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Searching for multi-nucleon processes in neutrino interactions by proton identification in the fine-grained… Kim, Jiae 2018

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Searching for Multi-nucleon Processesin Neutrino Interactions by ProtonIdentification in the Fine-GrainedDetectors for T2KbyJiae KimB.Sc., The Yonsei University, 2008M.Sc., The University of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES AND POSTDOCTORAL STUDIES(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2018c© Jiae Kim 2018The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Searching for Multi-nucleon Processes in Neutrino Interac-tions by Proton Identification in the Fine-Grained Detectorsfor T2Ksubmitted by Jiae Kim in partial fulfillment of the requirements for thedegree of Doctor of Philosophy in Physics.Examining Committee:Hirohisa Tanaka, PhysicsSupervisorScott Oser, PhysicsCo-supervisorDavid Morrissey, PhysicsSupervisory Committee MemberColin Gay, PhysicsUniversity ExaminerTakamasa Momose, ChemistryUniversity ExaminerAdditional Supervisory Committee Members:Janis McKenna, PhysicsSupervisory Committee MemberJeremy Heyl, PhysicsSupervisory Committee MemberiiAbstractT2K is an accelerator-based neutrino experiment designed to observe neu-trino oscillations with a baseline of 295 km across Japan from Tokai toKamioka. Its main goal is to measure oscillation parameters (θ23, ∆m232and θ13) through νµ (νµ) disappearance and νe (νe) appearance channels. Ithas also begun to provide measurements of CP violation in the neutrino sec-tor combining all four neutrino oscillation channels. However, the precisionrequired for these experiments has resulted in the need to reduce challengingsystematic uncertainties. Among all the uncertainties, the largest contribu-tion comes from the neutrino interaction model, where nuclear effects arepoorly understood. As neutrinos typically interact on nucleons that arealmost always contained within nuclei, one immediately has to confront nu-clear effects. Nuclear effects alter the kinematics of out-going particles fromneutrino scatterings, and hence affect the neutrino oscillation measurements.Therefore it is crucial to understand the nuclear effects in these neutrino in-teractions.This dissertation describes the measurement of neutrino-nucleus interac-tions with no final state pion and at least one final state proton. Differentialcross sections are measured as a function of kinematic variables, which uti-lize both muon and protons. An iterative unfolding technique is used toextract the cross sections. By providing unfolded and efficiency correctedresults, this measurement can be more readily compared to theoretical mod-els to allow a better understanding of nuclear effects in neutrino interactions,thereby providing valuable constraints on the systematic uncertainties as-sociated with neutrino oscillation measurements for both T2K and otheraccelerator-based neutrino experiments.iiLay SummaryIn neutrino oscillation experiments, the incident neutrino energies and fla-vors are unknown. Instead, they are reconstructed with a choice of typesof interactions occurring in a detector. Therefore, the poor understand-ing of the neutrino-nucleus interaction can lead to wrong measurements ofneutrino oscillation parameters. The neutrino-nucleus interaction is yet tobe fully understood due to the complex inner structure of a nucleus andnucleon-nucleon correlations. This dissertation details the measurementsof neutrino-nucleus interaction rate as a function of experimental observ-ables. The measurements are compared to various theoretical models. Theresearch will provide invaluable information to understand neutrino-nucleusinteractions better.iiiPrefaceThis dissertation is based on the collaborative work of the T2K collabo-ration. Figures in Chapters 2 and 3 are taken from previously publishedpapers with permission and the corresponding papers are cited accordingly.The detectors described in Chapter 4 were designed and built by the T2Kcollaboration. Figures in Chapter 4, which are taken from internal T2Kwebsite 1, are made by many T2K collaborators. Figures in Section 4.5are taken from the public Suker-Kamiokande gallery 2. The event selectionand detector systematics described in Sections 5.2 and 5.4.1 are based onwork by T2K νµ working group with the extent to the proton reconstruc-tion done by A. Cervera, A. Izmaylov, and L. Monfregola. I was respon-sible to quantify a detector systematic uncertainties related to the Michelelectron tagging in the Fine-Grained Detectors. In addition, the flux pre-diction and uncertainties described in Section 5.4.2 were done by the beamgroup and cross-section model systematics were provided by the NeutrinoInteraction Working Group (NIWG). The software, T2KReWeight, used toestimate these model systematics was originally written by K. Mahn anddeveloped by other collaborators. The software, xsTool, used to extractcross sections shown in Chapter 7 was originally written by M. Hierholzerand developed by T. Yuan, E. Scantamburlo and other collaborators. xs-Tool relys on the existing ROOT libraries to implement various unfoldingmothods. The D’Agostini’s unfolding method is implented in ROOT by TimAdye 3 and adopte to xsTool. I modified this code to be compatible withT2K data output and my analysis structure. I conducted all the testingand plots in Chapter 7, but the plots in Appendix H were produced by S.Dolan 4. The results of Chapter 7 and Appendix H are published in [1], andthe manuscript was mainly written by other T2K collaborators.1http://www.t2k.org2http://www-sk.icrr.u-tokyo.ac.jp/sk/detector/index-e.html3T.J.Adye@rl.ac.uk4stephen.dolan@llr.in2p3.frivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Neutrino Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1 History of Neutrinos . . . . . . . . . . . . . . . . . . . . . . . 22.2 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . 42.3 Evidence of Neutrino Flavor Change . . . . . . . . . . . . . . 72.4 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . 102.5 Neutrino Oscillation Measurements . . . . . . . . . . . . . . . 132.6 Future Measurements: Beyond the Standard Model . . . . . . 153 Neutrino Interactions . . . . . . . . . . . . . . . . . . . . . . . . 183.1 Basics of Neutrino Interactions . . . . . . . . . . . . . . . . . 183.1.1 Neutrino-Lepton Scattering . . . . . . . . . . . . . . . 183.1.2 Neutrino-Nucleus Scattering . . . . . . . . . . . . . . . 193.2 Theoretical Models and Calculations . . . . . . . . . . . . . . 22vTable of Contents3.2.1 Nuclear Models . . . . . . . . . . . . . . . . . . . . . . 223.2.2 Free Nucleon Cross-Section Models . . . . . . . . . . . 273.2.3 Final State Interaction . . . . . . . . . . . . . . . . . . 313.2.4 Discrepancy in the CCQE Measurements and Multi-nucleon Effects . . . . . . . . . . . . . . . . . . . . . . 323.3 Neutrino Interaction Generators . . . . . . . . . . . . . . . . 373.4 The CCQE Cross-Section Measurements . . . . . . . . . . . . 393.5 Other Cross-Section Measurements . . . . . . . . . . . . . . . 443.5.1 The CC Inclusive Measurements . . . . . . . . . . . . 443.5.2 The CC1pi Measurements . . . . . . . . . . . . . . . . 454 T2K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1 Off-Axis Configuration . . . . . . . . . . . . . . . . . . . . . . 494.2 Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3 On-Axis Detector: INGRID . . . . . . . . . . . . . . . . . . . 534.4 Off-Axis Near Detector: ND280 . . . . . . . . . . . . . . . . . 544.4.1 Time Projection Chambers . . . . . . . . . . . . . . . 564.4.2 Fine-Grained Detectors . . . . . . . . . . . . . . . . . 584.4.3 pi0 Detector . . . . . . . . . . . . . . . . . . . . . . . . 594.4.4 Electromagnetic Calorimeter . . . . . . . . . . . . . . 604.4.5 Side Muon Range Detector . . . . . . . . . . . . . . . 604.4.6 The ND280 Data Acquisition System . . . . . . . . . . 614.5 Far Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.6 The Details of the ND280 Tracker . . . . . . . . . . . . . . . 664.6.1 FGDs . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.6.2 TPCs . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 Event Selection and Analysis Variables . . . . . . . . . . . . 795.1 Data Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2.1 Event Reconstruction . . . . . . . . . . . . . . . . . . 815.2.2 Selection Cuts . . . . . . . . . . . . . . . . . . . . . . 815.2.3 Selection Efficiency and Phase-space Limits . . . . . . 855.2.4 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . 895.3 Analysis Variables . . . . . . . . . . . . . . . . . . . . . . . . 905.4 Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.4.1 Detector Uncertainties . . . . . . . . . . . . . . . . . . 925.4.2 Model Uncertainties . . . . . . . . . . . . . . . . . . . 995.5 Distributions of Analysis Variables . . . . . . . . . . . . . . . 100viTable of Contents6 Cross-section Extraction and Validation . . . . . . . . . . . . 1046.1 Iterative Bayesian Unfolding . . . . . . . . . . . . . . . . . . . 1046.1.1 Background Treatment . . . . . . . . . . . . . . . . . . 1056.1.2 Simultaneous Unfolding . . . . . . . . . . . . . . . . . 1086.1.3 The Propagation of the Uncertainties . . . . . . . . . 1086.2 Validations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2.1 Reproducing MC Truth . . . . . . . . . . . . . . . . . 1106.2.2 NEUT-based Fake Data Samples . . . . . . . . . . . . 1116.2.3 GENIE Studies . . . . . . . . . . . . . . . . . . . . . . 1127 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.1 Unfolded Results . . . . . . . . . . . . . . . . . . . . . . . . . 1187.1.1 NEUT Comparisons in Neutrino Interaction Modes . . 1187.1.2 Various Model Comparisons . . . . . . . . . . . . . . . 1247.2 Extracted Total Cross Sections . . . . . . . . . . . . . . . . . 1298 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132A The Selection Efficiencies . . . . . . . . . . . . . . . . . . . . . 144B The List of Model Parameters . . . . . . . . . . . . . . . . . . 148C Sideband Distributions . . . . . . . . . . . . . . . . . . . . . . . 154D Reproducing MC Truth . . . . . . . . . . . . . . . . . . . . . . 157E NEUT-based Fake Data Studies . . . . . . . . . . . . . . . . . 161E.1 Various CCQE Fractions . . . . . . . . . . . . . . . . . . . . . 161E.2 No 2p2h Components . . . . . . . . . . . . . . . . . . . . . . 168F Covariance Matrices . . . . . . . . . . . . . . . . . . . . . . . . 172G Correlation Matrices . . . . . . . . . . . . . . . . . . . . . . . . 188H Various Model Comparisons . . . . . . . . . . . . . . . . . . . 192H.1 Different Models of Fermi Motion . . . . . . . . . . . . . . . . 192H.2 Different Models of FSI . . . . . . . . . . . . . . . . . . . . . 196H.3 RFG Models in Different Generators . . . . . . . . . . . . . . 200viiList of Tables2.1 Best-fit values of the oscillation parameters . . . . . . . . . . 153.1 MC generators comparisons . . . . . . . . . . . . . . . . . . . 395.1 MC/data POT . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2 Number of selected events and signals of topologies with aµTPC track . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3 Number of selected events and signals of µFGD+pTPC topology 855.4 Background compositions . . . . . . . . . . . . . . . . . . . . 895.5 Michel electron tagging efficiency in FGD1 . . . . . . . . . . . 985.6 Fractional uncertainties the CC0piNp selection from each de-tector systematic source . . . . . . . . . . . . . . . . . . . . . 996.1 Mean bias and χ2 over broad range of CCQE variation inNEUT MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2 Mean bias and χ2 . . . . . . . . . . . . . . . . . . . . . . . . . 1116.3 The χ2 value of each variable with the chosen number of it-eractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.4 The extracted and true cross-sections . . . . . . . . . . . . . . 1147.1 The χ2 of the results unfolded with NEUT, relative to variousMC predictions . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.2 The χ2 of the results unfolded with GENIE, relative to variousMC predictions . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.3 The extracted cross-sections . . . . . . . . . . . . . . . . . . . 1297.4 Total uncertainties for each source . . . . . . . . . . . . . . . 129B.1 List of the flux model parameters . . . . . . . . . . . . . . . . 149B.2 List of the cross-section model parameters (FIS) . . . . . . . 151B.3 List of the cross-section model parameters (Nuclear and CC-QE/CCRES interaction models) . . . . . . . . . . . . . . . . 152viiiList of TablesB.4 List of the cross-section model parameters (Other interactionmodels) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153ixList of Figures2.1 Particles in the Standard Model . . . . . . . . . . . . . . . . . 52.2 Solar neutrino result . . . . . . . . . . . . . . . . . . . . . . . 82.3 Atmospheric neutrino result . . . . . . . . . . . . . . . . . . . 92.4 Two different mass hierarchy . . . . . . . . . . . . . . . . . . 143.1 Neutrino-electron scattering Feynman diagram . . . . . . . . 193.2 Cross sections for different neutrino interactions . . . . . . . . 213.3 CCQE Feynman diagram . . . . . . . . . . . . . . . . . . . . 223.4 Resonance pion production Feynman diagram . . . . . . . . . 233.5 Energy spectrum of the electron scattering off H2O . . . . . . 243.6 Differential cross section of electron scattering . . . . . . . . . 253.7 Comparison between the SF and FG models of the differentialcross section for electron scattering . . . . . . . . . . . . . . . 263.8 CCQE cross section measurements on carbon as a function ofneutrino energy . . . . . . . . . . . . . . . . . . . . . . . . . . 333.9 Martini’s np-nh computation compared to the MiniBooNE data 343.10 Nieves np-nh computation compared to the MiniBooNE data 353.11 Multi-Nucleon prediction for different neutrino energies . . . 363.12 Comparison of T2K data to oscillated and unoscillated MCspectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.13 The double differential cross sections of CCQE-like events oncarbon from T2K . . . . . . . . . . . . . . . . . . . . . . . . . 413.14 MINERνA CCQE differential cross section measurement . . . 423.15 The differential cross sections of CCQE-like interactions onvarious nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.16 Inclusive CC cross section measurement . . . . . . . . . . . . 443.17 The double differential cross sections of neutrino-carbon in-teractions from MINERνA . . . . . . . . . . . . . . . . . . . . 453.18 CCQE cross section measurement . . . . . . . . . . . . . . . . 463.19 MINERνA resonance pion production cross-section . . . . . . 473.20 Differential cross section from MiniBooNE and MINERνA . . 47xList of Figures3.21 Resonance pion production cross-section measurements . . . . 484.1 Neutrino flux for difference off-axis angles . . . . . . . . . . . 494.2 Simplified schematic of the T2K beamline . . . . . . . . . . . 504.3 History of protons on target . . . . . . . . . . . . . . . . . . . 514.4 The neutrino flux predictions at ND280 for different parentdecay modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.5 An overview of INGRID . . . . . . . . . . . . . . . . . . . . . 534.6 A single INGRID module . . . . . . . . . . . . . . . . . . . . 544.7 Daily event rate and beam profile on INGRID . . . . . . . . . 544.8 All elements in ND280 . . . . . . . . . . . . . . . . . . . . . . 554.9 A single TPC . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.10 TPC momentum resolution . . . . . . . . . . . . . . . . . . . 574.11 Cross-section of a single scintillating bar for FGDs . . . . . . 584.12 The scintillating bars for P0D . . . . . . . . . . . . . . . . . . 594.13 Sliced view of a SMRD slab . . . . . . . . . . . . . . . . . . . 614.14 The DAQ system architecture . . . . . . . . . . . . . . . . . . 624.15 The electronic system architecture . . . . . . . . . . . . . . . 634.16 Structure of the Super-Kamiokande detector . . . . . . . . . . 644.17 Example event displays at SK . . . . . . . . . . . . . . . . . . 654.18 Wave-length shifting fiber . . . . . . . . . . . . . . . . . . . . 684.19 The absorption and emission spectra of WLS fibers . . . . . . 694.20 The MPPC sensitivity as a function of wavelength . . . . . . 694.21 A photo of the MPPCs with the WLS fibers . . . . . . . . . . 704.22 The MPPC gain . . . . . . . . . . . . . . . . . . . . . . . . . 704.23 dE/dx measurements in FGDs . . . . . . . . . . . . . . . . . 724.24 Momentum measurements in FGDs . . . . . . . . . . . . . . . 734.25 TPC laser pattern . . . . . . . . . . . . . . . . . . . . . . . . 754.26 TPC clustering scheme . . . . . . . . . . . . . . . . . . . . . . 764.27 TPC track fitting scheme . . . . . . . . . . . . . . . . . . . . 774.28 dE/dx measurement in TPCs . . . . . . . . . . . . . . . . . . 785.1 Event topologies . . . . . . . . . . . . . . . . . . . . . . . . . 825.2 Selection efficiencies . . . . . . . . . . . . . . . . . . . . . . . 865.3 Selection efficiencies (limited p phase-space) . . . . . . . . . . 875.4 Event populations for CCQE, CCRES and 2p2h as functionof µ p-θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.5 The muon phase-space bins . . . . . . . . . . . . . . . . . . . 885.6 Analysis variables . . . . . . . . . . . . . . . . . . . . . . . . . 905.7 TPC momentum resolution of data and MC . . . . . . . . . . 94xiList of Figures5.8 TPC PID pull mean and sigma of data and MC . . . . . . . . 955.9 TPC2 track reconstruction efficiency . . . . . . . . . . . . . . 965.10 FGD track reconstruction efficiency . . . . . . . . . . . . . . . 975.11 FGD1-TPC2 matching efficiency . . . . . . . . . . . . . . . . 985.12 Reconstructed distributions of ∆pp in reconstructed muonphase-space bins . . . . . . . . . . . . . . . . . . . . . . . . . 1015.13 Reconstructed distributions of ∆θp in reconstructed muonphase-space bins . . . . . . . . . . . . . . . . . . . . . . . . . 1025.14 Reconstructed distributions of |∆pp| in reconstructed muonphase-space bins . . . . . . . . . . . . . . . . . . . . . . . . . 1036.1 Purity correction diagram . . . . . . . . . . . . . . . . . . . . 1066.2 The scheme of using the sideband . . . . . . . . . . . . . . . . 1076.3 Simultaneous unfolding diagram . . . . . . . . . . . . . . . . 1096.4 Mean bias and χ2/ndof over the numbers of iterations . . . . 1136.5 Unfolded GENIE with NEUT MC (∆pp) . . . . . . . . . . . . 1156.6 Unfolded GENIE with NEUT MC (∆θp) . . . . . . . . . . . . 1166.7 Unfolded GENIE with NEUT MC (|∆pp|) . . . . . . . . . . . 1177.1 Unfolded results compared to NEUT in interaction modes (∆pp)1217.2 Unfolded results compared to NEUT in interaction modes (∆θp)1227.3 Unfolded results compared to NEUT in interaction modes(|∆pp|) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.4 Unfolded results (∆pp) compared to various models . . . . . . 1267.5 Unfolded results (∆θp) compared to various models . . . . . . 1277.6 Unfolded results (|∆pp|) compared to various models . . . . . 128A.1 Efficiencies for each µ bin (∆pp) . . . . . . . . . . . . . . . . 145A.2 Efficiencies for each µ bin (∆θp) . . . . . . . . . . . . . . . . . 146A.3 Efficiencies for each µ bin (|∆pp|) . . . . . . . . . . . . . . . . 147B.1 The covariance matrix of flux systematic . . . . . . . . . . . . 148B.2 The covariance matrix of cross-section systematic . . . . . . . 150C.1 The true distributions of the sideband predicted from NEUT 155C.2 The observed distributions of the sideband compared to theNEUT predictions . . . . . . . . . . . . . . . . . . . . . . . . 156D.1 Reproducing MC truth (∆pp) . . . . . . . . . . . . . . . . . . 158D.2 Reproducing MC truth (∆θp) . . . . . . . . . . . . . . . . . . 159D.3 Reproducing MC truth (|∆pp|) . . . . . . . . . . . . . . . . . 160xiiList of FiguresE.1 -20 % CCQE in NEUT (∆pp) . . . . . . . . . . . . . . . . . . 162E.2 -20 % CCQE in NEUT (∆θp) . . . . . . . . . . . . . . . . . . 163E.3 -20 % CCQE in NEUT (|∆pp|) . . . . . . . . . . . . . . . . . 164E.4 +20 % CCQE in NEUT (∆pp) . . . . . . . . . . . . . . . . . 165E.5 +20 % CCQE in NEUT (∆θp) . . . . . . . . . . . . . . . . . 166E.6 +20 % CCQE in NEUT (|∆pp|) . . . . . . . . . . . . . . . . 167E.7 No 2p2h in NEUT (∆pp) . . . . . . . . . . . . . . . . . . . . 169E.8 No 2p2h in NEUT (∆θp) . . . . . . . . . . . . . . . . . . . . . 170E.9 No 2p2h in NEUT (|∆pp|) . . . . . . . . . . . . . . . . . . . . 171F.1 Covariance matrix I (∆pp) . . . . . . . . . . . . . . . . . . . . 173F.2 Covariance matrix II (∆pp) . . . . . . . . . . . . . . . . . . . 174F.3 Covariance matrix III (∆pp) . . . . . . . . . . . . . . . . . . . 175F.4 Covariance matrix IV (∆pp) . . . . . . . . . . . . . . . . . . . 176F.5 Covariance matrix V (∆pp) . . . . . . . . . . . . . . . . . . . 177F.6 Covariance matrix I (∆θp) . . . . . . . . . . . . . . . . . . . . 178F.7 Covariance matrix II (∆θp) . . . . . . . . . . . . . . . . . . . 179F.8 Covariance matrix III (∆θp) . . . . . . . . . . . . . . . . . . . 180F.9 Covariance matrix IV (∆θp) . . . . . . . . . . . . . . . . . . . 181F.10 Covariance matrix V (∆θp) . . . . . . . . . . . . . . . . . . . 182F.11 Covariance matrix I (|∆pp|) . . . . . . . . . . . . . . . . . . . 183F.12 Covariance matrix II (|∆pp|) . . . . . . . . . . . . . . . . . . 184F.13 Covariance matrix III (|∆pp|) . . . . . . . . . . . . . . . . . . 185F.14 Covariance matrix IV (|∆pp|) . . . . . . . . . . . . . . . . . . 186F.15 Covariance matrix V (|∆pp|) . . . . . . . . . . . . . . . . . . 187G.1 Correlation matrix (∆pp) . . . . . . . . . . . . . . . . . . . . 189G.2 Correlation matrix (∆θp) . . . . . . . . . . . . . . . . . . . . 190G.3 Correlation matrix (|∆pp|) . . . . . . . . . . . . . . . . . . . 191H.1 The differential cross sections in ∆pp compared to variousmodels of Fermi motion . . . . . . . . . . . . . . . . . . . . . 193H.2 The differential cross sections in ∆θp compared to variousmodels of Fermi motion . . . . . . . . . . . . . . . . . . . . . 194H.3 The differential cross sections in |∆pp| compared to variousmodels of Fermi motion . . . . . . . . . . . . . . . . . . . . . 195H.4 The differential cross sections in ∆pp compared to modelswith various FSI strength . . . . . . . . . . . . . . . . . . . . 197H.5 The differential cross sections in ∆θp compared to modelswith various FSI strength . . . . . . . . . . . . . . . . . . . . 198xiiiList of FiguresH.6 The differential cross sections in |∆pp| compared to modelswith various FSI strength . . . . . . . . . . . . . . . . . . . . 199H.7 The differential cross sections in ∆pp compared to RFG mod-els in various generators . . . . . . . . . . . . . . . . . . . . . 201H.8 The differential cross sections in ∆θp compared to RFG mod-els in various generators . . . . . . . . . . . . . . . . . . . . . 202H.9 The differential cross sections in |∆pp| compared to RFGmodels in various generators . . . . . . . . . . . . . . . . . . . 203xivGlossary1p1h One-particle One-hole: An electromagnetic or weak interaction ona single nucleon in a nucleus, in which one particle is knocked out, leavinga hole in the nucleus.2p2h Two-particle Two-hole: An electromagnetic or weak interaction ontwo-nucleon in a nucleus, in which two particles are knocked out, leavingtwo holes in the nucleus as opposed to 1p1h.ANL Argonne National Laboratory.APD Avalanche photodiode: A semiconductor electronic device whichconverts light to electricity. It can be fired by a single photon and amplifiesthe resulting photocurrent when a reverse voltage is applied.BNL Brookhaven National Laboratory.CC Charged-current: A weak interaction mediated by W± bosons.CC0pi Neutrino interactions that leave one muon and no pion in the finalstate. The number of outgoing nucleons is not considered.CC0piNp Neutrino interactions that leave one muon, no pion and anynumber of protons in the final state.CKM Cabibbo-Kobayashi-Maskawa matrix: The matrix describing themixing between quarks.CP Charge Parity: The product of two symmetries, which are chargeconjugation (transforms a particle into its antiparticle) and parity (invertsthe spatial coordinates).xvGlossaryCCQE Charged-current quasi-elastic scattering: A charged-current weakinteraction that an incident neutrino interacts with a nucleon resulting inan outgoing charged lepton and a nucleon in the final state.CCRES Charged-current resonant single pion production: A charged-current weak interaction that an incident neutrino interacts with a nucleonresulting in an outgoing charged-lepton, a nucleon, and a pion from a baryonresonance in the nucleus.CTM Cosmic trigger modules: An electronic module of ND280 triggeringsystem to trigger on cosmic rays.DAQ Data acquisition.DIS Deep inelastic scattering: An interaction on a constituent quark of anucleon that emits many new particles in the final state.ECal Electromagnetic calorimeter: A subdetector of ND280 that assiststo detect particle exiting the detectors located at the center of ND280.FG Fermi gas model: A nuclear model that treats nucleons in a nucleusas two independent degenerate Fermi gas systems (neutrons and protons)and assumes the nucleons can freely move inside the nucleus.FGD Fine-Grained Detector: The central part of ND280 which providestarget mass and tracking. There are two FGDs.FPN Front-end processor node: Electronics to communicate the data ac-quisition system and the ND280 subdetectors.FSI Final state interaction: An interaction between a particle emergingfrom a lepton-nucleus interaction with the nuclear medium that occurs be-fore it exits the target nucleus.FV Fiducial volume: The region of a detector where the reconstructionand event selection are reliable.GENIE Generate Events for Neutrino Interaction Experiments: A neu-trino event generator.xviGlossaryGFMC Green’s function Monte Carlo: A Monte Carlo calculation ofground states of nuclei.IA Impulse approximation: An incoherent sum of interactions on a singlenucleon in a nucleus to calculate a total cross section.INC Intra-nuclear cascade: A model to describe final state interactions.INGRID Interactive Neutrino GRID: The on-axis near detector for T2K.J-PARC Japan Proton Accelerator Research Center.LINAC Linear accelerator of J-PARC.MC Monte Carlo simulation.MCM Master Clock Module: A electronic board within the ND280 dataacquisition system that receives timing from the accelerator complex, ac-knowledges trigger requests, and provides a master clock signal for the elec-tronics.MSW Mikheyev-Smirnov-Wolfenstein (MSW) effect: The matter effectwhich causes an additional phase change in neutrino oscillation due to neu-trino interactions on electrons in matter.MPPC Multi-pixel photon counter: A photosensor that detects opticallight produced by particles passing through the scintillating bar detectorsin ND280.MUMON Muon monitor: A detector just downstream of the T2K beamline that detects the muons produced in the beam line to monitor the sta-bility of the beam line.NC Neutral-current: A weak interaction mediated by Z boson.ND280 The near detector at 280 m: The off-axis near detector for T2K.NEUT A neutrino event generator used for T2K experiment.xviiGlossaryNuWro A neutrino event generator created at the University of Wroclaw.P0D pi0 detector: A subdetector of ND280 to measure pi0 events.PID Particle identification: Determination of a particle type from detectorobservables.PMNS Pontecorvo-Maki-Nakagawa-Sakata matrix: The matrix describ-ing the mixing between flavor and mass states of the neutrino.PMT Photomultiplier tube: A photosensor used at SK to detector Cˇerenkovlight.POT Protons on target: The number of protons delivered to the T2Kneutrino beam line.RMM Readout merger module: Back-end electronics for the P0D, ECals,and SMRD subsystems of ND280.RPA Random phase approximation: A correction added to a nuclearmodel to account for the impact of the nuclear medium as the struck nucleonpropagates.SCM Slave Clock Module: A system that receives a trigger from MCMand distributes the trigger to the subdetectors of ND280.SF Spectral function: A nuclear model using non-relativistic Hamiltonianof nucleons in a nucleus, which includes nucleon-nucleon correlations.SK Super-Kamiokande detector: A large water Cˇerenkov detector used asthe far detector for T2K.SMRD Side Muon Range Detector: A subdetector of ND280 to vetomuons from rocks surrounding the ND280 hall.SRC Short range correlation: A correlation between pairs of initial statenucleons.xviiiGlossaryTEM Transverse Enhancement Model: A parametrization to account forthe enhancement in the transverse response function due to 2p2h.TFB TripT front-end board: Front-end electronics for the P0D, ECals,and SMRD subsystems of ND280.TPC Time Projection Chamber: Gas-filled detectors that are part of themain tracking system of ND280.WLS Wavelength shifting: Wavelength shifting fibers are used to shift thewavelengths of emitted scintillating lights to which MPPCs are sensitive.xixAcknowledgementsHow many helpless days had passed! How many sleepless nights had passed!It was a curse and a bliss. If I knew what I know now, then could my lifehave been better, worse, or just different? Being unable to see the end ofthe road, there were critical moments when I nearly gave up. I would nothave been able to complete, if there were all the supports.The greatest thank for my completion is sure to my supervisor, HiroTanaka. His knowledge and insight of physics never failed to guide mewhenever or wherever I lost. However, more than that, without his greatpatience, I would have never been able to finish this as his student.T2K-Canada, it has been a previlege to be in the great team. All thenames of T2K-Canada, in the past and present, will remain in the memory,but thre are a few names that I would like to particularly thank to: TomFeusels provided endless advices and insights on the analysis. Kendall Mahn,even after she left T2K-Canada, helped me not get lost or distracted, andpush the analysis forward. Mike Wilking and Thomas Lindner were the onesI should look for whenever I came up with practical questions.My analysis strongly relies on many different collaborated works:• The event selection inherited from the official numu selection was doneby Valencia group: Laura Monfregola, Anselmo Cervera, and Alexan-der Izmaylov. In particular, without Alexander Izmaylov, relative sys-tematic evaluation could have been almost impossible.• xsTool. Nevertheless, Martin Hierholzer gave it a birth, it was TianluYuan and Enrico Scantamburlo who raised it. Without their works,this analysis could have taken forever.• highland. Anselmo Cervera and Alexander Izmaylov, without theirparenting, I should have done all the complicate and detailed worksby hands from the scratch.• Sharing analysis with Stephen Dolan helped me not to be stuck in aloophole.xxAcknowledgementsAnd, my people, those whom I have come across during the journey,either at work or outside, those who colored my life with beautiful chaos:Elder, Javi, Lorena, Panos, Raquel, Sujeewa; Hanna, Jay, Jinhee Daniel,Kwangjin, Soo; and many other names I cannot think of at the very moment.For many of you, soon I will be just someone you used to know and adistant memory. But life is a series of accidents. I wish I will see you allagain soon on another occasion. So long, and thanks for all the love.xxiDedicationTo may parentsxxiiChapter 1IntroductionOver the past few decades, neutrino physics has expanded to include numer-ous experimental and theoretical efforts. While neutrinos were first assumedmassless in the Standard Model, it was proved that neutrinos can changeflavor through the oscillation process, and hence have mass.The history of neutrinos from their first theoretical postulation to its in-clusion in the Standard Model will be explained in Chapter 2. Subsequently,anomalies that were evidenced and now known to be the first indicationsof neutrino oscillations will be discussed, along with the physics of neutrinooscillations. Recent results from various neutrino oscillation experiments areincluded as well.In Chapter 3, neutrino interactions will be discussed. After discussingneutrino interactions on various targets, theoretical models and calculationswill be introduced, including traditional treatments and new approachesmotivated by discrepancies in recent neutrino cross-section measurements.Various nuclear effects and potential impacts on the measurements are in-cluded. Discussions on neutrino interaction generators and recent cross-section measurements will be also shown.Chapter 4 describes the Tokai-to-Kamiokande (T2K) experiment. Thebeam facilities (J-PARC), the near detector complexes (ND280 and INGRD)and the far detector (SK) are described including the detector design andoperating principles. This is followed by more detailed descriptions of themain tracking detectors of ND280 which are crucial for the analysis of thisdissertation.Chapter 5 details the event selection of neutrino interactions at ND280including selection efficiencies, potential backgrounds and relevant system-atics. Analysis variables are defined and the distributions are shown.In Chapter 6, the method of cross-section extraction is explained andvalidated.The extracted differential cross-section of the ND280 data will be shownand summarized in Chapter 7 and concluded in Chapter 8.1Chapter 2Neutrino Physics2.1 History of NeutrinosThe neutrino, named by Enrico Fermi, means “a little neutral object”. Asits name suggests, it was believed to be a chargeless fundamental particle.It was theoretically postulated for the first time by Pauli as a solution tomissing energy in beta decays. Pauli named it a “neutron”, but years laterChadwick found another neutral particle and called it a “neutron” as well.The postulated particle was renamed as a “neutrino” by Enrico to distin-guish from the massive particle Chadwick discovered.Beta decay is a radioactive process in which a nucleus of atomic numberZ transforms to a nucleus of atomic number (Z + 1) emitting an electronas:N(Z,A)→ N ′(Z + 1, A) + e− (2.1)where N and N′are two different nuclei. Then, by the energy conservationlaw, the released electron energy should be fixed when the three masses arespecified following the formula:Ee =m2N −m2N ′ +m2e2mN. (2.2)However, the observed spectrum was continuous, not constant, and noteven discrete [2]. To explain this spectrum of electron energy while satisfyingthe conservation law, Pauli thought that there should be another undetectedparticle in the final state and he postulated this neutral particle. In 1933,Fermi introduced a theory including this postulated particle, which is calledFermi’s interaction or Fermi’s theory, to explain beta decay of a neutron.He suggested direct couplings of a neutron with an electron, a neutrino anda proton. In his theory, the range of the force is zero, therefore the particlesinteract directly at a single point. Its interaction strength is characterizedby GF . The most exact measurement of GF is from the muon decay, which22.1. History of Neutrinoscan be also described as a 4-fermion coupling as:µ− → e− + ν¯e + νµ. (2.3)This was a precursor of the modern theory of the weak interaction [3, 4]. Itdescribed the weak interaction well in the relatively low-energy range suchas β-decay and µ decay. Later, it was replaced with the weak interaction inthe Standard Model mediated by the intermediate weak bosons, W± and Z.The coupling constant of the weak interaction g is directly related to GF as:GF√2=g28M2W(2.4)where MW is the mass of W boson.In 1956, Reines and Cowan confirmed the existence of this particle, whichwas identified later as the electron anti-neutrino (νe), one of six neutrinosand corresponding anti-neutrinos [5]. It was the first direct observation ofneutrinos in history. They used the inverse beta decay interaction in theirexperiment:νe + p→ e+ + n. (2.5)At this time, this neutrino was not yet named as electron anti-neutrino,which is now known to be coupled to a positron, as the concept of leptonflavor had not been developed yet. These neutrinos interact with protons,then turn into positrons and neutrons in the liquid scintillating detector witha target of water and CdCl2. The produced positrons encounter electronsand produce gamma rays from the annihilation. Each gamma ray scattersoff an electron which results in ionization, which produces a large numberof UV photons. The scintillator absorbs these UV photons and emits visiblelight which was detected by photomultiplier tubes. In addition, delayedphotons from the capture of the neutrons on Cadmium were also detected.In 1962, L. Lederman, M. Schwartz and J. Steinberger detected neutrinosproduced from pion decay as follows:pi+ → µ+ + ν (2.6)pi− → µ− + ν. (2.7)32.2. The Standard ModelUsing about 1014 anti-neutrinos from pi− decay, the physicists investigatedtwo reactions:ν + p→ µ+ + n (2.8)ν + p→ e+ + n. (2.9)If only one type of neutrino exists, the two reactions should happen equally.But they did not detect positrons. The absence of positron events suggestedthat the neutrinos produced by the pion decays in Equations 2.6 and 2.7were different from those produced by reactors, and coupled only with muonsand not electrons. This was the first observation of the muon neutrino(νµ) [6].Since it was found that there were two separate types of neutrinos (onecoupled to an electron and the other to a muon), a third neutrino wasinferred after the existence of the third lepton τ was confirmed in 1970s. Itwas named the tau neutrino (ντ ) and observed finally in 2000 by the DONUTexperiment [7]. ντ was particularly challenging to detect, because τ decaysrapidly, making the reconstruction of its track difficult. DONUT identified τby the kink formed when τ decays and other particles are emitted. DONUTused the nuclear emulsion technique which allowed to track τ with highresolution. Since no additional charged leptons have been observed, threegenerations of leptons have become widely accepted. Therefore, three speciesof fermions complete the Standard Model as shown in Figure 2.1.2.2 The Standard ModelThe Standard Model is a theory of particle physics that concerns propertiesof fundamental particles and forces in the nature. In the Standard Model,neutrinos are spin 1/2 fermions which do not have charge, colour or mass.Therefore, they only interact via the weak interactions at tree level. Unlikeother fundamental forces in the Standard Model, the weak interaction ischiral. Chirality, sometimes called handed-ness, is an intrinsic propertyof a particle associated with the fifth gamma matrix γ5. Right- and left-handed fields are eigenstates of γ5 with eigenvalues of ±1. In the weakinteractions, the charged weak gauge bosons (W±) only act on left-handedfields. Therefore only left-handed neutrinos are expected. In this case,neutrinos cannot have a Dirac mass term, since this requires a right-handedfield [9]. In the language of field theory, the left-handed leptons, which are42.2. The Standard ModelFigure 2.1: Fundamental particles and forces in the Standard Model areshown [8]. There are three generations of fermions (quarks (purple) andleptons (green)) and four bosons (red) which mediate fundamental forces.charged leptons and neutrinos, can be expressed as weak isospin doublets,while the right-handed leptons, which are only charged leptons, are singletsas [10]:EL =(lLνL), lR .52.2. The Standard ModelThe electroweak Lagrangian in the Standard Model is written as:LSM = Lgauge +Lkinetic +LEW +LHiggs +LY ukawa. (2.10)Lgauge and Lkinetic are kinematic terms of gauge fields and fermions. LEWcouples the gauge fields to the fermions. LHiggs describes the Higgs fields,φ, with a potential as [11]:V (φ) = µ2φ†φ+ λ(φ†φ)2. (2.11)LY ukawa describes how the fermion fields couple to the Higgs fields. It iswritten as:−LY ukawa = Y lijE LiφlRj + h.c. (2.12)where L/R denote left-handed and right-handed and i/j denote differentflavor of leptons. The Lagrangian includes its Hermitian conjugate (h.c.)to make the Lagrangian Hermitian. Here Y is a non-diagonal coefficientmatrix. After spontaneous symmetry breaking, when the Higgs field obtainsa non-zero vacuum expectation value v, the Higgs field is rewritten as:φ(x) =1√2(0v + h(x))(2.13)where v is a constant and h(x) describes the fluctuations of the field aroundthe ground state. Then, the Yukawa couplings break into terms with andwithout dependence in h(x):−LY ukawa = vY lijlLilRj + h.c.+ (couplings to h(x))= (ml)ijlLilRj + h.c.+ (couplings to h(x)). (2.14)A fermion mass term requires left and right-handed components of a field, asshown in the second line of Equation 2.14. The part of the Yukawa couplingsinvolving v describes masses of charged leptons. However, no mass term canbe formed for neutrinos in a similar way, because there is no right-handedneutrino in the Standard Model.Generally, a mass matrix ((ml)ij in Equation 2.14) is non-diagonal. Butit can be diagonalized by introducing transformation matrices, VlL and V†lR.Since right-handed neutrinos are not included in this theory, there is a free-dom to choose the transformation matrix V for neutrinos such that Vl = V†ν .62.3. Evidence of Neutrino Flavor ChangeThat means the couplings of leptons and weak interaction gauge bosons areisolated for each flavor after transformation. This is why neutrino flavormixing is not allowed in the Standard Model.The masslessness of neutrinos has been questioned since Fermi developedhis theory of neutrinos. Several experiments have attempted to measure themass of neutrinos directly via beta decays, pion decays, and tau lepton de-cays. However, they did not find any evidence of neutrino mass. Anotherway to probe neutrino mass is to search for neutrino flavor change. Sinceneutrino flavor mixing is forbidden with massless neutrinos as explainedabove, any evidence of flavor mixing may be an indication of massive neu-trinos. In the following section, evidence of neutrino flavor mixing observedat various experiments will be discussed in details.2.3 Evidence of Neutrino Flavor ChangeSolar Neutrino ProblemIn the 1960s, calculations detailing the expected flux and spectra of electronneutrinos produced in solar fusion processes were done [12]. However, themeasured electron neutrino flux traveling to the Earth was less than theprediction by the model. This is the solar neutrino problem.The Homestake experiment measured the solar neutrino flux and com-pared the result to the prediction. They concluded that the flux was onlyone-third of the prediction [13]. This was the first measurement of the solarneutrino problem. Afterward, the Kamiokande experiment [14] using a largewater Cˇerenkov detector, SAGE [15] and GALLEX [16] using liquid Galliumpublished similar results of the deficit in the solar electron neutrino flux.As those experiments were primarily sensitive to electron neutrinos, theyonly saw a fraction of the total flux. Later SNO measured the neutrinoflux via charged-current, neutral-current and elastic scattering channels [18],hence it could measure the neutrino flux regardless of flavor. It is confirmedthat the total neutrino flux summing over all flavors is consistent with theSSM expectation as shown in Figure 2.2 [17], but the neutrino flavors musthave changed.Atmospheric Neutrino ProblemPrimary cosmic rays entering the Earth’s atmosphere are mostly made ofprotons (about 90 %). When they enter the atmosphere, hadronic interac-tions produce hadrons such as pions and kaons which then decay to produce72.3. Evidence of Neutrino Flavor ChangeFigure 2.2: Solar neutrino result. The x axis is the flux of νe and the yaxis is the flux of νµ or ντ from SNO measurements. The red band is theSNO CC result. The blue is the SNO NC result and the light green is theSNO elastic scattering result. The dark green band is he Super-Kamiokandeelastic scattering result. The bands represent the 1σ error. The sum of theneutrino fluxes is consistent with the SSM expectation (dashed line). Plotis taken from [17] with permission.neutrinos. The pions decay into µ and νµ, then subsequently the µ decayinto e, νe, and νµ. These neutrinos are called atmospheric neutrinos.Since the dominant interaction producing neutrinos is the decay chainpi+ → µ+ + νµ and subsequently µ+ → e+ + νµ + νe (along with the chargeconjugate channels), the amount of νµ and νµ is approximately twice asmuch as νe. However, the ratio of (νµ + νµ)/(νe + νe) was measured to besmaller than the prediction, which implied a deficit of νµ flux. This is calledthe atmospheric neutrino problem [19].It was solved by the Super-Kamiokande experiment, which not only sep-arated νµ and νe, but also measured the direction of the neutrinos. An angu-lar dependence in the deficit of νµ was observed as shown in Figure 2.3 [20],which shows the zenith angles (the angle between the zenith and center ofthe Sun) separately for the observed νµ and νe interactions. For cos θ = 1,82.3. Evidence of Neutrino Flavor Changeneutrinos travel∼ 10 km, while for cos θ = −1, neutrinos travel∼ 10000 km.The zenith angle dependence demonstrated that the muon neutrino flux dif-fered depending on how far the neutrinos had traveled from production tointeraction.Figure 2.3: The numbers of neutrino events are shown as a function ofthe zenith angle. Non-oscillated Monte Carlo predictions are in the dottedhistograms and the best-fit expectations for νµ to ντ oscillations are in thesolid histograms. Plot is taken from [20] with permission.92.4. Neutrino Oscillations2.4 Neutrino OscillationsThe solar and atmospheric neutrino problems discussed in the previoussection were resolved by neutrino oscillations resulting in neutrino flavorchange. Neutrino oscillations are a consequence of non-degenerate neutrinomasses and flavor mixing. In this section, neutrino mixing and oscillationphysics will be discussed.Since the neutrino mass eigenstates can be different from the flavor eigen-states, the neutrino flavor states ( |να〉, α = e, µ, τ ) can be written as a linearsuperposition of mass eigenstates ( |νi〉, i = 1, 2, 3 ):|να〉 =∑U∗αi|νi〉 (2.15)where U is a unitary matrix. This mixing matrix is referred to as thePontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [21], similar to the quarkmixing matrix, Cabibbo-Kobayashi-Maskawa (CKM) matrix [22]. Whilethere are many ways to parametrize the PMNS matrix. it is usually writtenin the following form:U = 1 0 00 c23 s230 −s23 c23 c13 0 s13e−iδ0 1 0−s13e−iδ 0 c13 c12 s12 0−s12 c12 00 0 1= c12c13 s12c13 s13e−iδ−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13 (2.16)where cij = cos θij , sij = sin θij , and δ is a phase. There are two additionalphases, if neutrinos are Majorana particles. The Majorana neutrinos will bediscussed later.If a neutrino is created at t = 0 as a |να(t = 0)〉, its state after time tcan be expressed as (assuming c = 1 in the following):|να(t)〉 =∑Uiβe−i(Eit−piL)U∗αi|νβ〉 (2.17)where L is the distance the neutrino travels during time t. Ei and pi are theenergy and momentum of the neutrino mass eigenstate i.Since neutrinos travel close to the speed of light and so are relativis-tic, one can approximate t ≈ L. In addition, neutrino masses are smallenough that one can approximate E =√p2i +m2i ≈ pi+m2i /2pi. Therefore,102.4. Neutrino OscillationsEquation 2.17 can be rewritten as:|να(L)〉 =∑Uiβe−i(pi+m2i /2pi−pi)LU∗αi|νβ〉 =∑Uiβe−im2iL/2piU∗αi|νβ〉.(2.18)Lastly, the energies are assumed to be all the same, Ei = E:|να(L)〉 =∑Uiβe−im2iL/2EU∗αi|νβ〉. (2.19)From Equation 2.19, the probability for the flavor change from να to νβ is:Prob(να → νβ) = |〈νβ|να(t)〉|= δαβ − 4∑Re(U∗αiUβiUαjU∗βj) sin2∆m2ijL4E−2∑i>jIm(U∗αiUβiUαjU∗βj) sin∆m2ijL2E(2.20)where ∆m2ij = m2i − m2j . For anti-neutrinos, the imaginary term has theopposite sign.There are seven oscillation parameters: three mixing angles, three neu-trino mass masses, and a single phase, δ. If the phase δ is non-zero, thematrix U (Equation 2.16) for neutrinos and anti-neutrinos will be different(opposite sign of δ), hence the oscillation probabilities of CP conjugate chan-nels. As this CP asymmetry arises from δ, it is called the CP phase (δCP ).Among three neutrino mass-squared splitting terms, only two of them areindependent: ∆m221 + ∆m231 + ∆m232 = 0. Therefore, there are six inde-pendent oscillation parameters, which are ∆m232, ∆m221, θ12, θ23, θ13, andδCP .∆m221 and θ12 are associated with the oscillations which result in thesolar neutrino deficit, therefore sometimes these parameters are called solarneutrino parameters. Likewise, ∆m232 and θ23 are atmospheric neutrino pa-rameters related to the atmospheric neutrino deficit. These parameters canbe measured in accelerator-based (νµ beam) and reactor-based (νe beam)experiments through the νµ disappearance and νe appearance with the prob-abilities as:112.4. Neutrino OscillationsProb(νµ → νµ) ' 1− sin2 2θ23 sin2 ∆32 (2.21)Prob(νµ → νe) ' sin2 2θ13 sin2 θ23 sin2[(1− x)∆31](1− x)2+∣∣∣∆m221∆m231∣∣∣ sin 2θ13 sin 2θ12 sin 2θ23 sin(x∆31)xsin[(1− x)∆31](1− x)×(− sin δCP sin ∆31 + cos δCP cos ∆31)(2.22)where ∆ij =∆m2ijL4Eν, x = 2√2GFNeEν∆m231. These equations are approximated forbaselines and neutrino energies where ∆m232 is dominant. For anti-neutrinos,the νµ disappearance is the same, but νe appearance has (1 + x) instead of(1 − x) and sin δCP instead of − sin δCP . While ∆m232, ∆m221, θ12 and θ23was being measured at various experiments, θ13 remained unknown as itwas relatively small. In 2011, T2K had first indications of non-zero θ13from νe appearance. Meanwhile, reactor experiments definitively observedthe disappearance of νe from non-zero θ13 in 2012 [23] and have since madeprecision measurements of θ13.Most of the oscillation parameters have been measured with varyingprecision. The only missing pieces are the mass hierarchy and CP violation.Mass Hierarchy and CP ViolationWhile the sign of the solar mass-squared splitting (∆m221) is determined, thesign of the atmospheric mass-squared splitting (∆m232) is unknown. This iscalled mass hierarchy problem, and one of the missing pieces in the under-standing of neutrinos.The sign of ∆m221 was determined using the propagation of neutrinos inmatter which gives an additional phase shift in the oscillation probabilitydue to the interactions between neutrinos and the matter. When electronneutrinos and anti-neutrinos travel in matter, they interact with electronsin matter through the charged weak current, while other flavors of neutrinosonly interact through the neutral weak current. Therefore, there will be aphase difference between the νe component and the other flavors. This iscalled the Mikheyev-Smirnov-Wolfenstein (MSW) effect or simply the mat-ter effect [24]. Including the matter effect, the effective mixing angle andmass-squared splitting in matter can be written as:122.5. Neutrino Oscillation Measurements∆m2M = ∆m2√sin2 2θ + (cos θ −A)2sin2 2θM =sin2 2θsin2 2θ + (cos θ −A)2A =2√2GFNeE∆m2(2.23)where GF is the Fermi constant, Ne is number density of electrons in matter,and E is neutrino energy. Depending on the sign of ∆m2, the effective mixingangle and mass-squared splitting will be different. The phase shift A hasopposite sign for anti-neutrinos. Since the phase shift affects the oscillationprobability, the sign of the mass squared difference can be determined bymeasuring the phase shift. However, it is challenging for ∆m232, because inorder to see the matter effect in the Earth, the baseline should be very long.There are two ways of arranging these neutrino masses depending on thehierarchy, conventionally named as normal (m1 < m2 < m3) and inverted(m3 < m1 < m2) hierarchy, as shown in Figure 2.4. If the normal hierarchyis correct, m3 is larger than m1 so that ∆m231 > 0. However, if the invertedhierarchy is correct, ∆m231 < 0.The other missing piece in the study of neutrino oscillations is the CPviolating phase δCP . As shown in Equations 2.21 and 2.22, the imaginarypart due to the CP phase has opposite sign for neutrinos and anti-neutrinos.That means the physics is not invariant under Charge and Parity (CP) con-jugations. This is called the CP violation. If sin δCP is zero, the imaginarypart becomes zero as well, therefore the oscillation probability of ν and νwill be the same.The sign of the atmospheric mass-squared splitting plays an importantrole to measure δCP , because the matter effect can mimic the CP violationin the oscillation. The phase shift by the matter effect has different signbetween neutrinos and anti-neutrinos, and will give an asymmetry in theoscillation probability independent of the CP violation. Therefore, botheffects should be considered simultaneously.2.5 Neutrino Oscillation MeasurementsOver the last decades, many experiments have measured the neutrino pa-rameters. Solar neutrino experiments have measured θ12 and ∆m221, whileatmospheric neutrino experiments measured θ23 and |∆m232|. In addition,132.5. Neutrino Oscillation MeasurementsFigure 2.4: Two different mass hierarchies. The left is normal hierarchy,and the right is inverted. ν1, ν2 and ν3 are mass eigenstates and coloursrepresent the fraction of each flavor contained in each mass eigenstate (|Uαi|2in Equation 2.16). Image is reused from [25] with permission.KamLAND measured θ12 and ∆m221 as well via νe oscillations from an reac-tor. There are also accelerator neutrino experiments, which have measuredθ23 and |∆m232| from νµ disappearance, and θ13 from νe appearance. Themost recently measured parameter is θ13. Initially, T2K reported indica-tions of non-zero θ13 from νe appearance [26], then reactor experiments(Daya Bay [23], Double CHOOZ [27] and RENO [28]) conclusively foundθ13 to be non-zero and measured it precisely. This brought light to the mea-surement of δCP which remains undetermined with the other problem, themass hierarchy.The most recent global fit over various neutrino oscillation experiments ispresented in Table 2.1. θ12 and ∆m221 have been measured by a combinationof the solar neutrino experiments (SNO and SK) and KamLAND [29]. Theaccelerator-based and reactor-based experiments provided comparable mea-surements of the magnitude of the atmospheric mass-squared splitting [30–32]. sin2 θ23 has been most precisely measured by T2K data.142.6. Future Measurements: Beyond the Standard ModelParameter best fit∆m221(10−5 eV2) 7.37+0.60−0.44|∆m232|(10−3 eV2) 2.50+0.13−0.13 (2.46+0.14−0.13)sin2 θ12 0.297+0.057−0.047sin2 θ23 0.437+0.1790.058 (0.569+0.068−0.186)sin2 θ13 0.0214+0.0032−0.0029 (0.0218+0.0030−0.0032)δCP /pi 1.35+0.64−0.4 (1.32+0.67−0.49)Table 2.1: The best-fit values of the oscillation parameters within three-flavor oscillation framework. The ranges are given at 3σ level except for δCP(2σ). The values in parentheses correspond to inverted mass hierarchy [33].2.6 Future Measurements: Beyond the StandardModelAs discussed in the previous section, many neutrino experiments have mea-sured neutrino oscillation parameters and confirmed that physics shouldmove forward beyond the Standard Model. Because the Standard Modeldoes not explain how neutrinos get their masses and why the masses aresmall with respect to other lepton masses, a new theory incorporating neu-trino mass must be formulated.The minimal extension of the Standard Model to accommodate neutrinomasses is adding right-handed neutrino singlets to the Standard Model La-grangian, so there will be Yukawa couplings as in Equation 2.12:Y lijE LiφNRj (2.24)where NRj is a right-handed neutrino singlet. It will lead to a Dirac massterm as other fermions as:vY lijνLiφNRj . (2.25)152.6. Future Measurements: Beyond the Standard ModelHowever, to get sub-eV masses as neutrino oscillation parameters indi-cate, the Yukawa coupling should be very small (order of 10−13). This isnot consistent with the scale of other particles in the Standard Model.One scenario to explain the small neutrino masses is the see-saw mecha-nism [34, 35] which introduces a set of heavy Majorana neutrinos in additionto the Dirac mass terms. A Majorana neutrino forms a Majorana mass termcoupling to its right-handed anti-particle as follow:MRνRνcR (2.26)where c denotes charge conjugation. The right-handed neutrino is a sin-glet in all the Standard Model gauge groups, so has no conserved charge.Therefore there will be both Dirac mass and Majorana mass terms. How-ever, the Higgs in the Standard Model can only generate Dirac masses. TheMajorana masses could be generated by some new physics beyond the Stan-dard Model at the Grand Unification energy scale (between 1014 and 1019GeV) [36], which have much heavier masses than the Dirac masses. Then,the eigenvalues after diagonalizing the mass matrix will be as:m′L =m2DMR(2.27)where m′L denotes the mass for left-handed neutrinos, m2D is Dirac mass forleft-handed neutrinos, and MR is heavy mass of the right-handed neutrino.The left-handed neutrino mass is suppressed by the heavy right-handed neu-trino mass. Assuming mD ∼ 100 GeV and MR ∼ 1015 GeV, there will beone light Majorana mass (m1 ∼ 10−2 eV like atmospheric mass-squaredsplitting) and one heavy Majorana mass (m2 ∼ 1015 GeV, too heavy to beseen).The Yukawa coupling in Equation 2.24 can violate the CP symmetry,therefore the heavy right-handed neutrino decays to leptons or anti-leptons(along with associated the Higgs fields respectively) with different decayrates. These lead to the lepton number violation which can be convertedto baryon number violation by B-L conserving and B+L violating. TheCP violating decays of the heavy Majorana neutrinos produced in the earlyuniverse could be a source of the asymmetry [37]. This CP violation in aleptonic sector may be a key ingredient to understand the matter-antimatterasymmetry, because CP violation in the quark sector is not enough to explainthe asymmetry.However, the see-saw mechanism is valid only when the neutrinos areMajorana particles. This means they are their own anti-particles. Neu-162.6. Future Measurements: Beyond the Standard Modeltrinoless double beta decay can test if neutrinos are Majorana or Diracfermions [38]. As the decay rate depends on the neutrino masses, the masshierarchy plays a key role to demonstrate the hypothesis. Therefore, theneutrino mass hierarchy also has ramifications for experiments looking forthis process.In addition, there is still the CP violating phase due to the neutrinomixing and oscillation. The accurate measurement of CP violation phase isone of the major goals for future neutrino oscillation experiments.17Chapter 3Neutrino Interactions3.1 Basics of Neutrino InteractionsIn the previous chapter, an overview of neutrino physics was presented in-cluding discussions on the importance of neutrino oscillation measurementsand neutrino properties that are still unknown. To address some of themissing pieces, such as the mass hierarchy and the CP phase, sensitive os-cillation measurements are required. In neutrino oscillation measurements,it is important to identify the flavor and reconstruct the energy of an inci-dent neutrino to construct the neutrino oscillation probability as a functionof neutrino energy. Since experiments use neutrino beams with a mix offlavors and a range of energies, the neutrino flavor and energy of an inter-acting neutrino are not known, but are reconstructed from the products ofthe neutrino interaction. Therefore, neutrino interactions and their crosssections need to be understood well to interpret the oscillation data.Since neutrinos interact via the weak interaction, they can interact througheither W± bosons or Z boson. If a neutrino interacts through W±, it iscalled a charged-current (CC) interaction. If it happens through Z, it isa neutral-current (NC) interaction. After a neutrino interaction occurs, acharged lepton emerges in a CC interaction, while a neutrino emerges (usu-ally undetected) in an NC interaction.The simplest case of neutrino interactions is neutrino-lepton scattering,where the scattering matrix can be accurately evaluated at tree level. Thiswill be discussed in Section 3.1.1. A neutrino also can interact with anucleus. This is more complicated, because a nucleus is a bound systemof nucleons, which in turn is a bound system of quarks. More details ofneutrino-nucleus scatterings will be discussed in Section 3.1.2.3.1.1 Neutrino-Lepton ScatteringNeutrino-lepton scattering is comparatively simple. Figure 3.1 shows a tree-level Feynman diagram of νµ-e scattering, which is an example of neutrino-lepton scattering. In this case, the cross section can be precisely evaluated183.1. Basics of Neutrino Interactionsat tree level by the Standard Model. Therefore, neutrino-electron scatteringmeasurements played a key role in confirming the Standard Model based onthe good agreement of experimental measurements with model predictions.Figure 3.1: A tree-level Feynman diagram of neutrino-electron scattering.The first measurement was done in 1973 by a bubble chamber experi-ment where ν¯µ-e scattering was measured [39]. This was the first time theweak neutral current was directly probed and the electroweak mixing an-gle, θW , was measured. θW is the parameter in the electroweak theory thatcharacterizes the mixing of two neutral electroweak bosons (W 3 and B) toform Z and γ.3.1.2 Neutrino-Nucleus ScatteringAs well as neutrino-lepton scattering, a neutrino can interact with a nucleon.Since nucleons are almost always contained within nuclei except for freeprotons in hydrogen, neutrino-nucleon scattering has to confront nucleareffects. Therefore, in this section, neutrino interactions will be more broadlydiscussed with a picture of neutrino-nucleus scattering.Neutrinos interact with a nucleus by different modes depending on energyand momentum transferred to the nucleus through a weak gauge boson.With very low energy and momentum transfers, the wavelength of the bosonis long so that it probes the whole nucleus coherently. A neutrino interactionthen will be as (A denotes a nucleus):193.1. Basics of Neutrino Interactionsνl +A→ νl +Aνl +A→ νl + pi0 +Aνl +A→ l− + pi+ +A.The nucleus remains unchanged in the interaction, hence there is no CCinteraction without pion. The last two interactions are CC and NC coherentpion production, respectively.As the transferred energy and momentum increases, the wavelength getsshorter and a weak gauge boson starts to see individual nucleons in a nu-cleus. Neutrino-nucleon interactions are dominant at neutrino energy ofmost recent accelerator-based neutrino experiments (∼ 1 GeV). Neutrino-nucleon interactions can either be quasi-elastic, where no additional particlesare produced, or accompanied by meson production, sometimes through theexcitation of an intermediary resonance.At higher energy, with more energy and momentum transfer, the proberesolves the nucleon into its constituent quarks. This is called deep inelasticscattering (DIS). For DIS, an interaction is described as scattering off aquark, which carries an unknown fraction of the nucleon momentum:pquark = xpnucleon. (3.1)In most neutrino experiments, neutrino cross section measurements arefocused on charged-current (CC) interactions where a charged lepton emerges.By measuring the energy of the outgoing particles and identifying the chargedlepton, the neutrino flavor can be determined and the energy estimated,while in NC interactions, the neutrino carries away an unknown amount ofmomentum. Thus, neutrino energy and flavor reconstruction is impossiblein the latter case. Figure 3.2 shows the total cross section of CC neutrinointeractions on H2 or D2 as a function of the incident neutrino energy [40].Three different neutrino interaction modes are shown - quasi-elastic (labeledas “qel”), single pion production (labeled as “1pi”), and DIS.At T2K neutrino energies (∼0.6 GeV), charged current quasi-elastic(CCQE) interactions are dominant, though there is also a significant contri-bution from charged current single pion production, mainly from the pro-duction of resonances (CCRES).CCQE interaction is a two-body interaction ν + n → l + p as shownin Figure 3.3. There is a lepton and a proton in the final state. Then the203.1. Basics of Neutrino InteractionsFigure 3.2: Cross sections for different charged-current neutrino interactionsover a range of energies. Plot is taken from [40] with permission.energy of the neutrino can be calculated from the energy (El) and direction(cos θl) of the final state lepton relative to the incident neutrino, assumingtwo-body kinematics with the neutron at rest:Erecν ≈m2p − (mn −ml)22(mn − El +√E2l −m2l cos θl)(3.2)where mn denotes the neutron mass, mp denotes the proton mass, and mldenotes the mass of the charged lepton.For CCRES interactions, a neutrino excites the interacting nucleon, re-sulting in a baryon resonance (for example N* or ∆) that quickly decaysback to a nucleon, often paired with a pion. There are different kinds ofresonance processes and the most common process is single pion productionthrough the ∆-resonance as shown Figure 3.4. This interaction is a signifi-cant background to CCQE events if the pion from a CCRES interaction isabsorbed in the target nucleus or not reconstructed. For CCRES interac-tions misidentified as CCQE, the energy reconstruction using Equation 3.2is biased.The details of theoretical models and calculations will be discussed inthe following section.213.2. Theoretical Models and CalculationsFigure 3.3: CCQE Feynman diagram [41].3.2 Theoretical Models and CalculationsIn the impulse approximation (IA), the total cross section of neutrino scat-tering on the nucleons in the nucleus is calculated as an incoherent sum ofneutrino scatterings on a free nucleon. To describe the initial state of a nu-cleus, nuclear models are required. This will be explained in Section 3.2.1.Then, neutrino-nucleon scattering on a free nucleon will be discussed inSection 3.2.2. After neutrino interactions occur in a nucleus, secondaryinteractions can happen until the particles produced from the interactionexit the nucleus. This is called final state interactions (FSI), which will bediscussed in Section 3.2.3. In Section 3.2.4, comparisons between existingmodels and neutrino experiments will be shown and new approaches will beintroduced.3.2.1 Nuclear ModelsSince nucleons are bound in a nucleus, a neutrino scattering off a nucleon ina nucleus is different from scattering off a free nucleon. This was seen fromelectron scattering on nuclei, which allows nuclear structure to be studiedusing the photon as a probe.Figure 3.5 shows data from an early electron scattering measurement onH2O [42]. It shows the number of electrons (e′) scattered off a thin H2Otarget at an angle of 148.5◦ as a function of the outgoing electron energy.With a fixed incoming electron energy and scattering angle, electron elasti-cally scattering off free nucleons can be identified by the outgoing electron223.2. Theoretical Models and CalculationsFigure 3.4: Resonance pion production Feynman diagram.energy (E′e) without reconstructing the outgoing hadronic system. The scat-tered electron has a definite energy, which corresponds to the sharp peakat around 160 MeV. However, it is overlaid on a wider distribution of scat-tered electron energies. This distribution can be understood as electronsscattered off individual nucleons in the oxygen nucleus. These nucleons arebound in the oxygen nucleus unlike the free proton in the hydrogen atom,and have Fermi momentum and binding energy which results in a broad dis-tribution at slightly lower E′e. These results suggested that nucleons movequasi-freely within the nucleus instead of being static within the nucleus.The peak at around 250 MeV corresponds to elastic scattering off the wholeoxygen nucleus, while the peaks at lower energies (between 200 MeV and250 MeV) correspond to scatterings with excitation of the oxygen nucleus.This demonstrated the importance of nucleonic motion within the nucleusin describing lepton-nucleus interactions.Fermi Gas ModelThe Fermi gas (FG) model was introduced in the quasi-elastic scatteringcalculation to describe the nucleons in the nucleus [43], and is broadly usedto describe the initial state of a nucleus in the scattering process. In thismodel, nucleons are modeled as weakly interacting fermions. As spin 1/2fermions, neutrons and protons obey Fermi-Dirac statistics. The nucleons ina nucleus can be viewed as two independent degenerate Fermi gas systems.The Fermi gas model assumes the nucleons can freely move inside the nuclear233.2. Theoretical Models and CalculationsFigure 3.5: Energy spectrum of electrons scattering off H2O targets with246 MeV incoming electrons and 148.5◦ scattering angle. Plot is reusedfrom [42] with permission.volume with a mean potential.In the nuclear ground state, the lowest states are all occupied up to amaximum momentum. This is called the Fermi momentum, and is given by:pF =~R0(9pi8)3(3.3)where R0 is determined experimentally from the electron scattering data.Figure 3.6 shows the differential cross section measurement (data points)in lepton energy loss for electron scattering on Ni. The theoretical calcula-tions used the Fermi gas model for the initial nucleus which is shown as asolid line. In the calculation, interactions on individual nucleons are summedincoherently over all nucleons in Ni. The left peak is quasi-elastic scatter-ing and the right is for resonance scattering. There is a transition wherethe two peaks overlap. The theoretical calculations showed good agreementwith electron scattering data at the QE peak.A few improvements have been considered in the FG model to incorpo-rate nuclear effects, including Pauli blocking and nucleon-nucleon correla-tions. In Pauli blocking, a neutrino interaction is forbidden if the outgoingnucleon from an interaction has lower momentum than the Fermi momen-243.2. Theoretical Models and Calculationstum, based on Pauli’s exclusion principle. Short-range correlations (SRC)are due to correlations between the wavefunctions of the nucleons, increasingtheir relative momentum.Figure 3.6: Differential cross section electron scattering on Ni as a functionof the outgoing lepton energy loss. The incoming electron energy is 0.5 GeVand the scattering angle is 60◦. Measurements are shown with dots alongwith the predictions from the FG model in solid line. The left curve is QEinteractions and the right curve is resonance interactions. Plot is taken from[43] with permission.Spectral FunctionThe Spectral function (SF) model is another model to describe the initialstate of a nucleus. In the Spectral Function model, instead of describingnucleons with Fermi-Dirac statistics, the system of A-nucleons is consideredwith a non-relativistic Hamiltonian as:HA =A∑i=1p2i2m+A∑j>i=1vij (3.4)where pi is the momentum of the ith nucleon and vij is the nucleon-nucleonpotential between the ith and jth nucleons. For neutrino-nucleus inclusiveinteractions, the spectral function P (p, E), is the probability for removing anucleon with momentum p and leaving the residual nucleus with excitation253.2. Theoretical Models and Calculationsenergy E, is given as:P (p, E) =∑n|〈Ψ(A−1)n |ap|ΨA0 〉|2∏δ(E + E0 − En) (3.5)where |ΨA0 〉 is the nuclear ground state (eigenfunction of the given Hamil-tonian in Equation 3.4 with eigenvalue E0), while |Ψ(A−1)n 〉 and En denotesthe n-th eigenstate and eigenvalue of the (A− 1)-nucleon system. ap is thelowering operator of a particle with momentum p. The spectral function in-tegrated over the energy (E) is the distribution of the momenta of nucleonsin the nucleus and the inclusive cross section can be obtained by integratingthe spectral function over the momentum and energy.Figure 3.7 shows comparisons of the differential cross section for theelectron scattering between different models. The SF model shows bet-ter agreement with electron scattering data than the Fermi gas model [44].While the SF model is more sophisticated in its treatment of the nuclearground state, it is largely limited to a non-relativistic framework.Figure 3.7: Comparison between the SF and FG models of the differentialcross section for electron scattering on oxygen. “SP” in the legend corre-sponds to “SF”. Plot is taken from [45] with permission.263.2. Theoretical Models and Calculations3.2.2 Free Nucleon Cross-Section ModelsIn this section, quasi-elastic scattering and resonant single pion productionon a free nucleon will be explained.As discussed in Section 3.1.2, at few-GeV neutrino energies, the dom-inant form of neutrino-nucleus interactions is scattering off quasi-free nu-cleons in a nucleus. Hence, neutrino-nucleon interactions are consideredand incorporated into a nuclear model to account for the nuclear effects.Among different types of neutrino-nucleon interactions, the dominant inter-action channels at T2K neutrino energy (0.6 GeV) are quasi-elastic (QE)scattering and single pion production through resonance (RES).Quasi-elastic ScatteringThe CCQE cross section on nucleons was formulated by Llewellyn-Smith interms of form factors [46]. A form factor describes the spatial extent of thenucleon and depends on transferred four-momentum (q2). It is the Fouriertransform of the charge function f(~x) defined as:f(x) = ρ(x)/Ze (3.6)where ρ(x) is a charge density. Then, the form factor can be written as:F (q) =∫eiq·x/~f(x)d3x. (3.7)Furthermore, since the nucleon is assumed as a spherically symmetric sys-tem, there is no specific spatial orientation dependence. Therefore, the formfactor only depends on |q|2. In the Breit frame where there is no energytransfer, it can be written as :|q|2 = −q2 (3.8)which is Lorentz invariant. Often, Q2, defined as −q2, is introduced.The scattering amplitude of this interaction is :A = −iGF√2Vud uµγρ(1− γ5)uν〈p|hρW |n〉 (3.9)where GF is Fermi constant and Vud is the element of the quark mixingmatrix corresponding to d to u transitions. The last term is a hadronictransition matrix, where hρW is the quark current (d → u transition) forthe charged weak interaction. This hadronic transition matrix includes the273.2. Theoretical Models and Calculationsstrong interactions of the valence quarks bound in the target nucleon. hρWcan be split into vector and axial vector parts as:hρW = vρW − aρW . (3.10)In the Llewellyn-Smith formula, the differential cross-section for CCQEcan be written as:dσν,ν¯dQ2=G2FVudM28piE2ν[A(Q2)±B(Q2)(s− uM2)+ C(Q2)(s− uM2)2](3.11)whereA(Q2) =(m2 +Q24M2)[(4 +Q2M2)|FA|2 −(4− Q2M2)|F 1V |2+Q2M2|ξF 2V |2(1− Q24M2)+4Q2Re(F 1V ξF2V )M2− Q2M2(4 +Q2M2)|F 3A|2 −m2M2(|F 1V + ξF 2V |2 + |FA + 2Fp|2−(4 +Q2M2)(|F 3V |2 + |FP |2))](3.12)B(Q2) =Q2M2Re(FA(F1V + ξF2V ))−m2M2Re[(F 1V −Q24M2ξF 2V)F 3V −(FA − Q2FP2M2)F 3A](3.13)C(Q2) =14(|FA|2 + |F 1V |2 +Q2M2|ξF2V2|2 + Q2M2|F 3A|2), (3.14)where M is a nucleon mass and m is a lepton mass. There are six formfactors, F 1V , F2V , F3V , F3A, FA, FP .F 1V and F2V are related to vρW as:vρW = γρF 1V +12MiσρµqµF2V +qρMF 3V , (3.15)283.2. Theoretical Models and Calculationsand vρW can be written as:vρW = γρT+ (3.16)where T+ is the isospin raising operator defined as:T± = T1 ± iT2 (3.17)where Ta(a = 1, 2, 3) are generators of the SU(2) isospin group. ThereforeEquation 3.16 can be written as:vρW = vρ1 + ivρ2 (3.18)where vρa is an isovector current defined as:vρa = γρTa. (3.19)The isovector current is conserved:∂ρvρa = 0. (3.20)This is called the conserved vector current (CVC) hypothesis, which is aconsequence of isospin invariance of strong interaction. It implies:〈p|vρW |n〉 = 〈p|vρ3 |p〉 − 〈n|vρ3 |n〉. (3.21)This difference between the proton and neutron matrix elements is equal tothe difference of the proton and neutrino electromagnetic matrix elements:〈p|vρW |n〉 = 〈p| jρ |p〉 − 〈n| jρ |n〉(3.22)where jρ is an electromagnetic quark current:jρ = vρ0 + vρ3 (3.23)and vρ0 is the isoscalar part of the current. Therefore, F1V and F2V canbe obtained from electromagnetic vector form factors from electron-nucleonquasi-elastic scattering. F 3V and F3A are scalar and tensor form factors whichare not included for neutrino interactions assuming the strong interaction isinvariant under isospin transformation.293.2. Theoretical Models and CalculationsThe other two form factors, FA and FP , are the axial form factors fromthe axial vector current aρW :aρW = γργ5FA(Q2) +qρMγ5FP (Q2) (3.24)and FP can be written as:FP =g0fpim2pi +Q2. (3.25)Here, g0 is the pion-nucleon coupling constant and fpi is the pion decay con-stant. The axial vector current, unlike the vector current, is not conserved,but approximately conserved in mpi → 0 limit as:limmpi→0∂ρaρW = 0. (3.26)This is called the partially conserved axial vector current (PCAC) hypoth-esis. Therefore,g0fpi = 2M2FA (3.27)and FP can be written as a function of FA as:FP (Q2) =2M2FA(Q2)m2pi +Q2. (3.28)In the end, the only unknown form factor for neutrino interactions is FA.There are different way of parametrizing the form factor. One way is to usea dipole form, which corresponds to an exponential charge distribution as:FA(Q2) =FA(0)(1 + Q2(MQEA )2)2. (3.29)FA(0) has been determined by β-decay.Resonant Single Pion ProductionAs discussed, given enough energy, a neutrino can excite a nucleon to aresonance state, which decays promptly to produce a pion. Rein and Se-hgal formulated single pion productions through various modes of baryonresonance [47].303.2. Theoretical Models and CalculationsThe cross section of single pion production is calculated as a superposi-tion of every possible resonance contributions as:dσdq2dW=1128pi2mNE2ν∑spins|T (νN → lN ′)|2 Γ(W −MR)2 + Γ2/4 (3.30)where theMR is a resonance mass, q2 is the squared four-momentum transferfrom the lepton, W is the invariant mass of the resonance, mN is the nucleonmass, Eν is the incident neutrino energy, and Γ is the resonance width. Tis a matrix describing the transition from a ground state nucleon (N) to aresonance (N′). Similarly to the CCQE cross section, T can be expressedwith vector and axial vector currents between a nucleon and a baryonicresonance. Evaluating the transition amplitude (|T |) [48], the form factoris assumed to have a dipole form with one parameter, the resonance axialmass (MRESA ) as:FRESA (q2) =(1− q24m2N) 12−n( 11− q2/(MRESA )2)2. (3.31)Here, n is the number of oscillator quanta in the framework of treatingquarks as non-relativistic harmonic oscillators. This additional factor isintroduced to resolve unphysical aspects of the model [48].3.2.3 Final State InteractionWhile secondary particles produced from a neutrino interaction are travelingin the nuclear medium, they can interact strongly, resulting in not only theirmomentum changing, but also being absorbed or producing other particles.These are called Final State Interactions (FSI).To predict FSI, the most commonly used model is the intra-nuclear cas-cade (INC) model which was first proposed by Serber in 1947 [49]. Mostneutrino interaction generators use this model, though their implementa-tions differ.INC considers interactions of the secondary particles on individual nu-cleons in the nuclear medium. The secondary particles can interact multipletimes traveling through the nucleus, and all the interactions that occur untilit exits the nucleus are integrated.To start, a spatial point where an incident particle is produced is deter-mined. Then, free particle-nucleon cross sections and nucleon densities areused to determine the mean free path for the projectile particle. The mean313.2. Theoretical Models and Calculationsfree path is the average distance which a particle travels before undergoinga collision. If a collision happens, a reaction type is selected along with themomentum of the struck nucleon to determine the kinematics of the out-going particles from the collision. The interactions considered in the modelare elastic scattering, pion production, pion absorption, and pion chargeexchange.3.2.4 Discrepancy in the CCQE Measurements andMulti-nucleon EffectsAs described previously, CCQE interactions with a nucleus are traditionallyviewed as scattering off a single nucleon and creating a hole in the recoilingnucleus (from A to A− 1). However, in a realistic situation, it is not alwaystrue that a neutrino only scatters off a single nucleon, but can scatter offmore than one nucleon due to many-body effects. Neutrino scattering onmulti-nucleon can result in multi-nucleon excitation, in which n-particle areknocked out leaving n-holes (np-nh), instead of a single nucleon excitation(1p1h). Direct evidence of this process was obtained earlier from electronscattering data [50]. The electron scattering process can be viewed asvirtual photon absorption and can be separated into absorption of photonswhich are either transversely or longitudinally polarized with respect to themomentum transfer. Existing nuclear models such as the Fermi gas modeldid not reproduce the observed longitudinal and transverse nuclear responsefunctions simultaneously. Instead, studies showed that only the longitudinalpart of the CCQE cross section could be described in terms of independentnucleons bound in the nuclear potential. On the other hand, there wasa significant enhancement in the transverse part, which turned out to becaused by multi-nucleon effects. As the vector form factors of neutrino QEcross section can be obtained from electromagnetic form factors of electronQE scattering, there should be also a transverse enhancement in neutrinoQE scattering.Figure 3.8 shows CCQE cross section measurements as a function ofneutrino energy by MiniBooNE. MiniBooNE reported an excess in the mea-surement over the model prediction with MQEA = 1.03 GeV (dotted green),which was obtained from the expectation based on the free nucleon cross sec-tion and impulse approximation. This was explained by a larger axial massof MQEA = 1.35±0.17 GeV (solid blue) resulting in a larger CCQE cross sec-tion. However, the higher axial mass did not agree with the measurementsat higher neutrino energy from NOMAD (blue cross). This discrepancymight arise from the fact that MiniBooNE measured events with one muon323.2. Theoretical Models and Calculationsand no pion as CCQE interactions, while NOMAD explicitly required aproton track. From the reconstructed kinematics of two tracks, NOMADconstructed missing transverse momentum, proton emission angle, and theangle between the two reconstructed tracks as likelihood variables, whichwere used to suppress the CC-non-QE events in the sample. Then, bothmuon and proton kinematics were used to reconstruct the neutrino energy.NOMAD also had a sample of events with one muon and no pions, whereno proton was reconstructed either. For this sample, as only the muon kine-matics were measured, other kinematic variables such as neutrino energyand proton kinematics were calculated from the muon measurement. Inparticular, the proton angle predicted from the measured muon kinematicswas used to suppress the CC-non-QE events similar to the two-track sample.By analyzing both one-track and two-track samples, NOMAD was able tohave an enriched CCQE selection. This might indicate that multi-nucleoneffects could affect the analyses differently depending on the details fo theevent selection.Figure 3.8: CCQE cross section measurements on carbon as a function ofneutrino energy and model fits. Three different experiments are shown:MiniBooNE in red squares, NOMAD [51] in blue crosses, and LSND [52]in light green triangles. Red dashed line and blue solid line represent RFGmodel with two different axial mass value. The green dotted line is freenucleon model prediction with MQEA = 1.03 GeV. Plot is taken from [53]with permission.One of the first attempts to resolve this discrepancy was the calculationof neutrino-nucleus interactions including multi-nucleon effects to reproducethe MiniBooNE data [53]. This section will focus on various models toexplain the excess observed by MiniBooNE through multi-nucleon effects.333.2. Theoretical Models and CalculationsOne approach is based on the Marteau model and upgraded later byMartini, Ericson, Chanfray and Marteau. This approach uses the non-relativistic Fermi gas model. In the calculation, QE, single pion productionvia ∆-resonance and multi-nucleon knock-out through np-nh are included. Itignores multi pion production and single pion production via other resonanceexcitation [54]. Figure 3.9 shows the predictions of νµ CCQE cross sectionon carbon as a function of neutrino energy with and without multi-nucleoncontribution. The prediction with np-nh (red) shows a good agreement withthe MiniBooNE data.Figure 3.9: The Martini predictions of νµ CCQE cross section on carbon asa function of neutrino energy with and without multi-nucleon contributioncompared to the MiniBooNE data. Plot is taken from [54] with permission.Another approach was proposed by Nieves [56]. It adapted and ex-tended the inclusive electron-nucleus scattering model described in [57] toneutrino-nucleus scattering. This model is relativistic and incorporates sim-ilar nuclear effects as the one by Martini in the kinematic region containingboth QE and ∆ excitation. Nieves included more multi-nucleon processes,but limited the calculation to the three momentum transfer less than (1.2GeV/c). In [58], the calculation is extended to 10 GeV, where it is shownthat multi-nucleon contribution saturates at 30 % of the CCQE cross sectionas the energy increases.In both models, it is found that long-range correlation should be in-cluded to reproduce cross sections on heavy nuclei, known as the randomphase approximation (RPA) correction. The RPA correction accounts for343.2. Theoretical Models and CalculationsFigure 3.10: The Nieves predictions of νµ CCQE cross section on carbon asa function of neutrino energy with and without multi-nucleon contributioncompared to the MiniBooNE data and Martini prediction. Plot is takenfrom [55] with permission.the impact of the nuclear medium as the struck nucleon propagates. Thisimpact can alter cross sections of neutrino interactions. Figure 3.10 showsthe computed CCQE cross section on carbon with multi-nucleon processescompared to the MiniBooNE data and the model by Nieves.Later, the Transverse Enhancement Model (TEM) [59] was also pro-posed. It adopted the parametrization of the electron scattering from quasi-free nucleons, which was used to describe the QE electron scattering onnuclear targets. A large enhancement in the transverse response was ob-served, but not in the longitudinal response. Thus, the model includesthe parametrization of enhancement of the transverse electron QE responsefunction keeping all other ingredients as in the free nucleon target case.There are also more general approaches to obtain the enhanced trans-verse component. One approach is ab initio calculation [60]. It uses non-relativistic Hamiltonian including many-body potentials between nucleons.The response functions are derived using the Green’s Function Monte Carlo(GFMC) method [61] and compared to those extrapolated from experimen-tal data. The calculation predicts the transverse axial response should alsohave an enhancement, in addition to the transverse vector response. Thiscontrasts with the TEM where only transverse vector response is enhanced.Another approach is super-scaling [62]. This normalizes the response func-353.2. Theoretical Models and Calculationstions with nucleon form factors weighted by atomic numbers. Both calcu-lations describe the data where a significant enhancement in the transverseresponse function by two-body components.Martini [63] demonstrated neutrino energy (E¯ν) reconstruction assum-ing the two-body kinematics of CCQE (1p1h) at three different incidentneutrino energies, which is shown in Figure 3.11. The CCQE and multi-nucleon contributions are also shown separately. The multi-nucleon effectssignificantly enhance the cross section at T2K neutrino energy (0.6 GeV).This contribution will cause a bias in neutrino energy reconstruction.This bias in the reconstructed neutrino energy is crucial for oscillationanalyses. Figure 3.12 shows νµ disappearance data from T2K [64]. It showsthe ratio of oscillated neutrino data to unoscillated data. The oscillationparameter for νµ disappearance comes from measuring the depth of theoscillated curve. But if the neutrino energy is underestimated, the depth willbe different. This can cause a bias in the oscillation parameter extraction.Hence, it is important to understand multi-nucleon effects to correctly modelthe reconstructed neutrino energy in oscillation analyses.Figure 3.11: The distribution of reconstructed neutrino energy for QE(dashed) and multi-nucleon (dotted) at three different true neutrino energy- 0.2 , 0.6 , and 1.0 GeV. The Martini calculation is used. Plot is takenfrom [63] with permission.363.3. Neutrino Interaction GeneratorsFigure 3.12: T2K data of νµ disappearance in ν-mode is compared tooscillated and unoscillated MC energy spectra. The ratio of the best fit to theunoscillated spectra is also shown. Plot is taken from [64] with permission.3.3 Neutrino Interaction GeneratorsNeutrino generators aim to simulate the kinematics of particles emergingfrom neutrino interactions on electrons, nucleons, and nuclei. Most neutrinogenerators are based on the common models described in Section 3.2, butmodel implementations and parameters are different between the generators.There are various neutrino generators, but the discussion in this section willbe focused on NEUT (v.5.3.2) [65], GENIE (v.2.8.0) [66] and NuWro [67].NEUT was initially developed to study atmospheric neutrinos in a waterCˇerenkov detector. The main application was neutrino interactions withincident neutrino energy range from tens of MeV to hundreds of TeV onhydrogen or oxygen. It has been extended to neutrino interactions on variousnuclei such as carbon, iron, and argon and used for various experimentsincluding T2K. NEUT is the default neutrino generator for T2K.GENIE, which stands for Generate Events for Neutrino Interaction Ex-periments, was designed to become a canonical generator covering all nucleartargets and neutrino flavors over wide energy spectrum. For T2K, GENIEis used as an alternate model for comparison.NuWro is another generator to make comparisons. NuWro is created373.3. Neutrino Interaction Generatorsat the University of Wroclaw to simulate neutrino-nucleon and neutrino-nucleus reactions.In the following paragraph, the different implementations and param-eters of nuclear and neutrino interaction models in NEUT, GENIE andNuWro will be explained and summarised in Table 3.1.Quasi-Elastic Scattering The CCQE cross section can differ dependingon the value of the axial mass and parameterization of the form factors, andthe different treatment of nuclear effects. NEUT includes the relativistic FG(RFG) model by Smith and Moniz [43] and includes the RPA correction,and the SF model based on [68]. Also, 2p2h contributions have been addedbased on Nieves’ model [69].GENIE uses Bodek-Ritchie modification of the RFG model adding alarge momentum tail. This is to incorporate short range correlation [70].NuWro implements the global and local FG models with the RPA cor-rection, as well as the SF model. For 2p2h contributions, two models areincluded: TEM and Nieves’ models. For the axial form factor, all the gener-ators assume dipole form for the axial form factor, but use different valuesfor the axial mass: MQEA = 1.21 GeV (NEUT), MQEA = 0.99 GeV (GENIE),and MQEA = 1.00 GeV (NuWro).Resonant Single Pion Production This interaction produces a pionpaired with a nucleon through ∆ resonances. The vector form factors ofpion production are determined from photoproduction and electroproduc-tion data. However, axial form factors are not well known.The simulation of resonance pion production in NEUT and GENIE areboth based on the Rein-Sehgal model (described 3.2.2) but include differentnumber of resonances. GENIE contains 16 contributions among the listedones in [47] with MRESA = 1.12 GeV, while NEUT includes 18 contributionswith MRESA = 1.21 GeV. NuWro has the similar model as NEUT, but witha different value of the axial mass (MRESA = 0.94 GeV).Final State Interaction Although NEUT, GENIE and NuWro rely onthe INC model to simulate FSI, the detailed implementations are quite dif-ferent.NEUT uses a full cascade model based on [71], using a Local FermiGas (LFG). LFG is a Fermi gas model using spatial dependent momentumdistribution. The mean free path is density dependent for low momentum383.4. The CCQE Cross-Section Measurementshadrons (p < 500 MeV/c), but density independent for high momentumhadrons (p > 500 MeV/c).GENIE uses a data-driven model called hA model that does not fully sim-ulate a cascade inside the nucleus. Instead, it extrapolates hadron-nucleuscross sections as a function of incident momentum based on hadron-Fe dataand uses it to determine whether an interaction happens or not.In NuWro, FSI is simulated with the cascade model as in NEUT.Other Neutrino Interactions In addition to resonance pion produc-tions, there is also coherent pion production where a neutrino scatters witha whole nucleus coherently producing a pion while the nucleus stays the sameafter the interaction. Coherent pion production is modeled according to theRein-Sehgal model (different from the resonance production model) [72].For DIS, interactions are considered on point-like particles with momen-tum described by patron distribution functions [73], as the fundamentalinteraction target is not a nucleon but its quark constituents. A partondistribution function describes the probability of finding a particle within agiven momentum range. Since these standard patron distributions are notapplicable to low Q2, modifications to this momentum transfer range havebeen made fitting external data to extract correction factors [74].NEUT/NuWro GENIECCQESF RFGMQEA = 1.21 GeV (NEUT) MQEA = 0.99 GeVMQEA = 1.00 GeV (NuWro)2p2h Nieves N/ASingle Rein-Seghal Rein-SeghalPion Production18 resonance modes 16 resonance modesMRESA = 1.21 GeV (NEUT) MRESA = 1.1 GeVMRESA = 0.94 GeV (NuWro)FSI Standard cascade hA (effective cascade)Table 3.1: Models and parameters in NEUT, GENIE and NuWro.3.4 The CCQE Cross-Section MeasurementsMost of existing data on CCQE interactions were collected from old bubblechamber experiments. They analyzed νµd → µ−pp reactions and extracted393.4. The CCQE Cross-Section Measurementsthe axial mass. The world average value from the bubble chamber data isMA = 1.026± 0.021 GeV [75].Later, some accelerator-based experiments measured CCQE interactionson heavy targets using proton information. K2K SciFi [76] and SciBar [77]detectors, which measured neutrino interactions on oxygen and carbon re-spectively, analyzed samples of one-track and two-track events as CCQEinteractions. For the one-track sample, the recoil proton or a pion is absentor below reconstruction threshold, hence only the muon is reconstructed.For the two-track sample, there are two charged particles reconstructed: onemust be a muon and the other can be either a proton or a pion. To identifythe second track as a proton, the angle of the recoil proton with respect tothe beam direction was predicted from the muon measurement, and com-pared to the angle of the second track. Then, a cut on the difference betweenthe predicted and measured angle of the second track was introduced andused to separate CCQE from CC-non-QE events in the two-track sample.SciFi and SciBar analyzed both one-track and CCQE-enhanced two-tracksamples to measure the CCQE cross sections and the axial mass, but theEν and Q2 reconstruction used only the muon kinematics. SciBooNE [78]did a similar measurement on carbon, but used a different technique for theCCQE/CC-non-QE separation in the two-track sample. They reconstructedneutrino energy from the CCQE assumption (two-body kinematics), whichwas compared to the total energy deposited in the detector to suppress CC-non-QE backgrounds in the selection. The axial mass determined at K2Kwere 1.20 ± 0.12 GeV (SciFi) and 1.144 ± 0.077 GeV (SciBar). SciBooNEinstead compared the measured total cross section to the NEUT predictionusing 1.21 GeV and claimed a good agreement. Although above experi-ments measured CCQE interactions including a proton in the selections,but on heavier target, the values of the axial mass did not quite agree withdeuterium data.Having various models including nuclear effects available, there havebeen many attempts to provide measurements sensitive to nuclear effects.Figure 3.13 shows the double differential cross section in muon kinematicsof CCQE-like events on carbon from T2K compared to models with (dashedred) and without (dashed black) 2p2h [79]. However, it was hard to tell inwhich model the data was favored.403.4. The CCQE Cross-Section MeasurementsFigure 3.13: The double differential cross sections in muon kinematics ofCCQE-like events on carbon. The data (point) compared with NEUT with(dashed red) and without (dashed black) 2p2h. Nieves’ model is used toimplement 2p2h components. Plot is taken from [79] with permission.MINERνA reported a CCQE differential cross section measurement withµ+p topology. In Figure 3.14, the CCQE differential cross section as a func-tion of Q2QE,p along with model predictions and the ratio between the dataand predictions are shown. Q2QE,p is reconstructed four-momentum transferusing the most energetic proton instead of a muon, with the assumption ofQE scattering from a neutron at rest as:Q2QE,p = (mn − V )2 −m2p + 2(mn − V )(Tp +mp −mn + V ) (3.32)where mn and mp are masses of a neutron and a proton, V is the bindingenergy of 34 MeV, and Tp is the proton kinetic energy. Q2QE,p can be alteredby the fact that the nucleon is bound in a nucleus. Therefore, it provides413.4. The CCQE Cross-Section Measurementsadditional information sensitive to nuclear effects such as Fermi motion orFSI effects. For the predictions, GENIE and NuWro were used.Figure 3.14: QE cross section versus Q2QE,p compared to different pre-dictions (right). Ratio between the data and predictions. All models arenormalized to the data. Plot is taken from [80] with permission.423.4. The CCQE Cross-Section MeasurementsFigure 3.15: The differential cross sections in momentum transfer (Q2)of CCQE-like interactions on various nuclei. The data (point) is comparedto model predictions with and without FSI: GENIE with FSI (solid red),GENIE without FSI (dashed pink), NuWro with FSI (dashed blue) andNuWro without FSI (dashed black). Plot is taken from [81] with permission.As a result of the need for better understanding of nuclear effects, manyexperiments have published measurements including hadronic kinematics inthe measurements. MINERνA has published cross sections in momentumtransfer (Q2) of CCQE-like interactions on various nuclei to explore thenuclear dependence of the measurements [81]. Figure 3.15 shows the dif-ferential cross sections on C (top-left), Fe (top-right) and Pb (bottom-left).GENIE and NuWro with and without FSI are shown. The FSI effect getsstronger as the nucleus becomes heavier.433.5. Other Cross-Section Measurements3.5 Other Cross-Section Measurements3.5.1 The CC Inclusive MeasurementsAs well as the CCQE measurements, many neutrino experiments have mea-sured cross sections of the inclusive neutrino interaction on various targetsover a broad range of neutrino energies. Figure 3.16 shows the inclusive CCcross-section as a function of neutrino energy as measured by many differentexperiments. At high energy, the total cross section has a linear dependenceon neutrino energy, which is expected for neutrino scattering off quarks.At lower neutrino energy, there is a complex combination of QE and RESinteractions as neutrino-nucleon interactions are dominant.Figure 3.16: Measurements of νµ and ν¯µ CC inclusive scattering crosssections as a function of neutrino energy. Plot is taken from [33] with per-mission.443.5. Other Cross-Section MeasurementsFigure 3.17: The double differential cross sections of neutrino-carbon in-teractions from MINERνA. The cross sections are measured in a pair ofvariables: three-momentum transfer (q3) and reconstructed energy (Eavail).GENIE is used as a default MC prediction (dotted) along with variation:with RPA (dashed), with RPA and 2p2h (solid), with 2p2h (shade). Plot istaken from [82] with permission.Recently MINERνA published the inclusive cross sections of neutrino-carbon interactions at low three-momentum transfer [82]. Figure 3.17 showsthe double differential cross sections in the transferred three-momentum(q3) and reconstructed available energy (Eavail). Eavail is reconstructed asthe calorimetric sum of energy near the interaction vertex. Including RPAsuppresses the cross sections at low Eavail, and the MC predictions agreewell with the data. However, there is the discrepancy seen at higher Eavail(∼ 0.2 GeV) which is the transition region between QE and ∆ resonance.In addition to the inclusive cross section, individual measurements oneach interaction channel have been reported as well.3.5.2 The CC1pi MeasurementsThe cross-section measurements of single pion production in the 1 GeV rangewere conducted decades ago on hydrogen and deuterium bubble chambersat ANL [85] and BNL [86]. Nowadays, as in the CCQE interactions, thesignificance of nuclear effects has surfaced in pion production channels.Figure 3.19 shows the CC1pi+ differential cross section from MINERνAas a function of pion kinetic energy in the few-GeV neutrino energy region.453.5. Other Cross-Section Measurements [GeV]νE1 10 210]2cm-38(E) [10σ 0.511.522.5ND280 stat+systNEUT MCMiniBooneNOMADINGRIDMINERvANEUT (binned)Figure 3.18: Measurements of νµ CCQE scattering cross sections on carbonfrom various neutrino experiments. The NEUT prediction with MQEA =1.2 GeV is shown in pink-dashed line. The binned NEUT prediction isalso shown in thick pink solid line to compare with the T2K near detectormeasurement (labeled as ND280). Plot is taken from [83] with permission.The data is compared to different model predictions [84]. It shows thatthe GENIE prediction with the pion FSI treatment moves closer to thedata. However, the prediction still appears to be high with the overallnormalization difference exceeding 1σ.Figure 3.20 shows the comparison of the differential cross section in pionkinetic energy between MiniBooNE and MINERνA [87], where the disagree-ment at low energy is shown. It is challenging to find a model which describesthe measurements from MiniBooNE and MINERνA, which are performedin different neutrino energy region: ∼ 1 GeV for MiniBooNE and few-GeVfor MINERνA.In addition, T2K has also reported the differential cross section on wa-ter [88]. Figure 3.21 shows the differential cross section in pion momentumwhere a good agreement with NEUT is shown, but the data is overall lowcomparing to GENIE.463.5. Other Cross-Section MeasurementsFigure 3.19: Differential cross section versus pion kinetic energy for CC1pi+sample from MINERνA. The data (black dots) is compared to model pre-dictions from different neutrino generators. Plot is taken from [84] withpermission.Figure 3.20: Differential cross section in pion kinematics. The MiniBooNEdata is from [89] and the MINERνA data is from [84]. Plot is takenfrom [87] with permission.473.5. Other Cross-Section Measurements / GeVpip0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 / nucleon / GeV)2 cm-38 10× (pi/dpσd 00.020.040.060.080.10.120.140.160.180.2NEUTGENIET2K dataFigure 3.21: Differential cross section of single pion production on wateras a function of pion momentum. The inner error bars show the statisticaluncertainty on the data and the outer error bars are total uncertainty. Thedashed line shows the NEUT prediction and the solid line is the GENIEprediction. Plot is taken from [88] with permission.The cross section of CC single pion production still requires better mod-els and more measurements. There is no single model that can explain allthe existing data. As discussed, there is a discrepancy between MINERνAand MiniBooNE data at low pion kinetic energy. T2K and MINERνA havedifferent favored model predictions. CC1pi+ measurement on carbon fromMINERνA shows better agreement with GENIE, while CC1pi+ measure-ment on oxygen from T2K agrees with NEUT better. It is hard to combinethe existing measurements from various experiments because they were mea-sured at different neutrino energies on different targets.48Chapter 4T2KT2K (Tokai to Kamioka) in Japan is a “long-baseline experiment” in whichneutrino oscillations are studied as neutrinos travel a long distance. Theneutrinos produced at the J-PARC accelerator facility in Tokai are measuredat the near detector (ND280) and the Super-Kamiokande detector (SK) inKamioka after traveling 280 m and 295 km, respectively.4.1 Off-Axis ConfigurationFigure 4.1: Neutrino flux (bottom) for three different off-axis angles andcorresponding νµ disappearance (top) and νe appearance probabilities. Thisfigure is taken from http://www.t2k.org .494.2. BeamT2K uses an “off-axis” configuration, in which neutrino detectors arelocated off the beam direction, to maximize the oscillation probability withT2K neutrino flux. The bottom plot in Figure 4.1 shows the neutrino flux fordifferent angles along with the neutrino oscillation probability. The neutrinoflux is narrower and peaks at lower energy at the off-axis angles than theon-axis angle. 2.5 degrees was chosen, because it is peaked at the energy(0.6 GeV) where the neutrino oscillation probability is maximum with theT2K baseline (295 km).Figure 4.2: Simplified schematic of T2K beamline. Primary protons enterthe target station from the left and produce secondary pions and kaons.The particles decay into charged leptons and neutrinos in the decay pipe.The neutrinos are observed by ND280 and SK. This figure is taken fromhttp://www.t2k.org .4.2 BeamJ-PARC is a high-intensity proton accelerator complex located in Japan.There are three accelerators: the 180 MeV linear accelerator (LINAC), the3 GeV rapid-cycling synchrotron (RCS) and the 30 GeV main ring [90].Figure 4.2 shows an overall schematic [91]. The extracted beam from theMR is delivered to the graphite target, where secondary particles (mainlypions and some kaons) are produced from the proton interactions. Thetarget itself is a thin carbon rod, ∼ 90 cm long.To produce the neutrino beam, the positively charged particles amongthe produced secondary particles are focused by three magnetic horns whichgenerate toroidal magnetic fields. The focused particles produce the neutrinobeam in a 100 m decay volume. The dominant processes occurring in thedecay volume are:504.2. Beampi+ → µ+ + νµK+ → µ+ + νµK+ → µ+ + νµ + pi0K+ → e+ + νe + pi0µ+ → e+ + νe + νµ.Among these processes, the dominant one to produce νµ beam is pi+ →µ+ + νµ. Figure 4.3 shows the history of protons on target (POT) and theprotons-per-pulse [92]. Since 2015, T2K has been operating also with anti-neutrino beam. For the anti-neutrino beam, the horn current is reversed tofocus negatively charged particles, mainly pi−, instead. Their decays resultin νµ via pi− → µ− + νµ.Accumulated POT0510152025302010 2011 2012 2013 2014 2015 2016 2017Run1 Run2 Run3 Run4 Run5 Run6 Run7 Run8 Run9Beam Power (kW)010020030040050020 10×Total Accumulated POT for Physics-Mode Beam Powerν-Mode Beam PowerνFigure 4.3: History of protons on target (POT). Solid line is the integratednumber of delivered protons from the beginning of the experiment. The scaleis on the left axis. Red dots are the number of protons per beam pulse, andthe scale is on the right axis. This figure is taken from http://www.t2k.org.Figure 4.4 shows the neutrino energy flux predictions in neutrino modeat ND280 for different parent decay modes. The νµ is mostly from pi+514.2. BeamFigure 4.4: The neutrino flux predictions at ND280 for different parentdecay modes [93]. The left column is of νµ (top-left) and νe (bottom-left).The right column is of νµ (top-right) and νe (bottom-right).decays and most of the νe is from kaon and muon decays. At low energy(below 1.5 GeV), the νe flux is ∼ 1 % and νµ flux is ∼ 5 %. However,at higher energy, the relative fraction of νe flux increases because the kaondecay becomes dominant. The νµ flux increases as these anti-neutrinos arefrom high energy pions which are produced in the forward direction and notdefocussed by the horn.Since the neutrino energy differs depending on the off-axis angle, it isimportant to measure the beam direction precisely. At the end of the decaypipe, the muon flux monitor (MUMON) sits to measure the properties ofthe muons penetrating the beam dump, including the profile center andintensity to ensure that the beam axis is correct.524.3. On-Axis Detector: INGRID4.3 On-Axis Detector: INGRIDINGRID, the Interactive Neutrino GRID, sits on-axis with respect to thebeam line and measures the beam direction and intensity by detecting neu-trino interactions.INGRID is a cross-shaped detector, made of 16 identical modules asshown in Figure 4.5. Each module is composed of alternating layers of ironand active scintillator. This is surrounded by veto scintillator planes toreject background events occurring outside the detector. Figure 4.6 showsthe structure within an INGRID module.Figure 4.5: An overview of the INGRID detector. Fourteen identical mod-ules are arranged as a cross and 2 additional modules located outside of themain cross. The beam direction is defined as z. This figure is taken fromhttp://www.t2k.org .INGRID is designed to provide enough statistics for daily measurements.Figure 4.7 shows daily event rate of the neutrino events normalized by POT.The profile of the beam in x and y directions (as the beam direction is definedas z) is reconstructed with the number of neutrino interactions accumulatedon a monthly basis. The history of the neutrino beam centers measured byINGRID is shown in bottom two plots of Figure 4.7 [94].534.4. Off-Axis Near Detector: ND280Figure 4.6: A single INGRID module. The module consists of a sandwichstructure of nine iron plates and 11 tracking scintillator planes surroundedby veto scintillator planes. This figure is taken from http://www.t2k.org .Day[events/1e14 POT]0.40.60.811.21.41.61.8Event rate Horn250kAHorn205kAHorn-250kA[mrad]1−0.5−00.5 Horizontal beam direction INGRIDMUMONDay[mrad]1−0.5−00.5 Vertical beam directionINGRIDMUMONT2K Run1Jan.2010-Jun.2010T2K Run2Nov.2010-Mar.2011T2K Run3Mar.2012-Jun.2012T2K Run4Oct.2012-May.2013T2K Run5May.2014-Jun.2014T2K Run6Oct.2014-June.2015T2K Run7Feb.2016-Figure 4.7: Daily event rate of the neutrino events normalized by POT onthe top and history of the neutrino beam centers on the bottom [64].4.4 Off-Axis Near Detector: ND280For the oscillation analysis, it is important to understand the neutrino beamand interaction properties before oscillation. ND280 is designed to serve thisrole.The central tracking system of ND280 consists of three time projection544.4. Off-Axis Near Detector: ND280Figure 4.8: All elements in ND280. P0D, FGDs, TPCs, and DsECalare held inside a stainless steel frame, called the basket. BrECal andP0DECal are attached directly to the magnet. The magnet encloses allof the detectors. The beam enters from the left. This figure is taken fromhttp://www.t2k.org.chambers (TPCs) and two fine-grained detectors (FGDs). The pi0 detector(P0D) is used to understand pi0 production channels. The electromagneticcalorimeter (ECal) and the side muon range detector (SMRD) surroundboth the tracker and the P0D. The ECal is designed to detect photons, andthe SMRD measures the range of charged particles exiting the tracker.All of the trackers, P0D, ECal, and the SMRD are surrounded by amagnet. The magnet provides a horizontal uniform 0.2 T magnetic fieldperpendicular to the beam direction. Because of the magnetic field, particleshave curved trajectories in the detectors, so that the momentum and sign canbe measured by the curvature of the particles. Figure 4.8 shows the overallconfiguration of ND280. ND280 uses a right-handed coordinate system withz along the neutrino beam direction. The most upstream detector withrespect to the beam is the P0D which is followed by the three TPCs andtwo FGDs. The TPCs and FGDs are cardinally labeled from the upstreamto the downstream. The ECals sit at the downstream end of the tracker andalso surround the sides of the P0D and tracker.554.4. Off-Axis Near Detector: ND280The data taken from the detector electronics are collected by the dataacquisition (DAQ) server. Each subdetector has front-end electronics lo-cated on the detector to communicate to the back-end electronics placedoutside of the detector and magnet. A trigger system tells the electronicsto start the data reading process. There are two triggers: an external beamtrigger and internal trigger by the detector components. When the detectoris triggered, the DAQ checks if all fragments from subdetectors are present,then saves an event when it completes reading all the fragments.In this section, each subdetector will be reviewed in terms of the physicalprinciples behind their design and roles. The FGDs and TPCs will be de-scribed in more detail later in 4.6.1 and 4.6.2. In addition, the global DAQsystem and event triggering will be briefly explained in 4.4.6.4.4.1 Time Projection ChambersFigure 4.9: An ND280 TPC module. The TPC is a double box, where theinner box is filled with a gas mixture [95].A Time Projection Chamber (TPC) is a gas-filled detector with a gasmixture of Ar, CF4, and C4H10. Figure 4.9 shows a single TPC. Along itswidth, a TPC volume is divided by a central high-voltage cathode plane,which makes a uniform electric field between the cathode and the end planes.When charged particles pass through the TPC volume, they produce ion-564.4. Off-Axis Near Detector: ND280ization in the gas. The uniform electric field across a TPC drifts ionizedelectrons to the end plates where the Micromegas modules lie.Micromegas is a micromesh gaseous detector to amplify primary elec-trons from ionization. Each end plate of a TPC has 12 Micromegas modulesand each module contains 1728 pads arranged in 48 rows of 36 pads [95, 95].Charge deposit in a single pad represents a hit and is used to reconstructa track from pattern recognition of activated pads. This 2D grid of padsprovides 2D tracking in the detectors. The third coordinate is given by drifttime with the initial time given by an FGD.The TPC is a slow detector in the sense that it must wait for all theionization to drift to the readout planes. The faster this happens, the fasterthe detector can start taking data again [96]. The ratio of gases is chosen tomaximize the drift velocity of ionized electrons. For ND280 TPCs, the driftvelocity is 78.5 mm/µs.T2K uses three identical TPCs for tracking. The primary purpose of theTPCs is 3D tracking of particles with a relative resolution (σp⊥/p⊥) of about0.1p⊥/(GeV/c) as shown in Figure 4.10. p⊥ is momentum perpendicular themagnetic field.Figure 4.10: Momentum resolution for a single TPC is shown as a func-tion of momentum perpendicular to the magnetic field. The dashed linesrepresents the momentum resolution goal [95].Another important measurement of TPCs is particle identification by574.4. Off-Axis Near Detector: ND280measuring the energy loss of particles along their trajectories. The energyloss resolution in TPCs is better than 10 %. More details will be discussedin Section 4.6.4.4.2 Fine-Grained DetectorsFigure 4.11: A cross-section of a single scintillating bar for FGDs [97]. Thescintillator part has rounded corners and is coated by TiO2. At the center,there is a hole for a WLS fiber. Each bar is 2 m long.The Fine-Grained Detectors are the other part of the tracker system ofND280, interleaved with TPCs to give target mass. The two FGDs havethe same size, but different internal structure. The upstream FGD (FGD1)has 30 layers of scintillating bars, while the downstream FGD (FGD2) hasfewer scintillator layers and instead has water layers between the scintillatorlayers. This allows the determination of neutrino cross sections on water bycomparing the rates in the two detectors.In each FGD, scintillating bars are arranged in layers along the x ory axis alternatively, perpendicular to the beam direction. This structureprovides two kinds of 2D tracking by alternating measurements of x and yas a track travels in the z-direction: xz and yz. Combining them together,3D tracking is achieved.Figure 4.11 shows a cross section of an FGD scintillating bar. Each barhas a wavelength shifting (WLS) fiber at the center coupled to a Multi-Pixel584.4. Off-Axis Near Detector: ND280Photon Counter (MPPC).MPPCs are chosen to be able to count photons down to the single pho-toelectron level and work in 0.2 T magnetic field. Each pixel is an avalanchephotodiode (APD), a semiconductor electronic device which converts lightto electricity. It can be fired by a single photon and amplifies the resultingphotocurrent when a reverse voltage is applied. So the number of fired pixelsis proportional to the number of incident photons.In addition to tracking, the FGDs can tag delayed activity of Michelelectrons which are produced from the decay of stopped muons. It is usedfor tagging pions which often decay to muons and then Michel electrons inthe FGDs. Pion tagging is important for classifying neutrino interactions,in particular those with and without pions. More details of FGDs will bediscussed in Section 4.6.4.4.3 pi0 DetectorFigure 4.12: The close-up view of a P0D module: scintillating bars, watercell and water layer. The scintillating bars for P0D are triangular to improveposition detection. The water cell is removable. This figure is taken fromhttp://www.t2k.org .The pi-zero detector (P0D) is used to measure neutrino interactionswhich produce pi0. The measurement is mostly focused on Neutral-Currentpi0 production (NCpi0).The P0D consists of active scintillator layers, brass and lead layers toinduce photon conversions, and target water layers. By taking data with thewater layers full and empty, neutrino interaction measurements on water canbe made. Active layers are built of triangular scintillating bars with a fiber594.4. Off-Axis Near Detector: ND280coupled to an MPPC. Figure 4.12 shows the P0D bars and how they arelaid. Because of the structure, a particle most likely passes through two barsinstead of one, which can improve the resolution of particle positions.There are two other types of inactive layers: targets and radiators. Tar-gets are water cells providing oxygen target mass. Radiator layers are madeof two different materials, lead and brass. These are using to contain decayphotons from pi0. Since the radiation length of lead is shorter than brass,lead will absorb more of the particles from the photon shower when thethickness is the same, reducing the information available for reconstruction.For this reason, lead radiator layers are used in the outer layers (as a radia-tor on). Otherwise, in the inner layer, brass is used as radiator rather thanlead. This is because with brass a radiator layer can be made thinner thanwith lead, so that it can give better energy resolution [98]. For the P0D, alead layer is 4.5 mm thick, while a brass layer is 1.5 mm thick.4.4.4 Electromagnetic CalorimeterThe ECal is an electromagnetic calorimeter consisting of three sections: theBarrel ECal (BrECal) surrounding the tracker, the P0D ECal surroundingthe P0D and the Downstream ECal placed downstream of TPC3. The ECalassists to detect particle exiting the tracker and the P0D.Each ECal module consists of layers of scintillating bars of 40 mm ×10 mm cross section bonded to lead sheets of 1.75 mm (4 mm for the P0DECal) thickness. The scintillating bars reconstruct tracks and showers. Eachlayer is oriented perpendicular to its neighboring layer to allow 3D tracking.The BrECal consists of four modules, and each module has 31 scintillator-lead layers. The DsECal has 34 layers and the P0D ECal has 6 layers withthinner lead sheets. The number of layers and thickness of lead sheets weredetermined to have enough radiation lengths to contain showers of photonsfrom pi0 decay [99]. The BrECal and DsECal have 1.75 mm of lead sheets(equivalent to 10 and 11 radiation lengths respectively), while the P0D ECalhas 4 mm of lead sheets (equivalent to 4.3 radiation lengths).4.4.5 Side Muon Range DetectorThe Side Muon Range Detector (SMRD) detects particles that exit theinner detector region at high angle. It also identifies background neutrinointeractions on the magnet and the surrounding walls.Whereas the other detectors are placed in the inner volume enclosed bythe magnet, the SMRD is installed between the steel yokes of the magnet.604.4. Off-Axis Near Detector: ND280Figure 4.13: Sliced view of a SMRD slab. A scintillator with s-bent WLSfiber is wrapped in a reflective layer and a lightproof stainless steel container.Endcaps are black epoxy glue [100].Therefore the size of the SMRD is determined by the magnet structure. Themagnet yoke is split into C-shaped segments on either side of the detector.A single C-like segment consists of 18 iron layers and each layer is separatedby an air gap 17 mm thick. The SMRD is held in this air gap [100]. Thesmallest building block of the SMRD is a scintillator slab coupled to thefiber. The slab is embedded with s-bent wavelength shifting fibers as shownin Figure 4.13. The s-bent fiber collects light evenly over the broad surfaceand the light is read out an MPPC coupled to the fiber end.4.4.6 The ND280 Data Acquisition SystemThe ND280 data acquisition system (DAQ) uses the MIDAS framework tocollect data fragments recorded by each subdetector after a trigger signal andstores the data on an event-by-event basis. The individual subdetectors havea standalone DAQ and electronic system. The DAQ system architecture isshown in Figure 4.14. The P0D, ECals and SMRD use identical electronicsto read out the signals based on the TripT ASIC. The signals integrated onthe TripT front-end boards (TFB) are transmitted to the back-end electronicsystem called readout merger modules (RMMs). Therefore, the P0D, ECalsand SMRD are called TripT detectors. The Trip-T detectors are read out bythe front-end processor nodes (FPNs). The FGDs and TPCs have additionalfront-ends (Cascade Front End). The light injection control system, whichis used for calibrations, is an additional client to the DAQ for the ECal andP0D.Data acquisition is initiated by the trigger system. At the center of the614.4. Off-Axis Near Detector: ND280Figure 4.14: The DAQ system architecture is shown. TripT front-endelectronics (TFBs) are transmitted to the back-end electronics (RMMs),then to the front-end processor nodes (FPNs) by optical Gigabit Ethernetlinks. The DAQ system gathers fragments from FPNs and the TPC andFGD front-ends (Cascade Front End). Light injection control system is tocontrol light injection to the ECal and P0D for calibration [101].system, there is a Master Clock Module (MCM). When the MCM receivestiming signals from the accelerator and a GPS-based clock indicating thedelivery of protons to the beamline, it generates trigger signals for the globalND280 system. When a trigger is issued, it distributes the trigger to thesubsystems through Slave Clock Modules (SCM). Each subdetector has oneSCM.MCM also connects to Cosmic Trigger Modules (CTM) for cosmic events.CTM decides whether there was a cosmic event in the detector and triggersthe electronic system. There are two independent CTMs - TripT CTM andFGD CTM. TripT CTM uses all TripT detectors and requires activity on twodifferent sides of the TripT detectors. FGD CTM requires a certain numberof fired channels in specific geometries consistent with a muon passing orstopping in the FGDs. When the condition is satisfied, the CTM asks theMCM to generate the cosmic trigger. The system architecture includingevent triggering and electronics of subdetectors is summarised in Figure 4.15.624.5. Far DetectorWhen a trigger is sent to the subdetectors, each subdetector front-endelectronics saves their event fragments. These fragments are transmitted tothe back-end electronics, and then to the global DAQ. An event-buildingprocess to merge the transmitted fragments and assembles them to write todisk.Figure 4.15: The electronic system architecture is shown. This describesthe event triggering and communication between electronics. At the center,there is MCM which receives triggers from beam (left) and cosmic (aboveand below). When MCM receives a trigger, it talks to subdetectors (right)so they can save the event [101].4.5 Far DetectorThe Super-Kamiokande detector (SK) is the world’s largest contained waterCˇerenkov detector, located 1000 m under Mt. Ikeno in Japan [103]. Thislocation provides clean water, hard rock, and 1000 m of overburden, allnecessary for the construction and operation of the experiment. As SKdetects particles via light produced by neutrino interactions in water, itis important to have transparent water. The 1000 m overburden reducescosmic muon backgrounds to neutrino measurements. Figure 4.16 showsthe structure of SK. It is 39.3 m in diameter and 41.4 m in height, filledwith 50,000 tones of ultra-pure water. On the site, there is clean water nearthe detector and the water is supplied to the detector through the waterpurification system. The purpose of water purification is to filter small634.5. Far DetectorFigure 4.16: Structure of the Super-Kamiokande detector [102]. The de-tector consists of two parts: inner detector and outer detector. Its locationunder Mt. Ikeno is shown as well.dust or metal ions to increase the transparency. It also removes radioactivematerials such as radon, which is the main background of solar neutrinomeasurements.The SK volume is divided into two sections, the inner-detector (ID) andthe outer-detector (OD). The ID is used to detect and reconstruct events,while the OD is a veto for background events such as cosmic muons or otherinteractions in the surrounding material. In addition, the OD is also usedfor identifying particles which are produced in the inner region, but exitthe detector [104]. The ID and OD walls are covered with photomultipliertubes (PMTs) to detect Cˇerenkov radiation from charged particles producedin the SK. A PMT is a very sensitive photosensor with high gain and goodtime resolution of 2 ns [105]. There are 11,146 20-inch PMTs are installedin the ID and 1,885 8-inch PMTs are in the OD.Cˇerenkov radiation is electromagnetic radiation which occurs when chargedparticles passing through a medium travel faster than the speed of light inthat medium [106]. The radiation is emitted in a cone shape, therefore anevent on the detector looks like a ring. Since T2K must separate νµ and νeinteractions, the SK detector must reconstruct and identify muons and644.5. Far DetectorFigure 4.17: Example event displays at SK: µ-like event (top) and e-likeevent (bottom). When a PMT collected charge, it is shown as a pointon an event display. Colour codes correspond to the amount of chargecollected by a PMT. These figures are taken from http://www-sk.icrr.u-tokyo.ac.jp/sk/detector/display-e.html.654.6. The Details of the ND280 Trackerelectrons. Muons and electrons are distinguished by looking at how well-defined the edge of the ring is. Particles with heavy mass will not radiatesignificantly in the water, so that they create a well-defined ring. In otherwords, massive particles such as muons make sharp rings. However, lightparticles such as electrons will lose energy radiatively and scatter, so thatsecondary particles are released. They make fuzzy rings. As a result, a ringfrom a muon has a much sharper edge than an electron ring. The top figurein Figure 4.17 is an example of a µ-like event, and the bottom figure is ane-like event.4.6 The Details of the ND280 TrackerAs discussed in the previous section, ND280 is a complex of many sub-detectors. However, most T2K analyses rely on the tracker - TPCs andFGDs - for event reconstruction and selection. This section will provide ad-ditional information on the tracker regarding its physical principles, hard-ware, and software.4.6.1 FGDsDetector Composition and GeometryTarget materials are mainly carbon and hydrogen from scintillating bars,small amounts of TiO2 from scintillator coating, and water. The FGD2contains scintillating bars and water modules, while FGD1 is made of onlyscintillating bars. By comparing interaction rates in the two FGDs, thecross section of neutrino interactions on water can be estimated. Since thefar detector is a water Cˇerenkov detector, it is important to measure crosssection on water target.Though the inner structure of the two FGDs is a little bit different,basically they have the same geometry and electronic architecture. EachFGD measures 280 × 240 × 36.5 cm3. Its depth along the beam directionis relatively thin so that charged leptons produced inside the FGDs canpenetrate into the TPCs, which can then provide a measurement of theparticle momentum. Each scintillator module hangs inside a light-tight boxmade of aluminum.When charged particles pass through the scintillating bars, they producelight, which is channeled by the WLS to the MPPC. The MPPC converts thelight into an electrical signal. Since the FGD electronics need to provide both664.6. The Details of the ND280 Trackertiming and charge measurements, they rely on digitizing the photosensorwaveform to extract time and charge.Every single scintillating bar has an MPPC, and this MPPC is attachedto a photosensor daughter board. FGD1 contains 5760 MPPCs and associ-ated daughter boards, and FGD2 contains 2688 MPPCs and boards. 16 ofthe boards and photosensors are together on a photosensor bus board.The front end electronic boards cannot be accessed during data runningso it is important to build a robust slow control system which is able tomonitor and control the condition of the boards.Furthermore, the temperature must be carefully controlled in order toensure proper operation of the photosensors. The bias voltage of each pho-tosensor must also be set individually in order to obtain a uniform detectorresponse. Thus, a slow control system must be built to monitor the elec-tronics and photosensor as well as to control the photosensor bias voltage.Water Module The water modules in FGD2 are a series of layers ofwater in polycarbonate vessels. The ends of each vessel are sealed by epoxy.Since the detectors have lots of water-sensitive electronics, it is importantto make sure the water never leaks out. A negative pressure system using avacuum pump provides water leak protection in the case of minor leaks inthe module.Scintillator Module The basic building block of a scintillator module isa scintillator layer. The layer consists of 192 scintillating bars alternatingbetween x and y in the xy plane. One X layer and one Y layer are gluedtogether and form a single XY module. The dimensions of each module are186.4× 186.4× 2.2025 cm3. A scintillating bar contains a fiber in its centralhole. The fiber is not glued to the bar, but coupled to the bar through theair gap. One end of a fiber is mirrored to reflect light back along the fiberand improve light collection efficiency. The other end is coupled to a Multi-Pixel Photon Counter to measure light and its timing. The FGD1 has 15XY modules, 30 layers in total. The FGD2 has 7 XY modules, alternatingwith water modules.Wavelength Shifting Fiber Light from scintillating bars travels througha fiber inside the bar. This fiber is a double-clad wavelength shifting fiber(WLS). Figure 4.18 is the cross-section of the fiber. As shown, the doublecladding improves capture and transmission of light in the fiber. The pro-duced light in the scintillator is in the UV range, 100 ∼ 400 nm, while the674.6. The Details of the ND280 TrackerFigure 4.18: Wave-length shifting fiber comparing single cladding and dou-ble cladding fibers [97].MPPC is sensitive at the range of green light, ∼ 500 nm. For this reason,WLS fibers are used. WLS fibers shift shorter wavelengths to longer onesas shown in Figure 4.19. Figure 4.20 shows the detection efficiency of theMPPC as a function of wavelength.Multi-Pixel Photon Counter For choosing proper photosensors for theFGDs, a number of requirements were considered. The sensors should becompact and robust. They should also have high gain with low voltageand high detection efficiency. Moreover, since the photosensors will be usedinside a magnet, they must be insensitive to magnetic field. The MPPCssatisfy these features, so T2K selected them.The MPPC is a pixellated avalanche photodiode that operates in Geigermode with 1.3 mm2 sensitive area, so it has single photon sensitivity. Anavalanche photodiode (APD) is a semiconductor electronic device whichconverts light to electricity. It can amplify the resulting photocurrent whena reverse voltage is applied. If the APD is operated in a reverse voltageabove its breakdown voltage, a very high gain(∼ 105) is obtained. This iscalled “Geiger mode”. Figure 4.21 shows the MPPC with the WLS fibers.Each pixel of the MPPC works independently in Geiger mode, and allthe pixels are connected in parallel. Since a single incident photon will fire684.6. The Details of the ND280 TrackerFigure 4.19: The absorption and emission spectra of WLS fibers [107]. Itshifts shorter wavelengths to longer ones where MCCPs are sensitive.Figure 4.20: The MPPC sensitivity as a function of wavelength [108].a single pixel, the number of fired pixels is proportional to the number ofincident photons. This device counts the number of photons. But when thenumber of incident photons is more than a significant fraction of the totalnumber of pixels in an MPPC, the device is saturated. In other words, thenumber of fired pixels are not proportional to the number of the photonsanymore. So the device becomes non-linear at higher light levels.The gain depends on the operating voltage and the ambient tempera-ture. Figure 4.22 shows the relations. In the left plot, the gain linearly694.6. The Details of the ND280 TrackerFigure 4.21: A photo of the MPPCs with the WLS fibers. This photo istaken from http://t2k-experiment.org.changes with varying reverse voltage. In the right plot, the gain decreasesas temperature increases at fixed reverse voltage. It is important to measurethe temperature and operation voltage during data-taking to calibrate theMPPCs.Figure 4.22: The MPPC gain as a function of the reverse voltage (left) andtemperature (right) [108].704.6. The Details of the ND280 TrackerMPPC signals have three important effects that must be accounted for:cross-talk, dark noise, and after-pulsing. Cross-talk is when 1 pixel fires anadjacent pixel in the same MPPC resulting in more than one pixel fired bya single photon. Dark noise is from avalanches induced by thermal excita-tion without an optical signal. Lastly, after-pulsing is caused by a secondavalanche from the same photon.Charge CalibrationThe raw data of charge is a waveform with pulse height (PH) from MPPC.This needs to convert to a normalized value representing the digitized charge(i.e. energy deposit) from the MPPC channels. At the first, the number ofpixels is calculated from PH, which is basically given by PH divided by PHof a single-pixel avalanche. After applying the light loss correction in thescintillating bars are applied to the number of pixels, it is converted to theenergy deposition in the scintillating bars, which is well described by Birks’formula [109]. The conversion rate of MPPCs is 0.382 fC/ch [108].ReconstructionParticles traversing through the FGDs with some energy and direction arerepresented as a set of hits which record point-like information along itspath based on the position of the bars which are hit. In the data, wecannot know any true information of particles. Instead, we have methodsto reconstruct the path, energy, or direction based on the hit information.The reconstruction algorithm finds 2D tracks in the xz or yz planes first bymatching reconstructed hits, then 3D tracks are found by matching the 2Dtracks.The FGD reconstruction has two goals; finding hits in the FGDs whichmatch to tracks reconstructed in the TPCs, typically muon tracks, andfinding short tracks that start and stop in FGDs such as proton tracks.Since the FGD reconstruction strongly depends on the TPC reconstruc-tion, they share a unified algorithm. Before doing the FGD reconstruction,the TPC reconstruction is done first. Once TPC tracks are reconstructed,the tracks are extended into FGDs and matched hits identified using theKalman filter.Hits which were not used in the FGD-TPC matching process are savedand reconstructed separately. This algorithm identifies short tracks whichstart and stop in an FGD and therefore are not matched to TPC tracks.A standalone hit clustering algorithm, called a Cellular Automaton, is used714.6. The Details of the ND280 Trackerfor isolated reconstruction. The method creates segments, which are setsof hits in adjacent layers. If the segments in different layers form a quasi-straight line, they are connected together. The sets of connected segmentsform tracks [110].FGD reconstruction has several types of objects. A cluster is a collectionof hits in space, represents a blob of energy. A vertex is also a collection ofhits, but differs in that hits in a vertex are at the same point in space. Ifall particles are emerging from a vertex, it could be a primary vertex. If atleast one particle enters a vertex and some particles are emerging from it, itcould be a secondary vertex. A track is a collection of clusters or hits, whichdefines a path of particles. It contains information of position, direction,charge, and curvature.For the short track fully contained in an FGD, since no TPC informationis available, FGD only information should be used for both PID and momen-tum measurement. dE/dx in an FGD is used to identify the particle typefor FGD only tracks similarly to TPC PID discussed in the previous section.Figure 4.23 shows comparison of dE/dx measurements of FGD tracks to thetheoretical predictions.Figure 4.23: Comparison of dE/dx measurements to the theoretical predic-tions - protons (black), muons (red), and pions (pink) [111].For the momentum measurement, an alternate method, called momentum-by-range, is used. Momentum for the FGD only tracks is computed using theinformation of energy loss in the detector materials for a given length. Fig-ure 4.24 describes how this works. The momentum is computed by addingstep by step the momentum lost inside the detector volume to the track final724.6. The Details of the ND280 Trackermomentum, taking into account the properties of the traversed material andthe length traversed by the particle at each step. The trajectory is assumedto be a straight line. The calculation starts from the end position wherethe particle momentum is supposed to be zero. The step size is chosen ateach step in such a way that the momentum variation is smaller than 10 %.The energy loss at each step is computed for each particle type based on theenergy reached the previous step.Figure 4.24: A scheme for calculating momentum-by-range for FGD-onlytracks [112].4.6.2 TPCsGas MixtureThe TPCs use a gas mixture made of 95 % argon, 3 % CF4 and 2 % isobu-tane. This specific gas mixture was chosen to have high drift velocity ofionized electrons and low transverse diffusion.Argon, which is the majority of the mixture, has low ionizing energythat allows electrons to be ionized easily. CF4 was added to increase thedrift velocity and decrease the transverse diffusion. CF4 and isobutane pre-vent quenching, a process in which there is continuous electric discharge.Quenching results when the energy transfer to an argon atom is not enoughto ionize it, but high enough to produce an excited state, resulting in pho-ton emission. The photon can produce additional electrons that will end up734.6. The Details of the ND280 Trackercontinuously discharging the detector, possibly blinding, and even damagingthe Micromegas. CF4 and isobutane are added to absorb these photons.Detector Configurations and Read-out SystemEach TPC is divided into an inner volume (IV) and outer volume (OV). TheOV is filled with CO2 to provide electrical insulation for the IV, becauseCO2 has a high breakdown voltage. The IV is divided into two section by acathode at the center and a high voltage is applied between the cathode andthe anode planes on the side of the detector. It creates a uniform electricfield along the drift direction. Since the differential pressure between IVand OV can cause changes to the electric field and distort electron drift,the differential pressure is kept at 0.4 ± 0.1 mbar. The OV is operatedwith slightly higher pressure relative to atmosphere, so that air is not drawninto the OV. The differential pressure between the atmosphere and OV ismaintained to be less than 1 mbar.At each end of a TPC, there are 12 Micromegas positioned in twocolumns. The electrical signal produced by the small amount of charge fromthe ionization electrons alone would be highly susceptible to electrical noise,so amplifying the signal in the gas is desirable. The ND280 TPCs accom-plish this amplification and subsequent readout using Micromegas modules.Each Micromega module consists of fine wire mesh and charge read-out padswith a gap of 128 µm and has 48 rows of 36 pads for a total of 1728 pads.The wire is biased to -350 V, therefore there is a 27.4 kV/cm electric fieldbetween the mesh and the pads. In this region, the ionized electrons areaccelerated and produce a cascade of ionization with a gain of 2000.Laser Calibration SystemThe TPCs have a laser calibration system that generates photoelectrons onthe cathode. These photoelectrons allow the drift velocity and absolute gainof the read-out electrons to be determined.This calibration system consists of aluminum targets on the cathodethat are illuminated by a laser to release photoelectrons. The targets arefixed on the cathodes in a pattern duplicated for each Micromega as shownin Figure 4.25. Emitted photoelectrons were read-out at the Micromegasand used to calculate the drift velocity and the gain because the time andintensity of the later is well controlled.744.6. The Details of the ND280 TrackerFigure 4.25: The pattern of aluminum targets for each Micromegas moduleis shown. Grid represents pads on a Micromegas module and red dots arealuminum targets with 8 mm diameter located at the corners of 4 pads tooptimize the spatial resolution. Two 4 mm wide strips are included in thepatter to measure the transverse size of the ionization [95].Reconstruction and Particle IdentificationAs a charged particle passes through the TPCs, it will leave hits along itstrajectory. A track of the particle, hence, needs to be reconstructed fromthe hits collected by the Micromegas. The raw data is saved in waveformsof charges as a function of time with the sample rate of 25 MHz.The zero suppression and peak finding are applied to the waveforms,then the peaks are used for the track reconstruction. The first step of re-construction is clustering of the waveforms. The waveforms from the samecolumn of Micromega pads are grouped into a vertical cluster if the wave-forms overlap in time and the pads are consecutive in space. The same isdone for waveforms from the same row as well. Figure 4.26 shows a simplesketch of the clustering. On the right, the waveforms which correspond toeach of the yellow pads in the column on the left are shown. A track fittingalgorithm connects the clusters to create a track. Two consecutive clustersare joined to a segment if they overlap in time and are contiguous in they direction. Then the algorithm navigates all the segments to look for thelongest possible track. Figure 4.27 briefly shows how the algorithm works.For a reconstructed track, track momentum and charge are determinedby its curvature. As particles are curved in the detector due to the magnetic754.6. The Details of the ND280 TrackerFigure 4.26: Sketch of the vertical clustering [113]. On the left, padson the Micromegas are shown and three yellow pads are consecutive. Thecorresponding waveforms of these pads are shown on the right and three ofthem overlap in time. Therefore, all yellow pads are grouped in the samecluster.fields, the trajectory is not straight but helical. In addition, the energy loss(dE/dx) is measured, which is used for particle identification (PID). Thebasic building block of dE/dx measurement of a track is the total energyof each cluster (CC) constituting the track. CC is computed as the sumof all the charged deposited on all the pads in a cluster. This is used toestimate the linear charge density of the track (i.e. dE/dx) dividing CC bythe length of the track segment corresponding to the cluster. The dE/dxfor a specific particle is modeled by the Bethe formula, which takes intoaccount the velocity, material, and magnitude of the charge of the particle.Therefore, a particle type can be determined by comparing dE/dx measure-ments to predictions because it is usually different for each particle type ata given momentum. Figure 4.28 shows comparisons of dE/dx measurementsin TPCs to the predictions with each particle hypothesis.The quantity used to determine which particle hypothesis should be764.6. The Details of the ND280 TrackerFigure 4.27: Sketch of the track forming algorithm [113]. On the left,four clusters are connected as shown in yellow and two blue clusters forminganother track. On the right, the joining of two clusters as a segment iszoomed to show how the pattern recognition works in 3D. They overlap notonly in space, but also in time.chosen is the pull, defined as:Pulli =(dE/dx)measured − (dE/dx)expected,iσ(dE/dx)expected,i(4.1)where i denotes a given particle type which can be either muon, electron,proton or pion. σ is a standard deviation of the expected dE/dx distribution.Then, the pull for each hypothesis is normalized as:Li =e−(Pulli)2Σle−(Pulll)2 (4.2)where l denotes a particle type (muon, electron, electron, and pion). PIDcuts are determined by these likelihood values based on the ability to prop-erly identify simulated particles.774.6. The Details of the ND280 TrackerFigure 4.28: Comparison of dE/dx measurements to the theoretical predic-tions - muons (black), electrons (pink dashed), protons (red dashed), pions(white dashed) [95].78Chapter 5Event Selection and AnalysisVariablesAs discussed in Chapter 3, neutrino oscillation analyses require the recon-struction of the incoming neutrino energy to be modeled accurately. Thishas always been challenging because there is a bias in the neutrino energyreconstruction due to assumptions about the neutrino interaction. Neutrinointeractions are typically categorized by final state topology and these cate-gories contain contributions from different underlying neutrino interactionsthat are kinematically different. Since these kinematic assumptions are usedin inferring the neutrino energy, it is essential that this is modeled accurately.In T2K, CCQE interactions are defined by the final topology of one muonand no pion (CC0pi) without explicitly reconstructing the proton. Thereforeonly the lepton kinematics is reconstructed and used for analyses. Given thebroad incident neutrino energy spectrum, it is difficult to isolate the nucleareffects that may distinguish interaction mode from another with just thelepton kinematics. If both lepton and hadron kinematics are accountedfor, nuclear effects can be quantified in the measurement. If there are nonuclear effects, muon and proton kinematics should be balanced. Therefore,any observed imbalance between muon and proton kinematics will be anindication of nuclear effects.This analysis measures differential cross sections which quantify the kine-matic imbalance in events where one muon, no pion and any number ofprotons (CC0piNp (N ≥ 1)) are measured at ND280. Since the kinemat-ics are obfuscated due to the limited detector resolution and efficiency, thisanalysis will present unfolded and efficiency corrected results, using MonteCarlo simulation to establish the relationship between the true and measuredquantities.Following a brief description of the ND280 data and MC samples in Sec-tion 5.1, the selection criteria of CC0piNp will be introduced in Section 5.2,including selection efficiencies and potential backgrounds. In Section 5.3,the analysis variables will be defined. Section 5.4 details relevant systemat-ics sources. Section 5.5 presents distributions of the analysis variables for795.1. Data Samplesthe selection including uncertainties.5.1 Data SamplesFor the analysis, T2K Runs 2-4 were used with 5.73 × 1020 POT. A run isa data taking period with continuous beam operation. Runs 2-4 were takenbetween 2010 winter and 2013 spring. The T2K Computing group pro-duced MC simulations with 10 times more statistics than the data. The MCsamples used for ND280 analyses in 2016 are called Production 6 and wereproduced with two different generators - NEUT and GENIE. The equivalentPOT of the MC samples produced with each generator, and the POT of thedata in each run period are listed in Table 5.1. There are two configurationscalled air and water depending on the P0D water in or out. The MC samplesused for the analysis is production 6B unless otherwise specified.NEUT/GENIE data(POT/1021) (POT/1020)Run 2 water 1.2015 0.428Run 2 air 0.0922 0.355Run 3b air 0.0448 0.215Run 3c air 2.6300 1.348Run 4 water 3.4990 1.6248Run 4 air 3.4965 1.7624Table 5.1: MC and data POT for each Run and its configuration.5.2 Event SelectionThis analysis looks for post-FSI topology with one muon, no pion, andany number of protons as the signal. This topology will be referred as theCC0piNp selection. When there is more than one proton reconstructed, themost energetic proton is used for the analysis. This selection is expectedto contain not only CCQE events, but also CCRES interactions followedby pion absorption, multi-nucleon processes, and any other process where amuon, proton, but no pion emerges from the interaction.805.2. Event Selection5.2.1 Event ReconstructionAs discussed in Chapter 4, ND280 is composed of several subdetectors. TheND280 reconstruction finds an interaction vertex and the associated tracksfor a given event, mainly using the tracker - TPCs and FGDs. There aretwo types of tracker tracks depending on which components of tracker areused: TPC-FGD tracks, which will be referred to as “TPC tracks” in whatfollows, and FGD-only tracks, which will be referred to as “FGD tracks”.TPC track This is a track starting from an FGD which crosses at leastone TPC and produces more than 18 hits. Using the curvature of the trackunder the magnetic field, the particle momentum can be measured via theLorentz force law, as well as its charge.FGD-only track This is a track which is fully contained in an FGD.Therefore it is not matched to any of TPC segments. Since it does not haveTPC information, momentum is reconstructed using the particle energy lossin the FGD, which is called momentum-by-range.5.2.2 Selection CutsThe selection used in this analysis only includes interactions occurring inthe FGD1 fiducial volume (FV). The FV is geometrically defined as 5 barsdistant from the edge in x and y directions and excluding the first upstreamXY module in z direction. All the reconstructed tracks should be asso-ciated with the vertex found in the FV. These events are split into fourreconstructed topologies based on which detectors are used and how manyprotons are detected. There are 4 sub-topology considered in this analysis:• µ-like TPC track + p-like TPC track• µ-like TPC track + p-like FGD track• µ-like TPC track + Np tracks• µ-like FGD track + p-like TPC track.Figure 5.1 shows the four topologies in the tracker (TPC1 - FGD1 - TPC2).Three of the samples identify the muon as a TPC track. The proton can beeither a TPC track or an FGD track. The other identifies the proton as aTPC track with the muon as an FGD track.815.2. Event SelectionFigure 5.1: A simple breakdown of the different event topologies, splitdepending on the detectors were used to reconstruct the events and thenumber of protons identified.• µ-like TPC track: The following describes three samples that requirethe muon as a TPC track.1. Identify a vertex from the highest momentum negative(HMN) TPC track: Find the HMN track among TPC tracks.This selected track is considered as the muon candidate.2. TPC Muon PID: The muon candidate should have a PID con-sistent with a muon hypothesis.3. Only one negative track: There should not be any other neg-ative track beside the muon candidate.4. No Michel electron: The Michel electron is an electron result-ing from a pion stopping and decaying in the detector. If thereis a Michel electron identified, it indicates a pion emitted to thefinal state, but stopped and decayed in the detector. Therefore,it is required that there are no Michel electrons to reject anyinteractions with pions.5. pi0 Veto: Since a pi0 decays immediately to photons, pi0 inducedevents can be rejected by identifying photons in ECal (as ener-getic objects).Once the muon is identified, there are three categories dependingon a number of proton reconstructed and its detector use.825.2. Event Selection– p-like TPC track6. Only one TPC positive track: There is only one posi-tively charged TPC track, which is considered as the protoncandidate.7. TPC Proton PID: The proton candidate should have aTPC PID consistent with a proton hypothesis.– p-like FGD track6. Only one FGD track: There are no positive TPC tracksin the event, and only one FGD track, which is considered asthe proton candidate.7. FGD Proton PID: The proton candidate should have anFGD PID consistent with a proton hypothesis.– Np6. More than one track: There are other positive TPC tracksor FGD tracks. These are considered as proton candidates.7. Proton PID: All of the proton candidates should pass eitherTPC PID or FGD PID cuts depending on the track type.If any of them fails the proton PID criteria, the event isrejected.• µ-like FGD track + p-like TPC track: The following describes thesample that requires the proton as a TPC track and the muon as anFGD track.1. Identify a vertex from the highest momentum positive(HMP) TPC track: If there are no negative TPC tracks, pos-itive TPC tracks are considered and the HMP is identified. Thisselected track is a proton candidate.2. Only one FGD track: There should be one FGD-only trackwhich will be a muon candidate.3. FGD Muon PID: The muon candidate should have an FGDPID consistent with a muon hypothesis.4. Common Vertex: The muon and proton candidates shouldshare a common vertex to make sure both of tracks are fromthe same interaction.5. Stopping Muon: The muon candidate should be fully containedin the FGD. This is required to reject a case of broken muon track835.2. Event Selectionwhich actually crosses a TPC but not correctly reconstructed dueto the reconstruction failure.6. TPC Proton PID: The proton candidate should have a TPCPID consistent with a proton hypothesis.Tables 5.2 and 5.3 summarize the numbers of selected events and signalsover the selection cuts. The numbers of total events are the events passedeach cut and the numbers of signal events are the selected events whichhave true CC0piNp topology. For the analysis, all the four topologies arecombined as a complete CC0piNp selection.Cut DataMCTotal SignalPurityEvents EventsHNM Vertex 12648 13763 2284 0.17Muon PID 12648 13763 2284 0.17Only 1 Negative TPC Track 11970 12849 2284 0.18No Michel Electron 11226 12167 2270 0.19pi0 Veto 10129 11003 2144 0.19Only 1 Positive TPC Track 2769 2964 2067 0.70Proton PID 1447 1696 1234 0.73Only 1 FGD Track 2769 2964 2067 0.70Proton PID 1322 1267 833 0.66More than one Proton 108 155 77 0.50Table 5.2: Numbers of selected events and signals (true CC0piNp) of topolo-gies with a µTPC track over the selection cuts. NEUT MC is used andnormalized to data POT.845.2. Event SelectionCut DataMCTotal SignalPurityEvents EventsHNP Vertex 1428 1473 1148 0.78Only 1 FGD Track 1428 1473 1148 0.78Muon PID 1428 1473 1148 0.78Common Vertex 1263 1282 1036 0.81Stopping Muon 948 914 739 0.81Proton PID 928 889 725 0.82Table 5.3: Numbers of selected events and signals (CC0piNp) ofµFGD+pTPC topology over the selection cuts. NEUT MC is used andnormalized to data POT.5.2.3 Selection Efficiency and Phase-space LimitsThe extraction of a cross section requires knowledge of the selection efficiencyin order to account for signal events which were not included in the selection:σ =NselectedMCΦNnucleons. (5.1)A cross section(σ) is extracted as the number of the selected events (Nselected)corrected by the selection efficiency predicted from MC (MC), then normal-ized by the neutrino flux (Φ) and the number of nucleons in the target massof the detector (Nnucleons). A small efficiency results in a large correction,with the possibility that any model dependent errors can be greatly ampli-fied. This can be mitigated by making a signal definition that is restrictedto a kinematic phase space region where the efficiency is high. The efficiencyis defined as:Efficiency =True Signal Events in the CC0piNp SelectionTrue Signal Events. (5.2)As the measurement includes both protons and muons in the selection,the restricted phase-space should be chosen as a region where protons andmuons reconstruction efficiencies are similar. As most of the protons are lessenergetic, with energies close to the detection threshold, and therefore aremore likely to be missed, it is reasonable to consider protons only above thedetector threshold.855.2. Event SelectionFigure 5.2: Selection efficiencies as function of true µ (top) and p (bottom)kinematics.Figure 5.2 shows the selection efficiencies as function of true muon andproton kinematics for NEUT (black) and GENIE (red) for CC0piNp. Thebottom two plots show that low momentum and backward-going protonshave very low efficiencies. These low efficiencies at large angles, and alsolow momentum, stems from the detector acceptance. Hence, the final phase-space limits are: pp > 450 MeV/c and cos θp > 0.4. Figure 5.3 shows theefficiencies as function of true p kinematics after these phase-space limits.865.2. Event SelectionFigure 5.3: Selection efficiencies after p phase-space constraints as functionof true µ (top) and p (bottom) kinematics.Muon phase-space BinningFigure 5.4: Event populations for CCQE (top), CCRES (bottom left) and2p2h (bottom right) as function of µ p-θ.875.2. Event SelectionIt is also important to retain interesting signal regions that are likely tobe sensitive to nuclear effects. For this reason, unlike the p phase-spacewhere limits are applied, the muon phase-space is binned in momentum andangle. The muon phase-space bins are determined mainly based on NEUTselection efficiencies, GENIE/NEUT efficiency differences, and CCQE/CC-non-QE distributions across the phase-space. Figure 5.4 shows how eventsfrom specific neutrino interactions are populated in the µ p− θ plane. CC-non-QE events are enhanced at low momentum and very forward angles.Figure 5.5 shows the true muon phase-space for true CC0piNp. The finalmuon phase-space bins are:• Bin 0: cos θµ < −0.6• Bin 1: −0.6 < cos θµ < 0.0 & pµ < 250 MeV/c• Bin 2: −0.6 < cos θµ < 0.0 & pµ > 250 MeV/c• Bin 3: cos θµ > 0.0 & pµ < 250 MeV/c• Bin 4: 0.0 < cos θµ < 0.8 & pµ > 250 MeV/c• Bin 5: 0.8 < cos θµ < 1.0 & 250 < pµ < 750 MeV/c• Bin 6: 0.8 < cos θµ < 1.0 & pµ > 750 MeV/c.Figure 5.5: The true number of events are shown in the true muon phase-space, taken from NEUT prod6B MC. 7 muon phase-space bins are indicatedwith red lines.885.2. Event Selection5.2.4 BackgroundsTo understand the limitations of the selection, it is useful to consider back-grounds by selection failure mode. There are three main backgrounds: pi0tagging failure, pi+ mis-identification and pi+ tagging failure.pi0 tagging failure When a pi0 is produced in a neutrino interaction, itdecays immediately to photons which are identified in the ECals. If thistagging fails, then the pi0 is not identified, and background events with pi0can migrate into the signal.pi+ mis-identification For low momentum pions and protons, dE/dxcurves are not so distinguishable. Therefore, low momentum pions are oftenmis-reconstructed as protons. In this case, a CC1pi event would be acciden-tally tagged as a CC0pi.pi+ tagging failure In cases where the pion cannot be tracked and isidentified instead by the Michel electron, if the Michel electron detectionfails, then these CC1pi background events will migrate into the signal.The breakdown of these backgrounds are shown for each generator in Ta-ble 5.4. From this, it can be seen that over two-thirds of the backgroundscome from failing to tag a pi+. The background treatment will be discussedin Section 6.1.MC Generator NEUT GENIETotal 3,898 3,649BackgroundsTotal 476 555pi0 Tagging Failure 87 101pi+ Mis-identification 15 19pi+ Tagging Failure 341 414other 33 22Table 5.4: Background compositions in the CC0piNp selection for threegenerators. The main contribution is from pion tagging failure which is72% (NEUT) and 75% (GENIE) of the total background. The numbers arescaled to data POT.895.3. Analysis Variables5.3 Analysis VariablesThis analysis employs a comparison between the expected kinematics of aproton in a CC0piNp signal event inferred from the measured muon kinemat-ics, assuming it is a CCQE interaction and four-momentum conservation,and the measured proton kinematics. While nuclear effects, such as Fermimotion and FSI impact the kinematics of the outgoing program, causing itto deviate from what would be inferred, CC-non-QE interactions will addi-tionally have imbalances resulting from the different underlying kinematics(e.g. ∆ production or multi-nucleon scattering) that may induce furtherdeviations. Thus it can be expected that a comparison of the inferred andreconstructed kinematics of the proton may be sensitive to the underlyinginteractions.Figure 5.6 illustrates how this may work for CCQE and multi-nucleonejection events. In the left diagram, for a CCQE event with no nuclear ef-fects, we expect that the reconstructed and inferred kinematics will match.On the right, for a multi-nucleon interaction, the assumption of a two-bodyCCQE interaction is incorrect, hence one expects that the inferred and re-constructed quantities will not match. Therefore variables are defined toquantify this difference.Figure 5.6: A diagram to show the analysis variable calculation. Theleft diagram shows CCQE dynamics where only two bodies are in the finalstate, while CC-non-QE has another particle even though it is not detected(on the right). Black arrows indicate reconstructed momentum that thedetector actually measures and blue arrows are for inferred momentum frommeasured µ kinematics under CCQE assumption. Then we expect to see thebias only in CC-non-QE cases.In the calculation of inferred proton kinematics, the neutrino energy isfirst calculated using the muon kinematics only, assuming a CCQE interac-905.3. Analysis Variablestion:Erecν ≈m2p −m2µ + 2Eµ(mn − V )− (mn − V )22[(mn − V )− Eµ + pµ cos θµ] (5.3)where n denotes the neutron, p denotes the proton, V is the mean bindingenergy (25 MeV for carbon).Setting the neutrino direction as the z-axis, the inferred proton kinemat-ics from four-momentum conservation would be as:pν + pn = pµ + pp. (5.4)With four-momenta defined as follows:pν = (Eν , 0, 0, pν,z)pn = (mn, 0, 0, 0)pµ = (Eµ, pµ,x, pµ,y, pµ,z)pp = (Ep, pp,x, pp,y, pp,z). (5.5)Comparing component by component, the following can be obtained:Ep = Eν − Eµ +mppp =√(E2p −m2p)cos θp =(E2ν + p2p − p2µ)2Eνpp(5.6)(5.7)pp = (−pµ,x,−pµ,y,−pµ,z + Eν). (5.8)With the proton kinematics completely inferred, analysis variables aredefined to quantify the difference between these inferred kinematics andthose actually reconstructed as:• ∆pp = |pmeasuredp | − |pinferredp |• ∆θp = θmeasuredp − θinferredp• |∆pp| = |pmeasuredp − pinferredp |.915.4. UncertaintiesNote that |∆pp| is considered because it contains the momentum-angle cor-relation.Since the event distributions of these variables will be unfolded and ef-ficiency corrected, it is important to understand the CC0piNp selection ef-ficiencies as functions of these variables in binned muon phase-space bins.The detailed efficiency distributions can be found in Appendix A.5.4 UncertaintiesThe cross-section extraction relies on MC predictions where relevant physicsis modeled. If the underlying physics or detector response is not modeledproperly, it can introduce systematic uncertainties. Variations which repre-sent potential errors in the physics or response are propagated through theanalysis to assess their impact on analysis observables.There are two main systematics sources: detector and model systematicuncertainties.5.4.1 Detector UncertaintiesThe observed distributions of the analysis variables for the selection dependon the ND280 detector resolution and efficiency. Therefore, uncertainties re-lated to the detector modeling must be estimated. The detector systematicsconsidered for this analysis follows the one officialized by a T2K systematicsworking group [114, 115]. The uncertainties are estimated by comparingrelevant quantities in Monte Carlo simulation to data in control samples toassess how well the simulation matches the data. Discrepancies between thedata and MC are the basis for setting the magnitude of the uncertainty.There are two main ways to assign the systematic uncertainties from thedata/MC differences: observable variation and efficiency weighting.• Observable Variation: This treats the systematics where the re-constructed quantities have uncertainties on themselves. The differ-ence of the observables in data and MC are assigned as systematics. Toestimate the systematics, pseudo-experiments (toys) are used varyingthe reconstructed quantities of MC as:x′toy = xMC + α∆x (5.9)where xMC is a reconstructed variable of MC, α is a random variable,and ∆x is the difference between data and MC. A covariance matrix925.4. Uncertaintiesof a binned distribution is calculated over pseudo-experiments (toys)varying the reconstructed quantities of MC as:covij =1NtoysNtoys∑s=1(N sj − µi)(N sj − µj) (5.10)where N si is a number of selected events in bin i of sth toy and µi isan average of all the toys.– TPC Momentum resolution: The momentum of a muonpassing through more than one TPC is measured in each TPC,accounting for energy loss in passing through the FGDs. For agiven TPC track, the inverse of the transverse momentum to themagnetic field (1/pt) is used, because fluctuation on 1/pt due tothe energy loss in FGDs is relative small to the TPC momen-tum resolution. The difference of 1/pt measured in each TPCquantifies the resolution, since they constitute independent mea-surements of the same quantity. It should follow a Gaussian dis-tribution centered at 0 and its standard deviation is the momen-tum resolution. It is done over a range of momenta and angles.The resolution is compared between data and MC as shown inFigure 5.7. Since data is found to have a worse resolution, asmearing factor is calculated to correct the data/MC difference.The data/MC difference mostly comes from the magnetic fielddistortion and Micromegas alignment.935.4. UncertaintiesFigure 5.7: TPC momentum resolution (1/σ(∆1/pt)) as a function of trans-verse momentum pt [114].– TPC PID: TPC PID depends on dE/dx measurements. Mis-identification of PID can cause events migration between the sig-nal and background. The systematic is calculated over the muon,proton, and electron hypotheses. High purity samples of muonsand protons are used to determine the TPC PID systematics.Pull distributions are calculated for both data and MC and MCis corrected to match the data, with the difference taken as anuncertainty. Figure 5.8 shows the mean and sigma of the pull formuon candidates.945.4. UncertaintiesFigure 5.8: TPC PID pull mean (circle) and sigma (triangle )of data (black)and MC (data) for muon candidates as a function of the momentum areshown [114].– FGD PID: Since FGD tracks do not contain TPC segments,energy loss in the FGD is used instead for particle identification.As with the TPC PID systematics, pure muon- and proton- sam-ples are used to estimate the systematics and the systematic iscalculated over the muon and proton hypotheses. The data/MCdifference comes from the improper simulation of FGD dead chan-nels and coating thickness.• Efficiency Weighting: This treats the systematics correspondingto a detection probability. An efficiency is calculated in both dataand MC, and a weight is calculated as the ratio of the two: weff =data/MC . A covariance matrix is calculated over pseudo-experiments(toys) weighting the efficiencies in MC as:covij =1NtoysNtoys∑s=1(N si −NMCi )(N sj −NMCj ) (5.11)whereN si =∑wseff,i (5.12)955.4. Uncertaintiesand NMCi is a number of the selected MC events in bin i.– Charge identification: Muon candidates are found by look-ing for a negative track which depends on charge identificationof TPC segment. The charge of a track is determined by itscurvature in the magnetic field. Therefore the probability ofcharge mis-identification depends on the momentum, length andthe number of hits of the reconstructed track. The probability isparametrized by the errors on the momentum measurement andevaluated both in data and MC.– TPC track reconstruction efficiency: The analysis selectionheavily relies on TPC-FGD tracks. Reconstruction failures canlead to mis-identification of the signal, therefore event migrationfrom and to the backgrounds. A control sample of through-goingmuons is used to estimate the tracking efficiency for both dataand MC. Figure 5.9 shows the TPC2 reconstruction efficiency asa function of the momentum in binned track angle.Figure 5.9: TPC2 track reconstruction efficiency using through-going muonsis calculated as a function in binned momentum and angle for data (black)and MC (red) [114].965.4. Uncertainties– FGD track reconstruction efficiency: For FGD tracks,where the track does not reach a TPC, a control sample of stop-ping muons are protons are used separately to evaluate the FGDtracking efficiency.Figure 5.10: FGD track reconstruction efficiency of protons fully containedin FGD1 is shown. The efficiency is calculated as a function of the trueangle between the proton and muon candidates. Data is shown in red andMC is in blue [114].– TPC-FGD matching efficiency: Once TPC tracks are found,Matching failure can lead to localize a wrong position of the eventvertex, hence to miscount the event selection. TPC-FGD match-ing efficiency is evaluated using control samples of through-goingmuons from cosmic and sand (muons from interactions on thesand surrounding the ND280 hall) in data and MC.975.4. UncertaintiesMC DataEfficiency (%) 56.5 ± 0.94 56.4 ± 0.16Table 5.5: Michel electron tagging efficiencies of data and MC in FGD1 withstatistical uncertainties are shown [114].Figure 5.11: FGD1-TPC2 matching efficiency is shown as a function of themomentum for data (black) and MC (red) [114].– Michel electron tagging: Since pions are rejected by taggingMichel electrons, it is important to understand the Michel elec-tron tagging efficiency. There are mainly two kinds of uncertaintysources - tagging failure of Michel electron and mistaken of non-electron. The efficiencies are calculated using a control sample ofstopping cosmic muons in data and MC.• Secondary interactions: Secondary interactions (SI) are interac-tions occurring within the detector after leaving the nuclear medium.Pion and proton SI are considered to alter the selection efficiency andkinematics significantly. The modeling of the detector geometry andthe proton/pion interactions are found to be different from the avail-able external data. This difference is used to generate an event-by-event weight to vary the normalization of the events undergoing SI985.4. UncertaintiesFractional Uncertainties (%)TPC momentum resolution 0.37TPC PID 0.46FGD PID 0.20TPC charge ID 0.35TPC reconstruction 0.34FGD reconstruction 1.48TPC-FGD matching 0.20Michel electron tagging 0.03Secondary interactions 5.65Total 5.90Table 5.6: Fractional uncertainties on the CC0piNp selection from eachdetector systematic source is shown. The dominant systematic is from sec-ondary interactions. The total detector uncertainty is 5.90 %.as:we = 1 + α · δe (5.13)where α is a random variable and δe is a systematic associated to thedata/MC difference for the SI events.Table 5.6 summarizes the fractional uncertainties of each detector sys-tematic source on the CC0piNp selection.5.4.2 Model UncertaintiesModel uncertainties include systematic uncertainties due to flux, cross-sectionand FSI models. This analysis follows the systematics studies by variousT2K working group, described in [116] (flux) and [117] (cross-section andFSI models).Cross section measurements of neutrino interaction depend on neutrinoenergy, flavor, and direction which are determined by neutrino beam flux.Given a flux prediction, a neutrino generator simulates the neutrino interac-tion, the resulting particles and kinematics in the detector according to thecross-section models. Changes in the models can change the properties ofthe neutrino interactions and the resulting particles and kinematics. It willaffect the event selection as the efficiency and backgrounds will be altered.995.5. Distributions of Analysis VariablesThe systematics due to the poor understanding of the models in theneutrino generators are estimated by reweighting model parameters usingT2KReWeight, a software package that estimates the effect of model pa-rameter changes on observables at the detectors. Instead of running thewhole MC change with different parameter values, T2KReWeight calculateschanges to the observables for different sets of theoretical parameter values.The ratio is calculated with respect to the nominal distributions. This ratiois called a weight for a given event, which represents how much changes oftheoretical parameters affect the observables. The weights are applied to theevents, already generated in MC, then the event selection and correspondingobservable distributions are altered.The full list of the parameters is shown in Appendix B.5.5 Distributions of Analysis VariablesFigures 5.12 – 5.14 show the distributions in binned muon phase-space ofreconstructed kinematics. Data is shown in black and NEUT predictionsare shown separated by primary neutrino interaction modes. Detector andmodel uncertainties are shown on NEUT predictions as well in red.1005.5. Distributions of Analysis VariablesFigure 5.12: Reconstructed distributions (black data points) of ∆pp inmuon phase-space bins are compared to NEUT predictions in primary neu-trino interaction modes: CCQE (red), 2p2h (purple), CCRES (green), DIS(blue), coherent scattering (cyan) and others. CCQE, 2p2h and CCRESare most dominant interactions contributed to this selection with small con-tributions from DIS and COH. Red data points with error bars representsystematic uncertainties on the NEUT predictions.1015.5. Distributions of Analysis VariablesFigure 5.13: Reconstructed distributions (black data points) of ∆θp inmuon phase-space are compared to NEUT predictions in primary neutrinointeraction modes: CCQE (red), 2p2h (purple), CCRES (green), DIS (blue),coherent scattering (cyan) and others. CCQE, 2p2h and CCRES are mostdominant interactions contributed to this selection with small contributionsfrom DIS and COH. Red data points with error bars represent systematicuncertainties on the NEUT predictions.1025.5. Distributions of Analysis VariablesFigure 5.14: Reconstructed distributions (black data points) of |∆pp| inreconstructed muon phase-space bins are compared to NEUT predictions inprimary neutrino interaction modes: CCQE (red), 2p2h (purple), CCRES(green), DIS (blue), coherent scattering (cyan) and others. CCQE, 2p2hand CCRES are most dominant interactions contributed to this selectionwith small contributions from DIS and COH. Red data points with errorbars represent systematic uncertainties on the NEUT predictions.103Chapter 6Cross-section Extraction andValidationWith the selection and observables described in Chapter 5, the differentialcross sections are extracted in the binned muon phase-space. The measureddistributions at the detector are smeared due to the detector resolutionand inefficiency. Hence a comparison to a prediction will require runninga full MC simulation in order to determine the observable distributions fora given theoretical model. Instead, by correcting for efficiency, resolution,and background, the prediction can be directly compared. This process iscalled unfolding. In this section, the unfolding method used to extract thecross sections for this analysis and its validation will be discussed.First, in Section 6.1, the iterative Bayesian unfolding method by D’Agostiniwill be explained including the propagation of the systematic uncertainties.The method is validated with various fake data samples as described inSection 6.2.6.1 Iterative Bayesian UnfoldingTo extract the cross section, the reconstructed distributions will be unfoldedto the true distributions to account for detector inefficiency and resolution.The relation between the true and measured spectrum can be written as:Ej =Nt∑i=1SjiCi (6.1)where Ci is the number of events in true bin i, Ej is the number of eventsin measured bin j, Sji is the smearing matrix, and Nt is the number of truebins.The smearing matrix is constructed from the MC prediction which givesthe information about event migrations and selection efficiencies. There areseveral ways to construct an unsmearing matrix. The most straight-forwardway is a matrix inversion in which the unsmearing matrix is simply the1046.1. Iterative Bayesian Unfoldinginverted smearing matrix. Other methods like Singular Value Decomposi-tion (SVD) [118] or least square fitting [119] have been introduced. Theone used in this analysis is the iterative Bayesian unfolding, proposed byD’Agostini [120, 121]. It uses Bayes’ theorem to derive an unsmearing ma-trix from the smearing matrix, and then iterates the procedure to updatethe prior assumptions. The unsmearing matrix is defined as:Uij =Peff (Ej |Ci)P0(Ci)∑Nti=1 P(Ej |Ci)P0(Ci). (6.2)P(Ej |Ci) is the probability of true events in bin i measured in bin j:P(Ej |Ci) = NjiCi(6.3)where Nji is the number of true events in bin i measured in bin j, predictedin MC. Peff (Ej |Ci) is normalized by the selection efficiency of bin i as:Peff (Ej |Ci) = P(Ej |Ci)∑Nmj=1NjiCi(6.4)where Nm is number of measured bins. P0(Ci) is a prior which is the pre-dicted number of events in bin i. For the first iteration, it directly comesfrom MC as:P0(Ci) =Ci∑Nti=1Ci. (6.5)The unfolded spectrum is given by:C′i =Nm∑j=1UijEdataj . (6.6)where Edataj is a measured spectrum of data. After each iteration, P0(Ci) isupdated with the posterior resulting from the previous iteration.6.1.1 Background TreatmentTo treat backgrounds, D’Agostini introduces an extra bin in the true spec-trum, which is called a trash bin, where all the backgrounds fall into. It takesinto account the event migrations between the signal and backgrounds.This is called the purity correction. Figure 6.1 describes the matrix mul-tiplication in the case of unfolding with the purity correction. The “unfold-ing Matrix” is built as a Nt×Nm matrix, one extra row for the background1056.1. Iterative Bayesian UnfoldingFigure 6.1: Diagram of how the purity correction works.is added as shown separately in Figure 6.1. The resulting matrix, hence, isa (Nt+ 1)×Nm matrix. The matrix is applied to the measured spectrum Eto get the true spectrum C′. The background events will be moved to thetrash bin in the true spectrum which is labeled as B′. As an example, thesimplest case of unfolding one reconstructed bin to one true bin is consideredas an example. For the first iteration, the unfolding matrix will be:U11 =P (E1|C1)P (E1|C1)+P (E1|Cb)P0(C1)P (E1|C1)P0(C1) + P (E1|Cb)P0(Cb)=N11N11 +N1b1N11C1(6.7)where N1b is the number of background events in the measured bin. This ispredicted from MC. Therefore, the unfolded spectrum is written as:C′1 =N11N11 +N1bC1N11Edata1 . (6.8)The first term of Equation 6.8 is the purity of the selection. In the casewhere there is no trash bin, Equation 6.8 will become:U11 =1P (E1|C1) (6.9)=C1N11, (6.10)1066.1. Iterative Bayesian Unfoldingwhich corrects the measured spectrum only by the selection efficiency.Sidebands As the purity correction uses MC predictions to estimate thebackgrounds in the measurement, it introduces another source of model de-pendence. To minimize the model dependence, instead of relying on the MCprediction, an additional selection, called a sideband sample, is introduced.Sideband samples are chosen to describe the background distributions well.Since pion tagging failure background due to Michel electron tagging fail-ures is the largest contribution, CC0piNp events with a successfully taggedMichel electron sample can be a good sideband sample for the given back-ground. As the tagging efficiency is largely independent of the kinematics,the kinematics of the sideband sample should be representative of the back-ground events where the Michel electron tagging failed. The distributionsof the sideband sample can be found in Appendix C.Figure 6.2: Diagram of how the sideband sample is unfolded simultaneously.SEL denotes the selection spectrum and SB is the sideband spectrum.The scheme of using the sideband is described in Figure 6.2. It is sim-ilar to adding a trash bin in the purity correction. The difference is thatinstead of adding only one trash bin in the true spectrum for the back-ground, the matrix is extended to the sideband samples accordingly. Inthis analysis, there is only one sideband sample considered with the same1076.1. Iterative Bayesian Unfoldingnumber of true bins and reconstructed bins as the CC0piNp signal events.Hence, the resulting unfolding matrix is a (2Nt + 1) × (2Nm) matrix. Inthis way, one can properly take into account the event smearing betweenthe main selection and sidebands without relying on MC predictions of thebackground. Hence it will give us less model-dependent background predic-tions. The backgrounds without specific sideband samples will be migratedto the background bins in the unfolding matrix as described in Figure 6.1.6.1.2 Simultaneous UnfoldingHaving binned muon phase space gives the benefit of separately examiningregions where the expected contributions of CCQE and CC-non-QE eventsare different. In the previously mentioned schemes, each muon phase spacebin would be unfolded independently, with its own smearing matrix andpurity corrections, hence events with true kinematics in another muon phasespace bin that migrates into another bin are considered background. Theseevents will be corrected as backgrounds in that bin, even though they areCC0piNp events. In addition, the migration and correlations between muonphase-space bins will have to be properly included. In order to account for allthis, all muon phase-space bins will be unfolded simultaneously. The muonphase-space bins are added consecutively, followed by the sideband sampleand the trash bin. This unfolding scheme, which is used in the analysis, isshown in Figure 6.3. Each E(i) and C(i) are measured and true spectra ofith muon phase-space bin. E(SB) and C′(SB) are spectra of the sidebandsample (integrated over the muon phase-space). B′is the trash bin whereother background events are unfolded to. Hence, the resulting unfoldingmatrix is a (7Nt +NSB,t + 1)× (7Nm +NSB,m) matrix.6.1.3 The Propagation of the UncertaintiesTo estimate the systematics, the muon and proton kinematics (both trueand reconstructed) in MC are varied according to the systematic sources(explained in Section 5.4). A varied MC (a toy MC) accounts for impactsof the model systematics and is used to unfold a given data. By repeatedly(500 toys) unfolding the same data with unfolding matrices defined by sys-tematically varied MC, the uncertainties are propagated into the unfoldeddistribution and a covariance matrix, defined as:covij =1NN∑k=1(resultk,i − resultnominal,i)(resultk,j − resultnominal,j). (6.11)1086.2. ValidationsFigure 6.3: Diagram of how the simultaneous unfolding works. C denotestrue, E denotes measured spectra.where N is the number of toys. The indices i, j run across all the bins ofa given analysis variable, which sequentially accounts for the muon phase-space bins as well. In addition to the systematic uncertainties, the statisticaluncertainties for a data and MC are separately estimated. Statistical uncer-tainties are estimated varying the reconstructed distribution of a data (2000toys) and MC (2000 toys). The covariance matrix is calculated separatelyfor each systematic source and added together to get the total covariancematrix.6.2 ValidationsTo validate the machinery discussed in Section 6.1, fake data samples areused as “data” to be unfolded with the machinery, defined with an inputMC. Each fake data sample represents a situation where the “data” differsfrom the assumptions in the input MC, but its true distribution is known. Invalidation, the input MC is varied by generating toys, each of which resultsin a varied unfolding that is applied to the fake data sample. The resultingvaried unfolded distributions define a covariance matrix which represents thesystematic uncertainties (see 6.1.3). Separately, the reconstructed distribu-tions of the fake data sample are also varied with a Poissonian distributionto account for the statistical fluctuation and unfolded with the nominal un-1096.2. Validationsfolding. The resulting covariance matrix represents statistical uncertaintieson the “data”. The systematic and statistical covariance matrices are addedtogether to get the final covariance matrix.To quantify the validation, two statistical metrics accounting for thedifferences and covariances are considered: mean bias and χ2. The biasand χ2 are calculated by comparing the unfolded fake data sample with itsexpectation (no statistical fluctuation is considered on the unfolded resultor expectation), using the covariance matrix to represent the uncertainties.The mean bias is the average of a bias weighted by an uncertainty of abin, defined as:〈bias〉 = 1Nbins∑i|unfoldedi − Fake data truthi|σi(6.12)where σi is the error on the unfolded results, derived from the diagonalentries of the covariance matrix. It represents the bias of the result relativeto the uncertainties.The χ2, which quantifies the estimated uncertainties and correlations onthe results, is defined as:χ2 =∑i∑j(unfoldedi − Fake data truthi)cov−1ij×(unfoldedj − Fake data truthj). (6.13)The mean bias and χ2 are considered together as complementary tests.The mean bias tests whether the machinery reproduces the expected valuesof the target distribution. However, a small value of the mean bias can mis-interpret the validity of the machinery. Even if the machinery reproducesthe expectation close enough, there can be the large bin-to-bin correlation.The χ2 tests whether the bias is compatible with the systematic uncertain-ties. In the χ2 calculation, the bias is hidden, because it can be canceledout by large bin-to-bin anti-correlations.6.2.1 Reproducing MC TruthAs a closure check, the nominal MC generated with NEUT is unfolded byitself to ensure that the machinery can recover the true distributions withoutbias. Results are shown in Appendix D.1106.2. Validations∆pp ∆θp |∆pp|-20 % +20 % -20 % +20 % -20 % +20 %Mean bias 0.15 0.12 0.13 0.11 0.15 0.15χ2 1.23 0.78 0.58 0.35 1.27 0.77(ndof) (33) (35) (23) (23) (42) (42)Table 6.1: Mean bias and χ2 for the fake data samples where CCQEvariations is varied in NEUT MC. The number of degrees of freedom (ndof)is the number of non-zero bins.∆pp ∆θp |∆pp|Mean bias 0.36 0.20 0.39χ2 4.97 1.10 6.73(ndof) (35) (23) (42)Table 6.2: Mean bias and χ2 for the fake data samples where the 2p2hcontribution has been removed. The number of degrees of freedom (ndof)is the number of non-zero bins.6.2.2 NEUT-based Fake Data SamplesIn this study, nominal NEUT MC is reweighted to have different fractionsof CCQE contributions, as well as the 2p2h component is removed entirely.In these fake data samples, the underlying model for each component isunchanged, but the contribution from the CCQE and 2p2h are varied. Thefake data samples are unfolded with the nominal assumptions, that is theresponse matrix from the default NEUT MC without varying the componentcontributions is used to construct the unfolding matrix. The differentialcross section for each fake data sample is extracted with 2 iterations andshown in Appendix E.2.In Table 6.1 and 6.1, the mean bias and χ2 are summarized for thesefake data samples. Since the input MC and fake data samples are built withthe same models, the bin-by-bin bias is very small resulting in small meanbias and χ2.These studies show that the machinery can recover the true distributionwith average bin bias smaller than the error in that bin and a small χ2regardless of the fraction of CCQE or 2p2h in the NEUT model.1116.2. Validations6.2.3 GENIE StudiesIn this study, GENIE is used as a fake data sample which will be unfoldedwith the nominal NEUT. As described in Section 3.3 (see Table 3.1), GE-NIE has different models from NEUT: the RFG model, no 2p2h interactions,and different FSI implementation. These model differences result in differ-ences in the unfolding matrix relative to the nominal assumptions of theunfolding as seen in the efficiencies as functions of the analysis variables(see Appendix A). This study tests wheather the machinery is preventedfrom recovering the truth due to the bias from the input MC. If the unfold-ing is significantly impacted by reasonable model variations, the results areinvalid as the data itself may be different from any of the assumed models.The Number of IterationsThe Bayesian unfolding is regularized by choosing the number of iterations,which can induce a bias towards the prior distributions of the nominal MCused to derive the smearing matrix. These are determined through thisstudy where the differences in the unfolding matrix resulting from NEUTand GENIE are significant.Figure 6.4 shows the mean bias and χ2/ndof over the numbers of iter-ations. As the number of iterations increases, the mean bias and χ2/ndofconverge to small numbers. The number of iterations is chosen to have theχ2/ndof close to 1. Therefore, two iterations is chosen for ∆pp, six itera-tions for ∆θp, and four iterations for |∆pp| are chosen, where the χ2/ndofapproaches one. With the chosen iterations, the mean bias is reasonably lowas well. The mean bias is not used in the determination, because it does notrepresent the bin-to-bin correlations. The χ2 value of each variable with thechosen number of iterations is summarized in Table 6.3.∆pp ∆θp |∆pp|χ2 32.5 33.5 47.3(ndof) (46) (34) (49)Table 6.3: The χ2 value of each variable with the chosen number of iterac-tions is shown with the number of degrees of freedom.1126.2. ValidationsFigure 6.4: Mean bias (left) and χ2/ndof (right) of unfolded GENIE withNEUT over the numbers of iterations. The three variables are shown sepa-rately: ∆pp (top), ∆θp (middle), and |∆pp| (bottom).1136.2. ValidationsUnfolded ResultsThe extracted differential cross sections simulated in GENIE are shown inFigures 6.5 ∼ 6.7, ∆pp, ∆θp and |∆pp| respectively. The machinery worksgenerally well with a given number of iterations determined through thestudies and the results of the different variables are consistent. There arecases where the unfolding does not seem to work so well (for example, bin3 of ∆pp and |∆pp|), but the bias is within the uncertainties.The extracted total cross-sections and true cross-sections from NEUTand GENIE are summarized in Table 6.4 to confirm the overall normalizationworks correctly as well.Total cross section σ (×10−38cm2/Nucleon)Extracted NEUT GENIEMC MC∆pp ∆θp |∆pp|0.191 0.192 0.1930.215 0.182± 0.005 (stat) ± 0.005 (stat) ± 0.005± 0.020 (syst) ± 0.020 (syst) ± 0.020Table 6.4: The extracted total cross-sections are shown with true cross-sections: NEUT (MC) and GENIE (fake data sample).1146.2. ValidationsFigure 6.5: Unfolded GENIE with NEUT MC for ∆pp are shown (datapoints) compared to the MC prior (dashed black) and its truth (red).1156.2. ValidationsFigure 6.6: Unfolded GENIE with NEUT MC for ∆θp are shown (datapoints) compared to the MC prior (dashed black) and its truth (red).1166.2. ValidationsFigure 6.7: Unfolded GENIE with NEUT MC for |∆pp| are shown (datapoints) compared to the MC prior (dashed black) and its truth (red).117Chapter 7ResultsIn this chapter, the differential cross section in ∆pp, ∆θp and |∆pp| forCC0piNp events in ND280 are presented in different muon phase-space bins.These results are the primary results of the analysis. Models of CC0piNpevents can be compared to the data by calculating these differential crosssections by selecting a muon phase space bin, and integrating the expectedproton kinematics over the muon phase space.In Section 7.1.1, the results unfolded with NEUT and GENIE are com-pared to NEUT predictions in primary neutrino interaction modes. Com-parisons to NEUT and GENIE are shown in Section 7.1.2, as examples ofthe comparisons that can be done. For the targeted CC0piNp, the two mod-els differ particularly in the choice of MQEA (1.2 GeV in NEUT/NuWro and0.99 GeV in GENIE) and the 2p2h modeling (model based on Nieves inNEUT/NuWro and no 2p2h events in GENIE).The extracted total cross sections are summarized in Section 7.2.7.1 Unfolded Results7.1.1 NEUT Comparisons in Neutrino Interaction ModesFigures 7.1 – 7.3 present the differential cross sections unfolded with NEUT(full circle) and GENIE (open circle), compared to the NEUT predictionsbroken into primary neutrino interaction modes. If the kinematic assump-tion used to infer the proton kinematics (two-body interaction on a nucleonat rest) is correct, each variable should be zero. By comparing the unfoldedresults to NEUT predictions in primary neutrino interaction modes, it canbe seen which interactions contribute to the deviation from zero. Over-all, there is an excess in MC in the bins where CCQE is dominant, whilethe CC-non-QE dominant bins do not show consistent tendency across themuon phase-space bins. This may indicate that including multi-nucleon ef-fects is not sufficient to describe heavy-nuclei data, but the existing CCQEmodel requires deeper understanding. Hence, the result comparisons to the1187.1. Unfolded Resultsmodel predictions should be done not only by each contribution of interac-tion modes, but also shape and normalization.The two results unfolded with different MC priors (NEUT and GENIE)are consistent over all the variables and the muon phase-space bins withslight excesses in NEUT results. The bin-by-bin comparisons to the NEUTpredictions between the variables are as follow:Bin 0 (cos θµ < −0.6): Overall, there is an excess in MC at large ∆ppwhere nuclear effects contribute. According to the NEUT predictions, theCCQE component is dominant even at the large non-zero values. This mightindicate that the kinematic deviation of the CCQE events is over-predictedin the models.Bin 1 (−0.6 < cos θµ < 0.0 & pµ < 250 MeV/c): The CC-non-QE com-ponents are dominant at large non-zero positive values in this bin. Thepredictions agree with data within the uncertainties.Bin 2 (−0.6 < cos θµ < 0.0 & pµ > 250 MeV/c): There is an excessin the first bin of ∆pp and the last bin of ∆θp, which may be related toan excess in data |∆pp| at the bins where CC-non-QE is dominant. Itis difficult to separate the CC-non-QE from CCQE, since the CC-non-QEseems to reside where CCQE peaks. As discussed for Bin 0, it is CCQE-dominant bin. This might indicate that nuclear effects for CCQE events forbackward-going muons could be modeled better.Bin 3 (cos θµ > 0.0 & pµ < 250 MeV/c): This bin is the most CC-non-QE dominant bin. It seems data and MC agree reasonably well withinthe uncertainties, but there appears to be an excess in MC in |∆pp| whereCC-non-QE is dominant.Bin 4 (0.0 < cos θµ < 0.8 & pµ > 250 MeV/c): This is the higheststatistics bin and also the most CCQE dominant bin. There is an excess inMC which may be due to the high axial mass (MQEA = 1.21 GeV) assumedin NEUT relative to values measured on free nucleon targets (MQEA ∼ 1.0GeV).Bin 5 (0.8 < cos θµ < 1.0 & 250 < pµ < 750 MeV/c): The excess inMC is seen in ∆pp and |∆pp|, but hidden in ∆θp. Like in Bin 3, it appears1197.1. Unfolded Resultsthat there is an excess in MC for both CCQE and the CC-non-QE region,especially in |∆pp|.Bin 6 (0.8 < cos θµ < 1.0 & pµ > 750 MeV/c): There is an excess inMC. This is likely due to the same reason which is discussed for Bin 4.1207.1. Unfolded ResultsFigure 7.1: Unfolded results with NEUT (full circle) and GENIE (opencircle) of ∆pp are compared to the NEUT predictions in primary neutrinointeraction modes: CCQE (red), 2p2h (purple), CCRES (green), DIS (blue),coherent scattering (cyan) and others. CCQE, 2p2h and CCRES are mostdominant interactions contributed to this selection. The distributions in alogarithmic scale are presented alongside.1217.1. Unfolded ResultsFigure 7.2: Unfolded results with NEUT (full circle) and GENIE (opencircle) of ∆θp compared to the NEUT predictions in primary neutrino in-teraction modes: CCQE (red), 2p2h (purple), CCRES (green), DIS (blue),coherent scattering (cyan) and others. CCQE, 2p2h and CCRES are mostdominant interactions contributed to this selection. The distributions in alogarithmic scale are presented alongside.1227.1. Unfolded ResultsFigure 7.3: Unfolded results with NEUT (full circle) and GENIE (opencircle) of |∆pp| are compared to the NEUT predictions in primary neutrinointeraction modes: CCQE (red), 2p2h (purple), CCRES (green), DIS (blue),coherent scattering (cyan) and others. CCQE, 2p2h and CCRES are mostdominant interactions contributed to this selection. The distributions in alogarithmic scale are presented alongside.1237.1. Unfolded Results7.1.2 Various Model ComparisonsIn Figures 7.4 – 7.6, the results unfolded with NEUT (full circle) and GE-NIE (open circle) are compared to three different model predictions: nom-inal NEUT (MC prior, red), GENIE (blue), and NuWro (green). NEUTand GENIE, in particular, have different MQEA values, 1.21 GeV in NEUTand 0.99 GeV in GENIE. NuWro has a similar value (1.00 GeV) as GE-NIE. In terms of multi-nucleon effects, NEUT and NuWro generate 2p2hcontributions based on the Nieves’ model, while GENIE has no 2p2h con-tribution. Complementary comparisons to different underlying models inthese generators are also shown in Appendix H.Bin 0 (cos θµ < −0.6): In ∆pp, GENIE has lower cross section than NEUTand NuWro. Data agrees better with NEUT/NuWro (with 2p2h) than GE-NIE (without 2p2h). This difference between models is rather subtle in∆θp, where there is an excess in data (also in |∆pp|, although the size issmaller). This excess is not seen in ∆pp. As the excess is located near zeroin the |∆pp| distributions unlike in ∆θp, this can be an indication of protonre-scattering after CCQE interactions which only changes its angle withoutloosing not much of momentum.Bin 1 (−0.6 < cos θµ < 0.0 & pµ < 250 MeV/c): This bin shows asimilar behaviour as Bin 0.Bin 2 (−0.6 < cos θµ < 0.0 & pµ > 250 MeV/c): This bin shows asimilar behaviour as Bin 0 except that there is no clear discrepancy in ∆θp.Instead, in |∆pp|, the data is lower than any of models at zero, but higherat the tail.Bin 3 (cos θµ > 0.0 & pµ < 250 MeV/c): This bin has the enhancedCC-non-QE components. NEUT has higher cross section that GENIE andeven NuWro. There is an excess at zero only in ∆θp.Bin 4 (0.0 < cos θµ < 0.8 & pµ > 250 MeV/c): NEUT and NuWropredictions are similar, while GENIE is lower. It seems that the data agreesbetter with the GENIE prediction where there is no 2p2h component.Bin 5 (0.8 < cos θµ < 1.0 & 250 < pµ < 750 MeV/c): This bin shows asimilar behaviour as Bin 4.1247.1. Unfolded ResultsBin 6 (0.8 < cos θµ < 1.0 & pµ > 750 MeV/c): GENIE and NuWro aresimilar and describe the data better NEUT which has slightly higher crosssection.In Tables 7.1 and 7.2, the χ2 and the number of degrees of freedom (ndof)of the results relative to different MC predictions are shown. To calculate theχ2 of the results unfolded with GENIE, the covariance matrix computed withNEUT MC is used. For the results unfolded with NEUT, the χ2 relative toNEUT and GENIE are similar with a slight preference of NEUT in ∆pp and∆θp, but of GENIE in |∆pp|. The results unfolded with GENIE, instead,are clearly in a favour of GENIE in ∆pp and |∆pp|, but slightly of NEUTin ∆θp.Variable ∆pp ∆θp |∆pp|NEUT 53.9 55.2 87.4GENIE 58.6 60.6 53.5NuWro 134.6 93.3 188.3(ndof) (35) (24) (42)Table 7.1: The χ2 of the results unfolded with NEUT, relative to variousMC predictions (NEUT, GENIE, and NuWro) are shown.Variable ∆pp ∆θp |∆pp|NEUT 124 64.6 994.5GENIE 34.5 76.6 196.8NuWro 230.3 122 1373.7(ndof) (35) (24) (42)Table 7.2: The χ2 of the results unfolded with GENIE, relative to variousMC predictions (NEUT, GENIE, and NuWro) are shown.1257.1. Unfolded ResultsFigure 7.4: Unfolded results of ∆pp with nominal NEUT (full circle) andnominal GENIE (open circle) are compared to various MC predictions:NEUT (red), GENIE (blue), and NuWro (green). The distributions in alogarithmic scale are presented alongside.1267.1. Unfolded ResultsFigure 7.5: Unfolded results of ∆θp with nominal NEUT (full circle) andnominal GENIE (open circle) are compared to various MC predictions:NEUT (red), GENIE (blue), and NuWro (green). The distributions in alogarithmic scale are presented alongside.1277.1. Unfolded ResultsFigure 7.6: Unfolded results of |∆pp| with nominal NEUT (full circle) andnominal GENIE (open circle) are compared to various MC predictions:NEUT (red), GENIE (blue), and NuWro (green). The distributions in alogarithmic scale are presented alongside.1287.2. Extracted Total Cross Sections7.2 Extracted Total Cross SectionsTable 7.3 shows the total cross sections extracted with NEUT and GENIE.Total statistics and systematic uncertainties are shown only for the resultsunfolded with NEUT. True cross section of NEUT and GENIE MC areshown as well. Table 7.4 summarizes relative uncertainties for each system-atic source on the results unfolded with NEUT.Total cross section σ (×10−38cm2/Nucleon)Extracted MC∆pp ∆θp |∆pp|NEUT0.227 0.230 0.2250.222± 0.005 (stat) ± 0.006 (stat) ± 0.006 (stat)± 0.023 (syst) ± 0.024 (syst) ± 0.024 (stat)GENIE 0.205 0.214 0.209 0.182Table 7.3: The extracted total cross-sections are shown with NEUT andGENIE separately. Total uncertainties on the extracted cross sections areshown only for the results unfolded with NEUT. True cross section of NEUTand GENIE are shown as well.Uncertainties (%)stat data stat mc flux fsi xs det syst∆pp 2.34 0.63 8.39 2.02 2.34 5.64∆θp 2.52 0.68 8.40 2.11 1.98 5.63|∆pp| 2.59 0.70 8.45 2.21 2.25 5.64Table 7.4: Total uncertainties for each source: statistics on data (stat data),statistics on MC (stat mc), neutrino flux model (flux), FSI model (fis), neu-trino interaction model (xs), and the detector model (det syst).129Chapter 8ConclusionsTo achieve precision measurements of the mass hierarchy and CP violationphase, accurate knowledge of neutrino interactions plays an important roleto reduce the systematic uncertainties on oscillation analyses, as discussed inChapters 2 and 3. Poor understanding of neutrino-nucleus interactions canbias neutrino energy reconstruction, hence measurements of the oscillationparameters. However, neutrino-nucleus interactions have been challengingto understand due to complicated nuclear effects. In particular, the dis-crepancy between neutrino cross section measurements on free nucleon andrecent measurements on heavier nuclei has led to more fundamental studiesof nuclear effects in neutrino-nucleus interactions.This dissertation has presented a measurement of νµ charged-current in-teractions on carbon. Neutrino interactions where one muon, no pion, andat last one proton have been identified are selected as CC0piNp events. Thedifferential cross sections of the CC0piNp selection are measured in analysisvariables which utilize kinematic imbalance between the muon and proton.To this end, the analysis variables are defined as the differences betweeninferred and measured proton kinematics. Inferred proton kinematics areexpected kinematics from the measured muon kinematics when CCQE ona nucleon at rest is assumed. If there are no nuclear effects, the differenceshould be zero (that is, the muon and proton kinematics are balanced). Evenif the selected CC0piNp events are all CCQE on a nucleus, a deviation fromzero will be seen due to Fermi motion or proton re-scattering, which alterthe kinematics. There will be further deviation due to CCRES followed bypion absorption in which a pion does not emerge, and multi-nucleon pro-cesses which eject multi-nucleons in the final state. Therefore, any deviationfrom zero in these distributions indicate nuclear effects. A study of thesedistributions can provide further information of nuclear effects in modelingwhich are uncertain not only in the normalization, but also the shape ofpredicted distributions.The differential cross sections are extracted by the iterative Bayesianunfolding method. The muon phase-space bins are unfolded simultaneouslyto avoid large efficiency correction and calculate the correlations between130Chapter 8. Conclusionsthe muon phase-space bins properly. 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A696 (2012) 1–31, arXiv:1204.3666.[112] T2K Collaboration , “CC0pi multi-topology selection andsystematics in FGD1”, T2K Internal Document, unpublished.[113] T2K Collaboration , “ND280 reconstruction”, T2K InternalDocument, unpublished.[114] T2K Collaboration , “νµ CC event selections in the ND280 trackerusing run 2+3+4 data”, T2K Internal Document, unpublished.[115] T2K Collaboration , “Measuring the flux-integrated CC0pidifferential cross section at ND280”, T2K Internal Document,unpublished.142References[116] T2K Collaboration , “Flux prediction and uncertainty updateswith NA61 2009 thin target data and antineutrino modepredictions”, T2K Internal Document, unpublished.[117] T2K Collaboration , “Cross section parameters for 2014oscillation analysis”, T2K Internal Document, unpublished.[118] A. Hocker and V. Kartvelishvili, “SVD approach to data unfolding”,Nucl. Instrum. Meth. A372 (1996) 469–481,arXiv:hep-ph/9509307.[119] S. Schmitt, “Tunfold: an algorithm for correcting migration effects inhigh energy physics”, JINST 7 (2012) T10003, arXiv:1205.6201.[120] G. D’Agostini, “A multidimensional unfolding method based onBayes’ theorem”, Nuclear Instruments and Methods in PhysicsResearch Section A: Accelerators, Spectrometers, Detectors andAssociated Equipment 362 (1995), no. 23, 487 – 498. URLhttp://www.sciencedirect.com/science/article/pii/016890029500274X.[121] G. D’Agostini, “Improved iterative Bayesian unfolding”,arXiv:1010.0632.[122] T2K Collaboration , “Constraining the flux and cross-sectionmodels with data from the ND280 detector using FGD1 and FGD2for the 2017 joint oscillation analysis”, T2K Internal Document,unpublished.143Appendix AThe Selection EfficienciesFigures A.1, A.2, and A.3 show the selection efficiencies as a function ofeach analysis variable for each muon phase-space bin.144Appendix A. The Selection EfficienciesFigure A.1: Selection efficiencies as function of ∆pp for 7 muon phase-spacebins. Two different generators are shown - NEUT (black) and GENIE (red).145Appendix A. The Selection EfficienciesFigure A.2: Selection efficiencies as function of ∆θp for 7 muon phase-spacebins. Two different generators are shown - NEUT (black) and GENIE (red).146Appendix A. The Selection EfficienciesFigure A.3: Selection efficiencies as function of |∆pp| for 7 muon phase-space bins. Two different generators are shown - NEUT (black) and GENIE(red).147Appendix BThe List of ModelParametersFigure B.1: The covariance matrix of flux systematic is shown. The binindices represent model paramters used in the flux prediction and systematicestimation, which are summarized in Table B.1.148Appendix B. The List of Model ParametersIndex Parameter Note0 JEnu2013a nd5numu0 νµ flux at ND280 (0.0 - 0.4 GeV)1 JEnu2013a nd5numu1 νµ flux at ND280 (0.4 - 0.5 GeV)2 JEnu2013a nd5numu2 νµ flux at ND280 (0.5 - 0.6 GeV)3 JEnu2013a nd5numu3 νµ flux at ND280 (0.6 - 0.7 GeV)4 JEnu2013a nd5numu4 νµ flux at ND280 (0.7 - 1.0 GeV)5 JEnu2013a nd5numu5 νµ flux at ND280 (1.0 - 1.5 GeV)6 JEnu2013a nd5numu6 νµ flux at ND280 (1.5 - 2.5 GeV)7 JEnu2013a nd5numu7 νµ flux at ND280 (2.5 - 3.5 GeV)8 JEnu2013a nd5numu8 νµ flux at ND280 (3.5 - 5.0 GeV)9 JEnu2013a nd5numu9 νµ flux at ND280 (5.0 - 7.0 GeV)10 JEnu2013a nd5numu10 νµ flux at ND280 (7.0 - 30.0 GeV)11 JEnu2013a nd5numub0 νµ flux at ND280 (0.0 - 0.7 GeV)12 JEnu2013a nd5numub1 νµ flux at ND280 (0.7 - 1.0 GeV)13 JEnu2013a nd5numub2 νµ flux at ND280 (1.0 - 1.5 GeV)14 JEnu2013a nd5numub3 νµ flux at ND280 (1.5 - 2.5 GeV)15 JEnu2013a nd5numub4 νµ flux at ND280 (2.5 - 30.0 GeV)16 JEnu2013a nd5nue0 νe flux at ND280 (0.0 - 0.5 GeV)17 JEnu2013a nd5nue1 νe flux at ND280 (0.5 - 0.7 GeV)18 JEnu2013a nd5nue2 νe flux at ND280 (0.7 - 0.8 GeV)19 JEnu2013a nd5nue3 νe flux at ND280 (0.8 - 1.5 GeV)20 JEnu2013a nd5nue4 νe flux at ND280 (1.5 - 2.5 GeV)21 JEnu2013a nd5nue5 νe flux at ND280 (2.5 - 4.0 GeV)22 JEnu2013a nd5nue6 νe flux at ND280 (4.0 - 30.0 GeV)23 JEnu2013a nd5nueb0 νe flux at ND280 (0.0 - 2.5 GeV)24 JEnu2013a nd5nueb1 νe flux at ND280 (2.5- 30.0 GeV)Table B.1: List of the flux model parameters [122]. The mean value of allthe paramters are 1 with σ from the covariance matrix shown in Figure B.1.“Index” corresponds to the bin index of the matrix.149Appendix B. The List of Model ParametersFigure B.2: The covariance matrix of cross-section systematic is shown.The bin indices represent model paramters used in the flux prediction andsystematic estimation, which are summarized in Tables B.2, B.3 and B.4.150Appendix B. The List of Model ParametersIndex Parameter Mean σ (%)0 NCasc FrInelLow pi 0.0 50inelastic rescattering probabilities (low momentum pions)1 NCasc FrInelHigh pi 0.0 30inelastic rescattering probabilities (high momentum pions)2 NCasc FrPiProd pi 0.0 50pion production probabilities3 NCasc FrAbs pi 0.0 50pion absorption probabilities4 NCasc FrCExLow pi 0.0 50charge exchange probabilities (low momentum pions)5 NCasc FrCExHigh pi 0.0 30charge exchange probabilities (high momentum pions)Table B.2: List of the cross-section model parameters (FIS) [122] is shownwith the mean values and σ. “Index” corresponds to the bin index of thematrix in Figure B.2.151Appendix B. The List of Model ParametersIndex Parameter Mean σ (%)6 NXSec MaCCQE 1.21 GeV 16.52CCQE axial mass7 NIWG2014a pF C12 217 MeV/c 7.37carbon Fermi momentum8 NIWGMEC Norm C12 0.0 100carbon 2p2h normalization9 NIWG2014a Eb C12 25 MeV 36carbon binding energy10 NIWG2014a pF O16 225 MeV/c 7.37oxygen Fermi momentum11 NIWGMEC Norm O16 0.0 100oxygen 2p2h normalization12 NIWG2014a Eb O16 27 MeV 33.33oxygen binding energy13 NXSec CA5RES 1.01 24.75CA5 for single pion interaction14 NXSec MaNFFRES 1.21 GeV 15.79Resonant single pion production axial mass15 NXSec BgSclRES 1.30 15.38I = 1/2 background scale factor for resonant single pion productionTable B.3: List of the cross-section model parameters (Nuclear and CC-QE/CCRES interaction models) [122] is shown with the mean values andσ. “Index” corresponds to the bin index of the matrix in Figure B.2.152Appendix B. The List of Model ParametersIndex Parameter Mean σ (%)16 NIWG2012a ccnueE0 1.0 6.0CC νe normalization17 NIWG2012a dismpishp 1.0 40DIS multi pion normalization18 NIWG2012a cccoh C E0 1.0 100CC Coherent on carbon normalization19 NIWG2012a cccoh O E0 1.0 100CC Coherent on oxygen normalization20 NIWG2012a nccohE0 1.0 30NC Coherent normalization21 NIWG2012a ncotherE0 1.0 30NC Other normalizationTable B.4: List of the cross-section model parameters (Other interactionmodels) [122] is shown with the mean values and σ. “Index” corresponds tothe bin index of the matrix in Figure B.2.153Appendix CSideband Distributions154Appendix C. Sideband Distributions (GeV/c)p p∆-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0Number of events020406080100120140160180 SignalBackgroundsSidebandpθ ∆-150 -100 -50 0 50 100 150Number of events020406080100120140160180200220SignalBackgroundsSideband| (GeV/c)p p∆|0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Number of events020406080100120140160180SignalBackgroundsSidebandFigure C.1: The true distributions of the sideband predicted from NEUT areshown: ∆pp (top), ∆θp (middle) and |∆pp| (bottom). Three event categoriesare shown separately: signal in the selection (dashed black), background inthe selection (black) and the sideband (red). The sideband distributionsagree well with the background distributions. The numbers of events arescaled to data POT.155Appendix C. Sideband DistributionsFigure C.2: The observed distributions (data point) of the sideband com-pared to the NEUT predictions are shown: ∆pp (top), ∆θp (middle) and|∆pp| (bottom). The NEUT predictions are scaled to data POT and totalwith statistics and detectors uncertainties are shown in red.156Appendix DReproducing MC TruthFor a sanity check of the machinery, the MC is unfolded by itself propagatingall the systematic uncertainties. Figures D.1 – D.3 show the unfoldeddistributions on the left column. Unfolded distributions are well matchedto the truth.157Appendix D. Reproducing MC TruthFigure D.1: Reproducing MC truth for ∆pp. The unfolded result (datapoints) with a MC prior (black) are identical with the expectation of thefake data sample (red), which is the MC itself.158Appendix D. Reproducing MC TruthFigure D.2: Reproducing MC truth for ∆θp. The unfolded result (datapoints) with a MC prior (black) are identical with the expectation of thefake data sample (red), which is the MC itself.159Appendix D. Reproducing MC TruthFigure D.3: Reproducing MC truth for |∆pp|. The unfolded result (datapoints) with a MC prior (black) are identical with the expectation of thefake data sample (red), which is the MC itself.160Appendix ENEUT-based Fake DataStudiesE.1 Various CCQE Fractions161E.1. Various CCQE FractionsFigure E.1: -20 % CCQE in NEUT as a fake data sample unfolded withnominal NEUT MC for ∆pp. The expectation of the fake data sample(reweighted NEUT) is shown in red, the MC prior is in dashed black, andthe unfolded result is in data point.162E.1. Various CCQE FractionsFigure E.2: -20 % CCQE in NEUT as a fake data sample unfolded withnominal NEUT MC for ∆θp. The expectation of the fake data sample(reweighted NEUT) is shown in red, the MC prior is in dashed black, andthe unfoled result is in data point.163E.1. Various CCQE FractionsFigure E.3: -20 % CCQE in NEUT as a fake data sample unfolded withnominal NEUT MC for |∆θp|. The expectation of the fake data sample(reweighted NEUT) is shown in red, the MC prior is in dashed black, andthe unfolded result is in data point.164E.1. Various CCQE FractionsFigure E.4: +20 % CCQE in NEUT as a fake data sample unfolded withnominal NEUT MC for ∆pp. The expectation of the fake data sample(reweighted NEUT) is shown in red, the MC prior is in dashed black, andthe unfolded result is in data point.165E.1. Various CCQE FractionsFigure E.5: +20 % CCQE in NEUT as a fake data sample unfolded withnominal NEUT MC for ∆θp. The expectation of the fake data sample(reweighted NEUT) is shown in red, the MC prior is in dashed black, andthe unfolded is in data point.166E.1. Various CCQE FractionsFigure E.6: +20 % CCQE in NEUT as a fake data sample unfolded withnominal NEUT MC for |∆pp|. The expectation of the fake data sample(reweighted NEUT) is shown in red, the MC prior is in dashed black, andthe unfolded result is in data point.167E.2. No 2p2h ComponentsE.2 No 2p2h Components168E.2. No 2p2h ComponentsFigure E.7: No 2p2h in NEUT as a fake data sample unfolded with nominalNEUT MC for ∆pp. The expectation of the fake data sample (reweightedNEUT) is shown in red, the MC prior is in dashed black, and the unfoldedresult is in data points.169E.2. No 2p2h ComponentsFigure E.8: No 2p2h in NEUT as a fake data sample unfolded with nominalNEUT MC for ∆θp. The expectation of the fake data sample (reweightedNEUT) is shown in red, the MC prior is in dashed black, and the unfoldedresult is in data points.170E.2. No 2p2h ComponentsFigure E.9: No 2p2h in NEUT as a fake data sample unfolded with nominalNEUT MC for |∆θp|. The expectation of the fake data sample (reweightedNEUT) is shown in red, the MC prior is in dashed black, and the unfoldedresult is in data points.171Appendix FCovariance Matrices172Appendix F. Covariance MatricesFigure F.1: The top plot is a full covariance matrix for ∆pp including all thesystematics. Every seven bins correspond to one muon phase-space bin as:Bin 0 (0-6), Bin 1 (7-13), Bin 2 (14-20), Bin 3 (21-27), Bin 4 (28-34), Bin 5(35-41), Bin 6 (42-48). The bottom plot is a subset of the covariance matrixin a numerical form for the bin indices [0-15:0-15]. The z-axis is covariancein (×10−38cm2/Nucleon)2.173Appendix F. Covariance MatricesFigure F.2: Subsets of the covariance matrix in a numerical form for the binindices [16-31:0-15] (top) and [32-48:0-15] (bottom) are shown. The z-axisis covariance in (×10−38cm2/Nucleon)2.174Appendix F. Covariance MatricesFigure F.3: Subsets of the covariance matrix in a numerical form for the binindices [0-15:16-31] (top) and [16-31:16-31] (bottom) are shown. The z-axisis covariance in (×10−38cm2/Nucleon)2.175Appendix F. Covariance MatricesFigure F.4: Subsets of the covariance matrix in a numerical form for the binindices [32-48:16-31] (top) and [0-15:32-48] (bottom) are shown. The z-axisis covariance in (×10−38cm2/Nucleon)2.176Appendix F. Covariance MatricesFigure F.5: Subsets of the covariance matrix in a numerical form for thebin indices [16-31:32-48] (top) and [32-48:32-48] (bottom) are shown. Thez-axis is covariance in (×10−38cm2/Nucleon)2.177Appendix F. Covariance MatricesFigure F.6: The top plot is a full covariance matrix for ∆pp including all thesystematics. Every seven bins correspond to one muon phase-space bin as:Bin 0 (0-6), Bin 1 (7-13), Bin 2 (14-20), Bin 3 (21-27), Bin 4 (28-34), Bin 5(35-41), Bin 6 (42-48). The bottom plot is a subset of the covariance matrixin a numerical form for the bin indices [0-10:0-10]. The z-axis is covariancein (×10−38cm2/Nucleon)2.178Appendix F. Covariance MatricesFigure F.7: Subsets of the covariance matrix in a numerical form for the binindices [11-21:0-10] (top) and [22-34:0-10] (bottom) are shown. The z-axisis covariance in (×10−38cm2/Nucleon)2.179Appendix F. Covariance MatricesFigure F.8: Subsets of the covariance matrix in a numerical form for the binindices [0-10:11-21] (top) and [11-21:11-21] (bottom) are shown. The z-axisis covariance in (×10−38cm2/Nucleon)2.180Appendix F. Covariance MatricesFigure F.9: Subsets of the covariance matrix in a numerical form for the binindices [22-34:11-21] (top) and [0-10:22-34] (bottom) are shown. The z-axisis covariance in (×10−38cm2/Nucleon)2.181Appendix F. Covariance MatricesFigure F.10: Subsets of the covariance matrix in a numerical form for thebin indices [11-21:22-34] (top) and [22-34:22-34] (bottom) are shown. Thez-axis is covariance in (×10−38cm2/Nucleon)2.182Appendix F. Covariance MatricesFigure F.11: The top plot is a full ovariance matrix for |∆pp| including allthe systematics. Every seven bins correspond to one muon phase-space binas: Bin 0 (0-6), Bin 1 (7-13), Bin 2 (14-20), Bin 3 (21-27), Bin 4 (28-34),Bin 5 (35-41), Bin 6 (42-48). The bottom plot is a subset of the covariancematrix in a numerical form for the bin indices [0-15:0-15]. The z-axis iscovariance in (×10−38cm2/Nucleon)2.183Appendix F. Covariance MatricesFigure F.12: Subsets of the covariance matrix in a numerical form for the binindices [16-31:0-15] (top) and [32-48:0-15] (bottom) are shown. The z-axisis covariance in (×10−38cm2/Nucleon)2.184Appendix F. Covariance MatricesFigure F.13: Subsets of the covariance matrix in a numerical form for thebin indices [0-15:16-31] (top) and [16-31:16-31] (bottom) are shown. Thez-axis is covariance in (×10−38cm2/Nucleon)2.185Appendix F. Covariance MatricesFigure F.14: Subsets of the covariance matrix in a numerical form for thebin indices [32-48:16-31] (top) and [0-15:32-48] (bottom) are shown. Thez-axis is covariance in (×10−38cm2/Nucleon)2.186Appendix F. Covariance MatricesFigure F.15: Subsets of the covariance matrix in a numerical form for thebin indices [16-31:32-48] (top) and [32-48:32-48] (bottom) are shown. Thez-axis is covariance in (×10−38cm2/Nucleon)2.187Appendix GCorrelation Matrices188Appendix G. Correlation MatricesFigure G.1: Correlation matrix for ∆pp including all the systematics. Everyseven bins correspond to one muon phase-space bin as: Bin 0 (0-6), Bin 1(7-13), Bin 2 (14-20), Bin 3 (21-27), Bin 4 (28-34), Bin 5 (35-41), Bin 6(42-48).189Appendix G. Correlation MatricesFigure G.2: Correlation matrix for ∆θp including all the systematics. Everyseven bins correspond to one muon phase-space bin as: Bin 0 (0-6), Bin 1(7-13), Bin 2 (14-20), Bin 3 (21-27), Bin 4 (28-34), Bin 5 (35-41), Bin 6(42-48).190Appendix G. Correlation MatricesFigure G.3: Correlation matrix for |∆pp| including all the systematics. Ev-ery seven bins correspond to one muon phase-space bin as: Bin 0 (0-6), Bin1 (7-13), Bin 2 (14-20), Bin 3 (21-27), Bin 4 (28-34), Bin 5 (35-41), Bin 6(42-48).191Appendix HVarious Model ComparisonsIn this appendix, the results presented in Chapter 7 will be compared tovarious models across different neutrino generators.H.1 Different Models of Fermi Motion192H.1. Different Models of Fermi MotionFigure H.1: The differential cross sections in ∆pp compared to various mod-els of Fermi motion: NuWro 11q with a Benhar Spectral Function nuclearmodel both with (solid light blue) and without (dotted light blue) an addi-tional ad hoc 2p2h contribution based on the Nieves model; NEUT 5.4.0 withan LFG+RPA nuclear model and the Nieves 1p1h/2p2h predictions (red);and NuWro 11q with an LFG+RPA nuclear model and a 2p2h predictionbased on the Nieves model (dotted red).193H.1. Different Models of Fermi MotionFigure H.2: The differential cross sections in ∆θp compared to various mod-els of Fermi motion: NuWro 11q with a Benhar Spectral Function nuclearmodel both with (solid light blue) and without (dotted light blue) an addi-tional ad hoc 2p2h contribution based on the Nieves model; NEUT 5.4.0 withan LFG+RPA nuclear model and the Nieves 1p1h/2p2h predictions (red);and NuWro 11q with an LFG+RPA nuclear model and a 2p2h predictionbased on the Nieves model (dotted red).194H.1. Different Models of Fermi MotionFigure H.3: The differential cross sections in |∆pp| compared to variousmodels of Fermi motion: NuWro 11q with a Benhar Spectral Function nu-clear model both with (solid light blue) and without (dotted light blue)an additional ad hoc 2p2h contribution based on the Nieves model; NEUT5.4.0 with an LFG+RPA nuclear model and the Nieves 1p1h/2p2h predic-tions (red); and NuWro 11q with an LFG+RPA nuclear model and a 2p2hprediction based on the Nieves model (dotted red).195H.2. Different Models of FSIH.2 Different Models of FSI196H.2. Different Models of FSIFigure H.4: The differential cross sections in ∆pp compared to model predic-tions with various FSI strength. NEUT 5.3.2.2 with various FSI strengths.SF is used as a nuclear model and 2p2h components are varied as well.The histograms in light blue shows the prediction with the default FSIstrength with (solid) and without (dotted) 2p2h components. The dou-bled FSI strength with 2p2h is shown in gray and no FSI with 2p2h is indotted gray.197H.2. Different Models of FSIFigure H.5: The differential cross sections in ∆θp compared to model predic-tions with various FSI strength. NEUT 5.3.2.2 with various FSI strengths.SF is used as a nuclear model and 2p2h components are varied as well.The histograms in light blue shows the prediction with the default FSIstrength with (solid) and without (dotted) 2p2h components. The dou-bled FSI strength with 2p2h is shown in gray and no FSI with 2p2h is indotted gray.198H.2. Different Models of FSIFigure H.6: The differential cross sections in |∆pp| compared to modelpredictions with various FSI strength. NEUT 5.3.2.2 with various FSIstrengths. SF is used as a nuclear model and 2p2h components are var-ied as well. The histograms in light blue shows the prediction with thedefault FSI strength with (solid) and without (dotted) 2p2h components.The doubled FSI strength with 2p2h is shown in gray and no FSI with 2p2his in dotted gray.199H.3. RFG Models in Different GeneratorsH.3 RFG Models in Different Generators200H.3. RFG Models in Different GeneratorsFigure H.7: The differential cross sections in ∆pp compared to RFG modelsin various generators: NEUT 5.3.2.2, GENIE 2.12.4 and NuWro 11q.201H.3. RFG Models in Different GeneratorsFigure H.8: The differential cross sections in ∆θp compared to RFG modelsin various generators: NEUT 5.3.2.2, GENIE 2.12.4 and NuWro 11q.202H.3. RFG Models in Different GeneratorsFigure H.9: The differential cross sections in |∆pp| compared to RFG modelsin various generators: NEUT 5.3.2.2, GENIE 2.12.4 and NuWro 11q.203

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