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UBC Theses and Dissertations

An analysis of the oscillation of atmospheric neutrinos Tobayama, Shimpei 2016

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Vn Vnvlysis of thz dsxillvtion ofVtmosphzrix czutrinosbyShimpei TobayamaB.Sc., Keio University, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2016c© Shimpei Tobayama 2016VwstrvxtThis thesis presents an analysis of the oscillation of atmospheric neutrinosobserved in Super-Kamiokande, a large underground water Cherenkov detec-tor in Japan. The observed atmospheric neutrino events are reconstructedand selected using a newly developed maximum likelihood event reconstruc-tion algorithm, and a Markov chain Monte Carlo technique is employedto present the results on neutrino oscillation parameters as marginalizedBayesian posterior probabilities.The result of analyzing the SK-IV data of 2520 days exposure shows apreference for normal mass hierarchy with the posterior probability of 85.9%,and the mode and the 68% credible interval of each oscillation parameter’smarginalized 1D posterior probability distribution for normal hierarchy aresin2 2= = 0:60650:044−0:118 and ∆m2=2 = 2:1350:17−0:=8 × 10−= eV2.iierzfvxzThis thesis is ultimately based on the work by the past and present membersof the Super-Kamiokande(SK) and T2K collaborations, which are both largeinternational collaborations comprising hundreds of members. My specificcontributions are below.For the Photosensor Test Facility described in Section 3.7, I performedthe early-stage designing of the magnetic field compensation system usingfinite element analysis, and also designed, constructed and tested the water-proof optical head unit in collaboration with Philip Lu and Chapman Limat TRIUMF.For the atmospheric neutrino simulation in Chapter 4, using the lat-est simulation softwares developed by the SK and T2K collaborations, Iproduced the atmospheric neutrino MC dataset for the SK collaborationand performed extensive validation, making detailed comparisons to the oldsimulation to understand its behaviour. This MC dataset was used for theatmospheric neutrino analysis presented in Chapter 8.I developed the SK event reconstruction algorithm described in Chap-ter 6 in collaboration with the T2K collaborators in Canada and US. Thebase software framework was developed by myself and Michael Wilking (for-merly at TRIUMF, now at Stony Brook U.), who also developed and op-timized the .0 reconstruction in Section 6.9. The time window algorithmdescribed in Section 6.6.2 was developed by Patrick de Perio (formerly atU. Toronto, now at Columbia U.) and Andrew Missert at CU Boulder.The T2K ,e appearance analysis described in Chapter 7 has been pub-lished by the T2K collaboration as: K. Abe ds ak-, “Observation of Elec-tron Neutrino Appearance in a Muon Neutrino Beam”, Phys. Rev. Lett.,112:061802, 2014. I performed the validation studies for the far detector(SK) data in addition to my involvement in the development of the new .0rejection method using the event reconstruction mentioned above.The SK atmospheric neutrino oscillation analysis presented in Chapter 8was performed largely by myself. I studied event selection methods usingthe new event reconstruction mentioned above. Systematic uncertainties onflux and cross section described in Section 8.2 are mostly inherited from theiiidryfuwyexisting atmospheric neutrino analysis framework at SK, while I introducednew methods to propagate FSI uncertainties as described in Section 8.2.3.Uncertainties on detector response and reconstruction was evaluated by thecollaborators in the SK energy scale working group which I was part of,and main contributions were made by Miao Jiang at Kyoto U. and YusukeSuda at U. Tokyo. Based on the basic Markov chain Monte Carlo (MCMC)analysis software developed in T2K, I developed an analysis framework toperform Bayesian oscillation analysis of atmospheric neutrinos using MCMCand performed sensitivity studies and the final data analysis. This is thefirst atmospheric neutrino analysis at SK which uses MCMC.ivivwlz of ContzntsVwstrvxt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iierzfvxz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiivwlz of Contznts . . . . . . . . . . . . . . . . . . . . . . . . . . . . vaist of ivwlzs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixaist of Figurzs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xVxknofilzygzmznts . . . . . . . . . . . . . . . . . . . . . . . . . . . xviF Introyuxtion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Standard Model and Neutrinos . . . . . . . . . . . . . . 11.2 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Neutrino Mass and Mixing . . . . . . . . . . . . . . . 31.2.2 Neutrino Oscillations in Vacuum . . . . . . . . . . . . 41.2.3 CP Violation in Neutrino Oscillations . . . . . . . . . 61.2.4 Neutrino Oscillations in Matter . . . . . . . . . . . . 61.3 Experimental Status . . . . . . . . . . . . . . . . . . . . . . . 91.3.1 Solar and Reactor Neutrinos . . . . . . . . . . . . . . 91.3.2 Atmospheric and Accelerator Neutrinos . . . . . . . . 111.3.3 Measurement of 1= . . . . . . . . . . . . . . . . . . . 141.4 Unresolved Issues . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Probing the Unknowns with Atmospheric Neutrinos . . . . . 161.6 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . 19G hupzrBKvmiokvnyz . . . . . . . . . . . . . . . . . . . . . . . . . 202.1 Detector Overview . . . . . . . . . . . . . . . . . . . . . . . . 202.2 A Water Cherenkov Detector . . . . . . . . . . . . . . . . . . 222.3 Detector Phases and SK-IV . . . . . . . . . . . . . . . . . . . 232.4 Inner Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 24vhuvly of Wontynts2.5 Outer Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 Water and Air Purification . . . . . . . . . . . . . . . . . . . 262.7 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 263 Dztzxtor Cvliwrvtion . . . . . . . . . . . . . . . . . . . . . . . . 283.1 Relative PMT Gain Calibration . . . . . . . . . . . . . . . . 283.2 Absolute PMT Gain Calibration . . . . . . . . . . . . . . . . 293.3 PMT Quantum Efficiency . . . . . . . . . . . . . . . . . . . . 303.4 Relative Timing Calibration . . . . . . . . . . . . . . . . . . 313.5 Water Property Measurement . . . . . . . . . . . . . . . . . 333.6 PMT and Black Sheet Reflectivity . . . . . . . . . . . . . . . 373.7 Photosensor Test Facility . . . . . . . . . . . . . . . . . . . . 38I Vtmosphzrix czutrino himulvtion . . . . . . . . . . . . . . . 434.1 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . . . . . 434.2 Neutrino Interaction . . . . . . . . . . . . . . . . . . . . . . . 454.2.1 Elastic and Quasi-Elastic Scattering . . . . . . . . . . 464.2.2 Meson Exchange Current . . . . . . . . . . . . . . . . 474.2.3 Single Meson Production . . . . . . . . . . . . . . . . 484.2.4 Deep Inelastic Scattering . . . . . . . . . . . . . . . . 484.2.5 Final State Interaction . . . . . . . . . . . . . . . . . 494.3 Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . 505 Dvtv gzyuxtion . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.1 First and Second Reduction . . . . . . . . . . . . . . . . . . 525.2 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . . 535.3 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . 545.4 Fifth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . 545.5 Final FC Selection . . . . . . . . . . . . . . . . . . . . . . . . 55K Evznt gzxonstruxtion . . . . . . . . . . . . . . . . . . . . . . . 566.1 Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . 566.2 Predicted Charge . . . . . . . . . . . . . . . . . . . . . . . . 576.2.1 Predicted Charge from Direct Light . . . . . . . . . . 576.2.2 Cherenkov Emission Profile . . . . . . . . . . . . . . . 586.2.3 Solid Angle Factor . . . . . . . . . . . . . . . . . . . . 606.2.4 Light Transmission Factor . . . . . . . . . . . . . . . 616.2.5 PMT Angular Acceptance . . . . . . . . . . . . . . . 616.2.6 Predicted Charge from Indirect Light . . . . . . . . . 616.2.7 Parabolic Approximation . . . . . . . . . . . . . . . . 64vihuvly of Wontynts6.3 Unhit Probability and Charge Likelihood . . . . . . . . . . . 666.4 Time Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . 676.4.1 Direct Light Time Likelihood . . . . . . . . . . . . . . 676.4.2 Indirect Light Time Likelihood . . . . . . . . . . . . . 696.4.3 Merging Direct and Indirect Light Time Likelihoods . 696.5 Vertex Pre-fit . . . . . . . . . . . . . . . . . . . . . . . . . . 746.6 Subevent Algorithm . . . . . . . . . . . . . . . . . . . . . . . 756.6.1 Peak Finder . . . . . . . . . . . . . . . . . . . . . . . 756.6.2 Defining Time Windows and Final Subevents . . . . . 776.6.3 Performance of the Subevent Algorithm . . . . . . . . 786.7 Single-Ring Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 786.7.1 Single-Ring Electron & Muon Fit . . . . . . . . . . . 796.7.2 Performance of the Single-Ring e &  Fitter . . . . . 796.7.3 e/ Particle Identification . . . . . . . . . . . . . . . 806.7.4 In-Gate Decay Electron Fit . . . . . . . . . . . . . . . 856.8 Upstream-Track .5 Fit . . . . . . . . . . . . . . . . . . . . . 876.8.1 R.5 Identification . . . . . . . . . . . . . . . . . . . 886.9 .0 Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.9.1 zR.0 Identification . . . . . . . . . . . . . . . . . . . . 916.10 Multi-Ring Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 926.10.1 Initial Multi-Ring Fit . . . . . . . . . . . . . . . . . . 926.10.2 Sequential Multi-Ring Fit . . . . . . . . . . . . . . . . 956.10.3 Performance of the Multi-Ring Fitter . . . . . . . . . 966.11 fiTQun in SK-I to III . . . . . . . . . . . . . . . . . . . . . . 102L First Vpplixvtion of ifunO iGK ,e Vppzvrvnxz . . . . . 1057.1 T2K Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 1057.2 ,e Appearance Analysis . . . . . . . . . . . . . . . . . . . . . 1087.3 ,e Event Selection . . . . . . . . . . . . . . . . . . . . . . . . 1107.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112M Vtmosphzrix czutrino dsxillvtion Vnvlysis . . . . . . . . . 1168.1 Event Selection and Binning . . . . . . . . . . . . . . . . . . 1168.1.1 Selection Criteria . . . . . . . . . . . . . . . . . . . . 1178.1.2 ,R,¯ Separation for Single-Ring Events . . . . . . . . . 1188.1.3 Multi-Ring ,eR,¯e Selection . . . . . . . . . . . . . . . 1198.1.4 Sample Statistics and Purities . . . . . . . . . . . . . 1238.1.5 Binning . . . . . . . . . . . . . . . . . . . . . . . . . . 1238.1.6 Observing the Oscillation Effects . . . . . . . . . . . . 1278.2 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . 132viihuvly of Wontynts8.2.1 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . 1328.2.2 Neutrino Interaction . . . . . . . . . . . . . . . . . . . 1348.2.3 Final State Interaction . . . . . . . . . . . . . . . . . 1358.2.4 Detector Response and Reconstruction . . . . . . . . 1378.3 Event Rate Prediction . . . . . . . . . . . . . . . . . . . . . . 1398.3.1 Oscillation Weighting . . . . . . . . . . . . . . . . . . 1398.3.2 Propagating Variations of Systematics . . . . . . . . . 1408.4 Bayesian Analysis using Markov Chain Monte Carlo . . . . . 1428.4.1 Likelihood Function . . . . . . . . . . . . . . . . . . . 1438.4.2 Bayesian Posterior Probability . . . . . . . . . . . . . 1438.4.3 Marginalization and Parameter Estimation . . . . . . 1448.4.4 Markov Chain Monte Carlo . . . . . . . . . . . . . . . 1458.5 Sensitivity Studies . . . . . . . . . . . . . . . . . . . . . . . . 1478.6 Data Analysis Results . . . . . . . . . . . . . . . . . . . . . . 1548.6.1 Posterior Distributions . . . . . . . . . . . . . . . . . 1558.6.2 Zenith Angle Distributions . . . . . . . . . . . . . . . 1608.6.3 Goodness of Fit . . . . . . . . . . . . . . . . . . . . . 1628.6.4 Result Comparison to Other Experiments . . . . . . . 162N Conxlusions vny dutlook . . . . . . . . . . . . . . . . . . . . . 168Biwliogrvphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171VppznyixzsV aist of hystzmvtix evrvmztzrs . . . . . . . . . . . . . . . . . . 180B Vtmosphzrix czutrino hzlzxtion Improvzmznt wy ifun 182viiiaist of ivwlzs1.1 Fermions in the Standard Model . . . . . . . . . . . . . . . . 12.1 Definition of the event triggers . . . . . . . . . . . . . . . . . 276.1 Decay electron detection efficiencies for stopping cosmic-raymuons for data and MC . . . . . . . . . . . . . . . . . . . . . 787.1 Assumed values for the oscillation parameters . . . . . . . . . 1117.2 Expected numbers of signal and background events passingeach selection stage . . . . . . . . . . . . . . . . . . . . . . . . 1128.1 Interaction mode breakdown and the total event rates for eachevent sample . . . . . . . . . . . . . . . . . . . . . . . . . . . 1268.2 Default values for the oscillation parameters which are usedfor event rate calculation . . . . . . . . . . . . . . . . . . . . . 1288.3 True oscillation parameters for the Asimov dataset production 1478.4 Priors for the oscillation parameters . . . . . . . . . . . . . . 1488.5 The 1D posterior mode and the 68% HPD credible intervals . 1558.6 Posterior probabilities for the mass hierarchy and the 2= octant1608.7 Oscillation parameters which are used to calculate the eventrate predictions shown in Figures 8.26 to 8.28. . . . . . . . . 1629.1 Expected posterior probabilities for for favouring the correctmass hierarchy for different SK exposure . . . . . . . . . . . . 169A.1 List of systematic parameters. Continues to Table A.2. . . . . 180A.2 List of systematic parameters. Continues from Table A.1. . . 181B.1 Interaction mode breakdown and the total MC event rates foreach event sample categorized by the APFIT-based selection 183ixaist of Figurzs1.1 Examples of weak interaction processes involving neutrinos . 31.2 Coherent scattering of neutrinos in matter . . . . . . . . . . . 61.3 8B solar neutrino flux measured by SNO through differentchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Zenith angle distributions observed at Super-Kamiokande . . 131.5 The ordering of the neutrino mass . . . . . . . . . . . . . . . 161.6 The oscillation probability e (, → ,e) and the ratio of oscil-lated to unoscillated ,e flux for atmospheric neutrinos reach-ing the Super-Kamiokande detector . . . . . . . . . . . . . . . 172.1 A sketch of the Super-Kamiokande detector . . . . . . . . . . 212.2 The supporting structure and the PMTs . . . . . . . . . . . . 212.3 A schematic drawing of Cherenkov radiation . . . . . . . . . . 222.4 An event display of a single electron event . . . . . . . . . . . 232.5 A schematic drawing of the Hamamatsu 20-inch PMT . . . . 242.6 Quantum efficiency of the photocathode as a function of wave-length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 The observed charge distribution for single p.e. hits obtainedfrom the nickel source calibration data . . . . . . . . . . . . . 303.2 A schematic view of the timing calibration system . . . . . . 313.3 The TQ distribution for an ID PMT . . . . . . . . . . . . . . 323.4 The timing distribution of ID PMTs at a charge bin ∼1 p.e. . 333.5 A schematic view of the laser injector system for measuringthe water property and the reflectivity of the PMTs . . . . . 343.6 The TOF-subtracted hit time distributions for the laser in-jector data taken at 405 nm . . . . . . . . . . . . . . . . . . . 353.7 Light absorption and scattering coefficients as a function ofwavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.8 Time variation of the light absorption and scattering coeffi-cients measured at various wavelengths . . . . . . . . . . . . . 37xList of Figurys3.9 A schematic view of the apparatus for black sheet reflectivitymeasurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.10 A schematic drawing of the PTF and the external view of thePTF magnetic field compensation system . . . . . . . . . . . 393.11 A SK PMT submerged in the tank filled with water . . . . . 403.12 The interior of the water-tight optical head box . . . . . . . . 413.13 Local variations of the gain of a SK PMT . . . . . . . . . . . 424.1 Zenith angle distributions of the atmospheric neutrino flux atSuper-K averaged over the azimuth . . . . . . . . . . . . . . . 444.2 The direction-averaged atmospheric neutrino flux at Super-Kas a function of energy . . . . . . . . . . . . . . . . . . . . . . 454.3 , and ,¯ CC cross sections per nucleon divided by neutrinoenergy, plotted as a function of energy . . . . . . . . . . . . . 474.4 .5-12C scattering cross sections . . . . . . . . . . . . . . . . . 504.5 Low momentum .5 absorption cross sections compared be-tween experimental data and the variations of the NEUT pioncascade model . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.1 Schematic diagram describing the variables relevant to pre-dicted charge calculation . . . . . . . . . . . . . . . . . . . . . 586.2 Cherenkov emission profile g(pP sP cos ) for electrons . . . . . 596.3 Cherenkov emission profile g(pP sP cos ) for muons . . . . . . . 606.4 Angular acceptance of the PMT . . . . . . . . . . . . . . . . 626.5 A schematic diagram describing the variables relevant to thescattering table . . . . . . . . . . . . . . . . . . . . . . . . . . 636.6 The photon acceptance factor J(s) overlaid with an approxi-mating parabola . . . . . . . . . . . . . . . . . . . . . . . . . 656.7 The unhit probability e (unhit|) with and without the cor-rection for the PMT threshold effect . . . . . . . . . . . . . . 666.8 The normalized charge likelihood fq(q|) . . . . . . . . . . . . 686.9 The direct light residual time likelihood fnsrt (tros) for 300 MeVRxelectrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.10 The direct light residual time likelihood fnsrt (tros) for 2000 MeVRxelectrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.11 The direct light residual time likelihood fnsrt (tros) for 450 MeVRxmuons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.12 The direct light residual time likelihood fnsrt (tros) for 2000 MeVRxmuons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73xiList of Figurys6.13 Distribution of the scanned goodness G(xP t) as a function oftime t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.14 Single-ring electron and muon vertex resolution for FC true-fiducial CCQE events in atmospheric neutrino MC, comparedbetween APFIT and fiTQun . . . . . . . . . . . . . . . . . . . 816.15 Single-ring electron and muon vertex resolution plotted asa function of true momentum, for FC true-fiducial CCQEevents in atmospheric neutrino MC, compared between AP-FIT and fiTQun . . . . . . . . . . . . . . . . . . . . . . . . . 816.16 Single-ring electron and muon direction resolution for FCtrue-fiducial CCQE events in atmospheric neutrino MC, com-pared between APFIT and fiTQun . . . . . . . . . . . . . . . 826.17 Single-ring electron and muon direction resolution plotted asa function of true momentum, for FC true-fiducial CCQEevents in atmospheric neutrino MC, compared between AP-FIT and fiTQun . . . . . . . . . . . . . . . . . . . . . . . . . 826.18 Single-ring electron and muon momentum resolution plottedas a function of true momentum, for FC true-fiducial CCQEevents in atmospheric neutrino MC, compared between AP-FIT and fiTQun . . . . . . . . . . . . . . . . . . . . . . . . . 836.19 Event displays of simulated single electron and muon events . 836.20 Likelihood separation of single-ring electron and muon eventsin the FC true-fiducial CCQE event sample in the atmo-spheric neutrino MC . . . . . . . . . . . . . . . . . . . . . . . 846.21 Misidentification rate of single-ring electron and muon eventsin the FC true-fiducial CCQE event sample in the atmo-spheric neutrino MC, plotted as a function of true momentum 846.22 Event displays of an MC event which has an in-gate decayelectron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.23 Charge distribution for hits that are associated with each in-gate subevent, for the same event which was shown in Fig-ure 6.22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.24 A schematic diagram of a charged pion which undergoes ahadronic scattering on a nucleus and the rings it produces,and an event display of a simulated single .5 event . . . . . . 876.25 A schematic diagram describing how an upstream-track hy-pothesis is constructed in fiTQun . . . . . . . . . . . . . . . . 886.26 Likelihood separation of ,CCQE and NC.5 events . . . . . 896.27 A schematic diagram showing how the .0 hypothesis is con-structed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90xiiList of Figurys6.28 CC single electron and NC single .0 events in the FC true-fiducial atmospheric neutrino MC sample, separated using the.0 fit variables . . . . . . . . . . . . . . . . . . . . . . . . . . 916.29 Misidentification rate for the .0 rejection cut Equation 6.28,for the CC single electron and NC single .0 events in the FCtrue-fiducial atmospheric neutrino MC sample, plotted as afunction of the true particle momentum . . . . . . . . . . . . 926.30 A tree diagram showing how the tree of the multi-ring hy-potheses evolve as the number of rings is increased . . . . . . 936.31 Distribution of the square root of the log likelihood ratio be-tween the best-fit single-ring electron hypothesis and the 2Rhypothesis assuming electron as the first ring, for FC sub-GeVatmospheric neutrino MC events with no decay electrons de-tected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.32 Reconstructed event categories of ,eCC events with no visiblepions in final state in the FC atmospheric neutrino MC forfiTQun and APFIT . . . . . . . . . . . . . . . . . . . . . . . . 976.33 Reconstructed event categories of NC events with a single.0 and no visible .± in final state in the FC atmosphericneutrino MC for fiTQun and APFIT . . . . . . . . . . . . . . 976.34 Reconstructed event categories of ,eCC events with a single.0 and no visible .± in final state in the FC atmosphericneutrino MC for fiTQun and APFIT . . . . . . . . . . . . . . 986.35 Reconstructed event categories of ,CC events with no visiblepions in final state in the FC atmospheric neutrino MC forfiTQun and APFIT . . . . . . . . . . . . . . . . . . . . . . . . 996.36 Reconstructed event categories of ,CC events with a single.0 and no visible .± in final state in the FC atmosphericneutrino MC for fiTQun and APFIT . . . . . . . . . . . . . . 996.37 Reconstructed invariant mass calculated using the 2nd andthe 3rd rings for events reconstructed as 3z ring by fiTQunand APFIT in the FC atmospheric neutrino MC . . . . . . . 1006.38 Reconstructed invariant mass calculated using the 2nd andthe 3rd rings for events reconstructed as 1 + 2z ring byfiTQun and APFIT in the FC atmospheric neutrino MC . . . 1016.39 Distributions of the log likelihood ratio between the fiTQunsingle-ring electron and muon fits for the FC sub-GeV single-ring atmospheric neutrino events in each SK phase . . . . . . 1036.40 Time variation of the light attenuation length in water . . . . 104xiiiList of Figurys7.1 Schematic drawing of the T2K beam production site . . . . . 1057.2 T2K , beam spectrum at different off-axis angles and the ,survival probability at SK . . . . . . . . . . . . . . . . . . . . 1067.3 A schematic drawing of ND280 . . . . . . . . . . . . . . . . . 1077.4 Predicted T2K neutrino flux at SK . . . . . . . . . . . . . . . 1087.5 .0 rejection cut which was used in the previous ,e appearanceanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.6 2D distribution of the fiTQun .0 cut variables after the Zrom, Q1250MeV cut, for the dataset shown in Table 7.2 . . . . . . . 1147.7 The best-fit (for normal hierarchy) pe−e distribution plottedwith the observed data . . . . . . . . . . . . . . . . . . . . . . 1158.1 Distribution of the variables which are used for the Multi-GeVMulti-ring E-like(MME) likelihood selection . . . . . . . . . . 1218.2 Distribution of the sum of ln(assgRalug) for all four MMEselection variables . . . . . . . . . . . . . . . . . . . . . . . . 1228.3 Distribution of the variables which are used for the multi-ring,eR,¯e likelihood separation . . . . . . . . . . . . . . . . . . . . 1248.4 Distribution of the sum of ln(a,eRa,e) for all three ,eR,¯e sep-aration variables . . . . . . . . . . . . . . . . . . . . . . . . . 1258.5 Binning for each event sample . . . . . . . . . . . . . . . . . . 1278.6 The effect of sin2 2= variations on the distributions of thecosine of the zenith angle for sub-GeV samples . . . . . . . . 1298.7 The effect of sin2 2= variations on the distributions of thecosine of the zenith angle for Multi-GeV samples . . . . . . . 1298.8 The effect of MZ variations on the distributions of the cosineof the zenith angle for Sub-GeV samples . . . . . . . . . . . . 1308.9 The effect of the mass hierarchy on the distributions of thecosine of the zenith angle for Multi-GeV e-like samples . . . . 1318.10 Uncertainty on the flux normalization as a function of energy 1328.11 The fractional error matrix sgn(kij)×√|kij | for the sub-GeVsamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368.12 The fractional error matrix sgn(kij) ×√|kij | for the multi-GeV samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.13 Reconstructed vertex Z position of stopping cosmic-ray muonsentering from the top of the detector . . . . . . . . . . . . . . 1388.14 The response coefficient f ji for the MQEA systematic for the,eCC events in each bin in the sub-GeV one-ring e-like 0decaysample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142xivList of Figurys8.15 ∆m2=2 and − lne (o⃗P f⃗ |Y), the negative-log of the posteriorprobability, plotted against the number of steps taken in eachchain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1498.16 The marginalized 2D posterior distribution for ∆m2=2 andsin2 2= given the Asimov dataset . . . . . . . . . . . . . . . . 1508.17 The 2D marginalized posterior distributions for sin2 2= andMZ given the Asimov dataset . . . . . . . . . . . . . . . . . . 1528.18 The marginalized 1D posterior distributions for sin2 2=, |∆m2=2|and MZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.19 The mean and the  of the marginalized 1D posterior foreach systematic parameter in the case of assuming normalhierarchy, for the Asimov dataset . . . . . . . . . . . . . . . . 1548.20 The marginalized 2D posterior distribution for ∆m2=2 andsin2 2= given the SK-IV FC data . . . . . . . . . . . . . . . . 1568.21 The marginalized 2D posterior distributions for |∆m2=2| andsin2 2= for the cases of assuming normal and inverted hierarchy1578.22 The marginalized 2D posterior distributions for sin2 2= andMZ for the cases of assuming normal and inverted hierarchy . 1588.23 The marginalized 1D posterior distributions for sin2 2=, |∆m2=2|and MZ for the cases of assuming normal and inverted hierarchy1598.24 The marginalized 1D posterior distributions for sin2 2= andMZ after marginalizing over the mass hierarchy . . . . . . . . 1608.25 The mean and the  of the marginalized 1D posterior foreach systematic parameter in the cases of assuming normaland inverted hierarchy . . . . . . . . . . . . . . . . . . . . . . 1618.26 Distributions of the cosine of the zenith angle for the sub-GeVevent samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1638.27 Distributions of the cosine of the zenith angle for the multi-GeV e-like event samples . . . . . . . . . . . . . . . . . . . . . 1648.28 Distributions of the cosine of the zenith angle for the multi-GeV -like and other event samples . . . . . . . . . . . . . . 1658.29 Distribution of the log likelihood ratio between the observeddata and fake data which is used for the goodness of fit test . 1668.30 90% credible and confidence regions for ∆m2=2 and sin2 2= forthe results from various experiments . . . . . . . . . . . . . . 167B.1 90% HPD credible regions for ∆m2=2 and sin2 2= comparedbetween the cases of using APFIT or fiTQun for event selec-tion, on Asimov datasets . . . . . . . . . . . . . . . . . . . . . 184xvVxknofilzygzmzntsFirst and foremost, I thank my supervisor, Hiro Tanaka, for all of his sup-port. His generosity and patience for letting me pursue my interests and theguidance throughout were essential for completing the long PhD journey.I also thank Akira Konaka, for stimulating my interests in Super-K andatmospheric neutrinos through a number of inspiring discussions. Manythanks to Mike Wilking, without whom the most important part of mywork, fiTQun, would have never happened. Working with Patrick de Periowas always fun, and I thank him for showing me what it is to be a successfulgraduate student and also for broadening my horizons in many aspects.I am grateful to the members of the SKanada group: Sophie Berkman,John Martin, Corina Nantais and Roman Tacik, and the US collaboratorsincluding Andy Missert, Eric Zimmerman and Cris Vilela, for all the progressmade over the years in fiTQun and various other new efforts at Super-K.I also thank Scott Oser, whose deep insights in statistics helped me formthe basis of my knowledge in statistics. I thank all the past and presentmembers of the T2K-Canada group: not only was it a great pleasure workingwith this extraordinary group of talent, the collaboration “meatings” we hadin Tokai were the highlights of my life as a graduate student.It was a great privilege working with the collaborators on the Super-Kexperiment. Special thanks to the SK E-scale team: Shunichi Mine, YusukeSuda and Miao Jiang, without whose help I would have not been able tofinish my analysis in such a timely manner. I also thank Roger Wendell forhelping me figure out the various details in atmospheric neutrino analysis,and Richard Calland for the great help on MCMC.I’d like to offer my special thanks to Tomonobu Tomura and my friendsin Kamioka for making the life in Kamioka such a wonderful experience.I am grateful to Haruka Haruno, Minami Kaido, Kirara Amanogawaand Towa Akagi who, during the most difficult times, gave me hope andenlightened the path to complete my PhD: a long dream since childhood.Finally, I’d like to sincerely thank my parents, grandparents, sister andmy entire family. I am where I am only because of their continuous supportand encouragement. I say to them all: gnmsnt-mh arhfasn-fnyahlart!xviChvptzr FIntroyuxtionNeutrino physics has achieved remarkable progress in the past two decades.Once thought to be massless, neutrinos are now confirmed to have nonzeromass by the discovery of neutrino oscillations which necessitates the revi-sion of the established theory of the Standard Model of particle physics.Despite the recent progress, many of the properties of neutrinos still remainunknown, and improving our knowledge of neutrinos leads the path towardthe fundamental theories governing the universe.FCF ihz htvnyvry boyzl vny czutrinosThe Standard Model of particle physics includes six quarks and six leptonsas shown in Table 1.1, which are all spin 1/2 Dirac fermions and each fermionhas a corresponding antiparticle which has opposite charge.Charge 1st generation 2nd generation 3rd generationquarks+2R3 up charm top−1R3 down strange bottomleptons−1 electron muon tau0 electron neutrino muon neutrino tau neutrinoTable 1.1: Fermions in the Standard Model.There are three generations of quarks and each generation has two typesof quarks, one with the electric charge of +2R3 and the other with −1R3.Similar to quarks there are three generations of leptons, and each generationhas a charged lepton with −1 electric charge and a neutrino which is elec-trically neutral. The six quarks and the three charged leptons are massive,and the mass gets larger as the generation increases, for instance, a charmquark is heavier than an up quark, and a muon is heavier than an electron.Neutrinos are assumed to be massless in the Standard Model, and until thediscovery of neutrino oscillations which is explained in the following section,experimental results were consistent with zero neutrino mass.1EBEB hhy gtunxurx aoxyl unx byutrinosThere are three fundamental interactions in the Standard Model whichare the strong, the electromagnetic and the weak interactions, and they aremediated by gauge bosons which are spin 1 particles. The strong interactionis mediated by gluons which are massless, and they only interact with quarksand other gluons but not with leptons. The electromagnetic interaction ismediated by photons which are also massless, and they interact with allelectrically charged particles; i.e. all fermions except for neutrinos interactelectromagnetically. Finally, weak interactions occur to all fermions andare mediated by the weak bosons: l± which have ±1 electric charge ando0 which is neutral. Unlike the other gauge bosons the weak bosons aremassive.The above interactions in the Standard Model are fundamentally definedby local gauge symmetries which also require the gauge bosons to be mass-less, and the strong interaction is described by a hj(3) symmetry in theinteractions between the quark and the gluon fields. The electromagneticand the weak interactions arise from the fundamental electroweak interac-tion which is described by a hj(2) × j(1) symmetry. The correspondinggauge fields interact with the Higgs field which spontaneously breaks thesymmetry and results in the massive weak bosons mediating the weak inter-action and the massless photons mediating the electromagnetic interaction.The symmetry breaking caused by the Higgs field also gives rise to the massof the quarks and the charged leptons, and introduces a massive spin 0 Higgsboson.As stated earlier, neutrinos interact by the weak interaction only and areinsusceptible to the electromagnetic or the strong interactions, and thus theyrarely interact with matter. The three generations, which are also called theflavours of neutrinos are labeled as ,e, , and , , and each of them respec-tively couples via the charged current(CC) weak interaction mediated bythe l± boson to the charged lepton of the corresponding flavour: elec-tron, muon and tau. A CC interaction involves a neutrino and a chargedlepton of the same flavour only, or rather, neutrino flavours are defined bythe flavour of the charged lepton with which each neutrino partners via theCC interaction. The CC interaction is responsible for various weak decayprocesses involving neutrinos such as the muon decay illustrated on the leftof Figure 1.1. The interaction mediated by the o0 boson as on the right ofFigure 1.1 is called the neutral current(NC) weak interaction, and neutrinosdo not change their type by NC interactions.2EBFB byutrino cswillutionsµe⌫µ ⌫¯eWe⌫eZ0e⌫eTimeFigure 1.1: Examples of weak interaction processes involving neutrinos. Theleft diagram is the decay of a muon into an electron and neutrinos viathe charged current interaction, and the right diagram is a neutral currentelectron-neutrino scattering.FCG czutrino dsxillvtionsThe neutrino mass is experimentally known to be small. The observationthat neutrinos are produced with definite helicity[1] suggests that neutrinosare massless, and the results of any direct mass measurements to date byprecision kinematic measurement of various weak decay processes are con-sistent with zero mass[2]. However, the discovery of neutrino oscillationshas proven that neutrino mass is, in fact, not zero, which contradicts theassumption made in the Standard Model. This section describes the basictheories behind neutrino oscillations.FCGCF czutrino bvss vny biflingSuppose now that neutrinos do have nonzero mass, and consider free neu-trinos propagating in vacuum. Since there are three neutrino flavours thereare three orthogonal states |,〉 ( = zP P ) which are the eigenstates of theweak interaction. One can also define the mass eigenstates |,i〉 (i = 1P 2P 3)which are the eigenstates of the free particle Hamiltonian with the possiblydifferent mass eigenvalues mi (i = 1P 2P 3), and since each of them form acomplete basis the two bases are related by a unitary matrix j :|,〉 ==∑ij∗i|,i〉: (1.1)A general c × c unitary matrix has c2 real free parameters, c(c −1)R2 of which can be considered as rotation angles as in the case for areal orthogonal matrix of the same size, and the remaining c(c + 1)R23EBFB byutrino cswillutionsreal parameters can be attributed to complex phases. Therefore, the 3 × 3unitary matrix j can be characterized by 3 rotation(mixing) angles and6 complex phases. However, assuming that neutrinos are standard Diracfermions, one can freely define the phase of each of the flavour and themass eigenstates without changing any physical consequences, and 5 of thephase parameters can be therefore factored out from the mixing matrix asthey are physically not meaningful. Thus, the fundamental matrix j whichdescribes the mixing of the flavour eigenstates and the mass eigenstates hasthree mixing angles and a single complex phase. The matrix is analogousto the Cabibbo-Kobayashi-Maskawa(CKM) matrix[3] which describes thequark mixing in weak interactions, and it is known as the Pontecorvo-Maki-Nakagawa-Sakata(PMNS) matrix[4][5] which is usually parameterized as:j =0B@ 1 0 00 cos 2= sin 2=0 − sin 2= cos 2=1CA0B@ cos 1= 0 sin 1=z−iCP0 1 0− sin 1=ziCP 0 cos 1=1CA×0B@ cos 12 sin 12 0− sin 12 cos 12 00 0 11CA P (1.2)where 12, 1= and 2= are the mixing angles and MZ is the complex phase.Two extra complex phases 1 and 2, known as the Majorana phases, needto be introduced in the mixing matrix as j → j × diag(1P zi1 P zi2) ifneutrinos are Majorana fermions as opposed to Dirac fermions, i.e. if aneutrino is its own antiparticle. In such case, one can no longer freely choosethe phase of the mass eigenstates and the corresponding degrees of freedommust remain in the unitary matrix.FCGCG czutrino dsxillvtions in kvxuumConsider now the time evolution of a flavour eigenstate in Equation 1.1as it propagates in vacuum. The transition amplitude between the flavoureigenstates , and , after time t is (xP h¯ = 1 in the following discussions):〈,|z−i hHt|,〉 ==∑iz−iEitjij∗iP (1.3)where Hˆ is the free particle Hamiltonian and Zi ≡√p2 +m2i is the energyeigenvalue of the free particle with momentum p. It can be seen from theequation that the transition amplitude between different flavour eigenstates4EBFB byutrino cswillutionscan be nonzero as the time elapses, and the transition probability from thestate , to , is:e (, → ,)(t) = |〈,|z−i hHt|,〉|2 ==∑i;jj∗ijijjj∗jz−i2Ei−Ej3t: (1.4)Note that the Majorana phases 1 and 2 mentioned in the previous sectiondo not affect the transition probability at all, as the phases cancel when thematrix elements are multiplied by their complex conjugates. Thus, 12, 1=,2= and MZ are the only parameters in the mixing matrix which affect thetransition probability, regardless of whether neutrinos are Dirac or Majoranafermions.In the ultra-relativistic limit of p2 ≫ m2i ,Zi − Zj =√p2 +m2j −√p2 +m2i ≈∆m2ij2ZP (1.5)where Z ≈ |p| is the characteristic energy and∆m2ij ≡ m2i −m2j : (1.6)Using Equation 1.5 and also replacing the time t with the distance a theneutrino traveled, Equation 1.4 now reads:e (, → ,) ==∑i;jj∗ijijjj∗j exp(−i∆m2ija2Z)P (1.7)and the unitarity condition of j further leads the equation to:e (, → ,) =  − 4∑i>jRe[j∗ijijjj∗j]sin2(∆m2ija4Z)+ 2∑i>jIm[j∗ijijjj∗j]sin(∆m2ija2Z): (1.8)From this equation it can be seen that neutrinos can change its flavouras it travels in space, and since the transition probability varies periodicallyaccording to the phase ∝ aRZ the phenomenon is referred to as the neutrinooscillation. In order for the oscillation to happen, the following two condi-tions are required, one of which is that the mixing matrix j ̸= I, i.e. theflavour and the mass eigenstates are not identical. The other condition isthat ∆m2ij ̸= 0, i.e., at least one of three mass eigenvalues has to be differentfrom the others. Thus, an observation of neutrino oscillations is an evidenceof nonzero neutrino mass.5EBFB byutrino cswillutionsW±e⌫ee ⌫eTimeZ0⌫↵p, n, ep, n, e⌫↵Figure 1.2: Coherent scattering of neutrinos in matter. The left diagram isa NC scattering which involves all neutrinos, and the right diagram is a CCscattering which involves electron neutrinos only. Note that such scatteringalso occurs for antineutrinos, while in such case the CC scattering is de-scribed by an “annihilation-creation-type” diagram, i.e., the right diagramrotated by 90 degrees.FCGC3 Ce kiolvtion in czutrino dsxillvtionsFor antineutrinos, mixing as in Equation 1.1 is described by the complexconjugate of the mixing matrix:|,¯〉 ==∑iji|,¯i〉P (1.9)and since the mixing matrix is different between neutrinos and antineutrinosif MZ is neither 0 nor ., MZ is referred to as the CP violating phase. In suchcase, it immediately follows that the sign of the third term in Equation 1.8flips for antineutrinos, which suggests that the oscillation probability willbe different between neutrinos and antineutrinos. Thus, CP violation in thelepton sector can be probed through neutrino oscillation measurements.FCGCI czutrino dsxillvtions in bvttzrWhen neutrinos propagate in matter, the interactions between the neutri-nos and the matter change the oscillation behaviour. The phenomenon,which was first introduced by Wolfenstein[6] and subsequently elaboratedby Mikheev and Smirnov[7], is known as the matter effect or the MSWeffect of neutrino oscillations.When neutrinos propagate in matter, all three flavours of neutrinos un-dergo neutral current coherent scattering with protons, neutrons and elec-trons as shown on the left of Figure 1.2. Such interaction introduces an6EBFB byutrino cswillutionseffective potential kNM which simply shifts the energy levels of all types ofneutrinos by a same amount. Since ordinary matter contains electrons butnot muons or taus, for electron neutrinos only the coherent charged currentscattering as on the right of Figure 1.2 also occurs in addition to the NCscattering, and the process introduces an additional potential kMM whichonly affects electron neutrinos:kMM = ±√2GFce: (1.10)In the equation, GF is the Fermi coupling constant, ce is the number densityof electrons in the matter and the positive and the negative signs are forneutrinos and antineutrinos respectively.In order to see the effect on the oscillations, for simplicity, consider nowa case of two flavour oscillations between two flavour eigenstates ,e and ,x,where ,x is a non-electron state which is insusceptible to kMM and can beconsidered as some combination of , and , . The flavor states are relatedto the mass eigenstates ,i with energies Zi by a mixing matrix j :(,e,x)= j(,1,2)≡(cos  sin − sin  cos )(,1,2): (1.11)Let H ≡ diag(Z1P Z2) ≡ H0+H1 be the free particle Hamiltonian in themass basis where H0 ≡ (Z1 + Z2)R2× diag(1P 1) andH1 ≡ Z2 − Z12× diag(−1P 1) = ∆m24Z× diag(−1P 1)P (1.12)where Equation 1.5 was used and ∆m2 ≡ m22 − m21. Since adding a termproportional to the unit matrix to the Hamiltonian merely adds a commonphase to all neutrino types and does not affect oscillations, H0 can be ig-nored, and the vacuum Schro¨dinger equation in the flavour basis becomes:iyyt(,e,x)= jH1jT(,e,x)=∆m24Z(− cos 2 sin 2sin 2 cos 2)(,e,x)P (1.13)which describes the oscillation of flavour states in vacuum. Solving theequation, the transition probability is:e (,e → ,x) = sin2 2 sin2 ∆m2a4ZP (1.14)7EBFB byutrino cswillutionswhere the substitution t→ a was done as before.Consider now the oscillations in matter, in which case the Hamiltonianhas an additional potential term due to matter. The potential is diagonal inthe flavour basis in the form diag(kMM + kNMP kNM), and shifting the matrixby a multiple of unit matrix the Schro¨dinger equation in matter becomes:iyyt(,e,x)=∆m24Z(− cos 2 + x sin 2sin 2 cos 2 − x)(,e,x)P (1.15)wherex ≡ 2ZkMM∆m2: (1.16)One can then define the effective mixing angle M and the mass splitting∆m2M such that:sin 2M ≡ sin 2√sin2 2 + (cos 2 − x)2P (1.17)∆m2M ≡ ∆m2√sin2 2 + (cos 2 − x)2P (1.18)and Equation 1.15 now becomes:iyyt(,e,x)=∆m2M4Z(− cos 2M sin 2Msin 2M cos 2M)(,e,x): (1.19)This is of the same form as Equation 1.13 but the original mixing angle andthe mass splitting  and ∆m2 are now replaced by the effective values inmatter M and ∆m2M. This is the matter effect, and the oscillation amplitudeand the frequency are changed from the case in vacuum due to the CCinteraction with matter shifting the energy levels.A dramatic consequence follows when:x = cos 2: (1.20)In such case, sin 2M = 1, and the mixing becomes maximal regardless ofthe original value of the mixing angle . Such effect is known as the MSWresonance, and it occurs when the energy level splitting in matter ∆m2Mbecomes minimal. Note that the resonance is sensitive to the sign of ∆m2as well as whether it is a neutrino(kMM S 0) or an antineutrino(kMM Q 0): if∆m2 S 0 the resonance only happens for neutrinos, whereas if ∆m2 Q 0 itonly happens for antineutrinos.8EBGB Efipyrimyntul gtutusFC3 Zflpzrimzntvl htvtusThrough the observation of neutrinos from various sources, neutrino oscil-lation is now an established fact, and the majority of the oscillation param-eters have been measured. This section summarizes the different neutrinooscillation experiments which have been performed to date.FC3CF holvr vny gzvxtor czutrinosThe sun produces electron neutrinos through various nuclear fusion and -decay processes at its interior, with their energy generally ranging in a fewto ten MeV. The first measurement of solar neutrinos was made by theHomestake experiment in 1968[8], which detected the neutrinos by countingthe decays of the radioisotope of argon produced in a chlorine tank via thereaction:,e +=7Cl→ =7Ar + z−: (1.21)The observed neutrino flux was ∼ 1R3 of the prediction from the standardsolar model(SSM)[9], and this was the first observed indication of neutrinoflavour transitions in history. Several experiments which followed also ob-served a significant deficit of the solar neutrino flux compared to the SSMprediction. However, the effect which was eventually known as the solar neu-trino problem remained unresolved for more than 30 years since its initialobservation.The first conclusive measurement to resolve the problem was made by theSNO experiment, which was capable of measuring both the electron neutrinoflux and the total neutrino flux from the sun using heavy water[10][11]. InSNO, neutrinos can be detected by the CC and NC interactions on deuteriumand also by elastic scattering (ES) on electron as the following:CC : ,e + y→ z− + p+ pP (1.22)NC : ,x + y→ ,x + p+ nP (1.23)ES : ,x + z− → ,x + z−: (1.24),x above represents any of the three neutrino flavours, and while the NCinteraction on deuterium occurs equally for all neutrino flavours, only ,eundergoes the CC interaction, and the experiment is thus capable of mea-suring the total flux and the ,e flux separately through the two channels.For the elastic scattering on electron, the channel is sensitive to all neutrinoflavours via NC interactions, however, since ,e can also undergo CC inter-actions with electron the cross section of this process for ,e is larger by a9EBGB Efipyrimyntul gtutus)-1 s-2 cm6 10× (eq0 0.5 1 1.5 2 2.5 3 3.5)-1  s-2 cm6  10× ( oµq0123456 68% C.L.CCSNOq 68% C.L.NCSNOq 68% C.L.ESSNOq  68% C.L.ESSKq 68% C.L.SSMBS05q 68%, 95%, 99% C.L.oµNCqFigure 1.3: 8B solar neutrino flux measured by SNO through different chan-nels. The horizontal axis is the ,e flux and the vertical axis is the sum ofthe , and , flux. 68% allowed regions from the CC, NC and ES measure-ments are shown as the red, blue and green bands respectively, and the ESmeasurement result from Super-Kamiokande is shown as a gray band. Thedashed lines represents the SSM prediction of the total neutrino flux. Thepoint and the contours represent the best estimate and the allowed regionsat 68, 95 and 99% confidence levels by combining the CC and NC results.Figure taken from [12].factor of ∼6 compared to ,; (“ES” refers to all elastic scattering on elec-tron via CC and NC interactions). As shown in Figure 1.3, the result of theexperiment showed a clear deficit of the ,e flux by a factor of ∼ 1R3 relativeto the total flux which agreed with the SSM prediction, which is a decisiveevidence for the flavour transition of solar neutrinos. Combining the SNOresults with another solar neutrino measurement by Super-Kamiokande[13]as well as the reactor neutrino measurement by KamLAND[14], it is nowconfirmed that the flavour transition of the solar neutrinos occurs via themechanism known as the large mixing angle(LMA) MSW solution[6][7].According to the LMA-MSW solution, due to the high matter density atthe core of the sun, the matter effect described in Section 1.2.4 significantlychanges the energy levels of the neutrinos such that at energies ∼ 10 MeVthe second mass eigenstate ,2 becomes nearly equal to the ,e state, i.e.,10EBGB Efipyrimyntul gtutusneutrinos at that energy are produced in a nearly pure ,2 state at the core.As the neutrinos propagate through the sun and exit to space, the matterdensity gradually decreases such that the neutrino state changes adiabati-cally, keeping the neutrinos produced at the core in a nearly-pure ,2 stateuntil they exit the sun while the flavour composition of the neutrino statechanges according to the change in the matter density. The neutrinos thenpropagate until they get detected on earth spending most of the time invacuum, during which the neutrino flavours do not oscillate as it is alreadyin a mass eigenstate. An observer on earth then projects the mass-state neu-trinos onto flavour states, and the flavour change ,e → ,x of solar neutrinosis thus observed. The observation of the solar neutrino flavour transitionmeasures the parameters 12 and ∆m221.Another method to measure 12 and ∆m221 is to study the “disappear-ance” of ,¯e’s coming from nuclear reactors, by measuring the deficit in the,¯e flux due to the transition ,¯e → ,¯x in a O(100 km) path in the air,which is effectively a vacuum. The KamLAND experiment[14] measured asignificant deficit of the ,¯e flux from the various nuclear reactors in Japancompared to the expectations assuming no oscillations, measuring the os-cillation parameters precisely and thus contributing to resolving the solarneutrino problem.From the global analysis of various solar and reactor neutrino data, thecurrent known values for 12 and ∆m221 are[15]:sin2 12 = 0:30450:014−0:01=P (1.25)∆m221 = 7:53± 0:18× 10−5 eV2: (1.26)FC3CG Vtmosphzrix vny Vxxzlzrvtor czutrinosAtmospheric neutrinos are produced by the cosmic ray particles impingingon earth’s atmosphere. Primary cosmic ray particles are mostly protons andhelium nuclei, and when they interact with the nuclei in the atmosphere,they produce hadronic showers which consist mostly of pions as well assome kaons. Neutrinos are produced when the produced hadrons decay, forexample, through the decays of charged pions and the subsequent decays ofmuons:.5 → 5 + , → , + z5 + ,¯ + ,eP (1.27).− → − + ,¯ → ,¯ + z− + , + ,¯e: (1.28)Thus, the atmospheric neutrino flux contains neutrinos and antineutrinos ofboth electron and muon type with the flavour ratio (,+ ,¯)R(,e+ ,¯e) ∼ 2,11EBGB Efipyrimyntul gtutuswhile the ratio increases at higher energy since ± have a higher chanceto reach the ground without decay. Since the cosmic ray flux is close toisotropic the neutrinos come from all directions, and while the ones producedin the atmosphere right above a detector and coming downward only travelfor ∼ 10 km before reaching the detector, the ones produced on the otherside of the earth coming upward travel S 10P 000 km across the earth. Thepath length of atmospheric neutrinos therefore ranges between 10− 104 kmdepending on the zenith angle of neutrino’s direction, and together withenergies ranging from 10−1 to 10= GeV, atmospheric neutrinos cover a widerange of aRZ to which the oscillation phase depends on, making them aperfect source for studying neutrino oscillations.In the late 1980’s and early 1990’s, experiments using large undergroundwater Cherenkov detectors such as Kamiokande[16][17] and IMB-3[18][19]observed a large deficit in the flavour ratio (,+ ,¯)R(,e+ ,¯e): the measuredratio was nearly half of what was expected. The results suggested neutrinooscillations of a different kind from that involving solar neutrinos.In 1998, Super-Kamiokande[20] reported a large deficit in the upward-going , events compared to the expectations assuming no oscillations whilethe downward-going ,’s as well as ,e’s from all directions were rather con-sistent with expectations, as shown in Figure 1.4. This is a strong evidencefor , oscillating to , or to unknown sterile states, and the data was, infact, consistent with a two flavour , → , mixing described by:e (, → , ) = sin2 2 sin2 ∆m2a4ZP (1.29)with a maximal mixing of sin2 2 = 1( = 45◦) and |∆m2| ≈ O(10−= eV2).The oscillation effect observed in atmospheric neutrinos were then con-firmed by experiments using artificial neutrinos produced from accelerators.In such experiments, a high energy proton beam is collided with a tar-get producing hadronic showers which subsequently produce an intense ,beam, and the neutrinos are then observed at a large far detector locatedat O(100 km) distance. Unlike atmospheric neutrinos, accelerator neutrinobeams are very pure in ,’s with small contaminations from ,e’s or an-tineutrinos, and the neutrino energy is tuned ∼ 1 GeV such that the ,disappearance effect , → ,x becomes maximal at the far detector based onthe ∆m2 observed at Super-Kamiokande. K2K[21] and MINOS[22] are thefirst of such long-baseline accelerator neutrino experiments, and they bothobserved significant deficit in the observed , events as well as distortions inthe , energy spectrum, producing results consistent with the atmosphericneutrino observation at Super-Kamiokande.12EBGB Efipyrimyntul gtutusFigure 1.4: Zenith angle distributions for electron-like(top) and muon-like(bottom) events in sub-GeV(left) and multi-GeV(right), observed atSuper-Kamiokande. cos  Q 0 and cos  S 0 represent upward- anddownward-going events respectively. The hatched bands show the expec-tations assuming no oscillations, while the black lines are the best-fit todata assuming , → , oscillations. Figure taken from [20].There are many ongoing experiments which are currently measuring theparameters 2= and ∆m2=2, which govern , → , oscillations, with increas-ing precision, and their results are in general still consistent with the maxi-mal mixing of 2= = 45◦, and |∆m2=2| is measured to be ∼ 2:5×10−= eV2 with∼ 5% uncertainty[15]. The latest results from the long-baseline experimentT2K[23] is (assuming ∆m2=2 S 0):sin2 2= = 0:53250:046−0:068P (1.30)|∆m2=2| = 2:54550:081−0:084 × 10−= eV2: (1.31)Latest results on 2= and ∆m2=2 from the atmospheric and accelerator neu-trino experiments are discussed again in Section 8.6.4.Note that oscillation experiments are often insensitive to the sign of∆m2=2 as can be seen from the expressions for oscillation probabilities Equa-13EBGB Efipyrimyntul gtutustions 1.8 and 1.29, and although the absolute value of ∆m2=2 has been mea-sured rather precisely, its sign is still not determined. This is not the case for∆m221, since the matter effect observed in solar neutrinos are only possibleif ∆m221 S 0 as discussed in Section 1.2.4. Similarly, one way to determinethe sign of ∆m2=2 is to observe the oscillations driven by ∆m2=2 for neutrinoswhich travel through significant amount of matter, which will be discussedmore in Section 1.5.FC3C3 bzvsurzmznt of 1=The mixing angle 1= was considered to be small, as the solar and atmo-spheric neutrino results were able to be explained assuming it is zero. Thefirst attempt to directly measure 1= was made by CHOOZ, by observing the,¯e disappearance ,¯e → ,¯x from a nuclear reactor at ∼ 1 km, which is muchshorter than KamLAND. At such distance and the ,¯e energy of ∼ 3 MeV,rather than the oscillation driven by the “solar” oscillation parameters 12and ∆m221, the oscillation driven by 1= and ∆m2=1:e (,¯e → ,¯e) ≈ 1− sin2 21= sin2 ∆m2=1a4ZP (1.32)becomes dominant, and 1= can be measured by such short-baseline reactorantineutrino experiments. The result showed no evidence for nonzero 1=,giving un upper limit of sin2 21= Q 0:15 at 90% confidence level[24].In 2012, new generations of short-baseline reactor experiments DoubleChooz[25], Daya Bay[26] and RENO[27] separately reported that sin2 21= ≈0:1 by observing clear deficit in the reactor ,¯e flux, where Daya Bay andRENO excluded 1= = 0 at 5:2 and 4:9 respectively.1= can also be measured via the appearance of ,e in a pure accelerator, beam, whose probability is expressed to the leading order as:e (, → ,e) ≈ sin2 21= sin2 2= sin2 ∆m2=1a4Z: (1.33)As in the equation, the leading term in the appearance probability is propor-tional to sin2 21= and ,e appearance is thus greatly suppressed if 1= = 0. In2011, T2K reported the first indication of ,e appearance excluding 1= = 0at 2:5 significance[28], and the updated result in 2013 showed the firstdecisive evidence for ,e appearance at 7:3[29].The current best measurements for 1= come from the short-baselinereactor experiments, and the averaged values from them are[15]:sin2 21= = 8:5± 0:5× 10−2: (1.34)14EB4B inrysolvyx IssuysFCI Unrzsolvzy IssuzsAlthough many of the parameters involved in neutrino oscillations have beenmeasured, there are still important open questions which need to be an-swered in order to fully understand the nature of neutrinos.First of all, it is not known whether 2= mixing is exactly maximal (45◦),and if not, whether the angle is larger or smaller than 45◦. This is known asthe 2= octant, and the cases 2=Q45◦ and 2=S45◦ are referred to as the firstand the second octant respectively. , disappearance is the primary oscilla-tion mode which is sensitive to 2=, however, since the oscillation probabilityhas a parameter dependence of the form sin2 22=, this mode alone cannotdetermine the octant. Therefore, one needs to look at other subdominantoscillation effects such as the ones involving ,e in order to resolve the octant,for instance, the , → ,e oscillation whose leading term depends on sin2 2=as in Equation 1.33.While the squared mass differences between the neutrino mass stateshave been measured rather precisely, the sign of ∆m2=2(or ∆m2=1) is stillyet to be determined. As illustrated in Figure 1.5, the case ∆m2=2 S 0 isreferred to as normal hierarchy(NH) in which case the mass eigenstate m=has the largest mass, whereas ∆m2=2 Q 0 is called inverted hierarchy(IH) inwhich case m= has the smallest mass. The absolute mass is also unknownsince oscillation experiments can only measure the mass differences. Fromcosmological measurements, the sum of the mass of the three neutrinos aremeasured to be less than 0:3 eV[2], and combining the knowledge of themass hierarchy from oscillation experiments we may be able to fully revealthe mass structure of the neutrinos in near future. Furthermore, the originof the neutrino mass and the reason for its smallness compared to otherparticles is still not known. Some of the appealing theories which can explainthis assume that neutrinos are Majorana fermions, i.e. a neutrino is itsown antiparticle, which is a fundamentally new type of particles which havenever been observed. The next generation of neutrino-less double beta decayexperiments can determine whether or not neutrinos are Majorana fermionsin case the mass hierarchy is inverted, but not necessarily if it is normalhierarchy[30]. Therefore, information regarding the mass hierarchy providesvaluable input in order to unveil the fundamental nature of the massiveneutrinos. Knowledge of the mass hierarchy also resolves the degeneracies inthe oscillation probabilities such that the sensitivity of various ongoing andfuture neutrino oscillation experiments to MZ can be significantly improved.Finally, and probably most importantly, it is still an open questionwhether CP is violated for leptons, i.e., whether MZ is nonzero. Some lead-15EBIB droving thy inknowns with Utmosphyriw byutrinosνe νμ ντm232m221m22m23m23m2Normal Invertedm21m22m21?0Figure 1.5: The ordering of the neutrino mass. For normal hierarchy m= isthe heaviest state and ∆m2=2 S 0, whereas for inverted hierarchy m= is thelightest and ∆m2=2 Q 0. The absolute mass is also still unknown and onlythe relative mass differences have been measured.ing theories suggest that CP violation in the lepton sector can produce thematter-antimatter asymmetry which is essential to the formation of the cur-rent matter-dominated universe[31]. Although the observable CP violationin the currently known leptons itself may not necessarily be directly relatedto such process[32][33], it is nonetheless of fundamental interest whether CPviolation can be observed in neutrino oscillations.FC5 erowing thz Unknofins fiith VtmosphzrixczutrinosThe primary oscillation effect which is seen in atmospheric neutrinos is the, disappearance, and from the probability e (, → ,) one can measuresin2 22= and |∆m2=2|. However, due to the wide range of energies and thelong path length in matter, atmospheric neutrinos provide other informationon the oscillation parameters through various sub-leading oscillation effects,allowing us to probe the remaining unknown properties of neutrinos.As mentioned in Section 1.3.2, the path length of atmospheric neutrinosvaries as a function of the zenith angle of neutrinos’ direction. Since neutrinooscillation is characterized by the oscillation phase proportional to aRZ,oscillation effects for atmospheric neutrinos manifest in the two-dimensionaldistributions of zenith angle versus energy. Figure 1.6 shows the oscillationprobability e (, → ,e) and the ratio of oscillated to unoscillated ,e flux16EBIB droving thy inknowns with Utmosphyriw byutrinosP(νµ → νe)-1-0.8-0.6-0.4- 10 10200.ν (GeV)cosΘννe flux ratio  Φosc / Φ0-1-0.8-0.6-0.4- 10 1020.ν (GeV)cosΘνFigure 1.6: The oscillation probability e (, → ,e)(left) and the ratio of os-cillated to unoscillated ,e flux(right) for atmospheric neutrinos reaching theSuper-Kamiokande detector. The horizontal axes are the neutrino energyand the vertical axes are the cosine of the zenith angle of neutrino direc-tion where cos  = 1 is downward-going and cos  = −1 is upward-going.Oscillation probabilities are calculated by assuming the following oscillationparameters: (∆m221, ∆m2=2, sin2 12, sin2 2=, sin2 1=, MZ) = (7.7×10−5eV2,+2.1×10−=eV2, 0.3, 0.5, 0.04, 0◦). Taken from [36].for atmospheric neutrinos reaching the Super-Kamiokande detector. Theoscillation probabilities are calculated by fully considering the three-flavouroscillation effects in matter, using the methods by Barger ds ak-[34] andassuming the radial density structure of the earth described in [35]. Inthe figure, discontinuities can be seen at cos  ≈ −0:45 and cos  ≈ −0:84which correspond to the upper and the lower mantle boundary and thelower mantle and the outer core boundary respectively, which is due to thediscrete change in the matter density causing the matter induced oscillationprobabilities to change discretely. Some of the prominent oscillation effectsare described in the following.The enhancement of the , → ,e oscillation probability seen in 2 −10 GeV upward going region is due to the MSW resonance effect whichhappens while the neutrinos traverse the earth’s core. Following a discussionsimilar to Section 1.2.4, when ∆m2=2 ≫ ∆m221, the transition probability inconstant density matter is approximated as[37]:e (, → ,e) ≈ sin2 21=;M sin2 2= sin2∆m2=2;Ma4ZP (1.35)17EBIB droving thy inknowns with Utmosphyriw byutrinoswhere the effective mixing angle and mass splitting in matter are:sin 21=;M =sin 21=√sin2 21= + (cos 21= − x=2)2P (1.36)∆m2=2;M = ∆m2=2√sin2 21= + (cos 21= − x=2)2P (1.37)whilex=2 ≡ 2ZkMM∆m2=2P (1.38)and kMM = ±√2GFce as before. MSW resonance happens when the reso-nance conditionx=2 = cos 21= (1.39)is satisfied, and ,e appearance due to 1= becomes maximal despite thesmall value of the actual 1= parameter. Since the sign of kMM flips betweenneutrinos and antineutrinos, the enhancement happens for neutrinos only if∆m2=2S0 and for antineutrinos only if ∆m2=2Q0. Thus, mass hierarchy canbe probed by observing the excess in multi-GeV upward-going ,e and ,¯e flux.Furthermore, since the appearance probability Equation 1.35 is proportionalto sin2 2= and not sin2 22=, this oscillation effect also provides sensitivityto the 2= octant.The oscillation patterns seen below 1 GeV on the left plot are the os-cillations driven by 12 and ∆m221 which are relevant for solar neutrino os-cillations. Assuming 1= = 0 and constant matter density, the oscillationprobabilities in such region are given as[38]:e (,e → ,e) = 1− eexP (1.40)e (,e → ,) = e (, → ,e) = eex cos2 2=P (1.41)whereeex = sin2 212;M sin2∆m221;Ma4Z: (1.42)sin 212;M and ∆m221;M are expressed simply by replacing the mixing angleand mass splitting in Equation 1.36 and Equation 1.37 by 12 and ∆m221.Consider now the effect of this oscillation on the net observed ,e flux. Lettingr ≡ Φ0RΦ0e (1.43)18EB6B hhysis cvyrviywbe the ratio of , and ,e flux before oscillation, the fractional difference inthe observed ,e flux due to oscillation, considering the contributions fromboth ,e and , produced in the atmosphere, is expressed as:ΦeΦ0e− 1 = eex(r cos2 2= − 1): (1.44)As mentioned previously, for low energy atmospheric neutrinos the fluxflavour ratio r is roughly 2, and the terms in the parenthesis cancel whencos2 2= = 0:5, i.e. when 2= mixing is maximal. When 2=Q45◦ an excess ofsub-GeV ,e flux is observed, while if 2=S45◦ a deficit will be seen instead,and this oscillation effect therefore provides sensitivity to the 2= octant.When the ∆m221 driven oscillations depending on 1= is also considered,i.e. when not taking the approximation 1= = 0 in the discussion above,the interference between the 1=-dependent oscillations and the oscillationsdescribed by Equation 1.41 introduces a dependence of oscillation probabil-ities on MZ. The interference gives an overall ∼3% effect on the observedtotal ,e flux in sub-GeV depending on the value of MZ[39], and althoughthis is not particularly a large effect this provides sensitivity to MZ.The oscillation effects discussed here will be revisited in Section 8.1.6.FCK ihzsis dvzrvizfiIn this thesis, atmospheric neutrino data observed at the Super-Kamiokandedetector is analyzed in order to the measure the neutrino oscillation parame-ters, namely, sin2 2=, ∆m2=2 including its sign and MZ, to which atmosphericneutrino data has sensitivity. Chapter 2 describes of the Super-Kamiokandedetector, and its calibration methods are summarized in Chapter 3. Chap-ter 4 describes the atmospheric neutrino simulation which is used in con-junction with the observed data in the oscillation analysis. Chapter 5 givesan overview of the data reduction process to select atmospheric neutrinosfrom the data acquired at Super-Kamiokande. Chapter 6 details the maxi-mum likelihood event reconstruction algorithm “fiTQun” which was recentlydeveloped for Super-Kamiokande, which provides input information to theoscillation analysis. As an example of the application of fiTQun, Chapter 7summarizes the T2K ,e appearance analysis which was the first physicsanalysis in which fiTQun was used. Chapter 8 details the methods for theatmospheric neutrino oscillation analysis and presents the results of analyz-ing Super-Kamiokande data. The conclusions and the future outlook of theanalysis are summarized in Chapter 9.19Chvptzr GhupzrBKvmiokvnyzSuper-Kamiokande(Super-K, SK) is a large underground water Cherenkovdetector located in the Kamioka mine in Mt. Ikenoyama, Gifu Prefecture,Japan. The detector consists of a water tank holding 50 kt of ultra-purewater, and the photomultiplier tubes (PMTs) lining the tank interior detectthe Cherenkov radiation which is produced by charged particles which prop-agate in the water. The primary purposes of the detector are nucleon decaysearches and the detection of neutrinos from various sources: solar neutri-nos, astrophysical neutrinos such as the ones from supernovae, atmosphericneutrinos and accelerator neutrinos in long-baseline neutrino oscillation ex-periments such as K2K and T2K.GCF Dztzxtor dvzrvizfiFigure 2.1 shows the sketch of the detector and the surrounding experimentalarea. The water tank is a vertical cylinder of 41.4 m in hight and 39.3m in diameter holding 50 kt of ultra-pure water in total, and the volumeis optically separated into two concentric cylinders: the inner detector(ID)which is the main detector region holding 32 kt of water with 36.2 m in heightand 33.8 m in diameter, and the outer detector(OD) which is a cylindricalshell volume surrounding the ID with its thickness being 2.05 m on the topand the bottom and 2.2 m on the side. The OD serves as a veto detectorfor cosmic ray backgrounds entering from outside of the detecter as well asa passive shield against other entering backgrounds such as neutrons andgamma rays produced in the surrounding rock. The two detector regionsare separated by a 55 cm wide stainless steel supporting structure coveredwith opaque materials as shown in Figure 2.2, and the inner side of thesupporting structure is mounted with 11,129 20-inch PMTs facing inwardviewing the ID volume, while the outer side is mounted with 1,885 8-inchPMTs facing outward viewing the OD. Since the PMTs are sensitive tomagnetic fields, 26 Helmholtz coils line the wall of the tank and reduce thegeomagnetic field to ∼50 mG.The detector has a ∼1000 m overburden of rock which has a shielding20FBEB Dytywtor cvyrviywFigure 2.1: A sketch of the Super-Kamiokande detector. Taken from [40].Figure 2.2: The supporting structure and the PMTs. Taken from [40].21FBFB U kutyr Whyrynkov DytywtorθCβctct/nFigure 2.3: A schematic drawing of Cherenkov radiation.effect against cosmic rays equivalent to 2700 m of water. The rate of cosmicrays reaching the detector is reduced to ∼3 Hz which is low enough that var-ious nucleon decay and neutrino studies can be performed without concernsabout cosmic ray muon backgrounds.GCG V lvtzr Chzrznkov DztzxtorCherenkov radiation is an electrodynamic phenomenon which is analogousto the sonic boom produced by an object travelling faster than the speedof sound. When a charged particle travels in a dielectric medium withrefractive index n at velocity v=x which exceeds the phase velocity xRn oflight in the medium, the local disturbances to the dielectric caused by themoving charge radiate electromagnetic waves which form an electromagneticshock wave as schematically shown in Figure 2.3. This is the Cherenkovlight, and the light is emitted as a cone with its opening angle M relativeto the particle’s direction satisfying:cos M =1n: (2.1)Since n≈1.34 in water at the typical visible light wavelength the detector issensitive to, for a particle traveling at =1 Cherenkov light is therefore emit-ted at the angle M≈42◦. The number of emitted photons per wavelengthper unit travel distance of the particle is given as:y2cyxy=2.2(1− 1n22)P (2.2)where  is the wavelength and  is the fine-structure constant.22FBGB Dytywtor dhusys unx gKAIVSuper-Kamiokande IVRun 73002 Sub 376 Event 7838959214-09-22:19:45:35Inner: 3647 hits, 9221 peOuter: 3 hits, 2 peTrigger: 0x10000007D_wall: 575.8 cmEvis: 979.9 MeVe-like, p = 979.9 MeV/cCharge(pe)    >26.723.3-26.720.2-23.317.3-20.214.7-17.312.2-14.710.0-12.2 8.0-10.0 6.2- 8.0 4.7- 6.2 3.3- 4.7 2.2- 3.3 1.3- 2.2 0.7- 1.3 0.2- 0.7    < 0.200 mu-edecays0 500 1000 1500 2000168336504672840Times (ns)Figure 2.4: An event display of a single electron event in the atmosphericneutrino data, where the unrolled view of the ID is shown in the figure. Eachdot represents a PMT which detected Cherenkov photons, and the colourscale indicates the integrated charge which corresponds to the number ofphotons detected at each PMT.Since S1Rn is required in order for a particle to emit Cherenkov ra-diation, the momentum threshold at which this occurs for various particletypes assuming n=1.34 are roughly: 0:57 MeVRx for an electron, 118 MeVRxfor a muon, 156 MeVRx for a .5 and 1052 MeVRx for a proton.In Super-Kamiokande, neutrinos are detected when they interact withthe water and produce charged particles above the Cherenkov threshold,and the cone of light emitted by a charged particle is observed as a ring bythe PMTs on the detector wall as shown in Figure 2.4. For each event thetime of the first photon arrival as well as the integrated charge at each PMTis recorded, and from that information the type and the kinematics of theparticles are inferred as described in Chapter 6.GC3 Dztzxtor ehvszs vny hKBIkSince Super-K started its operation in 1996, there have been four distinctdetector phases. The initial phase, referred to as SK-I, started in April 1996and continued until the detector was shut down for a scheduled maintenancein July 2001. On November 12, 2001 when the detector was being refilledwith water after the completion of the maintenance, one of the PMTs atthe bottom imploded, causing a shock wave which destroyed more than half23FB4B Innyr DytywtorFigure 2.5: A schematic drawing of the Hamamatsu 20-inch PMT used forthe ID. Taken from [40].of the PMTs in the detector. After the accident the detector was oper-ated temporarily by redistributing the surviving PMTs evenly, while eachID PMT was enclosed in a cover made of acrylic and fibre-reinforced plas-tic(FRP) in order to prevent another implosion accident. The second phaseof the detector, SK-II, ran in such configuration with less than half of theoriginal photo-coverage between October 2002 and October 2005. The de-tector was then brought back to its original full capacity after installingadditional PMTs, and the SK-III phase started in July 2006. Finally, inSeptember 2008 the detector underwent a brief shut down in order to up-grade the electronics, and the SK-IV phase continues since then until thepresent day.This thesis focuses on analyzing the data from SK-IV only, and theconfigurations of the detector in this phase are described in the following.GCI Innzr DztzxtorFigure 2.5 shows a schematic drawing of the Hamamatsu 20-inch PMT whichis used in the ID. 11,129 of the PMTs are evenly placed on the wall of theID on a 70 cm square grid realizing 40% coverage of the ID surface area bythe photocathodes, and Cherenkov photons produced by the particles in the24FBIB cutyr DytywtorFigure 2.6: Quantum efficiency of the photocathode as a function of wave-length. Taken from [40].ID are detected with high efficiency and resolution. The photocathodes arecomposed of a bialkali (Sb-K-Cs) material which has quantum efficiency asshown in Figure 2.6 with a peak efficiency being around 22% at 360 nm wave-length. The 11-stage dynode in the PMT has a gain of 107 when suppliedwith high voltage ranging in 1700-2000V, and the collection efficiency of thefirst dynode is over 70%. The transit time spread for a single photoelectronsignal is 2.2 ns.In order to protect the detector against PMT implosions as happened in2001, each ID PMT is enclosed in a FRP housing with an acrylic windowcovering the photocathode area so that the shock wave does not occur evenif one of the PMTs imploded. The spaces on the ID wall between the PMTsare covered with opaque black sheets made of polyethylene terephthalateensuring the ID and the OD are optically well separated.GC5 dutzr DztzxtorThe outer detector consists of evenly-spaced 1,885 8-inch PMTs facing out-ward to the OD water. Unlike the ID, the spaces between the PMTs arecovered with a reflective material Tyvek which has ∼90% reflectivity at 400nm in order to increase the photon collection efficiency in the OD. To fur-ther enhance the light collection, each PMT is also surrounded by a 60 cmsquare wavelength-shifting plates which absorbs the ultra-violet Cherenkovphotons and reemits blue-green visible light to which the PMTs are sensitive.25FB6B kutyr unx Uir duriwutionAlthough the timing resolution of the OD PMTs slightly degrade from 13ns to 15 ns due to the reemission process, the wavelength shifter increasesthe light collection efficiency in the OD by 60% which is more importantthan the timing resolution considering the role of the OD as a veto detectorfor entering backgrounds. In order to better distinguish between entering,exiting and through-going particle events, the barrel region of the OD isoptically separated from the top and the bottom regions by Tyvek.GCK lvtzr vny Vir eurixvtionAs Cherenkov light needs to travel tens of metres in the detector water be-fore reaching the PMTs to be detected, maintaining high transparency ofthe water is crucial for this experiment. The water which is originally takenfrom the spring water in the mine undergoes multiple filtering, sterilizingand degassing processes in order to remove particle, bacterial and radioac-tive contaminants from the water. The water is continuously purified andcirculated at a rate of ∼50 t/hr, and its temperature is kept at ∼13 ◦Cwith temporal variations within 0:01 ◦C using a heat exchanger in order tosuppress bacterial growth as well as stabilizing the detector response.As the rock in the mine has high radon content, the air in the mine isnaturally rather high in radioactivity. In order to reduce the radon level inthe air especially so that it does not contaminate the detector water, filteredRn-free air is continuously supplied from outside of the mine into the exper-imental area, keeping the area at positive pressure. Furthermore, the rocksurrounding the experimental hall is entirely coated with a polyurethanematerial in order to prevent the radon in the rock from being released intothe air.GCL Dvtv VxquisitionWhen a photon hits the photocathode of a PMT and produces a photoelec-tron(p.e.), the signal is amplified by the dynodes in the PMT and producesa pulse current signal. In SK-IV, such analog signal from a PMT is firstfed into a charge-to-time convertor(QTC) which was specially designed forSuper-K[41]. When the input signal from a PMT exceeds a discriminatorthreshold(we will call this a “hit” for brevity), the QTC starts integratingthe charge from the signal over the following 400 ns, and then outputs arectangular pulse signal whose rising edge representing the time the inputanalog signal crossed the threshold and the width of the pulse representing26FBKB Dutu Uwquisitionthe integrated charge. The output rectangular signal from the QTC is thendigitized by a time-to-digital convertor(TDC), and the digitized charge andtime, which represent the number of the detected p.e.’s and the the pro-duction time of the first p.e. at the PMT respectively, are then sent forfurther processing and are eventually used in physics analyses. The QTCdemonstrates an overall dynamic range of 0.2-2500 pC which corresponds toa maximum signal of ∼1000 p.e., and the non-linearity of charge is measuredto be below 1% across the entire dynamic range. Such high dynamic rangeand linearity enables precise analyses of neutrinos events across a wide rangeof energy.Since it is not feasible to store the information of every single hit inthe detector, an event trigger is issued when the number of PMTs thatare simultaneously hit exceeds a certain threshold, and only the hits thatare accompanied by a trigger are saved. Event triggering in SK-IV is doneentirely by software using the digitized hit time information from the de-tector, and depending on the purpose there are five different types of eventtriggers which are summarized in Table 2.1. A triggering computer contin-uously monitors the observed data and calculates the variable c200 whichrepresents the number of hits in a sliding 200 ns time window at a giventime. When c200 exceeds the defined threshold of a trigger, the trigger isissued and the charge and time information for the hits which are containedin the entire event time window around the trigger is recorded. For theOD trigger c200 is calculated using the hits in the OD PMTs only, whilefor the other triggers hits in the ID are used. The thresholds for the SLEand SHE triggers were lowered in May 2015 and September 2011 respec-tively. The events accompanying the high-energy(HE) trigger are used inthis atmospheric neutrino analysis.Trigger Type c200 Threshold Event Time Window (s)SLE 34→31 [−0:5P+1:0]LE 47 [−5P+35]HE 50 [−5P+35]SHE 70→58 [−5P+35]OD 22 (in OD) [−5P+35]Table 2.1: Definition of the event triggers. The triggers based on ID hits are:Super Low Energy(SLE), Low Energy(LE), High Energy(HE) and SpecialHigh Energy(SHE) triggers. The OD trigger is based on OD hits.27Chvptzr 3Dztzxtor CvliwrvtionThis chapter describes the procedures for calibrating the Super-Kamiokandedetector. The calibration results are used for producing the detector simu-lation which is described in Section 4.3 as well as for analyzing the observeddata. Further details on the calibration methods can be found in [42].As mentioned in Chapter 2, what Super-K records as data is ultimatelythe charge and the time information from the PMTs through the electronics,and their charge and time response therefore needs to be calibrated precisely.The dominant factors which characterize the charge response are the quan-tum efficiency and the gain of the PMTs, and as the characteristics of eachPMT have non-negligible variations, such properties need to be calibratedon an individual PMT basis. The time response also has significant varia-tions between different PMT channels. In addition to the response of thePMTs and the electronics, the optical properties of water and other compo-nents in the detector also need to measured in order to precisely model theCherenkov photon propagation in the detector simulation.3CF gzlvtivz ebi Gvin CvliwrvtionThe supply high voltage for the PMTs are set individually for each PMTso that the charge response for a given light intensity is approximately thesame across all PMTs. In order to measure and correct for the remainingindividual variations in the PMT gain, the following measurement is done.The measurement uses the same apparatus as the timing calibrationwhich is described in detail in Section 3.4. As schematically shown in Fig-ure 3.2, pulsed laser light is injected isotropically at a fixed position nearthe centre of the ID, and the hits and the charge at the ID PMTs are thenmeasured repeatedly by flashing the laser at two different intensities.One set of measurements is performed by flashing the laser at high in-tensity IH such that every PMT detects several photons at a time. In suchcase, the average measured charge f(i) at the i-th PMT is proportional tothe intensity IH of the injected light as well as the individual gain of the28GBFB Uvsoluty dah Guin WulivrutionPMT G(i):f(i) ∝ IH × v(i)× ϵ(i)×G(i): (3.1)In this equation, the measured charge is also proportional to the light accep-tance v(i), which depends on the PMT’s geometrical configuration withinthe detector, and the quantum efficiency ϵ(i) of the PMT.Another set of measurements is made by flashing the laser at very lowintensity IL so that only a few PMTs get hit at a time. At such low intensitya PMT detects at most one photon at a time, and the average number of hitsc(i) at each PMT, which is equivalent to the hit probability, is proportionalto the intensity of the injected light but is almost independent of the PMTgain due to the low hit discriminator threshold:c(i) ∝ IL × v(i)× ϵ(i): (3.2)Since the position of the light source is unchanged and the only differencebetween the two sets of measurements is the intensity of the injected light,by taking the ratio f(i)Rc(i) the factors which vary between individualPMTs cancel except for the gain G(i), and the relative individual variationof the PMT gain is thus obtained from the measured ratio:G(i) ∝ f(i)Rc(i): (3.3)According to the measurement, the variation of the gain across all PMTs hasa standard deviation of 5.9%, and the individual relative gain factor obtainedabove is used to correct the observed charge at each PMT to effectively makethe charge response of the PMTs uniform across the entire detector whenthe data is analyzed.3CG Vwsolutz ebi Gvin CvliwrvtionThe absolute gain, which relates the measured charge to the number ofdetected p.e.’s at a PMT, is obtained by a calibration measurement us-ing a “nickel source”: a low energy gamma ray source made of a sphereof nickel which emits 9 MeV gamma rays isotropically upon capturing theneutrons emitted from a 252Cf neutron source which is placed at the cen-tre. The source is placed at the centre of the ID and produces on average0.004 p.e./event at each PMT, and more than 99% of the produced hits willtherefore be single p.e. hits.After correcting for the relative gain variation for each PMT as describedin Section 3.1, the observed charge distribution for the single p.e. hits pro-duced by the nickel source is obtained by accumulating the hits from all29GBGB dah euuntum EffiwiynwflpC0 5 10 15 20 25 30 35 40 45 50110210310410510610Figure 3.1: The observed charge distribution for single p.e. hits obtainedfrom the nickel source calibration data. Figure taken from [42].PMTs as is shown in Figure 3.1. From the average of this distribution, theconversion factor between the observed charge and the number of p.e.’s isdetermined to be 2.658 pC/p.e. The obtained single p.e. charge distributionis also used in the detector simulation which is discussed in Section 4.3.3C3 ebi fuvntum ZffixiznxyIn addition to the gain, the quantum efficiency(Q.E.) also varies betweendifferent PMTs and the efficiency for each PMT needs to be measured. Themeasurement is done using the same nickel source as described in Section 3.2.When the intensity of light reaching a PMT is small enough that theexpected number of produced p.e. is significantly smaller than one as inthe nickel source data, the average number of hits at a PMT, i.e. the hitprobability, is proportional to the quantum efficiency as well as the intensityof the light as described by Equation 3.2. Therefore, the hit probabilitiesmeasured by the nickel source data can be used to calibrate the Q.E. foreach PMT. Since it is not possible to express the light acceptance v(i) inEquation 3.2 in closed form and accurately correct for it to extract the Q.E.ϵ(i) directly from the measured hit probability, the nickel source events aresimulated assuming no individual variations for the Q.E. so that it can becompared to the observed data. The hit probability obtained from the sim-ulated events, which accounts for the factor v(i) via simulation but does notinclude the variations in ϵ(i), is compared to the hit probability calculated30GB4B fylutivy himing Wulivrution0.CUGT&[G.CUGT&KHHWUGTDCNN64)/QP2/6PORWNUGYKFVJPUGEPORWNUGYKFVJPUGE8CTKCDNGHKNVGT5-+&Figure 3.2: A schematic view of the timing calibration system. Figure takenfrom [42].for each PMT using the real nickel source data and the Q.E. for each PMTϵ(i) is obtained. The resulting Q.E. for each individual PMT is used in thedetector simulation.3CI gzlvtivz iiming CvliwrvtionThe time interval between the time a photoelectron is initially producedat a PMT and the time the signal from the PMT is registered as a hit bythe electronics varies between the PMTs due to many factors including thedifference in the length of the cables connecting the PMTs to the electronics.Also, since the time it takes for an analog signal from a PMT to cross thehit discriminator threshold depends on the pulse height, the time responsedepends on the observed charge.In order to calibrate such variations in the time response, a fast pulsedlaser light is injected isotropically at the centre of the ID as schematicallyshown in Figure 3.2. The time of the laser injection is monitored directlyfrom the light source by a fast 2-inch monitoring PMT, and using that31GB4B fylutivy himing WulivrutionQBin0 20 40 60 80 100 120 140 160 180T (ns)1170118011901200121012201230Entries  5616790. of cable # 000100 1 2 3 4 5 6 7 8 9 10Q (p.C.)210310Figure 3.3: The TQ distribution for an ID PMT. The horizontal axis rep-resents the charge where the scale in pC is shown at the top: the scale islinear up to 10pC and then goes to log scale. The vertical axis representsthe TOF-corrected hit time where a larger value corresponds to earlier hits.Figure taken from [42].injection time as a reference the relative offsets between the hit time ofthe ID PMTs are obtained from the time-of-flight(TOF) corrected hit timewhich is defined as the hit time subtracted by the time required for a photonto travel from the light source to the PMT. Since the timing also depends onthe charge as stated before, calibration data is taken at various laser intensityand for each PMT the 2D distribution of the observed TOF-corrected timeand charge is produced which is referred to as the “TQ distribution”.Figure 3.3 shows an example of the TQ distribution for a PMT. In thefigure, the horizontal axis represents the charge and the vertical axis rep-resents the TOF-corrected hit time. Once the TQ distribution is obtainedfor each PMT, the distribution is fitted by an asymmetric Gaussian in eachcharge bin, and the peak position of the Gaussians is then fitted by a polyno-mial as a function of charge. The fitted peak position of the TQ distribution,named the TQ map, is saved into a database for each PMT and is then usedto correct the hit timing as a function of charge on an individual PMT basis.This ensures that the time response of the PMTs in the entire detector issynchronized.32GBIB kutyr dropyrtfl ayusurymyntT (ns)-50 -40 -30 -20 -10 0 10 20 30 40 50Entries per 0.52 ns0100200300400500600700310Figure 3.4: The timing distribution of ID PMTs at a charge bin ∼1 p.e.,produced using hits from all PMTs. The horizontal axis represents thetiming after TQ map correction, where a larger value corresponds to earlierhits. The red data points represent the data obtained from calibration, andthe blue curve is the fitted asymmetric Gaussian. Figure taken from [42].The same dataset is also used to obtain the PMT timing resolutionwhich is put in to the detector simulation. At each charge bin, a timingdistribution is produced by accumulating the hits from all PMTs after TQmap correction, and the distribution is fitted by an asymmetric Gaussianas shown in Figure 3.4. The detector simulation then simulates the PMTtime response by an asymmetric Gaussian using the widths as a function ofcharge which is obtained from the calibration.3C5 lvtzr eropzrty bzvsurzmzntIn order to accurately model the Cherenkov photon propagation in water, anempirical model for light scattering and absorption in the detector simulationis tuned based on a calibration measurement using a laser injector.In the detector simulation, the attenuation of light in water is character-ized as exp (−lRa()), where l is the distance the light traveled and a() is33GBIB kutyr dropyrtfl ayusurymyntFigure 3.5: A schematic view of the laser injector system for measuring thewater property and the reflectivity of the PMTs. Figure taken from [42].a wavelength-dependent attenuation length defined as:a() = (sym() + ksy() + kls())−1 : (3.4)The coefficient sym() represents the amplitude of the “symmetric scat-tering” which takes into account the effects from Rayleigh scattering andsymmetric Mie scattering, and it has the angular distribution of the form1 + cos2  where  is the photon scattering angle. ksy() represents the“asymmetric scattering” which accounts for forward Mie scattering, and ithas the angular dependence of cos  in the forward direction while there isno backward scattering. Finally, kls() is for absorption. These coefficientsare empirical functions of the wavelength  which are determined based onthe calibration data from a laser injector system as shown in Figure 3.5.In the setup, a collimated laser beam is injected at the top of SK verti-cally down toward the bottom, and the light scattered in water and reflectedfrom the bottom is detected by the PMTs on the barrel and the top wallof the detector. The TOF-subtracted hit time distribution, i.e. the PMThit timing distribution after subtracting the time required for photons totravel from the beam spot at the bottom to each PMT, for the PMTs ineach detector region as indicated in Figure 3.5 is shown in Figure 3.6. Thesharp peaks on the right between 1830-1900 ns are due to the photons re-flected at the beam spot from the bottom of the detector, whereas the hitsat earlier times are caused by photons which scattered in water before reach-ing the bottom and arrived at the barrel or the top wall. The scattering34GBIB kutyr dropyrtfl ayusurymyntns1500 1600 1700 1800 19000.0010.0020.0030.0040.0010.0020.0030.0040.0010.0020.0030.0040.0010.0020.0030.0040.0010.0020.0030.0040.0010.0020.0030.004Figure 3.6: The TOF-subtracted hit time distributions for the laser injectordata taken at 405 nm. The top plot is for the PMTs on the top wall, andthe following five plots correspond to the the five regions on the barrel asindicated in Figure 3.5. The black circles represent the measured data andthe red histograms are the result from the laser injector simulation aftertuning the water properties and PMT reflectivity to data. The data before1830 ns and between the two vertical blue lines on the left is used for thewater measurement, and the data in the range 1830-1900 ns is used fortuning the PMT reflectivity. Figure taken from [42].35GBIB kutyr dropyrtfl ayusurymyntWavelength (nm)300 325 350 375 400 425 450 475 500Inverse water transparency (1/m)-410-310-210-110Total AbsorptionSymmetric AsymmetricFigure 3.7: Light absorption and scattering coefficients as a function ofwavelength. The data points are the data taken in April 2009, and the red,blue and magenta lines represent the tuned kls(), sym() and ksy()respectively. The black line represents the sum of the three, which is theinverse of the attenuation length a(). Figure taken from [42].and absorption coefficients sym(), ksy() and kls() are tuned using thelatter, the data before 1830 ns, such that the tuned simulation best agreeswithin the model parameter space with the measured data taken at variouswavelengths. Figure 3.7 shows the scattering and the absorption coefficientsas functions of wavelength which are tuned using the data taken in April2009, and the red histograms in Figure 3.6 are the timing distributions forthe best-tuned laser injector simulation. The measurement leads to an over-all attenuation length of ∼120 m at 400 nm wavelength. The apparatus issituated permanently in the detector and constantly takes data during theSK operation so that the water parameters can be constantly monitored.Figure 3.8 shows the time variation of the measured coefficients at variouswavelengths.36GB6B dah unx Vluwk ghyyt fyywtivitflFigure 3.8: Time variation of the light absorption and scattering coefficientsmeasured at various wavelengths. The coefficients sym, ksy and kls areshown in purple, black and blue respectively. Some of the laser units werereplaced in 2009 with some changes in the wavelength, and the vertical blacklines indicate when the replacement took place. Figure taken from [42].3CK ebi vny Wlvxk hhzzt gzzxtivityThe laser injector data used in Section 3.5 for water measurement is alsoused to tune the reflectivity of the PMTs in the detector simulation. Ignor-ing the reflections at the PMT internal structure, the detector simulationmodels light reflections at the PMT surface by considering the interfaceof four different layers: the water, the PMT glass, the photocathode andthe vacuum inside the PMT. Each layer is characterized by a wavelength-dependent refractive index, and since the photocathode layer absorbs light acomplex refractive index is assigned to the photocathode which is treated asan effective tuning parameter for PMT reflectivity. Using the laser injectordata in the time range 1830-1900 ns which contains the peaks from the lightreflected at the tank bottom as shown in Figure 3.6, the complex refractiveindex of the photocathode is tuned such that the laser injector simulation37GBKB dhotosynsor hyst FuwilitflFigure 3.9: A schematic view of the apparatus for black sheet reflectivitymeasurement. Figure taken from [42].best agrees with the measured data in this region. The tuning is performedat the wavelengths 337, 365, 400 and 420 nm, and the result is used whensimulating physics events such as neutrinos.The reflectivity of the black sheet is measured in the SK detector using adedicated apparatus. As shown in Figure 3.9, a laser injector unit to whicha specimen of the black sheet is attached is placed at the centre of SK,and the reflected laser from the black sheet is projected horizontally to thebarrel of the ID. The intensity and the profile of the reflected light is thenmeasured for each of the the incident angles 30◦, 45◦ and 60◦ and for thewavelengths 377, 400 and 420 nm. The intensity of the injected light directlyprojected onto the wall without reflection is also measured for normalizingthe reflected light data, and the reflectivity calculated from the measuredresult is used in the detector simulation.3CL ehotosznsor izst FvxilityProperties of the SK PMTs have been measured over the years throughvarious in-situ calibration measurements in SK as described above. Whilesuch methods have served well in calibrating the overall characteristics ofthe PMTs so far, the increasing physics data statistics and the growingcomplexities of physics analyses require more detailed understanding of the38GBKB dhotosynsor hyst FuwilitflFigure 3.10: Left: A schematic drawing of the PTF, showing the watertank, mechanical arms, optical heads and the PMT which is being measured.Right: External view of the PTF magnetic field compensation system.detector, so that we can improve the detector simulation for better data-MCagreement and also reduce the systematic uncertainties in the detector.The Photosensor Test Facility (PTF) at TRIUMF (Vancouver, Canada)is a facility which was constructed to perform detailed ex-situ measurementsof the responses and the passive optical properties of large photosensors inwater. The left of Figure 3.10 shows the schematic drawing of the facility.The facility consists of a large water tank in which a 20-inch SK PMT canbe placed upright and fully submerged in filtered water provided by a waterpurification system. An optical head unit equipped with a collimated lightsource is mounted on each of the two automated mechanical arms whichmove the optical heads with three translation and two rotation degrees offreedom with a position and direction accuracy of 1 mm and 1◦. This allowsthe detailed measurement of the PMT response for light incident at variouspositions on the PMT. Each optical head is also equipped with a smallreceiver PMT which is used to detect reflected light, and by using bothmechanical arms at the same time and receiving the light emitted by theother optical head, PMT reflectivity measurements can also be performed.Figure 3.11 shows the interior of the tank with a SK PMT submerged in39GBKB dhotosynsor hyst FuwilitflFigure 3.11: A SK PMT submerged in the tank filled with water. The twoblack boxes are the water-tight optical head units mounted on the mechan-ical arms.water, along with the optical heads.The optical head unit is a water-tight box. Light from a pulsed laser ora Xe lamp is delivered via an optical fibre which is connected in the box to alight source unit, which collimates and linearly polarizes the light as shownin Figure 3.12. The collimated beam is then split in two by a beam splitter,one of which is projected onto the target PMT outside of the box while theother is detected by a small monitoring PMT in the box which monitorsthe intensity variation of the incident light. Another 18 mm square receiverPMT is placed next to the light source unit, which views out of the box andis used for reflectivity measurements.The response of large photosensors are greatly affected by magneticfields. In order to compensate the geomagnetic field and the field from thecyclotron at TRIUMF, active field cancellation using three-axis Helmholtzcoils and passive magnetic shielding are employed as shown on the right ofFigure 3.10. During a measurement, local magnetic fields are constantlymonitored by a magnetometer placed in the optical head unit. Studies areongoing to automatically tune the coil current based on real time field mea-surements so that the magnetic field can be reduced to ∼10 mG level, whichis lower than the ∼50 mG field in SK.40GBKB dhotosynsor hyst FuwilitflFigure 3.12: Top: The interior of the water-tight optical head box. Light isdelivered via an optical fibre which is connected to the light source unit. Areceiver PMT is placed next to the light source and receives the light fromoutside. Bottom: The light source unit, which consists of a collimator, apolarizer, a beam splitter and a monitoring PMT.41GBKB dhotosynsor hyst FuwilitflFigure 3.13: Local variations of the gain of a SK PMT. The plot showsthe map of the gain variation viewing the PMT vertically down. This isa relative gain variation measurement and the colour scale is in arbitraryunits. Note that the horizontal and vertical band structure crossing nearthe centre is produced by bands which are placed on the surface of the PMTfor calibrating the position and directions of incident beam and is not theresult of the characteristics of the PMT itself.Finally, the entire apparatus is surrounded by two layers of dark curtainswhich produce a dark environment adequate for measuring PMT responses.The facility has been making various measurements of different photo-sensors including the SK PMT, such as the measurement of local variationsin the photon detection efficiencies and the gain. Figure 3.13 shows an exam-ple result of a relative gain variation measurement for a SK PMT performedin air, in which the peak position of the 1 p.e. distributions are measuredby shining laser at different positions on the SK PMT. Although the resultshave not been incorporated into SK analyses yet, it is expected that the re-sults from the PTF measurements will be used to improve the PMT modelsin the SK detector simulation and also to better understand the systematicsrelated to detector response in near future.42Chvptzr IVtmosphzrix czutrinohimulvtionPhysics analyses at Super-K are done by comparing the observed data to thepredictions obtained from the Monte Carlo(MC) simulations. This chapterdescribes the atmospheric neutrino MC simulation which is used in the os-cillation analysis. Producing the MC involves simulating the following threemajor components: the atmospheric neutrino flux reaching the detector, theinteraction of the neutrinos with water and the response of the detector tothe particles produced by the neutrino interaction.ICF Vtmosphzrix czutrino FluflAs the analysis of atmospheric neutrino oscillations is essentially done byinvestigating the change in the observed atmospheric neutrino flux at thedetector due to oscillations, it is essential to have an accurate model for theproduction of neutrinos in the atmosphere. Super-K adopts the flux modelby Honda ds ak-[43] for various analyses concerning atmospheric neutrinos,and the model is briefly described below.The primary cosmic ray flux, which consists mostly of hydrogen nuclei,is modeled based on the measurements by AMS[44] and BESS[45][46]. Theprimary flux is affected by the solar wind and depending on whether the solaractivity is at maximum or minimum the primary flux at ∼1 GeV varies bynearly a factor of two. The effect of the geomagnetic field manifests in theup-down and east-west asymmetry of the neutrino flux at Super-K as well asthe low-energy cutoff in the flux due to the deflection of low energy primarycosmic rays out to space.The US-standard atmosphere ’76[47] is adopted as the model for thedensity structure of the atmosphere, and for the production of secondarycosmic ray particles in the atmosphere, DPMJET-III[48] is used for hadronicinteractions above 32 GeV while for energy below 32 GeV JAM[49] is usedfor better agreement with low energy cosmic ray muon data. The hadron434BEB Utmosphyriw byutrino FlufiFigure 4.1: Zenith angle distributions of the atmospheric neutrino flux atSuper-K averaged over the azimuth at 0.32 GeV(left), 1.0 GeV(middle) and3.2 GeV(right). cos =1 represents vertically downward going and cos =-1represents vertically upward going neutrinos. Figure taken from [43].production models are further tuned based on the cosmic ray muon mea-surements at various altitudes such as [46], [50] and [51]. During the fluxcalculation the trajectories of the cosmic ray particles in the atmosphere arefully simulated in three-dimensions.Figure 4.1 shows the zenith angle distributions of the atmospheric neu-trino flux at Super-K calculated by the Honda model. The flux is peaked atthe horizon largely since the cosmic ray particles can travel longer distancesin the atmosphere horizontally rather than vertically, allowing more time fordecay. The large up-down asymmetry as seen in the left plot is due to thegeomagnetic field, and the effect gets smaller at higher energies since higherenergy cosmic rays are affected less by the geomagnetic field.Figure 4.2 shows the direction-averaged neutrino flux at Super-K as afunction of energy. The plots compare the predictions from the Honda modelwith other flux models such as the Fluka model[52] and the Bartol model[53],which have differences in hadronic interaction simulations and particle tra-jectory calculations as well as the cosmic ray dataset the models are tunedto. The differences in the flux predictions between the models are used toestimate the systematic uncertainties related to the atmospheric neutrinoflux.The MC events are generated using the predicted flux at Super-K with-out considering any neutrino oscillations, i.e., assuming that all neutrinos444BFB byutrino IntyruwtionFigure 4.2: The direction-averaged atmospheric neutrino flux at Super-K asa function of energy. The left plot is the absolute flux and the right plotis the various flux ratios. Red represents the Honda flux model which isused in the Super-K MC, and predictions from other models are used forestimating the systematic uncertainties for the flux. Figure taken from [43].reach the detector in their original flavour at the production in the atmo-sphere. Oscillations are applied at the time of performing an analysis byreweighting each MC event according to its oscillation probability based onthe neutrino energy and direction, and the procedure is discussed in moredetail in Section 8.3.1.ICG czutrino IntzrvxtionAs stated in Section 2.2, neutrinos are detected at Super-K when they in-teract with the water in the detector. Neutrino interactions at Super-K aresimulated by the neutrino event generator NEUT[54]. The software simu-lates the interaction of neutrinos with oxygen and hydrogen nuclei in water,while the interactions with electrons are neglected in atmospheric neutrinosimulations due to the three orders of magnitude smaller cross sections rel-ative to that for nuclei.Neutrino interactions can be categorized into Charged Current (CC)and Neutral Current (NC) interactions. A CC interaction happens whena neutrino interacts with a target by exchanging a l± boson producing acharged lepton whose flavour corresponds to that of the neutrino. Since the454BFB byutrino Intyruwtionflavour of the charged lepton, i.e. electron or muon, can be identified inSuper-K as it will be discussed in Chapter 6, one can identify the flavourof the incoming neutrino in a CC event, which is important in observingthe effects from neutrino flavour change in an oscillation analysis. On theother hand, a NC interaction is mediated by a o0 boson in which case theoutgoing lepton is also a neutrino and does not leave any clear signal inthe detector concerning the flavour. The NC events are therefore in generalconsidered as backgrounds in atmospheric neutrino oscillation analyses.Although the original neutrino flux produced in the atmosphere onlycontains electron and muon neutrinos, tau neutrinos are also detected atSuper-K because of oscillations. Due to the large 1:78 GeVRx2 rest mass ofthe tau lepton ,CC interactions only occur at energies above several GeV,resulting in a significantly smaller number of produced events compared to,e and ,. Nevertheless, , interactions are simulated in the same manneras ,e and , CC interactions by NEUT. The produced tau leptons undergocomplex decays into multiple mesons and leptons which are simulated byTAUOLA[55].The neutrino interaction modes which are simulated by NEUT are sum-marized in the following, and the cross sections as functions of energy forthe dominant interaction modes are shown in Figure 4.3. Except for mesonexchange currents, each interaction mode is modeled for both CC and NC.In addition to the initial interaction of the incident neutrino with the targetnucleus, NEUT also simulates the subsequent interaction of the producedparticles within the nucleus, which is described in Section 4.2.5.ICGCF Zlvstix vny fuvsiBZlvstix hxvttzringNC elastic scattering is a process in which a neutrino simply scatters offa nucleon target by transferring some momentum without producing anynew particles. Quasi-elastic (QE) scattering is an equivalent process fora CC interaction in which case the incident neutrino is converted into acorresponding charged lepton while the target nucleon also converts in orderto preserve the total electric charge. For example, a , CCQE interactionis the following process:, + n→ − + p: (4.1)In NEUT, such interactions on a free nucleon target are simulated by theLlewellyn-Smith model[58], and for a nucleon bound in an oxygen nucleusthe relativistic Fermi gas model by Smith and Moniz[59] is used in order totake into account the Fermi motion and the Pauli blocking of the nucleon.The model contains a form factor which is characterized by the axial vector464BFB byutrino IntyruwtionFigure 4.3: ,(left) and ,¯(right) CC cross sections per nucleon dividedby neutrino energy, plotted as a function of energy. Experimental datasummarized in [56] is compared to the predictions from NEUT, and thecontributions from quasi-elastic scattering(QE), resonance production(RES)and deep inelastic scattering(DIS) are shown separately in different colours.Figure taken from [57].mass parameter MfEA , and the shape of the differential cross section as afunction of f2(square of the four-momentum transfer) as well as the totalcross section depend on the parameter. MfEA is set to 1:21 GeVRx2 basedon the results from K2K[60] and MiniBooNE[61].ICGCG bzson Zflxhvngz CurrzntIn the CCQE interaction on a bound nucleon described above, an approx-imation is made such that a neutrino interacts with a single nucleon onlywhile neglecting any direct correlations between the nucleons in the nucleus.However, in reality the interactions among the nucleons must introduce somedirect correlations such that an incident neutrino interacts simultaneouslywith multiple nucleons. The tension observed between the existing CCQEmodel and the data from experiments such as MiniBooNE[61] also suggeststhe existence of such process, and multi-nucleon correlations have been agrowing interest in the field of neutrino research in the past several years.The Meson Exchange Current(MEC) model by Nieves ds ak-[62] is one ofthe prominent models which attempt to go beyond the preexisting simplisticmodel of CCQE, and their model has been recently introduced to NEUT.In the simulated process, the incident neutrino interacts with a pair of twonucleons as opposed to a single nucleon target as:, +cc′ → l +c ′′c ′′′P (4.2)474BFB byutrino Intyruwtionwhere cc ′ and c ′′c ′′′ are the nucleon pair before and after the interaction.Although multi-nucleon correlations should in principle be present inany neutrino interactions on a bound nucleon, the CC MEC interactionmentioned above is the only such process which is considered in NEUT atthe present moment.ICGC3 hinglz bzson eroyuxtionProduction of a single meson in a neutrino interaction is modeled in NEUTprimarily as a resonance production, in which a baryon resonance excitationproduces a single meson in the final state as in:, +c → l +c∗ → l +c ′ +mP (4.3)where c and c ′ are the nucleon in the initial and the final state, c∗ is theintermediate baryon resonance and m is a meson such as ., K and . l isthe outgoing lepton which is a charged lepton for CC interaction while it is aneutrino for NC. Such interaction is considered in NEUT forl Q 2 GeVRx2,where l is the invariant mass of the hadronic final state. The simulationis done adopting the Rein-Sehgal model[63] with the revised form factor byGraczyk and Sobczyk[64], and the parameters characterizing the form factorwere determined using bubble chamber data[65][66].Coherent pion production is also considered, in which the incident neu-trino interacts with the entire oxygen nucleus producing a pion:, +16O→ l + 16O+ .: (4.4)Due to the low momentum transfer in this process the outgoing lepton andpion produced through this mechanism are peaked in the forward direction.The interaction is simulated according to the model by Rein and Sehgal[67].ICGCI Dzzp Inzlvstix hxvttzringIn deep inelastic scattering (DIS), the incident neutrino interacts with aconstituent quark in the target nucleon and often produce multiple hadrons.In NEUT the process is considered for the cases with hadronic invariantmass l S 1:3 GeVRx2 and becomes increasingly dominant in multi-GeVenergies. The nucleon structure functions used in the model are based onthe GRV98 parton distribution functions[68], and the corrections by Bodekand Yang[69] are applied in order to use the model in the low f2 region.484BFB byutrino IntyruwtionFor l Q 2 GeVRx2, only pions are considered as outgoing hadrons, andpion multiplicity in this region is estimated from bubble chamber experi-ments [70][71]. Since there is an overlap with the resonance pion productionin this region, only the cases of producing two or more pions are consideredin the DIS channel for l Q 2 GeVRx2. For l S 2 GeVRx2, production ofheavier mesons such as K and  are also considered and the hadronic finalstates are determined by PYTHIA/JETSET[72].ICGC5 Finvl htvtz IntzrvxtionWhen a neutrino interaction happens in a nucleus, the hadrons which areproduced at the initial interaction must propagate through the nuclearmedium before getting detected outside of the nucleus, and during suchtime the hadrons often interact hadronically with the nuclear medium. Suchinteractions are referred to as final state interactions (FSI) and it is an im-portant effect for neutrino analyses at Super-K since it directly alters theobservable particle final state from what is initially produced at the neu-trino interaction. NEUT considers the FSI for mesons and nucleons, andfor nucleons[73] and heavy mesons such as K[74][75][76] and [77] the FSIis simulated based on the cascade models and external data.The hadrons which are most frequently seen at Super-K are, by far,the pions, and it is therefore important to accurately model the FSI for pi-ons. Pion FSI is simulated by the NEUT pion cascade model[78] in whichpions are propagated classically in finite steps through a nuclear mediumdescribed by the Woods-Saxon nuclear density profile[79]. At each step, apion may undergo the following processes according to the interaction prob-abilities given in the model: quasi-elastic scattering in which a single pion ofthe same charge remains after the interaction, charge exchange in which acharged pion is converted into a .0 and vice versa, pion absorption in whichno pions remain after the interaction and pion production in which multiplepions are produced. The model has been tuned based on various pion-nucleon and pion-nucleus scattering data, and Figure 4.4 shows the tuned.5-12C scattering cross section simulated in NEUT compared with experi-mental data. The uncertainty in the model is evaluated by comparing themodel variations against pion scattering data. As shown in Figure 4.5, by si-multaneously varying the FSI model parameters and comparing the resultingcross sections against various pion scattering data, representative parametersets are chosen which span a “1 error surface” on the multi-dimensionalFSI model parameter space. There are in total six model parameters whichcharacterize the interaction probabilities for quasi-elastic scattering, charge494BGB Dytywtor gimulutionFigure 4.4: .5-12C scattering cross sections, where the cross section foreach interaction process is shown separately in the colour described in thelegend. The data points are the measured results from .5-12C scatteringexperiments and the solid lines are the cross sections calculated by the tunedNEUT pion cascade model. The dashed lines are for the previous NEUTpion FSI model. Figure taken from [78].exchange, absorption and pion production in two energy ranges(pion mo-mentum of below or above 500 MeVRx), and 16 representative parametersets are obtained as the 1 variations[80].IC3 Dztzxtor himulvtionThe particles which are produced by NEUT are then propagated in thedetector by a GEANT3[81]-based detector simulation called SKDETSIM.GEANT simulates the interaction of the particles with water, particle decaysas well as the production of Cherenkov photons. Hadronic interactions aresimulated by GCALOR[82] except for the pions below 500 MeVRx for whichthe NEUT pion cascade model described in Section 4.2.5 is used.The Cherenkov photons produced by the particles are then propagated inthe detector using a custom code which models the scattering and absorption504BGB Dytywtor gimulutionFigure 4.5: Low momentum .5 absorption cross sections compared betweenexperimental data and the variations of the NEUT pion cascade model. Inthe figure, the interaction probabilities for quasi-elastic(qe), absorption(ab)and charge exchange(cx) for pions Q 500 MeVRx are simultaneously variedin NEUT, and the resulting cross sections for eight representative variationsof the parameter set are shown in different colours. Figure taken from [78].of light in water based on the calibration results in Section 3.5. Reflectivityof the PMT and the black sheet are also modeled using the results obtainedin Section 3.6.The charge and time response of the PMTs and the electronics are alsosimulated based on the calibration measurements, and the detector simula-tion at the end provides the same data structure as the observed data sothat the two can be analyzed in the same manner.51Chvptzr 5Dvtv gzyuxtionWhile SK detects roughly 10 atmospheric neutrino events each day, the high-energy event trigger is fired O(106) times per day, most of which are dueto background events caused by cosmic ray muons, radioactivity and noisefrom the electronics. In order to remove such backgrounds and select theneutrino events for physics analyses, a series of reduction cuts are appliedto the observed data.The events which are targeted in the neutrino oscillation analysis pre-sented in Chapter 8 are the atmospheric neutrinos which interact with thewater in the ID, whose daughter particles are then detected as Cherenkovrings by the ID PMTs with high resolution. Such neutrino events can beclassified into two categories based on the activity in the OD: an event iscategorized as Fully-Contained(FC) if all produced particles stop within theID and leave no signals in the OD, whereas it is categorized as Partially-Contained(PC) if any of the daughter particles exit to the OD and leavesignals there. The FC event sample has large statistics and since all theCherenkov photons are deposited in the ID the events can be precisely re-constructed and used in physics analyses. The oscillation analysis in Chap-ter 8 therefore uses the FC atmospheric neutrino data, and the proceduresto select the FC events are described in the following. The selection processconsists of a series of five data reduction stages which gradually eliminatethe backgrounds in data.5CF First vny hzxony gzyuxtionThe first two stages of the reduction process are designed to quickly filterout the obvious backgrounds from cosmic ray muons, electrical noise andlow energy backgrounds such as those caused by radioactivity.In the first reduction, events whose total charge from the ID PMTs isless than 200 p.e. are rejected as low energy backgrounds. This thresholdroughly corresponds to the charge deposited by a 22 MeVRx electron. Aswill be mentioned later, only the events with energy greater than that of a30 MeVRx electron are used in the oscillation analysis, and this cut removes52IBFB hhirx fyxuwtionthe majority of low energy backgrounds while keeping the targeted neutrinoevents. In addition, in order to reject cosmic ray muons which usually leavesignals in the OD, events with more than 50 OD hits within a 800 ns timewindow around the event trigger are rejected.In the second reduction, the threshold for the above OD activity cut islowered to 30 OD hits for events with the total ID charge less than 100,000p.e. so that lower energy muons can be further reduced. Furthermore, anevent is rejected if more than half of the total ID charge originates froma single PMT. This cut is intended to reject low energy events which arecaused by electrical noise such as a discharge in a PMT.After the first and the second reduction the number events reduces to∼200 events/day, which is already more than three orders of magnitudereduction from the initial dataset defined by the high-energy triggers.5CG ihiry gzyuxtionThe third reduction targets several specific background sources which cannotbe removed by the first two reduction stages.One of the major remaining backgrounds are the cosmic ray muons whichenter from outside of the detector but leave low activity in the OD. Inorder to remove through-going muons which enter and exit the detector, theremaining events are processed by the through-going muon fitter which isa fast algorithm to reconstruct the entering and the exiting point on theID wall assuming a muon. Once the events are reconstructed, a goodnessof fit is evaluated based on the deviation of the observed PMT hit timingsfrom the predictions assuming a through-going muon, and events whichagree well with the muon hypothesis get rejected. In addition to the cutbased the goodness of fit, events which have more than 9 OD hits within8m from the estimated entrance or exit points are also rejected. Similarrejection methods are also applied assuming a stopping muon which is amuon entering the detector but stopping within the ID.Another type of a muon background is the cable hole muons. At thetop of the SK detector there are twelve holes where the bundles of signaland high voltage cables for the PMTs go through, and comic ray muons canenter the detector through the holes without leaving any signals in the OD.Such muons are rejected based on the signals from the plastic scintillationcounters installed on top of the holes and whether the estimated enteringpoint is within 4 m from a cable hole.In order to reduce the remaining low energy backgrounds, events are53IBGB Fourth fyxuwtionrejected if they have less than 50 hits within a 50 ns time window in the thetime-of-flight corrected hit time distribution. The time-of-flight correction isdone using a vertex roughly estimated using the hit time information. Thecut threshold corresponds to the energy equivalent to a 9 MeVRx electron.A flasher PMT event is an event caused by an electrical discharge ina PMT. To reduce such background, an event is rejected if the minimumnumber of ID hits in a 100 ns time window sliding from 300 ns to 800 nsafter the event trigger is greater than 19. Since flasher PMT events typicallyhave broader hit timing distributions compared to real particle events, eventshaving a long tail in the hit timing distribution are rejected by this cut.After the third reduction the number events reduces to ∼45 events/day.5C3 Fourth gzyuxtionThe fourth reduction employs a pattern matching algorithm to remove theremaining flasher PMT events. As flasher PMT events tend to repeat withsimilar PMT hit patterns, correlations of the charge pattern is calculatedbetween different events. For a given event the charge correlation is calcu-lated between the 10,000 events neighboring in time, and events with highcorrelations are rejected.The event rate after the fourth reduction is ∼18 events/day.5CI Fifth gzyuxtionThe fifth reduction is designed to reject the remaining cosmic ray muon andflasher PMT events.A cosmic ray muon whose momentum is near the Cherenkov thresholdmay not issue an event trigger by itself, but instead its decay electron canbe detected as an event. Such events are rejected if there are more than 9OD hits per 200 ns during any time between 8800 ns and 100 ns before theevent trigger.Further reduction of flasher PMT events are done by requiring tightercriteria for the tails in the hit timing distribution compared to the cut inthe third reduction. This is done by using the time-of-flight corrected hittime based on the reconstructed vertex rather than the raw hit times.Approximately 16 events/day remain after the fifth reduction.54IBIB Finul FW gylywtion5C5 Finvl FC hzlzxtionAfter the five reduction stages are applied, the final FC event sample forphysics analyses are selected by requiring the following:• Number of hit PMTs in the largest OD hit cluster is less than 16.• Visible energy is greater than 30 MeV.• Reconstructed vertex position is more than 2 m away from the ID wall.The first cut ensures that the OD activity is low enough that it can beclassified as an FC event rather than PC. Atmospheric neutrino analysesonly use the events with energy greater than that of a 30 MeVRx electron asthe atmospheric neutrino flux falls off at such low energy, and the second cutis applied to reject any potential low energy events which are not relevant.The third cut is the fiducial volume(FV) cut which defines a 22.5 kt fiducialvolume of the SK detector. The visible energy and the vertex is obtainedusing the event reconstruction algorithm described in Chapter 6 and moredetails can be found there as well as in Section 8.1.1.The remaining backgrounds after the five reduction stages are mostlycosmic ray muons and flasher events, both of which are then largely rejectedby the FV cut since they both originate on the ID wall. The backgroundcontamination in the final selected FC sample is estimated to be ∼0.1%based on the eye-scan of the selected events.The overall efficiency of the FC sample selections for the true neutrinoevents which happen in the FV is estimated to be greater than 98% basedon the simulation, and the average observation rate of the final FC sampleis ∼8 events/day.55Chvptzr KZvznt gzxonstruxtionIn order to use the observed events in physics analyses, one needs to recon-struct the type of the particles within an event as well as their kinematics.This chapter describes the event reconstruction algorithm which is used asan input for the oscillation analysis. The algorithm, named fiTQun, employsa maximum likelihood method using the charge and time information thatis observed by the PMTs to reconstruct particle types and kinematics in thedetector. The algorithm is based on the methods which are developed forthe MiniBooNE experiment[83], but is developed from scratch for Super-Kwith additional features such as the advanced multi-ring reconstruction.In the following, Sections 6.1 to 6.4 describe the core algorithm of thefiTQun maximum likelihood event reconstruction, and the subsequent sec-tions detail the practical procedures of reconstructing Super-K events.KCF aikzlihooy FunxtionThe basis of the fiTQun reconstruction algorithm is the likelihood function.Let x denote a particle event hypothesis, which specifies the type of theparticles in the event as well as their vertices and kinematics. An event inSuper-K contains the information on which PMTs registered a hit, and forPMTs that were hit the integrated charge as well as the time of the firstphoton arrival are also provided. Using such observed information, one canconstruct the following likelihood function given the hypothesis x:a(x) =uxrst∏jej(unhit|x)rst∏i{1− ei(unhit|x)}fq(qi|x)ft(ti|x): (6.1)In this equation, the index j runs over all PMTs which did not register ahit, and for each such PMT the unhit probability ej(unhit|x) which is theprobability the PMT does not register a hit given hypothesis x is multiplied.For the PMTs which did register a hit, the hit probability as well as thecharge likelihood and the time likelihood are multiplied, where qi and tirepresent the observed charge and time respectively. The charge likelihood566BFB dryxiwtyx Whurgyfq(qi|x) is a conditional probability density function of observing charge qiat the i-th PMT given hypothesis x provided that the PMT registered ahit. Similarly, the time likelihood ft(ti|x) is the probability density for a hitbeing produced at time ti, given x. Once the likelihood function is defined,one can look for the hypothesis x which maximizes a(x) and use that inphysics analyses as the estimate for the particle configuration in the event.KCG erzyixtzy ChvrgzIn practice, when calculating the likelihood, the process of particle and op-tical photon propagation are decoupled from the response of the PMTs andthe electronics by introducing the predicted charge i, which is the meannumber of photoelectrons that are produced at the i-th PMT given a hy-pothesis. i explicitly depends on the hypothesis x, and Equation 6.1 isthen rewritten using i asa(x) =uxrst∏jej(unhit|j)rst∏i{1− ei(unhit|i)}fq(qi|i)ft(ti|x)P (6.2)where the unhit probability and the charge likelihood now depend on xonly through the predicted charge. ei(unhit|i) and fq(qi|i) are purely theproperties of the PMTs and the electronics and do not explicitly depend onthe processes of Cherenkov photon emission and propagation.Likelihood evaluation is then performed in two steps: for a given hypoth-esis x the predicted charge i for each PMT is first calculated, and usingthe calculated i and the observed information the likelihood function Equa-tion 6.2 is evaluated. For a given single particle hypothesis the predictedcharge contribution from direct and indirect light are calculated separately,and the sum of the two is used in likelihood calculation. For a multi-particlehypothesis, predicted charge is first calculated separately for each particlein the hypothesis, and then the sum of the predicted charge from all theparticles are used in evaluating Equation 6.2. The method for predictedcharge calculation for a single particle is described in the following.KCGCF erzyixtzy Chvrgz from Dirzxt aightThe predicted charge from direct light produced by a single particle is ex-pressed as an integral along the particle track:nsr = Φ(p)∫ys g(pP sP cos )Ω(g)i (g)ϵ()P (6.3)576BFB dryxiwtyx WhurgyRsPMTFigure 6.1: Schematic diagram describing the variables relevant to predictedcharge calculation. The white dot is the initial position of the particle.where s is the distance the particle traveled along the track from its initialposition. p is the initial momentum of the particle, and variables gP P characterize the relative orientation of the particle and the PMT in concernwhich are all functions of s, as described in Figure 6.1. The following sectionsdescribe each of the following factors which appear in the above equation:Φ which represents the photon yield, the Cherenkov emission profile g, thePMT solid angle factor Ω, the light transmission factor i and the PMTangular acceptance ϵ.KCGCG Chzrznkov Zmission erolzThe Cherenkov emission profile g(pP sP cos ) describes the number of pho-tons emitted per unit track length per unit solid angle at angle  with respectto the particle direction, at the instance when the particle with initial mo-mentum p has traveled a distance s along the particle track. g(pP sP cos ) isnormalized such that ∫g(pP sP cos )ysyΩ = 1: (6.4)For each particle type, the profiles are generated at a range of discreteinitial momentum values using particle gun MC, which is produced by inject-ing a single particle in the GEANT3-based detector simulation(SKDETSIM)mentioned in Section 4.3. Representative distributions of the Cherenkovprofile for electrons and muons are shown in Figures 6.2 and 6.3. Whenproducing the profile, event-by-event randomness due to multiple scattering586BFB dryxiwtyx Whurgyθcos0.6 0.8 1s (cm)010020030040000. 50MeV/cθcos0.6 0.8 1s (cm)010020030040000. 500MeV/cθcos0.6 0.8 1s (cm)010020030040000. 200MeV/cθcos0.6 0.8 1s (cm)010020030040000.050.1e 1000MeV/cFigure 6.2: Cherenkov emission profile g(pP sP cos ) for electrons at differentinitial momentum. Horizontal axes represent the cosine of the angle fromthe particle direction, and vertical axes are the distance traveled from theinitial position of the particle.dsb- is averaged over. It can be seen in the figure that photon emission forelectrons are peaked at cos  ≈ 0:75 which corresponds to the opening angleof the Cherenkov cone from a particle travelling at the velocity  = 1 inwater, regardless of the momentum. Also, electromagnetic showers causedby electrons produce rather broad angular distributions. For muons on theother hand, the angular distribution of photon emission is much sharperand depends much stronger on momentum compared to electrons, and wesee the collapse of the Cherenkov cone as the particle travels and loses itsmomentum. Such differences in the emission profile between different par-ticle types provide particle identification capability, which will be discussedin more detail in later sections.The factor Φ(p) is a normalization factor which is proportional to the596BFB dryxiwtyx Whurgyθcos0.6 0.8 1s (cm)010020030040001234 160MeV/cµθcos0.6 0.8 1s (cm)010020030040000. 600MeV/cµθcos0.6 0.8 1s (cm)010020030040000. 300MeV/cµθcos0.6 0.8 1s (cm)010020030040000.050.10.15 1000MeV/cµFigure 6.3: Cherenkov emission profile g(pP sP cos ) for muons at differentinitial momentum. Horizontal axes represent the cosine of the angle fromthe particle direction, and vertical axes are the distance traveled from theinitial position of the particle.average total number of photons that are emitted by a particle with initialmomentum p. Φ(p) also absorbs the constant factors that are not accountedfor by other factors in Equation 6.3 such as the quantum efficiencies of thePMTs, and enforces proper normalization to the predicted charge.KCGC3 holiy Vnglz FvxtorΩ(g) represents the solid angle subtended by a PMT viewed from its normaldirection at distance g. In order to reduce computation time, the factor issimply approximated as the area of a circle normalized by the distance:Ω(g) =.v2g2 + v2P (6.5)606BFB dryxiwtyx Whurgywhere v = 25:4 cm is the radius of the PMT. The approximation holdssufficiently well at distance g S 1 m.KCGCI aight irvnsmission Fvxtori (g) is the attenuation factor of direct light due to absorption and scatteringin water, and it is written asi (g) = exp(−gRaktt)P (6.6)where aktt is the wavelength-averaged attenuation length for Cherenkovradiation. The attenuation length is obtained from the detector simulationand its value for SK-IV is 7496.46 cm.KCGC5 ebi Vngulvr Vxxzptvnxzϵ() is the PMT angular acceptance, which is a function of the angle between the PMT normal and the direction of the particle position viewedfrom the PMT. Figure 6.4 is the acceptance curve obtained from the detectorsimulation, which also includes the effect of the shadowing by neighboringPMTs at high incident angle. The distribution is fitted with a joint poly-nomial as indicated by the solid line in the figure, and the fit function isused as ϵ() in event reconstruction. We adopt a normalization conditionϵ( = 0) = 1.KCGCK erzyixtzy Chvrgz from Inyirzxt aightIn addition to direct light which was just discussed, indirect light must alsobe taken into account in order to properly predict the amount of chargedeposited at each PMT. This includes the light scattered in water as wellas the reflected light coming from detector components such as the blacksheet and PMTs themselves. Similar to Equation 6.3 for direct light, thepredicted charge from indirect light is written assmt = Φ(p)∫ys14./(pP s)Ω(g)i (g)ϵ()V(s)P (6.7)where/(pP s) ≡∫g(pP sP cos ) yΩ (6.8)is the fraction of photons emitted per unit track length, at position s alongthe particle trajectory. When the factor V(s) is removed from Equation 6.7,616BFB dryxiwtyx Whurgyηcos0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 105001000150020002500Figure 6.4: Angular acceptance of the PMT plotted as a function of cos which is obtained from detector simulation, with a fitted curve overlaid as asolid line. Note that the vertical axis in the figure is in arbitrary unit; afterthe shape of the curve is extracted, the fitted function is renormalized sothat its value is 1 at  = 0.the integral represents the amount of direct light deposited from an imagi-nary isotropic light source whose trajectory and the total light intensity arethe same as the charged particle described by Equation 6.3. In other words,photon emission described in Equation 6.3 is averaged over all directions ateach point on the particle track. The objectV(s) = V(xZMTP zvtxP gvtxP <P P ϕ) ≡ ysmtyssy;nsrP (6.9)which is referred to as the scattering table, is the ratio of the differentialpredicted charges deposited from point s along the track, by the indirectCherenkov light from the concerned charged particle and the direct lightfrom the imaginary isotropic source which was discussed above. Assumingthat the Cherenkov opening angle does not change as a function of momen-tum, the momentum dependence is factored out from the scattering table bytaking the ratio. The scattering table then depends only on the positionaland directional alignment of the particle and the PMT, and a common table626BFB dryxiwtyx WhurgyzPMTzvtxRvtx(θ,ϕ)φFigure 6.5: A schematic diagram describing the variables relevant to thescattering table. For PMTs on the top or the bottom wall, the distancegZMT of the PMT position from the vertical axis at the tank center is used,instead of zZMT.can be used for particles at any momentum. Note, again, that this is an ap-proximation which assumes the Cherenkov opening angle to be independentof momentum and is fixed at the opening angle for particles at  = 1, whencalculating the distribution of the indirect light.Assuming the azimuthal symmetry of the detector, the scattering tableis sufficiently expressed as a function of six variables which describe theorientations of the PMT and the particle in the detector, as schematicallyshown in Figure 6.5. Note that the figure is for the PMTs on the side wall,in which case the variable xZMT in Equation 6.9 corresponds to the PMTz position zZMT. For the PMTs on the top or the bottom wall, the PMT’sdistance gZMT from the vertical axis at the tank center is used, instead ofzZMT.The scattering table is produced by generating a particle gun MC where3 MeVRx electrons are placed randomly within the detector while multiplescattering is turned off. Such electrons can be considered as point Cherenkovsources with  = 1. During the simulation, the number of reflection andscattering of each photon which arrives at a PMT is recorded, and directand indirect photons arriving at the PMTs are counted and filled separatelyinto six-dimensional histograms, each axis corresponding to one of the six636BFB dryxiwtyx Whurgyvariables in Equation 6.9. Then, at each bin in the six-dimensional pa-rameter space, the numerator ysmt in Equation 6.9 is obtained from thehistogram for indirect photons, while the denominator yssy;nsr is calculatedby averaging the histogram for direct photons over the particle direction.The scattering table is thus obtained as a six-dimensional histogram, and itis linearly interpolated in terms of the six variables at the time of evaluatingthe predicted charge.KCGCL evrvwolix VpproflimvtionAs in Equations 6.3 and 6.7, evaluation of the predicted charge involvesintegrals along the particle trajectory, and it is not computationally feasibleto perform the integrals at the time of reconstructing an event. The followingapproximation is therefore made in order to avoid the need of evaluating theintegral at the time event reconstruction.Let J(s) denote the last three factors in Equation 6.3, which describeshow the photons are received by the PMT:J(s) ≡ Ω(g)i (g)ϵ() ≈ j0 + j1s+ j2s2: (6.10)This factor varies slowly as a function of s, and it is therefore reasonablyapproximated by a parabola, an example of which is shown in Figure 6.6.The coefficients jn are obtained by evaluating J(s) at three points along theparticle trajectory, namely, the particle’s initial position, the point at which90% of Cherenkov light is emitted, and their midpoint.Once J(s) is approximated as a parabola, Equation 6.3 reduces tonsr = Φ(p)∫ys g(s)J(s) ≈ Φ(p)(I0j0 + I1j1 + I2j2)P (6.11)whereIn ≡∫ys g(s)sn: (6.12)The integral in Equation 6.12 is performed in advance, and In is tabulatedas a function of three parameters which specify the initial condition of theparticle and its relative orientation to the PMT; namely, the initial valuesof the momentum p, the distance g from the particle to the PMT, and theangle  between the particle direction and the line connecting the particleand the PMT positions. Once the integral Equation 6.12 is performed atdiscrete bins in g,  and p, In is fitted as a function of p at each gP  bin inorder to smooth the likelihood surface. When reconstructing an event, In is646BFB dryxiwtyx Whurgys (cm)0 20 40 60 80 100 120 140 1600.001060.001080.00110.001120.001140.001160.001180.0012J(s)FullParabolicFigure 6.6: The photon acceptance factor J(s) (black) overlaid with anapproximating parabola (red). The coefficients for the parabola are obtainedby evaluating J(s) at the three points indicated by the hollow circles. Theplot is for the initial conditions g = 500 cm,  = 90◦,  = 0◦.linearly interpolated in terms of g and , after the fit function is evaluatedat momentum p at the relevant neighboring gP  bins.Since the scattering table V(s) also tends to vary slowly as a function ofs, for indirect light the approximationJ(s)V(s) ≈ k0 + k1s+ k2s2 (6.13)is made, which reduces Equation 6.7 tosmt = Φ(p)∫ys14./(s)J(s)V(s) = Φ(p)14.(k0 +K1k1 +K2k2)P (6.14)whereKn ≡∫ys /(s)sn: (6.15)Kn is a function of p only, and it is therefore fitted as a function of p. Thecoefficients kn are obtained in the same way as jn in the direct light case.656BGB inhit drovuvilitfl unx Whurgy LikylihooxKC3 Unhit erowvwility vny Chvrgz aikzlihooyOnce the direct and indirect predicted charge is evaluated for all the par-ticles involved in the event hypothesis, the predicted charge is summed upand the unhit probability ei(unhit|i) and the charge likelihood fq(qi|i) inEquation 6.2 are evaluated for each PMT.Since the predicted charge  is the mean number of photoelectrons pro-duced at a PMT, the actual number of the produced photoelectrons shouldobey a Poisson distribution with mean . Therefore, the probability of nophotoelectron being produced given the predicted charge  is z−. However,there are cases in which the signal from a photoelectron does not cross thePMT hit threshold such that it does not produce a hit, and in order to takeinto account such PMT threshold effects, a correction up to the third orderis applied in the expression of the unhit probability:e (unhit|) ≈ (1 + v1+ v22 + v==)z−: (6.16)The coefficients vn are obtained from the detector simulation, and Figure 6.7shows the unhit probability with and without the correction, compared withthe actual values obtained from the simulation.(p.e.)µ0 2 4 6 8 10 12)µP(unhit| -510-410-310-210-1101w/o corr.w/ 3rd order corr.MC trueFigure 6.7: The unhit probability e (unhit|) with(red) and without(blue)the correction for the PMT threshold effect. The data points show the valuesobtained from detector simulation.666B4B himy LikylihooxThe charge likelihood fq(q|) is obtained by directly generating pho-toelectrons following Poisson statistics with mean  at the PMTs in thedetector simulation, and taking the distribution of the measured charge forthe hit PMTs. Figure 6.8 shows the obtained normalized charge likelihooddistribution at a range of predicted charge values. In order to smooth thelikelihood surface, the normalized charge distributions are then fitted by apolynomial as a function of  at each fixed value of q. When evaluatingthe likelihood, the fit parameters are linearly interpolated in terms of theobserved charge q, and the resulting polynomial is evaluated at .KCI iimz aikzlihooyThe time likelihood ft(ti|x) in Equation 6.2 depends on the event hypoth-esis x as well as the position of the PMT in a complex way. Since it isimpractical to accurately account for all such dependencies, the followingapproximations are made for the time likelihood calculation.The approximation starts by expressing the time likelihood in terms ofthe residual hit time trosi which is defined based on the raw hit time ti astrosi := ti − t− smsnRx− |RiZMT − x− smsnd|RxnP (6.17)where xP t are the particle vertex position and time, d is the particle direc-tion, RiZMT is the position of the i-th PMT and smsn represents half of theparticle track length.xn := xRnP n = 1:38 (6.18)is the group velocity of Cherenkov light in water. trosi is therefore the residualhit time after subtracting the expected direct photon arrival time from theraw hit time, assuming all the photons are emitted when the particle is atthe track midpoint.As described below, time likelihoods for direct photon hits and indirectphoton hits are calculated separately using trosi , and the two likelihoods aremerged based on the relative intensities of the direct and indirect light toobtain the final time likelihood that is used in Equation 6.2. The directand indirect light time likelihoods for a single particle is calculated as thefollowing.KCICF Dirzxt aight iimz aikzlihooyIn order to calculate the direct light time likelihood, an approximation ismade so that the time likelihood depends only on trosi , the predicted charge676B4B himy Likylihooxq (p.e.)0 5 10 15 20 2500. 0.30 p.e.µq (p.e.)0 5 10 15 20 2500. 1.50 p.e.µq (p.e.)0 5 10 15 20 2500. 6.00 p.e.µq (p.e.)0 5 10 15 20 2500. 0.70 p.e.µq (p.e.)0 5 10 15 20 2500. 3.00 p.e.µq (p.e.)0 5 10 15 20 2500. p.e.µFigure 6.8: The normalized charge likelihood fq(q|) at a range of predictedcharge . The data points are obtained from the detector simulation, andthe solid lines indicate the fitted function, which is used when evaluatingthe likelihood.686B4B himy Likylihooxfrom direct light nsr and the particle’s momentum p. Since a hit is pro-duced by the first photon arriving at a PMT, the width of the residual timedistribution decreases as photon statistics increases, and such effect is char-acterized well by the predicted charge. The momentum also translates tothe width of the residual time distribution since the photon time-of-flightcorrection in Equation 6.17 becomes less accurate as the particle track be-comes longer, resulting in a wider distribution. All other dependencies onthe configurations of the particle and the PMT are averaged out in thisapproximation, and part of such information is embedded in the predictedcharge.The residual time likelihood distribution for direct light fnsrt (tros) is as-sumed to be a Gaussian, whose peak position and  depending on nsr andmomentum p. In order to produce the residual time distribution, particlegun MC samples are generated at various fixed momentum, and at eachmomentum a 2D histogram of tros-lognsr is filled for every hit caused bya direct photon. The tros distribution is then fitted by a Gaussian at eachlognsr bin, and its mean and  are fitted by a sixth-order polynomial inlognsr. Finally, each of the polynomial fit parameters is fitted as a functionof momentum. The direct light residual time likelihood fnsrt (tros) is thusparameterized as a smooth function of nsr and p. The resulting likelihoodas well as the original distribution obtained from the MC are shown forelectrons and muons in Figures 6.9 to 6.12.KCICG Inyirzxt aight iimz aikzlihooyThe residual time PDF for indirect light is currently modeled in fiTQun as:f smtt (tros) = 1R(√.2 + 2)×{exp(−2R22) ( Q 0)(R + 1) exp(−R) ( S 0) P (6.19)where  = tros− 5 ns,  = 8 ns and  = 25 ns. This is an empirical functionwhich was determined so that it reproduces a typical shape of the indirectlight residual time distribution, and it has a characteristic long right sidetail due to reflected light. No smt or momentum dependence is consideredat the present moment.KCIC3 bzrging Dirzxt vny Inyirzxt aight iimz aikzlihooysThe direct and indirect light time likelihoods obtained above are then mergedto produce the final overall time likelihood function which appears in Equa-tion 6.2. A further assumption is made here that all indirect photons arrive696B4B himy Likylihoox [ns]rest-20 0 20 4000.050.1-ep=300MeV/c= 0.26 p.e.µ [ns]rest-20 0 20 4000.050.10.15-ep=300MeV/c= 1.05 p.e.µ [ns]rest-20 0 20 4000. -ep=300MeV/c= 3.80 p.e.µ [ns]rest-20 0 20 4000.10.20.3-ep=300MeV/c=15.14 p.e.µFigure 6.9: The direct light residual time likelihood fnsrt (tros) for 300 MeVRxelectrons, at a range of predicted charge nsr. The data points indicate theoriginal distributions obtained from the MC simulation, and the red curvesare the Gaussian likelihood function which is parameterized as a function ofnsr and p.706B4B himy Likylihoox [ns]rest-20 0 20 4000.050.1-ep=2000MeV/c= 0.26 p.e.µ [ns]rest-20 0 20 4000.050.1-ep=2000MeV/c= 1.05 p.e.µ [ns]rest-20 0 20 4000. 3.80 p.e.µ [ns]rest-20 0 20 4000.10.20.3-ep=2000MeV/c=15.14 p.e.µFigure 6.10: The direct light residual time likelihood fnsrt (tros) for2000 MeVRx electrons, at a range of predicted charge nsr. The data pointsindicate the original distributions obtained from the MC simulation, andthe red curves are the Gaussian likelihood function which is parameterizedas a function of nsr and p.716B4B himy Likylihoox [ns]rest-20 0 20 4000.050.1-µp=450MeV/c= 0.26 p.e.µ [ns]rest-20 0 20 4000.050.10.15 -µp=450MeV/c= 1.05 p.e.µ [ns]rest-20 0 20 4000. -µp=450MeV/c= 3.80 p.e.µ [ns]rest-20 0 20 4000.10.20.3-µp=450MeV/c=15.14 p.e.µFigure 6.11: The direct light residual time likelihood fnsrt (tros) for 450 MeVRxmuons, at a range of predicted charge nsr. The data points indicate theoriginal distributions obtained from the MC simulation, and the red curvesare the Gaussian likelihood function which is parameterized as a function ofnsr and p.726B4B himy Likylihoox [ns]rest-20 0 20 4000. -µp=2000MeV/c= 0.26 p.e.µ [ns]rest-20 0 20 4000.050.1-µp=2000MeV/c= 1.05 p.e.µ [ns]rest-20 0 20 4000.050.10.15-µp=2000MeV/c= 3.80 p.e.µ [ns]rest-20 0 20 4000.10.20.3-µp=2000MeV/c=15.14 p.e.µFigure 6.12: The direct light residual time likelihood fnsrt (tros) for2000 MeVRx muons, at a range of predicted charge nsr. The data pointsindicate the original distributions obtained from the MC simulation, andthe red curves are the Gaussian likelihood function which is parameterizedas a function of nsr and p.736BIB Vyrtyfi dryAtafter any of the direct photons; i.e., an indirect photon can produce a hit onlyif there were no direct photons observed at the PMT. The time likelihoodcan then be written as:ft(trosi ) = wfnsrt (trosi ) + (1− w)f smtt (trosi )P (6.20)wherew ≡ 1− z−dir1− z−dirz−sci P (6.21)and fnsrt (trosi ), fsmtt (trosi ) are the normalized residual time likelihoods for di-rect light and indirect light respectively, at the given predicted charge andmomentum values. The time likelihood is thus constructed on-the-fly bycombining the time likelihoods for direct and indirect light based on theirpredicted charge values.This method is extended to a multi-particle hypothesis by ordering theinvolving particles by the calculated residual time at each PMT, and thenassuming that all the direct photons from a particle with smaller tros alwaysarrive earlier than any of the direct photons from a particle with larger tros.KC5 kzrtzfl erzBtAs it was mentioned at the beginning of this chapter, fiTQun is a maximumlikelihood event reconstruction which searches for a particle event hypoth-esis x which maximizes the likelihood function defined in Equation 6.1. Inpractice, this is done by minimizing the negative log likelihood − lna(x)with respect to x, using the minimization package MINUIT[84]. In order toensure the minimizer properly locates and converges at the global minimumrather than being stuck at a local minimum, it is important to seed theminimizer with reasonably accurate parameter values before execution.The vertex pre-fitter is a fast algorithm which uses only the hit timeinformation from the PMTs to roughly estimate the vertex position andtime which can then be used as the parameter seed for − lna minimization,and it is run at the beginning of processing an event before the minimizationis run. The estimation is done by searching for the vertex position x andtime t which maximizes the vertex goodness which is defined as:G(xP t) :=rst∑iexp(−(i irosR)2R2)P (6.22)wherei iros := ti − t− |RiZMT − x|Rxn: (6.23)746B6B guvyvynt Ulgorithmi iros is the residual hit time which is similar to Equation 6.17, however, apoint light source is assumed to be located at the initial particle vertex inthis case. When the vertex position and time get close to their true values,i iros distribute near zero, which results in a large value of the goodness. Thepre-fitter first maximizes the goodness by performing an iterative coarsegrid-search in space and time, gradually shrinking the grid size and . Afterthe grid search is done, a MINUIT minimization of −G(xP t) is performedwith  = 4 ns, which provides the final seed values for vertex and time.KCK huwzvznt VlgorithmThe term “subevent” refers to particle activities in the detector which hap-pen closely in time. An event in Super-K, which is defined as detectoractivities in a O(10 s) time window around an event trigger, may con-tain multiple subevents; for example, in case there is a neutrino interactionwhich produces a decay electron, the initial particles produced at the neu-trino interaction and the subsequent decay electron are counted as separatesubevents since they are well separated in time due to the relatively longmuon lifetime of ∼ 2 s. A subevent algorithm is a mechanism to searchfor such subevent activities around the event trigger and associate the PMThits produced by each of them so that precise maximum likelihood recon-struction can be applied to each subevent. The procedures are applied atthe very beginning of processing an event by fiTQun, prior to any of thelikelihood reconstruction.KCKCF ezvk FinyzrThe subevent algorithm starts by searching and locating in time the subeventactivities around the event trigger, which is done by the peak finder.As stated in Section 6.5, the vertex goodness Equation 6.22 takes a largevalue when it is evaluated at the vertex position x and time t which are closeto where the actual particle vertex is located, since in such case the residualhit time distributes near zero for hits that are produced by direct light emit-ted by the particle. Using this fact, the peak finder searches for subeventsby fixing the vertex position x at the value provided by the vertex pre-fitter,and scanning the goodness while varying the time t. Assuming that all thevertex positions of the primary particles and the decay electrons lie close tothe pre-fit vertex, subevents appear as large peaks in the distribution of thegoodness scanned as a function of t.756B6B guvyvynt Ulgorithm14001300120011001000900800700t (ns)8007006005004003002001000Figure 6.13: Distribution of the scanned goodness G(xP t) as a function oftime t, for an example event with a muon primary and a decay electron.The vertical dashed lines indicate true particle time, the black dots are thegoodness scan points, and the blue and green curves represent the thresholdcurves which are used for subevent peak selection. The red vertical linesindicate the time of the subevent peaks which is found by the algorithm.First, a rough interaction vertex search is done by running the ver-tex pre-fitter described in Section 6.5 using the PMT hits that are in the[−100 nsP+400 ns] time window around the event trigger. Then, the peakfinder scans and evaluates the goodness every 8 ns step in t while x is fixed atthe pre-fit vertex, starting at ∼300 ns before the event trigger. For this scan,all the hits within the time window of [−200 nsP 15000 ns] around the eventtrigger is used. The parameter  in Equation 6.22 is set to 6.3 ns, which wasoptimized for the step size. Figure 6.13 is an example of the distribution ofthe goodness for an event with a single muon and a decay electron, wherethe vertical dashed lines indicate true simulated particle time and black dotsshow the scan points.In order to select subevent peaks in this distribution, a goodness thresh-old curve F (t) is defined as:F (t) := 0:25maxi∈b{G(xP ti)f(t− ti)}+ P (6.24)wheref() :=11 + (R)2P  ={25 ns ( Q 0)70 ns ( S 0):(6.25)766B6B guvyvynt UlgorithmM represents all local maxima of goodness scan points, i.e. a scan pointwhich is higher than the neighboring two points, which has a goodness valuelarger than the constant threshold  = 9. The blue curve in the figurerepresents F (t), and the green curve is 0:6F (t). The first subevent peakis defined as the first local maximum scan point which lies above the bluecurve. The scan then continues onto later time searching for another peak,and a local maximum is ignored even if it is above the blue curve if thereis no scan point after the previous found peak lying below the green curve.This double-threshold criterion is employed in order not to double count asubevent which produces multiple peaks, which often occurs since the photontime-of-flight subtraction assuming a point light source in Equation 6.23 doesnot work perfectly and a single subevent may produce multiple peaks.KCKCG Dzning iimz linyofis vny Finvl huwzvzntsOnce the subevent candidates are found by the peak finder, a time windowis defined around each of the peaks which contains the hits associated withthe peak, in which likelihood reconstruction can be performed. Within atime window a PMT can produce a hit at most once, and only the hits thatlie within the time window will be used for likelihood reconstruction; i.e.,the PMT hits that lie outside of the time window are considered as unhit inEquation 6.2.In order to define the time window for a given peak, hit times are firstconverted to residual time with respect to the peak using Equation 6.23, andthe earliest and the latest hits in raw hit time ti which satisfy −180 ns Qi iros Q 800 ns define the start and the end time of the time window respec-tively. The procedure is repeated for all the peaks identified by the peakfinder, and time windows that overlap are merged into a single time window.After the time windows are defined, the vertex pre-fitter and the peakfinder are run once again in each time window, this time, by only using thehits that are contained within the respective time window. This procedureis necessary especially for the cases where the distance between the primaryand the decay electron vertices is large, since in such case the original pre-fitvertex is far from the actual decay electron vertex and the photon time-of-flight correction in Equation 6.23 works poorly, which results in the peakfinder finding fake subevent peaks around the true decay electron subevent.The remaining peaks after re-running the peak finder are considered asthe final subevents found in fiTQun, and precise likelihood reconstruction isapplied to each of them as discussed in the following sections. The numberof decay electrons, which is a variable that is often used in various physics776BKB ginglyAfing Fitanalyses, is calculated as the number of the found subevents minus one.KCKC3 ezrformvnxz of thz huwzvznt VlgorithmThe performance of the subevent algorithm has been studied using the dataand MC samples of stopping cosmic-ray muons, i.e., cosmic-ray muons thatenter the detector and stop in the ID. Stopping muon events are clean singlemuon events which almost always leaves a single decay electron, and theyare therefore an ideal source for studying the performance of the algorithm.Table 6.1 shows the detection efficiencies of decay electrons from stoppingmuons for data and MC in sub-GeV and multi-GeV energy range, where theenergy range is defined by whether the muon momentum is below or above1:33 GeVRx. The efficiency here is defined as the number of stopping muonevents in which multiple subevents are found divided by the total numberof events. Note that the detection efficiency gets lower at higher energy,since the more PMTs are hit by the primary muon, the harder it becomesto detect decay electrons that are produced shortly after the parent muon.The difference in efficiencies between data and MC is taken as the systematicuncertainty on the decay electron detection efficiency.Energy range Data MCSub-GeV 88:41± 0:08% 87:81± 0:15%Multi-GeV 67:77± 0:05% 68:02± 0:09%Table 6.1: Decay electron detection efficiencies for stopping cosmic-raymuons for data and MC. The energy range is defined by whether the muonmomentum is below or above 1:33 GeVRx.KCL hinglzBging FitAfter the subevents are defined by the subevent algorithm, precise maximumlikelihood event reconstruction as discussed in Sections 6.1 to 6.4 is appliedto each of the subevents. The most basic reconstruction which is appliedto every subevent is the single-ring fitter which is a fit to obtain the single-particle hypothesis that maximizes the likelihood, which is described in thissection.fiTQun considers three types of single-ring hypotheses: electron, muonand .5, and the three fits are applied to every subevent that was foundby the subevent algorithm. The electron and muon hypotheses contain786BKB ginglyAfing Fitseven parameters, which are the vertex position x, time t, zenith angle andazimuth of the direction P ϕP and momentum p. The .5 hypothesis hasone additional parameter and it is therefore an eight parameter hypothesis,which is described separately in detail in Section 6.8.KCLCF hinglzBging Zlzxtron & buon FitThe single-ring electron fit starts by setting the vertex initially at the po-sition determined by the vertex pre-fit which was run in the time windowto which the subevent belongs, and the time to the peak time. In order todetermine the initial value for the direction, a likelihood scan of the direc-tion is performed by sampling the likelihood at 400 points that are equallyspaced on the unit sphere. During the direction scan, momentum is fixed atthe value which is roughly estimated using the total observed charge. Af-ter the direction is determined by the grid scan, the momentum seed valueis set by performing another likelihood scan, this time, by only varying themomentum. Once all the seven track parameters are seeded, the parametersare then simultaneously fit by minimizing the − lna in Equation 6.2 usingthe SIMPLEX algorithm in the MINUIT package, which provides the finalmaximum likelihood estimation of the particle track parameters assuming asingle electron.Following the single-ring electron fit is the single-ring muon fit, whichtakes the vertex, time and direction from the electron fit result as their seedvalues. The momentum is then estimated by a likelihood scan as it wasdone in the electron fit, after which the full simultaneous minimization ofthe − lna is performed.KCLCG ezrformvnxz of thz hinglzBging z &  FittzrIn this section, the performance of the single-ring electron and muon fitsdescribed above are compared to APFIT, an event reconstruction algorithmwhich preexisted at Super-K before fiTQun. The algorithm of APFIT isfundamentally different from fiTQun which is detailed in [85].Figure 6.14 shows the distribution of the distance between the recon-structed and true vertices for single electron and muon events in the FCsingle-ring CCQE event sample in the atmospheric neutrino MC whose trueinteraction vertex lies within the fiducial volume(as opposed to the recon-structed vertex as in Section 5.5), compared between APFIT and fiTQun.The resolution is defined as the 68 percentile of the respective distributions.Figure 6.15 shows the resolution plotted as a function of true momentum.796BKB ginglyAfing FitThe angle between the reconstructed and true directions as well as directionresolution are shown similarly in Figures 6.16 and 6.17.Figure 6.18 shows the momentum resolution for the same set of events.For these plots, the distribution of the fractional difference between thereconstructed and true momenta are fitted by a Gaussian at each true mo-mentum bin, and the width of the fitted Gaussian is plotted.It can be seen in the figures that fiTQun has higher resolution comparedto APFIT, especially for vertex and momentum reconstruction.KCLC3 zD evrtixlz IyzntixvtionAs mentioned in Section 6.2.2, Cherenkov light emission happens differentlybetween particle types, and the ring pattern they produce will thereforebe different. As shown in Figure 6.19, electrons produce fuzzy rings dueto electromagnetic shower while muons which do not shower produce clearrings with sharp edges, and the difference arises from the differences in theCherenkov emission as seen in Figures 6.2 and 6.3.In fiTQun, particle identification (PID) is done by comparing the best-fit likelihood values between different particle hypotheses. For example,electrons and muons are distinguished by making a cut on ln (aeRa), thelog likelihood ratio between the best-fit electron and muon hypotheses.Figure 6.20 shows the likelihood separation of the single-ring electron andmuon events in the FC true-fiducial CCQE event sample in the atmosphericneutrino MC. In order to properly evaluate the PID performance for lowmomentum muons, for a muon event it is also required that a decay electronis found by the subevent algorithm, in order to remove the low momentummuon events which was triggered on the decay electron. The vertical axesin the plots are ln (aeRa), and the horizontal axes are the reconstructedsingle-ring electron fit momentum. As it can be seen in the figure, electronevents are clearly identified and separated from muon events by making acut:ln (aeRa) S −10P (6.26)as indicated by the magenta line. The cut threshold was determined basedon the MC considering the misidentification rate for both electrons andmuons.The misidentification rate of the same electron and muon events as aboveare shown as a function of true momentum in Figure 6.21. As shown in thefigure, for both particle types the misidentification rate is well below 1% andis smaller for fiTQun than APFIT.806BKB ginglyAfing FitDist. btw. tru/rec vtx. [cm]0 50 1000100200300APfit : 29.4cmfiTQun: 20.1cmResolution: Dist. btw. tru/rec vtx. [cm]0 50 1000100200300APfit : 22.7cmfiTQun: 19.3cmResolution: Figure 6.14: Single-ring electron(left) and muon(right) vertex resolution forFC true-fiducial CCQE events in atmospheric neutrino MC, compared be-tween APFIT(black) and fiTQun(red). The resolution is defined as the 68percentile of the respective distributions.True momentum [MeV/c]0 500 1000 1500Vertex Resolution [cm]050100fiTQunAPFITTrue momentum [MeV/c]500 1000 1500Vertex Resolution [cm]050100fiTQunAPFITFigure 6.15: Single-ring electron(left) and muon(right) vertex resolutionplotted as a function of true momentum, for FC true-fiducial CCQEevents in atmospheric neutrino MC, compared between APFIT(black) andfiTQun(red).816BKB ginglyAfing FitAng. btw. tru/rec dir. [deg]0 5 10050100150200°APfit : 2.91°fiTQun: 2.81Resolution: Ang. btw. tru/rec dir. [deg]0 5 100100200300°APfit : 1.75°fiTQun: 1.73Resolution: Figure 6.16: Single-ring electron(left) and muon(right) direction resolutionfor FC true-fiducial CCQE events in atmospheric neutrino MC, comparedbetween APFIT(black) and fiTQun(red). The resolution is defined as the68 percentile of the respective distributions.True momentum [MeV/c]0 500 1000 1500Direction Resolution [deg]051015fiTQunAPFITTrue momentum [MeV/c]500 1000 1500Direction Resolution [deg]051015fiTQunAPFITFigure 6.17: Single-ring electron(left) and muon(right) direction resolu-tion plotted as a function of true momentum, for FC true-fiducial CCQEevents in atmospheric neutrino MC, compared between APFIT(black) andfiTQun(red).826BKB ginglyAfing FitTrue momentum [MeV/c]0 500 1000 1500Momentum Resolution [%]0246810fiTQunAPFITTrue momentum [MeV/c]500 1000 1500Momentum Resolution [%]0246810fiTQunAPFITFigure 6.18: Single-ring electron(left) and muon(right) momentum resolu-tion plotted as a function of true momentum, for FC true-fiducial CCQEevents in atmospheric neutrino MC, compared between APFIT(black) andfiTQun(red).Super-Kamiokande IVRun 999999 Sub 0 Event 4011-10-30:15:38:38Inner: 3974 hits, 8035 peOuter: 2 hits, 1 peTrigger: 0x03D_wall: 678.5 cm Charge(pe)    >26.723.3-26.720.2-23.317.3-20.214.7-17.312.2-14.710.0-12.2 8.0-10.0 6.2- 8.0 4.7- 6.2 3.3- 4.7 2.2- 3.3 1.3- 2.2 0.7- 1.3 0.2- 0.7    < 0.200 mu-edecays0 500 1000 1500 200026052078010401300Times (ns)Super-Kamiokande IVRun 999999 Sub 0 Event 2111-10-30:15:41:52Inner: 2526 hits, 7078 peOuter: 5 hits, 6 peTrigger: 0x03D_wall: 560.0 cm Charge(pe)    >26.723.3-26.720.2-23.317.3-20.214.7-17.312.2-14.710.0-12.2 8.0-10.0 6.2- 8.0 4.7- 6.2 3.3- 4.7 2.2- 3.3 1.3- 2.2 0.7- 1.3 0.2- 0.7    < 0.201 mu-edecay0 500 1000 1500 2000100200300400500Times (ns)Figure 6.19: Event displays of simulated single electron(left) andmuon(right) events. Electrons produce fuzzy rings due to electromagneticshower, while muons, which do not shower, produce clear rings with sharpedges.836BKB ginglyAfing FitRec. e momentum (MeV/c)0 200 400 600 800 1000) µ/L eln(L-2000020000246Rec. e momentum (MeV/c)0 200 400 600 800 1000) µ/L eln(L-2000020000102030Figure 6.20: Likelihood separation of single-ring electron(left) andmuon(right) events in the FC true-fiducial CCQE event sample in the atmo-spheric neutrino MC. The vertical axes are ln (aeRa), and the horizontalaxes are the reconstructed single-ring electron fit momentum. The magentalines indicate the cut criteria for electron-muon separation.True momentum [MeV/c]0 500 1000 1500Miss-ID rate [%]012345-eTrue momentum [MeV/c]0 500 1000 1500Miss-ID rate [%]012345-µFigure 6.21: Misidentification rate of single-ring electron(left) andmuon(right) events in the FC true-fiducial CCQE event sample in the at-mospheric neutrino MC, plotted as a function of true momentum. The redmarkers indicate the performance of fiTQun, and the black markers are forAPfit.846BKB ginglyAfing FitKCLCI InBGvtz Dzxvy Zlzxtron FitAs stated in Section 6.6.2, the initial time window which was defined aroundeach subevent peak found by the peak finder can be merged with anothertime window in cases where two subevents happen closely in time, andin such case, the time window will contain multiple subevents. The decayelectrons which are contained in the same time window as the parent primaryparticles are referred to as “in-gate” decay electrons, and fitting such eventsrequires a special treatment since the hits that are contained in the timewindow have to be distributed among the subevents. By employing the hitseparation scheme described below, the single-ring electron, muon and .5fits are applied to all subevents including the in-gate ones.For a given in-gate subevent, a PMT hit is considered to be associatedwith the subevent if the hit satisfies:−30 ns Q i iros Q 60 nsP (6.27)where i iros is the residual time defined in Equation 6.23 calculated using thepre-fit vertex and the peak time of the subevent. In order to reconstructthe in-gate subevent, single-ring fits are performed by using the hits thatsatisfy Equation 6.27 only, while all the other hit PMTs are ignored. Thisprocedure effectively masks the PMT hits that are caused by the particleactivities that are not contained in the subevent in concern. This maskingand fitting procedure is applied to each of the in-gate subevents in the timewindow except for the first, primary subevent.The primary subevent is reconstructed by masking all the hits that areassociated with the other in-gate subevents, i.e., a hit PMT is considered tobe unhit if the hit satisfies Equation 6.27 for any of the in-gate subevents.In order to see how the procedures described above work, event displaysof an MC event which has an in-gate decay electron are shown in Figure 6.22.In this event, in addition to the rings from the primary particles, there isa ring from an in-gate decay electron in the upper half of the figures whichwas observed considerably later compared to the other rings, as it is clearfrom the hit time distribution on the right. The peak finder finds two in-gatesubevents in this event, and after the hit allocation among the subevents isperformed, the charge distributions for hits that are associated with eachof the two subevents become as shown in Figure 6.23. As shown in thefigures, hits from the primary particles and the in-gate decay electron areclearly separated. Single-ring fits are performed on each of the subeventsusing these allocated hits.856BKB ginglyAfing Fit-4000 -2000 0 2000 4000-4000-200002000400005101520253035Charge (p.e.)-4000 -2000 0 2000 4000-4000-2000020004000900950100010501100115012001250130013501400Time (ns)Figure 6.22: Event displays of an MC event which has an in-gate decayelectron. The left plot shows the distribution of the charge of each hit in thetime window, and the right plot shows the hit time. In addition to the ringsdue to the primary particles, a ring from an in-gate decay electron is visiblein the upper half of the figures, and they are clearly separated in time as itis seen in the right figure.-4000 -2000 0 2000 4000-4000-200002000400005101520253035Charge (p.e.)-4000 -2000 0 2000 4000-4000-200002000400002468101214Charge (p.e.)Figure 6.23: Charge distribution for hits that are associated with each in-gate subevent, for the same event which was shown in Figure 6.22. The leftfigure shows the hits allocated to the first subevent, and the right is for thein-gate second subevent. As it can be seen from the figures, rings from theprimary particles and the in-gate decay electron are clearly separated.866BLB ipstryumAhruwk .5 FitKCM UpstrzvmBirvxk + FitAnother single-ring hypothesis which has not been discussed yet is the .5 hy-pothesis. As schematically shown on the left of Figure 6.24, when a chargedpion propagates in water it often interacts hadronically with the nuclei inwater, which results in an abrupt change in its direction due to hard scatter-ing or even absorption. The “upstream” portion of the .5 trajectory, i.e.,the part before the hadronic interaction happened, thus produces a thin andsharp ring pattern which is quite distinct from that of electrons or muonsas seen in the event display on the right of Figure 6.24.Upstream ringDownstream ringSuper-Kamiokande IVRun 999999 Sub 0 Event 511-10-31:20:15:13Inner: 1180 hits, 2736 peOuter: 3 hits, 2 peTrigger: 0x03D_wall: 270.6 cm Charge(pe)    >26.723.3-26.720.2-23.317.3-20.214.7-17.312.2-14.710.0-12.2 8.0-10.0 6.2- 8.0 4.7- 6.2 3.3- 4.7 2.2- 3.3 1.3- 2.2 0.7- 1.3 0.2- 0.7    < 0.202 mu-edecays0 500 1000 1500 20004896144192240Times (ns)Figure 6.24: On the left is a schematic diagram of a charged pion whichundergoes a hadronic scattering on a nucleus and the rings it produces. Thethin cyan ring represents the “upstream” ring which is produced by theparticle track before the scattering. On the right is an event display of asimulated single .5 event. The pion underwent multiple hadronic scatteringand several upstream rings are seen as a result.The “upstream-track” .5 hypothesis is designed to reconstruct such athin upstream ring which is produced from a truncated charged pion track.Figure 6.25 schematically shows how an upstream-track hypothesis is con-structed in fiTQun. The “anti-track” starts at the assumed position of thehadronic interaction where the .5 track gets truncated, and the predictedcharge deposited by the anti-track is subtracted from the predicted chargefrom the full track which originates at the initial .5 vertex. The remain-ing charge will then correspond to the predicted charge from the upstreamtrack which is represented by the outer cyan ring in the figure. Predictedcharges for both the full track and the anti-track are calculated using thesame framework as in the regular single-ring hypothesis.876BLB ipstryumAhruwk .5 FitFull trackAnti-trackFigure 6.25: A schematic diagram describing how an upstream-track hy-pothesis is constructed in fiTQun. The predicted charge from the anti-track(dashed red arrow) is subtracted from the predicted charge from the fulltrack (solid blue arrow), in order to produce the predicted charge distribu-tion for the upstream track which is represented by the outer cyan ring.In practice, the upstream-track hypothesis has an additional parameterZvyss which represents the kinetic energy that was lost in the upstream-track,in addition to the seven parameters which the regular single-ring hypothesishas. The momentum of the anti-track is determined by the energy left aftersubtracting Zvyss from the initial total energy. By assuming constant kineticenergy loss per unit track length, the vertex of the anti-track is calculated byconverting Zvyss to the distance using a constant conversion factor. Finally,for time likelihood, residual time as in Equation 6.17 is calculated using themidpoint of the upstream track, not the full track.The upstream-track single-ring .5 fit is performed after the single-ringmuon fit is done, taking the muon fit results as parameter seeds. This is aneight-parameter fit and all parameters are simultaneously fitted by MINUIT.KCMCF =5 IyzntixvtionAs muon and .5 have similar mass, in absence of any hadronic interactionthe two produce similar ring pattern. .5’s are therefore often hard to distin-guish from muons, and they often become backgrounds to muon signals inSuper-K. The upstream-track .5 fitter, however, provides means to separatethe two particle types.Figure 6.26 shows the likelihood separation of ,CCQE and NC.5 eventsin the T2K accelerator neutrino MC simulation, which has a narrow-bandneutrino flux spectrum peaked at ∼ 600 MeV. The vertical axes are the886BLB ipstryumAhruwk .5 FitFigure 6.26: Likelihood separation of ,CCQE(left) and NC.5(right)events, in the T2K accelerator neutrino MC simulation. The vertical axesare ln(a.+Ra) and the horizontal axes are Zvyss.log likelihood ratio ln(a.+Ra) between the best-fit upstream-track .5 fitand the muon fit, and the horizontal axes are the Zvyss parameter fromthe .5 fit. It can be seen in the figures that, although there are someindistinguishable events, a significant fraction of .5 events can be clearlyseparated from muons. A study shows that introducing a cut as indicated bythe black lines can reduce the NC.5 background contamination in the T2K, signal sample by ∼ 60%. Although the R.5 identification is not usedin the atmospheric neutrino analysis at the present moment, it is expectedthat introducing such selection in future analyses will reduce the NC.5backgrounds in the signal single muon event samples.It is interesting to note that the log likelihood ratio for muon events dis-tribute near zero as in the left of Figure 6.26, which means that muons arefit almost equally well by the muon and the .5 hypotheses (although thereconstructed kinematics may be biased). As muons and .5’s are both non-showering particles and have similar mass, the Cherenkov emission profileof the two are similar, and the .5 hypothesis can therefore reproduce muonrings well by increasing the Zvyss parameter to the point where the momen-tum at the assumed upstream-track endpoint drops below the Cherenkovthreshold. The fact that muon rings can be fit by the upstream-track .5hypothesis is used in the multi-ring fitter, which will be described later inthis chapter.896BMB .0 Fit0Conversion pointFigure 6.27: A schematic diagram showing how the .0 hypothesis is con-structed. Rings from two electron tracks, as indicated by the green arrows,are combined in this hypothesis, and the two tracks point back to a singlevertex while the actual origin of each track has some displacement along thetrack which corresponds to the photon conversion length.KCN 0 FitA .0 in most cases decays into two gamma rays, and as a gamma ray pro-duces a ring pattern which is almost indistinguishable to an electron in awater Cherenkov detector, a decayed .0 produces two electron-like ringswhich point back to a common single vertex. The .0 fitter in fiTQun is adedicated two-ring fitter which is designed to fit such .0 events.Figure 6.27 schematically shows how a .0 hypothesis is constructed infiTQun. Two electron rings are combined in this hypothesis, and the ringsare placed so that the two tracks point back to a single vertex which cor-responds to the assumed .0 vertex. Since it takes some finite conversionlength for a  to convert into a z5 + z− pair and produce electromagneticshower, the vertex of each electron ring is shifted along its track by a free pa-rameter. The .0 hypothesis therefore has twelve parameters: the directioniP ϕi, momentum pi and the conversion length ymyxvi for each ring (i = 1P 2),and the common vertex xP yP zP t.The .0 fit starts by making a rough estimate for the twelve fit parametersusing a dedicated seeding procedure. Taking the result of the single electronfit as the seed and fixing the parameters there for the first ring, the secondring is searched by placing the second ring and looking for the direction of thesecond ring which minimizes the − lna at 100 points uniformly distributedon the unit sphere surface. During this search, the momentum of the secondring is set arbitrarily to 50 MeVRx, and the conversion lengths of the tworings are set to 50 cm. After the second ring search is done, the momentaof the rings p1P p2 are fit simultaneously while the other ten parameters arefixed at the seed values to improve the momentum estimation. Finally, all906BMB .0 Fit)2 (MeV/c0piRec. m0 100 200 300 400) e/L 0piln(L0100200300400020406080)2 (MeV/c0piRec. m0 100 200 300 400) e/L 0piln(L010020030040000.511.5Figure 6.28: CC single electron(left) and NC single .0(right) events in theFC true-fiducial atmospheric neutrino MC sample, separated using the .0fit variables. The vertical axes are the log likelihood ratio ln(a.0Rae) andthe horizontal axes are the reconstructed invariant mass of the .0 fit.the twelve parameters are fit simultaneously to get the final best-fit values forthe parameters. As in the single-ring fit case, the fit is done by minimizingthe − lna using the SIMPLEX algorithm in MINUIT.KCNCF e=0 Iyzntixvtion.0s were known as one of the main backgrounds to electron signals in waterCherenkov detectors as a .0 becomes indistinguishable to electrons if oneof the two gamma rings are missed by event reconstruction, which happensoften for high momentum .0s which have significant Lorentz boost. Oneof the main applications of the .0 fitter is to efficiently separate .0s fromelectrons and reduce the background contamination in electron signal eventsamples.Figure 6.28 shows the CC single electron and NC single .0 events in theFC true-fiducial atmospheric neutrino MC sample, separated using the .0 fitvariables. In the plots, the vertical axes are ln(a.0Rae), the log likelihoodratio between the best-fit .0 and electron hypotheses, and the horizontalaxes m.0 are the reconstructed invariant mass calculated from the .0 fitresult. As shown in the figures, the two particle types are separated well inthe 2D distributions. By introducing a .0 rejection line cut:ln(a.0Rae) Q 175− 0:875×m.0(MeVRx2)P (6.28)as indicated by the magenta lines in the plots, the two event categories canbe separated by the misidentification rate shown in Figure 6.29.916BEDB aultiAfing FitTrue momentum [MeV/c]500 1000 1500Miss-ID rate [%]01020304050-eTrue momentum [MeV/c]500 1000 1500Miss-ID rate [%]05101520250piFigure 6.29: Misidentification rate for the .0 rejection cut Equation 6.28, forthe CC single electron(left) and NC single .0(right) events in the FC true-fiducial atmospheric neutrino MC sample, plotted as a function of the trueparticle momentum. For electrons the misidentification rate is calculated asthe fraction of events above the cut line in Figure 6.28, while for .0’s thefraction below the cut line is plotted.KCFE bultiBging FitIn atmospheric neutrino analyses, it is essential to reconstruct and use theevents with multiple particles, since a large fraction of events in multi-GeVhave multi-particle final states. This section describes the multi-ring fitter,which is designed to identify and reconstruct up to six rings in an event.The multi-ring fitter is applied only to the first subevent in an event; i.e.,the detector activities caused by the particles which immediately follow theinitial neutrino interaction, and not the subsequent decay electrons.KCFECF Initivl bultiBging FitThe process of reconstructing a multi-ring event starts by doing an exhaus-tive search for any possible ring candidates. This is done by doing an iter-ative search for an additional ring by increasing the number of rings one-by-one, considering electron and upstream-track .5 ring hypotheses. Theelectron rings are intended to fit the electron or gamma-ray rings, while theupstream-track .5 rings are expected to fit muon or .5 rings since they canboth be fit by .5 rings as mentioned in Section 6.8.1.The tree diagram in Figure 6.30 shows how the multi-ring hypothesesevolve as the number of rings is increased in this ring search procedure.Starting from the single-ring electron fit result, an additional electron ring926BEDB aultiAfing Fiteeee eeeπ eeπe eeππ eπee eπeπ eππe eπππeee eeπ eπe eππee eπe4R3R2R1RFigure 6.30: A tree diagram showing how the tree of the multi-ring hypothe-ses evolve as the number of rings is increased. The diagram is for the caseassuming the first ring as electron, and equivalent procedure is done for thecase of assuming .5 as the first ring.is attached at the same vertex as the first electron ring, and the likelihood isevaluated at different directions for the second ring at 400 points which areuniformly distributed on the unit sphere surface. For this scan the secondring momentum is arbitrarily assumed to be 50 MeVRx. Then, the scanpoint with the minimum − lna is chosen and the momentum of the secondring is fitted while all the other parameters are fixed. After the momentumof the second ring is roughly estimated, the direction and the momentum ofthe second ring are simultaneously fitted, while the common vertex and thedirection and the momentum of the first ring are fixed. The momenta ofthe two rings are then fit simultaneously while the vertex and the directionsof the two rings are fixed in order to more properly distribute the observedvisible energy among the rings, as the single-ring fit momentum tends tooverestimate the momentum of the first ring due to the influence of thecharge from the additional rings. Finally, the directions and momenta ofthe two rings as well as the rings’ common vertex are all simultaneously fitto get the final best-fit two-ring result. After the two-ring fit assuming anelectron second ring is done, the procedure above is repeated using a .5 ringas the second ring, this time, also fitting the Zvyss parameter simultaneouslywith the other fitted parameters.Once the two-ring fits are done, for each of the two particle hypothesesfor the new ring, whether the fitted second ring is an actual true ring ischecked by comparing the likelihoods between the hypotheses before andafter adding the new ring. Figure 6.31 shows the square root of the loglikelihood ratio between the best-fit single-ring electron hypothesis and the936BEDB aultiAfing FitfQ RC likelihood0 10 20 30 40 500200400600SK4 DataCC 1eeνCC oth.eνµCC 1µνCC oth.µνNCsgn⇥p| ln (L2Rex/L1Re) |Figure 6.31: Distribution of the square root of the log likelihood ratio be-tween the best-fit single-ring electron hypothesis and the 2R hypothesis as-suming electron as the first ring, for FC sub-GeV atmospheric neutrino MCevents with no decay electrons detected. MC distribution is subdivided by fi-nal state categories and shown in different colours. The data points overlaidare the SK-IV FC data.better of the 2R hypotheses assuming electron as the first ring(for brevity,such variable will be called the “ring-counting likelihood” henceforth), forFC sub-GeV atmospheric neutrino MC events with no decay electrons de-tected. In the plot, MC events are categorized by the final states, basedon whether there are any .0’s or visible .±’s or protons in the final state.As shown in the figure, true single-ring events such as the CC single elec-tron events(blue) have relatively small value for the ring-counting likelihood,while the value is large for multi-ring events, for example for NC events(grey)which is dominated by single .0 events. A cut on the ring-counting likeli-hood is made at a constant threshold(9.35 in this case), and if the likelihoodis below the threshold the new ring is assumed to be fake and no furtheraction is done. If the likelihood exceeds the threshold, on the other hand,it is assumed that the new ring is a true ring, and another ring is searchedbased on the two-ring fit result and examined using the procedures describedabove.As schematically shown in Figure 6.30, the above procedures of adding,946BEDB aultiAfing Fitfitting and examining the new ring is iterated until either the newly addedring fails the true ring criterion or reach six rings which is the maximumnumber of rings fiTQun can handle. The simultaneous fit of all parametersincluding the direction of the rings and the common vertex is only donefor the two-ring fit case, and for anything higher, only the simultaneousmomentum fit of all rings is done. This is done mainly to save computationtime, and it is assumed that once the simultaneous two-ring fit is done thefit vertex gives a reasonable estimate for the true interaction vertex. Also,for the fifth and the sixth ring, only electron hypothesis is considered to savecomputation time.The entire procedure above is also repeated assuming .5 as the first ring,and once all the branches of the multi-ring hypothesis tree are terminated,the hypothesis with the smallest − lna is chosen as the “seeding multi-ring hypothesis” among the hypotheses that terminates a branch, i.e., thehypotheses which do not contain any fake rings. The chosen hypothesis willbe the seed hypothesis for the next step of the multi-ring fit.KCFECG hzquzntivl bultiBging FitThe final multi-ring fit hypothesis obtained in Section 6.10.1 often containsfake rings as well misidentified rings due to the limitations of the seedingprocedure. In order to improve the fit result, each ring is refitted sequentiallyand its particle type is reevaluated following the procedures below.The rings in the final seeding multi-ring hypothesis obtained in Sec-tion 6.10.1 are first reordered in terms of the visible energy of the ring, i.e.,the reconstructed kinetic energy above Cherenkov threshold. Then, for themost energetic ring, the angle between the directions of the ring and eachlower energy ring is calculated, and if the angle for a given lower energy ringis smaller than 20◦, the lower energy ring is merged to the highest energyring by adding the visible energy. After that, the highest energy ring isre-fitted assuming three particle types: electron, muon and .5, while all theother remaining rings which were not merged in the ring-merge procedureare fixed at their original configurations. This is essentially a single-ring fitin the presence of other fixed rings, and all the ring parameters are fit simul-taneously including the vertex, which will in the end provide a multi-ringhypothesis with the vertex position of the rings not necessarily being at asingle location. The type of the most energetic ring, whether it is a show-ering(such as electron and gamma) or a non-showering(such as muon and.5) ring, is then examined by comparing the best-fit likelihoods betweenthe hypotheses assuming the most energetic ring as electron or .5. The956BEDB aultiAfing Fitcriterion for selecting it as a showering(electron) ring is:ln (aeRa.+) S −10P (6.29)which is analogous to Equation 6.26, but instead of the muon the .5 ringhypothesis is used here since this way both muons and .5’s can be dis-tinguished from electrons. The ring type selected by this cut is acceptedas the final type of the ring(showering or non-showering), and the ring inthe original multi-ring hypothesis is replaced by the selected re-fitted hy-pothesis (the cases for muon and .5 is described below in more detail).After the most energetic ring is re-fitted, the above procedure is repeatedfor the subsequent lower energy rings which remained after ring merging,in the descending order of the ring energy until all the remaining rings arere-fitted.At the present moment, the above zR.5 cut is the only particle identi-fication cut that is applied in the sequential fit procedure, and the R.5identification as described in Section 6.8.1 is not applied. If the mostenergetic ring is determined to be non-showering(.5-like) as opposed toshowering(electron-like), the ring is always assumed to be a muon, sincesuch case usually happens only for ,CC events with a true muon ring andrarely happens for NC events with a true .5 ring being most energetic. Forother rings, a ring which is identified as .5-like rather than electron-like isalways assumed as a .5.The fit result obtained after the entire sequential fit procedure above isthe final result which will be used as fiTQun’s best estimate for the event’sfinal state and is used in the atmospheric neutrino analysis.KCFEC3 ezrformvnxz of thz bultiBging FittzrIn this section, the ring-counting and particle identification performance ofthe fiTQun multi-ring fitter is compared to APFIT, the preexisting Super-Kreconstruction which was mentioned in Section 6.7. The dataset used hereis the FC atmospheric neutrino MC normalized to 2519.89 days exposurewhich is the same as the data exposure in SK-IV.Figures 6.32 and 6.33 show the reconstructed event categories for the,eCC single electron events and the NC single .0 events respectively, com-pared between fiTQun and APFIT. As shown in the figures, fiTQun hasa higher fraction of NC1.0 events which are properly reconstructed as twoelectron ring events(shown in green), while the fraction of reconstructing CCsingle electron events properly as single electron ring event(shown in grey)is at a similar level between fiTQun and APFIT. Furthermore, Figure 6.34966BEDB aultiAfing Fit [GeV])vis(E10Log-1 -0.5 0 0.5 1010002000: fiTQun Ctg.±pi00piCC0eν [GeV])vis(E10Log-1 -0.5 0 0.5 1010002000: APfit Ctg.±pi00piCC0eνFigure 6.32: Reconstructed event categories of ,eCC events with no visiblepions in final state in the FC atmospheric neutrino MC, for fiTQun(left)and APFIT(right). The horizontal axes are the log of the visible energy inGeV. Reconstructed single electron ring event category, which correspondsto the true category, is shown in grey. [GeV])vis(E10Log-1 -0.5 0 0.5 10200400600800: fiTQun Ctg.±pi00piNC1 [GeV])vis(E10Log-1 -0.5 0 0.5 10200400600800: APfit Ctg.±pi00piNC1Figure 6.33: Reconstructed event categories of NC events with a single .0and no visible .± in final state in the FC atmospheric neutrino MC, forfiTQun(left) and APFIT(right). The horizontal axes are the log of thevisible energy in GeV. Reconstructed two electron ring event category, whichcorresponds to the true category, is shown in green.976BEDB aultiAfing Fit [GeV])vis(E10Log-1 -0.5 0 0.5 1050100150200: fiTQun Ctg.±pi00piCC1eν [GeV])vis(E10Log-1 -0.5 0 0.5 1050100150200: APfit Ctg.±pi00piCC1eνFigure 6.34: Reconstructed event categories of ,eCC events with a single.0 and no visible .± in final state in the FC atmospheric neutrino MC,for fiTQun(left) and APFIT(right). The horizontal axes are the log of thevisible energy in GeV. Reconstructed three electron ring event category,which corresponds to the true category, is shown in blue.shows that fiTQun’s efficiency for properly reconstructing ,eCC 1.0 eventsas three electron ring events(shown in blue) is significantly higher comparedto APFIT. Similarly, reconstructed event categories for ,CC single muonand ,CC 1.0 events are shown in Figures 6.35 and 6.36, which lead to asimilar conclusion.In order to see how well the multi-ring kinematics are reconstructed,Figure 6.37 shows the reconstructed invariant mass calculated from the 2ndand the 3rd rings in the events categorized as 3z ring events by fiTQun andAPFIT. The plots show that the true ,eCC 1.0 events(shown in blue) whichare targeted by the ring selection are selected with much higher efficiency andpurity by fiTQun than APFIT, and a clear .0 mass peak is seen. Figure 6.38shows the equivalent plots for ,CC 1.0 events.From these distributions, it is clearly seen that fiTQun’s multi-ring fit-ter has a significantly better performance in properly identifying and recon-structing the rings in multi-ring events compared to the preexisting APFITreconstruction.986BEDB aultiAfing Fit [GeV])vis(E10Log-1 -0.5 0 0.5 1050010001500: fiTQun Ctg.±pi00piCC0µν [GeV])vis(E10Log-1 -0.5 0 0.5 1050010001500: APfit Ctg.±pi00piCC0µνFigure 6.35: Reconstructed event categories of ,CC events with no visiblepions in final state in the FC atmospheric neutrino MC, for fiTQun(left)and APFIT(right). The horizontal axes are the log of the visible energy inGeV. Reconstructed single muon ring event category, which corresponds tothe true category, is shown in brown. [GeV])vis(E10Log-1 -0.5 0 0.5 1050100150200: fiTQun Ctg.±pi00piCC1µν [GeV])vis(E10Log-1 -0.5 0 0.5 1050100150200: APfit Ctg.±pi00piCC1µνFigure 6.36: Reconstructed event categories of ,CC events with a single.0 and no visible .± in final state in the FC atmospheric neutrino MC,for fiTQun(left) and APFIT(right). The horizontal axes are the log of thevisible energy in GeV. Reconstructed 1 + 2z ring event category, whichcorresponds to the true category, is shown in cyan.996BEDB aultiAfing Fit]2Rec. Invar. Mass[MeV/c0 100 200 30001020:0piCC1eν 100.77 evts. Purity: 61.2 % < 1.00GeV : fiTQun sel.vis0.32 < E]2Rec. Invar. Mass[MeV/c0 100 200 30001020 < 1.00GeV : APfit sel.vis0.32 < E:0piCC1eν 58.65 evts. Purity: 44.1 %]2Rec. Invar. Mass[MeV/c0 100 200 30005101520:0piCC1eν 71.81 evts. Purity: 62.9 % < 3.16GeV : fiTQun sel.vis1.00 < E]2Rec. Invar. Mass[MeV/c0 100 200 30005101520 < 3.16GeV : APfit sel.vis1.00 < E:0piCC1eν 23.73 evts. Purity: 45.6 %Figure 6.37: Reconstructed invariant mass calculated using the 2nd and the3rd rings for events reconstructed as 3z ring by fiTQun(left column) andAPFIT(right column), in the FC atmospheric neutrino MC. The upper rowis for visible energy in the range 0:32 − 1:00 GeV, and the lower row is for1:00 − 3:16 GeV. Each true final state is shown in different colour in theplots, and the true ,eCC 1.0 events which are targeted by the ring selectionis shown in blue. As shown in the plots, a clearer and larger peak is seenin the fiTQun plots near the .0 mass ∼ 135MeVRc2, which indicates thatthe .0 was reconstructed properly with higher efficiency. The event rateand the purity of the ,eCC 1.0 events which pass the invariant mass cut asindicated by the magenta arrows are also shown in the plots.1006BEDB aultiAfing Fit]2Rec. Invar. Mass[MeV/c0 100 200 300010203040 :0piCC1µν 124.43 evts. Purity: 57.4 % < 1.00GeV : fiTQun sel.vis0.32 < E]2Rec. Invar. Mass[MeV/c0 100 200 300010203040 < 1.00GeV : APfit sel.vis0.32 < E:0piCC1µν 57.81 evts. Purity: 44.8 %]2Rec. Invar. Mass[MeV/c0 100 200 3000102030:0piCC1µν 114.15 evts. Purity: 72.4 % < 3.16GeV : fiTQun sel.vis1.00 < E]2Rec. Invar. Mass[MeV/c0 100 200 3000102030 < 3.16GeV : APfit sel.vis1.00 < E:0piCC1µν 27.27 evts. Purity: 49.3 %Figure 6.38: Reconstructed invariant mass calculated using the 2nd and the3rd rings for events reconstructed as 1 + 2z ring by fiTQun(left column)and APFIT(right column), in the FC atmospheric neutrino MC. The upperrow is for visible energy in the range 0:32 − 1:00 GeV, and the lower rowis for 1:00 − 3:16 GeV. Each true final state is shown in different colourin the plots, and the true ,CC 1.0 events which are targeted by the ringselection is shown in cyan. As shown in the plots, a clearer and larger peakis seen in the fiTQun plots near the .0 mass ∼ 135MeVRc2, which indicatesthat the .0 was reconstructed properly with higher efficiency. The eventrate and the purity of the ,CC 1.0 events which pass the invariant masscut as indicated by the magenta arrows are also shown in the plots.1016BEEB heun in gKAI to IIIKCFF ifun in hKBI to IIIAs mentioned in Section 2.3, there are four distinct detector phases in Super-K, and only the data from the latest SK-IV phase is used in the atmosphericneutrino oscillation analysis which is presented in Chapter 8. The data fromthe old SK-I to III phases is not used in the analysis at the present momentdue to the large data-MC discrepancies observed in the reconstructed vari-able distributions from fiTQun, and usage of such data requires improve-ments in the detector simulation in those phases.As an example of such data-MC discrepancy, Figure 6.39 shows the data-MC comparison of the distribution of the log likelihood ratio between thesingle-ring electron and muon fits ln (aeRa) discussed in Section 6.7.3, forthe FC sub-GeV (i.e. visible energy below 1330 MeV) single-ring atmo-spheric neutrino events. It can be seen from the plots that while the dis-tribution agrees well in SK-IV, there are rather large discrepancies betweendata and MC for SK-I and III, namely, the single electron events in dataare shifted to the left such that the electron-muon separation appears to beworse in data compared to MC.In general, SK-IV is the most stable of the four detector phases due tothe various hardware improvements such as the electronics and the watersystem, and because of the sophisticated calibration methods described inChapter 3 which have significantly improved over the years the detectorsimulation is also more detailed and better tuned in SK-IV. The data-MCagreement is therefore overall better in SK-IV compared to the other phases.One of the potential improvements one can make in the detector sim-ulation for SK-I to III is the time response of the PMTs. As discussed inSection 3.4, the time resolution of the ID PMTs is obtained from the thetiming calibration data using a laser system, and the time response is asym-metric having a longer tail in later time as Figure 3.4 shows. For SK-IV, suchasymmetric time response is simulated by an asymmetric Gaussian based onthe measured time resolution, however, for SK-I to III the response is sim-ulated by a simple symmetric Gaussian. As the fiTQun likelihood uses thedetailed time information, such differences in the time response between thereality and the simulation can directly manifest as the data-MC discrepan-cies in the fitted likelihoods. By reprocessing the old TQ calibration data,it is possible to improve the simulation for SK-I to III to incorporate theasymmetric time response.Differences in the light attenuation length in water also cause differencesin the fiTQun outputs. Figure 6.40 shows the variation of the measuredlight attenuation length over time, and as the figure shows the water quality1026BEEB heun in gKAI to III)µ/Leln(L-2000 0 20000200400600800 SK1 DataCC 1eeνCC oth.eνµCC 1µνCC oth.µνNC)µ/Leln(L-2000 0 20000100200SK3 DataCC 1eeνCC oth.eνµCC 1µνCC oth.µνNC)µ/Leln(L-1000 0 10000200400SK2 DataCC 1eeνCC oth.eνµCC 1µνCC oth.µνNC)µ/Leln(L-2000 0 200005001000SK4 DataCC 1eeνCC oth.eνµCC 1µνCC oth.µνNCFigure 6.39: Distributions of the log likelihood ratio between the fiTQunsingle-ring electron and muon fits ln (aeRa) for the FC sub-GeV single-ringatmospheric neutrino events in each SK phase. The plot on the top-left, top-right, bottom-left and bottom-right is for SK I, II, III and IV respectively.Points with error bars represent the data, and MC expectations are brokendown by final state categories where 1e and 1 represent the events in whichthe only visible particle being an electron or a muon.1036BEEB heun in gKAI to IIIYear2000 2005 2010 2015Attenuation length (m)6080100120Figure 6.40: Time variation of the light attenuation length in water. Thefour detector phases are shown in different colours, where SK I, II, III andIV are shown in black, green, blue and red respectively.in SK-IV is much more stable compared to the other phases. The varia-tions in water quality such as the changes in absorption or scattering affectsthe charge pattern of the Cherenkov rings and hence affects the likelihood.Currently, atmospheric neutrino MC is produced without considering suchvariations in the detector properties over time, and the simulation is doneusing the calibration constants measured at a single point in time in eachdetector phase. For instance, the water in SK-IV is simulated based onthe water calibration data taken in April 2009. When the property of thewater significantly deviates in data compared to the simulation, the dif-ference manifests as data-MC discrepancies in the fiTQun likelihoods. Onepotential solution to improve the data-MC agreement is to simulate the timevariations in the MC based on the available calibration data.It is thus expected that the detector simulation in SK-I to III can beimproved in near future by using the available calibration data so that onecan benefit from the advanced reconstruction performance of fiTQun whenanalyzing the data from the old SK phases.104Chvptzr LFirst Vpplixvtion of ifunOiGK e VppzvrvnxzAs an example application of fiTQun, this chapter briefly describes the T2K,e appearance analysis which was the first application of fiTQun in a physicsanalysis. In the analysis, fiTQun is used to reduce the NC .0 background inthe signal ,e event sample observed at Super-K, resulting in ∼62% reductionof the NC background compared to the previous .0 rejection method whilethe loss in the signal efficiency is only ∼2%. The analysis result has beenpublished by the T2K collaboration in [29], which reports the discovery ofthe , → ,e oscillation.LCF iGK ZflpzrimzntT2K is a long-baseline accelerator neutrino oscillation experiment located inJapan. A nearly pure , beam produced at the J-PARC accelerator facilityin Tokai is detected at Super-K at the distance of 295 km, and the flavourchange of the neutrinos are measured precisely.As schematically shown in Figure 7.1, at the beam production site, pro-tons accelerated to 30 GeV at the J-PARC main ring are collided with agraphite target producing mainly charged pions with some kaons, and thepions are subsequently focused by three magnetic horns. The focused pionsFigure 7.1: Schematic drawing of the T2K beam production site. Figuretaken from [23].105KBEB hFK EfipyrimyntFigure 7.2: T2K , beam spectrum at different off-axis angles. The ,survival probability at SK is also shown on the top. Figure taken from [86].then undergo the decay .5 → 5 + , in a decay volume, producing anintense , beam. The produced 5’s are mostly stopped at the beam dumpso that their decay ,¯ and ,e, which become backgrounds to the oscillationmeasurements, are at very low energies and are not focused towards thebeam direction. By reversing the current to the magnetic horns, one caneither focus .5 and defocus .− or vice versa and thus choose whether toproduce a , or ,¯ beam. SK is located 2:5◦ off-axis from the beam direc-tion, which results in a narrow neutrino energy spectrum peaked at 0.6 GeVat SK as shown in Figure 7.2. The peak energy is where the ∆m2=2-inducedoscillation effects become maximal given the known value for ∆m2=2 and the295 km baseline. The neutrino beam is pulsed, and a precise GPS timingsystem allows one to only select the T2K beam neutrino events at SK withbeam-unrelated backgrounds being ∼0.002%. The neutrinos before oscil-lation are measured at a near detector complex located at 280 m from thetarget, which consists of an on-axis detector(INGRID) and an off-axis detec-tor(ND280) at 2:5◦. Figure 7.3 shows a schematic drawing of ND280 whichis a suite of several sub-detectors enclosed in a magnet producing magneticfield of 0.2 T. The primary components of ND280 used in the oscillationanalysis are the fine-grained detectors (FGDs) which are plastic scintillator106KBEB hFK EfipyrimyntFigure 7.3: A schematic drawing of ND280. Figure taken from [87].trackers and also serve as the target for neutrino interaction, and the timeprojection chambers (TPCs) which are placed next to the FGDs and areused to determine the type and the momentum of the particles produced inthe FGDs. The near detectors are used to precisely measure the propertiesof the neutrino beam and neutrino cross sections in order to constrain thesystematic uncertainties in oscillation analyses.The initial goals of T2K are the discovery of , → ,e oscillation inducedby nonzero 1= and the precision measurement of 2= and ∆m2=2 through ,disappearance, i.e. the measurement of the survival probability , → ,.Given the value of 1= which is recently precisely measured by the reactorexperiments, CP violation can also be probed by T2K through a CP-oddsub-leading term in the , → ,e oscillation probability which depends onsin MZ:e (, → ,e) ≈ sin2 2= sin2 21= sin2 ∆m2=1a4Z− sin 212 sin 22=2 sin 1=sin∆m221a4Zsin2 21= sin2 ∆m2=1a4Zsin MZ+ (CP evenP solar and matter effect terms): (7.1)107KBFB ,e Uppyurunwy UnulflsisFigure 7.4: Predicted T2K neutrino flux at SK. Figure provided by the T2Kbeam working group.LCG z Vppzvrvnxz VnvlysisThe initial goal of the T2K ,e appearance analysis is to discover the , → ,eoscillation and precisely measure its oscillation probability, whose leadingterm depends on sin2 21= as shown in Equation 7.1. This is done by observ-ing beam-induced ,e events appearing at SK in a beam which is originallyproduced consisting almost entirely of ,. For a precise measurement, it isessential to accurately predict the neutrino flux at SK and also constrain theuncertainties related to neutrino interactions at SK, so that neutrino eventscan be simulated accurately and properly compared to the observed datafor oscillation analyses.For the neutrino flux prediction, hadron production at the graphite tar-get is simulated using FLUKA[88] and the data from a dedicated hadronproduction experiment NA61/SHINE[89][90], and the propagation of thesecondary particles are simulated by GEANT3[81] with GCALOR[82]. Fig-ure 7.4 shows the predicted neutrino flux at SK based on the simulation.The flux consists mostly of ,, however, there are some contaminations from,¯ and also ,e and ,¯e which become irreducible backgrounds to ,e appear-ance analysis. The overall uncertainty on the absolute flux is estimated to be10−15%, and it is dominated by the uncertainties in the hadron production.108KBFB ,e Uppyurunwy UnulflsisOther than the neutrino flux, T2K neutrino event simulation at SK is donefollowing the same procedures as the atmospheric neutrino MC described inChapter 4, using NEUT neutrino event generator and the GEANT3-baseddetector simulation SKDETSIM.In order to constrain the systematic uncertainties related to the neutrinoflux and cross sections, the data from ND280 is analyzed, namely, by per-forming a fit to the ,CC interaction data in the FGDs. Taking the prioruncertainties on the flux simulation and the prior constraints on cross sec-tion model parameters estimated from external datasets such as [91][92][93],the neutrino energy spectrum and the NEUT interaction models are furtherconstrained based on a simultaneous fit of the flux and cross section modelparameters to the ND280 data. The resulting uncertainties on the modelparameters after the fit are then propagated to the event predictions at SKand used in oscillation analysis. After the ND280 fit, the fractional totaluncertainty on the expected number of ,e candidate events observed at SKreduces from 27.2% to 8.8% when assuming sin2 21= = 0:1.The signal for ,e appearance at SK is observed as ,eCC events whichare coincident with the beam. As CCQE is the dominant neutrino interac-tion mode at ∼ 600 MeV where the energy of the expected ,e appearancesignal peaks at, single electron events are selected as candidate events for,e appearance following the event selection criteria detailed in Section 7.3.According to the selection and assuming sin2 21= = 0:1, 21.6 ,e candidateevents are expected to be observed at the given beam exposure, out of which4.3 events are backgrounds from beam-intrinsic ,e CC and NC events. Thus,the observation of a significant number of the selected ,e candidate eventswill be a clear evidence of ,e appearance.Once ,e candidate events at SK are selected, an extended maximumlikelihood fit is performed to obtain the neutrino oscillation parameters fromthe observed SK data. The likelihood is defined as:a(cylsP x⃗|o⃗P f⃗) = axyrm(cyls|o⃗P f⃗)× asrkpo(x⃗|o⃗P f⃗)× asyst(f⃗)P (7.2)where cyls is the number of observed ,e candidate events, x⃗ represents thereconstructed kinematic variables for the candidate events, o⃗ is the oscilla-tion parameters and f⃗ represents the parameters for systematic uncertain-ties which include the parameters for the flux, neutrino interaction and theSK selection efficiencies. axyrm is the likelihood of observing cyls eventsgiven the model parameters o⃗ and f⃗ , and it is expressed as a Poisson dis-tribution at the predicted event rate calculated with o⃗ and f⃗ . For eachobserved event, its reconstructed momentum pe and the angle e between109KBGB ,e Evynt gylywtionthe directions of the neutrino beam and the observed electron are used as“shape” information in the likelihood term asrkpo which is a 2D probabilitydistribution for pe − e given the model parameters. This is to improve thesensitivity to ,e appearance by taking advantage of the fact that the shape ofthe pe− e distribution differs between the appearance signal and the back-ground events. The predicted event rate for axyrm and pe − e distributionfor asrkpo are both calculated using the T2K MC simulation. asyst is theconstraint on the parameters for systematic uncertainties, and the resultsfrom the ND280 fit mentioned above are used for the applicable parameters.During an oscillation fit, the parameters for systematic uncertainties are nu-merically integrated over at each point in the oscillation parameter space,and the resulting marginalized likelihood is then maximized with respect tothe oscillation parameters of interest to obtain the best fit values.LC3 z Zvznt hzlzxtionBelow summarizes the full event selection criteria for the ,e appearance anal-ysis. In order to achieve high sensitivity to ,e appearance via 1=-inducedoscillation, it is essential to keep the backgrounds unrelated to the oscilla-tion signal as low as possible. The targeted event category for the signalis ,e CCQE which is a single electron event with no decay electrons. Asdiscussed in Section 6.9.1, in SK, .0’s are one of the major backgrounds tosingle electron events, and fiTQun is used in order to reduce the .0 back-ground which contaminate the ,e candidate event sample. Other than the.0 rejection cut, the ,e event selection described below is done based onthe information from APFIT[85], the preexisting reconstruction algorithmat SK before fiTQun was introduced as mentioned in Chapter 6.T2K neutrino events at SK are first selected by a reduction process usinga precise GPS timing system mentioned earlier, and then the FC selectionas described in Section 5.5 using APFIT information is applied to selectneutrino events whose interaction occur in the fiducial volume(FV) and arecontained in the ID (T2K neutrino events passing such selection is referredto as FCFV). Then, single-ring events whose ring type is determined to beelectron-like as opposed to muon-like are selected. In addition, in order toreject the events in which only a decay electron is seen, the reconstructedmomentum of the electron ring is required to be S 100 MeVRx. Since thetargeted ,e CCQE events do not produce any decay electrons, it is alsorequired that the number of detected decay electrons following the primaryinteraction is zero. Furthermore, only the events with the reconstructed110KBGB ,e Evynt gylywtionParameter Fixed valuesin2 12 0.306sin2 2= 0.50∆m221 7:6× 10−5 eV2|∆m2=2| 2:4× 10−= eV2MZ 0Table 7.1: Assumed values for the oscillation parameters.neutrino energy Zrom, being less than 1250 MeV are selected, since the eventswith larger Zrom, are mostly beam-intrinsic ,e background events. Zrom, isdefined as:Zrom, =m2p − (mn − Zb)2 −m2e + 2(mn − Zb)Ze2(mn − Zb − Ze + pe cos e) P (7.3)where mp and mn are the proton and the neutron mass respectively, Zb =27 MeV is the neutron binding energy in oxygen nucleus, me is the electronmass and Ze is the reconstructed total energy of the electron. Ignoring theFermi momentum of the target neutron, the above equation gives the energyof the incident ,e based on the two body kinematics of ,e+n→ z+p, usingthe observed pe and e.After the above selection cuts based on APFIT are applied, the finalcut to reduce the remaining NC .0 backgrounds are applied using fiTQun,namely, the cut described by Equation 6.28. Table 7.2 shows the expectednumber of signal and background events after each stage of the selection cuts,assuming sin2 21= = 0:1, normal hierarchy and other oscillation parametersset to the values in Table 7.1. As in the table, the NC background remainingafter the single-ring electron-like selections, which is dominated by NC .0events, is drastically reduced after applying the fiTQun .0 cut, while the lossin the ,→,e CC appearance signal events by the cut is small. The finalsample selected after all the cuts above has a ,→,e CC signal purity of80.2 %, while the background contamination from NC events is only 4.4 %.The remaining backgrounds are predominantly the irreducible backgroundsfrom CC interactions from the beam-intrinsic ,e and ,¯e, i.e. the ,e and ,¯ecomponents at neutrino flux production as shown in Figure 7.4.In the previous T2K ,e appearance analysis[28], a different method wasemployed for .0 rejection. Using a dedicated .0 reconstruction algorithmwhich assumes two electron-like rings and force-reconstructs an event asa .0 based on the light patterns[94], an invariant mass is calculated fromthe reconstructed momenta and the directions of the two rings, and .0111KB4B fysultsExpected Data,+,¯ ,e+,¯e NC BG ,→,eSelection CC CC total CCFCFV 247.75 15.36 83.02 346.13 26.22 377Single-ring 142.44 9.82 23.46 175.72 22.72 193Electron-like PID 5.63 9.74 16.35 31.72 22.45 60pe S 100MeVRx 3.66 9.68 13.99 27.32 22.04 57No decay-e 0.69 7.87 11.84 20.40 19.63 44Zrom, Q 1250MeV 0.21 3.73 8.99 12.94 18.82 39Previous .0 cut 0.13 3.41 2.55 6.08 17.69 31fiTQun .0 cut 0.07 3.24 0.96 4.27 17.32 28Fraction [%] 0.3 15.0 4.4 19.8 80.2 -Efficiency [%] 0.0 21.1 1.2 1.2 66.0 -Table 7.2: Expected numbers of signal and background events passing eachselection stage with the assumption sin2 21= = 0:1, normal hierarchy andother oscillation parameters set to the values in Table 7.1. The fractionof each event category in the final selected sample is also shown, and theefficiency of the overall selection is calculated taking the FCFV event ratesas the denominator. The shaded row also shows the event rates in the caseof applying the previous .0 rejection method[28] as opposed to the fiTQun.0 cut. Data taken from January 2010 to May 2013 which has the neutrinobeam exposure of 6:57× 1020 POT(protons on target) is also shown on therightmost column, and the MC expectations are normalized to the sameexposure.rejection is done by selecting the events with the invariant mass being lessthan 105 MeVRx2 as shown in Figure 7.5. The predicted event rates for thesignal and backgrounds in the case of employing the previous .0 rejectionmethod, instead of the fiTQun .0 cut, are shown in the shaded row inTable 7.2. As in the table, the remaining NC background in the final selectedsample is reduced by ∼62% by replacing the previous method with thefiTQun cut, while the efficiency loss for the appearance signal is only ∼2%.LCI gzsultsThe results of analyzing the data taken from January 2010 to May 2013which has the neutrino beam exposure of 6:57× 1020 POT(protons on tar-get) is the following. As shown in Table 7.2, 28 events are observed as ,e112KB4B fysults)2Invariant mass (MeV/c0 100 200 300Number of events0102030RUN1-4 data)POT2010×(6.570  CCeνOsc.  CCµν+µν CCeν+eνNC=0.1)13θ22(MC w/ sinFigure 7.5: .0 rejection cut which was used in the previous ,e appearanceanalysis[28]. The reconstructed invariant mass calculated by a dedicated.0 reconstruction algorithm is required to be less than 105 MeVRx2 for anevent to be selected as a ,e candidate. As in the plot, the NC backgroundswhich are dominated by NC .0 events produce a peak near the .0 massof ∼ 135 MeVRx2 and are largely rejected by the cut. The data and theexpectations shown in the plot are for the dataset shown in Table 7.2.candidates, and Figure 7.6 shows the 2D distribution of the fiTQun .0 cutvariables for the observed data after the Zrom, Q 1250MeV cut.Using the observed data, a maximum likelihood fit as described in Sec-tion 7.2 is performed by fitting for sin2 21= only, while other oscillationparameters are fixed at the values in Table 7.1. The fit results in the best-fitvalue and the 68% confidence interval of sin2 21= = 0:14050:0=8−0:0=2 assumingnormal hierarchy and sin2 21= = 0:17050:045−0:0=7 assuming inverted hierarchy,and the best-fit prediction of the pe − e distribution overlaid with the ob-served data are shown in Figure 7.7.The significance for nonzero 1= was calculated by generating a largenumber of toy experiments assuming 1= = 0 and calculating a p-value.While generating the toy experiments, other oscillation parameters are fixedat the values on Table 7.1, and systematic uncertainties are randomly thrown113KB4B fysults)2 (MeV/cγγfiTQun m0 100 200 300) e/L 0pifiTQun ln(L0200400 MC w/=0.113θ22sinFigure 7.6: 2D distribution of the fiTQun .0 cut variables after the Zrom, Q1250MeV cut, for the dataset shown in Table 7.2. The horizontal axis isthe reconstructed invariant mass from the .0 fit, and the vertical axis is thelog likelihood ratio between the .0 and the single-ring electron fits. Theblack markers indicate the observed data and the gray box histograms areMC expectations. The events below the blue line are selected as the final ,ecandidate events.according to their prior uncertainties. Then, a test statistic defined as:∆2 := −2× (lna′(sin2 21= = 0)− lna′(sin2 2lost1= ))P (7.4)is calculated for each toy experiment, where a′ is the likelihood marginal-ized over systematics and sin2 2lost1= is the best-fit value obtained from datamentioned above. The p-value is then defined as the fraction of toy exper-iments which have ∆2 S 53:64, where 53.64 is the ∆2 for the observeddata. After producing 1015 fake experiments the p-value is calculated to be1:0 × 10−1=, which is equivalent to the exclusion of 1= = 0 at 7:3 signifi-cance and shows a clear discovery ,e of appearance.114KB4B fysults51050306090120150180Angle (degrees)DataBest fitBackground componentMomentum (MeV/c)0 500 1000 150000. fitFigure 7.7: The best-fit (for normal hierarchy) pe − e distribution plottedwith the observed data. The 1D plots on the top and the left are the 1D peand e distributions respectively. Figure taken from [29].115Chvptzr MVtmosphzrix czutrinodsxillvtion VnvlysisIn this chapter, a neutrino oscillation analysis using the SK atmosphericneutrino data is described in detail. As mentioned in Chapter 1, in at-mospheric neutrino data, oscillation effects manifest in the energy and thezenith angle distributions of the observed neutrino events. In this analysis,the atmospheric neutrino data is first selected into several subsamples inorder to separate the events in terms of neutrino type, and for each samplethe events are binned in the 2D distribution of the observed momentumand the zenith angle of the direction. The oscillation parameters are thenextracted by comparing the binned data to the expectation calculated fromMC simulation at different oscillation parameters.The dataset used in this analysis is the FC data in SK-IV, and recon-structed particle information from fiTQun described in Chapter 6 is usedfor selecting and binning the data. For oscillation parameter estimation, aMarkov Chain Monte Carlo technique is employed to make Bayesian pre-dictions on the oscillation parameters based on the observed data. Thefollowing sections describe each of the analysis elements in detail.MCF Zvznt hzlzxtion vny WinningThis section describes the event selection and the binning which is usedas an input for the oscillation analysis. FC data is categorized into sub-samples based on the reconstructed particle information from fiTQun, andthen binned in reconstructed momentum and zenith angle. Since the oscil-lation effects manifest differently between the flavours of atmospheric neu-trinos as described in Chapter 1, it is important to implement an eventselection scheme which is capable of categorizing the observed neutrinoevents by neutrino flavour. In addition to separating the events by neutrinoflavour(electron or muon) based on the reconstructed ring type, statisticalseparations of neutrinos and antineutrinos are done in order to improve the116LBEB Evynt gylywtion unx Vinningsensitivity to the oscillation effects such as the ones from the mass hierarchy,which happen differently for neutrinos and antineutrinos.MCFCF hzlzxtion CritzrivThe events which are used in the analysis are the FC events as described inChapter 5 which have the visible energy greater than 30 MeV. The visibleenergy, Zvss, is defined as the sum of the reconstructed kinetic energy aboveCherenkov threshold for all rings, taking into account the assumed particletype (electron, muon or .5) of each ring. The fiducial volume cut is alsoapplied, which requires that the distance from the reconstructed vertex ofthe most energetic ring to the nearest point on the wall to be greater than200 cm. The events which pass these basic selections are further divided into13 event categories based on the reconstructed information from fiTQun asfollows:1. huwBGzk onzBring zBlikz E yzxvyZvssQ1330MeV, one e-like ring with momentum S100MeVRx, nnmy=0.2. huwBGzk onzBring zBlikz ≥F yzxvyZvssQ1330MeV, one e-like ring with momentum S100MeVRx, nnmy≥1.3. huwBGzk onzBring Blikz E yzxvyZvssQ1330MeV, one -like ring with momentum S200MeVRx, nnmy=0.4. huwBGzk onzBring Blikz F yzxvyZvssQ1330MeV, one -like ring with momentum S200MeVRx, nnmy=1.5. huwBGzk onzBring Blikz ≥G yzxvyZvssQ1330MeV, one -like ring with momentum S200MeVRx, nnmy≥2.6. huwBGzk tfioBring .0ZvssQ1330MeV, two e-like rings, nnmy=0, reconstructed invariant masscalculated from the two rings in the range 85-215MeVRx2.7. bultiBGzk onzBring zBlikz ,eZvssS1330MeV, one e-like ring, nnmy≥18. bultiBGzk onzBring zBlikz ,¯eZvssS1330MeV, one e-like ring, nnmy=09. bultiBGzk onzBring BlikzZvssS1330MeV, one -like ring117LBEB Evynt gylywtion unx Vinning10. bultiBGzk multiBring zBlikz ,eZvssS1330MeV, two or more rings, highest momentum ring is e-like,passes the MME selection cut, passes the multi-ring ,e-like cut.11. bultiBGzk multiBring zBlikz ,¯eZvssS1330MeV, two or more rings, highest momentum ring is e-like,passes the MME selection cut, fails the multi-ring ,e-like cut.12. bultiBGzk multiBring BlikzZvssS600MeV, two or more rings, highest momentum ring is -like,momentum of the -like ring S600MeVRx.13. bultiBGzk multiBring zBlikz othzrZvssS1330MeV, two or more rings, highest momentum ring is e-like,fails the MME selection cut.In the selection criteria above, nnmy is the number of decay electrons,which is defined as the number of subevents minus 1 as described in Sec-tion 6.6.2. As discussed in Section 8.1.2, neutrinos and antineutrinos can bestatistically separated using the decay electron information. In addition tothe e-like and -like samples which are the enriched samples of ,eCC and,CC events respectively, the two-ring .0 sample is included in the anal-ysis in order to constrain the NC background in the e-like samples. Thedefinitions of the MME selection cut and the multi-ring ,e-like cut whichare applied to the multi-GeV multi-ring e-like samples are detailed in Sec-tion 8.1.3. In multi-GeV, while the e-like samples are statistically separatedinto ,e and ,¯e enriched samples, no neutrino-antineutrino separation is doneto the -like samples, mainly due to the lack of statistics to give any noti-ciable improvement in the mass hierarchy sensitivity.MCFCG = hzpvrvtion for hinglzBging ZvzntsA large fraction of the events selected as single-ring are CCQE events. In aCCQE event, the particles which are left after the neutrino interaction arethe outgoing charged lepton and a nucleon, a proton or a neutron dependingon whether it is a neutrino or antineutrino interaction. As the lepton chargeis indistinguishable in a water Cherenkov detector and the emitted proton isusually below Cherenkov threshold, separating neutrinos and antineutrinosin the CCQE channel is therefore difficult.The situation is, however, different for CC single charged pion eventswhich are categorized as single-ring due to the pion momentum being below118LBEB Evynt gylywtion unx VinningCherenkov threshold. Consider the CC1.± interaction for a neutrino andan antineutrino:, +c → l− +c + .5P (8.1),¯ +c → l5 +c + .−: (8.2)For the case of neutrino, the produced .5 often decays into a 5, whichsubsequently produces a decay positron which will be detected. For an-tineutrino, on the other hand, the emitted .− in most cases gets absorbedby the nuclei in water, and the interaction therefore leaves no decay elec-tron. Thus, a single-ring event with an extra decay electron being detectedhas a higher chance of being a neutrino event, rather than antineutrino.The event selection criteria in Section 8.1.1 thus subdivides the single-ringevents further into neutrino-enriched and antineutrino-enriched samples bythe number of decay electrons. For instance, the “Sub-GeV one-ring e-like≥1 decay” sample has high concentration of ,e with significantly less ,¯e, andthe “Sub-GeV one-ring -like ≥2 decay” sample, in which in most cases thedecay electrons from the primary muon and a .5 are both detected, consistsmostly of , with small ,¯ content.MCFC3 bultiBging e=e hzlzxtionAs the fraction of multi-ring events increases at higher energy, multi-ringevent samples are also included in the analysis in multi-GeV. Due to thecomplexity of multi-ring events, it is in general more difficult to achievehigh purity of the targeted event types in a multi-ring event category com-pared to the single-ring case. The multi-ring e-like event sample, as selectedby requiring the highest momentum ring to be e-like, has rather high con-tamination of ,CC and NC events.In order to reduce such backgrounds, a likelihood selection is appliedto subdivide the Multi-GeV Multi-ring E-like(MME) sample into a ,e +,¯eCC enriched sample and a background enriched sample. Below are thedescriptions of the four variables which are input to the likelihood selection,whose distributions are shown in Figure 8.1:1. Enzrgy frvxtion of thz bost Enzrgztix gingThe fraction of the visible energy of the Most Energetic Ring(MER) tothe total visible energy. The definition of the visible energy is statedin Section 8.1.1. The variable tends to be large for ,e + ,¯eCC eventssince the outgoing electron often carries a large fraction of the event’senergy, whereas the value is small for , + ,¯CC and NC events in119LBEB Evynt gylywtion unx Vinningthe same visible energy range since often in such events the MER is agamma ring from a .0 and the hadronic system carrying large energyproduces large number of rings and deposit larger energy outside ofthe MER.2. Enzrgy frvxtion of thz bost Enzrgztix .5Blikz gingThe fraction of the visible energy of the Most Energetic .5-like(ME.5)ring to the total visible energy. The value is set to zero if there is no.5-like ring found. The , + ,¯CC events which are categorized asmulti-ring e-like usually have a rather low momentum muon ring whosevisible energy is lower than another ring such as the  ring from a .0decay. For such events the muon ring is often reconstructed as a .5-like ring, and the ring tends to have higher energy than a true .5ring which is produced in ,e + ,¯e and NC events. The variable thuseffectively discriminates , + ,¯CC events from the rest.3. cumwzr of yzxvy zlzxtronsNumber of decay electrons as defined in Section 8.1.1. The value tendsto be larger for , + ,¯CC and NC events since the hadronic systemcarrying large energy produces large number of decay electrons forsuch events.4. Distvnxz to thz fvrthzst yzxvy zlzxtronThe distance between the vertices of the most energetic ring and themost distant decay electron, normalized by the visible energy of themost energetic ring. The variable is set to zero if there is no decayelectron detected. The variable tends to be larger for ,+ ,¯CC eventssince the muon in such events travel longer distance than charged pionsin ,e + ,¯eCC events.As shown in Figure 8.1, the variables provide power to discriminate between,e + ,¯eCC events and the backgrounds. For event selection, the normalizeddistributions for signal(,e+ ,¯eCC) and background(,+ ,¯CC and NC) foreach variable are produced and used as likelihood functions. For a givenevent, ln(assgRalug), the log likelihood ratio of assuming the event as asignal and a background, is evaluated for each of the four variables usingthe likelihood functions, and the sum of the four log likelihood ratios isthen used to make a selection cut. Figure 8.2 is the distribution of the cutvariable for inclusive multi-GeV multi-ring e-like events where the signal andthe background are shown separately, and the signal(,e + ,¯eCC) enrichedsample is selected by applying a cut as S-0.25 which was determined based120LBEB Evynt gylywtion unx VinningEnergy fraction of MER0 0.2 0.4 0.6 0.8 10102030 < 5.01GeVvisMRe : 2.51GeV < E+piEnergy fraction of ME0 0.2 0.4 0.6 0.8 1-210-110110210 < 5.01GeVvisMRe : 2.51GeV < E# of decay-e0 2 4 6 8 10050100 < 5.01GeVvisMRe : 2.51GeV < EMERDistance to decay-e / E0 0.2 0.4 0.6 0.8 1-210-110110210 < 5.01GeVvisMRe : 2.51GeV < EFigure 8.1: Distribution of the variables which are used for the Multi-GeV Multi-ring E-like(MME) likelihood selection, for the multi-ring e-likeevents with visible energy in the range 2.51-5.01 GeV. Distributions areshown separately for ,eCC(green), ,¯eCC(blue), ,CC(red), ,¯CC(yellow)and NC(gray).121LBEB Evynt gylywtion unx Vinning)Bkg/LSigLklhd. sep. by all variables: ln(L-10 -5 0 5 10050100Total pass: 1217.8 eventsMode: Fraction (Eff.) CC: 56.9% (77.8%)eν CC: 21.3% (93.3%)eν CC: 3.8% (16.4%)µν CC: 0.3% (14.5%)µνNC: 17.6% (49.1%)Figure 8.2: Distribution of the sum of ln(assgRalug) for all four MME selec-tion variables, for inclusive multi-GeV multi-ring e-like events. The distribu-tion is shown separately for signal(,e+,¯eCC) and background(,+,¯CC andNC) in cyan and magenta respectively. The MME selection cut is appliedat -0.25 as indicated by the red arrow, and the interaction mode breakdownof the selected sample is shown in the figure. The efficiency for passing thecut is shown in the parenthesis for each interaction mode. For comparison,the fraction of ,eCC and ,¯eCC events before the selection cut is applied are46.4% and 14.5% respectively.on sensitivity. The events which are rejected by this cut are categorized asthe “Multi-GeV multi-ring e-like other” sample.Once the ,e + ,¯eCC enriched sample is selected by the MME selectioncut, another likelihood selection is applied, this time, to separate the ,eCCevents from ,¯eCC events. The selection is done in a similar manner as aboveusing the following three variables:1. cumwzr of rings2. cumwzr of yzxvy zlzxtrons3. irvnsvzrsz momzntum frvxtion122LBEB Evynt gylywtion unx VinningDefined as:ptrkx :=∑iZivss sin iRZvssP (8.3)where i is the angle between each ring and the most energetic ring,Zivss is the visible energy of the ring and Zvss is the sum of the visibleenergies of all rings.The variables are motivated by the fact that ,e scattering tends to be lessforward peaked and have larger energy transferred to the hadronic systemcompared to ,¯e, and as shown in Figure 8.3, the variables tend to be largerfor ,e than ,¯e. Similar to what was done for the MME selection, likelihoodfunctions for ,eCC and ,¯eCC are produced using the distributions, andthen for a given event ln(a,eRa,e), the likelihood ratio between the ,e and,¯e hypotheses, is evaluated for each variable. The distribution of the sum ofln(a,eRa,e) for all variables are shown in Figure 8.4, and the events passingthe cut at S0 are selected as the “Multi-GeV multi-ring e-like ,e” sample,while the ones rejected are categorized as “Multi-GeV multi-ring e-like ,¯e”.MCFCI hvmplz htvtistixs vny euritizsTable 8.1 shows the statistics and the interaction mode breakdown of eachevent sample selected by the selections in Section 8.1.1. Event rates for dataand MC expectation are both shown in the table, and the expected eventrates are calculated assuming two-flavour , → , oscillations at sin2 2 =1:0 and ∆m2 = 2:5×10−= eV2 and are normalized to 2519.89 days exposurewhich is the same as data. Note that the expectations for , + ,¯CC eventsare not shown on the table, however, they are properly considered in theactual oscillation analysis. It can be seen from the table that the eventselection achieves the intended separation of observed events by neutrinotype, namely, the ,e-enriched samples as selected in Sections 8.1.2 and 8.1.3have high purity of ,e with low ,¯e contamination.As demonstrated in Section 6.10.3, fiTQun achieves a significantly betterperformance in reconstructing multi-ring events compared to the preexistingAPFIT algorithm. In order to see the improvement in the atmosphericneutrino selection made by fiTQun, the event sample breakdown for anequivalent event selection using APFIT is shown in Appendix B.MCFC5 WinningFigure 8.5 shows the binning scheme for each event sample. The events ineach sample are binned in the observed momentum and zenith angle in order123LBEB Evynt gylywtion unx Vinning# of rings0 2 4 600.20.4 < 5.01GeVvisMRe : 2.51GeV < E# of decay-e0 2 4 6 8 1000.51 < 5.01GeVvisMRe : 2.51GeV < ETransverse momentum fraction0 0.2 0.4 0.6 0.8 < 5.01GeVvisMRe : 2.51GeV < EFigure 8.3: Distribution of the variables which are used for the multi-ring,eR,¯e likelihood separation, for the multi-ring e-like events passing the MMEselection cut with visible energy in the range 2.51-5.01 GeV. Distributionsare shown separately for ,eCC(green) and ,¯eCC(blue). Each distribution isarea normalized for easier comparison of the shape.124LBEB Evynt gylywtion unx Vinning)eν/LeνLklhd. sep. by all variables: ln(L-4 -2 0 2 4050100150Total pass: 602.9 eventsMode: Fraction (Eff.) CC: 60.1% (52.3%)eν CC: 12.4% (28.7%)eν CC: 5.4% (69.6%)µν CC: 0.4% (59.1%)µνNC: 21.7% (61.0%)Figure 8.4: Distribution of the sum of ln(a,eRa,e) for all three ,eR,¯eseparation variables, for multi-GeV multi-ring e-like events passing theMME selection. The distribution is shown separately for ,eCC, ,¯eCC andbackground(, + ,¯CC and NC) in green, blue and black respectively. The,e-like selection cut is applied at 0 as indicated by the red arrow, and theinteraction mode breakdown of the selected sample is shown in the figure.The efficiency for passing the cut is shown in the parenthesis for each inter-action mode. For comparison, the fraction of ,eCC and ,¯eCC events beforethe selection cut is applied are 56.9% and 21.3% respectively.125LBEB Evynt gylywtion unx Vinning% 1Re 0nmy 1Re ≥1nmy 1R 0nmy 1R 1nmy 1R ≥2nmy 2R.0,eCC 73.22 91.30 0.92 0.02 0.00 5.00,¯eCC 23.96 2.14 0.35 0.00 0.00 1.49,CC 0.10 2.46 69.82 68.15 94.86 0.81,¯CC 0.02 0.60 10.08 29.38 3.57 0.08NC 2.70 3.50 18.83 2.45 1.57 92.61Total event rateMC 4876.2 450.7 851.7 4378.1 356.5 1130.5Data 4677 450 814 4412 378 1095% 1Re ,e 1Re ,¯e 1R MRe ,e MRe ,¯e MR MRe oth.,eCC 75.39 56.11 0.02 60.15 53.70 0.92 28.21,¯eCC 8.26 38.35 0.02 12.38 30.14 0.11 2.68,CC 4.55 0.22 59.71 5.36 2.30 72.56 33.95,¯CC 0.93 0.05 40.20 0.38 0.26 24.07 3.27NC 10.87 5.27 0.05 21.73 13.61 2.34 31.89Total event rateMC 161.2 839.1 763.0 602.9 614.9 1627.0 699.0Data 174 773 721 585 536 1562 807Table 8.1: Interaction mode breakdown and the total event rates for eachevent sample, where the upper and the lower table is for sub-GeV and multi-GeV samples respectively. Event rates for MC and data are both shown inthe tables. MC expectations are calculated with two-flavour oscillations atsin2 2=1.0 and ∆m2=2:5×10−= eV2 and is normalized to the same 2519.89days exposure as data.126LBEB Evynt gylywtion unx Vinning-0.6-0.4-ν̄eMReνe1Reν̄e1ReνeMReOth.MRμ1Rμ1Re≧1dcy1Re0dcy1Rμ1dcy1Rμ0dcy 2Rπ01Rμ≧2dcySub-GeVMulti-GeV-0.8Log10P[GeV/c]Log10P[GeV/c)]Figure 8.5: Binning for each event sample. In the figure, the momentum binedges are shown in the log10 of the momentum in GeVRx, and the highestand the lowest momentum bins in each sample are overflow and underflowbins respectively. The unshaded samples are further subdivided into 10 binsin the cosine of the zenith angle equally spaced in [−1P+1], while there isno zenith angle subdivision for the shaded samples.to extract the oscillation pattern as mentioned before, and for single-ringevents the momentum and the zenith angle of the direction of the observedring are used. For multi-ring events, the total visible energy Zvss is used asthe “generalized momentum”, and the direction of the reconstructed total3-momentum vector is used for the zenith angle. The zenith angle is binnedin 10 bins equally spaced in the cosine of the zenith angle, except for somesamples whose statistics are too low such that the statistical fluctuation ofMC expectation becomes problematic, or have no directional sensitivity.MCFCK dwszrving thz dsxillvtion ZffzxtsOnce the events are selected and binned, the binned data is compared againstthe expectations and the oscillation parameters are estimated. In order tosee how the oscillation effects are observed in each sample, the effects ofvarying each oscillation parameter on the predicted zenith distributions are127LBEB Evynt gylywtion unx VinningParameter Default Valuesin2 12 0.304sin2 1= 0.0219sin2 2= 0.50∆m221 7:53× 10−5 eV2∆m2=2 +2:39× 10−= eV2 (NH)MZ 0Table 8.2: Default values for the oscillation parameters which are used forevent rate calculation.shown below. Full three-flavour oscillations including the matter effect areconsidered, with the base values for the oscillation parameters set to what areshown in Table 8.2. The event rates are normalized to the data-equivalent2519.89 days exposure. In the zenith distributions, cos dQ0 is for upward-going events and cos dS0 is for downward-going. The discussions belowfollow the discussions of atmospheric neutrino oscillations in Section 1.5.Figure 8.6 shows the zenith distributions for the sub-GeV e-like and -like samples when sin2 2= is varied as 0.4, 0.5 and 0.6. Since the -likesample mainly observes , disappearance which depends on sin2 22=, theobserved event rate is minimized when 2==45◦, i.e. sin2 2==0.5, and anydeviation from that will increase the event rates. For the e-like sample onthe other hand, the ∆m221-induced oscillation which depends on cos2 2= isimportant in sub-GeV, and thus an excess of event rate is seen when 2=Q45◦while a deficit is seen when 2=S45◦. Thus, the sub-GeV e-like sample issensitive to the 2= octant.Figure 8.7 shows the effect of sin2 2= on the multi-GeV e-like samples.These samples observe the enhanced ,e appearance by the MSW resonanceeffect which is proportional to sin2 2=. Therefore, we see a correspondingexcess or deficit of the upward-going e-like events depending on whethersin2 2= is greater or smaller than 0.5.The most noticeable effect from MZ is the enhancement of sub-GeV ,eevents when −.QMZQ0, as seen in Figure 8.8. Since this effect somewhatcompetes with the effect from sin2 2= shown in Figure 8.6, the correlationsbetween the two parameters will be seen in the oscillation analysis discussedin Section 8.6.Finally, Figure 8.9 compares the zenith distributions for the multi-GeVe-like samples for normal and inverted hierarchy. The MSW resonance en-hancement of electron type neutrino appearance only happens for neutrinoswhen it is normal hierarchy, while for inverted hierarchy it only happens for128LBEB Evynt gylywtion unx VinningZθcos-1 -0.5 0 0.5 10200400600Sub-GeV 1Re 0dcyZθcos-1 -0.5 0 0.5 10200400600 1dcyµSub-GeV 1RFigure 8.6: The effect of sin2 2= variations on the distributions of the cosineof the zenith angle for sub-GeV samples. The left is for one-ring e-like 0decayand the right is for one-ring -like 1decay. The plots show the expectationsat sin2 2= set to 0.4(solid green), 0.5(solid black) and 0.6(dashed red).Zθcos-1 -0.5 0 0.5 101020eνMulti-GeV 1Re Zθcos-1 -0.5 0 0.5 1050100eνMulti-GeV 1Re Figure 8.7: The effect of sin2 2= variations on the distributions of the cosineof the zenith angle for Multi-GeV samples. The left is for one-ring e-like ,eand the right is for one-ring e-like ,¯e. The plots show the expectations atsin2 2= set to 0.4(solid green), 0.5(solid black) and 0.6(dashed red).129LBEB Evynt gylywtion unx VinningZθcos-1 -0.5 0 0.5 10200400600Sub-GeV 1Re 0dcyZθcos-1 -0.5 0 0.5 102004001dcy≥Sub-GeV 1Re Figure 8.8: The effect of MZ variations on the distributions of the cosine ofthe zenith angle for Sub-GeV samples. The left is for one-ring e-like 0decayand the right is for one-ring e-like ≥1decay. The plots show the expectationsat MZ set to −.R2(solid green), 0(solid black) and +.R2(dashed red).antineutrinos. As in the left plots, the multi-GeV ,e-enriched event sam-ples, selected by the ,eR,¯e separation scheme as described in Sections 8.1.2and 8.1.3, therefore observe an excess of upward-going events in the caseof normal hierarchy compared to inverted hierarchy. On the other hand,the ,¯e-enriched samples on the right, which have much closer to equal mix-ture of ,e and ,¯e as in Table 8.1, show almost no sensitivity to the masshierarchy, since the effect of flipping the hierarchy almost cancels betweenneutrinos and antineutrinos. Employing the ,eR,¯e separation and havingthe ,e-enriched samples thus makes the effect from the mass hierarchy moreprominent and improves the sensitivity.130LBEB Evynt gylywtion unx VinningZθcos-1 -0.5 0 0.5 101020eνMulti-GeV 1Re Zθcos-1 -0.5 0 0.5 1050100eνMulti-GeV MRe Zθcos-1 -0.5 0 0.5 1050100eνMulti-GeV 1Re Zθcos-1 -0.5 0 0.5 1050100eνMulti-GeV MRe Figure 8.9: The effect of the mass hierarchy on the distributions of the cosineof the zenith angle for Multi-GeV e-like samples. The left side is for the ,e-enriched samples and the right side is for the ,¯e-enriched samples. The solidgreen lines are the expectations for normal hierarchy while the dashed redlines are for inverted hierarchy.131LBFB gflstymutiw inwyrtuintiysMCG hystzmvtix UnxzrtvintizsThis section describes the sources of systematic uncertainties which are con-sidered in the oscillation analysis. While comparing the observed data toprediction in order to extract the oscillation parameters, the parameter foreach systematic is also varied and its effect is propagated to the predictedevent rate for each bin. In total there are 82 model parameters considered inthe analysis, six of which are the oscillation parameters. Out of the remain-ing 76 parameters for systematics, 18 are for the atmospheric neutrino flux,19 are for neutrino interaction, 27 are for FSI and 12 are for the detectorresponse and reconstruction, and the full list of the systematic parameterscan be found in Appendix A. The prior uncertainties on the systematic pa-rameters are all treated as Gaussians where the 1 widths are given below,and apart from the FSI parameters they are assumed to be uncorrelated.MCGCF Vtmosphzrix czutrino FluflVwsolutz cormvlizvtionThe uncertainty on the absolute flux normalization is given by Honda dsak-[95] as an energy-dependent error as in Figure 8.10. Scaling parametersfor below and above 1 GeV independently scale the flux according to thefunction shown. In order to account for the additional differences betweenthe base Honda flux model[95] and other flux models such as FLUKA[52]and Bartol[53] above 10 GeV, an additional 5% normalization error on multi-GeV events is considered as an independent parameter. (GeV)νE-110 1 10 210 310Normalization Error (%)0510152025Figure 8.10: Uncertainty on the flux normalization as a function of energy.132LBFB gflstymutiw inwyrtuintiysFlux gvtiosThe uncertainties on the various flux ratios are estimated by the comparisonof the Honda flux[95] to the FLUKA[52] and the Bartol[53] flux models. Forthe flavour ratio (,+ ,¯)R(,e+ ,¯e), the uncertainty is 2% for Z, Q 1 GeV,3% for 1 GeV Q E, Q 10 GeV, 5% for 10 GeV Q E, Q 30 GeV and then lin-early increases in log10Z, to 30% at 1 TeV. For ,¯eR,e ratio the uncertaintyis 5% for Z, Q 10 GeV, 8% for 10 GeV Q E, Q 100 GeV and then linearlyincreases in log10Z, to 30% at 1 TeV. Finally, the uncertainty on ,¯R,ratio is 2% for Z, Q 1 GeV, 6% for 1 GeV Q E, Q 50 GeV and then linearlyincreases in log10Z, to 60% at 1 TeV. For these ratio uncertainties, the fluxin the following three energy regions are varied independently according toa scaling parameter in each region: Z, Q 1 GeV, 1 GeV Q E, Q 10 GeVand Z, S 10 GeV. The uncertainty on the up/down ratio and the hor-izontal/vertical ratio of the flux is also estimated by comparing the threeflux models, and the differences in the zenith distributions between the fluxmodels are taken as the uncertainty.KR. gvtioWhile the atmospheric neutrino flux below 10 GeV is mainly from pion de-cays, contributions from kaons become increasingly dominant as the energyincreases. Based on the KR. production ratio measurement by the SPYexperiment[96], the uncertainty on the ratio of the flux induced by kaonsand pions is estimated to be 5% for Z, Q 100 GeV and then linearly in-creases with energy to 20% at 1 TeV.holvr VxtivityThe activity of the sun cycles with 11 year period, and the resulting varia-tions in the magnetic field affects the cosmic-ray flux. The variation in theneutrino flux caused by a ±1 year variation in the solar cycle is taken as theuncertainty on the flux.czutrino evth azngthThe production height of atmospheric neutrinos depends on the densitystructure of the atmosphere, and the uncertainty on that needs to be prop-agated to the oscillation probability which depends on the path length ofneutrinos. The uncertainty in the density of the atmosphere is estimatedto be 10% based on the comparison between the US-standard 76 and theMSISE-90 models for the atmosphere[97], and the resulting variations in theproduction height of atmospheric neutrinos are propagated to the oscillation133LBFB gflstymutiw inwyrtuintiysprobability calculation.bvttzr EffzxtAlthough the matter density structure of the earth is rather well known[35],there is an uncertainty on the electron density which depends on the exactchemical composition of the core, and such uncertainty needs to be takeninto account when calculating the matter-induced oscillation probabilitiesthrough the Earth. As the core is assumed to consist of heavy elementssuch as iron whose electron density is about 6.8% less than that of lightelements, the uncertainty on the electron density in the core is estimated tobe 6.8% and is considered during oscillation probability calculation.MCGCG czutrino IntzrvxtionCCfEFor CCQE interaction the uncertainties on the absolute total cross sectionas well as the cross section ratios ,¯R, and (, + ,¯)R(,e + ,¯e) are assignedaccording to the differences between the base model of relativistic Fermi gasand the model by Nieves ds ak-[98]. In addition, 20% uncertainty is assignedon the axial vector mass MfEA .bECAs there are still large uncertainties on the MEC model and the cross sectionhas not been explicitly measured at the present moment, 100% uncertaintyis assigned to the total cross section of the MEC process.hinglz bzson eroyuxtionThe uncertainties on the parameters which characterize the Graczyk-Sobczykform factor[64] were obtained from the data from bubble chamber experi-ments [65][66]. Additional uncertainties in the total cross sections are as-signed based on the differences between the base model in NEUT to themodel by Hernandez ds ak-[99].DIhFor DIS interaction, 10% uncertainty is assigned to the total cross section,and in order to account for the larger uncertainties in the energy below 10GeV the differences between the base model of GRV98 with the Bodek-Yangcorrection and the CKMT model[100] are considered as an additional uncer-tainty. Also, the uncertainty on the f2 distributions of the DIS interactionis assigned by the differences between the GRV98 model with and without134LBFB gflstymutiw inwyrtuintiysthe Bodek-Yang correction. Separate parameters for l Q 1:3 GeVRx2 andl S 1:3 GeVRx2 vary the f2 distributions in the two regions independently.Cohzrznt eion eroyuxtionAs there was no evidence for ,CC coherent pion production observed bythe SciBooNE experiment[101], 100% uncertainty is assigned to the totalcross section of coherent pion production.cCDCC gvtio20% uncertainty is assigned to the ratio of the inclusive NC and CC crosssections.,CC Cross hzxtion25% uncertainty is assigned to the inclusive ,CC total cross section.MCGC3 Finvl htvtz IntzrvxtionAs final state interactions(FSI) change the observable final state of particleswhich exit the nucleus, information such as the number of rings and decayelectrons which the event selection is based on is directly affected by theFSI. It is therefore important to consider the uncertainties in the FSI modelin the oscillation analysis.As described in Section 4.2.5, based on the external pion scattering datathe uncertainties on the NEUT pion cascade model parameters are givenas representative sets of the FSI model parameters which span the 1 errorsurface in the multi-dimensional FSI parameter space. In order to incorpo-rate this uncertainty in the analysis, a covariance matrix is formed amongthe event rates of the selected event samples based on the allowed FSI vari-ations. Figure 8.11 shows the error matrix for the fractional event ratevariations for the sub-GeV samples by the simultaneous variations of theFSI parameters on the 1 surface. In addition to the FSI which happenswithin the initial target nucleus, secondary interactions(SI) of pions outsidethe target nucleus are also varied simultaneously since they are simulatedin the detector simulation using the same NEUT pion cascade model. Inthe oscillation analysis, instead of directly manipulating the underlying FSImodel parameters, a systematic parameter to scale the event rate is intro-duced for each sample, and the prior uncertainty on the scaling parametersare treated as a multivariate Gaussian with the errors and the correlationsgiven in Figure 8.11. For the sub-GeV event samples, each selected sampleis further subdivided into two momentum bins and there are thus 12 scaling135LBFB gflstymutiw inwyrtuintiys-0.15-0.1-0.0500.050.10.151Re 0dcy #11Re 0dcy #21Re >0dcy #11Re >0dcy #2 0dcy #1µ1R 0dcy #2µ1R 1dcy #1µ1R 1dcy #2µ1R >1dcy #1µ1R >1dcy #2µ1R #10pi2R #20pi2R1Re 0dcy #11Re 0dcy #21Re >0dcy #11Re >0dcy #2 0dcy #1µ1R 0dcy #2µ1R 1dcy #1µ1R 1dcy #2µ1R >1dcy #1µ1R >1dcy #2µ1R #10pi2R #20pi2RFigure 8.11: The fractional error matrix sgn(kij)×√|kij | for the sub-GeVsamples, where kij is the covariance for the fractional variations in the eventrate of each sample by FSI and SI parameter variations. Each sample isfurther subdivided into two momentum bins, for below(#1) and above(#2)400 MeVRxmomentum. The diagonal elements represent the fractional erroron the event rate of each sample which can be up to 13% depending on thesample.parameters in total.For multi-GeV, while there is no subdivision of the samples by momen-tum, the e-like samples are subdivided into ,eCC, ,¯eCC and background(,+,¯CC and NC) based on the true interaction mode. Since the multi-GeV samples are not as pure as the sub-GeV samples, the composition ofeach sample is also allowed to change this way as well as the total eventrate during the oscillation analysis. When forming the covariance matrixfor multi-GeV samples, in addition to the parameters for FSI and SI, pionmultiplicity is also varied based on the differences between NEUT and theresult from the CHORUS experiment[102] in order to account for the un-certainties which may not be covered by the FSI/SI variations. Figure 8.12is the resulting error matrix for the multi-GeV samples. There are thus 15scaling parameters for multi-GeV samples which are introduced to the anal-136LBFB gflstymutiw inwyrtuintiys-0.1-0.0500.050.1CC eν eν1RCC eν eν1R bkg.eν1RCC eν eν1RCC eν eν1R bkg.eν1Rµ1R CCeν eνMRCC eν eνMR bkg.eνMRCC eν eνMRCC eν eνMR bkg.eνMRµMRMR oth.CCeν eν1RCCeν eν1R bkg.eν1RCCeν eν1RCCeν eν1R bkg.eν1Rµ1RCCeν eνMRCCeν eνMR bkg.eνMRCCeν eνMRCCeν eνMR bkg.eνMRµMRMR oth.Figure 8.12: The fractional error matrix sgn(kij) ×√|kij | for the multi-GeV samples, where kij is the covariance for the fractional variations in theevent rate of each sample by FSI, SI and pion multiplicity variations. Thediagonal elements represent the fractional error on the event rate of eachsample which can be up to 11% depending on the sample.ysis whose prior correlations are given by the figure. No prior correlation isintroduced between the sub-GeV and the multi-GeV parameters.MCGCI Dztzxtor gzsponsz vny gzxonstruxtionFC gzyuxtionThe uncertainty on the FC reduction efficiency is estimated to be 1.3% basedon the comparison of the relevant cut variables between data and MC. Theuncertainty on the efficiency of the FC/PC separation cut is estimated to be0.2%. Contaminations from non-neutrino backgrounds such as the flasherPMT events and cosmic-ray muon events are estimated by eye-scanning tobe within 0.1%.Fiyuxivl kolumzThe fiducial volume uncertainty is estimated to be 2%, which corresponds137LBFB gflstymutiw inwyrtuintiysZ (cm)1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400Number of Events010000200003000040000500006000070000SK4: Muon vertex (after dwall cut, top-entering) 0.01%±Error: 0.10 0.110±MC:   1808.400 0.066±Data: 1806.634Mean:   Data: 1806.63±0.07 cm   MC:1808.40±0.11 cmFigure 8.13: Reconstructed vertex Z position of stopping cosmic-ray muonsentering from the top of the detector. The red histogram is for MC and theblack data points are for data, and the mean of each distribution is shown onthe top-right of the plot. The top wall of the ID is located at o = 1810 cm.to a systematic inward/outward shift of the vertex position by ∼10 cm nearthe fiducial boundary. The reconstructed vertex distributions of stoppingcosmic-ray muons for data and MC agree well within this uncertainty asshown in Figure 8.13.ging CountingThe uncertainty on the ring-finding efficiency is evaluated by comparingthe distributions of ring-counting likelihood as described in Section 6.10.1between the FC data and MC. The uncertainty ranges from 0.3% to 2.8%depending on the event category, and it is in general larger in multi-GeV.evrtixlz IyzntixvtionParticle identification uncertainties are estimated by comparing the distri-butions of zR.5 likelihood ratio as discussed in Section 6.10.2 between theFC data and MC. The errors are estimated separately for single-ring andmulti-ring events and vary from 0.1% to 0.4% depending on the event cate-gory.Enzrgy Cvliwrvtion138LBGB Evynt futy dryxiwtionThe uncertainty on the absolute energy scale is estimated by the data-MCcomparison of the energy scales in three control samples: stopping cosmic-ray muons, decay electrons and the mass peak of NC.0. The largest differ-ence between data and MC across all samples is taken as the uncertainty onthe absolute energy scale and is estimated to be 2.1%. The up/down asym-metry of the energy scale is also estimated from the momentum of decayelectrons with different directions and is estimated to be 0.3%.Dzxvy Elzxtron ivggingAs in Section 6.6.3, the uncertainty on decay electron detection efficiency isestimated by the data-MC comparison of the detection efficiency in stoppingcosmic-ray muon samples. The uncertainty is estimated to be 0.8% acrossall energies.MC3 Zvznt gvtz erzyixtionThis section describes the procedures to calculate the predicted event ratesin the analysis bins for a given set of oscillation and systematic parameters.The calculation is done by manipulating the base atmospheric neutrino MCwhich was described in Chapter 4, and the method to reflect the variationsof each parameter depends on the type of the parameter.MC3CF dsxillvtion lzightingThe predicted the event rates at a given oscillation parameter set and nom-inal systematic parameters are calculated as the following.The atmospheric neutrino MC is produced assuming no oscillations. Theoscillation is reflected to the event rate predictions by reweighting the MCfrom unoscillated flux to oscillated flux on event-by-event basis. For eachMC event, the ratio of the calculated oscillated flux over the original un-ocsillated flux is taken as a weight factor, and the event is filled into areconstructed momentum and zenith bin as is done for the observed databut with the calculated weight. When reweighting a MC event of neutrinotype (=zP P ), the following weight factor is applied:! =iNktkiMM1Φ0{Φee (,e → ,) + Φe (, → ,)} : (8.4)The first factor in the equation adjusts the exposure of the original MCto the data exposure where iNktk and iMM are the exposure for the dataand the MC respectively. iMM is 500 years for the MC which is used in139LBGB Evynt futy dryxiwtionthis analysis. Φ0 is the atmospheric neutrino flux from which the MC wasgenerated, evaluated at the energy and the direction of the neutrino in theevent, and the Honda flux at middle solar activity before oscillations wasused for each neutrino type. For , and ,¯ event generation the flux for ,and ,¯ respectively was used, since tau neutrinos are not produced in theatmosphere via pion and kaon decays and only appear after oscillations. Theterms in the middle parentheses represent the total flux of neutrino type after oscillations where Φe and Φ are the electron and muon neutrino fluxrespectively before oscillations, and as the original atmospheric neutrinoflux consists of electron and muon neutrinos there are contributions fromboth once the oscillation is considered. The flux used here is the Hondaflux as described in Section 4.1 weighted according to the solar activitycorresponding to the data period as:Φ = (1− )Φmsx + Φmkx P (8.5)where Φmsx and Φmkx are the flux at the minimum and maximum solaractivity respectively. Φ is thus linearly interpolated between the two witha weight factor , and for this analysis the weight is set to 0.35 whichrepresents the average solar activity during the data period. Finally, theobject e (, → ,) is the transition probability from a neutrino of type to type  calculated at the energy and the direction of the neutrino in theevent.The oscillation probability is calculated using the Prob3++ softwarepackage developed by the Super-Kamiokande collaboration[103] which ispublicly available and has been widely used in various neutrino oscillationanalyses. Following the methods given by Barger ds ak-[34] the softwarealgebraically calculates the full three-flavour oscillation probabilities for at-mospheric neutrinos including the matter effect in the earth based on theradial density structure of the earth given by the PREM model[35].MC3CG eropvgvting kvrivtions of hystzmvtixsVariations in the atmospheric neutrino flux parameters are applied on event-by-event basis at the time of filling the MC events into analysis bins, bymanipulating the weight for each MC event according to the variations inthe flux systematics. For example, if the absolute flux at the energy fora given event is supposed to be increased by 5%, the weight for the eventis increased by 5%. The weights from all flux parameters are multipliedtogether to get the overall weight for the flux systematic variations for eachevent. The solar activity systematic is propagated by accordingly changing140LBGB Evynt futy dryxiwtionthe mixture of the fluxes at minimum and maximum solar activity whenlinearly combining them as described in Section 8.3.1. For the neutrinopath length and matter effect systematics, the variations are reflected to theoscillation probabilities by directly manipulating the path length and thematter density which are used in Prob3++ for probability calculation.Unlike the flux uncertainties which can be parameterized rather simplybased on the neutrino type, energy and direction, the cross section uncer-tainties have complex dependencies on the event configurations and it istherefore not computationally feasible to do an event-by-event reweighting.The cross section systematic variations are propagated instead by directlymanipulating the predicted event rate at each analysis bin, after the MCevents are reweighted by flux and oscillation parameters and binned. As-suming a linear response of the bin contents to each cross section systematicparameter, the cross section parameter variations are applied to each bin as:i → i∏j(1 + f ji ϵj)P (8.6)where i is the predicted event rate in the i-th bin(before the systematicvariations are applied), ϵj is the variation of the j-th systematic from itsnominal value in the unit of : the magnitude of the uncertainty as describedin Section 8.2. The object f ji is a linear response coefficient which gives thefractional change in the content of the i-th bin by the variation of the j-th systematic by 1, and it is calculated for each systematic and analysisbin by reweighting the MC events event-by-event using the detailed MCtruth information, prior to executing the software for oscillation parameterestimation. Figure 8.14 shows an example of such pre-calculated responsecoefficients. In order to allow the interaction mode composition of each binto change, the MC events are first binned separately for ,eCC, ,¯eCC, ,CC,,¯CC, ,+,¯CC and NC, and after the response is applied as in Equation 8.6separately for the event rates from each interaction mode, the total predictedevent rate in each bin is obtained by combining the contributions from allinteraction modes.Most of the detector response and reconstruction systematic parametersare treated in the same way as the cross section systematics, by applyingthe response linearly according to Equation 8.6. The exceptions are theenergy scale parameters, which are applied event-by-event by manipulatingthe reconstructed momentum before the MC events are binned, in order tomore properly treat the migration of events between the momentum binsdue to energy scale change.141LB4B Vuflysiun Unulflsis using aurkov Whuin aonty Wurlo(P[GeV/c])10Log-1 -0.8 -0.6 -0.4 -0.2 0 0.2Zθcos-1-0.500.51 CC : neut_axial_masseνSGeV 1Re 0dcy , 8.14: The response coefficient f ji for the MQEA systematic for the,eCC events in each bin in the sub-GeV one-ring e-like 0decay sample. Theentries represent the fractional variations in the expected event rate at eachbin caused by a +1 variation in the systematic parameter.Finally, for the FSI parameters, the predicted rate at each bin is simplyscaled according to the scaling parameter for the event sample in concern.The correlations between the scaling parameters for different samples arethen enforced using the covariance which was disucussed in Section 8.2.3,and the details will be discussed again later in Section 8.4.2.MCI Wvyzsivn Vnvlysis using bvrkov Chvinbontz CvrloOnce the events in the observed data are binned and the predicted eventrate at each bin is ready to be calculated, the oscillation parameters can beobtained by searching for the set of the parameters for which the predic-tion best matches with the observed data. This analysis employs a MarkovChain Monte Carlo method to make Bayesian predictions on the oscillationparameters based on the observed data, and the procedures are detailed inthe following.142LB4B Vuflysiun Unulflsis using aurkov Whuin aonty WurloMCICF aikzlihooy FunxtionIn order to compare the observed binned data to MC expectations, this anal-ysis defines the following likelihood function which is based on the observeddata and the model parameters in the MC simulation:a(Y|o⃗P f⃗) :=n∏i=1Niici!exp(−i)P (8.7)where Y represents the observed data, o⃗ is the vector of model parameterswhich we are interested in such as the oscillation parameters and f⃗ is thevector of nuisance parameters: the model parameters which we are notinterested in such as the parameters for systematic uncertainties. ci is theobserved number of events in the i-th bin where there are n bins in total,and i = i(o⃗P f⃗) is the expected event rate in the i-th bin given the modelparameters o⃗P f⃗ which is calculated as described in Section 8.3. A likelihoodfunction is the probability for obtaining the observed data given an assumedmodel, and as the data is binned in this analysis, the likelihood is definedas the product of the probability at each bin of observing ci events giventhe mean event rate i based on the assumed model, which is given by thePoisson distribution with mean i.MCICG Wvyzsivn eostzrior erowvwilityThe Bayes’ theorem states that the probability of a hypothesis being truegiven data is given by:e (H|YP I) = e (Y|HP I)e (H|I)e (Y|I) P (8.8)where H represents the hypothesis, I is the prior knowledge on the hypoth-esis and Y represents the observed data. e (H|I) is the prior probability forthe hypothesis H, and the object e (Y|HP I) is the likelihood, which is theprobability for observing the data given the hypothesis and the prior knowl-edge. The denominator e (Y|I) can simply be considered as a normalizationterm:e (Y|I) =∑{H}e (Y|HP I)e (H|I)P (8.9)where the sum runs over all possible hypotheses in the given model space,as is clear from the normalization condition of applying the model sum toeach side of Equation 8.8.143LB4B Vuflysiun Unulflsis using aurkov Whuin aonty WurloThe discussions above is directly applicable to the problem of estimatingthe oscillation parameters from the atmospheric neutrino data in this anal-ysis. The probability distribution for the model parameters o⃗P f⃗ given theobserved data Y, i.e. the posterior probability given data, is written as:e (o⃗P f⃗ |Y) = a(Y|o⃗P f⃗)p(o⃗P f⃗)∫a(Y|o⃗P f⃗)p(o⃗P f⃗) yo⃗ yf⃗ P (8.10)where a(Y|o⃗P f⃗) is the likelihood function defined in Equation 8.7. p(o⃗P f⃗)is the prior probability distribution for the model parameters o⃗P f⃗ , and inthis analysis, except for some oscillation parameters for which flat priors areused, it is assumed that the prior is described as a multivariate Gaussian:p(o⃗P f⃗) = p(o⃗)p(f⃗)P (8.11)p(o⃗) ∝ exp(−12y⃗oik −1o y⃗o)P (8.12)p(f⃗) ∝ exp(−12y⃗fik −1f y⃗f )P (8.13)where y⃗o and y⃗f are the deviations from the nominal values for o⃗ and f⃗respectively. ko and kf are the covariance matrices for the prior constraintson the parameters o⃗ and f⃗ , and apart from the systematic parameters relatedto the FSI uncertainties which are correlated as described in Section 8.2.3,the parameter priors are assumed to be uncorrelated, i.e., only the diagonalelements in the matrices are nonzero for the majority of the parameters.Once the Bayesian posterior probability Equation 8.10 is defined, onecan obtain the joint probability distribution for the model parameter setso⃗P f⃗ given the observed data in two steps: by calculating the expected eventrate i at each bin for the assumed values of o⃗ and f⃗ , and then evaluatingEquation 8.10.MCIC3 bvrginvlizvtion vny evrvmztzr ZstimvtionIn the oscillation analysis, we are only interested in certain oscillation pa-rameters while not caring about the rest of the model parameters. Therefore,rather than the joint probability e (o⃗P f⃗ |Y), we want the probability for theinteresting parameters o⃗ only, disregarding the values of the nuisance pa-rameters f⃗ . This is done by marginalization, i.e., by integrating out thenuisance parameters from the posterior probability e (o⃗P f⃗ |Y):e (o⃗|Y) =∫e (o⃗P f⃗ |Y) yf⃗ : (8.14)144LB4B Vuflysiun Unulflsis using aurkov Whuin aonty WurloThe probability e (o⃗|Y) is now simply the probability for the interestingparameters o⃗ given the observed data, taking into consideration all the pos-sibilities for the values for the nuisance parameters based on their prioruncertainties and data. Obtaining such marginalized posterior probabili-ties for the oscillation parameters based on the SK data is the goal of thisoscillation analysis.MCICI bvrkov Chvin bontz CvrloIn this analysis there are 82 model parameters in total, and performing theintegration in Equation 8.14 in such high dimensionality numerically can bechallenging. Markov Chain Monte Carlo provides means to perform suchhigh dimensional integrals efficiently.Markov Chain Monte Carlo(MCMC) is essentially a directed randomwalk in a multi-dimensional parameter space which is used to sample from amulti-dimensional probability distribution. After randomly sampling largenumber of points from the multi-dimensional parameter space following thisprocedure, the density distribution of the sampled points will eventuallyapproximate the multi-dimensional probability distribution in concern.Metropolis-Hastings algorithm[104] is the method which is used in thisanalysis to produce a Markov chain, and the procedures to produce sampledpoints {mt} from a probability distribution e (m) for a multi-dimensionalparameter set m follow as below:1. Randomly chose the initial state m0 for the parameters.2. Define a proposal function f(m → n ) which gives the probabilitydistribution for proposing the next state n given the current state m.3. After t steps when the chain is currently at the state mt, propose anew state n using f(mt → n ).4. Accept the proposed state n with the acceptance probability:(mt → n ) := min(1Pe (n )e (mt)f(n → mt)f(mt → n ))P (8.15)i.e., move to the proposed new state(mt51 = n ) with the probability(mt → n ), discard the proposed state and remain in the currentstate(mt51 = mt) with the probability 1− (mt → n ).5. Repeat the procedures from 3.145LB4B Vuflysiun Unulflsis using aurkov Whuin aonty WurloIn the case when the proposal function is symmetric, i.e.,f(m → n ) = f(n → m)P (8.16)Equation 8.15 reduces to:(mt → n ) = min(1Pe (n )e (mt)): (8.17)It is interesting to note that although the number of steps required forthe chain to reasonably converge to the sampled distribution depends onthe choice for the proposal function f(m → n ), given sufficient numberof sampled points the chain will eventually approximate the distributionregardless of the choice of the functional form of f(m → n ), as long as thetransition probability to any point where e (n ) ̸= 0 is nonzero.For the oscillation analysis, the distribution which is sampled usingthe above algorithm is the posterior probability distribution e (o⃗P f⃗ |Y) inEquation 8.10, and a Gaussian proposal function is chosen which drawsa Gaussian-distributed random number centered at the current parametervalue. Since this is a symmetric proposal function, proposed steps are ac-cepted with the probability:(o⃗P f⃗ → o⃗′P f⃗ ′) = min(1Pe (o⃗′P f⃗ ′|Y)e (o⃗P f⃗ |Y)): (8.18)Note that since the denominator in Equation 8.10 is a constant which doesnot depend on o⃗ and f⃗ , the above equation further reduces to:(o⃗P f⃗ → o⃗′P f⃗ ′) = min(1Pa(Y|o⃗′P f⃗ ′)p(o⃗′P f⃗ ′)a(Y|o⃗P f⃗)p(o⃗P f⃗))P (8.19)which eliminates the need of evaluating the integral in Equation 8.10.Once a large number of points are sampled from the probability distri-bution, marginalization as in Equation 8.14 is readily done: one can simplyproject the sampled points onto the axes of interesting parameters and binthem in a histogram. In this oscillation analysis, the posterior probabilitydensity distribution for the model parameters Equation 8.10 is sampled usingMCMC, and the sampled points are then binned into histograms of oscilla-tion parameters which, after normalizing, gives the marginalized posteriorprobability distributions for the oscillation parameters which is presented asthe final result.146LBIB gynsitivitfl gtuxiysMC5 hznsitivity htuyizsIn order to test the oscillation analysis framework and understand its be-haviour before analyzing the real data, a sensitivity study was performed byrunning the analysis on an Asimov dataset, i.e., taking the expected eventrates without any statistical fluctuations as a dataset. The expectations arecalculated at the oscillation parameters in Table 8.3, and all the system-atic parameters are set to the nominal values. The exposure is taken to be2519.89 days which is the same as the real data.Parameter True valuesin2 12 0.304sin2 1= 0.0219sin2 2= 0.50∆m221 7:53× 10−5 eV2∆m2=2 +2:39× 10−= eV2 (NH)MZ −.R2Table 8.3: True oscillation parameters for the Asimov dataset production.The MCMC oscillation analysis as described in Section 8.4 is performedon the above dataset, which in the end provides Bayesian predictions on theoscillation parameters estimated from the Asimov data. During the analysisthe priors for the oscillation parameters are taken as shown in Table 8.4,while the priors for the systematic parameters are all Gaussian with theuncertainties assigned as in Section 8.2. Prior constraints on sin2 12, ∆m221and sin2 1= are based on the PDG2015 average values[15]. Flat priors areassumed for sin2 2=, ∆m2=2 and MZ to which the atmospheric neutrino datais sensitive, and the results on the three parameter will therefore be purelydriven by sensitivity of the Super-K data. Since we expect a large gapbetween the normal and inverted hierarchy peaks in the ∆m2=2 posterior, aspecial treatment is made for the proposal function for ∆m2=2 so that the signof the proposed ∆m2=2 flips by 50% probability at each step, in addition tothe Gaussian proposal for its absolute value. This ensures that the MarkovChain properly samples the posterior probability in both hierarchies withinthe limited number of MCMC steps. The resulting posterior can be used topredict the mass hierarchy as well as the value for ∆m2=2.For this sensitivity study, 50 MCMC chains were produced, each having120,000 steps. Figure 8.15 shows the evolution of ∆m2=2 and − lne (o⃗P f⃗ |Y),the negative-log of the posterior probability, as the MCMC processes progress,where the horizontal axes are the number of steps taken within each chain.147LBIB gynsitivitfl gtuxiysParameter Priorsin2 12 Gaussian, 0:304± 0:014sin2 1= Gaussian, 0:0219± 0:0012sin2 2= Flat in [0P 1]∆m221 Gaussian, 7:53± 0:18× 10−5 eV2∆m2=2 Flat, P(NH)=P(IH)=0.5MZ Flat in [−.P+.]Table 8.4: Priors for the oscillation parameters which are used in the anal-ysis. Constraints on sin2 12, ∆m221 and sin2 1= are based on the global fitand averaged results listed in [15].It can be seen especially from the − lne (o⃗P f⃗ |Y) plot that for the first ∼5000steps the distribution of the variable depends on the number of steps taken,which indicates that the random walk has not reached an equilibrium dur-ing this period. Such period is referred to as the “burn-in” period, andsince the steps taken during this time do not properly represent the pos-terior distribution, certain number of steps at the beginning of the chainsneed to be discarded. In this analysis, the initial 20,000 steps in each chainare discarded as burn-in, and as it can be seen from the plots the MCMCprocess reaches an equilibrium after 20,000 steps and is evenly sampling theposterior distribution. Thus, there are 5×106 post-burn-in steps used forthis sensitivity study.Once the posterior probability is sampled by MCMC, the marginalizedposterior distributions for the oscillation parameters are readily obtainedsimply by projecting the sampled points into histograms for the oscillationparameters. Figure 8.16 shows the marginalized 2D posterior distribution for∆m2=2 and sin2 2= where all the other oscillation and systematic parametershave been marginalized over. The blue cross indicates the mode of this2D posterior, i.e. the point with the highest posterior probability densitywhich is analogous to the “best fit” point in a more-common frequentistanalysis, and the cyan star is the true point where the Asimov dataset wasproduced. The dashed and solid contours indicate the 68% and 90% HighestPosterior Density(HPD) credible regions respectively. A % HPD credibleregion is defined such that the integrated probability within the region is% while the posterior density in any point within the region is higherthan any point outside. Thus, the point estimate and the allowed regionsfor the oscillation parameters can be obtained from the Bayesian posteriorprobability distributions produced using MCMC.148LBIB gynsitivitfl gtuxiys steps310×0 20 40 60 80 100 120delm2_23-0.005-0.004-0.003-0.002-0.00100.0010.0020.0030.0040.0050100020003000 steps310×0 20 40 60 80 100 120LogL-57110-57100-57090-57080-57070-57060-57050-57040-57030050010001500200025003000Figure 8.15: ∆m2=2 (upper plot) and − lne (o⃗P f⃗ |Y) (lower plot), thenegative-log of the posterior probability, plotted against the number of stepstaken in each chain. Steps from all the produced MCMC chains are com-bined in the plots.149LBIB gynsitivitfl gtuxiys23θ2sin0.3 0.4 0.5 0.6 0.7)2 eV-3 (10322m∆-5-4-3-2-101234500.511.522.53P(NH)=P(IH)=0.5Figure 8.16: The marginalized 2D posterior distribution for ∆m2=2 andsin2 2= given the Asimov dataset, where the blue cross indicates the modeof this 2D posterior and the cyan star is the true point where the Asimovdataset was produced. The dashed and solid contours indicate the 68% and90% HPD credible regions respectively.150LBIB gynsitivitfl gtuxiysAs in Figure 8.16, the posterior is distributed for both positive and neg-ative signs for ∆m2=2, and one can obtain the posterior probabilities for themass hierarchy by marginalizing over all other information, i.e., by simplyintegrating the ∆m2=2 posterior for above and below zero which is equivalentto counting the number of steps taken in the normal and inverted hierar-chies. For this Asimov dataset the ratio of the number of steps taken in eachhierarchy is NH:IH=2901738:2098262, which yields the posterior probabilityfor normal hierarchy to be 58.0%.By selecting the steps in either one of the hierarchies only, one can alsoobtain the posteriors in the case of assuming one of the hierarchy to be true.For example, Figure 8.17 shows the 2D posteriors for sin2 2= and MZ in thecase of fixed mass hierarchy. Figure 8.18 is the 1D posteriors for the threeparameters for each hierarchy where the 68% and 90% HPD credible regionsare indicated by the dark gray and light gray regions respectively.It is interesting to note that, although the mode of the 2D posterioris located near the true point as in the upper plot of Figure 8.17, once thedistribution is further marginalized over sin2 2= the mode of the 1D posteriorfor MZ drifts away from the true value. This is due to the correlationsbetween the two variables as seen in Figure 8.17, namely, although the localdensity is higher near the true point at MZ = −0:5. and sin2 2= = 0:5,the distribution is wider in sin2 2= at MZ ≈ −0:7.. Being able to fullymarginalize over the nuisance parameters with complex correlations in suchway without having to assume the Gaussian-ness of the posterior is one ofthe distinct features of a Bayesian analysis such as this analysis, as opposedto the more common frequentist analysis by 2 minimization.Figure 8.19 shows the mean and the standard deviation() of the 1Dmarginalized posterior for each systematic parameter in the case of assum-ing normal hierarchy. The enumeration of the systematic parameters can befound in Appendix A. In the plot the markers and the error bars representthe mean and the  of the posterior respectively, where the nominal value ofeach systematic parameter is taken to be zero and the deviation from nom-inal is normalized by the prior uncertainty. Although it is a small effect,the posterior mean for some of the parameters deviate from zero due to thecorrelations with other parameters. This plot essentially shows how eachsystematic parameter can be constrained by the atmospheric neutrino dataitself, and for the parameters whose posterior is significantly narrower thanthe prior, i.e. the parameters whose width of the error bar is significantlysmaller than one, the constraint from the data is strong. For instance, theflux normalization below 1 GeV (parameter #1) which has ∼10% prior un-certainty is strongly constrained by the large statistics of sub-GeV samples.151LBIB gynsitivitfl gtuxiys)pi (CPδ-1 -0.5 0 0.5 123θ2sin0. (CPδ-1 -0.5 0 0.5 123θ2sin0. 8.17: The 2D marginalized posterior distributions for sin2 2= andMZ given the Asimov dataset, where the upper plot is for assuming normalhierarchy to be true while the lower plot is for inverted hierarchy. Theblue cross indicates the mode of this 2D posterior and the cyan star is thetrue point where the Asimov dataset was produced. The dashed and solidcontours indicate the 68% and 90% HPD credible regions respectively.152LBIB gynsitivitfl gtuxiys23θ2sin0.2 0.4 0.6 0.80246NH)2 eV-3| (10322m∆|1 2 3 4 500.51NH)pi (CPδ-1 -0.5 0 0.5θ2sin0.2 0.4 0.6 0.80246IH)2 eV-3| (10322m∆|1 2 3 4 500. (CPδ-1 -0.5 0 0.5 8.18: The marginalized 1D posterior distributions for sin2 2= (top),|∆m2=2|(middle) and MZ (bottom) for the cases of assuming normal (left)and inverted (right) hierarchy, given the Asimov dataset. The vertical bluelines indicate the mode of the 1D posteriors while the cyan lines indicate thetrue point where the Asimov dataset was produced. The dark gray and lightgray regions indicate the 68% and 90% HPD credible regions respectively.153LB6B Dutu Unulflsis fysultsSystematic parameter index10 20 30 40 50 60 70σIn units of prior -2-1.5-1-0.500.511.52Posterior Mean & RMS of Systematic Parameters : NHFigure 8.19: The mean and the  of the marginalized 1D posterior foreach systematic parameter in the case of assuming normal hierarchy, forthe Asimov dataset. The enumeration of the systematic parameters can befound in Appendix A. The markers and the error bars represent the meanand the  of the posterior respectively, where the nominal value of eachsystematic parameter is taken to be zero and the deviation from nominal isnormalized by the prior uncertainty.A similar sensitivity study was performed using APFIT for event selec-tion instead of fiTQun, and the comparison of the sensitivity between thetwo cases can be found in Appendix B.MCK Dvtv Vnvlysis gzsultsThe result of analyzing the Super-K data is presented in this section. Thedataset which is used in this analysis is the FC data from SK-IV with 2519.89days exposure. The observed number of events in each selected event cate-gory has been summarized in Table 8.1.The analysis is performed in the same manner as in the sensitivity studyin Section 8.5, taking the priors for the oscillation parameters as describedin Table 8.4. 200 MCMC chains of 120,000 steps were produced where theinitial 20,000 steps in each chain are discarded as burn-in, which leaves intotal 2×107 post-burn-in steps for this analysis.154LB6B Dutu Unulflsis fysultsMCKCF eostzrior DistriwutionsFigure 8.20 shows the marginalized 2D posterior distribution for ∆m2=2 andsin2 2=. It can be seen in the figure that the posterior is distributed morein the upper half which indicates that there is a larger preference for normalhierarchy, which is due to the observed overall excess in the upward-goingmulti-GeV ,e events as shown in Figure 8.27. The posteriors for the caseof assuming one of the hierarchy to be true are also shown in Figure 8.21.The mode of the posteriors are located at sin2 2= = 0:606 and ∆m2=2 =+2:15×10−= eV2 for normal hierarchy, and at sin2 2= = 0:406 and ∆m2=2 =−1:85× 10−= eV2 for inverted hierarchy.Figure 8.22 shows the 2D posteriors for sin2 2= and MZ for each of thehierarchy assumption. As was discussed in Section 8.1.6, since decreasingsin2 2= from 0.5 and decreasing MZ from 0 both have the effect of increasingthe sub-GeV e-like events, the competing effects introduce high correlationsin the posterior for the two parameters and the preference for MZ is differentdepending on the 2= octant.Figure 8.23 shows the marginalized 1D posteriors for the three oscilla-tion parameters plotted separately for normal and inverted hierarchy, andthe mode and the 68% HPD credible region for each 1D posterior are summa-rized in Table 8.5. The 1D posteriors in the case mass hierarchy is marginal-ized are also shown in Figure 8.24.Parameter Normal Hierarchy Inverted Hierarchysin2 2=0.606 0.406[0.488,0.650] [0.375,0.475]∪[0.538,0.625]|∆m2=2|(10−=eV2)2.13 1.88[1.75,2.30] [1.7,2.3]MZ(.)-0.92 0.92[-1.00,-0.17]∪[0.72,1.00] [-1.00,-0.17]∪[0.61,1.00]Table 8.5: The 1D posterior mode and the 68% HPD credible intervals.From the posterior in Figure 8.20, the posterior probabilities for normal(∆m2=2S0) and inverted (∆m2=2Q0) hierarchy as well as the first (sin2 2=Q0.5)and the second (sin2 2=S0.5) 2= octant are calculated by integrating the2D posterior in each of the four quadrants. The results are summarized inTable 8.6, and the data shows mild preferences for the normal hierarchy andthe second 2= octant.155LB6B Dutu Unulflsis fysults23θ2sin0.3 0.4 0.5 0.6 0.7)2 eV-3 (10322m∆-5-4-3-2-10123450123456P(NH)=P(IH)=0.5Figure 8.20: The marginalized 2D posterior distribution for ∆m2=2 andsin2 2= given the SK-IV FC data, where the blue cross indicates the modeof this 2D posterior. The dashed and solid contours indicate the 68% and90% HPD credible regions respectively.156LB6B Dutu Unulflsis fysults23θ2sin0.3 0.4 0.5 0.6 0.7)2 eV-3| (10322m∆|123450246NH23θ2sin0.3 0.4 0.5 0.6 0.7)2 eV-3| (10322m∆|123450246IHFigure 8.21: The marginalized 2D posterior distributions for |∆m2=2| andsin2 2= for the cases of assuming normal (top) and inverted (bottom) hier-archy, given the SK-IV FC data. The blue cross indicates the mode of eachposterior. The dashed and solid contours indicate the 68% and 90% HPDcredible regions respectively.157LB6B Dutu Unulflsis fysults)pi (CPδ-1 -0.5 0 0.5 123θ2sin0. (CPδ-1 -0.5 0 0.5 123θ2sin0. 8.22: The marginalized 2D posterior distributions for sin2 2= andMZ for the cases of assuming normal (top) and inverted (bottom) hierarchy,given the SK-IV FC data. The blue cross indicates the mode of each 2Dposterior. The dashed and solid contours indicate the 68% and 90% HPDcredible regions respectively.158LB6B Dutu Unulflsis fysults23θ2sin0.2 0.4 0.6 0.80246NH)2 eV-3| (10322m∆|1 2 3 4 500.511.5NH)pi (CPδ-1 -0.5 0 0.5θ2sin0.2 0.4 0.6 0.8024IH)2 eV-3| (10322m∆|1 2 3 4 500.511.5IH)pi (CPδ-1 -0.5 0 0.5 8.23: The marginalized 1D posterior distributions for sin2 2= (top),|∆m2=2| (middle) and MZ (bottom) for the cases of assuming normal (left)and inverted (right) hierarchy, given the SK-IV FC data. The vertical bluelines indicate the mode of the 1D posteriors. The dark gray and light grayregions indicate the 68% and 90% HPD credible regions respectively.159LB6B Dutu Unulflsis fysults23θ2sin0.2 0.4 0.6 0.80246MH Marginalized)pi (CPδ-1 -0.5 0 0.5 MarginalizedFigure 8.24: The marginalized 1D posterior distributions for sin2 2= andMZ after marginalizing over the mass hierarchy, given the SK-IV FC data.The vertical blue lines indicate the mode of the 1D posteriors. The darkgray and light gray regions indicate the 68% and 90% HPD credible regionsrespectively.1st Oct. 2nd Oct. SumNormal 0.275 0.584 0.859Inverted 0.076 0.065 0.141Sum 0.351 0.649 1.000Table 8.6: Posterior probabilities for the mass hierarchy and the 2= octant.The 1D posterior distributions for the other oscillation parameters havethe mean and the  of the following: sin2 12 = 0:304 ± 0:014, sin2 1= =0:0219 ± 0:0012 and ∆m221 = 7:53 ± 0:18 × 10−5 eV2. These values arenearly identical to the prior constraints listed in Table 8.4 due to the lackof sensitivity of atmospheric neutrino data to these parameters.Figure 8.25 shows the mean and the  of the marginalized 1D posteriorfor each systematic parameter where the posteriors for normal and invertedhierarchy are shown separately. The enumeration of the systematic param-eters can be found in Appendix A.MCKCG oznith Vnglz DistriwutionsFigures 8.26 to 8.28 show the zenith angle distributions for each event sam-ple comparing the data with the expectations calculated with normal andinverted hierarchy. For each hierarchy, the expectations are calculated bysetting each oscillation and systematic parameter to the value at the modeof the parameter’s 1D posterior for the given choice of mass hierarchy. For160LB6B Dutu Unulflsis fysultsSystematic parameter index10 20 30 40 50 60 70σIn units of prior -2-1.5-1-0.500.511.52Posterior Mean & RMS of Systematic Parameters : NHSystematic parameter index10 20 30 40 50 60 70σIn units of prior -2-1.5-1-0.500.511.52Posterior Mean & RMS of Systematic Parameters : IHFigure 8.25: The mean and the  of the marginalized 1D posterior foreach systematic parameter in the cases of assuming normal(top) and in-verted(bottom) hierarchy, for the SK-IV FC data. The enumeration of thesystematic parameters can be found in Appendix A. The markers and theerror bars represent the mean and the  of the posterior respectively, wherethe nominal value of each systematic parameter is taken to be zero and thedeviation from nominal is normalized by the prior uncertainty.161LB6B Dutu Unulflsis fysultsthe oscillation parameters sin2 2=, ∆m2=2 and MZ only, since there are largecomplex correlations between the parameters, their values are set to whatare shown in Table 8.7 which are close to the mode of the 2D posteriors. Theobserved overall excess in the upward-going multi-GeV ,e events contributesto the preference for the normal hierarchy.Parameter Normal Hierarchy Inverted Hierarchysin2 2= 0.61 0.41∆m2=2(10−=eV2) +2.15 -1.85MZ(.) -0.9 +0.7Table 8.7: Oscillation parameters which are used to calculate the event ratepredictions shown in Figures 8.26 to 8.28.MCKC3 Gooynzss of FitIn order to test whether or not the models assumed in this analysis disagreewith the observed data, a goodness of fit test is performed according to theprescriptions described in [105]. First, 2000 points are randomly sampledfrom the MCMC chains which were used in the data analysis, which producesa reduced set of sample points which represents the posterior distribution.For each sampled point, a fake dataset is thrown using the predictions at thatpoint, and the likelihood Equation 8.7 is compared between the fake data andthe real data. Figure 8.29 shows the distribution of the log likelihood ratiobetween the real and the fake data for all points. A p-value is then calculatedas the fraction of points at which the real data having a larger likelihoodvalue than the fake data. The calculated p-value, which the authors of[105] refer to as the posterior predictive p-value, is essentially the classicalp-value as used in the standard goodness of fit test in frequentist analysesaveraged over the entire posterior. This test is therefore assessing the fitnessof the entire Bayesian posterior model to the observed data, rather than ata single best-fit point. The posterior predictive p-value for this analysis iscalculated to be 10.3%, which is large enough that the assumed models arenot excluded by the data according to this test.MCKCI gzsult Compvrison to dthzr ZflpzrimzntsFigure 8.30 compares the results of this analysis for ∆m2=2 and sin2 2=with the latest results from other experiments. In the plots, 90% HPDcredible regions are shown for Bayesian analyses, whereas for frequentist162LB6B Dutu Unulflsis fysultsZθcos-1 -0.5 0 0.5 10200400600Sub-GeV 1Re 0dcyZθcos-1 -0.5 0 0.5 1050100 0dcyµSub-GeV 1RZθcos-1 -0.5 0 0.5 101002003004002dcy≥ µSub-GeV 1RZθcos-1 -0.5 0 0.5 102004001dcy≥Sub-GeV 1Re Zθcos-1 -0.5 0 0.5 10200400600 1dcyµSub-GeV 1RZθcos-1 -0.5 0 0.5 1050010000piSub-GeV 2RFigure 8.26: Distributions of the cosine of the zenith angle for the sub-GeVevent samples. The black markers indicate the data, and the solid greenand the dashed red lines represent the expectations calculated with normalhierarchy and inverted hierarchy respectively. For each hierarchy, the ex-pectation is calculated by setting the oscillation and systematic parametersat the mode of the posteriors for the respective hierarchy.163LB6B Dutu Unulflsis fysultsZθcos-1 -0.5 0 0.5 10102030eνMulti-GeV 1Re Zθcos-1 -0.5 0 0.5 1050100eνMulti-GeV MRe Zθcos-1 -0.5 0 0.5 1050100eνMulti-GeV 1Re Zθcos-1 -0.5 0 0.5 1020406080eνMulti-GeV MRe Figure 8.27: Distributions of the cosine of the zenith angle for the multi-GeV e-like event samples. The black markers indicate the data, and thesolid green and the dashed red lines represent the expectations calculatedwith normal hierarchy and inverted hierarchy respectively. For each hierar-chy, the expectation is calculated by setting the oscillation and systematicparameters at the mode of the posteriors for the respective hierarchy.164LB6B Dutu Unulflsis fysultsZθcos-1 -0.5 0 0.5 1050100µMulti-GeV 1RZθcos-1 -0.5 0 0.5 1050100Multi-GeV MRe oth.Zθcos-1 -0.5 0 0.5 1050100150200µMulti-GeV MRFigure 8.28: Distributions of the cosine of the zenith angle for the multi-GeV-like and other event samples. The black markers indicate the data, and thesolid green and the dashed red lines represent the expectations calculatedwith normal hierarchy and inverted hierarchy respectively. For each hierar-chy, the expectation is calculated by setting the oscillation and systematicparameters at the mode of the posteriors for the respective hierarchy.165LB6B Dutu Unulflsis fysultsThrow-lnLDatalnL-60 -40 -20 0 20 400204060 p = 206/2000= 10.3%Figure 8.29: Distribution of the log likelihood ratio between the observeddata and fake data which is used for the goodness of fit test. The posteriorpredictive p-value is calculated to be 10.3%.analyses 90% confidence regions are shown instead. The plots show thelatest preliminary results from the conventional SK atmospheric neutrinoanalysis[106][107](labeled as “SK1-4 Freq.” in the plots) which employs afrequentist analysis technique by 2 minimization and event selection basedon APFIT, the preexisting event reconstruction which was mentioned inChapter 6. The analysis uses the full SK-I to IV data of 5326 days exposurewhich is more than double the data statistics compared to the SK-IV-onlydata which is used in the Bayesian analysis with fiTQun event selectionpresented in this chapter, and the difference in sensitivity between the twoanalyses is dominantly due to the additional data in the APFIT-based anal-ysis. The data from the old SK-I to III phases is not included in the fiTQun-based analysis due to the observed large data-MC discrepancies as discussedin Section 6.11, and inclusion of such data requires improvements in the de-tector simulation which is expected to be achievable in near future. TheT2K result[108] is from a Bayesian analysis which uses a similar MCMCtechnique described in Section 8.4 and produces Bayesian credible regions.The results from MINOS[109] and NO,A[110] are both from frequentistanalyses which report their results as confidence regions. One should notethat the Bayesian credible regions and frequentist confidence regions are dif-ferent and are not comparable in a strict sense, although such comparisonis interesting as a rough measure.166LB6B Dutu Unulflsis fysults23θ2sin0.3 0.4 0.5 0.6 0.7)2 eV-3 (10322m∆1234Normal HierarchySK4 Bayes.SK1-4 Freq.T2KMINOSAνNO23θ2sin0.3 0.4 0.5 0.6 0.7)2 eV-3 (10322m∆-4-3-2-1Inverted HierarchySK4 Bayes.SK1-4 Freq.T2KMINOSAνNOFigure 8.30: 90% credible and confidence regions for ∆m2=2 and sin2 2= forthe results from various experiments. The red contours labeled as “SK4Bayes.” are the results from the analysis presented in this chapter whichis a Bayesian analysis using SK-IV atmospheric neutrino data selected byfiTQun event reconstruction. The upper and the lower plots are the casesof assuming normal and inverted hierarchy respectively.167Chvptzr NConxlusions vny dutlookAn oscillation analysis of the Super-Kamiokande atmospheric neutrino datawas presented. The analysis employed a new maximum likelihood eventreconstruction algorithm to reconstruct and select atmospheric neutrinoevents, and oscillation parameters were estimated using a Markov chainMonte Carlo technique which produces marginalized Bayesian posterior prob-ability distributions for neutrino oscillation parameters based on the ob-served data.Analyzing the data of 2520 days exposure from SK-IV, the result prefersnormal hierarchy with the posterior probability of 85.9%, and the modeand the 68% highest posterior density credible interval of each oscillationparameter’s 1D posterior probability distribution for normal hierarchy aresin2 2= = 0:60650:044−0:118 and ∆m2=2 = 2:1350:17−0:=8 × 10−= eV2. The analysisalso has small sensitivity to MZ, however, nothing conclusive can be saidregarding whether CP violation is observed.The Bayesian analysis method using MCMC allows one to fully marginal-ize over nuisance parameters and report the result on the parameters of in-terest as marginalized Bayesian posterior probabilities. When parametershave highly non-Gaussian behaviours, which is the case for the oscillationparameters in this analysis, being able to actually marginalize over nuisanceparameters is advantageous compared to the more common frequentist tech-nique of profiling the likelihood, i.e. to minimize the negative-log-likelihoodor the 2 over nuisance parameters, which becomes equal to marginalizationonly when the parameters have Gaussian behaviours.The new event reconstruction algorithm “fiTQun” significantly improvesparticle identification and vertex and kinematic reconstruction performancescompared to the preexisting event reconstruction at SK. The new algorithmwas first used in the T2K ,e appearance analysis for .0 rejection, whichresulted in more than 60% reduction of the NC background in the signal,e event sample compared to the previous .0 rejection method. However,the application of fiTQun in T2K has been limited to the .0 rejection only,and the SK atmospheric neutrino analysis presented in this thesis is thefirst physics analysis which makes an extensive use of the new algorithm168Whuptyr MB Wonwlusions unx cutlookby employing an entirely fiTQun-based event selection. There are currentlyongoing efforts to extend the usage of fiTQun in various other studies suchas the nucleon decay analyses at SK[111], T2K oscillation analyses[112] andstudies for future water Cherenkov detectors[113].Only the data from SK-IV was used in the analysis presented in thisthesis due to the large data-MC discrepancy observed in SK-I to III as men-tioned in Section 6.11. However, by improving the detector simulation, it isexpected that the data from SK-I to III can also be included in the oscil-lation analysis using fiTQun in near future. Table 9.1 shows the expectedsensitivity to the mass hierarchy at each of the following SK exposure: 2520days which is the the current SK-IV exposure used in the presented dataanalysis, 5300 days which is roughly the current total exposure from SK-I toIV, and 8500 days which is the expected accumulated exposure in 10 yearswhen the next generation neutrino oscillation experiments such as Hyper-Kamiokande[114] and DUNE[115] are expected to start. As in the table,sensitivity to the mass hierarchy continues to improve significantly as thedata statistics increase.Exposure 2520 days 5300 days 8500 daysTrue NH 67.2% 81.6% 91.0%True IH 61.8% 71.4% 80.9%Table 9.1: Expected posterior probabilities for for favouring the correct masshierarchy for different SK exposure, based on the Asimov dataset producedat either normal or inverted hierarchy, sin2 2= = 0:6 and other oscillation pa-rameters set to the values in Table 8.3. The middle row shows the posteriorprobabilities for favouring NH for the dataset produced with NH, and thebottom row are the probabilities for favouring IH for the datasets producedwith IH.Long baseline accelerator neutrino experiments such as T2K and NO,Aare rapidly accumulating data, and the experiments have sensitivity to MZand mass hierarchy. However, the competing effects from the unknownsproduce degeneracies in the oscillation probabilities which limit the exper-iments’ sensitivities. 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Acciarri ds ak-. “Long-Baseline Neutrino Facility (LBNF) and DeepUnderground Neutrino Experiment (DUNE) Conceptual Design Re-port Volume 1: The LBNF and DUNE Projects”. arXhu905/0-/4360,2016.179Vppznyifl Vaist of hystzmvtixevrvmztzrs1 Absolute flux EQ1GeV 26 DIS model difference2 Absolute flux ES1GeV 27 DIS xsec.3 Flux ,R,e ratio EQ1GeV 28 DIS f2 shape low-W4 Flux ,R,e ratio 1QEQ10GeV 29 DIS f2 shape high-W5 Flux ,R,e ratio ES10GeV 30 Coherent . xsec.6 Flux ,¯eR,e ratio EQ1GeV 31 NC/CC ratio7 Flux ,¯eR,e ratio 1QEQ10GeV 32 KR. ratio8 Flux ,¯eR,e ratio ES10GeV 33 Up/down ratio9 Flux ,¯R, ratio EQ1GeV 34 Horizontal/vertical ratio10 Flux ,¯R, ratio 1QEQ10GeV 35 Multi-GeV normalization11 Flux ,¯R, ratio ES10GeV 36 Hadron simulation12 , path length 37 CC , xsec.13 Matter effect 38 FC/PC separation14 Solar activity 39 FC reduction15 CCQE xsec. 40 Fiducial volume16 CCQE ,R,¯ ratio 41 Non-, bkg. e-like17 CCQE ,R,e ratio 42 Non-, bkg. mu-like18 MfEA 43 Ring separation19 MEC xsec. 44 PID 1-ring20 Single meson xsec. 45 PID multi-ring21 Single meson MA 46 Energy scale22 Single meson XA5 (0) 47 Up/down energy scale asym.23 Single meson bkg. 48 Decay-e tagging eff.24 Single meson .0R.± ratio 49 2R.0 eff.25 Single meson ,¯R, ratio 50 Sub-GeV FSI parameter #00Table A.1: List of systematic parameters. Continues to Table A.2.180Uppynxifi UB List of gflstymutiw durumytyrs51 Sub-GeV FSI parameter #01 64 Multi-GeV FSI parameter #0252 Sub-GeV FSI parameter #02 65 Multi-GeV FSI parameter #0353 Sub-GeV FSI parameter #03 66 Multi-GeV FSI parameter #0454 Sub-GeV FSI parameter #04 67 Multi-GeV FSI parameter #0555 Sub-GeV FSI parameter #05 68 Multi-GeV FSI parameter #0656 Sub-GeV FSI parameter #06 69 Multi-GeV FSI parameter #0757 Sub-GeV FSI parameter #07 70 Multi-GeV FSI parameter #0858 Sub-GeV FSI parameter #08 71 Multi-GeV FSI parameter #0959 Sub-GeV FSI parameter #09 72 Multi-GeV FSI parameter #1060 Sub-GeV FSI parameter #10 73 Multi-GeV FSI parameter #1161 Sub-GeV FSI parameter #11 74 Multi-GeV FSI parameter #1262 Multi-GeV FSI parameter #00 75 Multi-GeV FSI parameter #1363 Multi-GeV FSI parameter #01 76 Multi-GeV FSI parameter #14Table A.2: List of systematic parameters. Continues from Table A.1.181Vppznyifl WVtmosphzrix czutrinohzlzxtion Improvzmznt wyifunIn the previously published Super-K atmospheric neutrino oscillation anal-yses, APFIT was used for event selection instead of fiTQun. The impact ofemploying fiTQun for event selection is discussed in this appendix.The event selection criteria for APFIT-based analysis, as described indetail in [106], are very similar to the fiTQun-based event selection intro-duced in Section 8.1.1, with some differences in the sample definitions. Inparticular, the events selected as sub-GeV one-ring e-like with 0 decay elec-tron by APFIT are further subdivided into two categories using the fiTQun.0 fit variables discussed in Section 6.9.1: out of the selected events, theones which satisfy ln(a.0Rae) S 130 and 95 Q m.0(MeVRx2) Q 185 arecategorized as “Sub-GeV one-ring .0” sample, and the rest are categorizedas “Sub-GeV one-ring e-like 0 decay” sample. The additional cut usingfiTQun was introduced to reduce the rather large NC.0 background con-tamination in the events selected as one-ring e-like by APFIT. Other thanthe .0 cut above, all selection cuts are done using APFIT information onlyin the APFIT-based analysis. Such additional .0 rejection cut is not em-ployed for the fiTQun-only selection since the one-ring e-like selection byfiTQun reduces the .0 background to a sufficiently low level.Table B.1 shows the expected event rates and the interaction modebreakdown of each event sample selected by the APFIT-based event selec-tion. MC expectations are calculated using the same oscillation parametersand normalization as described in Section 8.1.4. Comparing this to Ta-ble 8.1 for fiTQun-only selection, it can be seen that fiTQun event selectionachieves higher fraction of the targeted event category and lower fractionof backgrounds across all event samples. In particular, , contamination inthe e-like samples in multi-GeV is significantly reduced by fiTQun.In order to see the impact of the above improvement in event selection on182Uppynxifi VB Utmosphyriw byutrino gylywtion Improvymynt vfl heun% 1Re 0nmy 1Re≥1nmy 1R 0nmy 1R 1nmy 1R≥2nmy 1R.0 2R.0,eCC 72.77 81.20 4.36 0.11 0.05 9.97 7.04,¯eCC 23.92 1.80 1.32 0.00 0.00 3.25 2.44,CC 0.19 8.24 68.07 68.21 94.75 1.23 0.99,¯CC 0.04 2.31 8.03 29.11 3.29 0.23 0.08NC 3.08 6.44 18.23 2.57 1.91 85.32 89.44Total 4899.5 563.0 909.0 4185.9 367.0 234.6 889.9% 1Re ,e 1Re ,¯e 1R MRe ,e MRe ,¯e MR MRe oth.,eCC 63.70 54.96 0.27 53.43 53.66 2.74 29.31,¯eCC 9.44 37.68 0.12 10.03 26.60 0.42 3.26,CC 8.98 0.72 62.19 14.70 4.73 70.50 31.72,¯CC 1.55 0.20 37.26 1.98 0.58 21.25 3.54NC 16.33 6.44 0.15 19.86 14.43 5.09 32.17Total 345.8 1040.4 1358.7 431.9 427.7 1190.2 681.5Table B.1: Interaction mode breakdown and the total MC event rates foreach event sample categorized by the APFIT-based selection, where the up-per and the lower table is for sub-GeV and multi-GeV samples respectively.MC expectations are calculated with two-flavour oscillations at sin2 2=1.0and ∆m2=2:5× 10−= eV2 and is normalized to 2519.89 days exposure.the sensitivity to oscillation parameters, a sensitivity study was performedfollowing the same procedures described in Section 8.5. The Asimov datasetsfor this study were produced by taking sin2 2= = 0:55 and ∆m2=2 = +2:39×10−= eV2 (normal hierarchy), and the rest of the oscillation parameters setto the values listed in Table 8.3. The exposure was taken to be 2519.89 days.Figure B.1 compares resulting 90% HPD credible regions for ∆m2=2 andsin2 2= between the cases of using APFIT or fiTQun for event selection.The posterior probability for normal hierarchy is 61.5% for APFIT selectionand 63.0% for fiTQun selection.183Uppynxifi VB Utmosphyriw byutrino gylywtion Improvymynt vfl heunNormal Hierarchy23θ2sin0.3 0.4 0.5 0.6 0.7)2 eV-3 (10322m∆1234APFITfiTQunInverted Hierarchy23θ2sin0.3 0.4 0.5 0.6 0.7)2 eV-3 (10322m∆4−3−2−1−APFITfiTQunFigure B.1: 90% HPD credible regions for ∆m2=2 and sin2 2= comparedbetween the cases of using APFIT or fiTQun for event selection, on Asimovdatasets. The black contours are for APFIT selection and the red contoursare for fiTQun. The upper and the lower plots are the cases of assumingnormal and inverted hierarchy respectively.184


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