Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Berkman, Sophie Sattell
Doctor of Philosophy - PhD
University of British Columbia
T2K is an experiment designed to make precision measurements of neutrino oscillation parameters. Neutrinos produced in the Japan Proton Accelerator Complex are sent 295 km across Japan to the far detector Super-Kamiokande. Charged current quasi-elastic neutrino interactions in which the neutrino produces a lepton and a nucleon are dominant at T2K beam energies, and these are the signal events used in the measurement of muon disappearance parameters θ₂₃ and ∆m²₃₂. The second most dominant neutrino interaction mode at T2K is the charged current single pion interaction (CC1π+) in which both a lepton and a charged pion are produced in addition to the nucleon. This thesis presents a method to identify charged pions and CC1π+ neutrino interactions at Super-Kamiokande. This is used to develop a selection method for muon neutrino CC1π+ interactions at Super-Kamiokande in the T2K beam. There are 93 events expected based on Monte Carlo predictions which would increase the number of muon neutrino events in the T2K analysis by 40%. Methods developed to evaluate the systematic errors associated with explicitly selecting charged pions at Super-Kamiokande for the first time are also discussed.
νµ CC1pi+ Events Produced in the T2K Beam atSuper-KamiokandebySophie Sattell BerkmanB. A., Reed College, 2010a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoralstudies(Physics)The University of British Columbia(Vancouver)April 2018c© Sophie Sattell Berkman, 2018AbstractT2K is an experiment designed to make precision measurements of neu-trino oscillation parameters. Neutrinos produced in the Japan Proton Ac-celerator Complex are sent 295 km across Japan to the far detector Super-Kamiokande. Charged current quasi-elastic neutrino interactions in whichthe neutrino produces a lepton and a nucleon are dominant at T2K beamenergies, and these are the signal events used in the measurement of muondisappearance parameters θ23 and ∆m232. The second most dominant neu-trino interaction mode at T2K is the charged current single pion interaction(CC1pi+) in which both a lepton and a charged pion are produced in ad-dition to the nucleon. This thesis presents a method to identify chargedpions and CC1pi+ neutrino interactions at Super-Kamiokande. This is usedto develop a selection method for muon neutrino CC1pi+ interactions atSuper-Kamiokande in the T2K beam. There are 93 events expected basedon Monte Carlo predictions which would increase the number of muon neu-trino events in the T2K analysis by 40%. Methods developed to evaluatethe systematic errors associated with explicitly selecting charged pions atSuper-Kamiokande for the first time are also discussed.iiLay AbstractAt the beginning of the universe equal parts matter and anti-matter werecreated, but if this were the full story nothing in the universe would existtoday. It is possible that the pervasive tiny neutral particles called neutrinoscould contribute to this discrepancy if neutrinos and anti-neutrinos act dif-ferently. The T2K experiment hopes to measure the extent that neutrinosand anti-neutrinos are different. To do this it sends a beam of neutrinosproduced in an accelerator from one side of Japan to the other and mea-sures the neutrinos produced in a 50 kTon tank of pure water. Neutrinoscan interact in the water in several ways, but until now only the dominantinteraction was used in T2K measurements. This thesis presents a methodto identify the second most dominant neutrino interaction which is predictedto increase the total number of neutrinos used in the analysis by 40%.iiiPrefaceAs the member of two large collaborations, T2K and Super-Kamiokande(SK), the work presented in this thesis is ultimately the culmination ofyears of effort on behalf of many people.Chapter 1 summarizes the history, formalism and current state of neu-trino oscillation physics, as well as discusses future open questions the fieldhopes to address. None of the work is original, but the summary is my own.Chapter 2 describes the T2K experiment. I worked to update the modelused to simulate charged pions in the SK detector simulation, mentionedbriefly in Section 2.5.5, and described in more detail in [1] [2], and par-ticipated in detector calibrations and shifts throughout my time as a PhDstudent. The summary of the existing methods and literature is my own.I led the work on the PICCOLO detector described in Chapter 3, includ-ing detector design, construction, operation, and analysis. All of the resultspresented are my own.Chapter 4 describes the fiTQun reconstruction algorithm for SK. Thealgorithm described in Sections 4.1-4.6 was generally adapted from the Mini-BooNE experiment, and coded and developed for SK by Shimpei Tobayama,Michael Wilking, Patrick de Perio and Andrew Missert [3] [4]. The workpresented from 4.7 onward describes my implementation of a technique toidentify charged pions originally used at MiniBooNE and that I adaptedfor SK [5]. Although not discussed elsewhere in this thesis, I was the firstperson to look at the fiTQun algorithm on the T2K data in preparationfor the addition of a pi0 rejection cut which reduced the background andcontributed to the observation of electron neutrino appearance made by theivT2K experiment in 2013 [6] [7].Chapter 5 describes a selection of νµ CC1pi+ events for T2K at SK. Allof the work is my own.Chapter 6 describes the systematic errors associated with CC1pi+ eventsin T2K. The multi-dimensional weights used in 6.1.2 to analyze the corre-lated differences between the neutrino generator output were calculated byCris Vilela. The multi-pion and deep inelastic scattering weights used in6.1.3 were calculated by Christophe Bronner. I applied these weights to theselected CC1pi+ samples, and otherwise, the work presented in 6.1 on neu-trino interactions is my own. The fit to the world pion hadronic interactiondata described in 6.2 was done by Elder Pinzon [8]. The method for evaluat-ing the systematic errors associated with pion final state and secondary in-teractions using fixed hadronic interaction variations was initially developedby Patrick de Perio and myself. The method for evaluating the systematicerrors using covariance matrix throws was developed by Elder Pinzon. Thework to evaluate the effect of the pion final state and secondary interactionsystematic uncertainties on the selected CC1pi+ samples is my own. Thehybrid CC1pi+ sample was developed by Guang Yang, Shunichi Mine andThomas Mueller [9].The conclusions presented in Chapter 7 are my own.This analysis is blind. There has recently been continued developmentof the reconstruction algorithm described in Chapter 4 in the context ofseveral SK analyses. These have not yet been applied to T2K. After theyare incorporated and new Monte Carlo is generated for T2K this analysis willbe unblinded. The exact timescale for this is unknown, but is estimated tobe sometime in 2019. Work has been done to ensure that the developmentspresented in this thesis will be fully integrated into the new framework sothat they can be used in the future.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv1 Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . 41.3.1 Evidence of Oscillations . . . . . . . . . . . . . . . . . 41.3.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Neutrino Oscillation Measurements . . . . . . . . . . . . . . . 121.5 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 142 T2K Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 17vi2.1 Neutrino Beam . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Near Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 INGRID . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 ND280 . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Super-Kamiokande . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 Detector Design . . . . . . . . . . . . . . . . . . . . . 262.3.2 Cherenkov Radiation . . . . . . . . . . . . . . . . . . . 282.3.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . 302.4 Neutrino Interactions at SK . . . . . . . . . . . . . . . . . . . 352.4.1 Charged Current Quasi Elastic . . . . . . . . . . . . . 352.4.2 Charged Current Single Pion . . . . . . . . . . . . . . 372.4.3 Deep Inelastic Scattering . . . . . . . . . . . . . . . . 372.4.4 Neutral Current Interactions . . . . . . . . . . . . . . 382.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.5.1 Flux Prediction . . . . . . . . . . . . . . . . . . . . . . 392.5.2 Neutrino Interaction Generators . . . . . . . . . . . . 392.5.3 NEUT . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.5.4 ND280 Detector Simulation . . . . . . . . . . . . . . . 422.5.5 SK Detector Simulation . . . . . . . . . . . . . . . . . 423 PICCOLO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 SK Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 514.1 Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . 514.2 Charge Likelihood . . . . . . . . . . . . . . . . . . . . . . . . 534.2.1 Predicted Charge . . . . . . . . . . . . . . . . . . . . . 554.2.2 Calculation of Charge Likelihood . . . . . . . . . . . . 614.3 Time Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . 644.3.1 Time Likelihood from Direct Light . . . . . . . . . . . 654.3.2 Time Likelihood from Indirect Light . . . . . . . . . . 664.4 Sub-Event Algorithm . . . . . . . . . . . . . . . . . . . . . . . 674.4.1 Vertex Pre-Fitter . . . . . . . . . . . . . . . . . . . . . 684.4.2 Subevents . . . . . . . . . . . . . . . . . . . . . . . . . 68vii4.5 One Ring Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.5.1 Upstream pi+ Fit . . . . . . . . . . . . . . . . . . . . . 694.6 Generic Multi-Ring Fit . . . . . . . . . . . . . . . . . . . . . . 714.7 Scattered Charged Pion Fit . . . . . . . . . . . . . . . . . . . 744.7.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . 744.7.2 Fit Performance . . . . . . . . . . . . . . . . . . . . . 794.8 νµ CC1pi+ Reconstruction . . . . . . . . . . . . . . . . . . . . 904.8.1 Performance . . . . . . . . . . . . . . . . . . . . . . . 975 νµ CC1pi+ Event Selection . . . . . . . . . . . . . . . . . . . 1035.1 Neutrino Energy Calculation . . . . . . . . . . . . . . . . . . 1045.2 Visible Event Topologies . . . . . . . . . . . . . . . . . . . . . 1065.3 Selection Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.3.1 One Ring νµ CC1pi+ Events . . . . . . . . . . . . . . . 1105.3.2 Multi-Ring νµ CC1pi+ Events . . . . . . . . . . . . . . 1155.4 Summary of CC1pi+ Samples . . . . . . . . . . . . . . . . . . 1345.5 Background from ν¯µ, νe and ν¯e Interactions . . . . . . . . . . 1345.6 Neutrino Energy Calculation Resolution . . . . . . . . . . . . 1355.7 Oscillation Variations . . . . . . . . . . . . . . . . . . . . . . 1406 Charged Pion Systematic Error Evaluation . . . . . . . . . 1486.1 Neutrino Interaction Uncertainties . . . . . . . . . . . . . . . 1486.1.1 Generator Comparisons of Visible CC1pi+ Signal . . . 1496.1.2 Reweighting to Different Signal Models . . . . . . . . 1506.1.3 Multi-Pion Variations . . . . . . . . . . . . . . . . . . 1566.2 Pion Hadronic Interactions . . . . . . . . . . . . . . . . . . . 1616.2.1 Fit to World Data . . . . . . . . . . . . . . . . . . . . 1616.3 Pion FSI and SI Cross Section Variations . . . . . . . . . . . 1636.3.1 Covariance Matrix Throws . . . . . . . . . . . . . . . 1646.4 Detector Response Uncertainties . . . . . . . . . . . . . . . . 1686.4.1 Hybrid CC1pi+ Sample . . . . . . . . . . . . . . . . . . 1686.5 Systematic Error Summary . . . . . . . . . . . . . . . . . . . 1697 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171viiiBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174ixList of TablesTable 1.1 Best fit parameters for neutrino oscillations [10]. . . . . . . 14Table 2.1 Cherenkov thresholds for particles in water. Calculatedusing masses from [10]. . . . . . . . . . . . . . . . . . . . . 28Table 4.1 Fraction that the µ ring is better matched to the higher orlower energy ring of the two ring pion fit. . . . . . . . . . . 94Table 4.2 Fraction that the µ ring is correctly assigned in the bestfit configuration when the likelihood cut is used to selectthe best fit. . . . . . . . . . . . . . . . . . . . . . . . . . . 97Table 5.1 Oscillation parameters used in this section. . . . . . . . . . 110Table 5.2 One ring µ-like events, after each selection cut . . . . . . . 116Table 5.3 Percent of one ring µ-like events after each selection cut . 118Table 5.4 Number of two ring signal CC1pi+ events broken up acrossthe possible two ring fit configurations. . . . . . . . . . . . 120Table 5.5 Number of two ring events after each selection cut. . . . . 122Table 5.6 Sample purity after each selection cut. . . . . . . . . . . . 122Table 5.7 Efficiency of the two ring CC1pi+ samples relative to theFCFV and the two ring cut. . . . . . . . . . . . . . . . . . 122Table 5.8 Number of three ring events after each selection cut . . . . 129Table 5.9 Efficiency of selecting three ring events from the FCFVand the three ring cut. . . . . . . . . . . . . . . . . . . . . 130Table 5.10 Percent of sample, 3 ring events, after each selection cut . 133Table 5.11 Number of events in each of the CC1pi+ samples. . . . . . 135xTable 5.12 Number of events in each of the CC1pi+ samples below1.33 GeV in true neutrino energy. . . . . . . . . . . . . . . 135Table 5.13 Background from non-νµ beam neutrino interactions. . . . 136Table 5.14 Mean and standard deviations of the neutrino energy res-olution when the ∆ formula is used compared to when theCC1pi+ formula is used to calculate neutrino energy. . . . . 137Table 6.1 Number of events when the total multi-ring signal sampleis reweighted in one dimension for each kinematic variableseparately. . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Table 6.2 Variations in the number of events as a result of changingthe multi-pion DIS cross section to that of new models. . . 159Table 6.3 Variations in the number of events as a result of changingthe multi-pion DIS kinematics to that of new models. . . . 161Table 6.4 Best fit values of for the parameters in the FSI fit. . . . . . 164Table 6.5 Fractional RMS of each of the samples from covariancematrix throw variations. . . . . . . . . . . . . . . . . . . . 166Table 6.6 Systematic error summary for sum of multi-ring samplesand total CC1pi+ samples as a fraction of the nominal num-ber of events. . . . . . . . . . . . . . . . . . . . . . . . . . 169xiList of FiguresFigure 1.1 Kamland result showing neutrino interaction measure-ments of the neutrino energy spectrum as a function ofL0/E. The ratio of the best fit to the energy spectrumgiven oscillation, decay and decoherence hypotheses areplotted. The data best match the oscillation hypothesis.Figure from [11]. . . . . . . . . . . . . . . . . . . . . . . . 7Figure 1.2 Latest Kamland result showing ν¯e oscillations. Figurefrom [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Figure 1.3 Contours of allowed fit regions for ∆m232 and sin2 θ23.Data from T2K [13], NOvA [14], SK [15], MINOS+ [16],and IceCube [17] are compared. Figure from [15]. . . . . . 13Figure 1.4 Illustration of the normal mass hierarchy. . . . . . . . . . 15Figure 2.1 Path that T2K neutrino travels from production to de-tection. Figure from [18]. . . . . . . . . . . . . . . . . . . 17Figure 2.2 The T2K beamline. Protons enter from the bottom right.They are then sent into the primary beam line whichbends them left towards the secondary beam line. Fig-ure from [18]. . . . . . . . . . . . . . . . . . . . . . . . . . 18Figure 2.3 An illustration of the probability of νµ survival (top), νeappearance (middle), and neutrino flux from the T2Kbeamline as a function of neutrino energy for several offaxis angles (bottom). Figure from [6]. . . . . . . . . . . . 20xiiFigure 2.4 An illustration of the INGRID detector. The beam passesstraight through the center of the module at the center ofthe cross. Figure from [18]. . . . . . . . . . . . . . . . . . 21Figure 2.5 An illustration of the ND280 detector. The neutrino beampasses from left to right through the center of the detector.Figure from [18]. . . . . . . . . . . . . . . . . . . . . . . . 22Figure 2.6 An illustration of the SK detector. Figure from [6]. . . . . 26Figure 2.7 Simulated Cherenkov rings produced by an electron, muonand charged pion. . . . . . . . . . . . . . . . . . . . . . . 29Figure 2.8 Illustration of the five vertical slices B1-B5 that the SKtank is broken up into to calibrate the properties of thewater. The sixth corresponds to the PMTs on the top ofthe tank. Figure from [19]. . . . . . . . . . . . . . . . . . 33Figure 2.9 World neutrino interaction cross section data. Figurefrom [20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.10 Feynman diagrams of common neutrino interactions atT2K neutrino energies. . . . . . . . . . . . . . . . . . . . . 36Figure 3.1 Momentum distribution of true CC1pi+ events in the T2K-SK Monte-Carlo. . . . . . . . . . . . . . . . . . . . . . . . 45Figure 3.2 The PICCOLO detector. . . . . . . . . . . . . . . . . . . 45Figure 3.3 Schematic of the PICCOLO detector. . . . . . . . . . . . 46Figure 3.4 M11 beam composition at each of a range of momentumsettings based on particle time of flight. The time differ-ence between the accelerator and a scintillator in front ofthe PICCOLO detector is plotted. Because of the waythe difference is taken, electrons appear as the peak atthe highest times, muons in the middle and pions at theshortest time. . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 3.5 The muon lifetime as measured in the PICCOLO detector. 48Figure 3.6 Time that light emission begins for muons in PICCOLO. 48Figure 3.7 PICCOLO results. . . . . . . . . . . . . . . . . . . . . . . 50xiiiFigure 4.1 Charge probability distribution functions fq(q;µ) at a va-riety of predicted charges µ. The data points are the out-put of the simulation, and the line is the fitted functionused to evaluate the likelihood. Diagram from [4]. . . . . 54Figure 4.2 Variables used in calculation of predicted charge. Theinitial particle production is indicated by the white dot,and the point of light emission, s, is indicated by a blackdot. Diagram from [4]. . . . . . . . . . . . . . . . . . . . . 56Figure 4.3 Example charge profiles for electrons (left) and muons(right) as a function of the position at which light is emit-ted s, and the Cherenkov angle. Figure adapted from [4]. 58Figure 4.4 PMT acceptance output from SKDETSIM. The verticalaxis has arbitrary units, and the normalization is set sothat (cos η = 1) = 1. Diagram from [4]. . . . . . . . . . . 60Figure 4.5 Variables used to parameterize the six dimensional scat-tering table for PMTs on the walls of the detector. Dia-gram is the same for PMTs on the top and bottom of thetank except that RPMT, which is not illustrated, is usedinstead of zPMT . Diagram from [4]. . . . . . . . . . . . . 61Figure 4.6 Time likelihood from direct light for 450 MeV/c muonsand a range of predicted charge. The data points indicatethe raw simulation output and the red curve is the finalfit output assuming that the likelihood depends only onthe predicted charge and particle momentum. Diagramfrom [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.7 Pion hadronic scatter. . . . . . . . . . . . . . . . . . . . . 70Figure 4.8 Cartoon of the three configurations of the charge like-lihood for the upstream pion fit. The blue cones repre-sent the Cherenkov light emitted by each charge likelihoodconfiguration. The red dot indicates the pion scatter point. 72Figure 4.9 Comparison of muon and pion ranges in water at low energy. 77xivFigure 4.10 Vertex resolution for the scattered pion fit. The left col-umn shows the production point and the right columnshows the scatter point. All charged pions are shown inthis plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 4.11 Kinematic resolution of scattered pion fitter for the up-stream pion track. All charged pions are shown in thisplot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Figure 4.12 Kinematic resolution of scattered pion fitter for the down-stream pion track. All charged pions are shown in this plot. 83Figure 4.13 Vertex resolution for the scattered pion fit. The left col-umn shows the production point and the right columnshows the scatter point. Charged pions that scatter ex-actly once are shown in this plot. . . . . . . . . . . . . . . 84Figure 4.14 Kinematic resolution of the scattered pion fitter for theupstream pion track. Charged pions that scatter exactlyonce are shown in this plot. . . . . . . . . . . . . . . . . . 85Figure 4.15 Kinematic resolution of the scattered pion fitter for thedownstream pion track. Charged pions that scatter ex-actly once are shown in this plot. . . . . . . . . . . . . . . 86Figure 4.16 Vertex resolution for the scattered pion fit on NC1pi+ eventsin the T2K-SK MC for all outgoing pions. The left col-umn shows the neutrino interaction vertex and the rightcolumn shows the scatter point. . . . . . . . . . . . . . . . 87Figure 4.17 Kinematic resolution of the scattered pion fitter on NC1pi+ eventsin the T2K-SK MC for all outgoing pions. . . . . . . . . . 88Figure 4.18 Vertex resolution for the scattered pion fit on NC1pi+ eventsin the T2K-SK MC with outgoing pions that scatter ex-actly once. The left column shows the neutrino interac-tion vertex and the right column shows the scatter pointvertex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 4.19 Kinematic resolution of the scattered pion fitter on NC1pi+events in the T2K-SK MC with outgoing pions that scat-ter exactly once. . . . . . . . . . . . . . . . . . . . . . . . 90xvFigure 4.20 Cartoon of the CC1pi+ hypothesis. . . . . . . . . . . . . . 91Figure 4.21 Pion and muon momentum distributions for samples usedto study the CC1pi+ fitter. . . . . . . . . . . . . . . . . . 93Figure 4.22 CC1pi+ particle gun broken into samples in which themuon is better matched to the higher energy ring fromthe generic two ring fit (top) and the lower energy ringfrom the generic two ring fit (bottom). The muon/pionPID is the black line. . . . . . . . . . . . . . . . . . . . . . 95Figure 4.23 T2K-SK MC broken into samples in which the muon isbetter matched to the higher energy ring from the generictwo ring fit (left row) and the lower energy ring from thegeneric two ring fit (right row). The top row shows theunoscillated MC and the bottom row shows the oscillatedMC. The muon/pion PID is the black line. . . . . . . . . 96Figure 4.24 Example event display outputs from the CC1pi+ fit. Thetop row is the true PMT hit distribution, the middle rowis the seed charge distribution, and the bottom row isthe charge distribution after the CC1pi+ fit is run. Thepost-fit charge distribution is visibly the same as the truedistribution of hit PMTs in both of these events. . . . . . 98Figure 4.25 Vertex resolution for the CC1pi+ fit on the CC1pi+ par-ticle gun, oscillated and unoscillated T2K-SK MC. Theleft column shows the neutrino interaction vertex and theright column shows the scatter point vertex. . . . . . . . . 99Figure 4.26 Fit resolution of the muon kinematics compared for CC1pi+ par-ticle gun sample, unoscillated CC1pi+ events from theT2K-SK MC, and oscillated CC1pi+ events from the T2K-SK MC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Figure 4.27 Fit resolution of the upstream pion kinematics comparedfor CC1pi+ particle gun sample, unoscillated CC1pi+ eventsfrom the T2K-SK MC, and oscillated CC1pi+ events fromthe T2K-SK MC. . . . . . . . . . . . . . . . . . . . . . . . 101xviFigure 4.28 Fit resolution of the downstream pion kinematics com-pared for CC1pi+ particle gun sample, unoscillated CC1pi+ eventsfrom the T2K-SK MC, and oscillated CC1pi+ events fromthe T2K-SK MC. . . . . . . . . . . . . . . . . . . . . . . . 102Figure 5.1 Flow chart describing CC1pi+ event selection. . . . . . . . 109Figure 5.2 Initial quality cuts performed for νµ CC1pi+ selection onthe T2K-SK MC as a function of true (left) and recon-structed (right) neutrino energy. . . . . . . . . . . . . . . 111Figure 5.3 Number of rings for all events passing the FCFV and min-imum energy cuts. . . . . . . . . . . . . . . . . . . . . . . 112Figure 5.4 Cut sequence for preliminary event quality cuts. . . . . . 112Figure 5.5 Number of decay electrons for one ring muon-like events. 114Figure 5.6 Cut sequence for one ring muon-like events after the pre-liminary quality cuts. . . . . . . . . . . . . . . . . . . . . 115Figure 5.7 One ring events after ring counting (a,b) and PID cuts(c,d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Figure 5.8 One ring µ-like events with two decay electrons. . . . . . 117Figure 5.9 Ring PID of best multi-ring fit output for two ring events. 119Figure 5.10 Number of decay electrons for two ring pion-like events. . 121Figure 5.11 Cut sequence for two ring events. The black line indicatesthe final samples . . . . . . . . . . . . . . . . . . . . . . . 123Figure 5.12 Two ring events after ring counting (a, b) and PID cuts(c, d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Figure 5.13 Two ring pi+-like events with one decay electron. . . . . . 125Figure 5.14 Two ring pi+-like events with two decay electrons. . . . . . 126Figure 5.15 Ring PID of best multi-ring fit output for three ring events.127Figure 5.16 Number of decay electrons for three ring events. . . . . . 128Figure 5.17 Cut sequence for three ring CC1pi+ events. The blackarrow indicates the final samples. . . . . . . . . . . . . . . 129Figure 5.18 Three ring events after ring counting (a, b)and PID cuts(c, d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Figure 5.19 Three ring one decay electron pi+-like events. . . . . . . . 131xviiFigure 5.20 Three ring two decay electron pi+-like events. . . . . . . . 132Figure 5.21 Energy resolution using ∆ formula and CC1pi+ formulafor selected two ring samples. . . . . . . . . . . . . . . . . 138Figure 5.22 Energy resolution using ∆ formula and CC1pi+ formulafor selected three ring samples. . . . . . . . . . . . . . . . 139Figure 5.23 Number of events in the four multi-ring samples as a func-tion of reconstructed neutrino energy. Each plot is at afixed value of sin2 θ23, and illustrates each of the ∆m232variations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Figure 5.24 Number of events in all five νµ CC1pi+ samples as a func-tion of reconstructed neutrino energy. Each plot is at afixed value of sin2 θ23, and illustrates each of the ∆m232variations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Figure 5.25 Number of events in the multi-ring samples and all sam-ples illustrated in the grid of calculated oscillation param-eters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Figure 5.26 Number of events in the multi-ring samples and all sam-ples illustrated in the grid of calculated oscillation pa-rameters for events with true neutrino energy below 1.33GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Figure 5.27 CC1pi+ efficiency in the multi-ring samples and all sam-ples illustrated in the grid of calculated oscillation param-eters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Figure 5.28 CC1pi+ efficiency in the multi-ring samples and all sam-ples illustrated in the grid of calculated oscillation pa-rameters for events with true neutrino energy below 1.33GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Figure 6.1 Comparisons between different generators in muon andpion kinematics for signal events with one muon and onepion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151xviiiFigure 6.2 Ratio of different generators in muon and pion kinematicsfor signal events with one muon and one pion to the de-fault NEUT. NEUT is black, Genie default is pink, Geniealternate is red and NUWRO is blue. . . . . . . . . . . . 153Figure 6.3 Sum of true signal events in the four multi-ring samplesvaried to match the kinematics of other neutrino interac-tion generators. NEUT is black, NUWRO is blue, GenieDefault is red and Genie Alternate is pink. . . . . . . . . 154Figure 6.4 Sum of true signal events in the total CC1pi+ sample var-ied to match the kinematics of other neutrino interactiongenerators. NEUT is black, NUWRO is blue, Genie de-fault is red and Genie alternate is pink. . . . . . . . . . . 155Figure 6.5 Error from neutrino interaction variations. . . . . . . . . . 156Figure 6.6 Weights applied to total multipion cross section. M0 isred, M1 is black and M2 is blue [21]. The plot is zoomedin so that the differences between M0 and M2 are visible. 158Figure 6.7 Variations in CC1pi+ samples when the different multi-pion cross section models are applied. . . . . . . . . . . . 158Figure 6.8 Weights applied to leading pion momentum and invarianthadronic mass for M0 [21]. . . . . . . . . . . . . . . . . . 160Figure 6.9 Variations in CC1pi+ samples when the different multi-pion models are applied. . . . . . . . . . . . . . . . . . . . 160Figure 6.10 Comparison of FSI fit results to pi+-C external cross sec-tion measurements as a function of pion momentum. FSIfit best fit is indicated by the black line, and the 1σ errorband is indicated in red. The dashed line and the blueerror band are the result of an earlier fit to the externalpion cross section data. Figure from [22]. . . . . . . . . . 163Figure 6.11 Covariance matrix between the parameters in the FSI fit.Figure from [22]. . . . . . . . . . . . . . . . . . . . . . . . 164Figure 6.12 Visualization of 1000 covariance matrix throw variationsof FSI and SI parameters. . . . . . . . . . . . . . . . . . . 166Figure 6.13 Error from FSI and SI covariance matrix throws. . . . . . 167xixFigure 6.14 Relation between the five CC1pi+ samples in terms of theFSI and SI variations. . . . . . . . . . . . . . . . . . . . . 167xxGlossaryCC1pi+ Charged current single charged pion neutrino interactionCCOther Other charged current neutrino interactionsCCQE Charged current quasi-elastic neutrino interactionCMB Cosmic microwave backgroundCP Charge-parityCSDA Continuous slowing down approximation for particle rangecalculationCX Charge exchange∆ Delta baryonDIS Deep inelastic scattering neutrino interactionECAL Electromagnetic calorimeterFC Fully containedFCFV Fully contained within the Super-Kamiokande fiducial volumeFGD Fine grained scintillating detectorsfiTQun Maximum likelihood reconstruction algorithm used atSuper-KamiokandeFSI Final state interactions pions undergo in a nucleusxxiINGRID Interactive Neutrino GRIDJ-PARC Japan Proton Accelerator Research ComplexLINAC Linear acceleratorMC Monte-CarloMPPC Multi-pixel photon counterMR sum Multi-ring sum. Sum of two and three ring CC1pi+ events.NC Neutral current neutrino interactionNC1pi+ Neutral current single charged pion neutrino interactionNC1pi0 Neutral current single neutral pion neutrino interactionND280 T2K off axis near detector located 280 m from the targetNEUT Neutrino interaction generator used by the T2K experimentot Over Cherenkov thresholdP0D Neutral pion (pi0) DetectorPDG Particle Data Groupp.e. Photo-electronsPICCOLO Pion Integrating Cylinder for Cherenkov Optics and LightObservationPID Particle identificationPMT Photo-multiplier tubePOT Protons on targetRMS Root mean squaredSI Secondary interactions pions undergo in a detectorxxiiSK Super-KamiokandeSKDETSIM SK Detector SimulationSMRD Side muon range detectorsub-GeV Below 1.33 GeV in true neutrino energyT2K Tokai to KamiokaT2K-SK MC Simulated neutrino interactions with the T2K beam flux atthe SK detectorTPC Time projection chamberut Under Cherenkov threshold1R+2de One ring CC1pi+ events with two decay electrons2R+1de Two ring CC1pi+ events with one decay electron2R+2de Two ring CC1pi+ events with two decay electrons3R+1de Three ring CC1pi+ events with one decay electron3R+2de Three ring CC1pi+ events with two decay electronsxxiiiAcknowledgmentsI am grateful to the many people who have made this work possible.As a supervisor, Hiro Tanaka has been an extraordinary example of whatit means to be a scientist. I have always appreciated his insights as well ashis careful and deliberate approach to problem solving. He has given meincredible opportunities and independence.Mike Wilking has been an amazing source of information and encour-agement. I am appreciative that he was always willing to talk about pionswith me.Akira Konaka has an infectious joy about physics and it has been apleasure to get to share many productive discussions with him.Kendall Mahn has always made the time to talk with me about physicsand otherwise. I was lucky to get to sit across the trailer from her.Thanks to Christophe Bronner for his help with the deep inelastic scat-tering uncertainties and to Cris Vilela for his help with the neutrino inter-action uncertainties.The SKanada group a was a special place to work thanks to: Eric Zim-merman, John Martin, Andy Missert, Shimpei Tobayama, Corina Nantais,Xiaoyue Li, Patrick de Perio, Roman Tacik, and Trevor Towstego.I have also been fortunate to work within the T2K-Canada group andI would like to thank everyone who has been a part of it, in particular:Sampa Bhadra, Scott Oser, Thomas Lindner, Tom Feusels, Elder Pinzon,Mark Scott, Khalid Gameil, Jiae Kim, John Walker and Christine Nielsen.The T2K and Super-Kamiokande collaborations have given me manyextraordinary colleagues, and I am glad to have had the opportunity to workxxivwith all of them. Thanks in particular to Roger Wendell, Okumura-san andHayato-san for their many useful comments over the years.Thanks to my PhD committee who have all been incredibly helpful andsupportive: Janis McKenna, John Behr, and David Morrissey.Thank you to my friends who have kept me grounded and stuck by me,no matter which continent I happened to be on.And finally, thank you to my family, for your love, support, and every-thing else.xxvChapter 1NeutrinosWhat are we made of, and why? Particle physics is one of the ways thatwe address these fundamental questions. We know that the universe ismade of matter, as described by the standard model of particle physics, aswell as dark matter and dark energy. The nature of dark matter and darkenergy is currently unknown. This chapter will introduce neutrinos and theirplace in particle physics, give a brief overview of the history of neutrinomeasurements, describe the current understanding of the field today, andfinally mention the open questions that still need to be addressed.1.1 Standard ModelThe Standard Model of particle physics describes the known matter in theuniverse and what we currently believe to be its fundamental building blocks.There are two categories of particle in the Standard Model: bosons withinteger spins and fermions with half integer spins.The vector bosons with spin 1 mediate the forces of nature. The elec-tromagnetic force is mediated by the photon. The strong force is mediatedby gluons. The weak force is mediated by the W± and the Z0. The Higgsboson is a spin 0 particle that mediates the mechanism that gives the otherparticles mass.The fermions contain three generations and are separated into two cat-1egories: quarks, which interact strongly, and leptons which do not. Thequarks are fractionally charged and not observed freely. The strong forcegluons bind them into mesons like pions, which are made of quark-antiquarkpairs, or baryons like protons and neutrons, which are made up of threequarks. Only the up and down quarks are prevalent in ordinary matter; theother quarks are made, for example, in colliders.Leptons only interact electromagnetically or through the weak inter-action. The charged leptons are electrons, muons and taus. Each chargedlepton has a corresponding neutrino (νe, νµ, ντ ), which are neutral masslessleptons. The neutrinos are related to the charged leptons by their flavor, asindicated by the subscript above. In charged current interactions neutrinosare produced in a given flavor state. For example, in the decay of a posi-tively charged pion decay, an anti-muon and muon neutrino are produced.Neutrinos can be produced in many ways: in the sun, the atmosphere, re-actors, and supernovas among others. It is now known that while neutrinoshave masses that are much smaller than those of the other standard modelparticles, they are not massless as was formerly assumed [23]. Neutrinos willbe discussed in more detail in the following section.For the most part the Standard Model agrees incredibly well with ex-perimental measurements, however there are several missing pieces. In par-ticular the standard model does not include gravity, or describe neutrinoscorrectly. It also neglects dark matter and dark energy.1.2 NeutrinosNeutrinos were initially postulated by Pauli to solve the problem that energywas apparently not conserved in beta decay measurements.As understood at the time, beta decay occurred when a nucleus decayedto produce a lighter nucleus and an electron. As this is a two body decay,using conservation of energy and momentum it is possible to calculate theenergy of the electron based on the masses of the three particles. Thismeans that because the masses are constant, beta decay was expected toproduce mono-energetic electrons. In 1918 Chadwick measured the energy2of outgoing electrons produced by beta nuclear decays and instead observeda spectrum of electron energies [24]. To explain these results some suggestedthat perhaps energy was not conserved.In 1930 Pauli wrote a letter addressed to his colleagues attending aconference in which he suggested that a third particle might also be producedin beta decay. This would make it possible to maintain conservation ofenergy in light of the experimental measurements of the electron energyspectrum [25]. Based on this proposal Fermi reformulated beta decay as:n→ p+ e− + ν¯e, (1.1)in which both the neutron and proton remain bound within the nucleus [26].Neutrinos were first detected in 1956 by Reines and Cowan using a sourceof reactor neutrinos from the Savannah River reactor. They placed a 400L liquid scintillator detector comprised of water doped with 40 kg of CdCl2nearby. It was instrumented with phototubes to observe the scintillationlight produced. They detected the reactor electron anti-neutrinos using theinverse beta decay signal:ν¯e + p→ e+ + n. (1.2)The CdCl2 made it possible to observe a delayed coincidence signal betweenthe positron and a photon produced by neutron capture on cadmium [27].The muon neutrino was discovered by Lederman et al. in 1962. Theyused a proton accelerator to produce pions, and looked at the beam of neu-trinos produced when those pions decayed. Charged pions decay to producea muon and a muon neutrino. This is similar to the neutrino beam techniqueused in modern experiments such as T2K, and which will be discussed inmore detail in Chapter 2. At the end of the beam they placed a detector thatcould distinguish electrons and muons, and as they saw muons instead ofelectrons in their detector they could determine that the neutrinos observedwere different than those that Reines and Cowan had seen [28].The tau neutrino was discovered in 2000 by the DONUT collaboration.3They used the Fermilab Tevatron to produce Ds mesons and looked forthem to decay into a τ and ντ with a detector that could separately detectelectron, muon, and tau leptons. With this detector they could identify theτ leptons produced by ντ charged current interactions [29].1.3 Neutrino OscillationsAs neutrinos travel over a distance they “oscillate”, or change flavor. Thismeans that a beam of neutrinos that begins as one flavor can be detectedwith a different composition when observed at a distance L away from pro-duction. This can result in observing fewer of the initial flavor of neutrinosthan expected, which is called “disappearance”, or detecting neutrinos of ad-ditional flavors than were initially produced, which is called “appearance”.1.3.1 Evidence of OscillationsThis section describes the evidence for neutrino oscillations.Solar NeutrinosThe first indications for neutrino oscillations came from solar neutrino mea-surements. Neutrinos are produced through fusion reactions in the sun.Solar models give a precise prediction of the neutrino flux from the sunbased on the relative frequency of these fusion reactions, and experimentsset out to measure these.The first experiment to measure solar neutrinos was the Homestake ex-periment performed by Ray Davis in 1968. This detector was made of a 390kL tank of C2Cl4 and located 1.5 km underground. It measured neutrinocapture on the chlorine:νe + Cl37 → e− + Ar37. (1.3)To measure the neutrino interactions the number of argon atoms in thechlorine were counted. The overall rate of solar neutrinos was found to beabout 30% that of the expected flux [30].4Discrepancies between the number of expected and measured neutrinoswere also observed in the GALLEX and Sage experiments which looked forneutrino capture on gallium, as well as in the Kamiokande water Cherenkovexperiment [31] [32] [33] [34].The SNO experiment ultimately solved the solar neutrino problem. SNOwas a 1 kTon Cherenkov experiment filled with heavy water. It could detectthree types of neutrino interactions from solar neutrinos:Charged Current : νe + d→ p+ p+ e− (1.4)Neutral Current : νx + d→ p+ n+ νx (1.5)Elastic Scattering : νx + e− → νx + e−. (1.6)The neutral current and elastic scattering interactions are both sensitiveto all flavors of neutrinos. The charged current interactions are only sensitiveto electron neutrinos because any νµ or ντ would not be high enough energyto produce the corresponding µ or τ .By looking at the neutral current and elastic scattering channels it waspossible for the SNO experiment to measure the total neutrino flux of solarneutrinos, independent of neutrino flavor. These measurements were foundto agree with the solar models. Then, using the charged current channel itwas possible for them to measure the ratio of the electron neutrino flux tothe total solar flux. This was found to be smaller than the total flux, whichindicated that neutrinos were oscillating as they travelled from the sun [35].Atmospheric NeutrinosNeutrino oscillations were first observed in atmospheric neutrinos by theSuper-Kamiokande (SK) experiment. Neutrinos are produced in the Earth’satmosphere when the charged pions and muons that are created by highenergy protons decay. Muon neutrinos are produced by pion and muondecays and electron neutrinos are produced by muon decays. SK is a 50 kTonwater Cherenkov detector, which has good particle direction resolution andelectron/muon particle identification which means that they are sensitive5to νe and νµ charged current interactions. The detector is described inmore detail in Chapter 2. SK measured atmospheric neutrino oscillationsby separating the atmospheric neutrinos that they observed into upwardgoing and downward going electron-like and muon-like samples. The upwardgoing sample consisted of neutrinos that were produced in the atmosphereon the other side of the earth and had traveled approximately 13,000 kmto reach the detector. The downward going neutrinos were produced in theatmosphere above the detector at an average distance of approximately 15km. The SK collaboration found that there were fewer upward going muonneutrinos than downward going, which indicated that the muon neutrinosthat had travelled through the Earth had disappeared. Only the muonneutrinos were affected, and a relative surplus of electron neutrinos wasnot observed, which suggested that muon neutrinos were changing into tauneutrinos as they travelled through the Earth [36].Reactor NeutrinosOne of the clearest pictures of neutrino oscillations comes from the Kamlandexperiment in 2005. This experiment measured the reactor anti-neutrinosproduced by 53 nuclear reactors at varying distances from the detector. Thedata were compared to the hypotheses of neutrino oscillation, neutrino de-cay and neutrino decoherence, each of which have a different dependence onthe ratio of production distance to neutrino energy. The neutrino oscillationdependence on the L/E ratio will be discussed more explicitly in the follow-ing section on oscillation formalism. As illustrated in Fig. 1.1 the data bestfit the neutrino oscillation hypothesis, with the oscillatory behavior directlyobserved [11].1.3.2 FormalismThis section describes the formalism of neutrino oscillations, which comeabout because the flavor states that undergo weak interactions are linearcombinations of the neutrino mass states.The three neutrinos can be described in one of two ways, either as definite6Figure 1.1: Kamland result showing neutrino interaction measure-ments of the neutrino energy spectrum as a function of L0/E.The ratio of the best fit to the energy spectrum given oscilla-tion, decay and decoherence hypotheses are plotted. The databest match the oscillation hypothesis. Figure from [11].mass states, called |νi〉 where i = 1, 2, 3, or as definite flavor states |νf 〉 wheref = e, µ, τ . Both the flavor and mass states can be written as a superpositionof the other, and they are related by a unitary matrix U . The flavor statesare written in terms of the mass states as:|νf 〉 =∑iU∗fi |νi〉 (1.7)As a neutrino that begins in a definite flavor state propagates the massstates will interfere so that the neutrino is in a mixed flavor state. When themixed neutrino flavor state is measured it is forced to be in a definite flavorstate. These transitions from definite flavor state to mixed flavor statesare called neutrino oscillations. The probability of one neutrino of flavor fcreated at (x, t) = (0, 0) will be measured as another flavor g at (x, t) can7be calculated as:P(νf → νg) = | 〈νg| νf (x, t)〉|2. (1.8)Here it is assumed that ~ = c = 1.The flavor states in Eq. 1.7 evolve in time according to the Schrodingerequation:|νf (x, t)〉 =∑iU∗fie−i(Eit−pix) |νi〉 . (1.9)This means that the probability can be written as:P(νf → νg) =|∑i∑j〈νj |UgjU∗fie−i(Eit−pix) |νf 〉 |2=|∑iUgiU∗fie−i(Eit−pix)|2 (1.10)=∑i∑jUgiU∗fiU∗gjUfje−i((Ei−Ej)t−(pi−pj)x).The momentum of the ith mass state pi can be calculated in terms of theneutrino energy as:pi =√E2i −m2i (1.11)= Ei(1− m2iE2i) 12. (1.12)Assuming that the neutrino’s mass is much less than its momentumEq. 1.12 can be Taylor expanded as:pi ≈ Ei + mi22Ei. (1.13)For ultra-relativistic particles like the neutrino the time t, and distancetraveled x are approximately the same. The distance the neutrino travelswill from here be denoted as L. It is assumed that all of the initial neutrinostates have the same energy E. Using these assumptions the oscillation8probability in Eq. 1.10 can be written as:P(νf → νg) =∑i∑jUgiU∗fiU∗gjUfje−i∆m2jiL2E (1.14)where ∆m2ji = m2i −m2j is the difference in the squares of the ith and jthmass states.By expanding these sums the probability that a neutrino will oscillate istypically written as:Pα→β =| 〈νβ(x)|να(0)〉 |2=δαβ − 4∑i>jRe[UβiU∗αiU∗βjUαj ] sin2(∆m2ijL4E)(1.15)+ 2∑i>jIm[UβiU∗αiU∗βjUαj ] sin(∆m2ijL2E).The probability for anti-neutrinos to oscillate is the same except that thesign of the last term is flipped. This term is the one that makes it possiblefor neutrinos and anti-neutrinos to oscillate differently, which may give riseto charge-parity (CP) violation in the lepton sector. It is of note that theoscillation probability depends only on the differences in the squares of themass states, which means that they are not sensitive to the absolute scaleof the neutrino masses.The matrix U is called the Pontocorvo-Maki-Nakagawa-Sakata (PMNS)matrix and is traditionally written as the product of three unitary transfor-mations which are parameterized in terms of three mixing angles (θ12, θ13, θ23),9and one overall phase (δCP ),U =e1 e2 e3µ1 µ2 µ3τ1 τ2 τ3 (1.16)=1 0 00 c23 s230 −s23 c23c13 0 s13e−iδCP0 1 0−s13eiδCP 0 c13c12 s12 0−s12 c12 00 0 1 ,where cij = cos(θij) and sij = sin(θij), for i, j = 1, 2, 3 and δCP is responsiblefor CP violation in the neutrino sector. θ23 in the first term was originallymeasured using atmospheric neutrinos, and now measurements can also bemade using accelerator produced neutrinos. θ13 in the second term wasoriginally measured using reactor neutrinos and is currently measured withreactor neutrinos and using accelerator produced neutrino beams. θ12 ismeasured using solar and reactor neutrinos. The mass splittings can befully described by just two differences in the mass states: ∆m221, ∆m232,where ∆m231 = ∆m221 + ∆m232. An overview of the measurements of theseparameters will be discussed in Section 1.4.Current accelerator-based neutrino oscillation experiments, like T2Kwhich will be discussed in more detail in Chapter 2, typically begin witha beam initially comprised of νµ and measure either neutrino appearance ordisappearance, where the oscillation probability is approximated with theleading order terms. One type of appearance corresponds to finding a neu-trino of flavor νe in a beam comprised of νµ, and this occurs with probabilityP(νµ → νe) ≈ sin2(2θ13) sin2(θ23) sin2(∆m231L4E)+ sin(2θ12) sin(2θ13) sin(2θ23) cos θ13 sin(∆m221L4E)sin(∆m231L4E)×[cos(∆m232L4E)cos δCP − sin(∆m232L4E)sin δCP](1.17)+ solar and matter effect terms.10Disappearance corresponds to finding fewer νµ than predicted by theneutrino flux alone because they have oscillated into other neutrino flavors,and to first order the probability of seeing a νµ after oscillations is:P(νµ → νµ) ≈1−[cos2 θ13 sin2(2θ23) + sin4 θ23 sin2 θ23]sin2(∆m232L4E)+ solar and matter effect terms. (1.18)For antineutrinos Eq. 1.17 is the same except that, as mentioned earlier,the term that is sensitive to δCP switches signs. The survival probabilityfor muon neutrinos is the same for neutrinos and anti-neutrinos. Theseoscillation probability formulas also show that oscillations can only occur ifneutrinos have non-zero mass.Oscillations in MatterNeutrino oscillations often do not actually occur in a vacuum as assumedin the formalism described above. Solar neutrinos travel through the sun,atmospheric neutrinos travel through the atmosphere, and perhaps throughthe Earth as in the Super-Kamiokande oscillation measurement, and beamsof accelerator neutrinos travel through the Earth’s crust. Interactions theneutrinos undergo as they travel in matter affect the properties of the oscilla-tions. The formalism to describe these effects was developed by Wolfenstein,Mikheev and Smirnov [37] [38]. Most charged current neutrino interactionsproduce a lepton, not a neutrino, and these do not affect the oscillationphysics except through a small reduction in the overall expected neutrinoflux. Neutral current interactions affect all neutrino flavors equally and soacts as an overall phase shift that can be factored out of the oscillationphysics. Charged current scattering interactions of an electron neutrino offof an electron in matter can impact the oscillations. This is because mattercontains electrons, but not muons or taus, which means that only electronneutrinos can undergo these interactions. These interactions add an extraterm to the electron neutrino Hamiltonian: VCC = ±√2GFNe, where GFis the Fermi coupling constant, Ne is the electron density of the material11and the sign depends on whether neutrinos or anti-neutrinos are described.This means that the measured flavor and mass states are slightly differentthan the ones described above and the interactions in matter can mimic thesignature of CP violation. Similarly the oscillation probabilities in matterdepend on the mass states squared, ∆m2s, which gives experiments sen-sitivity to the neutrino mass hierarchy [10]. Experiments with longer dis-tances between production and detection, such as the Super-Kamiokandeatmospheric neutrino measurement, are more sensitive to the mass hierar-chy because neutrinos travel from one side of the Earth to the other whichincreases the size of the matter effects.1.4 Neutrino Oscillation MeasurementsThis section describes the current world measurements of the neutrino os-cillation parameters. As illustrated in the oscillation probability formulasabove, all of the parameters are highly correlated and fits are done to the ex-isting experimental data by phenomenologists and the Particle Data Group(PDG) to extract as much information from the data as possible.θ12 and ∆m221 are measured using a fit to experiments that measure solarneutrinos, like SNO, Super-Kamiokande, Borexino, and Kamland with longbaseline reactor neutrinos at a distance of 180 km on average [12] [39] [40] [41].The latest results for the Kamland experiment is shown in Fig. 1.2.θ23 and ∆m232 can be measured using accelerator neutrinos such as inMINOS, NOvA and T2K, as well as with atmospheric neutrinos such as inIceCube and SK [13] [14] [15] [16] [17]. The world measurements of theseparameters by each experiment are compared in Fig. 1.3.θ13 and ∆m231 can be measured with short baseline reactor neutrinoswith a baseline of about 1 km like Daya Bay, Double Chooz and Reno, orwith accelerator neutrinos like with T2K and NOvA [13] [14] [42] [43] [44].The best fit parameters as measured by the PDG fit are given in Ta-ble 1.1. This result assumes that ∆m231 ≈ ∆m232 because ∆m221 has beenmeasured to be much smaller than ∆m231 or ∆m232.12Figure 1.2: Latest Kamland result showing ν¯e oscillations. Figurefrom [12].Figure 1.3: Contours of allowed fit regions for ∆m232 and sin2 θ23.Data from T2K [13], NOvA [14], SK [15], MINOS+ [16], andIceCube [17] are compared. Figure from [15].13Table 1.1: Best fit parameters for neutrino oscillations [10].Parameter Normal Hierarchy Inverted Hierarchysin2 θ12 0.307+0.013−0.012 0.307+0.013−0.012sin2 θ13(×10−2) 2.10+0.11−0.11 2.10+0.11−0.11sin2 θ23 0.51+0.04−0.04 0.50+0.04−0.04∆m221(×10−5 eV2) 7.53+0.18−0.18 7.53+0.18−0.18∆m232(×10−3 eV2) 2.45+0.05−0.05 2.52+0.05−0.05δCP /pi 0-2 0-21.5 Open QuestionsAlthough at this point all of the neutrino mixing angles and mass splittingshave been measured there are still many important open questions.First, there has, to this point, been no statistically significant observationof δCP , so it is unknown to what extent there is CP violation in neutrinos,and to what extent this might explain the CP violation necessary to createthe universe we have today given our models of the Big Bang and earlyuniverse. There are some hints from the T2K experiment that there maybe CP violation in neutrinos, because the CP conserving values of δCP = 0and δCP = pi are excluded at the two sigma level [45]. Confirmation andprecision measurements of δCP will require more data, such as proposed incontinued running of the T2K experiment as T2K-II [46] and the NOvAexperiment [14], or in future experiments such as DUNE [47] [48] or Hyper-Kamiokande [49]. These measurements will also require continued effort toprecisely measure the other mixing parameters because, as mentioned ear-lier, δCP dependence is a second order oscillation effect, and any uncertaintyin the first order term will make the measurements more difficult.Another open question is if θ23 is exactly maximal at 45◦, and if not,which octant the angle is in. At this point, as illustrated in Fig. 1.3, the14Neutrinos• Neutrinos:+leptons,+small+mass,+neutral+• Classify+them+in+two+ways+– Definite+masses:+ν1,+ν2,+ν3++– Definite+flavors+(interact+to+produce+corresponding+lepton):+νe,+νµ,+ντ+• Flavor+states+are+a+superposiFon+of+mass+states+related+by+unitary+transform+– Parameterize+in+terms+of+3+mixing+angles+(θ13,+θ23,+θ12)+and+one+phase+(δcp)+• As+neutrinos+propagate+the+mass+states+interfere+and+neutrino+oscillaFons+occur+• OscillaFons:+transiFon+between+flavor+states++ν3+(m32)++ νe+νµ+ντ+Normal+Hierarchyν2+(m22)++ν1+(m12)++m322m212February+15,+2013 WNPPC+SS+S.+Berkman 0@ ⌫e⌫µ⌫⌧1A = U(✓12)U(✓13, )U(✓23)0@ ⌫1⌫2⌫31AFigure 1.4: Illustration of the normal mass hierarchy.NOvA experiment has excluded maximal mixing at 2.6 sigma [14], but thelatest results from other experiments are still consistent with maximal mix-ing.There are also many open questions associated with neutrino mass. Asmentioned earlier, oscillations are only sensitive to the differences in themass states squared, not the absolute scale of the neutrino masses. There isa limit on the neutrino mass from cosmological measurements of the cosmicmicrowave background (CMB) because the mass of the neutrinos wouldaffect the early universe. These measurements indicate that the sum of thethree neutrino mass states must be less than 0.23 eV [50]. The fact thatneutrinos oscillate mean that at least two mass states must be nonzero [51].The ordering of the mass states, referred to as the neutrino mass hierarchy, isalso unknown. Neutrinos could follow the normal hierarchy (m1 < m2 < m3)or the inverted hierarchy (m3 < m1 < m2). An illustration of the normalhierarchy is shown in Fig. 1.4. The neutrino mass hierarchy will be measuredby oscillation experiments. As shown earlier in the PDG global fit valuesof the neutrino mixing parameter measurements are currently degeneratewithout knowing the neutrino mass hierarchy [10].Neutrinos may be Majorana fermions which would mean that they aretheir own anti-particles. This type of particle has never before been ob-served. It is possible to measure if neutrinos are Majorana by looking forneutrino-less double beta decay:(A,Z)→ (A,Z + 2) + e− + e−. (1.19)15Experiments are searching for this process. If the neutrino mass hierarchyis inverted then the predicted decay rates will be high enough that thesecurrent or next generation experiments will be sensitive to neutrino-lessdouble beta decay. If the mass hierarchy is normal then the decay ratescan be almost two orders of magnitude lower which makes the process moredifficult to detect. Measurements of the neutrino mass hierarchy will informthe future of these experiments [51].16Chapter 2T2K ExperimentT2K is a “long baseline” neutrino experiment designed to perform preci-sion measurements of the neutrino mixing parameters and neutrino crosssections. The experiment consists of three main components: the neu-trino beam, a near detector complex located 280 m away from the tar-get in the neutrino beam line (ND280 and INGRID), and a far detectorSuper-Kamiokande (SK) located 295 km from the target, as illustrated inFig. 2.1 [18].Figure 2.1: Path that T2K neutrino travels from production to detec-tion. Figure from [18].2.1 Neutrino BeamThe T2K experiment measures neutrinos that are produced in a neutrinobeam at the Japan Proton Accelerator Research Complex (J-PARC) inTokai, Ibaraki, Japan. Protons are accelerated through a series of accel-17erators and then sent to create a neutrino beam as illustrated in Fig. 2.2.0 50 100 mMain RingSecondary beamline(1) Preparation section(2) Arc section(3) Final focusing section(4) Target station(5) Decay volume(6) Beam dumpND280(1)(2)(3)(4)(5)(6)Figure 2.2: The T2K beamline. Protons enter from the bottom right.They are then sent into the primary beam line which bendsthem left towards the secondary beam line. Figure from [18].First, a linear accelerator (LINAC), accelerates negative hydrogen ions(H−) up to 400 MeV in kinetic energy. Then charge stripping foils are usedto change it to a H+ beam. These protons are then accelerated up to 3 GeVin a rapid cycling synchrotron, and finally to the main ring synchrotron inwhich they are accelerated to 30 GeV in eight bunches. Five kicker magnetseject the eight proton bunches, called a spill, from the main ring within asingle turn. Spills happen every 2-3 seconds and there are about 3 ×1014protons per spill at the current J-PARC beam power of 475 kW.These protons are sent to the primary neutrino beamline in which theprotons are pointed towards Super-Kamiokande. First the beam is tunedwith a series of 11 magnets called the preparation section. This beam isthen curved towards SK at an angle of 80.7◦ with a radius of 104 m using aseries of 14 additional magnets, which is called the arc section. Finally theprotons are focused onto a graphite target using a series of ten magnets in18the final focusing section.The secondary neutrino beamline begins with the graphite target, whichis a cylindrical graphite rod of 2.6 cm diameter and 91.4 cm long locatedin the target station. As the protons from the primary beamline interact inthe target they produce charged pions, which are focused using a series ofthree electromagnetic horns. The polarity of these horns can be switched tofocus either positively or negatively charged pions which makes it possibleto create either a neutrino or anti-neutrino beam. The pions are then sentinto a 96 m long steel decay volume where they decay into muons and muonneutrinos. At the end of the beam line is shielding, called a beam dump,which consists of 75 tons of graphite followed by 2.4 m of iron plates. Thisstops most remaining decay products except the neutrinos. The primaryand secondary beamlines are illustrated in Fig. 2.2.The neutrino beam is directed 2.5◦ off-axis from SK to position the peakof the neutrino energy spectrum at Eν = 0.6 GeV where the neutrino oscil-lation effect is largest for a baseline of 295 km [18]. This off-axis techniquemakes it possible to “focus” the energy of the neutrino beam because theneutrinos are produced in two body pion decay [52]. The neutrino energycan be calculated in terms of the pion kinematics as:Eν =m2pi −m2µ2(Epi − ppi cos θνpi) (2.1)where mpi and mµ are the masses of the pion and muon, Epi and ppi arethe energy and momentum of the pion, and θνpi is the angle between theneutrino and pion directions. For θνpi = 0 neutrino energy is proportionalto the pion momentum, but for larger off axis angles the energy dependenceon pion momentum almost disappears.Fig. 2.3 shows that the chosen 2.5◦ off-axis angle for T2K correspondswith maximum oscillation at the peak of the neutrino flux [6].19 (GeV)νE0 1 2 3 (A.U.)295kmµνΦ00.51 °OA 0.0°OA 2.0°OA 2.50 1 2 3) eν → µνP(0.050.1 = 0CPδNH, = 0CPδIH, /2pi = CPδNH, /2pi = CPδIH, 0 1 2 3) µν → µνP(0.51 = 1.023θ22sin = 0.113θ22sin2 eV-3 10× = 2.4 322m∆Figure 2.3: An illustration of the probability of νµ survival (top), νeappearance (middle), and neutrino flux from the T2K beam-line as a function of neutrino energy for several off axis angles(bottom). Figure from [6].2.2 Near DetectorThe T2K near detector consists of an on-axis and an off-axis component,both located 280 m from the target. The on-axis part of the detector mea-sures the neutrino beam direction and profile as well as the neutrino rate perprotons on target. The off-axis part of the detector measures the neutrinoflux and energy spectrum, as well as neutrino cross sections.2.2.1 INGRIDThe INGRID detector (Interactive Neutrino GRID) is the on-axis part ofthe near detector that measures the beam direction and profile. It is madeof 16 identical modules which each consist of nine iron plates of 1.24 ×1.24 m2 interspersed with 11 tracking scintillator plates. Fourteen of these20modules are stacked and arranged as a cross as illustrated in Fig. 2.4, wherethe center of the cross corresponds to the center of the neutrino beam. Thisallows INGRID to measure a 10 m by 10 m section of the beam. Twoadditional modules are also placed off-axis on either side of the cross tomeasure the beam symmetry. INGRID measures the beam center to within10 cm [53].Figure 2.4: An illustration of the INGRID detector. The beam passesstraight through the center of the module at the center of thecross. Figure from [18].2.2.2 ND280ND280 is the off-axis component of the T2K near detector, positioned 2.5◦away from the beam center. This detector is used to measure the neutrinoenergy spectrum as well as the beam contamination from electron neutri-nos. Many cross section measurements can also be made in this detector.ND280 is made up of multiple sub-detectors as illustrated in Fig 2.5 [18].The center of the detector contains two fine grained scintillating detectors(FGDs), sandwiched with three time projection chambers (TPCs). A pi0detector (P0D) is located on the upstream end of the first TPC, and anelectromagnetic calorimeter (ECAL) is located downstream of the trackingdetectors. These detectors are surrounded by barrel ECALs and a magnet.21Scintillator was attached to the magnet to act as a side muon range detector(SMRD).Figure 2.5: An illustration of the ND280 detector. The neutrino beampasses from left to right through the center of the detector.Figure from [18].MagnetThe magnet that surrounds the ND280 detector was first used in the CERNUA1 and NOMAD experiments. It was refurbished and shipped to Tokaifor T2K. The magnet is made of water cooled aluminum coils. This innervolume is surrounded by eight magnet yokes. The total magnet dimensionsare 7.6 m × 5.6 m × 6.1 m. A current of 2900 A is used to produce a dipolemagnetic field of 0.2 T. The field is constant to within 10−3 of the nominalvalue. The magnet makes it possible to measure particle momentum anddetermine the sign of charged particles from neutrino interactions in ND280.This is because the charged particles bend in the magnetic field.22Fine Grained DetectorsThe FGDs are at the center of the ND280 detector. They are made of longnarrow scintillator bars (9.61 mm × 9.61 mm × 1.8643 m) with wavelengthshifting fibers in the center. One end of the fiber has a mirror to reflectthe light back along the fiber, and the other is attached to a multi-pixelphoton counter (MPPC). The MPPCs are photon counters that work inhigh magnetic fields. They each contain 667 pixels arranged in a 1.3 ×1.3 mm2 active region. Each of the pixels is a Geiger micro-counter whichcreates a signal when electron-hole pairs are created by exciting electrons ina semiconductor above the band gap. These pixels are binary devices, butthe sum of the total charge from each of the pixels is proportional to thenumber of incident photons assuming that the number of photons is smallcompared to the total number of pixels [54].The scintillating bars are arranged into layers by placing 192 of thembeside each other. Then 192 more bars are glued perpendicularly to thebars of the first layer to make a module. This design makes it possible totrack particles granularly in both x and y. When neutrinos interact in thedetector they produce charged particles. As these charged particles travelthrough the scintillator bars light is produced in each of the scintillator barsthrough which the particle passes. Using the information from the MPPCsto read out which bars have collected scintillation light and how much lightthere is, it is possible to reconstruct the path of the particle in the detector.The first FGD is made of 15 scintillator modules. The second FGD alter-nates six water modules that are 2.5 cm thick with seven scintillator modules.This makes it possible to measure neutrino interactions on water as well ason carbon, which is important as the far detector, Super-Kamiokande, iswater [54].Time Projection ChambersThe three TPCs are positioned around the FGDs within the ND280 detector.They are made of two concentric boxes. The inner box is filled with argongas and has copper-clad G10 walls instrumented with twelve micromegas23detectors on either end. There is a central cathode in the middle to providean electric field. The outer box contains CO2 to insulate the inner box. Asa charged particle travels through the TPC it ionizes and produces elec-trons which drift away from the central cathode, which is connected to ahigh voltage, and towards the readout planes. Information from the readoutplanes combined with timing information and the ionization drift velocitymakes it possible to reconstruct the three dimensional path of the parti-cles. The TPC is used to determine the number of particles produced in aneutrino interaction as well as their momenta, which is possible because ofthe surrounding magnetic field. Particle identification can be done in theTPC using the energy loss of charged particles along the track. The TPCstypically measure neutrino interactions with vertices in the more massivesurrounding detectors such as the FGDs [55].P0DThe pi0 detector (P0D) was designed to measure neutral current single pi0(NC1pi0) cross sections on water and carbon. The NC1pi0 neutrino inter-action is defined as: ν + n/p → ν + pi0 + n/p. This is one of the mainbackgrounds for a νe appearance measurement. This detector contains threeregions. The center region is made of alternating scintillator modules andwater modules with brass sheets which can produce electromagnetic show-ers. The scintillator modules are made of perpendicular arrays of triangularscintillator bars read out by a wavelength shifting fiber and MPPC. Thewater modules each contain two bags that can either be filled with water orbe empty. On either end of this central detector are alternating scintillatorplates and lead sheets. This outer region is designed to produce electromag-netic showers in the lead as well as provide a veto. Neutrinos will interactin the scintillator and metal as well as the water, so to determine a crosssection, the P0D is operated with and without water so that a subtractionmethod can be used [56].24ECalThe ECal surrounds the P0D, TPC and FGD in 13 separate modules. Eachmodule is made of layers of scintillator interspersed with 1.75 mm of lead.The scintillator is read out with a wavelength shifting fiber and MPPC.There are six modules parallel to the beam around the FGD and TPC. Thesecomplement the FGD and TPC by reconstructing pi0 and photons producedin the FGD and TPC, that the FGD and TPC are not well designed toreconstruct because photons are not charged. Each of the modules alongthe beam direction contain 31 layers of lead and scintillator. The modulefarthest downstream along the beam direction is made of 34 layers of leadand scintillator. Six dedicated ECal modules surround the P0D parallel tothe beam. These have only six layers of lead and scintillator each becausethey are primarily used to reconstruct energy missed by the P0D, or as aveto [57].SMRDThe SMRD instruments the magnet surrounding the ND280 detectors. Eachof the eight magnet yokes is made of 16 iron plates with 1.7 cm of air betweenthem. The SMRD contains 440 scintillator modules placed in these gaps.The first five magnet yokes contain three layers of scintillator, the sixth yokecontains four layers of scintillator and the final two yokes contain six layersof scintillator. The scintillator is in all cases placed in the innermost gapsto measure particles escaping from the inner detectors. This detector canbe used to measure muons escaping from the inner detector, as a trigger fora cosmic muon control sample, and as a way to identify beam interactionsin the magnet [58].2.3 Super-KamiokandeSuper-Kamiokande is a 50 kTon cylindrical water Cherenkov detector lo-cated in the Mozumi mine 1 km under Mt. Ikenoyama in Gifu, Japan.It has been operating since April 1996. Over the years the detector haschanged, which has lead to several distinct analysis periods. As the T2K25experiment is taking place during the SK period that began in 2008 andhas continued to the present, this is the detector configuration that will bedescribed here.2.3.1 Detector DesignThe SK detector is composed of optically separated inner and outer detectorsthat fill a cylindrical stainless steel tank 41.4 m tall, and 39.3 m in diameteras illustrated in Fig. 2.6 [59]. It is designed to detect Cherenkov radiationproduced in the inner detector water.xyzInner Outer Detector 1,000m Control roomAccess TunnelPhoto multipliers41m Detector hallBeam Direction39mDetectorFigure 2.6: An illustration of the SK detector. Figure from [6].The inner detector is 36.2 m tall, and 33.8 m in diameter, which corre-sponds to 32 kTon of water. This means that the diagonal is approximately50 m, or on the order of the attenuation and scattering lengths for ultra-pure water. It is lined with 11,129 photo-multiplier tubes (PMTs) with a 20inch diameter, which corresponds to 40% photocathode coverage by area.These PMTs have 20% quantum efficiency and 70% collection efficiency atthe first dynode [19]. They are protected by fiber reinforced plastic coverswith acrylic front windows which have 96% transparency for photons. ThePMTs are mounted in modules composed of 3 by 2 PMTs which are placedin rows to cover the inner detector. The space between the tubes is coveredwith black plastic sheet to prevent reflections [59].26The outer detector is designed to veto backgrounds from cosmic muonsand radioactivity in the mountain rock surrounding the tank. It is 2.6 mthick on the top and bottom, and 2.7 m wide on the sides, which corre-sponds to 18 kTon of water. It has 1885 outward facing 20 inch photo tubesattached to wavelength shifting plates, which corresponds to 20% photo-cathode coverage by area. The wavelength shifting plates, along with whiteTyvek coating on the walls between the PMTs maximizes the light collectionand veto efficiency of the detector [59]. The PMTs and wavelength shiftingfibers in the outer detector were previously used in the IMB experiment.The outer detector has almost 100% efficiency for rejecting entering muons.There is a dead region of about 55 cm between these two detector regionsdue to the support structure that holds the PMTs in place. This space isoptically separated from the inner and outer detectors so that light will notescape into the active regions.The entire detector is filled with ultra-pure water which is constantlybeing processed by a water system to maintain the transparency of thewater and minimize background from naturally occurring radiation.The registered charges, and associated hit times are sent from each PMTto custom boards in electronics huts located on top of the SK tank. Thethreshold for recording a signal is 0.25 photo electrons (p.e.) which corre-sponds to -0.69 mV [19]. The boards generate a square wave that begins atthe first time light is recorded at the PMT, with the length of the squarewave proportional to the integrated PMT charge. Then it uses a time todigital converter to digitize the analog signals and is sent by ethernet tocomputers that process the data with a software trigger. Each board canaccommodate 8 PMT channels [18].These hits are sorted into event candidates using a series of computersand software triggers. For T2K beam data all times and charges within a1 ms time window of the T2K beam spill time, as determined by GPS, arerecorded. Events in these time windows are categorized in one of three ways.Fully contained (FC) events are above 30 MeV in reconstructed electronenergy, and contain tracks that begin and end in the SK inner detector andhave fewer than 16 hits in the outer detector. Outer detector events have 1627Table 2.1: Cherenkov thresholds for particles in water. Calculatedusing masses from [10].Particle Mass (MeV/c2) Threshold (MeV/c)e 0.511 0.583µ 105.7 120.5pi+ 139.6 159.2p 938.3 1070or more hits in the outer detector. Low energy events are defined as eventsthat are fully contained within the detector, but are too low energy to fallin the fully contained sample. Typically only the fully contained events areused in physics analyses [18].2.3.2 Cherenkov RadiationCherenkov radiation is produced when charged particles move faster thanthe speed of light in a medium [60]. The speed of light in a medium is relatedto the index of refraction of the medium, n. In the SK detector, chargedparticles produce light when they have velocityv >cn, (2.2)where n = 1.33 for water.This limit is often expressed in terms of the minimum momentum acharged particle must have in order to produce Cherenkov light. This isreferred to as the Cherenkov threshold, and calculated in terms of the indexof refraction, n, and the particle’s mass m aspmin =mc√n2 − 1 . (2.3)Cherenkov thresholds for some typical particles in the SK detector aregiven in Table 2.1.As a particle travels in the water, Cherenkov light is produced as a coneof light around the trajectory of the particle, so in the SK tank particles28CAP Congress 2013Neutrino Experiments13T2K Signal & BackgroundsSignal & background for νμ → νe oscillations νμ → νee-p (undetected)Oscillation Signal:ν ee-p (undetected)Beam νe Background:νlN+others (undetected)γ γπ0Neutral Current π0:Details of work to improve reconstruction in S. Tobayama's talk, later this session npe- νeW+16OQuasi-elastic Scattering(a) Electron ring. (b) Muon ring. (c) pi+ ring.Figure 2.7: Simulated Cherenkov rings produced by an electron,muon and charged pion.are seen as rings of light projected on the walls. The angle of emission is afunction of the speed the particle is moving:cos θc =1βn, (2.4)where β = vc .Faster particles will emit Cherenkov radiation at larger angles, with amaximum of 41.2◦ in water, and as a particle slows while traveling througha medium, the Cherenkov cone will slowly collapse until no light is emit-ted [61]. Different particles will produce different patterns of Cherenkovlight. Muon and electron Cherenkov rings are distinct because electronswill shower in the water producing “fuzzy rings”, while muons travel in astraighter trajectory which produces a sharper ring. Due to their similarmasses charged pions and muons produce similar rings, except that the pi-ons may interact hadronically. Simulated electron, muon and charged pionrings are displayed in Fig 2.7.The Cherenkov light produced by charged particles from neutrino inter-actions is detected in the SK PMTs and used to determine what happenedin a given interaction. This will be described in detail in Chapter 4.292.3.3 CalibrationCalibration of the Super-Kamiokande detector is essential for understandingthe detector output and making the physics measurements of interest toT2K.Relative PMT Gain CalibrationThe PMT gain is used to convert the output PMT charge to photoelectrons,and needs to be calibrated for each PMT. This is done by first calibrating therelative gain difference between each PMT. A nitrogen laser driven dye laseris used as a stable light source with constant intensity. The laser initiallyemits light at 337 nm, and then this light is shifted to 398 nm with a dye.Light is sent into the detector with an optical fiber connected to a diffuserball placed in the center of the SK tank to produce isotropic light. The laseris run in two modes, high intensity and low intensity which are selected bychanging the intensity of the laser light with a variable optical filter. Inthe high intensity mode each PMT has enough light to calculate an averagecharge Q(i) for each PMT. In the low intensity mode only a few PMTs arehit so it is possible to assume that all of the hits are single photoelectron hits,and the number of times a PMT is hit N(i) can be measured. If these twoquantities Q(i) and N(i) are measured in the same place, they are mostlyidentical except for the PMT gain for the ith PMT:Q(i) ∝ Ihigh a(i) QE(i) G(i) (2.5)N(i) ∝ Ilow a(i) QE(i) (2.6)where Ihigh and Ilow are the intensities of the high and low intensity lightfrom the laser, a(i) is the PMT acceptance QE is the quantum and collectionefficiency and G(i) is the PMT gain.Then the PMT gain can be calculated asG(i) = Q(i)N(i) , (2.7)30using the measurements of Q(i) and N(i). This ratio is tabulated for eachPMT and is used to convert charge to photoelectrons on a PMT by PMTbasis. The root mean squared (RMS) of the PMT gains in the detectoris 5.9%, which is assumed to be due to differences in the quantum andcollection efficiencies [19].Absolute Gain CalibrationThe absolute gain of the detector is measured using a nickel source thatproduces a uniform source of light at the one photo electron level. Thenickel source is a ball of NiO and polyethylene with a californium-252 sourceinserted in the center. The californium source produces neutrons whichcapture on the nickel and result in the reaction Ni58 + n → Ni59 + γ. Thisprovides an isotropic source of 9 MeV gammas which produce an average of0.004 photoelectrons per PMT per event when placed at the center of theSK detector, and means that more than 99% of the event signals are at theone photoelectron level.All of the one photoelectron distributions from each PMT are added to-gether in PMT charge to find the absolute gain of the SK detector. ThePMT by PMT corrections as determined in the relative gain calibration areincluded to account for differences between the PMTs. The average of thissummed single photoelectron distribution over the range of output chargesis the absolute gain of the SK detector. It is found to be 2.658 pC perphotoelectron as measured at the beginning of the latest SK run period.This summed single photoelectron distribution is used in the detector sim-ulation [19].Relative Quantum EfficiencyFor low intensity light, as described in the relative gain calibration, the hitprobability is proportional to the PMT quantum and collection efficiency,QE . Using Eq. 2.6 the quantum and collection efficiency can be written as:QE =NiIlowa(i). (2.8)31The number of hitsNi is calculated as the ratio of the number of observedhits to the number of predicted hits where the predicted hits are calculatedwith the detector simulation. Corrections are applied to take into accountthe PMT position relative to the source, the PMT acceptance a(i), and theproperties of light propagation in the detector. Any remaining differencesare attributed to the quantum and collection efficiencies. These are tab-ulated relative to the average for each PMT and are used in the detectorsimulation [19].Timing CalibrationThe timing of the detector also needs to be calibrated. The timing responseof different channels can vary based on various factors such as the length ofthe PMT cables and the processing time of the readout electronics. It alsodepends on the observed charge, as the amount of time it takes a large pulseto cross the discriminator threshold is shorter than for a small pulse.This calibration is done with a fast pulsing nitrogen laser, and an opticalfilter is used to vary the light intensity. The filtered light goes into a diffuserball in the center of the SK tank via an optical fiber, and this producesan isotropic source of light. The time that the laser light is emitted ismeasured with a dedicated 2 inch PMT, and then the channel response timeis corrected with this emission time as well as the time it will take light totravel from the center of the tank. Using the optical filter to vary the pulseheight of the light, the two dimensional distribution of timing and chargeare measured for each channel. Each charge bin is fit with an asymmetricGaussian distribution to reproduce the timing distribution of the data, andthese are used in the SK detector simulation [19].Water Quality CalibrationsThe optical properties of the water that Cherenkov light travels through arecalibrated using a laser injected vertically into the SK tank. The propertiesof light transmission are measured as a distance from the bottom of thetank. The tank is broken up vertically into five PMT regions, and a sixth32Figure 2.8: Illustration of the five vertical slices B1-B5 that the SKtank is broken up into to calibrate the properties of the water.The sixth corresponds to the PMTs on the top of the tank.Figure from [19].region is defined as the PMTs on the top of the tank. These are illustratedin Fig. 2.8.The time of flight corrected timing distributions are calculated for thePMTs in each detector region B1 to B5 and the top PMTs, and Monte-Carloparameters are generated to match the data. These timing distributions arebroken into a “scattering and absorption” region as well as a reflection region.In the scattering and absorption region the laser light is modeled as afunction of wavelength as:I(λ) = I0(λ) exp[ −lL(λ)](2.9)where I0 is the initial intensity of the laser, l is the distance the light travels,and L(λ) is the attenuation length due to scattering and absorption.The attenuation length is described in terms of three components: ab-33sorption, and two angular regions of scattering defined by the scattered pho-ton angle θ. Symmetric scattering is due to Rayleigh and symmetric Miescattering with an angular distribution of 1 + cos θ. Asymmetric scatteringis primarily due to forward Mie scattering where the scattering is describedby cos θ in the forward direction, and there is no backward scattering.This means the attenuation length may be written asL(λ) = 1αabs(λ) + αsym(λ) + αasym(λ)(2.10)where the α parameters are determined from fits to the laser data.The water transparency is measured as the amount of scattering andabsorption of light in the SK water. This is done by injecting laser light ofvarying wavelengths into the SK tank. Combining the attenuation lengthsfrom each of these components leads to a total attenuation length of 120 mfor light with a wavelength of 400 nm.The reflection portion of the timing distribution from the laser data isused to measure the effective complex index of refraction at the interface ofthe water, PMT glass, photocathode, and vacuum. The index of refractionof water and vacuum are set separately to 1.33 and 1.0 respectively. Thelaser data for wavelengths from 337-420 nm is fit to determine the effectiveindex of refraction for each wavelength. These are used in the detectorsimulation.Reflectivity of Black SheetA specialized laser injection system is used to measure the reflectivity of theblack sheet surrounding the inner detector PMTs. This is done with lightinjector placed in the center of the SK tank. The injector points a laserwith a variety of wavelengths at various incident angles on a sample of blacksheet. The reflected light enters the SK tank and is measured by the detectorPMTs. The laser injector can also be operated without the sample of blacksheet to normalize the measurement. The ratio of the measured charge withand without the black sheet is the reflectivity of the black sheet, and is usedin the detector simulation [59].342.4 Neutrino Interactions at SKBeam neutrinos are detected when they interact in the SK detector andproduce charged particles. Neutrinos can interact either through chargedcurrent weak interactions mediated by the W±, or neutral current weakinteractions, mediated by the Z0. The cross section for charged currentneutrino interactions as a function of energy is shown in Fig. 2.9 along withthe world measurements of these cross sections [20]. As illustrated in thisfigure, there are three main types of charged current neutrino interactions:charged-current quasi-elastic, resonance, and deep inelastic scattering. Eachof these will be discussed in more detail below.Figure 2.9: World neutrino interaction cross section data. Figurefrom [20].2.4.1 Charged Current Quasi ElasticCharged-current quasi-elastic (CCQE) interactions are dominant in the en-ergy range of the T2K beam, around 600 MeV. These interactions occurwhen a neutrino of flavor l interacts with a neutron in the water to produce35(a) CCQE interaction. (b) CC1pi interaction.Figure 2.10: Feynman diagrams of common neutrino interactions atT2K neutrino energies.a lepton and a proton:νl + n→ l− + p, (2.11)as illustrated in Fig. 2.10a. At T2K l can be either an electron or a muonbecause the neutrino energy is typically below the tau production threshold.The proton in the CCQE interaction is almost always below the Cherenkovthreshold, so these events are selected by looking for single electron or muonrings in the SK water. Since CCQE is a two body interaction, the neutrinoenergy Eν can be calculated using only the lepton kinematics, asEν =m2p − (mn − Eb)2 −m2l + 2(mn − Eb)El2(mn − Eb − El + pl cos θlν) (2.12)where mn and mp are the masses of the neutron and proton, Eb = 27 MeVis the binding energy of a nucleon in an 16O nucleus, and El, pl, θlν are,respectively, the energy, momentum, and angle between the direction of theincoming neutrino and outgoing lepton. This calculation assumes that theinteraction happens with a bound nucleon at rest, and neglects the Fermimomentum of the nucleon within the nucleus.362.4.2 Charged Current Single PionThe second most dominant interaction at T2K beam energies is the charged-current single pion interaction (CC1pi), also known as “RES” in Fig. 2.9because they typically come from resonances in the nucleus that decay intopions. The most common resonance at these energies is the ∆(1232). In-teractions on protons produce the ∆++ resonance which decays to a protonand pi+. Interactions on neutrons produce the ∆+ resonance which maydecay into either a neutron and pi+ or a proton and pi0. This means thatneutrinos can produce charged pions or neutral pions in these interactions:νl + p/n→ l− + pi+ + p/n (2.13)νl + n→ l− + pi0 + p, (2.14)as illustrated in Fig. 2.10b, but are dominated by pi+ production. Negativepions are produced in anti-neutrino interactions.Charged current resonance events that produce pi0s are unlikely to pro-duce backgrounds to the CCQE samples. These events will have three rings,one from the electron or muon and two from decay of the pi0 into photons.This means that it will be unlikely that the event will be single ring-likeas the lepton and one of the photon rings will usually be visible. As thephotons come from pi0 decay one of them is required to have energy that isat least half of the mass of the pi0, which is a minimum of about 67 MeV.The CC1pi+ interactions can also be explicitly selected at SK to increasethe number of signal events and reduce the backgrounds in the CCQE sam-ple by using a combination of ring particle identification and decay electroncounting. The focus of this thesis is the νµ CC1pi+ interaction which pro-duces positively charged pions and the details of the selection methods willbe discussed in more detail in the following chapters.2.4.3 Deep Inelastic ScatteringDeep inelastic scattering (DIS) events happen at high energy and occur onindividual quarks within the nucleons as opposed to the nucleons themselves.37These interactions often produce multiple pions as well as other mesonsin addition to the final state lepton. These are typically backgrounds tothe CCQE and CC1pi+ interaction channels. If all of the pions are belowCherenkov threshold these events may look CCQE-like or CC1pi+-like de-pending on how many decay electrons are produced. If only one pion isabove Cherenkov threshold these events may look CC1pi+-like. These mayalso contribute to CCQE and CC1pi+ backgrounds because of the final stateinteractions that pions undergo in the nucleus which may result in only zeroor one pion leaving the nucleus after the interaction.2.4.4 Neutral Current InteractionsNeutral current (NC) interactions occur through exchange of a Z0. Anelastic NC interaction is:νl + p/n→ νl + p/n. (2.15)Neutral current interactions also produce pions from excitation of ∆ reso-nances within the nucleus:νl + p/n→ νl + pi+ + p/n (2.16)νl + n→ νl + pi0 + p, (2.17)as in the CC1pi interactions. There are also high energy neutral currentinteractions that may produce multiple pions or other mesons.Neutral current single pion (NC1pi+ ) events are a background to theνµ CCQE selection because muon and pion rings can be very similar atSK, as illustrated in Fig. 2.7. These may also be a background to theCC1pi+ selection if the pion scatters in the SK water and both rings arevisible so that the event looks two ring-like despite the fact that only oneparticle is present. They may also be a background to the CC1pi+ selectionif both the pion and the proton are above Cherenkov threshold.382.5 SimulationThe T2K experiment is simulated using a series of different models andpackages described in the following sections.2.5.1 Flux PredictionThe T2K flux as produced by the beam is simulated from the protons enter-ing the secondary beamline to the target and finally to the decays in whichthe neutrinos are produced. The protons from the primary beamline andtheir interactions in the target are simulated using FLUKA [62]. The restof the secondary beamline is simulated using a Geant3 simulation calledJNUBEAM and with the GCALOR package from Geant3 used for hadronicinteractions [63]. All of the hadronic interactions from JNUBEAM are savedfor any hadron that eventually produces a neutrino. The hadron productionthat occurs in the target is reweighted to hadron production data from theNA61/SHINE experiments at CERN [64]. This hadron production data wastaken on a 2 cm thin graphite target, and with the same beam energy asT2K. Data from a replica of the target used in the T2K secondary beamlinehas been taken and will be used in the future. Additional data, such as fromthe HARP experiment, is used to tune pion and nucleon interactions afterthe primary production process, such as from scattering in the horns. Thisreweighting improves upon the GCALOR model predictions by matchingthe flux prediction to hadron production measurements [65] [66].2.5.2 Neutrino Interaction GeneratorsNeutrino interactions are simulated for a given neutrino flux and cross sec-tion as a function of neutrino energy using neutrino interaction generators.Each of a variety of neutrino interactions as defined in Section 2.4 are simu-lated with different probabilities depending on the neutrino flux and energy.There are several common neutrino interaction generators, each with differ-ent implementations of various neutrino interaction models. Some of thesestandard generators are NEUT [67], GENIE [68] and NuWro [69].392.5.3 NEUTThe T2K experiment primarily uses the NEUT neutrino interaction gener-ator to simulate neutrino interactions in the ND280 and SK detectors [67].The neutrino cross sections calculated in NEUT assume that interactionsoccur on single nucleons within a nucleus and then these are summed tocalculate the cross section. This requires the kinematics of the nucleonswithin the nucleus.There are several nuclear models used in NEUT. The relativistic Fermigas model is the simplest model of the nucleus. It assumes a flat distributionof initial nucleon momentum up to the maximum of the Fermi momentum.This corresponds to a constant nuclear density. The Spectral Function modeldescribes the nuclear density as a two dimensional distribution of momentumand binding energy which is integrated over the volume of the nucleus.NEUT by default uses the spectral function nuclear model to calculateCCQE interactions and the relativistic Fermi gas nuclear model to calculatecharged current interactions that produce pions and all neutral current inter-actions. Due to discrepancies between the CCQE cross section calculated inNEUT with the spectral function nuclear model and external measurementsof the CCQE cross sections, however, T2K currently uses the relativisticFermi gas model for all neutrino interactions.The relativistic Fermi gas model requires a value of the Fermi momen-tum and nuclear binding energy in the medium in which the neutrinos areinteracting. The nuclear binding energy is the energy required to eject anucleon from the nucleus, and is subtracted from the energy available afterthe neutrino interaction. These are obtained by fits to electron scatteringdata interpreted using the relativistic Fermi gas model. The reference valuesfor binding energy and Fermi momentum on oxygen are found to be 27 MeVand 225 MeV/c respectively [18] [21].The CCQE differential cross section is calculated using the Llewellyn-Smith formalism [70]. Multi-nucleon correlations and nucleon-nucleon cor-relations such as 2p2h are simulated using the Nieves model [71].CC1pi+ interactions can happen either on nucleons or coherently with the40nucleus. Nucleon interactions are simulated with an implementation of theRein-Seghal model that assumes the pions come from excitation of a baryonin the nucleus [72]. There are 18 resonances included in this model and theprobability of a final state is proportional to the sum of these contributionsalong with the probability that each nucleon will decay into a final statewith a single nucleon and pion. NEUT uses the Rein-Sehgal implementa-tion of the resonant axial mass and the value of the axial form factor at zerotransferred four-momentum. The non-resonant contribution is modeled assuggested by Rein-Sehgal, using a Breit-Wigner amplitude. Modified formfactors that focus on the ∆(1232) are used, because at T2K neutrino energiesa ∆(1232) is the most commonly produced resonance. The angular distri-bution of pions is calculated using the Rein-Sehgal method for the ∆(1232)resonance and assumes an isotropic distribution in the resonance rest framefor the other resonances [73]. Some of the higher resonances may generatemulti-pion production processes in addition to single pion production. Thismodel is only used below 2 GeV of hadronic invariant mass. Coherent in-teractions are simulated with a different Rhein-Seghal model assuming thatthe interactions happen on the nucleus as a whole instead of an individualnucleon [74].There is a transition region above the ∆(1232) resonance in which thereis a mixture of resonance production and deep inelastic scattering. Thismeans that deep inelastic scattering interactions are simulated separatelyabove and below 2 GeV of invariant hadronic mass. Above 2 GeV thePythia/JetSet model is used [75]. As this model is tuned to high ener-gies a combination of models are used below 2 GeV. Exclusive modes, suchas the single pion discussed earlier, single kaon, and single eta are calcu-lated separately. Events with more than one pion are generated with theRein-Sehgal model for resonance production described earlier [18].Particles that are produced in simulated neutrino interactions are propa-gated through the nuclear medium assuming a Woods-Saxon potential usinga cascade model. The particles are simulated at a location within the nu-cleus, and then this model calculates the interaction probability of eachparticle at each step it takes towards exiting the nucleus. For the analysis41presented in this thesis these final state interactions (FSI) are most impor-tant in the T2K energy range for charged pions as they have the highestinteraction probabilities. Pions with momentum below 500 MeV/c can in-teract via absorption, charge exchange or scattering. These probabilities arecalculated with the Oset et. al. model which uses a many body calculation inan infinite nuclear matter combined with a local density approximation [76].Pions with momentum above 500 MeV/c can also produce multiple hadronsor double charge exchange in a secondary interaction. These are modeledusing fits to pion-proton and pion-deuteron scattering data [10] [18].2.5.4 ND280 Detector SimulationThe ND280 Detector is simulated using a GEANT4 representation of the de-tector geometry. Models within GEANT4 are also used to simulate particlespropagating in the detector and the detector response, such as scintillationlight and MPPC response and associated electronics. All of the subdetectorsare simulated in a single framework. Individual detectors may have theirown reconstruction packages which are then integrated into a global eventreconstruction.2.5.5 SK Detector SimulationThe SK detector is simulated using a custom software package written bythe Super-Kamiokande collaboration called SKDETSIM [18]. This packagemodels the SK tank, water and PMTs and uses a combination of GEANT3and custom models to propagate particles from their production at a neu-trino interaction vertex and through the detector geometry [63]. The pro-duction and propagation of Cherenkov light is also modeled and takes intoaccount processes such as absorption, Rayleigh scattering and Mie scat-tering. The laser calibration data as described earlier is used to calculateparameter inputs and tune the detector simulation so that it matches theSK water [18].42Pion Hadronic InteractionsCharged pions may re-interact in the SK water. These are called secondaryinteractions (SI). The SI that pions undergo in the water are modeled withthe NEUT cascade model as described in Section 2.5.3 for pions with momen-tum less than 500 MeV/c. They are modeled with the GCALOR packagefor pions with momentum greater than 500 MeV/c [65] . Although FSI isa property of the neutrino interaction, and SI describes what happens inthe detector, the NEUT cascade model is used in both the NEUT neutrinointeraction generator and SKDETSIM to model pion hadronic interactions.The hadronic interactions that charged pions undergo either in the nu-cleus or in the SK water can be described by the number and type of pionsafter the interaction. For positively charged pions these are defined as:• Absorption: no pions after the interaction.• Charge exchange: one pi0 after the interaction.• Scattering: one pi+ after the interaction. There is both elastic andinelastic scattering, that, while modeled differently, produce the sametopology in the SK detector.• Double charge exchange: one pi− after the interaction.• Hadron production: multiple pions of any charge after the interaction.These definitions will be used throughout this thesis to discuss the in-teractions pions may undergo in the nucleus or in the SK water.43Chapter 3PICCOLOThis thesis will focus on the identification of charged pions at SK. Most ofthe charged pions produced in neutrino interactions at SK are close to theCherenkov threshold of 160 MeV/c in momentum, as illustrated in the truemomentum distribution for charged pions produced by CC1pi+ interactionsin Fig. 3.1.The PICCOLO (Pion Integrating Cylinder for Cherenkov Optics andLight Observation) detector was built to better understand the amount andproperties of the Cherenkov light produced by pions close to the Cherenkovthreshold, which may be different than for electrons and muons becauseof the pion hadronic interactions. This detector is an integrating waterCherenkov cylinder made out of PVC pipe with a 12 inch diameter and 1 mlength displayed in Fig. 3.3. The inside of the cylinder is painted with whiteAvianD reflectance coating so the Cherenkov light produced will reflect offthe walls of the detector, and be collected in one of four PMTs. Three 2 inchPMTs are located on top of the detector, and one 5 inch PMT is locatedopposite a 1 cm beam window. The detector set up is illustrated in Fig. ??.Data was collected in the TRIUMF M11 secondary beam line whichcontains electrons, muons and charged pions. To scan the pion Cherenkovthreshold data was collected in momentum steps of 5 MeV/c from 130 to200 MeV/c, and then more coarsely in 20-30 MeV/c steps up to 300 MeV/c.Several data sets were collected at some of these momentum settings. The44Figure 3.1: Momentum distribution of true CC1pi+ events in the T2K-SK Monte-Carlo.Figure 3.2: The PICCOLO detector.451 cm beam window 1 m 12 inch 5 inch PMT 2 inch PMTs Figure 3.3: Schematic of the PICCOLO detector.beam has approximately 2% momentum resolution. Each of the four PMTwaveforms were read simultaneously on a LeCroy oscilloscope with 2 nssampling of the PMT waveform over 5 µs. Within this time it is possibleto see primary PMT hits from the initial particle as well as decay electronslater in the waveform. Particles were detected in the PMT waveform usinga custom peak finder. The single photoelectron peak of each of the fourphototubes was calibrated for each data run by looking at all peaks in thewaveforms after 4 µs where there is the least contamination from decayelectrons.Particle identification (PID) was done using particle time of flight fromproduction to the detector, as illustrated in Fig. 3.4. It becomes more diffi-cult to separate the three particle types at higher energies, but the electronand muon fraction of the beam also drops off at higher energies, so above200 MeV/c it is assumed that all of the beam particles are pions.To test the efficacy of the detector and the data collection technique themuon lifetime was measured. As the beam was run with positively chargedparticles, no effects from muon capture are expected. Using the time of flightparticle identification muons were identified, and then the peak finder was46(a) 130 MeV/c. (b) 160 MeV/c.(c) 190 MeV/c. (d) 220 MeV/c.Figure 3.4: M11 beam composition at each of a range of momentumsettings based on particle time of flight. The time differencebetween the accelerator and a scintillator in front of the PIC-COLO detector is plotted. Because of the way the difference istaken, electrons appear as the peak at the highest times, muonsin the middle and pions at the shortest time.used to select a decay electron peak. A plot of the decay times for muons isshown in Fig. 3.5, and when fit with an exponential the lifetime is measuredas 2.21±0.04 µs, which is consistent with the PDG lifetime of 2.197 µs [10].In order to measure only the Cherenkov light that comes from the pri-mary pion or muon, any contribution from late decay electron or dark noiselight needs to be removed. This is done by looking at the starting time ofa peak in the waveform that produces light. Below Cherenkov thresholdthe only visible light is from the decay electron, and the peak start times47s)µDecay Time (0 1 2 3 4-610×Number of Events5001000Figure 3.5: The muon lifetime as measured in the PICCOLO detector.s)µPeak Start Times (0 1 2 3 4 5-610×Number of Events050100(a) Time that light emission begins for130 MeV/c muons, at Cherenkov threshold.s)µPeak Start Times (0 1 2 3 4 5-610×Number of Events0500100015002000(b) Time that light emission begins for180 MeV/c muons, above Cherenkov thresh-old.Figure 3.6: Time that light emission begins for muons in PICCOLO.roughly follow the decay electron spectrum as illustrated in Fig. 3.6a. AboveCherenkov threshold there is a sharp peak in the distribution of the timethat light is produced which corresponds to the light from the muon or pion,as illustrated in Fig. 3.6b. To look at only the light from the initial particlea cut is made on a 20 ns time window in light production times.48Looking only at this 20 ns time window, the mean amount of light pro-duced by the initial muons and pions over the momentum range in whichdata was collected are shown in Fig. 3.7. It is possible to see evidence ofthe Cherenkov threshold in both cases. In particular, charged pions expendapproximately as much Cherenkov light as expected, and there is evidence ofthe threshold at expected location of 160 MeV/c in pion momentum. Therewas not enough light produced in this detector to do more detailed stud-ies of the Cherenkov light emission model in SKDETSIM compared to thedata and more extensively understand any threshold effects that come frompion hadronic interactions. It may be possible to study these effects moreextensively in a future water Cherenkov beam test experiment such as theproposed NuPRISM Phase 0 beam test at Fermilab [77]. Future beam testsmay also make use of the explicit charged pion reconstruction described inthe next chapter to further understand the impact of hadronic interactionson the Cherenkov light production of charged pions in water.49(a) Light emitted by muons.(b) Light emitted by charged pions.Figure 3.7: PICCOLO results.50Chapter 4SK ReconstructionTo extract the physics we are interested in from the SK detector we needto use the data recorded by the PMTs to determine which particles wereproduced and their kinematics. An event at SK is a set of times and chargesrecorded by each of the PMTs. These charges are clustered into sets ofsimilar times and then arranged into “sub-events” which contain exactlyone charge and one time per PMT. We then use a maximum likelihoodalgorithm called fiTQun to extract the particle identification and kinematicquantities of interest.4.1 Likelihood FunctionfiTQun is a maximum likelihood algorithm for the SK detector. It wasinspired by the event reconstruction used for the MiniBooNE detector [3],and uses the times and charges recorded on each of the 11,129 PMTs in theSK inner detector to determine what happened in an event.The quantities that we are interested in reconstructing are the numberof particles in an event, which particles there are in an event (particle iden-tification), and the location and kinematics of each of these particles. It issimplest to start by looking at how to reconstruct a single particle, and thenthe method will be generalized to include multiple particles. One particlecan be defined in terms of eight track parameters: vertex position (x, y, z)51and time, momentum, direction (θ, φ), and the energy lost along the particletrack. Collectively these track parameters are called x.A likelihood function can be written in terms of these track parametersas:L(x) =Nunhit∏i=1Pi(unhit;x)Nhit∏j=1Pj(hit;x)fq(qj ;x)ft(tj ;x). (4.1)Here, Nhit is the number of hit PMTs in the event, and Nunhit is the numberof unhit PMTs in the event. The likelihood is constructed as the product ofthe probability that a given PMT will be hit or unhit for the hypothesizedparticle type and track parameters. Each of the 11,129 inner detector PMTsis assumed to be either hit or unhit such that Nhit+Nunhit=11,129 PMTs.Pi(unhit;x) is the probability that PMT i will be unhit for a given set oftrack parameters x and Pj(hit;x) is the probability that PMT j will behit for the same set of track parameters. The unhit probability can bewritten in terms of the hit probability as Pi(unhit;x) = 1 − Pi(hit;x). Forthe PMTs that are hit there is a charge and time likelihood fq(qj ;x) andft(tj ;x) specific to the track parameters x. These charge and time profilesare evaluated at the measured charge qj and time tj at PMT j respectively.It is easiest to find the best likelihood, and the track parameters that goalong with this likelihood, by working with the negative of the log likelihoodinstead of the likelihood itself. Working with the negative of the likelihoodmakes it possible for us to use existing minimization algorithms. Using thelogarithm of the likelihood makes it possible to break it up into time andcharge components.In this formulation the time and charge likelihoods can be fully separatedas:Lq(x) =Nunhit∑i=1log(Pi(unhit;x)) +Nhit∑j=1log(1− Pj(unhit;x)fq(qj ;x)) (4.2)Lt(x) =Nhit∑j=1log(ft(tj ;x)), (4.3)52and the full likelihood written as L = −Lq−Lt. The negative log likelihoodis minimized for a given set of track parameters x, as well as recorded timesand charges to determine the type of particle and its track parameters for agiven event.4.2 Charge LikelihoodIt is possible to calculate the charge likelihood by decoupling the propa-gation of particles in the detector and the emission of Cherenkov photonsfrom the PMT response and associated electronics by introducing a quantitycalled the predicted charge. The predicted charge is defined as the averagenumber of photoelectrons that a PMT will observe for a given particle hy-pothesis and set of track parameters and is written at the ith PMT as µi.It is possible to calculate the predicted charge for a given set of track pa-rameters x using what is known from simulation and calibration data aboutthe particle propagation, Cherenkov light emission, and detector response.This is done separately for each type of particle that is reconstructed, andthe details will be described in the next section.Using this mapping between track parameters x and predicted charge µithe charge likelihood can be written only in terms of the detector response:Lq(µ) =Nunhit∑i=1log(Pi(unhit;µi)) +Nhit∑j=1log(1− Pj(unhit;µj)fq(qj ;µj)). (4.4)With this definition of the likelihood, the unhit probability of a givenPMT should be Poisson with a mean of the predicted charge µ:Pi(unhit;µi) = e−µ. (4.5)Due to the PMT threshold, however, not all photoelectrons producedwill necessarily result in a PMT hit. To correct for this effect the unhitprobability is expanded in predicted charge as:Pi(unhit;µi) ≈ (1 + a1µi + a2µ2i + a3µ3i )e−µi , (4.6)53Figure 4.1: Charge probability distribution functions fq(q;µ) at a va-riety of predicted charges µ. The data points are the outputof the simulation, and the line is the fitted function used toevaluate the likelihood. Diagram from [4].with the a coefficients extracted from the detector simulation. This is foundto better replicate the detector simulation than the Poisson probabilityalone.Finally the charge probability distribution function fq(q;µ) is calculatedby simulating photons in the detector that have a range of predicted chargeµ, and looking at the distribution of measured charge for the hit PMTs asillustrated in Fig. 4.1. These charge distributions are normalized and fitwith a polynomial that is a function of predicted charge µ at each value ofmeasured charge q. The likelihood is evaluated for a given measured charge qby linearly interpolating between the values closest to the measured charge,and evaluating the result at the predicted charge µ.This method requires calculating the predicted charge for a given particlehypothesis and set of track parameters before the charge likelihood can beevaluated with the measured charges.544.2.1 Predicted ChargeThe predicted charge contains two main components, light that travels di-rectly from emission to a PMT, and indirect light which scatters or reflectsin the detector before reaching a PMT. For a given particle track the pre-dicted charges for direct and indirect light are calculated separately, andthen added together to put into the likelihood calculation. The calculationof the direct and indirect predicted charges for a particle hypothesis and setof track parameters is described here.The predicted charge from direct and indirect light are both a functionof the light emission point relative to the PMT that the light hits. Both areexpressed as integrals along the track length of the particle, s.The predicted charge for direct light is:µdir = Φ(p)∫g(p, s, cos θ) Ω(R) T (R) (η) ds. (4.7)As illustrated in Fig. 4.2, R is the distance from the light emission point tothe PMT, θ is the angle between the particle direction relative to R, and η isthe angle between the PMT normal and R. p is the initial momentum of theparticle. The functions that make up this integral define the light emissionand propagation along with the phototube acceptance and response. Thelight that is emitted by a particle traveling in the SK detector is describedwith the Cherenkov emission profile g(p, s, cos θ). The amount that lightattenuates due to absorption and scattering in the water is described byT (R). Finally the Cherenkov light acceptance by the PMT is described bytwo factors: the solid angle covered by the PMT face Ω(R), and the angularacceptance of the PMT (η). There is also a normalization term Φ(p) whichrepresents the number of photons and absorbs the constant factors acrossthe Cherenkov light emission, propagation and PMT acceptance such as theaverage number of photons produced by a particle with initial momentump and the PMT quantum efficiency.The predicted charge for indirect light is:µindir = Φ(p)∫Ω(R)T (R)(η)A(s)[∫ 14pig(p, s, cos θ) sin θdθdφ]ds. (4.8)55Figure 4.2: Variables used in calculation of predicted charge. Theinitial particle production is indicated by the white dot, and thepoint of light emission, s, is indicated by a black dot. Diagramfrom [4].Note that the expression for the direct and indirect predicted charges are thesame except for the factor A(s) which describes the light that is producedby scattering in the water or reflections from the PMT glass and black sheet.The calculation of each of the factors in the integrals in Eq. 4.7 and 4.8will be described in the following sections.Cherenkov Emission ProfileThe Cherenkov emission profile g(p, s, cos θ) describes the light that is emit-ted by a particle traveling in the SK detector. More specifically it is afunction that gives the number of photons that are emitted at angle θ alongthe path of a particle with initial momentum p and that has traveled adistance s. The emission profile is normalized so that∫g(p, s, cos θ)ds(sin θdθdφ) = 1. (4.9)The normalization corresponding to the total number of detected photons isabsorbed into Φ(p) in Eq. 4.7 along with the other normalizations described56earlier.The Cherenkov emission profile is calculated by generating particles ofeach type that we plan to reconstruct: electrons, muons, charged pions, pro-tons, kaons, and gammas, at discrete initial momenta using the SK detectorsimulation. Plots of these emission profiles for electrons and muons at sev-eral momenta are shown in Fig. 4.3. The emission profiles are displayed as afunction of the position at which light is emitted, s, and the Cherenkov an-gle, θ. These plots illustrate that the Cherenkov light emitted by electronsand muons looks very different. For electrons the peak of the Cherenkovangle emission tends to be the same as that of a particle with v = c. Muonshave a more obvious cone collapse as they travel in the detector. Electronsalso tend to have a wider angular profile compared to muons, as they pro-duce electromagnetic showers in the detector. Differences like these in theCherenkov emission profile are used to do particle identification on the ringsthat the reconstruction identifies.Light TransmissionThe function T (R) corresponds to the amount that light attenuates as aresult of absorption and scattering in the water. It is parameterized interms of the attenuation length in the SK water, Latt. The attenuationlength was calculated in the detector simulation by generating samples of 100simultaneously produced 3 MeV electrons from a single vertex and at randomdirections to approximate a point source of Cherenkov light. This was donein two special configurations of the detector simulation. One mode consistsof direct light only, so any light that would have scattered or reflected isinstead absorbed, and the other is perfect transmission, which does notallow any light to be scattered, absorbed, or reflected. These samples weregenerated in a line across the vertical center of the detector and plottedas a ratio of the number of photons in the direct light sample to those inthe perfect transmission sample. The results are fit to an exponential, and576.2. Predicted Chargeθcos0.6 0.8 1s (cm)010020030040000.20.40.60.8e 50MeV/cθcos0.6 0.8 1s (cm)010020030040000.050.10.150.20.25e 500MeV/cθcos0.6 0.8 1s (cm)010020030040000.10.20.30.40.5e 200MeV/cθcos0.6 0.8 1s (cm)010020030040000.050.1e 1000MeV/cFigure 6.2: Cherenkov emission profile g(p, s, cos θ) for electrons at differentinitial momentum. Horizontal axes represent the cosine of the angle fromthe particle direction, and vertical axes are the distance traveled from theinitial position of the particle.etc. is averaged over. It can be seen in the figure that photon emission forelectrons are peaked at cos θ ≈ 0.75 which corresponds to the opening angleof the Cherenkov cone from a particle travelling at the velocity β = 1 inwater, regardless of the momentum. Also, electromagnetic showers causedby electrons produce rather broad angular distributions. For muons on theother hand, the angular distribution of photon emission is much sharperand depends much stronger on momentum compared to electrons, and wesee the collapse of the Cherenkov cone as the particle travels and loses itsmomentum. Such differences in the emission profile between different par-ticle types provide particle identification capability, which will be discussedin more detail in later sections.The factor Φ(p) is a normalization factor which is proportional to the596.2. Predicted Chargeθcos0.6 0.8 1s (cm)010020030040000.20.40.60.8e 50MeV/cθcos0.6 0.8 1s (cm)010020030040000.050.10.150.20.25e 500MeV/cθcos0.6 0.8 1s (cm)010020030040000.10.20.30.40.5e 200MeV/cθcos0.6 0.8 1s (cm)010020030040000.050.1e 1000MeV/cFigure 6.2: Cherenkov emission profile g(p, s, cos θ) for electrons at differentinitial momentum. Horizontal axes represent the cosine of the angle fromthe particle direction, and vertical axes are the distance traveled from theinitial position of the particle.etc. is averaged over. It can be seen in the figure that photon emission forelectrons are peaked at cos θ ≈ 0.75 which corresponds to the opening angleof the Cherenkov cone from a particle travelling at the velocity β = 1 inwater, regardless of the momentum. Also, electromagnetic showers causedby electrons produce rather broad angular distributions. For muons on theother hand, the angular distribution of photon emission is much sharperand depends much stronger on momentum compared to electrons, and wesee the collapse of the Cherenkov cone as the particle travels and loses itsmomentum. Such differences in the emission profile between different par-ticle types provide particle identification capability, which will be discussedin more detail in later sections.The factor Φ(p) is a normalization factor which is proportional to the596.2. Predicted Chargeθcos0.6 0.8 1s (cm)010020030040001234 160MeV/cµθcos0.6 0.8 1s (cm)010020030040000.050.10.150.20.250.3 600MeV/cµθcos0.6 0.8 1s (cm)010020030040000.20.40.60.8 300MeV/cµθcos0.6 0.8 1s (cm)010020030040000.050.10.15 1000MeV/cµFigure 6.3: Cherenkov emission profile g(p, s, cos θ) for muons at differentinitial momentum. Horizontal axes represent the cosine of the angle fromthe particle direction, and vertical axes are the distance traveled from theinitial position of the particle.average total number of photons that are emitted by a particle with initialmomentum p. Φ(p) also absorbs the constant factors that are not accountedfor by other factors in Equation 6.3 such as the quantum efficiencies of thePMTs, and enforces proper normalization to the predicted charge.6.2.3 S lid Angle FactorΩ(R) represents the solid angle subtended by a PMT viewed from its normaldirection at distance R. In order to reduce computation time, the factor issimply approximated as the area of a circle normalized by the distance:Ω(R) =pia2R2 + a2, (6.5)606.2. Predicted Chargeθcos0.6 0.8 1s (cm)010020030040001234 160MeV/cµθcos0.6 0.8 1s (cm)010020030040000.050.10.150.20.250.3 600MeV/cµθcos0.6 0.8 1s (cm)010020030040000.20.40.60.8 300MeV/cµθcos0.6 0.8 1s (cm)010020030040000.050.10.15 1000MeV/cµFigure 6.3: Cherenkov emission profile g(p, s, cos θ) for muons at differentinitial momentum. H rizont l axes represent the cosine of the angle fromth p rticle direction, and vertical axes are the distance traveled from theinitial position of the particle.averag total number f photon that are emitted by a particle with initialmomentum p. Φ(p) also absorbs the constant factors that are not accountedfor by other fac ors in Equation 6.3 such as the quantum efficiencies of thePMTs and enforces proper normalization to the predicted charge.6.2.3 S lid Angle FactorΩ(R) represents the solid angle subtended by a PMT viewed from its normaldirection at distance R. In order to reduce computation time, the factor issimply approximated as the area of a circle normalized by the distance:Ω(R) =pia2R2 + a2, (6.5)60Figure 4.3: Example charge profiles for electrons (left) and muons(right) as a function of the position at which light is emitteds, and the Cherenkov angle. Figure adapted from [4].58defined as the transmission function given byT (R) = exp(−RLatt). (4.10)The attenuation length as measured by this method was determined asLatt = 72.96 m.Solid AngleThe solid angle factor Ω(R) corresponds to the geometric area presentedby the active detector surface as viewed from a point R, assuming that allof the light is perpendicularly incident. It is calculated as the solid anglecovered by a PMT when viewed perpendicularly at a distance R from thecenter of the phototube, assuming that the PMT is a flat circular object:Ω(R) = pia2R2 + a2 (4.11)where a is the radius of a PMT (a=25.4 cm). The assumption that the PMTis a flat circular object holds sufficiently well at distances R>1 m.PMT Angular AcceptanceThe function (η) is the PMT angular acceptance. This corresponds to howmuch the light a phototube collects is reduced if the light is not perpendic-ularly incident on the phototube as assumed in Ω(R). The incident angle ηis between the direction perpendicular to the PMT surface and the particledirection. The acceptance function as calculated from the detector simula-tion is shown in Fig. 4.4. The detector simulation output is fit to a piecewisecontinuous polynomial and the result is used as (η). It is approximated thatthe angular acceptance can only reduce the total amount of light detectedby a PMT, so the angular acceptance function is normalized to 1 for theperpendicular light calculated by the solid angle factor.59Figure 4.4: PMT acceptance output from SKDETSIM. The verticalaxis has arbitrary units, and the normalization is set so that(cos η = 1) = 1. Diagram from [4].Scattering TableThe additional factor present in the predicted charge integral for scatteredlight is the scattering table, which is calculated as the ratio of the differentialpredicted charge from scattered light to that of isotropic direct light:A(s) = A(xPMT , zvtx, Rvtx, ϕ, θ, φ) =dµscatdµdir. (4.12)The isotropic direct light in the denominator is the same as that con-sidered in the direct light predicted charge. Assuming that the Cherenkovangle does not change as the particle propagates and loses energy, and takingthe ratio as above, it is possible to factor out the momentum dependence inthe scattering table. This allows one table to be used for all particles acrossall momenta.The scattering table is parameterized in terms of six variables that de-scribe the PMT and particle locations relative to each other. These areillustrated in Fig. 4.5. It was calculated by simulating 3 MeV/c electrons atrandom positions in the SK detector. The output is filled in a six dimen-sional histogram. When a fit is run A(s) is calculated for the indirect chargeprediction by linearly interpolating between the six dimensions of this table.60Figure 4.5: Variables used to parameterize the six dimensional scat-tering table for PMTs on the walls of the detector. Diagram isthe same for PMTs on the top and bottom of the tank exceptthat RPMT, which is not illustrated, is used instead of zPMT .Diagram from [4].4.2.2 Calculation of Charge LikelihoodWith the predicted charge fully defined as described above, it is possible tocalculate the charge likelihood using the measured charges for each photo-tube.Parabolic Approximation of Predicted Charge CalculationThe integrals required to calculate the predicted charge must be evaluatedmultiple times for every PMT in an event. This is computationally imprac-tical in a reconstruction algorithm, so these integrals are pre-calculated andstored in lookup tables. This is done using the fact that the part of thepredicted charge integral that describes the PMT acceptance usually variesslowly as a function of the light emission point relative to the PMT, s. Thismeans that it can be approximated with a parabola as follows:J(s) ≈ j0 + j1s+ j2s2. (4.13)61The function J(s) combines the PMT solid angle, Ω(R), the water trans-mission function T (R) and the PMT angular acceptance (η) into a singlefunction in terms of the distance s.With this approximation the predicted charge from direct light in Eq. 4.7simplifies to:µdir = Φ(p)∫ ∞−∞g(p, s, cos θ)J(s)ds≈ Φ(p)[j0∫ ∞−∞g(p, s, cos θ)ds+ j1∫ ∞−∞g(p, s, cos θ)s ds+ j2∫ ∞−∞g(p, s, cos θ)s2ds](4.14)The coefficients j0, j1, j2 can be calculated by evaluating J(s) at threedifferent points along the track: the initial position (s = 0), the point atwhich 90% of the particle’s Cherenkov light has been emitted, and halfwaybetween these two points. This approximation is found to well approximatethe full integral.The scattering table A(s) also varies slowly in s, so this approximationcan be extended for indirect light. This is done by approximating:J(s)A(s) ≈ k0 + k1s+ k2s2 (4.15)This makes it possible to write the indirect predicted charge in Eq. 4.862as:µdir = Φ(p)∫ ∞−∞g(p, s, cos θ)J(s)A(s)ds≈ Φ(p)4pi[k0∫ ∞−∞g(p, s, cos θ) sin θdθdφ ds+ k1∫ ∞−∞g(p, s, cos θ) sin θdθdφ s ds+ k2∫ ∞−∞g(p, s, cos θ) sin θdθdφ s2 ds](4.16)and k0, k1, k2 can be calculated in the same way as the direct light jicoefficients.The remaining integrals are over the Cherenkov profile for direct andindirect light as:Idir(n) =∫ ∞−∞g(p, s, cos θ)snds (4.17)Iscat(n) =∫ ∞−∞g(p, s, cos θ)sin θdθdφ4pi snds. (4.18)These integrals are calculated in advance for a set of parameters thatdefine the initial conditions of the particle and the relative orientation tothe PMT. These are, as illustrated in Fig. 4.2, the initial momentum of theparticle, the initial distance R from the particle to the PMT, and the angleθ between the direction of the particle and the line connecting the initialposition of the particle and the PMT. The results of these integrals fordiscrete sets of these three parameters are fit as a function of initial particlemomentum for each value of R and θ. During the event reconstruction theintegral calculation is done by first evaluating it at a given initial momentumfor the two closest tabulated values of R and θ, and then the results arelinearly interpolated. The results can then be used as the predicted chargeneeded to calculate the charge likelihood.634.3 Time LikelihoodThe time likelihood Lt(ti|x) depends on the track parameters x as well asthe position of the ith PMT. Calculation can be simplified by introducinga corrected time tci that depends on the initial position and direction of theparticle and the relative location of the hit PMT. This corrected time re-moves some of the dependence on the track parameters x that comes fromjust using the raw hit time ti, because it makes it possible to parameter-ize the time likelihood in three dimensions as a function of only the trackmomentum, and the direct and indirect predicted charge.The corrected time is calculated by assuming that all of the light fromthe particle comes from the center of the particle track. Then the correctedtime is defined by subtracting the starting time of the track, the time forthe particle to propagate from the start point to the midpoint of the track,and the time for light to travel from the midpoint of the track to the PMTfrom the measured time ti:tci = ti − t0 −smidc− |xPMTi − (x0 + smidd)|cn. (4.19)Here, t0 is the starting time of the particle track. The term smidc is the timeit takes the particle to travel from its starting point to the track midpointsmid, assuming that the particle’s velocity is close to c. The midpoint ofthe track is calculated using the track energy. The final term is the time ittakes light to travel from the track midpoint to the PMT. This depends onthe distance of the ith PMT from the track midpoint, which is calculatedusing the midpoint of the particle track smid, the distance from the startingpoint of the track to the ith PMT xPMTi , the starting position of the trackx0, and the direction of the particle d. Light travels from the center of thetrack to the PMT at the group velocity of the Cherenkov photons in water,or cn = c/n where n = 1.38.Calculation of the time likelihood can be simplified by assuming thatthe corrected time likelihood only depends on the momentum of the particle64and the predicted charge. The dependence on which particle is producinglight and the properties of the PMT response are included in the predictedcharge. The predicted charge dependence is broken into direct and indirectcomponents where the direct component comes from the direct charge pre-diction and indirect from the indirect charge prediction. The direct lightarrives promptly, whereas indirect light arrives later due to the fact that ithas been scattered or reflected. Varying these separately makes it possibleto vary the ratio of prompt and late times that light reaches the PMT in thetime likelihood. After being calculated separately the direct and indirecttime likelihoods are combined to produce the final time likelihood.Assuming that an indirect photon can only provide a hit at PMT i if nodirect light reached this PMT then the direct and indirect time likelihoodscan be combined asLt(tci ) = wLdirt (tci ) + (1− w)Lsctt (tci ). (4.20)Here w is the weight given to the direct light distribution and 1−w is theweight given to the late light distribution. The probability that there are nodirect or indirect light hits is the Poisson probability of drawing zero witha mean value of the predicted charge. This means that w, the probabilitythat a hit has at least one prompt photoelectron is calculated asw = 1− e−µdir1− e−µdire−µsct . (4.21)The time likelihood is calculated at each step in the likelihood minimiza-tion.4.3.1 Time Likelihood from Direct LightThe direct time likelihood for the corrected time tci defined above is assumedto be a Gaussian and depend only on the predicted charge from direct lightµdir and the initial momentum of the particle. A hit typically incorporatesall the light that reaches a PMT in association with an event, but onlyone time, which corresponds to when the first photon to reach a PMT is65registered. This means that as there are more photons the width of thecorrected time distribution will decrease. Similarly, the particle momentumis related to the width of this distribution because the longer the track themore possible photon production times there are, which means that thecalculation of the amount of time that it takes a photon to reach a PMTbuilt into the corrected time definition will become less accurate.Corrected time distributions can be generated using MC samples of eachparticle type at a range of fixed momentum values. Using these samples atwo dimensional histogram in corrected time tc and predicted charge fromdirect light log(µdir) is filled for all PMT hits from direct light at eachmomentum. Each bin of predicted charge is fit as a Gaussian distributionin corrected time. The resulting mean and standard deviations for eachpredicted charge bin are fit as a sixth order polynomial. Then each of theresulting polynomial fit parameters are fit as a function of initial particlemomentum. This parameterizes the direct time likelihood as a function thatdepends only on the predicted charge and particle momentum. An exampleof this parameterization compared to the raw output of the simulation isshown for 450 MeV/c muons and a range of predicted charge in Fig. 4.6.4.3.2 Time Likelihood from Indirect LightThe indirect time likelihood is modeled as the following formula that followsthe typical shape of the corrected time distribution from indirect light:Lsctt =1√pi/2σ + 2γ× e−τ2/2σ for τ < 0 (4.22)1√pi/2σ + 2γ× (τγ+ 1)e−τ/γ for τ > 0. (4.23)Here, τ = tc − 5 ns, σ = 8 ns, and γ = 25 ns. The long tail at latetimes corresponds to reflected light. The predicted charge and momentumdependence are not taken into account in this calculation.66Figure 4.6: Time likelihood from direct light for 450 MeV/c muonsand a range of predicted charge. The data points indicate theraw simulation output and the red curve is the final fit out-put assuming that the likelihood depends only on the predictedcharge and particle momentum. Diagram from [4].4.4 Sub-Event AlgorithmA SK event is defined as any PMT hits in a time window of order 10 µsaround an event trigger. These hits are clustered into groups of similartimes, and then these clusters are arranged into subevents which contain onecharge and one time per PMT. The likelihood algorithm described above isrun on each of these subevents. Typically the first subevent contains theprimary particles from a neutrino interaction, and subsequent subeventscontain activity that is separate or delayed. This delayed activity is oftena result of decay electrons that come from muons and pions produced inneutrino interactions. These are well separated in time as the muon lifetime67is about 2 µs. The timing resolution of the SK detectors is about 3 ns [19].4.4.1 Vertex Pre-FitterThe vertex prefitter uses the hit times of the PMTs to estimate the vertexposition and time (x, t). This is done by looking for the (x, t) that maximizesthe vertex goodness assuming that a point light source is emitted at thevertex:G(x, t) =hit∑iexp(− tc(point)iσ)2/2 , (4.24)wheretc(point)i = ti − t0 −|xPMTi − x0|cn. (4.25)For values close to the true vertex tc(point)i is close to zero, which results inlarge values of the goodness. Maximization of the goodness is done with agrid search in x and t. The grid search calculates the goodness for a grid ofallowed vertex positions and times, and the grid point at which the goodnessis best is considered the best fit. Then the goodness is minimized using theSimplex algorithm in MINUIT [78]. This is done on all of the hits in theevent to quickly find a seed for the vertex position and time.4.4.2 SubeventsPossible subevents are first identified with a peak finder. This is done byfixing the vertex position at the output of the vertex prefitter and thenscanning the goodness function while varying the time. Subevents appear aspeaks in the goodness distribution, assuming that the true vertex positionsare close to the prefit values.Time windows are defined around each peak. Only one hit per PMT isallowed within the peak. To calculate the time windows first the correctedtime as defined in Eq. 4.25 is calculated for each measured time ti within a68peak. The hits that are earliest and latest in terms of corrected time, andthat also fall within a window of 180 ns < tc(point)i <800 ns are called a timewindow.The time windows are defined by calculating the corrected time as de-fined in Eq. 4.25 for each measured time ti within a peak and looking for theearliest and latest hits that fall within a window of -180 ns < tc(point)i <800ns. Any time windows that overlap are combined.Finally the vertex pre-fitter and the peak finder are rerun on each ofthe time windows to define the subevents on which the full likelihood recon-struction algorithm is applied.4.5 One Ring FitsThe full likelihood as defined in Eq. 4.1 is minimized for each subevent to findthe track parameters x for a given subevent. This is done sequentially foreach of the one ring particle hypotheses: electron, muon, and charged pion,with a separate Cherenkov profile used for each. The SIMPLEX algorithmin MINUIT is used to do the minimization [78].The electron fit is run first. The vertex is seeded with the output of thevertex pre-fitter. The direction is seeded with a likelihood scan of pointsequally spaced on a unit sphere with the momentum fixed to an estimatethat comes from the total event charge. The momentum is seeded withanother likelihood scan at the direction found in the earlier step. Oncethese seeds have been determined the fit is done by minimizing the negativelog likelihood.The muon fit is seeded with the vertex and direction result from theelectron fit, and uses a grid search to seed the momentum. Fit results areobtained by minimizing the negative log likelihood with these seeds.4.5.1 Upstream pi+ FitCharged pions and muons have similar masses, 139 MeV/c2 and 105 MeV/c2respectively, which means that their charge profiles in SK will be similar [10],as illustrated by the rings in Fig. 2.7. Unlike muons, however, charged pions69can undergo hadronic interactions, and these can be used to separate themfrom muons. These hadronic interactions mean that a pion may abruptlystop emitting light because it has either been absorbed in the water orlost enough energy in a hadronic scatter to fall below Cherenkov thresh-old. When this occurs a pion will produce a sharp thin ring. Muons, how-ever, tend to generate Cherenkov light continuously until they fall belowCherenkov threshold. Pion tracks are defined in relation to their hadronicinteractions, where the portion before the interaction is called the “upstreamtrack”, and after the interaction is called the “downstream track”, as illus-trated in Fig. 4.7. The upstream pi+ fit only considers light that is producedby the upstream segment, before a hadronic interaction.π+ Downstream Track Scatter Point Production Point π+ Upstream Track Figure 4.7: Pion hadronic scatter.This effect, in which pions abruptly stop generating light, can be builtinto the one ring fit algorithm by defining a parameter that indicates howmuch energy the pion has lost from where it started to where it interacts.Using this parameter, and assuming that a pion expends energy at a con-stant rate along the track length, it is possible to calculate the position andmomentum of the pion at its interaction point. Then two separate chargepredictions are combined to create a pion track. The first, which is calledthe “full track”, uses the initial momentum of the pion and assumes thatit expends all of its energy traveling in the water. The second, called the“anti-track”, uses the pion momentum at its interaction point and repre-sents the energy the pion would have expended if it continued to travel inthe water. A charge prediction for the pion before an interaction is created70by subtracting these two charge predictions. These three components of thetrack are illustrated in Fig. 4.8. The upstream pion fit is done for all of thestandard track parameters x, as well as for the energy loss along the piontrack. This fit is done after the one ring electron and muon fits, and usesthe one ring muon fit as a seed.4.6 Generic Multi-Ring FitThe one ring fits described above can be generalized to fit for topologieswith multiple rings by adding their charge predictions. fiTQun contains ageneric multi-ring fitter that can look for events with up to six rings.Initially the rings in multi-ring events are assumed to be either electron-like or upstream pion-like. The sharp rings produced by muons and pionsare distinct from the “fuzzy” rings produced by electron showers in the SKwater. The upstream pion-like hypothesis mostly encompasses the muonhypothesis because of the similarity of their Cherenkov profiles. In theabsence of hadronic interactions, when the energy loss along the track of thepion is the maximum allowed, pions and muons will appear to be almostidentical.The multi-ring fit starts with either the one ring electron-like or the onering pion-like fit output. Then another ring that is electron-like or pion-like,and comes from the same vertex is added. A fit is done to calculate thelikelihood for the new hypothesis. All possible two ring combinations aretested: ee, epi, pie, pipi. For each of these two ring fits the likelihood ischecked to see if it improves with the addition of the new ring. This is donewith a quantity called the ring counting likelihood defined asLring = log(L(n+1)R/LnR), (4.26)where n corresponds to the number of rings. When comparing the one ringand two ring fits, n = 1 and n+ 1 = 2.There are several issues to keep in mind when comparing the likelihoodsof an event after adding a new ring. First, when additional degrees offreedom are added to the fit, such as with a new ring, the likelihood will71Full$Track$Upstream$Pion$Track$An56Track$Figure 4.8: Cartoon of the three configurations of the charge likeli-hood for the upstream pion fit. The blue cones represent theCherenkov light emitted by each charge likelihood configuration.The red dot indicates the pion scatter point.72typically improve because the result cannot do any worse than the originalfit. That is, if the second ring is turned off it is possible to recover theoriginal prediction. This means that when adding a new ring it is necessaryto define a threshold over which a change in the likelihood due to a new ringis determined to be improving the event prediction.The two ring likelihood for the ee and epi configurations are comparedto the one ring electron fit likelihood. The two ring likelihood for the pieand pipi configurations are compared to the one ring pion fit likelihood. Asecond ring is considered to improve the event prediction if the ring countinglikelihood is greater than 87.5 for a new electron-like ring and 140 for a newpion-like ring. These values were determined empirically by looking at thering counting distributions from SK atmospheric Monte Carlo (MC), whichconsists of simulated neutrino events produced with the flux of atmosphericneutrino events at SK .Fits with three or more rings are constructed by adding an electron-likeor pion-like ring to each of the two ring fit configurations. The fit result withan additional ring n+ 1 is compared to the best fit of the previous fit stagewith n rings using the ring counting likelihood. A new ring is determined toimprove the event prediction if the ring counting likelihood is greater than70 for these higher ring multiplicities, independent of ring type. This is doneuntil either the addition of another ring does not improve the ring countinglikelihood, or the fit reaches six rings. The multi-ring fit is currently onlyconfigured to work for up to six rings. The best fit is the fit hypothesis withthe best likelihood among the final fits.Once this preliminary multi-ring fitting is done the results are refinedwith a series of ring merging and refitting.The ring merging is done by looking at the angle between the highestenergy ring in the initial multi-ring fit, and each of the lower energy rings.If this angle is less than 20◦ the rings are added together.The result of the ring merging is refit ring by ring. First the highestenergy ring is refit assuming either an electron or pion hypothesis with allof the other rings fixed to their values from the result of the ring merging.The fit result of the electron ring hypothesis is compared to the pion ring73hypothesis to determine if, based on the likelihood, the highest energy ringis electron-like or pion-like. If the highest energy ring is determined to beelectron-like then the fit with the electron hypothesis is used as the best fit.If the highest energy ring is determined to be pion-like then the event is refitwith a muon hypothesis for the most energetic ring. The muon hypothesisis used because it is more likely that the highest energy ring in an eventwill be muon-like, from a charged current interaction, than pion-like froma neutral current interaction. Each of the lower energy rings are similarlyrefit using just an electron or a pion hypothesis and their particle type isreevaluated. The result of this refitting is the final output of the multi-ringfitter.4.7 Scattered Charged Pion FitIn addition to the generic multi-ring fit described above, fiTQun also makesit possible to combine rings to search for particular event topologies, suchas pion hadronic interactions. The simplest hadronic interaction topology isa charged pion that scatters exactly once in the SK detector, and produceslight before and after the hadronic scatter, as illustrated in Fig 4.7. A scat-tered charged pion fitter was written to look for this charged pion topology.The method is based on one used at MiniBooNE, but the implementationis unique to SK [5].This scattered charged pion fitter was studied to understand the recon-struction performance for charged pions individually before looking at themore complicated topology of CC1pi+. This fitter may also be used forphysics analyses such as a NC1pi+ cross section measurement at SK.4.7.1 MethodEach ring of a fiTQun track is characterized by its particle identification aswell as its vertex position and time (x, y, z, t), direction of motion (θ, φ),momentum, and energy lost along the track as the particle propagates inthe water.The scattered pi+ hypothesis is constructed by placing two single ring74pi+ tracks together. The downstream vertex position and time is calculatedusing kinematics of the upstream track as:xDS = xUS + lUS sin(θUS) cos(φUS)yDS = yUS + lUS sin(θUS) sin(φUS)zDS = zUS + lUS cos(θUS)tDS = tUS + t(lUS),(4.27)where lUS is the distance that the upstream track traveled, and t(lUS) is thetime that it took the upstream track to travel its length.The pion is also allowed to elastically or inelastically scatter such that thepion can lose energy in the scatter by restricting the downstream momentumto less than or equal to that of the upstream track. To make this constrainteasier to work with, given the fact that the upstream momentum variesin the fit, the downstream momentum is fit as a fraction of the upstreammomentum.Pion Range TableTo approximate the energy lost as a pion travels in water, tables of thecontinuous slowing down approximation (CSDA) range were produced forfiTQun. The CSDA is defined as:CSDA =∫ 0E0dE−dEdx(4.28)[79].The energy lost as a particle passes through matter dEdx can be estimatedusing the Bethe-Bloch equation:− dEdx= kz2(ZA) 1β2[12 ln(2mec2β2γ2WmaxI2)− β2 − δ(βγ)2](4.29)75[10] [79]. The constant k is defined ask = 4piNAremec2 (4.30)where NA = 6.022141 × 1020 mol−1 is Avogadro’s number, re = e24pi0mec2 =2.817940285 fm is the classical electron radius, and mec2 = 0.510998902MeV is the electron rest energy.z2 is the charge of the incident particle. The relativistic kinematic quan-tities are β = v/c and γ = 1/√1− β2, as calculated for a particle with agiven kinetic energy.Wmax is the maximum kinematically allowed electron recoil kinetic en-ergy in matter for an incident particle of mass M , given by:Wmax =2mec2β2γ21 + 2γmeM +m2eM2(4.31)[79].Z/A is the ratio of the atomic number to the atomic mass and I is themean excitation energy of the material. For water these are 0.55509 and79.7 eV respectively [10].The final term of the Bethe-Bloch equation, δ(βγ), is the density effectterm which is a function of the energy of the incident particle. The pions itis possible to reconstruct at SK fall into the region where the density effectmatters, but not high enough in energy to be well described by an asymptoticapproximation based on the overall plasma energy of the material. Thisregion is defined by: x0 < log(p/M) < x1. For water x0 = 0.2400 andx1 = 2.8004 which corresponds to a range in pion momentum from 242MeV/c to 87.8 GeV/c. Below this range in pion momentum the densityeffect is zero in a nonconductor like water. In the intermediate region thedensity effect function is defined as a function of the Sternheimer coefficientsa, k, and c¯ as:δ(βγ) = 2ln(10) log10(βγ)− c¯+ a(x1 − log10(βγ))k (4.32)76For water, these constants are: a=0.0912, k=3.4773, c¯=3.5017 [10].The PDG calculates tables of the CSDA range as a function of kineticenergy for muons [10] [79]. As other particles, such as charged pions, arenot included, the Bethe-Bloch equation was numerically integrated using theinputs above to calculate the CSDA range as a function of momentum forcharged pions. This calculation neglects the pion hadronic interactions sincethe Bethe-Bloch equation only considers ionization loss. As the hadronicinteractions are explicitly reconstructed these assumptions are compatible.The results of these calculations are shown in Fig. 4.9 for the momentumrange of most pions expected at T2K neutrino energies.100 200 300 400 500 600Momentum050100150200RangeCalculated π PDG µ (MeV/c) (cm) Figure 4.9: Comparison of muon and pion ranges in water at low en-ergy.A table of pion range as a function of pion momentum was producedup to 4 GeV/c in pion momentum to use in the scattered charged pionreconstruction. The range at a given momentum is calculated by linearlyinterpolating between the calculated points.With this table, the distance that the upstream pion travels lUS is cal-culated as the difference in the range of a pion with given momentum pUSat the beginning of the upstream track, and momentum pf at the end of the77upstream track:lUS = Range(pUS)− Range(pf ). (4.33)The range is linearly interpolated between the calculated points in Fig. 4.9.pf is calculated in terms of the initial upstream pion momentum, pUS ,and the upstream energy loss along the pion track as:pf =√(EUS/c− Eloss/c)2 −m2pic2, (4.34)where EUS is the energy of the pion at the beginning of the upstream track,Eloss is the energy the pion loses along the upstream track, and mpi is themass of the pion.The time at which the upstream track ends, t(lUS) is calculated as:t(lUS) =ρUSc(pUSc− pDSc). (4.35)ρUS is an estimate of the range as a function of energy for the upstreamtrack in terms of the distance the pion travels lUS and the energy loss of thepion along the upstream track Eloss:ρUS =lUSEloss. (4.36)In the one ring fits, and the generic multi-ring fit an approximation ofthe range as a function of energy is given by ρ = 1dE/dx , where dE/dx isassumed to be constant.Seeding ConfigurationsThe starting point of all seeding configurations for the scattered pion fit isthe output of the generic multi-ring fit with two pion rings. The scatteredpion algorithm is applied after the generic multi-ring fit. This fit hypothesisassumes that there are two pion rings with independent kinematics producedat a single vertex, as described in Sec. 4.6. The scattered pion fit is donein two configurations, separately assuming that each of the rings from the78generic multi-ring fit with two pion rings is the upstream ring. The down-stream ring is seeded by a grid search at the end point of the upstreampion track. The scattered pion fit outputs the kinematics outlined above, aswell as the fit negative log likelihood for these two configurations (L1, L2).The best fit is selected by comparing the likelihood ratio between these twoconfigurations. If L1 − L2 > 0, then L1 is the best fit, and if L2 − L1 > 0,then L2 is the best fit.Studies were also done assuming that the two ring pion fit had identifiedthe upstream and downstream tracks and using each of those to seed the fitinstead of the grid search. The fit performance was better when the generictwo ring fit was combined with a grid search even though this discards theinformation from one of the rings in the generic two ring fit.For comparison the fitter is seeded with the true Monte Carlo informa-tion which gives a representation of the theoretical limit of how well the fitcan perform. In this case if the fit gets stuck in a local minimum or failsto converge this will be a result of the fit function itself as opposed to im-provements that may be made to the seeding algorithm. Similarly the fitresults, including the kinematic resolution will be representative of the bestthat the algorithm can return.4.7.2 Fit PerformanceValidation of the scattered pion fitter was done by testing a variety of differ-ent seeding configurations, and fixing and floating the upstream and down-stream portions of the fit individually to isolate the fit performance.The fitter performance was studied in different seeding configurationson Monte Carlo charged pions. Single charged pions with a flat momentumdistribution from 0-1 GeV/c were simulated in random positions and direc-tions in the SK tank. Only basic selection cuts requiring the events to havea vertex position more than two meters from the walls of the inner detectorand fewer than 16 hits in the outer detector, were applied. Events werecategorized depending on the type of true hadronic interactions the pionsunderwent. The resolution of the vertex position and scatter point for pions79that undergo any hadronic interactions is shown in Fig. 4.10. The kinematicresolution of the upstream pion tracks for any pion hadronic interactions isshown in Fig. 4.11. The kinematic resolution for the downstream track forpions that undergo any hadronic interaction are shown in 4.12.The ideal topology for this fitter is that of a charged pion scatteringexactly once in the SK detector and continuing to emit light before andafter the scatter, and the fitter performance was also explicitly studied onthese events. The vertex position and scatter point for pions that scatterexactly once is shown in Fig. 4.13. The upstream kinematic resolution forpions that scatter exactly once is shown in Fig. 4.14 and the downstreamkinematic resolution is in Fig. 4.15. In all of these plots the fitter perfor-mance when seeded with the true Monte Carlo kinematics is compared tothe reconstructed seeding configuration described in Sec. 4.7.1. In general,the fitter is able to well reconstruct the kinematics, in particular for theupstream pion. Fit performance on pion events that scatter exactly once isalmost as good when seeded with reconstructed quantities as when seededwith true quantities.This fitter was also studied on the T2K-SK νµ Monte-Carlo (T2K-SKMC), and the performance is shown to be similar on true NC1pi+ events.These events are, as described in Section 2.5, simulated to come from theT2K beam flux and contain other outgoing particles from neutrino interac-tions such as nucleons which could affect the reconstructed topology. Theyare also more realistic interactions in terms of event topologies that we couldexpect to reconstruct from neutrino interactions. The vertex and scatterpoint for all NC1pi+ events is shown in Fig. 4.16, and the kinematic resolu-tion for all NC1pi+ events is in Fig. 4.17. The vertex and scatter point forthe ideal topology of NC1pi+ events in which the pion scatters exactly onceare shown in Fig. 4.18 and the kinematic resolution for these events is shownin Fig. 4.19. The resolution is similar in both all pion topologies and thesingle scatter topology to the results on the simulated charged pions witha flat momentum distribution, except that the statistics for NC1pi+ eventsare more limited.In summary the scattered pi+ fit is able to well reconstruct the up-80(a) Vertex resolution in x direction. (b) Scatter point resolution in x direc-tion.(c) Vertex resolution in y direction. (d) Scatter point resolution in y direc-tion.(e) Vertex resolution in z direction. (f) Scatter point resolution in z direction.Figure 4.10: Vertex resolution for the scattered pion fit. The leftcolumn shows the production point and the right column showsthe scatter point. All charged pions are shown in this plot.81(a) Momentum Resolution.(b) Angle between true and reconstructed direc-tions.(c) Resolution of energy loss along the track.Figure 4.11: Kinematic resolution of scattered pion fitter for the up-stream pion track. All charged pions are shown in this plot.82(a) Momentum Resolution.(b) Angle between true and reconstructed directions.Figure 4.12: Kinematic resolution of scattered pion fitter for thedownstream pion track. All charged pions are shown in thisplot.83(a) Resolution in x direction. (b) Resolution in x direction.(c) Resolution in y direction. (d) Resolution in y direction.(e) Resolution in z direction. (f) Resolution in z direction.Figure 4.13: Vertex resolution for the scattered pion fit. The leftcolumn shows the production point and the right column showsthe scatter point. Charged pions that scatter exactly once areshown in this plot.84(a) Momentum Resolution.(b) Angle between true and reconstructed direc-tions.(c) Resolution of energy loss along the track.Figure 4.14: Kinematic resolution of the scattered pion fitter for theupstream pion track. Charged pions that scatter exactly onceare shown in this plot. 85(a) Momentum Resolution.(b) Angle between true and reconstructed directions.Figure 4.15: Kinematic resolution of the scattered pion fitter for thedownstream pion track. Charged pions that scatter exactlyonce are shown in this plot.86(a) Neutrino interaction vertex resolu-tion in x direction.(b) Scatter point resolution in x direc-tion.(c) Neutrino interaction vertex resolu-tion in y direction.(d) Scatter point resolution in y direc-tion.(e) Neutrino interaction vertex resolu-tion in z direction.(f) Scatter point resolution in z direction.Figure 4.16: Vertex resolution for the scattered pion fit onNC1pi+ events in the T2K-SK MC for all outgoing pions. Theleft column shows the neutrino interaction vertex and the rightcolumn shows the scatter point.87(a) Momentum resolution for the up-stream track.(b) Angle between true and recon-structed directions for the upstream track.(c) Momentum resolution for the down-stream track.(d) Angle between true and recon-structed directions for the downstreamtrack.Figure 4.17: Kinematic resolution of the scattered pion fitter onNC1pi+ events in the T2K-SK MC for all outgoing pions.stream and downstream track parameters for simulated charged pions thatare placed at random position and directions in the SK detector as well asfor single charged pions that come from neutrino interactions with the T2Kbeam flux. The kinematics are best for the target topology of a charged pionscattering exactly once, but can also be reconstructed for charged pions thatundergo any secondary interactions in the SK water.88(a) Neutrino interaction vertex resolu-tion in x direction.(b) Scatter point resolution in x direc-tion.(c) Neutrino interaction vertex resolu-tion in y direction.(d) Scatter point resolution in y direc-tion.(e) Neutrino interaction vertex resolu-tion in z direction.(f) Scatter point resolution in z direction.Figure 4.18: Vertex resolution for the scattered pion fit onNC1pi+ events in the T2K-SK MC with outgoing pions thatscatter exactly once. The left column shows the neutrino in-teraction vertex and the right column shows the scatter pointvertex.89(a) Momentum resolution for the up-stream track.(b) Angle between true and recon-structed directions for the upstream track.(c) Momentum resolution for the down-stream track.(d) Angle between true and recon-structed directions for the downstreamtrack.Figure 4.19: Kinematic resolution of the scattered pion fitter onNC1pi+ events in the T2K-SK MC with outgoing pions thatscatter exactly once.4.8 νµ CC1pi+ ReconstructionIt is possible to combine the scattered pion fit with a muon ring to create adedicated νµ CC1pi+ hypothesis, where the muon and the upstream pion areassumed to originate from the same vertex, and the downstream pion trackbegins at the endpoint of the upstream pion track. The topology that is usedis illustrated in Fig. 4.20. The charged pion portion of the fit is constructedin exactly the same way as the scattered charged pion fit described earlier,but a single muon track is added.90Upstream π+ µ Downstream π+ Scatter Point Neutrino Interaction Vertex Figure 4.20: Cartoon of the CC1pi+ hypothesis.Like the scattered pion fit, this fit is seeded with the two ring pion fitresult from the generic multi-ring fitter and is applied to an event after themulti-ring fit has already been run. In this case, however, the fit is seededassuming that one of the rings is the muon and the other is the upstreampion. A seed for the downstream pion is found using a grid search on theend of whichever ring is called the “upstream pion” for seeding purposes.The fit is done twice, assuming that the two rings from the generic multi-ring fit can correspond to either the muon or the upstream pion. In thiscase, the fit will output 19 parameters: the muon kinematics (momentumand direction), upstream pion kinematics (momentum, direction and energylost along the track), downstream pion kinematics (momentum, direction,and energy lost along the track), as well as the neutrino interaction vertex,the scattered pion vertex. The likelihoods for each fit configuration are alsoreturned.The fit resolution and seeding configuration were optimized on a CC1pi+ par-ticle gun sample in which a Monte Carlo muon and pion were generated tocome from a single vertex randomly placed in the SK tank, and to be emit-ted at random angles relative to each other. This sample is called a “particlegun” sample because the particles are placed in the tank at a position andwith some designated kinematics without assuming that they come from aneutrino interaction, as if it were possible to place a muon and pion source atrandomly distributed points in the detector. The muons and pions were uni-formly distributed in momentum between 0.1 and 1 GeV/c, and completely91uncorrelated. The momentum distribution of the events generated is shownin Fig 4.21a. These are quite different from the muon and pion momentumdistributions from events in the T2K-SK MC, in which the muon tends to behigher momentum than the pion for both unoscillated and oscillated eventsas illustrated in Fig. 4.21b and 4.21c. As illustrated in these plots the effectis larger for oscillated events.Each time that the CC1pi+ fit is run, there are two output likelihoods,one for each seeding combination of muon and pion rings. In most cases oneof the two seeding configurations is able to reconstruct the events, but unlikein the scattered pion fit, simply picking the result with the best likelihooddid not typically return the result with the best kinematic resolution.The two output fits from the CC1pi+ fit can be classified as one wherethe reconstructed momentum of the muon track is higher than that of thepion track, L(pµ > ppi), and one where the reconstructed momentum of themuon track is lower than that of the pion track, L(ppi > pµ). These twocases correspond almost exactly to which particle is seeded with the higherenergy ring from the generic two ring fit.The result with the best kinematic resolution is usually the case wherethe muon is seeded with the higher energy of the two rings from the generictwo ring fit. This is because even if the pion energy is initially higher thanthat of the muon, the muon will appear to be higher energy than the piondue to the fact that the charged pion may lose energy in hadronic interac-tions. The fraction of times that the muon is better matched to the higherenergy ring of the generic multi-ring fit with two pion rings is displayed inTable 4.1. Despite this, the likelihood prefers to assign the higher energyring to the fit configuration with more degrees of freedom. As describedin Section 4.5.1, the pion track has an additional degree of freedom dueto the energy loss parameter that defines the distance the pion travels be-fore a hadronic interaction. This additional fit parameter makes it easierto describe other complexities of high energy muon rings such as multiplescattering or decay in flight, and means that typically the muons are betterdescribed by the charged pion fit.The likelihood ratio of these two outputs from the CC1pi+ fit, L(pµ >92True π+ Momentum (MeV/c) True µ Momentum (MeV/c) (a) CC1pi+ particle gun.True π+ Momentum (MeV/c) True µ Momentum (MeV/c) (b) Unoscillated CC1pi+-like T2K-SKMC events.True π+ Momentum (MeV/c) True µ Momentum (MeV/c) (c) Oscillated CC1pi+-like T2K-SK MCevents.Figure 4.21: Pion and muon momentum distributions for samplesused to study the CC1pi+ fitter.93Table 4.1: Fraction that the µ ring is better matched to the higher orlower energy ring of the two ring pion fit.µ better matched to: Higher energy ring Lower Energy Ringof 2R pi fit (%) of 2R pi fit (%)CC1pi+ Particle Gun 73.3 26.7T2K-SK MC, no oscillation 81.8 18.2T2K-SK MC, with oscillation 88.5 11.5ppi) − L(ppi > pµ), was studied to look for a better place to put the muon-pion particle identification cut to improve the efficiency at which the correctconfiguration is selected. To do this, the CC1pi+ particle gun sample wasseparated into cases where the angle between the true muon direction bettermatches the higher energy ring and where the true muon direction bettermatches the lower energy ring. For these two samples the likelihood ratiobetween the two fit outputs were plotted in terms of a variety of kinematicvariables. The likelihood ratio as a function of the momentum of the higherenergy ring from the generic multi-ring fit with two pion rings is shown inFig. 4.22. These plots show significant overlap between these samples, aswell as that if the likelihood cut is placed at zero, the fit assuming ppi > pµwill be selected almost all of the time, despite the fact that this is trueonly about 25% of the time. Instead of this more basic cut at L(pµ >ppi)− L(ppi > pµ) = 0, a cut in the likelihood space was made to follow thepeak of the distribution that matches the muon to the higher energy ring,and to go through the center of the distribution that matches the muon tothe lower energy ring. This cut is placed at: L(pµ > ppi) − L(ppi > pµ) =2p2R high−500, where p2R high is the momentum of the higher energy ring ofthe generic multi-ring fit with two pion rings in MeV/c. It was optimized onthe CC1pi+ particle gun sample. This cut is shown for the CC1pi+ particlegun in Fig. 4.22, and for the oscillated and unoscillated T2K-SK MC inFig. 4.23. The fraction that this line cut selects the correct configurationfor the muon ring is shown in Table 4.2. This cut selects the correct fitconfiguration 88.6% of the time for the oscillated T2K-SK MC. This couldpotentially be improved in the future by applying machine learning methods.94True µ direction closer to higher energy ring Momentum of higher energy ring of 2R fit (MeV/c) L(higher E π/higher E µ) (a) True muon direction closer to higher energyring of 2R fit.True µ direction closer to lower energy ring Momentum of higher energy ring of 2R fit (MeV/c) L(higher E π/higher E µ) (b) True muon direction closer to lower energyring of 2R fit.Figure 4.22: CC1pi+ particle gun broken into samples in which themuon is better matched to the higher energy ring from thegeneric two ring fit (top) and the lower energy ring from thegeneric two ring fit (bottom). The muon/pion PID is the blackline.95True µ direction closer to higher energy ring Momentum of higher energy ring of 2R fit (MeV/c) L(higher E π/higher E µ) (a) True muon direction closer to higherenergy ring of 2R fit for the unoscillatedT2K-SK MC.True µ direction closer to lower energy ring Momentum of higher energy ring of 2R fit (MeV/c) L(higher E π/higher E µ) (b) True muon direction closer to lowerenergy ring of 2R fit for the unoscillatedT2K-SK MC.True µ direction closer to higher energy ring Momentum of higher energy ring of 2R fit (MeV/c) L(higher E π/higher E µ) (c) True muon direction closer to higherenergy ring of 2R fit for the oscillated T2K-SK MC.True µ direction closer to lower energy ring Momentum of higher energy ring of 2R fit (MeV/c) L(higher E π/higher E µ) (d) True muon direction closer to lowerenergy ring of 2R fit for the oscillated T2K-SK MC.Figure 4.23: T2K-SK MC broken into samples in which the muon isbetter matched to the higher energy ring from the generic tworing fit (left row) and the lower energy ring from the generictwo ring fit (right row). The top row shows the unoscillatedMC and the bottom row shows the oscillated MC. The muon/-pion PID is the black line.96Table 4.2: Fraction that the µ ring is correctly assigned in the best fitconfiguration when the likelihood cut is used to select the bestfit.% Correct % IncorrectCC1pi+ Particle Gun 73.3 26.7T2K-SK MC, no oscillation 81.8 18.2T2K-SK MC, with oscillation 88.6 11.44.8.1 PerformanceTwo example post-fit charge distributions are shown compared to event dis-plays for a CC1pi+ particle gun event in Fig. 4.24. In these events themuon ring is visible, along with an upstream and downstream ring from ascattered charged pion. The upstream pion ring is thin because the pionabruptly changes direction and stops emitting light in its initial directionwhen it scatters. The muon ring is more filled in because it travels in asingle direction and emits light until it drops below Cherenkov threshold.The CC1pi+ fitter performance is compared using the CC1pi+ particlegun sample as well as the oscillated and unoscillated T2K-SK MC. The ver-tex resolution for the neutrino interaction point and the position of the pionscatter is shown in Fig. 4.25. The muon momentum and direction resolutionare shown in Fig. 4.26. The upstream pion kinematics are shown in Fig. 4.27and the downstream pion kinematics are shown in Fig. 4.28. These plotsdemonstrate that the CC1pi+ fitter is able to successfully reconstruct themuon and pion kinematics needed to calculate the neutrino energy of anevent.97(a) Left-most ring is the muon. Twothin rings to the right are the upstream anddownstream pion rings.(b) Right-most ring is the muon. Twothin rings to the left are the upstream anddownstream pion rings.Figure 4.24: Example event display outputs from the CC1pi+ fit. Thetop row is the true PMT hit distribution, the middle row isthe seed charge distribution, and the bottom row is the chargedistribution after the CC1pi+ fit is run. The post-fit chargedistribution is visibly the same as the true distribution of hitPMTs in both of these events.98(a) Neutrino interaction vertex resolu-tion in x direction.(b) Scatter point resolution in x direc-tion.(c) Neutrino interaction vertex resolu-tion in y direction.(d) Scatter point resolution in y direc-tion.(e) Neutrino interaction vertex resolu-tion in z direction.(f) Scatter point resolution in z direction.Figure 4.25: Vertex resolution for the CC1pi+ fit on the CC1pi+ par-ticle gun, oscillated and unoscillated T2K-SK MC. The leftcolumn shows the neutrino interaction vertex and the rightcolumn shows the scatter point vertex.99(a) Muon momentum resolution.(b) Angle between true and reconstructed muon direction.Figure 4.26: Fit resolution of the muon kinematics compared forCC1pi+ particle gun sample, unoscillated CC1pi+ events fromthe T2K-SK MC, and oscillated CC1pi+ events from the T2K-SK MC.100(a) Upstream pion momentum resolution.(b) Angle between true and reconstructed upstream pion direction.Figure 4.27: Fit resolution of the upstream pion kinematics comparedfor CC1pi+ particle gun sample, unoscillated CC1pi+ eventsfrom the T2K-SK MC, and oscillated CC1pi+ events from theT2K-SK MC.101(a) Downstream pion momentum resolution.(b) Angle between true and reconstructed downstream pion direc-tion.Figure 4.28: Fit resolution of the downstream pion kinemat-ics compared for CC1pi+ particle gun sample, unoscil-lated CC1pi+ events from the T2K-SK MC, and oscillatedCC1pi+ events from the T2K-SK MC.102Chapter 5νµ CC1pi+ Event SelectionAs described in Section 2.4, CC1pi+ interactions have the second largest crosssection in the energy range of the T2K neutrino beam. Identifying theseevents can increase the total number of charged current neutrino interactionsobserved at SK from the T2K beam line, and can provide an additional signalto measure neutrino oscillations. This chapter will present the selection ofone ring and multi-ring νµ CC1pi+ events.As introduced in Chapter 1, CC1pi+ events are defined as the interactionwhere a neutrino interacts on a nucleon to produce a lepton, charged pionand nucleon. For νµ CC1pi+ : νµ + n/p → µ− + pi+ + n/p. Due to finalstate interactions, however, the pion may absorb or charge exchange in thenucleus, which means that not all CC1pi+ events as defined by neutrinointeraction mode will have a pi+ that exits the nucleus. This chapter willfocus on visible CC1pi+ events, which are defined as having exactly onepion and one muon exiting the nucleus after a neutrino interaction. Thiscorresponds to what we will be able to see at SK, independent of what mayhave occurred in the neutrino-nucleon interaction.CC1pi+ events primarily have either one, two, or three rings. One ringevents occur when only the muon is above Cherenkov threshold and visible inthe detector. Two ring events occur when the muon and pion are both aboveCherenkov threshold. Three ring events occur when the muon and pion areboth above Cherenkov threshold, and the pion scatters in the detector and103continues to produce light after the interaction.Decay electrons can also be used to help tag CC1pi+ events. Positivelycharged pions, such as those produced in CC1pi+ neutrino interactions, decay99.988% of the time as:pi+ → µ+ + νµ→ e+ + νe + ν¯µ(5.1)with a mean lifetime for the initial pion decay of 2.603 × 10−8 s [10]. Theother 1.234 × 10−4 % of the time they decay directly into a positron andelectron neutrino as pi+ → e+ + νe. Hadronic interactions in the water suchas charge exchange and absorption can also affect whether or not a chargedpion produces a decay electron.Muons produced in CCQE and CC1pi+ interactions decay as:µ− → e− + ν¯e + νµ (5.2)with a mean lifetime of 2.197µs [10].Negatively charged muons may also capture in the water, in which casethey will not produce a decay electron. Capture will also impact the effectivelifetime of the muon [10].This means that a νµ CC1pi+ event may have zero decay electrons ifneither the muon nor the pion produces a decay electron, one decay electronif one of the muon or pion produces a decay electron, or two decay electronsif both the muon and the pion produce a decay electron. The decay electrondetection efficiency at SK is about 88%, and this may also cause migrationsbetween these decay electron categories.The CC1pi+ selection presented here is built around the number of decayelectrons and the number of rings.5.1 Neutrino Energy CalculationFor a high purity sample of CC1pi+ events it is possible to calculate theneutrino energy. This gives more detailed information about the oscillations104neutrinos undergo.If both the muon and pion are above Cherenkov threshold and theirkinematics are reconstructed as discussed in Chapter 4, the neutrino energycan be calculated using four momentum conservation. Assuming that thetarget nucleon is at rest and that the unmeasured final state particle is anucleon, the reconstructed neutrino energy isEν =m2µ +m2pi+ − 2mN (Eµ + Epi+) + 2pµ · ppi+2(Eµ + Epi+ − |pµ| cos θνµ − |ppi+ | cos θνpi+ −mN ), (5.3)where mµ, mpi+ , mN are the masses of the muon, pion and nucleon,Eµ and Epi+ are the energy of the muon and pion, pµ and ppi+ are the 4-momentum of the muon and pion, |pµ| and |ppi+ | are the magnitude of the3-momentum of the pion and muon, θνµ is the angle between the neutrinoand muon directions, and θνpi+ is the angle between the neutrino and piondirections. This calculation neglects the Fermi motion of the nucleons withinthe nucleus.If the pion is below Cherenkov threshold and its kinematics can not bedirectly reconstructed, the neutrino energy can still be calculated by assum-ing that the pion produced comes from the decay of a ∆ in the nucleus asthis is the primary method of pion production. With this assumption it ispossible to calculate neutrino energy again using four-momentum conserva-tion. In this case the neutrino energy only depends on the muon kinematicsand the mass of the ∆ which is assumed to be the peak of the resonance at1232.0 MeV/c2:Eν =2mpEµ +m∆ −m2p −m2µ2(mp − Eµ + pµ cos θνµ) , (5.4)where Eµ, pµ and cos θνµ are the muon energy, momentum and directionrelative to the neutrino beam, and mp,m∆,mµ are the masses of the proton,∆ resonance and muon respectively.These formulas will be used to calculate neutrino energy in the selectedCC1pi+ samples discussed in the rest of this chapter. Either formula may beused for events with multiple rings, but the neutrino energy resolution for105the signal is found to be better using the muon and pion kinematics thanthe assumption of a ∆ resonance. In the following sections Eq. 5.4 is usedfor the one ring events and Eq. 5.3 is used for the multi-ring events. Thiswill be discussed in more detail in Section 5.6.5.2 Visible Event TopologiesAt each stage of the selection the events remaining are broken up into vis-ible event topologies based on the true particles exiting the nucleus afterthe neutrino interaction. These particles correspond to what we will be ableto actually see in the SK detector, independent of what may have actuallyoccurred in a neutrino interaction. For example, these categories may varyfrom the true neutrino interaction topologies because of final state interac-tions of pions or nucleons within the nucleus.A particle that exits the nucleus after a neutrino interaction is called aprimary particle. These primary particles may also re-interact in the waterin the SK tank, and these are called secondary interactions, with the par-ticles produced in these interactions called secondary particles. Secondaryinteractions may also change the visible event topology, and what is re-constructed in the detector. In particular these affect whether we are ableto reconstruct an event as the CCQE or CC1pi+ events used as neutrinooscillation signal in the T2K analysis.As mentioned earlier we are interested in CC1pi+ events with pions bothabove and below Cherenkov threshold. One ring CC1pi+ events with a pionbelow Cherenkov threshold can be tagged if both the muon and the pionproduce a decay electron. The number of decay electrons can, however, beaffected by secondary interactions in the water. Pions that absorb or chargeexchange will not produce decay electrons, and if they are below thresholdwill not be detected. On the other hand, if a pion is above Cherenkovthreshold and absorbs in the water the pion ring may still be detected beforethe absorption even if it does not produce a decay electron. Furthermore weare interested in reducing the background due to events with pi0 as much aspossible because the pion kinematics are not well understood and will have106a large systematic uncertainty.It is also important to note that particles just at the Cherenkov thresholdmay produce little or no light that is detectable in the SK PMTs, so for aparticle to be considered visible it is required to have a momentum that is30 MeV/c above the Cherenkov threshold.Using these considerations the visible event categories defined in theT2K-SK MC are as follows:• CC0pi: one muon, no pions, any number of nucleons and photonsfrom nuclear de-excitations (called nuclear gammas), no other primaryparticles. The proton is below Cherenkov threshold.• CC0pi, proton over threshold (ot): one muon, no pions, any number ofnucleons or nuclear gammas, no other primary particles. The protonis at least 30 MeV/c over Cherenkov threshold.• CC1pi+, pi+ over threshold (ot): one primary muon, one primary pi+with any number of nucleons and nuclear gammas and no other pri-mary particles. The pion is at least 30MeV/c over Cherenkov thresholdand is allowed to scatter any number of times, or absorb. It can nothave a secondary interaction charge exchange, double charge exchangeor hadron production.• CC1pi+, pi+ under threshold (ut): one primary muon, one primarypi+ with any number of nucleons and nuclear gammas and no otherprimary particles. The pion is below Cherenkov threshold and is al-lowed to scatter any number of times, or absorb. It can not havea secondary interaction charge exchange, double charge exchange orhadron production.• CC1pi+, charge exchange (CX): one primary muon, one primary pi+that undergoes charge exchange in the water.• CC1pi+, double charge exchange or hadron production (mpi): one pri-mary muon, one primary pi+ that either undergoes double charge ex-change or produces multiple pions in the water.107• CCpi0: one primary muon, at least one primary pi0.• CCmpi: one primary muon, multiple pi+ or pi−, or both a pi+ and pi−.No pi0s.• CCother: one muon, and anything not in the other categories.• NC0pi: no primary muon, no pions, any number of nucleons and nu-clear gammas, no other primary particles.• NC1pi+: no primary muon, one primary pi+ with any number of nu-cleons and nuclear gammas, and no other primary particles.• NCpi0: no primary muon, one primary pi0 with any number of nucleonsand nuclear gammas, and no other primary particles.• NCmpi: no primary muon, multiple pi+ or pi−, or both a pi+ and pi−.No pi0s.• NCother: no primary muon, and not in any of the other categoriesThese detailed categories may then be more coarsely grouped into thefollowing categories based on the type of neutrino interaction that may havecreated these visible topologies. These are defined as:• CCQE= CC0pi + (CC0pi, proton over threshold)• CC1pi+ = (CC1pi+,pi+ over threshold) + (CC1pi+, pi+ under threshold)+ (CC1pi+, charge exchange) + (CC1pi+, double charge exchange orhadron production)• CCother = CCpi0 + CCmpi + CCother• NC = NC0pi + NC1pi+ +NCpi0 + NCmpi +NCother.These coarser visible categories are used to illustrate the selection, whereCC1pi+ is signal and the other three categories (CCQE, CCother and NC)are backgrounds. The more detailed categories will be used to further discussfeatures of the selections shown in the following sections.108Fully%Contained%and%Ee>30%MeV%25Ring%All%rings%are%µ/π5like%35Ring%All%rings%are%µ/π5like%1%decay%electron%2%decay%electrons%1%decay%electron%2%decay%electrons%15Ring%Ring%is%µ5like%2%decay%electrons%Figure 5.1: Flow chart describing CC1pi+ event selection.5.3 Selection CutsThe CC1pi+ selection is built around the number of rings, the particle iden-tification of these rings, and the number of decay electrons in an event. Thisleads to several categories of CC1pi+ events, which are illustrated schemat-ically in the flow chart in Fig. 5.1. Each of these selection stages will bedescribed in more detail in the remainder of this chapter.The first cuts made are standard SK quality cuts that require eventsbe fully contained within the fiducial volume of the SK detector. These re-quire that there be fewer than 16 hits in the SK outer detector, and that theCC1pi+ event vertex be reconstructed more than 2 m from the SK tank wall.There is also a minimum energy event quality cut. The visible reconstructedenergy assuming that a ring is one ring electron-like is required to be greaterthan 30 MeV. The events remaining after each of these cuts are shown as afunction of true and reconstructed neutrino energy in Fig. 5.2. In these plotsthe neutrino energy is calculated using the CC1pi+ reconstruction formula inEq. 5.3. The plots are stacked histograms with each color area showing thenumber of events in each of the four visible categories defined in Section 5.2:CCQE, CC1pi+, CCother and NC. This comparison illustrates how the sig-nal and background events shift when the neutrino energy is calculated usingthe CC1pi+ formula and particle kinematics are reconstructed with fiTQunas opposed to the true neutrino energy from the simulation. These eventsare scaled to T2K Run 1-8 protons on target (POT), which is 14.7 × 1020109Table 5.1: Oscillation parameters used in this section.sin2 θ12 sin2 θ13 sin2 θ23 ∆m221 ∆m232 δCP Hierarchy0.32 0.0257 0.5 7.6×10−5eV2 2.4×10−3eV2 0 NormalPOT, and this data normalization is used throughout unless otherwise spec-ified. After these preliminary cuts the events are broken down by numberof rings using the generic fiTQun multi-ring fitter, and by number of decayelectrons using the number of fiTQun subevents to optimize CC1pi+ purityand efficiency.All the plots in this section are shown assuming oscillation parametersas displayed in Table 5.1. These are selected as the nominal values becausethey are close to the current world best fit parameters.The number of rings broken down by visible interaction topology for allevents that pass the fully contained fiducial volume (FCFV) and minimumenergy cuts are shown in Fig. 5.3. The plot has a limit of six rings as thisis the maximum number of rings that fiTQun reconstructs. Most of theCC1pi+ events are one, two, or three ring-like. The cut sequence, or thenumber of events remaining after each cut, for these preliminary cuts isshown in Fig. 5.4.These events will be discussed both over all neutrino energy regions, andbelow 1.33 GeV in true neutrino energy, called the “sub-GeV” region, whereneutrino oscillations are most important. This 1.33 GeV cut is the same asthe one used in the SK measurements of atmospheric neutrinos.5.3.1 One Ring νµ CC1pi+ EventsThe selection for one ring muon events has been optimized for the standardCCQE event selection used in the current analysis [45] [80]. The selectioncuts used are:1. The event is fully contained within the SK inner detector.2. The event is within the SK fiducial volume.3. The event is one ring-like.110(a) After FCFV cut. (b) After FCFV cut.(c) After minimum energy cut. (d) After minimum energy cut.(e) Fewer than four rings. (f) Fewer than four rings.Figure 5.2: Initial quality cuts performed for νµ CC1pi+ selection onthe T2K-SK MC as a function of true (left) and reconstructed(right) neutrino energy.111Figure 5.3: Number of rings for all events passing the FCFV and min-imum energy cuts.Figure 5.4: Cut sequence for preliminary event quality cuts.1124. The ring is muon-like, not electron-like as defined by Le/Lµ < 0.2pe,where Le and Lµ are the likelihoods of the one ring electron and muonfits and pe is the one ring electron momentum.5. The reconstructed muon momentum is greater than 200 MeV/c.6. The event has zero or one decay electrons.7. The event is muon-like, not pi+-like as defined by Lpi+/Lµ < 0.15pµwhere Lpi+ is the likelihood of the upstream charged pion, Lµ is thelikelihood of the one ring muon fit, and pµ is the one ring muon mo-mentum. This cut removes NC1pi+ background.This selection can be modified to select one ring νµ CC1pi+ events byrequiring events to have exactly two decay electrons. In these events thepion is below Cherenkov threshold, so is not directly visible, but the decayelectron produced will be visible, and acts as an effective way to tag pionswithout seeing the ring. Requiring two decay electrons means that both themuon and the pion produce a decay electron. A similar strategy has beenused to selecct one ring νe CC1pi+ events [81].The selection cuts used for this one ring νµ CC1pi+ sample are:1. The event is fully contained within the SK inner detector.2. The event is within the SK fiducial volume.3. The event is one ring-like.4. The ring is muon-like, not electron-like.5. The reconstructed muon momentum is greater than 200 MeV/c6. The event is muon-like, not pi+-like.7. The event has two decay electronsAll of the cuts are the same as in the CCQE sample described earlier exceptfor the decay electron cut. In Fig. 5.1, and throughout this section, cutsfour, five and six are collectively referred to as muon-like PID cuts.113Figure 5.5: Number of decay electrons for one ring muon-like events.The number of decay electrons for one ring muon events is shown inFig. 5.5. Most CC1pi+ events have either one or two decay electrons. Thetwo decay electron sample is dominantly CC1pi+. As the zero and one decayelectron events are dominated by CCQE interactions and already part of theT2K selection, the one ring CC1pi+ sample will be the one ring µ-like 2 decayelectron sample (1R+2de).The event breakdown after each of these cuts, starting with the 1-ringevents, is shown in Fig. 5.6.For these one ring events where the pion kinematics can not be directlyreconstructed, neutrino energy is calculated assuming that the pions pro-duced come from the decay of a ∆ in the nucleus using Eq. 5.4.The event breakdown in true and reconstructed neutrino energy afterthe one ring-like cut and the particle identification cuts which correspondto cuts four to six above (muon-like ring, not a pion-like ring, reconstructedmuon momentum greater than 200 MeV/c) are shown in Fig. 5.7. The finalone ring CC1pi+ sample is selected by requiring that there be two decayelectrons in an event, and the sample after this criteria is applied is shownin Fig. 5.8 as a function of true and reconstructed neutrino energy. The114Figure 5.6: Cut sequence for one ring muon-like events after the pre-liminary quality cuts.number of events remaining after each cut is shown in Table 5.2.This one ring CC1pi+ sample contains 28.90 events, with 85% purity.Table 5.3 shows the sample purity for events over the entire possible rangeof neutrino energy, and for events with true neutrino energy less than 1.33GeV in the region around the peak of the neutrino oscillation probability.The CC1pi+ purity increases to 91% in this lower energy region. Table 5.2also shows the efficiency of the one CC1pi+ sample events from the FCFVcut and the one ring cut. This sample is 17% efficient for CC1pi+ eventsfrom the FCFV and 57% efficient for CC1pi+ events from the one ring cut.5.3.2 Multi-Ring νµ CC1pi+ EventsThe multi-ring CC1pi+ selection was developed by breaking events that passthe standard SK quality cuts into categories based on the number of ringsand number of decay electrons.As described in Chapter 4, fiTQun has a generic multi-ring fitter thatreconstructs up to six rings, and will assign either an electron-like or a pion-like PID to each of the rings that it finds. In this framework, true muon rings115(a) One ring events. (b) One ring events.(c) One ring µ-like events. (d) One ring µ-like events.Figure 5.7: One ring events after ring counting (a,b) and PID cuts(c,d).Table 5.2: One ring µ-like events, after each selection cutCCQE CC1pi+ CCother NC TotalFCFV 290.40 142.69 213.50 643.15 1289.75evis>30 261.25 138.99 212.93 210.44 823.611R Events 231.84 43.34 9.04 43.68 327.901R µ Events 225.67 40.55 7.17 10.48 283.881R+2de 1.86 24.73 1.98 0.33 28.90Efficiency from FCFV(%) 0.64 17.33 0.93 0.05 2.24Efficiency from 1R(%) 0.80 57.05 21.93 0.75 8.81116(a) True neutrino energy.(b) Reconstructed neutrino energy.Figure 5.8: One ring µ-like events with two decay electrons.117Table 5.3: Percent of one ring µ-like events after each selection cutCut CCQE CC1pi+ CCother NC TotalFCFV 22.52 11.06 16.55 49.87 100.00evis>30 31.72 16.88 25.85 25.55 100.00All Events 1 R Events 70.71 13.22 2.76 13.32 100.001R µ Events 79.50 14.28 2.53 3.69 100.001R+2de 6.45 85.55 6.86 1.14 100.00FCFV 26.21 6.25 1.95 65.59 100.00True evis>30 53.21 13.68 4.48 28.63 100.00Sub-GeV 1 R Events 78.08 10.29 0.81 10.82 100.00Events 1R µ Events 85.47 10.40 0.66 3.47 100.001R+2de 6.51 91.16 1.50 0.82 100.00will typically be classified as pion-like because of the similar charge profileof muons and pions. This means that νµ CC1pi+ events can be selectedby looking for events with rings that are all classified as pion-like. Thisfitter is used to select multi-ring CC1pi+ events, and then the output of thededicated CC1pi+ fitter is used to calculate the neutrino energy for each ofthe selected events. The CC1pi+ fitter is also used to determine which ofthe rings is the muon and which is the pion.For these events, because both the muon and pion are above Cherenkovthreshold and their kinematics can be reconstructed, neutrino energy is cal-culated using Eq. 5.3.The sequence of cuts for multi-ring CC1pi+ events is:1. The event is fully contained within the SK inner detector.2. The event is within the SK fiducial volume.3. The event is two or three ring-like.4. All of the rings in the event are pi+-like5. The event has one or two decay electrons.118Figure 5.9: Ring PID of best multi-ring fit output for two ring events.Two ring eventsThe two ring events are selected using a combination of ring PID and numberof decay electrons, as described earlier.There are three possible combinations of ring PID for these events. Theycan both be pion-like (pipi), one can be pion-like and one can be electron-like (epi), or they can both be electron-like (ee). The two ring events arebroken down by best multi-ring fit configuration in Fig. 5.9. There are someCC1pi+ events in the epi and ee categories, but these also have a significantCCother and neutral current backgrounds.In Table 5.4 the signal composition is broken down into the definitionsfrom Section 5.2 for both all events and true sub-GeV events below 1.33GeV in true neutrino energy where neutrino oscillations are most impor-tant. The ideal signal topology for two ring events is CC1pi+ with the pionabove Cherenkov threshold. These are the events for which we expect to beable to reconstruct the muon and the pion rings simultaneously. For thisdefinition of signal the highest fraction of these true visible signal eventsare in the two ring pipi category, with 24.85 events, compared to the pie andee, which contain 6.24 and 0.37 of these events respectively. When a cut119Table 5.4: Number of two ring signal CC1pi+ events broken up acrossthe possible two ring fit configurations.CC1pi+ ot CC1pi+ ut CC1pi+, CX CC1pi+, mpiAll 2R 31.46 3.07 0.21 12.89All Events pipi 24.85 1.05 0.04 8.98epi 6.24 1.95 0.13 3.72ee 0.37 0.07 0.04 0.19True All 2R 11.31 0.50 0.09 3.26Sub-GeV pipi 9.79 0.20 0.02 2.61Events epi 1.31 0.26 0.03 0.60ee 0.21 0.03 0.04 0.06is made on true sub-GeV events the relative fraction of these ideal signaltopology events that fall into the epi and ee categories decreases. The num-ber of CC1pi+ events with hadronic interactions that produce multiple pionsalso decreases because lower energy neutrino events typically produce lowerenergy pions and these are less likely to undergo pion producing hadronicinteractions.The number of decay electrons for the two ring pion-like events are shownin Fig. 5.10. The one and two decay electron events are both dominantlyCC1pi+ events. The zero decay electron sample is dominated by neutralcurrent events and has relatively few true CC1pi+ events, so this samplewill not be included in the multi-ring CC1pi+ selection. Similarly the threedecay electron sample is dominated by CCother events.The number of events and composition in terms of neutrino interactionmode after each of these cuts, starting from the two ring events, is shown inFig. 5.11.True and reconstructed neutrino energy for two ring events with andwithout the requirement that both rings be pion-like are shown in Fig. 5.12.The final two ring CC1pi+ samples are the one decay electron sampleand the two decay electron sample. The two ring one decay electron sample(2R+1de) is shown as a function of true and reconstructed neutrino energyin Fig. 5.13. Similarly the two ring two decay electron (2R+2de) sample120Figure 5.10: Number of decay electrons for two ring pion-like events.is shown in Fig. 5.14. The number of events in each of these samples arein Table 5.5. The 2R+1de and 2R+2de samples are both predominantlyCC1pi+ events with a purity of 52% and 81% respectively, as shown inTable 5.6. The purity increases to 55% for the 2R+1de and 96% for the2R+2de events in the true sub-GeV neutrino energy region.The main background for the 2R+1de sample comes from CCQE events.These are predominantly (77%) events where the proton is over Cherenkovthreshold. The largest background for the 2R+2de sample, and the secondlargest for the one decay electron sample, is CCother. In both cases this ismostly from CCmpi events in which multiple pions exit the nucleus after theneutrino interaction.The efficiency for the two ring CC1pi+ samples is shown in Table 5.7relative to the number of events in the FCFV and the number of two ringevents. The two ring CC1pi+ samples are 22% efficient for CC1pi+ eventsfrom the FV and 66% efficient for CC1pi+ events from the two ring cut.121Table 5.5: Number of two ring events after each selection cut.CCQE CC1pi+ CCother NC TotalFCFV 290.40 142.69 213.50 643.15 1289.75evis>30 261.25 138.99 212.93 210.44 823.612R Events 21.71 47.63 32.18 102.52 204.042R pi Events 8.59 34.92 7.54 7.95 58.992R+1de 6.06 13.53 2.72 3.88 26.202R+2de 0.84 17.93 2.88 0.60 22.25Table 5.6: Sample purity after each selection cut.Cut CCQE CC1pi+ CCother NC TotalFCFV 22.52 11.06 16.55 49.87 100.00evis>30 31.72 16.88 25.85 25.55 100.002R Events 10.64 23.34 15.77 50.24 100.00All Events 2 R pi Events 14.56 59.20 12.77 13.48 100.002R+1de 23.13 51.65 10.40 14.82 100.002R+2de 3.76 80.59 12.95 2.70 100.00FCFV 26.21 6.25 1.95 65.59 100.00True evis>30 53.21 13.68 4.48 28.63 100.00Sub-GeV 2R Events 6.83 18.33 4.67 70.17 100.00Events 2R pi Events 16.03 69.36 2.50 12.11 100.002R+1de 28.51 54.73 2.46 14.01 100.002R+2de 2.01 95.76 2.08 0.14 100.00Table 5.7: Efficiency of the two ring CC1pi+ samples relative to theFCFV and the two ring cut.CCQE CC1pi+ CCother NC TotalEfficiency 2R+1de 2.09 9.48 1.28 0.60 2.03from 2R+2de 0.29 12.56 1.35 0.09 1.72FCFV(%) Total 2.38 22.04 2.63 0.69 3.75Efficiency 2R+1de 27.91 28.41 8.46 3.79 12.84from 2R+2de 3.86 37.64 8.95 0.59 10.902R (%) Total 31.77 66.05 17.41 4.38 23.74122Figure 5.11: Cut sequence for two ring events. The black line indi-cates the final samples123(a) Two ring events. (b) Two ring events.(c) Two ring pi+-like events. (d) Two ring pi+-like events.Figure 5.12: Two ring events after ring counting (a, b) and PID cuts(c, d).124(a) True neutrino energy.(b) Reconstructed neutrino energy.Figure 5.13: Two ring pi+-like events with one decay electron.125(a) True neutrino energy.(b) Reconstructed neutrino energy.Figure 5.14: Two ring pi+-like events with two decay electrons.126Figure 5.15: Ring PID of best multi-ring fit output for three ringevents.Three ring eventsThree ring events are selected similarly to the two ring events. Here, theideal interaction topology is that of a CC1pi+ event in which the muon andpion are both above Cherenkov threshold, and the pion scatters in the waterand continues to produce light after the hadronic interaction.Three ring events have four possible combinations of best ring PID. Ringscan be either all pion like (pipipi), two pion like rings and one electron-like ring(epipi), one pion-like ring and two electron-like rings (piee), or all electron-like (eee). The distribution of which events are best fit in each category isshown in Fig. 5.15.As with the two ring events, the three ring events which are all identifiedas pion-like contain the highest fraction of the target sample of events withone muon and one pion above Cherenkov threshold. There are still relativelysignificant fractions of these signal events in the other categories, but thesecome with much higher backgrounds of multi-pion events, and other chargedcurrent interactions, and so will not be included in the final selection.The events that are best fit with three rings and three pion-like rings are127Figure 5.16: Number of decay electrons for three ring events.shown in Fig. 5.18 as a function of true and reconstructed neutrino energy.The dominant background in these events are CCother interactions. Threering events are more common at higher energy because the cross sectionfor multi-particle producing CCother events is higher, and because higherenergy particles tend to produce more rings.The number of decay electrons for events with three pion-like rings isshown in Fig. 5.16. As with the two ring events, the zero decay electronevents have a large neutral current component, but the one and two decayelectron events both have relatively high CC1pi+ fractions. The events withthree decay electrons are primarily CCother.The cut sequence of the three ring events is shown in Fig. 5.17.True and reconstructed neutrino energy for three ring events with andwithout the requirement that both rings be pion-like are shown in Fig. 5.18.The final three ring CC1pi+ samples are the one decay electron sampleand the two decay electron sample. The three ring one decay electron sample(3R+1de) is shown in Fig. 5.19 as a function of true and reconstructedneutrino energy. The three ring two decay electron sample (3R+2de) issimilarly shown in Fig. 5.20.128Figure 5.17: Cut sequence for three ring CC1pi+ events. The blackarrow indicates the final samples.Table 5.8: Number of three ring events after each selection cutCCQE CC1pi+ CCother NC TotalFCFV 290.40 142.69 213.50 643.15 1289.75evis>30 261.25 138.99 212.93 210.44 823.613R Events 5.53 28.14 70.80 32.03 136.503R pi Events 1.53 10.53 5.55 2.70 20.303R+1de 0.93 3.66 1.56 1.39 7.543R+2de 0.29 5.50 2.13 0.34 8.26As shown in Table 5.8, there are 7.54 events in the 3R+1de sampleand 8.26 events in the 3R+2de sample. These have a CC1pi+ fraction of48% and 66% respectively, as shown in Table 5.10. The purity increasesto 59% for the 3R+1de sample and 95% for the 3R+2de sample when onlysub-GeV events are considered. Table 5.9 shows the efficiency for selectingevents of each topology relative to the total number of events in the fiducialvolume, and the number of events in the three ring sample. The three ringCC1pi+ samples are 6% efficient for selecting CC1pi+ events from the FCFVand 33% efficient for selecting CC1pi+ events from the three ring cut.129(a) Three ring events. (b) Three ring events.(c) Three ring pi+-like events. (d) Three ring pi+-like events.Figure 5.18: Three ring events after ring counting (a, b)and PID cuts(c, d).Table 5.9: Efficiency of selecting three ring events from the FCFV andthe three ring cut.CCQE CC1pi+ CCother NC TotalEfficiency 3R+1de 0.32 2.57 0.73 0.22 0.58from 3R+2de 0.10 3.86 1.00 0.05 0.64FCFV(%) Total 0.42 6.43 1.73 0.27 1.22Efficiency 3R+1de 16.76 13.02 2.20 4.33 5.52from 3R+2de 5.17 19.56 3.01 1.05 6.053R (%) Total 21.93 32.58 5.21 5.38 11.05130(a) True neutrino energy.(b) Reconstructed neutrino energy.Figure 5.19: Three ring one decay electron pi+-like events.131(a) True neutrino energy.(b) Reconstructed neutrino energy.Figure 5.20: Three ring two decay electron pi+-like events.132Table 5.10: Percent of sample, 3 ring events, after each selection cutCut CCQE CC1pi+ CCother NC TotalFCFV 22.52 11.06 16.55 49.87 100.00evis>30 31.72 16.88 25.85 25.55 100.00All Events 3R Events 4.05 20.61 51.87 23.47 100.003R pi Events 7.54 51.86 27.32 13.28 100.003R+1de 12.30 48.62 20.69 18.39 100.003R+2de 3.47 66.65 25.81 4.08 100.00FCFV 26.21 6.25 1.95 65.59 100.00evis>30 53.21 13.68 4.48 28.63 100.00True 3R Events 4.14 26.90 41.75 27.22 100.00Sub-GeV 3 R pi Events 8.85 76.30 3.64 11.21 100.00Events 3R+1de 17.29 59.64 3.32 19.75 100.003R+2de 2.15 95.46 2.39 0.00 100.001335.4 Summary of CC1pi+ SamplesThere are five CC1pi+ samples defined in the earlier sections: one ring withtwo decay electrons (1R+2de), two rings with one decay electron(2R+1de),two rings with two decay electrons(2R+2de), three rings with one decayelectron (3R+1de) and three rings with two decay electrons (3R+2de). TheMC expectation is 28.90 events in the 1R+2de sample, 26.20 events in the2R+1de sample, 22.25 events in the 2R+2de sample, 7.54 events in the3R+1de sample, and 8.26 events in the 3R+2de sample. These range in pu-rity from 48-85%. These final samples, along with the relative event fractionof each type of visible neutrino interaction, are summarized in Table 5.12.Two additional summary samples are also defined here, the multi-ring sum(MR sum), which is the sum of the two and three ring events, and the totalsample which is the sum of all five samples. The multi-ring sum MC ex-pectation is 64 events with 60% purity. The MC expectation for the totalCC1pi+ sample, including all the events from the five previously discussedsamples, is 93 events with 70% purity, which is similar to the purity of thecurrent T2K CCQE signal samples. The purity below 1.33 GeV increasesto 82% across all samples, and 90-95% for the two decay electron samples,which means that these samples are especially pure in the region where neu-trino oscillations are the most important. The multi-ring sample is 28.38%efficient for selecting CC1pi+ events from the FCFV, and the total sampleis 45.71% efficient for selecting CC1pi+ events from the FCFV.There are currently 240 CCQE-like νµ events in the beam data usedin the T2K analysis [45]. According to the MC prediction the addition ofthese five CC1pi+ samples will increase the number of νµ signal events byapproximately 40%.5.5 Background from ν¯µ, νe and ν¯e InteractionsThere are also additional backgrounds to these muon neutrino CC1pi+ sam-ples that come from beam neutrinos of other flavors interacting at SK. Thepossible backgrounds are ν¯µ, νe and ν¯e interactions. These backgrounds arecompared to the νµ CC1pi+ samples in Table 5.13. The ν¯µ come from the134Table 5.11: Number of events in each of the CC1pi+ samples.Event Category CCQE CC1pi+ CCother NC Total1R+2de 1.86 24.73 1.98 0.33 28.902R+1de 6.06 13.53 2.72 3.88 26.202R+2de 0.84 17.93 2.88 0.60 22.253R+1de 0.93 3.66 1.56 1.39 7.543R+2de 0.29 5.50 2.13 0.34 8.26MR sum 8.12 40.62 9.29 6.21 64.25Total 9.98 65.35 11.27 6.54 93.15MR sum efficiency from FCFV(%) 2.80 28.38 4.36 0.96 4.97Total efficiency from FCFV(%) 3.44 45.71 5.29 1.01 7.21Table 5.12: Number of events in each of the CC1pi+ samples below1.33 GeV in true neutrino energy.Event Category CCQE CC1pi+ CCother NC Total1R+2de 0.89 12.52 0.21 0.11 13.742R+1de 2.24 4.31 0.19 1.10 7.872R+2de 0.16 7.70 0.17 0.01 8.043R+1de 0.18 0.61 0.03 0.20 1.033R+2de 0.03 1.55 0.04 0.00 1.62MR sum 2.61 14.17 0.43 1.31 18.56Total 3.50 26.69 0.64 1.42 32.30wrong sign background of pi− in the beam. The beam νe and beam ν¯e comefrom the small components of the T2K beam that are not muon neutrinos.The signal νe come from muon neutrino oscillations into electron neutrinos.This background is 4.54 events in the multi-ring samples, and 5.25 eventsin the total sample, mostly coming from the ν¯µ events. This is about a 6%background on the sum of the four multi-ring samples, and 5% on the sumof all five CC1pi+ samples.5.6 Neutrino Energy Calculation ResolutionThere are two possible ways to calculate neutrino energy for CC1pi+ events.One assumes that the pion is created in the nucleus by the decay of a ∆, and135Table 5.13: Background from non-νµ beam neutrino interactions.νµ ν¯µ Beam νe Beam ν¯e Signal νe Signal ν¯e1R+2de 28.90 0.69 0.008 0.002 0.001 0.0122R+1de 26.19 2.78 0.136 0.018 0.014 0.2512R+2de 22.24 0.37 0.023 0.000 0.002 0.0253R+1de 7.53 0.58 0.047 0.006 0.005 0.0663R+2de 8.25 0.19 0.010 0.001 0.001 0.015MR sum 64.23 3.92 0.215 0.025 0.021 0.356Total 93.14 4.61 0.223 0.027 0.022 0.368therefore assumes that the recoiling hadronic system has the ∆ mass, alongwith the reconstructed muon kinematics to calculate neutrino energy as inEq 5.4. The second uses both the muon and the pion reconstructed kinemat-ics to calculate neutrino energy. In the case of the one ring CC1pi+ sampleonly the ∆ formula may be used because the pion is below Cherenkov thresh-old and its kinematics cannot be reconstructed. To decide between use ofthe ∆ reconstruction formula or the muon and pion kinematics formula forthe multi-ring events, comparisons of the energy resolution are made for allof the two and three ring selected samples, shown in Fig. 5.21 and 5.22respectively. The energy resolution is defined as (Ecalculated − Etrue)/Etrue.These results are quantified by calculating the standard deviation and meanfor both the total sample distribution and the signal CC1pi+ events individ-ually. From the results in Table 5.14, the mean appears to be better whenthe CC1pi+ formula is used for both the signal alone, and the signal plusbackground distributions. The standard deviation is about the same in bothcases. This means that it would be best to use the CC1pi+ formula withthe muon and pion kinematics to reconstruct neutrino energy because theneutrino energy measurement will be better.136Table 5.14: Mean and standard deviations of the neutrino energy res-olution when the ∆ formula is used compared to when theCC1pi+ formula is used to calculate neutrino energy.Mean Standard Deviation∆ CC1pi+ ∆ CC1pi+Sample Formula Formula Formula FormulaSignal CC1pi+2R+1de −0.093 0.00035 0.19 0.192R+2de −0.067 0.026 0.18 0.203R+1de −0.16 0.0082 0.22 0.223R+2de −0.14 0.0077 0.22 0.22Signal CC1pi+ 2R+1de 0.12 −0.062 0.34 0.332R+2de −0.11 −0.035 0.23 0.25+ Background 3R+1de −0.25 −0.13 0.32 0.343R+2de −0.23 −0.11 0.26 0.27137(a) 2R+1de sample with neutrino energycalculated using Delta formula.(b) 2R+1de sample with neutrino energycalculated using CC1pi+ formula.(c) 2R+2de sample with neutrino energycalculated using Delta formula.(d) 2R+2de sample with neutrino energycalculated using CC1pi+ formula.Figure 5.21: Energy resolution using ∆ formula and CC1pi+ formulafor selected two ring samples.138(a) 3R+1de sample with neutrino energycalculated using Delta formula.(b) 3R+1de sample with neutrino energycalculated using CC1pi+ formula.(c) 3R+2de sample with neutrino energycalculated using Delta formula.(d) 3R+2de sample with neutrino energycalculated using CC1pi+ formula.Figure 5.22: Energy resolution using ∆ formula and CC1pi+ formulafor selected three ring samples.1395.7 Oscillation VariationsAll of the initial sample studies done and presented until this point haveassumed the default oscillation parameters built into the T2K-SK MC asshown in Table 5.1. The CC1pi+ samples were also studied under a largervariety of oscillation parameters, as defined by a grid of muon neutrinooscillation parameters: sin2 θ23={0.4, 0.5, 0.6} and ∆m232 (eV2)={0.0020,0.0024, 0.0025, 0.0030}. The focus here is on sin2 θ23 and ∆m232 because thisis a sample of additional muon neutrino events. The oscillation probabilitywas calculated on an event by event basis for the T2K-SK MC. This is afunction of the neutrino energy once the oscillation parameters are specified.These weights are applied to scale to these additional oscillation possibilities.This will demonstrate that these selected CC1pi+ samples are sensitiveto the neutrino oscillation parameters.The variation in the number of events in the sum of the four multi-ringsamples as a function of reconstructed neutrino energy are illustrated inFig. 5.23. The same variations for the total CC1pi+ sample are shown inFig. 5.24. There is almost no dependence on the oscillation parameters above1.33 GeV in reconstructed neutrino energy because the neutrino oscillationprobability is much smaller at these higher energies for the T2K baseline.To help illustrate these variations the total number of events in each ofthese samples is plotted in Fig. 5.26. It is of note that there is a degeneracyin the oscillation probability at sin2 θ23 ≈ 0.511 which means that points oneither side of this value have nearly the same oscillation probability, whichresults in the same number of events. This comes from the fact that thedisappearance probability increases to a maximum at sin2 θ23 ≈ 0.511 andthen decreases again.The number of events varies about 20% from the nominal oscillationparameters in ∆m232, with more events at lower values and fewer events athigher values. This is relatively consistent for both the multi-ring samplesalone and the one ring and multi-ring samples combined.The CC1pi+ efficiency was also calculated as a function of these differentoscillation parameters and is shown in Fig. 5.28. Here the efficiency is defined140as the total number of true visible CC1pi+ events in the sample divided bythe total number of events in the sample. The efficiency is relatively flat insin2 θ23. Depending on the value of ∆m232 the purity of the multi-ring samplevaries from 59-67%, while the total sample varies from 66-74%. In the regionbelow 1.33 GeV in true neutrino energy where neutrino oscillations occur thepurity of the multi-ring sample is 66-80%, and for the total sample is 76-86%.The efficiency is better for higher values of ∆m232.141(a) sin2 θ23 = 0.4.(b) sin2 θ23 = 0.5.(c) sin2 θ23 = 0.6.Figure 5.23: Number of events in the four multi-ring samples as afunction of reconstructed neutrino energy. Each plot is at afixed value of sin2 θ23, and illustrates each of the ∆m232 varia-tions. 142(a) sin2 θ23 = 0.4.(b) sin2 θ23 = 0.5.(c) sin2 θ23 = 0.6.Figure 5.24: Number of events in all five νµ CC1pi+ samples as a func-tion of reconstructed neutrino energy. Each plot is at a fixedvalue of sin2 θ23, and illustrates each of the ∆m232 variations.143(a) Sum of the four multi-ring CC1pi+ samples.(b) Sum of the one ring and multi-ring CC1pi+ samples.Figure 5.25: Number of events in the multi-ring samples and all sam-ples illustrated in the grid of calculated oscillation parameters.144(a) Sum of four multi-ring CC1pi+ samples.(b) Sum of one ring and multi-ring CC1pi+ samples.Figure 5.26: Number of events in the multi-ring samples and all sam-ples illustrated in the grid of calculated oscillation parametersfor events with true neutrino energy below 1.33 GeV.145(a) Sum of four multi-ring CC1pi+ samples.(b) Sum of one ring and multi-ring CC1pi+ samples.Figure 5.27: CC1pi+ efficiency in the multi-ring samples and all sam-ples illustrated in the grid of calculated oscillation parameters.146(a) Sum of four multi-ring CC1pi+ samples.(b) Sum of one ring and multi-ring CC1pi+ samples.Figure 5.28: CC1pi+ efficiency in the multi-ring samples and all sam-ples illustrated in the grid of calculated oscillation parametersfor events with true neutrino energy below 1.33 GeV.147Chapter 6Charged Pion SystematicError EvaluationIt is necessary to evaluate the systematic errors associated with the sam-ples presented in Chapter 5. The main errors can be grouped into threecategories: those associated with the neutrino interaction on water, thoseassociated with the pion interactions in the target nucleus and detector, andthose associated with the detector response.6.1 Neutrino Interaction UncertaintiesThe error associated with the neutrino interaction addresses how sensitivethe selection is to the properties of the neutrino interaction. In particularwe are interested in how the number of particles produced in a neutrinointeraction and their kinematics affect what is visible in the SK detector.This comes in two parts: the properties of the particles directly producedby the neutrino interaction, and any re-interaction these particles undergobefore exiting the nucleus.The error associated with the properties of the particles directly pro-duced by the neutrino interaction can be evaluated before any re-interactionin the nucleus or the detector when there is just a muon and a pion. Dif-ferent neutrino interaction models predict different pion momentum spectra148from CC1pi+ interactions which may affect whether or not a pion is aboveCherenkov threshold, and will change the overall detection efficiency. Thiseffect can also result in migrations between the one ring and multi-ring sam-ples. Similarly, different neutrino interaction models also predict differentpion multiplicities will be produced by neutrino interactions. The momen-tum of these pions, and whether they produce decay electrons could changethe size of the backgrounds in the CC1pi+ samples.Pions produced in the nucleus by neutrino interactions may also undergoFSI in the nucleus. FSI may also affect the total number of visible positivelycharged pions produced, because if a pion absorbs or charge exchanges insidethe nucleus then there will be no positively charged pion to detect. Similarlya pion that was initially produced above Cherenkov threshold may scatterbelow threshold by the time it exits the nucleus. These effects will be dis-cussed along with pion re-interactions in the SK water in Section 6.2, as asimilar methodology is used to evaluate them both.6.1.1 Generator Comparisons of Visible CC1pi+ SignalThe output of different neutrino interaction generators are compared tounderstand how the properties of the particles produced may vary whendifferent pion production models are used.NEUT is the nominal neutrino interaction generator used to generatethe T2K-SK MC [67]. The pion production model in NEUT, as describedin Chapter 2, is compared to two different pion production models withinthe GENIE generator, referred to as the default and the alternate GENIEmodels, as well as to the NUWRO pion production model. The defaultGENIE model is an implementation of the Rhein-Sehgal model which usesonly 16 of the resonances and does not take into account the lepton massterm [68]. The alternate GENIE model is an implementation of the BergerSeghal model which is similar to the default GENIE model except thatit takes the lepton mass term into account [82]. The NUWRO model onlyincludes an implementation of the ∆(1232) resonance, and then extrapolatesthe DIS model down to cover pion production from higher resonances [83].149Events were generated with each of these neutrino interaction genera-tors using the T2K beam flux at SK. True visible CC1pi+ signal events areselected by looking for events with exactly one true muon and true pionafter the neutrino interaction, but before any final state or secondary re-interaction. Any number of nucleons are allowed, but no other types ofparticles. These signal events can be characterized by six kinematic param-eters: neutrino energy, the momentum of the muon and the pion, and theangles between the neutrino and muon, neutrino and pion, and muon andpion. The kinematics from each generator are compared in Fig. 6.1. Thesekinematics are of interest because the Cherenkov threshold at SK affectswhat events have muons and pions with visible rings. If the muon and pionrings are overlapping because the angle between the direction of the twoparticles is very small then it is also more difficult to well reconstruct theparticles and this introduces a systematic error.6.1.2 Reweighting to Different Signal ModelsUsing these comparisons between generators it is possible to study how muchthe selected CC1pi+ samples will vary if the underlying signal distributionwere those of the other generators instead of that of the NEUT generatedT2K-SK MC. Instead of regenerating the full MC with each of these gener-ators, which is very computationally intensive, it is possible to weight thesignal events in the selected samples to these underlying kinematic distribu-tions to see the effect on the size of the sample.One Dimensional ReweightingThe simplest way to do the reweighting is to take the ratio of the differentgenerator models to the NEUT default in each of the kinematic variablesshown in Fig. 6.1. These ratios are one dimensional weights as a functionof each of kinematic variable and are illustrated in Fig. 6.2. These weightscan then be applied to the selected signal events to see a general indicationof how the models differ. The result of applying these weights is shown inTable 6.1 for the sum of all multi-ring events. This table illustrates that150(a) Neutrino energy. (b) Muon momentum.(c) Pion momentum. (d) Angle between neutrino and muon.(e) Angle between neutrino and pion. (f) Angle between muon and pion.Figure 6.1: Comparisons between different generators in muon andpion kinematics for signal events with one muon and one pion.151Table 6.1: Number of events when the total multi-ring signal sampleis reweighted in one dimension for each kinematic variable sepa-rately.Eν pµ ppi θνµ θνpi θµpiNEUT 17.94 17.94 17.94 17.94 17.94 17.94NuWRO 18.50 19.83 16.23 17.96 18.10 17.64Genie Default 21.08 20.07 18.81 18.49 17.89 18.42Genie Alt 21.29 20.25 18.95 18.30 17.91 18.37the largest variations are about 23%, and happen in neutrino energy in theGENIE models.Multi-Dimensional ReweightingThe kinematic variables presented earlier are all very correlated, so it ispossible to make the size of the variations more realistic by looking at corre-lated weights. Weights that correlated the six kinematic variables illustratedabove were generated event by event for the signal events with exactly onetrue muon and pion in the T2K-SK MC using a gradient boosted decisiontree reweighting tool from hep_ml [84].These weights were applied to the selected samples, and the size of thevariations are shown for each of the six kinematic variables for the multi-ringsamples and the total CC1pi+ sample in Fig. 6.3 and 6.4 respectively.These variations across these three generators can be turned into a co-variance matrix binned in three bins of reconstructed neutrino energy. Thesebins are defined as: 0-1 GeV, 1-2 GeV and 2-30 GeV to more finely spanlower neutrino energies where there are the most events. This also morefinely spans the region in which the neutrino oscillation probability is thelargest, and the entire tail region at higher energies where there are fewevents and a small probability of oscillation can be treated together. Thecovariance matrix and fractional error for the sum of the multi-ring samples,and the total CC1pi+ sample are shown in Fig. 6.5. These variations look thesame with and without oscillation, as well as for each sample individually so152(a) Neutrino energy. (b) Muon momentum.(c) Pion momentum. (d) Angle between neutrino and muon.(e) Angle between neutrino and pion. (f) Angle between muon and pion.Figure 6.2: Ratio of different generators in muon and pion kinematicsfor signal events with one muon and one pion to the defaultNEUT. NEUT is black, Genie default is pink, Genie alternateis red and NUWRO is blue.153(a) Neutrino energy. (b) Muon momentum.(c) Pion momentum. (d) Angle between neutrino and muon.(e) Angle between neutrino and pion. (f) Angle between muon and pion.Figure 6.3: Sum of true signal events in the four multi-ring samplesvaried to match the kinematics of other neutrino interactiongenerators. NEUT is black, NUWRO is blue, Genie Default isred and Genie Alternate is pink.154(a) Neutrino energy. (b) Muon momentum.(c) Pion momentum. (d) Angle between neutrino and muon.(e) Angle between neutrino and pion. (f) Angle between muon and pion.Figure 6.4: Sum of true signal events in the total CC1pi+ sample var-ied to match the kinematics of other neutrino interaction gen-erators. NEUT is black, NUWRO is blue, Genie default is redand Genie alternate is pink.155(a) Covariance matrix for sum of fourmultiring samples.(b) Fractional error for sum of four mul-tiring samples.(c) Covariance matrix for all samples. (d) Covariance matrix for all samples.Figure 6.5: Error from neutrino interaction variations.it is possible to apply a flat 14% error to the signal due to the uncertainty ofthe kinematics of the neutrino interaction. The error from each of the Geniemodels individually is at the 15% level, and at the 13% level for NuWro.6.1.3 Multi-Pion VariationsOne of the larger backgrounds of the CC1pi+ samples described in Chapter 5comes from events with multiple pions, and similarly to the signal eventsthese may change depending on the model used in the simulation. Thereare known issues in the implementation of the deep inelastic cross sectionmodel that is used in the current T2K-SK MC production of NEUT, andthese impacted the calculation of the cross section. Now fixed, these model156changes have an effect on the total DIS cross section as well as the propertiesof the events, and may therefore impact the CC1pi+ samples [21]. In par-ticular, the number of DIS and multi-pion events will affect the size of thebackgrounds in the CC1pi+ samples. For example, an event with multiplepi+ exiting the nucleus may appear to have only one pi+ if only one of themis above Cherenkov threshold. The updated NEUT model is referred to asM0. Two additional models have also been added to NEUT as alternativeways to model events that produce multiple pions. One is based on deu-terium bubble chamber data [85], and one is the AGKY model which is usedin GENIE [86]. These will be referred to as M1 and M2 respectively.These different models may produce different total cross sections as afunction of neutrino energy, and the particle production from each of thesemodels is different. These effects will be discussed separately below.Reweighting in Total DIS Cross SectionThe effect of the different model cross sections on the CC1pi+ samples canbe studied by reweighting to the difference in the total cross section as afunction of neutrino energy. This is done by taking the ratio of these differentmodels to the default NEUT model. This is done separately depending ifthe target nucleon is a neutron or proton. An example of these weights isshown in Fig. 6.6 for neutrino interactions on neutrons [21].The effect of this reweighting on the multi-ring and total CC1pi+ samplesis illustrated in Fig. 6.7. As shown in Table 6.3, the largest variations ineach sample are 5-8%, and typically come from M1, the model from fittingto deuterium bubble chamber data. It is also of note that changing to theupdated NEUT model, M0, has very little effect on the overall sample.Reweighting in W and Leading Pion MomentumChanges to the multi-pion production models can also change the kinematicsof the pions produced by multi-pion production neutrino interactions. Theseeffects are studied by reweighting to the invariant hadronic mass of thesystem, W , and the true momentum of the most energetic charged pion157Figure 6.6: Weights applied to total multipion cross section. M0 isred, M1 is black and M2 is blue [21]. The plot is zoomed in sothat the differences between M0 and M2 are visible.(a) Sum of multiring samples. (b) Total CC1pi+ sample.Figure 6.7: Variations in CC1pi+ samples when the different multi-pion cross section models are applied.158Table 6.2: Variations in the number of events as a result of changingthe multi-pion DIS cross section to that of new models.NEUT M0 M1 M21R+2de 28.79 28.87 30.46 29.082R+1de 26.11 26.06 26.95 26.002R+2de 22.16 22.20 23.55 22.593R+1de 7.48 7.45 7.95 7.463R+2de 8.18 8.17 8.87 8.34Multi-ring sum 63.93 63.89 67.33 64.39Total 92.72 92.75 97.79 93.46after final state interactions. The invariant hadronic mass of the system iscalculated as:W 2 = (pν + pp/n − pl)2 (6.1)where pν , pp/n, pl are respectively the four momentum of the neutrino, thetarget nucleon and the outgoing lepton.The weights were calculated by looking at the model output for W andpion momentum in each of the models described earlier (M0, M1 and M2)for interactions on water and assuming the T2K beam flux at SK. They weregenerated separately for proton and neutron target nucleons. An exampleof the weights for muon neutrino interactions for M0 on neutrons in wateris shown in Fig. 6.8.The results of applying these weights to the multi-ring CC1pi+ sampleand the total CC1pi+ sample are shown in Fig. 6.9. As with the cross sec-tion variations the largest variations come from M1 deuterium fits, and theserange from 3-11%. The large effect on the one ring two decay electron sam-ple may come from the larger pion multiplicities predicted at low hadronicmasses in M1. In this case there is less momentum to give to each pion indi-vidually. As there are lower momentum pions that come from these highermultiplicity events the reweighting generally results in lower momentum pi-ons. This means that events are more likely to migrate into the one ringsample which does not require that the pions be visible. As with the cross159Figure 6.8: Weights applied to leading pion momentum and invarianthadronic mass for M0 [21].(a) Sum of multiring samples. (b) One ring and multiring samples.Figure 6.9: Variations in CC1pi+ samples when the different multi-pion models are applied.section variations, the differences between the current NEUT model and thefixed model are minor.160Table 6.3: Variations in the number of events as a result of changingthe multi-pion DIS kinematics to that of new models.NEUT M0 M1 M21R+2de 28.79 28.99 31.91 28.912R+1de 26.11 26.12 26.99 25.882R+2de 22.16 22.22 23.69 22.473R+1de 7.48 7.43 7.79 7.393R+2de 8.18 8.14 8.71 8.26Multi-ring sum 63.93 63.91 67.17 64.00Total 92.72 92.89 99.09 92.916.2 Pion Hadronic InteractionsAs described in Chapter 4, fiTQun relies on the hadronic interactions pi-ons undergo in the SK water to reconstruct pions. This is explicit in thepion hypothesis which assumes that a pion may produce a thin sharp ringfrom absorption or scattering below Cherenkov threshold, as well as in theCC1pi+ fitter which reconstructs the pion before and after a scatter. Asthere is no relatively pure sample of charged pions at SK, external mea-surements of pion hadronic interactions are used to evaluate the systematicerrors associated with charged pion interactions in the nucleus, FSI, and SIin the SK water. The external data give us an idea of the uncertainties in theNEUT cascade model parameters, and these are then propagated throughthe analysis.6.2.1 Fit to World DataA fit to the world pion scattering data was done to tune the NEUT cascademodel to match the data [22]. The result of this fit is a best fit centralvalue for the FSI parameters, and a correlation matrix of FSI uncertain-ties that can be used to calculate the systematic error associated with theCC1pi+ samples.As described in Chapter 2, the NEUT cascade model propagates theparticles produced in a neutrino interaction from their production point tothe point at which they exit the nucleus, assuming a Woods-Saxon nuclear161density. The interaction probability is calculated for the particle at eachstep it takes towards exiting the nucleus.The model is parameterized by the following FSI parameters which scalethe interaction probabilities for the pion as it exits the nucleus:• FEFABS: Absorption, for pions with momentum below 500 MeV/c.• FEFQE: Quasi-elastic scattering, for pions with momentum below 500MeV/c.• FEFCX: Single charge exchange, for pions with momentum below 500MeV/c.• FEFQEH: Quasi-elastic scattering, for pions with momentum above400 MeV/c.• FEFCXH: Single charge exchange, for pions with momentum above400 MeV/c.• FEFINEL: Hadron production in which nucleons and multiple pionsare produced, for pions with momentum above 400 MeV/c.The momentum overlap is done to transition between low and high momen-tum region models in NEUT.The fit was done to absorption, quasi-elastic scattering, charge exchange,absorption + charge exchange and reactive channel total cross section data.The reactive channel is a sum of all interactions except elastic scattering.It uses data of pion cross sections measured on carbon, oxygen, aluminum,iron, copper and lead to span light and heavy nuclei.The best fit parameters were found by minimizing a χ2 of the differencebetween the NEUT cross section predictions and the cross section measure-ments. The NEUT cross section was determined by performing simulationsof pions at set energies from the edge of a target nucleus and using the NEUTcascade model to propagate the pion through the nucleus. The cross sectioncould then be calculated as proportional to the ratio between the number162Figure 6.10: Comparison of FSI fit results to pi+-C external cross sec-tion measurements as a function of pion momentum. FSI fitbest fit is indicated by the black line, and the 1σ error bandis indicated in red. The dashed line and the blue error bandare the result of an earlier fit to the external pion cross sectiondata. Figure from [22].of events in a given interaction channel to the total number of incident pi-ons. These were tabulated to minimize computation time during the fit, andvalues were interpolated to compare to the cross section measurements.Fig. 6.10 shows the existing measurements of pi+-carbon cross sectionsfor each interaction type compared to the best fit and 1σ fit error band fromthe FSI fit.The resulting best fit values of the FSI fit parameters and covariancematrix for all of the data sets are displayed in Table 6.4 and Fig. 6.11respectively.6.3 Pion FSI and SI Cross Section VariationsIn order to calculate a systematic error from the effect of FSI the micro-scopic interaction probability for different hadronic interaction modes arevaried separately. At low energy, below 500 MeV/c in pion momentum,the interaction probabilities are varied in absorption, charge exchange and163Table 6.4: Best fit values of for the parameters in the FSI fit.Parameter Best fit ± 1σFEFQE 1.07 ± 0.31FEFABS 1.40 ± 0.43FEFCX 0.70 ± 0.30FEFINEL 1.00 ± 1.10FEFQEH 1.82 ± 0.86FEFCX 1.8 (Fixed)Figure 6.11: Covariance matrix between the parameters in the FSIfit. Figure from [22].elastic scattering. At high energy, above 500 MeV/c in pion momentum thecross section is parameterized in terms of elastic scattering, charge exchangeand other inelastic interactions including inelastic scattering and productionof multiple pions.6.3.1 Covariance Matrix ThrowsUsing the result of the FSI fit it is possible to use the Cholesky decomposi-tion method to take correlated throws of possible FSI parameter set valuesto calculate a FSI and SI error on the CC1pi+ samples. For secondary in-teractions each of these variations can be turned into a constraint on thetotal cross section by interpolating the cross section tables as a function ofFSI parameters that were used in the FSI fit while keeping the total pion164interaction cross section constant. Then just the pion interaction modelwithin the water is rerun assuming these new secondary interaction crosssection, σthrowi (pi, fFSI), which is a function of the pion momentum and theFSI parameter. This method makes it possible to calculate a weight as:wSI =∏iσthrowi (pi, fFSI)σnomi (pi)(6.2)The selected CC1pi+ samples are reweighted for each of the covariancematrix throws in both FSI and SI simultaneously.1000 throws were made from the FSI covariance matrix. Results of thesevariations are shown in Fig. 6.12.From these variations it is possible to calculate a covariance matrix foreach of the samples. The covariance is calculated as:Vij =1nn∑k=0(Nik −Ni nom)(Njk −Nj nom)Ni nomNj nom(6.3)where n is the number of parameter set variations. The covariance matrixbinning is defined by i and j with Nik events in the ith bin for the kthvariation. The nominal values are denoted with “nom”.The covariance and fractional error for the sum of the four multi-ringsamples, and all of the samples are shown in Fig. 6.13. The fractional erroris defined as√Vij The covariance is binned in true neutrino mode fromthe interaction generator. The backgrounds: CCQE, CCother and NC eachhave one bin. The signal CC1pi+ is broken into three reconstructed neutrinoenergy bins: 0-1 GeV, 1-2 GeV and 2-30 GeV. The largest errors are about12% on the neutral current background events. The error on the signal isabout 7% across all energy bins.The relationship between these five samples can also be calculated asthe fractional RMS between the number of events in these samples, and thecorrelation between the events in the samples. The fractional RMS is shownin Table 6.5, and is about the same for most of the samples. The correlationbetween these samples is shown in Fig. 6.14.165(a) Variations on the reconstructed neu-trino energy distribution for the sum of themulti-ring samples.(b) Visualization of error envelope as ra-tio to nominal parameter set for sum of themultiring samples.(c) Variations on the reconstructed neu-trino energy distribution for the sum of allsamples.(d) Visualization of error envelope as ra-tio to nominal parameter set for total eventsample.Figure 6.12: Visualization of 1000 covariance matrix throw variationsof FSI and SI parameters.Table 6.5: Fractional RMS of each of the samples from covariancematrix throw variations.Sample Number of Events Fractional RMS (%)1R+2de 29.31 3.732R+1de 27.51 6.072R+2de 22.81 8.983R+1de 8.14 9.023R+2de 8.65 11.27MR Sum 67.11 5.54Total 96.42 4.67166(a) Covariance matrix for sum of fourmultiring samples.(b) Fractional error for sum of four mul-tiring samples.(c) Covariance matrix for all samples. (d) Fractional error for all samples.Figure 6.13: Error from FSI and SI covariance matrix throws.(a) Covariance between the fiveCC1pi+ samples.(b) Correlation between the fiveCC1pi+ samples.Figure 6.14: Relation between the five CC1pi+ samples in terms ofthe FSI and SI variations.1676.4 Detector Response UncertaintiesDetector systematic errors also need to be evaluated to determine the differ-ence between the MC detector simulation and the actual detector response.There are existing methods used to evaluate the systematic errors associatedwith electrons and muons at T2K using control samples. Cosmic muons areused to constrain the kinematic reconstruction of muons and electron/muonparticle identification. Decay electrons are used to constrain the kinematicreconstruction of electrons and electron/muon particle identification. Atmo-spheric neutrinos, which produce electrons and muons in the same energyrange as those produced by the T2K beam neutrinos are also used. Thereis no relatively pure pion only control sample of SK data, however, unlikefor electrons and muons.6.4.1 Hybrid CC1pi+ SampleIt is possible to address the detector uncertainties for charged pions throughuse of a hybrid sample. This is a technique that has been used to evaluateerrors associated with pi0 in previous T2K analyses. In the hybrid pi0 sampleone atmospheric data electron-like ring is combined with a MC photon toconstrain the efficiency of the pi0 background rejection. A similar techniqueof combining an atmospheric neutrino data muon with a MC pion can beused to evaluate detector uncertainties for CC1pi+ events, and in particu-lar to address the question of how charged pions are reconstructed in thepresence of a muon.DevelopmentThere are two parts to the hybrid sample, the MC and the data sample.The MC sample contains a MC muon matched with a MC pion. The datasample contains an atmospheric data muon matched with a MC pion.The data sample is constructed by first selecting single ring muons fromCCQE atmospheric neutrino interactions. Then the true muon kinematicsand vertex position are extracted from each T2K CC1pi+ event in the MC.The muon momentum of the atmospheric neutrino data is matched to the168Table 6.6: Systematic error summary for sum of multi-ring samplesand total CC1pi+ samples as a fraction of the nominal number ofevents.MR Sum (%) Total (%)Neutrino CC1pi+ Signal 14 14Interaction DIS in Eν 5 5Error DIS in ppi −W 5 6CCQE 2 2CC1pi+ Eν [0-1 GeV] 4 3FSI/SI CC1pi+ Eν [1-2 GeV] 5 4Error CC1pi+ Eν [2-30 GeV] 7 6CCother 8 6NC 12 12true muon momentum of the T2K CC1pi+ MC events. The data muon isrotated to the direction of the MC muon so that the muon kinematics matchexactly. Next a charged pion ring is generated based on the true T2K-SKMC pion information with dark noise turned off. Finally the muon and thepion rings are merged.The MC sample is generated in the same way except that the single ringmuon is selected from true CCQE events in the atmospheric MC instead offrom the atmospheric data.The event reconstruction can then be run on both of these samples, anddifferences in the detector response between data and MC can be determinedby comparing the data and MC hybrid CC1pi+ samples. The sample con-struction is largely complete, but a systematic has not yet been evaluated.6.5 Systematic Error SummaryThe systematic errors associated with the neutrino interaction on water aswell as those associated with pion interactions in the target nucleus anddetector have been evaluated. These are summarized for the sum of themulti-ring samples and the total CC1pi+ sample in Table 6.6.The neutrino interaction error associated with the properties of neutrino169pion production were found to be 14% across all five CC1pi+ samples and inthe sum of the multi-ring samples and the total CC1pi+ sample. This erroris also constant in neutrino energy.The systematic associated with the uncertainty in the amount of multi-pion and DIS background in the selected CC1pi+ samples was found to rangefrom 5-8% across each of the five CC1pi+ samples and to be 5% in both thesum of the multi-ring samples and the total CC1pi+ sample when evaluatedby varying the total multi-pion and DIS cross section. This error was foundto range from 3-11% across the five CC1pi+ samples and to be 5% in thesum of the multi-ring samples and 6% in the total CC1pi+ sample whenevaluated in W and leading pion momentum.The error associated with pion FSI and SI was found to be 2-12% acrossall five CC1pi+ samples and in the sum of the multi-ring samples and thetotal CC1pi+ sample depending on the true neutrino interaction mode.These neutrino interaction and pion interaction systematics will be fur-ther constrained by a fit to the T2K near detector data.170Chapter 7ConclusionsThis thesis presents a method for reconstructing and selecting charged pionsat SK. It applies these methods to the CC1pi+ interaction to look for eventsin the T2K neutrino beam. This is the first multi-ring sample of neutrinoevents that will be used at T2K, and the first time that pion kinematics havebeen explicitly reconstructed and used in a SK analysis. The MC predictsthat there will be 93 additional T2K νµ events with 70% purity, and thesewill increase the T2K νµ statistics by approximately 40%. These events aresensitive to oscillations as illustrated in Chapter 5, which means that theywill provide an additional constraint on the neutrino oscillation parameterswhen included in a T2K analysis. This thesis also examines the systematicerrors necessary to include this new sample in a T2K oscillation analysis.The systematic error uncertainty on the total number of predicted events isexpected to be about 14 % from the neutrino-nucleus interaction and 2-12%depending on the true neutrino interaction mode and reconstructed energyfor FSI and SI. These will be constrained by a fit to the T2K near detectordata which may reduce these uncertainties.There are many possible extensions to this work. The first is to completea T2K oscillation analysis with this sample that uses the full reconstructedneutrino energy spectrum information as opposed to simply looking at thevariation of the number of events as a function of the oscillation parametersas presented in Chapter 5. These samples will include information about171pions above and below Cherenkov threshold in the T2K neutrino oscillationanalysis. Events may migrate between the above and below threshold sam-ples depending on the true momentum spectrum of charged pions producedby neutrino interactions, and on the effect of FSI and SI interactions on thepion momentum spectrum. With events that are sensitive to both above andbelow threshold samples these migration issues will be mitigated. This isnot yet the case at T2K, such as in the current T2K result which includes aone ring νe CC1pi+ sample where the pion is below Cherenkov threshold andthe number of measured events is much higher than the expectation [45].This work can also be easily extended to fit for an electron and a chargedpion simultaneously which will make it possible to have explicit above andbelow Cherenkov samples of νe events.Nucleon decay is one of the only measurements that can currently con-strain grand unified theories, as most of these theories include proton decay.The CC1pi+ fitter described in Chapter 4 can also be used at SK to searchfor new nucleon decay modes, such as:n→ e/µ+ pi+ (7.1)p→ νe/νµ + pi+. (7.2)This fitter may make it possible to improve the SK sensitivity to these modeswith an improved selection.The scattered charged pion fitter could also be used in the future tomake a NC1pi+ measurement at SK.These analysis techniques can also be applied to the future proposedwater Cherenkov experiment Hyper-Kamiokande (HK) [49]. Identificationof CC1pi+ events will be even more important for the HK experiment if adetector is placed at a longer baseline in Korea to measure the first oscillationmaximum. A smaller off axis angle, about 1.5 degrees, is chosen in thiscase so that the neutrino flux covers both the first and second oscillationmaxima. 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νμ CC1π+ Events Produced in the T2K Beam at Super-Kamiokande
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