oneofthees-sentialingredientstoperformingnext-to-next-to-leadingorder(NNLO)calculationsarethepartondistributionfunctions.Recently,globalfitstothedataforthepartondistributionfunctionshavebeenperformed[24]withinanapproximateNNLOframework[25].Thereare,howeversizableuncertaintiesassociatedwiththeexperimentalinputtothepartondistributionfunctions[26].Nevertheless,anexactNNLOcomputationofjetproductionrateswouldbeverywelcome.Besidesreducingthescaleuncertaintiesforjetrates,suchacomputationwillallowabetterunderstandingofenergyflowwithinjets,asajetmaycon-sistofuptothreepartonsatthisorder.Forverylargemomentumtransfer,improvementscanalsobemadebyresummingthresholdlogarithms[27].Besidesthetwoloopmatrixelements,anNNLOcalculationoftwojetproductionathadroncollidersrequiresthetreeamplitudesforsixexternalpartons[28,29]andtheone-loopamplitudesforfiveexternalpartons[30,31],whichhavebeenknownforsometimenow.Anastasiou,Glover,Oleari,andTejeda-YeomanshaveprovidedtheNNLOinterferencesofthetwo-loopamplitudeswiththetreeamplitudes,forallQCDfour-partonprocesses,summedoverallexternalhelicitiesandcolors[4,5,6].Thehelicityamplitudesforgg→gghavealsobeencomputed,usingthespinorhelicityformalism[32].Inthispaper,wepresentthe¯qq→ggandqg→qgtwo-loophelicityamplitudes.Thefour-quarkhelicityamplitudeswillbepresentedelsewhere.Ausefulpropertyofhelicityamplitudesisthattheyexposethefulldependenceoncolorandspin.Manyformalpropertiesofscatteringamplitudesaresimplerinahelic-itybasisand/oraftercolordecomposition.SuchpropertiesincludesupersymmetryWardidentities[33,34],collinearlimits[29,35,36],andhigh-energybehavior[37,38].Thefullcolordependenceisalsousefulforunderstandingthestructureoftheinfrareddiver-gences[19,9,39].Thisadditionalformalinformationisnotnecessaryforthemainphenomenologicalap-plication,NNLOjetproductionincollisionsofunpolarizedhadrons.Ontheotherhand,thehelicityamplitudescanhavephenomenologicalapplicationsforjetproductionincollisionsofpolarizedprotons,asarebeingcarriedoutattherelativisticheavyioncollider(RHIC)atBrookhaven.Thismay,forexample,helptodeterminethepoorly-knownpolarizedgluondistributionintheproton[40].Theoreticalpredictionsoftherelevantobservablesrequire∗ResearchsupportedbytheUSDepartmentofEnergyundergrantDE-FG03-91ER40662.†ResearchsupportedbytheUSDepartmentofEnergyundergrantDE-AC03-76SF00515.–1–scatteringamplitudesforpolarizedpartons.Currently,predictionsareavailablethroughNLO[41];thehelicityamplitudespresentedhereareaprerequisiteforimprovingthepre-dictionstoNNLOaccuracy.Ourresultsalsoserveasacheckoftheresultsofref.[5]forunpolarizedq¯q→ggandqg→qgscattering.Whilepreparingourresultsforpublication,webecameawareofasimilarcomputationbeingcompletedsimultaneously[42].Ref.[42]computestheone-andtwo-loophelicityamplitudesforq¯q→ggandqg→qgscattering,inthe’tHooft-Veltman(HV)scheme[45](seebelow).Aslightlydiffere
本文标题:Helicity Amplitudes for Quark-Gluon Scattering in
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