EFFECTOFNEWSAMPLESINTHET2KOFF-AXISNEARDETECTORFORTHET2K OSCILLATIONANALYSIS By JacobAlexanderMorrison ADISSERTATION Submittedto MichiganStateUniversity inpartialful˝llmentoftherequirements forthedegreeof PhysicsDoctorofPhilosophy 2019 ABSTRACT EFFECTOFNEWSAMPLESINTHET2KOFF-AXISNEARDETECTORFORTHET2K OSCILLATIONANALYSIS By JacobAlexanderMorrison TheTokai-to-Kamioka(T2K)experimentisalongbaselineneutrinooscillationexperiment.T2K usesabeamofmuonneutrinos(neutrinobeammode)orantineutrinos(antineutrinobeammode) producedattheJapanProtonAcceleratorResearchComplexanddirectedtowardstheSuper- Kamiokandedetectortostudyneutrinooscillationsintwoways.Oneisthedisappearanceof muonneutrinosastheyoscillatetoother˛avorsofneutrinos,whiletheotheristheappearanceof electronneutrinosthathaveoscillatedfrommuonneutrinos.Inadditiontothefardetector,Super- Kamiokande,asuiteofdetectorsissetclosetotheneutrinosourcetoprobethebeamcomposition priortotheneutrinososcillating. Withintheneutrinooscillationanalysis,uncertaintiesduetotheneutrinobeam˛uxandthe crosssectionofneutrinosserveasthelargestsourcesoferrorontheoscillationparameters.By includingdatafromtheNearDetectorat280m(ND280),theuncertaintiesonthe˛uxandcross sectioncanbeconstrainedbeyondwhatthedataatthefardetectorcandoonitsown.Thiswork describestheneardetectormaximumlikelihood˝tandhowitisusedtoconstrainuncertaintiesfor theoscillationanalysis. Forthisthesis,newdatasampleswereincludedintheneardetector˝tsothattheantineutrino beammodesampleswouldbetreatedinthesamewayastheneutrinobeammodesamples.The resultsareconsistentwiththoseseenbefore;however,theyalsoindicatethatcertainchecksshould beupdatedwhenthenewneutrinointeractionmodelisavailablebeforefullytransitioningtothe newsamples.Additionally,testswereperformedtostudythee˙ectofalternativecrosssection modelsontheneardetector˝t.Thesestudiesshowedthatthereisnotenoughfreedominthe currentcrosssectionmodeltofullydescribeanye˙ectsonthedataiftheunderlyingcrosssection di˙eredfromthecurrentmodel. Copyrightby JACOBALEXANDERMORRISON 2019 For, Kaleigh Iloveyou! iv ACKNOWLEDGMENTS Asanyonewhohaswrittenadissertationwouldtellyou,researchingandwritingsuchanextensive documentisnoeasytask.Overthepastfouryears,agreatnumberofpeoplehavehelpedand supportedmewhileIhavebeenworkingonmine. TothoseatMichiganState,thankyouforallthefunwehavehadovertheyears.Kendall, thankyouforbeinganawesomethesisadvisor.Yourinsightsintophysicshaveintroducedmeto anewwayoflookingatthings.MattandLuke,thankyouforyourhelpwithprogrammingand forthetimespentdiscussingthevariousprojectsIhaveworkedon.Andrew,thankyouforyour invaluablediscussionsoncoding,frustrationswithROOT,traveling,and,mostimportantly,sports. OurtimespenttravelingaroundJapantogether,fromFukuokaandHiroshimatoToyamaandSK toalloverTokyo,thosetimeswillstaywithmefortheyearstocome.Jake,thankyouforthetime spentworkingonhomeworkandstudyingfor˝nalsaswewentthroughclasses.Dan,thankyou forbeingwillingtogooverpresentationsandapplicationessaysandforyourthoughtfulinsights intoneutrinophysics. Tothesupportsta˙atMichiganState,thankyouforallthehardworkthatyoudoeveryday tomakethisdepartmentrunsmoothly.Kim,thankyouforalwayshavingasmilereadyandfor keepingeverything(andeveryone)inline.Brenda,thankyouformakingtheprocessofsettingup travelandgettingrefundsabreezeandforalwaysknowingwhereKendallisintheworldonany givenweek(especiallyinthesummer). ToallthoseIhavehadachancetoworkwithonT2K,thankyouforallthatyoudoinadvancing ourknowledgeofneutrinos.WhilethereamanypeopleIhaveworkedwithdirectlyoverthelastfew years,Iwanttohighlightafew.Hayato-san,thankyouforhostingmeatIPMUinthesummerof 2016.TheopportunitytoliveandworkinJapanoverthatsummerwasoneofthehighlightsofmy timeingraduateschool.Christophe,thankyouforyourhelp,bothduringthatsummerwhileIwas workingonNEUTandwhileIwaspreparingmythesisbygettingmesetupwiththeP-Thetacode toproduceSKeventdistributions.Ciro,thankyouforyourhelpingettingHighLANDrunningand v foryourquickrepliestoallmyquestions.Mark,thankyouforyourkeeninsightsintotheBANFF andfortheusefuldiscussionscoveringeverythingfromworkingontheBANFFcodetoanalyzing ˝tresults.John,thankyouforyoursupportinthetrenchesofworkingontheBANFF.Beingable toworkalongsidesomeoneontheBANFFcodemadeitmucheasiertohandle,especiallyduring thepressureofpreparinginputsfortheoscillationanalysis. Ialsowanttothankmyfamilyfortheirsupportandprayersovertheyears,especiallythelast eightattheUniversityofAlabamaandMichiganState.Ithasbeenhardbeingsofaraway,but Iknowyouarealwaysaphonecallortextmessageaway.Thankyou,Dad,foralwaysbeing agoodsoundingboardandforeditinganythingfromundergraduatepaperstomyresumétomy dissertation.Mom,thankyouforyourloveandsupportandalwaysbeingavailabletotalkwhenever Ineedyou.AbbeyandClaire,thankyouforbeingthebestsistersthatabrothercouldhave. TothenewfamilythatIgainedwhileIwasworkingonmydoctorate,thankyouforopening yourhometomeandlettingmejoinyourfamily. And,mostespecially,thankyoutomybeautifulwife,Kaleigh,foryourloveandencouragement overthelasttwoyears.Iwouldnothavemadeittothispointwithoutyoutherebesideme.Ilove you,Kaleigh. vi TABLEOFCONTENTS LISTOFTABLES .......................................ix LISTOFFIGURES ......................................xii CHAPTER1EXECUTIVESUMMARY .........................1 CHAPTER2NEUTRINOPHYSICS ...........................3 2.1AHistoryofNeutrinos ................................3 2.1.1InitialPostulation ...............................3 2.1.2FurtherEvidence ...............................4 2.1.3TheFirstObservation .............................5 2.1.4ProposalofDi˙erentNeutrinoTypes .....................7 2.1.5DetectingtheMuonandTauNeutrinos ....................8 2.1.6CurrentUnderstanding ............................10 2.2NeutrinoInteractions .................................11 2.3NeutrinoOscillations .................................14 2.3.1Motivation ..................................14 2.3.2Theory ....................................19 2.4TheCurrentKnowledgeofNeutrinos .........................25 CHAPTER3THET2KLONGBASELINENEUTRINOEXPERIMENT .......28 3.1JapanProtonAcceleratorResearchComplex .....................29 3.1.1TheT2KNeutrinoBeamline .........................30 3.1.2TheO˙-AxisNeutrinoBeam .........................33 3.2TheT2KNearDetectors ...............................35 3.2.1TheInteractiveNeutrinoGRIDDetector ...................35 3.2.2TheNearDetectorat280Meters .......................37 3.3TheT2KFarDetectorSuperKamiokande .....................41 CHAPTER4THEOSCILLATIONANALYSISATT2K ................44 4.1TheT2KOscillationAnalysis .............................44 4.1.1OverviewandMotivation ...........................44 4.1.2DataSelectionatSuper-Kamiokande .....................47 4.1.3TheOscillationFitatSuper-Kamiokande ..................50 4.2TheNearDetectorFit .................................51 4.2.1TheMaximumLikelihoodFitMethod ....................51 4.3FitParametersintheNearDetectorFit ........................53 4.3.1FluxParameters ................................54 4.3.2CrossSectionParameters ...........................57 4.3.2.1CC 0 ˇ Parameters ..........................60 4.3.2.2CCResonantParameters ......................64 4.3.2.3OtherChargedCurrentParameters .................64 vii 4.3.2.4NeutralCurrentParameters ....................66 4.3.2.5FinalStateInteractions .......................66 4.3.3TheObservableNormalizationParameters ..................68 CHAPTER5THENEARDETECTORSELECTIONS .................70 5.1SelectionConsiderations ...............................70 5.2DataandMonteCarloInputstotheSelections ....................71 5.3DescriptionofSelections ...............................73 5.3.1ForwardHornCurrentMultiPiSelections ..................75 5.3.1.1ChargedCurrentInclusiveSelection ................75 5.3.1.2SelectionsforMultiPiTopologies .................78 5.3.2ReverseHornCurrentMultiTrackSelections .................81 5.3.2.1ChargedCurrentInclusiveSelection ................82 5.3.2.2SelectionsforMultiTrackTopologies ...............87 5.3.3ReverseHornCurrentMultiPiSelections ..................88 5.3.3.1ChargedCurrentInclusiveSelection ................88 5.3.3.2SelectionsforMultiPiTopologies .................89 5.3.3.3ComparisonofMultiTrackandMultiPisamples .........96 CHAPTER6ND280SYSTEMATICUNCERTAINTIESINTHENEARDETEC- TORFIT ...................................103 6.1DetectorSystematicUncertaintyDetails .......................103 6.2PropagationoftheDetectorSystematicUncertainties ................117 6.2.1ObservableVariationSystematics ......................118 6.2.2WeightSystematics ..............................119 6.3SummaryoftheIndividualSystematicUncertainties .................121 6.4TheObservableNormalizationMatrix ........................121 CHAPTER7RESULTS ...................................130 7.1ComparingRHCMultiTrackandRHCMultiPiSamples ...............130 7.1.1Results ....................................130 7.2E˙ectofAdditionalCCResonantEventswithLowPionMomentum ........140 7.2.1Results ....................................140 7.3 Q 2 SuppressioninSinglePionProductionEvents ..................142 7.3.1Results ....................................143 CHAPTER8CONCLUSIONANDSUMMARY .....................150 APPENDICES .........................................152 APPENDIXACOMPARISONSOFDIFFERENTPSYCHEVERSIONSON THENEARDETECTORFIT ................... 153 APPENDIXBOBSERVABLENORMALIZATIONCOVARIANCEMATRIX BINNINGSTUDIES ......................... 170 APPENDIXCFINE-GRAINEDDETECTOR2RELATEDPLOTS ...... 185 BIBLIOGRAPHY .......................................202 viii LISTOFTABLES Table2.1:StandardModelofParticlePhysics.Thetopportionundertheermions headingshowsthequarks,whilethebottomshowstheleptons. ..........11 Table2.2:Typesofneutrinoandantineutrinointeractions. l canbeeither e , ,or ˝ . Thecategorycoversanyhighenergyinteractionsthatproducemore thanonepion.Thepionsintheseinteractionscanbechargedorneutralor both.Note,thisdoesnotcoverallinteractionsthatmayfallwithinagiven interactiontype.Tablefrom[ 1 ] ..........................12 Table2.3:Currentunderstandingoftheneutrinooscillationparametersandtheir 3 ˙ allowedranges.Parametervaluesarederivedfromaglobal˝ttocurrent neutrinooscillationdata[ 2 ].Inthecaseof CP ,the 2 ˙ allowedrangeisshown. Thevalues(valuesinparentheses)arefor m 1 < m 2 < m 3 ( m 3 < m 1 < m 2 ). m 2 ,asde˝nedin[ 2 ],is m 2 = m 2 3 ¹ m 2 2 + m 2 1 ºš 2 .Underthisde˝nition, m 2 > 0 for m 1 < m 2 < m 3 and m 2 < 0 for m 3 < m 1 < m 2 .Valuesare givenof m 2 31 > 0 for m 1 < m 2 < m 3 and m 2 32 < 0 for m 3 < m 1 < m 2 . Tablefrom[ 3 ]. ...................................25 Table4.1:UncertaintyonthenumberofeventsineachSKsampleseparatedbyerror sourcewithandwithouttheconstraintprovidedbytheND280data.The SK+FSI+SIuncertaintiesarenotconstrainedbytheND280data.Tablefrom[ 4 ]. 45 Table4.2:Priorsusedfortheoscillationparametersduringthemarginalizationprocess. denotesnormalhierarchy,whiledenotesinvertedhierarchy.Table from[ 5 ]. ......................................51 Table4.3:Relationbetweenthe˛uxparametersandtheirbinnumberinthe˛uxcorrela- tionmatrix. .....................................59 Table4.4:CC 0 ˇ parametersintheBANFF˝t.Includedinthetablearethepriorvalue anderror,thetypeofparameteritis,andwhetherornotthe˝nalvalueis passedtoSKornot. ................................63 Table4.5:CCresonantparametersintheBANFF˝t.Includedinthetablearetheprior valueanderror,thetypeofparameteritis,andwhetherornotthe˝nalvalue ispassedtoSKornot. ...............................64 Table4.6:OtherchargedcurrentparametersintheBANFF˝t.Includedinthetableare thepriorvalueanderror,thetypeofparameteritis,andwhetherornotthe ˝nalvalueispassedtoSKornot. .........................65 ix Table4.7:NeutralcurrentparametersintheBANFF˝t.Includedinthetablearethe priorvalueanderror,thetypeofparameteritis,andwhetherornotthe˝nal valueispassedtoSKornot. ............................66 Table4.8:FSIparametersintheBANFF˝t.Includedinthetablearethepriorvalueand error,thetypeofparameteritis,andwhetherornotthe˝nalvalueispassed toSKornot. ....................................67 Table5.1:ThedataandMonteCarloPOT,aswellasthebeammode,foreachrunperiod usedintheanalysis. .................................72 Table5.2:Breakdownofreconstructedeventsintosamplesunderthepreviousparadigm. ..74 Table5.3:Breakdownofreconstructedeventsintosamplesunderthenewparadigm. ....74 Table5.4:ObservedandpredictedeventratesforthepreviousND280sampleset.The predictedeventratesareMonteCarloeventsweightedbyPOT,˛ux,detector, andcrosssectionweights.TheleftcolumnshowstheratesforFGD1,while therightcolumnshowsFGD2. ...........................75 Table5.5:ObservedandpredictedeventratesforthenewND280sampleset.The predictedeventratesareMonteCarloeventsweightedbyPOT,˛ux,detector, andcrosssectionweights.TheleftcolumnshowstheratesforFGD1,while therightcolumnshowsFGD2. ...........................75 Table5.6:PercentageofeventswithagivennumberofMichelelectrons(ME)and isolatedFGDtracks(isotracks).Backgroundrepresentseventsfallinginto anybackgroundsamples(whicharenotusedinthisanalysis),whileOOFV includesanyeventsoccurringoutsideoftheFGD˝ducialvolume.Tablefrom[ 6 ]. 95 Table6.1:Resultsforthedi˙erenceintheTPCclustere˚ciencybetweendata( data ) andMC( MC ).Tablefrom[ 7 ]. ..........................107 Table6.2:TPCtrackreconstructione˚cienciesfordataandMonteCarlo.Tablefrom[ 8 ]. .108 Table6.3:E˚cienciesforFGD-TPCmatchedtrackswithtwoorlessreconstructedhits inthecorrespondingFGDfordataandMonteCarlo.Tablefrom[ 9 ]. .......109 Table6.4:E˚cienciesforthedetectionofMichelelectronsfordataandMonteCarlo (MC).Tablefrom[ 10 ]. ...............................111 Table6.5:DataandMonteCarlo(MC)ratesforfalseMichelelectronidenti˝cation.The rateisde˝nedasthenumberofexpectedfalseMichelelectronsperspill.Table from[ 7 ]. ......................................112 x Table6.6:DataMonteCarlo(MC)di˙erence,theuncertaintyduetodirectmeasure- ments,andthetotaluncertaintyfortheXYandwatermodulesintheFGDs. Tablefrom[ 11 ]. ..................................113 Table6.7:Correlationsinuncertaintiesbetweenmasscomponents.water-like andXY-likerefertothewatermoduleandXYmoduleportionsof FGD2,respectively.Tablefrom[ 7 ]. ........................113 Table6.8:UncertaintiesontheOOFVratesbasedontheirsubdetectororigin.Tablefrom [ 12 ]. .........................................115 Table6.9:ReconstructionuncertaintiesforOOFVbasedontheirreconstructioncategory. Tablefrom[ 12 ]. ..................................116 Table6.10:ThedetectorsystematicuncertaintiesusedwithintheBANFF˝t.The˝nal columnshowswhichsetsofsamplestheuncertaintiesapplyto.Thesamples listedasapplytoMultPisamples,whileTracappliestotheMultiTrack samples. .......................................122 Table6.11:IntegrateduncertaintyforeachofthesystematicuncertaintiesinFGD1.The protonsecondaryinteractionssystematicuncertaintyisnotincludedasithas asmalle˙ectonthesample.Tablefrom[ 7 ]. ...................123 Table6.12:IntegrateduncertaintyforeachofthesystematicuncertaintiesinFGD2.The protonsecondaryinteractionssystematicuncertaintyisnotincludedasithas asmalle˙ectonthesample.Tablefrom[ 7 ]. ...................124 Table7.1:Eventratesforthemodi˝edandnominalMonteCarlosets.Thenominalevent ratesareMonteCarloeventsweightedbyPOT,˛ux,detector,andcrosssection weights.Themodi˝edrateincludesthesameweightsasthenominal,plusthe additionalmodi˝cationduetothelow Q 2 eventsuppression.Thedi˙erence isgivenby ¹ modified nominal ºš nominal .Theleftcolumnshowsthe ratesforFGD1,whiletherightcolumnshowsFGD2. ...............147 TableA.1:ComparisonofeventratesbetweenPsychev1andPsychev3duringthesummer of2018. .......................................154 TableA.2:ComparisonofdetectorweightsappliedbytheBANFFandNuMugroupsto speci˝cevents. ...................................155 TableB.1:Thetotalnumberofbinsalongthediagonaloftheobservablenormalization covariancematrix. .................................173 xi LISTOFFIGURES Figure2.1:Theshapeoftheenergyspectrumoftheoutgoingelectronfromthebeta decayoftritiumcomparedwiththeexpectedvaluefromatwo-bodybeta decaygivenbyEquation 2.2 .Figurefrom[ 1 ]. ..................4 Figure2.2:Totalneutrino(top)andantineutrino(bottom)chargedcurrentcrosssection pernucleon(foranisoscalartarget)dividedbyneutrinoenergyandplotted asafunctionofenergy.Additionally,thetotalcrosssectionisseparatedinto CCQE(labeledasQE),CCresonant(RES),anddeepinelasticscattering (DIS)crosssections.Figurefrom[ 13 ]. ......................8 Figure2.3:Varioustypesof˝nalstateinteractionsoccurringwithinanucleus.Figure from[ 14 ]. .....................................13 Figure2.4:Theratioofthenumberofdatainteractionstothenumberofpredictedinter- actionsassumingnoneutrinooscillations(fromMonteCarlo)asafunction of L š E inSuper-Kamiokande.Thepointsshowtheratio,whilethedashed linesshowtheexpectedshapewhenincludingoscillationsof to ˝ .Figure from[ 15 ]. .....................................19 Figure3.1:SchematicoftheT2Kexperiment.Figurefrom[ 16 ]. ...............28 Figure3.2:Themuonneutrinosurvivalprobabilityat295km(top)andneutrino˛uxes fordi˙erento˙-axis(listedasOAinthe˝gure)angles(bottom).Itshouldbe notedthatthe˛uxpredictionsarenormalizedsothattheunitsonthey-axis arearbitrary.Realistically,thetotal˛uxdecreasesforhighero˙-axisangles. Figurefrom[ 17 ]. .................................29 Figure3.3:ThePOTcollectedatT2KbetweenJanuary2010andMay2018.Thered shadedregionsshowwhentheT2Kbeamwasbeingproduced.Figurefrom[ 18 ]. 31 Figure3.4: Top: OverviewoftheT2Kbeamline. Bottom: Sideviewofthesecondary beamline.Figurefrom[ 19 ]. ............................32 Figure3.5:Neutrinoenergyasafunctionofthepionenergyforneutrinosproducedfrom thetwo-bodydecayofpionsintoamuonandaneutrino.Predictionsare shownforvariousanglesbetweentheneutrinoandpiondirections. .......34 Figure3.6: Top: TheINGRIDdetector. BottomLeft: AnINGRIDmoduleshowing thescintillatorplanes(blue)andtheironplates(gray). BottomRight: An INGRIDmodulewiththevetoplanes(black)shown.Figurefrom[ 16 ]. .....36 xii Figure3.7:ThelayoutofND280.Figurefrom[ 16 ]. .....................38 Figure3.8:Simpli˝eddiagramofasingletimeprojectionchamber.Figurefrom[ 16 ]. ...41 Figure3.9:TheSuper-Kamiokandedetector.Figurefrom[ 16 ]. ...............42 Figure4.1:Exampleeventdisplayofamuon-likeeventatSuper-Kamiokande.Photo- multipliertubesthathavechargedepositedinthemduringtheeventareshown ascoloredcircles,wherethecolorrepresentshowmuchchargewasdeposited. Thetimedistributionofhitscanbeseeninthebottomrightcorner.Figure from[ 20 ]. .....................................48 Figure4.2:Exampleeventdisplayofanelectron-likeeventatSuper-Kamiokande.Pho- tomultipliertubesthathavechargedepositedinthemduringtheeventare shownascoloredcircles,wherethecolorrepresentshowmuchchargewas deposited.Thetimedistributionofhitscanbeseeninthebottomrightcorner. Figurefrom[ 20 ]. .................................49 Figure4.3:T2Ktuned˛uxprediction(left)andratiooftheT2Ktuned˛uxprediction tothenominalprediction(right)forND280andSKinbothneutrinomode (FHC)andantineutrinomode(RHC). .......................55 Figure4.4:ThetotaluncertaintiesontheND280˛uxprediction.The13av3uncertainty (solidblackline)isthecurrentversion.The11bv3.2uncertainty(dashed blackline)isanearlierversion.Figurefrom[ 21 ]. ................56 Figure4.5:Correlationmatrixfor˛uxparametersusedintheBANFF˝t.Thelabels denotewhichdetectorandbeammodethatregioncovers.Eachbininthe matrixcorrespondstoanenergyrangegiveninthetext. .............58 Figure4.6:2p2hdiagrams.Singlelinesrepresentnucleons,doublelinesrepresentthe , dashedlinesrepresentpions,andcurlylinesrepresenttheWboson.Adapted from[ 22 , 23 ]. ...................................62 Figure4.7:Thetotalcrosssection(left)comparedwiththe2p2hcrosssection(right). Figurefrom[ 24 ]. .................................62 Figure4.8:Correlationsbetweenthecrosssectionparameters. ................68 xiii Figure4.9:Thecorrelationmatrixfortheobservablenormalizationparameters.Theshort dashedlinesdi˙erentiatebetweentheCC 0 ˇ ,CC 1 ˇ ,andCCOthersamples, whilethelongdashedlinesseparateFGD1andFGD2samples.Thesolid blacklinesseparatetheFHCMultiPi,RHC MultiPi,andRHC MultiPi samples.Withineachsample,theparametersareorderedfrombackward goingtoforwardgoingangularbins.Eachcompletesetofangularbins shareacommonmomentumbin,whichareorderedinfromlowesttohighest momentum. ....................................69 Figure5.1:DistributionofthenumberofMichelelectronsinFGD1(top)andFGD2 (bottom)categorizedbydi˙erentinteractiontopologies.Theleftplotineach pairshowsthedistributionwhennosecondarytracksareseenintheTPC, whiletherightplotshowstheCCinclusiveselectionwithnosuchconstraints. Figurefrom[ 7 ]. ..................................82 Figure5.2:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD1FHC MultiPiselections.Themomentumdistributionsareshownontheleft,while theangulardistributionsareshownontheright. .................83 Figure5.3:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD1FHCMultiPiselections. ................84 Figure5.4:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD1FHCMultiPiselections. .....85 Figure5.5:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD1RHC MultiTrackselections.Themomentumdistributionsareshownontheleft, whiletheangulardistributionsareshownontheright. ..............89 Figure5.6:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD1RHC MultiTrackselections. .............90 Figure5.7:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD1RHC MultiTrackselections. ..91 Figure5.8:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD1RHC MultiTrackselections.Themomentumdistributionsareshownontheleft, whiletheangulardistributionsareshownontheright. ..............92 Figure5.9:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD1RHC MultiTrackselections. .............93 Figure5.10:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD1RHC MultiTrackselections. ..94 xiv Figure5.11:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD1RHC MultiPiselections.Themomentumdistributionsareshownontheleft,while theangulardistributionsareshownontheright. .................97 Figure5.12:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD1RHC MultiPiselections. ..............98 Figure5.13:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD1RHC MultiPiselections. ...99 Figure5.14:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD1RHC MultiPiselections.Themomentumdistributionsareshownontheleft,while theangulardistributionsareshownontheright. .................100 Figure5.15:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD1RHC MultiPiselections. ..............101 Figure5.16:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD1RHC MultiPiselections. ...102 Figure6.1:Thefractionalerrorincludedintheobservablenormalizationcovariance matrixduetotheMonteCarlostatisticaluncertainty.Thebluedashedlines di˙erentiatebetweentheCC 0 ˇ ,theCC 1 ˇ ,andtheCCOthersamples,while thereddashedlinesseparatesamplesinFGD1andFGD2.Thesolidredline demarcatestheFHCMultiPi,RHC MultiPi,andRHC MultiPisamples. ...127 Figure6.2:ThedetectorcovariancematrixwithouttheMCstatisticaluncertaintiesor theuncertaintiesfromthe1p1he˙ects,plottedassgn ¹ V ij º p j V ij j foreasier viewing.Theshortdashedlinesdi˙erentiatebetweentheCC 0 ˇ ,CC 1 ˇ ,and CCOthersamples,whilethelongdashedlinesseparateFGD1andFGD2 samples.ThesolidblacklinesseparatetheFHCMultiPi,RHC MultiPi, andRHC MultiPisamples.Withineachsample,theparametersareordered frombackwardgoingtoforwardgoingangularbins.Eachcompletesetof angularbinsshareacommonmomentumbin,whichareorderedfromlowest tohighestmomentum. ...............................128 Figure6.3:ThefulldetectorcovariancematrixasinputtotheBANFF˝t,plottedas sgn ¹ V ij º p j V ij j foreasierviewing.Theshortdashedlinesdi˙erentiate betweentheCC 0 ˇ ,CC 1 ˇ ,andCCOthersamples,whilethelongdashedlines separateFGD1andFGD2samples.ThesolidblacklinesseparatetheFHC MultiPi,RHC MultiPi,andRHC MultiPisamples.Withineachsample, theparametersareorderedfrombackwardgoingtoforwardgoingangular bins.Eachcompletesetofangularbinsshareacommonmomentumbin, whichareorderedfromlowesttohighestmomentum. ..............129 xv Figure7.1:Comparisonof˝tstothenominalMonteCarlofortheRHCMultiTrack samples(blue)andtheRHCMultiPisamples(red).Theinputparameter errorbarscanbeseeninthebackground.Note,theCCQEcrosssection parametersdonothaveanypriorconstraintappliedinthe˝t,soaninputerror bandisnotdisplayedfortheseparameters. ....................132 Figure7.2:Comparisonof˝tstoT2KdatafortheRHCMultiTrack(blue)andRHC MultiPisamples(red).Theinputparametererrorbarscanbeseeninthe background.Note,theCCQEcrosssectionparametersdonothaveanyprior constraintappliedinthe˝t,soaninputerrorbandisnotdisplayedforthese parameters. .....................................134 Figure7.3:Ratiosofthedata p cos distributiontothepost-˝tdistributionforFGD1 fortheRHCMultiTracksamples(left)andtheRHCMultiPisamples(right). The ˜ 2 pernumberofbinscanbeseenforeachdistribution. ..........135 Figure7.4:Themomentum(left)and cos (right)distributionsfortheRHCMultiTrack (black)andtheRHCMultiPi(red)samples.The ˜ 2 pernumberofbinsis shownforeachsampleset. ............................136 Figure7.5:ComparisonoftheSKneutrinoenergydistributionsfortheRHCMultiTrack (red)andRHCMultiPi(black)samples.Theerrorbarsaretheapproximate uncertaintyfromthe˛uxpluscrosssectionsystematicuncertainties. ......137 Figure7.6:ComparisonoftheSKneutrinoenergydistributionsfortheRHCMultiTrack (red)andRHCMultiPi(black)samples.Theerrorbarsaretheapproximate uncertaintyfromthe˛uxpluscrosssectionsystematicuncertainties. ......138 Figure7.7:Constant ˜ 2 68%and90%intervalsforthehybridfrequentist-Bayesianthe fullyBayesiananalysesonT2K,assumingthenormalmasshierarchy.These intervalsarebasedonaneardetector˝tusingtheRHCMultiTracksamples. Figurefrom[ 25 ]. .................................139 Figure7.8:Ratiosofthemodi˝ed p cos distribution,whichincludesadditionalan- tineutrinosinglepionproductionevents,tothenominaldistributionforFGD1. TheCC 0 ˇ samples(left),CC 1 ˇ samples(middle),andCCOthersamples (right)areshown. .................................141 xvi Figure7.9:Comparisonof˝tstothenominalMonteCarlo(blue)andtheMonteCarlo includingadditionalsinglepionproductionevents(red).TheND280neutrino ˛uxparameters(topleft)arecharacteristicoftheparametershiftsforthefull ˛uxparametersetandtheFSIparameters.Ontheotherhand,someshiftsare seeninthecrosssectionparameters(topright).Littledi˙erenceisseenfor themajorityoftheobservablenormalizationparameters(bottomleft),while slightchangesareseenintheRHC CC 1 ˇ samples(bottomright).Theinput parametererrorbarscanbeseeninthebackground. ...............143 Figure7.10:Themodi˝cationappliedtothenominalT2KMonteCarlousingtheMINOS parameterization. .................................144 Figure7.11:Ratiosofthemodi˝ed p cos distribution,whichincludesthelow Q 2 suppressionweight,tothenominaldistributionforFGD1.TheCC 0 ˇ samples (left),CC 1 ˇ samples(middle),andCCOthersamples(right)areshown. ....145 Figure7.12:Comparisonof˝tstothenominalMonteCarlo(blue)andtheMonteCarlo includingthelow Q 2 eventsuppressionmodi˝cation(red). ...........146 Figure7.13:Ratiosofthemodi˝ed p cos distribution,whichincludesthelow Q 2 suppressionweight,tothepost-˝tdistributionforFGD1.The ˜ 2 per numberofbinscanbeseenforeachdistribution. .................148 FigureA.1:MomentumdistributionscomparingPsychev1andPsychev3.Theerrorbars onthev3distributionarefromthePsychev3detectoruncertainties.More oftenthannot,thedi˙erencebetweenPsycheversionsisnotcoveredbythe errorbars. .....................................154 FigureA.2:ComparingBANFF(left)andNuMu(right)reconstructedmuonmomentum distributionsforPsychev1andPsychev3.TheBANFFplotsincludean additionalweightrelatedtothenon-Gaussianityofsomeofthedetector parameters. .....................................156 FigureA.3:E˙ectofdetectorweightbuginPsychev3.TheleftsideshowstheBANFF usingPsychev3withoutthebug˝x,whiletherightsideincludesthebug˝x. Inbothcases,theNuMucurveincludesthebug˝x. ...............157 FigureA.4:ComparingselectedeventsinPsychev3forBANFFandNuMuwithonlythe detectorvariationsapplied. ............................158 FigureA.5:ComparingselectedeventsinPsychev3forBANFFandNuMuwiththe detectorvariations,the˛uxweight,andalldetectorweightsapplied. ......159 FigureA.6:ComparingselectedeventsinPsychev3forBANFFandNuMuwiththe detectorvariationsandthe˛uxweightapplied. ..................160 xvii FigureA.7:BANFF˝tresultsforthe˛uxparameterscomparingthee˙ectofthedetector selectionsandsystematicuncertaintiesforPsychev1(red)andPsychev3 (blue).Theinputvaluesvaluecanbeseeninthebackground. ..........162 FigureA.8:BANFF˝tresultsfortheFSIandcrosssectionparameterscomparingthe e˙ectofthedetectorselectionsandsystematicuncertaintiesforPsychev1 (red)andPsychev3(blue).Theinputvaluesvaluecanbeseeninthebackground. 163 FigureA.9:BANFF˝tresultsfortheobservablenormalizationparameterscomparingthe e˙ectofthedetectorselectionsandsystematicuncertaintiesforPsychev1 (red)andPsychev3(blue).Theinputvaluesvaluecanbeseeninthebackground. 164 FigureA.10:Comparisonofthee˙ectofthedetectorselectionsandsystematicuncertain- tiesontheBANFFpost-˝tdistributionsforPsychev3(left)andPsychev1 (right). .......................................165 FigureA.11:BANFF˝tresultsforthe˛uxparameterscomparingthee˙ectofthedetector selectionsandsystematicuncertaintiesandtheMCstatisticalerrorforPsyche v1(red)andPsychev3(blue).Theinputvaluesvaluecanbeseeninthe background. ....................................166 FigureA.12:BANFF˝tresultsfortheFSIandcrosssectionparameterscomparingthe e˙ectofthedetectorselectionsandsystematicuncertaintiesandtheMC statisticalerrorforPsychev1(red)andPsychev3(blue).Theinputvalues valuecanbeseeninthebackground. .......................166 FigureA.13:BANFF˝tresultsfortheobservablenormalizationparameterscomparing thee˙ectofthedetectorselectionsandsystematicuncertaintiesandtheMC statisticalerrorforPsychev1(red)andPsychev3(blue).Theinputvalues valuecanbeseeninthebackground. .......................167 FigureA.14:Comparisonofthee˙ectofthedetectorselectionsandsystematicuncertain- tiesandtheMCstatisticalerrorontheBANFFpost-˝tdistributionsforPsyche v3(left)andPsychev1(right). ..........................168 FigureA.15:SKoscillatedeventpredictionsforthe (upperleft), (upperright), e (lowerleft), e (lowerright)˛uxes.TheratioisofPsychev1toPsychev3(v1 /v3)andthe š plotisof( j v1-v3 j /(v1error))andmeasuresthedi˙erence betweenthetwoversionswithrespecttothev1errorbar. ............169 FigureB.1:MonteCarloeventdistributionofreconstructedmuonmomentumandangle. ..170 FigureB.2:Correlationmatriceswherethebinsaregroupedbyangularbins(left)or momentumbins(right). ..............................171 xviii FigureB.3:Correlationmatrixbeforecombiningbins(left)andwithredlinesdemarcating theregionstobecombined(right). ........................172 FigureB.4:CombiningtheCC 0 ˇ angularbins.Beforecombiningbinscanbeseeninthe topleft,thebinstobecombined(seto˙bytheredlines)areinthetopright, andtheresultingmatrixaftercombiningbinsisonthebottom. .........173 FigureB.5:CombiningtheCC 0 ˇ momentumbins.Beforecombiningbinscanbeseen inthetopleft,thebinstobecombined(seto˙bytheredlines)areinthetop right,andtheresultingmatrixaftercombiningbinsisonthebottom. ......174 FigureB.6:TheFGD1FHCCC 0 ˇ p cos distributionsforthe˝tbinning(topleft), thebinningwithacombinedangularbinningandthefullmomentumbinning (topright),andthe˝naldetectorbinning(bottom).Eachbinisscaledbythe binarea. ......................................175 FigureB.7:The˝nalcorrelationmatricesfortheFGD1FHCCC 0 ˇ sample.Thebinsare groupedbycommonangularbinsontheleftandcommonmomentumbins (whichistheorderusedinthelikelihood˝t)ontheright. ............176 FigureB.8:CombiningtheCC 1 ˇ angularbins.Beforecombiningbinscanbeseeninthe topleft,thebinstobecombined(seto˙bytheredlines)areinthetopright, andtheresultingmatrixaftercombiningbinsisonthebottom. .........177 FigureB.9:CombiningtheCC 1 ˇ momentumbins.Beforecombiningbinscanbeseen inthetopleft,thebinstobecombined(seto˙bytheredlines)areinthetop right,andtheresultingmatrixaftercombiningbinsisonthebottom. ......178 FigureB.10:TheFGD1FHCCC 1 ˇ p cos distributionsforthe˝tbinning(topleft), thebinningwithacombinedangularbinningandthefullmomentumbinning (topright),andthe˝naldetectorbinning(bottom).Eachbinisscaledbythe binarea. ......................................179 FigureB.11:The˝nalcorrelationmatricesfortheFGD1FHCCC 1 ˇ sample.Thebinsare groupedbycommonangularbinsontheleftandcommonmomentumbins (whichistheorderusedinthelikelihood˝t)ontheright. ............180 FigureB.12:CombiningtheCCOtherangularbins.Beforecombiningbinscanbeseenin thetopleft,thebinstobecombined(seto˙bytheredlines)areinthetop right,andtheresultingmatrixaftercombiningbinsisonthebottom. ......181 FigureB.13:CombiningtheCCOthermomentumbins.Beforecombiningbinscanbeseen inthetopleft,thebinstobecombined(seto˙bytheredlines)areinthetop right,andtheresultingmatrixaftercombiningbinsisonthebottom. ......182 xix FigureB.14:TheFGD1FHCCCOther p cos distributionsforthe˝tbinning(topleft), thebinningwithacombinedangularbinningandthefullmomentumbinning (topright),andthe˝naldetectorbinning(bottom).Eachbinisscaledbythe binarea. ......................................183 FigureB.15:The˝nalcorrelationmatricesfortheFGD1FHCCCOthersample.Thebins aregroupedbycommonangularbinsontheleftandcommonmomentum bins(whichistheorderusedinthelikelihood˝t)ontheright. ..........184 FigureC.1:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD2FHC MultiPiselections.Themomentumdistributionsareshownontheleft,while theangulardistributionsareshownontheright. .................185 FigureC.2:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD2FHCMultiPiselections. ................186 FigureC.3:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD2FHCMultiPiselections. .....187 FigureC.4:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD2RHC MultiTrackselections.Themomentumdistributionsareshownontheleft, whiletheangulardistributionsareshownontheright. ..............188 FigureC.5:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD2RHC MultiTrackselections. .............189 FigureC.6:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD2RHC MultiTrackselections. ..190 FigureC.7:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD2RHC MultiTrackselections.Themomentumdistributionsareshownontheleft, whiletheangulardistributionsareshownontheright. ..............191 FigureC.8:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD2RHC MultiTrackselections. .............192 FigureC.9:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD2RHC MultiTrackselections. ..193 FigureC.10:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD2RHC MultiPiselections.Themomentumdistributionsareshownontheleft,while theangulardistributionsareshownontheright. .................194 FigureC.11:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD2RHC MultiPiselections. ..............195 xx FigureC.12:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD2RHC MultiPiselections. ...196 FigureC.13:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD2RHC MultiPiselections.Themomentumdistributionsareshownontheleft,while theangulardistributionsareshownontheright. .................197 FigureC.14:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD2RHC MultiPiselections. ..............198 FigureC.15:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD2RHC MultiPiselections. ...199 FigureC.16:Ratiosofthemodi˝ed p cos distribution,whichincludesadditionalan- tineutrinosinglepionproductionevents,tothenominaldistributionforFGD2. TheCC 0 ˇ samples(left),CC 1 ˇ samples(middle),andCCOthersamples (right)areshown. .................................200 FigureC.17:Ratiosofthemodi˝ed p cos distribution,whichincludesthelow Q 2 suppressionweight,tothenominaldistributionforFGD2.TheCC 0 ˇ samples (left),CC 1 ˇ samples(middle),andCCOthersamples(right)areshown. ....201 xxi CHAPTER1 EXECUTIVESUMMARY Neutrinoshavebeenastoundingphysicistsforalmostacentury.Overthelasttwentyyears,they haveprovidedoneofthebestviewsofphysicsbeyondtheStandardModelduetotheirnon-zero mass,whichwascon˝rmedthroughtheabilityforneutrinostooscillate.Sincethediscoveryof neutrinooscillations,eachofthefundamentalparametersgoverningtheprobabilityforaneutrino tooscillatehasbeenmeasured,withtheexceptionofone.Thisparameterisrelatedtocharge- parityviolationanditsmeasurementremainsoneofthehighestprioritiesforcurrentandfuture experimentsstudyingneutrinooscillations.Similarly,whileneutrinooscillationsdemonstrate thatneutrinoshavemass,thevaluesandorderingofthesemasseshaveyettobedetermined. OneexperimentmakingmeasurementsofneutrinooscillationsistheTokai-to-Kamioka(T2K) experiment. Since2010,T2Khasbeenstudyingneutrinooscillationsusingabeamofmuonneutrinos(or antineutrinos)generatedontheeastcoastofJapanwhicharedirectedtowardsthewestcoastofJapan andaredetectedintheSuper-Kamiokande(SK)detector.Thedistancebetweentheproduction pointandSKis295km,whichischosentomaximizetheprobabilitythatthemuon(anti)neutrinos oscillatetotheother˛avorsofneutrinos.Atabasiclevel,theT2Koscillationanalysiscounts thenumberofmuon(anti)neutrinointeractionsclosetotheproductionpoint,andthencompares thatwiththenumberofmuon(anti)neutrinointeractionsatSK.Thistypeofanalysisiscalleda muon(anti)neutrinoanalysis.Additionally,thenumberofelectron(anti)neutrino interactionscanbecountedatSK,whichiscalledtheelectronneutrinoanalysis. ThelargestuncertaintiesintheT2Kneutrinooscillationanalysisarerelatedtotheneutrino beam˛uxandtheneutrinocrosssection.Inordertoreducetheseuncertainties,datafromtheNear Detectorat280m(ND280)isincludedintheanalysis,whichallowstheuncertaintiesontheneutrino ˛uxandthecrosssectiontobeconstrainedsimultaneously.Bysimultaneouslyconstrainingthese uncertainties,correlationsbetweenthemcanbeusedtoreducetheuncertaintiesmorethanthey 1 couldbeindividually.Theprocessforconstrainingtheseuncertaintieswiththeneardetectordata usesabinnedmaximumlikelihood˝t,whichbinstheobservedinteractions,orevents,basedon theirtopologicalsample,whichseparatestheeventsbasedonwhatoutgoingparticlesareobserved, andtheoutgoinglepton'smomentumandanglewithrespecttotheneutrinobeam. Currently,theneutrinodatasamplesusedatthedetectorarecategorizedinasimilar waytothoseattheardetector.However,theantineutrinosamplesarecategorizedinasimpler wayfortheneardetectorcomparedtothoseatthefardetector.Forthisthesis,theantineutrino sampleswereupdatedtomorecloselymatchboththeneutrinosamplesattheneardetectorand thefardetectorsamples.Studiestocompareresultsusingthenewsamplestothoseusingtheold sampleswereperformedandaredescribedhere.Theseincludelookingatchangestothenear detectorportionoftheoscillationanalysisandhowthismighta˙ecttheoverallresultsfortheT2K oscillationanalysis. Chapter 2 willprovideanoverviewofthehistoryofneutrinophysics,includingthediscovery andtheoryofneutrinooscillationsandanintroductiontothetypesofneutrinointeractions.The chapterwillconcludewithasummaryofthecurrentunderstandingofneutrinooscillationsand someoftheopenquestionsthatremain,includingtheorderoftheneutrinomasses,theexistenceof charge-parityviolationinneutrinointeractions,andtheexistenceofsterileneutrinos.InChapter 3 , theT2Kexperimentwillbedescribedindetail,includingtheproductionofneutrinosattheJapan ProtonAcceleratorResearchComplexandthedetectorsusedtodetecttheseneutrinos.Chapter 4 willdescribetheprocedureusedintheoscillationanalysis,aswellashowtheworkpresented inthisthesis˝tsintotheoverallanalysis.Chapter 5 willdiscussthedatausedandtheprocessof selectingdataintosamplesfortheoscillationanalysis,whileChapter 6 willcoverthesystematic uncertaintiesrelatedtotheneardetector,ND280.Chapter 7 willpresenttheresultsfromthestudies performed.Finally,Chapter 8 willgiveasummaryoftheworkpresentedhere,aswellasproviding insightintofurtherstudiesthatcanbeperformedtoimprovetheneutrinooscillationanalysisat T2K. 2 CHAPTER2 NEUTRINOPHYSICS 2.1AHistoryofNeutrinos 2.1.1InitialPostulation Intheearly1930s,aproblemhadariseninnuclearbetadecay,inwhicharadioactivenucleus, X , decaysintoaslightlylighternucleus, Y ,emittinganelectronintheprocess: X ! Y + e : (2.1) Inthesetypesoftwo-bodydecays,theenergiesoftheoutgoingparticlesarekinematicallyde- terminedinthecenter-of-massframe,wheretheparentparticle, X ,isstationaryandthedecay products, Y and e ,comeoutwithequalandoppositemomenta.Inthiscase,theenergyofthe electronisdeterminedby E e = m 2 X m 2 Y + m 2 e 2 m X ! c 2 : (2.2) Oncethemasseshavebeendetermined,theenergyoftheelectronshouldbe˝xed.However,in experimentsstudyingnuclearbetadecay,theenergyoftheelectronvariedconsiderably,asseenin Figure 2.1 . Inordertorectifythisproblem,WolfgangPauliproposedthatathirdparticlewasemitted, alongwiththedaughternucleusandelectron,duringbetadecay.Inordertoconservecharge,the particlehadtobeneutral.Furthermore,becausetheenergyoftheelectroncouldrangeuptoalimit ofEquation 2.2 ,theparticlehadtobeverylight.Paulicalledhistheoreticalparticlethe However,Chadwick'sdiscoveryoftheparticleknowntodayastheneutronpreemptedthename, thoughthisparticlewastoomassivetobePauli'sproposedparticle. In1933,EnricoFermiproposedanewtheoryforbetadecay[ 26 ]whichsuccessfullyincluded Pauli'sparticle,whichhecalledtheino.Fermi'stheorythatincludedtheneutrino,denoted 3 Figure2.1:Theshapeoftheenergyspectrumoftheoutgoingelectronfromthebetadecayof tritiumcomparedwiththeexpectedvaluefromatwo-bodybetadecaygivenbyEquation 2.2 . Figurefrom[ 1 ]. bytheGreekletter, ,gavethebetadecayprocess: n ! p + e + ; (2.3) wheretheneutron, n ,iscon˝nedinnucleus X andtheproton, p ,iscon˝nedinnucleus Y .This processisveryclosetothecurrentunderstandingofthebetadecayprocess,withoneslight di˙erencethatwillbeseenshortly. 2.1.2FurtherEvidence Intheearly1930s,cosmicraysprovidedoneofthesigni˝cantsourcesofparticlemeasurements. Cosmicraysarehighenergychargedparticleswhichoriginateinastrophysicalprocesses.Primarily, cosmicraysareprotons,buttheycanalsobeheavierions.Whentheseparticlesapproachthe Earth,theycaninteractwithmoleculesfoundintheupperatmosphere.Theseinteractionsproduce secondaryparticles,mostofwhichdecaypriortoreachingthesurfaceoftheEarth.However,due 4 totimedilation,someofthesesecondaryparticlesareabletoreachthesurface.Ineithercase, scientistsareabletostudycosmicraysthroughtheparticlesdetectedontheground. OnesecondaryparticlewhichoftenreachestheEarth'ssurfaceisthemuon.Becauseofthis, cosmicraymuonshavebeenwell-studied.Similartobetadecay,thedecayofthemuonproduces anelectronwitharangeofenergies,whichmeanstheremustbeatleastoneadditionalneutral particleoriginatinginthedecayprocess.Theacceptedexplanation[ 27 ]forthisisthattheremust betwoneutrinosproducedwiththeelectron: ! e + 2 : (2.4) Inthelate1940s,C.F.Powellperformedanexperimentthatobservedtracksfromcharged particlesinvolvedincosmicrayinteractionsthroughtheuseofphotographicemulsions[ 28 ]. Usingthetracksproducedbyparticlestravelingthroughtheemulsion,theirinteractionsinthe medium,andtheirdecay,thenatureofcosmicrayscouldbestudied.Theresultsofhisexperiment showedclearevidencethatanothersecondaryparticle,nowknowntobethepion,existed.The piondecaysintoamuonand,duetoconservationofmomentum,anotherparticle.Thisother particlehadtobeneutral,astherewasnotrackleftintheemulsionasittraveled.Therefore,itwas postulatedthatthepionmustdecayvia: ˇ ! + : (2.5) 2.1.3TheFirstObservation Bythe1950s,increasingtheoreticalevidencefortheneutrinoexisted,beitfrommuon,pion, orunstablenucleidecay,buttherewasnoexperimentalveri˝cation.In1953,Konopinskiand Mahmoud[ 29 ]introducedaconceptwhichwouldhelpinthedeterminationofwhetheragiven reactionwaspossibleornot.Thisconceptbecameknownastheleptonnumber,wherealepton numberof + 1 wasassignedtotheelectron,negativelychargedmuon,andneutrino,whilethe positron,positivelychargedmuon,andantineutrinoweregivenaleptonnumberof 1 .Anyother particleswouldreceivealeptonnumberof 0 .Theyproposedthatthetotalleptonnumbermust 5 beconservedwithinparticleinteractions.Duetothis,itwasdeterminedthattheneutrinoinbeta decaymust,inreality,beanantineutrino,andthebetadecayprocessgivenbyEquation 2.3 ,must actuallybe n ! p + e + : (2.6) Bythistime,neutrinoswereunderstoodtohaveasmallcrosssection,orprobabilitytointeract, withmatter.Becauseofthis,anintensesourceofeitherneutrinosorantineutrinosisneeded toachieveastatisticallysigni˝cantresultinareasonableamountoftime.Inordertostudy antineutrinos,nuclearreactorsprovidedthebestoption,astheywerethoughttobethemostintense sourceofantineutrinosavailable. Underthisassumption,ReinesandCowanconductedanexperimentattheSavannahRiver nuclearreactorinSouthCarolina,whichhadanantineutrino˛uxofof 1 : 2 10 13 particlesper squarecentimeterpersecond[ 30 ].TheyusedadetectorcontainingwaterdopedwithCdCl 2 andinstrumentedwithscintillatorcoupledtophotomultipliertubes,allowingthemtodetectlight producedinsidethedetector.Throughinversebetadecay( + p ! e + + n ),thepresenceof antineutrinoscouldbedetectedinoneoftwoways: 1. Thepositronwouldannihilatewithanelectroninthewater,producinglightvia e + + e ! 2 . 2. Theneutronwouldbecapturedonacadmiumnucleus,whichwouldproduceanexcitedstate thatwoulddecaybyemittingaphoton.Anexampleofthiswouldbe, n + 113 Cd ! 114 Cd ! 114 Cd + . Becauseneutroncapturetakesmoretimethanelectron-positronannihilation,acharacteristic delayedcoincidencesignalcouldbelookedforintheexperiment.In1956,CowanandReines con˝rmedtheexistenceoftheneutrinobyobservingtwoorthreeinteractionsperhourwhilethe reactorwasrunning[ 31 , 32 ]. 6 2.1.4ProposalofDi˙erentNeutrinoTypes Aroundthistime,anissuewiththeconceptofconservationofleptonnumberarose.Aruleof thumbinparticlephysics,isthatythingthatisnotcompulsoryisforbidden[ 33 ];speci˝cally, ifaninteractionisnotobserved,thenitisexpresslyforbiddenbyaconservationlaw.Basedonthe conservationofleptonnumber,themuoncandecayvia ! e + .However,thisdecayisnever observed,whichiscontradictorytothelawoftheconservationofleptonnumber. Oneanswertothisproblempositedthattheleptonnumberisactuallydividedamongthelepton ˛avors,ortypes[ 34 , 35 ].Underthisnewassumption,therewouldbeanelectronandanelectron neutrino(eachhavinganelectronnumberof + 1 ),whilethemuonwouldhaveacorresponding muonneutrino(bothwithamuonnumberof + 1 ).Correspondingly,therewouldbematching antiparticles,butwithleptonnumbersof 1 .Inthiscase,themuondecay, ! e + ,would beforbidden,asthemuonandelectronnumbersarenotconservedacrosstheinteraction.Using thismodi˝edconservationofleptonnumber,themuondecayshowninEquation 2.4 wouldnow become ! e + + e (2.7) + ! e + + + e (2.8) formuonsandantimuons,respectively.Furthermore,thepiondecayinEquation 2.5 nowbecomes ˇ + ! + + (2.9) ˇ ! + : (2.10) Theequationforbetadecay(Equation 2.3 )becomes n ! p + e + e ; (2.11) whichmatchesthecurrentunderstandingofbetadecay. 7 Figure2.2:Totalneutrino(top)andantineutrino(bottom)chargedcurrentcrosssectionper nucleon(foranisoscalartarget)dividedbyneutrinoenergyandplottedasafunctionofenergy. Additionally,thetotalcrosssectionisseparatedintoCCQE(labeledasQE),CCresonant(RES), anddeepinelasticscattering(DIS)crosssections.Figurefrom[ 13 ]. 2.1.5DetectingtheMuonandTauNeutrinos Inordertoverifythissolution,experimentsbeganlookingforevidenceofadistinctmuontype neutrino.Becausetheneutrinocrosssectionissmall(Figure 2.2 showsthesizeofthecrosssection), eitheranintensesourceofneutrinosoralargedetectorandlotsoftimeisneededtosearchfor muonneutrinos.Thankfully,natureprovidesalargesourceofmuonneutrinosandantineutrinos throughthedecayofcosmicraymuons.However,thesealsoprovidealargesourceofelectron neutrinosandantineutrinos,makingithardtodistinguishbetweenthetwo˛avorsofneutrinos. Ratherthanrelyingonnaturallyproducedmuonneutrinos,experimentalistscanarti˝ciallyproduce 8 theminalaboratorybyacceleratingprotonsinaparticleacceleratorandcollidingthemwitha target.Thiswouldcreateasimilarsituationtothatofcosmicrayscollidingwithparticlesinthe upperatmosphere,producingpionswhichdecayintomuonsandmuonneutrinosorantineutrinos. Usingthistechnique,Lederman,Schwartz,andSteinberger[ 36 ]designedanexperimentthat acceleratedprotonsto15GeVbeforecollidingthemwithaberylliumtargetatBrookhavenNational Laboratory.Thesecollisionsproducedcopiousamountsofpionstravelingapproximatelyinthe samedirectionastheprotons.Thepionswouldthendecayintomuonsandmuonneutrinosand antineutrinos.Athickironwallwasstationeddownstreamfromthetargettostopthemuonsand preventthemfromtravelingthroughthedetector.Neutrinosproducedfromthemuonsdecaying intheironwallwouldbefocusedinthedirectionthemuonsweretraveling,ratherthanthebeam direction,givingamuchlowerbackgroundofelectronneutrinosandantineutrinoscomparedto neutrinoscomingfromcosmicrays. Onthedownstreamsideoftheironwall,asparkchamberdetectorwassituated.Thedetector consistedoften1tonmodules,eachofwhichcontainedaseriesofparallelaluminumplates. Thevolumebetweentheplateswas˝lledwithagas,which,whenachargedparticlepassed throughit,wouldionizealongtheparticle'strack,producingavisiblespark.Muonstraveling throughthedetectorcouldbedistinguishedfromelectronsbythefactthattheycouldtravelthrough severalaluminumplatesbeforeinteracting,whereaselectronscouldnot.Theneutrinointeractions observedinthisexperimentwereconsistentwithneutrinosproducedfromthedecayofpionsinto muonsandmuonneutrinos,thelatterofwhichwouldinteractinthedetectortoproducemuonsand notelectrons.Duetothisfact,theseneutrinosareadistinct˛avorfromthoseseenbyReinesand Cowan,establishingtheexistenceofthemuonneutrino. In1975,anewlepton,thetau, ˝ ,wasdiscoveredbyMartinLewisPerlandtheSLA group[ 37 ].Asithadalreadybeendeterminedthattherewasaneutrinopartnerfortheelectron andmuon,thisdiscoverycreatedtheneedforathirdneutrino˛avor, ˝ . Inordertodirectlyobservetauneutrinos,theDONUT(DirectObservationoftheNUTau) experimentwascreated[ 38 ].Thisexperimentusedprotonsacceleratedto800GeVattheTevatron 9 atFermiNationalAcceleratorLaboratorywhichweredirectedtowardsa1mlongtungstentarget. Throughthedecayof D S mesons( D + S ! ˝ + + ˝ ),tauneutrinoscouldbeproduced.Usinga detectorwhichcoulddi˙erentiatebetweenelectrons,muons,andtaus,theexperimentanalyzed 203neutrinointeractionsandfoundfour ˝ interactions,thuscon˝rmingtheexistenceofthetau neutrino[ 38 ]. 2.1.6CurrentUnderstanding Today,theneutrinohasbeenincorporatedintotheStandardModel,whichdescribestheelementary particlesandhowtheyinteractwithoneanother.TheStandardModelhasbeenindevelopment sincethemiddleofthetwentiethcenturyandprovidesthetheoreticaldescriptionoftheelectroweak andstrongforces.TwotypesofparticlesareincludedintheStandardModel,thespin- 1 2 fermions andtheintegerspinbosons,whichmediatetheelectroweakandstrongforces[ 27 ]. Thefermionsaredividedintotwogroups,theleptonsandthequarks.Theleptonsincludethe electron,muon,andtauparticles,aswellasthecorrespondingneutrinosdescribedinSection 2.1 . Therearesixdi˙erenttypesofquarks,theup( u ),charm( c ),andtop( t ),whichhaveachargeof + 2 š 3 ,whilethedown( d ),strange( s ),andbottom( b )quarkshaveachargeof 1 š 3 .Thequarksare abletointeractwiththechargedleptonsthroughboththeweakandelectromagneticforces,while theycaninteractwithneutrinosviatheweakforce. Thebosonsincludethephoton, ,whichmediatestheelectromagneticforce,andthegluon, g , whichisthemediatorofthestrongforce.Fortheweakforce,therearetwomediators,the W and the Z 0 ,whicharedescribedinmoredetailinSection 2.2 .TheHiggsboson, H ,isthemostbasic manifestationoftheBrout-Englert-Higgsmechanism,whichgivesthe W and Z bosonstheirmass throughtheirinteractionwiththeHiggs˝eld.AsummaryoftheparticlescanbefoundinTable 2.1 . 10 Fermions Bosons u c t d s b g e ˝ W ; Z e ˝ H Table2.1:StandardModelofParticlePhysics.Thetopportionundertheermionsheading showsthequarks,whilethebottomshowstheleptons. 2.2NeutrinoInteractions AccordingtotheStandardModel,neutrinosareonlyabletointeractwithmatterviatheweak force.Weakinteractionsaremediatedbyeitherthe Z 0 orthe W .The Z 0 isaneutralparticle withamassof 91 : 1876 0 : 0021 GeV/c 2 [ 3 ],whilethe W isachargedparticlewithamass of 80 : 385 0 : 015 GeV/c 2 [ 3 ].Interactionsmediatedbythe Z 0 arecalledneutralcurrent(NC) interactions,whilethosemediatedbythe W arecalledchargedcurrent(CC)interactions.Ina neutralcurrentinteraction,aneutrinoorantineutrino(ofany˛avor)interactswithaquarkorlepton (ofany˛avor),transferringenergyandmomentumviaa Z 0 .Asidefromtransferringenergyand momentum,theparticlesareunchangedaftertheinteraction.Chargedcurrentinteractionsoccur whenaneutrinoorantineutrinointeractsviathe W toproducealeptonofthesame˛avor.Because theseinteractionsproduceachargedleptoninthe˝nalstate,theyareeasiertodetectthanneutral currentinteractions. WhileinteractionsonindividualquarksareallowedbytheStandardModel,quarksdonot appearasfreeparticles,butareboundinnucleons 1 ,suchastheproton( uud )andneutron( udd ). Interactionsbetweenneutrinosorantineutrinosandnucleonsareclassi˝edintomultiplecategories. 1 Generically,quarksareboundinhadrons,whichincludesmesons,suchasthepion,and baryons,suchastheprotonandneutron. 11 Name Abbreviation Neutrinos Antineutrinos CCQuasi-Elastic CCQE l + n ! l + p l + p ! l + + n CC2particle2hole 2p2h l + np ! l + p + p l + np ! l + + n + n CCresonant CCRES l + p ! l + ++ ! l + p + ˇ + l + n ! l + + ! l + + n + ˇ PionProduction l + n ! l + + ! l + p + ˇ 0 l + p ! l + + 0 ! l + + n + ˇ 0 l + n ! l + + ! l + n + ˇ + l + p ! l + + 0 ! l + + p + ˇ CCCoherent CCCoh l + A ! l + A + ˇ + l + A ! l + + A + ˇ PionProduction (whereAisanucleus) (whereAisanucleus) CCOther CCOther l + ¹ n or p º! l + ¹ p or n º + pions l + ¹ p or n º! l + + ¹ n or p º + pions NCResonant NCRES l + n ! l + 0 ! l + n + ˇ 0 l + n ! l + 0 ! l + n + ˇ 0 PionProduction l + p ! l + + ! l + p + ˇ 0 l + p ! l + + ! l + p + ˇ 0 l + n ! l + 0 ! l + p + ˇ l + n ! l + 0 ! l + p + ˇ l + p ! l + + ! l + n + ˇ + l + p ! l + + ! l + n + ˇ + NCOther NCOther l + ¹ n or p º! l + ¹ p or n º + pions l + ¹ n or p º! l + ¹ p or n º + pions Table2.2:Typesofneutrinoandantineutrinointeractions. l canbeeither e , ,or ˝ .The categorycoversanyhighenergyinteractionsthatproducemorethanonepion.Thepionsinthese interactionscanbechargedorneutralorboth.Note,thisdoesnotcoverallinteractionsthatmay fallwithinagiveninteractiontype.Tablefrom[ 1 ] Furthermore,experimentstendtousenucleiwhichcontainmorethanonenucleon,openingup additionalprocessesinwhichtheneutrinocaninteractwithmultiplenucleonsortheentirenucleus. SomeoftheseinteractionscanbeseeninTable 2.2 . Thechargedcurrentquasi-elastic(CCQE)processisofparticularinterest.AsaCCprocess, theoutgoingchargedleptoncanbedetectedandidenti˝edtodeterminethe˛avoroftheincoming neutrino.Theenergyfromtheneutrinocanbecalculatedusingthemomentumoftheoutgoing lepton( p )andtheanglebetweentheneutrinopathandtheleptonpath( l ),whichisfairlywell knownforaneutrinobeam: E = m 2 p ¹ m n E b º 2 m 2 l + 2 ¹ m n E b º E l 2 ¹ m n E b E l + p l cos l º ; (2.12) where m p istheprotonmass, m n istheneutronmass,and m l and E l arethemassandenergyofthe lepton,respectively.The E b termisrelatedtotheenergyittakestoremovetheneutronfromthe nucleus. Interactionsbetweenneutrinosandnucleiposeanumberofexperimentalproblems.Inthe caseofinteractionsoccurringonmultiplenucleons,like2p2h,describedinmoredetailinSection 12 Figure2.3:Varioustypesof˝nalstateinteractionsoccurringwithinanucleus.Figurefrom[ 14 ]. 4.3.2.1 ,orontheentirenucleus,likeCCcoherent,theoutgoingparticlesmimicothertypesof interactions(CCQEandCCresonant,respectively)intermsofwhatismeasuredinadetector.These processescanbemisidenti˝edforanumberofreasons,includingoutgoingprotonsandpionshaving energiesbelowthethresholdfordetectionorthedetectorbeinginsensitivetooutgoingparticles, likeneutrons.Furthermore,whileleptonscaneasilyescapethenucleus,otherparticlessuchas protonsandpions,mayinteractinsidethenucleus.Thisprocessiscalleda˝nalstateinteraction (FSI),andadiagramofsomeofthetypesof˝nalstateinteractionscanbeseeninFigure 2.3 . Whentheseinteractionsoccur,theparticleswhichthedetectorcoulddetectmaybedi˙erentfrom theparticlesproducedintheinitialinteraction.Theseprocessescreateaddedlevelsofdi˚culty whenitcomestomeasuringneutrinointeractionsbyalteringthereconstructedenergyspectrum. Forexample,iftheoutgoingpionfromaCCresonantinteractswithinthenucleusandisabsorbed, theinteractioncouldbemisidenti˝edasaCCQEinteraction.Inthisinstance,theneutrinoenergy wouldbecalculatedviaEquation 2.12 ,whichdoesnotaccountforthepion,resultinginanincorrect reconstructedenergy. 13 2.3NeutrinoOscillations Theconceptofneutrinooscillationsarosetodescribethesolarneutrinoproblemandthe atmosphericneutrinoanomaly(bothofwhicharedescribedinmoredetailinSection 2.3.1 ).In neutrinooscillations,aneutrinoofone˛avorchangesintoadi˙erent˛avorasittravelsthrough spaceandtime.Throughtheseoscillations,someneutrinosmaychangetoa˛avorwhichan experimentisnotsensitiveto,causingfewerneutrinosoftheoriginal˛avortobemeasuredthan expected,referredtoasinodisappearance.Conversely,inoappearanceiswhen neutrinosofone˛avoraredetectedfromaneutrinosourceofadi˙erent˛avor. 2.3.1Motivation Asthestudyofneutrinosfromvarioussourcesprogressed,di˙erencesbetweenthetheoretical predictionsandtheobservednumberofneutrinosarose.Thetwomaindiscrepancieswerethesolar neutrinoproblemandtheatmosphericneutrinoanomaly. TheSolarNeutrinoProblem Asthe1960sprogressed,modelsofthesunweredevelopedbasedontheresultsofexperiments lookingforinteractionsthoughttooccurinsidethesun.Asmanyoftheseinteractionsproduceneu- trinosacrossdi˙erentenergyranges,thestudyofsolarneutrinoswouldprovidecriticalinformation onhowthesunworks. TheHomestakesolarneutrinoexperimentwasbuiltinthemid-1960s[ 39 ]andwasplaced nearly1500mbelowthesurfaceintheHomestakeGoldMineinLead,SouthDakota.Byplacing theexperimentatsuchadepth,thecosmicraymuonbackgroundcouldbegreatlyreduced.The detectorwasmadeofatankholding615tonsoftetrachloroethylene(C 2 Cl 4 ),whichwasusedto detectneutrinosthroughtheinteractionsofelectronneutrinoswithchlorineatoms, e + 37 Cl ! 37 Ar + e : (2.13) Toseeifthisinteractionoccurred,heliumgaswasbubbledthroughthetetrachloroethylene,ex- tractingthegaseousargon,whichwasthenpassedoveranabsorberthatabsorbedtheargon. 37 Ar 14 isaradioactiveisotopeofargon;therefore,thenumberofdecayscouldbecountedintheabsorber todeterminethenumberofneutrinointeractionsthatoccurredinthedetector. OneoftheearlyresultsoftheHomestakeexperiment[ 40 ]setanupperlimitof0.5interactions perdaywheretheyhadexpectedtoseebetweentwoandseveninteractionsperdayinthedetector. Atthetime,itwasassumedthattherewasaproblemwiththesolarmodelandatheoreticalpaper [ 41 ]waspublishedthatarguedthemodelscouldbesu˚cientlymodi˝edtomakethemagreebetter withtheHomestakeresults. Intheyearsfollowingthe˝rstHomestakeresult,thesolarmodelcontinuedtobedeveloped. Alongsidethemodeldevelopment,Homestakecontinuedtooperate,measuringthesolarneutrino ˛uxdowntoenergiesof0.814MeV,whichistheminimumenergyrequiredofaneutrinotointeract viaEquation 2.13 .However,bythelate1970s,attemptstoreconciletheHomestakeresultwiththe theoreticalsolarmodelswerethwartedtothepointthattheissuewasdeclaredtheNeutrino Problem. Additionalexperimentsprobingdi˙erentneutrinoenergyregionsonlydeepenedtheprob- lem.OnesuchexperimentwasKamiokande-II,awaterCherenkovdetectorandprecursortothe Super-Kamiokandeexperiment(describedinSection 3.3 ).WaterCherenkovdetectorsworkby instrumentingavolumeofwaterwithsensitivelightdetectors,whichdetectlightproducedby particlestravelingfasterthanthespeedoflightinwater,calledCherenkovradiation[ 42 ].The Kamiokande-IIexperimentworkedbylookingfortheelasticscatteringofelectronneutrinoswith electronsvia: e + e ! e + e ; (2.14) whichcanoccureitherthroughtheexchangeofaW oraZboson.Thekinematicsofthisin- teraction,inadditiontotheabilitytoreconstructthedirectionoftheelectron,providedenough informationtodeterminethedirectionoftheincomingneutrino,whichcon˝rmedtheseneu- trinoswerecomingfromthesun.Inordertodistinguishsignalinteractionsfrombackground, Kamiokande-IIrequiredthattheoutgoingelectronsmetacertainenergythreshold.Byenforc- ingthisminimumenergy,Kamiokande-IIwasonlysensitivetothehighestenergysolarneutrino 15 productionchannels,primarily 8 B ! 8 Be + e + + e decays.In1990,theexperimentobserveda neutrino˛uxof 0 : 46 0 : 05 (stat.) 0 : 06 (syst.)relativetothesolarneutrinoprediction[ 43 ].While thisresultcon˝rmedthede˝citseenbyHomestake,thetwode˝citswereindisagreementwhen consideringtheneutrinoproductionchannelstowhichHomestakewassensitive. Duringitstenure,Kamiokande-IIwasrestrictedtolookingforelasticscatteringofneutrinos, whichwasduetotheuseofwater(H 2 O).Inorderforanelectronneutrinotointeractvia achargedcurrentprocess,aneutronisrequiredand,whileoxygencontainsanumberofneutrons, theenergythresholdfortheinteraction 16 O + e ! 16 F + e is15.4MeV.Thisisabovethe energyofmostsolarneutrinos,andisnearingtheexpectedmaximumenergyofsolarneutrinos, 18.8MeV[ 44 ].Becauseofthehighthresholdandtherelativelylownumberofsolarneutrinos abovethisenergy,theseinteractionsaresuppressed.Therefore,eventhemostenergeticofelectrons wouldnotbeabletoreachthethresholdforsignal-backgrounddiscrimination. TheSolarNeutrinoProblemwasresolvedbytheSudburyNeutrinoObservatory(SNO)[ 45 ]. SNOisaonektonwaterCherenkovdetectorlocatedatthe6800footleveloftheCreightonMine inSudbury,Ontario,Canada.Ratherthanusingwater,SNOemployedtheuseofheavy water(D 2 O),wherethehydrogenatomshavebeenexchangedfordeuterium( d )atoms,whichisan isotopeofhydrogenwithoneneutron.ThisallowsSNOtobesensitivetobothachargedcurrent interactionchannel: e + d ! p + p + e (2.15) andaneutralcurrentinteractionchannel: e + d ! p + n + e : (2.16) In2001,apotentialsolutiontotheSolarNeutrinoProblemwasproposedinajointmeasurement bySNOandSuper-Kamiokande(thesuccessortoKamiokande-II)[ 46 ].Bythistime,SNOhad observedenough e chargedcurrentinteractionstomakeaprecisemeasurementofthesolar neutrino˛ux.WhencombinedwiththeSuper-Kamiokandeelasticscatteringresults,whichwere sensitivetoallthree˛avors,thisindicatedtherewasalsoa˛uxfrom and ˝ .Thecombined resultfellwithinthesolarmodelpredictionforthetotalneutrino˛ux. 16 Usingneutralcurrentinteractions,SNOwasabletoimproveitsmeasurements.Inorderto measuretheseinteractions,theexperimentreliedondetectingtheoutgoingneutronfromEquation 2.16 ,whichisbeyondthetraditionalabilitiesofawaterCherenkovdetector.Todetecttheneutron, SNOusedavarietyofneutroncapturemethodsoverthecourseofitsthreeoperationalphases. Basedondatafromits˝rstphase,whichendedinMay2001,SNOindependentlycon˝rmed thepresenceofmuonneutrinosandtauneutrinos,inadditiontotheexpectedelectronneutrinos,in thesolarneutrino˛ux[ 47 ].Thisresultshowedatotalneutrino˛uxconsistentwiththesolarmodel prediction.Therefore,thesolutiontotheSolarNeutrinoProblemrequiredthatsolarneutrinosbe abletooscillatebetween˛avors. TheAtmosphericNeutrinoAnomaly Oneoftheareasofphysicsthatgrewininterestduringthe1980swasthesearchforproton decay.WhilethisprocessisforbiddenintheStandardModel,manytheoriesbeyondtheStandard Modelincludeprotondecay.Duetoitstheoreticallylonghalflife,searchesforprotondecayrequire alargedetectorthatisheavilyshieldedfromexternalbackgrounds,likecosmicraymuons,and holdsamultitudeofprotons.Whileplacingthisdetectordeepundergroundwouldgreatlyreduce thecosmicraymuonbackground,atmosphericneutrinoswouldcontinuetostreamthroughthe detector.Therefore,itwasnecessaryfortheatmosphericneutrino˛uxtobewellunderstoodwhen searchingforprotondecay. Atthetime,thereweremanytheoreticalmodelsoftheatmosphericneutrino˛ux.These modelswerebasedonthecosmicraymuon˛ux,which,incombinationwiththeunderstanding oftheproductionofatmosphericneutrinosfromthesemuons,providedagoodpredictionofthe neutrino˛ux.Additionally,theprocessestoproduceatmosphericneutrinosformedaprediction oftheratiobetweenmuonneutrinosandelectronneutrinos,asafunctionofenergy.Forlower energies,bothpiondecayandmuondecaywouldoccur,givingaratioofapproximately 2:1 for ¹ + º : ¹ e + e º .However,astheenergiesincreased,moreandmoremuonswouldreach thesurfacebeforedecaying,reducingthenumberofelectronneutrinosorantineutrinos,thereby increasingthisratio. 17 TheIrvine-Michigan-Brookhaven(IMB)andKamiokaNucleonDecay(KamiokaNDE)exper- imentsweretwoprotondecayexperimentsstudyingtheatmosphericneutrinobackgroundinmore detail.Bothdetectorswereabletodistinguishbetweenmuonandelectronneutrinos 2 ,allowing themtomakemeasurementsoftheatmosphericneutrino˛avorratio.Eachexperimentcompared theobserved˛avorratio( R obs )tothepredictionfromtheory( R theor y ),whichdi˙eredsigni˝cantly fromoneanother.TheresultfromIMBshowedaratioof R obs š R theor y thatwas 0 : 54 0 : 05 0 : 11 [ 48 ],whileKamiokaNDEhadaresultof 0 : 60 + 0 : 07 0 : 06 0 : 05 [ 49 ].Bothresultsshowedade˝citof observedresultscomparedtothetheoreticalprediction,whichwasdesignatedtheAtmospheric NeutrinoAnomaly. ThisanomalywassolvedbySuper-Kamiokandein1998[ 15 ].Inadditiontobeingableto distinguishbetween˛avorsofneutrinos,thedirectionoftheoutgoingleptonallowedtheexperiment todeterminethedirectionoftheincomingneutrino.Whenthisdirectionwascomparedwiththe cosineoftheanglemadewithrespectthevertical,aclearangulardependencewasseeninthe di˙erencebetweenthemodelpredictedinteractionrateandthemeasuredrate. Thisangulardependencecanbeshowntotieintohowfartheneutrinotraveled.When cos = 1 ,theneutrinoiscomingfromdirectlyoverheadandonlytraversesafewtensofkilometers ofatmosphere.Conversely,if cos = 1 ,thentheneutrinoiscomingupfrombelowandhad totravelthroughthousandsofkilometersofEarth.Inneutrinooscillationtheory(discussedin Section 2.1.6 ),theprobabilityforaneutrinotooscillatehasasinusoidaldependenceon L š E , where L isthedistancetraveledbytheneutrinoand E istheneutrinoenergy.Figure 2.4 showsan L š E dependenceconsistentwiththeoscillationof to ˝ .Therefore,theAtmosphericNeutrino Anomalywassolvedbytakingintoaccountneutrinooscillations. 2 Theseexperimentswereunabletodeterminethechargeoftheoutgoinglepton,sotheycould notdistinguishbetweenneutrinosandantineutrinos. 18 Figure2.4:Theratioofthenumberofdatainteractionstothenumberofpredictedinteractions assumingnoneutrinooscillations(fromMonteCarlo)asafunctionof L š E in Super-Kamiokande.Thepointsshowtheratio,whilethedashedlinesshowtheexpectedshape whenincludingoscillationsof to ˝ .Figurefrom[ 15 ]. 2.3.2Theory TheabilityforneutrinostooscillateprovidedasolutiontoboththeSolarNeutrinoProblemand theAtmosphericNeutrinoAnomaly.Thissectionwilldescribethetheorybehindhowneutrinos oscillate 3 . NeutrinoOscillationsinVacuum IntheStandardModel,neutrinosareassumedtobemassless.However,forneutrinooscillations tooccur,itisrequiredthatneutrinoshaveanonzeromass 4 .Therefore,neutrinooscillationsprovide auniquewindowintostudyingphysicsbeyondtheStandardModel. Toseetherelationbetweentheneutrinoshavingmassandneutrinooscillations,considerthe quantummechanicalstatesofthetheneutrino˛avors(calledthe˛avoreigenstates), j e i , j i ,and 3 Throughoutthissection,naturalunits( ~ = c = 1 )willbeused. 4 Technically,atleasttwoofthethreemassstatesmustbenon-zero.Thelightestmassstateis allowedtobemassless. 19 j ˝ i .UsingtheHamiltonianoperator, H ,thetotalenergyofastate, j i ,anditstimeevolutioncan bedeterminedusingtheSchrödingerEquation i @ @ t j ¹ t ºi = H j ¹ t ºi : (2.17) If H istimeindependent,thesolutiontakestheform j ¹ t ºi = e iHt j ¹ 0 ºi ; (2.18) where j ¹ 0 ºi canbewrittenasasuperpositionoforthonormaleigenstatesoftheHamiltonian, j j i . Applyingthesesuperpositions,Equation 2.18 becomes j ¹ t ºi = Õ j a j e iE j t j j i : (2.19) Equation 2.19 showsthatthetimedependentphaseineacheigenstateisafunctionofthe eigenstate'senergy(theeigenvalueoftheHamiltonian).Anychangesinphasebetweenthevarious eigenstatesofthesystemproducechangesthatcanbeobservedexperimentally.Thisisbecausethe probabilitytoobservestate j ¹ 0 ºi aftertime, t ,is jh ¹ 0 ºj ¹ t ºij 2 . Thetotalenergy, E ,ofamassiveneutrinothatdoesnotsitinapotential˝eldis E = q j ® p j 2 + m 2 ; (2.20) where ® p isthemomentumand m isthemassoftheneutrino.Therefore,theHamiltonianof theneutrinopropagatingthroughtimeismassdependent.Theeigenstatesofthefreeparticle Hamiltonianarecalledthemasseigenstatesand,currently,thereisbelievedtobethreestates, j 1 i , j 2 i ,and j 3 i .ThePontecorvo-Maki-Nakagawa-Sakata(PMNS)matrix,namedforthetheorists who˝rstdevelopedthetheoryofneutrinooscillations[ 50 , 51 ],showstherelationshipbetween the˛avoreigenstatesandmasseigenstates.UsingthePMNSmatrix,neutrinooscillationscanbe parameterizedusingthreemixingangles, 12 , 23 ,and 13 ,andaCP(charge-parity)violating phase, CP .Thematrix 5 canbewrittenasaproductofthreerotationmatriceswithacomplex 5 Inadditiontothethreematricesshown,afourthmatrixcontainingtheso-calledjorana 20 phase U = © « U e 1 U e 2 U e 3 U 1 U 2 U 3 U ˝ 1 U ˝ 2 U ˝ 3 ª ® ® ® ® ® ¬ = © « 100 0 c 23 s 23 0 s 23 c 23 ª ® ® ® ® ® ¬ © « c 13 0 s 13 e i CP 010 s 13 e i CP 0 c 13 ª ® ® ® ® ® ¬ © « c 12 s 12 0 s 12 c 12 0 001 ª ® ® ® ® ® ¬ ; (2.21) where c ij = cos ij and s ij = sin ij . FromthePMNSmatrix,thecorrespondencebetweenthe˛avor(subscript f )andmass(subscript m )eigenstatesis j f i = 3 Õ m = 1 U fm j m i (2.22) j f i = 3 Õ m = 1 U fm j m i (2.23) forneutrinosandantineutrinos,respectively. Nowthatarelationshipbetweenthe˛avorandmasseigenstateshasbeendetermined,thetime evolutionofthe˛avoreigenstatescanbedetermined.Asimpli˝edderivation,liketheonefound in[ 55 ],providesasu˚cientresult,butamoredetailedderivationcanbefoundinthearticle, NeutrinoMasses,Mixing,andOscillations of[ 3 ].Inthissimplerderivation,startbytakingthe ultrarelativisticlimitofEquation 2.20 ,whichisavalidapproximationasneutrinoshaveverysmall massesandtravelnearlyatthespeedoflight.Assumingallmasseigenstatestravelatthesame momentum( p ˇ E ,inthislimit),theenergyeigenstatescanbeTaylorexpandedas E j ˇ E + m 2 j 2 E : (2.24) Becauseneutrinosaretravelingclosetothespeedoflight,thenthetraveltime, t ,isapproximately thedistancetraveled, L ,dividedbythespeedoflightandcanbeusedintheoscillationprobabilities CP-violatingphasescanbeincluded.Thesephasesonlyhavephysicale˙ectsifneutrinosare Majoranaparticles,wheretheneutrinoisitsownantiparticle.However,evenifneutrinosare Majoranaparticles,thesephasesdonota˙ecttheprobabilityforoscillation,asthephasescancel whencalculatingtheprobability.Therefore,itisnotpossibletodetermineifneutrinosareMajorana particlesornotfromoscillationexperiments,whichmeansotherexperimentsareneededtostudy this.SuchexperimentsincludeEXO[ 52 ],MAJORANA[ 53 ],andXMASS[ 54 ]. 21 seeninEquation 2.25 (neutrinos)andEquation 2.26 (antineutrinos): P ¹ ! º = 4 Õ k > j R kj sin 2 © « 1 : 267 m 2 kj L E ª ® ¬ (2.25) + 2 Õ k > j I kj sin © « 2 : 534 m 2 kj L E ª ® ¬ P ¹ ! º = 4 Õ k > j R kj sin 2 © « 1 : 267 m 2 kj L E ª ® ¬ (2.26) 2 Õ k > j I kj sin © « 2 : 534 m 2 kj L E ª ® ¬ ; where R kj = Re » U k U k U j U j ¼ , I kj = Im » U k U k U j U j ¼ ,and m 2 kj = m 2 k m 2 j .In Equations 2.25 and 2.26 ,theconstantswithinthesinefunctionsarisebyrequiring m 2 kj , L ,and E haveunitsofeV,kilometers,andGeV,respectively. Speci˝cally,onT2K,Equation 2.25 (Equation 2.26 )canbebeusedtocalculatethemuon (anti)neutrinodisappearanceprobabilityandtheelectron(anti)neutrinoappearanceprobability.To ˝rstorder,thedisappearanceprobabilityis P ¹ ¹ º! ¹ ººˇ 1 4cos 2 13 sin 2 23 ¹ 1 cos 2 13 sin 2 23 º sin 2 m 2 32 L 4 E ! (2.27) + » solarandmattere˙ectterms ¼ ; whiletheappearanceprobabilityis P ¹ ¹ º! e ¹ e ººˇ sin 2 23 sin 2 2 13 sin 2 m 2 31 L 4 E ! (2.28) + sin 2 2 23 sin2 13 cos 13 sin m 2 21 L 4 E ! sin m 2 31 L 4 E ! " cos m 2 32 L 4 E ! cos CP ¹ + º sin m 2 32 L 4 E ! sin CP # + » solarandmattere˙ectterms ¼ ; 22 wherethesolartermshavebeenleftout,asT2Kisnotsensitivetotheseoscillations.The probabilitiesforneutrinoandantineutrinodisappearanceinEquation 2.27 arethesameifmatter e˙ects,whicharedescribedbelow,areneglected.However,evenwiththeinclusionofmattere˙ects, attheT2Kpeakneutrinoenergyandbaseline,theprobabilityonlydi˙ersbyabout0.1%fromthe probabilityseeninEquation 2.27 [ 25 ] 6 .ThroughEquation 2.27 ,T2Kisabletomeasure j m 2 32 j and sin 2 2 23 ,buttheoctant 7 inwhich 23 livescannotbedeterminedthroughthedisappearance channel. However,throughtheappearancechannel,T2Kissensitivetotheoctantof 23 astheleading termisproportionalto sin 2 23 .Thediscoveryoftheoctantof 23 hasimportanttheoretical rami˝cationsfortheunderstandingofneutrinomixingandmasses[ 56 ].If 23 weretohave mixing( 23 = 45 ),therecouldbeapotentialsymmetryinthemixingofmuon neutrinoswithtauneutrinos[ 57 ].Ontheotherhand,theorderoftheneutrinomassesmaybetied towhichoctant 23 livesin[ 58 ]. Inadditiontobeingsensitivetotheoctantof 23 ,theneutrinoandantineutrinoappearance probabilityinEquation 2.28 di˙ersbyanoppositesigninthethirdterm.IfCPviolationexists inneutrinooscillations(implying CP isnot 0 or ˇ ),thenadi˙erencebetween P ¹ ! e º and P ¹ ! e º wouldbepresentandpotentiallymeasurablethroughneutrinoappearancemeasure- mentsatT2K. NeutrinoOscillationsinMatter Whilethetheorydescribedpreviouslyissu˚cientforneutrinooscillationsinvacuum,exper- imentsstudyingoscillationsdonotexiststrictlyinavacuum.Whethertheneutrinosarecoming fromtheatmosphere,travelingthroughthematterthatcomprisestheEarth,orstartingsomewhere intheultradensecoreoftheSun,itisimportanttoconsiderthee˙ectmatterhasonneutrino oscillations.Thise˙ectiscalledtheMikheyev-Smirnov-Wolfenstein(MSW)e˙ect,andisnamed afterthetheoristswhodevelopedthetheory[ 59 , 60 ].Colloquially,theMSWe˙ectisknownby 6 ItshouldbenotedthatT2Kincludesalltermsintheprobabilityfortheiroscillationanalysis. 7 Theoctantdescribeswhether 23 > 45 or 23 < 45 . 23 thenamemattere˙ects. Asneutrinostravelthroughmatter,thereisanon-zeroprobabilitytheyinteract.Whencon- sideringthetimeevolutionofneutrinostravelingtothedetector,processeswhichdonothave anyneutrinosinthe˝nalstate,suchasmanychargedcurrentprocesses,arenotrelevant,asthey servetoonlyslightlyreducetheneutrino˛ux.Furthermore,neutralcurrentinteractionsoccur independentlyofthe˛avorofneutrino,a˙ectingeachmasseigenstateequally.Neglectingthese varioustypesofinteractionsonlyleavesonetypeofinteraction,elasticscatteringo˙ofelectrons viathe W boson.Astheelectronneutrinoistheonly˛avorthatcanoccurinthe W mediated interaction,thise˙ectis˛avordependentandwillhaveane˙ectonneutrinooscillation. Inordertodeterminethesizeofthemattere˙ects,startbyconsideringtheHamiltonianofthe neutrinotravelinginavacuum.Here,theenergyoftheneutrinoisthesumofthemassenergyof theneutrinoanditskineticenergy.However,inmatter,ane˙ectivepotentialenergytermmustbe includedintheHamiltonianduetotheabilityforelectronneutrinosandantineutrinostoelastically scatter.Thepotentialenergytermisdependentonhowfartheneutrinotravelsthroughmatter andtheelectrondensityofthatmatter.Becauseoftheadditionalterm,thenewHamiltonianwill havedi˙erentmasseigenstatesthantheHamiltonianforneutrinosinvacuum,whichimpliesthe e˙ectivemassforeachmasseigenstatewillbedi˙erentinmatter.Therefore,theprobabilityfor neutrinooscillationstooccurisdependentonthesedi˙erentmasseigenstatesandthePMNSmatrix modi˝edtogivetherelationshipbetweenthe˛avoreigenstatesandthemattermasseigenstates. Theresultantprobabilityisacomplicatedcombinationofthevariousmixingparametersandthe masssquareddi˙erences.Whiletheprobabilityisnotshownhere,afulltreatmentofthee˙ectcan befoundinmanysources,suchas[ 61 ]and[ 62 ].Forthisthesis,itissu˚cienttoknowthatmatter e˙ectshaveimportante˙ectsonthemeasurementofneutrinooscillationsand,therefore,mustbe takenintoaccount. 24 Parameter BestFitValue 3 ˙ Band m 2 21 [10 5 eV 2 ] 7.37 6.937.96 m 2 31 ¹ 32 º [10 5 eV 2 ] 2.56(2.54) 2.452.69(2.422.66) sin 2 12 0.297 0.2500.354 sin 2 23 , 2 31 ¹ 32 º > 0 0.425 0.3810.615 sin 2 23 , 2 32 ¹ 31 º < 0 0.589 0.3840.636 sin 2 13 , 2 31 ¹ 32 º > 0 0.0215 0.01900.0240 sin 2 13 , 2 31 ¹ 31 º < 0 0.0216 0.01900.0242 CP š ˇ 1.38(1.31) 2 ˙ :1.01.9( 2 ˙ :0.921.88) Table2.3:Currentunderstandingoftheneutrinooscillationparametersandtheir 3 ˙ allowed ranges.Parametervaluesarederivedfromaglobal˝ttocurrentneutrinooscillationdata[ 2 ].In thecaseof CP ,the 2 ˙ allowedrangeisshown.Thevalues(valuesinparentheses)arefor m 1 < m 2 < m 3 ( m 3 < m 1 < m 2 ). m 2 ,asde˝nedin[ 2 ],is m 2 = m 2 3 ¹ m 2 2 + m 2 1 ºš 2 .Underthis de˝nition, m 2 > 0 for m 1 < m 2 < m 3 and m 2 < 0 for m 3 < m 1 < m 2 .Valuesaregivenof m 2 31 > 0 for m 1 < m 2 < m 3 and m 2 32 < 0 for m 3 < m 1 < m 2 .Tablefrom[ 3 ]. 2.4TheCurrentKnowledgeofNeutrinos Overthelastfewdecades,manyexperimentshavemeasuredtheparametersinvolvedinneutrino oscillations,includingthethePMNSmixingangles,theCPviolatingphase,andthemasssquared di˙erences.Asitstands,thecurrentunderstandingofeachoftheseparameterscanbefoundin the2018editionoftheParticleDataGroup'sReviewofParticlePhysics[ 3 ]andissummarizedin Table 2.3 . ThedatashowninTable 2.3 comesfromavarietyofexperiments: 1. Solarneutrinoexperimentsaresensitiveto sin 2 12 and m 2 21 throughthedisappearanceof electronneutrinos. 2. Reactorneutrinoexperimentsaresensitiveto sin 2 12 , sin 2 13 ,and m 2 21 throughthedis- appearanceofelectronantineutrinos. 3. Atmosphericneutrinoexperimentsaresensitiveto sin 2 23 and j m 2 32 j or j m 2 31 j throughthe 25 disappearanceofmuonneutrinos. 4. Accelerator-basedneutrinoexperimentsaresensitiveto sin 2 23 and j m 2 32 j or j m 2 31 j through thedisappearanceofmuonneutrinos,aswellas sin 2 13 and CP throughtheappearanceof electronneutrinos. Whilemanyquestionsregardingneutrinooscillationshavebeenanswered,threeparticular questionshaveyettobeanswered.First,doneutrinooscillationsexhibitCPviolation?Until recently,withthemeasurementofanon-zero sin 2 13 ,thepotentialforCPviolationinthelepton sectorwasnotwellknown.WiththeresultsfromDayaBayandT2K, sin 2 13 wasfoundtobenot onlynon-zero,butlargeenoughthatameasurementof CP wasexperimentallypossible.Results fromT2K[ 63 ]havehintedatavalueof CP ,butthisremainsanareaofparticularinterest,both nowandintothefuture. Anotheropenquestioninneutrinooscillationphysicsconcernstheorderofthemasseigenstates, whichpertainstothesignof m 2 kj m 2 k m 2 j .Typically,neutrinooscillationexperimentsstudy oscillationsthroughadisappearancechannel,whichdependsoncalculating P ¹ ! º .Inthe caseofneutrinodisappearance, I kj = 0 inEquations 2.25 and 2.26 for P ¹ ! º ,andthe oscillationprobabilityisnotdependentonthesignof m 2 kj .However,whentakingmattere˙ects intoconsideration,theprobabilitycanbesu˚cientlymodi˝edsuchthatthesignof m 2 kj canbe measured.Withsolarneutrinos,theextremeelectrondensityinthecoreofthesuncausessigni˝cant e˙ectsduetotheMSWe˙ect[ 64 ].Duetothis,measurementshavebeenabletoestablishthat m 1 < m 2 .Furthermore,ithasbeendeterminedthat j m 2 32 jˇj m 2 31 j >> m 2 21 ,butithasyet tobeestablishedwhethertheorderofmassesis m 1 < m 2 < m 3 (calledthenormalhierarchy) or m 3 < m 1 < m 2 (calledtheinvertedhierarchy).Thedeterminationofthemasshierarchyis consideredoneoftheprimarygoalsforcurrentandfutureneutrinoexperiments. Athirdquestionthatremainsopentoconsiderationistheexistenceoftheso-calledterile neutrino.IntheStandardModel,thethreeneutrinosareallleft-handed,wherethespinisantiparallel toitsmomentum,andtheantineutrinosareallright-handed.Becausetherearenoright-handed 26 neutrinos,orleft-handedantineutrinos,neutrinosarenotabletogaintheirmassthroughtheHiggs mechanismand,therefore,shouldbemassless.However,becauseneutrinooscillationsimplythat neutrinosdo,infact,havemass,thismeansthattheStandardModelmustbeextended.Oneof thesimplestanswerstorectifythisissuewouldbetointroduceright-handedneutrinos.These neutrinoswouldonlyinteractviagravityandwouldbeconsideredterile.Astheseneutrinos donotinteractviatheweakforce,theycouldnotbedetectedinatypicalneutrinooscillation experiment.However,duetotheirbeingmorethanthreeneutrinos,sterileneutrinoscouldbe detectedindirectlythroughtheirmodi˝cationtotheoscillationprobabilities.Whileanumberof experimentshaveseenanomaliesintheirdata[ 65 , 66 , 67 ]consistentwithsterileneutrinos,there remainsheavytensionastotheirexistence.Thisisduetoseveralresultsfromsearchesforsterile neutrinosthatshownoevidenceforsterileneutrinosexisting[ 68 , 69 ].Inparticular,T2Khas studiedtheimpactoftheexistenceofasterileneutrinointheoscillationanalysisandtherewasno evidencefortheexistenceofasterileneutrinoseenintheirresults[ 70 , 71 ].Astheworkpresented inthisthesisisbasedonthedataseenbyT2K,theanalysispresentedherepresumesthatsterile neutrinosdonotexist. 27 CHAPTER3 THET2KLONGBASELINENEUTRINOEXPERIMENT TheTokai-to-Kamioka(T2K)experiment[ 16 ]isalongbaselineneutrinooscillationexperiment. Itwasdesignedtoobserveandmeasurethemixingofmuonneutrinoswithelectronneutrinos usinganintensebeamofmuonneutrinoscreatedviaa30GeVprotonbeamattheJapanProton AcceleratorResearchComplex(J-PARC)facilityinTokai,Japan.Theprotonbeamisguidedintoa graphitetarget,producinghadronsthroughprotoninteractionsinthetarget.Theresultinghadrons arefocusedbyasetofmagnetichorns,whichcanselecteither ˇ + ,forabeamprimarilycomposed of ,or ˇ ,forabeamprimarilyconsistingof .Thefocusedhadronsarepointedtowards a96mlongtunnel,whereneutrinosareproducedthroughthehadronsdecaying.Theneutrino beamisthenobservedbytwoneardetectors,whichare280mfromthetarget,andafardetector, Super-Kamiokande(SK),whichis295kmfromthetarget(seeFigure 3.1 ). Thedistancebetweenthetargetandthefardetector,calledthebaseline,ischosensuchthat thepeakoftheneutrinoenergyspectrum,0.6GeV,sitsinthe˝rstoscillationmaximum.This correspondstothe˝rstminimumofthe survivalprobability,whichcanbeseeninFigure 3.2 . Bychoosingthislocation,T2Kisabletomeasureneutrinooscillationsthroughthedisappearance ofmuonneutrinosandthroughtheappearanceofelectronneutrinos.Oscillationintotauneutrinos alsooccursintheneutrinobeam;however,chargedcurrent ˝ interactionsarenotmeasuredbecause oftheunlikelychanceforproducinga ˝ leptonattheneutrinoenergiesoftheT2Kbeam,asthe Figure3.1:SchematicoftheT2Kexperiment.Figurefrom[ 16 ]. 28 Figure3.2:Themuonneutrinosurvivalprobabilityat295km(top)andneutrino˛uxesfor di˙erento˙-axis(listedasOAinthe˝gure)angles(bottom).Itshouldbenotedthatthe˛ux predictionsarenormalizedsothattheunitsonthey-axisarearbitrary.Realistically,thetotal˛ux decreasesforhighero˙-axisangles.Figurefrom[ 17 ]. neutrinoneedstohavemorethan1.75GeVofenergytocreatea ˝ atrest. 3.1JapanProtonAcceleratorResearchComplex TheJ-PARCbeamlinewasconstructedforT2Kandconsistsofthreeaccelerators,alinear accelerator(LINAC),arapid-cyclingsynchrotron(RCS),andamainring(MR).Initially,abeam ofH isacceleratedto400MeVintheLINACbeforebeingconvertedtoabeamofH + bycharge- strippingfoilsattheinjectionpointtotheRCS.Theparticlesarethenacceleratedupto3GeV intheRCS,whichhasacycleof25Hzandholdstwobunches,orgroups,ofparticlespercycle. Approximately5%ofthebunchesaresuppliedtotheMR[ 16 ],whiletherestaresenttothemuon andneutronbeamlinesusedbyotherexperimentsatJ-PARC.Theparticleswhicharesuppliedto theMRareacceleratedupto30GeVbeforebeingusedtoproducetheT2Kbeam. InordertogeneratetheT2Kneutrinobeam,theprotonbeamisextractedfromtheMRusing 29 asetof5kickermagnets,whichareusedtoredirectthebeamthroughasingleturn.Asingle extractionoftheprotonbeamiscalledaandcontains8bunchesofprotonsseparatedby about560nanosecondsbetweenbunches,foratotaldurationofapproximately5microseconds. Byhavingasolidunderstandingofthetimestructurefortheextractedprotonbeam,theT2Kbeam triggerisabletodiscriminatebetweenvariousbackgrounds,suchascosmicrays,andthebeam signalintheT2Kdetectors. T2Khasbeencollectingdatasince2010.Theamountofdatacollectediscalculatedbycounting thenumberofprotonsimpingingonthetarget(protonsontarget,orPOT).UsingthePOTcollected todeterminetheamountofdataacquiredisdonebecausetheamountofPOTdirectlycorrelates tothebeampowerandthenumberofspillscollected.TheamountofPOTcollectedandthe correspondingT2KrunperiodscanbeseeninFigure 3.3 . 3.1.1TheT2KNeutrinoBeamline ThespillsextractedfromtheMRaredirectedtowardstheT2Kneutrinobeamline,whichhastwo parts,theprimaryandsecondarybeamline,asseeninFigure 3.4 .Thepurposeoftheprimary beamlineistobendtheextractedprotonbunchestopointinthedirectionofthesecondarybeamline, whichisinlinewiththeT2Kdetectors,aswellasfocusingthebeamtohavethedesiredpro˝le atthetarget.Furthermore,theintensity,position,pro˝le,andprotonbeamlossaremeasuredin theprimarybeamline.Thecharacteristicsoftheprotonbeamaremeasuredbecauseawell-tuned beamisessentialforproducingastableneutrinobeam. Astheprotonbeamentersthesecondarybeamline,theprotonsimpingeuponatargettoproduce secondarymesons,whichconsistmostlyofpionsandkaons.Thetargetisagraphiterodwitha diameterof2.6cmandalengthof91.4cm,whichisequivalentto1.9interactionlengthsforthe protonbeam.Thetargetiscooledbyheliumgastoo˙settheheatloadfromthepulsedbeam. Thesecondarymesonsarefocusedwiththreemagnetichorns[ 72 ].Thetargetsitsinsidethe ˝rsthorn,whoseobjectiveistocollectthemesonsproducedviatheprotoninteractions,while thesecondandthirdhornsfocusthemesonsintoabeam.Eachhornismadeupoftwocoaxial 30 Figure3.3:ThePOTcollectedatT2KbetweenJanuary2010andMay2018.Theredshaded regionsshowwhentheT2Kbeamwasbeingproduced.Figurefrom[ 18 ]. conductorswhichformaclosedvolume.Atoroidalmagnetic˝eldisgeneratedwithinthevolume betweentheconductors,whichvariesas1/(distancefromthebeamaxis).Themagnetichornsare operatedwitha250kApulsedcurrentandproduceamaximummagnetic˝eldof1.7Tesla.By usingthemagnetichorns,theneutrino˛uxisincreasedbyapproximatelyafactorof16forthepeak energyatthefardetector[ 16 ]. Thecurrentwhichpowersthemagnetichornscanrunintwomodestofocuseitherpositively ornegativelychargedmesons,producingaprimarilyneutrinoorantineutrinobeam.Thecurrent usedtofocuspositivelychargedmesonsandcreatesabeamprimarilymadeofneutrinos,iscalled neutrinomodeorForwardHornCurrent(FHC).Ontheotherhand,thecurrentwhichfocuses 31 Figure3.4: Top: OverviewoftheT2Kbeamline. Bottom: Sideviewofthesecondarybeamline. Figurefrom[ 19 ]. negativelychargedmesons,toproduceabeammostlycomposedofantineutrinos,isreferredtoas antineutrinomodeorReverseHornCurrent(RHC). Thefocusedmesonsentera96mlongtunnel,whichprovidesavolumeforthemesonstodecay toproducemuonneutrinosandantineutrinos: ˇ + ! + + K + ! + + (FHCprimarily) ˇ ! + K ! + .(RHCprimarily) Theresultingbeamisdominatedbymuonneutrinos(orantineutrinos),butasmallcontribution ofelectronneutrinosandantineutrinoscomefromdecaysincluding K + ! ˇ 0 + e + + e and + ! e + + e + 32 inFHC. Abeamdumpissituatedattheendofthedecayvolume,whereallhadronsfromthebeamand anymuonswithenergiesbelowapproximately5GeVarestopped.Muonsabovethatenergypass throughthebeamdumpandaremeasuredbymuonmonitors[ 73 ],whicharedirectlybehindthe beamdump.Thesemeasurementsareusedtovalidatethestabilityoftheneutrinobeambunch-by- bunch,becausemuonsareproducedalongwithneutrinosprimarilythroughtwo-bodypiondecay. TheneutrinosproducedfrommesondecaypassthroughthebeamdumpandbecometheT2K neutrinobeam. 3.1.2TheO˙-AxisNeutrinoBeam OneoftheT2Kneardetectorsandthefardetectorareset2.5 o˙-axiswithrespecttotheprimary protonbeam.Thisisdonetomakeuseofthefactthattheenergyofneutrinosemittedatlarge anglestotheparentmesoninatwo-bodypionorkaondecaydependsweaklyonthemomentumof theparentmeson. Toshowthisweakdependence,considerneutrinosproducedviathetwo-bodydecayofapion intoamuonandamuonneutrino 1 .Usingenergyandmomentumconservation,anexpressioncan bewrittenwhichrelatestheneutrinoenergy, E ,tothepionenergy, E ˇ ,thepionmass, m ˇ ,and themuonmass, m : E = m 2 ˇ m 2 2 ¹ E ˇ p ˇ cos º ; (3.1) where istheanglebetweenthedirectionthepionistravelingandthedirectionoftheemitted neutrino.Di˙erentiatingthepreviousequationwithrespectto E ˇ andholding constant,a maximumfor E isfoundfor E ˇ = E max ˇ = p ˇ š cos .Bysubstituting E max ˇ intoEquation 3.1 ,the maximumneutrinoenergy E max ,foragivenangle ,is: E max j = m 2 ˇ m 2 2 E max ˇ sin 2 : (3.2) 1 Thetreatmentdescribedhereisbasedontheo˙-axisneutrinobeamdescriptionin[ 74 ]. 33 Figure3.5:Neutrinoenergyasafunctionofthepionenergyforneutrinosproducedfromthe two-bodydecayofpionsintoamuonandaneutrino.Predictionsareshownforvariousangles betweentheneutrinoandpiondirections. Whenpionenergiesareeithergreaterthanorlessthan E max ˇ ,theenergyoftheneutrinoisless than E max .Becauseofthis,whentheemissionangle, ,islarge,therangeofpossibleneutrino energiesisdecreased.Thismeansthatpionswithawiderangeofenergieswillproduceneutrinos withsimilarlaboratoryenergieswhentheydecay.Figure 3.5 showstheneutrinoenergyatvarious pionenergiesandangles. Bysettingthedetectorsatanangletothebeam,T2Ktakesadvantageofthispropertytogenerate aneutrinobeamwithanarrowspreadinenergies.Figure 3.2 showsthat,foranangleof2.5 ,the ˛uxismoresharplypeakedthantheon-axis˛uxasafunctionofneutrinoenergy.Theenergyof theprotonbeamandtheo˙-axisanglewerechoseninsuchawaythatthepeakneutrinoenergy isapproximately0.6GeV,whichcorrespondstothe˝rstoscillationmaximumandmaximizesthe e˙ectofneutrinooscillationatthefardetector. 34 3.2TheT2KNearDetectors T2Kusesapairofneardetectorsituated280mfromthetargettomeasuretheneutrinobeam priortotheneutrinososcillating.Thisallowsthepropertiesofthebeam,includingitspositionand thenumberofmuonneutrinointeractions,tobemeasuredpriortoneutrinooscillationsoccurring. ThetwoneardetectorsaretheInteractiveNeutrinoGRID(INGRID)detectorandtheNearDetector at280m(ND280). 3.2.1TheInteractiveNeutrinoGRIDDetector TheprimarytaskoftheINGRIDdetector[ 16 , 75 ],whichispositionedinlinewiththeproton beam,istomonitorthedirectionandpro˝leofthebeam.Becausetheenergyspectrumofthebeam isdependentontheo˙-axisangle(asshowninFigures 3.2 and 3.5 ),itisessentialtomeasurethe directionoftheneutrinobeamveryprecisely. TheINGRIDdetector,seeninFigure 3.6 ,consistsof16identicalmodules,14ofwhichare arrangedinacrossformation,whiletheother2aresetato˙-axispositionsfromthecross.It wasdesignedsothecrossformationcoversa10m 10mrangeinthetransversedirectiontothe neutrinobeam.Themodulesoftheverticalandhorizontalarmsofthecrossoverlapatthecenterof thecrossandaredirectlyalignedwiththedirectionoftheprotonbeam.EachmoduleinINGRIDis madeofalternatinglayersofironplatesandplanesofscintillatorbars.Theselayersaresurrounded byvetoscintillatorplanestorejectanyinteractionswhichtakeplaceoutsidethemodule.Each scintillatorbarcontainsawavelength-shifting(WLS)˝ber,whichissetina3mm-diameterholein thecenterofthebarandisconnectedtoamulti-pixelphotoncounter(MPPC)[ 76 ]atoneend.The MPPCconvertsthelightfromtheWLS˝berintoanelectronicsignal,whichallowsthedatatobe readoutfromthedetector.INGRIDdetectsenoughneutrinointeractionstomeasuretheneutrino interactionratewith4%precisiondaily,whilealsoprovidingmonthlymeasurementsofthecenter oftheneutrinobeamtoanaccuracybetterthan0.4milliradians. 35 Figure3.6: Top: TheINGRIDdetector. BottomLeft: AnINGRIDmoduleshowingthe scintillatorplanes(blue)andtheironplates(gray). BottomRight: AnINGRIDmodulewiththe vetoplanes(black)shown.Figurefrom[ 16 ]. 36 3.2.2TheNearDetectorat280Meters ND280[ 16 ]sits280mfromtheprotontarget.However,unlikeINGRID,ND280sits2.5 o˙-axis. Itconsistsof˝vedi˙erentdetectors,ofwhichthemainportioniscalledthetracker.Thetracker ismadeofthreetimeprojectionchambers(TPCs)andtwo˝ne-graineddetectors(FGDs).The Pi-ZeroDetector(PØD)sitsupstreamofthetrackerandisusedformeasuringthe ˇ 0 background fromneutralcurrentinteractions.ThePØDandthetrackersitina6.5m 2.6m 2.5mmetal frame,whichissurroundedbyanelectromagneticcalorimeter(ECal).TherecycledUA1magnet encapsulatestheECalandisinstrumentedwiththesidemuonrangedetector,whichdetectsmuons thatescapeoutthesidesofND280.ThelayoutofND280canbeseeninFigure 3.7 .Thisthesis focusesoninteractionsinthetracker,whiletheotherdetectorsprovidesupplementalinformation whenperformingtheanalysis. UA1Magnet PriortobeingusedbyT2K,themagnetwasusedbytheUA1/NOMADexperiment.It providesanearconstantdipolemagnetic˝eldwithinND280,whichallowsforaccurate momentummeasurementsandparticlechargeidenti˝cation.Themagnetic˝eldisgenerated bypassinga2900Acurrentthroughasetofwater-cooledaluminumcoilswithinthemagnet [ 16 ].Theinnerdimensionofthecoilsmeasures7.0m 3.5m 3.6m,whiletheouter dimensionis7.6m 5.6m 6.1m.Inbetweentheinnerandouterdimensions,thereare sixteen˛uxreturnyokes.Thealuminumcoilsandreturnyokesareseparatedintotwomirror- symmetrichalves.Thisallowsthemagnettobeopenedforaccesstotheotherdetectors. Additionally,themagnetisinstrumentedwithscintillatortofunctionasasidemuonrange detector(SMRD). TheSideMuonRangeDetector TheSMRDisincorporatedintothemagnetyokethatsurroundstheinnerdetectorsofND280. Itsobjectivesaretodetectmuonsthatescapethedetectorwithlargeanglesrelativetothe 37 Figure3.7:ThelayoutofND280.Figurefrom[ 16 ]. beamdirectionandtoserveasavetoforparticlesenteringtheND280volume.Furthermore, itidenti˝esbeaminteractionswhichoccurinthemagnetandthesurroundingdetectorpit. TheSMRDisinstalledintheairgapsbetweenthesteelplatesthatserveasthemagnet returnyoke.Intotal,thereare440scintillatormodules,whichconsistofeitherfouror˝ve scintillationcounters,dependingonwhetherthemodulesitsinahorizontalorverticalair gap.Thescintillationcountersareoptimizedinsuchawayastomaximizetheactiveareain eachgap. TheElectromagneticCalorimeter TheECalisasamplingelectromagneticcalorimeterthatprovidesalmostcompletecoverage forparticlesexitingthetrackerorthePØD.Becausethetrackerisunabletodetectneutral 38 pions,theECalisusedtoreconstructany ˇ 0 'sproducedwithinthetrackervolume.It consistsoflayeredscintillatorbarsandlayersoflead,whichprovidesaninteractiontarget forneutrinosandfunctionsasaradiatorforproducingelectromagneticshowers. WithintheECal,therearethreetypesofmodules[ 77 ].ThesixBarrelECalmodulesare placedparalleltothebeamaxisalongthesidesofthetracker.EachmoduleintheBarrel ECalconsistsof31layersof50scintillatorbarseach,interwovenwith1.75mmthicklead sheets.TheDownstreamECalsitsattheendofthetrackerandissimilarincompositionto theBarrelECals,butwith34scintillatorlayers,insteadof50.Finally,thesixPØDECal modulessurroundthePØDparalleltothebeamaxis.BecausethePØDwasdesignedto detect ˇ 0 's,theECalisusedtoprovideadditionalenergyinformation.Therefore,itonlyhas sixlayersofscintillator,butisalternatedwith4mmthickleadsheets,ratherthanthe1.75 mmthicksheetsusedintheBarrelandDownstreamECals. ThePi-ZeroDetector ThePØDwasdesignedtomeasureneutralcurrentinteractionsonwater,whichisthesame targetusedatSK,andconsistsofscintillatormodulesalternatedwithleadandbronzesheets and˝llablewatertargetbags.Whilethebeamisrunning,thesebagscaneitherbe˝lledwith water,orleftempty.Byusingdatacollectedinbothcon˝gurations,asubtractionanalysiscan beperformedtodeterminetheneutrino-watercrosssection.ThePØDcontains40scintillator modules,whichconsistoftwoperpendicularlayersoftriangularscintillatorbarswithaWLS ˝berrunningthroughthemiddleofeachbar.Thetotalactivevolumeis2103mm 2239 mm 2400mm,andhasatotalmassof13.3tonswhenthewaterbagsareempty,or16.1 tonswhenthebagsare˝lled[ 78 ]. Fine-GrainedDetectors ThetwoFGDs[ 79 ]areusedtoprovidealargetargetmassforneutrinointeractionsinND280 aswellastotrackshort-ranged,chargedparticlesclosetothepointofinteraction.These short-rangedparticlesareimportanttoaccuratelymeasure,astheyaidinidentifyingthetype 39 ofneutrinointeractionthatoccurred.Therefore,theFGDsmustbeabletomeasurecharged particleswith˝negranularity,suchthattheindividualparticletracksanddirectionscanbe resolved. EachFGDisinstrumentedwithscintillatorbarsthatare1864.3mmlongwithasquare crosssectionof9.6mm.EachbarcontainsaWLS˝berrunningdownitscenterandis mirroredononeend.TheoppositeendcontainsaMPPCtoregisterhitsoccurringinthebar. Becausetheymustbeoperatedinthemagnetic˝eldofND280,MPPCsareused,ratherthan photomultipliertubes.Thescintillatorbarsarearrangedintomodules,whereeach modulecontains192barsinboththehorizontalandverticaldirections.FGD1containsonly scintillatorbars,whicharearrangedin15XYmodules.Ontheotherhand,FGD2contains sevenXYscintillatormodules,whicharealternatedwithsix2.5cmthicklayersofwater. TimeProjectionChambers TheTPCsprovide3Dtrackingandparticleidenti˝cationforchargedparticles,whichis crucialforidentifyingneutrinointeractionsoccurringintheFGDs.Furthermore,themag- netic˝eldinND280allowstheparticlechargeandmomentumtobemeasured.Because theTPCsareabletodistinguishbetweenparticleswithoppositecharges,ND280isableto di˙erentiatebetweenneutrinoandantineutrinointeractions,astheoutgoingleptonshave di˙erentcharges.TherearethreeidenticalTPCs,whichsandwichtheFGDs,suchthatTPC1 sitsupstreamofFGD1,TPC2sitsbetweenFGD1andFGD2,andTPC3sitsdownstreamof FGD2. EachTPChastwochambers(seeFigure 3.8 ),wheretheinnerchamberis˝lledwithan argon-baseddriftgasandtheouterchamberis˝lledwithcarbondioxideforinsulation[ 80 ]. Thewallsoftheinnerboxaremadeofcompositepanelswithcoppercladskins.Thepanels, inconjunctionwithacentralcathodepanelinthemiddleoftheinnerbox,createauniform electricdrift˝eldinthevolumeoftheTPC[ 80 ].AsachargedparticletraversestheTPC,it producesionizationelectronsinthegas,whichdriftawayfromthecentralcathodetowards 40 Figure3.8:Simpli˝eddiagramofasingletimeprojectionchamber.Figurefrom[ 16 ]. readoutplanesattheendsofthedetector.Thereadoutplanesconsistoftwelvemodulesthat sampleandmultiplytheelectrons[ 80 , 81 ]. 3.3TheT2KFarDetectorSuperKamiokande TheT2Kfardetector,Super-Kamiokande[ 82 ],isa50ktonwaterCherenkovdetector.Itis located295kmwestoftheinteractiontargetandburied1kmdeepinMt.Ikenoyama. SK,asseenin 3.9 ,isacylindricaltank,42mtallwitha39mdiameter,andis˝lledwith purewater.Thetankisdividedintotwovolumes,theinnerandouterdetectors,andareseparated byacylindricalstainless-steelframework.Theouterdetectorisprimarilyusedasanactiveveto forbackgrounds,suchascosmicraymuonsandinteractionsoccurringinthesurroundingrock, andhasane˚ciencyofalmost100%inrejectingcosmicraybackgrounds.Theinnerdetector isinstrumentedwith11,129inward-facing50cm-diameterphotomultipliertubes(PMTs).This providesabout40%coverageonthesurroundingwallsandgivesenoughspatialresolutionto reconstructthepositionoftheproductsfromneutrinointeractions. NeutrinointeractionsinSKtendtoproducechargedparticlesabovetheenergythresholdto 41 Figure3.9:TheSuper-Kamiokandedetector.Figurefrom[ 16 ]. produceCherenkovradiationastheytravelthroughthewater.ThephotonsfromtheCherenkov radiationareemittedinaconeandaredetectedbythePMTsinahitpatternshapedlikearing 2 . Thepatternofhits,plustiminginformation,allowsinformationtobeextractedfromtheinteraction, suchastheitslocation,themomentaofoutgoingparticles,andthe˛avorofthechargedlepton. TheoscillationanalysisatT2Kreliesonidentifyingneutrinointeractionsanddeterminingthe energyand˛avoroftheincomingneutrino.IftheinteractionisaCCQEinteraction,thenthe energyoftheneutrinocanbereconstructedfrommomentumanddirectionoftheoutgoinglepton. However,iftheinteractionisaCCresonantpionproductioninteraction,notonlyisthemomentum anddirectionoftheoutgoingleptonneeded,butthekinematicsoftheoutgoingpionareneededto properlyreconstructtheenergyoftheneutrino.Inbothcases,the˛avorcanbeinferredfromthe ˛avorofthedetectedlepton. ItisimportanttokeepinmindthattheSKdetectordoesnothaveamagnetic˝eld.Withouta magnetic˝eld,SKisunabletodistinguishbetweenpositiveleptonsfromantineutrinointeractions 2 SeeFigures 4.1 and 4.2 forexamplesonwhattheseringslooklike. 42 andnegativeleptonsfromneutrinointeractions.Therefore,itisimportantthattheantineutrinoand neutrinocomponentsofthebeambemeasuredatND280toaccuratelypredictthecomponentsat SK. SKhasbeenrunningsince1996,andbecauseofthelongtimeinwhichishasbeenrunning, thebehaviorofthedetectoriswellunderstood.Datacollectedfromcosmicraymuonsand atmosphericneutrinointeractionsgiveanumberofcontrolsamplesunrelatedtotheT2Kbeam. Thesesamplesareusedtoassessthedetectorresponseandsystematicuncertainties.Bywayof example,atmosphericneutrinodatacanbeusedtoquantifytheuncertaintyforcorrectlycounting thenumberofringsseeninaninteraction[ 83 ]. 43 CHAPTER4 THEOSCILLATIONANALYSISATT2K 4.1TheT2KOscillationAnalysis 4.1.1OverviewandMotivation ThemaingoaloftheT2KexperimentistomeasureneutrinooscillationsatSuper-Kamiokande (SK).Atabasiclevel,thisisdonebycomparingthenumberofcandidateneutrinointeractions, orevents,atND280withthenumberobservedatSK.BecauseND280sitsonly280mfromthe target,theneutrinoshavenothadachancetooscillate,sothemeasurementisoftheunoscillated neutrinobeam.Ontheotherhand,withSKbeingnearly300kmfromthesource,theneutrinos havehadtimetooscillateintootherneutrino˛avors,allowingtheoscillatedbeamofneutrinosto bemeasured. Thenumberofpredictedeventscanbenaivelycalculatedas[ 84 ] N ! ND 280 ¹ p reco º = Õ i ¹ E true º ˙ i ¹ p true º ¹ p true º R i ¹ p true ; p reco º (4.1) forND280,and N ! SK ¹ p reco º = Õ i ¹ E true º P ¹ E true º ˙ i ¹ p true º ¹ p true º R i ¹ p true ; p reco º (4.2) forSK.Here, N ND 280 ¹ SK º ¹ p reco º isthepredictedeventrateasafunctionofthereconstructed muonkinematics,where p reco isthereconstructedmuonfour-momentum.Theprimarydi˙erence between N ND 280 and N SK istheoscillationprobability, P ,includedinthefardetectorevent rate.Otherthan P ,theotherfourcomponentsaresimilarbetweenND280andSK.Theneutrino ˛uxasafunctionoftrueneutrinoenergy, ¹ E true º ,combinedwiththeneutrinocrosssection, ˙ i ¹ º ¹ p true º ,providesapredictiononthenumberofeventsexpectedinthedetector.Thedetector e˚ciency, ¹ º ¹ p true º ,representshowwellthedetectorcandetectinteractionsoccurringinthe detector.Finally,theeventrateisdependentonthereconstructedkinematicvariables,asthisis 44 -likeEvents e -likeEvents ErrorSource -mode -mode -mode -mode -modeCC 1 ˇ Beam(withoutND280data) 8.0%7.3% 8.0%8.1%8.9% Beam(withND280data) 4.3%4.1% 4.4%4.2%4.4% Crosssection(withoutND280data) 12.3%10.3% 12.3%10.1%8.7% Crosssection(withND280data) 5.6%4.4% 8.4%6.2%5.6% Beam+Crosssection(withoutND280data) 14.5%12.6% 14.5%13.0%12.6% Beam+Crosssection(withND280data) 4.4%2.9% 7.7%5.7%5.6% SK+FSI+SI 3.3%2.9% 4.1%4.3%16.6% Total(withoutND280data) 15.0%13.0% 15.0%13.7%20.1% Total(withND280data) 5.5%4.4% 8.8%7.3%17.8% Table4.1:UncertaintyonthenumberofeventsineachSKsampleseparatedbyerrorsourcewith andwithouttheconstraintprovidedbytheND280data.TheSK+FSI+SIuncertaintiesarenot constrainedbytheND280data.Tablefrom[ 4 ]. whatismeasuredinthedetector.However,theothercomponentsinEquations 4.1 and 4.2 are dependentonthetruekinematicvariables.Therefore,afunction, R i ¹ p true ; p reco º ,isincluded torepresenttheprobabilityaneventwithatruefour-momentum, p true ,isreconstructedwitha four-momentum, p reco .Thisfunctionisdependentonthetypeofinteraction, i ,aswellasdetector andnucleare˙ects. P includestheinformationregardingthefundamentalneutrinooscillationparameters,which arethesignalparametersinanoscillationanalysis.Systematicuncertaintiesontheseparameters comefromtheothercomponentsfoundinEquations 4.1 and 4.2 .ForT2K,thelargestsourcesof uncertaintycomefromtheneutrino˛uxandcrosssections.Theseuncertaintiescanbereduced,or constrained,byusingdatafromotherexperiments,suchastheNA-61/SHINEexperiment[ 85 ]for the˛uxorMINER A[ 86 ]forthecrosssections.Additionally,datafromND280canbeusedto constraintheseuncertainties.Byincludingdatafromtheneardetector,theT2Kneutrino˛uxand crosssectioncanbesimultaneouslyconstrained.Thisreducestheuncertaintybeyondwhatdata fromexternalexperimentscoulddoontheirown.Table 4.1 showsthee˙ectofincludingthenear detectordataonarecentT2Koscillationanalysis. TheT2Koscillationanalysisframeworkisathreestepprocess. 45 1. Externaldataisusedtoproducepriorconstraintsonthe˛uxandcrosssection,aswellasfor thedetectorsystematicuncertainties.The˛uxandcrosssectionconstraintsarediscussedin moredetailinSection 4.3 ,whiletheND280systematicuncertaintiesarecoveredinChapter 6 . 2. TheND280dataisincludedbyperformingabinnedmaximumlikelihood˝t,wherethe dataisdividedinbinsbasedontheirtopologicalsample(describedinChapter 5 )andtheir reconstructedmomentum, p ,andanglewithrespecttotheneutrinobeam, cos .These kinematicvariableswerechosenasND280wasdesignedtomeasurethekinematicsofthe outgoingleptonfromaneutrinointeraction.Theycanthenbeusedtocalculatetheneutrino energy,whichisneededtodeterminetheoscillationparameters.Finally,asSKprimarily usesleptonkinematics,itisbene˝cialtousethesamevariablesineachdetector. Thepriorconstraintsdeterminedintheprevioussteparetreatedasnuisanceparametersin the˝t.Theoutputfromthis˝tisatunedsetof˛uxandcrosssectionparametersandtheir correspondingcovariance,andispassedontothe˝nalstepofoscillationanalysis.The likelihood˝tisdescribedinSection 4.2 . 3. The˝nalstepintheprocessisthedeterminationoftheneutrinooscillationparameters themselves.Thisisdonethroughabinnedlikelihood˝ttotheSKdata(describedinSection 4.1.3 ),usingtheresultsofthe˝ttotheND280dataasapriorconstraintterm.Thedata selectionprocessforSKisdescribedinSection 4.1.2 . Thisthreestepprocessiscalledasequentiallikelihoodmaximization.Ratherthan˝ttinga globallikelihoodtoallthreedatasets,whichiscomputationallyexpensive,eachdatasetreceives itsowntreatment.Thisgreatlydecreasesthecomplexityofeachstep,makingtheoverallprocess computationallytractable. 46 4.1.2DataSelectionatSuper-Kamiokande AsdescribedinSection 3.3 ,eventsatSKareobservedbycharacteristicringsseeninthedetector. TheseringscomefromtheCherenkovradiationemittedbyparticlesastheytravelthroughthe detector.SK,inparticular,looksforringsthatcouldbetheresultofamuonorelectronproducedin achargedcurrentneutrinointeraction.Thissectionwillprovideanoverviewfortheeventselection atSK. Atthefardetector,thedata(andsimulation)issplitinto˝veselections:single-ringmuon- like(1R )andsingle-ringelectron-like(1R e )forbothneutrinoandantineutrinobeammodes, plusanelectron-likeCC 1 ˇ + selectioninneutrinomode.Thereisnodi˙erenceinthe1R and 1R e selectionsbetweenbeammodes,asSKcannotdistinguishbetweenthechargeofthelepton. FormoredetailsontheselectionprocessatSK,see[ 87 ]. The1R selectionmustpassthefollowingcriteria: ‹ FiducialVolume: Theeventmustbefullycontainedwithinthe˝ducialvolumeofSuper- Kamiokande.Furthermore,theeventmustbereconstructedwithinthe˝ducialvolume. ‹ CherenkovRings: Onlyeventswhichhaveoneringfoundbytheringcountingalgorithm areused. ‹ RingPID: Theringmustbeidenti˝edasmuon-likebythePIDalgorithm. ‹ ReconstructedMomentum: Themuonreconstructedmomentummustbegreaterthan200 MeV/c. ‹ DecayElectrons: Thenumberofdecayelectronsmustbelessthanorequaltoone. ‹ ˇ + RejectionCut: Theeventmustpassthe ˇ + rejectioncut: ln ¹L ˇ + šL º < 0 : 15 p ,where ln ¹L ˇ + šL º isthelog-likelihoodratioofthe ˇ + and hypothesesand p isthemomentum ofthemuon. 47 Figure4.1:Exampleeventdisplayofamuon-likeeventatSuper-Kamiokande.Photomultiplier tubesthathavechargedepositedinthemduringtheeventareshownascoloredcircles,wherethe colorrepresentshowmuchchargewasdeposited.Thetimedistributionofhitscanbeseeninthe bottomrightcorner.Figurefrom[ 20 ]. BasedonaMonteCarlodataset,thepurityforthe1R sampleis82.86%inneutrinobeammode and79.72%inantineutrinobeammode[ 88 ].Anexampleofamuon-likeeventcanbefoundin Figure 4.1 . The1R e selectionisasfollows: ‹ FiducialVolume: Theeventmustbefullycontainedandreconstructedwithinthe˝ducial volumeofSuper-Kamiokande. ‹ CherenkovRings: Onlyeventswhichhaveoneringfoundbytheringcountingalgorithm areused. ‹ RingPID: Theringmustbeidenti˝edaselectron-likebythePIDalgorithm. ‹ VisibleEnergy: Theremustbeatleast100MeVofvisibleCherenkovlightinthedetector. 48 Figure4.2:Exampleeventdisplayofanelectron-likeeventatSuper-Kamiokande. Photomultipliertubesthathavechargedepositedinthemduringtheeventareshownascolored circles,wherethecolorrepresentshowmuchchargewasdeposited.Thetimedistributionofhits canbeseeninthebottomrightcorner.Figurefrom[ 20 ]. ‹ DecayElectrons: Therecanbenodecayelectronswithintheevent. ‹ ReconstructedNeutrinoEnergy: Thereconstructedenergyoftheneutrinomustbeless than1250MeV. ‹ ˇ 0 RejectionCut: Theeventmustpassacutforneutralpions: ln ¹L ˇ 0 šL e º < 175 0 : 875 m ˇ 0 ,where ln ¹L ˇ 0 šL e º isthelog-likelihoodratioofthe ˇ 0 and e hypothesesand m ˇ 0 isthemassofthe ˇ 0 . Thepurityforthe1R e sampleis81.18%inneutrinobeammodeand61.99%inantineutrinobeam mode[ 88 ].Anexampleofanelectron-likeeventcanbefoundinFigure 4.2 . Theelectron-likeCC 1 ˇ + selectionissimilartothe1R e selection,withonekeymodi˝cation. Whereas,the1R e samplerequirestheretobenodecayelectronsintheevent,theCC 1 ˇ + selection 49 requirestheretobeonedecayelectron.Thiselectronisconsideredtobetheresultofadecaying ˇ + ;therefore,theseeventsareselectedintotheelectron-likeCC 1 ˇ + sample.Thepurityinthis sampleis78.75%[ 88 ]. 4.1.3TheOscillationFitatSuper-Kamiokande TheT2Koscillationanalysisemploystwomainmethods,ahybridfrequentist-Bayesianmethodto dealwithnuisanceparametersandafullyBayesiananalysis,whichusesaMarkovchainMonte Carlomethod[ 89 ].Itshouldbenoted,thefullyBayesianmethod˝tsthenearandfardetectordata simultaneously,combiningthesecondandthirdstepsofthesequentiallikelihood. Theoscillation˝tusesabinnedlikelihood,givenby: L¹ N obs e ; N obs ; ® o ; ® f º = L main ¹ N obs e ; N obs ; ® o ; ® f ºL s y s ¹ ® f º ; (4.3) where N obs e istheobservednumberofelectron-likeevents, N obs isthenumberofobservedmuon- likeevents, ® o aretheoscillationparameters( sin 2 23 , j m 2 32 j , sin 2 2 13 ,and CP )and ® f arethe nuisanceparameters.Thesenuisanceparametersincludethecrosssectionmodelparameters, the˛uxparameters,theSKdetectorsystematics,andthe˝nalstateinteractionandsecondary interactionmodeluncertainties.Thelikelihoodisdividedintotwoparts, L main and L s y s . L main isthetermwhichcomparesthedatawiththepredictedeventratesusingtheoscillationandnuisance parameters,while L s y s isaGaussianpenaltytermwhichonlydependsonthenuisanceparameters. Forthehybrid˝tapproach,thenuisanceparametersaremarginalizedbyintegratingthelikeli- hoodoverthevaluesofthenuisanceparameters.Thisisdonebygenerating10,000throwsofthe nuisanceparametersbasedontheirpriorsandthen˝ttingthesedistributions.Thisproducesagrid oflikelihoodsforthevariousparametersets,wherebythe˝tismarginalizedbychoosingthesetof parameterswiththelowest ˜ 2 .Furthermore,theoscillationparameterscanbemarginalizedover toproduce2Dintervals.Here,theparametersaremarginalizedbythrowingtheparametervalues accordingtheparametertypeandrangegiveninTable 4.2 .Oneexceptiontothemarginalization processisthemasshierarchy,whereitis˝xedtobeeithernormalorinvertedhierarchy.After 50 Parameter PriorTypeBounds sin 2 23 Uniform[0.3,0.7] sin 2 2 13 w/reactorconstraint Gaussian0.0830 0.0031 sin 2 2 13 w/oreactorconstraint Uniform[0.03,0.2] sin 2 2 12 Gaussian0.851 0.020 j m 2 32 j (NH)or j m 2 13 j (IH) Uniform[2.3,2.7] 10 3 eV 2 /c 4 m 2 21 Gaussian(7.53 0.18) 10 5 eV 2 /c 4 CP Uniform[ ˇ , + ˇ ] MassHierarchy FixedNHorIH Table4.2:Priorsusedfortheoscillationparametersduringthemarginalizationprocess. denotesnormalhierarchy,whiledenotesinvertedhierarchy.Tablefrom[ 5 ]. thenuisanceparametershavebeenmarginalizedover,the ˜ 2 forthemarginalizedlikelihoodis minimizedtodeterminethebest˝toscillationparameters. Becausethenuisanceparametersaremarginalized,theresultisnotanexactsetofbest˝t parameterscorrespondingtothebest˝tspectrum.Rather,thebest˝tspectrumisproducedby usingthethrownnuisanceparameters: N mar g j = Í n i = 1 L¹ N obs i ; ® o ; ® f i º N pred j ¹ ® o bf ; ® f i º Í n i = 1 L¹ N obs i ; ® o bf ; ® f i º ; (4.4) where n isthenumberofnuisanceparameterthrows, ® o bf isthebest˝toscillationparameters,and ® f i isthe i -thsetofthrownnuisanceparameters. 4.2TheNearDetectorFit 4.2.1TheMaximumLikelihoodFitMethod Theneardetector˝t,alsocalledtheBANFF( B eam A nd N D280 F luxextrapolationtask F orce) ˝t,usesthemaximumlikelihoodmethodtodeterminetheoptimalsetof˛ux,crosssection,and detectorparametersgiventheND280data.By˝ttingalltheparameterssimultaneously,correlations betweenthe˝tted˛uxandcrosssectionparameterscanbedetermined. AsmentionedinSection 4.1.1 ,theBANFF˝tusesabinnedlikelihood.Bybinningthedatain 51 p cos ,the˝tisabletoconstraintheoveralleventrateaswellastheneutrinoenergydistribution atSK.The p cos binswerechoseninsuchawayastoprovidesu˚cientstatisticsfora ˜ 2 ˝t, whilealsohaving˝nerbinninginregionswithhighereventratestogainmoreinformationabout theshapeofthedistribution.Foragivenbin,thedataandMonteCarloobservedeventratesare expectedtofollowaPoissondistribution.Therefore,theprobabilitytoobserve N obs i eventsinbin i ,givenapredictedeventrate, N pred i ,whichdependsonthe˛ux( ® b ),crosssection( ® x ),anddetector parameters( ® d )describedinSection 4.3 ,is: P ¹ N obs i j N pred i º = ¹ N pred i ¹ ® b ; ® x ; ® d ºº N obs i e N pred i ¹ ® b ; ® x ; ® d º N obs i ! : (4.5) ThefullPoissonlikelihoodtermisaproductoftheindividualbinprobabilities. Theremainingtermsinthelikelihoodarerelatedtotheconstrainttermsforthebeam,cross section,anddetectorparameters.TheseareeachmodeledasindependentmultivariateGaussian distributionsand,therefore,treatedasseparatetermsinthelikelihood.Foravectorofparameter values ® y (whichcanbeoneof f ® b ; ® x ; ® d g )with n parameters,anassociatedcovariancematrixgenerated frompriorinputs, ¹ V y º i ; j ,andthedi˙erencebetweenthecurrentparametervalueanditsnominal value, ® y ,theprobabilitycanbecalculatedas: ˇ ¹ ® y º = ¹ 2 ˇ º n š 2 j¹ V y º i ; j j 1 š 2 e ® y ¹¹ V y º 1 i ; j º ® y T 2 : (4.6) Theequationmaximizedbythe˝tisthelikelihoodratio,wherethenumeratorisde˝nedusing Equations 4.5 and 4.6 andthedenominatoristhelikelihoodfunctionevaluatedat N pred i = N obs i , whichisthemaximumvalueforthenumerator, L ND 280 = ˇ ¹ ® b º ˇ ¹ ® x º ˇ ¹ ® d º Î i ¹ N pred i ¹ ® b ; ® x ; ® d ºº N obs i e N pred i ¹ ® b ; ® x ; ® d º š N obs i ! ˇ ¹ ® b nom º ˇ ¹ ® x nom º ˇ ¹ ® d nom º Î i ¹ N obs i º N obs i e N obs i š N obs i ! : (4.7) Byusingtheratio,comparisonscanbemadebetweenthelikelihoodofthepredictedvaluesand themaximumpossiblevalue.Furthermore,severaltermscancelout,suchasthedeterminantsof thecovariancematricesinEquation 4.6 . 52 Theminimizedquantityis ˜ 2 ND 280 2ln L ND 280 ,whichcontainstwoindependentparts, aPoissoncontributionfromthe˝ttedobservablesandaGaussiancontributionfromthe˝tted parametersandtheircovariance.Thequantity ˜ 2 ND 280 isde˝nedas ˜ 2 ND 280 = 2 Nbins Õ i N pred i ¹ ® b ; ® x ; ® d º N obs i + N obs i ln » N obs i š N pred i ¹ ® b ; ® x ; ® d º¼ + B Õ B Õ b ¹ V 1 b º ; b + X Õ X Õ x ¹ V 1 x º ; x + D Õ ˆ D Õ ˝ d ˆ ¹ V 1 d º ˆ;˝ d ˝ ; (4.8) where B , X ,and D arethenumberof˛ux,crosssection,anddetectorparameters.Thenumber ofpredictedeventsinagivenbiniscalculatedusingdetectorweights,crosssectionresponse functions,and˛uxweights(describedinSection 4.3 ),whichdependonthesample p cos bin, theneutrinointeractionmode,andthetrueneutrinoenergy. Thebasic˝ttingroutinereliesontheMinuitminimizationpackage[ 90 ]asimplementedinthe ROOT[ 91 ]framework.TheMIGRADalgorithm,whichisagradientdescentmethod,isused intheneardetector˝ttominimizethe ˜ 2 showninEquation 4.8 .Thepost-˝tuncertainties arethenestimatedusingtheHESSEalgorithm,whichcalculatesthesecondderivativesofthe parametersaroundtheminimizedvaluestoproducetheHessianmatrix,theinverseofwhichgives thecovariancematrixofthe˝ttedparameters. 4.3FitParametersintheNearDetectorFit Thenuisanceparametersoftheneardetector˝tcomefromthreesources,theT2Kneutrino beam˛ux,theneutrinointeractioncrosssectionmodel,andthedetectorsystematicuncertainty parameters.Thissectiondescribestheparametersthatareusedforthisanalysis. 53 4.3.1FluxParameters Manyoftheelementsrelatedtotheneutrinobeamproductionhaveane˙ectontheneutrino˛ux. Theseincludepropertiesoftheprotonbeam,suchastheenergy,intensity,anddirectionofthe beam,andthecrosssectionforprotoninteractionsonthetargetandtheinteractionsofoutgoing particleswithinthetargetorinthedecayvolume,suchasthepionsthatlaterdecayintoneutrinos. T2Khasproducedacomplex˛uxmodelthatusestechnicaldesignspeci˝cationsandstandard particlephysicssimulationpackages,whicharetunedwithdatatakenfrommonitoringtheT2K beamandexternalexperiments[ 17 ]. Twooftheprimarysourcesofuncertaintyonthebeammodelcomefromhadronproductionin thetarget,whicha˙ectsthepredictiononthenumberofneutrinosproduced,andtheinteractionsof hadronsoutsidethetarget,whichcontributesasigni˝cantfractionofthebackgroundcomponentof theneutrinobeam.Withregardstohadronproductioninthetarget,thecrosssectionforproducing pions,thenumberofpionsproducedinthetarget,andanyoutgoingnucleonsfromtheinteractions providethedominantsourceoferroratlowenergies.Athigherenergies,theuncertaintyinkaon productionisthedominantsource.Interactionswhichoccuroutsidethetargetarelesslikelytobe focused(ordefocused)bythemagnethorns,increasingthebackgroundcomponentoftheneutrino ˛ux.These,alongwiththerestofthesystematicuncertainties,areincludedinthe˛uxmodel throughacovariancematrixgeneratedthroughaseriesofsimulationstudies. Usingthemodel,a˛uxpredictionisproducedfor , , e ,and e atND280andSuper- Kamiokande,andisbinnedinneutrinoenergy.ByincorporatingdatafromtheNA61/SHINE experiment[ 85 ],thenominal˛uxpredictioncanbetuned,reducingtheuncertaintyduetohadron productioninthetarget.Inordertoaccountfordi˙erentbeamconditionsineachrunperiod,the ˛uxpredictionisdoneseparatelyforeachrun.Figure 4.3 showsthetuned˛uxpredictionandthe ratioofthetuned˛uxpredictiontothenominalpredictionforthedatasetusedinthisanalysis.The combineduncertaintyontheND280predictioncanbeseeninFigure 4.4 .Thetotaluncertaintyas afunctionofenergyatSKisverysimilartoND280,andcanbefoundin[ 21 ]. WithintheBANFF˝t,the˛uxpredictionistunedbyminimizingEquation 4.8 usingasetof 54 Figure4.3:T2Ktuned˛uxprediction(left)andratiooftheT2Ktuned˛uxpredictiontothe nominalprediction(right)forND280andSKinbothneutrinomode(FHC)andantineutrino mode(RHC). 55 Figure4.4:ThetotaluncertaintiesontheND280˛uxprediction.The13av3uncertainty(solid blackline)isthecurrentversion.The11bv3.2uncertainty(dashedblackline)isanearlier version.Figurefrom[ 21 ]. 56 nuisanceparameters.Theseparameterscorrespondtobinsofneutrinoenergy(inGeV)anddepend onthebeammode,neutrino˛avor,anddetector.Thebinsare: ‹ ( -mode)/ ( -mode):0.0,0.4,0.5,0.6,0.7,1.0,1.5,2.5,3.5,5.0,7.0,30.0 ‹ ( -mode)/ ( -mode):0.0,0.7,1.0,1.5,2.5,30.0 ‹ e ( -mode)/ e ( -mode):0.0,0.5,0.7,0.8,1.5,2.5,4.0,30.0 ‹ e ( -mode)/ e ( -mode):0.0,2.5,30.0 Thisbinningcorrespondsto25parametersineachbeammode,rangingfrom030GeVfor thefourrelevantneutrino˛avors.UsingbothbeammodesforbothND280andSK,atotalof 100parametersareincludedintheanalysis.Eachparameterisde˝nedrelativetothetuned˛ux prediction,sothenominalvalueissetat1.The˛uxparametershavehighcorrelationsbetween thetwodetectors,thedi˙erentenergybins,neutrino˛avors,andbeammodes.Thecorrelation matrixfortheseparameterscanbeseeninFigure 4.5 ,withTable 4.3 describinghowtheparameters correspondtobinnumbers. 4.3.2CrossSectionParameters Thenextsetofnuisanceparametersisrelatedtohowneutrinointeractionsaremodeled.The determinationoftheoscillationparametersreliesoncorrectlydeterminingtheneutrinoenergy spectrum.Becauseneutrinosareneutral,theenergyofaneutrinomustbecalculatedusingthe kinematicinformationoftheoutgoingparticlesandtheconservationofenergy.Theoreticalmodels havebeendevelopedtoaidinpredictingthekinematicsoftheoutgoingparticlesinvarioustypes ofinteractions.Whilethesemodelsareabletopredictexactlywhatadetectorshouldexpecttosee foragiveninteractiontype,inreality,thee˙ectsofdetectorthresholdsand˝nalstateinteractions canalterwhatisobserved. Practically,neutrinointeractionsintheT2KdetectorsaresimulatedusingtheNEUTneutrino eventgenerator[ 92 ],whichusesneutrinocrosssectionmodelstosimulatedi˙erenttypesof 57 Figure4.5:Correlationmatrixfor˛uxparametersusedintheBANFF˝t.Thelabelsdenote whichdetectorandbeammodethatregioncovers.Eachbininthematrixcorrespondstoan energyrangegiveninthetext. interactionsandtheiroutgoingparticles.Thekinematicinformationfromtheoutgoingparticles inthesesimulatedinteractionscanthenbeusedtobuildapredictionforhowmanyeventsofeach interactiontypeshouldbeseeninthenearandfardetectorsforagivenneutrinobeam˛ux. Intheneardetector˝t,twomethodsareusedtotunetheNEUTcrosssectionmodelprediction. First,amultiplicativefactorcanbeappliedtoanyeventsinagiveninteractionchannel,suchas allCCQEeventsorallneutrinoevents.Thisservestoincreaseordecreasetherateatwhicha givenchanneloccurs,andisapplieduniformlyacrossneutrinoenergiesandinteractionchannel observables.Thesemultiplicativefactorsarereferredtoas normalizationparameters .Theneutral currentparametersandchargedcurrentparametersnotassociatedwithCC 0 ˇ orCC 1 ˇ interactions arenormalizationparameters. Second,parametersassociatedwithhowneutrinocrosssectionaremodeledareappliedas 58 DetectorBeamModeFlavor BinNumbers ND280Neutrino 010 ND280Neutrino 1115 ND280Neutrino e 1622 ND280Neutrino e 2324 ND280Antineutrino 2529 ND280Antineutrino 3040 ND280Antineutrino e 4142 ND280Antineutrino e 4349 SKNeutrino 5060 SKNeutrino 6165 SKNeutrino e 6672 SKNeutrino e 7374 SKAntineutrino 7579 SKAntineutrino 8090 SKAntineutrino e 9192 SKAntineutrino e 9399 Table4.3:Relationbetweenthe˛uxparametersandtheirbinnumberinthe˛uxcorrelation matrix. responseparameters .Becausetheseparametersarerelatedtotheunderlyingtheoreticalcalculation oftheneutrinocrosssection,theyaremorecomplicatedthanthenormalizationparameters.When changingtheseparameters,itispossibletosigni˝cantlyalterthekinematicsoftheevent,and,in turn,howtheeventsaredistributedinthesamplesdescribedinChapter 5 .Themostaccurateway todeterminethee˙ectsofalteringtheseparameterswouldbetoreruntheentireanalysisfrom eventsimulationinNEUT,toeventreconstructionandselection,foreacheventateachstepofthe ˝tminimization.Forobviousreasons,thisiscomputationallyunfeasible.Therefore,weightsfor eachparameterarecalculatedforeveryeventpriortothe˝tbeingrun.Bysavingtheprecalculated weightsassplines,themostcomputationallyintensivepartcanberunpriortothe˝tandrelevant weightscanbeinterpolatedduringtheminimizationprocess.ManyoftheCC 0 ˇ parameters,the CC 1 ˇ parameters,theBeRPAparameters,andtheFSIparametersareresponseparameters. 59 4.3.2.1CC 0 ˇ Parameters TheCC 0 ˇ parameterization,whichdescribeseventswhereonlyamuonisobservedinthedetector, canbeorganizedintotwogroupsofparameters.Onegroupisrelatedtothemodelofthenuclear medium,whiletheotherdescribesinteractionsrelevanttoCC 0 ˇ events.Thenuclearmediumin theCC 0 ˇ parameterizationisbasedontherelativisticFermigas(RFG)nuclearmodel,wherethe initialnucleonsareconsideredtohavea˛atmomentumdistributionuptoamaximummomentum, theso-calledermimomentum, p F .Withintheneardetector˝t,theFermimomentumistreated asavariableparameterand,becauseitisdependentonthesizeofthenucleus,aseparateparameter hasbeenincludedinthe˝tforinteractionsoccurringoncarbonandoxygen. TheLocalFermiGas(LFG)modelisanalternativetotheRFGnuclearmodel.Ratherthan treatingthemomentumdistributionoftheinitialnucleonas˛at,inaLFGmodel,e˙ectsdueto the˝nitesizeofthenucleusareincluded.Inthismodel,themomentumoftheinitialnucleon dependsonitsradialposition.Anadditionaluncertaintyisincludedintheneardetector˝tdueto di˙erencesinleptonkinematicsbetweentheNEUTeventgenerator,whichusestheRFGmodel, andtheNievesgenerator,whichusestheLFGmodel[ 93 ].StudiescomparingtheNieves1p1h modelagainstthemodelusedbyT2Khaveshownthatthechoiceofmodelcanhaveasigni˝cant impactonthe˝ttedparametervaluesatND280andintheoscillationanalysis[ 94 ].Unliketheother crosssectionuncertainties,thisuncertaintyistreatedasanadditionalcovarianceintheobservable normalizationcovariancematrix,describedinSection 4.3.3 . Therandomphaseapproximation(RPA)isanon-perturbativemethodfordescribingtheinter- actionsandcorrelationsofnucleonswithinthenucleus.Becauseoftheseinteractions,theoverall neutrino-nucleuscrosssectionismodi˝edandcanbeparameterizedasafunctionofthetransfer four-momentum, Q 2 ,whichisconstrainedbypion-nucleusscatteringdata[ 24 ].Onemethodof parameterizingthismodi˝cationusestheBernsteinpolynomialsofdegree n [ 95 ],whichforma basisforthepowerpolynomialsoforder n ,andaregivenby B i ; n ¹ x º = n i x i ¹ 1 x º n i ; (4.9) 60 where x runsfrom 0 to 1 and n i isabinomialcoe˚cient. UsingtheseBernsteinpolynomials,aresponseparameterization,calledtheBeRPAparameter- ization,isgivenby: f ¹ x º = 8 > > > >< > > > > : A ¹ 1 x 0 º 3 + 3 B ¹ 1 x 0 º 2 x 0 + 3 p 1 ¹ 1 x 0 º x 0 2 + Dx 0 3 ; x < 1 : 2 1 + p 2 exp E ¹ x U ºº ; x > 1 : 2 (4.10) where x = Q 2 and x 0 = Q 2 š 1 : 2 and p 1 and p 2 arede˝nedas p 1 = D + UE ¹ D 1 º 3 (4.11) p 2 = D 1 : (4.12) CCQEeventsarethedominanttypeofinteractionsintheCC 0 ˇ model,aswellasbeing theprimaryinteractiontypeforT2K.Thecrosssectionfortheseeventscanbeparameterized usingthreenucleonformfactors:theelectric,magnetic,andaxialformfactors.Theelectricand magneticformfactorshavebeenstronglyconstrainedthroughelectron-nucleonscattering[ 24 ],but theaxialformfactor,whichisonlyfoundinneutrino-nucleoninteractions,hasnotbeenastightly constrained.Theaxialformfactorisgenerallyparameterizedusingthedipoleformof: F A ¹ Q 2 º = g A » 1 + Q 2 š¹ M QE A º 2 ¼ 2 : (4.13) Here, g A isthenormalizationfor Q 2 = 0 andhasbeendeterminedtobe 1 : 2670 0 : 0035 from neutronbeta-decay.Withintheneardetector˝t, M QE A istreatedasane˙ectiveparametertodescribe thenucleare˙ectsofCCQEinteractions 1 . InadditiontoCCQEinteractions,theCC 0 ˇ modelincludesmulti-nucleon,or2particle2hole (2p2h),interactions.Within2p2hinteractions,therearetwomainmodesofinteractions,plusan interferencebetweenthetwo.Mesonexchangecurrentinteractions(MEC)eventsarethosewhich Feynmandiagramsincludea propagatorand pion-lessdecay,asseeninthelowerportion 1 Itshouldbenotedthat,inthecaseofanantineutrinointeractionwithahydrogennucleus ( + p ! X ), M QE A is˝xedat1.03GeV,whichisbasedonresultsfrombubblechamberdata[ 24 ]. Inallothercases, M QE A isfreetomove. 61 Figure4.6:2p2hdiagrams.Singlelinesrepresentnucleons,doublelinesrepresentthe ,dashed linesrepresentpions,andcurlylinesrepresenttheWboson.Adaptedfrom[ 22 , 23 ]. Figure4.7:Thetotalcrosssection(left)comparedwiththe2p2hcrosssection(right).Figure from[ 24 ]. ofFigure 4.6 .Ontheotherhand,theotherprimarymodecomesfromnucleon-nucleon(NN) correlations,likethoseseenintheupperportionofFigure 4.6 .Whencomparingthedistributions ofthesetwomodes,adistinctshapedi˙erenceisseeninthetransfermomentumandenergycross section,asshowninFigure 4.7 . 62 InNEUT,theNievesmodelfornp-nhinteraction[ 93 , 96 ]isincludedtomodel2p2hinteractions intheCC 0 ˇ model.Themodelemployedbytheneardetector˝tuses˝veparameterstodescribe theseinteractions.Twonormalizationparametersareincludedtoscalethe2p2hcrosssectionfor neutrinosandantineutrinosseparately.Largeuncertaintiesareappliedtotheseparametersinorder tocoverdi˙erencesinalternatemodels[ 25 ].Becauseoftheshapedi˙erenceseeninthecross section,ashapeparameterisincludedtoshifteventsbetweenthetwoextremesforinteractions occurringoncarbonandoxygen,separately.Finally,asthe2p2hmodelhasbeendevelopedfor isoscalarnuclei(likecarbonandoxygen),aparameterisincludedtoestimatetheuncertaintyin extrapolatingthemodelfromcarbontooxygen.Thisuncertaintyisbasedonelectronscattering measurementsofandpairs[ 97 ]. AsummaryoftheCC 0 ˇ parameterscanbefoundinTable 4.4 .Note,theBeRPAUparameter is˝xedintheBANFF˝tbecauseitintroducescomplicatedcorrelationsbetweentheparameters; therefore,itdoesnothaveanerrorlisted. Parameter PriorValuePriorError Type PassedtoSK pFC(MeV) 21731 Response NO pFO(MeV) 22531 Response YES BeRPAA 0.590.118 Response YES BeRPAB 1.050.21 Response YES BeRPAD 1.130.1695 Response YES BeRPAE 0.880.352 Response YES BeRPAU 1.20 Response YES M QE A (GeV) 1.200.025 Response YES 2p2hNorm 1.001.00 Norm YES 2p2hNorm 1.001.00 Norm YES 2p2hNormCtoO 1.000.20 Norm YES 2p2hShapeC 0.001.00 Response NO 2p2hShapeO 0.001.00 Response YES Table4.4:CC 0 ˇ parametersintheBANFF˝t.Includedinthetablearethepriorvalueanderror, thetypeofparameteritis,andwhetherornotthe˝nalvalueispassedtoSKornot. 63 4.3.2.2CCResonantParameters Inthepast,CCresonantinteractionsservedasoneoftheprimarybackgroundsintheoscillation analysis.However,withtherecentinclusionofthe1R e CC 1 ˇ sample,CCresonantinteractions arenowasignalinteraction.InNEUT,theRein-Sehgalmodel[ 98 ]isusedtomodelCCresonant interactions.Thismodelisparameterizedbythreeparameters.Theaxialmass, M RES A ,istheaxial massofCCresonantinteractions,whiletheaxialformfactor, C 5 A ¹ Q 2 = 0 º ,isthedominantaxial formfactorforresonantpionproduction.Theisopin= 1 š 2 backgroundscalesthenon-resonant backgroundforsinglepionprocesses,whichisassumedtobemadeupentirelyof I = 1 š 2 events inNEUT.AsummaryoftheCCresonantparameterscanbeseeninTable 4.5 . Parameter PriorValuePriorError Type PassedtoSK M RES A 1.070.15 Response YES IsoscalarBackground 0.960.40 Response YES CA5 0.960.15 Response YES Table4.5:CCresonantparametersintheBANFF˝t.Includedinthetablearethepriorvalueand error,thetypeofparameteritis,andwhetherornotthe˝nalvalueispassedtoSKornot. 4.3.2.3OtherChargedCurrentParameters WhileCC 0 ˇ andCCresonantinteractionsserveasthesignalforT2K,interactionsproducing multiplepions(N ˇ interactions),deepinelasticscattering(DIS),andCCcoherentinteractions serveastwoofthebackgroundinteractions.WhiletheCCcoherentcrosssectionismuchsmaller thantheCCresonantcrosssection,itssignalinthedetectorcanbeconfusedforaCCresonant interaction.Inordertoaccountforthisinteraction,twoparametersareincludedinthe˝ttocover CCcoherentinteractionsoncarbonandoxygennuclei.Theprioruncertaintiesfortheseparameters weredeterminedusingMINER Adata[ 99 ].ForDISandN ˇ interactions,onlyoneparameter, calledCCDIS,iscurrentlyusedintheanalysis.ThisparameterisbasedonresultsfromMINOS [ 24 ]andisdependentontheenergyoftheinteraction. 64 Twonormalizationparametersareincludedinthe˝ttoaccountforpotentialdi˙erencesinthe e š andthe e š crosssectionratios[ 100 ].Theseareincludedduetoanumberofdi˙erences arisinginelectron(anti)neutrinoandmuon(anti)neutrinointeractions.Twoofthesedi˙erences includee˙ectsduetothemassoftheoutgoingleptonandradiativecorrections.Withregardsto theformer,e˙ectsofthenucleonformfactorscanbecomeconvolutedwiththedi˙erenceinthe massoftheoutgoingmuonandelectron.Forthelatter,radiativecorrectionsareappliedtoboth electronsandmuons;however,becauseofthelightermassoftheelectron,thesecorrectionshave alargere˙ectonelectronneutrinoandantineutrinointeractions. Additionally,acorrectionisappliedduetotheCoulomb˝eldinsidethenucleus.Theso-called correctiondecreases(increases)themomentumoftheoutgoingnegatively(positively) chargedleptonduetotheelectrostaticattraction(repulsion)oftheremnantnucleusanddependson thechargedistributionofthenucleusandthepositionoftheinteractioninsidethenucleus[ 101 ]. Thiscorrectionisnotaparameterinandofitself,butitdoesshiftthereconstructedmomentum byafewMeV,whichallowseventstomovebins,buttheycannotmovebetweensamples.Related totheCoulombcorrections,anormalizationparameterisappliedtochargedcurrentneutrinoand antineutrinoevents,separately.Theseparametersallowforscalingofthetotalcrosssectiondueto thee˙ectsoftheCoulombinteraction. AsummaryofthesechargedcurrentparameterscanbeseeninTable 4.6 . Parameter PriorValuePriorError Type PassedtoSK CCnorm 1.000.020 Norm YES CCnorm 1.000.010 Norm YES e š 1.000.028 Norm YES e š 1.000.028 Norm YES CCDIS 0.000.40 Response YES CCCoherentC 1.000.30 Norm NO CCCoherentO 1.000.30 Norm YES Table4.6:OtherchargedcurrentparametersintheBANFF˝t.Includedinthetablearetheprior valueanderror,thetypeofparameteritis,andwhetherornotthe˝nalvalueispassedtoSKor not. 65 4.3.2.4NeutralCurrentParameters Neutralcurrenteventsareasourceofbackgroundcontaminationinthenearandfardetector samples.Inthecaseofneutralcurrentpionproductionevent,lowmomentumpionscanbe incorrectlyidenti˝edasmuoncandidates,whichwillcontaminatethesamplesforCC 0 ˇ events (describedinChapter 5 ).Toaccountforthesebackgroundevents,fournormalizationparameters areincludedintheneardetector˝t.TheNCcoherentparameterappliesanormalizationtoall neutralcurrentcoherentevents.Similarly,theNC1 parameterappliesanormalizationfactorto NC1 interactions,whichareabackgroundatSK.Finally,theNCothernearandfarparameters catchtherestoftheneutralcurrentinteractionsforND280andSK,respectively.Asummaryof theneutralcurrentparameterscanbeseeninTable 4.7 .ItshouldbenotedthattheNC1 and theNCotherfarparametersare˝xedintheneardetector˝t,astheneardetectordoesnothave sensitivitytotheseparameters.Theyareincludedintheneardetector˝tbecausethecrosssection parameterizationforthefardetectorconsistsofthoseparameterswhicharepassedtoSKfromthe neardetector˝t. Parameter PriorValuePriorError Type PassedtoSK NCCoherent 1.000.30 Norm YES NC1 2.002.00 Norm YES NCOthernear 1.000.30 Norm NO NCOtherfar 1.000.30 Norm YES Table4.7:NeutralcurrentparametersintheBANFF˝t.Includedinthetablearethepriorvalue anderror,thetypeofparameteritis,andwhetherornotthe˝nalvalueispassedtoSKornot. 4.3.2.5FinalStateInteractions The˝nalstateinteractionparameterscontrolthetransportofpionswithinthenucleusaftera neutrinointeractionoccurs.Theseinteractionsaresimulatedasacascade,wherethepionis propagatedthroughthenucleusstep-wise,allowingformultipleinteractionstooccurbeforethe particleexitsthenuclearmedium.FSIalloweventstomovebetweendi˙erenttopologiesandfor 66 theeventkinematicstochange.Forexample,ifonlyonepionisobservedfromaneventthat producesmultiplepions,thenthiseventwouldbeclassi˝edasaCC 1 ˇ eventtopology,ratherthan aCCOthereventtopology. Previously,sixparameterswereusedtoparameterizeFSI.However,recentwork[ 102 ],which usedanimproved˝ttingmethodtoDUETdata[ 103 ],hasreducedthenumberofparametersto ˝ve.Furthermore,the˝nalresultsoftheseparametersarenowpassedontoSK. TheFSIparametersare: ‹ FSIinelasticscattering hasa˝nalstatepionofthesamechargeastheinitialpion.Low andhighenergyeventsaretreatedastwoseparateparameters.Inbothcases,theparameters areappliedtoinelasticandelasticscatteringprocesses,becausethemodelonT2Kdoesnot di˙erentiatebetweenthese[ 104 ]. ‹ FSIpionproduction actsonhighenergyinteractionswhichhavetwoormorepionsinthe ˝nalstate. ‹ FSIpionabsorption appliestolowenergyeventswithnopionsinthe˝nalstate. ‹ FSIchargeexchange actsonlowenergyeventswhichhaveasinglepionoftheopposite chargefromtheinitialpion( ˇ + ! ˇ or ˇ ! ˇ + )ora ˇ 0 .Previously,therewasa highenergyversionofthisparameter.However,therecent˝tsshowedlittleresponsetothis parameter,soitisnotincludedintheparameterization. AsummaryoftheFSIparameterscanbefoundinTable 4.8 .Thecorrelationmatrixforthe Parameter InputValueInputError TypePassedtoSK Quasi-ElasticLowEnergy 1.0690.313 ResponseYES Quasi-ElasticHighEnergy 1.8240.859 ResponseYES PionProduction 1.0021.101 ResponseYES PionAbsorption 1.4040.432 ResponseYES ChargeExchangeLowEnergy 0.6970.305 ResponseYES Table4.8:FSIparametersintheBANFF˝t.Includedinthetablearethepriorvalueanderror,the typeofparameteritis,andwhetherornotthe˝nalvalueispassedtoSKornot. 67 Figure4.8:Correlationsbetweenthecrosssectionparameters. parametersdescribedintheSections 4.3.2.1 4.3.2.5 canbeseeninFigure 4.8 . 4.3.3TheObservableNormalizationParameters ThecomplexstructureofND280requiresacomplicatedsetofdetectorsystematicparameters, discussedindetailinChapter 6 ,tobeincludedintheneardetector˝t.However,includingthese parametersinthe˝tiscomputationallyexpensive,astheyhavetoberecalculatedforeveryeventat eachstepintheminimizationprocess.Inordertoreducethecomputationalexpense,thee˙ectofthe detectorsystematicuncertaintiesareencapsulatedinasetofnormalizationparameters,described inmoredetailinSection 6.4 ,intheneardetector˝t.Theseparametersarecalledtheobservable normalizationparametersandtheydescribethee˙ectsofthedetectorsystematicuncertaintiesof theeventratesfora p cos binineachsample.Inadditiontotheparameters,acovariancematrix describingtheuncertaintiesandcorrelationsoftheparametersareincludedinthe˝t. 68 Figure4.9:Thecorrelationmatrixfortheobservablenormalizationparameters.Theshortdashed linesdi˙erentiatebetweentheCC 0 ˇ ,CC 1 ˇ ,andCCOthersamples,whilethelongdashedlines separateFGD1andFGD2samples.ThesolidblacklinesseparatetheFHCMultiPi,RHC MultiPi,andRHC MultiPisamples.Withineachsample,theparametersareorderedfrom backwardgoingtoforwardgoingangularbins.Eachcompletesetofangularbinsshareacommon momentumbin,whichareorderedinfromlowesttohighestmomentum. Aswiththe˛uxandcrosssectionparameters,theobservablenormalizationparametersand theircovariance(Figure 4.9 )areassumedtobehaveinaGaussianmannerandareincludedinthe likelihoodgiveninEquation 4.8 .Withinthe˝t,theseparametersareallowedtocorrelatewiththe ˛uxandcrosssectionparameters.Astheobservablenormalizationparametersprovideconstraints ontheneardetectoreventrates,theyarenotusedintheoscillation˝tsatSK,whereasetofdetector systematicparametersforthefardetectorareusedinstead. 69 CHAPTER5 THENEARDETECTORSELECTIONS Theinteractionsthataremostrelevanttotheneardetector˝tarechargedcurrentquasi-elastic (CCQE),chargedcurrent(CC)resonantpionproduction,anddeepinelasticscatteringandmultiple pionproduction(DIS+N ˇ ). TheabilitytoselectthesetypesofeventsatND280is,therefore,ofutmostimportance.Event selectionatND280reliesonthreetimeprojectionchambers(TPCs)andtwo˝negraindetectors (FGDs).TheFGDsserveasthetargetvolumefortheneardetector,withFGD2providinganoxygen targetsimilartoSuper-Kamiokande(SK).Inaddition,theTPCsallowforparticleidenti˝cation andmomentum.ThischapterwillfocusonhoweventsareselectedusingtheFGDsandTPCsto createsamplesrelatingthethreeinteractionsmentionedpreviouslytowhatisseeninthedetector. 5.1SelectionConsiderations Becausetheselectionsexisttohelpconstrainthe˛uxandcrosssectionuncertainties,the samplesmustbesensitivetotheneutrinobeam˛uxandtheneutrinocrosssection.Therefore, whendesigningthesampleselections,thereareanumberofconsiderationsthatmustbetakeninto account. Withregardtotheneutrinobeam˛ux,the˛avorcompositionofneutrinosinneutrinoand antineutrinobeammodesarehighlycorrelated(seeSection 4.3.1 )aswellasbetweenthevarious neutrino˛avorsandenergies.Becauseofthislargeamountofcorrelation,anylargesampleof neutrinoorantineutrinointeractionswithahighpurityisenoughtoconstraintheuncertaintieson theneutrino˛uxparameters. Theprimarygoaloftheneardetector˝tistoconstraintheuncertaintyonCC 0 ˇ events,which encompasseseventswithnoobservedpions,andCCresonantinteractions,asthesearethesignal processes.Secondarytothat,itmustalsodecreaseuncertaintiesontheprimarybackgrounds,CC coherentandDIS+N ˇ interactions.Whileneutralcurrent(NC)interactionsalsooccurinND280, 70 theydonotprovidethesamelevelofconstraintprovidedbychargedcurrentevents.Therefore,by focusingonCCinteractions,itbecomesmucheasiertoconstrainsignalandbackgroundeventsin theneardetector. Additionally,thesamplesmusttakeintoaccountwhetherthedataiscollectedwhenthebeam isinneutrinoorantineutrinobeammode.Becausethebeamutilizesprotonscollidingwithatarget consistingofprotonsandneutrons,thereisapreferenceforpositivepions( ˇ + )tobecreatedover negativepions( ˇ ).Becauseofthispreference,thereishigher ˛uxinneutrinomodethan ˛uxinantineutrinomode.Thisalsomeansthereexistsasigni˝cantamountofhigherenergy ˇ + whichcannotbebentoutofthebeaminantineutrinomode.Therefore,theamountofneutrinos ismuchhigherinantineutrinomodethanthenumberofantineutrinosinneutrinomode.Neutrinos alsohavealargercrosssectionthanantineutrinosformatterinteractions,ascanbeseeninFigure 2.2 .Becauseantineutrinosinteractlessofteninmatter,theseeventsaresuppressedwhencompared toneutrinoevents. Becauseofthesetwoe˙ects,theneutrino˛uxinantineutrinobeammodeprovidesasigni˝cant background,whereastheantineutrino˛uxinneutrinobeammodeisnegligible.Furthermore,the backgroundneutrino˛uxinantineutrinomodeisanirreduciblebackgroundatSKbecausethe leptonchargeisunabletobedeterminedinchargedcurrentinteractions.Therefore,itisuseful tocreateasetofsampleswhichseparatesneutrinoandantineutrinoeventsinantineutrinomode, whilethesamplesforneutrinomodeonlyneedstoincorporateneutrinoevents. 5.2DataandMonteCarloInputstotheSelections ThedatacollectedatT2Kisdividedbyrunperiod,whichcorrespondstoaboutayearof runningthedetector.Thesecanbefurthercategorizeddependingonthedetectorcon˝guration, suchaswhetherthePØDwaterbagswere˝lledwithwaterorairorbasedonthemagnetcurrent. Forthisanalysis,Runs28wereused,whichcorrespondstodatatakenbetweenNovember2010 andApril2017.ThedatafromRun1wasnotusedbecauseofcalibrationissuesinthedetectoras wellasthetoptrackermissingfromtheECal.BecausetheECalwasoriginallymeanttobeusedin 71 theselection,itcouldposedi˚cultiesinusingthisdata.AsRun1representsonlyasmallamount ofstatisticswhencomparedwithlaterruns,itisreasonabletoleaveitoutbecauseitshouldhave littleimpactontheresults.WhileadditionaldatahasbeentakensinceApril2017,ithasnotbeen fullyprocessedand,therefore,wasnotincludedinthisanalysis. TheND280MonteCarloisgeneratedusingtheNEUTneutrinoeventgeneratortosimulate neutrinointeractions,whicharepropagatedthroughtheND280geometryusingGEANT4[ 105 ]. Thesimulationisrunseparatelyforeachrunperiodandcon˝gurationofthePØDwaterbags,as theinteractionrateincreaseswhenthebagsare˝lledwithwater.FortheMonteCarlo,twotypes ofeventsaregenerated:neutrinointeractionsoccurringinthemagnetvolumeandsandmuon events.Sandmuoneventscorrespondtoeventsoccurringinthesandsurroundingthedetectorand representtheoutsidebackgroundeventsinthedata.ThesearedescribedinmoredetailinSection 6.1 . Table 5.1 showswhethertherunisinneutrinoorantineutrinomodeandtheamountofprotons ontarget(POT)foreachrun. RunPeriod BeamMode DataPOT( 10 20 )MCPOT( 10 21 ) Run2a,WaterOut Neutrino 0.3590.924 Run2w,WaterIn Neutrino 0.4341.203 Run3b,WaterOut Neutrino 0.2170.448 Run3c,WaterOut Neutrino 1.3642.632 Run4a,WaterOut Neutrino 1.7833.500 Run5c,WaterIn Antineutrino 0.4352.296 Run6b,WaterOut Antineutrino 1.2731.417 Run6c,WaterOut Antineutrino 0.5080.528 Run6d,WaterOut Antineutrino 0.7750.688 Run6e,WaterOut Antineutrino 0.8510.859 Run7c,WaterIn Antineutrino 2.4373.371 Run8a,WaterOut Neutrino 4.1493.631 Run8w,WaterIn Neutrino 1.5812.641 Table5.1:ThedataandMonteCarloPOT,aswellasthebeammode,foreachrunperiodusedin theanalysis. 72 5.3DescriptionofSelections The˝rststepintheselectionsintheneardetectoristocreateasamplethatincludesthefull setofneutrinoandantineutrinoevents,calledtheCCinclusivesample.Thiscutremovesany eventsidenti˝edasnon-CCinclusive,suchasneutralcurrentevents.Then,theremainingevents arecategorizedby˝naleventtopology,whichisde˝nedasthesetofparticlesleavingthenucleus. Becausethedetectorisonlyabletoreconstructtheparticlesleavingthenucleus,˝naleventtopology ischoseninsteadofthetypeofinteraction. FinaleventtopologiescorrespondtothethreemaininteractionsseenatND280andSK.For CCQEevents,thereareonlytwooutgoingchargedparticles,aprotonandamuon.Thismakesit rathersimpletolookforthistopology.Inpractice,italsotendstobeevenmoresimplebecause theoutgoingprotontendstonotbevisibleduetodetectorthresholde˙ects.Theeventtopology associatedwithCCQEeventsisCC 0 ˇ ,whereonechargedleptontrackandnosecondarypionsare visibleinthedetector.Thetwootherinteractions,CCresonantandDIS+N ˇ ,alsohavetheirown topologies,CC 1 ˇ andCCOther,respectively,andwillbedescribedinmoredetailinSection 5.3.1 . Thesetopologiesaresimilartowhatisseeninthefardetectorandarecomparabletothesamples usedbySK,asdescribedinSection 4.1.2 . Inordertobeconsideredanevent,theremustbeamuon-liketrackwhichstartsinthe˝ducial volumeofeitherFGD1orFGD2andthenpassesintoaneighboringTPC.Therefore,thequality oftheselectionreliesoncorrectlyreconstructingthedirectionandvertexpositionofeacheventin theneardetector.EachselectiontakesintoaccountthetargetFGD,thebeammode,andwhether themuonispositivelyornegativelycharged.Withthisinmind,thecutsusedforneutrinomode andantineutrinomodemaybedi˙erent.However,eachselectionusesasimilarcut˛owforthe CCinclusiveselectionandasimilarstructurefordividingtheCCinclusivesampleintotopological samples. Becauseofthelargeneutrinobackgroundintheantineutrinobeammode,therearetwoCC inclusiveselectionsinantineutrinomode,onefor andoneforthe background.Ontheother hand,thereisonlyoneCCinclusiveselectionperformedforneutrinomode.Theneutrinomode 73 (alsoknownasforwardhorncurrent,orFHC)CCinclusiveselectionisthenseparatedintoCC 0 ˇ , CC 1 ˇ ,andCCOthersamples.Previously,theantineutrinomode(alsocalledreversehorncurrent, orRHC)CCinclusiveselectionswereseparatedinto1-trackandN-trackssamples.Thiswas duetodi˚cultyseparatingnon-CC 0 ˇ topologiesinthereconstructionandwillbediscussedin Section 5.3.3.2 .Table 5.2 showsthebreakdownofselectionsunderthisparadigm.Recently,the reconstructionhasimprovedsuchthatnon-CC 0 ˇ topologiescanbedistinguishedTherefore,the RHCCCinclusiveselectionisnowsubdividedintoCC 0 ˇ ,CC 1 ˇ ,andCCOthersamples,likethe FHCCCinclusiveselection.Table 5.3 showshowtheeventsarefunnelednow.Theexactcuts foreachselectionaredescribedinthefollowingsections.Thenumberofselectedandpredicted AllReconstructedEvents w + u FHC CCInclusiveRHC CCInclusiveRHC CCInclusive +++ CC 0 ˇ CC1-trackCC1-track CC 1 ˇ CCN-tracksCCN-tracks CCOther Table5.2:Breakdownofreconstructedeventsintosamplesunderthepreviousparadigm. AllReconstructedEvents w + u FHC CCInclusiveRHC CCInclusiveRHC CCInclusive +++ CC 0 ˇ CC 0 ˇ CC 0 ˇ CC 1 ˇ CC 1 ˇ CC 1 ˇ CCOtherCCOtherCCOther Table5.3:Breakdownofreconstructedeventsintosamplesunderthenewparadigm. eventsineachsamplecanbeseeninTables 5.4 and 5.5 forthepreviousantineutrinobeammode selectionsandthenewselections,respectively. 74 Sample DataND280Prediction FGD1FHC CCInclusive 42,20540,874.71 FGD1FHC CC 0 ˇ 28,67127,475 FGD1FHC CC 1 ˇ 6,6387,106.14 FGD1FHC CCOther 6,8966,293.57 FGD1RHC CCInclusive 7,9267,875.1 FGD1RHC 1-Track 6,1596,135.82 FGD1RHC N-Tracks 1,7671,739.28 FGD1RHC CCInclusive 4,4374,145.47 FGD1RHC 1-Track 2,2392,076.37 FGD1RHC N-Tracks 2,1982,069.1 Sample DataND280Prediction FGD2FHC CCInclusive 40,12137,867.46 FGD2FHC CC 0 ˇ 28,44426,826.1 FGD2FHC CC 1 ˇ 5,1495,498.52 FGD2FHC CCOther 6,5285,542.84 FGD2RHC CCInclusive 7,8727,601.02 FGD2RHC 1-Track 6,1745,934.79 FGD2RHC N-Tracks 1,6981,666.23 FGD2RHC CCInclusive 4,1603,895.55 FGD2RHC 1-Track 2,0991,970.47 FGD2RHC N-Tracks 2,0611,925.08 Table5.4:ObservedandpredictedeventratesforthepreviousND280sampleset.Thepredicted eventratesareMonteCarloeventsweightedbyPOT,˛ux,detector,andcrosssectionweights. TheleftcolumnshowstheratesforFGD1,whiletherightcolumnshowsFGD2. Sample DataND280Prediction FGD1FHC CCInclusive 42,40540,821.52 FGD1FHC CC 0 ˇ 28,67127,429.5 FGD1FHC CC 1 ˇ 6,6387,095.14 FGD1FHC CCOther 6,8966,296.88 FGD1RHC CCInclusive 8,0057,839.886 FGD1RHC CC 0 ˇ 6,3686,245.19 FGD1RHC CC 1 ˇ 535542.636 FGD1RHC CCOther 1,1021,052.06 FGD1RHC CCInclusive 4,5694,264.058 FGD1RHC CC 0 ˇ 2,7072,547.87 FGD1RHC CC 1 ˇ 847875.593 FGD1RHC CCOther 1,015840.595 Sample DataND280Prediction FGD2FHC CCInclusive 40,12137,810.02 FGD2FHC CC 0 ˇ 28,44426,772 FGD2FHC CC 1 ˇ 5,1495,493.58 FGD2FHC CCOther 6,5285,544.44 FGD2RHC CCInclusive 7,9487,630.104 FGD2RHC CC 0 ˇ 6,4516,211.62 FGD2RHC CC 1 ˇ 465486.26 FGD2RHC CCOther 1,032932.224 FGD2RHC CCInclusive 4,2734,030.805 FGD2RHC CC 0 ˇ 2,6482,565.66 FGD2RHC CC 1 ˇ 693675.982 FGD2RHC CCOther 932789.163 Table5.5:ObservedandpredictedeventratesforthenewND280sampleset.Thepredictedevent ratesareMonteCarloeventsweightedbyPOT,˛ux,detector,andcrosssectionweights.Theleft columnshowstheratesforFGD1,whiletherightcolumnshowsFGD2. 5.3.1ForwardHornCurrentMultiPiSelections 5.3.1.1ChargedCurrentInclusiveSelection The˝rsttwocutsfortheCCinclusiveselectionare: ‹ EventQuality: EventsmusthavethefullspillwithallND280subdetectorsingoodworking order(theprocessfordeterminingtheworkingconditionofeachdetectorisdescribedin [ 106 ]).Furthermore,theeventmustoccurwithinthetimewindowforeachprotonbunchin 75 theneutrinobeam.Inordertoavoideventpileup,eventswhichfallinthesamespill,but di˙erentbunches,aretreatedasindependentevents.Thiscutensuresthebeamwasonand runningproperlyandtheND280subdetectorswereallworkingcorrectly.Itshouldbenoted thatthiscutisfordataevents,sinceitisassumedtheMCeventshavegoodbeamanddata quality. ‹ TotalMultiplicityCut: Thiscutrequiresthataneventhasatleastonereconstructedtrack thatcrossesaTPC.Thiseliminatesanytriggeredeventsthathavelittleornoreconstructable information. These˝rsttwocutsarethesameforallCCinclusiveselections.Thecut˛owfortherestofthe cuts,describedbelow,isalsothesameforeachoftheCCinclusiveselections. ‹ FiducialCut: The˝ducialcutrequirestheretobeatleastonereconstructedtrackinthe ˝ducialvolumeofFGD1orFGD2.Furthermore,itrequiresthattherebeatleastonetrack withportionsinFGD1(orFGD2)andaTPC,withthevertexofthetrackstartinginsidethe FGD's˝ducialvolume.The˝ducialvolumesare: j x j < 874 : 51 mm, 819 : 51 < y < 929 : 51 mm, 136 : 875 < z < 446 : 955 mmforFGD1, j x j < 874 : 51 mm, 819 : 51 < y < 929 : 51 mm, 1481 : 45 < z < 1807 : 05 mmforFGD2. Cutsinthe x and y directionsremoveinteractionswhichhaveavertexthatisatleast5bars fromtheXYmoduleintheFGD.The z cutremovesthemostupstreamXYmodule,but keepstheremainingXYmodules.The˝nalpartofthiscutremovesshorttracks,which makestheTPCreconstructionlessreliable. ‹ UpstreamBackgroundVeto: Theupstreamvetocutistoeliminatereconstructionfailures thatleadtoamuontrackstartinginoneoftheFGD˝ducialvolumes,eventhoughthe realmuoninteractedfurtherupstream.Bywayofexample,amuonthatoriginatedinthe PØDthatunderwentalargescatterinFGD1maybereconstructedastwotracks,ratherthan one.Theseeventsareexcludedbycuttingeventswherethesecondhighestmomentumtrack 76 starts150mmupstreamofthemuontrack.Furthermore,inthecaseofFGD2,theeventcan alsobevetoedifthesecondarytrackstartsintheFGD1˝ducialvolume. ‹ BrokenTrackCut: Thebrokentrackcutwascreatedtoeliminateeventswithincorrectly reconstructedtracks,wherethereconstructionprocedurebreaksthemuoncandidatetrack intotwocomponents.Typically,the˝rsttrackisfullycontainedintheFGDandthesecond trackbeginsinthelastfewlayersoftheFGDandpassesintotheTPC.Inthiscase,thesecond trackwouldbeconsideredthemuoncandidate,ratherthanthe˝rsttrack.Therefore,toreject thesetypesofevents,iftheeventhasatleastoneFGD-onlytrack,thenthestartpositionof themuoncandidatetrackmustbelessthan425mmawayfromtheFGDupstreamedge. ‹ MuonParticleIdenti˝cationCut: Anyeventsthathavepassedtheabovecuts,havethe highestmomentumtrackidenti˝edasanegativeparticle,originatesinsideanFGD's˝ducial volume,andcrossesaTPCisconsideredthemuoncandidate.Theparticleidenti˝cation (PID)processforthistrackisbasedonthe dE š dx measurementintheTPCandcomparesthe energydepositedintheTPCtotheexpectedenergydepositassumingacertainparticletype, eithermuon,electron,orproton.Fromthisinformation,pullsanddiscriminationfunctions arecalculated. First,electronsarerejectedbyenforcing L + L ˇ 1 L P > 0 : 8 (5.1) andisonlyappliedforamomentumlessthan500MeV/c.Thenextcutremovestheremaining protonsandpionsbyconstraining L > 0 : 5 : (5.2) InEquations 5.1 and 5.2 , L isde˝nedas L i = e Pull 2 i Í l e Pull 2 l ; (5.3) 77 where Pull is Pull i = dE š dx measured dE š dx expected ; i ˙ ¹ dE š dx measured dE š dx expected ; i º : (5.4) InordertobreaktheCCinclusivesampleintothethreesubsamples,pioninformationfromthe TPCsandFGDsisused.Thisbreakdownwillbediscussedinthenextsection. 5.3.1.2SelectionsforMultiPiTopologies OnceeventshavebeenselectedintotheCCinclusivesampleandarereadytobedividedinto subsamples,nomoreeventsarecut.ThisisbecausealleventsthathavemadeitintotheCC inclusiveselectionshouldhaveoneoftheaforementionedtopologies,CC 0 ˇ ,CC 1 ˇ ,orCCOther. Inordertobreakdowntheselectedeventsintotopologicalsamples,informationaboutsecondary tracksandtheidenti˝cationofpionsintheFGDsandTPCsisused. Thethreetopologicalsamplesarede˝nedasfollows: ‹ TheCC 0 ˇ Sample hasnoobservedsecondarypions.ThissamplehasnoTPC-identi˝ed pions,electronsorpositronsasde˝nedbytheTPCPID,andnoMichelelectronsorcharged pionsfoundintheFGD. ‹ TheCC 1 ˇ Sample hasonenegativemuontrack,onepositivelychargedreconstructedpion, andnootherpionsselected.Inordertobeselectedintothissample,itisrequiredthatthere beonlyonereconstructedMichelelectronintheFGDandnopionsintheTPCoronepositive pionintheTPCorFGDandnoreconstructedMichelelectrons.Furthermore,eventswhich haveareconstructednegativepionoranelectronorpositron(whichsignalsthedecayofa ˇ 0 )intheTPCarerejected. ‹ TheCCOtherSample containstherestoftheCCinclusiveeventsthatdidnotmeetthe criteriatofallintooneoftheothersamples.Thismeansthateventswithasinglenegative muonandeithermultiplepositivepiontracksoratleastonenegativeorneutralpion. Additionally,anyeventswhichincludeotherparticles,includingkaonsoretas,fallintothis sample. 78 ThethreecutsusedtoseparatethesesamplesaretheTPCsecondarytrackPIDcut,theFGD- onlyreconstructedtrackcut,andtheMichelelectroncut.Eachofthesecutsissimilarbetween FGD1andFGD2,withslightmodi˝cationsforFGD2,asneeded.WithintheBANFF˝t,the p cos binningsforthesesamplesare: ‹ CC 0 ˇ ˝tbinedges: p [MeV/c]:0,200,300,400,450,500,550,600,650,700,750,800,850,900,950,1000, 1050,1100,1200,1300,1400,1500,1600,1700,1800,2000,2500,3000,5000,30000 cos :-1.0,0.5,0.6,0.7,0.76,0.78,0.8,0.83,0.85,0.88,0.89,0.9,0.91,0.92,0.925,0.93, 0.935,0.94,0.945,0.95,0.955,0.96,0.965,0.97,0.975,0.98,0.985,0.99,0.995,1.0 ‹ CC 1 ˇ ˝tbinedges: p [MeV/c]:0,300,350,400,500,600,650,700,750,800,900,1000,1100,1200,1500, 2000,3000,5000,30000 cos :-1.0,0.6,0.7,0.8,0.85,0.88,0.9,0.92,0.93,0.94,0.95,0.96,0.97,0.98,0.99,0.995, 1.0 ‹ CCOther˝tbinedges: p [MeV/c]:0,300,400,500,600,650,700,750,800,900,1000,1100,1250,1500,1750, 2000,3000,5000,30000 cos :-1.0,0.6,0.7,0.76,0.8,0.85,0.88,0.89,0.9,0.91,0.92,0.93,0.94,0.95,0.96,0.97, 0.98,0.99,0.995,1.0 ThebinningforFGD1andFGD2isthesame.Thesebinningsarechosensuchthatthereisatleast onedataeventperbin.Furthermore,thebinningis˝neraroundthepeakkinematicregionsto utilizeinformationabouttheshapeofthedistributionwithintheBANFF˝t. 79 ‹ TheTPCSecondaryTrackPIDCut: Inordertodeterminethepresenceofpionsinthe TPC,thepullsandlikelihoodsfromthemeasured dE š dx ,givenbyEquations 5.3 and 5.4 ,are used.However,insteadofusingtheprimarytrack,therelevanttracksarenowthesecondary tracksintheevent.ThesesecondarytracksmustbegininthesameFGD˝ducialvolumeas themuontrackandhaveamatchingTPCsegment.Forpiontrackcuts,thepion,positron,and protoncandidatesareconsideredforpositivetracksandthepionandelectronareconsidered fornegativetracks. Inordertoidentifyachargedpion,acutissetsuchthatanysecondaryFGD-TPCtrackis consideredtobeapioncandidateif L + L ˇ 1 L p > 0 : 8 ; (5.5) ifthemomentumislessthan500MeV/c.Otherwise,thetrackisapionif L ˇ > 0 : 3 : (5.6) ‹ TheFGD-onlyReconstructedTrackCut: Fortracksthatbothstartandendinthesame FGDasthemuoncandidateoriginates,acutisplacedtoidentifythesetracksassecondary ˇ + 's.InsteadofusingthegeneralFGD˝ducialvolume,thiscutusesa˝ducialvolume de˝nedby 887 < x < 888 mm(FGD1)or 843 < x < 848 mm(FGD2), 834 < y < 942 mm(FGD1)or 820 < y < 938 mm(FGD2), az-positionthatfallsbetweenthe˝rstandlastactivelayersofthecorrespondingFGD. InordertobeconsideredfortheFGD-onlyreconstructedtrackcut,theentiretrackunder considerationmustfallwithinthisvolume.Furthermore,thetrackmustoccurinthesame timebunchasthemuoncandidate.InordertodeterminethePID,apullcanbede˝nedusing theenergydepositedasafunctionofthetracklength,whichwillallowfortheselectionof themostprobableparticleforthetrack.Apositivepionisde˝nedashavinganFGD-only 80 trackwith j cos j > 0 : 3 ,whichistheregionthathassu˚cientreconstructione˚ciency,and havingapionpullbetween 2 and 2 : 5 . ‹ TheMichelElectronCut: Whenpionswithoutenoughenergytoreachtheneighboring TPCdecay,thereisthepotentialtocontaminatetheCC 0 ˇ sample.Inordertoreducethis contamination,anadditionalcutusingdelayedactivityfromMichelelectronscanbeused. Michelelectronsoriginatefrompiondecayandcansignallowmomentumpionsthatwere notidenti˝edbytheFGD-onlyreconstructedtrackcut. Delayedactivityisde˝nedasdetectoractivitythatoccursmorethan100nsaftertheinitial neutrinointeractionandisusedtoidentifyMichelelectrons.Ingeneral,themajorityof eventsdonothaveanydelayedactivity,asseeninFigure 5.1 .ThesignalfortheMichel electronmustoccuroutsidethebeambunchwindow,becausethereisnotawaytodistinguish betweenactivityduetoMichelelectronsandactivityduetothepresenceofthebeam.A Michelelectronhitclusterisrequiredtohaveatleast7hitsinFGD1and6hitsinFGD2 tobede˝nedasaMichelelectron.Figure 5.1 showsthatthemajorityofeventswithone reconstructedMichelelectronfallintheCC 1 ˇ sample,whichgenerallyaretheproductof CCresonantinteractions. Themomentumand cos distributionsforFGD1canbeseeninFigure 5.2 ,whilethecorre- spondinge˚ciencyandpurityplotsarefoundin 5.3 and 5.4 .PlotsforFGD1canbefoundin Figures C.1 C.3 ofAppendix C . 5.3.2ReverseHornCurrentMultiTrackSelections Previously,thereversehorncurrentCCinclusiveselectionwasdividedinto1-trackandN-tracks (theMultipleTrack,orMultiTrack)samplesbecauseofearlydi˚cultiesreconstructingsecondary tracksintheantineutrinobeammode.However,thecurrentprocessistodividethereversehorn currentCCinclusivesampleintoCC 0 ˇ ,CC 1 ˇ ,andCCOthersamples(MultiplePion,orMultiPi). 81 FGD1 FGD2 Figure5.1:DistributionofthenumberofMichelelectronsinFGD1(top)andFGD2(bottom) categorizedbydi˙erentinteractiontopologies.Theleftplotineachpairshowsthedistribution whennosecondarytracksareseenintheTPC,whiletherightplotshowstheCCinclusive selectionwithnosuchconstraints.Figurefrom[ 7 ]. Thissectionwillprovideanoverviewoftheselectionprocessforthepreviousselectionsforboth and ,whilethenewselectionswillbediscussedinSection 5.3.3 . 5.3.2.1ChargedCurrentInclusiveSelection Ingeneral,boththeRHC andRHC CCinclusiveselectionsusethesameinitialqualitycutsas describedinSection 5.3.1.1 .Thetwomaindi˙erencesbetweentheFHCCCinclusiveselectionand theseistheselectionofthemuoncandidatetrackandthemuonPIDcut.Withregardstothemuon candidatetrack,theRHC selectionrequiresthehighestmomentumtrackbepositive,because 82 FGD1FHCCC0 ˇ FGD1FHCCC1 ˇ FGD1FHCCCOther Figure5.2:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD1FHCMultiPi selections.Themomentumdistributionsareshownontheleft,whiletheangulardistributionsare shownontheright. 83 FGD1FHCCC0 ˇ FGD1FHCCC1 ˇ FGD1FHCCCOther Figure5.3:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD1FHCMultiPiselections. 84 FGD1FHCCC0 ˇ FGD1FHCCC1 ˇ FGD1FHCCCOther Figure5.4:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD1FHCMultiPiselections. 85 antineutrinointeractionswillproduceapositivelychargedmuon,comparedwiththenegatively chargedmuonproducedinneutrinointeractions.BecausetheRHC selectionislookingforthe neutrinobackgroundinantineutrinobeammode,thehighestmomentumtrackisrequiredtobe negative. Similarly,themuonPIDcutisslightlydi˙erentfromtheFHCCCinclusiveselectionforboth RHCCCinclusiveselections.Inthiscase,thepullcutsaremorestrictforRHCthantheywerefor FHC,whichhelpseliminatebackgroundsforthetwoselections. FortheRHC selection,thecutsare L + L ˇ 1 L p > 0 : 9 (5.7) forlowmomentum( < 500 MeV/c)eventsand 0 : 1 < L < 0 : 7 (5.8) fortherestofthetrackcandidates.Theupperboundfor L istoeliminateanymisidenti˝edlow energy tracks,becausetheseoftenarereconstructedwiththeincorrectcharge.Byincluding thiscut,the contaminationisreduced. FortheRHC selection,thecutsare L + L ˇ 1 L p > 0 : 7 (5.9) forlowmomentum( < 500 MeV/c)eventsand 0 : 1 < L < 0 : 8 (5.10) fortherestofthetrackcandidates.ThecutinEquation 5.9 isslightlylowerthanintheFHC selection,inordertoavoidcuttingsignalevents.Additionally,thiscutissettoreducethefraction ofbackgroundelectronsintheoverallselection.Ontheotherhand,inEquation 5.10 ,thestricter boundistoaidinrejectingprotonsandincorrectlyreconstructed,lowenergy + . 86 5.3.2.2SelectionsforMultiTrackTopologies UnliketheFHCMultiPiselection,theRHCMultiTrackselectionusestwosimplertopological samples: ‹ TheCC1-TrackSample containsoneprimarymuoncandidatetrack.OnlyoneFGD-TPC matchedtrackisallowedinthissample.However,unliketheCC 0 ˇ sample,nocutismade onFGD-onlytrackmultiplicity.Additionally,delayedMichelelectronswouldfallintothis sample.Thisisbecauselowenergy ˇ arequicklyabsorbed,makingithardertoreconstruct theseevents. ‹ TheCCN-TracksSample includesanyeventsthathaveanobservedsecondaryFGD-TPC matchedtrack.WhereastheCC 1 ˇ andCCOthersamplescanincludeFGD-onlytracks andnosecondaryFGD-TPCmatchedtracks,theCCN-Trackssampledoesnotcontainany FGD-onlytracks,unlessthereisatleastonesecondarytrack. WhendeterminingwhetheraneventfallsintotheCC1-TrackorCCN-Trackssample,thesame processisusedforbothRHC andRHC CCinclusiveselections.TheBANFF˝tbinningfor theRHC samplesare: ‹ CC1-Track˝tbinedges: p [MeV/c]:0,400,500,600,700,800,900,1100,1400,2000,10000 cos :-1.0,0.6,0.7,0.8,0.85,0.88,0.91,0.93,0.95,0.96,0.97,0.98,0.99,1.0 ‹ CCN-Tracks˝tbinedges: p [MeV/c]:0,700,950,1200,1500,2000,3000,10000 cos :-1.0,0.75,0.85,0.88,0.91,0.93,0.95,0.96,0.97,0.98,0.99,1.0 TheBANFF˝tbinningfortheRHC samplesare: ‹ CC1-Track˝tbinedges: 87 p [MeV/c]:0,400,600,800,1100,2000,10000 cos :-1.0,0.7,0.8,0.85,0.9,0.93,0.95,0.96,0.97,0.98,0.99,1.0 ‹ CCN-Tracks˝tbinedges: p [MeV/c]:0,500,700,1000,1250,1500,2000,3000,10000 cos :-1.0,0.7,0.8,0.85,0.90,0.93,0.95,0.96,0.97,0.98,0.99,1.0 Momentumandangulardistributions,aswellase˚ciencyandpurityplots,forthe MultiTrack samplescanbefoundinFigures 5.5 5.7 forFGD1.Withregardstothe MultiTrackselections, themomentumandangulardistributionsande˚ciencyandpurityplotsforFGD1canbefound inFigures 5.8 5.10 .ThecorrespondingFGD2plotscanbefoundinFigures C.4 C.6 forthe MultiTracksamplesandFigures C.7 C.9 forthe MultiTrackselections. 5.3.3ReverseHornCurrentMultiPiSelections OneofthekeychangestotheBANFF˝tpresentedinthisworkwasthetransitionfromusingthe RHCMultiTracksamplestotheRHCMultiPisamples.Thischangeputstheneutrinomodeand antineutrinomodesamplesonthesamefootingandtreatstheRHCsamplesinawaythatbetter mirrorstheprimaryneutrinointeractionsinND280andSK.Foramorein-depthdiscussiononthe di˙erencesbetweenthetwotypesofselections,seeSection 5.3.3.3 . 5.3.3.1ChargedCurrentInclusiveSelection ThesamemethodastheRHCMultiTrack(whichitselfisaslightmodi˝cationoftheFHCMultiPi CCinclusiveselection)isused.Themostimportantpartofthetheinclusiveselectionisthe determinationofaleadingmuon,a forthe backgroundanda + for . 88 FGD1RHC CC1-Track FGD1RHC CCN-Tracks Figure5.5:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD1RHC MultiTrackselections.Themomentumdistributionsareshownontheleft,whiletheangular distributionsareshownontheright. 5.3.3.2SelectionsforMultiPiTopologies AswiththeFHCMultiPisamples,oncetheCCinclusiveselectionhasbeenperformed,thesample isdividedintothreetopologicalsamplesbasedonthepresence(orlackthereof)ofpions. ‹ TheCC 0 ˇ Sample containsnoobservedpiontracks. ‹ TheCC 1 ˇ Sample hasonenegativepion(for interactions)oronepositivepion(for interactions)andnootherpiontracks. ‹ TheCCOtherSample containstherestoftheeventsthatdonotfallintoeitherofthe˝rst twosamples. 89 FGD1RHC CC1-Track FGD1RHC CCN-Tracks Figure5.6:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD1RHC MultiTrackselections. TheBANFF˝tbinningfortheRHC samplesare: ‹ CC 0 ˇ ˝tbinedges: p [MeV/c]:0,300,400,500,550,600,650,700,750,800,900,1000,1100,1200,1500, 2000,4000,30000 cos :-1.0,0.6,0.7,0.8,0.85,0.9,0.92,0.93,0.94,0.95,0.96,0.965,0.97,0.975,0.98, 0.985,0.99,0.995,1.0 ‹ CC 1 ˇ ˝tbinedges: 90 FGD1RHC CC1-Track FGD1RHC CCN-Tracks Figure5.7:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD1RHC MultiTrackselections. p [MeV/c]:0,500,700,900,1300,2500,30000 cos :-1.0,0.7,0.8,0.9,0.94,0.96,0.98,0.99,1.0 ‹ CCOther˝tbinedges: p [MeV/c]:0,600,800,1000,1250,1500,2000,4000,30000 cos :-1.0,0.7,0.8,0.85,0.9,0.93,0.95,0.97,0.98,0.99,1.0 TheBANFF˝tbinningfortheRHC samplesare: ‹ CC 0 ˇ ˝tbinedges: 91 FGD1RHC CC1-Track FGD1RHC CCN-Tracks Figure5.8:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD1RHC MultiTrackselections.Themomentumdistributionsareshownontheleft,whiletheangular distributionsareshownontheright. p [MeV/c]:0,300,500,700,800,900,1250,1500,2000,4000,30000 cos :-1.0,0.7,0.8,0.85,0.88,0.9,0.92,0.94,0.96,0.97,0.98,0.99,1.0 ‹ CC 1 ˇ ˝tbinedges: p [MeV/c]:0,600,800,1500,30000 cos :-1.0,0.7,0.8,0.86,0.9,0.94,0.96,0.97,0.98,0.99,1.0 ‹ CCOther˝tbinedges: p [MeV/c]:0,600,1000,1250,2000,4000,30000 92 FGD1RHC CC1-Track FGD1RHC CCN-Tracks Figure5.9:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD1RHC MultiTrackselections. cos :-1.0,0.7,0.8,0.86,0.9,0.93,0.95,0.97,0.99,1.0 Basedonthisbreakdownofsamples,itisimportanttobeabletotagpionsinevents.Inthe caseofaneutralpion,boththeRHC andRHC selectionslookfortracesofelectromagnetic cascadesfromphotonsintheTPCs.Additionally,ifanelectron-likenegativetrackoranelectron- likepositivetrack(withmomentumgreaterthan0.9MeV/c)ispresent,thenatleastoneneutral pionisassumedtobeintheevent.Whenitcomestochargedpions,ifthereisanegativelycharged pion-liketrackorapositivelychargedpion-liketrackintheTPC,thenitisassumedtheeventhas eitheranegativeorpositivepion,respectively. 93 FGD1RHC CC1-Track FGD1RHC CCN-Tracks Figure5.10:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD1RHC MultiTrackselections. However,piontaggingbecomesmoredi˚cultwhenthereareMichelelectronsorisolatedFGD tracks.FortheRHC selection,thefocusisonnegativepions,becausetheyaremoreabundant inantineutrinointeractions.Negativepionsareespeciallyproblematicbecausetheytendtobe capturedinthedetectormediumpriortodecaying,whichmakestaggingnegativepionsviaMichel electronsdi˚cult.Therefore,isolatedFGDtracksmustberelieduponinordertotagnegative pions.Itshouldbenotedthat,eventhoughthechargeoftheisolatedtrackcannotbedetermined currently,itisreasonabletoassumethechargeisthatofthemorecommonpion,whichisthe negativepioninthecaseofantineutrinointeractions.Withthisinmind,thefollowingstatements canbemadefordeterminingthepresenceofnegativeorpositivepionsintheRHC sample: 94 ‹ AMichelelectronintheFGD(inmostcases)istheresultofapositivepionintheevent, whereas ‹ anisolatedpion-liketrackintheFGDismorelikelytheresultofanegativepion. Table 5.6 showsthattheCC 1 ˇ samplepreferseventswithoneisolatedFGDtrackandnoMichel electrons,whereaswhenaMichelelectronispresent,theCCOthersampleisthepreferablechoice. Thisprovidescon˝rmationofthepreviousstatements,aspositivelychargedpionswouldfallinthe CCOthersampleandasinglenegativelychargedpionwouldfallintheCC 1 ˇ sample. #isotracks=0 #isotracks=1 FGD1FGD2 FGD1FGD2 #ME=0 CC 0 ˇ 61.2%59.7% 4.0%7.6% CC 1 ˇ 12.9%12.0% 46.8%37.0% CCOther 9.0%8.9% 22.6%21.0% Background 10.8%11.8% 18.0%26.3% OOFV 6.1%7.7% 8.7%8.1% #ME=1 CC 0 ˇ 6.5%4.0% 1.1%0.0% CC 1 ˇ 5.2%4.3% 13.8%10.3% CCOther 20.8%19.0% 32.3%25.0% Background 33.7%35.0% 41.0%47.1% OOFV 33.8%37.6% 11.8%17.6% Table5.6:PercentageofeventswithagivennumberofMichelelectrons(ME)andisolatedFGD tracks(isotracks).Backgroundrepresentseventsfallingintoanybackgroundsamples(whichare notusedinthisanalysis),whileOOFVincludesanyeventsoccurringoutsideoftheFGD˝ducial volume.Tablefrom[ 6 ]. Ontheotherhand,theRHC selectionfocusesonpositivepionidenti˝cation,ratherthan negativepionidenti˝cation.Eventhoughselectingpositivepionsisofmorebene˝tforthis selection,itisstillimportanttocorrectlyidentifynegativepions,especiallybecausetheycanbe misidenti˝edasnegativemuonsiftheycrosstheTPC.Inthecaseofpositivepions,thesignalfrom Michelelectronsoccurringoutsidethebeambunchtimewindowcanbeusedtotagtheexistence ofapositivelychargedpion. 95 Momentumandangulardistributions,aswellase˚ciencyandpurityplots,forthe MultiPi samplesinFGD1canbefoundinFigures 5.11 5.13 .Withregardstothe MultiPiselections inFGD1,themomentumandangulardistributionsande˚ciencyandpurityplots,canbefoundin Figures 5.14 5.16 .ThecorrespondingFGD2plotscanbefoundinFigures C.10 C.12 forthe MultiPisamplesandFigures C.13 C.15 forthe MultiPisamples. 5.3.3.3ComparisonofMultiTrackandMultiPisamples UsingtheRHCMultiPisamplesservesoneprimarypurposeintheoscillationanalysis,inthat thesesamplesmoreaccuratelymatchwhatisseenatSuper-Kamiokande(SK).Bywayofexample, asecondtrackinaneventintheCCN-trackssamplemaybetheresultofaprotoncrossingthe TPC.However,thefardetectorisunabletodetectprotonsbecausetheydonotdecayandtendnot tohaveenoughenergytomeettheCherenkovthreshold.WhereastheRHCMultiTracksamples havenocutsbasedonthenumberofpionsinanevent,theRHCMultiPisamplesaredesignedto bedi˙erentiatedbasedonthenumberofpionsobservedintheevent.TheSKsamplesaresimilarly categorized,asthepionscanbedetectedeitherthroughtheirdecayorthroughthepionhaving enoughenergytomeettheCherenkovthreshold. Additionally,thethreeRHCMultiPiaredesignedtobetterre˛ecthowthethreeprimary typesofinteractionsareobservedinthedetector.Whiletherearedetectore˙ectsthatcancause discrepanciesbetweenthetrueinteractiontypeandtheselectedsample,generally,CCQEevents areobservedasamuonwithnopionsinthedetector,CCresonantpionproductionhaveamuon andonepion,andDIS+N ˇ containstherestoftheobservedevents.Usingthesesamples,studies haveshowngoodagreementbetweenthetrueinteractionandtheselectedsample[ 6 ]. 96 FGD1RHC CC0 ˇ FGD1RHC CC1 ˇ FGD1RHC CCOther Figure5.11:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD1RHC MultiPiselections.Themomentumdistributionsareshownontheleft,whiletheangular distributionsareshownontheright. 97 FGD1RHC CC0 ˇ FGD1RHC CC1 ˇ FGD1RHC CCOther Figure5.12:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD1RHC MultiPiselections. 98 FGD1RHC CC0 ˇ FGD1RHC CC1 ˇ FGD1RHC CCOther Figure5.13:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD1RHC MultiPiselections. 99 FGD1RHC CC0 ˇ FGD1RHC CC1 ˇ FGD1RHC CCOther Figure5.14:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD1RHC MultiPiselections.Themomentumdistributionsareshownontheleft,whiletheangular distributionsareshownontheright. 100 FGD1RHC CC0 ˇ FGD1RHC CC1 ˇ FGD1RHC CCOther Figure5.15:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD1RHC MultiPiselections. 101 FGD1RHC CC0 ˇ FGD1RHC CC1 ˇ FGD1RHC CCOther Figure5.16:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD1RHC MultiPiselections. 102 CHAPTER6 ND280SYSTEMATICUNCERTAINTIESINTHENEARDETECTORFIT AstheeventselectionsdescribedinChapter 5 aredependentontheidenti˝cationofparticlesand theirkinematics,itisimportanttounderstandtheuncertaintyinmakingthesemeasurementsin thedetector.If,forexample,aparticleismisidenti˝edinthetimeprojectionchamber(TPC) oritsmomentumisincorrectlydetermined,thepredictedeventrateattheneardetectorcouldbe impacted,whichwouldthenbepropagatedtotheoscillationanalysisthroughtheBANFF˝tresults. ThesystematicuncertaintiesonND280willbedescribedindetailinthischapter. 6.1DetectorSystematicUncertaintyDetails Eventhoughmanypotentialsourcesofuncertaintyexistattheneardetector,thedetector systematicuncertaintiesassociatedwiththeeventselectionsdescribedinChapter 5 arethemost relevant.Themajorityofthesystematice˙ectsdescribedoverthecourseofthischapterareapplied tobothFHCandRHCsamples.However,aselectnumberarenotappliedtotheRHCMultiTrack samples,asthesesampleshavefewercutsrelatedtothe˝ne-graineddetectors(FGDs).Asummary oftheindividualuncertaintiescanbefoundinSection 6.3 ,aswellastheirimpactontheoverall detectorsystematicuncertaintyintheneardetector˝t.Thespeci˝cdetailsofeachuncertaintyare describedbelow. TPCFieldDistortions: Becausethemagnetic˝eldappliedinND280isnotcompletely uniform,˝elddistortionscanariseintheTPCs.Mappingdata,takenusingHallprobes intheneardetector,areusedtosetthemagnetic˝eldusedintheND280reconstruction [ 107 ].Byincludingthismapping,themagnetic˝elddistortionscanbeaccountedforduring thereconstructionprocessbyapplyingthe˝eldmaptotheTPChitpositionsfoundinthe reconstructionandevaluatingthechangeinthe y and z positions[ 108 ]. Furthermore,thevalidityofthemapistestedbycomparingtheresultstothosefoundusing 103 alasercalibrationsystemintheTPC.Thissystemusestheexpectedandmeasuredpositions ofphotoelectronsfromaluminumdotsilluminatedbyalaser.Thistestingisdoneboth whenthemagnetisturnedo˙andwhenitisturnedon.Basedonthesemeasurements,a secondcorrectionisappliedtothehitpositionsasafunctionofthedriftdistanceofthe photoelectrons. BothcorrectionswerecrosscheckedusingTPC2andTPC3,asTPC3isexpectedtohave largedistortionsduetoitspositionintheyoke.Thecrosschecksshowedthatincludingthe initialmagnetic˝eldcorrectionreducedtherelativemomentumbiasbetweenTPCs,whereas thesecondcorrectionincreasedthebias[ 108 ].The˝rstcorrectionisappliedduringthe reconstructionprocess,whilethesecondcorrectionisusedasasystematicuncertaintyon themagnetic˝elddistortionsinordertomitigatetheobservedbias. TPCMomentumScale: Inadditiontothesystematicuncertaintyforthemagnetic˝eld distortions,theTPCmomentumscalereliesuponthemagnetic˝eldatND280.Bychanging themagnitudeofthemagnetic˝eld,changescanariseintheTPCtrackmomenta,potentially resultinginmigrationbetween p cos binsintheselections.FourHallprobesinstalled atND280providescalingfactorsforthe˝eldstrengthintheMonteCarlo.Thesescaling factorsarenecessarybecausethenominalMCmagnetic˝eldissigni˝cantlydi˙erentfromthe measuredmapping[ 107 ].Theuncertaintyonthescalingfactorscomefromseveralsources includingtheintrinsicresolutionoftheprobes,potentialrelativemisalignmentbetweenthe probes,andthenonlinearnatureoftherelationshipbetweenthemagnetcurrentandthe˝eld magnitude. TPCMomentumResolution: Duetoanumberoffactors,includingdi˙erencesinthe electricandmagnetic˝elddistortionsattheneardetector,thereissomeuncertaintyinthe momentumresolutionfortracksintheTPCreconstruction.Whilethereisanuncertainty alreadyincludedforthemagnetic˝elddistortions,thereasonsbehindthedataMonteCarlo discrepancyinmomentumresolutionarenotwellunderstood.Forthisreason,aconservative 104 approachistakenandbothuncertaintiesareincluded[ 7 ]. InordertobetterunderstandthemomentumresolutionforthedataandMonteCarlo,astudy wasperformedwithacontrolsampleofeventsthathadtrackswhichcrossedmultipleTPCs [ 109 ].ByusingeventswithtracksinmultipleTPCs,thetrackmomentaineachTPCcan bedirectlycomparedwithoutusingtruthinformation.Themomentumuncertaintyisthen calculatedforasingleTPCsegment.GoodagreementisseenbetweensingleTPCsegments andtracksthatcrossmultipleTPCs;therefore,thesamefractionaldi˙erencesareusedfor trackscrossingmultipleTPCs. BecausetheMonteCarloshowsbettermomentumresolutionthanthedata[ 7 ],theinverse momentumintheMonteCarloissmearedtomatchthatseeninthedata.Thisisdoneusing thefractionaldi˙erenceinthemomentumresolutionbetweenthedataandMonteCarlo, binnedin x -positionalongtheTPC[ 7 , 109 ],asthisisthedirectionofthemagnetic˝eldin theTPC.Anuncertaintyof 0 : 1 wasusedforallscalingfactors,regardlessof x -position. TPCParticleIdenti˝cation: InordertodeterminetheparticleIDusingtheTPCs,a truncatedaverageofthechargecollectedintheTPCisusedtoperformparticlehypothesis test.TwoissuesareinvolvedinthedataMonteCarlodi˙erencesseenintheTPCPID.The ˝rstisduetothedi˙erencebetweenthemeanpullvalues,whilethesecondisduetotheratio betweenpullwidths.Theuncertaintiesontheseratiosarecalculatedforeachtypeofparticle andeachTPCandthenarebinnedinparticlemomentum.Anestimationofthesystematic biasisachievedthroughthedi˙erenceinthepullmeans,whileanestimateofthesmearing needingtobeappliedtotheMonteCarlocomesfromtheratioofthepullmeans. Inordertomeasurethesystematicuncertainties,eventsampleswithahighpurityofmuon, electron,andprotontracksarechosen.Theresultsfromthemuonstudiescanalsobeused forpions,duetomuonsandpionshavingsimilarenergylossintheTPCs.Betweenparticle types,nocorrelationisneeded,whileuncertaintiesbetweenthemomentumbinsandTPCs are100%correlatedforeachparticle.Furthermore,uncertaintiesonthepullmeanare100% 105 correlatedwiththeuncertaintyonthepullwidthforeachtypeofparticle. FGDParticleIdenti˝cation: Becausenon-interactingparticlesthatstopintheFGDslose alltheirkineticenergythroughionization,amassfortheparticlecanbedeterminedbyusing theenergydepositedalongtheFGDsegment.Apullfortheparticlehypothesiscanbe generatedbycalculatingthedi˙erencebetweenthemeasuredandpredictedenergydeposit atagivenrange,normalizedbythepredictedenergyspreadforthisrange.Theuncertainty onthemeasuredparticlepullcomesfromcorrectlychoosingtheparticlebasedonitspull. Furthermore,measuringthedepositedenergydependsontheuncertaintyfromthethickness ofthescintillatorcoatingandthechargereconstruction.AsthedatatoMonteCarlocharge reconstructionanddi˙erencesinthescintillatorcoatingthicknessarethesameforFGD1and FGD2,thissystematicuncertaintyiscorrelatedbetweentheFGD1andFGD2selections. StudiesfortheFGDPIDsystematicuncertaintyusedcontrolsampleswitheventscontaining asinglemuonandprotontrackswhichstoppedineitherFGD1orFGD2[ 110 ].Thecontrol sampleswerecreatedusingtheTPCPIDandthencomparedwiththeFGDPIDtocalculate thee˚ciencyforcorrectlyidentifyingamuonorproton.Thesystematicuncertaintyfor chargedpionidenti˝cationusesthesamevaluesasthosemeasuredformuons,becausethe FGDPIDdoesnotstronglydistinguishbetweenthetwo. FGDTimeofFlight: InordertodeterminethetrackdirectionandselectwhichFGDvolume theneutrinointeractedin,theaveragehittimebetweenFGD1andFGD2iscompared.A trackwhichcrossesbothFGDsisreconstructedasbackwardsgoingwhentheaveragehit timeinFGD1isatleastthreenanosecondsgreaterthantheaveragehittimeinFGD2[ 7 ]. Thesystematicuncertaintyonthetimeof˛ight,therefore,directlya˙ectswhetheranevent isreconstructedashavingoccurredinFGD1orFGD2. Thissystematicuncertaintydirectlyusestheanalysisneutrinosample,insteadofacontrol sample.EventswithtrackspassingthroughFGDsarethenselectedfromthissample.The ˝naluncertaintyforthetimeof˛ightis t 12 = 0 : 78 nsandisaddeddirectlytothe 106 reconstructedtimeof˛ightas t 12 ,where isarandomvariableassociatedwiththe systematicvariations.BecausethisuncertaintycomesfromtimingscalculatedinFGD1and FGD2,itisconsideredtobe100%correlatedbetweenselectionsforFGD1andFGD2. TPCClusterE˚ciency: Theuncertaintyoncorrectlyreconstructingacluster,orseries ofhits,intheTPCsarisesfromdi˙erencesinthereconstructione˚ciencyinthedataand MonteCarlo.Ifnotenoughclustersarereconstructedinanevent,thenthateventwillfailto passthe˝ducialvolumecutdescribedinChapter 5 .Whenthishappens,thedataandMonte Carlocangivedi˙erentfractionsofeventspassingtheinitialselectioncuts.Thee˚ciency anditsuncertaintyarecalculatedforreconstructedclustersinboththehorizontalandvertical direction.Becausetheuncertaintyforbothhorizontalandverticalclusterscomesprimarily fromtheunderlyinghite˚ciency,theuncertaintyiscorrelatedbetweenthehorizontaland verticalclusters. Todeterminetheclusterreconstructione˚ciencydi˙erence,twostudiesusingcontrolsam- pleswererun.Forthehorizontalclustere˚ciency,controlsamplesofcosmictriggerevents withverticaltrackscrossingTPC2wereused.Ontheotherhand,fortheverticalcluster e˚ciency,acontrolsampleofCCinclusiveeventsoriginatinginFGD1wasused.Thiswas toensuremostlyhorizontaltrackswerereconstructedwithverticalclusters.The˝nalcluster e˚cienciesarelistedinTable 6.1 . ClusterType ¹ MC data ºš MC data š MC Vertical 0.0011 0.00020.9989 0.0002 Horizontal 0.0007 0.00010.9993 0.0001 Table6.1:Resultsforthedi˙erenceintheTPCclustere˚ciencybetweendata( data )andMC ( MC ).Tablefrom[ 7 ]. TPCTrackReconstructionE˚ciency: Thissystematicuncertaintydescribesthee˚ciency oftheTPCreconstructionalgorithmsuccessfullyreconstructingthetracksfromparticles 107 crossingtheTPCs.Failingtoproperlydothiscanleadtochoosingtheincorrecteventtopol- ogyforanevent,resultinginamiscountingofthetotalnumberofeventsintheCCinclusive sampleandmigrationofeventsbetweenthesubsamples.TheTPCtrackreconstruction e˚ciencyisfullycorrelatedbetweenthethreeTPCs. Thee˚ciencyiscalculatedfromstudiesusingcontrolsamplesofthrough-goingmuonsfrom beamandcosmicevents.Theseareusedtomeasurethereconstructione˚ciencywhenno othertracksareintheTPC.Inordertostudythee˙ectoftracklengthonthereconstruction e˚ciency,thee˚ciencyfromcosmicmuonswithbarrelelectromagneticcalorimetertracks areused[ 7 ].Thee˚ciencyisthencalculatedseparatelyforeachTPCandareshownin Table 6.2 .Thestudiesofthereconstructione˚ciencyrevealahighoveralle˚ciencywith nomomentumorangulardependence. TPC1TPC2TPC3 DataE˚ciency 99.9 + 0 : 1 0 : 1 %99.7 + 0 : 2 0 : 7 %99.3 + 0 : 5 0 : 2 % MCE˚ciency 99.6 + 0 : 2 0 : 3 %99.5 + 0 : 3 0 : 4 %99.8 + 0 : 1 0 : 2 % Table6.2:TPCtrackreconstructione˚cienciesfordataandMonteCarlo.Tablefrom[ 8 ]. TrackChargeIdenti˝cation: InordertodeterminethechargeofthetracksintheTPC, thereconstructionusesinformationaboutthetrackfromthefulldetector,whilecharge identi˝cationintheTPCandadditionalinformationfromtheoveralltrackreconstruction isusedtodeterminethesignofthecharge[ 111 ].Twodi˙erentsourcescontributetothe uncertaintyontheTPCchargeidenti˝cation:theprobabilityofgettingtheTPCcharge incorrectandtheprobabilityofincorrectlyswitchingtheTPCchargesignintheoverall chargeidenti˝cation.Thee˚ciencyiscalculatedastheprobabilitythattheoverallchargeis di˙erentfromtheTPCreconstructedcharge. Studiesofthissystematicuncertaintyweredoneusingacontrolsampleoftracksstartingin thePØDandhavingatleastoneTPCsegment[ 111 ].Thesestudiesshowedtheuncertainty 108 onthechargeidenti˝cationdependsonthereconstructedmomentumerror.Duetothis dependence,theuncertaintyis100%correlatedbetweenthedi˙erentcasesaswellasbeing 100%correlatedbetweentheFGD1andFGD2selections. FGD-TPCMatchingE˚ciency: ThissystematicuncertaintycharacterizeshowwellFGD tracksarematchedtoTPCtracksduringreconstruction.Iftracksareincorrectlymatched, itispossibletolocalizethetrackvertexatthewrongposition,leadingtoamiscountinthe numberofCCinclusiveinteractionsoccurringintheFGDs.Thee˚ciencydependsonthe abilitytomatchaTPCtracktoasinglehitinanadjacentFGDaswellastheabilityto correctlymatchaFGD-TPCmatchedtrackwithanupstreamTPCtrack.Uncertaintydueto incompletematchingfromoutof˝ducialvolume(OOFV)eventmigrationisnotincluded, butreceivesitsownsystematicuncertaintyandisdiscussedbelow. Thee˚ciencyforthisuncertaintywascalculatedusingacontrolsampleofcosmicevents withlargeangleswithrespecttotheneutrinobeamthathadonlyoneFGD1TPC2FGD2 track.Thestudiesshowedthee˚ciencyforFGD-TPCmatchedtrackstobe100%fortracks thathadmorethantwohitsintheFGDandhadanupstreamTPCtrack.Thiswasconsistent betweenthetwoFGDs.FortrackswithtwoorlessmatchedhitsintheFGDs,thee˚ciencies anduncertaintiesareshowninTable 6.3 .Thissystematicuncertaintyis100%correlated betweenFGD1andFGD2,becausethee˚ciencyonlypertainstothe˝naltwolayersineach FGD,whichsharethesameuncertaintiesinthescintillatorbarproperties. FGD1FGD2 DataE˚ciency 96.9 0.8%96.5 0.85% MCE˚ciency 97.6 0.45%97.6 0.50% Table6.3:E˚cienciesforFGD-TPCmatchedtrackswithtwoorlessreconstructedhitsinthe correspondingFGDfordataandMonteCarlo.Tablefrom[ 9 ]. FGDHybridTrackE˚ciency: TheFGDhybridtracke˚ciencysystematicuncertainty characterizesthee˚ciencyofcorrectlyreconstructingatrueFGD-onlytrackinthepresence 109 ofaFGD-TPCmatchedtrack(anypertinentFGD-onlytracksintheselectionshaveanFGD- TPCmatchedtrack,byde˝nition).FGD-onlytrackswithoutanaccompanyingFGD-TPC matchedtrackarenotconsideredinthise˚ciency.BecauseFGD-onlytracksareassumed tobeeitherprotonsorpions,thestudiesforthisuncertaintywereperformedseparatelyfor eachparticletype. Inordertostudythee˚ciency,acontrolsamplewitheitheronereconstructedtrackentering theTPCortwotrackswhichbothentertheTPCwasused.Theybrportionofthe namecomesfromtheuseofaparticleguninGEANT4togeneratestoppingprotonand piontracksfortheFGD-onlytracks[ 110 ].Theseparticlesweregeneratedwithauniform energydistributionandscatteredisotropicallythroughoutthedetector.Alowerboundof 400MeV/cwassetfortheprotonmomentumduetodi˚cultiesreconstructingeventsbelow thatmomentum.Thehitsfromthesetrackswerethenaddedtothecontrolsampleeventsfor bothdataandMonteCarloandtheFGD-onlytrackreconstruction.Afterthereconstruction hadbeenrun,thee˚ciencywasde˝nedasthenumberofselectedeventshavingatleastone reconstructedFGD-onlytrackdividedbythetotalnumberofselectedeventsinthecontrol sample.BecausetheFGD-onlytrackreconstructionhasastrongdependenceontheangle ofthetrack,thee˚ciencyisbinnedin cos ,where isthetrueanglebetweenthemuon candidateandtheFGD-onlytrack.Theoverallsystematicuncertaintyforthepionandproton studieswaslessthan8%forthetracke˚ciency[ 110 ]. MichelElectronE˚ciencyandPurity: ThesystematicuncertaintyontheMichelelectron selectioncomesfromthee˚ciencyofdetectingtheMichelelectronandthepurityofthe cut.Thee˚ciencytodetectaMichelelectronisdependentontheprobabilitytheelectron producesenoughhitsintheFGDtopassthecut.CosmicmuonsstoppingineitherFGD1 orFGD2wereusedasacontrolsampletostudythise˚ciency.Becausemuonswithlarge anglesdonotalwayspassthroughaTPC,aPIDcutbasedonthelengthofthetrackandits momentumwasusedtoidentifythemuons[ 10 ]. 110 Inthesestudies,thee˚ciencywascalculatedseparatelyforT2Kruns24usingdatawith cosmictriggers.Thee˚ciencyisthende˝nedastheprobabilitytodetectanexpectedMichel electronduetotherebeingastoppedmuonintheFGD.WhereastheMichelpurityrelieson thebeampowerofeachrun,thee˚ciencydoesnot;therefore,theaveragee˚ciencyofthe samplesisusedfordataatallbeampowers.Finale˚ciencyvaluesareshowninTable 6.4 . FGD1FGD2 Data 56.4 1.6%42.8 1.1% MC 56.5 1.9%41.4 1.4% Table6.4:E˚cienciesforthedetectionofMichelelectronsfordataandMonteCarlo(MC).Table from[ 10 ]. ThepurityoftheMichelelectronsamplesislargelydependentonbackgroundfromout- of-FGDbeaminteractionsinthedetectorandmagnetvolumes,sandmuoninteractions, andcosmicmuonsbeingidenti˝edasMichelelectrons.BecausetheMichelelectroncut isusedtoseparateCC 0 ˇ eventsfromnon-CC 0 ˇ events,misidentifyingaMichelelectron cancauseaCC 0 ˇ eventtobelabeledasaCC 1 ˇ orCCOtherevent.Inordertostudythe externalbackgroundrate,emptybeamspillswereselectedtoremoveanyproductsfrom interactiondecaysintheFGDs.Furthermore,themeasurementwasdoneseparatelyforeach run,becausetherateisdependentontheaveragebeampower,whichcandi˙erbetween runs.Thisresultsinacleanselectionoftrueexternalbackgroundevents,whichwillgivea falseMichelelectronrateproportionaltothatwhichisinthephysicsselections.Thesame selectionswereusedforbothdataandMonteCarlo,becausenotruthinformationisneeded. Astheaveragebeampowerchangesfromruntorun,theoverallexternalbackgroundratein bothdataandMonteCarloalsochanges.Betweenruns3and4inthedata,thebeampower increasedfordata,whereasitremainedfairlyconsistentfortheMonteCarloleadingtothe di˙erenceinthebackgroundeventrateshowninTable 6.5 .Itshouldbenotedtheproduction ofMonteCarlooccursinfrequently,soanaveragebeampowerischosenduringproduction. 111 Ingeneral,thebackgroundratesfordataarehigherthaninMonteCarloforbothFGDs.This isduetothrough-goingcosmicmuonsleavingmanyhitsintimebinsbetweenbunches.In thebeamandsandMonteCarlo,cosmicmuonsarenotsimulated,leadingtoadiscrepancy inthehitdistributionbetweenthedataandMonteCarloandresultsinconsistentlylower overallratesinsimulationthanindata. FGD1 FGD2 MC( 10 3 º Data( 10 3 º MC( 10 3 º Data( 10 3 º Run2 0.603 0.0081.61 0.05 0.314 0.0061.21 0.05 Run3 0.714 0.0091.80 0.04 0.397 0.0061.31 0.03 Run4 0.729 0.0092.00 0.07 0.387 0.0061.50 0.06 Table6.5:DataandMonteCarlo(MC)ratesforfalseMichelelectronidenti˝cation.Therateis de˝nedasthenumberofexpectedfalseMichelelectronsperspill.Tablefrom[ 7 ]. FGDMasses: ThesystematicuncertaintyoftheFGDmassesiscausedbydi˙erencesinthe simulatedandrealdetectorvolumesofthescintillatorandwatermodulearealdensities[ 11 ]. BecausethewatermodulesareonlyinFGD2,theuncertaintyonthewatervolumedensity onlyappliestoFGD2,whiletheuncertaintyinthescintillatorXYmodulesisapplicableto bothFGDs.TheFGDmassuncertaintyisimplementedasanuncertaintyonthedensitiesof themodules,includingthecoatingofthescintillatormodulesandthepolycarbonatevessels forthewatermodules. ThetotaluncertaintyfortheXYmodulesisacombinationofthedataMCdi˙erencein massanddirectmassmeasurementsoftheXYmodules,whichcanbeseeninTable 6.6 .On theotherhand,thetotaluncertaintyinthewatermodulemassisacombinationofthedata MCdi˙erenceandanuncertaintyofeachwatermodule,whichisdominatedbythemasses oftheplasticandglue,andcanalsobeseeninTable 6.6 .Thewaterandscintillatormass uncertaintiesaretreatedasseparate,becausetheuncertaintyisassignedbasedonthetrue vertexlocation[ 7 ].Thecorrelationsbetweenthedi˙erentcomponentscanbeseeninTable 6.7 . 112 Module Di˙erenceUncertainty Total XY 0.41%0.38% 0.6% Water 0.26%0.46% 0.55% Table6.6:DataMonteCarlo(MC)di˙erence,theuncertaintyduetodirectmeasurements,and thetotaluncertaintyfortheXYandwatermodulesintheFGDs.Tablefrom[ 11 ]. FGD1totalFGD2totalFGD2water-likeFGD2XY-like FGD1total 10.7000.84 FGD2total 0.7010.500.91 FGD2water-like 00.5010.26 FGD2XY-like 0.840.910.261 Table6.7:Correlationsinuncertaintiesbetweenmasscomponents.water-likeand XY-likerefertothewatermoduleandXYmoduleportionsofFGD2,respectively.Table from[ 7 ]. PionSecondaryInteractions: WheneverpionsareproducedininteractionsatND280,there isachancetheyinteractwithinthedetectorproducingdetectionine˚cienciesforpionsinthe FGDs.Thisprocesscanoccurthroughabsorption,decay,quasi-elasticscattering,andother methods.Theseprocessesarecollectivelyknowasyinteractions(SI).Because pionsarethedistinguishingfeaturewhende˝ningwhichtopologyaneventfallsunderduring theselectionprocessdescribedinChapter 5 ,itisvitalsecondaryinteractionsareproperly modeledintheMonteCarlo.IthasbeenshownthatthemodelusedbyGEANT4forpion interactionsdoesnotmatchwellwithexistingdataofpioninteractionsonnuclei[ 112 ]. Therefore,acorrectionweightisappliedtotheseevents.Theuncertaintyiscalculatedusing studiesthatcompareND280dataandMonteCarloaswellasuncertaintiesfromexternaldata ofpioninteractions.Thecalculatedeventweightisbasedontheprobabilitiesofeachtrue piontrajectoryinaneventinteractingfordi˙erentcrosssectionmodels.Theprobabilities areenergydependentandcalculatedinstepsof0.1mmalongthetrajectory.Thisisdone separatelyforeachtargetnucleus,wheretheuncertaintyistreatedasindependentforeach target,but100%correlatedbetweenFGD1andFGD2foreachtarget. 113 ProtonSecondaryInteractions: Similartopions,anoutgoingprotonfromaninteraction inND280hastheabilitytogothroughsecondaryinteractionsinthedetector.Studiesusing acontrolsampleofalltrueprotonsoriginatingfromtheneutrinointeractionandhaving undergoneaninteractioninthevolumeofinterestwereperformed.Theuncertaintyin thecrosssectionfortheprotonsecondaryinteractionsistakenintoaccountbyapplyinga weightdependentonhowfarthetheprotontraveledbeforeinteractingandthedensityofthe scatteringmedium.Thee˙ectofthissystematicuncertaintyissmall[ 113 ],butaconservative approachwastaken,soitisincludedinthisanalysis. SandMuonBackground: TheprimaryMonteCarloforND280onlyincludesbeamevents occurringintheND280subdetectorsandthesurroundingmagnet.However,interactions frombeamneutrinoscanoccurinthesandoutsidethedetectorpitaswellasinthepitwalls whereND280issituated.Theseinteractionscanlooksimilartointeractionsoccurringinthe FGDsandprovideabackgroundtotheanalysis.Sucheventsarecalledsandmuons,because theprimarysourceoftheseeventsisfrominteractionsinthesurroundingsand.Adedicated MonteCarlosampleisproducedforsandmuoneventstoestimatethee˙ectsontheanalysis samples,astheseeventsarenotincludedintheprimaryMonteCarlo. TheeventrateofsandmuonsisestimatedbyrunningtheselectionsdescribedinChapter 5 overthesandmuonMonteCarlo,wherethedataPOTisscaledtogetthepredictedsand muoneventrate.Theuncertaintyonthisbackgroundisestimatedasthedi˙erencebetween theMonteCarloanddataratesandisdoneseparatelyforpositivelyandnegativelycharged tracks.Thesystematicuncertaintywasdeterminedtobe10%forneutrinomodeand30% forantineutrinomode[ 7 ].Nocorrelationisappliedbetweenbeammodesaswellasbetween FGD1andFGD2,astheunderlyingcausefortheuncertaintyisduetotheinteractions occurringoutsidethedetectorvolume. OOFVBackground: Theout-of-˝ducialvolumebackgroundsystematicuncertaintycovers interactionsthatoccuroutsideoftheFGD˝ducialvolumes,butisreconstructedasbeing 114 insideeithertheFGD1orFGD2˝ducialvolume.Thisincludesinteractionsinoneofthe othersubdetectorsorinthe˝rsttwolayersorFGD1orthe˝rstlayerofFGD2.Therearetwo maincontributinguncertaintiesonthebackgroundrates,theuncertaintyontheinteraction rateswithintheothersubdetectorsandtheuncertaintyonspeci˝cclassesofreconstructed eventsintheFGDs. Theuncertaintyontherateisdividedintofourcategoriesbasedonwherethetruevertex oftheinteractionoccurred,whetheritwasinthePØD,theECal,theSMRD,orother subdetectors.Therelativedi˙erenceintheinteractionratebetweenthedataandMonte Carlowasstudiedusingneutrinobeamdata,insteadofcontrolsamples.Thedi˙erenceis thenusedastheuncertaintyforeachoftheinteractionrates,whichareshowninTable 6.8 . Aseachinteractionisunrelatedtotheothers,theyareuncorrelatedwitheachother.Dueto theinteractionsoccurringoutsideoftheFGDs,theunderlyingsourcesoftheuncertainties arethesameforFGD1andFGD2,sothesystematicuncertaintyisfullycorrelatedbetween thetwoFGDs. BackgroundOrigin FHC RHC RHC PØD 5.1%8.4%5.4% ECal 11.6%8.8%6.7% SMRD 4.9%6.7%4.8% Other 13.6%13.5%24% Table6.8:UncertaintiesontheOOFVratesbasedontheirsubdetectororigin.Tablefrom[ 12 ]. Thereareanumberofdi˙erentcategoriesofthereconstructionuncertainties;however,notall ofthecategorieshaveasigni˝cantuncertainty,eveniftheycontributetothetotalbackground rate.Anexampleofthisisaneutralparticle(whichwillnotleaveatrackinthedetector) enterstheFGD,theninteractsandcreateschargedsecondaryparticles.Thecategorieswhich haveasigni˝cantsourceofuncertaintyincludeeventsstartinginthetrackercomponents downstreamoftheFGD,highangleevents,eventswithahardscatterintheFGD,events 115 wherehitsarenotdepositedfortwoconsecutivelayers,andtmodulefailureevents, whereeventsarenotmatchedbecausemosthitsweremissingintheFGD. Typically,thereconstructionrateanduncertaintyarenotdependentonthetrackmomentum, sothesystematicuncertaintycanbecalculatedseparatelyforeachcategory[ 12 ].Each categoryissu˚cientlydi˙erentfromtheothersthatnocorrelationisassumedbetweenthem. Withineachcategory,FGD1andFGD2arefullycorrelated,becausethecategoriesrely mainlyonthehite˚cienciesoftheFGDs.Theuncertaintieswerecalculatedusingevents whichwereknowntohavestartedoutsidetheFGD˝ducialvolume,suchascosmicevents, andcomparingtheratesforthedataandMonteCarlo.Thenon-zerocategoriesareshownin Table 6.9 . Category FGD1FGD2 DownstreamEvent 5%5% HighAngleEvent 33%28% LastModuleFailure 35%17% ConsecutiveSkippedLayers 55%82% HardScattering 32%21% Table6.9:ReconstructionuncertaintiesforOOFVbasedontheirreconstructioncategory.Table from[ 12 ]. EventPileUp: Eventpileupistheresultofanout-of-˝ducialvolumeeventbeingcoincident withanin-˝ducialvolumeCCinclusiveeventineitherFGD.ThiscanleadtoCCinclusive eventsbeingthrownoutfromtheselectionduetotheexternalvetocutdescribedinSection 5.3.1.1 .TheprimarysourceofpileupcomesfromsandmuonsforbothFHCandRHC selections[ 7 ].Theuncertaintyontheeventpileupisdependentonthebeamintensityand mode.Whilethesourceoftheuncertaintyisconsistentfordi˙erentintensitiesforneutrino orantineutrinomode,itisnotnecessarilysimilarbetweenthetwomodes.Therefore,thetwo beammodesaretreatedasuncorrelated,whilethebeamintensityis100%correlatedwithin agivenbeammode. 116 Aswiththesandmuonsystematicuncertainty,acorrectionisappliedtotheMonteCarlo toaccountforeventpileup.Becauseofthe10%uncertaintyforFHCand30%uncertainty forRHCfromthesandmuonbackgrounduncertainty,aswellasotherpotentialdi˙erences betweenthesandmuonMonteCarloanddata,asystematicuncertaintyisappliedonthe correction.Thisuncertaintyisbasedonthedi˙erencebetweendataandsimulation,which iscalculatedbycomparingthenumberofTPC1orTPC2eventsperbunchinthedataand simulation. 6.2PropagationoftheDetectorSystematicUncertainties InordertoperformtheselectionsdescribedinChapter 5 ,evaluatedetectorsystematicuncer- tainties,andstorerelevanteventinformation,ND280analysesuseaspecializedsoftwarepackage calledPsyche(PropagationofSYstematicsandCHaracterizationofEvents).ThePsychepackage wasdevelopedspeci˝callyforusewithND280events.Itisabletouseboththereconstructed andtrueeventinformationfromMonteCarloaswellasthereconstructedinformationfromdata events.Whiletheselectionsonlyusereconstructedinformationfromtheevents,thecalculationof thesystematicuncertaintiesusestrueeventinformationandisonlyusedwithMonteCarloevents. Psychedoesnotperformany˝tsitself,ratheritisdesignedtobeusedwithintheND280˝tsoftware andisoptimizedforlowmemoryuseandspeed.ThissectionwilldiscusshowPsychegoesabout calculatingandpropagatingthedi˙erentdetectorsystematicuncertainties. Therearetwotypesofsystematicuncertainties,dependingonhowtheuncertaintyispropagated totheeventspassingtheselectioncuts.Observablevariationsystematicuncertainties,described inSection 6.2.1 ,varythereconstructedquantitiesinevents,allowingeventstomoveinandoutof selections.Thesearepropagatedbyrerunningtheselectionusingthevariedquantitiesinorderto determinethee˙ectonthenumberofselectedevents. Ontheotherhand,weightsystematicuncertainties,asdescribedinSection 6.2.2 ,applyaweight toeachsimulatedevent,withoutalteringthereconstructedquantitiesthemselves.Theweight uncertaintiesaresplitintotwogroups,e˚ciency-likeandnormalizationweights.E˚ciency-like 117 weightsarerelatedtoareconstructionordetectorprobability,wheretheweightiscomputedfrom theprobabilityafterrunningtheselectiononce.Normalizationweightsgenerallyareassociated withthesubsamplesoftheselectionandscaletheeventrateupanddown. 6.2.1ObservableVariationSystematics Asmentionedpreviously,observablevariation,orvariation,systematicuncertaintiesarethose whichalloweventstomoveintoandoutofselectionsand,inrarecases,areabletoshiftaneventout oftheinclusiveselection.WhencomparingthedataandMonteCarlo,di˙erencesbetweenthemean valuesofthesedistributionscanbeseen.Thevariationsalterthereconstructedquantities,suchas thetrackmomentumandangle,inordertocorrectfortheobserveddi˙erence.Theuncertaintieson thesystematicsarepropagatedbysmearingtherelevantreconstructedvariablesandthenrerunning theselectiononthesmearedevent.Dependingonthefeaturesoftheuncertainties,thesmearing canbeimplementeddi˙erently.Therearefourmainclassesofuncertaintiesfortheobservable variations: 1. Whenthetrueobservableisknown,thedi˙erencebetweenthereconstructedvariable, x MC rec , anditstruevalue, x true ,isrescaled.Thissmearedvalueisgivenby, x 0 rec = x true + ¹ x MC rec x true º¹ s + s º ; (6.1) where isarandomvariablecorrespondingtotheprobabilitydistributionfunction(PDF) ofthesystematicuncertainty, s isascalefactorde˝nedby ˙ data x š ˙ MC x ,and s isde˝nedas s = s ˙ data x ˙ data x ˙ MC x ˙ MC x : (6.2) Thisisusedforthemomentumresolutionandtheresolutionofthemeasureddeposited charge. 2. WhenacorrectionmustbeappliedtotheMonteCarloobservablevaluetomatchthemean di˙erencebetweenthedataandMonteCarlovalues, x .Thealteredvalue,inthiscase, 118 becomes x 0 rec = x MC rec + x + x : (6.3) isarandomvariableassociatedwiththecorrespondingPDFand x isthesystematic uncertainty,de˝nedas x = q x 2 + ¹ x data rec º 2 + ¹ x MC rec º 2 ; (6.4) where x data ¹ MC º rec isthemeanerrorinthedataorMonteCarlosample.Thisapproachis usedforthemeandi˙erenceofthedepositedchargepull.Inthecaseofthetimeof˛ight systematicuncertainty,asimilarprincipleisused.However,theMCisnotcorrectedby x .Theuncertaintyforthissystematicisthequadraticsumbetween x andtheerrorofthe timingdi˙erencedistribution, x = q x 2 + ¹ ˙ data x ˙ MC x º 2 : (6.5) 3. Whenthereconstructionisaltered, x alter rec ,toaccountforachangeintheunderlyingparameter, thenewvalueis x 0 rec = x MC rec + ¹ x alter rec x MC rec º : (6.6) Thismethodisusedforthemagnetic˝elddistortions.Forthemagnetic˝elddistortions, isassumedtobearandomvalueinauniformdistributionbetween 0 and 1 . 4. Whentheobservabledependsonascale, s ,thealteredvalueis x 0 rec = x MC rec + x MC rec s ; (6.7) where x MC rec s istheerroroftheobservable.Thisprimarilydealswiththemomentumscale systematicuncertainty,whichreliesonthemagnetcurrentasitserror. 6.2.2WeightSystematics Whereasvariationsystematicshavetheabilitytomoveeventsbetweenselections,weightsystem- aticsonlya˙ecttheweightgiventoaparticulareventanddoesnotchangeanyofthereconstructed 119 eventvalues.Inthecaseofe˚ciency-likesystematics,studiescomparingdataandMCpredictions ofwellknowncontrolsamplesareusedtodeterminetheeventweights.Forexample,tracking andmatchinge˚cienciescanbecomputedusingredundancybetweendetectors,suchasusing trackswithsegmentsinFGD1andFGD2tocalculatetheTPC2tracke˚ciency.Ratherthan usinganalysissamples,whichtypicallydonotmeetthespecialrequirementsneededtocalculate thesesystematics,controlsamplesareusedinstead.However,thecontrolsamplesdonotalways accuratelyre˛ectthecomplexityoftheanalysissamplesandtendtocoveralimitedphasespace. Therefore,thee˚cienciescalculatedusingthecontrolsamplesdonotmatchexactlythoseofthe analysissamples,soamodelmustbeusedtoextrapolatethecontrolsamplee˚ciencytotheanalysis sample.AreasonableassumptiontouseinthismodelisthattheratiobetweenthedataandMC e˚cienciesisthesameforbothtypesofsamples[ 7 ]. Thepredictede˚ciencyofthedataanalysissampleis = CS data CS MC MC ; (6.8) wherethe CS data ¹ MC º arethee˚cienciesdeterminedfromthecontrolsamplesand MC isthe e˚ciencyoftheMCanalysissample.TheMCanalysissamplee˚ciencycanbedeterminedusing truthinformation.Becausethereisalimitedprecisiononthee˚cienciesofthecontrolsamples, theirstatisticalerrormustbetakenintoaccountwhenpropagatingtheseuncertainties.Therefore, thesystematicuncertaintyrepresentsthestatisticalerrorofthesamplesaswellasthedi˙erence betweendataandMonteCarlo.Ingeneral,thepredictede˚ciencycanthenbegivenby 0 data = ¹ r CS + r CS º MC ; (6.9) where r CS = CS data š CS MC and r CS = q ¹ 1 r CS º 2 + ¹ r CS stat º 2 ,where r CS stat isthestatisticalerror onthee˚ciencyratio. isarandomvariablefromaGaussiandistributionofmean, 0 ,andstandard deviation, 1 .Fordeterminingtheeventweight,therearetwode˝nitionsthatareused,depending onwhetheritisappropriatetoapplyane˚ciencyoranine˚ciency.Thee˚ciencyweight, w eff , is w eff = 0 data MC ; (6.10) 120 whiletheine˚ciencyweight, w ineff ,is w ineff = 1 0 data 1 MC : (6.11) Normalizationsystematicuncertainties,ontheotherhand,arerelatedtothetotaleventnormal- ization.Examplesoftheseuncertaintiesaretheeventpileuporsandmuonbackground,which correspondtoout-of-˝ducialvolumeeventscoincidentwithin-˝ducialvolumeeventsorevents occurringinthesandsurroundingthedetector,respectively.Theeventsareweightedaccordingto variationssuggestedbysystematicerrorstudiesandaregivenby w e = 1 + e cat ; (6.12) where w e istheeventweightforthegivensystematicuncertainty, istherandomvariablegiven bythePDFoftheuncertainty,and e cat isthesystematicerrorassociatedwiththecategoryof events.Ifaneventisnotpartofthecategoryofeventsinquestion,itisgivenaweightof 1 : 0 . 6.3SummaryoftheIndividualSystematicUncertainties Theneardetector˝tusessixobservablevariationandtwelveweightsystematicuncertainties. Theseeighteenuncertaintieswillbrie˛ybedescribedSection 6.1 .Table 6.10 liststheseuncertain- ties,theirtype,thetypeofPDFused,theirpropagationmodel,thetypeofdatausedtocalculate theuncertainty,andwhichsetofsamplestheuncertaintiesareappliedto. Ingeneral,manyoftheseuncertaintiescontributesmallamountstotheoveralldetectorsys- tematicuncertainty,ascanbeseeninTables 6.11 and 6.12 .Byfar,thelargestcontributortothe overalluncertaintyisthepionsecondaryinteractionsuncertainty,duetothelargedi˙erenceseen inthedataandMonteCarlo. 6.4TheObservableNormalizationMatrix AsdiscussedinSection 4.3.3 ,theneardetector˝tdoesnotusethedetectorsystematicun- certaintiesdirectlyforcomputationalreasons.Rather,theseparametersare˝tusingobservable normalizationparameters,whicharebinnedin p cos foreachsample,andacovariancematrix 121 Uncertainty TypePDFPropagationModelDataUsedSamples TPCFieldDistortions VariationUniformObservableVariationCalibrationPi/Track TPCMomentumScale VariationGaussianObservableVariationCalibrationPi/Track TPCMomentumResolution VariationGaussianObservableVariationControlSamplesPi/Track TPCPID VariationGaussianObservableVariationControlSamplesPi/Track FGDPID VariationGaussianObservableVariationControlSamplesPionly FGDTimeofFlight VariationGaussianObservableVariationPhysicsSamplesPi/Track TPCClusterE˚ciency WeightGaussianE˚ciency-likeControlSamplesPi/Track TPCTrackReconstructionE˚ciency WeightGaussianE˚ciency-likeControlSamplesPi/Track TrackChargeIdenti˝cation WeightGaussianE˚ciency-likeControlSamplesPi/Track FGD-TPCMatchingE˚ciency WeightGaussianE˚ciency-likeControlSamplesPi/Track FGDHybridTrackE˚ciency WeightGaussianE˚ciency-likeControlSamplesPionly MichelElectronE˚ciencyandPurity WeightGaussianE˚ciency-likeControlSamplesPionly FGDMasses WeightGaussianNormalizationExternalDataPi/Track PionSecondaryInteractions WeightGaussianNormalizationExternalDataPi/Track ProtonSecondaryInteractions WeightGaussianNormalizationExternalDataPi/Track SandMuonBackground WeightGaussianNormalizationSimulationPi/Track OOFVBackground WeightGaussianNormalizationPhysicsSamplesPi/Track EventPileUp WeightGaussianNormalizationSimulationPi/Track Table6.10:ThedetectorsystematicuncertaintiesusedwithintheBANFF˝t.The˝nalcolumn showswhichsetsofsamplestheuncertaintiesapplyto.ThesampleslistedasapplytoMultPi samples,whileTracappliestotheMultiTracksamples. whichdescribestheuncertaintiesandcorrelationsinthesebins.Thebinningfortheobservable normalizationparametersandcovarianceismorecoarsethanthe˝tbinningdescribedinChapter 5 forcomputationalandtimeconsiderations.Theselectionbinningusedinthe˝tisdesignedto haveallnon-zerobinsfortheMonteCarloprediction,whichwouldgiveover800normalization parametersfortheFGD1FHCCC 0 ˇ samplealone.Ifthebinningfortheobservablenormalization parameterswasthesameasthe˝tbinning,therewouldbeover4000nuisanceparametersjust forthedetectorportion.Thetimerequiredto˝tsuchalargeparameterspacequicklybecomes prohibitive.Therefore,acoarserbinningisusedtoreducethenumberofparametersinthe˝t. Fortheneutrinomodesamples,the p cos binningsare: ‹ CC 0 ˇ binedges: p [MeV/c]:0,200,400,550,600,650,700,750,900,1050,2500,5000,30000 cos :-1.0,0.6,0.83,0.89,0.94,0.975,0.985,0.99,0.995,1.0 122 SystematicUncertainty InclusiveCC 1 ˇ CC 0 ˇ CCOther DETECTORVARIATIONSYSTEMATICS TPCFieldDistortions 0.03930.02450.06300.0720 TPCMomentumScale 0.08770.06210.07370.2295 TPCMomentumResolution 0.08230.05490.09450.2861 TPCPID 0.34280.31600.79230.6163 FGDPID 0.00020.01100.03390.0149 FGDTimeofFlight 0.03810.03440.07040.0178 EFFICIENCY-LIKESYSTEMATICS TPCClusterE˚ciency 0.00060.00040.00060.0019 TPCTrackReconstructionE˚ciency 0.42210.25920.43981.7861 TrackChargeIdenti˝cation 0.12760.17820.27040.4732 FGD-TPCMatchingE˚ciency 0.22980.14800.27030.6046 FGDHybridTrackE˚ciency 0.03850.10620.09990.5327 MichelElectronE˚ciencyandPurity 0.00110.06220.25290.0076 NORMALIZATIONSYSTEMATICS FGDMasses 0.59260.59450.58180.5967 PionSecondaryInteractions 2.12451.43323.17316.1183 OOFVBackground 0.39750.39070.54090.2858 EventPileUp 0.11170.11170.11170.1117 ALL AllMagnetUncertainties 2.29271.65823.32766.4665 SandMuonBackground 0.06710.06920.08490.0309 TOTAL 2.29371.65973.32876.4666 Table6.11:IntegrateduncertaintyforeachofthesystematicuncertaintiesinFGD1.Theproton secondaryinteractionssystematicuncertaintyisnotincludedasithasasmalle˙ectonthe sample.Tablefrom[ 7 ]. ‹ CC 1 ˇ binedges: p [MeV/c]:0,400,500,600,650,700,800,900,1000,3000,5000,30000 cos :-1.0,0.6,0.85,0.88,0.9,0.92,0.93,0.97,0.995,1.0 ‹ CCOtherbinedges: p [MeV/c]:0,500,600,650,700,900,1100,3000,500,30000 cos :-1.0,0.6,0.95,0.98,0.99,0.995,1.0 Thebinningsfortheantineutrinomodesamplesare: 123 SystematicUncertainty InclusiveCC 1 ˇ CC 0 ˇ CCOther DETECTORVARIATIONSYSTEMATICS TPCFieldDistortions 0.09710.08220.15810.0989 TPCMomentumScale 0.07910.04780.07800.2343 TPCMomentumResolution 0.10820.07860.13540.3404 TPCPID 0.45380.42731.21790.7854 FGDPID 0.00030.00880.03220.0185 FGDTimeofFlight 0.07830.07350.07210.1130 EFFICIENCY-LIKESYSTEMATICS TPCClusterE˚ciency 0.00060.00040.00050.0014 TPCTrackReconstructionE˚ciency 0.52310.46240.69840.6801 TrackChargeIdenti˝cation 0.09350.12200.07660.0777 FGD-TPCMatchingE˚ciency 0.28500.22130.31860.6063 FGDHybridTrackE˚ciency 0.00300.00840.03200.0864 MichelElectronE˚ciencyandPurity 0.00430.09160.42940.0065 NORMALIZATIONSYSTEMATICS FGDMasses 0.38930.38790.38880.3984 PionSecondaryInteractions 2.05181.43503.61265.5949 OOFVBackground 0.46990.52550.45080.2053 EventPileUp 0.12190.12190.12180.1218 ALL AllMagnetUncertainties 2.27341.69093.88185.9080 SandMuonBackground 0.03320.03640.02110.0281 TOTAL 2.27371.69133.88185.9081 Table6.12:IntegrateduncertaintyforeachofthesystematicuncertaintiesinFGD2.Theproton secondaryinteractionssystematicuncertaintyisnotincludedasithasasmalle˙ectonthe sample.Tablefrom[ 7 ]. ‹ CC 0 ˇ binedges: p [MeV/c]:0,300,400,500,550,2000,4000,30000 cos :-1.0,0.6,0.7,0.8,0.85,0.9,0.96,1.0 ‹ CC 1 ˇ binedges: p [MeV/c]:0,500,30000 cos :-1.0,0.7,1.0 ‹ CCOtherbinedges: 124 p [MeV/c]:0,600,800,30000 cos :-1.0,0.7,1.0 ‹ CC 0 ˇ binedges: p [MeV/c]:0,300,500,700,800,30000 cos :-1.0,0.7,0.8,1.0 ‹ CC 1 ˇ binedges: p [MeV/c]:0,600,800,30000 cos :-1.0,0.7,1.0 ‹ CCOtherbinedges: p [MeV/c]:0,600,30000 cos :-1.0,0.7,1.0 ThesamebinningsareusedforFGD1andFGD2.Inthecasewheremultiple˝tbinsfallwithina singlenormalizationbin,thesameobservablenormalizationweightisusedforall˝tbins,because thechoiceofbinisdependentonthereconstructedmomentumandangle. Inordertocalculatethecentralvaluesfortheobservablenormalizationparameters,2000 variationsofthedetectorsystematicuncertaintiesarecreated.Theseuncertaintiesarethenapplied tothenominalMonteCarlo p cos distributiontocreateasetof2000varieddistributions.The centralvaluesarethencalculatedforeach p cos bin: d i = N mean i N nom i ; (6.13) where N mean i istheaveragenumberofeventsinbin i and N nom i isthenominalnumberofevents inthatbin.Thecovariancematrixelementsarecalculatedvia ¹ V d º ij = 1 1999 Õ ¹ N k i N mean i º¹ N k j N mean j º N mean i N mean j : (6.14) 125 Thisgivesacorrelatedsetofparameterstousewithintheneardetector˝t.Theseparametersand theircovarianceareusedundertheassumptionthattheoveralle˙ectofthevariationsisGaussian inallbins,which,to˝rstorder,iscorrectforbinswithalargenumberofevents.Itshouldbenoted, thatbinswithfewerthan20events,onaverage,tendtohaveamorenon-Gaussianshape. Thecovariancematrixalsoincludesuncertaintiesfromothersourcesthatcanbebinnedin p cos .TheseincludetheMonteCarlostatisticalerrorandanuncertaintyrelatedtodi˙erences intherelativisticFermigasandlocalFermigasmodels,termedoneparticleonehole(1p1h) e˙ects,discussedinSection 4.3.2.1 .TheMonteCarlostatisticalerrorisrelatedtotheuncertainty ofusinga˝niteamountofMonteCarlowhenrunningthe˝tandingeneratingthecovariance matrix.Ratherthanusingthecovariancebinning,thestatisticaluncertaintiesarecalculatedfor each˝tbin.Thelargest˝tbinuncertaintywithinagivencovariancebinisthenusedasthe statisticaluncertaintyforthatbin.Thisisdonetoavoidunderestimatingtheuncertaintyduetothe MonteCarlostatistics.Theseerrorsareincludedasuncorrelatederrortermsonthediagonalofthe observablenormalizationcovariancematrix.ThesizeoftheMCstatisticalerrorscanbeseenin Figure 6.1 . AsmentionedinSection 4.3.2.1 ,studieshaveshownthatthechoiceofnuclearmodelcanhave asigni˝cantimpactontheoscillationanalysis.Therefore,anadditionaluncertaintyisincludedto accountforthise˙ect.Fortheneardetector˝t,thisistreatedasanadditionalcovarianceterm addedtotheobservablenormalizationcovariancematrix.Thiscovarianceiscalculatedasthe bin-by-bindi˙erenceinmuonkinematicsbetweentheNEUTandNievesmodelpredictions: V 1 p 1 h ij = ¹ N Nie v es i N NEUT i º¹ N Nie v es j N NEUT j º ; (6.15) where N Nie v es i isthepredictedeventrateforthe i -thbinoftheNieves1p1hmodel,while N NEUT i isthepredictedeventrateforthe i -thbinofthenominalNEUTmodel. Theobservablenormalizationcovariance,withouttheMCstatisticalerrorsortheuncertainty duetothe1p1he˙ects,canbeseeninFigure 6.2 ,whilethefullcovariancematrixcanbeseenin Figure 6.3 . 126 Figure6.1:Thefractionalerrorincludedintheobservablenormalizationcovariancematrixdueto theMonteCarlostatisticaluncertainty.Thebluedashedlinesdi˙erentiatebetweentheCC 0 ˇ ,the CC 1 ˇ ,andtheCCOthersamples,whilethereddashedlinesseparatesamplesinFGD1andFGD2. ThesolidredlinedemarcatestheFHCMultiPi,RHC MultiPi,andRHC MultiPisamples. 127 Figure6.2:ThedetectorcovariancematrixwithouttheMCstatisticaluncertaintiesorthe uncertaintiesfromthe1p1he˙ects,plottedassgn ¹ V ij º p j V ij j foreasierviewing.Theshort dashedlinesdi˙erentiatebetweentheCC 0 ˇ ,CC 1 ˇ ,andCCOthersamples,whilethelongdashed linesseparateFGD1andFGD2samples.ThesolidblacklinesseparatetheFHCMultiPi,RHC MultiPi,andRHC MultiPisamples.Withineachsample,theparametersareorderedfrom backwardgoingtoforwardgoingangularbins.Eachcompletesetofangularbinsshareacommon momentumbin,whichareorderedfromlowesttohighestmomentum. 128 Figure6.3:ThefulldetectorcovariancematrixasinputtotheBANFF˝t,plottedas sgn ¹ V ij º p j V ij j foreasierviewing.Theshortdashedlinesdi˙erentiatebetweentheCC 0 ˇ , CC 1 ˇ ,andCCOthersamples,whilethelongdashedlinesseparateFGD1andFGD2samples. ThesolidblacklinesseparatetheFHCMultiPi,RHC MultiPi,andRHC MultiPisamples. Withineachsample,theparametersareorderedfrombackwardgoingtoforwardgoingangular bins.Eachcompletesetofangularbinsshareacommonmomentumbin,whichareorderedfrom lowesttohighestmomentum. 129 CHAPTER7 RESULTS Inthisthesis,oneofthebiggestchangesmadewasthetransitionfromtheRHCMultiTrackto theRHCMultiPisamplesinthelikelihood˝t.InSection 7.1 ,comparisonsbetween˝tsrunwith theRHCMultiTracksamplesandtheRHCMultiPisamplesareprovidedinordertoshowthefull impactofthenewsamples.Furthermore,twoadditionalstudies,describedinSections 7.2 and 7.3 ,wereperformedtotestalternatecrosssectionmodels.Atthistime,workisbeingdoneon T2Ktoresolvedi˙erencesseenbetweendataandMonteCarloforsinglepionproductionevents, particularlyintheoutgoingpionspectrum.Eachstudycomparedtheresultsofa˝ttothenominal MonteCarlowiththeresultsfroma˝ttoMonteCarlowiththealternatecrosssectionmodel.This comparisonprovidesinsightintowhetherthereisfreedominthecurrentcrosssectionmodelto coverthedi˙erencesintheunderlyingmodel. 7.1ComparingRHCMultiTrackandRHCMultiPiSamples AstheresultsfromtheBANFF˝tareincludedintheT2Koscillationanalysis,itisimportant tounderstandhowmovingfromtheRHCMultiTracksamplestotheRHCMultiPisamplesa˙ects the˝tresults.Forexample,dotheCC 1 ˇ parameterschangedramaticallywhenincludingthe RHCMultiPisamples,duetotheextrapioninformationfoundinthesesamples?Additionally, itisusefultoseehowthechangestotheBANFF˝ta˙ecttheSuper-Kamiokande(SK)event distributionpredictions,aslargechangestotheeventpredictionscanprovidehintsatpotentially largechangesinthe˝nalvaluesoftheoscillationparametersdeterminedbyT2K. 7.1.1Results The˝rststepincomparingthetwosamplessetsistoseehowtheirsensitivitytotheinputparameters di˙ers.Inordertotestthis,eachsamplesetis˝ttothenominalMonteCarlo,whichshowsthe sensitivityofthe˝ttothemodelparameterization.Forexample,ifthereweretwodegenerate 130 parametersinthemodel,the˝twouldnotbeabletoconstrainthoseparametersaswellasa non-degenerateparameter.AscanbeseeninFigure 7.1 ,thetwosamplesetshaveroughlythe samesensitivitytothecurrentparameterization 1 .Becausethereisvirtuallynodi˙erenceinthe sensitivity,bothsetsofsamplesshouldprovideasimilaramountofconstrainttotheoscillation analysis.However,itisbene˝cialtousetheRHCMultiPisamplesovertheRHCMultiTrack samples,astheyaremoreconsistentwiththesamplesusedatSK.Additionally,whencomparedto theRHCMultiTracksamples,theRHCMultiPisampleshaveanimprovedpuritywithrespectto theneutrinointeractiontypesthatarebeingmodeledinthecrosssectionmodel[ 7 ]. Afterobservingthatthetwosamplesetshaveasimilarsensitivitytothecurrentmodelparam- eterization,aseparate˝tforeachsamplesetwasperformedusingtheT2Kdata.Theresultsof thesetwo˝tscanbeseeninFigure 7.2 .Overall,the˝tparametersagreeverywell,particularlythe parametersassociatedwiththe˛ux.Generally,di˙erencesbetweenthetwo˝tsareexpecteddue totheuseoftheRHCMultiPisamples,whichmorecloselymatchtheunderlyinginteractionsthan theRHCMultiTracksamples.Inparticular,thedi˙erencesintheFSIparameterscanbedescribed bythedi˙erencesintheeventtopologiesforthetwosamplesets.TheFSIparametersa˙ectthe migrationofeventsbetweensamplesthroughtheinteractionofpionswithinthenucleus.Because thenumberofoutgoingpionsismoreimportantfortheRHCMultiPisamples,thedatainthese samplescouldpreferdi˙erent˝nalvaluesfortheFSIparameters. Inasimilarmanner,thedi˙erencesobservedinthe2p2hnormalizationparametersarereason- able.Theshiftoftheantineutrinonormalizationparametertobeclosertotheneutrinonormalization parametershowsevidencethattheRHCMultiPisamples,whencomparedwiththeRHCMulti- Tracksamples,arebetterabletoseparate2p2heventsfromeventswheretheoutgoingpionwas missedinthereconstruction.Finally,theparametersrelatedtosinglepionproduction,likethe CCresonantpionproductionparameters,aswellastheparameterfordeepinelasticscatteringand 1 Acomparisonoftheobservablenormalizationparametersisnotshowninthisthesis,asthe twosamplesetsdidnothavethesamebinning,makingadirectcomparisonnotastraightforward process.However,theywerecheckedinordertoverifytheparameterswerenotpulledfarfrom theirinputvalueandthatnosystemicshiftsupwardordownwardoccurredasaresultofthe˝t. 131 ND280FluxParameters SKFluxParameters FSIandCrossSectionParameters Figure7.1:Comparisonof˝tstothenominalMonteCarlofortheRHCMultiTracksamples (blue)andtheRHCMultiPisamples(red).Theinputparametererrorbarscanbeseeninthe background.Note,theCCQEcrosssectionparametersdonothaveanypriorconstraintappliedin the˝t,soaninputerrorbandisnotdisplayedfortheseparameters. 132 multiplepionproduction(DIS+N ˇ )interactionsexhibitsomedi˙erences.However,becausethe newRHCMultiPisamplesaredesignedinawaytobemoreconsistentwithhowtheseinteractions wouldnominallyappearinthedetector,thefactthattheseparametersmoveinthenewsampleset isunderstandable. Inadditiontocomparingthe˝tparameters,the p cos post-˝tdistributionswerecompared fortheneutrinobeammodesamples,asthesehadthesamebinninginboth˝ts.Thesedistributions, seeninFigure 7.3 ,showtheneutrinobeammodesamplesrespondinaverysimilarwayforboth ˝ts.FurtherevidenceofthiscanseeninFigure 7.4 ,wherethemomentumand cos distributions arecomparedseparately. Becausethetwosamplesetshadsimilarsensitivitytothe˝tparametersandproducedsimilar post-˝tdistributions,thefardetectoreventdistributions,whichareusedtodeterminetheoscil- lationparametersvalues,shouldbeconsistent.However,whencomparingtheunoscillatedevent distributions,asseeninFigure 7.5 ,thisisnotthecase,particularlyforthe1R sample.Likewise, whenexaminingtheoscillatedeventdistributionsinFigure 7.6 ,thedistributionsarenotconsistent withoneanother.Whilethedi˙erencesareonlyaboutonestandarddeviation,previousstudieson T2Khaveshownthatsimilarshiftsareabletochangetheoscillationparametervaluesfoundinthe oscillationanalysis[ 114 ]. Oneparticularwaythe˝naloscillationparametervaluescouldbea˙ectedwouldbeashiftinthe valueof sin 2 23 .Qualitatively,changesinthevalueof sin 2 23 canbedeterminedbycomparing thesizeofthedipintheoscillatedeventdistributions.BecausetheSKdistributionthatusesthe RHCMultiPi˝tresultsdecreasesrelativetothedistributionusingtheRHCMultiTrack˝tresults, thisimpliesthevalueof sin 2 23 willshiftclosertomaximalmixing( sin 2 23 = 0 : 5 ).Whenusing theRHCMultiTracksamples,the ˜ 2 intervalsfor sin 2 23 ,whichcanbefoundinFigure 7.7 ,are largeenoughtocovermaximalmixing.DuetotheRHCMultiPisamplesprovidingasimilarlevel ofconstraintastheRHCMultiTracksamples,thesizeofthesecontoursshouldbesimilarinthe future.Therefore,while sin 2 23 likelywillshifttowardsmaximalmixing,T2Kwillnothavethe constrainttomakeastatisticallysigni˝cantstatementastothetruevalueof sin 2 23 . 133 ND280FluxParameters SKFluxParameters FSIandCrossSectionParameters Figure7.2:Comparisonof˝tstoT2KdatafortheRHCMultiTrack(blue)andRHCMultiPi samples(red).Theinputparametererrorbarscanbeseeninthebackground.Note,theCCQE crosssectionparametersdonothaveanypriorconstraintappliedinthe˝t,soaninputerrorband isnotdisplayedfortheseparameters. 134 FGD1FHCCC0 ˇ FGD1FHCCC1 ˇ FGD1FHCCCOther Figure7.3:Ratiosofthedata p cos distributiontothepost-˝tdistributionforFGD1forthe RHCMultiTracksamples(left)andtheRHCMultiPisamples(right).The ˜ 2 pernumberof binscanbeseenforeachdistribution. 135 FGD1FHCCC0 ˇ FGD1FHCCC1 ˇ FGD1FHCCCOther Figure7.4:Themomentum(left)and cos (right)distributionsfortheRHCMultiTrack(black) andtheRHCMultiPi(red)samples.The ˜ 2 pernumberofbinsisshownforeachsampleset. 136 Figure7.5:ComparisonoftheSKneutrinoenergydistributionsfortheRHCMultiTrack(red)and RHCMultiPi(black)samples.Theerrorbarsaretheapproximateuncertaintyfromthe˛uxplus crosssectionsystematicuncertainties. 137 Figure7.6:ComparisonoftheSKneutrinoenergydistributionsfortheRHCMultiTrack(red)and RHCMultiPi(black)samples.Theerrorbarsaretheapproximateuncertaintyfromthe˛uxplus crosssectionsystematicuncertainties. 138 Figure7.7:Constant ˜ 2 68%and90%intervalsforthehybridfrequentist-Bayesianthefully BayesiananalysesonT2K,assumingthenormalmasshierarchy.Theseintervalsarebasedona neardetector˝tusingtheRHCMultiTracksamples.Figurefrom[ 25 ]. Currently,T2Kisintheprocessofoverhaulingthecrosssectionmodel,includingtheparameters relatedtoCCQEandCCresonantpionproductioninteractions.Assuch,the˝nalimpactonthe fundamentaloscillationparameterswillbeaconvolutionofthenewsampleswiththenewcross sectionmodel.Untilthecrosssectionmodelisready,comparisonsoftheSKeventdistributions provideinsightintowhetherdi˙erencesmightbeexpected.However,di˙erencesinthedistributions mayvanishwiththeinclusionofthenewmodel;therefore,theyshouldbeviewedassuggestive ofpotentialchangeandnotassubstantialevidenceforchange.Italsoprovidesanecessarystep invalidatingthethe˝nalresults.Anydi˙erencesobservedhereareduetoe˙ectsfromthenew samples,asthenewcrosssectionmodelhasnotbeenincluded,andprovidesabenchmarkto comparetowhenthenewmodelisadded. Whilethecombinede˙ectofthenewsamplesandmodelcannotbeascertaineduntilthenew 139 crosssectionmodelisavailable,thetransitiontotheRHCMultiPisamplescanbedoneatthistime. Indoingso,thesamplesforthenearandfardetectorwillbemoreconsistentwithoneanother. Additionally,thenewsamplesprovideabettercomparisonbetweentheunderlyinginteractionsand thecorrespondingtopologiesseeninthedetector. 7.2E˙ectofAdditionalCCResonantEventswithLowPionMomentum Currently,thenon-resonantpionproductionmodelusedbytheNEUTneutrinoeventgenerator istunedtoonlyneutrinodata[ 24 ].Itisassumedthatthistuningissu˚cientforantineutrino events;however,themodeldevelopedbyKabirnezhad(themodel)hasshowndi˙erencesin theCCresonantpionproductioncrosssectionforneutrinosandantineutrinosduetothecoupling ofresonantandnon-resonantbackgroundinteractions[ 115 ].Forthisthesis,asimplestudywas performedtotestthee˙ectofadditionalantineutrinoeventsinthenewRHCMultiPisamplesthat haveoutgoingchargedpions,astheMKmodelpredictsahigherantineutrinocrosssectionthan currentlyassumed.Thisstudywasperformedtoverifythatthecurrentcrosssectionmodelused intheneardetector˝tissu˚cientforhandlingachangeinthenumberofantineutrinosinglepion productioneventsinthenewsamples.Furthermore,becauseeventswhichhavepionsthatfallbelow thedetectionthresholdatSKcancontaminatetheoneringeventsamplesfortheoscillationanalysis, thisstudysolelylookedatsimulatedeventswithpionmomentabelowthedetectionthreshold. 7.2.1Results Inordertotestthee˙ectoftheadditionalantineutrinoevents,auniformnormalizationfactorwas appliedtosimulatedantineutrinosinglepionproductioneventswithachargedpionescapingthe nucleus.Inadditiontoneedingachargedpioninthe˝nalstate,themomentumofthepionhadto belessthan200MeV/c,whichisapproximatelythethresholdforapiontobedetectedatSK 2 .The e˙ectofthisadditionalfactorcanbeseeninFigure 7.8 forFGD1(FGD2canbeseeninAppendix 2 Thethresholdcanbecalculatedas E T = m ˇ c 2 š p 1 1 š n 2 ,where n istheindexofrefraction forwaterand m ˇ isthemassofthepion. 140 FHCMultiPiSamples RHC MultiPiSamples RHC MultiPiSamples Figure7.8:Ratiosofthemodi˝ed p cos distribution,whichincludesadditionalantineutrino singlepionproductionevents,tothenominaldistributionforFGD1.TheCC 0 ˇ samples(left), CC 1 ˇ samples(middle),andCCOthersamples(right)areshown. C ).Asexpectedthelargeste˙ectcanbeseenintheRHC samples,withlittlechangeseenin eithertheFHCMultiPiorRHC MultiPisamples. Overall,thereisverylittledi˙erenceintheparameterswhen˝ttingtheMonteCarlodistribution withadditionallowenergypionswhencomparedagainstthe˝ttothenominalMonteCarlo distribution(seeFigure 7.9 ).Ingeneral,the˛uxparameters,FSIparameters,andthevastmajority oftheobservablenormalizationparameters,whichcorrespondtotheresponseofthedetector, showshiftsoflessthan1%.Nominally,changesintheCCresonantpionproductionparameters 141 (CA5, M RES A ,andtheisoscalarbackground)wouldhavebeenexpectedasthea˙ectedeventsare CCresonantevents.However,thisisnotthecase.TheresultsindicatetheCCresonantpion productionmodelforantineutrinoeventsdoesnothavesu˚cientfreedominittocompensatefor theinclusionofmoresinglepionproductioneventswithlowmomentumpions. RatherthantheCCresonantpionproductionparametersmoving,someofthe2p2hparameters showedsmallincreases.Theobservedshiftintheseparametersisreasonable,asthemodi˝ed eventshavesimilarkinematicstothepionproductionregionofthe2p2hcrosssection(seeFigure 4.7 ).Becauseofthis,the2p2hshapeparametersarecompensatingwithavaluewheretherewould bemorepionproduction-likeeventsoverQE-likeevents 3 . WithinthenewcrosssectionmodelmentionedinSection 7.1 ,anewCCresonantpionproduc- tionmodelwillbeimplemented,onewhichincludesthee˙ectsofcouplingbetweentheresonant andnon-resonantbackgrounds[ 115 ].Whenthenewmodelbecomesavailable,theresultsofthis studycanbecomparedwithasimilarstudydoneusingthenewCCresonantpionproductionmodel anduncertaintiestoseeifthenewmodelhasthefreedomtocoverthechangesthatthecurrent modeldoesnot. 7.3 Q 2 SuppressioninSinglePionProductionEvents Recently,resultsatMINER A[ 116 , 117 ]andMINOS[ 118 ]haveindicatedadisagreement betweentheirdataandMonteCarloforsinglepionproductioneventswithlowtransfermomentum, Q 2 (belowabout0.4GeV 2 ).Inordertoaccountforthisdiscrepancy,theNO A[ 119 ]collaboration includesamodi˝cationtotheirnominalMonteCarlo[ 120 ].Furthermore,theuncertaintyrelated tothismodi˝cationhasbeenshowntobeadominantsystematicuncertaintyintheircrosssection model[ 121 ].T2Kdoesnotincludeanysuchmodi˝cationtotheirMonteCarlo;however,forthis thesis,astudywasperformedtodetermineifthecurrentT2Kcrosssectionmodelhasthefreedom tocoversuchamodi˝cationifitweretobeincludedintheMonteCarlo. 3 A2p2hvalueof1.0isanequalmixofbothdistributions,while2.0isonlypionproduction-like eventsand0.0isonlyQE-likeevents. 142 Figure7.9:Comparisonof˝tstothenominalMonteCarlo(blue)andtheMonteCarloincluding additionalsinglepionproductionevents(red).TheND280neutrino˛uxparameters(topleft)are characteristicoftheparametershiftsforthefull˛uxparametersetandtheFSIparameters.Onthe otherhand,someshiftsareseeninthecrosssectionparameters(topright).Littledi˙erenceis seenforthemajorityoftheobservablenormalizationparameters(bottomleft),whileslight changesareseenintheRHC CC 1 ˇ samples(bottomright).Theinputparametererrorbarscan beseeninthebackground. 7.3.1Results Themodi˝cationappliedforthisstudyisbasedontheMINOSdiscrepancy[ 118 ]: 1 : 01 1 + exp ¹ 1 p Q 2 š 0 : 156 º ; (7.1) whichisappliedtoallsimulatedsinglepionproductioneventswith Q 2 below0.7GeV.The modi˝cation,asafunctionof Q 2 ,canbeseeninFigure 7.10 .Theratioofthemodi˝edMonte CarlotothenominalMonteCarloforFGD1canbeseeninFigure 7.11 (FGD2plotscanbefound inAppendix C ).Asexpected,themodi˝cationtargetseventswithlow Q 2 ,which,ina p cos 143 Figure7.10:Themodi˝cationappliedtothenominalT2KMonteCarlousingtheMINOS parameterization. distribution,isconcentratedaroundthepeakenergyofT2Kandinthehigheranglebins.Whilethe reductionisprimarilyintheCC 1 ˇ samples,thereducednumberofeventsseenintheCC 0 ˇ and CCOthersamplesisduetothemigrationoftruesinglepionproductioneventsintooneofthese samples. Figure 7.12 showstheresponseofthe˛uxandcrosssectionparameterstothesuppressionof thelow Q 2 singlepionproductionevents.Onefeatureoftheseresultsisthepreferencefora˛ux tuningthatisapproximately3%belowthenominaltuning.Asthe˛uxparametersservetoshift theoverallnumberofeventsupordowninthe˝tbasedonthedata,thisisareasonableshiftto expect.WhencomparingthenumberofeventsinTable 7.1 ,thereisapproximately3%fewerevents predictedatND280inthemodi˝edMonteCarlothaninthenominalMonteCarlo.Therefore,the ˛uxparametersarecompensatingforthedecreaseinthepredictedeventrate. Withregardstotheresponseofthecrosssectionparameterstothemodi˝edMonteCarlo, twoprimaryaspectsshouldbehighlighted.First,thecurrentCCresonantpionproductionmodel showssigni˝cantshiftswhencomparedwiththenominal˝t.Thisshowsthatthecurrentmodel hassomefreedomtocoverthechangesappliedtotheMonteCarlo.Forthisstudy,theidealresult 144 FHCMultiPiSamples RHC MultiPiSamples RHC MultiPiSamples Figure7.11:Ratiosofthemodi˝ed p cos distribution,whichincludesthelow Q 2 suppression weight,tothenominaldistributionforFGD1.TheCC 0 ˇ samples(left),CC 1 ˇ samples(middle), andCCOthersamples(right)areshown. 145 ND280FluxParameters SKFluxParameters FSIandCrossSectionParameters Figure7.12:Comparisonof˝tstothenominalMonteCarlo(blue)andtheMonteCarloincluding thelow Q 2 eventsuppressionmodi˝cation(red). 146 Sample Modi˝edNominalDi˙erence FGD1FHC CCInclusive 39,529.9240,821.52-0.032 FGD1FHC CC 0 ˇ 26,667.727,429.5-0.028 FGD1FHC CC 1 ˇ 6,653.397,095.14-0.062 FGD1FHC CCOther 6,208.836,296.88-0.014 FGD1RHC CCInclusive 7,602.5387,839.886-0.03 FGD1RHC CC 0 ˇ 6,065.546,245.19-0.029 FGD1RHC CC 1 ˇ 509.598542.636-0.061 FGD1RHC CCOther 1,027.41,052.06-0.023 FGD1RHC CCInclusive 4,142.4134,264.058-0.029 FGD1RHC CC 0 ˇ 2,478.092,547.87-0.027 FGD1RHC CC 1 ˇ 833.492875.593-0.048 FGD1RHC CCOther 830.831840.595-0.012 Sample Modi˝edNominalDi˙erence FGD2FHC CCInclusive 36,583.0137,810.02-0.032 FGD2FHC CC 0 ˇ 25,946.126,772-0.031 FGD2FHC CC 1 ˇ 5,173.275,493.58-0.058 FGD2FHC CCOther 5,463.645,544.44-0.015 FGD2RHC CCInclusive 7,403.6157,630.104-0.03 FGD2RHC CC 0 ˇ 6,032.986,211.62-0.029 FGD2RHC CC 1 ˇ 459.37486.26-0.055 FGD2RHC CCOther 911.265932.224-0.022 FGD2RHC CCInclusive 3,911.1854,030.805-0.03 FGD2RHC CC 0 ˇ 2,490.352,565.66-0.029 FGD2RHC CC 1 ˇ 641.985675.982-0.05 FGD2RHC CCOther 778.85789.163-0.013 Table7.1:Eventratesforthemodi˝edandnominalMonteCarlosets.Thenominaleventratesare MonteCarloeventsweightedbyPOT,˛ux,detector,andcrosssectionweights.Themodi˝edrate includesthesameweightsasthenominal,plustheadditionalmodi˝cationduetothelow Q 2 eventsuppression.Thedi˙erenceisgivenby ¹ modified nominal ºš nominal .Theleft columnshowstheratesforFGD1,whiletherightcolumnshowsFGD2. wouldbeforonlytheseparameterstomove,asthesuppressedeventsareonlyCCresonantpion productionevents.Thiswouldshowthemodelhassu˚cientfreedomtofullydescribetheapplied suppressionoflow Q 2 events.However,therearechangesintheothercrosssectionparameters thatareconsistentwithasuppressionoflow Q 2 singlepionproductionevents.Whensuppressing theseevents,therewillbeareductionintheeventspopulatingthepionproduction-likeregionof 2p2hevents(seeFigure 4.7 );therefore,theparameterscompensateforthisbypreferringamore quasi-elastic-likevalueforthe2p2hshapeparametersandanoverallreductioninthenumberof events,asisseeninFigure 7.12 .Likewise,theBeRPAAandBparameters,whicharethedominant BeRPAparametersatlow Q 2 ,showmovementtocompensateforthereductionoflow Q 2 events. Finally,DIS+N ˇ eventsgenerallyhavelarger Q 2 values,whichmeanstheseeventswillnotbe a˙ectedbythemodi˝cation.Consequently,theupwardshiftintheCCDISparameterimplies thatthe˝twouldexpecttoseemoreDIS+N ˇ eventswhenthemodi˝cationisapplied,whichis consistentwithDIS+N ˇ eventsnotbeingsuppressed.ThesecompensationsshowthecurrentCC resonantpionproductionmodeldoesnothavethefreedomnecessarytofullydescribethemodi˝ed distributions. Inadditiontoobservinghowthecurrentmodeldoesnothavethefreedomnecessarytocover 147 FGD1FHCCC 0 ˇ FGD1FHCCC 1 ˇ FGD1FHCCCOther Figure7.13:Ratiosofthemodi˝ed p cos distribution,whichincludesthelow Q 2 suppression weight,tothepost-˝tdistributionforFGD1.The ˜ 2 pernumberofbinscanbeseenforeach distribution. theappliedmodi˝cationbycomparingthepost-˝tparametervalues,thepost-˝tdistributionscan alsohighlightthelackoffreedom.Figure 7.13 showshowthemodi˝ed p cos distribution comparestothepost-˝tdistribution.Ifthecurrentmodelchoicehadthefreedomtofullycover themodi˝cationinthe˝t,thenthe ˜ 2 valuesforeachdistributionshouldbeclosetozero,asthe onlydi˙erencebetweentheMonteCarlosetsistheadditionalweighttosuppresslow Q 2 events. However,becausethe ˜ 2 valuefortheCC 0 ˇ andCC 1 ˇ samplesarenotclosetozero,thisyields furtherevidencethattherecurrentlyisnotenoughfreedomtofullycoverthismodi˝cation. Theuncertaintyassociatedwiththismodi˝cationisoneofthedominantcrosssectionun- certaintiesonNO A.WhiletheT2KCCresonantpionproductionmodelappearstohavesome 148 freedomtoabsorbchangesduetothesuppressionoflow Q 2 singlepionproductionevents,itdoes notfullycoverthechanges.Therefore,withthepotentialforT2KandNO Atoperformajoint oscillationanalysis,itwouldbebene˝cialforT2Ktofurtherstudytheimpactofthedi˙erences observedbyMINER AandMINOS.OnepotentialstudywouldbetousetheMKmodel,whenit becomesavailable,tostudyifmorefreedomexistsinthatmodelparameterizationthanwhatisseen inthecurrentmodel.AnotherstudywouldbetouseMINER A'sparameterizationoftheevent suppression,ratherthantheMINOSparameterization.Inthiscase,theimpactontheneardetector ˝tcanbedirectlycomparedtoverifyconsistencyinhowtheT2Kcrosssectionmodelrespondsto eachparameterization. 149 CHAPTER8 CONCLUSIONANDSUMMARY Thisthesisshowedthee˙ectsofusinganewsetofsamplesfordatacollectedintheantineutrino beammodeatT2K.Previously,theantineutrinobeammodesampleswerecategorizedbasedonthe numberofparticletracksseeninthetimeprojectionchambers(TPCs),calledtheRHCMultiTrack samples.Onesamplecollectedeventswithonlyamuon-liketrackthatcrossedtheTPC(the CC1-tracksample),whiletheotherselectedalleventswithamuon-liketrackandanynumber ofothertracks,betheyelectron,pion,orprotontracks(theCCN-trackssample).Ontheother hand,thesamplesarenowseparatedbasedonthenumberofpionsvisibleinthedetector,the RHCMultiPisamples,andareabletouseinformationfromtheTPCsaswellasthe˝ne-grained detectors(FGDs).Eventsthatonlyhaveamuon-liketrackfallintotheCC 0 ˇ sample,whileevents withamuon-liketrackandanappropriatelychargedpion-liketrack(anegativelychargedpionfor antineutrinointeractionsandapositivelychargedpionforneutrinointeractions)areplacedinthe CC 1 ˇ sample.Finally,anyeventsthatdonotfallintothe˝rsttwocategoriesareselectedintothe CCOthersample. Usingthenewsamplesservesoneprimarypurposeintheoverarchingoscillationanalysis,in thatthenewsamplesmoreaccuratelymatchwhatisseenatSuper-Kamiokande(SK).Forexample, intheRHCMultiTracksamples,asecondtrackinaneventintheCCN-trackssamplecouldbe fromaproton.However,ifasimilareventweretooccurinthefardetector,theprotonwouldbe unabletobedetectedasitdoesnotdecayorhaveenoughenergytomeettheCherenkovthreshold. Ontheotherhand,theRHCMultiPisamplesusepionstodistinguishbetweensamples.Atthefar detector,samplesaresimilarlycategorizedasthepionscanbedetectedeitherthroughtheirdecay orthroughthepionbeingenergeticenoughtomeettheCherenkovthreshold. ThestudypresentedinSection 7.1 discussesthee˙ectsofusingtheRHCMultiPisamples, ratherthantheRHCMultiTracksamples.Usinga˝ttothenominalMonteCarlo,itwasshown thatthetwosamplesetshavecomparablesensitivitytothe˝tparameters,whichimpliesthat˝ts 150 totheT2Kdatawillyieldapproximatelythesamelevelofconstraintonthe˛uxandcrosssection parameters.Whencomparingtheresultsof˝tstothedata,di˙erencesbetweenthetwosamplesets wereobserved.However,thesedi˙erencescanbedescribedeitherthroughthepriorconstraints placedontheparametersortheresultofmovingtotheRHCMultiPisamples,whicharemore closelyrelatedtothecrosssectionmodelsusedinthe˝t.Aftercomparingthee˙ectsonthenear detector,theneardetector˝tresultswerepropagatedtothefardetectoreventdistributions,which showeddi˙erencesinthepredictedeventratesforSK.Becausetheimpactofthenewsamplesis coupledtothecrosssectionmodel,whichwillbeupdatedsoon,the˝nalimpactofthesedi˙erences remainstobeseen.Nevertheless,thesecomparisonsareausefulvalidationforunderstandinghow thenewsamplesa˙ecttheSKeventdistributions,withoutadditionallyincludingthee˙ectsofthe newcrosssectionmodel. Additionalstudieswereperformedtostudythee˙ectofchangestothesinglepionproduction interactionchannelsattheneardetector.Onestudy(coveredinSection 7.2 )focusedonthee˙ectof anincreasednumberofeventswithanoutgoingchargedpionthatisnotenergeticenoughtomeet theCherenkovthreshold.Theresultsshowedthat,atthistime,theCCresonantpionproduction modelforantineutrinoeventsdoesnothavethefreedomtorespondtothesechanges.However, withtheinclusionoftheMKmodelanduncertaintiesinthenewcrosssectionmodel,itwouldbe bene˝cialtorevisitthisstudyanddetermineifthenewmodelhasfreedomwheretheoldmodel didnot.Theotherstudy(describedinSection 7.3 )focusedonthesuppressionoflow Q 2 single pionproductioneventsseenbyMINER AandMINOS.ThecurrentCCresonantpionproduction modelwasshowntohavesomefreedomtocoverchangestotheMonteCarlowhenamodi˝cation isappliedtosuppresslow Q 2 singlepionproductionevents.However,itwasnotabletocompletely coverthemodi˝cationappliedtotheMonteCarlo.DuetothepotentialforT2KandNO Ato performajointoscillationanalysis,itwouldbene˝tbothexperimentstocontinuestudyingthe impactofthethisuncertainty.TwopotentialextensionstothisstudyaretostudywhethertheMK modelanduncertaintieshavemorefreedomthanwhatisseenforthecurrentmodelandtousethe MINER Aparameterizationoftheeventmodi˝cation,ratherthantheMINOSparameterization. 151 APPENDICES 152 APPENDIXA COMPARISONSOFDIFFERENTPSYCHEVERSIONSONTHENEARDETECTOR FIT ThePsychesoftwarepackageisusedformakingselectionsinND280samplesandforapplying detectorvariationsystematicuncertainties,detectorsystematicweights,andprotonontarget(POT) weightsinND280.In2014,Psychewasupdatedfromversion1(v1)toversion3(v3) 1 .Inthe meantime,theBANFFneardetector˝tcontinuedtousePsychev1asthecodeforapplyingdetector variationsandweightsandforloadingeventsattheoutsetofaBANFFmaximumlikelihood˝t. Startingin2017,workbeganonportingtheBANFFsoftwaretoworkwithPsychev3.This appendixwillcoverthebroaddi˙erencesbetweenPsychev1andv3andtheirimpactonthenear detector˝t.Oneofthemaindi˙erencesbetweentheversionsisanupdatetothedetectorselections andsystematicuncertainties.Section A.1 willprovidebackgroundonthestudiespresentedin thisAppendix.Additionally,itwillshowthevalidationprocessthatensuredtheBANFFcode achievedthesamedistributionsandweightsastheNuMugroup,whichistheT2Kgrouptasked withdevelopingtheeventselections.InSection A.2 ,acomparisonofthe˝tresultsbetween Psychev1andPsychev3andhowthoseresultspropagatetotheSuper-Kamiokande(SK)event ratepredictionswillbeshown. A.1ComparisonofDetectorWeightApplication Duringthesummerof2018,theBANFFgroupbeganlookingathowthePsychev1andPsyche v3versionsoftheBANFFcodedi˙ered.First,thedi˙erencebetweentheeventratesforeach versionwerecompared.Table A.1 showsthebreakdownofeventswiththevariouseventweights applied.Whileitwasnotsurprisingtheratesdi˙ered,theBANFFgroupwantedtodelvemoreinto thereasonswhythetworatesweredi˙erent. Initially,itwasthoughtthatthedi˙erencesinthedetectorsystematicweightswouldbeable 1 Version2ofPsychesu˙eredfrommanyissuesandwasusedasasteppingstonetov3. 153 Sample V1DataV3Data V1RawMCV3RawMC V1POTonlyV3POTonly FHCFGD1 CC 0 ˇ 17136.0017135.00 337436.00302866.00 16090.8316101.55 FHCFGD1 CC 1 ˇ 3954.003955.00 84982.0076491.00 4058.364075.15 FHCFGD1 CCOther 4149.004159.00 65286.0058819.00 3107.043123.00 Sample V1POT+˛uxV3POT+˛ux V1POT+xsecV3POT+xsec V1POT+detV3POT+det FHCFGD1 CC 0 ˇ 17535.9217543.64 15340.1115340.70 15905.2415625.75 FHCFGD1 CC 1 ˇ 4606.614621.26 3819.313833.63 4011.583911.90 FHCFGD1 CCOther 3703.693719.39 3078.523092.76 3071.212993.17 Sample V1Pre˝tV3Pre˝t FHCFGD1 CC 0 ˇ 16723.6916408.21 FHCFGD1 CC 1 ˇ 4381.484258.71 FHCFGD1 CCOther 3943.953834.01 TableA.1:ComparisonofeventratesbetweenPsychev1andPsychev3duringthesummerof 2018. todescribethedi˙erencesseen.However,whencomparingmomentumdistributionsforPsyche v1andPsychev3(seeFigure A.1 ),thedetectorsystematicerrorbarsdidnotcoverthedi˙erences seeninthetwodistributions.Therefore,theBANFFandNuMugroupsbegancomparingresultsin ordertoverifytheyweregettingthesameresults. FigureA.1:MomentumdistributionscomparingPsychev1andPsychev3.Theerrorbarsonthe v3distributionarefromthePsychev3detectoruncertainties.Moreoftenthannot,thedi˙erence betweenPsycheversionsisnotcoveredbytheerrorbars. 154 Oneofthe˝rsttestsperformedwastocomparethenumberofselectedeventsandthatthe detectorvariationsandweightswerebeingappliedinthesamewayforPsychev3.TheNuMu grouphasbeenusingPsychev3foranumberofyearsandverifyingtheBANFFgroupisgettingthe sameselectedeventsandweightsservedasavalidationprocessthattheBANFFimplementationis workingproperlyforPsychev3.Inordertoperformthesecomparisons,thetwogroupslookedat T2Krun4air(4a). TheBANFFandNuMugroupscomparedreconstructedmuonmomentumdistributionsin Psychev1andPsychev3(seeFigure A.2 ).Whilethetwohaveapproximatelythesameshape,it wasnoticedthattheeventratesdi˙eredquiteabitbetweenBANFFandNuMu.Afterdiscussing withtheNuMugroup,abugwasdiscoveredinhowoneofthesystematicuncertaintieswas calculated,whichresultedinsomeofthedi˙erencesseen.Figure A.3 showsthee˙ectofthebug ˝xonthemomentumdistributions. Next,thetwogroupscomparedtheweightsbeingappliedtospeci˝ceventstoverifytheweights wereappliedconsistently.Table A.2 showsanabbreviatedlookattheweightscomparison.The resultsbetweenthetwogroupsmatchupverywell,showingthetwogroupsarecalculatingthe samedetectorweightforeachevent. RunSubrunEventSample NuMuWeightBANFFWeight 4a026FGD1FHCCC0 ˇ 0.98711990.98712 4a0883FGD1FHCCC0 ˇ 0.98947280.989473 4a01304FGD1FHCCC0 ˇ 0.98456220.984562 4a02078FGD1FHCCC0 ˇ 0.98711990.98712 4a02107FGD1FHCCC0 ˇ 0.98711990.98712 4a02555FGD1FHCCC0 ˇ 0.97789620.977896 4a03291FGD1FHCCC0 ˇ 0.98711990.98712 4a04539FGD1FHCCC0 ˇ 0.98858300.988583 4a04732FGD1FHCCC0 ˇ 0.88627850.886279 4a04736FGD1FHCCC0 ˇ 0.98711990.98712 4a04781FGD1FHCCC0 ˇ 0.98947280.989473 TableA.2:ComparisonofdetectorweightsappliedbytheBANFFandNuMugroupstospeci˝c events. 155 FGD1FHCCC0 ˇ FGD1FHCCC1 ˇ FGD1FHCCCOther FigureA.2:ComparingBANFF(left)andNuMu(right)reconstructedmuonmomentum distributionsforPsychev1andPsychev3.TheBANFFplotsincludeanadditionalweightrelated tothenon-Gaussianityofsomeofthedetectorparameters. 156 FGD1FHCCC0 ˇ FGD1FHCCC1 ˇ FGD1FHCCCOther FigureA.3:E˙ectofdetectorweightbuginPsychev3.TheleftsideshowstheBANFFusing Psychev3withoutthebug˝x,whiletherightsideincludesthebug˝x.Inbothcases,theNuMu curveincludesthebug˝x. Onewaytodoublechecktheeventweightsarebeingappliedcorrectly,aswellasverifying thesameeventsarebeingselected,istoplottheweighteddistributions.UsingPsychev3,the 157 twogroupscompareddistributionswithonlythedetectorvariationsapplied,thevariationsandall weightsapplied,andthevariationswitheachindividualweightappliedseparately.Thevariations onlyandvariationsplusallweightsappliedplotscanbeseeninFigures A.4 A.5 .Figure A.6 showsthee˙ectofapplyingthevariationsonlyplusonedetectorweight(the˛uxweight,inthis case)andisrepresentativeofthecomparisonforeachindividualweight.Overall,thereisexcellent agreementbetweenBANFFandNuMuinthenumberofselectedeventsandweightvaluesbeing appliedtotheRun4aevents. FGD1FHCCC0 ˇ FGD1FHCCC1 ˇ FGD1FHCCCOther FigureA.4:ComparingselectedeventsinPsychev3forBANFFandNuMuwithonlythedetector variationsapplied. AfterlookingatboththeexactvaluesforweightsappliedandtheplotscomparingtheBANFF andNuMuselectedevents,foreachindividualweightandthecombinedweight,theBANFFand NuMugroupsarecon˝dentthatPsychev3isworkingasitshouldfortheBANFFgroup. 158 FGD1FHCCC0 ˇ FGD1FHCCC1 ˇ FGD1FHCCCOther FigureA.5:ComparingselectedeventsinPsychev3forBANFFandNuMuwiththedetector variations,the˛uxweight,andalldetectorweightsapplied. A.2FitResultComparison Whileunderstandingthedi˙erencesbetweenPsychev1andPsychev3areimportant,ultimately, thegoalistounderstandhowthesedi˙erencesa˙ecttheneardetector˝tresults.Inordertocompare howthedi˙erencesa˙ecttheBANFFresults,asystematicapproachwastakentousethesame inputsandchangeonethingatatimetoseethee˙ectsofthechange.Inadditiontothedi˙erences inthedetectorsystematicweightsanduncertaintiesbetweenPsycheversions,abugwasdiscovered intheapplicationoftheMCstatisticaluncertaintyandthe1p1hmodeluncertaintytotheobservable normalizationcovariancematrix.Thesethreedi˙erencesweretakenintoaccountbyproducing observablenormalizationcovariancematriceswithdi˙erentinputs.First,acovariancematrixwas 159 FGD1FHCCC0 ˇ FGD1FHCCC1 ˇ FGD1FHCCCOther FigureA.6:ComparingselectedeventsinPsychev3forBANFFandNuMuwiththedetector variationsandthe˛uxweightapplied. generatedusingonlyvariationsofthedetectorparametersforbothPsychev1andPsychev3.This matrixallowsforacomparisonoftheunderlyingselectionsandsystematicuncertaintychanges betweenPsycheversions.Second,acovariancematrixwasgeneratedusingdetectorparameter variationsandaddingtheMCstatisticalerrortotheentriesalongthediagonalofthecovariance matrix.ByincludingtheMCstatisticalerror,theimpactoftheadditionaluncertaintyonthe˝tcan beobserved.Inasimilarmanner,acovariancematrixwiththe1p1hmodeluncertaintyaddedto thecovariancematrixwasgenerated.Thisshowsthedi˙erenceintheresultsgivenbyincluding the1p1hcorrection.Finally,anobservablenormalizationcovariancematrixincludingboththe MCstatisticalerrorandthe1p1hcorrectionshowsthee˙ectofincludingbothuncertaintiesonthe ˝t. 160 Initially,studieswereperformedcomparingPsychev1resultswithdi˙erentdetectorcovariance matricestooneanother,andsimilarlyforPsychev3.Fromthesestudies,anumberofobservations weremade[ 122 ]: 1. Thereislittlee˙ectonthe˝tresultswhenthe1p1hmodeluncertaintyisincludedinthe detectorcovariancematrix. 2. WhentheMCstatisticalerrorisincluded,bothPsychev1andPsychev3exhibitupwardshifts inthe˛uxparametersanddownwardshiftsintheobservablenormalizationparameters. 3. TheFSIandcrosssectionparametershavesimilarresultsregardlessofwhethertheMC statisticalerrorisincludedornot. 4. WhenboththeMCstatisticalerrorandthe1p1hmodelerrorareincluded,the˝tresultshave verysimilarresultstothoseseenwhenonlytheMCstatisticalerrorisincluded. Oncethesestudieshadbeenperformed,comparisonsweremadebetweenthePsychev1and Psychev3results.First,whenlookingatresultsduetothedetectorweightsonly,signi˝cant di˙erencesbetweenPsychev1andPsychev3wereseen(Figures A.7 A.9 ).Becausetheonly di˙erenceintheinputsisduetotheversionofPsycheused,thesedi˙erencesmustarisefromthe changesinselectionandsystematicuncertaintiesbetweenPsychev1andPsychev3.However, whencomparingthepost-˝tdistributionsbetweenthetwoversionsinFigure A.10 ,theywere foundtobelargelysimilar.So,eventhoughtheparametervalueshavelargerdi˙erences,theyare counterbalancedinsuchawaythatthe˝naldistributionsremainalike. Becausethee˙ectofthe1p1hmodelerrorshowedlittlemovementinthe˝tresults[ 122 ], comparisonsofPsychev1toPsychev3withthe1p1hmodelerroradded,bothonitsownandwith theMCstatisticalerror,willnotbelookedatinthisthesis.However,becausetheMCstatistical errorhadmoremovementinbothPsychev1andPsychev3,itisbene˝cialtoseehowthesecompare tooneanother.WhenlookingatFigures A.11 and A.13 ,thereisanapproximatelyconstantupward shiftforthe˛uxparametersandacorrespondingdownwardshiftinthedetectorparameterswhen 161 FigureA.7:BANFF˝tresultsforthe˛uxparameterscomparingthee˙ectofthedetector selectionsandsystematicuncertaintiesforPsychev1(red)andPsychev3(blue).Theinputvalues valuecanbeseeninthebackground. comparedtoFigures A.7 and A.9 ,respectively.Thisshiftisapproximatelythesameforboth Psychev1andPsychev3.Therefore,itwasdeterminedthatthee˙ectoftheMCstatisticalerroris approximatelyconstantbetweenthetwoversionsofPsyche,whichimpliesthatthedi˙erencesseen betweenPsychev1andPsychev3mustbecomingfromthedi˙erencesinselectionsandsystematic uncertainties.Finally,whenlookingatthepost-˝tdistributionsforPsychev1andPsychev3,little di˙erenceisfoundbetweenthedistributions. Inordertotesttheimpactofthechangesintheneardetector˝tresultstotheoscillation analysis,predictionsoftheoscillatedeventdistributionsatSKwereproducedusingthe˝tresults fromthefullobservablenormalizationcovariance(detectorweightsplusMCstatisticalerrorplus 1p1hmodelerror).Becausethe1p1hmodeluncertaintyhasverylittlee˙ect,Figures A.11 A.13 162 FigureA.8:BANFF˝tresultsfortheFSIandcrosssectionparameterscomparingthee˙ectofthe detectorselectionsandsystematicuncertaintiesforPsychev1(red)andPsychev3(blue).The inputvaluesvaluecanbeseeninthebackground. providesanideaofthe˝tresultsusedasinputstotheeventprediction.LookingatFigure A.15 , theeventpredictionforeachtypeofneutrinoisapproximatelythesamebetweenthetwoPsyche versions.Becausetheeventpredictionsareconsistent,the˝naloscillationresultsshouldbesimilar, givingcredencethatmovingfromPsychev1toPsychev3isaviableoption. Insummary,themaincauseofthedi˙erencesinthepost-˝tparametervaluesbetweenPsychev1 andPsychev3isduetotheupdatesinthesampleselectionsanddetectorsystematicuncertainties. Withregardstothedi˙erencesseenintheMCstatisticalerrorand1p1hmodeluncertainty,itwas foundthattheMCstatisticalerrorresultsinashiftupwardforthe˛uxparametersandanddownward forthedetectorparameters.Ontheotherhand,the1p1hcorrectiondidnotresultinmuchchange whencomparedtotheshiftsfromtheMCstatisticalerror.Ultimately,itwasdeterminedthat,even thoughtheparametervaluesdi˙er,thepost-˝tdistributionsbetweenPsychev1andPsychev3di˙er little.ThisisalsoseenwhencomparingPsychev1toitselforPsychev3toitself[ 122 ].When translatingtheseresultstotheoscillationanalysis,theresultsshowedthattheSKpredictionofthe oscillatedeventdistributionseachfallwithinuncertainties,implyingthe˝naloscillationresults wouldbeconsistent,regardlessofwhetherPsychev1orPsychev3isused. 163 FigureA.9:BANFF˝tresultsfortheobservablenormalizationparameterscomparingthee˙ect ofthedetectorselectionsandsystematicuncertaintiesforPsychev1(red)andPsychev3(blue). Theinputvaluesvaluecanbeseeninthebackground. 164 FigureA.10:Comparisonofthee˙ectofthedetectorselectionsandsystematicuncertaintieson theBANFFpost-˝tdistributionsforPsychev3(left)andPsychev1(right). 165 FigureA.11:BANFF˝tresultsforthe˛uxparameterscomparingthee˙ectofthedetector selectionsandsystematicuncertaintiesandtheMCstatisticalerrorforPsychev1(red)andPsyche v3(blue).Theinputvaluesvaluecanbeseeninthebackground. FigureA.12:BANFF˝tresultsfortheFSIandcrosssectionparameterscomparingthee˙ectof thedetectorselectionsandsystematicuncertaintiesandtheMCstatisticalerrorforPsychev1 (red)andPsychev3(blue).Theinputvaluesvaluecanbeseeninthebackground. 166 FigureA.13:BANFF˝tresultsfortheobservablenormalizationparameterscomparingthee˙ect ofthedetectorselectionsandsystematicuncertaintiesandtheMCstatisticalerrorforPsychev1 (red)andPsychev3(blue).Theinputvaluesvaluecanbeseeninthebackground. 167 FigureA.14:Comparisonofthee˙ectofthedetectorselectionsandsystematicuncertaintiesand theMCstatisticalerrorontheBANFFpost-˝tdistributionsforPsychev3(left)andPsychev1 (right). 168 FigureA.15:SKoscillatedeventpredictionsforthe (upperleft), (upperright), e (lower left), e (lowerright)˛uxes.TheratioisofPsychev1toPsychev3(v1/v3)andthe š plotisof ( j v1-v3 j /(v1error))andmeasuresthedi˙erencebetweenthetwoversionswithrespecttothev1 errorbar. 169 APPENDIXB OBSERVABLENORMALIZATIONCOVARIANCEMATRIXBINNINGSTUDIES Ideally,whengeneratingtheobservablenormalizationcovariancematrix,eachentryalongthe diagonalshouldcorrespondtoasingle p cos ˝tbin.However,whentallyingthenumberof˝t bins,thenumberquicklyreachesanamountthatbecomesimpracticalwhentryingtominimizethe likelihoodfunctiondescribedinSection 4.2.1 usingMinuit.Therefore,abinningischosenforthe detectorcovariancematrixthatcombinesseveral˝tbins. B.1ThoughtsBehindCombiningBins Whencombiningbins,thereareanumberofconsiderationstobearinmind.First,thebins shouldapproximatelyfollowtheshapeofthedata.Therefore,itispreferabletonotcombinebinsin thepeakkinematicregion,which,forT2K,isareconstructedmuonmomentumofapproximately 600MeV/candintheveryforwarddirection,asseeninFigure B.1 .Furthermore,aseachdetector bincorrespondstoa p cos bin,itisbesttolookatthecovariance(orcorrelation)matrixwhere thebinsaregroupedintoblockswiththesameangularbinsor,conversely,wherethebinsare groupedintoblockswiththesamemomentumbins,asshowninFigure B.2 . FigureB.1:MonteCarloeventdistributionofreconstructedmuonmomentumandangle. 170 FigureB.2:Correlationmatriceswherethebinsaregroupedbyangularbins(left)ormomentum bins(right). Inthisbinningstudy,thedetectorcorrelationmatrixwasused,ratherthanthecovariancematrix, asitiseasiertodistinguishregionsthatarehighlycorrelatedusingacorrelationmatrix.Here, regionswithrelativelyhighcorrelationswerecombined,pendingtheywereforbinsoutsidethe peakkinematicregion.Itshouldbenotedthevaluethatde˝nescorrelationisfairlyarbitrary, ascombiningbinsbasedpurelyonthecorrelationisusuallynotsu˚cienttoreducethenumberof binstoareasonablenumber.Therefore,aftercombiningbinsbasedonhighcorrelations,regions ofthecovariancematrixwithcorrelationsofapproximatelythesameamountwerecombinedinto singlebins,asseeninFigure B.3 . B.2ResultsfromtheBinningStudy Inthisstudy,onlytheforwardhorncurrentsampleswerestudied,asthe˝tbinningforthese sampleschangeddramaticallycomparedtopreviousanalyses,whilethereversehorncurrent˝t binninghasnotbeenchangedfrompreviousanalyses.Therefore,onlyresultsfromtheFHC sampleswillbeshown. CC 0 ˇ BinningandCorrelation FortheFGD1FHCCC 0 ˇ sample,theprocessofcombiningbinscanbeseeninFigures B.4 and B.5 ,whiletheeventdistributionsateachstepareshowninFigure B.6 .The˝nalcorrelation 171 FigureB.3:Correlationmatrixbeforecombiningbins(left)andwithredlinesdemarcatingthe regionstobecombined(right). matrixcanbeseeninFigure B.7 .Whilenotshown,theFGD2FHCCC 0 ˇ sampleshowssimilar resultstothoseseenhere. CC 1 ˇ BinningandCorrelation FortheFGD1FHCCC 1 ˇ sample,theprocessofcombiningbinscanbeseeninFigures B.8 and B.9 ,whiletheeventdistributionsateachstepareshowninFigure B.10 .The˝nalcorrelation matrixcanbeseeninFigure B.11 .Whilenotshown,theFGD2FHCCC 1 ˇ sampleshowssimilar resultstothoseseenhere. CCOtherBinningandCorrelation FortheFGD1FHCCCOthersample,theprocessofcombiningbinscanbeseeninFigures B.12 and B.13 ,whiletheeventdistributionsateachstepareshowninFigure B.14 .The˝nal correlationmatrixcanbeseeninFigure B.15 .Whilenotshown,theFGD2FHCCCOthersample showssimilarresultstothoseseenhere. B.3ConclusionandFurtherStudy Inordertoreducethenumberofobservablenormalizationparameters,astudywasperformed tocombinesimilarlycorrelatedbins.Thetotalnumberofbins(alongthediagonal)forthereduced observablenormalizationcovariancematrixcanbeseeninTable B.1 . 172 FigureB.4:CombiningtheCC 0 ˇ angularbins.Beforecombiningbinscanbeseeninthetopleft, thebinstobecombined(seto˙bytheredlines)areinthetopright,andtheresultingmatrixafter combiningbinsisonthebottom. SampleSet NumberofBins FHCMultiPiplusRHCMultiTrack 834 FHCMultiPiplusRHCMultiPi 690 TableB.1:Thetotalnumberofbinsalongthediagonaloftheobservablenormalization covariancematrix. WhilethenumberofbinsshowninTable B.1 wassu˚cientforthisthesis,itisontheupper endofwhatisreasonablewhenperformingstudiesfromtheneardetector˝t.Additionally,a numberofassumptionsweremadewhencombiningbins,suchasonlyusingthecorrelationmatrix 173 FigureB.5:CombiningtheCC 0 ˇ momentumbins.Beforecombiningbinscanbeseeninthetop left,thebinstobecombined(seto˙bytheredlines)areinthetopright,andtheresultingmatrix aftercombiningbinsisonthebottom. forcombiningbinsandnotusingthecovariancematrix.Therefore,amorethoroughstudyof combiningbinsshouldbedonetoensuretheBANFF˝tisprovidingaresulttotheT2Koscillation analysisthatisindependentofthechoiceofbinning. Inordertoperformamorethoroughstudy,twopotentialroutescouldbetaken.First,the combinationprocesscanbedoneinawaythattakesintoaccountdi˙erencesintheerrorbetween adjacentbinsandtheoverallcorrelationofgroupsofbinsthatsharethesameangularormomentum bin.Thisprocesscaneitherbedonebyeye,orthroughtheuseofanalgorithmtoappropriately combinebinsmeetingcriteriadeterminedeitherthroughmachine-learningorthroughhuman means. 174 FigureB.6:TheFGD1FHCCC 0 ˇ p cos distributionsforthe˝tbinning(topleft),thebinning withacombinedangularbinningandthefullmomentumbinning(topright),andthe˝nal detectorbinning(bottom).Eachbinisscaledbythebinarea. Anothermethodtoreducethenumberofbinswouldbetoperformaprincipalcomponent analysis(PCA)oftheobservablenormalizationcovariancematrix,whichwouldshowthesetof parametersthatprovidethelargestfreedomtomoveparametersduringthelikelihoodmaximization. OncethePCAhasbeenperformed,thenumberofparameterscanbereducedinanumberofways: 1. ThelargestXvaluescanbechosen,whereXisaconstantvalue. 2. Plotthevaluesfromlargesttosmallest,cuttingwherethereissharpdropo˙fromonevalue tothenext. 3. Integrateoverthevalues,cuttingafterasetintegralhasbeenmet. 175 FigureB.7:The˝nalcorrelationmatricesfortheFGD1FHCCC 0 ˇ sample.Thebinsare groupedbycommonangularbinsontheleftandcommonmomentumbins(whichistheorder usedinthelikelihood˝t)ontheright. Withregardstothe˝rstpoint,onebene˝twouldbethateach˝twouldcontainthesamenumber ofparameters,regardlessofthenumberofinputparameters.However,dependingonwherethis numberisset,toomuchinformationabouttheobservablenormalizationcovariancematrixcan bethrownaway,likelyreducingthee˙ectivenessofthelikelihood˝t.Astothesecondoption, thiswouldkeepthesetofparameterswhichhaveasmoothtransitionbetweenthem.However, thede˝nitionofpcouldbehardtodetermine.Furthermore,thereisapotentialto keeptoomanyparametersdependingonwherethedropoccurs.Finally,thethirdoptionwould determinewhichparameterscorrespondtosomepercentageoftheinformationcontainedinthe detectorcovariancematrix.Thischoicewouldensurethataminimumamountofinformationis passedintothe˝tregardingthedetector.However,dependingonthepercentagethatischosen,too manyparameterscouldbepassedintothe˝t,makingthe˝ttakeanunreasonableamountoftime. WhetherthenumberofbinsisreducedthroughaPCAorthroughthecarefulcombinationof bins,withtheincreasingnumberof˝tbins,itisessentialthenumberofdetectorcovariancebins bekeptatareasonablelevel,sothattheBANFF˝tcontinuestoruninareasonableamountoftime andcontinueitscontributionstotheT2Koscillationanalysis. 176 FigureB.8:CombiningtheCC 1 ˇ angularbins.Beforecombiningbinscanbeseeninthetopleft, thebinstobecombined(seto˙bytheredlines)areinthetopright,andtheresultingmatrixafter combiningbinsisonthebottom. 177 FigureB.9:CombiningtheCC 1 ˇ momentumbins.Beforecombiningbinscanbeseeninthetop left,thebinstobecombined(seto˙bytheredlines)areinthetopright,andtheresultingmatrix aftercombiningbinsisonthebottom. 178 FigureB.10:TheFGD1FHCCC 1 ˇ p cos distributionsforthe˝tbinning(topleft),the binningwithacombinedangularbinningandthefullmomentumbinning(topright),andthe˝nal detectorbinning(bottom).Eachbinisscaledbythebinarea. 179 FigureB.11:The˝nalcorrelationmatricesfortheFGD1FHCCC 1 ˇ sample.Thebinsare groupedbycommonangularbinsontheleftandcommonmomentumbins(whichistheorder usedinthelikelihood˝t)ontheright. 180 FigureB.12:CombiningtheCCOtherangularbins.Beforecombiningbinscanbeseeninthetop left,thebinstobecombined(seto˙bytheredlines)areinthetopright,andtheresultingmatrix aftercombiningbinsisonthebottom. 181 FigureB.13:CombiningtheCCOthermomentumbins.Beforecombiningbinscanbeseeninthe topleft,thebinstobecombined(seto˙bytheredlines)areinthetopright,andtheresulting matrixaftercombiningbinsisonthebottom. 182 FigureB.14:TheFGD1FHCCCOther p cos distributionsforthe˝tbinning(topleft),the binningwithacombinedangularbinningandthefullmomentumbinning(topright),andthe˝nal detectorbinning(bottom).Eachbinisscaledbythebinarea. 183 FigureB.15:The˝nalcorrelationmatricesfortheFGD1FHCCCOthersample.Thebinsare groupedbycommonangularbinsontheleftandcommonmomentumbins(whichistheorder usedinthelikelihood˝t)ontheright. 184 APPENDIXC FINE-GRAINEDDETECTOR2RELATEDPLOTS FGD2FHCCC0 ˇ FGD2FHCCC1 ˇ FGD2FHCCCOther FigureC.1:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD2FHCMultiPi selections.Themomentumdistributionsareshownontheleft,whiletheangulardistributionsare shownontheright. 185 FGD2FHCCC0 ˇ FGD2FHCCC1 ˇ FGD2FHCCCOther FigureC.2:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD2FHCMultiPiselections. 186 FGD2FHCCC0 ˇ FGD2FHCCC1 ˇ FGD2FHCCCOther FigureC.3:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD2FHCMultiPiselections. 187 FGD2RHC CC1-Track FGD2RHC CCN-Tracks FigureC.4:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD2RHC MultiTrackselections.Themomentumdistributionsareshownontheleft,whiletheangular distributionsareshownontheright. 188 FGD2RHC CC1-Track FGD2RHC CCN-Tracks FigureC.5:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD2RHC MultiTrackselections. 189 FGD2RHC CC1-Track FGD2RHC CCN-Tracks FigureC.6:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD2RHC MultiTrackselections. 190 FGD2RHC CC1-Track FGD2RHC CCN-Tracks FigureC.7:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD2RHC MultiTrackselections.Themomentumdistributionsareshownontheleft,whiletheangular distributionsareshownontheright. 191 FGD2RHC CC1-Track FGD2RHC CCN-Tracks FigureC.8:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD2RHC MultiTrackselections. 192 FGD2RHC CC1-Track FGD2RHC CCN-Tracks FigureC.9:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD2RHC MultiTrackselections. 193 FGD2RHC CC0 ˇ FGD2RHC CC1 ˇ FGD2RHC CCOther FigureC.10:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD2RHC MultiPiselections.Themomentumdistributionsareshownontheleft,whiletheangular distributionsareshownontheright. 194 FGD2RHC CC0 ˇ FGD2RHC CC1 ˇ FGD2RHC CCOther FigureC.11:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD2RHC MultiPiselections. 195 FGD2RHC CC0 ˇ FGD2RHC CC1 ˇ FGD2RHC CCOther FigureC.12:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD2RHC MultiPiselections. 196 FGD2RHC CC0 ˇ FGD2RHC CC1 ˇ FGD2RHC CCOther FigureC.13:KinematicdistributionsusedasinputstotheBANFF˝tfortheFGD2RHC MultiPiselections.Themomentumdistributionsareshownontheleft,whiletheangular distributionsareshownontheright. 197 FGD2RHC CC0 ˇ FGD2RHC CC1 ˇ FGD2RHC CCOther FigureC.14:Selectione˚ciencyasafunctionoftruemuonmomentum(left)ortruemuon cos (right)fortheFGD2RHC MultiPiselections. 198 FGD2RHC CC0 ˇ FGD2RHC CC1 ˇ FGD2RHC CCOther FigureC.15:Selectionpurityasafunctionofreconstructedmuonmomentum(left)or reconstructedmuon cos (right)fortheFGD2RHC MultiPiselections. 199 FHCMultiPiSamples RHC MultiPiSamples RHC MultiPiSamples FigureC.16:Ratiosofthemodi˝ed p cos distribution,whichincludesadditionalantineutrino singlepionproductionevents,tothenominaldistributionforFGD2.TheCC 0 ˇ samples(left), CC 1 ˇ samples(middle),andCCOthersamples(right)areshown. 200 FHCMultiPiSamples RHC MultiPiSamples RHC MultiPiSamples FigureC.17:Ratiosofthemodi˝ed p cos distribution,whichincludesthelow Q 2 suppression weight,tothenominaldistributionforFGD2.TheCC 0 ˇ samples(left),CC 1 ˇ samples(middle), andCCOthersamples(right)areshown. 201 BIBLIOGRAPHY 202 BIBLIOGRAPHY [1] J.Myslik,MeasurementofmuonantineutrinodisappearanceintheT2KExperiment,Ph.D. thesis,UniversityofVictoria(2016),( https://t2k.org/docs/thesis/074 ) [2] F.Capozzietal.,constraintsonabsoluteneutrinomassesandtheirorder Phys.Rev.Dvol.95p.096014(2017),doi:10.1103/PhysRevD.95.096014,URL https: //link.aps.org/doi/10.1103/PhysRevD.95.096014 [3] M.Tanabashietal.(ParticleDataGroup),eviewofParticlePhyPhys.Rev.D vol.98p.030001(2018),doi:10.1103/PhysRevD.98.030001,URL https://link.aps. org/doi/10.1103/PhysRevD.98.030001 [4] S.Bienstocketal.,iveSampleJointOscillationAnalysiswithT2KRun1-9 Tech.Rep.367,T2K(2018),InternalT2KTechnicalNote, https://t2k.org/docs/ technotes/367 [5] C.Andreopoulosetal.,T2KNeutrinoandAnti-Neutrino3-FlavourJointAnalysisofRun 1-9( 1 : 4938 10 21 -POT 1 : 6346 10 21 -POT )dataTech.Rep.360,T2K(2018), InternalT2KTechnicalNote, https://t2k.org/docs/technotes/360 [6] M.G.Catanesietal.,multiplepion and backgroundeventselectionsintheND280 trackerusingRun5ctoRun7bTech.Rep.273,T2K(2018),InternalT2KTechnical Note, https://t2k.org/docs/technotes/273 [7] P.BartetandO.Thers, CCeventselectionsintheND280trackerusingRun2+3+4 Tech.Rep.212,T2K(2015),InternalT2KTechnicalNote, https://www.t2k.org/ docs/technotes/212 [8] Y.PetrovandA.Hillairet,TPCTrackReconstructionTech.Rep.163, T2K(2016),InternalT2KTechnicalNote, https://www.t2k.org/docs/technotes/ 163 [9] A.Hillairet,T.Lindner,J.Myslik,andP.Stamoulis,trackertracking Tech.Rep.075,T2K(2012),InternalT2KTechnicalNote, https://www.t2k.org/docs/ technotes/075 [10] J.Kim,C.Nielsen,andM.Wilking,helElectronTaggingintheTech.Rep.104, T2K(2015),InternalT2KTechnicalNote, https://www.t2k.org/docs/technotes/ 104 [11] K.Mahn,S.Oser,andT.Lindner,masschecTech.Rep.122,T2K(2012),Internal T2KTechnicalNote, https://www.t2k.org/docs/technotes/122 [12] F.Dufour,L.Haegel,T.Lindner,andS.Oser,stematicsonOut-of-Fiducial-Volume BackgroundsintheND280TrackTech.Rep.098,T2K(2015),InternalT2KTechnical Note, https://www.t2k.org/docs/technotes/098 203 [13] J.A.FormaggioandG.P.Zeller,eVtoEeV:Neutrinocrosssectionsacrossenergy Rev.Mod.Phys.vol.84(3)pp.(2012),doi:10.1103/RevModPhys.84. 1307,URL https://link.aps.org/doi/10.1103/RevModPhys.84.1307 [14] S.Gollapinni,eutrinoCrossSectionF(2016), https://arxiv.org/pdf/1602. 05299.pdf [15] Y.Fukudaetal.(Super-KamiokandeCollaboration),forOscillationofAtmo- sphericNeutrPhys.Rev.Lett.vol.81pp.(1998),doi:10.1103/PhysRevLett. 81.1562,URL https://link.aps.org/doi/10.1103/PhysRevLett.81.1562 [16] K.Abeetal.,TheT2KexperNuclearInstrumentsandMethodsinPhysicsResearch SectionA:Accelerators,Spectrometers,DetectorsandAssociatedEquipmentvol.659(1) pp.106135(2011),doi:https://doi.org/10.1016/j.nima.2011.06.067,URL http://www. sciencedirect.com/science/article/pii/S0168900211011910 [17] K.Abeetal.(T2Kcollaboration),neutrino˛uxPhys.Rev.Dvol.87(1) p.012001(2013),doi:10.1103/PhysRevD.87.012001 [18] T2KCollaboration,andfrequentlyupdatedPrivatecommunication(2018) [19] K.Suzukietal.,ofthemuonbeamdirectionandmuon˛uxfortheT2Kneu- trinoexperProgressofTheoreticalandExperimentalPhysicsvol.2015(5)p.053C01 (2015),doi:10.1093/ptep/ptv054,URL http://dx.doi.org/10.1093/ptep/ptv054 [20] J.Imber,X.Li,andJ.Palomino,-KamiokandeeventdisplaysforT2KRun5-6 analysisTech.Rep.219,T2K(2015),InternalT2KTechnicalNote, https: //www.t2k.org/docs/technotes/219/tn219supplements/ [21] A.Fiorentinietal.,PredictionandUncertaintyUpdateswithNA612009ThinTarget DataandNegativeFocussingModeTech.Rep.217,T2K(2018),InternalT2K TechnicalNote, https://www.t2k.org/docs/technotes/217 [22] W.M.Alberico,M.Ericson,andA.Molinari,Theroleoftwoparticle-twoholeexcitations inthespin-isospinnuclearAnnalsofPhysicsvol.154(2)pp.356395(1984), doi:https://doi.org/10.1016/0003-4916(84)90155-6,URL http://www.sciencedirect. com/science/article/pii/0003491684901556 [23] I.RuizSimoetal.,elativisticmodelof2p-2hmesonexchangecurrentsin(anti)neutrino scatterJournalofPhysicsG:NuclearandParticlePhysicsvol.44(6)p.065105(2017), URL http://stacks.iop.org/0954-3899/44/i=6/a=065105 [24] S.Bolognesietal.,Gmodelanduncertaintiesfor2017oscillationanalyTech. Rep.315,T2K(2017),InternalT2KTechnicalNote, https://www.t2k.org/docs/ technotes/315 [25] K.Abeetal.(TheT2KCollaboration),ofneutrinoandantineutrinooscilla- tionsbytheT2Kexperimentincludinganewadditionalsampleof e interactionsatthefar Phys.Rev.Dvol.96p.092006(2017),doi:10.1103/PhysRevD.96.092006,URL https://link.aps.org/doi/10.1103/PhysRevD.96.092006 204 [26] F.L.Wilson,ermi'sTheoryofBetaDecaAmericanJournalofPhysicsvol.36(12) pp.(1968),doi:10.1119/1.1974382, https://doi.org/10.1119/1.1974382 , URL https://doi.org/10.1119/1.1974382 [27] D.Gri˚ths,IntroductiontoElementaryParticles,Wiley-VCH,second,revisededn.(2008) [28] C.M.G.Lattes,H.Muirhead,G.P.S.Occhialini,andC.F.Powell,OCESSESINVOLV- INGCHARGEDNaturevol.159pp.(1947),doi:10.1038/159694a0, [,42(1947)] [29] E.J.KonopinskiandH.M.Mahmoud,TheUniversalFermiPhys.Rev. vol.92pp.(1953),doi:10.1103/PhysRev.92.1045,URL https://link.aps. org/doi/10.1103/PhysRev.92.1045 [30] F.Reines,Theneutrino:frompoltergeisttoparRev.Mod.Phys.vol.68pp. (1996),doi:10.1103/RevModPhys.68.317,URL https://link.aps.org/doi/10.1103/ RevModPhys.68.317 [31] F.ReinesandC.L.Cowan,TheneutrNaturevol.178pp.(1956) [32] C.L.Cowanetal.,oftheFreeNeutrino:aCon˝rSciencevol.124(3212) pp.(1956),doi:10.1126/science.124.3212.103, http://science.sciencemag. org/content/124/3212/103.full.pdf ,URL http://science.sciencemag.org/ content/124/3212/103 [33] M.Gell-Mann,TheinterpretationofthenewparticlesasdisplacedchargeIl NuovoCimento(1955-1965)vol.4(2)pp.(1956),doi:10.1007/BF02748000,URL https://doi.org/10.1007/BF02748000 [34] B.Pontecorvo,andMuonNeutrSov.Phys.JETPvol.10pp. (1960),[Zh.Eksp.Teor.Fiz.37,1751(1959)] [35] T.D.Lee,mediatebosonhypothesisofweakin10th InternationalConferenceonHigh-EnergyPhysics(ICHEP60):Rochester,NY,USA,25 Aug-1Seppp.(1960) [36] G.Danbyetal.,vationofHigh-EnergyNeutrinoReactionsandtheExistenceofTwo KindsofNeutrPhys.Rev.Lett.vol.9pp.(1962),doi:10.1103/PhysRevLett.9.36, URL https://link.aps.org/doi/10.1103/PhysRevLett.9.36 [37] M.L.Perletal.,forAnomalousLeptonProductionin e + e Phys.Rev.Lett.vol.35pp.(1975),doi:10.1103/PhysRevLett.35.1489,URL https://link.aps.org/doi/10.1103/PhysRevLett.35.1489 [38] K.Kodamaetal.,vationoftauneutrinoPhysicsLettersBvol.504(3) pp.218224(2001),doi:https://doi.org/10.1016/S0370-2693(01)00307-0,URL http: //www.sciencedirect.com/science/article/pii/S0370269301003070 [39] B.T.Clevelandetal.,ofthesolarelectronneutrino˛uxwiththeHomestake chlorineAstrophys.J.vol.496pp.(1998) 205 [40] R.Davis,D.S.Harmer,andK.C.Ho˙man,hforNeutrinosfromthePhys. Rev.Lett.vol.20pp.(1968),doi:10.1103/PhysRevLett.20.1205,URL https: //link.aps.org/doi/10.1103/PhysRevLett.20.1205 [41] J.N.Bahcall,N.A.Bahcall,andG.Shaviv,StatusoftheTheoreticalPre- dictionsforthe 37 Cl Solar-NeutrinoExperPhys.Rev.Lett.vol.20pp. 1212(1968),doi:10.1103/PhysRevLett.20.1209,URL https://link.aps.org/doi/10. 1103/PhysRevLett.20.1209 [42] P.A.Cherenkov,isibleemissionofcleanliquidsbyactionof Doklady AkademiiNaukSSSRvol.2pp.451+(1934),URL http://ufn.ru/en/articles/2007/ 4/g/ [43] K.S.Hirataetal.,esultsfromonethousanddaysofreal-time,directionalsolar-neutrino Phys.Rev.Lett.vol.65pp.(1990),doi:10.1103/PhysRevLett.65.1297, URL https://link.aps.org/doi/10.1103/PhysRevLett.65.1297 [44] J.N.BahcallandP.I.Krastev,hepneutrinosa˙ectthesolarneutrinoenergyspec- trPhysicsLettersBvol.436(3)pp.243250(1998),doi:https://doi.org/10.1016/ S0370-2693(98)00920-4,URL http://www.sciencedirect.com/science/article/ pii/S0370269398009204 [45] N.Jelley,A.B.McDonald,andR.G.H.Robertson,TheSudburyNeutrinoObservator AnnualReviewofNuclearandParticleSciencevol.59(1)pp.(2009),doi: 10.1146/annurev.nucl.55.090704.151550, https://doi.org/10.1146/annurev.nucl. 55.090704.151550 ,URL https://doi.org/10.1146/annurev.nucl.55.090704. 151550 [46] Q.R.Ahmadetal.(SNOCollaboration),oftheRateof e + d ! p + p + e InteractionsProducedby 8 B SolarNeutrinosattheSudburyNeutrinoObservatorPhys. Rev.Lett.vol.87p.071301(2001),doi:10.1103/PhysRevLett.87.071301,URL https: //link.aps.org/doi/10.1103/PhysRevLett.87.071301 [47] Q.R.Ahmadetal.(SNOCollaboration),EvidenceforNeutrinoFlavorTrans- formationfromNeutral-CurrentInteractionsintheSudburyNeutrinoObservatorPhys. Rev.Lett.vol.89p.011301(2002),doi:10.1103/PhysRevLett.89.011301,URL https: //link.aps.org/doi/10.1103/PhysRevLett.89.011301 [48] E.Kearns,imentalmeasurementsofatmosphericneutrNuclearPhysicsB- ProceedingsSupplementsvol.70(1)pp.315323(1999),doi:https://doi.org/10.1016/ S0920-5632(98)00441-1,proceedingsoftheFifthInternationalWorkshopontopicsinAs- troparticleandUndergroundPhysics,URL http://www.sciencedirect.com/science/ article/pii/S0920563298004411 [49] K.S.Hirataetal.,vationofasmallatmospheric / e ratioin PhysicsLettersBvol.280(1)pp.146152(1992),doi:https://doi.org/10.1016/ 0370-2693(92)90788-6,URL http://www.sciencedirect.com/science/article/ pii/0370269392907886 206 [50] B.Pontecorvo,eutrinoExperimentsandtheProblemofConservationofLeptonic ChargSov.Phys.JETPvol.26pp.(1968),[Zh.Eksp.Teor.Fiz.53,1717(1967)] [51] Z.Maki,M.Nakagawa,andS.Sakata,emarksontheUni˝edModelofEle- mentaryParProgressofTheoreticalPhysicsvol.28(5)pp.(1962), doi:10.1143/PTP.28.870, http://oup.prod.sis.lan/ptp/article-pdf/28/5/870/ 5258750/28-5-870.pdf ,URL https://dx.doi.org/10.1143/PTP.28.870 [52] J.B.Albertetal.(EXO-200Collaboration),hfor 2 decayof 136 Xe tothe 0 + 1 excitedstateof 136 Ba withtheEXO-200liquidxenonPhys.Rev.Cvol.93p. 035501(2016),doi:10.1103/PhysRevC.93.035501,URL https://link.aps.org/doi/ 10.1103/PhysRevC.93.035501 [53] W.Xuetal.,TheMajoranaDemonstrator:ASearchforNeutrinolessDouble-betaDe- cayJournalofPhysics:ConferenceSeriesvol.606p.012004(2015),doi: 10.1088/1742-6596/606/1/012004,URL https://doi.org/10.1088%2F1742-6596% 2F606%2F1%2F012004 [54] K.Abeetal.,NuclearInstrumentsandMethodsinPhysicsRe- searchSectionA:Accelerators,Spectrometers,DetectorsandAssociatedEquipment vol.716pp.7885(2013),doi:https://doi.org/10.1016/j.nima.2013.03.059,URL http: //www.sciencedirect.com/science/article/pii/S0168900213003690 [55] G.C.Branco,R.GonzálezFelipe,andF.R.Joaquim, CP Rev. Mod.Phys.vol.84pp.(2012),doi:10.1103/RevModPhys.84.515,URL https: //link.aps.org/doi/10.1103/RevModPhys.84.515 [56] C.R.Das,J.Pulido,J.Maalampi,andS.Vihonen,minationofthe 23 octantin longbaselineneutrinoexperimentswithinandbeyondthestandardPhys.Rev.D vol.97p.035023(2018),doi:10.1103/PhysRevD.97.035023,URL https://link.aps. org/doi/10.1103/PhysRevD.97.035023 [57] G.AltarelliandF.Feruglio,˛avorsymmetriesandmodelsofneutrino Rev.Mod.Phys.vol.82pp.(2010),doi:10.1103/RevModPhys.82.2701,URL https://link.aps.org/doi/10.1103/RevModPhys.82.2701 [58] A.E.CárcamoHernández,S.Kovalenko,J.W.F.Valle,andC.A.Vaquera-Araujo, dictivePati-SalamtheoryoffermionmassesandJournalofHighEnergyPhysics vol.2017(7)p.118(2017),doi:10.1007/JHEP07(2017)118,URL https://doi.org/10. 1007/JHEP07(2017)118 [59] S.P.MikheyevandA.Yu.Smirnov,esonanceAmpli˝cationofOscillationsinMatter andSpectroscopyofSolarNeutrSov.J.Nucl.Phys.vol.42pp.(1985), [,305(1986)] [60] L.Wolfenstein,eutrinooscillationsinPhys.Rev.Dvol.17pp. (1978),doi:10.1103/PhysRevD.17.2369,URL https://link.aps.org/doi/10.1103/ PhysRevD.17.2369 207 [61] E.K.Akhmedov,eutrinophyinSummerSchoolinParticlePhysics: Trieste,Italy,June21-July9,pp.(1999), hep-ph/0001264 [62] M.Freund,Analyticapproximationsforthreeneutrinooscillationparametersandprobabil- itiesinPhys.Rev.Dvol.64p.053003(2001),doi:10.1103/PhysRevD.64.053003, URL https://link.aps.org/doi/10.1103/PhysRevD.64.053003 [63] K.Abeetal.(T2KCollaboration),ofNeutrinoOscillationinAppearance andDisappearanceChannelsbytheT2KExperimentWith 6 : 6 10 20 ProtonsonTarg Phys.Rev.Dvol.91(7)p.072010(2015),doi:10.1103/PhysRevD.91.072010 [64] S.P.MikheyevandA.Yu.Smirnov,esonantampli˝cationof oscillationsinmatter andsolar-neutrinospectroscopIlNuovoCimentoCvol.9(1)pp.(1986),doi: 10.1007/BF02508049,URL https://doi.org/10.1007/BF02508049 [65] J.N.Abdurashitovetal.,oftheresponseofaGasolarneutrinoexperi- menttoneutrinosfroma 37 Ar Phys.Rev.Cvol.73p.045805(2006),doi:10. 1103/PhysRevC.73.045805,URL https://link.aps.org/doi/10.1103/PhysRevC. 73.045805 [66] C.Giuntietal.,pdateofshort-baselineelectronneutrinoandantineutrino Phys.Rev.Dvol.86p.113014(2012),doi:10.1103/PhysRevD.86.113014,URL https: //link.aps.org/doi/10.1103/PhysRevD.86.113014 [67] A.A.Aguilar-Arevaloetal.(MiniBooNECollaboration),ExcessofElectronlike EventsintheMiniBooNEShort-BaselineNeutrinoExperPhys.Rev.Lett.vol.121 p.221801(2018),doi:10.1103/PhysRevLett.121.221801,URL https://link.aps.org/ doi/10.1103/PhysRevLett.121.221801 [68] Ade,P.A.R.etal.(PlanckCollaboration),k2015results-XIII.Cosmologi- calA&Avol.594p.A13(2016),doi:10.1051/0004-6361/201525830,URL https://doi.org/10.1051/0004-6361/201525830 [69] M.G.Aartsenetal.(IceCubeCollaboration),hforsterileneutrinomixingusing threeyearsofIceCubeDeepCorePhys.Rev.Dvol.95p.112002(2017),doi:10. 1103/PhysRevD.95.112002,URL https://link.aps.org/doi/10.1103/PhysRevD. 95.112002 [70] K.Abeetal.(T2KCollaboration),hforshortbaseline e disappearancewiththeT2K nearPhys.Rev.Dvol.91p.051102(2015),doi:10.1103/PhysRevD.91.051102, URL https://link.aps.org/doi/10.1103/PhysRevD.91.051102 [71] K.M.Tsui,SterileneutrinooscillationstudieswiththeT2KfardetectorSuper-Kamiokande, Ph.D.thesis,TheUniversityofTokyo(2019),( https://t2k.org/docs/thesis/095 ) [72] T.Sekiguchietal.,velopmentandoperationalexperienceofmagnetichornsys- temforT2KexperNuclearInstrumentsandMethodsinPhysicsResearchSec- tionA:Accelerators,Spectrometers,DetectorsandAssociatedEquipmentvol.789 208 pp.5780(2015),doi:https://doi.org/10.1016/j.nima.2015.04.008,URL http://www. sciencedirect.com/science/article/pii/S0168900215004672 [73] K.Suzukietal.,ofthemuonbeamdirectionandmuon˛uxfortheT2K neutrinoexperProgressofTheoreticalandExperimentalPhysicsvol.2015(5)p. 053C01(2015),doi:10.1093/ptep/ptv054 [74] D.Beavisetal.,BaselineNeutrinoOscillationExperimentattheA(1995), E889PhysicsDesignReport,BNL-52459,URL http://puhep1.princeton.edu/ ~mcdonald/nufact/e889/chapter3a.pdf [75] K.Abeetal.,oftheT2KneutrinobeampropertiesusingtheIN- GRIDon-axisnearNuclearInstrumentsandMethodsinPhysicsResearch SectionA:Accelerators,Spectrometers,DetectorsandAssociatedEquipmentvol.694 pp.211223(2012),doi:https://doi.org/10.1016/j.nima.2012.03.023,URL http://www. sciencedirect.com/science/article/pii/S0168900212002987 [76] M.Yokoyamaetal.,velopmentofMulti-PixelPhotoneConfvol.C0604032 p.0126(2006), arXiv:physics/0605241[physics.ins-det] [77] D.Allanetal.,TheelectromagneticcalorimeterfortheT2KneardetectorJour- nalofInstrumentationvol.8(10)p.P10019(2013),URL http://stacks.iop.org/ 1748-0221/8/i=10/a=P10019 [78] S.Assylbekovetal.,TheT2KND280o˙-axispi-zeroNuclearInstruments andMethodsinPhysicsResearchSectionA:Accelerators,Spectrometers,Detectorsand AssociatedEquipmentvol.686pp.4863(2012),doi:10.1016/j.nima.2012.05.028 [79] P.-A.Amaudruzetal.,TheT2KFine-GrainedNuclearInstrumentsandMeth- odsinPhysicsResearchSectionA:Accelerators,Spectrometers,DetectorsandAssociated Equipmentvol.696pp.131(2012),doi:10.1016/j.nima.2012.08.020 [80] N.Abgralletal.,TimeProjectionChambersfortheT2KNearNuclearInstru- mentsandMethodsinPhysicsResearchSectionA:Accelerators,Spectrometers,Detectors andAssociatedEquipmentvol.637(1)pp.2546(2011),doi:10.1016/j.nima.2011.02.036 [81] G.Charpak,J.Derre,Y.Giomataris,andP.Rebourgeard,OMEGAS,amul- tipurposegaseousNucl.Instrum.Meth.vol.A478pp.(2002),doi: 10.1016/S0168-9002(01)01713-2 [82] S.Fukudaetal.,TheSuper-KamiokandeNuclearInstrumentsandMeth- odsinPhysicsResearchSectionA:Accelerators,Spectrometers,DetectorsandAs- sociatedEquipmentvol.501(2)pp.418462(2003),doi:https://doi.org/10.1016/ S0168-9002(03)00425-X,URL http://www.sciencedirect.com/science/article/ pii/S016890020300425X [83] J.Kameda,pdatedstudyofthesystematicerrorin disappearanceanalysisfromSuper- Tech.Rep.159,T2K(2013),InternalT2KTechnicalNote, https://www. t2k.org/docs/technotes/163 209 [84] L.Alvarez-Rusoetal.,uSTEC11NeutrinoScatteringTheoryExperimentCollaboration http://nustec.fnal.gov.WhitePaper:Statusandchallengesofneutrscatter ProgressinParticleandNuclearPhysicsvol.100pp.168(2018),doi:https://doi.org/10. 1016/j.ppnp.2018.01.006,URL http://www.sciencedirect.com/science/article/ pii/S0146641018300061 [85] N.Abgralletal.,A61/SHINEfacilityattheCERNSPS:beamsanddetectorsys JournalofInstrumentationvol.9(06)p.P06005(2014),URL http://stacks.iop.org/ 1748-0221/9/i=06/a=P06005 [86] L.Aliagaetal.,,calibration,andperformanceoftheMINERvANu- clearInstrumentsandMethodsinPhysicsResearchSectionA:Accelerators,Spectrometers, DetectorsandAssociatedEquipmentvol.743pp.130159(2014),doi:https://doi.org/10. 1016/j.nima.2013.12.053,URL http://www.sciencedirect.com/science/article/ pii/S0168900214000035 [87] R.Akutsuetal.,-KamiokandeeventsanddataqualitystudiesforT2KRun Tech.Rep.317,T2K(2017),InternalT2KTechnicalNote, https://www.t2k.org/docs/ technotes/317 [88] X.LiandM.Wilking,iTQunEventSelectionTech.Rep.319,T2K(2017), InternalT2KTechnicalNote, https://www.t2k.org/docs/technotes/319 [89] A.Kaboth,R.Calland,andD.Payne,AJointND280-SK1R -SK1R e Fitusing Tech.Rep.171,T2K(2014),InternalT2KTechnicalNote, https://www.t2k.org/docs/ technotes/171 [90] F.JamesandM.Roos,-asystemforfunctionminimizationandanalysisofthe parametererrorsandcorComputerPhysicsCommunicationsvol.10(6)pp. 343367(1975),doi:https://doi.org/10.1016/0010-4655(75)90039-9,URL http://www. sciencedirect.com/science/article/pii/0010465575900399 [91] R.BrunandF.Rademakers,OOTAnobjectorienteddataanalysisframewNu- clearInstrumentsandMethodsinPhysicsResearchSectionA:Accelerators,Spectrometers, DetectorsandAssociatedEquipmentvol.389(1)pp.8186(1997),doi:https://doi.org/ 10.1016/S0168-9002(97)00048-X,URL http://www.sciencedirect.com/science/ article/pii/S016890029700048X [92] Y.Hayato,AneutrinointeractionsimulationprogramlibraryActaPhys.Polon. vol.B(40)pp.(2009) [93] J.Nieves,I.RuizSimo,andM.J.VicenteVacas,echarged-currentneutrino-nucleus Phys.Rev.Cvol.83p.045501(2011),doi:10.1103/PhysRevC.83.045501,URL https://link.aps.org/doi/10.1103/PhysRevC.83.045501 [94] S.Bolognesietal.,Assessingthee˙ectofcross-sectionmodeluncertaintiesonthe T2KoscillationanalyseswithfakedatastudiesusingtheBANFF,MaCh3andVALOR ˝tframewTech.Rep.285,T2K(2016),InternalT2KTechnicalNote, https: //t2k.org/docs/technotes/285 210 [95] S.Dulatetal.,ewpartondistributionfunctionsfromaglobalanalysisofquantumchro- Phys.Rev.Dvol.93p.033006(2016),doi:10.1103/PhysRevD.93.033006, URL https://link.aps.org/doi/10.1103/PhysRevD.93.033006 [96] R.Gran,J.Nieves,F.Sanchez,andM.J.VicenteVacas,eutrino-nucleusquasi-elastic and2p2hinteractionsupto10GePhys.Rev.Dvol.88p.113007(2013),doi:10. 1103/PhysRevD.88.113007,URL https://link.aps.org/doi/10.1103/PhysRevD. 88.113007 [97] C.Colleetal.,themassdependenceandquantumnumbersofshort-range correlatedpairsfrom A ¹ e ; e 0 p º and A ¹ e ; e 0 pp º scatterPhys.Rev.Cvol.92p. 024604(2015),doi:10.1103/PhysRevC.92.024604,URL https://link.aps.org/doi/ 10.1103/PhysRevC.92.024604 [98] D.ReinandL.M.Sehgal,eutrino-excitationofbaryonresonancesandsinglepionpro- AnnalsofPhysicsvol.133(1)pp.79153(1981),doi:https://doi.org/10.1016/ 0003-4916(81)90242-6,URL http://www.sciencedirect.com/science/article/ pii/0003491681902426 [99] L.Fieldsetal.(MINERvACollaboration),ofMuonAntineutrinoQuasielas- ticScatteringonaHydrocarbonTargetat E ˘ 3 : 5GeV Phys.Rev.Lett.vol.111 p.022501(2013),doi:10.1103/PhysRevLett.111.022501,URL https://link.aps.org/ doi/10.1103/PhysRevLett.111.022501 [100] MelanieDayandKevinS.McFarland,inquasielasticcrosssectionsofmuon andelectronneutrPhys.Rev.Dvol.86p.053003(2012),doi:10.1103/PhysRevD.86. 053003,URL https://link.aps.org/doi/10.1103/PhysRevD.86.053003 [101] K.S.KimandM.K.Cheoun,inalstateinteractionandCoulombe˙ectforneutrino-nucleus scatteringinthequasielasticAIPConferenceProceedingsvol.1189(1)pp. (2009),doi:10.1063/1.3274148, https://aip.scitation.org/doi/pdf/10.1063/1. 3274148 ,URL https://aip.scitation.org/doi/abs/10.1063/1.3274148 [102] T.Feuselsetal.,TuningoftheNEUTCascadeModelusing ˇ -AScatteringExternalData toImproveFinalStateInteractionandSecondaryInteractionSystematicUncer Tech.Rep.325,T2K(2017),InternalT2KTechnicalNote, https://www.t2k.org/docs/ technotes/325 [103] E.S.PinzonGuerraetal.(DUETCollaboration),of ˙ ABS and ˙ CX of ˇ + oncarbonbytheDualUseExperimentatTRIUMFPhys.Rev.Cvol.95p. 045203(2017),doi:10.1103/PhysRevC.95.045203,URL https://link.aps.org/doi/ 10.1103/PhysRevC.95.045203 [104] P.dePerio,Y.Hayato,andR.Tacik,nuclearTech.Rep.033,T2K(2012), InternalT2KTechnicalNote, https://www.t2k.org/docs/technotes/033 [105] S.Agostinellietal.,asimulationNuclearInstrumentsandMethods inPhysicsResearchSectionA:Accelerators,Spectrometers,DetectorsandAssociated 211 Equipmentvol.506(3)pp.(2003),doi:10.1016/S0168-9002(03)01368-8,URL https://www.sciencedirect.com/science/article/pii/S0168900203013688 [106] M.Bassetal.,assessmentstrategyofthe2010bdatasetatND280.Tech.Rep.021, T2K(2011),InternalT2KTechnicalNote, https://www.t2k.org/docs/technotes/ 021 [107] E.Frank,A.Marchionni,andM.Messina,calibrationandsystematicer Tech.Rep.081,T2K(2011),InternalT2KTechnicalNote, https://www.t2k.org/docs/ technotes/081 [108] C.Bojechkoetal.,andCorrectionofMagneticFieldDistortionsinthe TimeProjectionTech.Rep.061,T2K(2012),InternalT2KTechnicalNote, https://www.t2k.org/docs/technotes/061 [109] A.CerveraandL.Escudero,tudyofmomentumresolutionandscaleusingtracksthat crossmultipleTech.Rep.222,T2K(2015),InternalT2KTechnicalNote, https: //www.t2k.org/docs/technotes/222 [110] W.OryszczakandW.Warzycha,systematics:PIDandIsoReconhybrid Tech.Rep.223,T2K(2015),InternalT2KTechnicalNote, https://www.t2k.org/docs/ technotes/223 [111] F.SanchezandJ.voMedina,globalchargeidenti˝cationsystematicerTech. Rep.229,T2K(2016),InternalT2KTechnicalNote, https://www.t2k.org/docs/ technotes/229 [112] J.Myslik,minationofpionsecondaryinteractionsystematicsfortheND280tracker analyTech.Rep.125,T2K(2013),InternalT2KTechnicalNote, https://www. t2k.org/docs/technotes/125 [113] A.Cerveraetal.,Multi-TopologySelectionandSystematicsinTech. Rep.216,T2K(2015),InternalT2KTechnicalNote, https://www.t2k.org/docs/ technotes/216 [114] S.Bienstocketal.,Assessingthee˙ectofcross-sectionmodeluncertaintiesontheT2K oscillationanalyseswithsimulateddatastudiesusingtheBANFF,MaCh3,P-Thetaand VALOR˝tframewTech.Rep.331,T2K(2018),InternalT2KTechnicalNote, https: //www.t2k.org/docs/technotes/331 [115] M.Kabirnezhad,pionproductioninneutrino-nucleonPhys.Rev.D vol.97p.013002(2018),doi:10.1103/PhysRevD.97.013002,URL https://link.aps. org/doi/10.1103/PhysRevD.97.013002 [116] C.L.McGivernetal.(MINERvACollaboration),sectionsfor and induced pionproductiononhydrocarboninthefew-GeVregionusingMINERvAPhys.Rev.D vol.94p.052005(2016),doi:10.1103/PhysRevD.94.052005,URL https://link.aps. org/doi/10.1103/PhysRevD.94.052005 212 [117] O.Altinoketal.,of charged-currentsingle ˇ 0 productiononhydrocarbon inthefew-GeVregionusingMINERvAPhys.Rev.Dvol.96p.072003(2017),doi:10. 1103/PhysRevD.96.072003,URL https://link.aps.org/doi/10.1103/PhysRevD. 96.072003 [118] P.Adamsonetal.(MINOSCollaboration),tudyofquasielasticscatteringusingcharged- current -ironinteractionsintheMINOSnearPhys.Rev.Dvol.91p. 012005(2015),doi:10.1103/PhysRevD.91.012005,URL https://link.aps.org/doi/ 10.1103/PhysRevD.91.012005 [119] D.S.Ayresetal.(NOvA),NOvATechnicalDesignRepor(2007),doi:10.2172/ 935497 [120] M.Sanchez,OvAResultsand(2018),doi:10.5281/zenodo.1286758,URL https://doi.org/10.5281/zenodo.1286758 [121] J.Wolcott,ofcrosssectionmodellingonNO Aoscillationanaly(2018),URL https://indico.cern.ch/event/703880/contributions/3159021/ [122] J.Morrison,M.Scott,andJ.Walker,ANFFPsychev1/v3ComparT2K CollaborationMeetingParallelTalk, https://t2k.org/asg/oagroup/meeting/2018/ 2018-12-03/v1v3banff 213