MEASUREMENTOFTHECHARGEDCURRENTMUONNEUTRINO DIFFERENTIALCROSSSECTIONONSCINTILLATORWITHZEROPIONSINTHE FINALSTATEWITHTHET2KON/OFF-AXISNEARDETECTORS. By AndrewBruceCudd ADISSERTATION Submittedto MichiganStateUniversity inpartialful˝llmentoftherequirements forthedegreeof PhysicsDoctorofPhilosophy 2020 ABSTRACT MEASUREMENTOFTHECHARGEDCURRENTMUONNEUTRINO DIFFERENTIALCROSSSECTIONONSCINTILLATORWITHZEROPIONSINTHE FINALSTATEWITHTHET2KON/OFF-AXISNEARDETECTORS. By AndrewBruceCudd TheTokaitoKamioka(T2K)isalong-baselineneutrinooscillationexperimentinJapan producingprecisemeasurementsofneutrinooscillationsandneutrinointeractionswithnu- cleartargets.T2Kutilizesamuon(anti-)neutrinobeamproducedattheJ-PARCproton acceleratorfacilitywhichismeasuredatasuiteofneardetectors,ND280andINGRID,and thefardetector,Super-Kamiokande.Oneofthedominantsystematicuncertaintiesforthe oscillationanalysisisfromtheuncertaintyinneutrinointeractionmodelingwithcomplex nuclei,whichwilleventuallybecomethelimitinguncertaintyforthenextgenerationofneu- trinooscillationexperiments.Thereforemeasurementsofneutrinocrosssectionsonnuclear targetsisessentialforunderstandinghowtomodelthesecomplicatednuclearinteractions. Thisthesispresentsanovelneutrinocrosssectionmeasurementutilizingbothofthe T2Kneardetectors,ND280andINGRID,inajointstatistical˝t.BecausetheT2Knear detectorsareexposedtoneutrinosfromthesamebeamline,theuncertaintiesintheneutrino ˛uxpredictionwillbecorrelated.Thisfactcombinedwiththedi˙erentneutrinoenergy spectraseenateachdetectorwillallowforsomeseparationof˛uxandcrosssectione˙ects, andpresentsanopportunitytostudytheneutrinocrosssectionasafunctionofenergy usingthesameneutrinobeam.Thisanalysisisthe˝rstcrosssectionmeasurementon T2Ktousesamplesfrommultipledetectorsinthesamebeamline.Thisthesispresents adescriptionofthestatisticalanalysisframework,theeventselection,thetreatmentof systematicuncertainties,andtheextracted CC- 0 ˇ doubledi˙erentialcrosssectionin binsofmuonkinematicsforND280andINGRID,includingthecorrelationsbetweenthe detectors. Copyrightby ANDREWBRUCECUDD 2020 ACKNOWLEDGEMENTS Despitethefactthatthereisonlyasingleauthoronthisthesis,therearenumerouspeople whohelpedmewritethistitanicdocumentwhichcapturesthelast˝veyearsorsoofmy workandlife(whichwhendividedevenlyequatestoroughly6.8daysofmylifeperpageat thetimeofwriting).TrulythisworkandIstandontheshouldersofgiants. Iwanttothankmyparents,LisaandSteven,andsister,Megan,forsupportingmefrom thebeginningandallthroughoutschool.Thankyoufortheneverendingencouragementto followmyinterestsandideas,especiallywhentheyincludedbuildinganarcadecabinetthat remainsinyourgarage,andforalwaysbeingtherewhenlifewasbothgreatandawful.I nowwill˝nallyjointherestofthefamilyinholdinganadvanceddegreeandnolongerbeing astudent. Thisthesiswouldnothavebeenpossibleintheslightestwithouttheguidanceofmy advisor,Kendall.Thankyoufortakingtheriskofhiringmeasyour˝rstgraduatestudent andpushingmetosucceed,evenwhenIhadzeroideawhatIwasdoing.Iappreciate immenselyhowyoumanagedto˝tmeinyourschedulenearlywheneverIneededtotalk aboutmyanalysis,ormorerecently,myjobsearch,andalltheexcitementyouhavefor sciencewhichhelpedremindmethatmyworkismorethanjustmakingprettyplots.You arethebestadvisor. AhugethankstoallofmyfellowT2Kcollaboratorswhohaveallcontributedinsome waytomythesisanalysis,fromhelpingruntheexperimenttoprovidingnecessarysoftware, datainputs,anddiscussion.InparticularIwanttothank(innoparticularorder),Stephen, Ciro,Margherita,Callum,Benjmain,andDanforalltheirknowledgeandexpertiseoncross sectionanalysesandstatistical˝tting.Lukeforbeingmypostdoc/T2Kcollaboratorand helpingmeoutwithwhateverIneededwithmyanalysis,andforbeingamodelofgoodcode. Luciefortestingmy˝tsoftwareand˝ndingallsortsofbugs,andthenpatientlywaitingfor metowakeupand˝xthemduetothetimezonedi˙erence.YueandMitchellandtherest iv oftheUS/CanadaT2KgroupformakingMinouchiafunplacetostay.MarkHartzand Nakaya-sanforhelpingmeapplyforfellowshipsandMarkhostingmewhenIsucceededthat onetime.ThankyoutorestofT2Kformakingthisanincredibleexperimentandwelcoming experience(andforgivingmeareasontotraveltoJapan). ThankyoutotheMSUT2K/DUNEgroupformakingtheo˚ceafunplaceworkall theseyears.ThankyouJacobforthevaluablediscussionsonT2Kwork,ROOTissues, andsports,andforaccompanyingmewhiletravelingJapan.ThankyouJakeandDanfor participatinginourteambuildingexercises(i.e.playingSmash),forprovidinganon-T2K perspectivewhenIneededitformyanalysis,andfortrustingmetogiveyouadviceonyour analyses(evenifitwasbad). Agoodwbalancewasessentialforsurvivingmytimeingraduateschool.Thelife partwouldnothavebeennearlyasfunandsatisfyingwithoutBryan,Lisa,Sean,Zak,and (math)Brandonbeingmyregulargaminggroup.Theweekendsplayingboardgamesand rollingdicewasanecessaryescapefromthestressofgraduateschoollife,andagreatspace tocomplainaboutallthatstress.Iwillmissourusualfalltraditionsofwillinglygettinglost inamazeofcornandpickingapples. ThankyoutotheMSUsupportsta˙,particularlyBrendaandKim,formakingnearly alltheadministrativeworkinvisibletome.Brenda,thanksforhelpingmeschedulemy numerous˛ightsandtravelarrangements.Kim,thanksforhelpingmeandKendall˝gure outwhatweweresupposedtobedoingtogetmegraduated.DeanandMike,thanksfor takingthetimetohelpmeoutwheneverIneededitinthelab,especiallywhenitincluded detonatingabarrelfullofwaterusingLN2.ThankyouCarlforgettingmekickstartedat MSUandwritingallthosereferenceletters. Finally,DOESupportprovidedbyU.S.DOEAwardDE-SC0015903. v TABLEOFCONTENTS LISTOFTABLES..................................ix LISTOFFIGURES.................................xi CHAPTER1EXECUTIVESUMMARY....................1 1.1Analysisoverview.................................1 1.2Speci˝ccontributions...............................2 CHAPTER2NEUTRINOTHEORY......................4 2.1Introductiontoneutrinos.............................4 2.2Neutrinooscillations...............................7 2.2.1Neutrinooscillationtheory........................7 2.2.2Statusofneutrinooscillations......................10 2.3Neutrinointeractions...............................15 2.3.1Interactionsonfreenucleons.......................15 2.3.2Nuclearmediume˙ects..........................17 2.3.3Coherentscattering............................20 2.3.4Resonantpionproduction........................21 2.3.5Shallowanddeepinelasticscattering..................21 2.3.6Finalstateinteractions..........................22 2.3.7Neutrinoeventgenerators........................23 2.3.8Currentstatusandmotivation......................24 CHAPTER3T2KLONGBASELINENEUTRINOEXPERIMENT...27 3.1Beamsetup....................................28 3.2Fluxprediction..................................32 3.3TheInteractiveNeutrinoGRIDdetector....................34 3.4TheNearDetectorat280meters........................36 3.4.1Fine-GrainedDetectors..........................39 3.4.2TimeProjectionChambers........................41 3.4.3ElectromagneticCalorimeters......................43 3.4.4UA1MagnetandSideMuonRangeDetectors.............44 CHAPTER4EVENTSELECTION.......................46 4.1Signalde˝nition..................................46 4.1.1ND280signalde˝nition..........................47 4.1.2INGRIDsignalde˝nition.........................47 4.2MonteCarloanddatasamples..........................48 4.3Signaleventselection...............................50 4.3.1ND280eventselection..........................51 4.3.2INGRIDeventselection.........................64 4.4Sidebandselection................................70 vi 4.4.1ND280sidebandselection........................70 4.4.2INGRIDsidebandselection.......................76 4.5Analysisbinning.................................80 4.5.1ND280...................................80 4.5.2INGRID..................................82 4.6E˚ciencyandpurity...............................84 4.6.1ND280...................................84 4.6.2INGRID..................................85 CHAPTER5SYSTEMATICUNCERTAINTIES...............87 5.1Fluxsystematicuncertainties..........................87 5.1.1ND280integrated˛ux..........................91 5.1.2INGRIDintegrated˛ux.........................91 5.2Detectorsystematicuncertainties........................91 5.2.1ND280detectorsystematics.......................92 5.2.2INGRIDdetectorsystematics......................97 5.3Neutrinointeractionsystematicuncertainties..................100 5.4Numberoftargetssystematicuncertainty....................104 5.4.1ND280targetuncertainty........................105 5.4.2INGRIDtargetuncertainty.......................105 5.5E˚ciencyuncertainty...............................105 CHAPTER6STATISTICALFIT.........................107 6.1Crosssectionde˝nition..............................107 6.2Fitmethod....................................108 6.3Errorpropagation.................................113 6.4Integrated˛ux..................................115 6.5Numberoftargets.................................117 CHAPTER7FITVALIDATION.........................120 7.1Asimov˝ts....................................121 7.1.1Nominalpriors..............................121 7.1.2Randomtemplatepriors.........................127 7.1.3ND280onlyAsimov˝t..........................132 7.1.4INGRIDonlyAsimov˝t.........................137 7.2Statistical˛uctuations..............................140 7.3Systematicparametervariations.........................144 7.4Degreesoffreedom................................149 7.5Neutrinoenergyweights.............................152 7.6Alteredsignalweights..............................157 7.7Low Q 2 suppressionofresonantevents.....................162 7.8MINERvAdata-drivenweightsofsignalevents.................167 7.9AlternateRPAmodel...............................172 7.10Horncurrentvariation..............................179 7.11Hornalignmentvariation.............................185 vii 7.12Fitfailuremodes.................................190 7.13Summary.....................................190 CHAPTER8RESULTS..............................193 8.1Reconstructedeventrate.............................193 8.1.1ND280eventsamples...........................193 8.1.2INGRIDeventsamples..........................195 8.2ND280onlydata˝t................................199 8.3INGRIDonlydata˝t...............................206 8.4Jointdata˝t...................................210 8.5Futurework....................................219 8.6Conclusions....................................219 APPENDICES....................................221 APPENDIXAFITSOFTWARE..................... 222 APPENDIXBBARLOW-BEESTON................... 223 APPENDIXCREGULARIZATION................... 225 APPENDIXDPRINCIPALCOMPONENTANALYSIS....... 228 APPENDIXEND280SAMPLERECONSTRUCTEDBINNING.. 231 APPENDIXFEVENTDISTRIBUTIONSIN Q 2 ........... 234 APPENDIXGEXTRACTEDCROSSSECTIONADDITIONAL PLOTS............................ 238 BIBLIOGRAPHY..................................241 viii LISTOFTABLES Table2.1:NeutrinooscillationparametervaluesfromthePDG[1],whicharecal- culatedusingaglobal˝tofneutrinodata...................11 Table3.1:Neutrinoproductiondecaymodesforneutrinobeam(FHC)mode.....30 Table3.2:ListofND280subdetectorsandtheirprimaryfunction...........39 Table4.1:Data-takingperiodsandthePOTusedinthisanalysisfordataandMC forND280....................................49 Table4.2:Data-takingperiodsandthePOTusedinthisanalysisfordataandMC forINGRID...................................49 Table4.3:SummaryoftheND280data/MCcorrections(left)andINGRIDdata/MC corrections(right)................................50 Table4.4:PurityofeachND280signalsampleandthepurityofthecombinedtotal.58 Table4.5:PurityoftheINGRIDsignalsampleforbothstoppingonlyandstopping plusthrough-goingtracks............................68 Table4.6:PurityofeachND280sidebandsample....................72 Table4.7:PurityoftheINGRIDsidebandsampleforbothstoppingonlyandstop- pingplusthrough-goingtracks.........................78 Table4.8:ND280binningusedfortheextractedcrosssectionanddatadistribution inmuonkinematics p ; cos .........................81 Table4.9:INGRIDbinningusedfortheextractedcrosssection(left)anddata distribution(right)inmuonkinematics.Notethatthedataismeasured usingmuondistanceinironandthecrosssectionismeasuredinmuon momentum....................................82 Table5.1:Theneutrinoenergybinningusedforthe˛uxsystematicparameters. BoththeND280andINGRID˛uxparametersusethesameenergybin- ning,andaretreatedasseparateparametersinthe˝t............90 Table5.2:ListofND280detectorsystematicparametersandtheirpropagationtype (looselygroupedbydetectororgeneral).Eachparameterisvariedinthe simulationtoproducethedetectorcovarianceamtrixusedinthe˝t....96 ix Table5.3:ListofINGRIDdetectorsystematicparameters,looselygroupedbyef- fect.Eachparameterisvariedinthesimulationtoproducethedetector covariancematrixusedinthe˝t........................99 Table5.4:Neutrinointeractionmodelingparametersusedinthisanalysisalong withtheirindex,type,prior,anderror.ValuestakenfromRef.[2,3]...102 Table6.1:Informationusedtocomputethetotalnumberofnucleonsforeachchem- icalelementoftheFGD1˝ducialvolume[4].................118 Table6.2:Informationusedtocomputethetotalnumberofnucleonsforeachchem- icalelementoftheProtonModule˝ducialvolume[5].Siliconisconsid- eredtohaveanegligiblecontribution.....................119 Table7.1:Listofstudiesusedtovalidatetheanalysis..................122 Table7.2:ParametervaluesusedfortheBeRPAweights................173 Table8.1:Breakdownofthepost-˝t ˜ 2 contributionfortheND280onlydata˝tat thereconstructedeventlevelandforthesystematicparameterpenalty...200 Table8.2:Breakdownofthepost-˝t ˜ 2 contributionfortheINGRIDonlydata˝t atthereconstructedeventlevelandforthesystematicparameterpenalty.207 Table8.3:Breakdownofthepost-˝t ˜ 2 contributionforthejointdata˝tatthe reconstructedeventlevelandforthesystematicparameterpenalty.....211 TableE.1:ND280reconstructedbinningusedforthe FGDsample(left)and FGD+pTPC sample(right)..................................232 TableE.2:ND280reconstructedbinningforallnon FGDsamples...........233 x LISTOFFIGURES Figure1.1:Simple˛owchartdiagramofthemajorstepsoftheanalysis.Chapter4 detailsthedataandMCinput,Chapter6detailsthelikelihood˝tand crosssectioncalculation,andChapter8showstheresults.........1 Figure2.1:Schematicoftheproposedexperimenttouseanatomicbombasan intensesourceofneutrinos.FigurefromRef.[6]..............5 Figure2.2:Left:Comparisonofmeasured90%con˝dencelevelcontoursfor m 2 32 vs sin 2 23 forT2K(2017)[7],NOvA(2018),SK(2018)[8],IceCube (2018)[9],andMINOS(2014)[10];Fig.fromRef.[11].Right:Up- dated2018T2Kcontoursincludingadditionaldataandanalysisim- provements;Fig.fromRef.[12]........................12 Figure2.3:TheneutrinoenergyspectrumfortheT2Kexperiment(bothon-/o˙- axisneardetectorsandfardetectorafteroscillation),plustheneutrino energyspectrumforNOvAandtheMINERvAlowenergymode.Over- laidaretheneutrinocrosssectionpredictionsfordi˙erentcombinations ofinteractionchannels.Notethatthedi˙erentenergyspectracovera wideenergyrangewherethecrosssectionpredictionschangerapidly...13 Figure2.4:ReconstructedenergydistributionsatSKforthe (left)and (right) -enrichedsampleswiththetotalpredictedeventrateshowninred. Ratiostothepredictionsunderthenooscillationhypothesisareshown inthebottom˝gures.FigurefromRef.[12].................13 Figure2.5:E˙ectofthe 1 ˙ variationsofthesystematicuncertaintiesonthepre- dictedeventrateforT2K(left)andNOvA(right).T2KtablefromRef. [7]andNOvAtablefromRef.[11]......................14 Figure2.6:Left:The WN ! Nˇ verticesconsideredbytheNievesmodelfor multinucleonprocesses.Onlyonenucleonlineisshown,thesecond nucleonisimpliedtobecoupledtothevirtualpion.Fig.fromRef.[13]. Right:Thedistributionofthestrengthofthemultinucleonprocesses aspredictedbytheNieveset.al.modelinenergy( q 0 )andmomentum ( q 3 )transferspace...............................20 xi Figure2.7:Neutrinoenergyreconstructioncalculatedforamonoenergeticsourceof 600MeVneutrinos,showingthecontributionforwithandwithoutmult- inucleonprocesses.Ifthereconstructionwasperfect,thereconstructed energywouldbeadeltafunctionat600MeV.Insteadthereconstructed energyisspreadout,andshowsabiastowardlowerenergyfromthe multinucleonprocesses.Fig.fromRef.[7]..................26 Figure3.1:SchematicoftheT2KexperimentshowingthelocationofJ-PARC,the nearandfardetectors,andthe295kmbaseline...............27 Figure3.2:Neutrinoenergyasafunctionofparentpionenergyforvariouso˙-axis angles.Astheo˙-axisangleincreases,theneutrinoenergybecomes increasinglyindependentofthepionenergy.T2Kusesa2.5degrees o˙-axisbeamcorrespondingtoaneutrinoenergypeakof0.6GeV.....31 Figure3.3:Theneutrino˛uxshownatdi˙erento˙-axisangles(arbitrarilynormal- ized)comparedtotheneutrinooscillationprobability.T2Kusesa2.5 degreeo˙-axisbeamcorrespondingtotheoscillationmaximumat0.6 GeV.FigurefromRef.[14]..........................31 Figure3.4:Accumulatedprotonsontargetforbothneutrinoandanti-neutrino beammodesandbeampowerforeachT2Kdatarun(theshadedred regions).ThisanalysisusesdatafromRun2,3,4and8...........32 Figure3.5:Thepredicted˛uxatND280andSKforforwardhorncurrentrunning separatedbyneutrino˛avoraveragedoverT2Kruns1-9[15].......33 Figure3.6:TheINGRIDdetectorcon˝gurationshowingthehorizontalandvertical planes(top)andadepictionofanINGRIDstandardmodule(bottom). TheINGRIDstandardmoduleisshownhighlightingtheironplates andinnerscintillatorpanelsontheleftandtheoutervetopanelsonthe right.FigurefromRef.[16]..........................35 Figure3.7:ViewoftheProtonModule.Similartothestandardmodulesbutcon- taining˝nergrainedscintillatorbarsandnoironplates.Figurefrom Ref.[16]....................................36 Figure3.8:Neutrinobeamdirectioninboththeverticalandhorizontalposition measuredbyINGRIDandMUMONforeachT2Krunperiod.Ingeneral INGRIDandMUMONmatchquitewellandtheneutrinobeamposition isfairlystable.TheeventratemeasuredbyINGRIDisalsoplottedand verystableacrosstheT2Krunperiods....................37 xii Figure3.9:ExplodedviewoftheND280detectorshowingtheinnertrackingregion, witheachsub-detectorvisible.TheSMRDsareinterleavedwiththe magnetyoke.Takenfrom[16].........................38 Figure3.10:ND280eventdisplayshowingthePØD,FGDs,andTPCs.Aneutrino interactionoccurredinFGD1producingmanytrackswithanunrelated muontraversingthePØDandTPCs.ThedownstreamECALisshown, whilethebarrelECALandSMRDsarenotshown.............38 Figure3.11:ViewoftheFGD1scintillatorbarsshowingtheorientationofthelayers. Eachpairofhorizontalandverticallayersofscintillatorbarsisde˝ned asasingleXYmodule.ThedimensionsgivetheoverallsizeoftheFGD andits˝ducialvolume.............................40 Figure3.12:Cut-awayschematicviewofaTPCmoduleshowingthemainaspects oftheTPCdesign...............................41 Figure3.13:Measuredenergylossversusmomentumforpositively(top)andnega- tively(bottom)chargedparticlestraversingtheTPC.Plottedarethe expectedenergylosscurvesforelectron,muons,protons,andpions. FiguresfromRef.[17]............................43 Figure4.1:Examplesignalandbackgroundeventsketchesforagenericdetector. Aneutrinointeractsinthedetectorandseveralparticlesareproduced. Signaleventshavezeropions,whilebackgroundeventshaveoneormore pionsinthe˝nalstate.............................47 Figure4.2:EventdisplaycartoonfortheND280signalsamples............53 Figure4.3:Chartshowingtheselectioncutsusedtode˝neeachND280signalsample.57 Figure4.4:Eventdistributionforreconstructedmuonmomentumandangleforthe ND280signalsampleswithamuontrackintheTPCstackedbytrue topology.Thepurityofeachtopologyislistedinthelegend.Thelast binformuonmomentumcontainsalleventswithmomentumgreater than5GeV/c..................................59 Figure4.5:Eventdistributionforreconstructedmuonmomentumandangleforthe ND280signalsampleswithamuontrackintheFGDstackedbytrue topology.Thepurityofeachtopologyislistedinthelegend.Thelast binformuonmomentumcontainsalleventswithmomentumgreater than5GeV/c..................................60 xiii Figure4.6:Eventdistributionforreconstructedmuonmomentumandangleforthe ND280signalsampleswithamuontrackintheTPCstackedbytrue reaction.Thepurityofeachreactionislistedinthelegend.Thelast binformuonmomentumcontainsalleventswithmomentumgreater than5GeV/c..................................61 Figure4.7:Eventdistributionforreconstructedmuonmomentumandangleforthe ND280signalsampleswithamuontrackintheFGDstackedbytrue reaction.Thepurityofeachreactionislistedinthelegend.Thelast binformuonmomentumcontainsalleventswithmomentumgreater than5GeV/c..................................62 Figure4.8:Twodimensionaleventdistributionforreconstructedmuonmomentum vsanglefortheND280signalsampleswithamuontrackintheTPC (left)oramuontrackintheFGD(right).Thecombinationisshown inthebottomplot...............................63 Figure4.9:EventdisplayfortheProtonModuleshowingthedi˙erentINGRID samples.Greenistrackingscintillator,blueisvetoscintillator,and grayareironplates.FigurefromRef.[5]..................65 Figure4.10:Eventdistributionforreconstructedmuonequivalentdistanceiniron andanglefortheINGRID(early)stoppingsignalsamples(top)andall INGRIDsignalsamples(bottom)stackedbytruetopology.........68 Figure4.11:Eventdistributionforreconstructedmuonequivalentdistanceiniron andanglefortheINGRID(early)stoppingsignalsample(top)andall INGRIDsignalsamples(bottom)stackedbytruereaction.........69 Figure4.12:EventdisplaycartoonfortheND280sidebandsamples...........71 Figure4.13:Chartshowingtheselectioncutsusedtode˝neeachND280sideband sample.....................................72 Figure4.14:Eventdistributionforreconstructedmuonmomentumandanglefor theND280sidebandsamplesstackedbytruetopology.Thepurityof eachtopologyislistedinthelegend.Thelastbinformuonmomentum containsalleventswithmomentumgreaterthan5GeV/c.........73 Figure4.15:Eventdistributionforreconstructedmuonmomentumandanglefor theND280sidebandsamplesstackedbytruereaction.Thepurityof eachreactionislistedinthelegend.Thelastbinformuonmomentum containsalleventswithmomentumgreaterthan5GeV/c.........74 xiv Figure4.16:Twodimensionaleventdistributionforreconstructedmuonmomen- tumvsanglefortheND280sidebandsamples.CC- 1 ˇ inthetopleft, CC-Otherinthetopright,CC-Michelatthebottom............75 Figure4.17:Areanormalizedeventdistributionsasafunctionof Q 2 fortheback- groundeventscomparedbetweenthesignalsamplesandthecorrespond- ingsidebands.TheCC- 1 ˇ + distributionisontheleft,andmatches quitewell.TheCC-Otherdistributionisontherightandisquitedif- ferentbetweenthesignalandsideband....................75 Figure4.18:Eventdistributionforreconstructedmuonequivalentdistanceiniron andanglefortheINGRID(early)stoppingsidebandsamplesandall INGRIDsidebandsamplesstackedbytruetopology.Theroughness ofthedistributionisduetothecombinationoflowstatisticsandthe binningchoicetomatchthesignalregion..................78 Figure4.19:Eventdistributionforreconstructedmuonequivalentdistanceiniron andanglefortheINGRID(early)stoppingsidebandsamplesandall INGRIDsidebandsamplesstackedbytruereaction.Theroughness ofthedistributionisduetothecombinationoflowstatisticsandthe binningchoicetomatchthesignalregion..................79 Figure4.20:SmearingmatrixbetweentrueandreconstructedvariablesfortheIN- GRIDCC- 0 ˇ selectionusingtheINGRID(early)stoppingsamples(top) andallINGRIDsignalsamples(bottom)..................83 Figure4.21:ND280signale˚ciencyforeachsampleasafunctionoftruemuon momentum(left)andtruemuonangle(right)................85 Figure4.22:ND280totalsignale˚ciency(left)andpurity(right)asafunctionof bothtruemuonmomentumandangle....................85 Figure4.23:INGRIDsignale˚ciencyforthestoppingsample(top)andallsamples combined(bottom)asafunctionoftruemuonmomentum(left)andtrue muonangle(right).Notethee˚ciencyaxisscaleisdi˙erentbetween thetopandbottomplots...........................86 Figure5.1:Fluxcovariance(left)andcorrelation(right)matrices.Thebinnumber correspondstotheparameternumberinTab.5.1,andND280isthe ˝rst20binsandINGRIDthesecond20bins................89 Figure5.2:Therelative˛uxuncertaintyforINGRID(right)andND280(left)asa functionofneutrinoenergybins,separatedbytheuncertaintysource. The˛uxuncertaintyforbothdetectorsisdominatedbytheuncertainty inthehadroninteractionmodel.The10to30GeVbinisnotshown...89 xv Figure5.3:ND280nominal˛uxprediction(left)usingtheneutrinoenergybinning forthe˛uxuncertainties(notethatthe10to30GeVbinisnotshown). Thedistributionofintegrated˛uxthrows(right)whichgivetheinte- grated˛uxerrorforthecrosssectionextraction...............91 Figure5.4:INGRIDnominal˛uxprediction(left)usingtheneutrinoenergybin- ningforthe˛uxuncertainties(notethatthe10to30GeVbinisnot shown).Thedistributionofintegrated˛uxthrows(right)whichgive theintegrated˛uxerrorforthecrosssectionextraction..........92 Figure5.5:ND280detectorcorrelationmatrixbinnedinmuonkinematicsforeach sample.Thedetectormatrixisgenerallystronglycorrelatedasthe detectorsystematicstendtoa˙ectthesameregionsofkinematicphase spaceandsamples...............................97 Figure5.6:INGRIDdetectorcorrelationmatrixbinnedinmuonkinematicsforeach sample.Thedetectormatrixisgenerallystronglycorrelatedasthe detectorsystematicstendtoa˙ectthesameregionsofkinematicphase space......................................98 Figure5.7:Crosssectionsystematicparametercovariancematrix(top)andcorre- lationmatrix(bottom).Thecovariancevalueshavebeennormalizedto betherelativevariance,andheparametersarearrangedfollowingtheir indiceslistedinTab.5.4............................101 Figure5.8:CovariancematrixofthearealdensitiesforFGD1usedtodetermine theuncertaintyinthenumberoftargetsforND280.FigurefromRef.[3]106 Figure6.1:Nominal˛uxpredictionatND280(left)andINGRID(right)byneutrino ˛avor.The˛uxpredictioniscorrectedforthebeamconditionsforeach runindividually................................116 Figure7.1:Pre/post-˝tparameterplotsfortheAsimov˝twithnominalpriors....123 Figure7.2:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithnominal priors,ND280signalsamplesonly......................124 Figure7.3:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithnominal priors,ND280sidebandsamplesonly.....................125 Figure7.4:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithnominal priors,INGRIDsamplesonly.........................125 Figure7.5:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)fortheAsimov˝twithnominalpriors...............126 xvi Figure7.6:Pre/post-˝tparameterplotsfortheAsimov˝twithrandomtemplate priors......................................128 Figure7.7:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithrandom templatepriors,ND280signalsamplesonly.................129 Figure7.8:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithrandom templatepriors,ND280sidebandsamplesonly...............130 Figure7.9:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithrandom templatepriors,INGRIDsamplesonly....................130 Figure7.10:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)fortheAsimov˝twithrandomtemplatepriors..........131 Figure7.11:Pre/post-˝tparameterplotsfortheAsimov˝tusingonlytheND280 samples.....................................133 Figure7.12:Pre/post-˝treconstructedeventplotsfortheAsimov˝tusingonlythe ND280samples,signalsamplesonly.....................134 Figure7.13:Pre/post-˝treconstructedeventplotsfortheAsimov˝tusingonlythe ND280samples,sidebandsamplesonly....................135 Figure7.14:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)fortheAsimov˝tusingonlytheND280samples..........136 Figure7.15:Pre/post-˝tparameterplotsfortheAsimov˝tusingonlytheINGRID samples.....................................138 Figure7.16:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithrandom templatepriorspriors,INGRIDsamplesonly................138 Figure7.17:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)fortheAsimov˝tusingonlytheINGRIDsamples.........139 Figure7.18:Pre/post-˝tparameterplotsforthe˝ttostatistical˛uctuations......140 Figure7.19:Pre/post-˝treconstructedeventplotsforthe˝ttostatistical˛uctua- tions,ND280signalsamplesonly.......................141 Figure7.20:Pre/post-˝treconstructedeventplotsforthe˝ttostatistical˛uctua- tions,ND280sidebandsamplesonly.....................142 xvii Figure7.21:Pre/post-˝treconstructedeventplotsforthe˝ttostatistical˛uctua- tions,INGRIDsamplesonly..........................142 Figure7.22:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)forthe˝ttostatistical˛uctuations.................143 Figure7.23:Pre/post-˝tparameterplotsforthe˝ttosystematicparametervariations.145 Figure7.24:Pre/post-˝treconstructedeventplotsforthe˝ttosystematicparameter variations,ND280signalsamplesonly....................146 Figure7.25:Pre/post-˝treconstructedeventplotsforthe˝ttosystematicparameter variations,ND280sidebandsamplesonly...................147 Figure7.26:Pre/post-˝treconstructedeventplotsforthe˝ttosystematicparameter variations,INGRIDsamplesonly.......................147 Figure7.27:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)forthe˝ttosystematicparametervariations............148 Figure7.28:Distributionof ˜ 2 valuesbetweenthepost-˝tandthenominalMC crosssectionformanystatisticalandsystematic˛uctuations.Thesolid redcurvecorrespondstoatheoretical ˜ 2 distributionwith70degreesof freedom,andthedashedbluecurvecorrespondstothe˝tted ˜ 2 distribution.150 Figure7.29:Boxplotsshowingthedistributionofpost-˝tcrosssectionvaluesfor eachbin(top)andthedistributionofrelativeerrorsforeachbin(bot- tom)forthestatisticalandsystematic˛uctuations.Thecirclemarkis theaveragevalue,thedashisthemedianvalue,theboxcontains25% aboveandbelowthemedian,thewhiskersextendto1.5timestheinner quartilerange,andoutliersaremarkedwiththex's.............151 Figure7.30:Pre/post-˝tparameterplotsforthe˝ttoneutrinoenergyvariations...153 Figure7.31:Pre/post-˝treconstructedeventplotsforthe˝ttoneutrinoenergyvari- ations,ND280signalsamplesonly......................154 Figure7.32:Pre/post-˝treconstructedeventplotsforthe˝ttoneutrinoenergyvari- ations,ND280sidebandsamplesonly.....................155 Figure7.33:Pre/post-˝treconstructedeventplotsforthe˝ttoneutrinoenergyvari- ations,INGRIDsamplesonly.........................155 Figure7.34:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)forthe˝ttoneutrinoenergyvariations...............156 xviii Figure7.35:Pre/post-˝tparameterplotsforthe˝ttosignaleventvariations.....158 Figure7.36:Pre/post-˝treconstructedeventplotsforthe˝ttosignaleventvaria- tions,ND280signalsamplesonly.......................159 Figure7.37:Pre/post-˝treconstructedeventplotsforthe˝ttosignaleventvaria- tions,ND280sidebandsamplesonly.....................160 Figure7.38:Pre/post-˝treconstructedeventplotsforthe˝ttosignaleventvaria- tions,INGRIDsamplesonly..........................160 Figure7.39:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)forthe˝ttosignaleventvariations.................161 Figure7.40:Pre/post-˝tparameterplotsforthe˝ttolowmomentumtransfersup- pressedresonantevents............................163 Figure7.41:Pre/post-˝treconstructedeventplotsforlowmomentumtransfersup- pressedresonantevents,ND280signalsamplesonly.............164 Figure7.42:Pre/post-˝treconstructedeventplotsforlowmomentumtransfersup- pressedresonantevents,ND280sidebandsamplesonly...........165 Figure7.43:Pre/post-˝treconstructedeventplotsforlowmomentumtransfersup- pressedresonantevents,INGRIDsamplesonly...............165 Figure7.44:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)forthe˝ttolowmomentumtransfersuppressedresonantevents.166 Figure7.45:Pre/post-˝tparameterplotsforthe˝ttolowmomentumtransfersup- pressedsignalevents..............................168 Figure7.46:Pre/post-˝treconstructedeventplotsforlowmomentumtransfersup- pressedsignalevents,ND280signalsamplesonly..............169 Figure7.47:Pre/post-˝treconstructedeventplotsforlowmomentumtransfersup- pressedsignalevents,ND280sidebandsamplesonly............170 Figure7.48:Pre/post-˝treconstructedeventplotsforlowmomentumtransfersup- pressedsignalevents,INGRIDsamplesonly.................170 Figure7.49:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)forthe˝ttolowmomentumtransfersuppressedsignalevents..171 xix Figure7.50:ThenominalNievesrelativisticRPAcorrectionfactorrelativetothe unmodi˝edCCQEcrosssectionisshown(thesolidline)withthe ˙ uncertainties(thedottedlines).[2,13]...................173 Figure7.51:Pre/post-˝tparameterplotsforthe˝ttoanalternateRPAmodel....175 Figure7.52:Pre/post-˝treconstructedeventplotsforthe˝ttoanalternateRPA model,ND280signalsamplesonly......................176 Figure7.53:Pre/post-˝treconstructedeventplotsforthe˝ttoanalternateRPA model,ND280sidebandsamplesonly.....................177 Figure7.54:Pre/post-˝treconstructedeventplotsforthe˝ttoanalternateRPA model,INGRIDsamplesonly.........................177 Figure7.55:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)forthe˝ttoanalternateRPAmodel................178 Figure7.56:Horncurrentweighthistogram,correspondingtoa 3 ˙ increaseofthe nominalhorncurrent.The˝rst20binsareND280,thesecond20are INGRID....................................179 Figure7.57:Pre/post-˝tparameterplotsforthe˝ttoincreasedhorncurrent.....181 Figure7.58:Pre/post-˝treconstructedeventplotsforthe˝ttoincreasedhorncur- rent,ND280signalsamplesonly.......................182 Figure7.59:Pre/post-˝treconstructedeventplotsforthe˝ttoincreasedhorncur- rent,ND280sidebandsamplesonly......................183 Figure7.60:Pre/post-˝treconstructedeventplotsforthe˝ttoincreasedhorncur- rent,INGRIDsamplesonly..........................183 Figure7.61:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)forthe˝ttoincreasedhorncurrent.................184 Figure7.62:Hornalignmentweighthistogram,correspondingtoa 3 ˙ shiftofthe nominalhorn2and3alignment.The˝rst20binsareND280,the second20areINGRID.............................185 Figure7.63:Pre/post-˝tparameterplotsforthe˝ttovariedhornalignment......186 Figure7.64:Pre/post-˝treconstructedeventplotsforthe˝ttovariedhornalign- ment,ND280signalsamplesonly.......................187 xx Figure7.65:Pre/post-˝treconstructedeventplotsforthe˝ttovariedhornalign- ment,ND280sidebandsamplesonly.....................188 Figure7.66:Pre/post-˝treconstructedeventplotsforthe˝ttovariedhornalign- ment,INGRIDsamplesonly.........................188 Figure7.67:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)forthe˝ttovariedhornalignment.................189 Figure7.68:Examplesofagoodlikelihoodscan(left)andabadlikelihoodscan (right)onasingleparameter.Agoodlikelihoodscanshouldbeapprox- imatelyGaussianwithnodiscontinuities...................191 Figure8.1:EventdistributionformeasureddataandMCpredictioninreconstructed muonmomentumandanglefortheND280signalsampleswithamuon trackintheTPCstackedbytruetopology.Thepurityofeachtopology islistedinthelegend.Thelastbinformuonmomentumcontainsall eventswithmomentumgreaterthan5GeV/c................194 Figure8.2:EventdistributionformeasureddataandMCpredictioninreconstructed muonmomentumandanglefortheND280signalsampleswithamuon trackintheFGDstackedbytruetopology.Thepurityofeachtopology islistedinthelegend.Thelastbinformuonmomentumcontainsall eventswithmomentumgreaterthan5GeV/c................195 Figure8.3:EventdistributionformeasureddataandMCpredictioninreconstructed muonmomentumandanglefortheND280sidebandsamplesstackedby truetopology.Thepurityofeachtopologyislistedinthelegend.The lastbinformuonmomentumcontainsalleventswithmomentumgreater than5GeV/c..................................196 Figure8.4:EventdistributionformeasureddataandMCpredictioninreconstructed muonequivalentdistanceinironandanglefortheINGRID(early) stoppingsignalsamples(top)andallINGRIDsignalsamples(bottom) stackedbytruetopology............................197 Figure8.5:EventdistributionformeasureddataandMCpredictioninreconstructed muonequivalentdistanceinironandanglefortheINGRID(early)stop- pingsidebandsamples(top)andallINGRIDsidebandsamples(bottom) stackedbytruetopology............................198 Figure8.6:Pre/post-˝tparameterplotsfortheND280onlydata˝t.Blueispre˝t andredispost˝t,andthefractionalchangesanderrorsarepresented...201 xxi Figure8.7:Pre/post-˝treconstructedeventplotsfortheND280onlydata˝t,signal samplesonly..................................202 Figure8.8:Pre/post-˝treconstructedeventplotsfortheND280onlydata˝t,side- bandsamplesonly...............................203 Figure8.9:Extractedcross-sectionplotshowingallanalysisbins(intruekinemat- ics)fortheND280onlydata˝t.Approximatingthenumberofbins (58)asthedegreesoffreedomgives1.992 ˜ 2 /DOF............204 Figure8.10:ExtractedcrosssectionfortheND280onlydata˝tcomparedtothe nominalMCpredictionasafunctionofmuonmomentumforslicesof muonangle.Thelastmomentumbinto30GeVisnotshown,andnote they-axisisnotthesameacrossalltheplots................205 Figure8.11:Pre/post-˝tparameterplotsfortheINGRIDonlydata˝t.Blueispre˝t andredispost˝t,andthefractionalchangesanderrorsarepresented...207 Figure8.12:Pre/post-˝treconstructedeventplotsfortheINGRIDonlydata˝tin bothlogscale(top)andlinearscale(bottom)forthey-axis........208 Figure8.13:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekine- matics)fortheINGRIDonlydata˝t.Approximatingthenumberof bins(12)asthedegreesoffreedomgives1.967 ˜ 2 /DOF..........209 Figure8.14:ExtractedcrosssectionfortheINGRIDonlydata˝tcomparedtothe nominalMCpredictionasafunctionofmuonmomentumforslicesof muonangle.Thelastmomentumbinto30GeVisnotshown,andnote they-axisisnotthesameacrossalltheplots................209 Figure8.15:Pre/post-˝tparameterplotsforthejointdata˝t.Blueispre˝tandred ispost˝t,andthefractionalchangesanderrorsarepresented.......212 Figure8.16:Pre/post-˝treconstructedeventplotsforthejointdata˝t,ND280signal samplesonly..................................213 Figure8.17:Pre/post-˝treconstructedeventplotsforthejointdata˝t,ND280side- bandsamplesonly...............................214 Figure8.18:Pre/post-˝treconstructedeventplotsforthejointdata˝t,INGRID samplesonlyinbothlogscale(top)andlinearscale(bottom)forthey-axis.215 xxii Figure8.19:Extractedcross-sectionplotshowingallanalysisbins(intruekinemat- ics)forthejointdata˝twhereND280isthe˝rst58binsandINGRID arethelast12bins.Thenumberofdegreesoffreedomis66.4(cf.Sec. 7.4)whichgives2.48 ˜ 2 /DOF........................216 Figure8.20:ND280extractedcrosssectionbinsforjointdata˝tcomparedtothe nominalMCpredictionasafunctionofmuonmomentumforslicesof muonangle.Thelastmomentumbinto30GeVisnotshown,andnote they-axisisnotthesameacrossalltheplots................217 Figure8.21:INGRIDextractedcrosssectionbinsforthejointdata˝tcomparedto thenominalMCpredictionasafunctionofmuonmomentumforslices ofmuonangle.Thelastmomentumbinto30GeVisnotshown,and notethey-axisisnotthesameacrossalltheplots.............218 FigureC.1:ExampleL-curveplot.Eachpointislabeledwiththeregularization strength ,andthepointofmaximumcurvatureforthisplotoccursat ˘ 1 : 0 .....................................227 FigureF.1:Eventdistributionfortruemomentumtransfer( Q 2 )fortheND280 signalsamplesstackedbytruetopology.Thepurityofeachtopologyis listedinthelegend.Thelastbincontainsalleventswith Q 2 > 2 : 5 GeV 2 .234 FigureF.2:Eventdistributionfortruemomentumtransfer( Q 2 )fortheND280 signalsamplesstackedbytruereaction.Thepurityofeachreactionis listedinthelegend.Thelastbincontainsalleventswith Q 2 > 2 : 5 GeV 2 .235 FigureF.3:Eventdistributionfortruemomentumtransfer( Q 2 )fortheND280 sidebandsamplesstackedbytruetopology.Thepurityofeachtopology islistedinthelegend.Thelastbincontainsalleventswith Q 2 > 2 : 5 GeV 2 .236 FigureF.4:Eventdistributionfortruemomentumtransfer( Q 2 )fortheND280 sidebandsamplesstackedbytruereaction.Thepurityofeachreaction islistedinthelegend.Thelastbincontainsalleventswith Q 2 > 2 : 5 GeV 2 .237 FigureG.1:ND280extractedcrosssectionbinsforjointdata˝tcomparedtothe nominalMCpredictionasafunctionofmuonmomentumforslicesof muonangle.Notethey-axisisnotthesameacrossalltheplots......239 FigureG.2:INGRIDextractedcrosssectionbinsforthejointdata˝tcomparedto thenominalMCpredictionasafunctionofmuonmomentumforslices ofmuonangle.Notethey-axisisnotthesameacrossalltheplots....240 xxiii CHAPTER1 EXECUTIVESUMMARY 1.1Analysisoverview Theanalysispresentedinthisthesisisameasurementoftheneutrinocrosssection onplasticscintillatorinasimultaneous˝tofmuonneutrinointeractionsintheT2Knear detectors,ND280andINGRID.Theprimarygoaloftheanalysisistoprovidethe˝rst correlatedmeasurementbetweenthetwoneardetectors,andtoprovideanewanalysis frameworkforfuturecombineddetectormeasurementsonT2K.Chapter2andChapter3 providebackgroundonneutrinotheory,themotivation,andanintroductiontotheT2K experiment.Chapter4describesthedataandMCinputtothestatistical˝t,Chapter 5describesthemodelassumptionsusedforthepredictionandhowtheyareincludedin the˝t,andChapter6describesthe˝tandtheoutputusedtocalculatethecrosssection. FinallyChapter7describesthevalidationoftheanalysisandChapter8presentstheresults. Asimple˛owchartoftheanalysisisshowninFig.1.1toemphasizethedesignofthe measurement. Figure1.1:Simple˛owchartdiagramofthemajorstepsoftheanalysis.Chapter4details thedataandMCinput,Chapter6detailsthelikelihood˝tandcrosssectioncalculation, andChapter8showstheresults. Thisisthe˝rstT2Kanalysistoperformajointcrosssectionmeasurementusingdetectors atdi˙erentpositionsintheneutrinobeam.Thedi˙erentpositionsintheneutrinobeam producedi˙erentneutrino˛uxdistributionsthatarehighlycorrelated.The˛uxcorrelations canbeusedtoreducetheoverall˛uxuncertaintiesandproduceacorrelatedcrosssection 1 measurement.Theresultingcrosssectionwillbeauniquedatasettocomparetotheoretical andempiricalmodelsincludingtheT2KtunedmodelusedfortheT2Koscillationanalysis. 1.2Speci˝ccontributions Theworkpresentedinthisthesisisbuiltuponexistinganalysesande˙ortfromthe T2Kcollaborationandisproperlyreferencedandcitedwhereapplicable.Inadditionto performingtheanalysis,Icontributednewandoriginalcomponentstomakethisanalysis possiblewhicharebrie˛ysummarizedhere.Thisgoalofthisanalysisistoperformajoint statistical˝toftwoexistinganalyses(onefromND280andonefromINGRID)whichare largelyusedintheiroriginalformsbutwereupdatedinseveralwaysbymyself.TheND280 analysiswasprocessedusingthelatestMCupdates/corrections,newerversionsofinternal T2Ksoftware,andusesnewlycollecteddata.TheINGRIDanalysiswasupdatedtomatch thesystematicuncertaintytreatmentastheND280analysis.Ialsohelpeddevelopthe multinucleonshapeuncertaintyandparameterizationusedinthisanalysisandusedinthe T2Koscillationanalysis. Thejointmeasurementrequiresauni˝edtreatmentofthe˛uxmodelbetweenthetwo analyses.IrantheT2Kbeamand˛uxsimulationtocreatethenecessaryinputsforthis analysis,andIproducedtheo˚cialT2K˛uxresultforthe2017winteroscillationanalysis. Additionally,I˝xedseveralbugsinthe˛uxsimulation,improvedthescriptsusedtorunthe simulationchain,andwrotethenew˛uxsimulationdocumentation. Themajorityofmycontributionsareindevelopingthenextversionofthestatistical ˝ttingsoftwarewhichmakesthejoint˝tpossible.Anewversionofthe˝tsoftwarewas writtentoaccommodatemultipledetectorsandisdesignedtobeagenerictoolforallT2K crosssectionanalyses,andprovidesimprovedperformanceasanalysesincreaseincomplexity. The˝tsoftwarewasparallelizedusingOpenMP,madecon˝gurableviainputtext˝les, nowincludesanimprovedtreatmentofMonteCarlostatisticalerror,andprovidesbuilt-in principlecomponentanalysistolistthemajorupdates.Newexecutableswerealsowritten 2 toperformthetransformationofanalysisspeci˝cinputsintoagenericformatusedinthe main˝tsoftware.ThenewcodeisversioncontrolledviaGitandisnowbeingusedbyother T2Kcrosssectionanalyses.Finally,Ihavebeenexploringnewwaystovalidatecrosssection analysesandpresentthedataanddiagnosticsinthemosthelpfulway. 3 CHAPTER2 NEUTRINOTHEORY 2.1Introductiontoneutrinos Neutrinosareneutralleptonicfermionsthatarethesecondmostabundantparticleinthe universe.Despitethesheernumberofneutrinoseverywhere,theyhaveremainedmysterious withseveraloftheirfundamentalpropertiesstillunknown.Neutrinosinteractwithother particlesonlyviatheweakforceandgravity,leadingtoafrustratinglysmallprobabilityto haveaninteractionwhichcanbestudied. Neutrinoswereoriginallypostulatedin1930byWolfgangPauliasapossiblesolutionto explainthecontinuousspectrumobservedfromnuclearbetadecay.Theprevailingtheoryat thetimewasbetadecaywasatwo-bodydecayanucleusproducinganelectronandsmaller daughternucleus.Thekinematicsofatwo-bodydecayaresuchthateachdecayproductis mono-energeticduetoenergyandmomentumconservationandcanbecalculatedexactly. However,acontinuousenergyspectrumwasobservedfortheelectronwhenmeasuredwhich contradictedthetwo-bodyassumption.Acontinuousenergyspectrumwouldbepossible ifthedecaycontainedmorethantwoparticles,or,assuggestedatthetime,thatenergy andmomentumconservationmaynotbeasuniversalasoncethought.Pauliproposeda thirdinvisibleparticlewasalsoproducedinbetadecaytokeepenergyandmomentum conservation;however,itwasthoughttobeundetectable. In1934EnricoFermipublishedhistheoryofbetadecayinwhichaneutronwould decayintoaproton,electron,andPauli'sproposedparticle,whichFerminamedaneutrino (meaningneutralAdditionally,experimentalevidencehadbeencollectedthat refutedtheideathatenergyconservationwasinvalidforbetadecay,furthernecessitating theneedforthisthirdinvisibleparticle,theneutrino. In1951ReinesandCowanstartedProjectPoltergeisttode˝nitivelydetectthesignature 4 ofaneutrino.Theproposedsignaturewasaninversebetadecayreactionwhereaneutrino interactswithaprotontoproduceapositronandaneutron,andthegammaraysfrom positron-electronannihilationwouldbedetected.Thegoalwastoshowanincreasein annihilationgammarayswhenthedetectorwasexposedtoaneutrinosourcecomparedto withoutaneutrinosource,howeverthemainissuewaswhereto˝ndanintensesourceof neutrinos.ReinesandCowan'sinitialideaforaneutrinosourcewastousetheexplosion ofanatomicbombandplaceadetectorrelativelyclosetothedetonation[6](sketchofthe setupshowninFig.2.1).Theatomicbombwouldproducealarge˛uxofneutrinosina shortpulseminimizingthenumberofbackgroundcountsduringdetection.Theproposalto useanatomicbombwasapprovedbythedirectorofLosAlamosNationalLaboratoryand wasconsideredasthebestshotforanexperimentatthetime. Figure2.1:Schematicoftheproposedexperimenttouseanatomicbombasanintense sourceofneutrinos.FigurefromRef.[6]. EventuallyReinesandCowanwerepersuadedtoinsteadusethe˛uxofneutrinosfrom theSavanahRivernuclearreactor,andin1956hadcollectedconclusiveevidenceofneutrino detection[18].Thedetectorobservedtheinversebetadecayreactionanddetectedthe gammaraysfrompositron-electronannihilationandneutroncaptureoncadmium.This two-pulsesignaturefrompromptannihilationanddelayedcapturegavearobustsignalof 5 neutrinodetectioncomparedtobackgroundsources. Lessthanadecadelater,themuonneutrinowasdiscoveredbyLederman,Schwartz,and Steinberger et.al. usingtheAGSatBrookhavenNationalLab[19].Notonlydidthisshow thatneutrinosarepairedwithaspeci˝clepton,muonneutrinosareseparatefromelectron neutrinos,itwasthe˝rstdemonstrationofproducingneutrinosusinganaccelerator.The acceleratorwasusedtocreateabeamofchargedpionsthatwoulddecayintomuonsand muonneutrinos.Itwouldthentakeuntiltheyear2000fortheDONUTcollaborationat Fermilabtodirectlydetectthetauneutrino[20],completingthethreeknowngenerationsof leptons.Precisionmeasurementsofthe Z 0 bosondecaywidthfromtheLEPcollidergave evidencethatthereexistonlythreeactivelightneutrinos[21],thethreeneutrinosknown todaytheelectron,muon,andtauneutrinos. Paralleltothediscoveryofeachtypeofneutrino,severalexperimentswereobserving anomaliesinthe˛uxofneutrinosexpectedfromthesunandtheatmospherewhencompared totheory.RayDavisandtheHomestakeexperimentmeasuredthesolarneutrino˛uxand measuredarateofaboutonethirdoftheexpectedvaluefromsolarmodels.Boththe experimentandthesolarmodelswereinvestigatedforde˝cienciesorerrors,butnomajor errorswerefound,andthediscrepancyremained.SeveralmoreexperimentssuchasSAGE [22]andGALLEX[23]continuedtoobservedthede˝citofexpectedsolarneutrinos,andit wasnotuntiltheSNOexperimentwhichwasabletosolvethepuzzleofthemissingneutrino ˛ux.In1969GribovandPontecorvoproposedtheelectronneutrinosfromthesunwere oscillatingintoother˛avorsofneutrinosduetoneutrinomixing[24],anideaPontecorvo proposedinthelate1950's[25]andMaki,Nakagawa,andSakataalsoproposedin1962[26]. MeasurementsfromtheSNOexperiment[27]andtheSuper-Kamiokande[28]experiment in2002and1998respectivelyshowedconclusiveevidenceofneutrinooscillation,withthe SNOresultshowingthelowerthanexpectedsolarneutrino˛uxwasduetooscillations. ThediscoveryofneutrinooscillationsfrombothSKandSNOwouldeventuallyearnthe 2015NobelPrizeinPhysics.Themeasurementofneutrinooscillationsalsogaveconclusive 6 evidencethatneutrinosmusthavesomenon-zeromass,otherwisetheycouldnotoscillate, insteadofbeingmasslessasoriginallyincludedintheStandardModel. 2.2Neutrinooscillations 2.2.1Neutrinooscillationtheory ThecurrentformalismofneutrinooscillationsisthePMNSformalism,originallyproposed anddevelopedbyPontecorvo,Maki,Nakagawa,andSakata[24,26],andhasbeenhighly successfulinpredictingthee˙ectsofneutrinooscillations.ThekeyhypothesisinthePMNS formalismisthatneutrinosinteractwiththeweakforceintheir˛avorstates,whilethey propagatethroughspaceintheirmassstate,andthemassand˛avorstatesaresuperpositions ofeachother.ThissectionfollowsasimilarproceduretotheonepresentedinGiuntiand Kim[29].De˝ningthemasseigenstatesas j k i ;k =1 ; 2 ; 3 andthe˛avoreigenstatesas j i ; = e;˝ thesuperpositionof˛avorstatesasmassstatescanbewrittenas: j i = X k U k j k i (2.1) where U k isaunitarymatrixthatdescribesthemixingbetweenthedi˙erenteigenstates, commonlycalledthePMNSmatrix(orPMNSmixingmatrix).Theunitarityconditionis toensuretotalprobabilityisconserved,orthatneutrinooscillationdoesnotcauseanet gainorlossofneutrinos.ThePMNSmatrixiswrittenasa 3 3 matrixwithelements correspondingtoeachofthethree˛avorandmasseigenstates: U k = 0 B B B B @ U e 1 U e 2 U e 3 U 1 U 2 U 3 U ˝ 1 U ˝ 2 U ˝ 3 1 C C C C A (2.2) whichcloselyresemblestheCabbibo-Kobayashi-Maskawa(CKM)quarkmixingmatrixin QCDinform[29].Thesuperpositionofeigenstatesallowsforaneutrinoofcreatedasone ˛avortointeractasadi˙erent˛avoraftersometimeordistanceofpropagation(e.g.an electronneutrinobeingdetectedasamuonneutrinoafterpropagation).Theneutrinomass 7 statesareeigenstatesoftheHamiltonian, Hj k i = E k j k i (2.3) E k = q ~p 2 + m 2 k ; (2.4) where E k aretheenergyeigenvaluesinnaturalunitswhere ~ = c =1 (whichareused throughouttherestofthischapter).Themasseigenstatesevolveintimeaccordingtothe Schrödingerequationwhichgives j k ( t ) i = e iE k t j k i : (2.5) Thetimeevolutionofthe˛avoreigenstatescanbewrittenusingthemasseigenstatesand themixingmatrix.Assumeaneutrinocreatedwithade˝nite˛avor attime t =0 ,using Eqs.2.5and2.1thetimeevolutionofa˛avorstateisgivenby j ( t ) i = X k U k e iE k t j k i : (2.6) Equation2.1canbeinvertedtoexpressmassstatesasasuperpositionof˛avorstatesand appliedtoEq.2.6togive j ( t ) i = X X k U k e iE k t U k j i : (2.7) Thusaneutrinothatstartsout( t =0 )aspure˛avorstate becomesasuperpositionof di˙erent˛avorstatesasitpropagatesthroughspace( t> 0 ).Thestrengthofthemixingis determinedbythevaluesofthePMNSmixingmatrix,andifthematrixwerediagonalno mixing(i.e.oscillation)wouldtakeplace.Theprobabilityofaneutrinototransition(or oscillate)from˛avor ! isgivenbythefollowing P ( ! )( t )= jh j ij 2 = X k;j U k U k U j U j e i ( E k E j ) t : (2.8) 8 Assumingtheneutrinosareultrarelativistic( j ~p j >>m k ),thedispersionrelationcanbe approximatedbyexpandingthesmallmasstermas E k ' E + m 2 k 2 E (2.9) where E = j ~p j istheneutrinoenergy,neglectingthemasscontribution.Thedi˙erencein energybetween˛avorstatesthencanbewrittenas E k E j ' m 2 kj 2 E ; (2.10) where m 2 kj isthesquaredmassdi˙erence m 2 kj m 2 k m 2 j : (2.11) Sincetheneutrinosareultrarelativistictheytravelessentiallyatthespeedoflight,allowing foronemoreapproximationof t = L ,whichwhenputtogetherwiththepreviousapprox- imationsgivesthefamiliarneutrinotransitionprobabilityasafunctionofneutrinoenergy andtraveldistance: P ( ! )( L;E )= X k;j U k U k U j U j exp i m 2 kj L 2 E ! : (2.12) Thederivationforanti-neutrinosfollowsthesamepatternandonlydi˙ersfromtheneutrino oscillationprobabilitybythecomplexconjugatesofthemixingmatrixelements.Adi˙erent waytowritetheoscillationprobabilityistoseparatetherealandimaginarypartsofthe mixingmatrixelements( U i )asfollows P ( ! )( L;E )= 4 X k>j Re [ U k U k U j U j ]sin 2 m 2 kj L 4 E ! (2.13) 2 X k>j Im [ U k U k U j U j ]sin m 2 kj L 2 E ! ; (2.14) wherethepositiveimaginarytermisforneutrinosandthenegativeimaginarytermisfor anti-neutrinos.ThePMNSmatrixiscommonlywrittenasaproductofthreematriceswith 9 asetofmixingangles 12 ; 13 ; 23 andacomplexphase : U = 0 B B B B @ 100 0 c 23 s 23 0 s 23 c 23 1 C C C C A 0 B B B B @ c 13 0 s 13 e 010 s 13 e 0 c 13 1 C C C C A 0 B B B B @ c 12 s 12 0 s 12 c 12 0 001 1 C C C C A (2.15) where c ij =cos ij ;s ij =sin ij .Theunitarityofthemixingmatrixandinvarianceunder phasetransformationsoftheLagrangianresultsinrequiringonlythreeanglesandaphaseto completelyparameterizethemixingmatrix.The(1,2)parametersarecommonlyreferredto astheparameters,the(1,3)asparameters,andthe(2,3)as parameters.The iscommonlyreferredtoastheDiracCPviolatingphase, CP .Themixing matrixparametersalongwiththemassdi˙erencesbetweenthethreeactiveneutrinostates formtheparameterswhichgovernneutrinooscillations,whicharethetargetmeasurements forpast,current,andfutureneutrinooscillationexperiments. Theabovederivationisforneutrinooscillationsinavacuum.Whenneutrinospropagate throughmatter,theyaresubjecttoanadditionale˙ectivepotentialduetocoherentinterac- tionswiththemedium[29].ThisextrapotentialwhenincludedintheHamiltonianaltersthe neutrinooscillationprobabilityandisknownastheMikheyolfenstein(MSW) e˙ect(oroftensimplycalledmattere˙ects).Thematterpotentialchangessignbetween neutrinosandanti-neutrinoscausingtheoscillationsinmattertobedi˙erentbetweenneu- trinosandanti-neutrinos.Bymeasuringaresonanceintheoscillationspectruminduced bymattere˙ectsforeitherneutrinosoranti-neutrinos,thesignof m 2 32 andthusneutrino massorderingcanbedetermined.Thederivationofthemattere˙ecttermsintheneutrino oscillationprobabilitycanbefoundinChapters9and13inRef.[29]. 2.2.2Statusofneutrinooscillations Thestudyandmeasurementofneutrinooscillationshasrapidlyprogressedsincede˝nitive evidencewaspresentednearlytwentyyearsago.Mostoftheparametersthatgovernneutrino 10 oscillationshavebeenmeasuredto ˘ 5 %precision,summarizedinTable2.1,withthemain exceptionbeingthecharge-parityviolatingphase CP . ParameterBest-˝t 3 ˙ range m 2 21 [10 5 eV 2 ] 7.376.93-7.96 j m 2 32 j [10 3 eV 2 ] ; m 2 32 > 0 2.562.45-2.69 j m 2 32 j [10 3 eV 2 ] ; m 2 32 < 0 2.542.42-2.66 sin 2 12 0.2970.250-0.354 sin 2 23 ; m 2 32 > 0 0.4250.381-0.615 sin 2 23 ; m 2 32 < 0 0.5890.384-0.636 sin 2 13 ; m 2 32 > 0 0.02150.0190-0.0240 sin 2 13 ; m 2 32 < 0 0.02160.0190-0.0242 =ˇ; m 2 32 > 0 1.38 2 ˙ :1.0-1.9 =ˇ; m 2 32 < 0 1.31 2 ˙ :0.92-1.88 Table2.1:NeutrinooscillationparametervaluesfromthePDG[1],whicharecalculated usingaglobal˝tofneutrinodata. Currentgenerationlong-baseline(e.g.T2K,NOvA)andatmosphericexperiments(e.g. IceCube,SK)focusonpreciselymeasuring sin 2 23 , m 2 32 ,searchingforevidenceofCPvio- lationthroughmeasuring CP ,anddeterminingtheneutrinomassordering.Acomparison ofthe sin 2 23 ; m 2 32 contoursusingrecentresultsfromavarietyofexperimentsisshownin Fig.2.2.Outofthethreemixingangles, sin 2 23 istheleastwellknown,andisafocusof long-baselineexperiments.Aninterestingquestionisif sin 2 23 ismaximal(equalto0.5), whichcouldimplysomenewunderlyingsymmetry,orifitisnon-maximal,whichoctantit fallsin(whetheritisgreaterorlessthan0.5). Long-baselineoscillationexperimentsmeasuretheneutrinoeventrateattheirfardetector andcompareittotheexpectedeventrateundertheneutrinooscillationhypothesistoextract 11 Figure2.2:Left:Comparisonofmeasured90%con˝dencelevelcontoursfor m 2 32 vs sin 2 23 forT2K(2017)[7],NOvA(2018),SK(2018)[8],IceCube(2018)[9],andMINOS(2014) [10];Fig.fromRef.[11].Right:Updated2018T2Kcontoursincludingadditionaldataand analysisimprovements;Fig.fromRef.[12]. theoscillationparameters.Theeventratecanbeparameterizedinthefollowingway, N ( ~x )= Z E ) ˙ ( E ; ~x ) R ( E ; ~x ) P ( i ! j ; E )d E (2.16) where ~x arethemeasuredobservables, istheneutrino˛ux, ˙ istheneutrinocrosssection, R isthedetectorresponse(whichincludesthee˚ciency),and P ( i ! j ) istheoscillation probabilitybetweenneutrinosof˛avor i and j (whichcanbethesame).Cruciallyeachterm intherateisafunctionoftheneutrinoenergy,whichisnotknownapriori.ShowninFig. 2.3istheneutrinoenergyspectra(orneutrino˛ux)forseveraldi˙erentexperimentsandde- tectors.Insteadtheneutrinoenergyforeachinteractionmustbeestimatedorreconstructed usingtheinteractionmodelandfromsomeotherobservablesinthedetector,suchasthe outgoingmuonkinematics. Thereconstructionoftheneutrinoenergyislimitedbyboththeresolutionofthede- tector,andtheaccuracyoftheinteractionmodel.ShowninFig.2.4istheneutrinoevent distributionatSKfortheT2Kexperiment(fromRef.[12]),whereacleardipinmuon (anti-)neutrinoeventscanbeseen.Roughlyspeaking,thedepthandpositionofthisdip givesthevaluesfor sin 2 23 and m 2 32 respectively.Iftheneutrinoenergyspectrumwas systematicallylower,forexample,itwoulddirectlybiasthemeasuredvaluefor m 2 32 . 12 Figure2.3:TheneutrinoenergyspectrumfortheT2Kexperiment(bothon-/o˙-axisnear detectorsandfardetectorafteroscillation),plustheneutrinoenergyspectrumforNOvA andtheMINERvAlowenergymode.Overlaidaretheneutrinocrosssectionpredictionsfor di˙erentcombinationsofinteractionchannels.Notethatthedi˙erentenergyspectracover awideenergyrangewherethecrosssectionpredictionschangerapidly. Figure2.4:ReconstructedenergydistributionsatSKforthe (left)and (right)-enriched sampleswiththetotalpredictedeventrateshowninred.Ratiostothepredictionsunder thenooscillationhypothesisareshowninthebottom˝gures.FigurefromRef.[12]. 13 ForT2KandNOvAthecurrentlimitingfactortotheprecisionofneutrinooscillation measurementsisthecollectedstatistics,howeveroneofthelargestsystematicuncertainty sourcesisrelatedtothemodellingofneutrinointeractions(showninFig.2.5).Thenext generationlong-baselineexperimentswillcollectvastamountsofevents,whichwillrequire thereductionoftheoverallimpactofthesystematicuncertaintiestolevelofafewpercent. Figure2.5:E˙ectofthe 1 ˙ variationsofthesystematicuncertaintiesonthepredictedevent rateforT2K(left)andNOvA(right).T2KtablefromRef.[7]andNOvAtablefromRef. [11]. 14 2.3Neutrinointeractions 2.3.1Interactionsonfreenucleons Thedominantinteractionmodefortheanalysispresentedinthisthesisischarged-current quasi-elastic(CCQE)scattering.TheinteractionconsistsofW-bosonexchangebetweenan incomingneutrinoandanucleonproducingachargedleptonandiso-spin˛ippednucleon, ` + n ! ` + p (2.17) ` + p ! ` + + n (2.18) where ` = e;˝ .Thenucleonisacompositeobject,comprisedofthreevalencequarks, whichpreventstheanalyticcalculationofthecrosssectionforthisprocess.Insteadthe crosssectionisparametrizedusingaseriesofparametersknownasformfactors,whichcan bemeasuredthroughprocessessuchasbetadecayandelectronscattering[29,30].These formfactorscharacterizetheinternalchargedistributionandstructureofanucleon.Using theseformfactors,thedi˙erentialcrosssectionasafunctionthefour-momentumtransfer squared( Q 2 )inthelaboratoryframeisgivenby: d ˙ d Q 2 = m 2 N G 2 F j V ud j 2 8 ˇE 2 " A ( Q 2 ) B ( Q 2 ) s u m 2 N + C ( Q 2 ) ( s u ) 2 m 4 N # (2.19) where G F istheFermiconstant, V ud isanelementoftheCKMmatrix, m N isthenucleon mass, E istheneutrinoenergy,theMandelstamvariables s and u ,andthe isforneutrinos andanti-neutrinosrespectivelyintheLlewellynSmithformalism[31].Thefunctions A;B;C of Q 2 aregivenby: 15 A = m 2 ` + Q 2 m 2 N ( 1+ Q 2 4 m 2 N ! F 2 A 1+ Q 2 4 m 2 N ! F 2 1 + Q 2 4 m 2 N F 2 2 ! + Q 2 m 2 N F 1 F 2 m 2 ` 4 m 2 N " ( F 1 + F 2 ) 2 +( F A +2 F P ) 2 1 4 1+ Q 2 4 m 2 N ! F 2 P #) (2.20) B = Q 2 m 2 N F A ( F 1 + F 2 ) (2.21) C = 1 4 F 2 A + F 2 1 + Q 2 4 m 2 N F 2 2 ! (2.22) where F 1 ;F 2 ;F A ;F P arethechargedcurrentnucleonformfactorsasafunctionof Q 2 .For electronandmuonneutrinointeractionsthetermin A proportionalto m 2 ` = 4 m 2 N iscommonly neglectedsincethenucleonmassismuchlargerthantheelectronormuonmass.The F 1 ;F 2 formfactorsarevectorformfactorsintheelectroweakcurrent,whicharedeterminedthrough electronscattering.Theelectromagneticformfactorsareassumedtohaveadipolefunctional formwhen˝ttingtodata,withextensionstotheformfactorstodescribediscrepanciesseen athigh Q 2 [32].The F A ;F P termsaretheaxialvectorandpseudoscalarformfactorswhich arepresentonlyforneutrinointeractions.Thepseudoscalarformfactorcanbewrittenin termsintheaxialvectorformfactor,andanalogoustotheelectromagneticformfactors, theaxialvectorformfactoriscommonlyassumedtohaveadipolefunctionalformwhichis givenby F A ( Q 2 )= g A 1+ Q 2 =M 2 A 2 (2.23) F P ( Q 2 )= 2 m 2 N m 2 ˇ + Q 2 F A ( Q 2 ) (2.24) where m ˇ isthepionmass,and g A and M A arefreeparameters.The g A parametercan bepreciselydeterminedthroughbetadecaymeasurements,while M A mustbemeasured throughneutrinoscattering.Thedipolefunctionalformisanassumptionwhichhasworked moreorlesswellinthepast,howeverthismaynolongerbethecaseasmoreprecise measurementsareperformed.Di˙erentfunctionalformsfortheaxialvectorformfactor havebeenexplored,suchastheZ-expansionmodelpresentedinRef.[33]. 16 Neutralcurrentquasi-elasticscatteringinvolvestheexchangeofa Z 0 bosonwithanu- cleon, ` + N ! ` + N (2.25) ` + N ! ` + N (2.26) where ` = e;˝ ,and N iseitheraprotonorneutron.Theformalismtodescribetheneutral currentinteractionisverysimilartothechargedcurrentinteraction,withslightlydi˙erent formfactorsforthenucleons. 2.3.2Nuclearmediume˙ects Neutrinointeractionsonnucleonswithinanucleusincuramyriadofadditionale˙ectsdue tobeinginaboundstateandthepresenceofothernucleons.Theseextrae˙ectsmustbe consideredforprecisemodelingofneutrinointeractions.Additionally,thenuclearmedium allowsforfurtherscatteringandinteractionsoftheoutgoingparticlesandarefurtherdis- cussedinSection2.3.6.Thissectionpresentsashortoverviewofthekeynuclear-medium e˙ectsthatareincludedintheinteractionmodelingforthisanalysis. Initialstate TheLlewellynSmithformuladetailedinEq.2.19describesneutrinosscatteringwithfree nucleonswhichhaveasimpleinitialstate,thefreeparticle.Forboundnucleonsinanucleus, aninitialstatemodelmustbeusedtoprovidetheinitialwavefunctionfortheinteracting nucleon.TheinitialstateconditionsarecomprisedofFermimotionofthenucleons,the bindingenergyofthenucleons,andbothshort-andlong-rangeinteractionsbetweennucleons. Fermimotion Fermimotionreferstothemotionofthenucleonsrelativetothenucleusasawholethe nucleonsarenotallmovingtheinthesamedirectionwiththesamemomentum.Thisinitial momentumcauseseachnucleontohaveadi˙erentandunknownLorentzboostinthelab frameforeachneutrinointeraction[34].Modelswhichattempttopredictthemomentum 17 distributionofthenucleonsarereferredtoasspectralfunctions.Themostcommononesused inneutrinointeractionsimulationsarevariationsoftheFermigasmodel,andthespectral functionfromBenharet.al.AgoodsummaryofthesemodelscanbefoundinRef.[35]. Fermigasmodelstreatthenucleonsinsidethenucleusasagasofnon-interactingfermions containedinsidesomenuclearpotentialwell.Thenucleons˝llmomentumstatesfromthe lowestmomentumstateuptosomemaximummomentum,knownastheFermimomentum ( p F ).TheRelativisticFermiGas(RFG)model[36]assumesthenuclearpotentialisconstant forallnucleons.Theassumptionoftheconstantpotentialgivesrisetoasharpcuto˙ofthe momentumdistributionattheFermimomentum(sometimesreferredtoastheFermicli˙ feature).ThemoresophisticatedLocalFermiGas(LFG)model[13,37]insteadtreatsthe nuclearpotentialasafunctionoftheradiusofthenucleus,usingthelocalnucleardensity ˆ ( r ) where r istheradialpositionofagivennucleon(commonlyreferredtoasthelocaldensity approximation).Thissmoothsoutthemomentumdistributionofthenucleons,removing thesharpcuto˙seenintheRFGmodel.However,likeintheRFGmodel,theLFGmodel stilltreatsthenucleonsasnon-interacting. Thespectralfunction(SF)fromBenharet.al.[38]insteadallowsfortwoandthree bodynucleoninteractions.TheBenharSFstartswithashellmodelofthenucleons andconsidersmodi˝cationstotheorbitalsfromtheinteractionsoftwoandthreenucleons andtheshortrangecorrelationsofpairsofnucleons[34,38].Thisresultsinasmooth initialmomentumdistributionofnucleons.Moresophisticatedmodelsfortheinitialstate nucleonsexist,suchasrelativisticmean˝eldmodelsandtherelativisticplanewaveimpulse approximation,butarestillintheearlystagesofbeingimplementedinneutrinosimulations [39,40,41]. Bindingenergy InadditiontotheFermimotion,theboundnucleonshaveanassociatedbindingenergy whichisrequiredtoejectthenucleonfromthenucleus.Thede˝nitionofthebinding energyforagivennucleonisdependentonthemodelbeingused[42].TheBenharspectral 18 functionusestheshellmodeltocalculatethebindingenergyofeachnucleonasaprobability distributionofFermimomentumandmissingenergy.Adi˙erentmethodistohaveaconstant bindingenergyforeachnucleonbasedonthemassdi˙erencebetweentheinitialanddaughter nucleusandhasbeenusedinconjunctionwithFermigasmodels. Wbosonselfenergy Asmentionedabove,itisknownthattheboundnucleonswithinanucleusarecorrelated withinthenuclearpotential.Onewaytodescribethelong-rangecorrelationsbetweennu- cleonsistherandomphaseapproximation(RPA)[43].Therandomphaseapproximation includesthee˙ectofthenuclearmediumprovidingascreeningoftheelectroweakpropagator, whichmanifestsasa Q 2 dependentcorrectionontheneutrinocrosssection. MultinucleonProcesses Multinucleonprocessesrefertointeractionswhichinvolvemultiplecorrelatednucleons attheinteractionvertex(atypeofshort-rangecorrelation).Theneutrinointeractionmodel fromNieveset.al.[13]includesadescriptionofmultinucleonprocesses,whichissummarized hereasanexample.TheNieveset.al.modelformultinucleonprocesses,speci˝callytwo- particletwo-hole(2p2h)interactions,considerstheseveninteractionverticesshowninFig. 2.6wherethevirtualpionisconnectedtoasecondcorrelatednucleon(themodelalso accountsfordiagramswherethe ˇ isreplacedbya ˆ propagator). Theverticesarecommonlyclassi˝edintotwocategoriesbasedonthestimulationofa baryonresonance(suchasinthetoptwodiagramsinFig.2.6);andtheycanbecalculated independentlytoseparatetheimpactofeachonthetotal2p2hcrosssection.Theseparation of resonanceproductionisduetotheregionsofkinematicphasespacetheinteractions occupy,duetothehigherenergyrequiredtoexcitethe resonance.Thetotalstrengthof themultinucleonprocessaspredictedbytheNieveset.al.modelisshownintherightofFig. 2.6,whichclearlyshowstwodi˙erentpopulations. resonanceeventslargelycontributeto thepopulationathigherenergyandmomentumtransfer. 19 Figure2.6:Left:The WN ! Nˇ verticesconsideredbytheNievesmodelformultinucleon processes.Onlyonenucleonlineisshown,thesecondnucleonisimpliedtobecoupledtothe virtualpion.Fig.fromRef.[13].Right:Thedistributionofthestrengthofthemultinucleon processesaspredictedbytheNieveset.al.modelinenergy( q 0 )andmomentum( q 3 )transfer space. 2.3.3Coherentscattering Coherentscatteringoccurswhenaneutrinoscatterscoherentlywiththenucleus,thatisthe neutrinointeractswiththeentirenucleusinsteadofscatteringo˙ofaparticularnucleon. Thisisonlypossibleatverylowvaluesof Q 2 andcanbecompletelyelastic(knownas coherentelasticscattering)orproduceapionintheinteraction(knownascoherentpion production).Coherentpionproductionischaracterizedbythefollowingmodes,including bothchargedcurrentandneutralcurrentinteractions: ` + A ! ` + A + ˇ 0 (2.27) ` + A ! ` + A + ˇ + (2.28) where A isthetargetnucleus.Coherentpionproductioniscommonlymodeledinneutrino interactionsimulationsusingtheRein-Sehgalmodel[44],butothermodelssuchasthe Berger-Sehgalmodel[45]andmicroscopicmodelsareavailable[46]. 20 2.3.4Resonantpionproduction Resonantpionproductionoccurswhentheincidentneutrinohasenoughenergytoexcite anucleonresonance,whichcansubsequentlydecayproducingoneormorepions.The thresholdforresonantpionproductionistheenergyneededtoexciteaDeltabaryonand isthedominantinteractionchannelforthesingletofewGeVenergyregion.Manynucleon resonantstates( N or )contributetothecrosssectionforresonantpionproduction, withastrongcontributionfromthe (1232)resonantstate.Thesinglepionproduction charged-currentchannelsare: ` + n ! ` + n + ˇ + (2.29) ` + n ! ` + p + ˇ 0 (2.30) ` + p ! ` + p + ˇ + (2.31) wheretheintermediatebaryonresonancedecaysbacktoaprotonorneutronandapionwhile stillinthenuclearmedium.Theintermediatebaryonresonancecanalsodecayintoother mesonsotherthanpions,mostnotablykaons,whichisknownmoregenerallyasresonant mesonproduction. 2.3.5Shallowanddeepinelasticscattering Eventuallytheneutrinohasenoughenergytoresolveandscattero˙ofindividualquarks insidethenucleons,aprocessknownasdeepinelasticscattering(DIS).Thereisnoprecise cut-o˙betweenresonantanddeepinelasticscatteringprocesses,howeveragoodruleof thumbistheregionof W> 2 : 0 GeVand Q 2 > 1 : 0 GeV 2 isprimarilyonlydeepinelastic scattering[47].Indeepinelasticscatteringaneutrinointeractswithaquarkinsidethetarget nucleon(N)producingaleptonandahadronicjet(X): ` + N ! ` + X (2.32) 21 Ifthenucleonisboundwithinanucleus,theinteractionisa˙ectedbyseveralnuclearmedium e˙ectssuchasinitialFermimotion,nucleonnucleoncorrelations,nuclearshadowing,etc. Deepinelasticprocessesaredescribedusingasetofstructurefunctions,whichareinturnex- pressedusingpartondistributionfunctions(PDFs).Apartondistributionfunctiondescribes themomentumdistributionofthequarksandgluonsinsidethenucleon. Shallowinelasticscattering(alsoreferredtoasmulti-pionproduction)referstothetran- sitionregionbetweensingleresonantproductionanddeepinelasticscattering.Thisregion isverychallengingtomodelasitmarksthetransitionfromthepion-nucleondescription tothequark-gluondescriptionofneutrinointeractions[47],andispoorlyunderstoodboth theoreticallyandexperimentally.Comparedtoquasi-elasticscatteringandresonantpion production,DISprocessesarecomparativelywellmodeledathighenergy(5to10GeVand higher)[47]. 2.3.6Finalstateinteractions Particlescreatedinaneutrinointeractionmustescapethenucleus,propagatingthrough thenuclearmedium.Theoutgoingparticlescaninteractwithothernucleonsbeforeexiting thenucleus,undergoingscattering,beingabsorbed,orstimulatingadditionalproductionof hadrons,whicharereferredto˝nalstateinteractions(FSI) 1 . Whendetectinganeutrinointeraction,onlytheparticleswhichescapethenucleusare seenastheneutrino,beinganeutral,weaklyinteractingparticle,isnotvisibletothedetector. Therefore,alltheinformationabouttheneutrinointeractionmustbeinferredfromthe producedparticles.Forexample,intheabsenceofanyotherinteractions,apionproduced inaneutrinointeractionwouldindicatenon-quasi-elasticscattering(e.g.singleresonant production).Interactionsthatalterthekinematicsorstateoftheseparticlesbeforedetection willobscuretheinformationabouttheinitialneutrinointeraction.Tocontinuetheexample, 1 Forclari˝cation,inthisanalysis˝nalstateinteractionsonlyrefertointeractionswithin thenucleusbeforetheoutgoingparticleshaveescaped(ifatall). 22 ifthepionproducedwassubsequentlyabsorbedinthenucleus,theinteractionwouldthen appeartobeaquasi-elasticscatterratherthanpionproduction.Theejectedprotonfrom quasi-elasticscatteringcouldstimulatepionproductionwhileexitingthenucleus,again changingwhatinteractionappearedtohavetakenplaceatthevertex.Inferringtheoriginal vertexinteractionwouldrequireamodeltopredictthe˝nalstateinteractionsinsidethe nucleus. Forthisreason,modernneutrinointeractionexperimentsde˝nethesignalasafunction ofthe˝nalstateparticlesinanevent,whicharetheparticlesthatthedetectorcanmeasure afterany˝nalstateinteractionshaveoccurred.Thecombinationof˝nalstateparticlesin aneventiscalledthetopologyofanevent.Forexample,aneventwhichproducesasingle muonandzeropionsisreferredtoasaCC- 0 ˇ event,whichstandsforcharged-currentzero pionevent.Similarly,aneventwithasinglemuonandasinglechargedpositivepionwould bereferredtoasaCC- 1 ˇ + event,acharged-currentsinglepositivelychargedpionevent. De˝ningthesignalordesiredeventtypebythe˝nalstateparticlesislessinteractionmodel dependentasthereisnoattempttocorrectfortheunknownprocesseswhichoccurredinside thenucleus. 2.3.7Neutrinoeventgenerators ThesimulationofaneutrinointeractioniscommonlyperformedwithaMonteCarlo(MC) simulationprogramknownasaneutrinointeraction(orevent)generator.Theseneutrino eventgeneratorsuseavarietyoftheoreticalandempiricalmodelstosimulatetheinteraction ofaneutrinoonsomenucleartarget(e.g.afreenucleonoracarbonnucleus).Theoutput ofaneventgeneratorcontainsthekinematicsofeachincidentneutrino,thekinematicsof alloutgoingparticles,andinformationabouttheeventsuchasthesimulatedreaction(e.g. quasi-elasticorresonantproduction).Givenaninput˛uxofneutrinos(e.g.monoenergetic orbasedonanexperiment),thecrosssectionforagivenneutrinoprocessoreventtopology canbepredicted,alongwithkinematicdistributionsoftheoutgoingparticles. 23 Aneutrinoeventgeneratoristheinitialstepinachaintoproducetheexpecteddistribu- tionsforexperiment,andoftenusedtoperformvalidationchecksofanalysesandevaluate systematicuncertainties.Itisimportanttounderstandthelimitationsofthecurrentiter- ationoftheeventgenerators.Themodelsofneutrinointeractionsareoftenonlyinclusive models,onlypredictingtheleptonkinematics.However,apredictionoftheoutgoinghadron kinematicsisoftennecessary,butthefullcalculationofthedesiredsemi-inclusiveprocess eitherdoesnotexistormaybetooexpensivetosimulateenoughevents.Modernneutrino generatorsaddressthisproblembyfactorizingthesimulationofaneutrinoeventintosep- arate,butmanageablepieces.Forasimpli˝edexample,theneutrinoreactiononaninitial statenucleonissimulatedandleptonkinematicsareproduced,thenthehadronproductionis simulatedseparately,and˝nallytheoutgoingparticlesarepropagatedthroughthenucleus. Currently,therearefourmainneutrinoeventgeneratorsusedinexperimentforsimula- tion:NEUT[48],GENIE[49],NuWro[50],andGiBUU[51].Eachisprimarilydevelopedby aseparategroupofresearchersandusestheirownimplementationsoftheincludedtheoret- ical/empiricalmodels.Thiscan,andoftenwill,givedi˙erentresultsforthesamepredicted interactionprocessbetweenthedi˙erentgenerators. 2.3.8Currentstatusandmotivation Themodernpictureofneutrinointeractiontheoryandmeasurementwouldbestbedescribed ascomplicatedneithertheorynormeasurementcanpaintaconsistentpictureofneutrino interactions.Infact,anentireworkshopseriesnamedTENSIONS[52]isdedicatedto discussinghowmeasurementsnotonlydisagreewiththeory,butareintensionbetween di˙erentexperiments. Thedisagreementonthevaluefortheaxialmass M A inthedipolefunctionalformforthe axialformfactorillustratesthesituationnicely.EarlymeasurementsoftheneutrinoCCQE crosssectionusinghydrogenanddeuteriumbubblechambersresultedinanextractedvalue of M A ˘ 1 : 0 GeV,whichmatchedthetheorywellwithintheexperimentaluncertainties[53]. 24 Howevermorerecentandmodernneutrinoexperimentsusecomplextargets,suchascarbon oroxygennuclei,forwhichnucleare˙ectsaremuchmoreimportant.TheK2Kexperiment extractedavalueof M A =1 : 21 0 : 12 GeVfrominteractionsonawater(oxygen)target [54],theMiniBooNEexperimentextractedavalueof M A =1 : 35 0 : 17 GeVfromaCCQE measurementonascintillator(carbon)target[55],and˝nallytheNOMADexperiment extractedavalueof M A =1 : 05 0 : 05 GeVfromaCCQEmeasurementonacarbontarget [56]. Suddenlythemeasurementsnolongeragreeonthevaluefortheaxialmass,butonething theyallhaveincommonistheuseofacomplextarget(comparedtohydrogen/deuterium). Oneprevailingideaisthatthelackofmultinucleone˙ectsintheprediction(orfurthernuclear mediume˙ects)canexplainthein˛atedvaluesfor M A seenbyMiniBooNEasmultinucleon e˙ectsshouldenhancethecrosssectionandallowforadecreased M A value.Boththe Martiniet.al.groupandtheNieveset.al.grouphaveshowngoodagreementbetween theirmodelswhichincludemultinucleone˙ectsandtheMiniBooNEdatawhileusingavalue of M A ˘ 1 : 0 GeV[13,57].Thisisoneexampleoftensionbetweenexperimentaldataand theory,andthereareplentyofothersourcesoftension. Theprecisevalueof M A andthenatureofmultinucleonprocesses(bothnormaliza- tionandshape)areacoupleofthemostpressingquestionsforT2Kastheyheavilya˙ect quasi-elasticscattering,whichistheprimarysourceofsignaleventsfortheT2Koscillation analysis.T2Kreliesontheneutrinointeractionmodelforthemappingfrommeasuredmuon kinematicstoreconstructedenergyandanybiaswilla˙ecttheextractedoscillationparame- ters.Multinucleonprocesseshavethepotentialtobiastheeneryreconstruction(asshownin Fig.2.7)wheremultinucleoneventsaresystematicallybiasedtolowerreconstructedenergy. AsT2Kcontinuestotakemoredata(includingtheproposedphasetwoofrunning)theun- certaintiesinthemodelingofneutrinointeractionswillbecomeincreasinglyimportant[58], andeventuallybecomethelimitinguncertaintyforthenextgenerationofneutrinooscillation experiments,suchasHyperK[59].Neutrinoscatteringandcrosssectionmeasurementsare 25 criticalforimprovingtheunderstandingofneutrinointeractionswithmatter,whichisthe mainmotivationfortheanalysispresentedinthisthesis. Figure2.7:Neutrinoenergyreconstructioncalculatedforamonoenergeticsourceof600 MeVneutrinos,showingthecontributionforwithandwithoutmultinucleonprocesses.If thereconstructionwasperfect,thereconstructedenergywouldbeadeltafunctionat600 MeV.Insteadthereconstructedenergyisspreadout,andshowsabiastowardlowerenergy fromthemultinucleonprocesses.Fig.fromRef.[7]. 26 CHAPTER3 T2KLONGBASELINENEUTRINOEXPERIMENT TheokaitoKamioka(T2K)experimentisalong-baselineneutrinooscillationexperiment designedtoprovidemeasurementsofneutrinooscillationparameters.T2Kconsistsofa relativelypure(anti-)muonneutrinobeamproducedattheJ-PARCfacilitywhichtravels ˘ 295 kmacrossJapantotheSuper-Kamiokande(SK)detectorthroughaseriesofdetectors positioned280metersfromthetarget(showninFig.3.1).T2Kstudiesfourchannelsof neutrinooscillationstomeasureneutrinooscillationparameters:muonneutrinodisappear- ance( ! ),muonanti-neutrinodisappearance( ! ),electronneutrinoappearance ( ! e ),andelectronanti-neutrinoappearance( ! e ). Figure3.1:SchematicoftheT2KexperimentshowingthelocationofJ-PARC,thenearand fardetectors,andthe295kmbaseline. The disappearancechannelisusedtomakeprecisionmeasurementsof 23 and m 2 32 , whilethe e appearancechannelisusedtomakemeasurementsofalltheoscillationpa- rameterswhichareaccessiblebyT2K.Bycomparingtheratesbetweentheneutrinoand anti-neutrinochannelsameasurementofthecharge-parityviolatingphase, CP ,canbeper- formed[7,12].Thedistancetothefardetector,SK,isreferredtoasthebaseline.Itwas chosensincethepeakoftheneutrinoenergyspectrum(0.6GeV)sitsatthe˝rstoscillation maximumwhichcoincideswherethe survivalprobabilityisataminimum.The e ap- 27 pearancechannelwasusedtogivethe˝rstindicationofanon-zerovalueof 13 [60]andthe ˝rst e appearancemeasurementinan beam[61]. Inadditiontotheneutrinooscillationmeasurements,T2Khasarichneutrinocross sectionprogramproducingavarietyofhigh-qualitymeasurementsinpartduetothewell constrained˛uxfromexternalmeasurementsandacapablewell-understooddetector.Cross sectionmeasurementsareimportantforbothunderstandingtheneutrinointeractionmodel usedintheoscillationanalysisandcollaboratingwiththeorygroupstoimproveneutrino modeling.TheT2Kneardetectorcomplexprovidesahigheventrateenvironmentand detectorcon˝gurationstoproducemeasurementsonmultipletargets,multiple˝nalstates, andmultiple˛avorsofneutrinos.Neutrinocrosssectionsonbothcarbonandwaterhave beenpublished[62,63].Anti-neutrinocrosssectionsoncarbonandtheratiowiththe correspondingneutrinocrosssectionhavebeenperformed[64].Newmethodsofinvestigating theneutrinocrosssectionhavebeenperformedwiththesingle-transversevariablesanalysis [65]withcurrentanalyseslookingatusingthevertexactivitytoperformmeasurements.T2K hasproducedcrosssectionsbothwiththeon-ando˙-axisdetectors[66],andisinvestigating methodstouseallthedetectorsinasimultaneousanalysis,oneofwhichwillbepresented inthisthesis. 3.1Beamsetup TheT2KneutrinobeamisproducedattheJapanprotonacceleratorresearchcomplex (J-PARC)inokai-mura,Japan.Theinitialprotonbeamiscreatedbyaccelerating H ionsinalinearaccelerator(LINAC)to400MeV/cbeforepassingthemtotheRapidCycling Synchrotron(RCS).Theionspassthroughachargestrippingfoilconvertingtheminto H + ions(i.e.protons)astheyentertheRCSandacceleratedto3GeV/c.Finally,theprotons arepassedtotheMainRing(MR)synchrotronandacceleratedtoa˝nalmomentumof 31GeV/cbeforebeingfastextractedintotheneutrinobeamline.The˝rstsectionofthe neutrinobeamline,theprimarybeamline,bendstheprotonbeamtothedirectionofSKand 28 provides˝nalfocusingandbeammeasurementsbeforetheprotonsimpingeonagraphite target.Aseriesofbeammonitorsmeasurepropertiesoftheprotonbeamline,suchasthe beampro˝leandposition,tomonitortheconditionoftheprotonbeamandprovideinput parameterstotheneutrino˛uxsimulation. Theprotoncollisionswiththegraphitetargetproduceprimarilymesonswhicharethen focused(orde˛ected)bythreemagnetichorns.Thetargetisa91.4cmlonggraphiterod2.6 cmindiametercorrespondingto1.9interactionlengthsfortheincidentprotonswhichensures mostprotonswillinteractwithinthetarget.Thethreemagnetichornsaredrivenat 250 kA toproducealargemagnetic˝eld(1.7T)whichfocusorde˛ectchargedparticlesdepending onthepolarityofthecurrent.ThepositivecurrentmodeisknownasForwardHornCurrent (FHC)modeandproducesaprimarilyneutrinobeamfromthedecayofpositivelycharged particleswhilethenegativecurrentmodeisknownasReverseHornCurrent(RHC)and producesaprimarilyanti-neutrinobeamfromthedecayofnegativelychargedparticles. Thehornsde˛ectparticleswiththeincorrectsignforthegivencon˝gurationreducingthe backgroundcontamination,butthisisnotperfectandsomewrong-signbackgroundremains. Additionally,muonspresentinthebeamlinewilldecay;and,alongwithotherkaondecay modes,produceelectronneutrinosaddingtothebackgroundcontaminants.Afterexitingthe magnetichorns,thesecondaryparticlestravelthroughahelium˝lleddecayvolumeabout 96metersinlengthanddecaytoneutrinos.Themainneutrinoproductionchannelsfor FHC(neutrinomode)arelistedinTab.3.1,andthecorrespondinganti-neutrinoproduction channelsarethechargeconjugatesoftheneutrinoproductionchannels.Secondarybeamline particleswhichdonotdecayintoneutrinosarestoppedbythebeamdumpattheendofthe decayvolumeexceptforhighmomentummuons.Muonswithmomentumabove5GeV/cwill penetratethroughthebeamdumpandpassthroughamuonmonitor(knownasMUMON) whichprovidesmeasurementsofthebeampositionandintensityfromtheincomingmuon ˛ux.UsingMUMON,theneutrinobeamdirectionisdeterminedtobethedirectionfrom thetargettothecenterofthemuonbeampro˝le. 29 DecaymodeBranchingfraction(%) ˇ + ! + + 99.9877 ˇ + ! e + + e 1 : 23 10 4 K + ! + + 63.55 K + ! ˇ 0 + + + 3.353 K + ! ˇ 0 + e + + e 5.07 K 0 L ! ˇ + + + 27.04 K 0 L ! ˇ + e + + e 40.55 + ! e + + + e 100 Table3.1:Neutrinoproductiondecaymodesforneutrinobeam(FHC)mode. TheT2Kneutrinobeamlinewasbuiltusingatechniqueknownasano˙-axisneutrino beamwherethefardetector,Super-Kamiokande,isnotatthecenteroftheneutrinobeam. Thiswaschosenfortwomainbene˝ts:1)itproducesanarrowneutrinoenergyspectra atsomeknownenergy;and2),itreducestherateofwrong-sign e contaminationforap- pearancesearches.Mostneutrinosinthebeamcomefrompiondecaywhichisatwo-body kinematicprocess.Thisapproachutilizesthefactthattheoutgoingneutrinoenergybe- comesincreasinglyindependentofparentpionenergyastheoutgoingneutrinoangle(with respecttotheparentpion)increases(asshowninFig.3.2).Theneutrinoenergy( E )can beapproximatedusingtheparentpionenergy( E ˇ )andtheneutrino-pionangle( ˇ )as E = m 2 ˇ m 2 2( E ˇ p ˇ cos ˇ ) ; (3.1) where m ˇ ;m arethepionandmuonmass,and p ˇ isthepionmomentum.TheT2K experimentusesano˙-axisangleof2.5degrees(43.6mrad)fromthebeamcenterchosen tomaximizetheneutrino˛uxattheoscillationmaximumatSKwhichoccursat E ˘ 0 : 6 GeV(asshowninFig.3.3). T2Khasbeencollectingdatasince2010andhascollectedaboutonethirdoftheplanned amountofdata.TheamountofdatacollectedisgivenintermsofthenumberofProtons OnTarget,orPOT.POTiscalculatedbymeasuringthecurrentoftheprotonbeamwhich isusedtodeterminethenumberofprotonsinthebeamwhichthencollidewiththetarget. Thenumberofprotonsontargetisagoodmetricforquantifyingthedatacollectedsinceit 30 Figure3.2:Neutrinoenergyasafunctionofparentpionenergyforvariouso˙-axisangles. Astheo˙-axisangleincreases,theneutrinoenergybecomesincreasinglyindependentofthe pionenergy.T2Kusesa2.5degreeso˙-axisbeamcorrespondingtoaneutrinoenergypeak of0.6GeV. Figure3.3:Theneutrino˛uxshownatdi˙erento˙-axisangles(arbitrarilynormalized) comparedtotheneutrinooscillationprobability.T2Kusesa2.5degreeo˙-axisbeamcorre- spondingtotheoscillationmaximumat0.6GeV.FigurefromRef.[14] 31 isdirectlyproportionaltotheamountofbeamdeliveredanditspowerwhichcanbeeasily comparedtootherexperiments.ThetotalPOTcollectedovertimeandthecorresponding T2KrunperiodsaredisplayedinFig3.4. Figure3.4:Accumulatedprotonsontargetforbothneutrinoandanti-neutrinobeammodes andbeampowerforeachT2Kdatarun(theshadedredregions).Thisanalysisusesdata fromRun2,3,4and8. 3.2Fluxprediction Thepredictionoftheneutrino˛uxattheT2Kdetectorsisbasedonasimulationthat beginswiththeprimaryprotonbeamcollidingwiththegraphitetargetandendingwith thedecayofthesecondarymesonsintoneutrinos[14].FLUKA2011[67,68]isusedto simulatetheinitialinteractionsoftheprotonsinsidethetargetandba˜ewhichproduce themajorityofthesecondaryparticles.FLUKAisusedatthisstagebecauseitachieves thebestagreementtoexternalhadronproductiondata.Thekinematicinformationofeach secondaryparticleissavedandpassedontoJNUBEAMthenextstageofthesimulation chain.JNUBEAMisaGeant3basedMonteCarlosimulationwhichpropagatesallthe 32 secondaryparticlesfromFLUKAthroughthesecondarybeamline(magnetichorns,decay volume,etc.),includinganyfurtherinteractionswiththematerialinthesecondarybeamline whicharesimulatedusingGCALOR.Theneutrinosproducedinthesecondarybeamlineare forcedtopointtoSKorarandompointinaneardetectorplanetosavesimulationtime,and thekinematicinformationandprobabilitytotravelinagivendirectionforeachneutrinoare savedinthe˝naloutput.The˝nal˛uxandenergyspectrumarepredictedfromthesaved eventsweightedbypreviouslymentionedprobabilities.ThelatestT2Kneutrinoprediction forforwardhorncurrent,separatedbyneutrino˛avor,isshowninFig.3.5. Figure3.5:Thepredicted˛uxatND280andSKforforwardhorncurrentrunningseparated byneutrino˛avoraveragedoverT2Kruns1-9[15]. Theneutrino˛uxsimulationusesseveralin-situmeasurementsfromboththeprimary andsecondarybeamlineasinputstotunethesimulationandverifytheoutput.Theprimary beamlineusesacollectionofbeammonitorstomeasuretheprotonbeampro˝le,beamcenter, andintensityasitcollideswiththegraphitetargetasinputstotheFLUKAsimulation. Measurementsoftheaveragecurrentprovidedtothethreemagnetichornsareprovided toJNUBEAMtoprovideaccuratemagnetic˝eldstrengths.Measurementsofthemuon ˛uxfromMUMONanddirectmeasurementsoftheneutrino˛uxbytheon-axisdetector (INGRID)arebothusedtoverifytheoutputofthesimulationandtrackthestabilityofthe neutrinobeam. 33 Externaldataonhadronproductionandscatteringisusedtotuneandimprovethe interactionmodelusedbyFLUKAandJNUBEAM.The˛uxsimulationprimarilyrelies onexternaldatafromtheNA61/SHINEexperimentforhadronproductionandtheHARP experimentforpionre-scattering[69]alongwithseveralotherdatasetsforthekinematic phasespacenotcoveredbyNA61/SHINE.TheNA61/SHINEexperimentattheCERNSPS isadedicatedhadronproductionexperimentwhichusesa31GeV/cprotonbeamandboth athincarbontarget(2cmthickness)andareplicaoftheT2Ktarget.NA61/SHINEmakes measurementsofboththetotalanddi˙erentialproduction/multiplicityofhadronsfromthe target,primarilyprotons,pions,andkaons,whicharethemostrelevantparticlesfortheT2K ˛uxsimulation[70,71,72].Thehadronproductiondataareusedtotunethemultiplicityof exitinghadronsaswellashowoftenhadronsinteractwithinthetargetandothermaterial inthebeamline.Thethin-targetdatahasbeenfullyanalyzedandincludedintheT2K˛ux simulationusedinthisthesis,whilethe˝rstroundofreplica-targetdatawillbeincludedin upcomingT2Kanalyses. 3.3TheInteractiveNeutrinoGRIDdetector TheInteractiveNeutrinoGRID(INGRID)detectorispositionedon-axisat280meters downstreamofthegraphitetarget[16].TheprimaryfunctionofINGRIDistoprovidepre- cisemeasurementsoftheneutrinobeamcenter,pro˝le,andrate,andmakemeasurements ofvariousneutrinocross-sections.INGRIDconsistsofsixteenidenticalmoduleswith14ar- rangedinacrosspatternand2diagonalo˙-axismodulesextending10metersbothvertically andhorizontally(showninFig.3.6)tospantheexpectedneutrinobeampro˝le. Thecentermodulesofthehorizontalandverticalplanesoverlapwitheachotherand areplaceddirectlyon-axiswiththeprimaryprotonbeam.AstandardINGRIDmodule isconstructedfromalternatinglayersofscintillatingbarsandironplateswithscintillating vetopanelssurroundingthemoduleasshowninFig.3.6.Theironplatesprovidealarge targetmassforneutrinointeractions,theinnerscintillatingbarsprovideparticletracking, 34 Figure3.6:TheINGRIDdetectorcon˝gurationshowingthehorizontalandverticalplanes (top)andadepictionofanINGRIDstandardmodule(bottom).TheINGRIDstandard moduleisshownhighlightingtheironplatesandinnerscintillatorpanelsontheleftandthe outervetopanelsontheright.FigurefromRef.[16]. andouterpanelsvetoparticlesenteringfromoutsidethemodule.Thevetoplanesprimarily eliminatecosmicmuons(INGRIDsitsnearthesurface)andmuonsfromneutrinointeractions inthesurroundingconcreteandrockofthedetectorhall.Thescintillationlightisreadout bywavelengthshifting˝bersinthecenterofeachbarconnectedtoamulti-pixelphoton countertoconvertthelighttoanelectricalsignal.TwospecialINGRIDmodulesexistto providecross-sectionmeasurementsattheon-axisposition:theProtonModuleandthe WaterModule.Bothmodulescollectdataplacedinbetweenthecentermodulesofthe verticalandhorizontalplane.TheProtonModuleisdesignedtomeasureneutrinocross- 35 sectionsonscintillator,andcontains34scintillatorbars(somewithdi˙eringdimensions fromthestandardmodule)andthesameoutervetoplanesasshowninFig.3.7tocreatea ˝nergrainedandfullyactivetrackingmodule. Figure3.7:ViewoftheProtonModule.Similartothestandardmodulesbutcontaining ˝nergrainedscintillatorbarsandnoironplates.FigurefromRef.[16]. TheWaterModuleisdesignedtoprovideawatertargettomeasureneutrinocrosssections containingalatticeofscintillatorbarsandwavelengthshifting˝bersfortracking.INGRID measuresthepositionoftheneutrinobeamcenterbycomparingtheinteractionratesofeach standardmoduletoaprecisionbetterthan10cm,correspondingto0.4mradprecisionat 280metersdownstreamfromthegraphitetarget. BoththehorizontalandverticalbeampositionmeasuredbyINGRIDmatchesquitewell withthemeasurementfromMUMONandisstableacrossalltheT2Krunperiodsasshown inFig.3.8.TheneutrinointeractionrateismeasuredbyINGRIDdailytoaprecisionof4% [14]asshowninFig.3.8.Thedropineventrateduringnegativehorncurrentmode(RHC mode)isduetothelowerinteractioncrosssectionforanti-neutrinos. 3.4TheNearDetectorat280meters TheNearDetectorat280Meters(ND280)isplacedat280metersdownstreamofthe graphitetargetandunlikeINGRID,itiscenteredat2.5degreeso˙-axis[16].ND280iscom- 36 Figure3.8:Neutrinobeamdirectioninboththeverticalandhorizontalpositionmeasured byINGRIDandMUMONforeachT2Krunperiod.IngeneralINGRIDandMUMON matchquitewellandtheneutrinobeampositionisfairlystable.Theeventratemeasured byINGRIDisalsoplottedandverystableacrosstheT2Krunperiods. prisedof˝veseparatesub-detectorsanditsprimaryfunctionistomeasureuclei interactionsbytrackingtheparticlesinvolvedintheinteractions.The˝vesub-detectors includethefollowing:theTimeProjectionChambers(TPCs),theFine-GrainedDetectors (FGDs),thePi-ZeroDetector(PØD),theElectromagneticCalorimeters(ECAL),andthe Side-MuonRangeDetectors(SMRDs).Asummaryoftheprimaryfunctionsofeachsub- detectorusedintheanalysisispresentedinTab.3.2.Thesub-detectorsareplacedinside theUA1magnetexceptfortheSMRDs,whichareinterleavedwiththemagnetyoke,inthe con˝gurationshowninFig.3.9.TheUA1magnetproducesa0.2Tmagnetic˝eldtoenable signselectionandprecisemomentummeasurementsofchargedparticles.ThePØDisde- signedtoidentify ˇ 0 events,totrackchargedparticles,andtomeasureneutrinointeractions oncarbon,water,andleadastargets.ThePØDisnotusedintheanalysispresentedinthis thesismoredetailscanbefoundinRef.[73]. 37 Figure3.9:ExplodedviewoftheND280detectorshowingtheinnertrackingregion,with eachsub-detectorvisible.TheSMRDsareinterleavedwiththemagnetyoke.Takenfrom [16]. Figure3.10:ND280eventdisplayshowingthePØD,FGDs,andTPCs.Aneutrinointerac- tionoccurredinFGD1producingmanytrackswithanunrelatedmuontraversingthePØD andTPCs.ThedownstreamECALisshown,whilethebarrelECALandSMRDsarenot shown. 38 Sub-detector PrimaryFunctions Fine-GrainedDetector(FGD) Chargedparticletracking,targetmaterial TimeProjectionChamber(TPC) Chargedparticletracking,particleID, momentummeasurement ElectromagneticCalorimeter(ECal) Chargedparticletracking,photonand ˇ 0 ID SideMuonRangeDetector(SMRD) Vetoplanesforcosmicmuons,highangle tracking Table3.2:ListofND280subdetectorsandtheirprimaryfunction. 3.4.1Fine-GrainedDetectors TwoFine-GrainedDetectors(FGDs)providethemaintargetinND280,consistingoffully activescintillatorbarsinthe˝rstFGDwithacombinationofscintillatorbarsandwater targetinthesecondFGD.Thisprovidesahighmassofcarbon(fromplasticscintillator)and waterasatargetmaterialforanalyseswitheachFGDsupplying1.1tonsoftargetmaterial andtrackingofparticlesfromtheinitialinteractionvertex.Chargedparticlestraversingthe FGDscreatescintillationlight;andbycombiningwhichXandYbarswerehit,theparticle canbetrackedinthreedimensions.IftheparticlestopsintheFGD,thelengthofthetrack fromtheinteractionvertexisusedtodetermineitsmomentumknowingtheenergylossper unitdistanceinthescintillator.Theamountoflightcollectedineachbarisproportional todepositedenergywhichisusedincertainanalysistechniquessuchasquantifyingvertex activity.Eachscintillatorbarhasawavelengthshifting˝berthreadedthroughthecenter whichisattachedtoamulti-pixelphotoncounteratoneendtoreadoutthelightsignal. Theotherendofthescintillatorbarismirroredwithaluminumandthewholebarhasa re˛ectivecoatingof TiO 2 toincreasetheamountoflightcaptured. The˝rstFGDcontains5760scintillatorbarsarrangedinto30layersof192barswhere eachlayerisorientedinalternatingXandYdirectionswhichareperpendiculartothe incomingneutrinobeam(showninFig.3.11).AnXYmoduleconsistsofonelayerof192 scintillatorbarsinthehorizontaldirectionand192scintillatorbarsintheverticaldirection, 39 Figure3.11:ViewoftheFGD1scintillatorbarsshowingtheorientationofthelayers.Each pairofhorizontalandverticallayersofscintillatorbarsisde˝nedasasingleXYmodule. ThedimensionsgivetheoverallsizeoftheFGDandits˝ducialvolume. whichareperpendiculartoeachother.Thisseparatesthe5760scintillatorbarsinto15XY modulesalongthez-directionoftheFGD.TheXandYlayersaregluedtogetherwiththin sheetsofG10ontheoutsidetoprovidestructuralstability.ThesecondFGDuses7ofthe sameXYmodulesasthe˝rstFGDalternatingwithsix2.5cmwaterlayers(givingatotal of2688activescintillatorbars).Thewaterlayersareconstructedusingthin-walledhollow corrugatedpolycarbonate˝lledwithpurewatersealedatbothends.Bycomparingthe interactionratesofthe˝rstandsecondFGD,orbyidentifyingneutrinointeractionsthat likelyoccurredinthewaterlayers,thecross-sectiononoxygen(water)canbedetermined. TheFGDsareeachcontainedinalight-tightdarkboxthatcontainsthescintillatorbars, ˝bers,andMPPCswhiletherestoftheelectronicsaremountedoutsideofthedarkbox. BothFGDswerebuiltwiththesamegeometryandreadoutforinteroperability. 40 3.4.2TimeProjectionChambers ND280hasthreegaseousargonTimeProjectionChambers(TPCs)whichprovidehigh resolutiontrackingandidenti˝cationofchargedparticles.TheTPCsmeasurethecurvature oftracksleftbychargedparticlesinthemagnetic˝eldalongwiththeirenergylossallowing formomentummeasurementsandparticleidenti˝cation.EachTPCisconstructedofan innerboxholdinganargon-baseddriftgasandanouterboxwitha CO 2 insulatinggas volume(theoutergap).Thewallsoftheinnerboxaremadefrompanelswithcopper-clad G10skinsthathavebeenmachinedtohavea11.5mmstrippatternthatwhencombinedwith thecentralcathodeplaneformsanuniformelectricdrift˝eldwithintheTPC.Aschematic ofthelayoutofasingleTPCisshowninFig.3.12. Figure3.12:Cut-awayschematicviewofaTPCmoduleshowingthemainaspectsofthe TPCdesign. ChargedparticlestraversingtheTPCgasvolumewillionizethedriftgasproducingelec- tronsalongthepaththatdriftawayfromthecentralcathodetowardthereadoutplanes.The driftelectronsaremultipliedandsampledwithbulkmicromegasdetectorsonthereadout planes.Thesixreadoutplanes(twoperTPC)contain12micromegaspanelsmeasuring342 mm 359mm(vertical horizontal)foratotalof72micromegaspanels.Eachmicromegas 41 panelcontains1728padsarrangedin48rowsof36padswhereeachpadcoversanaareaof 7mm 9.8mm.Thetimeofarrivalcombinedwiththepatternofthesignalsinthereadout planecombinetogivecomplete3DreconstructionoftheparticletrackswithintheTPCs. Thesignalsfromthemicromegasarecollectedbyasetofsixfrontendcardscontaining customintegratedcircuitspermicromegaspanel. ThedriftgasintheTPCsisprimarilyanargonmixture,chosenforitslowdi˙usion, highspeed,andgoodcompatibilitywiththemicromegas.EachTPCcontains3000liters ofthedriftgas,whichisamixtureofAr:CF 4 :iC 4 H 10 ina95:3:2ratio(whereiC 4 H 10 is isobutane),andtheoutergapvolumescontain3300litersofCO 2 asabu˙ergas.Themajor contaminantsinthedriftgasareO 2 ,CO 2 andH 2 O,whichcanchangethedriftvelocityof theelectronsandgainofthemicromegas.Boththedriftgasandbu˙ergasarepumped continuouslythroughtheirrespectivevolumestorefreshthegasandpreventabuildupof contaminants.Theoutergapvolumeis˛ushed˝vetimesevery1.5dayswhereaseachTPC volumeis˛ushed˝vetimeseverydaywithapproximately90+%ofthegasbeingpuri˝ed andrecycledinthesystem. TheTPCmeasuresthemomentumofchargedparticlesbymeasuringthecurvatureof thetrackleftbytheparticle.CombinedwiththespatialresolutionoftheTPC,theTPC achievesthedesigngoalofa p ? =p ? < 0 : 1 momentumresolution.Inadditiontomeasuring momentumbytheradiusofcurvature,thedirectionofcurvature(curvingupordown)can beusedtodeterminethesignofthechargedparticleallowingfortheseparationofpositive andnegativelychargedparticles.Neutrinoandanti-neutrinointeractionscanbeseparated byidentifyingthechargeoftheoutgoinglepton,forexampleinmuonneutrinocharged currentinteractions, + X ! + X 0 (3.2) + X ! + + X 0 (3.3) where X;X 0 isthetargetbeforeandaftertheinteraction.Particleidenti˝cationisperformed bymeasuringtheenergyloss( d E= d x )ofaparticleversusitsmomentumandcomparingit 42 toknownvaluesforagivenparticletypefromcalibrationstudies.Theenergyresolutionfor aminimumionizingparticleis 7 : 8 0 : 2% andallowsforexcellentseparationbetweenmuons andelectronsbelowamomentumof1.0GeV/c,seeFig.3.13. Figure3.13:Measuredenergylossversusmomentumforpositively(top)andnegatively (bottom)chargedparticlestraversingtheTPC.Plottedaretheexpectedenergylosscurves forelectron,muons,protons,andpions.FiguresfromRef.[17] 3.4.3ElectromagneticCalorimeters TheND280innerdetectors(PØD,FGD,TPC)aresurroundedbyasamplingElectromag- neticCalorimeter(ECAL).TheECALisdesignedtoaidinfulleventreconstructionby detectingphotonsandmeasuringtheirenergyanddirectioninadditiontomeasuringany chargedparticlesthatexitthetrackingdetectors.Akeyfunctionofphotondetectionisthe identi˝cationofpi-zeroparticlesproducedinneutrinointeractionsinsidethetracker. 43 TheECALisconstructedfromlayersofplasticscintillatorbarsasactivedetectionma- terialandleadsheetsasradiatorsprovidingnearhermeticcoverageoftheinnerdetector volume.Itiscomprisedof13individualmodulesthatarearrangedintothreecon˝gurations: 6barrel-ECALmodulessurroundingtheinnertrackingdetectoronallsides,6PØD-ECAL modulessurroundingthePØD,and1downstreamECALmoduletocoverforwardgoing particles.Thescintillatingpanelsuseawavelengthshifting˝berandmulti-pixelphoton counterforlightdetectionandreadout.EachscintillatorbarhasacoatingofTiO 2 and oneendofthebarmirroredwithaluminumtoprovideinternalre˛ectiontoincreaselight collection.ThedownstreamECALmoduleconsistsof34layerswith1.75mmthicklead sheetscorrespondingto10.6radiationlengths( X 0 ).DuetospaceconstraintsfromtheUA1 magnet,thebarrel-ECALmoduleshave31layersofthesameleadsheetsforatotalof9.7 radiationlengths( X 0 ).ThescintillatorbarsareagainconstructedinalternatingXYlayers orientedperpendiculartoeachother. 3.4.4UA1MagnetandSideMuonRangeDetectors TheND280detectorusestherecycledUA1magnetwhichprovidesa0.2Tdipolemagnetic ˝eldtomeasurethemomentumandsignofchargedparticlesproducedinneutrinointerac- tions.Themagnetisbuiltfromwater-cooledaluminumcoils,whichprovidethemagnetic ˝eld,andamagnetic˛uxreturnyoke.Duringtheinstallationofthemagnet,adetailed magnetic˝eldsurveywascarriedouttomapoutthemagnetic˝eldwithinthedetector volume.Themagnetic˝eldstrengthwasdeterminedusinganarrayofHallprobesandcar- riesanuncertaintyof2mTofeach˝eldcomponentatthenominalstrengthof0.2T.This precisemeasurementofthemagnetic˝eld,particularlythetransversecomponent,reduces theuncertaintyinthemomentumdeterminationofparticles. Thesidemuonrangedetectorprovidestwomainfunctions:itprovidesavetotriggerfrom cosmicmuonsormuonsfrominteractionsinthemagnetorthesurroundingwallsthatenter ND280,anditmeasureshigh-anglemuons(withrespecttothebeamdirection)thatescape 44 fromthetracker.TheSMRDiscomprisedof440scintillatormoduleswhichareinsertedin theairgapbetweenthesteelplatesofthemagnet˛uxreturnyokes.Themagnetconsists of16individualC-shaped˛uxreturnyokeelements,eachofwhichcontain16steelplates providing15airgapsbetweenthem.EachmoduleoftheSMRDcontainsseveralrectangular scintillatingpanelsutilizingawavelengthshifting˝berandmulti-pixelphotoncounterfor lightreadout.Intotalthereare4016channelsacrosstheSMRD. 45 CHAPTER4 EVENTSELECTION 4.1Signalde˝nition Thesignalde˝nitionconsistsofneutrinointeractioneventsonplasticscintillator(C 8 H 8 ) thatproduceonenegativelychargedmuon,zeropions,andanynumberofprotonsasthe ˝nalstateparticleswheretheinitialvertexoccurredinthe˝ducialvolumeofthedetector, collectivelycalledtheCC- 0 ˇ topology(examplesketchesshowninFig.4.1).Signalevents aredescribedusingthemeasuredmuonkinematics, p ; cos wheretheangle isthepolar anglebetweentheincidentneutrinoz-axis 1 andtheoutgoingmuondirection.Itisworth notingthattheneutrinoz-axisandthedetectorz-axisarethesameatINGRIDwhereasat ND280theyareslightlyoutofalignment(whichiscorrectedforinthisanalysis).Sincethe signalisde˝nedintermsofthe˝nalstateparticles,interactionswhereapionwasproduced intheinitialinteractionandsubsequentlyabsorbedinthenucleusareincludedinthesignal de˝nition. Thesignalisde˝nedtoremoveasmuchdependenceontheinteractionmodelingas possiblebyspecifyingthede˝nitionintermsofthe˝nalstateparticlesthatareobservedin thedetector.Whilehardertopredictusingneutrinotheory,itislessmodeldependentto de˝nethesignalbasedonwhatparticlesthedetectorwasabletomeasureinsteadofrelying onthenuclearmodelandMonteCarlogeneratortocorrectbacktotheinitialinteraction (cf.Section2.3.6).Thecollectionof˝nalstateparticleswhichde˝neaneventisdenoted asthetopologyinthecontextofthisthesis.TheCC- 0 ˇ topologywaschosenasthesignal de˝nitionbecauseitisthemostcommoneventtypefortheT2Kenergyspectrum,anditis theprimarysignaleventtopologyfortheT2Koscillationanalysis.Asummaryofthesignal de˝nitionforeachdetectorispresentedbelow. 1 Thez-axisisde˝nedtobethedirectionoftheneutrinopropagation. 46 Figure4.1:Examplesignalandbackgroundeventsketchesforagenericdetector.Aneutrino interactsinthedetectorandseveralparticlesareproduced.Signaleventshavezeropions, whilebackgroundeventshaveoneormorepionsinthe˝nalstate. 4.1.1ND280signalde˝nition Topology: onenegativelychargedmuon,zeropions,andanynumberofprotonsas the˝nalstateparticles. Observables: Muonmomentumandangle: p ; cos Flux: T2K ˛ux,version13av2.0(thin-targettuning)[15]. Target: plasticscintillator(C 8 H 8 )intheFGD1˝ducialvolume Phasespace: Norestrictions. 4.1.2INGRIDsignalde˝nition Topology: onenegativelychargedmuon,zeropions,andanynumberofprotonsas the˝nalstateparticles. 47 Observables: Muonmomentumandangle: p ; cos Flux: T2K ˛ux,version13av2.0(thin-targettuning)[15]. Target: plasticscintillator(C 8 H 8 )intheProtonModule˝ducialvolume Phasespace: Eventswith p > 0 : 35 GeV/cand cos > 0 : 50 . NotethatINGRIDisnotamagnetizeddetectoranddoesnothavethecapabilitytosign- selectmuons.Bothpositiveandnegativemuonsfrom and eventswillbeselectedby thedetectorassignalcandidates,howeveronly eventsaretreatedassignaleventsinthe analysis.ThekinematicphasespacerestrictionofthemuonfortheINGRIDsignalde˝nition isplacedbasedonthee˚ciencyfortheProtonModuletodetecttheoutgoingmuon(see Section4.6forthediscussionofe˚ciency).GiventhegeometryandcapabilityoftheProton Module,itisnotwellsuitedtomeasuringlowmomentumorbackwardgoingmuons.Thisis thesamerestrictionusedintheoriginalINGRID-onlyanalysisdetailedinT2K-TN-204[5]. 4.2MonteCarloanddatasamples ThisanalysisusesT2KdatacollectedatINGRIDusingtheProtonModuletakenduring Run2,Run3,andRun4correspondingto 5 : 9 10 20 protons-on-target(POT),andND280 datatakenduringRun2,Run3,Run4,andRun8correspondingto 11 : 53 10 20 POT.The di˙erenceindeliveredPOTisduetoboththeuptimeofeachdetectorandtheadditional runforND280(theProtonModulewasmovedbeforeRun8).Tables4.1and4.2reportthe datastatisticscollectedandtheMonteCarlo(MC)statisticsusedforND280andINGRID respectively.TheMCsamplesareweightedtomatchthePOTofthedatasamplesusing thedata/MCratioforeachrunindividually.Eachrunrepresentsadi˙erentdatataking periodincludingthee˙ectofdi˙erentprotonbeamconditions,horncurrent,etc.andis furthersubdividedbasedonwhetherthePØDwatertargetwasin(water)orout(air).This analysisonlyusesdatatakingrunsinneutrinobeam(FHC)mode. 48 T2KRunDataPOTMCPOTData/MCRatio Run2a(air) 0.359 10 20 9.239 10 20 0.0389 Run2w(water) 0.433 10 20 12.034 10 20 0.0360 Run3b(air) 0.217 10 20 4.478 10 20 0.0485 Run3c(air) 1.364 10 20 26.323 10 20 0.0518 Run4a(air) 1.783 10 20 34.996 10 20 0.0509 Run4w(water) 1.643 10 20 22.622 10 20 0.0726 Run8a(air) 1.581 10 20 36.305 10 20 0.0435 Run8w(water) 4.149 10 20 26.412 10 20 0.1571 Total 11.529 10 20 172.409 10 20 0.0669 Table4.1:Data-takingperiodsandthePOTusedinthisanalysisfordataandMCfor ND280. T2KRunDataPOTMCPOTData/MCRatio Run2-4 5.9 10 20 2.77 10 23 0.0021 Total 5.9 10 20 2.77 10 23 0.0021 Table4.2:Data-takingperiodsandthePOTusedinthisanalysisfordataandMCfor INGRID. TheND280MCisproducedinseveralstepsstartingwiththeNEUTneutrinoevent generator[48]tosimulateneutrinointeractionswithintheND280geometry.Theoutput fromtheNEUTsimulationispassedtoaGEANT4[74]simulationoftheND280detector, whichpropagatesthe˝nalstateparticlesthroughthedetectoraccountingforenergyloss,the externalmagnetic˝eld,secondaryinteractions,etc.The˝nalstepistoprocesstheGEANT4 simulationthroughtheND280dataacquisitionandelectronicsimulationproducing˝lesand objectswhichsimulaterealdata.MonteCarlosamplesaregeneratedforeachdatataking runseparatelytoaccountforthedetectorconditions,suchasthecon˝gurationofthePØD. ThedefaultnuclearmodelforthisanalysisistheBenharet.al.spectralfunction(SF)with M QE A =1 : 21 GeV = c 2 andcontributionsfromtheNieveset.al.multinucleon(2p2h)model [13]producedwithNEUT5.3.2. TheINGRIDMCisproducedinasimilarfashionstartingwiththeNEUTneutrino 49 eventgeneratorfortheinitialneutrinointeractions,thenproceedingwithaGEANT4based simulationforthedetector.INGRIDhasaseparateNEUT,GEANT4,andelectronicsimu- lationfromND280,andproducesslightlydi˙erent˝lesandobjectsforthesimulateddata. TheINGRIDMCsoftwarestackisfurtherdocumentedinT2K-TN-204[5].TheINGRID MCusesNEUT5.3.3toproducetheneutrinointeractionsusingthesamedefaultnuclear modeldescribedabove,howeverthedi˙erencesbetweenNEUTversions5.3.2and5.3.3have minimaltonoimpactonthisanalysis. ThedataandMCsamplesforND280havesomecorrectionsappliedtoreduceknown data/MCdi˙erences.Thesecorrectionsarebasedonknownhardwarefailuresanddiscrep- ancies,orfromstudiesusingprecisecontrolsamples.ThecorrectionsarelistedinTable4.3 andareappliedduringeventselectionforND280.ThedataandMCsamplesforINGRID undergoasimilarprocedurewithadi˙erentsetofcorrections,whicharelistedinTable4.3. ND280Corrections DataQuality dE=dx Datacorrection dE=dx MCcorrection Momentumresolution Momentumbyrange TimeofFlight(TOF) TPCexpected dE=dx TPCparticleidenti˝cation(PID) FGDparticleidenti˝cation(PID) IgnoreDeadChannels INGRIDCorrections DataQuality IgnoreDeadChannels Pile-upCorrection VetoE˚ciency Table4.3:SummaryoftheND280data/MCcorrections(left)andINGRIDdata/MCcor- rections(right). 4.3Signaleventselection ThesignalforthisanalysisistheCC- 0 ˇ topology,whereamuoncandidate,anynumber ofprotoncandidates,andzeropioncandidatesaredetectedinthe˝nalstateonaCHtarget (subjectto˝ducialvolumecuts)ineitherFGD1(ND280)ortheProtonModule(INGRID). ThisanalysisusespreviouslywelldevelopedandtestedCC- 0 ˇ selectionsforINGRID[5] 50 andND280[3],whichhavebeenusedintheneardetectorconstraintfortheT2Koscillation analysisandpreviouscrosssectionanalyses.Abriefdescriptionoftheselectionforeach detectorispresentedhere;thissectiondetailsthesignalsampleswhileSection4.4details thesidebandsamples. 4.3.1ND280eventselection TheND280eventselectionisthenearlysameasthemostrecentCC- 0 ˇ analyses,detailed inT2K-TN-337[3]andT2K-TN-338[75],whicharethemselvesbuiltupontheselectioncuts presentedinT2K-TN-216[76]andRefs.[34,77].FGD1isusedbothasaCHtargetanda trackeralongwiththethreeTPC'sandFGD2providingtrackingandparticleidenti˝cation. Eventswithonenegativelychargedmuon,anynumberofprotons,andnootherreconstructed trackswherethevertexwasfoundtobeintheFGD1˝ducialvolumeareselectedassignal events.Thesignaleventsarethencategorizedintosignalsamplesbywhichsub-detectors wereusedinthereconstruction,andbythepresenceofareconstructableproton.This separationofthesamplesallowsformoreaccuratetreatmentofdetectorsystematicsbased onthecapabilitiesofeachdetector.Thesesamplesarede˝nedasthefollowing(andshown aseventdisplaysinFig.4.2): Sample0 ( TPC):asinglemuoncandidatewasdetectedasanFGD-TPCtrackand noothertrackcandidatespresent. Sample1 ( TPC+pTPC):asinglemuoncandidateandprotoncandidatewerede- tectedasFGD-TPCtracks. Sample2 ( TPC+pFGD):asinglemuoncandidatedetectedasanFGD-TPCtrack andasingleprotoncandidateasanFGDtrack. Sample9 ( TPC+Np):asinglemuoncandidatedetectedasanFGD-TPCtrackand multipleprotoncandidatespresent. 51 Sample3 ( FGD+pTPC):asinglemuoncandidatedetectedasanFGDtrackand asingleprotoncandidateasanFGD-TPCtrack. Sample4 ( FGD):asinglemuoncandidatewasdetectedasanFGDtrackandno othertrackcandidatespresent. Additionalsamplesarede˝nedandusedassidebandstoconstrainbackgroundevents,and aredescribedinSection4.4.1,andforthisanalysisthe TPC+pFGDandthe TPC+Np samples(samples2and9)arecombinedwhenperformingthe˝tduetothelowstatisticsof the TPC+Npsample.Othersamplede˝nitionsarepossible(e.g. FGD+pFGD),however thesesampleseitherhavetoofewexpectedeventsorhaveverypoorreconstruction.This resultsin˝veseparatesignalsamplesasinputforthe˝t. Thecut˛owforeachsignalsampleisshowninFig.4.3.Aneventmustpassallcutsfor agivensamplede˝nitiontobeplacedinthatsample,andeventswhichfailacutmaystill meetthecriteriatobeplacedinadi˙erentsample(signalorsideband).Thesamplesare mutuallyexclusive,aneventcanbelongtoonlyasinglesample(signalorsideband)ormay notpassenoughcutstobeusedintheanalysisatall. EventQuality :Eventsarerequiredtopassaqualitycutensuringthedetectorand beamwereingoodworkingcondition.Beamrunsandbeamspillswhicharenot consideredtobeofgoodqualitytobeanalyzedarerejected.Thiscutonlyappliesto dataeventsasMCeventsareassumedtopassthequalitycut. FiducialVolumeCut :Allselectedeventsarerequiredtohavetheirvertexplaced intheFGD1˝ducialvolume,whichispartofthevertexcutsbelow.The˝ducial volumeofFGDisde˝nedtobe: j x j < 874 : 51 mm, 819 : 51 0 : 8 (4.4) whichisonlyappliedfortrackswithmomentumlessthan500MeV/c.Toidentifyas amuon,proton,orpionthecutsare L > 0 : 05 ; L p > 0 : 5 ; L ˇ > 0 : 3 (4.5) ThecutvalueswerechosenfromMCstudiesandtest-beamdataandcanbetunedfor agivenanalysis[78,79]. FGDMuonPID :TheFGDPIDalgorithmisconstructedinasimilarfashiontothe TPCPIDalgorithmasitde˝nesasetofpulls i foreachparticletypetocalculate alikelihood.Thedi˙erenceisinthefunctionusedtocalculatethepull,wherethe expectedenergydepositedisproportionaltothemeasuredtracklength.Thepullsfor agivenparticletype i arecalculatedby i = E obs E exp i ( x ) ˙ exp i ( x ) (4.6) where E istheobservedorexpectedenergydeposited, x isthemeasuredtracklength, and ˙ istheexpectedenergyresolutionforaparticletype.Basedondetectorstudies andMC,thefunctionusedtocalculatetheexpectedenergyasafunctionoftrack lengthisgivenby E i = A i x B i + C i x (4.7) where A i ;B i ;C i areconstantsforeachparticletypeextractedfrom˝tstodata[80]. Thelikelihoodforaparticletype i isbasedontheGaussiandistributionsofthepulls andisde˝nedas L i = G i ( i ) P n G n ( n ) (4.8) wherethesumisovertheparticletypes:muon,electron,pion,proton. 55 Onenegativetrack :Thiscutisfailedifthereareextranegativetracks(identi˝ed bytheTPC)whichshareacommonvertexwiththeHMNtrack.Thiscutidenti˝es eventswithasinglemuonandnootherparticles. Onlymuon/protontracks :Thiscutisfailedifthereisatrackpresentwhichwas identi˝edtobeneithermuon-likenorproton-likebythePIDalgorithmwhichsharesa commonvertexwiththeHMNorHMPtrack.Thiscutidenti˝eseventswithasingle muonandanynumberofprotons. NoMichelelectron :Thiscutrequiresnoidenti˝ableMichelelectrons(theelectron frommuondecay)arefoundintheFGD.Thiscutisappliedafterthemuoncandidate hasbeenidenti˝ed,thusaMichelelectroncouldhavebeenproducedbyalowenergy piondecayingintoamuon,whichitselfsubsequentlydecaysintoanelectron.AMichel electronisidenti˝edbyseeingachargedepositionconsistentwithanelectroninthe FGDoccurringaboutonemuonlifetime(2.2 s)aftertheeventstarttime. Zero/One/Multipleprotons :Thiscutseparateseventsintodi˙erentsamplesbased onhowmanyprotontrackswereidenti˝edbythePIDalgorithmsharingacommon vertexwiththeHMNtrack. FGD/TPCproton :Thiscutseparateseventswithasingleprotontrackintodi˙erent samplesbasedoniftheprotonstoppedintheFGDorwasidenti˝edasaTPCproton track. HMPvertex :SimilartotheHMNvertexcut,thehighestmomentumpositive(HMP) chargedvertexpositionissetbythehighestmomentumpositivechargedtrackwhich reachesandisidenti˝edbytheTPC.Requiringthetracktobereconstructedinthe TPCincreasestheaccuracyofthevertexplacement.Tracksarerequiredtohaveat least18segmentsintheTPCtobereconstructedasatrack. 56 Leadingprotontrack :ThiscutrequiresthetrackusedtosettheHMPvertexis identi˝edasproton-likebytheTPCPIDalgorithm. LongFGDtrack :ThiscutrequirestheFGDmuoncandidatetobelongenough (greaterthan500mm)tohaveareliablemeasureofitskinematics. Stoppingmuon :ThiscutrequirestheFGDmuoncandidatetobewithintheactive regionofND280.ThemuonmustbefullycontainedasthePIDalgorithmandmomen- tummeasurementsarebasedontherangetraversedintheFGDandtheECAL/SMRD. HAmuon :Thiscutsearchesforahighangle(HA)trackwhichisidenti˝edasa muoncandidate(bytheFGDPIDalgorithm)wherethevertexisintheFGD˝ducial volume.AhighangletrackisonewhereitdoesnotentertheTPC,whichgenerally requiresahighanglerelativetotheforwarddirection. ECALPID1&2 :IftheHAmuoncandidatereachestheECAL,the d E= d x and lengthofthetrackintheECALareusedtofurtherverifytheparticleidentity.This cutlooksfor d E= d x consistentwithamuon(ECALPID1),andfortheratiolength toestimatedenergyofthetracktobeconsistentwithamuonhypothesis(ECALPID 2). Figure4.3:Chartshowingtheselectioncutsusedtode˝neeachND280signalsample. ThemuoncandidatekinematicdistributionsforeachND280signalsamplearedisplayed inFig.4.4andFig.4.5splitbythetruetopology,showingtheproportionoftruesignal 57 andbackgroundevents.Inaddition,Fig.4.6andFig.4.7showthekinematicdistributions splitbytruereaction,anddistributionsasafunctionofmomentumtransfercanbefound inAppendixF.Forbothsetsofplots,the`outFV'referstoeventswhichactuallyoccurred outsidethe˝ducialvolume,butwerestillselected,andthe`BKG'(or`other')categoryis acatch-allforeveryothertypeofeventnotbelongingtoalistedcategory.Thesamples wherethemuoncandidatewasreconstructedintheTPChavesimilarmomentumandangle distributionswithsampleswhereaprotoncandidatewerefoundtohavemoremuonsat higherangles.ThesampleswherethemuoncandidatewasreconstructedintheFGDhave veryfewforwardgoingmuonsandlowermomentummuons.Thesharpdropofeventsata cosineofarondzeroisduetoalowreconstructione˚ciencyformuonswhicharetraveling (nearly)straightupwardsastheytravelparalleltothescintillatorbarsanddonotcross enoughbarstobereconstructed.ThesampleshaveahighpurityoftrueCC- 0 ˇ events,with themainbackgroundsplitfairlyevenlybetweenCC- 1 ˇ andCC-Otherevents.The˝nal CC- 0 ˇ signalisextractedbyaddingthecontributionsofeachsignalsampletogether,but theseparationisvaluablewhenrunningthe˝tbecauseeachsampleisa˙ectedbydi˙erent systematicsandbackgrounds.ThepurityforeachsampleandthetotalisalsolistedinTab. 4.4. SampleCC- 0 ˇ CC- 1 ˇ CC-OtherBackgroundOutofFVEvents Sample0: TPC86.664.833.981.173.3615526.90 Sample1: TPC+pTPC73.589.8711.433.032.093480.27 Sample2: TPC+pFGD82.758.485.361.312.092332.07 Sample9: TPC+Np57.8816.0317.573.365.17317.41 Sample3: FGD+pTPC84.254.192.782.136.661533.94 Sample4: FGD63.191.951.333.2730.252220.59 CombinedTotal81.935.725.001.715.6425411.18 Table4.4:PurityofeachND280signalsampleandthepurityofthecombinedtotal. 58 Figure4.4:EventdistributionforreconstructedmuonmomentumandanglefortheND280 signalsampleswithamuontrackintheTPCstackedbytruetopology.Thepurityofeach topologyislistedinthelegend.Thelastbinformuonmomentumcontainsalleventswith momentumgreaterthan5GeV/c. 59 Figure4.5:EventdistributionforreconstructedmuonmomentumandanglefortheND280 signalsampleswithamuontrackintheFGDstackedbytruetopology.Thepurityofeach topologyislistedinthelegend.Thelastbinformuonmomentumcontainsalleventswith momentumgreaterthan5GeV/c. 60 Figure4.6:EventdistributionforreconstructedmuonmomentumandanglefortheND280 signalsampleswithamuontrackintheTPCstackedbytruereaction.Thepurityofeach reactionislistedinthelegend.Thelastbinformuonmomentumcontainsalleventswith momentumgreaterthan5GeV/c. 61 Figure4.7:EventdistributionforreconstructedmuonmomentumandanglefortheND280 signalsampleswithamuontrackintheFGDstackedbytruereaction.Thepurityofeach reactionislistedinthelegend.Thelastbinformuonmomentumcontainsalleventswith momentumgreaterthan5GeV/c. 62 Figure4.8:Twodimensionaleventdistributionforreconstructedmuonmomentumvsangle fortheND280signalsampleswithamuontrackintheTPC(left)oramuontrackinthe FGD(right).Thecombinationisshowninthebottomplot. 63 4.3.2INGRIDeventselection TheINGRIDCC- 0 ˇ selectionisthesameonedescribedinT2K-TN-204[5]andisbrie˛y describedhere.TheselectionusesthescintillatorintheProtonModule(PM)asaCHtarget andboththePMandtheadjacentdownstreamstandardINGRIDmodulefortracking.The signalselectionisseparatedintosamplesbasedonwherethemuoncandidatestoppedinthe detectors,orwhereitexitedthedetectors,listedasfollowsandillustratedinFigure4.9: PMearlystopping :thetrackstoppedandisfullycontainedinthePMscintillator. PMescaping :thetrackescapesthePMbutdoesnotdepositanyenergyinthe INGRIDmodule. INGRIDearlystopping :thetrackescapesthePManddepositsenergyinthe INGRIDmodule,butisnotreconstructedasanINGRIDtrack. INGRIDstopping :thetrackescapesthePMandisreconstructedasanINGRID trackwhichstopsinthemodule. INGRIDside-escaping :thetrackescapesthePMandisreconstructedasan INGRIDtrackwhichhashitsinanINGRIDsidechannel. INGRIDthrough-going :thetrackescapesthePMandisreconstructedasan INGRIDtrackwhichhashitsinthelastlayer. ThesampleswherethemuoncandidatecouldhaveexitedINGRIDcanonlybeusedto placealowerlimitonthemuonmomentumofapproximately1GeV/c.Allsamplesare used,butalleventsthatarenotfullycontainedwillfallinthelastmomentumbinfrom1 to30GeV/c.Theremainingeventsthatarefullycontainedprovideameasurementofthe muonmomentum(determinedbyrange).Alleventshavereliableanglereconstructionand measurements. TheCC- 0 ˇ sampleselectseventswhichsatisfythefollowingcriteria.NotethatINGRID isnotamagnetizeddetectoranddoesnothaveanycapabilitytosign-selectmuons.Both 64 Figure4.9:EventdisplayfortheProtonModuleshowingthedi˙erentINGRIDsamples. Greenistrackingscintillator,blueisvetoscintillator,andgrayareironplates.Figurefrom Ref.[5] positiveandnegativemuonsfrom and eventsareselectedascandidatesignalevents andthe eventsarebackgroundconstrainedwithdatawhenperformingtheanalysis(and aregenerallyanegligiblecontamination). Eventquality :Eventsarerequiredtopassaqualitycutensuringthedetectorand beamwereingoodworkingcondition.ThiscutisonlyappliedtodataeventsasMC eventsareassumedtopassthequalitycut. Beamtiming :Eventsarerequiredtobe 100 nsoftheexpectedbeambuncharrival timetoreducethecosmicbackground.Thetimingoftheeventisde˝nedasthetiming ofthelargestchargedepositionhitintheevent. Trackmultiplicity :Eventsarerequiredtohaveonlyoneortworeconstructedtracks. ACC- 0 ˇ eventingeneralwillhaveamuonandasingleprotonknockout,thuslarge majorityofCC- 0 ˇ eventsareoneortwotrackevents.Howeverthiswillrejecttrue CC- 0 ˇ eventswhichejectmultiplereconstructedprotons.Thisisasmallimpactonthe analysisgivenhowoftentwoprotontracksareabletobesuccessfullyreconstructed. 65 MuonPID :ThePIDalgorithmusedforthePM/INGRIDtracksisbasedonamul- tivariateboosteddecisiontree(BDT),whichusesaBayesiancon˝dencelevelasa preliminaryPIDplustheenergydepositiondistributionneartheendofthetrack(the last˝vehits).TheBayesiancon˝dencelevelusesBayes'theoremtoidentifyaparticle basedonits d E= d x measurement,forexamplethemuoncon˝dencelevelis CL = P ( j d E d x )= P (d E= d x j ) P ( ) P (d E= d x ) (4.9) wherethe d E= d x informationisfromeachhitinthetrack.Sincetrackhitscanoverlap duetothegranularityofthescintillator,onlyhitswhichareexclusivetoagiventrack areusedinthecon˝dencelevel.Assumingtheenergylossperhitisindependent,the probabilityforatracktobemuon-like( )ornotmuon-like( )canbewrittenas CL = ( Q nhits i P ([ d E d x ] i j ) P ( ) ( Q nhits i P ([ d E d x ] i j ) P ( )+( Q nhits i P ([ d E d x ] i j ) P ( ) (4.10) Alltracksreconstructedinagiveneventwhichshareavertexhavetheir d E= d x dis- tributionmeasuredandthennormalizedtogenerateprobabilitiesforatracktobe muon-likeandnotmuon-like,usingthecon˝dencelevelconstructedasabove. Thisinformationcombinedwiththechargedepositiondistributionneartheendofthe trackaregivenasinputstotheBDT.Thechargedistributionneartheendofthetrack isusedbecausethe d E= d x foragivenparticledependsonitsmomentum.Protons aremuchmorelikelytodepositmostoftheirenergyattheendofthetrack,whereas theenergydepositionformuonsandpionswillbemoreevenalongthetrack.The boosteddecisiontreewastrainedusingAdaBoost[81]throughtheTMVApackage [82],andtwocon˝dencelevelswerebuilt:onetoidentifymuontracks( CL ),andone toseparateprotonsfrompions( p CL )thatwererejectedbythemuoncon˝dencelevel. ForfurtherdetailsseeRef.[5]. Singlemuontrack :Oneandonlyonemuoncandidatetrack(positiveornegative). IfthecandidateisanINGRIDstoppingtrack,itismuon-likeif CL > 0 : 1 ,andifitis anINGRIDearlystoppingtrackthemuoncutis CL > 0 : 05 . 66 Zeropiontracks :Ifthereisasecondtrack,itmustbeproton-like, p CL > 0 : 17 . Timeclusteringcut :Requiresmorethan3hitsinan100nstimewindow.Inthis caseallhitswithin 50nsoftheaveragetimingoftheseeventsareselectedasahit clustertoremovemostofthehitscreatedbyrandomnoise. Activeplanescut :Requiresatleastthreeactivetrackingscintillatorplanes,andthe trackhastohaveareconstructeddistanceofatleast15cmtopassthiscut.Thisfurther reducestherandomnoiseandexternalbackgroundcreatedbyphotonsorneutrons. Upstreamveto :Eventshavingtheirmostupstreamhitinthevetoplanes,de˝ned asthe˝rstfourlayersofthePM,arerejected.AdditionallythePMhasasetofedge vetoplaneswhereaneventshowingahitintheedgevetoatadistancelessthan80 mmtotheextrapolatedtrackupstreampositionarerejected. Fiducialvolumecut :The˝ducialvolumeisde˝nedasatransversecentral ( 50) ( 50) cm 2 regionofthePM(totaling100cmineachdirection).Eventsinwhichthe mostupstreamhitisnotcontainedinthe˝ducialvolumearerejected. ThemuoncandidatekinematicdistributionsfortheINGRIDsamplearedisplayedinFig. 4.10splitbytruetopology,showingtheproportionoftruesignalandbackgroundevents. TheCC- 0 ˇ topologyinFig.4.10hasbeensubdividedintocategoriesbyhowmanyprotons wereintheevent,allofwhichcombinetogethertoformthetotalCC- 0 ˇ signal.Theother backgroundcategoryisacombinationofcontributionsfrom , e ,outof˝ducialvolume, andotherdetectorbackgroundevents.TheINGRIDsamplehasadecentpurityoftrue CC- 0 ˇ eventsconsideringthelimitationscomparedtoND280,whichisshowninTab.4.5. Inaddition,Fig.4.11showsthesamekinematicdistributionssplitbytruereaction. 67 INGRIDCC- 0 ˇ CC- 1 ˇ CC-OtherBackgroundEvents Stopping62.8115.444.8616.891634.37 Stop+Escaping67.1615.925.0411.889921.24 Table4.5:PurityoftheINGRIDsignalsampleforbothstoppingonlyandstoppingplus through-goingtracks. Figure4.10:Eventdistributionforreconstructedmuonequivalentdistanceinironandan- glefortheINGRID(early)stoppingsignalsamples(top)andallINGRIDsignalsamples (bottom)stackedbytruetopology. 68 Figure4.11:Eventdistributionforreconstructedmuonequivalentdistanceinironandangle fortheINGRID(early)stoppingsignalsample(top)andallINGRIDsignalsamples(bottom) stackedbytruereaction. 69 4.4Sidebandselection Theprimarybackgroundsforthisanalysisareeventswhereoneormultiplepionswere misidenti˝edornotreconstructed(eitherthroughdetectorerrororFSI),causingtheeventto appearlikeaCC- 0 ˇ event.Toimprovetheperformanceoftheanalysis,severalsideband(or control)sampleshavebeendevelopedtoselecteventswithoneormultiplepions.Thisgives the˝ttheabilitytoconstraintherateofcharged-currentsingleormultiplepionproduction rates. 4.4.1ND280sidebandselection TheND280CC- 0 ˇ analysishasthreemainsidebandsamplestoconstrainthebackgrounds byselectingeventswithoneormorechargedpionsinthe˝nalstateorthepresenceofa Michelelectron.Thesidebandsamplesareconstructedinasimilarmethodtothesignal samples,butseveralcutshavebeen˛ippedorchangedtoselectbackgroundeventsinstead ofsignalevents.Thethreesidebandsamplesare(shownaseventdisplaysinFig.4.12): Sample5 (CC- 1 ˇ + ):asinglemuonwithanFGD-TPCtrackandasinglepositively chargedpioncandidateweredetectedwithnoothertrackcandidatespresent(onlytwo trackspresent). Sample6 (CC-Other):asinglemuonwithanFGD-TPCtrackandmultipletracks detectedintheTPCwherethehighestmomentumpositivetrackisidenti˝edasapion candidate. Sample7 (CC-Michel):asinglemuonwithanFGD-TPCtrack,atleastoneMichel electrondetected,andnopiontracks. Thecut˛owforeachsidebandsampleisshowninFig.4.13.Aneventmustpassallcuts foragivensamplede˝nitiontobeplacedinthatsample,andeventswhichfailacutmay stillmeetthecriteriatobeplacedinadi˙erentsample.Thesidebandsamplesusemany 70 Figure4.12:EventdisplaycartoonfortheND280sidebandsamples. ofthesamecutswhicharedescribedinSec.4.3.1,andsidebandspeci˝ccutsaredescribed here. HMPPionTrack :Thiscutrequiresthehighestmomentumpositive(HMP)track tobe ˇ + -likeaccordingtotheTPCPIDalgorithmdescribedearlier.Resonantpion productionfromaneutrinoeventshouldproduceapositiveorneutralchargedpionin theabsenceofanyFSIe˙ects. TPCTracks :ThiscutisbasedonthenumberoftracksthatreachtheTPCwhich shareacommonvertextheFGD˝ducialvolume. MichelElectron :ThiscutrequiresthepresenceofasingleMichelelectronidenti˝ed intheFGD. 71 Figure4.13:Chartshowingtheselectioncutsusedtode˝neeachND280sidebandsample. SampleCC- 0 ˇ CC- 1 ˇ CC-OtherBackgroundOutofFVEvents Sample5:CC- 1 ˇ 7.0269.2914.956.991.751715.49 Sample6:CC-Other0.7210.7071.7110.866.002825.08 Sample7:CC-Michel10.3060.8710.501.9116.421424.66 Table4.6:PurityofeachND280sidebandsample. ThemuoncandidatekinematicdistributionsaredisplayedinFig.4.14splitbythetrue topology,showingtheproportionoftruesignalandbackgroundevents.Inaddition,Fig.4.15 showsthesamekinematicdistributionssplitbytruereaction,anddistributionsasafunction ofmomentumtransfercanbefoundinAppendixF.Forbothsetsofplots,the`outFV'refers toeventswhichactuallyoccurredoutsidethe˝ducialvolume,butwerestillselected,and the`BKG'(or`other')categoryisacatch-allforeveryothertypeofeventnotbelongingtoa listedcategory.Thethreesidebandsampleshavegoodpurity,andimportantlyhavefewto nosignalevents.ThepurityforeachsidebandsampleisalsolistedinTab.4.6.Figure4.17 showsthedistributionofthedesiredbackgroundevents(CC- 1 ˇ + andCC-Other)inboth thesignalandsidebandsamplesasafunctionoftrue Q 2 .ThedistributionofCC- 1 ˇ + events matchesverywellbetweenboththesignalandsidebandsamples,andthedistributionof CC-Othereventsshowssomeoverlapbetweenthesignalandsidebandsamplesatlowmuon momentum.Thesameplotsasafunctionofmuonmomentumandpionmomentumcanbe foundinT2K-TN-380[83]. 72 Figure4.14:EventdistributionforreconstructedmuonmomentumandanglefortheND280 sidebandsamplesstackedbytruetopology.Thepurityofeachtopologyislistedinthe legend.Thelastbinformuonmomentumcontainsalleventswithmomentumgreaterthan 5GeV/c. 73 Figure4.15:EventdistributionforreconstructedmuonmomentumandanglefortheND280 sidebandsamplesstackedbytruereaction.Thepurityofeachreactionislistedinthelegend. Thelastbinformuonmomentumcontainsalleventswithmomentumgreaterthan5GeV/c. 74 Figure4.16:Twodimensionaleventdistributionforreconstructedmuonmomentumvsangle fortheND280sidebandsamples.CC- 1 ˇ inthetopleft,CC-Otherinthetopright,CC-Michel atthebottom. Figure4.17:Areanormalizedeventdistributionsasafunctionof Q 2 forthebackground eventscomparedbetweenthesignalsamplesandthecorrespondingsidebands.TheCC- 1 ˇ + distributionisontheleft,andmatchesquitewell.TheCC-Otherdistributionisontheright andisquitedi˙erentbetweenthesignalandsideband. 75 4.4.2INGRIDsidebandselection TheINGRIDCC- 0 ˇ asdevelopedinT2K-TN-204hasasingleCC- 1 ˇ sideband[5].Un- fortunatelythereisnoINGRIDCC-OtherorCC-Michelsidebandtoincludesimilartothe ND280selection.TheINGRIDCC- 1 ˇ sampleselectseventswhichsatisfythefollowing criteria. Eventquality :Eventsarerequiredtopassaqualitycutensuringthedetectorand beamwereingoodworkingcondition.ThiscutisonlyappliedtodataeventsasMC eventsareassumedtopassthequalitycut. Beamtiming :Eventsarerequiredtobe 100 nsoftheexpectedbeambuncharrival timetoreducethecosmicbackground.Thetimingoftheeventisde˝nedasthetiming ofthelargestchargedepositionhitintheevent. TrackMultiplicity :Exactlytwoorthreereconstructedtracks.TheCC- 1 ˇ sample requiresamuonandpioncandidate,sotwotracksminimumarerequired. Muontrack :Asinglemuoncandidatetrackwith CL > 0 : 1 (sameastheCC- 0 ˇ selection). Piontrack :Asinglepioncandidatetrackwith p CL < 0 : 06 .Thiscutvaluehas beentunedtoincreasee˚ciencyofselectingpioncandidates. Optionalprotontrack :Ifathirdtrackisreconstructed,itshouldbeproton-like with p CL > 0 : 06 . Timeclusteringcut :Requiresmorethan3hitsinan100nstimewindow.Inthis caseallhitswithin 50nsoftheaveragetimingoftheseeventsareselectedasahit clustertoremovemostfhtehitscreatedbyrandomnoise. 76 Activeplanescut :Requiresatleastthreeactivetrackingscintillatorplanes.This furtherreducestherandomnoiseandexternalbackgroundcreatedbyphotonsorneu- trons. Upstreamveto :Eventshavingtheirmostupstreamhitinthevetoplanes,de˝ned asthe˝rstfourlayersofthePM,arerejected.AdditionallythePMhasasetofedge vetoplaneswhereaneventshowingahitintheedgevetoatadistancelessthan80 mmtotheextrapolatedtrackupstreampositionarerejected. Fiducialvolumecut :The˝ducialvolumeisde˝nedasatransversecentral ( 50) ( 50) cm 2 regionofthePM(totaling100cmineachdirection).Eventsinwhichthe mostupstreamhitisnotcontainedinthe˝ducialvolumearerejected. UnlikeND280,INGRIDhasnomagnetic˝eldtodeterminethechargeofaparticle/track,so theCC- 1 ˇ sidebandwillselectboth ˇ + ;ˇ aspioncandidates.Theeventsinthesampleare categorizedbasedonwhere(if)themuonstoppedinthedetectorsliketheCC- 0 ˇ selection. ThemuoncandidatekinematicdistributionsfortheINGRIDsidebandsamplearedis- playedinFig.4.18splitbytruetopology,showingtheproportionoftruesignalandback- groundevents.TheCC- 0 ˇ topologyinFig.4.18hasbeensubdividedintocategoriesby howmanyprotonswereintheevent,allofwhichcombinetogethertoformthetotalCC- 0 ˇ category.Theotherbackgroundcategoryisacombinationofcontributionsfrom , e ,out of˝ducialvolume,andotherdetectorbackgroundevents.InadditionFig.4.19showsthe kinematicdistributionssplitbytruereaction.WhilethepurityofCC- 1 ˇ eventsisnotgreat, thesidebandsamplescontainrelativelyfewtruesignaleventsallowingthesidebandsample tobequiteusefulprovidingadatadrivenconstraintonthetotalbackgroundeventrate. 77 INGRIDCC- 0 ˇ CC- 1 ˇ CC-OtherBackgroundEvents Stopping10.2756.5020.5012.73560.18 Stop+Escaping11.2152.6026.939.262454.82 Table4.7:PurityoftheINGRIDsidebandsampleforbothstoppingonlyandstoppingplus through-goingtracks. Figure4.18:Eventdistributionforreconstructedmuonequivalentdistanceinironandangle fortheINGRID(early)stoppingsidebandsamplesandallINGRIDsidebandsamplesstacked bytruetopology.Theroughnessofthedistributionisduetothecombinationoflowstatistics andthebinningchoicetomatchthesignalregion. 78 Figure4.19:Eventdistributionforreconstructedmuonequivalentdistanceinironandangle fortheINGRID(early)stoppingsidebandsamplesandallINGRIDsidebandsamplesstacked bytruereaction.Theroughnessofthedistributionisduetothecombinationoflowstatistics andthebinningchoicetomatchthesignalregion. 79 4.5Analysisbinning Thechoiceofanalysisbinningmustbeconsideredcarefullytoprovideapreciseanduseful cross-sectionmeasurement.Ifthebinsaretoowide,theninformationabouttheshapeof thedistributionwillbelost,whileifthebinsaretoothin,thensomebinscouldhavefew tozeroeventscausingminimizationissuesandlargestatisticalerrorsforthosebins.The followingcriteriawereconsideredwhenchoosingtheanalysisbinning: Statistics :Eachbinshouldhaveasu˚cientnumberofeventstopreventminimization issues. Detectorresolution :Theanalysisshouldhavebinsofasimilarsizeorlargerthan themeasureddetectorresolution.Measuredvariablesinthedetectorcannotprovide moreprecisionthanthedetectoriscapableofinamodelindependentway. E˚ciency :Thee˚ciencyacrossbinsideallyshouldberelativelyhighandconsistent. Binswithlowe˚ciencywillrelymoreontheMCtoperformthee˚ciencycorrection. Inthisanalysisthebinningforeachdetectorhasbeenoptimizedseparatelyandisdescribed foreachdetectorbelow.Theanalysisbinningreferstothebinningusedforthesignal parametersinthestatistical˝t(describedinChapter6andthe˝nalcrosssectionresult. Thesamplebinningusedinotherpartsofthe˝t(e.g.detectorsystematics)maybedi˙erent thanthebinningpresentedhere. 4.5.1ND280 TheanalysisbinningforND280usesmuonkinematicvariables p ; cos forboththeex- tractedcrosssectionandmeasureddatabins.Thechoiceofbinning,asshowninTab.4.8, isthesameasT2K-TN-337[3].The FGDand FGD+pTPCsamplesuseareduceddata binningwhichcanbefoundinAppendixEtoavoidemptybinswhichisdrivenbytheevent kinematics. 80 Binindex cos p [GeV/c] 0 -1,0.2 0,30 1 0.2,0.6 0,0.3 2 0.2,0.6 0.3,0.4 3 0.2,0.6 0.4,0.5 4 0.2,0.6 0.5,0.6 5 0.2,0.6 0.6,30 6 0.6,0.7 0,0.3 7 0.6,0.7 0.3,0.4 8 0.6,0.7 0.4,0.5 9 0.6,0.7 0.5,0.6 10 0.6,0.7 0.6,0.8 11 0.6,0.7 0.8,30 12 0.7,0.8 0,0.3 13 0.7,0.8 0.3,0.4 14 0.7,0.8 0.4,0.5 15 0.7,0.8 0.5,0.6 16 0.7,0.8 0.6,0.8 17 0.7,0.8 0.8,30 18 0.8,0.85 0,0.3 19 0.8,0.85 0.3,0.4 20 0.8,0.85 0.4,0.5 21 0.8,0.85 0.5,0.6 22 0.8,0.85 0.6,0.8 23 0.8,0.85 0.8,1.0 24 0.8,0.85 1.0,30 25 0.85,0.9 0,0.3 26 0.85,0.9 0.3,0.4 27 0.85,0.9 0.4,0.5 28 0.85,0.9 0.5,0.6 Binindex cos p [GeV/c] 29 0.85,0.9 0.6,0.8 30 0.85,0.9 0.8,1.0 31 0.85,0.9 1.0,1.5 32 0.85,0.9 1.5,30 33 0.9,0.94 0,0.4 34 0.9,0.94 0.4,0.5 35 0.9,0.94 0.5,0.6 36 0.9,0.94 0.6,0.8 37 0.9,0.94 0.8,1.25 38 0.9,0.94 1.25,2.0 39 0.9,0.94 2.0,30 40 0.94,0.98 0,0.4 41 0.94,0.98 0.4,0.5 42 0.94,0.98 0.5,0.6 43 0.94,0.98 0.6,0.8 44 0.94,0.98 0.8,1.0 45 0.94,0.98 1.0,1.25 46 0.94,0.98 1.25,1.5 47 0.94,0.98 1.5,2.0 48 0.94,0.98 2.0,3.0 49 0.94,0.98 3.0,30 50 0.98,1.0 0,0.5 51 0.98,1.0 0.5,0.7 52 0.98,1.0 0.7,0.9 53 0.98,1.0 0.9,1.25 54 0.98,1.0 1.25,2.0 55 0.98,1.0 2.0,3.0 56 0.98,1.0 3.0,5.0 57 0.98,1.0 5.0,30 Table4.8:ND280binningusedfortheextractedcrosssectionanddatadistributioninmuon kinematics p ; cos 81 4.5.2INGRID TheanalysisbinningforINGRIDusesmuonkinematicvariables p ; cos fortheextracted crosssectionmeasurement,andmuonkinematicvariables d ; forthemeasureddatavari- ableswhere d isthereconstructedequivalentdistanceinironforthemuon.Thechoiceof binning,asshowninTab.4.9,ismostlythesameasT2K-TN-204[5].Thereconstructed binninghasangularbinwidthsof10degreesfrom0to60degrees,anddistancebinwidths of 5 cmfrom0to 80 cm(exceptforthe˝rstdistancebin,whichisfrom0to10). Binindex cos p [GeV/c] 0 0.5,0.82 0.35,0.5 1 0.5,0.82 0.5,0.7 2 0.5,0.82 0.7,1.0 3 0.5,0.82 1.0,30.0 4 0.82,0.94 0.35,0.5 5 0.82,0.94 0.5,0.7 6 0.82,0.94 0.7,1.0 7 0.82,0.94 1.0,30.0 8 0.94,1.00 0.35,0.5 9 0.94,1.00 0.5,0.7 10 0.94,1.00 0.7,1.0 11 0.94,1.00 1.0,30.0 Binindex d [cm] 0 0,10 0,10 1 0,10 10,15 2 0,10 25,20 . . . . . . . . . 13 0,10 70,75 14 0,10 75,80 15 10,20 0,10 16 10,20 10,15 17 10,20 15,20 . . . . . . . . . 88 50,60 70,75 89 50,60 75,80 Table4.9:INGRIDbinningusedfortheextractedcrosssection(left)anddatadistribution (right)inmuonkinematics.Notethatthedataismeasuredusingmuondistanceinironand thecrosssectionismeasuredinmuonmomentum. ThemuonmomentumisdeterminedbytherangetraveledintheboththeProtonModule andthestandardINGRIDmodulelocateddirectlybehindtheProtonModulefollowingthe sameprocedurefromT2K-TN-204[5].Thedataisbinnedindistancetraveledinsteadof momentumsincethedistanceisthedirectlymeasuredquantity.Thedetectorsmearing(or transfer)matrixhandlesthemappingbetweenmeasureddistancetomomentum(shownin Fig.4.20)whencalculatingthecrosssection.MuonsintheProtonModuleandINGRID travelthroughbothplasticscintillatorandironplatesbeforestopping,andbothmaterials mustbeaccountedfortoaccuratelycalculatethemomentum.Thus,anequivalentdistance 82 inironiscalculatedfromthedistancetraveledintheplasticandironasfollows: d = d Fe + ˆ CH ˆ Fe d CH (4.11) where d X istheactualdistancetravelediniron(Fe)orplastic(CH)and ˆ X isthecorre- spondingmaterialdensity.Thisanalysisuses ˆ CH =1 : 03 g = cm 3 and ˆ Fe =7 : 87 g = cm 3 from T2K-TN-204[5]. Figure4.20:SmearingmatrixbetweentrueandreconstructedvariablesfortheINGRID CC- 0 ˇ selectionusingtheINGRID(early)stoppingsamples(top)andallINGRIDsignal samples(bottom). Oncemuonsreachanenergyofabout1.0GeV/corgreatertheytraversetheentire detectorandescape.Dependingontheanglethroughthedetector,thiscorrespondstoa maximumofapproximately65to75cmequivalentdistanceiniron,whichcanbeeasilyseen inthebottom-rightplotofFigure4.20.Thepopulationofeventsatatruemomentumand 83 angleofzeroareduetoreconstructionfailureswheretheselectedmuonwasnotatruemuon (e.g.apion),whichhasadefaultvaluefortruemomentumandangleequaltozero. 4.6E˚ciencyandpurity Thee˚ciencyandpurityoftheanalysisarecalculatedseparatelyforeachdetectorse- lectionandarede˝nedas: E˚ciency = Numberofselectedsignalevents Totalnumberofsignalevents (4.12) Purity = Numberofselectedsignalevents Numberofselectedevents (4.13) Thee˚ciencyissimplytheratioofselectedsignaleventstothetotalnumberofsignalevents, whilethepurityistheratiotoselectedsignaleventstoallselectedevents.Sincethee˚ciency correctionrequiresknowingthetotalnumberofsignalevents,itisestimateddirectlyfrom theMCsimulation.Thisaddsadditionalmodeldependencetothemeasurementwhichis quanti˝edasanuncertaintyonthee˚ciency(seeSection5.5). 4.6.1ND280 Thesignale˚ciencyforeachsampleandthetotalfortheND280selectionisshowninFig. 4.21inonedimensionalprojectionsofmuonmomentumandangle.Thetwodimensional totalsignale˚ciencyandpurityareshowninFig.4.22.Thee˚ciencyishigherforforward goingmuonsastheseeventsareusuallyreconstructedintheTPCs,whichhaveexcellent reconstructioncapabilities.Backwardgoingeventsarehardtoreconstructasaccuratetime of˛ightinformationisneededtoseparateeventswhichoriginatedintheFGD˝ducialvolume versuseventsthatwereoutsideandstoppedintheFGD.Thee˚ciencyinmomentumisfairly ˛at,withthesampleswherethemuonisreconstructedintheFGDdroppingo˙past1GeV wheretheyhaveenoughenergytoescape.Therelativelylowe˚ciencyforthesampleswith protonsisrelatedtobeingabletodi˙erentiateandreconstructprotonsversuspions. 84 Figure4.21:ND280signale˚ciencyforeachsampleasafunctionoftruemuonmomentum (left)andtruemuonangle(right). Figure4.22:ND280totalsignale˚ciency(left)andpurity(right)asafunctionofbothtrue muonmomentumandangle. 4.6.2INGRID Thesignale˚ciencyfortheINGRIDstoppingsampleandthesignale˚ciencyforallINGRID samplescombinedarebothshowninFig.4.23.INGRIDhasessentiallyzeroe˚ciencypast about60to70degreesasthemuonsaretravelingnearlyparalleltothescintillatorbarsand donotcrossenoughbarstobereconstructable.Additionally,theminimummuonmomentum tocrossenoughscintillatorbarsisabout300to400MeV,andoncemuonsreachabout900 to1000MeVtheydonotstopinthedetectorswhichshowsupasadropine˚ciencyforthe stoppedsample. 85 Figure4.23:INGRIDsignale˚ciencyforthestoppingsample(top)andallsamplescombined (bottom)asafunctionoftruemuonmomentum(left)andtruemuonangle(right).Note thee˚ciencyaxisscaleisdi˙erentbetweenthetopandbottomplots. 86 CHAPTER5 SYSTEMATICUNCERTAINTIES Thischapterdescribesthesystematic(ornuisance)parameterswhichareincludedinthe ˝ttorepresentplausiblevariationsofthedetector,˛ux,andneutrinointeractionmodels usedintheMonteCarlosimulation.Thesystematicparametersarevariedalongsidethe signalparameters,butareconstrainedbyanaccompanyingcovariancematrix.Whenmoved fromtheirnominalvalues,thesystematicparametersaddapenaltytermtothelikelihood proportionaltothecovariancematrixasde˝nedinEq.6.5andreprintedhere: 2ln L syst = ˜ 2 syst = X p ~p ~p prior V syst cov 1 ~p ~p prior (5.1) where ~p isthevectorofsystematicparameters, ~p prior isthepriorornominalvalueofthe systematicparameters,and V cov isthecovariancematrixwhichdescribesthecon˝denceand errorforeachparameteraswellasthecorrelationsbetweeneachparameter.Thismethod approximatesandtreatsthesystematicuncertaintyasGaussiandistributedinthe˝t,and thefulldetailsaredescribedinChapter6. Therearealsoseveralsourcesofsystematicuncertaintywhicharenotincludedas˝t parameters,butareincorporatedwhenpropagatingtheerrorsasdescribedinSec6.3,such astheuncertaintyinthenumberoftargets,andaredescribedinthischapter. 5.1Fluxsystematicuncertainties Theuncertaintiesforthe˛uxsimulationincludeuncertaintiesfromtheinitialproton beam,uncertaintiesfromthebeamlinesetup(e.g.horncurrent,hornalignment),anduncer- taintiesintheinteractionmodelsusedtosimulatethehadroninteractions.Theuncertainties onthe˛uxpredictionareevaluatedbyvaryingtheinputstothesimulationandcalculating thee˙ectonthepredicted˛ux.Therearetwomainmethodstoevaluatingtheuncertainties. The˝rstmethodisusedwhenanerrorsourceincludesanumberofcorrelatedunderlying 87 parametersthatcanbecontinuouslyvaried.Theparametersarevariedrandomlyaccording totheircovariancematrixandmanydi˙erent˛uxpredictionsareproduced(typicallyseveral hundredsets).Theuncertaintyonthe˛uxiscalculatedbyconstructingacovariancematrix binnedinneutrinoenergyfromthe N ˛uxpredictions.Thesecondmethodisusedwhen thereisonlyasingleparametertovaryorfore˙ectsthatareeitheronoro˙.Theparameter ischangedby 1 ˙ (orturnedonoro˙)andthe˛uxsimulationisruntoproduceavaried˛ux prediction.Theuncertaintyonthe˛uxissimilarlycalculatedasacovariancematrixbinned inneutrinoenergyusingthe 1 ˙ variations.Finallythetotal˛uxuncertaintyisgivenby addingthecovariancematrixforallerrorsourcestogether,treatingeacherrorsourceas independent.The˛uxuncertaintymatrixiscalculatedasacovariancematrixbetweenbins inneutrinoenergy,neutrino˛avor,neutrinodetector,andneutrinobeammode. Thisanalysisusesa˛uxcovariancewith20neutrinoenergybinsfortheforwardhorn current(FHC) ˛avorpredictionforeachdetector(seeFig.5.1).Onlythe ˛avoris includedinthe˝tandcovariancematrixduetothelowcontaminationfromthebackground ˛avors( , e , e )andtopreventanunrealisticconstraintonthe˛uxmodel.Therelatively fewbackgroundeventscouldprovideaverytightconstraintonthe˛uxfromthebackground ˛avorswhichwoulda˙ectthe ˛uxduetothestrongcorrelationsinthe˛uxmodel.The ND280andINGRID˛uxpredictionswerevariedsimultaneouslytoproduceacovariance matrixwhichincludescorrelationsbetweenthe˛uxateachdetector.Thisgivesa˛ux covariancematrixwith40totalbins:the˝rst20forND280andthesecond20forINGRID. Bothdetectorsusethesame˛uxbinning(showninTab.5.1),buthaveseparate˛ux parameters,allowingthe˛uxtovaryateachdetector(accordingtothecovariancematrix), giving40total˛uxnuisanceparametersinthe˝t.The˛uxsystematicparameters( f n )are scalefactorsonthenumberofeventsinagivenneutrinoenergybinforeachdetector,and haveanominalvalueofone(seeEq.6.9). The˛uxintegralusedtonormalizethecrosssection(seeSec.6.1andSec.6.4)hassome systematicuncertaintywhichisdescribedbythe˛uxcovariancematrix.Theuncertaintyon 88 Figure5.1:Fluxcovariance(left)andcorrelation(right)matrices.Thebinnumbercorre- spondstotheparameternumberinTab.5.1,andND280isthe˝rst20binsandINGRID thesecond20bins. Figure5.2:Therelative˛uxuncertaintyforINGRID(right)andND280(left)asafunction ofneutrinoenergybins,separatedbytheuncertaintysource.The˛uxuncertaintyforboth detectorsisdominatedbytheuncertaintyinthehadroninteractionmodel.The10to30 GeVbinisnotshown. the˛uxintegraliscalculatedbyproducingmanytoythrowsofthe˛uxparametersaccording tothecovariancematrix,andcalculatingtheintegralforthatthrowasfollows = E X n ˚ n f n (5.2) where isthetotalintegrated˛ux, ˚ n isthe˛uxforbin n ,and f n isthe˝ttedorthrown ˛uxparameterforbin n .Thedistributionofthesetoythrowsgivestheuncertaintyonthe ˛uxintegral. 89 ND280 INGRID Energybin[GeV] f 0 f 20 0.0-0.1 f 1 f 21 0.1-0.2 f 2 f 22 0.2-0.3 f 3 f 23 0.3-0.4 f 4 f 24 0.4-0.5 f 5 f 25 0.5-0.6 f 6 f 26 0.6-0.7 f 7 f 27 0.7-0.8 f 8 f 28 0.8-1.0 f 9 f 29 1.0-1.2 f 10 f 30 1.2-1.5 f 11 f 31 1.5-2.0 f 12 f 32 2.0-2.5 f 13 f 33 2.5-3.0 f 14 f 34 3.0-3.5 f 15 f 35 3.5-4.0 f 16 f 36 4.0-5.0 f 17 f 37 5.0-7.0 f 18 f 38 7.0-10.0 f 19 f 39 10.0-30.0 Table5.1:Theneutrinoenergybinningusedforthe˛uxsystematicparameters.Both theND280andINGRID˛uxparametersusethesameenergybinning,andaretreatedas separateparametersinthe˝t. The˝twillreturnasetofpost-˝t˛uxparametersanderrors(intheformofacovariance matrix)thatcanbeusedtoupdatethenominal˛uxpredictionandinthecrosssection extraction.Thepost-˝t˛uxparametersingeneralwillhavereducederrorscomparedto thenominalmodelandcaremustbetakensuchthatthe˝tdoesnotoverconstrainthe ˛uxparameters.Anoverconstraintismostcommonlycausedwhenthe˝tdoesnothave su˚cientfreedomtovarythebackgroundandsidebandsamplesusingnon-˛uxparameters, andincorrectlyusesthe˛uxparameterstocorrectmostofthedi˙erencebetweendataand MC.Thebasesetofneutrinointeractionmodelsystematicsisadaptedfromtheoscillation analysis,andadditionalneutrinointeractionmodelsystematicparametershavebeenadded totheanalysistopreventthe˛uxparametersfrombeingoverconstrained(seeSection5.3). 90 5.1.1ND280integrated˛ux Thenominal˛uxpredictionfortheFHC modeatND280isshowninFigure5.3inthe binningusedfortheanalysis(highestenergybinnotshown,andthefullrangeisfrom0to 30GeV).Theestimatedfractionalerrorontheintegrated˛uxusingthepost-˝terrorsis 0.067,andthedistributionofintegrated˛uxthrowsisshowninFigure5.3.Thefractional erroristhewidthofaGaussian˝ttedtothedistributionofintegrated˛uxthrows. Figure5.3:ND280nominal˛uxprediction(left)usingtheneutrinoenergybinningfor the˛uxuncertainties(notethatthe10to30GeVbinisnotshown).Thedistribution ofintegrated˛uxthrows(right)whichgivetheintegrated˛uxerrorforthecrosssection extraction. 5.1.2INGRIDintegrated˛ux Thenominal˛uxpredictionfortheFHC modeatINGRIDisshowninFigure5.4inthe binningusedfortheanalysis(highestenergybinnotshown,andthefullrangeisfrom0to 30GeV).Theestimatedfractionalerrorontheintegrated˛uxusingthepost-˝terrorsis 0.065,andthedistributionofintegrated˛uxthrowsisshowninFigure5.4.Thefractional erroristhewidthofaGaussian˝ttedtothedistributionofintegrated˛uxthrows. 5.2Detectorsystematicuncertainties Thedetectorsystematicuncertaintiesrepresenttheuncertaintyontheperformanceof thedetector(suchastheparticleidenti˝cation).Thisanalysisusestwodi˙erentdetectors 91 Figure5.4:INGRIDnominal˛uxprediction(left)usingtheneutrinoenergybinningfor the˛uxuncertainties(notethatthe10to30GeVbinisnotshown).Thedistribution ofintegrated˛uxthrows(right)whichgivetheintegrated˛uxerrorforthecrosssection extraction. whichrequireseparatetreatmentofthedetectorsystematicsandaretreatedascompletely uncorrelated.Thedetectorssharesomesimilaritiesintheunderlyingsimulationandpartsof theconstruction,forexamplethescintillatorbarsareofasimilarconstruction,andthesys- tematicuncertaintiescouldbecorrelatedinthoseaspects.Howeverthedetectorcalibrations couldbedi˙erentandthiswouldbeasmalle˙ectrelativetomoredominantuncertainties, suchasthestatisticsor˛uxuncertainties,andhasnotbeenconsideredforthisanalysis. Thedetectorcovariancematrixistherelativeuncertaintyonthenumberofreconstructed eventsinabincorrespondingtoreconstructedkinematicvariables.Finally,adetectornui- sanceparameter( d j )foreachdetectorsamplebinisincludedinthe˝t,whichisallowedto alterthenumberofeventsinareconstructedbinsubjecttoconstraintsfromthecovariance matrix(asshowninSec.6.2). 5.2.1ND280detectorsystematics ThedetectorsystematicsforND280havebeenstudiedindetailforbothcrosssectionanalysis andoscillationanalysis,anddetailedinRefs.[7,84]andinT2K-TN-212[85]andT2K- TN-216[76].Abriefdescriptionofthedetectorsystematicuncertaintieswhichhavebeen accountedforispresentedhere,primarilyfollowingtheinformationinRef.[76],andlistedin 92 Table5.2.Ideally,thedetectorsystematicsshouldbeparameterizedintheunderlyingphysics variablesintheMC(suchasBirks'constantforthescintillator),howeverthisisnotpossible inallcases.Thus,inpractice,somederivedparameters(suchasthetrackreconstruction e˚ciencyorthemeanenergydeposition)thatcanbecomputedforbothdataandMCare usedtopropagatethedetectoruncertainties.TheND280detectorsystematicuncertainties canbecategorizedintothreedi˙erenttypesbasedonhowthearepropagated:e˚ciency- likesystematicswhichsimplyaltereventweights,normalization-likesystematicswhichalter alleventweights,andobservablevariation-likesystematicswhichalterthereconstructed variablesofaneventandcanchangethenumberofselectedevents.Inallcasesallthe detectorsystematicparametersareassumedtobeGaussiandistributedwiththeexception ofthemagnetic˝elddistortions,whichareassumedtohaveauniformdistribution. E˚ciency-likesystematics :E˚ciency-likesystematicsarecomputedthroughstudies comparingdataandMCpredictionsinwell-knowncontrolsamples.Forexample,acontrol samplescontainingmuonswhichoriginateinthesand/rocksurroundingthedetectorand leaveasingletrackwhichtraversestheentiretracker.Thetrackreconstructione˚ciency ormatchingcanbenicelyassessedbytheredundancybetweendetectorsusingthecontrol sampleofmuonsthatcrosstheallofthetracker.Howeveritispossiblethatthee˚ciency computedusingthecontrolsampledoesnotcorrespondexactlytothesamplesusedinan analysis,soamodeltoextrapolatethee˙ectseeninthecontrolsampletotheanalysis sampleisnecessary.Thisanalysisassumestheratiobetweenthee˚cienciesindataandMC isthesameinboththeanalysisandcontrolsamples[76].Thepredictede˚ciencyinan analysissamplecanbecomputedasfollows data = CS data CS MC MC (5.3) where data and MC arethee˚cienciesfordataandMCrespectivelyfortheanalysissample, andCSreferstothecontrolsample.Avariationofthepredicteddatae˚ciency( 0 data )is 93 givenby 0 data = CS data CS MC + ˘ ˙ CS ! MC (5.4) where ˙ CS isthestatisticaluncertaintyofthee˚ciencyratiousingthecontrolsample(one standarddeviation),and ˘ isaGaussianrandomnumbercenteredatzerowithawidthof one.Thiswillgenerateanewpredicteddatae˚ciencywhichhasbeenvariedaccordingto thestatisticaluncertaintyofthecontrolsample.Eventweightsareappliedbasedonifthe eventwassuccessfullyreconstructedinthesimulation.Foreventsthatweresuccessfully reconstructedthefollowingweightisappliedtotheevent w e˙ = 0 data MC (5.5) andiftheeventwasnotreconstructed,thenthefollowingweightisappliedtotheevent w ine˙ = 1 0 data 1 MC (5.6) Normalization-likesystematics :Normalization-likesystematicsaresimplercasesof e˚ciency-likesystematicswheretheeventweightisappliedtoalleventsasanoverallnor- malization.Avariationofanormalization-likesystematiccanbegeneratedasfollows w = w 0 (1+ ˘ ˙ w ) (5.7) where w isthevariedeventweight, w 0 isthenominaleventweight, ˙ w isthesystematic uncertaintyforthenormalization(onestandarddeviation),and ˘ isaGaussianrandom numbercenteredatzerowithawidthofone.Thenewvariedeventweightisappliedtoall eventsinthesimulation. Observablevariation-likesystematics :Observablevariation-likesystematicschange observablevariables(suchasthemomentumscaleorresolution)ofagiveneventratherthan theeventweight,andareappliedtovariableswhichhaveadi˙erentmeanorresolution betweendataandMC.Thisgivesthepotentialtochangethereconstructedtopologyfor aneventorcauseeventstomigrateinandoutofagivensample/selectionastheirnew 94 reconstructedvariablesmaypass/failoneoftheselectioncuts.Forexample,ifthereisacut onmomentum,thenaneventmaystarttopassorfailacutifitsmomentumissu˚ciently changed.Thusforeachvariationofasystematic,theeventselectionisrunagainwiththe neweventreconstructedvariables.Ingeneralthereconstructedvaluesarevariedinthe followingway x 0 = x + x + ˘ ˙ x (5.8) where x and x 0 aretheoriginalandnewvalueofthereconstructedvariablerespectively, x isthecorrectionthatshouldbeappliedtotheMCtomatchdata, ˙ x isthestatistical uncertaintyonthecorrection,and ˘ isaGaussianrandomnumbercenteredatzerowitha widthofone. Thedetectorsystematicuncertaintyisevaluatedbygeneratingmanyvariationsorthrows ofeachsystematicparameterandexaminingthetotale˙ectofallparametersontherecon- structedeventdistributionseparatelyforeachsample.Eachthrowofthesystematicparam- eterswillingeneralchangethenumberofeventsineachreconstructedbinoftheanalysis variables( p ; cos ),andtheeventselectionisrunforeachtoythrowtoallowforevent migration.Theeventdistributionineachbinisusedtogenerateacovariancematrixforthe e˙ectofallsystematicparametersgivenby V ij = 1 N X t ( x it x i )( x jt x j ) (5.9) where V ij isthe ij th elementofthecovariancematrix, N isthenumberoftoythrows, x it is the i th reconstructedkinematicbinforthrow t ,and x i isthenominalvalueinreconstructed kinematicbin i .Theprocedurecanalsobeusedtoevaluatethee˙ectofeachsystematic parameterindividually.Thecovariancematrixcontainsseparatebinsforeachsamplesince thesamplesareseparatedbytheparticletopologyandwhichsub-detectorswereusedin theevent.Forexample,systematicvariationsintheTPCmagnetic˝eldwillnothavemuch e˙ectoneventsinthe FGDsamplebecausetheTPCwasnotusedintheevent. 95 Ideallythebinningforeachsampleinthedetectorcovariancewouldusethesamebinning fortheextractedcrosssection.Howeverthispresentstwomainproblems:alargenumberof detectorparameters,andsampleshavingbinswithafewtozeroevents.Thesecauseissues withstabilityofinvertingthedetectormatrix,andcancauseissueswithconvergenceofthe ˝t.Thussomeofthesampleshaveareducedbinningtoreducetheimpactoftheseissues,but allsamplesarestillseparate.ThebinningforeachsamplemaybefoundinAppendixE.The TPC+pFGDandthe TPC+Npsampleswerecombinedwhenrunningthe˝tduetothelow statisticsofthe TPC+Npsampleandthusarecombinedinthedetectorcovariance.Further more,principalcomponentanalysisisusedtoremoveanumberofdetectorparameterswhen runningthe˝t(seeAppendixD).Thedetectorfractionalcovarianceandcorrelationmatrices forND280areshowninFigure5.5. SystematicName SystematicType TPCPID Variation TPCB-˝elddistortions Variation TPCmomentumscale Variation TPCmomentumresolution Variation TPCclustere˚ciency E˚ciency TPCtracke˚ciency E˚ciency TPC/FGDchargeconfusion E˚ciency FGDPID Variation FGDtracke˚ciency E˚ciency FGDhybridtracke˚ciency E˚ciency Michelelectrone˚ciency E˚ciency TPC-FGDmatchinge˚ciency E˚ciency Vertexmigration Variation Momentumbyrangeresolution Variation Neutrinoparentdecayposition Variation Eventpileup Normalization OOFVbackground Normalization Pionsecondaryinteractions Normalization Protonsecondaryinteractions Normalization Table5.2:ListofND280detectorsystematicparametersandtheirpropagationtype(loosely groupedbydetectororgeneral).Eachparameterisvariedinthesimulationtoproducethe detectorcovarianceamtrixusedinthe˝t. 96 Figure5.5:ND280detectorcorrelationmatrixbinnedinmuonkinematicsforeachsample. Thedetectormatrixisgenerallystronglycorrelatedasthedetectorsystematicstendtoa˙ect thesameregionsofkinematicphasespaceandsamples. 5.2.2INGRIDdetectorsystematics ThedetectorsystematicsfortheProtonModule(PM)andstandardINGRIDmodulehave beenevaluatedandpresentedinpreviousT2Kanalysis,T2K-TN-204[5]andT2K-TN-352 [86],andthisanalysisusesthesamedetectorcovariancematrix.Thedetectorsystematic variationsconsideredforthisanalysisarelistedinTable5.3(looselygroupedbytype),and abriefsummaryofhowthedetectorsystematicuncertaintiesarecalculatedispresented here.Thee˙ectofchangingaparameteronthecrosssectionisestimatedusingthedetector simulation.Eachdetectorsystematicparameterisvariedandproducesadi˙erentweighted MCsample,includingavariede˚ciency.ThisnewMCsampleisusedintheanalysisto measureacrosssectionusingthestandardanalysisprocedure(asdetailedinT2K-TN-204 97 andT2K-TN-352)andthesystematicerroriscalculatedas X = ˙ 0 X ˙ X ˙ X (5.10) where ˙ 0 X and ˙ X istheextractedcrosssectionfromthevariedMCandthenominalMC respectivelyforsystematic X ,evaluatedineachtruemuonmomentumandanglebin.The errorfromeachsystematicvariationisaddedinquadraturetoformthe˝nalcovariance matrix.ThedetectorfractionalcovarianceandcorrelationmatricesforINGRIDareshown inFigure5.6.TheINGRIDdetectorcovarianceusedinthisanalysiswasproducedby numericallypropagatingtheoriginaldetectorcovariancematrixintruekinematicbinsto reconstructedkinematicbinsbythrowingmanytoys.TheINGRIDsignalandsideband samplessharethesamecovariancematrixforthedetectorsystematicsasanapproximation duetothesimplicityoftheINGRIDanalysisanddetectoruncertaintytreatment. Figure5.6:INGRIDdetectorcorrelationmatrixbinnedinmuonkinematicsforeachsample. Thedetectormatrixisgenerallystronglycorrelatedasthedetectorsystematicstendtoa˙ect thesameregionsofkinematicphasespace. 98 Num. SystematicName 1 Darknoise 2 Hite˚ciency 3 Scintillatorlightyield 4 Scintillatorquenching 5 Externalbackgrounds 6 2Dtrackreconstruction 7 PM-INGRIDtrackmatchingangle 8 PM-INGRIDtrackmatchingposition 9 3Dtrackmatching 10 Vertexlongitudinalcut 11 Vertextransversecut 12 Edgevetocut 13 Fiducialvolumecut 14 Eventpileup 15 Pionsecondaryinteractions 16 Protonsecondaryinteractions Table5.3:ListofINGRIDdetectorsystematicparameters,looselygroupedbye˙ect.Each parameterisvariedinthesimulationtoproducethedetectorcovariancematrixusedinthe ˝t. 99 5.3Neutrinointeractionsystematicuncertainties ThenominalMCpredictionforthesignalandbackgroundisproducedusinganeutrino interactionmodelwhichisknowntobeincomplete.Thisanalysisusesasetofsystematic parametersdesignedtoparametrizetheuncertaintyintheneutrinointeractionmodelsused tocalculatetheMCprediction,andisbasedonthetreatmentusedintheT2Koscilla- tionanalysis.Theseparameterscorrespondtoscalingfactorsonthecorrespondingphysics parameterinthemodel,andhaveacovariancematrixdescribingtheuncertaintyandcorre- lationbetweentheparameters.Thecovariancematrixanduncertaintyforeachparameter hasbeenmotivatedfromworkperformedbytheT2KNeutrinoInteractionsWorkingGroup (NIWG)groupandpresentedinT2K-TN-315[2].Thesystematicparametersarebrie˛y describedhere(listedinTable5.4),andtheinputcovariancematrixisshowninFig.5.7. CCQEmodeling :ThenominalmodelforCCQEinteractionsusedforthisanalysisis theBenharet.al.spectralfunctionmodel[38]withadditionalcontributionsfromtheNieves et.al.multinucleonmodel[13]asimplementedinNEUT[48],whichhasthreesystematic parameters.Thevalueoftheaxialmass, M QE A ,asdiscussedinSec.2.3.1canbevariedwhich willa˙ecttheshapeandnormalizationofthesignaleventdistribution.The2p2hevents havebothanoverallnormalizationparameter,whichuniformlyincreasesordecreasesthe numberof2p2h events,andashapeparameter.The2p2h shapeparameterchanges theproportionof2p2heventswhichare -likeversusnon- -like(seeSec.2.3.2)keeping thenormalizationconstant.The2p2hshapeparameterappliesamultiplicativeeventweight basedontheneutrinoenergy,energytransfer,andmomentumtransfer( E ;q 0 ;q 3 )onan event-by-eventbasis. Resonantsinglepionmodeling :Themodelusedforresonantsinglepionproduction istheRein-Sehgalmodel[87,88],includingleptonmasse˙ectsandmodi˝edformfactors focusingonthe contribution.TherearethreeparametersfortheNEUTimplemen- tationoftheRein-Sehgalmodelwhichcanbevaried, M RES A theaxialmassfortheresonant interaction, C A 0 oneoftheaxialformfactorsintheGraczyk-Sobczykparameterizationat 100 Figure5.7:Crosssectionsystematicparametercovariancematrix(top)andcorrelation matrix(bottom).Thecovariancevalueshavebeennormalizedtobetherelativevariance, andheparametersarearrangedfollowingtheirindiceslistedinTab.5.4. 101 Index Parameter Type Prior Error 0 M QE A Signalshape 1.21 0.3 1 2p2h norm. Signalnormalization 1.0 1.0 2 2p2h shape Signalshape 1.0 1.0 3 M Res A Backgroundshape 0.95 0.15 4 C 5 A Backgroundshape 1.01 0.12 5 I 1 = 2 BkgResonant Backgroundnormalization 1.3 0.2 6 DISMultiplepion Backgroundshape 1.0 0.4 7 CC-1 ˇE < 2.5GeV Backgroundnormalization 1.0 0.5 8 CC-1 ˇE > 2.5GeV Backgroundnormalization 1.0 0.5 9 CCDIS Backgroundnormalization 1.0 0.5 10 CCMulti- ˇ Backgroundnormalization 1.0 0.5 11 CCCoherentonC Backgroundnormalization 1.0 1.0 12 NCCoherent Backgroundnormalization 1.0 0.3 13 NCOther Backgroundnormalization 1.0 0.3 14 CC e Backgroundnormalization 1.0 0.03 15 FSIInelastic,LE Backgroundshape 1.0 0.41 16 FSI ˇ absorption Backgroundshape 1.1 0.41 17 FSIChargeexchange,LE Backgroundshape 1.0 0.57 18 FSIInelastic,HE Backgroundshape 1.8 0.34 19 FSI ˇ production Backgroundshape 1.0 0.50 20 FSIChargeexchange,HE Backgroundshape 1.8 0.28 Table5.4:Neutrinointeractionmodelingparametersusedinthisanalysisalongwiththeir index,type,prior,anderror.ValuestakenfromRef.[2,3]. Q 2 =0 ,andthescaleofthe I 1 = 2 non-resonantbackgroundcontribution.Thepriorvalues anderrorsweresetbytuningtheNEUTpredictiontoselectANLandBNLbubblecham- berexperimentaldata(forthedetailedprocedureseeT2K-TN-315[2]).Additionallythis analysisincludestwooverallnormalizationparametersforCC- 1 ˇ productiontopreventover constraintsonthe˛uxparameters. Coherentpionmodeling :Themodelusedforcoherentsinglepionproductionisthe Rein-Sehgalmodel[44]forcoherentinteractions.Twosystematicparametersareusedto varytheoverallnormalizationofbothchargedcurrentandneutralcurrentcoherentevents oncarbon.TheRein-Sehgalmodeliswellknowntooverestimatethecrosssectionwhen comparedtorecentmeasurements[2].Thisisasimplertreatmentforcoherentpionevents 102 comparedtotheT2Koscillationanalysis. DISandmulti-pimodeling :Deepinelasticscattering( W> 2 : 0 GeV/c 2 )ismodeled inNEUTusingthePYTHIA5.7generator[89].NEUTusesthestructurefunctionsfrom theGRV98partondistributions[90]withBodek-Yangcorrections[91].Forthetransition region( W< 2 : 0 GeV/c 2 ),multiplemodelsaresuperimposedproducingbothsignalresonant productionandmultiplepionproduction[2].Therearefoursystematicparameterswhichcan varyDISandmulti-pievents.Threeareoverallnormalizationparameters,oneforcharged currentDISevents,oneforchargedcurrentmulti-pievents,andoneforallneutralcurrent eventsthatarenotsinglepionorcoherentproduction.Thelastparameteriscanchange theshapeoftheDISdistribution,andisappliedasanevent-by-eventweightbasedonthe neutrinoenergy. Pion˝nalstateinteractionmodeling :NEUTusesanintra-nuclearcascademodel tomodelthe˝nalstateinteractions(FSI)ofpionsastheytraversethenucleus.Following itscreationfromaneutrinointeraction,thestartingpositionofthepionischosenbased onanucleardensitypro˝le(aWoods-Saxonpotential)whichisnucleusdependent,andthe initialpionkinematicsareprovidedbytheinteractionmodel.Thepionisthen"classically" propagatedthroughthenucleusin˝nitesteps,wheretheprobabilitytohaveaninteraction atthatstepiscalculated.Thepionissteppedthroughthenucleusuntilitiseitherabsorbed orhasescapedthenucleus.Theproductoftheinteractionprobabilitiesforallstepsis de˝nedastheescapeprobability[2]. Forpionswithlowmomentum( p ˇ < 500 MeV/c),tablescomputedfromtheOsetet.al. model[92]areusedtodeterminetheinteractionprobabilities.Forpionswithhighmomen- tum( p ˇ > 500 MeV/c)theinteractionprobabilitiesarecalculatedfrompionsscatteringo˙ offreeprotonanddeuterondatacompiledbythePDG[1].Thetwomodelsareblended togetherbetween400and500MeV/ctoavoiddiscontinuities. TheNEUTFSImodelisparameterizedbysixscalefactors,listedinTab.5.4.Each parameterscalesthecorrespondingprobabilityforpioninteractionateachstepinthenu- 103 cleusforabsorption,pionproduction,inelasticscattering,andchargeexchangeprocesses (whereLEandHErefertolowenergyandhighenergyrespectively).Neweventweights arecalculatedbyvaryingtheFSIparametersandre-runningtheNEUTcascadealgorithm producinganewescapeprobability,wheretheneweventweightistheratiobetweenthe newescapeprobabilityandthenominalescapeprobability.Thenominalvaluesanderror fortheFSIparametersaretunedby˝ttingtoexternalpionscatteringdata. Parametrization :Thee˙ectonaneventfromchangingasystematicparameterisen- codedinasetofresponsefunctions,orsplines,foreachparameter.Rerunningthesimulation tochangetheunderlyingphysicsparametersiscomputationallyprohibitiveinmostcasesso thesplinetreatmentisusedasanapproximationforspeed.Basedontheeventtruereaction (e.g.CCQE),truetopology(e.gCC- 0 ˇ ),andtruekinematicvariablesasinput,thecorre- spondingsplinereturnsaneventweightforanarbitraryvalueofthesystematicparameter. Thesplineresponseisnormalizedsuchthatthevalueofthesplineatthenominalvalueofa parameterisde˝nedtobeone.Splinesaregeneratedbycalculatingtheneweventweights whenvaryingasystematicparameteratspeci˝cvalues(suchasthe 0 ˙; 1 ˙; 2 ˙; 3 ˙ val- ues)andbinningtheeventweightsasfollows r i;j;k ( x 0 )= P n w n ( x 0 ) T i;j;k ( x ) (5.11) where x;x 0 arethenominalandalteredcrosssectionvalues, w n istheweightforevent n as calculatedabove,and T i;j;k isthesumoftheweightsforthenominalMCforbin i;j;k .For anarbitraryvalueofadial,theresponsecanbecalculatedbyinterpolatingthegenerated splinepoints.Eachsystematicparameterhasitsownsetofsplines,andthesplinesare generatedseparatelyforeachdetector. 5.4Numberoftargetssystematicuncertainty Thecrosssectionisnormalizedbythenumberoftargetnucleonsineachdetector,which hassomesystematicuncertaintyprimarilyduetocontaminantsinthematerials.Thishas beenestimatedforeachdetectorandpropagatedtothecrosssectionbyvaryingthenumber 104 oftargetsforeachtoythrowofthecrosssectionasdescribedinSection6.3.Thenumberof targetsforeachthrowisvariedaccordingtoaGaussiancenteredatthenominalnumberof targetswithawidthequaltotheerrorpresentedbelow. 5.4.1ND280targetuncertainty TheuncertaintyonthenumberofnucleonsinFGD1hasbeencalculatedbeforeanda summaryoftheresultfromT2K-TN-337[3]ispresentedhere.Theuncertaintyinthe numberofnucleonsiscalculatedusingthecovariancematrixofthearealdensitiesofeach element(showninFigure5.8,correlationsobtainedfromT2K-TN-91[4]).Thecovariance matrixisusedtogenerate10,000toythrowsofthearealdensities,whichthenareusedto calculatethenumberofnucleonsforthatthrow(usingEq.6.19).Theratiobetweenthe RMSandthemeanoftheresultingdistributiongivesthefractionalerroronthenumberof targets.Thisproceduregivesa 0 : 67% erroronthenumberoftargetsinFGD1. 5.4.2INGRIDtargetuncertainty TheuncertaintyinthenumberofnucleonsintheProtonModulehasbeencalculatedbefore andtheresultfromT2K-TN-352[86]isquotedhere.ThescintillatormassintheProton Modulehasbeenmeasuredtobe 291 : 5 1 : 1 kg,whichresultsina 0 : 38% erroronthenumber oftargets. 5.5E˚ciencyuncertainty Thee˚ciencyiscalculatedusingtheMCsimulationforeachdetectorwhichaddsaddi- tionalmodeldependencetotheanalysis.Toaccountforthismodeldependence,anuncer- taintyonthee˚ciencycorrectioninincludedintheanalysis.TheMCsimulationisvaried accordingtothe˝tparametersandthee˚ciencyiscalculatedformanyvariations.Thedis- tributionofe˚ciencycurvesisrepresentativeoftheuncertaintyinthee˚ciencycorrection 105 Figure5.8:CovariancematrixofthearealdensitiesforFGD1usedtodeterminetheuncer- taintyinthenumberoftargetsforND280.FigurefromRef.[3] andincludedintheuncertaintyofthecrosssectionresult.Theoveralluncertaintyonthe e˚ciencyisestimatedusinganAsimov˝t 1 . 1 AnAsimov˝tisde˝nedasa˝twherethe"data"isthesameastheinputsimulation the˝tstartsatthetruebest˝tpoint. 106 CHAPTER6 STATISTICALFIT 6.1Crosssectionde˝nition Themeasurementofthedouble-di˙erential CC- 0 ˇ crosssectionsonscintillator(C 8 H 8 ) forINGRIDandND280areextractedinasimultaneous˝tofbothINGRIDandND280 dataasafunctionofthemuonmomentum( p )andcosineofthemuonangle( cos ) 1 .The ˛ux-integratedcrosssectionisextractedratherthana˛ux-unfoldedcrosssectiontoavoid modeldependencefromassumptionsontheshapeoftheneutrinoenergyspectrum.The ˛ux-integratedcrosssectionisde˝nedas: d˙ dx i = ^ N CC- 0 ˇ i MC i N FV nucleons 1 x i (6.1) where x i isthevariableusedforthecrosssectionextraction, ^ N CC- 0 ˇ i isthenumberof selectedCC- 0 ˇ signaleventsinagivenbin i asdeterminedfromthe˝t(describedinthe followingsections), MC i isthee˚ciencyineachbin, istheintegrated˛ux 2 , N FV nucleons is thenumberofnucleonsinthe˝ducialvolume,and x i isthebinwidth(inthisanalysis x i = p ).AsindicatedinEq.6.1,wedividetheeventratebythetotalintegrated ˛uxanddonotcorrectforthe˛uxineachseparatebinofmuonkinematics.Theresults areexperiment-dependentsincenobin-by-bincorrectionforthe˛uxisapplied,butitis completelymodel-independentasnoassumptionneedstobemadeontheparticularneutrino energydistributionineachmuonkinematicbin.Thecomparisonofamodelwiththe˝nal resultcanbemadebyconvolvingthemodelwiththeproper˛ux. 1 Theangleiswithrespecttotheincomingneutrinodirection 2 Integratedoverarangeof0to30GeV. 107 6.2Fitmethod Thisanalysisusesabinnedmaximumlikelihoodmethodtoperforma˝ttothenumber ofselectedeventsinFGD1(ND280)andtheProtonModule(INGRID)˝ducialvolumes asafunctionofoutgoingmuonkinematics( p ; cos ).Themaximumlikelihoodmethod variesasetofsignalandnuisanceparametersto˝ndthevalueswhichbestdescribethe dataproducingasetofpost-˝tvalues,errors,andcorrelationsforeach˝tparameter.The inputsforthe˝tarethesignalandbackgroundsamplesforeachdetectorasdescribedin Sections4.3and4.4.Forthisanalysis,anewversionofthe˝ttingcodewaswrittentohandle theadditionalcomplexitiesof˝ttingmultipledetectorswithdi˙erentinputsandsystematic uncertainties.MoredetailsonthesoftwareitselfcanbefoundinAppendixA. Theoutputofthe˝tisthesetofparametervalues ~ whichmaximizethefollowing likelihoodgiventhedata ~y : L ( ~y ; ~ )= L stat ( ~y ; ~ ) L syst ( ~y ; ~ ) : (6.2) Oftenitismoreconvenienttoworkwithtwicethenegativelog-likelihoodbecauseitap- proximatesa ˜ 2 distributioninthelimitoflargestatistics(Wilks'Theorem[93]).Sincethe log-likelihoodisamonotonicallyincreasingfunction,theparametervalueswhichmaximize thelikelihoodwillalsomaximizethelog-likelihood[93].Thelog-likekihoodfunctiontobe maximizedor ˜ 2 functiontobeminimizedisgivenby: ˜ 2 = ˜ 2 stat + ˜ 2 syst = 2ln L stat 2ln L syst (6.3) wherethedependenceontheparameters ~ andthedata ~y isimplied.Thestatisticalterm, L stat ,isthePoissonlikelihood(usingStirling'sapproximation 3 )whichhasbeenmodi˝edto includethestatisticaluncertaintyof˝niteMonteCarlosamplesusingtheBarlow-Beeston method[94](seeAppendixBforthederivation): 2ln L stat = ˜ 2 stat = bins X j 2 j N exp j N obs j + N obs j ln N obs j j N exp j + ( j 1) 2 2 ˙ 2 j ! (6.4) 3 Stirling'sapproximationis ln n != n ln n n + O (ln n ) 108 where N exp j isthenumberofexpectedeventsestimatedfromtheMonteCarlo(alsocommonly denoted N MC ), N obs j isthenumberofobservedeventsfromdatainagivenbin j , j is theBarlow-Beestonscalingparameter,and ˙ 2 j istherelativevarianceoftheMonteCarlo predictionforbin j .ThisimplementationoftheBarlow-BeestonmethodassumestheMonte CarlopredictionisGaussiandistributedineachbin.Thesystematicterm, L syst ,isapenalty termforthesystematicuncertainties(e.g.˛uxuncertainties)beingpulledawayfromtheir prior(ornominal)values: 2ln L syst = ˜ 2 syst = X p ~p ~p prior V syst cov 1 ~p ~p prior (6.5) where ~p isthevectorofsystematicparameters, ~p prior isthepriorornominalvalueofthe systematicparameters,and V cov isthecovariancematrixwhichdescribestheerrorand correlationsforeachsystematicparameter.Thesystematicpenaltytermallowsfortheuse ofpriorknowledgeinthe˝tfromboththeoryandexternalexperimentaldata.Byusing theforminEq.6.5andacovariancematrixtodescribetheuncertainty,thesystematic parametersareapproximatedbyandtreatedasGaussiandistributionsinthe˝t.Finally, thelikelihood˝tisunregularizedtoavoidanyadditionalbiasfromtheregularizationterm attheexpenseofhighervarianceinthesolution.Adiscussiononusingregularizationinthe likelihood˝tcanbefoundinAppendixC. The˝tperformsaspeci˝ctypeoflikelihood˝tcalledatemplatelikelihood˝ttohandle theunfoldingormappingfromreconstructedvariables(withdetectore˙ects)totruevari- ables(withoutdetectore˙ects).Theunfoldingisperformedtoremovethedetectore˙ects fromthedataandallowforcomparisonsbetweenexperimentsandforeasycomparisonsto theory 4 .Therearedi˙erentmethodsandtypesofunfoldingtechniques,suchasiterativema- trixinversion(anexamplebeingD'Agostiniunfolding[95]),whichallhavetheirrespective positivesandnegatives.ForfurtherdiscussionsonunfoldingseeRefs.[93,96,97,98,99]. 4 Analternativeforcomparisonstotheoryistoconvolvethetheorymodelwiththede- tectorsimulationforagivenexperiment,producingatheorypredictionwhichisinthe reconstructedspace. 109 Theunfoldingtechniqueemployedbythetemplatelikelihoodmethodisimplementedby assigningafreeparameterknownasatemplateweight(denotedas c i 's)toeachbinofthe truedistributionwhichactsasanormalizationconstantforthatbin.Thetemplateweight canincreaseordecreasethenumberofselectedsignaleventsfromtheMCinagivenbinof truevariables(notalteringtheweightofbackgroundevents).Eachiterationthe˝troutine changesthetemplateweightsandobserveshowthee˙ectispropagatedtothereconstructed distributionbycomparingtheresulttothedatadistribution.The˝t˝ndsthecombination ofvaluesforthetemplateweightsandthenuisanceparameterswhichminimizesthe ˜ 2 describedaboveinEq.6.3. Thepost-˝tparametervaluesareusedtocalculatethenumberofsignaleventsinthe truedistributionwhichbest˝tsthedata,andsubsequentlyusedintheextractionofthe crosssection.Thisissimilartoiterativeunfoldingbymatrixinversionwherethetemplate parametersactasthematrixelements.Anadvantageofthismethodincontrasttoother typesofunfoldingora˝tformodelparametersisthat,with˝neenoughbins,theresult isnotinherentlybiased(orthebiasissmall)totheshapeoftheinputsimulationasthe templateweightsarecompletelyfreetomovethesignalmodelsincetheyhavenoprior constraint[34].Theinputsimulationisstillusedtomodelthebackgroundprocesses,which canbevalidatedbyassessingthegoodnessof˝tinsidebandregions. Thetemplateweightsarewrittenas c i where i runsoverthenumberoftruebins,giving thenumberofexpectedsignaleventsinabin: N expsignal i = c i N MCsignal i (6.6) Thee˙ectofchangingthenumberofeventsinatruebinonthenumberofeventsina reconstructedbinneedstobeknowntocomparetothedatadistribution.Thetruebins andreconstructedbinsarerelatedbyasmearingmatrix t ij ,whichisdeterminedfromthe simulation.Thesmearingmatrixcontainsthetemplatesusedtorelatetrueandreconstructed bins,whichcanbethoughtofasthecolumnsorrowsofthematrix.Thenumberofevents 110 inareconstructedbin j fromtruebins i canbethenwrittenas: N expsignal j = true X i c i N MCsignal i t ij (6.7) where t ij isthedetectorsmearingmatrixfromtruebin i toreconstructedbin j .Thetotal numberofselectedeventsinareconstructedbin j includingbackgroundeventscanbewritten as: N exp j = true X i 0 @ c i N MCsignal i + bkg X k N MCbkg ik 1 A t ij (6.8) where k runsovereachbackgroundtype,typicallycategorizedbytopology(e.g.CC- 1 ˇ ). Inadditiontothetemplateweights,the˝tincludesnuisance(orsystematic)parametersfor crosssectionmodel,detector,andneutrino˛uxvariations.Theseareincludedasadditional eventweightsasfollows: N exp j = true X i 2 4 c i 0 @ N MCsignal i model Y a w ( a;~x ) 1 A + bkg X k N MCbkg ik model Y a w ( a;~x ) 3 5 t ij d j E X n v in f n (6.9) Theparametersvariedinthe˝tarethetemplateweights( c i )andthesystematicparameters: crosssection( a ),detector( d j ),and˛ux( f n ). Theindex i runsovertruemuonkinematicanalysisbins.Theindex j runsover reconstructedmuonkinematicanalysisbins. The t ij termisthedetectorsmearingmatrixfromtruebin i toreconstructedbin j . Theproduct Q model a runsoverthesystematicparametersrelatedtothetheoretical modelingofthesignalandbackground.The w ( a;~x ) termsareweightstoaccountfor thee˙ectofchangingthevalueofsometheoreticalmodelparameter, a ,(e.g. M RES A ) foragiveneventbasedonkinematics,truereaction,andtruetopologyencodedin ~x (see Section5.3).Eachmodelparameterhasanassociatednominalvalueanduncertainty, whichareobtainedfrom˝tstoexternaldata. 111 The d j termsaresystematicparametersdescribingthedetectoruncertaintywhich actasweightstoincreaseordecreasethetotalnumberofeventsinareconstructed bin.Thenominalvaluesarede˝nedtobe1(thenominaldetectorMC)andhavea covariancematrixwhichiscalculatedusingthedetectorsimulation(seeSection5.2). The f n termsaresystematicparametersdescribingthe˛uxuncertaintyforatruebin i asafunctionoftrueneutrinoenergybin n .The v in termsaretheweightswhich describethe˛uxweightsfromneutrinoenergybins n correspondingtotruebin i .The ˛uxparametersactasweightsincreasingordecreasingthenumberofeventsintrue neutrinoenergybins,whichmaybelongtoanumberoftrueanalysisbins.Thenominal valuesof f n arede˝nedtobe1(thenominal˛uxMC)andhaveacovariancematrix whichissuppliedbytheT2KBeamMCgroup(seeSection5.1). ThecovariancematrixforeachsetofsystematicparametersisusedinEq.6.5tocalculatethe contributionfrommovingthesystematicparametersfromtheirnominalvalues.Equation 6.9givesthetotalnumberofeventsinreconstructedbin j forthecurrentvalueofallthe˝t parameters. TheMinuit2minimizer[100]isusedwiththeMIGRADandHESSEalgorithmstoper- formthemultidimensionalminimizationofthe ˜ 2 function.MIGRADusesavariablemetric methodbasedontheDavidon-Fletcher-Powellalgorithm[101,102]toexploretheparame- terspaceand˝ndtheminimumofthe ˜ 2 function,accordingtosomespeci˝edtolerance. HESSEusesthemethodof˝nitedi˙erencestocalculatethematrixofsecondderivatives(the Hessianmatrix)aroundthebest-˝tpointfoundbyMIGRAD,whichistheninvertedtogive thecovariancematrixforthe˝tparameters.TheHESSEmethodassumesthelikelihood surfacearoundthebest-˝tpointcanbedescribedbyamulti-variateGaussian.The˝nal outputofthe˝tisavectorofthebest-˝tparametersandacovariancematrixdescribing theirpost-˝terrorandcorrelations.Thenumberofselectedsignaleventsinatruebincan 112 becalculatedusing: N expsignal i = reco X j 0 @ c i N MCsignal i model Y a w ( a;~x ) 1 A ( t ij ) 1 d j E X n v in f n (6.10) wherethepost-˝tvaluesareusedforthe˝tparameters.Thepost-˝tnumberofselected signaleventsisusedinEq.6.1withabin-by-bine˚ciencycorrection(seeSection4.6), ˛uxnormalization(seeSection6.4),numberoftargetsnormalization(seeSection6.5),and bin-widthcorrectiontocalculatethe˛ux-integrateddi˙erentialcrosssection. Particularcarehasbeentakentoimprovetheperformanceandtoallowthe˝ttoconverge inalmostallcases.Speci˝callyitisimportanttoavoidorlimitthenumberofdegeneracies betweenparameters,andoverallminimizethetotalnumberof˝ttedparameters,evenat theexpenseoflargersystematicerrors.Dimensionalityreductionisperformedonthede- tectorparametersthroughprincipalcomponentanalysis,keeping99%ofthetotaldetector information.Allotherparametersareretainedduetotheirlimitednumbercomparedto thenumberofdetectorparameters.Thetechniquesusedtoperformprincipalcomponent analysisanddimensionalityreductionaredescribedinAppendixD. 6.3Errorpropagation Theoutputofthe˝tisthebest-˝tvaluefortheparametersandacovariancematrix describingtheerrorsontheparameters(calculatedusingtheHESSEmethod).The˝nal stepiscalculatingthecrosssectionusingthebest-˝tparameters,andpropagatingthepost-˝t errorstothecross-section.Howeverthedependenceofthecrosssectiononthe˝tparameters isingeneralsomehighdimensionalunknownfunction,soananalyticalexpressionforthe errorpropagationisalsounknown.Insteadtheerrorsfromthe˝tarepropagatednumerically tothecrosssectionthroughtheMonteCarloproceduredescribedbelow. The˝tisrunandallowedtoconverge,producingasetofpost-˝tparametersandtheir associatedcovariancematrix.Thepost-˝tcovariancematrixisCholeksydecomposedinto itslowertriangularform(anditstranspose)whichrepresentstheparametererrorsand 113 hasthesamemultivariatedistributionandcorrelationsastheoriginalcovariancematrix [103].Arandomdeviationofthe˝tparametersisgeneratedbymultiplyingtheCholesky decompositionandavectorofrandomnumbersdistributedaboutaGaussianofmeanzero andwidthone.Thisrandomdeviationisaddedtothepost-˝tparameterstocreatearandom variationofthe˝tparametersthatisdistributedaccordingtothecovariancematrix[103], asshownbelow: = LL (6.11) ~ t = ~ f +( L ~r t ) (6.12) where isthepost-˝tcovariancematrix, LL istheCholeskydecomposition, ~ t isthenew variedparametervector, ~ f isthebest-˝tparameters,and ~r t isavectorofrandomnumbers distributedaboutaGaussianofmeanzeroandwidthone.Thevariedparametervector, ~ t ,representsaplausiblevariationoftheparametersaccordingtothecalculatedcovariance matrix,andtheprocedureiscommonlyreferredasgeneratingatoythrowoftheparameters. Thisprocessisrepeatedformanyvariationsofthe˝tparameters(oforder1000ormore) tosamplethelikelihoodspaceencodedinthepost-˝tcovariancematrix.Thecrosssectionis calculatedforeachvariationofthe˝tparametersandrepresentsthedistributionofplausible crosssectionvaluesaccordingthestatisticalandsystematicuncertainties.Thedistribution isusedtocalculatethecrosssectioncovariancematrixasfollows: V ij = 1 N X t ( x it x i )( x jt x j ) (6.13) where V ij isthe ij th elementofthecrosssectioncovariancematrix, N isthenumberof throws, x it isthe i th crosssectionbinforthrow t ,and x i isthe i th crosssectionbincal- culatedusingthebest-˝tparameters.Thenumberoftargets,˛uxintegral,ande˚ciency normalizationarealsovariedwitheachthrowtocalculatethecrosssection.Thusthe˝nal crosssectioncovarianceincludestheuncertaintyfromthe˝tparametersandthenumberof targets,˛uxintegral,ande˚ciencynormalizations. 114 Theprocedureusedtogeneraterandomcorrelatedvectorsoftheparameterscaninsome situationsgenerateanunphysicalvalueforaparameter.Forexample,ifaparameterwhich isonlymeaningfulwhenpositivehasalargeenougherror,thentheGaussianthrowswill eventuallyproduceanegativevaluefortheparameterwithenoughrandomthrows.This analysishandlesthisissuebylimitingtherangeofsomeparameterstotheirvalidregions, clippingthevariationstobewithintheirrespectivelimits.Thishasthee˙ectoftruncating thetailsoftheGaussiandistributionfortheseparameters.Ifthetruncatedareaissu˚ciently small,thenthedistributioncanstillbeapproximatedasGaussianandthetruncationisa negligiblee˙ectontheresult. ThismethodmakestwoseparateassumptionsthattheerrorsareGaussiandistributed. The˝rstisaresultoftheHESSEmethod,whichassumesthelikelihoodsurfacearoundthe best-˝tpointcanbedescribedbyamulti-variateGaussian.Thesecondisaresultofpa- rameterizingtheerroronthecrosssectionasacovariancematrixfromthetoythrows.Both assumptionsofGaussianerrorsarerequiredtopropagateandpresenterrorsasdescribed, andtoavoideitherorbothwouldrequireamoreadvanced˝tmethodormorecomplicated datarelease.Ifthesearevalidassumptions,thenthecovariancematrixcalculatedbyHESSE describesanN(numberofanalysisbins)dimensionalcontourwithconstantlog-likelihood aroundthebest-˝tpointwhichrepresentstheprobablespreadofthe˝tparameters.The propagatedcovariancematrixthendescribesacontourwithconstantlog-likelihoodaround thebest-˝tcross-sectionineachanalysisbin.Suchacontourisanapproximationofa Bayesiancredibleinterval,speci˝edtohavethebest-˝tpointatthecenterandlimitedto besymmetric[34]. 6.4Integrated˛ux Thisanalysismeasuresthe˛ux-integratedcrosssectionandneedstheintegrated˛ux foreachdetector.Theneutrino˛uxpredictionisperformedbytheBeamMCgroupas describedinSection3.2.Thisanalysisusesthe13av2versionofthesimulationand˛ux 115 tuningwhichusesNA61thin-targetdataforthehadrontuning,andonlyusestheforward horncurrent(FHC)prediction.INGRIDandND280areexposedtothesameneutrino beam,butareplacedatdi˙erentanglesrelativetothebeamcenterwhichgivesadi˙erent integrated˛uxand˛uxshapeforeachdetector.WhenrunninginFHCmode,theneutrino beamisprimarilycomposedof withsmallcontributionsfrom , e , e whichgivesa lowbackgroundratefromthenon-signal˛avors.AtINGRIDthebeamis95.3% ,3.9% ,and0.8% e / e ,whileatND280thebeamis92.9% ,5.9% ,and1.2% e / e .The lowbackgroundrateisparticularlyimportantfortheINGRIDselectionasINGRIDlacks theabilitytoseparate and interactionsandhastorelyonbackgroundsubtractionto remove events. Figure6.1:Nominal˛uxpredictionatND280(left)andINGRID(right)byneutrino˛avor. The˛uxpredictioniscorrectedforthebeamconditionsforeachrunindividually. Theneutrino˛uxpredictionsarecalculatedfor 10 21 POTandneedtoberescaledfor eachdatarunindividually.The˝nal˛uxpredictionisthePOTaverageoftheindividual runscalculatedby: = 1 10 21 run X r P r 0 @ E X n ˚ nr 1 A (6.14) where P isthecollectedPOTforrun r and ˚ nr istheneutrino˛uxforagivenenergybin n andrun r . 116 The˛uxatND280isshowninFig.6.1,andtheintegrated ˛uxforRun2-4andRun8 is: ND =2 : 19 10 13 cm 2 (6.15) The˛uxatINGRIDisshowninFig.6.1,andtheintegrated ˛uxforRun2-4is: ING =3 : 01 10 13 cm 2 (6.16) TheProtonModulefortheINGRIDselectionwasmovedtoadi˙erentlocationduring Run8,soonlytheintegrated˛uxforRun2-4isneededforINGRID. 6.5Numberoftargets Thenumberoftargetsinsidethe˝ducialvolumeforeachdetectormustbeknownto calculatethedi˙erentialcrosssectionpernucleon(asshowninEq.6.1).Thetotalnumber ofnucleonscanbecalculatedinthefollowingway: N t = N A B X i M i A i i (6.17) where N A isAvogadro'sconstant, B isthenumberofdetectormodules, M i isthetotal massofeachelement i , A i isthenumberofnucleonsaveragedovernaturalabundancesfor eachelement,and i istheatomicweightofeachelement.Thisassumesthateachdetector moduleisidentical.Thetotalmassofeachelementinthedetectorcanbewrittenas: M i = ˆ i V FV = ˆ areal i X Y (6.18) where V FV isthe˝ducialvolume, X; Y aretheXandYdimensionsofthe˝ducial volume,and ˆ areal i = ˆ Z isthearealdensity 5 foreachelement i .Thisallowsustorewrite thepreviousequationusingthearealdensityas: N t = N A B X Y ) X i ˆ areal i A i i (6.19) 5 Thearealdensityisthesurfaceorareadesityofanobjectwithunitsofkg/m 2 (or g/cm 2 ). 117 ThetargetvolumeforND280isFGD1,anditiscomposedof14XYsupermoduleswithan xy-plane˝ducialvolumeof X = Y =174 : 902cm (the˝rstXYsupermoduleisnotpart ofthe˝ducialvolume).Usingthearealdensities,theaveragenumberofnucleons,andthe atomicweightofeachelementgiveninTable6.1thetotalnumberofnucleonsinFGD1is: N FGD t =5 : 53 10 29 nucleons(6.20) ElementA N i Nat.abundance(%) A i i ( g = mol ) ˆ areal i ( g = cm 2 ) C 12698.9 12.01112.010781.849 1371.1 O 16899.762 16.00415.999430.07941790.038 18100.2 H 1099.985 1.0021.0079470.1579 210.015 Ti 46248 48.02447.86710.0355 47257.5 482673.8 49275.5 50285.4 Si 281492.22 28.10628.08550.021829154.68 30163.09 N 14799.634 14.00414.006720.0031 1580.366 Table6.1:Informationusedtocomputethetotalnumberofnucleonsforeachchemical elementoftheFGD1˝ducialvolume[4]. ThetargetvolumeforINGRIDistheProtonModule,anditiscomposedofasingle moduleforthepurposeofthemathinthissectionwitha˝ducialvolumeof X = Y = 100 cm.Usingthearealdensities,theaveragenumberofnucleons,andtheatomicweightof eachelementgiveninTable6.2thetotalnumberofnucleonsintheProtonModuleis: N PM t =1 : 76 10 29 nucleons(6.21) wherethearealdensityforelementfortheProtonModulewascalculatedusingthetotal massoftheProtonModule˝ducialvolume(292.1kg)andthefractionofeachelementby 118 massfoundinRef.[5](whichcontainsanalternatemethodforcalculatingthenumberof nucleons). ElementA N i Nat.abundance(%) A i i ( g = mol ) ˆ areal i ( g = cm 2 ) C 12698.9 12.01112.0107826.57 1371.1 O 16899.762 16.00415.999430.17231790.038 18100.2 H 1099.985 1.0021.0079472.223 210.015 Ti 46248 48.02447.86710.222 47257.5 482673.8 49275.5 50285.4 N 14799.634 14.00414.006720.02 1580.366 Table6.2:Informationusedtocomputethetotalnumberofnucleonsforeachchemical elementoftheProtonModule˝ducialvolume[5].Siliconisconsideredtohaveanegligible contribution. 119 CHAPTER7 FITVALIDATION Thischapterdescribestheproceduretovalidatetheperformanceoftheanalysisand˝t framework.Theanalysisistestedbyrunningaseriesof˝tstoavarietyof"data"inputs (colloquiallyreferredtoasfake/mock/pseudodata)producedbyalteringthenominalMC simulationthataredesignedtotestaparticularaspectoftheanalysis.Thetestsareim- portanttoquantifytherobustnessofthe˝ttobiasesinthepriors,abilitytoaccurately˝t underlyingphysics,andidentifypossiblefailuremodesofthe˝t.Thestudiesperformedare listedinTable7.1,andingeneraleach˝tisperformedwiththedefaultMIGRADstrategy, ane˙ectivetoleranceof 2 10 4 ,andwith4000toythrowsfortheerrorpropagation.For readingtheparameterplotsinthischapter,theparametershaveallbeennormalizedsuch thattheirpriorisatone,andanydeviationfromoneisthefractionalchange.Correspond- inglytheerrorbarsarethefractionalorrelativeerrorbarsandaresymmetricaboutthe parametervalue(byconstruction). Themetricsusedtoevaluatetheperformanceofagiventestarethefollowing:pre/post- ˝tcrosssectiondistributions,pre/post-˝treconstructedeventdistributions,pre/post-˝t˝t parameters,andthe ˜ 2 calculatedbetweenthepost-˝tandthenominalandpseudodata distributions.Sincethepseudodataismodi˝edMonteCarlo,thecrosssectionforthepseudo dataisknownandisreferredtoasthe"truth"crosssection.The˝tshouldideally˝ndand reportthetruthcrosssectionasthebest-˝tcrosssectionassumingithasenoughfreedom intheparameterization(whichwouldresultina ˜ 2 ofzero).Therearetwodi˙erent ˜ 2 metricsusedtojudgetheperformanceofthe˝twhicharede˝nedasfollows: forcrosssectiondistributions: ˜ 2 = N X i =1 ( ˙ ˝t i ˙ true i ) V 1 ˝t ( ˙ ˝t i ˙ true i ) (7.1) 120 where ˙ i isthecrosssectionforbin i forthepost-˝tandtruthcrosssection, N is thetotalnumberofbins,and V ˝t isthepost-˝tcovariancematrix.Thisisusedto quantifythecompatibilitybetweenthepost-˝tresultandachosentruedistribution (e.g.NEUTtruthorGENIE). reconstructedeventdistribution: ˜ 2 = N X i =1 2 N ˝t i N obs i + N obs i ln N obs i N ˝t i ! (7.2) where N ˝t istheexpectednumberofeventsinreconstructedbin i fromthe˝tresult and N obs istheobservednumberofeventsinreconstructedbin i fromthedata.This isusedtoquantifythecompatibilitybetweenthepost-˝tresultandthedatadirectly inthemeasuredquantities.Thisissimplythestatistical ˜ 2 betweentheexpected numberofeventsfromthepost-˝tresultandtheobserveddata.Itdoesnotinclude anycontributionfromthesystematic ˜ 2 andisintendedasasimplesanitycheckon theperformanceofthe˝t. 7.1Asimov˝ts AnAsimov˝t 1 isthemostbasictestofthe˝tmachinerysincethe˝tstartsatthetrue best-˝tpoint.Themovementofanyparameterwillraisethe ˜ 2 ,andthe˝tshouldexplore theparametersspaceanddeterminethebest-˝tpointisthenominalvalues.Thisisboth usefulasasanitycheckthatthe˝tfunctionscorrectlyandwillestimatethesizeofthe uncertaintyonthecrosssectionand˝tparameters.ThecorrectbehaviorfortheAsimov ˝tsisforthepost-˝tvaluestomatchthenominalvaluesforallparameters. 7.1.1Nominalpriors TheAsimov˝twasperformedwithallparametersstartingattheirnominalvalueandwith the˝tstartingatthetruebest-˝tpoint.Thepost-˝tvaluesforallparameterscorrectly 1 AnAsimov˝tisde˝nedasa˝twherethe"data"isthesameastheinputsimulation. 121 FitName Description Asimov˝t BasicAsimov˝ttotestthemachinery. Randomtemplatepriors Asimov˝twithrandomtemplatepriors. ND280onlyAsimov Asimov˝tusingonlyND280samples. INGRIDonlyAsimov Asimov˝tusingonlyINGRIDsamples. Statistical˛uctuations Fittostatistical˛uctuations. Systematic˛uctuations Fittosystematicparametervariations. Degreesoffreedom 200˝tswithstat+syst˛uctuations. Neutrinoenergyweights Fittovariationsbasedon E weights. Alteredsignalweights FittoincreasedINGRIDordecreasedND280sig- nalweights. Low Q 2 pionsuppression FittosuppressedRESeventweightsbasedon Q 2 . Low Q 2 CC- 0 ˇ weights Fittomodi˝edCC- 0 ˇ eventweightsbasedon Q 2 . AlternateRPAmodel FittoanalternateRPAmodelusingtheBeRPA parameterization. Horncurrentvariation Fittovaried E weightsbasedonan 3 ˙ increase ofhorncurrent. Hornalignmentvariation Fittovaried E weightsbasedonan 3 ˙ shiftof horn2and3alignment. Table7.1:Listofstudiesusedtovalidatetheanalysis. matchtheirnominalvaluesasshowninFig.7.1andindicatethatthe˝tisfunctioning correctly.TheAsimov˝tgivesanestimateoftheexpectedsizeoftheerrorsandsensitivity foreachparameterfortherealdata˝t.Thetemplate(signal)parametersareproportional tothestatisticalerrorandareapproximatelyequaltothestatisticalerrorfromthenumber ofeventsineachbin.Thetemplateparameterscorrespondingtothebackwardorhighangle binshavelargererrorscomparedtotheforwardgoingbinsduetotherelativepopulationof events. Thepost-˝terroreachsystematicparameterisexpectedingeneraltobesmallerorequal totheprioruncertaintyfortheAsimov˝t.Ifthepost-˝terrorisaboutequaltotheprior error,thenthatparameterdoesnothavemuchofanimpactonthe˝t.The˛uxparameter errorswereroughlyuniformlyreducedbyoneortwopercentagepointscomparedtotheir priorerrors.Thissmallanduniformreductionisexpectedfromthehighlycorrelated˛ux uncertaintiesandfrommeasurestakentolimittheconstraintonthe˛uxparameters.Most 122 ofthecrosssectionparameterssawareductioninerrorcomparedtotheirpriorswithafew remainingapproximatelyequaltotheirprior.Thisisexpectedasthepriorerrorsforthe crosssectionparametersareconservativeestimatestopreventanoverconstraint.Finallythe detectorparametersalsosawafairlyuniformreductioncomparedtotheirpriorerror. ThereconstructedeventdistributionsforeachsampleareshowninFigs.7.2,7.3,7.4.The pre-˝t,post-˝t,andinputpseudodatareconstructedeventdistributionsallmatchperfectly asexpected(resultingininazero ˜ 2 )sincetheyareidenticaltobeginwith.Finallythe extractedcrosssectionisshowninFig.7.5,andthepost-˝tcrosssectionperfectlymatching theinputpseudodata(resultinginazero ˜ 2 ).ThisisthecorrectbehaviorforanAsimov ˝tandshowsthatthe˝tisworkingforthissimplecase. Figure7.1:Pre/post-˝tparameterplotsfortheAsimov˝twithnominalpriors. 123 Figure7.2:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithnominalpriors, ND280signalsamplesonly. 124 Figure7.3:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithnominalpriors, ND280sidebandsamplesonly. Figure7.4:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithnominalpriors, INGRIDsamplesonly. 125 Figure7.5:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekinematics)for theAsimov˝twithnominalpriors. 126 7.1.2Randomtemplatepriors ThisAsimov˝tisaslightvariationonthe˝rst,andwasperformedwiththetemplate (signal)parameterssettorandompriorvalueswhilekeepingallsystematicparametersat theirnominalvalues.Thetemplateparameterpriorsweresettoarandomnumberdrawn fromaGaussiandistributionwithameanofoneandwidthoftenpercent.Thepseudo datainputisstillthesameastheinputsimulation,butthe˝tdoesnotstartexactlyatthe best-˝tpoint.Thisistotestthatthe˝tcanaccurately˝ndthecorrectbest-˝tpointfrom arandompointinparameterspace. Thepost-˝tvaluesforallparameterscorrectlyremainatorreturntotheirnominalvalues asshowninFig.7.6.Thepost-˝tparametererrorsshowsimilarbehaviortothenominal Asimov˝twithmostparametersseeingasmallererrorcomparedtotheirpriorerror.The reconstructedeventdistributionsforeachsampleareshowninFigs.7.7,7.8,7.9.The pre-˝t,post-˝t,andinputpseudodatareconstructedeventdistributionsallmatchperfectly asexpected(resultingininazero ˜ 2 ).FinallytheextractedcrosssectionisshowninFig. 7.10,withthepost-˝tcrosssectionperfectlymatchingtheinputpseudodata(resultingina zero ˜ 2 ).Thisisthecorrectbehaviorandshowsthe˝tcancorrectly˝ndthebest-˝tpoint wherethebest-˝tpointisaknownvalue. 127 Figure7.6:Pre/post-˝tparameterplotsfortheAsimov˝twithrandomtemplatepriors. 128 Figure7.7:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithrandomtemplate priors,ND280signalsamplesonly. 129 Figure7.8:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithrandomtemplate priors,ND280sidebandsamplesonly. Figure7.9:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithrandomtemplate priors,INGRIDsamplesonly. 130 Figure7.10:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekinematics)for theAsimov˝twithrandomtemplatepriors. 131 7.1.3ND280onlyAsimov˝t ThisAsimov˝tusesonlytheND280sampleswithalltheparametersattheirpriorvalues, anddoesnotincludeanyoftheINGRIDspeci˝cparametersorsamples..Thisistotest howtheanalysisrespondstoonlyusingonedetectorandhowtheresultsaredi˙erentfrom thecombination,isolatingpotentialfeatureswhicharedrivenbythedatafromND280or INGRID. TheresultsoftheND280onlyAsimov˝tareverysimilartothefullcombinedAsimov ˝t,whichisencouraging.Thetemplateparametersarenearlyidentical,andisexpectedas theND280templateparameterscanonlya˙ectND280eventsbyconstruction.The˛ux parametersstillshowaslightreductioninerror,howeveritisnotaspronouncedcompared tothecombinedAsimov˝t,particularlyinthesomeofhigherenergybins.Thisisanice exampleofhowthedataandcorrelationswiththeINGRID˛uxcanreducethe˛uxerrors inthecombinedanalysis.Thecrosssectionparametersshowsimilarlargereductionsfor thenormalizationparameters,withmostparametershavingslightlyhighererrorcompared tothecombined˝t.HowevermostofthepionFSIparameters(parameters15through20), endupwithalargererrorcomparedtothecombined˝t. 132 Figure7.11:Pre/post-˝tparameterplotsfortheAsimov˝tusingonlytheND280samples. 133 Figure7.12:Pre/post-˝treconstructedeventplotsfortheAsimov˝tusingonlytheND280 samples,signalsamplesonly. 134 Figure7.13:Pre/post-˝treconstructedeventplotsfortheAsimov˝tusingonlytheND280 samples,sidebandsamplesonly. 135 Figure7.14:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekinematics)for theAsimov˝tusingonlytheND280samples. 136 7.1.4INGRIDonlyAsimov˝t ThisAsimov˝tusesonlytheINGRIDsampleswithalltheparametersattheirpriorvalues, anddoesnotincludeanyoftheND280speci˝cparametersorsamples. TheresultsoftheINGRIDonlyAsimov˝tarealsoverysimilartothefullcombined Asimov˝t,againwhichisencouraging.SimilartotheND280only˝t,theINGRIDtem- plateparametersarenearlythesameasthecombined˝t.The˛uxparametersshowlittle constraintatthelowenergypartofthespectrumwiththeonlyreductioncomingatthe higherenergy.ThisislargelyexpectedastheINGRID˛uxpeaksat1GeVandhasahigher proportionofhighenergyneutrinos.Thecrosssectionparametersstillshowasimilartrend asthecombined˝tandmostparametersseeareductioninerror,howeverthereductionin errorismuchsmallercomparedtotheND280only˝torthecombined˝t.Thissuggests mostoftheconstraintonthecrosssectionparametersisduetotheND280samples.Thisis expectedasND280hasmuchhigherstatisticstoprovideastrongerconstraint.Additionally therearethreededicatedsidebandsamplesforND280toconstrainthebackgroundprocesses comparedtotheonesidebandsampleforINGRID.HoweverthemostpionFSIparameters forINGRIDdoshowanoticeablereduction,similartothatofthecombined˝t,indicating thatmostoftheconstraintinthecombined˝tisprovidedbytheINGRIDsample. 137 Figure7.15:Pre/post-˝tparameterplotsfortheAsimov˝tusingonlytheINGRIDsamples. Figure7.16:Pre/post-˝treconstructedeventplotsfortheAsimov˝twithrandomtemplate priorspriors,INGRIDsamplesonly. 138 Figure7.17:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekinematics)for theAsimov˝tusingonlytheINGRIDsamples. 139 7.2Statistical˛uctuations Thisstudyusespseudodatathatisastatistical˛uctuationofthenominalMCsimu- lation.ThepseudodatawasbuiltbygeneratingaPoissonian˛uctuationforeachbinof thereconstructedeventdistributionforeachsample,usingthenominalMCpredictionas theaverageforthePoissondistribution.The˝twasthenrunwithallparametersattheir nominalprior,butnow˝ttingtoapseudodatasetthatisnotidenticaltotheinputMC. UnlikethepreviousAsimov˝ts,thetruebest-˝tvalueforalltheparametersisunknown andthe˝tshouldendwithanon-zero ˜ 2 . Figure7.18:Pre/post-˝tparameterplotsforthe˝ttostatistical˛uctuations. 140 Figure7.19:Pre/post-˝treconstructedeventplotsforthe˝ttostatistical˛uctuations, ND280signalsamplesonly. 141 Figure7.20:Pre/post-˝treconstructedeventplotsforthe˝ttostatistical˛uctuations, ND280sidebandsamplesonly. Figure7.21:Pre/post-˝treconstructedeventplotsforthe˝ttostatistical˛uctuations, INGRIDsamplesonly. 142 Figure7.22:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekinematics)for the˝ttostatistical˛uctuations. 143 7.3Systematicparametervariations Thisstudyusespseudodatathatisbothastatistical˛uctuationandsystematicvari- ationofthenominalMCsimulationaccordingtothechangingthesystematicparameters. Thepseudodatawasbuiltbygeneratingacorrelatedrandomvectorofallthesystematic parameters,andweightingthenominalMCsimulationusingtheeventweightsthatcorre- spondtothevaluesofthethrownsystematicparametersandthen˝nallyapplyingstatistical ˛uctuationsontheresult.The˝twasrunwithallparametersattheirnominalprior,but now˝ttingtoapseudodatasetthatisnotidenticaltotheinputMC. Overallthe˝tperformsdecentlywithmostofthesystematicparametershavingpost- ˝tvaluesclosetothetruevariedvalue,oratleastbeingmovedinthatdirection.The ˝tparametersareshowninFig.7.23withthepre-˝t,post-˝t,andtruevaluesforeach parameter.Thetruevaluesfortheparametersarethevaluesusedtogeneratethepseudo data.Giventhatmovingthesystematicparametersawayfromtheirnominalvalueincurs apenaltytothe ˜ 2 ,itisexpectedthatasubsetofparameterswillnotendupattheir truevaluesasthe˝twillmovethetemplateparameterstocoversomeofthedi˙erence. Ingeneralthe˛uxparametersshowanoveralldecreaseinvaluetomatchthetruevalue, followingtheshapeofthethrownparameters.Finally,thestatistical˛uctuationscanresult intheparametershavingpost-˝tvaluesawayfromthetruevaluesasthe˝tisminimizing overthecombinationofstatisticalandsystematicvariations. Mostofthecrosssectionparametersmovedinthecorrectdirection,withseveralhaving apost-˝tvaluenearthetruevalue.Howevertherearesomenotableexceptionswhere theparametersmovedintheoppositedirectionofthetruevalue.TheDISshapeparameter (index6)was˝tintheoppositedirection,butthisatleastispartiallyexpectedgiventhe˛ux parameterswerealldecreasedastheDISparametertendstobehighlyanti-correlatedwith the˛uxparameters.Allthesystematicparametersaremarginalizedoverwhencalculating thecrosssectionsothe˝ttedvaluesnotmatchingthetruevaluesisnotaproblemaslong astheanalysisisunbiased. 144 Thereconstructedeventdistributionsallshowlargereductionsinthe ˜ 2 movingfrom thenominalMCtothepost-˝tdistributions.TheND280CC-Michelsampleshowstheleast improvement,howeveringeneraltheCC-Michelsamplehastheworstimprovementoutof thesidebandsamples.The˝nalcrosssectionisshowninFig.7.27withthepost-˝textracted crosssectionandthetruecrosssectionvalueusingthenominalMCandthepseudodata. The ˜ 2 showsanimprovement,withtheextractedcrosssectionpreferringthevaluesused pseudodatacomparedtothenominalMC.Ofparticularnoteisthenegativemeasurementin bin66ofthedistribution,correspondingtooneoftheINGRIDlowmomentum(350to500 MeV)bins.The 1 ˙ errorbarscomfortablycoverzero,andcoverthetruevalueifextended tothe 2 ˙ errorbars.Thisbehavioriscausedbyadownwardstatistical˛uctuationinthe INGRIDsamples. Figure7.23:Pre/post-˝tparameterplotsforthe˝ttosystematicparametervariations. 145 Figure7.24:Pre/post-˝treconstructedeventplotsforthe˝ttosystematicparametervari- ations,ND280signalsamplesonly. 146 Figure7.25:Pre/post-˝treconstructedeventplotsforthe˝ttosystematicparametervari- ations,ND280sidebandsamplesonly. Figure7.26:Pre/post-˝treconstructedeventplotsforthe˝ttosystematicparametervari- ations,INGRIDsamplesonly. 147 Figure7.27:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekinematics)for the˝ttosystematicparametervariations. 148 7.4Degreesoffreedom Thenextsteptostudytheperformanceoftheanalysisistoperformseveralhundred˝ts topseudodataandanalyzethedistributionofresults.Severalhundredpseudodatasets areproducedbystatisticallyandsystematically˛uctuatingthenominalsimulation(asin Sec7.2and7.3).Theanalysisisrunusingeachoneofthesepseudodatasetsandgivesa distributionoftheexpectedperformance. Thedistributionof˝toutcomescanbeusedtoestimatethenumberofdegreesoffreedom ( N )intheanalysis.The ˜ 2 pernumberofdegreesoffreedomisanimportantmetricto judgethegoodnessof˝toragreementfortheanalysiswhere ˜ 2 =N ' 1 isareasonable ˝tgiventhedataisastatistical˛uctuationoftheparentdistribution[104].To˝rstorder thenumberofdegreesoffreedominastatistical˝tisthenumberofdatapointsminus thenumberofestimated/˝ttedparameters[93,104].Thisisaccurateiftheparametersare uncorrelatedandtheerrorsareapproximatelyGaussian.Howeverforcorrelatedparameters andnon-Gaussianerrors,amoreaccuratemethodistosimulatemanypseudoexperiments andusetheresulting ˜ 2 distributionfromthe˝tstoestimatethenumberofdegreesof freedomempirically. ThedistributioninFig.7.28wasbuiltfrom220pseudoexperimentswithstatistical andsystematic˛uctuationstothenominalsimulationandrepresentsthe ˜ 2 distribution betweentheextractedcrosssectionandthenominalsimulation.Performinga˝ttothe ˜ 2 PDFgivesanestimateofthedegreesoffreedomof 66 : 38 1 : 21 fortheextractedcross sectiondistribution(andisshownasthedottedbluecurve). TheboxplotsinFig.7.29showthedistributionofcrosssectionvaluesandtheirrelative errorsforeachbin.Thecirclemarkistheaveragevalue,thedashisthemedianvalue,the boxcontains25%aboveandbelowthemedian,thewhiskersextendto1.5timestheinner quartilerange,andoutliersaremarkedwiththex's.ForGaussiandistributeddatathe meanandmedianwillbethesame,placedatthecenterofthebox,andtheboxrepresents 0 : 6745 ˙ where ˙ isthestandarddeviation.Mostofthecrosssectionbinshaveamedian 149 andmeanthatarefairlyclosetogetherwiththelargestseparationinthelowmomentum (350to500MeV/c)INGRIDbins(bins58and62).Thebinswithlowerstatisticsshow widerdistributionsoftheir˝ttedcrosssectionvalueswhichisexpectedduetotheincreased statisticalerrorandisveryevidentintheINGRIDbins.Thisisfurtherre˛ectedinthe distributionsofrelativeerrorswherethelowstatisticbinshaveawidespreadofpost-˝t error. Figure7.28:Distributionof ˜ 2 valuesbetweenthepost-˝tandthenominalMCcrosssection formanystatisticalandsystematic˛uctuations.Thesolidredcurvecorrespondstoathe- oretical ˜ 2 distributionwith70degreesoffreedom,andthedashedbluecurvecorresponds tothe˝tted ˜ 2 distribution. 150 Figure7.29:Boxplotsshowingthedistributionofpost-˝tcrosssectionvaluesforeachbin (top)andthedistributionofrelativeerrorsforeachbin(bottom)forthestatisticaland systematic˛uctuations.Thecirclemarkistheaveragevalue,thedashisthemedianvalue, theboxcontains25%aboveandbelowthemedian,thewhiskersextendto1.5timesthe innerquartilerange,andoutliersaremarkedwiththex's. 151 7.5Neutrinoenergyweights ThisstudyusespseudodatawherethenominalMCsimulationwasweightedtoan arbitrarydistributionintrueneutrinoenergy.Thepseudodatawasbuiltbyweighting thenominalMCsimulationevent-by-eventasfunctionoftrueneutrinoenergywherethe weightisaccordingtotheEquation7.3.Thechosenweightingfunctionisnotbasedon anyphysicalprocess,andistotesthowwellthe˝thandlesanextremechangecompared tothenominalMC.Sincetheweightingfunctionvariesonlywithtrueneutrinoenergy,the expectedbehaviorwouldbetoseethetemplateand˛uxparameterschangingthemost,with the˛uxparametersattemptingtofollowtheshape. w ( E )= 8 > > > > > > > > > > < > > > > > > > > > > : 1+0 : 5 = 500 E if E < 500 MeV 2 E = 1000 if 500 2000 MeV (7.3) The˝tresultingeneralshowstheexpectedbehavior,withmostofthemovementinthe templateand˛uxparameters,asseeninFig.7.30.Boththetemplateand˛uxparameters showtheshapeofenergyweightfunction,witheventsatlowerenergy(whichcorresponds tolowermomentum)gettingahigherweightandcorrespondinglywithhigherenergyevents receivingalowerweight.Mostofthecrosssectionparametersshowlittlemovement,with theexceptionofthe2p2hshapedial(index2)andthelowenergypionFSIparameters(index 15,16,17).Finallymostofthedetectorparametersshowlittlemovement,withafewbeing pulledrelativelyfarduetoalargerprioruncertainty. Thereconstructedeventdistributions,showninFigs.7.31,7.32,7.33,giveexcellent agreementbetweenthepost-˝tdistributionandtheinputpseudodatapoints.The˝nal crosssectiondistribution,showninFig.7.34,alsogivesexcellentagreementbetweenthe post-˝tandpseudodatadistribution. 152 Figure7.30:Pre/post-˝tparameterplotsforthe˝ttoneutrinoenergyvariations. 153 Figure7.31:Pre/post-˝treconstructedeventplotsforthe˝ttoneutrinoenergyvariations, ND280signalsamplesonly. 154 Figure7.32:Pre/post-˝treconstructedeventplotsforthe˝ttoneutrinoenergyvariations, ND280sidebandsamplesonly. Figure7.33:Pre/post-˝treconstructedeventplotsforthe˝ttoneutrinoenergyvariations, INGRIDsamplesonly. 155 Figure7.34:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekinematics)for the˝ttoneutrinoenergyvariations. 156 7.6Alteredsignalweights ThisstudyusespseudodatawherethesignaleventsinthenominalMCsimulationwere weightedtoseeifthe˝tcouldcorrectlyandexactlyrecoverthechange.Thepseudodatawas builtbyweightingonlythesignaleventsinthenominalMCsimulationevent-by-eventwhere ND280eventsweredecreasedby20%andINGRIDeventswereincreasedby20%.Sincethe templateparametershavenopriorerrororpenaltyformoving,theyshouldbeableto exactlyrecoverasimplenormalizationvariationwherenoneofthesystematicparameters shouldmove.Iftheweightingfunctionwassomethingmorecomplicated,forexamplebased onthemomentumtransferofthesignalevents,thenslightmovementwouldbeexpectedin thesystematicparameters. The˝tresultsshowtheexpectedbehavior,withthetemplateparametersmovingto thecorrectvaluesandthesystematicparametersessentiallynotmovingfromtheirnominal values.Thereareafewexceptions,namelytheINGRIDparametersshowslightdeviations fromthecorrectvalue.ThelowmomentumINGRIDbins(350to500MeV)showlarge di˙erencesofabout4%,comparedtoatmost1%fortherestofthetemplateparameters. Thisismostlikelyduetothe˝tstoppingearlybyusingtoolooseofatolerance,those INGRIDbinshaveissueswithlowstatistics,oritcouldberelatedtotheslightmovement inthe2p2hshapeparameter.Therestofthesystematicparametersshowverylittleto nomovement,howeverthe2p2hshapeparameter(index2)doesendupslightlybelowits nominalvalueforabouta2%deviation.Overallthepost-˝terroriscomparabletothe resultsfromtheAsimov˝ts. Thereconstructedeventdistributionsshowexcellentagreementbetweenthepost-˝tdis- tributionandthepseudodatapoints,withalltheND280samplesachievingaperfect˝t. TheINGRIDsampleshaveaslightnon-zero ˜ 2 ,butthisisexpectedgiventheINGRID templateparametersnot˝ttingtotheexactcorrectvalue.Theextractedcrosssectiondis- tributionalsoshowsexcellentagreement,withthenon-zero ˜ 2 aresultoftheinaccuracies intheINGRIDparametersasalreadydiscussed. 157 Figure7.35:Pre/post-˝tparameterplotsforthe˝ttosignaleventvariations. 158 Figure7.36:Pre/post-˝treconstructedeventplotsforthe˝ttosignaleventvariations, ND280signalsamplesonly. 159 Figure7.37:Pre/post-˝treconstructedeventplotsforthe˝ttosignaleventvariations, ND280sidebandsamplesonly. Figure7.38:Pre/post-˝treconstructedeventplotsforthe˝ttosignaleventvariations, INGRIDsamplesonly. 160 Figure7.39:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekinematics)for the˝ttosignaleventvariations. 161 7.7Low Q 2 suppressionofresonantevents ThisstudyusespseudodatawheretrueresonanteventsinthenominalMCsimulation wereweightedtoseehowthe˝twouldhandleachangeinthedistributionofresonantpion productionevents.Resonantpioneventsmostoftenproduceeventswithapioninthe˝nal statewhichwillcontributetothesidebandsamples,howeverthepioncanbeabsorbedor missedandtheeventcanbeselectedinthesignalsamples.Theexpectedbehavioristhen toseefewerCC- 1 ˇ eventsmostlyinthesidebandsample,andseeaslightreductionin eventsinthesignalsamples.Thepseudodatawasbuiltbyweightingtrueresonantevents inthenominalMCsimulationevent-by-eventasafunctionof Q 2 accordingtothefollowing equation: w ( Q 2 )= 1 : 01 1+ exp (1 p Q 2 = 0 : 156) if Q 2 < 0 : 7 GeV 2 (7.4) ThesuppressionisbasedonresultsfromMINERvA[105]andMINOS[106]showinga disagreementbetweentheirdataandMCforlowmomentumtransferpionproductionevents, withthespeci˝cequationfortheweightsfromRef.[106]. The˝tperformsasexpected,withthelargestparametermovementcorrespondingtothe pionproductionsystematicparameters.Thesuppressionoftrueresonanteventswillpri- marilycauseade˝citofeventsintheCC- 1 ˇ andCC-Michelsidebands,andthe˝tresponds bydecreasingthestrengthofthepionnormalizationparameters(index8and9)andthe coherentpionproductionnormalization(index11).AdditionallythepionFSIparameters correspondingtolowenergyprocesses(index15and17)wereincreased,whichalsoresults indecreasingtheamountofpioneventsaspionswillescapethenucleuslessoften.The2p2h parametersweredecreasedwhichisprobablydueto2p2heventscommonlybeingmisclassi- ˝edasCC- 1 ˇ events.Ingeneralthe˝treducedthemostcommonsourcesofpioneventsto achievethesuppressionseeninthepseudodata.Mostofthetemplateand˛uxparameters showaslightincreasewhichislargelytobalanceloweringthepionnormalizations. Thereconstructedeventdistributionsshowexcellentagreementbetweenthepost-˝tdis- 162 tributionandthepseudodatapointsoverall.Mostoftheremainingdiscrepancyisinthe CC- 1 ˇ andCC-Michelsamples,whichislargelyexpectedasthe Q 2 suppressionhadthemost e˙ectonthosesamples.Theextractedcrosssectionmatchesquitewellwiththepseudodata distribution,andshowstheexpectedslightlylowercrosssectionduetothelossofresonant eventswhichendupmis-taggedasCC- 0 ˇ events. Figure7.40:Pre/post-˝tparameterplotsforthe˝ttolowmomentumtransfersuppressed resonantevents. 163 Figure7.41:Pre/post-˝treconstructedeventplotsforlowmomentumtransfersuppressed resonantevents,ND280signalsamplesonly. 164 Figure7.42:Pre/post-˝treconstructedeventplotsforlowmomentumtransfersuppressed resonantevents,ND280sidebandsamplesonly. Figure7.43:Pre/post-˝treconstructedeventplotsforlowmomentumtransfersuppressed resonantevents,INGRIDsamplesonly. 165 Figure7.44:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekinematics)for the˝ttolowmomentumtransfersuppressedresonantevents. 166 7.8MINERvAdata-drivenweightsofsignalevents ThisstudyusespseudodatawheretruesignaleventsinthenominalMCsimulation wereweightedtoseehowthe˝twouldhandleaphysicallymotivatedchangetothesignal distribution.ThepseudodatawasbuiltbyweightingtruesignaleventsinthenominalMC simulationevent-by-eventasafunctionof Q 2 basedonresultsfromMINERvA[107],which showedadisagreementbetweentheirdataandMCforlowmomentumtransfer. The˝tresultsareasexpectedwithessentiallyonlythetemplateparametersmoving fromtheirnominalvalue.Thealterationisonsignalevents,the˝twillprefertomove thetemplateparameterssincetheyincurzero ˜ 2 penaltyformovementasopposedtothe systematicparameters.Ingeneralmostofthetemplateparametersshowaslightincrease, whichisfromtheslightenhancementatmediummomentumtransferwhichcorresponds tolessforwardtohighangleevents.Themostforwardgoingevents,particularlyatlow momentum,showasharpdecreaseinthetemplateparameterswhichisexpectedaslow momentumtransfereventswillprimarilybeforwardgoing.Therestofthesystematic parametersshowlittletonomovement,withtheonlynotableexceptionbeingthe2p2h shapeparameter,whichiscommontomanyoftheother˝ts.Thepost-˝terrorsonthe parametersaresimilartotheAsimov˝ts. Thereconstructedeventdistributionsmatchwellbetweenthepost-˝tdistributionand thepseudodatapoints.Thesidebandsamplesshowlittledi˙erence,whichisexpectedasthe variationonlya˙ectedsignaleventsandthesamplesarefairlypure.Similartothetemplate parameters,thesignalsamplesshowaslightincreaseineventrateforlessforwardtohigh anglebins,andanoticeabledecreaseinthemostforwardgoingbins.Theextractedcross sectiondistributionmatchesverywellwiththepseudodatadistribution,andagainshows theexpectedenhancementorsuppressionofbinsbasedontheirangleandmomentum. 167 Figure7.45:Pre/post-˝tparameterplotsforthe˝ttolowmomentumtransfersuppressed signalevents. 168 Figure7.46:Pre/post-˝treconstructedeventplotsforlowmomentumtransfersuppressed signalevents,ND280signalsamplesonly. 169 Figure7.47:Pre/post-˝treconstructedeventplotsforlowmomentumtransfersuppressed signalevents,ND280sidebandsamplesonly. Figure7.48:Pre/post-˝treconstructedeventplotsforlowmomentumtransfersuppressed signalevents,INGRIDsamplesonly. 170 Figure7.49:Pre/post-˝tcross-sectionplotshowingallanalysisbins(intruekinematics)for the˝ttolowmomentumtransfersuppressedsignalevents. 171 7.9AlternateRPAmodel ThisstudyusespseudodatawheretrueCCQEeventsinthenominalMCsimulation wereweightedtoseehowthe˝twouldhandleaphysicallymotivatedchangetothesignal distribution.ThepseudodatawasbuiltbyweightingtrueCCQEeventsinthenominalMC simulationevent-by-eventasafunctionof Q 2 accordingtothefollowingequation w ( Q 2 )= 8 > > < > > : A (1 x 0 ) 3 +3 B (1 x 0 ) 2 x 0 +3 C (1 x 0 ) x 0 2 + Dx 0 3 ;xU (7.5) C = D + 1 3 U E ( D 1) (7.6) where x = Q 2 and x 0 = Q 2 =U .Thecoe˚cientsandformoftheequationweredesignedto mimictheshapeoftheNieveset.al.[13]modelRPAcorrectionforCCQEevents(shown inFig.7.50),andallowforshapevariationsbychangingthevaluesforthecoe˚cients.The valuesusedfor A;B;D;E;U arethenominalvaluesusedintheT2Koscillationanalysis, andaregiveninTab.7.2.TheparameterizationisdenotedastheBeRPAparameterization duetotheuseofBernsteinpolynomialstoconstructthepolynomialformusedfor x 2 : 5 GeV 2 . 234 FigureF.2:Eventdistributionfortruemomentumtransfer( Q 2 )fortheND280signalsam- plesstackedbytruereaction.Thepurityofeachreactionislistedinthelegend.Thelast bincontainsalleventswith Q 2 > 2 : 5 GeV 2 . 235 FigureF.3:Eventdistributionfortruemomentumtransfer( Q 2 )fortheND280sideband samplesstackedbytruetopology.Thepurityofeachtopologyislistedinthelegend.The lastbincontainsalleventswith Q 2 > 2 : 5 GeV 2 . 236 FigureF.4:Eventdistributionfortruemomentumtransfer( Q 2 )fortheND280sideband samplesstackedbytruereaction.Thepurityofeachreactionislistedinthelegend.The lastbincontainsalleventswith Q 2 > 2 : 5 GeV 2 . 237 APPENDIXG EXTRACTEDCROSSSECTIONADDITIONALPLOTS Thisappendixcontainsadditionalplotstoshowtheextractedcrosssection. 238 FigureG.1:ND280extractedcrosssectionbinsforjointdata˝tcomparedtothenominal MCpredictionasafunctionofmuonmomentumforslicesofmuonangle.Notethey-axis isnotthesameacrossalltheplots. 239 FigureG.2:INGRIDextractedcrosssectionbinsforthejointdata˝tcomparedtothe nominalMCpredictionasafunctionofmuonmomentumforslicesofmuonangle.Notethe y-axisisnotthesameacrossalltheplots. 240 BIBLIOGRAPHY 241 BIBLIOGRAPHY [1] M.Tanabashietal.(ParticleDataGroup),ofParticlePhPhys.Rev. 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