Two-loop QCD helicity amplitudes for massless quar

arXiv:hep-ph/0401119v32Sep2004PreprinttypesetinJHEPstyle-PAPERVERSIONDCTP/04/04,IPPP/04/02,hep-ph/0401119Two-loopQCDhelicityamplitudesformasslessquark-quarkscatteringE.W.N.GloveraaDepartmentofPhysics,UniversityofDurham,DurhamDH13LE,EnglandE-mail:[e-mail]:Wepresentthetwo-loophelicityamplitudesforthescatteringofmasslessquarksinQCD.Weuseprojectortechniquestocomputethecoefficientsofthegeneralfour-quarkamplitudeatuptotwo-loopsinconventionaldimensionalregularisationandusethesecoefficientstoderivethehelicityamplitudesinthe’tHooft-Veltmanscheme.ThestructureoftheinfrareddivergencesagreeswiththatpredictedbyCataniwhileexpressionsforthefiniteremaindersforq¯q→Q¯Qandtheprocessesrelatedbycrossingsymmetryaregivenintermsoflogarithmsandpolylogarithmsthatarerealinthephysicalregion.Wehavecheckedthattheinterferenceoftreeandtwo-loophelicityamplitudes,summedoverhelicitiesandcolours,reproducesthepreviousresultsforthefiniteremaindersforinterferenceoftreeandtwo-loopamplitudesgiveninRef.[1,2].Keywords:QCD,Jets,LEPHERAandSLCPhysics,NLOandNNLOComputations.1.IntroductionInthepastdecade,QCDhasbecomeaquantitativescienceandcomparisonsofhighqualityexper-imentaldatafromhighenergycolliderexperimentswithQCDpredictionsatnext-to-leadingorder(NLO)inαshavebecomethedefactostandard.Insuchstudies,thepartonlevelpredictionsservetoprovideanestimateoftheuncalculatedtheoreticaluncertaintiesviathevariationintherenormal-isation(andfactorisation)scale.However,therearemanywellestablishedreasonswhyextendingperturbativecalculationstonext-to-next-to-leadingorder(NNLO)isvitalinreducingthetheoreticaluncertaintyforthedominant2→1and2→2processessuchasDrell-Yanorjetproduction.Mostoftheanticipatedimprovementsareeitherdue(a)toincludingthenexttermintheperturbationseriesor(b)toincludingtheeffectofmorepartonsinthefinalstate.Here,welistsixoftheimprovementsexpectedforNNLOpredictionsathadroncollidersandreferthereadertoRef.[3]foramoredetaileddiscussion;1.(a)Reductionoftherenormalisationscaleuncertainty;2.(a)Reductionofthefactorisationscaleuncertainty;3.(a)Reducedpowercorrections;4.(b)Bettermatchingbetweenparton-levelandhadronleveljetalgorithms;5.(b)Betterdescriptionofthetransversemomentumoftheinitialstate;6.(b)Extendedkinematiccoverageduetoenlargedphasespace.AllofthesewillbeunderpinnedbyabetterknowledgeofthepartondistributionsusingglobalfitstoNNLOobservablesfromDIS,Drell-Yanandjetproductionandbyabetterdeterminationofαsobtainedfrom3jeteventshapedataatelectron-positroncolliders.Takentogether,oneexpectsthatthetheoreticaluncertaintyonagivenobservablewilldropfromO(20%)atNLOtoO(10%)atNNLO.Inthispaper,wefocusonjetproductioninhadron-hadroncollisions.ThefullNNLOpredictionrequiresaknowledgeofthetwo-loop2→2partonicmatrixelements,aswellastheone-loop2→3andtree-level2→4amplitudesandNNLOpartondistributions.Muchworkhasbeeninvestedinthethree-loopsplittingfunctions[4,5,6,7,8,9,10,11,12,13,14]neededtoevolvethepartondistributionfunctionsatthisorder.Theavailablemomentshavebeenusedtoprovideapproximatefitsinx-spaceandNNLOpartondistributionsarebeginningtobecomeavailable[15,16].Theexactthree-loopsplittingfunctionshavebeeneagerlyanticipatedandtheunpolarisedsingletandnon-singletsplittingfunctionsarenowavailable[17,18].Ontheotherhand,thetree-levelsix-pointamplitudes[19,20,21,22,23,24,25,26]andone-loopfivepointamplitudes[27,28,29]havebeenknownforsometimewhiletheinterferenceoftreeandtwo-loopamplitudes[1,2,30,31]andself-interferenceoftheone-loopamplitudes[30,32,33]arealsoavailable.Subsequently,two-loophelicityamplitudeshavebeenevaluatedforthegluon-gluon[34,35]andquark-gluon[36,37]processesandhaveconfirmedtheearlier”squared”matrixelements.1Here,wecompletethesetoftwo-loopQCD1Notethathelicityamplitudesforthephenomenologicallyimportantgg→γγprocess[38],e+e−→3jets[39]andγγ→γγ[40,41]havealsobeencomputed.–1–helicityamplitudesforparton-partonscatteringbystudyingprocessesinvolvingtwopairsofmasslessquarks,q¯q→Q¯Q,andtheprocessesrelatedbytimereversalandcrossingsymmetrywhenqandQareeitherdistinctoridentical.WeextractthehelicityamplitudesusingthesamemethodasthatemployedinRefs.[41,39,37].Firstwewritedownageneraltensorthatdescribesthefourquarkamplitude.BecausetheDiracalgebraisinfinitedimensionalfornon-integerd,weuseabasissetofDiracstringsthataresufficienttodescribetheamplitudeuptotwo-loops.Atthree-loopsnewstructureswillappear.Thecoefficientsarethenextractedusingasetofprojectorsthatarevalidatuptotwo-loops.TheseprojectorsturntheDiracstringsintotraces(whicharetakenind-dimensions)andtherebyreducetheproblemtoevaluatingscalarintegrals.Themethodologyofreducingthesescalarintegralstomasterintegralsisexactlythesameasforthecalculationofthespinsummedtwo-loopmatrixelements[1,2].Bytakingthetraceind-dimensionsandkeepingallLorentzindicesind-dimensions,weensurethattensorcoefficientsareevaluatedinconventionaldimensionalregularisation(CDR).Wethenprovidetheperturbativeexpansionfortheone-andtwo-looptensorcoefficientsandremoveultraviolet(UV)divergencesateachorderinαsbyrenormalisationwithintheMSscheme.Theinfrared(IR)divergentstructureisshown