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Measurement of an off-axis neutrino beam energy spectrum Kirby, Brian 2012

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Measurement of an Off-Axis Neutrino Beam Energy Spectrum by Brian Kirby  B.ASc., University of Toronto, 2004 M.Sc., York University, 2006  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Physics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2012 c Brian Kirby 2012  Abstract The T2K long baseline neutrino oscillation experiment is designed to measure the neutrino flavour mixing parameter θ13 , as well as θ23 and ∆m223 with twenty times greater precision than previous measurements. A neutrino beam is produced using the Japan Proton Accelerator Research Complex (J-PARC) proton accelerator in Tokai, Japan and is incident on the Super-Kamiokande water Cherenkov detector 295 km away beam at an offaxis angle of 2.5◦ . A suite of near detectors 280 m away from the proton target (ND280) provides additional constraints on beam flux estimates as well as measures neutrino interaction cross sections. A critical component of ND280 is a Fine Grained Detector (FGD) that provides an active target for neutrino interactions, with sufficient granularity to reconstruct short ranged particle tracks. This thesis describes the T2K experiment, the design and calibration of the FGD, and its online data reduction system. A cut-based selection for charged current neutrino interactions is used to produce quasi-elastic and non-quasi-elastic enhanced samples. These samples are used in a maximum likelihood fit to measure the T2K neutrino beam energy spectrum. The fit determines the flux scale factors that best reproduce the kinematic distribution of muons produced in selected interactions, and accounts for all relevant neutrino interaction model and detector systematic uncertainties. The fitted flux factors f (Eν ) in the energy ranges defined for the analysis are as follows: f (0 < Eν < 0.5GeV ) = 1.10+0.28 −0.24 , +0.17 f (0.5 < Eν < 1.0GeV ) = 0.93+0.17 , f (1.0 < E < 3.5GeV ) = 0.85 ν −0.14 −0.14 . These flux factors are consistent with and f (Eν > 3.5GeV ) = 0.92+0.28 −0.23 the default neutrino beam flux prediction and the flux measurement used in the primary oscillation analysis of the T2K collaboration, and provide an independent confirmation that the neutrino beam flux model is reliable.  ii  Preface This dissertation was completed while working on the Tokai-to-Kamioka (T2K) experiment, and with aid from other members of the T2K collaboration. The text of this document has not been taken directly from previously published articles or other documents. The Fine Grained Detector (FGD) described in Chapter 3 was designed by members of the T2K FGD group and TRIUMF staff. I assisted in its assembly and commissioning but not its design. The FGD readout electronics and slow control system were designed by members of the FGD electronics group other than myself, based on a design by D. Calvet and other members of the T2K Time Projection Chamber electronics group. The firmware code implemented on the FGD CMB FPGA was primarily written by D. Calvet, with major modifications made by K. Mizouchi, C. Gutjahr and myself. The data acquisition program running on the FGD DCC was primarily implemented by K. Olchanski and K. Mizouchi, with modifications made by C. Gutjahr and myself. The FGD calibration process was designed by many members of the FGD group. I specifically contributed the analysis of the scintillator bar light yield based upon cosmic data, as well as studies of MPPC performance. The FGD online data compression scheme described in Chapter 4 makes extensive use of the digital logic framework implemented by D. Calvet. I replaced the previously implemented data reduction scheme with the process described in the chapter. The pulse identification and data reduction algorithm was designed, implemented and tested by myself. I also redesigned the FGD output data format, and the corresponding modifications to the CMB FPGA process creating data packets. The ND280 software and analysis framework presented in Chapter 5 was designed by other ND280 collaborators. The FGD cosmic trigger electronics described in Chapter 6 was implemented by FGD electronics group members other than myself, and the CMB digital trigger logic implemented by K. Mizouchi. The Corsika flux was simulated by M. George, and the simulation of the ND280 pit was by J. Lagoda and K. Kowalik. I implemented and tested the simulation of the FGD trigiii  Preface ger signals and logic, which was then integrated into the ND280 software framework by M. Wilking. The inclusive charged current event selection presented in chapter 7 was developed primarily by F. Sanchez and M. Ravonel, with contributions from other ND280 collaborators. The two additional selection cuts for CCQE interactions were designed and implemented by myself. The selected muon track binning was defined by K. Mahn. The systematic effects relevant to the event selection that are defined in Chapter 8 were identified and measured by members of the ND280 collaboration. I personally measured the size of the systematic error introduced into the selection by external background in the Michel electron and track multiplicity CCQE selection cuts. I also implemented the weight-based method for propagating systematic effects through the event selection. The measurement of the ND280 neutrino flux energy spectrum using a maximum likelihood fit presented in Chapter 9 was devised by members of the ND280 BANFF group, including S. Oser, K. Mahn, M. Hartz and P. dePerio. The version of the fit where only ND280 information is used and prior constraints on the beam flux ignored was implemented and validated by myself. The T2KReWeight package that was used to regenerate the prediction for the selected event distributions used in the fit was created by other members of the ND280 collaboration. The analysis of the neutrino energy spectrum measured with only ND280 information is my original work.  iv  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xi  Abstract Preface  List of Tables  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 1 Neutrino Theory . . . . . . . . . . . . . . . . 1.1 History . . . . . . . . . . . . . . . . . . . . 1.2 Neutrinos in the Standard Model of Physics 1.3 Neutrino Oscillations and Mass . . . . . . . 1.3.1 Neutrino Oscillation Probability . . 1.3.2 Mass Hierarchy . . . . . . . . . . . 1.3.3 Matter Effects . . . . . . . . . . . . 1.3.4 CP Violation . . . . . . . . . . . . . 1.3.5 Neutrino Interaction Cross Sections  . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  1 1 2 3 5 7 8 9 10  2 The T2K Experiment . . . . . . . . . . 2.1 Neutrino Physics at T2K . . . . . . . 2.2 The T2K Neutrino Beam . . . . . . . 2.2.1 Proton Accelerator . . . . . . 2.2.2 Neutrino Beamline . . . . . . 2.2.3 Off-Axis Neutrino Beam . . . 2.2.4 Neutrino Beam Flux Prediction 2.3 Near Detectors . . . . . . . . . . . . . 2.3.1 The INGRID On-Axis Detector 2.3.2 The Off-Axis Detector Suite .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  13 13 16 16 17 19 19 22 23 24  . . . . . .  . . . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  v  Table of Contents 2.4  Super-Kamiokande . . . . . . . . . . . . . . . . . . . . . . . .  26  ND280 Fine Grained Detector . . . . FGD Physical Description . . . . . . . . . Scintillators and Light Collection . . . . . . Multi-Pixel Photon Counters . . . . . . . . Readout Electronics . . . . . . . . . . . . . 3.4.1 Backplanes and Mini-Crates . . . . 3.4.2 Front End Boards . . . . . . . . . . 3.4.3 Crate Master Boards . . . . . . . . 3.4.4 Data Concentrator Cards . . . . . . 3.4.5 CMB-DCC Data Transfer Protocol Slow Control . . . . . . . . . . . . . . . . . Calibration . . . . . . . . . . . . . . . . . . 3.6.1 Gain Calibration . . . . . . . . . . . 3.6.2 Gain Dependence on Temperature . 3.6.3 Gain Dependence on Overvoltage . 3.6.4 Ratio of Low to High Channel Gain 3.6.5 MPPC Gain Saturation . . . . . . . 3.6.6 Bar-to-Bar Light Variation . . . . . 3.6.7 Timing Calibration . . . . . . . . . 3.6.8 Other Calibrations . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  31 31 34 35 36 36 38 39 40 40 41 42 42 43 44 44 45 45 45 46  4 FGD Online Data Compression . . . . . . . . . . . . 4.1 Data Compression . . . . . . . . . . . . . . . . . . . 4.2 The FGD Pulse Identification Algorithm . . . . . . 4.2.1 Pulse Finder Design . . . . . . . . . . . . . . 4.2.2 Efficiency Measurements . . . . . . . . . . . 4.3 Firmware Implementation of the FGD Pulse Finder 4.3.1 ASIC Channel Readout and Processing . . . 4.4 Pulse Height Based Data Reduction . . . . . . . . . 4.5 Validation of the Simulated Pulse Finder Algorithm  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  47 47 48 49 52 53 54 55 57  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  58 58 58 59 63 64 68  3 The 3.1 3.2 3.3 3.4  3.5 3.6  5 The Near Detector Analysis Framework 5.1 Data Acquisition . . . . . . . . . . . . . 5.2 The ND280 Software Package . . . . . . 5.2.1 Simulation of ND280 Data . . . 5.2.2 Calibration . . . . . . . . . . . . 5.2.3 Reconstruction . . . . . . . . . . 5.2.4 Event Summary with oaAnalysis  . . . . . .  . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . .  . . . . . . .  vi  Table of Contents 6 The FGD Cosmic Trigger . . . . . . . 6.1 Cosmic Rays in FGD Analysis . . . 6.2 The Trigger System . . . . . . . . . 6.2.1 FGD Cosmic Trigger . . . . 6.3 Cosmic Flux Simulation . . . . . . . 6.4 FGD Cosmic Trigger Simulation . . 6.5 Validation of the Trigger Simulation 6.6 Trigger Simulation Applications . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  70 70 70 71 72 73 75 77  7 The CCQE and CCnQE Interaction Selection . . . 7.1 CC Inclusive Selection . . . . . . . . . . . . . . . . . 7.1.1 Data Format . . . . . . . . . . . . . . . . . . 7.1.2 Data Samples . . . . . . . . . . . . . . . . . 7.1.3 Neutrino Beam Flux and Interaction Models 7.1.4 Beam Summary and Data Quality Flags . . 7.1.5 Global Reconstruction Tracks and Variables 7.1.6 Beam Bunch Time Association . . . . . . . . 7.1.7 Track Topology . . . . . . . . . . . . . . . . 7.1.8 Steps in Muon-Like Track Selection . . . . . 7.1.9 External Background Pileup Correction . . . 7.2 CCQE and CCnQE Enhanced Samples . . . . . . . 7.3 Selection Results in Real and Simulated Data . . . 7.4 Selected Muon Track Binning . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  79 79 79 80 81 81 81 82 84 84 86 87 89 90  8 Detector Systematic Studies . . . . . . . . . . . . . . . 8.1 Identification of Systematic Effects . . . . . . . . . . . 8.2 Systematic Error Propagation . . . . . . . . . . . . . 8.2.1 Systematic Error Covariance Matrices . . . . . 8.2.2 Systematic Propagation through Simulation . 8.2.3 Systematic Propagation through Reweighting 8.2.4 Systematic Propagation through Bin Migration 8.3 Systematic Error Covariance Matrices . . . . . . . . . 8.3.1 TPC Tracking Efficiency . . . . . . . . . . . . 8.3.2 Track Charge Misidentification . . . . . . . . . 8.3.3 Track Quality Cut . . . . . . . . . . . . . . . . 8.3.4 TPC Particle Identification . . . . . . . . . . . 8.3.5 Magnetic Field Distortions . . . . . . . . . . . 8.3.6 Momentum Resolution . . . . . . . . . . . . . 8.3.7 Momentum Scale . . . . . . . . . . . . . . . . 8.3.8 FGD-TPC Tracking Efficiency . . . . . . . . .  . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  99 99 101 101 102 103 105 105 105 106 107 109 110 112 113 114  . . . . . . . . .  vii  Table of Contents 8.3.9 8.3.10 8.3.11 8.3.12 8.3.13 8.3.14 8.3.15  8.4  FGD-TPC Broken Tracks . . . . . . . . . . . . . . . . FGD Target Mass . . . . . . . . . . . . . . . . . . . . FGD Out of Fiducial Volume Interactions . . . . . . Michel Detection Efficiency . . . . . . . . . . . . . . . External Background in the Michel Electron Cut . . . Track Multiplicity External Background . . . . . . . External Background from Sand Muons and Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.16 Sand Muon Pileup . . . . . . . . . . . . . . . . . . . . Total Detector Systematic Covariance Matrix . . . . . . . . .  9 Neutrino Energy Spectrum Measurement . . . . . . . 9.1 Data Analysis Tools . . . . . . . . . . . . . . . . . . . . 9.2 Maximum Likelihood Fit of Neutrino Energy Spectrum 9.2.1 BANFF Neutrino Flux Fit . . . . . . . . . . . . 9.3 Neutrino Interaction Modes . . . . . . . . . . . . . . . . 9.4 Neutrino Energy Binning . . . . . . . . . . . . . . . . . 9.5 Systematic Errors . . . . . . . . . . . . . . . . . . . . . 9.5.1 Detector Systematic Errors . . . . . . . . . . . . 9.5.2 Neutrino Interaction Model Systematic Errors . 9.5.3 Overall Systematic Covariance Matrix . . . . . . 9.5.4 Interaction Model Reweighting . . . . . . . . . . 9.6 Fit Reliability Studies . . . . . . . . . . . . . . . . . . . 9.6.1 Fit to Default Simulated Data . . . . . . . . . . 9.6.2 Fake Data Studies . . . . . . . . . . . . . . . . . 9.7 Fit of the Observed Data . . . . . . . . . . . . . . . . . 9.7.1 Statistics Only Fit . . . . . . . . . . . . . . . . . 9.7.2 Fit with Neutrino Interaction Model Systematics 9.7.3 Fit with All Systematics . . . . . . . . . . . . . 9.7.4 Comparison with Beam Flux Prediction . . . . . 9.8 Alternative Relative Energy Spectrum Likelihood Fit . 9.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  116 118 119 122 123 124 125 126 127 133 133 133 135 138 139 142 143 143 145 146 148 148 148 161 162 163 164 166 173 176  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178  viii  List of Tables 1.1 1.2  Current experimentally derived values of neutrino oscillation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of current ∆m2 measurements . . . . . . . . . . . . .  5 8  4.1  Simulated pulse finder efficiency and purity measurements . .  55  5.1  Primary neutrino interactions modelled by NEUT . . . . . .  61  7.1 7.2 7.3 7.4 7.5  Real and simulated data sample summary . . . . . Real and Simulated beam bunch times . . . . . . . Neutrino interaction selection results . . . . . . . . pµ − θµ bin index definitions . . . . . . . . . . . . . Number of bin entries for real and simulated data .  82 84 90 96 97  8.1  Near detector event selection systematic effects and the measured uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . 100 Number of selected event in normal and modified event selections related to the track multiplicity external background systematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Number of selected event in normal and modified event selections related to the track multiplicity external background systematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125  8.2  8.3  9.1 9.2 9.3 9.4 9.5 9.6  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  Definition of variables used in the maximum likelihood fit for the spectral measurement. . . . . . . . . . . . . . . . . . . . . Neutrino interaction categories . . . . . . . . . . . . . . . . . Number of selected neutrino interactions by mode. . . . . . . Neutrino energy binning . . . . . . . . . . . . . . . . . . . . . Neutrino interaction systematic errors . . . . . . . . . . . . . Results of fit to default simulated data, including flux factors and neutrino interaction model parameters . . . . . . . . . .  136 139 139 142 146 149  ix  List of Tables 9.7 9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17 9.18 9.19 9.20 9.21 9.22 9.23 9.24 9.25  9.26  9.27  Results of fit to default simulated data, pµ − θµ parameters only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Effective 5% significance probability thresholds for different sized samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Mean and pulls of fitted flux factors measured in fake data results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Probability of fitted flux factors’ pull distributions mean and RMS values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Fitted flux factors with neutrino interaction model systematics included fake data results . . . . . . . . . . . . . . . . . . 154 Fitted flux factor with neutrino interaction model systematics included pull probabilities . . . . . . . . . . . . . . . . . . . . 155 Fitted neutrino interaction model scale factors from fake data fits with neutrino interaction model systematics included . . . 156 Pull probabilities fitted neutrino interaction model from fake data fits with neutrino interaction model systematics included 156 Fitted flux factors fake data results . . . . . . . . . . . . . . . 158 Fitted flux factor pull probabilities when all systematic effects are included . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Fitted neutrino interaction model factors fake data results . . 160 Pull probabilities fitted neutrino interaction model from fake data fits with all systematics included . . . . . . . . . . . . . 160 Fitted pµ − θµ bin scale factors fake data results when all systematics are included, bins 0 to 19 . . . . . . . . . . . . . 162 Fitted pµ − θµ bin scale factors fake data results, bins 20 to 39 163 Fitted flux factors in the observed data distribution for the statistics only fit. . . . . . . . . . . . . . . . . . . . . . . . . . 164 Fitted flux factors in the observed data distribution where neutrino interaction model parameters are included. . . . . . 166 Fitted flux factors in the observed data distribution when all systematic parameters are included . . . . . . . . . . . . . . . 166 Comparison of fit results and beam flux prediction. . . . . . . 169 Fitted flux factors fake data results where all systematics nuisance parameters are included in the relative flux factor version of the fit . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Fitted flux factor pull probabilities where all systematics nuisance parameters are included in the relative flux factor version of the fit . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Fitted relative flux factors in the observed data distribution when all systematic parameters are included . . . . . . . . . . 175 x  List of Figures 1.1 1.2 1.3 1.4 1.5 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16  P (νµ → νµ ) for different neutrino energies . . . . . . . . . . . Charged current neutrino cross sections vs experimental measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutrino interactions in the Standard Model . . . . . . . . . Charged current neutrino cross sections vs experimental measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutrino interaction modes relevant to the T2K experiment . T2K design sensitivity to sin2 2θ13 for different 30 GeV proton beam exposures . . . . . . . . . . . . . . . . . . . . . . . . . . T2K design sensitivity to sin2 2θ13 for different 30 GeV proton beam exposures . . . . . . . . . . . . . . . . . . . . . . . . . . Charged current neutrino cross sections vs experimental measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T2K neutrino beamline overview . . . . . . . . . . . . . . . . Beam monitor position along in the T2K primary beamline . Correlation between off-axis angle and neutrino energies produced by pion decay . . . . . . . . . . . . . . . . . . . . . . . Neutrino flux energy spectrum at various off-axis angles . . . Predicted muon neutrino energy spectrum at Super-Kamiokande Predicted neutrino flux at Super-Kamiokande for different neutrino flavours . . . . . . . . . . . . . . . . . . . . . . . . . Predicted neutrino flux at N280 for different neutrino flavours Muon neutrino flux weights provided by scaling FLUKA pion production prediction to NA61 measurements. . . . . . . . . . Systematic errors in ND280 muon neutrino flux prediction due to various beam-related uncertainties . . . . . . . . . . . The INGRID on-axis detector . . . . . . . . . . . . . . . . . . The ND280 off-axis detector . . . . . . . . . . . . . . . . . . . The TPC detector . . . . . . . . . . . . . . . . . . . . . . . . The Super-Kamiokande detector . . . . . . . . . . . . . . . .  6 8 10 11 12 14 15 16 17 18 20 21 22 23 24 25 26 27 28 29 30  xi  List of Figures 2.17 Muon and electron neutrino signals at the Super-Kamiokande detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  30  3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11  Diagram of FGD frame and module supports. . . . . . . . . . FGD module with cover plate attached. . . . . . . . . . . . . FGD scintillator bar cross section . . . . . . . . . . . . . . . . FGD example MPPC image . . . . . . . . . . . . . . . . . . . FGD MPPCs coupled to wavelength shifting fibres . . . . . . Schematic overview of FGD readout electronics system . . . . Example image of FGD mini-crate and backplane . . . . . . . Example image of FGD FEB, with important features shown Example image of FGD CMB, with important features shown Example image of FGD DCC, with important features shown MPPC dark noise distribution . . . . . . . . . . . . . . . . . .  32 33 34 36 37 37 38 39 40 41 43  4.1 4.2 4.3 4.4 4.5 4.6  Example MPPC channel digitized waveform. . . . . . Distribution of ADC samples about baseline mean. . . Schematic overview of pulse finder algorithm operation Example of a simulated MPPC pulse . . . . . . . . . . Pulse detection efficiency vs simulated pulse time . . . Cosmic MPPC pulse spectrum . . . . . . . . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  48 49 51 53 54 56  5.1 5.2 5.3 5.4  Schematic overview of ND280 software package Schematic of SBCAT track pattern recognition. TPC CT vs. momentum in simulated data. . . Example of FGD-TPC matched track . . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  59 66 67 68  6.1 6.2 6.3 6.4 6.5 6.6 6.7  ND280 trigger and clock distribution system. . . . . ND280 subdetectors at the base of the ND280 pit. . ASUM group charge signals . . . . . . . . . . . . . . ASUM group charge signals efficiency . . . . . . . . FGD cosmic trigger TPC2 direction in Cosθ . . . . . FGD cosmic trigger TPC2 direction in φ . . . . . . . FGD calibrated charge deposit per layer produced by ple of real and simulated cosmic tracks . . . . . . . .  . . . . . . a .  . . . . . . . . . . . . . . . . . . . . . . . . sam. . . .  71 73 74 75 76 77  7.1 7.2 7.3 7.4 7.5  . . . .  . . . .  Example beam bunch time distribution. . . . . . . . . . . . . Muon pulls in simulated CC-inclusive sample . . . . . . . . . Electron pulls in simulated CC-inclusive sample . . . . . . . . Division of CC-inclusive selection into CCQE/CCnQE samples Selected muon momentum distribution . . . . . . . . . . . . .  78 83 87 88 88 91 xii  List of Figures 7.6  Selected muon cos θ distribution for real and default simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Selected muon momentum vs. cos θ distribution for real and default simulated data . . . . . . . . . . . . . . . . . . . . . . 7.8 Selected CCQE enhanced sample muon momentum distribution 7.9 Selected CCnQE enhanced sample muon momentum distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Default p − θ distribution . . . . . . . . . . . . . . . . . . . . 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19 8.20 8.21 8.22 8.23 8.24  Example TPC2 tracking efficiency systematic measurement . TPC tracking efficiency systematic change in selected event pµ − θµ distribution . . . . . . . . . . . . . . . . . . . . . . . TPC tracking efficiency covariance matrix . . . . . . . . . . . Charge misidentification covariance matrix . . . . . . . . . . . TPC hit efficiency in real and simulated data with efficiency systematic fit . . . . . . . . . . . . . . . . . . . . . . . . . . . Quality cut covariance matrix . . . . . . . . . . . . . . . . . . Particle identification covariance matrix . . . . . . . . . . . . Example reduction in reconstructed position residuals due to empirical field correction . . . . . . . . . . . . . . . . . . . . . Magnetic field distortion systematic covariance matrix . . . . Magnetic field distortion systematic statistical error covariance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . Momentum resolution systematic covariance matrix . . . . . . Momentum scale systematic covariance matrix . . . . . . . . FGD-TPC tracking efficiency covariance matrix . . . . . . . . FGD-TPC track breaking covariance matrix . . . . . . . . . . FGD mass systematic covariance matrix . . . . . . . . . . . . FGD out-of-fiducial volume systematic covariance matrix . . Michel hit cluster detection efficiency covariance matrix . . . Michel electron systematic covariance matrix . . . . . . . . . Track multiplicity external background systematic covariance Sand muon external background systematic covariance matrix Total systematic covariance matrix . . . . . . . . . . . . . . . Fractional error introduced by each systematic effect for the CCQE sample, −1 ≤ cos θ < 0.84. . . . . . . . . . . . . . . . . Fractional error introduced by each systematic effect for the CCQE sample, 0.84 ≤ cos θ < 0.9. . . . . . . . . . . . . . . . . Fractional error introduced by each systematic effect for the CCQE sample, 0.9 ≤ cos θ < 0.94. . . . . . . . . . . . . . . . .  92 93 94 95 98 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 126 127 128 129 129 130 xiii  List of Figures 8.25 Fractional error introduced by each CCQE sample, 0.94 ≤ cos θ < 1. . . 8.26 Fractional error introduced by each CCnQE sample, −1 ≤ cos θ < 0.84. 8.27 Fractional error introduced by each CCnQE sample, 0.84 ≤ cos θ < 0.9. 8.28 Fractional error introduced by each CCnQE sample, 0.9 ≤ cos θ < 0.94. 8.29 Fractional error introduced by each CCnQE sample, 0.94 ≤ cos θ < 1. . 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17 9.18 9.19  systematic effect . . . . . . . . . . systematic effect . . . . . . . . . . systematic effect . . . . . . . . . . systematic effect . . . . . . . . . . systematic effect . . . . . . . . . .  for . . for . . for . . for . . for . .  the . . . the . . . the . . . the . . . the . . .  130 131 131 132 132  BANFF fit neutrino flux factor constraints . . . . . . . . . . . 137 BANFF analysis fitted covariance matrix. . . . . . . . . . . . 138 Selected neutrino interaction momentum distribution with neutrino mode contributions . . . . . . . . . . . . . . . . . . . 140 Selected neutrino interaction angular distribution with neutrino mode contributions . . . . . . . . . . . . . . . . . . . . . 140 Selected neutrino interaction pµ − θµ distribution with neutrino modes contributions . . . . . . . . . . . . . . . . . . . . 141 Selected neutrino interaction energy spectrum . . . . . . . . . 143 Energy templates in selected events pµ − θµ distribution . . . 144 Energy bin contributions to the selected event muon momentum distribution . . . . . . . . . . . . . . . . . . . . . . . . . 145 Overall systematic covariance matrix, including detector, relevant neutrino model and statistical errors. . . . . . . . . . . 147 Fake data study statistics only fit flux scale factor pulls . . . 153 Fake data study with neutrino interaction model systematics included flux scale factor pulls . . . . . . . . . . . . . . . . . . 155 Fake data study with neutrino interaction model systematics fitted neutrino interaction scale factor pulls . . . . . . . . . . 157 Fake data study with all systematics fitted scale flux scale factor pulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Fake data study with all systematics fitted neutrino interaction scale factor pulls . . . . . . . . . . . . . . . . . . . . . . . 161 Fake data study with all systematics fitted pµ − θµ nuisance parameter pulls . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Distribution of mean values of fitted parameter pull distribution165 Distribution of RMS values of fitted parameter pull distribution165 Muon neutrino flux prediction with coarse binning . . . . . . 168 Neutrino flux prediction with binning used in fit . . . . . . . 168 xiv  List of Figures 9.20 Selected events with contributions shown by neutrino type . . 9.21 Selected neutrino interaction energy spectrum measurement, statistical fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.22 Selected neutrino interaction energy spectrum measurement, neutrino interaction model parameters included . . . . . . . . 9.23 Selected neutrino interaction energy spectrum measurement, all systematics included . . . . . . . . . . . . . . . . . . . . . 9.24 Selected neutrino interaction energy spectrum measurement, all systematics included, fitted number of interactions divided by bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.25 Selected neutrino interaction energy spectrum measurement correlation between fitted flux and neutrino interaction model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.26 Pull distribution for relative flux factor fake data fits with all systematic parameters included . . . . . . . . . . . . . . . . .  169 170 170 171  171  172 175  xv  Acknowledgements I am grateful for the guidance of Prof. Scott Oser while producing this thesis and throughout my time at UBC. Additionally Prof. Sampa Bhadra provided valuable advice while I was working on my Master’s degree. I’d also like to acknowledge the various TRIUMF employees, T2K researchers and graduate students with whom I collaborated. Finally I’d like to thank my friends, family and Camillia for their support.  xvi  Chapter 1  Neutrino Theory This chapter introduces the history of neutrino phenomenology and summarizes the present state of knowledge with an emphasis on neutrino oscillations. Current experimental efforts and potential future research are also described.  1.1  History  The existence of neutrinos was originally posited by Pauli in order to explain the continuous energy spectrum observed in electrons produced by nuclear beta decay [1]. If only an electron were emitted, conservation of energy would require it possess discrete energies similar to spectral lines seen in atomic or nuclear transitions. The continuous spectrum can be explained if the decay energy is shared with another particle emitted in addition to the electron. This explanation was expanded by Fermi following the discovery of the neutron [2], and confirmed when anti-neutrino interactions were observed by Reines and Cowan [3]. The existence of more than one type of neutrino was established in 1962 [4]. It was observed that neutrinos produced in muon decay would only produce muons in subsequent interactions, suggesting the existence of muon neutrinos distinct from electron (anti-)neutrinos produced by nuclear reactors. Following the discovery of the tau lepton [5] in 1975 the existence of the tau neutrino was predicted and subsequently verified by the DONUT collaboration [6]. Experimental measurements of the Z boson decay width and comparison to the decay rate into visible particles suggest there are only three light neutrino flavors that couple to the Z [7]. Measurements of the cosmic microwave background interpreted within the standard cosmological model are also compatible with the existence of three generations of neutrinos [8]. Following the observation of parity violation in weak interactions [9], the apparent very low mass of neutrinos suggested that they were eigenstates of the helicity operator. This was corroborated by experimental evidence when Goldhaber, Grodzins, and Sunyar demonstrated that γ particles pro1  1.2. Neutrinos in the Standard Model of Physics duced by Sm152 (1-) decay were uniformly circularly polarized, suggesting that neutrinos were entirely left-handed and similarly anti-neutrinos righthanded [10].  1.2  Neutrinos in the Standard Model of Physics  The Standard Model of particle physics was developed throughout the 20th century as a quantum field theoretical description of the electroweak and strong interactions, as well as the spontaneous symmetry breaking mechanism to generate gauge boson masses [11]. In the Standard Model neutrino fields are spin 1/2 particles represented by four-component bispinors. The chiral representation has these bispinors take the general form ψ=  ψL ψR  (1.1)  where ψL is a spinor transforming under the “left-handed” two-dimensional representation of the Lorentz group, and ψR transforming by the “righthanded” representation [12]. In the Glashow-Weinberg-Salam theory of electroweak interactions, lefthanded neutrinos and their corresponding leptons are combined into a twocomponent field ΨLl =  ψLνl ψLl  , l = e, µ, τ  (1.2)  with the free-field Lagrangian given by ¯ L l γ µ ∂µ ΨL l + ψ¯R l γ µ ∂µ ψR l + ψ¯R ν γ µ ∂µ ψR ν ] [13]. L0 = i[Ψ l l  (1.3)  The field ΨLl is an isoweak doublet, and initially the theory is invariant under the combined SUL (2)×U(1) symmetry transformation. Following spontaneous symmetry breaking through the Higgs mechanism the coupling of the individual neutrino bispinor fields ψνl , lepton fields ψl and the W and Z gauge bosons is given by g g ψ¯ν γ α (1 − γ5 )ψνl Zα . − √ [ψ¯νl γ α (1 − γ5 )ψl Wα + c.c.] − 4 cos θW l 2 2  (1.4)  Constants g and cos θW are parameters of the Standard Model that can be related to experimentally observed variables. 2  1.3. Neutrino Oscillations and Mass As can be seen in Eq. 1.4, only the left-handed components of the lepton fields participate in the weak interaction due to the presence of the helicity projection operator 1 − γ5 , incorporating the experimental observation of maximal parity violation into the theory. Right-handed neutrino fields (left-handed anti-neutrino fields) do not appear anywhere within the Standard Model with massless neutrinos. Neutrinos only interact through the massive weak gauge bosons, and do not participate in electromagnetic or strong interactions. This is equivalent to stating that neutrinos do not possess electric charge or colour. Charged current interactions occur through the first term in Eq. 1.4, where the neutrino and the lepton fields of the same flavour couple to a charged W boson. Neutral current interactions are governed by the second term where the neutrino field couples with the Z boson. In either case, the large gauge boson masses make neutrinos interact only weakly compared to other particles. In the original formulation of the Standard Model neutrinos were considered to be massless, or effectively massless. This was supported by the maximal parity violation of the weak interaction and the lack of experimental observations of right-handed neutrinos. Cosmological observations limit the neutrino mass scale to below ∼ 1 eV [14], while tritium decay provides an upper limit on m(νe ) to less than 2 eV at 95% CL [15]. However, the existence of neutrino mass was indirectly corroborated through the observation of neutrino flavour oscillations by the Super-Kamiokande collaboration [16] and the Sudbury Neutrino Observatory (SNO) [17]. Neutrinos can gain mass through the Standard Model Higgs mechanism, in which case they are Dirac fermions with distinct particle and anti-particle fields. However the fact that neutrinos possess no electric charge allows the possibility that they are their own anti-particles and hence Majorana fermions. In this case additional Majorana mass terms can be introduced to the theory to generate neutrino mass. The see-saw mechanism is a possible explanation for the origin of these additional Majorana mass terms [18]. Experimental evidence cannot currently distinguish between the Dirac and Majorana fermion hypotheses, and the problem is a subject of research. The small size of neutrino masses in comparison with other fundamental particles and how they are generated in the Majorana fermion hypothesis are open questions.  1.3  Neutrino Oscillations and Mass  Neutrino oscillations were proposed as an explanation for the observed deficit in the solar electron neutrino flux [19], commonly referred to as the “solar 3  1.3. Neutrino Oscillations and Mass neutrino problem”. If neutrinos flavour eigenstates were in fact superpositions of mass eigenstates, there is some probability to change flavours in the case that the masses are different. Consequently solar electron neutrinos might oscillate into other neutrino flavours when they arrive at earth and consequently are not observable by experiments detecting only electron neutrinos. Neutrino oscillations were first observed using neutrinos produced by interactions in the upper atmosphere and detected with the Super-Kamiokande detector [16]. The SNO collaboration was able to resolve the solar neutrino problem by directly measuring the total neutrino flux for all flavours through neutral current interactions and verifying that oscillations accounted for the observed reduction in the electron neutrino flux [17]. Neutrino masses can be introduced into the Standard Model Lagrangian by redefining the left-handed flavour neutrino fields to be a linear combination of mass eigenstates [18] as shown by νl L =  Ulj νj L (x) ,  (1.5)  j  where l = e, µ, τ . The fields νj L possess mass mj = 0 and Ulj is a unitary matrix commonly referred to as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix 1 . A common parametrization is shown by   c12 c13  s13 e−iδ  s12 c13   U =  −s12 c23 − c12 s23 s13 eiδ c12 c23 − s12 s23 s13 eiδ s12 s23 − c12 c23 s13 eiδ −c12 s23 − s12 c23 s13 eiδ    s23 c13   ,(1.6) c23 c13  where cij = cos θij , sij = sin θij , θij are neutrino mixing angles, and δ is the Dirac charge-parity violating phase. In the case that neutrinos are Majorana fermions and hence their own antiparticles an additional diagonal matrix is required of the form   1  0  α21  0 ei 2 UM ajorana =   0 0  0 0 i  e  α31 2       (1.7)  1 The lightest neutrino mass state can have zero mass, but the other two states must have mass > 0 due to the observed mass differences.  4  1.3. Neutrino Oscillations and Mass sin2 (θ12 ) sin2 (θ23 ) sin2 (θ13 )  0.306+0.018 −0.015 0.42+0.08 −.03 0.0251 ± 0.0034  Table 1.1: Current experimentally derived values of neutrino oscillation parameters [18].  where αij are additional CP violating phases that cannot be absorbed by Majorana neutrino fields as they are for Dirac fields. These matrices fully describe the mixing between neutrino mass and flavour states, and contain three mixing angles and 1 to 3 CP violating phases. The current experimentally derived values of these parameters are shown in Table 1.1.  1.3.1  Neutrino Oscillation Probability  Neutrino flavour oscillation is a direct consequence of quantum mechanics and the different neutrino masses. The derivation shown here assumes the neutrino states can be approximated as plane waves. A complete description would use wave-packet or quantum field theoretic formalism, although the final expressions for the oscillation probabilities are very similar. Additionally, it is assumed that the neutrino states can be expressed as coherent superpositions of mass eigenstates. It is assumed that the neutrinos are propagating through a vacuum, although the effect of the presence of matter on oscillation probabilities will be described later. The equation describing the oscillation amplitude for neutrino flavour states is shown by A(νl → νl′ ) =  Ul′ j Dj Ujl† ,  (1.8)  j  where l = e, µ, τ . Ulj is the neutrino mass and flavour mixing matrix and Dj is the propagation amplitude between the initial and final points in the neutrino’s path. Using the plane wave approximation, the propagation amplitude is given by Dj = e−ipj (xf −x0 )  (1.9)  where the initial and final space-time positions of the neutrino are specified by x0 and xf . The distance between detectors is often referred to as the baseline, and can be expressed as L = k(xf − x0 ), where k is the unit 5  1.3. Neutrino Oscillations and Mass momentum vector. The neutrino flavor oscillation probability can then be directly calculated as P (νl → νl′ ) =  j  |Ul′ j |2 |Ulj |2 + 2 cos  ∆m2jk 2p  j>k  |Ul′ j Ulj∗ Ulk Ul∗′ k |  L − φl′ l;jk  , l = e, µ, τ  (1.10)  where φl′ l;jk = arg(Ul′ j Ulj∗ Ulk Ul∗′ k ). The anti-neutrino oscillation probability is similar except for the sign of φl′ l;jk . Eq. 1.10 shows that the oscillation probability depends only on the square of the eigenstate mass-splitting ∆m2jk . An example plot of oscillation probability for different muon neutrino energies is shown in Figure 1.1, where the baseline is chosen to be 295 km, ∆m232 = 2.43 × 10−3 eV 2 and θ23 = 45◦ . 1  0.8  Pµ µ  0.6  0.4  0.2  0 0  0.2  0.4  0.6  0.8  1 1.2 Eν (GeV)  1.4  1.6  1.8  2  Figure 1.1: P (νµ → νµ ) for different neutrino energies (assuming only two neutrino flavors are involved in the oscillation), assuming the baseline is chosen to be 295 km, ∆m232 = 2.43 × 10−3 eV 2 and θ23 = 45◦ . The expression for neutrino oscillation probability can be simplified if the mixing is restricted to occur between two lepton flavours. This is approximately true as the value of |∆m232 | is much greater than |∆m221 |, such that the states closer in mass can be treated as effectively the same. Additionally the θ13 neutrino mixing angle is relatively small, allowing terms 6  1.3. Neutrino Oscillations and Mass proportional to sin θ13 to be neglected. Considering only muon to tau neutrino oscillations induced by the mass splitting between ν2 and ν3 , as in the T2K experiment, the unitary mixing matrix can be expressed as:  U=  cos θ23  sin θ23  − sin θ23 cos θ23  .  (1.11)  Assuming a pure muon neutrino beam, the oscillation probability for relativistic neutrinos of energy E travelling over a distance L can be easily derived from 1.8: P(νµ → νµ ) = 1 − sin2 (2θ23 ) sin2  ∆m232 L 4E  ,  (1.12)  where ∆m232 = m23 − m22 . The muon neutrino disappearance probability is similarly expressed by  P(νµ → νx ) = 1 − P(νµ → νµ ) = sin2 (2θ23 ) sin2  ∆m232 L 4E  .  (1.13)  If terms containing sin θ13 are no longer neglected, the oscillation probability from muon to electron neutrino flavours is given by P(νµ → νe ) = sin2 (2θ13 ) sin2 (θ23 ) sin2  1.3.2  ∆m231 L 4E  .  (1.14)  Mass Hierarchy  Previous neutrino oscillation experiments only contrained the difference in mass between neutrino states, and the current experimental results are summarized in Table 1.2. The square of the mass differences makes the relative masses of the neutrino flavour eigenstates ambiguous. |∆m232 | is much larger than |∆m221 | and its sign is unknown, consequently different theoretical orderings of neutrino masses are distinguished by whether this larger masssquared difference makes ν3 larger in mass than ν1 or ν2 (normal hierarchy) or smaller (inverted hierarchy). This distinction has consequences for theories of neutrino mass generation and is the subject of current experimental measurements.  7  1.3. Neutrino Oscillations and Mass  Figure 1.2: Diagram showing splitting of mass states and fractional flavour content in the normal and inverted neutrino mass hierarchies assuming δ = 0. |∆m221 | |∆m232 | |∆m231 |  7.59 ± 0.21 × 10−5 eV 2 2.43 ± 0.13 × 10−3 eV 2 ≈ |∆m232 |  Table 1.2: Results of current ∆m2 measurements [18].  1.3.3  Matter Effects  Neutrino oscillation probabilities are affected when the neutrinos propagate through matter, and by a process usually referred to as the Mikheyev, Smirnov, Wolfenstein (MSW) effect. Interactions with matter can be described as a modification of the Hamiltonian governing neutrino propagation. The resulting system of equations describing the evolution of the neutrino flavour state probabilities Ae and Aµ in a two neutrino flavour mixing model is given by d i dt  Ae (t, t0 Aµ (t, t0 )  =  −ǫ(t) ǫ′  ǫ′  Ae (t, t0 )  ǫ(t)  Aµ (t, t0 )  .  (1.15)  The modification to the neutrino propagation Hamiltonian is a consequence of the fact that electron neutrinos can interact with electrons in ordinary 8  1.3. Neutrino Oscillations and Mass matter through both W and Z exchange, while muon neutrinos can only interact through Z exchange since ordinary matter does not contain muons. The factor ǫ(t) is defined as ǫ(t) =  1 2  √ ∆m2 cos 2θ − 2GF Ne (t) 2E  ,  (1.16)  and ǫ′ is defined as ǫ′ =  ∆m2 sin 2θ . 4E  (1.17)  √ The term 2GF Ne (t) includes the effect of matter on the neutrino oscillation, where GF is Fermi’s constant and Ne (t) the electron number density in matter. This lead to a modification in the effective mixing angles and resulting oscillation probabilities. The sign of the term changes when considering antineutrino oscillations.  1.3.4  CP Violation  Irrespective of whether neutrinos are Dirac of Majorana fermions, the recent experimental measurement of non-zero θ13 raises the possibility of experimentally measuring CP violation in the lepton sector [20]. Longbaseline neutrino oscillation measurements carried out for neutrinos and anti-neutrinos provide one method of searching for leptonic CP violation. The asymmetry in neutrino-antineutrino oscillation probabilities is given by ′  ll = P (νl → νl′ ) − P (ν l → ν l′ ) ACP  Im(Ul′ j Ulj∗ Ulk Ul∗′ k ) sin  =4 j>k  ∆m2jk 2p  L , l, l′ = e, µ, τ ,  (1.18)  where matter effects have been neglected. Measuring this asymmetry requires a determination of the sign of ∆m231 and accurate estimates of the neutrino mixing parameters. These measurments are the goals of current neutrino oscillation experiments such as T2K (which will be described in Chapter 2) and potential future oscillation experiments using megaton scale neutrino detectors.  9  1.3. Neutrino Oscillations and Mass  1.3.5  Neutrino Interaction Cross Sections  Neutrino interactions with other elementary particles are well understood through the Standard Model and summarized in Figure 1.3. However, in practice neutrino experiments must use atomic nuclei as targets for neutrino interactions. This leads to complications as the dynamics of the nuclear medium are complicated and often only approximately understood, and differ significantly between different nuclei. As well the parton structure of the nucleons themselves can affect the mode of interaction. At the 1 GeV energy scales that long baseline neutrino oscillation experiments typically operate, charged current neutrino interactions with matter can be separated into quasi-elastic, resonant and deep-inelastic scattering exclusive modes. Similar modes exist for neutral current interactions, and the processes most relevant to the T2K experiment are shown in Figure 1.5. The energy dependence of these different modes with energy is shown in Figure 1.4. Additionally the products of neutrino interactions can interact with the nuclear medium, leading to significant modification of observed particle kinematics and interaction topologies. Uncertainties in neutrino interaction models are one of the main causes of systematic error in neutrino oscillation experiments and significant effort is currently directed towards better measurements and improved models. Further details on the neutrino interaction models used in the T2K experiment are provided in Chapter 5.  Figure 1.3: Neutrino interactions in the Standard Model with vertex factor for charged current and neutral current processes. Charged current quasi-elastic interactions have been important in previous neutrino oscillation experiments as the two-body final state allows the neutrino energy to be reconstructed from the measured lepton energy and kinematic variables. A sample of CCQE interactions can be used in principle to reconstruct the neutrino energy spectrum, and directly observe the energy-dependence of the oscillation probability. The equation summarizing  10  1.3. Neutrino Oscillations and Mass  Figure 1.4: Charged current neutrino cross sections vs experimental measurements [21], c Nuclear Physics B, 2002, by permission.  this relationship, neglecting Fermi momentum is given by: Eν =  mN Eµ − m2µ /2 mN − Eµ + Pµ cos θµ  (1.19)  where mN is the mass of the nucleon, Eµ the muon energy, Pµ the muon momentum, and θµ the muon angle with respect to the beam direction. In practice nuclear effects such as Fermi momentum and the momentum resolution of the detector limit the resolution with which neutrino energy can be reconstructed. Several other neutrino interaction modes are relevant at T2K energy scales. Resonant pion production occurs when the neutrino interaction excites the nucleon into an excited state that later decays to produce a pion. At higher neutrino energies different mesons can be produced. A related process is coherent pion production, where the neutrino interacts with the nucleus as a whole to produce the final state pion. Deep-inelastic scattering becomes the dominant interaction mode at higher energy scales where the neutrino interacts with individual partons as opposed to the entire nucleon, leading to a high-multiplicity final state. There are similar neutral current processes, with the neutral current π 0 being an important background in the electron neutrino appearance oscillation analysis. 11  1.3. Neutrino Oscillations and Mass  Figure 1.5: Neutrino interaction modes relevant to the T2K experiment, including charged current quasi elastic, resonant pion production and deepinelastic scattering. Neutral current π 0 production is also shown as it is an important background in the electron-neutrino appearance measurement.  12  Chapter 2  The T2K Experiment The T2K (Tokai-to-Kamioka) experiment [22] is a long baseline neutrino oscillation experiment and successor to K2K [23]. It is designed to measure the neutrino mixing angles θ13 and θ23 , as well as ∆m223 . To do this it will measure the muon neutrino disappearance and electron neutrino appearance oscillation probabilities in a neutrino beam as it travels between the production point and a far detector. The T2K neutrino beam is produced using the Japan Proton Accelerator Research Complex (J-PARC) proton accelerator in Tokai, Japan and is incident on the Super-Kamiokande detector [24] 295 km away beam at an off-axis angle of 2.5◦ relative to the beam center. The number of interactions predicted at the Super-Kamiokande detector under various oscillation probabilities is compared to the observed number to produce an estimate of the mixing parameters. A suite of near detectors 280 m away from the production point provides additional constraints on the beam flux estimates used to calculate the oscillation probabilities and measures neutrino interaction cross sections. The T2K experiment finished construction and successfully began operation in Fall 2009.  2.1  Neutrino Physics at T2K  T2K is primarily designed to measure the muon-to-electron neutrino oscillation probability, and hence the value of the neutrino mixing parameter θ13 . While muon neutrino disappearance is observed in atmospheric neutrino oscillations, the small size of the muon to electron neutrino oscillation probability (as shown in Eq. 1.14) made it difficult historically to measure the mixing angle θ13 . The design sensitivity of T2K to sin2 2θ13 for different proton beam exposures is shown in Figure 2.1, while the sensitivity for different ∆m223 is shown in Figure 2.2 . The precise value of θ13 is of theoretical interest as in the non-zero case the Dirac CP-violating phase present in Eq. 1.6 can contribute to oscillation probabilities (the Majorana phases do not contribute to oscillation measurements). This would make CP-violation in leptonic interactions possible, which has important cosmological implications [25]. At present the value of the Dirac CP-violating 13  2.1. Neutrino Physics at T2K phase is experimentally undetermined.  90% CL θ13 Sensitivity  Systematic Error Fraction 5% sys error  sin2 2 θ13 Sensitivity  10% sys error  10-1  20% sys error  Normal Hierarchy  10-2  10-3  1021 Protons on Target  1022  Figure 2.1: T2K design sensitivity to sin2 2θ13 vs. the number of 30 GeV protons delivered to the T2K target, assuming normal mass hierarchy, ∆m223 = 2.4 × 10−3 eV2 , and δCP = 0. Another major T2K physics goal is the precise measurement of neutrino mixing parameters θ23 and ∆m223 . The approximate formula for muon neutrino disappearance is shown in Eq. 1.13 and the oscillation probability is seen to depend on both neutrino mixing parameters. T2K will estimate the values of these parameters through measurements of the muon neutrino flux at the Super-Kamiokande far detector. Previous experiments such as MINOS and Super-Kamiokande have constrained these parameters [26] [27], the results of which are summarized in Figure 2.3 along with the results of the initial T2K muon-neutrino disappearance measurement [28]. T2K is designed to ultimately measure sin2 θ23 with a precision of ∼ 0.01 while ∆m223 will be measured to 10−4 eV 2 at 90% CL. Precise measurements of neutrino mixing parameters provide useful constraints on unknown values 14  2.1. Neutrino Physics at T2K  90% CL θ13 Sensitivity  10-1  2 ∆ m23 (eV 2)  10-2  10-3 Systematic Error Fraction 5% sys error 10% sys error 20% sys error CHOOZ Excluded  -4  10 -3 10  Normal Hierarchy  10-2 10-1 sin2 2 θ13 sensitivity  1  Figure 2.2: T2K design sensitivity to sin2 2θ13 for different values of ∆m223 , with the CHOOZ limits also shown for reference.  such as δCP and in conjunction with other neutrino experiments can help resolve the neutrino mass hierarchy problem. Recently reactor neutrino experiments and the MINOS and T2K longbaseline experiments have observed significant evidence for non-zero θ13 . T2K and MINOS saw evidence of electron neutrino appearance with T2K observing a 2.5σ deviation from the sin2 2θ13 = 0 hypothesis [29] [30]. More recent T2K results find a 3.2σ deviation with a best fit value of sin2 2θ13 = 2 −3 2 0.104+0.060 −0.045 for ∆m32 = 2.4 × 10 eV . The Daya Bay reactor neutrino experiment was the first experiment to observe a 5σ deviation from the null hypothesis through a measurement of electron anti-neutrino disappearance and measured sin2 2θ13 = 0.092 ± 0.016(stat) ± 0.005(syst) [20], which was corroborated by the RENO reactor neutrino experiment [31]. The discovery of non-zero θ13 makes searches for CP-violation in leptons possible, meaning that precision measurements of neutrino mixing angles will be the main focus 15  2.2. The T2K Neutrino Beam  Figure 2.3: 90% confidence regions for sin2 θ23 and ∆m223 as measured by MINOS, Super-Kamiokande and the initial T2K results [28], c APS, 2012, reprinted with permission.  of the T2K physics program in the future. T2K will also be able to make measurements of several neutrino interaction cross sections at the near detector. These cross sections will be useful in the design and interpretation of future neutrino oscillation experiments, and will provide useful constraints for neutrino interaction models.  2.2 2.2.1  The T2K Neutrino Beam Proton Accelerator  The T2K neutrino beam is produced at J-PARC using a 30 GeV proton beam designed with an eventual beam power of 750kW [32]. The J-PARC proton accelerator contains three accelerators: a linear accelerator (LINAC), a rapid-cycling synchrotron (RCS), and the main ring synchrotron (MR). At the LINAC stage an H− beam is accelerated to an energy of 181 MeV (with a future upgrade to the design energy of 400 MeV planned in 2013) and injected into the RCS after being converted to an H+ beam through 16  2.2. The T2K Neutrino Beam the use of charge-stripping foils. The RCS accelerates the particles to a design energy of 3 GeV before injecting them into the MR where they are accelerated to an energy of 30 GeV. The proton beam in the MR contains 8 beam bunches (6 before June 2010) spaced roughly 580 ns apart. The proton beam is then extracted from the MR to the neutrino beamline through the use of five kicker magnets. The extraction process repeated every 3.5 seconds before June 2010, and then every 3.2 seconds for the running period between November 2010 and March 11th, 2011.  2.2.2  Neutrino Beamline  The T2K neutrino beamline was constructed at J-PARC specifically for the T2K experiment. It contains two sections; a primary beamline and a secondary beamline. The primary beamline curves the proton beam in the direction of Kamioka, while the secondary beamline directs the beam into a graphite target to produce hadronic particles such as pions and kaons. These mesons are focused by a magnetic horn system into a decay volume where they decay into neutrinos, producing the T2K neutrino beam. An overview of the system can be seen in Figure 2.4.  Main Ring  Secondary beamline  ND280  (3) (6)  0  50  100 m  (5)  (1) (2) (3) (4) (5) (6)  (4)  (2)  Preparation section Arc section Final focusing section Target station Decay volume Beam dump  (1)  Figure 2.4: T2K neutrino beamline overview [22], c Nuclear Instruments and Methods in Physics Research, 2011, reprinted with permission.  17  2.2. The T2K Neutrino Beam The primary beamline contains a preparation section, an arc section and a focusing section. The preparation section tunes the extracted proton beam with a series of magnets in order to guide it to the arc section, which turns the beam by 80.7◦ so as to point towards Kamioka. The focusing section provides the final tuning of the beam and directs it into the production target at a downward angle of 3.637◦ with respect to horizontal. The primary beamline is instrumented with a large number of monitors that measure the beam’s position, intensity, width and loss as it passes through the various sections. The position of these beam monitors is shown in Figure 2.5.  Figure 2.5: Beam monitor positions in the T2K primary beamline [22], c Nuclear Instruments and Methods in Physics Research, 2011, reprinted with permission. The secondary beamline consists of a target station, a decay volume and the beam dump. The target station is a self-contained helium vessel that houses the graphite production target and is separated from the primary beamline atmosphere by a titanium window through which the beam passes. The Optical Transition Radiation (OTR) beam monitor is placed directly in the path of the beam in front of the production target to provide a final measurement of the beam profile and position. The T2K target is a graphite rod 2.6 cm in diameter and 91.4 cm long, corresponding to 1.9 interaction lengths. The proton beam is directed into the target and interacts almost completely to produce mesons, which are focused by a magnetic horn system. Three magnetic horns are used each producing toroidal magnetic fields of 18  2.2. The T2K Neutrino Beam strength up to 2.1 T. The first horn collects the meson beam and the other two provide additional focusing of the mesons produced in the target. A baffle placed upstream of the production target prevents the edges of the proton beam from degrading the horns. The focused meson beam is directed into the decay volume, which is a 96m long steel tunnel completely filled with helium gas. The beam dump is placed at the end of the decay volume, and consists of 75 tons of graphite designed to absorb any remaining protons or undecayed mesons.  2.2.3  Off-Axis Neutrino Beam  The Super-Kamiokande detector is 2.5◦ off-axis from the T2K neutrino beam’s primary direction. This off-axis configuration results in a significantly narrower neutrino energy spectrum due to the correlation of off-axis angle and neutrino energy. This correlation is shown in Figure 2.6, where it can be seen that pions over a wide range of energy will decay and produce similar neutrino energies at a given angle. A simulation of the resulting flux for a 50 GeV version of the J-PARC proton beam is shown in Figure 2.7, where its clear that the energy spectrum narrows with off-axis angle. There is a trade-off between the width of the neutrino energy spectrum and the neutrino flux. The T2K neutrino flux incident on Super-Kamiokande is peaked at 600 MeV, which is designed to maximize the oscillation probability at 295 km for a specific set of mixing parameters and reduces background relevant to electron neutrino appearance. The predicted unoscillated neutrino flux at the Super-Kamiokande detectors can be seen in Figure 2.8. The off-axis angle can be adjusted between 2.0◦ and 2.5◦ by varying the incident proton beam direction to allow for some tuning of the neutrino energy spectrum. The neutrino beam is predicted to contain mainly muon neutrinos, with a small electron neutrino component and very small anti-neutrino components. One of the tasks of the T2K near detectors is to test this prediction and the size of different flavor components in the neutrino beam.  2.2.4  Neutrino Beam Flux Prediction  The neutrino flux incident on the Super-Kamiokande detector is predicted using a set of software packages that simulate the T2K proton beamline, the production of pions and kaons in the graphite target, and the decay of pions and kaons in the decay volume. The simulation also extrapolates the resulting neutrino flux to the Super-Kamiokande detector position. Systematic 19  2.2. The T2K Neutrino Beam  Off-Axis Angle °  12  0 ° 0.1 ° 0.5 ° 1.0 ° 2.5  Eν (GeV)  10 8 6 4 2 0  0  5  10  15  20  25  30  Eπ (GeV) Figure 2.6: Correlation between off-axis angle and neutrino energies produced by pion decay. Higher off-axis angles result in a narrower range of possible neutrino energy.  uncertainties in the simulation and production cross sections are propagated through to the flux prediction and the results are shown in Figure 2.9. A similar plot of the flux prediction at ND280 is shown in Figure 2.10. One of the primary physics goals of the T2K near detectors is to reduce these systematic errors in the flux prediction. It can be clearly seen that at ND280 the vast majority of the flux are muon neutrinos, with a small contribution from electron neutrinos and a very small anti-neutrino component. The proton beam simulation provided by the JNUBEAM package is implemented using the GEANT3 simulation library [34]. This simulation includes the propagation of the protons along the JPARC secondary beamline and focusing into the graphite target. Real horn current measurements are used to improve the accuracy of the simulation. The simulation of the primary proton and resulting meson production uses measurements provided by the NA61/Shine experiments [35] to predict the resulting pion and kaon flux and kinematic distributions. These measurements were made using a proton beam of similar energy to T2K and the resulting pion and kaon  20  2.2. The T2K Neutrino Beam  Figure 2.7: Simulated neutrino flux energy spectrum at various off-axis angles produced by a 50 GeV version of the J-PARC proton beam [33], unpublished T2K internal document, 2003, reprinted with permission. The distribution with the black solid line corresponds to an off-axis angle of 1.0◦ , the red dashed distribution to 2.0◦ and the blue dashed line 3.0◦ . The narrowing of the energy spectrum at higher off-axis angles is clearly visible.  momentum and angular distributions cover much of the region relevant to T2K. The FLUKA hadron interaction simulation library provides the initial simulation of the hadronic interactions that are tuned by the NA61 measurements [36]. Interactions and neutrino decays that that take place outside of the production target and secondary interactions that occur within are simulated using a combination of GEANT3 and GCALOR with cross sections tuned to experimental data. The NA61 pion and kaon production measurements are used to tune FLUKA simulation provided by the JNUBEAM package and the resulting T2K flux prediction. The NA61 experiment measured the production rate of  21  2.3. Near Detectors  Figure 2.8: Predicted muon neutrino energy spectrum at Super-Kamiokande [22], c Nuclear Instruments and Methods in Physics Research, 2011, reprinted with permission.  these mesons using a thin carbon target and a replica of the T2K production target for improved accuracy. The pion momentum and angular distribution measured by the NA61 experiment very closely matches the region relevant to the T2K experiment. The measurements were used to directly tune the FLUKA prediction of the neutrino as shown in Figure 2.11 for muon neutrinos produced by pions. The coverage of the NA61 measurement for kaons is not as complete as the pion case, and so additional flux tuning was applied to the high momentum range using measurements from other hadronic production experiments [37] [38]. As can be seen in Figure 2.12 the pion and kaon production uncertainties are major systematic errors in the muon neutrino flux prediction for the ND280 detectors. Improvements in the uncertainty are expected as the the full NA61/Shine dataset is used.  2.3  Near Detectors  The 280m detectors (ND280) are divided into two types, the on-axis and off-axis detectors with respect to the neutrino beam primary direction. INGRID is the only on-axis 280m detector, while the off-axis detectors are at 22  2.3. Near Detectors  Flux /(cm2 ⋅ 1021 POT ⋅ 100 MeV)  SK Flux Prediction (All Runs) with Systematic Errors  νµ νµ νe νe  106 105 104 103 0  1  2 3 4 Neutrino Energy (GeV)  5  Figure 2.9: Predicted neutrino flux at Super-Kamiokande for different neutrino flavours.  the same 2.5◦ as Super-Kamiokande and contains several integrated subdetectors. The ND280 detectors are placed in a roughly 37m deep pit that allows them to intersect the neutrino beam at the desired angle.  2.3.1  The INGRID On-Axis Detector  The INGRID on-axis detector is designed to measure the neutrino beam direction and intensity with a sufficient number of events to provide daily measurements at the design beam intensity. INGRID consists of fourteen identical modules arranged in a cross centred on the T2K neutrino beam center and two modules placed off-axis as shown in Figure 2.13. Each INGRID module contains eleven tracking scintillator planes and nine iron plates, with each module providing 7.1 tonnes of target mass. The scintillator planes contain extruded scintillator bars threaded with a wavelength shifting fibre that are coupled to Multi-Pixel Photon Counters (MPPC). Scintillator veto planes surround the tracking and target planes, and the whole assembly is contained in a light-tight box.  23  2.3. Near Detectors  Flux /(cm2 ⋅ 1021 POT ⋅ 100 MeV)  ND280 Flux Prediction (Runs 1 & 2) with Systematic Errors  νµ νµ νe νe  1012 1011 1010 109 0  1  2 3 4 Neutrino Energy (GeV)  5  Figure 2.10: Predicted neutrino flux at ND280 for different neutrino flavours.  2.3.2  The Off-Axis Detector Suite  The off-axis detectors (Figure 2.14) are contained within the reused CERN UA1 magnet, which provides a nominal 0.2 T horizontally oriented dipole magnetic field to allow momentum and charge reconstruction. The subdetectors within the magnet include a pi-zero detector (P0D) designed to measure neutral current processes that produce a π 0 , a tracker consisting of the three Time Projection Chambers (TPCs) and two Fine Grained Detectors (FGD), and electromagnetic calorimeters (ECal) surrounding the P0D and tracker to detect escaping photons. The magnet yoke itself is instrumented with scintillator to measure exiting muon ranges (SMRD). Collectively these integrated subdetectors are designed to measure the T2K unoscillated neutrino beam off-axis energy spectrum and flavour components, as well as measure neutrino interaction cross-sections. The Tracker The tracker consist of the three TPCs interlayered with two FGDs, with the FGDs providing ∼ 2.1 tonnes of active target mass for neutrino interactions and the TPCs precisely reconstructing the resulting charged particle tracks. 24  2.3. Near Detectors  Figure 2.11: Muon neutrino flux weights provided by scaling FLUKA pion production prediction to NA61 measurements using different tuning methods, taken from T2K Technical Note 99 [39].  The FGD modules are segmented scintillator detectors that are described in Chapter 3. The TPC consists of a gas volume holding an argon-based drift gas, with Micromegas detector modules on either side as shown in Figure 2.15 [40]. An electric field applied across the gas allows e− produced by a particle track to drift towards Micromegas detector modules, which amplify and sample the ion charge signals. TPC charge and time measurements are combined with scintillator detector time measurements to provide three-dimensional reconstruction of the particle track’s passage through the detector. The Micromegas modules provide a 0.7mm point spatial resolution, allowing reconstruction of track position and providing better than 10% momentum resolution in the momentum region of interest. The size of the Micromegas charge signals provide dE/dx measurements that can be used in particle identification techniques. The tracker subdetector signals are combined to provide excellent neutrino interaction reconstruction which are the basis of the analysis in this thesis.  25  Fractional Error  2.4. Super-Kamiokande  0.3  Run 1+2 ND280 νµ Flux Total Pion Production Kaon Production Secondary Nucleon Production Hadronic Interaction Length Proton Beam, Alignment and Off-axis Angle Horn Current & Field  0.2  0.1  0  10-1  1  10 Eν (GeV)  Figure 2.12: Systematic errors in ND280 muon neutrino flux prediction due to various beam-related uncertainties, taken from T2K Technical Note 99 [39].  2.4  Super-Kamiokande  Super-Kamiokande is a water Cherenkov detector containing over 50 kton of pure water. It is used for a variety of physics studies, including T2K. A schematic overview of the detector is shown in Figure 2.16. The water is contained in a cylinder 41 m in height and 39 m in diameter, and is located in a mine 1 km underground to reduce the rate of cosmic ray induced background. It is located 295 km away from J-PARC and measures the T2K neutrino beam flavour composition and intensity at a position where the oscillation probability is predicted to be maximum. The Super-Kamiokande detector contains roughly 13,000 photomultiplier tubes that are used to reconstruct neutrino interactions in the 22.5 kton fiducial volume, with an instrumented outer volume providing a means to exclude external background. Water Cherenkov light is projected onto the detector walls in ring-like patterns. Multiple-scattering of electrons in water degrades the sharpness of the ring’s edges, which provides a means to differentiate between muon and electron neutrino interactions (Figure 2.17). Combined with the beam flux model and constraints from ND280, the Super-Kamiokande detector provides the crucial measurement of neutrino oscillation probability used to 26  2.4. Super-Kamiokande  Figure 2.13: The INGRID on-axis detector [22], c Nuclear Instruments and Methods in Physics Research, 2011, reprinted with permission.  estimate the corresponding neutrino mixing angles that is the goal of the T2K physics program.  27  2.4. Super-Kamiokande  Figure 2.14: The ND280 off-axis detector suite [22], c Nuclear Instruments and Methods in Physics Research, 2011, reprinted with permission. The ND280 magnet outer dimensions are 7.6 m × 5.6 m × 6.1 m and weighs 850 tons.  28  2.4. Super-Kamiokande  Figure 2.15: The T2K Time Projection Chamber (TPC) [40], c Nuclear Instruments and Methods in Physics Research, 2011, reprinted with permission. The outer dimensions of each TPC are are approximately 2.3 m × 2.4 m × 1.0 m.  29  2.4. Super-Kamiokande  Figure 2.16: The Super-Kamiokande detector [22], c Nuclear Instruments and Methods in Physics Research, 2011, reprinted with permission. The cylinder is approximately 41 m in height and 39 m in diameter, and contains 50 kton of purified water.  Figure 2.17: Muon and electron neutrino signals at the Super-Kamiokande detector [22], c Nuclear Instruments and Methods in Physics Research, 2011, reprinted with permission. The distinction between rings produced by muons and electrons is clearly visible, and is the basis of the particle identification algorithm.  30  Chapter 3  The ND280 Fine Grained Detector The Fine Grained Detector is an active target for neutrino interactions, with sufficient granularity to track short ranged recoil protons [41]. Combining with the TPCs in the ND280 tracker results in a tracking detector capable of precisely measuring particle track kinematic variables and identifying neutrino interaction vertices. The FGDs are designed to provide sufficient mass to ensure a reasonable number of interactions for physics studies while also being thin enough that charged leptons can travel into the TPCs for precise particle identification. The FGDs were also designed with the capacity to reconstruct short-ranged particles produced in neutrino interactions by using individually instrumented segmented scintillator bars. This provides additional information to identify the type of interaction and study its kinematic properties. The energy resolution of the detector is sufficient that dE/dx measurements have some ability to identify particle type. This section describes the FGD’s physical design, instrumentation and readout electronics system. It also describes how the detector has been calibrated following its successful installation in the fall of 2009. Technical details are fully described in an instrumentation paper written by the FGD group [41], and are briefly summarized here.  3.1  FGD Physical Description  The FGD is a segmented scintillator bar detector with two modules, referred to as FGD1 and FGD2. The FGD contains 8448 extruded polystyrene square scintillator bars of dimension 9.61 cm × 9.61 cm by 186.4 cm. Layers are composed of 192 bars glued together on the sides, and two layers stacked with perpendicular orientations compose an XY panel. There are fifteen XY panels in FGD1, while FGD2 contains seven XY panels and six similarly sized polycarbonate panels filled with water. FGD panel modules are held in a frame by stainless steel straps. This frame combined with front and 31  3.1. FGD Physical Description back covers enclose the scintillator modules within a dark box that prevents ambient light from being identified as originating in the scintillator. The exterior of the support frame provides mechanical support and power for readout electronics through “mini-crates”, of which there are six on each side of the frame. The outer dimensions of the frame are 230 cm wide, 240 cm in height and 36.5 cm thick (Figure 3.1). An image of a completed detector module with cover plate attached is shown in Figure 3.2.  Figure 3.1: Diagram of FGD frame and XY module supports, reprinted with permission [41].  32  3.1. FGD Physical Description  Figure 3.2: FGD module with cover plate attached. The six water modules used in FGD2 are constructed from polycarbonate panels 2.5 cm thick and sealed to prevent leaking. They are filled with water that is maintained at sub-atmospheric pressure, preventing water from escaping from small leaks that might develop. The panels are placed exclusively in the FGD2 module, and interleaved with regular plastic XY modules so that interactions that begin in the water panels can be reconstructed within the detector. The water modules allow the comparison of interaction rates on carbon and oxygen, which is relevant given that Super-Kamiokande is a water Cherenkov detector.  33  3.2. Scintillators and Light Collection  3.2  Scintillators and Light Collection  The FGD scintillator bars were made of extruded polystyrene doped with PPO and POPOP. The outer surfaces of the polystyrene are coated with an opaque layer of polystyrene doped with TiO2 , allowing segmented light collection. A 1.85 mm hole runs along the centre of each scintillator bar, through which an optical fibre is threaded. A cross-section of a typical scintillator bar is shown in Figure 3.3. The bar widths were measured and found to be reasonably uniform with an RMS in the width of 0.013 mm and in the height of 0.019 mm. The light yield of a random subset of bars was also tested using a bar scanner that placed a radioactive source at specific positions along the length of each bar. Using this method the light yield of bars was found to be Gaussian with a standard width of 4.5%.  Figure 3.3: FGD scintillator bar cross section, reprinted with permission [41].  34  3.3. Multi-Pixel Photon Counters The scintillators are threaded with wavelength shifting fibres that direct photons produced in scintillator to the photosensor at one of its ends. The fibres were double-clad Kuraray Y11 (200) S-35 J-type fibre with a diameter of 1 mm. The peak absorption of the fibres in the electromagnetic spectrum is at 430 nm and peak emission at 476 nm. Light collection is improved by coating the exposed end of the fibre with aluminium. The fibres were tested for mechanical faults, light yield and for uniformity in attenuation with length. Following these these tests the fibres were threaded through the scintillator bars as the modules were constructed.  3.3  Multi-Pixel Photon Counters  The wavelength shifting fibres direct photons to Multi-Pixel Photon Counters (MPPC), which are the photosensors used in the FGD. The MPPC is a segmented self-quenching avalanche photodiode, with each of the pixels able to discharge independently allowing single photon resolution in principle [42]. The FGD uses Hamamatsu MPPCs of 1.3mm by 1.3mm surface area and 667 pixels (Figure 3.4). Each pixel can produce ∼ 5 × 105 electrons in an avalanche. The photosensors were used for their ability to work within an 0.2 T magnetic field, their small size and high optical photon resolution. The MPPCs are coupled to wavelength shifting fibres using a plastic frame supported on a busboard (Figure 3.5). MPPCs have two contacts used to both provide power and read out charge signals, which are carried out of the dark box through ribbon cables and routed through a backplane to the frontend electronics. MPPCs have a few non-ideal features, including the incidence of random dark noise pulses, cross-talk between pixels, and after-pulsing. Dark noise results from discharges of MPPC pixels due to random thermal processes. While it degrades the single photon resolution, the effect on measured physics quantities is minor and they also provide a useful calibration tool. Cross-talk is a phenomena where an MPPC pixel avalanche has some probability of causing another pixel to discharge without any incident photon. After-pulsing is a repeated discharge of an MPPC some time after an initial discharge due to the decay of excited charge carrier states in the semiconductor. Cross-talk and after-pulsing irreducibly degrade the charge and time resolution of the device.  35  3.4. Readout Electronics  Figure 3.4: FGD example MPPC image, reprinted with permission [41].  3.4  Readout Electronics  The FGD readout electronics system is used to detect MPPC charge avalanche signals. The system samples and shapes the signal pulses and temporarily stores them using an ASIC, followed by digitization and recording the resulting digital waveforms in memory located on the frontend boards. These waveforms are read out at a later time on demand by off-magnet electronic boards as part of the data acquisition process. A schematic overview of the readout electronics are shown in Figure 3.6.  3.4.1  Backplanes and Mini-Crates  The frontend readout electronics boards are placed within the ND280 magnet on the exterior of the FGD frame. These boards are contained in minicrates placed on the sides of the FGD frame, which provide mechanical support for the cards and cooling. Frontend cards are attached to a backplane within each mini-crate, which routes MPPC charge signals from inside the detector volume. The backplanes are part of the dark box, and prevent ambient light from entering the inner detector volume. The backplane also provides all the necessary power and electronic connections associated with 36  3.4. Readout Electronics  Figure 3.5: FGD MPPCs coupled to wavelength shifting fibres.  Figure 3.6: Schematic overview of FGD readout electronics system, reprinted with permission [41].  37  3.4. Readout Electronics the frontend cards. An example image of a mini-crate and backplane is shown in Figure 3.7.  Figure 3.7: Example image of FGD mini-crate and backplane, with prototype readout cards inserted.  3.4.2  Front End Boards  The Front End Board (FEB) reads in up to 64 MPPCs and divides the sampled charge between high and low attenuation channels. Each FGD minicrate contains up to four FEBs. Each FEB contains two AFTER ASICs [43], which samples, shapes and stores the MPPC analog signals in a 511bin switch capacitor array (SCA). The ASIC preamplifier and shaper time constant is set to 30ns, reulting in a peaking time of 180ns for effective sampling at 50MHz (though 33MHz is also possible). This long time constant also allows the pulse height to be reliably used as an estimator proportional to the MPPC avalanche charge. The SCA acts as a circular buffer until the write signal is turned off due to a trigger signal, allowing the samples collected over the previous 10.24 µs to be read out. Dual pipeline 12-bit ADCs digitize the outputs of both ASICs on the FEBs, and the serialized samples are sent out of the FEB through an LVDS backplane connection to the Crate Master Board (CMB). The FEB also contains comparators that 38  3.4. Readout Electronics meaure the analog sum of MPPC charge signals (ASUM) as part of the FGD cosmic trigger system. An image of the FEB is shown in Figure 3.8.  Figure 3.8: Example image of FGD FEB, with important features shown. Reprinted with permission [41].  3.4.3  Crate Master Boards  A single Crate Master Board (CMB) is located in each mini-crate and receives and deserializes ADC samples from up to 4 FEBs simultaneously at 20 Msps. A Xilinx Virtex-II PRO FPGA implements ADC receiver logic that synchronizes the samples in time before storing them in one of two external 9Mb SRAM memory chips. The FPGA also controls the FEB ASIC acquisition signals and clocks, and is used to set related ASIC control registers. On request the CMB retrieves digitized channel data from storage and sends a data packet through a 2 Gb/s RocketIO optical connection to the Data Concentrator Card (DCC), located outside of the detector area and magnet. Data reduction processes can optionally be performed at this stage and are described in detail in Chapter 4. This optical link is also used for global trigger and clock distribution to the mini-crates via the DCC, as well as configuration of CMB control registers. The CMB also receives cosmic trigger signals provided by ASUM comparators on FEBs and applies trigger logic. If a signal consistent with a cosmic ray event is found the CMB can send a trigger signal through a CAT-5 connection to the Cosmic Trigger Module (CTM). An image of a CMB is shown in Figure 3.9.  39  3.4. Readout Electronics  Figure 3.9: Example image of FGD CMB, with important features shown. Reprinted with permission [41].  3.4.4  Data Concentrator Cards  Each DCC collects data from 4 CMBs and transfers it through a 1 Gb/s Ethernet connection to the data acquisition backend computer following a trigger signal. The maximum data transfer rate of this Ethernet connection was measured to be 7MB/s. The DCC controls the channel readout process of its associated CMBs through data packet requests and setting CMB firmware control registers through the optical connection. The DCC was implemented on the Xilinx ML405 Virtex-IV evaluation board, using the built in optical link as well as 3 additional links placed on a custom board that is connected to the evaluation board by two SATA cables. The Virtex-IV PowerPC (PPC405) runs an embedded Linux operating system (kernel 2.6.23-rc2) at 300MHz, allowing the processor to run a MIDAS Data Acquisition frontend application and the Ethernet connection used by the data acquisition system. Global trigger and clock signals for the 280m detectors are produced on the Master Clock Module (MCM) and sent through a Slave Clock Module (SCM) to each DCC. The DCC passes these signals to the CMBs through the optical connection, beginning the MPPC sample readout process. An image of a DCC is shown in Figure 3.10.  3.4.5  CMB-DCC Data Transfer Protocol  The CMB channel readout must finish well in advance of the 50ms target readout time to allow the DCC to completely transfer the received data over the Ethernet network before the next trigger signal. A custom data protocol  40  3.5. Slow Control  Figure 3.10: Example image of FGD DCC, with important features shown. Reprinted with permission [41].  was implemented that starts when the DCC requests a data packet from the CMB, which automatically processes a list of channels. It continues until it finds a data packet produced by the data reduction process or it reaches the end of the list. At this point the data packet is sent by the CMB to the DCC, and the small size of the compressed data packets allows the DCC to read out all four CMBs in parallel. This transfer is most efficient when a large fraction of the FGD channels are completely zero-suppressed, as the number of DCC data requests and CMB idle time is reduced. This is generally found to be the case in real detector operation.  3.5  Slow Control  The status of the FGD detector and readout electronics are monitored by a slow control system co-ordinated by an instance of the MIDAS framework. The slow control system uses a parallel data bus (MIDAS Slow Control Bus or MSCB) to access and record measurements made by various probes, including temperature sensors, power supply voltage and current measure41  3.6. Calibration ments and the status of the target water system. Importantly the system monitors the bias voltage of each MPPC sensor and the temperature of the local busboard. The MSCB is integrated into all the frontend electronics and readout boards through an independent system of connectors. The slow control system is also used to switch the readout electronic boards off and on and other system control tasks. Measurements made by the system are logged in a database and play a major role in detector calibration.  3.6  Calibration  The FGD online data processing system identifies MPPC pulses in the digitized waveform and records height and time information in a summary data packet. Pulses with heights greater than what is typical for dark noise are saved with the waveform samples surrounding the measured peak position. These waveform segments are fit offline to provide raw MPPC height measurements. The fitting function used to measure pulse time and height offline is given as y(t > t0 ) = B + A  t − t0 τ  4  1−  t − t0 λτ  ×e  t−t0 τ  ,  (3.1)  where B is the baseline value, A is the pulse height constant, τ is the pulse shape time constant and λ is the MPPC recovery time constant. More information is provided about the online data processing system in Chapter 4. The FGD calibration process converts these raw MPPC pulse height and time measurements into calibrated quantities with individual detector gain variation and time-dependent effects removed. This section describes the steps used to calibrate FGD real and simulated data for the sample used in this thesis.  3.6.1  Gain Calibration  The number of electrons produced by each MPPC due to a pixel avalanche is expected to vary from device to device. As the only information that is available is pulse height proportional to the total MPPC charge signal, more information is needed to infer the number of pixels that discharged and similarly the number of incident photoelectrons. The MPPC dark noise pulses provide this information, as the total charge measured should be proportional to the charge from an single avalanche multiplied by the number  42  3.6. Calibration of avalanches. As a result the measured pulse height (PH) is given by P H = Nav P H1 ,  (3.2)  where PH Nav is the number of MPPC pixel avalanches and P H1 is the average pulse height for single pixel avalanches. P H1 can be measured for each MPPC from a histogram of dark noise pulse heights, as shown in Figure 3.11. The single avalanche average pulse height is apparent, and is measured by fitting the distribution with a Gaussian. Dark noise pulse height information is regularly saved and the gain of each MPPC monitored over time, while also providing time-dependent gain measurements. Eq. 3.2 can be inverted to infer Nav from the measured charge and P H1 for the MPPC.  Figure 3.11: MPPC dark noise distribution with single avalanche peak fit by a Gaussian distribution, reprinted with permission [41].  3.6.2  Gain Dependence on Temperature  In principle the dark noise pulse distribution could be measured often enough to include the effect of temperature changes in the average single avalanche pulse height. In practice it is simpler to measure the dark noise distribution less frequently and correct the measured number of avalanches to account for temperature variation in the gain. Busboard temperatures recorded through 43  3.6. Calibration the slow control system are used to calculate the effective value of P H1 . The empirical function shown by V − CT (T − T0 ) = Vbd,0 + (1/G) P H1 + CG P H1  2  (3.3)  is used, where V is the operating voltage, T the measured temperature, Vbd,0 the estimated breakdown voltage, T0 a reference temperature. G, CG and CT are empirically determined values, where G and CG are determined for each MPPC (along with Vbd,0 ) and CT is a global parameter found to be 57±3 mV/deg. This function and temperature measurements are used to correct the value of P H1 used to calculate the measured pulse height in 3.2.  3.6.3  Gain Dependence on Overvoltage  The bias voltage applied to the MPPC gain affects the photodetection efficiency, cross-talk and after-pulsing probabilities. This effect was directly measured using the TRIUMF electron test beam and cosmic ray events at J-PARC, where the MPPC operating voltages were varied in order to measure the resulting change in MPPC charge signals produced by the selected events and from dark noise. The function Nav =  PH ∝ C0 + Cǫ P H 1 P H1  (3.4)  was found to empirically describe the effect, where Nav is the average number of MPPC avalanches, P H the mean pulse height of the selected event distribution, P H1 the mean dark noise single-avalanche pulse height. C0 is an empirical constant found to be 0.0885 ± 0.0071, while Cǫ was found to be 0.0338 ± 0.0007 ADC counts−1 . This equation can be used to correct the value of P H1 measured at a particular overvoltage for use in 3.2.  3.6.4  Ratio of Low to High Channel Gain  Each MPPC is sampled by a high and low gain channel in order to extend the dynamic range of the readout electronics. The low gain channel is not sensitive to single photoavalanches and an alternative calibration process is required. Instead, cosmic ray events where a pulse is identified in high and low gain channels are used to establish a conversion factor between the measurements for each MPPC. The conversion factors measured for all the channels in the system are on average 8.81 and distributed with an RMS of 0.27. 44  3.6. Calibration  3.6.5  MPPC Gain Saturation  MPPC gain is non-linear due to the finite number of pixels on the device. As the number of avalanches increases, the chances of an additional photon hitting an undischarged pixel is reduced. This results in progressively more light required to increase the observed MPPC charge. A related effect is the coupled fibre does not uniformly illuminate all MPPC pixels, resulting in an “effective” number of pixels smaller than the actual number on the device. Finally, the effective number of pixels depends on the number of pixels fired, as at higher light intensities discharged pixels can recharge quickly enough to fire again within the 7ns light decay time of the fibre. Bench tests using a variable intensity laser light source were used to show that the number of incident photons could be adequately related to the observed charge by the relation Nav,corr = Npix (1 − e−ǫ(1+nsec ) Nph  /Npix  ),  (3.5)  where Nav,corr is the temperature corrected number of MPPC avalanches, Npix the number of pixels on the device, ǫ the MPPC photon detection efficiency, nsec the number of cross-talk or afterpulse avalanches that occur after a primary avalanche, and Nph the number of incident photons. The value of the constants used in this expression are found using a fit to the bench test data. Eq. 3.5 can be inverted to yield an expression for Nph . In real data analysis Npix is taken to be 396 for all MPPCs.  3.6.6  Bar-to-Bar Light Variation  An additional empirical correction is applied to calibrated charges to take into account hardware effects like variations in the scintillator bar light yield and fibre coupling to the MPPCs. This is accomplished by using a large number of cosmic ray events and estimating in each case the particle’s path length through each bar. The distribution of the calibrated charge divided by the estimated path length is produced for each channel. These distributions are used to infer a correction constant that results in the distribution means becoming equal. The distribution of these correction factors over the entire detector has an RMS of roughly 7%, which is comparable in magnitude to the 4.5% variation in light yield observed during scintillator bar tests.  3.6.7  Timing Calibration  MPPC pulse signals are fit to provide an initial time estimate. The fit function is shown in Eq. 3.1, where B is the baseline estimate, A an amplitude 45  3.6. Calibration constant, t0 the pulse start time, τ the pulse shape time constant and λ the pulse undershoot time constant. Only the leading edge of the pulse is fit in order to reduce the effect related to late photons or after-pulsing. The fitted time is corrected using timing markers, which are simultaneously and asynchronously inserted into an unused channel on every FEB in the FGD during a trigger by the MCM. By fitting the average timing marker time variations between FEB channels due to electronic path lengths and clock signal skew can be efficiently removed. In practice each individual fitted time is corrected by the average time offset of the timing marker of the FEB producing the signal with respect to the timing marker in a single reference FEB used for the whole detector for that run. Residual timing offsets are detected and removed using a large number of cosmic tracks were the individual hit time is compared to that of the average time of all hits in the selected track. This last correction removes timing offsets due to imperfections in the timing marker distribution hardware, and is found to be constant over the entire period of detector operation.  3.6.8  Other Calibrations  In principle it is possible to perform further corrections to the MPPC charge measurement to account for light-attenuation along the length of the bar or for scintillator saturation. Additionally the signal time estimate could be corrected to account for light propagation time along the wavelength shifting fibre. In practice both of these corrections require detailed knowledge about the position and direction of the particle track as it passes through the scintillator bar, which is generally inaccessible at the calibration stage. Consequently these corrections are not included in the standard calibration process.  46  Chapter 4  FGD Online Data Compression The uncompressed size of a full FGD data acquisition is estimated to be 13.5 MB per trigger, and at the target data rate of 20Hz would produce 270 MB/s. This is a significant amount of data and would be costly to store without any reduction. An example of a digitized waveform produced by the frontend readout electronics is shown in Figure 4.1, where MPPC pulses are clearly visible. The same image shows that much of the digitized waveform is empty, suggesting that a significant reduction in data size is possible. Data compression on frontend electronics is desirable as it reduces the bandwidth required through the remainder of the frontend readout system. This allows the use of simpler and cheaper data acquisition systems. More generally, reducing the data size allows similar reductions in required offline storage capacity and data processing time. This chapter describes the design and validation of an online data reduction system based around a pulse detection algorithm that was successfully implemented on the FGD frontend electronics firmware.  4.1  Data Compression  Data size reduction schemes can be generally separated into lossless and “lossy” compression, depending on whether the compressed data can be used to reproduce the original data. Lossless data compression has advantages in the context of frontend readout systems, as they reduce the amount of data transferred while ensuring the unmodified data sample is available for offline analysis. However the data reduction factor is generally not as high as for lossy compression. In the case of the FGD a physics interaction will likely only deposit significant amounts of energy in 100 of the scintillator bars. Consequently the large majority of channels will contain no useful information and their waveform data can be discarded entirely. This suggests that alternative data compression systems that save only sections of 47  4.2. The FGD Pulse Identification Algorithm  MPPC Channel Sample (ADC counts)  700 690 680 670 660 650 640 630 620 0  100  200 300 ADC Sample Number  400  500  Figure 4.1: Example MPPC channel digitized waveform. MPPC pulse shapes are clearly visible and large compared to baseline noise.  digitized waveforms containing information relevant to particle interactions will likely provide larger reductions in data size. Identifying sections of digital data that are relevant to physics analysis can be performed in several ways. If a trigger signal is provided, the interesting waveform sections can be identified through time information leading to a very simple data reduction system. However, this would be problematic for the FGD as the multiple bunch structure produces multiple regions of interest and it is necessary to be able to detect delayed energy deposit for the purpose of identifying Michel electrons. As well, trigger jitter in time and other electronic effects could reduce the pulse detection efficiency of such a system. More generally, trigger signals are not available and the data itself must be used to identify regions of interest. A common method is to identify detector pulse shapes in the recorded data, which is the approach used in the FGD online data reduction system.  4.2  The FGD Pulse Identification Algorithm  The low noise of the FGD frontend readout electronics system and the high gain of MPPCs are ideal for pulse signal detection. The magnitude of electrical noise in the MPPC signal can be estimated from the distribution of 48  4.2. The FGD Pulse Identification Algorithm samples about the baseline mean as shown in Figure 4.2, with a measured RMS of 3.2 ADC counts. Dark noise pulse heights measured at the nominal MPPC operating voltage are generally above 25 ADC counts with the peak of the single photoelectron pulse height distribution at 37 ADC counts, meaning that the signal-to-noise ratio is always greater than 7.8. This is adequate to consistently identify MPPC pulse signals.  Number of Entries / ( ADC count)  ×106 45 40 35 30 25 20 15 10 5 0  -40  -20 0 20 40 Baseline Sample - Baseline Mean (ADC counts)  Figure 4.2: Distribution of ADC samples about baseline mean, obtained from uncompressed waveform data with the MPPCs turned off. The mean of the uncompressed samples was calculated for each waveform, and the deviation of individual samples from the mean recorded in the histogram. This provides an estimate of the strength of the electronic noise signal.  4.2.1  Pulse Finder Design  The design of the FGD pulse finder algorithm was subject to the following constraints: • It must not significantly increase the time required for data acquisition. • It must have 100% detection efficiency for pulses produced by the passage of a charged particle, defined here as pulses > 10 MPPC avalanches at the nominal operating voltage.  49  4.2. The FGD Pulse Identification Algorithm • It must not distort the single pixel avalanche pulse height spectrum used in MPPC gain calibration over the expected range of operating voltages. • Ideally it should not require the measurement and monitoring of channelspecific constants, such as baseline values. These various requirements favour a simple pulse detection algorithm that does not require channel specific calibration. The selected algorithm is a local peak detector that compares the sum of consecutive rising and falling edges to a threshold to identify a pulse. This is similar to identifying the rising edge of a pulse by comparing the change in value between successive samples to a threshold, but including the falling edge allows some ability to identify overlapping pulses. A threshold value of 33 was found to provide optimal purity and acceptable efficiency for dark noise pulses produced at the typical operating voltage. In order to enhance purity a three-point average is used to smooth the waveform and reduce the effect of electronic noise. In the current implementation of the pulse finding algorithm, each sample of the digitized waveform is considered sequentially as it is read from RAM. The three-point average is calculated using the two previous samples to produce a test value. The test value is compared to the test value calculated for the previous ADC sample; if the test value is greater the algorithm is in the “rising-edge” state, while if it is less than the previous test value the algorithm is in the “falling-edge” state. In the case where the current test sample is exactly equal to the the previous test value the algorithm remains in its current state. A pulse candidate is identified by a rising-edge state immediately followed by a falling-edge state. A rising edge starts following a local minimum in the smoothed waveform, and ends at the subsequent local maximum. Similarly, the falling-edge starts at the local maximum and finishes at the subsequent local minimum. The absolute change in waveform height is measured for both the rising edge and the falling edge to produce a test statistic T S, as summarized by T S = |Rising Edge Height| + |Falling Edge Height|  = |Speak − S0 | + |Speak − Sf inal |  (4.1)  where S0 is the smoothed waveform value at the start of the rising edge, Speak is the value at the peak of the rising edge and Sf inal is the final smoothed sample value in the falling edge. This test statistic is then compared to the 50  4.2. The FGD Pulse Identification Algorithm pulse finder threshold and if it is exceeded a pulse is identified. As the test statistic is the sum of the rising and falling edges of the pulse it is clear that the algorithm selects pulses with height roughly half of the threshold value. The basic operation of the pulse finder algorithm is shown in Figure 4.3.  Figure 4.3: Schematic overview of pulse finder algorithm operation. Samples are processed sequentially in time (left-to-right in the diagram) and rising and falling edges are identified. The total change in the 3-point average of the waveform is measured across edges, and a test statistic is formed by the absolute sum of rising and falling edge heights. If the test statistic exceeds the threshold value, a pulse is identified. The MPPC pulse height is estimated as the digitized sample value at the pulse peak minus the value of the baseline, estimated from a sample acquired a fixed number of clock cycles previously. The smoothed waveform values are not used to measure the pulse height. The pulse time is estimated using the time of the peak sample. Offline calibration processes fit the pulse shape to obtain much more accurate estimates of pulse height and time, as 51  4.2. The FGD Pulse Identification Algorithm described in Chapter 3. The pulsefinder algorithm described here was selected in part because of its simple implementation, but also for its ability to efficiently identify overlapping pulses that would be difficult to detect for simple threshold or template fitting algorithms. Additionally the algorithm is robust and can tolerate large fluctuations in the baseline noise level between data acquisitions and moderately sized fluctuations within the data acquisition sampling period. It is also handle changes in the MPPC gain by adjusting the threshold to maintain the same efficiency to detect smaller pulses if necessary. The current threshold value of 33 was chosen to reduce to an acceptable level the number of baseline noise fluctuations misidentified as MPPC pulses, while maximizing the dark noise efficiency. In practice the pulse heights measured for these misidentified baseline fluctuations are small enough that they do not significantly affect the gain calibration process described in Chapter 3. As described previously, a threshold of 33 roughly corresponds to rejecting pulses with height less than 16. This corresponds approximately to a ∼ 4.7σ fluctuation in the baseline electronic noise for any given sample assuming a Gaussian distribution. This ensures fake pulse signals are sufficiently rare that they are not a significant fraction of the recorded data.  4.2.2  Efficiency Measurements  The efficiency of the pulse detection to identify MPPC pulses can be measured with simulated data, using the pulse shape shown in Eq. 3.1. Simulated pulse shapes are randomly placed in digitized waveforms and processed by the pulse identification algorithm. If a pulse is identified within 100ns (5 time bins) of the simulated pulse’s true time it is considered found. The number of incorrectly identified false pulses is also recorded in order to provide an estimate of the algorithm’s purity. The test is made more realistic by using waveforms recorded from the real FGD, so that the effects of actual electronic noise are included. An example of a simulated pulse placed in a waveform from real data is shown in Figure 4.4, although for the purpose of evaluating the pulse finding algorithm only waveforms containing no real MPPC pulses are used. These waveforms were obtained by recording waveform data when the MPPCs were turned off, accomplished by reducing the applied bias voltage to below the minimum required for Geiger mode discharges. The measured efficiency for the algorithm to identify pulses that are 37 ADC counts in height when the threshold parameter is set to 33 ADC counts is shown for different simulated pulse times in Figure 4.5. The ef52  4.3. Firmware Implementation of the FGD Pulse Finder ficiency measurement was performed for pulses that are 37 ADC counts in height as this is the average single pixel avalanche pulse height observed in the real detector. The threshold parameter value of 33 also corresponds to the threshold used in the real system. As can be seen, the efficiency drops off rapidly at the edges of the waveform. This is expected as some minimum number of samples is required to identify the peak of the pulse. To simplify the evaluation of the algorithm’s performance only pulses simulated in the range starting at sample number 10 to 490 are considered in the efficiency measurement. The measured efficiency and purity of the algorithm for various pulse heights in this restricted range are summarized in Table 4.1 using the default value of the pulse finder threshold of 33 ADC count.  Figure 4.4: Example of a waveform with a simulated MPPC pulse starting in time bin 416, and real MPPC pulses marked with blue squares. As can be seen the shape of the simulated pulse is very similar to the real pulse shapes, supporting the use of simulated pulses to measure pulse finding efficiency.  4.3  Firmware Implementation of the FGD Pulse Finder  The online data compression scheme implemented on the CMB FPGA uses a firmware implementation of the pulse finding algorithm previously described. 53  4.3. Firmware Implementation of the FGD Pulse Finder  1  Efficiency  0.8 0.6 0.4 0.2 0 0  100  200  300  400  500  Simulated Pulse Time (Sample Number) Figure 4.5: Pulse detection efficiency vs simulated pulse time, for pulses 37 ADC counts in height.  It is used to retain only interesting sections of waveform and makes additional compression decisions based on the measured pulse height. The data packet produced by the CMB for each processed channel includes height and time measurements for each identified pulse. The ADC samples for the section of waveform containing the pulse can also be retained. In the current implementation of the system the waveforms are retained for pulses with a measured height above some threshold designed to exclude dark noise pulses.  4.3.1  ASIC Channel Readout and Processing  ADC samples are read from CMB memory and processed by the firmwarebased pulse finder. When a pulse is identified the pulse finder writes the pulse height and time information into a “first-in first-out” (FIFO) data storage structure implemented in FPGA firmware. This FIFO is used by a separate compression process to zero-suppress a delayed sequence of ADC samples. The number of delays depends on the desired number of samples in the data packet prior to the pulse rising edge, as well as the time required 54  4.4. Pulse Height Based Data Reduction True Pulse Height 0 7.4 14.8 22.2 29.7 37.1 55.6 74.1 111.2  Measured Mean Pulse Height 15.4 16.1 17.7 22.6 29.6 37.0 55.4 73.8 110.7  Efficiency  Purity  0.018 0.444 0.964 0.999 0.999 0.999 0.999 1.000  0.0 0.819 0.991 0.996 0.996 0.996 0.996 0.996 0.996  Table 4.1: Simulated pulse finder efficiency and purity measurements. Efficiency is the rate at which simulated pulses are identified, while purity is the fraction of identified pulses that are also true simulated pulses.  for the pulse finder to identify a pulse and calculate its height. Additional data-reduction decisions are made by the compression process based on the identified pulse height. The results of the compression process are output into a results FIFO, and include pulse height and time pairs and optionally waveform sections. After all the samples for the channel are read from RAM the contents of the output FIFO are moved into the RocketIO clock domain and used to form the data packet that is sent over the optical link to the DCC.  4.4  Pulse Height Based Data Reduction  The pulse height spectrum obtained from acquisitions following a cosmic trigger (Figure 4.6) shows that the vast majority of pulses produced by the FGD are from random MPPC pixel avalanches. Dark noise pulses dominate the spectrum for pulse heights below 200 ADC units, while larger pulses are primarily produced by cosmic muons interacting in the scintillator bars. Dark noise information is useful for gain calibrations as well as monitoring the detector status over time, but the size of the dark noise data must be limited. For the purpose of data compression, dark noise pulses are defined as pulses with heights less than the dark-noise threshold of 200 ADCs. After identifying a pulse and applying zero-suppression, the compression algorithm uses the measured pulse height to determine whether the pulse 55  4.4. Pulse Height Based Data Reduction  Figure 4.6: Cosmic MPPC pulse spectrum with important features shown, reprinted with permission [41]. The saturation peak includes all pulses with height that exceeds the range of the high-gain MPPC channels.  is due to dark noise. If it is a dark noise pulse, the process can either discard the pulse information entirely or retain only the pulse height and time. This decision is controlled by a single firmware register bit that can be changed between acquisitions. If the pulse is larger than the dark-noise threshold the pulse height and time are retained along with ADC samples from the section of waveform containing the pulse. The total size in data for dark noise pulse information processed using this scheme is 300 kB per acquisition, compared to 40 kB for pulses due to particle interactions. As dark noise information can be discarded or retained on an event-by-event basis, the total data rate can be adjusted by only reading it out for some fraction of the the acquisitions. In the current implementation the dark noise data is typically saved only 10% of the time, which provides sufficient reduction in data size and recorded pulses for calibration purposes.  56  4.5. Validation of the Simulated Pulse Finder Algorithm  4.5  Validation of the Simulated Pulse Finder Algorithm  The agreement between the real and simulated pulse identification process can be directly tested using a special dataset where both the uncompressed waveform and the compressed data packets are recorded. The simulated algorithm can be applied to the uncompressed waveforms and compared to the output of the online data compression process. This test was performed using a dataset containing only dark noise data taken at the nominal operating voltage, and the simulated compression process identified 100% of the pulses found by the online compression. This provides confidence that the efficiency measurements made for the simulated pulse finder are applicable to real data, and the online data compression system can measure signals from both MPPC dark noise and particle interactions with sufficient accuracy.  57  Chapter 5  The Near Detector Analysis Framework This chapter describes the data-analysis framework developed by the ND280 collaboration to analyze data recorded during data acquisitions following a proton beam spill trigger signal provided by the accelerator system. It describes low-level analysis tools that converts raw detector data into calibrated energy deposit measurements. Higher-level track reconstruction and particle identification algorithms are also described. The simulation of ND280 is also described, along with the neutrino interaction generators that are of central importance for neutrino physics analyses.  5.1  Data Acquisition  The near detector data acquisition system (DAQ) collects data recorded by each ND280 subdetector following a trigger signal, and combines the resulting data packets into a single file on an event by event basis. Each individual subdetector has a standalone DAQ system that is based on the MIDAS DAQ software package [44] running in a Linux operating system. These subdetector-specific MIDAS systems transfer data to the global DAQ system through a Gigabit Ethernet network. The global DAQ system uses an event-building process to merge all subdetector data recorded after a specific trigger into a MIDAS data structure. The global DAQ is flexible and can be configured to include or exclude specific subdetector data from the output file. The data produced during regular operation is saved in the MIDAS file format to a local raid array and archived.  5.2  The ND280 Software Package  The ND280 offline software package provides a framework to access real and simulated ND280 detector data and run calibration and reconstruction processes. Detector data is converted from the MIDAS format using the “oaUn58  5.2. The ND280 Software Package pack” library and is stored and manipulated using a general “oaEvent” library. The oaEvent library is based on the ROOT software library [45], and makes extensive use of ROOT data structures and geometry tools. The simulation, calibration and reconstruction packages that are implemented within the oaEvent framework are described here as they were defined in version v9r9p1. A schematic overview of the ND280 software package is shown in Figure 5.1  Figure 5.1: Schematic overview of ND280 software package. The conversion of raw MIDAS data into the oaEvent format and the subsequent application of calibration, reconstruction and summarization processes are shown. A similar process is depicted for simulated data starting with the neutrino interaction generators and the nd280mc and elecSim packages.  5.2.1  Simulation of ND280 Data  Several steps are required to simulate neutrino interactions in the near detectors. The beam flux predictions described in Chapter 2 are passed to the two neutrino interaction packages used in ND280 analysis, NEUT [46] and GENIE [47]. The neutrino interaction generators are used to simulate the number and types of neutrino interactions throughout the near detectors during a simulated beam spill. A ROOT-based representation of the ND280 59  5.2. The ND280 Software Package detectors is used by the generators to reproduce a realistic spatial distributions of neutrino interactions and to account for variations in detector material. The neutrino generator’s output for each simulated interaction is an ordered set of summary arrays, describing the final states of particles produced from the interaction and subsequent processes in the atomic nucleus. These arrays include position and momentum four-vectors and other relevant truth information. Neutrino Interaction Generators Two neutrino interaction generators are typically used in ND280 physics analyses, NEUT and GENIE [46] [47]. As the analysis presented in this thesis only makes use of the NEUT package, GENIE will not be described. Both packages use a given neutrino flux and energy spectrum to produce a list of neutrino interactions and the product particles. The neutrino interaction generators are standalone packages that include all neutrino interaction model and cross section information used to simulate neutrino interactions in detector material. Both NEUT and GENIE are independent projects separate from T2K and have been used in other neutrino experiments. The NEUT package was developed to describe atmospheric neutrino interactions observed by the Super-Kamiokande experiment [48]. It has been expanded to describe a wide range of energies and interaction processes. NEUT models neutrino interactions in a two step process; a primary neutrino-nucleon interaction that produces a set of particles, and the secondary interactions of these product particles as they travel through the nuclear material. Most primary interactions of interest for the ND280 experiment are with nucleons bound within a nucleus or with the nucleus itself, which include quasi-elastic scattering, meson production through nuclear resonances and deep inelastic scattering for both charged current and neutral current processes. Neutrino interactions with hydrogen protons in water molecules are considered to be with free nucleons, which involves a different set of models. A full listing of NEUT primary interaction modes is shown in Table 5.1. Quasi-elastic primary interactions with free nucleons are simulated by NEUT using the Llewellyn-Smith model [49], with experimentally derived values for the form factors used in cross sections. In the case of bound nucleons the effect of the nuclear medium is accounted for through a relativistic Fermi gas model [50]. This model simulates the nucleon momentum distribution as flat up to the Fermi momentum of pF = 225 MeV/c, with Pauli blocking accounted for by requiring the recoil nucleon to have momentum 60  5.2. The ND280 Software Package Interaction Mode CC quasi-elastic scattering CC single π production CC coherent π production CC multiple π production CC single η production CC single kaon production CC single γ production CC deep inelastic scattering NC single π production NC coherent π production NC multiple π production NC single η production NC single kaon production NC single γ production NC deep inelastic scattering NC elastic scattering Equivalent anti-neutrino modes  Process νN → lN ′ νN → lN ′ π νO(16) → lO(16)π νN → lN ′ multi-π νN → lN ′ η νN → lN ′ K νN → lN ′ γ νN → lN ′ mesons νN → νN ′ π νO(16) → νO(16)π νN → νN ′ multi-π νN → νN ′ η νN → νN ′ K νN → νN ′ γ νN → νN ′ mesons νN → νN ′ -  Table 5.1: Primary neutrino interactions modelled by NEUT. N and N ′ are the initial and final state nucleons (either protons or neutrons) and l is the charged current lepton.  greater than the Fermi momentum. The primary parameter for modelling quasi-elastic cross sections is the axial mass MA constant used in a dipole expansion of the axial form factor. Several non-quasi-elastic primary processes are also modelled in detail by NEUT. Resonant pion production processes are simulated using the ReinSehgal model [51], which includes a treatment of baryonic resonances that can be produced by the initial neutrino interaction and decay to produce pions (or other mesons). Coherent pion production is similarly modelled as the neutrino interacting with the entire nucleus [52]. Deep inelastic scattering cross sections are calculated directly for invariant mass energies greater than 1.3 GeV using parton structure functions provided by [53] with additional corrections suggested by Bodek and Yang. Particles produced by deep inelastic scattering are modelled using PYTHIA/JetSet for interactions. There are some empirical corrections applied for deep inelastic scattering interactions below 2 GeV, as well as restricting the simulated in-  61  5.2. The ND280 Software Package teraction to pion multiplicities greater than one to avoid double-counting production probabilities with the resonant pion production processes. Secondary interactions in the nucleus, also referred to as final state interactions, modify the observed event topology and are of central importance for experiments with nuclear targets. A cascade model tracks the propagation of each particle produced by the primary interaction and any subsequent reinteractions as it propagates out of the nucleus. A Woods-Saxon model of the nucleon density distribution is used [54], and Fermi momentum and Pauli blocking effects are also modelled. Pion reinteraction with nuclear material is a significant source of systematic uncertainty in the simulated final state of the interaction, and NEUT uses a combination of direct calculation and fits to pion and proton cross section data produced by other experiments. The propagation of other mesons is similarly modelled. nd280mc The neutrino generator output is processed by the nd280mc simulation package, based on the Geant4 library. The nd280mc simulates the trajectory of particles produced by neutrino interaction through the near detectors, using the final-state particle summary arrays to define the types of particle simulated and their initial kinematic variables. The simulation estimates the energy deposit in active detector material due to the particle’s passage, which is used when simulating the detector response and electronic system. nd280mc uses the same ROOT geometry representation of ND280 used by the neutrino interaction generator. This geometry includes details like the UA1 magnetic field and dead material around the detector to ensure the simulation is accurate. The time structure of the beam spill is included at this point by randomly choosing the time of simulated neutrino interactions so as to reproduce the expected beam bunch structure. The simulated energy deposits of all particles produced during a simulated beam spill are recorded in the same output event, along with data structures containing truth information. For simplicity the ROOT-geometry is also saved with the output file to ensure that subsequent software processes use the appropriate version of the detector geometry. elecSim Following the nd280mc process, the “elecSim” package simulates the detector response to simulated energy deposit. There are several different simulations run at this point specific to each subdetector. In the case of the FGD, 62  5.2. The ND280 Software Package energy deposit in scintillator bars is used to predict the number of photons produced in the bar and transferred by the fibres to the MPPCs. The MPPC response and frontend electronics are also simulated, as are signal processing and data reduction processes implemented within the readout electronics. Similarly the TPC electronics simulation includes a model of ion drift in the TPC gas volume, the operation of the Micromegas modules and the related readout electronics. The results of these simulations are saved using a “digits” class, which is the same format used to save real data. At this point real and simulated data are saved in the same format. The calibration and reconstruction packages treat real and simulated data in the same way, with the only difference between the output files is the truth information stored in the simulated data case.  5.2.2  Calibration  The ND280 calibration package “oaCalib” converts detector signal measurements (saved using the “digits” class), to energy-deposit estimates (saved using the “hits” class). In general hits are simplified expressions of detector data that contain only the calibrated charge or energy deposit measurement, position and time. Different subdetectors use significantly different calibration methods, but in general the calibration process attempts to convert measurements in terms of ADC units to physically meaningful values, and to remove non-linearity between the measured quantity and the amount of energy deposited. Additionally calibration processes remove time dependent effects or differences between individual sensors. Simulated data is also calibrated, although in this case there are no time-dependent effects. FGD Calibration FGD calibration is described in detail in Chapter 3. MPPC signal charge and time measurements are corrected to account for gain variation between sensors, temperature and overvoltage gain dependence and gain non-linearity due to MPPC pixel saturation. Time measurements are corrected for electronic signal delays and also with reference to a timing marker included in the detector readout of every event. In principle further calibrations could be applied, including a correction for fibre-attenuation and non-linearity in scintillator light production, but in practice this would require accurate estimates of the position in three-dimensions and is not required for reconstruction purposes.  63  5.2. The ND280 Software Package Trip-T Detector Calibration The common use of scintillator bar and Trip-T based electronics readout of the P0D, ECal and SMRD detectors allows a unified approach to calibration [55]. The pedestal value of each electronics channel is constantly monitored and subtracted from the ADC value measured during a triggered integration cycle. The measured charge is linearized and corrections made for variations in gain between individual MPPCs. The measured time is also calibrated, including corrections for different signal propagation times due to cable lengths and charge-dependent effects. TPC Calibration The TPC Micromegas gain is monitored by using a laser to emit photoelectrons from aluminium discs glued to the central cathode [40]. This allows monitoring of distortions in the electric and magnetic fields and can be used to measure the gain of the readout electronics. Charge measurements are also corrected to account for variations in gas density and temperature. Two dedicated monitor chambers with similar readout electronics to the main TPCs provide additional measurements of drift velocity and gain constants used in calibration.  5.2.3  Reconstruction  Reconstruction of ND280 data is carried out in several stages, starting with algorithms applied to specific subdetectors only (“local” reconstruction). The local algorithms are followed by “global” reconstruction processes that attempt to combine hits and local reconstruction results from different subdetectors. The global reconstruction algorithm is based on the RecPack package (RecPack reference), which provides a general framework for track reconstruction, propagation and matching. The only local reconstruction algorithm relevant to this thesis is “tpcRecon”, which is applied to TPC hits only. “trackerRecon” matches the tracks found by tpcRecon with hits in the FGDs, while “oaRecon” attempts to match the trackerRecon tracks with hits in the ECals, P0D and SMRD to produce the final reconstruction result. RecPack RecPack is an external reconstruction library used extensively in ND280 reconstruction packages [56]. It provides general track fitting, propagation 64  5.2. The ND280 Software Package and matching methods. RecPack uses a simplified version of the ND280 ROOT geometry in order to correctly account for energy loss in detector material and similar effects. A crucial component of RecPack in ND280 is the Kalman filter fitting algorithm that allows incremental matching of seed tracks with additional hits. This matching algorithm allow for flexible reconstruction as instead of using pattern recognition it allows any hit to be matched as long as the χ2 variable calculated for the resulting track does not become too large. Another major feature is a merging algorithm that connects separately reconstructed tracks into one longer track, with a similar requirement on a χ2 variable. These features allow hits and local reconstruction results across different ND280 subdetectors to be combined to provide a more complete reconstruction of particle interactions. tpcRecon The local TPC reconstruction algorithm is applied to each TPC individually [56]. An initial hit clustering step is followed by a pattern recognition algorithm that attempts to identify tracks. tpcRecon uses a modified version of the SBCat algorithm developed for K2K near detectors, and looks for overlapping clusters of TPC hits between adjacent rows of the Micromegas readout pad. Individual Micomegas pad elements are are 9.8 × 7.0 mm2 in size and the sampling rate is typically 25 MHz. The longest segment of overlapping hits defines the track, as shown schematically in Figure 5.2. The identified track requires a reference time T0 before it can be fit in three dimensions. Typically the TPC track’s T0 is deduced by using RecPack to match it to hits in the surrounding scintillator detectors and using the matched hit time as T0 . Alternatively if the track crosses the central cathode the value of T0 can be estimated using the average drift time over the maximum drift length. After estimating T0 a likelihood function is used to determine the track trajectory that best fits the observed hit distribution. The fit provides track kinematic variable estimates, most importantly a momentum estimate. The TPC charge deposit measurement can be combined with the momentum estimate to provide a powerful method of particle identification. Charge deposit is measured using a truncated mean estimate CT , which is the average charge of the lowest 70% of hit cluster charges. The distribution of CT with momentum is shown for simulated data in Figure 5.3. As expected there is clear separation for much of the momentum range of interest between types of particles due to differences in dE/dx.  65  5.2. The ND280 Software Package  Figure 5.2: Schematic of SBCAT track pattern recognition, taken from T2K Technical Note 72 [56]. Hit clustering is performed in both spatial co-ordinates and sampling time. The Micomegas pad sizes are 9.8 × 7.0 mm2 and the sampling rate is typically 25 MHz.  FGD-TPC Track Matching The local TPC reconstruction results provide seed tracks that RecPack can match with FGD hits to identify particle tracks crossing or originating within an FGD. This is a crucial feature used to identify neutrino interactions that originate within the FGD. It also provides a powerful method to reject external background and obtain more accurate estimates of kinematic variables. The “fgdRecon” package implements the FGD-TPC track matching algorithm, using the output of the tpcRecon package and FGD hits clustered by time. The FGD hit time clustering algorithm sorts all the hits in a recorded event by time. If the time difference between two consecutive hits exceeds a threshold the previous cluster of hits is completed and a new time cluster created for subsequent hits. This process is repeated until all hits in the event have been processed. Clusters of FGD hits created by this process are also referred to as FGD reconstruction time bins. The FGD-TPC track matching process begins by attempting to match the TPC seed track’s T0 to a FGD hit cluster in time. If no such match is found the FGD-TPC track matching algorithm does not proceed any further. If a hit cluster is found to match in time the TPC track is extrapolated to 66  CT  5.2. The ND280 Software Package Entries  2000  32630  Exp µ Exp ele Exp pro Exp kaon  1800 1600 1400  300 250 200  1200 1000  150  800 100  600 400  50  200 0  200  400  600  800  1000  1200  1400  1600  1800  2000  0  p (MeV/c)  Figure 5.3: TPC measured truncated charge CT vs. momentum in simulated data. Expected charge deposit for different particle hypotheses are shown with solid curves, including muons, electrons, protons and kaons.  the closest layer containing hits from the cluster, and the incremental track matching process is started. The track matching algorithm calculates a chisquared value for the trajectory produced by matching the seed track to different hits, which takes into account uncertainties in the seed track direction and multiple scattering effects. A hit is matched to the seed track if the calculated chi-squared value is below a predefined threshold. Each time a hit is successfully matched to the propagated track the result is refit and becomes the new seed track for the next iteration of the incremental fit. When no further hits can be matched to the track it is refit one final time and output as a FGD-TPC matched track. The fit includes an momentum estimate that takes into account the energy loss of the particle passing through the FGD and dead material, leading to more accurate estimates than can be provided by the local TPC reconstruction. Figure 5.4 shows an example FGD-TPC matched track. The hit position lowest in the z co-ordinate or most upstream with respect to the proton beam direction is considered to be the initial track position or vertex. The fgdRecon algorithm only matches TPC seed tracks with hits in an adjacent FGD. The “trackerRecon” package attempts to combine FGD-TPC matched tracks together, providing a more complete reconstruction of the event and a consistent way to combine momentum estimates from different 67  5.2. The ND280 Software Package  Figure 5.4: Example of a FGD-TPC matched track. The blue line indicates where a TPC track was successfully matched to FGD hits. Hits from a separate non-matched track are shown in red.  TPCs. In a similar fashion the “oaRecon” package matches the results of the trackerRecon algorithm to hits in the P0D, ECals and SMRD. The results of the oaRecon package are the final output of the reconstruction package.  5.2.4  Event Summary with oaAnalysis  After all calibration and reconstruction tasks are performed, the results are summarized in a more convenient format for analysis and physics studies. Detector data and algorithm results stored in the oaEvent format are converted and stored in a standard ROOT tree using the “oaAnalysis” package. This process retains only the summarized results of high-level reconstruction algorithms, as well as a reduced number of hits from select detects. The most important results for the analysis presented in this thesis are the global reconstruction results, including TPC and FGD-TPC matched tracks. The detector signal measurements stored in the digits format and most hit measurements are not included in the oaAnalysis output. The oaAnalysis package also integrates information related to the J-PARC beam, including whether the triggered beam spill is suitable for use in analysis and the associated number of proton on target. The output can be analyzed using standard root or python scripts supplemented by the oaAnalysis library, and is used to select neutrino interactions and compare measured quantities 68  5.2. The ND280 Software Package between real and simulated data.  69  Chapter 6  The FGD Cosmic Trigger Cosmic radiation is a useful source of particle tracks that are used to calibrate the ND280 detectors. A trigger system was implemented to detect cosmic particle tracks and record their interactions. Different subdetectors provide trigger primitives derived from sensor signals to a Cosmic Trigger Module control board (CTM). When the trigger primitives satisfy the criteria for a cosmic event, the CTM board sends a trigger request signal that may result in a data acquisition. This chapter describes the design, calibration and simulation of the FGD cosmic trigger system.  6.1  Cosmic Rays in FGD Analysis  Cosmic radiation is used at several stages in the FGD calibration process, specifically to evaluate bar-to-bar variations in light yield and high-to-low gain channel calibration. Additionally cosmic samples can be used to determine constants used in simulation, specifically the scintillator bar light yield. Finally cosmic samples can be used to identify and measure systematic effects, such as track reconstruction efficiency or the size of external background in a selection. The importance of the cosmic sample has resulted in cosmic triggers being included in regular ND280 operation and in the development of a simulated sample.  6.2  The Trigger System  The ND280 electronics and trigger system is shown in Figure 6.1. The FGD cosmic trigger begins with Crate Master Board (CMB) triggers asserted in response to MPPC signal monitors. The FGD CMB trigger signals are passed to the FGD CTM, which applies trigger logic to detect a coincidence between both FGD modules. This tends to select horizontal throughgoing tracks. If a coincidence is found a trigger signal is sent by the FGD CTM to the ND280 Master Clock Module (MCM). If there is no beam trigger and the FGD cosmic trigger is accepted by the DAQ an ND280 data acquisition 70  6.2. The Trigger System process is initiated. Alternative trigger paths are also shown, including the Trip-T cosmic trigger.  Figure 6.1: ND280 trigger and clock distribution system [22], c Nuclear Instruments and Methods in Physics Research, 2011, reprinted with permission. The MCM produces the global clock, which is distributed to all ND280 subdetectors. The FGD cosmic trigger signal originates on the FGD CMBs and is passed to the FGD CTM, which applies additional trigger logic to coincident signals. If the trigger conditions are satisfied a data acquisition request signal is sent to the MCM.  6.2.1  FGD Cosmic Trigger  The FGD cosmic trigger primitives are referred to as Analog Sums (ASUM). Each ASUM trigger primitive is formed from the charge signals by up to eight MPPCs, this being an ASUM group. The eight MPPCs that form an ASUM group are adjacent to each other on an FGD busboard, with two ASUM groups formed from the sixteen MPPCs placed on each busboard. Consequently each ASUM group detect light signals produced by eight bars spaced one bar apart in a single scintillator bar layer. If the ASUM group signal exceeds the comparator’s reference voltage in response to photosensor signals it outputs a voltage signal that indicates the ASUM trigger is activated. All the ASUM comparator signals in a given mini-crate are monitored by digital logic implemented in the CMB firmware. 71  6.3. Cosmic Flux Simulation If the CMB detects that an ASUM group has triggered, a trigger signal is sent to the FGD CTM through a CAT-5 cable. The trigger logic can be configured to require up to 4 different ASUM groups from different FEBs to be activated within a specified time period before the CMB trigger signal is sent. This can be used to reduce the incidence of triggering due to electrical noise. The FGD CTM monitors the FGD trigger signals and requires at least four trigger signals before sending a trigger request signal to the MCM. The trigger logic requires that each FGD module produce a trigger signal from a CMB in a vertical row of crates and a horizontal row of crates. These trigger signals must be coincident in time. Optionally the CMB trigger signals can be prescaled to adjust the rate at which different parts of the detector produce a trigger, although in practice this functionality has not been used. Trigger Calibration ASUM trigger primitives are calibrated using a program that continuously monitors the trigger rate. The trigger threshold is varied for each ASUM group and the resulting change in the group’s trigger rate recorded. The trigger rate is observed to increase dramatically below some threshold presumably due to the effect of triggering on electronic noise and dark noise MPPC pulses. The operating threshold is set just above this noise threshold.  6.3  Cosmic Flux Simulation  Cosmic flux incident at Tokai was simulated using the Corsika package [57]. Protons with energy up to 1 × 105 GeV are simulated as interacting with air molecules in the upper atmosphere and the resulting products propagated down to different elevations. The simulation accounts for energy loss and unstable particle decay, as well as scattering processes due to the effect of the earth’s magnetic field. The result of the simulation is a set of vertices specifying the position, angular direction and momentum of particles that survive to sea-level at the ND280 site in Tokai. Only muon trajectories are recorded and included in the simulation at this time, although electrons may be retained in future. The effect of the pit walls on the cosmic flux was included in the simulation by developing a Geant4 representation of the ND280 pit and surrounding earth. Muons predicted by the Corsika package are propagated through the pit geometry and any incident on the ND280 detector element 72  6.4. FGD Cosmic Trigger Simulation are recorded. The pit simulation is intended to reproduce the additional screening provided by the pit walls for low-angle cosmic particles as well as the effect of the off-centre position of the ND280 detectors in the pit. The position of the detectors after being installed in the ND280 pit can be seen in Figure 6.2.  Figure 6.2: ND280 subdetectors installed within the magnet shown in its open configuration at the base of the ND280 pit.  6.4  FGD Cosmic Trigger Simulation  A simulation of the FGD cosmic trigger was developed for the ND280 software package to provide a simulated data sample that could be directly compared to real FGD cosmic trigger data. This simulation includes the behaviour of the ASUM trigger primitives and digital trigger logic implemented on the frontend electronic boards. Trigger Thresholds The ASUM trigger thresholds can be measured directly from cosmic data. A binary representation of the ASUM trigger state over time can be read 73  6.4. FGD Cosmic Trigger Simulation out along with MPPC waveform data used to identify MPPC signals that activate each trigger primitive. One issue is that ASUM trigger primitives monitor the output of eight MPPCs. Consequently the ASUM charge signal can only be approximately estimated by summing measured pulse heights from all eight MPPCs. The distribution of ASUM group charge signals that result in a trigger can be compared to the total sample of charge signals in Figure 6.3. As can be seen the average ASUM trigger threshold is somewhat higher than the peak of the overall distribution, suggesting that the trigger may be biased against selecting minimum ionizing particles. The efficiency for triggering on MPPC charge signals for an example ASUM group is shown in Figure 6.4.  Figure 6.3: ASUM group charge signals measured during cosmic triggers, showing signals that trigger an ASUM trigger primitive. The ASUM group signal is estimated as the sum of MPPC pulse heights measured during the trigger time window, with the cosmic MIP peak corresponding to ∼ 1200 ADC counts. The peak at ∼ 3500 ADC counts is due to the saturation of the 12-bit ADCs by the MPPC pulse and baseline electronic noise signals. In the cosmic simulation the ASUM group charge signal is once again modelled as the sum of MPPC pulse signals within the group. A linear piecewise function is used to approximate the trigger efficiency for ASUM group charge signals. Trigger threshold measurements from real cosmic data are 74  6.5. Validation of the Trigger Simulation  Figure 6.4: Example group charge signal trigger efficiency vs summed charge of the 8 MPPCS in an ASUM group, measured for one sample ASUM group. One photoelectron is equivalent to ∼ 38 ADC counts, meaning that the ASUM only begins to trigger for ∼ 26 photoelectron signals. used to provide constants used in the simulation, specifically the slope of the efficiency curve and the charge signals at which the trigger efficiency is 0% and 100%. The group charge signal is simulated every 20 ns over the entire simulated waveform, which is used to determine the ASUM trigger primitive states in these time periods. Realistic trigger logic is simulated with these ASUM trigger states as inputs, and reproduces the logic implemented on the CMBs and CTM. At the CMB level, ASUM trigger primitives from two different layers must be activated within a certain time period in order to assert the crate level trigger. The simulated CTM requires a CMB trigger from a horizontal and vertical crate on both FGDs within a separate time range. If the trigger conditions are fulfilled a trigger flag is set to true.  6.5  Validation of the Trigger Simulation  Ideally the simulated cosmic sample possesses the same angular correlations and the momentum spectrum as the real data, simplifying calibration and evaluation of detector response. These distributions are compared to demonstrate the extent to which the simulation reproduces real data, and identify 75  6.5. Validation of the Trigger Simulation discrepancies. The angular distribution for TPC2 tracks reconstructed from FGD cosmic triggers is shown in Figure 6.5 and Figure 6.6. Here the polar angle θ is defined with respect to the downward vertical direction, such that cosmic particle moving straight down would correspond to cos θ = 1. The φ angular direction is defined about the vertical axis, with φ = 0 corresponding to the x-axis direction, perpendicular to the z-axis in line with the neutrino beam direction. As can be seen the simulated data reproduces the real distribution fairly well. The FGD cosmic trigger acceptance has a strong dependence on angular φ as the particle must pass through both FGDs for a trigger signal to occur.“Sideways” particles that only hit one FGD module are ignored. The simulation also reproduces the asymmetry in the φ angular distribution observed in real data due to the effect of the ND280 pit.  Number of Entries / (0.01)  0.03 0.025 0.02 0.015 0.01 0.005 0 0  0.1 0.2  0.3 0.4  0.5 0.6  0.7  0.8  0.9  1  TPC2 Track Direction Cosθ Figure 6.5: FGD cosmic trigger TPC2 direction in cos θ. The polar coordinate is defined such that cos θ = 1 corresponds to a downward going particle. Horizontal particles are not selected due to cosmic flux distribution and screening effect of the pit walls. The points with blue error bars are simulated cosmic ray tracks, while the points with no error bars are data.  76  Number of Entries / (0.628 rad)  6.6. Trigger Simulation Applications  0.03  0.025 0.02  0.015 0.01  0.005 0  -3  -2  -1  0  1  2  TPC2 Track Direction φ (rad)  3  Figure 6.6: FGD cosmic trigger TPC2 direction in φ. Sideways going particles corresponding to φ = 0 are not found due to trigger requiring energy deposit in both FGDs. The φ > 0 peak is smaller than the φ < 0 peak due to the screening effect of the pit walls and the off-center position of the ND280 detectors in the pit. The points with blue error bars are simulated cosmic ray tracks, while the points with no error bars are data.  6.6  Trigger Simulation Applications  As seen in Figures 6.3 and 6.4, the ASUM trigger threshold is greater than the average signal charge resulting from the passage of minimum ionizing particles through scintillator bars. This introduces the potential for bias in the momentum spectrum of the FGD cosmic trigger sample as compared to a sample of minimum ionizing particles, underscoring the need for an accurate simulation. As the plots in the validation section show, the current implementation of the trigger simulation combined with the Corsika cosmic flux and pit geometry simulations reproduces all the major features in the observed cosmic distributions. An important application of the cosmic simulation was the tuning of FGD scintillator bar light yield parameter simulation using real data. This was accomplished by selecting a real and simulated sample of minimum 77  6.6. Trigger Simulation Applications ionizing cosmic tracks from the overall cosmic trigger sampled, and requiring that they pass straight through the FGD1 module. The distribution of hit charge deposited in each layer divided by the reconstructed track direction within the layer was produced for both real and simulated data, accounting for the effect of light attenuation in the wavelength shifting fibre. The light yield parameter was varied until the real and simulated distributions matched. The distribution of calibrated charge deposit measured for cosmic tracks following the tuning process is shown in Figure 6.7, where it can be seen that the real and simulated distributions are in good agreement.  Figure 6.7: FGD calibrated charge deposit per layer produced by a sample of real and simulated cosmic tracks, reprinted with permission [41]. The FGD cosmic trigger simulation has been used for a variety of ND280 analyses, including light yield calibration in various subdetectors, and to perform low-level cross-checks of various reconstructed quantities. It has also been used extensively to compare track reconstruction and energy deposit efficiency between real and simulated data, and has proven useful for measuring the size of systematic effects as discussed in Chapter 8.  78  Chapter 7  The CCQE and CCnQE Interaction Selection Inclusive charged current neutrino interactions are identified using a cutbased selection applied to reconstructed particle tracks in the ND280 tracker detectors. The CC-inclusive selection is further divided by additional cuts into CCQE and CCnQE samples. The selected muon momenta and direction distributions are measured from both samples and fitted to measure the unoscillated neutrino beam flux energy spectrum as described in Chapter 9.  7.1  CC Inclusive Selection  The inclusive CC selection uses the results of the ND280 global reconstruction algorithms summarized in the standard ND280 event summary format. The global reconstruction algorithm identifies CC-inclusive interactions using tracks matched between the FGD and TPC subdetectors, with a further requirement that the TPC charge deposit measurements are compatible with the passage of a muon. The details and performance of the reconstruction algorithms and the “oaAnalysis” event summary format are summarized in Chapter 5 and in a T2K technical note [56].  7.1.1  Data Format  Near detector data was processed with version v9r9p1 of the ND280 software package. Near detector data is organized into a ROOT n-tuple, with each entry containing summary data and reconstruction results for a specific proton beam spill or alternative triggered data acquisition. The data relevant to this analysis are summarized versions of the TPC and FGDTPC matched tracks produced by the global reconstruction algorithm, and saved on a event-by-event basis. The track summary information includes the track’s reconstructed initial position, time, direction, momentum, charge and the ND280 subdetectors it passes through. Also used in this analysis  79  7.1. CC Inclusive Selection are summaries of FGD hit clusters that are used here as indicators of energy deposit in the FGD at a certain time. Files in the oaAnalysis format are accessed and analyzed either with a standard root macro or with version v3r0 of the CC-inclusive selection and analysis package developed by ND280 collaborators.  7.1.2  Data Samples  Simulated data was used to test the event selection and predict neutrino interaction rates and kinematic distributions. Neutrino beam flux predictions provided by the T2K beam group [58] were combined with a neutrino interaction simulation to produce the simulated sample, after which a Geant 4 based software package (nd280mc) was used to propagate particle trajectories and simulate energy deposit. A simulation of the ND280 subdetector response is provided by an additional software package (elecSim) and the resulting simulated data is saved in exactly the same format as real data. Real and simulated data are processed in the same way by the ND280 calibration, reconstruction and oaAnalysis summary packages (oaCalib, oaRecon, oaAnalysis). There were two different neutrino interaction simulations used in T2K near detector studies, NEUT [59] and GENIE [60], although only the results of the NEUT simulation were used in this analysis. An additional data sample was produced to account for neutrino interactions occurring outside the ND280 magnet, predominantly in the sand surrounding the ND280 pit. “Sand muons” produced by these interactions enter the ND280 detectors and are a source of external background. They were simulated using a Geant4 simulation of the ND280 pit geometry and the resulting muon trajectories recorded [61]. These simulated interactions are then processed by the event selection in the same manner as the other data samples so the effect of external background can be accounted for directly in the simulated event distributions. The size of the simulated sand muon sample is relatively small compared to other simulated data, but as the contribution of sand muons to the selected CC-inclusive sample is also small the effect of the statistical error in flux measurements is minor. This analysis uses oaAnalysis files produced in ND280 MC production 4-C, specifically “Magnet” type files where the neutrino interactions were simulated throughout the volume containing the near detector magnet. Different simulated datasets were produced for Run 1 and Run 2 proton beam configurations, and the Run 2 sample was further divided to correspond to periods where the P0D detector contained water or was empty [62].  80  7.1. CC Inclusive Selection  7.1.3  Neutrino Beam Flux and Interaction Models  The simulated data is generated with reference to a neutrino beam flux model provided by the TK2 beam group as described in T2K-TN-99 [39]. This model predicts the neutrino flux spectrum incident on the ND280 detectors during the Run 1 and Run 2 beam runs. MC production 4C was generated with beam flux version v11a though selected interactions were reweighted so that the simulated sample used in this analysis is defined according to the recommended v11b version of the beam flux model. The predicted neutrino beam flux incident on the ND280 detectors is shown in Figure 2.10. The neutrino interaction generator used to define the simulated data sample is NEUT version 5.1.1. The NEUT package is defined in greater detail in Chapter 2 and Chapter 5. Details of the simulated neutrino interaction such as neutrino energy and interaction mode are saved in the event summary format, allowing the reweighting of events according to modifications in the neutrino interaction model in analyses.  7.1.4  Beam Summary and Data Quality Flags  Data quality flags included in each oaAnalysis file entry summarize the near detector state and quality of the proton beam spill data. The ND280 offaxis global flag indicates whether all ND280 subdetectors were well calibrated and recording data during the beam spill under consideration and that there was no error detected by the data acquisition or slow control monitoring systems. The criteria used to determine that the data is of good quality is summarized in T2K-TN-50 [63]. A separate flag indicates whether the proton beam spill occurred and was usable. The inclusive CC selection requires both of these flags to be in the “on” state for a real data candidate event, though they are not required when analyzing simulated data. Table 7.1 summarizes the measured number of protons on target (POT) as reported by proton beam monitors for real candidate beam spills, as well as the number of POT equivalent to the size of the simulated data samples.  7.1.5  Global Reconstruction Tracks and Variables  Particle tracks identified by reconstruction algorithms are stored in an array of reconstruction objects. Each track object summarizes the initial position, direction and momentum inferred by the reconstruction, recording subdetectors the track passed through and constituent objects that contain  81  7.1. CC Inclusive Selection Data Sample Run 1 Run 2 MC Run 1 MC Run 2 Sand Muon  ND280 Production 4D 4D NEUT Magnet 4C NEUT Magnet 4C See reference [61]  POT 2.939 ×1019 7.857 ×1019 54.5 ×1019 110.5 ×1019 7.0 ×1019  Table 7.1: Summary of real and simulated data samples, with run 1 and run 2 contributions separated.  subdetector specific information. Only tracks reconstructed as originating within the tracker are used in the inclusive CC analysis. Several errors were discovered in the default “global” reconstruction algorithm used in production 4-C, including incorrect track kinematic variable estimates that could introduce systematic error into the results of the selection. The error rate varied with the recontructed track angle, with higherangle tracks being misreconstructed at a rate up to roughly 5%. These errors are avoided by requiring the event selection to use more accurate track position, time and reconstructed momentum estimates constructed from sub-detector specific reconstruction results. As described in Chapter 5 reconstruction algorithms are first applied to data produced by a specific subdetector to create “locally” reconstructed objects specific to that detector. These local results are then combined by the global reconstruction algorithm to produce the final result. This analysis uses tracks identified by the global reconstruction algorithm, but the position and direction of the track at its initial position is provided by the corresponding local FGD1 reconstruction. The track momentum is estimated using the momentum measured by the local TPC reconstruction result and adding a correction to account for the energy loss of the particle as it passed through FGD1 before it could be reconstructed in the TPC. All measured track parameters such as momentum and direction used in this analysis are these subdetector-derived “local” variables.  7.1.6  Beam Bunch Time Association  The proton beam spill is divided into several separate sections (bunches) which are approximately 580 ns apart. Run 1 beam spills contained six bunches while run 2 had eight spills. A reconstructed track is associated with a specific beam bunch if its time estimate is consistent with the bunchs time 82  7.1. CC Inclusive Selection distribution. The bunch time structure is inferred by recording the estimated time for globally reconstructed tracks over many events in the sample. The resulting time distribution clearly shows the separate bunches as seen in Figure 7.1. A Gaussian fit applied to each peak in the distribution provides the mean time and a constant σ of 15 ns was used for all bunches in real and simulated data. Changes to the configuration of ND280 subdetectors, the proton beam accelerator and the data acquisition system resulted in different beam bunch time estimates over the total data taking period. The beam bunch times were stable for all of Run 1, while they changed part of the way through Run 2 leading to different beam running periods when considering bunch times. For simplicity the first part of Run 2 with stable beam bunch times is referred to as Run 2-A and covers run numbers before 7000, while the latter period Run 2-B includes all run numbers greater than 7000. The results of the bunch fitting procedure performed for each separate period are summarized in Table 7.2. A similar beam bunch time structure was included in simulated data and is also shown.  Figure 7.1: Beam bunch time distribution for Run 2-B period.  A track is associated with a bunch if the tracks time estimate is within four standard deviations of the bunch mean time defined for that datataking period. The error in the reconstructed track time estimate is not considered. The inclusive CC selection is applied separately to the set of tracks associated with each bunch, reducing the effects of event pile-up and 83  7.1. CC Inclusive Selection  Bunch Bunch Bunch Bunch Bunch Bunch Bunch Bunch  0 1 2 3 4 5 6 7  MC (ns) 2750.2 3332.0 3914.7 4497.0 5078.4 5659.7 6243.4 6824.2  Run 1 (ns) 2839.7 3423.5 4005.4 4588.6 5172.2 5745.6 -  Run 2-A (ns) 2853.95 3444.15 4030.41 4620.34 5180.28 5770.12 6343.77 6924.67  Run 2-B (ns) 3019.11 3597.74 4180.73 4763.93 5346.49 5927.83 6508.5 7093.56  Table 7.2: Real and simulated beam bunch fitted mean times.  external background. Tracks not associated with a beam bunch are not included in any event selection, and are generally due to external interactions or electrons decaying from muons or pions produced in the neutrino interaction.  7.1.7  Track Topology  As described in the reconstruction section, tracks passing through the TPC are first reconstructed using TPC information only. The resulting TPC tracks are then matched to hits in the FGDs by a Kalman-filter based algorithm and if successful create a matched FGD-TPC reconstructed track. Reconstructed FGD-TPC tracks can include any number of adjacent subdetectors and provide a natural way to combine momentum and charge deposition measurements from the lower-level TPC-only tracks. The CC-inclusive selection uses FGD-TPC matched tracks, as well as TPC-only tracks. Tracks reconstructed only within FGD2 are not used in any way.  7.1.8  Steps in Muon-Like Track Selection  As mentioned the inclusive CC selection is performed on a set of reconstructed FGD-TPC matched tracks associated with a particular beam bunch, and repeated separately for all bunches present in a beam trigger. The selection contains the following steps: • Identify tracks that possess a valid reconstructed momentum estimate, these for the initial set of candidate muon-like tracks. • Require the candidate tracks possess a TPC segment with greater than 18 TPC hits. 84  7.1. CC Inclusive Selection • Require the candidate tracks have a reconstructed position starting in the FGD1 fiducial volume. • Require the candidate tracks have negative charge. • Of the candidate tracks, select the one with the highest reconstructed momentum as the muon candidate. If no muon candidate is found the bunch is rejected. • Require that the muon candidate tracks reconstructed starting position is upstream of the final position with respect to the neutrino beam direction. • Remove external background by vetoing events where the highest momentum non-muon candidate track originates more than 150 mm upstream of the muon candidate tracks start position. • Require that the muon track candidate measured charge deposit in the TPC is consistent with that expected from muons of the same momentum. FGD1 Fiducial Volume Muon-like track candidates are required to start in the FGD1 fiducial volume, which is defined in the ND280 coordinate system as follows: • -874.51 < x < 874.51, excludes the five outermost vertical bars on the sides of FGD1 • -819.51 < y < 929.51, excludes the five outermost horizontal bars on the sides of FGD1 • 136.875 < z < 446.955, excludes the two most upstream layers in FGD1 in order to remove external background Tracks starting in FGD2 are not used in any way in this analysis, in order to avoid the need to consider cross section differences between carbon and oxygen and associated reconstruction effects. Likelihood-Based Particle Identification Reconstructed TPC tracks possess a charge-deposit measurement using associated hits described in [64]. The track path length and momentum are 85  7.1. CC Inclusive Selection used to predict the expected charge deposit measurement for different types of particles, as shown in Figure 5.3. The charge estimate uses a truncated measurement that sorts all TPCs by charge and sums the lowest 70%, resulting in a more robust estimate less sensitive to fluctuations. The expected and measured charge deposits are used to produce pull measurements as defined in Equation 7.1. The simulated muon pulls are shown with the pulls measured in the CC-inclusive selection in Figure 7.2, while the electron pulls are shown in Figure 7.3. The CC-inclusive selection uses these pull values in a maximum likelihood function to determine the most likely particle hypotheses. Particle hypotheses included in the likelihood calculation are muons, electrons, protons and pions. The equation used to calculate the likelihood is shown in equation 7.2, with the muon selection cut requiring Lµ > 0.05. Additionally for tracks with momentum below 500 MeV/c, the combined muon and pion likelihood must be greater than 0.8. This second cut selects minimum ionizing tracks and helps remove electrons from the selection. P ulli =  dE/dxmeasured − dE/dxexpected,i σdE/dxmeasured −dE/dxexpected,i  (7.1)  P ulli j P ullj  (7.2)  Li =  7.1.9  External Background Pileup Correction  The efficiency of the simulated data sample is corrected to account for the larger amount of external background seen in real data, primarily due to sand muons. External background can affect the CC-inclusive selection by producing tracks in TPC1 that cause the external background cut to veto the selection of any potential neutrino interactions starting in FGD1. This introduces a potentially large systematic effect if not corrected. In order to account for the additional external background observed in real data, the number of TPC1 tracks per bunch was separately measured in the real and simulated data samples (including the simulated sand muon sample). The discrepancy between the number of reconstructed TPC1 tracks in real and simulated data was found to be 0.43% for the Run 1 beam running period and 0.78% for Run 2 [65]. The difference between the two runs is likely due to changes in the average beam power over the run periods as compared to the value that was used in the simulation. These data/MC discrepancies were used to correct the number of selected interactions in the simulated CC-inclusive sample for the corresponding run period, as well 86  7.2. CCQE and CCnQE Enhanced Samples  Figure 7.2: Muon pulls in real and simulated CC-inclusive sample. Contributions from different interaction modes are shown in the case of the simulated data sample using MC truth information.  as the muon momentum and angular distributions. All values presented in this analysis include this external background pileup correction. The error in the correction factor is treated as a systematic effect that is described in Chapter 8.  7.2  CCQE and CCnQE Enhanced Samples  Two additional cuts were applied to the CC-inclusive event selection to divide it into CCQE and CCnQE enhanced samples. The first is a cut on the number of FGD-TPC matched tracks reconstructed in time with the selected CC-inclusive interaction, while the second is a Michel electron cut. The division of the CC-inclusive selection into CCQE/CCnQE samples is illustrated diagrammatically in Figure 7.4. This division was made so that analyses making use of the event selection have additional sensitivity to neutrino interaction parameters such as MA , or to constrain the fraction of CC-inclusive interactions that are CCQE as a function of energy. CCQE interactions have been used in previous experiments to estimate neutrino energy from the muon’s reconstructed track, as described in Chapter 1. Retaining the CCnQE interactions makes full use of the CC-inclusive sample 87  7.2. CCQE and CCnQE Enhanced Samples  Figure 7.3: Electron pulls in real and simulated CC-inclusive sample.  and allows them to further constrain parameters used in neutrino interaction models.  Figure 7.4: Division of CC-inclusive selection into CCQE/CCnQE samples.  Ideally CCQE interactions are identified by requiring both a muon and proton track originating from the same vertex, while other processes such as resonant pion production or deep inelastic scattering produce more than two tracks. In practice the protons produced by CCQE interactions often don’t travel far enough to produce a reconstructed track in the FGD or TPC. Most CCQE events produce only a reconstructed track for the muon, while multi-track events are predominantly CCnQE as pions often travel far enough to produce a reconstructed track. Consequently CCQE events are 88  7.3. Selection Results in Real and Simulated Data selected using a track multiplicity cut that requires a candidate event have exactly one FGD1-TPC matched track in the beam bunch time window. If there are more than one matched tracks the event is classified as CCnQE. Pions produced in resonant pion production processes or deep inelastic scattering have some chance to stop within FGD1. In this case the presence of the pion can be inferred by detecting the Michel electron produced at the end of the decay chain. A Michel electron cut uses the presence of delayed FGD1 hits to identify Michel electrons, identifying the event as CCnQE. In order to remove low energy background (most likely due to late photons and neutrons produced by neutrino interactions outside the FGD) the delayed hit cluster’s total charge must exceed 200 pixel avalanches in order to classify the event as CCnQE. Additionally the cut ignores delayed hits that occur during a beam bunch time window after the candidate event in order to prevent identifying beam correlated activity as a Michel electron.  7.3  Selection Results in Real and Simulated Data  The results of the event selection applied to the Run 1 and Run 2 data samples are summarized in Table 7.3. Figures 7.5, 7.6 and 7.7 show the measured muon momentum and angular distributions for CC-inclusive events, where θ is the angle between the muon’s initial direction and the ND280 co-ordinate system’s z-axis. The z-axis closely corresponds to the neutrino beam direction at the near detector. The simulated and observed distributions are qualitatively similar, providing some assurance that the simulation can be used to successfully fit for the neutrino energy spectrum. The predicted contribution from different neutrino interaction modes to the muon momentum distribution of the CCQE enhanced sample is shown in Figure 7.8, and the CCnQE sample’s distribution is shown in Figure 7.9. As can be seen the CCQE sample is dominated by true CCQE interactions, especially around peak of the momentum distribution that roughly corresponds to the peak of the neutrino energy spectrum. It is apparent that the major backgrounds in the CCQE sample are resonant pion interactions and external interactions. The CCnQE sample’s momentum distribution is not as sharply peaked due to the larger number of final state particles decreasing the correlation between the lepton momentum and neutrino energy, as well as due to the energy dependence of the CCnQE cross sections.  89  7.4. Selected Muon Track Binning Data Sample Run 1 Run 2 Real Data Total MC Run 1 MC Run 2 Sand Muons MC Total  CC-inclusive 1201 3283 4484 1261.21 3433.23 49.35 4743.79  CCQE 619 1733 2352 677.09 1854.55 38.56 2570.20  CCnQE 582 1550 2132 584.116 1578.68 10.80 2173.59  Table 7.3: Number of selected CC-inclusive, CCQE and CCnQE events in real and simulated data with contributions separated by run period and data type.  7.4  Selected Muon Track Binning  Selected CC-inclusive muon tracks are binned by reconstructed momentum and angle to produce a pµ − θµ distribution. This distribution can be fit to constrain neutrino interaction and flux parameters. The following bin boundaries were used to define the distribution: • Momentum: 0-400 MeV/c, 400-500 MeV/c, 500-700 MeV/c, 700-900 MeV/c, >900 MeV/c • Angle: -1 < cosθ < 0.84, 0.84 < cosθ < 0.9, 0.9 < cosθ < 0.94, 0.94 < cosθ < 1 There are 5 momentum bins and 4 angular bin for each of the CCQE and CCnQE samples, resulting in a total of 40 bins. Each pµ − θµ bin is assigned an index number which are summarized in Table 7.4. These bin boundaries were chosen to have a non-zero number of events in each bin for real data and to roughly correspond in terms of muon momentum to the binning provided by the beam group in the their flux parametrization. These index numbers run from 0 to 19 for the CCQE sample and 20 to 39 for the CCnQE sample. The pµ − θµ distribution predicted by the default simulation is shown in Figure 7.10 with the observed data superimposed for comparison. Table 7.5 summarizes the number of entries in each pµ − θµ bin for the real and simulated data samples.  90  Number of Entries / (50 MeV/c)  7.4. Selected Muon Track Binning  250 200 150 100 50 0 0  500 1000 1500 2000 2500 3000 3500 4000 4500 5000  Selected Muon Momentum (MeV/c)  Figure 7.5: Selected muon momentum distribution for real and default simulated data. Simulated data has been normalized to match the number of protons on target in the real data sample.  91  7.4. Selected Muon Track Binning  Number of Entries / (0.02)  800 700 600 500 400 300 200 100 0 -1  -0.8  -0.6 -0.4  -0.2  0  0.2  0.4  0.6  0.8  1  Selected Muon Cosθ  Figure 7.6: Selected muon cos θ distribution for real and default simulated data. Several backwards going track are included in the selection due to a rare reconstruction error. Simulated data has been normalized to match the number of protons on target in the real data sample.  92  7.4. Selected Muon Track Binning  Figure 7.7: Selected muon momentum vs. cos θ distribution for real and default simulated data. The number of entries in shown in the case of simulated data using solid colour, while in the real data case boxes are used. Simulated data has been normalized to match the number of protons on target in the real data sample.  93  7.4. Selected Muon Track Binning  Figure 7.8: Selected CCQE enhanced sample muon momentum distribution for real (points) and simulated data (line). Simulated data has been normalized to match the number of protons on target in the real data sample.  94  7.4. Selected Muon Track Binning  Figure 7.9: Selected CCnQE enhanced sample muon momentum distribution for real (points) and simulated data (line). Simulated data has been normalized to match the number of protons on target in the real data sample.  95  7.4. Selected Muon Track Binning  CCQE Bins 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  CCnQE Bins 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  pµ Range 0-400 MeV/c 0-400 MeV/c 0-400 MeV/c 0-400 MeV/c 400-500 MeV/c 400-500 MeV/c 400-500 MeV/c 400-500 MeV/c 500-700 MeV/c 500-700 MeV/c 500-700 MeV/c 500-700 MeV/c 700-900 MeV/c 700-900 MeV/c 700-900 MeV/c 700-900 MeV/c >900 MeV/c >900 MeV/c >900 MeV/c >900 MeV/c  cosθµ Range -1 to 0.84 0.84 to 0.9 0.9 to 0.94 0.94 to 1 -1 to 0.84 0.84 to 0.9 0.9 to 0.94 0.94 to 1 -1 to 0.84 0.84 to 0.9 0.9 to 0.94 0.94 to 1 -1 to 0.84 0.84 to 0.9 0.9 to 0.94 0.94 to 1 -1 to 0.84 0.84 to 0.9 0.9 to 0.94 0.94 to 1  Table 7.4: pµ − θµ bin index definitions.  96  7.4. Selected Muon Track Binning  Bin 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  Default MC CCQE 334.85 44.40 27.61 31.77 265.20 45.19 29.33 33.39 342.80 105.92 73.22 92.26 114.31 62.92 53.34 72.09 96.24 90.71 107.42 547.23  Real Data CCQE 314 41 27 36 244 35 28 25 315 100 59 81 100 40 38 72 98 68 95 536  Bin 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  Default MC CCnQE 219.37 33.23 26.03 32.78 102.73 15.60 12.99 20.05 158.11 33.07 25.29 40.02 102.88 30.89 20.02 35.86 212.92 159.12 187.97 704.67  Real Data CCnQE 215 31 18 36 116 22 16 13 173 40 19 40 102 33 20 31 196 155 171 685  Table 7.5: Number of entries in each pµ − θµ bin in real and simulated data.  97  Number of Entries / (pµ-θµ Bin)  7.4. Selected Muon Track Binning  700 600 500 400 300 200 100 0 0  5  10  15  20  25  p µ-θµ Bin Number  30  35  40  Figure 7.10: Default MC p − θ distribution (blue) with observed data also shown.  98  Chapter 8  Detector Systematic Studies The data selection described in Chapter 7 is used as the basis for studies of cross-sections and energy spectra measurements. However systematic errors in the simulated data potentially introduce biases and incorrect error estimates. The systematic effects relevant to this analysis were identified, measured and propagated through the event selection to quantify their impact on the resulting pµ − θµ distribution. The errors in the final pµ − θµ distribution due to each systematic effect were summarized by a fractional covariance matrix, and the total systematic error was the sum of these matrices.  8.1  Identification of Systematic Effects  The systematic errors introduced by the detector model to the event selection are summarized in Table 8.1. The cut based event selection allowed a clear identification of relevant systematic effects and it is assumed that the list presented here covers all major sources of error. The size of the systematic errors were measured by various T2K collaborators and the appropriate references are also included in the table. Most of the systematic effects are related to track reconstruction efficiency and momentum measurements, but there are also errors introduced by detector mass and unsimulated external background.  99  8.1. Identification of Systematic Effects  Systematic Effect TPC Hit Efficiency in the Track Quality Cut TPC Reconstruction Efficiency Broken FGD-TPC Tracks Rate TPC Pull Widths in the PID Cut B-Field Strength  Reference T2K-TN-079  Uncertainty 0.07%  T2K-TN-075 T2K-TN-075 T2K-TN-078 T2K-TN-081  B-Field Distortion Correction TPC Momentum Resolution  T2K-TN-061 T2K-TN-095  FGD1 Fiducial Mass Charge Misidentification Rate Out of Fiducial Volume Rate FGD-TPC Matching Efficiency FGD-TPC Track Multiplicity Rate Michel Tagging Efficiency  T2K-TN-091 T2K-TN-048 T2K-TN-098 T2K-TN-075 T2K-TN-093  0.5% 0.6% 1.5% δBx = 0.5% δBT = 1.3G 100% 2.8 ± 0.6 × 10−5 (c/M eV ) 0.7% Varied, < 0.3% 20% Varied, < 1.0% 0.56%  T2K-TN-104  0.5%  Michel Electron External Background Rate Sand Muons External Background Rate Cosmic Ray External Background Fake Rate Sand Muon Pileup Rate  T2K-TN-104  0.112%  T2K-TN-077  15%  T2K-TN-112  < 0.03%  T2K-TN-093  Varied, ∼ 0.2%  Table 8.1: Near detector event selection systematic effects and the measured uncertainties, documented in T2K technical notes and Chapter 8  100  8.2. Systematic Error Propagation  8.2  Systematic Error Propagation  After estimating the size of a systematic error, its effect was propagated through the event selection in order to measure the change in the pµ − θµ distribution. Systematic errors were propagated through three different methods; the first method directly simulates the effect of a systematic by randomly modifying or dropping elements used in the event selection, the second assigns a weight to an event based on the magnitude of the systematic error and the number of affected elements in the selection, and the third corresponds to some fraction of events moved between specific pµ − θµ bins. The effect of each propagated systematic on the pµ − θµ bin distribution is summarized by a 40×40 element covariance matrix. This matrix takes into account correlations between the numbers of entries in different bins due to the systematic error.  8.2.1  Systematic Error Covariance Matrices  The systematic covariance matrices are defined so that element (i, j) contains the fractional covariance between pµ −θµ bin i and bin j. The fractional covariance vij is defined as vij =  sign(corr(i, j))σi σj , Ni Nj  (8.1)  where σi is the square root of the variance measured in pµ − θµ bin i as the parameter associated with a systematic effect is varied, and corr(i, j) is the correlation between the two bins. The bin indices i and j run from 0 to 39, and the matrix is symmetric under their exchange. It was assumed that the systematic uncertainty in each of the simulation model parameters was small and followed a Gaussian distribution centred on the default value. This allows a linear approximation in the calculation of the fractional covariance as shown by vij =  ∆i ∆j , Ni Nj  (8.2)  where ∆i is the difference in the number of entries in pµ − θµ bin i when the simulation model parameter λ is varied one standard deviation away from the default value, as given by ∆i = Ni (λ0 ± 1σλ ) − Ni (λ0 ) .  (8.3)  101  8.2. Systematic Error Propagation The resulting covariance does not depend on the sign of the variation in the systematic parameter. Under this definition of the covariance matrix the correlations between the CCQE and CCnQE enhanced samples are organized as follows: • Rows 0-19 and columns 0-19: CCQE sample pµ − θµ bins and their correlations. • Rows 20-39 and columns 20-39: CCnQE sample pµ − θµ bins and their correlations. • Rows 0-19 and columns 20-39, rows 20-39 and columns 0-19: Correlations between CCQE and CCnQE sample pµ − θµ bins.  8.2.2  Systematic Propagation through Simulation  The most direct way to propagate a systematic error is to simulate its effect on the pµ − θµ distribution resulting from the event selection. Ideally the systematic effect is associated with a parameter that can be directly varied in simulated data, in which case it is straightforward to vary the parameter according to its probability distribution and measure the resulting variance in the number of bin entries as in Equation 8.2. When the systematic parameter is assumed to be small and its probability distribution Gaussian it is sufficient to only vary the systematic parameter by one standard deviation and estimate the number of bin entries in that specific case, as expressed in Eq. 8.3. An example of this method of evaluation would be a systematic error in track reconstruction efficiency where the simulated efficiency can be directly adjusted when producing a new dataset and its associated pµ − θµ distribution. In practice the reconstruction efficiency can be effectively lowered in simulation by randomly dropping tracks at a rate equal to the size of the efficiency systematic error. Although it is not possible to add additional tracks through this method, the resulting uncertainty is assumed to be symmetric. Another practical consideration is that it would take too much time and computing resources to regenerate a simulated dataset for each variation in the value of the systematic parameter. Instead the event selection is rerun on the existing simulated dataset with the effective variation of the systematic parameter. In the example of the track reconstruction efficiency this would involve rerunning the event selection but ignoring the presence of some fraction of the tracks when applying selection cuts. Using the same data also reduces the statistical error due to random fluctuations in the dataset to the 102  8.2. Systematic Error Propagation estimated variation in the number of bin entries under the systematic effect. The event selection can be rerun on the same simulated dataset many times in order to further reduce statistical error in the estimated change in number of bin entries. In this case the contribution of each selected event to the final pµ − θµ distribution is reduced by a factor corresponding to the total number of times the event selection is applied. This approach is limited in practice by computing time and the size of the simulated data sample.  8.2.3  Systematic Propagation through Reweighting  Simulating a systematic effect is conceptually simple but limited by the amount of time required to rerun the event selection. In practice significant statistical error can be introduced into the systematic covariance matrix if the size of the systematic error is small. In this case estimating ∆i is limited by the precision in the number of bin entries Ni (λ0 ± 1σλ ) predicted under a one standard deviation variation in the systematic parameter λ. An event weighting method was devised to circumvent this problem by calculating how much each simulated event would change the pµ − θµ distribution in the limit of an infinite number of throws of the event selection where the systematic effect is directly simulated. The weights are estimated by identifying the set of elements used in the selection and removing each one in turn from the event, then weighting the selected pµ − θµ bins in each pass of the event selection by the probability of the element being dropped under the systematic effect. As a concrete example, if the systematic effect is the TPC track reconstruction efficiency the set of elements to drop includes all TPC tracks identified in each specific event. The event weight wk under a variation in the systematic parameter calculated for the pµ − θµ bin k originally selected by the default version of the event selection is shown by wk = 1 − N pdrop + nk pdrop ,  (8.4)  wl = nl pdrop ,  (8.5)  where N is the number of elements relevant to the systematic effect in the event, pdrop is the probability of the element being removed from the event selection under the variation in the systematic effect, and nk is the number of times the originally selected pµ − θµ bin k was selected during the multiple passes of the event selection where the elements were dropped sequentially. Similarly, the expression for the weight in other bins due to the systematic effect is shown by  103  8.2. Systematic Error Propagation where nl is the number of times the pµ − θµ bin l was selected during the multiple passes of the event selection where the elements were dropped sequentially. Applying this weighting method to each event in the simulated dataset produces an accurate estimate for the number of entries in each bin under a variation in the systematic parameter assuming a Gaussian probability distribution. In practice this method can only be used for systematic effects in measurement efficiencies, such as track reconstruction or hit efficiency where the elements in the event can be removed and the event selection repeated. The event weighting method is implemented as follows, in the specific case where the set of elements are reconstructed tracks and the systematic error is in the reconstruction efficiency. • Identify the set of entities in each event affected by the systematic effect. In the case of track reconstruction, the set of entities relevant to the systematic error are tracks reconstructed in the event. • Define N as the number of affected tracks and pdrop as the probability of dropping the track due to a change in reconstruction efficiency under a variation in the systematic error. • Rerun the event selection on the same event N + 1 times. • For each of the first N iterations of the event selection ignore one of the tracks in the set when applying selection cuts. Weight the modified event’s contribution to the pµ − θµ distribution by a factor of pdrop . • For the last iteration run the event selection with no tracks dropped and weight the contribution to the pµ − θµ distribution by a factor of 1 − N × pdrop In the case where the systematic error does not affect the results of the event selection each of the N + 1 iterations will contribute to the same pµ − θµ bin with weights that sum to 1, reproducing the default selection result. However if dropping entities changes the selected pµ − θµ bin, the weight assigned to each bin corresponds to the average that would be obtained after rerunning the event selection a very large number of times with the systematic effect directly simulated. In principle all possible cases where multiple elements are dropped simultaneously should be considered, but as the factor p is generally small higher order contributions would be insignificant.  104  8.3. Systematic Error Covariance Matrices  8.2.4  Systematic Propagation through Bin Migration  In some cases the systematic effect can be associated with a specific change in the number of entries in a pµ − θµ bin or with a migration of some fraction of events from one set of bins to another. In this case the pµ −θµ distribution predicted by the default simulation can be used to calculate the new pµ − θµ distribution due to the variation in the systematic parameter. For example if a systematic effect results in a misidentification of some fraction of CCQE events as CCnQE, the change in the pµ −θµ due to the systematic error can be measured by moving a fraction of the CCQE bin entries to the corresponding CCnQE bins.  8.3  Systematic Error Covariance Matrices  The systematic error was propagated through the event selection for each relevant parameter to produce the following covariance matrices. All of these matrices were produced using the production 4-C simulated dataset with the standard event selection.  8.3.1  TPC Tracking Efficiency  TPC tracks are used by the ND280 reconstruction to produce the matched FGD-TPC tracks used in the CC-inclusive event selection as described in Chapters 5 and 7. The systematic error in tracking efficiency for each subdetector was measured using a cosmic control sample, where thoroughgoing muons tracks were selected using a different set of subdetectors. For example, efficiency measurements in TPC1 used events with a track in the P0D and TPC2, while TPC2 measurements used events with a track in TPC1 and TPC3. In each of these cases, the efficiency for reconstructing a track in the specified TPC was calculated for these selected events and compared between real and simulated data. The data/MC discrepancy was found to be small with no obvious angular dependence, resulting in an overall 0.5% systematic error in the efficiency. An example of the TPC efficiency systematic measurement is shown in Figure 8.1. The track selection used in this study results in an muon track angular distribution that is different from selected CC-inclusive interactions, but the large number of cosmic events allows for the efficiency to be measured independently in the different angular ranges used in the CC-inclusive event selection. This systematic error was propagated through the event selection using the reweighting method to produce a new pµ − θµ distribution, and the 105  8.3. Systematic Error Covariance Matrices  Figure 8.1: TPC2 tracking efficiency systematic measurement, showing the rate at which selected tracks in the angular range 0.94 < cos θ < 1.0 are reconstructed in TPC2 for real and simulated data, taken from T2K Technical Note 75 [66].  difference between the new and original pµ − θµ distributions ∆i is shown in Figure 8.2. The covariance matrix corresponding to this set of ∆i shown in Figure 8.3. There are off-diagonal correlations between the number of entries in pµ − θµ bins due to two effects, the first being the loss of 0.5% of TPC tracks should to first order lead to a reduction in the number of selected events in every bin in the sample resulting in correlations. Secondly there is some probability for CCnQE events to be identified as CCQE when one of the TPC tracks is dropped and only a single selected track remains in the event. This transfers a significant number of events from CCnQE bins to the corresponding CCQE bin and ensures the number of entries in the pµ − θµ bins are correlated under variations of the systematic effect.  8.3.2  Track Charge Misidentification  There is some probability for the TPC standalone reconstruction algorithm to misidentify the charge of a track, which is estimated from its curvature due to magnetic field. This charge misidentification rate was measured using globally reconstructed tracks with segments in all three TPCs, allowing comparison of the charge estimates produced by the TPC standalone re106  8.3. Systematic Error Covariance Matrices  Figure 8.2: Fractional change in the number of entries in each pµ − θµ bin due to the TPC tracking efficiency systematic effect.  construction. Different charge estimates between segments must be due to reconstruction charge misidentification, and a probabilistic method was used to infer the misidentification rate based on how often all three segments had the same charge estimate. Similar studies with two TPC-segment samples were performed as a cross-check and found to agree. The charge misidentification was observed to vary with momentum and was included in the systematic effect measurement. The measured systematic effect in charge misidentification was propagated through the event selection using a simulation where track charge would be randomly flipped at a specified rate. To improve statistical accuracy 500 passes of the event selection were performed. In each pass the track flipping probability was randomly drawn from a Gaussian probability distribution with width equal to the measured systematic error in the relevant momentum range. The resulting covariance matrix is shown in Figure 8.4.  8.3.3  Track Quality Cut  The track quality cut removes tracks from the event selection with less than 18 TPC hits. The systematic effect introduced by the cut is assumed to be 107  8.3. Systematic Error Covariance Matrices  ×10-3 0.06  40  0.05  30 25  0.04  20  0.03  15  vij  p µ-θµ Bin Index  35  0.02  10 0.01  5 0 0  5  10  15  20  25  p µ-θµ Bin Index  30  35  40  Figure 8.3: TPC tracking efficiency fractional covariance matrix. Offdiagonal correlations are due to common changes in the number of selected events and event migration due to the systematic effect.  mainly due to the error in the simulated hit efficiency [67]. The hit efficiency is inferred by comparing between real and simulated data the number of hits in the TPC track segment closest to the selected muon-like track’s front position. The sample of tracks used in this comparison are selected by CCinclusive selection except with track quality cut removed, and with greater than 62 hits. The simulated distribution can be modified assuming different values of the hit efficiency, with the further assumption that hits at the edge of the Micromegas modules are a factor of 10 more likely to be dropped. These simulated distributions were fit to the observed data and resulted in a measured hit efficiency systematic of 0.07 ± 0.01%, with real, simulated and fitted distributions shown in Figure 8.5. This effect was propagated by direct simulation where tracks with 18 hits are randomly dropped at a rate equal to the systematic error in the hit efficiency. The resulting covariance matrix is shown in Figure 8.6. The relatively small number of selected tracks affected by this systematic results in relatively few entries in the covariance matrix, leading to the white space correponding to zero covariance observed  108  8.3. Systematic Error Covariance Matrices  ×10-3  40  0.12 30  0.1  25  0.08  20  0.06  vij  p µ-θµ Bin Index  35  15 0.04 10 0.02  5 0 0  5  10  15  20  25  p µ-θµ Bin Index  30  35  40  0  Figure 8.4: Track charge misidentification covariance matrix. The systematic effect is largest for straight tracks with high momentum, as seen by the large uncertainties in bins 20 and 40 and the correlation between them.  in large portions of the matrix.  8.3.4  TPC Particle Identification  The TPC particle identification method uses pull values derived from charge deposit measurements to calculate a likelihood for different particle hypotheses. Discrepancies in TPC gas or electronics gain can cause systematic effects in the truncated charge measurement and the resulting pull values. The size of the systematic effect was directly measured using a high purity muon sample in real and simulated data, where it was found that the MC pull distribution is slightly narrower than the real data case with minor variations between TPCs. The discrepancies between the means of the pull distributions are negligible. Consequently pull values considered in the simulated event selection were smeared out using a Gaussian distribution so that the real and simulated widths would agree, and the results were used in the default simulated data prediction. The systematic error was taken to be equal to the uncertainty of the 109  8.3. Systematic Error Covariance Matrices  Figure 8.5: TPC hit efficiency in real and simulated data with results of hit efficiency systematic fit also shown, taken from T2K Technical Note 79 [67].  difference in the width of the pull distributions between real and simulated data. The error was propagated by direct simulation, where the smearing applied to pull measurements was smeared by an additional factor corresponding to the systematic error. These smearing factors are different for different TPCs and simulated running periods but are generally on the order of 1.5%. The resulting change in the number of selected events was used produce the covariance matrix shown in Figure 8.7.  8.3.5  Magnetic Field Distortions  The simulated data takes into account distortions in the TPC’s magnetic field through the use of a magnetic field map. The validity of the field map is tested using TPC calibration targets, which are aluminium dots placed at precise positions on the interior of the TPC that can be stimulated by an in situ laser to emit photoelectrons. The Micromegas measurement of the calibration target positions can be used to determine if the field map correctly describes the magnetic field. Calibration measurements demonstrate that the simulated field map does not completely reproduce the magnetic field distortion effects seen in real data and an additional empirical correction is required. 110  40  ×10-9  35  0.3  30  0.25  25  0.2  vij  p µ-θµ Bin Index  8.3. Systematic Error Covariance Matrices  20 0.15 15 0.1  10  0.05  5 0 0  5  10  15  20  25  p µ-θµ Bin Index  30  35  40  0  Figure 8.6: Track quality cut covariance matrix, assuming a TPC hit efficiency systematic error of 0.07% ± 0.01. The relatively small number of selected tracks affected by this systematic results in relatively few entries to define the covariance matrix, leading to the white space observed in the matrix.  The empirical correction is derived using calibration target positions recorded during normal running condition and when the magnetic field is turned off. The difference between the measured target positions when the magnetic field is on or off provides a measure of the magnetic field distortion. These field distortion maps are used in the reconstruction of real and simulated data. The difference between the real and simulated field distortion maps provides the empirical correction. The empirical correction is used to evaluate position distortions which are accounted for during TPC reconstruction, with the correction between at any given position derived from an interpolation of the empirical correction map. The empirical correction was validated through several methods, including a simulated sample of tracks where the residuals in the measured inverse momentum were shown to decrease after applying the correction as shown in Figure 8.8. The systematic error in magnetic field distortions was assumed to be 100% of the size of the empirical correction. The systematic was propagated 111  40  ×10-6 40  35  35  30  30 25  25  20 20  15  15  10  10  5  5  0  0 0  vij  p µ-θµ Bin Index  8.3. Systematic Error Covariance Matrices  -5 5  10  15  20  25  p µ-θµ Bin Index  30  35  40  Figure 8.7: Particle identification systematic covariance matrix calculated using event selections where the pull measurements were smeared out in proportion to the systematic error in muon pull width.  by rerunning the reconstruction on selected with and without using the empirical correction, and measuring the change in the number of bin entries. The resulting covariance matrix is shown in Figure 8.9. In this case the systematic error estimate has significant statistical error due to the limited number of selected MC events that could be reprocessed. The estimated statistical error were calculated and propagated separately in an additional covariance matrix shown in Figure 8.10.  8.3.6  Momentum Resolution  The systematic error in TPC momentum resolution was measured directly using a control sample of tracks crossing multiple TPCs in real and simulated beam spill data. The difference in a track’s momentum measurements in different TPCs is due in part to the TPC momentum resolution, assuming that the difference has been corrected to account for energy loss in intervening material. The distribution of the difference in momentum transverse to the magnetic field 1/pT should be approximately Gaus112  8.3. Systematic Error Covariance Matrices  Figure 8.8: Example reduction in reconstructed inverse momentum residuals due to empirical field correction, taken from T2K Technical Note 61 [68].  sian, and fitting these distributions for different TPC combinations and pT ranges allows the momentum resolution of each TPC to be extracted. The systematic error in the momentum resolution of TPC2 was found to be δσ1/pT = 2.8 ± 0.6 × 10−5 (c/M eV ), which is relatively small compared to the measured TPC2 momentum resolution of 8 × 10−5 (c/M eV ). The systematic error in the TPC momentum resolution was propagated through the event selection by randomly smearing track momentum measurements according to a Gaussian distribution with width equal to the estimated data/MC difference in momentum resolution in quadrature with the uncertainty in the estimate. Momentum from different TPC segments were smeared by the systematic measured for the specific TPC. The resulting covariance matrix is shown in Figure 8.11.  8.3.7  Momentum Scale  Separate from distortions in the magnetic fields described in the previous section, the uncertainty in the magnitude of the magnetic field strength is a potential source of systematic error. The measurement errors arise from magnetic field probe resolution, errors in relative probe alignment and nonlinear corrections in the relationship between magnetic coil current and field 113  8.3. Systematic Error Covariance Matrices  40 0.003 30  0.002  25 0.001  vij  p µ-θµ Bin Index  35  20 15  0  10 -0.001  5 0 0  5  10  15  20  25  p µ-θµ Bin Index  30  35  40  Figure 8.9: Magnetic field distortion systematic covariance matrix.  strength. The final uncertainty in the magnetic field strengths is ∼ 0.5% in the Bx component and ∼ 1.3 G in the transverse components [69]. The magnetic field systematic errors were propagated through the event selection by way of direct simulation. The event selection was rerun 500 times with the field strength of each magnetic field component randomly varied according to Gaussian probability distributions with widths equal to the associated systematic errors. The modified magnetic field strengths are used to rescale track momentum measurements and can lead to migration between pµ − θµ bins. The resulting covariance matrix is shown in Figure 8.12. High and low momentum bins are anti-correlated in the CCQE sample (and to a lesser extent in the CCnQE sample) due to variations in the magnetic field strength migrating tracks between momentum bins.  8.3.8  FGD-TPC Tracking Efficiency  The FGD-TPC track matching efficiency was measured using a combined cosmic and beam spill control sample for real and simulated data in 20 different pµ − θµ regions corresponding to those used in the event selection. Throughgoing tracks are selected in beam spills by identifying events with 114  8.3. Systematic Error Covariance Matrices  40 0.002  30  0.0015  25  0.001  vij  p µ-θµ Bin Index  35  20  0.0005  15 10  0  5 0 0  -0.0005 5  10  15  20  25  p µ-θµ Bin Index  30  35  40  Figure 8.10: Covariance matrix for statistical error introduced by magnetic field distortion systematic propagation. The band structure reflects the fact that the systematic effect mainly affects track momentum estimates, such that tracks that migrate are most likely to change momentum bins without changing the angular bin.  exactly one TPC segment reconstructed in TPC1, TPC2 and TPC3. In cosmic data, high angle tracks are selected by requiring exactly one TPC2 segment and no tracks reconstructed in TPC1 or TPC3. As the cosmic trigger requires hits in both FGDs to occur, these cosmic tracks are assumed pass through FGD1. The efficiency with which these identified tracks are matched to hits in FGD1 is measured in real and simulated data, with the results binned by TPC track angle and momentum and are summarized in [66]. The discrepancy in the measured reconstruction efficiency between real and simulated data was combined with the statistical error in the measurements to produce the systematic error estimate. These systematic errors were propagated using the reweighting method for each of the 20 pµ − θµ bin cases separately to produce the combined covariance matrix seen in 8.13. Similar to the TPC reconstruction efficiency covariance matrix, off-diagonal  115  8.3. Systematic Error Covariance Matrices  ×10-3 0.6  40 35  0.2  25  vij  p µ-θµ Bin Index  0.4 30  20  0  15 -0.2  10 5 0 0  -0.4 5  10  15  20  25  p µ-θµ Bin Index  30  35  40  Figure 8.11: Momentum resolution systematic covariance matrix.  correlations between pµ − θµ bins are induced due to the common change in the number of bin entries under a variation in the track matching efficiency, as well as migration of selected events from the CCnQE to the CCQE sample under the track multiplicity cut. The off-diagonal correlations within each sample correspond to common changes in the number of bin entries when the systematic effect is varied. This is possible due to the systematic effect being defined for different TPC track momentum ranges, which does not exactly correspond to the FGD-TPC matched track reconstructed momentum due to the energy loss correction in the latter.  8.3.9  FGD-TPC Broken Tracks  The FGD-TPC track matching algorithm has failure modes that are not covered by the matching efficiency systematic. Most importantly is the “broken track” failure mode, where a single track entering the TPC from FGD1 can be reconstructed as 2 separate matched tracks. This introduces a potential systematic error that specifically affects the CCQE selection track multiplicity cut. The rate at which the broken tracks error occurs was measured in real 116  8.3. Systematic Error Covariance Matrices  ×10-3  40  0.5 0.4  30  0.3  25  0.2  20  0.1  vij  p µ-θµ Bin Index  35  0  15  -0.1 10  -0.2  5 0 0  -0.3 5  10  15  20  25  p µ-θµ Bin Index  30  35  40  -0.4  Figure 8.12: Momentum scale systematic covariance matrix. High and low momentum bins are anti-correlated in the CCQE sample (and to a lesser extent in the CCnQE sample) due to variations in the magnetic field strength migrating tracks between momentum bins.  and simulated data using a set of specialized cuts applied to beam spill data. These include cuts on general event topology, such as requiring at least two tracks with TPC segments with similar reconstructed front positions and the same charge. An additional set of cuts that identify broken-track events due to the specific manner in which the track matching algorithm fails are also applied. The final systematic error was calculated by multiplying the measured rate of broken-tracks in the selected sample by the proportion of the CC-inclusive sample that would be selected under the broken-track selection cuts. This yields a systematic error of 0.63% in the rate of broken track events. This systematic was propagated by migrating a fraction of CCQE events equal to the size of the systematic error to the corresponding CCnQE bin. This method could be applied directly to the selected pµ − θµ distribution, and no simulation or event selection propagation was required. The resulting covariance matrix is shown in Figure 8.14. Broken track events are never  117  8.3. Systematic Error Covariance Matrices ×10-3  40  0.08 35 0.07  p µ-θµ Bin #  30  0.06  25  0.05  20  0.04  15  0.03  10  0.02 0.01  5  0 0 0  5  10  15  20  25  p µ-θµ Bin #  30  35  40  Figure 8.13: FGD-TPC tracking efficiency fractional covariance matrix. The off-diagonal correlations within each sample correspond to common changes in the number of bin entries when the systematic effect is varied. This is possible due to the systematic effect being defined for different TPC track momentum ranges instead of FGD-TPC matched track momentum ranges.  classified as CCQE, there is an overall common change between all CCQE bins as the systematic effect is varied causing the lower-left quadrant to be flat.  8.3.10  FGD Target Mass  The systematic error in FGD1 mass relevant to the event selection is the uncertainty in the active scintillator target mass, which is estimated to be 0.67%. This systematic was estimated by comparing the measured masses and densities of the active detector components inside the FV with the assumed masses in the Monte Carlo model [70]. The systematic only affects events that occur within FGD1, this being determined through simulated data truth information. The systematic is propagated by rescaling the number of true FGD1 events by a factor equal to the systematic error, then producing a covariance matrix using the new event distribution. The result118  8.3. Systematic Error Covariance Matrices  ×10-3 0.4  40  0.3  30 25  0.2  vij  p µ-θµ Bin Index  35  20 0.1  15 10  0  5 0 0  -0.1 5  10  15  20  25  p µ-θµ Bin Index  30  35  40  Figure 8.14: FGD-TPC track breaking fractional covariance matrix. Since broken track events are never classified as CCQE, the bottom left quadrant is flat.  ing covariance matrix is shown in figure 8.15.  8.3.11  FGD Out of Fiducial Volume Interactions  Some fraction of the selected interactions occur outside of the FGD1 fiducial volume, and are erroneously reconstructed as fiducial. A systematic is introduced as the rate at which events migrate into the fiducial volume may differ between real and simulated data due to differences in the reconstruction or in the amount of external background. Several relevant sources of out of fiducial volume external background are identified and associated with reconstruction failures. All of these out-of-fiducial volume interactions are identified in the simulated data sample using truth information and their respective contributions to the overall number of events and to the pµ − θµ distribution are measured. External neutrino interactions that are reconstructed within the FGD1 fiducial volume include several different sources. Neutral particles produced by interactions outside the detector have some chance of propagating unde119  8.3. Systematic Error Covariance Matrices  ×10-6  40  42 40  30  38  25  36  20  34  vij  p µ-θµ Bin Index  35  32  15  30 10  28  5 0 0  26 5  10  15  20  25  p µ-θµ Bin Index  30  35  40  Figure 8.15: FGD target mass systematic fractional covariance matrix. The variation is due to the systematic effect only affecting interactions that originate within the FGD, and consequently the contribution from external interactions remains unchanged. As the low-momentum CCQE sample bins have the highest proportion of external interactions they are least affected by this systematic effect.  tected into the detector fiducial volume and interacting there to produce a reconstructed track. Charged particles that originate outside or inside the tracker can also be reconstructed within the fiducial volume due to a failure in the Reconstruction algorithms or selection cuts. The primary systematic error introduced by these types of events are through uncertainties in their interaction cross sections and the associated rates. Reconstruction failures are separated into six categories based on visual inspection of simulated events. These categories include events that occur in the two most upstream FGD1 layers, interactions in upstream tracker dead material and similarly interactions in downstream material such as the cover plate or TPC walls. Additionally backwards going tracks can enter and stop in FGD1 and are erroneously reconstructed as forward going. Neutral particles entering FGD1 can interact and produce a track, but is not directly  120  8.3. Systematic Error Covariance Matrices due to a reconstruction failure and no systematic is applied. Finally the FGD-TPC track matching process can fail in many different ways resulting in a track incorrectly reconstructed within the FGD1 fiducial volume. Using truth information selection cuts were developed for use with the simulated data sample to identify each type of failure mode so that their the pµ − θµ distribution can be measured. An overall 20% systematic error in the normalization of out-of-tracker external background events is estimated from comparisons between event generators and from event rates in the ECAL and SMRD regions of ND280. This error is propagated to the event selection by scaling the contribution of external interactions to the pµ − θµ distribution and calculating the resulting covariance matrix. Similarly the pµ − θµ distribution for each type of out-of-fiducial volume selected event is rescaled by a factor equal to the estimated systematic uncertainty in the frequency of occurrence between real and simulated data [71]. The total resulting covariance matrix is shown in Figure 8.16.  35  0.007  30  0.006  25  0.005  20  0.004  15  0.003  vij  p µ-θµ Bin Index  40  0.002  10  0.001  5 0 0  0 5  10  15  20  25  p µ-θµ Bin Index  30  35  40  Figure 8.16: FGD out of fiducial volume covariance matrix.  121  8.3. Systematic Error Covariance Matrices  8.3.12  Michel Detection Efficiency  40  ×10-3  35  0.2  30 0.1  25 20  0  vij  p µ-θµ Bin Index  The Michel electron hit cluster detection efficiency was measured using a control sample of stopping FGD1 cosmic events in both real and simulated data. Selection cuts were applied to the track stopping position, the TPC track quality and the muon pull. The muon pull was required to remove electron-like events that are under-represented in the simulated cosmic sample. Using these stopping muon-like tracks, the identification rate for Michel electrons was measured in real data as 0.593 ± 0.004 and in simulated data as 0.642 ± 0.020. The large discrepancy between the Michel detection efficiency in real and simulated data is not understood. Consequently the systematic error is taken to be the discrepancy in the Michel efficiency measurement between real and simulated data added in quadrature with the statistical error in the difference. This systematic error was propagated through the event selection using the reweighting method applied to delayed hit clusters in FGD1. The resulting covariance matrix is shown in Figure 8.17  15 -0.1  10 5 0 0  -0.2 5  10  15  20  25  p µ-θµ Bin Index  30  35  40  Figure 8.17: Michel hit cluster detection efficiency fractional covariance matrix.  122  8.3. Systematic Error Covariance Matrices  8.3.13  External Background in the Michel Electron Cut  The Michel electron cut is used to separate CCQE and CCnQE events on the basis of delayed energy deposit in FGD1. External background is defined as particle interactions outside FGD1 that generate energy deposit that can be misidentified as a Michel electron. The effect of external background is to erroneously classify a random fraction of true CCQE events as CCnQE. External background introduces a systematic error for several reasons, most importantly simulated data does not include interactions outside of the ND280 magnet. Other contributing systematic effects include the beam power causing a different amount of delayed activity in FGD1, and the neutron transport simulation. The number of unrelated background signal events that pass the Michel electron cut can be measured directly from real and simulated data control samples. The measurement indicates that external background creates a Michel like signal for 0.15% of events in real data and 0.034% of simulated events, resulting in a 0.012% systematic excess of CCQE events in real data. This error was propagated using the event migration method to produce the covariance matrix seen in Figure 8.18. ×10-6  40  14 12  30  10  25  8  20  6 4  15  2  10  0  5  -2  0 0  vij  p µ-θµ Bin Index  35  -4 5  10  15  20  25  p µ-θµ Bin Index  30  35  40  Figure 8.18: Michel electron cut external background covariance matrix.  123  8.3. Systematic Error Covariance Matrices  8.3.14  Track Multiplicity External Background  The FGD-TPC track multiplicity cut identifies selected CC-inclusive interactions as CCnQE if there is more than one FGD1-TPC reconstructed track present in the bunch of the selected event. As the simulated data sample does not include interactions outside of the ND280 magnet such as cosmic events or sand muons, the track multiplicity cut introduces a systematic error in the number of CCQE and CCnQE events. The magnitude of the systematic error is increased from failing to require the tracks that define an event as CCnQE to start in the FGD1 fiducial volume, resulting in a larger acceptance for external events to become associated with a selected interaction. In order to estimate the size of the systematic error the 2011 event selection was applied to the real and simulated data samples twice, once in its default configuration and again where the implementation of the track multiplicity cut was changed so that only fiducial tracks were considered. The results of these two selection are shown in Table 8.2, and Table 8.3 summarizes the percentage change in the total number of CCQE events due to the modified event selection with appropriate binomial errors. Data Sample Run 1 Run 2 MC Run 1 MC Run 2 Total Real Total MC  Default CCQE 632 1740 11307 46203 2372 57510  Default CCnQE 590 1557 9141 36812 2147 45953  Modified CCQE 645 1797 11591 47408 2442 58999  Modified CCnQE 577 1500 8857 35607 2077 44464  Table 8.2: Number of real and selected CCQE and CCnQE events in the normal event selection and when requiring the track multiplicity cut to only consider fiducial tracks. The size of the systematic error introduced by the cut is estimated as the difference between real and simulated data for the percentage change in the number of CCQE events between the two event selections, added in quadrature with the statistical error of the percentage difference. The total discrepancy between data and MC is 0.362% and the statistical error is conservatively estimated as 0.395%, resulting in the total systematic error estimated as 0.56%. Additional contributions to the systematic error from external interactions that are reconstructed as fiducial are assumed to be 124  8.3. Systematic Error Covariance Matrices Data Sample Run 1 Run 2 MC Run 1 MC Run 2 Total Real Total MC  Change in CCQE 13 57 284 1205 70 1489  % Change 2.057 3.276 2.512 2.608 2.951 2.589  68% CL Lower Bound 1.631 2.901 2.373 2.536 2.642 2.524  68% CL Upper Bound 2.786 3.759 2.668 2.684 3.340 2.657  Table 8.3: Number of real and selected CCQE and CCnQE events in the normal event selection and when requiring the track multiplicity cut to only consider fiducial tracks.  insignificant. The resulting covariance matrix is shown if Figure 8.19.  8.3.15  External Background from Sand Muons and Cosmic Rays  The effect of external interactions due to sand muons was simulated using a dedicated MC sample that is included in the standard simulated data sample. However this simulation introduces several systematic effects including differences between the real and simulated beam power, neutrino interaction cross-sections and pit geometry. The size of the systematic error was measured by comparing the number of tracks entering the upstream face of the P0D subdetector in real and simulated data, resulting in an estimated 15% uncertainty in the sand muon rate. The uncertainty in the event selection due to this systematic error was propagated by increasing the sand muon MC sample’s predicted pµ − θµ distribution by a factor equal to the systematic error, resulting in the covariance matrix shown in Figure 8.20. Cosmic rays are not included in the simulated data samples used in the analysis, leading to a possible systematic excess in the number of events selected in real data. The rate at which cosmic rays are accepted by the event selection was estimated using control samples in real and simulated data. Based on these measurements the total number of cosmic ray events expected to be misidentified as a CC-inclusive event in the real data sample was predicted to be 1.72±0.25. This amounts to correcting the total number of events by ∼ 0.025%, which was insignificant compared to statistical and other systematic errors and was neglected.  125  8.3. Systematic Error Covariance Matrices  ×10-3 0.3  40  0.25  30  0.2  25  0.15  20  0.1  15  0.05  10  0  5 0 0  vij  p µ-θµ Bin Index  35  -0.05 5  10  15  20  25  p µ-θµ Bin Index  30  35  40  -0.1  Figure 8.19: Track multiplicity external background systematic covariance matrix.  8.3.16  Sand Muon Pileup  While the systematic error in the sand muon simulation is accounted for in the external background systematic, there is an additional systematic effect on the event selection through the ∆ − z external background selection cut. Tracks that enter TPC1 in coincidence with a muon-like track starting in FGD1 can cause the event to be rejected. An error in the rate at which simulated tracks enter TPC1 will introduce systematic error in the number of events vetoed due to unrelated external background. The fraction of bunches with a TPC1 track is measured in real and simulated data, and the difference is measured to be 0.23% for the Run 1 period and 0.19% for the Run 2 period. The discrepancy between the Run 1 and 2 rates is due to how well the simulated beam power matches the average beam power over the running periods and the simulation of the ND280 detector geometry. Another systematic error introduced by the sand muon simulation is through the efficiency correction described in Chapter 7. The efficiency correction was calculated using the sand muon simulation, in which there is a 15% uncertainty in the overall rate as measured by the number of track 126  8.4. Total Detector Systematic Covariance Matrix  ×10-3 0.12  40  0.1  30 0.08  25 20  0.06  15  vij  p µ-θµ Bin Index  35  0.04  10 0.02  5 0 0  5  10  15  20  25  p µ-θµ Bin Index  30  35  40  Figure 8.20: Sand muon external background covariance matrix.  entering the most upstream layer of the P0D. The systematic error in the efficiency correction is assumed to 15% of its calculated value, resulting in an error of 0.1% for Run 1 and 0.06% for Run 2. The efficiency correction systematic errors are added in quadrature with the TPC1 external background systematic, resulting in an overall 0.24% systematic uncertainty in the number of events. This uncertainty is added to every element of the covariance matrix as the pµ − θµ are fully correlated under these systematic effects.  8.4  Total Detector Systematic Covariance Matrix  The final covariance matrix for the p − θ bins is the linear sum of all the detector systematic covariances described in this chapter. The resulting matrix can be seen in Figure 8.21. The overall uncertainty in pµ −θµ bins due to systematic effects varies from 1.9% to 9.5%. The covariance matrix used in analyses may also include systematic effects due to neutrino interaction model parameters and statistical error in the simulated data sample, as described in Chapter 9. 127  8.4. Total Detector Systematic Covariance Matrix  40 0.008  30  0.006  25 0.004  vij  p µ-θµ Bin Index  35  20 15  0.002  10  0  5 0 0  5  10  15  20  25  p µ-θµ Bin Index  30  35  40  -0.002  Figure 8.21: Total pµ − θµ bin covariance matrix for detector systematic effects.  The relative importance of the different systematic effects is qualitatively understood by plotting the fractional error they introduce into the number of entries for each pµ − θµ bin. For ease of viewing these contributions are seperately plotted for each of the different angular ranges and CCQE/CCnQE samples in Figure 8.22, Figure 8.23, Figure 8.24, Figure 8.25, Figure 8.26, Figure 8.27, Figure 8.28 and Figure 8.29. These plots also show the contribution due to the systematic error in modeling pion nuclear secondary interactions, which will be discussed in Chapter 9. These plots do not show important correlations between pµ − θµ bins introduced by systematic effects. However they demonstrate that the statistical uncertainty is the largest single source of error in the analysis, with the largest systematic error introduced by the detector model due to the magnetic field distortion correction. In future the detector model systematic uncertainties will become relatively more important as the experiment continues to monitor neutrino interactions.  128  8.4. Total Detector Systematic Covariance Matrix  Figure 8.22: Fractional error introduced by each systematic effect into the number of entries in different pµ − θµ bins within the CCQE sample with angular range −1 ≤ cos θ < 0.84.  Figure 8.23: Fractional error introduced by each systematic effect into the number of entries in different pµ − θµ bins within the CCQE sample with angular range 0.84 ≤ cos θ < 0.9. 129  8.4. Total Detector Systematic Covariance Matrix  Figure 8.24: Fractional error introduced by each systematic effect into the number of entries in different pµ − θµ bins within the CCQE sample with angular range 0.9 ≤ cos θ < 0.94.  Figure 8.25: Fractional error introduced by each systematic effect into the number of entries in different pµ − θµ bins within the CCQE sample with angular range 0.94 ≤ cos θ < 1. 130  8.4. Total Detector Systematic Covariance Matrix  Figure 8.26: Fractional error introduced by each systematic effect into the number of entries in different pµ − θµ bins within the CCnQE sample with angular range −1 ≤ cos θ < 0.84.  Figure 8.27: Fractional error introduced by each systematic effect into the number of entries in different pµ − θµ bins within the CCnQE sample with angular range 0.84 ≤ cos θ < 0.9. 131  8.4. Total Detector Systematic Covariance Matrix  Figure 8.28: Fractional error introduced by each systematic effect into the number of entries in different pµ − θµ bins within the CCnQE sample with angular range 0.9 ≤ cos θ < 0.94.  Figure 8.29: Fractional error introduced by each systematic effect into the number of entries in different pµ − θµ bins within the CCnQE sample with angular range 0.94 ≤ cos θ < 1. 132  Chapter 9  Neutrino Energy Spectrum Measurement The simulated sample of CCQE and CCnQE selected events is used to predict the muon momentum and angular distributions for different values of the neutrino interaction model parameters and neutrino flux factors. The parameter values most favoured by observed data are found using a maximum likelihood fit. The fitted parameters constrain the neutrino beam flux and neutrino interaction model, and can be used to cross-check similar analyses.  9.1  Data Analysis Tools  The real and simulated data samples used in this analysis are described in Chapter 7. Both samples are processed with the standard event selection [65], producing a root tree summarizing the selected events in a format suited for analysis. An additional processing step is applied to simulated data, in which the reweighting application T2KReWeight [72] produces weights for each event corresponding to different values of the neutrino interaction model parameters. This reweighting package allows reevaluation of event distributions for different sets of neutrino interaction model parameters without having to regenerate the Monte Carlo sample. The output event summary trees include the selected muon track momentum and direction measurements along with the event weights, which are required to predict the pµ − θµ distributions used in analyses.  9.2  Maximum Likelihood Fit of Neutrino Energy Spectrum  This analysis applies the CCQE and CCnQE event selection described in Chapter 7 to the real and simulated data samples in order to measure the neutrino flux energy spectrum at the ND280 detectors. This is accomplished 133  9.2. Maximum Likelihood Fit of Neutrino Energy Spectrum by using the simulated event selection to predict the pµ − θµ distribution for different neutrino interaction model and beam flux hypotheses. A maximum likelihood fit is used to determine the hypothesis that best reproduces the observed data distribution. This fit is similar in design to that used by the K2K collaboration for near detector data [23]. It is also similar in scope and implementation to the analysis performed by the Beam and ND280 Flux Measurement Task Force (BANFF) described below and in T2K-TN-106 [73], with the main distinguishing feature being that this analysis does not include prior constraints on the beam flux factors as derived from the beam group, but instead attempts to fit for the flux factors using only ND280 data in order to provide a completely independent cross-check of the beam flux prediction The fitting algorithm accepts a pµ − θµ distribution as input data to compare with various simulated distributions during the fitting process. The input data is obtained directly from the event selection applied to real or simulated data samples and is not modified in any way during the fit. The parameters used in the fit are flux scaling factors fk , neutrino interaction model parameters xj , and scaling factors dm associated with each of the forty pµ − θµ bins to account for systematic uncertainties related to detector performance and certain neutrino interaction model parameters. This provides a measurement of the neutrino energy spectrum that is independent of the beam group’s flux prediction and that can be compared to that prediction as a cross check. The actual fit to data is performed by minimizing the negative natural logarithm of the likelihood function − ln L =  i  + j  nexp i − nobs i ln(nexp i )  (9.1)  1 1 (xj − Xj )2 −1 (dm − Dm )Vm,n (dn − Dn ) , + 2 2 2 σj m,n  (9.2)  with the expected number of pµ − θµ bin entries nexp i defined as MC (x) fk Nk,i  nexp i = di  (9.3)  k  and the observed number of entries taken from the input data. The fit parameters and constraint terms used in the likelihood function are listed in Table 9.1. The variables used in the fit are scale factors defined with respect to the simulated data prediction. As an example the neutrino flux scale factors 134  9.2. Maximum Likelihood Fit of Neutrino Energy Spectrum express how much the flux model prediction needs to be scaled from their default values to best fit the observed data. A fitted flux scale factor of 1.2 indicates that the observed neutrino flux is 20% higher than what was used as in the simulation. The actual value of the observed neutrino flux is inferred by scaling the simulated flux by the fitted scale factor. Similarly the neutrino interaction model and pµ − θµ bin scale factors are nuisance parameters that express how much the neutrino model parameters and number of pµ − θµ bin entries should be scaled in order to best agree with the fitted data. Prior errors that constrain these nuisance parameters in the fitting process (i.e. Bayesian-style priors on the values of these nuisance parameters) are also expressed as fractional errors on the MC default value. The fit constrains nuisance parameters associated with systematic errors in two ways, either through dedicated constraint terms or through the pµ −θµ bin covariance matrix. The neutrino interaction model parameters xj have Gaussian constraint terms with mean equal to one and width equal to the parameter’s prior error expressed as a fraction of the default value. For example, if a parameter has a default value of 1.21 GeV /c2 and a prior error of 0.24 GeV /c2 , the constraint is a Gaussian with width equal to 0.198. The pµ −θµ bin scale factors dm are constrained about their default value of 1 by a covariance matrix term. Fitting for the nuisance parameters simultaneously with the flux factors ensures the effect of the systematics uncertainties are automatically propagated to the flux factor values and error estimates. The end result of the fit, after marginalizing over the cross-section and detector systematics nuisance parameters, is a set of flux rescalings fk and the uncertainties on these parameters, incorporating all prior information on the cross-section and detector systematics. Multiplying these scaling factors fk by the beam flux prediction yields an absolute measurement of the flux of muon neutrinos in ND280.  9.2.1  BANFF Neutrino Flux Fit  The neutrino flux measurement presented here is best understood in comparison with the BANFF flux measurement [73] that is used in T2K’s most recent oscillation analyses. The BANFF analysis uses a maximum likelihood function to measure the neutrino energy spectrum in order to provide constraints on the flux model used in the analyses. The BANFF fit makes full use of prior constraints on the neutrino beam flux provided by the T2K beam group, which includes measurements from T2K beam monitors and uncertainties in pion and kaon production provided by the NA61 collaboration [39]. In contrast this analysis does not apply any constraints on the 135  9.2. Maximum Likelihood Fit of Neutrino Energy Spectrum Parameter nexp i nobs i fk M C (x) Nk,i  xj Xj σj dm Dm Vm,n  Definition Expected number of interactions in pµ − θµ bin i Observed number of interactions in pµ − θµ bin i Neutrino flux scale factor for energy bin k Number of predicted interactions in pµ − θµ bin i from neutrinos in energy bin k for the set of cross-section parameters x Neutrino interaction model parameters Neutrino interaction model parameter constraint constant Neutrino interaction model parameter prior error pµ − θµ bin m scale factor pµ − θµ bin m scale factor constraint constant Detector systematics covariance matrix element (m, n)  Table 9.1: Definition of variables used in the maximum likelihood fit for the spectral measurement. Included are flux factors, variables related to the simulated data templates, and nuisance parameters related to neutrino interaction model and detector model systematic errors.  neutrino beam flux factors, which is equivalent to measuring the neutrino beam energy spectrum using ND280 information only. Without constraints on the neutrino beam flux factors the neutrino interaction model systematic uncertainties introduce large errors into the measured neutrino flux. The beam constraints used by the BANFF group are shown in Figure 9.1 in error matrix form. The figure shows the correlated errors between the different energy bins used in the BANFF analysis. Bins defined for T2K Run 3 are also shown, although this running period is not included in the analysis presented here. These uncertainties directly constrain the neutrino beam flux factors in the BANFF fit in exactly the same manner that the pµ − θµ bin scale factors dm are constrained by the ND280 pµ − θµ covariance matrix. The fitted errors on the neutrino beam flux factors, along with the fitted errors on the neutrino interaction model parameters are used as constraints in the T2K oscillation analysis. Specifically, the constraints provided by the fit to ND280 data reduce the systematic error in the flux predicted at Super-K, and by extension the predicted number of events that occur in the far detector. Additionally, the BANFF fitted flux and neutrino interaction model parameters are anti-correlated, as shown in Figure 9.2. As the number of predicted events at the far detector is the convolution of 136  9.2. Maximum Likelihood Fit of Neutrino Energy Spectrum the flux and neutrino interaction cross sections, this anti-correlation leads to a further reduction in the systematic error in the predicted number of events.  Figure 9.1: BANFF fit neutrino flux factor constraints, taken from T2K Technical Note 106 [73]. The BANFF group calculates the fluxes of muon and electron neutrinos and antineutrinos in several neutrino energy bins, in both the near and far detectors, as well as the correlated uncertainties between these bins. The bins are ordered as follows: (0-10) = ND280 Run 1+2 νmu , (11-12) = ND280 Run 1+2 ν¯mu , (13-19) = ND280 Run 1+2 νe , (20-21) = ND280 Run 1+2 ν¯e , (22-32) = SK Run 1+2 νmu , (33-34) = SK Run 1+2 ν¯mu , (35-41)) = SK Run 1+2 νe , (42-43) = SK Run 1+2 ν¯e , (44-87) = SK Run 3 flux energy bins.  The goal of the BANFF fit is to reduce the size of the uncertainties for the predicted event rates at Super-K. It effectively uses the beam flux prediction as a prior, and then uses the ND280 data to constrain the convolution of the flux and cross-section model to match the ND280 data. While the resulting uncertainties on the flux or cross-sections individually can be large, their product (convolution) is tightly constrained by the ND280 data. In contrast, in this thesis the approach is to ignore the far detector entirely and to use the ND280 data to independently check the beam flux prediction. Without including any prior on the flux, the resulting uncertainties are larger than the BANFF fit approach. However, the result is independent of the beam flux model, and is a useful cross-check 137  9.3. Neutrino Interaction Modes  Figure 9.2: BANFF fit correlations between flux and neutrino interaction model parameters, taken from T2K Technical Note 106 [73]. The bins are ordered as follows: (0-21) = SK neutrino beam flux factors, (22-26) = neutrino interaction model parameters. The large negative correlations between the flux and cross-section parameters reflects the fact that the BANFF fit constrains the convolution of flux with cross-section to match data, and so increases in flux scalings can be accommodated by decreasing relevant cross-sections.  9.3  Neutrino Interaction Modes  It is useful to identify the relative contribution from different neutrino interaction modes to the selected event sample when defining the maximum likelihood fit and relevant neutrino interaction model systematic parameters. This analysis uses the NEUT neutrino interaction simulation described in Chapter 5 and consequently categorizes neutrino interactions in terms of NEUT interaction modes. These categories are summarized in Table 9.2. There are a few categories that combine several interactions together such as “CC-other”. Additionally interactions that originate outside of the FGD1 fiducial and detector volumes are identified in separate categories. The predicted number of selected CC-inclusive events of each type are summarized in Table 9.3, and their contributions to the selected muon momentum and angular distributions shown in Figure 9.3 and 9.4. The contribution of different neutrino modes to the predicted pµ − θµ distribution is shown in Figure 138  9.4. Neutrino Energy Binning 9.5. Category CCQE CC1π CC coherent CC other NC Anti-Neutrino Out of FGDFV Out of FGD  Interactions (NEUT Codes) Charged-current quasi-elastic (1) Charged-current resonant pion production (11,12,13) Charged-current coherent pion production (16) Charged-current DIS (21,26), resonant γ (17) η (22) and K production (23) Neutral current (> 31) Anti neutrino interactions (< 0) Out of FGD fiducial volume Out of FGD  Table 9.2: Neutrino interaction categories used in neutrino flux spectrum measurement and associated NEUT interaction codes.  Interaction Type CCQE CC1π CC coherent CC other NC Anti-Neutrino Out of FGDFV Out of FGD Total  Number Selected 2101.1 996.96 133.8 897.3 149.9 33.3 45.7 385.8 4743.8  Table 9.3: Number of selected neutrino interactions by mode in the simulated CC-inclusive sample.  9.4  Neutrino Energy Binning  The neutrino energy spectrum is divided into four bins, with each of the flux factors used in the maximum likelihood fit corresponding to a particular bin. The bin boundaries are defined in Table 9.4 and Figure 9.6 shows energy spectrum of selected interactions distributed in these bins. The boundaries were chosen so that all bins were predicted to have more than 200 entries, 139  Number of Entries / (50 MeV/c)  9.4. Neutrino Energy Binning  Event Types CCQE CC Resonant Pion CC Coherent CC Other NC Anti Neutrino Out of FGD FV Out of FGD  250 200 150 100 50 0 0  500 1000 1500 2000 2500 3000 3500 4000 4500 5000  Muon Momentum (MeV/c)  Figure 9.3: Selected CC inclusive neutrino interaction momentum spectrum with contributions from different interaction types shown.  Number of Entries / (0.02)  800 700 600 500  Event Types CCQE CC Resonant Pion CC Coherent CC Other NC Anti Neutrino Out of FGD FV Out of FGD  400 300 200 100 0 -1  -0.8  -0.6 -0.4  -0.2  0  0.2  0.4  0.6  0.8  1  Muon Cosθ  Figure 9.4: Selected CC inclusive neutrino interaction angular spectrum with contributions from different interaction types shown.  140  Number of Entries / (pµ-θµ Bin)  9.4. Neutrino Energy Binning  700 600 500  Event Types CCQE CC Resonant Pion CC Coherent CC Other NC Anti Neutrino Out of FGD FV Out of FGD  400 300 200 100 0 0  5  10  15  20  25  p µ-θµ Bin Number  30  35  40  Figure 9.5: Selected neutrino interaction pµ − θµ distribution with contributions from different interaction types shown. CCQE dominates bins 0-19 (the CCQE-like selection), while other processes dominate the CCnQE sample (bins 20-39).  while in particular the boundaries of bin 0 were chosen so that it would contain almost entirely CCQE interactions. Additionally the boundaries of bin 1 where chosen so as to contain the peak of the simulated neutrino energy spectrum, and bin 3 was defined to contain the entire high energy range where the tracker has relatively poor momentum resolution. The contributions from different neutrino energy bins to the simulated pµ − θµ distribution of selected events using the default MC is shown in Figure 9.7, and similarly the selected muon momentum distribution is shown in Figure 9.8. Both plots show that the contribution from different energy bins to measured momentum and angular distributions are different, but do overlap. This suggests that the fit will be able to resolve the relative fluxes of neutrinos in each neutrino energy bin, but that there will be statistical correlations due to the partial overlap of the p, θ distributions of neutrinos in different energy bins. In principle if the purity of the selected CCQE sample was 100% the energy spectrum could be measured directly with resolution limited by the tracking resolution of the TPCs and the Fermi momentum, using the formula shown in Eq. 1.19. In practice the presence of CCnQE interactions smears 141  9.5. Systematic Errors out the contribution from different neutrino energy bins to muon observables, requiring a more sophisticated measurement of the energy spectrum such as the maximum likelihood fit presented here. This fitting approach encounters difficulty due to the large correlations between energy bins and cross section systematic parameters, especially the CCQE normalization systematics. These correlations can prevent the fitting algorithm from finding the minimum value of the likelihood, thereby failing to estimate the flux and nuisance parameter values. The convergence of the fitter to the minimum value is improved by reducing the number of parameters in the fit, such as neutrino energy bin flux factors. The binning presented here was found to allow the fitter to converge consistently. The number of flux factors could be increased and the width of the energy bins decreased if beam flux prior constraints were used, as this increases the sensitivity of the likelihood function to changes in the values of the flux factor parameters. The beam flux constraint was successfully implemented by the T2K collaboration and described in section 9.2.1 and in T2K-TN-106 [73]. Bin 0 1 2 3  Energy Range (GeV) 0-0.5 0.5-1 1-3.5 >3.5  Table 9.4: Selected neutrino interaction energy binning.  9.5  Systematic Errors  The systematic uncertainties relevant to this analysis fall into two categories, neutrino interaction model uncertainties and detector simulation systematic uncertainties. Systematic uncertainties are propagated through the maximum likelihood fit by introducing nuisance parameters and simultaneously fitting for them with appropriate constraint terms. This ensures that systematic errors contribute to the fitted uncertainty in the neutrino flux factors. Table 9.5 summarizes the neutrino interaction model uncertainties relevant to this event selection which are described in detail in T2K-TN-113 [74]. The detector model systematic uncertainties are described in Chapter 8 and are listed in Table 8.1.  142  Number of Entries / (Energy Bin)  9.5. Systematic Errors  Event Types CCQE CC Resonant Pion CC Coherent CC Other NC Anti Neutrino Out of FGD FV Out of FGD  1600 1400 1200 1000 800 600 400 200 0 0  1000 2000 3000 4000 5000 6000 7000 8000 9000 10000  True Neutrino Energy (MeV)  Figure 9.6: Selected neutrino interaction energy spectrum with contributions from different interaction types shown.  9.5.1  Detector Systematic Errors  Detector systematics uncertainties are propagated through the maximum likelihood fit using the covariance matrix that was defined in Chapter 8. A set of 40 nuisance parameters dm are defined and each acts as a scale factor for a corresponding pµ − θµ bin. These systematic nuisance parameters are fitted at the same time as the other parameters while being constrained by the covariance matrix Vm,n . This allows the fit to account for the uncertainty in the number of bin entries due to detector model systematic uncertainties. The covariance matrix corresponding to detector model uncertainties is shown in Figure 8.21.  9.5.2  Neutrino Interaction Model Systematic Errors  Most neutrino interaction model uncertainties are propagated by including a specific nuisance parameter in the maximum likelihood fit with appropriate constraints. This neutrino interaction model nuisance parameter is a scale factor that specifies the fractional change from the default value of the parameter that is to be used when generating the predicted pµ − θµ distribution. Several neutrino interaction model systematic effects were propagated through pµ − θµ bin covariance matrices in a manner similar to detector 143  Number of Entries / (pµ-θµ Bin)  9.5. Systematic Errors  700 600 500  Neutrino Energy 0 - 0.5 GeV/c 0.5 - 1.0 GeV/c 1.0 - 3.5 GeV/c > 5.0 GeV/c Other  400 300 200 100 0 0  5  10  15  20  25  p µ-θµ Bin Number  30  35  40  Figure 9.7: pµ − θµ distribution of selected events with contributions from different energy bins shown. Contributions from sand muons and selected interactions that are not correctly matched to a true neutrino interaction are labelled as “other”.  systematic effects. These include NEUT parameters related to final state interaction, pion absorption and charge exchange processes in nuclear material, the width of the ∆ mass Breit-Wigner distribution, the CC-coherent interaction cross-section normalization, the combined NC-other modes crosssection normalization and pion absorption and charge exchange. The systematic errors associated with these effects were propagated by reweighting selected events in a way that accounts for variations in the model parameter, and then measuring the resulting change in the pµ − θµ distribution [65][73]. Of the neutrino interaction model parameter errors, the MAQE and MARES parameters vary the value of the axial mass used in the axial-vector form factor to parametrize the CCQE and resonant cross sections respectively. The binding energy parameter EB adjusts the strength of the nuclear potential, which is used by some of the cross section models to estimate the final nucleon momentum. The Fermi momentum parameter pF sets the maximum nucleon momentum used in the relativistic Fermi gas model. The spectral function parameter is a binary value that determines whether a spectral function model is used in place of the relativistic Fermi gas model. The CC Other parameter adjusts the shape of the total cross section for 144  Number of Entries / ( 50 MeV/c )  9.5. Systematic Errors  Muon Momentum  250  0 - 0.5 GeV/c 0.5 - 1.0 GeV/c  200  1.0 - 3.5 GeV/c > 5.0 GeV/c  150 100 50 0 0  500 1000 1500 2000 2500 3000 3500 4000 4500 5000  Selected Muon Momentum  Figure 9.8: Energy bin contributions to the selected event muon momentum distribution.  the corresponding class of categories. There are several cross section normalization parameters, and a parameter that adjust the width of the ∆ mass resonance. Finally the systematics in the final state interactions (FSI) include a set of nuclear processes modelled in the NEUT secondary interactions cascade. Most important are the pion absorption and charge exchange processes, which were given separate categories.  9.5.3  Overall Systematic Covariance Matrix  The covariance matrix Vm,n used in this analysis is the sum of all the covariances corresponding to detector model and selected neutrino interaction model parameter systematic errors. It also includes the statistical error introduced by the size of the simulated data sample in to the predicted pµ − θµ distribution for the default values for the neutrino interaction model parameters. The overall covariance matrix used in the maximum likelihood fit is shown in Figure 9.9, where it can be seen that the final uncertainties in the number of pµ − θµ bin entries varies between 3.0% and 18.9%. This covariance matrix is not updated during the fitting process to reflect changes in the predicted pµ − θµ distribution as the change in the size of the MC statistical errors are negligible.  145  9.5. Systematic Errors Cross-section Systematic CCQE Axial Mass MAQE Res. Axial Mass MARES Binding Energy EB Fermi Momentum pF Spectral Function CC other shape CCQE Norm., Eν < 1.5 CCQE Norm., 1.5 < Eν < 3.5 CCQE Norm., Eν > 3.5 CC1π Norm., Eν < 2.5 CC1π Norm., Eν > 2.5 N C1π 0 Norm. CC coherent NC other FSI ∆ Mass Width π Absorption π Charge Exchange  Prior Value 1.21 GeV /c2 1.16 GeV /c2 25 M eV 217 M eV /c Off 0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 87.7 MeVc2 -  Prior Error 0.24 0.11 9 30 On 0.4 0.11 0.30 0.30 0.35 0.40 0.37 1.0 0.3 45.3 -  Propagation Method A A A A A A B B B B B B C C C C C C  Table 9.5: Neutrino interaction model systematic errors relevant to neutrino beam flux measurement. Propagation method A corresponds to rescaling the events contribution to the pµ − θµ distribution using T2KReWeight, B is an energy dependent scale factor, and C means the systematic effect is propagated through the pµ − θµ bin covariance matrix.  9.5.4  Interaction Model Reweighting  Most cross-section and neutrino interaction model systematic uncertainties are propagated to the energy spectrum measurement by simultaneously fitting for the model parameters that best fit the data with appropriate constraints. This is accomplished by producing pµ −θµ distributions for different values of the interaction model parameters, which are then used when calculating the logarithm of the likelihood function for a specific set of parameters. Instead of producing different versions of the simulated data with different sets of model parameters, the default simulated data is reweighted on an event by event basis in such a way that the effect is equivalent to a variation of the model parameter. This event by event reweighting was performed 146  9.5. Systematic Errors  40  0.035  35  0.03  p µ-θµ Bin #  30  0.025  25  0.02  20  0.015  15 0.01 10 0.005  5 0 0  0 5  10  15  20  25  p µ-θµ Bin #  30  35  40  Figure 9.9: Overall systematic covariance matrix, including detector systematic errors, relevant neutrino interaction model systematic errors and statistical errors due to the size of the simulated data sample.  using the T2KReWeight package which is described in T2K-TN-007 [72]. One exception to this procedure is the spectral function model parameter, which is a binary value that is fixed as “on” or “off” when running the fit. The difference in the fitted values between the two versions of the final fit to real data are treated as an additional systematic error that is combined with the fitted errors to produce the final error estimates. An additional processing step was introduced to allow the event-by-event reweighting to proceed more quickly in the fit application. The simulated data selected event trees were processed by a separate application that used T2KReWeight to calculate new event weights for a set of variations in each neutrino interaction model parameter. All of these alternative event weights were stored in an output tree along with the other details from the event needed to produce the pµ −θµ distribution. This new event summary tree was used in the fitting application to produce the predicted pµ − θµ distribution for a given set of neutrino interaction model parameter values. The weights used in the fit were calculated by interpolating between the appropriate  147  9.6. Fit Reliability Studies values stored in the new summary tree. Each neutrino interaction model parameter variation produces an individual weight, and the total weight for the event is the product of these individual weights. Reweighting with these predefined weights in the new event summary tree was significantly faster than reweighting each event with T2KReWeight for every step in the fitter minimization process.  9.6  Fit Reliability Studies  The maximum likelihood fit was tested with simulated data sets as a basic test of functionality and to determine if the expected true values were returned. If the true values are not returned than these fake data studies can be used to measure the extent of bias in fitted parameter and error estimates. These studies are necessary to demonstrate that the fitting algorithm produces accurate results and can be used to measure the neutrino energy spectrum using the observed ND280 data.  9.6.1  Fit to Default Simulated Data  As a first test the maximum likelihood fit was applied to the pµ −θµ distribution predicted by the simulation using the default set of neutrino interaction model parameters, as seen in Figure 9.5. The fitting algorithm uses the simulated data to predict pµ − θµ distribution, so fitting the simulated data distribution itself should result in fitted scale factor values consistent with 1. Table 9.6 and 9.7 summarizes the fit results and shows the fitted values agree with the expected value of 1 within the fitted errors. Interestingly for some nuisance parameters the fitted errors were smaller than the prior errors, demonstrating that the fit has some power to constrain systematics uncertainties.  9.6.2  Fake Data Studies  The performance of the maximum likelihood fit was tested using simulated data in order to demonstrate consistent convergence of the fitter and to test for bias in the parameter estimates and errors. Fitting multiple simulated datasets allows direct measurements of parameter bias as the true values of the parameters are known. Bias is detected by producing “pull” distributions for each fitted parameterwhere the pull g is calculated from the fitted  148  9.6. Fit Reliability Studies Parameter f0 f1 f2 f3 MACCQE MARES EB pF CC Other Shape CCQE Bin 0 CCQE Bin 1 CCQE Bin 2 CC1π Bin 0 CC1π Bin 1 N Cπ0 Bin 0  Fitted Value 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00  Fitted Error -0.22, 0.25 -0.12, 0.15 -0.17, 0.21 -0.23, 0.28 0.113 0.0934 0.345 0.079 0.377 0.104 0.251 0.261 0.244 0.311 0.370  Prior Error 0.198 0.0948 0.360 0.138 0.400 0.110 0.300 0.300 0.350 0.400 0.370  Table 9.6: Fit to default MC prediction of the pµ − θµ distribution, fitted flux scale factor and neutrino interaction model parameters.  parameter value τ and estimated error σ according to τ − τtrue [75]. (9.4) σ The true value of the fitted parameters in simulated data is 1 by definition. When the fitted parameter error estimates are asymmetric the definition of the pull is modified as shown by g=  If τ > τtrue then g =  τ − τtrue τtrue − τ , otherwise g = . σ− σ+  (9.5)  “Fake” datasets generated from the default MC pµ − θµ distribution are used to simulate the effect of numerical fluctuations on experimentally measured values. This is accomplished by randomly varying each pµ −θµ bin according to a Poisson distribution independently for every fake data set. The number of entries in a pµ − θµ bin predicted by the default simulated is used as the mean for the Poisson probability distribution used to set the number of entries in the corresponding bin for the fake data sets. Fitting multiple fake data sets is equivalent to running the same pseudo-experiment 149  9.6. Fit Reliability Studies Parameter D0 D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14 D15 D16 D17 D18 D19  Fitted Value 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00  Fitted Error 0.05 0.08 0.10 0.10 0.03 0.06 0.07 0.08 0.03 0.04 0.04 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.04 0.04  Parameter D20 D21 D22 D23 D24 D25 D26 D27 D28 D29 D30 D31 D32 D33 D34 D35 D36 D37 D38 D39  Fitted Value 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00  Fitted Error 0.06 0.09 0.13 0.10 0.05 0.11 0.13 0.12 0.04 0.09 0.12 0.10 0.05 0.08 0.10 0.09 0.04 0.04 0.04 0.05  Table 9.7: Results of fit to default MC prediction of the pµ −θµ distribution, pµ − θµ bin scale factors.  multiple times such that the variation in fitted values provides an estimate of the accuracy of the fit and error estimates. A subtlety when producing pull distributions using fake data sets is that the constraint term constants used in the maximum likelihood fit must also be varied according to the corresponding prior probability distribution [75]. Importantly the underlying neutrino interaction model parameters are not varied when producing the new pµ − θµ distribution, as the uncertainty in the model parameters is already included through the variations in the constraint terms. It is straightforward to vary the constraint constants for the individually fitted neutrino interaction model parameters using a Gaussian distribution with width equal to the parameter prior error, while the pµ − θµ bin nuisance parameter constraint constants are randomly generated from  150  9.6. Fit Reliability Studies the Cholesky decomposition of the overall covariance matrix. The fake data sets are equivalent to the default MC pµ − θµ distribution up to random numerical fluctuations and consequently represent the same underlying energy spectrum. The average of the fitted parameter values should therefore agree with the default simulation values within errors. Furthermore, the fitted values should be randomly distributed about the true value with a standard deviation equivalent to the estimated error. This can be directly tested by checking if the pull distributions for each parameter have means consistent with 0 and an RMS consistent with 1. The standard error of the mean is defined as 1 σg¯ = √ N and the standard error in the RMS estimate is defined as σs =  1 , 2(N − 1)  (9.6)  (9.7)  where N is the number of values used to define each pull distribution and is equal to the number of fitted fake datasets. These standard errors are used to determine if the measured mean of the pull distribution is consistent with the expected value of 0 and the RMS with 1. This is done by calculating the probability of measuring a deviation from the expected values as large or larger than those corresponding to the measured mean and RMS values. The probability of measuring a pull mean as large or larger than a given value g¯ is defined as P (|g| ≥ |¯ g|) = 1 − Erf √  g¯ , 2σg¯  (9.8)  and a similar probability for the pull RMS measurement sg is defined as 1 − sg P (|s| ≥ |sg |) = 1 − Erf √ . (9.9) 2σs These probabilities are calculated separately for the pull distribution of every fitted parameter. The significance level for this analysis is 5%, so if any of the probabilities are less than 5% the null hypothesis is rejected and the fitter is considered as biased for that parameter. One issue is that there are multiple fitted parameters and testing them all in the same manner increases the probability that normal fluctuations in the estimated values lead to erroneously declaring a parameter to be biased (a Type I error). To account for this the probability 151  9.6. Fit Reliability Studies threshold used in the test is adjusted depending on the number of parameters being compared. Toy MC is used to determine the appropriate probability threshold corresponding to the 5% significance level when considering different numbers of parameters, the results of which are summarized in Table 9.8. Number of Comparisons 1 4 11 40 55  5% Significance Probability 0.050 0.012 0.0045 0.0012 0.0009  Table 9.8: Effective 5% significance probability thresholds for different sized samples.  As an additional check, the distribution of all fitted parameter pull means and RMS should have variances should be consistent with 0 and 1 respectively. In this case the standard errors are calculated using Eq. 9.6 and Eq. 9.7 where N is the number of fitted parameters used in the fit. Statistics Only Fit The pull bias tests are initially applied to fits where only the flux scale factors are fit and systematic parameters and constraint terms are fixed to the default MC values. This is equivalent to including only statistical errors in the fitted results. The rationale for this test is that if the fit to fake data is biased when ignoring systematic errors it will likely be biased when they are included. It will also provide a sense of the size of the statistical error component for the fitted flux factors. The pull distributions resulting from this fit are shown in Figure 9.10, the measured mean and RMS values summarized in Table 9.9 and the probabilities summarized in Table 9.10. The calculated probabilities show that the fitted flux factors have pull distributions with mean and RMS values consistent with 0 and 1 respectively. It can be concluded that the statistics only version of the fit produces appropriate error estimates and is not significantly biased in any of the parameters.  152  9.6. Fit Reliability Studies Fitted Scale Factor Pull 0  Fitted Scale Factor Pull 1  180  Number of Entries / (0.5)  Number of Entries / (0.5)  180 160 140 120 100 80 60 40 20 0 -5  160 140 120 100 80 60 40 20  -4 -3  -2 -1  0  1  2  3  4  0 -5  5  -4 -3  Fitted Scale Factor Pull 0 Fitted Scale Factor Pull 2  1  2  3  4  5  4  5  180  Number of Entries / (0.5)  Number of Entries / (0.5)  0  Fitted Scale Factor Pull 3  180 160 140 120 100 80 60 40 20 0 -5  -2 -1  Fitted Scale Factor Pull 1  160 140 120 100 80 60 40 20  -4 -3  -2 -1  0  1  2  3  4  5  0 -5  -4 -3  Fitted Scale Factor Pull 2  -2 -1  0  1  2  3  Fitted Scale Factor Pull 3  Figure 9.10: Fitted flux scale factor pulls for fake data statistics only fits. The fake data pulls are shown in black with normal distribution shown in blue for comparison. Parameter f0 f1 f2 f3  Mean Value 0.990 1.002 1.001 1.003  Mean Error 0.140 0.039 0.089 0.084  Pull Mean 0.083 -0.041 -0.012 -0.032  Pull RMS 1.013 1.048 0.941 1.010  Table 9.9: Average of fitted flux scale factors, errors and mean and RMS of the pull distributions. The simulated data sample is 500 fake data statistics only fits.  Fit with Neutrino Interaction Model Systematics The pull bias tests were applied to the version of the fit where only flux scale factors and the neutrino interaction model nuisance parameters were fitted. The pµ − θµ bin nuisance parameters were held fixed at the default value of 1 153  9.6. Fit Reliability Studies Parameter  Pull Mean  f0 f1 f2 f3  0.083 -0.041 -0.012 -0.032  Pull Mean Probability 0.064 0.357 0.784 0.471  Pull RMS 1.013 1.048 0.941 1.010  Pull RMS Probability 0.689 0.130 0.063 0.742  Table 9.10: Probability of measuring deviations from the expected pull and RMS values greater than or equal to the measured pull mean and RMS values for the flux factor pull distributions. The data sample is 500 fake data statistics only fits.  and the associated constraint terms were not varied. The pull distributions resulting from this fit are show in Figure 9.11, the measured mean and RMS values of the fitted parameter pull distributions summarized in Table 9.11 and the corresponding probabilities summarized in Table 9.12. The calculated probabilities show that the fitted flux factors have pull distributions with mean and RMS values consistent with 0 and 1 respectively, and so this version fit also does not introduce bias or produce incorrect error estimates for the flux scale factors. Energy Bin f0 f1 f2 f3  Mean Value 1.011 1.022 1.019 1.016  Mean Error 0.198 0.128 0.180 0.246  Pull Mean 0.031 -0.040 0.012 0.035  Pull RMS 1.000 1.038 1.003 1.003  Table 9.11: Average of fitted flux scale factors, errors and mean and RMS of the pull distributions. The data sample is 500 fake data fits with neutrino interaction model systematics included.  Table 9.13 shows the average fitted scale factor and pull distribution mean and RMS measurements for neutrino interaction model parameters included in this version of the fit, and Figure 9.12 the associated pull distributions. The neutrino interaction scale factors are all consistent with the expected value of 1 within the average fitted errors. The large increase in the size of the error estimates when neutrino interaction model parameters are included compared to the statistical fit shows the effects of the systematic uncertainty induced by uncertainties in the neutrino interaction model. 154  9.6. Fit Reliability Studies  180  Number of Entries / (0.5)  Number of Entries / (0.5)  180 160 140 120 100 80 60 40 20 0 -5  160 140 120 100 80 60 40 20  -4 -3  -2 -1  0  1  2  3  4  0 -5  5  Fitted Scale Factor Pull 0  0  1  2  3  4  5  4  5  180  Number of Entries / (0.5)  Number of Entries / (0.5)  -2 -1  Fitted Scale Factor Pull 1  180 160 140 120 100 80 60 40 20 0 -5  -4 -3  160 140 120 100 80 60 40 20  -4 -3  -2 -1  0  1  2  3  4  5  0 -5  Fitted Scale Factor Pull 2  -4 -3  -2 -1  0  1  2  3  Fitted Scale Factor Pull 3  Figure 9.11: Fitted flux scale factor pulls with normal distribution shown in blue for fake data fits with neutrino interaction model systematics included. Parameter  Pull Mean  f0 f1 f2 f3  0.031 -0.040 0.012 0.035  Pull Mean Probability 0.488 0.377 0.788 0.439  Pull RMS 1.000 1.038 1.003 1.003  Pull RMS Probability 0.994 0.230 0.933 0.931  Table 9.12: Probability of measuring deviations from the expected pull and RMS values greater than or equal to the measured pull mean and RMS values for the flux factor pull distributions. The data sample is 500 fake data fits with neutrino interaction model systematics included.  In all four bins this neutrino interaction model systematic dominates over the purely statistical uncertainty. Additionally Table 9.14 shows the measured pull mean and RMS values are consistent with the values expected for normal distributions within standard errors. 155  9.6. Fit Reliability Studies Parameter MACCQE MARES EB pF CC Other Shape CCQE Bin 0 CCQE Bin 1 CCQE Bin 2 CC1π Bin 0 CC1π Bin 1 N Cπ0 Bin 0  Mean Fitted Value 1.004 0.997 0.968 0.997 0.988 1.003 1.022 0.994 0.998 1.015 0.978  Mean Error 0.132 0.089 0.342 0.079 0.372 0.103 0.239 0.256 0.225 0.297 0.370  Mean Pull -0.082 -0.008 -0.093 0.003 -0.039 0.022 0.058 -0.035 -0.058 -0.010 -0.059  Pull RMS 1.123 1.034 1.024 1.045 1.049 1.045 0.963 0.993 0.982 0.989 0.934  Table 9.13: Fitted neutrino interaction model scale factors fake data from fake data fits with neutrino interaction model systematics included. Parameter  Pull Mean  MACCQE MARES EB pF CC Other Shape CCQE Bin 0 CCQE Bin 1 CCQE Bin 2 CC1π Bin 0 CC1π Bin 1 N Cπ0 Bin 0  -0.0820 -0.0077 -0.0934 0.0035 -0.0394 0.0224 0.0578 -0.0353 -0.0577 -0.0098 -0.0588  Pull Mean Probability 0.0677 0.8634 0.0375 0.9386 0.3801 0.6179 0.1978 0.4312 0.1989 0.8281 0.1902  Pull RMS 1.1239 1.0338 1.0236 1.0456 1.0493 1.0447 0.9625 0.9935 0.9823 0.9888 0.9379  Pull RMS Probability 0.0001 0.2878 0.4575 0.1517 0.1207 0.1594 0.2375 0.8383 0.5780 0.7245 0.0508  Table 9.14: Probability of measuring deviations from the expected pull and RMS values greater than or equal to the measured pull mean and RMS values for the fitted neutrino interaction model scale factor pull distributions. The data sample is 500 fake data fits with neutrino interaction model systematics included.  156  9.6. Fit Reliability Studies  160 140 120 100 80 60 40 20 0 -5  180  Number of Entries / (0.5)  180  Number of Entries / (0.5)  Number of Entries / (0.5)  180  160 140 120 100 80 60 40 20  -4  -3  -2  -1  0  1  2  3  4  0 -5  5  -3  120 100 80 60 40 20 -2  -1  0  1  2  3  4  2  3  4  100 80 60 40 20  120 100 80 60 40  0  1  2  -4  -3  3  4  -3  -2  -1  0  1  2  3  4  0  1  2  3  4  5  3  4  5  3  4  5  140 120 100 80 60 40  -4  -3  -2  -1  0  1  2  Fitted Scale Factor Pull 5  180  140 120 100 80 60 40  160 140 120 100 80 60 40 20  -4  -3  Fitted Scale Factor Pull 6  -2  -1  0  1  2  3  4  5  Fitted Scale Factor Pull 7  0 -5  -4  -3  -2  -1  0  1  2  Fitted Scale Factor Pull 8  180  Number of Entries / (0.5)  180  -1  160  0 -5  5  160  0 -5  5  -2  20 -4  20 -1  40  180  Number of Entries / (0.5)  120  -2  60  Fitted Scale Factor Pull 2  180  140  -3  80  Fitted Scale Factor Pull 4  160  -4  100  0 -5  5  140  0 -5  5  Number of Entries / (0.5)  Number of Entries / (0.5)  1  20  180  Number of Entries / (0.5)  0  160  Fitted Scale Factor Pull 3  160 140 120 100 80 60 40 20 0 -5  -1  Number of Entries / (0.5)  Number of Entries / (0.5)  Number of Entries / (0.5)  140  0 -5  -2  180  160  -3  120  Fitted Scale Factor Pull 1  180  -4  140  20 -4  Fitted Scale Factor Pull 0  0 -5  160  160 140 120 100 80 60 40 20  -4  -3  -2  -1  0  1  2  3  4  5  Fitted Scale Factor Pull 9  0 -5  -4  -3  -2  -1  0  1  2  3  4  5  Fitted Scale Factor Pull 10  Figure 9.12: Fitted neutrino interaction parameter scale factor pulls with normal distribution shown in blue for fake data fits with neutrino interaction model systematics included.  Fit with All Systematics The fake data studies were repeated for the version of the fit where all systematic nuisance parameters are included, with 500 fake datasets fitted in total. The average fitted flux scale factors and errors are shown in Table 9.15 along with the mean and RMS of the pull distributions. The error estimates produced when all systematic parameters are included in the fit are only slightly larger than for the version of the fit where only the neutrino interaction model parameters are included as shown in 9.11, with the exception of the lowest energy flux bin. Figure 9.13 shows the pull distribution, where pull values are calculated according to 9.5. Table 9.26 shows the probability of observing values equal to or greater than the measured pull mean and RMS values under the assumption that they are distributed 157  9.6. Fit Reliability Studies by a normal distribution about 0 and 1 respectively with a width equal to the standard error. These probabilities are all greater than that required at the 5% significance and therefore the flux scale factor pull distributions’ mean and RMS values are consistent with the values expected for normal distributions. Energy Bin 0 1 2 3  Mean Value 1.020 1.014 1.031 1.018  Mean Error 0.240 0.134 0.191 0.259  Pull Mean 0.019 0.033 -0.031 0.028  Pull RMS 1.024 1.026 1.029 0.962  Table 9.15: Average of fitted flux scale factors, errors and mean and RMS of the pull distributions. The data sample is 500 fake data fits where all systematics nuisance parameters are included in the fit.  Energy Bin  Pull Mean  0 1 2 3  0.019 0.033 -0.031 0.028  Pull Mean Probability 0.679 0.456 0.493 0.533  Pull RMS 1.024 1.026 1.029 0.962  Pull RMS Probability 0.444 0.416 0.358 0.228  Table 9.16: Probability of the measured pull mean and RMS values for the flux factor pull distributions assuming a normal distribution with width equal to the standard errors. The data sample is 500 fake data fits where all systematics nuisance parameters are included in the fit.  Table 9.17 shows the average fitted scale factor and pull distribution mean and RMS measurements for neutrino interaction model parameters included in the fit, and Figure 9.14 the associated pull distributions. The neutrino interaction scale factors are all consistent with the expected value of 1 within the average fitted errors. The measured pull mean and RMS values are consistent with the values expected for normal distributions within standard errors as shown by the calculated probabilities in Table 9.18. Tables 9.19 and 9.19 show the average pµ − θµ bin nuisance parameter values from the 500 fake data studies where all the systematic parameters were included. The fitted values are consistent with the value expected when 158  9.6. Fit Reliability Studies  180  Number of Entries / (0.5)  Number of Entries / (0.5)  180 160 140 120 100 80 60 40 20 0 -5  160 140 120 100 80 60 40 20  -4 -3  -2 -1  0  1  2  3  4  0 -5  5  Fitted Scale Factor Pull 0  0  1  2  3  4  5  4  5  180  Number of Entries / (0.5)  Number of Entries / (0.5)  -2 -1  Fitted Scale Factor Pull 1  180 160 140 120 100 80 60 40 20 0 -5  -4 -3  160 140 120 100 80 60 40 20  -4 -3  -2 -1  0  1  2  3  Fitted Scale Factor Pull 2  4  5  0 -5  -4 -3  -2 -1  0  1  2  3  Fitted Scale Factor Pull 3  Figure 9.13: Fitted scale flux scale factor pulls with normal distribution shown in blue for fake data fits with all systematics included.  fitting to the default simulation prediction of the pµ − θµ distribution, and the pull distribution mean and RMS values are consistent with 0 and 1 as expected for normal distributions. The flux scale factors, neutrino interaction model parameters and pµ −θµ bin nuisance parameter average fitted values were all consistent with the expected default MC predicted values, and the pull distributions universally had means and RMS consistent those expected for normal distributions. All of the fitted parameter pull mean and RMS values had reasonable probabilities under the hypothesis that they were distributed about 0 and 1 respectively. Figure 9.16 shows the distribution of the mean values measured in all fitted parameter pull distributions, and Figure 9.17. The distribution of pull mean values has width comparable to the standard error expected in a sample with 55 measurements (0.13), which is further confirmation that the fitted parameter pulls are distributed about the expected value of 0. While distribution of pull RMS values is slightly asymmetric the values are distributed around 1 within the standard error. These results demonstrate 159  9.6. Fit Reliability Studies Parameter MACCQE MARES EB pF CC Other Shape CCQE Bin 0 CCQE Bin 1 CCQE Bin 2 CC1π Bin 0 CC1π Bin 1 N Cπ0 Bin 0 Table 9.17: results.  Mean Value 1.000 0.993 1.009 1.004 0.978 1.009 1.007 0.994 1.010 1.031 0.998  Mean Error 0.144 0.090 0.346 0.084 0.380 0.107 0.254 0.262 0.258 0.318 0.370  Mean Pull -0.078 -0.074 0.030 0.073 -0.063 0.081 -0.002 -0.034 -0.014 0.037 -0.004  Pull RMS 1.037 0.960 0.952 0.982 0.980 0.961 0.964 0.973 0.991 1.014 0.971  Fitted neutrino interaction model scale factors fake data fit  Parameter  Pull Mean  MACCQE MARES EB pF CC Other Shape CCQE Bin 0 CCQE Bin 1 CCQE Bin 2 CC1π Bin 0 CC1π Bin 1 N Cπ0 Bin 0  -0.078 -0.074 0.030 0.073 -0.063 0.081 -0.002 -0.034 -0.014 0.037 -0.004  Pull Mean Probability 0.081 0.096 0.502 0.103 0.162 0.071 0.972 0.446 0.754 0.411 0.921  Pull RMS 1.037 0.960 0.952 0.982 0.980 0.961 0.964 0.973 0.991 1.014 0.971  Pull RMS Probability 0.238 0.207 0.133 0.560 0.518 0.220 0.261 0.393 0.787 0.655 0.363  Table 9.18: Probability of measuring deviations from the expected pull and RMS values greater than or equal to the measured pull mean and RMS values for the fitted neutrino interaction model scale factor pull distributions. The data sample is 500 fake data fits with all systematics included.  that fitting algorithm does not introduce significant bias and the error estimates are realistic when all systematic parameters are propagated. It is 160  9.7. Fit of the Observed Data  160 140 120 100 80 60 40 20 0 -5  180  Number of Entries / (0.5)  180  Number of Entries / (0.5)  Number of Entries / (0.5)  180  160 140 120 100 80 60 40 20  -4  -3  -2  -1  0  1  2  3  4  0 -5  5  -3  140 120 100 80 60 40 20 -2  -1  0  1  2  3  4  2  3  4  120 100 80 60 40  140 120 100 80 60 40  -1  0  1  2  40  -4  -3  3  4  -3  -2  -1  0  1  2  3  4  Number of Entries / (0.5)  140 120 100 80 60 40 20  1  2  3  4  5  3  4  5  3  4  5  160 140 120 100 80 60 40  -4  -3  -2  -1  0  1  2  180  120 100 80 60 40  160 140 120 100 80 60 40 20  -4  -3  -2  -1  0  1  2  3  4  5  Fitted Scale Factor Pull 7  160  0  Fitted Scale Factor Pull 5  140  Fitted Scale Factor Pull 6  180  -1  180  0 -5  5  160  0 -5  5  -2  20 -4  20 -2  60  Fitted Scale Factor Pull 2  Number of Entries / (0.5)  140  -3  80  0 -5  5  180  160  -4  100  Fitted Scale Factor Pull 4  Number of Entries / (0.5)  Number of Entries / (0.5)  1  160  0 -5  5  20  Number of Entries / (0.5)  0  20  180  0 -5  -1  180  Fitted Scale Factor Pull 3  0 -5  -2  Number of Entries / (0.5)  Number of Entries / (0.5)  Number of Entries / (0.5)  160  -3  120  Fitted Scale Factor Pull 1  180  -4  140  20 -4  Fitted Scale Factor Pull 0  0 -5  160  0 -5  -4  -3  -2  -1  0  1  2  Fitted Scale Factor Pull 8  180 160 140 120 100 80 60 40 20  -4  -3  -2  -1  0  1  2  Fitted Scale Factor Pull 9  3  4  5  0 -5  -4  -3  -2  -1  0  1  2  3  4  5  Fitted Scale Factor Pull 10  Figure 9.14: Fitted neutrino interaction parameter scale factor pulls with normal distribution shown in blue for fake data fits with all systematics included.  possible that the fitted parameters values are biased at a level smaller than the standard errors associated with the fake data samples, but this is bias is much less than the fitted error estimates and not significant. The fit can therefore be applied to the observed pµ −θµ distribution and the results used to provide an unbiased estimate of the real neutrino flux energy spectrum in the near detector.  9.7  Fit of the Observed Data  The previous section presented three versions of the maximum likelihood fit, a statistics only version, a statistics and neutrino interaction model systematics version and a version where all systematics are included. The versions of the maximum likelihood fit that include neutrino interaction model pa161  9.7. Fit of the Observed Data Bin 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  Mean Value 0.998 0.998 1.008 1.003 1.001 0.994 0.997 1.006 1.001 1.003 0.998 1.001 1.004 1.001 0.998 1.002 0.998 1.000 1.001 1.001  Mean Error 0.050 0.076 0.097 0.096 0.032 0.062 0.072 0.084 0.028 0.037 0.045 0.055 0.049 0.055 0.054 0.053 0.051 0.051 0.045 0.041  Pull Mean 0.031 0.032 -0.065 -0.014 -0.037 0.103 0.049 -0.059 -0.047 -0.065 0.041 -0.009 -0.083 -0.004 0.048 -0.026 0.041 0.012 -0.014 -0.018  Pull RMS 1.014 0.943 1.035 1.010 0.977 1.047 0.983 1.031 0.947 1.005 1.013 0.974 0.940 1.058 0.939 0.949 1.045 0.937 1.010 0.953  Table 9.19: Fitted pµ − θµ bin scale factors fake data fit results when all systematics are included for bins 0 to 19.  rameters are applied to the observed data distribution twice, first with the spectral function parameter set to “off” and the second with parameter “on”. The reported results correspond to the case where the spectral function was not used, while the difference between the fitted parameter values obtained through the two versions of the fit are taken to be an additional systematic error, and combined with the fitted errors appropriately.  9.7.1  Statistics Only Fit  The results of the statistics only fit to real data are shown in Table 9.21. Note that in this case the spectral function systematic error is not included in the final error estimates. The results show significantly less neutrino interactions in energy bin 1 (500 to 1000 MeV) than predicted, while the 162  9.7. Fit of the Observed Data Bin 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  Mean Value 1.000 0.998 1.001 1.000 1.002 1.002 1.005 0.995 1.001 1.002 0.997 1.000 1.001 0.999 1.002 1.001 0.999 1.001 1.000 1.000  Mean Error 0.060 0.095 0.127 0.096 0.053 0.111 0.135 0.119 0.042 0.088 0.118 0.107 0.052 0.077 0.098 0.094 0.043 0.040 0.039 0.049  Pull Mean 0.019 0.043 0.028 0.025 -0.023 -0.004 -0.010 0.075 -0.014 -0.006 0.058 0.039 -0.020 0.025 -0.003 0.017 0.020 -0.011 -0.002 0.034  Pull RMS 0.979 0.952 1.051 0.980 0.941 0.972 0.968 0.986 0.957 0.993 1.015 0.975 0.992 1.029 1.071 0.950 0.957 0.994 1.017 0.994  Table 9.20: Fitted pµ − θµ bin scale factors fake data fit results when all systematics are included for bins 20 to 39.  other energy bin flux factors are roughly consistent with the MC prediction. The statistical error in fitted flux factors is generally 10%.  9.7.2  Fit with Neutrino Interaction Model Systematics  The results of the fit to real data where neutrino interaction model parameters are included is shown in Table 9.22. As expected the flux factor error estimates are larger after including neutrino interaction model parameters, and consequently all fitted flux factors are consistent with the default MC prediction.  163  9.7. Fit of the Observed Data  40 20 -4  -3  100 80 60 40 20 -4  -3  -2  -1  0  1  2  3  4  5  120 100 80 60 40 20 -3  -2  -1  0  1  2  3  4  140 120 100 80 60 40  80 60 40 20 -2  -1  0  1  2  3  4  -4  -3  0  1  2  3  4  5  120 100 80 60 40  160  120 100 80 60 40  -3  -2  -1  0  1  2  -3  3  4  -2  -1  0  1  2  3  4  120 100 80 60 40  -2  -1  0  1  2  3  4  140 120 100 80 60 40  100 80 60 40 20  -4  -3  -2  -1  0  1  2  3  4  -1  0  1  2  3  4  140  100 80 60 40  -3  -2  -1  0  1  2  3  4  80 60 40 20  160  120 100 80 60 40 20  -4  -3  -2  -1  0  1  2  3  4  0 -5  5  -3  -2  -1  0  1  2  3  4  120 100 80 60 40 20  160 140 120 100 80 60 40 20  -2  -1  0  1  2  Fitted Scale Factor Pull 35  3  4  5  0-5  0  1  2  3  4  160  120 100 80 60 40  -3  -2  -1  0  1  2  3  4  -3  -2  -1  0  1  2  -3  3  4  5  Fitted Scale Factor Pull 36  -1  0  1  2  3  4  120  80 60 40  -2  -1  0  1  2  3  4  120 100 80 60 40  140  100 80 60 40  -3  -2  -1  0  1  2  3  4  -3  -2  -1  0  1  2  3  4  120 100 80 60 40  160  120 100 80 60 40  -3  -2  -1  0  1  -3  -2  -1  0  1  2  3  4  2  3  4  160  120 100 80 60 40  -2  -1  0  1  2  3  4  5  Fitted Scale Factor Pull 37  140 120 100 80 60 40  3  4  5  2  3  4  5  3  4  5  3  4  5  3  4  5  3  4  5  3  4  5  80 60 40  -4  -3  -2  -1  0  1  160 140 120 100 80 60 40  -4  -3  -2  -1  0  1  2  160 140 120 100 80 60 40 20  -4  -3  -2  -1  0  1  2  3  4  0-5  5  -4  -3  -2  -1  0  1  2  Fitted Scale Factor Pull 24  180  160 140 120 100 80 60 40  160 140 120 100 80 60 40 20  -4  -3  -2  -1  0  1  2  3  4  0-5  5  -4  -3  -2  -1  0  1  2  Fitted Scale Factor Pull 29  180  160 140 120 100 80 60 40  160 140 120 100 80 60 40 20  -4  -3  -2  -1  0  1  2  3  4  0 -5  5  -4  -3  -2  -1  0  1  2  Fitted Scale Factor Pull 34  180  160 140 120 100 80 60 40  0-5  1  Fitted Scale Factor Pull 19  20 -3  0  100  Fitted Scale Factor Pull 33  140  -1  180  160  0 -5  5  -2  120  0-5  5  180  -4  -3  20 -4  20 -4  -4  160 140  Fitted Scale Factor Pull 14  180  160  2  60 40  Fitted Scale Factor Pull 28  140  5  180  140  0-5  5  4  80  0 -5  5  20 -4  3  20 -4  180  120  2  100  Fitted Scale Factor Pull 23  160  1  Fitted Scale Factor Pull 9  160  0-5  5  0  180  20 -3  -1  140  0-5  5  180  100  -2  120  Fitted Scale Factor Pull 18  140  0-5  -2  140  0-5  5  160  -4  -3  20 -4  20 -4  -4  160  Fitted Scale Factor Pull 13  20 -4  60 40  0 -5  5  180  Number of Entries / (0.5)  Number of Entries / (0.5)  160  -1  40  Fitted Scale Factor Pull 4  80  Fitted Scale Factor Pull 32  180  140  -2  80 60  0-5  5  180  Fitted Scale Factor Pull 31  180  -3  -3  140  0 -5  5  4  20 -4  20 -4  Fitted Scale Factor Pull 30  -4  60 40  180  140  3  100  Fitted Scale Factor Pull 27  Number of Entries / (0.5)  Number of Entries / (0.5)  120 100  2  Fitted Scale Factor Pull 8  80  0-5  5  180  160  5  100  Fitted Scale Factor Pull 26  140  4  20 -4  Fitted Scale Factor Pull 25  180  3  180  120  1  140  Fitted Scale Factor Pull 22  160  0-5  5  2  120  0-5  5  20 -2  1  20  Number of Entries / (0.5)  Number of Entries / (0.5)  140  0  180  160  0  180  Fitted Scale Factor Pull 17  180  120  -1  160  0-5  5  -1  120  0-5  20 -3  -2  160  Fitted Scale Factor Pull 12  160  -4  -3  180  Fitted Scale Factor Pull 21  180  -2  140  0 -5  5  140  0-5  5  160  -3  -3  120 100  20 -4  20  180  Fitted Scale Factor Pull 20  -4  60 40  -4  160 140  Fitted Scale Factor Pull 3  20 -4  20 -4  0-5  5  Fitted Scale Factor Pull 7  Number of Entries / (0.5)  140  20  Number of Entries / (0.5)  -1  180  Number of Entries / (0.5)  Number of Entries / (0.5)  180  4  80  Fitted Scale Factor Pull 16  160  3  180  Fitted Scale Factor Pull 15  Number of Entries / (0.5)  -2  140  0-5  5  2  100  0-5  20 -3  1  140  20  Number of Entries / (0.5)  Number of Entries / (0.5)  Number of Entries / (0.5)  120 100  0  120  0-5  180  160  -1  180  Fitted Scale Factor Pull 11  140  -2  160  20  0 -5  5  180  -4  -3  Fitted Scale Factor Pull 2  20 -4  -4  180  Fitted Scale Factor Pull 10  Number of Entries / (0.5)  0-5  5  Number of Entries / (0.5)  Number of Entries / (0.5)  Number of Entries / (0.5)  160  0-5  4  180  140  0 -5  3  Fitted Scale Factor Pull 6  180  0-5  2  160  Fitted Scale Factor Pull 5  0-5  1  Number of Entries / (0.5)  140 120  0-5  0  180  Number of Entries / (0.5)  Number of Entries / (0.5)  180  0 -5  -1  Fitted Scale Factor Pull 1  160  0-5  -2  Number of Entries / (0.5)  0-5  5  40  Number of Entries / (0.5)  4  80 60  Number of Entries / (0.5)  3  120 100  Number of Entries / (0.5)  2  Number of Entries / (0.5)  1  Number of Entries / (0.5)  0  Fitted Scale Factor Pull 0  180  160 140  20  Number of Entries / (0.5)  -1  40 20  Number of Entries / (0.5)  -2  80 60  Number of Entries / (0.5)  -3  120 100  Number of Entries / (0.5)  -4  180  160 140  Number of Entries / (0.5)  80 60  Number of Entries / (0.5)  0-5  120 100  Number of Entries / (0.5)  40 20  180  160 140  Number of Entries / (0.5)  80 60  Number of Entries / (0.5)  120 100  Number of Entries / (0.5)  160 140  Number of Entries / (0.5)  180  Number of Entries / (0.5)  Number of Entries / (0.5)  180  160 140 120 100 80 60 40 20  -4  -3  -2  -1  0  1  2  3  4  5  0-5  -4  Fitted Scale Factor Pull 38  -3  -2  -1  0  1  2  Fitted Scale Factor Pull 39  Figure 9.15: Fitted pµ − θµ bin nuisance parameter pulls with normal distributions shown in blue for fake data fits with all systematics included. Flux Factor f0 f1 f2 f3  Fitted Value SF = 0 1.18 0.89 0.93 0.96  Fitted Error -0.14,+0.14 -0.04,+0.04 -0.09,+0.09 -0.08,+0.08  Table 9.21: Results of the statistics only fit to the observed data distribution, where the spectral function interaction model parameter (SF) is turned off.  9.7.3  Fit with All Systematics  The results of the fit where all systematic parameters are included is shown in Table 9.23. The final error estimates are slightly larger after including the pµ − θµ bin nuisance parameters as compared to the version of the 164  9.7. Fit of the Observed Data  16  Number of Entries / (0.02)  14 12 10 8 6 4 2 0 -0.2  -0.15  -0.1 -0.05 0 0.05 0.1 0.15 Mean of Paramater Fit Pull Distribution  0.2  Figure 9.16: Distribution of mean values of fitted parameter pull distributions for fake data fits with all systematics included.  Number of Entries / (0.02)  12 10 8 6 4 2 0 0.8  0.85  0.9 0.95 1 1.05 1.1 1.15 RMS of Paramater Fit Pull Distribution  1.2  Figure 9.17: Distribution of RMS values of fitted parameter pull distributions for fake data fits with all systematics included.  165  9.7. Fit of the Observed Data Flux Factor f0 f1 f2 f3  Fitted Value SF = 0 1.08 0.90 0.85 0.91  Fitted Error -0.19,+0.21 -0.10,+0.12 -0.13,+0.15 -0.22,+0.26  Fitted Value SF = 1 1.01 0.98 0.87 0.89  Final Error -0.20,+0.22 -0.13,+0.14 -0.14,+0.16 -0.22,+0.26  Table 9.22: Results of the fit to the observed data distribution where neutrino interaction model parameters are included.  fit with only neutrino interaction model parameters included. The fitted flux factors are consistent with the default MC prediction within the final estimated errors. Flux Factor f0 f1 f2 f3  Fitted Value SF = 0 1.10 0.93 0.85 0.92  Fitted Error -0.24,+0.28 -0.11,+0.14 -0.14,+0.17 -0.23,+0.28  Fitted Value SF = 1 1.04 1.02 0.88 0.90  Final Error -0.24,+0.28 -0.14,+0.17 -0.14,+0.17 -0.23,+0.28  Table 9.23: Results of fit to the observed data distribution when all systematic parameters are included.  9.7.4  Comparison with Beam Flux Prediction  The predicted muon neutrino beam flux at the ND280 detectors is shown in Figure 2.10, and versions with coarser neutrino energy bins is shown in 9.18 and 9.19. Only the muon neutrino flux is shown as the contribution from other types of neutrinos to the selected event sample is negligible, as shown in Figure 9.20. The neutrino flux prediction provided by the beam group is compared with the results of the three version of the fit in Table 9.24. The selected neutrino energy spectrum for the statistics only fit is shown in Figure 9.21, where the black curve shows the spectrum predicted by the default simulation parameters. The blue curve shows the spectrum predicted when using the neutrino interaction model parameters selected by the fit, but still using the default flux factor values. The difference between the blue curve and the fitted spectrum gives a sense of the size of the dis166  9.7. Fit of the Observed Data crepancy between data and MC due to the flux model, while the difference between the black and the blue curve shows the extent to which neutrino interaction model parameter systematics affect the predicted energy spectrum. As expected, since the neutrino interaction model parameters are fixed in the statistics only fit the black and blue curves overlap. Similarly the fitted energy spectrum from the version of the fit including neutrino interaction model parameters is shown in Figure 9.22, and the version with all systematic parameters included is shown in Figure 9.23. An alternative plot of the fitted neutrino energy spectrum where the number of selected events is divided by the width of the energy bin is shown in Figure 9.24. The fitted correlation between neutrino beam flux factors and the neutrino interaction model parameters constrained by the fit is shown in Figure 9.25. There are significant correlations between the fitted neutrino flux factors, while the flux factors are anti-correlated with many of the neutrino interaction model parameters. Interestingly the uncertainties in the neutrino interaction model parameters themselves are relatively uncorrelated. These anti-correlations reduce the uncertainty in the predicted number of events at the far-detectors when the full set of constraints are propagated in an oscillation analysis. As was done with the BANFF fit this can significantly reduce the size of the systematic error in the number of far detector events and improves the precision of neutrino oscillation parameter measurements. The fitted flux factors measured in this analysis are consistent with the beam flux predictions and the results of the BANFF fit, although the fractional uncertainties are significantly larger at the lowest and highest energies, but roughly comparable around the beam’s peak energy. While the results presented here do not provide significantly better constraints for the neutrino beam flux, it provides an independent cross-check of the beam model and BANFF analysis results. Only ND280 data was used to measure the beam flux factors in this analysis, and finding that they are consistent with the existing flux predictions provides additional evidence that the beam model is accurate and suitable for use in neutrino oscillation analyses.  167  Flux (/cm2/1021POT/Energy Bin)  9.7. Fit of the Observed Data  3500  ×109  3000 2500 2000 1500 1000 500 0 0  5000  10000  15000  20000  25000  30000  Neutrino Energy (MeV)  Figure 9.18: Predicted νµ flux at the ND280 detectors using flux tuning version 11b with coarser bins.  Flux (/cm2/1021POT/Energy Bin)  ×1012 12 10 8 6 4 2 0 0  1000 2000 3000 4000 5000 6000 7000 8000 900010000  Neutrino Energy (MeV)  Figure 9.19: Predicted νµ flux at the ND280 detectors using flux tuning version 11b with neutrino energy binning used in fit.  168  Number of Entries / (Sample)  9.7. Fit of the Observed Data  ν Type νµ νµ νe νe  2500 2000 1500 1000 500 0  CCQE Sample  CCnQE Sample  Figure 9.20: Selected event CCQE and CCnQE sample sizes with contributions shown by neutrino type.  Flux Factor f0 f1 f2 f3  Beam Flux Prediction 1.00 ± 0.132 1.00 ± 0.117 1.00 ± 0.111 1.00 ± 0.120  Statistics Only Fit 1.18+0.14 −0.14 0.89+0.04 −0.04 0.93+0.09 −0.09 0.96+0.08 −0.08  Interaction Systematics Fit 1.08+0.22 −0.20 0.90+0.14 −0.13 0.85+0.16 −0.14 0.91+0.26 −0.22  All Systematics Included in Fit 1.10+0.28 −0.24 0.93+0.17 −0.14 0.85+0.17 −0.14 0.92+0.28 −0.23  Table 9.24: Comparison of fit results and beam flux prediction.  169  9.7. Fit of the Observed Data  2000 1800  Number of Entries  1600 1400 1200 1000 800 600 400 200 0 0  1000 2000 3000 4000 5000 6000 7000 8000 9000 10000  Neutrino Energy (MeV)  Figure 9.21: Measured neutrino energy spectrum of selected events in each energy bin for the statistics only version of the fit. The neutrino energy spectrum predicted by the default simulation is shown in black, the one with fitted neutrino interaction model parameters shown in blue, and the neutrino flux measurement marked with error bars. As the neutrino model parameters were held fixed, the black and blue curves overlap.  2000 1800  Number of Entries  1600 1400 1200 1000 800 600 400 200 0 0  1000 2000 3000 4000 5000 6000 7000 8000 9000 10000  Neutrino Energy (MeV)  Figure 9.22: Measured neutrino energy spectrum of selected events in each energy bin for the version of the fit that includes neutrino interaction model parameters. The neutrino energy spectrum predicted by the default simulation is shown in black, the one with the fitted neutrino interaction model parameters shown in blue, and the neutrino flux measurement marked with error bars.  170  9.7. Fit of the Observed Data  2000 1800  Number of Entries  1600 1400 1200 1000 800 600 400 200 0 0  1000 2000 3000 4000 5000 6000 7000 8000 9000 10000  Neutrino Energy (MeV)  Figure 9.23: Measured neutrino energy spectrum of selected events in each energy bin for the version of the fit that includes neutrino interaction model parameters and detector systematic parameters. The neutrino energy spectrum predicted by the default simulation is shown in black, the one with fitted neutrino interaction model parameters shown in blue, and the neutrino flux measurement marked with error bars.  Number of Entries / (MeV)  3.5 3 2.5 2 1.5 1 0.5 0 0  1000 2000 3000 4000 5000 6000 7000 8000 9000 10000  Neutrino Energy (MeV)  Figure 9.24: Measured neutrino energy spectrum of selected events divided by the width of each energy bin for the version of the fit that includes all systematic parameters. The neutrino energy spectrum predicted by the default simulation is shown in black, the one with fitted neutrino interaction model parameters shown in blue, and the neutrino flux measurement marked with error bars.  171  9.7. Fit of the Observed Data  1  Fit Parameters  14  0.8  12  0.6  10  0.4  8  0.2  6  0 -0.2  4  -0.4 2 0 0  -0.6 2  4  6  8  10  12  14  Fit Parameters  Figure 9.25: Selected neutrino interaction energy spectrum measurement correlation between fitted flux and neutrino interaction model parameters. The bins are ordered as follows: (0-3) = neutrino flux factor, (4) = MAQE , (5) = MARES , (6) = EB , (7) = pF , (8) = CC-other shape, (9) = CCQE Norm. 0, (10) = CCQE Norm. 1, (11) = CCQE Norm. 2, (12) = CC1π Norm. 0, (13) = CC1π Norm. 1, (14) = NC1π 0 Norm.  172  9.8. Alternative Relative Energy Spectrum Likelihood Fit  9.8  Alternative Relative Energy Spectrum Likelihood Fit  An alternative form of the neutrino energy spectrum fit was implemented that fits for the flux factor values relative to a normalization parameter, as opposed to fitting for the absolute flux factors. This version of the fit is similar in principle to that used in the final K2K analysis and requires only a small adjustment to the definition of the likelihood fit. The fit is implemented by fixing the value of a specific flux factor to 1, and the other flux factors are fit relative to the fixed flux factor. All of the neutrino energy bins are also scaled by a common normalization parameter P . In this analysis the fixed neutrino energy bin is the one ranging from 500 MeV/c to 1000 MeV/c. The number of predicted events in pµ − θµ bin i under this definition n′exp i is modified to Equation 9.10. n′exp i = di P  MC fk Nk,i (x)  (9.10)  k  This version of the fit is useful for a long-baseline neutrino oscillation experiment with a near and far detector, as is the case for T2K. The largest sources of uncertainty in a neutrino energy spectrum measurement are the neutrino interaction model systematics, which introduce an error in the overall normalization and in the shape of the spectrum. Using a dedicated normalization parameter in the fit can be used to separately evaluate the effect of these two categories of errors. In a long-baseline experiment the effect of normalization error can be removed through the use of a near-to-far flux ratio, and ideally this decreases the overall size of the systematic errors in the beam flux. The remaining uncertainties included energy dependent neutrino interaction systematic effects and statistical fluctuations. This relative flux factor version of the fit was tested in a similar manner to absolute version. 500 fake data sets were generated with appropriate variations in the pµ − θµ bins and constraint terms and fit with all systematic parameters included. The measured mean and RMS values of the pull distributions for the fitted parameters are summarized in Table 9.25. Interestingly flux factor f3 seems to be slightly biased in this version of the fit. While the mean fitted value of f3 is very close to the expected value of 1, the mean of the pull distribution is 0.135 with a standard error of the mean of 0.045. Figure 9.26 shows the pull distributions for the flux factors and normalization parameter obtained from this fake data study. The pull distribution for the f3 parameter is visibly shifted with respect to the ex173  9.8. Alternative Relative Energy Spectrum Likelihood Fit pected normal distribution. In principle this pull bias could be corrected but currently no such correction is made. Parameter P f0 f1 f2 f3  Mean Value 1.026 0.998 ≡1 0.998 0.998  Mean Error 0.135 0.218 0.167 0.279  Pull Mean -0.055 0.045 0.075 0.135  Pull RMS 1.024 0.957 0.943 1.000  Table 9.25: Average of fitted flux scale factors, errors and mean and RMS of the pull distributions. The data sample is 500 fake data fits where all systematics nuisance parameters are included in the relative flux factor version of the fit.  Parameter  Pull Mean  P f0 f1 f2 f3  -0.055 0.045 0.075 0.135  Pull Mean Probability 0.220 0.312 0.095 0.002  Pull RMS 1.024 0.957 0.943 1.000  Pull RMS Probability 0.456 0.180 0.07 0.995  Table 9.26: Probability of the measured pull mean and RMS values for the flux factor pull distributions assuming a normal distribution with width equal to the standard errors. The data sample is 500 fake data fits where all systematics nuisance parameters are included in the fit. The results of applying this relative flux factor fit to the real data sample are summarized in Table 9.27. The errors in the normalization parameter are relatively small compared to fitted flux factors. However the error in f3 is significantly larger than in the absolute flux factor version of the fit. More work is required to determine how to interpret these results while accounting for the bias measured in parameter f3 . These results suggest that the contribution of energy-dependent flux uncertainties are not small compared to simply an energy-independent scaling of the flux.  174  9.8. Alternative Relative Energy Spectrum Likelihood Fit  180  Number of Entries / (0.5)  Number of Entries / (0.5)  180 160 140 120 100 80 60 40 20 0 -5  160 140 120 100 80 60 40 20  -4 -3  -2 -1  0  1  2  3  4  0 -5  5  -4 -3  Fitted Scale Factor Pull 0  0  1  2  3  4  5  180  Number of Entries / (0.5)  Number of Entries / (0.5)  180 160 140 120 100 80 60 40 20 0 -5  -2 -1  Normalization Parameter Pull  160 140 120 100 80 60 40 20  -4 -3  -2 -1  0  1  2  3  4  5  0 -5  -4 -3  Fitted Scale Factor Pull 2  -2 -1  0  1  2  3  4  5  Fitted Scale Factor Pull 3  Figure 9.26: Pull distribution for relative flux factor fake data fits with all systematic parameters included.  Flux Factor P f0 f1 f2 f3  Fitted Value SF = 0 0.93 1.19 ≡1 0.92 0.99  Fitted Error -0.11,+0.14 -0.25,+0.28 -0.14,+0.17 -0.26,+0.32  Fitted Value SF = 1 0.95 1.16 0.88 0.99  Final Error -0.12,+0.15 -0.25,+0.28 -0.15,+0.18 -0.26,+0.32  Table 9.27: Results of the relative flux factor version of the fit to the observed data distribution when all systematic parameters are included.  175  9.9. Conclusion  9.9  Conclusion  The maximum likelihood fit for the muon neutrino flux described in this chapter provides a way to measure the neutrino beam energy spectrum without making use of prior beam constraints. This fit was shown to be unbiased using fake data studies, and so could be applied to the ND280 Run 1 and Run 2 dataset. When the neutrino interaction model and detector systematic uncertainties are accounted for the fitted flux factors are found to be consistent with the default neutrino beam flux prediction. This is an important confirmation that the primary T2K analysis’ method of using hadron production data to constrain the neutrino beam flux model is appropriate. In the T2K primary oscillation analysis these prior constraints are used in a fit to ND280 data to reduce the size of the uncertainties in the beam flux measurements. If the ND280-only measurement was inconsistent with the flux prediction, it would indicate that the constraints provided by other experiments are not reasonable and cannot be used in the primary T2K analysis. However the consistency in the flux values demonstrates that the method is sound, and can be used to increase the precision of the neutrino oscillation measurement. The large uncertainties in the flux factors measured with the ND280only fit are mainly due to the systematic errors in the neutrino interaction model parameters, and to a lesser extent the detector model systematics and statistical errors. However these cross-section systematics also affect neutrino interactions at the Super-Kamiokande detector. Consequently the joint constraint provided by ND280 measurements on the flux factors and neutrino interaction model parameters affects the uncertainty in the number of predicted events more than considering the errors in the flux alone. The anti-correlation between the neutrino flux and neutrino interaction model parameters observed in both the ND280-only measurement and in the primary T2K analysis can be used to reduce the systematic uncertainty in the predicted number of events at Super-Kamiokande, as this is obtained from the convolution of the neutrino flux and interaction cross-sections. While this reduction was achieved using the ND280 flux measurement that made use of prior constraints, it could also be done using the ND280-only fit presented here. In either case, the reduction of systematic errors in the T2K oscillation analysis using measurements made with the ND280 detectors is a major achievement for the T2K collaboration. A preliminary exploration of an alternative fit where the uncertainties were divided between an energy-independent normalization term and remaining energy-dependent flux factors suggest that there are important en176  9.9. Conclusion ergy dependent effects in the flux model, but additional work is needed to understand these results.  177  Bibliography [1] Wolfgang Pauli. Pauli letter collection: letter to Lise Meitner. Typed copy. [2] E. Fermi. Versuch einer Theorie der β-Strahlen. I. Zeitschrift fur Physik, 88:161–177, March 1934. [3] C. L. Cowan, Jr., F. Reines, F. B. Harrison, H. W. Kruse, and A. D. McGuire. Detection of the Free Neutrino: A Confirmation. Science, 124:103–104, July 1956. [4] G. Danby, J-M. Gaillard, K. Goulianos, L. M. Lederman, N. Mistry, M. Schwartz, and J. Steinberger. 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