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Measurement of the Inclusive Jet Cross Section Using the Midpoint Algorithm in Run II at the Collidor Detector at Fermil...

HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Phenomenology of quantum chrom...
 Experimental apparatus
 Jet definition
 Inclusive jet measurements
 Data sample and event selectio...
 Jet corrections
 System uncertainties
 Theoretical predictions and...
 Results
 Comparison with the KT algorit...
 Conclusions
 Appendix A: Jet triggers at...
 Appendix B: Relative correctio...
 Appendix C: Simulation of detector...
 References
 Biographical sketch
 

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Iwouldliketothankmywife,Nicole,forherunderstandingofmytimeawayfromhomeduringthiswork.Herencouragementandcondenceinmyabilitycarriedoverintomyprofessionaldemeanor;mysuccessisdirectlycorrelatedwithhersupport.Iwouldliketothankmyadvisorsfortheirguidanceoverthelastfouryears.ProfessorField'stirelessdedicationtoanswerthetoughestquestionsofQCDkeptmeworkinghardtokeepup.ProfessorMatchev'srequirementtogivemanypublicpresentations,andtravelsupportfromhisOutstandingJuniorInvestigatoraward,helpedmetoconquermyfearofpublicspeaking.Also,partialsupportformystipendthissemesterfromDr.Matchev,madeitpossibleformetoconcentrateonthisdissertationandobtainingemployment.WithoutprofessorField'sandprofessorMatchev'scombinedeortsonmybehalf,IdonotknowwhereIwouldhaveendedupafterUF.IalsowanttothankDr.KenichiHatakeyama,oftheTheRockefellerUniversity,forhispatiencewhileteachingmehowtodoanexperimentalanalysis.Withouthisconstantcollaboration,IwouldhavehadnochancetocompleteaCDFanalysiswithoutbeinglocatedatFermilab.Iamproudoftheworkwedidtogetherovertheselasttwoyears.Finally,IwanttothankProfessorField,ProfessorMatchev,andDr.HatakeyamaforwritingtherecommendationlettersthathelpedmeobtainmoreemploymentoptionsthanIcouldpossiblyhaveexpected.ProfessorPaulAveryandDr.JorgeRodriguezallowedmetoworkwiththeUFGridcomputingprojectmyrstsummeratUF.ThisallowedmywifeandmetomovetoGainesvilleearlyandbuyahouse.Iamthankfulforthis.Iappreciate iv

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page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. ix LISTOFFIGURES ................................ x ABSTRACT .................................... xiii CHAPTER 1INTRODUCTION .............................. 1 2PHENOMENOLOGYOFQUANTUMCHROMODYNAMICS ..... 5 2.1TheParticleContentoftheStandardModel ............. 5 2.2FeynmanRulesofQCD ........................ 7 2.3ColorConnementandAsymptoticFreedom ............. 10 2.4TheFactorizationTheorem ...................... 15 2.5JetProductionCrossSections ..................... 16 2.6StructureofHadronicCollisions .................... 18 3EXPERIMENTALAPPARATUS ...................... 23 3.1CoordinatesandConventions ..................... 23 3.2TheTevatron .............................. 24 3.3TheCDFDetector ........................... 27 4JETDEFINITION .............................. 36 4.1TheCDFMidpointJetClusteringAlgorithm ............ 36 4.2OtherJetClusteringAlgorithms .................... 39 4.3JetDenitionIssues .......................... 41 5INCLUSIVEJETMEASUREMENTS ................... 45 6DATASAMPLEANDEVENTSELECTION ............... 50 7JETCORRECTIONS ............................ 55 7.1MonteCarloSimulation ........................ 57 7.2RelativeCorrection .......................... 59 7.3PileupCorrection ........................... 61 vii

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.......................... 61 7.5UnfoldingCorrection .......................... 62 7.6HadrontoPartonCorrection ..................... 63 8SYSTEMATICUNCERTAINTIES ..................... 65 9THEORETICALPREDICTIONSANDUNCERTAINTIES ....... 68 10RESULTS ................................... 72 11COMPARISONWITHTHEKTALGORITHM .............. 83 12CONCLUSIONS ............................... 88 APPENDIX AJETTRIGGERSATCDF .......................... 91 A.1JetTriggerEciency .......................... 93 A.2JetTriggerPrescales .......................... 94 BRELATIVECORRECTIONS ........................ 103 B.1Eventselectionforrelativecorrectionstudies ............. 103 B.2RelativeCorrectionswiththeMidpointAlgorithm .......... 105 CSIMULATIONOFDETECTORRESPONSEANDRESOLUTION ... 108 C.1JetEnergyResolution:BisectorMethod ............... 108 C.2JetEnergyResponse:DijetPTBalance ................ 117 REFERENCES ................................... 121 BIOGRAPHICALSKETCH ............................ 125 viii

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Table page 2{1Somepropertiesofthequarkcontentofthestandardmodel. ....... 6 2{2Somepropertiesoftheleptoncontentofthestandardmodel. ....... 6 2{3Somepropertiesofthegaugebosoncontentofthestandardmodel. ... 7 A{1Datasamplesusedtostudytheeciencyofeachjettrigger ....... 94 B{1Selectioncutsappliedtorequirethedijeteventtopologyusedtoderivetherelativecorrections ............................ 104 C{1DijetbalancecorrectionappliedtothePYTHIAMCsimulationforeachrapidityregion. ................................ 118 ix

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Figure page 2{1FeynmanrulesforQCDinacovariantgauge. ............... 8 2{2One-loopVacuumpolarizationdiagramofQED. .............. 11 2{3One-loopvacuumpolarizationdiagramofQCD. .............. 12 2{4One-loopVacuumpolarizationdiagramofQCDwhicharisesfromthegluonselfcoupling. .............................. 12 2{5RunningofthetheQEDandQCDcouplingconstants. .......... 13 2{6SchematicoftheQCDfactorizationtheorem. ............... 16 2{7Diagramswhichcontributetoleadingorderjetproductionatahadroncollider. .................................... 17 2{8Diagramswhichcontributetoe+e)]TJ/F1 11.95 Tf 10.98 -4.34 TD[(annihilationtohadronsatNLO. .. 18 2{9Componentsofatypicalhadroncollidereventatthepartonlevel. .... 19 2{10DescriptionofthedierentlevelsofajeteventatCDF. ......... 22 3{1AschematicoftheacceleratorcomplexusedforRunIIatFermilab. ... 26 3{2Developmentofanelectromagneticshower. ................. 30 3{3ColliderDetectoratFermilab(CDF). .................... 34 3{4LongitudinalviewoftheCDFIITrackingSystemandplugcalorimeters. 35 4{1Conealgorithmsensitivitytosoftradiation. ................ 42 4{2DarktowersobservedbytheoriginalMidpointalgorithm ......... 43 5{1Uncertaintyonthepartondistributionfunctionfortheup-quarkandthegluonatQ=500GeV. ......................... 46 5{2Dominantprocessindeepinelasticscatteringexperiments. ........ 47 6{1JetyielddistributionsasafunctionofPTinthecentralregion. ..... 53 6{2Measuredrawjetcrosssectionfortheverapidityregions. ........ 54 7{1DierencebetweencalorimeterjetPTandhadronleveljetPT. ...... 55 x

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.. 58 7{3DegreeofdijetbalanceobservedintheCDFcalorimeter. ......... 60 7{4Hadrontopartonlevelcorrectionappliedinthecentralregion. ...... 64 9{1EectofvaryingtheparameterRsep. .................... 71 10{1InclusivejetcrosssectioncorrectedtothehadronlevelandratiototheNLOpQCDpredictionsfortherapidityregion0:1
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...... 100 A{7TriggerecienciesasafunctionofjetPTforthejet100trigger. ..... 101 A{8Crosssectionratiosusedasacross-checkofthejet20,jet50,andjet70prescales. ................................... 102 B{1versusDdistributionsforjetsclusteredbytheMidpointalgorithmafterapplyingtherelativecorrectionsderivedfromjetsclusteredbyJetClu. 106 B{2versusDforjetsclusteredbytheMidpointalgorithmusingalargerDbinning. .................................. 107 C{1Bisectorvariablesarelabeledinthediagramofthetransverseplane. .. 110 C{2ResultsofthebisectorstudyfortherapidityregionjYj<0:1. ...... 112 C{3Resultsofthebisectorstudyfortherapidityregion0:71:1. ................................... 120 xii

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TheseadvancesandcountlessothersoverthelastonehundredyearshaveledtothedevelopmentoftheStandardModel(SM)ofparticlephysics.Thismodeldescribestheelectromagnetic,weak,andstronginteractionsbetweenparticles.UndertheStandardModel,thestrongandelectroweakinteractionsareuniedunderthegaugegroupSU(3)SU(2)U(1).TheobservationbyLouisdeBrogliethatmovingbodieshaveawavenaturehasprofoundconsequencesinparticlephysics.Theresolutionofanopticalmicroscopeisapproximatelyproportionaltothewavelengthoftheincidentlight.Assumingtheprobingbeamconsistsofparticles,thentheresolutionislimitedbythedeBrogliewavelengthoftheseparticles p;(1{1)whereisthedeBrogliewavelength,hisPlank'sconstant,andpisthemomentumoftheparticlebeam.Thisprediction,thatanobject'smomentumisinverselyproportionaltoitswavelength,impliesthatasthemomentumtransfer,Q,oftheprobingbeamisincreaseditispossibletoresolvesmallerdistancescales.Lowenergyparticlesonlyprobelargespatialregions,whilehighenergyparticlescanresolveshortdistanceeects.ThisobservationbydeBrogliemotivatestheuseofhighenergyparticleacceleratorsasthelaboratoryofparticlephysicsinthemodernage.Throughtheuseofparticlecolliders,theStandardModelhascompiledanimpressivehistoryofexperimentalsuccess.Forexample,theWandZbosonswerepredictedbeforetheirdiscoverybytheelectroweaktheoryofGlashow,WeinbergandSalam.Therunningoftheelectromagneticandthestrongcouplingconstantshavebeenveriedbyexperiment.Thenalquark,thetopquark,oftheStandardModelwasdiscoveredattheTevatroncolliderbythetwocolliderexperiments,CDFandD0.TheSMalsosurvivedtheplethoraofprecision

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electroweakmeasurementsofLEP,theCERNe+e)]TJ/F1 11.95 Tf 10.98 -4.34 TD[(collider.Insummary,theStandardModelisasuccessfulmodelofhighenergyparticlephysicsatallenergiesaccessibletotheexperimentalcommunitysofar.NodeviationsfromtheSMhavebeenobserved;however,thereareimperfectionswiththeStandardModel.SofartheHiggsbosonhasnotbeenobserved.ThisparticlemustexistintheStandardModeltoallowthebasicbuildingblocksofthemodeltoobtainmass.Also,therearemanytheoreticalargumentsthatsuggestthattheSMbreaksdownathigherenergyscales.OnesuchargumentisrelatedtothemassoftheHiggsbosonitself.ThishierarchyproblemisrelatedtothesensitivityoftheHiggsmasstophysicsathighenergyscalesandrequiresanetuning(i.e.,cancellationtoaprecisionof1032)oftheStandardModelwhichisundesirabletomanytheorists.Supersymmetric(SUSY)models,whichrequireasymmetrybetweenfermionsandbosons,canprovideanelegantsolutiontothisproblem.However,sofar,noSUSYparticleshavebeendiscovered.ManymodelsofphysicsbeyondtheSM(BSM)suchasSUSYrequireadditionalparticleswhichareheavierthantheSMparticles.Theparticlesmustbeheavyortheywouldhavealreadybeendiscoveredinpreviousmeasurements.Highenergycollisionsarerequiredtoproduceheavyparticlesinthelaboratory.Thisneedtosearchforheavyparticlesfurthermotivatestheneedforhighenergycollidersinparticlephysics.Theoristshaveusedcompellingarguments,suchashierarchy,formanyyearstomotivatetheneedfornewparticlesornewforcestobeobservedattheTeVscale.WithoutexperimentalguidanceattheTeVscale,theoristswhostudyphysicsbeyondtheStandardModelhavehadfreedomtopursuecountlesspossibilities.TheTevatronatFermilabisonlyjustbeginningtothreatentheTeVscalewithacenterofmassenergyof1:96TeV,buthasthepotentialtoconstrainmanyofthesetheories.Atthesametime,theparticlephysicscommunityiswaitingeagerlyfor

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therstcollisionsattheLargeHadronCollider(LHC)whichwillbegincollidingprotonsinthenextfewyearsatacenterofmassenergyof14TeV.PrecisionmeasurementsathighenergycollidershavebeenanimmenselyvaluabletoolbothtovalidatetheSMandtoconstrainitsproperties.Ashigherenergycollidersbecomeavailable,itispossibletomakediscoveriesofBSMphysicsortovalidateandconstrainourunderstandingoftheSM.Inthisdissertation,ameasurementwhichprobesthesmallestdistancescaleseverprobedbystudyingthecollisionsofthehighestenergyparticleacceleratorintheworldwillbediscussed.Thismeasurementprovidesvalidationofquantumchromodynamics(QCD),thetheoryofthestrongforce,atthehighestenergyeverdirectlyprobed,andatthesametimeprovidesconstraintsonthequantumnatureoftheprotonwhichwillimprovetheoreticalpredictionsforthehighenergycollidersofthefuture.

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1 ]andthestandardmodelofparticlephysics[ 2 3 ]isbeyondthescopeofthisexperimentaldissertation.Instead,theparticlecontentofthestandardmodelwillbereviewed,andonlyaspectsofQCDphenomenologywhicharerelevanttotheinclusivejetcrosssectionmeasurementwillbeaddressedindetail.AmorecompletediscussionofQCDcanbefoundinmanyreferences[ 4 5 6 7 ]. 2fermions.ThequarksandleptonsoftheSMcaneachbearrangedintothreedoublets.Eachleptondoubletincludesachargedleptonpartneredwithaneutralneutrino.Thequarkandleptoncontentofthestandardmodelislistedintable 2{1 andtable 2{2 alongwithsomeofthefermionmeasuredpropertiesaslistedintheParticleDataBook[ 8 ].Thefermionsofthestandardmodelinteractthroughtheexchangeoftheintegerspingaugebosons.Thefourgaugebosonsareshownintable 2{3 withsomeoftheirproperties.Themass-lessphotonisthepropagatoroftheelectromagnetic 5

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Table2{1. Somepropertiesofthequarkcontentofthestandardmodel.QuarkpropertiesaretakenfromtheParticleDataBook. FlavorSymbolElectricCharge(e)Mass(GeV=c2) Upu+2 31:5)]TJ/F1 11.95 Tf 11.95 0 TD[(310)]TJ/F4 7.97 Tf 6.59 0 TD[(3Downd)]TJ/F4 7.97 Tf 10.49 4.71 TD[(1 33)]TJ/F1 11.95 Tf 11.95 0 TD[(710)]TJ/F4 7.97 Tf 6.59 0 TD[(3 31:250:09Stranges)]TJ/F4 7.97 Tf 10.49 4.7 TD[(1 3952510)]TJ/F4 7.97 Tf 6.59 0 TD[(3 3174:23:3Bottomb)]TJ/F4 7.97 Tf 10.49 4.7 TD[(1 34:20:07 force,theWandtheZbosonsaretheforcecarriersoftheweakinteractions,andtheeightmass-lessgluons(giwherei=1::8correspondtothe32)]TJ/F1 11.95 Tf 12.44 0 TD[(1generatorsoftheSU(3)symmetrygroup)mediatethestronginteraction.Amajordierencebetweenquarksandleptonsisthatquarkscarryanadditionalinternaldegreeoffreedomcalledcolor.Thisisthechargeofthestrongforceandiscommonlydenotedasred,green,orblue(RGB).Ofthefermions,onlyquarksparticipateinthestronginteractionsofQCD. Table2{2. Somepropertiesoftheleptoncontentofthestandardmodel.LeptonpropertiesaretakenfromtheParticleDataBook. FlavorSymbolElectricCharge(e)Mass(MeV=c2) Electrone-10:511ElectronNeutrinoe0<310)]TJ/F4 7.97 Tf 6.59 0 TD[(6 Tau-11777TauNeutrino0<18:2 Gravityisnotmentionedintheabovediscussion.Althoughallmassiveparticlescoupletogravity,itistheweakestforceandistypicallyonlyimportantonmacroscopicscales.Gravityisnotincludedinthestandardmodel.Itisnotimportantfortheresearchdiscussedhereandwillnotbediscussedfurther.ItisaremarkabletriumphoftheSMthatalloftheinteractionsofmatterintheobserveduniverse(barringgravity)canbedescribedwithamazingprecisionbasedonthesimpleparticlecontentdiscussedhere.

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Table2{3. Somepropertiesofthegaugebosoncontentofthestandardmodel.ThegaugebosonpropertiesaretakenfromtheParticleDataBook. BosonSymbolElectricCharge(e)Mass(GeV=c2) Photon00WW180:4030:029ZZ091:18760:0021Gluong00 2quarksandthespin1gluons.RequiringthatQCDbeagaugetheorybasedonthegroupSU(3)withthreecolorchargesxestheLagrangiandensitytobe 48XA=1FAFA+nfXj=1qj(iD=)]TJ/F2 11.95 Tf 11.95 0 TD[(mj)qj;(2{2)whereqjarethequarkeldsofnfdierentavorsandmassmj.ThearetheDiracmatricesandD(D=D)isthecovariantderivativedenedby (D)ab=@ab+ig(tCAC)ab;(2{3)wheregisthegaugecouplingofQCD,andtCarethematricesofthefundamentalrepresentationofSU(3).Thesegeneratorsobeythecommutationrelations [tA;tB]=ifABCtC;(2{4)wherefABCarethecompleteantisymmetricstructureconstantsofSU(3).Thenormalizationofthestructureconstantsandofgisspeciedby

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ThequantityFAistheeldstrengthtensorderivedfromthegluoneldAA FeynmanrulesforQCDinacovariantgauge.Thesolidlines(black)representfermions,curlylinesgluons,anddottedlines(black)ghostpropagators. TheFeynmanrulesofQCDcanbecalculateddirectlyfromtheQCDLagrangianafterxingagauge.Thegaugemustbexedinordertodenethegluonpropagator.TheFeynmanrulesinacovariantgaugearegiveninFig 2{1 [ 9 ].ThephysicalverticesinQCDincludethegluon-quark-antiquarkvertex.ThisvertexandthephysicalpropagatorsofthequarkandgluonareanalogoustothecouplingandpropagatorsoftheelectronandphotonofQED.However,thereis

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alsothethree-gluonandfour-gluonvertices,ofordergandg2respectively.ThesegluonselfcouplingshavenoanalogueinQEDsincethephotondoesnotcarryelectricchargeandthereforedoesnotinteractwithotherphotons.TheyarisefromthethirdtermofEq. 2{7 whichisnotpresentintheQEDeldstrength.TheFeynmanrulesdiscussedabovecanbeusedtomakemanypredictionsofQCD.Forexample,onecancomputetheprobabilitythatagiveninitialstatewillinteracttoyieldanalstate,P(A+B!C+D).Inparticlephysics,theseprobabilitiesarecalledcrosssections,.Crosssectionsareexpressedinunitsofareacalledbarns,b.Onebarnisequalto10)]TJ/F4 7.97 Tf 6.58 0 TD[(24cm2.Inthisanalysis,crosssectionresultsareexpressedinnano-barns(1nb=10)]TJ/F4 7.97 Tf 6.59 0 TD[(9b).Onceacrosssectionforaprocesshasbeencalculated,andthetotalnumberofcollisionswhichhaveoccurredinsometimeperiod(integratedluminosity,L)hasbeenmeasured,itispossibletopredictthenumberofeventsofthatprocessthathaveoccurred

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(i.e.,theintegratedluminosityfora24hourperiodatCDFcanbeontheorderof2000nb)]TJ/F4 7.97 Tf 6.59 0 TD[(1). 2{1 aremadeupoftree-leveldiagrams(i.e.,diagramswhichdonotcontainloops).MostcalculationsmadeinQFTareanapproximationbasedonanexpansioninpowersofthecouplingconstant.Thisiscalledaperturbativeexpansion,andisonlyvalidinthelimitthatthecouplingconstantissmall(i.e.,<<1).Perturbativepredictionsforobservablessuchasscatteringamplitudesareaectedbyhigherorderloopcorrections.ThevacuumpolarizationdiagramforQEDshowningure 2{2 isanexampleofsuchacorrection.Thiscorrectiontothephotonpropagatordivergeslogarithmicallyattheonelooplevelasthefour-momentumsquaredofthevirtualphoton(q2=)]TJ/F2 11.95 Tf 9.3 0 TD[(Q2)increases.Inthehighenergylimit(Q2>>m2e)thecontributionis 3g(2{9)whereisanultravioletcutoand0isknownasthebareelectriccharge(0=e20=4).Aneectivecouplingmaybedenedwhichsumsthevacuumbubblestoallorders 137:(2{11)

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Inthehighenergylimit(largeQ2) 2{12 ,weseethatthecouplingconstantofQEDincreaseswithenergy.Atlowenergy,QEDissmallandaperturbativeexpansioninthecouplingisrelevant.Athighenergy,thecouplinggetslargeandeventuallydiverges(Landaupole).Perturbationtheoryisnotvalidinthisregime.Luckily,thisoccursataveryhighenergyscaleforQED,anditisexpectedthatphysicsbeyondthestandardmodelshouldmodifytherunningofthecouplingatanenergyfarbelowtheLandaupole. Figure2{2. One-loopVacuumpolarizationdiagramofQED.Diagramslikethisaecttherateforelectron-positronscatteringinQED.Thisdiagramdivergeslogarithmically. TheQCDdiagramanalogoustotheQEDdiagramshowningure 2{2 isshowningure 2{3 .InQCD,thereisanadditionalcontributiontothepropagatorofthegluonduetoitsselfcoupling.Thisextradiagram(showningure 2{4 )leadstoprofoundconsequenceswithrespecttotherunningofthecouplingconstant.ByfollowingsimilarargumentsasthoseappliedtoQEDandincludingtheextraQCDdiagrams,onearrivesattheanalogousequationto 2{12 1)]TJ/F3 7.97 Tf 13.15 5.69 TD[(0(2r) 4log(Q2

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Thequantity0istheone-loop-functionofQCD 3nf;(2{14)wherenfisthenumberofquarkavors.InQED,theone-loop-functionwas 3:(2{15)Thesignofthe-functionisdierentforQEDandQCDifnf<16.ThishastheconsequencethattheeectivecouplingofQCDrunsintheoppositedirectionoftheQEDeectivecoupling.Equation 2{13 representsonlytheleadingorder Figure2{3. One-loopvacuumpolarizationdiagramofQCD.ThisdiagramisanalogoustothevacuumpolarizationdiagramofQED. Figure2{4. One-loopVacuumpolarizationdiagramofQCDwhicharisesfromthegluonselfcoupling.Contributionsfromthisdiagramareresponsibleinthesignipforthe-functionofQCDwithrespecttoQED.

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behaviorofthecouplingconstantinQCD,wheretheexperimentalcharge(2r)hasbeendenedatanarbitraryrenormalizationscale(Q2=2r).ThelowenergylimitisnotausefulscaleinQCD.Thisisbecausethecouplingsdivergesinthelowenergylimit.TypicallythearbitraryscaleristakentobethemassoftheZbosonwhere 2{12 andEq. 2{13 aresketchedingure 2{5 .ForQED,weseethedynamicsthatwerementionedabove;forlowQ2(largedistances)thecouplingconstantissmallandthecouplingincreaseswithQ2untilatsomeveryhighenergyscale(1034GeV)itdiverges.ForQCD,thedynamicsarevery Figure2{5. RunningofthetheQEDandQCDcouplingconstants.InQEDtheeectivecouplingissmallatlargedistances,butdivergesatveryhighenergy(\Landaupole").InQCDthecouplingdivergesatlargedistances(\colorconnement")andgoestozeroasymptoticallyatlargeenergy(\asymptoticfreedom").ColorconnementandasymptoticfreedomareimportantqualitiesofQCD.

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dierent.AtlowQ2thecouplingdiverges.Thisisknownascolorconnementandisthereasonwhyfreequarksandfreeglounsarenotobservedinnature.Asobjectsconnectedbycoloreldsareseparatedtheeldstrengthbecomessostrongthatnewquark-antiquarkpairsarepulledfromthevacuum.Thesequark-antiquarkpairsformcolorneutralsingletscalledhadrons.Thisprocessofcoloredpartons(i.e.,quarksandgluons)formingcolorsinglethadronsisreferredtoashadronization.Throughthisprocessacoloredpartoncanhadronizeintomanyhadronswhichareroughlyco-linearwithrespecttothemomentumvectoroftheoriginalparton.Theseclustersofroughlyco-linearhadronsarecalledjets.AthighQ2(particlesresolvesmalldistances)thecouplingconstantofQCDbecomessmallandevenvanishesasymptotically.Thisisthephenomenaknownasasymptoticfreedom.AconsequenceofasymptoticfreedomisthatperturbativemethodsarevalidathighenergiesinQCD.BecauseofthispropertyofQCD,thelongdistance(lowQ2)andshortdistance(highQ2)behaviorofQCDmaybeseparated(i.e.,factorized).ThisfactorizationallowsthemethodsofperturbativeQCD(pQCD)tobeappliedtothelargeQ2componentandphenomenologicalmodelstobeappliedtolowQ2component.ThisfactorizationpropertymeansthatthepartoniccrosssectioncalculatedwiththemethodsofpQCDisusefulforhadroniccollisionsinthehighenergylimit.AnotherimportantresultthatmaybeobtaineddirectlyfromEq. 2{3 istheenergyscaleatwhichthecouplingconstantinQCDdiverges(QCD).Solvingfortheenergyscale(Q2)wherethedenominatorofEq. 2{3 vanishesyields 0(2r)+log(2r):(2{17)ThisresultcanbeusedtoobtaintheeectivecoupingintermsofQCD 0log(Q2=2QCD):(2{18)

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ThequantityQCDhasbeendeterminedexperimentallytoberoughly200MeV.Therefore,theQCDeectivecouplinggetslargeforQ21GeV2.Perhaps,itismorethancoincidencethatthisisroughlythemassoftheproton. 10 ].AfewyearslaterDrellandYanextendedthesepartonmodelideastosomehadron-hadronprocesses[ 11 ].Theideaofthefactorizationmodelcanbeseenpictoriallyingure 2{6 [ 9 ].Itmeansthatthehadroniccrosssectionmaybewrittenas

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SchematicoftheQCDfactorizationtheorem.Thepartoniccrosssectionmustbefoldedinwiththepartondensityfunctionsofthehadron. 2{7 [ 9 ].Thesediagramsmaybereadfromlefttoright,orbottomtotop.Forexample, 2{7 (c)canbeinterpretedasqq!ggwhenreadfromlefttoright,oritmaybeinterpretedasgq!gqwhenreadfrombottomtotop.Lowestorder(LO)calculationshaveuncertaintiesformultiplereasons.Theleadingorderresultquiteoftenhasalargedependenceonrenormalizationandfactorizationscales.Thisdependenceisreducedbygoingtohigherorderintheperturbativeexpansion.AnothersourceofuncertaintyonLOpredictionsisthatadditionalprocessesmaybecomepossibleonlywhengoingbeyondleadingorder.Atnexttoleadingorder(NLO),allFeynmandiagramswhichcontributeanadditionalfactorofstothescatteringamplitudemustbeconsideredwhencalculatingthescatteringcrosssection.Extrafactorsofthestrongcoupling

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Diagramswhichcontributetoleadingorderjetproductionatahadroncollider. constantcanbeaddedintwoways.Realradiationmaybeaddedtotheinitialornalstate,ordiagramsmaycontainoneloop.Asasimpleexample,considerthecaseofelectron-positronannihilationtohadronsthroughavirtualphotonexchange(e+e)]TJ/F6 11.95 Tf 11.14 -4.34 TD[(!qq).ThediagramswhichcontributetothisprocessatNLOareshowningure 2{8 [ 9 ].Thediagramsingure 2{8 (a)includethetree-leveldiagramfortheprocessaswellasone-loopdiagramswithavirtualgluonemission.Althoughthevirtualgluondiagramshavetwoextrafactorsofgsduetothetwoextravertices,theystillcontributeatNLO.Sincethesediagramshavethesamenalstateasthetree-leveldiagram,thematrixelementforthesumofthefourdiagramsmustbesquared.CrosstermsfromthissquaredmatrixelementareatNLOins(i.e.,

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Diagramswhichcontributetoe+e)]TJ/F1 11.95 Tf 10.99 -4.34 TD[(annihilationtohadronsatNLO.Thedivergencesintherealgluonemissiondiagramscancelthedivergencesintheloopdiagrams. 2{8 (b)showstheNLOdiagramswitharealgluonemission.Therealgluonemissiondiagramsdivergeinthelimitthatthegluoniscollineartothequark.AverypowerfultheoremofQFTstatesthatsoftandcollineardivergencescanceltoallordersinperturbationtheory[ 12 13 14 ].Thiscancellationmeansthatinclusivequantitieswillbefreeofdivergences.Theinclusiverequirementmeansthattheobservablecannotincludeonlythediagramsingure 2{8 (a)oronlythediagramsingure 2{8 (b)becausecontributionsfromallofthesediagramsmustbeincludedtoensurethatthedivergencesarecanceled.Itisfromargumentsofthistypethatthejetcrosssectionmustbeinclusive,andjetalgorithmsmustbedenedinsuchawayastonotbesensitivetoinfraredandco-lineareects.

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relativetothebeamdirection.Sometimeshowever,thereisahardscatteringwhereparticleswithlargetransversemomentumaregeneratedinthecollision.ThefactorizationtheoremofQCDdiscussedaboveallowsonetofactoroutthehardscatteringcomponentofthehadroncollidereventandcalculateitperturbatively.However,thereareothercomponentsthatmustbeincludedforacompletemodelofthehadroncolliderevent. Figure2{9. Acartoondescriptionofatypicalhadroncollidereventatthepartonlevel(beforehadronizationofcoloredpartonsintocolorsinglethadrons).Thehardscattering,initialstateradiation,nalstateradiation,multiplepartoninteractions,pileup,andthebeamremnantsarethecomponentsofaneventatahadroncollider. Figure 2{9 showsasimplieddescriptionofahardscatteringeventatahadroncollider.Theschematicshowncanbeconsideredtorepresentwhatgoesonwithintheradiusofaprotonaroundthehardcollision.Oncethecoloredpartonsmoveoutsideoftheradiusoftheproton,theymusthadronizeintocolorneutralhadronsduetotherequirementofcolorconnement.Thestateoftheeventbeforehadronizationisnotaphysicalobservable,butisusefulwhendiscussingthephenomenologyofhadroncollidereventsandwillbereferredtoasthepartonlevel.Thissimplepartonlevelmodelofhadron-hadronscatteringisusedasthebasisfortheQCDMonte-Carlo(MC)eventgeneratorprograms.

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Aspartonsbegintofeeltheeectofotherpartons,theyradiatequarksandgluons.Thesequarksandgluonscanalsoradiatemorequarksandgluonsandthisseriesofpartonsradiatingmorepartonsleadstoanavalancheorshowerofpartons.Thischainofradiationiscalledapartonshower.InthecontextofaQCDeventgenerator,apartonshowerisanapproximateperturbativetreatmentofQCDpartonsplittingwhichisvalidabovesomecut-ovalue(Q01GeV).Thepartonshowerisbasedonidentifyingandsummingtoallordersthelogarithmicenhancementsduetosoftgluonemissionandgluonsplittingfunctions.Becausethepartonshowerisbasedonenhancementsduetosoftgluonemission(smallangles)itisonlyanapproximationofthehardgluonemissioncomponent(largeangles).Partonshowersareusedtomodelinitialstateradiation(ISR),andnalstateradiation(FSR)inQCDMCgenerators.Thesemodelsforthepartonshowercanbecombinedwithphenomenologicalmodelsofhadronizationwhichtakeoverforenergyscalesbelowthecut-oscale(Q
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averagepropertiesoftheUEcontribution.AnanalysisbasedonthisstrategywaspublishedinRunIatCDF[ 15 16 ],andhasbeencontinuedandimprovedinRunIIatCDF[ 17 18 19 ].BystudyingdistributionswhicharesensitivetotheunderlyingeventtheMCprogramsaretunedtottheeectsobservedindata.Theleadingordermatrixelements,PDFs,partonshowers,hadronizationmodels,theunderlyingevent,andpileuparerequiredcomponentsofaQCDeventgeneratorforcompletegenerationofhadroncolliderevents.Afterallofthepartonsintheeventhadronize,theparticlecontentoftheeventisreferredtoasthehadronlevel.Theparticlesatthehadronlevelareobservable,andtheyarethestateswhichinteractwiththedetector.Aftertheparticlesinteractwiththedetector,theresultingdescriptionoftheeventisreferredtoasthedetectorlevel.Thesethreelevelsoftheevent(parton,hadron,anddetector)willbereferredtothroughoutthisdraftandaredepictedingure 2{10 [ 20 ].Experimentalmeasurementsareonlyavailableatthedetectorlevel.However,theMCgenerators,whencombinedwithadetectorsimulationprogram,canbeusedtomakepredictionsatallthreelevels

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Figure2{10. AcartoondescriptionofthedierentlevelsofajeteventatCDF.Thepartonlevelisthestatebeforethepartonshadronizeandisnotphysicalobservable.Thehadronorparticlelevelisthestateafterhadronizationbutbeforetheparticleshaveinteractedwiththedetector.Finally,thedetectorlevelistheresultoftheeventasreportedbythedetector.

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2lnE+Pz 23

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vectorperpendiculartothebeamdirectionandisdenedby XET;(3{6)where6ETisthemagnitudeof6~ET.f6ETisausefulvariabletoremovebackgroundswhichdonotoriginatesymmetricallyfromthecenterofthedetector(e.g.,cosmicrays).

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discoverieswouldnotbepossibleatCDFwithoutthebeamprovidedtotheexperimentbyFermilab.ThecollidingbeamsattheCDFandD0experimentsaretheresultofthecomplexacceleratorchainshowningure 3{1 [ 21 ].ACockroft-WaltonacceleratorstartstheprocessbyacceleratingHydrogenions(Hydrogenatomswithoneextraelectron)to750keV.Theionsaretheninjectedintothe500ftlongLinacwheretheirenergyisboostedto400MeVbyoscillatingelectricelds.Theelectronsarethenstrippedfromtheionsbyacarbonfoil.Theremainingprotonsthenenterafast-cyclingsynchrotronringcalledtheBooster.Herethebeamisacceleratedbyradiofrequency(RF)cavitiesateachrevolutionuntiltheyreachanenergyof8GeV.Bunchesofprotons,eachcontainingabout51010protons,arepassedontotheMainInjector.ProtonbunchesfromtheMainInjectorarealsousedtocreateantiprotons(p)bycollisionswithanickel-coppertarget.Thistechniqueproducesantiprotonswithawiderangeofmomentumwhichmustbecooledintoamono-energeticbeam.TheantiprotonsarerstfocusedwithalithiumcollectorlensandthenpassedintothetheDebuncher.TheDebuncherappliescomplexcomputer-controlledRFtechniquestocooltheantiprotonbeamasmuchaspossible.Correctionsignalsareappliedtoindividualparticlesinordertofurtherstocasticallycooltheantiprotonbeam.An8GeVbeamemergesandispassedontotheAccumulatorwherepbunchesarestacked(i.e.,accumulated)atratesashighas1012antiprotonsperhour.ThebeamisthenpassedontotheRecyclerring,whichisan8GeVmagneticstorageringthatutilizesstochasticcoolingsystems.Asitsnamesuggests,theRecyclerisalsocapableofrecoveringantiprotonsleftoverattheendofastore(i.e.,periodofcollidingbeamtime).Oncetheaccumulatedantiprotonbeamreaches8GeVitcanbeextractedintotheMainInjectorandacceleratedto150GeV.

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Figure3{1. AschematicoftheacceleratorcomplexusedforRunIIatFermilab.Theacceleratorprocessmaybedividedintoeightsteps.Eachstepinthisprocessissummarizedinthetext. TheMainInjectorisasynchrotronringlocatednexttotheTevatron.ItwasaRunIIupgradetoreplacetheMainRing.TheMainRingwaslocatedintheTevatrontunnelandwasreplacedwiththeMainInjectorbecauseitcausedbeambackgroundsinthecolliderdetectors.ProtonbunchesexitingtheBoosterarecombinedbytheMainInjectorintoasinglehighintensitybunchofapproximately1012protons.ProtonsaretransferredtotheTevatronafterreachinganenergyof150GeV.TheTevatronisthelargestoftheFermilabaccelerators,withacircumferenceofapproximately4miles.Itisacircularsynchrotronwitheightacceleratingcavities.TheTevatronacceptsbothprotonsandantiprotonsfromtheMain

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Injectorandacceleratesthemfrom150GeVto980GeVinoppositedirections.Oncethebeamsenergyhasreached980GeV,theyaresqueezedtosmalltransversedimensionsbyquadrupolemagnetsattheinteractionpoints(thecentersoftheCDFandD0detectors).ThebeamcanbestoredintheTevatronwhilecollidingformanyhours.Typically,collisionscontinueuntilthereissomefailure,ortheremainingcolliderluminosityislowandtheantiprotonstackislargeenoughtobeginanewstore. 22 23 ].Here,thosecomponentsofthedetectorwhicharecrucialtothismeasurementarebrieydiscussed.AdetailedschematicdrawingontheCDFdetectorisshowningure 3{3 [ 23 ].Althoughitisnotshowninthegure,theCerenkovLuminosityCounter(CLC)isacriticalcomponentofthisanalysis[ 24 ].Whenchargeparticlestravelfasterthanthespeedoflightinamediumtheradiationthattheyemitbecomescoherent.Thisisasimilarphenomenatoasonicboomwhichoccurswhensomethingtravelsfasterthanthespeedofsound.ThiseectisusedtomeasuretheaveragenumberofinelasticppcollisionsperbunchcrossinginordertocalculatetheinstantaneousluminositydeliveredbytheTevatron.TheinstantaneousluminosityprovidedbytheCLCmustbeintegratedwithrespecttotimetocalculatetheintegratedluminosity.Thetotalintegratedluminosityincludedinthismeasurementisapproximately1000pb)]TJ/F4 7.97 Tf 6.59 0 TD[(1or1fb)]TJ/F4 7.97 Tf 6.58 0 TD[(1,andisusedforthenormalizationofthecrosssection.TheCerenkovcountersarelocatedintheregion3:7
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byanelectriceldtowardsthenearestpositivelychargedwiretoenergieshighenoughtocausesecondaryionization.Theelectronsarisingfromthischainofionizationarecollectedonthewireandanelectronicpulseisreadout.TheCOTprovideschargedparticlereconstructionuptojj=1:0.Surroundingthetrackingdetectorsisasuperconductingsolenoidwhichprovidesa1.4Tmagneticeld.Tracksarethereconstructedpathsofchargedparticlesinthemagneticeldbasedonthewiresthatcollectedelectronicsignals.Thesetrackscanbetracedbacktotheirpointofclosestapproachtothebeamline(i.e.,impactparameter).Aplacealongthebeamlinewheremultipletracksintersectiscalledaninteractionpointorvertex.Verticesaresignsthatappinteractionoccurredatthatposition.AtCDFtheremaybemultipleinteractionpointsinthesameeventduetopileup.Foreacheventaprimaryvertexisreconstructed.ThisisdenedasthevertexwiththehighestsumPT(sumofthePTofalltrackspointingtothevertex).ThepositionoftheprimaryvertexcanvarysignicantlywithrespecttothecenterofthedetectoratCDFduetothelengthofthecollidingbunches.Thelengthofthebunchesisroughly50cminthez-direction.Forcalculatingjetproperties,theprimaryvertexisusedastheoriginofthecoordinatesystem.Shiftingtheoriginofthecoordinatesystemalongthez-directionchanges()andthereforethevaluesofPTandETaremodied.Thenumberofextraverticesintheeventisagoodindicatorofthenumberofmultipleinteractionswithinabunchcrossing(pileup).Insummary,thetracksreconstructedbytheCOTareusedintheinclusivejetanalysistoreconstructtheverticesineacheventfortworeasons:todeterminetheprimaryvertexintheeventwhichisusedtodenetheoriginoftheeventcoordinatesystem,andtodeterminethenumberofsecondaryverticesintheeventwhichisusedtocountthetotalnumberofinteractionsthatoccurredinthebunchcrossing.

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Thetrackingchambersaboveonlydetectchargedparticles.However,onaverage,approximately40%oftheenergyinaneventiscarriedbyunchargedparticles.Calorimetersareusedtodeterminetheenergyandpositionofbothchargedandunchargedparticlesbytheirtotalabsorption.Electromagneticparticles(photonsandelectrons)andhadronicparticlesinteractwiththedetectormaterialdierently.Theelectromagneticandhadroniccalorimetersareallsamplingcalorimeters(i.e.,alternatinglayersofabsorberanddetectormaterial);however,eachcomponentiscomposedofdierentmaterialandhasdierentdepthbasedontheinteractionpropertiesoftheparticlesitwasdesignedtomeasure.Athighenergies,whenelectronsorpositronsinteractwithmatter,thedominantwayinwhichtheyloseenergyisthroughradiationofphotons(i.e.,bremsstrahlung:e)]TJ/F6 11.95 Tf 12.06 -4.34 TD[(!e)]TJ/F2 11.95 Tf 7.08 -4.34 TD[().Forhighenergyphotonsthedominantinteractionprocessispairproduction(i.e.,!e+e)]TJ/F1 11.95 Tf 7.08 -4.34 TD[().Aninitialelectronorphotonwillinteractthroughthesetwoprocessestoproduceashowerofphotonsandelectronsinthedetector.Thisphenomenaissketchedingure 3{2 andisreferredtoasanelectromagnetic(EM)shower.Theshowerdevelopsuntiltheenergyreachesacriticalenergy(Ec600MeV=c)andionizationlossesequalthoseofbremsstrahlung.Thedepthisgovernedbytheradiationlength(X0)ofthematerialandonlyincreaseslogarithmicallywiththeenergyoftheparticlewhichinitiatedtheshower.Theenergyresolutionoftheelectromagneticcalorimeterislimitedbystatisticaluctuationsinshowerdevelopment[ 25 ].HadronsinteractwithmatterthroughmuchdierentinteractionsthantheoneswhichleadtotheEMshowerdescribedabove.However,asimilarphenomenaoccurswhichisreferredtoasahadronicshower.Anincidenthadronundergoesaninelasticcollisionwithnuclearmatterinthedetectorresultinginsecondaryhadrons.Thesehadronsalsoundergoinelasticcollisions.Becausemanydierentprocessescontributetothedevelopmentofahadronicshower,themodelingofthe

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Figure3{2. Developmentofanelectromagneticshower.ElectronsandphotonsshowerwhentheyinteractwiththedetectorthroughtheprocessesofBremsstrahlungandpairproduction. showerismuchmorecomplexthananEMshower.Forexample,neutralmesons(0's)maybeproduced.NeutralmesonsprimarilydecayintotwophotonswhichinstigateanEMshower.Fluctuations,suchasthenumberof0'swhichareproducedearlyoninthehadronicshower,leadtoanenergyresolutionwhichingeneralismuchworsethantheresolutionofEMcalorimeters.Thedepthofahadronicshowerisgovernedbythenuclearinteractionlengthofthedetectormaterial.Formostmaterialsthenuclearinteractionlengthismuchlargerthantheradiationlength.Thismeansthathadronicshowerstypicallypassthroughmorematerialbeforestartingtoshower,andtheshowerstypicallytakeupmoredetectorvolume.ThisisthereasonthathadroniccalorimetersarelocatedoutsideofEMcalorimetersandaretypicallymuchthicker.Asmentionedabove,theCDFcalorimeterconsistofalternatingabsorberanddetectorlayers.Theabsorberconsistsofadensematerial(leadoriron)withalargeradiationorinteractionlengthforthepurposeofinstigatingthe

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showersdiscussedabove.Thismaterialisdeadinthesensethatithasnodetectioncapabilities.Whenchargedparticlesreachenergiesbelowsomecriticalenergytheyloseenergytothedetectormaterialthroughexcitationandionizationofatoms.Insomematerials,calledscintillators,afractionofthisexcitationenergyemergesasvisiblelightastheexcitedatomsreturntotheirgroundstate.Someofthislightcanbetransferredtophotomultipliertubeswhichconvertsthelighttoelectronicsignalsthroughthephotoelectriceect.ThedetectionlayersoftheCDFcalorimetersarecomposedofvariousscintillatingmaterial.Theenergyofaparticleabsorbedbythecalorimeterisproportionaltotheamountoflightmeasuredbythescintillatingmaterial,andthisproportionalityconstantmustbedeterminedthroughcalibration.Theprocessofconvertingchargedparticleinteractionsintoelectronicsignalsdescribedaboveisveryfast.InformationfromthecalorimeterisavailableveryquicklyatCDFandisusedtomaketherstdecisionsonwhetheraneventisinterestingornot(i.e.,itisusefulinthelevel-1trigger).Thesize,granularity,location,andresolutionofthevariousCDFcalorimetermoduleswillnowbedescribed.Thecentralcalorimeterislocatedoutside(i.e.,fartherawayfromtheinteractionpoint)ofthesolenoidmagnetandisdividedintoelectromagnetic(CEM)andhadronic(CHA)sections.ThecentralcalorimeterissegmentedinY)]TJ/F2 11.95 Tf 13.26 0 TD[(spaceinto480towerswhichpointbacktowardstheinteractionpoint.Thegranularityofthetowersis0:10:26.Thecentralcalorimetercoversapseudo-rapidityrangeupto1.1.TheCEMisalead-scintillatorcalorimeterwithadepthofabout18radiationlengths;theCHAisaniron-scintillatorcalorimeterwithadepthofapproximately4.7interactionlengths.TheenergyresolutionoftheCEMforelectronsis

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whiletheaverageenergyresolutionoftheCHAforchargedpionsis 3{4 [ 23 ].ThePEMandPHAareidenticallysegmentedinto480towersofsizewhichvarieswith(0:10:13atjj<1:8andincreasesto0:60:26atjj=3:6).TheenergyresolutionofthePEMforelectronsis

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However,fortheinclusivejetcrosssection,nodistinctionbasedonparticletypeisnecessary.OnlythetotalenergydepositedwithinaY)]TJ/F2 11.95 Tf 12.63 0 TD[(regionofthecalorimeterisimportanttothisjetmeasurement.Therefore,theenergydepositedinelectromagneticandhadronicsectionsofeachtowerarecombinedintophysicstowers.Thepositionofeachsectionisdenedbythevectorjoiningtheinteractionpointtothegeometricalcenterofthesection.Eachsectionisassumedtohavenomass(i.e.,E/P),andthesumofthemomentumfour-vectorfortheEMandhadronicsectionaretakenasthemomentumvectorofthephysicstowers.Physicstowersarethedetectorobjectswhichareclusteredintojets.

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Figure3{3. CDFdetector.TheCLC,COT,andthecalorimetersaretheimportantcomponentsforthisanalysis.Theyaredescribedbrieyinthetext.

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Figure3{4. LongitudinalviewoftheCDFIITrackingSystemandplugcalorimeters.Thepositionofthecalorimetermoduleswithrespecttodetectorisshowninthegure.

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2.6 ),itisnotclearwhatphysicalobservables(ifany)willyieldaclearinterpretationoftheoriginalhardinteraction.Jetclusteringalgorithmsaredesignedtoclusterthecomplexstructureofnalstateobjectsfromeachcollidereventintojets.Thesejetsmustbeamaptothephysicalpropertiesofthepartonsfromthehardscatteringtobeausefulconstructforcomparisonwiththeoreticalpredictions.CurrentlyatCDFtherearethreejetclusteringalgorithmsinuse:JetClu:JetClu[ 26 ]istheconealgorithmusedinRunIatCDF.ConealgorithmscombineobjectsbasedonrelativeseparationinY)]TJ/F2 11.95 Tf 12.94 0 TD[(space,Rp 27 ].InthischaptertheMidpointjetclusteringalgorithmwillbedescribedindetail.Forcompleteness,theKTalgorithmandJetClualgorithmwillalsobesummarized.Afterdeningthealgorithmssometechnicalissuesonthetopicofjetdenitionwillbediscussed. 36

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particlescanhaveinY)]TJ/F2 11.95 Tf 11.97 0 TD[(spaceandstillbecombinedintoajet.AtCDF,jetsarereconstructedwiththreedierentconesizes:0.4,0.7,and1.0.Whatconesizeisusefuldependsonthedetailsoftherolethatjetsplayinananalysis.ThisanalysisusesaconesizeofRcone=0:7.Therststepinanyjetalgorithmistoidentifythelistofobjectstobeclustered.Inthisanalysis,jetswillbeclusteredatfourdierentlevels(seegure 2{10 ).Thelistofobjectstobeclusteredisdierentineachcase:Detectorlevel(dataorMCwithdetectorsimulation):four-vectorsofthecalorimeterphysicstowersareusedasthebasicelementsoftheclustering.Toreducetheeectofelectronicnoise,onlytowerswithPT>100MeV=careincludedinthelist.Particleorhadronlevel(MC):four-vectorsofthestableparticles(i.e.,hadrons)arethebasicelementstobeclustered.Partonlevel(MC):four-vectorsofthepartonsbeforehadronizationareclusteredintojets.ForMCsuchasHERWIGorPYTHIAthiswillincludethemanyquarksandgluonsfromthepartonshowerandmultiplepartoninteractions.Partonlevel(NLOpartonlevel):four-vectorsofthepartonsareclusteredintojets.ThereareatmostthreepartonsinthelistatNLO.Thenextstepistoidentifyalistofseedobjects.ThisisasubsetofthelistofobjectstobeclusteredwiththeextrarequirementthatthePToftheobjectbeabovesomethreshold(1GeV=c).ItwouldbepreferabletheoreticallytoincludeseedscorrespondingtoeverypointinY)]TJ/F2 11.95 Tf 12.39 0 TD[(space;however,bysearchingforjetsonlyatseedlocationstheCPU-timetorunthealgorithmisgreatlyreduced.Ateachseedlocationaconeofradius,R=Rcone=2,inY)]TJ/F2 11.95 Tf 12.49 0 TD[(spaceisconstructed.Thisreducedconesize,orsearchcone,isnotafeatureofthestandardMidpointalgorithm.ThestandardMidpointalgorithmusesR=Rconeforallclusteringsteps.CDFusesamodiedversionoftheMidpointalgorithmwhichisoftencalledtheSearchConealgorithm.Themomentumfour-vectorsofallobjectslocatedinthe

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searchconearesummed.Thisfour-vectorsumiscalledthecentroidofthecluster.Thefour-vectorofthecentroidisthenusedasanewconeaxis.Fromthisaxisanewconeisdrawnandtheprocessofsummingupthefour-vectorsofallparticlesintheconeisconductedagain.Thisprocessisiterateduntiltheconeaxisandthecentroidcoincide,indicatingthatthecongurationisstable.Oncethestablecongurationisfound,theconeaxisisexpandedtothefullconesize(R=Rcone),andthefour-vectorofaprotojetisformedbyaddingupallofthefour-vectorsoftheobjectsintheexpandedcone.Theexpandedconeisnotiteratedforstability.Thisprocedureofndingstableconesisappliedtoeveryobjectintheseedlist.Thenextstepinthealgorithmistheoneforwhichitisnamed.AdditionalseedsareaddedatthemidpointbetweenallprotojetswhoseseparationinY)]TJ/F2 11.95 Tf 12.38 0 TD[(spaceislessthantwotimesRcone(i.e.,ifR<2Rcone).AconeofradiusR=Rconeisthendrawnaroundthemidpointseedanditerateduntilastablecongurationisfound.Ifthiscongurationisnotalreadyinthelistofprotojets,itisaddedtothelist.Afterallmidpointseedshavebeeniteratedtostableconecongurations,thelistofprotojetsiscomplete.Theprocessofaddingseedsatthemidpointbetweenallstableconesreducesthesensitivityofthealgorithmtosoftradiation.Itispossiblethatmanyoftheprotojetswilloverlap(i.e.,objectsmayappearinmorethanoneprotojet).Overlappingprotojetsmustbesplitormergedtomakesurethatthesameobjectisnotincludedinmorethanonejet.Beforesplittingandmergingbegins,protojetsaresortedaccordingtotheirPT.Ifthesum-PT(four-vectorsum)ofsharedobjectsbetweentwoprotojetsismorethanthefraction,fmerge=0:75,oftheprotojetwithlowerPT,thenthetwoprotojetsaremerged.Ifthesum-PTofsharedobjectsbetweentwoprotojetsislessthanfmerge,thenthesharedobjectsbetweenthetwoprotojetsaresplitandassignedtothecloserconeinY)]TJ/F2 11.95 Tf 11.96 0 TD[(space.TheMidpointalgorithmmaybesummarized:

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1.Alistofallobjectstobemergedintojetsisconstructed.2.Aseedlistwhichincludesonlyobjectswithtransversemomentumgreaterthan1GeV=cisgenerated.3.Stableconesareconstructedaroundeachseed(R=Rcone=2).4.TheradiusofallstableconesisextendedtoR=Rcone.5.Anadditionalseedisaddedformidpointsbetweeneachpairofstableconesseparatedbylessthantwicetheconeradius.Eachadditionalseedissearchedforstablecones(R=Rcone)thathavenotalreadybeendiscovered.6.ThestableconesarePT-orderedandsplittingandmergingisperformedforoverlappingcones.AtNLOpartonlevel,thereareatmostthreeparticlesintheevent.Inthiscase,theideaofajetalgorithmbecomesverysimple.ThealgorithmmustdecideifthetwoparticleswhichareclosestinY)]TJ/F2 11.95 Tf 12.53 0 TD[(spaceshouldbecombinedornot.Thereisnocomplicatedsplittingandmergingstepneeded.Here,thedefaultMidpointalgorithmwouldmergeanytwoparticleswhichareseparatedbylessthan2RconeinY)]TJ/F2 11.95 Tf 12.66 0 TD[(space.AtthedetectorlevelitisobservedthatparticlesarealmostnevermergedbytheMidpointalgorithmiftheyareseparatedby2Rcone.Forthisreason,RsephasbeenintroducedforNLOcalculationswiththeMidpointalgorithm.AtNLO,thealgorithmismodiedbyRsepsothatparticlesareonlymergediftheirseparationininY)]TJ/F2 11.95 Tf 12.41 0 TD[(spaceislessthanRsepRcone.AvalueofRsep=1:3isconsistentwithdetectorlevelstudies.RsepplaystheroleofsplittingandmergingattheNLOpartonlevel[ 26 ]. 28 ].ThestepsforjetclusteringatthedetectorlevelusedbyJetCluaredescribedbelow:1.AnETorderedlistofseedtowerswithET>1.0GeViscreated.

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2.BeginningwiththehighestETtower,preclustersareformedfromadjacentseedtowers,providedthatthetowersarewithina0.7x0.7windowcenteredattheseedtower.Anytoweroutsideofthiswindowisusedtoformanewprecluster.Thisclusteringstepisdependentondetectorgeometry,anditcannotbeconductedinthesamewayatthehadronorparticlelevel.3.ThepreclustersareorderedindecreasingETandtheETweightedcentroidisformedbyaddingtheenergyfromalltowerswithmorethan100MeVwithinR=0.7ofthecentroid.4.Anewcentroidiscalculatedfromthesetoftowerswithintheconeandanewconedrawnaboutthisposition.Steps3and4areiterateduntilthesetoftowerscontributingtothejetisstable.ThepropertyoftheJetClualgorithm,thatalltowersincludedintheoriginalclusterremainintheclusterevenwhentheynolongerliewithintheconeradius,iscalledRatcheting.5.ClustersarereorderedindecreasingETandoverlappingjetsaremergediftheyshare75%ofthesmallerjet'senergy.Iftheyshareless,thetowersintheoverlapregionareassignedtothenearestjet.TheKTalgorithmhandlesparticlecombinationmuchdierentlythantheconealgorithmsdescribedsofar.Theprocedureforcombiningobjectsintojetsisexactlythesameattheparton,hadron,anddetectorlevelsfortheKTalgorithmandtheyaredescribedbelow:1.ThequantitiesY,,andPTareconstructedforeachobjectinthelistofobjectstobecombined.2.Foreachobjectinthelistdi=P2Tiiscalculated,andforeachpairofpartonsthequantitydij=min((di)2;(dj)2)(Rij)2=D2isdened.TheDparameterplaysasimilarroleintheKTalgorithmasRconedoesintheconealgorithms.3.Findtheminimumofalldianddij.4.Iftheminimumisoneofthedijremoveparticlesiandjfromthelistandreplacethemwithanobjectdenedbythesumoftheirmomentumfour-vectors.5.Iftheminimumisoneofthedithenremoveitfromthelisttobecheckedformergingandaddittothelistofjets.6.Ifanyparticlesremaininthelist,gotosecondstepabove.

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26 ].Thereasonthisistrueisillustratedingure 4{1 [ 29 ].Inthecongurationontheleft,twoseedsmayhaveleadtotwostableconecongurations.Ifaseedhadbeenconstructedbetweenthesetwocones,adierentcongurationmayhavebeenfound.Astableconemayhavebeenfoundwhichincludedtheothertwoobjectssothatonlyonejetwasconstructedintheevent.Softradiationcanpushobjectsjustbeloworjustaboveseedthreshold.Inthisway,theeventtopologyissensitivetosoftradiation.TheMidpointalgorithmreducesthissensitivitybyaddinganextraseedatthemidpointbetweenallstablecones.Theresultofthemidpointseedisthattheeventtopologyisnotas

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sensitivetouctuationsaboveorbelowseedthresholdcausedbysoftradiation.Forthisreason,andthereduceddependenceondetectorgranularity,theMidpointalgorithmhasreplacedJetCluastheconeclusteringalgorithmusedatCDFinRunII. Figure4{1. TheMidpointjetclusteringalgorithmchecksforastableconecongurationatthemidpointofallstableconeslocatedfromsearchingatseedlocations.Inthiswaythealgorithmislesssensitivetouctuationsduetosoftradiation. AtCDF,itwasobservedthatinsomeevents,afterclusteringjetswiththestandardMidpointalgorithm,thereweresignicantclustersofenergywhichwerenotincludedinanyjet.Aneventdisplayofaneventinwhichthisoccursisshowningure 4{2 [ 26 ].Thegureshowstheenergydepositedineachtowerofthedetectoronthe)]TJ/F2 11.95 Tf 12.52 0 TD[(plane.Theclustersofenergyshowninblackwerenotclusteredintoanyjetinthisevent.Theseclustersofenergywhicharenotincludedinajetarereferredtoasdark-towers.Darktowersoccurbecauseaconethatstartedfromaseedwithinthedark-towerclustermovesawayfromthisclusterofenergytowardsalargerclusterofenergy(i.e.,theconemigratestoaneighboringclusterofenergy).Theeectofdarktowersissignicant.Forexample,approximatelytwopercentofeventswitha400GeV=cjethavemorethan50GeV=cofun-clusteredtransversemomentum.Becausethisdark-towereectisnotincludedatNLO,itwasdecidedthattheissueofun-clusteredenergyneededtobeaddressed.The

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Figure4{2. DarktowersobservedbytheoriginalMidpointalgorithm.Notallclustersofenergyinthedetectorwerebeingincludedinajet. solutionappliedatCDFistouseasmallerinitialsearchcone(Rcone=2)(i.e.,usetheSearchConealgorithm).ThenetresultofusingtheSearchConealgorithmisavepercentincreaseintheinclusivejetcrosssection,whichisroughlyindependentofjetPT[ 30 ].TheCDFSearchConealgorithmappliedinthisanalysisisnotperfect.ItisslightlymoresensitivetotowerandseedthresholdsthanthestandardMidpointalgorithm.Also,thereisasensitivitytothesizeofthethesearchcone.VaryingthesearchconeradiusfromR=2toR=p 30 ].

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AlthoughtheSearchConeisnotperfect,andconealgorithmsingeneralhavemoreissuestheoreticallythantheKTalgorithm,conealgorithmshavesomeadvantages.WithastandardconealgorithmtheuserhascompletecontroloverwhatisincludedintheconethroughthevariableRcone.Thefactthatthejetiscomposedofaspeciedconesizeinthedetectorisausefulpropertyformakingcorrections.Forexample,whencorrectingformultipleppinteractionsinthesamebunchcrossingaconelocatedrandomlyinthedetectorcanbeusedtostudytheextraenergythatisincludedonaverageinajetduetotheextrainteraction.ThiscannotbedonewithaKTalgorithmsinceitdoesnotuseaxedconesize.Itisalsousefultomakemeasurementswithtwodierenttypesofalgorithms.ThedierenceinresultsfromdierentalgorithmsshouldbepredictablebyMCandcanbeusedtolearnaboutthejetclusteringpropertiesofeachalgorithm.

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31 32 33 ].Therstinclusivejetcrosssectionmeasurementatahadroncollider,aswellastherstdirectobservationofacleartwo-jeteventtopology,camefromtheSppScollideratCERN[ 34 35 ]withacenter-of-massenergyof540GeV(p 36 ].ThemeasurementofthedierentialinclusivejetcrosssectionatCDFreachesthehighestmomentumtransferseverstudiedincolliderexperiments.Thus,itispotentiallysensitivetophysicsbeyondthestandardmodel[ 37 ].StudyingthehighestenergyeventsattheTevatronisequivalenttoprobingdistancesontheorderof10)]TJ/F4 7.97 Tf 6.59 0 TD[(19m.Thismeasurementisprobingdistancescalesmorethanonethousandtimessmallerthantheradiusoftheprotonandissensitivetowhetherornotthequarkhassubstructure[ 38 ].TheinclusivejetcrosssectionmeasurementisalsoafundamentaltestofpredictionsofperturbativeQCD[ 39 40 ]overeightordersofmagnitudeincrosssection.Jeteventsinthecentralregionofthedetectorwithtransversemomentumhigherthan530GeV=chaveacrosssectionofapproximately30fb(3010)]TJ/F4 7.97 Tf 6.59 0 TD[(6nb).Theseareamongthesmallestcrosssectionsevermeasuredatacolliderexperiment.Thecurrentmeasurementspans600GeV/cinjettransversemomentum,anditcanthereforebeusedtoobservetherunningofsinasinglemeasurement.ItisacommonmisconceptionthattheQCDforce 45

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getssmallatsmalldistances.Thisiscertainlynotthecase.Theeectivecouplingdoesdecreaseathighmomentumtransfer,buttheeectiveforceisinverselyproportionaltosquareofthedistance(F/s 41 ].Infact,thegluonPDFisthedominantsourceoftheoreticaluncertaintyintheinclusivejetcrosssectionandmanyotherprocessesathadroncolliders.TheuncertaintyonthequarkandgluonPDFsforQ=500GeVisshownasafunctionofBjorken-xingure 5{1 (b)Figure5{1. Uncertaintyontheupquark, 5{1(a) ,andgluon, 5{1(b) ,PDFsforQ=500GeVasafunctionofBjorken-x(momentumfractioncarriedbytheparton).TheuncertaintyismuchlargeronthegluonPDFbecausethiscomponentoftheprotonisnotprobeddirectlyinDISexperiment.

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Thestrongestconstraintsonthepartondensityfunctionscomefromdeepinelasticscatteringexperiments(DIS).TheDISexperimentsscatterelectronsoofprotons.Thisprocessisdepictedingure 5{2 [ 9 ].Theelectroninteractswiththequarksintheprotonthroughtheexchangeofavirtualphoton.Becausephotonsdonotcoupledirectlytogluons,theDISexperimentscannotmakestrongconstraintsonthegluoncontentoftheproton.ThestrongestconstraintsonthegluonPDFcomefromjetmeasurementsathadroncolliders,suchastheinclusivejetcrosssectionmeasurementdiscussedhere. Dominantprocessindeepinelasticscatteringexperiments.Thevirtualphotonprobesthequarkcontentoftheproton.Sincephotonsdonotcoupledirectlytogluons,thisprocesscannotbeusedtoplacestrongconstraintsonthegluonPDF. Becausetheinclusivejetmeasurementintheforwardregion(largeY)probesakinematicrangewhichisnotexpectedtobesensitivetonewphysics,itshouldleadtoapowerfulconstraintonthegluonPDF.DijeteventsproducehighPTjetsathighrapiditywhenthemomentumfractionofthetwoincomingpartonsdonotbalance.Inthistopology,thehigh-xcomponentoftheprotonisprobedatalowermomentumtransferthanforanequivalentenergyjetinthecentralregion.Inotherwords,thehigh-xcomponentoftheprotoncanbeprobedatalowerenergyscalewithforwardjets.Becauselowerenergyscaleshavebeenstudiedextensivelyinthecentralregion,andagreewiththepredictionsofpQCD,onecanbecondentthatphysicsbeyondtheSMisnotaectingtheresult.Sincetheforwardregion

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jetsisnotassensitivetonewphysicsasthehighPTjetsinthecentralregion,measurementsintheforwardregionarecrucialtoconrmingthatobservationscanbeattributedtoPDFeectsandnotphysicsbeyondthestandardmodel.TheCDFexperimenthasahistoryofmakingimportantinclusivejetmeasurements.InRunI(p 42 ].Thisresultalludedattheneedforlargergluoncontentathigh-xintheproton.TheRunIBCDFcentralregionresult,with87pb)]TJ/F4 7.97 Tf 6.58 0 TD[(1ofdatacollectedduringtheperiod1994-1995[ 36 ],stillshowedanexcessathighjetPT.WhentheseresultswerecombinedwiththeD0resultincludingthehigherrapidityregion[ 43 ],with95pb)]TJ/F4 7.97 Tf 6.58 0 TD[(1ofdataupto=3:0,itwasconrmedthatthegluoncontentathigh-xhadbeenunderestimatedintheprotonstructurefunctions.ThisexperiencefromRunIrevealedaneedtobeabletoquantifytheeectofPDFuncertaintyoncolliderobservablessuchascrosssection[ 44 45 ].Theincreasedcenter-of-massenergyinRunII(p 46 ].CDFalsorecentlypublishedaninclusivejetmeasurementinthecentralregionforjetsclusteredbythekTalgorithm[ 27 ].Inthisdocument,theRunIconejetanalysisisupdatedwithapproximatelytentimestheintegratedluminosityresultinginover1fb)]TJ/F4 7.97 Tf 6.58 0 TD[(1ofdata.Thisisthe

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rstinclusivejetmeasurementatCDFtoincludetheforwardregion(jYj<2:1)usingaconealgorithmforjetclustering.Thetechniquesappliedinthismeasurementwillbefullymotivatedandbrieydescribed.Forcompletedetailsandallrelevantdistributionspleaseseereference[ 47 ].

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48 ].TherateatwhichcollisionsoccurwithintheCDFdetectorismuchhigherthantherateatwhichdatacanbecollectedandstored.BunchcrossingsoccuratCDFatarateofapproximately1:7MHz,whiledatacanonlybewrittentotapeatabout75Hz.VarioustriggersaredesignedtoextracteventsthatareusefulforphysicsanalysisatCDF.Thetriggerissplitintothreelevels,andateachleveltheeventswhichpassthetriggerrequirementsarepassedontothenextlevel.Thetriggerrequirementsateachlevelmustproducearatereductionlargeenoughforprocessingatthenextleveltobepossible.TheLevel1jettriggerconsistsoftwotriggerstreams;requiringacalorimetertriggertowertohaveET>5GeVforthejet20andjet50triggers,andET>10GeVforthejet70andjet100triggers.AtLevel2,thecalorimetertowersareclusteredusinganearestneighboralgorithm.FourtriggerpathswithclusterET>15,40,60,and90GeVareused.EventsinthesepathsarerequiredtopassjetET>20,50,70,and100GeVthresholdsatLevel3,wheretheclusteringisperformedusingtheJetCluconealgorithmwithaconeradiusRcone=0:7.TheCDFtrigger,andalltriggerstudiesrequiredforthisanalysisaredescribedinmoredetailinAppendix A .Theselectioncriteriafortheinclusivejetcrosssectionisminimal,sincealljeteventsareincludedintheanalysis.Sincecosmicraybackgroundsoriginatefromoutsidethedetector,thetransverseenergydepositedinthedetectorisnot 50

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balanced.CosmicrayscanthereforebeecientlyremovedbyapplyingamissingETsignicancerequirement(f6ET).Thef6ETselectioncriteriavariesaccordingtothejetsampleanditis4,5,5,and6GeV1=2forjet20,50,70,and100triggersrespectively.Withanyselectioncriteriaitispossiblethatsomejeteventswhichcamefromhardscatteringeventsareremoved.Thepercentageofrealjeteventswhichsurvivetheselectioncriteriaiscalledtheeciency.Thef6ETrequirementisapproximately100%ecientatlowPTandvariestoapproximately90%atthehighestjetPTincludedinthemeasurement.Inordertoensurethatparticlesfromtheppinteractionsareinaregionofthedetectorwithgoodtrackingcoverage,primaryverticesarerequiredtobewithin60cmofthecenterofthedetector(jZj<60cm).Theeciencyofthisrequirementismeasuredtobe95:8%atCDF[ 49 ].Thejet20,50,and70triggersareprescaledtoavoidsaturatingthebandwidthofthetriggeranddataacquisitionsystem.Prescalingbyafactorofnmeansthatonly1outofneventssatisfyingthetriggerrequirementarestoredtotape.Thejet70triggerisprescaledbyaconstantfactorof8foralldatausedinthisanalysis.Theprescalesforthejet20and50triggershavechangedduringthedatatakingperiodconsidered.Theeectiveprescalesofthejet20and50triggersforallthedatawerefoundtobe776:8and33:6,respectively,byluminosity-weightingtheinverseofprescalefactors 1

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tobeconsistenttobetterthan1%betweenthetwomethods.Thedistributionsforthenumber(i.e.,jetyield)ofjetsasfunctionofPTinthecentralregion(0:1
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(b)Figure6{1. JetyielddistributionsasafunctionofPTinthecentralregionbefore, 6{1(a) ,andafter, 6{1(b) ,correctingfortriggerprescales.

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Figure6{2. Measuredrawjetcrosssectionfortheverapidityregions.Therawjetcrosssectionhasnotbeencorrectedtoremovedetectoreects.

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Figure7{1. DierencebetweencalorimeterjetPTandhadronleveljetPTforthreedierentjetPTranges.ThecalorimeterjetPTissystematicallylowerthanthehadronleveljetPT.Also,thereisasmearingeectduetotheuctuationintheenergymeasuredinthecalorimeterforagivenhadronleveljet. 55

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Twodetectoreects,whichthejetdatamustbecorrectedfor,areillustratedinFigure 7{1 .Thedierencebetweencalorimeterjettransversemomentum(PCalT)andhadronleveljettransversemomentum(PHadT)forthreedierentjetPTrangesisshowninthegure.HadronleveljetswerematchedtocalorimeterjetsbytheirseparationinY)]TJ/F2 11.95 Tf 12.74 0 TD[(space(R<0:7)usingMCgeneratorresultswhichwerepassedthroughtheCDFdetectorsimulation.ThedistributionspeakbelowzerobecausePCalTissystematicallylowerthanPHadT.Thisreectsthenon-compensatingnatureofthesamplingcalorimeters.Thehadroniccalorimeterwascalibratedbasedonchargedpionsfromatestbeamwithatransversemomentumof57GeV=c.OnlypionswhichdidnotinteractwiththeEMcalorimeterwereincludedinthecalibration.Inrealjetshowever,alargefractionofhadronsdointeractwiththeEMcalorimeter.BecausetheEMcalorimeteriscalibratedbasedonelectronsitsresponsetohadronsislower.Thislowerstheoverallresponsetosinglehadrons.ThiseectislargerforlowtransversemomentumparticlesbecausetheyinteractintheEMcalorimetermoreoften,anditcanthereforecontributetothenon-linearityofthecalorimeterresponsetohadrons.Hadronicshowershavealargerfractionofneutralpionswhentheincidenthadronhasahighertransversemomentum.BecausethecalorimeterhasahigherresponsetoEMshowers(i.e.,0decays),thisalsocontributestothenonlinearnatureofthecalorimeterresponsetochargedhadrons.Ingeneralthecalorimeterresponsegoesupastransversemomentumoftheincidenthadronincreasesandisnotlinear.Thiscausesasystematicshiftdownintheenergyresponsetojetsbecausetheyincludemultiplehadronswithlowertransversemomentum,ratherthanonehadronwiththefulljetPT.Thejetenergysmearingeectiscausedbythelimitedjetenergyresolutionofthecalorimeters,andisreectedingure 7{1 bythewidthofthepeak.Fluctuationsinshowerdevelopmentduetotheprobabilisticnatureofthe

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interactionsbetweentheparticlesinthejetandthedetectormaterialcausethedetectorresponsetoparticlejetswithaxedenergytovary.Figure 7{2 illustratestheowofthejetcorrectionschemeusedtoobtainresultscorrectedtothehadronorpartonlevel.First,an-dependentrelativecorrectionisappliedtothedataandMCinordertoequalizetheresponseoftheCDFcalorimeterstojetsinY.TheequalizedjetPTisthencorrectedforthepileupeect.Then,theabsolutecorrectionisappliedtocorrectonaverageforthehadronenergythatisnotmeasuredbythecalorimeter.Afterthat,thehadronandcalorimeterleveljetPTdistributionsarecomparedinMonteCarlotoderiveabin-by-bincorrectioninordertoremoveresolutioneects.Thisiscalledunfolding.Atthispoint,thedatahavebeencorrectedtothehadronlevel.InordertocomparedirectlywithpQCDpredictions,theeectsoftheunderlyingeventandhadronizationneedtoberemovedfromthedata.Afterthisnalcorrection,thedatahavebeencorrectedtothepartonlevel.TheMonteCarlosimulationusedtoderivethecorrections,andthedetailsofeachcorrectionstepwillnowmedescribed. 50 ]andHERWIG6.4[ 51 ],alongwiththeCDFdetectorsimulation,areusedtoderivethevariouscorrectionswhichareappliedtothedata,andtoestimatesystematicuncertainties.Structurefunctions(i.e.,partondistributionfunctions)fortheprotonandanti-protonaretakenfromCTEQ5L[ 52 ].TheCDFdetectorsimulationisbasedonGEANT3[ 53 ]inwhichaparametrizedshowersimulation,GFLASH[ 54 ],isusedtosimulatetheenergydepositedinthecalorimeter[ 55 ].TheGFLASHparametersaretunedtotest-beamdataforelectronsandhighmomentumchargedpionsandtothein-situcollisiondataforelectronsfromZdecaysandlowmomentumchargedhadrons.However,theCDFsimulationdoes

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Figure7{2. Flowdiagramforthejetcorrectionsusedintheinclusivejetanalysis.Correctionstepsareshowninred,whilethehadronlevelandpartonlevelcorrectedstatesareshowninblue.Relative,pileupandaverage(absolute)correctionsareapplieddirectlytothejetPTbeforebinning. notdescribeenergydepositioninthecalorimetersperfectly,especiallyintheregionscorrespondingtotheplugcalorimetersandcracksbetweencalorimetermodules.SincetheMCsimulationisusedtoderivevariousjetcorrectionstobemadeonthedata,dierencesbetweentherealcalorimeterresponsetojetsandthecalorimetersimulationneedtobewellunderstood.DierencesintherelativejetenergyresponseandjetenergyresolutionbetweenthecollisiondataandMCsimulationeventswereinvestigatedusingdijetPTbalancingindijetevents[ 55 ]andthebisectormethod[ 56 ],respectively.ThedijetPTbalanceandbisectorstudiesarebrieydescribedbelow.MoredetailscanbefoundinAppendix C .ComparisonsofdijetPTbalancebetweendataandMCrevealthattherelativejetenergyscaleversusisdierentbetweendataandMC,andthatthedierencedependsonjetPTathighrapidity(jYj>1:1).Forexample,thejetenergyscaleintheplugcalorimeterregionishigherinMCthanindataby2%andthe

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dierenceincreasesslightlywithjetPT.Thisdierenceisaccountedforbytherelativecorrectionswhicharedescribedindetailinsection 7.2 .ThebisectormethodallowsonetocomparetheenergyresolutionoftheCDFdetectorandtheCDFdetectorsimulation.Inthecentralregion(0:1
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Figure7{3. Degreeofdijetbalance()observedintheCDFcalorimeterandandMCwithCDFdetectorsimulationasafunctionofdetectorfortheprobejetintheevent.TherelativecorrectionisappliedtoequalizetheresponseinoftheCDFcalorimetertojets. 7{3 ),correctionsarederivedseparatelyfordataandMC.Therelativecorrectionsforthisanalysiswerederivedataxedvalue(PT=117:5GeV=C)ofjetPTsothatthedierenceinthePTdependenceoftheresponseobservedindataandMCsimulationcouldbehandledmoredirectly.MoredetailsregardingtherelativejetenergycorrectionscanbefoundinAppendix B .Asmentionedearlier,thedata-MCdierenceintherelativejetenergyscaledependsonjetPTforjYj>1:1.Therefore,aPT-dependentcorrectionderivedforthetwocorrespondingrapidityregionsisalsoappliedtotheMCinordertoforcethedistributionstoagreewithdataforalljetPT.Requiringadijetevent

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topologyforhighPTjetsintheforwardregiongreatlyreducesstatistics.ThelackofstatisticsindataandsimulatedMCleadtoasignicantuncertaintyinthisPT-dependentcorrection.DetailsofthePT-dependentcorrectiontothedijetPTbalanceintheMCareincludedinAppendix C 55 57 ].

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ofthecalorimeterjetenergyineachrapidityregion.ThisdistributionisttoafourthorderpolynomialandthetisappliedasacorrectiontothePTofeachjetinthedatasample.Thecorrectionisoftheorderof20%forPCalT50GeV=canddecreasestotheorderofafewpercentforPCalT600GeV=c.Thiscorrectionisslightlydierentforeachrapidityregion. 36 43 ].IfPYTHIAisusedtocorrectthedatabacktothehadronlevel,theshapesofthepredictedPTdistributionsshouldbethesameasthedatainordertoavoidintroducinganybiaswiththecorrections.TheratiosofdatacorrectedtothehadronleveltothePYTHIApredictionsarettopolynomials.Thesetsareusedtore-weightthePYTHIAPTdistributions;thereby,forcingtheshapesofthePTdistributionsoftheMCto

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agreewiththedata.TheunfoldingcorrectionfactorsobtainedfromtheweightedPYTHIAdistributionsareappliedtothedata.Themodicationduetothere-weightingofPYTHIAislessthan1%inallrapidityregionsexceptfortheregion0:7
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Figure7{4. Hadrontopartonlevelcorrectionappliedinthecentralregion.ThedierencebetweenHERWIGandPYTHIApredictionsforthiscorrectionisconservativelytakenasthesystematicuncertainty(shadedbands).

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55 ].ThecorrectedjetPT,inthePYTHIAMCwithdetectorsimulation,wasvariedupanddownaccordingtothisparametrization.TheresultingdistributionswerecomparedwiththecentralvalueinordertoderivethesystematicuncertaintyduetotheJES.Eventhoughthejetenergyscaleisknowtobetterthan3%,whenconvolutedwiththesteeplyfallingjetPTdistributions,theuncertaintiesonthecrosssectionarelarge.Theyvaryfromapproximately10%atlowjetPTuptoashighas60%athightransversemomentuminsomerapidityregions.Thereisanadditionaluncertaintyonthejetenergyscaleinthehigherrapidityregions(jYj>1:1).Statisticsarelimitedwhenadijettopologyisrequiredathighjettransversemomentum;asaresult,thePT-dependentcorrectiontotheMC,basedondijetPTbalance,isnotverywellconstrainedathighjetPT.Thisuncertaintyisapproximately40%inthehighesttransversemomentumbins,andisdiscussedinmoredetailinAppendix C .Theremainingsourcesofsystematicuncertaintyinthisanalysisaresummarizedbelow:Unfolding:Theunfoldingcorrectionissensitivetothemomentumdistributionsofparticleswithinjets.HERWIGandPYTHIArelyondierentfragmentationmodels;therefore,thedierencebetweentheirunfoldingcorrectioncanbeusedasameasureofthesensitivityofthiscorrectiontothefragmentation 65

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model.ThesystematicuncertaintyonthemeasuredcrosssectionistakenfromtheratiooftheunfoldingfactorsobtainedfromPYTHIAandHER-WIG.Thedierenceintheunfoldingfactors,asobtainedwithweightedandun-weightedPYTHIA,istakenasadditionalsystematicuncertainty.Jetenergyresolution:Duetothesharplyfallingspectrumoftheinclusivejetcrosssection,anyimperfectioninthejetenergysmearingofthedetectorsimulationwillaecttheunfoldingcorrection.ThebisectorstudyrevealedthattheresolutiondierencebetweendataandMCvariedbyapproximately10%,withnodependenceonthetransversemomentumofthejets.ThisvariationwastakenastheuncertaintyontheresolutionoftheMCresponsetojetenergy.ThecalorimeterleveljetsinthePYTHIAsimulationhavebeensmearedbyadditionalGaussianofwidthequalto10%ofthenominalsimulatedresolution.Inotherwords,thecalorimeterjetPTwasmodiedby 55 ].A30%errorinthepileupcorrectionresultsinanuncertaintyoflessthan3%onthecrosssectionmeasurement.Luminosity:Thereisa6%uncertaintyinthenormalizationifthecrosssection.ThisisadirectconsequenceoftheuncertaintyinthemeasurementoftheluminosityatCDF[ 24 ].Hadrontopartonlevelcorrection:ThesystematicuncertaintyonthehadrontopartonlevelcorrectionisestimatedfromthedierenceinthepredictionsforthiscorrectionfromHERWIGandPYTHIA.HERWIGdoesnotincludemultiplepartoninteractionsinitsunderlyingeventmodel,andinsteadreliesoninitialstateradiationandbeamremnantstopopulatetheUE.PYTHIAincludesMPIaswellasthecomponentsincludedinHERWIG.PYTHIApredictsalargercorrectionatlowPTduetoMPI.ThedierencebetweentheHERWIGandPYTHIApredictionforthehadrontopartonlevelcorrection

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isaconservativeestimateofthesystematicuncertaintysincePYTHIAisknowntoreproducetheUEobservablesatCDFbetterthanHERWIG[ 18 ].Thesizeofthisuncertaintyissimilarinallrapidityregions,andisrepresentedbytheyellowbandingure 7{4 forthecentralregion.Itisontheorderof15%atlowjettransversemomentumandisnegligibleforhigherjetPT.Theuncertaintyinthejetenergyscaleleadstothelargestsystematicerrorontheinclusivejetcrosssectionandislimitedbythesimulationofthecalorimeterresponse.Thetotalsystematicuncertaintyisthequadraticsumofalluncertaintieslistedabove.Whenresultsarecorrectedtothepartonlevelthehadrontopartonlevelsystematicmustbeincluded,andthiserrorissignicantforlowjettransversemomentum.

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59 ],JETRAD[ 60 ],andFastNLO[ 61 62 63 64 ].Theseprogramsdonotincludetheeectsofhadronization,theunderlyingevent,orpartonshowers.TheyincludealldiagramsthatcontributetotheNLOcrosssection.AtNLOitispossibletohave2!2processeslikethoseshowningure 2{7 ,andtheone-loopdiagramswithequivalentnalstates.ItisalsopossibleatNLOtohave2!3processes,whereanadditionalpartonhasbeenradiatedfromoneofthelegsorpropagatorsinanyofthediagramsofgure 2{7 .Theseprogramshaveatmostthreeparticlesinthenalstateforeachevent.Thecalculationsaredoneinthemass-lesslimitwithvequarkavors(u,d,s,c,b,andtheiranti-particles),anddonotincludeanyprocessesbesidestheonesoutlinedabove.JetsfromotherprocessessuchasZbosondecaystohadrons,Wbosonplusjets,andtopquarkdecaysallcontributejetswhichareincludedintheinclusivejetmeasurement.However,theQCDcrosssectionissolargeincomparisontotheseprocessesthatitisagoodapproximation 68

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toneglectthem.Forexample,thettproductioncomponentofthejetsampleisapproximately0:01%[ 65 ].Contributionsfromotherprocessesshouldevenbesmaller[ 36 ].Thepredictionsoftheprogramslistedabovedependonmanyinputparameterssuchas:themethodforclusteringpartons,factorizationandrenormalizationscale,andpartondistributionfunctions.Thereisanuncertaintyonthetheoreticalpredictionrelatedtoeachoftheseinputchoices.AtCDF,theMidpointalgorithmismodiedforNLOpartonlevelcalculationsinordertomimicthesplittingandmergingstepoftheMidpointalgorithm(Rsep=1:3).Thevalueof1:3isconsideredareasonablechoicebyCDF;however,othervaluesforRsepclosetothisvaluearealsoreasonable.Theeect,ontheNLOcrosssectionpredictionofFastNLO,duetovaryingtheparameterRsepisshowningure 9{1 .IncreasingRsepfrom1:3to2increasesthecrosssection,whiledecreasingRsepto1(equivalenttotheKTalgorithmatNLOpartonlevel)decreasesthecrosssection.Thisvariationfromof1to2representsthemaximalreasonablerangewhichRsepcanbevaried.Thesizeofthiseectissimilarinallrapidityregions,andisnevergreaterthan5%awayfromthepredictionofRsep=1:3.Thefactorizationscale,f,isoftentakentobeonehalfofthetransversemomentumofthejet.ThisisconvenientbecausethisisthescaleusedtodeterminetheprotonstructurefunctionsbytheCTEQgroup.Usingf=PjetTorf=2PjetTgivesapproximately10%and20%smallerpredictionsforthecrosssection,respectively[ 46 ].However,ifoneextractedpartondensityfunctiontsbasedonf=PjetTorf=2PjetTinordertouseaconsistentvalueoffforthePDFandthecalculation,thedependencewouldbereduced.Thedominanttheoreticaluncertainty,ontheinclusivejetcrosssectionsatNLO,isduetotheerrorintroducedintothecalculationbytheuncertaintyon

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theprotonstructurefunctions.TheRunIinclusivejetexperiencerevealedtheneedfortoolswhichcouldquantifythiseect.Experimentalconstraintsmustbeincorporatedintotheuncertaintiesofpartondistributionfunctionsbeforetheycanbepropagatedthroughtoerrorsonthepredictionsofobservables.Inrecentyears,tools[ 66 ]suchastheHessianMethod[ 67 ],havebeendevelopedtomakeerrorpropagationtonalstateobservablespossible.TheCTEQ6.1M[ 37 ]errorsetshave41PDFmembers:acentralvalue,and40errorsetmembers.Theerrorsetmemberscorrespondto20eigenvectordirectionswhichhavebeenvariedinthepositiveandnegativedirections.TheeigenstatesareobtainedfromdiagonalizingtheHessianerrormatrix.Thematrixisobtainedbyvaryingeachparameterusedintheglobaltwithinthetolerancesoftheexperimentaldataincluded.InordertoapproximatethePDFuncertaintyonthepredictionofaphysicalobservablewiththismethod,theobservablemustbecalculatedwitheachPDFsetmember(i.e.,theobservablemustbecalculated41times).Afterthepredictionhasbeencalculatedwitheachmember,theuncertaintyontheobservableiscalculatedwiththefollowingequations[ 68 69 70 ]: X+max=vuut X)]TJ/F3 7.97 Tf -0.94 -7.89 TD[(max=vuut 10

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Figure9{1. EectofvaryingtheparameterRsepontheNLOcrosssectionpredictionofFastNLOforthecentralrapidityregion.IncreasingRsepfrom1:3to2increasesthecrosssection,whiledecreasingRsepto1(theequivalentoftheKTalgorithm)decreasesthecrosssection.

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10{1 10{5 .Ineachgure,theinclusivedierentialjetcrosssectioncorrectedtothehadronlevelisshownin(a),andtheratioofdata(correctedtothepartonlevel)totheNLOpartonlevelpredictionsofEKS,withCTEQ6.1Mpartondistributionfunctions,isshownin(b).Theyellowbandshowstheexperimentalsystematicuncertainty.Allsourcesofsystematicerrorareconsideredindependentandhavebeenaddedinquadrature.Thebluebandalsoincludesthemodelinguncertaintyassociatedwiththehadronizationandunderlyingeventcorrections.Thissystematicassociatedwiththehadrontopartonlevelcorrectionisaddedinquadraturetothetotalexperimentalsystematic.Thereisanadditional6%normalizationuncertaintyduetotheuncertaintyontheintegratedluminositywhichhasnotbeenincludedinthegures.Theuncertaintyonthetheoreticalpredictionduetoestimatederrorontheprotonstructurefunctionsisdrawninredontheratioplots.Figure 10{1(a) showstheinclusivejetcrosssectioncorrectedtothehadronlevelinthewellunderstoodcentralregionofthedetector.Theverticalaxisisplottedonalogscale.Thecrosssectionvariesbymorethaneightordersofmagnitudeasthejettransversemomentumincreasesfrom55GeV=ctoapproximately650GeV=c.ThedierencesbetweenthemeasuredresultandthetheoreticalpredictionsofEKSarenotresolvableonthelogscale.Ingure 10{1(b) ,theratioofthemeasuredcrosssectioncorrectedtothepartonleveltotheNLOpredictionisshown.Exceptionallygoodagreementisobserved.Thesystematicuncertaintyvariesfromapproximately20%atlowjetPTupto80%inthehighest 72

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transversemomentumbin.Thesystematicuncertaintyislargerthanstatisticalerrorsoneverybin.ThesystematicerrorsareslightlysmallerthanthePDFerrorsforthecentralPTrangeindicatingthatthismeasurementwillbeusefultoconstrainthepartondensities.Figure 10{2(a) showstheinclusivejetcrosssectioncorrectedtothehadronlevelinthecentralcrackregion.Thecrosssectionvariesbyapproximatelysevenordersofmagnitudeasthejettransversemomentumincreasesfrom55GeV=ctoapproximately650GeV=c.Ingure 10{2(b) ,theratioofthemeasuredcrosssectioncorrectedtothepartonleveltotheNLOpredictionisshown.Theagreementisgoodinmostbins.Thereisaslightexcessinthehighesttwotransversemomentumbins.Itispossiblethattheeectofthecrackneedsmoreattentioninthisregion,ortheseuctuationscouldbestatistical.Regardless,themeasuredresultisconsistentwiththetheoreticalpredictionwhentheerroronthepartondistributionsfunctionsisalsoconsidered.Thesystematicuncertaintyvariesfromapproximately20%atlowjetPTupto100%inthehighesttransversemomentumbin.ThesystematicuncertaintyislargerthanstatisticalerrorsoneverybinexceptforthehighestjetPTbin.PDFerrorandsystematicerrorareroughlyofthesameorderinthisrapidityregion.Thetworegionswherethecentralcalorimeterandtheplugcalorimetermodulesoverlaphavesimilarfeaturesandwillbediscussedintandem.Figures 10{3(a) and 10{4(a) showtheinclusivejetcrosssectioncorrectedtothehadronlevelinintherapidityregions0:7
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measuredresult.ThisisasmalleectandtheNLOpredictionsareconsistentwiththemeasuredresultforalljettransversemomenta.InbothregionsthesystematicuncertaintyisslightlysmallerthanthePDFuncertainty.Theseresultswillbeusefultoconstrainprotonstructurefunctions.Figure 10{5(a) showstheinclusivejetcrosssectioncorrectedtothehadronlevelforthehighestrapiditybin.Themeasurementonlygoesuptoapproximately300GeV=cinjettransversemomentumbecausethejetcrosssectionfallsomuchmorerapidlyathighrapidity.Infact,thecrosssectionvariesbyapproximatelysevenordersofmagnitudeasthejettransversemomentumvariesfrom55GeV=ctoapproximately300GeV=c.Ingure 10{5(b) ,theratioofthemeasuredcrosssectioncorrectedtothepartonleveltotheNLOpredictionisshown.ThetheNLOpredictionofEKSissystematicallyhigherthanthemeasuredcrosssectionoverthefullrangeofjetPT;however,whensystematicandPDFuncertaintiesareconsidereditisstillconsistentwiththemeasuredresult.Thesystematicerrorsarelargestinthisregionandapproach170%forthehighestbininjetPT.ThesteepershapeofthePTdistributioninthisregion,combinedwiththeadditionalsystematiconthejetenergyscaleduetothePT-dependentcorrection,areresponsibleforthisincreaseduncertainty.Evenso,thePDFuncertaintyisstillsignicantlylargerthanthesystematicerrorsformostjetPT.Theresultinthisrapidityregionwillleadtothestrongestconstraintonpartondensityfunctionsoutofalltheregionsincludedinthismeasurement.ThereisatrendinthedataforthelastbintouctuatehigherthantheNLOprediction.Thereisasimpleexplanationforthissystematiceect.Thereareveryfeweventsinthelastbins.Ifthisnumbeructuateddownsignicantlythentherewouldnotbeeventsinthebin,andthereforitwouldnotbeshownonthegure.Bythisreasoning,binswithveryfeweventsaremorelikelytobe

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statisticaluctuationswhichaddedtothebincontentsratherthanuctuationswhichsubtractedfromthecontentsofthebin.Thecrosssectionsforthevariousrapidityregionsarepresentedongure 10{6 wheretheyhavebeenscaledbydierentfactorssotheywouldbedistinguishablewhenplottedonthesameaxis.Theregion0:7
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(b)Figure10{1. MeasuredinclusivejetcrosssectionwiththeMidpointalgorithmintheregion0:1
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(b)Figure10{2. MeasuredinclusivejetcrosssectionwiththeMidpointalgorithmintheregionjYj<0:1.Thedistributionforthehadronlevelcrosssectionisshowningure 10{2(a) .TheratioofdatacorrectedtothepartonleveltothepartonlevelpQCDpredictionofEKSisshowningure 10{2(b)

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(b)Figure10{3. MeasuredinclusivejetcrosssectionwiththeMidpointalgorithmintheregion0:7
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(b)Figure10{4. MeasuredinclusivejetcrosssectionwiththeMidpointalgorithmintheregion1:1
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(b)Figure10{5. MeasuredinclusivejetcrosssectionwiththeMidpointalgorithmintheregion1:6
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Figure10{6. MeasuredinclusivejetcrosssectionatthehadronlevelwiththeMidpointalgorithmforallrapidityregions.

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Figure10{7. RatioofthemeasuredinclusivejetcrosssectionatthepartonlevelwiththeMidpointalgorithmtothepQCDpredictionofEKSinthedierentrapidityregions.

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71 27 ].Onlyrecentlyhasthisalgorithmbeenappliedtojetsinthemorechallenginghadron-hadroncolliderenvironmentoftheTevatron:studiedrstbyD0[ 72 ]inRunI,andmorerecentlybyCDFinRunII.D0reportedonlymarginalagreementwithNLOprediction;however,themorerecentCDFresultreportsgoodagreement.TheratiooftheCDFresulttotheNLOpredictionoftheJETRADprogramisshowningure 11{1 [ 73 ].Inthegure,theJETRADresulthasbeencorrectedtothehadron-leveltoincludetheeectsofjetfragmentationandtheunderlyingevent.Thiscorrectionisshowningure 11{2 [ 73 ].Figure 11{3 showstheratiooftheinclusivejetcrosssectionmeasurementforjetsclusteredwiththeKTalgorithmtotheresultforjetsclusteredwiththeMidpointalgorithm(black).Onlystatisticalerrorsareshown.Thedatausedinthe 83

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9{1 ).ThePTdependenceandmagnitudeofthemeasuredratioobservedinthedataisclosetotheNLOprediction.ThisresultprovidescondencethatthealgorithmdenitionsareconsistentatcalorimeterlevelandNLOpartonlevel.Figure 11{3 showstheratioofthehadrontopartonlevelcorrectionderivedwiththeKTalgorithm(inversionofthedistributionshownin 11{2 )totheonederivedwiththeMidpointalgorithm(gure 7{4 ).ThesecorrectionswerederivedfromPYTHIATUNE-A,asdescribedinsection 7 .Themultiplicativecorrectionsarebothlessthanone,sotheratiomeansthatthesizeofthecorrectionderivedwiththeKTalgorithmislarger(i.e.,fartherawayfromone)thanthecorrectionderivedwiththeMidpointalgorithm.ThisresultindicatesthattheKTalgorithmisslightlymoresensitivetotheunderlyingevent.TheconsistencyoftheKTinclusivejetcrosssectionmeasurementwithNLOpredictions,combinedwithonlyasightlylargerunderlyingeventcorrection,supportstheuseofKT-typealgorithmsattheTevatronandhadron-hadroncollidersofthefuture,providedthatonehasagoodunderstandingoftheUE.The

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agreement,betweentheNLOpredictionandthemeasuredresultfortheratioofthethetwojetclusteringalgorithms,addscredencethatthejetdenitionsaredenedconsistentlyatthepartonanddetectorlevels. Figure11{1. MeasuredinclusivejetcrosssectionwiththeKTalgorithmintherapidityregion0:1
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Figure11{2. Parton-to-hadroncorrectionusedbytheKTinclusivejetcrosssectionanalysistocorrecttheNLOpredictiontothehadronlevel. Figure11{3. RatiooftheinclusivejetcrosssectionmeasuredwiththeKTalgorithmtothatmeasuredbytheMidpointalgorithm(black).ThepredictionofthisratiofromtheNLOprogramFastNLOisalsoshowninthegure(blue).

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Figure11{4. RatioofthehadrontopartonlevelcorrectionderivedwiththeKTalgorithmtothatderivedwiththeMidpointalgorithm.Themultiplicativecorrectionsarebothlessthanone.ThecorrectionderivedwiththeKTalgorithmislarger(fartherawayfromone).ThecorrectionswerederivedfromPYTHIATUNE-A.

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74 ].UncertaintiesofthisorderdonotdetermineiftheHiggsbosonwillbediscoveredornot;however,theydolimittheprecisionwithwhichthecouplingoftheHiggstootherparticlescanbemeasuredafteritsdiscovery.Inallrapidityregions,andoverthefullrangeofjetPTexcludingthehighestbins,thesystematicuncertaintyislargerthanthestatisticalerroronthismeasurement.Thismeansthattheprecisionofthemeasurementislimitedbysystematicsandnotbystatistics.Simplyobtainingmoredatawillnotimprovethe 88

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accuracyofthemeasurementsignicantly.ThesystematicuncertaintyduetothejetenergyscalecanbereducedbyimprovingthesimulationoftheCDFdetector.Currently,CDFisworkingtoimprovethedetectorsimulation.Thiswillleadtoareduceduncertaintyonthejetenergyscale,andtoasmallersystematicerroronfutureinclusivejetmeasurements.ThereareaspectsoftheinclusivejetcrosssectionwhichhaveimplicationsfortheLHC.ItisimportantfortheLHCtochooseaconsistentsetofjetalgorithmsearlyonbetweenexperimentsandstickwiththesejetdenitionsthroughoutthelifeoftheexperiment.Nojetalgorithmisperfect,andsomealgorithmsarebettersuitedforcertaintypesofanalyses.Makingastrongeorttounderstandtheawsofanalgorithmmaybeabetterapproachthantotrytocorrectalloftheimperfectionsofanalgorithmafteritisinuse.Perhaps,thislessonshouldbelearnedfromTevatronexperiments,whereaneorttoimprovetheMidpointalgorithmonlyintroducedanewsetofimperfectionsandledtothetwocolliderexperimentsusingdierentjetdenitions.ThefactthatnosignicantdeviationsfromthepredictionsofpQCDhavebeenobservedintheinclusivejetcrosssectionisausefulresultforLHCexperiments.IntheearlyrunningoftheLHCitislikelythatmanydeviationsfromthestandardmodelpredictionswillbeobserved.Anydeviationsobservedinjetswithtransversemomentumbelow600GeV=carelikelydetectoreectswhicharenotcompletelyunderstood,ratherthansomesignofphysicsbeyondthestandardmodel.Inotherwords,condenceinQCDpredictionsatthisenergyscaleprovidesakinematicregionwhereitissafetocalibratethedetector.SomeofthetechniquesappliedinthisanalysiswillalsobeusefulattheLHC.Forexample,themethodofdijetPTbalancingandthebisectormethodcanbothbeusedtounderstandthedetectorresponse,andtonetunethedetectorsimulationforthe

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LHCexperiments.Ingeneral,theexperiencegainedattheTevatronimprovestechniquesandtoolsthatwillbeappliedattheLHC.Sometimes,itisassumedthattherangeofdiscoveryatanexperimentisonlylimitedbytheintegratedluminosityandcenterofmassenergyprovidedbytheaccelerator.Theinclusivejetcrosssectionprovidesanexampleinwhichtheprecisionisnotlimitedbystatistics.Inmanycasesthelevelatwhichthedetectorisunderstoodisjustascrucialtomakingimportantdiscoveriesastheperformanceoftheaccelerator.Overthenextfewyears,thephysicistsoftheLHCexperimentswillbeworkingtounderstandtheirdetectors.Thiseortshouldleadtoexcitingdiscoveriesregardingthemostfundamentallawsoftheuniverse.Theinterpretationoftheseresultswouldnotbepossiblewithoutthewealthofpreviousmeasurementsofstandardmodelproperties,suchastheinclusivejetcrosssection.TheresultsfromtheLHCwilldenethefutureofthefascinatingeldofhighenergyparticlephysics,andwillbringhumanbeingsonestepclosertoacompleteunderstandingoftheuniverseinwhichtheylive.Hopefully,itwillbeagiantstep!

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A{1 .AdetaileddescriptionoftheCDFtriggerisavailableinreference[ 48 ].EachofthethreelevelsoftheCDFjettriggerarebrieydescribedbelow:Level-1:Thelevel-1triggerhas132nstodecideifaneventwillpassontolevel-2ornot.ThistimeconstraintlimitsthehardwareaccesstoinformationfromthecalorimeterandtheCOT.Thelevel-1jettriggersrelyonapplyingthresholdstothesumofhadronicandelectromagneticenergyinindividualcalorimetertowers.Becausejetsaremadeofmorethenonecalorimetertower,thethresholdsatlevel-1mustbelessthanthejetenergyrequirementsatthehighertriggerlevels.Asshowningure A{1 STT5(SingleTriggerTower5),andSTT10(SingleTriggerTower10)arethe2triggersrelevantforjetsatlevel-1.STT5andSTT10require5GeVand10GeVinasingletower.Thelevel-1triggermustreducethe1:7MHzbunchcrossingratetothelevel-2 91

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FigureA{1. Triggerowdiagramforthefourjettriggers.Prescalesaregiveninparentheses.Ifmultipleprescalesareshownthentheprescaleforthattriggerwaschangedduringthedatatakingperiod. designrateof40kHz.Forthelevel-1STT5triggeraprescaleisrequiredinadditiontotheabovementionedrequirementstomakethisratereduction.AtCDFjettriggersareprescaledbyacceptingaxedfractionofeventswhichpassthetriggerrequirements.Theprescalerequiredmayneedtobechangedbasedonluminosity.Duetoincreasesintheinstantaneousluminosity,thejet20andjet50L1andL2prescaleshavechangedduringtheperiodofdatatakingforthisanalysis.Thedetailsoftheseprescalechangesaregiveninsection A.2 .Level-2:Atlevel-2a\nearestneighbor"clusterndingalgorithmisusedtoclusteradjacentcalorimetertowerswithnon-trivialenergy.Clustersareformedaroundallseedtowerswithenergyaboveacertainthreshold.Allneighboringtowerswithenergyaboveasecond,slightlylowerthreshold,arethenaddedtotheclusteruntilnotowersadjacenttotheclusterhaveenergyabovethissecondthreshold.ThetotalEMandHADenergiesarerecordedforeachclusteraswellasthenumberoftowersandthe(;)oftheseedtower.Thecalorimetertowersaresummedintotriggertowers,whoseenergiesareweightedbysin.Asshowningure A{1 ,the4level-2jettriggersareCL15,CL40,CL60,andCL90forjet20,50,70,and100respectively.Itisthejoboftheselevel-2triggerstoreducethe40kHzratepassedbylevel-1toapproximately300Hzwhichcanbehandledbythelevel-3system.CL15,andCL60haveprescalesthathavevariedoverthedataincludedinthisanalysisandaregiveninthetriggerowdiagram.Seesection A.2 fordetailsoncalculatingtheireectiveprescales.

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4 andinmoredetailinreference[ 36 ].The300Hzpassedfromlevel-2mustbereducetoapproximately75Hzbeforethedataiswrittentotape. A{1 .SincetheL1triggerbecomesecientatmuchlowerPTthanL2orL3(seegures A{2 and A{3 ),itscontributiontothetriggereciencyinthebinsofinterestisnegligible.Therefore,L2andL3eciencieshavebeencombinedandtheresultsareshowningures A{4 A{7 .Forcompleteness,thetriggerturn-oncurves

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werettothefunction 1+e)]TJ/F3 7.97 Tf 6.58 0 TD[(P0(PT+P1);(A{2)andthetparameters(P0andP1)areincludedonthegures. TableA{1. Datasamplesusedtostudytheeciency(middle)ofeachtrigger.Thedatasamples,whicheachtriggercontributes,toisalsoshown(right). TriggerSampleusedDatasetaected L1:ST05HighPTJet20andJet50L2:J15ST05Jet20L3:J20ST05Jet20L2:J40Jet20Jet50L3:J50Jet20Jet50L1:ST10Jet20Jet70andJet100L2:J60Jet50Jet70L3:J70Jet50Jet70L2:J90Jet70Jet100L3:J100Jet70Jet100 A{8 .Thestudyofprescalesforthejet20andjet50triggersarecomplicatedbythefactthatthesamplesareprescaledatmultiplelevelsandtheseprescaleshavenotremainedconstantduringthethedataperiodconsidered.Thejet20andjet50L1prescalechangedfrom20to50atrunnumber184444.Thejet20L2triggerincreasedfrom12to25afterrun153068,whilethejet50L2isonlyprescaledforrunsafterrun194917withinitialinstantaneousluminositygreater

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than1501030cm)]TJ/F4 7.97 Tf 6.58 0 TD[(2sec)]TJ/F4 7.97 Tf 6.58 0 TD[(1.Inordertodealwiththesechangesandtotreatalleventsequally,an`eective'prescalewascalculated.Theeectiveprescalewascalculatedbyluminosity-weightingtheinverseofprescalefactors.Eq. 6{1 fromChapter 6 clariesthismethod.Thismethodrequirescalculatingtheintegratedluminosityseparatelyforeachprescaleperiod.Forjet20theeectiveprescaleis 2012+259:7 2025+659:9 5025=776:8:(A{3)Similarly,forjet50theeectiveprescalecanbecalculatedas 201+639:3 501+20:7 505=34:(A{4)Asacrosscheck,theeectiveprescalewerealsobeobtainedfromtheratioofjet70tojet50forthejet50trigger,andtheratioofjet70tojet20forthejet20trigger.InFigure A{8 thesedistributionsarettoaconstantvalueintheregionwherebothrelevantdistributionsareecient.Inordertotakethejet70prescaleintoaccounttheseratiosarescaledbyafactorofeight.Theprescalesobtainedarealsoshowninthegures.Theeectiveprescalesobtainedfromthetwomethodsareconsistenttobetterthan1:0%.

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FigureA{2. TriggerecienciesasfunctionofuncorrectedjetPTfortheL1-ST5trigger.Thisisthelevel-1triggerforthejet20andjet50triggersamples.

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FigureA{3. TriggerecienciesasfunctionofuncorrectedjetPTfortheL1-ST10trigger.Thisisthelevel-1triggerforthejet70andjet100triggersamples.

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FigureA{4. TriggerecienciesasafunctionofjetPTforthejet20trigger.L2andL3turn-oncurveshavebeencombinedtocalculatetheeciencyofthejet20trigger.

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FigureA{5. TriggerecienciesasafunctionofjetPTforthejet50trigger.L2andL3turn-oncurvesarecombinedtocalculatetheeciencyofthejet50trigger.

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FigureA{6. TriggerecienciesasafunctionofjetPTforthejet70trigger.L2andL3turn-oncurvesarecombinedtocalculatetheeciencyofthejet70trigger.

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FigureA{7. TriggerecienciesasafunctionofjetPTforthejet100trigger.L2andL3turn-oncurvesarecombinedtocalculatetheeciencyofthejet100trigger.

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(b) (c)FigureA{8. Crosssectionratioofthejet100tojet70datasamples, A{8(a) ;theratioofthejet70tojet50samplesmultipliedby8, A{8(b) ;andtheratioofthejet70tojet20samplesmultipliedby8, A{8(c) .Thetsareconsistentwiththejet70,jet50andjet20prescalesobtainedfromtheluminosityweightingmethodtobetterthan1%.

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55 75 ].Atriggerjetisrequiredtobeinthewellunderstoodcentralregionofthecalorimeter(0:2
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22,2004.ForthedataandMCcomparisons,dijetMCsamplesgeneratedwithPYTHIAtuneAandprocessedwiththe5.3.3releaseCDFsimulationareused.Thefollowingeventselectionwasappliedtothedatausedinthisstudyunlessstatedotherwise:Thegoodrunlistfor\QCDnosilicon"wasappliedforgoodrunselection PET2:7radiansAveragePToftheleadingtwojets,PaveTmin
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B{1 showstheversusDforjetsclusteredbyMidpointwithRcone=0:7aftertherelativecorrectionsderivedfromjetsclusteredbyJetCluhavebeenapplied.TherelativecorrectionsaredeterminedtomaketheversusDatat1,thereforedeviationsoffromoneindicatestheimperfectionofthecorrections.ItisclearthatthecorrectionsderivedfromjetsclusteredbyJetCludoareasonablejobinjetsclusteredbyMidpointatPT55GeV/c.Intheplotfor25PaveT<55GeV/c,thedistributionisreasonablyatuptojDj=2;however,startstodeviatefromoneupwardwithincreasingjDjatjDj>2.ThisrisingtrendatjDj>2ismoresignicantindatathaninMC.Figure B{2 showstheversusDplotscombiningbinstoimprovestatisticsaftertherelativecorrectionshavebeenapplied.PointsareshownforthedefaultPTcutonthethirdjet(black),andatighter(red)andlooser(blue)cuts.Inthisgure,thesystematictrendofvsDisbetterpresented.Intheplotsfor25PaveT<55GeV/c,whilesignicantvariationfromoneisobservedinthehighDregions,therelativecorrectionsdoareasonablejobmakingdijetbalancingdistributionsatintheregionofjDj<2.TherelativecorrectionsderivedfromjetsclusteredbyJetCluwithRcone=0:7arevalidforjetsclusteredbyMidpointwithRcone=0:7uptojdj=2.ThetrendobservedatjDj>2andPT<55GeV/cmaybeattributedtothefactthatthephysicalsizeofcalorimetertowersissmallerintheregionofjDj>2thaninjDj<2.However,eventhoughthephysicalsizeissmaller,individualcalorimetertowersoccupylarger(Y;)spacesthantowersinjDj<2.

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JetsatjDj>2maybesensitivetodetailsofthejetclusteringalgorithm,especiallysplittingandmergingprocedures,andinMCjetclusteringintheforwardregionwouldbesensitivetothelateralproletuningofparticleshowersinthecalorimeters.AnalysesthatusejetsmuchabovejDj>2willrequiremorestudies. (b) (c) (d)FigureB{1.

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C{1 thefollowingobjectsaredenedforusewiththebisectorstudyandareconstructedforthejetdataandtheMCatthedetectorlevel: 108

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C{2 C{5 showtheresultofthebisectorstudyinallotherrapidityregions.InthetworegionswheretheCDFdetectorsimulationisseentounderestimatethejetenergyresolution(0:7
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FigureC{1. Bisectorvariables,describedinthetext,arelabeledinthediagramofthetransverseplane.PT?isdenedtobethecomponentofthesumofthejetPTalongthebisectoraxis. wheremcreferstothenominalresolutionoftheCDFsimulationappliedtotheMC.TheresolutionoftheMCwasstudiedineachrapidityregion.ForbinsofPhadT,thedistributionofPhadT)]TJ/F2 11.95 Tf 12.22 0 TD[(PcalT(hadronanddetectorleveljetswerematchedbasedonseparationinY)]TJ/F2 11.95 Tf 12.15 0 TD[(space)wastbyaGaussianandthewidthofthetwastakenasmcforeachPhadTbin.BysmearingtheMCjetsintheregions0:7
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Fortheregion1:1
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FigureC{2. ResultsofthebisectorstudyfortherapidityregionjYj<0:1.InthisYregiontheCDFdetectorsimulationsmearsthejetenergytoomuch.

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FigureC{3. Resultsofthebisectorstudyfortherapidityregion0:7
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FigureC{4. Resultsofthebisectorstudyfortherapidityregion1:1
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FigureC{5. Resultsofthebisectorstudyfortherapidityregion1:6
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FigureC{6. ModicationstothebincorrectionstoaccountforimperfectionsinsimulationofthejetPTresolution.

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B ,Equations B{1 and B{2 .BecauseaPTindependentcorrectionisappliedfortherelativecorrection,thedijetbalancecorrectiontotheMCasafunctionofjetPTmaystillbeneededinsomerapidityregions.Figures C{7 and C{8 showtheresultsofthedijetbalancestudyaftertherelativecorrectionsareapplied.TherelativecorrectionsmakethedataandMCdijetbalanceagreeto0.5%orbetterfor(jYj<1:1).Theenergyscaleinthehigherrapidityregionsisnotaswellunderstoodasthecentralregionofthecalorimeter,andthereforemustbetreatedwithspecialcare.StatisticsforthedataandPYTHIAsamplesarelimitedathighjetPTintheseregions,andthismakesitverydiculttocheckthevalidityofthesimulationathighPT.Ingure C{8 arstorderpolynomialisusedtotthedijetbalanceratiobetweendataandMCatlowPT.ThetisthenmodiedforthehigherPTpoints.Thedottedlinesrepresentwhatistakenasanadditionaluncertaintyonthejetenergyscaleinthisregionduetothelimitedconstraintonthetsforthedijetbalancecorrection.Thiscorrespondstoasignicantcontributiontothesystematicuncertaintyonthecrosssectioninthehighrapidityregions(40%).InaneorttoimprovestatisticsathighPTforthisstudy,thethirdjetcutwasrelaxedfrom10GeV=cto10%ofthemeanPTofthedijetpair.Unfortunately,

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theerrorisnotgreatlyreducedfortherelaxedcut.ThebinerrorisproportionaltotheRMSwhichincreaseswitharelaxedthirdjetcut.Eventhoughstatisticsincreasedbyafactoroftwobyrelaxingthecut,thebinerrorsarelargelyunchanged.ThesamePTdependenceintheratioofdatatoMCinthehigherrapidityregionsisobserved.ThecorrectionstotheMCdijetbalanceforeachrapidityregionwerecalculatedwiththestandard10GeV=cPTcutonthethirdjet,andaresummarizedintable C{1 TableC{1. DijetbalancecorrectionappliedtothePYTHIAMCsimulationforeachrapidityregion.

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FigureC{7.

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FigureC{8.

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PAGE 138

RobertCraigGroupwasborninColumbia,SC,onMay28,1977.Hegrewupinrurallow-countrySouthCarolinainthetownofAllendale,andattendedhighschoolinBarnwell,SC.Hehastwolovingparents,CindyandBob,andaslightlyyoungerbrother,Eric.WhileatBarnwellHighSchoolCraigdatedNicoleL.Tyler,andletteredinacademicsandgolf.CraigattendedErskineCollegeinDueWest,SC.Heservedaspresidentofthemathematicshonorsociety(SigmaPiSigma)andalsovice-presidentandthenpresidentofthePhilomatheanLiterarySociety.CraigearnedaB.A.degreeinphysicswithaminorinmathematicsfromErskine.WhileaJunioratErskine,CraigengagedinasummerresearchprojectforundergraduatesatFloridaStateUniversitystudyingexperimentalnuclearphysics.ThisexperiencecontributedtohischoicetoattendFloridaStateforgraduateschooldirectlyafterhisgraduationfromErskineCollege.CraigspenttwosummersattheFermiNationalAcceleratorLaboratory(FNAL)andstudiedhighenergyparticlephysicswhileatFSU.HereceivedhisMSinphysicsfromFSUandthentooktimeofromschooltobereunitedwithNicole.TheyweremarriedinAugustof2002.Oneweeklater,Craigreturnedtograduateschool,thistimeattheUniversityofFlorida.AtUF,Craigtooktheupperlevelgraduatecourseshisrsttwosemesters.Aftercompletionofthecourserequirements,heengagedhimselfcompletelywithresearchforthenextthreeyears.Hishardworkanddeterminationledtothisdissertation.CraigwillcontinuehisresearchatFNALasaresearchassociatenextyear.Helooksforwardtothechallengesandrewardsthatlifewillsurelyprovide. 125


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Table of Contents
    Title Page
        Page i
        Page ii
    Dedication
        Page iii
    Acknowledgement
        Page iv
        Page v
        Page vi
    Table of Contents
        Page vii
        Page viii
    List of Tables
        Page ix
    List of Figures
        Page x
        Page xi
        Page xii
    Abstract
        Page xiii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
    Phenomenology of quantum chromodynamics
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
    Experimental apparatus
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
    Jet definition
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
    Inclusive jet measurements
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
    Data sample and event selection
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
    Jet corrections
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
    System uncertainties
        Page 65
        Page 66
        Page 67
    Theoretical predictions and uncertainties
        Page 68
        Page 69
        Page 70
        Page 71
    Results
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
    Comparison with the KT algorithm
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
    Conclusions
        Page 88
        Page 89
        Page 90
    Appendix A: Jet triggers at CDF
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
    Appendix B: Relative corrections
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
    Appendix C: Simulation of detector response and resolution
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
    References
        Page 121
        Page 122
        Page 123
        Page 124
    Biographical sketch
        Page 125
Full Text










MEASUREMENT OF THE INCLUSIVE JET CROSS SECTION USING THE
MIDPOINT ALGORITHM IN RUN II AT THE COLLIDER DETECTOR AT
FERMILAB (CDF)















By

ROBERT CRAIG GROUP















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006
































Copyright 2006

by

Robert Craig Group


































I dedicate this work to all who devote their careers to the study of science and the

selfless pursuit of knowledge.















ACKNOWLEDGMENTS

I would like to thank my wife, Nicole, for her understanding of my time away

from home during this work. Her encouragement and confidence in my ability

carried over into my professional demeanor; my success is directly correlated with

her support.

I would like to thank my advisors for their guidance over the last four years.

Professor Field's tireless dedication to answer the toughest questions of QCD kept

me working hard to keep up. Professor Matchev's requirement to give many public

presentations, and travel support from his Outstanding Junior Investigator award,

helped me to conquer my fear of public speaking. Also, partial support for my

stipend this semester from Dr. Matchev, made it possible for me to concentrate

on this dissertation and obtaining employment. Without professor Field's and

professor Matchev's combined efforts on my behalf, I do not know where I would

have ended up after UF. I also want to thank Dr. Kenichi Hatakeyama, of the

The Rockefeller University, for his patience while teaching me how to do an

experimental analysis. Without his constant collaboration, I would have had

no chance to complete a CDF analysis without being located at Fermilab. I am

proud of the work we did together over these last two years. Finally, I want to

thank Professor Field, Professor Matchev, and Dr. Hatakeyama for writing the

recommendation letters that helped me obtain more employment options than I

could possibly have expected.

Professor Paul Avery and Dr. Jorge Rodriguez allowed me to work with the

UF Grid computing project my first summer at UF. This allowed my wife and me

to move to Gainesville early and buy a house. I am thankful for this. I appreciate









the patience and time extended to me by Dr. Dimitri Bourilkov. The technical

skills he helped me to develop will be invaluable to me in my future.

The High Energy Theory Group supported me for travel to the TeV4LHC

workshop, and the TASI summer school. These experiences helped me build

exposure and confidence in my place in the high energy physics community. The

High Energy Experimental Group gave me partial support this last semester so

that I could search for a job, and finish my degree. I appreciate this support and

hope to take full advantage of the freedom it allows me. The HEE group also

made it possible for me to spend three months at FNAL last Winter to fulfill my

Ace shift responsibility. I took advantage of this time to make great progress on

my research, and meet many of the people at CDF I had only worked with over

email. I am truly thankful for this opportunity. I also thank Yvonne Dixon for

her patience with travel paperwork. She .i.l i-x made it easy to travel. The four

year full support from the Alumni Fellowship offered to me by the University of

Florida allowed me to pursue additional research projects outside of the scope of

this dissertation. The generous support from all of the above groups has made my

experience at UF an exciting and successful one.

I also have had the benefit of many great friends over my graduate career

whom I have been able to share my experiences with. At FSU, Jose Lazoflores,

and Jorge O'Farrill helped me find my place in the department and have remained

good friends over the years. At UF, Wayne Bomstad, Ethan Siegal, Leanne Duffy,

Alberto Cruz, Bo.1 .iv Scurlock and many others contributed indirectly to my

success. The selfless support extended to me by other students in the UF advanced

graduate program has been exceptional.

I want to thank my mother and father for alvb-- i being there. Without their

encouragement and support throughout my lifetime, I could never have dreamed









I would obtain a PhD in physics. They ahv-- said I was capable of anything I

wanted to do, and never showed surprise when I was successful.















TABLE OF CONTENTS


ACKNOWLEDGMENTS ..........

LIST OF TABLES ..............

LIST OF FIGURES .............

ABSTRACT ..... ............

CHAPTER

1 INTRODUCTION ...........

2 PHENOMENOLOGY OF QUANTUM


page

iv

ix

x

xiii


CHROMODYNAMICS


2.1 The Particle Content of the Standard Model .......
2.2 Feynman Rules of QCD ..................
2.3 Color Confinement and Asymptotic Freedom .......
2.4 The Factorization Theorem ................
2.5 Jet Production Cross Sections ...............
2.6 Structure of Hadronic Collisions . . . .

3 EXPERIMENTAL APPARATUS . . . .

3.1 Coordinates and Conventions . . . .
3.2 The Tevatron . . . . . .
3.3 The CDF Detector . . . . . .

4 JET DEFINITION . . . . . .

4.1 The CDF Midpoint Jet Clustering Algorithm . .
4.2 Other Jet C'!u i1i. ng Algorithms . . . .
4.3 Jet Definition Issues . . . . .

5 INCLUSIVE JET MEASUREMENTS . . . .

6 DATA SAMPLE AND EVENT SELECTION . . .

7 JET CORRECTIONS . . . . . .

7.1 Monte Carlo Simulation . . . . .
7.2 Relative Correction . . . . .
7.3 Pileup Correction . . . . . .









7.4 Absolute Correction . . . . . . 61
7.5 Unfolding Correction . . . . . . 62
7.6 Hadron to Parton Correction . . . . ... 63

8 SYSTEMATIC UNCERTAINTIES . . . . . 65

9 THEORETICAL PREDICTIONS AND UNCERTAINTIES . ... 68

10 RESU LT S . . . . . . . . 72

11 COMPARISON WITH THE KT ALGORITHM ............. 83

12 CONCLUSIONS ............................... 88

APPENDIX

A JET TRIGGERS AT CDF .......................... 91

A.1 Jet Ti i-.- i Efficiency ... .. .. .. ... .. .. .. .. ... .. 93
A .2 Jet Ti l-.-, i Prescales . . . . . . 94

B RELATIVE CORRECTIONS ........................ 103

B.1 Event selection for relative correction studies . . . 103
B.2 Relative Corrections with the Midpoint Algorithm . .... 105

C SIMULATION OF DETECTOR RESPONSE AND RESOLUTION 108

C.1 Jet Energy Resolution: Bisector Method . . 108
C.2 Jet Energy Response: Dijet PT Balance ... . . 117

REFERENCES . . . . . . . . 121

BIOGRAPHICAL SKETCH . . . . . . . 125















LIST OF TABLES


Table page

2-1 Some properties of the quark content of the standard model. . 6

2-2 Some properties of the lepton content of the standard model. . 6

2-3 Some properties of the gauge boson content of the standard model. 7

A-1 Data samples used to study the efficiency of each jet tri -.-r . .. 94

B-1 Selection cuts applied to require the dijet event topology used to derive
the relative corrections . . . . . . . 104

C-1 Dijet balance correction applied to the PYTHIA MC simulation for each
rapidity region . . . . . . . . 118















LIST OF FIGURES
Figure page

2-1 Feynman rules for QCD in a covariant gauge. . . . 8

2-2 One-loop Vacuum polarization diagram of QED. .... . 11

2-3 One-loop vacuum polarization diagram of QCD. .... . 12

2-4 One-loop Vacuum polarization diagram of QCD which arises from the
gluon self coupling . . . . . . . 12

2-5 Running of the the QED and QCD coupling constants. . .... 13

2-6 Schematic of the QCD factorization theorem. ..... . 16

2-7 Diagrams which contribute to leading order jet production at a hadron
collider . . . . . . . . . 17

2-8 Diagrams which contribute to e+e- annihilation to hadrons at NLO. 18

2-9 Components of a typical hadron collider event at the parton level. . 19

2-10 Description of the different levels of a jet event at CDF. . .... 22

3-1 A schematic of the accelerator complex used for Run II at Fermilab. 26

3-2 Development of an electromagnetic shower. . . . . 30

3-3 Collider Detector at Fermilab (CDF). . . . . . 34

3-4 Longitudinal view of the CDF II Tracking System and plug calorimeters. 35

4-1 Cone algorithm sensitivity to soft radiation. . . . . 42

4-2 Dark towers observed by the original Midpoint algorithm . . 43

5-1 Uncertainty on the parton distribution function for the up-quark and
the gluon at Q = 500 GeV . . . . . . 46

5-2 Dominant process in deep inelastic scattering experiments. . . 47

6-1 Jet yield distributions as a function of PT in the central region. . 53

6-2 Measured raw jet cross section for the five rapidity regions. . . 54

7-1 Difference between calorimeter jet PT and hadron level jet PT .. . 55









7-2 Flow diagram for the jet corrections used in the inclusive jet analysis. 58

7-3 Degree of dijet balance observed in the CDF calorimeter. . .... 60

7-4 Hadron to parton level correction applied in the central region. .. . 64

9-1 Effect of varying the parameter Rsep ..................... 71

10-1 Inclusive jet cross section corrected to the hadron level and ratio to the
NLO pQCD predictions for the rapidity region 0.1 < |Y < 0.7 . 76

10-2 Inclusive jet cross section corrected to the hadron level and ratio to the
NLO pQCD predictions for the rapidity region Y < 0.1 . . 77

10-3 Inclusive jet cross section corrected to the hadron level and ratio to the
NLO pQCD predictions for the rapidity region 0.7 < Y| < 1.1 .. . 78

10-4 Inclusive jet cross section corrected to the hadron level and ratio to the
NLO pQCD predictions for the rapidity region 1.1 < |Y < 1.6 . 79

10-5 Inclusive jet cross section corrected to the hadron level and ratio to the
NLO pQCD predictions for the rapidity region 1.6 < |Y| < 2.1 ... 80

10-6 Measured inclusive jet cross section at the hadron level with the Midpoint
algorithm for all rapidity regions. . . . . . 81

10-7 Ratio of inclusive jet cross section corrected to the partron level to the
NLO pQCD predictions for all rapidity regions ... . 82

11-1 Measured inclusive jet cross section with the KT algorithm in the central
rapidity region . . . . . . . . 85

11-2 Parton-to-hadron correction used by the KT algorithm inclusive jet cross
section analysis . . . . . . . . 86

11-3 Ratio of the inclusive jet cross section measured with the KT algorithm
to that measured by the Midpoint algorithm. . . . 86

11-4 Ratio of the hadron to parton level correction derived with the KT algorithm
to that derived with the Midpoint algorithm. . . . . 87

A-1 T i flow diagram for the four jet triggers. . . . 92

A-2 T i--, i efficiencies as function of uncorrected jet PT for the L1-ST5 tri ,v--vr. 96

A-3 Tii i efficiencies as function of uncorrected jet PT for the L1-ST10 trigger. 97

A-4 Ti:--:. i efficiencies as a function of jet PT for the jet20 trigger. .. . 98

A-5 Tii-:.-, i efficiencies as a function of jet PT for the jet50 trigger. .. . 99









A-6 Tii-.-, i efficiencies as a function of jet PT for the jet70 trigger. . 100

A-7 T i--.- efficiencies as a function of jet PT for the jetl00 t-ri. ..... .101

A-8 Cross section ratios used as a cross-check of the jet20, jet50, and jet70
prescales . . . . . . . . 102

B-1 3 versus UTD distributions for jets clustered by the Midpoint algorithm
after applying the relative corrections derived from jets clustered by JetClu. 106

B-2 3 versus UTD for jets clustered by the Midpoint algorithm using a larger
T]D inning . . . . . . . . 107

C-1 Bisector variables are labeled in the diagram of the transverse plane. 110

C-2 Results of the bisector study for the rapidity region |Y| < 0.1 . 112

C-3 Results of the bisector study for the rapidity region 0.7 < Y| < 1. 113

C-4 Results of the bisector study for the rapidity region 1.1 < |Y| < 1.6. 114

C-5 Results of the bisector study for the rapidity region 1.1 < |Y| < 1.6. 115

C-6 Modifications to the bin corrections to account for imperfections in simulation
of the jet PT resolution . . . . . . 116

C-7 3 as functions of PTMEAN observed in data and MC for the rapidity region
|Y | < 1.1. . . . . . . ... 119

C-8 3 as functions of PTMEAN observed in data and MC for the rapidity region
|Y | > 1.1. . . . . .. ... . .... . 120















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MEASUREMENT OF THE INCLUSIVE JET CROSS SECTION USING THE
MIDPOINT ALGORITHM IN RUN II AT THE COLLIDER DETECTOR AT
FERMILAB (CDF)

By

Robert Craig Group

December 2006

C'! ,i: Richard D. Field
Major Department: Physics

A measurement is presented of the inclusive jet cross section using the

Midpoint jet clustering algorithm in five different rapidity regions. This is the

first analysis which measures the inclusive jet cross section using the Midpoint

algorithm in the forward region of the detector. The measurement is based on more

than 1 fb-1 of integrated luminosity of Run II data taken by the CDF experiment

at the Fermi National Accelerator Laboratory. The results are consistent with with

the predictions of perturbative quantum chromodynamics.















CHAPTER 1
INTRODUCTION

Since the dawn of intellectual thought, human beings have questioned the

origin of the matter that composes the world in which they live. The earliest

well documented attempts to explain the physical world were constructed by

philosophers. Democritus (ca 460-370 BC) postulated that invisible and indivisible

,Ii. is:" made up everything around us. In our modern world, we have the

technology to go beyond mere speculation and design experiments to directly

probe the nature of matter. Thanks to experimental studies science has advanced

immeasurably since the time of Democritus; however, to this d4 i, one of the

i i, i"r goals of the physical sciences is still to identify and understand the most

fundamental building blocks of nature. This is the defining quest of particle

physics.

Many profound observations in the late 19th and early 20th centuries led to a

revolution in the perceptions which guided microscopic physics. Modern particle

physics was born with the discovery of the electron in 1897 by J.J. Thompson.

This discovery demonstrated that what we call ii. .~- in the modern world,

are actually not fundamental. Einstein's explanation for the photoelectric effect

through the quantization of light led to the quantization of the electric field. In

1911, the scattering experiments of Rutherford led to the Bohr model of the atom.

Compton's scattering experiments in 1923 demonstrated the particle nature of

light. In 1927, Dirac wrote down an equation which predicted that all fermionic

particles should also have antiparticles and, in 1931, the positron was discovered

by Anderson. A theory of strong interactions mediated by massive particles was

introduced by Yukawa in 1934 to describe the binding of nucleons.









These advances and countless others over the last one hundred years have led

to the development of the Standard Model (SM) of particle physics. This model

describes the electromagnetic, weak, and strong interactions between particles.

Under the Standard Model, the strong and electroweak interactions are unified

under the gauge group SU(3) & SU(2) 0 U(1).

The observation by Louis de Broglie that moving bodies have a wave nature

has profound consequences in particle physics. The resolution of an optical

microscope is approximately proportional to the wavelength of the incident light.

Assuming the probing beam consists of particles, then the resolution is limited by

the de Broglie wavelength of these particles


A (1-1)


where A is the de Broglie wavelength, h is Plank's constant, and p is the momentum

of the particle beam. This prediction, that an object's momentum is inversely

proportional to its wavelength, implies that as the momentum transfer, Q, of the

probing beam is increased it is possible to resolve smaller distance scales. Low

energy particles only probe large spatial regions, while high energy particles can

resolve short distance effects. This observation by de Broglie motivates the use of

high energy particle accelerators as the laboratory of particle physics in the modern

age.

Through the use of particle colliders, the Standard Model has compiled an

impressive history of experimental success. For example, the W and Z bosons

were predicted before their discovery by the electroweak theory of Glashow,

Weinberg and Salam. The running of the electromagnetic and the strong coupling

constants have been verified by experiment. The final quark, the top quark, of

the Standard Model was discovered at the Tevatron collider by the two collider

experiments, CDF and DO. The SM also survived the plethora of precision









electroweak measurements of LEP, the CERN e+e- collider. In summary, the

Standard Model is a successful model of high energy particle physics at all energies

accessible to the experimental community so far.

No deviations from the SM have been observed; however, there are imperfections

with the Standard Model. So far the Higgs boson has not been observed. This

particle must exist in the Standard Model to allow the basic building blocks of the

model to obtain mass. Also, there are many theoretical arguments that -i--::. -1

that the SM breaks down at higher energy scales. One such argument is related

to the mass of the Higgs boson itself. This I,,'. .,.;; problem is related to the

sensitivity of the Higgs mass to physics at high energy scales and requires a fine

tuning (i.e., cancellation to a precision of 1032) of the Standard Model which is

undesirable to many theorists. Supersymmetric (SUSY) models, which require a

symmetry between fermions and bosons, can provide an elegant solution to this

problem. However, so far, no SUSY particles have been discovered. Many models

of physics beyond the SM (BSM) such as SUSY require additional particles which

are heavier than the SM particles. The particles must be heavy or they would

have already been discovered in previous measurements. High energy collisions are

required to produce heavy particles in the laboratory. This need to search for heavy

particles further motivates the need for high energy colliders in particle physics.

Theorists have used compelling arguments, such as hierarchy, for many years

to motivate the need for new particles or new forces to be observed at the TeV

scale. Without experimental guidance at the TeV scale, theorists who study physics

beyond the Standard Model have had freedom to pursue countless possibilities.

The Tevatron at Fermilab is only just beginning to threaten the TeV scale with a

center of mass energy of 1.96 TeV, but has the potential to constrain many of these

theories. At the same time, the particle physics community is waiting eagerly for









the first collisions at the Large Hadron Collider (LHC) which will begin colliding

protons in the next few years at a center of mass energy of 14 TeV.

Precision measurements at high energy colliders have been an immensely

valuable tool both to validate the SM and to constrain its properties. As higher

energy colliders become available, it is possible to make discoveries of BSM physics

or to validate and constrain our understanding of the SM. In this dissertation, a

measurement which probes the smallest distance scales ever probed by studying the

collisions of the highest energy particle accelerator in the world will be discussed.

This measurement provides validation of quantum chromodynamics (QCD), the

theory of the strong force, at the highest energy ever directly probed, and at the

same time provides constraints on the quantum nature of the proton which will

improve theoretical predictions for the high energy colliders of the future.














CHAPTER 2
PHENOMENOLOGY OF QUANTUM CHROMODYNAMICS

The Standard Model of particle physics is a quantum field theory (QFT) based

on the principle of local gauge invariance of the gauge group

SU(3)c 0 SU(2)L 0 U(1)y (2-1)

where SU(3)c is the symmetry group of the strong interactions, called quantum

chromodynamics, and SU(2)L 0 U(1)y represents the symmetry group of the

electroweak theory. A complete discussion of QFT [1] and the standard model

of particle physics [2, 3] is beyond the scope of this experimental dissertation.

Instead, the particle content of the standard model will be reviewed, and only

aspects of QCD phenomenology which are relevant to the inclusive jet cross section

measurement will be addressed in detail. A more complete discussion of QCD can

be found in irn' i references [4, 5, 6, 7].

2.1 The Particle Content of the Standard Model

The particle content of the standard model of particle physics includes six

quarks, six leptons, and four gauge bosons (an anti-particle also exists for each

particle). Quarks and leptons are spin { fermions. The quarks and leptons of the

SM can each be arranged into three doublets. Each lepton doublet includes a

charged lepton partnered with a neutral neutrino. The quark and lepton content

of the standard model is listed in table 2-1 and table 2-2 along with some of the

fermion measured properties as listed in the Particle Data Book [8].

The fermions of the standard model interact through the exchange of the

integer spin gauge bosons. The four gauge bosons are shown in table 2-3 with some

of their properties. The mass-less photon is the propagator of the electromagnetic









Table 2-1. Some properties of the quark content of the standard model. Quark
properties are taken from the Particle Data Book.

Flavor Symbol Electric C' -ge (e) Mass (GeV/
Up u +T 1.5-3 x 1(
Down d 1 3 7 x 10


c2)
0-3
-3


('1 .,ii,, c +j 1.25 0.09
Strange s -3 95 25 x 10-3
Top t +j 174.2 3.3
Bottom b 1 4.2 0.07
3


force, the W and the Z bosons are the force carriers of the weak interactions, and

the eight mass-less gluons (gi where i = 1..8 correspond to the 32 1 generators

of the SU(3) symmetry group) mediate the strong interaction. A in i ir difference

between quarks and leptons is that quarks carry an additional internal degree

of freedom called color. This is the charge of the strong force and is commonly

denoted as red, green, or blue (RGB). Of the fermions, only quarks participate in

the strong interactions of QCD.

Table 2-2. Some properties of the lepton content of the standard model. Lepton
properties are taken from the Particle Data Book.

Flavor Symbol Electric C('i -ge (e) Mass (MeV/c2)
Electron e -1 0.511
Electron Neutrino Ve 0 < 3 x 10-6
Muon p -1 105.7
Muon Neutrino V 0 < 0.19
Tau 7 -1 1777
Tau Neutrino VT 0 < 18.2


Gravity is not mentioned in the above discussion. Although all massive

particles couple to gravity, it is the weakest force and is typically only important

on macroscopic scales. Gravity is not included in the standard model. It is not

important for the research discussed here and will not be discussed further.

It is a remarkable triumph of the SM that all of the interactions of matter in

the observed universe (barring gravity) can be described with amazing precision

based on the simple particle content discussed here.









Table 2-3. Some properties of the gauge boson content of the standard model. The
gauge boson properties are taken from the Particle Data Book.

Boson Symbol Electric CI ,rge (e) Mass (GeV/c2)
Photon 7 0 0
W W +1 80.403 + 0.029
Z Z 0 91.1876 + 0.0021
Gluon g 0 0


2.2 Feynman Rules of QCD

The theory of QCD describes the interactions of the spin 1 quarks and the

spin 1 gluons. Requiring that QCD be a gauge theory based on the group SU(3)

with three color charges fixes the Lagrangian density to be
8 nf
S FAaFA + q(i- mj)qj, (2-2)
A= 1 j 1

where qj are the quark fields of nf different flavors and mass mj. The 7'a are the

Dirac matrices and Da (/1 = Daf7) is the covariant derivative defined by


(Da)ab a 9ab + g(CA)ab, (2-3)

where g is the gauge coupling of QCD, and tC are the matrices of the fundamental

representation of SU(3). These generators obey the commutation relations


[tA,tB] ifABCtc, (2-4)

where fABC are the complete antisymmetric structure constants of SU(3). The

normalization of the structure constants and of g is specified by


Tr [tAB 6AB/2. (2-5)


In analogy with Quantum Electrodynamics (QED) we may also make the definition


as 4g- (2-6)
47










The quantity F(A is the field strength tensor derived from the gluon field AA

FA a a9A ap A ABC B c. (2-7)
Fa3 a3- ag fAB A (2 7)


A,a p BE^ P e p A
p +ie p +ie
A p B
(p2+ie)
ai p bj ab i
(lP-m+ie)ji


) r' -g f'c[(p-q) g" +(q-r)"g '+(r-p) g"]
S(all moment incoming)
A,a C,'y

A,a Bp _ig2 fACfXBD [ga gy_ ad y]
ig2 ffADfXBC Ir a# y6 gagpd]

C,y X D,5 ig2 fXffXCD [g g g gag g]

SA,a g



B C



-ig (tA)cb (7Ya )ji
b,i c,j

Figure 2-1. Feynman rules for QCD in a covariant gauge. The solid lines (black)
represent fermions, curly lines gluons, and dotted lines (black) ghost
propagators.


The Feynman rules of QCD can be calculated directly from the QCD

Lagrangian after fixing a gauge. The gauge must be fixed in order to define the

gluon propagator. The Feynman rules in a covariant gauge are given in Fig 2-1 [9].

The physical vertices in QCD include the gluon-quark-antiquark vertex. This

vertex and the physical propagators of the quark and gluon are analogous to the

coupling and propagators of the electron and photon of QED. However, there is









also the three-gluon and four-gluon vertices, of order g and g2 respectively. These

gluon self couplings have no analogue in QED since the photon does not carry

electric charge and therefore does not interact with other photons. They arise from

the third term of Eq. 2-7 which is not present in the QED field strength.

The Feynman rules discussed above can be used to make many predictions

of QCD. For example, one can compute the probability that a given initial state

will interact to yield a final state, P(A + B -+ C + D). In particle physics, these

probabilities are called cross sections, a. Cross sections are expressed in units of

area called barns, b. One barn is equal to 10-24 cm2. In this analysis, cross section

results are expressed in nano-barns (1 nb = 10-9 b). Once a cross section for a

process has been calculated, and the total number of collisions which have occurred

in some time period (integrated Lam.-.:/ ) has been measured, it is possible to

predict the number of events of that process that have occurred


N = L.1 (2-8)


The instantaneous luminosity is a measure at a specific time of the number of

collisions per unit time occurring in the collider. The integrated luminosity is a

measure of the number of pp collisions that have occurred over some period of time.

Instantaneous luminosity is usually measured in units of cm-2 -1. Recently, the

Tevatron has achieved instantaneous luminosities greater than 200 x 1030 cm-2 -1.

Integrated luminosity is measured in nb-'. A process with a cross section of 1 nb

will be created approximately 2000 times per d4i at current CDF luminosities



1 Due to the substructure of the proton, the cross section calculated from the
Feynman rules must be convoluted with the quark and gluon density functions of
the proton before it can be used to predict the number of events expected at pp
colliders like the Tevatron.









(i.e., the integrated luminosity for a 24 hour period at CDF can be on the order of

2000 nb-1).

2.3 Color Confinement and Asymptotic Freedom

The Feynman rules shown in figure 2-1 are made up of tree-level diagrams

(i.e., diagrams which do not contain loops). Most calculations made in QFT

are an approximation based on an expansion in powers of the coupling constant.

This is called a perturbative expansion, and is only valid in the limit that the

coupling constant is small (i.e., a << 1). Perturbative predictions for observables

such as scattering amplitudes are affected by higher order loop corrections. The

vacuum polarization diagram for QED shown in figure 2-2 is an example of such

a correction. This correction to the photon propagator diverges logarithmically at

the one loop level as the four-momentum squared of the virtual photon (q2 _Q2)

increases. In the high energy limit (Q2 >> m2) the contribution is

aoB(Q2) {log(2/Q2) + } (2-9)

where A is an ultraviolet cutoff and a0 is known as the bare electric charge (ao =

e0/47). An effective coupling may be defined which sums the vacuum bubbles to all

orders


aeff(Q2) ao( + aoB(q2) + aoB(q2aoB(q2) + ...) -ao (2-10)
1 aoB(q2)

This procedure for defining an effective coupling to absorb the ultraviolet diverges

in the theory into the unobservable bare coupling is called renormalization. The

effective coupling now varies (i.e., runs) with the energy scale of the problem. The

long distance behavior of the effective coupling is used to define the experimental

coupling
1
a ~ff (Q2 0) 2 (2 11)
137









In the high energy limit (large Q2)


aQED(Q) a fQ2) t a (2-12)
1 alog(?)

From Eq. 2-12, we see that the coupling constant of QED increases with energy. At

low energy, aQED is small and a perturbative expansion in the coupling is relevant.

At high energy, the coupling gets large and eventually diverges (Landau pole).

Perturbation theory is not valid in this regime. Luckily, this occurs at a very high

energy scale for QED, and it is expected that physics beyond the standard model

should modify the running of the coupling at an energy far below the Landau pole.



e








e

Figure 2-2. One-loop Vacuum polarization diagram of QED. Diagrams like this
affect the rate for electron-positron scattering in QED. This diagram
diverges logarithmically.


The QCD diagram analogous to the QED diagram shown in figure 2-2 is

shown in figure 2-3. In QCD, there is an additional contribution to the propagator

of the gluon due to its self coupling. This extra diagram (shown in figure 2-4) leads

to profound consequences with respect to the running of the coupling constant. By

following similar arguments as those applied to QED and including the extra QCD

diagrams, one arrives at the analogous equation to 2 12


a(Q2) aff (Q2) = (2 13)
1 'log( 2)









The quantity /o is the one-loop /3-function of QCD


00- t 2 f
3f~


(2-14)


where nf is the number of quark flavors. In QED, the one-loop 3-function was


/30QED


(2-15)


The sign of the /3-function is different for QED and QCD if nf < 16. This has

the consequence that the effective coupling of QCD runs in the opposite direction

of the QED effective coupling. Equation 2-13 represents only the leading order

q


Figure 2-3. One-loop vacuum polarization diagram of QCD. This diagram is
analogous to the vacuum polarization diagram of QED.


Figure 2-4. One-loop Vacuum polarization diagram of QCD which arises from the
gluon self coupling. Contributions from this diagram are responsible in
the sign flip for the /3-function of QCD with respect to QED.









behavior of the coupling constant in QCD, where the experimental charge a(pt2)
has been defined at an arbitrary renormalization scale (Q2 = p ). The low energy
limit is not a useful scale in QCD. This is because the coupling as diverges in the
low energy limit. Typically the arbitrary scale pr is taken to be the mass of the Z
boson where


a,(Mz) ~ 0.12.


(2-16)


The main features of Eq. 2-12 and Eq. 2-13 are sketched in figure 2-5. For QED,
we see the dynamics that were mentioned above; for low Q2 (large distances) the
coupling constant is small and the coupling increases with Q2 until at some very
high energy scale (~ 1034 GeV) it diverges. For QCD, the dynamics are very


04
G)
^i
a


Q2


Figure 2-5. Running of the the QED and QCD coupling constants. In QED the
effective coupling is small at large distances, but diverges at very
high energy ("Landau 1p'! ). In QCD the coupling diverges at large
distances ("color olli. ii, ,i ) and goes to zero .,-,iiii ,ll ically
at large energy ( ,-i_,_iiiI freedom"). Color confinement and
.,~-ii ,i, 1i ic freedom are important qualities of QCD.


CQCD(Q2)


Q):
(QED(2





: Landau
; pole



... asymptotic
.---- .. I .. .


confinement









different. At low Q2 the coupling diverges. This is known as color confinement

and is the reason why free quarks and free glouns are not observed in nature.

As objects connected by color fields are separated the field strength becomes

so strong that new quark-antiquark pairs are pulled from the vacuum. These

quark-antiquark pairs form color neutral singlets called hadrons. This process of

colored partons (i.e., quarks and gluons) forming color singlet hadrons is referred to

as hadronization. Through this process a colored parton can hadronize into many

hadrons which are roughly co-linear with respect to the momentum vector of the

original parton. These clusters of roughly co-linear hadrons are called jets.

At high Q2 (particles resolve small distances) the coupling constant of QCD

becomes small and even vanishes .,-i~,, i illy. This is the phenomena known
as .,-vmptotic freedom. A consequence of .,i-mptotic freedom is that perturbative

methods are valid at high energies in QCD. Because of this property of QCD, the

long distance (low Q2) and short distance (high Q2) behavior of QCD may be

separated (i.e., factorized). This factorization allows the methods of perturbative

QCD (pQCD) to be applied to the large Q2 component and phenomenological

models to be applied to low Q2 component. This factorization property means

that the partonic cross section calculated with the methods of pQCD is useful for

hadronic collisions in the high energy limit.

Another important result that may be obtained directly from Eq. 2-3 is the

energy scale at which the coupling constant in QCD diverges (AQCD). Solving for

the energy scale (Q2) where the denominator of Eq. 2-3 vanishes yields


log(A2CD) = 4o- + log(Pt). (2-17)

This result can be used to obtain the effective couping in terms of AQCD

47
8(Q2) (2 18)









The quantity AQCD has been determined experimentally to be roughly 200 MeV.

Therefore, the QCD effective coupling gets large for Q2 ~ 1 GeV2. Perhaps, it is

more than coincidence that this is roughly the mass of the proton.

2.4 The Factorization Theorem

The observation by J.D. Bjorken in 1969 from deep inelastic scattering

experiments, that protons when probed with sufficiently high momentum behaved

like free partons, had profound implications [10]. A few years later Drell and Yan

extended these parton model ideas to some hadron-hadron processes [11]. The idea

of the factorization model can be seen pictorially in figure 2-6 [9]. It means that

the hadronic cross section may be written as

o7(Pl,P2) /dxidx2fi(xl, )fj (2,)ij(pl,p2,' s(/), 2/P2), (2-19)


where the moment of the partons which engaged in the partonic scattering is

pi = x1P1 and p2 x2P2, x1 and x2 are the momentum fraction of the hadron
carried by the interacting partons, fi p(p)(xi) are the quark and gluon parton

density functions (PDFs) defined at the arbitrary factorization scale pif, and rij is

the partonic (short-distance) cross section for the scattering of partons of type i

and j. At leading order in QCD the parton cross section is directly calculated from

the leading order tree diagrams. However, at higher order, there are long-distance

contributions which must be factored out and absorbed into the PDFs of the

incoming hadrons. This factorization is possible to all orders in perturbation

theory, and is the property of QCD which makes it a useful tool in calculations for

hadron collisions.

In principle, the factorization scale (p/9) and the renormalization scale (P/) can

differ. However, in practice, it is convenient to set both scales equal to the hard

scattering scale (p = / r = pi = Q). For the inclusive jet cross section, the hard

scattering scale (Q) is often taken to be one half of the jet transverse moment.












f ( '

X11










P2 f( )




Figure 2 6. Schematic of the QCD factorization theorem. The partonic cross
section must be folded in with the parton density functions of the
hadron.

2.5 Jet Production Cross Sections

Diagrams contributing to jet production at leading order are shown in

figure 2 7 [9]. These diagrams may be read from left to right, or bottom to top.

For example, 2 7(c) can be interpreted as qq gg when read from left to right, or

it may be interpreted as gq gq when read from bottom to top.

Lowest order (LO) calculations have uncertainties for multiple reasons. The

leading order result quite often has a large dependence on renormalization and

factorization scales. This dependence is reduced by going to higher order in the

perturbative expansion. Another source of uncertainty on LO predictions is that

additional processes may become possible only when going beyond leading order.

At next to leading order (NLO), all Feynman diagrams which contribute

an additional factor of a8 to the scattering amplitude must be considered when

calculating the scattering cross section. Extra factors of the strong coupling













(a)





(b)





(C)





(d)
Figure 2-7. Diagrams which contribute to leading order jet production at a hadron
collider.

constant can be added in two v-,. Real radiation may be added to the initial
or final state, or diagrams may contain one loop. As a simple example, consider

the case of electron-positron annihilation to hadrons through a virtual photon
exchange(e~e -+ qq). The diagrams which contribute to this process at NLO are

shown in figure 2-8 [9].
The diagrams in figure 2-8(a) include the tree-level diagram for the process
as well as one-loop diagrams with a virtual gluon emission. Although the virtual
gluon diagrams have two extra factors of gs due to the two extra vertices, they
still contribute at NLO. Since these diagrams have the same final state as the
tree-level diagram, the matrix element for the sum of the four diagrams must be
squared. Cross terms from this squared matrix element are at NLO in a8 (i.e.,














(a)


Pi

S k +

Pu
(b)

Figure 2-8. Diagrams which contribute to e+e- annihilation to hadrons at NLO.
The divergences in the real gluon emission diagrams cancel the
divergences in the loop diagrams.


O(af)). These cross terms have infrared (Q2 -+ 0) divergences related to the
integral over the loop moment. Figure 2-8(b) shows the NLO diagrams with a

real gluon emission. The real gluon emission diagrams diverge in the limit that the

gluon is collinear to the quark. A very powerful theorem of QFT states that soft

and collinear divergences cancel to all orders in perturbation theory [12, 13, 14].

This cancellation means that inclusive quantities will be free of divergences. The

inclusive requirement means that the observable can not include only the diagrams

in figure 2-8(a) or only the diagrams in figure 2-8(b) because contributions from

all of these diagrams must be included to ensure that the divergences are canceled.

It is from arguments of this type that the jet cross section must be inclusive, and

jet algorithms must be defined in such a way as to not be sensitive to infrared and

co-linear effects.

2.6 Structure of Hadronic Collisions

When protons of equal and opposite momentum collide at high energy there

is a large probability that the protons will break up and the resulting hadrons

will continue in roughly the same direction with very little transverse momentum









relative to the beam direction. Sometimes however, there is a hard scattering

where particles with large transverse momentum are generated in the collision. The

factorization theorem of QCD discussed above allows one to factor out the hard

scattering component of the hadron collider event and calculate it perturbatively.

However, there are other components that must be included for a complete model

of the hadron collider event.


"Hard" Scattering Event Outgoing Parton
Outgoing Parton
Initial-State (PT Hard)
_.. Radiation ( Hr

Proton ^ AntiProton

Multiple Parton Interaction

Beam Remnants
Outgoing Parton / Final-State
(PT Hard) 4 i Radiation

Figure 2-9. A cartoon description of a typical hadron collider event at the parton
level (before hadronization of colored partons into color singlet
hadrons). The hard I I, -ii:.- initial state radiation, final state
radiation, multiple parton interactions, pileup, and the beam remnants
are the components of an event at a hadron collider.


Figure 2-9 shows a simplified description of a hard scattering event at a

hadron collider. The schematic shown can be considered to represent what goes

on within the radius of a proton around the hard collision. Once the colored

partons move outside of the radius of the proton, they must hadronize into color

neutral hadrons due to the requirement of color confinement. The state of the event

before hadronization is not a physical observable, but is useful when discussing the

phenomenology of hadron collider events and will be referred to as the parton level.

This simple parton level model of hadron-hadron scattering is used as the basis for

the QCD Monte-Carlo (\ C) event generator programs.









As partons begin to feel the effect of other partons, they radiate quarks and

gluons. These quarks and gluons can also radiate more quarks and gluons and

this series of partons radiating more partons leads to an avalanche or shower of

partons. This chain of radiation is called a parton shower. In the context of a

QCD event generator, a parton shower is an approximate perturbative treatment of

QCD parton splitting which is valid above some cut-off value (Qo ~ 1 GeV). The

parton shower is based on identifying and summing to all orders the logarithmic

enhancements due to soft gluon emission and gluon splitting functions. Because

the parton shower is based on enhancements due to soft gluon emission (small

angles) it is only an approximation of the hard gluon emission component (large

angles). Parton showers are used to model initial state radiation (ISR), and final

state radiation (FSR) in QCD MC generators. These models for the parton shower

can be combined with phenomenological models of hadronization which take over

for energy scales below the cut-off scale (Q < Qo).

As shown in figure 2-9, it is also possible that there is a second parton-parton

interaction within the same proton-antiproton collision. This is referred to as

multiple parton interactions (\!PI). Because the protons and antiprotons collide in

bunches, it is possible that multiple proton-antiproton collisions occur in the same

bunch crossing. This is commonly referred to as pileup. The rate of pileup collisions

is proportional to the luminosity and can be studied be by looking at the number

of secondary vertices in the event.

Beam-beam remnants and multiple parton interactions define the under-

lying event (UE). The underlying event is alv--- present at hadron colliders

and increases the difficulty of resolving the properties of the hard scattering

process. Separating particles from the UE and particles which come directly

from the hard scattering is not possible on an event by event basis. However, the

topological structure of hadron-hadron collisions can be used to study the the









average properties of the UE contribution. An analysis based on this strategy was

published in Run I at CDF[15, 16], and has been continued and improved in Run II

at CDF[17, 18, 19]. By studying distributions which are sensitive to the underlying

event the MC programs are tuned to fit the effects observed in data.

The leading order matrix elements, PDFs, parton showers, hadronization

models, the underlying event, and pileup are required components of a QCD event

generator for complete generation of hadron collider events.

After all of the partons in the event hadronize, the particle content of the

event is referred to as the hadron level. The particles at the hadron level are

observable, and they are the states which interact with the detector. After

the particles interact with the detector, the resulting description of the event

is referred to as the detector level. These three levels of the event (parton,

hadron, and detector) will be referred to throughout this draft and are depicted in

figure 2-10 [20].

Experimental measurements are only available at the detector level. However,

the MC generators, when combined with a detector simulation program, can be

used to make predictions at all three levels 2 MC generators combined with

detector simulations are a useful tool for deriving corrections to the data.















2 The hadron level in the Monte Carlo generators is defined as all final-state
particles with lifetime above 10-11 s.














calorimeter jet


Calorimeter /
shower /



SDecays
\ T ; 1interac
Yi mater
S Magne

had biation
1q Outofc
Spartons


CI-.


Figure 2-10.


q e Ln Icl;
f en


A cartoon description of the different levels of a jet event at CDF.
The parton level is the state before the partons hadronize and is
not physical observable. The hadron or particle level is the state
after hadronization but before the particles have interacted with
the detector. Finally, the detector level is the result of the event as
reported by the detector.


tions in
al,
tic field


one


(














CHAPTER 3
EXPERIMENTAL APPARATUS

3.1 Coordinates and Conventions

The CDF detector is naturally described by a cylindrical coordinate system

(p, z, Q). The z-coordinate is taken along the proton beam direction with the

origin at the center of the detector. Defining the x-axis to point away from the

center of the accelerator ring fixes the azimuthal angle Q. The p-coordinate is the

perpendicular distance from the z-axis. It is also useful to define the polar angle 0,

which is usually expressed through the pseudorapidity


q- lntan(0/2). (3-1)


A value of q = 0 corresponds to 0 = 900, and a value of q = 1 corresponds to

0 ~ 400. The rapidity Y is defined as


Y In {E+P, (3-2)
2 E P,

where E denotes the energy and Pz is the momentum component along z-direction.

Rapidity is a useful quantity at a hadron collider because it is invariant under

boosts in the z-direction up to a constant. In the limit that the momentum of

a particle is much larger than its mass, Y and q] are equivalent. One difference

between Y and q] is that Y does not correspond to a definite 0 value.

Because the interacting partons need not balance in momentum along the

z-direction, the colliding system can have an arbitrary boost in the lab frame.

Momentum of the interacting partons is not known in the direction parallel to

the beam; however, the initial parton moment is roughly zero in the direction

perpendicular to the beam. The transverse momentum vector is a two dimensional









vector perpendicular to the beam direction and is defined by


PrT PXi+Pj, (3-3)

where Px and Py are the x and y components of the momentum. The sum of the

transverse momentum vectors of all particles in the event should sum to zero in

the absence of any particles escaping detection. The magnitude of the transverse

momentum is given by PT -P sin 0, where P is the magnitude of the momentum

vector. Similarly, the transverse energy vector is defined as

ET = E sin 0 cos fi + E sin 0 sin j -- T(cos fi + sin (j). (3-4)

The magnitude of the transverse energy is ET = E sin 0.

Some other useful quantities are the missing transverse energy vector (4r),

and the missing ET significance. The missing transverse energy vector is defined by



VT E^fi, (3-5)

where fij is a unit vector perpendicular to the beam axis and pointing at the ith

calorimeter tower. Missing ET significance is defined by


T -ET/ PET, (3-6)

where iT is the magnitude of 7T. fET is a useful variable to remove backgrounds

which do not originate symmetrically from the center of the detector (e.g., cosmic

rays).

3.2 The Tevatron

The Tevatron accelerator at the Fermi National Accelerator Laboratory

(FNAL or Fermilab) provides the highest energy proton-antiproton collisions

available in the world with a center of mass energy of 1.96 TeV. Experimental









discoveries would not be possible at CDF without the beam provided to the

experiment by Fermilab.

The colliding beams at the CDF and DO experiments are the result of the

complex accelerator chain shown in figure 3-1 [21]. A Cockroft-Walton accelerator

starts the process by accelerating Hydrogen ions (Hydrogen atoms with one extra

electron) to 750 keV. The ions are then injected into the 500 ft long Linac where

their energy is boosted to 400 MeV by oscillating electric fields. The electrons are

then stripped from the ions by a carbon foil. The remaining protons then enter

a fast-cycling synchrotron ring called the Booster. Here the beam is accelerated

by radio frequency (RF) cavities at each revolution until they reach an energy of

8 GeV. Bunches of protons, each containing about 5 x 1010 protons, are passed on

to the Main Injector.

Proton bunches from the Main Injector are also used to create antiprotons (p)

by collisions with a nickel-copper target. This technique produces antiprotons with

a wide range of momentum which must be cooled into a mono-energetic beam. The

antiprotons are first focused with a lithium collector lens and then passed into the

the Debuncher. The Debuncher applies complex computer-controlled RF techniques

to cool the antiproton beam as much as possible. Correction signals are applied to

individual particles in order to further stocastically cool the antiproton beam. An

8 GeV beam emerges and is passed on to the Accumulator where p bunches are

stacked (i.e., accumulated) at rates as high as 1012 antiprotons per hour.

The beam is then passed on to the Recycler ring, which is an 8 GeV magnetic

storage ring that utilizes stochastic cooling systems. As its name --.:: -i- the

Recycler is also capable of recovering antiprotons left over at the end of a store

(i.e., period of colliding beam time). Once the accumulated antiproton beam

reaches 8 GeV it can be extracted into the Main Injector and accelerated to

150 GeV.





















ABORT / L" RF CDF DETECTOR
150 GeV p lNJ & LOW BETA
p 150GeV p bNJ



TEVATRON p( TV)/

\ (1 TeAf)

DO DETECTOR pABORT
& LOW BETA
DO

Figure 3-1. A schematic of the accelerator complex used for Run II at Fermilab.
The accelerator process may be divided into eight steps. Each step in
this process is summarized in the text.

The Main Injector is a synchrotron ring located next to the Tevatron. It was

a Run II upgrade to replace the Main Ring. The Main Ring was located in the

Tevatron tunnel and was replaced with the Main Injector because it caused beam

backgrounds in the collider detectors. Proton bunches exiting the Booster are

combined by the Main Injector into a single high intensity bunch of approximately

1012 protons. Protons are transferred to the Tevatron after reaching an energy of

150 GeV.

The Tevatron is the largest of the Fermilab accelerators, with a circumference

of approximately 4 miles. It is a circular synchrotron with eight accelerating

cavities. The Tevatron accepts both protons and antiprotons from the Main









Injector and accelerates them from 150 GeV to 980 GeV in opposite directions.

Once the beams energy has reached 980 GeV, they are squeezed to small transverse

dimensions by quadrupole magnets at the interaction points (the centers of the

CDF and DO detectors). The beam can be stored in the Tevatron while colliding

for many hours. Typically, collisions continue until there is some failure, or the

remaining collider luminosity is low and the antiproton stack is large enough to

begin a new store.

3.3 The CDF Detector

The CDF detector is described in detail in [22, 23]. Here, those components of

the detector which are crucial to this measurement are briefly discussed. A detailed

schematic drawing on the CDF detector is shown in figure 3-3 [23].

Although it is not shown in the figure, the Cerenkov Luminosity Counter

(CLC) is a critical component of this analysis [24]. When charge particles

travel faster than the speed of light in a medium the radiation that they emit

becomes coherent. This is a similar phenomena to a sonic boom which occurs

when something travels faster than the speed of sound. This effect is used to

measure the average number of inelastic pp collisions per bunch crossing in

order to calculate the instantaneous luminosity delivered by the Tevatron. The

instantaneous luminosity provided by the CLC must be integrated with respect

to time to calculate the integrated luminosity. The total integrated luminosity

included in this measurement is approximately 1000 pb-1 or 1 fb-1, and is used for

the normalization of the cross section. The Cerenkov counters are located in the

region 3.7 < T|11 < 4.7.

The central tracking system consists of a silicon vertex detector inside a

cylindrical drift chamber. The drift chamber is referred to as the central outer

tracker or COT. C!i r'ge particles ionize atoms in the gas (argon-ethane 50 : 50) of

the COT as they pass through the detector. The liberated electrons are accelerated









by an electric field towards the nearest positively charged wire to energies high

enough to cause secondary ionization. The electrons arising from this chain of

ionization are collected on the wire and an electronic pulse is read out. The COT

provides charged particle reconstruction up to |T1 = 1.0. Surrounding the tracking

detectors is a superconducting solenoid which provides a 1.4 T magnetic field.

Tracks are the reconstructed paths of charged particles in the magnetic field

based on the wires that collected electronic signals. These tracks can be traced

back to their point of closest approach to the beam line (i.e., impact parameter).

A place along the beam line where multiple tracks intersect is called an interaction

point or vertex. Vertices are signs that a pp interaction occurred at that position.

At CDF there may be multiple interaction points in the same event due to pileup.

For each event a 1 .':,i,1 .i vertex is reconstructed. This is defined as the vertex

with the highest sum PT (sum of the PT of all tracks pointing to the vertex). The

position of the primary vertex can vary significantly with respect to the center

of the detector at CDF due to the length of the colliding bunches. The length of

the bunches is roughly 50 cm in the z-direction. For calculating jet properties, the

primary vertex is used as the origin of the coordinate system. Shifting the origin of

the coordinate system along the z-direction changes 0 (q]) and therefore the values

of PT and ET are modified. The number of extra vertices in the event is a good

indicator of the number of multiple interactions within a bunch crossing (pileup).

In summary, the tracks reconstructed by the COT are used in the inclusive jet

analysis to reconstruct the vertices in each event for two reasons: to determine

the primary vertex in the event which is used to define the origin of the event

coordinate system, and to determine the number of secondary vertices in the event

which is used to count the total number of interactions that occurred in the bunch

crossing.









The tracking chambers above only detect charged particles. However, on

average, approximately II' .- of the energy in an event is carried by uncharged

particles. Calorimeters are used to determine the energy and position of both

charged and uncharged particles by their total absorption. Electromagnetic

particles (photons and electrons) and hadronic particles interact with the detector

material differently. The electromagnetic and hadronic calorimeters are all sampling

calorimeters (i.e., alternating I~ i,-S of absorber and detector material); however,

each component is composed of different material and has different depth based on

the interaction properties of the particles it was designed to measure.

At high energies, when electrons or positrons interact with matter, the

dominant way in which they lose energy is through radiation of photons (i.e.,

tb in- i,,1.,ii,,j: e- e-i7). For high energy photons the dominant interaction

process is pair production (i.e., 7 -- e+e-). An initial electron or photon

will interact through these two processes to produce a shower of photons and

electrons in the detector. This phenomena is sketched in figure 3-2 and is referred

to as an electromagnetic (EM) shower. The shower develops until the energy

reaches a critical energy (Ec ~ 600 MeV/c) and ionization losses equal those of

bremsstrahlung. The depth is governed by the radiation length (Xo) of the material

and only increases logarithmically with the energy of the particle which initiated

the shower. The energy resolution of the electromagnetic calorimeter is limited by

statistical fluctuations in shower development [25].

Hadrons interact with matter through much different interactions than the

ones which lead to the EM shower described above. However, a similar phenomena

occurs which is referred to as a hadronic shower. An incident hadron undergoes

an inelastic collision with nuclear matter in the detector resulting in secondary

hadrons. These hadrons also undergo inelastic collisions. Because many different

processes contribute to the development of a hadronic shower, the modeling of the





























Figure 3-2. Development of an electromagnetic shower. Electrons and photons
shower when they interact with the detector through the processes of
Bremsstrahlung and pair production.

shower is much more complex than an EM shower. For example, neutral 7 mesons

(7's) may be produced. Neutral 7 mesons primarily decay into two photons

which instigate an EM shower. Fluctuations, such as the number of 7r's which

are produced early on in the hadronic shower, lead to an energy resolution which

in general is much worse than the resolution of EM calorimeters. The depth of

a hadronic shower is governed by the nuclear interaction length of the detector

material. For most materials the nuclear interaction length is much larger than the

radiation length. This means that hadronic showers typically pass through more

material before starting to shower, and the showers typically take up more detector

volume. This is the reason that hadronic calorimeters are located outside of EM

calorimeters and are typically much thicker.

As mentioned above, the CDF calorimeter consist of alternating absorber

and detector l-. rs. The absorber consists of a dense material (lead or iron)

with a large radiation or interaction length for the purpose of instigating the









showers discussed above. This material is dead in the sense that it has no detection

capabilities. When charged particles reach energies below some critical energy

they lose energy to the detector material through excitation and ionization of

atoms. In some materials, called scintillators, a fraction of this excitation energy

emerges as visible light as the excited atoms return to their ground state. Some of

this light can be transferred to photomultiplier tubes which converts the light to

electronic signals through the photoelectric effect. The detection 1lv. rS of the CDF

calorimeters are composed of various scintillating material. The energy of a particle

absorbed by the calorimeter is proportional to the amount of light measured by

the scintillating material, and this proportionality constant must be determined

through calibration. The process of converting charged particle interactions into

electronic signals described above is very fast. Information from the calorimeter is

available very quickly at CDF and is used to make the first decisions on whether an

event is interesting or not (i.e., it is useful in the level-1 trigger).

The size, granularity, location, and resolution of the various CDF calorimeter

modules will now be described. The central calorimeter is located outside (i.e.,

farther away from the interaction point) of the solenoid magnet and is divided into

electromagnetic (CEM) and hadronic (CHA) sections. The central calorimeter

is segmented in Y 0 space into 480 towers which point back towards the

interaction point. The granularity of the towers is AT7 x A 0.1 x 0.26.

The central calorimeter covers a pseudo-rapidity range up to 1.1. The CEM is a

lead-scintillator calorimeter with a depth of about 18 radiation lengths; the CHA

is an iron-scintillator calorimeter with a depth of approximately 4.7 interaction

lengths. The energy resolution of the CEM for electrons is

o(Er) 13.5 (37)
ET /Er(GeV)









while the average energy resolution of the CHA for charged pions is

7 -)(ET) 50 ED 3 (3-8)
ET /Er(GeV)

In Run II at CDF, a new forward scintillating calorimeter replaced the original

gas calorimeter. The forward region, 1.1 < IT]\ < 3.6, is covered by the Plug

Calorimeters which consist of lead-scintillator for the electromagnetic section

(PEM) and the region, 1.3 < ]\1 < 3.6 is covered by the iron-scintillator for the

hadronic section (PHA). The positions of the various plug calorimeter modules

with respect to detector T] are shown in figure 3-4 [23]. The PEM and PHA are

identically segmented into 480 towers of size which varies with T] (Aq x AO

0.1 x 0.13 at 1 < 1.8 and increases to AT/ x AO m 0.6 x 0.26 at T]1 = 3.6). The

energy resolution of the PEM for electrons is

-(ET) 16 .
E7 G-1 (3-9)
ET ET (GeV)

while the average energy resolution of the PHA for charged pions is

F -) 85, (3-10)
ET 7E7(GeV)

The region between the central and forward calorimeters, 0.7 < T]1 < 1.3, is covered

by an iron-scintillator hadron calorimeter (WHA) with similar segmentation to

the central calorimeter. The WHA has a depth of approximately 4.5 interaction

lengths, and a resolution for charged pions of

7(Er) 75 .
a-) 3 (3-11)
ET 7ET(GeV)

EM calorimeters are designed to contain most EM showers. However, some

EM shower energy may spill over into the hadronic calorimeter. It is also possible

that hadrons may begin to interact in the EM calorimeter. In some analyses the

features of the EM and hadronic showers are important for particle identification.









However, for the inclusive jet cross section, no distinction based on particle

type is necessary. Only the total energy deposited within a Y Q region of the

calorimeter is important to this jet measurement. Therefore, the energy deposited

in electromagnetic and hadronic sections of each tower are combined into 1 -

towers. The position of each section is defined by the vector joining the interaction

point to the geometrical center of the section. Each section is assumed to have no

mass (i.e., E oc P), and the sum of the momentum four-vector for the EM and

hadronic section are taken as the momentum vector of the physics towers. Physics

towers are the detector objects which are clustered into jets.




































ELECTROMAGNE
CALORIMETER
EM SHOWER
MAXIMUM CHAM


II3


31111 I


0 1m 2m 3m 4m 5m / / k -

0 2 4 6 8 10 12 14 16 ft.


Figure 3-3. CDF detector. The CLC, COT, and the calorimeters are the

important components for this analysis. They are described briefly

in the text.


II




















CDF Tracking Volume


, 300


...n = 2.0





-n = 3.0


-30


I .5 \ 1.0 1.5
SVX II INTERMEDIATE
5 LAYERS SILICON LAYERS


Figure 3-4. Longitudinal view of the CDF II Tracking System and plug
calorimeters. The position of the calorimeter modules with respect
to detector T is shown in the figure.















CHAPTER 4
JET DEFINITION

Cross sections of the hard partonic scattering processes can be calculated

to a fixed order in perturbation theory. However, due to hadronization of the

partons, and other aspects of the hadron collider environment (see section 2.6),

it is not clear what physical observables (if any) will yield a clear interpretation

of the original hard interaction. Jet clustering algorithms are designed to cluster

the complex structure of final state objects from each collider event into jets.

These jets must be a map to the physical properties of the partons from the hard

scattering to be a useful construct for comparison with theoretical predictions.

Currently at CDF there are three jet clustering algorithms in use:

* JetClu: JetClu [26] is the cone algorithm used in Run I at CDF. Cone
algorithms combine objects based on relative separation in Y 0 space,
AR VAY2 + A52.

* Midpoint: The Midpoint algorithm is a cone algorithm similar to JetClu. It
has certain advantages over JetClu and is the cone algorithm of CDF in Run
II.

* KTr: The KT algorithm combines objects based on their relative transverse
momentum as well as their relative separation in Y 0 space [27].

In this chapter the Midpoint jet clustering algorithm will be described in

detail. For completeness, the KT algorithm and JetClu algorithm will also be

summarized. After defining the algorithms some technical issues on the topic of jet

definition will be discussed.

4.1 The CDF Midpoint Jet Clustering Algorithm

When clustering jets with a cone algorithm a cone size, Rcone, must be

specified. The cone size determines the maximum amount of angular separation









particles can have in Y 0 space and still be combined into a jet. At CDF, jets are

reconstructed with three different cone sizes: 0.4, 0.7, and 1.0. What cone size is

useful depends on the details of the role that jets p1 : in an analysis. This analysis

uses a cone size of Rc,,e 0.7.

The first step in any jet algorithm is to identify the list of objects to be

clustered. In this analysis, jets will be clustered at four different levels (see

figure 2-10). The list of objects to be clustered is different in each case:

* Detector level (data or MC with detector simulation): four-vectors of the
calorimeter physics towers are used as the basic elements of the clustering. To
reduce the effect of electronic noise, only towers with PT > 100 MeV/c are
included in the list.

* Particle or hadron level (\IC): four-vectors of the stable particles (i.e.,
hadrons) are the basic elements to be clustered.

* Parton level (M\ C): four-vectors of the partons before hadronization are
clustered into jets. For MC such as HERWIG or PYTHIA this will include
the many quarks and gluons from the parton shower and multiple parton
interactions.

* Parton level (NLO parton level): four-vectors of the partons are clustered into
jets. There are at most three partons in the list at NLO.

The next step is to identify a list of seed objects. This is a subset of the list

of objects to be clustered with the extra requirement that the PT of the object be

above some threshold (1 GeV/c). It would be preferable theoretically to include

seeds corresponding to every point in Y 0 space; however, by searching for jets

only at seed locations the CPU-time to run the algorithm is greatly reduced. At

each seed location a cone of radius, R = Roe,/2, in Y 0 space is constructed.

This reduced cone size, or search cone, is not a feature of the standard Midpoint

algorithm. The standard Midpoint algorithm uses R = Rcce for all clustering steps.

CDF uses a modified version of the Midpoint algorithm which is often called the

Search Cone algorithm. The momentum four-vectors of all objects located in the









search cone are summed. This four-vector sum is called the centroid of the cluster.

The four-vector of the centroid is then used as a new cone axis. From this axis a

new cone is drawn and the process of summing up the four-vectors of all particles

in the cone is conducted again. This process is iterated until the cone axis and

the centroid coincide, indicating that the configuration is stable. Once the stable

configuration is found, the cone axis is expanded to the full cone size (R = Rcoe,),

and the four-vector of a protojet is formed by adding up all of the four-vectors of

the objects in the expanded cone. The expanded cone is not iterated for stability.

This procedure of finding stable cones is applied to every object in the seed list.

The next step in the algorithm is the one for which it is named. Additional

seeds are added at the midpoint between all protojets whose separation in Y -

space is less than two times Rcone (i.e., if AR < 2Rcone). A cone of radius

R = Rcone is then drawn around the midpoint seed and iterated until a stable

configuration is found. If this configuration is not already in the list of protojets,

it is added to the list. After all midpoint seeds have been iterated to stable cone

configurations, the list of protojets is complete. The process of adding seeds at the

midpoint between all stable cones reduces the sensitivity of the algorithm to soft

radiation.

It is possible that many of the protojets will overlap (i.e., objects may appear

in more than one protojet). Overlapping protojets must be split or merged to

make sure that the same object is not included in more than one jet. Before

splitting and merging begins, protojets are sorted according to their PT. If the

sum-PT (four-vector sum) of shared objects between two protojets is more than the

fraction, fmerge = 0.75, of the protojet with lower PT, then the two protojets are

merged. If the sum-PT of shared objects between two protojets is less than fmerge,

then the shared objects between the two protojets are split and assigned to the

closer cone in Y 0 space. The Midpoint algorithm may be summarized:









1. A list of all objects to be merged into jets is constructed.

2. A seed list which includes only objects with transverse momentum greater
than 1 GeV/c is generated.

3. Stable cones are constructed around each seed (R = R.ee/2).

4. The radius of all stable cones is extended to R = Rcme.

5. An additional seed is added for midpoints between each pair of stable cones
separated by less than twice the cone radius. Each additional seed is searched
for stable cones (R = Rme) that have not already been discovered.

6. The stable cones are PT-ordered and splitting and merging is performed for
overlapping cones.

At NLO parton level, there are at most three particles in the event. In this

case, the idea of a jet algorithm becomes very simple. The algorithm must decide

if the two particles which are closest in Y 0 space should be combined or not.

There is no complicated splitting and merging step needed. Here, the default

Midpoint algorithm would merge any two particles which are separated by less than

2 x Rme in Y 0 space. At the detector level it is observed that particles are

almost never merged by the Midpoint algorithm if they are separated by 2 x Rme.

For this reason, Rsep has been introduced for NLO calculations with the Midpoint

algorithm. At NLO, the algorithm is modified by Rsep so that particles are only

merged if their separation in in Y 0 space is less than Rsep x Rcme. A value of

Rsep = 1.3 is consistent with detector level studies. Rsep p v the role of splitting

and merging at the NLO parton level [26].

4.2 Other Jet Clustering Algorithms

JetClu is a cone algorithm similar to the Snowmass parton clustering

algorithm [28]. The steps for jet clustering at the detector level used by JetClu

are described below:

1. An ET ordered list of seed towers with ET >1.0 GeV is created .









2. Beginning with the highest ET tower, preclusters are formed from .,i.] I,'ent
seed towers, provided that the towers are within a 0.7x0.7 window centered
at the seed tower. Any tower outside of this window is used to form a new
precluster. This clustering step is dependent on detector geometry, and it
cannot be conducted in the same way at the hadron or particle level.

3. The preclusters are ordered in decreasing ET and the ET weighted centroid is
formed by adding the energy from all towers with more than 100 MeV within
R=0.7 of the centroid.

4. A new centroid is calculated from the set of towers within the cone and a new
cone drawn about this position. Steps 3 and 4 are iterated until the set of
towers contributing to the jet is stable. The property of the JetClu algorithm,
that all towers included in the original cluster remain in the cluster even when
they no longer lie within the cone radius, is called R.l/. I/. l:in,

5. C'!u-I, i are reordered in decreasing ET and overlapping jets are merged if
they share >7 .' of the smaller jet's energy. If they share less, the towers in
the overlap region are assigned to the nearest jet.

The KT algorithm handles particle combination much differently than the

cone algorithms described so far. The procedure for combining objects into jets is

exactly the same at the parton, hadron, and detector levels for the KT algorithm

and they are described below:

1. The quantities Y, 4, and PT are constructed for each object in the list of
objects to be combined.

2. For each object in the list di PT is calculated, and for each pair of partons
the quantity dij im((di)2, (d)2) (Rj)2/D2 is defined. The D parameter
pl-,i- a similar role in the KT algorithm as Rcoe does in the cone algorithms.

3. Find the minimum of all di and dij.

4. If the minimum is one of the dij remove particles i and j from the list
and replace them with an object defined by the sum of their momentum
four-vectors.

5. If the minimum is one of the di then remove it from the list to be checked for
merging and add it to the list of jets.


6. If any particles remain in the list, go to second step above.









4.3 Jet Definition Issues

The JetClu algorithm used in Run I at CDF has several flaws. One in i i"r

problem with JetClu is that the first step of the algorithm, when applied at

the detector level, is to "form preclusters from .,.li ,.:ent seed to .- i~ This step

is dependent on the detector granularity. This type of dependence makes it

impossible to define an equivalent algorithm at the parton and hadron level. The

Midpoint algorithm clusters towers based on their separation in Y 0 space and is

therefore less sensitive to the detector granularity.

As mentioned above, it is preferable to place seeds at every point in Y -

space. This is because the use of seeds adds an infrared sensitivity. Fluctuations

in tower energy due to soft radiation can push the energy just above or below the

seed threshold. In this way the jets clustered in a given event may depend on soft

radiation. This sensitivity can be minimized if the seed threshold is low, and only

high PT jets are studied. If high energy jets ahv-b have at least one physics tower

with PT far above the seed threshold, then recognizing the seed is not sensitive to

fluctuations due to soft radiation. The sensitivity introduced to this measurement

is not significant, since only jets with PT greater than 54 GeV/c are included.

The CDF Midpoint jet clustering algorithm is less sensitive to soft radiation

than JetClu [26]. The reason this is true is illustrated in figure 4-1 [29]. In

the configuration on the left, two seeds may have lead to two stable cone

configurations. If a seed had been constructed between these two cones, a different

configuration may have been found. A stable cone may have been found which

included the other two objects so that only one jet was constructed in the event.

Soft radiation can push objects just below or just above seed threshold. In this

way, the event topology is sensitive to soft radiation. The Midpoint algorithm

reduces this sensitivity by adding an extra seed at the midpoint between all

stable cones. The result of the midpoint seed is that the event topology is not as









sensitive to fluctuations above or below seed threshold caused by soft radiation.

For this reason, and the reduced dependence on detector granularity, the Midpoint

algorithm has replaced JetClu as the cone clustering algorithm used at CDF in

Run II.












Figure 4-1. The Midpoint jet clustering algorithm checks for a stable cone
configuration at the midpoint of all stable cones located from searching
at seed locations. In this way the algorithm is less sensitive to
fluctuations due to soft radiation.


At CDF, it was observed that in some events, after clustering jets with the

standard Midpoint algorithm, there were significant clusters of energy which

were not included in any jet. An event display of an event in which this occurs is

shown in figure 4-2 [26]. The figure shows the energy deposited in each tower of

the detector on the TI 0 plane. The clusters of energy shown in black were not

clustered into any jet in this event. These clusters of energy which are not included

in a jet are referred to as dark-towers. Dark towers occur because a cone that

started from a seed within the dark-tower cluster moves away from this cluster of

energy towards a larger cluster of energy (i.e., the cone migrates to a neighboring

cluster of energy).

The effect of dark towers is significant. For example, approximately two

percent of events with a 400 GeV/c jet have more than 50 GeV/c of un-clustered

transverse momentum. Because this dark-tower effect is not included at NLO, it

was decided that the issue of un-clustered energy needed to be addressed. The



















01

-2
-1
0 300
S1 200
2 100





Figure 4-2. Dark towers observed by the original Midpoint algorithm. Not all
clusters of energy in the detector were being included in a jet.


solution applied at CDF is to use a smaller initial search cone (Rcre,/2) (i.e., use

the Search Cone algorithm). The net result of using the Search Cone algorithm is a

five percent increase in the inclusive jet cross section, which is roughly independent

of jet PT [30].

The CDF Search Cone algorithm applied in this ,, in -i- is not perfect. It is

slightly more sensitive to tower and seed thresholds than the standard Midpoint

algorithm. Also, there is a sensitivity to the size of the the search cone. Varying

the search cone radius from R/2 to R/v2 leads to a variation in the cross section

by less than 2 percent. When comparing to NLO predictions the imperfections with

the Search Cone algorithm are minor when compared with the five percent shift in

the cross section caused by the dark towers. However, the Search Cone algorithm

has serious theoretical issues if applied to NNLO predictions. Finding a stable

configuration with the reduced cone size, and then expanding the cone without

further iteration yields an infrared sensitivity at NNLO [30].









Although the Search Cone is not perfect, and cone algorithms in general

have more issues theoretically than the KT algorithm, cone algorithms have some

advantages. With a standard cone algorithm the user has complete control over

what is included in the cone through the variable Rc,,,. The fact that the jet is

composed of a specified cone size in the detector is a useful property for making

corrections. For example, when correcting for multiple pp interactions in the same

bunch crossing a cone located randomly in the detector can be used to study the

extra energy that is included on average in a jet due to the extra interaction. This

cannot be done with a KT algorithm since it does not use a fixed cone size. It

is also useful to make measurements with two different types of algorithms. The

difference in results from different algorithms should be predictable by MC and can

be used to learn about the jet clustering properties of each algorithm.














CHAPTER 5
INCLUSIVE JET MEASUREMENTS

The prediction in the early 1970's, that jets of hadrons whose momentum

when summed up would be equal to the momentum of the initially scattered

partons, has lead to a rich history of theoretical predictions and jet measurements

at hadron colliders [31, 32, 33]. The first inclusive jet cross section measurement

at a hadron collider, as well as the first direct observation of a clear two-jet event

topology, came from the SppS collider at CERN [34, 35] with a center-of-mass

energy of 540 GeV (/ = 540 GeV). A detailed history of the evolution of

theoretical predictions and experimental measurements of the inclusive jet cross

section can be found in reference [36].

The measurement of the differential inclusive jet cross section at CDF reaches

the highest momentum transfers ever studied in collider experiments. Thus, it

is potentially sensitive to physics beyond the standard model [37]. Studying the

highest energy events at the Tevatron is equivalent to probing distances on the

order of 10-19 m. This measurement is probing distance scales more than one

thousand times smaller than the radius of the proton and is sensitive to whether or

not the quark has substructure [38]. The inclusive jet cross section measurement

is also a fundamental test of predictions of perturbative QCD [39, 40] over eight

orders of magnitude in cross section. Jet events in the central region of the

detector with transverse momentum higher than 530 GeV/c have a cross section of

approximately 30 fb (30 x 10-6 nb). These are among the smallest cross sections

ever measured at a collider experiment. The current measurement spans 600 GeV/c

in jet transverse momentum, and it can therefore be used to observe the running

of a8 in a single measurement. It is a common misconception that the QCD force











gets small at small distances. This is certainly not the case. The effective coupling

does decrease at high momentum transfer, but the effective force is inversely

proportional to square of the distance(F oc N ). In fact, the forces occurring in

high energy dijet events are of among the largest ever observed in a laboratory

environment.

Perhaps the most useful aspect of these measurements, is that they can

be used to constrain the proton PDF, which in turn improves the theoretical

predictions in all physics channels for experiments at the Tevatron and the future

experiments at the LHC. This is important because the probability of a gluon

carrying a large fraction of the momentum of the proton (i.e. large Bjorken-x or

high-x) is not well known [41]. In fact, the gluon PDF is the dominant source of

theoretical uncertainty in the inclusive jet cross section and many other processes

at hadron colliders. The uncertainty on the quark and gluon PDFs for Q =

500 GeV is shown as a function of Bjorken-x in figure 5-1.


Up Quark PDF Uncertainty (Q=500GeV) Gluon PDF Uncertainty (Q=500GeV)

I- I-
.21.4 .21.4
2 1.2 1.2 r
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7
x x

(a) (b)


Figure 5-1. Uncertainty on the up quark 5-1(a), and gluon 5-1(b), PDFs for
Q = 500 GeV as a function of Bjorken-x (momentum fraction carried
by the parton). The uncertainty is much larger on the gluon PDF
because this component of the proton is not probed directly in DIS
experiment.









The strongest constraints on the parton density functions come from deep

inelastic scattering experiments (DIS). The DIS experiments scatter electrons off of

protons. This process is depicted in figure 5-2 [9]. The electron interacts with the

quarks in the proton through the exchange of a virtual photon. Because photons do

not couple directly to gluons, the DIS experiments can not make strong constraints

on the gluon content of the proton. The strongest constraints on the gluon PDF

come from jet measurements at hadron colliders, such as the inclusive jet cross

section measurement discussed here.

k

k /
q






Figure 5-2. Dominant process in deep inelastic scattering experiments. The virtual
photon probes the quark content of the proton. Since photons do not
couple directly to gluons, this process can not be used to place strong
constraints on the gluon PDF.


Because the inclusive jet measurement in the forward region (large Y) probes

a kinematic range which is not expected to be sensitive to new physics, it should

lead to a powerful constraint on the gluon PDF. Dijet events produce high PT jets

at high rapidity when the momentum fraction of the two incoming partons do not

balance. In this topology, the high-x component of the proton is probed at a lower

momentum transfer than for an equivalent energy jet in the central region. In other

words, the high-x component of the proton can be probed at a lower energy scale

with forward jets. Because lower energy scales have been studied extensively in

the central region, and agree with the predictions of pQCD, one can be confident

that physics beyond the SM is not affecting the result. Since the forward region









jets is not as sensitive to new physics as the high PT jets in the central region,

measurements in the forward region are crucial to confirming that observations can

be attributed to PDF effects and not physics beyond the standard model.

The CDF experiment has a history of making important inclusive jet

measurements. In Run I ( = 1.8 TeV), CDF made several measurements of

the inclusive jet cross section in the central region of the detector (0.1 < Tr1 < 0.7)

using the JetClu cone jet clustering algorithm. In Run IA, with 19.6 pb-1 of data

collected during the period 1991-1993, an excess of data over theoretical predictions

was observed at high jet transverse momentum [42]. This result alluded at the

need for larger gluon content at high-x in the proton. The Run IB CDF central

region result, with 87 pb-1 of data collected during the period 1994-1995 [36], still

showed an excess at high jet PT. When these results were combined with the DO

result including the higher rapidity region [43], with 95 pb-1 of data up to T = 3.0,

it was confirmed that the gluon content at high-x had been underestimated in

the proton structure functions. This experience from Run I revealed a need to be

able to quantify the effect of PDF uncertainty on collider observables such as cross

section [44, 45].

The increased center-of-mass energy in Run II ( = 1.96 TeV) significantly

increases the jet cross section at high jet PT. Quantitatively, the cross section for

dijet production is approximately five times higher for jet PT of approximately

600 GeV/c. A new inclusive jet cross section measurement by CDF (with 385 pb-1

of data collected during the period 2001-2004), in the central region (0.1 < |Y <

0.7), using the Midpoint jet algorithm in Run II, has been accepted for publication

recently [46]. CDF also recently published an inclusive jet measurement in the

central region for jets clustered by the kr algorithm [27].

In this document, the Run I cone jet analysis is updated with approximately

ten times the integrated luminosity resulting in over 1 fb-1 of data. This is the






49


first inclusive jet measurement at CDF to include the forward region (|Y| <

2.1) using a cone algorithm for jet clustering. The techniques applied in this

measurement will be fully motivated and briefly described. For complete details

and all relevant distributions please see reference [47].















CHAPTER 6
DATA SAMPLE AND EVENT SELECTION

This analysis includes data taken from Summer 2001 until November 2005

and corresponds to an integrated luminosity of 1.04 fb-1. The jet data used in

this analysis were collected using four different paths of the CDF three-level trial '--r

system [48]. The rate at which collisions occur within the CDF detector is much

higher than the rate at which data can be collected and stored. Bunch crossings

occur at CDF at a rate of approximately 1.7 MHz, while data can only be written

to tape at about 75 Hz. Various triggers are designed to extract events that are

useful for physics analysis at CDF. The trigger is split into three levels, and at each

level the events which pass the trigger requirements are passed on to the next level.

The trigger requirements at each level must produce a rate reduction large enough

for processing at the next level to be possible.

The Level 1 jet trigger consists of two trigger streams; requiring a calorimeter

tri.-1--r tower to have ET > 5 GeV for the jet20 and jet50 tri.-. -. irS, and ET >

10 GeV for the jet70 and jetlOO triggers. At Level 2, the calorimeter towers are

clustered using a nearest neighbor algorithm. Four trigger paths with cluster

ET > 15, 40, 60, and 90 GeV are used. Events in these paths are required to pass

jet ET > 20, 50, 70, and 100 GeV thresholds at Level 3, where the clustering is

performed using the JetClu cone algorithm with a cone radius Rc,", = 0.7. The

CDF trigger, and all trigger studies required for this analysis are described in more

detail in Appendix A.

The selection criteria for the inclusive jet cross section is minimal, since all

jet events are included in the analysis. Since cosmic ray backgrounds originate

from outside the detector, the transverse energy deposited in the detector is not









balanced. Cosmic rays can therefore be efficiently removed by applying a missing

ET significance requirement (gzT). The ]ZT selection criteria varies according to

the jet sample and it is 4, 5, 5, and 6 GeV1/2 for jet20, 50, 70, and 100 tri.. i-

respectively. With any selection criteria it is possible that some jet events which

came from hard scattering events are removed. The percentage of real jet events

which survive the selection criteria is called the eff. ",. :, The YT requirement is

approximately 1(1 'O. efficient at low PT and varies to approximately 91i at the

highest jet PT included in the measurement. In order to ensure that particles from

the pp interactions are in a region of the detector with good tracking coverage,

primary vertices are required to be within 60 cm of the center of the detector

(I ZI < 60 cm). The efficiency of this requirement is measured to be 9. -'". at

CDF [49].

The jet20, 50, and 70 triggers are prescaled to avoid saturating the bandwidth

of the trigger and data acquisition system. Prescaling by a factor of n means that

only 1 out of n events satisfying the trigger requirement are stored to tape. The

jet70 trigger is prescaled by a constant factor of 8 for all data used in this analysis.

The prescales for the jet20 and 50 tri --. r-S have changed during the data taking

period considered. The effective prescales of the jet20 and 50 tri .-.r- i for all the

data were found to be 776.8 and 33.6, respectively, by luminosity-weighting the

inverse of prescale factors

1t1
S( ) (6-1)
Effective ttotal Pi

where Peffective is the effective prescale, total is the total integrated luminosity,

and Li is the integrated luminosity of a period when a prescale factor Pi is used.

As a cross-check of the jet20 and jet50 effective prescales, the cross section ratios

(before prescale correction) of jet70 to jet20 triggers and of jet70 to jet50 tri --. 1-

in the jet PT region where jet70 is efficient were studied. The results were found









to be consistent to better than 1 between the two methods. The distributions

for the number (i.e., jet ,;/'. /,1) of jets as function of PT in the central region

(0.1 < |Y| < 0.7) before and after correcting for trigger prescales are shown in

figure 6-1.

The inclusive differential jet cross section is determined as follows:

(6-2)
dPTdY AY f Cdt APT '

where Njet is the number of jets in the PT range APT, c is the trii._ r, ITL cut, and

vertex cut efficiency, f Cdt is the effective integrated luminosity which is corrected

for trigger prescales, and AY is the rapidity interval.

Because the jet t-r-i. S do not use the Midpoint algorithm they are not 10l'

efficient near the trigger threshold. A trigger efficiency greater than 99.5 is

required to include the jets collected by a given jet trigger. A complete study

of the trigger efficiency (i.e., turn-on curves) in all rapidity regions and the

study of the effective prescales is included in Appendix A. Figure 6-2 shows the

uncorrected (i.e., raw) inclusive differential jet cross section for the five rapidity

regions: |Y| < 0.1, 0.1 < |Y| < 0.7, 0.7 < |Y| < 1.1, 1.1 < |Y < 1.6, and

1.6 < |Y| < 2.1. This rapidity inning is roughly based on detector segmentation.

The region Y| < 0.1 corresponds to central crack region where the detector

modules meet, 0.1 < |Y| < 0.7 corresponds to the well understood central region

of the detector, 0.7 < |Y < 1.1 and 1.1 < |Y < 1.6 correspond to the region

where central calorimeter and the plug calorimeter are connected by the WHA, and

1.6 < |Y| < 2.1 corresponds to the plug (forward) region of the calorimeter.































0 100 200


10 10
109
108
107
106
105
104

103
102
10
1-
0


0.1<|Y|<0.7
CDF Run II Prelir


300 400 500 600 700
pUncorrected (GeV/c)


L=1.04 fb-1 Midpoint (R cone=0.7)


- Jet20 (prescale=776.8)
- Jet50 (prescale=33.6)
Jet70 (prescale=8)
- Jet100 (prescale=1)






minary


100 200 300 400 500 600 700
pUncorrected (GeV/c)


Figure 6-1. Jet yield distributions as a function of PT in the central region
before, 6- (a), and after, 6- (b), correcting for tri --. r prescales.


L=1.04 fb' Midpoint (Rone=0.7)
S-- Jet20 (prescale=776.8)
Jet50 (prescale=33.6)
Jet70 (prescale=8)
-Jet10 (prescale=1)






0.1<|Y|<0.7
CDF Run II Preliminary



























-Uncorrected Cross Section (statistical errors only)

f L=1.04 fb-1 Midpoint (Rcone=0.7)


.- *.. |Y|<0.1
0.1<|Y|<0.7
S--- 0.7<|Y|<1.1
S-* -_, 1.1<|Y|<1.6
-.--i
,-*---,--__ 1.6<|Y|<2.1




CDF Run Preliminary -
-- -r-v


-C--
CDF Run II Preliminary I
. I . . . . .


100 200 300


r 102


0 -


N
. 10
1

.10-1


10 -

10


10-7

10-8


Figure 6-2. Measured raw jet cross section for the five rapidity regions. The raw jet
cross section has not been corrected to remove detector effects.


400 500 600 700
pUncorrected (GeV/c)
T





m


.















CHAPTER 7
JET CORRECTIONS

The jet energy measured by the calorimeters must be corrected for detector

effects, such as calorimeter non-linearity and energy --i. ii:1-. before comparing

experimental measurements with theoretical predictions. In addition to detector

effects, corrections must also be made for some physics effects such as pileup and

the underlying event before the measurement may be compared with NLO parton

level perturbative predictions.


-100 -80 -60 -40 -20 0 20
pCalpHad
T --T


Figure 7-1. Difference between calorimeter jet PT and hadron level jet PT for three
different jet PT ranges. The calorimeter jet PT is systematically lower
than the hadron level jet Pr. Also, there is a smearing effect due to
the fluctuation in the energy measured in the calorimeter for a given
hadron level jet.









Two detector effects, which the jet data must be corrected for, are illustrated

in Figure 7-1. The difference between calorimeter jet transverse momentum (Pcal)

and hadron level jet transverse momentum (PFHad) for three different jet PT ranges

is shown in the figure. Hadron level jets were matched to calorimeter jets by their

separation in Y 0 space (AR < 0.7) using MC generator results which were

passed through the CDF detector simulation. The distributions peak below zero

because P/al is systematically lower than PTHad. This reflects the non-compensating

nature of the sampling calorimeters. The hadronic calorimeter was calibrated based

on charged pions from a test beam with a transverse momentum of 57 GeV/c.

Only pions which did not interact with the EM calorimeter were included in the

calibration. In real jets however, a large fraction of hadrons do interact with the

EM calorimeter. Because the EM calorimeter is calibrated based on electrons its

response to hadrons is lower. This lowers the overall response to single hadrons.

This effect is larger for low transverse momentum particles because they interact in

the EM calorimeter more often, and it can therefore contribute to the non-linearity

of the calorimeter response to hadrons. Hadronic showers have a larger fraction

of neutral pions when the incident hadron has a higher transverse momentum.

Because the calorimeter has a higher response to EM showers (i.e., 7 decays),

this also contributes to the nonlinear nature of the calorimeter response to charged

hadrons. In general the calorimeter response goes up as transverse momentum of

the incident hadron increases and is not linear. This causes a systematic shift down

in the energy response to jets because they include multiple hadrons with lower

transverse momentum, rather than one hadron with the full jet PT.

The jet energy smearing effect is caused by the limited jet energy resolution

of the calorimeters, and is reflected in figure 7-1 by the width of the peak.

Fluctuations in shower development due to the probabilistic nature of the









interactions between the particles in the jet and the detector material cause

the detector response to particle jets with a fixed energy to vary.

Figure 7-2 illustrates the flow of the jet correction scheme used to obtain

results corrected to the hadron or parton level. First, an 9l-dependent relative

correction is applied to the data and MC in order to equalize the response of

the CDF calorimeters to jets in Y. The equalized jet PT is then corrected for

the pileup effect. Then, the absolute correction is applied to correct on average

for the hadron energy that is not measured by the calorimeter. After that, the

hadron and calorimeter level jet PT distributions are compared in Monte Carlo to

derive a bin-by-bin correction in order to remove resolution effects. This is called

unfolding. At this point, the data have been corrected to the hadron level. In order

to compare directly with pQCD predictions, the effects of the underlying event

and hadronization need to be removed from the data. After this final correction,

the data have been corrected to the parton level. The Monte Carlo simulation

used to derive the corrections, and the details of each correction step will now me

described.

7.1 Monte Carlo Simulation

The parton shower MC programs PYTHIA 6.2 [50] and HERWIG 6.4 [51],

along with the CDF detector simulation, are used to derive the various corrections

which are applied to the data, and to estimate systematic uncertainties. Structure

functions (i.e., parton distribution functions) for the proton and anti-proton are

taken from CTEQ5L [52]. The CDF detector simulation is based on GEANT3 [53]

in which a parametrized shower simulation, GFLASH [54], is used to simulate the

energy deposited in the calorimeter [55].

The GFLASH parameters are tuned to test-beam data for electrons and high

momentum charged pions and to the in-situ collision data for electrons from Z

decays and low momentum charged hadrons. However, the CDF simulation does




























Figure 7-2. Flow diagram for the jet corrections used in the inclusive jet analysis.
Correction steps are shown in red, while the hadron level and parton
level corrected states are shown in blue. Relative, pileup and average
(absolute) corrections are applied directly to the jet PT before inning.

not describe energy deposition in the calorimeters perfectly, especially in the

regions corresponding to the plug calorimeters and cracks between calorimeter

modules. Since the MC simulation is used to derive various jet corrections to be

made on the data, differences between the real calorimeter response to jets and

the calorimeter simulation need to be well understood. Differences in the relative

jet energy response and jet energy resolution between the collision data and MC

simulation events were investigated using dijet PT balancing in dijet events [55] and

the bisector method [56], respectively. The dijet PT balance and bisector studies are

briefly described below. More details can be found in Appendix C.

Comparisons of dijet PT balance between data and MC reveal that the relative

jet energy scale versus T] is different between data and MC, and that the difference

depends on jet PT at high rapidity (|Y| > 1.1). For example, the jet energy scale

in the plug calorimeter region is higher in MC than in data by ~ '' and the









difference increases slightly with jet PT. This difference is accounted for by the

relative corrections which are described in detail in section 7.2.

The bisector method allows one to compare the energy resolution of the CDF

detector and the CDF detector simulation. In the central region (0.1 < |Y| < 0.7)

the detector simulation reproduces the detector jet energy resolution well. In the

other rapidity regions, small differences were found between data and MC. To

account for these differences, modifications to the unfolding factors were derived

and can be found in Appendix C. The corrections are less than ,'. in most bins,

and less than 10''. in the most extreme cases.

7.2 Relative Correction

The calorimeter response to jets is not flat in detector-q (Tld). The non-uniformity

in qd arises from cracks between calorimeter modules and also from the different

energy responses of the central and plug calorimeters. The leading two jets in dijet

events are expected to be nearly balanced in PT in absence of QCD radiation.

This dijet PT balance provides a useful tool to study the jet energy response as a

function of Urd, and to derive the relative correction. To study the dijet balance,

a dijet event topology is required. The CDF calorimeter response to jets is well

understood and almost flat in Ud in the central region. For this reason, a jet with

0.2 < Ird| < 0.6 is required in the event, and this jet is defined as the trigger jet.

The other jet in the event is defined as a probe jet. Figure 7-3 [55] shows a measure

of the dijet balance, 3 = pT o / Ptr9, observed in the CDF calorimeter and in

detector simulation, as a function of detector-q of the probe jet. If the response of

the calorimeter was the same in all regions of the detector, this distribution would

be approximately flat and very close to one. The relative correction is applied to

equalize the response in T] of the CDF calorimeter to jets.

To determine the qd-dependent relative jet energy correction, the PT balance

of the probe and tri .,r jet is studied as a function of the probe jet Tr [55]. The





60


S ptProbe/pttrig CDF Run 2 Preliminary







0.9




0.6 --------------i-- --dijet5O MC 5.3.1pre2
0 .7 .................................................
: I- jet50 data 5.3.1 pre2
0.6 .................... dijet50 M C 5.3.1 pre2 I................

0.51 11 1 1 1 .... i .... .... I,,
-3 -2 -1 0 1 2 3
Jet

Figure 7-3. Degree of dijet balance (/3) observed in the CDF calorimeter and and
MC with CDF detector simulation as a function of detector TI for the
probe jet in the event. The relative correction is applied to equalize the
response in TI of the CDF calorimeter to jets.

T/d-dependent relative corrections are obtained by making a fit to the 3 distribution
at a fixed jet transverse momentum. Since the relative jet energy response is
different between data and MC (see figure 7 3), corrections are derived separately
for data and MC. The relative corrections for this analysis were derived at a fixed
value (PT = 117.5GeV/C) of jet PT so that the difference in the PT dependence
of the response observed in data and MC simulation could be handled more
directly. More details regarding the relative jet energy corrections can be found in
Appendix B.
As mentioned earlier, the data-MC difference in the relative jet energy scale
depends on jet PT for IY > 1.1. Therefore, a Pr-dependent correction derived
for the two corresponding rapidity regions is also applied to the MC in order to
force the 3 distributions to agree with data for all jet PT. Requiring a dijet event









topology for high PT jets in the forward region greatly reduces statistics. The lack

of statistics in data and simulated MC lead to a significant uncertainty in this

Pr-dependent correction. Details of the Pr-dependent correction to the dijet Pr

balance in the MC are included in Appendix C.

7.3 Pileup Correction

Hard scattering events with additional pp interactions in the same bunch

crossing produce additional particles which can contribute to the jet energy in an

event. The number of reconstructed vertices is a good estimator of the number of

inelastic collisions occurring in a bunch crossing. The correction for the additional

pp interactions is derived by studying the Pr measured in a randomly chosen cone

in Y 0 space as a function of the number of vertices reconstructed in minimum

bias events. The minimum bias trigger only requires that there be coincidence

in the CLC on both sides of the detector. This trigger criteria is effectively only

requiring that there was an inelastic scattering within the bunch crossing. The

PT in the randomly located cone scales linearly with the number of additional

vertices in the event. The pileup correction is derived from the slope of this line,

and estimates the average amount of transverse momentum to be subtracted from

the jet PT per additional vertex reconstructed in the event. The pileup correction is

approximately 1 GeV/c for each additional vertex in the event [55, 57].

7.4 Absolute Correction

As hadrons pass through the CDF calorimeter, all of their energy is not

collected. This effect is mostly due to the non-compensating nature of the

calorimeters [discussed above]. The absolute correction corrects the jet for the

average energy loss and is derived by comparing hadron level and calorimeter

level jets using PYTHIA and the CDF detector simulation. Hadron level

and calorimeter level jets are matched by their position in the Y 0 space

(AR < 0.7). The average hadron level jet energy is then studied as a function









of the calorimeter jet energy in each rapidity region. This distribution is fit to a

fourth order polynomial and the fit is applied as a correction to the PT of each jet

in the data sample. The correction is of the order of 21 '. for P /l ~ 50 GeV/c and

decreases to the order of a few percent for P ~ 600 GeV/c. This correction is

slightly different for each rapidity region.

7.5 Unfolding Correction

The next step in correcting the jet PT distribution to the hadron level is the

unfolding correction. It removes smearing effects due to the finite energy resolution

of calorimeter and accounts for the efficiency of the event selection cuts. After

the absolute correction has been applied, PT distributions from PYTHIA MC at

hadron level and calorimeter level are compared on a bin-by-bin basis to derive the

multiplicative correction. The selection criteria, Z vertex position and lZT cuts,

are applied at the calorimeter level, but not at the hadron level. These unfolding

factors are slightly different for each rapidity bin; they vary from roughly 1.4 at low

jet PT to just below 2.5 at high jet PT.

After this final correction is applied to the data, the measurement has been

corrected to the hadron level. It is now possible to compare the data directly

with MC at the hadron level. In all rapidity regions, the PYTHIA transverse

momentum distributions fall off slightly faster than the data at high PT. This is

due to the fact that the PYTHIA samples were generated with the CTEQ5L PDF,

which do not include the enhanced high-x gluon distribution that was required to

fit the Run I inclusive jet data [36, 43]. If PYTHIA is used to correct the data

back to the hadron level, the shapes of the predicted PT distributions should be the

same as the data in order to avoid introducing any bias with the corrections.

The ratios of data corrected to the hadron level to the PYTHIA predictions

are fit to polynomials. These fits are used to re-weight the PYTHIA PT

distributions; thereby, forcing the shapes of the PT distributions of the MC to









agree with the data. The unfolding correction factors obtained from the weighted

PYTHIA distributions are applied to the data. The modification due to the

re-weighting of PYTHIA is less than 1 in all rapidity regions except for the

region 0.7 < |Y| < 1.1. In this region at high PT the correction is still less than ,'.

7.6 Hadron to Parton Correction

Before hadron level results can be directly compared to theoretical predictions

of pQCD at the parton level, the effects of the UE and fragmentation must be

removed. The hadron to parton level correction is obtained from comparing

PYTHIA-Tune A [58] MC results at the hadron and the parton level to derive a

bin-by-bin correction.

Tune A was tuned to fit the underlying event observables measured at CDF

in Run I 1 It is used for all PYTHIA calculations mentioned in this text, but is

especially important for the UE correction [15, 18]. Multiple parton interactions are

turned off in PYTHIA to generate the parton level distributions. The hadron to

parton level correction for the central region is shown in figure 7-4. The corrections

for the other regions are similar in magnitude.


















1 PYTHIA-Tune A implies that the following parameters are set in PYTHIA
(CTEQ5L): PARP(67)-4, MSTP(82)-4, PARP(82)-2, PARP(84)-0.4,
PARP(85)-0.9, PARP(86)-0.95, PARP(89)-1800, PARP(90)-0.25.




















1.5
1 1.4- Midpoint Rcone=0.7, fmerge=0.75, 0.1<|Y|<0.7
13 Hadron- to Parton-level Correction
1.2-
Uncertainty
1.1 -
1
0.9
0.8 -
0.7-
0.6 O- CDF Run II Preliminary
0 .5 .. . I . I . I . .- 1
"0 100 200 300 400 500 600 700
pJET (GeV/c)

Figure 7-4. Hadron to parton level correction applied in the central region. The
difference between HERWIG and PYTHIA predictions for this
correction is conservatively taken as the systematic uncertainty (shaded
bands).














CHAPTER 8
SYSTEMATIC UNCERTAINTIES

The uncertainty of the jet energy scale (JES) is the dominant source of

systematic error on this measurement. Imperfections in the tuning of the detector

simulation for the central calorimeter (0.2 < |rd| < 0.6) energy response make

the largest contribution to this uncertainty. In this region, the jet energy scale is

known to better than :'-. and has been expressed in a functional form [55]. The

corrected jet PT, in the PYTHIA MC with detector simulation, was varied up and

down according to this parametrization. The resulting distributions were compared

with the central value in order to derive the systematic uncertainty due to the JES.

Even though the jet energy scale is know to better than ::'. when convoluted with

the steeply falling jet PT distributions, the uncertainties on the cross section are

large. They vary from approximately 10't. at low jet PT up to as high as ,i i'. at

high transverse momentum in some rapidity regions.

There is an additional uncertainty on the jet energy scale in the higher

rapidity regions (|Y| > 1.1). Statistics are limited when a dijet topology is required

at high jet transverse momentum; as a result, the PT -dependent correction to the

MC, based on dijet PT balance, is not very well constrained at high jet PT This

uncertainty is approximately 40'. in the highest transverse momentum bins, and is

discussed in more detail in Appendix C.

The remaining sources of systematic uncertainty in this analysis are summarized

below:

* Unfolding: The unfolding correction is sensitive to the momentum distributions
of particles within jets. HERWIG and PYTHIA rely on different fragmentation
models; therefore, the difference between their unfolding correction can be
used as a measure of the sensitivity of this correction to the fragmentation









model. The systematic uncertainty on the measured cross section is taken
from the ratio of the unfolding factors obtained from PYTHIA and HER-
WIG. The difference in the unfolding factors, as obtained with weighted and
un-weighted PYTHIA, is taken as additional systematic uncertainty.

* Jet energy resolution: Due to the sharply falling spectrum of the inclusive
jet cross section, any imperfection in the jet energy smearing of the detector
simulation will affect the unfolding correction. The bisector study revealed
that the resolution difference between data and MC varied by approximately
1(0'. with no dependence on the transverse momentum of the jets. This
variation was taken as the uncertainty on the resolution of the MC response
to jet energy. The calorimeter level jets in the PYTHIA simulation have
been smeared by additional Gaussian of width equal to 10' of the nominal
simulated resolution. In other words, the calorimeter jet PT was modified by

Psys P1 + ,mc x Gauss(0, v2 -), (8-1)

where F was taken as 1.10, cr is the nominal resolution for jets in the CDF
simulation, and Gauss(0, F2- 1) is a number randomly pulled from a
Gaussian distribution of width F2- 1 and centered at zero. The resulting
PSYS distribution was then compared with the nominally smeared result, Pr",
to obtain the systematic uncertainty on the cross section. This uncertainty is
small all rapidity regions. It is less than 5'. for most jet transverse momentum
and still less than 10' in the highest jet PT bins.

* Pileup correction: The pileup correction applied in this analysis was obtained
from minimum bias data. The energy away from jets in dijet events, electrons
in (W -- ev) events, and photons and jets in photon-jet events, as a
function of the number of reconstructed vertices was also studied. The
quoted uncertainty of 31i '. on the pileup correction covers variations from all
of these cross-checks [55]. A Si ', error in the pileup correction results in an
uncertainty of less than 2;'. on the cross section measurement.

* Luminosity: There is a ,'. uncertainty in the normalization if the cross
section. This is a direct consequence of the uncertainty in the measurement of
the luminosity at CDF [24].

* Hadron to parton level correction: The systematic uncertainty on the hadron
to parton level correction is estimated from the difference in the predictions for
this correction from HERWIG and PYTHIA. HERWIG does not include
multiple parton interactions in its underlying event model, and instead relies
on initial state radiation and beam remnants to populate the UE. PYTHIA
includes MPI as well as the components included in HERWIG. PYTHIA
predicts a larger correction at low PT due to MPI. The difference between the
HERWIG and PYTHIA prediction for the hadron to parton level correction









is a conservative estimate of the systematic uncertainty since PYTHIA is
known to reproduce the UE observables at CDF better than HERWIG [18].
The size of this uncertainty is similar in all rapidity regions, and is represented
by the yellow band in figure 7-4 for the central region. It is on the order of
15'. at low jet transverse momentum and is negligible for higher jet PT .

The uncertainty in the jet energy scale leads to the largest systematic error on

the inclusive jet cross section and is limited by the simulation of the calorimeter

response. The total systematic uncertainty is the quadratic sum of all uncertainties

listed above. When results are corrected to the parton level the hadron to parton

level systematic must be included, and this error is significant for low jet transverse

momentum.














CHAPTER 9
THEORETICAL PREDICTIONS AND UNCERTAINTIES

In order to use the inclusive jet cross section to extract information on the

structure of the proton, the partonic cross section must be known. Any uncertainty

on this perturbative calculation will limit the precision with which the parton

distribution functions can be extracted. For this reason, it is important to go

beyond the leading log predictions of PYTHIA. The NLO predictions have a

smaller dependence on the factorization and renormalization scale, and should also

be more precise since they are at a higher order in perturbation theory.

Several programs are available which make predictions for the inclusive

jet cross section at next-to-leading order parton level. Three similar examples

which will be mentioned in this document are: EKS [59], JETRAD [60],

and FASTNLO [61, 62, 63, 64]. These programs do not include the effects

of hadronization, the underlying event, or parton showers. They include all

diagrams that contribute to the NLO cross section. At NLO it is possible to have

2 -+ 2 processes like those shown in figure 2-7, and the one-loop diagrams with

equivalent final states. It is also possible at NLO to have 2 3 processes, where

an additional parton has been radiated from one of the legs or propagators in any

of the diagrams of figure 2-7. These programs have at most three particles in the

final state for each event. The calculations are done in the mass-less limit with

five quark flavors (u, d, s, c, b, and their anti-particles), and do not include any

processes besides the ones outlined above. Jets from other processes such as Z

boson decays to hadrons, W boson plus jets, and top quark decays all contribute

jets which are included in the inclusive jet measurement. However, the QCD cross

section is so large in comparison to these processes that it is a good approximation









to neglect them. For example, the tt production component of the jet sample is

approximately 0.01 [65]. Contributions from other processes should even be

smaller [36].

The predictions of the programs listed above depend on many input

parameters such as: the method for clustering partons, factorization and

renormalization scale, and parton distribution functions. There is an uncertainty on

the theoretical prediction related to each of these input choices.

At CDF, the Midpoint algorithm is modified for NLO parton level calculations

in order to mimic the splitting and merging step of the Midpoint algorithm

(Rsep = 1.3). The value of 1.3 is considered a reasonable choice by CDF; however,

other values for Rse, close to this value are also reasonable. The effect, on the

NLO cross section prediction of FASTNLO due to varying the parameter Rsep

is shown in figure 9-1. Increasing Rsep from 1.3 to 2 increases the cross section,

while decreasing Rse, to 1 (equivalent to the KT algorithm at NLO parton level)

decreases the cross section. This variation from of 1 to 2 represents the maximal

reasonable range which Rsep can be varied. The size of this effect is similar in

all rapidity regions, and is never greater than 5'. away from the prediction of

R sep 1.3.

The factorization scale, tif, is often taken to be one half of the transverse

momentum of the jet. This is convenient because this is the scale used to determine

the proton structure functions by the CTEQ group. Using [if = PT or [if = 2P

gives approximately 1' and 20 smaller predictions for the cross section,

respectively [46]. However, if one extracted parton density function fits based on

Pf = PT or if 2P f'in order to use a consistent value of pf for the PDF and the
calculation, the dependence would be reduced.

The dominant theoretical uncertainty, on the inclusive jet cross sections at

NLO, is due to the error introduced into the calculation by the uncertainty on









the proton structure functions. The Run I inclusive jet experience revealed the

need for tools which could quantify this effect. Experimental constraints must be

incorporated into the uncertainties of parton distribution functions before they

can be propagated through to errors on the predictions of observables. In recent

years, tools [66] such as the Hessian Method [67], have been developed to make

error propagation to final state observables possible. The CTEQ6.1M [37] error

sets have 41 PDF members: a central value, and 40 error set members. The error

set members correspond to 20 eigenvector directions which have been varied in the

positive and negative directions. The eigenstates are obtained from diagonalizing

the Hessian error matrix. The matrix is obtained by varying each parameter used

in the global fit within the tolerances of the experimental data included. In order

to approximate the PDF uncertainty on the prediction of a physical observable

with this method, the observable must be calculated with each PDF set member

(i.e., the observable must be calculated 41 times). After the prediction has been

calculated with each member, the uncertainty on the observable is calculated with

the following equations [68, 69, 70]:

N
AXL, [max(X Xo, X Xo, 0)]2 (9-1)


and
N
AXmax [max(Xo X ,Xo X 0)]2, (9-2)

where the Xo is the central value of the observable; X, and X,- are the values

obtained from the plus and minus variations along the &th eigenvector direction;

and AXmax and AXax are the positive and negative uncertainties on the physical

observable. The PDF error, calculated in this way, is included on all comparisons of

data with the NLO predictions in section 10.






71



























1.05Midpoint (R l =.0)/Midpoint (Rse13)
(0.1




Midpoint (Rp =2.0) / Midpoint (Rp =1.3)
0.85 ...... ............ Midpoint (R eP=1.0) / Midpoint (R eP=1.3)

0.8. i .. . .l . . .. .. i .
0"O 100 200 300 400 500 600 700
PT (GeV/c)

Figure 9-1. Effect of varying the parameter Rsep on the NLO cross section
prediction of FASTNLO for the central rapidity region. Increasing
Rsep from 1.3 to 2 increases the cross section, while decreasing Rsep to
1 (the equivalent of the KT algorithm) decreases the cross section.















CHAPTER 10
RESULTS

The results of the inclusive jet cross section measurement with over 1 fb-1

of Run II integrated luminosity are shown in figures 10-1 10-5. In each figure,

the inclusive differential jet cross section corrected to the hadron level is shown

in (a), and the ratio of data (corrected to the parton level) to the NLO parton

level predictions of EKS, with CTEQ6.1M parton distribution functions, is shown

in (b). The yellow band shows the experimental systematic uncertainty. All

sources of systematic error are considered independent and have been added in

quadrature. The blue band also includes the modeling uncertainty associated with

the hadronization and underlying event corrections. This systematic associated

with the hadron to parton level correction is added in quadrature to the total

experimental systematic. There is an additional 6 normalization uncertainty due

to the uncertainty on the integrated luminosity which has not been included in the

figures. The uncertainty on the theoretical prediction due to estimated error on the

proton structure functions is drawn in red on the ratio plots.

Figure 10-1(a) shows the inclusive jet cross section corrected to the hadron

level in the well understood central region of the detector. The vertical axis

is plotted on a log scale. The cross section varies by more than eight orders

of magnitude as the jet transverse momentum increases from 55 GeV/c to

approximately 650 GeV/c. The differences between the measured result and the

theoretical predictions of EKS are not resolvable on the log scale. In figure 10-1(b),

the ratio of the measured cross section corrected to the parton level to the NLO

prediction is shown. Exceptionally good agreement is observed. The systematic

uncertainty varies from approximately 21' at low jet PT up to i' in the highest









transverse momentum bin. The systematic uncertainty is larger than statistical

errors on every bin. The systematic errors are slightly smaller than the PDF

errors for the central PT range indicating that this measurement will be useful to

constrain the parton densities.

Figure 10-2(a) shows the inclusive jet cross section corrected to the hadron

level in the central crack region. The cross section varies by approximately seven

orders of magnitude as the jet transverse momentum increases from 55 GeV/c

to approximately 650 GeV/c. In figure 10-2(b), the ratio of the measured

cross section corrected to the parton level to the NLO prediction is shown.

The agreement is good in most bins. There is a slight excess in the highest two

transverse momentum bins. It is possible that the effect of the crack needs more

attention in this region, or these fluctuations could be statistical. Regardless, the

measured result is consistent with the theoretical prediction when the error on

the parton distributions functions is also considered. The systematic uncertainty

varies from approximately 21'. at low jet PT up to 1(11',. in the highest transverse

momentum bin. The systematic uncertainty is larger than statistical errors on

every bin except for the highest jet PT bin. PDF error and systematic error are

roughly of the same order in this rapidity region.

The two regions where the central calorimeter and the plug calorimeter

modules overlap have similar features and will be discussed in tandem. Figures 10-3(a)

and 10-4(a) show the inclusive jet cross section corrected to the hadron level in

in the rapidity regions 0.7 < IYI < 1.1 and 1.1 < IY < 1.6, respectively.

The measurement includes jet PT up to approximately 500 GeV/c in region

0.7 < |Y < 1.1, and only up to 400 GeV/c in region 1.1 < |Y| < 1.6. In

figures 10-3(b) and 10-4(b), the ratio of the measured cross section corrected to

the parton level to the NLO prediction is shown. The agreement is good in most

bins; however, the NLO prediction seems to be systematically higher than the









measured result. This is a small effect and the NLO predictions are consistent with

the measured result for all jet transverse moment. In both regions the systematic

uncertainty is slightly smaller than the PDF uncertainty. These results will be

useful to constrain proton structure functions.

Figure 10-5(a) shows the inclusive jet cross section corrected to the hadron

level for the highest rapidity bin. The measurement only goes up to approximately

300 GeV/c in jet transverse momentum because the jet cross section falls off much

more rapidly at high rapidity. In fact, the cross section varies by approximately

seven orders of magnitude as the jet transverse momentum varies from 55 GeV/c

to approximately 300 GeV/c. In figure 10-5(b), the ratio of the measured cross

section corrected to the parton level to the NLO prediction is shown. The the

NLO prediction of EKS is systematically higher than the measured cross section

over the full range of jet PT; however, when systematic and PDF uncertainties are

considered it is still consistent with the measured result. The systematic errors are

largest in this region and approach 17"'-. for the highest bin in jet PT. The steeper

shape of the PT distribution in this region, combined with the additional systematic

on the jet energy scale due to the Pr-dependent correction, are responsible for this

increased uncertainty. Even so, the PDF uncertainty is still significantly larger than

the systematic errors for most jet PT. The result in this rapidity region will lead to

the strongest constraint on parton density functions out of all the regions included

in this measurement.

There is a trend in the data for the last bin to fluctuate higher than the

NLO prediction. There is a simple explanation for this systematic effect. There

are very few events in the last bins. If this number fluctuated down significantly

then there would not be events in the bin, and therefore it would not be shown

on the figure. By this i .- ,iii.- bins with very few events are more likely to be






75


statistical fluctuations which added to the bin contents rather than fluctuations

which subtracted from the contents of the bin.

The cross sections for the various rapidity regions are presented on figure 10-6

where they have been scaled by different factors so they would be distinguishable

when plotted on the same axis. The region 0.7 < |Y| < 1.1 has note been scaled.

The regions |Y| < 0.1, 0.1 < |Y < 0.7, 1.1 < |Y < 1.6, and 1.6 < |Y| < 2.1 have

been scaled by the factors of 106, 103, 10-3, and 10-6, respectively. Figure 10-7

shows the ratios to NLO pQCD predictions for the different rapidity regions.
















c D


IL



























2.


0 100 200 300 400


500 600 700
PJET (GeV/c)
T


Figure 10-1. Measured inclusive jet cross section with the Midpoint algorithm in
the region 0.1 < |Y| < 0.7. The distribution for the hadron level cross
section is shown in figure 10-1(a). The ratio of data corrected to the
parton level to the parton level pQCD prediction of EKS is shown in
figure 10-1(b).


10


1
10-1


10-3

o-4 -

0-5

10-6
10-7


SData corrected to the hadron level
Systematic uncertainty
NLO EKS CTEQ 6 1M P=P ET/2, Rsep=1 3
Midpoint Rone=0 7, f merge=0 75

CDF Run II Preliminary
0.1<|Y|<0.7

L L=1.04 fb1

FL

, I I I . t


I

I

I

I

I

I


0 100 200 300 400 500 600 700
ET (GeV/c)


(a)


Data corrected to the parton level
5 NLO pQCD EKS CTEQ 6 1M =Pe /2, Rsep=1 3
Midpoint R =0 7, f =0 75
cone merge
3 0.1<|Y|<0.7 jL=1.04 fb-1
-- PDF Uncertainty on pQCD
5 Data / NLO pQCD
S- Systematic uncertainty
2 Systematic uncertainty including
hadronization and UE

5-




5 -CDF Run II Preliminary
.. .I .I l. lI. I. I. ..I ..I
















w


Data corrected to the hadron level
Systematic uncertainty
NLO EKS CTEQ 6 1M P=P ET/2, Rsep=1 3
Midpoint R one=0 7, f mege=0 75

CDF Run II Preliminary
IYI<0.1

f-1 JL=1.04 ftb


i i






















OC3.
C 1-
Cao

























2.




1.



0.
I


0J


0 100 200 300 400


500 600 700
PJET (GeV/c)


Figure 10-2. Measured inclusive jet cross section with the Midpoint algorithm in
the region |Y| < 0.1. The distribution for the hadron level cross
section is shown in figure 10-2(a). The ratio of data corrected to the
parton level to the parton level pQCD prediction of EKS is shown in
figure 10-2(b).


10
1

10-1

10-2

10-3

10-4

10-5

10-6

10-7


0 100 200 300 400 500 600 700
ET (GeV/c)


(a)




Data corrected to the parton level
5 NLO pQCD EKS CTEQ 6 1M =P e'/2, Rsep=1 3
Midpoint R =0 7, f =0 75
cone merge
3- |YI<0.1 f L=1.04 fb-1
-- PDF Uncertainty on pQCD
5 Data / NLO pQCD
- Systematic uncertainty
2 Systematic uncertainty including
hadronization and UE

5-




5 CDF Run II Preliminary
-, I l. l I I .


, ,







78







Data corrected to the hadron level
.> 10
Systematic uncertainty
1
NLO EKS CTEQ 6 1M P=P ET/2, Rsep=1 3
S101 Midpoint R on=0 7, f merg=0 75

c 10-2 CDF Run II Preliminary
.r -0.7<|Y|<1.1
T 10 r
0 L=1.04 fb1

10 -4


10-6 -
1 0 -7 ,_._. .. ,
0 100 200 300 400 500 600 700
PT (GeVlc)


(a)




Data corrected to the parton level
O 3.5 NLO pQCD EKS CTEQ 6 1M i=PT /2, Rsep=1 3
SMidoiMidpoint R =0 7, f =0 75
cone merge
3 0.7<|Y|<1.1 L=1.04 fb
-- PDF Uncertainty on pQCD
2.5 Data / NLO pQCD
S- Systematic uncertainty
2 Systematic uncertainty including
Q 2 hadronization and UE

1.5

1

0.5 -CDF Run II Preliminary

0 100 200 300 400 500 600 700
pJET (GeV/c)


(b)


Figure 10-3. Measured inclusive jet cross section with the Midpoint algorithm in
the region 0.7 < |Y < 1.1. The distribution for the hadron level cross
section is shown in figure 10-3(a). The ratio of data corrected to the
parton level to the parton level pQCD prediction of EKS is shown in
figure 10-3(b).







79







Data corrected to the hadron level
.> 10
1 0 LSystematic uncertainty
1
NLO EKSCTEQ6 1M P=PET 12, Rse=1 3
10"1 Midpoint R one=07, f erge=075

C V 102 CDF Run II Preliminary
io- ~1.1<|Y|<1.6
S 10. r
SL=1.04 fb-1
10-4

10-5 r

10-6 r

1 0 7 . ,
0 100 200 300 400 500 600 700
ET (GeV/c)


(a)




Data corrected to the parton level
0 3.5 NLO pQCD EKS CTEQ 6 1M p=P /2, Rsep=1 3
Midpoint R =0 7, f =0 75
cone merge
3- 1.1<|Y|<1.6 L=1.04 fb-1
-- PDF Uncertainty on pQCD
2.5 Data / NLO pQCD
| Systematic uncertainty
( 2 Systematic uncertainty including
Shadronization and UE

1.5-




0.5 -CDF Run II Preliminary

0 100 200 300 400 500
pJET (GeV/c)


(b)


Figure 10-4. Measured inclusive jet cross section with the Midpoint algorithm in
the region 1.1 < |Y| < 1.6. The distribution for the hadron level cross
section is shown in figure 10-4(a). The ratio of data corrected to the
parton level to the parton level pQCD prediction of EKS is shown in
figure 10-4(b).















Data corrected to the hadron level
.Q ^ 10 -
S10 Systematic uncertainty

-- NLO EKS CTEQ 6 1M P=P ET/2, Rsep=1 3
10"1 Midpoint R one=07, f merge=0 75

C V 102 CDF Run II Preliminary
S1.6<|Y|<2.1
10- rr

10-4 L=1.04 fb -1

10-5

10-6 r

1 0 7 .. . ,
0 100 200 300 400 500 600 700
ET (GeV/c)


(a)




Data corrected to the parton level
03.5- NLO pQCD EKS CTEQ 6 1M =PT /2, Rsep=1 3
Midpoint R =0 7, f =0 75
cone merge
3- 1.6<|Y|<2.1 L=1.04 fb1
-- PDF Uncertainty on pQCD
2.5 Data / NLO pQCD
| Systematic uncertainty
( 2 Systematic uncertainty including
Q -~
hadronization and UE

1.5 -




0.5 -CDF Run II Preliminary

0 50 100 150 200 250 300
pJET (GeV/c)


(b)


Figure 10-5. Measured inclusive jet cross section with the Midpoint algorithm in
the region 1.6 < |Y| < 2.1. The distribution for the hadron level cross
section is shown in figure 10-5(a). The ratio of data corrected to the
parton level to the parton level pQCD prediction of EKS is shown in
figure 10-5(b).































r 1015
10 13
10
1>



a710
107


>. 104




10-2


10-s5


10-8

10-11

10-14


* Data corrected to the hadron level

Systematic uncertainty

S NLO: EKS CTEQ 6.1M P=PrT /2, R =1.3
T sep


*
*,

* -U-
* -U-.





- --_._
* -U -


Midpoint Rcone=0.7, fmerge=0.75


-4-

-U-
-4-


SL=1.04 fb-1


-U-


=8=


pY<~ X1'


-U-


4. -m-
-U-
-U-
-U-
-~
-U-


-C-


1

-2 6 C
1.6<|Y|<2.1 (x10-)
I . I . I . I


0.1
0.7<|Y|<1.1
-3)


DF Run II Preliminary
. I . I . I ,


0 100 200 300 400 500 600 700
PET (GeV/c)



Measured inclusive jet cross section at the hadron level with the
Midpoint algorithm for all rapidity regions.


Figure 10-6.


"


---
































CDF Run II Preliminary J L=1.04 fb'
3.5
0 3-
25 |IYI<0.1 0.1<|Y|<0.7
S2.5-






0
1. *3:5 -4------------------ -=-'----------------
3







0.5
3. -- 100 200 300 400 500 600 700
S.- 6<|Y|<2.1 1.1 (GeV/c)
S2.5 PDF Uncertainty on pQCD
2 Data (parton level) / NLO pQCD
Systematic uncertainty
1.5- Systematic uncertainty including
1 hadronization and UE
0.5r- Midpoint R .7, 0.75
100 200 300 400 500 600 700 cone ferge
S(GeV) NLO pQCD: EKS CTEQ 6.1 M JET/2, R(GeV/c=1.3
I--










Figure 1-7. Ratio of the measured inclusive jet cross section at the parton level


different rapidity regions.
different rapidity regions.














CHAPTER 11
COMPARISON WITH THE KT ALGORITHM

CDF has made inclusive jet cross section measurements with the Midpoint

and the KT jet clustering algorithms in Run II. Here, a simple comparison of these

measurements in the central region (0.1 < |Y| < 0.7 ) is presented. In order to

require that the algorithms use a similar scale for clustering in y Q space, the cone

size for the Midpoint algorithm and the D-parameter of the KT algorithm will both

be taken as 0.7 (Rcoe = D = 0.7). It is important to note that different algorithms

correspond to different observables. The inclusive jet cross section is not expected

to be exactly the same for jets clustered with these two different algorithms (even

though Rc,,e = D = 0.7). The NLO predictions for the cross sections, as well as the

hadron to parton level corrections, will also be compared for the two algorithms.

Historically, the KT jet clustering algorithm has been used successfully at

electron-positron colliders, and electron-proton colliders [71, 27]. Only recently

has this algorithm been applied to jets in the more challenging hadron-hadron

collider environment of the Tevatron: studied first by DO [72] in Run I, and more

recently by CDF in Run II. DO reported only marginal agreement with NLO

prediction; however, the more recent CDF result reports good agreement. The

ratio of the CDF result to the NLO prediction of the JETRAD program is shown

in figure 11-1 [73]. In the figure, the JETRAD result has been corrected to the

hadron-level to include the effects of jet fragmentation and the underlying event.

This correction is shown in figure 11-2 [73].

Figure 11-3 shows the ratio of the inclusive jet cross section measurement

for jets clustered with the KT algorithm to the result for jets clustered with the

Midpoint algorithm (black). Only statistical errors are shown. The data used in the









KT measurement is a subset of the data included in the Midpoint measurement,

so the statistics are clearly correlated; however, the errors were propagated based

on no correlation because all information required for full error propagation was

not available. This results in a slight over-estimate of the true statistical error on

the ratio. Systematic errors are not included, and could provide an additional large

contribution to the uncertainty. The prediction of this ratio from the NLO program

FASTNLO is also shown in the figure (blue). At the NLO parton level, where there

are at most three objects in the event, the Midpoint algorithm with Rsep = 1 is

equivalent to the KT algorithm. However, the parameter Rsep is taken as 1.3 for

the Midpoint algorithm at CDF. The standard value of Rsep being equal to 2 would

result in a prediction of a larger difference between the two algorithms at NLO

(i.e., setting Rsep = 2 results in a larger jet cross section as seen in figure 9-1).

The PT dependence and magnitude of the measured ratio observed in the data is

close to the NLO prediction. This result provides confidence that the algorithm

definitions are consistent at calorimeter level and NLO parton level.

Figure 11-3 shows the ratio of the hadron to parton level correction derived

with the KT algorithm (inversion of the distribution shown in 11-2) to the one

derived with the Midpoint algorithm (figure 7-4). These corrections were derived

from PYTHIA TUNE-A, as described in section 7. The multiplicative corrections

are both less than one, so the ratio means that the size of the correction derived

with the KT algorithm is larger (i.e., farther away from one) than the correction

derived with the Midpoint algorithm. This result indicates that the KT algorithm

is slightly more sensitive to the underlying event.

The consistency of the KT inclusive jet cross section measurement with NLO

predictions, combined with only a sightly larger underlying event correction,

supports the use of KT -type algorithms at the Tevatron and hadron-hadron

colliders of the future, provided that one has a good understanding of the UE. The










agreement, between the NLO prediction and the measured result for the ratio of

the the two jet clustering algorithms, adds credence that the jet definitions are

defined consistently at the parton and detector levels.


100 200 300


Systematic errors -
PDF uncertainties -
. I . I . I I I


100 200 300 400


am==


p = 2 x po
MRST2004
. I . I .


500 600 700

p ET [GeV/c]


Figure 11 -1.


Measured inclusive jet cross section with the KT algorithm in the
rapidity region 0.1 < |Y| < 0.7 The ratio of the hadron level
cross section to the NLO prediction of JETRAD is shown. JE-
TRAD has been corrected to the hadron-level to include the effects of
fragmentation and the underlying event.


3


2.5


2


1.5


- 1.2


- 1


- 0.8


0.5


0
0


: ....... --- .= .=


*


-- Data













S1.5 CDF RUN II Preliminary
:JET
r 1.4 KT D=0.7 0.1 1.3 -- Parton to hadron level correction

1.2 Monte Carlo modeling uncertainty
1.1

0.91 -
0 100 200 300 400 500 600 700
P JET [GeV/c]

Figure 11-2. Parton-to-hadron correction used by the KT inclusive jet cross
section analysis to correct the NLO prediction to the hadron level.


0
S1.6

1.4


1.21--


Cross Section Ratio: kT(385 pb-1) / Midpoint(1 fb-1)


0 100 200 300 400 500 600 700
PT (GeV/c)


Figure 11 -3.


Ratio of the inclusive jet cross section measured with the KT
algorithm to that measured by the Midpoint algorithm (black). The
prediction of this ratio from the NLO program FASTNLO is also
shown in the figure (blue).


FastNLO: NLO, CTEQ61,=P JET/2
Data corrected to the parton level
* Only statistical errors included
* Errors considered uncorrelated


S---- ---- ---

- Midpoint: f =0.75 R =1.3
merge sep
Rcone = D = 0.7 0.1<|Y|<0.7
- CDF Run II Preliminary


I ,I





























Hadron to Parton Correction Ratio: kT/Midpoint


- PYTHIA CTEQ5L (0.1<|Y|<0.7)

SRcone= D =0.7

Midpoint: f =0.75
merge
- -------------------------------------------------- ---------------------.
+ + +
S+++ + + + +
4-
- +
CDF Run II Preliminary
-No errors included on ratio.
. I . lI . I . I . lI I .


100 200 300 400 500 600 700
PT (GeV/c)


Figure 11 -4.


Ratio of the hadron to parton level correction derived with the
KT algorithm to that derived with the Midpoint algorithm. The
multiplicative corrections are both less than one. The correction
derived with the KT algorithm is larger (farther away from one). The
corrections were derived from PYTHIA TUNE-A.


1.1
1.08

1.06

1.04
1.02

1

0.98

0.96

0.94

0.92

0.90


-