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Constraining the flux and cross section models using carbon and oxygen targets in the off-axis near detector… Nielsen, Christine 2017

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Constraining the Flux and Cross Section Models usingCarbon and Oxygen Targets in the Off-Axis Near Detectorfor the 2016 Joint Oscillation Analysis at T2KbyChristine NielsenB.Sc. Physics, University of California Santa Barbara, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)The University of British Columbia(Vancouver)March 2017c© Christine Nielsen, 2017AbstractThe T2K experiment is a long baseline neutrino experiment designed to measure various neu-trino oscillation parameters, particularly θ13 and θ23, at world-leading precisions. T2K usesmuon neutrino and antineutrino beams produced at the Japan Proton Accelerator ResearchComplex. The off-axis beam strikes near detectors 280 m from the source (ND280) and thewater Cherenkov detector, Super-Kamiokande, 295 km away. The far detector performs theprimary oscillation analysis, while ND280 provides cross-section and flux constraints to re-duce uncertainties in the oscillation analyses. ND280 consists of several different detectors,with the main target mass being the Fine Grained Detectors, one of which provides an activecarbon target and one which provides a combined water-carbon target. This thesis describesthe fitting methods used for T2K’s oscillation analyses and near detector constraints, with afocus on the near detector fit. The addition of water target data in the near detector fit improvescross-section uncertainties at both near and far detectors. The near detector fit is a maximumlikelihood fit to observed neutrino interaction rates in ND280 to constrain the cross-section andflux at the far detector. Event cuts identify and separate charged-current neutrino interactionsinto topology-based samples, with selection uncertainties and correlations handled through ex-tensive detector systematic studies. The near detector fit to data including a water target showsconsistent fitted cross-section and flux with previous ND280 fits, and significantly reduces theuncertainty on the neutrino energy spectrum predictions at the far detector. In addition, nuclearmodel dependencies in the cross-section model used at T2K were investigated using the neardetector fit, and the T2K model was shown to be sufficient in accounting for nuclear effects.iiPrefaceThis dissertation is based on the experimental apparati and data of the T2K experiment, and wascompleted with the aid of other members of the T2K collaboration. The text of this dissertationhas not been taken from previously published articles or other collaborative documents.The detectors and selections described in Chapter 2 were designed and built by members ofthe T2K collaboration. The fit software used for the analysis at Super-Kamiokande in Chapter 3was created and used by members of the T2K VALOR and p–Theta groups for both the previousresults and the portion of the results in this thesis at SK. The first implementation of the neardetector fit software was created by M. Hartz and the current software implementation usingonly FGD 1 was created by J. Myslik with some help from myself and optimization by M.Scott. The addition of the ability to run the fit using both FGD 1 and FGD 2, as well asallowing correlations between detector systematic parameters in toy parameter throws, was mywork. The creation of the observable normalization parameter covariance matrix was doneby myself, building on previous work by M. Scott, with the addition of detector systematiccorrelations being mine.The reconstruction methods described in Chapter 4.4.1 were developed by the members ofthe various reconstruction groups on the T2K project. The inclusive charged event selectionfor the Forward Horn Current interactions was primarily developed by members of the ND280Numu Analysis group. I personally did the initial selection migration, tuning and testing forthe CC inclusive FGD 2 selection, adapting cuts originally developed for FGD 1. The FHC CCinclusive multi-pion samples and cuts were defined and developed by members of the ND280iiiNuMu analysis group. I personally developed and studied the current Michel electron cut usedin defining the sample selections. The RHC CC inclusive samples and cuts for neutrino andantineutrino interactions were developed by the ND280 Numu Analysis group.The systematics effects described in Chapter 5 were identified, studied and measured bymembers of the T2K ND280 Numu Analysis group. In addition, the correlations described inthat Chapter were defined by M. Ravonel and implemented by myself. I personally measuredthe uncertainty associated with the purity of the Michel electron cut for the FHC multipionselection. The work and studies shown in Chapter 6 were done entirely by myself.The analysis of the near detector fit using FGD 1 + FGD 2 shown in Chapters 7 and 8is my original work. The prediction and prediction uncertainties at SK using the ND280 fitresult were produced by the T2K VALOR group. The plots of the oscillation contours shownin this Chapter were also created by the T2K VALOR group. The model dependency studiesin Chapter 8 was my original work, with the SK spectra and total event numbers produced byRaj Shah from the results.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix1 Physics of Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Standard Model Neutrino Theory . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Basic Neutrino Formalism . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.1 Neutrino Oscillation Theory . . . . . . . . . . . . . . . . . . . . . . . 91.2.2 Oscillation Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.3 Matter Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Previous Oscillation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Neutrino Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.1 Neutrino Interaction Modes . . . . . . . . . . . . . . . . . . . . . . . . 161.4.2 Nucleon-neutrino Interactions . . . . . . . . . . . . . . . . . . . . . . 17v1.4.3 CC-0pi Neutrino Interactions on Nuclei . . . . . . . . . . . . . . . . . 211.4.4 Current state of Neutrino Interaction Modeling . . . . . . . . . . . . . 272 The T2K Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1 The History of T2K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 The Neutrino Beam at T2K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.1 The Off-Axis Neutrino Beam . . . . . . . . . . . . . . . . . . . . . . . 322.3 Super-Kamiokande . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.1 The Super-Kamiokande Detector . . . . . . . . . . . . . . . . . . . . . 332.3.2 Event Reconstruction and Selection . . . . . . . . . . . . . . . . . . . 352.4 The T2K Near Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.1 The Interactive Neutrino GRID Detector . . . . . . . . . . . . . . . . . 372.4.2 The ND280 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5 The Fine-Grained Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.5.2 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.5.3 Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.5.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.6 Limitations of the T2K Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 503 Oscillation Analysis at T2K and the ND280 Fit . . . . . . . . . . . . . . . . . . . 523.1 The T2K Oscillation Analysis Structure . . . . . . . . . . . . . . . . . . . . . 523.1.1 General Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1.3 Oscillation Fit At Super-Kamiokande . . . . . . . . . . . . . . . . . . 553.2 The ND280 Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.1 The Maximum Likelihood Fit Method . . . . . . . . . . . . . . . . . . 573.2.2 Changes from Previous Fit . . . . . . . . . . . . . . . . . . . . . . . . 60vi3.3 Fit Parameters and Prior Constraints . . . . . . . . . . . . . . . . . . . . . . . 613.3.1 Flux Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.2 Cross Section Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 673.3.3 The Observable Normalization Matrix . . . . . . . . . . . . . . . . . . 724 The Tracker Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.2 Data and Monte Carlo Inputs to the Selections . . . . . . . . . . . . . . . . . . 774.3 ND280 Detector Tracker Reconstruction . . . . . . . . . . . . . . . . . . . . . 774.3.1 Time Projection Chamber Track Reconstruction . . . . . . . . . . . . . 784.3.2 Fine Grained Detector Reconstruction . . . . . . . . . . . . . . . . . . 804.3.3 Global Reconstruction and Charge Determination . . . . . . . . . . . . 834.4 The ND280 Detector Tracker Selection . . . . . . . . . . . . . . . . . . . . . . 844.4.1 Selection Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.4.2 Fine Grained Detector 2 Considerations . . . . . . . . . . . . . . . . . 864.4.3 Antineutrino Subsamples . . . . . . . . . . . . . . . . . . . . . . . . . 884.5 Selection Cuts and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.5.1 General Data Quality Cuts . . . . . . . . . . . . . . . . . . . . . . . . 894.5.2 The Neutrino Mode Charged Current Inclusive Selection . . . . . . . . 904.5.3 The Neutrino Mode Multipion Selection . . . . . . . . . . . . . . . . . 964.5.4 Reverse Horn Current Antineutrino Charged Current Inclusive Selection 1094.5.5 Reverse Horn Current Antineutrino Charged Current Multitrack Selection1104.5.6 Wrong Sign in Reverse Horn Current Charged Current Inclusive Selection1174.5.7 Wrong Sign in Reverse Horn Current Charged Current Multitrack Se-lection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185 Systematics and Systematic Correlations . . . . . . . . . . . . . . . . . . . . . . 1255.1 Overview of ND280 Detector Systematics . . . . . . . . . . . . . . . . . . . . 125vii5.1.1 List of Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.1.2 Variation Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.1.3 Weight Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.2 Systematic Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.2.1 Correlations Between the Fine Grained Detector 1 and Fine GrainedDetector 2 Selections . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.3 Systematic Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.3.1 B Field Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.3.2 Time Projection Chamber Momentum Scale . . . . . . . . . . . . . . . 1325.3.3 Time Projection Chamber Momentum Resolution . . . . . . . . . . . . 1325.3.4 Time Projection Chamber Particle Identification . . . . . . . . . . . . . 1335.3.5 Time Projection Chamber Cluster Efficiency . . . . . . . . . . . . . . . 1345.3.6 Time Projection Chamber Track Reconstruction Efficiency . . . . . . . 1355.3.7 Charge Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.3.8 Time Projection Chamber – Fine Grained Detector Matching Efficiency 1365.3.9 Fine Grained Detector Particle Identification . . . . . . . . . . . . . . . 1375.3.10 Fine Grained Detector Time of Flight . . . . . . . . . . . . . . . . . . 1385.3.11 Fine Grained Detector Hybrid Track Efficiency . . . . . . . . . . . . . 1385.3.12 Michel Electron Efficiency and Purity . . . . . . . . . . . . . . . . . . 1395.3.13 Sand Muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.3.14 Event Pile Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.3.15 Fine Grained Detector Mass . . . . . . . . . . . . . . . . . . . . . . . 1445.3.16 Pion Secondary Interactions . . . . . . . . . . . . . . . . . . . . . . . 1455.3.17 Out of Fiducial Volume Background . . . . . . . . . . . . . . . . . . . 1465.4 The Observable Normalization Covariance Matrix . . . . . . . . . . . . . . . . 1475.4.1 Systematic Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1495.4.2 Detector Correlations in the ND280 Fit . . . . . . . . . . . . . . . . . 150viii6 Fitter Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.1 Nominal Monte Carlo Validation Fit . . . . . . . . . . . . . . . . . . . . . . . 1536.2 Parameter Pulls and p-value Calculation . . . . . . . . . . . . . . . . . . . . . 1557 Final Fit Results at the ND280 Detector . . . . . . . . . . . . . . . . . . . . . . 1627.1 Finalized Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.2 Results at the ND280 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.2.1 The p-value Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 1687.3 Impact at Super-Kamiokande . . . . . . . . . . . . . . . . . . . . . . . . . . . 1697.3.1 Oscillation Fit Results at Super-Kamiokande . . . . . . . . . . . . . . 1788 Further Parameterization Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 1858.1 Fitting MQEA for Oxygen and Carbon Targets . . . . . . . . . . . . . . . . . . . 1878.1.1 Additional Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1878.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1888.2 Adding Carbon-Oxygen Separation for C5A(0) and MRESA . . . . . . . . . . . . . 1918.2.1 Additional Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1928.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1928.3 Effect on the Predictions at Super-Kamiokande . . . . . . . . . . . . . . . . . 1959 Conclusions and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204A Additional Flux Prediction Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . 212B Momentum and Angle Distributions for Selected ND280 Events . . . . . . . . . 219C Tables of Fitted Parameter Values at ND280 . . . . . . . . . . . . . . . . . . . . 224ixList of TablesTable 1.1 Summary of current measured values for the various oscillation parameters.Values taken from [14] and are shown for the normal mass hierarchy. ∆m232is shown as an absolute value as the sign is not currently known. . . . . . . . 16Table 3.1 Fractional error on the prediction for the number of νµ events at SK, withand without the ND280 constraint from the previous neutrino mode analysisin 2014. Uncertainties shown are calculated from the RMS of the event rateswhen 10000 parameter variations are thrown[58]. ND280-unconstrainedparameters are the 2p-2h, pF , EB, CC coherent, Isospin=12 Background andνe/νµ cross-section parameters. ND280-constrained parameters are the fluxparameters and all other cross-section parameters. . . . . . . . . . . . . . . 55Table 3.2 Fractional error on the prediction for number of ν¯µ events at SK, with andwithout the ND280 constraint from the previous antineutrino mode analysisin 2015. Uncertainties shown are calculated from the RMS of the event rateswhen 10000 parameter variations are thrown [59]. ND280-unconstrainedparameters are the 2p-2h, pF , EB, CC coherent, Isospin=12 Background andνe/νµ cross-section parameters. ND280-constrained parameters are the fluxparameters and all other cross-section parameters. . . . . . . . . . . . . . . 55Table 3.3 Priors used for oscillation parameters when marginalizing. From [61]. . . . . 57xTable 3.4 Cross section parameters used for the near detector fit, showing the validrange of the parameter, prior mean, nominal value in NEUT, and prior erroras provided by the Neutrino Interaction Working Group (NIWG).[67] Thetype of systematic (response or normalization) is also shown. The CC-0piparameters (MQEA through EB16O) use the NEUT nominal value for thenominal MC tuning and are fit without prior constraints, with the exceptionof the binding energy EB.[67] Not all listed parameters are used in the fit atSK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Table 4.1 Total protons on target (PoT) for data and Monte Carlo, broken down byrun period and P0D water status. Monte Carlo PoT includes POT from sandmuon Monte Carlo. Data PoT is after the basic data quality cuts describedin Sec. 4.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Table 4.2 List of CC Inclusive Cuts for the three main selections. Major differencesbetween cuts are listed here. . . . . . . . . . . . . . . . . . . . . . . . . . . 86Table 4.3 List of the FHC multipion selection cuts and sample criteria. . . . . . . . . . 87Table 4.4 List of the RHC multipion selection cuts and sample criteria. The multipionselection cuts are the same for the ν¯µ and νµ , as there is no cut on the chargeof the secondary tracks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Table 4.5 Observed and predicted event rates for the different ND280 samples in theND280 fits. Predicted event rates include PoT, flux, detector and cross sec-tion reweighting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Table 4.6 Event reduction fractions for each cut step for both FGD 1 and FGD 2[81] . 96Table 4.7 Fractional breakdown of the CC-Inclusive sample by true topology for bothFGD 1 and FGD 2[81] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Table 4.8 Fraction of correctly identified Michel electrons with hits in delayed timebins.102xiTable 4.9 Efficiencies and purities for the FHC CC-0pi , CC-1pi and CC-Other samplesfor both FGDs[81].Efficiencies shown are with respect to the total numberof generated true events with that topology. . . . . . . . . . . . . . . . . . . 104Table 4.10 Fractional breakdown of the RHC ν¯µ CC-Inclusive sample by true topologyfor both FGD 1 and FGD 2[83]. . . . . . . . . . . . . . . . . . . . . . . . . 110Table 4.11 Efficiencies for the RHC samples for both FGD 1 and FGD 2[83] [84] . . . 112Table 4.12 Fractional breakdown of the RHC ν¯µ CC-1 Track sample by true topologyfor both FGD 1 and FGD 2[83]. Distributions shown use the T2K MonteCarlo before the ND280 fit tuning. . . . . . . . . . . . . . . . . . . . . . . 112Table 4.13 Fractional breakdown of the RHC ν¯µ CC-N tracks sample by true topologyfor both FGD 1 and FGD 2[83]. Distributions shown use the T2K MonteCarlo before the ND280 fit tuning. . . . . . . . . . . . . . . . . . . . . . . 112Table 4.14 Fractional breakdown of the RHC νµ CC-Inclusive sample by true topologyfor both FGD 1 and FGD 2[84]. . . . . . . . . . . . . . . . . . . . . . . . . 118Table 4.15 Fractional breakdown of the RHC νµ CC-1 Track sample by true topologyfor both FGD 1 and FGD 2[84] . . . . . . . . . . . . . . . . . . . . . . . . 120Table 4.16 Fractional breakdown of the RHC νµ CC-N tracks sample by true topologyfor both FGD 1 and FGD 2[84] . . . . . . . . . . . . . . . . . . . . . . . . 120Table 5.1 List of near detector systematic error sources and types for the T2K neardetector selection.[81] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Table 5.2 Efficiency differences for the TPC cluster matching for data and MonteCarlo[81]. εdata is the measured data cluster efficiency and εMC is the mea-sured Monte Carlo cluster efficiency. . . . . . . . . . . . . . . . . . . . . . 134Table 5.3 Efficiencies for the TPC track reconstruction for data and Monte Carlo.[88] . 135Table 5.4 Efficiencies for the TPC – FGD track matching for tracks with few recon-structed hits for data and Monte Carlo.[90] . . . . . . . . . . . . . . . . . . 137Table 5.5 Michel detection efficiencies for Monte Carlo and data[82]. . . . . . . . . . 140xiiTable 5.6 False Michel electron identification rates for data and Monte Carlo. Ratesare defined as the number of expected false Michel electrons per spill. . . . . 141Table 5.7 Rate uncertainties for OOFV events by origin.[95] . . . . . . . . . . . . . . 146Table 5.8 Reconstruction uncertainties (MC/data difference) for OOFV events by origin.[95]147Table 7.1 Actual and predicted event totals for the different ND280 samples in theND280 fit. The MC predictions are shown both before and after the ND280fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Table 7.2 Fractional error on the prediction for number of events at SK, broken downby source, using the FGD 1 only ND280 fit results. FHC numbers use datafrom Runs 1 – 4, and RHC numbers use data from Runs 5 – 6. Flux andcross-section are shown separately for the pre-ND280 case. Uncertaintiesshown are calculated from the RMS of the event rates when 10000 parame-ter variations are thrown[99] from the final ND280 fit covariance matrix. . . 173Table 7.3 Fractional error on the prediction for number of events at SK, broken downby source. Pre-ND280 indicates that the ND280 fit result was not used asthe prior, while post-ND280 indicates that the ND280 fit using FGD 1 andFGD 2 events as described in this thesis was used as the prior. Flux andcross-section are shown separately for the pre-ND280 case. Uncertaintiesshown are calculated from the RMS of the event rates when 10000 parame-ter variations are thrown[99] from the final ND280 fit covariance matrix. . . 174Table 7.4 Fractional error on the prediction for number of events at SK from cross-section and flux, broken down to show the contributions from parametersunconstrained by ND280. The combination of these uncertainties gives theflux × cross-section uncertainties shown in Table 7.3. Uncertainties shownare calculated from the RMS of the event rates when 10000 parameter vari-ations are thrown[61]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174Table 7.5 Expected Fractional Uncertainty on sin2 2θ23 for Runs 1 – 7. . . . . . . . . . 182xiiiTable 8.1 Fitted parameter values for the three different data fits for the tested param-eters. If a listed fit did not have separate parameters for oxygen and carbontargets, the value from the general parameter is used as the entry for both. . . 195Table 8.2 Predicted number of events at SK using various ND280 postfit values andtheir uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Table C.1 Prefit and ND280 postfit values for the ND280 FHC flux parameters. Fluxvalues are relative to the nominal T2K flux prediction. . . . . . . . . . . . . 225Table C.2 Prefit and ND280 postfit values for the ND280 RHC flux parameters. Fluxvalues are relative to the nominal T2K flux prediction. . . . . . . . . . . . . 226Table C.3 Prefit and ND280 postfit values for the SK FHC flux parameters. Flux valuesare relative to the nominal T2K flux prediction. . . . . . . . . . . . . . . . . 227Table C.4 Prefit and ND280 postfit values for the SK RHC flux parameters. Flux val-ues are relative to the nominal T2K flux prediction. . . . . . . . . . . . . . . 228Table C.5 Prefit and ND280 postfit values for the cross-section parameters . . . . . . . 229xivList of FiguresFigure 1.1 Diagrams for the primary charged-current interactions relevant to the T2Kexperiment[31]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Figure 1.2 Diagram of protons and neutrons in a Fermi gas potential. Here EF is theFermi energy, related to the Fermi momentum pF and B is the nucleonbinding energy. The proton potential differs from the neutron potential dueto electrostatic repulsion. Image reused from [35] with permission. . . . . . 22Figure 1.3 Plots of the cross-sections d2σ/dΩdε for electron scattering on varioustargets versus electron energy loss ω = ε1−ε2, with solid line representingthe Fermi gas model predictions. Plot a is on carbon, plot b is on nickeland plot c on lead. Diagrams taken from [36] with permission. . . . . . . . 23Figure 1.4 Example diagrams for 2 particle – 2 hole interactions in nuclear medium.Diagrams taken from [42] with permission. . . . . . . . . . . . . . . . . . 26Figure 1.5 Measured CCQE neutrino cross-sections for neutrino (black) and antineu-trino (grey). Plot taken from [14] with permission. . . . . . . . . . . . . . 28Figure 2.1 Plot showing the T2K beam run periods. Light red regions indicate periodswhere the T2K neutrino beam was being produced. Plot from [46]. . . . . . 30Figure 2.2 Overview of the T2K beamline. Figure taken from [45] with permission. . . 32xvFigure 2.3 The neutrino energy spectra at the T2K far detector at various choice ofoff-axis angles, along with the probability of νµ disappearance and νe ap-pearance as a function of neutrino energy. The third plot shows the neutrinospectrum as a function of angle at the far detector. Figure taken from [47]with permission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 2.4 Diagram of the SK detector and facility. Figure taken from [45] with per-mission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.5 The INGRID detector. Figure taken from [45] with permission. . . . . . . . 38Figure 2.6 Picture of the ND280 layout. Figure taken from [45] with permission. . . . 39Figure 2.7 Simplified diagram of an individual TPC. Figure taken from [45] with per-mission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 2.8 Cross-section of a scintillator bar used in the FGDs. Figure taken from [53]with permission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 2.9 Diagram of structure of FGD water module. Not to scale. Figure takenfrom [53] with permission. . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 3.1 Correlations between the flux normalization parameters. The binning is:ND280 FHC νµ : 0 -10, ND280 FHC ν¯µ : 11 – 15, ND280 FHC νe: 16 –22, ND280 FHC ν¯e: 23 – 24, ND280 RHC νµ : 25 – 29, ND280 RHC ν¯µ :30 – 40, ND280 RHC νe: 41 – 42, ND280 RHC ν¯e: 43 – 49, SK FHC νµ :50 – 60, SK FHC ν¯µ : 61 – 65, SK FHC νe: 66 – 72, SK FHC ν¯e: 73 –74, SK RHC νµ : 75 – 79, SK RHC ν¯µ : 80 – 90, SK RHC νe: 91 – 92, SKRHC ν¯e: 93 – 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 3.2 Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 3.3 Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66xviFigure 3.4 Correlations between cross section parameters listed in Table 3.4. Each bincorresponds to a single cross section parameter, and bins are in the orderlisted in Table 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Figure 3.5 Detector covariance matrix, plotted as sign(Vi j)×√|Vi j| for clarity. Max-imum values are truncated at 0.45 for display purposes. Labels indicatewhere bins for a given sample start. The bins within each sample are or-dered in increasing momentum intervals, each containing all angular binsfrom backward going to forward going. . . . . . . . . . . . . . . . . . . . 74Figure 4.1 Momentum and θµ distributions for data and Monte Carlo, broken down bytrue topology, for the FGD 1 FHC CC Inclusive selection before the ND280fit tuning[81]. The “No Truth” category indicates cases where there was nospecific truth information associated with the selected vertex. . . . . . . . . 94Figure 4.2 Momentum and θµ distributions for data and Monte Carlo, broken down bytrue topology, for the FGD 2 FHC CC Inclusive selection before the ND280fit tuning[81]. The “No Truth” category indicates cases where there was nospecific truth information associated with the selected vertex. . . . . . . . . 95Figure 4.3 Total number of delayed out-of-bunch FGD timebins in selected CC Inclu-sive events. Plots are normalized to data POT, and were created for a subsetof the run 4 data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Figure 4.4 Example diagram of how FGD hits are grouped into timebins. The hori-zontal axis is time and each point represents the time of an individual FGDhit. Each red ellipse represents an FGD time bin and encompasses hitsplaced into it.[82] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Figure 4.5 Total charge seen in out-of-bunch timebins for selected CC Inclusive events.Plots show the breakdown by true particle associated with the delayed timebin. Plots are normalized to data POT, and were created for a subset of therun 4 data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102xviiFigure 4.6 Number of hits seen in out-of-bunch timebins for selected CC Inclusiveevents. Plots show the breakdown by true particle associated with the de-layed time bin. Plots are normalized to data POT, and were created for asubset of the run 4 data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 4.7 Momentum distributions for each of the three FHC samples broken downby true topology for FGD 1[81]. Distributions shown use the T2K MonteCarlo before the ND280 fit tuning. The “No Truth” category indicates caseswhere there was no specific truth information associated with the selectedvertex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Figure 4.8 Momentum distributions for each of the three FHC samples broken downby true topology for FGD 2. Distributions shown use the T2K Monte Carlobefore the ND280 fit tuning. . . . . . . . . . . . . . . . . . . . . . . . . . 106Figure 4.9 cosθ distributions for each of the three FHC samples broken down by truetopology for FGD 1[81]. Distributions shown use the T2K Monte Carlobefore the ND280 fit tuning. . . . . . . . . . . . . . . . . . . . . . . . . . 107Figure 4.10 cosθ distributions for each of the three FHC samples broken down by truetopology for FGD 2[81]. Distributions shown use the T2K Monte Carlobefore the ND280 fit tuning. . . . . . . . . . . . . . . . . . . . . . . . . . 108Figure 4.11 Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 1 RHC ν¯µ CC 1 Track selection[83]. Dis-tributions shown use the T2K Monte Carlo before the ND280 fit tuning.The “No Truth” category indicates cases where there was no specific truthinformation associated with the selected vertex. . . . . . . . . . . . . . . . 113xviiiFigure 4.12 Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 2 RHC ν¯µ CC 1-Track selection[83]. Dis-tributions shown use the T2K Monte Carlo before the ND280 fit tuning.The “No Truth” category indicates cases where there was no specific truthinformation associated with the selected vertex. . . . . . . . . . . . . . . . 114Figure 4.13 Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 1 RHC ν¯µ CC N-Track selection[83]. Dis-tributions shown use the T2K Monte Carlo before the ND280 fit tuning.The “No Truth” category indicates cases where there was no specific truthinformation associated with the selected vertex. . . . . . . . . . . . . . . . 115Figure 4.14 Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 2 RHC ν¯µ CC N-Track selection[83]. Dis-tributions shown use the T2K Monte Carlo before the ND280 fit tuning.The “No Truth” category indicates cases where there was no specific truthinformation associated with the selected vertex. . . . . . . . . . . . . . . . 116Figure 4.15 Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 1 RHC νµ CC 1-Track selection[84]. Dis-tributions shown use the T2K Monte Carlo before the ND280 fit tuning.The “No Truth” category indicates cases where there was no specific truthinformation associated with the selected vertex. . . . . . . . . . . . . . . . 121Figure 4.16 Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 2 RHC νµ CC 1-Track selection[84]. Dis-tributions shown use the T2K Monte Carlo before the ND280 fit tuning.The “No Truth” category indicates cases where there was no specific truthinformation associated with the selected vertex. . . . . . . . . . . . . . . . 122xixFigure 4.17 Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 1 RHC νµ CC N-Track selection[84]. Dis-tributions shown use the T2K Monte Carlo before the ND280 fit tuning.The “No Truth” category indicates cases where there was no specific truthinformation associated with the selected vertex. . . . . . . . . . . . . . . . 123Figure 4.18 Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 2 RHC νµ CC N-Track selection[84]. Dis-tributions shown use the T2K Monte Carlo before the ND280 fit tuning.The “No Truth” category indicates cases where there was no specific truthinformation associated with the selected vertex. . . . . . . . . . . . . . . . 124Figure 5.1 Total number of hits in out-of-bunch timebins from empty spills. Plots arenormalized to data POT, and were created for a subset of the run 3 data. . . 142Figure 5.2 The fractional error due to the statistical uncertainty in the MC predictionfor each of the ND280 detector systematic p – θ bins. The solid black linesseparate the CC0pi , CC1pi and CC Other sample, the dashed lines separatethe RHC ν¯ CC 1-Track, RHC ν¯ CC N-Tracks, RHC ν CC 1-Track andRHC ν CC N-Tracks samples, and the solid red line separates FGD1 andFGD2 samples[67]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150Figure 5.3 Detector covariance matrix calculated using the correlated systematic un-certainty variations. This does not include MC statistical errors, or the1p−1h shape covariance. . . . . . . . . . . . . . . . . . . . . . . . . . . 151xxFigure 5.4 Detector correlation matrices without the 1p-1h errors included. This com-pares the current treatment of detector correlations with how detector cor-relations were implemented in the last analysis. Labels indicate the binnumber; starting bins for each sample are: 0 - FGD1 CC0pi , 70 - FGD1CC1pi , 140 - FGD1 CC Other, 210 - FGD1 RHC ν¯ CC 1-Track, 230 -FGD1 RHC ν¯ CC N-Tracks, 250 - FGD1 RHC ν CC 1-Track, 270 - FGD1RHC ν CC N-Tracks, 290 - FGD2 CC0pi , 360 - FGD2 CC1pi , 430 - FGD2CC Other, 500 - FGD2 RHC ν¯ CC 1-Track, 520 - FGD2 RHC ν¯ CC N-Tracks, 540 - FGD2 RHC ν CC 1-Track and 560 - FGD2 RHC ν CC N-Tracks. The bins within each sample are ordered in increasing momentumintervals, each containing all angular bins from backward going to forwardgoing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Figure 6.1 Nominal Monte Carlo dataset validation for the ND280 fit, showing fluxparameters. Axis labels show the first bin of each category of flux parameters.154Figure 6.2 Nominal Monte Carlo dataset validation for ND280 fit, showing cross-section parameters. Axis labels show the name of each parameter; pa-rameters are typically a ratio to NEUT nominal values, excepting the CCOther Shape parameter and the FSI parameters. There is no prior used inthe fit for the CCQE parameters; the prior error bands shown for these pa-rameters are the error bands produced as potential cross section inputs andare included here for comparison purposes only. . . . . . . . . . . . . . . . 155Figure 6.3 Nominal Monte Carlo dataset validation for the ND280 fit, showing detec-tor normalization parameters. Axis labels show the first bin of each cate-gory of detector parameters. Values are fit to the normalization weights foreach p – θ bin and are generally not expected to equal to 1. . . . . . . . . . 156xxiFigure 6.4 ∆χ2 distributions for toys with observable normalization parameters thrownin red and Psyche detector systematic parameters thrown in black. The∆χ2 distribution for the observable normalization throws is from 114 toyfits and the distribution for the fits with Psyche throws has been scaled tomatch overall area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Figure 6.5 Pull means and widths for all flux parameters. The pull distributions aremade from the results of fits to 461 toy datasets. . . . . . . . . . . . . . . . 160Figure 6.6 Pull means and widths for all cross section parameters. The pull distribu-tions are made from the results of fits to 461 toy datasets. . . . . . . . . . . 161Figure 6.7 Pull means and widths for all observable normalization parameters. Thepull distributions are made from the results of fits to 461 toy datasets. . . . 161Figure 7.1 The pre-fit and post-fit ND280 flux parameters and their uncertainties. . . . 165Figure 7.2 The pre-fit and post-fit SK flux parameters and their uncertainties. . . . . . 166Figure 7.3 The pre-fit and post-fit cross section parameters and their uncertainties.Axis labels show the name of each parameter. Parameters are typically aratio to NEUT nominal values, excepting the CC Other Shape parameterand the FSI parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Figure 7.4 The pre-fit and post-fit observable normalization parameters and their un-certainties. Axis labels show the name of each parameter. . . . . . . . . . . 167Figure 7.5 The parameter correlations prior to the ND280 fit. The parameters are 0-24 SK FHC flux, 25-49 SK RHC flux, 50 MQEA , 51 pF16O, 52 MEC 16O,53 EB 16O, 54 CA5RES, 55 MRESA , 56 Isospin=12 Background, 57 CC OtherShape, 58 CC Coh 16O, 59 NC Coh, 60 NC 1 γ , 61 NC Other, 62 MEC ν¯normalization, 63 σνe , 64 σν¯e . . . . . . . . . . . . . . . . . . . . . . . . . 168xxiiFigure 7.6 The parameter correlations included after the ND280 fit. The parametersare 0-24 SK PF flux, 25-49 SK NF flux, 50 MQEA , 51 pF16O, 52 MEC 16O,53 EB 16O, 54 CA5RES, 55 MRESA , 56 Isospin=12 Background, 57 CC OtherShape, 58 CC Coh 16O, 59 NC Coh, 60 NC 1 γ , 61 NC Other, 62 MEC ν¯normalization, 63 σνe , 64 σν¯e . . . . . . . . . . . . . . . . . . . . . . . . . 169Figure 7.7 Total ∆χ2 distribution for 444 toy fit results (black), with the value from thefit to the data superimposed in red. The total is the sum of the contributionfrom the Poisson data term and the contribution from the prior constraintterm. The total ∆χ2 from the fit to the data is 1448.05, which correspondsto a p-value of 0.086. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170Figure 7.8 Contribution from the Poisson data terms to total ∆χ2 distribution for 444toy fit results (black), with the value from the fit to the data superimposedin red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170Figure 7.9 Contribution from prior constraint term to total ∆χ2 distribution for 444toy fit results (black), with the value from the fit to the data superimposedin red. This is the sum of the prior constraint terms from the flux, cross-section and detector parameters. . . . . . . . . . . . . . . . . . . . . . . . 171Figure 7.10 Contribution from flux constraint term to the prior constraint term ∆χ2distribution from 444 toy fit results (black), with the value from the fit tothe data superimposed in red. . . . . . . . . . . . . . . . . . . . . . . . . . 171Figure 7.11 Contribution from cross section constraint term to the prior constraint term∆χ2 distribution from 444 toy fit results (black), with the value from the fitto the data superimposed in red. . . . . . . . . . . . . . . . . . . . . . . . 172Figure 7.12 Contribution from detector constraint term to the prior constraint term ∆χ2distribution from 444 toy fit results (black), with the value from the fit tothe data superimposed in red. . . . . . . . . . . . . . . . . . . . . . . . . . 172xxiiiFigure 7.13 Predicted energy spectrum for FHC νµ events at SK, with and withoutND280 constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Figure 7.14 Predicted energy spectrum for FHC νe events at SK, with and withoutND280 constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Figure 7.15 Predicted energy spectrum for RHC ν¯µ events at SK, with and withoutND280 constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Figure 7.16 Predicted energy spectrum for RHC ν¯e events at SK, with and withoutND280 constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Figure 7.17 2D contours for sin2θ13 and δCP for the Run 1 – 7c joint analysis and theRun 1 – 4 joint analysis for normal mass hierarchy. [61] . . . . . . . . . . . 178Figure 7.18 2D contours for sin2θ13 and δCP for the Run 1 – 7c joint analysis and theRun 1 – 4 joint analysis for inverted mass hierarchy. [61] . . . . . . . . . . 179Figure 7.19 2D contours for sin2θ23 and ∆m232 for the Run 1 – 7c joint analysis and theRun 1 – 4 joint analysis for normal mass hierarchy. [61] . . . . . . . . . . . 179Figure 7.20 2D contours for sin2θ23 and ∆m213 for the Run 1 – 7c joint analysis and theRun 1 – 4 joint analysis for inverted mass hierarchy. [61] . . . . . . . . . . 180Figure 7.21 2D contours for sin2θ23 and ∆m232 for the Run 1 – 4 joint analysis for nor-mal mass hierarchy using the FGD 1 + FGD 2 ND280 fit result comparedwith other T2K contours. 2D contours for sin2θ23 and ∆m213 for the Run1 – 7c joint analysis and the Run 1 – 4 joint analysis for normal masshierarchy are taken from [61]. The difference between the red and bluecontours shows the improvement due to the inclusion of the FGD 2 datain the ND280 fit, for an oscillation fit to the Run 1 – 4 SK data. The 1Dsin2θ23 range is reduced by ∼3% and the 1D ∆m213 range is reduced by∼11%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183xxivFigure 7.22 2D contours for sin2θ23 and ∆m213 for the Run 1 – 4 joint analysis for in-verted mass hierarchy using the FGD 1 + FGD 2 ND280 fit result comparedwith other T2K contours. 2D contours for sin2θ23 and ∆m213 for the Run1 – 7c joint analysis and the Run 1 – 4 joint analysis for inverted masshierarchy are taken from [61]. The difference between the red and bluecontours shows the improvement due to the inclusion of the FGD 2 datain the ND280 fit, for an oscillation fit to the Run 1 – 4 SK data. The 1Dsin2θ23 range is reduced by∼2% and the 1D ∆m213 range is reduced by∼8%.184Figure 8.1 Fitted parameter values and uncertainties for the flux for the fit to the nom-inal MC with split MQEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188Figure 8.2 Fitted parameter values and uncertainties for the cross section parametersfor the fit to the nominal MC with split MQEA . . . . . . . . . . . . . . . . . 189Figure 8.3 Fitted parameter values and uncertainties for the observable normalizationparameters for the fit to the nominal MC with split MQEA . . . . . . . . . . . 190Figure 8.4 Fitted parameter values and uncertainties for the cross section parametersfor the fake data fit with MQEA12C = 1 and MQEA16O = 1.2. Parametervalues shown are relative to nominal, so MQEA12C is expected to fit to 1and MQEA16O to 0.833. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Figure 8.5 Fitted parameter values and uncertainties for the cross section parametersfor the fake data fit with MQEA12C = 1.2 and MQEA16O = 1 . . . . . . . . . 192Figure 8.6 The post-fit cross section parameters from the MQEA carbon and oxygenfit and their uncertainties and the cross section parameters from the datafit described in Chapter 7. Axis labels show the name of each parameter.Parameters are typically a ratio to NEUT nominal values, excepting the CCOther Shape parameter and the FSI parameters. As the original data fit didnot separately fit MQEA for carbon and oxygen targets, the same fitted MQEAvalue is shown for both targets. . . . . . . . . . . . . . . . . . . . . . . . . 193xxvFigure 8.7 The postfit correlations between cross section parameters for the MQEA car-bon and oxygen fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194Figure 8.8 The post-fit cross section parameters from the MQEA , C5A(0) and MRESA car-bon and oxygen fit and their uncertainties and the cross section parametersfrom the data fit described in Chapter 7. Axis labels show the name ofeach parameter. Parameters are typically a ratio to NEUT nominal values,excepting the CC Other Shape parameter and the FSI parameters. As theoriginal data fit did not separately fit MQEA , C5A(0) and MRESA for carbon andoxygen targets, the same fitted MQEA , C5A(0) and MRESA values are shown forboth targets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Figure 8.9 The postfit correlations between cross section parameters for the MQEA ,C5A(0) and MRESA carbon and oxygen fit. . . . . . . . . . . . . . . . . . . . 197Figure 8.10 Predicted event spectrum for FHC νµ signal events at SK. Red lines showthe prediction, as a solid line, and error envelopes, as dashed lines, fromthe near detector fit with MQEA , C5A(0) and MRESA split by target, and bluelines show the prediction and error envelopes from the near detector fitwith MQEA split by target. . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Figure 8.11 Predicted event spectrum for FHC νe signal events at SK. Red lines showthe prediction, as a solid line, and error envelopes, as dashed lines, fromthe near detector fit with MQEA , C5A(0) and MRESA split by target, and bluelines show the prediction and error envelopes from the near detector fitwith MQEA split by target. . . . . . . . . . . . . . . . . . . . . . . . . . . . 199Figure 8.12 Predicted event spectrum for RHC νµ signal events at SK. Red lines showthe prediction, as a solid line, and error envelopes, as dashed lines, fromthe near detector fit with MQEA , C5A(0) and MRESA split by target, and bluelines show the prediction and error envelopes from the near detector fitwith MQEA split by target. . . . . . . . . . . . . . . . . . . . . . . . . . . . 199xxviFigure 8.13 Predicted event spectrum for RHC νe signal events at SK. Red lines showthe prediction, as a solid line, and error envelopes, as dashed lines, fromthe near detector fit with MQEA , C5A(0) and MRESA split by target, and bluelines show the prediction and error envelopes from the near detector fitwith MQEA split by target. . . . . . . . . . . . . . . . . . . . . . . . . . . . 200Figure A.1 Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213Figure A.2 Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Figure A.3 Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Figure A.4 Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216Figure A.5 Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Figure A.6 Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218Figure B.1 Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 1 RHC ν¯µ CC Inclusive selection[83]. Dis-tributions shown use the T2K Monte Carlo before the ND280 fit tuning. . . 220Figure B.2 Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 2 RHC ν¯µ CC Inclusive selection[83]. Dis-tributions shown use the T2K Monte Carlo before the ND280 fit tuning. . . 221Figure B.3 Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 1 RHC νµ CC Inclusive selection[84]. . . . . 222xxviiFigure B.4 Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 2 RHC νµ CC Inclusive selection[84]. . . . . 223xxviiiGlossary2p-2h Two particle – two holeAPD Avalanche photodiodeASIC Application-specific integrated circuitASUM Analog sumCC Charged weak currentCCQE Charged-current quasi-elasticCKM Cabibo-Kobayashi-Maskawa matrixCMB Crate master boardCP Charge-parityDCC Data concentrator cardECal Electromagnetic CalorimeterFEB Front End BoardFGD Fine Grained DetectorFHC Forward horn current mode in which positive particles are focused to produce the T2Kbeam. Also referred to as neutrino mode.xxixFPGA Field-programmable gate arrayFSI Final state interactionGEANT4 Simulation package for particle detectors and particle propagation and interactionID Inner detectorIH Inverted heirarchyINGRID Interactive Neutrino GRIDJ-PARC Japan Proton Accelerator Research CenterLINAC Linear acceleratorLPB Light pulser boardMC Monte CarloMPPC Multi-pixel Photon CounterMR Main ring synchrotronMSW Mikheyev-Smirnov-Wolfenstein effectNC Neutral weak currentND280 The Near Detector at 280 metersNH Normal heirarchyOD Outer detectorOOFV Out of fiducial volumeP0D Pi-0 DetectorPID Particle identificationxxxPMNS Pontecorvo-Maki-Nakahata-Sakata matrixPMT Photomultiplier tubePOT Protons on targetRCS Rapid-cycling synchrotronRFG Relativistic Fermi Gas modelRHC Reverse horn current mode, in which negative particles are focused to produce the T2Kbeam. Also referred to as antineutrino mode.ROOT C++ Libraries designed for use with data analysis and simulationRPA Random phase approximationSCA Switched capacitor arraySF Spectral functionSI Secondary interactionsSK Super-KamiokandeSMRD Side Muon Range DetectorT2K The Tokai to Kamiokande ExperimentTPC Time Projection ChamberWLS Wavelength shiftingxxxiChapter 1Physics of Neutrinos1.1 Standard Model Neutrino TheoryThe first suggestions of the existence of neutrinos came from experiments with β decay, wherecurrent theory predicted that the energy of the electron produced in the decay A→ B+e wouldhave a single constant value. Instead, the emitted electron energy was found to be a continuousspectrum of energies, lower than the predicted value[1]. This “missing energy” was in fact dueto the production of a light neutral particle in the decay, now known to be the neutrino. The firstto postulate the neutrino’s existence was Pauli in 1930 as some additional neutral particle, andhis theory was extended later by Fermi for β decay[1]. Direct experimental confirmation of theneutrino’s existence came in 1953, from the work of Reines and Cowan[2]. A few years later in1959, Davis and Harmer showed that the neutrino and antineutrino were distinct particles[1].It was not until 1962 that the neutrino (and antineutrino) was discovered to come in morethan one flavor; so far theories had assumed that all neutrinos were the same and the onlydistinction was between particle and antiparticle. An experiment at Brookhaven NationalLaboratory[3] had observed that neutrinos produced from muon decay would only go on toproduce muons, rather than electrons. This indicated that there were at least two types of neu-trino: the muon neutrino and the electron neutrino to match the two known charged leptons.Once the tau was discovered in 1975, the existence of the electron and muon neutrino sug-1gested that there should be a third neutrino, one related to the τ . Its existence was verifiedmore recently by the DONUT collaboration, using data from 1997[4]. So far, there have onlybeen three kinds of neutrino discovered, which is consistent both with the measurements of thecosmic microwave background and its constraints on the number of neutrino generations[5].The measurements of the Z boson decay also suggest that there are only three neutrino flavorsthat can couple to the Z[6].1.1.1 Basic Neutrino FormalismThe neutrino is part of the Standard Model, which describes the various types of fundamentalparticles and their interactions with each other. The Standard Model itself has been developedover the past century and serves as the theoretical description of the electroweak and stronginteractions. The particles are composed of the fermions, with spin-12 , and the bosons whichmediate the various forces in the Standard Model[1].The fermions are split into the lepton and quark sectors. In the lepton sector, there are threelepton flavors: the electron (e), the muon (µ) and the tau (τ) with negative (matter) or positive(antimatter) charge. Each lepton flavor is associated with a neutrino and antineutrino of thesame flavor: νe, νµ and ντ for neutrinos and ν¯e, ν¯µ and ν¯τ for antiparticles. Each flavor hasits own lepton number (+1 for matter and −1 for antimatter), which is conserved in the weakinteraction. The quark sector consists of six different flavours of quark: the up (u), charm (c)and top (t) quarks which have a charge of +23 , and the down (d), strange (s) and bottom (b)quarks with charge of −13 . The quarks interact with the charged leptons and neutrinos throughthe weak force[1]; quarks and charged leptons can also interact through the electromagneticforce.Standard Model BasicsThe Standard Model is built off of the principle of gauge invariance, and depends upon varioussymmetries. Gauge invariance means that the Lagrangian must remain invariant under trans-formations, which can be either global or local. In particular, the Standard Model rests on the2assumption of local gauge invariance, which imposes the symmetries from which we derivethe properties of the gauge bosons.To start, we look at the simple case of the Dirac equation, which is a quantum mechanicalequation that satisfies the relativistic energy equation. The Dirac Lagrangian for a free particleisL =Ψ(i6∂ −m)Ψ (1.1)with Ψ = Ψ†γ0, 6∂ = γµ∂µ and γµ are the Dirac matrices[1]. Solutions to the Dirac equationrepresent spin-12 particles such as the neutrino. This is invariant under the global transformationΨ→ eiαΨ, where α is any real number. This global invariance corresponds to a conservedcurrent. However, we additionally require that the Lagrangian be invariant under local gaugetransformations.To guarantee invariance under local transformationsΨ(x)→ eiα(x)Ψ(x) (1.2)a correction must be added to the derivative givingL =Ψ(i6D−m)Ψ (1.3)where the covariant derivative Dµ = ∂µ− iqAµ , with Aµ being some new field that transforms asAµ → Aµ + ∂µq. Local invariance additionally requires that Aµ be massless. Aµ is in fact theelectromagnetic potential, generated through the requirement of local invariance. Invarianceunder the global phase transformation described previously also still holds; as eiα is a unitary1 × 1 matrix, the group of all such matrices is U(1). With the addition of a kinetic term, wecan then construct the QED Lagrangian[1]:LQED = iΨ6∂Ψ−mΨΨ− 14FµνFµν +qΨγµΨAµ (1.4)3where Fµν = ∂ µAν −∂ νAµ .This principle can be extended to SU(2) gauge group describing the weak force as well:starting with the Lagrangian for two spin-12 fields ψ1 and ψ2 with equal mass:L = ih¯cψγµ∂µψ−mc2ψψ (1.5)where ψ =ψ1ψ2 – this is just the sum of two Dirac Lagrangians. We can see this is invariantunder the global transformation ψ→ eiτ·aψ , where τ are the Pauli matrices τ1, τ2 and τ3[1] anda is three real numbers a1, a2 and a3. This is a global SU(2) gauge transformation and rotatesthe ψ1 and ψ2 fields into each other – meaning that the weak force acts equally on the ψ1 andψ2 components. Much like with the initial Dirac case, we promote this to a local invariance:ψ → e−iqτ·a(x)ψ (1.6)As before, the Lagrangian is no longer invariant under this transformation and a correctionmust be added to the derivative:Dµ = ∂µ + iqτ ·Aµ (1.7)where Aµ = (Aµ1 ,Aµ2 ,Aµ3 ). These Aµ are three gauge fields which transform such thatDµψ → e−iqτ·a(x)Dµψ (1.8)fulfilling the local gauge invariance. The three vector gauge fields are massless, just as withthe U(1) case. These three gauge fields give rise to the W+, W− and Z0 bosons, which acquiretheir masses through electroweak symmetry breaking. In moving from a simple SU(2) gaugetheory to the electroweak, we can identify ψ1 and ψ2 with either the charged lepton – neutrinoor up – down quark weak isospin doublets. In these pairs, the neutrino and up quark have aweak isospin of +12 , while the lepton and down quark have a weak isospin of −12 . The weak4isospin, often written as I3, is the conserved charge of the weak interaction, and is assigned toonly to left-handed particles.To describe the electroweak Lagrangian, we want to combine the electromagnetic andSU(2) gauge symmetries. This leads to a Lagrangian composed of one part which describes in-teractions between the three SU(2) gauge bosons and the U(1) boson, and one which containsthe kinetic terms for the fermions. This forms the electroweak group with SU(2)L × U(1)Ygauge symmetry, where the subscript L indicates that the SU(2) transformations are only onleft-handed particles, and Y is hypercharge. There are four generators in total: one from U(1),corresponding to the hypercharge and three from SU(2). These correspond to massless gaugeboson fields, as the mass term is not locally invariant. Of course, the bosons in the weak in-teraction – W+, W− and Z0 – are massive; to correctly describe the electroweak Lagrangian,there needs to be some way to give the gauge bosons mass.In order to generate the massive gauge boson fields in the electroweak theory, we employthe Higgs mechanism and spontaneous symmetry breaking. This allows the local gauge invari-ance to be preserved by way of choosing a vacuum ground state and expanding the fluctuationsaround it, with the addition of a gauge-invariant scalar field to the Lagrangian. In the case ofthe electroweak symmetries, this causes the original SU(2) × U(1) to be broken to U(1)QED,the electromagnetic subgroup. This gives three massless Goldstone boson fields, which witha bit of math can be transformed away to generate the masses of the gauge bosons W+, W−and Z0 as well as ending up with a single additional massive scalar. This is more specificallyknown as the Higgs mechanism, and also serves to give mass to the leptons and quarks[1].From the electroweak Lagrangian we can find the various propagators and vertex factorswhich are used to describe the weak interactions. This includes both the vertex factors forthe interaction of quarks and leptons with the gauge bosons as well as boson self-interactionvertices. For leptons and neutrinos, the vertex factor for the W±[1] is:− igW2√2γµ(1− γ5) (1.9)5where gW is the weak coupling constant for the W . The quark coupling is similar with anadditional factor from the Cabibo-Kobayashi-Maskawa (CKM) matrix which depends on thequark flavour. The vertex factor for the Z0 interacting with a quark or lepton is−igZ2γµ(c fV − c fAγ5) (1.10)where c fV and cfA are constants that depend on the interacting particle, and gZ is the weakcoupling constant for the Z[1]. The propagator for both the W and the Z0 is−igµν − qµqνM2q2−M2 (1.11)where gµν is the metric tensor, qµ is the momentum transfer of the particles and M is the massof the W or Z respectively. As M is very large, this propagator looks like 1M2 at lower energies;this applies to many neutrino experiments such as T2K. This is why weak interactions are somuch weaker than electromagnetic interactions, which have similar coupling constants but nomass term in the denominator of the propagator due to the photon being massless. This formsthe basis of how we calculate neutrino interactions and cross-sections for free partons.The Standard Model NeutrinoThe Standard Model neutrino is represented by four-component spinors, and is generally as-sumed to be massless. The chiral representation of these bispinors is :Ψ=ΨLΨR (1.12)where ΨL is the left-handed Weyl spinor and ΨR is the right-handed Weyl spinor. Here left-handed and right-handed refer not to the helicity of the fields, which relates the particle spinand direction, but to the chirality of the particle. This means that ψL transforms under theleft-handed representation of the Lorentz group, and ψR transforms under the right-handed6representation[7].In the Glashow-Weinberg-Salam theory of weak interactions, the left-handed fermion fieldsare SU(2) doublets:ψL =νLllL (1.13)with the right-handed U(1) singlet being lR for each lepton-neutrino pair. The free-field La-grangian for neutrinos isL = i(Ψ¯Llγµ∂µΨLl + Ψ¯Rlγµ∂µΨRl) (1.14)and is invariant under SU(2)×U(1) symmetry[8].The neutrino, as a neutral particle with no color, only interacts with other particles throughthe weak force. In the charged current interactions a neutrino and lepton of the same flavourcouple to the W±, and for the neutral current interactions a neutrino field couples to the Z bosonwithout any change of flavor. However, due to chirality only left-handed neutrino fields andright-handed antineutrino fields participate in weak interactions. The coupling for the neutrinofield and lepton field ψνl and ψl to the W± is :− gW2√2(ψ¯νlγµ(1− γ5)ψl)W µ (1.15)and the neutrino coupling to the Z boson is:− gW4cosθWψ¯νlγµ(1− γ5)ψνl Zµ (1.16)where cosθW is the weak mixing angle. The 12(1− γ5) ensures that only the left-handed fieldsparticipate in the weak interaction. Additionally, both types of weak interaction can be sepa-rated into vector (γµ ) and axial (γµγ5) current parts.So far, we have assumed that the neutrino, as in the Standard Model formulation, is massless7– this avoids attempting to construct a mass term without evidence for a right-handed neutrinofield. This assumption was also initially corroborated by the limits on the neutrino mass scalefrom cosmology[9] and maximal parity violation of the weak interaction. However, through theinitial observations of neutrino flavor oscillation by the Super-Kamiokande collaboration[10]and the Sudbury Neutrino Observatory[11], as well as subsequent oscillation measurements, itis now evident that neutrinos do have some mass, however tiny. It is not yet know where theneutrinos derive their masses from, and there are various potential theories.Neutrinos can gain mass in the Standard Model in a few ways: the first would be theassumption that like their partner leptons, neutrinos are Dirac particles and thus have Diracmass terms[12]. This does require both that neutrinos and antineutrinos are distinct particlesas well as requiring the existence of a right-handed neutrino field. This right-handed neutrinowould be a “sterile neutrino” – unable to interact through the weak force or electromagneticforce – making it very difficult to observe. In addition, the neutrino mass scale would requirea very weak Yukawa coupling, more than a million times smaller than that of the electron.There is also the possibility for the neutrino to be a Majorana particle, and have its massgenerated through a Majorana mass term. A Majorana particle is one which is its own an-tiparticle, requiring that there are no charges to be conserved[12]. While the neutrino is aneutral particle and does not carry electric charge, this would require that lepton number notbe conserved. Unlike the Dirac neutrino, a Majorana neutrino does not need an additionalright-handed sterile neutrino to generate a mass term, as the Majorana mass term only involvesfields of the same chirality. The Majorana mass term also can be used to explain the small sizeof the neutrino masses by way of the seesaw mechanism. In the seesaw mechanism, light Ma-jorana neutrinos are paired with extremely heavy neutrinos, with the observed neutrinos beingsuperpositions of these[13].81.2 Neutrino OscillationsThe discovery of massive neutrinos introduced a new effect to the Standard Model neutrino:oscillations similar to those seen in the quark sector. Because the flavor eigenstates and masseigenstates are not part of the same basis, neutrinos propagate in mass states that are a mixof the flavor states that interact through the weak force. In experimental terms, this explainsthe “solar neutrino problem”, where a deficit of νe was seen in the observed flux of neutri-nos generated by the Sun, by allowing electron neutrinos to be converted into other flavorsthrough interactions with matter, producing an observed deficit in electron neutrinos. Oscil-lation also explains the second big neutrino problem: the “atmospheric neutrino anomaly”,where a deficit in atmospheric νµ was observed in two different experiments. While neutrinooscillations require that neutrinos do have mass, they do not give any indication to how thatmass is generated.1.2.1 Neutrino Oscillation TheoryOnce neutrino mixing is included, the left-handed neutrino flavor states can be represented aslinear combination of the mass eigenstates:νl(x) =3∑j=1Ul jν j(x) (1.17)where l = e, µ and τ , νl is the flavor eigenstate and ν j is the mass eigenstate with mass m j 6= 01.U is the neutrino mixing matrix, shown in Equation 1.18. Current experimental evidencepoints towards 3-flavor mixing, and thus three mass states ν1, ν2 and ν3 with differing masses.This does not exclude the possibility of more than three types of massive neutrino, such asin the case of potential sterile neutrinos, which do not interact through the weak force. Themixing between the three neutrino flavor states and the three mass eigenstates is described bya unitary mixing matrix called the Pontecorvo-Maki-Nakahata-Sakata (PMNS) matrix shown1At least two of the three neutrino mass states must have non-zero mass. However, it is allowed for the lightestneutrino to be massless.9in Equation 1.18. This is analogous to the Cabibo-Kobayashi-Maskawa (CKM) matrix thatgoverns mixing in the quark sector, and is characterized by the three mixing angles: θ13, θ12and θ23 and a Dirac phase δCP, which is a complex phase governing CP violation in the neutrinosector. It is written as the product of three rotation matrices, with the complex phase includedin the matrix for the 1-3 plane rotation.U =1 0 00 cosθ23 sinθ230 −sinθ23 cosθ23cosθ13 0 sinθ13e−iδ0 1 0−sinθ13eiδ −0 cosθ13cosθ12 sinθ12 0−sinθ12 cosθ12 00 0 1(1.18)which then can be written as :U =Ue1 Ue2 Ue3Uµ1 Uµ2 Uµ3Uτ1 Uτ2 Uτ3=c12c13 c13s12 s13e−iδCP−s12c23− c12s13s23eiδCP c12c23− s12s13s23eiδCP c13s23s12s23− c12s13c23eiδCP c12s23− s12s13c23eiδCP c13c23(1.19)where si j = sinθi j and ci j = cosθi j. From this, it is apparent that the CP violating phaseδCP is associated here with sinθ13 – meaning that if θ13 = 0, there can be no CP violation inneutrino oscillations. However, the mixing matrix can be reparametrized such that δCP canbe associated with any of the the three mixing angles, meaning that all three mixing anglesmust be non-zero for CP violation to occur. In the case that neutrinos are Majorana particles,with ν = ν¯ , there are two additional CP-violating phases that must be included to preserveinvariance of the Majorana mass term: φ12 and φ13. These comprise an additional diagonalmatrix used in calculating U from the other rotation matrices[12], though they have no impact10on the neutrino oscillation probabilities:UMa jorana =1 0 00 eiφ21/2 00 0 eiφ31/2 (1.20)1.2.2 Oscillation ProbabilityWith the PMNS mixing matrix, and the understanding that the flavor neutrinos are superpo-sitions of the mass eigenstates as in Equation 1.17, it is possible to find the probability thata neutrino will oscillate from one flavor to another over some known distance. Here we willtake the assumption that the neutrinos can be approximated as plane waves – a fully completedescription would require using a wave packet description – and that the neutrino is travelingin a vacuum.The vacuum oscillation amplitude for a neutrino of flavor a to flavor b, traveling a distanceL is :A(νa→ νb) =∑kUbkDkU†ka (1.21)where Dk is the function that describes the propagation of the neutrino mass state νk over adistance L:Dk = e−iEkT+ipkL (1.22)where T is the travel time, and Ek and pk are the energy and momentum of the neutrino massstate2. To find the oscillation probability, we first assume that all mass eigenstates were gener-ated with the same momentum p and different energies Ek. 3 As the neutrino is ultrarelativisticdue to its very low mass, the propagation time, T , can be approximated as ≈ L/c and the time2Here units are h¯ = c = 13Performing this calculation starting with equal energies E and differing momenta pk provides the same an-swer.11dependence can be written out of Dk through Taylor expansion:EkT − pkL≈ (Ek−√E2−m2k)L≈m2k2EL (1.23)where E is the energy of the neutrino in the massless limit. The probability of oscillation canthen simply be calculated as the oscillation amplitude squared:Pνa→νb(L) = |A(νa→ νb)|2 =∣∣∣∣∣∑k U∗bke−im2kL/2EUka∣∣∣∣∣2(1.24)This can then be simplified to:Pνa→νb(L,E) =∣∣∣∣∣∑j,k U∗a jUb jUakU∗bke−i∆m2jkL2E∣∣∣∣∣ (1.25)where ∆m2i j = m2i −m2j . This means that the only intrinsic property, other than the mixingmatrix itself, that the oscillation probability depends on is the square of the mass splittings.Expanding out the terms separates the probability into contributions from the real and imagi-nary terms, and makes apparent how the probability differs between neutrino and antineutrinooscillation:P(νa→ νb)= δab−2∑i>kRe[UbiU∗aiU∗bkUak](1− cos(∆m2kiL2E))±2∑i>kIm[UbiU∗aiU∗bkUak]sin(∆m2kiL2E)(1.26)where the third term is + for neutrinos and − for antineutrinos. The final term in this equationallows for the observation of CP violation in the lepton sector, so long as a and b are differ-ent flavors. By looking for the asymmetry between oscillation probabilities of neutrinos andantineutrinos, CP violation has the potential to be measured experimentally.AabCP = P(νa→ νb)−P(ν¯a− ν¯b) (1.27)12With the probability given in Equation 1.26, this asymmetry depends only on the final term:AabCP = 4∑i>kIm[UbiU∗aiU∗bkUak]sin(∆m2kiL2E)(1.28)To properly measure this asymmetry, the values and signs of the mass splittings must be known– while the magnitudes of ∆m221, ∆m232 and ∆m231 have been measured[14], only the sign of ∆m221has been. This gives two potential orderings of the neutrino masses: the Normal Hierarchy(NH), with m1 lightest and m3 heaviest, and the Inverted Hierarchy (IH), with m3 lightest andm2 heaviest. Like CP violation, determining the mass ordering is the subject of current andfuture neutrino experiments.1.2.3 Matter EffectsWhile the oscillation probability derived in Equation 1.26 assumes that the neutrino travels ina vacuum, this is generally not the case in neutrino experiments. Before detection, neutrinosmay travel through the atmosphere or the Earth’s crust, as in neutrino beam experiments. Whena neutrino travels through matter, there is some probability of interaction before reaching itsdestination which can have an impact on the observed oscillations. These matter effects arereferred to as the Mikheyev-Smirnov-Wolfenstein (MSW) effect[15]. The primary effect onoscillations comes from νe scattering elastically off of electrons, specifically through charged-current interactions. While the other types of neutrino interactions can and do occur, interac-tions with no neutrino in the final state will show no effect on oscillation measurements, andneutral current processes occur at equal rates for all of the neutrino mass eigenstates. However,as there are only e− in ordinary matter, and no µ− or τ−, only νe and ν¯e can interact with matterthrough the W±. This leads to an effect on the overall flavour composition during propagation.This changes the Hamiltonian from the vacuum case, as the electron neutrino acquires anadditional potential energy VMSW =√2GFNe(t), where Ne is the electron density. This changein the Hamiltonian leads to a difference in the mass eigenstates when propagating throughmatter, and thus differences in the effective mixing angles and oscillation probabilities due to13this effect. These matter effects have only a small impact on experiments such as T2K, wherethe baseline is not long enough to see major differences, but are important when consideringneutrinos that propagate through the Earth’s core or are produced in the Sun[15].1.3 Previous Oscillation ExperimentsThe first observed effects of neutrino oscillation came in the form of the “solar neutrino prob-lem” at Homestake in 1965. While the experiment was looking for electron neutrinos producedby the sun, the observed rate of νe was much less than expected: they only observed 0.5 eventsper day compared to predictions of 2 – 7 per day[16]. At the time, however, neutrinos werealways assumed massless and neutrino oscillation unheard of. As a major unsolved problem inneutrinos, there were many various attempts to both reproduce the νe deficit and understand itssource, including Bahcall’s work on solar neutrino models[17]. However, it was not until the1990s that progress was made on this “solar neutrino problem”.The experiment to first solve the solar neutrino problem was the Sudbury Neutrino Obser-vatory (SNO), a 1 kton heavy water Cherenkov detector which was able to look at more thanjust the signal from νe. This, combined with the measurements from Super-Kamiokande[10],the follow-up to Kamiokande II, allowed SNO to both confirm the νe deficit as well as con-firm that the solar neutrino flux also contained νµ and ντ in 2001[11] by using neutral current,elastic scattering and charged current interactions to determine the electron and nonelectronneutrino components of the flux – clear evidence of neutrino flavor change.The “solar neutrino problem” was not the only historic issue that arose from neutrino os-cillations: equally key in discovering them was the “atmospheric neutrino anomaly”. In the1980s, measuring the decay of protons had become experimentally viable. However, the pre-dicted lifetime for the proton was very long, which required both a large detector volume witha large number of protons, and a low, well understood background. Part of this backgroundcomes in the form of muon and electron neutrinos produced by cosmic rays in the Earth’satmosphere, known as atmospheric neutrinos, which could then interact in the proton decay14detectors to produce a background.This neutrino background could be predicted using the cosmic ray muon flux, and knowl-edge about the neutrino production processes in the atmosphere – both of which had beenstudied by then. This allowed the prediction of the muon neutrino to electron neutrino produc-tion ratio, as a function of energy – something that could be measured at these proton decayexperiments. When the Kamioka Nucleon Decay Experiment (KamiokaNDE)[18], designedoriginally to measure proton decay in a large water Cherenkov detector, did so, it saw a signif-icant deficit in νµ compared to predictions of theνµνe flux ratio.As with the solar neutrino problem, the solution came later from measurements at Super-Kamiokande in 1998, which was able to both measure the νµ and νe flux along with directionof the outgoing lepton, which relates to the initial direction of the interacting neutrino[10]. Thisshowed that there was an angular dependence to the difference between the measured νµ rateand predicted – which for atmospheric neutrinos gives a measure of the distance the neutrinohas traveled, and can be related back to the distance and energy dependence in the neutrinooscillation probability.Since then, there have been many different neutrino experiments that sought to measure thevarious neutrino oscillation parameters. Several of these are reactor experiments, which use ν¯edisappearance to probe neutrino oscillation, such as Chooz[19] in 2003 and Double Chooz[20]in 2011, RENO[21] in 2012, KamLAND[22] and Daya Bay[23]. Other neutrino experimentshave used neutrino beams to look at both appearance and disappearance of various flavoursof neutrinos. Some of these are short baseline, where the detectors are placed close to theneutrino source, such as the NOMAD[24] experiment. Other major short baseline experimentswere LSND[25], and MiniBooNe[26]4. Other neutrino beam neutrino experiments are long-baseline, where there is a detector is placed many kilometers from the beam source, such asK2K[27], MINOS[28], NOνA[29] and T2K[30]. The current measured values for the neutrino4Both LSND and MiniBooNe saw some indications of neutrino oscillations. However, these results are con-troversial and are difficult to reconcile with the current neutrino model. As such, they will not be discussed furtherhere.15Parameter Valuesin2θ12 0.304±0.014sin2θ23 (NH) 0.50±0.05sin2θ13 (2.19±0.12)×10−2∆m221 (7.53±0.18)×10−5 eV2|∆m232| (2.44±0.06)×10−3 eV2Table 1.1: Summary of current measured values for the various oscillation parameters.Values taken from [14] and are shown for the normal mass hierarchy. ∆m232 is shownas an absolute value as the sign is not currently known.oscillation parameters are summarized in Table 1.1.1.4 Neutrino Interactions1.4.1 Neutrino Interaction ModesIn the Standard Model, the neutrino only interacts with other particles through the weak force,which is mediated by the Z0 and W± bosons. The Z0 is a neutral particle of mass 91.1876±0.0021 GeV/c2 and the W± is a charged particle of mass 80.385± 0.015 GeV/c2, with W+being positive and W− negative. Interactions mediated by Z0 are referred to as Neutral Current(NC) interactions, and interactions mediated by W± are referred to as Charged Current (CC)interactions. For the neutrino, neutral current interactions allow the neutrino to transfer energyand momentum to some other quark or lepton without any change of flavor. In Charged Currentinteractions, the neutrino interacts through the W to produce a lepton of the same flavour:νl + l→ νl + l, where the neutrino interacts with another lepton5 , and νl +d→ l−+u, whereit interacts with a quark. These produce a charged lepton in the final state, making them easierto detect than the Neutral Current interactions. Of course, quarks are not free particles in natureand instead exist in bound states inside of nucleons – p and n – and mesons. This complicatesthe possible interactions, and leads to classifying these interactions into multiple categories tobetter describe them.Of particular interest are the CC interactions, specifically the charged-current quasi-elastic5νl + l→ νl + l can also be a neutral current process, with the lepton and neutrino scattering off each other.16interaction. The main charged-current interactions relevant to the T2K experiment are shown inFigure 1.1. These are Charged-current Quasi-elastic (CCQE), where only a lepton and outgoingnucleon are produced, the charged-current resonant pion production, where a pion is producedin the final state via ∆ resonance, coherent pion production where the neutrino scatters off anentire nucleus and produces a single outgoing pion, and deep inelastic scattering, where thenucleon is broken into another nucleon and hadrons.Understanding these various cross-sections is important to correctly model neutrino inter-actions in experiment. In particular, the choice of neutrino interaction model and measurementsof the various model parameters has a significant impact on how well neutrino interaction rates– and interaction cross-sections – can be predicted. For this reason, there has been significanteffort to measure various neutrino interaction cross-sections and underlying parameters, and toconstruct a sufficiently robust model of neutrino interactions on nuclei.1.4.2 Nucleon-neutrino InteractionsCharged-current Quasi-elastic ScatteringCharged-current quasi-elastic neutrino scattering for nucleons is the simplest neutrino interac-tion, with simple two-body kinematics for the interaction:νl +n→ p+ l− (1.29)where vl and l are a neutrino and lepton of the same flavor. Because of its simplicity, the initialneutrino energy and momentum transfer can be determined using just the outgoing lepton’senergy and angle as in Equation 1.30 (assuming the nucleon is initially at rest), making thistype of interaction very useful for experiments. This is the primary signal interaction at T2K.The initial neutrino energy can be calculated as:Eν =m2p−m2n−m2l +2mnEl2(mn−El + pl cosθl) (1.30)17nWddupudulνl −(a) Charged-current quasielasticpWduupuuuduuudlνlπ+Δ++−(b) Charged-current resonant pion productionπAA+Wlνl −(c) Charged-current coherent pion productionlνlWNN−hadrons(d) Charged-current deep inelastic scatteringFigure 1.1: Diagrams for the primary charged-current interactions relevant to the T2Kexperiment[31].Here mp is the mass of the proton, mn the mass of the neutron, ml the mass of the outgoinglepton, which has an energy of El and momentum of pl and travels at an angle θ relative to thedirection of the incident neutrino.The cross-section for this interaction can be calculated by contracting the hadron currentwith the lepton current, using the parton model for the quarks[32], and the vertex function forthis interaction is written as a combination of a vector current and an axial current:JµQE =VµQE −AµQE (1.31)18with the vector current V µQE and axial current AµQEV µQE =−F1γµ +F22Miσµλqλ (1.32)AµQE =−FAγµγ5−FPMqµγ5 (1.33)where Q2 = −q2, the four-momentum transfer, M is the mass of the nucleon, F1 and F2 thevector form factors, FA the axial form factor and FP the pseudoscalar form factor[33]. Thevector current can be well measured using results from electron scattering experiments, as theform factors are directly related to the electromagnetic form factors[33]. The form factors forthe axial current AµQE are less well known as they are more difficult to measure experimentally,though the form factor FP can be related to FA by way of pion pole dominance[33]:FP(Q2) =2M2Q2+m2piFA(Q2) (1.34)In general, the axial form factor is expressed as a dipole when modeling the CCQE interactionin experiments:FA(Q2) =gA(1+ Q2MA)2(1.35)The choice of the dipole approximation comes from the assumption that the weak axial chargefollows an exponential distribution in the nucleus. Here the axial form factor shape is nowdetermined only by the axial mass, MA, as gA can be experimentally measured from β -decay.So in the neutrino-nucleon case for the CCQE interaction, we only need to determine the valueof MA to calculate the interaction cross-section.Charged-current Resonant Pion ProductionIn addition to the CCQE scattering, another important neutrino-nucleon interaction channel isthe charged-current resonant pion production. Here the neutrino is high enough in energy,above 200 MeV, such that pion production through nucleon resonances becomes possible.19These resonances are mostly either spin-1/2 or spin-3/2. In particular, at experimental energiesthe spin-3/2 resonances where the pion is produced through delta resonances are dominant:νl + p→ l−+∆++→ l−+pi++ p (1.36)For the spin-1/2 resonances, the currents are the same form as the quasielastic interactiondescribed previously for the nucleon in the CCQE interaction, though the incident neutrinoenergy cannot be calculated in the same way. However, the currents for the spin-3/2 resonancesare more complicated in structure. Like the CCQE interaction, the vertex can be written asa combination of a vector current and an axial current. The axial current for the spin-3/2resonance isAαµ3/2 =−(CA3M(gαµ 6q−qαγµ)+ CA4M2(gαµqp˙−qα pµ)+CA5 gαµ +CA6M2qαqµ) (1.37)where CA3 , CA4 , CA5 and CA6 are the axial N−∆ (Nucleus to ∆) transition form factors, q is themomentum transfer, gαµ is the metric tensor and M is the nucleon mass in the case of the freenucleon. The form factor C5A(Q2) can be expressed as a modified dipole:CA5 (Q2) =CA5 (0)(1+Q2M2RES)−2(1+Q23M2A)−1(1.38)where MRES is the axial mass for resonant processes. For the other form factors, we use theAdler model, where CA4 can be related back to CA5 :CA4 (Q2) =−CA5 (Q2)4(1.39)and CA3 (Q2) is equal to 0. For CA6 , this form factor can be related back to CA5 through pion pole20dominance similar to the QE case[33], giving:CA6 (Q2) =CA5 (Q2)M2Q2+m2pi(1.40)where mpi is the mass of the pion and M is the mass of the nucleon. This allows there toonly be one independent form factor, CA5 , for the axial current for the ∆ resonance. This isthe Llewellyn-Smith formalism[32], and is the standard description used for neutrino-nucleoninteractions in current neutrino generators. In this case there are two parameters that must bemeasured in order to determine the cross-section: the axial mass MRES, and the form factor CA5 .1.4.3 CC-0pi Neutrino Interactions on NucleiWhile the description of the charged-current neutrino scattering off of nucleons can be reducedto equations that depend only on a select few form factors and masses, these equations assumethat the nucleon is free, and not bound in a nucleus. Just as quarks are found in bound states,the nucleons in neutrino experiments are bound in nuclei. So what needs to be considered isnot just neutrino-nucleon interactions, but neutrino-nucleus interactions – introducing nucleareffects to the interaction cross-sections. Once a nucleon is bound in a nucleus, it becomessubject to various other effects which present various experimental problems. Interactions ofthe final state products (known as Final State Interactions or FSI), where the produced pions andnucleons interact in the nucleus itself, can lead to different particles leaving the nucleus thanthose from the original neutrino interactions – leading to difficulties in correctly identifying theinteraction type.To correctly calculate the cross sections for interactions with a nucleon inside a nucleus,other effects such as the Fermi motion of the initial nucleons, Pauli blocking of final nucleonsand final state interactions, where produced particles continue to interact inside the nucleus,must be considered. Because of the complexity and variety of nuclear effects on the interac-tions, neutrino experiments use a variety of different models to account for these. In additionto modeling nuclear effects for the CCQE interaction, this section will also discuss modeling21multi-nucleon-neutrino interactions, also known as n particle – n hole. These interactions sharethe same observed final state topology as CCQE interactions, and therefore form an irreducibleexperimental background for a CCQE signal.Relativistic Fermi Gas ModelThe Relativistic Fermi Gas model (RFG) is the most commonly used model for simple CCQEinteractions on nuclei in neutrino generators, and allows nuclear effects to be accounted foralongside the neutrino-nucleon dipole model described previously. The RFG model which ismost commonly used by neutrino interaction generators is the Smith-Moniz model[34]. Thismodel has been in use for several decades. The RFG model depends on just two nucleus-dependent parameters: the Fermi momentum pF and the nucleon binding energy EB. Thenuclear ground state is thus modeled as a Fermi gas of non-interacting nucleons, both protonand neutron, as shown in Figure 1.2. The Fermi momentum and binding energy are global forall nucleons and constant for a given material. The effect on modeled cross-sections is shownfor electron scattering in Figure 1.3, and is modeled the same as for neutrino interactions.Figure 1.2: Diagram of protons and neutrons in a Fermi gas potential. Here EF is theFermi energy, related to the Fermi momentum pF and B is the nucleon bindingenergy. The proton potential differs from the neutron potential due to electrostaticrepulsion. Image reused from [35] with permission.The RFG model is generally applied only for CCQE interactions, rather than also for CCresonant interactions. The reason for this is that the Fermi momentum and the binding energy22Figure 1.3: Plots of the cross-sections d2σ/dΩdε for electron scattering on various tar-gets versus electron energy loss ω = ε1− ε2, with solid line representing the Fermigas model predictions. Plot a is on carbon, plot b is on nickel and plot c on lead.Diagrams taken from [36] with permission.should have little effect on these types of interactions. In resonant interactions, the Fermi mo-mentum would apply to the nucleon produced from the decay of the intermediate resonances.These nucleons are generally above the Fermi momentum, and thus are rarely affected by Pauliblocking[34]. Pauli blocking comes from the Pauli exclusion principle, which bars identicalfermions (in this case neutrons and protons) from occupying the same quantum state in a nu-cleus. In practice, the Fermi gas model treats all of its possible particle states as filled, and anyproduced outgoing nucleon cannot occupy them. This gives a lower bound to the momentumof the outgoing nucleon, the Fermi momentum pF . Additionally, the binding energy is much23smaller than the resonance mass, M∆, and so only has a small effect on the resonance decay.Spectral Function ModelThe RFG model, while simple to implement and use in generators, does not provide a verycomplete description of nuclear effects: it assumes the neutrino interacts with a single nucleonin the nucleus, without correlations to other nucleons. The spectral function (SF) model, wherespectral function refers to some function that describes the momentum and energy distributionsof the initial nucleons in the nucleus, is made to be a more physically motivated model. Like theRFG model, the SF model does depend on the target nucleus: separate spectral functions mustbe built for each target nucleus. Additionally, the SF model is used only for CCQE interactions.These functions are relatively easy to calculate for light nuclei, but for medium nuclei, such ascarbon or oxygen, the spectral function will be an approximation[37].The spectral functions are made up of two terms: a mean field term for single particles,and a term from correlated nucleon-nucleon pairs. This second term leads to a tail for themomentum and binding energy. Unlike the RFG model, where Pauli blocking arises from theFermi momentum parameter pF , the spectral functions used in the SF do not implicitly havePauli blocking. While the correct way to include this would be to determine the cutoff usingthe local density, this is computationally impractical; instead a simple cutoff is applied to thespectral function at the appropriate energy, effectively forcing Pauli blocking. The spectralfunction is used as an alternative to the RFG model, and is presented here to give a morecomplete idea of the state of currently used neutrino models.Random Phase ApproximationWhile the RFG model includes important nuclear effects such as Pauli blocking, it is still asomewhat basic model and does not cover many other nuclear effects, particularly those relatedto correlations between nucleons within a nucleus. At low and intermediate energies, one ofthese additional nuclear effects is that of long-range correlations between nucleons[38]. TheCCQE peak energy lies in these energy regions, and therefore such long-range correlations can24have a significant effect on the shape of the differential cross-section for the neutrino.The Random Phase Approximation (RPA) model specifically accounts for the nuclear-medium polarization effects on the 1 particle – 1 hole contributions and is used as a correc-tion to the RFG nuclear model. The RPA model can be calculated using either relativistic ornon-relativistic nucleon kinematics, with T2K and other neutrino experiments commonly us-ing relativistic kinematics. At lower energies, relativistic and non-relativistic treatments of thekinematics shows similar behavior, In particular, adding the RPA effects reduces the CCQEcross-section at lower energies Additionally, while RPA models a nuclear effect, the correc-tions from the model do not depend strongly on the choice of nucleus[38].Multi-nucleon Correlation ModelsA major contributor to additional nuclear effects on the interaction cross-sections, and theobserved topologies, are multi-nucleon-neutrino interactions,, where the incoming neutrinointeracts with multiple nucleons simultaneously. These interactions are generally referredto as n particle – n hole (np-nh) interactions, as multiple particle-hole pairs are propagatedin the nucleus. These models were introduced to help resolve the disagreement betweenthe MiniBooNE cross-section results published in 2010 and its best-fit MA = 1.35± 0.66GeV/c2 [39], and the best-fit values from other experiments such as NOMAD, with a bestfit MA = 1.05±0.02(stat)±0.06(syst) GeV/c2[24]. MiniBooNE was not the only experimentthat measured a high value for MA, as K2K had also measured the axial mass and found a valueof 1.144± 0.077( f it)+0.078−0.072(syst)[40] although this was in better agreement with the globalresults. One possibility for these measurements is that otherwise unmodelled nuclear effectswere being absorbed into other CCQE parameters, such as pF , EB or MA. MiniBooNE did notuse a multi-nucleon-neutrino interaction model, and was not able to subtract such events fromthe CCQE sample. However, the MiniBooNE data was found to agree with global MA valueswhen fit with a model using multi-nucleon-neutrino effects[41].To predict the contribution from these interactions, the cross-sections can be computed with25a multi-body expansion of the weak propagator in the nuclear medium[42]. In this multi-bodyexpansion, the first-order expansion gives the standard CCQE interaction, where one nucleonin involved in the interaction and there is a single hole in the nucleus. Once one looks atthe second order terms, Feynman diagrams for two possible 2 particle – 2 hole (2p-2h) W -self energy interactions are shown in Figure 1.4. In Figure 1.4a, the W couples to a secondnucleon, and in Figure 1.4b, the 2p2h process is driven by ρ exchange through short rangecorrelations [42]. However, there are many more possible diagrams from the various types of2p-2h interactions.(a) Diagram of W -selfenergy from nucleon-nucleon correlations.(b) Diagram of W -selfenergy driven by ρ ex-change.Figure 1.4: Example diagrams for 2 particle – 2 hole interactions in nuclear medium.Diagrams taken from [42] with permission.These are not quasi-elastic interactions, as more than one nucleon is included in the inter-action through correlations. This means that the CCQE neutrino energy calculation in Equa-tion 1.30 is no longer applicable[42], and complete energy calculation requires more than justthe outgoing lepton momentum. However, these multi-nucleon interactions still share the CC-0pi final state topology with CCQE interactions and are a irreducible experimental background,26as the final state protons are both difficult to detect and can be affected by various final stateinteractions. Because of this, neglecting to model the higher order terms in the expansion canlead to bias in the reconstructed energy spectrum for CCQE events.One frequently used model for 2p-2h interactions is the Nieves 2p-2h model[43], whichis the model currently used in the T2K experiment. In the Nieves model, the many bodyexpansion of the W-boson self energy contribution in the calculation for the differential cross-section for neutrino charged-current interactions, up to the third order terms. As this model isdifficult to implement in generators, the hadronic level model parameters cannot be changedin a fit. Another major 2p-2h model is the Martini model[44]. Like the Nieves model, theMartini model uses a a multi-body expansion for the n particle – n hole terms. However,Nieves also accounts for higher order terms in the meson-exchange current (MEC) and axial-vector interference terms, which are not included in the Martini model.1.4.4 Current state of Neutrino Interaction ModelingThe neutrino interaction models used today are by no means complete – because nuclear mod-eling is a complicated and difficult process, all models include significant assumptions thatmay not always be valid. Of course, a nuclear model with poor assumptions can have a nega-tive effect on any oscillation measurements, as it can lead to incorrectly assigning energies tointeractions and incorrectly predicting the relative amounts of the various interaction channels,among other things. In addition, it is difficult to simply scale the modeling of nuclear effectsbetween different nuclei, which is a major source of uncertainty for experiments with differingnear and far detector materials such as T2K.In part, this is due to relatively few cross-section measurements on oxygen targets relative tocarbon – as Figure 1.5 shows, most of the cross-section measurements for neutrinos have beenon carbon and deuterium, rather oxygen. Outside of measurements at T2K and its predecessorK2K, only MINERνA has used a water target to study neutrino interactions. As the far detectortarget of T2K is oxygen[45], this presents some issues in using previous cross-section data to27 (GeV)νE1−10 1 10 210 / nucleon)2 cm-38 (10QEσ00.20.40.60.811.21.41.6T2K, CMINERvA, CHMiniBooNE, CBr3CF8H3GGM, CNOMAD, CSerpukhov, AlBr3SKAT, CF2ANL, D2BEBC, D2, D2BNL, H2FNAL, DLSND, C)νNUANCE ()νNUANCE (Figure 1.5: Measured CCQE neutrino cross-sections for neutrino (black) and antineutrino(grey). Plot taken from [14] with permission.understand interaction rates at the far detector due to difficulty in scaling between targets. Inpractical terms, this means that oscillation measurements at T2K must either incorporate un-certainties due to potential differences in carbon and oxygen interactions or be able to providesome form of information on neutrino interactions on oxygen itself.28Chapter 2The T2K Experiment2.1 The History of T2KThe T2K experiment is a long-baseline neutrino oscillation experiment designed to study var-ious neutrino oscillation parameters through νµ disappearance and νe appearance, using a νµbeam. T2K was built in order to measure the mixing angle θ13, which had not been measuredat the time. Additionally, the best limits on θ13 from the CHOOZ experiment[19] did notexclude θ13 = 0. The design goal of T2K was to improve on these limits and achieve a sensi-tivity a factor of 20 better than that of the CHOOZ experiment. In addition to measurementsof θ13, T2K is designed to be able to study other oscillation parameters with the precisionof δ (∆m223) ∼ 10−4eV2 and δ (sin2 2θ23) ∼ 0.01 using νµ disappearance. As the T2K beamcan run in either neutrino or antineutrino mode, δCP can also potentially be investigated. TheT2K near detectors also provides for a variety of neutrino interaction studies and cross-sectionmeasurements.The T2K experiment is made up of the neutrino beamline at Japan-Proton Accelerator Re-search Center (J-PARC), the near detectors 280 m from the beam source, and the far detector atSuper-Kamiokande (SK), 295 km from the beam source. The far detector at SK has been oper-ational since 1996 and both the T2K near detectors and the neutrino beamline were completedin 2009. T2K has been accumulating neutrino beam data since 2010; the total data collected29is counted in numbers of protons on target as that directly corresponds to the beam energy andnumber of spills collected. The T2K neutrino beam was run in neutrino mode from 2010 to thespring of 2013, and in antineutrino mode from 2014 to 2016. The total protons on target (POT)and running periods are shown in Figure 2.1.      𝛎-mode POT: 7.57×1020 (50.14%)      𝛎-mode POT: 7.53×1020 (49.86%)27 May 2016POT total: 1.510×10212011    2012    2013    2014    2015    2016Figure 2.1: Plot showing the T2K beam run periods. Light red regions indicate periodswhere the T2K neutrino beam was being produced. Plot from [46].Both the near detectors and far detector sit at 2.5◦ off-axis from the muon neutrino beam,which gives a narrow-band energy spectrum with a peak energy of 600 MeV. This energyis chosen to maximize the probability of νµ oscillation at the far detector 295 km away.This is possible due to the fact that the probability of oscillation, as described previouslyin Section 1.2.2, depends on the neutrino energy and distance traveled as sin2∆m2i jL4E , where∆m2i j = m2i −m2j for the relevant neutrino flavors. Because the energy spectrum of the T2Kneutrino beam has a broad spread from the energy of the parent pions, situating the detectorsoff-axis is necessary to achieve both a narrow energy band, and a peak energy around the os-cillation maximum, as is discussed in Section 2.2.1. The near detectors are positioned close30enough to the beam source to have effectively no oscillations, allowing measurement of theunoscillated beam flux.2.2 The Neutrino Beam at T2KThe T2K neutrino beam is produced using the 30 GeV proton beam produced at the Japan-Proton Accelerator Research Center (J-PARC). The beam facilities are located in Tokai, onthe eastern coast of Japan. J-PARC comprises three proton accelerators for the beam: a lin-ear accelerator (LINAC), a rapid-cycling synchrotron (RCS) and the main ring synchrotron(MR)[45]. The initial proton beam is produced by injecting H− ions into the LINAC andaccelerating them up to 400 MeV. The H− beam is passed through charge stripping foils toproduce a H+ beam. This proton beam is then injected into the RCS, and accelerated to 3GeV. The RCS has a 25 Hz cycle and has two bunches in each cycle. Around 5% of the beambunches are supplied to the MR, with the rest used in the muon and neutron beamline at J-PARC. In the MR, the proton beam is accelerated to 30 GeV, its final kinetic energy. The MRbeam has eight bunches and these are extracted in a single turn for use by the T2K neutrinobeamline.The neutrino beamline, where the proton beam is extracted from the MR, consists of aprimary beamline and secondary beamline. The primary beamline transports and focuses theextracted proton beam. This is done in three separate sections: the preparation section, thearc section and the final focusing section. The preparation section is 54 m long, and tunes theextracted proton beam for the arc section using magnets. The arc section is 147 m long, andis where the beam is redirected towards Kamioka and the far detector. The tuned proton beamis bent by 80.7◦ with a 104 m radius of curvature using superconducting combined functionmagnets. The final focusing section is 37 m and serves to focus and guide the proton beam onthe target, as well as aim the beam downwards by 3.637◦ with respect to the horizontal[45].The secondary neutrino beam line consists of the target station, decay volume and thebeam dump. The target station contains the proton beam target, a 91.4 cm long graphite tube31Figure 2.2: Overview of the T2K beamline. Figure taken from [45] with permission.surrounded by another 2mm thick graphite tube in a titanium case, and three magnetic horns.The proton beam is incident on the target, producing primarily pions along with kaons, and themagnetic horns are used to collect and focus the pions. The horns can be run in Forward HornCurrent (FHC) mode, which collects the pi+ to produce a neutrino beam, and in Reverse HornCurrent (RHC) mode, which collects the pi− to produce an antineutrino beam. The pions gothrough the decay volume, which is a 96 m long steel tunnel filled with helium[47], and decayprimarily into muons and muon neutrinos. The muons and other decay products are stoppedby the beam dump, producing a muon neutrino beam.2.2.1 The Off-Axis Neutrino BeamIn order to achieve a narrow-band neutrino beam, T2K uses the off-axis method to tune theneutrino beam energy spectrum. This places the detectors at an angle relative to the primaryfocusing axis for the beam. The neutrinos in the T2K beam are produced from two-body piondecay, where the energy of the emitted neutrino at a given angle relative to the direction of the32parent meson has a weak dependence on the parent momentum. While the neutrino energy hasa linear relation to the parent pion energy when on-axis, at high off-axis angles, the producedneutrino energy instead levels out at higher pion energies [48]. This means that the neutrinoenergy spectrum has a strong angular dependence, and the energy peak and spread can be tunedby choice of angle relative to the primary beam focusing axis. On axis, the T2K neutrino beampeaks at around 1 GeV. At T2K, the off-axis angle is set to be 2.5◦, setting the peak neutrinoenergy at around 0.6 GeV. This gives a neutrino spectrum peaked near the first oscillationmaximum at SK, as shown in Fig. 2.3. The lower peak energy also reduces the backgroundfrom non-CCQE processes at both the near and far detectors.2.3 Super-Kamiokande2.3.1 The Super-Kamiokande DetectorThe Super-Kamiokande detector functions as the far detector for the T2K experiment, and islocated in the Mozumi mine beneath Mt. Ikenoyama, 295 km from the beam source. Themountain provides 1000m of rock, equivalent to 2700 m water, mean overburden to reduce thebackground from cosmic ray muons [49]. The detector itself is a water Cherenkov detector,consisting of a stainless-steel tank 39 m in diameter and 42 m tall, with a total volume of 50ktons. Inside the tank, there is a 55 cm thick stainless-steel framework 2.5 m inside the wallsthat support arrays of Photomultiplier Tubes (PMTs), which split the tank volume into theOuter Detector (OD) and Inner Detector (ID). These are optically isolated by two light-proofsheets on the surfaces of the stainless-steel framework, which is uninstrumented dead space, toprevent light leaking between the two volumes. A diagram of the SK detector and its placementis shown in Figure 2.4.The OD is the volume from the outer edge of the tank to the PMTs and framework 2.5m into the tank, and is instrumented with 1885 Hamamatsu 20 cm PMTs. These PMTs faceoutwards towards the wall of the SK tank. As the OD is sparsely instrumented, the walls of thevolume are lined with a reflective coating of Tyvek[49] in order to improve the light collection33 (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: The neutrino energy spectra at the T2K far detector at various choice of off-axis angles, along with the probability of νµ disappearance and νe appearance asa function of neutrino energy. The third plot shows the neutrino spectrum as afunction of angle at the far detector. Figure taken from [47] with permission.efficiency. The OD serves as a veto for incoming particles for the ID, as well as providingpassive shielding from neutrons and other particles.The ID is the main detector volume for SK, with dimensions of 33.8 m in diameter and 36.2m tall with a total volume of 32 kton of water. It is instrumented with inwards-facing 11,146PMTs of 50 cm diameter which are mounted on a 70 cm grid[49]. To achieve an effectivephotocoverage of 40%, 7650 PMTs are located on the sides of the ID, 1748 on the top, and34Figure 2.4: Diagram of the SK detector and facility. Figure taken from [45] with permis-sion.1748 on the bottom. Unlike the OD, the area between the ID PMTs is lined with opaque blacksheets in order to minimize the number of reflected photons detected, as this can impact eventreconstruction. The ID is used for the T2K oscillation analysis and detects leptons producedfrom neutrino interactions in the inner volume.2.3.2 Event Reconstruction and SelectionThe SK event selection used in the oscillation analysis looks for charged leptons produced bycharged-current quasielastic neutrino interactions: µ for νµ interactions and e for νe interac-tions. The event selection is based around correctly identifying these leptons. To do this, theSK detector is designed to detect Cherenkov radiation produced by these particles. Cherenkovradiation is produced when a charged particle moves through a medium such as water at aspeed greater than the speed of light in that medium. This is given as v > cn , where v is thespeed of the particle, c the speed of light and n the index of refraction. This light is emittedat an angle cosθ = cnv relative to the direction the particle is traveling. This produces a lightcone around the particle, which can then be detected and reconstructed in the SK detector as a35ring-shaped hit pattern.The event selection at SK looks for CCQE interactions from νµ and νe events. Due todetector design and lack of magnetic field, there is no simple way to differentiate betweenneutrino and antineutrino events at the SK detector. The event selection looks at events whichare fully contained in the SK inner detector, as the event selection and oscillation analysis relyon the reconstructed neutrino energy and it is difficult to correctly reconstruct that energy forevents where the produced particles leave the detector volume. Events are considered to befully contained if there are less than 16 hit PMTs in the OD, indicating that the event depositedno energy outside the ID volume. The reconstructed event vertex must also be at least 200 cmfrom the ID walls for all selected events.The primary signal channel at SK is the CCQE interaction, described in Section 1.4. Asthis produces a single lepton and proton for the final state, CCQE events should only haveone detectable Cherenkov ring in the ID, as the proton will be below the Cherenkov thresh-old. This means the event selections for both νe and νµ events require that there is only onereconstructed ring in the event. Selected events are then separated based on the particle ID ofthe single reconstructed ring; muons are heavier and scatter less than electrons as they travelthrough the detector volume. This produces a sharply defined ring as compared to the fuzzierCherenkov ring from an electron, which scatters more easily due to its lower mass and alsoinduces electromagnetic showers as it travels. For the νe event selection, the cuts are:1. The single reconstructed ring must be identified as electron-like by the SK particle iden-tification.2. Evis > 100MeV. Evis is the total visible light from the reconstructed rings in the event.This removes background from NC processes and Michel electrons from other particledecays, as the CCQE signal process is unlikely at low energies.3. There must be no decay electrons detected. These show up as secondary events shortlyafter the primary neutrino interaction, and indicate the presence of a µ or pi .364. The reconstructed neutrino energy must be less than 1250 MeV. Detected νe with higherenergies are likely to be contamination from the beam due to the lower energy peak andspread of the off-axis beam.5. pi0 identified events are rejected based on a comparison of the likelihoods for νe and pi0particle ID hypotheses.The νµ event selection cuts are:1. The single reconstructed ring must be identified as muon-like by the SK particle identi-fication.2. The reconstructed muon momentum must be greater than 200 MeV/c, in order to removelow energy background.3. There must be one or no decay electrons in the event, as events with 2 or more arepredominantly CC non-QE events, while some CCQE events will have no detected decayelectrons.2.4 The T2K Near DetectorsThe T2K near detectors sit 280 m from the beam source at J-PARC, and consist of two sepa-rate parts: the Interactive Neutrino GRID (INGRID) Detector and the Near Detector at 280m(ND280). Unlike the SK detector and ND280, INGRID is situated on-axis, while ND280 is setat 2.5◦ off-axis.2.4.1 The Interactive Neutrino GRID DetectorThe Interactive Neutrino GRID (INGRID) detector at J-PARC is centered around the neutrinobeam axis and is primarily used to monitor the beam direction and intensity. The INGRIDdetector is composed of 16 separate identical modules arranged as shown in Figure 2.5 andsamples the beam in a 10 m× 10 m transverse section where the center of INGRID corresponds37to the neutrino beam center[45]. Each module is composed of 9 iron plates, which provide thetarget material, and 11 tracking scintillator planes arranged in a sandwich structure. The totaliron target mass is 7.1 tons per module. Each module has veto scintillator planes placed aroundit for reducing external background.Figure 2.5: The INGRID detector. Figure taken from [45] with permission.2.4.2 The ND280 DetectorsThe near detector (ND280) for T2K is situated 280 meters from the T2K beam target and is 2.5◦off-axis. As mentioned at the start of this chapter, this gives a narrower energy spectrum forneutrinos, and matches the off-axis angle at SK. ND280 is composed of five different detectors,which are used together to perform the ND280 analysis. The main section used for analysisis called the tracker, which is made of three time projection chambers (TPCs) and two FineGrain detectors (FGDs). The two FGDs are the primary target volume for ND280, and aredescribed in more detail in Section 2.5. Upstream from the track, closer to the beam source,is the Pi-0 Detector (P0D), for measuring the pi0 background from neutral current interactions.The P0D and the tracker are situated in a 6.5 m × 2.6 m × 2.5 m metal frame. The frameis surrounded by an electromagnetic calorimeter (ECal), which is in turn surrounded by the38recycled UA1 magnet. This magnet was previously used by the UA1/NOMAD experimentat CERN, and provides a 0.2 T dipole magnetic field for ND280, which is near constant inthe detector volumes. The magnetic field allows for accurate momentum measurements andparticle charge identification in the near detectors. This field is generated by a 2900 A currentpassed through the water-cooled aluminum coils that make up the magnet[45]. The innerdimension of these coils is 7.0 m × 3.5 m × 3.6 m, and they are surrounded by 16 flux returnyokes, arranged in pairs, with outer dimensions of 7.6 m× 5.6 m× 6.1 m with a total weight of850 tons. The coils and yoke are divided into two mirror-symmetric halves to allow for openingand closing the magnet to access the detectors. The magnet is additionally instrumented withscintillator to function as a muon range detector (SMRD).Figure 2.6: Picture of the ND280 layout. Figure taken from [45] with permission.39The Pi-0 DetectorThe Pi-0 Detector (P0D) is designed to measure the neutral current process νµ+N→ νµ+N+pi0+X on water, and uses the same material as the target at SK. To do this, the P0D is composedof scintillator modules alternated with fillable water target bags and lead and bronze sheets. Thewater bags can be filled or left empty during running; this allows for subtraction analysis todetermine the water cross-sections. There are 40 scintillator modules, each composed of an Xand a Y layer of triangular scintillator bars with wavelength shifting (WLS) fibers through themiddle of each bar. There are 134 vertical bars and 126 horizontal bars in each module. Thetotal active target volume is 2103 mm × 2239 mm × 2400 mm, and has a total mass of 16.1tons with the water bags filled, and 13.3 tons with the water bags empty [50].The Side Muon Range DetectorThe Side Muon Range Detector (SMRD) is not part of the inner detectors in ND280, butinstead is incorporated with the magnet yoke that surrounds the rest of ND280. It serves severalpurposes: the detection of escaping high-angle muons relative to the beam direction, cosmicray trigger and veto for particles entering the ND280 volume, and identifying beam interactionsfrom the surrounding magnet and detector pit. The SMRD is not set up separately from themagnet yoke but is instead installed in the air gaps between the steel plates that make up theflux return yoke. Each yoke has 15 air gaps in the radial direction, where the SMRD layersare placed. The detector consists of 440 scintillator modules, with two kinds of module, asthe horizontal and vertical gaps are of different sizes: horizontal, which are composed of fourscintillation counters, and vertical, which are composed of five scintillation counters. Thecounters themselves are optimized to maximize the active area in each gap. There are threelayers of scintillator modules on the top and bottom of all flux return yokes. There are 3 layersof scintillator modules on the sides of the first five flux return yoke pairs, four layers on thesides of the sixth pair and six layers of scintillator modules on the last two pairs of flux returnyokes.40The Electromagnetic CalorimetersThe Electromagnetic calorimeter (ECal) is a sampling electromagnetic calorimeter which sur-rounds the TPCs, FGDs and the P0D. It provides near-complete coverage for particles exitingthe inner detector volume, by detecting photons and measuring their energy and direction. TheECals can also provide reconstruction of pi0s produced inside the tracker, as unlike the P0D,the tracker is not designed to detect these. To do this, the ECal is made up of various config-urations of layers of scintillator bars and layers of lead, which provide a neutrino-interactiontarget and additionally act as a radiator to produce electromagnetic showers.There are three types of modules in the ECal: six Barrel ECal modules, which are placedaround the sides of the tracker volume parallel to the beam axis, one Ds-ECal module, posi-tioned downstream at the exit of the tracker volume and six P0D-ECal modules around the P0Dparallel to the beam axis[51]. The Barrel ECal modules consist of 31 layers with 50 scintillatorbars each, interspersed with 1.75 mm thick lead sheets. The Ds-Ecal module is similar to theBarrel ECal module, with 34 layers of scintillator interspersed with 1.75 mm thick lead sheets.The P0D-ECal modules have only six scintillator layers each, which alternate with five 4 mmthick lead sheets, as the P0D is designed for initial pi0 detection and the ECal serves to provideadditional energy information.The Time Projection ChambersThe Time Projection Chambers (TPCs) provide several important functions for the ND280tracker, and are an integral part of the near detector analysis. They provide 3-D tracking forcharged particles, as well as particle identification, which is important for identifying neutrinointeractions originating from the neighboring detectors with more target mass. Additionally,the magnetic field the near detectors sit in allows the TPCs to make measurements of particlemomentum and charge, as well as aiding particle identification. This is a key part of the ND280analysis and allows ND280 to differentiate between neutrino and antineutrino events. Thereare three TPCs, all of the same design. The TPCs are placed around the FGDs, so that TPC 141sits upstream of FGD 1, TPC 2 between FGD 1 and FGD 2 and TPC 3 downstream of FGD 2.Each TPC consists of an inner chamber filled with an argon-based drift gas and an outerchamber filled with CO2 for insulation[52]. The inner and outer walls of the inner box arecomposite panels with copper clad skins. The panels have a 11.5 mm pitch copper strip pattern,giving a uniform electric drift field in the active volume of the TPC in conjunction with a centralcathode panel in the middle of the inner box[52], as shown in Figure 2.7, which is a simplifieddiagram of the TPC structure. When a charged particle passes through a TPC, it producesionization electrons in the gas, which drift in the electric field towards the readout planes at theends of the detector away from the central cathode. The readout planes are composed of 12micromegas modules, which sample and multiply the electrons [52].Figure 2.7: Simplified diagram of an individual TPC. Figure taken from [45] with per-mission.42The Fine Grained DetectorsThe Fine Grained Detectors (FGDs) are two scintillator based detectors that serve as the mainneutrino target for ND280 and provide vertexing and tracking for events. There are two FGDs:FGD 1, which functions as the carbon-only target volume, and FGD 2, which provides bothcarbon and oxygen targets. The FGDs are described in more detail in the following section.2.5 The Fine-Grained Detectors2.5.1 OverviewThe T2K tracker, which consists of the three TPCs and two FGDs, is designed to measure thecharged current neutrino interactions, particularly the CCQE interaction process as it is themost common interaction at the T2K beam energies. As described in Section 1.4.2, the energyfrom these interactions can be reconstructed from the outgoing lepton momentum and anglealone, making it convenient for determining the neutrino energy spectrum. While the TPCsprovide momentum measurements, charge identification and particle identification, they areinsufficiently dense to function as a neutrino target. Instead, the FGDs provide the main targetmaterials for ND280.As the main target volume for ND280, the FGDs are designed to fulfill several analysisrequirements. To be useful in measuring neutrino interactions, the FGDs must be able to detectall charged particles at the neutrino interaction vertex, and also be thin enough that the chargedleptons can travel to the adjacent TPCs for particle identification and momentum measure-ments. The FGDs are also designed to be able to perform basic particle identification using thedE/dx of particles contained in the FGD volume. Additionally, as the ND280 target volume,the FGDs must be able to provide measurements of neutrino interaction on water, the sametarget material as at SK.The basic functional unit of the FGDs is the scintillator module, referred to as an XYmodule, made up of two layers of 192 scintillator bars each. These layers are perpendicular43to the beam, with one layer horizontal and the other vertical, giving each module the totaldimensions of 186.4 cm × 186.4 cm × 2.02 cm. The scintillator bars have a square 9.6 mmcross section, as shown in Figure 2.8. Each bar has a reflective TiO2 coating on the outside, anda wavelength shifting (WLS) fiber through the center of the bar. To improve light collection,one end of the fiber is mirrored and the other mechanically coupled to a photosensor. Thescintillator itself is made out of polystyrene and provides the carbon-based target material forthe FGDs [53]. There are 15 scintillator modules in FGD 1 and 7 in FGD 2.Figure 2.8: Cross-section of a scintillator bar used in the FGDs. Figure taken from [53]with permission.Unlike FGD 1, which is only composed of scintillator modules, FGD 2 must include a watertarget mass and therefore includes separate water modules. There are six water modules in FGD2, alternated with the scintillator modules so that FGD 2 starts and ends with an scintillatormodule. This is because while the scintillator modules function as both active detector materialand a carbon target simultaneously, the water modules are uninstrumented. This also meansthat FGD 2 has less active material, which has an impact on track reconstruction in FGD 2. The44water modules are each 25.4 mm thick and 1809 mm wide and are build from rigid, hollowpolycarbonate panels with an internal structure to maintain panel shape when filled with water.The internal structure of a water panel is shown in Figure 2.9. In addition, the oxygen content ofthe polycarbonate used allows each panel’s elemental composition to match the compositionof a water and scintillator mixture. The polycarbonate composition is C16H14O3, which iseffectively C16H8 plus water. To match the C:H ratio of this remainder to that of polystyrenein the scintillator, two 0.8 mm thick sheets of polypropylene, CH2 are attached to the frontand back surfaces of each water panel. The end result is that the water module compositionis equivalent to 490 mg/cm2 of scintillator1 and 2297.2 mg/cm2 of water, enabling subtractionanalysis for measurements on water[53].Figure 2.9: Diagram of structure of FGD water module. Not to scale. Figure taken from[53] with permission.FGD 2 has an additional water system to maintain the pressure in the water modules as wellas monitor the flow rates for potential leaks. This system maintains an operating pressure of300 mbarA. This is below atmospheric pressure in order to prevent water emission in the caseof leaks; instead, air will enter the system in the case of a leak, which does not have as muchimpact on effective operation of the FGDs. The water level in the modules is monitored throughthe upwards flow rates into the modules, as there was insufficient space for sensors in the panelsthemselves. The water levels are maintained by a closed loop with a pump that returns waterto an overhead reservoir. Each panel has its own independent flow loop that draws on the same1Valid assuming a negligible amount of Ti, as the water modules do not contain any.45reservoir, so that each panel can be drained or filled separately from each other, such as in thecase of a major leak. The water used in the panels contains a small amount of antimicrobial andanticorrosive agents in order to prevent biological growth in the water system. This composes0.25% by volume of the water and has little impact on the target composition.The scintillator modules for FGD 1 and the water panels and scintillator modules for FGD 2hang in a light-tight box that provides the structural framework for the FGD. Each box is 2300mm × 2300 mm × 365 mm, and also contains the photosensors attached to the scintillatorbars. The electronic readout cards are outside this volume, and are attached to the photosensorbusboards through twenty-four backplanes mounted on the four sides of the dark box. Thesebackplanes then attach to the minicrates that house the FGD readout electronics.2.5.2 ElectronicsThe electronics in the FGDs are designed to take a snapshot of the detector activity, includebefore and after a beam spill. Each beam spill is composed of 6 - 8 bunches, with 580 nsseparation between them and 3 s between beam spills. As the FGD also needs to be able tolook for delayed activity related to pion decays, the FGD electronics must be active betweenbunches and for several µs after a spill. To achieve this, the FGD electronics trigger on beamspills and record for a total of around 10 µs. For the vertex reconstruction, there must also bea timing resolution of < 3 ns for each hit.In order to detect optical signals from charged particles, the FGDs use Multi-Photon PixelCounters (MPPCs) as the readout. These photosensors are able to count photons at the levelof a few photoelectrons, and are small enough to be used as readout for the scintillator bars.Additionally, MPPCs are not affected by strong magnetic fields like the field at ND280, whilePMTs like those used at SK have reduced sensitivity in magnetic fields. Because of this,MPPCs are used both for the two FGDs and for the other scintillator detectors in ND280.The MPPCs are pixelated avalanche photodiodes (APD) with an outer dimension of 5 mm× 6 mm and active area of 1.3 mm × 1.3 mm, and are attached to the WLS fibers in the scin-46tillator bars with a custom coupler in order to minimize light loss. Each MPPC has 667 APDpixels, which each operate independently with an applied voltage slightly above the breakdownvoltage so that the output charge of the pixel is independent of the number of produced photo-electrons in the pixel itself. Because of this, the number of incident photons from a WLS fiberwill be roughly proportional to the number of observed discharged pixels, so long as there aremany fewer incident photons than pixels in the MPPC, allowing it to be an effective photoncounter. To read out and digitize the data, the MPPCs are attached to busboards which aggre-gate the signals from the MPPCs in order to pass the data to the readout electronics that sitoutside of the dark box. Each busboard is connected to 16 MPPCs, and also provides tem-perature and humidity sensors to monitor the dark box. The busboards are then connected byribbon cables to the electronics backplanes which provide the interface between the inside andoutside of the dark box. There are fifteen busboards connected per backplane in FGD 1, andseven per backplane in FGD 2 due to lower numbers of MPPCs [53].The FGD readout electronics are situated outside the dark box, and connect to the MPPCsand busboards via the electronics backplane. The electronics are placed in minicrates andcontain the frontend boards (FEBs), the light pulser boards (LPBs) and the crate master boards(CMB). There are four FEBs in each FGD 1 minicrate, and two for FGD 2. Each minicratealso contains one CMB, with an Field-Programmable Gate Array (FPGA) to control the readoutprocedure, and LPB for calibration. The LPB controls LEDs mounted on the busboards, whichare used for testing WLS fiber integrity and calibrating the MPPC gains.The FEBs take the MPPC signals from the busboards and digitize them, as well as providingthe MPPCs with power. Each FEB contains two AFTER ASICs[54], with 32 MPPCs readout per ASIC. The AFTER ASIC provides a preamplifier and shaper to amplify the electricalpulses from the MPPCs and extend the pulse lengths, and can record both low-gain and high-gain signals. The ASIC also provides a switched capacitor array (SCA) which can store analogsamples of the shaped MPPC pulses. The SCA has a sampling frequency of 50 MHz, with atotal readout time of 10.1 µs. Using the ASICs allows for a lower sampling frequency versus47directly digitizing the MPPC signals. Once an event is triggered, the contents of the SCA arethen digitized by a 12-bit flash ADC, which runs at 20 MHz[53], and transferred from the FEBto the CMB.The CMBs handles the trigger signal processing, the LPB triggering and setting the con-figuration of the ASICs on top of controlling the data acquisition for the FGDs. This is doneusing an FPGA, which controls the record and readout phases of the attached ASICs in theminicrate. The digitized data from the SCAs is stored on the CMB in a pair of SRAMs. Thereadout process takes around 2 ms, which does not make a significant contribution to the totalFGD readout time[53]. The data on the CMBs is then transmitted through an optical link to theData Concentrator Cards (DCCs) outside the minicrates, which provide the interface betweenthe FGD electronics and the rest of the ND280 control systems.2.5.3 OperationThe triggering for ND280 is controlled by the master clock module, so called because it pro-vides the global clock signal for the ND280 detectors. The primary trigger type is the beamspill trigger, which does not depend on activity at ND280 and is produced outside the ND280system. This provides the trigger for the ND280 beam data, which is used for the near detectoranalysis. Other types of trigger are Trip-T cosmic trigger and the FGD cosmic trigger. TheTrip-T cosmic trigger comes from coincidence in the ECal, P0D and SMRD, which are all runby the Trip-T readout electronics. The FGDs and TPCs do not use the Trip-T system. As therate of cosmics triggered by the Trip-T detectors is low, the FGD also provides a trigger forcosmic events.The FGD uses FGD 1 – FGD 2 coincidence to trigger on cosmic rays, which can be usedfor various measurements, calibration studies and systematic studies. To do this, the FGDelectronics look for coincident activity in both FGD 1 and FGD 2. For the trigger, the MPPCsin each minicrate are grouped in sets of 8 and the signals from these 8 MPPCs are summed.These analog sum groups (ASUM) are considered to have fired when the signal sum exceeds a48set threshold, which is tuned for each individual group of MPPCs. If two or more of the ASUMgroups in a minicrate fire, the minicrate is also considered to have fired. To have coincidencebetween FGD 1 and FGD 2, the trigger looks for two crates firing in FGD 1 coincident withtwo crates firing in FGD 2. This ensures that these cosmic events have energy deposits in FGD1 and FGD 2, as well as at least one TPC track.2.5.4 CalibrationThere are two main tasks for the FGD calibration: calibration of measured charge and calibra-tion of measured time. The charge calibration converts the raw pulse height (PH) measuredof the digitized waveform from an MPPC to a normalized value that represents the energy de-posited in the scintillator bar. The timing calibrations provide corrections to resolve differencesbetween timing on different FEBs and the CMB timings, as well as account for the travel timeof light in the WLS fibers. This gives the FGD a timing resolution of < 3 ns between hits ina single FGD, and a resolution on the timing between hits in FGD 1 and FGD 2, discussed inlater chapters.There are many steps in the charge calibration chain, several of which are temperature-dependent effects. Calibration starts with high- and low-gain channel response calibration, tofind the MPPC response independent of which channel was used. The high-gain and low-gainchannel responses have a linear relationship, measured for each MPPC using cosmic rays trig-gered in the FGDs. These pulse height values are then normalized using the average pulseheight from a single pixel firing. This is measured using the pulse height distribution of theMPPC dark noise, which is where a pixel spontaneously fires due to electrons thermally gen-erated in the pixel. This effect depends on the temperature, which can vary by up to ±2◦C.This temperature variation also affects the breakdown voltage of the MPPCs. An additionaltemperature-dependent calibration effect is the average numbers of pixels firing for a givennumber of photons incident on an MPPC. This depends on photodetection efficiency, crosstalk and after-pulsing probabilities, all of which have some temperature dependence. This ef-49fect was studied at TRIUMF using an electron beam[53]. The MPPC response is also correctedfor the effects of MPPC pixel saturation, where the measured pulse heights eventually saturateat increasing light levels due to multiple photons incident on individual pixels.Other charge calibration corrections come from the scintillator bars, rather than the MPPCs.As there can be minor variations in things such as the MPPC to fiber coupling, the exactposition of the WLS fiber in the scintillator bar, and minor variations in the scintillator material,the calibration uses an additional correction constant specific to each bar which modifies theefficiency of energy deposition to photons incident on the MPPC. Additionally, as there is somelight loss along the fiber, measured by cosmic ray studies, there is a correction to account forthe position of the particle track relative to the MPPC, as there can be up to a 25% loss atthe far end. All these calibration steps yield a number proportional to the number of photonsproduced in the scintillator, which is then corrected using Birks’ formula[53] with an additionalefficiency normalization to yield the deposited energy, which is stored as a hit for that channeland can be used in the ND280 reconstruction.2.6 Limitations of the T2K AnalysisPart of the T2K oscillation analysis, as detailed in the following chapter, uses selected eventsin ND280 to improve the overall uncertainty on the results. However, so far the near detectorcomponent of the oscillation analyses has only used neutrino interactions occurring in FGD 1,which only provides a carbon target. As the difference in target material between ND280 andSK limits how well ND280 can be used to understand the neutrino cross-section on water atSK, this puts a limit on how much the ND280 contribution can reduce the uncertainties at SK.While FGD 2 was designed specifically to provide an oxygen-based target to avoid this issue,neutrino interactions in FGD 2 have not been used before for a few reasons.As described in the previous section, the water modules in FGD 2 are not active, and aredesigned to be used as part of a subtraction analysis to isolate information about the watercross-section in conjunction with FGD 1 rather than using FGD 2 on its own. This allows the50carbon target in FGD 1 to be used to improve the measurements on water in FGD 2. However,this introduces the issue of possible detector systematic correlations between the two detectors,as only using FGD 2 would not give a better reduction in uncertainty. Not only would detectorsystematics need to be remeasured for a second detector but there would need to be some stud-ies and understanding of how each systematic source correlates. Additionally, the selection,described in Chapter 4, was initially only designed and tested for FGD 1; as the amount ofactive material differs between the two FGDs, the selection could potentially need changes toaccount for reconstruction differences. For these reasons, the initial near detector selectionsand analysis only used events occuring in FGD 1. The intent of this thesis is to extend theprevious T2K analyses to include FGD 2 along with FGD 1, in order to directly determineinteraction rates on water.51Chapter 3Oscillation Analysis at T2K and theND280 Fit3.1 The T2K Oscillation Analysis Structure3.1.1 General OverviewThe primary goal of the T2K experiment is to measure neutrino oscillations at the far detector.As oscillation parameters are not directly measurable, T2K instead fits the neutrino oscillationparameters using the number and energy spectrum of observed events at the far detector. TheT2K oscillation fit is a maximum likelihood fit on the data measured at the Super-Kamiokandedetector, which compares the predicted event distribution to the observed event distributionat SK. Predicting the event rate requires three main components: a calculation of the flux, aparameterization of the cross-section, and a model of the detector efficiencies. The fit itself fitsboth the oscillation parameters and these other parameters, which have prior constraints andcan be regarded as nuisance parameters in the result. Selected νµ and νe event rates are used tofit the oscillation parameters while marginalizing over the various nuisance parameters. Theseare the parameters that describe the flux from the neutrino beam at SK, the various cross sectionparameters that govern neutrino interactions with oxygen nuclei and the detector systematics52from SK itself and are used to tune the initial predictions at SK. The detector systematics canbe characterized using SK information alone, while constructing prior constraints for flux andcross section parameters is more complicated. The initial set of inputs come from experimentsand information external to the near and far T2K detectors. The nuisance parameters in ouroscillation fit are the neutrino interaction cross section parameters and the parameters govern-ing the beam flux at Super-Kamiokande (SK) from J-PARC and the prior mean values anduncertainties are calculated from external experiments such as NA-61[55] and MINERνA[56].3.1.2 MotivationThe primary source of uncertainty on the predicted number and energy distribution of eventsat SK comes from the flux and cross-section, which are some of the nuisance parameters in theSK oscillation fits. Without using the data at the near detector, reducing uncertainty on the fluxand cross-section parameters relies on using external inputs such as measurements of hadronproduction and neutrino interaction cross-section measurements. Additionally, this means thatthere is no correlation between the flux and cross-section inputs, as the priors are constructedcompletely independently. Adding in data from the near detector allows for the simultaneousconstraint of the T2K beam flux and the cross section. This allows a global likelihood forthe oscillation parameters to be constructed using the external data (for constraining nuisanceparameters), the ND280 data, and the SK data. This depends on the nuisance parametersfrom the flux and cross-section models, as well as detector systematics at the near and fardetectors. As maximizing this global likelihood is computationally impractical, sequentiallikelihood maximization is used instead. This allows each set of data to be handled as separatelikelihoods, greatly decreasing the complexity of each individual fit.The sequential likelihood maximization has essentially three separate steps, one each forthe external, ND280 and SK data:1. The external data is analyzed to produce the prior constraints on the various parame-ters. For the detector parameters, this is in the form of various studies on the systematic53uncertainties, described for ND280 in Chapter 5.2. The ND280-specific likelihood is fit using the near detector data, which can be maxi-mized separately from the oscillation fit at SK and is described in more detail this section.This uses the prior constraints found in the previous step as the prior constraint terms.The fit output is the fitted flux and cross section parameters, along with their covariance,to be used as a prior at SK.3. The fitted values from the ND280 fit are then used as part of the constraint term for theoscillation likelihood which uses the SK data.The uncertainty on the predicted number of events at SK is shown in Table 3.1 and Ta-ble 3.2 for the previous FGD-1 only analyses[57]. The uncertainty improvement to the fluxand ND280-constrained cross-section parameters for the ν¯µ event rate is due to significantimprovements in the flux model and prediction – where the ν¯µ prediction had a 9.2% uncer-tainty from the flux and ND280-constrainable cross-sections pre-ND280 fit, the νµ predictionhas an uncertainty of 21.8% from the same source. However, both the νµ and ν¯µ predictionsshow significant improvement to the overall uncertainty once the ND280 constraint is included.This prediction comes from the improved constraints provided by the near detector fit, as theND280 fit constrains the convolution of the flux and cross-section as a function of energy andthis constraint can be carried over to SK due to similar energy spectra1.However, even with the ND280 constraint to the flux and some cross-section parameters,there remains a large contribution that cannot be reduced with the near detector. This givesan uncertainty of 5.0% for the νµ prediction and an uncertainty of 10% for the ν¯µ prediction,larger than the contribution from the ND280 constrained parameters. This is due to the effectof oxygen-only cross section uncertainties, which could not be constrained with the previousnear detector fits due to differences in target material. For these parameters, relatively largeuncertainties based on theoretical predictions and external experimental measurements are used1Even for the ν¯µ prediction, where the flux uncertainty is significantly reduced prior to the ND280 fit, theadditional constraint provided by the near detector fit gives an significant improvement on the prediction.54no ND280 Constraint With ND280 ConstraintND280-Constrained Parameters 21.8% 2.7%ND280-Unconstrained Parameters 5.0%SK FSI + SI 3.0%SK Systematics 4.0%Total 23.5% 7.7%Table 3.1: Fractional error on the prediction for the number of νµ events at SK, with andwithout the ND280 constraint from the previous neutrino mode analysis in 2014.Uncertainties shown are calculated from the RMS of the event rates when 10000parameter variations are thrown[58]. ND280-unconstrained parameters are the 2p-2h, pF , EB, CC coherent, Isospin=12 Background and νe/νµ cross-section parameters.ND280-constrained parameters are the flux parameters and all other cross-sectionparameters.no ND280 Constraint With ND280 ConstraintND280-Constrained Parameters 9.2% 3.4%ND280-Unconstrained Parameters 10.0%SK FSI + SI 2.1%SK Systematics 3.8%Total 14.4% 11.6%Table 3.2: Fractional error on the prediction for number of ν¯µ events at SK, with andwithout the ND280 constraint from the previous antineutrino mode analysis in 2015.Uncertainties shown are calculated from the RMS of the event rates when 10000parameter variations are thrown [59]. ND280-unconstrained parameters are the 2p-2h, pF , EB, CC coherent, Isospin=12 Background and νe/νµ cross-section parameters.ND280-constrained parameters are the flux parameters and all other cross-sectionparameters.instead to set the uncertainty and prior values. The major change for the analysis presented hereis the ability to fit the oxygen-only cross-section parameters by way of samples that includeoxygen as a target material in order to reduce that uncertainty.3.1.3 Oscillation Fit At Super-KamiokandeThere are two main oscillation analysis methods in use at SK: a hybrid frequentist-Bayesianmethod to deal with nuisance parameters, and MaCh3, which is a fully Bayesian analysis anduses Markov chain Monte Carlo[60]. MaCh3 fits the near and far detector data simultaneouslyand will not be discussed here, as it does not use the sequential ND280 fit – SK fit method that55is the main focus of this thesis. This Markov chain fit was developed later than the originalnear-to-far extrapolation method discussed here and must use the hybrid method for validationdue to computational reasons. While this method fits a global likelihood rather than sequentiallikelihoods, this does not have a significant impact on the fit results. Additionally, the FGD 2selections described in this thesis have been incorporated in to the MaCh3 fit as well.The SK fits use the 1-ring e-like sample for the νe appearance fits and the 1-ring µ-likesample for νµ disappearance fits. As the SK selection does not distinguish between neutrinoand antineutrino events, the samples are also split into FHC and RHC subsamples. The SKdata itself can be binned in two ways: in Ereco, the reconstructed neutrino energy, and θl , thereconstructed lepton angle, or p, the lepton momentum, and θ , the scattering angle with respectto the beam direction.The oscillation fit maximizes a binned likelihood:L(Nobse ,Nobsµ ,~o, ~f ) = Lmain(Nobse ,Nobsµ ,~o, ~f )×Lsys(~f ) (3.1)where ~o are the oscillation parameters (sin2(θ23), |∆m232|, sin2(2θ13) and δCP) and ~f are thenuisance parameters – this encompasses the various cross-section model parameters, the fluxparameters, the SK detector systematics, and final state interaction and secondary interactionmodel uncertainties. The likelihood is separated into two parts, where Lmain is the term com-paring the data with the event rates predicted with the oscillation and nuisance parametersand Lsys is a Gaussian penalty term that only depends on the nuisance parameters. The actualminimized quantity in the fit is −2lnL, separating the two likelihood contributions.In the hybrid fit approach, the nuisance parameters are marginalized by integrating the like-lihood over the nuisance parameter values. In practice, this means creating 10000 throws of thenuisance parameters according to their priors and fitting these to produce a grid of likelihoodsfor the various parameter sets. The fit is then marginalized by picking the set of parameterswith the lowest χ2. In addition to this, the oscillation parameters can be marginalized over,56Parameter Prior Type Boundssin2θ23 Uniform [0.3;0.7]sin2 2θ13 reactor Gaussian 0.085±0.005sin2 2θ13 T2K Uniform [0;0.4]sin2 2θ12 Gaussian 0.846±0.021|∆m232| (NH) or |∆m213| (IH) Uniform [2;3]×10−3 eV2/c4∆m221 Gaussian (7.53±0.18)×10−5 eV2/c4δCP Uniform [−pi;+pi]Mass Hierarchy Fixed NH or IHTable 3.3: Priors used for oscillation parameters when marginalizing. From [61].in order to produce 2D intervals. In that case, the oscillation parameters being marginalizedare thrown according to the parameter type and range given in Table 3.3 with the exception ofmass hierarchy, which is instead fixed to either normal or inverted hierarchy. Once the nuisanceparameters have been marginalized over, the ∆χ2 for the marginalized likelihood is minimizedto fit the best fit oscillation parameters.As the nuisance parameters are marginalized, there is no exact set of best fit parametersthat correspond to the best fit spectra. Instead, the best fit spectra is produced using the thrownnuisance parameters:N jmarg =∑ni=1 L(Nobsi ,~o, ~fi) ·N jpred(~θb f , ~fi)∑ni=1 L(Nobsi ,~ob f , ~fi)(3.2)where n is the number of nuisance parameter throws, ~ob f is the best fit oscillation parameters,and ~fi is the ith set of thrown nuisance parameters..3.2 The ND280 Fit3.2.1 The Maximum Likelihood Fit MethodThe near detector fit maximizes the likelihood for the flux, cross-section and detector param-eters given the ND280 data. All parameters are fit simultaneously, which allows there to becorrelations between the fitted flux and cross-section parameters. This uses a binned likelihood,where the ND280 data is separated into bins which depend on topological sample and the re-57constructed kinematics of an event. The samples used are described in detail in Chapter 4;there are three samples for neutrino mode data and four for antineutrino mode data. Withineach sample, the data is binned by the reconstructed momentum, p, and the reconstructed an-gle, cosθ .The ND280 data is binned by sample type, described in Chapter 4, and reconstructed kine-matic variables p and cosθ , allowing the fit to constrain both the overall event rate at SK andthe neutrino energy distribution. The p – cosθ bins were chosen in order to ensure that eachbin had sufficient statistics for a χ2 fit and have finer binning in regions with a high event ratein order to obtain more shape information. For the individual p – cosθ bins in the fit, the ob-served event rate for both data and MC is expected to follow a Poisson distribution. Therefore,the probability of observing Nobsi events in a bin i, given a predicted event rate Npredi whichdepends on the flux, cross-section and observable normalization parameters isP(Ndatai |N predi ) =(N predi )Ndatai e−NprediNdatai !(3.3)The Poisson likelihood term is a product of this probability over all bins.The flux, cross-section and detector normalization parameters are all modeled as multi-variate Gaussian distributions, with the probability shown in Eq. 3.4, where ~y is the vector ofparameter values, n is the number of parameters, Vy is the associated covariance matrix and ∆~yis the difference between current parameter value and nominal parameter values.pi(~y) = (2pi)n2 |Vy| 12 e− 12∆~y(V−1y )∆~y (3.4)As the flux, cross-section and detector parameters are all independent from each other, eachtype of parameter contributes a separate term to the likelihood.The maximized quantity is the likelihood ratio, where the denominator is evaluated at58N predi = Ndatai , as this corresponds to the maximum value for the numerator:LrND280 =pi(~b)pi(~x)pi(~d)∏i[Npi (~b,~x, ~d)]Ndi e−Npi (~b,~x,~d)/Ndi !pi(~bnom)pi(~xnom)pi(~dnom)∏i[Ndi ]Ndi e−Ndi /Ndi !(3.5)where~b is the flux parameters,~x is the cross-section parameters, and ~d is the detector observ-able normalization parameters at ND280. This allows for comparison between the likelihoodof the predicted values and the maximum possible value. Several of the terms in this ratio can-cel out, such as the determinants of the covariance matrices in Eq. 3.4. The actual minimizedquantity is −2ln(LrND280), which has an approximately χ2 distribution and is referred to as∆χ2ND280 in the near detector fit.For ND280, this log likelihood is composed of two separate independent parts: a Poissoncontribution from the fitted observables and a Gaussian contribution from the fitted parame-ters and their covariance. The full ∆χ2ND280 for the near detector, defined as 2 times the loglikelihood with constant terms dropped is defined as:∆χ2ND280 = 2Nbins∑iN predi (~b,~x, ~d)−Ndatai ln(Ndatai /N predi (~b,~x, ~d))+Eνbins∑iEνbins∑j∆bi(V−1b )i, j∆b j +xsec∑ixsec∑j∆xi(V−1x )i, j∆x j+obsbins∑iobsbins∑j∆di(V−1d )i, j∆d j(3.6)Here Nobsi is the number of observed events in the ith p – θ sample bin, Npredi the predictednumber of events in the ith p – θ bin, bi the flux parameters, xi the cross section parameters, anddi the detector observable normalization parameters. Each parameter set has its own covariancematrix V from the prior inputs. N predi is calculated from the nominal prediction for bin i usingthe detector observable normalization weights, the cross section response functions and theflux weights, which depend on the sample p – cosθ bin, the neutrino interaction mode and thetrue neutrino energy.59The basic fitting routine uses the Minuit algorithm package[62] as part of the ROOT[63]framework. The near detector analysis uses the MIGRAD algorithm for minimizing the ∆χ2described in Eq. 3.6. The postfit uncertainties are estimated using the HESSE algorithm fromMinuit, which computes the second derivatives of the parameters around the minimized valuesto construct the Hessian matrix. The inverse of this gives the covariance matrix for the fittedparameters.3.2.2 Changes from Previous FitThis fit has several differences relative to the old ND280 fit: most significant is the addition ofthe FGD 2 target samples for both neutrino and antineutrino mode selections. As described inSec. 2.5, FGD 1 is comprised of only scintillator target, while FGD 2 is alternating scintillatorand water modules, thus providing a target more similar to that at SK. The samples addedfor FGD 2 do not separate between the carbon and oxygen target interactions, as the watervolumes are uninstrumented and non-statistical differentiation is difficult. Instead, the fit usesthe carbon and carbon + oxygen target samples to effectively do a subtraction analysis to fitthe oxygen cross section parameters. The purpose of this is to reduce the remaining crosssection uncertainties in the previous analysis, as this is the largest source of uncertainty in theoscillation analysis as shown in Tables 3.1 and 3.2.In addition to the new samples, the detector observable normalization parameters have hadseveral changes, including updated detector systematic studies for FGD 2. Because the fit nowincludes two separate targets with potential systematic correlations, the detector parametersmust include FGD 1 to FGD 2 correlation considerations. For the FGD 1 systematics, therehave also been intra-systematic correlations added. While the previous analysis assumed thatthere were no correlations between p – θ or other binning choices within a given detectorsystematic, these have now been estimated and incorporated in the covariance matrix. Thesesystematics and correlations will be described in more detail in Chapter 5.The new oxygen target sample additions allow for some changes to the cross section inputs60as well. In the previous analysis, the fit results for any of the oxygen cross section parameterswere not passed to the oscillation fits as SK, and also the 2p-2h parameter was not separatedinto different versions for oxygen and carbon targets. Because the fit is to effective cross sec-tion parameters, there could potentially be a difference in response between different targetmaterials so for this analysis the 2p-2h parameter has been fit as two separate parameters de-pending on target. The model now also includes a 2p-2h ν¯ normalization parameter, motivatedby several neutrino cross section models having different scaling for neutrino and antineutrinoevents.3.3 Fit Parameters and Prior ConstraintsThe near detector fit parameters come from three sources: the T2K beam flux, the neutrinointeraction cross-section model and the detector systematic parameters. Of these, the neutrinoand antineutrino flux, as well as relevant cross-section parameters at SK, are passed on to theoscillation fits. This section describes the various inputs that are used in the analysis.3.3.1 Flux ParametersA key part of the near detector fit is its ability to constrain and measure the flux to improve theevent rate predictions at SK. The flux itself depends on many factors, such as the beam directionand intensity, and proton and secondary particle interactions in the target volume; because ofthis, the beam flux simulation is complex and involves multiple steps. The beam flux predictionis based on the Monte Carlo simulation of the T2K beam, tuned with external information fromthe NA61/SHINE experiments. The beam simulation uses FLUKA 2011.2b for the hadronicinteractions in the target[64] and JNUBEAM, a GEANT3 simulation of the T2K beam and neardetectors[47] developed by T2K, for the rest of the simulation. JNUBEAM handles particlepropagation and all interactions outside of the beam target volume, while FLUKA is usedbecause it shows better agreement with NA61 data than the GEANT3 interaction model[64].Hadronic interactions are then reweighted using thin target data for various pion and kaon61interactions, primarily from the NA61/SHINE experiments at CERN[55][65].The uncertainties on the flux prediction come from uncertainties on the hadron models aswell as uncertainty on the beam direction. For the flux at low energy, the dominant uncertaintycomes from uncertainties on the production cross sections, pion production multiplicities andsecondary nucleon production, while for high energies, the predominant uncertainties comefrom kaon production. The uncertainty on the secondary nucleon production is the largestcontributor to the flux uncertainty at low energy. As the primary source of wrong-sign neutrinosin the RHC flux comes from interactions outside the target, the uncertainties on the secondarynucleon production are reduced by using the NA61/SHINE 2009 data[64]. The target scalinguses data from the HARP collaboration[66], which helps constrain the uncertainties on pionproduction for pion rescattering.The near detector fit uses a flux model that is binned in neutrino energy and into νµ , ν¯µ , νeand ν¯e at ND280 and SK. As the beam power and beam conditions have differed over the T2Krun time, the flux model predictions are also split by run period. The forward horn current fluxand reverse horn current flux have separate flux parameters. In the fit, these are normalizationparameters which allow the flux prediction to vary around the nominal value from the sim-ulations and a covariance matrix constructed from the uncertainties and correlations of eachflux normalization parameter. The flux model uses separate parameters for the flux at ND280and the flux at SK to account for differences in the spectra between the near and far detector.ND280 is positioned close enough to the T2K beam source that the neutrino beam does notlook like a point source, while SK is far enough away that it is reasonable to model the beamas one. There are correlations between the ND280 flux parameters and the SK flux parameters,allowing measurements of the flux at ND280 to change the flux at SK. The source of thesestrong correlations is that the uncertainties on the underlying production parameters, such asproduction cross-sections, affect the flux spectra at near and far detector in similar ways.The flux parameters are binned in energy, with units of GeV, for each flavor and beam modeas follows[67][57] and truncate at 30 GeV for both ND280 and SK flux parameters. The bin62boundaries for the flux parameters are:• FHC νµ and RHC ν¯µ : 0.0, 0.4, 0.5, 0.6, 0.7, 1.0, 1.5, 2.5, 3.5, 5.0, 7.0, 30.0• FHC ν¯µ and RHC νµ : 0.0, 0.7, 1.0, 1.5, 2.5, 30.0• FHC νe and RHC ν¯e : 0.0, 0.5, 0.7, 0.8, 1.5, 2.5, 4.0, 30.0• FHC ν¯e and RHC νe : 0.0, 2.5, 30.0There are 100 flux normalization parameters in total, with 50 for the flux at ND280 and 50 forthe flux at SK. The correlations between flux energy bins is shown in Figure 3.1 – this showsthe flux as highly correlated between neutrino and antineutrino as well as between fluxes atND280 and SK. Predicted flux and fractional uncertainties on the flux at ND280 are shownin Figures 3.2 and 3.3. Predicted flux and fractional uncertainties for SK are shown in Ap-pendix A.63Parameter number0 10 20 30 40 50 60 70 80 90 100Parameter number0102030405060708090100Correlation-1-0.8-0.6-0.4-0.200.20.40.60.81Figure 3.1: Correlations between the flux normalization parameters. The binning is:ND280 FHC νµ : 0 -10, ND280 FHC ν¯µ : 11 – 15, ND280 FHC νe: 16 – 22,ND280 FHC ν¯e: 23 – 24, ND280 RHC νµ : 25 – 29, ND280 RHC ν¯µ : 30 – 40,ND280 RHC νe: 41 – 42, ND280 RHC ν¯e: 43 – 49, SK FHC νµ : 50 – 60, SK FHCν¯µ : 61 – 65, SK FHC νe: 66 – 72, SK FHC ν¯e: 73 – 74, SK RHC νµ : 75 – 79, SKRHC ν¯µ : 80 – 90, SK RHC νe: 91 – 92, SK RHC ν¯e: 93 – 9964 (GeV)νE-110 1 10Fractional Error00.10.20.3µνND280: Positive Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal ErrorPrevious T2K Results(a) ND280 FHC νµ flux uncertainty and prediction. (GeV)νE-110 1 10Fractional Error00.10.20.30.4µνND280: Positive Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal ErrorPrevious T2K Results(b) ND280 FHC ν¯µ flux uncertainty and prediction.Figure 3.2: Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy.65 (GeV)νE-110 1 10Fractional Error00.10.20.3µνND280: Negative Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal Error(a) ND280 RHC νµ flux uncertainty and prediction.[64] (GeV)νE-110 1 10Fractional Error00.10.20.3µνND280: Negative Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal Error(b) ND280 RHC ν¯µ flux uncertainty and prediction.Figure 3.3: Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy.663.3.2 Cross Section ParametersThe cross section model parameters are implemented in two different ways: normalizationparameters which apply a normalization weight to certain events, and response parameters,which can alter the particles produced in an interaction, and the pµ and cosθµ distributions.The normalization parameters work much like the flux parameters – a multiplicative factor isapplied to all events in the applicable interaction channel, without any dependence on energyor other observables. The response parameters are more complicated, as these are cross sectionparameters that are part of the theoretical cross section calculation. The most exact way toincorporate changes to the cross section calculations would be to rerun the entire T2K analysisstructure, from generating neutrino interactions in NEUT to reconstruction and selection, foreach event on each step of the fit minimization. This is of course not computationally feasible;instead weights for the parameters are calculated individually for each event and saved as cubicsplines, with the weight as a function of the parameter value. This allows the computationallyintensive part to be run prior to the fit itself and for the appropriate weights to be quicklyinterpolated from the saved splines during minimization.The cross section parameters used in the T2K near detector fit are listed in Table 3.4. Pa-rameters are listed along with their range of validity, NEUT nominal values and the prior valuesas calculated by the Neutrino Interaction Working Group on T2K[68]. The prior constraintsand values on the CCQE and CC resonant parameters comes from fits to cross section data fromMiniBooNE[69] and MINERνA[70][56], though the CCQE prior constraints are not used inthe fit for reasons described below.The final state interaction parameters control hadron transport within the nucleus after aneutrino interaction, which is simulated as a cascade where the pion is propagated through thenucleus in steps, allowing for multiple interactions before the particle exits the nucleus. Thesefinal state interactions (FSI) allow events to migrate between different topologies and for par-ticle kinematics in the event to change. There are six FSI parameters used in the near detectorfit for T2K, for pion final state interactions and are split into parameters that apply for high en-67Parameter Validity Range Prior Mean NEUT Nominal Error TypeFSI Inel. Low E all 0.0 0.0 0.41 responseFSI Inel. High E all 0.0 0.0 0.34 responseFSI Pion Prod. all 0.0 0.0 0.50 responseFSI Pion Abs. all 0.0 0.0 0.41 responseFSI Ch. Exch. Low E all 0.0 0.0 0.57 responseFSI Ch. Exch. High E all 0.0 0.0 0.28 responseMQEA (GeV/c2) all 1.20 1.20 – responsepF 12C (MeV/c) 200 - 275 217 217 – response2p-2h 12C all 100% 100% – normEB 12C (MeV) 12 - 42 25 25 9 responsepF 16O (MeV/c) 200 - 275 225 225 – response2p-2h 16O all 100% 100% – normEB 16O (MeV) 12 - 42 27 27 9 responseC5A(0) all 1.01 1.01 0.12 responseMRESA (GeV/c2) all 0.95 0.95 0.15 responseIsospin=12 Background all 1.30 1.30 0.20 responseCC νe/νµ all 1.0 1.0 0.04 normCC ν¯e/ν¯µ all 1.0 1.0 0.04 normCC Other Shape all 0.0 0.0 0.40 responseCC Coherent all 1.0 1.0 0.30 normNC Coherent all 1.0 1.0 0.30 normNC 1γ all 1.0 1.0 1.00 normNC Other (near) all 1.0 1.0 0.30 normNC Other (far) all 1.0 1.0 0.30 norm2p-2h ν¯ > 0 1.0 1.0 0.50 normTable 3.4: Cross section parameters used for the near detector fit, showing the valid rangeof the parameter, prior mean, nominal value in NEUT, and prior error as providedby the Neutrino Interaction Working Group (NIWG).[67] The type of systematic(response or normalization) is also shown. The CC-0pi parameters (MQEA through EB16O) use the NEUT nominal value for the nominal MC tuning and are fit withoutprior constraints, with the exception of the binding energy EB.[67] Not all listedparameters are used in the fit at SK.68Parameter number0 2 4 6 8 10 12 14 16 18 20 22 24Parameter number024681012141618202224Correlation-1-0.8-0.6-0.4-0.200.20.40.60.81Figure 3.4: Correlations between cross section parameters listed in Table 3.4. Each bincorresponds to a single cross section parameter, and bins are in the order listed inTable 3.4.ergy and ones that apply for low energy. These are implemented as response parameters whichmodify the various interaction cross-sections and are not used in the oscillation fits at SK2. Thecurrent implementation uses pion scattering data from the DUET experiment[71] to estimatethe prior and prior constraints for each FSI parameter. The FSI parameters which apply in thehigh energy region are inelastic scattering (high energy), pion production and charge exchange(high energy). The low energy region uses the inelastic scattering (low energy), pion absorp-tion, and charge exchange (high energy) parameters, as the charge exchange and scattering aretuned separately for the two regions.• FSI Inelastic Scattering: final state has a single pion of the same charge as the initialpion. These parameters apply to both inelastic and elastic scattering, as the model used2The FSI effects are instead dealt with as part of the SK detector covariance.69at T2K does not differentiate between these[72]. There are separate high energy and lowenergy versions for this parameter.• FSI Charge Exchange: final state has a single pion of the opposite charge from the initialpion (pi+ from pi− and pi− from pi+) or a single pi0. There are separate high energy andlow energy versions for this parameter.• FSI Pion Absorption: Final state contains no pion. This parameter only applies to lowenergy interactions.• FSI Pion Production: Final state includes 2 or more pions. This parameter only appliesto high energy interactions.The CCQE parameterization uses the relativistic Fermi gas nuclear model (RFG), and in-cludes the Fermi momentum pF and the binding energy Eb as response parameters. As pFand Eb depend on the nuclear target, there are separate parameters for oxygen and carbon in-teractions. The CCQE model also includes the Nieves model for np-nh interactions[43][42];the 2p-2h parameters for 12C and 16O are overall normalizations for these interactions; theseare also split by interaction target. The 2p-2h 12C parameter is used for all non-oxygen 2p-2h interactions. In addition, this analysis adds an extra normalization for antineutrino 2p-2hinteractions[68]. Of the CCQE cross section parameters, the near detector fit passes MQEA , 2p-2h 16O, pF 16O and Eb 16O as inputs to the oscillation analysis, as the far detector does notcontain carbon as a target material.The CC-1pi interactions are parameterized by three main parameters: MRESA , C5A(0) and theIsospin = 12 background scaling, all three of which are response parameters. The MRESA param-eter is a correction to the axial mass for resonant 1pi interactions and C5A(0) is the dominantaxial form factor for resonant pion production. The Isospin = 12 background is the scaling ofthe nonresonant background for single pion processes which in NEUT is assumed to consistentirely of I = 12 events. These parameters use prior constraints from the MiniBooNE and70MINERνA cross section data[68]. The near detector fit results for all three parameters areused in the oscillation fits at SK.Of the cross section parameters that apply to interactions other than CCQE or CC-1pi , onlythe CC other shape, from deep inelastic scattering, is implemented as a response parameter.The rest are normalization parameters for various interaction types. Unlike the CCQE andCC-1pi model parameters, these do not have priors calculated from external data fits. Instead,the NEUT prior value is used and an uncertainty estimated for each parameter by the T2Kneutrino interaction modelling working group. The CC Coherent normalization parameter is anormalization on carbon and oxygen CC coherent interactions; the prior uncertainty is derivedfrom comparisons with MINERνA data[56]. Of the parameters listed in Table 3.4, the NCOther (far), NC 1γ , CC νe/νµ and CC ν¯e/ν¯µ parameters are included by the near detector fit,but due to lack of sensitivity there is no change in parameter values and widths. The NC Other(far) and NC Other (near) parameters are overall normalizations for NC interactions at SK andND280 respectively. Due to differences in the NC Other contribution to samples at SK andND280, the NC Other (near) fitted parameter is not propagated to the oscillation fit[68]. TheNC1γ parameter is an overall normalization on the NC1γ interaction, separate from the CCOther normalization, and is a background at SK. The CC νe/νµ and CC ν¯e/ν¯µ parameters areincluded to account for potential differences in the νe/νµ cross-section ratio and ν¯e/ν¯µ cross-section ratio. The priors and uncertainties listed in the table are those used in the oscillationfits at SK.Unconstrained Charged-Current Quasielastic ParametersThe external data fits using MiniBooNE and MINERνA provide constraints on the variouscross section parameters. However, the constraints for the CC0pi cross section parameters arenot used in either the data fit or the various validation studies. The reasoning behind this is:1. The MiniBooNE covariance matrix used in the external fits was incomplete due to lackof bin correlations in the data release, and changes to this covariance were found to cause71significant changes in the external fit.2. The signal definitions for MiniBooNE and MINERνA were not consistent with eachother, as MiniBooNE subtracts pi-less ∆ decay, a CC0pi process, from the data usingNUANCE while MINERνA does not.3. There is poor agreement between the MiniBooNE and MINERνA data sets, and diffi-culty in consistently fitting both with the current models and parameters.4. The ν¯ 2p-2h normalization parameter in the ND fit is not implemented in the externaldata fit and therefore cannot provide a prior.Therefore, the CC0pi cross section parameters MAQE , pF12C, pF 16O , 2p-2h 12C and 2p-2h 16O are left unconstrained in the fit (that is, using a flat prior) and use the NEUT nominalvalues for calculating the nominal Monte Carlo rates. The ν¯ 2p-2h normalization parameteris also unconstrained in the fit and uses a nominal value of 1, corresponding to equal 2p-2hcontributions for neutrinos and antineutrinos. When unconstrained, a parameter is allowed tovary in the fit, and is included in the covariance calculations after minimization but does notcontribute to the overall ∆χ2 term, allowing the parameter to vary across its entire valid rangewithout penalty.3.3.3 The Observable Normalization MatrixThe complex structure of the near detector and selection leads to a complicated set of detectorsystematic parameters; these must be accounted for and marginalized over in the near detectorfit. These detector systematic parameters and their implementation are described in detail inChapter 5. However, the current implementation of the detector systematic parameters is com-putationally expensive to fit on an event-by-event basis, where the various detector weightsand variations for each individual event would need to be recalculated at every step of mini-mization. For this reason, the near detector fits a set of observable normalization parametersdescribing the effect of the systematic variations on the event rates for each sample.72The input consists of a set of normalization parameters for each fitted sample binned in p–θ and an associated covariance matrix describing the overall uncertainties and correlations onthe observable normalization parameters. These normalization parameters and their covarianceare calculated from 2000 variations of the detector systematic parameters and how these affectthe event rates in each bin at ND280. The covariance also includes shape uncertainties fromother sources that can be binned in p–θ , such as the Monte Carlo statistical error and a shapeuncertainty for 1p-1h effects. Monte Carlo statistical errors are the uncertainty associated withthe size of the generated Monte Carlo which is reweighted in the fit, due to the finite size ofthe Monte Carlo used in generating the matrix. These errors are added as independent errorterms on the diagonal of the observable normalization covariance matrix. The previous neardetector fit had a total of 290 detector observable normalization parameters, while the currentfit has 580, due to the addition of the FHC νµ and RHC ν¯µ and νµ samples for FGD 2. Thishas the effect of slowing down the fit simply due to the added difficulty of fitting such a largeparameter space. The binning details are described later in Section 5.4.1.The uncertainty on the model used for 1p-1h kinematics is also included as an additionalindependent covariance added to the observable normalization covariance matrix. This uncer-tainty covers the lepton kinematic model differences between the NEUT generator, which iswhat is used to generate the T2K Monte Carlo, and the Nieves generator. NEUT uses a Rela-tivistic Fermi Gas (RFG) model to describe these interactions while Nieves uses a Local FermiGas model. Other models include Spectral Function and Relativistic Mean Field, as describedin Section 1.4. In this case, the 1p-1h Nieves model was tested against the current modelused for fitting at T2K. As fake data studies comparing the two different 1p-1h implementa-tions have shown model choice to have a significant impact on the fit parameters at ND280and in the oscillation fits[73], this model uncertainty is now incorporated into the normaliza-tion parameters as an additional shape covariance. The 1p-1h covariance was calculated asthe bin-by-bin difference between the Nieves model prediction and the current NEUT model73, FHCpiFGD 1 CC-0, FHCpiFGD 1 CC-1FGD 1 CC-Other, FHC CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1  CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1 , FHCpiFGD 2 CC-0, FHCpiFGD 2 CC-1FGD 2 CC-Other, FHC CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2  CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2 , FHCpiFGD 1 CC-0, FHCpiFGD 1 CC-1FGD 1 CC-Other, FHC CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1  CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1 , FHCpiFGD 2 CC-0, FHCpiFGD 2 CC-1FGD 2 CC-Other, FHC CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2  CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2 ) ijsign(V×| ij|V-0.100.10.20.30.4Figure 3.5: Detector covariance matrix, plotted as sign(Vi j)×√|Vi j| for clarity. Maxi-mum values are truncated at 0.45 for display purposes. Labels indicate where binsfor a given sample start. The bins within each sample are ordered in increasing mo-mentum intervals, each containing all angular bins from backward going to forwardgoing.prediction:V1p−1h,i j = (NNieves,i−NNEUT,i)× (NNieves, j−NNEUT, j) (3.7)where NNieves,i is the predicted event rate in bin i using the Nieves 1p-1h model, and NNeut,i isthe predicted event rate in bin i using the nominal NEUT model.Like the flux and cross section parameters, the observable normalization parameters areassumed to behave in a Gaussian manner and are included in the likelihood term shown in74Eq. 3.6, and are allowed to correlate with flux and cross section parameters in the fit. Asthe detector normalization parameters are constraints on the near detector event rates, theseparameters are not used in the oscillation fits at SK. Instead, SK uses its own set of detectorsystematic parameters.75Chapter 4The Tracker Selection4.1 Introduction and MotivationThe strength of the near detector fit depends on the events selected at ND280, and how theneutrino interaction types can be identified. The primary signal for neutrino interactions atthe T2K beam energies is the CCQE interaction, along with other charged current interactionssuch as CC Resonant and Deep Inelastic Scattering. To properly constrain the cross sectionsand the beam flux, being able to identify neutrino events and their interaction modes well iskey. Unlike the SK event selections, ND280 is able to include specific samples for non-CCQEtopologies as well as separate selections for neutrino and antineutrino events.The selection used for the near detector fits is based on the tracker volume, comprised of thetwo FGDs and three TPCs. As described previously, the FGDs serve as the target volume, withFGD 2 providing oxygen targets similar to SK, and the TPCs allow for momentum and particleidentification. The previous ND280 fit, used as the input for the previous ν¯µ disappearance[74]and the joint νµ disappearance and νe appearance results[30], was FGD 1-only and did nothave any selection for events occurring in FGD 2.764.2 Data and Monte Carlo Inputs to the SelectionsThe data collected at T2K is separated by run period as well as detector configuration. For thisanalysis, neutrino beam data from Runs 2, 3, and 4 and antineutrino beam data from Runs 5and 6 were used. Data from Run 1 had issues with calibration and is missing the top trackerECal. As the ECal was originally intended to be used as part of this selection, this could presentdifficulties in using this data. Therefore data from this beam period is not used in the fit, asRun 1 only represents a small amount of statistics compared to later run periods and shouldhave only a small impact on results.The ND280 Monte Carlo is generated using the NEUT generator[75] for the neutrino inter-actions and simulated using the ND280 geometry in GEANT4. The MC is generated separatelyfor each run period and P0D configuration, as the P0D water bags have been either filled withwater or air at various points during ND280 running. As this increases the interaction ratesin the P0D due to higher density, these must be simulated separately. There are two types ofMonte Carlo events generated: νµ or ν¯µ interactions in the magnet volume and sand muonevents. Sand muon events are interactions of the neutrino beam with the sand surrounding thenear detector. This is intended to simulate potential outside background events in the data.Both the MC and the data are initially saved as files which contain the raw readout infor-mation from the detectors, real or simulated. Before reconstruction, gain calibration constantsare applied, and dead and noisy channels removed from these files. The now calibrated filesare then processed into ROOT file format compatible with the ND280 reconstruction software.4.3 ND280 Detector Tracker ReconstructionThe ND280 reconstruction uses an external reconstruction toolkit, RecPack, for various fitting,propagation and matching routines[76]. RecPack provides two fitting algorithms, the leastsquares fit method and the Kalman filter method [77]. RecPack is designed to work with com-plex multi-volume geometries like ND280 and interfaces with the ND280 oaRecPack package,which provides RecPack with a simplified version of the ND280 geometry implemented in77Run Period Data PoT MC PoTFHCRun 2, Water In 4.2858e19 1.2015e21Run 2, Water Out 3.55045e19 9.215e20Run 3b, Water Out 2.146e19 4.48e20Run 3c, Water Out 1.24821e20 2.63e21Run 4, Water In 1.62818e20 3.499e21Run 4, Water Out 2.84438e20 3.4965e21RHCRun 5c, Water In 4.29779e19 2.0825e21Run 6b, Water Out 1.27374e20 1.4103e21Run 6c, Water Out 5.02038e19 5.321e20Run 6d, Water Out 6.39851e10 6.941e20Total FHC PoT 5.82e20 1.22e22Total RHC PoT 2.84e20 4.72e21Table 4.1: Total protons on target (PoT) for data and Monte Carlo, broken down by runperiod and P0D water status. Monte Carlo PoT includes POT from sand muon MonteCarlo. Data PoT is after the basic data quality cuts described in Sec. 4.5.ROOT. This simplified geometry models all volumes as boxes, and combines parts into singlevolumes where appropriate. The simple geometry also differentiates between active and deadmaterial for reconstruction. The material properties used are the average properties in a givenvolume, so the FGD water module material would be an average of the material of the moduleitself and the properties of the water it is filled with.The samples used in the near detector fit are designed using only information from thetracker volume at ND280 - the TPCs and FGDs. While the ND280 reconstruction does includeinformation from other detectors such as the ECal and SMRD, these are not used in selectionsfor the near detector fits and will not be described here.4.3.1 Time Projection Chamber Track ReconstructionThe TPC track reconstruction starts by clustering hit waveforms in a TPC by way of connec-tivity within the Micro Mega (MM) pad columns or rows of a TPC. Each cluster is made up ofhit waveforms that are consecutive in both position and time. TPC tracks consist of a series ofthese overlapping row or column clusters.78To identify TPC hit clusters, the reconstruction algorithm starts with the raw hit informationin the form of waveforms, as a single hit is stored as a set of continuous charges on a TPC padin time. Clusters are grouped by rows or columns of pads and are considered to be eitherhorizontal or vertical clusters respectively. Clustering is done in XY planes, and hit waveformsare required to be overlapping in time and consecutive in position. To be considered to overlapin time, the ADC values above the pedestal threshold must be close in time. The hits mustalso be within a distance d of each other, which by default is the TPC pad pitch plus the gapbetween the pads in the cluster direction. This distance also takes into account whether or notwaveforms are separated by the cathode, and is increased for clusters which already have alarge spread. This allows clusters to pick up potentially isolated waveforms from longer tracks.Once the clusters are grouped, TPC tracks are constructed using pattern recognition. Firstoverlapping column or row clusters are matched into TPC segments. Segments are defined asa pair of clusters in two consecutive columns or rows; clusters are not required to be uniqueto segments and can be considered part of multiple segments. The reconstruction looks forconsecutive clusters that are close in position to each other, and overlapping in time. Fornon-consecutive clusters, segments can be made from clusters with waveforms that overlapin time. Once segments have been constructed, the reconstruction attempts to pair segmentsby checking for segments that share clusters. This process can be extended to more than twosegments, and produces the TPC tracks. Tracks made through combining segments often misshigh angle tracks crossing the vertical gap between pads. For this reason, once the patternmatching has constructed shorter tracks out of the TPC segments, the RecPack Kalman filteralgorithm is used to look for matches among the TPC tracks and construct longer ones. Newtracks constructed this way are refit with the likelihood algorithm.The TPC provides the main particle identification (PID) for the ND280 tracker and allowsthe selection to differentiate between muon, pion, electron and proton tracks, with muons andpions being difficult to differentiate in the TPCs[78]. To do this, the reconstruction uses theenergy loss due to ionization of charged particles to provide a likelihood for each particle79hypothesis. Each track can consist of up to 72 clusters for a track that fully crosses a TPCand each cluster has a total energy computed from the sum of the charges of all hits in thatcluster. The expected distribution of the energy for a cluster is very wide, with a long very highenergy tail due to each TPC being thin. To account for this, the total energy is computed usinga truncated mean consisting of the lower 70% of the clusters in that track[78][79]. In addition,a TPC track needs to have at least 18 nodes for this to be a reliable method of calculating theenergy deposition. This total energy deposit is compared with the expected energy deposit fora given particle to create a particle ID likelihood for that particle. This takes track momentuminto account and length into account. These likelihoods are used in the selection steps describedlater in this chapter.4.3.2 Fine Grained Detector ReconstructionThe FGD reconstruction consists of two separate major parts: reconstructing tracks containedwithin the FGDs and reconstructing FGD tracks with a TPC component. For this reason, theFGD reconstruction is run after the TPC reconstruction is finished and the fitted TPC tracks aresaved. This allows the FGD reconstruction routines to use the TPC tracks for matching FGDhits into TPC-FGD tracks.The basic information in the FGD consists of saved hits, each with a time and XZ or XYposition. The first step of the FGD reconstruction is to sort these hits into timebins, as de-scribed in more detail in Section 4.5.3. Each timebin consists of a set of FGD hits where eachconsecutive hit is less than 100 ns from the previous hit and are reconstructed independentfrom other time bins. This means that no hits are shared between time bins. Once the hits areall sorted into time bins, the FGD reconstruction steps are performed on each timebin individ-ually, starting with the TPC-FGD matched track reconstruction. FGD-only reconstruction andparticle identification are performed once the TPC-FGD reconstruction is finished.80The Time Projection Chamber – Fine Grained Detector Matched ReconstructionThe TPC-FGD reconstruction functions by looping over all reconstructed TPC tracks and at-tempting to incrementally match FGD hits to the TPC track to produce a FGD track segment.This uses the Kalman filter routine[77] provided by RecPack. To ensure that the FGD hitsbeing matched are correct, the reconstruction checks that the initial time for the TPC track iswithin the time window of the FGD timebin. This initial time for the TPC track is determinedin two ways, depending on whether the TPC track crosses the cathode or not. In the casethat the TPC track crosses the cathode, T0 can be calculated from the time of crossing and themaximal drift time. Otherwise, the initial TPC track time is found by comparing with hits inadjacent detectors using RecPack. If the initial time is outside that, the TPC-FGD matching isnot performed for that track. If the track starts within the time window for that timebin, theTPC track is extrapolated back to the closest layer of hits in the FGD using RecPack.The hit matching is processed layer-by-layer. Starting with the FGD hit with the lowestresidual between the extrapolated TPC track and its position, the reconstruction computes aχ2 using the distance between the extrapolated track and each hit using RecPack routines;this cutoff is the same for X and Y layers. Each time this calculated χ2 is below a set cutoffvalue, the hit is matched to the TPC track, and the Kalman filter seed state recomputed withthe new track information[76]. This is repeated for each hit in the layer, and repeats for thenext FGD layer once done. The extrapolation continues even if there are no matched hitsin a given layer. However, if two or more FGD layers have no matched hits in them, theextrapolation is terminated in order to avoid incorrectly matching hits from other particles.Once the extrapolation is finished either by termination or all hits have been examined, thefinal TPC track and the associated FGD hits are refit to give a consistent reconstructed output.This does allow for a TPC 2 track to be matched to hits in both FGD 1 and FGD 2, creatingan FGD 1 + TPC 2 + FGD 2 track. However, the hits matched with this track cannot be matchedwith tracks in any other TPCs. To allow for multi-TPC tracks, additional reconstruction mustbe performed.81The Fine Grained Detector-Only ReconstructionWhile the primary track from a neutrino interaction in the FGDs is generally high enoughmomentum to leave the target volume and enter other detectors, this is not necessarily true forall secondary particles such as protons. The tracks left by these particles can be fully containedin the target FGD, with no TPC segments to match to. As there can be no dE/dx or curvatureinformation from the TPCs, the FGD must be able to reconstruct all the needed information forthese particles.The input to the FGD-only reconstruction is the unmatched FGD hits from the timebin usedin the TPC-FGD matched reconstruction stage. The initial step takes the unmatched hits anduses a pattern recognition method called SBCAT[76] to create XZ and YZ tracks from segmentsof hits in adjacent layers, similar to how clusters are matched in the TPC reconstruction. Thesereconstructed two-dimensional tracks are fit with straight lines and associated hits with highresiduals are removed. In addition, tracks are checked to ensure no more than one consecutivelayer is skipped, as with the TPC-FGD matched reconstruction. To match XZ and YZ trackprojections together to create a full three-dimensional FGD track, the reconstruction requires:• The most upstream X and Y layers in each track must be adjacent.• The most downstream X and Y layers from each track must also be adjacent.• Both tracks must cross at least four layers.Once an XZ and YZ projection are matched together, the resulting track is saved as an FGD-only track and the particle identification pulls are calculated.For 3D tracks contained in the FGDs, particle identification pulls can be computed throughcomparing track length versus the total track energy [80]. As FGD 2 has less active materialand therefore less complete information about energy deposition, this method is less effectivethan in FGD 1. First the track length is calculated for the 3D FGD-only track by fitting astraight line to each 2D projection (XZ or YZ). This differs slightly for FGD 1 versus FGD2, as FGD 1 uses the middle of the scintillator bar for the z-position, while FGD 2 uses the82middle point of the scintillator bar and adjacent water layer for hit z-position. This is becausethe track could potentially have passed through the uninstrumented water layer next to the barwhere the hit was seen. This gives a worse resolution for FGD 2 than for FGD 1. The totalenergy in the track is calculated from the sum of the charge deposited in the hits that make upthe track, converted from photoelectron units (PEU) measured by the FGD readout to MeV,with corrections for WLS fiber attenuation and Birk’s saturation. As with the track lengthestimation, FGD 2 is less accurate than FGD 1 in calculating the total charge, as energy isdeposited in water layers but cannot be included in the calculation.In general, the momentum estimation for FGD-only tracks is poor, and is not used forselection cuts. After these reconstruction steps, the tracks and associated information are savedand passed to the global tracker reconstruction.4.3.3 Global Reconstruction and Charge DeterminationThe final reconstruction step allows for longer tracks than are created with the initial TPC-FGDreconstruction, as well as track direction correction. While the initial TPC-FGD reconstructionallows for hits in the FGDs to be matched with a track from a single TPC, there are many caseswhere higher energy particles can travel through multiple TPCs before exiting the ND280volume. For example, the plain TPC-FGD reconstruction would miss tracks starting in oneTPC and traveling through an adjacent FGD and TPC, which could be a potential background.To do this, the tracker reconstruction loops through sets of tracks in adjacent TPCs – tracksin TPC 1 and TPC 2, or TPC 2 and TPC 3 – and uses RecPack to extrapolate pairs of tracks.Tracks formed in this manner can have a single TPC track component in each TPC, alongwith associated FGD segements. After this step, the final TPC-FGD and FGD-only tracksare all refit using the Kalman filter from RecPack to ensure that all tracks use the energy-losscorrections from RecPack.The final step in the tracker reconstruction is to apply FGD timing information to determinethe matched TPC-FGD track direction. By default, both TPC and FGD tracks are assumed to83be downstream tracks, with tracks with FGD 1 and FGD 2 components starting from the FGD1 volume. Using the average time of the hits in each FGD for tracks which cross both FGD 1and FGD 2 volumes, a time difference can be computed:∆tFGDs = tavg,FGD1− tavg,FGD2 (4.1)When ∆tFGDs > 3ns, the track is considered to be backwards-going and the saved reconstruc-tion track is flipped. A difference of at least 3 ns is used to prevent accidental flipping offorward going tracks due to intrinsic timing resolution, as there are more forward going tracksthan backwards. At this point, the tracker reconstruction is done and the reconstructed infor-mation is processed into the file format used in the selection and fit.4.4 The ND280 Detector Tracker Selection4.4.1 Selection BasicsAll selected events and samples used in the fit share a few characteristics and are designedto select signal νµ and ν¯µ interactions. Initially, Charged Current Inclusive (CC Inclusive)neutrino or antineutrino interactions are selected, removing all non-CC Inclusive identifiedevents. These selected events are then sorted by final event topology – no additional eventsare removed from the selection at this point. The final event topology is defined as the set ofparticles that leave the interaction nucleus after the interaction. Event topology is chosen overevent mode as it better matches what can be reconstructed in the near detectors, as ND280 onlysees the final state that leaves the nucleus.The main signal for the analysis, and the main signal interaction at SK is the CCQE in-teraction, due to the ease of energy reconstruction. As these types of events only have twooutgoing charged particles, they have a relatively simple topology to look for as the proton isoften not visible due to low energy. This topology is called the CC-0pi topology, where onecharged lepton track is seen and there are no visible secondary pions. Non-CCQE interactions84can also be detected and are categorized into the CC-1pi , and CC Other, which includes allother topologies.To perform the selection, evaluate systematics and store relevant event information, ND280uses a specialized software tool called Psyche. Psyche was made specifically for use withND280 events and uses the reconstructed event information and truth information from theMonte Carlo, and can also use the reconstructed data events. The selections only use recon-structed information from the events, while the systematic calculations can use true event in-formation and are only used for Monte Carlo events. Psyche does not use any fitting routinesitself but is designed to be used within the ND280 fit software in order to select events and isoptimized for speed and low memory use.Each event must have a muon-like track that starts in the defined fiducial volume in eitherFGD and passes into a neighboring TPC. Because of this, the efficacy of the selections dependson how well direction and vertex position can be reconstructed in the near detectors. Selectionsare divided by target FGD, neutrino mode and muon track sign. While the selection cuts usedfor forward horn current beam and reverse horn current mode are different, all selections sharethe same basic structure and similar CC Inclusive cuts.For forward horn current, the ND280 selection differentiates between the CC0pi , the CC1piand the CC Other topologies for νµ , while for reverse horn current (RHC) the selection thereare two separate CC-Inclusive selections for the ν¯µ signal and the wrong-sign νµ background.These are both separated into 1-Track and multi-track samples, as the lower antineutrino modestatistics for CC Inclusive and other reconstruction considerations make it difficult to separatebetween non-CC0pi topologies.• Forward Horn Current Selections:– CC Inclusive∗ CC0Pi∗ CC1Pi85∗ Other Charged Current Interactions• Reverse Horn Current– CC Inclusive Antineutrino Interactions∗ Charged Current Single Track Topologies∗ Charged Current Multi-track Topologies– CC Inclusive Wrong-sign Interactions∗ Charged Current Single Track Topologies∗ Charged Current Multi-track TopologiesCuts used for each selection are shown in Tables 4.2, 4.3 and 4.4 and are described in detailin the following sections. The selected numbers of events in data and Monte Carlo for theseselections are shown in Table 4.5. The numbers shown for the Monte Carlo are using the flux,PoT and cross section reweighting. CCQE weights use the NEUT nominal values.Cut FHC νµ RHC ν¯µ RHC νµData Quality Yes Yes YesTrack Multiplicity Yes Yes YesFiducial Volume Cut Negative track Positive track Negative trackUpstream Veto Yes Yes YesBroken Track Cut Yes Yes YesMuon PID Negative track Positive track, stricter criteria Negative track, stricter criteriaTable 4.2: List of CC Inclusive Cuts for the three main selections. Major differencesbetween cuts are listed here.4.4.2 Fine Grained Detector 2 ConsiderationsWhile the two FGDs are very similar in their makeup and electronics, there are several differ-ences in FGD 2 that impact event reconstruction and sample selections. The largest differencecomes from the fact that FGD 2 contains water modules alternated with the XY scintillatormodules, while FGD 1 only contains scintillator modules. While this allows for interactions86Cut Cut DetailsTPC Positive Pion Tracks n1 = # of TPC-FGD pi+ TracksMichel Electrons n2 = # of Michel electrons identifiedFGD-Only Pion Track n3 = # of FGD-Only Pion TracksTPC Negative Pion Tracks n4 = # of TPC-FGD pi− TracksTPC Electron & Positron Tracks n5 = # of TPC-FGD e+ and e− TracksCC 0pi CC 1pi CC OtherSample Criteria5∑i=1ni = 03∑i=1= 1 and n4+n5 = 03∑i=1> 1 and/or n4+n5 > 0Table 4.3: List of the FHC multipion selection cuts and sample criteria.Cut Cut DetailsTPC-FGD Matched Track n = # of TPC-FGD Matched Tracks1-Track Multi-TrackSample Criteria n = 0 n > 0Table 4.4: List of the RHC multipion selection cuts and sample criteria. The multipionselection cuts are the same for the ν¯µ and νµ , as there is no cut on the charge of thesecondary tracks.on oxygen in FGD 2, determining which events are on oxygen is not necessarily possible. Thewater modules in FGD 2 are uninstrumented and so can only act as target material. In addition,due to this reduction in vertex resolution, determining the exact start position of a track in FGD2 is difficult - not only do tracks go through less active material, but any events occurring in thewater volume will appear to have started in the closest scintillator layer. This means that theselection can only distinguish between carbon and oxygen events statistically rather than on anevent-by-event basis. This reduction in active material also changes the efficiency of FGD-onlyreconstruction as short tracks will leave fewer hits.One improvement that adding in the second FGD offers is inter-detector timing. While pre-vious selections using only FGD 1 primarily used momentum reconstruction for determiningthe direction of tracks in order to select out backwards going events originating in FGD 2 andterminating in FGD 1, the hit timing between the two FGDs can be used instead. As the hittiming between the FGDs is precise to 3 ns, it is accurate enough to determine the direction ofa track by comparing the time of the first hit in FGD 1 to the first hit in FGD 2. This allows87Sample Data ND280 prefit MC predictionFGD1 νµ CC Inclusive (ν mode) 25558 25420.74FGD1 νµ CC0pi (ν mode) 17354 16950.81FGD1 νµ CC1pi (ν mode) 3984 4460.15FGD1 νµ CC Other (ν mode) 4220 4009.78FGD1 ν¯µ CC Inclusive (ν¯ mode) 3438 3506.38FGD1 ν¯µ CC 1-Track (ν¯ mode) 2663 2708.65FGD1 ν¯µ CC N-Tracks (ν¯ mode) 775 797.73FGD1 νµ CC Inclusive (ν¯ mode) 1990 1933.36FGD1 νµ CC 1-Track (ν¯ mode) 989 938.13FGD1 νµ CC N-Tracks (ν¯ mode) 1001 995.33FGD2 νµ CC Inclusive (ν mode) 25151 24454.89FGD2 νµ CC0pi (ν mode) 17650 17211.71FGD2 νµ CC1pi (ν mode) 3383 3616.62FGD2 νµ CC Other (ν mode) 4118 3626.56FGD2 ν¯µ CC Inclusive (ν¯ mode) 3499 3534.33FGD2 ν¯µ CC 1-Track (ν¯ mode) 2762 2729.88FGD2 ν¯µ CC N-Tracks (ν¯ mode) 737 804.45FGD2 νµ CC Inclusive (ν¯ mode) 1916 1860.51FGD2 νµ CC 1-Track (ν¯ mode) 980 943.90FGD2 νµ CC N-Tracks (ν¯ mode) 936 916.61Table 4.5: Observed and predicted event rates for the different ND280 samples in theND280 fits. Predicted event rates include PoT, flux, detector and cross sectionreweighting.both better removal of the background in FGD 2 for forward going FGD 1 events and inclusionof backwards going FGD 2 events in the selected samples.4.4.3 Antineutrino SubsamplesFor the antineutrino beam mode, there are two major considerations to take into account relativeto neutrino beam mode selections. The first is that unlike the neutrino beam, the antineutrinobeam at T2K is has a significant wrong-sign component from neutrinos. While ND280 is ca-pable of using charge identification through curvature in the magnetic field, as described inSection 4.3, in order to distinguish between µ+ and µ− for the outgoing lepton, the selectionat SK is insensitive to charge and relies on the near detector fit to properly constrain this wrongsign background. This means that a CC Inclusive selection for antineutrinos alone is not suffi-88cient for use in an oscillation analysis and instead there are two CC Inclusive selections used,one for signal ν¯µ and the other for the wrong sign νµ events.Additionally, reconstructing all the final state particles in an ν¯µ interaction is not as easyas for a νµ interaction – while secondary particles from the νµ interactions are often highenough in energy to be reliably reconstructed and used in a cut, this is not always the casefor antineutrino events. The RHC samples are instead separated into 1-track and multi-tracktopologies rather than CC-0pi , CC-1pi and CC Other topologies used for the FHC selection.For the RHC ν¯µ selection, this is because while the FHC νµ samples use reconstructed pi+to distinguish between topologies, the analogous pi− tend to be absorbed and are difficult toreconstruct in the FGDs. The wrong-sign νµ events in RHC could use the same selection asFHC νµ however the RHC ν¯µ and νµ selections both use the same sample topologies to reducesystematic effects between the two selections.4.5 Selection Cuts and Results4.5.1 General Data Quality CutsThe most basic cuts are common to all selections at ND280, and are not specific to beam mode,as their purpose is to verify that the events being used are from a time where beam and detectorswere operational and have enough information to run the selection on.Event QualityThis check applies only to events from data, as all MC events are assumed to have good beamand data quality. The main concerns are whether the beam was on and working correctly, andall ND280 detectors were operational. To check this, there are Boolean values saved in the rawdata files which indicate beam status and data acquisition system status for each spill. Thisensures that there are no potential data collection issues, either in the detectors or the datacollection software, that could bias selected events.89Multiplicity CheckBecause the majority of beam-triggered events saved have little to no reconstructible informa-tion, we cut out all events without reconstructed tracks. Events are required to have at leastone reconstructed track in any of the three TPCs and in order to reconstruct a track in a TPC,there must be at least 5 nodes to use for the track. As the event selection depends on having aTPC-FGD matched track, events without any are not useful.4.5.2 The Neutrino Mode Charged Current Inclusive SelectionThe Charge Current Inclusive (CC Inclusive) cuts for the neutrino mode FGD 1 and FGD 2samples are roughly the same, with most differences arising from geometry considerations. Inaddition, the Charged Current Inclusive selection cuts are very similar for all the selectionsused in the ND280 fit, with the main differences being track polarity and highest momentumtrack choice.The Fiducial Volume CutThe initial identifier for a CC Inclusive neutrino event is the reconstruction of a matched TPC-FGD track with negative curvature, where the highest momentum negative matched track isconsidered the lepton candidate for that event. In the case there are two neutrino interactionsclose in time enough to be reconstructed as one event, only the highest momentum one will beconsidered, effectively discarding the second, lower momentum event. The TPC segment ofthe lepton candidate is also required to have at least 18 nodes, rather than the minimum of 5 forbasic reconstruction. While tracks can be reconstructed in the TPCs with fewer nodes, havingat least 18 nodes gives more accurate momentum and particle identification.This track is required to have its starting point in the fiducial volume of either FGD 1 orFGD 2, which is a subset of the total FGD volume. The fiducial volume for FGD 1 is definedin ND280 detector coordinates as :• |x|< 874.51 mm90• −819.51 < y < 929.51 mm• 136.875 < z < 446.955 mmThe offset in the y-coordinate is due to the FGD XY modules being shifted off-axis by 55 mmrelative to the ND280 detector coordinate system. This volume excludes the first layer of theinitial XY module in FGD 1, and the outer 5 bars of a given XY module.The fiducial volume for FGD 2 is defined as:• |x|< 874.51 mm• −819.51 < y < 929.51 mm• 1481.45 < z < 1807.05 mmThe fiducial volume for FGD 2 does not include the first scintillator module, but allows forinteractions originating in the first water module to be selected, as forward-going interactionsin the first water module will appear to have started in the second scintillator module. Thefiducial volume cut is used as the first pass to remove interactions that occur outside of theFGD volumes, such as cosmic muons or interactions in the TPCs or ECALs.The Upstream Veto Track CutThe second cut for the FHC mode selection uses the highest momentum global track that hasa segment in a TPC, but is not the lepton candidate described for the previous cut, referred toas the veto track. The reconstruction can sometimes fail to properly link tracks, for examplea muon that starts in the P0D or TPC and goes through FGD 1 and TPC 2 may fail to linkback and instead will be reconstructed as two separate tracks, one in TPC 1 and FGD 1, andanother in FGD 1 and TPC 2. For this reason, the selection cuts on high-momentum upstreamtracks. For both FGD 1 and FGD 2, events are removed from the selection if the veto trackstart position is more than 150 mm upstream from the muon candidate start position. The FGD2 selection also cuts events which have the veto track starting in the FGD 1 fiducial volume91in order to remove potential upstream background from FGD 1 which might make topologyseparation difficult.Broken Track CutThis cut is used to reject another class of broken track events where the reconstruction breaksa potential candidate track into two separate tracks. Here the target is tracks which are brokeninto a short track fully contained in an FGD, and a FGD-TPC matched track starting in the lastlayers of the FGD. Such events would both have incorrectly reconstructed overall momentumas well as potentially be classified as the wrong topology due to an apparent FGD-only trackoriginating from the selected vertex. To remove such events, this cut requires that an event witha candidate muon track and a reconstructed FGD-only track must have the muon candidate startposition be less than 309.05 mm away from the upstream edge of the relevant FGD, removingany events where the muon candidate starts in the last two layers of the FGD. This distance isthe same in FGD 1 and FGD 2 and there are no additional requirements for FGD 2.Muon Particle Identification CutThe final cut for the FHC CC Inclusive selection is the µ particle identification cut, wherethe candidate highest momentum negative track is required to be identified as a muon throughreconstruction. The particle identification (PID) procedure uses dE/dx measurements fromthe TPC modules, as different particle types, such as muons, electrons or protons, will leavedifferent energy deposits. The reconstruction calculates the pulls, shown in Eq. 4.2 for var-ious different particle hypothesis using the measured energy deposits; for these calculations,only segments which pass the TPC track quality cuts contribute to the measured dE/dx. Thelikelihood for a given particle i is defined as in Eq. 4.3.Pulli =dE/dxmeas−dE/dxexpected,iσdE/dxmeas−dE/dxexp(4.2)92Li =e−Pull2iΣ je−Pull2j(4.3)and j is all particle hypotheses.For the muon hypothesis, there are two specific requirements imposed in order to removeelectrons, protons and pions using PID. The electron cut requires :Lµ +Lpi1−Lp > 0.8 (4.4)for tracks with p < 500MeV/c, as higher momentum tracks are less likely to be electrons.Finally, all candidate tracks must have a sufficiently high muon pid likelihood:Lµ > 0.05 (4.5)Similar particle identification methods are also used for cuts on secondary tracks for theFHC topologies.Forward Horn Current Charged Current Inclusive ResultsThe event reduction for each cut is shown in Table 4.6. The CC-Inclusive selection is effectiveat cutting out non-CC Inclusive events for both FGDs, with a final CC Inclusive purity of90.54% in FGD 1 and 89.29% in FGD 2. Momentum and cosθ distributions are shown inFigs. 4.1 and 4.2 for MC and data, with a breakdown by event topology. Total composition byevent topology is shown in Table 4.7. For FGD 2, Fig. 4.2b shows that the selection containsbackwards-going muon events, unlike the FGD 1 selection. This is enabled by using the timingdifferences between FGD 1 and FGD 2 to identify the track direction. As there is no equivalentfor FGD 1, there is instead a backward track veto. This is due to poorer timing resolutionbetween the P0D and FGD 1 than between FGD 1 and FGD 2.93 (MeV/c)µP0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000500100015002000250030003500DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(a) Momentum distributionµθ0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60200400600800100012001400 DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(b) θµ distributionFigure 4.1: Momentum and θµ distributions for data and Monte Carlo, broken down bytrue topology, for the FGD 1 FHC CC Inclusive selection before the ND280 fittuning[81]. The “No Truth” category indicates cases where there was no specifictruth information associated with the selected vertex.94 (MeV/c)µP0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000500100015002000250030003500DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(a) Momentum distributionµθ0 0.5 1 1.5 2 2.5 3020040060080010001200DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(b) θµ distributionFigure 4.2: Momentum and θµ distributions for data and Monte Carlo, broken down bytrue topology, for the FGD 2 FHC CC Inclusive selection before the ND280 fittuning[81]. The “No Truth” category indicates cases where there was no specifictruth information associated with the selected vertex.95Cut Level FGD 1 FGD 2Quality and fiducial cut 100% 100%Upstream veto cut 77.6% 70.4%Broken track cut 76.5% 69.5%Muon PID cut 56.4% 50.7%Table 4.6: Event reduction fractions for each cut step for both FGD 1 and FGD 2[81]Event topology FGD 1 FGD 2CC-0pi 49.47% 49.26%CC-1pi 17.91% 17.57%CC-Other 23.16% 22.46%In-FV Background 4.09% 4.00%Out of FGD FV Background 5.73% 6.69%Table 4.7: Fractional breakdown of the CC-Inclusive sample by true topology for bothFGD 1 and FGD 2[81]4.5.3 The Neutrino Mode Multipion SelectionOnce the CC Inclusive events are selected, they are separated by interaction topology into threedifferent samples. At this point, no events leave the selection as all events should belong to oneof the defined topologies: CC0pi , CC1pi and CC Other.The key to selecting these topologies is pion identification and detection of secondarytracks. Like the CC Inclusive cuts, the information used in the selection comes from the TPCsand FGDs. Unlike the long high momentum tracks used to define the muon candidate track,secondary tracks are shorter and more difficult to reconstruct and understand. These topologiesare defined as:• The CC 0pi Sample: This sample is characterized by no observed secondary pions; thiscorresponds closely with the CCQE interaction vertex and the signal definition at SK.It contains no TPC-identified pion, electron or positron tracks and does not have anyMichel electrons or charged pions contained in the FGD.• The CC 1pi sample: This sample is characterized by a negative muon and a single pi+with no other pions (pi0 or pi−). To select these, we require that there be either a single96reconstructed Michel electron and zero positive pion tracks in the TPC or no Michelelectrons and a single reconstructed positive pion in the TPCs or FGD. Additionally,there must be no reconstructed negative pion tracks in the event as well as no electron orpositron tracks in the TPC, which identify pi0 through their decay.• The CC Other Sample: This sample includes all the selected CC Inclusive events that donot meet the criteria defined above for the CC0pi or CC1pi samples. These events have asingle negative muon, and either at least one pi0 or pi− or more than one pi+. This alsoincludes any topologies with other particles such as kaons.The cuts used in separating the samples are the Michel electron cut, the FGD-containedreconstructed track cut and the TPC secondary track PID cut. All cuts are extended to the FGD2 selection and modified where appropriate. The p – θ binning for these samples, as used inthe fit, is given as:• FHC CC-0pi sample bin edges for the fit:p (MeV/c) : 0, 300, 400, 500, 600, 700, 800, 900, 1000, 1250, 1500, 2000, 3000, 5000,30000cosθ : -1.0, 0.6, 0.7, 0.8, 0.85, 0.90, 0.92, 0.94, 0.96, 0.98, 0.99, 1• FHC CC-1pi sample bin edges for the fit:p (MeV/c) : 0, 300, 400, 500, 600, 700, 800, 900, 1000, 1250, 1500, 2000, 5000, 30000cosθ : -1.0, 0.6, 0.7, 0.8, 0.85, 0.90, 0.92, 0.94, 0.96, 0.98, 0.99, 1• FHC CC-Other sample bin edges for the fit:p (MeV/c) : 0, 300, 400, 500, 600, 700, 800, 900, 1000, 1250, 1500, 2000, 3000, 5000,30000cosθ : -1.0, 0.6, 0.7, 0.8, 0.85, 0.90, 0.92, 0.94, 0.96, 0.98, 0.99, 1where the FGD 1 and FGD 2 samples use the same binning.97Time Projection Chamber Pion Track CutsThe TPC pion track cuts use the pulls and likelihoods calculated from measured dE/dx asdefined in Eq. 4.2 and Eq. 4.3. However, for the pion track cuts, the relevant tracks are definedas secondary tracks in the event that start in the same FGD fiducial volume as the selected muoncandidate, and have a matched TPC segment with at least 18 clusters in the TPC, similar to themuon candidate cuts. Instead of checking the electron and muon pull hypotheses, the threeparticle types used in the cuts are pion, positron and proton for positive tracks. For negativetracks, the pion and electron hypotheses are considered instead.For charged pion identification, the selection considers a secondary FGD-TPC track to bea pion ifLµ +Lpi1−Lp > 0.8 (4.6)if p < 500 MeV/c, andLpi > 0.3 (4.7)for all other cases.Fine Grained Detector Isolated Reconstruction Track CutThe FGD isolated reconstruction cut applies to tracks originating in the same FGD fiducialvolume as the candidate muon track, but do not leave the FGD. This cut serves to identifysecondary pi+ that stop in the FGDs. The definition of fiducial volume used to define trackcontainment is slightly different from the general fiducial volume definition used in the CC-Inclusive cuts:• −887 < x < 888 mm• −834 < y < 942 mm• z-position between the first active layer and last active layer in the FGD98FGD-only tracks must both start and end within this volume. As described in Sec. 4.3.2, theFGD-only track reconstruction can calculate a particle ID pull using the energy deposited asa function of track length. Like the TPC PID pulls, this is used to identify the most probableparticle for that track. To be considered a contained positive pion track, the FGD-only trackmust have cosθ > 0.3 due to reconstruction efficiencies and a pion pull greater than -2 and lessthan 2.5.Determining the Michel cutAlong with the other updates to various parts of the selection, the Michel electron cut must bedefined for both FGD 1 and FGD 2 selections, as it is used to differentiate between the CC0pisample and CC1pi sample for both detector selections. The intention is to select out eventsthat have a delayed Michel electron originating from decaying pions that were produced byneutrino interactions. As not all pions produced in such a way will have sufficient energy toreach a neighboring detector, thus stopping within the volume of the FGD, these events canlead to contamination of the CC0pi sample. In the case of Michel electrons, these particlesoften do not deposit enough energy to be reconstructed as an FGD-only track as described forthe previous cut. Therefore, an additional cut that does not use reconstructed tracks is required.To identify these Michel electrons, the basic procedure is to look for delayed activity inthe FGD containing the muon track, with delayed activity being defined as occurring at least100 ns after the initial neutrino interaction for that event. As shown in Fig. 4.3, the majority ofselected CC Inclusive events do not have any delayed time bins. For events with at least onedelayed time bin, there is a larger fraction of non-CC0pi events than for the events with none.Hits in the FGDs are sorted into FGD time bins by distance in time between each concurrenthit. To be considered to be in the same FGD time bin, a hit must be have a incident timedifference less than 100 ns from the last hit seen in the FGD; if the difference is greater than100 ns, the hit is then placed in the next FGD time bin, as shown in Fig. 4.4. The main purposeof this sorting, along with being used for the Michel tagging as described here, is to separate99(a) FGD1(b) FGD 2Figure 4.3: Total number of delayed out-of-bunch FGD timebins in selected CC Inclusiveevents. Plots are normalized to data POT, and were created for a subset of the run 4data.100neutrino interactions in different bunches from each other. As the muon decay lifetime is onthe order of 2.2 µs, particles such as pions and Michel electrons should be sorted into timebins separate from their generating neutrino interaction. In addition to this, the delayed FGDtimebin must not occur during any beam bunch windows, to be considered for a Michel electroncandidate, as there is no way to separate the Michel activity from similar beam-induced activityin the FGD.Figure 4.4: Example diagram of how FGD hits are grouped into timebins. The horizontalaxis is time and each point represents the time of an individual FGD hit. Each redellipse represents an FGD time bin and encompasses hits placed into it.[82]The original Michel electron cut used in the FHC FGD 1 samples uses the total charge inthe candidate time bin to determine whether or not the event had a Michel candidate; in FGD1, due to the fully instrumented volume, there is a clear cutoff between background and chargedeposits left by true Michel electrons. However, the charge deposition has less separationfor FGD 2 events, as shown in Fig. 4.5. Instead, the cut now looks at number of hits in thedelayed timebin, shown in Fig. 4.6. This allows for a higher Michel electron identificationrate in FGD 2 and FGD 1 compared to the previous charge-based cut, shown in Table 4.8.While there is still more difficulty in selecting FGD 2 Michel e−, due to the less instrumentedgeometry, a hit-based cut gives significantly higher efficiency. One feature that shows up inboth implementations of the cut is the discrepancy between data and MC at low hits. At lownumbers of hits or amount of charge deposited, there is a significant excess in MC consistingof background events, while the data is lower by a factor of 5. This indicates that there issome over-simulation of background and is discussed further in Section 5.3.12. This excess insimulated background limits the power of the Michel cut and would need to be understood tofurther improve it.101(a) FGD1(b) FGD 2Figure 4.5: Total charge seen in out-of-bunch timebins for selected CC Inclusive events.Plots show the breakdown by true particle associated with the delayed time bin.Plots are normalized to data POT, and were created for a subset of the run 4 data.Michel Cut Definition FGD 1 FGD 2Previous Cut on Total Charge > 200 photoelectrons 80.6% 31.9%Current FGD-Specific Hit Cut 83.9% 52.7%Table 4.8: Fraction of correctly identified Michel electrons with hits in delayed timebins.102(a) FGD1(b) FGD 2Figure 4.6: Number of hits seen in out-of-bunch timebins for selected CC Inclusiveevents. Plots show the breakdown by true particle associated with the delayed timebin. Plots are normalized to data POT, and were created for a subset of the run 4data.103Forward Horn Current Multipion Selection ResultsThe selected momentum and cosθ distributions for the data and nominal MC are shown inFigs. 4.7, 4.8, 4.9 and 4.10; the “No Truth” category indicates cases where there was no specifictruth information associated with the selected vertex. Overall purities and efficiencies for eachsample in the respective signal topology are shown in Table 4.9. Of the three samples, theCC-0pi sample has a significantly higher efficiency than the others. This is due in part to thefact that failure to correctly tag the non-CC-0pi events will place those events in other sampleseven when selected as CC Inclusive. As the efficiency is calculated on a per-sample basis forthe multipion selection, this can lower the efficiency for those samples. As CC-0pi events areless likely to be misidentified, this has less impact on that sample’s overall efficiency. Theefficiencies and purities for each sample are roughly similar for FGD 1 and FGD 2, with theexception of CC-1pi . This sample has a lower efficiency in FGD 2 due to the reduced activevolume, which lowers the efficacy of FGD-only reconstruction and the Michel electron cut.Sample Efficiency PurityFGD1 νµ CC0pi (ν mode) 47.61 70.4FGD1 νµ CC1pi (ν mode) 27.5 54.1FGD1 νµ CC Other (ν mode) 27.61 72.9FGD2 νµ CC0pi (ν mode) 48.45 67.4FGD2 νµ CC1pi (ν mode) 23.69 53.5FGD2 νµ CC Other (ν mode) 28.23 72.8Table 4.9: Efficiencies and purities for the FHC CC-0pi , CC-1pi and CC-Other samplesfor both FGDs[81].Efficiencies shown are with respect to the total number of gener-ated true events with that topology.104 (MeV/c)µP0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000050010001500200025003000 DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(a) FGD 1 CC-0pi selection. (MeV/c)µP0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000100200300400500DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(b) FGD 1 CC-1pi selection. (MeV/c)µP0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000050100150200250300 DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(c) FGD 1 CC-Other selection.Figure 4.7: Momentum distributions for each of the three FHC samples broken down bytrue topology for FGD 1[81]. Distributions shown use the T2K Monte Carlo beforethe ND280 fit tuning. The “No Truth” category indicates cases where there was nospecific truth information associated with the selected vertex.105 (MeV/c)µP0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000050010001500200025003000DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(a) FGD 2 CC-0pi selection. (MeV/c)µP0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000050100150200250300350400450DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(b) FGD 2 CC-1pi selection. (MeV/c)µP0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000050100150200250DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(c) FGD 2 CC-Other selection.Figure 4.8: Momentum distributions for each of the three FHC samples broken down bytrue topology for FGD 2. Distributions shown use the T2K Monte Carlo before theND280 fit tuning.106µθ0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60100200300400500600700800DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(a) FGD 1 CC-0pi selection.µθ0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6050100150200250DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(b) FGD 1 CC-1pi selection.µθ0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6050100150200250300350DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(c) FGD 1 CC-Other selection.Figure 4.9: cosθ distributions for each of the three FHC samples broken down by truetopology for FGD 1[81]. Distributions shown use the T2K Monte Carlo before theND280 fit tuning.107µθ0 0.5 1 1.5 2 2.5 30100200300400500600700800DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(a) FGD 2 CC-0pi selection.µθ0 0.5 1 1.5 2 2.5 3050100150200250 DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(b) FGD 2 CC-1pi selection.µθ0 0.5 1 1.5 2 2.5 3050100150200250300350DatapiCC-0piCC-1CC-OtherBKGExternalNo truth(c) FGD 2 CC-Other selection.Figure 4.10: cosθ distributions for each of the three FHC samples broken down by truetopology for FGD 2[81]. Distributions shown use the T2K Monte Carlo beforethe ND280 fit tuning.1084.5.4 Reverse Horn Current Antineutrino Charged Current InclusiveSelectionThe antineutrino CC Inclusive selection starts with the same initial quality cuts as the FHCsamples, and is used to identify the signal ν¯µ interactions in the FGDs when operating inantineutrino beam mode. The main difference in the ν¯µ preselection from the cuts describedin Section 4.5.2 is that where the νµ selection looks for a µ−, antineutrino interactions willproduce µ+. So instead of requiring the highest momentum track in the event to be negative,the ν¯µ selection requires the highest momentum track to have a positive charge.In addition, the µ PID cut is slightly different: the selection still checks for a muon candi-date, but requires stricter PID pull cuts on the highest momentum positive track in the event.The bounds on the pulls are :Lµ +Lpi1−Lp > 0.9 (4.8)if p < 500 MeV and0.1 < Lµ < 0.7 (4.9)for all candidate tracks. The upper bound on Lµ serves to remove misidentified low energyµ− tracks to reduce the νµ contamination, as these are frequently reconstructed with the wrongcharge. As the ν¯µ fraction in the FHC beam flux is insignificant, this upper limit is not requiredfor the FHC selections.Reverse Horn Current Antineutrino Charged Current Inclusive ResultsThe CC Inclusive antineutrino selection has an overall purity of 81.3% in FGD 1 and 80.7%in FGD 2, with topology breakdown shown in Table 4.10, and momentum and cosθ distri-butions shown in Appendix B. FGD 2 has a higher overall efficiency than FGD 1 due to theability to select backwards-going events in FGD 2, as mentioned in Section 4.5.2. The loweroverall purity of the RHC ν¯µ selections compared to the FHC νµ CC Inclusive events is due to109background contamination by wrong-sign νµ events. As the FHC beam flux does not have asignificant fraction of ν¯ , this effect is much smaller.Event topology FGD 1 FGD 2CC-0pi 59.7% 59.7%CC-Multiple pi 21.6% 21.0%νµ Background 10.6% 11.0%Other In-FV Background 1.9% 2.2%Out of FGD FV Background 6.2% 6.1%Table 4.10: Fractional breakdown of the RHC ν¯µ CC-Inclusive sample by true topologyfor both FGD 1 and FGD 2[83].4.5.5 Reverse Horn Current Antineutrino Charged Current MultitrackSelectionAlthough the CC Inclusive selection for antineutrino events is similar to the CC Inclusive se-lection for FHC neutrino events, the CC Multitrack selection for RHC ν¯µ does not split eventsinto the CC-0pi , CC-1pi and CC Other samples. Instead, a simpler set of cuts are used to sortevents into two topological samples:• The CC 1-Track Sample: This sample is characterized by no observed secondary pionsor other particles; this is considered a CCQE-enhanced sample. It contains no secondaryFGD-TPC matched tracks, but may contain events with short FGD-only tracks or withdelayed Michel electrons leading to reduced purity compared to the CC-0pi sample inFHC.• The CC N-Tracks Sample: This sample is characterized by events containing observedsecondary particles; unlike the CC-1pi and CC-Other samples from the FHC selection,it does not contain any events where there are FGD-only tracks, but no secondary FGD-TPC tracks. For this reason this is considered to be a charged-current non-quasielastic-enhanced sample, where the fraction of non-quasielastic events is increased..The single track selection for ν¯µ is similar to the CC-0pi selection for the FHC νµ , and isdesigned to primarily select CC-0pi events. However, due to the difficulty of reconstructing the110short range pi−, this sample imposes a single additional cut on top of the CC Inclusive selection.For an antineutrino event to be considered a 1-Track event, there must only be a single TPC-FGD matched track. In addition, all TPC-FGD matched tracks are counted as secondary tracks,as there is no additional PID requirements. As the µ+ candidate is a TPC-FGD matched track,this effectively requires that there are no secondary tracks that reach the TPCs for an event tobe in the CC 1-Track sample.Unlike the FHC selection, there are no cuts on FGD-only track multiplicity, as it is moredifficult to reconstruct the contained secondary tracks due to low energy. When low energypi− are produced from ν¯µ interactions, they are quickly absorbed and therefore are not easilyreconstructed. As non-CC-0pi events with Michel electrons or other short-range secondaryparticles are not separated out, this is therefore considered to be a CCQE-enhanced sample.The multi-track sample for RHC ν¯µ contains all selected charged current ν¯µ events that donot pass the 1-Track cut. When using in the ND280 fit, each sample has 20 p – θ bins in total:• RHC 1-track sample bin edges for the fit:p (MeV/c) : 0, 500, 900, 1200, 2000, 10000cosθ : -1.0, 0.8, 0.92, 0.98, 1and• RHC N-tracks sample bin edges for the fit:p (MeV/c) : 0, 600, 1000, 1500, 2200, 10000cosθ : -1.0, 0.8, 0.9, 0.97, 1As with the FHC samples, the FGD 1 and FGD 2 selections use the same binning.ResultsThe overall efficiencies for the RHC ν¯µ samples are shown in Table 4.11, along with the RHCνµ efficiencies. The efficiencies are similar to those for the FHC samples, taking CC-1pi andCC-Other together. As these samples are used to understand the overall composition of the111RHC flux at SK, it is useful to know the fraction of νµ contamination in each sample, shownin Tables 4.12 and 4.13. These tables also show the overall purities for the samples, and as noν¯µ cuts depend on FGD-only tracks, the purities for FGD 1 and FGD 2 are similar for both the1-track and multitrack samples. The CC 1-Track sample has low wrong-sign νµ contaminationin both FGDs, while around 1/3 of the ν¯µ CC N-Track sample is comprised of backgroundνµ events. This results in a much lower purity for the CC N-Tracks sample. The selectedmomentum and θ distributions for the data and nominal MC are shown in Figs. 4.11, 4.12,4.13 and 4.14.RHC ν¯µ RHC νµSelection FGD 1 FGD 2 FGD 1 FGD 2CC 1-Track 66% 68% 46.3% 46.4%CC N-Tracks 29% 31% 36.5% 36.7%Table 4.11: Efficiencies for the RHC samples for both FGD 1 and FGD 2[83] [84]Event topology FGD 1 FGD 2CC-0pi 74.4% 74.5%CC-Multiple pi 14.5% 13.8%νµ Background 4.1% 4.2%Other In-FV Background 1.3% 1.4%Out of FGD FV Background 5.7% 6.1%Table 4.12: Fractional breakdown of the RHC ν¯µ CC-1 Track sample by true topologyfor both FGD 1 and FGD 2[83]. Distributions shown use the T2K Monte Carlobefore the ND280 fit tuning.Event topology FGD 1 FGD 2CC-0pi 8.8% 9.4%CC-Multiple pi 46.4% 45.6%νµ Background 32.9% 34.4%Other In-FV Background 4.1% 4.4%Out of FGD FV Background 7.9% 6.2%Table 4.13: Fractional breakdown of the RHC ν¯µ CC-N tracks sample by true topologyfor both FGD 1 and FGD 2[83]. Distributions shown use the T2K Monte Carlobefore the ND280 fit tuning.112Muon momentum (MeV/c)0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Events/(100 MeV/c)050100150200250300350400450Integral    2836DatapiCC-0piCC-NBKGExternalNo truthIntegral    2600(a) Momentum distribution)θMuon cos(0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Events02004006008001000Integral    2903DatapiCC-0piCC-NBKGExternalNo truthIntegral    2681(b) cosθµ distributionFigure 4.11: Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 1 RHC ν¯µ CC 1 Track selection[83]. Distributionsshown use the T2K Monte Carlo before the ND280 fit tuning. The “No Truth”category indicates cases where there was no specific truth information associatedwith the selected vertex.113 candidate reconstructed momentum (MeV)+µ0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Normalised to Run 5c + Run 6 POT050100150200250300350400450Integral    2882DatapiCC-0piCC-NBKGExternalNo truthIntegral    2689(a) Momentum distribution)θ candidate cos(+µ-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Normalised to Run 5c + Run 6 POT02004006008001000120014001600Integral    2949DatapiCC-0piCC-NBKGExternalNo truthIntegral    2784(b) cosθµ distributionFigure 4.12: Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 2 RHC ν¯µ CC 1-Track selection[83]. Distributionsshown use the T2K Monte Carlo before the ND280 fit tuning. The “No Truth”category indicates cases where there was no specific truth information associatedwith the selected vertex.114Muon momentum (MeV/c)0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Events/(250 MeV/c)020406080100Integral   763.1DatapiCC-0piCC-NBKGExternalNo truthIntegral     723(a) Momentum distribution)θMuon cos(0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Events050100150200250300350400450Integral     839DatapiCC-0piCC-NBKGExternalNo truthIntegral     799(b) cosθµ distributionFigure 4.13: Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 1 RHC ν¯µ CC N-Track selection[83]. Distributionsshown use the T2K Monte Carlo before the ND280 fit tuning. The “No Truth”category indicates cases where there was no specific truth information associatedwith the selected vertex.115 candidate reconstructed momentum (MeV)+µ0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Normalised to Run 5c + Run 6 POT051015202530354045Integral     790DatapiCC-0piCC-NBKGExternalNo truthIntegral     684(a) Momentum distribution)θ candidate cos(+µ-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Normalised to Run 5c + Run 6 POT0100200300400500600700 Integral   869.1DatapiCC-0piCC-NBKGExternalNo truthIntegral     761(b) cosθµ distributionFigure 4.14: Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 2 RHC ν¯µ CC N-Track selection[83]. Distributionsshown use the T2K Monte Carlo before the ND280 fit tuning. The “No Truth”category indicates cases where there was no specific truth information associatedwith the selected vertex.1164.5.6 Wrong Sign in Reverse Horn Current Charged Current InclusiveSelectionAs with the previously described CC Inclusive selections, the νµ in ν¯µ mode CC Inclusiveselection requires the basic quality cuts. In order to reduce the effect of systematics on under-standing the νµ contamination in the ν¯µ beam, the wrong-sign selection uses mostly the samecuts as the antineutrino mode ν¯µ selections, described in the previous subsection, but with themain muon track required to be negative as in the FHC selection. In addition, the muon PIDcut now requires:Lµ +Lpi1−Lp > 0.7 (4.10)for tracks with pµ < 500 MeV/c. This is to reduce the fraction of background e− in the overallselection. The threshold is set slightly lower than the FHC selection, in order to avoid cuttingout signal events. In addition, rather than requiring the condition in Eqn. 4.5 a more strict cuton the muon PID pull is used:0.1 < Lµ < 0.8 (4.11)This helps reject both protons and low energy µ+ from antineutrino interactions that are recon-structed with the incorrect charge, similar to the cuts used for the ν¯µ selection.ResultsThis gives an efficiency of 55.17% in FGD 1 and 54.6% in FGD 2 for CC Inclusive νµ events.As with the RHC ν¯µ and FHC νµ CC Inclusive selections, FGD 1 and FGD 2 select roughlythe same number of events. The CC Inclusive wrong-sign νµ selection has a purity of 80.0%for FGD 1 and a purity of 79.2% for FGD 2. This is lower than the purities seen for the FHCνµ CC Inclusive sample and similar to the purities for the RHC ν¯µ CC Inclusive selection. Thiscomes from the increased ν¯µ backgrounds compared to the FHC selection, with FGD 1 havinga 10.6% ν¯µ background and FGD 2 11.0%. As with the FHC selections, FGD 2 has a largerbackwards-going component due to inter-FGD timing. Momentum and angle distributions forthe RHC νµ selection are shown in AppendixB.117Event topology FGD 1 FGD 2CC-0pi 33.0% 32.3%CC-Multiple pi 47.0% 45.8%ν¯µ Background 7.3% 8.5%Other In-FV Background 3.7% 3.2%Out of FGD FV Background 9.0% 10.2%Table 4.14: Fractional breakdown of the RHC νµ CC-Inclusive sample by true topologyfor both FGD 1 and FGD 2[84].4.5.7 Wrong Sign in Reverse Horn Current Charged CurrentMultitrack SelectionThe wrong sign νµ events are split into two different track-multiplicity samples. The basicsample definition is the same as the samples used for the RHC ν¯µ CC Inclusive events, but forevents with a highest-momentum negative muon track.• The CC 1-Track Sample: This sample is characterized by no observed secondary pionsor other particles; this is considered a CCQE-enhanced sample. It contains no secondaryFGD-TPC matched tracks, but may contain events with short FGD-only tracks or withdelayed Michel electrons leading to reduced purity compared to the CC-0pi sample inFHC.• The CC N-Tracks Sample: This sample is characterized by events with observed sec-ondary particles; unlike the CC-1pi and CC-Other samples from the FHC selection, itdoes not contain any events where there are FGD-only tracks, but no secondary FGD-TPC track. For this reason this is considered to be a CCnQE-enhanced sample.Both the sample definitions and the cuts used in distinguishing the samples are analogous tothe cuts and definitions for RHC ν¯µ events.The RHC νµ 1-Track selection uses the same separation criteria as the ν¯µ selection. Thisis intended to reduce the overall effects of the systematic errors on the fit predictions, as the νµ– ν¯µ separation in the near detector is important for the antineutrino oscillation studies at SK.118Therefore this sample is defined as all selected CC Inclusive νµ events with a single TPC-FGDmatched track and no requirements on FGD-contained secondary tracks.The RHC νµ multi-track sample contains all selected RHC νµ CC Inclusive events that donot pass the track multiplicity cut for the 1-Track sample. These events all have at least twoFGD-TPC matched tracks.When using in the ND280 fit, each RHC νµ sample has 20 p – θ bins in total, and thebinning is identical to the RHC ν¯µ samples:• RHC 1-track sample bin edges for the fit:p (MeV/c) : 0, 500, 900, 1200, 2000, 10000cosθ : -1.0, 0.8, 0.92, 0.98, 1and• RHC N-tracks sample bin edges for the fit:p (MeV/c) : 0, 600, 1000, 1500, 2200, 10000cosθ : -1.0, 0.8, 0.9, 0.97, 1As with the FHC samples and the RHC ν¯µ samples, the FGD 1 and FGD 2 selections use thesame binning.ResultsEfficiencies for both samples are shown in Table 4.11 and purities in Tables 4.15 and 4.16.The CC-1 Track sample has a similar number of events selected to the CC-Multi Track samplefor RHC νµ , unlike the RHC ν¯µ 1-Track and the FHC νµ CC-0pi samples which both selectsignificantly more events than the other samples. The purity of the N-Tracks sample is higherfor the RHC νµ selection than the RHC ν¯µ CC N-Track sample due to higher energy secondaryparticles from the neutrino interactions being able to enter the TPCs. The selected momentumand cosθ distributions for the data and nominal MC are shown in Figs. 4.15, 4.16, 4.17 and4.18.119Event topology FGD 1 FGD 2CC-0pi 50.5% 49.1%CC-Multiple pi 27.7% 26.2%ν¯µ Background 6.6% 7.4%Other In-FV Background 2.9% 2.7%Out of FGD FV Background 12.3% 14.6%Table 4.15: Fractional breakdown of the RHC νµ CC-1 Track sample by true topologyfor both FGD 1 and FGD 2[84]Event topology FGD 1 FGD 2CC-0pi 15.3% 14.3%CC-Multiple pi 66.1% 67.0%ν¯µ Background 8.1% 8.6%Other In-FV Background 4.4% 4.7%Out of FGD FV Background 6.1% 5.4%Table 4.16: Fractional breakdown of the RHC νµ CC-N tracks sample by true topologyfor both FGD 1 and FGD 2[84]120Muon momentum (MeV/c)0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Events/(200 MeV/c)020406080100120Entries  11827Mean     1358RMS      1097Integral   903.1DatapiCC-0piCC-NBKGExternalNo truthEntries  1016Mean     1418RMS      1144Integral     934(a) Momentum distribution)θMuon cos(-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Events0100200300400500600700Entries  11827Mean   0.8622RMS    0.1802Integral   988.9DatapiCC-0piCC-NBKGExternalNo truthEntries  1016Mean   0.8666RMS    0.1736Integral    1016(b) cosθµ distributionFigure 4.15: Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 1 RHC νµ CC 1-Track selection[84]. Distributionsshown use the T2K Monte Carlo before the ND280 fit tuning. The “No Truth”category indicates cases where there was no specific truth information associatedwith the selected vertex.121Muon momentum (MeV/c)0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Events/(200 MeV/c)020406080100120140160180Entries  11770Mean     1306RMS      1088Integral   911.7DatapiCC-0piCC-NBKGExternalNo truthEntries  1014Mean     1321RMS      1081Integral     920(a) Momentum distribution)θMuon cos(-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Events0100200300400500600700Entries  11770Mean    0.842RMS    0.2317Integral   986.1DatapiCC-0piCC-NBKGExternalNo truthEntries  1014Mean   0.8478RMS    0.2147Integral    1014(b) cosθµ distributionFigure 4.16: Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 2 RHC νµ CC 1-Track selection[84]. Distributionsshown use the T2K Monte Carlo before the ND280 fit tuning. The “No Truth”category indicates cases where there was no specific truth information associatedwith the selected vertex.122Muon momentum (MeV/c)0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Events/(250 MeV/c)020406080100Entries  11513Mean     1915RMS      1210Integral   857.4DatapiCC-0piCC-NBKGExternalNo truthEntries  1055Mean     1981RMS      1217Integral     873(a) Momentum distribution)θMuon cos(-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Events0100200300400500600700800900Entries  11513Mean   0.9058RMS    0.1296Integral    1016DatapiCC-0piCC-NBKGExternalNo truthEntries  1055Mean   0.9031RMS    0.1432Integral    1055(b) cosθµ distributionFigure 4.17: Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 1 RHC νµ CC N-Track selection[84]. Distributionsshown use the T2K Monte Carlo before the ND280 fit tuning. The “No Truth”category indicates cases where there was no specific truth information associatedwith the selected vertex.123Muon momentum (MeV/c)0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Events/(250 MeV/c)020406080100Entries  10995Mean     1902RMS      1189Integral   825.8DatapiCC-0piCC-NBKGExternalNo truthEntries  978Mean     1929RMS      1210Integral     841(a) Momentum distribution)θMuon cos(-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Events0100200300400500600700800Entries  10995Mean   0.9036RMS    0.1537Integral     982DatapiCC-0piCC-NBKGExternalNo truthEntries  978Mean   0.9123RMS    0.1351Integral     978(b) cosθµ distributionFigure 4.18: Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 2 RHC νµ CC N-Track selection[84]. Distributionsshown use the T2K Monte Carlo before the ND280 fit tuning. The “No Truth”category indicates cases where there was no specific truth information associatedwith the selected vertex.124Chapter 5Systematics and Systematic Correlations5.1 Overview of ND280 Detector SystematicsThe uncertainty on the ND280 reconstruction and selected events comes from various under-lying uncertainties on the properties of the near detector and the reconstruction methods used.These come from the various underlying physical properties of the detectors and are calculatedas effective systematic effects on the Monte Carlo. These are generally characterized as thedifference between the data and Monte Carlo simulation, such as the uncertainty in the mea-sured target mass of the FGDs versus the implementation in Monte Carlo. These systematicuncertainties are used to produce the observable normalization weights and covariance matrixdescribed in Chapter 3 and are a key part of constraining the cross section parameters with thenear detector fit. This includes both the systematic uncertainties and associated corrections tothe Monte Carlo.While there are many different potential sources of uncertainty at the near detectors, the rel-evant detector systematics are those that relate to the event selections described in Chapter 4.The majority of the identified systematic effects apply to both the FHC and RHC samples,with the only exceptions being effects specific to the FHC νµ multipion selection due to usingmore cuts to define samples. Each systematic effect was identified and studied by various T2Kcollaborators for this analysis. In addition, a first estimation of the correlations between the125systematic uncertainties of the FGD 1 and FGD 2 selections was made based on knowledge ofthe physical sources for the systematic uncertainties. These correlations and detector system-atic uncertainties were then used to generate the observable normalization parameter weightsand covariance matrix used in the near detector fit.5.1.1 List of SystematicsAt ND280, the detector systematic uncertainties are implemented as two different types: vari-ation detector systematics, which vary reconstructed quantities in events, and weight systemat-ics, which apply a weight to each event without modifying the event itself, such as in the case ofa detector efficiency. There are six variation systematics and eleven weight systematics. Withineach category, there are different methods for applying systematics to events. The systematicsrelated to tracks or other measured quantities in the FGDs, such as Michel electron efficiencyand FGD Particle identification, are implemented to allow for separate systematic calculationsfor FGD 1 and FGD 2 where necessary. Table 5.1 lists these systematics, the type of sys-tematic and the probability density function used for each systematic. Systematic uncertaintycalculations and propagation are done using the Psyche software, described in Section 4.4.1.5.1.2 Variation SystematicsVariation systematics are those detector systematics that can allow a selected event to migratebetween different selected samples, or in rare cases in and out of the overall inclusive selec-tion. These systematics modify reconstructed quantities such as track momentum and particleID pulls, which can potentially change the highest momentum track in an event. When a vari-ation systematic is applied, the relevant reconstructed observables are smeared and the eventselection is rerun on the modified event. This smearing is implemented differently depend-ing on the specifics of the systematic for each parameter. There are two main ways variationsystematic parameters are applied at ND280 :1. When the true observable for the systematic is known, the difference between the recon-126Systematic Source Type PDFTPC SystematicsMagnetic Field Distribution Variation UniformTPC Momentum Resolution Variation GaussianTPC Momentum Scaling Variation GaussianTPC Particle Identification Variation GaussianTPC Cluster Efficiency Weight GaussianTPC Track Reconstruction Weight GaussianCharge Confusion Weight GaussianFGD ReconstructionTPC – FGD Matching Weight GaussianFGD Particle Identification Variation GaussianParticle Time of Flight Variation GaussianFGD Hybrid Track Reconstruction Weight GaussianMichel Electron Detection Weight GaussianSelection BackgroundOut of Fiducial Volume Events Weight GaussianSand Muon Events Weight GaussianEvent Pile-Up Weight GaussianMonte Carlo ModelingPion Secondary Interactions Weight GaussianFGD Mass Weight GaussianTable 5.1: List of near detector systematic error sources and types for the T2K near de-tector selection.[81]structed observable and its true value is rescaled asx′rec = xtrue+(xMCrec − xtrue)(s+α×δ s) (5.1)where s is a scale factor, δ s its systematic error and α a Gaussian random variable with amean of 0 and width of 1. Here s will always equal 1, unless there is systematic differencebetween MC and data.2. When a correction is applied to the MC observable value using the mean differencebetween the data and Monte Carlo observable values, ∆x¯, the observable value becomes:x′rec = xMCrec +∆x¯+α×δ∆x¯ (5.2)127where ∆x¯ is the mean difference between the data and Monte Carlo values for the ob-servable, δ∆x¯ the uncertainty on that mean and α a Gaussian random variable with amean of 0 and width of 1. In this case, the systematic uncertainty on the observable isadded separately from the correction as its effect on the number of selected events differsbetween the nominal Monte Carlo and the corrected Monte Carlo.The only exception to these are for the systematics derived from the magnetic field maps andare described along with the systematic details.5.1.3 Weight SystematicsWeight systematics only have an effect on the overall event weight and do not modify anyof the reconstructed values for an event. Because of this, they are implemented as weightsapplied to relevant events and can not cause sample migration for events as underlying re-constructed values do not change. There are two different implementation types for weightsystematics: efficiency-like systematics and normalization systematics. Efficiency-like sys-tematics are those that correspond to specific reconstruction or detection probabilities and nor-malization systematics are those that control overall normalizations for specific subsamples ofthe selection.The efficiency-like systematics use studies comparing Monte Carlo predictions with datafor well-understood event control samples. This allows an efficiency for data to be calculatedas truth information is unnecessary. These control samples are specific subsets of events thatfulfill specific requirements so that the feature being studied is well understood. An exampleof this would be using stopping cosmic events for looking at Michel electron efficiency [82].These samples can also be constructed from beam triggered events as well. Control samplesdo not necessarily contain all the features that would be present in the T2K beam analysis,and may cover a more limited phase space. However, the control samples are designed to besufficiently representative of the selection for use in finding systematic efficiencies [81].In general, the efficiency on the data without taking control sample systematic error into128account isεdata =εCSdataεCSMCεMC (5.3)where the CS superscript denotes values calculated from the control samples[81]. As the con-trol sample efficiencies have some statistical error on them, these must also be taken into ac-count. There are two definitions for the individual event weights for the efficiency-like system-atics, depending on whether applying an efficiency or inefficiency is appropriate:we f f =ε ′dataεMC(5.4)for efficiency[81] andwine f f =1− ε ′data1− εMC (5.5)for inefficiency[81], whereε ′data = (rCS+α×δ rCS)εMC (5.6)where rCS = εCSdataεCSMCand δ rCS is the measured uncertainty on the ratio of the efficiencies measuredusing the control samples. As with the variation systematics, α is a Gaussian random variablewith a mean of 0 and width of 1.Normalization systematics are related to the overall event normalization and apply to spe-cific subsamples of events. Like the efficiency-like systematics, these are applied as individualweights on each event. This weight is calculated as :wevent = 1+α×δecategory (5.7)where α is the random variable for the systematic variations and δecategory is the systematicerror for that category of events. A weight of 1 is applied when events are not part of thesystematic category.1295.2 Systematic CorrelationsThe initial implementation of the detector systematics, and the implementation used in previousnear detector fits, assumes there are no correlations between bins within systematics, and wasonly applied for FGD 1 samples. However, the systematics as implemented for ND280 areeffective systematics – they depend on underlying physical quantities that may be shared. Asthe near detector fit now has been expanded to include FGD 2, a similar but not identical targetto FGD 1, it is not immediately obvious how a systematic is correlated between events inFGD 1 and events in FGD 2. For detector systematics this means identifying which systematicparameters apply to measured quantities in the FGDs and how they are correlated betweenthe two detectors. Allowing bins within a defined systematic uncertainty to be correlated witheach other is also a new addition to the ND280 analysis and relies on the underlying physicalquantities that contribute to the systematic, much like the FGD 1 – FGD 2 correlations. As thisis the first implementation of FGD1 – FGD 2 correlations, all systematics are approximated toeither be fully correlated (ρ = 1), fully anticorrelated (ρ =−1) or uncorrelated (ρ = 0). For thein-systematic correlations, bins are also assumed to fully correlated (ρ = 1), fully anticorrelated(ρ =−1) or uncorrelated (ρ = 0) with other bins in that systematic.5.2.1 Correlations Between the Fine Grained Detector 1 and FineGrained Detector 2 SelectionsThe CC Inclusive selections described in Chapter 4 are split into mutually exclusive FGD1 and FGD 2 selections, which both take the various systematic uncertainties into account.As the near detector fit uses FGD 1 to constrain the carbon-specific parameters for FGD 2,understanding how the systematic uncertainties are correlated between the two selections isvital to correctly constraining the oxygen-specific parameters. For now, an understanding ofthe underlying physical quantities that contribute to a systematic are used as a rationale forchoice of correlation; in the future more exact treatments of the underlying parameters andcorrelations should be done.130The primary concern for the FGD 1 – FGD 2 correlations are the FGD reconstruction sys-tematic uncertainties. While there are correlations between other parts of the tracker volume,such as correlations between the three TPCs for certain systematics, these do not depend onchoice of FGD for the target and are therefore considered to be fully correlated between thetwo selections.5.3 Systematic Details5.3.1 B Field DistributionThe magnetic field used in the ND280 reconstruction is taken from mapping data taken atthe ND280 detectors [85], performed using Hall probes at the near detector. This mappingallows for magnetic distortions to be accounted for in the reconstruction, as the field is notconstant throughout the magnet volume. The field map is applied to the cluster positions atreconstruction to evaluate change in y and z position[86]. In addition, the map validity is testedagainst the laser calibration system in the TPC, which uses the measured and expected positionsof photo-electrons emitted by aluminum dots in the cathode which are illuminated by a laser.This testing is done both when the magnet is off and during normal running for ND280. Asecond correction to the distortion from these measurements is applied to the cluster positionsas a function of drift distance [86].These corrections were cross checked using the reconstructed momentum in TPC 2 andTPC 3, as TPC 1 is not expected to have large distortions in the B-field. The studies foundthat adding the initial B-field corrections reduces the relative momentum bias between TPCs,while the second correction increases it[86]. For this reason, the primary mapping correctionis applied during reconstruction while the TPC calibration corrections are used as systematicuncertainty on the B field distortions. When the mappings for the magnetic field distortions areapplied, the reconstructed momentum is recalculated as:x′rec = xMCrec +α(xnewrec − xMCrec ) (5.8)131where xnewrec is the reconstructed momentum after the mapping changes. Unlike the other sys-tematics, the variation α for the B field distortions has a uniform distribution between 0 and1.5.3.2 Time Projection Chamber Momentum ScaleThe momentum scale is the second systematic uncertainty that arises from the measurementsof the magnetic field at ND280. There are four Hall probes installed at ND280 to providescaling factors for the magnetic field strength in the Monte Carlo, as the nominal MC B-fieldis significantly different from the measured mapping [85]. Changing the magnitude of theB-field can lead to changes in reconstructed TPC track momenta and migration between p– θ bins in the selections. The uncertainty on this scaling comes from several sources: theintrinsic resolution of the probes, potential relative misalignment between the probes, and non-linearities in the relationship between magnet current and field magnitude.The systematic uncertainty for the momentum scaling can be accounted for as a simplescaling of the error on the momentum:x′rec = xMCrec (1+α×δ s) (5.9)where α is a Gaussian random variable with a mean of 0 and width of 1 and δ s is the statisticalerror on the scaling, with δx = xMCrec δ s for the uncertainty on the momentum. The measuredscaling factor comes from the B-field studies described in [85] and the final uncertainty usedfor systematic propagation is measured as 0.57% [81].5.3.3 Time Projection Chamber Momentum ResolutionDue to factors such as the differences in magnetic field distortions and electric field distortionsat the near detector, there is some uncertainty on the resolution of the momentum reconstructionin the TPCs. While there is a separate systematic for the B field distortions, the sources forthe data – Monte Carlo discrepancy in resolution are not well understood and therefore the132conservative approach of including both errors is taken [81].To study the TPC momentum resolution for data and Monte Carlo, a study has been per-formed using a control sample of events with tracks that cross multiple TPCs[87]; this allowsthe momenta of the two TPC tracks to be compared without using truth information. Thisis done by comparing the difference between the reconstructed momentum of the two TPCtracks with each other. The momentum resolution uncertainty is calculated for a single TPCsegment; for global tracks, there is good agreement in the inverse momentum distributions be-tween global tracks and single TPC segments. Therefore the same fractional differences areused for global tracks.The Monte Carlo shows better TPC momentum resolution than data; this difference is ac-counted for by smearing the inverse momentum in the Monte Carlo to match that seen in data.This uses the fractional difference in momentum resolution between data and Monte Carlo,which is binned in x-position along the TPC[81][87]. While the scaling factor depends onx-position, an uncertainty of 0.1 was used for all scaling factors regardless of x-position. Addi-tionally, this uncertainty has 100% correlation between different x-position bins, as a differencein position does not change the underlying physical source.5.3.4 Time Projection Chamber Particle IdentificationAs described in Section 4.3.1, the TPC particle ID is calculated using a truncated mean of thecharge collected in the TPC to calculated particle hypothesis pulls. There are two contributingdata – Monte Carlo differences for the TPC particle ID: the difference between the mean pullvalues and the ratio between the pull widths. The uncertainties on these ratios is calculated foreach particle type and TPC, and binned in particle momentum. The difference in pull meansgives a estimation of the systematic bias and the ratio is used to estimate the smearing thatneeds to be applied to the Monte Carlo.To measure the systematic uncertainties, samples of events with a high purity of muon,electron and proton tracks are selected. The selections are different for each particle type, with133Cluster Type (εMC− εdata)/εMC εdata/εMCVertical 0.0011±0.0002 0.9989±0.0002Horizontal 0.0007±0.0001 0.9993±0.0001Table 5.2: Efficiency differences for the TPC cluster matching for data and MonteCarlo[81]. εdata is the measured data cluster efficiency and εMC is the measuredMonte Carlo cluster efficiency.high purity for the muon and electron samples and lower purity for the proton sample [79].The results from the muon studies are used for events identified as pions as well, as muonsand pions have very similar energy loss. There is no correlation between particle type, whilewithin the uncertainties for each particle, uncertainties for the momentum bins and TPCs are100% correlated. Additionally the uncertainties on the pull mean is 100% correlated with theuncertainty on the pull width for each particle case.5.3.5 Time Projection Chamber Cluster EfficiencyThe systematic uncertainty on reconstructing a cluster in the TPCs comes from the differencein reconstruction efficiency in data and Monte Carlo. As the TPC track quality cut describedin Chapter 4 uses the number of clusters, this can lead to differing fractions of Monte Carloand data events passing the initial selection cuts. The efficiency and its uncertainty are mea-sured for both horizontal and vertical reconstructed clusters. The uncertainty for horizontal andvertical clusters mostly comes from the underlying hit efficiency and therefore the systematicuncertainty is correlated between the horizontal and vertical clusters.To evaluate the difference in cluster reconstruction efficiency, two control sample studieswere performed. For the horizontal cluster efficiency, the control samples used consisted ofcosmic trigger events with vertical tracks crossing TPC 2. For vertical cluster efficiency, asample of CC inclusive events that originated in FGD 1 was used, to ensure mostly horizontaltracks that would be reconstructed with vertical clusters. The final cluster efficiencies areshown in Table 5.2.134TPC 1 TPC 2 TPC 3Data Efficiency 99.9+0.1−0.1% 99.7+0.2−0.7% 99.3+0.5−0.2%MC Efficiency 99.6+0.2−0.3% 99.5+0.3−0.4% 99.8+0.1−0.2%Table 5.3: Efficiencies for the TPC track reconstruction for data and Monte Carlo.[88]5.3.6 Time Projection Chamber Track Reconstruction EfficiencyThe event selection described in Chapter 4 uses TPC – FGD matched tracks for the basis of theselection. These tracks start as TPC-contained tracks, relying on only the TPC reconstructionmethods. Incorrectly reconstructing these can lead to missing or incorrectly classifying thetopology of events, and the main expected source of differences between data and Monte Carlois extraneous hits from δ ray production. The systematic uncertainty on these TPC tracks isapplied as an overall efficiency for successfully reconstructing a track in the TPC, which in-cludes potential effects from the likelihood fit and other sources. The TPC track reconstructionefficiency is fully correlated between the TPCs.The TPC track reconstruction efficiency is calculated using studies on control samplescomposed of through-going muons from the beam and cosmic events. Through-going cos-mic muons are used along with the beam events to measure the reconstruction efficiency whenthere are no other tracks in the TPC. To study the effect of track length on reconstruction effi-ciency, the reconstruction efficiency of cosmic muons with barrel ECal tracks is used[81]. Theefficiencies are computed separately for each TPC and the studies are described in [88]. Thestudies show high overall track reconstruction efficiencies and that there is no momentum orangle dependence. The efficiencies used for this analysis are listed in Table 5.3.5.3.7 Charge IdentificationThe reconstruction uses global track information to identify the charge on tracks in the TPC,described in [89], using local charge identification in the TPC and information from the globaltrack reconstruction to determine charge sign. There are two different sources of systematicuncertainty on the TPC track charge with some correlation between the two: the probability of135getting the local charge, or charge as determined from one TPC, incorrect, and the probabilityof the global charge identification incorrectly swapping the local charge sign. The correlationdepends on the number of TPC segments, as the charge for each segment is used in the globalcharge determination. For this reason, the systematic uncertainty is calculated differently de-pending on if the global track has one, two or three associated TPC segments[89]. However,the three cases are considered to be correlated with each other, as the reconstruction still usescommon information in each case. The efficiency is calculated as the probability that the globalcharge is different from the local reconstructed charge.Systematic studies were performed on a control sample of tracks starting in the P0D andcontaining at least one TPC segment[89]. These studies show that the uncertainty on the chargeidentification depends on the reconstructed momentum error; the parameters used in calculat-ing the uncertainty for the 1, 2 and 3 segment cases are listed in [89]. Because of this depen-dence, this systematic is considered 100% correlated between the different cases and 100%correlated between the FGD 1 and FGD 2 selections.5.3.8 Time Projection Chamber – Fine Grained Detector MatchingEfficiencyThe TPC – FGD track matching in the reconstruction creates tracks by matching FGD hits toa reconstructed TPC track as described in Section 4.3.1. The efficiency of this reconstructiondepends on both the efficiency of matching a TPC track to a single hit in an adjacent FGD,potentially failing to connect tracks to an FGD, and the efficiency for correctly matching a TPC– FGD track with a second TPC track, potentially incorrectly reconstructing the event vertexposition[90]. The uncertainty from incomplete matching causing out of fiducial volume eventmigration is not included here and is part of the out of fiducial volume (OOFV) backgroundsystematic described later in this sectionThe systematic uncertainty was computed using a control sample of high angle cosmicevents with one FGD 1 – TPC 2 – FGD 2 track in them. The systematics studies find that theefficiency for TPC – FGD matched tracks with more than two hits in the FGD to be 100%136FGD 1 FGD 2Data Efficiency 96.9±0.8% 96.5±0.85%MC Efficiency 97.6±0.45% 97.6±0.50%Table 5.4: Efficiencies for the TPC – FGD track matching for tracks with few recon-structed hits for data and Monte Carlo.[90]for when the events also has a second TPC track . This holds for both FGD 1 and FGD 2.The systematic uncertainty for the TPC – FGD matching is applied to tracks with two or lessmatched hits in the FGDs, with the efficiencies and uncertainties shown in Table 5.4. Becausethe matching efficiency only pertains to the last two layers in each FGD, FGD 1 and FGD 2share the same source of uncertainties on the matching efficiency from the uncertainty on thescintillator coating thickness and other scintillator bar properties. Therefore, this systematic is100% correlated between FGD 1 and FGD 2.5.3.9 Fine Grained Detector Particle IdentificationThe systematic uncertainty on the FGD particle ID described in Section 4.3.2 primarily comesfrom the difficulty in translating the measured deposited energy in an FGD into particle mass[91]. This results in an uncertainty on the measured particle pulls, which are derived from themeasured energy deposits. Measuring that deposited energy in turn depends on uncertaintyfrom the dead channels in the FGDs, the thickness of the scintillator coating and the chargereconstruction. Because the data to Monte Carlo charge reconstruction and scintillator coatingthickness differences are the same for FGD 1 and FGD 2, this systematic uncertainty is con-sidered to be correlated between the FGD 1 and FGD 2 selections. As there are no FGD-onlytrack cuts in the RHC ν¯µ and νµ selections, this systematic uncertainty does not apply.The systematic studies for the FGD particle ID used control samples of events with singlemuon and proton tracks which stop in either FGD 1 or FGD 2[91]. The TPC particle ID isused to create the control samples and can be compared with the FGD particle ID to find theefficiency of correctly identifying a muon or proton. The systematic error on the charged pionidentification uses the same numbers as measured for muons, as the FGD particle ID does not137strongly distinguish between the two. This systematic uses different uncertainties for tracks inFGD 1 and FGD 2.5.3.10 Fine Grained Detector Time of FlightThe average hit time between FGD 1 and FGD 2 is used to determine the track directionand select which FGD volume the initial neutrino interaction occurred in. As described inSection 4.3.3, a track that crosses both FGDs is reconstructed as backwards-going when theaverage hit time in FGD 1 is at least 3 ns greater than the average hit time in FGD 2. Thesystematic uncertainty on this time of flight directly impacts whether an event is reconstructedas occurring in FGD 1 or in FGD 2.To compute the systematic uncertainty, the standard neutrino interaction sample was usedinstead of a targeted control sample, selecting for events with tracks which passes through bothFGD 1 and FGD 2. The final uncertainty on the time of flight is found to be δ∆t12 = 0.78ns. This systematic variation is added directly to the reconstructed time of flight as δ∆t12×α ,where α is the random variable for the systematic variations. As this systematic uncertaintycomes from the timing in FGD 1 and FGD 2, the time of flight is considered 100% correlatedbetween the FGD 1 and FGD 2 selections.5.3.11 Fine Grained Detector Hybrid Track EfficiencyThe hybrid track efficiency is defined as the efficiency of reconstructing a true FGD-containedtrack in the presence of a TPC – FGD matched track, as all relevant FGD-contained tracks inthe selection are accompanied by a TPC – FGD matched track. As the selection does not lookfor solitary FGD-contained tracks, that efficiency is not used. This systematic only applies tothe FHC νµ selection as there are no FGD-contained track cuts in the RHC selections. As theselected FGD-only tracks are expected to be either protons or pions, the systematic uncertaintystudies were performed separately for each case. The studies were only performed for the FHCrunning mode, as the RHC selections do not include any FGD-contained track cuts.The control sample used to study the efficiency consists of events with either one recon-138structed track that enters a TPC, or two tracks which both enter a TPC. As these events do notcontain any FGD-only tracks, instead stopping pion and proton tracks were generated usinga particle gun in GEANT4 [91] for each event. These particles were generated as isotropicand with uniform energies, with further details in [91], with a lower momentum bound of 400MeV for protons due to reconstruction limitations below that momentum. The hits from thesetracks were injected into the control sample events, both data and Monte Carlo, and the FGD-only track reconstruction run. The efficiency is calculated as the ratio of the number of eventswith at least one reconstructed FGD-only track over the total number of selected events in thecontrol sample. As the FGD track reconstruction has a strong dependence on the track angle,the efficiency is binned in cosθ , with θ being the true angle between the FGD-only track andthe muon candidate. The efficiency and uncertainty are also calculated separately for pion andproton tracks [91], with overall systematic uncertainties on the track efficiency ratios being <8%.Additionally, there is a significant difference in data – Monte Carlo efficiency differencesbetween FGD 1 and FGD 2 due to geometry differences. The lower active volume of FGD 2leads to both a lower track efficiency and to lower data – Monte Carlo differences than seenin FGD 1. As this is not currently well understood, there is no correlation between FGD 1and FGD 2 for this systematic. However, as the tracking efficiency also depends on underlyinguncertainties on the scintillator coating and overall hit efficiencies, there is 100% correlationimplemented for all angular bins within each particle[81].5.3.12 Michel Electron Efficiency and PurityAs described previously in Section 4.5.3, Michel electrons from pi+ decays (pi → µ → e) areidentified by looking for sufficient delayed FGD hits after an interaction is seen. As this cutis only used in the FHC νµ selection, the systematic studies have only been performed forthe neutrino mode running period. The Michel electron systematic is two separate systematicsrelating to the Michel electron cut that are applied together; one is the difference in efficiency139FGD 1 FGD 2Monte Carlo 56.5±1.9% 41.4±1.4%Data 56.4±1.6% 42.8±1.1%Table 5.5: Michel detection efficiencies for Monte Carlo and data[82].of selecting the Michel electrons through this cut and the other the difference in the purity ofdelayed time bins selected by the Michel electron cut for data and Monte Carlo. In particular,the initial studies used to construct the Michel electron cut indicate there may be significantdata and Monte Carlo differences in the purity.Michel Electron EfficiencyThe Michel electron detection efficiency depends on how likely a produced electron is to leavesufficient hits to pass the cut and therefore be detected. To study this, a control sample ofcosmic muons stopping in FGD 1 and FGD 2 was used. As the high angle cosmic tracks donot necessarily pass through a TPC, a particle ID cut on the track length and momentum wasused[82] to identify muons. This particle ID cut was based on studies of the muon lifetimeand associated track lengths and momenta in the FGDs, as cuts using only the FGD particle IDmuon pulls gave insufficiently pure muon samples.The efficiency from these studies is defined as the probability of detecting an expectedMichel electron given a known stopping muon in the FGD. The efficiencies are calculatedseparately using cosmic triggers from runs 2, 3 and 4 in data as the Michel purity is separatedbased on beam power. Because the cosmic samples do not depend on beam power, the averageefficiency of these samples is used for data at all beam powers. The final efficiencies are shownin Table 5.5.Michel Electron PurityThe primary source of background for identified Michel electrons in the Monte Carlo comesfrom out-of-FGD beam interactions in the near detector and magnet volumes, sand muon in-teractions, and cosmic muons. These particles can enter from outside of the FGD and leave140FGD 1 FGD 2MC (×10−3) Data (×10−3) MC (×10−3) Data (×10−3)Run 2 0.602±0.008 1.61±0.05 0.314±0.006 1.21±0.05Run 3 0.714±0.009 1.80±0.04 0.397±0.006 1.31±0.03Run 4 0.729±0.009 2.00±0.07 0.387±0.006 1.50±0.06Table 5.6: False Michel electron identification rates for data and Monte Carlo. Rates aredefined as the number of expected false Michel electrons per spill.large enough energy depositions to be misidentified as a Michel electron. To isolate and studythe external background, empty beam spills from the neutrino beam data were selected, ratherthan events. Spills were required to have no reconstructed tracks within the beam bunches inorder to remove any true Michel electrons originating from beam νµ interactions in the FGDs.The same selection was used for both Monte Carlo and data, as no truth information is used.There are significant differences in the simulated background for out-of-bunch particles,both at the high energy and low energy regions. The Monte Carlo does not simulate the highenergy tail seen in the data – this is potentially due to coincident hits left by through goingcosmics, which is not included as a background in the ND280 Monte Carlo. This can be seenwhere the simulated hit distributions stop at 30 for FGD 1 and 25 for FGD 2 (due to loweractive volume) in Figure 5.1. These particles have roughly the same incidence rate in FGD 1and FGD 2, and no dependence on the beam power. This can be seen in the rates in Table 5.6,where the rates for FGD 1 are around double those for FGD 2 for Monte Carlo, but much closerfor data.Additionally, the false identification rate in data depends on the beam power for both MonteCarlo and data. The Monte Carlo rates for run 3 and run 4 are expected to be similar, as thesame beam configuration was used to simulate them. For data, the total protons on target perspill increases between run 3 and run 4 and thus shows an increase in false identification rate.Excess Simulated BackgroundThe primary background that affects the purity of the Michel electron identification is thehigh energy tail described previously, likely originating from cosmics. However, the external141(a) FGD1(b) FGD 2Figure 5.1: Total number of hits in out-of-bunch timebins from empty spills. Plots arenormalized to data POT, and were created for a subset of the run 3 data.background studies for the Michel electron cut also reproduce the large excess in the simulatedbackground at low numbers of hits seen previously in Section 4.5.3. There, it was a main factorin choosing the minimum number of hits to cut on due to this discrepancy between data andMonte Carlo.142As is shown in Fig. 4.6, and in Fig. 5.1, the shape of the simulated background at lowhits is similar to that seen in data, but with a significant increase in magnitude. A large partof this excess comes from neutrons which produce γs which can be detected. Various otherparticles such as protons and pions contribute as well, though as the Monte Carlo did not saveall relevant information for delayed hits it was not always possible to determine the simulatedsource of the delayed timebin hits. Because of this, it is difficult to understand exactly whatpart of the Monte Carlo is incorrectly tuned and therefore how to fix it. Until this is understoodin more detail and can be fixed, this places an artificial lower limit on the Michel electron cutdue to a lack of understanding of the actual cut purity below that point.5.3.13 Sand MuonsThe main ND280 Monte Carlo simulation only includes beam events that occur in the ND280volume itself, including the surrounding magnet. However, neutrinos from the beam can inter-act with the sand surrounding the detector pit, and the walls of the pit where ND280 is situated.Tracks from these interactions can appear similar to neutrino interactions in the FGDs, and area potential background for the analysis. As the primary source is from interactions with thesurrounding sand, these background events are referred to as sand muons. Because these arenot included in the main T2K Monte Carlo for the near detector, there is a separate dedicatedMonte Carlo sample of sand muon events to estimate the effects on the analysis samples.The sand muon Monte Carlo allows the total event rate from sand muons to be estimatedfor the analysis sample by running the selections described in Chapter 4 on the Monte Carloand scaling to the data protons on target to get the predicted sand muon event rate. To estimatethe systematic uncertainties on this background, the difference between rates in Monte Carloand data is used for positive tracks and negative tracks separately. The uncertainty on the sandmuon rate is 10% for neutrino mode and 30% for antineutrino mode [81] and is accountedfor as a normalization systematic. As the interactions occur outside the detector volume, theunderlying cause for uncertainties is the same for FGD 1 and FGD 2, with no correlations143between neutrino and antineutrino mode.5.3.14 Event Pile UpEvent pile up occurs when an out of fiducial volume event is coincident with a in-fiducial vol-ume CC Inclusive event in either FGD. These coincident events can lead to CC Inclusive eventsbeing removed from the selection sample by the external veto cut described in Section 4.5.2.For both the FHC and RHC selections, the primary source of this pile up comes from the sandmuons [81]. This uncertainty depends on the beam intensity and horn current – within FHCor RHC, the source of the systematic uncertainty is similar for different intensities, but notnecessarily between neutrino and antineutrino mode. Therefore a 100% correlation is usedbetween beam intensity, and 0% correlation between FHC and RHC. Like the sand muon ratesystematic, a correction is applied to the Monte Carlo to account for this.Because there is a 10% uncertainty on the sand muon simulation for νµ and 30% uncer-tainty for ν¯µ , along with other possible differences between the sand muon simulation and datasuch as beam intensity, there is a systematic uncertainty on the correction. This systematic usesthe difference between data and Monte Carlo, as calculated using a comparison of the numberof TPC 1 or TPC 2 events per bunch in data and Monte Carlo.5.3.15 Fine Grained Detector MassThe FGD mass systematic is the uncertainty on the scintillator and water module areal densi-ties due to differences in the simulated FGD volumes and the real detector volumes [92]. Theuncertainty of the scintillator XY modules affects both FGD 1 and FGD 2, while the uncer-tainty on the water volume density is only applicable to FGD 2. These are implemented asuncertainties on the densities of modules, including coating for the scintillator modules and thepolycarbonate water module vessels.The uncertainty on an XY module is calculated from the data – Monte Carlo difference inmass, 0.41% [92] and the spread in direct mass measurements of all of the XY modules, 0.38%,giving an overall 0.6% systematic uncertainty on the XY modules in the FGDs. The water144module mass has a difference of 0.26% between data and Monte Carlo, and an uncertainty ofthe water module mass of 0.46% mostly from the masses of the plastic and glue [93], leadingto an uncertainty of 0.55% on the FGD water module mass. The water and scintillator massuncertainties are independent from each other, as the uncertainty is assigned based on the truevertex position[81]. As the source of the uncertainty on the scintillator mass is based on themodule measurements and therefore the same for FGD 1 and FGD 2, the scintillator massvariation is considered to be 100% correlated between the FGD 1 and FGD 2 selections. Thewater mass variation is also considered to be 100% correlated between the FGD 1 and FGD 2selections.5.3.16 Pion Secondary InteractionsWhen pions are produced from interactions in ND280, they can interact elsewhere in the detec-tor; this is referred to pion secondary interactions (SI) and can result in detection inefficienciesfor pions in the FGDs. This can be through absorption, decay, quasi-elastic interactions andother methods[94]. As pion detection is used as the main criteria for topology definition andselection as described in Chapter 4, correctly modeling this in the Monte Carlo is important.The model using GEANT4 for the pion interactions does not agree well with existing dataon pion interactions on nuclei[94] and so a correction weight is applied to the events. As theuncertainty depends on the target nuclei, this systematic is uncorrelated between target typebut is considered 100% correlated between FGD 1 and FGD 2 for each target. The systematicuncertainty is calculated both from the studies comparing the ND280 Monte Carlo and dataand uncertainties on the external data on pion interactions. The event weight is calculated asa product of the probabilities for each true pion trajectory in an event either interacting or notinteracting for different cross section models. These probabilities are energy dependent andcalculated every 0.1 mm along a given trajectory. The process is detailed in [94].145Background Origin FHC νµ RHC ν¯µ RHC νµP0D 5.1% 8.4% 5.4%ECal 11.6% 8.8% 6/7%SMRD 4.9% 6.7% 4.8%Other 13.6% 13.5% 24%Table 5.7: Rate uncertainties for OOFV events by origin.[95]5.3.17 Out of Fiducial Volume BackgroundPart of the background for CC Inclusive for both the FHC and RHC selections originates fromoutside of the FGD fiducial volume (OOFV). In these cases, an interaction has occurred some-where else in the near detectors but has been reconstructed as starting in either the FGD 1 orFGD 2 fiducial volume. This also includes interactions in the first two layers of FGD 1 andthe first layer of FGD 2. There are two main contributing types of uncertainty on the back-ground OOFV rates: uncertainty on the interaction rates in the other detectors at ND280, anduncertainty on certain classes of reconstructed events in the FGDs.The rate uncertainty is separated into four categories: interactions where the true vertexoccurred in the P0D, the ECal, the SMRD or other near detector volumes. The relative dif-ference for the interaction rate for data and Monte Carlo was studied using the neutrino beamdata rather than control samples. The difference for each interaction source is used as the un-certainty on the interaction rate. The rate uncertainties are shown in Table 5.7. As these arefrom non-FGD sources, the underlying physical sources of uncertainties are the same for theFGD 1 and FGD 2 selections and this systematic is fully correlated between the two with nocorrelations between different background rate sources.The reconstruction uncertainties are the probabilities that some part of the reconstruction,generally the TPC – FGD matching, has failed and an OOFV event has been reconstructed asoccurring within the fiducial volume. The reconstruction uncertainties are split into several dif-ferent categories depending on the true vertex location and type of reconstruction failure. Notall types of OOFV events have a significant reconstruction uncertainty even if they contributeto the overall background, such as events occurring in the FGD but outside the fiducial volume146Background Category FGD 1 FGD 2Downstream event 5% 5%High Angle Event 33% 28%Last Module Failure 35% 17%Consecutive skipped layers 55% 82%Hard scattering 32% 21%Table 5.8: Reconstruction uncertainties (MC/data difference) for OOFV events byorigin.[95]or events originating in the upstream tracker. The OOFV background categories that do havesome reconstruction uncertainty are events originating in the tracker components downstreamof the FGD, high angle events, failure to match events with most hits missing in the FGD (re-ferred to as last module failure), events which hard scatter in the FGD and events where no hitsare deposited for two consecutive layers.In general, the reconstruction rate and uncertainty do not depend on the track momentum sothe systematic uncertainty is calculated for each category[95]; these categories are sufficientlydifferent that there is no correlation between the categories. As these rely mainly on the hitefficiencies in the FGDs, each category is 100% correlated between FGD 1 and FGD 2. Thereconstruction uncertainties are measured using events known to have started outside the FGDfiducial volume, such as cosmic events and comparing the reconstruction rate for data andMonte Carlo. The measured uncertainties are shown in Table 5.8; all other categories of OOFVevents have a reconstruction uncertainty of 0%.5.4 The Observable Normalization Covariance MatrixAs discussed in Section 3.3.3, the near detector fit does not directly use the underlying system-atic parameters as implemented in the Psyche framework for computational resource reasons.Instead, the detector systematic parameters are fit in the form of a set of observable normal-ization parameters, binned in p – θ and separated by sample, and a covariance matrix thatdescribes the uncertainties and correlations for these bins. The binning used for the observablenormalization parameters and covariance is more coarsely binned than the binning used for the147FHC samples as described in Section 4.5.3, primarily for computational reasons. The selectionbinning is not sufficiently coarse to use in the fit, as it is designed to have all non-zero binsfor the MC prediction, which would give 531 normalization parameters for the FGD 1 samplesalone. When FGD 2 samples are included, this would lead to fitting 1062 parameters alongwith the flux and cross section. For this reason, the FHC samples for both FGD 1 and FGD 2are rebinned to have ten bins in p and seven bins in cosθ for each sample :• FHC sample bin edges for the detector covariance:p (MeV/c) : 0, 300, 500, 600, 700, 900, 1000, 1500, 3000, 5000, 30000cosθ : -1.0, 0.6, 0.8, 0.85, 0.9, 0.94, 0.98, 1This gives 70 bins for each FHC sample in FGD 1 and in FGD 2 and cuts off at a momentumof 30000 MeV. All FHC samples have the same binning. The binning for the RHC samples isnot changed, as fewer bins are used for those samples in the fit selection and reducing thesewill have less effect on computational time. These bins are:• RHC 1-track sample bin edges for the detector covariance:p (MeV/c) : 0, 500, 900, 1200, 2000, 10000cosθ : -1.0, 0.8, 0.92, 0.98, 1and• RHC N-tracks sample bin edges for the detector covariance:p (MeV/c) : 0, 600, 1000, 1500, 2200, 10000cosθ : -1.0, 0.8, 0.9, 0.97, 1giving 20 bins in each RHC sample for both the ν¯µ and νµ selections. In the case that multipleselection bins are covered by a single covariance matrix bin, the same observable normalizationweight is used for all covered bins, as choice of bin depends on the reconstructed momentumand angle.1485.4.1 Systematic VariationsTo generate the prior values and the detector covariance matrix using the systematics and corre-lations described in this section and Chapter 5, 2000 sets of variations of the detector systematicparameters are first thrown with a Gaussian distribution with mean 0 and standard deviation of1. The correlations described previously are then applied using Cholesky decomposition[96].As the analysis now includes samples from both FGD 1 and FGD 2, systematic correlationsbetween the two detectors are somewhat complicated to correctly account for. To do so, twosets of detector systematic parameter variations are generated for each throw, with one set ofvariations applied for FGD 1 selected events and the other set of variations for FGD 2 selectedevents. In addition to the correlations between FGD 1 and FGD 2 samples, correlations be-tween detector systematic parameter bins are now also applied. Bin-to-bin correlations arealso new to this analysis, and are described along with the FGD 1 – FGD 2 correlations earlierin this chapter in Section 5.2.The observable normalization values are calculated using the mean values for the p – θbins as :dnomi =NmeaniNnomi(5.10)and the observable normalization covariance matrix elements as:(Vd)i j =11999Σ(Nki −Nmeani )(Nkj −Nmeanj )Nmeani Nmeanj(5.11)This gives a correlated set of p–θ parameters to use in the fit. The observable normalizationparameters and covariance are used in the fit with the assumption that the overall effect ofvariations in the systematics is Gaussian in all bins; this is roughly true for the p – θ binswith large numbers of events, while bins with less than 20 events on average show a morenon-Gaussian shape. In addition to the detector systematics information, we also include a fewother sources of uncertainties in the observable normalization parameter covariance matrix.149These are the Monte Carlo statistical uncertainties, from the finite size of the Monte Carlo, andan additional shape uncertainty from the 1p1h model differences, described in Section 3.3.3.The Monte Carlo statistical errors are included on the diagonal of the observable normalizationmatrix, and are calculated using the fit binning rather than the covariance matrix binning toavoid underestimating the uncertainties. This only affects the FHC samples as the binningis identical for RHC. The size of these is shown in Figure 5.2. The covariance is shown inFigure 5.3, without the Monte Carlo statistical errors on the diagonal.ND280 detector systematic bin0 100 200 300 400 500Fractional uncertainty00.050.10.150.20.250.3MC statistical errorFigure 5.2: The fractional error due to the statistical uncertainty in the MC prediction foreach of the ND280 detector systematic p – θ bins. The solid black lines separatethe CC0pi , CC1pi and CC Other sample, the dashed lines separate the RHC ν¯ CC 1-Track, RHC ν¯ CC N-Tracks, RHC ν CC 1-Track and RHC ν CC N-Tracks samples,and the solid red line separates FGD1 and FGD2 samples[67].5.4.2 Detector Correlations in the ND280 FitThe addition of FGD 1 – FGD 2 selection correlations and bin-to-bin correlations has a sig-nificant effect on the observable normalization parameter covariance matrix used to model thedetector systematics in the ND280 fit. The correlations between the p – θ bins for the selectedsamples are shown without the FGD 1 – FGD 2 and bin-to-bin correlations in Figure 5.4a andwith the new correlations described previously in Figure 5.4b.150, FHCpiFGD 1 CC-0, FHCpiFGD 1 CC-1FGD 1 CC-Other, FHC CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1  CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1 , FHCpiFGD 2 CC-0, FHCpiFGD 2 CC-1FGD 2 CC-Other, FHC CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2  CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2 , FHCpiFGD 1 CC-0, FHCpiFGD 1 CC-1FGD 1 CC-Other, FHC CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1  CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1 , FHCpiFGD 2 CC-0, FHCpiFGD 2 CC-1FGD 2 CC-Other, FHC CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2  CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2 ) ijsign(V×| ij|V-0.100.10.20.30.4Figure 5.3: Detector covariance matrix calculated using the correlated systematic uncer-tainty variations. This does not include MC statistical errors, or the 1p−1h shapecovariance.151Parameter Bin Number0 100 200 300 400 500Parameter Bin Number0100200300400500-1-0.8-0.6-0.4-0.200.20.40.60.81(a) Detector correlation matrix for all samples without bin-to-bin orinter-FGD correlations, without 1p-1h correlations.Parameter Bin Number0 100 200 300 400 500Parameter Bin Number0100200300400500-1-0.8-0.6-0.4-0.200.20.40.60.81(b) Detector correlation matrix with updated correlations used, without1p-1h correlations.Figure 5.4: Detector correlation matrices without the 1p-1h errors included. This com-pares the current treatment of detector correlations with how detector correlationswere implemented in the last analysis. Labels indicate the bin number; starting binsfor each sample are: 0 - FGD1 CC0pi , 70 - FGD1 CC1pi , 140 - FGD1 CC Other,210 - FGD1 RHC ν¯ CC 1-Track, 230 - FGD1 RHC ν¯ CC N-Tracks, 250 - FGD1RHC ν CC 1-Track, 270 - FGD1 RHC ν CC N-Tracks, 290 - FGD2 CC0pi , 360 -FGD2 CC1pi , 430 - FGD2 CC Other, 500 - FGD2 RHC ν¯ CC 1-Track, 520 - FGD2RHC ν¯ CC N-Tracks, 540 - FGD2 RHC ν CC 1-Track and 560 - FGD2 RHC νCC N-Tracks. The bins within each sample are ordered in increasing momentumintervals, each containing all angular bins from backward going to forward going.152Chapter 6Fitter ValidationBefore passing the ND280 fit to data to the oscillation analysis at SK to use as an input, thereis extensive validation of the fit machinery. This is done by performing multiple Monte Carlostudies and verifying the results from those. There are two types of validation studies done forthe near detector fit:1. Fits to Nominal Monte Carlo2. Parameter Pull StudiesEach type of study aims to validate specific aspects of the overall fit machinery and methods.Fits to nominal Monte Carlo inputs serve to check potential issues with the fitter code as wellas check the expected reduction in overall uncertainties. Finally, the parameter pull studies usean ensemble of toy fits to both check for potential code or input issues that were not seen in thenominal validation fit, as well as check fitted parameter biases. The parameter pull studies arealso used to provide the p-value for the data fit to verify that the data fit ∆χ2 falls within theexpected distribution for the fit model used.6.1 Nominal Monte Carlo Validation FitThe initial validation study done is a fit to the nominal MC; this is a check that the fitter returnsfitted parameter values identical to their input values. The nominal dataset is defined here as153the dataset where the fake data bin content is identical to the predicted bin content predicted bythe ND280 fit with all parameters set to their nominal values. To ensure that the machinery isworking, this should ideally return identical parameter values to within machine precision, andreduced errors on those parameters relative to their priors. The fitted errors indicate how mucherror reduction can be expected in the data fit.The flux, cross section and detector observable normalization parameter inputs and priorsused for this validation study are identical to those that will be used in the final data fit. Inaddition, the CCQE parameters are left unconstrained in this fit, as will be done for the data fit.The results of this study are shown in Figures 6.1, 6.2 and 6.3, and all flux, cross section andobservable normalization parameters behave as expected. All fitted parameter values, includ-ing the unconstrained CCQE parameter values, are fitted to within 10−11 of their initial inputvalues, which is close to machine precision. Overall uncertainties on parameters are reduced,and the fit introduces an overall anticorrelation between the flux and observable normalizationparameters., FHCµνND280 , FHCµνND280 , FHCeνND280 , FHCeνND280 , RHCµνND280 , RHCµνND280 , RHCeνND280 , RHCeνND280 , FHCµνSK , FHCµνSK , FHCeνSK , FHCeνSK , RHCµνSK , RHCµνSK , RHCeνSK , RHCeνSK Parameter Value0.80.850.90.9511.051.11.151.2 PrefitPostfitFigure 6.1: Nominal Monte Carlo dataset validation for the ND280 fit, showing flux pa-rameters. Axis labels show the first bin of each category of flux parameters.154MAQEpF_CMEC_CEB_CpF_OMEC_OEB_O CA5MANFFRESBgRESCCNUE_0CCOTHER_SHAPECCCOH_O_0NCCOH_0NCOTHER_NEAR_0MEC_NUBARFSI_INEL_LO_EFSI_INEL_HI_EFSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EParameter Value-0.500.511.52 PrefitPostfitFigure 6.2: Nominal Monte Carlo dataset validation for ND280 fit, showing cross-sectionparameters. Axis labels show the name of each parameter; parameters are typicallya ratio to NEUT nominal values, excepting the CC Other Shape parameter and theFSI parameters. There is no prior used in the fit for the CCQE parameters; the priorerror bands shown for these parameters are the error bands produced as potentialcross section inputs and are included here for comparison purposes only.6.2 Parameter Pulls and p-value CalculationIn order to check that the fitted model is in agreement with the data, and to check for anyparameter biases or other fitter issues, a large number of toy data sets are generated to coverpossible parameter variations given the parameter priors and are fit using the same flux, crosssection and observable normalization inputs as the final data fit. The main use for these toysis to conduct pull studies, where an ensemble of fitted parameter values can be compared withtheir priors to understand potential fit biases and other issues, and to construct a p-value for thefinal data fit from the ∆χ2 distribution of these fits. The two types of studies use different toydata sets, but throwing the parameters works the same for both.Parameter pulls are defined for a parameter x aspull =x−µσ(6.1)155, FHCpiFGD 1 CC-0, FHCpiFGD 1 CC-1FGD 1 CC-Other, FHC CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1  CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1 , FHCpiFGD 2 CC-0, FHCpiFGD 2 CC-1FGD 2 CC-Other, FHC CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2  CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2 Parameter Value0.50.60.70.80.911.11.21.31.41.5 PrefitPostfitFigure 6.3: Nominal Monte Carlo dataset validation for the ND280 fit, showing detectornormalization parameters. Axis labels show the first bin of each category of detectorparameters. Values are fit to the normalization weights for each p – θ bin and aregenerally not expected to equal to 1.and are valid when parameter value has been thrown as a Gaussian distribution with a mean µand width σ [97]. For a fitted parameter value d, the pulls are calculated aspull =d f it−dnominalσ f it(6.2)and thus calculated relative to this fixed prior toy value[97]. In this case the pulls comparethe fitted values to the prior mean of the parameter distribution. In the case of parameters fitwithout a prior constraint, the pulls are calculated the same way. However, as there is no priordistribution to throw the parameter with, and the parameter does not contribute directly to thefit ∆χ2, the nominal value is the fixed parameter value used in the toy throws.The parameter pull values from a set of toy parameter fits should have a standard nor-mal distribution with a mean value of 0, meaning the fitted parameter values were distributedaround the prior constraint value, and a width of 1, matching the thrown distribution of theseparameters. Parameters that are fit close to their physical boundaries or are poorly constrained156by the fit will often have a pull width of less than 1, and the mean may be shifted above 0. Ifthe pull distribution means are offset from 0, this can indicate that there is a bias in that fittedparameter value. Pull distribution widths greater than 1 can also indicate that there are issueswith toy fitter or ensemble. Pulls for parameters with no prior constraints are also expected tohave a Gaussian distribution with a mean of 0 and width of 1.The toy data set ensemble is generated by making Gaussian throws of the flux, cross sectionand detector systematic parameter variations according to their prior values and uncertainties.The exception to this are the CCQE parameters - as unconstrained parameters do not contributeto the overall ∆χ2 term minimized in the fit, throwing these parameters should not have an im-pact on the overall p-value. Instead, the CCQE parameters are fixed at their NEUT nominalvalues shown in Table 3.4 when throwing. Additionally, while the detector systematic param-eter variations described in Chapter 5 are each thrown using a Gaussian distribution, this doesnot necessarily translate to expected Gaussian distributions for the fitted detector observablenormalization parameters. Because the detector systematic parameters are thrown using thePsyche framework, the 1p-1h shape uncertainty that is included in the observable normaliza-tion covariance matrix is thrown and applied as a separate shape covariance.To make the Monte Carlo statistical throws, a Poisson throw with a mean of 1 is made andmultiplied with each individual toy event weight after all parameter throws are made and ac-counted for. This changes the overall number of reweighted events in the p – cosθ sample bins.The final generation step is to include the data statistical uncertainties, as the data is expectedto follow a Poisson distribution for each bin and is accounted for in the ND280 likelihood. Foreach bin, a Poisson throw is made using the total toy event weight in that bin as the mean. Theresult from this throw replaces the total event weight in that bin.While the treatment of the flux and cross section parameters is Gaussian in both the thrownvariations and the fit itself, this is not true of the detector systematic parameters. As described inSection 3.3.3, the ND280 fitter uses a Gaussian covariance matrix binned in p–cosθ to handlethe observable normalization parameters for computational reasons. However, this does not157account for non-Gaussian behavior in the detector systematic parameters when varied; for thisreason, we vary the detector systematic parameters separately in Psyche, rather than throwingfrom the observable normalization values and covariance. For these variations the correlationsas described in Sec. 5.2 are used.To perform the p-value studies, 500 toy data sets, each thrown independently from eachother, are produced and fit with the ND280 fitting software. These toys consist of sets of fakedata constructed using the thrown parameter values, which are then fit to the nominal priorused in the data fit. Of these 500 toy data sets, 461 fits successfully finished and converged forthe pull studies, and 444 fits successfully finished and converged for the p-value study. This ispartially due to computing issues, as these fits can potentially take over 48 hours to run, whilethe computing resources used for these studies have a runtime cutoff of 48 hours. This comesfrom the addition of the FGD 2 samples, doubling both the size of the Monte Carlo inputsand number of selected data events, and doubling the number of observable normalizationparameters. For the toys that failed, there were no obvious features in either the thrown eventrates or the thrown parameter values compared to successful toy fits.A comparison of the ∆χ2 from the toy data sets described previously with toy fits wherethe detector parameter variations are thrown using the observable normalization parametersinstead of the underlying Psyche parameters is shown in Fig 6.4. This shows that the toysthrown from the detector covariance matrix do not reproduce the non-Gaussian tail seen in the∆χ2 distribution when Psyche detector systematic variations are thrown and that throwing thePsyche detector systematic variations is necessary to accurately represent the ∆χ2 range forfinding the p-value.The parameter pulls are calculated using different toy data sets and priors that the ∆χ2distributions. In these toy data sets, parameters are varied in accordance to their prior valuesand uncertainties, and statistical uncertainty from data and Monte Carlo are accounted for inorder to fully represent the model used for fitting and the uncertainties on the priors. This isin place of throwing the various parameters to produce toy data sets and fitting to the nominal1582χ∆1000 1500 2000 2500 3000Number of toy experiments051015202530 Observable Norm ThrowsDetector Parameter ThrowsFigure 6.4: ∆χ2 distributions for toys with observable normalization parameters thrownin red and Psyche detector systematic parameters thrown in black. The ∆χ2 distri-bution for the observable normalization throws is from 114 toy fits and the distribu-tion for the fits with Psyche throws has been scaled to match overall area.prior.Figures 6.5, 6.6 and 6.7 show the mean and width from a Gaussian fit to the individualparameter pull distributions. For these toys, instead of throwing the parameter values used ingenerating toy data sets, the fit prior values are thrown instead. The parameters are thrownidentically to the parameters in the p-value toys, but are not applied to generate fake data. Thiscorresponds with the definition of a robust pull study[97]. Unlike the toy fits for determiningthe p-value, the underlying detector systematic parameters are not thrown, as they are not usedas priors in the fit. Instead, the observable normalization parameters are thrown – with theMonte Carlo statistical error and 1p-1h corrections in the covariance – and used as the prior forthe toy fits. Each toy fit has its own set of thrown prior values, and is fit to the nominal MonteCarlo with data statistical throws applied.Pull widths are distributed around 1 as expected for flux and detector normalization param-eters. For cross section parameters pF C and pF O, a width less than 1 due to low constraint159power in the fit and a biased mean due to physical fit boundaries is expected. Similarly, MECC and MEC O have fit boundaries that slightly affect the pull mean and width. Additionally,the pull means are distributed around 0, indicating that the model used for fitting has little orno bias. There is a slight increase in the pull means corresponding to an increase of 1% in theflux parameters; this may be due to some nonlinearities in the parameters but should not havea major impact on the validity of the data fit., FHCµνND280 , FHCµνND280 , FHCeνND280 , FHCeνND280 , RHCµνND280 , RHCµνND280 , RHCeνND280 , RHCeνND280 , FHCµνSK , FHCµνSK , FHCeνSK , FHCeνSK , RHCµνSK , RHCµνSK , RHCeνSK , RHCeνSK Pull mean or width00.20.40.60.811.2Pull meanPull widthFigure 6.5: Pull means and widths for all flux parameters. The pull distributions are madefrom the results of fits to 461 toy datasets.160FSI_INEL_LO_EFSI_INEL_HI_EFSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EMAQEpF_CMEC_CEB_CpF_OMEC_OEB_O CA5MANFFRESBgRESCCNUE_0CCOTHER_SHAPECCCOH_O_0NCCOH_0NC1GAMMA_0NCOTHER_NEAR_0NCOTHER_FAR_0MEC_NUBARPull mean or width-0.4-0.200.20.40.60.811.2FSI_INEL_LO_EFSI_INEL_HI_EFSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EMAQEpF_CMEC_CEB_CpF_OMEC_OEB_O CA5MANFFRESBgRESCCNUE_0CCOTHER_SHAPECCCOH_O_0NCCOH_0NC1GAMMA_0NCOTHER_NEAR_0NCOTHER_FAR_0MEC_NUBARPull mean or widthFSI_INEL_LO_EFSI_INEL_HI_EFSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EMAQEpF_CMEC_CEB_CpF_OMEC_OEB_O CA5MANFFRESBgRESCCNUE_0CCOTHER_SHAPECCCOH_O_0NCCOH_0NC1GAMMA_0NCOTHER_NEAR_0NCOTHER_FAR_0MEC_NUBARPull mean or widthPull meanPull widthFigure 6.6: Pull means and widths for all cross section parameters. The pull distributionsare made from the results of fits to 461 toy datasets., FHCpiFGD 1 CC-0, FHCpiFGD 1 CC-1FGD 1 CC-Other, FHC CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1  CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1 , FHCpiFGD 2 CC-0, FHCpiFGD 2 CC-1FGD 2 CC-Other, FHC CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2  CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2 Pull mean or width-1-0.500.51, FHCpiFGD 1 CC-0, FHCpiFGD 1 CC-1FGD 1 CC-Other, FHC CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1  CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1 , FHCpiFGD 2 CC-0, FHCpiFGD 2 CC-1FGD 2 CC-Other, FHC CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2  CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2 Pull mean or widthPull meanPull widthFigure 6.7: Pull means and widths for all observable normalization parameters. The pulldistributions are made from the results of fits to 461 toy datasets.161Chapter 7Final Fit Results at the ND280 Detector7.1 Finalized InputsThe final inputs to the data fit at ND280 are as described in Chapter 3, with flux and cross-section parameters using priors from external experiments and the detector observable nor-malization parameters using prior constraints from ND280 detector systematic studies. Whenfitting the ND280 samples, the flux, cross-section and detector observable normalization pa-rameters are allowed to vary. All of the flux and detector observable normalization parametersare constrained by input priors; additionally, all cross-section parameters are also constrainedusing prior inputs with the exception of the CCQE parameters: MAQE , 2p-2h12Carbon, 2p-2h16Oxygen, pF 12Carbon, pF 16Oxygen and the 2p-2h ν¯ normalization. The rationale for thisis explained in Section 3.3.2 – for these parameters, the data from other experiments is incon-sistent, so only ND280 data is used to constrain them. These parameters are fit without a priorvalue or uncertainty and do not contribute to the minimized ∆χ2. In the fit, data from the for-ward horn current and reverse horn current beam modes are considered simultaneously. Theselections and samples described in Chapter 4 for FGD 1 and FGD 2 are used.1627.2 Results at the ND280 DetectorThe selected data event totals, along with the prefit Monte Carlo and postfit Monte Carlo eventtotals, are shown in Table 7.1. Fitted parameters are shown in Figs. 7.1, 7.2, 7.3 and 7.4.Tables with the fitted parameter values are listed in Appendix C. The data fit shifts the fittedflux parameter values above their prior bounds for all parameters, by approximately 1.5 σ .This increased flux effect was seen in the previous analysis and does not appear to be relatedto the ∼ 1% bias seen in the flux parameters in Chapter 6. Additionally, for all flux types asidefrom the FHC νµ and νe, the shape of the flux parameters shows a slight increase towards thehigher energy flux bins for the flux at both SK and ND280.The near detector fit at T2K finds an axial mass of MA = 1.1113±0.033, which is consistentwith previous T2K measurements of MA as well the K2K measurement on oxygen with MA =1.20± 0.12[98]. As with the K2K measurement, this is above the global average value ofMA = 1.03±0.02[24], and lower than that measured by MiniBooNE[69].163Sample Data ND280 prefit MC ND280 postfit MCFGD 1 νµ CC Inclusive (ν mode) 25558 25420.74 25607.36FGD 1 νµ CC0pi (ν mode) 17354 16950.81 17344.83FGD 1 νµ CC1pi (ν mode) 3984 4460.15 4112.98FGD 1 νµ CC Other (ν mode) 4220 4009.78 4149.55FGD 1 ν¯µ CC Inclusive (ν¯ mode) 3438 3506.38 3424.03FGD 1 ν¯µ CC 1-Track (ν¯ mode) 2663 2708.65 2639.31FGD 1 ν¯µ CC N-Tracks (ν¯ mode) 775 797.73 784.72FGD 1 νµ CC Inclusive (ν¯ mode) 1990 1933.36 1955.57FGD 1 νµ CC 1-Track (ν¯ mode) 989 938.13 966.18FGD 1 νµ CC N-Tracks (ν¯ mode) 1001 995.33 989.39FGD 2 νµ CC Inclusive (ν mode) 25151 24454.89 25052.24FGD 2 νµ CC0pi (ν mode) 17650 17211.71 17638.36FGD 2 νµ CC1pi (ν mode) 3383 3616.62 3448.92FGD 2 νµ CC Other (ν mode) 4118 3626.56 3964.96FGD 2 ν¯µ CC Inclusive (ν¯ mode) 3499 3534.33 3542.56FGD 2 ν¯µ CC 1-Track (ν¯ mode) 2762 2729.88 2728.34FGD 2 ν¯µ CC N-Tracks (ν¯ mode) 737 804.45 814.22FGD 2 νµ CC Inclusive (ν¯ mode) 1916 1860.51 1924.22FGD 2 νµ CC 1-Track (ν¯ mode) 980 943.90 987.12FGD 2 νµ CC N-Tracks (ν¯ mode) 936 916.61 937.08Table 7.1: Actual and predicted event totals for the different ND280 samples in theND280 fit. The MC predictions are shown both before and after the ND280 fit.164 (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, µνND280  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, µνND280  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, eνND280  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, eνND280  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, µνND280  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, µνND280  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, eνND280  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, eνND280 Figure 7.1: The pre-fit and post-fit ND280 flux parameters and their uncertainties.165 (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, µνSK  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, µνSK  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, eνSK  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, eνSK  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, µνSK  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, µνSK  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, eνSK  (GeV)νE-110 1 10Flux Parameter Value0.60.70.80.911.11.21.31.4Prior to ND280 ConstraintAfter ND280 Constraint beam modeν, eνSK Figure 7.2: The pre-fit and post-fit SK flux parameters and their uncertainties.166MAQEpF_CMEC_CEB_CpF_OMEC_OEB_O CA5MANFFRESBgRESCCNUE_0CCOTHER_SHAPECCCOH_O_0NCCOH_0NCOTHER_NEAR_0MEC_NUBARFSI_INEL_LO_EFSI_INEL_HI_EFSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EParameter Value-1-0.500.511.522.5PrefitPostfitFigure 7.3: The pre-fit and post-fit cross section parameters and their uncertainties. Axislabels show the name of each parameter. Parameters are typically a ratio to NEUTnominal values, excepting the CC Other Shape parameter and the FSI parameters., FHCpiFGD 1 CC-0, FHCpiFGD 1 CC-1FGD 1 CC-Other, FHC CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1  CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1 , FHCpiFGD 2 CC-0, FHCpiFGD 2 CC-1FGD 2 CC-Other, FHC CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2  CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2 Parameter Value0.50.60.70.80.911.11.21.31.41.5PrefitPostfitFigure 7.4: The pre-fit and post-fit observable normalization parameters and their uncer-tainties. Axis labels show the name of each parameter.167The parameter correlations prior to the fit and after fitting are shown in Figs. 7.5 and 7.6.The fit introduces correlations and anticorrelations between the flux and cross section parame-ters as intended as well as reducing the correlations between flux parameters.Parameter number0 10 20 30 40 50 60Parameter number0102030405060Correlation-1-0.8-0.6-0.4-0.200.20.40.60.81Figure 7.5: The parameter correlations prior to the ND280 fit. The parameters are 0-24 SK FHC flux, 25-49 SK RHC flux, 50 MQEA , 51 pF16O, 52 MEC 16O, 53 EB16O, 54 CA5RES, 55 MRESA , 56 Isospin=12 Background, 57 CC Other Shape, 58 CCCoh 16O, 59 NC Coh, 60 NC 1 γ , 61 NC Other, 62 MEC ν¯ normalization, 63 σνe ,64 σν¯e7.2.1 The p-value CalculationThe p-value for the data fit is calculated as the percentage of toy data fits with fitted ∆χ2 greaterthan the observed ∆χ2 from the data. A p-value of 0.05 or greater indicates that the fit modelis considered sufficiently reasonable for fitting the data and the result can be passed to SK forthe oscillation fits; this value is determined prior to fitting. The data fit gives a fitted ∆χ2 of1448.05 at the minimum, which when compared with the distribution of the ∆χ2 from the toydata fits described in Section 6.2 gives a p-value of 0.086. As shown in Figure 7.7, the data168Parameter number0 10 20 30 40 50 60Parameter number0102030405060Correlation-1-0.8-0.6-0.4-0.200.20.40.60.81Figure 7.6: The parameter correlations included after the ND280 fit. The parameters are0-24 SK PF flux, 25-49 SK NF flux, 50 MQEA , 51 pF16O, 52 MEC 16O, 53 EB 16O,54 CA5RES, 55 MRESA , 56 Isospin=12 Background, 57 CC Other Shape, 58 CC Coh16O, 59 NC Coh, 60 NC 1 γ , 61 NC Other, 62 MEC ν¯ normalization, 63 σνe , 64 σν¯e∆χ2 is towards the upper tail of the distribution. Distributions for the ∆χ2 contributions fromthe Poisson data terms and prior constraint terms are shown in Figures 7.8 and 7.9 respectively.The primary source of tension between the data fit and the toys is from the ∆χ2 contributionfrom the detector normalization parameters, as shown in Figures 7.10, 7.11 and 7.12. Both theflux and cross section contributions fall within the main part of their corresponding distributionsand the overall sample contribution shown in Figure 7.8 does as well. This is not unexpected,as the detector observable normalization parameters use many approximations to the ND280systematics in order to work in the fit, such as being modeled as Gaussian parameters.7.3 Impact at Super-KamiokandeThe oscillation analysis at SK uses the ND280 fit result as the prior values and constraintsfor many parameters; as discussed in Chapter 3, the ND280 fit constrains the expected neu-169totalEntries  444Mean     1233RMS     145.42χ∆800 1000 1200 1400 1600 1800 2000 2200 2400Number of toy experiments051015202530354045totalFigure 7.7: Total ∆χ2 distribution for 444 toy fit results (black), with the value from thefit to the data superimposed in red. The total is the sum of the contribution fromthe Poisson data term and the contribution from the prior constraint term. The total∆χ2 from the fit to the data is 1448.05, which corresponds to a p-value of 0.086.samplesEntries  444Mean     1134RMS     139.12χ∆800 1000 1200 1400 1600 1800 2000 2200 2400Number of toy experiments051015202530354045samplesFigure 7.8: Contribution from the Poisson data terms to total ∆χ2 distribution for 444 toyfit results (black), with the value from the fit to the data superimposed in red.170priorEntries  444Mean      101RMS     16.642χ∆0 20 40 60 80 100 120 140 160 180 200Number of toy experiments0102030405060priorFigure 7.9: Contribution from prior constraint term to total ∆χ2 distribution for 444 toyfit results (black), with the value from the fit to the data superimposed in red. Thisis the sum of the prior constraint terms from the flux, cross-section and detectorparameters.fluxEntries  444Mean     31.5RMS     8.9852χ∆0 10 20 30 40 50 60 70 80Number of toy experiments010203040506070fluxFigure 7.10: Contribution from flux constraint term to the prior constraint term ∆χ2 dis-tribution from 444 toy fit results (black), with the value from the fit to the datasuperimposed in red.171xsecEntries  444Mean    4.629RMS     3.0152χ∆0 5 10 15 20 25 30 35 40 45 50Number of toy experiments020406080100120140xsecFigure 7.11: Contribution from cross section constraint term to the prior constraint term∆χ2 distribution from 444 toy fit results (black), with the value from the fit to thedata superimposed in red.obsNormEntries  444Mean    65.03RMS      9.512χ∆0 20 40 60 80 100 120 140Number of toy experiments0102030405060obsNormFigure 7.12: Contribution from detector constraint term to the prior constraint term ∆χ2distribution from 444 toy fit results (black), with the value from the fit to the datasuperimposed in red.172trino spectra at SK. Because of the addition of the FGD 2 sample containing events on water,the various oxygen cross section parameters used at SK (EB, pF and 2p-2h 16O) can now beconstrained with the near detector fit results. Table 7.2 shows the uncertainties on the predic-tion at SK using the previous FGD 1-only ND280 fitted constraint, and Table 7.3 shows theuncertainties on the prediction at SK using the new FGD 1 + FGD 2 ND280 constraint as de-scribed in this thesis, for the joint νµ – νe analysis. The addition of FGD 2 as a target in theND280 fit gives an oxygen target at ND280, thus allowing oxygen target-specific cross-sectionparameters to be constrained using ND280.FHC νµ FHC νe RHC νµ RHC νeND280-Constrained Parameters 2.7% 3.1% 3.4% 3.0%ND280-Unconstrained Parameters 5.0% 4.7% 10.0% 9.8%Flux × XSec (Post ND280) 5.7% 5.6% 10.6% 10.2%SK FSI + SI 3.0% 2.4% 2.1% 2.2%SK Systematics 4.0% 2.7% 3.8% 3.0%Total 7.7% 6.8% 11.6% 11.0%Table 7.2: Fractional error on the prediction for number of events at SK, broken downby source, using the FGD 1 only ND280 fit results. FHC numbers use data fromRuns 1 – 4, and RHC numbers use data from Runs 5 – 6. Flux and cross-section areshown separately for the pre-ND280 case. Uncertainties shown are calculated fromthe RMS of the event rates when 10000 parameter variations are thrown[99] fromthe final ND280 fit covariance matrix.As Table 7.2 shows, the largest source of uncertainty at SK when only FGD 1 is included inthe ND280 fit comes from uncertainties on cross-section parameters with carbon-oxygen dif-ferences: EB, pF and the 2p-2h normalization (see row labeled “ND280-Unconstrained Param-eters”) . As the FGD 1 only analysis only contains a carbon target and extrapolation betweentarget materials is not well understood, previous fits were not able to provide strong constraintson those cross-section parameters. In contrast, the FGD 1 + FGD 2 fit is able to constrain thesetarget-specific cross-section parameters as shown by the reduced contribution from ND280-unconstrained parameters in Table 7.3. This leaves only a few cross-section parameters thatcannot be constrained by the ND280 data: the NC γ normalization, the NC other normalizationand the CC νe/νµ and CC ν¯e/ν¯µ cross-section ratio normalizations which are mentioned in173νµ FHC νe FHC ν¯µ RHC ν¯e RHCFlux (Pre-ND280) 7.6% 8.9% 7.1% 8.0%XSec (Pre-ND280) 9.7% 7.2% 9.3% 10.1%Flux × XSec (Post-ND280) 2.9% 4.2% 3.4% 4.6%FSI + SI 1.5% 2.5% 2.1% 2.5%SK Detector Syst 3.9% 2.4% 3.3% 3.1%Total (Pre-ND280) 12.0% 11.9% 12.5% 13.7%Total (Post-ND280) 5.0% 5.4% 5.2% 6.2%Table 7.3: Fractional error on the prediction for number of events at SK, broken downby source. Pre-ND280 indicates that the ND280 fit result was not used as the prior,while post-ND280 indicates that the ND280 fit using FGD 1 and FGD 2 events asdescribed in this thesis was used as the prior. Flux and cross-section are shownseparately for the pre-ND280 case. Uncertainties shown are calculated from theRMS of the event rates when 10000 parameter variations are thrown[99] from thefinal ND280 fit covariance matrix.νµ FHC νe FHC ν¯µ RHC ν¯e RHCND280-Constrained Flux × XSec 2.82% 2.84% 3.27% 3.19%CC νe/νµ + CC ν¯e/ν¯µ 0.01% 2.60% 0.00% 1.51%NC γ 0.00% 1.47% 0.00% 2.93%NC Other 0.77% 0.16% 0.75% 0.33%Table 7.4: Fractional error on the prediction for number of events at SK from cross-section and flux, broken down to show the contributions from parameters uncon-strained by ND280. The combination of these uncertainties gives the flux × cross-section uncertainties shown in Table 7.3. Uncertainties shown are calculated fromthe RMS of the event rates when 10000 parameter variations are thrown[61].Chapter 3. These have a much smaller contribution to the overall uncertainty than EB, pF andthe 2p-2h normalization, as shown in Table 7.4.All four samples – FHC νµ and νe and RHC νµ and νe – benefit from the improvementto the cross-section constraints. In the case of the FHC νµ samples, this reduces a flux ×cross-section uncertainty of 2.7% and an ND280 unconstrained cross-section parameter un-certainty of 5%, which combines to give 5.7% flux × cross-section uncertainty as shown inrow 3 of Table 7.2, to a single flux × cross-section uncertainty of 2.9% as shown in row 3of Table 7.3. This allows the overall uncertainty on the prediction to be reduced from 7.7%using only FGD 1 in the ND280 fit, to 5%. The RHC samples show an even greater reductionin overall uncertainty, due to a larger contribution to the uncertainty from the oxygen-specific174cross-section parameters – 10% for νµ and 9.8% for νe respectively. With the ability to con-strain oxygen cross-sections, this combines with the flux× cross-section uncertainties to give atotal uncertainty 3.4% for νµ and 4.6% for νe from the flux and cross-section parameters. Theuncertainty on the prediction as a function of the neutrino energy for the four samples can beseen in Figures 7.13, 7.14, 7.15 and 7.16. Outside of the high energy bins for the νe spectra, theuncertainty is significantly reduced for the entire energy range. These show that not only doesthe improvement to the ND280 constraint improve the overall uncertainties on the predictionsbut also improves the uncertainty across the neutrino energy spectrum.Figure 7.13: Predicted energy spectrum for FHC νµ events at SK, with and withoutND280 constraints.175Figure 7.14: Predicted energy spectrum for FHC νe events at SK, with and withoutND280 constraints.Figure 7.15: Predicted energy spectrum for RHC ν¯µ events at SK, with and withoutND280 constraints.176Figure 7.16: Predicted energy spectrum for RHC ν¯e events at SK, with and withoutND280 constraints.1777.3.1 Oscillation Fit Results at Super-KamiokandeThe oscillation fit results shown here are the result of fitting both neutrino and antineutrinomode data looking at νµ disappearance and νe appearance and using the near detector fit resultsdescribed in this thesis as the prior constraints for flux and cross section parameters in the fit.The fitted 2D contours for sin2θ13 vs δCP are shown in Figures 7.17, 7.18 for normal andinverted mass hierarchy respectively, and sin2θ13 vs ∆m232 (∆m213 for IH) contours are shownin Figures 7.19 and 7.20. The other oscillation parameters are marginalized over according tothe priors in Table 3.3. These plots show both the Run 1 – 7c analysis results (fitting neutrinoand antineutrino mode data simultaneously), which are the latest T2K oscillation fit results andthe first to use the FGD 1 + FGD 2 ND280 fit results described in this thesis, and the Run 1– 4 joint oscillation fit results, which simultaneously fit νµ disappearance and νe appearanceusing neutrino mode data only[30]. The Run 1 – 4 fit uses the previous FGD 1-only ND280results, as can be seen from the larger 68% and 90% CL contours, with a significant portion ofthe changes coming from overall increase in statistics at SK.13θ 22sin0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 (radians)CPδ-3-2-10123T2K Run1-7c preliminary = 2.3)crit68.27%CL (-2ln L = 4.61)crit90%CL (-2ln LBest-fitT2K Run 1-7cT2K Run 1-4Normal HierarchyFigure 7.17: 2D contours for sin2θ13 and δCP for the Run 1 – 7c joint analysis and theRun 1 – 4 joint analysis for normal mass hierarchy. [61]17813θ 22sin0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 (radians)CPδ-3-2-10123T2K Run1-7c preliminary = 2.3)crit68.27%CL (-2ln L = 4.61)crit90%CL (-2ln LBest-fitT2K Run 1-7cT2K Run 1-4Inverted HierarchyFigure 7.18: 2D contours for sin2θ13 and δCP for the Run 1 – 7c joint analysis and theRun 1 – 4 joint analysis for inverted mass hierarchy. [61]23θ 2sin0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7)4/c2 eV-3| (10322 m∆|22.12.22.32.42.52.62.72.82.93T2K Run1-7c preliminary = 2.3)crit68%.27CL (-2ln L = 4.61)crit90%CL (-2ln LBest-fitT2K Run 1-7T2K Run 1-4 Normal HierarchyFigure 7.19: 2D contours for sin2θ23 and ∆m232 for the Run 1 – 7c joint analysis and theRun 1 – 4 joint analysis for normal mass hierarchy. [61]17923θ 2sin0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7)4/c2 eV-3| (10322 m∆|22.12.22.32.42.52.62.72.82.93T2K Run1-7c preliminary = 2.3)crit68.27%CL (-2ln L = 4.61)crit90%CL (-2ln LBest-fitT2K Run 1-7T2K Run 1-4 Inverted HierarchyFigure 7.20: 2D contours for sin2θ23 and ∆m213 for the Run 1 – 7c joint analysis and theRun 1 – 4 joint analysis for inverted mass hierarchy. [61]In order to get a sense of how the improvements to the ND280 constraint impact the oscil-lation contours, it is useful to perform a simple estimation of the uncertainty on the contours inorder to compare the previous FGD 1-only result, as described in Table 7.2 and the new FGD 1+ FGD 2 result shown in Table 7.3. However, the uncertainty in the contours comes from twosources: the systematic uncertainty on the SK rate prediction and the statistical uncertainty onthe total number of events at SK, which scales with the total accumulation of protons on target.The statistical uncertainty is larger than that from systematics, particularly in the case of νemeasurements, making it difficult to compare previous results. Instead, the overall uncertain-ties are best compared using the same SK statistics, as the systematics contribution is largelyunaffected by the increase in statistics. As discussed earlier, the most significant difference inuncertainty on the rate prediction at SK comes from the unconstrained oxygen cross-section pa-rameters, as previous to this analysis, the unconstrained cross-section uncertainties dominatedthe contributions from the SK detector systematics and flux.Of the current T2K analyses, the least statistics-dominated is the νµ disappearance results180measuring θ23. The current preliminary Run 1 – 7c results give an overall fractional uncertaintyof +0.086−0.128 on the value of sin2θ23[61] for normal hierarchy. To estimate the uncertainty on θ23,one can make a few simplifying assumptions: that the uncertainty on the expected number ofevents is an energy-independent normalization uncertainty, and that the oscillation probabilityfor νµ → νµ can be approximated asP(νµ → νµ)∼ 1− sin2 2θ23 (7.1)where sin2 2θ is close to 1. The observed number of events Nobs can be expressed asNobs =∫ dNdE(1− sin2 2θ sin2(∆m2LE))dE (7.2)By then treating the error on the number of expected events as energy independent, this can bewritten asNobs = Nexp(1− sin2 2θ[∫ ( 1Nexp)dNdEsin2(∆m2LE)dE])(7.3)The integral in the previous equation gives∫ ( 1Nexp)dNdEsin2(∆m2LE)dE = 1− NobsNexp,unosc(7.4)and so the approximate error on sin2 2θ can be found:δ (sin2 2θ) =11− NobsNexp,unosc√(dNobsNobs)2+(dNexp,unoscNexp,unosc)2(7.5)with Nobs the observed number of νµ events, Nexp,unocs the expected number of events assumingno oscillation, dNobs being the Poisson statistical error on the observed number of events anddNexp,unosc being the systematic uncertainty on the predicted number of events at SK assumingno oscillation. While this formula does not account for the energy dependence in the analysis,this is nonetheless a useful tool to estimate the effect of the new constraint.181FHC νmu RHC ν¯µUsing old ND280 Constraint 0.040 0.0939Using new ND280 Constraint 0.035 0.0742Table 7.5: Expected Fractional Uncertainty on sin2 2θ23 for Runs 1 – 7.For the Run 1 – 7 νµ disappearance, SK observes 135 FHC νµ events and 66 RHC ν¯µevents, with 521.8 and 184.8 events expected with no oscillations respectively[61]. Combinedwith the systematic uncertainties in Tables 7.3 and 7.2, an estimate of the overall uncertaintyon 1−sin2 2θ23 can be made using Run 1 – 7 statistics for both prior constraints. The fractionaluncertainties are shown in Table 7.5. For FHC νµ signal, we expect roughly a 15% reductionin overall uncertainty on the contour due to the improved prior constraint. The uncertaintyreduction for the RHC ν¯µ is even greater, due to the larger impact from oxygen cross-sectionuncertainties. Here the uncertainty is reduced by ∼ 25% compared to using the FGD 1-onlyconstraint. As an additional comparison, Figures 7.21 and 7.22 show the T2K oscillation con-tours from the Run 1 – 4 SK analysis, using the FGD 1-only Run 1 – 4 ND280 constraints, andthe Run 1 – 7 SK data joint oscillation analysis, using the new ND280 fit which uses Runs 1– 7 ND280 data, compared with Run 1 – 4 SK contours made using the new FGD 1 + FGD 2ND280 fit constraints made with Run 1 – 7 ND280 data. This additional contour is shown as ablue line. Due to time constraints, the Run 1 – 4 SK contour using the FGD 1 + FGD 2 priorconstraints is shown, rather than comparing FGD 1-only Run 1 – 7 SK fits; however, the effectof the systematics improvement is still visible. Additionally only the sin2θ23 – ∆m213 contoursare shown, as the other results use the reactor constraints, which are implemented differentlyfor the fits with FGD 1-only constraints and the fits with the FGD 1 + FGD 2 constraints. Thisshows an overall improvement in the contour size for both normal and inverted hierarchy fits,with the FGD 1 + FGD 2 constrained SK Run 1 – 4 fit contours sitting much closer to the Run1 – 7 contours. This shows that the new oxygen constraints from ND280 currently provide anoticeable reduction in the size of oscillation contours even at current statistics, and will showeven greater impact as more statistics are collected.182Figure 7.21: 2D contours for sin2θ23 and ∆m232 for the Run 1 – 4 joint analysis for normalmass hierarchy using the FGD 1 + FGD 2 ND280 fit result compared with otherT2K contours. 2D contours for sin2θ23 and ∆m213 for the Run 1 – 7c joint analysisand the Run 1 – 4 joint analysis for normal mass hierarchy are taken from [61].The difference between the red and blue contours shows the improvement due tothe inclusion of the FGD 2 data in the ND280 fit, for an oscillation fit to the Run1 – 4 SK data. The 1D sin2θ23 range is reduced by ∼3% and the 1D ∆m213 rangeis reduced by ∼11%.183Figure 7.22: 2D contours for sin2θ23 and ∆m213 for the Run 1 – 4 joint analysis for in-verted mass hierarchy using the FGD 1 + FGD 2 ND280 fit result compared withother T2K contours. 2D contours for sin2θ23 and ∆m213 for the Run 1 – 7c jointanalysis and the Run 1 – 4 joint analysis for inverted mass hierarchy are takenfrom [61]. The difference between the red and blue contours shows the improve-ment due to the inclusion of the FGD 2 data in the ND280 fit, for an oscillation fitto the Run 1 – 4 SK data. The 1D sin2θ23 range is reduced by ∼2% and the 1D∆m213 range is reduced by ∼8%.184Chapter 8Further Parameterization StudiesT2K currently splits several of the CCQE parameters into carbon and oxygen versions of theparameters for use in the analysis at SK, as discussed in Chapter 3 to account for nuclear ef-fects and remove possible model dependencies. This allows such parameters specific to carbontargets to be strongly constrained using only FGD 1, as it only contains carbon as a target ma-terial, and then use the strong constraints on carbon to constrain the oxygen versions of thoseparameters in FGD 2, as it contains both carbon and oxygen as target materials. This addi-tionally removes any dependence on modeling how these parameters scale from one nucleusto another. Splitting fit parameters by target is not directly equivalent to a cross-section studyusing the near detectors, as the fit is still a simultaneous constraint on the flux and cross sectionparameters. These parameters are the 2p-2h scaling, EB and pF parameters, all which primarilyaffect the CC-0pi samples. Chapter 3 discusses the T2K implementation of the parameters inmore detail, and a description of the neutrino interaction modeling is found in Chapter 1.There is currently no strong physical reasoning for differences in the axial mass MQEA thatdepend on the target nucleus, as it only relates to the calculation of the neutrino-nucleon in-teraction form factors. While T2K cannot measure neutrino interactions on free nucleons andnuclear effects need to be accounted for, the nucleus-based parameters, such as the Fermi mo-mentum, binding energy and the 2p-2h normalization, were introduced to account for such185effects and in a good model should absorb all such effects. However, the MQEA parameter usedin the T2K fit is necessarily an effective one and may nonetheless have some dependence onthe target nuclei that would otherwise be included in the 2p-2h parameters, as the 2p-2h modelwas introduced specifically to absorb nuclear effects that would otherwise affect MQEA . If 2p-2hhas been well modeled, there should not be any difference between target nuclei for MQEA . Inaddition, the value of MQEA has a significant impact on the expected number of CCQE events atND280, and the primary signal for the SK analysis. In this case, if our effective parameter waspicking up nuclear effects that caused carbon and oxygen versions of the axial mass to havesignificantly different values, this could change the overall predicted rates at SK.For CC-1pi interactions, the effective parameters for C5A(0) and MRESA described in Sec-tion 3.3.2 are the largest contributors to the resonant pion production at the T2K near detector.Like MQEA , these are nucleon-level parameters and in theory should not vary across nucleartargets. However, as neutrino – nucleus interactions are less well understood for pion pro-duction than for quasielastic processes, splitting these parameters into effective carbon andoxygen versions may allow potential interaction rate differences due to target be accountedfor. Additionally, the near detector fit is more sensitive to those parameters, which is importantfor sufficiently constraining oxygen target parameters. In general, because the measurementof oxygen parameters must be done through subtraction, the fitted uncertainties will be largerthan the uncertainties on their carbon counterparts.Two separate studies were done to look for residual dependencies of effective cross-sectionparameters on the nuclear target, in order to probe the robustness of the cross-section model:1. Splitting the quasielastic axial mass parameter MQEA by target.2. Splitting the resonant pion production form factor C5A(0) and the resonant axial massparameter MRESA by target.For both studies, all other parameters were implemented identically to the official near detectorfit described in the previous chapters of this thesis. These studies are not part of the official186T2K analysis, but are an additional extension of the analysis to serve as a consistency check.8.1 Fitting MQEA for Oxygen and Carbon TargetsWe first investigate splitting a single cross-section parameter into two separate parameters, oneon a carbon target and the other on oxygen. For this, the CCQE axial mass MQEA is chosen, as itaffects the primary signal at T2K. A split version of this parameter might allow the near detectorto more accurately represent how well the effective parameter is fit for use in the oscillationanalysis, as well as pick up any nuclear effects not accounted for in the 2p-2h, binding energyor Fermi momentum parameters. From a technical standpoint, separate versions of the axialmass parameter are simple to implement in the fit, as this is done by separately reweightingcarbon and oxygen events in the Monte Carlo in accordance to the new parameters. As wedo not use external constraints for the CCQE cross section parameters in the near detector fit,there are no prior assumptions or correlations to take into account when creating a separateoxygen-based parameter. The same response splines described in Section 3.3.2 are used forboth carbon and oxygen.8.1.1 Additional ValidationBefore running the data fit, several of the validation studies described in Chapter 6 were per-formed with the split MQEA parameters, as well as a few toy studies with the oxygen and carbonaxial mass varied separately to ensure the parameters were correctly implemented. Like thestandard near detector fit, parameters are fitted to their prior values with near machine preci-sion. Both MQEA12C and MQEA16O parameters have fit to the value set for MQEA in the nominalMC, with MQEA16O having a larger uncertainty due to lower statistics on oxygen as expected.This indicates that the parameters are both being fit and do not have unexpected effects on otherparameter values. Plots for these fits are shown in Figures 8.1, 8.2 and 8.3.Two fake data fits have been performed in order to validate the performance of the param-eter splitting code:187, FHCµνND280 , FHCµνND280 , FHCeνND280 , FHCeνND280 , RHCµνND280 , RHCµνND280 , RHCeνND280 , RHCeνND280 , FHCµνSK , FHCµνSK , FHCeνSK , FHCeνSK , RHCµνSK , RHCµνSK , RHCeνSK , RHCeνSK Parameter Value0.80.850.90.9511.051.11.151.21.251.3PrefitPostfitFigure 8.1: Fitted parameter values and uncertainties for the flux for the fit to the nominalMC with split MQEA .1. MQEA12C = 1 GeV/c2 and MQEA16O = 1.2 GeV/c22. MQEA12C = 1.2 GeV/c2 and MQEA16O = 1 GeV/c2where the nominal NEUT value is MQEA = 1.2. All other parameter values were set to nominal.As all previous fits were performed on fake data with a single set value for MQEA , these simpletests can verify that the fit machinery is able to correctly fit two separate values of MQEA . Plotsof the fitted cross-section parameters for these are shown in Figures 8.4 and 8.5 and show thatthe parameters fit separately and correctly.8.1.2 ResultsThe final ∆χ2 for the MQEA carbon and oxygen fit was found to be 1447.9. This is very close to,though slightly lower than, the original ∆χ2 for the data fit with a single MQEA parameter. Fig-ure 8.6 show the cross section parameter fit values compared with the original data fit discussedin Chapter 7. These are shifted slightly compared to the original data fit, on the sub-< 1% level188MAQE_CpF_CMEC_CEB_CpF_OMEC_OEB_O CA5MANFFRESBgRESDISMPISHPCCOTHER_SHAPENCCOH_0NCOTHER_NEAR_0MAQE_OMEC_NUBARFSI_INEL_LO_EFSI_INEL_HI_EFSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EParameter Value-1-0.500.511.522.5PrefitPostfitFigure 8.2: Fitted parameter values and uncertainties for the cross section parameters forthe fit to the nominal MC with split MQEA .– both the flux and observable normalization parameters show a small decrease. The fittedvalues are most easily compared using a simple paired test statistic to remove the correlatedcomponent of the uncertainties in order to evaluate significance. For a parameter P this isPnew−Pold√σ2new−σ2old(8.1)where σnew is the uncertainty on a given parameter from the new model-testing fit and σoldis the uncertainty from the original fit, and√σ2new−σ2old is the uncertainty on the shift inparameter value.The addition of a separate MQEA parameter for oxygen interactions has an impact on notonly the fitted MQEA value but also the fitted value for the 2p-2h normalization on oxygen.While MQEA12C, shown in Table 8.1, is found to be slightly higher than the combined fit valuefor the axial mass, the MQEA16O parameter is decreased relative to the combined fit value to a189, FHCpiFGD 1 CC-0, FHCpiFGD 1 CC-1FGD 1 CC-Other, FHC CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1  CC-1Track, RHCµνFGD 1  CC-NTracks, RHCµνFGD 1 , FHCpiFGD 2 CC-0, FHCpiFGD 2 CC-1FGD 2 CC-Other, FHC CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2  CC-1Track, RHCµνFGD 2  CC-NTracks, RHCµνFGD 2 Parameter Value0.50.60.70.80.911.11.21.31.41.5PrefitPostfitFigure 8.3: Fitted parameter values and uncertainties for the observable normalizationparameters for the fit to the nominal MC with split MQEA .value of 1.0747 ± 0.0590 from the original fit value of 1.1113 ± 0.0333 for the unsplit MQEAparameter, by less than 1 σ using Eq. 8.1. This is accompanied by a small overall increaseof around 0.8 σ in the oxygen 2p-2h normalization, while the normalization on carbon doesnot change significantly. This shift in the oxygen 2p-2h normalization is to compensate forthe lower observed CC-0pi event rates due to the lower MQEA value for oxygen. As shown inFigure 8.7, the two MQEA parameters are nearly uncorrelated with one another. While this maybe due to some nuclear effect on the event rates that can be absorbed by splitting the axial massresponse parameter into separate versions for carbon and oxygen targets, the shift for the 2p-2hnormalization and the MQEA values are not statistically significant and so there is no evidencethat the MQEA parameter is absorbing unmodeled nucleus-dependent physics.190MAQE_CpF_CMEC_CEB_CpF_OMEC_OEB_O CA5MANFFRESBgRESDISMPISHPCCOTHER_SHAPENCCOH_0NCOTHER_NEAR_0MAQE_OMEC_NUBARFSI_INEL_LO_EFSI_INEL_HI_EFSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EParameter Value-1-0.500.511.522.5PrefitPostfitFigure 8.4: Fitted parameter values and uncertainties for the cross section parameters forthe fake data fit with MQEA12C = 1 and MQEA16O = 1.2. Parameter values shownare relative to nominal, so MQEA12C is expected to fit to 1 and MQEA16O to 0.833.8.2 Adding Carbon-Oxygen Separation for C5A(0) and MRESAOutside of the axial mass parameter, there are a few other cross section parameters that mightsee some effect from the target nucleus. In particular, we can also look at C5A(0) and MRESA andsplit these into separate carbon and oxygen versions. Other cross section parameters such as theCC-other shape are not be expected to see any difference. Unlike MQEA , these two parameters douse a prior and prior uncertainty supplied by fits to external cross section measurements whenfitting. This means that any separated version of the parameters should also have some prior -as C5A(0) and MRESA are not correlated to other cross section parameters in the current versionof the inputs, it is relatively simple to choose the priors for carbon and oxygen. Thereforethe priors for MRESA16O and C5A(0)16O have identical prior values to the original parametersand are assumed to be uncorrelated from their 12C counterparts. These fits also split the MQEAparameter into oxygen and carbon versions as in the previous section.191MAQE_CpF_CMEC_CEB_CpF_OMEC_OEB_O CA5MANFFRESBgRESDISMPISHPCCOTHER_SHAPENCCOH_0NCOTHER_NEAR_0MAQE_OMEC_NUBARFSI_INEL_LO_EFSI_INEL_HI_EFSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EParameter Value-1-0.500.511.522.5PrefitPostfitFigure 8.5: Fitted parameter values and uncertainties for the cross section parameters forthe fake data fit with MQEA12C = 1.2 and MQEA16O = 18.2.1 Additional ValidationAs with the MQEA carbon and oxygen fit, the near detector fit framework was revalidated toensure the parameters were being fit properly. The original toy throws described in Chapter 6were used to rerun the parameter pull studies for fits with the new parameters. While thisessentially makes the assumption that the carbon and oxygen versions of the parameters are100% correlated with each other, as a single C5A(0) parameter is used where C5A(0)12C and 16Owould otherwise be, this should not have an impact on the validation process. These studiesshow similar results as the MQEA carbon and oxygen validation fits and show no obvious issueswith the fit machinery.8.2.2 ResultsThe total ∆χ2 from the data fit is 1449.86, slightly higher than both the original data fit andthe MQEA fit described in the previous section due to the addition of two additional constrained192MAQE_CMAQE_OpF_CMEC_CEB_CpF_OMEC_OEB_O CA5MANFFRESBgRESDISMPISHPCCOTHER_SHAPENCCOH_0NCOTHER_NEAR_0MEC_NUBARFSI_INEL_LO_EFSI_INEL_HI_EFSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EFitted Parameter Value-1-0.500.511.522.5 split by targetAQEFit with MOriginal data fitFigure 8.6: The post-fit cross section parameters from the MQEA carbon and oxygen fit andtheir uncertainties and the cross section parameters from the data fit described inChapter 7. Axis labels show the name of each parameter. Parameters are typicallya ratio to NEUT nominal values, excepting the CC Other Shape parameter and theFSI parameters. As the original data fit did not separately fit MQEA for carbon andoxygen targets, the same fitted MQEA value is shown for both targets.parameters. The fitted MQEA , C5A and MRESA values are shown in Table 8.1 along with the resultsfrom the other fits. Similar to the fit with only MQEA split, the flux and detector parametersare all lowered slightly compared to the original data fit, though less so than the fit with onlyMQEA split. For the flux parameters and most of the cross-section parameters, this shift wason the order of less than 1 σ from the original fit value, and not a statistically significantchange. Figure 8.8 show the cross section parameter fit values compared with the original datafit discussed in Chapter 7.Correlations between fitted cross section parameters are shown in Figure 8.9. As MQEA issplit, the effects seen when only splitting MQEA appear in this fit, though MQEA16O has beendecreased slightly less. As expected from the original fit, the C5A(0) and MRESA parameters areanticorrelated with each other. Similar to the MQEA oxygen and carbon fit described earlier in193FSI_INEL_LO_EFSI_INEL_HI_EFSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EMAQE_CpF_CMEC_CEB_CpF_OMEC_OEB_O CA5MANFFRESBgRESDISMPISHPCCCOH_O_0NCCOH_0NCOTHER_NEAR_0MAQE_OMEC_NUBARFSI_INEL_LO_EFSI_INEL_HI_EFSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EMAQE_CpF_CMEC_CEB_CpF_OMEC_OEB_OCA5MANFFRESBgRESDISMPISHPCCCOH_O_0NCCOH_0NCOTHER_NEAR_0MAQE_OMEC_NUBARCorrelation-1-0.8-0.6-0.4-0.200.20.40.60.81Figure 8.7: The postfit correlations between cross section parameters for the MQEA carbonand oxygen fit.this chapter, MQEA12C and MQEA16O are only slightly correlated with each other. The MRESA12Cand 16O parameters are fit within 1 σ of the previously fit MRESA value, with a larger uncertaintyon the 16O parameter as expected, indicating that MRESA is not sensitive to target nuclei in theT2K near detector fit, unlike the MQEA parameter. The C5A(0) parameters are an exception, inthat they do show a difference between target materials, with the carbon version fit close tothe C5A(0) value from the standard data fit, while the oxygen parameter is increased by ∼ 1.5σfrom the original fit value. C5A(0) and MRESA remain anticorrelated with each other for boththe oxygen and carbon parameters, as in the official fit. As with the fit with only MQEA splitby target, most parameters did not show statistically significant changes when split by target,which indicates that there are no obvious nuclear effects being absorbed by these parameters.The single exception to this is C5A16O, which does show a hint of a variation from the originalfitted C5A parameter. So unlike MRESA , there may be some unmodeled nuclear effect being seenhere, as unlike CCQE cross-sections, there is no modelling of 2p-2h or other such effects forresonant pion production.194Official MQEA Split MQEA , C5A and MRESA SplitMQEA12C1.1113±0.0338 1.1256±0.0384 1.1289±0.0389MQEA16O 1.0749±0.0590 1.0802±0.0593C5A(0)12C0.7860±0.0607 0.7854±0.0606 0.7896±0.0681C5A(0)16O 0.8943±0.0879MRESA12C0.8491±0.0384 0.8497±0.0383 0.8405±0.0457MRESA16O 0.8652±0.072072p-2h 12C 156.90±22.64 155.04±22.86 156.59±23.212p-2h 16O 155.89±34.24 166.80±36.70 147.35±39.57Table 8.1: Fitted parameter values for the three different data fits for the tested parame-ters. If a listed fit did not have separate parameters for oxygen and carbon targets,the value from the general parameter is used as the entry for both.8.3 Effect on the Predictions at Super-KamiokandeAs discussed in the previous sections, splitting the MQEA and C5A(0) parameters into separateversions for carbon and oxygen targets leads to different fitted values depending on targetmaterial, while MRESA does not show this behavior. However, while it is useful to look athow individual parameter values change in the ND280 fit itself, the primary purpose of theND280 fit is to provide flux and cross-section constraints for the oscillation analysis at SKand tune the predicted spectra and expected number of events. Splitting parameters by targetmaterial removes possible model dependence, even in the case where the parameter values donot change greatly. In the original fit output described in Chapter 7, only three parameterswere split into separate carbon and oxygen versions – 2p-2h normalization, EB and pF – andthe oscillation fit used the oxygen parameter values as the inputs. As there are now both moreparameters that have an oxygen-specific version and see a difference in value from both thecarbon version and the previous parameter values used for the fit, the next step is to test whatimpact this has on the predicted rates at SK. In particular, because parameters on oxygen targetsin the ND280 fit have larger uncertainties than their carbon counterparts, it is important to lookat the overall change in uncertainties on predicted spectra.Table 8.2 shows the predicted number of events at SK given the official near detector fit andthe two model dependence test fits described earlier in this chapter, for the FHC νµ , FHC νe,195MAQE_CMAQE_OpF_CMEC_CEB_CpF_OMEC_OEB_OCA5_CCA5_OMANFFRES_CBgRESDISMPISHPCCOTHER_SHAPENCCOH_0NCOTHER_NEAR_0FSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EFitted Parameter Value-1-0.500.511.522.5 split by targetARES, CA5 and MAQEFit with MOriginal FitFigure 8.8: The post-fit cross section parameters from the MQEA , C5A(0) and MRESA carbonand oxygen fit and their uncertainties and the cross section parameters from thedata fit described in Chapter 7. Axis labels show the name of each parameter.Parameters are typically a ratio to NEUT nominal values, excepting the CC OtherShape parameter and the FSI parameters. As the original data fit did not separatelyfit MQEA , C5A(0) and MRESA for carbon and oxygen targets, the same fitted MQEA , C5A(0)and MRESA values are shown for both targets.RHC νµ and RHC νe samples. The overall number of events predicted by the MQEA split fit isvery close to that of the original fit, though all predictions are slightly higher. This is possiblydue to the increase in the 2p-2h normalization on oxygen, which compensates for the reducedCCQE interactions in the observed CC-0pi samples. In addition, splitting the MQEA parameterin the near detector fit causes and increase in the uncertainties for the FHC samples but showslittle change in the RHC sample predictions. Such a small change is in line with the assumptionthat the axial mass MQEA should not pick up any strong dependence on nuclear target, as thechange in predicted number of events is not statistically significant.Unlike the fit with only MQEA split, the value of the 2p-2h normalization is lower relativeto the original near detector fit for the MQEA , C5A and MRESA split fit. Despite this, this fit shows196FSI_INEL_LO_EFSI_INEL_HI_EFSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EMAQE_CpF_CMEC_CEB_CpF_OMEC_OEB_OCA5_CMANFFRES_CBgRESDISMPISHPCCCOH_O_0NCCOH_0NCOTHER_NEAR_0MAQE_OCA5_OMANFFRES_OMEC_NUBARFSI_INEL_LO_EFSI_INEL_HI_EFSI_PI_PRODFSI_PI_ABSFSI_CEX_LO_EFSI_CEX_HI_EMAQE_CpF_CMEC_CEB_CpF_OMEC_OEB_OCA5_CMANFFRES_CBgRESDISMPISHPCCCOH_O_0NCCOH_0NCOTHER_NEAR_0MAQE_OCA5_OMANFFRES_OMEC_NUBARCorrelation-1-0.8-0.6-0.4-0.200.20.40.60.81Figure 8.9: The postfit correlations between cross section parameters for the MQEA , C5A(0)and MRESA carbon and oxygen fit.a larger increase in the predicted number of events for all samples compared to just removingmodel dependence on MQEA alone, though the difference is still not statistically significant. Thisappears to be due to the large increase in the value of the C5A form factor, and slight increasein the MRESA correction value. As these parameters control the background resonant pion pro-duction process, the increased rate is likely due to an increased non-CCQE background. Inaddition, the larger uncertainties from all three split parameters lead to a larger increase ofthe uncertainties on the prediction compared to both other fits. Plots of the predicted energyspectra at SK for each sample are shown in Figures 8.10, 8.11, 8.12 and 8.13.In adding additional degrees of freedom to the ND280 fit in order to account for unmodelednuclear effects, there is no significant indication that there are nuclear-target based inadequa-cies in the nominal T2K cross-section model. The axial mass MQEA and MRESA do not showsignificant change in the fit results, and there is no significant change in the overall predictedevent rates for any of the samples at SK. This indicates that the current nominal cross-sectionmodel in use is sufficient for the main T2K results.197FHC νµ FHC νe RHC νµ RHC νeOfficial 135.82±6.79 28.687±1.549 64.21±3.34 6.004±0.372MQEA Split 136.9±7.077 28.88±1.637 64.69±3.28 6.129±0.379MQEA , C5A and MRESA Split 138.20±7.49 30.12±1.823 65.31±3.48 6.474±0.412Table 8.2: Predicted number of events at SK using various ND280 postfit values and theiruncertainties. (GeV)recoE0 0.5 1 1.5 2 2.5 3Events per bin012345678Current Predicted Ereco Vs Q2 Spectrum: FCFV 1-ring mu-likeFigure 8.10: Predicted event spectrum for FHC νµ signal events at SK. Red lines showthe prediction, as a solid line, and error envelopes, as dashed lines, from the neardetector fit with MQEA , C5A(0) and MRESA split by target, and blue lines show theprediction and error envelopes from the near detector fit with MQEA split by target.198 (GeV)recoE0 0.2 0.4 0.6 0.8 1 1.2Events per bin00.511.522.53Current Predicted Ereco Vs Q2 Spectrum: FCFV 1-ring e-likeFigure 8.11: Predicted event spectrum for FHC νe signal events at SK. Red lines showthe prediction, as a solid line, and error envelopes, as dashed lines, from the neardetector fit with MQEA , C5A(0) and MRESA split by target, and blue lines show theprediction and error envelopes from the near detector fit with MQEA split by target. (GeV)recoE0 0.5 1 1.5 2 2.5 3Events per bin00.511.522.533.5Current Predicted Ereco Vs Q2 Spectrum: FCFV 1-ring mu-like (RHC running)Figure 8.12: Predicted event spectrum for RHC νµ signal events at SK. Red lines showthe prediction, as a solid line, and error envelopes, as dashed lines, from the neardetector fit with MQEA , C5A(0) and MRESA split by target, and blue lines show theprediction and error envelopes from the near detector fit with MQEA split by target.199 (GeV)recoE0 0.2 0.4 0.6 0.8 1 1.2Events per bin00.10.20.30.40.50.6Current Predicted Ereco Vs Q2 Spectrum: FCFV 1-ring e-like (RHC running)Figure 8.13: Predicted event spectrum for RHC νe signal events at SK. Red lines showthe prediction, as a solid line, and error envelopes, as dashed lines, from the neardetector fit with MQEA , C5A(0) and MRESA split by target, and blue lines show theprediction and error envelopes from the near detector fit with MQEA split by target.200Chapter 9Conclusions and SummaryAdding a water-carbon target to the T2K near detector maximum likelihood fit used to constrainthe neutrino flux and cross-sections in the oscillation fits provides a method to significantly re-duce the overall uncertainties on cross-sections at SK and thus the neutrino spectra predictionsand oscillation measurements by providing constraints on oxygen-specific cross-section pa-rameters. The various validation steps performed, such as pull studies, indicate that the fit issufficiently unbiased with this addition, and both neutrino mode and antineutrino mode datawas used in the fit.This was accomplished by developing neutrino and antineutrino CC selections for FGD 2,to match the selections previously used in FGD 1. To do this, the CC inclusive selection cuts forneutrinos and antineutrinos were adapted to use FGD 2 as a target. In addition, the topology-based sample selections were adapted for FGD 2 and implemented as a separate set of cuts, ascuts are not identical due to differences in geometry and active volume. These new samplesallow for subtraction analysis between the FGD 1 carbon target and the FGD 2 water-carbontarget to measure oxygen interactions.Along with the additional water-carbon samples, the near detector fit was updated withimproved treatment of the detector systematics. As the systematics between the two detec-tors may have some correlations due to similar geometries, the fit now allows for correlations201between the two sets of samples for appropriate systematics. This translates to correlationsbetween the angle and momenta bins used in the fit. Additionally, correlations were imple-mented within systematics, such as between momenta bins in order to more accurately modelsystematic effects.The near detector fit, with the exception of the oxygen target parameters which were notfit in the previous analyses and new parameters added for this analysis, is in agreement withthe results seen previously when fitting with only FGD 1. The fit gives an axial mass value ofMQEA = 1.113± 0.033, which is in agreement with the previous axial mass measurements atND280.The overall uncertainty on the neutrino spectrum predictions at SK, which includes un-certainties from the flux and SK-specific detector systematics as well, was reduced by around2.5% for the neutrino mode spectra and by around 5% for the antineutrino mode spectra, giv-ing an overall uncertainty of 5.0% for FHC νµ down from 7.7%, 5.4% for FHC νe down from6.8%, 5.2% for RHC ν¯µ down from 11.6% and 6.2% for RHC ν¯e down from 11.0%. Thisreduces the overall contribution from systematic sources to the uncertainty on the oscillationparameters significantly. Additionally, the 2D contours for sin2θ23 and ∆m232 (∆m213 for IH)show improvements to the 1D contour ranges between fits using the FGD 1-only constrainsand the FGD 1 + FGD 2 constraints described in the this thesis. The 1D sin2θ23 range is re-duced by 3% for NH and 2% for IH, while the ∆m2 1D range is reduced by 11% for NH and8% for IH.Additional studies were performed using the T2K near detector fit to check the nuclearmodel dependencies in how cross-sections are modelled at T2K, as unaccounted-for nucleareffects could be absorbed by model parameters that are assumed to have no nuclear targetdependence. To this end, the axial mass for CCQE interactions and the axial mass and C5A formfactor for resonant pion production were implemented as separate carbon and oxygen versionsto test this.These studies have shown that there is no significant change in the fitted parameter values202for these parameters when fit in a non-model dependent way, with the exception of the C5Aform factor, which shows a weak ∼ 1.5σ shift between carbon and oxygen. This indicates thatthe current neutrino-nucleus cross-section model used at T2K sufficiently models the nucleareffects such that the axial mass parameters do not pick up unaccounted-for effects, as theresonant pion production cross-section primarily contributes to background at SK.In addition, using the results of these model dependency tests to construct the spectra pre-dictions at SK shows no significant deviation in the predicted spectra from the original officialpredictions. The main difference in the predictions is the increase on the overall uncertaintyon the prediction due to increased uncertainty on the new oxygen target parameters. This indi-cates that the current cross-section model used adequately models the nuclear target-dependenteffects, and that additional uncertainties to account for deviations of this data from the modelare not required.While the studies performed and described here are for a few specific model parameters,there remain other cross-section parameters, such as those used to model final state interac-tions (FSI) that could benefit from similar model-dependency checking. Additionally, furtherinvestigation into the shift in C5A when model dependencies are removed may be important, aswhile the SK analyses use CCQE as the signal with resonant pion production as background,future studies may wish to include a CC 1pi signal as well. As less work has been done onmodelling nuclear effects for resonant pion production than for quasielastic processes, furtherstudy is needed.203Bibliography[1] David Griffiths. Introduction to Elementary Particles. Wiley-VCH, 2009.[2] F. Reines and C. L. Cowan. 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Minuit: A System for Function Minimization and Analysis ofthe Parameter Errors and Correlations. Comput. Phys. Commun., 10:343–367, 1975.doi:10.1016/0010-4655(75)90039-9.[63] R. Brun and F. Rademakers. ROOT: An object oriented data analysis framework. Nucl.Instrum. Meth., A389:81–86, 1997. doi:10.1016/S0168-9002(97)00048-X.[64] Megan Friend et al. Flux prediction and uncertainty updates with NA61 2009 thintarget data and negative focussing mode predictions. Technical Report 217, T2K, June2015.[65] N. Abgrall et al. Measurement of production properties of positively charged kaons inproton-carbon interactions at 31 GeV/c. Phys. Rev. C, 85:035210, Mar 2012.doi:10.1103/PhysRevC.85.035210. URLhttp://link.aps.org/doi/10.1103/PhysRevC.85.035210.[66] M. Apollonio et al. Forward production of charged pions with incident protons onnuclear targets at the CERN proton synchrotron. Phys. Rev. C, 80:035208, Sep 2009.doi:10.1103/PhysRevC.80.035208. URLhttp://link.aps.org/doi/10.1103/PhysRevC.80.035208.[67] Mark Hartz et al. Constraining the flux and cross section models with data from theND280 detector using FGD1 and FGD2 for the 2016 joint oscillation analysis.Technical Report 230, T2K, June 2016.[68] M. Dunkman et al. Updated recommendation of the 2014-5 oscillation parameters.Technical Report 265, T2K, April 2016.[69] A. A. Aguilar-Arevalo et al. First measurement of the muon antineutrinodouble-differential charged-current quasielastic cross section. Phys. Rev. D, 88:032001,Aug 2013. doi:10.1103/PhysRevD.88.032001. URLhttp://link.aps.org/doi/10.1103/PhysRevD.88.032001.[70] G. A. Fiorentini et al. Measurement of muon neutrino quasielastic scattering on ahydrocarbon target at Eν ∼ 3.5 GeV. Phys. Rev. Lett., 111:022502, Jul 2013.doi:10.1103/PhysRevLett.111.022502. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.111.022502.209[71] A. Bercellie et al. Cross section parameters for 2014 oscillation analysis. TechnicalReport 192, T2K, June 2015.[72] Patrick de Perio, Yoshinari Hayato, and Roman Tacik. NEUT nuclear effects (FSI).Technical Report 33, T2K, March 2012.[73] Sarah Bolognesi et al. Assessing the effect of cross-section model uncertainties on theT2K oscillation analyses with fake data studies using the BANFF, MaCh3 and VALORfit frameworks. Technical Report 285, T2K, June 2016.[74] K. Abe and others (T2K Collaboration). Measurement of muon antineutrinooscillations with an accelerator-produced off-axis beam. Phys. Rev. Lett., 116(181801),2016.[75] Yoshinari Hayato. A neutrino interaction simulation program library NEUT. Acta Phys.Polon., B40:2477–2489, 2009. Version 5.3.3 of the NEUT library is used that includes(i) the multinucleon ejection model of Nieves et al. [42] and (ii) nuclear long rangecorrelations for CCQE interactions, treated in the random phase approximation [100].[76] A. Hillairet, A. Izmaylov, B. Jamieson, and T. Lindner. ND280 reconstruction.Technical Report 072, T2K, November 2011.[77] Rudolph Emil Kalman. A new approach to linear filtering and prediction problems.Transactions of the ASME–Journal of Basic Engineering, 82(Series D):35–45, 1960.[78] Claudio Giganti and Marco Zito. Particle Identification with the T2K TPC. TechnicalReport 001, T2K, October 2009.[79] S Bordoni, Giganti C, Hillairet A, and Sanchez F. The TPC particle identificationalgorithm with production 6B. Technical Report 221, T2K, April 2015.[80] Caio Liccardi and Mauricio Barbi. Particle identification with the Fine GrainedDetectors. Technical Report 103, T2K, January 2012.[81] Pierre Bartet et al. νµ CC event selections in the ND280 tracker using Run 2+3+4 data.Technical Report 212, T2K, October 2015.[82] Jiae Kim, Christine Nielsen, and Michael Wilking. Michel Electron Tagging in theFGDs. Technical Report 104, T2K, Jan 2015.[83] V. Berardi et al. CC ν¯µ event selection in the ND280 tracker using Run 5c and Run 6anti-neutrino beam data. Technical Report 246, T2K, Oct 2015.[84] V. Berardi et al. CC νµ background event selection in the ND280 tracker using Run 5c+ Run 6 anti-neutrino beam data. Technical Report 248, T2K, October 2015.[85] A. Frank, A. Marchionni, and M. Messina. B-field calibration and systematic errors.Technical Report 081, T2K, September 2011.210[86] C. Bojechko et al. Measurement and correction of magnetic field distortions in the timeprojection chambers. Technical Report 061, T2K, June 2011.[87] A. Cervera and L Escudero. Study of momentum resolution and scale using tracks thatcross multiple TPCs. Technical Report 222, T2K, November 2014.[88] Yevgeny Petrov and Anthony Hillairet. ND280 TPC track reconstruction efficiency.Technical Report 163, T2K, February 2016.[89] Federico Sanchez and John vo Medina. ND280 global charge identification systematicerror. Technical Report 229, T2K, May 2016.[90] A. Hillairet et al. ND280 tracker tracking efficiency. Technical Report 075, T2K,January 2016.[91] Wojciech Oryszczak and Weronika Warzycha. FGD systematics: PID and IsoReconhybrid efficiencies. Technical Report 223, T2K, April 2015.[92] Kendall Mahn, Scott Oser, and Thomas Lindner. FGD mass checks. Technical Report122, T2K, May 2012.[93] Daniel Roberge. Elemental composition of FGD passive water modules. TechnicalReport 198, T2K, January 2010.[94] Jordan Myslik. Determination of pion secondary interaction systematics for the ND280tracker analysis. Technical Report 125, T2K, October 2013.[95] F. Dufour, L. Haegel, T. Lindner, and S. Oser. Systematics on out-of-fiducial-volumebackgrounds in the ND280 tracker. Technical Report 098, T2K, June 2015.[96] Graham Upton and Ian Cook. A Dictionary of Statistics. Oxford University Press, 3edition, 2014. ISBN 9780199679188.[97] Luc Demortier and Louis Lyons. Everything you always wanted to know about pulls.Technical Report 5776, CDF, August 2002.[98] R. Gran et al. Measurement of the quasi-elastic axial vector mass in neutrino-oxygeninteractions. Phys. Rev., D74:052002, 2006. doi:10.1103/PhysRevD.74.052002.[99] James Imber et al. Four sample joint oscillation analysis with T2K run1-6 data.Technical Report 267, T2K, July 2016.[100] J. Nieves, J. E. Amaro, and M. Valverde. Inclusive quasi-elastic neutrino reactions.Phys. Rev. C, 70:055503, 2004. URLhttp://journals.aps.org/prc/abstract/10.1103/PhysRevC.70.055503. [Erratum-ibid. C72 (2005) 019902].211Appendix AAdditional Flux Prediction PlotsThis appendix contains plots of the flux predictions and uncertainties on the predictions forthe neutrino fluxes at ND280 and SK, for both FHC and RHC beam modes. These plots wereproduced by the T2K beam group as described in Section 3.3.1 in Chapter 3.212 (GeV)νE-110 1 10Fractional Error00.10.20.30.4eνND280: Positive Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal ErrorPrevious T2K Results(a) ND280 FHC νe flux uncertainty and prediction. (GeV)νE-110 1 10Fractional Error00.10.20.30.40.5eνND280: Positive Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal ErrorPrevious T2K Results(b) ND280 FHC ν¯e flux uncertainty and prediction.Figure A.1: Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy.213 (GeV)νE-110 1 10Fractional Error00.10.20.3eνND280: Negative Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal Error(a) ND280 RHC νe flux uncertainty and prediction.(b) ND280 RHC ν¯e flux uncertainty and prediction.Figure A.2: Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy.214 (GeV)νE-110 1 10Fractional Error00.10.20.3µνSK: Positive Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal ErrorPrevious T2K Results(a) SK FHC νµ flux uncertainty and prediction. (GeV)νE-110 1 10Fractional Error00.10.20.30.4µνSK: Positive Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal ErrorPrevious T2K Results(b) SK FHC ν¯µ flux uncertainty and prediction.Figure A.3: Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy.215 (GeV)νE-110 1 10Fractional Error00.10.20.30.4eνSK: Positive Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal ErrorPrevious T2K Results(a) SK FHC νe flux uncertainty and prediction. (GeV)νE-110 1 10Fractional Error00.10.20.30.40.5eνSK: Positive Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal ErrorPrevious T2K Results(b) SK FHC ν¯e flux uncertainty and prediction.Figure A.4: Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy.216 (GeV)νE-110 1 10Fractional Error00.10.20.3µνSK: Negative Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal Error(a) SK RHC νµ flux uncertainty and prediction. (GeV)νE-110 1 10Fractional Error00.10.20.3µνSK: Negative Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal Error(b) SK RHC ν¯µ flux uncertainty and prediction.Figure A.5: Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy.217 (GeV)νE-110 1 10Fractional Error00.10.20.3eνSK: Negative Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal Error(a) SK RHC νe flux uncertainty and prediction. (GeV)νE-110 1 10Fractional Error00.10.20.30.4eνSK: Negative Focussing Mode, Hadron InteractionsProton Beam Profile & Off-axis AngleHorn Current & FieldHorn & Target Alignment, Arb. Norm.νE×ΦMaterial ModelingProton NumberTotal Error(b) SK RHC ν¯e flux uncertainty and prediction.Figure A.6: Predicted fractional uncertainties on the flux priors as a function of neutrinoenergy.218Appendix BMomentum and Angle Distributions forSelected ND280 EventsThis appendix contains plots relevant to Chapter 4. These plots show the momentum andangular distributions for the RHC ν¯µ CC Inclusive selection, described in Section 4.5.4, andthe RHC wrong-sign νµ CC Inclusive selection, described in Section 4.5.6.219Muon momentum (MeV/c)0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Events/(100 MeV/c)0100200300400500Integral    3600DatapiCC-0piCC-NBKGExternalNo truthIntegral    3323(a) Momentum distribution)θMuon cos(0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Events02004006008001000120014001600Integral    3742DatapiCC-0piCC-NBKGExternalNo truthIntegral    3480(b) cosθµ distributionFigure B.1: Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 1 RHC ν¯µ CC Inclusive selection[83]. Distributionsshown use the T2K Monte Carlo before the ND280 fit tuning.220 candidate reconstructed momentum (MeV)+µ0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Normalised to Run 5c + Run 6 POT050100150200250300350400450Integral    3672DatapiCC-0piCC-NBKGExternalNo truthIntegral    3373(a) Momentum distribution)θ candidate cos(+µ-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Normalised to Run 5c + Run 6 POT020040060080010001200140016001800200022002400Integral    3819DatapiCC-0piCC-NBKGExternalNo truthIntegral    3545(b) cosθµ distributionFigure B.2: Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 2 RHC ν¯µ CC Inclusive selection[83]. Distributionsshown use the T2K Monte Carlo before the ND280 fit tuning.221Muon momentum (MeV/c)0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Events/(200 MeV/c)020406080100120140160180Entries  23340Mean     1630RMS      1187Integral    1761DatapiCC-0piCC-NBKGExternalNo truthEntries  2071Mean     1690RMS      1213Integral    1807(a) Momentum distribution)θMuon cos(-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Events02004006008001000120014001600Entries  23340Mean   0.8843RMS    0.1581Integral    2004DatapiCC-0piCC-NBKGExternalNo truthEntries  2071Mean   0.8852RMS    0.1599Integral    2071(b) cosθµ distributionFigure B.3: Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 1 RHC νµ CC Inclusive selection[84].222Muon momentum (MeV/c)0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Events/(200 MeV/c)020406080100120140160180200Entries  22765Mean     1589RMS      1176Integral    1738DatapiCC-0piCC-NBKGExternalNo truthEntries  1992Mean     1612RMS      1184Integral    1761(a) Momentum distribution)θMuon cos(-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Events02004006008001000120014001600Entries  22765Mean   0.8727RMS    0.1991Integral    1968DatapiCC-0piCC-NBKGExternalNo truthEntries  1992Mean   0.8795RMS     0.183Integral    1992(b) cosθµ distributionFigure B.4: Momentum and cosθµ distributions for data and Monte Carlo, broken downby true topology, for the FGD 2 RHC νµ CC Inclusive selection[84].223Appendix CTables of Fitted Parameter Values atND280This appendix contains tables of the fitted parameter values from the ND280 fit described inChapter 7, for flux and cross-section parameters.224FHC flux parameter Prefit ND280 postfitND280 νµ Bin 0 1.0 ± 0.097467 1.1494 ± 0.060655ND280 νµ Bin 1 1.0 ± 0.10102 1.1745 ± 0.055057ND280 νµ Bin 2 1.0 ± 0.093831 1.1554 ± 0.046689ND280 νµ Bin 3 1.0 ± 0.087563 1.1412 ± 0.039376ND280 νµ Bin 4 1.0 ± 0.10743 1.1205 ± 0.042462ND280 νµ Bin 5 1.0 ± 0.10542 1.0945 ± 0.043416ND280 νµ Bin 6 1.0 ± 0.074043 1.15 ± 0.042205ND280 νµ Bin 7 1.0 ± 0.069071 1.1505 ± 0.041948ND280 νµ Bin 8 1.0 ± 0.082987 1.1574 ± 0.043407ND280 νµ Bin 9 1.0 ± 0.098196 1.1112 ± 0.045209ND280 νµ Bin 10 1.0 ± 0.1146 1.1046 ± 0.051429ND280 ν¯µ Bin 0 1.0 ± 0.10173 1.1015 ± 0.080716ND280 ν¯µ Bin 1 1.0 ± 0.08147 1.1024 ± 0.05676ND280 ν¯µ Bin 2 1.0 ± 0.078275 1.1175 ± 0.055378ND280 ν¯µ Bin 3 1.0 ± 0.082962 1.1584 ± 0.062253ND280 ν¯µ Bin 4 1.0 ± 0.086702 1.1694 ± 0.067138ND280 νe Bin 0 1.0 ± 0.090449 1.1485 ± 0.050188ND280 νe Bin 1 1.0 ± 0.092268 1.1492 ± 0.048532ND280 νe Bin 2 1.0 ± 0.090296 1.1746 ± 0.05036ND280 νe Bin 3 1.0 ± 0.087908 1.1399 ± 0.048494ND280 νe Bin 4 1.0 ± 0.088004 1.1293 ± 0.054468ND280 νe Bin 5 1.0 ± 0.082902 1.1442 ± 0.04385ND280 νe Bin 6 1.0 ± 0.096231 1.1216 ± 0.061689ND280 ν¯e Bin 0 1.0 ± 0.074039 1.13 ± 0.056243ND280 ν¯e Bin 1 1.0 ± 0.14013 1.181 ± 0.12721Table C.1: Prefit and ND280 postfit values for the ND280 FHC flux parameters. Fluxvalues are relative to the nominal T2K flux prediction.225NF flux parameter Prefit ND280 postfitND280 νµ Bin 0 1.0 ± 0.092802 1.1014 ± 0.072527ND280 νµ Bin 1 1.0 ± 0.080395 1.137 ± 0.05007ND280 νµ Bin 2 1.0 ± 0.083051 1.1476 ± 0.050325ND280 νµ Bin 3 1.0 ± 0.083992 1.1825 ± 0.051599ND280 νµ Bin 4 1.0 ± 0.082472 1.1475 ± 0.043341ND280 ν¯µ Bin 0 1.0 ± 0.1034 1.1323 ± 0.068888ND280 ν¯µ Bin 1 1.0 ± 0.098917 1.144 ± 0.055399ND280 ν¯µ Bin 2 1.0 ± 0.090715 1.1444 ± 0.049222ND280 ν¯µ Bin 3 1.0 ± 0.087484 1.1394 ± 0.043408ND280 ν¯µ Bin 4 1.0 ± 0.11988 1.1587 ± 0.059233ND280 ν¯µ Bin 5 1.0 ± 0.11453 1.1597 ± 0.059095ND280 ν¯µ Bin 6 1.0 ± 0.085223 1.1637 ± 0.051206ND280 ν¯µ Bin 7 1.0 ± 0.075007 1.1675 ± 0.051101ND280 ν¯µ Bin 8 1.0 ± 0.097631 1.188 ± 0.070358ND280 ν¯µ Bin 9 1.0 ± 0.091953 1.1752 ± 0.064477ND280 ν¯µ Bin 10 1.0 ± 0.1395 1.1845 ± 0.10848ND280 νe Bin 0 1.0 ± 0.068591 1.1303 ± 0.049596ND280 νe Bin 1 1.0 ± 0.089673 1.1493 ± 0.071727ND280 ν¯e Bin 0 1.0 ± 0.09542 1.1402 ± 0.055452ND280 ν¯e Bin 1 1.0 ± 0.086103 1.1363 ± 0.047436ND280 ν¯e Bin 2 1.0 ± 0.092835 1.1699 ± 0.058373ND280 ν¯e Bin 3 1.0 ± 0.090264 1.16 ± 0.052849ND280 ν¯e Bin 4 1.0 ± 0.090515 1.1423 ± 0.066818ND280 ν¯e Bin 5 1.0 ± 0.092819 1.1409 ± 0.070634ND280 ν¯e Bin 6 1.0 ± 0.1584 1.1966 ± 0.14206Table C.2: Prefit and ND280 postfit values for the ND280 RHC flux parameters. Fluxvalues are relative to the nominal T2K flux prediction.226PF flux parameter Prefit ND280 postfitSK νµ Bin 0 1.0 ± 0.098732 1.1465 ± 0.061691SK νµ Bin 1 1.0 ± 0.10349 1.1759 ± 0.05881SK νµ Bin 2 1.0 ± 0.096444 1.1659 ± 0.048881SK νµ Bin 3 1.0 ± 0.086696 1.1416 ± 0.040814SK νµ Bin 4 1.0 ± 0.11305 1.1177 ± 0.04471SK νµ Bin 5 1.0 ± 0.091748 1.111 ± 0.04374SK νµ Bin 6 1.0 ± 0.070174 1.1415 ± 0.041818SK νµ Bin 7 1.0 ± 0.07368 1.1446 ± 0.044229SK νµ Bin 8 1.0 ± 0.087373 1.1471 ± 0.043473SK νµ Bin 9 1.0 ± 0.097937 1.1065 ± 0.044603SK νµ Bin 10 1.0 ± 0.11436 1.1003 ± 0.055456SK ν¯µ Bin 0 1.0 ± 0.10258 1.1114 ± 0.080454SK ν¯µ Bin 1 1.0 ± 0.078533 1.1232 ± 0.04736SK ν¯µ Bin 2 1.0 ± 0.084454 1.1204 ± 0.059361SK ν¯µ Bin 3 1.0 ± 0.085568 1.1308 ± 0.068884SK ν¯µ Bin 4 1.0 ± 0.086428 1.1799 ± 0.067391SK νe Bin 0 1.0 ± 0.089698 1.1516 ± 0.049553SK νe Bin 1 1.0 ± 0.089952 1.1529 ± 0.046389SK νe Bin 2 1.0 ± 0.085965 1.153 ± 0.044459SK νe Bin 3 1.0 ± 0.080918 1.1373 ± 0.039562SK νe Bin 4 1.0 ± 0.078972 1.137 ± 0.041913SK νe Bin 5 1.0 ± 0.08385 1.1371 ± 0.043819SK νe Bin 6 1.0 ± 0.093894 1.1475 ± 0.061585SK ν¯e Bin 0 1.0 ± 0.074028 1.1362 ± 0.055155SK ν¯e Bin 1 1.0 ± 0.12842 1.1701 ± 0.11563Table C.3: Prefit and ND280 postfit values for the SK FHC flux parameters. Flux valuesare relative to the nominal T2K flux prediction.227NF flux parameter Prefit ND280 postfitSK νµ Bin 0 1.0 ± 0.093682 1.1102 ± 0.071174SK νµ Bin 1 1.0 ± 0.079343 1.1353 ± 0.050959SK νµ Bin 2 1.0 ± 0.076727 1.1426 ± 0.046598SK νµ Bin 3 1.0 ± 0.080559 1.173 ± 0.051551SK νµ Bin 4 1.0 ± 0.08029 1.139 ± 0.049424SK ν¯µ Bin 0 1.0 ± 0.10448 1.133 ± 0.069783SK ν¯µ Bin 1 1.0 ± 0.10153 1.144 ± 0.058132SK ν¯µ Bin 2 1.0 ± 0.096167 1.1323 ± 0.050464SK ν¯µ Bin 3 1.0 ± 0.084637 1.1337 ± 0.042324SK ν¯µ Bin 4 1.0 ± 0.12509 1.1708 ± 0.064084SK ν¯µ Bin 5 1.0 ± 0.10529 1.1478 ± 0.055719SK ν¯µ Bin 6 1.0 ± 0.079987 1.159 ± 0.049348SK ν¯µ Bin 7 1.0 ± 0.073938 1.1651 ± 0.049955SK ν¯µ Bin 8 1.0 ± 0.093992 1.179 ± 0.066625SK ν¯µ Bin 9 1.0 ± 0.092513 1.1637 ± 0.064092SK ν¯µ Bin 10 1.0 ± 0.13031 1.1138 ± 0.10591SK νe Bin 0 1.0 ± 0.068881 1.1358 ± 0.048551SK νe Bin 1 1.0 ± 0.084945 1.1352 ± 0.067841SK ν¯e Bin 0 1.0 ± 0.094695 1.1419 ± 0.056154SK ν¯e Bin 1 1.0 ± 0.091039 1.1425 ± 0.049143SK ν¯e Bin 2 1.0 ± 0.091012 1.1472 ± 0.050225SK ν¯e Bin 3 1.0 ± 0.083856 1.1478 ± 0.044191SK ν¯e Bin 4 1.0 ± 0.079578 1.1436 ± 0.054013SK ν¯e Bin 5 1.0 ± 0.089008 1.1409 ± 0.067854SK ν¯e Bin 6 1.0 ± 0.15581 1.1922 ± 0.1391Table C.4: Prefit and ND280 postfit values for the SK RHC flux parameters. Flux valuesare relative to the nominal T2K flux prediction.228Cross Section Parameter Prefit ND280 postfitMQEA (GeV/c2) 1.2 ± 0.069607 1.1113 ± 0.033281pF 12C (MeV/c) 217.0 ± 12.301 248.71 ± 16.048MEC 12C 100.0 ± 29.053 156.9 ± 22.635EB 12C (MeV) 25.0 ± 9.0 16.846 ± 7.5097pF 16O (MeV/c) 225.0 ± 12.301 239.22 ± 23.246MEC 16O 100.0 ± 35.228 155.89 ± 34.243EB 16O (MeV) 27.0 ± 9.0 24.262 ± 7.5922CA5RES 1.01 ± 0.12 0.78601 ± 0.060705MRESA (GeV/c2) 0.95 ± 0.15 0.84904 ± 0.038442Isospin=12 Background 1.3 ± 0.2 1.3633 ± 0.17371CC Other Shape 0.0 ± 0.4 -0.024697 ± 0.17809CC Coh 1.0 ± 0.3 0.85333 ± 0.22835NC Coh 1.0 ± 0.3 0.9308 ± 0.29816NC Other 1.0 ± 0.3 1.3066 ± 0.15511MEC ν¯ 1.0 ± 1.0 0.6109 ± 0.1672FSI Inel. Low E 0.0 ± 0.41231 -0.29976 ± 0.099264FSI Inel. High E 0.0 ± 0.33793 0.017027 ± 0.17908FSI Pion Prod. 0.0 ± 0.5 0.0077633 ± 0.26256FSI Pion Abs. 0.0 ± 0.41161 -0.2558 ± 0.16117FSI Ch. Exch. Low E 0.0 ± 0.56679 -0.071184 ± 0.39018FSI Ch. Exch. High E 0.0 ± 0.27778 0.004313 ± 0.14587Table C.5: Prefit and ND280 postfit values for the cross-section parameters229

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